diff --git "a/process_1/tokenized_finally.jsonl" "b/process_1/tokenized_finally.jsonl"
new file mode 100644--- /dev/null
+++ "b/process_1/tokenized_finally.jsonl"
@@ -0,0 +1,8928 @@
+{"id": "4258.png", "formula": "\\begin{align*} H _ k = \\textrm { s p a n } _ { \\R } \\{ u _ 1 , \\ldots , u _ k \\} . \\end{align*}"}
+{"id": "5760.png", "formula": "\\begin{align*} P _ p ( J _ X ^ p \\tau _ X ) = \\tilde P _ p ( J _ X ^ p ) e ^ { \\frac { \\mu _ p } { \\Delta _ p ^ 2 } J _ X ^ p \\tau _ X } , \\end{align*}"}
+{"id": "8437.png", "formula": "\\begin{align*} & \\rho _ L ( g ) ( h , \\phi ) = ( h g ^ { - 1 } , \\mathrm { A d } ^ * _ g ( \\phi ) ) , \\\\ & \\rho _ R ( g ) ( h , \\phi ) = ( g h , \\phi ) \\end{align*}"}
+{"id": "4556.png", "formula": "\\begin{align*} ( \\mathcal { B } _ 1 { \\mathbf V } _ { x _ 2 } , { \\mathbf V } _ { x _ 2 } ) | _ { x _ 1 = 0 } = 2 [ \\partial _ 2 \\dot { q } ( \\partial _ 2 \\dot { u } _ { N } - \\hat { \\lambda } \\partial _ 2 \\dot { H } _ { N } ) ] , \\end{align*}"}
+{"id": "1526.png", "formula": "\\begin{align*} \\frac { V ^ { m _ 1 } } { m _ 1 ! } = \\frac { V ^ { m _ 2 } } { m _ 2 ! } \\left ( 1 + O \\left ( \\frac { E L } V \\right ) \\right ) = \\frac { V ^ { m _ 2 } } { m _ 2 ! } \\left ( 1 + O \\left ( \\sqrt { \\frac E V } \\right ) \\right ) . \\end{align*}"}
+{"id": "5319.png", "formula": "\\begin{align*} n = \\ln \\left ( \\frac { L ( C ( N + 1 ) ) ^ { k - 1 } } { \\varepsilon } \\right ) ^ 2 \\end{align*}"}
+{"id": "4559.png", "formula": "\\begin{align*} \\partial _ 2 \\varphi & = \\frac { \\hat { H } ^ { + } _ 2 \\dot { H } ^ { + } _ { N } + \\hat { H } ^ { - } _ 2 \\dot { H } ^ { - } _ { N } + ( \\hat { H } ^ { + } _ 2 \\partial _ 1 \\hat { H } ^ { + } _ { N } - \\hat { H } ^ { - } _ 2 \\partial _ 1 \\hat { H } ^ { - } _ { N } ) \\varphi } { ( \\hat { H } ^ { + } _ 2 ) ^ 2 + ( \\hat { H } ^ { - } _ 2 ) ^ 2 } \\\\ & = \\hat d _ 1 \\dot H ^ + _ N + \\hat d _ 2 \\dot H ^ - _ N + \\hat d _ 3 \\varphi \\ , , \\quad \\mbox { o n } \\ , \\ , \\ , \\Gamma _ T \\ , , \\end{align*}"}
+{"id": "8416.png", "formula": "\\begin{align*} W _ { K , L } = \\sum _ { L < \\ell \\leq 2 L } b ( \\ell ) \\sum _ { \\substack { K < k \\leq 2 K \\\\ \\frac { P } { \\ell } < k \\leq \\frac { 2 P } { \\ell } } } a ( k ) e ( h ( k \\ell ) ) , \\end{align*}"}
+{"id": "3977.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) u = f ( { \\bf u } _ t , \\mu _ t ) \\zeta _ t + g ( u _ t , \\mu _ t ) \\end{align*}"}
+{"id": "6550.png", "formula": "\\begin{align*} R _ { B _ * } { H } ( z ) R _ { B _ * } & = R _ { B _ * } { D } ( z ) R _ { B _ * } + R _ { B _ * } ( \\varepsilon \\Delta + \\delta { T } _ \\phi ) R _ { B _ * } \\\\ & = R _ { B _ * } { D } ( z ) R _ { B _ * } + O ( \\varepsilon + \\delta ) , \\end{align*}"}
+{"id": "6730.png", "formula": "\\begin{align*} 2 \\operatorname { R i c } _ M ( \\nu , \\nu ) = R _ { M } - K _ { \\Sigma _ t } + H ^ { 2 } - | h | ^ { 2 } . \\end{align*}"}
+{"id": "1429.png", "formula": "\\begin{align*} \\frac { \\partial f _ i } { \\partial z } = f _ { i + 1 } + \\frac { \\partial } { \\partial z } \\log | f _ i | ^ 2 f _ i , \\ \\frac { \\partial f _ i } { \\partial \\bar z } = - \\frac { | f _ i | ^ 2 } { | f _ { i - 1 } | ^ 2 } f _ { i - 1 } , \\ \\langle f _ { i + 1 } , f _ i \\rangle = 0 . \\end{align*}"}
+{"id": "3720.png", "formula": "\\begin{align*} u ''' ( t ) + A u ( t ) = 0 , t > 0 , \\end{align*}"}
+{"id": "2728.png", "formula": "\\begin{align*} \\mathcal { A } _ n : = \\big \\{ F \\in [ \\mathbb { N } ] ^ { < \\omega } \\colon | F | \\leqslant n \\big \\} \\quad ( n \\in \\mathbb N ) \\end{align*}"}
+{"id": "3443.png", "formula": "\\begin{align*} u ^ * ( p ) = \\max _ { x \\in \\Delta } ( \\langle x , p \\rangle - u ( x ) ) , \\Delta = \\{ x \\in \\R ^ { m + 1 } _ { \\geq 0 } | \\sum _ 0 ^ m x _ i = 1 \\} , \\end{align*}"}
+{"id": "1669.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j , k \\leq n } H ^ { ( m , n ) } _ { \\texttt { a } ; j , k } ( \\boldsymbol { \\xi } ) x _ j x _ k = & \\sum _ { 1 \\leq j \\leq n } m x _ j ^ 2 + \\sum _ { 1 \\leq j < k \\leq n } u _ q ( \\xi _ j - \\xi _ k ) ( x _ j - x _ k ) ^ 2 \\\\ \\ge & \\ , m \\sum _ { 1 \\leq j \\leq n } x _ j ^ 2 . \\end{align*}"}
+{"id": "1501.png", "formula": "\\begin{align*} & d ( X _ { t } + A _ { i } ( t ) ) = \\sum _ { k = 1 } ^ { m } \\left ( \\frac { 1 } { \\vert B _ { t } \\vert } B _ { k } ( t ) + \\frac { 1 } { 2 } \\left ( U ^ { ( i ) } B _ { t } \\right ) _ { k } \\right ) d B _ { k } ( t ) , \\end{align*}"}
+{"id": "4078.png", "formula": "\\begin{align*} \\eta ^ p ( p _ 0 ) = \\int _ { \\sigma ( p ) } ^ p B ( p _ 0 , \\cdot ) , \\end{align*}"}
+{"id": "8280.png", "formula": "\\begin{align*} R = | q | ^ 2 \\sum _ { 1 \\leq \\beta < \\gamma \\leq s } ( | b _ \\beta | - | b _ \\gamma | ) ^ 2 ( | u _ { 2 \\beta - 1 } | ^ 2 + | u _ { 2 \\beta } | ^ 2 ) ^ 2 ( | u _ { 2 \\gamma - 1 } | ^ 2 + | u _ { 2 \\gamma } | ^ 2 ) ^ 2 . \\end{align*}"}
+{"id": "7443.png", "formula": "\\begin{align*} F _ 1 ( t ) : & = t ^ { \\nu + \\kappa _ 1 + \\kappa _ 2 + 2 } F _ { ( y _ 1 , y _ 2 ) } ^ { \\prime } ( t ) + ( - \\nu - \\kappa _ 1 - \\kappa _ 2 - 1 ) \\int _ { \\Omega } \\left [ a _ 1 \\vert y _ 1 \\vert ^ { 1 - \\nu } + a _ 2 \\vert y _ 2 \\vert ^ { 1 - \\nu } \\right ] d z . \\end{align*}"}
+{"id": "2039.png", "formula": "\\begin{align*} B = S _ { \\phi } u - A \\in \\mathrm { L } ^ { 1 } _ { \\mathrm { l o c } } ( \\mathbb { R } ^ { n - 1 } ) \\cap \\dot { \\mathrm { H } } ^ { 1 , p } ( \\mathbb { R } ^ { n - 1 } ) \\end{align*}"}
+{"id": "3604.png", "formula": "\\begin{align*} { \\bf { r } } = \\left [ { \\bf { r } } _ 1 , \\ , \\ldots , \\ , { \\bf { r } } _ { M _ { \\rm R } } \\right ] ^ T = { \\bf { H } } _ { \\rm D S } { \\bf F } _ { \\rm D S } { \\bf s } + { \\bf { n } } , \\end{align*}"}
+{"id": "5874.png", "formula": "\\begin{align*} T _ { S L } = \\inf \\Big \\{ t : \\max _ { k = t - s + 1 \\in \\mathcal { K } } l ( \\mathbf { p } _ { s t } ) \\geq C _ { \\gamma } \\Big \\} , \\end{align*}"}
+{"id": "1941.png", "formula": "\\begin{align*} G ( t , x ) = \\left ( \\sin ( x - t ) - 1 \\right ) \\cos ( x - t ) + \\epsilon \\ , \\sin ( x - t ) + \\left ( 1 . 4 9 3 6 4 8 2 6 5 6 2 4 8 5 4 \\right ) \\left ( 1 + \\sin ( x - t ) \\right ) \\sin ( x - t ) \\end{align*}"}
+{"id": "4796.png", "formula": "\\begin{align*} \\partial _ { v } f ( z ) = \\lim _ { s \\rightarrow 0 ^ + } \\frac { f ( z + s v ) - f ( z ) } { s } . \\end{align*}"}
+{"id": "227.png", "formula": "\\begin{align*} x \\to x + c \\rho _ L , \\rho _ L = \\sum _ { 1 \\leq j \\leq n } e _ j \\end{align*}"}
+{"id": "2242.png", "formula": "\\begin{align*} h _ r ( n , Q ) = h _ r ( n , \\overline Q ) \\overline Q = \\left \\{ \\left ( m , { m \\choose r } - f \\right ) : ( m , f ) \\in Q \\right \\} . \\end{align*}"}
+{"id": "36.png", "formula": "\\begin{align*} \\mathcal { Q } _ 2 ^ { 1 } \\Psi & = \\frac { 1 } { \\vert \\Lambda \\vert } \\sum _ { k \\neq 0 } \\widehat { g } ( k ) \\sum _ { i \\neq j } e _ { k } ( x _ { j } ) e _ { 0 } ( x _ { i } ) a _ { 0 } a _ { k } \\Psi \\\\ & = \\frac { 1 } { \\vert \\Lambda \\vert } \\sum _ { k \\neq 0 } \\widehat { g } ( k ) a _ { k } ^ { \\dagger } a _ { 0 } ^ { \\dagger } a _ { 0 } a _ { k } \\Psi . \\end{align*}"}
+{"id": "7093.png", "formula": "\\begin{align*} L ( n _ 1 , n _ 2 , k ) = \\sum _ { n _ 1 \\ll p _ 1 C r } \\ , \\frac { 1 } { n ^ { k - 2 / 3 } _ 1 } \\ , \\frac { 1 } { n ^ { 2 / 3 } _ 1 } \\ , \\left ( \\sum _ { n ^ 2 _ 1 n _ { 2 } \\ll N _ 0 } \\frac { | A _ { \\pi } ( n _ { 2 } , n _ { 1 } ) | ^ 2 } { n _ 2 ^ { 2 / 3 } } \\right ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "1505.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow \\infty } \\frac { \\lambda _ { 1 } ( m , n ) } { \\lambda _ { 1 } ^ { ( m ) } } = 1 , \\end{align*}"}
+{"id": "2842.png", "formula": "\\begin{align*} | T | & = \\Delta + 5 - | F _ { y x } \\cup ( F _ { x y } \\setminus S ) \\cup S ' | \\\\ & \\geq \\Delta + 5 - ( 1 + 3 + 2 ) \\\\ & = \\Delta - 1 \\end{align*}"}
+{"id": "6681.png", "formula": "\\begin{align*} W _ n ( x _ 1 , \\dots , x _ n ; N , \\beta , p , q ) : = \\Big \\langle G ( x _ 1 ) \\cdots G ( x _ n ) \\Big \\rangle , G ( x ) : = \\sum _ { j = 1 } ^ N { 1 \\over x - e ^ { i \\theta _ j } } . \\end{align*}"}
+{"id": "5361.png", "formula": "\\begin{align*} H ^ 1 ( \\O _ { L _ 1 \\cup \\ldots \\cup L _ { 2 b - 2 } } ( 1 ) ( - B _ { b - 1 } ) ) = 0 . \\end{align*}"}
+{"id": "8410.png", "formula": "\\begin{align*} V _ { z } ( v ) & = \\sum _ { P < p \\leq 2 P } ( \\log p ) \\left ( \\sum _ { | r | \\leq Z ( \\log N ) ^ { 4 } } g _ { z } ( r ) e ( r p ^ c ) \\right ) e \\left ( v \\left ( N + j - p ^ { c } + \\frac { z } { 2 Z } \\right ) ^ { \\gamma } \\right ) + O \\left ( N ^ { - 1 0 } \\right ) \\\\ & = \\sum _ { | r | \\leq Z ( \\log N ) ^ { 4 } } g _ { z } ( r ) U \\left ( N + j + \\frac { z } { 2 Z } , r , v \\right ) + O \\left ( N ^ { - 1 0 } \\right ) \\\\ & \\ll N ^ { - 1 0 } + \\frac { 1 } { Z } \\sum _ { | r | \\leq R } \\sup _ { T \\in [ N , N + 2 ] } | U ( T , r , v ) | , \\end{align*}"}
+{"id": "4476.png", "formula": "\\begin{align*} \\partial ^ j _ t \\mathbb { L } ( { \\mathbf U } ^ a , \\Psi ^ a ) | _ { t = 0 } = 0 \\Omega , j \\in \\{ 0 , \\cdots , \\mu - 1 \\} , \\end{align*}"}
+{"id": "3007.png", "formula": "\\begin{align*} H _ * \\left ( G , c \\right ) = H _ * \\left ( X , \\gamma \\right ) . \\end{align*}"}
+{"id": "8291.png", "formula": "\\begin{align*} V _ 1 ( \\bar u , \\bar u _ t ) = \\frac { 1 } { 2 } \\int _ 0 ^ 1 e ^ { \\mu x } ( \\bar u _ t + \\bar u _ x ) ^ 2 + e ^ { - \\mu x } ( \\bar u _ t - \\bar u _ x ) ^ 2 d x , \\end{align*}"}
+{"id": "466.png", "formula": "\\begin{align*} _ 2 \\tilde { F } _ 1 ( a , b ; c ; z ) = \\frac { 1 } { \\Gamma ( a ) \\Gamma ( b ) } \\sum _ { k = 0 } ^ \\infty \\frac { \\Gamma ( a + k ) \\Gamma ( b + k ) } { \\Gamma ( c + k ) k ! } z ^ k , | z | < 1 . \\end{align*}"}
+{"id": "7589.png", "formula": "\\begin{align*} \\zeta = \\beta ^ { \\frac { 1 } { 4 } } N ^ { \\frac { 1 } { 4 } } , \\end{align*}"}
+{"id": "6670.png", "formula": "\\begin{align*} \\rho _ { ( 1 ) , \\infty } ^ { ( \\rm c J ) } ( x ; \\beta , p , q ) \\Big | _ { \\beta = 4 \\atop q = 0 } = \\rho _ { ( 1 ) , \\infty } ^ { ( \\rm c J ) } ( 2 x ; \\beta , p , q ) \\Big | _ { \\beta = 2 \\atop q = 0 , p \\mapsto 2 p } - \\pi p { J _ { 2 p - 1 / 2 } ( 2 x ) \\over x ^ { 1 / 2 } } \\int _ 0 ^ x s ^ { - 1 / 2 } J _ { 2 p + 1 / 2 } ( 2 s ) \\ , d s , \\end{align*}"}
+{"id": "7090.png", "formula": "\\begin{align*} \\Theta = \\sum _ { n _ { 2 } \\ll N _ 0 / n ^ 2 _ 1 } ^ { \\infty } \\frac { | A _ { \\pi } ( n _ { 2 } , n _ { 1 } ) | ^ 2 } { n ^ { 2 / 3 } _ { 2 } } , \\end{align*}"}
+{"id": "1614.png", "formula": "\\begin{align*} | \\mathcal K _ k ( \\hat P ^ { ( k - 1 ) } ) | \\le ( 1 + \\eta ) \\frac { 1 } { a ^ k _ { k - 1 } } \\prod \\limits _ { \\ell = 1 } ^ { k - 2 } ( \\frac { 1 } { a _ { \\ell } } ) ^ { \\binom { k } { \\ell } } n ^ k \\le \\frac { 1 + \\eta } { 1 - \\eta } \\cdot \\frac { 1 } { a ^ k _ { k - 1 } } \\cdot | \\mathcal K _ k ( \\mathcal G _ T ) \\cap \\mathrm { C r o s s } _ { \\mathcal Y } | \\end{align*}"}
+{"id": "8484.png", "formula": "\\begin{align*} \\theta \\left ( \\{ g < M \\} \\cap E ; x \\right ) = 0 , \\mbox { f o r e v e r y } M > 0 ( s = + \\infty ) , \\end{align*}"}
+{"id": "6901.png", "formula": "\\begin{align*} \\liminf _ { ( t _ n , h _ n ) \\to ( 0 ^ + , h ) } d ( - J \\lambda ( p ) h _ n + v , t _ n ^ { - 1 } N \\Gamma ( p + t _ n h _ n ) ) = 0 . \\end{align*}"}
+{"id": "7664.png", "formula": "\\begin{align*} \\psi ( u ) = \\mathbb { P } ( \\tau _ u < \\infty ) . \\end{align*}"}
+{"id": "7387.png", "formula": "\\begin{align*} \\underline { \\rho _ { n } } : = \\inf _ { t \\in [ 0 , T ^ { * } ) } \\min _ { x \\in \\mathbb { T } } \\rho _ { n } ( t , x ) > 0 , \\end{align*}"}
+{"id": "354.png", "formula": "\\begin{align*} G r ( R ) = \\{ ( R ( h ) , h ) \\mid h \\in H \\} \\end{align*}"}
+{"id": "7013.png", "formula": "\\begin{align*} ( r _ 1 a _ 2 ) ^ 2 - ( r _ 2 \\sqrt { a _ 1 a _ 2 } ) ^ 2 = 4 a _ 2 ( a _ 2 - a _ 1 ) \\end{align*}"}
+{"id": "80.png", "formula": "\\begin{align*} \\sum _ { k \\in \\mathcal { P } _ H } \\mathcal T _ { \\rm { c o m } } ( k ) = - \\rho _ z \\sum _ { p \\in \\mathcal P _ L } \\big ( \\widehat { g \\omega } ( 0 ) + \\widehat { g \\omega } ( p ) \\big ) a _ p ^ \\dagger a _ p + \\mathcal E '' + \\mathcal E ' + \\mathcal E , \\end{align*}"}
+{"id": "3998.png", "formula": "\\begin{align*} S _ n ( X ) = \\frac { c } { C ^ m _ n } S _ m ( X ) + b _ t f , \\end{align*}"}
+{"id": "2349.png", "formula": "\\begin{align*} f _ { u g } ( \\tau _ { u h } ) & = f _ { u g } ( \\theta _ \\tau \\delta _ { u h } ) = \\zeta _ { u g } ( \\theta _ f \\theta _ \\tau \\delta _ { u h } ) = \\zeta _ { u g } \\left ( \\sum _ { v \\in G } \\eta ( v ) \\rho _ v \\delta _ { u h } \\right ) \\\\ & = \\sum _ { v \\in G } \\eta ( v ) \\zeta _ { u g } \\rho _ v \\delta _ { u h } = \\sum _ { v \\in G } \\eta ( v ) \\zeta _ { u g } \\delta _ { u h v ^ { - 1 } } = \\eta ( g ^ { - 1 } h ) = f _ g ( \\tau _ h ) , \\forall u , g , h \\in G . \\end{align*}"}
+{"id": "7701.png", "formula": "\\begin{align*} \\wedge ^ 2 \\psi _ 1 \\circ \\phi _ E = \\phi _ F \\circ \\widetilde { \\psi } . \\end{align*}"}
+{"id": "5134.png", "formula": "\\begin{align*} { \\sf P r } ( Y _ t = y _ t \\mid { \\sf W } _ k ^ { ( L ) } = w ) = \\mbox { t r } ( \\rho _ t ' M _ { t , y _ t } ) , ~ ~ \\forall t \\in [ T ] \\end{align*}"}
+{"id": "7011.png", "formula": "\\begin{align*} | a _ 2 m ( m + 1 ) - b n ( n + 1 ) | = r _ 2 ( m - n ) . \\end{align*}"}
+{"id": "8703.png", "formula": "\\begin{align*} \\int e ^ { \\frac { \\phi _ 0 y _ 0 + \\phi _ 1 y _ 1 + . . . + \\phi _ n y _ n } { \\hbar } } ~ \\psi ~ d \\phi _ 1 \\wedge . . . d \\phi _ n = ( 2 \\pi \\hbar ) ^ { \\frac { n } { 2 } } \\cdot y _ 0 ^ { - \\frac { n } { 2 } } ~ \\hat { g } \\Bigg ( \\frac { y _ 1 } { y _ 0 } , . . . , \\frac { y _ n } { y _ 0 } \\Bigg ) \\cdot e ^ { \\frac { y _ 0 } { \\hbar } \\hat { f } \\big ( \\frac { y _ 1 } { y _ 0 } , . . . , \\frac { y _ n } { y _ 0 } \\big ) } . \\end{align*}"}
+{"id": "7375.png", "formula": "\\begin{align*} \\begin{array} { l } \\mathbb { P } ( l o g ( R _ { j } ^ { k } ) = r _ { j } ^ { k } | Y _ { i j } ^ { ( k ( } < a _ 1 ~ o r ~ Y _ { i j } ^ { ( k ( } > a _ 2 , b _ k ^ { ( 2 ) } ) \\\\ = \\mathbb { E } [ ~ \\mathbb { I } ( l o g ( R _ { j } ^ { k } ) \\le r _ { j } ^ { * k } ) ~ \\vert ~ \\mathbb { I } _ { ( Y _ { i j } ^ { ( k ) } < a _ 1 ) } = 1 ~ o r ~ \\mathbb { I } _ { ( Y _ { i j } ^ { ( k ) } > a _ 2 ) } = 1 ~ , ~ b _ { k } ^ { 2 } ] \\end{array} \\end{align*}"}
+{"id": "7672.png", "formula": "\\begin{align*} \\varphi ( u ) = \\hat { \\varphi } ( u + 1 ) \\end{align*}"}
+{"id": "6174.png", "formula": "\\begin{align*} \\eta _ ( A ) & = \\sum _ { 1 \\leq i ' \\leq n } \\sum _ { j = d _ { i ' - 1 , 1 } , d _ { i ' - 1 , 2 } , \\ldots , d _ { i ' - 1 , i ' - 1 } } \\left ( \\sum _ { 1 \\leq j ' \\leq j } a _ { i ' j ' } \\right ) \\\\ & = \\sum _ { 1 \\leq \\tilde { j } \\leq i ' \\leq n - 1 } \\left ( \\sum _ { 1 \\leq j ' \\leq d _ { i ' \\tilde { j } } } a _ { i ' + 1 , j ' } \\right ) \\\\ & = \\sum _ { 1 \\leq j \\leq i \\leq n - 1 } \\left ( \\sum _ { 1 \\leq j ' \\leq d _ { i j } } a _ { i + 1 , j ' } \\right ) . \\end{align*}"}
+{"id": "3474.png", "formula": "\\begin{align*} & \\omega _ M + d d ^ c \\sum _ 0 ^ m - p _ i \\log | F _ i | _ { h ^ { \\otimes d _ i } } = d d ^ c \\phi _ h + d d ^ c \\sum _ 0 ^ m - p _ i \\log ( | F _ i | e ^ { - d _ i \\phi _ h } ) \\\\ = & ( 1 + \\sum d _ i p _ i ) \\omega _ M - \\sum p _ i d d ^ c \\log | F _ i | \\geq 0 , \\end{align*}"}
+{"id": "1693.png", "formula": "\\begin{align*} \\left . \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } \\right | _ { q , q _ 0 , q _ 1 = 0 } = \\frac { \\pi ( \\lambda + \\varrho _ { \\texttt { b } } ) } { n + m + 1 } ( \\lambda \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { b } } ) , \\end{align*}"}
+{"id": "159.png", "formula": "\\begin{align*} E _ { r + 1 } ( x ) = x ( 1 - x ) E ' _ r ( x ) + ( 1 + r x ) E _ r ( x ) . \\end{align*}"}
+{"id": "3031.png", "formula": "\\begin{align*} M _ 3 ( G ) & = M _ 3 ( T ) + M _ 3 ( C Q ) + \\nu _ 2 M _ 2 ( T ) + ( \\nu _ 1 - 1 ) M _ 2 ( C Q ) \\\\ & = \\frac { ( \\nu _ 1 + \\nu _ 2 ) ^ 3 - 1 2 ( \\nu _ 1 + \\nu _ 2 ) ^ 2 + 3 5 ( \\nu _ 1 + \\nu _ 2 ) + 6 ( x + y ) } { 6 } \\\\ & = f ( n ) + x + y . \\end{align*}"}
+{"id": "2818.png", "formula": "\\begin{align*} \\left | \\mathcal { I } _ s \\left ( z _ k , \\frac { 1 } { ( t + | x | ^ 2 ) ^ { \\frac { \\theta } { 2 } } } \\right ) ( x ) \\right | & = \\frac { c _ { N , s } } { 2 } \\left | \\int _ { \\Omega _ { 2 / k } } \\frac { - \\rho ( k \\delta _ \\Omega ( x ) ) \\big ( ( t ^ 2 + | x | ^ 2 ) ^ { - \\frac { \\theta } { 2 } } - ( t ^ 2 + | y | ^ 2 ) ^ { - \\frac { \\theta } { 2 } } \\big ) } { | x - y | ^ { N + 2 s } } d y \\right | \\\\ & \\leq \\bar { c } ( \\Omega ^ { \\prime } , \\Omega ^ { \\prime \\prime } , N , s , t ) \\ , \\textrm { v o l } ( \\Omega _ { 2 / k } ) , \\end{align*}"}
+{"id": "3880.png", "formula": "\\begin{align*} I _ 4 ( m , t ) = : T _ 1 ( m , t ) + T _ 2 ( m , t ) + T _ 3 ( m , t ) + T _ 4 ( m , t ) + T _ 5 ( m , t ) + T _ 6 ( m , t ) , \\end{align*}"}
+{"id": "4666.png", "formula": "\\begin{align*} s ^ { ( k + 1 ) } = ( M _ 1 + \\Omega _ { 1 } + I - L _ 1 ) ^ { - 1 } [ ( N _ { 1 } + I - L _ 1 ) s ^ { ( k ) } + ( \\Omega _ 1 - A _ { 1 } ) | s ^ { ( k ) } | - r q ] \\end{align*}"}
+{"id": "4242.png", "formula": "\\begin{align*} d T = \\frac { \\alpha ' } { 4 } ( { \\rm t r } \\ , \\Omega ^ { c } \\ ! \\wedge \\Omega ^ { c } - { \\rm t r } \\ , \\Omega ^ { A } \\ ! \\wedge \\Omega ^ { A } ) = - \\frac { \\alpha ' } { 4 } { \\rm t r } \\ , \\Omega ^ { A } \\ ! \\wedge \\Omega ^ { A } , \\end{align*}"}
+{"id": "3284.png", "formula": "\\begin{align*} \\frac { d ^ \\ell } { d t ^ \\ell } \\phi _ X ( t ) = \\int _ \\mathbb { R } x ^ \\ell f _ X ( x ) e ^ { \\mu t x } d x \\mu ^ \\ell . \\end{align*}"}
+{"id": "139.png", "formula": "\\begin{align*} { \\rm B r } ( \\phi ) : = { \\rm R e s } _ { ( G ^ \\sigma / K ^ \\sigma \\times G ^ \\sigma / K ^ \\sigma ) } ( \\phi ) \\end{align*}"}
+{"id": "7110.png", "formula": "\\begin{align*} T ( X , 3 \\lambda Z ) = | X | ^ { - 1 / 2 } \\ , e ( \\mathfrak { s } ) \\ , T _ 1 ( \\mathfrak { s } ) + O _ A ( N ^ { - A } ) , \\end{align*}"}
+{"id": "188.png", "formula": "\\begin{align*} & \\left \\{ e ^ { \\frac { 1 } { 2 4 } A _ 1 } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T _ C M } - \\widetilde { L _ C } - 8 + W _ i ) - e ^ { \\frac { 1 } { 2 4 } A _ 1 } A _ 1 \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 4 ) } \\\\ & = - 2 6 4 \\left \\{ e ^ { \\frac { 1 } { 2 4 } A _ 1 } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 4 ) } . \\end{align*}"}
+{"id": "1853.png", "formula": "\\begin{gather*} \\mathcal { E } _ 1 = \\bigcup _ { \\substack { ( m _ 0 , \\ldots , m _ N ) \\in \\mathbb { Z } ^ { N + 1 } \\setminus \\{ \\mathbf { 0 } \\} \\\\ \\forall n , ~ | m _ n | \\leq \\Delta } } \\left \\{ \\alpha \\in \\mathcal { A } ~ \\middle | ~ \\prod _ { n = 0 } ^ { N } ( n + \\alpha ) ^ { m _ n } = 1 \\right \\} . \\end{gather*}"}
+{"id": "8097.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j = N } ^ { \\infty } \\mu _ { j } b _ { j } \\right \\| _ { h _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = N } ^ { \\infty } \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } . \\end{align*}"}
+{"id": "925.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathcal { T } _ t ^ \\alpha u = u _ { x x } + \\frac { c } { x } u _ x + \\frac { m } { x } v _ x , \\\\ & \\mathcal { T } _ t ^ \\alpha v = v _ { x x } + \\frac { c } { x } v _ x + \\frac { n } { x } u _ x , ~ ~ x > 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7706.png", "formula": "\\begin{align*} & f ^ j ( X _ j ) _ x ( x ^ i ) = 0 \\forall \\ ; \\ ; i = 1 , \\ldots , m _ 0 , \\ ; \\implies f ^ j = 0 , \\\\ & g ^ \\nu ( Y _ \\nu ) _ x ( e ^ \\mu ) = 0 \\forall \\ ; \\mu = 1 , \\ldots , m _ 1 , \\ ; \\implies g ^ \\nu = 0 , \\\\ & h ^ J ( Z _ J ) _ x ( p ^ I ) = 0 \\forall \\ ; I = 1 , \\ldots , m _ 2 , \\ ; \\implies h ^ J = 0 , \\end{align*}"}
+{"id": "2573.png", "formula": "\\begin{align*} \\ell _ 0 ( \\bar { X } ) & : = \\frac { 1 } { ( n + \\sqrt { n } ) \\sqrt { N } } ( \\sqrt { n N } - \\sum _ { 1 \\le j \\le n } X _ j ) , \\\\ \\ell _ i ( \\bar { X } ) & : = \\frac { 1 } { ( n + \\sqrt { n } ) \\sqrt { N } } ( X _ i + \\sqrt { N } ) \\hbox { f o r } i = 1 , \\dots , n . \\end{align*}"}
+{"id": "5566.png", "formula": "\\begin{align*} h ( r ) = U _ k ( \\zeta _ 0 ) . \\end{align*}"}
+{"id": "1967.png", "formula": "\\begin{align*} \\sum _ { \\sigma = 0 } ^ { p ^ k - 1 } \\mathcal { A } ( \\mathbf { a } _ \\sigma ^ t ) ( \\tau ) = \\sum _ { r = 0 } ^ { p ^ { m } - \\tau - 1 } \\sum _ { \\sigma = 0 } ^ { p ^ { k } - 1 } \\omega _ q ^ { \\left ( a ^ t _ { \\sigma , r } - a ^ t _ { \\sigma , r + \\tau } \\right ) } = 0 , \\end{align*}"}
+{"id": "3864.png", "formula": "\\begin{align*} M ( s , t ) = ( t , y ( s ) , z ( s ) ) . \\end{align*}"}
+{"id": "8534.png", "formula": "\\begin{align*} | D \\ell | ( J ) = \\sup \\left \\{ \\sum _ { i = 1 } ^ { N - 1 } | \\ell ( z _ { i + 1 } ) - \\ell ( z _ { i } ) | : \\ a < z _ { 1 } < z _ { 2 } < \\cdots < z _ { N } < b \\right \\} , \\end{align*}"}
+{"id": "2730.png", "formula": "\\begin{align*} E \\Bigg ( \\sum _ { i = 1 } ^ { n } a _ i e _ i \\Bigg ) = \\sum _ { i \\in E } a _ i e _ i . \\end{align*}"}
+{"id": "8298.png", "formula": "\\begin{align*} S = \\prod _ i S _ i , \\end{align*}"}
+{"id": "1387.png", "formula": "\\begin{align*} \\left ( h ^ { j } \\right ) _ { \\delta } = \\left ( h _ { \\delta } \\right ) ^ { j } j = 1 , 2 , . . . \\end{align*}"}
+{"id": "2562.png", "formula": "\\begin{align*} S _ K ^ { \\frac { N ( r + 1 ) } { N + p r s } } & < C _ 2 \\varepsilon _ W ( \\ell ^ { - 1 } ( C _ 3 \\varepsilon _ W ) ^ { q _ 1 } + \\ell ^ { - 1 } ( C _ 3 \\varepsilon _ W ) ^ { q _ 2 } ) = O ( \\varepsilon _ W ^ { \\frac { p } { p - \\min \\{ q _ 1 , q _ 2 \\} } } ) \\end{align*}"}
+{"id": "8372.png", "formula": "\\begin{align*} \\tilde Y _ F ^ { 1 } ( \\omega _ 2 , x ) = S _ B ( t ) \\tilde Y _ F ^ { 1 } ( \\theta _ { - t } \\omega _ 2 , x ) + \\int _ { - t } ^ 0 S _ { B } ( - r ^ \\prime ) \\tilde g _ 1 ( x , \\tilde Y ^ 1 _ F ( \\theta _ { r ^ \\prime } \\omega _ 2 , x ) , \\theta _ { r ^ \\prime } \\omega _ 2 ) d r ^ \\prime . \\end{align*}"}
+{"id": "14.png", "formula": "\\begin{align*} \\varphi _ { \\mathbb { R } ^ 2 } : = 2 \\delta \\varphi ^ { 0 } _ { \\mathbb { R } ^ 2 } \\end{align*}"}
+{"id": "3327.png", "formula": "\\begin{align*} 1 - z _ i = \\prod _ { j = 1 } ^ N z _ j ^ { A _ { i j } } , \\end{align*}"}
+{"id": "5800.png", "formula": "\\begin{align*} w _ 0 = s _ n ( s _ { n - 1 } s _ n s _ { n - 1 } ) ( s _ { n - 2 } s _ { n - 1 } s _ n s _ { n - 1 } s _ { n - 2 } ) \\dots ( s _ 1 s _ 2 \\dots s _ n \\dots s _ 2 s _ 1 ) \\end{align*}"}
+{"id": "3687.png", "formula": "\\begin{align*} B ^ { T } _ { n , k - 1 , k - 2 } \\phi _ u ( \\eta ) = \\sum _ { \\substack { \\tau \\in \\binom { [ n ] } { k - 1 } , \\\\ \\eta \\subset \\tau } } \\phi _ u ( \\tau ) = 0 , \\end{align*}"}
+{"id": "2604.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\tilde { \\alpha } } ( q ( x ) ) = \\sum _ { i } q _ i \\tilde { \\alpha } _ i = \\sum _ { i } \\sum _ { j } p _ j q _ i \\alpha _ { i + j } = \\mathcal { L } _ { \\alpha } ( p ( x ) q ( x ) ) \\geq 0 . \\end{align*}"}
+{"id": "2084.png", "formula": "\\begin{align*} \\begin{aligned} & ( \\Lambda , \\phi , \\alpha , f ) \\hbox { s o l u t i o n , t h e n } \\\\ & \\left ( \\Lambda , \\phi + k \\pi , C _ 1 \\alpha , C _ 2 f \\right ) \\hbox { i s a l s o s o l u t i o n } , k \\in \\mathbb Z , C _ 1 , C _ 2 > 0 . \\end{aligned} \\end{align*}"}
+{"id": "4659.png", "formula": "\\begin{align*} k _ v \\cdot \\bigl ( ( \\mathbf { X } _ { v | s } \\otimes \\mathbf { Y } _ { v | t } ) \\cdot B \\bigr ) = \\bigl ( k _ v \\bullet _ v ( \\mathbf { X } _ { v | s } \\otimes \\mathbf { Y } _ { v | t } ) \\bigr ) \\cdot \\bigl ( k _ v \\cdot B \\bigr ) \\end{align*}"}
+{"id": "6660.png", "formula": "\\begin{align*} x ^ 3 R ''' ( x ) + 4 x ^ 2 R '' ( x ) + 2 x ( 1 - 2 p ^ 2 - 4 q x + 2 x ^ 2 ) R ' ( x ) - 4 ( p ^ 2 + q x ) R ( x ) = 0 . \\end{align*}"}
+{"id": "7049.png", "formula": "\\begin{align*} = \\sqrt { \\frac { } { } } = \\sqrt { \\frac { N } { ( p _ 1 Q K ) ^ 3 / N } } = \\frac { N } { ( p _ 1 Q K ) ^ { 3 / 2 } } . \\end{align*}"}
+{"id": "6094.png", "formula": "\\begin{align*} \\begin{aligned} \\nabla f ( \\omega _ { [ k ] } ) = \\left \\{ \\begin{array} { l l } \\nabla f _ j ( \\omega _ k ) , & \\forall j \\in S _ k \\\\ \\nabla f _ j ( \\omega _ { [ k - 1 ] } ) , & \\forall j \\notin S _ k \\\\ \\end{array} \\right . \\end{aligned} \\end{align*}"}
+{"id": "3715.png", "formula": "\\begin{align*} 2 \\alpha \\int _ { G _ 0 ^ + } \\frac { x _ 1 } { | x | ^ { 2 + 2 \\alpha } } d x = \\int _ 0 ^ 1 \\frac { d x _ 2 } { ( x _ 2 ^ 2 + 2 ^ 2 ) ^ \\alpha } + \\int _ 0 ^ 1 \\frac { 2 ^ { - \\alpha } } { ( ( x _ 2 + 1 ) ^ 2 + 1 ) ^ \\alpha } d x _ 2 = 2 ^ { 1 - 2 \\alpha } f \\left ( \\frac 1 2 \\right ) + 2 ^ { - \\alpha } ( f ( 2 ) - f ( 1 ) ) . \\end{align*}"}
+{"id": "4057.png", "formula": "\\begin{align*} \\rho _ { k \\left ( n - \\frac { 1 } { 2 } \\right ) + 1 } = k \\left ( n - \\frac { 1 } { 2 } \\right ) \\pi , \\ , n \\in \\mathbb { Z } ; \\end{align*}"}
+{"id": "1772.png", "formula": "\\begin{gather*} A ( s , t ) = \\{ e ^ { i \\theta } \\mid s < \\theta < t \\} . \\end{gather*}"}
+{"id": "3218.png", "formula": "\\begin{align*} \\Delta ^ { \\mathrm { P L S Q R } } _ { { \\ell : k } } \\equiv \\sum _ { j = { \\ell } } ^ { { k } } \\hat { \\phi } _ { j + 1 } ^ { \\ , 2 } \\approx \\| \\hat { x } - \\hat { x } _ { { \\ell } } \\| ^ 2 _ { \\hat { A } ^ T \\hat { A } } = \\| { x } - { x } _ { { \\ell } } \\| ^ 2 _ { { A } ^ T { A } } . \\end{align*}"}
+{"id": "763.png", "formula": "\\begin{align*} T _ { \\mathbf U } = \\sup \\left \\{ t \\in [ 0 , T ] \\colon \\norm { \\mathbf U } _ t ^ 2 < 2 R \\right \\} . \\end{align*}"}
+{"id": "1282.png", "formula": "\\begin{align*} A _ n = & \\int _ { \\R ^ 3 } [ I _ \\alpha \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } ( | ( g _ n ^ 1 ) ^ { - 1 } u _ n | ^ p - | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p ) ] | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n | ^ p d x \\\\ & - \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p ) | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n | ^ p d x . \\end{align*}"}
+{"id": "7853.png", "formula": "\\begin{align*} s = s ( \\Delta ) = ( h ; m _ 1 , \\ldots , m _ l ) , \\end{align*}"}
+{"id": "1898.png", "formula": "\\begin{align*} a _ h ( u _ h , \\phi _ h ) : = - \\sum _ { i = 1 } ^ { N _ x } \\int _ { I _ i } \\frac { u _ h ^ 2 } { 2 } \\partial _ x \\phi _ h \\ , { \\rm d } x - \\sum _ { i = 0 } ^ { N _ x - 1 } \\left ( \\frac { \\widehat { u _ h ^ 2 } [ \\ ! [ \\phi _ h ] \\ ! ] } { 2 } \\right ) _ { i + 1 / 2 } \\end{align*}"}
+{"id": "1711.png", "formula": "\\begin{align*} \\Delta ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } = \\frac { m } { n } \\left ( \\det \\left [ H ^ { ( m , n ) } _ { \\texttt { a } ; j , k } ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } ) \\right ] _ { 1 \\leq j , k \\leq n } \\right ) ^ { - 1 } , \\end{align*}"}
+{"id": "1774.png", "formula": "\\begin{gather*} \\mathbb { X } _ \\alpha ( n ) ( \\omega ^ 1 , \\omega ^ 2 ) = \\mathbb { X } _ \\alpha ( n ) ( \\omega ^ 1 ) \\quad \\mathbb { Y } _ \\alpha ( n ) ( \\omega ^ 1 , \\omega ^ 2 ) = \\mathbb { Y } _ \\alpha ( n ) ( \\omega ^ 2 ) \\end{gather*}"}
+{"id": "5787.png", "formula": "\\begin{align*} s _ { \\alpha } s _ { \\beta } s _ { \\alpha } = \\begin{cases} s _ { \\alpha + \\beta } ( \\alpha , \\beta ) = - \\frac { 3 } { 2 } , \\\\ s _ { \\alpha - \\beta } ( \\alpha , \\beta ) = \\frac { 3 } { 2 } . \\\\ \\end{cases} \\end{align*}"}
+{"id": "4489.png", "formula": "\\begin{align*} e ' _ k : = \\mathcal { L } ( { \\mathbf V } _ { k + 1 } , \\Psi _ { k + 1 } ) - \\mathcal { L } ( { \\mathbf V } _ { k } , \\Psi _ { k } ) - \\mathcal { L } ' ( { \\mathbf V } _ { k } , \\Psi _ { k } ) ( \\delta { \\mathbf V } _ k , \\delta \\Psi _ k ) , \\end{align*}"}
+{"id": "1457.png", "formula": "\\begin{align*} \\sigma _ { s } ^ { * } = \\sigma _ { s } + 2 \\sum \\limits _ { t \\in J } k ^ { s t } \\big ( \\mu _ t ^ { * } + \\mu _ { t ^ { * } } ^ { * } \\big ) = \\sigma _ { s } + 2 \\sum \\limits _ { q = j - 1 } ^ { s - 1 } \\left ( \\sum \\limits _ { k = j } ^ { 2 j + | J | - 2 - q } \\mu ^ { * } _ { k } - \\sum \\limits _ { k = j } ^ { q } \\mu ^ { * } _ k \\right ) , \\ \\ s \\in J . \\end{align*}"}
+{"id": "7474.png", "formula": "\\begin{align*} h _ { a , c } ( x , z ) = h _ { b , a } ( y , x ) = h _ { c , b } ( z , y ) , a + b + c = ( 1 , - 1 ) , x y z = 1 \\ , . \\end{align*}"}
+{"id": "4359.png", "formula": "\\begin{align*} \\tilde { \\omega } ( A ) = ( F _ \\omega \\Omega , \\pi _ \\omega ( A ) F _ \\omega \\Omega _ \\omega ) \\ \\ ( A \\in { \\tt C A R } ( \\mathcal { K } , \\Gamma ) ) \\end{align*}"}
+{"id": "5152.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\frac { { \\rm A o I } ( S _ { \\mathrm { z } } , Q _ { \\mathrm { u n i } } , F ^ { * } ) } { \\log _ 2 D } = - \\frac { 3 } { 4 } , \\end{align*}"}
+{"id": "1309.png", "formula": "\\begin{align*} - \\Delta u = P _ \\varepsilon ( \\mu _ t ^ { B , H } - \\mathfrak { m } ) . \\end{align*}"}
+{"id": "4278.png", "formula": "\\begin{align*} I ( i , J , p , n ) = \\prod _ { s = 1 } ^ { k } \\chi _ { [ 1 , n ] } ( i + \\sum _ { \\ell = 1 } ^ { 2 s - 1 } ( - 1 ) ^ \\ell j _ \\ell ) \\chi _ { [ 1 , p ] } ( i + \\sum _ { \\ell = 1 } ^ { 2 s } ( - 1 ) ^ \\ell j _ \\ell ) . \\end{align*}"}
+{"id": "7572.png", "formula": "\\begin{align*} \\kappa = \\begin{cases} \\beta ^ { \\frac { 1 } { 3 } } N ^ { \\frac { 1 } { 3 } } , & d = 1 , \\\\ \\beta ^ { \\frac { 1 } { 4 } } N ^ { \\frac { 1 } { 4 } } , & d = 2 , 3 , \\\\ \\end{cases} \\alpha = \\begin{cases} 0 , & \\ d = 1 , \\\\ - \\frac { 1 } { 4 } , & \\ d = 2 , 3 . \\end{cases} \\end{align*}"}
+{"id": "3388.png", "formula": "\\begin{align*} b ^ \\mu \\alpha ^ 0 _ { \\nu \\tau } = 0 . \\end{align*}"}
+{"id": "6354.png", "formula": "\\begin{align*} \\| \\partial _ x | u _ \\lambda | ^ 2 \\| _ { L ^ 2 } ^ 2 + \\| B _ \\lambda u \\| _ { L ^ 6 } ^ 6 \\lesssim \\epsilon ^ 4 c _ \\lambda ^ 4 \\lambda ^ { 1 - 4 s } . \\end{align*}"}
+{"id": "8219.png", "formula": "\\begin{align*} \\Phi _ a ^ * \\omega _ 2 = \\omega _ 2 + a ^ 2 d x _ 3 \\wedge d x _ 1 = \\omega _ 2 ^ a , \\end{align*}"}
+{"id": "5143.png", "formula": "\\begin{align*} { \\rm A o I } = \\frac { E [ ( L + Z ( S ) ) ^ 2 ] } { 2 E [ L + Z ( S ) ] } + E [ L ] , \\end{align*}"}
+{"id": "5675.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l } F _ 3 = x _ 0 y + B + x _ 1 ( y + Q ) = 0 , \\\\ G _ 4 = y ( y + Q ) - C = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "1434.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta u _ i + \\sum \\limits _ { j = 1 } ^ 3 k _ { i j } e ^ { u _ j } & = 4 \\pi \\alpha _ i \\delta _ 0 \\ \\ B _ 1 ( 0 ) \\subseteq \\mathbb { R } ^ 2 , \\ \\ \\forall \\ i = 1 , 2 , 3 , \\\\ u _ 1 + u _ 2 + 2 u _ 3 & \\equiv 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "8938.png", "formula": "\\begin{align*} \\int _ { U ^ - U ^ 0 U } \\varphi ( g ) \\ , d m _ G ( g ) = \\int _ { U ^ - \\times U ^ 0 \\times U } \\varphi ( u ^ - u ^ 0 u ) \\Delta ( u ^ 0 ) \\ , d \\nu ^ - ( u ^ - ) d \\nu ^ 0 ( u ^ 0 ) d \\nu ( u ) \\end{align*}"}
+{"id": "3343.png", "formula": "\\begin{align*} \\frac { D _ \\zeta ( \\zeta x ) } { D _ \\zeta ( x ) } = \\frac { ( 1 - x ) ^ m } { 1 - x ^ m } . \\end{align*}"}
+{"id": "6215.png", "formula": "\\begin{align*} V _ { \\hat { g } } \\hat { f } ( \\xi , - x ) = e ^ { i x \\cdot \\xi } V _ g f ( x , \\xi ) , \\end{align*}"}
+{"id": "2396.png", "formula": "\\begin{align*} H = \\bigcup _ { \\epsilon > 0 } H _ { \\epsilon } . \\end{align*}"}
+{"id": "6009.png", "formula": "\\begin{align*} \\left \\Vert \\psi \\right \\Vert _ { k _ { 0 } , \\infty } = \\sum _ { \\left \\vert \\alpha \\right \\vert \\leq k _ { 0 } } \\left \\Vert \\partial ^ { \\alpha } \\psi \\right \\Vert _ { \\infty } . \\end{align*}"}
+{"id": "7426.png", "formula": "\\begin{align*} c = \\frac { ( c x + 1 ) ( x + c ) } { x ^ 2 + \\left ( c + \\frac { 1 } { c } \\right ) x + 1 } , \\end{align*}"}
+{"id": "6105.png", "formula": "\\begin{align*} \\| T \\| _ A = \\sup _ { \\| h \\| _ A = 1 } \\| T h \\| _ A , \\end{align*}"}
+{"id": "1863.png", "formula": "\\begin{gather*} A = \\left \\{ g \\in H ( D ) ~ \\middle | ~ \\sup _ { s \\in K } | g ( s ) - p ( s ) | < \\frac { \\epsilon } { 2 } \\right \\} . \\end{gather*}"}
+{"id": "7982.png", "formula": "\\begin{align*} \\mathrm { J a c o b i a n } = \\det \\left ( \\frac { \\theta ^ { i - j } } { ( i - j ) ! } + \\partial _ { u _ i } \\partial ^ j _ { \\theta } R \\right ) _ { i \\ne \\mathfrak m _ 0 , j \\ne d _ 1 - 1 } + \\sum _ { k \\ne d _ 1 - 1 } \\det \\left ( a ^ { ( k ) } _ { i j } \\right ) _ { i \\ne \\mathfrak m _ 0 , j \\ne d _ 1 - 1 } , \\end{align*}"}
+{"id": "3496.png", "formula": "\\begin{align*} \\norm { s } _ { y , t } ^ 2 = \\int _ { U _ { y , t } } | \\sigma | ^ 2 | z _ 0 | ^ { 2 l _ 0 } \\ldots | z _ m | ^ { 2 l _ m } e ^ { - 2 l \\phi _ h } \\sqrt { - 1 } ^ { n ^ 2 } \\Omega _ t \\wedge \\overline { \\Omega } _ t . \\end{align*}"}
+{"id": "7632.png", "formula": "\\begin{align*} \\langle w , X \\rangle \\geq 0 \\mathbb { P } ^ * \\mbox { - a . s . } & \\Rightarrow \\langle w , X \\rangle = 0 \\mathbb { P } ^ * \\mbox { - a . s . } \\\\ & \\Rightarrow w \\in \\{ w \\in \\R ^ d : \\langle w , x \\rangle = 0 , \\forall x \\in \\mbox { s u p p } ( \\P ^ \\ast ) \\} \\Rightarrow w = 0 . \\end{align*}"}
+{"id": "3351.png", "formula": "\\begin{align*} K \\mathbf { e } ( \\lambda r / m ) = \\mu u _ \\zeta \\end{align*}"}
+{"id": "3171.png", "formula": "\\begin{align*} \\mathfrak { n } _ { k _ 1 , k _ 0 } = \\mathfrak { m } ^ { \\mathbf { L } } _ { k _ 1 + 1 + k _ 0 } . \\end{align*}"}
+{"id": "6619.png", "formula": "\\begin{align*} { S } _ N ( k ; \\beta ) = \\Big \\langle \\Big | \\sum _ { j = 1 } ^ N e ^ { i k \\lambda _ j } \\Big | ^ 2 \\Big \\rangle - \\bigg | \\Big \\langle \\sum _ { j = 1 } ^ N e ^ { i k \\lambda _ j } \\Big \\rangle \\bigg | ^ 2 , \\end{align*}"}
+{"id": "2354.png", "formula": "\\begin{align*} \\mathcal { X } = \\theta _ f ( \\mathcal { X } ) , \\pi = \\Delta , G = \\{ \\lambda _ g | { _ { \\theta _ f ( \\mathcal { X } ) } } : g \\in G \\} , \\tau = \\theta _ f \\theta _ \\tau \\delta _ e , f = \\zeta _ e ( \\theta _ f \\theta _ \\tau ) . \\end{align*}"}
+{"id": "4522.png", "formula": "\\begin{align*} | | D _ { k + \\frac { 1 } { 2 } } \\delta \\Psi _ k | | _ { s , \\ast , T } & \\lesssim | | \\delta \\Psi _ k | | _ { 4 , \\ast , T } | | R _ k | | _ { s , \\ast , T } + | | \\delta \\Psi _ k | | _ { s , \\ast , T } | | R _ k | | _ { 4 , \\ast , T } \\\\ & + | | \\delta \\Psi _ k | | _ { 4 , \\ast , T } | | R _ k | | _ { 4 , \\ast , T } | | \\Psi ^ a + \\Psi _ { k + \\frac { 1 } { 2 } } | | _ { s + 2 , \\ast , T } . \\end{align*}"}
+{"id": "3526.png", "formula": "\\begin{align*} 1 = ( a ' d ' - b ' c ' ) q ^ { 2 } \\pm ( a ' + d ' ) q + 1 , \\end{align*}"}
+{"id": "8025.png", "formula": "\\begin{align*} j _ { \\nu } ( x ) = \\Gamma ( \\nu + 1 ) \\sum _ { k = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ { k } } { k ! \\ , \\Gamma ( \\nu + k + 1 ) } ( \\frac { x } { 2 } ) ^ { 2 k } . \\end{align*}"}
+{"id": "8366.png", "formula": "\\begin{align*} \\lim _ { t \\to \\pm \\infty } \\frac { \\log ^ + X ( \\theta _ t \\omega ) } { | t | } = 0 . \\end{align*}"}
+{"id": "5447.png", "formula": "\\begin{align*} & \\int _ 0 ^ t \\frac { 2 \\rho ( e ^ { 2 \\rho t } - 1 ) } { 2 \\rho \\alpha e ^ { 2 \\rho s } ( e ^ { 2 \\rho t } - 1 ) + ( e ^ { 2 \\rho ( t - s ) } - 1 ) ( e ^ { - 2 \\rho ( t - s ) } - 1 ) } d s \\\\ & \\phantom { A A A A A A A A } = \\frac { \\sqrt { 1 - e ^ { - 2 \\rho t } } } { 2 \\sqrt { 2 \\rho \\alpha } } \\log \\left ( \\frac { ( e ^ { \\rho t } a _ t - 1 ) ( e ^ { \\rho t } a _ t + 1 + a _ t ^ 2 - e ^ { - 2 \\rho t } ) } { ( e ^ { \\rho t } a _ t + 1 ) ( - e ^ { \\rho t } a _ t + 1 + a _ t ^ 2 - e ^ { - 2 \\rho t } ) } \\right ) \\end{align*}"}
+{"id": "8718.png", "formula": "\\begin{align*} \\min \\Big ( \\alpha , \\ , \\frac { d } { \\alpha } T ^ { - \\frac { \\beta - 1 } { \\beta } } \\Big ) = \\min \\Big ( \\max ( \\alpha , T ^ { - 1 / 2 + 1 / \\beta } ) , \\frac { d } { \\sqrt { T } } , \\ , \\frac { d } { \\alpha } T ^ { - \\frac { \\beta - 1 } { \\beta } } \\Big ) \\enspace . \\end{align*}"}
+{"id": "4472.png", "formula": "\\begin{align*} \\sum ^ { \\mu } _ { j = 0 } ( | | \\tilde { { \\mathbf U } } _ j | | _ { H ^ { \\mu + 1 . 5 - j } ( \\mathbb R ^ 2 _ + ) } + | | \\varphi _ j | | _ { H ^ { \\mu + 1 . 5 - j } ( \\R ) } ) \\leq C ( M _ 0 ) , \\end{align*}"}
+{"id": "4228.png", "formula": "\\begin{align*} g ( \\nabla ^ { \\varepsilon , \\rho } _ X Y , Z ) = g ( \\nabla ^ { L C } _ X Y , Z ) + \\varepsilon \\ , T ( X , Y , Z ) + \\rho \\ , C ( X , Y , Z ) , \\quad \\ X , Y , Z \\in \\mathfrak { X } ( M ) \\end{align*}"}
+{"id": "2096.png", "formula": "\\begin{align*} u : = \\dfrac { t + x } { 2 } , \\underline { u } : = \\dfrac { t - x } { 2 } , \\end{align*}"}
+{"id": "5046.png", "formula": "\\begin{align*} \\kappa _ \\infty = u \\ , d r ^ 2 , u \\in L ^ 2 ( N ) . \\end{align*}"}
+{"id": "7288.png", "formula": "\\begin{align*} \\{ \\tau \\mid C ^ s ( \\tau ) < k \\} \\cup F \\cup \\{ \\sigma \\} & \\subseteq \\{ \\tau \\mid C ^ B ( \\tau ) < k \\} \\cup F \\cup \\{ \\sigma \\} \\\\ & = \\{ \\tau \\mid C ^ B ( \\tau ) < k \\} \\cup F \\end{align*}"}
+{"id": "4268.png", "formula": "\\begin{align*} c \\leftarrow J _ { \\lambda _ k } ( u _ j ) = & \\dfrac { 1 } { 2 } \\mathcal { B } _ { \\alpha } ( u _ j ^ + , u _ j ^ + ) + \\dfrac { 1 } { 2 } \\mathcal { B } _ { \\alpha } ( u _ j ^ - , u _ j ^ - ) - \\dfrac { \\lambda _ k } { 2 } \\int _ { \\Omega } \\left ( | u _ j ^ + ( x ) | ^ 2 + | u _ j ^ - ( x ) | ^ 2 \\right ) d x \\\\ & - \\int _ { \\Omega } \\left ( F ( x , u _ j ( x ) ) - F ( x , u _ j ^ 0 ( x ) ) \\right ) d x - \\int _ { \\Omega } F ( x , u _ j ^ 0 ( x ) ) d x . \\end{align*}"}
+{"id": "8065.png", "formula": "\\begin{align*} T _ { N } ( f ) ( x ) = \\sum \\limits _ { j \\in \\mathbb N } \\sum \\limits _ { Q \\in \\Pi _ { j + N } } \\vert Q \\vert \\psi _ { j } ( x - u _ { Q } ) ( \\psi _ { j } \\ast f ) ( u _ { Q } ) , \\end{align*}"}
+{"id": "849.png", "formula": "\\begin{align*} \\| T f _ 0 \\| & = \\| x ^ * ( f _ 0 ) h \\| \\\\ & = | x ^ * ( f _ 0 ) | \\| h \\| \\\\ & = \\| x ^ * \\| \\| h \\| \\\\ & = \\| T \\| , \\end{align*}"}
+{"id": "3613.png", "formula": "\\begin{align*} \\int \\nolimits _ 0 ^ \\infty { { e ^ { - \\mu x } } \\ln \\left ( { 1 + \\beta x } \\right ) d x } = - { \\frac { 1 } { \\mu } } e ^ { \\frac { \\mu } { \\beta } } { \\rm E i } \\left ( - { \\frac { \\mu } { \\beta } } \\right ) . \\end{align*}"}
+{"id": "6878.png", "formula": "\\begin{align*} f _ * \\sigma \\smallfrown \\ell = f _ * ( \\sigma \\smallfrown f ^ * \\ell ) \\in H ^ { B M } _ { \\bullet } ( V , \\mathbb { R } ) . \\end{align*}"}
+{"id": "286.png", "formula": "\\begin{align*} \\left \\langle u , v \\right \\rangle _ { \\omega } : = \\int _ { \\omega } u v \\quad \\left ( u , v \\right ) _ { L ^ { 2 } \\left ( \\omega \\right ) } = \\left \\langle u , \\overline { v } \\right \\rangle _ { \\omega } , \\end{align*}"}
+{"id": "7181.png", "formula": "\\begin{align*} I = \\big < \\{ \\prod _ { k _ j = 1 } ^ { n _ j } \\big ( \\sum _ { \\ell = 1 } ^ m \\alpha _ { \\ell k _ j } ^ j y _ { \\ell } \\big ) y _ j ~ | ~ j = 1 , \\dots , m \\} \\big > . \\end{align*}"}
+{"id": "7970.png", "formula": "\\begin{align*} { ( \\overline { F } \\cdot \\mu _ A ) _ { c , d , e } } _ { ( f , g , y ) } = { \\overline { F } _ { c , e , f g } } _ { ( y ) } , { ( ( \\overline { F } \\cdot T a ) \\circ ( b \\cdot T \\overline { F } ) ) _ { c , d , e } } _ { ( f , g , y ) } = ( \\overline { F } _ { c , d , f } \\circ A ( g ) ) _ y \\circ ( B ( f ) \\circ \\overline { F } _ { d , e , g } ) _ y . \\end{align*}"}
+{"id": "3574.png", "formula": "\\begin{align*} { { \\bf { H } } _ { k , { \\rm { R } } } } = \\sum \\nolimits _ { l \\in { \\mathcal L } _ { { \\rm { R } } } } { { \\alpha _ { { k , { \\rm { R } } } , l } } { { \\bf { a } } _ { \\rm { R } } } \\left ( { \\Phi _ { { k } , l } ^ { \\rm { A } } } \\right ) { \\bf { a } } _ { { \\rm { S , } } k } ^ H \\left ( { \\Phi _ { { k } , l } ^ { \\rm { D } } } \\right ) } , \\end{align*}"}
+{"id": "458.png", "formula": "\\begin{align*} ( | D | ^ \\alpha f ) ( x ) = \\frac { 2 ^ { \\alpha } \\Gamma \\left ( \\frac { 1 } { 2 } ( d + \\ell - \\sigma ) \\right ) \\Gamma \\left ( \\frac { 1 } { 2 } ( \\ell + \\sigma + \\alpha ) \\right ) } { \\Gamma \\left ( \\frac { \\ell + \\sigma } { 2 } \\right ) \\Gamma \\left ( \\frac { 1 } { 2 } ( d + \\ell - \\sigma - \\alpha ) \\right ) } | x | ^ { - \\alpha } \\cdot | x | ^ { - \\sigma } Y _ { \\ell , m } ( \\omega _ x ) \\end{align*}"}
+{"id": "8019.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ D \\sum _ { i \\in \\mathcal { C } _ k ^ { \\{ 1 , \\dots , D \\} } } \\sum _ { h = 0 } ^ { D + 1 - k } \\lambda ^ { D + 1 - ( h + k ) } \\Biggl [ \\binom { D + 1 - k } { h } - \\Bigl ( p _ 0 + \\sum _ { j \\not \\in i } p _ j \\Bigr ) \\binom { D - k } { h } \\Biggr ] \\frac { \\partial ^ { h + k } p } { \\partial t ^ { h } \\partial x _ { i _ 1 } \\cdots \\partial x _ { i _ k } } = 0 \\end{align*}"}
+{"id": "2146.png", "formula": "\\begin{align*} E [ \\Lambda ^ { ( 0 ) } , \\phi ^ { ( 0 ) } ; \\alpha ] ( t ) = \\int \\hat e _ 0 d x = \\int \\frac { 1 } { 2 } [ L ( \\ln \\alpha ) + \\underline { L } ( \\ln \\alpha ) ] d x < \\infty . \\end{align*}"}
+{"id": "7674.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\sum _ { k = 1 } ^ n X _ k ^ * \\leqslant \\sum _ { k = 1 } ^ n X _ k \\right ) = 1 , \\ , n \\in \\mathbb { N } . \\end{align*}"}
+{"id": "2744.png", "formula": "\\begin{align*} U ^ { - 1 } ( e _ { \\sigma ( i ) } ) = \\varepsilon _ i e _ i \\end{align*}"}
+{"id": "5376.png", "formula": "\\begin{align*} - \\cfrac { r } { s } = [ - a _ 1 , \\ldots , - a _ { 2 m } ] , \\end{align*}"}
+{"id": "4767.png", "formula": "\\begin{align*} \\sum _ { n = N } ^ { \\infty } \\mu _ { p } ( \\{ x \\mid v _ { \\phi } ( a ^ { - n } x ) > n \\} ) = \\sum _ { n = N } ^ { \\infty } \\mu _ { p } ( \\{ x \\mid v _ { \\phi } ( x ) > n \\} ) < \\epsilon . \\end{align*}"}
+{"id": "1137.png", "formula": "\\begin{align*} ( \\mathfrak { g } _ { c o m } ) ^ 0 = \\mathfrak { g } ^ 0 ~ ~ ~ ~ ~ ~ ~ ~ ( \\mathfrak { g } _ { c o m } ) ^ n = \\underbrace { \\mathfrak { g } ^ n \\oplus \\cdots \\oplus \\mathfrak { g } ^ n } _ { ( n + 1 ) } , ~ n \\geq 1 . \\end{align*}"}
+{"id": "3432.png", "formula": "\\begin{align*} P _ { E _ J } ( y ) = \\frac { 1 } { ( 1 - y ^ { d _ 0 } ) \\ldots ( 1 - y ^ { d _ m } ) } P _ M ( y ) . \\end{align*}"}
+{"id": "6793.png", "formula": "\\begin{align*} & \\underline { u } ' ( z _ 1 ^ - ) = - \\beta < 0 = \\underline { u } ' ( z _ 1 ^ + ) , \\\\ [ 0 . 2 c m ] & \\overline { v } ' ( 0 ^ + ) = 0 < q ( 1 ) \\lambda _ 1 = \\overline { v } ' ( 0 ^ - ) , \\\\ [ 0 . 2 c m ] & \\underline { v } ' ( z _ 2 ^ - ) = - q ( 1 ) \\varepsilon e ^ { \\lambda _ 1 z _ 2 } < 0 = \\underline { v } ' ( z _ 2 ^ + ) . \\end{align*}"}
+{"id": "6715.png", "formula": "\\begin{align*} \\mathfrak { m } _ { A D M } : = \\frac { 1 } { 1 6 \\pi } \\lim \\limits _ { r \\rightarrow \\infty } \\int _ { S _ r } \\left ( \\partial _ { j } g _ { i j } - \\partial _ { i } g _ { j j } \\right ) \\frac { x ^ i } { | x | } d A _ { \\bar g } , \\end{align*}"}
+{"id": "6828.png", "formula": "\\begin{align*} \\mathcal { Y } _ 4 ( M , g ) = \\mathcal { Y } _ 4 ^ * ( M , g ) \\leqslant \\mathcal { Y } _ 4 ( \\mathbb { S } ^ n , g _ 0 ) \\end{align*}"}
+{"id": "794.png", "formula": "\\begin{align*} c ^ * & = \\min \\left \\{ \\left ( \\frac { 1 } { p } - \\frac { 1 } { 2 \\cdot p ^ \\flat } \\right ) \\left ( \\frac { p ^ \\flat } { b } \\right ) ^ { \\frac { p } { 2 \\cdot p ^ \\flat - p } } ( a S ^ \\flat ) ^ { \\frac { 2 \\cdot p ^ \\flat } { 2 \\cdot p ^ \\flat - p } } , \\left ( \\frac { 1 } { p } - \\frac { 1 } { 2 \\cdot p ^ \\sharp } \\right ) \\left ( \\frac { p ^ \\sharp } { b } \\right ) ^ { \\frac { p } { 2 \\cdot p ^ \\sharp - p } } ( S ^ \\sharp ) ^ { \\frac { 2 \\cdot p ^ \\sharp } { 2 \\cdot p ^ \\sharp - p } } \\right \\} . \\end{align*}"}
+{"id": "7499.png", "formula": "\\begin{align*} \\Delta _ \\gamma = \\gamma ^ { 4 r + 1 } ( \\gamma - 4 ) G _ { m - 1 } ^ 2 ( \\gamma ) = \\gamma ( \\gamma - 4 ) ( \\gamma ^ { 2 r } G _ { m - 1 } ( \\gamma ) ) ^ 2 . \\end{align*}"}
+{"id": "2833.png", "formula": "\\begin{align*} \\ell ^ { ( p ) } ( r e s _ { s ' } ( \\mathfrak { q } ' ) ) - \\ell ^ { ( p ) } ( r e s _ { s } ( \\mathfrak { q } ) ) = r e s _ { s = 0 } \\Omega ^ { ( p ) } ( \\mathfrak { q } ' - \\mathfrak { q } ) . \\end{align*}"}
+{"id": "7080.png", "formula": "\\begin{align*} \\mathcal { V } _ 2 \\left ( \\frac { m } { p _ 2 ( p _ 1 q ) ^ 2 } \\right ) = 2 \\pi i ^ k \\int _ 0 ^ \\infty V _ 2 \\left ( \\frac { y } { N } \\right ) y ^ { - i ( t + \\nu ) } e \\left ( - \\frac { u y } { p _ 1 q Q } \\right ) \\ , J _ { k - 1 } \\left ( \\frac { 4 \\pi } { p _ 1 q } \\sqrt { \\frac { m y } { p _ 2 } } \\right ) \\ { d } y . \\end{align*}"}
+{"id": "6955.png", "formula": "\\begin{align*} 2 \\sum _ { k = 1 } ^ d \\sum _ { i = 1 } ^ { k - 1 } \\sum _ { j = i + 1 } ^ { k - 1 } & \\frac { 1 } { ( ( x _ k - x _ i ) + \\epsilon ^ 2 ) ( ( x _ k - x _ j ) + \\epsilon ^ 2 ) } \\\\ & = 2 \\sum _ { k = 1 } ^ d \\sum _ { i = k + 1 } ^ { d } \\sum _ { j = i + 1 } ^ { d } \\frac { 1 } { ( ( x _ j - x _ k ) + \\epsilon ^ 2 ) ( ( x _ j - x _ i ) + \\epsilon ^ 2 ) } . \\end{align*}"}
+{"id": "4082.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\oint _ { C ^ { \\mathfrak { p } } _ { \\frac 1 2 , 2 } } \\phi ( p ) \\cdot \\omega _ { \\frac 1 2 , 2 } ( p , p _ 1 ) = 0 . \\end{align*}"}
+{"id": "1506.png", "formula": "\\begin{align*} f ( x ) : = \\frac { \\lambda _ { 1 } ^ { ( m ) } } { \\sqrt { 1 - x } } + \\frac { \\lambda _ { 1 } ^ { ( n ) } \\sqrt { 1 - x } } { 4 x } = \\lambda _ { 1 } ^ { ( m ) } \\left ( \\frac { 1 } { \\sqrt { 1 - x } } + \\frac { c \\sqrt { 1 - x } } { 4 x } \\right ) > 0 \\end{align*}"}
+{"id": "3212.png", "formula": "\\begin{align*} \\| x - x _ { k - 1 } \\| ^ 2 - \\| x - x _ { k } \\| ^ 2 = \\zeta _ k ^ 2 \\end{align*}"}
+{"id": "6767.png", "formula": "\\begin{align*} \\bigl | F ( ( z ) , s + t ) | = ( 1 - \\frac { | s + t | } { \\widehat \\alpha } ) \\varphi ( z ) \\le \\frac { \\widehat \\alpha - | s + t | } { \\widehat \\alpha } \\le \\frac { | s | } { \\widehat \\alpha } . \\end{align*}"}
+{"id": "6493.png", "formula": "\\begin{align*} ( L \\Phi ' ) _ { e _ \\pm } = \\ell _ e \\Phi ' _ { e _ \\pm } . \\end{align*}"}
+{"id": "6958.png", "formula": "\\begin{align*} C _ P ^ { a s } ( d ) : = N ( N - 1 ) ( 2 N - 1 ) / 3 + ( 3 - ( - 1 ) ^ d ) N ^ 2 / 2 , \\end{align*}"}
+{"id": "8509.png", "formula": "\\begin{align*} \\mathcal { H } ^ { n - 1 } ( \\partial ^ { * } E \\cap \\{ z = \\bar { z } \\} ) = \\mathcal { H } ^ { n - 1 } ( \\partial ^ { * } F _ { \\ell } \\cap \\{ z = \\bar { z } \\} ) . \\end{align*}"}
+{"id": "2707.png", "formula": "\\begin{align*} \\begin{array} { r c l } [ w ] ^ { \\boxtimes a } & = & [ w ^ { \\boxtimes a } ] . \\end{array} \\end{align*}"}
+{"id": "6593.png", "formula": "\\begin{align*} \\omega _ l ^ { ( r ) } ( a ) = \\omega _ l ^ { ( 0 ) } + ( \\varepsilon + \\delta ) \\varphi _ l ^ { ( r ) } ( a ) \\ ( 1 \\leq l \\leq b ) , \\end{align*}"}
+{"id": "8147.png", "formula": "\\begin{align*} k _ { i j } = p _ { i j } ( 0 0 ) , m _ { i j } = q _ { i j } ( 0 0 ) \\ , , \\end{align*}"}
+{"id": "7087.png", "formula": "\\begin{align*} | z - y | \\ll \\frac { p _ 1 q Q } { N } = \\frac { q } { Q K } . \\end{align*}"}
+{"id": "8865.png", "formula": "\\begin{align*} \\Xi _ T : = d ^ { - \\frac { 2 ( \\beta - 1 ) } { 2 \\beta - 1 } } T ^ { - \\frac { \\beta } { 2 \\beta - 1 } } \\enspace . \\end{align*}"}
+{"id": "395.png", "formula": "\\begin{align*} \\mathcal { F } \\cap \\mathcal { W } & = \\mathsf { T F i b } , \\\\ \\mathcal { C } \\cap \\mathcal { W } & = \\mathsf { T C o f } , \\end{align*}"}
+{"id": "4534.png", "formula": "\\begin{align*} \\mathrm { d i v } \\dot { \\mathbf h } ^ { \\pm } = 0 \\Omega _ T \\quad \\mbox { a n d } \\quad \\hat { H } ^ { \\pm } _ 2 \\partial _ 2 \\varphi - \\dot { H } ^ { \\pm } _ { N } \\mp \\varphi \\partial _ 1 \\hat { H } ^ { \\pm } _ { N } = 0 \\Gamma _ T \\ , , \\end{align*}"}
+{"id": "8551.png", "formula": "\\begin{align*} P ( F _ { \\ell } ; J \\times \\mathbb { R } ^ { n - 1 } ) & \\leq P ( \\widetilde { E } ; J \\times \\mathbb { R } ^ { n - 1 } ) \\leq \\liminf _ { k \\rightarrow \\infty } P ( E ^ { k } ; J \\times \\mathbb { R } ^ { n - 1 } ) \\\\ & = \\liminf _ { k \\rightarrow \\infty } P ( F _ { \\ell ^ { k } } ; J \\times \\mathbb { R } ^ { n - 1 } ) = \\lim _ { k \\rightarrow \\infty } P ( F _ { \\ell ^ { k } } ; J \\times \\mathbb { R } ^ { n - 1 } ) \\\\ & = P ( F _ { \\ell } ; J \\times \\mathbb { R } ^ { n - 1 } ) , \\end{align*}"}
+{"id": "8589.png", "formula": "\\begin{align*} \\phi _ 0 = \\dfrac { \\cosh { ( ( z + 1 ) \\sqrt { \\mu } | \\mathrm { D } | ) } } { \\cosh { ( \\sqrt { \\mu } | \\mathrm { D } | ) } } \\psi = \\mathrm { F } _ 0 \\psi . \\end{align*}"}
+{"id": "8075.png", "formula": "\\begin{align*} B _ { i , 0 } = \\left \\{ P \\colon P \\in \\Pi _ { N } , \\vert P \\cap \\Omega _ { i , 0 } \\vert > \\frac { 1 } { 2 } \\vert P \\vert , \\vert P \\cap \\Omega _ { i + 1 , 0 } \\vert \\leq \\frac { 1 } { 2 } \\vert P \\vert \\right \\} \\end{align*}"}
+{"id": "1399.png", "formula": "\\begin{align*} \\rho _ n ^ { - ( 2 M _ 1 + 1 ) } \\Delta ( \\rho _ n ^ 2 ) = - \\sin \\rho _ n \\pi + \\varkappa _ { \\pi } ( \\rho _ n ) + O \\left ( n ^ { - 1 } \\right ) = 0 . \\end{align*}"}
+{"id": "15.png", "formula": "\\begin{align*} \\mathcal H _ N = \\sum _ { j = 1 } ^ N - \\Delta _ j + \\sum _ { 1 \\leq i < j \\leq N } v ( x _ i - x _ j ) , \\end{align*}"}
+{"id": "33.png", "formula": "\\begin{align*} \\mathcal { Q } _ 0 ^ { \\rm r e n } = \\frac { n _ 0 ( n _ 0 - 1 ) } { 2 | \\Lambda | } ( \\widehat { g } ( 0 ) + \\widehat { g \\omega } ( 0 ) ) , \\end{align*}"}
+{"id": "3094.png", "formula": "\\begin{align*} A _ n = \\frac { 1 } { | S _ n | } \\sum _ { \\tau \\in S _ n } \\tau ( D ) \\in P S D _ { \\binom { [ n ] } { 2 } } . \\end{align*}"}
+{"id": "8365.png", "formula": "\\begin{align*} \\phi ( t , \\omega , Y ( \\omega ) ) = Y ( \\theta _ t \\omega ) \\end{align*}"}
+{"id": "8168.png", "formula": "\\begin{align*} & [ Y ^ \\star , [ Y ^ \\star , Y ] ] = - 2 ( 1 - r ) ( Y ^ \\star ) ^ 2 + ( n ( 1 - r ) - X ) Y ^ \\star + Y , \\\\ & [ Y , [ Y ^ \\star , Y ] ] = - 2 ( r - 1 ) \\{ Y , Y ^ \\star \\} + Y ( X - n ( 1 - r ) ) - 2 ( k - n ) ( r - 1 ) ( X + k ( r - 1 ) ) \\\\ & - Y ^ \\star \\left ( X ^ 2 + 2 ( 1 - r ) ( n - 2 k - 1 ) X + ( r - 1 ) ^ 2 ( 2 n + ( n - 2 k ) ^ 2 ) \\right ) . \\end{align*}"}
+{"id": "681.png", "formula": "\\begin{align*} B = - b + 1 - \\varepsilon . \\end{align*}"}
+{"id": "4069.png", "formula": "\\begin{align*} \\begin{aligned} \\Theta _ { \\alpha , \\alpha } ( \\rho ) & = \\rho ^ { 2 - 2 \\alpha } \\Delta _ { \\alpha , \\alpha } - \\rho \\sin \\rho + W _ { \\alpha , \\alpha } ( a , \\rho ) , \\\\ \\Theta _ { \\alpha , \\beta } ( \\rho ) & = \\rho \\Delta _ { \\alpha , \\beta } - ( - 1 ) ^ { \\alpha } ( \\cos \\rho - W _ { \\alpha , \\beta } ( a , \\rho ) ) . \\end{aligned} \\end{align*}"}
+{"id": "2325.png", "formula": "\\begin{align*} f _ g = \\zeta _ g U , \\tau _ g = V \\delta _ g , \\forall g \\in G , \\end{align*}"}
+{"id": "1157.png", "formula": "\\begin{align*} & \\sum _ { i + j = n } \\big ( m _ { 2 , i } ( x , m _ { 1 , j } ( y , z ) ) + m _ { 1 , i } ( x , m _ { 2 , j } ( y , z ) ) \\big ) \\\\ & = \\sum _ { i + j = n } \\big ( m _ { 2 , i } ( m _ { 1 , j } ( x , y ) , z ) + m _ { 1 , i } ( m _ { 2 , j } ( x , y ) , z ) - m _ { 2 , i } ( m _ { 1 , j } ( x , z ) , y ) - m _ { 1 , i } ( m _ { 2 , j } ( x , z ) , y ) \\big ) . \\end{align*}"}
+{"id": "8600.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ z ^ 2 \\phi _ 2 = - \\dfrac { \\zeta } { h _ b } \\big ( 1 + \\dfrac { h } { h _ b } \\big ) \\Delta _ X \\psi \\ \\ \\mathrm { i n } \\ \\ \\mathcal { S } _ b , \\\\ \\phi _ 2 | _ { z = 0 } = 0 , \\ \\ \\partial _ z \\phi _ 2 | _ { z = - 1 + \\beta b } = 0 . \\end{cases} \\end{align*}"}
+{"id": "3316.png", "formula": "\\begin{align*} P _ { i + 1 } ^ { ( a + v ) } = \\frac { q ^ { a + v } - q ^ i } { q ^ m - q ^ i } , 1 - P _ { i + 1 } ^ { ( a + v ) } = \\frac { q ^ m - q ^ { a + v } } { q ^ m - q ^ i } . \\end{align*}"}
+{"id": "3339.png", "formula": "\\begin{align*} B ( F ) = A ( F ) / C ( F ) \\end{align*}"}
+{"id": "1072.png", "formula": "\\begin{align*} \\psi \\left ( x \\right ) = x _ { 1 } + A _ { 1 } + 2 , \\varphi _ { \\lambda , \\nu } \\left ( x \\right ) = e ^ { 2 \\lambda \\psi ^ { \\nu } } , \\end{align*}"}
+{"id": "156.png", "formula": "\\begin{align*} \\iota _ n = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathrm { i d } ^ { \\otimes ( i - 1 ) } \\otimes \\iota \\otimes \\mathrm { i d } ^ { \\otimes ( n - i ) } , \\end{align*}"}
+{"id": "2563.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } V | u _ n | ^ p \\eta _ R d x & = \\int _ { \\mathbb { R } ^ N } ( ( V - \\tau _ 0 ) _ { + } + \\tau _ 0 ) | u _ n | ^ p \\eta _ R d x - \\int _ { \\mathbb { R } ^ N } ( V - \\tau _ 0 ) _ { - } | u _ n | ^ p \\eta _ R d x \\\\ & \\geq \\int _ { \\mathbb { R } ^ N } \\tau _ 0 | u _ n | ^ p \\eta _ R d x - \\| ( V - \\tau _ 0 ) _ { - } \\| _ { L ^ { \\frac { N } { p s } } ( \\mathbb { R } ^ N \\setminus B _ R ( 0 ) ) } \\| u _ n \\| _ { p _ s ^ * } ^ { p } \\end{align*}"}
+{"id": "6334.png", "formula": "\\begin{align*} K ( \\xi , \\eta ) = \\int e ^ { i x \\xi - y \\eta } d x \\end{align*}"}
+{"id": "4625.png", "formula": "\\begin{align*} \\mathcal { H } ^ \\pi = \\mathcal { H } ^ \\pi _ { \\Gamma , G , S } \\end{align*}"}
+{"id": "2675.png", "formula": "\\begin{align*} _ { \\omega _ t } \\ = \\ - \\omega _ t + t \\omega _ { 0 } - \\frac { \\alpha } { 2 \\beta } d d ^ c \\varphi _ t . \\end{align*}"}
+{"id": "4639.png", "formula": "\\begin{align*} Q _ { L \\otimes \\underline { S } , N } ( T ) : = \\sum _ { j } T ( n _ j ^ { * } ) \\otimes n _ j , \\end{align*}"}
+{"id": "5890.png", "formula": "\\begin{align*} | E _ { 1 } Z ^ { n } | = | \\Delta | \\sqrt { k } \\geq \\sqrt { 2 ( 1 + \\epsilon ) ^ { \\frac { 1 } { 2 } } V ^ { - 1 } N ^ { \\zeta } } , \\end{align*}"}
+{"id": "1012.png", "formula": "\\begin{align*} I I = ( ( \\omega \\ , k + \\nabla \\varphi ) \\cdot \\nabla ) { \\bf E } & = ( \\omega \\ , k _ 1 + \\varphi _ { x _ 1 } ) { \\bf E } _ { x _ 1 } + ( \\omega \\ , k _ 2 + \\varphi _ { x _ 2 } ) { \\bf E } _ { x _ 2 } + ( \\omega \\ , k _ 3 + \\varphi _ { x _ 3 } ) { \\bf E } _ { x _ 3 } . \\end{align*}"}
+{"id": "815.png", "formula": "\\begin{align*} 1 = \\left ( \\frac { b } { p ^ \\sharp } \\right ) ^ { \\frac { p } { 2 \\cdot p ^ \\sharp } } ( S ^ \\sharp ) ^ { - 1 } ( B ( \\varepsilon _ g ) ) ^ { 1 - \\frac { p } { 2 \\cdot p ^ \\sharp } } + \\varepsilon _ g ^ { \\frac { p } { 2 \\cdot p ^ \\sharp } } ( S ^ * ) ^ { - \\frac { p _ s ^ * } { 2 \\cdot p ^ \\sharp } } ( B ( \\varepsilon _ g ) ) ^ { \\frac { p _ s ^ * - p } { 2 \\cdot p ^ \\sharp } } . \\end{align*}"}
+{"id": "7430.png", "formula": "\\begin{align*} J ( y _ 1 , y _ 2 ) = & \\frac { 1 } { p } \\| ( y _ 1 , y _ 2 ) \\| _ { 1 , p } + \\frac { 1 } { q } \\| ( \\nabla y _ 1 , \\nabla y _ 2 ) \\| _ { q , \\eta } - \\frac { 1 } { 1 - \\nu } \\int _ { \\Omega } \\left [ a _ 1 \\vert y _ 1 \\vert ^ { 1 - \\nu } + a _ 2 \\vert y _ 2 \\vert ^ { 1 - \\nu } \\right ] d z \\\\ & - \\lambda \\int _ { \\Omega } \\vert y _ 1 \\vert ^ { \\kappa _ 1 + 1 } \\vert y _ 2 \\vert ^ { \\kappa _ 2 + 1 } d z . \\end{align*}"}
+{"id": "6063.png", "formula": "\\begin{align*} { \\Phi } ( r , z , \\omega ) = \\vert \\nabla _ x { c } ( z , { X } ^ { M _ { { \\mathcal { P } } _ n } } _ { r - } ) - \\nabla _ x { c } ( z , { X } ^ { M _ { { \\mathcal { P } } _ m } } _ { r - } ) \\vert \\vert D ^ Z { X } ^ { M _ { { \\mathcal { P } } _ m } } _ { r - } \\vert _ { l _ 2 } , \\end{align*}"}
+{"id": "7116.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\Delta U = p & \\hbox { \\rm i n } \\ D \\\\ U _ { | \\partial D } = 0 & \\hbox { \\rm o n } \\ \\partial D \\\\ ( \\nabla U \\cdot n ) _ { | \\partial D } = 0 & \\hbox { \\rm o n } \\ \\partial D \\end{array} \\right . \\end{align*}"}
+{"id": "5175.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\frac { 3 } { 2 } H [ Q _ { \\mathrm { u n i } } ( X ) ] + \\frac { 3 } { 2 } \\log _ 2 { \\delta } - \\frac { 3 } { 2 } h ( X ) = 0 . \\end{align*}"}
+{"id": "7767.png", "formula": "\\begin{align*} \\rho _ { \\pm } ^ { } : = ( \\rho ^ { } \\pm \\rho _ 3 ^ { } ) / 2 \\ , . \\end{align*}"}
+{"id": "2210.png", "formula": "\\begin{align*} ( v , i ) ^ { j \\hat { \\rho } _ 1 } = ( v ^ j , i + 1 ) ^ { \\hat { \\rho } _ 1 } = ( v ^ { j \\rho _ 1 } , - ( i + 1 ) + 1 ) = ( v ^ { \\rho _ 1 j } , - i ) = ( v ^ { \\rho _ 1 } , - i ) ^ j = ( v , i ) ^ { \\hat { \\rho } _ 1 j } . \\end{align*}"}
+{"id": "2166.png", "formula": "\\begin{align*} \\alpha _ t = 0 , \\alpha _ r ^ 2 - \\alpha _ t ^ 2 = 1 > 0 , \\frac { \\alpha _ t } { \\alpha } = 0 . \\end{align*}"}
+{"id": "7031.png", "formula": "\\begin{align*} E [ v ^ \\Phi ] & \\leq F _ L [ \\rho ^ \\Phi ] - \\int _ { \\Omega } v ^ \\Phi \\ , d \\rho ^ \\Phi \\\\ & \\leq E [ v _ \\Phi ] + \\varepsilon \\\\ & \\leq F _ L [ \\rho ] - \\int _ { \\Omega } v ^ \\Phi \\ , d \\rho + \\varepsilon \\\\ & = F _ L [ \\rho ] - \\int _ { \\Omega } v \\ , d \\rho + \\int _ \\Omega ( v ^ \\Phi - v ) \\ , d \\rho + \\varepsilon \\\\ & \\leq E [ v ] + \\int _ \\Omega ( v ^ \\Phi - v ) \\ , d \\rho + 2 \\varepsilon . \\end{align*}"}
+{"id": "1867.png", "formula": "\\begin{align*} \\left \\| \\zeta _ N ( s , \\mathbb { X } _ \\alpha ) ( \\omega ) - \\zeta _ N ( s , \\mathbb { Y } _ \\alpha ) ( \\omega _ 0 ) \\right \\| & \\leq \\sum _ { n = 0 } ^ { N } \\left | \\mathbb { X } _ \\alpha ( n ) ( \\omega ) - \\mathbb { Y } _ \\alpha ( n ) ( \\omega _ 0 ) \\right | \\| ( n + \\alpha ) ^ { - s } \\| \\\\ & \\leq \\sum _ { n = 0 } ^ { N } 2 \\pi \\delta \\sqrt { M } ( n + c / 2 ) ^ { - 1 / 2 } \\\\ & \\leq \\delta B _ 1 \\sqrt { N } \\end{align*}"}
+{"id": "90.png", "formula": "\\begin{align*} b _ + = \\frac { 1 } { \\sqrt { 1 - \\alpha ^ 2 } } \\big ( a _ + + \\alpha a _ - ^ \\dagger + \\bar c _ 0 \\big ) , b _ - = \\frac { 1 } { \\sqrt { 1 - \\alpha ^ 2 } } \\big ( a _ - + \\alpha a _ + ^ \\dagger + c _ 0 \\big ) , \\end{align*}"}
+{"id": "7152.png", "formula": "\\begin{align*} \\| \\langle x \\rangle ^ { s } ( P ( h ) - E \\pm i \\varepsilon ) v \\| _ { L ^ 2 } & \\le \\| ( P ( h ) - E \\pm i \\varepsilon ) \\langle x \\rangle ^ { s } v \\| _ { L ^ 2 } + \\| [ P ( h ) , \\langle x \\rangle ^ { s } ] \\langle x \\rangle ^ { - s } \\langle x \\rangle ^ { s } v \\| _ { L ^ 2 } \\\\ & \\le C _ { E _ { \\max } , \\varepsilon , h } \\| \\langle x \\rangle ^ { s } v \\| _ { H ^ 2 } , \\end{align*}"}
+{"id": "5051.png", "formula": "\\begin{align*} \\chi _ j \\circ \\iota _ { x _ { 1 , j } } = \\chi _ j \\circ \\psi _ { x _ { 1 , j } } \\circ \\iota = \\psi _ { x _ { 2 , j } } \\circ \\chi ' _ j \\circ \\iota = \\psi _ { x _ { 2 , j } } \\circ \\iota = \\iota _ { x _ { 2 , j } } . \\end{align*}"}
+{"id": "3201.png", "formula": "\\begin{align*} - \\frac { \\theta _ k \\phi _ { k - 1 } } { \\rho _ k \\phi _ k } = - \\frac { ( s _ { k - 1 } \\alpha _ k ) ( c _ { k - 1 } \\bar { \\phi } _ { k - 1 } ) } { \\rho _ k \\phi _ k } = \\frac { \\bar { \\rho } _ k \\bar { \\phi } _ { k } } { \\rho _ k \\phi _ k } = 1 \\end{align*}"}
+{"id": "8414.png", "formula": "\\begin{align*} \\left | \\Sigma _ { 0 } \\right | + \\left | \\Sigma _ { 1 } \\right | \\ll \\frac { N ^ { 2 \\gamma - 1 } } { ( \\log N ) ^ { 2 } } + \\sum _ { i = 1 } ^ { 3 } \\Omega _ { i } , \\end{align*}"}
+{"id": "2713.png", "formula": "\\begin{align*} L = ( ( 1 , 1 , 1 , 3 , 2 ) , ( 1 , 0 , 2 , 3 , 2 ) , ( 1 , 0 , 2 , 2 , 3 ) , ( 1 , 0 , 1 , 3 , 3 ) , ( 0 , 1 , 1 , 3 , 3 ) ) \\end{align*}"}
+{"id": "1022.png", "formula": "\\begin{align*} S _ x M = \\left \\{ v \\in T _ x M ~ | ~ | v | ^ 2 _ x = 1 \\right \\} \\subset S M \\end{align*}"}
+{"id": "3306.png", "formula": "\\begin{align*} m _ 2 = \\frac { d ^ 2 } { d t ^ 2 } \\phi _ Y ( 0 ) ( - \\mu ) ^ 2 = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\frac { 1 } { 2 \\lambda _ { _ \\Sigma } } . \\end{align*}"}
+{"id": "2300.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ { t } S + i c _ { 1 } \\partial _ { x } S + \\partial ^ { 2 } _ { x } S = c _ { 2 } S L + c _ { 3 } | u | ^ 2 u , \\\\ \\partial _ { t } L + c _ { 4 } \\partial _ { x } L + P ( \\partial _ { x } ) L + c _ { 5 } L \\partial _ { x } L = c _ { 6 } | S | ^ 2 , \\end{cases} \\end{align*}"}
+{"id": "3841.png", "formula": "\\begin{align*} \\partial _ t \\rho + \\div ( \\rho \\tilde v ) = \\rho \\tilde r \\end{align*}"}
+{"id": "5360.png", "formula": "\\begin{align*} H ^ 1 ( \\O _ { L _ 1 \\cup L _ 2 \\cup L _ 3 \\cup L _ 4 } ( 1 ) ( - B _ 2 ) ) = H ^ 1 ( \\O _ { \\P ^ 1 } ( - 1 ) ^ { \\oplus 4 } ) = 0 \\end{align*}"}
+{"id": "3050.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { I ^ { k , m } } \\hat { \\xi } ^ { k , m , i } = \\xi ^ k \\ , . \\end{align*}"}
+{"id": "5603.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\lim _ { n \\to \\infty } \\int _ 0 ^ \\infty | ( u _ { n , R } ^ 0 ) ' | ^ p r ^ { p - 1 } \\mathrm d r = \\lim _ { R \\to \\infty } \\lim _ { n \\to \\infty } \\int _ 0 ^ R | u _ n ' | ^ p r ^ { p - 1 } \\mathrm d r \\end{align*}"}
+{"id": "5353.png", "formula": "\\begin{align*} h ^ 1 ( N _ Z ( K _ Z - ( k - 1 ) H ) ) = - \\chi ( N _ Z ( K _ Z - ( k - 1 ) H ) ) = s \\end{align*}"}
+{"id": "3633.png", "formula": "\\begin{align*} \\Delta \\langle H , P \\rangle = \\sum _ { i = 1 } ^ m e _ i \\langle \\overline { \\nabla } _ { e _ i } H , P \\rangle + \\sum _ { i = 1 } ^ m e _ i \\langle H , \\overline { \\nabla } _ { e _ i } P \\rangle . \\end{align*}"}
+{"id": "5710.png", "formula": "\\begin{align*} \\| ( z - A ) ^ { - 1 } \\| \\leq \\frac { 1 } { | z | } + \\sum \\limits _ { k = 1 } ^ { m _ 0 } \\frac { \\prod \\limits _ { j = 1 } ^ k w _ j } { | z | ^ { k + 1 } } + \\frac { 1 } { | z | } \\big ( e ^ { \\frac { 1 } { | z | } } - \\sum \\limits _ { k = 0 } ^ { m _ 0 } \\frac { 1 } { k ! | z | ^ { k } } \\big ) . \\end{align*}"}
+{"id": "3920.png", "formula": "\\begin{align*} d ( u , \\omega ) : = \\inf \\{ d ( u , v ) \\mid v \\in \\omega \\} \\end{align*}"}
+{"id": "6666.png", "formula": "\\begin{align*} h _ { n - 1 } ( \\beta , p , q ) | _ { \\beta = 2 } = \\pi d _ n { i ^ n \\over ( n - 1 ) ! } , n \\ge 1 . \\end{align*}"}
+{"id": "5332.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } ^ n } { \\mathrm { d } x ^ n } e ^ { - x ^ 2 } = H _ n ( x ) e ^ { - x ^ 2 } , \\end{align*}"}
+{"id": "1617.png", "formula": "\\begin{align*} U _ { t + 1 } = \\{ u \\in V _ { t + 1 } : T \\cup \\{ u \\} \\frac { \\rho } { 2 } \\prod _ { j \\in [ k ] \\setminus [ t + 1 ] } | V _ j | H \\} . \\end{align*}"}
+{"id": "4974.png", "formula": "\\begin{align*} b _ i = 0 \\mbox { f o r } 4 \\leq i \\leq r . \\end{align*}"}
+{"id": "6328.png", "formula": "\\begin{align*} \\delta x = \\sqrt { T } , \\delta \\xi = \\frac { 1 } { \\sqrt { T } } . \\end{align*}"}
+{"id": "2148.png", "formula": "\\begin{align*} \\partial _ t \\beta = \\partial _ x \\alpha , \\mbox { a n d } \\partial _ x \\beta = \\partial _ t \\alpha , \\end{align*}"}
+{"id": "6735.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow + \\infty } \\left ( 4 \\pi - \\mathfrak { c } _ { p } ^ { - 2 } t ^ { 2 a } ( 1 + \\frac { m } { 2 t } ) ^ { 4 a } \\int _ { \\Sigma _ { t } } | \\nabla u | ^ 2 \\right ) = 0 . \\end{align*}"}
+{"id": "1142.png", "formula": "\\begin{align*} & C ^ 1 _ { c o m } ( L , L ) = C ^ 1 ( L , L ) ; \\\\ & C ^ { n + 1 } _ { c o m } ( L , L ) = \\underbrace { C ^ { n + 1 } ( L , L ) \\oplus \\cdots \\oplus C ^ { n + 1 } ( L , L ) } _ { ( n + 1 ) } , ~ n \\geq 1 \\end{align*}"}
+{"id": "5607.png", "formula": "\\begin{align*} \\nu _ \\infty = \\lim _ { R \\to \\infty } \\lim _ { n \\to \\infty } \\int _ R ^ \\infty \\left ( \\exp _ p ( \\mu | u _ n | ^ { p ' } ) - \\dfrac { \\mu ^ { p - 1 } } { \\Gamma ( p ) } | u _ n | ^ { p } \\right ) r ^ { \\alpha _ 0 } \\mathrm d r + \\lim _ { R \\to \\infty } \\lim _ { n \\to \\infty } \\dfrac { \\mu ^ { p - 1 } } { \\Gamma ( p ) } \\| u _ n \\| ^ { p } _ { L ^ { p } _ { \\alpha _ 0 } ( R , \\infty ) } . \\end{align*}"}
+{"id": "3255.png", "formula": "\\begin{align*} \\mathcal { C } * \\widetilde { \\mathcal { D } } = \\langle \\mathcal { C } \\widetilde { \\mathcal { D } } \\rangle _ { _ 0 } = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } \\mathcal { C } _ { _ \\Sigma } \\mathcal { D } _ { _ \\Sigma } . \\end{align*}"}
+{"id": "2621.png", "formula": "\\begin{align*} J ( t ) = \\int _ 0 ^ t \\frac { U ( t - x , x ) } { 1 - F ( x ) } \\ , d F ( x ) \\ , . \\end{align*}"}
+{"id": "1214.png", "formula": "\\begin{align*} A _ n = & \\int _ { \\R ^ 3 } [ I _ \\alpha \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } ( | ( g _ n ^ 1 ) ^ { - 1 } u _ n | ^ p - | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p ) ] | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n | ^ p d x \\\\ & - \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p ) | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n | ^ p d x . \\end{align*}"}
+{"id": "48.png", "formula": "\\begin{align*} \\frac { \\widehat { g } ( 0 ) } { 2 \\pi | \\Lambda | } \\int _ { \\mathbb { C } } | z | ^ 2 \\vert z \\rangle \\langle z \\vert \\dd z = \\frac { \\widehat { g } ( 0 ) } { 2 | \\Lambda | } a _ 0 a ^ { \\dagger } _ 0 \\geq - C \\frac { n _ 0 + 1 } { | \\Lambda | } \\widehat { g } ( 0 ) . \\end{align*}"}
+{"id": "7805.png", "formula": "\\begin{align*} \\begin{pmatrix} m ' \\\\ n ' \\\\ \\end{pmatrix} = \\begin{pmatrix} a & b \\\\ c & d \\\\ \\end{pmatrix} ^ \\intercal \\begin{pmatrix} m \\\\ n \\\\ \\end{pmatrix} , \\end{align*}"}
+{"id": "1246.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left \\{ \\log [ \\frac { \\lambda _ n ^ j } { \\lambda _ n ^ k } ] + \\frac { | x _ n ^ j - x _ n ^ k | ^ 2 } { \\lambda _ n ^ j \\lambda _ n ^ k } + \\frac { | t _ n ^ j ( \\lambda _ n ^ j ) ^ 2 - t _ n ^ k ( \\lambda _ n ^ k ) ^ 2 | } { \\lambda _ n ^ j \\lambda _ n ^ k } \\right \\} = \\infty . \\end{align*}"}
+{"id": "8592.png", "formula": "\\begin{align*} \\phi _ 1 = - h _ b \\frac { \\sinh ( \\frac { z } { h _ b } \\sqrt { \\mu } | \\mathrm { D } | ) } { \\cosh ( \\sqrt { \\mu } | \\mathrm { D } | ) } \\frac { 1 } { \\sqrt { \\mu } | \\mathrm { D } | } \\nabla _ X \\cdot \\big ( \\mathcal { L } _ { 1 } ^ { \\mu } [ \\beta b ] \\nabla _ X \\psi \\big { ) } , \\end{align*}"}
+{"id": "6379.png", "formula": "\\begin{align*} \\chi ' & \\Delta _ { H ' } ( f ( x ) ) = ( f \\otimes f ) ( \\chi ) ( f \\otimes f ) ( \\Delta _ H ( x ) ) = ( f \\otimes f ) ( \\chi \\Delta _ H ( x ) ) \\\\ & = ( f \\otimes f ) ( \\Delta _ H ( x ) \\chi ) = ( f \\otimes f ) ( \\Delta _ H ( x ) ) ( f \\otimes f ) ( \\chi ) = \\Delta _ { H ' } ( f ( x ) ) \\chi ' \\end{align*}"}
+{"id": "5026.png", "formula": "\\begin{align*} u ^ 2 : = ( 4 \\pi ) ^ { - 1 } e ^ { - H } . \\end{align*}"}
+{"id": "6980.png", "formula": "\\begin{align*} f '' + A ( z ) f ' + B ( z ) f = H ( z ) , \\end{align*}"}
+{"id": "5847.png", "formula": "\\begin{align*} & s _ 2 s _ 1 s _ 2 = s _ { \\alpha _ 1 + \\alpha _ 2 } , s _ 1 s _ 2 s _ 1 = s _ { 3 \\alpha _ 1 + \\alpha _ 2 } , \\\\ & w _ 0 = s _ { \\alpha _ 1 + \\alpha _ 2 } s _ { 3 \\alpha _ 1 + \\alpha _ 2 } \\end{align*}"}
+{"id": "2785.png", "formula": "\\begin{align*} x ^ { n + 4 } \\cdot x ^ { n + 1 } \\cdot x ^ 5 = x ^ { 2 k + 1 0 } , \\end{align*}"}
+{"id": "8840.png", "formula": "\\begin{align*} \\delta _ { t } \\leq \\frac { 2 ( t _ { 0 } - 1 ) \\delta _ { t _ { 0 } } } { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { ( 2 - p _ { i } ) t ^ { p _ { i } } } \\enspace . \\end{align*}"}
+{"id": "6121.png", "formula": "\\begin{align*} { \\rm R e } \\frac { \\zeta '' } { \\zeta ' } ( s ) & = \\frac { \\zeta '' } { \\zeta ' } ( 0 ) - 2 + \\sum _ { \\rho _ 1 } \\frac { 1 } { \\rho _ 1 } + \\sum _ { \\rho _ 1 } \\frac { \\sigma - \\beta _ 1 } { ( \\sigma - \\beta _ 1 ) ^ 2 + ( t - \\gamma _ 1 ) ^ 2 } - \\sum _ { n \\geq 1 } \\frac { a _ n \\sigma + | s | ^ 2 } { a _ n | s + a _ n | ^ 2 } \\\\ & \\leq \\frac { \\zeta '' } { \\zeta ' } ( 0 ) - 2 + \\sum _ { \\rho _ 1 } \\frac { 1 } { \\rho _ 1 } - \\sum _ { n \\geq 1 } \\frac { \\sigma } { | s + a _ n | ^ 2 } - \\sum _ { n \\leq | s | } \\frac { | s | ^ 2 } { a _ n | s + a _ n | ^ 2 } . \\end{align*}"}
+{"id": "6431.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\sup _ { t \\geq t _ n } \\tilde { v } _ n ( c t , t ) = 0 . \\end{align*}"}
+{"id": "393.png", "formula": "\\begin{align*} ( \\mathcal { C } \\otimes \\delta ) ^ \\pitchfork \\ , = \\ , \\mathcal { F } . \\end{align*}"}
+{"id": "6474.png", "formula": "\\begin{align*} \\left < P _ R F _ 1 , P _ R F _ 1 ' \\right > = \\left < ( I - P _ N ) F _ 1 , ( I - P _ D ) F _ 1 ' \\right > = \\left < F _ 1 , F _ 1 ' \\right > , \\end{align*}"}
+{"id": "4643.png", "formula": "\\begin{align*} \\widetilde { \\Xi } : = \\sum _ j \\Xi ( n _ j ^ * ) \\otimes n _ j \\in \\bigl ( \\mathbf { M } ^ { \\pi _ { \\mathcal { S } ( v _ { n + 1 } ) } } \\otimes N \\bigr ) ^ { K _ { v _ { n + 1 } } } . \\end{align*}"}
+{"id": "4958.png", "formula": "\\begin{align*} \\Phi _ { 4 n } ( 2 ) = \\Phi _ { 8 n } ( \\sqrt { 2 } ) . \\end{align*}"}
+{"id": "3788.png", "formula": "\\begin{align*} H ( x _ 0 , s _ 0 , x _ 1 , s _ 1 ) & = \\sup \\big \\{ s _ 0 \\psi _ 0 + s _ 1 \\psi _ 1 \\ , \\mid \\ , R ^ * ( \\psi _ 0 ) + R ^ * ( \\psi _ 1 ) \\leq c ( x _ 0 , x _ 1 ) \\big \\} \\\\ & = \\sup \\big \\{ - s _ 0 F ^ * ( - \\phi _ 0 ) - s _ 1 F ^ * ( - \\phi _ 1 ) \\ , \\mid \\ , \\phi _ 0 + \\phi _ 1 \\leq c ( x _ 0 , x _ 1 ) \\big \\} . \\end{align*}"}
+{"id": "297.png", "formula": "\\begin{align*} \\gamma _ { \\mathbf { n } ; j } ^ { - } \\left ( \\left . \\mbox { \\boldmath $ \\psi $ } \\right \\vert _ { \\Omega _ { j } ^ { - } } \\right ) = - \\gamma _ { \\mathbf { n } ; j } ^ { + } \\left ( \\left . \\mbox { \\boldmath $ \\psi $ } \\right \\vert _ { \\Omega _ { j } ^ { + } } \\right ) \\end{align*}"}
+{"id": "8835.png", "formula": "\\begin{align*} \\frac { \\Gamma ( d + 1 ) } { \\Gamma ( d + \\beta ) } & = \\frac { \\Gamma ( d + 1 ) } { \\Gamma \\big ( d + \\underbrace { ( \\beta - \\ell ) } _ { \\in ( 0 , 1 ] } \\big ) \\prod _ { i = 1 } ^ { \\ell } \\big ( d + \\beta - i \\big ) } \\leq \\frac { ( d + \\beta - \\ell ) ^ { 1 - ( \\beta - \\ell ) } } { \\prod _ { i = 1 } ^ { \\ell } \\big ( d + \\beta - i \\big ) } \\leq \\frac { 1 } { d ^ { \\beta - 1 } } \\enspace , \\end{align*}"}
+{"id": "7284.png", "formula": "\\begin{align*} \\begin{cases} m X ( j ) = 0 \\\\ m X ( j ) = 1 . \\end{cases} \\end{align*}"}
+{"id": "2016.png", "formula": "\\begin{align*} \\dot { \\mathrm { H } } ^ { s , p } _ 0 ( \\Omega ) = \\dot { \\mathrm { H } } ^ { s , p } ( \\Omega ) \\end{align*}"}
+{"id": "2378.png", "formula": "\\begin{align*} E _ t ( d V ^ { p r e } ) = ( 1 - h d t ) \\mathcal { A } _ 1 V ^ { p r e } ( x , t ) + h d t \\left [ V ^ { a f t e r } ( x _ t ^ { a f t e r } , t ) - V ^ { p r e } ( x _ t , t \\right ] \\end{align*}"}
+{"id": "5556.png", "formula": "\\begin{align*} \\nabla ^ 2 U _ k ( \\zeta ) & = 0 \\mbox { i f } \\zeta \\in G _ m , \\\\ U _ k ( \\zeta ) & = 1 \\mbox { i f } \\zeta \\in C _ j , j = 1 , \\ldots , k - 1 , \\\\ U _ k ( \\zeta ) & = 1 \\mbox { i f } \\zeta \\in C ' _ k , \\\\ U _ k ( \\zeta ) & = 0 \\mbox { i f } \\zeta \\in C '' _ k , \\\\ U _ k ( \\zeta ) & = 0 \\mbox { i f } \\zeta \\in C _ j , j = k + 1 , \\ldots , m , \\end{align*}"}
+{"id": "3375.png", "formula": "\\begin{align*} \\begin{aligned} g ^ { - 1 } \\dd g + A d _ { g ^ { - 1 } } ( \\phi ^ * \\alpha _ V ) = \\alpha _ U & + \\frac { 1 } { n + 1 } \\textnormal { t r } \\left ( A d _ { g ^ { - 1 } } ( \\phi ^ * \\alpha _ V ) + g ^ { - 1 } \\dd g - \\alpha _ U \\right ) ( E _ j ) \\omega ^ i \\\\ & + \\frac { 1 } { n + 1 } \\textnormal { t r } \\left ( A d _ { g ^ { - 1 } } ( \\phi ^ * \\alpha _ V ) + g ^ { - 1 } \\dd g - \\alpha _ U \\right ) I _ n . \\end{aligned} \\end{align*}"}
+{"id": "5329.png", "formula": "\\begin{align*} n = O \\left [ \\ln \\left ( \\frac { 2 C m ( m - 1 ) } { \\varepsilon } \\right ) \\right ] \\end{align*}"}
+{"id": "8228.png", "formula": "\\begin{align*} \\bar { T } = d \\pi ( T - \\frac { g _ 0 ( T , \\xi _ 1 ) } { g _ 0 ( \\xi _ 1 , \\xi _ 1 ) } \\xi _ 1 ) = d \\pi ( T - 2 x _ 1 \\xi _ 1 ) . \\end{align*}"}
+{"id": "233.png", "formula": "\\begin{align*} \\mathbb { A } ^ { \\emph { r } } : = \\begin{cases} \\{ x \\in \\mathbb { R } ^ n \\mid x _ 1 > x _ 2 > \\cdots > x _ n > 0 \\} & \\ \\ \\emph { r } = \\emph { b c } , \\\\ \\{ x \\in \\mathbb { R } ^ n \\mid x _ n > 0 \\} & \\ \\ \\emph { r } = \\emph { t } , \\\\ \\{ x \\in \\mathbb { R } ^ n \\mid x _ 1 > x _ 2 > \\cdots > x _ n \\} & \\ \\ \\emph { r } = \\emph { c s } , \\end{cases} \\end{align*}"}
+{"id": "5809.png", "formula": "\\begin{align*} \\alpha ' _ i = \\alpha _ i i < n , \\alpha ' _ n = 2 \\alpha _ n \\end{align*}"}
+{"id": "4219.png", "formula": "\\begin{align*} \\rho = \\ , & \\ , a _ 1 e ^ { 1 2 3 } + a _ 2 e ^ { 1 2 4 } + a _ 3 e ^ { 1 2 5 } + a _ 4 e ^ { 1 3 4 } + a _ 5 e ^ { 1 3 5 } + a _ 6 e ^ { 2 3 4 } + a _ 7 e ^ { 2 3 5 } + a _ 8 ( - e ^ { 1 4 5 } + e ^ { 2 3 6 } ) \\\\ [ 2 p t ] & + a _ 9 ( e ^ { 1 3 6 } + e ^ { 2 4 5 } ) + a _ { 1 0 } ( - e ^ { 1 2 6 } + e ^ { 3 4 5 } ) + a _ { 1 1 } e ^ { 4 5 6 } , \\end{align*}"}
+{"id": "8383.png", "formula": "\\begin{align*} \\int _ { a } ^ { b } f \\mathrm { d } g = ( - 1 ) ^ { \\alpha } \\int _ { a } ^ { b } D _ { a + } ^ { \\alpha } f ( t ) D _ { b - } ^ { 1 - \\alpha } g _ { b - } ( t ) \\mathrm { d } t \\end{align*}"}
+{"id": "8490.png", "formula": "\\begin{align*} g = g ^ { a c } + g ^ { j } + g ^ { c } , \\end{align*}"}
+{"id": "3357.png", "formula": "\\begin{align*} g ( - \\frac { 1 } { \\tau } ) = S \\ g ^ { \\vee } ( \\frac { \\tau } { 2 } ) , g ^ { \\vee } ( - \\frac { 1 } { \\tau } ) = 2 S \\ g ( \\frac { \\tau } { 2 } ) , \\end{align*}"}
+{"id": "2576.png", "formula": "\\begin{align*} \\pi ( \\beta ( h , h ' ) ) & = \\pi ( \\alpha ( h ) + \\alpha ( h ' ) - \\alpha ( h + h ' ) ) = \\\\ & = \\pi ( \\alpha ( h ) ) + \\pi ( \\alpha ( h ' ) ) - \\pi ( \\alpha ( h + h ' ) ) = \\\\ & = h + h ' - ( h + h ' ) = 0 , \\end{align*}"}
+{"id": "4853.png", "formula": "\\begin{align*} \\check { R } ( u ) = F ( u ) ( I + G ( u ) \\xi E ) \\end{align*}"}
+{"id": "4054.png", "formula": "\\begin{align*} \\rho _ { 0 , 1 , n } = \\left ( n - \\frac { 1 } { 2 } \\right ) \\pi + \\frac { q _ 1 ( 0 ) } { n \\pi } \\sin \\left ( n - \\frac { 1 } { 2 } \\right ) \\pi a + \\frac { \\kappa _ { 0 , 1 , n } } { n } , \\ , \\ , \\{ \\kappa _ { 0 , 1 , n } \\} \\in l _ 2 , n \\in \\mathbb { Z } ; \\end{align*}"}
+{"id": "741.png", "formula": "\\begin{align*} \\mathcal H ( \\psi ^ \\mu ( t ) ) - \\mathcal H ( \\psi ^ \\mu ( 0 ) ) = 2 M _ { \\mathfrak K _ 1 } \\int _ 0 ^ t \\left ( \\norm { P _ { \\mu } ( P _ { \\mu } \\psi ^ \\mu ( s ) ) } _ { L _ 2 } ^ 2 - \\norm { P _ { \\mu } \\psi ^ \\mu ( s ) } _ { L ^ 2 } ^ 2 \\right ) \\ , d s , \\end{align*}"}
+{"id": "8673.png", "formula": "\\begin{align*} \\tilde { \\phi } _ i = \\frac { \\sum _ { j = 0 } ^ n a _ { i , j } \\phi _ j + a _ { i , n + 1 } } { \\sum _ { j = 0 } ^ n a _ { n + 1 , j } \\phi _ j + a _ { n + 1 , n + 1 } } , ~ i = 0 , . . . , n , \\end{align*}"}
+{"id": "795.png", "formula": "\\begin{align*} c ^ * = \\min \\left \\{ \\left ( \\frac { 1 } { p } - \\frac { 1 } { 2 \\cdot p ^ \\flat } \\right ) \\left ( \\frac { p ^ \\flat } { b } \\right ) ^ { \\frac { p } { 2 \\cdot p ^ \\flat - p } } ( a S ^ \\flat ) ^ { \\frac { 2 \\cdot p ^ \\flat } { 2 \\cdot p ^ \\flat - p } } , \\left ( \\frac { 1 } { p } - \\frac { 1 } { 2 \\cdot p ^ \\sharp } \\right ) A ( \\varepsilon _ g ) \\right \\} , \\end{align*}"}
+{"id": "517.png", "formula": "\\begin{align*} W ( t ) = \\sum _ { k = 1 } ^ \\infty B _ k ( t ) e _ k , \\end{align*}"}
+{"id": "1413.png", "formula": "\\begin{gather*} \\tilde \\psi _ { n i } ( x ) = \\psi ^ K _ { n i } ( x ) + \\sum _ { k = 1 } ^ { K } ( \\tilde H _ { n i , k 0 } ( x ) \\psi ^ K _ { k 0 } ( x ) + \\tilde H _ { n i , k 1 } ( x ) \\psi ^ K _ { k 1 } ( x ) ) , n = \\overline { 1 , K } , i = 0 , 1 . \\end{gather*}"}
+{"id": "1593.png", "formula": "\\begin{align*} \\bigl ( \\Delta _ { a } ( f _ 1 \\otimes f _ 2 ) \\bigr ) ( x _ 1 , x _ 2 ) & \\ , = \\ , f _ 1 ( x _ 1 + a ) f _ 2 ( x _ 2 ) - f _ 1 ( x _ 1 ) f _ 2 ( x _ 2 ) \\\\ & \\ , = \\ , ( f _ 1 ( x _ 1 + a ) - f _ 1 ( x _ 1 ) ) f _ 2 ( x _ 2 ) \\\\ & \\ , = \\ , \\bigl ( ( \\Delta _ { a } f _ 1 ) \\otimes f _ 2 \\bigr ) ( x _ 1 , x _ 2 ) \\ , . \\end{align*}"}
+{"id": "1237.png", "formula": "\\begin{align*} a ( x ) = \\begin{cases} | x | ^ 2 \\ \\ f o r \\ \\ | x | \\leq R \\\\ C R ^ 2 \\ \\ f o r \\ \\ | x | > 2 R , \\end{cases} \\end{align*}"}
+{"id": "7211.png", "formula": "\\begin{align*} \\mathcal { U } ( t _ 0 , m _ 0 ) = \\inf _ { ( \\mu , \\alpha ) } \\int _ { t _ 0 } ^ T \\int _ { \\R ^ d } L \\bigl ( x , \\alpha ( t , x ) \\bigr ) d \\mu ( t ) ( x ) d t + \\int _ { t _ 0 } ^ T \\mathcal { F } \\bigl ( \\mu ( t ) \\bigr ) d t + \\mathcal { G } \\bigl ( \\mu ( T ) \\bigr ) \\end{align*}"}
+{"id": "3445.png", "formula": "\\begin{align*} u ^ { * * } ( x ) = \\max _ { p \\in \\Delta ^ \\vee } \\langle p , x \\rangle - u ^ * ( p ) \\geq \\max _ { p \\in \\Delta ^ \\vee } \\langle p , x \\rangle - \\tilde { u } ( p ) = u ( x ) . \\end{align*}"}
+{"id": "848.png", "formula": "\\begin{align*} \\| T \\| = \\| x ^ * \\| \\| h \\| . \\end{align*}"}
+{"id": "3922.png", "formula": "\\begin{align*} d _ \\mu ( A , B ) : = \\mu ( A , B ) . \\end{align*}"}
+{"id": "8191.png", "formula": "\\begin{align*} g = \\bar { g } + V \\sum _ { i = 1 } ^ 3 ( d x _ i ) ^ 2 + \\frac { 1 } { V } \\eta ^ 2 . \\end{align*}"}
+{"id": "5703.png", "formula": "\\begin{align*} k _ { ( T , x ) } = \\limsup _ { z \\rightarrow 0 } \\frac { \\ln \\| ( z - T ) ^ { - 1 } x \\| _ p } { \\ln \\| ( z - T ) ^ { - 1 } \\| } & = \\limsup _ { z \\rightarrow 0 } \\frac { \\ln \\| ( z - A ) ^ { - 1 } x _ 2 \\| _ p } { \\ln \\| ( z - T ) ^ { - 1 } \\| } \\\\ & \\approx \\limsup _ { z \\rightarrow 0 } \\frac { \\ln \\| ( z - A ) ^ { - 1 } x _ 2 \\| _ p } { \\ln \\| ( z - A ) ^ { - 1 } \\| ^ 2 } = \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "5655.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l } \\alpha = - \\frac { B } { A } \\gamma + \\delta , \\\\ \\beta = - \\frac { C } { A } \\gamma \\end{array} \\right . \\ \\ \\ a n d \\ \\ \\ \\ \\left \\lbrace \\begin{array} { l } \\alpha = - \\delta , \\\\ \\beta = \\frac { C } { A } \\gamma - \\frac { B } { A } \\delta . \\end{array} \\right . \\end{align*}"}
+{"id": "2405.png", "formula": "\\begin{align*} \\L _ d ( W ( \\mathcal { S } , ( x _ n ) _ { n = 1 } ^ { \\infty } , h ) \\cap B ( z , r ) ) > \\L _ d ( B ( z , r ) ) / 2 \\end{align*}"}
+{"id": "4056.png", "formula": "\\begin{align*} \\rho _ { k n } = k n \\pi , \\ , n \\in \\mathbb { Z } \\ ! \\setminus \\ ! \\{ 0 \\} ; \\end{align*}"}
+{"id": "2.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ t \\frac { 1 } { n - i } & = H _ { n - 1 } - H _ { n - 1 - t } = \\log \\left ( \\frac { n - 1 } { n - 1 - t } \\right ) + o ( 1 ) . \\end{align*}"}
+{"id": "3656.png", "formula": "\\begin{align*} \\psi - \\phi _ 1 & = 0 \\ , \\\\ \\psi _ 1 \\phi _ { 2 2 } - { \\psi _ 2 } ^ 2 + \\psi _ 3 + \\phi _ { 2 4 } & = 0 \\ , \\end{align*}"}
+{"id": "8132.png", "formula": "\\begin{align*} d ( \\alpha ( t ) , \\alpha ( t ' ) ) = d ( x , y ) ~ | t ' - t | . \\end{align*}"}
+{"id": "3655.png", "formula": "\\begin{align*} \\phi _ { 1 1 } \\phi _ { 2 2 } - ( \\phi _ { 1 2 } ) ^ 2 + \\phi _ { 1 3 } + \\phi _ { 2 4 } = 0 \\ . \\end{align*}"}
+{"id": "955.png", "formula": "\\begin{align*} \\left \\{ \\ \\begin{aligned} \\frac { \\delta L } { \\delta u } | _ { \\{ p = \\psi _ 1 \\} } = \\lambda _ 1 ( \\mathcal { T } _ t ^ \\alpha u - x u _ { x x } - a v _ x ) + \\lambda _ 2 ( \\mathcal { T } _ t ^ \\alpha v - x v _ { x x } - b u _ x ) , \\\\ \\frac { \\delta L } { \\delta v } | _ { \\{ q = \\psi _ 2 \\} } = \\lambda _ 3 ( \\mathcal { T } _ t ^ \\alpha u - x u _ { x x } - a v _ x ) + \\lambda _ 4 ( \\mathcal { T } _ t ^ \\alpha v - x v _ { x x } - b u _ x ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "6486.png", "formula": "\\begin{align*} \\psi _ e ( x ) = a _ e + \\epsilon ( b _ e x + c _ e ) , \\end{align*}"}
+{"id": "1032.png", "formula": "\\begin{align*} F _ 0 \\left ( \\sum _ { i \\in N } A _ i \\right ) = \\sum _ { i \\in N } F _ 0 ( A _ i ) \\end{align*}"}
+{"id": "3301.png", "formula": "\\begin{align*} g _ Y ( x ) = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\sqrt { \\frac { \\lambda _ { _ \\Sigma } } { \\pi } } e ^ { - \\lambda _ { _ \\Sigma } x ^ 2 } , \\end{align*}"}
+{"id": "3953.png", "formula": "\\begin{align*} & \\rho _ { K ^ * } ( u ' ) u ' v \\leq h ( K ^ * , v ) = \\rho _ { K ^ * } ( u ) u v \\\\ & \\rho _ { L ^ * } ( u ' ) u ' v \\leq h ( L ^ * , v ) = \\rho _ { L ^ * } ( u ) u v \\\\ & \\rho _ { K ^ * } ( u ' ) u ' v ' = h ( K ^ * , v ' ) \\geq \\rho _ { K ^ * } ( u ) u v ' \\\\ & \\rho _ { L ^ * } ( u ' ) u ' v ' = h ( L ^ * , v ' ) \\geq \\rho _ { L ^ * } ( u ) u v ' . \\end{align*}"}
+{"id": "1326.png", "formula": "\\begin{align*} J ^ { w } ( \\textbf { X } _ { m i n R S S U } ^ { ( n ) } ) & = - \\frac { ( n ! ) ^ 2 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( \\int _ { 0 } ^ { 1 } ( 1 - u ) ^ { 2 i - 2 } w ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) d u \\right ) \\\\ & = - \\frac { ( n ! ) ^ 2 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( \\int _ { 0 } ^ { 1 } ( 1 - u ) ^ { 2 i - 2 } u ^ m d u \\right ) \\\\ & = - \\frac { ( n ! ) ^ 2 } { 2 } \\prod _ { i = 1 } ^ { n } \\beta \\left ( 2 m + 1 , 2 i - 1 \\right ) . \\end{align*}"}
+{"id": "7142.png", "formula": "\\begin{align*} \\begin{aligned} & r ^ { - 1 } m ^ 2 _ S ( r ) \\in L ^ 1 [ 1 , \\infty ) & & \\delta = 1 , \\\\ & m _ S ( r ) = 1 & & 0 < \\delta < 1 , \\\\ & m _ S ( r ) = ( \\log r + 1 ) ^ { - \\rho } & & \\delta = 0 . \\end{aligned} \\end{align*}"}
+{"id": "781.png", "formula": "\\begin{align*} - \\Delta u + u = ( K \\ast F ( u ) ) F ' ( u ) \\mathbb { R } ^ N , \\end{align*}"}
+{"id": "2593.png", "formula": "\\begin{align*} \\partial _ { k - 1 } \\partial _ j \\Delta ^ n = \\partial _ j \\Delta ^ n \\cap \\partial _ k \\Delta ^ n = \\partial _ j \\partial _ k \\Delta ^ n . \\end{align*}"}
+{"id": "8217.png", "formula": "\\begin{align*} d \\Phi _ a ( X ) & = ( d L _ { a ^ 2 { x _ 1 } } ) | _ m ( X _ 0 ) + a _ 0 T + a _ 1 ( a ^ 2 | T | ^ 2 + 1 ) I _ 1 T , \\\\ d \\Phi _ a ( Y ) & = ( d L _ { a ^ 2 { x _ 1 } } ) | _ m ( Y _ 0 ) + b _ 0 T + b _ 1 ( a ^ 2 | T | ^ 2 + 1 ) I _ 1 T . \\end{align*}"}
+{"id": "1857.png", "formula": "\\begin{align*} \\mathbf { E } _ { \\Omega _ 0 } \\left [ \\| \\zeta ( s , \\mathbb { X } _ \\alpha ) - \\zeta _ L ( s , \\mathbb { X } _ \\alpha ) \\| ^ 2 \\right ] & = \\mathbf { E } \\left [ \\mathbf { 1 } _ { \\Omega _ 0 } \\cdot \\| \\zeta ( s , \\mathbb { X } _ \\alpha ) - \\zeta _ L ( s , \\mathbb { X } _ \\alpha ) \\| ^ 2 \\right ] \\\\ & \\leq \\mathbf { E } \\left [ \\| \\zeta ( s , \\mathbb { X } _ \\alpha ) - \\zeta _ L ( s , \\mathbb { X } _ \\alpha ) \\| ^ 2 \\right ] . \\end{align*}"}
+{"id": "2957.png", "formula": "\\begin{align*} \\begin{bmatrix} n + 1 \\\\ s \\end{bmatrix} - \\begin{bmatrix} n + 1 \\\\ s + 1 \\end{bmatrix} = J _ s ^ { n + 1 } = J _ { s - 1 } ^ n + J _ s ^ n + J _ { s - 1 } ^ n = \\begin{bmatrix} n \\\\ s - 1 \\end{bmatrix} - \\begin{bmatrix} n \\\\ s + 2 \\end{bmatrix} \\end{align*}"}
+{"id": "4904.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { q ^ k ( 1 + q ^ { 2 k + 1 } ) } { ( 1 - q ^ { 2 k + 1 } ) ^ { 2 } } = \\frac { ( q ^ { 2 } ; q ^ { 2 } ) _ { \\infty } ^ { 4 } } { ( q ; q ^ { 2 } ) _ { \\infty } ^ { 4 } } , \\sum _ { k = 0 } ^ \\infty \\dfrac { q ^ k ( 1 + q ^ { 2 k + 1 } ) } { ( 1 - q ^ { 2 k + 1 } ) ^ 2 } = \\dfrac { ( q ^ 2 ; q ^ 2 ) _ \\infty ^ 4 } { ( q ; q ^ 2 ) _ \\infty ^ 4 } . \\end{align*}"}
+{"id": "3968.png", "formula": "\\begin{align*} [ ( 1 + e _ i ) N , ( 1 + e _ j ) N ] = 1 + e _ i e _ j - e _ j e _ i = 1 + \\varphi ( e _ i , e _ j ) . \\end{align*}"}
+{"id": "8488.png", "formula": "\\begin{align*} D g = D ^ { a c } g + D ^ { s } g , \\end{align*}"}
+{"id": "4504.png", "formula": "\\begin{align*} \\partial _ t ( \\varphi ^ a + \\psi _ { i + 1 / 2 } ) - ( u _ 1 ^ a + u _ { 1 , i + 1 / 2 } ) + ( u _ 2 ^ a + u _ { 2 , i + 1 / 2 } ) \\partial _ 2 ( \\varphi ^ a + \\psi _ { i + 1 / 2 } ) = 0 \\ , , \\quad \\mbox { o n } \\ , \\ , \\ , \\{ x _ 1 = 0 \\} \\ , . \\end{align*}"}
+{"id": "183.png", "formula": "\\begin{align*} & \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T _ C M } - \\widetilde { L _ C } + 8 + W _ i + W _ j ) \\right \\} ^ { ( 1 4 ) } \\\\ & = A \\left \\{ e ^ { \\frac { 1 } { 2 4 } A } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) - \\frac { e ^ { \\frac { 1 } { 2 4 } A } - 1 } { A } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T _ C M } - \\widetilde { L _ C } + 8 + W _ i + W _ j ) \\right \\} ^ { ( 1 0 ) } . \\end{align*}"}
+{"id": "8129.png", "formula": "\\begin{align*} u _ R : = \\chi _ { B _ R } u , v _ R : = \\chi _ { B _ R } v \\end{align*}"}
+{"id": "2247.png", "formula": "\\begin{align*} E _ g ( \\Gamma ) = \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\sum _ { j = 1 } ^ n \\sum _ { \\l \\in \\L \\backslash \\{ x _ j - x _ k \\} } g ( | \\l + x _ j - x _ k | ) . \\end{align*}"}
+{"id": "5533.png", "formula": "\\begin{align*} h ( r ) = \\omega ( z _ 0 , \\partial \\Omega \\cap \\overline { B ( z _ 0 , r ) } , \\Omega ) \\end{align*}"}
+{"id": "6554.png", "formula": "\\begin{align*} \\mathbb { D } _ l ( \\sigma ^ * ) = \\mathbb { D } _ { ( \\frac { 1 } { 1 0 } + \\frac { l } { 5 0 b } ) ( \\varepsilon + \\delta ) ^ { \\frac { 1 } { 8 b } } } ( \\sigma ^ * ) . \\end{align*}"}
+{"id": "2564.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } ( K \\ast g ( u _ n ) ) g ( u _ n ) d x = \\int _ { \\mathbb { R } ^ N } ( K \\ast g ( u _ 0 ) ) g ( u _ 0 ) d x \\end{align*}"}
+{"id": "2829.png", "formula": "\\begin{align*} \\ell ^ { ( p ) } ( r e s _ { s - x t ^ w } ( \\mathfrak { q } ' ) ) - \\ell ^ { ( p ) } ( r e s _ { s } ( \\mathfrak { q } ) ) = r e s _ { s = 0 } \\Omega ^ { ( p ) } ( \\mathfrak { q } ' - \\mathfrak { q } ) . \\end{align*}"}
+{"id": "5612.png", "formula": "\\begin{align*} B _ { p , \\alpha _ 0 } = \\lim J ( u _ n ) = \\lim \\| u _ n \\| ^ q _ { L ^ q _ { \\alpha _ 0 } } = \\| u _ 0 \\| _ { L ^ q _ { \\alpha _ 0 } } ^ q , \\end{align*}"}
+{"id": "4035.png", "formula": "\\begin{align*} s _ { ( 2 ) } [ s _ { ( 1 ^ 2 ) } ] = h _ 3 [ e _ 2 ] = h _ 2 ( x _ 1 x _ 2 , x _ 1 x _ 3 , \\ldots ) = x _ 1 ^ 2 x _ 2 ^ 2 + x _ 1 ^ 2 x _ 2 x _ 3 + 3 x _ 1 x _ 2 x _ 3 x _ 4 + \\cdots \\\\ = s _ { 2 , 2 } ( x _ 1 , x _ 2 , x _ 3 , \\ldots ) + s _ { 1 , 1 , 1 , 1 } ( x _ 1 , x _ 2 , x _ 3 , \\ldots ) , \\end{align*}"}
+{"id": "8208.png", "formula": "\\begin{align*} \\nabla _ X Y = ( \\nabla ^ S _ { X _ S } Y _ S + y _ r X _ S + x _ r Y _ S ) + ( X _ S ( y _ r ) + x _ r y _ r - g _ S ( X _ S , Y _ S ) ) \\rho \\frac { \\partial } { \\partial \\rho } . \\end{align*}"}
+{"id": "4025.png", "formula": "\\begin{align*} \\| \\phi _ m \\| _ { W ^ { 1 , \\infty } _ y } + \\| \\phi _ m \\| _ { \\mathcal M ^ { 2 - 1 / p , p } ( \\R ^ { 2 } ) } \\leq \\delta . \\end{align*}"}
+{"id": "7648.png", "formula": "\\begin{align*} A : = \\Big \\{ Z ^ \\ast \\neq - \\mathrm { s i g n } ( x _ 0 ) | Z ^ \\ast | \\frac { w } { | w | } \\Big \\} \\end{align*}"}
+{"id": "8441.png", "formula": "\\begin{align*} & e _ R = - d ^ 2 e _ L - c d h _ L + c ^ 2 f _ L , \\\\ & h _ R = - 2 b d e _ L - ( a d + b c ) h _ L + 2 a c f _ L , \\\\ & f _ R = b ^ 2 e _ L + a b h _ L - a ^ 2 f _ L , \\\\ & \\omega _ e = d \\cdot ( d b ) - b \\cdot ( d d ) , \\\\ & \\omega _ h = \\frac { 1 } { 2 } ( d \\cdot ( d a ) + c \\cdot ( d b ) - b \\cdot ( d c ) - a \\cdot ( d d ) ) , \\\\ & \\omega _ f = - c \\cdot ( d a ) + a \\cdot ( d c ) . \\end{align*}"}
+{"id": "8142.png", "formula": "\\begin{align*} x ^ m y ^ n \\leq x ^ i y ^ j \\Leftrightarrow \\begin{cases} m + n < i + j \\\\ \\\\ m + n = i + j \\quad n \\leq j \\ , . \\end{cases} \\end{align*}"}
+{"id": "6099.png", "formula": "\\begin{align*} \\Delta _ { k + 1 } = \\xi \\Delta _ k + \\zeta \\Delta _ 0 \\beta ^ k \\end{align*}"}
+{"id": "2783.png", "formula": "\\begin{align*} \\begin{multlined} x ^ { k + 3 } \\cdot x ^ { k + 1 } \\cdot 1 7 x ^ 4 + x ^ { k + 3 } \\cdot ( 2 ^ k + k ) x ^ k \\cdot x ^ 5 + ( 2 ^ { k + 2 } + k + 2 ) x ^ { k + 2 } \\cdot x ^ { k + 1 } \\cdot x ^ 5 \\\\ = ( 2 k + 2 ^ k + 2 ^ { k + 2 } + 1 9 ) x ^ { 2 k + 8 } , \\end{multlined} \\end{align*}"}
+{"id": "6089.png", "formula": "\\begin{align*} \\begin{aligned} s _ { X Y } & = \\frac { 1 } { | S | - 1 } \\sum _ { j \\in S _ k } ( X _ j - \\bar { X } ) ( Y _ j - \\bar { Y } ) \\\\ s _ { Y } ^ 2 & = \\frac { 1 } { | S | - 1 } \\sum _ { j \\in S _ k } ( Y _ j - \\bar { Y } ) ^ 2 \\end{aligned} \\end{align*}"}
+{"id": "7835.png", "formula": "\\begin{align*} ( T ) = T _ { 0 \\overline { 0 } } T _ { 1 \\overline { 1 } } - ( T _ { 0 \\overline { 1 } } ) ^ 2 = - \\frac { t ^ 4 } { 3 } - \\frac { t \\chi } { 4 ( 2 \\pi ) ^ 3 } \\sum _ { ( m , n ) \\in \\mathbb { Z } ^ 2 - ( 0 , 0 ) } \\left ( \\frac { 3 ( m \\tau _ 1 + n ) ^ 2 } { | m \\tau + n | ^ 5 } - \\frac { 1 } { | m \\tau + n | ^ 3 } \\right ) \\ , . \\end{align*}"}
+{"id": "5505.png", "formula": "\\begin{align*} O _ B ^ { ( b ^ k g ) } = O _ B ^ { ( g ) } , \\qquad \\textrm { f o r a l l } k \\in \\mathbb Z , g \\in F r e e _ 2 . \\end{align*}"}
+{"id": "8164.png", "formula": "\\begin{align*} & X ^ \\star p _ { i , j } ( x , y ) = i p _ { i , j } ( x , y ) , \\\\ & Y ^ \\star p _ { i , j } ( x , y ) = j p _ { i , j } ( x , y ) , \\end{align*}"}
+{"id": "2592.png", "formula": "\\begin{align*} F _ \\ell \\Delta ^ n : = S \\cup \\bigcup _ { j \\le \\ell } \\partial _ { i _ j } \\Delta ^ n . \\end{align*}"}
+{"id": "8165.png", "formula": "\\begin{align*} [ X , Y ] = 0 \\ , , [ X ^ \\star , Y ^ \\star ] = 0 \\ , . \\end{align*}"}
+{"id": "6756.png", "formula": "\\begin{align*} ( Q ^ { q _ 1 } \\circ R ^ { r _ 1 } \\circ \\dotsb \\circ Q ^ { q _ n } \\circ R ^ { r _ n } ) ( n ) = ( Q ^ { q _ 1 + \\dotsb + q _ n } \\circ R ^ { r _ 1 + \\dotsb + r _ n } ) ( n ) + C \\end{align*}"}
+{"id": "8139.png", "formula": "\\begin{align*} y ^ 2 = \\theta x ( x - 1 ) ( x - \\lambda ) ( x - \\mu ) \\left ( x - \\dfrac { \\lambda ( 1 - \\mu ) } { 1 - \\lambda } \\right ) \\end{align*}"}
+{"id": "5412.png", "formula": "\\begin{align*} \\begin{cases} | \\sigma _ y ( t ) | \\to R \\ \\ , \\ \\ , u ( \\sigma _ y ( t ) ) \\to 0 \\ \\ , \\ \\ , t \\to t _ y ^ - , \\ \\ & \\\\ | \\sigma _ y ( t ) - z | \\to 0 \\ \\ , \\ \\ , u ( \\sigma _ y ( t ) ) \\to u ( z ) \\ \\ , \\ \\ , t \\to t _ y ^ + . & \\end{cases} \\end{align*}"}
+{"id": "1478.png", "formula": "\\begin{align*} & \\rho \\left ( \\delta _ { \\lambda } ( x ) \\right ) = \\lambda \\rho ( x ) \\lambda > 0 x \\in \\mathbb { G } , \\\\ & \\rho ( x ) > 0 x \\not = 0 . \\end{align*}"}
+{"id": "8242.png", "formula": "\\begin{align*} f _ x ( m ) = \\frac { 1 } { 2 } \\rho ^ 2 ( e ^ { \\tau ( m , x ) } \\cdot m ) - 2 x _ 1 \\tau ( m , x ) . \\end{align*}"}
+{"id": "505.png", "formula": "\\begin{align*} \\mathcal L ( \\psi , \\phi , x , t ) = \\mathcal L _ { \\mathrm { D i r a c } } ( \\psi ) + \\mathcal L _ { \\mathrm { m e s o n } } ( \\phi ) + \\mathcal L _ { \\mathrm { Y u k a w a } } ( \\psi , \\phi ) + \\mathcal L _ { \\mathrm { n o i s e } } ( \\psi , \\phi , x , t ) \\end{align*}"}
+{"id": "6128.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & 0 & 1 & 0 \\\\ 1 & 0 & - 1 & 1 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "7057.png", "formula": "\\begin{align*} H ( y ) = 2 \\pi i ^ k \\int _ 0 ^ \\infty F ( x ) J _ { k - 1 } \\left ( 4 \\pi \\sqrt { x y } \\right ) d x . \\end{align*}"}
+{"id": "2810.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } I _ { k } ^ { 2 } = 0 = \\lim _ { k \\rightarrow + \\infty } I _ { k } ^ { 3 } . \\end{align*}"}
+{"id": "1488.png", "formula": "\\begin{align*} & \\langle U ^ { ( i ) } \\overline { x } , U ^ { ( j ) } \\overline { x } \\rangle = \\langle U ^ { ( j ) } U ^ { ( i ) } \\overline { x } , - \\overline { x } \\rangle = \\langle U ^ { ( i ) } U ^ { ( j ) } \\overline { x } , \\overline { x } \\rangle . \\end{align*}"}
+{"id": "759.png", "formula": "\\begin{align*} \\theta ( x ) - \\theta ( y ) = \\left ( \\int _ 0 ^ 1 \\theta ' \\left ( \\kappa ( t ) \\right ) \\ , d t \\right ) ( x - y ) = : I _ 1 ( x - y ) \\end{align*}"}
+{"id": "532.png", "formula": "\\begin{align*} ( \\psi _ + , \\psi _ - , \\phi _ + ) ( t ) = ( \\Psi _ + , \\Psi _ - , \\Phi _ + ) ( t ) . \\end{align*}"}
+{"id": "1556.png", "formula": "\\begin{align*} \\Delta = y ^ 3 x ^ 2 ( x - \\lambda y ) + z f _ 5 ( x , y , z ) \\end{align*}"}
+{"id": "3865.png", "formula": "\\begin{align*} e ^ { - \\lambda _ 2 z ( s ) } y ' ( s ) = \\cos \\theta ( s ) , z ' ( s ) = \\sin \\theta ( s ) , \\end{align*}"}
+{"id": "6712.png", "formula": "\\begin{align*} \\det \\left ( \\sum _ { s = 0 } ^ \\infty \\binom { i - j } { s } \\beta ^ s G ^ { [ [ f _ i / g _ j ] ] } _ { \\lambda _ i - \\mu _ j - i + j + s } ( x ) \\right ) _ { i , j } . \\end{align*}"}
+{"id": "5173.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\frac { { \\rm A o I } ( S _ { \\mathrm { z } } , Q _ { \\mathrm { u n i } } , F ^ * ) } { \\log _ 2 D } = - \\frac { 3 } { 4 } . \\end{align*}"}
+{"id": "7813.png", "formula": "\\begin{align*} N _ { K , \\epsilon } : = \\{ ( Z ^ 0 , z ^ a , \\zeta ^ i , \\widetilde { \\zeta } _ i ) \\in M \\times ( \\mathbb { R } / \\mathbb { Z } ) ^ { 2 n + 2 } \\ ; | \\ ; \\epsilon < \\tau _ 2 = | Z ^ 0 | , \\ ; \\ ; t ^ a > K , \\ ; a = 1 , . . . , n \\} \\ , . \\end{align*}"}
+{"id": "3515.png", "formula": "\\begin{align*} - \\log \\vert \\det ( 1 - A ) \\vert \\leqslant \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { k } \\vert \\mathrm { R e } ( \\mathrm { t r } ( A ^ k ) ) \\vert \\leqslant \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { k } \\vert \\mathrm { t r } ( A ^ k ) \\vert . \\end{align*}"}
+{"id": "1435.png", "formula": "\\begin{align*} ( k _ { i j } ) _ { 3 \\times 3 } = \\left ( \\begin{matrix} 1 & 0 & - 1 \\\\ 0 & 1 & - 1 \\\\ - \\frac 1 2 & - \\frac 1 2 & 1 \\end{matrix} \\right ) . \\end{align*}"}
+{"id": "133.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } W ( s ) W ^ T ( s ) T _ W X d s = T _ W \\tilde { X } P T _ W W ( t ) \\end{align*}"}
+{"id": "87.png", "formula": "\\begin{align*} \\delta _ k = \\sum _ { m \\in \\Z } \\big ( \\theta _ { \\mathcal M } ( m ) \\theta _ { \\mathcal M } ( m + k ) - \\theta _ { \\mathcal M } ( m ) ^ 2 \\big ) = - \\frac { 1 } { 2 } \\sum _ m \\big ( \\theta _ { \\mathcal M } ( m ) - \\theta _ { \\mathcal M } ( m + k ) \\big ) ^ 2 . \\end{align*}"}
+{"id": "8016.png", "formula": "\\begin{align*} X ^ H ( t ) = v _ { i _ k } ^ H t + \\sum _ { \\substack { h = 0 \\\\ h \\not = k } } ^ H ( v _ { i _ h } ^ H - v _ { i _ k } ^ H ) T _ { ( i _ h ) } ( t ) = g _ k ^ H \\Bigl ( T _ { ( - i _ k ) } ^ H ( t ) \\Bigr ) , \\ \\ \\ t \\ge 0 , \\ , k = 0 , \\dots , H . \\end{align*}"}
+{"id": "1681.png", "formula": "\\begin{align*} \\rho _ { \\texttt { b } } ( \\boldsymbol { \\xi } ) : = 2 ^ { n ( n + 1 ) } \\prod _ { 1 \\leq j \\leq n } \\bigl ( 1 - \\cos ^ 2 ( \\xi _ j ) \\bigr ) \\prod _ { 1 \\leq j < k \\leq n } \\bigl ( \\cos ( \\xi _ j ) - \\cos ( \\xi _ k ) \\bigr ) ^ 2 \\end{align*}"}
+{"id": "306.png", "formula": "\\begin{align*} \\left \\langle \\mathsf { L } _ { j } \\left ( s \\right ) u , \\overline { v } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } : = \\ell _ { j } \\left ( s \\right ) \\left ( u , v \\right ) \\qquad \\forall u , v \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) . \\end{align*}"}
+{"id": "596.png", "formula": "\\begin{align*} I I I _ + + I I I _ - = \\int _ 0 ^ t \\norm { P _ { \\mu } ( P _ { \\mu } \\psi ^ \\mu ( s ) ) \\mathfrak K _ 1 } _ { \\mathcal L _ 2 } ^ 2 d s , \\end{align*}"}
+{"id": "3095.png", "formula": "\\begin{align*} B _ 1 = \\{ \\xi ( 1 , 2 , i , j ) \\mid 3 \\le i < j \\} , ~ ~ B _ 2 = \\{ \\xi ( 1 , 2 , 3 , j ) - \\xi ( 1 , 2 , j , 3 ) \\mid j \\ge 4 \\} . \\end{align*}"}
+{"id": "6281.png", "formula": "\\begin{align*} \\frac { d } { d t } M _ \\lambda ( u , \\bar u ) = \\partial _ x P _ \\lambda ( u , \\bar u ) , \\frac { d } { d t } P _ \\lambda ( u , \\bar u ) = \\partial _ x E _ \\lambda ( u , \\bar u ) . \\end{align*}"}
+{"id": "651.png", "formula": "\\begin{align*} \\mathbb L ^ 2 ( \\Omega , Z ) = \\left \\{ u \\in L ^ 2 ( \\Omega , Z ) \\colon \\right \\} . \\end{align*}"}
+{"id": "4042.png", "formula": "\\begin{align*} \\begin{aligned} C ( x , \\rho ) = & \\cos \\rho ( x - a ) + \\frac { 1 } { \\rho } \\left ( q _ { 1 } ( x ) - q _ { 1 } ( a ) \\cos \\rho ( x - a ) \\right ) \\\\ & - \\frac { 1 } { \\rho } \\int \\limits _ { a } ^ { x } q _ { 1 } ^ { \\prime } ( t ) \\cos \\rho ( x - t ) d t + \\frac { 1 } { \\rho } \\int \\limits _ { a } ^ { x } q _ { 0 } ( t ) \\sin \\rho ( x - t ) d t . \\end{aligned} \\end{align*}"}
+{"id": "5840.png", "formula": "\\begin{align*} w _ 2 = s _ { \\alpha _ 2 } s _ { \\alpha _ 2 + 2 \\alpha _ 3 } s _ { \\alpha _ 2 + 2 \\alpha _ 3 + 2 \\alpha _ 4 } . \\end{align*}"}
+{"id": "2804.png", "formula": "\\begin{align*} \\psi _ N ( r ) & : = 2 \\ , \\mathrm { v o l } ( \\mathbb { S } ^ { N - 2 } ) \\int _ { - 1 } ^ 1 \\frac { ( 1 - h ^ 2 ) ^ { \\frac { N - 3 } { 2 } } } { \\big ( 1 + r ^ 2 - 2 r h \\big ) ^ { ( N + 2 s ) / 2 } } \\dd h & & , \\\\ \\psi _ N ( r ) & : = 2 \\left ( \\frac { 1 } { ( 1 - r ) ^ { 1 + 2 s } } + \\frac { 1 } { ( 1 + r ) ^ { 1 + 2 s } } \\right ) & & . \\end{align*}"}
+{"id": "5312.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - z } - \\prod _ { i = 0 } ^ n ( 1 + z ^ { 2 ^ i } ) & = \\frac { 1 } { 1 - z } - \\sum _ { i = 0 } ^ { 2 ^ { n + 1 } - 1 } z ^ i \\\\ & = \\sum _ { i = 2 ^ { n + 1 } } ^ \\infty z ^ i \\\\ & = \\frac { z ^ { 2 ^ { n + 1 } } } { 1 - z } \\end{align*}"}
+{"id": "967.png", "formula": "\\begin{align*} \\Pr [ X _ \\kappa = 1 ] \\geq ( 1 - \\gamma ) \\cdot \\left ( \\frac { \\beta - \\alpha \\cdot \\gamma } { ( 1 + \\rho ) \\cdot \\beta } \\right ) \\cdot \\left ( 1 - \\frac { 2 \\cdot \\beta \\cdot \\rho } { \\alpha \\cdot \\gamma } \\right ) \\end{align*}"}
+{"id": "2881.png", "formula": "\\begin{align*} \\Phi _ \\xi : = \\mathcal { J } \\phi _ \\xi \\quad \\phi _ \\xi : = \\sum _ { v \\in W _ 0 } T _ v e ^ { i \\xi } , \\end{align*}"}
+{"id": "8190.png", "formula": "\\begin{align*} d x _ i = - \\iota _ T \\omega _ i . \\end{align*}"}
+{"id": "8677.png", "formula": "\\begin{align*} \\frac { x _ { n + 1 } } { x _ 0 } - f \\Big ( \\frac { x _ 1 } { x _ 0 } , . . . , \\frac { x _ n } { x _ 0 } \\Big ) = 0 , ~ ~ ~ ~ ~ ~ ~ \\frac { y _ 0 } { y _ { n + 1 } } + \\hat { f } \\Big ( \\frac { y _ 1 } { y _ { n + 1 } } , . . . , \\frac { y _ n } { y _ { n + 1 } } \\Big ) = 0 . \\end{align*}"}
+{"id": "5199.png", "formula": "\\begin{align*} D + D ^ { - 1 } \\sum _ { i > D } | N _ i \\cap N _ v | < 2 D = : M . \\end{align*}"}
+{"id": "4581.png", "formula": "\\begin{align*} I _ { 1 , \\ast } ( t ) + \\Vert \\varphi ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } \\le & \\frac { C _ 3 } { \\varepsilon } \\int _ 0 ^ t ( I _ { 1 , \\ast } ( s ) + \\Vert \\varphi ( s ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } ) d s \\\\ & + C _ 1 \\varepsilon \\left \\{ I _ { 1 , \\ast } ( t ) + \\Vert { \\mathbf F } ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ^ 2 _ + ) } \\right \\} + \\frac { C _ 3 } { \\varepsilon } \\Vert { \\mathbf F } \\Vert ^ 2 _ { H ^ 1 _ \\ast ( \\Omega _ t ) } \\ , . \\end{align*}"}
+{"id": "7113.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta \\varphi = \\sigma & \\hbox { \\rm i n } \\ D \\\\ ( \\nabla \\varphi \\cdot n ) _ { | \\partial D } = \\tau & \\hbox { \\rm o n } \\ \\partial D \\end{array} \\right . \\end{align*}"}
+{"id": "7992.png", "formula": "\\begin{align*} \\tilde { f } _ { k } + \\sum _ { i = 1 } ^ { n } a _ { i } g _ { i } | _ { \\Omega _ { k } } \\notin \\bigcup _ { q \\in \\Gamma } L _ { q } ( \\Omega _ { k } ) a _ { 1 } , \\ldots , a _ { n } \\in \\mathbb { K } \\end{align*}"}
+{"id": "544.png", "formula": "\\begin{align*} - i d \\mathbf u ( t ) + \\mathbf h ( D ) \\mathbf u ( t ) \\ , d t = \\mathbf N ( \\mathbf u ( t ) ) \\ , d t + \\mathbf M ( \\mathbf u ( t ) ) \\ , d W ( t ) , \\mathbf u ( 0 ) = \\mathbf u _ 0 , \\end{align*}"}
+{"id": "1079.png", "formula": "\\begin{align*} \\lambda \\left ( \\delta \\right ) = \\ln \\left ( \\delta ^ { - \\left ( 2 A _ { 1 } + 2 \\right ) ^ { - \\nu _ { 1 } } / 4 } \\right ) . \\end{align*}"}
+{"id": "1544.png", "formula": "\\begin{align*} \\overline { p ( \\rho ) } = \\overline { \\rho \\ , F ' ( \\rho ) } - \\overline { F ( \\rho ) } + \\frac { 1 } { 2 \\eta ^ 2 } \\ , \\overline { \\rho ^ 2 } , \\end{align*}"}
+{"id": "2266.png", "formula": "\\begin{align*} E _ { \\Phi } ( \\Gamma _ 0 ) = \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\sum _ { j = 1 } ^ n \\theta _ \\alpha \\left ( \\ell + \\frac { j - k } { n } \\right ) = \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\sum _ { j = 1 } ^ n \\theta _ \\alpha \\left ( \\ell + \\frac { j } { n } \\right ) = \\sum _ { j = 1 } ^ n \\theta _ \\alpha \\left ( \\ell + \\frac { j } { n } \\right ) , \\end{align*}"}
+{"id": "858.png", "formula": "\\begin{align*} \\lambda ^ * = \\frac { \\alpha _ i } { 2 \\sum _ { j \\neq i } \\alpha _ j ^ 2 \\sigma ^ 2 + 2 \\sigma _ N ^ 2 } = \\frac { n } { 2 ( n - 1 ) \\sigma ^ 2 + 2 n \\sigma _ N ^ 2 } . \\end{align*}"}
+{"id": "6141.png", "formula": "\\begin{align*} \\varphi \\left ( ( s _ t , t ) \\right ) = \\begin{cases} ( \\varphi _ t ( s _ t ) , t ) & \\varphi _ t ( s _ t ) \\in T , \\\\ ( \\varphi _ t ( s _ t ) , \\psi ( t ) ) & \\varphi _ t ( s _ t ) \\in \\tilde { T } . \\end{cases} \\end{align*}"}
+{"id": "3493.png", "formula": "\\begin{align*} y = ( y _ 0 , \\ldots y _ m ) , \\sum _ 0 ^ m y _ i = 1 , y _ i \\gtrsim \\epsilon , \\end{align*}"}
+{"id": "4934.png", "formula": "\\begin{align*} I _ 2 ( k ) = \\dfrac { \\log 2 } { 2 ^ { k + 1 } \\pi } + \\dfrac { 1 } { 2 \\pi ^ 2 } \\sum _ { m , n \\geq 1 } \\dfrac { ( - 1 ) ^ m } { 2 ^ k m ^ 2 + n ^ 2 } . \\end{align*}"}
+{"id": "3403.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle \\sum _ { i = 1 } ^ 3 \\delta _ i v _ i ^ 2 = 1 , \\ \\sum _ { i = 1 } ^ 3 \\delta _ i v _ i V _ i = 0 , \\ \\sum _ { i = 1 } ^ 3 \\delta _ i V _ i ^ 2 = ( c - \\tilde { c } ) \\ \\mbox { a n d } \\ \\sum _ { i = 1 } ^ 3 { \\frac { V _ i } { v _ i } } = 0 \\end{array} \\end{align*}"}
+{"id": "2163.png", "formula": "\\begin{align*} \\sinh ^ 2 \\Lambda = \\left ( \\frac { ( \\alpha ^ { d } m + \\alpha ^ { - d } m ^ { - 1 } ) m ^ d + ( \\alpha ^ { d } m ^ { - 1 } + \\alpha ^ { - d } m ) m ^ { - d } } { m ^ { d } + m ^ { - d } } \\right ) ^ 2 - 1 . \\end{align*}"}
+{"id": "7841.png", "formula": "\\begin{align*} \\widetilde { N } : = \\{ ( \\tau _ 2 , b + \\mathrm { i } t , \\tau _ 1 , \\zeta ^ 1 , \\widetilde { \\zeta } _ 0 , \\widetilde { \\zeta } _ 1 , \\sigma ) \\in \\mathbb { R } _ { > 0 } \\times \\overline { M } ^ { } \\times \\mathbb { R } ^ 4 \\times \\mathbb { R } \\ ; | \\ ; R _ 1 ( t , \\tau ) > 0 , R _ 2 ( t , \\tau ) > 0 \\ ; \\} . \\end{align*}"}
+{"id": "3077.png", "formula": "\\begin{align*} { N _ { { \\rm { S , } } k } } = \\left \\lceil \\frac { { 1 6 { \\pi ^ 2 } { d _ { { \\rm T } , k } } d _ { { k , { \\rm R } } } ^ { \\rm { r } } } } { { { \\lambda ^ 2 } } } \\sqrt { \\frac { { \\left ( { { \\kappa _ { { \\rm { T } } , k } } + 1 } \\right ) { L _ { { k , { \\rm { R } } } } } { \\xi _ 0 ^ \\star } } } { { { \\kappa _ { { \\rm { T } } , k } } { N _ { \\rm { T } } } { N _ { \\rm { R } } } } } } \\right \\rceil , \\end{align*}"}
+{"id": "881.png", "formula": "\\begin{align*} ( \\tilde { u } _ \\epsilon ( x , t ) , \\tilde { v } _ \\epsilon ( x , t ) ) = \\rho ( \\mathrm { e x p } ( \\epsilon V ) ) ( u ( x , t ) , v ( x , t ) ) , \\end{align*}"}
+{"id": "3835.png", "formula": "\\begin{align*} \\eta ( ( X ^ 2 \\times \\R _ + ^ 4 ) \\setminus \\Omega ) = 0 , \\Omega = \\{ ( x _ 0 , x _ 1 , s _ 0 , s _ 1 , w _ 0 , w _ 1 ) \\in X ^ 2 \\times \\R _ + ^ 4 \\mid w _ 0 = w _ 1 \\} . \\end{align*}"}
+{"id": "6785.png", "formula": "\\begin{align*} \\lambda _ 1 = \\frac { 1 } { 2 d } \\left ( c - \\sqrt { c ^ 2 - 4 d s } \\right ) , \\lambda _ 2 = \\frac { 1 } { 2 d } \\left ( c + \\sqrt { c ^ 2 - 4 d s } \\right ) . \\end{align*}"}
+{"id": "8690.png", "formula": "\\begin{align*} \\sum _ { 0 \\leq i _ 1 < . . . < i _ k \\leq n } ( u _ { i _ 1 } + . . . + u _ { i _ k } ) \\partial _ { u _ { i _ 1 } } . . . \\partial _ { u _ { i _ k } } g + ( n + 2 - k ) \\sum _ { 0 \\leq j _ 1 < . . . < j _ { k - 1 } \\leq n } \\partial _ { u _ { j _ 1 } } . . . \\partial _ { u _ { j _ { k - 1 } } } g = 0 \\end{align*}"}
+{"id": "4121.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { k - 2 i + 2 s - a + 1 } \\binom { s + k - 2 i - \\ell } { \\ell } \\binom { k + 3 s - 2 a + 1 - 2 i - 2 \\ell } { k + 2 s - a + 1 - 2 i - \\ell } . \\end{align*}"}
+{"id": "7930.png", "formula": "\\begin{align*} 0 = \\mathrm { T r } ( \\theta _ { I ( g _ j ) } ) = 1 2 + \\sum _ k \\theta _ { Y _ k } \\mathrm { F P d i m } ( Y _ k ) \\end{align*}"}
+{"id": "6417.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } { \\frac { \\partial w } { \\partial t } ( x , t ) = d ( J \\ast w - w ) + r w ( x , t ) ( k - w ( x , t ) ) , x \\in \\mathbb { R } , t > 0 , } \\\\ { w ( x , 0 ) = \\chi ( x ) , x \\in \\mathbb { R } , } \\end{array} \\right . \\end{align*}"}
+{"id": "626.png", "formula": "\\begin{align*} \\mathrm { l . h . s . } \\eqref { d i f f e r e n c e 2 } = \\norm { \\theta _ R \\left ( \\norm { \\mathbf V } _ { t } ^ 2 \\right ) \\mathbf V } _ { T } \\le C \\norm { \\theta _ R \\left ( \\norm { \\mathbf V } _ { t } ^ 2 \\right ) \\mathbf V } _ { T _ { \\mathbf V } } \\le C \\sqrt { 2 R } , \\end{align*}"}
+{"id": "6232.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } i u _ t + g ( u ) \\partial _ x ^ 2 u = N ( u , \\partial _ x u ) , u : \\R \\times \\R \\to \\C , \\\\ \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\end{array} \\right . \\end{align*}"}
+{"id": "2391.png", "formula": "\\begin{align*} f ' ( t ) - h f ( t ) = - \\frac { \\left ( \\mu - r - h \\right ) ^ 2 ( 2 - \\sigma ^ 2 ) } { 2 \\sigma ^ 2 } - h r ( t - T ) - r \\end{align*}"}
+{"id": "639.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} d \\psi = ( - \\alpha ^ j \\partial _ j - i M \\beta ) \\psi \\ , d t + i \\phi \\beta \\psi \\ , d t - M _ { \\mathfrak K _ 1 } \\psi \\ , d t + i \\beta \\psi \\mathfrak K _ 1 \\ , d W , \\\\ d \\phi = \\dot \\phi \\ , d t , \\\\ d \\dot \\phi = ( \\partial _ x ^ 2 - m ^ 2 ) \\phi \\ , d t + \\psi ^ * \\beta \\psi \\ , d t + \\phi \\mathfrak K _ 2 \\ , d W , \\end{gathered} \\right . \\end{align*}"}
+{"id": "3032.png", "formula": "\\begin{align*} G = P _ { 2 , c + 3 } ^ 2 \\cup C _ { \\ell , 2 s } ^ { 0 } \\end{align*}"}
+{"id": "7077.png", "formula": "\\begin{align*} S _ 2 ( . . . ) = \\frac { N ^ { 3 / 4 - i ( t + \\nu ) } \\ , \\eta _ f ( p _ 2 ) } { ( p _ 1 q ) ^ { 1 / 2 } \\ , p _ 2 ^ { 1 / 4 } } \\sum _ { \\pm } \\sum _ { m \\ll M _ 0 } \\frac { \\lambda _ f ( m ) } { m ^ { 1 / 4 } } e \\left ( - m \\frac { \\overline { ( a + b q ) p _ 2 } } { q } \\right ) \\mathcal { I } ^ { \\pm } ( m , u , q , p _ 1 , p _ 2 ) , \\end{align*}"}
+{"id": "3286.png", "formula": "\\begin{align*} \\frac { d } { d t } \\phi _ X ( t ) & = \\frac { d } { d t } \\left ( \\int _ \\mathbb { R } f _ X ( x ) e ^ { \\mu t x } d x \\right ) \\\\ & = \\int _ \\mathbb { R } f _ X ( x ) \\frac { d } { d t } \\left ( e ^ { \\mu t x } \\right ) d x \\\\ & = \\int _ \\mathbb { R } f _ X ( x ) e ^ { \\mu t x } x d x \\mu . \\end{align*}"}
+{"id": "4431.png", "formula": "\\begin{align*} \\hat { u } _ { 2 } ^ + + \\hat { u } _ { 2 } ^ - = 0 , \\hat { u } _ { 2 } ^ + > 0 , \\hat { p } ^ + + \\frac { ( \\hat { H } ^ + _ 2 ) ^ 2 } { 2 } = \\hat { p } ^ - + \\frac { ( \\hat { H } ^ + _ { 2 } ) ^ 2 } { 2 } \\ , , \\end{align*}"}
+{"id": "3139.png", "formula": "\\begin{align*} \\phi _ * W _ { L _ X } = W ^ { Y \\backslash D _ Y } _ { L _ Y } + \\left ( W _ { L _ Y } ^ { D _ Y } \\right ) ^ r - r ! \\langle \\psi _ { r - 2 } \\emph { } \\rangle ^ Y _ r , \\end{align*}"}
+{"id": "6479.png", "formula": "\\begin{align*} f ( 0 ) = 0 , f ( \\epsilon ) = 0 . \\end{align*}"}
+{"id": "1223.png", "formula": "\\begin{align*} v _ { n , T } ( t ) & = g _ n [ \\chi _ n P _ n e ^ { i ( \\lambda _ n ^ { - 2 } t + t _ n ) \\Delta } \\phi ] \\\\ & = \\chi ( \\frac { x - x _ n } { \\lambda _ n } ) e ^ { i ( t + \\lambda _ n ^ 2 t _ n ) \\Delta } g _ n [ P _ n \\phi ] . \\end{align*}"}
+{"id": "3795.png", "formula": "\\begin{align*} { \\rm U O T } ( \\mu _ 0 , \\mu _ 1 ) = \\inf _ { \\alpha \\in \\overline { S } ^ p _ { = } ( \\mu _ 0 , \\mu _ 1 ) } ( H _ p , \\alpha ) , \\end{align*}"}
+{"id": "5725.png", "formula": "\\begin{align*} \\int _ 0 ^ { + \\infty } e ^ { - \\lambda x } W ^ { ( q ) } ( x ) d x = \\frac { 1 } { \\Psi ( \\lambda ) - q } , \\lambda > \\Phi ( q ) . \\end{align*}"}
+{"id": "5161.png", "formula": "\\begin{align*} E [ L ^ 2 ] = & \\sum _ i f ( x _ i ) \\delta \\big ( \\log _ 2 { \\big ( f ( x _ i ) \\delta \\big ) } \\big ) ^ 2 \\\\ = & \\sum _ i f ( x _ i ) \\delta \\big ( \\log _ 2 { f ( x _ i ) } + \\log _ 2 { \\delta } \\big ) ^ 2 \\\\ = & \\sum _ i f ( x _ i ) \\delta \\big ( \\log _ 2 ^ 2 { f ( x _ i ) } + 2 \\log _ 2 { f ( x _ i ) } \\log _ 2 { \\delta } \\big ) + \\log _ 2 ^ 2 { \\delta } . \\end{align*}"}
+{"id": "7442.png", "formula": "\\begin{align*} \\psi _ { ( y _ 1 , y _ 2 ) } ^ { \\prime \\prime } ( t ) = & ( \\kappa _ 1 + \\kappa _ 2 + 1 ) t ^ { \\kappa _ 1 + \\kappa _ 2 } \\left [ F _ { ( y _ 1 , y _ 2 ) } ( t ) - \\lambda ( \\kappa _ 1 + \\kappa _ 2 + 2 ) \\int _ { \\Omega } \\vert y _ 1 \\vert ^ { \\kappa _ 1 + 1 } \\vert y _ 2 \\vert ^ { \\kappa _ 2 + 1 } d z \\right ] \\\\ & + t ^ { \\kappa _ 1 + \\kappa _ 2 + 1 } F _ { ( y _ 1 , y _ 2 ) } ^ { \\prime } ( t ) . \\end{align*}"}
+{"id": "4912.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } A _ { k } ( x ) \\frac { z ^ { n } } { n ! } : = \\frac { x - 1 } { x - e ^ { ( x - 1 ) z } } . \\end{align*}"}
+{"id": "7411.png", "formula": "\\begin{align*} - \\int _ { 0 } ^ { t } \\int _ { \\mathbb { T } } \\rho _ { n } W _ { n } \\underbrace { \\left ( \\partial _ { t } W _ { n } + u _ { n } \\partial _ { x } W _ { n } \\right ) } _ { = 0 } ~ d x d s + \\int _ { \\mathbb { T } } \\rho _ { n } W _ { n } ^ { 2 } ( x , t ) - \\rho _ { n } W _ { n } ^ { 2 } ( x , 0 ) ~ d x = 0 , \\end{align*}"}
+{"id": "3778.png", "formula": "\\begin{align*} \\sup _ { x \\in E } - f ( - x ) - g ( A x ) = \\min _ { y ^ * \\in F ^ * } f ^ * ( A ^ * y ^ * ) + g ^ * ( y ^ * ) \\end{align*}"}
+{"id": "5496.png", "formula": "\\begin{align*} c _ + ^ { ( h , g ) } = & \\mathrm { p e x p } \\left ( \\dfrac { ( Q ^ 8 - Q ^ { - 8 } ) } { ( 1 + s ^ { - \\varphi ( g ) _ { 2 , 2 } } p ^ { \\varphi ( g ) _ { 2 , 1 } } ) ( 1 + s ^ { - \\varphi ( h g ) _ { 1 , 2 } } p ^ { \\varphi ( h g ) _ { 1 , 1 } } ) ( 1 + s ^ { \\varphi ( h g ) _ { 2 , 2 } } p ^ { - \\varphi ( h g ) _ { 2 , 1 } } ) } \\right ) \\qquad \\in \\quad \\mathbf R , \\end{align*}"}
+{"id": "716.png", "formula": "\\begin{align*} \\mathbf V ( t ) = \\mathbf u ( t ) , \\end{align*}"}
+{"id": "8661.png", "formula": "\\begin{align*} \\tilde { x } _ { n + 1 } = \\tilde { x } _ 0 \\tilde { f } \\Big ( \\frac { \\tilde { x } _ 1 } { \\tilde { x } _ 0 } , . . . , \\frac { \\tilde { x } _ n } { \\tilde { x } _ 0 } \\Big ) , \\end{align*}"}
+{"id": "7783.png", "formula": "\\begin{align*} \\frac { 1 + t _ { + } ^ { 0 , n } t } { t - t _ { + } ^ { 0 , n } } : = \\begin{cases} 1 / t , n < 0 \\\\ - t , n > 0 \\end{cases} , \\frac { 1 - t _ { + } ^ { 0 , n } t } { t - t _ { + } ^ { 0 , n } } : = \\begin{cases} 1 / t , n < 0 \\\\ t , n > 0 \\ , . \\end{cases} \\end{align*}"}
+{"id": "5240.png", "formula": "\\begin{align*} \\mathcal { X } _ { T } = \\{ x : ~ x \\in \\mathbb { X } , ~ g _ { t } ( x ) \\le { \\bf 0 } _ { m } , ~ \\forall t \\in [ T ] \\} \\end{align*}"}
+{"id": "4778.png", "formula": "\\begin{align*} H _ { + } ^ { n } = \\{ h \\in H _ { C } ^ { + } \\mid ( h x ) _ { g } = x _ { g } g \\neq a ^ { k } M < k < M + n x \\in X _ { p } \\} \\end{align*}"}
+{"id": "2155.png", "formula": "\\begin{align*} \\mu : = w - \\beta - \\sqrt { ( w - \\beta ) ^ 2 - \\alpha ^ 2 } , \\end{align*}"}
+{"id": "8508.png", "formula": "\\begin{align*} \\partial ^ { * } E \\cap \\{ z > \\bar { z } \\} & = \\partial ^ { * } \\left ( ( 0 , \\tau ) + F _ { \\ell } \\right ) \\cap \\{ z > \\bar { z } \\} \\\\ & = \\left ( ( 0 , \\tau ) + \\partial ^ { * } F _ { \\ell } \\right ) \\cap \\left ( ( 0 , \\tau ) + \\{ z > \\bar { z } \\} \\right ) \\\\ & = ( 0 , \\tau ) + \\left ( \\partial ^ { * } F _ { \\ell } \\cap \\{ z > \\bar { z } \\} \\right ) . \\end{align*}"}
+{"id": "2407.png", "formula": "\\begin{align*} S _ { \\i } \\left ( x _ { | \\i | } + Q ( \\i ) \\right ) \\cap S _ { \\i ' } \\left ( x _ { | \\i ' | } + Q ( \\i ' ) \\right ) = \\emptyset \\end{align*}"}
+{"id": "6731.png", "formula": "\\begin{align*} F ' ( t ) = & \\gamma ' ( t ) \\left \\{ \\int _ { \\Sigma _ t } | \\nabla u | ^ { - 2 } | \\nabla _ { \\Sigma _ t } | \\nabla u | | ^ 2 + \\frac { 1 } { 2 } R _ { M } + \\frac 1 2 | \\overset { \\circ } { h } | ^ 2 + ( 2 a + 1 ) \\left ( \\frac { H } { 2 } - \\eta ( u ) | \\nabla u | \\right ) ^ 2 \\right \\} \\\\ & \\gamma ' ( t ) \\left ( 4 \\pi - \\int _ { \\Sigma _ t } K _ { \\Sigma _ t } \\right ) . \\end{align*}"}
+{"id": "6371.png", "formula": "\\begin{align*} \\chi ^ { \\mathrm { o p } } = a ( 1 \\otimes 1 ) + n ( x \\otimes x ) + o ( x g \\otimes x ) + r ( x \\otimes x g ) + s ( x g \\otimes x g ) . \\end{align*}"}
+{"id": "4948.png", "formula": "\\begin{align*} f _ k ( \\tau ) : = q ^ { k + 1 } \\Psi ( q ^ 4 ) ^ 2 \\cdot \\Psi ( q ^ 2 ) ^ { 4 k } = : \\sum _ { n \\geq 0 } t _ { k } ( n ) q ^ { 2 n + k + 1 } , \\end{align*}"}
+{"id": "2789.png", "formula": "\\begin{align*} \\begin{multlined} x ^ { k + 1 } \\cdot x ^ { k + 1 } \\cdot 1 7 x ^ 4 + x ^ { k + 1 } \\cdot ( 2 ^ k + k ) x ^ k \\cdot x ^ 5 + ( 2 ^ { k } + k ) x ^ { k } \\cdot x ^ { k + 1 } \\cdot x ^ 5 \\\\ = ( 2 k + 2 ^ { k + 1 } + 1 7 ) x ^ { 2 k + 6 } , \\end{multlined} \\end{align*}"}
+{"id": "2656.png", "formula": "\\begin{align*} k ( x , t ) = - \\int _ 0 ^ x \\frac { k ( y , t ) } { 1 - y } \\ , d y + b ( x , t ) \\ , , \\end{align*}"}
+{"id": "1805.png", "formula": "\\begin{gather*} \\sum _ { 0 \\leq n \\leq V } \\frac { 1 } { ( n + \\alpha ) ^ { 2 \\sigma } } \\leq \\alpha ^ { - 1 } + \\sum _ { n = 1 } ^ { \\infty } n ^ { - 2 \\sigma } \\ll \\frac { 1 } { 2 \\sigma - 1 } , \\end{gather*}"}
+{"id": "5114.png", "formula": "\\begin{align*} S ( \\sigma _ 1 , \\sigma _ 2 ) = 0 . \\end{align*}"}
+{"id": "6531.png", "formula": "\\begin{align*} \\bigcup _ { i = 1 } ^ N I _ i = I , \\ \\sup _ { 1 \\leq i \\leq N } | I _ i | \\leq \\frac { \\tau } { 2 A } . \\end{align*}"}
+{"id": "3234.png", "formula": "\\begin{align*} S ^ { 1 } _ { \\alpha } ( \\epsilon , R ) - \\Delta _ { \\alpha } ( \\epsilon , R ) = \\sum _ { \\frac { R } { \\sqrt { 1 + \\alpha ^ { 2 } } } < m \\leq \\frac { R } { \\sqrt { 1 + ( \\alpha - \\epsilon ) ^ { 2 } } } } \\# \\{ n : m ( \\alpha - \\epsilon ) < n < m ( \\alpha + \\epsilon ) ) \\} , \\end{align*}"}
+{"id": "5170.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\frac { \\frac { 3 } { 2 } H [ Q _ { { \\mathrm { u n i } } } ( X ) ] } { \\log _ 2 D } = - \\frac { 3 } { 4 } . \\end{align*}"}
+{"id": "7099.png", "formula": "\\begin{align*} \\mathcal { J } ^ { + } _ 1 ( N _ 0 \\xi , m , q ) = \\int _ 0 ^ \\infty U ( y ) \\ , y ^ { - i t } \\ , e \\left ( \\frac { 2 \\sqrt { m N y } } { p _ 1 q \\sqrt { p _ 2 } } + \\frac { 3 ( N N _ 0 \\xi ( y + w ) ) ^ { 1 / 3 } } { p _ 1 q r ^ { 1 / 3 } } \\right ) d y . \\end{align*}"}
+{"id": "562.png", "formula": "\\begin{align*} \\mathbf I ( t ) = \\mathbf \\Lambda ^ { \\mathbf s } \\int _ 0 ^ t \\mathbf S ( - s ) \\mathbf M ( \\mathbf u ( s ) ) \\ , d W ( s ) = \\int _ 0 ^ t \\mathbf \\Lambda ^ { \\mathbf s } \\mathbf S ( - s ) \\mathbf M ( \\mathbf u ( s ) ) \\ , d W ( s ) , \\end{align*}"}
+{"id": "1821.png", "formula": "\\begin{gather*} \\left \\| \\sum _ { j = 1 } ^ { n } \\beta _ j x _ j - \\sum _ { j = 1 } ^ { n } \\gamma _ j x _ j \\right \\| ^ 2 \\leq 4 \\sum _ { j = 1 } ^ { n } \\| x _ j \\| ^ 2 \\end{gather*}"}
+{"id": "7259.png", "formula": "\\begin{align*} P _ s ^ { N } \\left ( f \\right ) ( x ) = \\sum _ { \\omega \\in \\Omega ^ { N } } \\mathbb { Q } \\left ( [ \\omega ] \\right ) P _ { s , \\omega , N } \\left ( f \\right ) ( x ) . \\end{align*}"}
+{"id": "1230.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } E ( u _ n ) = \\sum _ { j = 1 } ^ J \\lim _ { n \\to \\infty } E ( e ^ { i t _ n ^ j \\Delta } \\phi ^ j ) + \\lim _ { n \\to \\infty } E ( W _ n ^ J ) . \\end{align*}"}
+{"id": "3934.png", "formula": "\\begin{align*} I = \\bigcup _ { x \\in \\omega _ c } [ x - 1 , x + 1 ] . \\end{align*}"}
+{"id": "5363.png", "formula": "\\begin{align*} H ^ 1 ( \\O _ { L _ 1 \\cup \\ldots \\cup L _ { 2 b - 2 } } ( 1 ) ( - B _ { b - 1 } ) ) = 0 . \\end{align*}"}
+{"id": "266.png", "formula": "\\begin{align*} [ v ^ \\alpha ] T ( x , v ) \\prod \\limits _ { v \\in V } \\alpha ( v ) ! = X _ { G ^ \\alpha } . \\end{align*}"}
+{"id": "2232.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\to 0 ^ + } \\frac { G ( \\lambda t ) } { G ( t ) } = h _ G ( \\lambda ) , \\quad \\lambda > 0 . \\end{align*}"}
+{"id": "3472.png", "formula": "\\begin{align*} \\phi _ t = | \\log | t | | u ( - \\frac { \\log | F _ 0 | _ { h ^ { \\otimes d _ 0 } } } { | \\log | t | | } , \\ldots , - \\frac { \\log | F _ m | _ { h ^ { \\otimes d _ m } } } { | \\log | t | | } ) , \\end{align*}"}
+{"id": "8766.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d } r h ^ { \\beta } \\Big | \\eta ( z _ { i } h ^ { - 1 } ) - \\sum _ { 0 \\leq | m | \\leq \\ell } \\frac { 1 } { m ! } D ^ { m } \\eta ( x _ { i } h ^ { - 1 } ) ( z _ { i } h ^ { - 1 } - x _ { i } h ^ { - 1 } ) ^ { m } \\Big | & \\leq r L \\sum _ { i = 1 } ^ { d } | z _ { i } - x _ { i } | ^ { \\beta } \\\\ & \\leq r L \\Big ( \\sum _ { i = 1 } ^ { d } | z _ { i } - x _ { i } | ^ { 2 } \\Big ) ^ { \\beta / 2 } \\\\ & = r L \\norm { z - x } ^ { \\beta } , \\end{align*}"}
+{"id": "2460.png", "formula": "\\begin{align*} \\overline { H } \\left ( \\bigsqcup _ { a \\in \\mathcal { A } } ^ { P _ A } G _ a \\right ) = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } H _ \\chi \\left ( \\bigwedge _ { a \\in \\mathcal { A } } G _ a ^ { \\wedge n T _ { \\overline { a } ^ n } ( a ) } \\right ) . \\end{align*}"}
+{"id": "3223.png", "formula": "\\begin{align*} \\textnormal { A r e a } ( \\textnormal { S e c t } _ { \\alpha , \\epsilon } ( R ) ) & = \\frac { R ^ 2 } { 2 } \\left ( \\tan ^ { - 1 } ( \\alpha + \\epsilon ) - \\tan ^ { - 1 } ( \\alpha - \\epsilon ) \\right ) = \\frac { \\epsilon R ^ { 2 } } { 1 + \\alpha ^ { 2 } } + O \\left ( \\epsilon ^ { 3 } R ^ { 2 } \\right ) . \\end{align*}"}
+{"id": "4589.png", "formula": "\\begin{align*} [ L _ m , L _ n ] = ( m \\ , { - } \\ , n ) \\ , L _ { m + n } + \\frac { 1 } { 1 2 } \\ , ( m ^ 3 \\ , { - } \\ , m ) \\ , \\delta _ { m + n , 0 } \\ , C \\end{align*}"}
+{"id": "8293.png", "formula": "\\begin{align*} \\begin{cases} \\bar p _ \\tau = \\bar p _ { x x } , & ( x , \\tau ) \\in ( 0 , 1 ) \\times ( 0 , \\infty ) , \\\\ \\bar p _ x ( 0 , \\tau ) = c \\bar p ( 0 , \\tau ) , & \\tau \\in ( 0 , \\infty ) , \\\\ \\bar p _ x ( 1 , \\tau ) = 0 , & \\tau \\in ( 0 , \\infty ) , \\\\ \\bar p ( x , 0 ) = \\bar p _ 0 ( x ) , & x \\in ( 0 , 1 ) . \\end{cases} \\end{align*}"}
+{"id": "6642.png", "formula": "\\begin{align*} { \\rho } _ { ( 1 ) , \\infty } ^ { ( \\rm c J ) } ( x ; \\beta , p , q ) | _ { \\beta = 2 \\atop p = 1 , q = 0 } = 1 - \\bigg ( { \\sin \\pi x \\over \\pi x } \\bigg ) ^ 2 . \\end{align*}"}
+{"id": "8869.png", "formula": "\\begin{align*} \\eta _ t = \\min \\left ( \\frac { \\mathfrak { y } } { d } , \\ , d ^ { - \\frac { 2 ( \\beta - 1 ) } { 2 \\beta - 1 } } T ^ { - \\frac { \\beta } { 2 \\beta - 1 } } \\right ) \\qquad h _ t = \\mathfrak { h } \\ , T ^ { - \\frac { 1 } { 2 ( 2 \\beta - 1 ) } } \\enspace , \\end{align*}"}
+{"id": "4640.png", "formula": "\\begin{align*} \\bigl ( \\mathbf { M } ^ { \\pi _ { \\mathcal { S } ( v _ 1 ) } } \\otimes L \\bigr ) ^ { K _ { v _ 1 } } = \\bigl ( M ^ { \\pi _ 1 ^ * } \\otimes L \\bigr ) ^ H \\simeq \\textup { H o m } _ H \\bigl ( M ^ { \\pi _ 1 } , L ) , \\end{align*}"}
+{"id": "7419.png", "formula": "\\begin{align*} \\frac { \\gamma _ { n } + 1 } { \\gamma _ { n } } \\int _ { \\mathbb { T } } \\left ( \\pi _ { n } ( x , t ) - \\pi _ { n } ( x , 0 ) \\right ) ~ d x = - \\int _ { 0 } ^ { t } \\int _ { \\mathbb { T } } \\lambda _ { n } ( \\rho _ { n } ) \\partial _ { x } u _ { n } ~ d x d s . \\end{align*}"}
+{"id": "8889.png", "formula": "\\begin{align*} H ^ * \\big ( P ( E ) \\big ) = H ^ * ( M ) [ x ] / I , \\end{align*}"}
+{"id": "2239.png", "formula": "\\begin{align*} 0 \\leq \\eta _ k \\leq 1 , \\eta _ k = 1 \\in B ( 0 , k ) , s u p p ( \\eta _ k ) = B ( 0 , 2 k ) , \\quad \\quad | \\nabla \\eta _ k | \\leq \\frac { 2 } { k } . \\end{align*}"}
+{"id": "5997.png", "formula": "\\begin{align*} S ^ { - \\infty } ( T ^ * \\mathbb { R } ^ n ) = \\bigcup _ { m \\in \\mathbb { R } } S ^ m ( T ^ * \\mathbb { R } ^ n ) . \\end{align*}"}
+{"id": "4979.png", "formula": "\\begin{align*} \\langle x , y \\rangle : = x y + x R ( y ) + y R ( x ) . \\end{align*}"}
+{"id": "5662.png", "formula": "\\begin{align*} \\Gamma _ 0 \\cup \\Gamma _ 1 = ( x _ 0 = x _ 2 = 0 ) \\cup ( x _ 0 = x _ 3 = 0 ) \\end{align*}"}
+{"id": "5418.png", "formula": "\\begin{align*} A _ { M _ 1 , \\ldots , M _ d ; S _ 1 , \\ldots , S _ d } ^ { n _ 1 , \\ldots , n _ d } f = A _ { M _ 1 ; S _ 1 } ^ { n _ 1 } \\circ \\cdots \\circ A _ { M _ d ; S _ d } ^ { n _ d } f , \\end{align*}"}
+{"id": "4486.png", "formula": "\\begin{align*} \\sum ^ { i } _ { k = 0 } f _ k + S _ { \\theta _ i } E _ i = S _ { \\theta _ i } \\mathcal { F } ^ a , \\sum ^ { i } _ { k = 0 } g _ k + S _ { \\theta _ i } \\tilde { E } _ i = 0 . \\end{align*}"}
+{"id": "1673.png", "formula": "\\begin{align*} \\left . \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } \\right | _ { q = 0 } = \\frac { 2 \\pi ( \\lambda + \\varrho _ { \\texttt { a } } ) } { n + m } ( \\lambda \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { a } } ) , \\end{align*}"}
+{"id": "3852.png", "formula": "\\begin{align*} H = - \\theta ' ( s ) - ( \\lambda _ 1 + \\lambda _ 2 ) \\cos \\theta ( s ) . \\end{align*}"}
+{"id": "7942.png", "formula": "\\begin{align*} S _ 1 = ( S \\cap V _ 1 ) \\cup \\{ v _ 1 \\} \\ 1 \\end{align*}"}
+{"id": "4224.png", "formula": "\\begin{align*} \\tilde \\lambda ( \\rho ) = \\ , & \\ , ( 2 a _ 2 a _ 3 a _ 6 - 2 a _ 4 a _ 5 a _ 6 + a _ 1 a _ 6 ^ 2 + 4 a _ 3 a _ 4 a _ 7 + 4 a _ 5 ^ 2 a _ 7 - 4 a _ 4 ^ 2 a _ 9 - 4 a _ 2 a _ 5 a _ 9 + 4 a _ 1 a _ 7 a _ 9 + a _ 1 ^ 2 a _ { 1 1 } ) a _ { 1 1 } \\\\ [ 2 p t ] & + \\frac { 3 } { 2 } a _ { 1 0 } q _ 1 - 2 a _ 9 q _ 3 + \\frac { 5 } { 6 } a _ 3 q _ 4 + a _ 5 q _ 5 - a _ 7 q _ 6 + \\frac { 1 } { 2 } a _ 4 q _ 7 + \\frac { 2 } { 3 } a _ 2 q _ 8 + \\frac { a _ 6 } { 2 } q _ 9 . \\end{align*}"}
+{"id": "8108.png", "formula": "\\begin{align*} \\begin{aligned} \\uppercase \\expandafter { \\romannumeral 1 } = \\sum \\limits _ { j } \\frac { \\lambda _ { j } } { \\omega ( Q _ { j } ) ^ { 1 / p } } ( M ( T ( a _ { j } ) ) ( x ) \\chi _ { 2 \\sqrt { n } Q _ { j } ^ { * } } ( x ) + ( M ( \\chi _ { Q _ { j } } ) ( x ) ) ^ { \\gamma } \\chi _ { ( 2 \\sqrt { n } Q _ { j } ^ { * } ) ^ { c } } ( x ) ) \\end{aligned} \\end{align*}"}
+{"id": "5209.png", "formula": "\\begin{align*} ( I \\cap \\widetilde { w } ^ { - 1 } I \\widetilde { w } ) = \\begin{pmatrix} [ 0 , 0 ] & [ 1 , \\infty ] & [ 2 , \\infty ] \\\\ [ 1 , \\infty ] & [ 0 , 0 ] & [ 1 , \\infty ] \\\\ [ 1 , \\infty ] & [ 1 , \\infty ] & [ 0 , 0 ] \\end{pmatrix} \\end{align*}"}
+{"id": "5640.png", "formula": "\\begin{align*} y ( y + Q ) - C + x _ 3 \\bigl ( ( x _ 0 + x _ 1 ) y + x _ 1 Q + B \\bigr ) = 0 , \\end{align*}"}
+{"id": "8641.png", "formula": "\\begin{align*} \\cosh ( x ) & = 1 + \\frac { x ^ 2 } { 2 } \\int _ 0 ^ 1 \\cosh ( t x ) ( 1 - t ) \\ : \\mathrm { d } t . \\end{align*}"}
+{"id": "6286.png", "formula": "\\begin{align*} \\begin{aligned} c ^ 4 _ { \\lambda , m } ( \\xi _ 1 , \\xi _ 2 , \\xi _ 3 , \\xi _ 4 ) = \\frac { i } 4 \\left ( - ( c _ { \\lambda } ( \\xi _ 1 , \\xi _ 2 , \\xi _ 3 ) + c _ { \\lambda } ( \\xi _ 1 , \\xi _ 4 , \\xi _ 3 ) ) + ( \\bar c _ { \\lambda } ( \\xi _ 2 , \\xi _ 3 , \\xi _ 4 ) + \\bar c _ { \\lambda } ( \\xi _ 2 , \\xi _ 1 , \\xi _ 4 ) ) \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "1288.png", "formula": "\\begin{align*} J ( f ) = \\frac { P ( f ) } { \\| \\nabla f \\| _ { L ^ 2 } ^ { 2 p } } . \\end{align*}"}
+{"id": "8210.png", "formula": "\\begin{align*} \\alpha ( I _ j ^ a X ) & = \\alpha ( I _ j X - \\frac { a ^ 2 V } { a ^ 2 + V } \\theta ( X ) I _ j T + a ^ 2 d x _ j ( X ) T ) \\\\ & = i \\alpha ( X ) - \\frac { a ^ 2 V } { a ^ 2 + V } \\theta ( X ) i \\alpha ( T ) + a ^ 2 d x _ j ( X ) \\alpha ( T ) . \\end{align*}"}
+{"id": "3557.png", "formula": "\\begin{align*} L ( \\bar x ^ k , \\lambda ) - L ( x , \\bar \\lambda ^ k ) \\leq \\frac 1 k \\sum _ { i = 0 } ^ { k - 1 } L ( x ^ { i + 1 } , \\lambda ) - L ( x , \\lambda ^ { i + 1 } ) \\leq \\frac { 1 } { 2 k } \\| z ^ { 0 } - z \\| _ { P } ^ 2 \\ . \\end{align*}"}
+{"id": "2248.png", "formula": "\\begin{align*} g ( r ^ 2 ) = \\int _ 0 ^ \\infty e ^ { - \\alpha r ^ 2 } \\ , d \\mu ( \\alpha ) . \\end{align*}"}
+{"id": "3730.png", "formula": "\\begin{align*} \\mathbb { A } = \\begin{bmatrix} 0 & - I & 0 \\\\ 0 & 0 & - I \\\\ A & 0 & 0 \\end{bmatrix} \\end{align*}"}
+{"id": "5482.png", "formula": "\\begin{align*} & h _ { p , s } : \\mathbb C ^ \\times \\rightarrow \\mathbb C , h _ { p , s } = Q ^ { m _ 1 } p ^ { m _ 2 } s ^ { m _ 3 } , m _ 1 \\neq 0 , \\\\ & H _ { p , s } : \\mathbb C \\rightarrow \\mathbb C , H _ { p , s } ( z ) : = G _ { \\boldsymbol \\epsilon } ( z , p ^ { a _ 1 } s ^ { b _ 1 } , \\dots , p ^ { a _ n } s ^ { b _ n } ) . \\end{align*}"}
+{"id": "7968.png", "formula": "\\begin{align*} ( a \\cdot T a ) _ c ( f , g , x ) = a ( f , A ( g ) ( x ) ) = A ( f g ) ( x ) , \\end{align*}"}
+{"id": "6927.png", "formula": "\\begin{align*} c [ \\phi ( \\xi _ n ) - \\phi ( 0 ) ] - & d _ 1 \\int _ { 0 } ^ { \\xi _ n } \\mathcal { N } _ 1 \\left [ \\phi \\right ] ( \\xi ) d \\xi = \\int _ { 0 } ^ { \\xi _ n } \\phi ( \\xi ) f \\left ( \\phi , \\psi \\right ) ( \\xi ) d \\xi . \\end{align*}"}
+{"id": "8740.png", "formula": "\\begin{align*} \\norm { x _ { t + 1 } - x _ { p } } ^ { 2 } & \\leq \\norm { x _ { t } - \\eta _ { t } \\hat { g } _ { t } - x _ { p } } ^ 2 \\\\ & = \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 \\\\ & \\leq ( 1 - 2 \\eta _ { t } \\alpha ) \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } - \\nabla f ( x _ t ) , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 , \\end{align*}"}
+{"id": "616.png", "formula": "\\begin{align*} \\nu ( t ) - \\nu ( r ) = 2 \\int _ { \\R ^ d } \\int _ r ^ t \\int _ 0 ^ s \\frac { \\left [ V ( s , \\zeta ) - V ( \\sigma , \\zeta ) \\right ] \\overline { \\left [ W ( s , \\zeta ) - W ( \\sigma , \\zeta ) \\right ] } } { ( s - \\sigma ) ^ { 1 + 2 b } } \\ , d \\sigma \\ , d s \\ , d \\zeta , \\end{align*}"}
+{"id": "6809.png", "formula": "\\begin{align*} 0 & = d y ' ( z _ 2 ) - c y ( z _ 2 ) + s v ( z _ 2 ) \\left ( 1 - \\frac { v ( z _ 2 ) } { q ( u ( z _ 2 ) ) } \\right ) \\\\ [ 0 . 2 c m ] & \\geq s v ( z _ 2 ) \\left ( \\frac { q ( 1 ) } { q ( 0 ) } - \\frac { v ( z _ 2 ) } { q ( u ( z _ 2 ) ) } + 1 \\right ) > 0 , \\end{align*}"}
+{"id": "7012.png", "formula": "\\begin{align*} ( r _ 1 a _ 2 ) ^ 2 - r _ 2 ^ 2 ( a _ 1 a _ 2 ) = 4 a _ 2 ( a _ 2 - a _ 1 ) . \\end{align*}"}
+{"id": "4894.png", "formula": "\\begin{align*} E _ 2 ( z ) : = \\left \\{ \\begin{array} { l l } \\exp [ - \\frac { z - z _ 0 } { z _ 1 - z _ 0 } P _ 1 ] [ ( z - z _ 1 - ( z - z _ 0 ) P _ 1 ) [ \\frac { A ( z ) - A ( z _ 1 ) } { z - z _ 1 } ] + A ( z _ 1 ) ] , z \\neq z _ 1 \\\\ \\exp ( - P _ 1 ) [ A ( z _ 1 ) - ( z _ 1 - z _ 0 ) P _ 1 A ' ( z _ 1 ) ] , z = z _ 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "6075.png", "formula": "\\begin{align*} v _ { n + 1 } = \\theta _ { n } v _ { n } + \\gamma _ { n + 1 } , \\theta _ { n } = \\frac { \\gamma _ { n } ^ { \\alpha } } { \\gamma _ { n + 1 } ^ { \\alpha } } \\times e ^ { - \\rho \\gamma _ { n + 1 } } . \\end{align*}"}
+{"id": "5471.png", "formula": "\\begin{align*} \\int _ M f ( \\Delta f ) ^ 2 \\ , d V _ g = - \\Lambda \\int _ M f ^ 2 \\Delta f \\ , d V _ g = 2 \\Lambda \\int _ M f | \\nabla f | ^ 2 \\ , d V _ g . \\end{align*}"}
+{"id": "7083.png", "formula": "\\begin{align*} S _ 2 ( . . . ) = \\frac { N ^ { 3 / 4 - i ( t + \\nu ) } \\ , \\eta _ f ( p _ 2 ) } { ( p _ 1 q ) ^ { 1 / 2 } \\ , p _ 2 ^ { 1 / 4 } } \\ , \\sum _ { \\pm } \\sum _ { m \\ll M _ 0 } \\frac { \\lambda _ f ( m ) } { m ^ { 1 / 4 } } e \\left ( - m \\frac { \\overline { ( a + b q ) p _ 2 } } { q } \\right ) \\mathcal { I } ^ { \\pm } ( m , u , q , p _ 1 , p _ 2 ) . \\end{align*}"}
+{"id": "1970.png", "formula": "\\begin{align*} \\left ( a ^ t _ { \\sigma ^ j , r } - a ^ t _ { \\sigma ^ j , s } \\right ) - \\left ( a ^ t _ { \\sigma , r } - a ^ t _ { \\sigma , s } \\right ) = \\frac { j q } { p } \\left ( { r } _ { \\pi _ \\beta ( 1 ) } - { s } _ { \\pi _ \\beta ( 1 ) } \\right ) . \\end{align*}"}
+{"id": "7776.png", "formula": "\\begin{align*} M ^ { } : = \\{ ( Z ^ 0 , . . . , Z ^ n ) = Z ^ 0 \\cdot ( 1 , z ) \\in \\mathbb { C } ^ { n + 1 } \\ ; | \\ ; Z ^ 0 \\in \\mathbb { C } ^ { \\times } , z \\in \\overline { M } ^ { } \\} , \\ , \\mathfrak { F } ^ { } : = - \\frac { 1 } { 6 } k _ { a b c } \\frac { Z ^ a Z ^ b Z ^ c } { Z ^ 0 } \\ , . \\end{align*}"}
+{"id": "2932.png", "formula": "\\begin{align*} C _ { i j k l } & = \\lambda \\delta _ { i j } \\delta _ { k l } + \\mu ( \\delta _ { i k } \\delta _ { j l } + \\delta _ { i l } \\delta _ { j k } ) \\\\ & + \\{ \\delta _ { i j } D _ { k l } ^ 1 + \\delta _ { k l } D _ { i j } ^ 1 \\} + \\{ \\delta _ { i k } D _ { j l } ^ 2 + \\delta _ { i l } D _ { j k } ^ 2 + \\delta _ { j k } D _ { i l } ^ 2 + \\delta _ { j l } D _ { i k } ^ 2 \\} \\\\ & + D _ { i j k l } \\end{align*}"}
+{"id": "1268.png", "formula": "\\begin{align*} \\Big \\| ( \\sum _ { j = 1 } ^ { J } v _ n ^ j ) ^ 2 \\Big \\| _ { L _ { t } ^ { \\frac { q _ 0 } { 2 } } L _ x ^ { \\frac { r _ 1 } { 2 } } } ^ 2 \\leq \\sum _ { j = 1 } ^ { J } \\| v _ n ^ j \\| _ { S _ 0 ( \\R ) } ^ 2 + \\sum _ { j \\neq k } \\| v _ n ^ j v _ n ^ k \\| _ { L _ { t } ^ { \\frac { q _ 0 } { 2 } } L _ x ^ { \\frac { r _ 1 } { 2 } } } . \\end{align*}"}
+{"id": "3067.png", "formula": "\\begin{align*} { \\bf { H } } = { { \\bf { A } } _ { \\rm { R } } } { \\bf { \\Xi A } } _ { \\rm { T } } ^ H , \\end{align*}"}
+{"id": "4417.png", "formula": "\\begin{align*} \\mathcal { A } _ 1 | _ { x _ 1 = 0 } = \\mathrm { d i a g } ( \\mathcal { E } _ { 1 2 } , - \\mathcal { E } _ { 1 2 } ) . \\end{align*}"}
+{"id": "4019.png", "formula": "\\begin{align*} \\chi ^ n _ { \\varphi _ t } > e ^ { b _ t + b ' } g \\chi ^ m _ { \\varphi _ t } \\wedge \\omega ^ { n - m } . \\end{align*}"}
+{"id": "5585.png", "formula": "\\begin{align*} \\begin{cases} \\Delta u _ { * } = 0 & B _ 1 ^ + \\\\ u _ { * } = \\pm 1 & D _ 1 \\end{cases} \\end{align*}"}
+{"id": "7038.png", "formula": "\\begin{align*} H : = \\{ x \\in \\Omega \\mid \\nu ( x ) = \\tilde { \\nu } \\} \\end{align*}"}
+{"id": "449.png", "formula": "\\begin{align*} | ( \\nabla [ h \\cdot u ] _ { \\ell , m } ) ( x ) | ^ 2 & = \\left | | x | ^ { - \\sigma } u ' ( | x | ) - \\sigma | x | ^ { - \\sigma - 1 } u ( | x | ) \\right | ^ 2 | Y _ { \\ell , m } ( \\omega _ x ) | ^ 2 \\\\ & + | x | ^ { - 2 \\sigma - 2 } | u ( | x | ) | ^ 2 \\ , | x | ^ 2 | \\nabla Y _ { \\ell , m } ( \\omega _ x ) | ^ 2 . \\end{align*}"}
+{"id": "837.png", "formula": "\\begin{align*} \\mathcal { F } = \\{ \\widetilde { T g } ~ ~ | ~ ~ g \\in S _ { A ( \\mathbb { D } ) } \\} , \\end{align*}"}
+{"id": "6592.png", "formula": "\\begin{align*} \\omega _ l ^ { ( 2 ) } ( a ) = \\omega _ l ^ { ( 0 ) } + ( \\varepsilon + \\delta ) \\varphi _ l ^ { ( 2 ) } ( a ) , \\ , \\l = 1 , 2 , . . . , b , \\end{align*}"}
+{"id": "977.png", "formula": "\\begin{align*} ( a , b , c ) = \\left ( \\frac { l ^ 2 - n ^ 2 } { 2 l n } , \\frac { m ^ 2 - n ^ 2 } { 2 m n } , \\frac { l m - n ^ 2 } { ( l + m ) n } \\right ) , \\ \\left ( \\frac { l ^ 2 - n ^ 2 } { 2 l n } , \\frac { m ^ 2 - n ^ 2 } { 2 m n } , - \\frac { ( l + m ) n } { l m - n ^ 2 } \\right ) \\end{align*}"}
+{"id": "8657.png", "formula": "\\begin{align*} \\hat { f } ( \\vec { y } ) = ( f ( \\vec { x } ) + \\vec { x } \\cdot \\vec { y } ) \\Big \\rvert _ { \\vec { x } = \\vec { t } } \\end{align*}"}
+{"id": "6611.png", "formula": "\\begin{align*} \\Big \\langle f \\Big ( { 1 \\over 1 + e ^ { i \\theta _ 1 } } , \\dots , { 1 \\over 1 + e ^ { i \\theta _ N } } \\Big ) \\Big \\rangle ^ { ( \\rm c J ) } = \\langle f ( x _ 1 , \\dots , x _ N ) \\rangle ^ { ( \\rm J ) } \\Big | _ { \\lambda _ 1 = - \\beta ( N + p - 1 ) / 2 - 1 + i q \\atop \\lambda _ 2 = - \\beta ( N + p - 1 ) / 2 - 1 - i q } . \\end{align*}"}
+{"id": "5253.png", "formula": "\\begin{align*} & \\frac { 1 } { n } \\sum _ { i = 1 } ^ n q _ { i , t + 1 } ^ \\top g _ { i , t } ( x _ { i , t } ) + \\frac { 1 } { n } \\sum _ { i = 1 } ^ n ( l _ { i , t } ( x _ { i , t } ) - l _ { i , t } ( y ) ) \\le \\frac { 1 } { n } \\sum _ { i = 1 } ^ n q _ { i , t + 1 } ^ \\top g _ { i , t } ( y ) + \\frac { \\tilde { \\Delta } _ t } { n } + \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\Delta _ { i , t } ( y ) , \\end{align*}"}
+{"id": "644.png", "formula": "\\begin{align*} \\psi _ \\pm ( 0 ) = f _ \\pm \\in L ^ 2 \\left ( \\Omega , H ^ a ( \\R , \\C ) \\right ) , \\phi _ \\pm ( 0 ) = g _ \\pm \\in L ^ 2 \\left ( \\Omega , H ^ r ( \\R , \\C ) \\right ) , \\overline { g _ + } = g _ - . \\end{align*}"}
+{"id": "6143.png", "formula": "\\begin{align*} S \\mid _ { \\eta = a } \\ , = \\left ( \\{ s \\in S ^ + \\mid \\eta ( s ) = a \\} , \\{ s \\in S ^ - \\mid \\eta ( s ) = a \\} \\right ) . \\end{align*}"}
+{"id": "308.png", "formula": "\\begin{align*} \\mathsf { L } _ { j } ^ { \\sigma } \\left ( s \\right ) v ^ { \\sigma } = 0 \\quad \\Omega _ { j } ^ { \\sigma } . \\end{align*}"}
+{"id": "7811.png", "formula": "\\begin{align*} k ( Z ^ i , \\overline { Z } ^ i ) = - ( \\tau _ { i j } ) Z ^ i \\overline { Z } ^ j = - | Z ^ 0 | ^ 2 ( \\tau _ { i j } ) z ^ i \\overline { z } ^ j \\ , , z ^ i = \\frac { Z ^ i } { Z ^ 0 } \\ , . \\end{align*}"}
+{"id": "5095.png", "formula": "\\begin{align*} C ^ { l a } ( G , L ) & = \\prod _ { g \\in G / G _ 0 } C ^ { l a } ( g G _ 0 , L ) \\\\ & = \\prod _ { g \\in G / G _ 0 } \\varinjlim _ { h > 0 } C ^ h ( G _ 0 , L ) \\\\ & = \\varinjlim _ { \\substack { g \\in G / G _ 0 \\\\ h _ g > 0 } } \\prod _ { g \\in G / G _ 0 } C ^ { h _ g } ( G , L ) . \\end{align*}"}
+{"id": "5745.png", "formula": "\\begin{align*} \\mathcal { K } & : = \\exp \\bigg \\{ M ( \\log ( M ( 1 + \\| V \\| _ { 1 } ) N _ { 0 } ^ 2 e ^ { M _ { 1 } { ( 1 - \\tau ) } / { \\tau } } \\rho ^ { - 2 } \\Theta ^ { 1 / \\tau } ) ) + M \\| V \\| _ { 1 } ^ { 1 / 2 s } \\bigg \\} \\\\ & \\leq \\exp \\bigg \\{ M \\log ( M N _ { 0 } ^ 2 \\rho ^ { - 2 } \\Theta ^ { 1 / \\tau } \\big ) + M \\log ( 1 + \\| V \\| _ { 1 } ) + M M _ { 1 } ( 1 - \\tau ) / \\tau + M \\| V \\| _ { 1 } ^ { 1 / 2 s } \\bigg \\} . \\end{align*}"}
+{"id": "3698.png", "formula": "\\begin{align*} z _ 1 ^ { T _ { n _ 0 } , y ' } - z _ 1 ^ { T _ { n _ 0 } , y } \\in \\left [ n _ 0 ^ { - 2 } { a _ { n _ 0 } } , 3 n _ 0 ^ { - 2 } a _ { n _ 0 } \\right ] \\qquad z _ 2 ^ { T _ { n _ 0 } , y ' } = z _ 2 ^ { T _ { n _ 0 } , y } \\end{align*}"}
+{"id": "2788.png", "formula": "\\begin{align*} x ^ { n + 1 } \\cdot x ^ { n + 1 } \\cdot x ^ 5 = x ^ { 2 k + 7 } , \\end{align*}"}
+{"id": "1416.png", "formula": "\\begin{align*} \\sigma ^ K ( x ) & = - \\frac { 1 } { \\pi i } \\oint _ { \\Gamma _ N } { \\Bigg ( \\tilde \\varphi ( x , \\mu ) \\varphi ^ K ( x , \\mu ) - \\frac { 1 } { 2 } \\Bigg ) \\hat M ( \\mu ) } d \\mu \\\\ & - 2 \\sum _ { k = N + 1 } ^ { K } \\sum _ { j = 0 } ^ 1 ( - 1 ) ^ j \\alpha _ { k j } \\Bigg ( \\tilde \\varphi _ { k , j } ( x ) \\varphi ^ K _ { k , j } ( x ) - \\frac { 1 } { 2 } \\Bigg ) , \\end{align*}"}
+{"id": "3962.png", "formula": "\\begin{align*} \\Omega _ t ( G ) = \\{ x \\in G \\ , | \\ , x ^ { p ^ t } = 1 \\} \\textrm { \\ a n d \\ } \\mho _ t ( G ) = \\{ x ^ { p ^ t } \\ , | \\ , x \\in G \\} . \\end{align*}"}
+{"id": "3757.png", "formula": "\\begin{align*} \\begin{gathered} u ' ( t ) - a B u ( t ) = f ( t ) , \\\\ f ' ( t ) - b B f ( t ) = g ( t ) , \\\\ g ' ( t ) + B g ( t ) = 0 , t > 0 , \\end{gathered} \\end{align*}"}
+{"id": "2280.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 f ' ( x ) ^ 2 d x & = \\sum _ { 1 \\leq | j | \\leq \\frac { n - 1 } { 2 } } ( 2 \\pi j ) ^ 2 | a _ j | ^ 2 \\\\ & \\leq ( n - 1 ) ^ 2 \\pi ^ 2 \\sum _ { 1 \\leq | j | \\leq \\frac { n - 1 } { 2 } } | a _ j | ^ 2 = ( n - 1 ) ^ 2 \\pi ^ 2 \\int _ 0 ^ 1 f ( x ) ^ 2 \\ , d x . \\end{align*}"}
+{"id": "3002.png", "formula": "\\begin{align*} \\textsf { A } = \\begin{pmatrix} 1 & 1 \\\\ 2 & 0 \\end{pmatrix} \\textrm { a n d } \\textsf { C } = \\begin{pmatrix} 1 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ 2 & 2 & 0 \\end{pmatrix} \\end{align*}"}
+{"id": "7741.png", "formula": "\\begin{align*} \\Vert T + I \\Vert = 1 + \\Vert T \\Vert , \\end{align*}"}
+{"id": "8557.png", "formula": "\\begin{align*} \\partial _ { n _ { b } } = \\mathbf { n } _ { b } \\cdot I ^ { \\mu } \\nabla ^ { \\mu } _ { X , z } , \\mathbf { n } _ b = \\frac { 1 } { \\sqrt { 1 + \\beta ^ 2 | \\nabla _ X b | ^ 2 } } \\begin{pmatrix} - \\beta \\nabla _ X b \\\\ 1 \\end{pmatrix} , \\end{align*}"}
+{"id": "4136.png", "formula": "\\begin{align*} c ( k ) \\leq & d - \\sqrt { ( 2 - \\frac { 1 } { r } ) k } - ( 2 - \\frac { 1 } { r } ) ^ { \\frac { 1 } { 4 } } k ^ { \\frac { 1 } { 4 } } - 1 = \\left ( 1 - \\sqrt { 2 - \\frac { 1 } { r } } - o ( 1 ) \\right ) \\sqrt { k } , \\\\ c ( k ) \\leq & \\frac { ( \\frac { 1 } { r } - 1 ) k } { \\sqrt { ( \\frac { 1 } { r } - 1 ) k } + ( \\frac { 1 } { r } - 1 ) ^ \\frac { 1 } { 4 } k ^ \\frac { 1 } { 4 } + 1 } = ( 1 - o ( 1 ) ) \\sqrt { ( \\frac { 1 } { r } - 1 ) k } . \\end{align*}"}
+{"id": "8667.png", "formula": "\\begin{align*} \\delta ( x ) = \\frac { 1 } { 2 \\pi \\hbar } \\int e ^ { \\frac { x y } { \\hbar } } d y . \\end{align*}"}
+{"id": "7001.png", "formula": "\\begin{align*} W \\left ( e , \\bar { e } , \\bar { e } , n \\right ) = - W \\left ( n , \\bar { n } , \\bar { e } , n \\right ) , ~ ~ ~ W \\left ( e , \\bar { e } , e , n \\right ) = W \\left ( n , \\bar { n } , e , n \\right ) , \\end{align*}"}
+{"id": "2234.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\to + \\infty } \\frac { G ( \\lambda t ) } { G ( t ) } = \\lambda ^ p , \\qquad [ 0 , + \\infty ) , \\end{align*}"}
+{"id": "7675.png", "formula": "\\begin{align*} \\psi ( u ) & = \\mathbb { P } \\left ( \\sup _ { n \\geqslant 1 } \\left \\{ \\sum _ { k = 1 } ^ n X _ k - n \\right \\} > u \\right ) = \\mathbb { P } \\left ( \\bigcup _ { n = 1 } ^ \\infty \\left \\{ \\sum _ { k = 1 } ^ n X _ k > n + u \\right \\} \\right ) \\\\ & \\geqslant \\mathbb { P } \\left ( \\bigcup _ { n = 1 } ^ \\infty \\left \\{ \\sum _ { k = 1 } ^ n X _ k ^ * > n + u \\right \\} \\right ) = \\mathbb { P } \\left ( \\sup _ { n \\geqslant 1 } \\left \\{ \\sum _ { k = 1 } ^ n X ^ * _ k - n \\right \\} > u \\right ) \\\\ & = : \\psi _ { \\varepsilon } ^ * ( u ) , \\end{align*}"}
+{"id": "1926.png", "formula": "\\begin{align*} \\left ( \\Pi _ { \\lambda _ 1 , \\lambda _ 2 } g \\right ) ^ { \\lambda _ 1 , 1 - \\lambda _ 2 } _ { i + \\frac { 1 } { 2 } , j - \\frac { 1 } { 2 } } = g ^ { \\lambda _ 1 , 1 - \\lambda _ 2 } _ { i + \\frac { 1 } { 2 } , j - \\frac { 1 } { 2 } } , \\mbox { i f } \\left ( i , j \\right ) \\in ( \\Bbbk _ 2 ^ - , \\Bbbk _ 1 ^ + ) \\end{align*}"}
+{"id": "2947.png", "formula": "\\begin{align*} \\frac { n - 1 } { 2 n - 1 } a & = - b \\end{align*}"}
+{"id": "6126.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac { 1 } { T } \\int _ { 0 } ^ { T } \\mathrm { R e } \\frac { \\zeta '' } { \\zeta ' } ( \\sigma + i t ) \\mathrm { d } t = \\phi _ { \\zeta ' } ' ( \\sigma ) = - \\log 2 , \\quad \\sigma > E . \\end{align*}"}
+{"id": "1349.png", "formula": "\\begin{align*} N = \\nu ( W ) \\end{align*}"}
+{"id": "1715.png", "formula": "\\begin{align*} G _ { \\texttt { b } } ( \\xi _ 1 ) = \\sum _ { 0 \\leq \\mu _ 1 \\leq m } e ^ { i \\mu _ 1 \\xi _ 1 } , \\end{align*}"}
+{"id": "350.png", "formula": "\\begin{align*} \\left [ u \\right ] _ { \\operatorname * { D } ; j } \\left ( s \\right ) = - \\psi _ { \\operatorname * { D } ; j } \\quad \\left [ u \\right ] _ { \\operatorname * { N } , j } ^ { \\operatorname * { e x t } } \\left ( s \\right ) = - \\psi _ { \\operatorname * { N } ; j } \\end{align*}"}
+{"id": "1409.png", "formula": "\\begin{gather*} \\tilde H ^ K _ { n i , k j } ( x ) = \\left \\{ \\begin{array} { l l } 0 , & k > K , \\\\ \\tilde H _ { n i , k j } ( x ) , & k \\le K . \\\\ \\end{array} \\right . \\end{gather*}"}
+{"id": "3132.png", "formula": "\\begin{align*} \\mathbb { E } [ | \\mathbf { H } _ { k , } \\cdot \\mathbf { H } _ { l , } ^ H | ^ 2 ] = \\mathbb { E } _ { \\mathbf { H } _ { k , } } \\Bigl [ \\mathbf { H } _ { k , } \\cdot \\mathbb { E } _ { \\mathbf { H } _ { l , } } [ \\mathbf { H } ^ H _ { l , } \\cdot \\mathbf { H } _ { l , } ] \\cdot \\mathbf { H } ^ H _ { k , } \\Bigr ] . \\end{align*}"}
+{"id": "8072.png", "formula": "\\begin{align*} S ^ { 1 } ( h ) ( x ) = \\left \\{ \\sum \\limits _ { j \\geq 1 } \\sum \\limits _ { Q \\in \\Pi _ { j + N } } \\sup \\limits _ { u \\in Q } \\vert \\psi _ { j } \\ast h ( u ) \\vert ^ 2 \\chi _ { Q } ( x ) \\right \\} ^ { 1 / 2 } . \\end{align*}"}
+{"id": "7004.png", "formula": "\\begin{align*} Q ( f , f ) = \\begin{cases} \\int _ { T ^ 2 } 8 f ^ 2 d A _ x > 0 , ~ ~ ~ ~ ~ & k = 1 , \\\\ \\ , 0 , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ & k = 2 , \\\\ \\int _ { T ^ 2 } 8 0 f ^ 2 d A _ x > 0 , ~ ~ ~ ~ ~ & k = 3 , \\end{cases} \\end{align*}"}
+{"id": "7940.png", "formula": "\\begin{align*} S _ i = ( S \\cap V _ i ) \\cup ( \\{ u _ i \\} \\cup Y _ i ) \\ 1 \\end{align*}"}
+{"id": "2723.png", "formula": "\\begin{align*} v _ 1 = ( 0 , 0 , 0 , 0 ) , \\ , v _ 2 = ( 2 , 2 , 0 , 0 ) , \\ , v _ 3 = ( 2 , 0 , 2 , 0 ) , \\ , v _ 4 = ( 0 , 2 , 2 , 0 ) . \\end{align*}"}
+{"id": "2249.png", "formula": "\\begin{align*} \\frac { 1 } { r ^ s } = \\int _ 0 ^ \\infty e ^ { - \\alpha r ^ 2 } \\frac { \\alpha ^ { s / 2 - 1 } } { \\Gamma ( s / 2 ) } \\ , d \\alpha . \\end{align*}"}
+{"id": "4798.png", "formula": "\\begin{align*} \\| f \\| ^ \\perp _ \\alpha = \\| f \\| _ { C ^ 0 } + | f | ^ \\perp _ \\alpha . \\end{align*}"}
+{"id": "1169.png", "formula": "\\begin{align*} & m \\rq _ { 1 , 1 } ( x , y ) - m _ { 1 , 1 } ( x , y ) = m _ 1 ( \\phi _ 1 ( x ) , y ) + m _ 1 ( x , \\phi _ 1 ( y ) ) - \\phi _ 1 ( m _ 1 ( x , y ) ) = \\delta _ 1 \\phi _ 1 ( x , y ) . \\\\ & m \\rq _ { 2 , 1 } ( x , y ) - m _ { 2 , 1 } ( x , y ) = m _ 2 ( \\phi _ 1 ( x ) , y ) + m _ 2 ( x , \\phi _ 1 ( y ) ) - \\phi _ 1 ( m _ 2 ( x , y ) ) = \\delta _ 2 \\phi _ 1 ( x , y ) . \\\\ \\end{align*}"}
+{"id": "8802.png", "formula": "\\begin{align*} \\forall x \\in \\Theta : \\mathbb { E } \\sum _ { t = 1 } ^ { T } \\big ( f ( x _ t ) - f ( x ) \\big ) \\le G B T . \\end{align*}"}
+{"id": "319.png", "formula": "\\begin{align*} \\mathsf { L } _ { j } ^ { \\sigma } \\left ( s \\right ) u ^ { \\sigma } = 0 \\quad \\Omega _ { j } ^ { \\sigma } \\sigma \\in \\left \\{ + , - \\right \\} . \\end{align*}"}
+{"id": "1271.png", "formula": "\\begin{align*} a ( x ) = \\begin{cases} | x | ^ 2 \\ \\ f o r \\ \\ | x | \\leq R \\\\ C R ^ 2 \\ \\ f o r \\ \\ | x | > 2 R \\end{cases} \\end{align*}"}
+{"id": "7068.png", "formula": "\\begin{align*} \\eth _ 1 ^ { \\pm } ( n ^ 2 _ 1 n _ 2 , u , q , p _ 1 ) = d { \\pi } ^ { 3 } \\int _ { 0 } ^ \\infty V _ 1 ^ { \\pm } ( z ) z ^ { - 1 / 3 + i \\nu } \\ , e \\left ( \\frac { N u z } { p _ 1 q Q } \\pm \\frac { 3 ( N z n _ 1 ^ 2 n _ 2 ) ^ { 1 / 3 } } { p _ 1 q r ^ { 1 / 3 } } \\right ) d z , \\end{align*}"}
+{"id": "4540.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ 2 _ + } ( \\mathcal B _ 0 { \\mathbf V } \\cdot { \\mathbf V } ) ( t ) d { \\bf x } \\ge c _ 0 \\Vert { \\mathbf V } ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ^ 2 _ + ) } = c _ 0 I ( t ) \\ , , \\end{align*}"}
+{"id": "7323.png", "formula": "\\begin{align*} [ E ^ - ( M ^ n _ { ( 0 , \\lambda ) } ) ] = [ E ^ - ( M ^ n _ { ( 1 , \\lambda ) } ) ] , n \\geq n _ 1 , \\end{align*}"}
+{"id": "8375.png", "formula": "\\begin{align*} \\bigcap _ { x \\in D } \\Omega _ { 2 , x } = : \\tilde { \\Omega } _ 2 \\end{align*}"}
+{"id": "8227.png", "formula": "\\begin{align*} d d ^ c _ { I _ 1 ^ a } ( K _ 1 ^ a ) ^ \\alpha = \\alpha ( K _ 1 ^ a ) ^ { \\alpha - 2 } \\left [ ( \\alpha - 1 ) d K _ 1 ^ a \\wedge d ^ c _ { I _ 1 ^ a } K _ 1 ^ a + K _ 1 ^ a d d ^ c _ { I _ 1 ^ a } K _ 1 ^ a \\right ] > 0 \\end{align*}"}
+{"id": "5756.png", "formula": "\\begin{align*} { \\cal A } _ 1 = \\{ 0 \\} , \\ \\ \\ { \\cal B } _ 1 = \\Lambda _ L . \\end{align*}"}
+{"id": "6231.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } i u _ t + g ( u , \\partial _ x u ) \\partial _ x ^ 2 u = N ( u , \\partial _ x u ) , u : \\R \\times \\R \\to \\C , \\\\ \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\end{array} \\right . \\end{align*}"}
+{"id": "4382.png", "formula": "\\begin{align*} | | | u ( t ) | | | ^ 2 _ { m , \\ast } : = \\sum ^ m _ { j = 0 } | | \\partial ^ j _ t u ( t ) | | ^ 2 _ { H ^ { m - j } _ { \\ast } ( \\Omega ) } \\ , . \\end{align*}"}
+{"id": "6984.png", "formula": "\\begin{align*} \\frac { d ^ 2 y } { d x ^ x } + ( Q _ 1 ( x ) + P _ 1 ( x ) ) y = 0 , \\end{align*}"}
+{"id": "1874.png", "formula": "\\begin{gather*} \\tau _ { n + 1 } \\geq \\tau _ n + 2 r \\quad \\mathfrak { S } ( \\alpha ) \\subset \\bigcup _ { n = 1 } ^ { \\infty } [ \\tau _ n - r , \\tau _ n + r ] \\end{gather*}"}
+{"id": "7593.png", "formula": "\\begin{align*} \\begin{aligned} & \\hat J ( \\beta , N , T , \\hat { \\mathbf { R } } _ 1 ) \\\\ & \\le N \\iint _ { \\hat { \\mathbf { R } } _ 1 } \\left ( \\left ( \\frac { C } { \\sqrt { 2 \\pi ( t - s ) } } \\exp \\left ( - \\frac { 1 } { 4 } T ^ { - \\frac { 1 } { 2 } } \\zeta ^ 2 ( t - s ) \\right ) \\right ) \\wedge 1 \\right ) d s d t . \\end{aligned} \\end{align*}"}
+{"id": "3637.png", "formula": "\\begin{align*} ( j ^ 1 \\phi ) ^ * q ^ \\mu & = q ^ \\mu & ( j ^ 1 \\phi ) ^ * u & = \\phi & ( j ^ 1 \\phi ) ^ * p _ \\mu & = \\phi _ \\mu \\ , \\end{align*}"}
+{"id": "6815.png", "formula": "\\begin{align*} C ( u ) & = 1 - g ( u ) v + s ( v - q ( 0 ) ) h ( u ) / \\varrho q ( u ) . \\end{align*}"}
+{"id": "4426.png", "formula": "\\begin{align*} b _ 1 = \\frac { H ^ + } { H ^ - } \\left ( a ^ + - \\frac { [ u ] } { H ^ + } \\right ) \\quad \\mbox { a n d } b _ 2 = \\frac { H ^ + } { H ^ - } \\left ( - a ^ + - \\frac { [ u ] } { H ^ + } \\right ) . \\end{align*}"}
+{"id": "1525.png", "formula": "\\begin{align*} \\frac { V ^ m } { m ! } = \\exp \\left ( h ( m ) - \\frac 1 2 \\log ( 2 \\pi ) \\right ) \\left ( 1 + O \\left ( \\frac 1 m \\right ) \\right ) \\ll \\frac { e ^ V } { V ^ { 1 / 2 } } \\exp \\left ( - \\frac { E ^ 2 } { 2 V } \\right ) , \\end{align*}"}
+{"id": "7814.png", "formula": "\\begin{align*} \\overline { N } _ { K , \\epsilon } : = \\{ ( \\tau _ 2 , b ^ a + \\mathrm { i } t ^ a , \\zeta ^ i , \\widetilde { \\zeta } _ i , \\sigma ) \\in \\overline { N } \\ ; | \\ ; \\tau _ 2 > \\epsilon , \\ ; \\ ; t ^ a > K , \\ ; a = 1 , . . . , n \\} \\ , . \\end{align*}"}
+{"id": "494.png", "formula": "\\begin{align*} ( \\psi \\mathfrak K _ 1 ) f ( x ) = \\psi ( x ) \\int _ { \\R } \\mathfrak k _ 1 ( x - y ) f ( y ) d y \\end{align*}"}
+{"id": "6019.png", "formula": "\\begin{align*} A _ { q , \\kappa , p } ^ { \\delta , \\eta } ( \\gamma _ { N ( t ) } ^ { \\beta } ) = \\frac { \\delta ^ { q } } { \\Delta ^ { \\frac { 2 p q } { p + \\kappa } } } + 2 \\Delta ^ { \\frac { p \\kappa } { p + \\kappa } } . \\end{align*}"}
+{"id": "7130.png", "formula": "\\begin{align*} I _ k = - \\int _ 0 ^ \\infty \\Psi ^ { ( k - 1 ) } ( Z ) \\Psi ^ { ( k + 1 ) } ( Z ) \\omega _ k ( r Z ) \\ : d Z - r \\int _ 0 ^ \\infty \\Psi ^ { ( k - 1 ) } ( Z ) \\Psi ^ { ( k ) } ( Z ) \\omega _ k ' ( r Z ) \\ : d Z . \\end{align*}"}
+{"id": "1195.png", "formula": "\\begin{align*} ( \\lim _ n \\Sigma ^ s M _ n ) \\otimes _ A \\pi _ 0 ( A _ 1 ) = ( \\lim _ n \\Sigma ^ s M _ n \\otimes _ { A } A _ 1 ) \\otimes _ { A _ 1 } \\pi _ 0 ( A _ 1 ) = \\Sigma ^ s M _ 1 \\otimes _ { A _ 1 } \\pi _ 0 ( A _ 1 ) \\end{align*}"}
+{"id": "5301.png", "formula": "\\begin{align*} \\hat { x } _ { i + 1 } & \\coloneqq x _ { i - 1 } + \\frac { y _ { i + 1 } - y _ { i - 1 } } { a _ { i + 1 } } \\\\ & = x _ { i - 1 } + \\frac { y _ i - y _ { i - 1 } } { a _ { i + 1 } } + \\frac { y _ { i + 1 } - y _ i } { a _ { i + 1 } } \\\\ & = x _ { i - 1 } + \\frac { a _ i } { a _ { i + 1 } } ( x _ i - x _ { i - 1 } ) + ( x _ { i + 1 } - x _ i ) . \\end{align*}"}
+{"id": "6663.png", "formula": "\\begin{align*} d _ 1 = - { q \\over \\pi } , d _ 2 = - { p ^ 2 + q ^ 2 \\over 2 \\pi ^ 2 } , d _ { n + 2 } = { 1 \\over \\pi } { 2 n + 1 \\over n + 2 } q d _ { n + 1 } + { 1 \\over \\pi ^ 2 } { n - 1 \\over n + 2 } \\Big ( p ^ 2 - { n ^ 2 \\over 4 } \\Big ) d _ n \\ : \\ : ( n \\ge 1 ) . \\end{align*}"}
+{"id": "1836.png", "formula": "\\begin{gather*} \\left \\| \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + \\alpha ) ^ s } - \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + c ) ^ s } \\right \\| < \\epsilon \\end{gather*}"}
+{"id": "8683.png", "formula": "\\begin{align*} F _ 1 ( \\partial _ { y _ 0 } F _ 2 , . . . , \\partial _ { y _ { n + 1 } } F _ 2 ) = ( f _ 2 - 1 ) F _ 2 ( y _ 0 , . . . , y _ { n + 1 } ) H _ 2 ( y _ 0 , . . . , y _ { n + 1 } ) , \\end{align*}"}
+{"id": "8907.png", "formula": "\\begin{align*} T ( \\beta _ i ( \\iota _ x ( C _ 0 ) ) ) = x _ i + x _ { d - i } + W _ { k - 3 } ( C _ 0 ) . \\end{align*}"}
+{"id": "3205.png", "formula": "\\begin{align*} \\| x - x _ { \\ell } \\| ^ 2 _ { A ^ T A } = \\sum _ { j = \\ell } ^ { k } { \\phi } _ { j + 1 } ^ 2 + \\| x - x _ { k + 1 } \\| ^ 2 _ { A ^ T A } \\end{align*}"}
+{"id": "4774.png", "formula": "\\begin{align*} \\mu _ { C } ( h C _ { j } \\cap C _ { k } ) = \\prod _ { g \\in A _ { j } \\setminus B _ { N } } P _ { i _ { g } ^ { j } } \\prod _ { g \\in A _ { k } \\setminus B _ { N } } P _ { i _ { g } ^ { k } } = \\mu _ { C } ( C _ { j } ) \\mu _ { C } ( C _ { k } ) . \\end{align*}"}
+{"id": "670.png", "formula": "\\begin{align*} S _ { h ( \\xi ) } ( t ) = e ^ { - i t h ( D _ x ) } , \\end{align*}"}
+{"id": "4975.png", "formula": "\\begin{align*} D \\leq \\dim G ^ \\circ = r ( r + 2 ) < \\binom { r + 1 } { 4 } , \\end{align*}"}
+{"id": "7427.png", "formula": "\\begin{align*} \\frac { { \\rm d i v } \\vec { F } } { S } = { \\rm d i v } ( ( \\lvert \\nabla y \\rvert ^ { p - 2 } + \\eta ( z ) \\lvert \\nabla y \\rvert ^ { q - 2 } ) \\nabla y ) . \\end{align*}"}
+{"id": "6816.png", "formula": "\\begin{align*} C ( u ) & = \\left [ ( \\varrho q ( u ) - s q ( 0 ) h ( u ) ) + v ( s h ( u ) - \\varrho q ( u ) g ( u ) ) \\right ] / \\varrho q ( u ) \\\\ [ 0 . 2 c m ] & < \\left [ ( \\varrho q ( 1 ) - s q ( 0 ) h ( 0 ) ) + v ( s h ( 1 ) - \\varrho q ( 0 ) g ( u ) ) \\right ] / \\varrho q ( u ) \\\\ [ 0 . 2 c m ] & = \\left \\{ \\left [ \\varrho ( h ( 1 ) + \\mu ) - s \\mu h ( 0 ) \\right ] + v ( s h ( 1 ) - \\varrho \\mu g ( u ) ) \\right \\} / \\varrho q ( u ) . \\end{align*}"}
+{"id": "2074.png", "formula": "\\begin{align*} \\mathbf { I I } & \\leq \\frac { C _ n } { t } \\int ^ { \\sqrt { t } } _ { r _ 0 } \\exp \\left ( - \\frac { r ^ 2 } { C _ 1 t } \\right ) \\frac { k ( x _ 0 , r ) } { r } \\ , d r \\leq \\frac { C _ 4 } { t } \\int ^ { \\sqrt { t } } _ { r _ 0 } \\frac { k ( x _ 0 , r ) } { r } \\ , d r \\end{align*}"}
+{"id": "6363.png", "formula": "\\begin{align*} \\sigma _ { X , Y } \\circ t _ { X , Y } = t _ { Y , X } \\circ \\sigma _ { X , Y } \\end{align*}"}
+{"id": "6478.png", "formula": "\\begin{align*} - \\left < \\Psi _ 1 - \\Psi _ 2 , \\Lambda P _ R F \\right > = 0 , \\end{align*}"}
+{"id": "4856.png", "formula": "\\begin{align*} B ^ 2 = ( \\lambda _ 1 + \\lambda _ 2 + \\lambda _ 3 ) B - ( \\lambda _ 1 \\lambda _ 2 + \\lambda _ 1 \\lambda _ 3 + \\lambda _ 2 \\lambda _ 3 ) I + \\lambda _ 1 \\lambda _ 2 \\lambda _ 3 B ^ { - 1 } \\end{align*}"}
+{"id": "1084.png", "formula": "\\begin{align*} \\kappa : = \\left \\{ \\begin{array} { l l } \\displaystyle \\frac { 1 } { \\mu _ 0 \\sqrt { \\lambda _ 1 } } & \\mbox { i f \\eqref { a l f a 1 } h o l d s } \\\\ \\\\ { \\displaystyle \\frac { 1 } { \\mu _ 0 \\sqrt { \\lambda _ 1 } \\sqrt { 1 - \\displaystyle \\frac { 1 } { \\mu _ 0 ^ 2 \\lambda _ 1 } { \\displaystyle \\int _ D { | \\alpha ( x ) | } d \\mu } } } } & \\mbox { i f \\eqref { a l f a 2 } h o l d s } , \\end{array} \\right . \\end{align*}"}
+{"id": "4316.png", "formula": "\\begin{align*} \\begin{alignedat} { 3 } F _ { L } ( t ) = \\frac { 1 } { \\sqrt { M } } \\sum _ { x = 0 } ^ { M - 1 } e ^ { - \\mu 2 \\pi \\left ( \\frac { x t } { M } \\right ) } \\ ; f ( x ) \\end{alignedat} \\end{align*}"}
+{"id": "7271.png", "formula": "\\begin{align*} | | g | | _ { C ^ 3 } = \\max _ { i = 0 , 1 , 2 , 3 } \\max _ { x \\in [ - 1 - D , D ' + 1 ] } \\left | g ^ { ( i ) } ( x ) \\right | . \\end{align*}"}
+{"id": "1644.png", "formula": "\\begin{align*} \\frac { 2 f ( r ) f '' ( r ) - ( f ' ( r ) ) ^ 2 } { 4 ( f ( r ) ) ^ 2 } = \\lambda _ 0 ( X ^ { p , q } ) + \\frac { q ( q - 2 ) } { 4 \\sinh ^ 2 ( r ) } + \\frac { p ( p + 2 q - 2 ) } { 1 6 \\sinh ^ 2 ( r / 2 ) } . \\end{align*}"}
+{"id": "4533.png", "formula": "\\begin{align*} \\dot u ^ \\pm _ n \\equiv \\dot u ^ \\pm _ N : = \\dot u _ 1 ^ \\pm - \\partial _ 2 \\hat \\varphi \\dot u ^ \\pm _ 2 \\ , , \\quad \\dot H ^ \\pm _ n \\equiv \\dot H ^ \\pm _ N : = \\dot H _ 1 ^ \\pm - \\partial _ 2 \\hat \\varphi \\dot H ^ \\pm _ 2 \\ , , \\quad \\mbox { o n } \\ , \\ , \\ , \\Gamma _ T \\ , . \\end{align*}"}
+{"id": "5712.png", "formula": "\\begin{align*} k _ { ( T , \\widetilde { x } ) } & = \\limsup _ { z \\rightarrow 0 } \\frac { \\ln \\| ( z - T ) ^ { - 1 } \\widetilde { x } \\| _ p } { \\ln \\| ( z - T ) ^ { - 1 } \\| } \\\\ & \\approx \\limsup _ { z \\rightarrow 0 } \\frac { \\ln \\frac { 1 } { | z | } + \\ln \\| ( z - A ) ^ { - 1 } \\| } { \\ln \\| ( z - A ) ^ { - 1 } \\| ^ 2 } = \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "6401.png", "formula": "\\begin{align*} c _ { 1 2 } c _ { 1 3 } c _ { 2 3 } = c _ { 2 3 } c _ { 1 3 } c _ { 1 2 } \\end{align*}"}
+{"id": "6779.png", "formula": "\\begin{align*} & \\lim \\limits _ { t \\rightarrow \\infty } \\left \\{ \\sup \\limits _ { | x | > ( s ^ * + \\epsilon ) t } w ( x , t ) \\right \\} = 0 , \\epsilon > 0 , \\\\ [ 0 . 2 c m ] & \\liminf \\limits _ { t \\rightarrow \\infty } \\left \\{ \\inf \\limits _ { | x | < ( s ^ * - \\epsilon ) t } w ( x , t ) \\right \\} > 0 , \\epsilon \\in ( 0 , s ^ * ) . \\end{align*}"}
+{"id": "3571.png", "formula": "\\begin{align*} \\delta ( \\alpha R + \\beta S ) = \\alpha ( \\alpha - \\beta ) \\delta ( R ) + \\beta ( \\beta - \\alpha ) \\delta ( S ) + 4 \\alpha \\beta \\delta \\left ( \\frac { R + S } { 2 } \\right ) \\end{align*}"}
+{"id": "4541.png", "formula": "\\begin{align*} \\Vert f \\vert _ { x _ 1 = 0 } \\Vert ^ 2 _ { L ^ 2 ( \\Gamma _ t ) } \\le \\Vert f \\Vert ^ 2 _ { L ^ 2 ( \\Omega _ t ) } + \\Vert \\partial _ 1 f \\Vert ^ 2 _ { L ^ 2 ( \\Omega _ t ) } \\end{align*}"}
+{"id": "8609.png", "formula": "\\begin{align*} \\mathrm { F } _ 1 = \\dfrac { \\tanh { ( \\sqrt { \\mu } | \\mathrm { D } | ) } } { \\sqrt { \\mu } | \\mathrm { D } | } , \\mathrm { F } _ 2 = \\frac { 3 } { \\mu | \\mathrm { D } | ^ 2 } ( 1 - \\mathrm { F } _ 1 ) , \\mathrm { F } _ 3 = \\mathrm { s e c h } ( \\sqrt { \\mu } | D | ) , \\mathrm { F } _ 4 = \\frac { 2 } { \\mu | \\mathrm { D } | ^ 2 } ( 1 - \\mathrm { F } _ 3 ) . \\end{align*}"}
+{"id": "6377.png", "formula": "\\begin{align*} \\chi = a ( 1 \\otimes 1 - \\lambda x \\otimes x + \\lambda x g \\otimes x g ) \\end{align*}"}
+{"id": "2652.png", "formula": "\\begin{align*} A = \\bigcap _ { i \\in \\N } \\{ X : \\ , w ' _ { i } ( X , \\delta _ i ) \\le \\frac { 1 } { 2 ^ i } \\ , , X ( t _ i ) \\in K _ i \\} \\end{align*}"}
+{"id": "7983.png", "formula": "\\begin{align*} a ^ { ( k ) } _ { i j } = \\begin{cases} \\displaystyle \\partial _ { u _ i } \\partial ^ j _ { \\theta } R + \\frac { \\theta ^ { i - j } } { ( i - j ) ! } & j \\ne k , \\\\ \\displaystyle - \\frac { \\Psi _ { d _ 1 - 1 , j + 1 } } { \\Psi _ { d _ 1 - 1 , d _ 1 } } \\left ( \\frac { \\theta ^ { i - d _ 1 + 1 } } { ( i - d _ 1 + 1 ) ! } + \\partial _ { u _ i } \\partial ^ { d _ 1 - 1 } _ { \\theta } R \\right ) & j = k . \\end{cases} \\end{align*}"}
+{"id": "3221.png", "formula": "\\begin{align*} S \\left ( R \\right ) = \\left ( \\left ( R \\right ) \\right ) + E \\left ( R \\right ) , \\end{align*}"}
+{"id": "8246.png", "formula": "\\begin{align*} \\frac { \\partial \\Psi } { \\partial x _ 1 } = \\frac { \\partial \\tau ( m , x ) } { \\partial x _ 1 } ( - I _ 1 T ) = - V I _ 1 T . \\end{align*}"}
+{"id": "1476.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ { \\mathcal { D } _ g } & : = \\langle ( L _ { g ^ { - 1 } } ) _ { \\ast } u , ( L _ { g ^ { - 1 } } ) _ { \\ast } v \\rangle _ { \\mathcal { D } _ e } \\\\ & = \\langle ( L _ { g ^ { - 1 } } ) _ { \\ast } u , ( L _ { g ^ { - 1 } } ) _ { \\ast } v \\rangle _ { \\mathcal { H } } u , v \\in \\mathcal { D } _ g . \\end{align*}"}
+{"id": "5217.png", "formula": "\\begin{align*} 0 = k + \\frac { v - 1 } { 2 } ( \\lambda - \\mu ) \\end{align*}"}
+{"id": "1803.png", "formula": "\\begin{gather*} Z _ N ( s , \\mathbb { X } _ \\alpha ) ( \\omega ) = \\frac { 1 } { 2 \\pi i } \\int _ { - \\delta - i \\infty } ^ { - \\delta + i \\infty } \\zeta ( s + w , \\mathbb { X } _ \\alpha ) ( \\omega ) \\widehat { \\phi } ( w ) N ^ w \\ , d w + \\zeta ( s , \\mathbb { X } _ \\alpha ) ( \\omega ) \\end{gather*}"}
+{"id": "4047.png", "formula": "\\begin{align*} U _ { \\alpha , \\beta } ( t ) = \\frac { ( - 1 ) ^ { \\alpha \\beta } } { 2 } \\mathcal { Q } ^ { - 1 } A _ { \\alpha , \\beta } \\mathcal { R } q _ { 0 } ( t ) , \\ \\ V _ { \\alpha , \\beta } ( t ) = \\frac { ( - 1 ) ^ { \\gamma } } { 2 } \\mathcal { Q } ^ { - 1 } B _ { \\alpha , \\beta } \\mathcal { R } p ( t ) , \\end{align*}"}
+{"id": "2297.png", "formula": "\\begin{align*} 3 0 = n _ { 2 } + 3 n _ { 3 } + 3 t _ { 5 } . \\end{align*}"}
+{"id": "2798.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\frac { u ^ { p - 1 } x \\cdot \\nabla u } { | x | ^ { \\theta + 2 } } \\dd x & = - \\frac { N - 2 - \\theta } { p } \\int _ { \\Omega } \\frac { u ^ p } { | x | ^ { \\theta + 2 } } \\dd x . \\end{align*}"}
+{"id": "7815.png", "formula": "\\begin{align*} \\widetilde { N } _ S : = \\widetilde { N } _ { K , \\epsilon } \\cap S \\cdot \\widetilde { N } _ { K , \\epsilon } \\cap S ^ 2 \\cdot \\widetilde { N } _ { K , \\epsilon } \\cap S ^ 3 \\cdot \\widetilde { N } _ { K , \\epsilon } \\end{align*}"}
+{"id": "337.png", "formula": "\\begin{align*} - \\operatorname * { d i v } \\left ( \\mathbb { A } _ { j } ^ { \\operatorname * { e x t } } \\nabla d \\right ) + s ^ { 2 } p _ { j } ^ { \\operatorname * { e x t } } d & = 0 \\quad \\mathbb { R } ^ { 3 } \\backslash \\Gamma _ { j } , \\\\ \\left [ d \\right ] _ { \\operatorname * { D } ; j } \\left ( s \\right ) & = 0 \\left [ d \\right ] _ { \\operatorname * { N } ; j } ^ { \\operatorname * { e x t } } \\left ( s \\right ) = 0 . \\end{align*}"}
+{"id": "3793.png", "formula": "\\begin{align*} s _ * : = ( \\mu _ 0 ( X ) + \\mu _ 1 ( X ) ) ^ { \\frac 1 p } . \\end{align*}"}
+{"id": "3502.png", "formula": "\\begin{align*} h _ { C Y , t } = h e ^ { - 2 ( \\phi _ t + | \\log | t | | \\psi _ t ) } , \\end{align*}"}
+{"id": "5744.png", "formula": "\\begin{align*} & \\mathcal { F } _ 1 : = \\{ Z _ i : 1 \\leq i \\leq n _ 1 ; Z _ { i } = ( z _ { i } , 0 ) \\in \\mathbb { B } _ { \\rho / 2 } \\} , \\\\ & \\mathcal { F } _ 2 : = \\{ Y ^ i = ( y ^ { i } , y ^ i _ { n + 1 } ) : 1 \\leq i \\leq n _ 2 , Y ^ { i } \\in \\mathbb { B } _ { c _ { 2 } \\rho } ^ { + } \\} , \\end{align*}"}
+{"id": "5061.png", "formula": "\\begin{align*} \\ker ( d ( \\Pi | _ { P _ 1 } ) _ p ) = \\ker ( d ( \\Pi | _ { P _ 2 } ) _ p ) \\end{align*}"}
+{"id": "7104.png", "formula": "\\begin{align*} P '' ( y ) = - \\frac { t } { \\pi y ^ 2 } - \\frac { 2 \\lambda \\xi ^ { 1 / 3 } } { 3 \\ , y ^ { 4 / 3 } } \\gg { \\max } \\left \\{ { t } , \\ , \\lambda \\right \\} . \\end{align*}"}
+{"id": "1141.png", "formula": "\\begin{align*} [ ( \\alpha _ 1 , \\alpha _ 2 ) , ( \\alpha _ 1 , \\alpha _ 2 ) ] _ c = ( [ \\alpha _ 1 , \\alpha _ 1 ] , [ \\alpha _ 1 , \\alpha _ 2 ] + [ \\alpha _ 2 , \\alpha _ 1 ] , [ \\alpha _ 2 , \\alpha _ 2 ] ) = ( [ \\alpha _ 1 , \\alpha _ 1 ] , 2 [ \\alpha _ 1 , \\alpha _ 2 ] , [ \\alpha _ 2 , \\alpha _ 2 ] ) . \\end{align*}"}
+{"id": "7261.png", "formula": "\\begin{align*} | | \\varphi | | _ { C ^ 1 } = | | \\varphi | | _ \\infty + | | \\varphi ' | | _ { \\infty } . \\end{align*}"}
+{"id": "2811.png", "formula": "\\begin{align*} J _ k : & = \\int _ { \\Omega } \\frac { Y \\cdot \\nabla z _ k } { ( t ^ 2 + | x | ^ 2 ) ^ \\frac { \\theta } { 2 } } ( - \\Delta ) ^ { s } U _ { t , p } \\dd x = \\int _ { \\Omega } Y _ t \\cdot \\nabla z _ k ( - \\Delta ) ^ { s } U _ { t , p } \\dd x \\\\ & = - \\int _ { \\Omega } Y _ t \\cdot \\nabla U _ { t , p } ( - \\Delta ) ^ { s } z _ k \\dd x - \\iint _ { \\R ^ N \\times \\R ^ N } ( z _ k ( x ) - z _ k ( y ) ) ( U _ { t , p } ( x ) - U _ { t , p } ( y ) ) \\mathcal { K } _ { Y _ t } ( x , y ) \\dd x \\dd y , \\end{align*}"}
+{"id": "3036.png", "formula": "\\begin{align*} \\overline { \\Delta } = \\sum _ { i = 2 } ^ r \\binom { r } { i } m ^ i [ m ( m - 1 ) ] ^ { r - i } p ^ { 2 \\binom { r } { 2 } - \\binom { i } { 2 } } . \\end{align*}"}
+{"id": "8625.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\zeta & = \\delta _ { \\psi } H \\\\ \\partial _ t \\psi & = - \\delta _ { \\zeta } H , \\end{cases} \\end{align*}"}
+{"id": "4246.png", "formula": "\\begin{align*} K ( t , v , v _ j ) \\leq \\left ( \\frac { 4 e t } { D } \\right ) ^ { D / 4 } , \\end{align*}"}
+{"id": "6027.png", "formula": "\\begin{align*} \\mathcal { D } _ { l , p } = \\overline { \\mathcal { S } } ^ { \\left \\Vert \\circ \\right \\Vert _ { L , l , p } } , \\end{align*}"}
+{"id": "6060.png", "formula": "\\begin{align*} O _ { 3 , 1 } = \\mathbb { E } ( \\int _ 0 ^ t \\int _ { B _ { { M _ { { \\mathcal { P } } _ m } } ( r ) } \\backslash B _ { M _ { { \\mathcal { P } } _ n } ( r ) } } \\vert \\nabla _ x { c } ( z , { X } ^ { M _ { { \\mathcal { P } } _ m } } _ { { T } _ { i } ^ { k } - } ) \\vert \\vert D ^ Z { X } ^ { M _ { { \\mathcal { P } } _ m } } _ { { T } _ { i } ^ { k } - } \\vert _ { l _ 2 } N ( d z , d r ) ) ^ 2 , \\end{align*}"}
+{"id": "2034.png", "formula": "\\begin{align*} \\lVert f \\rVert _ { \\mathrm { L } ^ p ( \\partial \\Omega ) } ^ p : = \\int _ { \\partial \\Omega } \\lvert f ( x ) \\rvert ^ p ~ \\mathrm { d } \\sigma _ x \\qquad & \\lVert f \\rVert _ { \\dot { \\mathrm { B } } ^ { s } _ { p , p } ( \\partial \\Omega ) } ^ p : = \\int _ { \\partial \\Omega } \\int _ { \\partial \\Omega } \\frac { \\lvert f ( x ) - f ( y ) \\rvert ^ p } { \\lvert x - y \\rvert ^ { p s + n - 1 } } ~ \\mathrm { d } \\sigma _ x \\mathrm { d } \\sigma _ y \\end{align*}"}
+{"id": "4549.png", "formula": "\\begin{align*} \\mathcal { B } _ 0 \\partial _ t { \\mathbf V } _ \\sigma + \\mathcal { B } _ 1 \\partial _ 1 { \\mathbf V } _ \\sigma + \\mathcal { B } _ 2 \\partial _ 2 { \\mathbf V } _ \\sigma + ( \\mathcal { B } _ 3 + \\partial _ 1 \\mathcal B _ 1 ) { \\mathbf V } _ \\sigma = \\tilde { \\mathcal F } _ { \\sigma } \\ , , \\end{align*}"}
+{"id": "6409.png", "formula": "\\begin{gather*} q _ { k j } ^ { - 1 } q _ { b j } t _ { i k } ^ { a b } = q _ { i j } q _ { a j } ^ { - 1 } t _ { i k } ^ { a b } = q _ { j k } q _ { j b } ^ { - 1 } t _ { i k } ^ { a b } = q _ { j i } ^ { - 1 } q _ { j a } t _ { i k } ^ { a b } , \\\\ q _ { j i } ^ { - 1 } q _ { j a } t _ { i k } ^ { a b } t _ { j b } ^ { u v } - q _ { u i } ^ { - 1 } q _ { u a } t _ { i b } ^ { a v } t _ { j k } ^ { u b } = t _ { j k } ^ { b v } t _ { i b } ^ { a u } - t _ { i j } ^ { a b } t _ { b k } ^ { u v } = q _ { v u } q _ { k u } ^ { - 1 } t _ { b k } ^ { a v } t _ { i j } ^ { b u } - q _ { v j } q _ { k j } ^ { - 1 } t _ { i k } ^ { b v } t _ { b j } ^ { a u } . \\end{gather*}"}
+{"id": "8174.png", "formula": "\\begin{align*} c _ 1 = n ( n + 2 ) ( r - 1 ) ^ 2 + k ^ 2 r ^ 2 - 2 k ( r - 1 ) ( r ( n + 1 ) - 2 ) , c _ 2 = 2 ( k r - ( n - 1 ) ( r - 1 ) ) . \\end{align*}"}
+{"id": "7168.png", "formula": "\\begin{align*} \\| P _ S V x \\| & = \\left \\| \\sum _ { j \\in S } f _ j ( V x ) \\tau _ j \\right \\| = \\left \\| \\sum _ { j \\in S } f _ j \\left ( \\sum _ { k = 1 } ^ { n } g _ k ( x ) \\tau _ k \\right ) \\tau _ j \\right \\| = \\left \\| \\sum _ { j \\in S } \\sum _ { k = 1 } ^ { n } g _ k ( x ) f _ j ( \\tau _ k ) \\tau _ j \\right \\| \\\\ & = \\left \\| \\sum _ { j \\in S } g _ j ( x ) \\tau _ j \\right \\| = \\left ( \\sum _ { j \\in S } | g _ j ( x ) | ^ p \\right ) ^ \\frac { 1 } { p } = \\| x \\| _ { S , g } , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "8851.png", "formula": "\\begin{align*} \\norm { x _ { t + 1 } - x _ { p } } ^ { 2 } & \\leq \\norm { x _ { t } - \\eta _ { t } \\hat { g } _ { t } - x _ { p } } ^ 2 \\\\ & = \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 \\\\ & \\leq ( 1 - 2 \\eta _ { t } \\alpha ) \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } - \\nabla f ( x _ t ) , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 , \\end{align*}"}
+{"id": "2776.png", "formula": "\\begin{align*} x ^ { k + 2 } \\cdot x ^ { k + 1 } \\cdot x ^ 4 = x ^ { 2 k + 7 } , \\end{align*}"}
+{"id": "846.png", "formula": "\\begin{align*} \\mathcal { F } = \\{ x _ 0 ^ * ( x ) h : x \\in S _ X \\} , \\end{align*}"}
+{"id": "3951.png", "formula": "\\begin{align*} L ( \\gamma ) = \\sup _ { \\pi \\in \\Pi } l ( \\pi , \\gamma ) < + \\infty \\end{align*}"}
+{"id": "45.png", "formula": "\\begin{align*} \\rho ^ 2 | \\Lambda | \\widehat { g } ( 0 ) = \\rho ( n _ 0 + n _ + ) \\widehat { g } ( 0 ) . \\end{align*}"}
+{"id": "2274.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\theta _ { \\alpha } \\left ( x - \\frac { j } { n } \\right ) & = n \\sum _ { k \\in \\mathbb { Z } } e ^ { - \\pi \\alpha k ^ 2 n ^ 2 } e ^ { 2 \\pi i k n x } \\\\ & = n + 2 n e ^ { - \\pi \\alpha n ^ 2 } \\cos \\left ( 2 \\pi n x \\right ) + \\mathcal { O } ( n e ^ { - 4 \\pi \\alpha n ^ 2 } ) . \\end{align*}"}
+{"id": "7580.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf { R } _ 1 & = \\left \\{ ( s , t ) : 0 \\le s \\le t \\le T , t \\ge 4 \\right \\} , \\\\ \\mathbf { R } _ 2 & = \\left \\{ ( s , t ) : 0 \\le s \\le t \\le T , \\ , t < 4 , \\ , | D _ t ^ { ( 0 ) } - D _ s ^ { ( \\ell ) } | ^ 2 \\leq 8 \\right \\} , \\\\ \\mathbf { R } _ 3 & = \\left \\{ ( s , t ) : 0 \\le s \\le t \\le T , \\ , t < 4 , \\ , | D _ t ^ { ( 0 ) } - D _ s ^ { ( \\ell ) } | ^ 2 > 8 \\right \\} . \\end{aligned} \\end{align*}"}
+{"id": "1688.png", "formula": "\\begin{align*} \\varrho _ { \\texttt { b } ; j } : = n + 1 - j ( j = 1 , \\ldots , n ) . \\end{align*}"}
+{"id": "6386.png", "formula": "\\begin{align*} b ^ { 1 } \\left ( \\gamma \\right ) = 1 _ { H } \\otimes \\gamma - \\Delta ( \\gamma ) + \\gamma \\otimes 1 _ { H } = \\chi + \\tau ( S \\otimes S ) ( \\chi ) . \\end{align*}"}
+{"id": "1904.png", "formula": "\\begin{align*} \\sum _ { i , j } \\int _ { T _ { i j } } \\left ( \\frac { v ^ 2 } { 2 } \\frac { \\partial f _ h } { \\partial t } + v ^ 2 f _ h - u _ h v f _ h \\right ) \\ , { \\rm d } x \\ , { \\rm d } v = 0 . \\end{align*}"}
+{"id": "6570.png", "formula": "\\begin{align*} & \\min _ { k \\in \\Pi _ b \\Lambda , \\sqrt { N _ 1 } \\leq L \\leq N _ 1 , \\mathcal { E R } _ d ( L ) \\ni \\Lambda ' \\subset \\Pi _ d \\Lambda ( k ) , \\xi = \\pm 1 , 1 \\leq l \\leq \\# \\Lambda ' } | \\sigma + k \\cdot \\omega + \\xi \\sqrt { \\zeta _ l ( k ) } | \\\\ & > e ^ { - \\frac 1 4 L ^ { \\rho _ 2 } } , \\end{align*}"}
+{"id": "4481.png", "formula": "\\begin{align*} \\delta \\dot { { \\mathbf V } } _ i : = \\delta { \\mathbf V } _ i - \\frac { \\partial _ 1 ( { \\mathbf U } ^ a + { \\mathbf V } _ { i + \\frac { 1 } { 2 } } ) } { \\partial _ 1 ( \\Phi ^ a + \\Psi _ { i + \\frac { 1 } { 2 } } ) } \\delta \\Psi _ i \\end{align*}"}
+{"id": "3186.png", "formula": "\\begin{align*} e v _ 1 \\circ h _ { n + 1 } ( v ) & = \\sum _ { i \\geq 0 } \\alpha _ i ( v ) \\\\ & = f _ { n + 1 } ( v ) - \\sum _ { i \\geq 0 } \\frac { d \\beta _ i ( v ) + \\beta _ i ( d v ) } { i + 1 } \\\\ & = f _ { n + 1 } ( v ) - d \\beta _ 0 ( v ) - \\sum _ { i \\geq 0 } \\frac { \\beta _ i d ( v ) } { i + 1 } \\\\ & = f _ { n + 1 } ( v ) + \\left ( g _ { n + 1 } ( v ) - f _ { n + 1 } ( v ) + \\sum _ { i \\geq 0 } \\frac { \\beta _ i d ( v ) } { i + 1 } \\right ) - \\sum _ { i \\geq 0 } \\frac { \\beta _ i d ( v ) } { i + 1 } \\\\ & = g _ { n + 1 } ( v ) \\end{align*}"}
+{"id": "1144.png", "formula": "\\begin{align*} ( \\delta _ 1 ) ^ 2 = 0 ( \\delta _ 2 ) ^ 2 = 0 . \\end{align*}"}
+{"id": "5015.png", "formula": "\\begin{align*} \\gamma = d r ^ 2 + r ( d r \\otimes \\beta + \\beta \\otimes d r ) + r ^ 2 h \\end{align*}"}
+{"id": "4895.png", "formula": "\\begin{align*} E _ n ( z ) & = B _ n \\exp ( \\tilde { B } _ n ) E _ { n + 1 } ( z ) \\\\ & = G _ { n - 1 } ^ { - 1 } ( z ) G _ n ( z ) E _ { n + 1 } ( z ) \\\\ & = G _ { n - 1 } ^ { - 1 } ( z ) P ( z ) E ( z ) . \\end{align*}"}
+{"id": "1348.png", "formula": "\\begin{align*} U _ 6 = \\{ t \\in U _ 5 \\ , | \\ , \\mathcal { U } _ { \\bar { t } } \\} \\end{align*}"}
+{"id": "8676.png", "formula": "\\begin{align*} \\int g ( u _ 0 , . . . , u _ { m _ 1 } ) e ^ { \\frac { 1 } { \\hbar } \\sum _ { i = 0 } ^ { n + 1 } \\phi _ i ( u _ 0 , . . . , u _ { m _ 1 } ) y _ i } d u _ 0 . . . d u _ { m _ 1 } = \\end{align*}"}
+{"id": "1919.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } \\frac { { \\rm d } } { { \\rm d } t } \\| \\theta _ u \\| ^ 2 _ { 0 , I _ h } + \\| \\theta _ w \\| ^ 2 _ { 0 , I _ h } & \\leq C \\left ( h ^ { 2 k + 2 } + \\| f - f _ h \\| ^ 2 _ { 0 , \\mathcal { T } _ h } + \\| \\theta _ u \\| ^ 2 _ { 0 , I _ h } \\right . \\\\ & \\left . \\quad \\qquad \\qquad + \\ , \\ , h ^ { - \\frac { 3 } { 2 } } \\| \\theta _ u \\| ^ 3 _ { 0 , I _ h } + h ^ { - 1 } \\| \\theta _ u \\| ^ 4 _ { 0 , I _ h } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "8278.png", "formula": "\\begin{align*} N ( \\mathbb { C } ) = \\{ ( z , w ) \\in \\mathbb { C } ^ m \\times \\mathbb { C } ^ m | | z | ^ 2 = | w | ^ 2 = \\frac { 1 } { 2 } , \\sum _ { \\alpha = 1 } ^ m z _ \\alpha w _ \\alpha = 0 \\} . \\end{align*}"}
+{"id": "5964.png", "formula": "\\begin{align*} - u _ f ( x ) & = \\int _ M G ^ \\omega _ M ( x , z ) \\Delta _ g u _ f d \\mu _ g ( z ) + \\int _ { \\partial M } \\left ( u _ f ( z ) \\partial _ \\nu G ^ \\omega _ M ( x , z ) - G ^ \\omega _ M ( x , z ) \\partial _ \\nu u _ f ( z ) \\right ) d \\mu _ h ( z ) \\\\ & - \\int _ M u _ f ( z ) _ g ( F ( z ) G ^ \\omega _ M ( x , z ) ) d \\mu _ g ( z ) + \\omega ^ 2 \\int _ M u _ f ( z ) G ^ \\omega _ M ( x , z ) d \\mu _ g ( z ) . \\end{align*}"}
+{"id": "6651.png", "formula": "\\begin{align*} c _ { 2 } = - { 1 \\over 2 \\pi ^ 2 } p ^ 2 , c _ { 2 n } = - { 1 \\over \\pi ^ { 2 n } } { ( 2 n - 3 ) ! ! \\over ( 2 n ) ! ! } \\prod _ { l = 0 } ^ { n - 1 } ( p ^ 2 - l ^ 2 ) , ( n \\ge 2 ) . \\end{align*}"}
+{"id": "2080.png", "formula": "\\begin{align*} D = \\left ( \\begin{array} { c c } \\alpha e ^ { \\Lambda } & 0 \\\\ 0 & \\alpha e ^ { - \\Lambda } \\end{array} \\right ) , R = \\left ( \\begin{array} { c c } \\cos \\phi & - \\sin \\phi \\\\ \\sin \\phi & \\cos \\phi \\end{array} \\right ) . \\end{align*}"}
+{"id": "1703.png", "formula": "\\begin{align*} ^ { ( n ) } _ { \\texttt { b } } & : = \\left \\{ \\bigl ( X _ 1 ( \\boldsymbol { \\xi } ) , \\ldots , X _ n ( \\boldsymbol { \\xi } ) \\bigr ) \\mid \\boldsymbol { \\xi } \\in \\mathbb { A } ^ { ( n ) } _ { \\texttt { b } } \\right \\} , \\\\ \\boldsymbol { X } ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } & : = \\bigl ( X _ 1 ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } ) , \\ldots , X _ n ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } ) \\bigr ) . \\end{align*}"}
+{"id": "2017.png", "formula": "\\begin{align*} ( \\dot { \\mathrm { H } } ^ { s , p } ( \\Omega ) ) ' = \\dot { \\mathrm { H } } ^ { - s , p ' } _ 0 ( \\Omega ) ( \\dot { \\mathrm { H } } ^ { s , p } _ 0 ( \\Omega ) ) ' = \\dot { \\mathrm { H } } ^ { - s , p ' } ( \\Omega ) \\end{align*}"}
+{"id": "1723.png", "formula": "\\begin{align*} \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } = \\left ( \\xi ^ { ( m + n ) } _ { \\lambda _ 1 + n - 1 } , \\xi ^ { ( m + n ) } _ { \\lambda _ 2 + n - 2 } , \\ldots , \\xi ^ { ( m + n ) } _ { \\lambda _ { n - 1 } + 1 } , \\xi ^ { ( m + n ) } _ { \\lambda _ n } \\right ) \\end{align*}"}
+{"id": "5118.png", "formula": "\\begin{align*} \\nabla _ X D ^ r _ Y ( Z ) & = \\nabla _ X r ( \\nabla _ Y Z ) - \\nabla _ X \\nabla _ { r ( Y ) } Z , \\\\ D ^ { r } _ { [ X , Y ] } ( Z ) & = r ( \\nabla _ { [ X , Y ] } Z ) - \\nabla _ { r ( [ X , Y ] ) } Z , \\\\ D ^ { r , * } _ X ( g ^ \\flat ( \\nabla _ Y Z ) ) & = - g ^ \\flat ( \\nabla _ X \\ , r ( \\nabla _ Y Z ) ) - g ^ \\flat ( \\nabla _ { r ( X ) } \\nabla _ Y Z ) . \\end{align*}"}
+{"id": "2123.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde e = \\tilde e [ \\Lambda , \\phi , \\alpha ] : = & ~ { } \\kappa \\partial _ t \\alpha \\left [ ( \\partial _ t \\Lambda ) ^ 2 + ( \\partial _ x \\Lambda ) ^ 2 + 4 \\sinh ^ 2 ( \\Lambda ) \\Big ( ( \\partial _ t \\phi ) ^ 2 + ( \\partial _ x \\phi ) ^ 2 \\Big ) \\right ] \\\\ & ~ { } - 2 \\kappa \\partial _ x \\alpha \\left ( \\partial _ x \\Lambda \\partial _ t \\Lambda + 4 \\partial _ x \\phi \\partial _ t \\phi \\sinh ^ 2 ( \\Lambda ) \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "8221.png", "formula": "\\begin{align*} d ^ c _ { I _ i ^ a } K _ i ^ a = d ^ c _ { I _ i } K _ i + a ^ 2 ( x _ j d x _ k - x _ k d x _ j ) . \\end{align*}"}
+{"id": "1696.png", "formula": "\\begin{align*} N _ { \\texttt { b } ; \\mu } : = 2 ^ { _ 0 ( \\mu ) } \\prod _ { \\substack { 1 \\leq j < k \\leq n \\\\ \\mu _ j - \\mu _ k = 0 } } \\frac { 1 + k - j } { k - j } \\end{align*}"}
+{"id": "4183.png", "formula": "\\begin{align*} t _ { } = t _ { } + t _ { } , \\end{align*}"}
+{"id": "3097.png", "formula": "\\begin{align*} 3 \\alpha ^ 2 - 4 \\alpha \\beta + 4 ( n - 2 ) \\beta ^ 2 - 4 ( n - 3 ) \\beta \\gamma + \\left [ \\binom { n } { 2 } - 3 \\right ] \\gamma ^ 2 \\le \\delta ^ 2 . \\end{align*}"}
+{"id": "7458.png", "formula": "\\begin{align*} \\langle x ; y \\rangle = e ^ { 2 \\pi i ( \\dot x \\ddot y + \\dot y \\ddot x ) } . \\end{align*}"}
+{"id": "6830.png", "formula": "\\begin{align*} P _ 6 ( g ) & : = ( - \\Delta _ g + a _ n R _ g ) ( - \\Delta _ g + b _ n R _ g ) ( - \\Delta _ g + c _ n R _ g ) . \\end{align*}"}
+{"id": "4913.png", "formula": "\\begin{align*} f _ k ( \\tau ) = \\begin{cases} \\dfrac { 1 } { 2 ^ { 2 k } \\cdot E _ { 2 k } } \\displaystyle \\sum _ { n \\geq 1 } \\chi _ { - 4 } ( n ) \\dfrac { q ^ { n } P _ { k } ( q ^ { n } ) } { ( 1 - q ^ { 2 n } ) ^ { 2 k + 1 } } , & k \\equiv 0 \\pmod { 2 } , \\\\ - \\dfrac { 1 } { E _ { 2 k } } \\displaystyle \\sum _ { n \\geq 1 } \\chi _ { - 4 } ( n ) \\dfrac { q ^ { 2 n } A _ { 2 k } ( q ^ { 2 n } ) } { ( 1 - q ^ { 2 n } ) ^ { 2 k + 1 } } , & k \\equiv 1 \\pmod { 2 } , \\end{cases} + T _ { 2 k + 1 } ( \\tau ) \\end{align*}"}
+{"id": "4088.png", "formula": "\\begin{align*} \\epsilon _ 1 = \\hbar \\cdot \\beta ^ { \\frac 1 2 } , \\quad \\epsilon _ 2 = - \\hbar \\cdot \\beta ^ { - \\frac 1 2 } . \\end{align*}"}
+{"id": "2132.png", "formula": "\\begin{align*} d s ^ 2 = f ^ { ( 0 ) } ( - d t ^ 2 + d r ^ 2 ) + e ^ { - u ^ { ( 0 ) } } ( \\alpha d \\phi ) ^ 2 + e ^ { u ^ { ( 0 ) } } d z ^ 2 , \\end{align*}"}
+{"id": "5198.png", "formula": "\\begin{align*} X = X ( \\tau ) = | T \\cap Z | = \\sum _ { i \\in N _ v } X _ i , \\end{align*}"}
+{"id": "810.png", "formula": "\\begin{align*} c + o ( 1 ) & = \\left ( \\frac { 1 } { p } - \\frac { 1 } { 2 \\cdot p ^ \\flat } \\right ) ( [ u _ n ] _ { s , p } ^ p + a \\| u _ n \\| _ p ^ p ) \\\\ & \\geq \\left ( \\frac { 1 } { p } - \\frac { 1 } { 2 \\cdot p ^ \\flat } \\right ) a \\| u _ n \\| _ p ^ p . \\end{align*}"}
+{"id": "5817.png", "formula": "\\begin{align*} \\footnotesize & \\alpha ^ { d 4 } _ { m a x } , \\ ; \\alpha ^ { d 6 } _ { m a x } , \\ ; \\dots , \\ ; \\alpha ^ { d , n - 2 } _ { m a x } , \\ ; \\alpha _ { m a x } , \\\\ & \\alpha _ 1 , \\alpha _ 3 , \\dots , \\alpha _ { n - 5 } , \\alpha _ { n - 3 } , \\alpha _ { n - 1 } , \\alpha _ { n } . \\end{align*}"}
+{"id": "5039.png", "formula": "\\begin{align*} \\frac { \\triangle _ { V _ s } w } { w } - \\frac 1 2 = \\frac { \\triangle r } { r + A } - \\frac { d r ( V _ s ) } { r + A } - \\frac 1 2 = \\frac { \\triangle r } { r + A } - \\frac { d r ( V _ s + \\frac 1 2 r \\partial _ r ) } { r + A } + \\frac { r } { 2 ( r + A ) } - \\frac 1 2 . \\end{align*}"}
+{"id": "2206.png", "formula": "\\begin{align*} \\lambda _ 1 = \\frac { q ^ { k - 1 } - 1 } { q - 1 } \\cdot \\frac { q ^ { k + 1 } - 1 } { q - 1 } + \\frac { q ^ { k } - q ^ { k - 1 } } { q - 1 } = D _ L \\cdot D _ R = \\Theta ( q ^ { 2 k - 2 } ) \\end{align*}"}
+{"id": "1107.png", "formula": "\\begin{align*} \\| u \\| _ { \\alpha } : = \\sqrt { \\displaystyle \\langle u , u \\rangle - \\int _ D { \\alpha ( x ) } | u ( x ) | ^ { 2 } d \\mu } , \\end{align*}"}
+{"id": "6451.png", "formula": "\\begin{align*} \\widetilde { \\mu } \\left ( \\pi _ { \\mathrm { N } } ^ * ( [ X / u ] ) \\right ) & = \\widetilde { \\mu } \\left ( [ g _ { u , 1 } ] \\oplus \\pi _ { \\mathrm { N } } ^ * ( [ X / 1 ] ) \\right ) \\\\ & = \\widetilde { \\mu } \\left ( [ g _ { u , 1 } ] \\right ) + \\widetilde { \\mu } \\left ( [ \\pi ] \\right ) \\\\ & = \\mu \\left ( [ X / u ] \\right ) - \\mu \\left ( [ X / 1 ] \\right ) + \\mu \\left ( [ X / 1 ] \\right ) \\\\ & = \\mu \\left ( [ X / u ] \\right ) . \\end{align*}"}
+{"id": "6675.png", "formula": "\\begin{align*} g _ n ( \\beta , p , q ) | _ { \\beta = 1 } = ( - 2 ) ^ { n + 1 } g _ n ( \\beta , - p / 2 , - 2 q ) | _ { \\beta = 4 } . \\end{align*}"}
+{"id": "8379.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\frac { \\epsilon } { \\delta } = 0 . \\end{align*}"}
+{"id": "880.png", "formula": "\\begin{align*} \\tilde { x } ( 0 ) = x , ~ ~ \\tilde { t } ( 0 ) = t , ~ ~ \\tilde { u } ( 0 ) = u , ~ ~ \\tilde { v } ( 0 ) = v . \\end{align*}"}
+{"id": "5351.png", "formula": "\\begin{align*} H ^ 0 ( N _ Z ( K _ Z - ( k - 1 ) H ) ) = 0 . \\end{align*}"}
+{"id": "4634.png", "formula": "\\begin{align*} \\textup { T r } _ { M ^ { \\pi _ n } } ^ { \\mathbf { S } } ( ( A \\otimes \\textup { i d } _ { \\mathbf { S } } ) B ) = \\textup { T r } _ { M ^ { \\pi _ n } } ^ { \\mathbf { S } } ( B A ) \\end{align*}"}
+{"id": "3225.png", "formula": "\\begin{align*} S _ { \\alpha } ( \\epsilon , R ) = \\textnormal { A r e a } ( \\textnormal { S e c t } _ { \\alpha , \\epsilon } ( R ) ) + O \\left ( R \\right ) . \\end{align*}"}
+{"id": "1112.png", "formula": "\\begin{align*} \\begin{aligned} & \\mbox { t h e r e a r e p o s i t i v e c o n s t a n t s } \\ , \\ , \\ , M _ 0 \\ , \\ , \\mbox { a n d } \\ , \\ , \\sigma \\ , \\ , \\mbox { s u c h t h a t } \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\displaystyle \\max _ { ( x , s ) \\in \\mathop D \\limits ^ \\circ \\times [ - M _ 0 , M _ 0 ] } f _ + ( x , s ) \\leq \\mu _ 0 ^ 2 \\frac { M _ 0 } { ( \\sigma + 1 ) } \\frac { \\lambda _ 1 } { 2 } . \\end{aligned} \\end{align*}"}
+{"id": "7821.png", "formula": "\\begin{align*} ( v ^ a ) \\cdot ( \\rho ^ { } , \\widetilde { \\zeta } _ 0 ^ { } , \\widetilde { \\zeta } _ a ^ { } , \\sigma ^ { } ) = ( \\rho ^ { } , \\widetilde { \\zeta } _ 0 ^ { } - v _ a \\widetilde { \\zeta } _ a ^ { } , \\widetilde { \\zeta } _ a ^ { } , \\sigma ^ { } ) \\end{align*}"}
+{"id": "5619.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ 0 ^ \\infty | u _ n | ^ p r ^ { \\alpha _ 0 } \\mathrm d r = \\int _ 0 ^ \\infty | u | ^ p r ^ { \\alpha _ 0 } \\mathrm d r . \\end{align*}"}
+{"id": "6567.png", "formula": "\\begin{align*} \\min _ { \\sqrt { N _ 1 } \\leq L \\leq N _ 1 , \\mathcal { E R } _ d ( L ) \\ni \\Lambda ' \\subset \\Pi _ d \\Lambda ( k ) , \\xi = \\pm 1 , 1 \\leq l \\leq \\# \\Lambda ' } | \\sigma + k \\cdot \\omega + \\xi \\sqrt { \\zeta _ l } | > e ^ { - \\frac 1 4 L ^ { \\rho _ 2 } } . \\end{align*}"}
+{"id": "7790.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\hat { \\gamma } } ^ { ( \\nu ) } = \\sum _ { n \\in \\mathbb { Z } } \\sum _ { m \\in \\mathbb { Z } - \\{ 0 \\} } \\frac { e ^ { - S _ { \\hat { \\gamma } , m , n } } } { m ^ { \\nu } | m \\tau + n | } , \\end{align*}"}
+{"id": "767.png", "formula": "\\begin{align*} \\mathrm { l . h . s . } \\eqref { d i f f e r e n c e 2 } = \\norm { \\theta _ R \\left ( \\norm { \\mathbf V } _ { t } ^ 2 \\right ) \\mathbf V } _ { T } \\le C \\norm { \\theta _ R \\left ( \\norm { \\mathbf V } _ { t } ^ 2 \\right ) \\mathbf V } _ { T _ { \\mathbf V } } \\le C \\sqrt { 2 R } , \\end{align*}"}
+{"id": "2651.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\sup _ { n } P ( w ' _ L ( X _ n , \\delta ) > \\epsilon ) ^ { 1 / r _ n } = 0 \\ , . \\end{align*}"}
+{"id": "6952.png", "formula": "\\begin{align*} E T ( k ) : = & 2 \\sum _ { i = 1 } ^ { k - 1 } \\sum _ { j = i + 1 } ^ { k - 1 } \\frac { 1 } { ( ( x _ k - x _ i ) + \\epsilon ^ 2 ) ( ( x _ k - x _ j ) + \\epsilon ^ 2 ) } \\\\ & - 2 \\sum _ { i = 1 } ^ { k - 1 } \\sum _ { j = k + 1 } ^ { d } \\frac { 1 } { ( ( x _ k - x _ i ) + \\epsilon ^ 2 ) ( ( x _ j - x _ k ) + \\epsilon ^ 2 ) } \\\\ & + 2 \\sum _ { i = k + 1 } ^ { d } \\sum _ { j = i + 1 } ^ { d } \\frac { 1 } { ( ( x _ i - x _ k ) + \\epsilon ^ 2 ) ( ( x _ j - x _ k ) + \\epsilon ^ 2 ) } . \\end{align*}"}
+{"id": "5145.png", "formula": "\\begin{align*} \\lim _ { D \\to 0 } H [ Q ( X ) ] - H [ Q ^ * ( X ) ] = 0 \\end{align*}"}
+{"id": "5546.png", "formula": "\\begin{align*} c ^ { ( i ) } _ j = c ^ { ( i - 1 ) } _ j - ( w ^ { ( i ) } _ j - w _ j ) , r ^ { ( i ) } _ j = r ^ { ( i - 1 ) } _ j - \\frac { \\ , 1 \\ , } { 4 } ( L ^ { ( i ) } _ j - L ) , j = 1 , \\ldots , m . \\end{align*}"}
+{"id": "7520.png", "formula": "\\begin{align*} \\mathbf { D } ( r , \\rho , r u ) = \\frac { \\rho \\ , n p ' ( n ) } { r ^ 2 } \\begin{pmatrix} 0 & 0 & 0 \\\\ - 1 & 1 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} + \\theta \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & n ^ 2 / r ^ 2 & 0 \\\\ - { \\rho u } / { r } & 0 & { \\rho } / { r } \\end{pmatrix} \\end{align*}"}
+{"id": "8140.png", "formula": "\\begin{align*} V _ 1 \\colon & y ^ 2 = ( ( 3 a - b - 3 ) x - a + 3 ) ( 1 + ( a - 3 ) x ( 1 - x ) ) , \\\\ V _ 2 \\colon & x ^ 3 + y ^ 3 + 1 + a ( x ^ 2 y + x y ^ 2 + x ^ 2 + x + y ^ 2 + y ) + b x y = 0 , \\\\ V _ 3 \\colon & y ^ 2 = - ( ( a + 1 ) x ^ 3 + ( 2 a + b ) x ^ 2 + 4 a x + 4 ) ( x ^ 3 + a x ^ 2 + a x + 1 ) . \\end{align*}"}
+{"id": "1181.png", "formula": "\\begin{align*} \\begin{cases} & l _ 1 ( x , m ) = m ^ E _ 1 ( s ( x ) , i ( m ) ) \\\\ & r _ 1 ( m , x ) = m ^ E _ 1 ( i ( m ) , s ( x ) ) \\end{cases} ; \\begin{cases} & l _ 2 ( x , m ) = m ^ E _ 2 ( s ( x ) , i ( m ) ) \\\\ & r _ 2 ( m , x ) = m ^ E _ 2 ( i ( m ) , s ( x ) ) . \\end{cases} \\end{align*}"}
+{"id": "8460.png", "formula": "\\begin{align*} E _ { z } : = \\{ w \\in \\mathbb { R } ^ { n - 1 } : ( z , w ) \\in E \\} . \\end{align*}"}
+{"id": "8034.png", "formula": "\\begin{align*} h _ { \\omega } ^ { p } = \\{ f \\in \\mathcal { S } ^ { \\prime } ( \\mathbb R ^ { n } ) \\colon M _ { \\Phi } ( f ) \\in L ^ { p } _ { \\omega } ( \\mathbb R ^ { n } ) \\} \\end{align*}"}
+{"id": "5777.png", "formula": "\\begin{align*} J _ X ^ p = L ^ { ( 1 - p ) / 2 } \\Delta _ p g _ X ^ p + L ^ { 1 - p } \\mu _ p . \\end{align*}"}
+{"id": "4714.png", "formula": "\\begin{align*} P _ 1 + P _ 2 & = ( x _ { i _ 1 } + v _ { i _ 1 } ) ( x _ { i _ 2 } + v _ { i _ 2 } ) + ( x _ { i _ 1 } + z _ { i _ 1 } ) ( x _ { i _ 2 } + z _ { i _ 2 } ) \\end{align*}"}
+{"id": "6658.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty \\Big ( R ( x ) - { 1 \\over \\pi } \\Big ) \\ , d x = - p \\end{align*}"}
+{"id": "5734.png", "formula": "\\begin{align*} \\widehat { H ^ s f } ( \\xi , \\sigma ) = ( 4 \\pi ^ 2 | \\xi | ^ 2 + 2 \\pi i \\sigma ) ^ s \\ \\hat f ( \\xi , \\sigma ) , \\end{align*}"}
+{"id": "5345.png", "formula": "\\begin{align*} q ( S ) = 0 . \\end{align*}"}
+{"id": "2900.png", "formula": "\\begin{align*} e ^ { i c \\langle \\xi , - v \\alpha _ 0 \\rangle } = \\prod _ { \\alpha \\in R ^ + _ 0 } \\left ( \\frac { 1 - t _ { \\alpha } e ^ { i \\langle \\xi , \\alpha \\rangle } } { e ^ { i \\langle \\xi , \\alpha \\rangle } - t _ { \\alpha } } \\right ) ^ { { \\langle - v \\alpha _ 0 , \\hat \\alpha \\rangle } } , \\forall v \\in { W _ 0 } , \\end{align*}"}
+{"id": "2517.png", "formula": "\\begin{align*} \\boldsymbol { y } _ { N h } ( \\boldsymbol { x } , t ) & = \\sum _ { k = 0 } ^ N [ \\boldsymbol { y } _ { k h } ^ c ( \\boldsymbol { x } ) \\cos ( k \\omega t ) + \\boldsymbol { y } _ { k h } ^ s ( \\boldsymbol { x } ) \\sin ( k \\omega t ) ] , \\\\ \\boldsymbol { p } _ { N h } ( \\boldsymbol { x } , t ) & = \\sum _ { k = 0 } ^ N [ \\boldsymbol { p } _ { k h } ^ c ( \\boldsymbol { x } ) \\cos ( k \\omega t ) + \\boldsymbol { p } _ { k h } ^ s ( \\boldsymbol { x } ) \\sin ( k \\omega t ) ] . \\end{align*}"}
+{"id": "426.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left ( \\log \\frac { \\Gamma ( x ) } { \\Gamma ( x - a ) } \\right ) = \\frac { \\Gamma ' ( x ) } { \\Gamma ( x ) } - \\frac { \\Gamma ' ( x - a ) } { \\Gamma ( x - a ) } = \\sum _ { k \\geq 0 } \\left ( \\frac { 1 } { x - a + k } - \\frac { 1 } { x + k } \\right ) > 0 , \\end{align*}"}
+{"id": "8090.png", "formula": "\\begin{align*} \\begin{aligned} g ( a _ { j } ) ^ { 2 } ( x ) & = \\sum \\limits _ { i } \\left \\vert \\int _ { \\mathbb { R } ^ { n } } a _ { j } ( z ) [ \\phi _ { i } ( y - z ) - p ^ { s } _ { i } ( y , z , x _ { j } ) ] d z \\right \\vert ^ { 2 } \\\\ & \\leq C \\frac { \\omega ( Q _ { j } ) ^ { - \\frac { 2 } { p } } l ( Q _ { j } ) ^ { 2 ( n + s + 1 ) } } { \\vert y - x _ { j } \\vert ^ { 2 ( n + s + 1 ) } } . \\end{aligned} \\end{align*}"}
+{"id": "6362.png", "formula": "\\begin{align*} t _ { X , Y \\otimes Z } = t _ { X , Y } \\otimes \\mathrm { I d } _ { Z } + ( \\sigma ^ { - 1 } _ { X , Y } \\otimes \\mathrm { I d } _ { Z } ) \\circ ( \\mathrm { I d } _ { Y } \\otimes t _ { X , Z } ) \\circ ( \\sigma _ { X , Y } \\otimes \\mathrm { I d } _ { Z } ) , \\\\ t _ { X \\otimes Y , Z } = \\mathrm { I d } _ { X } \\otimes t _ { Y , Z } + ( \\mathrm { I d } _ { X } \\otimes \\sigma ^ { - 1 } _ { Y , Z } ) \\circ ( t _ { X , Z } \\otimes \\mathrm { I d } _ { Y } ) \\circ ( \\mathrm { I d } _ { X } \\otimes \\sigma _ { Y , Z } ) , \\end{align*}"}
+{"id": "1498.png", "formula": "\\begin{align*} d \\langle A _ { i } \\rangle _ { t } = \\frac { 1 } { 4 } \\sum _ { k = 1 } ^ { m } \\left ( U ^ { ( i ) } B _ { t } \\right ) ^ { 2 } _ { k } d t = \\frac { 1 } { 4 } \\vert U ^ { ( i ) } B _ { t } \\vert ^ { 2 } d t = \\frac { 1 } { 4 } \\vert B _ { t } \\vert ^ { 2 } d t . \\end{align*}"}
+{"id": "5094.png", "formula": "\\begin{align*} d ( v \\otimes Z _ 1 \\wedge \\ldots \\wedge Z _ k ) = \\sum _ { i = 1 } ^ k ( - 1 ) ^ { i + 1 } v Z _ i \\otimes Z _ 1 \\wedge \\ldots \\wedge \\widehat { Z _ i } \\wedge \\ldots \\wedge Z _ k + \\sum _ { i < j } ( - 1 ) ^ { i + j } v \\otimes [ Z _ i , Z _ j ] \\wedge \\cdots \\wedge \\widehat { Z } _ i \\wedge \\cdots \\wedge \\widehat { Z } _ j \\cdots \\wedge Z _ k , \\end{align*}"}
+{"id": "5381.png", "formula": "\\begin{align*} \\left \\langle ( \\mathbb I _ { [ 0 , t _ 1 ] } , \\cdot \\cdot \\cdot , \\mathbb I _ { [ 0 , t _ { d _ { 1 } } ] } ) , ( \\mathbb I _ { [ 0 , s _ 1 ] } , \\cdot \\cdot \\cdot , \\mathbb I _ { [ 0 , s _ { d _ { 1 } } ] } ) \\right \\rangle _ { \\mathcal { H } } = \\sum _ { i = 1 } ^ { d _ { 1 } } R _ H ( t _ i , s _ i ) . \\end{align*}"}
+{"id": "1489.png", "formula": "\\begin{align*} & \\langle U ^ { ( i ) } \\overline { x } , U ^ { ( j ) } \\overline { x } \\rangle = \\langle - \\overline { x } , U ^ { ( i ) } U ^ { ( j ) } \\overline { x } \\rangle = - \\langle U ^ { ( i ) } U ^ { ( j ) } \\overline { x } , \\overline { x } \\rangle . \\end{align*}"}
+{"id": "6096.png", "formula": "\\begin{align*} \\| E ( g _ k ) \\| ^ 2 = \\| \\nabla f ( \\omega _ k ) \\| ^ 2 \\ge 2 \\mu ( f ( \\omega _ k ) - f ( \\omega ^ * ) ) \\end{align*}"}
+{"id": "1769.png", "formula": "\\begin{gather*} ( y ) ^ h = \\mathfrak { p } _ 1 ^ { h a _ 1 } \\cdots \\mathfrak { p } _ k ^ { h a _ k } = ( \\varpi _ { \\mathfrak { p } _ 1 } ^ { a _ 1 } \\cdots \\varpi _ { \\mathfrak { p } _ k } ^ { a _ k } ) . \\end{gather*}"}
+{"id": "2667.png", "formula": "\\begin{align*} R _ H = r - \\dim ( H _ { f | _ { H } } ^ { \\perp _ f } ) . \\end{align*}"}
+{"id": "3933.png", "formula": "\\begin{align*} I = \\bigcup _ { x \\in \\omega } [ x - 1 , x + 1 ] \\end{align*}"}
+{"id": "2116.png", "formula": "\\begin{align*} \\begin{aligned} & ~ { } G = \\dfrac { 1 } { 2 \\alpha ^ 2 } ( \\tilde { \\alpha } ' _ 0 ( 2 u ) + \\alpha _ 1 ( 2 u ) ) ( \\alpha _ 1 ( - 2 \\underline { u } ) - \\tilde { \\alpha } ' _ 0 ( - 2 \\underline { u } ) ) \\\\ & ~ { } - \\dfrac { 1 } { 2 } ( ( \\partial _ t \\Lambda ) ^ 2 - ( \\partial _ x \\Lambda ) ^ 2 ) - 2 \\sinh ^ 2 \\Lambda ( ( \\partial _ t \\phi ) ^ 2 - ( \\partial _ x \\phi ) ^ 2 ) = : G _ 1 + G _ 2 . \\end{aligned} \\end{align*}"}
+{"id": "3748.png", "formula": "\\begin{align*} \\sin ( 3 \\theta \\pi ) & = \\sin ( \\theta \\pi + 2 \\theta \\pi ) = \\sin ( \\theta \\pi ) \\cos ( 2 \\theta \\pi ) + \\sin ( 2 \\theta \\pi ) \\cos ( \\theta \\pi ) \\\\ & = \\sin ( \\theta \\pi ) ( \\cos ( 2 \\theta \\pi ) + 2 \\cos ^ 2 ( \\theta \\pi ) ) \\\\ & = \\sin ( \\theta \\pi ) ( 2 \\cos ( 2 \\theta \\pi ) + 1 ) \\end{align*}"}
+{"id": "1785.png", "formula": "\\begin{gather*} \\sum _ { n = 0 } ^ { \\infty } \\left | \\frac { 1 } { ( n + \\alpha ) ^ s } \\right | ^ 2 ( \\log { n } ) ^ 2 = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( \\log { n } ) ^ 2 } { ( n + \\alpha ) ^ { 2 \\sigma } } < \\infty \\end{gather*}"}
+{"id": "2496.png", "formula": "\\begin{align*} P _ { 1 , t _ 1 } : = P _ \\sigma \\big ( K _ { X _ 1 } + ( B _ 1 + t _ 1 N _ 1 ) + ( M _ 1 + t _ 1 P _ 1 ) \\big ) = ( 1 + t _ 1 ) P _ 1 . \\end{align*}"}
+{"id": "5849.png", "formula": "\\begin{align*} w _ 0 = ( s _ 2 s _ { 3 \\alpha _ 1 + \\alpha _ 2 } s _ 2 ) s _ 1 . \\end{align*}"}
+{"id": "3420.png", "formula": "\\begin{align*} t = f _ 2 z _ 0 ^ { b _ 0 } \\ldots z _ k ^ { b _ k } , \\end{align*}"}
+{"id": "5665.png", "formula": "\\begin{align*} r ( D _ { X ^ \\prime } ) = 2 r ( \\xi ) + ( 3 - c _ 1 ) r ( L ) = 2 r ( \\xi ) + ( 3 - c _ 1 ) \\alpha = ( 6 - b _ 0 - b _ 1 ) \\alpha + 2 b _ 0 e _ 0 + 2 b _ 1 e _ 1 \\ . \\end{align*}"}
+{"id": "6890.png", "formula": "\\begin{align*} ( r _ * \\Omega _ { \\widetilde { X } } ^ 1 ( \\mathrm { l o g } \\ , \\widetilde { B } ) ) ^ { * * } = \\Omega _ X ^ { [ 1 ] } ( \\mathrm { l o g } \\ , B ) . \\end{align*}"}
+{"id": "6167.png", "formula": "\\begin{align*} \\textrm { S G T } ( \\mathbf { k } ) = \\bigsqcup _ { T \\in \\textrm { A P } _ n } \\textrm { G T } ( T ( \\mathbf { k } ) ) . \\end{align*}"}
+{"id": "5762.png", "formula": "\\begin{align*} \\mathbb E \\langle \\sigma _ X \\rangle _ \\beta = \\mathbb E \\langle \\sigma _ X \\rangle _ \\beta \\langle \\sigma _ X \\rangle _ { \\beta _ { \\rm N } } , \\end{align*}"}
+{"id": "7222.png", "formula": "\\begin{align*} \\mathcal H ^ h ( \\partial ^ * E \\cap \\partial ^ * F \\cap \\partial ^ * ( E \\cup F ) ) = 0 \\end{align*}"}
+{"id": "7991.png", "formula": "\\begin{align*} L _ { p } ( \\Omega _ { i } ) & = \\left ( \\bigcup _ { q \\in \\Gamma \\cap ( 0 , p ) } L _ { q } ( \\Omega _ { i } ) \\cap L _ { p } ( \\Omega _ { i } ) \\right ) + \\left ( \\bigcup _ { r \\in \\Gamma \\cap ( p , \\infty ] } L _ { r } ( \\Omega _ { i } ) \\cap L _ { p } ( \\Omega _ { i } ) \\right ) \\\\ & = \\operatorname { s p a n } \\left \\{ \\bigcup _ { q \\in \\Gamma } L _ { q } ( \\Omega _ { i } ) \\cap L _ { p } ( \\Omega _ { i } ) \\right \\} \\end{align*}"}
+{"id": "4822.png", "formula": "\\begin{align*} \\check { R } _ i ( u ) \\check { R } _ j ( v ) = \\check { R } _ j ( v ) \\check { R } _ i ( u ) \\ ; \\textrm { f o r } | i - j | \\geq 2 \\\\ \\check { R } _ i ( u ) \\check { R } _ { i + 1 } ( u + v ) \\check { R } _ i ( v ) = \\check { R } _ { i + 1 } ( v ) \\check { R } _ { i } ( u + v ) \\check { R } _ { i + 1 } ( u ) \\end{align*}"}
+{"id": "6700.png", "formula": "\\begin{align*} [ \\psi _ m , \\psi _ n ] _ + = [ \\psi ^ \\ast _ m , \\psi ^ \\ast _ n ] _ + = 0 , [ \\psi _ m , \\psi ^ \\ast _ n ] _ + = \\delta _ { m , n } , \\end{align*}"}
+{"id": "1876.png", "formula": "\\begin{gather*} U _ n = \\{ s \\in \\mathbb { C } \\mid | s - ( \\sigma _ 0 + i \\tau _ n ) | < r \\} \\end{gather*}"}
+{"id": "1603.png", "formula": "\\begin{align*} \\hat { P } ^ { ( i ) } _ J : = \\bigcup \\big \\{ P ^ { ( i ) } _ I : I \\in [ J ] ^ { i } \\big \\} . \\end{align*}"}
+{"id": "6784.png", "formula": "\\begin{align*} \\nu _ 1 ^ \\pm = \\frac { 1 } { 2 } \\left ( c \\pm \\sqrt { c ^ 2 + 4 \\tau } \\right ) , \\nu _ 2 ^ \\pm = \\frac { 1 } { 2 d } \\left ( c \\pm \\sqrt { c ^ 2 + 4 d \\tau } \\right ) . \\end{align*}"}
+{"id": "6558.png", "formula": "\\begin{align*} | \\sigma _ l - z _ l | = O ( ( \\varepsilon + \\delta ) ^ { \\frac 1 2 } ) . \\end{align*}"}
+{"id": "841.png", "formula": "\\begin{align*} T f = x ^ * ( f ) h , \\end{align*}"}
+{"id": "7967.png", "formula": "\\begin{align*} \\sigma ( u _ { 1 , 0 } ) = v _ { r , 1 } ~ ~ \\sigma ( v _ { 1 , 1 } ) = u _ { r , 0 } . \\end{align*}"}
+{"id": "7720.png", "formula": "\\begin{align*} \\mathbb { A } _ { E _ { q u o t } } ^ \\nabla : = \\mathbb { A } _ { E _ { q u o t } } / \\nabla ( F ) \\end{align*}"}
+{"id": "913.png", "formula": "\\begin{align*} \\xi _ 1 ( x , t ) = \\frac { 1 } { 2 } \\bigg ( \\tau _ t - \\frac { 1 - \\alpha } { t } \\tau \\bigg ) x + \\sigma _ 1 , \\end{align*}"}
+{"id": "938.png", "formula": "\\begin{align*} \\tilde { x } = X ( x , t , u , v ) , ~ ~ \\tilde { t } = Y ( x , t , u , v ) , ~ ~ \\tilde { u } = R ( x , t , u , v ) , ~ ~ \\tilde { v } = S ( x , t , u , v ) , \\end{align*}"}
+{"id": "890.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathcal { T } _ t ^ \\alpha u = x ^ m u _ { x x } + a ' v _ x , \\\\ & \\mathcal { T } _ t ^ \\alpha v = x ^ m v _ { x x } + b ' u _ x , \\end{aligned} \\right . \\end{align*}"}
+{"id": "6601.png", "formula": "\\begin{align*} b ( t , x , v ) = \\big ( F ( t , x , v ) , E ( t , x ) \\big ) \\ t \\in [ 0 , T ] , \\ x , v \\in \\R ^ d , \\end{align*}"}
+{"id": "347.png", "formula": "\\begin{align*} \\gamma _ { \\operatorname * { D } , j } \\left ( s \\right ) \\mathsf { E } _ { j } \\left ( s \\right ) \\varphi = \\varphi \\quad \\left \\Vert \\mathsf { E } _ { j } \\left ( s \\right ) \\varphi \\right \\Vert _ { H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) ; s } \\leq C \\left \\Vert \\varphi \\right \\Vert _ { H ^ { 1 / 2 } \\left ( \\Gamma _ { j } \\right ) } . \\end{align*}"}
+{"id": "6721.png", "formula": "\\begin{align*} \\begin{aligned} & 4 \\pi ( 1 + 2 a ) - 2 ^ { 4 a } ( I _ { a } ( 1 ) ) ^ { 2 } \\int _ { \\Sigma } | \\nabla u | ^ 2 \\le 8 \\pi a \\frac { \\mathfrak { m } _ { A D M } } { m } . \\end{aligned} \\end{align*}"}
+{"id": "1686.png", "formula": "\\begin{align*} \\sum _ { \\mu \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { b } } } P _ { \\texttt { b } ; \\mu } \\bigl ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } ; q , q _ 0 \\bigr ) \\overline { P _ { \\texttt { b } ; \\mu } \\bigl ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { b } ; \\kappa } ; q , q _ 0 \\bigr ) } \\delta ^ { ( m , n ) } _ { \\texttt { b } ; \\mu } ( q ) = 0 \\ \\kappa \\neq \\lambda \\end{align*}"}
+{"id": "8670.png", "formula": "\\begin{align*} \\hat { G } ( y _ 0 , . . . , y _ { n + 1 } ) = \\delta \\Big ( \\frac { y _ 0 } { y _ { n + 1 } } + \\hat { f } \\Big ( \\frac { y _ 1 } { y _ { n + 1 } } , . . . , \\frac { y _ n } { y _ { n + 1 } } \\Big ) \\Big ) \\hat { g } \\Big ( \\frac { y _ 1 } { y _ { n + 1 } } , . . . , \\frac { y _ n } { y _ { n + 1 } } \\Big ) y _ { n + 1 } ^ { - \\frac { n + 2 } { 2 } } \\end{align*}"}
+{"id": "2705.png", "formula": "\\begin{align*} \\begin{array} { r c c c l } ( x _ 1 , { \\dots } , x _ m ) \\times ( y _ 1 , { \\dots } , y _ n ) & : = & ( x _ 1 , { \\dots } , x _ m , y _ 1 , { \\dots } , y _ n ) & \\in & M ^ { \\times ( m + n ) } \\end{array} \\end{align*}"}
+{"id": "3869.png", "formula": "\\begin{align*} H = - \\sin \\theta \\lambda e ^ { - \\lambda _ 2 z } - \\sin \\theta \\mu \\lambda _ 2 y e ^ { - \\lambda _ 2 z } + \\cos \\theta \\mu . \\end{align*}"}
+{"id": "6777.png", "formula": "\\begin{align*} 0 \\geq \\phi '' _ \\infty ( 0 ) = - F ( 1 , \\psi _ \\infty ( 0 ) ) > 0 , \\end{align*}"}
+{"id": "2553.png", "formula": "\\begin{align*} \\xi & = ( K \\ast | u | ^ { p _ { r ; s } ^ { \\uparrow * } } ) | u | ^ { p _ { r ; s } ^ { \\uparrow * } } + \\sum _ { i \\in \\mathcal { I } } \\xi _ i \\delta _ { x _ i } ; \\\\ \\mu & \\geq \\int _ { \\mathbb { R } ^ N } \\frac { | u ( x ) - u ( y ) | ^ p } { | x - y | ^ { N + p s } } d y + \\sum _ { i \\in \\mathcal { I } } \\mu _ i \\delta _ { x _ i } ; \\\\ \\nu & = | u | ^ { p _ s ^ * } + \\sum _ { i \\in \\mathcal { I } } \\nu _ i \\delta _ { x _ i } ; \\end{align*}"}
+{"id": "7376.png", "formula": "\\begin{align*} & { } ^ { g ' } ( h \\otimes g ) = [ g ' , h ] \\otimes g + h \\otimes [ g ' , g ] , { } ^ g \\{ h \\} = \\{ [ g , h ] \\} \\\\ ( & { } ^ { g ' } ( h \\wedge g ) = [ g ' , h ] \\wedge g + h \\otimes [ g ' , g ] , { } ^ g \\{ h \\} = \\{ [ g , h ] \\} ) . \\end{align*}"}
+{"id": "8507.png", "formula": "\\begin{align*} \\partial ^ { * } E \\cap \\{ z < \\bar { z } \\} = \\partial ^ { * } F _ { \\ell } \\cap \\{ z < \\bar { z } \\} . \\end{align*}"}
+{"id": "6344.png", "formula": "\\begin{align*} v ^ h = \\frac { d } { d h } u ^ h , \\end{align*}"}
+{"id": "7486.png", "formula": "\\begin{align*} \\det ( A _ L ) = \\mathrm { N } _ { q ^ m / q } ( \\alpha _ 1 / \\alpha _ 2 ) ^ 2 u ^ { \\sigma ^ { - 1 } } ( G _ { m - 1 } ^ { \\sigma + 1 } - G _ m G ^ \\sigma _ { m - 2 } ) = \\mathrm { N } _ { q ^ m / q } ( \\alpha _ 0 \\alpha _ 1 / \\alpha _ 2 ) \\end{align*}"}
+{"id": "8220.png", "formula": "\\begin{align*} d ^ c _ { I _ j } K _ j ( T ) = 2 x _ j . \\end{align*}"}
+{"id": "2251.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n } \\left [ e ^ { t \\Delta } \\right ] ( x _ k ) & = \\left \\langle e ^ { t \\Delta } f , \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n } \\delta _ { x _ k } \\right \\rangle = \\left \\langle f , \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n } e ^ { t \\Delta } \\delta _ { x _ k } \\right \\rangle . \\end{align*}"}
+{"id": "2101.png", "formula": "\\begin{align*} \\alpha ( t , x ) = \\frac { 1 } { 2 } \\left ( 2 + \\tilde \\alpha _ 0 ( 2 u ) + \\tilde \\alpha _ 0 ( - 2 \\underline { u } ) + \\int _ 0 ^ { 2 u } \\alpha _ 1 ( s ) d s - \\int _ { 0 } ^ { - 2 \\underline { u } } \\alpha _ 1 ( s ) d s \\right ) , \\end{align*}"}
+{"id": "1329.png", "formula": "\\begin{align*} \\Lambda _ X ^ { w _ 1 } ( u ) & = w _ 1 ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) \\\\ & \\ge w _ 2 ( G ^ { - 1 } ( u ) ) g ( G ^ { - 1 } ( u ) ) \\\\ = & \\Lambda _ Y ^ { w _ 2 } ( u ) . \\end{align*}"}
+{"id": "2692.png", "formula": "\\begin{align*} \\alpha _ 1 = a _ 0 + a _ 2 , \\alpha _ 2 = a _ 1 + a _ 3 , & \\beta _ 1 = b _ 0 + b _ 2 , \\beta _ 2 = b _ 1 + b _ 3 , \\\\ \\gamma _ 1 = c _ 0 + c _ 2 , \\gamma _ 2 = c _ 1 + c _ 3 , & \\delta _ 1 = d _ 0 + d _ 2 , \\delta _ 2 = d _ 1 + d _ 3 . \\end{align*}"}
+{"id": "6372.png", "formula": "\\begin{align*} 1 ) \\ n - o - r - s + \\lambda a = \\lambda a + n + o + r - s ; \\ \\ 2 ) \\ o - n - s - r + \\lambda a = \\lambda a + n - o + r + s \\end{align*}"}
+{"id": "7339.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} J \\begin{pmatrix} u ' _ 1 \\\\ u ' _ 3 \\end{pmatrix} + \\begin{pmatrix} a _ \\lambda ( t ) & c _ \\lambda ( t ) \\\\ c _ \\lambda ( t ) & e _ \\lambda ( t ) \\end{pmatrix} \\begin{pmatrix} u _ 1 \\\\ u _ 3 \\end{pmatrix} & = 0 , t \\in \\mathbb { R } \\\\ \\lim _ { t \\rightarrow \\pm \\infty } u ( t ) & = 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "3827.png", "formula": "\\begin{align*} ( \\overline F ) ^ * ( \\phi ) = ( \\overline R ) ^ * ( \\phi ) = \\phi , \\end{align*}"}
+{"id": "1143.png", "formula": "\\begin{align*} d _ { ( m _ 1 , m _ 2 ) } : = [ ( m _ 1 , m _ 2 ) , ~ ] : C ^ n _ { c o m } ( \\mathfrak { g } , \\mathfrak { g } ) \\rightarrow C ^ { n + 1 } _ { c o m } ( \\mathfrak { g } , \\mathfrak { g } ) , n \\geq 1 \\end{align*}"}
+{"id": "4275.png", "formula": "\\begin{align*} \\pi ( i ) = j , \\ \\ \\mbox { i f $ i \\in B _ j $ } . \\end{align*}"}
+{"id": "3965.png", "formula": "\\begin{align*} G = \\Big \\{ 1 + \\sum _ { i < j } a _ { i j } E _ { i j } \\ , \\Big | \\ , a _ { i j } = 0 \\textrm { i f } j < \\lceil n / 2 \\rceil \\textrm { o r } i > \\lceil n / 2 \\rceil \\Big \\} \\leq G L ( n , p ) . \\end{align*}"}
+{"id": "3078.png", "formula": "\\begin{align*} { \\mathbb E } \\left \\{ R _ { \\rm P L } \\right \\} \\le \\sum \\limits _ { k = 1 } ^ { { N _ { \\rm { R } } } } { { { \\log } _ 2 } \\left ( { 1 + \\frac { { \\mathbb E } { \\left \\{ { { { \\left | { \\xi _ k ^ \\star } \\right | } ^ 2 } } { p _ k ^ \\star } \\right \\} } } { \\sigma ^ 2 } } \\right ) } \\buildrel \\Delta \\over = { R _ { { \\rm { P L , u p p e r } } } } . \\end{align*}"}
+{"id": "8319.png", "formula": "\\begin{align*} \\beta - a _ { d - 1 } = a _ { d - 2 } \\beta ^ { - 1 } + \\cdots + a _ { 1 } \\beta ^ { - d + 2 } + a _ { 0 } \\beta ^ { - d + 1 } = \\lambda ( 0 , \\cdots , 0 , 1 ) . \\end{align*}"}
+{"id": "5618.png", "formula": "\\begin{align*} I _ 3 & \\leq \\sum _ { j = 2 } ^ \\infty \\dfrac { ( \\mu C ^ { p ' } ) ^ { p - 1 + j } } { \\Gamma ( p + j ) } \\int _ R ^ \\infty r ^ { \\alpha _ 0 - \\frac { ( \\alpha _ 0 + 1 ) ( p - 1 + j ) } { p } } \\mathrm d r \\\\ & \\leq \\dfrac { p R ^ { \\alpha _ 0 + 1 } } { \\alpha _ 0 + 1 } \\exp _ { p + 2 } ( \\mu C ^ { p ' } R ^ { - \\frac { \\alpha _ 0 + 1 } { p } } ) \\overset { R \\to \\infty } \\longrightarrow 0 . \\end{align*}"}
+{"id": "896.png", "formula": "\\begin{align*} \\frac { c } { x } \\bigg ( - \\tau _ t + \\frac { 1 - \\alpha } { t } \\tau \\bigg ) + n x ^ k \\eta _ v - t ^ { 1 - \\alpha } \\xi _ t - ( 2 \\eta _ { x u } - \\xi _ { x x } ) + \\frac { c } { x ^ 2 } \\xi + \\frac { c } { x } \\xi _ x - m x ^ k \\phi _ u = 0 , \\end{align*}"}
+{"id": "1765.png", "formula": "\\begin{gather*} K _ \\nu = \\left \\{ s \\in \\mathbb { C } ~ \\middle | ~ \\frac { 1 } { 2 } + \\frac { 1 } { 5 \\nu } \\leq \\sigma \\leq 1 - \\frac { 1 } { 5 \\nu } , ~ - \\nu \\leq t \\leq \\nu \\right \\} \\end{gather*}"}
+{"id": "118.png", "formula": "\\begin{align*} A _ 0 : = \\frac { \\widehat { g } ( 0 ) } { 2 \\lvert \\Lambda \\rvert } \\sum _ { k \\in \\Lambda ^ * } ( b _ k ^ { \\dagger } b _ k + b _ { - k } ^ { \\dagger } b _ { - k } ) , \\end{align*}"}
+{"id": "6833.png", "formula": "\\begin{align*} P _ 6 ( { g } ) \\phi = \\mathrm { u } ^ { - \\frac { n + 6 } { n - 6 } } P _ 6 ( { g _ 0 } ) ( \\mathrm { u } \\phi ) \\mbox { f o r a l l } \\phi \\in \\mathcal { C } ^ { \\infty } ( \\mathbb { S } ^ n \\setminus \\Lambda ) . \\end{align*}"}
+{"id": "6560.png", "formula": "\\begin{align*} \\inf _ { \\xi = \\pm 1 , ( k , n ) \\in \\Lambda \\setminus B } | \\xi ( \\sigma + k \\cdot \\omega ) + \\mu _ n | \\geq \\frac { ( \\varepsilon + \\delta ) ^ { \\frac { 1 } { 8 b } } } { 5 } . \\end{align*}"}
+{"id": "8444.png", "formula": "\\begin{align*} \\iota _ { K Z B } \\circ j ^ { d R } ( X , b ) ( x ) = \\int _ { b } ^ { x } \\alpha A + \\sum _ { r , s \\geq 0 } g _ { r , s } \\sigma _ { r , s } - \\int _ { b } ^ { x } \\beta S \\left ( \\int _ { b } ^ { x } \\alpha \\mathrm { a d } _ { A } , \\int _ { b } ^ { x } \\beta \\mathrm { a d } _ { B } \\right ) \\left ( - \\int _ { b } ^ { x } \\alpha \\sigma _ { 0 , 0 } + \\sum _ { r , s \\geq 0 } g _ { r , s } \\sigma _ { r + 1 , s } \\right ) \\end{align*}"}
+{"id": "2364.png", "formula": "\\begin{align*} \\sum _ { \\mu \\in G \\times \\widehat { G } } c _ \\mu \\pi ( \\mu ) & = \\pi ( \\lambda ) ^ { - 1 } S _ { f , \\tau , \\Lambda } \\pi ( \\lambda ) = \\pi ( \\lambda ) ^ { - 1 } \\left ( \\sum _ { \\mu \\in G \\times \\widehat { G } } c _ \\mu \\pi ( \\mu ) \\right ) \\pi ( \\lambda ) \\\\ & = \\sum _ { \\mu \\in G \\times \\widehat { G } } c _ \\mu \\pi ( \\lambda ) ^ { - 1 } \\pi ( \\mu ) \\pi ( \\lambda ) = \\sum _ { \\mu \\in G \\times \\widehat { G } } c _ \\mu d _ { \\mu , \\lambda } \\pi ( \\mu ) . \\end{align*}"}
+{"id": "2610.png", "formula": "\\begin{align*} F _ 0 ( x ) = \\mu \\int _ 0 ^ x ( 1 - F ( y ) ) \\ , d y \\ , . \\end{align*}"}
+{"id": "2633.png", "formula": "\\begin{align*} \\hat G _ n ( \\alpha , t ) = ( e ^ { \\alpha / ( b _ n \\sqrt { n } ) } - 1 - \\frac { \\alpha } { b _ n \\sqrt { n } } ) \\sum _ { i = 1 } ^ { { { \\hat A } _ n } ( t ) } \\int _ 0 ^ { t - \\hat \\tau _ { n , i } } \\frac { d F ( u ) } { 1 - F ( u ) } \\ , . \\end{align*}"}
+{"id": "6455.png", "formula": "\\begin{align*} \\mu ^ { * } _ p : = \\frac { M ^ { * } _ p } { M ^ { * } _ { p - 1 } } = \\frac { m _ { p - 1 } } { m _ p } = \\frac { p ! M _ { p - 1 } } { ( p - 1 ) ! M _ p } = \\frac { p } { \\mu _ p } , \\end{align*}"}
+{"id": "2371.png", "formula": "\\begin{align*} S _ { f , \\tau , \\Lambda } x = \\frac { o ( \\Lambda ) } { o ( G ) } \\sum _ { \\mu \\in \\Lambda ^ 0 } f ( \\pi ( \\mu ) ^ { - 1 } \\tau ) \\pi ( \\mu ) x = \\frac { o ( \\Lambda ) } { o ( G ) } f ( \\tau ) x + \\frac { o ( \\Lambda ) } { o ( G ) } \\sum _ { \\mu \\in \\Lambda ^ 0 \\setminus \\{ ( 0 , 0 ) \\} } f ( \\pi ( \\mu ) ^ { - 1 } \\tau ) \\pi ( \\mu ) x = x , \\forall x \\in \\mathbb { C } ^ { o ( G ) } . \\end{align*}"}
+{"id": "6661.png", "formula": "\\begin{align*} { \\rho } _ { ( 1 ) , \\infty } ^ { ( \\rm c J ) } ( x ; \\beta , p , q ) | _ { \\beta = 2 } - 1 \\mathop { \\sim } \\limits _ { x \\to \\pm \\infty } \\sum _ { n = 1 } ^ \\infty { d _ { n } \\over x ^ { n } } , \\end{align*}"}
+{"id": "5930.png", "formula": "\\begin{align*} A _ { 2 } \\le R _ { 3 } - S _ { 2 } + d [ - a _ { 1 , 3 } b _ { 1 } ] \\le R _ { 3 } - S _ { 2 } + \\beta _ { 1 } = 0 - ( 1 - d ( c ) ) + 1 = d ( c ) = d [ a _ { 1 , 2 } b _ { 1 , 2 } ] . \\end{align*}"}
+{"id": "6946.png", "formula": "\\begin{align*} C _ L ^ { a s } ( d ) : = \\frac { ( d ^ 2 - 2 ) ^ 2 } { 4 } \\Bigg ( 1 + \\frac { c _ d } { C _ P ^ { a s } ( d ) } \\Bigg ) ^ { - 1 } , \\end{align*}"}
+{"id": "8622.png", "formula": "\\begin{align*} \\mu \\beta \\mathcal { L } _ 2 ^ { \\mu } [ \\beta b ] = \\mu \\beta R . \\end{align*}"}
+{"id": "1992.png", "formula": "\\begin{align*} \\left \\lVert { f } \\right \\rVert _ { \\mathrm { F } ( X , Y ) } : = \\max \\left ( \\underset { t \\in \\mathbb { R } } { \\mathrm { s u p } } \\left \\lVert { f ( i t ) } \\right \\rVert _ { X } , \\underset { t \\in \\mathbb { R } } { \\mathrm { s u p } } \\left \\lVert { f ( 1 + i t ) } \\right \\rVert _ { Y } \\right ) \\end{align*}"}
+{"id": "807.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } ( K \\ast | u _ n | ^ { p ^ \\sharp } ) | u _ n | ^ { p ^ \\sharp } d x = \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } ( K \\ast | u _ n | ^ { p ^ \\flat } ) | u _ n | ^ { p ^ \\sharp } d x = 0 . \\end{align*}"}
+{"id": "6407.png", "formula": "\\begin{align*} t _ { 1 2 } t _ { 2 3 } + c _ { 1 2 } t _ { 2 3 } c ^ { - 1 } _ { 1 2 } t _ { 2 3 } & = t _ { 2 3 } t _ { 1 2 } + t _ { 2 3 } c _ { 1 2 } t _ { 2 3 } c ^ { - 1 } _ { 1 2 } , \\quad [ t _ { 1 2 } + c _ { 1 2 } t _ { 2 3 } c ^ { - 1 } _ { 1 2 } , t _ { 2 3 } ] = 0 , \\\\ t _ { 2 3 } t _ { 1 2 } + c _ { 2 3 } t _ { 1 2 } c ^ { - 1 } _ { 2 3 } t _ { 1 2 } & = t _ { 1 2 } t _ { 2 3 } + t _ { 1 2 } c _ { 2 3 } t _ { 1 2 } c ^ { - 1 } _ { 2 3 } , \\quad [ t _ { 2 3 } + c _ { 2 3 } t _ { 1 2 } c ^ { - 1 } _ { 2 3 } , t _ { 1 2 } ] = 0 , \\end{align*}"}
+{"id": "192.png", "formula": "\\begin{align*} & Q ( M , P _ i , \\tau ) = \\left \\{ e ^ { \\frac { 1 } { 2 4 } E _ 2 ( \\tau ) A _ 1 } \\left ( \\prod _ { j = 1 } ^ { 5 } \\frac { x _ j \\theta ' ( 0 , \\tau ) } { \\theta ( x _ j , \\tau ) } \\right ) \\frac { \\sqrt { - 1 } \\theta ( u , \\tau ) } { \\theta _ 1 ( 0 , \\tau ) \\theta _ 2 ( 0 , \\tau ) \\theta _ 3 ( 0 , \\tau ) } \\right . \\\\ & \\left . \\frac { 1 } { 2 } \\left ( \\prod _ { l = 1 } ^ 8 \\theta _ 1 ( y _ l ^ i , \\tau ) + \\prod _ { l = 1 } ^ 8 \\theta _ 2 ( y _ l ^ i , \\tau ) + \\prod _ { l = 1 } ^ 8 \\theta _ 3 ( y _ l ^ i , \\tau ) \\right ) \\right \\} ^ { ( 1 0 ) } . \\end{align*}"}
+{"id": "5230.png", "formula": "\\begin{align*} \\Pr \\left [ \\sum _ { j = 1 } ^ t \\delta _ j ^ { ( i ) } > x \\right ] \\leq \\exp \\left ( - \\frac { x ^ 2 } { 3 2 D ^ 2 G ^ 2 \\sum _ { j = 1 } ^ t ( \\eta _ j ^ { ( i ) } ) ^ 2 } \\right ) . \\end{align*}"}
+{"id": "1103.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal J _ \\lambda ( u ) & : = \\frac { \\langle u , u \\rangle } { 2 \\lambda } - \\frac { 1 } { 2 \\lambda } \\int _ D { \\alpha ( x ) } | u ( x ) | ^ { 2 } \\ , d \\mu - \\int _ D F ( x , u ( x ) ) d \\mu , \\end{aligned} \\end{align*}"}
+{"id": "304.png", "formula": "\\begin{align*} \\left . \\left ( \\nabla _ { \\operatorname * { p w } ; j } w \\right ) \\right \\vert _ { \\Omega _ { j } ^ { \\sigma } } : = \\nabla \\left ( \\left . w \\right \\vert _ { \\Omega _ { j } ^ { \\sigma } } \\right ) \\ ; , \\sigma \\in \\left \\{ - , + \\right \\} \\end{align*}"}
+{"id": "8564.png", "formula": "\\begin{align*} \\mathcal { B } [ \\beta b ] \\nabla _ X \\psi = b \\nabla _ X ( \\nabla _ X \\cdot ( b \\nabla _ X \\psi ) ) + h _ b \\nabla _ X \\big { ( } b \\nabla _ X \\cdot ( b \\nabla _ X \\psi ) \\big { ) } + 2 h _ b ( \\nabla _ X b ) \\nabla _ X \\cdot ( b \\nabla _ X \\psi ) , \\end{align*}"}
+{"id": "4705.png", "formula": "\\begin{align*} f ( z _ 0 ) [ f ( x y z _ 0 ) - f ( x ) g ( y ) - f ( y ) g ( x ) ] = f ( x ) [ f ( y z _ 0 ^ 2 ) - f ( y ) g ( z _ 0 ) - f ( z _ 0 ) g ( y ) ] . \\end{align*}"}
+{"id": "3755.png", "formula": "\\begin{align*} u ( 0 ) = \\varphi , \\ u ' ( 0 ) = \\psi , \\ u '' ( 0 ) = \\xi , \\end{align*}"}
+{"id": "5772.png", "formula": "\\begin{align*} \\lim _ { L \\rightarrow \\infty } { \\mathbb E } \\langle ( m ^ p - { \\mathbb E } \\langle m ^ p \\rangle ) ^ 2 \\rangle = 0 , \\end{align*}"}
+{"id": "2063.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\lambda _ i - 2 \\lambda _ n & = \\sum _ { i = 1 } ^ { n - 1 } \\lambda _ i - \\lambda _ n \\\\ & = \\sum _ { i = 1 } ^ { n - 1 } \\sum _ { j = 1 } ^ n R _ { i j j i } - \\sum _ { i = 1 } ^ { n - 1 } R _ { i n n i } \\\\ & = \\sum _ { i , j = 1 } ^ { n - 1 } R _ { i j j i } . \\end{align*}"}
+{"id": "5661.png", "formula": "\\begin{align*} \\begin{array} { l } e _ 0 = \\{ x = x _ 0 = 0 \\} , \\\\ e _ 1 = \\{ x = x _ 1 = 0 \\} , \\\\ e ' _ 0 = \\{ x _ 0 = B x _ 3 + C x = 0 \\} , \\\\ e ' _ 1 = \\{ x _ 1 = B x _ 3 + C x = 0 \\} . \\end{array} \\end{align*}"}
+{"id": "8245.png", "formula": "\\begin{align*} \\Psi ( m , x ) = \\psi _ x ( m ) = q _ x \\psi _ { | x | , 0 , 0 } ( q _ x ^ { - 1 } m ) = q _ x e ^ { \\tau ( m , | x | ) } \\cdot ( q _ x ^ { - 1 } m ) . \\end{align*}"}
+{"id": "735.png", "formula": "\\begin{align*} \\norm { \\psi ( t ) } _ { L ^ 2 } ^ 2 = \\norm { \\psi _ 0 } _ { L ^ 2 } ^ 2 \\end{align*}"}
+{"id": "1293.png", "formula": "\\begin{align*} e ^ { l i n } _ { n , T } = & \\Delta [ \\chi _ n ( \\frac { x - x _ n } { \\lambda _ n } ) ] e ^ { i ( t + \\lambda _ n ^ 2 t _ n ) \\Delta } g _ n [ P _ n \\phi ] \\\\ & + 2 \\nabla [ \\chi _ n ( \\frac { x - x _ n } { \\lambda _ n } ) ] e ^ { i ( t + \\lambda _ n ^ 2 t _ n ) \\Delta } \\nabla g _ n [ P _ n \\phi ] \\end{align*}"}
+{"id": "8061.png", "formula": "\\begin{align*} { \\rm s u p p } \\psi \\subseteq \\{ x \\in \\mathbb R ^ { n } \\colon \\vert x \\vert \\leq 1 \\} ; \\ \\int { \\psi ( x ) } x ^ { \\alpha } d x = 0 , \\ { \\rm f o r \\ a l l } \\ \\vert \\alpha \\vert \\leq M , \\end{align*}"}
+{"id": "3393.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle \\sum _ { i = 1 } ^ 3 \\delta _ i v _ i ^ 2 = 1 , \\ \\sum _ { i = 1 } ^ 3 \\delta _ i v _ i V _ i = 0 , \\ \\sum _ { i = 1 } ^ 3 \\delta _ i V _ i ^ 2 = ( c - \\tilde { c } ) \\ \\mbox { a n d } \\ \\sum _ { i = 1 } ^ 3 { \\frac { V _ i } { v _ i } } = 0 \\end{array} \\end{align*}"}
+{"id": "7986.png", "formula": "\\begin{align*} \\mathfrak { d } _ 2 < \\mathfrak { d } _ 1 , \\ \\ \\mathfrak { s } _ 2 \\in \\begin{cases} \\{ 1 \\} , & \\mathfrak { d } _ 2 = 2 ; \\\\ [ 2 ^ { - \\frac { \\mathfrak { k } _ 1 } { \\mathfrak { d } _ 2 } } , \\frac { 1 } { 2 } ] & \\mathfrak { d } _ 2 > 2 , \\end{cases} \\ \\ 2 ^ { \\mathfrak { k } _ 2 } = 2 ^ { \\mathfrak { k } _ 1 } ( \\mathfrak { s } _ 2 ) ^ { \\mathfrak { d } _ 2 } . \\end{align*}"}
+{"id": "1521.png", "formula": "\\begin{align*} \\frac 1 x \\# \\left \\{ n \\le x : \\frac { \\log _ 2 P ^ { ( \\alpha ) } ( n ) - \\alpha \\log _ 2 x } { \\sqrt { \\log _ 2 x } } < t \\right \\} = \\Phi \\left ( \\frac { t } { \\sqrt { \\alpha ( 1 - \\alpha ) } } \\right ) + O _ \\alpha \\left ( \\frac { \\left ( \\log _ 3 x \\right ) ^ { 3 / 2 } } { \\sqrt { \\log _ 2 x } } \\right ) . \\end{align*}"}
+{"id": "8133.png", "formula": "\\begin{align*} d _ T ( \\gamma ( t ) , \\gamma ( t ' ) ) = d _ T ( \\psi ( 0 , 0 ) , \\psi ( 1 , 0 ) ) \\left | t - t ' \\right | . \\end{align*}"}
+{"id": "7206.png", "formula": "\\begin{align*} | I _ 2 ( D ^ { \\beta _ \\perp } \\sigma _ \\rho \\chi _ { \\cup _ { i = 0 } ^ { j - 1 } { \\bf C } _ i ^ \\rho } ) ( x ) | & \\le C \\rho ^ { - 2 - k _ \\perp } \\| \\sigma \\| _ { H ^ 1 ( \\Lambda ) } \\sum _ { i = j + 1 } ^ { J } ( 2 ^ i \\rho ) ^ { - 1 } ( 2 ^ i \\rho ) \\\\ & \\le C \\rho ^ { - 2 - k _ \\perp } \\| \\sigma \\| _ { H ^ 1 ( \\Lambda ) } J \\\\ & \\le C \\rho ^ { - 2 - k _ \\perp } | \\log ( \\rho ) | \\| \\sigma \\| _ { H ^ 1 ( \\Lambda ) } , \\end{align*}"}
+{"id": "1462.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\Delta u _ { s } + \\sum \\limits _ { t \\in J } k _ { s t } ^ { \\prime } e ^ { u _ t } = 4 \\pi \\alpha _ s \\delta _ 0 \\ \\ \\mathbb { R } ^ 2 , \\\\ & \\int _ { \\mathbb { R } ^ 2 } e ^ { u _ s ( y ) } \\mathrm { d } y < + \\infty , \\ \\forall \\ s \\in J . \\end{aligned} \\right . \\end{align*}"}
+{"id": "8946.png", "formula": "\\begin{align*} h ( \\Lambda ) & = h ( ( A , v ) \\Z ^ d ) \\\\ & = \\begin{pmatrix} A & v \\\\ 0 & 1 \\end{pmatrix} \\Z ^ { d + 1 } \\\\ & = \\{ ( x + r v , r ) \\in \\R ^ { d + 1 } : x \\in A \\Z ^ d , r \\in \\Z \\} \\\\ & = \\{ ( x + r v , r ) \\in \\R ^ { d + 1 } : x \\in \\Lambda ' , r \\in \\Z \\} . \\end{align*}"}
+{"id": "7052.png", "formula": "\\begin{align*} \\delta ( n , m ) = \\frac { 1 } { Q } \\sum _ { 1 \\leq q \\leq Q } \\frac { 1 } { q } \\ , \\ \\sideset { } { ^ \\star } \\sum _ { a \\bmod q } e \\left ( \\frac { ( n - m ) a } { q } \\right ) \\int _ { \\mathbb { R } } \\psi ( q , x ) e \\left ( \\frac { ( n - m ) x } { q Q } \\right ) d x , \\end{align*}"}
+{"id": "2117.png", "formula": "\\begin{align*} G _ 1 = \\dfrac { 1 } { 2 \\alpha ^ 2 } ( \\tilde { \\alpha } ' _ 0 ( 2 u ) + \\alpha _ 1 ( 2 u ) ) ( \\alpha _ 1 ( - 2 \\underline { u } ) - \\tilde { \\alpha } ' _ 0 ( - 2 \\underline { u } ) ) \\in S ( \\mathbb R ) . \\end{align*}"}
+{"id": "135.png", "formula": "\\begin{align*} \\Big ( X ^ T - F ^ T - m \\hat { H _ 1 } ^ T - m \\hat { H _ 2 } ^ T \\Big ) T _ W W ( t ) = 0 \\end{align*}"}
+{"id": "7901.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 \\varphi } { \\partial y _ i \\partial y _ j } = \\frac { - u } { u ^ { \\star } } \\left ( u _ { j k } - \\frac { u _ j x _ m u _ { m k } } { u ^ { \\star } } \\right ) \\left ( \\delta _ { k i } - x _ k \\frac { u _ i } { u ^ { \\star } } \\right ) \\end{align*}"}
+{"id": "279.png", "formula": "\\begin{align*} d ( x , z ) = \\log \\frac { 1 } { z ( x ) } \\end{align*}"}
+{"id": "5859.png", "formula": "\\begin{align*} \\frac { ( r ^ 6 - 1 ) ( r ^ 3 - r ) ( r ^ 6 - r ^ 3 ) } { ( r ^ 2 - 1 ) ( r ^ 2 - r ) ( r ^ 3 - r ^ 2 ) } = \\frac { r ( r ^ 6 - 1 ) ( r ^ 2 - 1 ) ( r ^ 3 - 1 ) } { ( r ^ 2 - 1 ) ( r - 1 ) ^ 2 } = \\frac { r ( r ^ 6 - 1 ) ( r ^ 2 + r + 1 ) } { ( r - 1 ) } . \\end{align*}"}
+{"id": "2616.png", "formula": "\\begin{align*} X _ n ( t ) & = ( 1 - F ( t ) ) X _ n ( 0 ) ^ + + X _ n ^ { ( 0 ) } ( t ) + \\int _ 0 ^ t X _ n ( t - s ) ^ + \\ , d F ( s ) + H _ n ( t ) + \\Theta _ n ( t ) \\ , . \\end{align*}"}
+{"id": "3625.png", "formula": "\\begin{align*} & { \\mathcal F } : = \\{ ( \\phi , D ) \\in { \\mathcal D } \\ : : \\ : { \\mathcal Y } ( \\phi , D ) \\not = \\emptyset \\} . \\end{align*}"}
+{"id": "4792.png", "formula": "\\begin{align*} \\Omega = \\Omega _ \\gamma = \\{ \\bar { z } = ( z _ 0 , z _ 1 , \\dots ) \\in \\Gamma ^ \\infty : \\gamma ( z _ { i + 1 } ) = z _ i \\mbox { f o r a l l } i \\geq 0 \\} . \\end{align*}"}
+{"id": "8059.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\lambda _ { j } a _ { j } \\right \\| _ { L _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\lambda _ { j } \\chi _ { Q _ { j } } } { \\omega ( Q _ { j } ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } . \\end{align*}"}
+{"id": "2611.png", "formula": "\\begin{align*} ( Q _ n ( t ) - n ) ^ + + { \\hat A } _ n ( t ) = ( Q _ n ( 0 ) - n ) ^ + + A _ n ( t ) \\ , . \\end{align*}"}
+{"id": "5980.png", "formula": "\\begin{align*} \\lambda _ j ( u _ j , u ^ \\psi ) _ { L ^ 2 ( M , e ^ \\phi d \\mu _ g ) } & = - \\int _ M \\Delta _ g ^ F u _ j ( x ) u ^ \\psi ( x ) e ^ { \\phi ( x ) } d \\mu _ g ( x ) \\\\ & = \\int _ M u _ j ( x ) \\Delta _ g ^ F u ^ \\psi ( x ) e ^ { \\phi ( x ) } d \\mu _ g ( x ) + \\langle u _ j e ^ \\phi \\arrowvert _ { \\partial M } , \\psi \\rangle . \\end{align*}"}
+{"id": "3464.png", "formula": "\\begin{align*} u ^ * ( p ) = \\max _ { x \\in \\Delta } \\langle x , p \\rangle - u ( x ) , p \\in \\Delta ^ \\vee , \\end{align*}"}
+{"id": "2207.png", "formula": "\\begin{align*} \\lambda _ 2 = \\lambda _ 3 = \\ldots = \\lambda _ { \\# L } = \\frac { q ^ { k } - q ^ { k - 1 } } { q - 1 } = q ^ { k - 1 } . \\end{align*}"}
+{"id": "746.png", "formula": "\\begin{align*} \\tau _ R ( \\omega ) = \\sup \\left \\{ t \\in [ 0 , \\tau ( \\omega ) ) \\colon f ( t , \\omega ) < R \\right \\} \\end{align*}"}
+{"id": "749.png", "formula": "\\begin{align*} \\norm { u } _ { X ^ { s , b } } = \\left ( \\int _ { \\R ^ d } \\norm { U ( t , \\xi ) } _ { H ^ b _ t ( \\R ) } ^ 2 \\ , d \\xi \\right ) ^ { 1 / 2 } , \\end{align*}"}
+{"id": "8586.png", "formula": "\\begin{align*} \\| \\nabla ^ { \\mu } _ { X , z } u \\| _ { H ^ { k , 0 } ( \\mathcal { S } _ b ) } \\leq M ( k + 1 ) ( | g | _ { H ^ k } + \\sum \\limits _ { j = 0 } ^ k \\| f \\| _ { H ^ { k - j , j } ( \\mathcal { S } _ b ) } ) . \\end{align*}"}
+{"id": "226.png", "formula": "\\begin{align*} g _ M ^ { ( c ) } ( g _ M ^ { ( c ) } - 1 ) = e ^ c ( g _ M ^ { ( c ) } > 0 ) , \\end{align*}"}
+{"id": "3979.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) u _ n = f ( { u } _ n ( t ) , \\mu _ t ) \\zeta ^ { \\epsilon _ n } _ t - c _ n f ( { u } _ n ( t ) , \\mu _ t ) f ( u _ n ( t ) , \\mu _ t ) ' + g ( u _ n ( t ) , \\mu _ t ) \\end{align*}"}
+{"id": "7470.png", "formula": "\\begin{align*} W _ { a , c } ( T , x ) = \\| x _ { 0 , 1 } x _ { 2 , 3 } \\ddot c \\| ^ { \\dot a } \\| x _ { 0 , 2 } x _ { 1 , 3 } \\| ^ { \\dot b } \\| x _ { 0 , 3 } x _ { 1 , 2 } / \\ddot a \\| ^ { \\dot c } \\delta _ { F } \\Big ( x _ { 0 , 1 } x _ { 2 , 3 } \\ddot c + \\frac { x _ { 0 , 3 } x _ { 1 , 2 } } { \\ddot a } - x _ { 0 , 2 } x _ { 1 , 3 } \\Big ) \\end{align*}"}
+{"id": "4164.png", "formula": "\\begin{align*} \\tilde y _ k = \\begin{cases} { \\sqrt { \\mu ( 1 + \\hat { \\rho } _ k ) } } / ( { \\hat { \\rho } _ { k } ^ { 2 } + 3 \\hat { \\rho } _ { k } + 1 + \\varepsilon } ) & \\\\ { \\sqrt { \\mu \\rho _ k } } / \\big ( { ( \\hat { \\rho } _ k + 1 ) ^ 2 + \\varepsilon } \\big ) & \\end{cases} \\end{align*}"}
+{"id": "765.png", "formula": "\\begin{align*} \\kappa = \\psi - \\chi \\end{align*}"}
+{"id": "4122.png", "formula": "\\begin{align*} \\sum _ { j = a } ^ { 2 s - a } \\binom { 2 s - 2 a } { j - a } \\cdot \\big ( \\big ) = 0 . \\end{align*}"}
+{"id": "1205.png", "formula": "\\begin{align*} \\| f \\| _ { L _ { t } ^ q L ^ r _ x ( I \\times \\R ^ 3 ) } = \\bigg ( \\int _ { I } \\| f ( t , x ) \\| _ { L ^ r _ x } ^ q d t \\bigg ) ^ \\frac { 1 } { q } \\end{align*}"}
+{"id": "7547.png", "formula": "\\begin{align*} P _ 0 \\times _ { \\mathbb { C } ^ * } R _ { \\chi } = \\omega _ { D , \\log } , \\mathrm { w h e r e } \\ , \\ , \\ , R _ { \\chi } : = \\chi \\circ R . \\end{align*}"}
+{"id": "2884.png", "formula": "\\begin{align*} C ( { { \\xi } } ) : = \\prod _ { \\alpha \\in R _ 0 ^ + } \\frac { 1 - t _ { \\alpha } e ^ { - i \\langle \\xi , \\alpha \\rangle } } { 1 - e ^ { - i \\langle \\xi , \\alpha \\rangle } } . \\end{align*}"}
+{"id": "3104.png", "formula": "\\begin{align*} \\mathbf { b } _ R ( \\psi _ 1 , \\theta _ 1 ) = \\frac { 1 } { \\sqrt { N _ 1 } } \\left [ e ^ { j 2 \\pi n _ 1 \\frac { d _ R } { \\lambda } \\cos ( \\theta _ 1 ) \\sin ( \\psi _ 1 ) } \\right ] ^ T _ { n _ 1 \\in \\mathcal { I } ( N _ 1 ) } \\otimes \\frac { 1 } { \\sqrt { N _ 2 } } \\left [ e ^ { j 2 \\pi n _ 2 \\frac { d _ R } { \\lambda } \\sin ( \\theta _ 1 ) } \\right ] ^ T _ { n _ 2 \\in \\mathcal { I } ( N _ 2 ) } \\end{align*}"}
+{"id": "715.png", "formula": "\\begin{align*} \\mathbf V ( t ) = \\mathbf S ( t - S ) \\mathbf U ( S ) + i \\int _ { S \\wedge \\tau _ R } ^ { t \\wedge \\tau _ R } \\mathbf S ( t - \\sigma ) \\mathbf N ( \\mathbf u ( s ) ) \\ , d s + i \\int _ { S } ^ t \\mathbf S ( t - s ) \\mathbf M ( \\mathbf V ( s ) ) \\ , d W ( s ) . \\end{align*}"}
+{"id": "1988.png", "formula": "\\begin{align*} \\left [ s < \\frac { n } { p } \\right ] \\left [ q = 1 s \\leqslant \\frac { n } { p } \\right ] \\end{align*}"}
+{"id": "8235.png", "formula": "\\begin{align*} f _ { x _ 1 , 0 , 0 } ( m ) = \\frac { 1 } { 2 } \\rho ^ 2 ( e ^ { \\tau ( m , x _ 1 ) } \\cdot m ) - 2 x _ 1 \\tau ( m , x _ 1 ) . \\end{align*}"}
+{"id": "2826.png", "formula": "\\begin{align*} \\ell i _ { 2 } ^ { ( p ) } = ( \\frac { 1 } { 2 } \\sum _ { 1 \\leq i < p } i \\cdot \\ell _ { p - i } \\wedge \\ell _ { i } ) \\circ \\delta . \\end{align*}"}
+{"id": "1764.png", "formula": "\\begin{gather*} \\mathcal { A } _ \\rho ( c ) = \\{ \\alpha \\in \\mathcal { A } \\mid | \\alpha - c | \\leq \\rho \\} , \\end{gather*}"}
+{"id": "1840.png", "formula": "\\begin{gather*} \\sup _ { \\alpha \\in \\mathcal { A } _ \\rho ( c ) } \\left \\| \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + \\alpha ) ^ s } - \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + c ) ^ s } \\right \\| < \\frac { \\epsilon } { 3 } , \\end{gather*}"}
+{"id": "2803.png", "formula": "\\begin{align*} b _ { N , s , \\theta } & : = c _ { N , s } \\int _ { 0 } ^ 1 r ^ { 2 s - 1 } ( 1 - r ^ \\theta ) ( 1 - r ^ { N - 2 s - \\theta } ) \\psi _ N ( r ) \\dd r , \\end{align*}"}
+{"id": "2968.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ M \\sum _ { j = 1 } ^ N \\Theta _ { x _ i \\otimes y _ j , x _ i \\otimes y _ j } = \\psi _ N ^ * \\circ T _ M \\circ \\psi _ N \\end{align*}"}
+{"id": "217.png", "formula": "\\begin{align*} & { Q _ 1 } ( M , P _ i , \\tau ) = 2 ^ 6 [ h _ 0 ' ( 8 \\delta _ 1 ) ^ { 5 } + h _ 1 ' ( 8 \\delta _ 1 ) ^ { 3 } \\varepsilon _ 1 + h _ 2 ' ( 8 \\delta _ 1 ) \\varepsilon _ 1 ^ 2 ] , \\end{align*}"}
+{"id": "2487.png", "formula": "\\begin{align*} H _ \\chi ( G ) & \\leq H ( c ^ * ( V ) ) \\\\ & = h \\Big ( \\sum _ { j \\leq N } P _ { i _ V } ( j ) P _ { c ^ * ( V _ j ) } \\Big ) \\\\ & = h \\Big ( \\sum _ { j \\leq N } P _ { i _ V } ( j ) P _ { c _ 1 ^ * ( V _ 1 ) } \\Big ) \\\\ & = H ( c ^ * _ 1 ( V _ 1 ) ) \\\\ & = H _ \\chi ( \\tilde { G } _ 1 ) , \\end{align*}"}
+{"id": "8402.png", "formula": "\\begin{align*} W _ { z } ( v ) = \\sum _ { P < p \\leq 2 P } ( \\log p ) \\theta _ { z } ( p ^ c ) e \\left ( v \\left ( N + j - \\left [ p ^ { c } \\right ] \\right ) ^ { \\gamma } \\right ) . \\end{align*}"}
+{"id": "1380.png", "formula": "\\begin{align*} g = g _ { \\lambda } : = \\sum _ { r = 1 } ^ { \\infty } \\lambda _ r R _ { 2 - 2 k } ^ { \\ell - 1 } \\left ( \\mathcal { F } _ { 2 - 2 k , - r } \\right ) \\in R _ { 2 - 2 k } ^ { \\ell - 1 } \\left ( M _ { 2 - 2 k } ^ ! \\right ) , \\end{align*}"}
+{"id": "4627.png", "formula": "\\begin{align*} \\mathcal { S } ( v ) = \\mathcal { S } ( v | s ) \\cup \\mathcal { S } ( v | t ) \\end{align*}"}
+{"id": "1.png", "formula": "\\begin{align*} w ( ( x , y ) ) = \\begin{cases} \\frac 1 2 & \\textrm { i f } x y = 0 ; \\\\ \\frac { 1 } { n - i } & \\textrm { i f } x + y = n - 1 + i \\textrm { f o r s o m e } i \\in [ t ] ; \\\\ 0 & \\textrm { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "5260.png", "formula": "\\begin{align*} & \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ n \\frac { ( G _ 1 + G _ 2 \\| q _ { i , t + 1 } \\| ) \\| \\epsilon ^ z _ { i , t } \\| } { \\gamma _ { t } } \\le \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ n \\Big ( \\frac { 2 \\gamma _ 0 ( G _ 1 ^ 2 + G _ 2 ^ 2 \\| q _ { i , t + 1 } \\| ^ 2 ) } { \\sigma \\gamma _ { t } ^ 2 } + \\frac { \\sigma \\| \\epsilon ^ z _ { i , t } \\| ^ 2 } { 4 \\gamma _ 0 } \\Big ) . \\end{align*}"}
+{"id": "4563.png", "formula": "\\begin{align*} \\mathcal { B } _ 0 \\partial _ t { \\mathbf V } _ { t } + \\mathcal { B } _ 1 \\partial _ 1 { \\mathbf V } _ { t } + \\mathcal { B } _ 2 \\partial _ 2 { \\mathbf V } _ { t } + ( \\partial _ t \\mathcal B _ 0 + \\mathcal { B } _ 3 ) { \\mathbf V } _ { t } = \\tilde { \\mathcal F } _ t \\ , , \\end{align*}"}
+{"id": "7348.png", "formula": "\\begin{align*} { \\rm M M S E } ( \\gamma ) = \\frac { 1 } { \\gamma } \\Big ( 1 - \\Phi ( \\sqrt { \\gamma } S + Z ) \\Big ) . \\end{align*}"}
+{"id": "5948.png", "formula": "\\begin{align*} G _ M ^ \\omega ( x , y ) = \\sum _ { j = 1 } ^ \\infty \\frac { u _ j ( x ) u _ j ( y ) } { \\lambda _ j - \\omega ^ 2 } e ^ { \\phi ( y ) } , \\end{align*}"}
+{"id": "49.png", "formula": "\\begin{align*} \\mathcal R _ 0 & = - \\frac { 1 } { 2 \\vert \\Lambda \\vert } \\big ( \\widehat g ( 0 ) + \\widehat { g \\omega } ( 0 ) \\big ) \\big ( 4 n _ { 0 } - 2 \\big ) - \\widehat { g \\omega } ( 0 ) \\frac { n _ { + } } { \\vert \\Lambda \\vert } \\\\ & - \\frac { 1 } { \\vert \\Lambda \\vert } \\sum _ { k \\in \\Lambda ^ * } \\Big ( ( \\widehat g ( k ) + \\widehat { g \\omega } ( k ) ) a _ k ^ \\dagger a _ k + \\frac 1 2 \\widehat g _ k ( a _ k a _ { - k } + h . c . ) \\Big ) , \\end{align*}"}
+{"id": "8051.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j } \\lambda _ { j } a _ { j } \\right \\| _ { h _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } \\end{align*}"}
+{"id": "6396.png", "formula": "\\begin{align*} S \\left ( a \\triangleright b \\right ) & = S \\left ( b \\right ) \\triangleleft S \\left ( a \\right ) ; \\\\ \\chi ( S \\left ( a _ { 1 } \\right ) \\otimes a _ { 2 } \\triangleright b ) & = - \\chi ( a \\otimes b ) . \\end{align*}"}
+{"id": "4812.png", "formula": "\\begin{align*} k = \\left \\lceil \\frac { \\log \\left ( \\frac { 8 C _ \\mu } { \\varepsilon } \\right ) } { ( \\alpha - \\Lambda - \\alpha ' ) \\log \\lambda _ 0 } \\right \\rceil \\end{align*}"}
+{"id": "750.png", "formula": "\\begin{align*} \\norm { u } _ { X ^ { s , b } } = \\left ( \\int _ { \\R ^ d } \\left ( \\norm { U ( t , \\xi ) } _ { L ^ 2 _ t ( \\R ) } ^ 2 + c _ b \\norm { U ( t , \\xi ) } _ { S ^ b _ t ( \\R ) } ^ 2 \\right ) \\ , d \\xi \\right ) ^ { 1 / 2 } , \\end{align*}"}
+{"id": "1639.png", "formula": "\\begin{align*} \\frac { f ' ( r ) } { f ( r ) } = \\frac { p } { 2 } \\coth ( r / 2 ) + q \\coth r = \\frac { p + 2 q } { 2 } \\coth ( r / 2 ) - \\frac { q } { \\sinh ( r ) } , \\end{align*}"}
+{"id": "1736.png", "formula": "\\begin{align*} \\int _ a ^ b f ( x ) ( x ) x = \\sum _ { 0 \\leq l \\leq m } f ( x ^ { ( m + 1 ) } _ l ) ^ { ( m + 1 ) } _ { l } \\end{align*}"}
+{"id": "8405.png", "formula": "\\begin{align*} W ( v ) & = \\sum _ { z = 1 } ^ { 2 Z - 1 } W _ { z } ( v ) + W _ { 0 } ( v ) \\\\ & = \\sum _ { z = 1 } ^ { 2 Z - 1 } V _ { z } ( v ) + \\O \\left ( \\frac { v N ^ { 2 \\gamma - 2 } } { d \\log ^ 7 N } \\sum _ { P < p \\leq 2 P } ( \\log p ) \\sum _ { z = 1 } ^ { 2 Z - 1 } \\theta _ { z } ( p ^ c ) \\right ) + \\O \\left ( \\frac { N ^ { 2 \\gamma - 1 } } { d \\log ^ 6 N } \\right ) \\end{align*}"}
+{"id": "4839.png", "formula": "\\begin{align*} B = \\lim _ { u \\to u _ 0 } \\check { R } ( u ) = \\sum _ { i = 1 } ^ { n } \\Lambda _ i ( u _ 0 ) P _ i = \\sum _ { i = 1 } ^ { n } \\lambda _ i P _ i \\end{align*}"}
+{"id": "640.png", "formula": "\\begin{align*} d X = A X d t + \\mathcal N ( X ) d t + \\mathcal M ( X ) \\circ d W , \\end{align*}"}
+{"id": "256.png", "formula": "\\begin{align*} \\lim _ { c \\to + \\infty } e ^ { - c \\langle \\xi , \\rho _ L \\rangle } \\Delta _ \\emph { U } ( \\xi ) \\phi _ \\xi ^ { \\emph { b c } } ( x + c \\rho _ L ; g ^ { ( c ) } ) = \\Delta _ \\emph { U } ( \\xi ) \\phi ^ { \\emph { c s } } _ \\xi ( x ; g ) . \\end{align*}"}
+{"id": "4462.png", "formula": "\\begin{align*} | | \\dot { { \\mathbf U } } | | ^ 2 _ { s , \\ast , T } & = | | { \\mathbf V } + J _ 0 { \\mathbf V } | | ^ 2 _ { s , \\ast , T } \\\\ & \\leq C ( K ) \\Big ( | | { \\mathbf V } | | ^ 2 _ { s , \\ast , T } + | | \\dot { { \\mathbf U } } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ T ) } | | \\hat { W } | | ^ 2 _ { s , \\ast , T } \\Big ) \\\\ & \\leq C ( K ) | | { \\mathbf V } | | ^ 2 _ { s , \\ast , T } + T C ( K ) | | \\dot { { \\mathbf U } } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ T ) } | | \\hat { W } | | ^ 2 _ { s + 1 , \\ast , T } , \\end{align*}"}
+{"id": "2418.png", "formula": "\\begin{align*} \\lim _ { j \\to + \\infty } \\frac { z _ j - 4 \\pi / 3 } { f ( z _ j ) - f ( 4 \\pi / 3 ) } = \\frac { 1 } { f ' ( 4 \\pi / 3 ) } = \\frac { 3 ^ { \\alpha - 1 } } { \\alpha ( 4 \\pi ) ^ { \\alpha - 1 } } . \\end{align*}"}
+{"id": "4991.png", "formula": "\\begin{align*} \\frac 1 2 \\bigl ( \\left ( A ( x ) ^ 2 + A ( x ^ 2 ) \\right ) & = \\frac 1 2 \\bigl ( ( x + x ^ 2 + x ^ 3 + x ^ 4 + x ^ 5 + x ^ 6 ) ^ 2 + ( x ^ 2 + x ^ 4 + x ^ 6 + x ^ 8 + x ^ { 1 0 } + x ^ { 1 2 } ) \\bigr ) \\\\ & = x ^ { 2 } + x ^ { 3 } + 2 x ^ { 4 } + 2 x ^ { 5 } + 3 x ^ { 6 } + 3 x ^ { 7 } + 3 x ^ { 8 } + 2 x ^ { 9 } + 2 x ^ { 1 0 } + x ^ { 1 1 } + x ^ { 1 2 } . \\end{align*}"}
+{"id": "6948.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } f ( x ) d x = d ! \\int _ { \\Omega } f ( x ) d x , \\end{align*}"}
+{"id": "4158.png", "formula": "\\begin{align*} \\mathcal N _ k ( \\tau ) = \\max \\left \\{ t ' _ { i k } : t ' _ { i k } \\le \\tau \\right \\} . \\end{align*}"}
+{"id": "7037.png", "formula": "\\begin{align*} \\mathop { \\inf } _ { \\begin{array} { c } X \\in \\mathcal S ^ K \\\\ A ( X ) = b \\\\ X \\succcurlyeq 0 \\\\ \\end{array} } \\langle C , X \\rangle , \\end{align*}"}
+{"id": "8770.png", "formula": "\\begin{align*} \\mathbf { E } _ { \\omega , T } \\big [ \\norm { z _ { T } - x ^ { * } _ { \\omega } } ^ { 2 } \\big ] & \\geq \\frac { 1 } { 4 } \\mathbf { E } _ { \\omega , T } \\big [ \\norm { x _ { \\omega } ^ { * } - x _ { \\hat { \\omega } } ^ { * } } ^ { 2 } \\big ] \\\\ & = \\alpha ^ { - 2 } r ^ { 2 } h ^ { 2 } \\mathbf { E } _ { \\omega , T } \\rho ( \\hat { \\omega } , \\omega ) , \\end{align*}"}
+{"id": "363.png", "formula": "\\begin{align*} \\phi _ \\ell ( h ) = k ' h \\end{align*}"}
+{"id": "838.png", "formula": "\\begin{align*} \\| f \\| _ 2 = \\Big ( \\sum \\limits _ { n = 0 } ^ { \\infty } | a _ n | ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } < \\infty . \\end{align*}"}
+{"id": "6259.png", "formula": "\\begin{align*} g _ { [ < \\lambda ] } : = P _ { < \\frac { \\lambda } { 4 } } g ( u _ { \\ll \\lambda } ) . \\end{align*}"}
+{"id": "170.png", "formula": "\\begin{align*} E _ 2 ( \\tau ) = 1 - 2 4 q - 7 2 q ^ 2 + O ( q ^ 3 ) . \\end{align*}"}
+{"id": "8000.png", "formula": "\\begin{align*} | K ( x ) | = | H ( \\pi _ { p _ 0 } ^ { - 1 } ( x ) ) - H ( p _ 0 ) | \\leq C | \\pi _ { p _ 0 } ^ { - 1 } ( x ) - p _ 0 | ^ 2 \\end{align*}"}
+{"id": "4206.png", "formula": "\\begin{align*} \\sigma ( ( x \\otimes x ) T + T ( x \\otimes x ) ) & = \\bigcap _ { \\alpha > 0 } \\sigma _ { \\frac { \\delta } { \\alpha } } ( ( x \\otimes x ) T + T ( x \\otimes x ) ) \\\\ & = \\bigcap _ { \\alpha > 0 } \\sigma _ { \\frac { \\delta } { \\alpha } } ( ( x \\otimes x ) \\psi ( T ) + \\psi ( T ) ( x \\otimes x ) ) \\\\ & = \\sigma ( ( x \\otimes x ) \\psi ( T ) + \\psi ( T ) ( x \\otimes x ) ) . \\end{align*}"}
+{"id": "2026.png", "formula": "\\begin{align*} \\dot { \\mathrm { B } } ^ { s _ 0 } _ { p _ 0 , q _ 0 } ( \\Omega ) \\cap \\dot { \\mathrm { B } } ^ { s _ 1 } _ { p _ 1 , q _ 1 } ( \\Omega ) = [ \\dot { \\mathrm { B } } ^ { s _ 0 } _ { p _ 0 , q _ 0 } \\cap \\dot { \\mathrm { B } } ^ { s _ 1 } _ { p _ 1 , q _ 1 } ] ( \\Omega ) \\end{align*}"}
+{"id": "4699.png", "formula": "\\begin{align*} f ( x y z _ { 0 } ) = f ( x ) f ( y ) , \\ , x , y \\in S , \\end{align*}"}
+{"id": "4496.png", "formula": "\\begin{align*} & | | \\mathbb { B } '' ( ( W _ 1 , \\psi _ 1 ) , ( W _ 2 , \\psi _ 2 ) ) | | _ { H ^ s ( \\Gamma _ T ) } \\\\ & \\leq C \\sum _ { i \\neq j } \\Big ( | | W _ i | | _ { H ^ s ( \\Gamma _ T ) } | | \\psi _ j | | _ { W ^ { 1 , \\infty } ( \\Gamma _ T ) } + | | W _ i | | _ { L ^ { \\infty } ( \\Gamma _ T ) } | | \\psi _ j | | _ { H ^ { s + 1 } ( \\Gamma _ T ) } \\\\ & \\quad \\quad + | | W _ i | | _ { H ^ s ( \\Gamma _ T ) } | | W _ j | | _ { L ^ { \\infty } ( \\Gamma _ T ) } \\Big ) . \\end{align*}"}
+{"id": "4331.png", "formula": "\\begin{align*} W ^ { ( 2 ) } _ \\omega ( { \\sf f } , { \\sf g } ) = \\omega ( B ( { \\sf f } ) B ( { \\sf g } ) ) \\ \\ ( { \\sf f } , { \\sf g } \\in \\mathcal { K } ) \\ , . \\end{align*}"}
+{"id": "2910.png", "formula": "\\begin{align*} ( T _ j f ) ( \\mu ) = \\begin{cases} t _ j f ( \\mu ) & \\ a _ j ( \\mu ) = 0 \\\\ f ( s _ j \\mu ) = f ( \\mu - \\alpha _ j ) & \\ a _ j ( \\mu ) = 1 \\\\ f ( \\mu - 2 \\alpha _ j ) - ( t _ j - 1 ) f ( \\mu - \\alpha _ j ) & \\ a _ j ( \\mu ) = 2 \\end{cases} . \\end{align*}"}
+{"id": "2479.png", "formula": "\\begin{align*} \\eta \\big ( \\beta P ' _ A + ( 1 - \\beta ) P '' _ A \\big ) = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } H _ \\chi \\left ( \\bigwedge _ { a \\in \\mathcal { A } } G _ a ^ { \\wedge n T _ { \\overline { a } ^ n } ( a ) } \\right ) , \\end{align*}"}
+{"id": "3792.png", "formula": "\\begin{align*} ( H _ p , \\alpha ) : = \\int _ { Y \\times Y } H _ p ( y _ 0 , y _ 1 ) d \\alpha ( y _ 0 , y _ 1 ) , H _ p ( y _ 0 , y _ 1 ) = H ( x _ 0 , s _ 0 ^ p , x _ 1 , s _ 1 ^ p ) . \\end{align*}"}
+{"id": "3773.png", "formula": "\\begin{align*} F ( s ) = s \\log s - s + 1 , \\Big ( \\implies F _ \\infty ' = + \\infty \\ ; \\Big ) \\end{align*}"}
+{"id": "5906.png", "formula": "\\begin{align*} \\varphi ( n x _ 2 ) = \\left \\{ \\begin{array} { c c c l } & 0 & & | x _ 2 | \\leq 1 / n \\\\ & 1 & & | x _ 2 | \\geq 2 / n . \\end{array} \\right . \\end{align*}"}
+{"id": "4396.png", "formula": "\\begin{align*} \\mathbb { B } ' _ e ( \\hat { { \\mathbf U } } , \\hat { \\varphi } ) ( \\dot { { \\mathbf U } } , \\varphi ) = \\left [ \\begin{array} { c } \\partial _ t \\varphi + \\hat { u } ^ + _ 2 \\partial _ 2 \\varphi - \\dot { u } ^ + _ N - \\varphi \\partial _ 1 \\hat { u } ^ + _ N \\\\ \\partial _ t \\varphi + \\hat { u } ^ - _ 2 \\partial _ 2 \\varphi - \\dot { u } ^ - _ N + \\varphi \\partial _ 1 \\hat { u } ^ - _ N \\\\ \\dot { q } ^ + - \\dot { q } ^ - + \\varphi ( \\partial _ 1 \\hat { q } ^ + + \\partial _ 1 \\hat { q } ^ - ) \\end{array} \\right ] , \\end{align*}"}
+{"id": "4240.png", "formula": "\\begin{align*} 2 \\ , F = i \\ , ( \\omega ^ { 1 \\bar 1 } + \\omega ^ { 2 \\bar 2 } + t ^ 2 \\omega ^ { 3 \\bar 3 } ) + u \\ , \\omega ^ { 1 \\bar 2 } - \\bar { u } \\ , \\omega ^ { 2 \\bar 1 } , \\end{align*}"}
+{"id": "4850.png", "formula": "\\begin{align*} & ( I + \\xi E _ i ) ( I + \\xi E _ { i + 1 } ) ( I + \\xi E _ i ) = ( I + \\xi E _ { i + 1 } ) ( I + \\xi E _ i ) ( I + \\xi E _ { i + 1 } ) \\\\ = & ( I + 2 \\xi E _ i + \\xi E _ { i + 1 } + \\xi ^ 2 E _ i ^ 2 + \\xi ^ 2 E _ i E _ { i + 1 } + \\xi ^ 2 E _ { i + 1 } E _ i + \\xi ^ 3 E _ i E _ { i + 1 } E _ i ) \\\\ = & ( I + \\xi E _ i + 2 \\xi E _ { i + 1 } + \\xi ^ 2 E _ { i + 1 } ^ 2 + \\xi ^ 2 E _ i E _ { i + 1 } + \\xi ^ 2 E _ { i + 1 } E _ i + \\xi ^ 3 E _ { i + 1 } E _ i E _ { i + 1 } ) \\end{align*}"}
+{"id": "7758.png", "formula": "\\begin{align*} \\theta _ 0 ^ P = - \\frac { 1 } { 2 } \\pi _ N ^ * \\mathrm { d } f , \\theta _ 3 ^ P : = \\eta + \\frac { 1 } { 2 } \\pi _ N ^ * \\iota _ V g _ N , \\theta _ 1 ^ P : = \\frac { 1 } { 2 } \\pi _ N ^ * \\iota _ V \\omega _ 2 , \\theta _ 2 ^ P : = - \\frac { 1 } { 2 } \\pi _ N ^ * \\iota _ V \\omega _ 1 \\ , , \\end{align*}"}
+{"id": "1944.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\partial _ t \\| u _ x \\| ^ 2 _ { L ^ 2 ( I ) } + \\epsilon \\ , \\| u _ { x x } \\| ^ 2 _ { L ^ 2 ( I ) } = - \\left ( \\rho V , u _ { x x } \\right ) + \\left ( \\rho u , u _ { x x } \\right ) + \\left ( u u _ x , u _ { x x } \\right ) . \\end{align*}"}
+{"id": "8290.png", "formula": "\\begin{align*} \\begin{cases} \\bar u _ { t t } = \\bar u _ { x x } , & ( x , t ) \\in ( 0 , 1 ) \\times ( 0 , \\infty ) , \\\\ \\bar u ( 0 , t ) = 0 , & t \\in ( 0 , \\infty ) \\\\ \\bar u _ x ( 1 , t ) = - a \\bar u _ t ( 1 , t ) + \\frac { b d } { c } \\bar u ( 1 , t ) , & t \\in ( 0 , \\infty ) \\\\ \\bar u ( x , 0 ) = \\bar u _ 0 ( x ) , \\bar u _ t ( x , 0 ) = \\bar u _ 1 ( x ) , & x \\in ( 0 , 1 ) . \\end{cases} \\end{align*}"}
+{"id": "8377.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , T ] } \\sum _ { k = 0 } ^ { \\lfloor t / \\delta \\rfloor - 1 } \\| ( - A ) ^ { \\sigma } S _ { A } ( t - k \\delta ) \\| \\le C \\delta ^ { - 1 } , \\ , \\sigma \\in ( 0 , 1 ) , \\end{align*}"}
+{"id": "8082.png", "formula": "\\begin{align*} \\begin{aligned} \\uppercase \\expandafter { \\romannumeral 2 } & = \\sum \\limits _ { j = 1 } ^ { \\infty } \\mu _ { j } \\vert g ( b _ { j } ) ( x ) \\vert \\chi _ { 4 P _ { j } } ( x ) + \\sum \\limits _ { j = 1 } ^ { \\infty } \\mu _ { j } \\vert g ( b _ { j } ) ( x ) \\vert \\chi _ { ( 4 P _ { j } ) ^ { c } } ( x ) \\\\ & = \\uppercase \\expandafter { \\romannumeral 1 } _ { 1 } + \\uppercase \\expandafter { \\romannumeral 1 } _ { 2 } . \\end{aligned} \\end{align*}"}
+{"id": "3215.png", "formula": "\\begin{align*} \\big ( L ^ { - 1 } A ^ T A L ^ { - T } \\big ) \\big ( L ^ T x \\big ) = L ^ { - 1 } A ^ T b \\mathrm { a n d } \\big ( L ^ { - 1 } A A ^ T L ^ { - T } \\big ) \\big ( L ^ T y \\big ) = L ^ { - 1 } b , \\ \\mbox { r e s p e c t i v e l y } . \\end{align*}"}
+{"id": "5589.png", "formula": "\\begin{align*} \\Sigma = \\{ x \\in \\partial ^ 0 U : \\Theta ^ { n - 1 } ( \\mu , x ) \\geq \\eta _ 0 \\} \\end{align*}"}
+{"id": "8526.png", "formula": "\\begin{align*} \\lim _ { \\rho \\rightarrow 0 ^ { + } } \\frac { \\mathcal { H } ^ { n } ( ( \\mathbb { R } ^ { n } \\backslash E ) \\cap B _ { \\rho } ( ( \\bar { z } , w ) \\cap \\{ z \\geq \\bar { z } ) \\} } { \\omega _ { n } \\rho ^ { n } } = 0 . \\end{align*}"}
+{"id": "6017.png", "formula": "\\begin{align*} \\varphi _ { \\delta } = \\varphi \\ast \\phi _ { \\delta } , \\end{align*}"}
+{"id": "5146.png", "formula": "\\begin{align*} \\lim _ { D \\to 0 } H [ Q _ { \\mathrm { u n i } } ( X ) ] - H [ Q ^ * ( X ) ] = 0 . \\end{align*}"}
+{"id": "7234.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ d f \\nu ( s _ i ) n _ i = [ R ' : R ] \\ , . \\end{align*}"}
+{"id": "1121.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal J _ \\lambda ( u ) & : = \\frac { 1 } { p } \\| u \\| _ { W ^ { m , p } _ 0 ( D ) } ^ p - \\lambda \\int _ D F ( x , u ( x ) ) d \\mu , \\end{aligned} \\end{align*}"}
+{"id": "1763.png", "formula": "\\begin{gather*} \\max _ { | s - s _ 0 | = r } | \\zeta ( s + i \\tau , \\alpha ) | \\frac { \\delta ^ N } { 1 - \\delta } \\leq \\frac { 1 } { 3 } ( 2 - e ^ { \\delta r } ) \\epsilon \\end{gather*}"}
+{"id": "7505.png", "formula": "\\begin{align*} ( x , x ^ \\sigma + a , x ^ { \\sigma ^ 2 } + b ) A = ( y , y ^ \\tau + a ' , y ^ { \\tau ^ 2 } + b ' ) . \\end{align*}"}
+{"id": "130.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } \\Phi ( \\tau ) d \\tau = P \\Phi ( t ) , t \\in [ 0 , 1 ) , \\end{align*}"}
+{"id": "7952.png", "formula": "\\begin{align*} \\Psi _ { i } ^ r = \\sum _ { u \\in U _ i } \\deg _ { G _ i } ( u _ i ) = r ^ 2 - r + 1 \\end{align*}"}
+{"id": "4202.png", "formula": "\\begin{align*} D ( - 2 , \\varepsilon ) & = \\sigma _ { \\varepsilon } ( - 2 I ) = \\sigma _ { \\varepsilon } ( [ I \\bullet i I , \\frac { i I } { 2 } ] _ { \\ast } ) \\\\ & = \\sigma _ { \\varepsilon } ( [ \\psi ( I ) \\bullet \\psi ( i I ) , \\psi ( \\frac { i I } { 2 } ) ) ] _ { \\ast } ) = \\sigma _ { \\varepsilon } ( 4 i \\psi ( \\frac { i I } { 2 } ) ) . \\end{align*}"}
+{"id": "6954.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ d E T ( k ) = & 2 \\sum _ { k = 1 } ^ d \\sum _ { i = 1 } ^ { k - 1 } \\sum _ { j = i + 1 } ^ { k - 1 } \\frac { 1 } { ( ( x _ k - x _ i ) + \\epsilon ^ 2 ) ( ( x _ k - x _ j ) + \\epsilon ^ 2 ) } \\\\ & - 2 \\sum _ { k = 1 } ^ d \\sum _ { i = k + 1 } ^ { d } \\sum _ { j = i + 1 } ^ { d } \\frac { ( x _ i - x _ k ) } { ( ( x _ i - x _ k ) + \\epsilon ^ 2 ) ( ( x _ j - x _ k ) + \\epsilon ^ 2 ) ( ( x _ j - x _ i ) + \\epsilon ^ 2 ) } . \\end{align*}"}
+{"id": "711.png", "formula": "\\begin{align*} \\mathbf u ( t ) = \\mathbf S ( t - S ) \\mathbf u ( S ) + i \\int _ S ^ t \\mathbf S ( t - s ) \\mathbf N \\left ( \\Theta _ R ^ { \\mathbf u } ( s ) \\mathbf u ( s ) \\right ) \\ , d s + i \\int _ S ^ t \\mathbf S ( t - s ) \\mathbf M \\left ( \\mathbf u ( s ) \\right ) \\ , d W ( s ) \\end{align*}"}
+{"id": "5233.png", "formula": "\\begin{align*} \\Big \\| \\Big [ \\sum _ { t = 1 } ^ T g ( x _ { t } ) \\Big ] _ + \\Big \\| , \\end{align*}"}
+{"id": "122.png", "formula": "\\begin{align*} & \\langle \\mathcal { Q } _ 4 ^ { } \\rangle _ { \\Psi } \\leq C \\rho N \\widehat { g } ( 0 ) \\sqrt { \\rho a ^ 3 } , \\\\ & b \\frac { K _ L ^ 2 } { 2 \\ell ^ 2 } \\langle n _ + ^ H \\rangle _ { \\Psi } \\leq C \\rho N \\widehat { g } ( 0 ) \\times \\begin{dcases} C \\sqrt { \\rho a ^ 3 } \\frac { K _ { L } } { K _ { \\ell } } , & d = 3 , \\\\ C \\widehat { g } ( 0 ) , & d = 2 , \\end{dcases} \\end{align*}"}
+{"id": "6042.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ d \\langle \\partial _ { z _ j } { c } ( z , x ) , \\zeta \\rangle ^ { 2 } \\geq \\underline { c } ( z ) \\vert \\zeta \\vert ^ 2 . \\end{align*}"}
+{"id": "8120.png", "formula": "\\begin{align*} \\aligned C | _ k = & \\Big ( ( \\textrm { t h e b o t t o m $ u _ \\alpha $ r o w s } ) \\bigcup ( \\textrm { t h e r i g h t m o s t $ [ u _ \\alpha - S _ k ^ \\alpha ] _ + $ c o l u m n s } ) \\Big ) \\\\ & \\bigcap \\Big ( ( \\textrm { t h e b o t t o m $ [ u _ \\beta - S _ k ^ \\beta ] _ + $ r o w s } ) \\bigcup ( \\textrm { t h e r i g h t m o s t $ u _ \\beta $ r o w s } ) \\Big ) . \\endaligned \\end{align*}"}
+{"id": "1076.png", "formula": "\\begin{align*} \\widetilde { v } \\left ( x , t \\right ) = \\frac { \\widetilde { u } \\left ( x , t \\right ) } { G _ { 1 } \\left ( x , t \\right ) } , \\widetilde { w } \\left ( x , t \\right ) = \\frac { \\widetilde { m } \\left ( x , t \\right ) } { G _ { 2 } \\left ( x , t \\right ) } . \\end{align*}"}
+{"id": "5136.png", "formula": "\\begin{align*} \\begin{bmatrix} y _ 1 \\\\ \\vdots \\\\ y _ N \\end{bmatrix} & = \\begin{bmatrix} M _ { 1 , 1 } & \\cdots & M _ { 1 , { 2 N } } \\\\ \\vdots & \\vdots & \\vdots \\\\ M _ { N , 1 } & \\cdots & M _ { N , { 2 N } } \\end{bmatrix} \\begin{bmatrix} x _ 1 \\\\ \\vdots \\\\ x _ { 2 N } \\end{bmatrix} \\end{align*}"}
+{"id": "3060.png", "formula": "\\begin{align*} { \\bf { H } } = { { \\bf U } } \\left [ { { \\bf { \\Lambda } } , { { \\bf { 0 } } _ { { { N _ { \\rm { R } } } } \\times \\left ( { { N _ { \\rm { T } } } - { N _ { \\rm { R } } } } \\right ) } } } \\right ] { \\hat { \\bf V } } ^ H = { \\bf { U \\Lambda } } { { \\bf { V } } ^ H } , \\end{align*}"}
+{"id": "1976.png", "formula": "\\begin{align*} \\begin{aligned} a ^ t _ { \\sigma , r } - a ^ t _ { \\sigma , r ^ j } & = g _ r - g _ { r ^ j } \\\\ & = j \\left ( \\frac { q } { p } r _ { \\pi _ { \\hat { \\beta } } ( \\hat { \\gamma } - 2 ) } + \\frac { q } { p } r _ { \\pi _ { \\hat { \\beta } } ( \\hat { \\gamma } ) } + \\lambda _ { \\pi _ { \\hat { \\beta } } ( \\hat { \\gamma } - 1 ) } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "7403.png", "formula": "\\begin{align*} \\left ( W _ { n } ^ { M } \\right ) ' ( t ) & = \\lim _ { h \\searrow 0 } \\left [ \\frac { W _ { n } ^ { M } ( t ) - W _ { n } ^ { M } ( t - h ) } { h } \\right ] = \\lim _ { h \\searrow 0 } \\left [ \\frac { W _ { n } ( x _ { t } , t ) - W _ { n } ( x _ { t - h } , t - h ) } { h } \\right ] \\\\ [ 1 e x ] & \\le \\lim _ { h \\searrow 0 } \\left [ \\frac { W _ { n } ( x _ { t } , t ) - W _ { n } ( x _ { t } , t - h ) } { h } \\right ] = \\partial _ { t } W _ { n } ( x _ { t } , t ) , \\end{align*}"}
+{"id": "164.png", "formula": "\\begin{align*} \\theta _ 1 ( v , \\tau ) = 2 q ^ { \\frac { 1 } { 8 } } { \\rm c o s } ( \\pi v ) \\prod _ { j = 1 } ^ { \\infty } [ ( 1 - q ^ j ) ( 1 + e ^ { 2 \\pi \\sqrt { - 1 } v } q ^ j ) ( 1 + e ^ { - 2 \\pi \\sqrt { - 1 } v } q ^ j ) ] , \\end{align*}"}
+{"id": "4815.png", "formula": "\\begin{align*} d ( \\bar { b } , \\bar { c } ) = \\lambda ^ { - k ( \\bar { b } , \\bar { c } ) } , \\end{align*}"}
+{"id": "3049.png", "formula": "\\begin{align*} [ v ] = R ^ { \\mathrm { T } } \\zeta \\qquad \\textrm { o n } \\mathsf { C } \\ , . \\end{align*}"}
+{"id": "5559.png", "formula": "\\begin{align*} V _ k ( \\zeta ) = U _ k ( \\zeta ) - \\Psi _ k ( \\zeta ) \\end{align*}"}
+{"id": "3861.png", "formula": "\\begin{align*} \\nabla _ { \\partial _ x } \\partial _ x & = \\lambda _ 1 e ^ { - 2 \\lambda _ 1 z } \\partial _ z \\\\ \\nabla _ { \\partial _ y } \\partial _ y & = \\lambda _ 2 e ^ { - 2 \\lambda _ 2 z } \\partial _ z \\\\ \\nabla _ { \\partial _ x } \\partial _ z & = \\nabla _ { \\partial _ z } \\partial _ x = - \\lambda _ 1 \\partial _ x \\\\ \\nabla _ { \\partial _ y } \\partial _ z & = \\nabla _ { \\partial _ z } \\partial _ y = - \\lambda _ 2 \\partial _ y . \\end{align*}"}
+{"id": "4244.png", "formula": "\\begin{align*} \\langle \\tilde { g } , h \\rangle & = \\langle e ^ { t \\Delta } \\tilde { g } , a \\rangle \\\\ & = \\langle g , e ^ { t \\Delta } a \\rangle \\\\ & = \\langle g , h \\rangle . \\\\ \\end{align*}"}
+{"id": "6931.png", "formula": "\\begin{align*} \\limsup \\limits _ { \\xi \\rightarrow - \\infty } \\psi ( \\xi ) = \\zeta . \\end{align*}"}
+{"id": "6764.png", "formula": "\\begin{align*} d _ X \\big ( \\big ( \\Phi _ T ( c _ { b , x } ) & \\big ) ( s ) , \\big ( \\Phi _ T ( c _ { b , \\pi _ { R ' } ( x ) } ) \\big ) ( s ) \\big ) \\\\ = & \\ ; d _ X \\big ( c _ { b , x } ( s + T ) , c _ { b , \\pi _ { R ' } ( x ) } ( s + T ) \\big ) = s + T - R ' = s - ( R ' - T ) . \\end{align*}"}
+{"id": "6091.png", "formula": "\\begin{align*} \\begin{aligned} g _ k & = \\nabla f _ { S _ k } ( \\omega ^ k ) - \\gamma ^ * ( \\nabla f _ { S _ k } ( \\phi _ k ) - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla f _ i ( \\phi _ i ^ k ) ) \\\\ \\gamma ^ * & \\approx \\frac { s _ { X Y } } { s _ Y ^ 2 } \\end{aligned} \\end{align*}"}
+{"id": "5739.png", "formula": "\\begin{align*} \\mu = \\frac { n - 1 + a } { 2 } \\end{align*}"}
+{"id": "5554.png", "formula": "\\begin{align*} \\omega ( r ) = \\sum _ { j = 1 } ^ { k } \\sigma _ j ( \\zeta _ 0 ) , k = 1 , 2 , \\ldots , m - 1 , \\end{align*}"}
+{"id": "5773.png", "formula": "\\begin{align*} \\lim _ { L \\rightarrow \\infty } { \\mathbb E } \\langle ( R _ { 1 , 2 } ^ p - { \\mathbb E } \\langle R _ { 1 , 2 } ^ p \\rangle ) ^ 2 \\rangle = 0 , \\end{align*}"}
+{"id": "7691.png", "formula": "\\begin{align*} \\rho _ { x \\sigma } ^ { \\sigma ' } : = \\rho _ { g \\sigma } ^ { \\sigma ' } \\end{align*}"}
+{"id": "1045.png", "formula": "\\begin{align*} \\langle A _ p ( u ) , h \\rangle + \\langle A _ q ( u ) , h \\rangle + \\lambda \\int _ \\Omega u ^ { p ( z ) - 1 } h d z = \\int _ \\Omega f ( z , u ) h d z \\end{align*}"}
+{"id": "499.png", "formula": "\\begin{align*} \\mathcal M _ k [ \\mathcal M _ k ( X ) ] = \\begin{pmatrix} ( i \\mathfrak K _ 1 e _ k \\beta ) ^ 2 \\psi \\\\ 0 \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} - ( \\mathfrak K _ 1 e _ k ) ^ 2 \\psi \\\\ 0 \\\\ 0 \\end{pmatrix} \\end{align*}"}
+{"id": "4671.png", "formula": "\\begin{align*} | S | & \\geq \\sum \\limits _ { i = 1 } ^ { 2 m + 1 } | S _ i | - \\sum \\limits _ { \\substack { i , j \\in [ 2 m + 1 ] \\\\ i \\neq j } } | S _ i \\cap S _ j | \\\\ & > ( 2 m + 1 ) \\cdot ( \\delta ( G ) - 2 ) - \\binom { 2 m + 1 } { 2 } \\cdot \\epsilon n . \\end{align*}"}
+{"id": "8124.png", "formula": "\\begin{align*} C _ 2 = C _ { \\max } ( S \\cup C _ 2 ) \\cap \\{ v _ { \\ell + 1 } , \\dots , v _ { | L | } \\} \\end{align*}"}
+{"id": "2507.png", "formula": "\\begin{align*} \\big ( \\sigma \\partial ^ { 1 / 2 } _ t \\boldsymbol { y } , \\partial ^ { 1 / 2 } _ t \\boldsymbol { v } \\big ) : = \\frac { T } { 2 } \\sum _ { k = 1 } ^ { \\infty } k \\omega ( \\sigma \\boldsymbol { y } _ k , \\boldsymbol { v } _ k ) _ { \\Omega } . \\end{align*}"}
+{"id": "976.png", "formula": "\\begin{align*} \\tilde \\alpha & = a - a ' \\sqrt d , \\\\ N ( \\alpha ) & = \\alpha \\tilde \\alpha = a ^ 2 - d a '^ 2 . \\end{align*}"}
+{"id": "625.png", "formula": "\\begin{align*} \\kappa ( t ) - \\kappa ( r ) = \\theta _ R ( \\norm { \\mathbf U } _ { t } ^ 2 ) - \\theta _ R ( \\norm { \\mathbf V } _ { t } ^ 2 ) - \\left [ \\theta _ R ( \\norm { \\mathbf U } _ { r } ^ 2 ) - \\theta _ R ( \\norm { \\mathbf V } _ { r } ^ 2 ) \\right ] \\end{align*}"}
+{"id": "5811.png", "formula": "\\begin{align*} w _ 0 = \\prod \\limits _ { i = 1 } ^ n s _ { \\varepsilon ' _ i } = s _ { \\alpha ' _ n } \\prod \\limits _ { i = 1 } ^ { n - 1 } s _ { \\alpha ^ { c , n - i + 1 } _ { m a x } } \\ ; \\ ; . \\end{align*}"}
+{"id": "3183.png", "formula": "\\begin{align*} a = 1 , b = 1 6 , c = 1 , \\ ; \\ ; d = 9 / 8 , \\end{align*}"}
+{"id": "545.png", "formula": "\\begin{align*} \\mathbf M ( \\mathbf f ) = \\begin{pmatrix} M _ 1 ( \\mathbf f ) \\\\ \\vdots \\\\ M _ n ( \\mathbf f ) \\end{pmatrix} \\mathbf N ( \\mathbf f ) = \\begin{pmatrix} N _ 1 ( \\mathbf f ) \\\\ \\vdots \\\\ N _ n ( \\mathbf f ) \\end{pmatrix} , \\end{align*}"}
+{"id": "4616.png", "formula": "\\begin{align*} \\mathfrak { g } ^ \\theta : = \\{ x \\in \\mathfrak { g } \\ , \\ , | \\ , \\ , \\theta ( x ) = x \\} . \\end{align*}"}
+{"id": "2359.png", "formula": "\\begin{align*} \\Lambda ^ 0 \\coloneqq \\{ \\mu \\in G \\times \\widehat { G } : \\pi ( \\lambda ) \\pi ( \\mu ) = \\pi ( \\mu ) \\pi ( \\lambda ) , \\forall \\lambda \\in \\Lambda \\} . \\end{align*}"}
+{"id": "3244.png", "formula": "\\begin{align*} \\Delta _ { \\alpha } ( \\epsilon , R ) = \\frac { \\epsilon R ^ { 2 } } { 1 + \\alpha ^ { 2 } } + O \\left ( \\epsilon ^ { \\frac { 1 } { 1 + \\eta } } R \\right ) . \\end{align*}"}
+{"id": "232.png", "formula": "\\begin{align*} \\mathbb { C } _ { \\emph { r e g } , + } ^ n : = \\{ \\xi \\in \\mathbb { C } ^ n \\mid 2 \\xi _ j \\not \\in \\mathbb { Z } _ { > 0 } , \\ , \\xi _ j \\pm \\xi _ k \\not \\in \\mathbb { Z } _ { > 0 } \\ , ( j < k ) \\} \\end{align*}"}
+{"id": "3761.png", "formula": "\\begin{align*} \\begin{aligned} f ( t ) & = e ^ { t b B } ( u ' ( 0 ) - a B u ( 0 ) ) \\\\ & + \\frac { 1 } { 1 + b } B ^ { - 1 } \\Big ( e ^ { t b B } - e ^ { - t B } \\Big ) ( u '' ( 0 ) - ( a + b ) B u ' ( 0 ) + a b B ^ 2 u ( 0 ) ) . \\end{aligned} \\end{align*}"}
+{"id": "4998.png", "formula": "\\begin{align*} - 2 K _ { \\Sigma } = - 2 R _ { 1 2 1 2 } + | h | ^ 2 - 4 H ^ 2 . \\end{align*}"}
+{"id": "3660.png", "formula": "\\begin{align*} 0 = \\lambda _ 0 ( t ) < \\lambda _ 1 ( t ) \\leq \\dots \\to \\infty \\end{align*}"}
+{"id": "8013.png", "formula": "\\begin{align*} Y ( t ) \\stackrel { d } { = } v _ 0 t + \\Bigl [ v _ h - v _ 0 \\Bigr ] _ { h = 1 , \\dots , D } X ( t ) , \\ \\ \\ t \\ge 0 . \\end{align*}"}
+{"id": "1392.png", "formula": "\\begin{align*} l y = \\lambda y , x \\in ( 0 , \\pi ) , \\end{align*}"}
+{"id": "3326.png", "formula": "\\begin{align*} g ( \\tau + 1 ) = \\begin{pmatrix} \\zeta _ { 6 0 } ^ { - 1 } & 0 \\\\ 0 & \\zeta _ { 6 0 } ^ { 1 1 } \\end{pmatrix} g ( \\tau ) , g ( - \\frac { 1 } { \\tau } ) = \\frac { 2 } { \\sqrt { 5 } } \\begin{pmatrix} \\sin \\frac { 2 \\pi } { 5 } & \\sin \\frac { \\pi } { 5 } \\\\ \\sin \\frac { \\pi } { 5 } & - \\sin \\frac { 2 \\pi } { 5 } \\end{pmatrix} g ( z ) . \\end{align*}"}
+{"id": "5383.png", "formula": "\\begin{align*} ( K _ H ^ { - 1 } g ) ( s ) = s ^ { H - \\frac { 1 } { 2 } } I _ { 0 + } ^ { \\frac { 1 } { 2 } - H } s ^ { \\frac { 1 } { 2 } - H } g ' , \\ \\ H \\in ( 0 , 1 / 2 ) . \\end{align*}"}
+{"id": "6429.png", "formula": "\\begin{align*} \\begin{aligned} & u _ { n } ( c \\tau _ n , \\tau _ n ) = \\frac { \\varepsilon _ 1 ( c ) } { 2 } , \\\\ & u _ { n } ( c t , t ) \\leq \\frac { \\varepsilon _ 1 ( c ) } { 2 } , t \\in ( \\tau _ n , t _ n ) , \\\\ & u _ { n } ( c ( \\tau _ n + t _ n - \\tau _ n ) , \\tau _ n + t _ n - \\tau _ n ) \\rightarrow 0 , n \\rightarrow \\infty . \\end{aligned} \\end{align*}"}
+{"id": "679.png", "formula": "\\begin{align*} \\tau < \\infty \\implies \\limsup _ { t \\nearrow \\tau } \\norm { ( \\psi _ + , \\psi _ - , \\phi _ + ) ( t ) } _ { \\mathbf H ^ { s , r } } = \\infty , \\end{align*}"}
+{"id": "8031.png", "formula": "\\begin{align*} T _ { \\alpha , m , \\sigma } \\varphi ( \\lambda ) = \\frac { 1 } { \\sigma ^ { 2 | \\alpha | + n } } \\int _ { \\R _ { + } ^ { n } } \\Theta _ { \\alpha } ( y , \\lambda , m ) \\ , \\varphi ( y ) \\ , d \\mu _ { \\alpha } ( y ) , \\end{align*}"}
+{"id": "335.png", "formula": "\\begin{align*} \\left \\langle \\left [ w \\right ] _ { \\operatorname * { D } ; j } \\left ( s \\right ) , \\gamma _ { \\mathbf { n } ; j } \\left ( s \\right ) \\overline { \\mathbf { m } } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } = \\left \\langle \\psi , \\gamma _ { \\mathbf { n } ; j } \\left ( s \\right ) \\overline { \\mathbf { m } } \\right \\rangle _ { \\Gamma _ { j } } \\quad \\forall \\mathbf { m } \\in \\mathbf { H } \\left ( \\mathbb { R } ^ { 3 } , \\operatorname * { d i v } \\right ) . \\end{align*}"}
+{"id": "2972.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { M _ k } \\sum _ { j = 1 } ^ { N _ k } \\Theta _ { x _ i \\otimes y _ j , x _ i \\otimes y _ j } T \\to T \\end{align*}"}
+{"id": "7294.png", "formula": "\\begin{align*} ( \\ 0 , \\N ) = ( s _ 0 , B _ 0 ) \\geq ( s _ 1 , B _ 1 ) \\geq ( s _ 2 , B _ 2 ) \\geq \\ldots \\end{align*}"}
+{"id": "8026.png", "formula": "\\begin{align*} { \\mathcal { F } } _ { \\alpha } ^ { - 1 } ( f ) ( \\lambda ) = { \\mathcal { F } _ { \\alpha } } ( f ) ( \\lambda ) , \\lambda \\in \\R _ { + } ^ { n } , \\end{align*}"}
+{"id": "7627.png", "formula": "\\begin{align*} \\mathrm { s u p p } ( \\Q ) & : = \\bigcap \\left \\{ A \\subset \\mathbb { R } ^ { d } , \\ ; \\mathrm { c l o s e d } , \\ ; \\Q ( A ) = 1 \\right \\} \\\\ \\mathrm { s u p p } ( \\mathcal { Q } ) & : = \\bigcap \\left \\{ A \\subset \\mathbb { R } ^ { d } , \\ ; \\mathrm { c l o s e d } , \\ ; \\Q ( A ) = 1 \\ ; \\forall \\mathbb { Q } \\in \\mathcal { Q } \\right \\} . \\end{align*}"}
+{"id": "5441.png", "formula": "\\begin{align*} \\int e ^ f d \\nu - 1 & = \\int \\Phi ( f ) \\ , d \\nu - \\Phi \\left ( \\int f d \\nu \\right ) \\\\ & = \\inf _ { t \\in \\mathbb { R } } \\int \\left ( \\Phi ( f ) - \\Phi ( t ) - \\Phi ' ( t ) ( f - t ) \\right ) d \\nu \\\\ & \\leq b \\ , \\inf _ { t \\in \\mathbb { R } } \\int \\left ( \\Phi ( f ) - \\Phi ( t ) - \\Phi ' ( t ) ( f - t ) \\right ) d \\mu \\ , = \\ , b \\ , \\left ( \\int e ^ f d \\mu - \\exp \\left \\{ \\int f d \\mu \\right \\} \\right ) . \\end{align*}"}
+{"id": "867.png", "formula": "\\begin{align*} \\begin{aligned} & \\left | m _ u \\left ( B \\left ( z , \\frac { R } { 2 ^ j } \\right ) \\right ) - m _ u \\left ( B \\left ( z , \\frac { R } { 2 ^ i } \\right ) \\right ) \\right | \\\\ & \\leq A [ u ] _ { W ^ { \\alpha , G } _ { s } ( X , d , \\mu ) } \\sum _ { l = i } ^ { j - 1 } G ^ { - 1 } \\left ( \\frac { R ^ { \\alpha p _ { 0 } - s } } { 2 ^ { l ( \\alpha p _ { 0 } - s ) } } \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "4527.png", "formula": "\\begin{align*} | | \\mathcal { B } ( { \\mathbf V } _ i | _ { x _ 1 = 0 } , \\Psi _ i ) | | _ { H ^ { s } ( \\Gamma _ T ) } \\leq \\delta \\theta ^ { s - \\alpha - 1 } _ i . \\end{align*}"}
+{"id": "3372.png", "formula": "\\begin{align*} f = f _ { \\chi } \\times \\psi \\end{align*}"}
+{"id": "8506.png", "formula": "\\begin{align*} E ^ { ( s ) } \\cap \\{ z < \\bar { z } \\} = \\left ( E \\cap \\{ z < \\bar { z } \\} \\right ) ^ { ( s ) } = \\left ( F _ { \\ell } \\cap \\{ z < \\bar { z } \\} \\right ) ^ { ( s ) } = F _ { \\ell } ^ { ( s ) } \\cap \\{ z < \\bar { z } \\} . \\end{align*}"}
+{"id": "3644.png", "formula": "\\begin{align*} \\chi \\mathbin { \\lrcorner } \\omega = 0 \\bot \\omega = 0 \\ . \\end{align*}"}
+{"id": "6716.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta _ { p } u : = { \\rm d i v } ( | \\nabla u | ^ { p - 2 } \\nabla u ) & = 0 \\ \\ { \\rm i n } \\ \\ M \\\\ u & = 0 \\ \\ { \\rm o n } \\ \\ \\partial M \\\\ u ( x ) & \\rightarrow 1 \\ \\ { \\rm a s } \\ \\ | x | \\rightarrow \\infty . \\end{aligned} \\right . \\end{align*}"}
+{"id": "463.png", "formula": "\\begin{align*} & \\frac { 1 } { \\sqrt { 2 \\pi } } \\cdot \\frac { t } { r s } \\cdot 2 ^ { - ( \\zeta - \\frac 1 2 ) } \\ , \\Gamma ( \\zeta + 1 ) \\cdot \\left ( \\frac { 2 r s } { r ^ 2 + s ^ 2 + t ^ 2 } \\right ) ^ { \\zeta + 1 } \\\\ & \\times \\ , _ 2 \\tilde { F } _ 1 \\left ( \\frac { \\zeta + 1 } { 2 } , \\frac { \\zeta + 2 } { 2 } ; \\zeta + \\frac 1 2 ; \\frac { 4 r ^ 2 s ^ 2 } { \\left ( r ^ 2 + s ^ 2 + t ^ 2 \\right ) ^ 2 } \\right ) . \\end{align*}"}
+{"id": "3297.png", "formula": "\\begin{align*} \\frac { d } { d t } ( \\phi _ X ) _ { _ \\Sigma } ( 0 ) = \\mu \\frac { \\beta _ { _ \\Sigma } ^ 2 - \\alpha _ { _ \\Sigma } ^ 2 } { 2 } , \\frac { d ^ 2 } { d t ^ 2 } ( \\phi _ X ) _ { _ \\Sigma } ( 0 ) = \\frac { \\alpha _ { _ \\Sigma } ^ 3 - \\beta _ { _ \\Sigma } ^ 3 } { 3 } . \\end{align*}"}
+{"id": "7391.png", "formula": "\\begin{align*} & \\| V _ { n } \\| ^ { \\frac { 1 0 } { 3 } } = \\| V _ { n } \\| ^ { 2 } \\| V _ { n } \\| ^ { \\frac { 4 } { 3 } } \\le \\frac { 2 } { 3 } \\| V _ { n } \\| ^ { 3 } + \\frac { 1 } { 3 } \\| V _ { n } \\| ^ { 4 } \\le \\frac { 1 } { 3 } \\| V _ { n } \\| ^ { 2 } + \\frac { 2 } { 3 } \\| V _ { n } \\| ^ { 4 } . \\end{align*}"}
+{"id": "6444.png", "formula": "\\begin{align*} \\begin{pmatrix} A & - V \\\\ B & U \\end{pmatrix} \\cdot \\begin{pmatrix} s _ 0 ( T ) \\\\ s _ 1 ( T ) \\end{pmatrix} = \\begin{pmatrix} A s _ 0 ( T ) - V s _ 1 ( T ) \\\\ B s _ 0 ( T ) + U s _ 1 ( T ) \\end{pmatrix} . \\end{align*}"}
+{"id": "4277.png", "formula": "\\begin{align*} \\epsilon _ { \\pi } ( i ) \\ , y _ { i } = \\epsilon _ { \\pi } ( j ) \\ , y _ { j } = x _ { \\pi ( i ) } \\end{align*}"}
+{"id": "4582.png", "formula": "\\begin{align*} I _ { 1 , \\ast } ( t ) + \\Vert \\varphi ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } \\le & \\frac { C _ 3 } { \\varepsilon } \\int _ 0 ^ t ( I _ { 1 , \\ast } ( s ) + \\Vert \\varphi ( s ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } ) d s + C _ 1 \\varepsilon I _ { 1 , \\ast } ( t ) + \\frac { C _ 3 } { \\varepsilon } \\Vert { \\mathbf F } \\Vert ^ 2 _ { H ^ 1 _ \\ast ( \\Omega _ t ) } \\ , , \\end{align*}"}
+{"id": "7660.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } \\mathop { { \\rm P f } } _ { 1 \\leq i , j \\leq N } \\big ( S ( x _ i , x _ j ) \\big ) \\mathop { { \\rm d e t } } _ { \\substack { 0 \\leq m \\leq N - 1 \\\\ 1 \\leq j \\leq N } } \\big ( f _ m ( x _ j ^ 2 ) \\big ) \\prod _ { i = 1 } ^ N \\rho ( x _ i ) \\dd x _ i . \\end{align*}"}
+{"id": "1798.png", "formula": "\\begin{gather*} \\zeta ( s , \\mathbb { X } _ \\alpha ) = Z _ N ( s , \\mathbb { X } _ \\alpha ) - \\frac { 1 } { 2 \\pi i } \\int _ { - \\delta - i \\infty } ^ { - \\delta + i \\infty } \\zeta ( s + w , \\mathbb { X } _ \\alpha ) \\widehat { \\phi } ( w ) N ^ w \\ , d w \\end{gather*}"}
+{"id": "4107.png", "formula": "\\begin{align*} 2 ^ { n ( n - 1 ) / 2 } \\prod _ { i = 0 } ^ { n - 1 } \\frac { ( 4 i + 2 ) ! } { ( n + 2 i + 1 ) ! } , \\end{align*}"}
+{"id": "7025.png", "formula": "\\begin{align*} \\Gamma = \\sum _ { i = 1 } ^ { + \\infty } \\alpha _ i | \\Psi _ i \\rangle \\langle \\Psi _ i | \\end{align*}"}
+{"id": "4159.png", "formula": "\\begin{align*} \\Delta _ k ( \\tau ) = \\tau - \\mathcal N _ k ( \\tau ) . \\end{align*}"}
+{"id": "8263.png", "formula": "\\begin{align*} & | \\Psi ^ * g _ a - ( a ^ 2 \\sum _ { j = 1 } ^ 3 ( d x _ j ) ^ 2 + g _ { 0 , 0 , 0 } + | x | H _ x + \\frac { 1 } { a ^ 2 } \\eta ^ 2 ) | _ { g _ 0 } \\\\ & \\leq C ( \\epsilon ) \\left ( \\frac { 1 } { \\rho ^ 2 ( m ) } + \\frac { | x | } { \\rho ^ 3 ( m ) } + \\frac { | x | ^ 2 } { \\rho ^ 4 ( m ) } \\right ) . \\end{align*}"}
+{"id": "4355.png", "formula": "\\begin{align*} \\omega ^ { ( \\beta ) } ( A ) = { \\rm T r } ( \\varrho _ \\beta \\pi ^ { P } ( A ) ) \\ \\ ( A \\in { \\tt C A R } ( \\mathcal { K } , \\Gamma ) ) \\end{align*}"}
+{"id": "4553.png", "formula": "\\begin{align*} \\mathcal { B } _ 0 \\partial _ t { \\mathbf V } _ { x _ 2 } + \\mathcal { B } _ 1 \\partial _ 1 { \\mathbf V } _ { x _ 2 } + \\mathcal { B } _ 2 \\partial _ 2 { \\mathbf V } _ { x _ 2 } + ( \\partial _ 2 \\mathcal B _ 2 + \\mathcal { B } _ 3 ) { \\mathbf V } _ { x _ 2 } = \\tilde { \\mathcal F } _ 2 \\ , , \\end{align*}"}
+{"id": "240.png", "formula": "\\begin{align*} c _ { w } ( z ) = \\begin{cases} \\frac { \\Gamma ( 2 z ) \\Gamma ( \\frac { 1 } { 2 } g _ S + z ) } { \\Gamma ( g _ S + 2 z ) \\Gamma ( \\frac { 1 } { 2 } g _ S + g _ L + z ) } & \\ \\ \\ , \\\\ \\frac { \\Gamma ( 2 z ) } { \\Gamma ( \\frac { 1 } { 2 } + g _ S + z ) } & \\ . \\end{cases} \\end{align*}"}
+{"id": "2888.png", "formula": "\\begin{align*} T _ w \\mathbf { e } ^ { i { \\xi } } = \\left ( \\prod _ { 1 \\leq k \\leq \\ell } c _ { j _ k } ( s _ { j _ k } \\cdots s _ { j _ 2 } s _ { j _ 1 } { \\xi } ) \\right ) { e } ^ { i w { \\xi } } + , \\end{align*}"}
+{"id": "2393.png", "formula": "\\begin{align*} f ( t ) = \\frac { - g + \\exp ( h ( t - T ) ( g - 2 r ) + r ( 2 + h ( t - T ) } { h } \\end{align*}"}
+{"id": "6810.png", "formula": "\\begin{align*} & w \\partial _ u H = \\frac { w ( u - u ^ * ) } { f ( u ) q ( u ) } \\leq \\frac { w ( u + u ^ * ) } { q ( 0 ) f ( u ) } \\leq \\frac { 2 w } { q ( 0 ) f ( u ) } < \\frac { 2 K } { q ( 0 ) } , \\\\ [ 0 . 2 c m ] & y \\partial _ v H = \\frac { y ( v - v ^ * ) } { s v } \\leq \\frac { 1 } { s } \\left ( \\frac { c } { d } v - \\frac { y } { v } v ^ * \\right ) \\leq \\frac { q ( 1 ) } { s } \\left ( \\frac { c } { d } + \\frac { 2 s v ^ * } { c q ( 0 ) } \\right ) . \\end{align*}"}
+{"id": "3196.png", "formula": "\\begin{align*} \\begin{bmatrix} B _ { k } & \\beta _ { 1 } e _ { 1 } \\end{bmatrix} = Q _ { k } ^ { T } \\begin{bmatrix} R _ { k } & f _ { k } \\\\ & \\bar { \\phi } _ { k + 1 } \\end{bmatrix} \\end{align*}"}
+{"id": "3003.png", "formula": "\\begin{align*} \\textsf { R } = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\\\ 2 & 0 \\end{pmatrix} \\textrm { a n d } \\textsf { S } = \\begin{pmatrix} 1 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} \\end{align*}"}
+{"id": "7428.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ p y - { \\rm d i v } ( \\eta ( z ) \\lvert \\nabla y \\rvert ^ { q - 2 } \\nabla y ) = f ( z , y ) , & z \\in \\Omega , \\\\ x = 0 , & z \\in \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "7471.png", "formula": "\\begin{align*} A _ { 4 _ 1 } ( L , M ) : = L + L ^ { - 1 } + ( M - M ^ { - 1 } ) ^ 2 - M - M ^ { - 1 } \\end{align*}"}
+{"id": "1916.png", "formula": "\\begin{align*} \\{ \\eta _ u \\} _ { i + 1 / 2 } = \\left ( \\frac { 2 \\lambda - 1 } { 2 } \\right ) [ \\ ! [ \\eta _ u ] \\ ! ] _ { i + 1 / 2 } . \\end{align*}"}
+{"id": "7892.png", "formula": "\\begin{align*} \\Phi _ t = \\log \\det D ^ 2 u _ 0 + k \\log u _ 0 ^ { \\star } - g ( x , u _ 0 ) \\{ 0 \\} \\times \\partial \\Omega . \\end{align*}"}
+{"id": "840.png", "formula": "\\begin{align*} \\| T f \\| _ 2 & = \\left ( \\sum _ { n = 0 } ^ { \\infty } \\left ( 1 - \\dfrac { 1 } { n + 1 } \\right ) ^ 2 | a _ n | ^ 2 \\right ) ^ { \\dfrac { 1 } { 2 } } \\\\ & < \\Big ( \\sum _ { n = 0 } ^ { \\infty } | a _ n | ^ 2 \\Big ) ^ { \\frac { 1 } { 2 } } \\\\ & = \\| f \\| _ 2 \\\\ & \\leq \\| f \\| _ { \\infty } . \\end{align*}"}
+{"id": "4253.png", "formula": "\\begin{align*} \\mathcal { J } _ { \\lambda } ( u ) : = \\dfrac { 1 } { 2 } \\int _ { \\Omega } | \\nabla u | ^ 2 \\ , d x + \\dfrac { \\alpha } { 2 } \\iint _ { \\mathbb { R } ^ { 2 n } } \\dfrac { | u ( x ) - u ( y ) | ^ 2 } { | x - y | ^ { n + 2 s } } \\ , d x d y - \\dfrac { \\lambda } { 2 } \\int _ { \\Omega } | u | ^ 2 \\ , d x - \\int _ { \\Omega } F ( x , u ) \\ , d x , \\end{align*}"}
+{"id": "391.png", "formula": "\\begin{align*} ( \\mathcal { C } \\otimes \\delta _ \\epsilon ) ^ \\pitchfork \\ = \\ \\mathcal { F } \\ , . \\end{align*}"}
+{"id": "8266.png", "formula": "\\begin{align*} \\mu _ \\lambda ( m , e ^ { i \\theta } , q ) = \\mu ( m ) - \\lambda q , \\end{align*}"}
+{"id": "507.png", "formula": "\\begin{align*} E = \\left \\{ ( s , \\omega ) \\in [ 0 , \\infty ) \\times \\Omega \\colon 0 \\le s < \\tau ( \\omega ) \\right \\} \\end{align*}"}
+{"id": "5887.png", "formula": "\\begin{align*} C _ { \\gamma } = \\log ( 4 \\gamma ^ { 2 } + 2 \\gamma ) \\sim 2 \\log ( \\gamma ) \\sim 2 N ^ { \\zeta } . \\end{align*}"}
+{"id": "201.png", "formula": "\\begin{align*} { Q _ 1 } ( M , P _ i , P _ j , \\tau ) = & \\left \\{ e ^ { \\frac { 1 } { 2 4 } E _ 2 ( \\tau ) A _ 2 } \\widehat { L } ( T M , \\nabla ^ { T M } ) { \\rm c h } \\left [ \\bigotimes _ { n = 1 } ^ { \\infty } S _ { q ^ n } ( \\widetilde { T _ C M } ) \\otimes \\bigotimes _ { m = 1 } ^ { \\infty } \\wedge _ { q ^ m } ( \\widetilde { T _ C M } ) \\right ] \\right . \\\\ & \\left . \\cdot \\varphi ( \\tau ) ^ { 1 6 } { \\rm c h } ( \\mathcal { V } _ i ) { \\rm c h } ( \\mathcal { V } _ j ) \\right \\} ^ { ( 1 2 ) } , \\end{align*}"}
+{"id": "181.png", "formula": "\\begin{align*} E _ 4 ( \\tau ) ^ 2 E _ 6 ( \\tau ) = 1 - 2 4 q - 1 9 6 6 3 2 q ^ 2 + \\cdots . \\end{align*}"}
+{"id": "3335.png", "formula": "\\begin{align*} \\mathbf { I } _ A [ f ] = \\frac { \\int _ { \\mathbb { R } ^ N } e ^ { - \\frac { 1 } { 2 } x ^ \\mathsf { T } A x } f ( x ) \\ , d x } { \\int _ { \\mathbb { R } ^ N } e ^ { - \\frac { 1 } { 2 } x ^ \\mathsf { T } A x } \\ , d x } . \\end{align*}"}
+{"id": "3987.png", "formula": "\\begin{align*} ( \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ) ^ n = g ( \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ) ^ m \\wedge \\omega ^ { n - m } , \\sup _ M \\varphi = 0 , \\end{align*}"}
+{"id": "7160.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ n a _ j \\tau _ j \\right \\| = \\left ( \\sum _ { j = 1 } ^ n | a _ j | ^ p \\right ) ^ \\frac { 1 } { p } . \\end{align*}"}
+{"id": "8783.png", "formula": "\\begin{align*} \\mathbb { E } [ \\norm { \\nabla f ( x _ { S } ) } ^ 2 ] \\leq \\frac { ( 7 2 L \\kappa + 4 ) \\delta ( 1 ) } { T \\gamma } + 2 \\mathcal { C } _ { 5 } \\sum _ { t = 1 } ^ { T } \\Big [ d h _ { t } ^ { 2 ( \\beta - 1 ) } + \\gamma d \\Big ( \\frac { \\sigma ^ 2 } { h _ { t } ^ 2 } + h _ { t } ^ 2 \\Big ) \\Big ] \\frac { 1 } { T } . \\end{align*}"}
+{"id": "730.png", "formula": "\\begin{align*} \\mathbf u ^ \\mu ( t ) - \\mathbf u ( t ) = \\Delta _ 1 ^ \\mu ( t ) + \\Delta _ 2 ^ \\mu ( t ) + \\Delta _ 3 ^ \\mu ( t ) , \\end{align*}"}
+{"id": "8169.png", "formula": "\\begin{align*} A ^ \\star _ { m n } = v ^ \\star _ { m n } ( A ^ \\star _ { 1 0 } , A ^ \\star _ { 0 1 } ) . \\end{align*}"}
+{"id": "7026.png", "formula": "\\begin{align*} \\mathop { \\lim } _ { R \\to + \\infty } \\sum _ { \\begin{array} { c } m \\in \\mathcal M \\\\ ( { \\rm S u p p } \\ ; \\varphi _ m ) \\cap B _ R ^ c \\neq \\emptyset \\\\ \\end{array} } \\int _ { \\mathbb { R } ^ 3 } \\rho \\varphi _ m = 0 , \\end{align*}"}
+{"id": "3926.png", "formula": "\\begin{align*} \\omega _ 1 \\tilde + _ t \\omega _ 2 : = \\{ P ( ( 1 - t ) u + t v ) \\mid u \\in \\omega _ 1 , v \\in \\omega _ 2 \\} , \\end{align*}"}
+{"id": "2773.png", "formula": "\\begin{align*} I ( T _ 1 ; x ) = \\ ! \\begin{multlined} [ t ] x ^ { 1 4 } + 5 1 x ^ { 1 3 } + 2 9 7 9 x ^ { 1 2 } + 1 8 6 8 3 x ^ { 1 1 } + 5 5 4 9 9 x ^ { 1 0 } + 1 0 0 1 4 4 x ^ 9 + \\\\ 1 2 1 3 7 6 x ^ 8 + 1 0 3 7 3 6 x ^ 7 + 6 3 9 3 3 x ^ 6 + 2 8 5 5 1 x ^ 5 + 9 1 4 2 x ^ 4 + 2 0 4 0 x ^ 3 + \\\\ 3 0 0 x ^ 2 + 2 6 x + 1 , \\end{multlined} \\end{align*}"}
+{"id": "5575.png", "formula": "\\begin{align*} \\mathcal { E } _ \\ell = \\sum _ { j = 1 } ^ { 2 0 } \\left ( h ( r _ j ) - C _ \\ell ( r _ j - 1 ) ^ { \\beta _ \\ell } \\right ) ^ 2 . \\end{align*}"}
+{"id": "3079.png", "formula": "\\begin{align*} { R _ { { \\rm { Q L , u p p e r } } } } \\buildrel \\Delta \\over = \\sum \\limits _ { k = 1 } ^ { { N _ { \\rm { R } } } } { { { \\log } _ 2 } \\left ( 1 + \\frac { { \\mathbb E } { \\left \\{ { { { \\left | { \\xi _ k } \\right | } ^ 2 } } { p _ k ^ \\star } \\right \\} } } { \\sigma ^ 2 } \\right ) } . \\end{align*}"}
+{"id": "6131.png", "formula": "\\begin{align*} \\bigsqcup _ { t \\in T } S _ t : = \\left ( \\bigsqcup _ { t \\in T ^ { + } } ( S _ t ^ + ) \\sqcup \\bigsqcup _ { t \\in T ^ { - } } ( S _ t ^ - ) , \\bigsqcup _ { t \\in T ^ { + } } ( S _ t ^ - ) \\sqcup \\bigsqcup _ { t \\in T ^ { - } } ( S _ t ^ + ) \\right ) . \\end{align*}"}
+{"id": "1091.png", "formula": "\\begin{align*} \\int _ V u ( x ) d \\mu : = \\sum _ { x \\in V } \\mu ( x ) u ( x ) , \\end{align*}"}
+{"id": "3116.png", "formula": "\\begin{align*} \\mathbb { E } [ g _ { n , m } g ^ * _ { n ' , m } ] = \\begin{cases} 1 , & n = n ' \\\\ 0 , & n \\neq n ' , \\end{cases} \\end{align*}"}
+{"id": "2283.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } e ^ { 2 \\pi i k \\varepsilon _ j } = n + \\sum _ { j = 1 } ^ { n } \\left ( e ^ { 2 \\pi i k \\varepsilon _ j } - 1 \\right ) = n + \\sum _ { j = 1 } ^ { n } \\left ( e ^ { 2 \\pi i k \\varepsilon _ j } - 1 - 2 \\pi i k \\varepsilon _ j \\right ) . \\end{align*}"}
+{"id": "8058.png", "formula": "\\begin{align*} M _ { \\alpha } ( f ) ( x ) = \\sup \\limits _ { Q } \\vert Q \\vert ^ { \\frac { \\alpha } { n } } \\left ( \\frac { 1 } { \\vert Q \\vert } \\int _ { Q } \\vert f ( y ) \\vert d y \\right ) \\chi _ { Q } ( x ) \\end{align*}"}
+{"id": "7205.png", "formula": "\\begin{align*} \\| I _ 2 ( D ^ { \\beta _ \\perp } \\sigma _ \\rho ) \\| _ { L ^ 2 _ { \\gamma , \\mu } ( C ( \\Lambda [ 0 , \\frac { L } { 2 } ] , 2 \\rho ) ) } & \\le \\| I _ 2 ( D ^ { \\beta _ \\perp } \\sigma _ \\rho ) \\| _ { L ^ 2 _ { \\gamma , \\mu } ( { \\bf C } _ 0 ^ { 2 \\rho } ) } + \\| I _ 2 ( D ^ { \\beta _ \\perp } \\sigma _ \\rho ) \\| _ { L ^ 2 _ { \\gamma , \\mu } ( \\cup _ { j = 1 } ^ { J - 1 } { \\bf C } _ j ^ { 2 \\rho } ) } = : I + I I . \\end{align*}"}
+{"id": "127.png", "formula": "\\begin{align*} x ( t ) = f ( t ) + \\int _ { 0 } ^ { t } k _ 1 ( s , t ) x ( s ) d s + \\int _ { 0 } ^ { t } k _ 2 ( s , t ) x ( s ) d B ( s ) \\end{align*}"}
+{"id": "5606.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\lim _ { n \\to \\infty } \\mathcal R _ { n , R } = 0 . \\end{align*}"}
+{"id": "1920.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( \\frac { \\partial e _ f } { \\partial t } , \\psi _ h \\right ) + a _ h ^ 0 ( e _ f , \\psi _ h ) + \\mathcal { N } ( u ; f , \\psi _ h ) - \\mathcal { N } ^ h ( u _ h ; f _ h , \\psi _ h ) = 0 \\quad \\forall \\ , \\psi _ h \\in \\mathcal { Z } _ h , \\end{aligned} \\end{align*}"}
+{"id": "7138.png", "formula": "\\begin{align*} \\Psi '' ( Z ) = \\left ( \\frac { \\Psi ( Z ) } { Z } Z \\right ) '' = 2 \\left ( \\frac { \\Psi ( Z ) } { Z } \\right ) ' + Z \\left ( \\frac { \\Psi ( Z ) } { Z } \\right ) '' . \\end{align*}"}
+{"id": "2417.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\frac { 4 \\pi } { f ( z ( t _ n ) ) } = ( 4 \\pi ) ^ { \\frac { 3 - 2 p } { 3 - p } } , & & \\lim _ { n \\to + \\infty } \\frac { f ( 4 \\pi ) - f ( z ( t _ n ) ) } { 4 \\pi - z ( t _ n ) } = f ' ( 4 \\pi ) = \\frac { p } { 3 - p } ( 4 \\pi ) ^ { \\frac { 2 p - 3 } { 3 - p } } , \\end{align*}"}
+{"id": "8786.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ { T _ 1 } ( 1 - a ) ^ { s } ( T _ 1 - s + 1 ) ^ { - \\frac { 2 \\beta - 1 } { \\beta } } & = \\sum _ { m = 1 } ^ { T _ { 1 } } \\rho ^ { T _ { 1 } - m + 1 } m ^ { - \\frac { 2 \\beta - 1 } { \\beta } } + ( T _ { 1 } + 1 ) ^ { - \\frac { 2 \\beta - 1 } { \\beta } } \\\\ & \\leq ( 2 4 \\frac { 2 \\beta - 1 } { \\beta } + 3 ( 2 ^ { \\frac { \\beta - 1 } { \\beta } } ) ) \\frac { T _ { 1 } ^ { - \\frac { 2 \\beta - 1 } { \\beta } } } { a } + ( T _ { 1 } + 1 ) ^ { - \\frac { 2 \\beta - 1 } { \\beta } } \\\\ & \\leq \\mathcal { A } _ 2 T _ { 1 } ^ { - \\frac { 2 \\beta - 1 } { \\beta } } ( 1 + \\frac { 1 } { a } ) . \\end{align*}"}
+{"id": "3828.png", "formula": "\\begin{align*} { \\rm U O T } ( \\mu _ 0 , \\mu _ 1 ) = \\sup _ { \\phi \\in \\Phi _ Y } \\sum _ i \\mu _ i ( \\phi _ i ) , \\end{align*}"}
+{"id": "7938.png", "formula": "\\begin{align*} V _ i = V ( G _ i ) \\setminus \\{ u _ i , v _ i \\} \\ 1 \\end{align*}"}
+{"id": "1065.png", "formula": "\\begin{align*} a _ { i j } \\left ( x \\right ) = a _ { j i } \\left ( x \\right ) , i , j = 2 , . . . n , \\end{align*}"}
+{"id": "5924.png", "formula": "\\begin{align*} d ( ( - 1 ) ^ { ( n + 2 ) / 2 } a _ { 1 , n + 2 } ) = d ( - a _ { n + 1 } a _ { n + 2 } ) = 1 - R _ { n + 2 } < 2 e . \\end{align*}"}
+{"id": "4448.png", "formula": "\\begin{align*} | | [ D ^ { \\alpha } _ { \\ast } , \\mathcal { A } _ 1 ] \\partial _ 1 { \\mathbf V } | | ^ 2 _ { L ^ 2 ( \\Omega _ t ) } & \\lesssim \\sum _ { 0 < \\beta \\leq \\alpha } | | D ^ { \\beta } _ { \\ast } \\mathcal { A } _ 1 D ^ { \\alpha - \\beta } _ { \\ast } ( \\partial _ 1 { \\mathbf V } ) | | ^ 2 _ { L ^ 2 ( \\Omega _ t ) } \\ , . \\end{align*}"}
+{"id": "5943.png", "formula": "\\begin{align*} & K _ a = \\frac { \\pi } { 2 } \\int _ 0 ^ { 2 \\pi } \\frac { a } { ( a ^ 2 \\cos ^ 2 \\theta + \\sin ^ 2 \\theta ) ^ { 1 / 2 } } d \\theta \\\\ & A = \\frac { 4 \\pi ^ 2 a } { K _ a } | u _ j ( x ^ * ) | ^ 2 e ^ { \\phi ( x ^ * ) } , \\\\ & B = \\frac { 4 \\pi ^ 3 a ^ 2 | u _ j ( x ^ * ) | ^ 2 e ^ { \\phi ( x ^ * ) } } { K _ a ^ 2 } ( H ( x ^ * ) - \\partial _ { \\nu } \\phi ( x ^ * ) ) \\end{align*}"}
+{"id": "2702.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { l } f _ { j } \\not \\equiv 0 ( q ) \\prod _ { j = 1 } ^ { l } b _ { j } ^ { c _ { j } f _ { j } } = d ^ { q } d . \\end{align*}"}
+{"id": "1779.png", "formula": "\\begin{align*} \\mathbf { E } \\left [ \\mathbb { Y } _ \\alpha ( n _ 1 ) ^ { m _ 1 } \\cdots \\mathbb { Y } _ \\alpha ( n _ k ) ^ { m _ k } \\right ] & = \\int _ { \\Omega _ 2 } \\prod _ { j = 1 } ^ { k } \\mathbb { Y } _ \\alpha ( n _ j ) ^ { m _ j } \\ , d \\mathbf { P } _ 2 \\\\ & = \\prod _ { j = 1 } ^ { k } \\int _ { S _ 1 } \\omega _ { n _ j } ^ { m _ j } \\ , d \\mathbf { m } ( \\omega _ { n _ j } ) \\end{align*}"}
+{"id": "1766.png", "formula": "\\begin{gather*} d ( f , g ) = \\sum _ { \\nu = 1 } ^ { \\infty } 2 ^ { - \\nu } \\frac { d _ \\nu ( f , g ) } { 1 + d _ \\nu ( f , g ) } , d _ \\nu ( f , g ) = \\sup _ { s \\in K _ \\nu } | f ( s ) - g ( s ) | . \\end{gather*}"}
+{"id": "7286.png", "formula": "\\begin{align*} C ^ { 0 ' } ( \\sigma ) = \\min _ { N } C ( \\sigma \\mid \\ ; \\geq N ) \\pm O ( 1 ) . \\end{align*}"}
+{"id": "6465.png", "formula": "\\begin{align*} \\lambda \\big ( H _ \\epsilon \\big ) = \\frac { \\lambda ( R _ \\epsilon ) } { \\epsilon ^ 2 } . \\end{align*}"}
+{"id": "5969.png", "formula": "\\begin{align*} 1 = \\sigma ( N ^ \\omega _ { \\partial M } ) \\# \\sigma ( \\Lambda _ { g , F } ^ \\omega ) ( x , \\xi ' ) + S ^ { - \\infty } , \\end{align*}"}
+{"id": "6604.png", "formula": "\\begin{align*} f _ 0 ( x , v ) = \\frac { \\mathbf 1 _ { ( - \\infty , 0 ] } \\left ( | v | ^ 2 - \\theta ( x ) ^ { \\frac 2 d } \\right ) } { | B _ 1 | \\ , \\| \\theta \\| _ { L ^ 1 } } , \\ x , v \\in \\R ^ d , \\end{align*}"}
+{"id": "2869.png", "formula": "\\begin{align*} \\chi _ \\lambda ( \\xi ) = \\delta ( \\xi ) ^ { - 1 } \\sum _ { w \\in W _ 0 } ( - 1 ) ^ { \\ell ( w ) } e ^ { i \\langle w \\xi , \\lambda + \\rho \\rangle } ( \\lambda \\in P ^ + ) , \\end{align*}"}
+{"id": "3893.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty | a _ n | ^ 2 = \\infty . \\end{align*}"}
+{"id": "4151.png", "formula": "\\begin{align*} \\exp ^ * ( \\alpha ) & = 1 + \\alpha + \\frac { 1 } { 2 ! } \\alpha * \\alpha + \\frac { 1 } { 3 ! } \\alpha * \\alpha * \\alpha + \\ldots \\\\ \\exp ^ { \\cdot } ( \\alpha ) & = 1 + \\alpha + \\frac { 1 } { 2 ! } \\alpha \\cdot \\alpha + \\frac { 1 } { 3 ! } \\alpha \\cdot \\alpha \\cdot \\alpha + \\ldots . \\end{align*}"}
+{"id": "4458.png", "formula": "\\begin{align*} & | | | \\varphi ( t ) | | | ^ 2 _ { H ^ { s - 1 } ( \\R ) } + | | \\nabla _ { t , x _ 2 } \\varphi | | ^ 2 _ { H ^ { s - 1 } ( \\Gamma _ t ) } \\\\ & \\quad \\leq C ( K ) \\Big ( | | { \\mathbf V } | | ^ 2 _ { s , \\ast , t } + | | \\varphi | | ^ 2 _ { H ^ { s - 1 } ( \\Gamma _ t ) } + ( | | \\dot { { \\mathbf U } } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ t ) } + | | \\varphi | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Gamma _ t ) } ) | | \\hat { W } | | ^ 2 _ { s + 2 , \\ast , t } \\Big ) . \\end{align*}"}
+{"id": "2956.png", "formula": "\\begin{align*} \\begin{bmatrix} n + 1 \\\\ n + 1 \\end{bmatrix} - \\begin{bmatrix} n + 1 \\\\ n + 2 \\end{bmatrix} = J _ { n + 1 } ^ { n + 1 } = J _ n ^ n = \\begin{bmatrix} n \\\\ n \\end{bmatrix} - \\begin{bmatrix} n \\\\ n + 1 \\end{bmatrix} \\end{align*}"}
+{"id": "8672.png", "formula": "\\begin{align*} \\tilde { f } = \\tilde { \\phi } _ 0 ( u _ 1 , . . . , u _ n ) , ~ ~ ~ x _ i = \\tilde { \\phi } _ i ( u _ 1 , . . . , u _ n ) , ~ i = 1 , . . . , n , \\end{align*}"}
+{"id": "7023.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 3 } \\varphi \\rho _ \\Gamma = \\int _ { \\mathbb { R } ^ 3 } \\varphi \\rho , \\forall \\varphi \\in { \\rm S p a n } \\{ \\Phi \\} . \\end{align*}"}
+{"id": "8056.png", "formula": "\\begin{align*} A _ { \\infty } ( \\mathbb { R } ^ { n } ) = \\bigcup \\limits _ { 1 \\leq p < \\infty } A _ { p } ( \\mathbb { R } ^ { n } ) . \\end{align*}"}
+{"id": "8899.png", "formula": "\\begin{align*} C _ 0 : = \\widetilde C / \\langle j \\rangle , C _ 1 : = \\widetilde C / \\langle j \\sigma \\rangle , \\ldots C _ { d - 1 } : = \\widetilde C / \\langle j \\sigma ^ { d - 1 } \\rangle . \\end{align*}"}
+{"id": "8401.png", "formula": "\\begin{align*} W ( v ) = \\sum _ { P < p \\leq 2 P } ( \\log p ) e \\left ( v \\left ( N + j - \\left [ p ^ { c } \\right ] \\right ) ^ { \\gamma } \\right ) \\sum _ { z = 0 } ^ { 2 Z - 1 } \\theta _ { z } ( p ^ c ) = \\sum _ { z = 0 } ^ { 2 Z - 1 } W _ { z } ( v ) , \\end{align*}"}
+{"id": "3010.png", "formula": "\\begin{align*} \\Phi ( \\bar { g } ) = f _ * \\left [ \\Sigma _ { k , 1 } \\right ] \\in H _ 2 \\left ( X , \\gamma ; \\mathbb { Z } \\right ) , \\end{align*}"}
+{"id": "5171.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\frac { { \\rm A o I } ( S _ { \\mathrm { z } } , Q _ { { \\mathrm { u n i } } } , F _ \\mathrm { s } ) } { H [ Q _ { { \\mathrm { u n i } } } ( X ) ] } = \\frac { 3 } { 2 } . \\end{align*}"}
+{"id": "1948.png", "formula": "\\begin{align*} \\| u \\| ^ 2 _ { X } = \\sup _ { t \\in [ 0 , T ] } \\left ( \\int _ I \\left ( u ^ 2 + | \\partial _ x u | ^ 2 \\right ) \\ , { \\rm d } x \\right ) . \\end{align*}"}
+{"id": "7777.png", "formula": "\\begin{align*} \\mathfrak { F } ( Z ^ i ) = \\mathfrak { F } ^ { } ( Z ^ i ) + \\mathfrak { F } ^ { } ( Z ^ i ) \\end{align*}"}
+{"id": "2721.png", "formula": "\\begin{align*} \\binom { r + t - n } { t - n } - \\binom { r - 2 + t - n } { t - n } & = \\binom { 4 + 1 7 1 - 5 } { 1 7 1 - 5 } - \\binom { 4 - 2 + 1 7 1 - 5 } { 1 7 1 - 5 } \\\\ & = 3 3 5 7 1 3 4 2 . \\end{align*}"}
+{"id": "1583.png", "formula": "\\begin{align*} \\int \\frac { h ^ 2 } { g } = a ^ 2 + \\beta ^ 2 + \\gamma ^ 2 + \\int \\frac { \\tilde { h } ^ 2 } { g } \\end{align*}"}
+{"id": "3694.png", "formula": "\\begin{align*} a _ n \\left ( 3 b _ n \\lambda _ \\beta ' ( 2 ^ { - 4 n - 1 } ) ^ { - 1 } \\right ) ^ { - 1 } = 3 ^ { - 1 } ( 2 - 2 \\beta ) ^ { \\beta / ( 1 - \\beta ) } 2 ^ { 2 ( 2 \\alpha - \\beta ) n / ( 1 - \\beta ) } \\to \\infty \\end{align*}"}
+{"id": "4371.png", "formula": "\\begin{align*} \\partial _ t \\varphi = u ^ { \\pm } _ { N } , H ^ { \\pm } _ { N } = 0 , \\quad [ q ] = 0 \\end{align*}"}
+{"id": "6472.png", "formula": "\\begin{align*} P _ D F _ 1 = 0 , P _ N F _ 1 ' = 0 , P _ R F _ 1 ' = \\Lambda P _ R F _ 0 , \\end{align*}"}
+{"id": "4221.png", "formula": "\\begin{align*} d \\big ( \\tilde J _ \\rho \\rho \\big ) = \\ , & \\ , q _ 1 e ^ { 1 2 3 4 } + q _ 2 e ^ { 1 2 3 5 } + q _ 3 e ^ { 1 2 4 5 } + q _ 4 ( e ^ { 1 2 4 6 } - e ^ { 1 3 5 6 } ) + q _ 5 e ^ { 1 2 5 6 } + q _ 6 e ^ { 1 3 4 5 } + q _ 7 ( e ^ { 1 3 4 6 } + e ^ { 2 3 5 6 } ) \\\\ [ 2 p t ] & + q _ 8 e ^ { 2 3 4 5 } + q _ 9 e ^ { 2 3 4 6 } , \\end{align*}"}
+{"id": "6272.png", "formula": "\\begin{align*} P _ { \\lambda } ( u _ \\mu g ' ( u _ { < \\lambda } ) \\partial _ x ^ 2 u _ { \\mu _ 1 } ) = P _ { \\lambda } ( u _ \\mu P _ { \\geq \\mu _ 1 } g ' ( u _ { < \\lambda } ) \\partial _ x ^ 2 u _ { \\mu _ 1 } ) , \\end{align*}"}
+{"id": "4111.png", "formula": "\\begin{align*} \\binom i \\ell : = \\begin{cases} \\displaystyle \\frac { i ( i - 1 ) \\cdots ( i - \\ell + 1 ) } { \\ell ! } , & \\ell \\ge 0 , \\\\ 0 , & \\ell < 0 , \\end{cases} \\end{align*}"}
+{"id": "6738.png", "formula": "\\begin{align*} \\begin{aligned} & - 4 \\pi \\left ( C _ { 2 } \\mathfrak { c } _ { p } \\frac { m } { a } + C _ { 1 } \\right ) - 2 C _ { 1 } \\int _ { \\Sigma } H | \\nabla u | + \\left ( C _ { 2 } \\mathfrak { c } _ { p } \\frac { m } { a } + C _ { 1 } \\right ) 2 ^ { 4 a } ( I _ { a } ( 1 ) ) ^ { 2 } \\int _ { \\Sigma } | \\nabla u | ^ 2 \\\\ \\ge & - C _ { 2 } \\mathfrak { c } _ { p } 8 \\pi ( \\mathfrak { m } _ { A D M } - m ) . \\end{aligned} \\end{align*}"}
+{"id": "4885.png", "formula": "\\begin{align*} \\mathcal { B } ( \\mathfrak { E } \\mathfrak { F } ) = E _ + \\mathcal { B } ( \\mathfrak { F } ) \\oplus F _ - \\mathcal { B } ( \\mathfrak { E } ) . \\end{align*}"}
+{"id": "7937.png", "formula": "\\begin{align*} \\Psi _ { 1 , 2 } = 4 + \\sum _ { i = 1 } ^ 2 | X _ i | + \\sum _ { i = 1 } ^ 2 | Y _ i | . \\end{align*}"}
+{"id": "6289.png", "formula": "\\begin{align*} f ^ { 4 , b a l } _ { \\lambda , m } = q ^ { 4 , b a l } _ { \\lambda , m } [ ( \\xi _ 1 - \\xi _ 2 ) ( \\xi _ 3 - \\xi _ 4 ) + ( \\xi _ 1 - \\xi _ 4 ) ( \\xi _ 2 - \\xi _ 3 ) ] , \\end{align*}"}
+{"id": "1933.png", "formula": "\\begin{align*} \\mathcal { K } ^ 1 _ { i j } ( v , \\eta _ f , \\theta _ f ) = \\mathcal { K } ^ 1 _ { i j } ( v - \\bar { v } , \\eta _ f , \\theta _ f ) + \\mathcal { K } ^ 1 _ { i j } ( \\bar v , \\eta _ f , \\theta _ f ) . \\end{align*}"}
+{"id": "8590.png", "formula": "\\begin{align*} \\phi _ 0 = \\dfrac { \\cosh { ( ( z + 1 ) \\sqrt { \\mu } | \\mathrm { D } | ) } } { \\cosh { ( \\sqrt { \\mu } | \\mathrm { D } | ) } } \\psi . \\end{align*}"}
+{"id": "2483.png", "formula": "\\begin{align*} H _ \\kappa ( G ) & \\leq I ( V ; W ^ * ) \\\\ & = I ( V , A ; W ^ * ) \\\\ & = I ( A ; W ^ * ) + \\sum _ { a \\in \\mathcal { A } } P _ A ( a ) I ( V ; W ^ * | A = a ) \\\\ & = \\sum _ { a \\in \\mathcal { A } } P _ A ( a ) I ( V _ a ; W _ a ^ * ) \\\\ & = \\sum _ { a \\in \\mathcal { A } } P _ A ( a ) H _ \\kappa ( G _ a ) ; \\end{align*}"}
+{"id": "7008.png", "formula": "\\begin{align*} v _ 0 & = z _ { ( 0 ) } , \\ v _ 1 = \\frac { 1 } { 2 } ( s _ 2 z _ { ( 0 ) } + c x _ { ( 0 ) } ) , \\ v _ { m + 2 } = s _ 2 v _ { m + 1 } - v _ m \\\\ w _ 0 & = z _ { ( 1 ) } , \\ w _ 1 = \\frac { 1 } { 2 } ( t z _ { ( 1 ) } + c y _ { ( 1 ) } ) , \\ w _ { n + 2 } = t w _ { n + 1 } - w _ n . \\end{align*}"}
+{"id": "934.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 y } \\bigg ( - \\frac { m } { \\sqrt { m n } } C + D \\bigg ) y ^ { 1 + \\sqrt { m n } - c } \\\\ ~ ~ = & \\frac { \\alpha x ^ { 1 + \\sqrt { m n } - c } } { 4 t ^ \\alpha } \\mathrm { e } ^ { - \\frac { \\alpha ( x ^ 2 + y ^ 2 ) } { t ^ \\alpha } } \\bigg ( \\frac y x \\bigg ) ^ { \\frac { 1 + \\sqrt { m n } - c } { 2 } } I _ { \\frac { 1 + \\sqrt { m n } - c } { 2 } } \\bigg ( \\frac { \\alpha \\sqrt { x y } } { 2 t ^ \\alpha } \\bigg ) . \\end{align*}"}
+{"id": "1807.png", "formula": "\\begin{gather*} \\mathbf { E } \\left [ | \\zeta ( s , \\mathbb { X } _ \\alpha ) | \\right ] ^ 2 \\leq \\mathbf { E } \\left [ | \\zeta ( s , \\mathbb { X } _ \\alpha ) | ^ 2 \\right ] = \\sum _ { m = 0 } ^ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\mathbf { E } [ \\mathbb { X } _ \\alpha ( m ) \\overline { \\mathbb { X } _ \\alpha ( n ) } ] } { ( m + \\alpha ) ^ s ( n + \\alpha ) ^ { \\overline { s } } } . \\end{gather*}"}
+{"id": "1305.png", "formula": "\\begin{align*} a ( x ) = \\begin{cases} | x | ^ 2 \\ \\ f o r \\ \\ | x | \\leq R \\\\ C R ^ 2 \\ \\ f o r \\ \\ | x | > 2 R \\end{cases} \\end{align*}"}
+{"id": "1573.png", "formula": "\\begin{align*} \\mathcal { K } _ 2 ( T ( x + y , y ) : = \\int _ { 0 } ^ { \\infty } \\frac { | v | e ^ { y / \\kappa | v | } } { \\kappa ( e ^ { 1 / \\kappa | v | } - 1 ) } \\mathcal { M } _ { T ( x + y ) } ( v ) \\mathrm { d } v . \\end{align*}"}
+{"id": "2299.png", "formula": "\\begin{align*} \\begin{cases} i \\partial _ { t } u + \\partial ^ { 2 } _ { x } u = \\alpha u v + \\beta | u | ^ 2 u , \\\\ \\partial _ { t } v + \\gamma \\partial ^ { 3 } _ { x } v - \\delta \\partial ^ { 5 } _ { x } v + v \\partial _ { x } v = \\varepsilon \\partial _ { x } | u | ^ 2 , \\\\ u ( x , 0 ) = u _ 0 ( x ) , v ( x , 0 ) = v _ 0 ( x ) , \\end{cases} & ( x , t ) \\in \\R ^ + \\times \\R ^ + , \\end{align*}"}
+{"id": "7158.png", "formula": "\\begin{align*} [ f , [ H _ 1 , H _ 2 ] - \\frac { \\partial H _ 1 } { \\partial t _ { 2 } } - \\frac { \\partial H _ 2 } { \\partial t _ { 1 } } ] = 0 , f = q _ i , p _ j . \\end{align*}"}
+{"id": "1064.png", "formula": "\\begin{align*} a _ { i j } \\left ( x \\right ) , b _ { j } \\left ( x \\right ) , c \\left ( x \\right ) \\in C ^ { 2 } \\left ( \\overline { \\Omega } \\right ) , i , j = 2 , . . . n , \\end{align*}"}
+{"id": "2955.png", "formula": "\\begin{align*} \\begin{bmatrix} n + 1 \\\\ 0 \\end{bmatrix} - \\begin{bmatrix} n + 1 \\\\ 1 \\end{bmatrix} = J _ 0 ^ { n + 1 } = J _ 1 ^ n = \\begin{bmatrix} n \\\\ 1 \\end{bmatrix} - \\begin{bmatrix} n \\\\ 2 \\end{bmatrix} \\end{align*}"}
+{"id": "996.png", "formula": "\\begin{align*} \\mathbb { P } ( Y _ j \\in U \\ \\big | \\ Y _ 1 = y _ 1 , \\dots , Y _ { j - 1 } = y _ { j - 1 } ) & = \\sum _ { y \\in U - W } \\mathbb { P } ( Y _ j = y \\ \\big | \\ Y _ 1 = y _ 1 , \\dots , Y _ { j - 1 } = y _ { j - 1 } ) \\le | U - W | \\beta \\\\ & \\le ( q ^ m - q ^ \\ell ) \\beta . \\end{align*}"}
+{"id": "1654.png", "formula": "\\begin{align*} _ x ( \\mathbf { x } ) : = | \\{ 1 \\leq j \\leq n \\mid x _ j = x \\} | . \\end{align*}"}
+{"id": "5831.png", "formula": "\\begin{align*} & s _ 4 s _ 7 s _ 6 s _ 5 ( s _ 2 s _ 4 s _ 3 s _ 8 s _ 1 s _ 3 s _ 4 s _ 2 ) s _ 5 s _ 6 s _ 7 s _ 4 = \\\\ & s _ 4 s _ 7 s _ 6 s _ 5 ( s _ 8 s _ { \\alpha _ 1 + \\alpha _ 3 + \\alpha _ 4 + \\alpha _ 2 } ) s _ 5 s _ 6 s _ 7 s _ 4 = \\\\ & s _ { \\alpha _ 7 + \\alpha _ 8 } s _ { \\alpha _ 1 + \\alpha _ 3 + 2 \\alpha _ 4 + \\alpha _ 2 + \\alpha _ 5 + \\alpha _ 6 + \\alpha _ 7 } . \\\\ \\end{align*}"}
+{"id": "5434.png", "formula": "\\begin{align*} \\sum _ r x _ r D _ r g & = 3 \\sum _ r x _ r D _ r \\left ( \\sum _ { i < j } d _ T ( i , j ) x _ i x _ j \\right ) \\\\ & = 3 \\sum _ r x _ r \\sum _ { j } d _ T ( r , j ) x _ j \\\\ & = 6 \\sum _ { r < j } d _ T ( r , j ) x _ r x _ j = 2 g . \\end{align*}"}
+{"id": "1422.png", "formula": "\\begin{gather*} \\varphi _ { n , 1 } ( x ) = \\psi _ { n 1 } ( x ) , \\varphi _ { n , 0 } ( x ) = \\xi _ n \\psi _ { n 0 } ( x ) + \\psi _ { n 1 } ( x ) . \\end{gather*}"}
+{"id": "1378.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { \\infty } \\lambda _ r r ^ { 2 k - 1 } \\mathcal { F } _ { 2 - 2 k , - 1 } | _ { 2 - 2 k } T _ r = \\sum _ { r = 1 } ^ { \\infty } \\lambda _ r \\mathcal { F } _ { 2 - 2 k , - r } \\in M _ { 2 - 2 k } ^ ! \\end{align*}"}
+{"id": "3382.png", "formula": "\\begin{align*} \\Upsilon _ 0 = \\frac { 1 } { n + 1 } \\left ( 2 a + \\partial _ \\mu b ^ \\mu + \\alpha ^ i _ { i \\nu } b ^ \\nu \\right ) . \\end{align*}"}
+{"id": "5012.png", "formula": "\\begin{align*} \\gamma = d r ^ 2 + r \\ , ( d r \\otimes \\beta + \\beta \\otimes d r ) + r ^ 2 h , \\end{align*}"}
+{"id": "3241.png", "formula": "\\begin{align*} \\frac { R } { q } = O \\left ( \\epsilon X R \\right ) . \\end{align*}"}
+{"id": "5433.png", "formula": "\\begin{align*} \\sum _ r x _ r D _ r p & = \\sum _ r x _ r ( g + s D _ r g ) \\\\ & = g \\sum _ r x _ r + s \\sum _ r x _ r D _ r g \\\\ & = s ( g + \\sum _ r x _ r D _ r g ) . \\end{align*}"}
+{"id": "5935.png", "formula": "\\begin{align*} A _ { i } \\le \\dfrac { R _ { i + 1 } - S _ { i } } { 2 } + e = \\dfrac { 0 - ( - 2 e ) } { 2 } + e = 2 e \\le d [ a _ { 1 , i } b _ { 1 , i } ] \\end{align*}"}
+{"id": "8069.png", "formula": "\\begin{align*} \\begin{aligned} \\left \\| f \\right \\| _ { h _ { \\omega } ^ { p } } & = \\left \\| T _ { N } ^ { - 1 } \\circ T _ { N } ( f ) \\right \\| _ { h _ { \\omega } ^ { p } } \\\\ & \\leq C \\left \\| T _ { N } ( f ) \\right \\| _ { h _ { \\omega } ^ { p } } \\\\ & \\leq C \\left \\| \\left \\{ \\sum \\limits _ { j \\in \\mathbb N } \\sum \\limits _ { Q \\in \\Pi _ { j + N } } \\vert ( \\psi _ { j } \\ast f ) ( u _ { Q } ) \\vert ^ { 2 } \\chi _ { Q } \\right \\} ^ { \\frac { 1 } { 2 } } \\right \\| _ { L _ { \\omega } ^ { p } } \\end{aligned} \\end{align*}"}
+{"id": "6621.png", "formula": "\\begin{align*} \\rho _ { ( 2 ) , \\infty } ^ T ( x , 0 ) = \\rho _ { ( 1 ) , \\infty } ^ { ( \\widetilde { \\rm c J } ) } ( x ; \\beta , 1 , 0 ) - 1 , \\end{align*}"}
+{"id": "2681.png", "formula": "\\begin{align*} G \\cdot \\left ( \\frac { \\omega _ { 0 } ^ { n } } { \\omega _ t ^ { n } } \\right ) ^ { \\frac { \\alpha } { 2 \\beta + \\alpha } } \\ = \\ G \\cdot \\left ( e ^ { - ( 1 + \\frac { \\alpha } { 2 \\beta } ) \\varphi _ t } \\right ) ^ { \\frac { \\alpha } { 2 \\beta + \\alpha } } \\ = \\ G \\cdot e ^ { - \\frac { \\alpha } { 2 \\beta } \\varphi _ t } . \\end{align*}"}
+{"id": "8812.png", "formula": "\\begin{align*} h ( t ) & \\leq \\rho ^ { 2 } \\sum _ { i = 1 } ^ { n } \\norm { x ^ { i } ( t - 1 ) - \\bar { x } ( t - 1 ) - \\eta _ { t } \\big ( g ^ { i } ( t ) - \\bar { g } ( t ) \\big ) } ^ { 2 } \\\\ & \\leq \\rho ^ { 2 } h ( t - 1 ) + \\rho ^ { 2 } \\eta _ { t } ^ { 2 } \\sum _ { i = 1 } ^ { n } \\norm { g ^ { i } ( t ) - \\bar { g } ( t ) } ^ { 2 } - 2 \\rho ^ { 2 } \\eta _ { t } \\sum _ { i = 1 } ^ { n } \\langle x ^ { i } ( t - 1 ) - \\bar { x } ( t - 1 ) , g ^ { i } ( t ) - \\bar { g } ( t ) \\rangle . \\end{align*}"}
+{"id": "1678.png", "formula": "\\begin{align*} \\hat { \\Delta } ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } : = \\Biggl ( \\sum _ { \\mu \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { a } } } \\left | P _ { \\texttt { a } ; \\mu } \\bigl ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } ; q \\bigr ) \\right | ^ 2 \\delta ^ { ( m , n ) } _ { \\texttt { a } ; \\mu } ( q ) \\Biggr ) ^ { - 1 } . \\end{align*}"}
+{"id": "2432.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\overline { \\nabla u _ n ( x ) } \\cdot \\nabla u _ m ( x ) \\ , { \\rm d } x = \\delta _ { n m } = \\begin{cases} 1 & m = n , \\\\ 0 & m \\ne n \\end{cases} , \\forall 1 \\leq n , m \\leq N . \\end{align*}"}
+{"id": "1451.png", "formula": "\\begin{align*} \\mathcal { P } _ 2 = \\pi \\sum \\limits _ { i = 1 } ^ { n } a ^ { i i } \\Big ( \\sum \\limits _ { t \\in I } k _ { i t } \\sigma ^ k _ t ( r _ k ) - 2 \\alpha _ i \\Big ) ^ 2 - 4 \\pi \\sum \\limits _ { i = 1 } ^ { n } a ^ { i i } \\alpha _ { i } ^ { 2 } + o ( 1 ) . \\end{align*}"}
+{"id": "6915.png", "formula": "\\begin{align*} 0 \\leq - d _ { 1 } \\mathcal { N } _ 1 [ \\phi ] ( \\xi _ 0 ) & = - \\displaystyle \\frac { m \\psi ( \\xi _ 0 ) } { 1 + a \\psi ( \\xi _ 0 ) } < 0 . \\end{align*}"}
+{"id": "3767.png", "formula": "\\begin{align*} { \\rm U O T } ( \\mu _ 0 , \\mu _ 1 ) = \\inf _ { \\alpha \\in S ( \\mu _ 0 , \\mu _ 1 ) } ( H , \\alpha ) , ( H , \\alpha ) : = \\int _ { Y \\times Y } H ( y _ 0 , y _ 1 ) d \\alpha ( y _ 0 , y _ 1 ) , \\end{align*}"}
+{"id": "5953.png", "formula": "\\begin{align*} - \\Delta _ g - g ( F , \\nabla _ g \\cdot ) - \\omega ^ 2 = D _ { t _ 3 } ^ 2 + i \\widetilde { E } ( t ) D _ { t _ 3 } + \\widetilde { Q ^ \\omega } ( t , D _ { t ' } ) , \\end{align*}"}
+{"id": "3808.png", "formula": "\\begin{align*} \\pi ^ i _ { \\# } \\nu _ X = \\mu _ j ( X ) \\mu _ i \\neq \\mu _ i , \\end{align*}"}
+{"id": "7549.png", "formula": "\\begin{align*} \\varkappa : P _ { 0 , F } \\times _ F \\chi = \\omega _ { D , \\log } . \\end{align*}"}
+{"id": "5748.png", "formula": "\\begin{align*} \\int _ { \\mathbb B _ { 1 / 2 } ^ + } U _ j ^ 2 ( X , 0 ) x _ { n + 1 } ^ a d X = 1 . \\end{align*}"}
+{"id": "3452.png", "formula": "\\begin{align*} M u ( E ) = \\int _ { \\nabla u ( E ) } W d p , E \\subset ( \\Delta ) . \\end{align*}"}
+{"id": "4442.png", "formula": "\\begin{align*} | | | ( \\tilde { \\mathcal { A } } \\mathcal { F } ) ( t ) | | | ^ 2 _ { s - 1 , \\ast } & \\lesssim | | \\tilde { \\mathcal { A } } J ^ T \\mathbf { F } | | ^ 2 _ { s , \\ast , t } \\\\ & \\leq C ( K ) \\Big ( | | \\mathbf { F } | | ^ 2 _ { s , \\ast , t } + | | \\mathbf { F } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ t ) } | | \\hat { W } | | ^ 2 _ { s , \\ast , t } \\Big ) \\\\ & \\leq C ( K ) \\mathcal { M } ( t ) . \\end{align*}"}
+{"id": "2227.png", "formula": "\\begin{align*} \\frac { b _ 1 } { \\alpha _ 1 } + \\frac { b _ 2 } { \\alpha _ 2 } + \\frac { b _ 3 } { \\alpha _ 3 } = 0 . \\end{align*}"}
+{"id": "1775.png", "formula": "\\begin{gather*} \\mathbf { E } [ \\mathcal { X } ] = \\int _ { \\Omega } \\mathcal { X } \\ , d \\mathbf { P } \\end{gather*}"}
+{"id": "7757.png", "formula": "\\begin{align*} N ' = \\{ p \\in N | f ( p ) \\neq 0 , f _ 3 ( p ) \\neq 0 , g _ N ( V _ p , V _ p ) \\neq 0 \\} , \\end{align*}"}
+{"id": "2315.png", "formula": "\\begin{align*} \\begin{cases} \\Gamma _ 1 ( u , v ) : = \\psi _ T S _ { \\lambda } ( t ) u _ { 0 , \\lambda } - i \\psi _ T \\int _ { 0 } ^ { t } S _ { \\lambda } ( t - t ' ) ( \\alpha \\lambda u v + \\beta \\lambda ^ { - 3 } | u | ^ 2 u ) d t ' , \\\\ \\Gamma _ 2 ( u , v ) : = \\psi _ T W _ { \\lambda } ( t ) v _ { 0 , \\lambda } + \\psi _ T \\int _ { 0 } ^ { t } W _ { \\lambda } ( t - t ' ) ( - v \\partial _ { x } v + \\varepsilon \\partial _ { x } | u | ^ 2 ) d t ' . \\end{cases} \\end{align*}"}
+{"id": "3952.png", "formula": "\\begin{align*} \\frac { h _ K ( v _ 1 ) } { h _ L ( v _ 1 ) } = \\frac { h _ K ( v _ 2 ) } { h _ L ( v _ 2 ) } \\end{align*}"}
+{"id": "7125.png", "formula": "\\begin{align*} f _ 3 ( t , x , z ) : = - \\sum _ { i = 1 , 2 } ( 1 + t ) ^ { - \\frac { i - 1 } { 2 } } ( a _ i ( x ) + z ( 1 + t ) ^ { 1 / 4 } b _ i ( x ) ) \\chi ( z ) . \\end{align*}"}
+{"id": "1158.png", "formula": "\\begin{align*} & \\sum _ { i + j = n } [ m _ { 1 , i } , m _ { 2 , j } ] = 0 . \\end{align*}"}
+{"id": "6641.png", "formula": "\\begin{align*} { \\rho } _ { ( 1 ) , \\infty } ^ { ( \\rm c J ) } ( x ; \\beta , p , q ) | _ { \\beta = 2 \\atop p = q = 0 } = 1 . \\end{align*}"}
+{"id": "2567.png", "formula": "\\begin{align*} \\tilde { I } [ u ] & \\coloneqq I _ { + } [ u ] - \\psi ( \\| u \\| _ E ^ p ) I _ { - } [ u ] \\\\ & = \\frac { 1 } { p } \\| u \\| _ { D ^ { s , p } } ^ p + \\frac { 1 } { p } \\int _ { \\mathbb { R } ^ N } V ( x ) | u | ^ p d x \\\\ & \\phantom { = } - \\frac { 1 } { 2 \\cdot p _ { r ; s } ^ { \\uparrow * } } \\psi ( \\| u \\| _ E ^ p ) \\int _ { \\mathbb { R } ^ N } ( K \\ast g ( u ) ) g ( u ) d x - \\varepsilon _ W \\psi ( \\| u \\| _ E ^ p ) \\int _ { \\mathbb { R } ^ N } W ( x ) f ( u ) d x . \\end{align*}"}
+{"id": "6002.png", "formula": "\\begin{align*} A _ { q , \\kappa , p } ^ { \\delta , \\eta } ( h ) = \\frac { \\delta ^ { q } } { \\eta ^ { 2 q } } + \\eta ^ { - p } h ^ { p } + \\eta ^ { \\kappa } , h > 0 . \\end{align*}"}
+{"id": "2303.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) : = \\int _ { \\R } \\alpha \\varepsilon | u | ^ { 2 } v - \\frac { \\alpha } { 6 } v ^ { 3 } + \\frac { \\beta \\varepsilon } { 2 } | u | ^ 4 + \\frac { \\alpha \\gamma } { 2 } | \\partial _ { x } v | ^ { 2 } - \\frac { \\alpha \\delta } { 2 } | \\partial ^ { 2 } _ { x } v | ^ { 2 } + \\varepsilon | \\partial _ { x } u | ^ { 2 } d x . \\end{align*}"}
+{"id": "7759.png", "formula": "\\begin{align*} g _ { \\overline { N } } : = - \\frac { 1 } { f } \\left ( \\frac { 2 } { f _ 3 } \\eta ^ 2 + \\pi ^ * _ N g _ N - \\frac { 2 } { f } \\sum _ { i = 0 } ^ 3 ( \\theta _ i ^ P ) ^ 2 \\right ) \\Bigg | _ { \\overline { N } } \\end{align*}"}
+{"id": "4585.png", "formula": "\\begin{align*} Y ( a , z ) \\ , b = \\sum _ { n \\in \\mathbb { Z } } a _ n b \\ , z ^ { - n - 1 } . \\end{align*}"}
+{"id": "7002.png", "formula": "\\begin{align*} & K \\left ( e , \\bar { e } , \\bar { e } , n \\right ) - K \\left ( n , \\bar { n } , \\bar { e } , n \\right ) = W \\left ( e , \\bar { e } , \\bar { e } , n \\right ) - W \\left ( n , \\bar { n } , \\bar { e } , n \\right ) = 2 W \\left ( e , \\bar { e } , \\bar { e } , n \\right ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
+{"id": "5090.png", "formula": "\\begin{align*} | \\nabla f | ^ 2 + S & = f \\ , . \\end{align*}"}
+{"id": "5208.png", "formula": "\\begin{align*} s _ 1 s _ 2 \\cdots s _ i ( \\bar \\alpha _ i ^ \\vee ) - \\eta = & s _ 1 \\cdots s _ { i - 1 } ( v _ i ) - s _ 1 \\cdots s _ i ( v _ i ) \\\\ & + s _ 1 \\cdots s _ i ( 0 ) - \\eta . \\end{align*}"}
+{"id": "440.png", "formula": "\\begin{align*} h _ { \\beta , \\gamma } ( r ) & : = \\int _ 0 ^ \\infty \\frac { d t } { t } \\ , t ^ \\beta \\int _ 0 ^ \\infty d s \\ , s ^ \\gamma p _ \\zeta ^ { ( \\alpha ) } ( t , r , s ) = C ^ { ( \\alpha ) } \\left ( \\beta , \\gamma , \\zeta \\right ) r ^ { \\alpha \\beta + \\gamma - 2 \\zeta } , r > 0 , \\end{align*}"}
+{"id": "4750.png", "formula": "\\begin{align*} { \\rm P e r } ( G _ { \\lambda } \\cap \\overline { T } _ n , \\overline { T } _ n ) & \\ge \\frac { 1 } { 2 } { \\rm P e r } _ { \\rm e u c } ( \\widetilde { G } _ { \\lambda , n } ; \\R ^ 2 ) \\\\ & \\ge C \\mathcal { L } ^ 2 ( \\widetilde { G } _ { \\lambda , n } ) ^ { \\frac { 1 } { 2 } } \\\\ & = C ' \\mathcal { L } ^ 2 ( G _ { \\lambda } \\cap T _ n ) ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
+{"id": "4691.png", "formula": "\\begin{align*} \\mu ^ n _ 5 ( S , W ) = \\frac { q ^ { 2 n - \\dim ( S + W ) - 2 } } { ( q - 1 ) } \\cdot \\begin{cases} \\left ( q ^ { 2 n - 2 } - q ^ { 2 n - \\dim ( S + W ) } \\right ) & \\Psi ( w _ S , u _ S ) = 0 , \\\\ \\left ( q ^ { 2 n - 2 } - q ^ { 2 n - \\dim ( S + W ) } - q ^ { 2 } + 1 \\right ) & \\Psi ( w _ S , u _ S ) \\neq 0 . \\end{cases} \\end{align*}"}
+{"id": "7697.png", "formula": "\\begin{align*} \\left ( { \\varphi ' } _ { \\sigma ' } ^ { \\sigma '' } [ x ' ] \\circ \\varphi _ { \\sigma } ^ { \\sigma ' } [ x ] \\right ) [ x '' ] \\not = 0 \\implies x '' \\subseteq x ' x . \\end{align*}"}
+{"id": "2500.png", "formula": "\\begin{align*} \\begin{aligned} \\nabla _ { \\boldsymbol { y } } \\mathcal { L } ( \\boldsymbol { y } , \\boldsymbol { u } , \\boldsymbol { p } ) = 0 , \\nabla _ { \\boldsymbol { u } } \\mathcal { L } ( \\boldsymbol { y } , \\boldsymbol { u } , \\boldsymbol { p } ) = 0 , \\nabla _ { \\boldsymbol { p } } \\mathcal { L } ( \\boldsymbol { y } , \\boldsymbol { u } , \\boldsymbol { p } ) = 0 , \\end{aligned} \\end{align*}"}
+{"id": "5440.png", "formula": "\\begin{align*} \\int e ^ f d \\mu \\leq F \\left ( \\int G \\left ( | \\nabla f | \\right ) d \\mu \\right ) \\mbox { f o r a l l s m o o t h } \\ f \\ \\mbox { w i t h } \\int f d \\mu = 0 , \\end{align*}"}
+{"id": "5144.png", "formula": "\\begin{align*} D ( Q ) = \\sum _ i \\int _ { a _ { i - 1 } } ^ { a _ i } ( x - c _ i ) ^ 2 f ( x ) d x . \\end{align*}"}
+{"id": "1397.png", "formula": "\\begin{gather*} \\varkappa _ a ( \\rho ) = \\int _ { - a } ^ { a } f ( x ) e ^ { i \\rho x } d x , f \\in L _ 2 ( - a , a ) . \\end{gather*}"}
+{"id": "3099.png", "formula": "\\begin{align*} \\alpha & = \\lambda _ 3 + o ( 1 ) = \\Theta ( 1 ) \\\\ \\beta & = O \\left ( \\frac { 1 } { n } \\right ) \\\\ \\gamma & = O \\left ( \\frac { 1 } { n ^ 2 } \\right ) . \\end{align*}"}
+{"id": "5474.png", "formula": "\\begin{align*} ( z ; p _ 1 , \\dots , p _ n ) : = \\prod _ { i _ 1 , \\dots , i _ n \\in \\mathbb Z _ { \\geq 0 } } ( 1 - z p _ 1 ^ { i _ 1 } \\dots p _ n ^ { i _ n } ) . \\end{align*}"}
+{"id": "4361.png", "formula": "\\begin{align*} { \\rm T r } ( \\varrho ( \\log ( \\varrho ) - \\log ( \\tilde { \\varrho } ) ) ) = { \\rm T r } ( \\varrho ( ( - \\boldsymbol { H } ) - F ( - \\boldsymbol { H } ) F ^ * ) = { \\rm T r } ( \\varrho F [ \\boldsymbol { H } , F ^ * ] ) \\end{align*}"}
+{"id": "1659.png", "formula": "\\begin{align*} \\omega _ j = e _ 1 + \\cdots + e _ j - { \\textstyle \\frac { j } { n } } ( e _ 1 + \\cdots + e _ { n } ) ( j = 1 , \\ldots , n - 1 ) \\end{align*}"}
+{"id": "900.png", "formula": "\\begin{align*} \\xi _ u = \\xi _ v = \\tau _ x = \\tau _ u = \\tau _ v = \\phi _ { u u } = \\phi _ { u v } = \\phi _ { v v } = \\eta _ { u u } = \\eta _ { u v } = \\eta _ { v v } = 0 . \\end{align*}"}
+{"id": "1029.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = 1 } ^ n \\lambda _ i x _ i \\right \\| \\ge \\frac { 1 } { 2 } \\left ( \\sum _ { i = 1 } ^ n | \\lambda _ i | \\right ) \\cdot \\min \\{ d ( x _ i , x _ j ) \\mid i \\neq j \\} \\end{align*}"}
+{"id": "3052.png", "formula": "\\begin{align*} W ( M ) : = \\frac { 4 } { \\sqrt { 3 } } \\Big [ \\frac { 1 } { 2 } \\Big ( \\psi _ 1 ( | M e _ 1 | ) + \\psi _ 1 ( | M \\nu | ) + \\psi _ 1 ( | M \\eta | ) \\Big ) + 3 \\ , \\psi _ 2 ( \\det M ) \\Big ] \\ , . \\\\ \\end{align*}"}
+{"id": "4908.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ n \\dfrac { q ^ n } { 1 - q ^ { 2 n + 1 } } = \\dfrac { ( q ^ 4 ; q ^ 4 ) _ \\infty ^ 2 } { ( q ^ 2 ; q ^ 4 ) _ \\infty ^ 2 } , \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ n \\dfrac { q ^ { 2 n } ( 1 + q ^ { 2 n + 1 } ) } { ( 1 - q ^ { 2 n + 1 } ) ^ 3 } = \\dfrac { ( q ^ 2 ; q ^ 4 ) _ \\infty ^ 2 ( q ^ 4 ; q ^ 4 ) _ \\infty ^ 6 } { ( q ; q ^ 2 ) _ \\infty ^ 4 } , \\end{align*}"}
+{"id": "285.png", "formula": "\\begin{align*} \\mathbf { H } \\left ( \\omega , \\operatorname * { d i v } \\right ) : = \\left \\{ \\mathbf { w } \\in \\mathbf { L } ^ { 2 } \\left ( \\Omega \\right ) \\mid \\operatorname * { d i v } \\mathbf { w } \\in L ^ { 2 } \\left ( \\omega \\right ) \\right \\} . \\end{align*}"}
+{"id": "6487.png", "formula": "\\begin{align*} P _ D \\Psi = 0 , P _ N \\Psi ' = 0 , P _ R \\Psi ' = \\epsilon \\Lambda P _ R \\Psi + O ( \\epsilon ^ 2 ) . \\end{align*}"}
+{"id": "131.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } \\Phi ( \\tau ) d B ( \\tau ) = P _ S \\Phi ( t ) , t \\in [ 0 , 1 ) \\end{align*}"}
+{"id": "6049.png", "formula": "\\begin{align*} \\Psi _ k ( y ) = \\psi ( \\vert y \\vert - ( k - \\tfrac { 1 } { 2 } ) ) , \\ \\forall k \\in \\mathbb { N } . \\end{align*}"}
+{"id": "7932.png", "formula": "\\begin{align*} F ( I ( g ) - Z _ 1 - Z _ 2 ) = 2 g \\oplus X \\oplus 3 Y . \\end{align*}"}
+{"id": "3891.png", "formula": "\\begin{align*} ( ( \\overline { D _ x } + \\overline { D _ y } ) \\circ \\det ) ( k ) & = ( \\overline { D _ x } + \\overline { D _ y } ) ( \\det ( k ) ) \\\\ & \\overset { } { = } \\overline { D _ x } ( \\det ( k ) ) + \\overline { D _ y } ( \\det ( k ) ) \\\\ & = ( \\overline { D _ x } \\circ \\det ) ( k ) + ( \\overline { D _ y } \\circ \\det ) ( k ) \\\\ & \\xrightarrow { \\alpha _ x + \\alpha _ y } D _ x ( k ) + D _ y ( k ) \\\\ & \\overset { } { = } ( D _ x + D _ y ) ( k ) \\end{align*}"}
+{"id": "7373.png", "formula": "\\begin{align*} Y ^ { ( k ) } _ { i j } = ( t _ i - t _ { i - 1 } ) \\mu ^ { ( k ) } _ { j } + \\varepsilon _ { i j k } ^ { ( 1 ) } , ~ ~ ~ ~ i = 1 , . . . , n ~ , ~ ~ j = 1 , . . . , p , \\end{align*}"}
+{"id": "5879.png", "formula": "\\begin{align*} e ^ { k \\Delta ^ { 2 } V } = o ( \\exp ( 2 N ^ { \\zeta } / 3 ) ) . \\end{align*}"}
+{"id": "481.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty d \\tau \\int _ 0 ^ \\infty d \\tau ' \\ , \\sigma _ t ^ { ( \\alpha / 2 ) } ( \\tau ) \\ , \\sigma _ { t ' } ^ { ( \\alpha / 2 ) } ( \\tau ' ) \\ , f ( \\tau + \\tau ' ) & = \\int _ 0 ^ \\infty d \\tau \\ , ( \\sigma _ t ^ { ( \\alpha / 2 ) } \\ast \\sigma _ { t ' } ^ { ( \\alpha / 2 ) } ) ( \\tau ) f ( \\tau ) \\\\ & = \\int _ 0 ^ \\infty d \\tau \\ , \\sigma _ { t + t ' } ^ { ( \\alpha / 2 ) } ( \\tau ) f ( \\tau ) , f \\in L ^ 1 . \\end{align*}"}
+{"id": "234.png", "formula": "\\begin{align*} L _ x ^ { \\emph { r } } \\phi ^ { \\emph { r } } _ \\xi ( x ; g ) = \\langle \\xi , \\xi \\rangle \\phi ^ { \\emph { r } } _ \\xi ( x ; g ) \\end{align*}"}
+{"id": "312.png", "formula": "\\begin{align*} v = \\mathsf { N } _ { j } \\left ( s \\right ) \\circ \\mathsf { L } _ { j } \\left ( s \\right ) v \\quad \\forall v \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) \\quad f = \\mathsf { L } _ { j } \\left ( s \\right ) \\circ \\mathsf { N } _ { j } \\left ( s \\right ) f \\quad \\forall f \\in H ^ { - 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) ; \\end{align*}"}
+{"id": "8567.png", "formula": "\\begin{align*} \\mathcal { L } _ 2 ^ { \\mu } [ \\beta b ] \\bullet = - \\frac { 1 } { 2 } b \\mathrm { F } _ 4 \\bullet + \\frac { \\beta ^ 2 } { 6 } b ^ 3 \\mathrm { F } _ 3 \\bullet , \\end{align*}"}
+{"id": "4236.png", "formula": "\\begin{align*} d \\omega ^ 1 \\ ! = \\omega ^ { 1 } \\ ! \\wedge ( \\omega ^ { 3 } + \\omega ^ { \\bar { 3 } } ) , \\ \\ d \\omega ^ 2 \\ ! = \\ ! - \\omega ^ { 2 } \\ ! \\wedge ( \\omega ^ { 3 } + \\omega ^ { \\bar { 3 } } ) , \\ \\ d \\omega ^ 3 \\ ! = \\omega ^ { 1 \\bar { 2 } } + \\omega ^ { 2 \\bar { 1 } } . \\end{align*}"}
+{"id": "1965.png", "formula": "\\begin{align*} g = \\frac { q } { p } \\sum _ { \\beta = 1 } ^ { k } \\sum _ { \\gamma = 1 } ^ { m _ { \\beta } - 1 } x _ { \\pi _ { \\beta } ( \\gamma ) } x _ { \\pi _ { \\beta } ( \\gamma + 1 ) } + \\sum _ { l = 1 } ^ { m } \\lambda _ l x _ l + \\lambda _ 0 , \\end{align*}"}
+{"id": "30.png", "formula": "\\begin{align*} a _ k ^ { \\dagger } : = \\frac { 1 } { | \\Lambda | ^ { 1 / 2 } } a ^ { \\dagger } ( e ^ { i k x } ) , a _ k : = \\frac { 1 } { | \\Lambda | ^ { 1 / 2 } } a ( e ^ { i k x } ) , \\end{align*}"}
+{"id": "7493.png", "formula": "\\begin{align*} Q ( X ) = Q _ 1 ( X ) Q _ 2 ( X ) , \\end{align*}"}
+{"id": "3630.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\Delta ( f ) + f | A | ^ 2 - f \\operatorname { \\overline { R i c } } ( \\eta , \\eta ) & = & 0 ; \\\\ \\\\ 2 A ( \\operatorname { g r a d } f ) + m f \\operatorname { g r a d } f - 2 f ( \\operatorname { \\overline { R i c c i } } \\eta ) ^ \\top & = & 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "6770.png", "formula": "\\begin{align*} d \\psi '' _ \\infty ( z ) - c \\psi ' _ \\infty ( z ) + \\psi _ \\infty ( z ) G ( 0 , \\psi _ \\infty ( z ) ) = 0 . \\end{align*}"}
+{"id": "4820.png", "formula": "\\begin{align*} \\check { R } ( u ) = P R ( u ) \\end{align*}"}
+{"id": "7988.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ J ( \\mathfrak { s } _ j ) ^ { d \\mathfrak { d } _ j } = 2 ^ { - d k } , \\end{align*}"}
+{"id": "4631.png", "formula": "\\begin{align*} \\phi \\bigl ( \\textup { T r } _ { M ^ { \\pi _ n } } ^ { \\mathbf { S } } ( B ) \\bigr ) = \\textup { T r } _ { M ^ { \\pi _ n } } \\bigl ( ( \\textup { i d } _ { M ^ { \\pi _ n } } \\otimes \\phi ) B \\bigr ) \\qquad \\forall \\ , \\phi \\in \\mathbf { S } ^ * , \\end{align*}"}
+{"id": "7491.png", "formula": "\\begin{align*} A ^ m _ 1 = \\begin{pmatrix} - ( m - 1 ) & - m \\\\ m & m + 1 \\end{pmatrix} \\end{align*}"}
+{"id": "5676.png", "formula": "\\begin{align*} x y + z ^ 3 + t ^ 3 = 0 \\end{align*}"}
+{"id": "5992.png", "formula": "\\begin{align*} \\langle L _ a ^ { - 1 } R _ { l o g , a } L _ a ^ { - 1 } \\tilde { u } _ j , \\tilde { u } _ j e ^ { \\tilde { \\phi } } \\rangle & = \\langle R _ { l o g , a } L _ a ^ { - 1 } \\tilde { u } _ j , L _ a ^ { - 1 } [ \\tilde { u } _ j e ^ { \\tilde { \\phi } } ] \\rangle \\\\ & = | u _ j ( x ^ * ) | ^ 2 e ^ { \\phi ( x ^ * ) } \\langle R _ { l o g , a } L _ a ^ { - 1 } [ 1 ] , L _ a ^ { - 1 } [ 1 ] \\rangle + O ( \\varepsilon ) . \\end{align*}"}
+{"id": "3935.png", "formula": "\\begin{align*} f ^ { - 1 } ( \\omega ) \\triangle g ^ { - 1 } ( \\omega ) = \\\\ \\Big ( \\bigcup _ { x \\in \\omega _ c } [ x - 1 , x + 1 ] \\Big ) \\setminus \\Big ( \\bigcup _ { x \\in \\omega } ( x - 1 , x + 1 ) \\Big ) \\subset \\\\ \\bigcup _ { x \\in \\omega _ c } \\{ x - 1 , x + 1 \\} \\end{align*}"}
+{"id": "2211.png", "formula": "\\begin{align*} \\left \\langle \\Phi , \\gamma _ { 2 } ^ { \\Psi } \\Phi \\right \\rangle = \\left \\Vert \\sum _ { k , l = 1 } ^ { \\infty } \\Phi _ { k , l } c _ { l } c _ { k } \\Psi \\right \\Vert ^ { 2 } \\end{align*}"}
+{"id": "5082.png", "formula": "\\begin{align*} E ( A z ) v ^ c = \\left ( \\begin{array} { c c c } E ( \\lambda _ { 1 } z ) v _ { 1 } & \\cdots & E ( \\lambda _ { n } z ) v _ n \\end{array} \\right ) \\left ( \\begin{array} { c c c } \\widetilde { v } _ { 1 } & \\cdots & \\widetilde { v } _ n \\end{array} \\right ) \\left ( \\begin{array} { c } v _ { 1 } ^ c \\\\ \\vdots \\\\ v _ n ^ c \\end{array} \\right ) , \\end{align*}"}
+{"id": "6500.png", "formula": "\\begin{align*} q ( k , n ) & = q ( - k , n ) \\in \\R , \\\\ q ( \\pm e _ l , n ^ { ( l ) } ) & = a _ l / 2 , \\ 1 \\leq l \\leq b , \\\\ \\sum _ { ( k , n ) \\notin \\mathcal { S } } | q ( k , n ) | e ^ { \\rho ( | k | + | n | ) } & < \\sqrt { \\varepsilon + \\delta } , \\ \\rho > 0 , \\end{align*}"}
+{"id": "8527.png", "formula": "\\begin{align*} | w - \\tau | + \\delta = r _ { \\ell } ^ { \\wedge } ( \\bar { z } ) . \\end{align*}"}
+{"id": "5889.png", "formula": "\\begin{align*} p ^ { n } \\leq 2 N ^ { \\zeta - 1 } \\mbox { a n d } p ^ { n } = o ( q ^ n ) . \\end{align*}"}
+{"id": "1787.png", "formula": "\\begin{gather*} \\mathcal { T } ^ k = \\prod _ { j = 1 } ^ { k } S ^ 1 . \\end{gather*}"}
+{"id": "5292.png", "formula": "\\begin{align*} g _ n \\left ( \\frac { 2 j - 1 } { 2 n + 1 } \\right ) & = 5 - \\frac { 3 } { 2 } ( 2 j - 2 ) + ( 2 j - 4 ) + \\sum _ { i = 4 } ^ j ( - 1 ) ^ i ( ( 2 j - 1 ) - ( 2 i - 1 ) ) \\\\ & = \\begin{cases} 0 & , \\\\ 1 & , \\end{cases} \\end{align*}"}
+{"id": "647.png", "formula": "\\begin{align*} \\mathcal L _ { \\mathrm { D i r a c } } ( \\psi ) = \\psi ^ * \\left ( i \\partial _ t + i \\alpha \\partial _ x - M \\beta \\right ) \\psi , \\mathcal L _ { \\mathrm { m e s o n } } ( \\phi ) = \\frac 1 2 ( \\partial _ t \\phi ) ^ 2 - \\frac 1 2 ( \\partial _ x \\phi ) ^ 2 - \\frac 1 2 m ^ 2 \\phi ^ 2 , \\end{align*}"}
+{"id": "2613.png", "formula": "\\begin{align*} \\hat \\tau _ { n , i } = \\inf \\{ t : \\ , \\hat A _ n ( t ) \\ge i \\} \\ , . \\end{align*}"}
+{"id": "4165.png", "formula": "\\begin{align*} R _ i = \\log \\left ( 1 + \\frac { | h _ { i i } | ^ 2 p _ i } { \\sum ^ L _ { j = 1 , j \\ne i } | h _ { i j } | ^ 2 p _ j + \\sigma ^ 2 _ i } \\right ) . \\end{align*}"}
+{"id": "860.png", "formula": "\\begin{align*} \\xi = \\prod _ { p \\in \\Pi } ~ p ^ { \\chi _ { \\xi } ( p ) } \\ , \\end{align*}"}
+{"id": "4618.png", "formula": "\\begin{align*} \\mathfrak { g } = \\mathfrak { g } ^ \\theta \\oplus \\mathfrak { p } . \\end{align*}"}
+{"id": "4279.png", "formula": "\\begin{align*} T ^ { ( 1 ) } _ { n \\times p } = \\sum _ { j = 0 } ^ { n - 1 } a ^ { ( 1 ) } _ { j } F _ { n \\times p } { ( j ) } + \\sum _ { j = 1 } ^ { p - 1 } a ^ { ( 1 ) } _ { - j } G _ { n \\times p } { ( j ) } , \\end{align*}"}
+{"id": "2462.png", "formula": "\\begin{align*} & \\overline { H } \\left ( \\bigwedge _ { a \\in \\mathcal { A } } G _ a \\right ) = \\sum _ { a \\in \\mathcal { A } } \\overline { H } ( G _ a ) = \\sum _ { a \\in \\mathcal { A } } H _ \\kappa ( G _ a ) , \\\\ & \\overline { H } \\left ( \\bigsqcup _ { a \\in \\mathcal { A } } ^ { P _ A } G _ a \\right ) = \\sum _ { a \\in \\mathcal { A } } P _ A ( a ) \\overline { H } ( G _ a ) = \\sum _ { a \\in \\mathcal { A } } P _ A ( a ) H _ \\kappa ( G _ a ) . \\end{align*}"}
+{"id": "6263.png", "formula": "\\begin{align*} N _ { \\lambda } ^ { n r } ( u ) = N _ \\lambda ^ { 3 , h h } + N _ { \\lambda } ^ { 3 , l h } + N _ \\lambda ^ { 5 , h h } + N _ { \\lambda } ^ { 5 , l h } . \\end{align*}"}
+{"id": "4751.png", "formula": "\\begin{align*} T = \\big \\{ ( x _ 1 , x _ 2 ) \\in \\mathbb R ^ 2 \\ , : \\ , x _ 2 > 0 , \\ , x _ 2 < x _ 1 < 1 - x _ 2 \\big \\} . \\end{align*}"}
+{"id": "6616.png", "formula": "\\begin{align*} \\rho _ { ( 1 ) , \\infty } ^ { ( \\widetilde { \\rm c J } ) } ( x ; \\beta , p , q ) : = \\lim _ { N \\to \\infty } { 2 \\pi \\over N } \\rho _ { ( 1 ) , N } ^ { ( \\widetilde { \\rm c J } ) } ( 2 \\pi \\chi _ { x < 0 } + 2 \\pi x / N ; \\beta , p , q ) . \\end{align*}"}
+{"id": "3617.png", "formula": "\\begin{align*} { \\mathbb E } \\left \\{ { \\left | { \\left [ { { { \\bf { \\Xi } } _ { { \\rm { B F } } } } } \\right ] _ { n , n } ^ * { { \\left [ { { { \\bf { \\Xi } } _ { { \\rm { B F } } } } } \\right ] } _ { m , m } } } \\right | } \\right \\} = \\frac { \\pi } { 4 } \\frac { { { N _ { \\rm { T } } } { N _ { \\rm { R } } } \\kappa } } { { { L _ { \\rm { R } } } \\left ( { \\kappa + 1 } \\right ) } } { N _ { { \\rm { S } } , n } } { N _ { { \\rm { S } } , m } } { \\rho _ n } { \\rho _ m } . \\end{align*}"}
+{"id": "5683.png", "formula": "\\begin{align*} ( z - T ) ^ { - 1 } e _ n & = \\sum \\limits _ { k = 0 } ^ { \\infty } \\frac { 1 } { z ^ { k + 1 } } T ^ k e _ n = \\sum \\limits _ { k = 0 } ^ { \\infty } \\frac { w _ n w _ { n + 1 } \\cdots w _ { n + k - 1 } } { z ^ { k + 1 } } e _ { n + k } \\\\ & = \\sum \\limits _ { k = 0 } ^ { \\infty } \\frac { \\beta _ { n + k } } { \\beta _ n z ^ { k + 1 } } e _ { n + k } = \\beta _ n ^ { - 1 } z ^ { n - 1 } \\sum \\limits _ { k = n } ^ { \\infty } \\frac { \\beta _ k } { z ^ k } e _ k . \\end{align*}"}
+{"id": "791.png", "formula": "\\begin{align*} I _ { \\tilde { a } } [ u ] & = \\frac { 1 } { p } [ u ] _ { s , p } ^ p + \\frac { 1 } { p } \\int _ { \\mathbb { R } ^ N } \\tilde { a } ( x ) | u | ^ p d x - \\frac { b } { 2 } \\int _ { \\mathbb { R } ^ N } ( K \\ast F ( u ) ) F ( u ) d x - \\frac { \\varepsilon _ g } { p _ g } \\int _ { \\mathbb { R } ^ N } | u | ^ { p _ g } d x . \\end{align*}"}
+{"id": "8141.png", "formula": "\\begin{align*} A _ i A _ j = A _ j A _ i = \\sum _ { k = 0 } ^ N p _ { i j } ^ k A _ k , \\end{align*}"}
+{"id": "2751.png", "formula": "\\begin{align*} U e _ i = \\left \\{ \\begin{array} { l l } \\hat { \\varepsilon } _ i e _ { \\pi ( i ) } , & 1 \\leq i \\leq \\theta ^ { - 1 } \\\\ \\hat { \\varepsilon } _ i e _ i , & i > \\theta ^ { - 1 } \\end{array} \\right . \\quad ( i \\in \\mathbb N ) , \\end{align*}"}
+{"id": "1999.png", "formula": "\\begin{align*} [ \\dot { \\mathrm { H } } ^ { s _ 0 , p _ 0 } ( \\mathbb { R } ^ n ) , \\dot { \\mathrm { H } } ^ { s _ 1 , p _ 1 } ( \\mathbb { R } ^ n ) ] _ { \\theta } = \\dot { \\mathrm { H } } ^ { s , p _ \\theta } ( \\mathbb { R } ^ n ) \\end{align*}"}
+{"id": "6366.png", "formula": "\\begin{align*} t _ { M , N } & ( h \\cdot ( m \\otimes n ) ) = t _ { M , N } ( h _ { 1 } \\cdot m \\otimes h _ { 2 } \\cdot n ) = t _ { M , N } ( l _ { m } ( h _ { 1 } ) \\otimes l _ { n } ( h _ { 2 } ) ) = t _ { M , N } ( l _ { m } \\otimes l _ { n } ) ( h _ { 1 } \\otimes h _ { 2 } ) \\\\ & = ( l _ { m } \\otimes l _ { n } ) t _ { H , H } ( h \\cdot ( 1 _ { H } \\otimes 1 _ { H } ) ) = h \\cdot ( l _ { m } \\otimes l _ { n } ) t _ { H , H } ( 1 _ { H } \\otimes 1 _ { H } ) = h \\cdot t _ { M , N } ( m \\otimes n ) \\end{align*}"}
+{"id": "351.png", "formula": "\\begin{align*} \\mathsf { C } _ { j } \\left ( s \\right ) \\mbox { \\boldmath $ \\psi $ } _ { j } = \\left ( \\begin{array} [ c ] { c } \\{ \\ ! \\ ! \\{ u \\} \\ ! \\ ! \\} _ { \\operatorname * { D } ; j } \\left ( s \\right ) \\\\ \\{ \\ ! \\ ! \\{ u \\} \\ ! \\ ! \\} _ { \\operatorname * { N } ; j } ^ { \\operatorname * { e x t } } \\left ( s \\right ) \\end{array} \\right ) . \\end{align*}"}
+{"id": "7820.png", "formula": "\\begin{align*} ( \\eta ^ i , \\widetilde { \\eta } _ i , \\kappa ) \\cdot ( \\rho ^ { } , \\widetilde { \\zeta } _ i ^ { } , \\sigma ^ { } ) = ( \\rho ^ { } , \\widetilde { \\zeta } _ i ^ { } , \\sigma ^ { } + \\widetilde { \\zeta } _ i ^ { } \\eta ^ i ) \\ , . \\end{align*}"}
+{"id": "608.png", "formula": "\\begin{align*} \\norm { u } _ { X ^ { s , b } } = \\left ( \\int _ { \\R ^ d } \\norm { U ( t , \\xi ) } _ { H ^ b _ t ( \\R ) } ^ 2 \\ , d \\xi \\right ) ^ { 1 / 2 } , \\end{align*}"}
+{"id": "7305.png", "formula": "\\begin{align*} \\mathcal { F S } ^ + ( H ) & : = \\{ T \\in \\mathcal { F S } ( H ) : \\ , \\sigma _ { e s s } ( T ) \\subset ( 0 , + \\infty ) \\} \\\\ \\mathcal { F S } ^ - ( H ) & : = \\{ T \\in \\mathcal { F S } ( H ) : \\ , \\sigma _ { e s s } ( T ) \\subset ( - \\infty , 0 ) \\} , \\end{align*}"}
+{"id": "477.png", "formula": "\\begin{align*} K ( x - y ) = \\sum _ { \\ell \\in L _ d } \\sum _ { m \\in M _ \\ell } K _ { \\ell } ( | x | , | y | ) Y _ { \\ell , m } ( \\omega _ x ) \\ , \\overline { Y _ { \\ell , m } ( \\omega _ y ) } \\end{align*}"}
+{"id": "2165.png", "formula": "\\begin{align*} \\alpha = 1 + \\tilde \\alpha _ 0 , \\beta = 1 + \\tilde \\beta _ 0 . \\end{align*}"}
+{"id": "3964.png", "formula": "\\begin{align*} | G : \\mho _ 1 ( G ) | & \\leq | G : \\mho _ 1 ( C _ G ( A ) ) | = | G : C | \\cdot | C : C _ G ( A ) | \\cdot | C _ G ( A ) : \\mho _ 1 ( C _ G ( A ) ) | \\\\ & \\leq 2 ^ { \\binom { k } 2 + 2 k + k } = 2 ^ { \\frac { k ( k + 5 ) } 2 } . \\end{align*}"}
+{"id": "8453.png", "formula": "\\begin{align*} \\omega = \\alpha ( A + \\sum _ { w } a _ w w ) + \\beta ( B + \\sum _ { w } b _ w w ) + \\sum _ { | w | = n } r _ w w + \\sum _ { | w | > n } r _ w w . \\end{align*}"}
+{"id": "3903.png", "formula": "\\begin{align*} \\norm { \\rho _ N } { 2 } ^ 2 = \\norm { \\sum _ { n = 1 } ^ N a _ n f ^ n - \\sum _ { k = 1 } ^ { Q _ N } ( \\xi _ k + \\eta _ k ) } { 2 } ^ 2 \\lesssim R _ N ^ 2 \\lesssim \\varphi ( N ) ^ { 1 / 8 } \\sigma _ N ^ 2 . \\end{align*}"}
+{"id": "7364.png", "formula": "\\begin{align*} g ( \\gamma ) = { \\rm M M S E } '' ( \\gamma ) - { \\rm M M S E } '' ( 1 ) \\end{align*}"}
+{"id": "1846.png", "formula": "\\begin{gather*} \\widetilde { U } _ { s , t } ( \\xi , \\Delta ) = \\int _ { \\mathbb { R } } U _ { s , t } ( x , \\Delta ) e ^ { - i x \\xi } \\ , d x , \\widetilde { K } _ { s , t } ( \\xi , \\Delta ) = \\int _ { \\mathbb { R } } K _ { s , t } ( x , \\Delta ) e ^ { - i x \\xi } \\ , d x \\end{gather*}"}
+{"id": "367.png", "formula": "\\begin{align*} \\psi _ 2 ( \\phi _ { R ( h _ 1 ) } ( h _ 2 ) h _ 2 ^ { - 1 } ) = \\varphi _ { S ( k _ 1 ) } ( k _ 2 ) k _ 2 ^ { - 1 } \\end{align*}"}
+{"id": "6694.png", "formula": "\\begin{align*} \\lim _ { q \\to 0 } { 1 \\over q } c _ \\infty ^ { ( \\widetilde { \\rm c J ) } } ( \\tau ; \\beta , 0 , q ) = - { 2 i \\over \\tau } \\Big ( 1 + c _ \\infty ^ { ( \\widetilde { \\rm c J ) } } ( \\tau ; \\beta , 1 , 0 ) \\Big ) . \\end{align*}"}
+{"id": "1620.png", "formula": "\\begin{align*} U ' _ { \\mathcal X _ 1 \\to y _ 1 } = \\{ u ' \\in \\mathcal P _ { \\mathcal X _ 1 } : \\{ u ' , \\alpha ^ { 1 } _ { \\mathcal X _ 2 } \\} \\frac { \\rho } { 2 } \\prod _ { j \\in [ k ] \\setminus \\{ 1 , 2 \\} } | \\mathcal P _ { \\mathcal X _ j } | \\mathcal A _ { \\mathcal Y } \\} . \\end{align*}"}
+{"id": "4357.png", "formula": "\\begin{align*} S ( \\omega _ { { \\sf f } _ 1 \\cdots { \\sf f } _ n } \\| \\omega ) = S ( \\tilde { \\omega } \\| \\omega ) = { \\rm T r } ( \\varrho ( \\log ( \\varrho ) - \\log ( \\tilde { \\varrho } ) ) ) \\end{align*}"}
+{"id": "5182.png", "formula": "\\begin{align*} a _ 2 ( f ) ( V \\otimes W ) = [ F ( f ) ( V ) \\otimes F ( f ) ( W ) ] [ F ( f ) ( V \\otimes W ) ] ^ * \\end{align*}"}
+{"id": "645.png", "formula": "\\begin{align*} \\psi ( t ) = S ( t ) \\psi _ 0 + \\int _ 0 ^ t S ( t - s ) \\ldots ( s ) d s + i \\int _ 0 ^ t S ( t - s ) \\psi \\Phi d W ( s ) - \\frac 1 2 \\int _ 0 ^ t S ( t - s ) \\psi F _ { \\Phi } d s , \\end{align*}"}
+{"id": "6292.png", "formula": "\\begin{align*} ( \\xi - \\eta ) ^ 2 = - [ ( \\xi _ 1 - \\xi _ 2 ) ( \\xi _ 3 - \\xi _ 4 ) + ( \\xi _ 1 - \\xi _ 4 ) ( \\xi _ 2 - \\xi _ 3 ) ] , \\end{align*}"}
+{"id": "2454.png", "formula": "\\begin{align*} \\overline { H } ( G ) = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } H _ { \\chi } ( G ^ { \\wedge n } ) , \\end{align*}"}
+{"id": "6823.png", "formula": "\\begin{align*} r _ - ( u ) = ( u + u ^ * + e _ 2 ) \\left ( \\sqrt { a + 1 } - 1 \\right ) ^ 2 , \\\\ [ 0 . 2 c m ] r _ + ( u ) = ( u + u ^ * + e _ 2 ) \\left ( \\sqrt { a + 1 } + 1 \\right ) ^ 2 . \\end{align*}"}
+{"id": "6546.png", "formula": "\\begin{align*} \\sigma ^ * = - k ^ * \\cdot \\omega ^ { ( 0 ) } - \\frac { 1 } { \\xi ^ * } \\mu _ { n ^ { * } } . \\end{align*}"}
+{"id": "1402.png", "formula": "\\begin{align*} \\lim _ { K \\to \\infty } \\sum _ { i = 0 } ^ 1 ( - 1 ) ^ i \\sum _ { k \\in I _ i , \\ , k \\le K } ( \\dots ) \\end{align*}"}
+{"id": "5358.png", "formula": "\\begin{align*} H ^ 1 ( N _ { \\Gamma / \\P ^ { c + 1 } } ( - 2 P _ 1 - \\ldots - 2 P _ s ) ) = 0 . \\end{align*}"}
+{"id": "1226.png", "formula": "\\begin{align*} \\| \\nabla e _ { n , T } ^ { n l } \\| _ { L _ t ^ { 2 } L _ x ^ { \\frac 6 5 } } \\lesssim & T ^ { 1 / 2 } \\big | \\frac { x _ n } { \\lambda _ n } \\big | ^ { - 2 b } \\| \\nabla \\Phi _ n \\| _ { L _ t ^ \\infty L _ x ^ r } ^ { 2 ( p - 1 ) } \\sum _ { j = 0 } ^ { 1 } \\big | \\frac { x _ n } { \\lambda _ n } \\big | ^ { - j } \\| \\partial ^ { 1 - j } \\Phi _ n \\| _ { L _ t ^ \\infty L _ x ^ r } \\\\ \\lesssim & T ^ { 1 / 2 } \\big | \\frac { x _ n } { \\lambda _ n } \\big | ^ { - 2 b + \\theta | 5 / 2 - 3 / r | } \\to 0 \\end{align*}"}
+{"id": "3879.png", "formula": "\\begin{align*} & \\lim _ { M \\to \\infty } \\P \\big ( \\tau _ M ^ 1 = T \\big ) = 1 , \\ \\lim _ { m \\to \\infty } \\P \\big ( \\tau _ { m } ^ 3 = T \\big ) = 1 , \\\\ & \\lim _ { M \\to \\infty } \\P \\big ( \\tau _ { M , m } ^ 2 = T \\big ) = 1 m \\in \\N . \\end{align*}"}
+{"id": "6620.png", "formula": "\\begin{align*} { S } _ \\infty ( \\tau ; \\beta ) : = \\lim _ { N , k \\to \\infty \\atop \\tau \\ : { \\rm f i x e d } } { S } _ N ( k ; \\beta ) \\Big | _ { \\tau = 2 \\pi k / N } = \\int _ { - \\infty } ^ \\infty \\Big ( \\rho _ { ( 2 ) , \\infty } ^ T ( x , 0 ) + \\delta ( x ) \\Big ) e ^ { i \\tau x } \\ , d x . \\end{align*}"}
+{"id": "6960.png", "formula": "\\begin{align*} F _ \\epsilon ^ k ( x ) = & \\alpha \\frac { \\sin x _ k } { \\omega + \\epsilon ^ 2 } - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { k - 1 } \\frac { \\cos ( x _ k / 2 ) } { ( \\sin ( x _ k / 2 ) - \\sin ( x _ i / 2 ) ) + \\epsilon ^ 2 } \\\\ & + \\frac { 1 } { 2 } \\sum _ { i = k + 1 } ^ d \\frac { \\cos ( x _ k / 2 ) } { ( \\sin ( x _ i / 2 ) - \\sin ( x _ k / 2 ) ) + \\epsilon ^ 2 } . \\end{align*}"}
+{"id": "3465.png", "formula": "\\begin{align*} u ^ { * * } ( x ) = \\max _ { p \\in \\Delta ^ \\vee } \\langle p , x \\rangle - u ^ * ( p ) . \\end{align*}"}
+{"id": "7845.png", "formula": "\\begin{align*} R _ 1 ( t , \\tau ) = \\frac { t ^ 3 } { 3 } - s ( \\tau ) > \\frac { t ^ 3 } { 3 } - \\frac { 9 } { 2 } s ( \\tau ) = R ( t , \\tau ) > 0 , \\ , \\end{align*}"}
+{"id": "4390.png", "formula": "\\begin{align*} & \\mathbb { L } ' ( \\hat { { \\mathbf U } } ^ { \\pm } , \\hat { \\Psi } ^ { \\pm } ) ( { \\mathbf U } ^ { \\pm } , \\Psi ^ { \\pm } ) \\\\ & = L ( \\hat { { \\mathbf U } } ^ { \\pm } , \\hat { \\Psi } ^ { \\pm } ) { \\mathbf U } ^ { \\pm } + C ( \\hat { { \\mathbf U } } ^ { \\pm } , \\hat { \\Psi } ^ { \\pm } ) { \\mathbf U } ^ { \\pm } - \\{ L ( \\hat { { \\mathbf U } } ^ { \\pm } , \\hat { \\Psi } ^ { \\pm } ) \\Psi ^ { \\pm } \\} \\frac { \\partial _ 1 \\hat { { \\mathbf U } } ^ { \\pm } } { \\partial _ 1 \\hat { \\Phi } ^ { \\pm } } , \\end{align*}"}
+{"id": "6282.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d t } M ( v _ \\lambda ) = & \\ 2 \\Re ( \\partial _ t v _ \\lambda \\cdot \\bar v _ \\lambda ) \\\\ = & \\ 2 \\Im ( \\partial _ x g _ { [ < \\lambda ] } \\partial _ x v _ \\lambda \\cdot \\bar v _ \\lambda ) + 2 \\Im ( f _ \\lambda \\bar u _ \\lambda ) \\\\ = & \\ - 2 \\partial _ x [ g _ { [ < \\lambda ] } \\Im ( \\partial _ x u _ \\lambda \\cdot \\bar u _ \\lambda ) ] + 2 \\Im ( f _ \\lambda \\bar u _ \\lambda ) \\end{aligned} \\end{align*}"}
+{"id": "2192.png", "formula": "\\begin{align*} x _ 1 y _ 1 + \\ldots + x _ k y _ k = 0 \\end{align*}"}
+{"id": "420.png", "formula": "\\begin{align*} L ^ 2 ( \\R ^ d ) = \\bigoplus _ { \\ell \\in L _ d } V _ { \\ell } V _ \\ell = \\bigoplus _ { m \\in M _ \\ell } V _ { \\ell , m } . \\end{align*}"}
+{"id": "7563.png", "formula": "\\begin{align*} q _ T ^ { ( < ) } & \\le \\sup _ { \\omega \\in A ^ { ( < ) } _ { T , r _ 1 ( T ) } } \\mathcal { E } _ T ( \\omega ) , \\\\ q _ T ^ { ( > ) } & \\le E ^ { P _ T } \\Big [ \\mathbf { 1 } _ { A ^ { ( > ) } _ { T , r _ 2 ( T ) } } \\Big ] = P _ T \\big ( A ^ { ( > ) } _ { T , r _ 2 ( T ) } \\big ) . \\end{align*}"}
+{"id": "629.png", "formula": "\\begin{align*} \\norm { u } _ { X ^ { s , b } _ { h ( \\xi ) } ( S , T ) } = \\norm { \\phi } _ { H ^ b ( S , T ) } \\norm { f } _ { H ^ s } \\norm { u } _ { \\widetilde X ^ { s , b } _ { h ( \\xi ) } ( S , T ) } = \\norm { \\phi } _ { \\widetilde H ^ b ( S , T ) } \\norm { f } _ { H ^ s } , \\end{align*}"}
+{"id": "5245.png", "formula": "\\begin{align*} h _ T ( x _ s ) & = \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\frac { q _ { i , t + 1 } ^ \\top g _ { i , t } ( x _ s ) } { \\gamma _ t } = \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T [ g _ { i , t } ( x _ { i , t } ) ] _ + ^ \\top g _ { i , t } ( x _ s ) \\\\ & \\le - \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\epsilon _ s [ g _ { i , t } ( x _ { i , t } ) ] _ + ^ \\top { \\bf 1 } _ { m _ i } = - \\epsilon _ s \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { i , t } ( x _ { i , t } ) ] _ + \\| _ 1 \\\\ & \\le - \\epsilon _ s \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { i , t } ( x _ { i , t } ) ] _ + \\| , \\end{align*}"}
+{"id": "7046.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\omega ( x ) u ( x , t ) d x = w ( t ) , t \\in \\lbrack 0 , T ] , \\end{align*}"}
+{"id": "3347.png", "formula": "\\begin{align*} h _ i = \\frac { \\tilde { \\theta } _ i ^ { b _ i / d _ i } } { \\theta _ i ^ { b _ i p / d _ i } } \\cdot \\frac { \\theta _ i ^ { p ( s ^ \\mathsf { T } A ) _ i } } { ( \\theta _ i ; \\zeta _ i ^ q ) _ { p s _ i } } \\cdot \\prod _ { t = 1 } ^ { p - 1 } \\frac { 1 } { ( \\theta _ i ; \\zeta _ i ) _ { q t } } \\in H ^ \\times . \\end{align*}"}
+{"id": "1354.png", "formula": "\\begin{align*} A _ { ( \\mathcal { X } _ s , \\mathcal { D } _ s + ( u - \\delta _ 0 ) D _ m ) } ( E ) & = A _ { \\mathcal { Y } _ s } ( E ) - \\mathrm { o r d } _ E \\left ( \\Gamma + \\left ( u - \\delta _ 0 \\right ) ( ( g _ s ^ * D _ m ) _ { \\mathrm { v e r t } } + ( g _ s ^ * D _ m ) _ { \\mathrm { h o r } } ) \\right ) \\\\ & \\ge ( \\delta _ 0 ' - ( u - \\delta _ 0 ) c \\epsilon ) A _ { \\mathcal { Y } _ s } ( E ) > 0 . \\end{align*}"}
+{"id": "2549.png", "formula": "\\begin{align*} \\| u \\| _ { D ^ { s , p } } ^ p + \\int _ { \\mathbb { R } ^ N } V | u | ^ p d x & = \\| u \\| _ { s , p , V _ { + } } ^ p - \\int _ { \\mathbb { R } ^ N } V _ { - } | u | ^ p d x \\\\ & \\geq \\| u \\| _ { s , p , V _ { + } } ^ p - \\| V _ { - } \\| _ { N / ( p s ) } \\| u \\| _ { p _ s ^ { * } } ^ p \\\\ & \\geq \\| u \\| _ { s , p , V _ { + } } ^ p - S _ { D ^ { s , p } } ^ { - 1 } \\| V _ { - } \\| _ { N / ( p s ) } \\| u \\| _ { D ^ { s , p } } ^ p \\\\ & \\geq ( 1 - S _ { D ^ { s , p } } ^ { - 1 } \\| V _ { - } \\| _ { N / ( p s ) } ) \\| u \\| _ { s , p , V _ { + } } ^ p . \\end{align*}"}
+{"id": "4429.png", "formula": "\\begin{align*} \\hat { \\mathbf U } ^ \\pm : = ( \\hat { \\rho } ^ \\pm , 0 , \\hat { u } ^ \\pm _ 2 , 0 , \\hat { H } _ 2 ^ \\pm , \\hat S ^ \\pm ) ^ T \\ , . \\end{align*}"}
+{"id": "1266.png", "formula": "\\begin{align*} & \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| u _ n ^ J \\| _ { S ( \\R ) } \\lesssim 1 , \\\\ & \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| e ^ { i t \\Delta } ( u _ { 0 , n } - u _ { n } ^ J ( 0 ) ) \\| _ { S ( \\R ) } = 0 \\\\ & \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| \\nabla [ ( i \\partial _ t + \\Delta ) u _ n ^ J + | x | ^ { - b } ( I _ { \\alpha } \\ast | u _ n ^ J | ^ p ) | u _ n ^ J | ^ { p - 2 } u _ n ^ J ] \\| _ { N ( \\R ) } = 0 . \\end{align*}"}
+{"id": "8180.png", "formula": "\\begin{align*} & X = A _ { 1 0 } - k ( r - 2 ) , \\\\ & Y = A _ { 0 1 } , \\\\ & X ^ \\star = \\frac { k ( r - 2 ) } { n ( r - 1 ) } \\left ( k + \\frac { n - k } { n - 1 } A ^ * _ { 0 1 } \\right ) - \\frac { k } { n ( r - 1 ) } A ^ * _ { 1 0 } , \\\\ & Y ^ \\star = \\frac { k ( n - k ) } { n } \\left ( 1 - \\frac { 1 } { n - 1 } A ^ * _ { 0 1 } \\right ) . \\end{align*}"}
+{"id": "7612.png", "formula": "\\begin{align*} J _ 1 ( \\beta , N , T , \\hat { \\mathbf { R } } _ 2 ) & \\leq C \\iint _ { \\hat { \\mathbf { R } } _ 2 } d s d t \\leq C \\beta ^ { - 1 / 3 } N ^ { 2 / 3 } T , \\end{align*}"}
+{"id": "4393.png", "formula": "\\begin{align*} ( Y , \\nabla _ y A ( \\hat { { \\mathbf U } } ^ { \\pm } ) ) = \\sum _ { i = 1 } ^ 6 y _ i ( \\frac { \\partial A ( Y ) } { \\partial y _ i } \\Big | _ { Y = \\hat { { \\mathbf U } } ^ { \\pm } } ) , Y = ( y _ 1 , \\cdots , y _ 6 ) . \\end{align*}"}
+{"id": "1936.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t f + v \\ , \\partial _ x f + \\partial _ v \\left ( \\left ( u - v \\right ) f \\right ) = F ( t , x , v ) , x \\in [ 0 , L ] = I , v \\in [ - M , M ] = J , t > 0 \\end{aligned} \\end{align*}"}
+{"id": "6669.png", "formula": "\\begin{align*} ( n + 3 ) ( n + 2 ) ( n + 1 ) e _ { n + 2 } + i q ( 2 n + 3 ) ( n + 1 ) e _ { n + 1 } + n \\Big ( p ^ 2 - { ( n + 1 ) ^ 2 \\over 4 } \\Big ) e _ n = 0 , ( n \\ge 0 ) . \\end{align*}"}
+{"id": "5519.png", "formula": "\\begin{align*} \\widetilde c _ { \\pm } ^ { ( a ^ { \\pm 1 } b ^ k , f a ) } ( p , s ) \\overset { ( \\ref { e q : c t i l d e D e f } ) } { = } c _ { \\pm } ^ { ( b ^ { \\pm 1 } a ^ k , \\sigma ( f ) b ) } ( s , p ^ { - 1 } ) \\overset { ( \\ref { e q : c p c m R i g h t M u l t i p l i c a t i o n } ) } { = } c _ { \\pm } ^ { ( b ^ { \\pm 1 } a ^ k , \\sigma ( f ) ) } ( s , p ^ { - 1 } s ^ { - 1 } ) \\overset { ( \\ref { e q : c t i l d e D e f } ) } { = } \\widetilde c _ { \\pm } ^ { ( a ^ { \\pm 1 } b ^ k , f ) } ( p s , s ) . \\end{align*}"}
+{"id": "1212.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left \\{ \\log \\left [ \\frac { \\lambda _ n ^ j } { \\lambda _ n ^ k } \\right ] + \\frac { | x _ n ^ j - x _ n ^ k | ^ 2 } { \\lambda _ n ^ j \\lambda _ n ^ k } + \\frac { | t _ n ^ j ( \\lambda _ n ^ j ) ^ 2 - t _ n ^ k ( \\lambda _ n ^ k ) ^ 2 | } { \\lambda _ n ^ j \\lambda _ n ^ k } \\right \\} = \\infty . \\end{align*}"}
+{"id": "3627.png", "formula": "\\begin{align*} \\tau ( \\varphi ) = \\operatorname { t r a c e } \\nabla d \\varphi = \\sum _ { i = 1 } ^ m \\nabla ^ \\varphi _ { e _ i } d \\varphi ( e _ i ) - \\sum _ { i = 1 } ^ m d \\varphi ( \\nabla ^ M _ { e _ i } e _ i ) = 0 , \\end{align*}"}
+{"id": "4435.png", "formula": "\\begin{align*} \\partial _ 1 { \\mathbf V } ^ + _ n = \\left [ \\begin{array} { c } \\mathcal { K } _ { 1 } \\\\ \\mathcal { K } _ { 2 } \\\\ - \\partial _ 2 ( \\dot { H } ^ + _ 2 \\partial _ 1 \\hat { \\Phi } ^ + ) \\end{array} \\right ] , \\partial _ 1 { \\mathbf V } ^ - _ n = \\left [ \\begin{array} { c } \\mathcal { K } _ { 7 } \\\\ \\mathcal { K } _ { 8 } \\\\ - \\partial _ 2 ( \\dot { H } ^ - _ 2 \\partial _ 1 \\hat { \\Phi } ^ - ) \\end{array} \\right ] \\ , . \\end{align*}"}
+{"id": "911.png", "formula": "\\begin{align*} m x ^ k \\bigg ( \\eta _ 1 - \\tau _ t + \\frac { 1 - \\alpha } { t } \\tau \\bigg ) - 2 \\eta _ { 2 x } - m k x ^ { k - 1 } \\xi - m x ^ k ( \\phi _ 1 - \\xi _ { 1 x } ) = 0 , \\end{align*}"}
+{"id": "3149.png", "formula": "\\begin{align*} \\alpha = \\beta _ 1 + \\dots + \\beta _ r , \\mu _ L ( \\beta _ k ) = 2 , \\beta _ k \\cdot D = 1 . \\end{align*}"}
+{"id": "1691.png", "formula": "\\begin{align*} 2 ( m + 1 ) \\xi _ j + v _ { q _ 0 } ( \\xi _ j ) + v _ { q _ 1 } ( \\xi _ j ) + \\sum _ { \\substack { 1 \\leq k \\leq n \\\\ k \\neq j } } \\Bigl ( v _ q ( \\xi _ j + \\xi _ k ) + v _ q ( \\xi _ j - \\xi _ k ) \\Bigr ) = 2 \\pi ( \\varrho _ { \\texttt { b } ; j } + \\lambda _ j ) , \\end{align*}"}
+{"id": "1704.png", "formula": "\\begin{align*} \\xi _ j - \\xi _ k & = \\frac { \\pi } { 2 } \\bigl ( 1 - ( q ) \\bigr ) \\pm i \\log ( | q | ) \\mod 2 \\pi \\\\ \\xi _ j + \\xi _ k & = \\frac { \\pi } { 2 } \\bigl ( 1 - ( q ) \\bigr ) \\pm i \\log ( | q | ) \\mod 2 \\pi \\end{align*}"}
+{"id": "8912.png", "formula": "\\begin{align*} [ x ] = [ \\sigma ^ 3 ( x ) ] = [ \\sigma ^ 6 ( x ) ] = [ j x ] = [ j \\sigma ^ 3 ( x ) ] = [ j \\sigma ^ 6 ( x ) ] . \\end{align*}"}
+{"id": "7478.png", "formula": "\\begin{align*} A : = \\frac { x _ { 0 , 2 } x _ { 1 , 3 } } { x _ { 0 , 3 } x _ { 1 , 2 } } , B : = \\frac { x _ { 0 , 3 } x _ { 1 , 2 } } { x _ { 0 , 1 } x _ { 2 , 3 } } \\ , . \\end{align*}"}
+{"id": "2140.png", "formula": "\\begin{align*} \\hat e = - \\kappa \\partial _ t \\alpha h _ 1 + 2 \\kappa \\partial _ x \\alpha h _ 2 \\geq ( \\partial _ t \\alpha - | \\partial _ x \\alpha | ) | \\kappa | h _ 1 . \\end{align*}"}
+{"id": "3818.png", "formula": "\\begin{align*} H \\left ( x _ 0 , \\frac { s _ 0 } { \\theta } , x _ 1 , \\frac { s _ 1 } { \\theta } \\right ) = \\frac { 1 } { \\theta } H ( x _ 0 , s _ 0 , x _ 1 , s _ 1 ) . \\end{align*}"}
+{"id": "7188.png", "formula": "\\begin{align*} \\alpha _ 2 & = ( y _ 1 + y _ 2 ) ^ { n _ 2 } y _ 2 \\\\ & = ( y _ 1 + y _ 2 ) ^ { n _ 2 - 1 } ( y _ 1 + y _ 2 ) y _ 2 \\\\ & = ( y _ 1 ^ { n _ 2 - 1 } + y _ 2 ^ { n _ 2 - 1 } ) ( y _ 1 + y _ 2 ) y _ 2 ~ ( \\mbox { a s } ~ \\binom { n _ 2 - 1 } { i } ~ \\mbox { i s e v e n f o r } ~ 0 < i < n _ 2 - 1 ) \\\\ & = ( y _ 1 + y _ 2 ) y _ 2 ^ { n _ 2 } ~ ( y _ 1 ^ { n _ 2 - 1 } = 0 , ~ \\mbox { s i n c e } ~ n _ 2 - 1 \\geq n _ 1 + 1 ~ \\mbox { a n d } ~ y _ 1 ^ { n _ 1 + 1 } = 0 ) \\\\ & = y _ 1 y _ 2 ^ { n _ 2 } + y _ 2 ^ { n _ 2 + 1 } . \\end{align*}"}
+{"id": "7298.png", "formula": "\\begin{align*} a ^ * a + b ^ * b = 1 . \\end{align*}"}
+{"id": "6534.png", "formula": "\\begin{align*} f ( m ) = k \\cdot \\omega ^ { ( 0 ) } ( m ) , \\ k \\cdot \\omega ^ { ( 0 ) } ( m ) + \\mu _ n ( m ) , \\ k \\cdot \\omega ^ { ( 0 ) } ( m ) + \\mu _ n ( m ) - \\mu _ { n ' } ( m ) \\end{align*}"}
+{"id": "4118.png", "formula": "\\begin{align*} \\frac { F ( n + 1 ) } { F ( n ) } & = 2 ^ n \\cdot \\frac { ( 4 n + 2 ) ! } { ( 3 n + 2 ) ! } \\prod _ { i = 0 } ^ { n - 1 } \\frac { ( n + 2 i + 1 ) ! } { ( n + 2 i + 2 ) ! } = 2 ^ n \\cdot \\frac { ( 4 n + 2 ) ! } { ( 3 n + 2 ) ! } \\prod _ { i = 0 } ^ { n - 1 } \\frac { 1 } { n + 2 i + 2 } \\\\ & = \\frac { 2 ^ n ( 4 n + 2 ) ! n ! ! } { ( 3 n + 2 ) ! ( 3 n ) ! ! } . \\end{align*}"}
+{"id": "4187.png", "formula": "\\begin{align*} I ( X ; Y ) & : = h ( Y ) - h ( Y | X ) = h ( Y ) - h ( X + N | X ) \\\\ & = h ( Y ) - h ( N | X ) = h ( Y ) - h ( N ) \\end{align*}"}
+{"id": "1481.png", "formula": "\\begin{align*} \\langle J _ { Z } X , Y \\rangle = \\langle Z , [ X , Y ] \\rangle \\end{align*}"}
+{"id": "6754.png", "formula": "\\begin{align*} C = W - \\frac { 3 ^ { \\sigma _ o } ( n + 1 ) - 2 ^ { \\sigma _ o } } { 2 ^ { \\sigma _ o + \\sigma _ e } } < W . \\end{align*}"}
+{"id": "1761.png", "formula": "\\begin{gather*} \\zeta ( s , \\alpha ) = \\frac { q ^ s } { \\phi ( q ) } \\sum _ { \\chi \\bmod q } \\overline { \\chi } ( a ) L ( s , \\chi ) \\end{gather*}"}
+{"id": "8030.png", "formula": "\\begin{align*} T _ { \\alpha , m , \\sigma } \\varphi ( x ) = { \\mathcal { F } } _ { \\alpha } ^ { - 1 } ( m _ { \\sigma } ) * _ { \\alpha } \\varphi ( x ) , \\ ; x \\in \\R _ { + } ^ { n } , \\end{align*}"}
+{"id": "3651.png", "formula": "\\begin{align*} \\bot ( \\omega \\wedge \\Omega ) = ( \\bot \\omega ) \\Omega - \\partial _ { q ^ \\mu } \\mathbin { \\lrcorner } \\omega \\wedge \\partial _ { p _ \\mu } \\mathbin { \\lrcorner } \\Omega + \\partial _ { p _ \\mu } \\mathbin { \\lrcorner } \\omega \\wedge \\partial _ { q ^ \\mu } \\mathbin { \\lrcorner } \\Omega + n \\omega \\ . \\end{align*}"}
+{"id": "7187.png", "formula": "\\begin{align*} y _ 1 ^ { n _ 1 } y _ 2 ^ { n _ 2 } = y _ 1 ^ { \\Bar { n } n _ 2 } y _ 2 ^ { n _ 2 } = y _ 1 ^ { \\Bar { n } n _ 2 } y _ 2 ^ { \\Bar { n } } y _ 2 ^ { n _ 2 - \\Bar { n } } = y _ 2 ^ { \\Bar { n } n _ 2 + \\Bar { n } } y _ 2 ^ { n _ 2 - \\Bar { n } } = y _ 2 ^ { \\Bar { n } n _ 2 + n _ 2 } = y _ 2 ^ { n _ 1 + n _ 2 } = y _ 2 ^ n . \\end{align*}"}
+{"id": "2747.png", "formula": "\\begin{align*} \\| U ( x _ k ) \\| < \\max \\big \\{ ( k - 1 ) ^ { - 1 } \\cdot \\theta ^ { - 1 } , \\ , \\theta \\cdot ( k - 1 ) \\cdot ( k - 1 ) ^ { - 1 } \\cdot \\theta ^ { - 1 } \\big \\} = 1 . \\end{align*}"}
+{"id": "4903.png", "formula": "\\begin{align*} ( E _ { 1 , 2 } , 0 , 0 ) t & = e _ { 1 , 1 , 1 } \\wedge e _ { 1 , 2 , 1 } \\wedge e _ { 1 , 1 , 2 } + e _ { 1 , 1 , 1 } \\wedge e _ { 1 , 1 , 2 } \\wedge e _ { 1 , 2 , 1 } = 0 , \\\\ ( 0 , E _ { 1 , 2 } , 0 ) t & = e _ { 1 , 1 , 1 } \\wedge e _ { 2 , 1 , 1 } \\wedge e _ { 1 , 1 , 2 } + e _ { 1 , 1 , 1 } \\wedge e _ { 1 , 1 , 2 } \\wedge e _ { 2 , 1 , 1 } = 0 , \\\\ ( 0 , 0 , E _ { 1 , 2 } ) t & = e _ { 1 , 1 , 1 } \\wedge e _ { 2 , 1 , 1 } \\wedge e _ { 1 , 2 , 1 } + e _ { 1 , 1 , 1 } \\wedge e _ { 1 , 2 , 1 } \\wedge e _ { 2 , 1 , 1 } = 0 . \\end{align*}"}
+{"id": "3071.png", "formula": "\\begin{align*} { \\Theta _ { { \\rm { T } } , k , 0 } ^ { \\rm { D } } } \\in \\left \\{ \\frac { 2 \\pi n } { N _ { \\rm T } } + \\Delta , n = 1 , 2 , \\ldots , N _ { \\rm T } \\right \\} , \\forall k , \\end{align*}"}
+{"id": "2501.png", "formula": "\\begin{align*} \\| \\boldsymbol { v } \\| _ { \\boldsymbol { H } ^ { \\textbf { c u r l } , 0 } } ^ 2 : = \\| \\boldsymbol { v } \\| ^ 2 + \\| \\textbf { c u r l } \\ , \\boldsymbol { v } \\| ^ 2 \\| \\boldsymbol { v } \\| _ { \\boldsymbol { H } ^ { \\textbf { c u r l } , 1 } } ^ 2 : = \\| \\boldsymbol { v } \\| ^ 2 + \\| \\textbf { c u r l } \\ , \\boldsymbol { v } \\| ^ 2 + \\| \\partial _ t \\boldsymbol { v } \\| ^ 2 , \\end{align*}"}
+{"id": "3414.png", "formula": "\\begin{align*} v _ x ( f ) = \\min \\{ \\langle x , \\alpha \\rangle | f _ \\alpha \\neq 0 \\} . \\end{align*}"}
+{"id": "6490.png", "formula": "\\begin{align*} \\psi _ e ( x ) = a _ e + b _ e x . \\end{align*}"}
+{"id": "6299.png", "formula": "\\begin{align*} \\begin{aligned} ( \\xi _ 1 + \\xi _ 2 ) ^ 2 + ( \\xi _ 3 + \\xi _ 4 ) ^ 2 - 2 ( \\xi _ 1 + \\xi _ 2 ) ( \\xi _ 3 + \\xi _ 4 ) = & \\ ( \\xi _ 1 + \\xi _ 2 - \\xi _ 3 - \\xi _ 4 ) ^ 2 \\\\ = & \\ 4 ( \\xi _ 1 - \\xi _ 4 ) ( \\xi _ 2 - \\xi _ 3 ) + ( \\Delta ^ 4 \\xi ) ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "8468.png", "formula": "\\begin{align*} \\mathcal { H } ^ { n - 1 } ( \\{ ( z , w ) \\in \\partial ^ { * } F _ { \\ell } : \\nu _ { w } ^ { F _ { \\ell } } ( z , w ) = 0 \\} \\cap ( \\Omega \\times \\mathbb { R } ^ { n - 1 } ) ) = 0 \\end{align*}"}
+{"id": "2109.png", "formula": "\\begin{align*} \\square \\partial _ x \\tilde { \\Lambda } = \\rho _ 1 + \\rho _ 2 , \\end{align*}"}
+{"id": "5370.png", "formula": "\\begin{align*} h ^ 1 ( N _ { Z '' } ( K _ { Z '' } - 3 H ) ) = 4 . \\end{align*}"}
+{"id": "1136.png", "formula": "\\begin{align*} & [ \\alpha + \\alpha ' , \\alpha + \\alpha ' ] = 0 \\\\ & [ \\alpha , \\alpha ] + [ \\alpha , \\alpha ' ] + [ \\alpha ' , \\alpha ] + [ \\alpha ' , \\alpha ' ] = 0 \\\\ & 2 [ \\alpha , \\alpha ' ] + [ \\alpha ' , \\alpha ' ] = 0 \\\\ & d _ \\alpha ( \\alpha ' ) + \\frac { 1 } { 2 } [ \\alpha ' , \\alpha ' ] = 0 . \\end{align*}"}
+{"id": "5382.png", "formula": "\\begin{align*} K _ H ( t , s ) : = \\Gamma \\left ( H + \\frac { 1 } { 2 } \\right ) ^ { - 1 } ( t - s ) ^ { H - \\frac { 1 } { 2 } } F \\left ( H - \\frac { 1 } { 2 } , \\frac { 1 } { 2 } - H , H + \\frac { 1 } { 2 } , 1 - \\frac { t } { s } \\right ) , \\end{align*}"}
+{"id": "2048.png", "formula": "\\begin{align*} \\left [ T _ \\phi u \\right ] _ { | _ { \\partial \\mathbb { R } ^ n _ + } } = S _ \\phi [ u _ { | _ { \\partial \\Omega } } ] \\end{align*}"}
+{"id": "1059.png", "formula": "\\begin{align*} \\hat { k } _ \\pm ( z , x ) = \\hat { k } ( z , \\pm x ^ { \\pm } ) . \\end{align*}"}
+{"id": "2876.png", "formula": "\\begin{align*} a _ j ( v _ k ) = \\delta _ { j , k } , j \\neq k . \\end{align*}"}
+{"id": "3337.png", "formula": "\\begin{align*} c ( Q ) = ( \\det \\widetilde { A } ) ^ { - \\frac { 1 } { 2 } } \\prod _ { i = 1 } ^ { N } \\theta _ i ^ { b _ i / d _ i } ( 1 - z _ i ) ^ { \\frac { 1 } { 2 } - \\frac { 1 } { m } } , \\end{align*}"}
+{"id": "6090.png", "formula": "\\begin{align*} \\gamma ^ * = \\frac { C o v ( \\bar { X } , \\bar { Y } ) } { V a r ( \\bar { Y } ) } \\approx \\frac { s _ { X Y } } { s _ Y ^ 2 } \\end{align*}"}
+{"id": "6445.png", "formula": "\\begin{align*} [ f ] \\oplus [ g ] & = ( [ f _ 0 ] \\oplus [ n \\pi ] ) \\oplus ( [ g _ 0 ] \\oplus [ m \\pi ] ) \\\\ & = ( [ f _ 0 ] \\oplus [ g _ 0 ] ) \\oplus [ ( n + m ) \\pi ] . \\end{align*}"}
+{"id": "712.png", "formula": "\\begin{align*} \\mathbf v ( t ) = \\mathbf S ( t - S ) \\mathbf u ( S ) + i \\int _ S ^ t \\mathbf S ( t - \\sigma ) \\mathbf N ( \\Theta _ R ^ { [ \\mathbf u , \\mathbf v ] } ( \\sigma ) \\mathbf v ( \\sigma ) ) \\ , d \\sigma \\\\ + i \\int _ S ^ t \\mathbf S ( t - \\sigma ) \\mathbf M ( \\mathbf v ( \\sigma ) ) \\ , d W ( \\sigma ) , \\end{align*}"}
+{"id": "2041.png", "formula": "\\begin{align*} \\dot { \\mathrm { B } } ^ { s } _ { p , q } ( \\partial \\Omega ) : = ( \\mathrm { L } ^ p ( \\partial \\Omega ) , \\dot { \\mathrm { H } } ^ { 1 , p } ( \\partial \\Omega ) ) _ { s , q } \\end{align*}"}
+{"id": "1700.png", "formula": "\\begin{align*} O _ { \\texttt { b } } ( \\boldsymbol { \\xi } ; q , q _ 0 ) : = \\prod _ { 1 \\leq j < k \\leq n } & ( 1 - 2 q \\cos ( \\xi _ j - \\xi _ k ) + q ^ 2 ) ( 1 - 2 q \\cos ( \\xi _ j + \\xi _ k ) + q ^ 2 ) \\\\ & \\times \\prod _ { 1 \\leq j \\leq n } ( 1 - 2 q _ 0 \\cos ( \\xi _ j ) + q _ 0 ^ 2 ) . \\end{align*}"}
+{"id": "2087.png", "formula": "\\begin{align*} h _ 1 ( t , x ) = ( \\partial _ t \\Lambda ) ^ 2 + ( \\partial _ x \\Lambda ) ^ 2 + 4 \\sinh ^ 2 ( \\Lambda ) ( ( \\partial _ x \\phi ) ^ 2 + ( \\partial _ t \\phi ) ^ 2 ) , \\end{align*}"}
+{"id": "5250.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { t } ( x _ { i , t } ) ] _ + \\| = \\mathcal { O } ( \\log ( T ) ) . \\end{align*}"}
+{"id": "3153.png", "formula": "\\begin{align*} r R = ( r - 1 ) \\phi ^ { - 1 } ( D _ Y ) , \\end{align*}"}
+{"id": "5446.png", "formula": "\\begin{align*} \\log P _ t ( e ^ f ) - P _ t f & \\leq \\log P _ { t } \\left ( e ^ { \\alpha \\Gamma ( f ) } \\right ) \\int _ 0 ^ t \\frac { 2 \\rho } { 2 \\rho \\theta ( s ) - 1 + e ^ { - 2 \\rho ( t - s ) } } d s \\\\ & = \\log P _ { t } \\left ( e ^ { \\alpha \\Gamma ( f ) } \\right ) \\log \\left ( \\frac { 2 \\rho \\alpha e ^ { 2 \\rho t } } { 2 \\rho \\alpha e ^ { 2 \\rho t } + 1 } \\times \\frac { 2 \\rho \\alpha + e ^ { - 2 \\rho t } } { 2 \\rho \\alpha - 1 + e ^ { - 2 \\rho t } } \\right ) \\end{align*}"}
+{"id": "5185.png", "formula": "\\begin{align*} \\omega _ i = \\sum _ { j \\leq i } L _ j \\end{align*}"}
+{"id": "8654.png", "formula": "\\begin{align*} \\int g ~ e ^ { \\frac { f + x _ 1 y _ 1 + . . . + x _ n y _ n } { \\hbar } } d x _ 1 . . . d x _ n = ( 2 \\pi \\hbar ) ^ { \\frac { n } { 2 } } e ^ { \\frac { \\hat { f } } { \\hbar } } \\det \\Bigg ( - \\frac { \\partial ^ 2 f } { \\partial x _ i \\partial x _ j } \\Bigg ) _ { i , j } ^ { - \\frac { 1 } { 2 } } \\Bigg \\rvert _ { \\vec { x } = \\vec { t } } ~ \\sum _ { k = 0 } ^ { \\infty } A _ k \\hbar ^ k \\end{align*}"}
+{"id": "1706.png", "formula": "\\begin{align*} \\omega _ { \\texttt { a } ; \\mu } : = \\sum _ { j = 1 } ^ { \\min ( _ { \\mu _ 1 } ( \\mu ) , _ { \\mu _ n } ( \\mu ) ) } \\bigl ( e _ { _ { \\mu _ 1 } ( \\mu ) + 1 - j } - e _ { n - _ { \\mu _ n } ( \\mu ) + j } \\bigr ) , \\end{align*}"}
+{"id": "5624.png", "formula": "\\begin{align*} \\begin{pmatrix} - 2 & 1 & 0 \\\\ 1 & - 2 & 0 \\\\ 0 & 0 & 4 \\end{pmatrix} \\end{align*}"}
+{"id": "1368.png", "formula": "\\begin{align*} f ( z ) \\big | U _ m : = m ^ { \\kappa - 1 } \\sum _ { j = 1 } ^ m f ( z ) \\Big | _ { 2 \\kappa } \\begin{pmatrix} 1 & j \\\\ 0 & m \\end{pmatrix} . \\end{align*}"}
+{"id": "7153.png", "formula": "\\begin{align*} [ p , q ^ { - 1 } ] = h q ^ { - 2 } , [ p ^ { - 1 } , q ] = h p ^ { - 2 } . \\end{align*}"}
+{"id": "6483.png", "formula": "\\begin{align*} P _ D = 0 , P _ N = \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} , P _ R = \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} , \\Lambda = \\begin{pmatrix} \\gamma & \\cdot \\\\ \\cdot & \\cdot \\end{pmatrix} , \\end{align*}"}
+{"id": "6084.png", "formula": "\\begin{align*} \\begin{aligned} V a r ( \\theta _ { \\gamma } ) & = V a r ( \\bar { X } - \\gamma ( \\bar { Y } - E ( \\bar { Y } ) ) ) \\\\ & = V a r ( \\bar { X } ) + V a r ( \\gamma ( \\bar { Y } - E ( \\bar { Y } ) ) ) - 2 C o v ( \\bar { X } , \\gamma ( \\bar { Y } - E ( \\bar { Y } ) ) ) \\\\ & = V a r ( \\bar { X } ) + \\gamma ^ 2 V a r ( \\bar { Y } ) - 2 \\gamma C o v ( \\bar { X } , \\bar { Y } ) \\end{aligned} \\end{align*}"}
+{"id": "5731.png", "formula": "\\begin{align*} \\operatorname { d i v } ( A ( x , t ) \\nabla u ) - u _ t = V u , \\end{align*}"}
+{"id": "1400.png", "formula": "\\begin{align*} \\sin \\rho _ n \\pi & = ( - 1 ) ^ m ( \\varepsilon _ n \\pi + O ( \\varepsilon _ n ^ 3 ) ) , m : = n - M _ 1 - 1 , \\\\ \\varkappa _ \\pi ( \\rho _ n ) & = \\int _ { - \\pi } ^ { \\pi } f ( x ) e ^ { i ( n + \\varepsilon _ n ) x } d x \\\\ & = \\int _ { - \\pi } ^ { \\pi } f ( x ) e ^ { i n x } d x + i \\varepsilon _ n \\int _ { - \\pi } ^ { \\pi } x f ( x ) e ^ { i n x } d x + O ( \\varepsilon _ n ^ 2 ) = \\varkappa _ n + o ( \\varepsilon _ n ) . \\end{align*}"}
+{"id": "1177.png", "formula": "\\begin{align*} & \\delta _ { 2 } ( m _ { 2 , n + 1 } ) = \\frac { 1 } { 2 } \\sum _ { \\substack { i + j = n + 1 \\\\ i , j > 0 } } [ m _ { 2 , i } , m _ { 2 , j } ] , \\end{align*}"}
+{"id": "669.png", "formula": "\\begin{align*} \\norm { u } _ { X ^ { s , b } _ { h ( \\xi ) } ( S , T ) } = \\inf \\left \\{ \\norm { v } _ { X ^ { s , b } _ { h ( \\xi ) } } \\colon \\right \\} . \\end{align*}"}
+{"id": "2954.png", "formula": "\\begin{align*} \\begin{bmatrix} n + 1 \\\\ s \\end{bmatrix} = \\begin{bmatrix} n \\\\ s - 1 \\end{bmatrix} + \\begin{bmatrix} n \\\\ s \\end{bmatrix} + \\begin{bmatrix} n \\\\ s + 1 \\end{bmatrix} , n \\ge 0 \\end{align*}"}
+{"id": "6116.png", "formula": "\\begin{align*} \\vartheta ( u ) : = \\int _ 0 ^ u \\kappa ( t ) \\d t . \\end{align*}"}
+{"id": "3635.png", "formula": "\\begin{align*} \\operatorname { \\overline { R i c } } ( \\eta , V ) = 0 . \\end{align*}"}
+{"id": "845.png", "formula": "\\begin{align*} T x = x _ 0 ^ * ( x ) h ( x \\in X ) . \\end{align*}"}
+{"id": "8214.png", "formula": "\\begin{align*} \\Phi _ a ( m ) = ( a ^ 2 x _ 1 ( m ) ) \\cdot m . \\end{align*}"}
+{"id": "6511.png", "formula": "\\begin{align*} { F } \\big | _ { \\mathcal { S } } ( q ) = 0 . \\end{align*}"}
+{"id": "6340.png", "formula": "\\begin{align*} N _ \\lambda ^ { l , 2 , l o } v _ \\lambda ^ { x _ 0 } = \\lambda L ( P _ { \\lambda } g ( u _ { < \\lambda } ) , \\partial v _ { < \\lambda } , v _ \\lambda ) . \\end{align*}"}
+{"id": "187.png", "formula": "\\begin{align*} E _ 4 ( \\tau ) E _ 6 ( \\tau ) = 1 - 2 6 4 q - 1 3 5 4 3 2 q ^ 2 + \\cdots . \\end{align*}"}
+{"id": "8849.png", "formula": "\\begin{align*} \\frac { \\eta _ { t } } { 2 } ( - 1 + 9 L \\kappa \\eta _ { t } ) 2 \\alpha = - \\frac { \\alpha } { 3 6 L \\kappa } \\leq - \\frac { 1 } { T _ { 1 } + 1 } . \\end{align*}"}
+{"id": "635.png", "formula": "\\begin{align*} \\xi _ j = \\frac { d \\mathcal W _ j } { d t } , \\mathcal W _ j = \\mathfrak K _ j W ( j = 1 , 2 ) , \\end{align*}"}
+{"id": "3766.png", "formula": "\\begin{align*} \\mu _ 0 ( X ) \\neq \\mu _ 1 ( X ) \\implies { \\rm O T } ( \\mu _ 0 , \\mu _ 1 ) = + \\infty . \\end{align*}"}
+{"id": "5832.png", "formula": "\\begin{align*} & s _ 2 s _ 4 s _ 5 s _ 3 ( s _ { \\alpha _ 7 + \\alpha _ 8 } s _ { \\alpha _ 1 + \\alpha _ 2 + \\alpha _ 3 + 2 \\alpha _ 4 + \\alpha _ 5 + \\alpha _ 6 + \\alpha _ 7 } ) s _ 3 s _ 5 s _ 4 s _ 2 = \\\\ & s _ { \\alpha _ 7 + \\alpha _ 8 } s _ { \\alpha _ 1 + 2 \\alpha _ 2 + 2 \\alpha _ 3 + 3 \\alpha _ 4 + 2 \\alpha _ 5 + \\alpha _ 6 + \\alpha _ 7 } . \\end{align*}"}
+{"id": "6004.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ d \\langle \\partial _ { z _ j } { c } ( z , x ) , \\zeta \\rangle ^ { 2 } \\geq \\underline { c } ( z ) \\vert \\zeta \\vert ^ 2 . \\end{align*}"}
+{"id": "6882.png", "formula": "\\begin{align*} r _ * { ( \\ell ( \\widetilde { X } , r ) \\smallfrown [ \\widetilde { X } ] ) } = \\sum _ { i = 1 } ^ r a _ i [ C _ i ] , \\end{align*}"}
+{"id": "4501.png", "formula": "\\begin{align*} \\Psi _ { i + \\frac { 1 } { 2 } } ^ \\pm = S _ { \\theta _ i } \\Psi ^ { \\pm } _ i \\ , , \\psi _ { i + \\frac { 1 } { 2 } } = ( S _ { \\theta _ i } \\Psi ^ { \\pm } _ i ) | _ { x _ 1 = 0 , } , \\end{align*}"}
+{"id": "2106.png", "formula": "\\begin{align*} \\partial _ x Q _ 0 ( \\phi , \\tilde { \\Lambda } ) = Q _ 0 ( \\partial _ x \\phi , \\tilde { \\Lambda } ) + Q _ 0 ( \\phi , \\partial _ x \\tilde { \\Lambda } ) . \\end{align*}"}
+{"id": "7061.png", "formula": "\\begin{align*} S _ r ( N ) & = \\mathop { \\mathop { \\sum \\sum } _ { \\substack { n , \\ , m = 1 } } ^ { \\infty } } \\delta _ { n = m } \\ , A _ { \\pi } ( n , r ) m ^ { - i t } \\ , \\lambda _ f ( m ) \\ , V _ 1 \\left ( \\frac { n } { N } \\right ) \\ , V _ 2 \\left ( \\frac { m } { N } \\right ) , \\end{align*}"}
+{"id": "2033.png", "formula": "\\begin{align*} S _ \\phi ^ { - 1 } v ( y ) : = v ( y ' ) y \\in \\partial \\Omega \\end{align*}"}
+{"id": "3803.png", "formula": "\\begin{align*} { \\rm O T } ( \\mu _ 0 , \\mu _ 1 ) = \\inf _ { \\alpha \\in S ^ p _ = ( \\mu _ 0 , \\mu _ 1 ) } \\int _ { Y \\times Y } H ( x _ 0 , s _ 0 , x _ 1 , s _ 1 ) d \\alpha . \\end{align*}"}
+{"id": "906.png", "formula": "\\begin{align*} t ^ { 1 - \\alpha } \\phi _ { 1 t } - \\phi _ { 1 x x } - \\frac { c } { x } \\phi _ { 1 x } - n x ^ k \\eta _ { 2 x } = 0 , \\end{align*}"}
+{"id": "1850.png", "formula": "\\begin{gather*} \\widetilde { U } _ { s , t } ( m , \\Delta ) = - \\frac { 1 } { i m } \\int _ { \\mathbb { R } } \\left ( \\frac { \\partial } { \\partial x } U _ { s , t } ( x , \\Delta ) \\right ) e ^ { - i m x } \\ , d x \\end{gather*}"}
+{"id": "8523.png", "formula": "\\begin{align*} \\mathcal { H } ^ { n - 1 } ( \\partial ^ { * } E \\cap \\{ z = \\bar { z } \\} ) = \\mathcal { H } ^ { n - 1 } ( \\partial ^ { * } F _ { \\ell } \\cap \\{ z = \\bar { z } \\} ) . \\end{align*}"}
+{"id": "5023.png", "formula": "\\begin{align*} e ^ { - F ( r ) } : = \\int _ N ( 4 \\pi ) ^ { - ( n - 1 ) / 2 } e ^ { - f ( r , \\cdot ) } d h . \\end{align*}"}
+{"id": "2019.png", "formula": "\\begin{align*} [ \\mathfrak { h } ^ { s _ 0 , p _ 0 } ( \\Omega ) , \\mathfrak { h } ^ { s _ 1 , p _ 1 } ( \\Omega ) ] _ { \\theta } & = \\mathfrak { h } ^ { s , p } ( \\Omega ) \\\\ [ \\mathfrak { h } ^ { - s _ 0 , p _ 0 ' } ( \\Omega ) , \\mathfrak { h } ^ { - s _ 1 , p _ 1 ' } ( \\Omega ) ] _ { \\theta } & = \\mathfrak { h } ^ { - s , p ' } ( \\Omega ) \\end{align*}"}
+{"id": "4864.png", "formula": "\\begin{align*} & I + f ( u ) E _ { i } + f ( u + v ) E _ { i + 1 } + f ( u ) f ( u + v ) E _ { i } E _ { i + 1 } + f ( v ) E _ { i } \\\\ & + f ( u ) f ( v ) E _ { i } ^ 2 + f ( u + v ) f ( v ) E _ { i + 1 } E _ { i } + f ( u ) f ( u + v ) f ( v ) E _ { i } E _ { i + 1 } E _ { i } \\\\ & = I + f ( v ) E _ { i + 1 } + f ( u + v ) E _ { i } + f ( v ) f ( u + v ) E _ { i + 1 } E _ { i } + f ( u ) E _ { i + 1 } \\\\ & + f ( v ) f ( u ) E _ { i + 1 } ^ 2 + f ( u + v ) f ( u ) E _ { i } E _ { i + 1 } + f ( v ) f ( u + v ) f ( u ) E _ { i + 1 } E _ { i } E _ { i + 1 } \\end{align*}"}
+{"id": "4141.png", "formula": "\\begin{align*} ( \\tilde \\nabla _ { H _ a } \\tilde T ) ( H _ b , H _ c ) & = H _ { H _ a \\overline { T } ( b , c ) } - \\xi _ { H _ a \\overline { R } ( b , c ) } \\\\ & = H _ { \\overline { \\nabla T } ( b , c ; a ) } - \\xi _ { \\overline { \\nabla R } ( b , c ; a ) } , \\end{align*}"}
+{"id": "3423.png", "formula": "\\begin{align*} \\Omega _ t = \\Omega _ { E _ J } \\wedge d \\log z _ 1 \\wedge \\ldots d \\log z _ k , \\end{align*}"}
+{"id": "7582.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { 1 } \\int _ { 0 } ^ { v } v ^ { - \\frac { 7 } { 8 } } u ^ { - \\frac { 3 } { 8 } } \\exp \\left ( - K v ^ { - \\frac { 3 } { 2 } } [ v - u ] ^ 2 \\right ) d u d v \\\\ & = \\int _ { 0 } ^ { 1 } \\int _ { 0 } ^ { v } v ^ { - \\frac { 7 } { 8 } } ( v - w ) ^ { - \\frac { 3 } { 8 } } \\exp \\left ( - K v ^ { - \\frac { 3 } { 2 } } w ^ 2 \\right ) d w d v . \\end{align*}"}
+{"id": "1322.png", "formula": "\\begin{align*} \\Lambda _ Z ^ { w } ( u ) = w ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) = \\frac { ( - 1 ) ^ m ( 1 - u ) ( l n ( 1 - u ) ) ^ m } { \\lambda ^ { m - 1 } } , 0 < u < 1 . \\end{align*}"}
+{"id": "7826.png", "formula": "\\begin{align*} h ( t ) = \\frac { t ^ 3 } { 6 } \\ , . \\end{align*}"}
+{"id": "832.png", "formula": "\\begin{align*} | \\Psi x _ 0 | & = | \\widetilde { T x _ 0 } ( e ^ { i \\theta _ 0 } ) | \\\\ & > 1 - \\dfrac { \\epsilon } { 3 } . \\end{align*}"}
+{"id": "7476.png", "formula": "\\begin{align*} \\Gamma ( z ) : = \\int _ 0 ^ \\infty t ^ { z - 1 } e ^ { - t } \\operatorname { d } \\ ! t . \\end{align*}"}
+{"id": "4437.png", "formula": "\\begin{align*} \\mathcal { M } ( t ) & = | | ( { \\mathbf V } , \\mathbf { F } ) | | ^ 2 _ { s , \\ast , t } + ( | | \\dot { { \\mathbf U } } | | ^ 2 _ { W ^ { 2 , \\infty } _ { \\ast } ( \\Omega _ t ) } + | | \\mathbf { F } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ t ) } ) | | \\hat { W } | | ^ 2 _ { s + 2 , \\ast , t } \\\\ \\end{align*}"}
+{"id": "5266.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ T \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n \\| [ g _ { i , t } ( x _ { j , t } ) ] _ + \\| ^ 2 = \\sum _ { t = 1 } ^ T \\sum _ { j = 1 } ^ n \\| [ g _ { t } ( x _ { j , t } ) ] _ + \\| ^ 2 . \\end{align*}"}
+{"id": "3261.png", "formula": "\\begin{align*} \\mathcal { F } ^ \\mu ( f ) ( \\xi ) = \\int _ \\mathbb { R } f ( x ) e ^ { \\mu x \\xi } d x \\end{align*}"}
+{"id": "3217.png", "formula": "\\begin{align*} \\| \\hat { x } - \\hat { x } _ { k } \\| ^ 2 _ { \\hat { A } ^ T \\hat { A } } = ( \\hat { x } - \\hat { x } _ { k } ) ^ T \\hat { A } ^ T \\hat { A } ( \\hat { x } - \\hat { x } _ { k } ) = ( x - x _ k ) ^ T L L ^ { - 1 } A ^ T A L ^ { - T } L ^ T ( x - x _ k ) = \\| { x } - { x } _ { k } \\| ^ 2 _ { { A } ^ T { A } } . \\end{align*}"}
+{"id": "1507.png", "formula": "\\begin{align*} & x ^ { \\ast } = \\frac { \\sqrt { c ^ { 2 } + 3 2 c } - 3 c } { 2 \\left ( 4 - c \\right ) } \\in ( 0 , 1 ) . \\end{align*}"}
+{"id": "5091.png", "formula": "\\begin{align*} V = \\sum _ i a _ i \\ , Y _ i \\ , , \\end{align*}"}
+{"id": "8714.png", "formula": "\\begin{align*} \\eta _ t = \\min \\left ( \\frac { \\mathfrak { y } } { d } , \\ , \\Xi _ T \\right ) \\qquad h _ t = \\mathfrak { h } \\cdot T ^ { - \\frac { 1 } { 2 ( 2 \\beta - 1 ) } } \\enspace , \\end{align*}"}
+{"id": "2569.png", "formula": "\\begin{align*} d _ 1 ( \\xi ) : = & \\frac { 1 } { 1 1 5 2 } \\left ( 8 4 6 1 6 8 + 9 6 4 1 8 6 e ^ { i ( \\xi _ 1 + \\xi _ 2 ) } - 5 8 1 9 3 6 e ^ { i \\xi _ 1 } + 6 4 8 9 7 3 e ^ { i ( \\xi _ 1 - \\xi _ 2 ) } - 6 5 e ^ { 2 i \\xi _ 2 } - 1 8 7 6 5 0 0 e ^ { i \\xi _ 2 } \\right . \\\\ & \\left . - 1 2 5 3 3 8 0 e ^ { - i \\xi _ 2 } + 9 1 2 6 5 7 e ^ { i ( \\xi _ 2 - \\xi _ 1 ) } + 5 7 6 e ^ { - i \\xi _ 1 } + 6 0 4 4 2 8 e ^ { - i ( \\xi _ 1 + \\xi _ 2 ) } + 2 6 3 9 5 5 e ^ { - 2 i \\xi _ 1 } \\right ) , \\\\ d _ 2 ( \\xi ) : = & ( 1 - d _ 1 ( \\xi ) / d _ 1 ( 0 ) ) ^ 2 , d _ 3 ( \\xi ) : = \\frac { d _ 2 ( \\xi ) ^ 2 - 1 } { d _ 1 ( \\xi ) } . \\end{align*}"}
+{"id": "3634.png", "formula": "\\begin{align*} \\langle \\overline { \\nabla } _ { \\omega ^ \\sharp _ M } H , P \\rangle + \\frac { 1 } { 2 } f \\langle \\operatorname { \\overline { R i c c i } } \\eta , V \\rangle = - f \\omega \\big ( A ( V ) \\big ) + \\phi \\ , \\omega ( \\operatorname { g r a d } f ) . \\end{align*}"}
+{"id": "2213.png", "formula": "\\begin{align*} \\sum _ { k , n = 1 } ^ { \\infty } \\left \\Vert \\sum _ { l , m = 1 } ^ { \\infty } A _ { k , l , m , n } c _ { l } ^ { \\ast } c _ { m } \\Psi \\right \\Vert ^ { 2 } \\leq N \\left \\Vert A \\right \\Vert _ { \\mathrm { H S } } ^ { 2 } . \\end{align*}"}
+{"id": "8311.png", "formula": "\\begin{align*} u ( X ) = ( p x _ 1 , p x _ 2 ) ~ \\ 2 ) \\ \\ u ( X ) \\equiv ( x _ 2 ^ { h _ 1 } , x _ 1 ^ { h _ 2 } ) \\ ( \\ p ) , \\end{align*}"}
+{"id": "1527.png", "formula": "\\begin{align*} \\sum _ { \\substack { P ^ + ( A ) \\le y \\\\ \\Omega ( A ) = J } } \\frac { 1 } { A } \\ll \\frac { J } { 2 ^ J } \\log ^ 2 y . \\end{align*}"}
+{"id": "5819.png", "formula": "\\begin{align*} \\begin{array} { l l l } & \\alpha ^ { d 4 } _ { m a x } < \\alpha ^ { d 6 } _ { m a x } < \\dots < \\alpha ^ { d , n - 2 } _ { m a x } & n = 2 k , \\\\ & \\alpha ^ { d 3 } _ { m a x } < \\alpha ^ { d 5 } _ { m a x } < \\dots < \\alpha ^ { d , n - 2 } _ { m a x } & n = 2 k + 1 . \\\\ \\end{array} \\end{align*}"}
+{"id": "8018.png", "formula": "\\begin{align*} Y ^ H ( t ) \\stackrel { d } { = } v _ { i _ 0 } ^ H t + \\biggl [ v _ { i } ^ H - v _ { i _ k } ^ H \\biggr ] _ { \\substack { i \\in I _ H \\\\ i \\not = i _ k } } X ^ H ( t ) . \\end{align*}"}
+{"id": "1943.png", "formula": "\\begin{align*} f ( 0 , x , v ) = \\frac { e ^ { \\frac { - v ^ 2 } { 2 } } } { \\sqrt { 2 \\pi } } \\left ( 1 + \\cos ( x ) \\right ) \\left ( 1 + 5 v ^ 2 \\right ) , u ( 0 , x ) = \\cos ( x ) + \\sin ( x ) . \\end{align*}"}
+{"id": "8184.png", "formula": "\\begin{align*} & A _ i = \\frac { ( i + \\alpha + \\beta + 1 ) ( i + \\alpha + 1 ) ( n - i ) } { ( 2 i + \\alpha + \\beta + 1 ) ( 2 i + \\alpha + \\beta + 2 ) } \\ , , C _ i = \\frac { i ( i + \\alpha + \\beta + n + 1 ) ( i + \\beta ) } { ( 2 i + \\alpha + \\beta ) ( 2 i + \\alpha + \\beta + 1 ) } . \\end{align*}"}
+{"id": "3314.png", "formula": "\\begin{align*} \\sigma ^ 2 = m _ 2 - m _ 1 ^ 2 = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\frac { 1 } { \\lambda ^ 2 _ { _ \\Sigma } } . \\end{align*}"}
+{"id": "8916.png", "formula": "\\begin{align*} \\tau _ V ( \\varphi ) : = \\tau ( \\varphi ) + d \\varphi ( V ) = 0 , \\end{align*}"}
+{"id": "386.png", "formula": "\\begin{align*} \\alpha ( y u ) = j \\big ( ( y u ) ^ * \\eta , y ' u ' \\big ) = j ( y ^ * \\eta , y ' ) \\circ u = \\alpha ( y ) u , \\end{align*}"}
+{"id": "4351.png", "formula": "\\begin{align*} S ( \\omega _ { { \\sf f } _ 1 \\cdots { \\sf f } _ m } \\| \\omega ) = \\frac { 1 } { 2 } { \\rm T r } \\left ( \\begin{array} { c c } - \\check { { \\bf I } } { \\bf a } ^ T & 0 \\\\ 0 & - \\check { { \\bf I } } { \\bf a } \\end{array} \\right ) \\ , . \\end{align*}"}
+{"id": "424.png", "formula": "\\begin{align*} \\index { $ \\kappa _ { \\rm c } ^ { ( \\alpha ) } ( \\ell ) $ } \\kappa _ { \\rm c } ^ { ( \\alpha ) } ( \\ell ) : = \\Phi _ { d , \\ell } ^ { ( \\alpha ) } \\left ( \\frac { d - \\alpha } { 2 } \\right ) = \\frac { 2 ^ { \\alpha } \\Gamma \\left ( \\frac { 1 } { 4 } ( d + 2 \\ell + \\alpha ) \\right ) ^ 2 } { \\Gamma \\left ( \\frac { 1 } { 4 } ( d + 2 \\ell - \\alpha ) \\right ) ^ 2 } . \\end{align*}"}
+{"id": "4176.png", "formula": "\\begin{align*} f _ \\mathbf { X } ( \\mathbf { x } ) = \\frac { \\Gamma \\left ( \\frac { d + 2 } { 2 } \\right ) } { \\pi ^ { \\frac { d + 2 } { 2 } } } \\frac { \\lambda } { \\left ( \\| \\mathbf { x } \\| ^ 2 + \\lambda ^ 2 \\right ) ^ { \\frac { d + 2 } { 2 } } } , \\end{align*}"}
+{"id": "396.png", "formula": "\\begin{align*} \\pi _ 0 ( K ^ X ) = \\hom ( X , K ) / \\ ! \\ ! \\sim \\ , . \\end{align*}"}
+{"id": "8643.png", "formula": "\\begin{align*} \\mathrm { F } _ 1 = \\dfrac { \\tanh { ( \\sqrt { \\mu } | \\mathrm { D } | ) } } { \\sqrt { \\mu } | \\mathrm { D } | } , \\mathrm { F } _ 2 = \\frac { 3 } { \\mu | \\mathrm { D } | ^ 2 } ( 1 - \\mathrm { F } _ 1 ) , \\mathrm { F } _ 3 = \\mathrm { s e c h } ( \\sqrt { \\mu } | D | ) , \\mathrm { F } _ 4 = \\frac { 2 } { \\mu | \\mathrm { D } | ^ 2 } ( 1 - \\mathrm { F } _ 3 ) . \\end{align*}"}
+{"id": "606.png", "formula": "\\begin{align*} 1 = \\frac 1 2 + \\frac { 1 - \\mu } { q } + \\mu , q = \\frac { 1 - \\mu } { 1 / 2 - \\mu } = \\frac { 2 b + 2 r - 1 - 2 \\varepsilon } { b + 2 r - 1 - 2 \\varepsilon } . \\end{align*}"}
+{"id": "4404.png", "formula": "\\begin{align*} \\partial _ t g ^ { \\pm } _ 3 + \\hat { u } ^ { \\pm } _ 2 \\partial _ 2 g ^ { \\pm } _ 3 + \\partial _ 2 \\hat { u } ^ { \\pm } _ 2 g ^ { \\pm } _ 3 = \\mathcal { G } ^ { \\pm } \\Gamma _ T , \\end{align*}"}
+{"id": "3293.png", "formula": "\\begin{align*} f _ X = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } f _ { _ \\Sigma } = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } g ^ 2 _ { _ \\Sigma } . \\end{align*}"}
+{"id": "1193.png", "formula": "\\begin{align*} E ^ 2 _ { p , q } = \\mathrm { T o r } _ { p , q } ^ { \\pi _ * A } ( k , \\pi _ * ( B ) ) \\Rightarrow \\pi _ { p + q } ( k \\otimes _ A B ) \\end{align*}"}
+{"id": "7859.png", "formula": "\\begin{align*} \\mu \\left ( \\{ \\underline { x } : \\sigma ^ { i + k } \\underline { x } \\in C _ { r _ n } ( \\sigma ^ i \\underline { x } ) \\} \\right ) = \\mu ( S _ k ( r _ n ) ) \\leq \\mu ( S _ { \\ell _ k } ( r _ n ) ) \\leq B ^ { 6 } \\sum _ { C _ { \\ell } \\in \\mathcal { C } _ { \\ell } } \\mu ( C _ { \\ell } ) ^ { \\omega _ { \\ell } + 1 } \\rho ^ { \\gamma _ { \\ell } } \\leq B ^ 6 Z _ { \\ell } ( \\omega _ { \\ell } ) , \\end{align*}"}
+{"id": "5349.png", "formula": "\\begin{align*} \\binom { k + c } { c + 2 } \\ge \\frac { d B } { 2 ( c + 2 ) ( c + 3 ) ( 3 c + 4 ) ( c + 1 2 ) } = \\frac { d ( k - 1 ) ( 8 c k + 1 2 k + 5 c ^ 2 + 7 c ) } { ( c + 2 ) ( c + 3 ) ( 3 c + 4 ) } . \\end{align*}"}
+{"id": "6551.png", "formula": "\\begin{align*} R _ { B _ * } { H } ( z ) R _ { \\Lambda ^ c } G _ { \\Lambda ^ c } ( z ) R _ { \\Lambda ^ c } { H } ( z ) R _ { B _ * } = O ( ( \\varepsilon + \\delta ) ^ { 2 - \\frac { 1 } { 4 b } } ) . \\end{align*}"}
+{"id": "6574.png", "formula": "\\begin{align*} \\bigcup _ { k \\in \\Z ^ b , \\ | k - k _ * | \\leq 2 N _ 1 } \\Theta _ k = \\bigcup _ { 1 \\leq l \\leq N _ 1 ^ { C _ 1 } } \\tilde I _ l , \\end{align*}"}
+{"id": "7283.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } f ( n ) = 0 . \\end{align*}"}
+{"id": "5570.png", "formula": "\\begin{align*} \\Psi _ k ( \\zeta ) \\approx P _ { k j } = \\Psi _ k ( c _ j ) , \\Phi _ k ( \\zeta ) \\approx Q _ { k j } = \\Phi _ k ( c _ j ) , \\zeta \\in C _ j , j = 1 , \\ldots , m , \\end{align*}"}
+{"id": "3123.png", "formula": "\\begin{align*} \\mathbb { E } [ \\| \\mathbf { H } _ { k , } \\| ^ 2 ] = \\frac { M L _ k } { \\kappa + 1 } . \\end{align*}"}
+{"id": "553.png", "formula": "\\begin{align*} \\mathbf U ( t ) = \\mathbf S ( t ) \\mathbf u _ 0 + i \\int _ 0 ^ { t \\wedge \\tau _ R } \\mathbf S ( t - s ) \\mathbf N ( \\mathbf u ( s ) ) \\ , d s \\\\ + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf M ( \\mathbf U ( s ) ) \\ , d W ( s ) \\end{align*}"}
+{"id": "7509.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 1 & 1 \\\\ ( q ^ { 2 } - 1 ) & ( q ^ { 3 } - 1 ) & ( q ^ { 4 } - 1 ) \\\\ ( q ^ { 2 } - 1 ) ( q - 1 ) & ( q ^ { 3 } - 1 ) ( q ^ { 2 } - 1 ) & ( q ^ { 4 } - 1 ) ( q ^ { 3 } - 1 ) \\end{pmatrix} \\end{align*}"}
+{"id": "3676.png", "formula": "\\begin{align*} \\Delta f = \\Delta _ N \\ , \\bar { f } + \\frac { 1 } { 4 } \\ , \\sum _ i \\Delta \\ , u _ i ^ 2 = \\Delta _ N \\ , \\bar { f } + \\frac { 1 } { 2 } \\ , \\sum _ i | \\nabla u _ i | ^ 2 = \\Delta _ N \\ , \\bar { f } + \\frac { k } { 2 } \\ , , \\end{align*}"}
+{"id": "8516.png", "formula": "\\begin{align*} \\lim _ { \\rho \\rightarrow 0 ^ { + } } \\frac { \\mathcal { H } ^ { n } \\left ( ( 0 , \\tau ) ) + E _ { 2 } ) \\cap B _ { \\rho } ( ( \\bar { z } , w ) ) \\right ) } { \\omega _ { n } \\rho ^ { n } } & \\leq \\frac { \\omega _ { n - 1 } } { \\omega _ { n } } \\lim _ { \\rho \\rightarrow 0 ^ { + } } \\frac { \\mathcal { H } ^ { 1 } \\left ( ( \\bar { z } , \\bar { z } + \\rho ) \\cap \\left \\{ r _ { \\ell } > \\frac { \\epsilon } { 2 } \\right \\} \\right ) } { \\rho } \\\\ & = 0 , \\end{align*}"}
+{"id": "8106.png", "formula": "\\begin{align*} \\left \\| \\uppercase \\expandafter { \\romannumeral 1 } _ { 1 } \\right \\| _ { L _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } . \\end{align*}"}
+{"id": "4652.png", "formula": "\\begin{align*} \\mathbf { k } \\cdot ( D f ) = ( \\mathbf { k } \\bullet D ) ( \\mathbf { k } \\cdot f ) \\end{align*}"}
+{"id": "678.png", "formula": "\\begin{align*} \\mathbf H ^ { s , r } = H ^ s ( \\R , \\C ) \\times H ^ s ( \\R , \\C ) \\times H ^ r ( \\R , \\C ) \\end{align*}"}
+{"id": "6596.png", "formula": "\\begin{align*} \\Sigma _ Q \\subset \\bigcup _ { j \\in Q , \\xi = \\pm 1 } J _ { j , \\xi } . \\end{align*}"}
+{"id": "3262.png", "formula": "\\begin{align*} \\mathcal { F } ^ \\mu ( f ) ( \\xi ) = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\mathcal { F } ^ \\mu ( f _ { _ \\Sigma } ) ( \\xi ) . \\end{align*}"}
+{"id": "1856.png", "formula": "\\begin{gather*} \\widetilde { K } _ { s _ n , t _ n } ( 0 , \\Delta ) = \\int _ { \\mathbb { R } } K _ { s _ n , t _ n } ( x , \\Delta ) \\ , d x \\ll \\frac { 1 } { \\Delta } \\end{gather*}"}
+{"id": "7740.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\lambda _ i S ( B _ X , x ^ * _ i , \\alpha _ i ) , \\end{align*}"}
+{"id": "6826.png", "formula": "\\begin{align*} P _ { 2 m } ( g ) ( \\phi ) = u ^ { - \\frac { n + 2 m } { n - 2 m } } P _ { 2 m } ( g ) ( u \\phi ) . \\end{align*}"}
+{"id": "7312.png", "formula": "\\begin{align*} S _ - ( T ) P _ - ( T ) T P _ - ( T ) S _ - ( T ) = - I _ { H _ - } . \\end{align*}"}
+{"id": "231.png", "formula": "\\begin{align*} a ^ { } _ \\nu ( \\xi ; g ) : = \\begin{cases} 1 & \\ \\nu = 0 , \\\\ 0 & \\ \\nu \\not \\geq 0 . \\end{cases} \\end{align*}"}
+{"id": "6959.png", "formula": "\\begin{align*} \\int _ { Q _ d } | \\nabla \\psi | ^ 2 = \\sum _ { n \\in \\Z ^ d } | n | ^ 2 | a ( n ) | ^ 2 = \\sum _ { | n | ^ 2 \\geq C _ P ^ { a s } ( d ) } | n | ^ 2 | a ( n ) | ^ 2 & \\geq C _ P ^ { a s } ( d ) \\sum _ { n \\in \\Z ^ d } | a ( n ) | ^ 2 \\\\ & = C _ P ^ { a s } ( d ) \\int _ { Q _ d } | \\psi | ^ 2 d x . \\end{align*}"}
+{"id": "6013.png", "formula": "\\begin{align*} \\overline { P } _ { s , t } = \\prod _ { i = N ( s ) } ^ { N ( t ) - 1 } \\overline { P } _ { \\gamma _ { i } } \\end{align*}"}
+{"id": "8497.png", "formula": "\\begin{align*} \\mathcal { H } ^ { n - 1 } ( \\partial ^ { e } E \\ \\backslash \\ \\partial ^ { * } E ) = 0 , \\end{align*}"}
+{"id": "4942.png", "formula": "\\begin{align*} H _ k ( \\tau ) = f _ k ( \\tau ) + \\sum _ { \\ell = 1 } ^ { k - 1 } \\alpha _ k ( \\ell ) \\cdot F ( \\tau ) ^ \\ell \\theta ( 2 \\tau ) ^ { 4 ( k - \\ell ) + 2 } + \\sum _ { \\ell = k } ^ { 2 k } \\beta _ k ( \\ell ) \\cdot F ( \\tau ) ^ { 2 k - \\ell } F ( 2 \\tau ) ^ { \\ell - k } \\theta ( 2 \\tau ) ^ 2 \\end{align*}"}
+{"id": "1968.png", "formula": "\\begin{align*} { a ^ t _ { \\sigma ^ j , r } - a ^ t _ { \\sigma , r } } = \\frac { j q } { p } { r } _ { \\pi _ \\beta ( 1 ) } , \\end{align*}"}
+{"id": "8585.png", "formula": "\\begin{align*} \\begin{cases} \\nabla ^ { \\mu } _ { X , z } \\cdot P ( \\Sigma _ b ) \\nabla _ { X , z } ^ { \\mu } u = f \\mathcal { S } _ b \\\\ u | _ { z = 0 } = 0 , \\partial _ { n _ { b } } ^ { P _ b } u | _ { z = - 1 + \\beta b } = g , \\end{cases} \\end{align*}"}
+{"id": "6067.png", "formula": "\\begin{align*} \\Tilde { X } ^ { M _ \\mathcal { P } } _ { t } ( x ) = x + \\int _ { 0 } ^ { t } \\int _ { B _ { M ( { \\gamma } ) } } c ( z , x ) d N ( z , s ) + \\int _ { 0 } ^ { t } b ( x ) d s . \\end{align*}"}
+{"id": "7670.png", "formula": "\\begin{align*} \\varphi ( u ) = \\mathbb { P } \\left ( \\sup _ { k \\geqslant 1 } \\sum _ { i = 1 } ^ k ( X _ i - c ) \\leqslant u \\right ) , \\ , u \\in \\mathbb { N } _ 0 . \\end{align*}"}
+{"id": "7337.png", "formula": "\\begin{align*} \\rho ( g ) = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & - 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & - 1 \\end{pmatrix} \\end{align*}"}
+{"id": "7461.png", "formula": "\\begin{align*} g ( z ) = \\frac { ( y - z ) ( \\frac { x } { z } - 1 ) } { z - 1 } - 1 \\end{align*}"}
+{"id": "1838.png", "formula": "\\begin{gather*} \\left \\| \\left ( f ( s ) - \\sum _ { n = 0 } ^ { M } \\frac { 1 } { ( n + c ) ^ s } \\right ) - \\sum _ { M < n \\leq N _ 0 } \\frac { \\beta _ n } { ( n + c ) ^ s } \\right \\| < \\frac { \\epsilon } { 3 } \\end{gather*}"}
+{"id": "8581.png", "formula": "\\begin{align*} 0 = \\partial _ { n _ b } \\Phi | _ { - h _ b } = \\mathbf { n } _ b \\cdot I ^ { \\mu } ( \\nabla _ { X , z } ^ { \\mu } \\Phi ) \\circ \\Sigma _ b . \\end{align*}"}
+{"id": "5521.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ { 3 } ( O ^ { ( 1 ) } _ B ) _ { i j } \\psi _ { k j } = ( X _ k + X _ k ^ { - 1 } ) \\psi _ { k i } \\end{align*}"}
+{"id": "7830.png", "formula": "\\begin{align*} M = \\{ ( Z ^ 0 , Z ^ 1 ) = Z ^ 0 ( 1 , z ^ 1 ) \\in \\mathbb { C } ^ 2 \\ ; | \\ ; Z ^ 0 \\in \\mathbb { C } ^ { \\times } , \\ ; \\ ; z ^ 1 \\in \\overline { M } ^ { } , \\ ; \\ ; \\frac { t ^ 3 } { 3 } > \\frac { \\chi \\zeta ( 3 ) } { 2 ( 2 \\pi ) ^ 3 } \\} \\ , . \\end{align*}"}
+{"id": "4190.png", "formula": "\\begin{align*} h _ 0 ( s ) & : = 2 \\log ( s ) - \\log ( 1 + s ) \\\\ & - s e ^ s \\left ( e \\cdot ( - 1 - s ) - 3 \\cdot ( - s ) \\right ) , s > 0 , \\end{align*}"}
+{"id": "3499.png", "formula": "\\begin{align*} \\frac { 1 } { l } \\log \\norm { s } _ { y , t } \\leq & \\frac { 1 } { 2 l } \\log \\int _ { X _ t } | s | ^ 2 _ { h ^ { \\otimes l } } e ^ { - 2 l ( \\phi _ t + | \\log | t | | \\psi _ t ) } \\sqrt { - 1 } ^ { n ^ 2 } \\Omega _ t \\wedge \\overline { \\Omega } _ t + | \\log | t | | u ( y ) + C | \\log | t | | \\epsilon . \\\\ \\leq & - ( \\phi _ t + | \\log | t | | \\psi _ t ) ( z ) + | \\log | t | | u ( y ) + \\frac { C } { l } | \\log | t | | + C | \\log | t | | \\epsilon . \\end{align*}"}
+{"id": "3415.png", "formula": "\\begin{align*} \\mathcal { X } = \\{ F _ 0 F _ 1 \\ldots F _ d + t F = 0 \\} \\subset M _ R \\end{align*}"}
+{"id": "7121.png", "formula": "\\begin{align*} \\| S \\| ^ 2 : = \\int _ 0 ^ 1 \\frac { S ( Z ) ^ 2 } { Z ^ 2 } \\ : d Z + \\int _ 0 ^ \\infty S ( Z ) ^ 2 e ^ { \\delta Z ^ { 4 / 5 } } \\ : d Z < + \\infty \\end{align*}"}
+{"id": "8712.png", "formula": "\\begin{align*} \\delta _ { t } \\leq \\frac { 2 ( t _ { 0 } - 1 ) \\delta _ { t _ { 0 } } } { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { ( 2 - p _ { i } ) t ^ { p _ { i } } } \\enspace . \\end{align*}"}
+{"id": "8392.png", "formula": "\\begin{align*} \\Gamma = \\sum _ { \\substack { P < p \\leq 2 P , m \\in \\mathbb { N } \\\\ \\left [ p ^ { c } \\right ] + \\left [ m ^ { c } \\right ] = N \\\\ ( m , P ( z ) ) = 1 } } ( \\log p ) . \\end{align*}"}
+{"id": "8650.png", "formula": "\\begin{align*} \\| \\nabla _ { X , z } ^ { \\mu } \\tilde u \\| _ { H ^ { k , 0 } ( \\mathcal { S } ) } & \\leq M ( k + 1 ) ( | g | _ { H ^ k } + \\| \\tilde { f } \\| _ { H ^ { k , 0 } } ) \\\\ & \\leq M ( k + 1 ) ( | g | _ { H ^ k } + \\sum \\limits _ { j = 0 } ^ k \\| \\partial _ z ^ j f \\| _ { H ^ { k - j , 0 } } ) . \\end{align*}"}
+{"id": "8253.png", "formula": "\\begin{align*} \\Psi ^ * g _ a = & V ( d x _ 1 ) ^ 2 + \\frac { 1 } { 4 x _ 1 ^ 2 } [ \\rho ^ 2 - 2 \\rho ^ 2 ( m ) + g _ { x _ 1 , 0 , 0 } ( \\xi _ 3 ( m ) , \\xi _ 3 ( m ) ) ] ( ( d x _ 2 ) ^ 2 + ( d x _ 3 ) ^ 2 ) + \\\\ & + a ^ 2 \\sum _ { j = 1 } ^ 3 ( d x _ j ) ^ 2 + \\beta _ 3 d x _ 2 - \\beta _ 2 d x _ 3 + g _ { x _ 1 , 0 , 0 } + \\frac { 1 } { V + a ^ 2 } \\eta ^ 2 . \\end{align*}"}
+{"id": "3593.png", "formula": "\\begin{align*} \\begin{aligned} { \\bf { a } } _ { { \\rm { R } } , n , { l _ n } } ^ H { { \\bf { a } } _ { { \\rm { R } } , m , { l _ m } } } = & \\left | { { \\bf { a } } _ { { \\rm { R } } , n , { l _ n } } ^ H { { \\bf { a } } _ { { \\rm { R } } , m , { l _ m } } } } \\right | \\\\ & \\times { e ^ { - j \\frac { { { N _ { \\rm { R } } } - 1 } } { 2 } \\Phi _ { n , l _ n } ^ { \\rm { A } } } } { e ^ { j \\frac { { { N _ { \\rm { R } } } - 1 } } { 2 } \\Phi _ { m , l _ m } ^ { \\rm { A } } } } , \\end{aligned} \\end{align*}"}
+{"id": "3152.png", "formula": "\\begin{align*} \\tau ^ D _ { \\alpha } ( L ) + r ! \\langle \\psi _ { r - 2 } \\rangle _ { \\alpha } = ( W ^ D _ { L } ) ^ r [ \\alpha ] , \\end{align*}"}
+{"id": "4375.png", "formula": "\\begin{align*} L ( { \\mathbf U } ^ \\pm , \\Psi ^ \\pm ) = A _ 0 ( { \\mathbf U } ^ \\pm ) \\partial _ t + \\tilde { A } _ 1 ( { \\mathbf U } ^ \\pm , \\Psi ^ \\pm ) \\partial _ 1 + A _ 2 ( { \\mathbf U } ^ \\pm ) \\partial _ 2 \\ , , \\end{align*}"}
+{"id": "2309.png", "formula": "\\begin{align*} & \\| S ( t ) g \\| _ { L ^ { q } _ { t } L ^ { r } _ { x } } \\lesssim \\| g \\| _ { L ^ { 2 } } , \\frac { 2 } { q } + \\frac { 1 } { r } = \\frac { 1 } { 2 } , 2 \\leq q , r \\leq \\infty , \\\\ & \\| S ( t ) g \\| _ { L ^ { 4 } _ { x } L ^ { \\infty } _ { t } } \\lesssim \\| g \\| _ { H ^ { \\frac 1 4 } } . \\end{align*}"}
+{"id": "2832.png", "formula": "\\begin{align*} \\ell ^ { ( p ) } ( r e s _ { s _ { p - 1 } } ( \\mathfrak { q } _ { p - 1 } ) ) - \\ell ^ { ( p ) } ( r e s _ { s } ( \\mathfrak { q } ) ) = r e s _ { s = 0 } \\Omega ^ { ( p ) } ( \\mathfrak { q } _ { p - 1 } - \\mathfrak { q } ) . \\end{align*}"}
+{"id": "853.png", "formula": "\\begin{align*} ( N _ 1 g ) ( z ) = \\eta ( z ) \\Psi _ 2 ( g ) , \\end{align*}"}
+{"id": "5132.png", "formula": "\\begin{align*} \\begin{bmatrix} x _ 1 \\\\ x _ 6 \\end{bmatrix} & = V _ { \\sf a b } ^ { \\sf a } { \\sf A } ^ { ( 4 ) } + V _ { \\sf a b } ^ { \\sf b } { \\sf B } ^ { ( 4 ) } . \\end{align*}"}
+{"id": "8545.png", "formula": "\\begin{align*} D r _ { \\ell } ^ { k } = \\sum _ { i = 1 } ^ { N _ { k } } \\left ( r _ { \\ell } ( z _ { i } ^ { k } ) - r _ { \\ell } ( z _ { i - 1 } ^ { k } ) \\right ) \\delta _ { z _ { i } ^ { k } } , \\end{align*}"}
+{"id": "1628.png", "formula": "\\begin{align*} m ( x _ 0 ) + \\sum _ { y } b ( x _ 0 , y ) E _ r ( y ) = E _ r ( x _ 0 ) \\sum _ { y } b ( x _ 0 , y ) \\end{align*}"}
+{"id": "6877.png", "formula": "\\begin{align*} H ^ { B M } _ k ( Y , \\mathbb { R } ) = H _ k ( \\overline { Y } , \\{ \\infty _ Y \\} , \\mathbb { R } ) , \\ H ^ { B M } _ k ( V , \\mathbb { R } ) = H _ k ( \\overline { X } , T \\cup \\{ \\infty _ X \\} , \\mathbb { R } ) . \\end{align*}"}
+{"id": "2784.png", "formula": "\\begin{align*} x \\cdot I ( G - N [ v _ 0 ] ) & = x \\cdot ( 2 x + 1 ) ^ { 2 k + 4 } \\cdot ( 3 x ^ 2 + 4 x + 1 ) \\\\ & = x ( 3 x ^ 2 + 4 x + 1 ) \\cdot \\bigg [ \\sum _ { i = 0 } ^ { 2 k + 4 } \\binom { 2 k + 4 } { i } ( 2 x ) ^ { i } ] \\\\ & = x ( 3 x ^ 2 + 4 x + 1 ) \\cdot [ ( 2 x ) ^ { 2 k + 4 } + \\dots ] \\\\ & = 3 \\cdot 2 ^ { 2 k + 5 } x ^ { 2 k + 8 } + \\dots \\end{align*}"}
+{"id": "1281.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( P ( u _ n ) - P ( u _ n - \\phi _ n ^ 1 ) - P ( \\phi _ n ^ 1 ) ) = 0 \\end{align*}"}
+{"id": "6235.png", "formula": "\\begin{align*} u = \\sum _ { k \\in \\N } u _ k , u _ k = P _ k u . \\end{align*}"}
+{"id": "831.png", "formula": "\\begin{align*} \\Psi x = \\widetilde { T x } ( e ^ { i \\theta _ 0 } ) ( x \\in X ) . \\end{align*}"}
+{"id": "585.png", "formula": "\\begin{align*} \\lim _ { \\mu \\to \\infty } \\norm { \\mathbf u ^ \\mu - \\mathbf u } _ { Z ^ { \\mathbf s , b } ( 0 , T ) } = 0 . \\end{align*}"}
+{"id": "8665.png", "formula": "\\begin{align*} f ( \\vec x ) = - { 1 \\over 2 } \\sum _ { i j } b _ { i j } x _ i x _ j + \\dots \\end{align*}"}
+{"id": "3602.png", "formula": "\\begin{align*} { \\bf F } _ { \\rm D S } = \\sqrt { \\frac { E } { N _ { \\rm R } } } { \\bf T } _ { \\rm D S } , { \\bf W } _ { { \\rm D S } , m } = { \\bf R } _ { { \\rm D S } , m } . \\end{align*}"}
+{"id": "5097.png", "formula": "\\begin{align*} C ^ { s m } ( G , L ) & = \\prod _ { g \\in G / G _ 0 } C ^ { s m } ( g G _ 0 , L ) \\\\ & = \\prod _ { g \\in G / G _ 0 } \\varinjlim _ { H \\subset G _ 0 } C ^ { s m } ( g G _ 0 / H , L ) \\\\ & = \\varinjlim _ { \\substack { g \\in G / G _ 0 \\\\ H _ g \\subset G _ 0 } } \\prod _ { g \\in G / G _ 0 } C ^ { s m } ( g G _ 0 / H _ g , L ) . \\end{align*}"}
+{"id": "5561.png", "formula": "\\begin{align*} \\Psi _ k ( \\zeta ) \\approx P _ { k j } = \\Psi _ k ( c _ j ) , \\zeta \\in C _ j , j = 1 , \\ldots , m , \\end{align*}"}
+{"id": "2006.png", "formula": "\\begin{align*} \\mathrm { \\mathrm { H } } ^ { s , p } ( \\Omega ) = \\mathrm { \\mathrm { H } } ^ { s , p } _ 0 ( \\Omega ) \\ , \\mathrm { B } ^ { s } _ { p , q } ( \\Omega ) = \\mathrm { B } ^ { s } _ { p , q , 0 } ( \\Omega ) \\end{align*}"}
+{"id": "5609.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ 0 ^ \\infty \\exp _ p ( \\mu | u _ n | ^ { p ' } ) r ^ { \\alpha _ 0 } \\mathrm d r & = \\nu _ 0 + \\nu _ \\infty = \\nu _ \\infty \\\\ & = \\lim _ { R \\to \\infty } \\lim _ { n \\to \\infty } \\dfrac { \\mu ^ { p - 1 } } { \\Gamma ( p ) } \\| u _ n \\| ^ p _ { L ^ p _ { \\alpha _ 0 } ( R , \\infty ) } \\leq \\dfrac { \\mu ^ { p - 1 } } { \\Gamma ( p ) } . \\end{align*}"}
+{"id": "7306.png", "formula": "\\begin{align*} \\mathcal { F S } ^ i ( H ) : = \\mathcal { F S } ( H ) \\setminus \\mathcal { F S } ^ \\pm ( H ) , \\end{align*}"}
+{"id": "1773.png", "formula": "\\begin{gather*} \\Lambda _ 2 = \\{ n \\in \\mathbb { Z } \\mid n \\geq 0 \\} . \\end{gather*}"}
+{"id": "7943.png", "formula": "\\begin{align*} S _ 1 = ( S \\cap V _ 1 ) \\cup \\{ u _ 1 , v _ 1 \\} \\ 1 \\end{align*}"}
+{"id": "8638.png", "formula": "\\begin{align*} L _ 2 ^ { \\mu } ( \\beta b ( X ) , \\xi ) & = b ( X ) \\big { ( } \\mathrm { s e c h } ( \\sqrt { \\mu } | \\xi | ) - 1 \\big { ) } \\frac { 1 } { \\mu | \\xi | ^ 2 } \\\\ & + \\frac { 1 } { 6 \\beta } \\Big ( \\beta ^ 3 b ( X ) ^ 3 \\int _ 0 ^ 1 \\cosh ( t \\beta b ( X ) \\sqrt { \\mu } | \\xi | ) ( 1 - t ) ^ 2 \\mathrm { d } t \\Big ) \\mathrm { s e c h } ( \\sqrt { \\mu } | \\xi | ) , \\end{align*}"}
+{"id": "5573.png", "formula": "\\begin{align*} h ( r ) = \\frac { 2 } { \\pi } \\tan ^ { - 1 } \\left ( \\sqrt { 2 } \\sqrt { \\frac { r - 1 } { 2 - r } } \\right ) \\end{align*}"}
+{"id": "5942.png", "formula": "\\begin{align*} \\Gamma _ { \\varepsilon , a } : = \\{ \\mathrm { e x p } _ { x ^ * ; h } ( \\varepsilon t _ 1 E _ 1 + \\varepsilon t _ 2 E _ 2 ) \\mid t _ 1 ^ 2 + a ^ { - 2 } t _ 2 ^ 2 \\leq 1 \\} . \\end{align*}"}
+{"id": "7904.png", "formula": "\\begin{align*} \\sum _ { x \\in B } \\chi _ \\varphi ( x ) \\chi _ \\rho ( x ^ \\ast ) = 0 . \\end{align*}"}
+{"id": "936.png", "formula": "\\begin{align*} E ( \\tilde { x } , \\tilde { t } , \\tilde { u } , \\tilde { v } ) = 0 , ~ \\tilde { x } \\in \\tilde { \\Omega } , ~ \\tilde { t } > 0 , \\end{align*}"}
+{"id": "2281.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } \\theta _ \\alpha \\left ( x - \\frac { j } { n } + \\varepsilon _ j \\right ) = \\sum _ { k \\in \\mathbb { Z } } e ^ { - \\pi \\alpha k ^ 2 } \\left ( \\sum _ { j = 1 } ^ { n } e ^ { 2 \\pi i k \\varepsilon _ j } e ^ { - 2 \\pi i k \\frac { j } { n } } \\right ) e ^ { 2 \\pi i k x } . \\end{align*}"}
+{"id": "7065.png", "formula": "\\begin{align*} S _ 1 ( . . . ) = \\sum _ { n = 1 } ^ { \\infty } A _ { \\pi } ( r , n ) e \\left ( \\frac { ( a + b q ) n } { p _ 1 q } \\right ) \\ , v _ 1 ( n ) , \\end{align*}"}
+{"id": "6267.png", "formula": "\\begin{align*} \\begin{aligned} N _ \\lambda ^ 2 = & \\ L ( P _ { \\lambda } g ( u _ { \\ll \\lambda } ) , \\partial ^ 2 u _ { < \\lambda } ) + L ( P _ \\lambda g ( u _ { \\ll \\lambda } ) , \\partial ^ 2 u _ { \\lambda } ) + \\sum _ { \\mu \\gg \\lambda } L ( P _ \\mu g ( u _ { \\ll \\lambda } ) , \\partial ^ 2 u _ \\mu ) \\\\ : = & \\ N _ \\lambda ^ { 2 , l o } + N _ \\lambda ^ { 2 , m e d } + N _ \\lambda ^ { 2 , h i } . \\end{aligned} \\end{align*}"}
+{"id": "6164.png", "formula": "\\begin{align*} A _ 1 = \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\begin{pmatrix} 0 & 1 \\end{pmatrix} , & & A _ 3 = \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\end{pmatrix} , & & A _ 5 = \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\begin{pmatrix} 1 & - 1 \\end{pmatrix} , & & O = \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\begin{pmatrix} 0 & 0 \\end{pmatrix} . & & \\end{align*}"}
+{"id": "7177.png", "formula": "\\begin{align*} \\lambda ( F _ 0 ^ j ) : = \\boldsymbol { \\alpha _ j } \\in \\mathbb { Z } _ 2 ^ n ~ ~ j = 1 , \\dots , m , \\end{align*}"}
+{"id": "5115.png", "formula": "\\begin{align*} l = ( r , r ^ * + g ^ \\flat ) , \\ , \\ , \\ , \\ , D = ( D ^ r , D ^ { r , * } + \\Sigma ) , \\end{align*}"}
+{"id": "117.png", "formula": "\\begin{align*} b _ k : = a _ 0 ^ { \\dagger } a _ k , b ^ { \\dagger } _ k : = a _ 0 a ^ { \\dagger } _ k , \\end{align*}"}
+{"id": "5852.png", "formula": "\\begin{align*} s _ k s _ { k + 1 } \\dots s _ { n - 1 } s _ n s _ { n - 1 } \\dots s _ { k + 1 } s _ k = s _ n s _ { n - 1 } \\dots s _ { k + 1 } s _ k s _ { k + 1 } \\dots s _ { n - 1 } s _ n . \\end{align*}"}
+{"id": "2637.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\ , Q _ n ( t ) - 1 = \\ , \\int _ 0 ^ t ( \\frac { 1 } { n } \\ , Q _ n ( t - s ) - 1 ) ^ + \\ , d F ( s ) + \\theta _ n ( t ) \\ , , \\end{align*}"}
+{"id": "3332.png", "formula": "\\begin{align*} Q ( n ) = \\frac { 1 } { 2 } n ^ { \\mathsf { T } } A D n + n ^ \\mathsf { T } b + c \\end{align*}"}
+{"id": "5299.png", "formula": "\\begin{align*} \\sum _ { d \\leq j \\leq \\frac { n + d } { 2 } } f _ { j , d } ( n ) & = \\sum _ { d \\leq j \\leq \\frac { n + d } { 2 } } { n - 2 j + 2 d - 1 \\choose d - 1 } + { n - 2 j + 2 d - 2 \\choose d - 1 } \\\\ & = \\sum _ { i = d - 1 } ^ { n - 1 } { i \\choose d - 1 } \\\\ & = { n \\choose d } . \\end{align*}"}
+{"id": "4345.png", "formula": "\\begin{align*} S ( \\omega _ { { \\sf f } _ 1 \\cdots { \\sf f } _ m } \\| \\omega ) = i \\left . \\frac { d } { d t } \\right | _ { t = 0 } { \\rm P f } ( { \\bf A } ( t ) ) = \\frac { 1 } { 2 } \\ , { \\rm T r } \\left ( { \\bf A } ^ { - 1 } \\left ( \\begin{array} { c c } 0 & - { \\bf a } \\\\ { \\bf a } ^ T & 0 \\end{array} \\right ) \\right ) \\end{align*}"}
+{"id": "2657.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\int _ 0 ^ 1 \\dot k ( x , t ) ^ 2 \\ , d x \\ , d t = \\int _ 0 ^ \\infty \\int _ 0 ^ 1 \\dot b ( x , t ) ^ 2 \\ , d x \\ , d t \\ , . \\end{align*}"}
+{"id": "5399.png", "formula": "\\begin{align*} z < z ' \\iff , \\end{align*}"}
+{"id": "1377.png", "formula": "\\begin{align*} D ^ { 2 k - 1 } \\left ( f \\big | _ { 2 - 2 k } T _ m \\right ) = { m ^ { 1 - 2 k } } D ^ { 2 k - 1 } ( f ) \\big | _ { 2 k } T _ m . \\end{align*}"}
+{"id": "8819.png", "formula": "\\begin{align*} b _ { t } < \\frac { 3 b _ { 3 } } { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { ( 2 - p _ { i } ) t ^ { p _ { i } } } . \\end{align*}"}
+{"id": "7708.png", "formula": "\\begin{align*} a ^ j ( X _ j ( x ^ i ) ) = 0 , \\ ; \\ ; \\ ; i = 1 , \\ldots , m _ 0 . \\end{align*}"}
+{"id": "8131.png", "formula": "\\begin{align*} d ^ 1 ( ( s , t ) , ( s ' , t ' ) ) = | s - s ' | + | t - t ' | . \\end{align*}"}
+{"id": "552.png", "formula": "\\begin{align*} \\mathbf u ( t ) = \\mathbf v ( t ) . \\end{align*}"}
+{"id": "2102.png", "formula": "\\begin{align*} S _ { t _ 0 } : = \\left \\{ ( t , x ) : \\ ; t = t _ 0 \\right \\} . \\end{align*}"}
+{"id": "4180.png", "formula": "\\begin{align*} & \\int _ 0 ^ \\infty r f ( r ) \\log f ( r ) ~ \\mathrm { d } r = \\int _ { | u | \\lambda } ^ \\infty \\frac { K _ { 3 / 2 } ( \\rho ) } { | u | ^ 2 \\rho ^ { 1 / 2 } } \\log \\left ( \\frac { K _ { 3 / 2 } ( \\rho ) } { \\rho ^ { 3 / 2 } } \\right ) \\mathrm { d } \\rho , \\end{align*}"}
+{"id": "4425.png", "formula": "\\begin{align*} \\lambda ^ + = \\frac { [ u ] + \\lambda ^ - H ^ - } { H ^ + } \\end{align*}"}
+{"id": "4737.png", "formula": "\\begin{align*} | D f | _ Y ( A \\cap B ) = | D f | _ B ( A ) . \\end{align*}"}
+{"id": "1872.png", "formula": "\\begin{gather*} \\sup _ { | s - \\sigma _ 0 | = r } | g ( s ) - f ( s ) | < \\inf _ { | s - \\sigma _ 0 | = r } | f ( s ) | \\end{gather*}"}
+{"id": "4806.png", "formula": "\\begin{align*} \\int _ { B _ T } f \\circ \\varphi _ t ( x ) \\ , d t = \\mathrm { V o l } ( B _ T ) \\mu ( f ) + \\sum _ { i = 2 } ^ { d ^ + } \\mathcal { D } _ i ^ + ( f ) G _ i ( T ) T ^ { d \\frac { \\log \\lambda _ i } { \\log \\lambda _ 1 } } + \\mathcal { O } ( | \\partial B _ T | ) , \\end{align*}"}
+{"id": "2473.png", "formula": "\\begin{align*} \\forall P _ A \\in \\Delta ( \\mathcal { A } ) , \\ ; \\overline { H } \\big ( \\textstyle \\bigsqcup _ { a \\in \\mathcal { A } } ^ { P _ A } G _ a \\big ) = \\textstyle \\sum _ { a \\in \\mathcal { A } } P _ A ( a ) \\overline { H } ( G _ a ) . \\end{align*}"}
+{"id": "1568.png", "formula": "\\begin{align*} g ( x , v ) & = \\int _ 0 ^ 1 \\frac { e ^ { - ( 1 - y ) / \\kappa | v | } } { \\kappa | v | ( 1 - e ^ { - 1 / \\kappa | v | } ) } \\rho _ g ( x + y ) \\left ( \\alpha \\mathcal { M } _ { T ( x + y ) } ( v ) + ( 1 - \\alpha ) \\mathcal { M } _ { \\tau ( x + y ) } ( v ) \\right ) \\mathrm { d } y . \\end{align*}"}
+{"id": "2759.png", "formula": "\\begin{align*} R _ { \\mu _ k } ( w ) \\approx \\frac { r _ A } { N } \\sum _ { j = 0 } ^ { N - 1 } \\xi _ N ^ { j } \\cdot d _ j ^ { ( k ) } \\cdot J _ { [ a _ k , b _ k ] } ' ( \\xi _ N ^ j ) \\cdot \\frac { J _ { [ a _ k , b _ k ] } ( \\xi _ N ^ j ) - 1 / c _ j ^ { ( k ) } } { c _ j ^ { ( k ) } - w } , k \\in \\{ 1 , 2 \\} . \\end{align*}"}
+{"id": "2586.png", "formula": "\\begin{align*} v a l ( a b ) & = ( v ( a _ { h _ { 1 } } b _ { h _ { 2 } } p ^ { - \\beta ( h _ { 1 } , h _ { 2 } ) } ) , h _ { 1 } + h _ { 2 } ) = \\\\ & = ( v ( a _ { h _ { 1 } } ) + v ( b _ { h _ { 2 } } ) - e \\beta ( h _ { 1 } , h _ { 2 } ) , h _ { 1 } + h _ { 2 } ) = \\\\ & = ( v ( a _ { h _ { 1 } } ) + v ( b _ { h _ { 2 } } ) - \\beta _ { e } ( h _ { 1 } , h _ { 2 } ) , h _ { 1 } + h _ { 2 } ) = \\\\ & = ( v ( a _ { h _ { 1 } } ) , h _ { 1 } ) + v ( a _ { h _ { 2 } } ) , h _ { 2 } ) = v a l ( a ) + v a l ( b ) ; \\end{align*}"}
+{"id": "4350.png", "formula": "\\begin{align*} { \\bf A } ^ { - 1 } = \\left ( \\begin{array} { c c } 0 & - \\check { { \\bf I } } \\\\ \\check { { \\bf I } } & 0 \\end{array} \\right ) = - { \\bf A } \\ , . \\end{align*}"}
+{"id": "1040.png", "formula": "\\begin{align*} \\hat { u } \\in [ 0 , \\overline { u } _ \\eta ] , \\ \\hat { u } \\not = 0 . \\end{align*}"}
+{"id": "4263.png", "formula": "\\begin{align*} \\left | \\int _ { \\Omega } F ( x , u ( x ) ) d x \\right | = \\left | \\int _ { \\Omega } \\int _ 0 ^ { u ( x ) } f ( x , t ) \\ , d t \\ , d x \\right | \\le M \\int _ { \\Omega } | u ( x ) | d x \\end{align*}"}
+{"id": "753.png", "formula": "\\begin{align*} f ( t ) = \\norm { u } _ { \\widetilde X ^ { s , b } ( 0 , t ) } ^ 2 . \\end{align*}"}
+{"id": "5674.png", "formula": "\\begin{align*} \\Gamma = \\left \\lbrace \\begin{array} { l } Q = y - x _ 1 L = 0 , \\\\ F = x _ 0 x _ 1 L ^ 2 + B L + C = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "5956.png", "formula": "\\begin{align*} - \\Delta _ g - g ( F , \\nabla _ g \\cdot ) - \\omega ^ 2 & = ( D _ { t _ 3 } + i \\widetilde { E } ( t ) - i A ^ \\omega _ F ( t , D _ { t ' } ) ) ( D _ { t _ 3 } + i A ^ \\omega _ F ( t , D _ { t ' } ) ) , \\end{align*}"}
+{"id": "5621.png", "formula": "\\begin{align*} I : = \\dfrac { \\mathrm d } { \\mathrm d t } \\left ( \\int _ 0 ^ \\infty \\exp _ p \\left ( \\mu | w _ t | ^ { p ' } \\right ) r ^ { \\alpha _ 0 } \\mathrm d r \\right ) \\Bigg | _ { t = 1 } \\neq 0 \\end{align*}"}
+{"id": "397.png", "formula": "\\begin{align*} \\gamma : = \\alpha _ * \\beta : \\Pi _ A { B } \\to X \\ , , \\end{align*}"}
+{"id": "5837.png", "formula": "\\begin{align*} & \\alpha ^ { c 2 } _ { m a x } : = \\alpha _ 2 + 2 \\alpha _ 3 , \\\\ & \\alpha ^ { c 3 } _ { m a x } : = \\alpha _ 2 + 2 \\alpha _ 3 + 2 \\alpha _ 4 , \\end{align*}"}
+{"id": "4935.png", "formula": "\\begin{align*} \\log g _ k = - \\dfrac { \\log 2 } { 2 } + \\dfrac { \\pi } { 3 \\cdot 2 ^ { k + 2 } } - \\dfrac { 2 ^ { k - 1 } } { \\pi } \\sum _ { m , n \\geq 1 } \\dfrac { ( - 1 ) ^ m } { 2 ^ k m ^ 2 + n ^ 2 } . \\end{align*}"}
+{"id": "1618.png", "formula": "\\begin{align*} U _ { \\mathcal X _ 2 \\to y _ 2 } = \\{ u \\in \\mathcal P _ { \\mathcal X _ 2 } : \\{ \\alpha ^ { 1 } _ { \\mathcal X _ 1 } , u \\} \\frac { \\rho } { 2 } \\prod _ { j \\in [ k ] \\setminus \\{ 1 , 2 \\} } | \\mathcal P _ { \\mathcal X _ j } | \\mathcal A _ { \\mathcal Y } \\} . \\end{align*}"}
+{"id": "3958.png", "formula": "\\begin{align*} \\lVert { ( x ' \\cdot v ) v ' - ( x ' \\cdot v ' ) v } \\rVert = \\\\ \\lVert ( x ' \\cdot v ) ( v ' - v + v ) - ( x ' \\cdot ( v ' - v + v ) ) v \\rVert = \\\\ \\lVert ( x ' \\cdot v ) ( v ' - v ) - ( x ' \\cdot ( v ' - v ) ) v \\rVert \\leq \\\\ 2 \\lVert x ' \\rVert \\lVert v \\rVert \\lVert v ' - v \\rVert . = \\\\ 2 \\lVert x ' \\rVert \\lVert v ' - v \\rVert . \\end{align*}"}
+{"id": "1151.png", "formula": "\\begin{align*} ( \\delta _ c ) ^ 2 ( m ) = \\delta _ c ( \\delta _ c m ) = ~ & ( \\delta _ 1 \\delta _ c m ~ , \\delta _ 2 \\delta _ c m ) \\\\ = ~ & ( \\delta _ 1 \\delta _ 1 m ~ , \\delta _ 2 \\delta _ 2 m ) = 0 . \\end{align*}"}
+{"id": "3341.png", "formula": "\\begin{align*} u = \\biggl ( \\prod _ { i = 1 } ^ { N } \\theta _ i ^ { b _ i / d _ i } D _ { \\zeta _ i } ( \\theta _ i ) ^ { - \\frac { 1 } { m } } \\biggr ) a _ \\zeta ( \\theta ) \\end{align*}"}
+{"id": "2304.png", "formula": "\\begin{align*} \\| \\partial _ { x } ( v _ 1 v _ 2 ) \\| _ { Y ^ { 0 , - c , \\infty } } \\lesssim \\| v _ 1 \\| _ { Y ^ { 0 , b , 1 } } \\| v _ 2 \\| _ { Y ^ { 0 , b , 1 } } , 0 < b , c < \\frac 1 2 , 2 b + c = 1 \\end{align*}"}
+{"id": "5536.png", "formula": "\\begin{align*} E _ \\ell = \\frac { 1 } { 3 } \\left ( E _ { \\ell - 1 } - \\frac { 1 } { 3 } \\right ) \\bigcup \\frac { 1 } { 3 } \\left ( E _ { \\ell - 1 } + \\frac { 1 } { 3 } \\right ) , \\ell \\ge 1 , \\end{align*}"}
+{"id": "7106.png", "formula": "\\begin{align*} \\mathcal { G } _ { 0 } ( . . . ) = & \\ , \\ , \\int _ 0 ^ \\infty \\ , \\int _ 0 ^ \\infty \\ , U ( y _ 1 ) U ( y _ 2 ) \\ , \\\\ & \\times \\int _ { \\mathbb { R } } \\ , V ( \\xi ) \\ , e \\left ( 3 \\ , \\lambda \\left ( { y ^ { 1 / 3 } _ 1 } - { y ^ { 1 / 3 } _ 2 } \\right ) \\xi ^ { 1 / 3 } \\right ) d \\xi \\\\ & \\times e \\left ( \\frac { - t } { 2 \\pi } ( \\log y _ 1 - \\log y _ 2 ) + \\frac { 2 \\sqrt { N } } { p _ 1 q \\sqrt { p _ 2 } } \\left ( { \\sqrt { m y _ 1 } } - { \\sqrt { m _ 1 y _ 2 } } \\right ) \\right ) d y _ 1 \\ , d y _ 2 . \\end{align*}"}
+{"id": "2774.png", "formula": "\\begin{align*} I ( T _ 2 ; x ) = \\ ! \\begin{multlined} [ t ] x ^ { 1 4 } + 4 8 x ^ { 1 3 } + 2 3 7 2 x ^ { 1 2 } + 1 5 4 9 8 x ^ { 1 1 } + 4 8 0 8 6 x ^ { 1 0 } + 9 0 1 7 8 x ^ 9 + \\\\ 1 1 2 8 7 0 x ^ 8 + 9 8 9 6 8 x ^ 7 + 6 2 1 8 3 x ^ 6 + 2 8 1 4 7 x ^ 5 + 9 0 8 9 x ^ 4 + 2 0 3 7 x ^ 3 + \\\\ 3 0 0 x ^ 2 + 2 6 x + 1 , \\end{multlined} \\end{align*}"}
+{"id": "2218.png", "formula": "\\begin{align*} \\left ( D _ { a + } ^ \\alpha y \\right ) ( x ) = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\frac { d } { d x } \\underset { a } { \\overset { x } { \\int } } \\frac { y ( t ) d t } { ( x - t ) ^ \\alpha } , \\ , ( x > a ) , \\end{align*}"}
+{"id": "2582.png", "formula": "\\begin{align*} g ( ( z , h ) + ( z ' , h ' ) ) & = g ( z + z ' - \\beta ( h , h ' ) , h + h ' ) = \\\\ & = ( z + z ' - \\beta ( h , h ' ) + \\psi ( h + h ' ) , h + h ' ) = \\\\ & = ( z + z ' - \\beta ' ( h , h ' ) + \\psi ( h ) + \\psi ( h ' ) , h + h ' ) = \\\\ & = ( z + \\psi ( h ) , h ) + ( z ' + \\psi ( h ' ) , h ' ) = \\\\ & = g ( z , h ) + g ( z ' , h ' ) \\end{align*}"}
+{"id": "2305.png", "formula": "\\begin{align*} \\hat { f } ( \\xi ) = \\int _ { \\R } e ^ { - i x \\xi } f ( x ) d x . \\end{align*}"}
+{"id": "1411.png", "formula": "\\begin{gather*} \\| \\tilde H ( x ) - \\tilde H ^ K ( x ) \\| _ { \\it m \\to \\it m } \\le C \\sup _ n \\sum _ { k = K + 1 } ^ \\infty \\frac { \\xi _ k } { | n - k | + 1 } \\le C \\sqrt { \\sum _ { k = K + 1 } ^ \\infty { \\xi ^ 2 _ k } } . \\end{gather*}"}
+{"id": "2508.png", "formula": "\\begin{align*} | \\boldsymbol { v } | _ { \\boldsymbol { H } ^ { \\emph { \\textbf { c u r l } } , \\frac { 1 } { 2 } } } ^ 2 : = T \\| \\emph { \\textbf { c u r l } } \\ , \\boldsymbol { v } _ 0 ^ c \\| _ { \\Omega } ^ 2 + \\frac { T } { 2 } \\sum _ { k = 1 } ^ { \\infty } \\Big ( k \\omega \\| \\boldsymbol { v } _ k \\| _ { \\Omega } ^ 2 + \\| \\emph { \\textbf { c u r l } } \\ , \\boldsymbol { v } _ k \\| _ { \\Omega } ^ 2 \\Big ) \\end{align*}"}
+{"id": "7699.png", "formula": "\\begin{align*} \\bigl ( ( \\rho \\otimes _ 0 \\varphi ) _ \\sigma ^ { \\sigma ' } \\bigr ) ( g ) = \\rho _ { g \\sigma } ^ { \\sigma ' } \\cdot s ( g ) . \\end{align*}"}
+{"id": "1799.png", "formula": "\\begin{align*} Z _ N ( s , \\alpha ) & = \\sum _ { n = 0 } ^ { \\infty } ( n + \\alpha ) ^ { - s } \\frac { 1 } { 2 \\pi i } \\int _ { c - i \\infty } ^ { c + i \\infty } \\widehat { \\phi } ( w ) \\left ( \\frac { n + \\alpha } { N } \\right ) ^ { - w } \\ , d w \\\\ & = \\frac { 1 } { 2 \\pi i } \\sum _ { n = 0 } ^ { \\infty } \\int _ { c - i \\infty } ^ { c + i \\infty } ( n + \\alpha ) ^ { - ( s + w ) } \\widehat { \\phi } ( w ) N ^ w \\ , d w . \\end{align*}"}
+{"id": "469.png", "formula": "\\begin{align*} p _ t ( r , s ) = p _ t ( s , r ) , r , s , t > 0 . \\end{align*}"}
+{"id": "2327.png", "formula": "\\begin{align*} \\| \\theta _ f x \\| = \\left \\| \\sum _ { { g \\in G } } f _ g ( x ) \\delta _ g \\right \\| = \\left \\| \\sum _ { { g \\in G } } \\zeta _ g ( U x ) \\delta _ g \\right \\| = \\| U x \\| = \\| x \\| , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "7420.png", "formula": "\\begin{align*} \\langle \\partial _ { t } \\rho _ { n } ( t ) , f \\rangle _ { * } & = \\langle - \\partial _ { x } ( \\rho _ { n } ( t ) u _ { n } ( t ) ) , f \\rangle _ { * } = ( \\rho _ { n } ( t ) u _ { n } ( t ) , \\partial _ { x } f ) _ { L ^ { 2 } _ { x } } \\\\ [ 1 e x ] & \\le \\| \\rho _ { n } ( t ) \\| _ { L ^ { 2 } _ { x } } \\| u _ { n } ( t ) \\| _ { L ^ { 2 } _ { x } } \\| f \\| _ { W ^ { 1 , \\infty } _ { 0 } } . \\end{align*}"}
+{"id": "1652.png", "formula": "\\begin{align*} \\mathfrak { s } = \\mathfrak { n } \\oplus \\mathfrak { a } , \\end{align*}"}
+{"id": "203.png", "formula": "\\begin{align*} & \\delta _ 1 ( \\tau ) = \\frac { 1 } { 8 } ( \\theta _ 2 ^ 4 + \\theta _ 3 ^ 4 ) , ~ ~ \\varepsilon _ 1 ( \\tau ) = \\frac { 1 } { 1 6 } \\theta _ 2 ^ 4 \\theta _ 3 ^ 4 , \\\\ & \\delta _ 2 ( \\tau ) = - \\frac { 1 } { 8 } ( \\theta _ 1 ^ 4 + \\theta _ 3 ^ 4 ) , ~ ~ \\varepsilon _ 2 ( \\tau ) = \\frac { 1 } { 1 6 } \\theta _ 1 ^ 4 \\theta _ 3 ^ 4 . \\end{align*}"}
+{"id": "882.png", "formula": "\\begin{align*} \\tilde { u } ^ i _ \\lambda ( x , 0 ) = \\mathrm { e } ^ { - \\lambda x } L _ i ^ u ( x ) , ~ ~ \\tilde { v } ^ i _ \\lambda ( x , 0 ) = \\mathrm { e } ^ { - \\lambda x } L _ i ^ v ( x ) , ~ ~ i = 1 , 2 . \\end{align*}"}
+{"id": "5503.png", "formula": "\\begin{align*} c _ + ^ { ( b a ^ k , g ) } = \\mathrm { p e x p } \\left ( \\frac { Q ^ 8 - Q ^ { - 8 } } { \\left ( 1 + p ^ { g _ { 2 , 1 } } s ^ { - g _ { 2 , 2 } } \\right ) \\left ( 1 + p ^ { - ( 1 - k ) g _ { 2 , 1 } - g _ { 1 , 1 } } s ^ { ( 1 - k ) g _ { 2 , 2 } + g _ { 1 , 2 } } \\right ) \\left ( 1 + p ^ { g _ { 1 , 1 } - k g _ { 2 , 1 } } s ^ { k g _ { 2 , 2 } - g _ { 1 , 2 } } \\right ) } \\right ) = \\frac 1 { c _ - ^ { ( b ^ { - 1 } , b a ^ k g ) } } = : \\lambda \\end{align*}"}
+{"id": "2506.png", "formula": "\\begin{align*} \\begin{aligned} \\boldsymbol { v } ^ { \\perp } ( \\boldsymbol { x } , t ) = \\sum _ { k = 1 } ^ { \\infty } \\left ( - \\boldsymbol { v } _ k ^ c ( \\boldsymbol { x } ) \\sin ( k \\omega t ) + \\boldsymbol { v } _ k ^ s ( \\boldsymbol { x } ) \\cos ( k \\omega t ) \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "4601.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\mathbf 1 \\otimes \\cdots \\otimes \\Big ( \\ ! \\sum _ { n \\gg - \\infty } \\ ! \\ ! a _ n ^ { ( i ) } J ^ a _ n \\Big ) \\otimes \\cdots \\otimes \\mathbf 1 \\ , , \\end{align*}"}
+{"id": "6170.png", "formula": "\\begin{align*} \\eta _ ( A ) = \\sum _ { 1 \\leq i < i ' \\leq n , \\ , 1 \\leq j ' \\leq j \\leq n } a _ { i j } a _ { i ' j ' } . \\end{align*}"}
+{"id": "3057.png", "formula": "\\begin{align*} { \\bf a } _ { X } ( Y ) = \\frac { 1 } { \\sqrt { N _ { X } } } { \\left [ 1 , e ^ { j Y } , \\ldots , e ^ { j ( N _ { X } - 1 ) Y } \\right ] ^ T } , \\end{align*}"}
+{"id": "5808.png", "formula": "\\begin{align*} & \\alpha ' _ i = \\varepsilon _ i - \\varepsilon _ { i + 1 } i = 1 , \\dots , n - 1 , \\alpha ' _ n = 2 \\varepsilon _ n , \\\\ & \\norm { \\alpha _ i } = \\sqrt { 2 } i = 1 , \\dots , n - 1 , \\norm { \\alpha _ n } = 2 . \\end{align*}"}
+{"id": "4482.png", "formula": "\\begin{align*} \\theta _ 0 \\geq 1 , \\ ; \\theta _ i = \\sqrt { \\theta ^ 2 _ 0 + i } \\ , . \\end{align*}"}
+{"id": "6314.png", "formula": "\\begin{align*} w _ \\lambda ^ { x _ 0 } \\partial _ x ( g _ { [ < \\lambda ^ { \\sigma } ] } - g _ { [ < \\lambda ] } ) \\partial _ x w _ \\lambda = \\lambda ^ 2 L ( w _ \\lambda , w _ \\lambda , u _ { < \\lambda } , u _ { [ \\lambda ^ \\sigma , \\lambda ) } ) , \\end{align*}"}
+{"id": "1695.png", "formula": "\\begin{align*} M _ { \\texttt { b } ; \\mu } ( \\boldsymbol { \\xi } ) : = \\frac { 1 } { N _ { \\texttt { b } ; \\mu } } \\sum _ { \\substack { \\sigma \\in S _ { n } \\\\ \\epsilon \\in \\{ 1 , - 1 \\} ^ n } } \\exp ( i \\epsilon _ 1 \\xi _ { \\sigma _ 1 } \\mu _ 1 + \\cdots + i \\epsilon _ n \\xi _ { \\sigma _ { n } } \\mu _ { n } ) \\end{align*}"}
+{"id": "3013.png", "formula": "\\begin{align*} 0 = \\Phi \\left ( \\theta ^ { - 1 } \\theta ' \\right ) = \\Phi \\left ( \\theta ^ { - 1 } \\right ) + \\Phi \\left ( \\theta ' \\right ) , \\end{align*}"}
+{"id": "6103.png", "formula": "\\begin{align*} \\begin{aligned} g _ { S _ l } ^ k ( \\gamma ) & = \\nabla f _ { S _ l } ( w _ { 1 0 1 } ) - \\gamma ( \\nabla f _ { S _ l } ( w _ { k } ) - \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\nabla f _ i ( w _ k ) ) \\\\ k & = 0 , 1 , 2 . . . 1 0 0 , l = 1 , 2 . . . 1 0 0 \\end{aligned} \\end{align*}"}
+{"id": "4215.png", "formula": "\\begin{align*} F = \\begin{pmatrix} \\lambda & A & B \\\\ A & \\mu & C \\\\ - B & - C & \\tau \\end{pmatrix} , \\end{align*}"}
+{"id": "3642.png", "formula": "\\begin{align*} \\Delta _ \\omega \\phi = 0 \\end{align*}"}
+{"id": "8458.png", "formula": "\\begin{align*} H _ { r , s } = \\frac { h _ { B } ^ { r + 1 } h _ { A } ^ { s + 1 } } { ( r + s + 2 ) r ! ( s + 1 ) ! } + \\sum _ { \\substack { 0 \\leq u \\leq r \\\\ 0 \\leq v \\leq s } } \\frac { h _ { u , v } h _ { B } ^ { r - u } h _ { A } ^ { s - v } } { ( r - u + s - v + 1 ) ( r - u ) ! ( s - v ) ! } . \\end{align*}"}
+{"id": "1847.png", "formula": "\\begin{gather*} \\mathbf { 1 } _ { A ( s _ n , t _ n ) } ( e ^ { i \\theta } ) = \\sum _ { k \\in \\mathbb { Z } } \\mathbf { 1 } _ { ( s _ n , t _ n ) } ( \\theta + 2 k \\pi ) \\end{gather*}"}
+{"id": "6788.png", "formula": "\\begin{align*} \\mathcal { V } \\left ( \\overline { u } , \\overline { v } \\right ) & = q ( 1 ) d \\lambda _ 1 ^ 2 e ^ { \\lambda _ 1 z } - q ( 1 ) c \\lambda _ 1 e ^ { \\lambda _ 1 z } + s \\lambda _ 1 e ^ { \\lambda _ 1 z } - s q ( 1 ) e ^ { 2 \\lambda _ 1 z } \\\\ [ 0 . 2 c m ] & = P ( c , \\lambda _ 1 ) q ( 1 ) e ^ { \\lambda _ 1 z } - s q ( 1 ) e ^ { 2 \\lambda _ 1 z } \\\\ [ 0 . 2 c m ] & = - s q ( 1 ) e ^ { 2 \\lambda _ 1 z } \\leq 0 . \\end{align*}"}
+{"id": "5165.png", "formula": "\\begin{align*} \\frac { E [ \\int _ { L _ j } ^ { L _ j + L _ { j + 1 } } t d t ] } { E [ L _ { j + 1 } ] } = \\frac { E [ L ^ 2 ] } { 2 E [ L ] } + E [ L ] . \\end{align*}"}
+{"id": "4370.png", "formula": "\\begin{align*} j ^ { \\pm } | _ { \\Gamma } = 0 , H ^ { \\pm } _ N | _ { \\Gamma } = 0 . \\end{align*}"}
+{"id": "1707.png", "formula": "\\begin{align*} Q _ { \\texttt { b } ; \\mu } ( \\boldsymbol { \\xi } ) : = & P _ { \\texttt { b } ; \\mu } ( \\boldsymbol { \\xi } ; q , q _ 0 ) \\\\ & - q ^ { \\frac { 1 } { 2 } _ { \\mu _ 1 } ( \\mu ) ( _ { \\mu _ 1 } ( \\mu ) - 1 ) } q _ 1 ^ { _ { \\mu _ 1 } ( \\mu ) } P _ { \\texttt { b } ; \\mu - \\omega _ { \\texttt { b } ; \\mu } } ( \\boldsymbol { \\xi } ; q , q _ 0 ) \\end{align*}"}
+{"id": "3939.png", "formula": "\\begin{align*} \\lambda ( K , \\omega \\setminus \\gamma _ n ) & \\longrightarrow 0 \\\\ \\lambda ( L , \\omega \\setminus \\gamma _ n ) & \\longrightarrow 0 . \\end{align*}"}
+{"id": "3882.png", "formula": "\\begin{align*} f ( u ( i _ 1 ( y _ 1 ) ) ) = i _ 1 ( y _ 1 ) \\end{align*}"}
+{"id": "7402.png", "formula": "\\begin{align*} \\left ( W _ { n } ^ { M } \\right ) ' ( t ) & = \\lim _ { h \\searrow 0 } \\left [ \\frac { W _ { n } ^ { M } ( t + h ) - W _ { n } ^ { M } ( t ) } { h } \\right ] = \\lim _ { h \\searrow 0 } \\left [ \\frac { W _ { n } ( x _ { t + h } , t + h ) - W _ { n } ( x _ { t } , t ) ) } { h } \\right ] \\\\ [ 1 e x ] & \\ge \\lim _ { h \\searrow 0 } \\left [ \\frac { W _ { n } ( x _ { t } , t + h ) - W _ { n } ( x _ { t } , t ) } { h } \\right ] = \\partial _ { t } W _ { n } ( x _ { t } , t ) . \\end{align*}"}
+{"id": "6414.png", "formula": "\\begin{align*} u ( x , 0 ) = u _ { 0 } ( x ) , v ( x , 0 ) = v _ { 0 } ( x ) , x \\in \\mathbb { R } , \\end{align*}"}
+{"id": "5616.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } I _ 1 = 0 , \\quad \\forall R > 0 \\end{align*}"}
+{"id": "5126.png", "formula": "\\begin{align*} \\Delta _ { t , s } \\geq \\frac { \\log _ d | \\mathcal { Q } _ { t , s } | } { L } = \\frac { \\log _ d \\delta _ { t , s } } { L } , & & \\forall t \\in [ T ] , s \\in \\mathcal { E } ( t ) . \\end{align*}"}
+{"id": "447.png", "formula": "\\begin{align*} & \\langle | F ( | D | ) | ^ { 1 / 2 } [ v ] _ { \\ell ' , m ' } , F ( | D | ) ^ { 1 / 2 } [ u ] _ { \\ell , m } \\rangle _ { L ^ 2 ( \\R ^ d ) } = \\langle ( [ v ] _ { \\ell ' , m ' } ) ^ \\wedge , F \\cdot ( [ u ] _ { \\ell , m } ) ^ \\wedge \\rangle _ { L ^ 2 ( \\R ^ d ) } , \\end{align*}"}
+{"id": "394.png", "formula": "\\begin{align*} \\mathsf { T C o f } \\ , = \\ , ^ { \\pitchfork } \\mathcal { F } . \\end{align*}"}
+{"id": "1643.png", "formula": "\\begin{align*} \\bigl ( \\alpha \\coth ( r / 2 ) + \\beta \\tanh ( r / 2 ) \\bigr ) ^ 2 = ( \\alpha + \\beta ) ^ 2 + \\frac { 4 \\beta ^ 2 } { \\sinh ^ 2 ( r ) } + \\frac { \\alpha ^ 2 - \\beta ^ 2 } { \\sinh ^ 2 ( r / 2 ) } . \\end{align*}"}
+{"id": "3175.png", "formula": "\\begin{align*} R \\begin{bmatrix} - S & 0 \\\\ I _ n & I _ n \\end{bmatrix} = \\begin{bmatrix} Y - X S & Y \\\\ 0 & I _ n \\end{bmatrix} . \\end{align*}"}
+{"id": "4260.png", "formula": "\\begin{align*} H _ k ^ { - } = \\textrm { s p a n } _ { \\R } \\{ u _ j \\ : \\ \\lambda _ j < \\lambda _ k \\} . \\end{align*}"}
+{"id": "3137.png", "formula": "\\begin{align*} \\mathbb { E } [ | \\mathbf { H } _ { k , } \\cdot \\mathbf { H } _ { l , } ^ H | ^ 2 ] & = \\frac { 1 } { N ^ 2 ( \\kappa + 1 ) ^ 2 } \\mathbb { E } [ ( x _ 1 + \\cdots + x _ M ) \\cdot ( x ^ * _ 1 + \\cdots + x ^ * _ M ) ] . \\end{align*}"}
+{"id": "1569.png", "formula": "\\begin{align*} g ( x , v ) & = \\int _ 0 ^ 1 \\frac { e ^ { - ( 1 - y ) / \\kappa | v | } } { \\kappa | v | ( 1 - e ^ { - 1 / \\kappa | v | } ) } \\rho _ g ( x - y ) \\left ( \\alpha \\mathcal { M } _ { T ( x - y ) } ( v ) + ( 1 - \\alpha ) \\mathcal { M } _ { \\tau ( x - y ) } ( v ) \\right ) \\mathrm { d } y . \\end{align*}"}
+{"id": "5111.png", "formula": "\\begin{align*} K ^ 2 = - \\mathrm { i d } _ { T E } \\Longleftrightarrow \\left \\{ \\begin{array} { l } r ^ { 2 } = - \\mathrm { i d } _ { T M } , \\\\ l ^ 2 = - \\mathrm { i d } _ { E } , \\\\ D _ { r ( X ) } ( \\sigma ) + l ( D _ X ( \\sigma ) ) = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "2289.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } \\varepsilon _ j ^ 2 & = \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n - 1 } \\left | \\sum _ { j = 1 } ^ { n } \\varepsilon _ j e ^ { - 2 \\pi i k \\frac { j } { n } } \\right | ^ 2 \\leq \\max _ { 1 \\leq k \\leq n - 1 } \\left | \\sum _ { j = 1 } ^ { n } \\varepsilon _ j e ^ { - 2 \\pi i k \\frac { j } { n } } \\right | ^ 2 . \\end{align*}"}
+{"id": "7335.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} J u ' ( t ) + A ( \\lambda , t ) u ( t ) & = 0 , t \\in \\mathbb { R } \\\\ \\lim _ { t \\rightarrow \\pm \\infty } u ( t ) & = 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7165.png", "formula": "\\begin{align*} \\| g _ j \\| & = \\displaystyle \\sup _ { x \\in \\mathcal { X } , \\| x \\| \\leq 1 } | g _ j ( x ) | = \\displaystyle \\sup _ { x \\in \\mathcal { X } , \\| x \\| \\leq 1 } | f _ j ( V ^ { - 1 } x ) | = \\displaystyle \\sup _ { V y \\in \\mathcal { X } , \\| V y \\| \\leq 1 } | f _ j ( y ) | \\\\ & = \\displaystyle \\sup _ { V y \\in \\mathcal { X } , \\| y \\| \\leq 1 } | f _ j ( y ) | = \\| f _ j \\| = 1 , \\forall 1 \\leq j \\leq n . \\end{align*}"}
+{"id": "4735.png", "formula": "\\begin{align*} \\Delta J ( X ) = \\sum \\limits _ { k = 1 } ^ D \\frac { d ^ 2 J ( X ) } { d x _ k ^ 2 } = \\sum \\limits _ { i = 1 } ^ n \\frac { - 2 D w _ i \\left [ \\sum \\limits _ { k = 1 } ^ D x _ { k i } ^ 2 \\right ] ^ 2 + 8 w _ i \\left [ \\sum \\limits _ { k = 1 } ^ D x _ { k i } ^ 2 \\right ] ^ 2 } { \\left [ \\sum \\limits _ { k = 1 } ^ D x _ { k i } ^ 2 \\right ] ^ 4 } = \\sum \\limits _ { i = 1 } ^ n \\frac { ( 8 - 2 D ) w _ i } { \\left [ \\sum \\limits _ { k = 1 } ^ D x _ { k i } ^ 2 \\right ] ^ 2 } . \\end{align*}"}
+{"id": "3875.png", "formula": "\\begin{align*} \\beta _ i ( t ) = \\left \\{ \\begin{aligned} & 0 , \\ & & \\ t \\leq 0 , \\\\ & \\beta _ i ( T ) , \\ & & \\ t \\geq T , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1974.png", "formula": "\\begin{align*} \\tau = s - r = \\sum _ { i = m - \\delta + 1 } ^ m \\left ( s _ i - r _ i \\right ) p ^ { i - 1 } \\geq p ^ { m - \\delta } , \\end{align*}"}
+{"id": "2492.png", "formula": "\\begin{align*} N _ \\sigma ( K _ X + B + M ) & = p _ * N _ \\sigma \\big ( p ^ * ( K _ X + B + M ) \\big ) \\\\ & = p _ * N _ \\sigma \\big ( q ^ * ( K _ { X ' } + B ' + M ' ) \\big ) + p _ * E , \\end{align*}"}
+{"id": "578.png", "formula": "\\begin{align*} \\mathbf U ( t ) - \\mathbf V ( t ) = \\Delta _ 1 ( t ) + \\Delta _ 2 ( t ) , \\end{align*}"}
+{"id": "6796.png", "formula": "\\begin{align*} & u ' ( z ) = e ^ { c ( z - \\varsigma ) } u ' ( \\varsigma ) - e ^ { c z } \\int _ z ^ \\varsigma e ^ { - c \\tau } \\left [ f ( u ( \\tau ) ) \\left ( v ( \\tau ) - p ( u ( \\tau ) ) \\right ) \\right ] d \\tau , \\end{align*}"}
+{"id": "6224.png", "formula": "\\begin{align*} C ( t ) \\coloneqq \\begin{cases} C e ^ { - t d } & ( t \\ge 1 ) \\\\ C t ^ { - d / \\tilde { p } } & ( 0 < t \\le 1 ) , \\end{cases} \\frac { 1 } { \\tilde { p } } \\coloneqq \\max \\Big \\{ \\frac { 1 } { p _ 2 } - \\frac { 1 } { p _ 1 } , 0 \\Big \\} , \\end{align*}"}
+{"id": "1732.png", "formula": "\\begin{align*} P _ { \\texttt { b } ; \\mu } ( \\boldsymbol { \\xi } ; 1 , q _ 0 ) = \\prod _ { 1 \\leq j \\leq n } p _ { \\mu _ j } ( \\xi _ j ; q _ 0 ) \\end{align*}"}
+{"id": "7483.png", "formula": "\\begin{align*} C _ L = \\begin{pmatrix} 0 & - \\alpha _ 0 / \\alpha _ 2 \\\\ 1 & - \\alpha _ 1 / \\alpha _ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "4720.png", "formula": "\\begin{align*} A = \\bigcup _ { ( x , y , z ) } A _ { ( x , y , z ) } , \\end{align*}"}
+{"id": "2605.png", "formula": "\\begin{align*} & \\sum _ { j = 0 } ^ { m } \\sum _ { k = 0 } ^ { m } \\tilde { \\alpha } _ { j + k + 1 } \\xi _ { j } \\xi _ { k } \\\\ & \\ \\ \\ \\ = \\int _ 0 ^ \\infty \\cdots \\int _ 0 ^ \\infty \\left ( \\sum _ { j = 0 } ^ { m } ( x _ 1 \\cdots x _ d ) ^ { j } \\xi _ { j } \\right ) ^ 2 ( x _ 1 \\cdots x _ d ) p ( { \\pmb x } ) d \\mu \\cdots d \\mu \\end{align*}"}
+{"id": "6561.png", "formula": "\\begin{align*} \\gamma _ N = \\gamma - N ^ { - \\kappa } , \\ \\kappa = \\kappa ( b , d , \\rho _ 2 , \\rho _ 3 ) > 0 . \\end{align*}"}
+{"id": "907.png", "formula": "\\begin{align*} t ^ { 1 - \\alpha } \\phi _ { 2 t } - \\phi _ { 2 x x } - \\frac { c } { x } \\phi _ { 2 x } - n x ^ k \\eta _ { 1 x } = 0 , \\end{align*}"}
+{"id": "2352.png", "formula": "\\begin{align*} \\lambda _ g \\rho _ h = \\rho _ h \\lambda _ g , \\forall g , h \\in G . \\end{align*}"}
+{"id": "972.png", "formula": "\\begin{align*} ( a - c ) ^ 2 ( b ^ 2 + 1 ) = ( b - c ) ^ 2 ( a ^ 2 + 1 ) \\end{align*}"}
+{"id": "671.png", "formula": "\\begin{align*} \\norm { u } _ { X _ { h ( \\xi ) } ^ { s , b } ( S , T ) } ^ 2 = \\int \\langle \\xi \\rangle ^ { 2 s } \\norm { e ^ { i t h ( \\xi ) } \\mathcal F _ x u ( t , \\xi ) } _ { H _ t ^ b ( S , T ) } ^ 2 d \\xi . \\end{align*}"}
+{"id": "7587.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\pi } \\exp \\left ( - \\frac { \\beta ^ { \\frac { 1 } { 2 } } N ^ { \\frac { 1 } { 2 } } s ^ { \\frac { 3 } { 4 } } t ^ { \\frac { 3 } { 4 } } \\theta ^ 2 } { 2 t } \\right ) d \\theta & \\le C \\beta ^ { - \\frac { 1 } { 4 } } N ^ { - \\frac { 1 } { 4 } } s ^ { - \\frac { 3 } { 8 } } t ^ { \\frac { 1 } { 8 } } \\\\ & \\le C \\beta ^ { - \\frac { 1 } { 4 } } N ^ { - \\frac { 1 } { 4 } } t ^ { - \\frac { 1 } { 4 } } . \\end{align*}"}
+{"id": "6285.png", "formula": "\\begin{align*} c ^ 4 _ { \\lambda , m } ( \\xi _ 1 , \\xi _ 2 , \\xi _ 3 , \\xi _ 4 ) = i c _ { \\lambda } ( \\xi _ 1 , \\xi _ 2 , \\xi _ 3 ) - i \\bar c _ { \\lambda } ( \\xi _ 2 , \\xi _ 3 , \\xi _ 4 ) . \\end{align*}"}
+{"id": "8610.png", "formula": "\\begin{align*} h _ b \\overline { V } _ 0 & = \\int _ { - 1 + \\beta b ( X ) } ^ 0 \\nabla _ X \\phi _ 0 \\ : \\mathrm { d } z + \\mu \\beta \\int _ { - 1 + \\beta b ( X ) } ^ 0 \\nabla _ X \\phi _ 1 \\ : \\mathrm { d } z \\\\ & = I _ 1 + I _ 2 . \\end{align*}"}
+{"id": "8323.png", "formula": "\\begin{align*} \\lfloor \\beta \\rfloor = \\left \\{ \\begin{array} { l l } a & \\mbox { i f $ ( a , b , c ) $ s a t i s f i e s ( I ) } \\\\ a - 1 & \\mbox { i f $ ( a , b , c ) $ s a t i s f i e s ( I I ) } \\\\ a - 2 & \\mbox { i f $ ( a , b , c ) $ s a t i s f i e s ( I I I ) . } \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "6649.png", "formula": "\\begin{align*} \\alpha _ 1 = { 1 \\over 2 } , \\alpha _ 2 = { 1 \\over 8 } , \\alpha _ 3 = { 1 \\over 1 6 } , \\alpha _ 4 = { 5 \\over 1 2 8 } , \\alpha _ 5 = { 7 \\over 2 5 6 } , \\alpha _ 6 = { 2 1 \\over 1 0 2 4 } . \\end{align*}"}
+{"id": "6540.png", "formula": "\\begin{align*} & \\ \\ \\ \\Sigma _ N \\\\ & = \\bigcup _ { \\Lambda \\in ( 0 , n ) + \\mathcal { E R } _ 0 ( N ) , \\ | n | \\leq 2 N } \\left \\{ \\sigma \\in \\R : \\ \\min _ { \\xi = \\pm 1 , ( k , n ) \\in \\Lambda } | \\xi ( \\sigma + k \\cdot \\omega ) + \\mu _ n | \\leq e ^ { - 2 N ^ { \\rho _ 1 } } \\right \\} . \\end{align*}"}
+{"id": "4314.png", "formula": "\\begin{align*} \\begin{alignedat} { 4 } & = ( \\theta , d , \\vec { l } , \\vec { m } ) \\\\ & = q _ r + \\epsilon q _ d \\\\ & = ( w _ r + \\vec { v } _ r ) + \\epsilon ( w _ d + \\vec { v } _ d ) \\end{alignedat} \\end{align*}"}
+{"id": "5737.png", "formula": "\\begin{align*} \\| V \\| _ 1 = \\| V \\| _ { C ^ 1 _ { ( x , t ) } ( Q _ 5 ) } + 1 . \\end{align*}"}
+{"id": "4332.png", "formula": "\\begin{align*} Q _ { ( \\beta ) } = ( { \\bf 1 } + { \\rm e } ^ { - \\beta \\boldsymbol { h } } ) ^ { - 1 } \\end{align*}"}
+{"id": "8587.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ { X , z } ^ { \\mu } \\phi _ 0 = 0 \\ \\ \\mathrm { i n } \\ \\ \\mathcal { S } _ b , \\\\ \\phi _ 0 | _ { z = 0 } = \\psi , \\ \\ \\big [ \\partial _ z \\phi _ 0 - \\mu \\beta \\nabla _ X b \\cdot \\nabla _ X \\phi _ 0 \\big ] \\big | _ { z = - 1 + \\beta b } = \\mu \\beta \\nabla _ X \\cdot \\mathcal { L } _ { 1 } ^ { \\mu } [ \\beta b ] \\nabla _ X \\psi , \\end{cases} \\end{align*}"}
+{"id": "444.png", "formula": "\\begin{align*} & \\int _ 0 ^ \\infty t ^ \\beta \\sigma _ t ^ { ( \\alpha / 2 ) } ( \\tau ) \\ , \\frac { d t } { t } = \\frac { \\Gamma ( \\beta ) } { \\Gamma ( \\alpha \\beta / 2 ) \\tau ^ { 1 - \\alpha \\beta / 2 } } , \\tau > 0 . \\end{align*}"}
+{"id": "821.png", "formula": "\\begin{align*} K = \\{ e ^ { i \\theta _ 0 } \\} . \\end{align*}"}
+{"id": "6780.png", "formula": "\\begin{align*} & \\lim \\limits _ { t \\rightarrow \\infty } \\left \\{ \\sup \\limits _ { | x | > ( 2 \\sqrt { d s } + \\epsilon ) t } v ( x , t ) \\right \\} = 0 , \\epsilon > 0 , \\\\ [ 0 . 2 c m ] & \\liminf \\limits _ { t \\rightarrow \\infty } \\left \\{ \\inf \\limits _ { | x | < ( 2 \\sqrt { d s } - \\epsilon ) t } v ( x , t ) \\right \\} > \\mu , \\epsilon \\in ( 0 , 2 \\sqrt { d s } ) , \\end{align*}"}
+{"id": "7496.png", "formula": "\\begin{align*} \\left ( \\frac { ( \\lambda + 1 ) ^ { \\sigma + 1 } } { \\lambda } \\right ) ^ { \\sigma - 1 } = 1 . \\end{align*}"}
+{"id": "3806.png", "formula": "\\begin{align*} { \\rm O T } ( \\mu _ 0 , \\mu _ 1 ) = \\inf _ { \\beta \\in \\widetilde S ^ p _ = ( \\mu _ 0 , \\mu _ 1 ) } \\int _ { X \\times X \\times \\R _ + } s \\ , c ( x _ 0 , x _ 1 ) d \\beta . \\end{align*}"}
+{"id": "1182.png", "formula": "\\begin{align*} { \\mu } ^ E _ 1 ( ( m , x ) , ( n , y ) ) = ~ & ( r _ 1 ( m , y ) + l _ 1 ( x , n ) + f ( x , y ) , ~ m _ 1 ( x , y ) ) , \\\\ { \\mu } ^ E _ 2 ( ( m , x ) , ( n , y ) ) = ~ & ( r _ 2 ( m , y ) + l _ 2 ( x , n ) + f ( x , y ) , ~ m _ 2 ( x , y ) ) . \\end{align*}"}
+{"id": "3830.png", "formula": "\\begin{align*} { \\rm U O T } ( \\mu _ 0 , \\mu _ 1 ) = \\inf _ { \\alpha \\in M ( Y \\times Y ) } \\int _ { Y \\times Y } \\tilde H ( x _ 0 , x _ 1 , s _ 0 , s _ 1 , \\rho _ 0 ( x _ 0 ) , \\rho _ 1 ( x _ 1 ) ) d \\alpha . \\end{align*}"}
+{"id": "4241.png", "formula": "\\begin{align*} 2 \\ , \\tilde { F } = i \\ , ( \\tilde { r } ^ 2 \\ , \\omega ^ { 1 \\bar 1 } + \\tilde { s } ^ 2 \\ , \\omega ^ { 2 \\bar 2 } + \\tilde { t } ^ { \\ , 2 } \\ , \\omega ^ { 3 \\bar 3 } ) , \\end{align*}"}
+{"id": "7346.png", "formula": "\\begin{align*} \\chi ( X ) & = \\iint \\ln | s - t | \\mu _ X ( s ) \\mu _ X ( t ) \\ , d s \\ , d t + \\frac { 3 } { 4 } + \\frac { 1 } { 2 } \\ln 2 \\pi , \\\\ \\Phi ( X ) & = \\frac { 4 \\pi ^ 2 } { 3 } \\int \\mu _ X ^ 3 ( s ) \\ , d s . \\end{align*}"}
+{"id": "3488.png", "formula": "\\begin{align*} \\frac { 1 } { | \\log | t | | } \\varphi = \\frac { 1 } { | \\log | t | | } \\phi _ h + \\frac { 1 } { | \\log | t | | } \\phi _ t + \\psi _ t \\geq - C \\epsilon . \\end{align*}"}
+{"id": "4909.png", "formula": "\\begin{align*} f _ k ( \\tau ) : = \\dfrac { \\eta ( 4 \\tau ) ^ { 8 k - 2 } \\eta ( 8 \\tau ) ^ 4 } { \\eta ( 2 \\tau ) ^ { 4 k } } \\in M _ { 2 k + 1 } ( 8 , \\chi _ { - 4 } ) , k \\in \\mathbb { Z } _ { \\geq 0 } , \\end{align*}"}
+{"id": "5512.png", "formula": "\\begin{align*} c _ { \\pm } ^ { ( b ^ { \\pm 1 } a ^ k , f ) } ( p , s ) = c _ { \\pm } ^ { ( b ^ { \\pm 1 } a ^ k , f ) } ( 1 / p , 1 / s ) \\qquad \\textrm { f o r a l l } f \\in F r e e _ 2 . \\end{align*}"}
+{"id": "4675.png", "formula": "\\begin{align*} | \\widetilde { S _ i } | = \\left | S _ i - \\bigcup _ { \\substack { j \\in [ 2 m + 1 ] \\\\ i \\neq j } } ( S _ i \\cap S _ j ) \\right | > \\frac { n } { 2 ( 2 l + 1 ) } - 2 - 2 m \\cdot \\epsilon n \\ge \\frac { n } { 2 ( 2 l + 1 ) + 1 } \\end{align*}"}
+{"id": "8684.png", "formula": "\\begin{align*} \\psi = g \\Big ( \\frac { x _ 1 } { x _ 0 } , . . . , \\frac { x _ n } { x _ 0 } \\Big ) ~ x _ 0 ^ { - \\frac { n } { 2 } } \\delta \\Big ( x _ { n + 1 } - x _ 0 f \\Big ( \\frac { x _ 1 } { x _ 0 } , . . . , \\frac { x _ n } { x _ 0 } \\Big ) \\Big ) \\end{align*}"}
+{"id": "705.png", "formula": "\\begin{align*} \\tau = \\sup _ { R } \\tau _ R , \\end{align*}"}
+{"id": "4403.png", "formula": "\\begin{align*} \\partial _ t R ^ { \\pm } + \\frac { 1 } { \\partial _ 1 \\hat { \\Phi } ^ { \\pm } } \\Big \\{ ( \\hat { \\mathbf { w } } ^ { \\pm } \\cdot \\nabla R ^ { \\pm } ) + R ^ { \\pm } \\mathrm { d i v } \\hat { \\mathbf { v } } ^ { \\pm } \\Big \\} = \\mathcal { F } ^ { \\pm } \\Omega _ T , \\end{align*}"}
+{"id": "5883.png", "formula": "\\begin{align*} P _ { \\infty } ( \\ell ( \\mathbf { p _ { s t } } ) \\geq C _ { \\gamma } ) \\leq e ^ { - C _ { \\gamma } } = ( 4 \\gamma ^ { 2 } + 2 \\gamma ) ^ { - 1 } . \\end{align*}"}
+{"id": "7543.png", "formula": "\\begin{align*} H ^ * _ { c , F _ 0 } ( X , \\rho _ * \\rho ^ * \\psi _ \\phi ) _ { l o c } = 0 . \\end{align*}"}
+{"id": "2843.png", "formula": "\\begin{align*} R _ { \\lambda } ( x ; t ) = \\sum _ { \\sigma \\in S _ { n } } C ( x _ { \\sigma _ 1 } , \\ldots , x _ { \\sigma _ { n } } ; t ) x _ { \\sigma _ 1 } ^ { \\lambda _ 1 } \\cdots x _ { \\sigma _ { n } } ^ { \\lambda _ { n } } , \\end{align*}"}
+{"id": "8777.png", "formula": "\\begin{align*} b _ { t } < \\frac { 2 ( t _ { 0 } - 1 ) b _ { t _ { 0 } } } { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { ( 2 - p _ { i } ) t ^ { p _ { i } } } . \\end{align*}"}
+{"id": "2799.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\frac { u ^ { p - 1 } \\Delta u } { | x | ^ \\theta } \\dd x & \\le \\int _ { \\Omega } \\frac { u ^ { p - 1 } } { | x | ^ \\alpha } \\ , \\frac { | \\Delta u | } { | x | ^ { \\theta - \\alpha } } \\dd x \\\\ & \\le \\left ( \\int _ { \\Omega } \\frac { u ^ p } { | x | ^ { \\frac { \\alpha p } { p - 1 } } } \\dd x \\right ) ^ { \\frac { p - 1 } { p } } \\left ( \\int _ { \\Omega } \\frac { | \\Delta u | ^ p } { | x | ^ { ( \\theta - \\alpha ) p } } \\dd x \\right ) ^ { \\frac { 1 } { p } } , \\end{align*}"}
+{"id": "1017.png", "formula": "\\begin{align*} q ( R ) = - \\sum _ s a _ s \\ , P _ s ^ * P _ s \\end{align*}"}
+{"id": "5931.png", "formula": "\\begin{align*} d [ - a _ { 1 , 3 } b _ { 1 } ] + d [ - a _ { 1 , 4 } b _ { 1 , 2 } ] \\le \\beta _ { 1 } + d ( - a _ { 1 , 4 } b _ { 1 , 2 } ) = 1 + ( 1 - S _ { 2 } ) \\le 2 e + S _ { 2 } - R _ { 4 } , \\end{align*}"}
+{"id": "4891.png", "formula": "\\begin{align*} A ( z ) = P ( z ) ~ E ( z ) = F ( z ) ~ Q ( z ) , \\end{align*}"}
+{"id": "2577.png", "formula": "\\begin{align*} \\beta ( h , h ' ) & = \\alpha ( h + h ' ) - \\alpha ( h ) - \\alpha ( h ' ) \\\\ \\beta ' ( h , h ' ) & = \\alpha ' ( h + h ' ) - \\alpha ' ( h ) - \\alpha ' ( h ' ) . \\end{align*}"}
+{"id": "5004.png", "formula": "\\begin{align*} \\int _ { B _ p ( R - 1 4 \\Lambda ) \\setminus B _ p ( R - 1 6 \\Lambda ) } e ^ { - b r } = 2 \\Lambda L ( \\xi ) e ^ { - b \\xi } . \\end{align*}"}
+{"id": "4932.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ^ { 2 \\pi i n t } e ^ { - 2 ^ k \\pi ( m + 1 ) t } d t & = \\int _ 0 ^ \\infty e ^ { 2 \\left ( \\pi i n - 2 ^ { k - 1 } \\pi ( m + 1 ) \\right ) t } d t = - \\dfrac { 1 } { 2 \\pi ( i n - 2 ^ { k - 1 } ( m + 1 ) ) } \\\\ \\int _ 0 ^ \\infty e ^ { - 2 \\pi i n t } e ^ { - 2 ^ k \\pi ( m + 1 ) t } d t & = \\int _ 0 ^ \\infty e ^ { - 2 \\left ( \\pi i n + 2 ^ { k - 1 } \\pi ( m + 1 ) \\right ) t } d t = \\dfrac { 1 } { 2 \\pi ( i n + 2 ^ { k - 1 } ( m + 1 ) ) } \\end{align*}"}
+{"id": "3310.png", "formula": "\\begin{align*} \\frac { d } { d t } ( \\phi _ Z ) _ { _ \\Sigma } ( t ) = \\frac { \\lambda _ { _ \\Sigma } \\mu } { ( \\lambda _ { _ \\Sigma } - \\mu t ) ^ 2 } , \\end{align*}"}
+{"id": "2444.png", "formula": "\\begin{align*} \\langle \\nabla \\zeta , \\nabla \\phi ^ x \\rangle _ { L ^ 2 ( \\Omega ) } = \\langle \\zeta , \\varphi ^ x \\rangle _ { L ^ 2 ( \\Omega ) } , \\forall \\zeta \\in H _ 0 ^ 1 ( \\Omega ) . \\end{align*}"}
+{"id": "3119.png", "formula": "\\begin{align*} \\mathbb { E } [ \\rho ^ 2 _ { i , j } ] & = \\mathbb { E } \\left [ \\frac { | \\mathbf { H } _ { k } \\cdot \\mathbf { H } _ { l } ^ H | ^ 2 } { \\| \\mathbf { H } _ { k } \\| ^ 2 \\| \\mathbf { H } _ { l } \\| ^ 2 } \\right ] \\\\ & \\approx \\frac { \\mathbb { E } \\left [ | \\mathbf { H } _ { k } \\cdot \\mathbf { H } _ l ^ H | ^ 2 \\right ] } { \\mathbb { E } \\left [ \\Vert \\mathbf { H } _ { k } \\Vert ^ 2 \\right ] \\mathbb { E } \\left [ \\Vert \\mathbf { H } _ { l } \\Vert ^ 2 \\right ] } . \\end{align*}"}
+{"id": "8598.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ { X , z } ^ { \\mu } \\phi _ 1 = \\mu \\beta F \\ \\ \\mathrm { i n } \\ \\ \\mathcal { S } _ b , \\\\ \\phi _ 1 | _ { z = 0 } = 0 , \\ \\ \\partial _ z \\phi _ 1 | _ { z = - 1 + \\beta b } = - \\nabla _ X \\cdot \\big ( \\mathcal { L } _ { 1 } ^ { \\mu } [ \\beta b ] \\nabla _ X \\psi ) , \\end{cases} \\end{align*}"}
+{"id": "3686.png", "formula": "\\begin{align*} \\sum _ { u = 1 } ^ n \\| \\phi _ { u } \\| ^ 2 = \\sum _ { u = 1 } ^ n \\sum _ { \\tau \\in \\binom { [ n ] } { k - 1 } } \\phi _ { u } ( \\tau ) ^ 2 = \\sum _ { \\tau \\in \\binom { [ n ] } { k - 1 } } \\sum _ { u \\in [ n ] \\setminus \\tau } \\phi ( \\tau \\cup \\{ u \\} ) ^ 2 = \\sum _ { \\sigma \\in \\binom { [ n ] } { k } } \\sum _ { u \\in \\sigma } \\phi ( \\sigma ) ^ 2 = k \\sum _ { \\sigma \\in \\binom { [ n ] } { k } } \\phi ( \\sigma ) ^ 2 = k \\| \\phi \\| ^ 2 . \\end{align*}"}
+{"id": "5627.png", "formula": "\\begin{align*} K _ W + D _ W = q ^ * ( K _ Y + D _ Y ) + \\sum a _ j E _ j , \\end{align*}"}
+{"id": "4902.png", "formula": "\\begin{align*} E _ + ( z ) ^ { - 1 } E _ - ( z ) & = S _ 2 ( z ) ^ { - 1 } N ^ { ( 1 ) } ( z ) ^ { - 1 } N ^ { ( 1 ) } ( z ) T _ 2 ( z ) \\\\ & = S _ 2 ( z ) ^ { - 1 } T _ 2 ( z ) . \\end{align*}"}
+{"id": "2862.png", "formula": "\\begin{align*} ( w f ) ( x ) : = f ( w ^ { - 1 } x ) ( w \\in W , \\ , f \\in \\mathcal { C } ( V ) , \\ , x \\in V ) . \\end{align*}"}
+{"id": "1891.png", "formula": "\\begin{align*} \\| w \\| _ { \\infty , \\mathcal { T } _ h } = \\sup _ { R \\in \\mathcal { T } _ h } \\| w \\| _ { L ^ { \\infty } ( R ) } , \\| w \\| _ { L ^ p ( \\mathcal { T } _ h ) } = \\left ( \\sum _ { R \\in \\mathcal { T } _ h } \\| w \\| ^ p _ { L ^ p ( R ) } \\right ) ^ \\frac { 1 } { p } , \\ , \\ , \\forall \\ , w \\in L ^ p ( \\mathcal { T } _ h ) , \\end{align*}"}
+{"id": "3054.png", "formula": "\\begin{align*} { \\bf { H } } = \\sum \\limits _ { k = 1 } ^ K { { \\rho _ k } { { \\bf { H } } _ { k , { \\rm { R } } } } { { \\bf { \\Gamma } } _ k } { { \\bf { H } } _ { { \\rm { T } } , k } } } , \\end{align*}"}
+{"id": "665.png", "formula": "\\begin{align*} \\int _ 0 ^ T F ( t ) \\ , d W ( t ) = \\lim _ { n \\to \\infty } \\sum _ { j , k = 1 } ^ n \\left ( \\int _ 0 ^ T F _ { j k } ( t ) \\ , d B _ k ( t ) \\right ) f _ j , \\end{align*}"}
+{"id": "452.png", "formula": "\\begin{align*} & \\frac { 1 } { ( 1 + \\beta ^ 2 ) ^ { ( d + \\alpha ) / 2 + \\ell } } \\ , _ 2 \\tilde F _ 1 \\left ( \\frac { ( d + \\alpha ) / 2 + \\ell } { 2 } , \\frac { ( d + \\alpha + 2 ) / 2 + \\ell } { 2 } ; \\frac d 2 + \\ell ; \\frac { 4 \\beta ^ 2 } { ( 1 + \\beta ^ 2 ) ^ 2 } \\right ) \\\\ & = \\ , _ 2 \\tilde F _ 1 \\left ( \\frac { d + \\alpha } { 2 } + \\ell , \\frac { \\alpha + 2 } { 2 } ; \\frac d 2 + \\ell ; \\beta ^ 2 \\right ) . \\end{align*}"}
+{"id": "743.png", "formula": "\\begin{align*} \\norm { \\psi _ + ( t ) } _ { L ^ 2 } ^ 2 + \\norm { \\psi _ - ( t ) } _ { L ^ 2 } ^ 2 = \\norm { \\psi _ 0 } _ { L ^ 2 } ^ 2 , \\end{align*}"}
+{"id": "5901.png", "formula": "\\begin{align*} \\| v _ 1 ( 0 , \\cdot ) \\| _ { L ^ 2 ( ( 0 , 1 ) ) } & \\leq N \\big ( \\| v _ 1 \\| _ { L ^ 2 ( ( 0 , 1 ) \\times ( 0 , 1 ) ) } + \\| \\nabla v _ 1 \\| _ { L ^ 2 ( ( 0 , 1 ) \\times ( 0 , 1 ) ) } \\big ) \\\\ & \\leq N \\| \\nabla v _ 1 \\| _ { L ^ 2 ( ( 0 , 1 ) \\times ( 0 , 1 ) ) } \\\\ \\end{align*}"}
+{"id": "3401.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } 2 c v _ 1 v _ 2 ( 1 + 3 v _ 2 ^ 2 ) ( 1 - v _ 1 ^ 2 ) ( v _ 1 ^ 2 + v _ 2 ^ 2 ) ^ 2 V _ 1 ^ 2 P _ 1 = 0 \\\\ \\\\ \\displaystyle \\frac { v _ 2 ( v _ 1 ^ 2 + v _ 2 ^ 2 ) } { v _ 1 ( v _ 2 ^ 2 + v _ 3 ^ 2 ) ^ 2 \\phi _ 3 } [ V _ 1 ^ 2 P _ 2 - c v _ 1 ^ 2 ( - 1 + 9 v _ 3 ^ 2 ) ( v _ 2 ^ 2 + v _ 3 ^ 2 ) ^ 2 ] = 0 \\\\ \\\\ v _ 1 ( v _ 1 ^ 2 + v _ 2 ^ 2 ) \\phi _ 1 [ V _ 1 ^ 2 P _ 3 + c v _ 1 ^ 2 ( v _ 2 ^ 2 + v _ 3 ^ 2 ) ^ 2 P _ 4 ] = 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "1445.png", "formula": "\\begin{align*} \\sum \\limits _ { i \\in I } \\sigma _ { i } \\Big ( \\sum \\limits _ { j \\in I } k _ { i j } \\sigma _ { j } - 2 \\mu _ { i } \\Big ) = 2 \\sum \\limits _ { i \\in I } \\mu _ { i } \\sigma _ { i } , \\end{align*}"}
+{"id": "1357.png", "formula": "\\begin{align*} D ^ { 5 } \\left ( \\frac { E _ 8 ( z ) } { \\Delta ( z ) } \\right ) = - f _ { 6 , i \\infty } ( z ) . \\end{align*}"}
+{"id": "3277.png", "formula": "\\begin{align*} | m _ \\ell | ^ 2 = E [ X ] \\overline { E [ X ] } = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } ( m _ \\ell ) _ { _ \\Sigma } ^ 2 . \\end{align*}"}
+{"id": "6345.png", "formula": "\\begin{align*} v ^ h ( 0 ) = P _ h u _ 0 , \\end{align*}"}
+{"id": "3721.png", "formula": "\\begin{align*} u ( 0 ) = u _ 0 \\in \\mathcal { H } ^ { \\frac { 2 } { 3 } } , \\ u '' ( 0 ) = u _ 1 \\in \\mathcal { H } ^ { \\frac { 1 } { 3 } } , \\ u ''' ( 0 ) = u _ 2 \\in \\mathcal { H } , \\end{align*}"}
+{"id": "92.png", "formula": "\\begin{align*} [ b _ + , b _ + ^ \\dagger ] = \\frac { [ a _ + , a _ + ^ \\dagger ] - \\alpha ^ 2 [ a _ - , a _ - ^ \\dagger ] } { 1 - \\alpha ^ 2 } , \\end{align*}"}
+{"id": "6892.png", "formula": "\\begin{align*} \\sum _ i \\left ( 1 - \\frac { 1 } { m _ i } \\right ) b _ i = 1 . \\end{align*}"}
+{"id": "5732.png", "formula": "\\begin{align*} \\int _ { B _ r \\times ( - 1 , 0 ) } u ^ 2 \\geq C r ^ { C ( 1 + \\| V \\| _ 1 ^ { 1 / 2 } ) } , \\ \\ r \\leq 1 / 2 , \\ C = C ( u ) . \\end{align*}"}
+{"id": "2609.png", "formula": "\\begin{align*} F ( 0 ) = 0 \\ , , F ( x ) < 1 \\ , , x \\ , . \\end{align*}"}
+{"id": "5340.png", "formula": "\\begin{align*} x \\mapsto \\begin{cases} \\frac { 1 + x } { \\varepsilon } & x \\in [ - 1 , - 1 + \\varepsilon ] , \\\\ 1 & x \\in [ - 1 + \\varepsilon , 1 - \\varepsilon ] , \\\\ \\frac { 1 - x } { \\varepsilon } & x \\in [ 1 - \\varepsilon , 1 ] , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "2434.png", "formula": "\\begin{align*} N = \\sum _ { n = 1 } ^ N \\| \\nabla u _ n \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 \\ge C _ d \\int _ { \\R ^ d } \\rho ( x ) ^ { \\frac { d } { d - 2 } } \\ , { \\rm d } x \\end{align*}"}
+{"id": "3219.png", "formula": "\\begin{align*} \\underbrace { L ^ { - 1 } A } _ { \\hat { A } } \\underbrace { A ^ T L ^ { - T } } _ { \\hat { A } ^ T } \\underbrace { L ^ T y } _ { \\hat { y } } = \\underbrace { L ^ { - 1 } b } _ { \\hat { b } } . \\end{align*}"}
+{"id": "1306.png", "formula": "\\begin{align*} \\frac { d } { d t } M ( t ) = & 8 \\int _ { \\R ^ 3 } | \\nabla u | ^ 2 - ( I _ \\alpha \\ast | \\cdot | ^ b | u | ^ p ) | x | ^ { - b } | u | ^ p d x \\\\ & + \\mathcal { O } \\left ( \\int _ { | x | > R } | \\nabla u | ^ 2 + | x | ^ { - 2 } | u | ^ 2 + ( I _ \\alpha \\ast | \\cdot | ^ b | u | ^ p ) | x | ^ { - b } | u | ^ p d x \\right ) , \\end{align*}"}
+{"id": "7812.png", "formula": "\\begin{align*} k & ( Z ^ 0 , z ^ a ) \\\\ & = | Z ^ 0 | ^ 2 \\left ( 4 h ( t ) - \\frac { \\chi \\zeta ( 3 ) } { ( 2 \\pi ) ^ 3 } + \\frac { 2 } { ( 2 \\pi ) ^ 3 } \\sum _ { q _ a \\gamma ^ a \\in \\Lambda ^ + } n _ { \\gamma } ( \\mathrm { L i } _ 3 ( e ^ { 2 \\pi i q _ a z ^ a } ) ) + \\frac { 2 } { ( 2 \\pi ) ^ 2 } \\sum _ { q _ a \\gamma ^ a \\in \\Lambda ^ + } n _ { \\gamma } ( \\mathrm { L i } _ 2 ( e ^ { 2 \\pi i q _ a z ^ a } ) ) q _ a t ^ a \\right ) \\ , , \\\\ \\end{align*}"}
+{"id": "3291.png", "formula": "\\begin{align*} \\sigma ^ 2 = m _ 2 - m _ 1 ^ 2 = E [ X ^ 2 ] - ( E [ X ] ) ^ 2 . \\end{align*}"}
+{"id": "1010.png", "formula": "\\begin{align*} A \\cdot ( \\omega \\ , k + \\nabla \\varphi ) = 0 ; \\end{align*}"}
+{"id": "2231.png", "formula": "\\begin{align*} [ X _ { \\sigma _ 1 ^ c } / K _ { \\beta _ 1 } ] = [ ( X _ { \\sigma _ 1 ^ c } / K _ \\Phi ) / K _ { \\beta _ 2 } ] \\to [ X _ { \\sigma _ 2 ^ c } / K _ { \\beta _ 2 } ] . \\end{align*}"}
+{"id": "138.png", "formula": "\\begin{align*} ( f _ 1 * f _ 2 ) ( x , z ) = \\sum _ { y \\in S } f _ 1 ( x , y ) f _ 2 ( y , z ) , \\end{align*}"}
+{"id": "8614.png", "formula": "\\begin{align*} R _ 2 = - \\frac { \\nabla _ X b } { \\mu } ( \\mathrm { F } _ 1 - 1 ) G . \\end{align*}"}
+{"id": "6454.png", "formula": "\\begin{align*} \\| f \\| _ { M , K , h } : = \\sup _ { p \\in \\N , x \\in K } \\frac { \\| f ^ { ( p ) } ( x ) \\| } { h ^ p M _ p } , \\end{align*}"}
+{"id": "7959.png", "formula": "\\begin{align*} \\phi ^ * _ { n , k } ( u _ { i , j } ) = u _ { \\phi _ { n , k } ( i ) , j } ~ ~ \\phi ^ * _ { n , k } ( v _ { i , j } ) = v _ { \\phi _ { n , k } ( i ) , j } , ~ i \\in \\Z _ n , ~ j \\in \\Z _ 2 . \\end{align*}"}
+{"id": "8447.png", "formula": "\\begin{align*} \\nabla _ { K Z B } = d - \\omega _ { K Z B } , \\omega _ { K Z B } = \\beta B + \\alpha \\exp \\left ( - \\sum _ { k = 2 } ^ { \\infty } \\frac { ( - 1 ) ^ k P _ { k } \\mathrm { a d } _ { B } ^ { k } } { k } \\right ) A \\end{align*}"}
+{"id": "1360.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\infty } \\lambda _ m c _ f ( m ) = 0 . \\end{align*}"}
+{"id": "4162.png", "formula": "\\begin{align*} \\rho _ k = \\frac { \\lambda _ k } { \\mu } \\quad \\quad \\hat { \\rho } _ { k } = \\sum _ { i = 1 } ^ { k - 1 } \\rho _ i . \\end{align*}"}
+{"id": "4377.png", "formula": "\\begin{align*} \\mathrm { d i v } \\mathbf { h } ^ { \\pm } = 0 , \\quad \\mbox { i n } \\ , \\ , \\ , \\mathbb R ^ 2 _ + \\end{align*}"}
+{"id": "7265.png", "formula": "\\begin{align*} R f ( z , t ) = \\frac { 1 } { \\chi } \\int _ { [ 0 , 1 ] } \\int _ { - t } ^ \\infty f ( y , u ) \\ , d u d \\nu ( y ) + \\lim _ { s \\rightarrow 0 ^ + } \\frac { 1 } { 2 \\pi } \\int e ^ { - i t \\theta } U ( s - i \\theta ) \\hat { f } ( z , \\theta ) \\ , d \\theta , \\end{align*}"}
+{"id": "2463.png", "formula": "\\begin{align*} & \\overline { H } ( G _ 1 \\overset { ( \\alpha , 1 - \\alpha ) } { \\sqcup } G _ 2 ) = \\alpha \\overline { H } ( G _ 1 ) + ( 1 - \\alpha ) \\overline { H } ( G _ 2 ) \\\\ \\Longleftrightarrow \\ : & \\overline { H } ( G _ 1 \\wedge G _ 2 ) = \\overline { H } ( G _ 1 ) + \\overline { H } ( G _ 2 ) . \\end{align*}"}
+{"id": "732.png", "formula": "\\begin{align*} \\Delta _ { 2 , 1 } ^ \\mu ( t ) & = i P _ \\mu \\int _ 0 ^ t \\mathbf S ( t - s ) \\left [ \\mathbf N \\left ( \\Theta _ R ^ { \\mathbf u ^ \\mu } ( s ) P _ \\mu \\mathbf u ^ \\mu ( s ) \\right ) - \\mathbf N \\left ( \\Theta _ R ^ { \\mathbf u } ( s ) \\mathbf u ( s ) \\right ) \\right ] \\ , d s , \\\\ \\Delta _ { 2 , 2 } ^ \\mu ( t ) & = i ( 1 - P _ \\mu ) \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf N \\left ( \\Theta _ R ^ { \\mathbf u } ( s ) \\mathbf u ( s ) \\right ) \\ , d s . \\end{align*}"}
+{"id": "4254.png", "formula": "\\begin{align*} F ( x , t ) : = \\int _ { 0 } ^ { t } f ( x , \\sigma ) \\ , d \\sigma , t \\in \\mathbb { R } . \\end{align*}"}
+{"id": "3451.png", "formula": "\\begin{align*} W ( p ) = ( 1 + \\sum _ 0 ^ { m } d _ i p _ i ) ^ { n - m } , \\end{align*}"}
+{"id": "8333.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ K \\alpha _ k ^ m ( 0 ) = 1 \\end{align*}"}
+{"id": "39.png", "formula": "\\begin{align*} \\mathcal H = \\frac { 1 } { \\pi } \\int _ { \\mathbb { C } } \\mathcal K ( z ) | z \\rangle \\langle z | \\dd z + \\mathcal Q _ 3 ^ { \\rm { r e n } } + \\mathcal Q _ 4 ^ { \\rm { r e n } } + \\mathcal R _ 0 , \\end{align*}"}
+{"id": "8914.png", "formula": "\\begin{align*} H ^ 0 ( \\widetilde C , \\omega _ { \\widetilde C } ) = \\bigoplus _ { i = 0 } ^ 8 H ^ 0 ( C , \\omega _ { C } \\otimes \\eta ^ i ) , \\end{align*}"}
+{"id": "1585.png", "formula": "\\begin{align*} \\mathcal { J } & = a ^ 2 + a \\sqrt { \\rho _ g } \\langle h ^ \\perp , \\mathcal { M } _ \\tau \\rangle - a ^ 2 - a \\frac { 1 } { \\sqrt { \\rho _ g } } \\int \\left ( 1 + \\rho _ g \\frac { \\mathcal { M } _ \\tau } { g } \\right ) h ^ \\perp - \\frac { 1 } { 2 } \\int \\left ( 1 + \\rho _ g \\frac { \\mathcal { M } _ \\tau } { g } \\right ) \\frac { ( h ^ \\perp ) ^ 2 } { g } \\\\ & = - \\frac { 1 } { 2 } \\int \\left ( 1 + \\rho _ g \\frac { \\mathcal { M } _ \\tau } { g } \\right ) \\frac { ( h ^ \\perp ) ^ 2 } { g } . \\end{align*}"}
+{"id": "251.png", "formula": "\\begin{align*} \\sum _ { \\substack { 1 \\leq j \\leq n \\\\ \\epsilon \\in \\{ 1 , - 1 \\} } } V ^ { } _ { \\epsilon j } ( \\xi ; g ) \\Bigl ( \\Phi ^ { } _ { \\xi + \\epsilon e _ j } ( x ; g ) - \\Phi ^ { } _ \\xi ( x ; g ) \\Bigl ) = E ^ { } _ 1 ( x ) \\Phi ^ { } _ \\xi ( x ; g ) \\end{align*}"}
+{"id": "4168.png", "formula": "\\begin{align*} y _ i & = \\frac { \\sqrt { | h _ { i i } | ^ 2 p _ i } } { \\sum ^ L _ { j = 1 , j \\ne i } | h _ { i j } | ^ 2 p _ j + \\sigma ^ 2 _ i } , \\\\ \\tilde y _ k & = \\frac { \\sqrt { \\sum ^ L _ { j = 1 } | \\tilde h _ { k j } | ^ 2 p _ j + \\tilde \\sigma ^ 2 _ k } } { | \\tilde h _ { k k } | ^ 2 p _ k + \\varepsilon } . \\end{align*}"}
+{"id": "4143.png", "formula": "\\begin{align*} { [ \\bar { x } , \\bar { y } ] } & = - \\bar { T } ( \\bar { x } , \\bar { y } ) + \\bar { R } ( \\bar { x } , \\bar { y } ) = - \\overline { T ( x , y ) } + \\overline { R ( x , y ) } , \\\\ { [ \\bar { E } _ 1 , \\bar { y } ] } & = \\bar { E } \\bar { y } = \\overline { E y } , \\\\ { [ \\bar { E } _ 1 , \\bar { E } _ 2 ] } & = - ( \\bar { E _ 1 } \\bar { E _ 2 } - \\bar { E _ 2 } \\bar { E _ 1 } ) = - ( \\overline { E _ 1 E _ 2 } - \\overline { E _ 2 E _ 1 } ) , \\end{align*}"}
+{"id": "1086.png", "formula": "\\begin{align*} \\begin{aligned} & \\mbox { t h e r e a r e p o s i t i v e c o n s t a n t s } \\ , \\ , \\ , M _ 0 \\ , \\ , \\mbox { a n d } \\ , \\ , \\sigma \\ , \\ , \\mbox { s u c h t h a t } \\\\ & \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\displaystyle \\max _ { ( x , s ) \\in \\mathop D \\limits ^ \\circ \\times [ - M _ 0 , M _ 0 ] } | f ( x , s ) | \\leq \\mu _ 0 ^ 2 \\frac { M _ 0 } { ( \\sigma + 1 ) } \\frac { \\lambda _ 1 } { 2 } , \\end{aligned} \\end{align*}"}
+{"id": "7716.png", "formula": "\\begin{align*} \\mathbb { A } _ E | _ { N } \\stackrel { \\pi '' } { \\longrightarrow } \\frac { \\mathbb { A } _ E | _ N } { K _ 1 \\wedge E | _ N } \\stackrel { \\pi ' } { \\longrightarrow } T M | _ { N } = \\frac { \\mathbb { A } _ E | _ N } { \\wedge ^ 2 E | _ N } , \\end{align*}"}
+{"id": "7292.png", "formula": "\\begin{align*} \\frac { \\delta N ' - N } { N ' } = \\delta - \\frac { N } { N ' } . \\end{align*}"}
+{"id": "4066.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathcal { R } _ { k - 1 } q _ 0 ( t ) = \\mathcal { R } _ { k - 1 } \\tilde { q } _ 0 ( t ) , & \\mathcal { R } _ { k } q _ 0 ( t ) & = \\mathcal { R } _ { k } \\tilde { q } _ 0 ( t ) \\\\ & \\mathcal { R } _ { k - 1 } p ( t ) = \\mathcal { R } _ { k - 1 } \\tilde { p } ( t ) , & \\mathcal { R } _ { k } p ( t ) & = \\mathcal { R } _ { k } \\tilde { p } ( t ) . \\end{aligned} \\end{align*}"}
+{"id": "5728.png", "formula": "\\begin{align*} | | u | | _ { L ^ { \\infty } ( B _ r \\times ( - r ^ 2 , 0 ] ) } = O ( r ^ k ) , \\ \\end{align*}"}
+{"id": "8354.png", "formula": "\\begin{align*} \\theta _ { - t } \\omega _ 2 ( t ) + & B \\int _ 0 ^ { t } S _ B ( t - r ) \\theta _ { - t } \\omega _ 2 ( r ) d r \\\\ & = B \\int _ 0 ^ { t } S _ B ( t - r ) \\omega _ 2 ( r - t ) d r - S _ B ( t ) \\omega _ 2 ( - t ) \\\\ & = B \\int ^ 0 _ { - t } S _ B ( - r ) \\omega _ 2 ( r ) d r - S _ B ( t ) \\omega _ 2 ( - t ) \\end{align*}"}
+{"id": "423.png", "formula": "\\begin{align*} \\Phi _ { d , \\ell } ^ { ( \\alpha ) } ( \\sigma ) : = \\frac { 2 ^ { \\alpha } \\Gamma \\left ( \\frac { \\ell + \\sigma + \\alpha } { 2 } \\right ) \\Gamma \\left ( \\frac { d + \\ell - \\sigma } { 2 } \\right ) } { \\Gamma \\left ( \\frac { \\ell + \\sigma } { 2 } \\right ) \\Gamma \\left ( \\frac { d + \\ell - \\sigma - \\alpha } { 2 } \\right ) } , \\sigma \\in ( - \\ell - \\alpha , d + \\ell ) , \\end{align*}"}
+{"id": "5122.png", "formula": "\\begin{align*} d _ { A ^ * } [ a , b ] = [ d _ { A ^ * } a , b ] + [ a , d _ { A ^ * } b ] , \\forall \\ , a , b \\in \\Gamma ( A ) . \\end{align*}"}
+{"id": "7016.png", "formula": "\\begin{align*} c - & 0 . 2 5 \\cdot 3 ^ 2 b ^ 3 = r _ { 3 j + 1 } ^ 2 - 4 - 2 . 2 5 ( r _ j ^ 2 - 4 ) ^ 3 = \\\\ & = 8 ( ( 2 - \\sqrt { 3 } ) ^ { 4 j + 2 } + ( 2 + \\sqrt { 3 } ) ^ { 4 j + 2 } ) - 3 4 ( ( 2 - \\sqrt { 3 } ) ^ { 2 j + 1 } + ( 2 + \\sqrt { 3 } ) ^ { 2 j + 1 } ) + \\frac { 1 2 8 } { 3 } > 0 \\end{align*}"}
+{"id": "7748.png", "formula": "\\begin{align*} \\chi ( C _ { 2 k } \\boxtimes C _ { 2 t + 1 } ) \\geq \\lceil ( 2 k ) ( 2 t + 1 ) / k t \\rceil = 5 . \\end{align*}"}
+{"id": "835.png", "formula": "\\begin{align*} \\left | \\widetilde { N y _ 0 } ( e ^ { i \\theta _ 0 } ) \\right | & = \\Big | \\eta ( e ^ { i \\theta _ 0 } ) \\Psi _ 2 ( y _ 0 ) + ( 1 - \\epsilon ) ( 1 - \\eta ( e ^ { i \\theta _ 0 } ) ) \\widetilde { T y _ 0 } ( e ^ { i \\theta _ 0 } ) \\Big | \\\\ & = | \\Psi _ 2 ( y _ 0 ) | \\\\ & = 1 . \\end{align*}"}
+{"id": "5896.png", "formula": "\\begin{align*} \\N _ \\infty ^ { \\oplus 2 } \\twoheadrightarrow P _ \\infty ^ 1 = \\N _ \\infty \\oplus _ \\N \\N _ \\infty , \\end{align*}"}
+{"id": "971.png", "formula": "\\begin{align*} \\Pr [ D _ i = x _ i ] \\geq ( 1 - \\exp ( - \\mu \\cdot ( q - 1 / 2 ) ^ 2 / ( 2 q ) ) ) \\cdot ( 1 - \\rho ^ * ) . \\end{align*}"}
+{"id": "2245.png", "formula": "\\begin{align*} f _ \\alpha ( x ) = \\sum _ { j = 1 } ^ n \\theta _ { \\alpha } ( x - x _ j ) \\end{align*}"}
+{"id": "5715.png", "formula": "\\begin{align*} D _ { 0 } ^ { \\delta } & = \\left ( \\begin{array} { c c c } 0 & E _ { 1 2 } ^ { \\delta } \\delta ^ 2 & E _ { 1 3 } ^ { \\delta } \\delta \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right ) \\ , . \\end{align*}"}
+{"id": "2891.png", "formula": "\\begin{align*} c \\xi _ \\mu + \\hat \\rho _ v ( \\xi _ \\mu ) = 2 \\pi ( \\hat \\rho + \\mu ) \\hat \\rho _ v ( \\xi ) : = \\sum _ { \\alpha \\in R _ 0 ^ + } v _ \\alpha ( \\langle \\xi , \\alpha \\rangle ) { \\hat \\alpha } . \\end{align*}"}
+{"id": "4434.png", "formula": "\\begin{align*} | | \\dot { { \\mathbf U } } | | _ { s , \\ast , T } + | | \\varphi | | _ { H ^ s ( \\Gamma _ T ) } \\leq C ( K _ 0 ) \\Big ( | | \\mathbf { F } | | _ { s , \\ast , T } + | | \\mathbf { F } | | _ { 6 , \\ast , T } | | \\hat { W } | | _ { s + 4 , \\ast , T } \\Big ) \\ , , \\end{align*}"}
+{"id": "4732.png", "formula": "\\begin{align*} \\begin{cases} F ( D ^ { 2 } u _ 0 ) = 1 & \\mathbb { R } _ + ^ n \\cap \\Omega _ 0 , \\\\ | \\nabla u _ 0 | = 0 & \\mathbb { R } _ + ^ n \\backslash \\Omega _ 0 , \\\\ u _ 0 = 0 & \\mathbb { R } _ + ^ { n - 1 } . \\end{cases} \\end{align*}"}
+{"id": "641.png", "formula": "\\begin{align*} X = \\begin{pmatrix} \\psi \\\\ \\phi \\\\ \\dot \\phi \\end{pmatrix} , A = \\begin{pmatrix} - \\alpha ^ j \\partial _ j - i M \\beta & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & \\Delta & 0 \\end{pmatrix} , \\mathcal M ( X ) = \\begin{pmatrix} i \\beta \\psi \\mathfrak K _ 1 \\\\ 0 \\\\ \\phi \\mathfrak K _ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "6563.png", "formula": "\\begin{align*} N _ 2 = N _ 1 ^ { \\widetilde C } , \\ N = N _ 2 ^ { \\widetilde C } , \\end{align*}"}
+{"id": "7479.png", "formula": "\\begin{align*} \\kappa ( X , K _ X + \\Delta ) & = \\kappa ( F , K _ F + \\Delta | _ F ) + \\kappa ( A , \\widehat { \\det } f _ * \\mathcal { O } _ X ( l D ) ) \\\\ & = \\kappa ( F , K _ F + \\Delta | _ F ) + \\dim V ^ 0 ( A , f _ * \\mathcal { O } _ X ( l D ) ) . \\end{align*}"}
+{"id": "4772.png", "formula": "\\begin{align*} h ^ { - } ( x ) _ { f } = \\begin{cases} x _ { h _ { f } } & f \\in ( B ' \\setminus B _ { N } ) \\cap ( F _ { \\ell } \\setminus W _ { a } ) \\\\ x _ { g } & f = h _ { g } g \\in ( B ' \\setminus B _ { N } ) \\cap ( F _ { \\ell } \\setminus W _ { a } ) \\\\ x _ { f } & \\end{cases} \\end{align*}"}
+{"id": "6904.png", "formula": "\\begin{align*} \\mathcal { N } _ i \\left [ w \\right ] ( \\xi ) = \\int _ { \\mathbb R } J _ i ( \\xi - y ) w ( y ) d y - w ( \\xi ) , \\ i = 1 , 2 , \\end{align*}"}
+{"id": "7602.png", "formula": "\\begin{align*} J _ 2 ( \\beta , N , T ) \\leq \\sum _ { i = 1 } ^ 2 J _ i ( \\beta , N , T , \\hat { \\mathbf { R } } _ i ) . \\end{align*}"}
+{"id": "6311.png", "formula": "\\begin{align*} i \\partial _ t v _ \\lambda + \\partial _ x g _ { [ < \\lambda ^ \\sigma ] } \\partial _ x v _ \\lambda + V _ { < \\lambda ^ \\sigma } \\partial _ x v _ \\lambda + W _ { < \\lambda ^ \\sigma } v _ \\lambda = f _ \\lambda , v _ \\lambda ( 0 ) = v _ { 0 , \\lambda } , \\end{align*}"}
+{"id": "1816.png", "formula": "\\begin{gather*} \\left \\langle f , g \\right \\rangle = \\iint _ { U } f ( s ) \\overline { g ( s ) } \\ , d \\sigma d t \\quad \\| f \\| = \\sqrt { \\left \\langle f , f \\right \\rangle } \\end{gather*}"}
+{"id": "5287.png", "formula": "\\begin{align*} \\mathcal { R } ( n _ 0 , \\dots , n _ L , n _ { L + 1 } ) = \\mathcal { R } ( n _ 0 , \\dots , n _ L , 1 ) . \\end{align*}"}
+{"id": "780.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } u + V ( x ) u = ( I _ \\alpha \\ast | u | ^ q ) | u | ^ { q - 2 } u \\quad \\mathbb { R } ^ N . \\end{align*}"}
+{"id": "5476.png", "formula": "\\begin{align*} \\Omega _ { \\boldsymbol \\epsilon } : = \\mathbb C \\times \\big ( \\mathbb D ^ \\times \\big ) ^ { \\epsilon _ 1 } \\times \\dots \\times \\big ( \\mathbb D ^ { \\times } \\big ) ^ { \\epsilon _ 1 } \\end{align*}"}
+{"id": "6462.png", "formula": "\\begin{align*} f _ { e ^ - } = f ( 0 ) , f _ { e ^ + } = f ( \\epsilon \\ell _ e ) , \\partial f _ { e ^ - } = f ' ( 0 ) , \\partial f _ { e ^ + } = - f ' ( \\epsilon \\ell _ e ) , \\end{align*}"}
+{"id": "5502.png", "formula": "\\begin{align*} b ^ { - 1 } a ^ 0 ( b a ^ k ( g ) ) = a ^ k ( g ) = b a ^ { 0 } ( b ^ { - 1 } a ^ k ( g ) ) , \\end{align*}"}
+{"id": "209.png", "formula": "\\begin{align*} & 3 2 \\left \\{ \\widehat { L } ( T M ) \\right \\} ^ { ( 1 2 ) } = \\left \\{ \\widehat { A } ( T M ) { \\rm c h } [ 2 2 4 0 + 3 0 9 \\widetilde { T _ C M } + 2 4 ( \\wedge ^ 2 \\widetilde { T _ C M } + W _ i + W _ j ) \\right . \\\\ & \\left . + \\widetilde { T _ C M } \\otimes \\widetilde { T _ C M } + \\widetilde { T _ C M } \\otimes ( W _ i + W _ j ) + \\wedge ^ 3 \\widetilde { T _ C M } ] \\right \\} ^ { ( 1 2 ) } . \\end{align*}"}
+{"id": "1194.png", "formula": "\\begin{align*} P _ * ( M ) = \\pi _ * ( M \\otimes _ A A / ( x _ 1 , \\ldots , x _ n ) ) . \\end{align*}"}
+{"id": "8576.png", "formula": "\\begin{align*} \\partial _ { n _ { b } } ^ { P _ b } \\phi _ b | _ { z = - h _ b } = \\mathbf { n } _ b \\cdot I ^ { \\mu } P ( \\Sigma _ b ) \\nabla ^ { \\mu } _ { X , z } \\phi _ b | _ { z = - h _ b } . \\end{align*}"}
+{"id": "7313.png", "formula": "\\begin{align*} S _ 0 ( T ) ^ \\ast T S _ 0 ( T ) = P _ + ( Q _ S ) - P _ - ( Q _ S ) = Q _ S , T \\in \\mathcal { U } . \\end{align*}"}
+{"id": "8092.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\lambda _ { j } a _ { j } \\right \\| _ { h _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\lambda _ { j } \\chi _ { Q _ { j } } } { \\omega ( Q _ { j } ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } \\end{align*}"}
+{"id": "3718.png", "formula": "\\begin{align*} 2 ^ { ( 4 n - 1 ) ( 2 \\alpha - \\beta ) / ( 1 - \\beta ) } a _ n ^ { \\delta } = a ^ { \\delta } 2 ^ { ( 3 n - 1 ) ( 2 \\alpha - \\beta ) / ( 1 - \\beta ) } , \\end{align*}"}
+{"id": "5112.png", "formula": "\\begin{align*} \\rho \\circ l & = r \\circ \\rho \\\\ \\rho ( D _ X ( a ) ) & = D ^ r _ X ( \\rho ( a ) ) \\\\ l ( [ a , b ] ) & = [ a , l ( b ) ] - D _ { \\rho ( b ) } ( a ) \\\\ D _ X ( [ a , b ] ) & = [ a , D _ X ( b ) ] + [ D _ X ( a ) , b ] + D _ { [ \\rho ( b ) , X ] } ( a ) - D _ { [ \\rho ( a ) , X ] } ( b ) , \\end{align*}"}
+{"id": "2532.png", "formula": "\\begin{align*} \\boldsymbol { y } ( \\boldsymbol { x } , t ) = ( 0 , 0 , e ^ t \\sin { t } \\sin { \\pi x _ 1 } \\sin { \\pi x _ 2 } ) ^ T . \\end{align*}"}
+{"id": "467.png", "formula": "\\begin{align*} & \\frac { \\Gamma ( \\zeta + ( 1 + \\alpha ) / 2 + 2 k ) } { k ! \\Gamma ( \\zeta + \\frac 1 2 + k ) } ( 2 \\beta ) ^ { - 2 k } \\\\ & = \\frac { \\Gamma ( \\zeta + ( 1 + \\alpha ) / 2 ) } { \\Gamma ( \\frac { \\zeta + ( 1 + \\alpha ) / 2 } { 2 } ) \\Gamma ( \\frac { \\zeta + ( 3 + \\alpha ) / 2 } { 2 } ) } \\frac { 2 ^ { 2 k } \\Gamma ( \\frac { \\zeta + ( 1 + \\alpha ) / 2 } { 2 } + k ) \\Gamma ( \\frac { \\zeta + ( 3 + \\alpha ) / 2 } { 2 } + k ) } { k ! \\Gamma ( \\zeta + \\frac 1 2 + k ) } ( 2 \\beta ) ^ { - 2 k } \\end{align*}"}
+{"id": "4005.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ l \\left ( S _ { n - 1 ; l } ( X ) - \\frac { c } { C ^ m _ n } S _ { m - 1 ; l } ( X ) \\right ) X ^ 2 _ { l \\bar l } \\\\ & = S _ n ( X ) \\left ( b _ t f S _ { n - 1 } ( X ^ { - 1 } ) + ( m + 1 ) \\frac { c } { C ^ m _ n } S _ { n - m - 1 } ( X ^ { - 1 } ) \\right ) \\\\ & \\leq C ( t ) S _ n ( X ) . \\end{aligned} \\end{align*}"}
+{"id": "8743.png", "formula": "\\begin{align*} \\norm { x _ { t + 1 } - x _ { p } } ^ { 2 } & \\leq \\norm { x _ { t } - \\eta _ { t } \\hat { g } _ { t } - x _ { p } } ^ 2 \\\\ & = \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 \\\\ & \\leq ( 1 - 2 \\eta _ { t } \\alpha ) \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } - \\nabla f ( x _ t ) , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 , \\end{align*}"}
+{"id": "1441.png", "formula": "\\begin{align*} \\Delta u _ s + \\sum \\limits _ { t \\in J } k _ { s t } ^ { \\prime } e ^ { u _ t } = 4 \\pi \\alpha _ { s } \\delta _ { 0 } \\ \\ \\mathbb { R } ^ 2 , \\ \\ \\int _ { \\mathbb { R } ^ 2 } e ^ { u _ s ( y ) } \\mathrm { d } y < + \\infty , \\ \\ \\forall \\ s \\in J \\subseteq I , \\end{align*}"}
+{"id": "4607.png", "formula": "\\begin{align*} \\mathcal { H } = \\bigoplus _ { \\chi \\in I ^ \\wedge } \\mathcal { H } _ \\chi , \\end{align*}"}
+{"id": "2233.png", "formula": "\\begin{align*} G ( t ) = t ^ p ( 1 + | \\log ( t ) | ) . \\end{align*}"}
+{"id": "4764.png", "formula": "\\begin{align*} ( T ^ 2 _ D T ^ 2 _ C - T ^ 2 _ C ) T ^ 2 _ C T _ { U _ { 2 , 4 } } ^ 2 = T ^ 2 _ D T ^ 2 _ { U _ { 2 , 3 } } - ( T ^ 2 _ { U _ { 2 , 3 } } T ^ 2 _ C - T ^ 2 _ D ) T _ L ^ 2 T ^ 2 _ { U _ { 2 , 3 } } + ( T ^ 2 _ { U _ { 2 , 3 } } T ^ 2 _ C - T ^ 2 _ D ) T ^ 2 _ { U _ { 1 , 3 } } . \\end{align*}"}
+{"id": "1375.png", "formula": "\\begin{align*} F = \\sum _ { n = 1 } ^ { n _ 0 } c ( n ) n ^ { 2 k - 1 } \\mathcal { F } _ { 2 - 2 k , - 1 } \\big | _ { 2 - 2 k } T _ n . \\end{align*}"}
+{"id": "627.png", "formula": "\\begin{align*} \\norm { \\theta _ R ( \\norm { \\mathbf V } _ { t } ^ 2 ) \\mathbf V } _ { T _ { \\mathbf v } } = \\norm { \\theta _ R ( \\norm { \\mathbf U } _ { t } ^ 2 ) \\mathbf U - \\theta _ R ( \\norm { \\mathbf V } _ { t } ^ 2 ) \\mathbf V } _ { T _ { \\mathbf V } } . \\end{align*}"}
+{"id": "4552.png", "formula": "\\begin{align*} \\mathcal B _ 1 = \\mathcal B ^ { ( 0 ) } _ 1 + \\mathcal B \\ , , \\quad \\mbox { w h e r e } \\ , \\ , \\ , \\mathcal B ^ { ( 0 ) } _ 1 \\vert _ { x _ 1 = 0 } = 0 \\ , , \\end{align*}"}
+{"id": "5362.png", "formula": "\\begin{align*} H ^ 1 ( \\O _ { L _ 1 \\cup \\ldots \\cup L _ { 2 b } } ( 1 ) ( - B _ b ) ) = 0 , \\ \\hbox { w h e r e } \\ B _ b = ( L _ 1 \\cup \\ldots \\cup L _ { 2 b } ) \\cap ( M _ 1 \\cup \\ldots \\cup M _ { b + 4 } ) . \\end{align*}"}
+{"id": "3805.png", "formula": "\\begin{align*} \\int _ X f ( x ) d \\pi ^ { x _ i } _ { \\# } ( s ^ p \\beta ) = \\int _ { X \\times X \\times \\R _ + } s ^ p f ( x _ i ) d \\beta . \\end{align*}"}
+{"id": "2112.png", "formula": "\\begin{align*} I _ { 1 , 1 } : = \\iint _ { D _ t } \\varphi ( \\underline { u } ) | \\underline { L } \\tilde { \\Lambda } | ^ 2 | L ( \\ln \\alpha ) | \\lesssim \\int _ { \\mathbb { R } } \\frac { K _ 1 \\gamma } { \\varphi ^ { 3 / 4 } ( u ) } \\underbrace { \\left [ \\int _ { C _ { t , u } } \\varphi ( \\underline { u } ) | \\underline { L } \\tilde { \\Lambda } | ^ 2 d s \\right ] } _ { \\lesssim \\mathcal { F } ( t ) } d u \\lesssim K _ 1 \\gamma \\varepsilon ^ 2 . \\end{align*}"}
+{"id": "4689.png", "formula": "\\begin{align*} 1 = \\Psi ( x , w ) \\Psi ( y , u ) - \\Psi ( x , u ) \\Psi ( y , w ) . \\end{align*}"}
+{"id": "4397.png", "formula": "\\begin{align*} [ \\partial _ 1 a ] : = \\partial _ 1 a ^ + _ { | x _ 1 = 0 } + \\partial _ 1 a ^ - _ { | x _ 1 = 0 } . \\end{align*}"}
+{"id": "812.png", "formula": "\\begin{align*} c + o ( 1 ) & = \\left ( \\frac { 1 } { p } - \\frac { 1 } { 2 \\cdot p ^ \\sharp } \\right ) [ u _ n ] _ { s , p } ^ p \\end{align*}"}
+{"id": "4859.png", "formula": "\\begin{align*} \\check { R } ( u ) = I + G ( u ) B + H ( u ) E + F ( u ) B \\end{align*}"}
+{"id": "4150.png", "formula": "\\begin{align*} \\frac { d } { d t } \\tilde \\Phi _ t = H _ { \\bar { f } } ( \\tilde \\Phi _ t ) , u ( 0 ) = u _ 0 , \\tilde \\Phi _ 0 = \\phi \\circ \\pi . \\end{align*}"}
+{"id": "1571.png", "formula": "\\begin{align*} \\mathcal { K } _ 1 ( T ( y + x ) , y ) = \\int _ 0 ^ \\infty \\frac { 1 } { \\kappa | v | } e ^ { - ( 1 - y ) / \\kappa | v | } ( 1 - e ^ { - 1 / \\kappa | v | } ) ^ { - 1 } \\mathcal { M } _ { T ( y + x ) } ( v ) \\mathrm { d } v . \\end{align*}"}
+{"id": "3441.png", "formula": "\\begin{align*} \\Delta ^ \\vee _ k = \\{ ( p _ 0 , \\ldots p _ m ) \\in \\R ^ { m + 1 } _ { \\leq 0 } | \\sum d _ i p _ i \\geq - 1 , p _ k = 0 \\} . \\end{align*}"}
+{"id": "8351.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { s \\in [ n , n + 1 ] } \\| \\omega _ 2 ( s ) \\| e ^ { - \\frac { 1 } { m } | s | } = 0 \\end{align*}"}
+{"id": "6316.png", "formula": "\\begin{align*} \\frac { \\partial y } { \\partial x } = \\frac { 1 } { \\sqrt { g _ { [ < \\lambda ^ { \\sigma } ] } } } . \\end{align*}"}
+{"id": "82.png", "formula": "\\begin{align*} \\int _ { \\{ \\lvert \\rho _ z - \\rho \\rvert \\leq \\rho \\varepsilon _ + \\} } ( \\rho _ z - \\rho ) \\langle n _ + \\rangle _ { \\Phi ( z ) } \\widehat { g } ( 0 ) \\ , \\dd z \\leq C \\rho \\varepsilon _ + \\widehat { g } ( 0 ) \\langle n _ + \\rangle _ { \\Psi } = o _ { d } ^ { } , \\end{align*}"}
+{"id": "7405.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial _ { x } w _ { n } } { \\rho _ { n } } \\right ) ( x , t ) \\le \\max _ { \\mathbb { T } } \\frac { \\partial _ { x } w _ { n } ^ { 0 } } { \\rho _ { n } ^ { 0 } } = : \\mathcal { M } _ { n } ^ { 0 } , ~ \\forall ~ ( x , t ) \\in \\mathbb { T } \\times [ 0 , T ^ { * } ) . \\end{align*}"}
+{"id": "4971.png", "formula": "\\begin{align*} \\Q ( \\varphi ) = \\Q ( \\zeta _ 4 ) . \\end{align*}"}
+{"id": "7661.png", "formula": "\\begin{align*} e ^ { - \\frac { 1 } { 2 } \\lambda _ i x ^ 2 } & = P _ { N - 1 } \\big ( - \\tfrac { 1 } { 2 } \\lambda _ i x ^ 2 \\big ) + \\frac { ( - \\frac { 1 } { 2 } \\lambda _ i x ^ 2 ) ^ m } { m ! } + R _ { N } \\big ( - \\tfrac { 1 } { 2 } \\lambda _ i x ^ 2 \\big ) , \\\\ R _ N ( \\xi ) & = \\frac { \\xi ^ { N } } { ( N - 1 ) ! } \\int _ { 0 } ^ { 1 } ( 1 - u ) ^ { N - 1 } e ^ { \\xi u } \\dd u \\end{align*}"}
+{"id": "7745.png", "formula": "\\begin{align*} \\chi _ { i } ( G ) = \\chi \\big { ( } \\mathcal { N } ( G ) \\big { ) } , \\end{align*}"}
+{"id": "5604.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\lim _ { n \\to \\infty } \\int _ 0 ^ \\infty | ( u _ { n , R } ^ \\infty ) ' | ^ p r ^ { p - 1 } \\mathrm d r = \\lim _ { R \\to \\infty } \\lim _ { n \\to \\infty } \\int _ R ^ \\infty | u _ n ' | ^ p r ^ { p - 1 } \\mathrm d r . \\end{align*}"}
+{"id": "5083.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c c } \\widetilde { v } _ { 1 } & \\cdots & \\widetilde { v } _ n \\end{array} \\right ) \\left ( \\begin{array} { c } v _ { 1 } ^ c \\\\ \\vdots \\\\ v _ n ^ c \\end{array} \\right ) = \\widetilde { v } _ { 1 } v _ 1 ^ c + \\cdots + \\widetilde { v } _ n v _ n ^ c , \\end{align*}"}
+{"id": "2892.png", "formula": "\\begin{align*} A = \\{ \\xi \\in V \\mid 0 < \\langle \\xi , \\alpha \\rangle < 2 \\pi , \\ , \\forall \\alpha \\in R _ 0 ^ + \\} . \\end{align*}"}
+{"id": "2397.png", "formula": "\\begin{align*} \\chi _ { \\cup _ { k = 1 } ^ { n } E _ k } ( x ) \\geq \\sum _ { k = 1 } ^ { n } \\chi _ { E _ k } ( x ) - \\sum _ { 1 \\leq k < k ' \\leq n } \\chi _ { E _ k \\cap E _ { k ' } } ( x ) . \\end{align*}"}
+{"id": "7095.png", "formula": "\\begin{align*} \\mathcal { C } ^ { + } ( n ^ 2 _ 1 n _ 2 , m ; q ) = p _ 1 \\ , \\sum _ { d | q } \\ , d \\ , \\mu \\left ( \\frac { q } { d } \\right ) \\ , \\mathop { \\mathop { \\sideset { } { ^ \\star } { \\displaystyle \\sum } _ { \\alpha \\ , \\mathrm { m o d } \\ , p _ 1 q r / n _ 1 } } _ { \\alpha n _ 1 \\ , \\equiv \\ , m \\bar { p _ 2 } \\ , \\ , \\mathrm { m o d } \\ , d } } _ { \\alpha n _ 1 \\ , \\equiv \\ , m \\bar { p _ 2 } \\ , \\ , \\mathrm { m o d } \\ , p _ 1 } \\ , e \\left ( \\frac { \\bar { \\alpha } n _ 2 } { p _ 1 q r / n _ 1 } \\right ) . \\end{align*}"}
+{"id": "7180.png", "formula": "\\begin{align*} x _ i = \\sum _ { \\ell = 1 } ^ m \\alpha ^ j _ { \\ell k _ j } y _ { \\ell } ~ \\mbox { w h e r e } ~ i = \\mathcal { N } _ { j - 1 } + k _ j , k _ j = 1 , \\dots , n _ j ~ \\mbox { a n d } ~ j = 1 , \\dots , m , \\end{align*}"}
+{"id": "389.png", "formula": "\\begin{align*} j ( u ^ * c , x \\circ c ^ * u , y \\circ u ) = j ( c , x , y ) \\circ u . \\end{align*}"}
+{"id": "5779.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\mathbb E \\langle ( m ^ 1 ) ^ 2 \\rangle _ \\beta = 0 , \\end{align*}"}
+{"id": "8795.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } \\mathbb { E } [ f ( x _ { t } ) - f ^ * ] \\leq \\frac { \\mathcal { B } _ 1 } { \\alpha } r _ { 1 } + \\mathcal { B } _ { 6 } \\frac { d } { \\alpha } + \\mathcal { B } _ { 4 } \\frac { d } { \\alpha } T ^ { \\frac { 1 } { \\beta } } , \\end{align*}"}
+{"id": "4097.png", "formula": "\\begin{align*} & \\sum _ { g _ 1 + g _ 2 = g } ^ { \\Delta } \\sum _ { J _ 1 \\sqcup J _ 2 = J } d _ p \\left ( \\omega _ { g _ 1 , n _ 1 + 1 } ( p , J _ 1 ) \\cdot \\omega _ { g _ 2 , n _ 2 + 1 } ( q , J _ 2 ) + \\omega _ { g - 1 , n + 2 } ( p , q , J ) \\right ) \\bigg | _ { p = q } \\\\ & = \\sum _ { g _ 1 + g _ 2 = g } ^ { \\Delta } \\sum _ { J _ 1 \\sqcup J _ 2 = J } d _ q \\left ( \\omega _ { g _ 1 , n _ 1 + 1 } ( p , J _ 1 ) \\cdot \\omega _ { g _ 2 , n _ 2 + 1 } ( q , J _ 2 ) + \\omega _ { g - 1 , n + 2 } ( p , q , J ) \\right ) \\bigg | _ { p = q } . \\end{align*}"}
+{"id": "8002.png", "formula": "\\begin{align*} \\nabla K ( x ) = \\nabla H ( \\pi _ { p _ 0 } ^ { - 1 } ( x ) ) [ \\nabla \\pi _ { p _ 0 } ^ { - 1 } ( x ) ] , \\end{align*}"}
+{"id": "4906.png", "formula": "\\begin{align*} \\beta ( s ) = \\sum _ { n \\geq 1 } \\dfrac { \\chi _ { - 4 } ( n ) } { n ^ s } = \\sum _ { n \\geq 0 } \\dfrac { ( - 1 ) ^ n } { ( 2 n + 1 ) ^ s } , \\operatorname { R e } ( s ) > 0 , \\chi _ { - 4 } = \\left ( \\frac { - 4 } { \\cdot } \\right ) . \\end{align*}"}
+{"id": "3257.png", "formula": "\\begin{align*} ( f , g ) _ { L ^ 2 ( \\mathbb { R } , C \\ell _ { p , q } ) } = \\int _ \\mathbb { R } f ( x ) \\overline { g ( x ) } d x . \\end{align*}"}
+{"id": "3716.png", "formula": "\\begin{align*} I = ( 2 - 2 ^ { 1 - 2 \\alpha } ) f ( 1 ) + 2 ^ { 2 - 2 \\alpha } f \\left ( \\frac 1 2 \\right ) - \\frac { 2 ^ { 1 - 2 \\alpha } ( 1 - 2 ^ { - \\alpha } ) } { 1 - 2 \\alpha } . \\end{align*}"}
+{"id": "7287.png", "formula": "\\begin{align*} A = \\{ n \\geq N \\mid & \\\\ & \\qquad \\qquad \\} . \\end{align*}"}
+{"id": "962.png", "formula": "\\begin{align*} \\begin{dcases*} T _ 1 f = & i n $ \\Sigma $ \\\\ \\frac { \\partial f } { \\partial \\nu } + \\alpha _ \\theta f = 0 & o n $ \\partial \\Sigma $ \\end{dcases*} . \\end{align*}"}
+{"id": "7705.png", "formula": "\\begin{align*} X _ x ( f | _ x ) = ( X ( f ) ) _ 0 ( x ) . \\end{align*}"}
+{"id": "5768.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\mathbb E \\langle ( R _ { 1 , 2 } ^ p - \\langle R _ { 1 , 2 } ^ p \\rangle ) ^ 2 \\rangle = 0 . \\end{align*}"}
+{"id": "6448.png", "formula": "\\begin{align*} g _ { u , v } \\oplus \\pi _ { \\mathrm { N } } ^ * \\left ( \\frac { X } { v } \\right ) = \\pi _ { \\mathrm { N } } ^ * \\left ( \\frac { X } { u } \\right ) . \\end{align*}"}
+{"id": "1797.png", "formula": "\\begin{gather*} \\zeta ( s , \\alpha ) = Z _ N ( s , \\alpha ) - \\frac { 1 } { 2 \\pi i } \\int _ { - \\delta - i \\infty } ^ { - \\delta + i \\infty } \\zeta ( s + w , \\alpha ) \\widehat { \\phi } ( w ) N ^ w \\ , d w - \\widehat { \\phi } ( 1 - s ) N ^ { 1 - s } \\end{gather*}"}
+{"id": "4174.png", "formula": "\\begin{align*} K _ { \\nu / 2 } ( \\sqrt { \\nu } | \\omega | ) & = \\frac { 2 ^ { \\frac { \\nu - 2 } { 2 } } \\Gamma \\left ( \\frac { \\nu + 1 } { 2 } \\right ) } { ( \\sqrt { \\nu } | \\omega | ) ^ { \\frac { \\nu } { 2 } } \\sqrt { \\nu \\pi } } \\int _ { \\mathbb { R } } \\left ( 1 + \\tfrac { n ^ 2 } { \\nu } \\right ) ^ { - \\tfrac { \\nu + 1 } { 2 } } e ^ { i \\omega n } ~ \\mathrm { d } n . \\end{align*}"}
+{"id": "2342.png", "formula": "\\begin{align*} & \\pi _ g \\Delta _ g = \\theta _ \\tau \\lambda _ { g } \\theta _ f \\theta _ \\tau \\lambda _ { g ^ { - 1 } } \\theta _ f = \\theta _ \\tau \\theta _ f \\theta _ \\tau \\lambda _ g \\lambda _ { g ^ { - 1 } } \\theta _ f = I _ \\mathcal { X } , \\\\ & \\Delta _ g \\pi _ g = \\theta _ \\tau \\lambda _ { g ^ { - 1 } } \\theta _ f \\theta _ \\tau \\lambda _ { g } \\theta _ f = \\theta _ \\tau \\lambda _ { g ^ { - 1 } } \\lambda _ g \\theta _ f \\theta _ \\tau \\theta _ f = I _ \\mathcal { X } . \\end{align*}"}
+{"id": "7122.png", "formula": "\\begin{align*} Z \\chi _ j ^ { ( 5 ) } - j \\chi _ j ^ { ( 4 ) } + 4 \\chi _ j = 0 ( 0 , + \\infty ) , \\end{align*}"}
+{"id": "5054.png", "formula": "\\begin{align*} | L _ { g '' } ( T ( \\gamma ^ * ) ) | _ { g '' } & \\leq C r ^ { - 1 } , \\\\ | L _ { g '' } ( \\psi ^ * ( T ( \\gamma ^ * ) ) ) | _ { g '' } & = | L _ { \\psi ^ * g ' } ( \\psi ^ * ( T ( \\gamma ^ * ) ) ) | _ { \\psi ^ * g ' } = | L _ { g ' } ( T ( \\gamma ^ * ) ) | _ { g ' } \\circ \\psi \\leq C r ^ { - 1 } . \\end{align*}"}
+{"id": "7347.png", "formula": "\\begin{align*} \\chi ( X ) = \\frac { 1 } { 2 } \\int _ 0 ^ { \\infty } \\Big ( \\frac { 1 } { 1 + t } - \\Phi ( X + \\sqrt { t } Z ) \\Big ) \\ , d t + \\frac { 1 } { 2 } \\ln 2 \\pi + \\frac { 1 } { 2 } \\end{align*}"}
+{"id": "2907.png", "formula": "\\begin{align*} ( L _ \\omega f ) ( \\lambda ) { = } & ( \\mathcal { J } L _ { \\omega ; 1 } \\mathcal { J } ^ { - 1 } f ) ( \\lambda ) { = } ( L _ { \\omega ; 1 } \\mathcal { J } ^ { - 1 } f ) ( \\lambda ) \\\\ { = } & \\sum _ { \\nu \\in W _ 0 \\omega } ( \\mathcal { J } ^ { - 1 } f ) ( \\lambda + \\nu ) \\\\ { = } & \\sum _ { \\nu \\in W _ 0 \\omega } t [ \\lambda + \\nu ] f ( \\lambda + \\nu ) + d _ { \\lambda , \\nu } ( 1 - t ^ { - 1 } _ { \\vartheta } ) f ( \\lambda ) . \\end{align*}"}
+{"id": "2735.png", "formula": "\\begin{align*} U e _ i = \\left \\{ \\begin{array} { l l } \\varepsilon _ i e _ { \\pi ( i ) } , & 1 \\leq i \\leq \\lfloor \\theta ^ { - 1 } \\rfloor \\\\ \\varepsilon _ i e _ i , & i > \\lfloor \\theta ^ { - 1 } \\rfloor \\end{array} \\right . \\quad ( i \\in \\mathbb N ) . \\end{align*}"}
+{"id": "8466.png", "formula": "\\begin{align*} \\mathcal { K } ( \\ell ) \\supset \\{ E \\subset \\mathbb { R } ^ { n } : \\mathcal { H } ^ { n } ( E \\triangle ( F _ \\ell + ( 0 , \\tau ) ) ) = 0 \\mbox { f o r s o m e } \\tau \\in \\mathbb { R } ^ { n - 1 } \\} , \\end{align*}"}
+{"id": "271.png", "formula": "\\begin{align*} ( i , j ) = ( i _ 0 , j _ 0 ) < ( i _ 1 , j _ 1 ) < \\cdots < ( i _ { \\alpha - 1 } , j _ { \\alpha - 1 } ) < ( i _ { \\alpha } , j _ \\alpha ) = ( k , \\ell ) \\end{align*}"}
+{"id": "5452.png", "formula": "\\begin{align*} G ^ * ( e ^ { t ^ 2 / 2 } ) e ^ { - t ^ 2 / 2 } - t & = W ^ { - 1 } ( t ^ 2 / 2 ) - \\frac { 1 } { V ' ( W ^ { - 1 } ( t ^ 2 / 2 ) ) } - t \\\\ & = x - \\frac { 1 } { V ' ( x ) } - \\sqrt { 2 W ( x ) } \\end{align*}"}
+{"id": "1094.png", "formula": "\\begin{align*} \\Gamma ( u , v ) ( x ) : = \\frac { 1 } { 2 \\mu ( x ) } \\sum _ { y \\sim x } w ( x , y ) ( u ( y ) - u ( x ) ) ( v ( y ) - v ( x ) ) , \\end{align*}"}
+{"id": "3923.png", "formula": "\\begin{align*} K \\hat { + } _ t L : = ( ( 1 - t ) K ^ * + t L ^ * ) ^ * . \\end{align*}"}
+{"id": "2032.png", "formula": "\\begin{align*} S _ \\phi u ( x ' ) : = u ( x ' , \\phi ( x ' ) ) x ' \\in \\mathbb { R } ^ { n - 1 } \\end{align*}"}
+{"id": "7854.png", "formula": "\\begin{align*} \\phi : S \\to S ' \\mbox { s u c h t h a t } \\phi \\psi ( H ) \\phi ^ { - 1 } = \\psi ' ( H ) \\end{align*}"}
+{"id": "2380.png", "formula": "\\begin{align*} X _ t = 1 _ { t \\leq \\tau } ( X _ 1 ) _ { t } + 1 _ { t \\geq \\tau } ( X _ 2 ) _ { t } + h d t 1 _ { d \\tau } \\left [ ( X _ 1 ) _ { \\tau } + ( X _ 2 ) _ { \\tau } - ( X _ 1 ) _ { \\tau } \\right ] \\end{align*}"}
+{"id": "4565.png", "formula": "\\begin{align*} \\int _ { \\Omega _ t } ( \\partial _ 1 \\mathcal B _ 1 + \\partial _ 2 \\mathcal B _ 2 - \\partial _ t \\mathcal B _ 0 - \\mathcal B _ 3 ) { \\mathbf V } _ { s } \\cdot { \\mathbf V } _ { s } d { \\bf x } d s \\le C _ 1 \\Vert { \\mathbf V } _ { s } \\Vert _ { L ^ 2 ( \\Omega _ t ) } ^ 2 = C _ 1 \\int _ 0 ^ t I _ 0 ( s ) d s \\ , , \\end{align*}"}
+{"id": "4708.png", "formula": "\\begin{align*} \\left ( \\dfrac { 1 - \\alpha ^ 2 } { \\beta ^ 2 } + \\dfrac { \\alpha - 1 } { \\beta } \\chi ( z _ 0 ) \\right ) \\chi ( x y ) + \\left ( \\dfrac { 1 - \\alpha } { \\beta ^ 2 } - \\dfrac { 1 } { \\beta } \\chi ( z _ 0 ) \\right ) \\chi ( x y ) A ( x y ) = 0 . \\end{align*}"}
+{"id": "8001.png", "formula": "\\begin{align*} \\tilde K ( x ) = \\sum _ { i = 1 } ^ \\infty a _ i K ( x - z _ i ) . \\end{align*}"}
+{"id": "4075.png", "formula": "\\begin{align*} K _ { \\Sigma } ^ { \\boxtimes n + 1 } = \\pi ^ * _ 0 ( K _ \\Sigma ) \\otimes \\cdots \\otimes \\pi ^ * _ n ( K _ \\Sigma ) . \\end{align*}"}
+{"id": "4685.png", "formula": "\\begin{align*} G & = \\mathrm { p r o j } ( D ) , \\\\ I & = \\{ x \\in G : \\ y \\in D _ x , F ( x , y ) = \\inf F _ x \\} , \\end{align*}"}
+{"id": "1270.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta ) u _ { n } ^ J + | x | ^ { - b } F ( u _ n ^ J ) = | x | ^ { - b } [ F ( \\sum _ { j = 1 } ^ J v _ n ^ j ) - \\sum _ { j = 1 } ^ J F ( v _ n ^ j ) ] . \\end{align*}"}
+{"id": "7752.png", "formula": "\\begin{align*} Z : = 2 \\omega _ M ( \\xi ^ { 1 , 0 } , - ) | _ { \\Gamma } \\ , . \\end{align*}"}
+{"id": "4773.png", "formula": "\\begin{align*} h C _ { j } \\cap C _ { k } = \\{ x \\mid x _ { h _ { g } } = i _ { g } ^ { j } g \\in A _ { j } \\setminus B _ { N } x _ { f } = i _ { f } ^ { k } f \\in A _ { k } \\} . \\end{align*}"}
+{"id": "8071.png", "formula": "\\begin{align*} S ^ { 0 } ( h ) ( x ) = \\left \\{ \\sum \\limits _ { P \\in \\Pi _ { N } } \\sup \\limits _ { u \\in P } \\vert \\psi _ { 0 } \\ast h ( u ) \\vert ^ 2 \\chi _ { P } ( x ) \\right \\} ^ { 1 / 2 } \\end{align*}"}
+{"id": "771.png", "formula": "\\begin{align*} \\norm { \\theta _ R \\left ( \\sum _ { i = 1 } ^ n \\norm { \\phi _ i } _ { \\widetilde H ^ { b } ( 0 , t ) } ^ 2 \\right ) \\phi _ j ( t ) } _ { H ^ { b } ( 0 , T ) } \\leqslant C \\sqrt { R } , \\end{align*}"}
+{"id": "6517.png", "formula": "\\begin{align*} | \\det { \\bf M } ( m ) | = \\prod _ { s = 1 } ^ \\beta | \\lambda _ s v _ s ^ { - 1 } ( m ) | ^ \\beta \\prod _ { 1 \\leq l _ 1 < l _ 2 \\leq \\beta } | v _ { l _ 1 } ^ { - 2 } ( m ) - v _ { l _ 2 } ^ { - 2 } ( m ) | . \\end{align*}"}
+{"id": "2440.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ N | u _ n \\rangle \\langle u _ n | \\le ( - \\Delta _ { \\Omega } ) ^ { - s } L ^ 2 ( \\Omega ) \\end{align*}"}
+{"id": "4077.png", "formula": "\\begin{align*} B ( p _ 0 , \\sigma ( p _ 1 ) ) = B ( \\sigma ( p _ 0 ) , p _ 1 ) , B ( p _ 0 , p _ 1 ) + B ( p _ 0 , \\sigma ( p _ 1 ) ) = \\frac { d x ( p _ 0 ) \\cdot d x ( p _ 1 ) } { ( x ( p _ 0 ) - x ( p _ 1 ) ) ^ 2 } , \\end{align*}"}
+{"id": "1776.png", "formula": "\\begin{gather*} \\mathbf { E } \\left [ \\mathbb { Y } _ \\alpha ( n _ 1 ) ^ { m _ 1 } \\cdots \\mathbb { Y } _ \\alpha ( n _ k ) ^ { m _ k } \\right ] = \\begin{cases} 1 & , \\\\ 0 & \\end{cases} \\end{gather*}"}
+{"id": "1124.png", "formula": "\\begin{align*} \\kappa _ { m , p } : = \\displaystyle \\frac { 1 } { \\mu _ 0 \\lambda _ { m , p } ^ { 1 / p } } , \\end{align*}"}
+{"id": "8619.png", "formula": "\\begin{align*} R _ 1 = - \\nabla _ X \\cdot ( \\zeta \\mathrm { F } _ 4 \\nabla _ X G ) , \\end{align*}"}
+{"id": "1356.png", "formula": "\\begin{align*} E _ { 2 k } ( z ) : = 1 - \\frac { 4 k } { B _ { 2 k } } \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { 2 k - 1 } ( n ) q ^ n , \\end{align*}"}
+{"id": "349.png", "formula": "\\begin{align*} u : = \\mathsf { S } _ { j } \\left ( s \\right ) \\psi _ { \\operatorname * { N } ; j } - \\mathsf { D } _ { j } \\left ( s \\right ) \\psi _ { \\operatorname * { D } ; j } . \\end{align*}"}
+{"id": "8563.png", "formula": "\\begin{align*} ( \\mathrm { A } ) \\ \\ \\begin{cases} \\partial _ t \\zeta + \\mathcal { N } _ 1 ( \\zeta , b , \\psi ) = 0 \\\\ \\partial _ t ( \\mathcal { T } [ \\zeta , b ] \\psi ) + \\mathcal { N } _ 2 ( \\zeta , b , \\psi ) = 0 , \\end{cases} \\end{align*}"}
+{"id": "8512.png", "formula": "\\begin{align*} & E _ { 1 } \\cap B _ { \\bar { \\rho } } ( ( \\bar { z } , w ) ) = F _ { \\ell } \\cap \\{ z < \\bar { z } \\} \\cap B _ { \\bar { \\rho } } ( ( \\bar { z } , w ) ) \\\\ & \\subset \\left \\{ ( z ' , w ' ) \\in \\mathbb { R } \\times \\mathbb { R } ^ { n - 1 } : z ' < \\bar { z } \\ \\mbox { a n d } \\ r _ { \\ell } ^ { \\vee } ( \\bar { z } ) + \\frac { \\delta } { 2 } < | w ' | < r _ { \\ell } ( z ' ) \\right \\} \\cap B _ { \\bar { \\rho } } ( ( \\bar { z } , w ) ) . \\end{align*}"}
+{"id": "4261.png", "formula": "\\begin{align*} \\mathcal { B } _ { \\alpha } ( u , u ) - \\lambda _ { k } \\| u \\| ^ 2 _ { L ^ 2 ( \\Omega ) } = \\mathcal { B } _ { \\alpha } ( u ^ - , u ^ - ) - \\lambda _ { k } \\| u ^ - \\| ^ 2 _ { L ^ 2 ( \\Omega ) } , \\end{align*}"}
+{"id": "1117.png", "formula": "\\begin{align*} - \\int _ { D } u ^ - _ \\infty ( x ) \\Delta _ \\mu u ^ + _ \\infty ( x ) d \\mu & = - \\sum _ { x \\in \\mathop D \\limits ^ \\circ } u ^ - _ \\infty ( x ) \\sum _ { y \\sim x } w ( x , y ) ( u ^ + _ \\infty ( y ) - u ^ + _ \\infty ( x ) ) \\\\ & = - \\sum _ { x \\in \\mathop D \\limits ^ \\circ } \\sum _ { y \\sim x } w ( x , y ) u ^ - _ \\infty ( x ) u ^ + _ \\infty ( y ) \\geq 0 \\ , . \\end{align*}"}
+{"id": "5851.png", "formula": "\\begin{align*} w _ 0 = s _ { 3 \\alpha _ 1 + 2 \\alpha _ 2 } s _ { \\alpha _ 1 } . \\end{align*}"}
+{"id": "8294.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d \\tau } W _ 2 ( \\bar p ) & = \\int _ 0 ^ 1 \\bar p \\bar p _ \\tau d x = \\int _ 0 ^ 1 \\bar p \\bar p _ { x x } d x = - \\int _ 0 ^ 1 \\abs { \\bar p _ x } ^ 2 d x - c \\abs { \\bar p ( 0 , \\tau ) } ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "7626.png", "formula": "\\begin{align*} \\P _ { \\tilde { x } } : = ( 1 - \\alpha ) \\P + \\alpha \\delta _ { \\tilde { x } } \\in B _ k ( \\P ) . \\end{align*}"}
+{"id": "255.png", "formula": "\\begin{align*} \\lim _ { c \\to + \\infty } e ^ { - c \\langle \\xi , \\rho _ M \\rangle } \\Delta _ \\emph { U } ( \\xi ) \\phi _ \\xi ^ { \\emph { b c } } ( x + c \\rho _ M ; g ^ { ( c ) } ) = \\Delta _ \\emph { U } ( \\xi ) \\phi ^ { \\emph { t } } _ \\xi ( x ; g ) . \\end{align*}"}
+{"id": "292.png", "formula": "\\begin{align*} \\mathbb { A } _ { j } ^ { \\sigma } : = \\left . \\mathbb { A } _ { j } ^ { \\operatorname * { e x t } } \\right \\vert _ { \\Omega _ { j } ^ { \\sigma } } \\quad p _ { j } ^ { \\sigma } : = \\left . p _ { j } ^ { \\operatorname * { e x t } } \\right \\vert _ { \\Omega _ { j } ^ { \\sigma } } \\sigma \\in \\left \\{ + , - \\right \\} . \\end{align*}"}
+{"id": "5384.png", "formula": "\\begin{align*} F = f ( B ^ H ( \\varphi _ 1 ) , \\cdot \\cdot \\cdot , B ^ H ( \\varphi _ n ) ) , \\end{align*}"}
+{"id": "1410.png", "formula": "\\begin{gather*} ( \\tilde H ^ K ( x ) f ) _ { n i } = \\sum _ { k = 1 } ^ { K } ( \\tilde H _ { n i , k 0 } f _ { k 0 } + \\tilde H _ { n i , k 1 } f _ { k 1 } ) , n \\ge 1 , i = 0 , 1 . \\end{gather*}"}
+{"id": "7747.png", "formula": "\\begin{align*} \\chi _ { i } ( G ) = \\chi \\big { ( } \\mathcal { N } ( G ) \\big { ) } = p _ { o } ( G ) , \\end{align*}"}
+{"id": "1009.png", "formula": "\\begin{align*} 0 = \\nabla \\cdot ( \\nabla \\times { \\bf H } ) = \\nabla \\cdot { \\bf J } + \\nabla \\cdot \\dfrac { \\partial { \\bf D } } { \\partial t } = \\nabla \\cdot { \\bf J } + \\dfrac { \\partial \\rho } { \\partial t } . \\end{align*}"}
+{"id": "4700.png", "formula": "\\begin{align*} g ( x y z _ { 0 } ) = g ( x ) g ( y ) - f ( x ) f ( y ) , \\ , x , y \\in S \\end{align*}"}
+{"id": "1026.png", "formula": "\\begin{align*} Z _ k \\pi _ k ^ * \\Psi = \\pi _ k ^ * P ^ E _ 3 \\Psi , Z _ k ^ * \\pi _ k ^ * ( w \\otimes \\Psi ) = \\pi _ k ^ * ( ( P ^ E _ 3 ) ^ * ( w \\otimes \\Psi ) ) \\end{align*}"}
+{"id": "7330.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} J u ' ( t ) + \\nabla _ u \\mathcal { H } _ \\lambda ( t , u ( t ) ) & = 0 , t \\in \\mathbb { R } \\\\ \\lim _ { t \\rightarrow \\pm \\infty } u ( t ) & = 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "8425.png", "formula": "\\begin{align*} X ^ \\star = \\Sigma ^ { 1 / 2 } Q \\psi ( \\Lambda ) Q ^ T \\Sigma ^ { 1 / 2 } . \\end{align*}"}
+{"id": "3620.png", "formula": "\\begin{align*} \\langle D \\phi , Y \\rangle & = \\langle D \\phi ^ { \\ell } , Y \\rangle = \\lim _ { n \\to \\infty } \\langle D \\phi _ n ^ { \\ell } , Y \\rangle = \\lim _ { n \\to \\infty } \\langle D \\phi _ n , Y \\rangle \\\\ & = \\lim _ { n \\to \\infty } \\langle V _ n , Q Y \\rangle = \\lim _ { n \\to \\infty } \\langle V _ n ^ { \\ell } , Q Y \\rangle = \\langle V ^ { \\ell } , Q Y \\rangle = \\langle V , Q Y \\rangle , \\end{align*}"}
+{"id": "1054.png", "formula": "\\begin{align*} g ( z , x ) = \\left \\{ \\begin{array} { l l } f ( z , u _ \\mu ( z ) ) + \\vartheta u _ \\mu ( z ) ^ { p ( z ) - 1 } , & \\hbox { i f } x < u _ \\mu ( z ) \\\\ f ( z , x ) + \\vartheta x ^ { p ( z ) - 1 } , & \\hbox { i f } u _ \\mu ( z ) \\leq x . \\end{array} \\right . \\end{align*}"}
+{"id": "4394.png", "formula": "\\begin{align*} \\dot { { \\mathbf U } } ^ { \\pm } : = { \\mathbf U } ^ { \\pm } - \\frac { \\partial _ 1 \\hat { { \\mathbf U } } ^ { \\pm } } { \\partial _ 1 \\hat { \\Phi } ^ { \\pm } } \\Psi ^ { \\pm } . \\end{align*}"}
+{"id": "5257.png", "formula": "\\begin{align*} & \\| q _ { i , t + 1 } \\| = \\gamma _ { t } \\| [ g _ { i , t } ( x _ { i , t } ) ] _ + \\| . \\end{align*}"}
+{"id": "1731.png", "formula": "\\begin{align*} \\delta ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } ( 1 ) = \\prod _ { \\substack { 1 \\leq j < k \\leq n \\\\ \\lambda _ j - \\lambda _ k = 0 } } \\frac { k - j } { 1 + k - j } \\prod _ { \\substack { 1 \\leq j < k \\leq n \\\\ \\lambda _ j - \\lambda _ k = m } } \\frac { n - k + j } { n + 1 - k + j } . \\end{align*}"}
+{"id": "8436.png", "formula": "\\begin{align*} I _ t ( \\Gamma \\setminus \\{ z \\} ) : ( x _ 1 ( z ) \\cdots x _ t ( z ) ) = I _ t ( \\Gamma ( z ) ) + ( x _ 0 ( z ) ) + \\sum \\limits _ { j = 0 } ^ { t - 1 } I _ { t - j } ( \\Delta _ j ^ { \\Gamma } ( z ) ) . \\end{align*}"}
+{"id": "2426.png", "formula": "\\begin{align*} c _ { a _ i , a _ 0 } = \\frac { u _ \\gamma ( a _ i ) ^ 2 } { c _ { \\gamma } ( a _ 0 ) ^ 2 } = \\frac { u _ \\gamma ( a _ i ) u _ { \\gamma } ( s _ i ^ { - 1 } a _ i ) } { u _ \\gamma ( s _ 0 ^ { - 1 } a _ 0 ) } = \\frac { u _ \\gamma ( s _ 0 ^ { - 1 } \\gamma _ i ) } { u _ { \\gamma } ( s _ 0 ^ { - 1 } a _ 0 ) } \\frac { u _ { \\gamma } ( a _ i ) } { u _ { \\gamma } ( a _ 0 ) } \\end{align*}"}
+{"id": "642.png", "formula": "\\begin{align*} d X = A X d t + \\left ( \\mathcal N ( X ) + \\frac 1 2 \\sum _ k \\mathcal M _ k [ \\mathcal M _ k ( X ) ] \\right ) \\ , d t + \\mathcal M ( X ) d W , \\end{align*}"}
+{"id": "1546.png", "formula": "\\begin{align*} K _ { X } = \\pi ^ * \\left ( K _ { \\mathbb { P } ^ 1 } + L \\right ) + \\sum _ { i } ( m _ i - 1 ) F _ i . \\end{align*}"}
+{"id": "7847.png", "formula": "\\begin{align*} F : = \\{ ( \\tau _ 2 , b + \\mathrm { i } t , \\tau _ 1 , \\zeta ^ 1 , \\widetilde { \\zeta } _ 0 , \\widetilde { \\zeta } _ 1 , \\sigma ) \\in \\widetilde { N } \\ ; | \\ ; \\tau \\in \\mathcal { F } _ { \\mathbb { H } } \\} \\end{align*}"}
+{"id": "7240.png", "formula": "\\begin{align*} \\begin{aligned} \\delta & = \\Bigl ( - \\frac { B } { G ^ 2 } ( \\cosh G - 1 ) + \\frac { 1 } { G } \\sinh G \\Bigr ) x \\\\ & = \\pm \\Bigl ( - R \\cosh G + R + \\sinh G \\Bigr ) \\sqrt { \\frac { 1 } { 2 } ( 1 - R ^ 2 ) } \\end{aligned} \\end{align*}"}
+{"id": "636.png", "formula": "\\begin{align*} \\mathfrak K _ j f ( x ) = \\int _ { \\R } \\mathfrak k _ j ( x - y ) f ( y ) d y \\end{align*}"}
+{"id": "879.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathrm { P r } ^ { ( \\alpha , n ) } V ( \\mathcal { T } _ t ^ \\alpha u - M ( x , u ^ { ( n ) } , v ^ { ( n ) } ) | _ { \\eqref { p a r a b o l i c e q u a t i o n s } } = 0 , \\\\ & \\mathrm { P r } ^ { ( \\alpha , n ) } V ( \\mathcal { T } _ t ^ \\alpha v - N ( x , u ^ { ( n ) } , v ^ { ( n ) } ) | _ { \\eqref { p a r a b o l i c e q u a t i o n s } } = 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2644.png", "formula": "\\begin{align*} \\mathcal { E } ^ { ( l ) } _ n ( \\alpha , s ) = e ^ { G ^ { ( l ) } _ n ( \\alpha , s ) } \\prod _ { 0 < s ' \\le s } ( 1 + \\Delta G ^ { ( l ) } _ n ( \\alpha , s ' ) ) e ^ { - \\Delta G ^ { ( l ) } _ n ( \\alpha , s ' ) } \\ , , \\end{align*}"}
+{"id": "2537.png", "formula": "\\begin{align*} s _ 1 = \\hdots = s _ u = t _ 1 = \\hdots = t _ v u = v , \\end{align*}"}
+{"id": "3710.png", "formula": "\\begin{align*} 2 ^ \\alpha ( I ( 1 ) + 2 g ( 1 ) ) = \\frac { 2 + 2 ^ { 1 - \\alpha } - 2 ^ { 1 + \\alpha } } { 1 - 2 \\alpha } - \\mu _ \\alpha - f ( 1 ) \\ge \\frac { 1 + 2 ^ { 1 - \\alpha } - 2 ^ { 1 + \\alpha } } { 1 - 2 \\alpha } - \\frac \\alpha { 1 + 2 \\alpha } . \\end{align*}"}
+{"id": "8486.png", "formula": "\\begin{align*} g ^ { \\vee } ( x ) = \\mbox { a p l i m } ( g , H _ { x , \\nu } ^ { + } , x ) \\mbox { a n d } g ^ { \\wedge } ( x ) = \\mbox { a p l i m } ( g , H _ { x , \\nu } ^ { - } , x ) . \\end{align*}"}
+{"id": "3085.png", "formula": "\\begin{align*} { \\mathbb E } \\left \\{ 1 - \\frac { { { N _ { { \\rm { S } } , k } } } } { \\pi } \\frac { { \\left | { \\tilde \\Theta _ { { k , { \\rm { R } } } } ^ { \\rm { D } } - \\Theta _ { { k , { \\rm { R } } } } ^ { \\rm { D } } } \\right | } } { 2 } \\right \\} = { 1 - { 2 ^ { - 1 - { x _ k } } } } . \\end{align*}"}
+{"id": "7398.png", "formula": "\\begin{align*} \\partial _ { t } \\partial _ { x } w _ { n } = - \\partial _ { x } u _ { n } \\partial _ { x } w _ { n } - u _ { n } \\partial _ { x } ^ { 2 } w _ { n } . \\end{align*}"}
+{"id": "3646.png", "formula": "\\begin{align*} \\omega _ { 1 \\epsilon } = k \\omega _ { 2 \\epsilon } \\end{align*}"}
+{"id": "6729.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ t } | \\nabla u | \\Delta _ { \\Sigma _ t } \\left ( \\frac { 1 } { | \\nabla u | } \\right ) = & \\int _ { \\Sigma _ t } | \\nabla u | ^ { - 2 } \\left | \\nabla _ { \\Sigma _ t } { | \\nabla u | } \\right | ^ { 2 } . \\end{align*}"}
+{"id": "4745.png", "formula": "\\begin{align*} \\phi _ i ( x ) = { \\rm m i n } \\big \\{ 0 , 1 - 2 i \\ , { \\rm d i s t } ( x , \\{ 0 \\} \\times \\R ) \\big \\} \\end{align*}"}
+{"id": "5142.png", "formula": "\\begin{align*} { { \\rm A o I } ^ * } = & \\min _ { S \\in \\mathcal { S } } \\limsup _ { T \\to \\infty } \\frac { 1 } { T } E \\bigg [ \\int _ { 0 } ^ { T } \\Delta _ t d t \\bigg ] , \\end{align*}"}
+{"id": "3927.png", "formula": "\\begin{align*} \\bigcup _ { t \\in ( 0 , 1 ) } P ( ( 1 - t ) v _ 1 + t v _ 2 ) = \\bigcup _ { t \\in ( 0 , 1 ) } P ( ( 1 - t ) \\frac { v _ 1 } { \\lVert v _ 1 \\rVert } + t \\frac { v _ 2 } { \\lVert v _ 2 \\rVert } ) \\end{align*}"}
+{"id": "6325.png", "formula": "\\begin{align*} i \\partial _ { \\tau } v _ { \\lambda } + i ( P _ { \\lesssim \\epsilon ^ 2 } ^ y V _ 2 ) \\partial _ y v _ { \\lambda } + \\frac { i } 2 ( \\partial _ y P _ { \\lesssim \\epsilon ^ 2 } ^ y V _ 2 ) v _ { \\lambda } + \\partial ^ 2 _ y v _ { \\lambda } = \\tilde { f } _ \\lambda , \\end{align*}"}
+{"id": "7114.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\Delta ^ 2 u = f & \\hbox { \\rm i n } \\ D \\\\ ( \\Delta u ) _ { | \\partial D } = g & \\hbox { \\rm o n } \\ \\partial D \\\\ ( \\nabla \\Delta u \\cdot n ) _ { | \\partial D } = h & \\hbox { \\rm o n } \\ \\partial D \\end{array} \\right . \\end{align*}"}
+{"id": "2929.png", "formula": "\\begin{align*} \\alpha = 0 , \\textbf { v } ^ 1 = \\frac { 5 } { 2 } \\textbf { v } ^ 3 - \\textbf { v } ^ 2 , D _ { i j } ^ 2 = - \\frac { 2 } { 3 } D _ { i j } ^ 1 . \\end{align*}"}
+{"id": "2515.png", "formula": "\\begin{align*} ( \\boldsymbol { v } , \\boldsymbol { q } ) = ( \\boldsymbol { y } - \\frac { 1 } { \\sqrt { \\alpha } } \\boldsymbol { p } - \\frac { 1 } { \\sqrt { \\alpha } } \\boldsymbol { p } ^ \\perp , \\boldsymbol { p } + \\sqrt { \\alpha } \\boldsymbol { y } - \\sqrt { \\alpha } \\boldsymbol { y } ^ \\perp ) \\end{align*}"}
+{"id": "3598.png", "formula": "\\begin{align*} \\begin{aligned} { { \\bar R } _ { { \\rm { B F , u p p e r } } } } = & { \\log _ 2 } \\left ( 1 + \\frac { { E { N _ { \\rm { T } } } { N _ { \\rm { R } } } \\kappa } } { { { \\sigma ^ 2 } K { L _ { \\rm { R } } } \\left ( { \\kappa + 1 } \\right ) } } \\right . \\times \\\\ & \\left . \\left ( { { \\rm { t } } { { \\rm { r } } _ 1 } \\left ( { { { \\bf { D } } _ { K } } } \\right ) + \\frac { \\pi } { 2 } { \\rm { t } } { { \\rm { r } } _ 2 } \\left ( { { \\bf { D } } _ { K } ^ { 1 / 2 } } \\right ) } \\right ) \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "4855.png", "formula": "\\begin{align*} L ( u ) = & 1 - y ( u ) \\\\ M ( u ) = & ( \\lambda _ 1 + \\lambda _ 2 ) ( \\lambda _ 2 + \\lambda _ 3 ) \\lambda _ 2 ^ { - 1 } y ( u ) \\\\ N ( u ) = & \\lambda _ 1 \\lambda _ 3 y ( u ) [ y ( u ) - 1 ] \\end{align*}"}
+{"id": "4816.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d + 1 } \\binom { d + 1 } { i } \\sf ( n - i ) & < \\sum _ { i = 1 } ^ { d + 1 } \\binom { d + 1 } { i } ( 2 d + 2 ) ^ { n - i } \\\\ & = ( 2 d + 2 ) ^ n \\sum _ { i = 1 } ^ { d + 1 } \\binom { d + 1 } { i } \\Big ( \\frac { 1 } { 2 d + 2 } \\Big ) ^ i \\\\ & = ( 2 d + 2 ) ^ n \\bigg ( \\Big ( 1 + \\frac { 1 } { 2 d + 2 } \\Big ) ^ { d + 1 } - 1 \\bigg ) . \\end{align*}"}
+{"id": "8359.png", "formula": "\\begin{align*} Z ( \\theta _ t \\omega _ 2 ) = Z ( t , \\omega _ 2 ) \\end{align*}"}
+{"id": "1942.png", "formula": "\\begin{align*} f ( 0 , x , v ) = \\left ( 1 + \\sin ( x ) \\right ) e ^ { - v ^ 2 } , u ( 0 , x ) = \\sin ( x ) . \\end{align*}"}
+{"id": "8949.png", "formula": "\\begin{align*} \\hat { f } ( a ^ s \\Lambda _ { u , v } ) = \\sum _ { ( \\bar { p } , \\bar { q } ) \\in \\Z ^ { m + n } \\setminus \\{ 0 \\} } f \\left ( e ^ { w _ 1 s } ( p _ 1 + \\langle \\bar { u } _ 1 , \\bar { q } \\rangle + v _ 1 ) , \\ldots , e ^ { w _ m s } ( p _ m + \\langle \\bar { u } _ m , \\bar { q } \\rangle + v _ m ) , e ^ { - s } \\bar { q } \\right ) \\end{align*}"}
+{"id": "4470.png", "formula": "\\begin{align*} \\varphi _ 1 = u ^ { \\pm } _ { 1 , 0 } - u ^ { \\pm } _ { 2 , 0 } \\partial _ 2 \\varphi _ 0 \\Big | _ { x _ 1 = 0 } . \\end{align*}"}
+{"id": "5387.png", "formula": "\\begin{align*} M ( t ) = S ( t ) \\Lambda ( t ) S ^ { - 1 } ( t ) , t \\geq 0 . \\end{align*}"}
+{"id": "2979.png", "formula": "\\begin{align*} \\textsf { A } = \\begin{pmatrix} 1 & 1 \\\\ 2 & 0 \\end{pmatrix} \\textrm { a n d } \\textsf { B } = \\begin{pmatrix} 1 & 0 & 1 \\\\ 1 & 0 & 1 \\\\ 1 & 1 & 0 \\end{pmatrix} , \\end{align*}"}
+{"id": "806.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon _ g \\to + 0 } A ( \\varepsilon _ g ) = \\left ( \\frac { p ^ \\sharp } { b C _ U } \\right ) ^ { \\frac { p } { 2 \\cdot p ^ \\sharp - p } } ( S ^ * ) ^ { \\frac { 2 \\cdot p ^ \\sharp } { 2 \\cdot p ^ \\sharp - p } } , \\end{align*}"}
+{"id": "1018.png", "formula": "\\begin{align*} q ( R ) ^ E = - \\sum _ s a _ s ( P _ s ^ E ) ^ * P _ s ^ E \\hbox { o n } C ^ \\infty ( M , V M \\otimes E M ) . \\end{align*}"}
+{"id": "5224.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ t \\frac { 1 } { j } & \\leq 1 + \\int _ 1 ^ t \\frac { 1 } { x } d x = 1 + \\ln x | _ 1 ^ t = 1 + \\ln t , \\\\ \\sum _ { j = 1 } ^ t \\frac { 1 } { \\sqrt { j } } & \\geq \\int _ 1 ^ { t + 1 } \\frac { 1 } { \\sqrt { x } } d x = 2 \\sqrt { x } | _ { 1 } ^ { t + 1 } = 2 ( \\sqrt { t + 1 } - 1 ) . \\end{align*}"}
+{"id": "708.png", "formula": "\\begin{align*} \\widetilde { \\mathbf I } ( \\tau , \\xi , \\omega ) = \\int _ 0 ^ T e ^ { - i t \\tau } \\ , \\widehat { \\mathbf I } ( t , \\xi , \\omega ) \\ , d t , \\end{align*}"}
+{"id": "7590.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { \\textbf { R } } _ 1 & = \\{ ( s , t ) \\in [ 0 , T ] ^ 2 \\ , : t ^ { \\frac { 3 } { 4 } } - s ^ { \\frac { 3 } { 4 } } > C _ { ( \\ref { r - i - j 1 } ) } \\zeta ^ { - 1 } \\} , \\\\ \\hat { \\mathbf { R } } _ 2 & = \\{ ( t , s ) \\in [ 0 , T ] ^ 2 : t ^ { \\frac { 3 } { 4 } } - s ^ { \\frac { 3 } { 4 } } < C _ { ( \\ref { r - i - j 1 } ) } \\zeta ^ { - 1 } \\} , \\end{aligned} \\end{align*}"}
+{"id": "7.png", "formula": "\\begin{align*} \\delta : = \\frac { 1 } { | \\log ( \\rho a ^ 2 | \\log ( \\rho a ^ 2 ) | ^ { - 1 } ) | } . \\end{align*}"}
+{"id": "8027.png", "formula": "\\begin{align*} f ( x ) = \\int _ { \\R _ { + } ^ { n } } { \\mathcal { F } } _ { \\alpha } ( f ) ( \\lambda ) \\prod _ { i = 1 } ^ { n } j _ { \\alpha _ { i } } ( \\lambda _ { i } x _ { i } ) d \\mu _ { \\alpha } ( \\lambda ) ; x \\in \\R _ { + } ^ { n } . \\end{align*}"}
+{"id": "3249.png", "formula": "\\begin{align*} \\varepsilon _ { _ \\ell } = \\left [ \\mathbb { 1 } _ { [ \\ ! [ 1 , p ] \\ ! ] } ( \\ell ) - \\mathbb { 1 } _ { [ \\ ! [ p + 1 , n ] \\ ! ] } ( \\ell ) \\right ] . \\end{align*}"}
+{"id": "8806.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb { R } ^ { d } } f ( x ) f ( x ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } f _ { i } ( x ) . \\end{align*}"}
+{"id": "1743.png", "formula": "\\begin{align*} \\rho _ \\epsilon ( \\xi ) : = 2 ^ { \\epsilon _ + + \\epsilon _ - } ( 1 + \\epsilon _ + \\cos ( \\xi ) ) ( 1 - \\epsilon _ - \\cos ( \\xi ) ) ( \\epsilon _ \\pm \\in \\{ 0 , 1 \\} ) . \\end{align*}"}
+{"id": "733.png", "formula": "\\begin{align*} R _ 1 ( \\mu , S ) = \\norm { \\mathbf u ^ \\mu - \\mathbf u } _ { \\mathbf X ^ { \\mathbf s , b } ( 0 , S ) } + \\norm { ( 1 - P _ \\mu ) \\mathbf u } _ { \\mathbf X ^ { \\mathbf s , b } ( 0 , T ) } \\to 0 , \\end{align*}"}
+{"id": "1642.png", "formula": "\\begin{align*} \\frac { \\alpha } { \\sinh ^ 2 ( r / 2 ) } - \\frac { \\beta } { \\cosh ^ 2 ( r / 2 ) } = \\frac { \\alpha - \\beta } { \\sinh ^ 2 ( r / 2 ) } + \\frac { 4 \\beta } { \\sinh ^ 2 ( r ) } \\end{align*}"}
+{"id": "4050.png", "formula": "\\begin{align*} \\begin{gathered} \\sum _ { j = 0 } ^ { k / 2 - 1 } ( - 1 ) ^ { j } U _ { 1 , 1 } ( 2 j a + t ) + \\sum _ { j = 1 } ^ { k / 2 } ( - 1 ) ^ { j } U _ { 1 , 1 } ( 2 j a - t ) = 0 \\ ; ( 0 , a ) , \\\\ \\sum _ { j = 0 } ^ { k / 2 - 1 } ( - 1 ) ^ { j } V _ { 1 , 1 } ( 2 j a + t ) = \\sum _ { j = 1 } ^ { k / 2 } ( - 1 ) ^ { j } V _ { 1 , 1 } ( 2 j a - t ) \\ ; ( 0 , a ) ; \\end{gathered} \\end{align*}"}
+{"id": "5775.png", "formula": "\\begin{align*} \\lim _ { L \\rightarrow \\infty } { \\mathbb E } \\langle ( R _ { 1 , 2 } ^ p - { \\mathbb E } \\langle R _ { 1 , 2 } ^ p \\rangle ) ^ 2 \\rangle = 0 , \\end{align*}"}
+{"id": "6937.png", "formula": "\\begin{align*} \\left \\lVert \\Psi _ { y } - K \\Psi _ { x } \\right \\rVert _ F ^ 2 , \\end{align*}"}
+{"id": "6030.png", "formula": "\\begin{align*} G _ { n } = \\sum _ { i = n } ^ { m _ { n } } \\gamma _ { i } ^ { n } \\times F _ { i } \\in \\mathcal { S } ^ d , \\end{align*}"}
+{"id": "498.png", "formula": "\\begin{align*} d X = A X d t + \\left ( \\mathcal N ( X ) + \\frac 1 2 \\sum _ k \\mathcal M _ k [ \\mathcal M _ k ( X ) ] \\right ) \\ , d t + \\mathcal M ( X ) d W , \\end{align*}"}
+{"id": "5508.png", "formula": "\\begin{align*} O _ A ^ { ( b ^ { - 1 } ) } = \\rho O _ A ^ { ( 1 ) } + \\frac { \\rho - \\rho ^ { - 1 } } { \\big ( Q ^ 2 - Q ^ { - 2 } \\big ) ^ 2 } O _ B ^ { ( 1 ) } O _ B ^ { ( 1 ) } O _ A ^ { ( 1 ) } , \\qquad \\textrm { w h e r e } \\rho = \\mathrm { p e x p } \\left ( - \\frac { Q ^ 8 - Q ^ { - 8 } } { ( 1 + p ^ { - 1 } ) ( 1 + s ^ { - 1 } ) ( 1 + p s ) } \\right ) . \\end{align*}"}
+{"id": "8391.png", "formula": "\\begin{align*} f ( s ) = \\frac { 2 e ^ G } { s } \\log ( s - 1 ) , F ( s ) = \\frac { 2 e ^ G } { s } ~ ~ 2 \\leq s \\leq 3 , \\end{align*}"}
+{"id": "6123.png", "formula": "\\begin{align*} \\phi _ f ( \\sigma ) = - \\sigma \\log n _ 0 + \\log | a _ { n _ 0 } | , \\quad \\sigma > \\sigma _ 0 . \\end{align*}"}
+{"id": "2358.png", "formula": "\\begin{align*} x = \\frac { 1 } { o ( G ) f ( \\tau ) } \\sum _ { \\lambda \\in G \\times \\widehat { G } } f ( \\pi ( \\lambda ) ^ { - 1 } x ) \\pi ( \\lambda ) \\tau , \\forall x \\in \\mathbb { C } ^ { o ( G ) } . \\end{align*}"}
+{"id": "6273.png", "formula": "\\begin{align*} N ( u , \\partial u ) = h ( u ) ( \\partial u ) ^ 2 . \\end{align*}"}
+{"id": "5425.png", "formula": "\\begin{align*} \\Xi _ j ^ s ( \\xi ) : = \\sum _ { a / q \\in \\Sigma _ { s } } \\eta _ { j ^ { \\tau } } ( \\xi - a / q ) . \\end{align*}"}
+{"id": "7949.png", "formula": "\\begin{align*} D = \\Big ( D _ 1 \\cup D _ 2 \\cup A \\Big ) \\setminus ( A _ 1 \\cup A _ 2 ) . \\end{align*}"}
+{"id": "3176.png", "formula": "\\begin{align*} \\abs { \\delta _ k } \\leq { } 1 , \\ , \\lambda _ k \\neq { } 0 , \\ , \\ , \\lambda _ k \\begin{bmatrix} x _ k & y _ k \\end{bmatrix} = \\begin{bmatrix} \\delta _ k & 1 \\end{bmatrix} \\begin{bmatrix} a & c \\\\ b & d \\end{bmatrix} \\begin{bmatrix} \\lambda _ k & 0 \\\\ 0 & 1 \\end{bmatrix} . \\end{align*}"}
+{"id": "6024.png", "formula": "\\begin{align*} \\mathbb { E } \\langle D F , D G \\rangle _ \\mathcal { H } = \\mathbb { E } ( F L G ) = \\mathbb { E } ( G L F ) , \\end{align*}"}
+{"id": "1264.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } E ( u _ n ) = \\sum _ { j = 1 } ^ J \\lim _ { n \\to \\infty } E ( e ^ { t _ n ^ j \\Delta } \\phi ^ j ) + \\lim _ { n \\to \\infty } E ( W _ n ^ J ) . \\end{align*}"}
+{"id": "5489.png", "formula": "\\begin{align*} \\widehat S : = \\widehat D _ A \\widehat D _ B \\widehat D _ A . \\end{align*}"}
+{"id": "4727.png", "formula": "\\begin{align*} 0 & = \\liminf _ { m } \\Big ( \\int _ 0 ^ 1 \\partial _ { e ^ m x _ 1 } \\tilde u _ { k _ m } ( s z _ { l _ m , k _ m } + ( 1 - s ) z _ { k _ m ( m ) } ) d s \\Big ) \\\\ & \\ge ( t ( R ) - t ( \\alpha ) ) e _ n a - T t ( \\alpha ) , \\end{align*}"}
+{"id": "3538.png", "formula": "\\begin{align*} w ( r , t ) & = w ( r _ { 0 } ( t ) , t ) - \\int _ { r } ^ { r _ { 0 } ( t ) } w _ { r } ( \\rho , t ) d \\rho \\\\ & \\leqslant K + \\int _ { r _ { 0 } ( t ) } ^ { r } \\vert w _ { r } ( \\rho , t ) \\vert d \\rho \\\\ & \\leqslant K + r _ { 0 } ^ { 1 - N } ( t ) \\int _ { r _ { 0 } ( t ) } ^ { r } \\rho ^ { N - 1 } \\vert w _ { r } ( \\rho , t ) \\vert d \\rho \\\\ & \\leqslant K + C K r _ { 0 } ^ { 1 - N } ( t ) . \\end{align*}"}
+{"id": "5998.png", "formula": "\\begin{align*} p ( x , D ) q ( x , D ) & = I + \\Psi ^ { - \\infty } ( \\mathbb { R } ^ n ) \\\\ \\tilde { q } ( x , D ) p ( x , D ) & = I + \\Psi ^ { - \\infty } ( \\mathbb { R } ^ n ) \\end{align*}"}
+{"id": "7649.png", "formula": "\\begin{align*} \\Big \\| \\frac { k } { \\| Z ^ \\ast \\| _ { L _ { \\mu } ^ p ( \\R ^ d ) } } | Z ^ \\ast | \\Big \\| _ { L _ { \\mu } ^ p ( \\R ^ d ) } = k , \\end{align*}"}
+{"id": "8189.png", "formula": "\\begin{align*} g = \\bar { g } + V \\sum _ { i = 1 } ^ 3 \\theta _ i ^ 2 + \\frac { 1 } { V } \\eta ^ 2 . \\end{align*}"}
+{"id": "3714.png", "formula": "\\begin{align*} 2 \\alpha \\int _ { G _ 0 } \\frac { x _ 1 } { | x | ^ { 2 + 2 \\alpha } } d x = 2 \\int _ 0 ^ 1 \\frac { d x _ 2 } { ( x _ 2 ^ 2 + 1 ) ^ \\alpha } = 2 f ( 1 ) . \\end{align*}"}
+{"id": "2421.png", "formula": "\\begin{align*} y ^ 2 = x ( x - 1 ) \\prod _ { i = 1 } ^ { 2 g - 1 } ( x - \\lambda _ i ) \\end{align*}"}
+{"id": "5256.png", "formula": "\\begin{align*} & \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ n ( G _ 1 + G _ 2 \\| q _ { i , t + 1 } \\| ) \\| \\epsilon ^ z _ { i , t } \\| \\le \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ n \\Big ( \\frac { 2 G _ 1 ^ 2 \\alpha _ { t } } { \\sigma } + \\frac { 2 G _ 2 ^ 2 \\alpha _ { t } \\| q _ { i , t + 1 } \\| ^ 2 } { \\sigma } + \\frac { \\sigma \\| \\epsilon ^ z _ { i , t } \\| ^ 2 } { 4 \\alpha _ { t } } \\Big ) . \\end{align*}"}
+{"id": "8128.png", "formula": "\\begin{align*} \\| u \\| _ { \\dot \\Lambda ^ { \\gamma } } = \\sup _ { x , y \\in \\R ^ N , x \\not = y } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { \\gamma } } . \\end{align*}"}
+{"id": "2556.png", "formula": "\\begin{align*} \\int _ { \\operatorname { s u p p } \\varphi } | \\Gamma _ n | ^ { 1 + \\vartheta } d x & \\leq \\left \\| \\int _ { \\mathbb { R } ^ N } \\kappa _ \\varphi ( x , y ) | v _ n ( y ) | ^ { p _ { r ; s } ^ { \\uparrow * } } d y \\right \\| _ { L ^ \\sigma _ x ( \\operatorname { s u p p } \\varphi ) } ^ { 1 + \\vartheta } \\\\ & \\phantom { = } \\times \\| | \\varphi v _ n | ^ { p _ { r ; s } ^ { \\uparrow * } } \\| _ { \\ell _ r } ^ { 1 + \\vartheta } \\leq C _ 5 . \\end{align*}"}
+{"id": "1960.png", "formula": "\\begin{align*} \\ell + 1 = \\deg _ { x _ j } ( { \\bf x } _ F u ) + \\deg _ { y _ j } ( { \\bf x } _ F u ) > \\deg _ { x _ j } ( { \\bf x } _ D v ) + \\deg _ { y _ j } ( { \\bf x } _ D v ) = \\ell . \\end{align*}"}
+{"id": "2927.png", "formula": "\\begin{align*} J _ 0 ^ 4 = 3 , J _ 1 ^ 4 = 6 , J _ 2 ^ 4 = 6 , J _ 4 ^ 4 = 3 , J _ 4 ^ 4 = 1 \\end{align*}"}
+{"id": "390.png", "formula": "\\begin{align*} \\tilde \\beta \\circ f ^ * \\eta _ { f _ * A } = \\epsilon _ A \\end{align*}"}
+{"id": "3611.png", "formula": "\\begin{align*} \\begin{aligned} & { R _ { { \\rm { D B } } } } \\\\ & \\approx \\frac { 1 } { { { M _ { \\rm { R } } } } } { \\log _ 2 } \\left ( { 1 + \\frac { { E { \\sum \\limits _ { m = 1 } ^ { { M _ { \\rm { R } } } } { \\sum \\limits _ { k = 1 } ^ K { \\sum \\limits _ { n = 1 } ^ K { { \\left | { \\left [ { { { \\bf { \\Xi } } _ { { \\rm { D B } } , m } } } \\right ] _ { k , k } ^ * { { \\left [ { { { \\bf { \\Xi } } _ { { \\rm { D B } } , m } } } \\right ] } _ { n , n } } } \\right | } } } } } } } { { K { \\sigma ^ 2 } } } } \\right ) \\end{aligned} \\end{align*}"}
+{"id": "897.png", "formula": "\\begin{align*} \\frac { c } { x } \\bigg ( - \\tau _ t + \\frac { 1 - \\alpha } { t } \\tau \\bigg ) + m x ^ k \\phi _ u - t ^ { 1 - \\alpha } \\xi _ t - ( 2 \\phi _ { x v } - \\xi _ { x x } ) + \\frac { c } { x ^ 2 } \\xi + \\frac { c } { x } \\xi _ x - n x ^ k \\eta _ v = 0 , \\end{align*}"}
+{"id": "1337.png", "formula": "\\begin{align*} \\mathrm { l c t } ( X , \\Delta ; M ) : = \\sup \\{ t \\in \\mathbb { Q } \\ , | \\ , ( X , \\Delta + t M ) \\ , \\textrm { i s s u b l c } \\} , \\end{align*}"}
+{"id": "7490.png", "formula": "\\begin{align*} A _ \\lambda = \\begin{pmatrix} 0 & - \\lambda \\\\ 1 & 1 + \\lambda \\end{pmatrix} , \\end{align*}"}
+{"id": "3259.png", "formula": "\\begin{align*} \\mathbb { C } _ \\mu = s p a n \\{ 1 , \\mu \\} = \\{ a + b \\mu \\ ; , a , b \\in \\mathbb { R } \\} \\end{align*}"}
+{"id": "8100.png", "formula": "\\begin{align*} \\left \\| T ( f ) \\right \\| _ { L _ { \\omega } ^ { p } } \\leq C _ { 1 } \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } + C _ { 2 } \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } . \\end{align*}"}
+{"id": "3311.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } ( \\phi _ Y ) _ { _ \\Sigma } ( t ) = \\frac { 2 \\mu \\lambda _ { _ \\Sigma } t - 2 \\lambda _ { _ \\Sigma } ^ 2 } { ( \\lambda _ { _ \\Sigma } - \\mu t ) ^ 4 } . \\end{align*}"}
+{"id": "868.png", "formula": "\\begin{align*} \\begin{aligned} | u ( z ) - u ( w ) | & \\leq \\left | u ( z ) - m _ u ( B ( z , R ) ) \\right | + \\left | u ( w ) - m _ u ( B ( w , R ) ) \\right | \\\\ & + \\left | m _ u ( B ( z , R ) ) - m _ u ( B ( w , R ) ) \\right | \\\\ & \\leq 2 \\widetilde { A } [ u ] _ { W ^ { \\alpha , G } _ { s } ( X , d , \\mu ) } d ( z , w ) ^ { \\alpha - \\frac { s } { p _ { 0 } } } \\\\ & + \\left | m _ u ( B ( z , R ) ) - m _ u ( B ( w , R ) ) \\right | . \\end{aligned} \\end{align*}"}
+{"id": "2595.png", "formula": "\\begin{align*} X \\to \\cdots \\to \\tau _ { \\le n } X \\to \\tau _ { < n } X \\to ^ \\sim \\tau _ { \\le n - 1 } X \\to \\cdots \\to \\tau _ { < 1 } X \\to ^ \\sim \\tau _ { \\le 0 } X = \\pi _ 0 X , \\end{align*}"}
+{"id": "3442.png", "formula": "\\begin{align*} u ( x ) = \\sup _ { p \\in \\Delta ^ \\vee } \\langle x , p \\rangle - \\tilde { u } ( p ) , \\end{align*}"}
+{"id": "7976.png", "formula": "\\begin{align*} \\det \\begin{bmatrix} \\partial _ x \\phi , \\ \\partial _ a \\phi , \\ \\partial _ b \\phi \\\\ \\partial _ x \\partial _ { \\theta } \\phi , \\ \\partial _ a \\partial _ { \\theta } \\phi , \\ \\partial _ b \\partial _ { \\theta } \\phi \\\\ \\partial _ x \\partial ^ 2 _ { \\theta } \\phi , \\ \\partial _ a \\partial ^ 2 _ { \\theta } \\phi , \\ \\partial _ b \\partial ^ 2 _ { \\theta } \\phi \\end{bmatrix} \\neq 0 \\end{align*}"}
+{"id": "3706.png", "formula": "\\begin{align*} I ( b ) = \\frac { 2 ^ { 1 - 2 \\alpha } - 2 } { 1 - 2 \\alpha } - 2 ^ { - \\alpha } ( f ( 1 ) + \\mu _ \\alpha ) + 2 b ^ { 1 - 2 \\alpha } \\left ( f \\left ( \\frac 1 b \\right ) + f ( 1 ) - \\frac { 2 ^ { - 1 - \\alpha } } { 1 - 2 \\alpha } + 2 ^ { - 1 - \\alpha } \\mu _ \\alpha \\right ) \\end{align*}"}
+{"id": "6018.png", "formula": "\\begin{align*} A _ { q , \\kappa , p } ^ { \\delta , \\eta } ( h ) = \\frac { \\delta ^ { q } } { \\eta ^ { 2 q } } + \\eta ^ { - p } h ^ { p } + \\eta ^ { \\kappa } , h > 0 . \\end{align*}"}
+{"id": "5141.png", "formula": "\\begin{align*} { \\rm A o I } = \\limsup _ { T \\to \\infty } \\frac { 1 } { T } E \\bigg [ \\int _ { 0 } ^ { T } \\Delta _ t d t \\bigg ] . \\end{align*}"}
+{"id": "7633.png", "formula": "\\begin{align*} A _ { n } : = \\left \\{ w \\in \\R ^ d : \\ ; | w | = 1 , \\ \\mathbb { P } ^ * \\left ( \\langle w , X \\rangle < - \\frac 1 n \\right ) < \\frac 1 n \\right \\} , n _ { 0 } : = \\inf \\{ n \\geq 1 : A _ { n } = \\emptyset \\} \\end{align*}"}
+{"id": "1049.png", "formula": "\\begin{align*} u _ \\eta ^ * \\in [ 0 , u ] , \\ ; u _ \\eta ^ * \\not = 0 . \\end{align*}"}
+{"id": "8464.png", "formula": "\\begin{align*} F _ { \\ell } : = \\left \\{ x = ( z , w ) \\in \\mathbb { R } \\times \\mathbb { R } ^ { n - 1 } : | w | < r _ { \\ell } ( z ) \\right \\} ; \\end{align*}"}
+{"id": "8663.png", "formula": "\\begin{align*} x _ i \\to x _ i ' = \\sum _ { j = 1 } ^ n a _ { i j } x _ j , x _ { n + i } \\to x _ { n + i } ' = \\sum _ { j = 1 } ^ n ( a ^ { - 1 } ) _ { j i } x _ { n + j } \\forall i = 1 , \\dots , n \\end{align*}"}
+{"id": "1597.png", "formula": "\\begin{align*} \\pi _ { 1 } ( F ) = \\sup \\{ d \\in [ 0 , 1 ] & : \\mu > 0 \\ n _ 0 \\in \\mathbb { N } , \\ F \\\\ & ( d , \\mu , 1 ) ~ 3 H \\ | V ( H ) | \\geq n _ 0 \\} . \\end{align*}"}
+{"id": "3265.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { R } } \\mathcal { F } ^ \\mu ( f ) ( \\xi ) e ^ { - \\mu \\xi x } d \\xi . \\end{align*}"}
+{"id": "1263.png", "formula": "\\begin{align*} u _ n ( 0 ) = \\sum _ { j = 1 } ^ { J } g _ n ^ j [ e ^ { t _ n ^ j \\Delta } \\phi ^ j ] + W _ n ^ J \\end{align*}"}
+{"id": "4962.png", "formula": "\\begin{align*} | A _ 1 | = | A _ 5 | = | B _ 7 | = | B _ { 1 1 } | , ~ | A _ 7 | = | A _ { 1 1 } | = | B _ 1 | = | B _ 5 | . \\end{align*}"}
+{"id": "7923.png", "formula": "\\begin{align*} & & d _ 1 ^ 2 & = 1 + n _ 1 d _ 1 + n _ 2 d _ 2 \\\\ \\Rightarrow & & 1 & = ( d _ 1 - n _ 1 ) d _ 1 + ( - n _ 2 ) d _ 2 \\end{align*}"}
+{"id": "7239.png", "formula": "\\begin{align*} G = \\log \\left ( \\frac { \\sqrt { \\Delta } \\pm \\sqrt { \\Delta - 1 + R ^ 2 } } { 1 - R } \\right ) ^ 2 . \\end{align*}"}
+{"id": "6696.png", "formula": "\\begin{align*} { } _ 1 F _ 1 ( a ; b ; z ) \\sim { e ^ z z ^ { a - b } \\over \\Gamma ( a ) } \\sum _ { s = 0 } ^ \\infty { ( 1 - a ) _ s ( b - a ) _ s \\over s ! } z ^ { - s } + { e ^ { \\pi i a } z ^ { - a } \\over \\Gamma ( b - a ) } \\sum _ { s = 0 } ^ \\infty { ( a ) _ s ( a - b + 1 ) _ s \\over s ! } ( - z ) ^ { - s } , z = i x \\end{align*}"}
+{"id": "7762.png", "formula": "\\begin{align*} r ^ 2 = | Z ^ 0 | ^ 2 \\frac { e ^ { - \\mathcal { K } } } { 2 } . \\end{align*}"}
+{"id": "7242.png", "formula": "\\begin{align*} R = \\frac { \\pm ( \\delta ^ 2 + 2 \\Delta - 2 \\Delta ^ 2 ) } { \\sqrt { 4 \\Delta ^ 4 + 4 \\Delta ^ 2 \\delta ^ 2 + \\delta ^ 4 - 8 \\Delta ^ 3 + 4 \\Delta \\delta ^ 2 + 4 \\Delta ^ 2 } } . \\end{align*}"}
+{"id": "1008.png", "formula": "\\begin{align*} \\langle \\rho ( \\cdot , t ) , \\varphi \\rangle = \\int _ { \\Omega } \\rho _ 0 ( x , t ) \\ , \\varphi ( x ) \\ , d x + \\int _ { \\Omega } \\varphi ( x ) \\ , d \\mu _ t ( x ) = \\int _ { \\Omega } \\rho _ 0 ( x , t ) \\ , \\varphi ( x ) d x + \\int _ { \\Gamma } \\varphi ( x ) \\ , d \\mu _ t ( x ) , \\end{align*}"}
+{"id": "120.png", "formula": "\\begin{align*} I ( d ) : = - \\frac { N } { 2 ( 2 \\pi ) ^ d } \\int _ { \\mathbb { R } ^ d } \\Big ( \\mathcal { A } _ k - \\sqrt { \\mathcal { A } _ k ^ 2 - \\widehat { g } _ k ^ 2 } - \\frac { N + 1 } { \\lvert \\Lambda \\rvert } G _ d ( k ) \\Big ) \\mathrm { d } k . \\end{align*}"}
+{"id": "8395.png", "formula": "\\begin{align*} \\Sigma _ { j } = \\Sigma _ { j } ^ { ( 1 ) } + \\O \\left ( \\Sigma _ { j } ^ { ( 2 ) } \\right ) , \\end{align*}"}
+{"id": "5538.png", "formula": "\\begin{align*} N ( s , t ) = \\frac { 1 } { \\pi } \\Im \\left ( \\frac { \\eta ' ( t ) } { \\eta ( t ) - \\eta ( s ) } \\right ) . \\end{align*}"}
+{"id": "4391.png", "formula": "\\begin{align*} \\mathbb { B } ' ( \\hat { { \\mathbf U } } ^ { + } , \\hat { { \\mathbf U } } ^ { - } , \\hat { \\varphi } ) ( { \\mathbf U } ^ { + } , { \\mathbf U } ^ { - } , \\varphi ) = \\left [ \\begin{array} { c } \\partial _ t \\varphi + \\hat { u } ^ + _ 2 \\partial _ 2 \\varphi - u ^ + _ N \\\\ \\partial _ t \\varphi + \\hat { u } ^ - _ 2 \\partial _ 2 \\varphi - u ^ - _ N \\\\ q ^ + - q ^ - \\end{array} \\right ] , \\end{align*}"}
+{"id": "68.png", "formula": "\\begin{align*} \\mathcal K _ H ^ { \\rm { d i a g } } = \\sum _ { k \\in \\mathcal P _ H } \\mathcal D _ k b _ k ^ \\dagger b _ k . \\end{align*}"}
+{"id": "2203.png", "formula": "\\begin{align*} E ( A , B ) = O \\left ( \\frac { 2 ^ { ( 2 k - 3 ) n } \\cdot \\# A \\cdot \\# B } { 2 ^ { ( 2 k - 1 ) n } } \\right ) = O \\left ( \\frac { \\# A \\cdot \\# B } { 2 ^ { 2 n } } \\right ) . \\end{align*}"}
+{"id": "320.png", "formula": "\\begin{align*} - \\left \\langle \\left [ u \\right ] _ { \\operatorname * { N } ; j } ^ { \\operatorname * { e x t } } \\left ( s \\right ) , \\gamma _ { \\operatorname * { D } ; j } \\left ( s \\right ) \\overline { w } \\right \\rangle _ { \\Gamma _ { j } } = \\left \\langle \\varphi , \\gamma _ { \\operatorname * { D } ; j } \\left ( s \\right ) \\overline { w } \\right \\rangle _ { \\Gamma _ { j } } \\quad \\forall w \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) . \\end{align*}"}
+{"id": "587.png", "formula": "\\begin{align*} \\Delta _ 2 ^ \\mu ( t ) = \\Delta _ { 2 , 1 } ^ \\mu ( t ) + \\Delta _ { 2 , 2 } ^ \\mu ( t ) , \\end{align*}"}
+{"id": "359.png", "formula": "\\begin{align*} \\psi \\ ; \\phi _ { R ( h ) } = \\varphi _ { S ( \\psi ( h ) ) } \\ ; \\psi . \\end{align*}"}
+{"id": "7326.png", "formula": "\\begin{align*} \\Gamma ( L , \\lambda ) [ u ] = \\langle \\dot { L } _ \\lambda u , u \\rangle , u \\in \\ker ( L _ \\lambda ) , \\end{align*}"}
+{"id": "6110.png", "formula": "\\begin{align*} \\sigma ( T ) = \\Big \\{ g ( T ) : g \\in \\mathcal { P } _ I ( T ) \\ \\Big \\} , \\end{align*}"}
+{"id": "5491.png", "formula": "\\begin{align*} O _ A ^ { ( 1 ) } : = \\left ( \\begin{array} { c c c } - \\mathrm i \\big ( Q ^ 2 - Q ^ { - 2 } \\big ) & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & i \\big ( Q ^ 2 - Q ^ { - 2 } \\big ) \\end{array} \\right ) \\end{align*}"}
+{"id": "937.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\mathcal { T } _ t ^ \\alpha u = h ( x , t ) u _ { x x } + f _ 1 ( x , t ) u _ x + g _ 1 ( x , t ) v _ x , \\\\ \\mathcal { T } _ t ^ \\alpha v = h ( x , t ) v _ { x x } + f _ 2 ( x , t ) v _ x + g _ 2 ( x , t ) u _ x , \\end{aligned} \\right . \\end{align*}"}
+{"id": "2272.png", "formula": "\\begin{align*} Z \\leq \\left [ \\min _ { 0 \\leq x \\leq 1 } \\sum _ { j = 1 } ^ { n } \\theta \\left ( x - \\frac { j } { n } \\right ) \\right ] - C e ^ { - \\pi \\alpha \\left ( \\frac { n - 1 } { 2 } \\right ) ^ 2 } \\left ( \\sum _ { j = 1 } ^ { n } \\varepsilon _ j ^ 2 \\right ) ^ { 1 / 2 } \\end{align*}"}
+{"id": "5578.png", "formula": "\\begin{align*} c ( G ) \\leq | S | + c ( G - S ) = O ( n ^ { 1 - \\alpha } ) \\end{align*}"}
+{"id": "6797.png", "formula": "\\begin{align*} \\lambda _ { 1 } & = \\frac { 1 } { 2 d } \\left ( c - \\sqrt { c ^ 2 - 4 d s } \\right ) , \\lambda _ { 3 } = \\frac { 1 } { 2 } \\left ( c + \\sqrt { c ^ { 2 } - 4 f ( 1 ) p ' ( 1 ) } \\right ) , \\\\ [ 0 . 2 c m ] \\lambda _ { 2 } & = \\frac { 1 } { 2 d } \\left ( c + \\sqrt { c ^ 2 - 4 d s } \\right ) , \\lambda _ { 4 } = \\frac { 1 } { 2 } \\left ( c - \\sqrt { c ^ { 2 } - 4 f ( 1 ) p ' ( 1 ) } \\right ) . \\end{align*}"}
+{"id": "4950.png", "formula": "\\begin{align*} t _ k ( n ) = [ q ^ { 2 n + k + 1 } ] H _ k ( \\tau ) + [ q ^ { 2 n + k + 1 } ] T _ k ( \\tau ) , \\end{align*}"}
+{"id": "1632.png", "formula": "\\begin{align*} q = Q ^ 2 = Q '^ 2 A '^ 2 , a = Q ^ 2 A ^ { - 1 } = Q '^ 2 A ' , t = T ^ 2 Q ^ { - 2 } = T '^ 2 Q '^ { - 2 } A '^ { - 2 } . \\end{align*}"}
+{"id": "6970.png", "formula": "\\begin{align*} \\sum _ { n \\in \\Z ^ d } | D u ( n ) | ^ 2 = 4 \\int _ { Q _ d } | \\Delta \\psi ( x ) | ^ 2 \\omega ( x ) d x , \\end{align*}"}
+{"id": "3592.png", "formula": "\\begin{align*} y = \\left \\| { { \\bf { H } } { { \\bf { f } } _ { { \\rm { B F } } } } } \\right \\| s + { \\bf w } ^ H _ { \\rm B F } { \\bf n } . \\end{align*}"}
+{"id": "6976.png", "formula": "\\begin{align*} - 2 M \\le w ( x ) : = u ( x ) - 2 M \\le - M \\end{align*}"}
+{"id": "3266.png", "formula": "\\begin{align*} 2 \\pi \\| f \\| _ 2 = \\| \\mathcal { F } ^ \\mu ( f ) \\| _ 2 . \\end{align*}"}
+{"id": "7999.png", "formula": "\\begin{align*} \\begin{cases} p = \\infty & m = 4 , \\\\ p > 2 & m = 5 , \\\\ p \\geq 2 & m \\geq 6 , \\end{cases} \\end{align*}"}
+{"id": "5862.png", "formula": "\\begin{align*} D _ { N } ^ { ( V ) } ( T ) = \\sup _ { 1 \\leq \\nu < \\infty } \\Big [ \\max _ { \\mathcal { N } : \\# \\mathcal { N } = V } E _ { \\nu , \\mathcal { N } } ( T - \\nu + 1 | T \\geq \\nu ) \\Big ] , \\end{align*}"}
+{"id": "3704.png", "formula": "\\begin{align*} A ^ \\pm _ x : = \\{ y \\in D ^ + \\ , | \\ , \\pm K _ 2 ( x , y ) > 0 \\} . \\end{align*}"}
+{"id": "7212.png", "formula": "\\begin{align*} v ^ N ( t , \\mathbf { x } ^ N ) & : = \\mathbb { P } \\bigl ( \\forall s \\in [ 0 , t ] , \\Psi ( \\hat { \\mu } _ s ^ N ) < 0 \\bigr ) . \\end{align*}"}
+{"id": "7432.png", "formula": "\\begin{align*} 0 = & ( \\kappa _ 1 + \\kappa _ 2 + 2 - p ) \\| ( y _ 1 , y _ 2 ) \\| _ { 1 , p } + ( \\kappa _ 1 + \\kappa _ 2 + 2 - q ) \\| ( \\nabla { y _ 1 } , \\nabla y _ 2 ) \\| _ { q , \\eta } \\\\ & - ( \\kappa _ 1 + \\kappa _ 2 + \\nu + 1 ) \\int _ { \\Omega } \\left [ a _ 1 \\vert y _ 1 \\vert ^ { 1 - \\nu } + a _ 2 \\vert y _ 2 \\vert ^ { 1 - \\nu } \\right ] d z . \\end{align*}"}
+{"id": "8408.png", "formula": "\\begin{align*} v = \\frac { h } { d } , 1 \\leq d \\leq D , 1 \\leq h \\leq H . \\end{align*}"}
+{"id": "3573.png", "formula": "\\begin{align*} { { \\bf { H } } _ { { \\rm { T } } , k } } = \\sum \\nolimits _ { l \\in { \\mathcal L } _ { { \\rm T } } } { { \\alpha _ { { \\rm { T } } , k , l } } { { \\bf { a } } _ { { \\rm { S } } , k } } \\left ( { \\Theta _ { k , l } ^ { \\rm { A } } } \\right ) { \\bf { a } } _ { \\rm { T } } ^ H \\left ( { \\Theta _ { k , l } ^ { \\rm { D } } } \\right ) } , \\end{align*}"}
+{"id": "6239.png", "formula": "\\begin{align*} S = L ^ \\infty _ t L ^ 2 _ x \\cap L ^ 4 _ t L ^ \\infty _ x , \\end{align*}"}
+{"id": "8218.png", "formula": "\\begin{align*} ( \\Phi _ a ^ * \\omega _ 2 ) ( X , Y ) = \\omega _ 2 ( X , Y ) + a ^ 2 | T | ^ 2 \\omega _ 2 ( X _ 0 , b _ 1 I _ 1 T ) + a ^ 2 | T | ^ 2 \\omega _ 2 ( a _ 1 I _ 1 T , Y _ 0 ) . \\end{align*}"}
+{"id": "438.png", "formula": "\\begin{align*} & _ 2 \\tilde F _ 1 \\left ( a , b ; c , z \\right ) = \\frac { \\Gamma ( c ) \\Gamma ( c - a - b ) } { \\Gamma ( c - a ) \\Gamma ( c - b ) } \\cdot \\ , _ 2 \\tilde F _ 1 \\left ( a , b ; a + b - c + 1 , 1 - z \\right ) \\\\ & + \\frac { \\Gamma ( c ) \\Gamma ( a + b - c ) ( 1 - z ) ^ { c - a - b } } { \\Gamma ( a ) \\Gamma ( b ) } \\cdot \\ , _ 2 \\tilde F _ 1 \\left ( c - a , c - b ; c - a - b + 1 , 1 - z \\right ) \\end{align*}"}
+{"id": "2902.png", "formula": "\\begin{align*} L _ \\omega = \\mathcal { J } L _ { \\omega ; 1 } \\mathcal { J } ^ { - 1 } . \\end{align*}"}
+{"id": "8427.png", "formula": "\\begin{align*} Y _ { i i } ^ \\star = \\psi ( \\lambda _ i ) = \\max \\Big \\{ \\frac { 1 - \\lambda _ i } { \\lambda _ i } , 0 \\Big \\} , \\end{align*}"}
+{"id": "7253.png", "formula": "\\begin{align*} c ( I , x ) = - \\log \\left | f ' _ I ( x ) \\right | . \\end{align*}"}
+{"id": "4789.png", "formula": "\\begin{align*} F _ R ( u ; \\sigma , \\nu ) { = } \\exp { \\left ( \\ ! - \\frac { \\nu ^ 2 } { 2 \\sigma ^ 2 } \\ ! \\right ) } \\sum _ { k _ 1 = 0 } ^ \\infty \\frac { \\psi ^ { 2 k _ 1 } } { 4 ^ { k _ 1 } k _ 1 ! \\beta ^ { k _ 1 } } \\ , { - } \\exp { \\left ( - \\frac { u ^ 2 } { 2 \\sigma ^ 2 } \\right ) } \\sum _ { k _ 1 = 0 } ^ \\infty \\sum _ { k = 1 } ^ { k _ 1 + 1 } \\frac { \\psi ^ { 2 k _ 1 } u ^ { 2 ( k _ 1 { + } 1 { - } k ) } } { 4 ^ { k _ 1 } k _ 1 ! ( k _ 1 { - } k { + } 1 ) ! \\beta ^ { k - 1 } } . \\end{align*}"}
+{"id": "459.png", "formula": "\\begin{align*} \\exp \\left ( - \\frac { b ^ 2 } { 4 p ^ 2 } \\right ) \\ , _ 1 \\tilde { F } _ 1 \\left ( \\frac { \\nu - \\mu + 2 } { 2 } ; \\nu + 1 ; \\frac { b ^ 2 } { 4 p ^ 2 } \\right ) = \\ , _ 1 \\tilde F _ 1 \\left ( \\frac { \\nu + \\mu } { 2 } ; \\nu + 1 ; - \\frac { b ^ 2 } { 4 p ^ 2 } \\right ) \\end{align*}"}
+{"id": "8249.png", "formula": "\\begin{align*} g _ 0 ( \\xi _ 3 , d \\psi _ { x _ 1 , 0 , 0 } ( Y ) ) = \\omega _ 3 ( \\rho \\frac { \\partial } { \\partial \\rho } , d \\psi _ { x _ 1 , 0 , 0 } ( Y ) ) . \\end{align*}"}
+{"id": "5386.png", "formula": "\\begin{align*} S ^ { - 1 } ( t ) = ( S ^ { - 1 } _ { j k } ( t ) ) _ { 1 \\leq j , k \\leq N } = ( \\L _ 1 ( t ) \\L _ 2 ( t ) \\cdots \\L _ N ( t ) ) ^ { \\rm t } = ( L _ { j k } ( t ) ) _ { 1 \\leq j , k \\leq N } , t \\geq 0 , \\end{align*}"}
+{"id": "7159.png", "formula": "\\begin{align*} [ H _ 1 , H _ 2 ] = 0 , \\frac { \\partial H _ 1 } { \\partial t _ { 2 } } - \\frac { \\partial H _ 2 } { \\partial t _ { 1 } } = 0 , \\end{align*}"}
+{"id": "1219.png", "formula": "\\begin{align*} \\Delta W + ( I _ \\alpha \\ast | \\cdot | ^ { - b } | W | ^ p ) | x | ^ { - b } | W | ^ { p - 2 } W = 0 . \\end{align*}"}
+{"id": "6939.png", "formula": "\\begin{align*} \\dot x = f + g u , \\end{align*}"}
+{"id": "7851.png", "formula": "\\begin{align*} K _ { \\nu } ( x ) = \\int _ { 0 } ^ { \\infty } d t \\exp ( - x \\cosh ( t ) ) \\cosh ( \\nu t ) , \\nu = 0 , 1 , 2 , . . . \\end{align*}"}
+{"id": "6786.png", "formula": "\\begin{align*} & h = \\lambda e ^ 2 / 2 , \\\\ [ 0 . 2 c m ] & 0 < \\beta < \\min \\left \\{ c , \\ \\lambda \\right \\} , \\\\ [ 0 . 2 c m ] & \\sigma > \\max \\left \\{ e ^ { 2 \\beta / \\lambda } , \\ \\frac { f ( 1 ) h q ( 1 ) } { \\left ( c - \\beta \\right ) \\left ( \\lambda - \\beta \\right ) \\beta e } \\right \\} , \\\\ [ 0 . 2 c m ] & r > \\max \\left \\{ h \\sqrt { 2 / \\lambda } , \\ \\frac { 4 s h ^ 2 q ( 1 ) } { d q ( 0 ) } \\left ( \\frac { 7 } { 2 e \\lambda } \\right ) ^ { 7 / 2 } \\right \\} , \\end{align*}"}
+{"id": "8769.png", "formula": "\\begin{align*} \\max _ { \\omega \\in \\Omega } \\mathbf { E } _ { \\omega , T } \\big [ \\norm { z _ { T } - x ^ { * } _ { \\omega } } ^ { 2 } \\big ] \\geq 0 . 0 0 1 \\times d \\alpha ^ { - 2 } r ^ { 2 } h ^ { 2 \\beta - 2 } = 0 . 0 1 \\times r ^ { 2 } \\min \\Big ( 1 , \\ , \\frac { d } { \\alpha ^ { 2 } } T ^ { - \\frac { \\beta - 1 } { \\beta } } \\Big ) . \\end{align*}"}
+{"id": "361.png", "formula": "\\begin{align*} \\eta ( g _ 1 g _ 2 ) = \\ ; & S ( \\psi ( h _ { g _ 1 } \\phi _ { R ( h _ { g _ 1 } ) } ( h _ { g _ 2 } ) ) ) \\\\ = \\ ; & S ( \\psi ( h _ { g _ 1 } ) \\varphi _ { S ( \\psi ( h _ { g _ 1 } ) ) } ( \\psi ( h _ { g _ 2 } ) ) ) \\\\ = \\ ; & S ( \\psi ( h _ { g _ 1 } ) ) S ( \\psi ( h _ { g _ 2 } ) ) \\\\ = \\ ; & \\eta ( g _ 1 ) \\eta ( g _ 2 ) . \\end{align*}"}
+{"id": "8531.png", "formula": "\\begin{align*} D \\ell = D \\ell ^ { c } = D ^ { c } \\ell . \\end{align*}"}
+{"id": "3669.png", "formula": "\\begin{align*} \\bar { \\lambda } _ 0 = 0 < \\frac { 1 } { 2 } < \\bar { \\lambda } _ 1 \\leq \\dots \\to \\infty \\ , , \\end{align*}"}
+{"id": "8733.png", "formula": "\\begin{align*} \\Big ( \\sum _ { t = 1 } ^ { T } \\eta _ t \\Big ) ^ { - 1 } = \\frac { 1 } { \\min \\big ( \\frac { T } { 1 6 \\kappa \\bar { L } d } , T \\gamma \\big ) } = \\max \\Big ( \\frac { 1 8 \\kappa \\bar { L } d } { T } , \\frac { 1 } { T \\gamma } \\Big ) \\leq \\frac { 1 6 \\kappa \\bar { L } d } { T } + \\frac { 1 } { T \\gamma } . \\end{align*}"}
+{"id": "2518.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\big ( k \\omega \\sigma \\ , \\boldsymbol { y } _ k \\cdot \\boldsymbol { v } _ k ^ { \\perp } + \\nu \\ , \\textbf { c u r l } \\ , \\boldsymbol { y } _ k \\cdot \\textbf { c u r l } \\ , \\boldsymbol { v } _ k \\big ) \\ , d \\boldsymbol { x } = \\int _ { \\Omega } \\boldsymbol { u } _ k \\cdot \\boldsymbol { v } _ k \\ , d \\boldsymbol { x } , \\end{align*}"}
+{"id": "182.png", "formula": "\\begin{align*} & \\left \\{ e ^ { \\frac { 1 } { 2 4 } A } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T _ C M } - \\widetilde { L _ C } - 1 6 + W _ i + W _ j ) - e ^ { \\frac { 1 } { 2 4 } A } A \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 4 ) } \\\\ & = - 2 4 \\left \\{ e ^ { \\frac { 1 } { 2 4 } A } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 4 ) } . \\end{align*}"}
+{"id": "6268.png", "formula": "\\begin{align*} N _ \\lambda ^ { 2 , l o } u _ \\lambda ^ { x _ 0 } = \\lambda L ( P _ { \\lambda } g ( u _ { < \\lambda } ) , \\partial u _ { < \\lambda } , u _ \\lambda ) , \\end{align*}"}
+{"id": "8257.png", "formula": "\\begin{align*} f _ { \\zeta , 0 , 0 } ( 1 , \\Theta ) = \\sum _ { \\nu \\geq 0 } k _ \\nu ( \\Theta ) \\zeta ^ \\nu , \\end{align*}"}
+{"id": "5147.png", "formula": "\\begin{align*} \\lim _ { D \\to 0 } H [ Q _ { { \\mathrm { u n i } } } ( X ) ] + \\log _ 2 \\sqrt { 1 2 D } = h ( X ) \\end{align*}"}
+{"id": "5285.png", "formula": "\\begin{align*} \\mathbf { n } = ( n _ 0 , n _ 1 , \\dots , n _ { L + 1 } ) \\in \\mathbb { N } ^ { L + 2 } . \\end{align*}"}
+{"id": "7047.png", "formula": "\\begin{align*} \\Delta _ { \\gamma } = \\sum _ { i = 1 } ^ { m } X _ { i } ^ { 2 } + \\sum _ { j = 1 } ^ { k } Y _ { j } ^ { 2 } = \\triangle _ { x } + | x | ^ { 2 \\gamma } \\triangle _ { y } , \\end{align*}"}
+{"id": "8055.png", "formula": "\\begin{align*} M f ( x ) = \\sup \\limits _ { x \\in Q } \\frac { 1 } { \\vert Q \\vert } \\int _ { Q } f ( u ) d u . \\end{align*}"}
+{"id": "6294.png", "formula": "\\begin{align*} h _ 1 ( \\xi , \\eta ) = h _ 3 ( \\xi + \\eta , ( \\xi - \\eta ) ^ 2 ) . \\end{align*}"}
+{"id": "2307.png", "formula": "\\begin{align*} & g ^ { \\vee } ( x ) : = \\frac { 1 } { 2 \\pi } \\int _ { \\R } e ^ { i x \\xi } g ( \\xi ) d \\xi , \\\\ & g ^ { \\vee } ( x , t ) : = \\frac { 1 } { ( 2 \\pi ) ^ 2 } \\int _ { \\R \\times \\R } e ^ { i ( x \\xi + t \\tau ) } g ( \\xi , \\tau ) d \\xi d \\tau . \\end{align*}"}
+{"id": "8623.png", "formula": "\\begin{align*} \\mathcal { B } [ \\beta b ] \\bullet & = b \\nabla _ X ( \\nabla _ X \\cdot ( b \\bullet ) ) + h _ b \\nabla _ X \\big { ( } b \\nabla _ X \\cdot ( b \\bullet ) ) + 2 h _ b ( \\nabla _ X b ) \\nabla _ X \\cdot ( b \\bullet ) , \\end{align*}"}
+{"id": "4129.png", "formula": "\\begin{align*} C ( k , s , S _ k ) \\sum _ { i = 0 } ^ { k - 1 } \\sum _ { \\ell = 0 } ^ { k - 2 i } U ( k , s , i , \\ell ) = 0 , \\end{align*}"}
+{"id": "7074.png", "formula": "\\begin{align*} \\mathcal { L } _ 1 ^ { \\pm } ( n ^ 2 _ 1 n _ 2 , u , q , p _ 1 ) = d { \\pi } ^ { 3 } \\int _ { 0 } ^ \\infty V _ 1 ^ { \\pm } ( z ) z ^ { - 1 / 3 + i \\nu } \\ , e \\left ( \\frac { N u z } { p _ 1 q Q } \\pm \\frac { 3 ( N z n _ 1 ^ 2 n _ 2 ) ^ { 1 / 3 } } { p _ 1 q r ^ { 1 / 3 } } \\right ) d z . \\end{align*}"}
+{"id": "534.png", "formula": "\\begin{align*} \\mathfrak K f ( x ) = \\int _ { \\R ^ d } \\mathfrak k ( x - y ) f ( y ) d y . \\end{align*}"}
+{"id": "5289.png", "formula": "\\begin{align*} J = \\{ ( j _ 1 , \\dots , j _ L ) \\in \\mathbb { Z } ^ L : 0 \\leq j _ l \\leq \\min \\{ n _ 0 , n _ 1 - j _ 1 , \\dots , n _ { l - 1 } - j _ { l - 1 } , n _ l \\} \\} . \\end{align*}"}
+{"id": "7157.png", "formula": "\\begin{align*} \\frac { d f } { d t _ { 1 } } = [ H _ { 1 } , f ] + \\frac { \\partial f } { \\partial t _ { 1 } } , f = x _ { 1 } , y _ { 1 } , x _ { 2 } , y _ { 2 } \\end{align*}"}
+{"id": "2296.png", "formula": "\\begin{align*} 4 8 = 6 4 - d _ { 1 } ( 8 - d _ { 1 } ) = n _ { 2 } + 4 n _ { 3 } + 5 t _ { 5 } . \\end{align*}"}
+{"id": "4324.png", "formula": "\\begin{align*} \\omega ( A ) = ( \\Omega _ \\omega , A \\Omega _ \\omega ) \\ \\ ( A \\in \\mathcal { N } _ \\omega ) \\ , . \\end{align*}"}
+{"id": "5210.png", "formula": "\\begin{align*} ( I \\cap \\widetilde { w } ^ { - 1 } I \\widetilde { w } ) \\backslash I = \\left \\{ \\begin{pmatrix} 1 & \\alpha _ 0 & \\delta _ 0 + \\delta _ 1 v \\\\ 0 & 1 & \\beta _ 0 \\\\ 0 & 0 & 1 \\end{pmatrix} \\right \\} \\end{align*}"}
+{"id": "2687.png", "formula": "\\begin{align*} \\det \\left ( \\rho ( F ( X , Y ) \\right ) & = \\det \\begin{pmatrix} f ( \\lambda ) & g ( \\lambda ^ { 2 ^ { n - 2 } - 1 } ) \\\\ g ( \\lambda ) & f ( \\lambda ^ { 2 ^ { n - 2 } - 1 } ) \\end{pmatrix} \\\\ & = f ( \\lambda ) f ( \\lambda ^ { 2 ^ { n - 2 } - 1 } ) - g ( \\lambda ) g ( \\lambda ^ { 2 ^ { n - 2 } - 1 } ) : = k ( \\lambda ) . \\end{align*}"}
+{"id": "5504.png", "formula": "\\begin{align*} O _ B ^ { ( g ) } : = \\widehat S ^ { - 1 } O _ A ^ { ( \\sigma ( g ) ) } \\widehat S \\end{align*}"}
+{"id": "7413.png", "formula": "\\begin{align*} \\pi _ { n } ( \\rho _ { n } ) = \\frac { \\gamma _ { n } } { \\gamma _ { n } + 1 } \\rho _ { n } ^ { \\gamma _ { n } + 1 } . \\end{align*}"}
+{"id": "6652.png", "formula": "\\begin{align*} R ( x ) - { 1 \\over \\pi } \\mathop { \\sim } \\limits _ { x \\to \\infty } { 1 \\over \\pi } \\sum _ { n = 1 } ^ \\infty { \\pi ^ { 2 n } c _ { 2 n } \\over x ^ { 2 n } } . \\end{align*}"}
+{"id": "6787.png", "formula": "\\begin{align*} \\mathcal { U } \\left ( \\underline { u } , \\overline { v } \\right ) = & e ^ { \\beta z } \\left [ \\beta \\sigma \\left ( c - \\beta \\right ) - f ( 1 - \\sigma e ^ { \\beta z } ) q ( 1 ) e ^ { ( \\lambda _ 1 - \\beta ) z } \\right ] + f ( 1 - \\sigma e ^ { \\beta z } ) p ( 1 - \\sigma e ^ { \\beta z } ) \\\\ [ 0 . 2 c m ] \\geq & e ^ { \\beta z } \\left [ \\beta \\sigma \\left ( c - \\beta \\right ) - f ( 1 ) q ( 1 ) \\right ] > 0 \\end{align*}"}
+{"id": "5326.png", "formula": "\\begin{align*} g ^ { ( m - 2 ) } = \\max \\{ 0 , g ^ { ( m - 3 ) } ( x ) + 2 ^ { - 2 m + 3 } ( h ^ { ( m - 1 ) } ( x ) - 1 ) + 1 \\} - 1 . \\end{align*}"}
+{"id": "5947.png", "formula": "\\begin{align*} u _ j ( x ) & = \\int _ M \\left ( - \\Delta _ g G _ M ^ \\omega ( x , y ) + _ y ( F ( y ) G _ M ^ \\omega ( x , y ) ) - \\omega ^ 2 G _ M ^ \\omega ( x , y ) \\right ) u _ j ( y ) d \\mu _ g ( y ) , \\\\ & = ( \\lambda _ j - \\omega ^ 2 ) \\sum _ { k = 1 } ^ \\infty \\int _ M v _ k ( x ) u _ k ( y ) u _ j ( y ) e ^ { \\phi ( y ) } d \\mu _ g ( y ) , \\\\ & = ( \\lambda _ j - \\omega ^ 2 ) v _ j ( x ) \\int _ M | u _ j ( y ) | ^ 2 e ^ { \\phi ( y ) } d \\mu _ g ( y ) . \\end{align*}"}
+{"id": "3844.png", "formula": "\\begin{align*} g = e ^ { - 2 \\lambda _ 1 z } d x ^ 2 + e ^ { - 2 \\lambda _ 2 z } d y ^ 2 + d z ^ 2 , \\end{align*}"}
+{"id": "1718.png", "formula": "\\begin{align*} P _ { \\texttt { a } ; 0 } ( \\boldsymbol { \\xi } ; q ) = P _ { \\texttt { b } ; 0 } ( \\boldsymbol { \\xi } ; q , q _ 0 ) = \\prod _ { 1 \\leq j < k \\leq n } \\frac { 1 - q ^ { 1 + k - j } } { 1 - q ^ { k - j } } = \\prod _ { 1 \\leq j \\leq n } \\frac { 1 - q ^ j } { 1 - q } . \\end{align*}"}
+{"id": "6092.png", "formula": "\\begin{align*} \\beta _ k ^ { P R P - F R } = m a x \\{ 0 , m i n \\{ \\beta ^ { P R P } , \\beta ^ { F R } \\} \\} \\end{align*}"}
+{"id": "2439.png", "formula": "\\begin{align*} \\frac { 1 } { | \\Omega | } \\int _ { \\Omega } \\exp \\left ( \\frac { \\alpha } { \\ln N } \\rho ( x ) \\right ) { \\rm d } x & \\geq \\exp \\left ( \\frac { \\alpha } { | \\Omega | \\ln N } \\int _ { \\Omega } \\rho ( x ) { \\rm d } x \\right ) \\\\ & = \\exp \\left ( \\frac { \\alpha } { 4 \\pi } \\left ( 1 + o ( 1 ) _ { N \\to \\infty } \\right ) \\right ) . \\end{align*}"}
+{"id": "171.png", "formula": "\\begin{align*} Q ( M , P _ i , P _ j , \\tau ) = & \\left \\{ e ^ { \\frac { 1 } { 2 4 } E _ 2 ( \\tau ) A } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } \\left [ \\bigotimes _ { n = 1 } ^ { \\infty } S _ { q ^ n } ( \\widetilde { T _ C M } ) \\otimes \\bigotimes _ { m = 1 } ^ { \\infty } \\wedge _ { - q ^ m } ( \\widetilde { L _ C } ) \\right ] \\right . \\\\ & \\left . \\cdot \\varphi ( \\tau ) ^ { 1 6 } { \\rm c h } ( \\mathcal { V } _ i ) { \\rm c h } ( \\mathcal { V } _ j ) \\right \\} ^ { ( 1 4 ) } . \\end{align*}"}
+{"id": "2099.png", "formula": "\\begin{align*} | | \\tilde \\alpha _ 0 | | _ { \\infty } < \\frac { \\gamma } { 2 } , | \\alpha _ 0 ^ { ( n ) } ( 2 u ) | \\leq \\frac { K _ 1 \\gamma } { \\varphi ^ { 3 / 4 } ( u ) } , n = 1 , 2 , \\end{align*}"}
+{"id": "419.png", "formula": "\\begin{align*} V _ { \\ell , m } : = \\{ [ u ] _ { \\ell , m } : \\ , u \\in L ^ 2 ( \\R _ + , r ^ { d - 1 + 2 \\ell } d r ) \\} , \\ell \\in L _ d , \\ ; m \\in M _ \\ell , \\end{align*}"}
+{"id": "5483.png", "formula": "\\begin{align*} \\widetilde G _ \\alpha ( Q , p , s ) = H _ { p , s } ( h _ { p , s } ( Q ) ) \\qquad \\textrm { f o r a l l } Q \\in \\mathbb C ^ \\times . \\end{align*}"}
+{"id": "2334.png", "formula": "\\begin{align*} \\pi _ g \\Delta _ g x & = \\pi _ g \\left ( \\sum _ { h \\in G } f _ { g h } ( x ) \\tau _ { h } \\right ) = \\sum _ { h \\in G } f _ { g h } ( x ) \\pi _ g \\tau _ { h } = \\sum _ { h \\in G } f _ { g h } ( x ) \\sum _ { u \\in G } f _ u ( \\tau _ h ) \\tau _ { g u } \\\\ & = \\sum _ { h \\in G } f _ { g h } ( x ) \\sum _ { u \\in G } f _ { g u } ( \\tau _ { g h } ) \\tau _ { g u } = \\sum _ { h \\in G } f _ { g h } ( x ) \\tau _ { g h } = x , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "1304.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta ) u _ { n } ^ J + | x | ^ { - b } F ( u _ n ^ J ) = | x | ^ { - b } [ F ( \\sum _ { j = 1 } ^ J v _ n ^ j ) - \\sum _ { j = 1 } ^ J F ( v _ n ^ j ) ] . \\end{align*}"}
+{"id": "8904.png", "formula": "\\begin{align*} \\Lambda _ N : = \\begin{pmatrix} 2 \\lambda _ { N } & \\lambda _ { N } \\circ \\gamma \\\\ \\lambda _ { N } \\circ \\gamma & 2 \\lambda _ { N } \\end{pmatrix} . \\end{align*}"}
+{"id": "1991.png", "formula": "\\begin{align*} ( X , Y ) _ { \\theta , q } : = \\left \\{ \\ , x \\in X + Y \\ , \\Big { | } \\ , t \\longmapsto t ^ { - \\theta } K ( t , x , X , Y ) \\in \\mathrm { L } ^ q _ \\ast ( \\mathbb { R } _ + ) \\ , \\right \\} \\end{align*}"}
+{"id": "1406.png", "formula": "\\begin{gather*} \\tilde \\psi _ { n 0 } = \\chi _ n \\left ( \\varphi _ { n , 0 } - \\varphi _ { n , 1 } + \\sum _ k ( \\tilde Q _ { n , 0 ; k , 0 } \\varphi _ { k , 0 } - \\tilde Q _ { n , 0 ; k , 1 } \\varphi _ { k , 1 } - \\tilde Q _ { n , 1 ; k , 0 } \\varphi _ { k , 0 } + \\tilde Q _ { n , 1 ; k , 1 } \\varphi _ { k , 1 } ) \\right ) , \\\\ \\tilde \\psi _ { n 1 } = \\varphi _ { n , 1 } + \\sum _ k ( \\tilde Q _ { n , 1 ; k , 0 } \\varphi _ { k , 0 } - \\tilde Q _ { n , 1 ; k , 1 } \\varphi _ { k , 1 } ) . \\end{gather*}"}
+{"id": "4532.png", "formula": "\\begin{align*} \\dot u ^ \\pm _ n : = \\dot u _ 1 ^ \\pm - \\partial _ 2 \\hat \\Psi ^ \\pm \\dot u ^ \\pm _ 2 \\ , , \\quad \\dot H ^ \\pm _ n : = \\dot H _ 1 ^ \\pm - \\partial _ 2 \\hat \\Psi ^ \\pm \\dot H ^ \\pm _ 2 \\ , ; \\end{align*}"}
+{"id": "7577.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\exp \\left ( - K \\theta ^ 2 \\right ) d \\theta = \\frac { \\sqrt { \\pi } } { 2 \\sqrt { K } } . \\end{align*}"}
+{"id": "5915.png", "formula": "\\begin{align*} & L _ { 1 } = \\langle 1 \\rangle ^ { \\perp n - 2 } \\perp \\langle 1 , - \\Delta \\rangle , L _ { 2 } = \\langle 1 \\rangle ^ { \\perp n - 2 } \\perp \\langle \\pi , - \\Delta \\pi \\rangle , \\\\ & L _ { 3 } = \\langle 1 \\rangle ^ { \\perp n - 2 } \\perp \\langle - 1 , - \\Delta \\pi \\rangle , L _ { 4 } = \\langle 1 \\rangle ^ { \\perp n - 2 } \\perp \\langle - \\Delta , - \\pi \\rangle . \\end{align*}"}
+{"id": "2690.png", "formula": "\\begin{align*} U _ 1 & = \\alpha _ 0 ^ 2 - \\alpha _ 1 ^ 2 + \\alpha _ 2 ^ 2 - \\alpha _ 3 ^ 2 , \\\\ V _ 1 & = \\alpha _ 0 \\alpha _ 1 + \\alpha _ 0 \\alpha _ 3 - \\alpha _ 1 \\alpha _ 2 + \\alpha _ 2 \\alpha _ 3 . \\end{align*}"}
+{"id": "5693.png", "formula": "\\begin{align*} \\| ( z - T ) ^ { - 1 } \\| \\leq \\frac { 1 } { | z | } + \\sum \\limits _ { k = 1 } ^ { \\infty } \\frac { \\| T ^ k \\| } { | z | ^ { k + 1 } } = \\frac { 1 } { | z | } + \\sum \\limits _ { k = 1 } ^ { \\infty } \\frac { \\prod \\limits _ { j = 1 } ^ { k } w _ { j } } { | z | ^ { k + 1 } } . \\end{align*}"}
+{"id": "5918.png", "formula": "\\begin{align*} \\alpha _ { i } = \\min \\{ ( R _ { i + 1 } - R _ { i } ) / 2 + e , R _ { i + 1 } - R _ { i } + d [ - a _ { i , i + 1 } ] \\} \\end{align*}"}
+{"id": "999.png", "formula": "\\begin{align*} \\langle \\nabla \\times { \\bf G } , \\varphi \\rangle = \\left ( \\langle G _ 2 , \\varphi _ { x _ 3 } \\rangle - \\langle G _ 3 , \\varphi _ { x _ 2 } \\rangle \\right ) { \\bf i } - \\left ( \\langle G _ 1 , \\varphi _ { x _ 3 } \\rangle - \\langle G _ 3 , \\varphi _ { x _ 1 } \\rangle \\right ) { \\bf j } + \\left ( \\langle G _ 1 , \\varphi _ { x _ 2 } \\rangle - \\langle G _ 2 , \\varphi _ { x _ 1 } \\rangle \\right ) { \\bf k } . \\end{align*}"}
+{"id": "3545.png", "formula": "\\begin{align*} t _ { 0 } \\leqslant \\frac { 1 } { \\left ( \\frac { 1 - \\theta } { \\theta } \\right ) \\left ( \\left ( \\frac { 2 \\theta } { 1 - \\theta } 2 ^ { \\frac { \\theta + 1 } { \\theta } } C _ { 1 } ^ { \\frac { 1 } { \\theta } } \\right ) ^ { \\frac { \\theta } { 1 - \\theta } } K ^ { \\frac { 2 } { 1 - \\theta } } \\right ) ^ { \\frac { 1 - \\theta } { \\theta } } 2 ^ { - \\frac { \\theta + 1 } { \\theta } } C _ { 1 } ^ { - \\frac { 1 } { \\theta } } K ^ { - \\frac { 2 } { \\theta } } } = \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "8659.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ 0 f \\Big ( \\frac { x _ 1 } { x _ 0 } , . . . , \\frac { x _ n } { x _ 0 } \\Big ) . \\end{align*}"}
+{"id": "5007.png", "formula": "\\begin{align*} - 4 \\int _ { 0 } ^ { R } \\phi ( r ) \\phi ' ( r ) \\int _ { B _ p ( r ) } v ^ 2 = 2 \\int _ { B _ p ( R ) } v ^ 2 \\phi ^ 2 ( r ) \\end{align*}"}
+{"id": "4313.png", "formula": "\\begin{align*} p ' = \\hat { \\zeta } p \\hat { \\zeta } ^ { - 1 } \\end{align*}"}
+{"id": "64.png", "formula": "\\begin{align*} \\sum _ { i \\neq j } ( P _ i \\overline { Q } _ { L , j } g Q _ i Q _ j + h . c . ) & = \\sum _ { i \\neq j } P _ i \\overline { Q } _ { L , j } g \\left [ Q _ i Q _ j + \\omega ( P _ i P _ j + P _ i Q _ j + Q _ i P _ j ) \\right ] + h . c . \\\\ & - \\sum _ { i \\neq j } P _ i \\overline { Q } _ { L , j } g \\omega ( P _ i P _ j + P _ i Q _ j + Q _ i P _ j ) + h . c . \\end{align*}"}
+{"id": "5009.png", "formula": "\\begin{align*} h _ { 1 1 } n H - \\sum _ { \\alpha = 1 } ^ n h _ { 1 \\alpha } ^ 2 \\geq - | h | ^ 2 + \\dfrac { n ^ 2 ( 5 - n ) H ^ 2 } { 4 } . \\end{align*}"}
+{"id": "5875.png", "formula": "\\begin{align*} D _ { N } ^ { ( V ) } ( T ) = \\sup _ { 1 \\leq \\nu < \\infty } \\Big [ \\max _ { \\mathcal { N } : \\# \\mathcal { N } = V } E _ { \\nu , \\mathcal { N } } ( T - \\nu + 1 | T \\geq \\nu ) \\Big ] . \\end{align*}"}
+{"id": "4305.png", "formula": "\\begin{align*} \\begin{alignedat} { 3 } & i j = k j k & & = i & & k i = j \\\\ & j i = - k k j & & = - i & & i k = - j \\\\ & i ^ 2 = j ^ 2 = k ^ 2 & & = & & i j k = - 1 \\end{alignedat} \\end{align*}"}
+{"id": "6081.png", "formula": "\\begin{align*} X _ j = \\nabla f _ j ( \\omega _ k ) , Y _ j = \\nabla f _ j ( \\phi _ k ) \\end{align*}"}
+{"id": "4415.png", "formula": "\\begin{align*} \\mathcal { B } _ 0 ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) \\partial _ t { \\mathbf V } + \\mathcal { B } _ 1 ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) \\partial _ 1 { \\mathbf V } + \\mathcal { B } _ 2 ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) \\partial _ 2 { \\mathbf V } + \\mathcal { B } _ 3 ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) { \\mathbf V } = \\tilde { \\mathcal { F } } ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) , \\end{align*}"}
+{"id": "8399.png", "formula": "\\begin{align*} \\Sigma _ { j } \\ll \\frac { N ^ { 2 \\gamma - 1 } } { ( \\log N ) ^ { 2 } } + \\sum _ { d \\leq D } \\sum _ { h \\leq H } \\frac { 1 } { h } \\left | W \\left ( \\frac { h } { d } \\right ) \\right | , j = 0 , 1 . \\end{align*}"}
+{"id": "5524.png", "formula": "\\begin{align*} \\max _ { z \\in K } \\left | \\frac { \\partial P } { \\partial z _ m } ( z ) \\right | \\le M \\deg ( P ) \\max _ { z \\in K } | P ( z ) | \\ \\mbox { f o r } z = ( z _ 1 , \\dots , z _ n ) . \\end{align*}"}
+{"id": "8472.png", "formula": "\\begin{align*} E _ { 1 } \\subset _ { \\mathcal { H } ^ { n } } E _ { 2 } \\mbox { i f } \\mathcal { H } ^ { n } ( E _ { 1 } \\backslash E _ { 2 } ) = 0 , \\end{align*}"}
+{"id": "6204.png", "formula": "\\begin{align*} \\mathcal H ^ 1 ( \\varphi ( [ 0 , 1 ] ) ) = \\int _ 0 ^ 1 \\norm { d \\varphi ( t ) } _ 2 \\d t . \\end{align*}"}
+{"id": "3231.png", "formula": "\\begin{align*} S _ { \\alpha } ( \\epsilon , R ) = \\textnormal { A r e a } ( \\textnormal { S e c t } _ { \\alpha , \\epsilon } ( R ) ) + O \\left ( R \\right ) , \\end{align*}"}
+{"id": "2243.png", "formula": "\\begin{align*} h _ 3 ( n , \\{ ( 4 , 0 ) , ( 4 , 1 ) , ( 4 , 3 ) \\} ) = \\begin{cases} \\frac { n } { 2 } & \\\\ \\lceil \\frac { n + 1 } { 2 } \\rceil & \\end{cases} \\end{align*}"}
+{"id": "3155.png", "formula": "\\begin{align*} \\pi _ 1 ( Y \\backslash D _ Y ) & = H _ 1 ( Y \\backslash D _ Y , \\mathbb { Z } ) = H ^ { 2 n - 1 } ( Y , D _ Y , \\mathbb { Z } ) \\\\ & = ( H ^ { 2 n - 2 } ( Y , \\mathbb { Z } ) \\rightarrow H ^ { 2 n - 2 } ( D _ Y , \\mathbb { Z } ) ) . \\end{align*}"}
+{"id": "7729.png", "formula": "\\begin{align*} \\delta \\varpi ( a _ 1 , a _ 2 ) = ( \\delta a _ 1 ) \\cdot a _ 2 + ( \\delta a _ 2 ) \\cdot a _ 1 , ( \\delta a ) \\cdot h = \\varpi ( a , \\delta h ) , \\end{align*}"}
+{"id": "512.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\tau _ R ( \\omega ) = \\tau ( \\omega ) , \\end{align*}"}
+{"id": "8487.png", "formula": "\\begin{align*} | D g | ( \\Omega ) = \\sup \\left \\{ \\int _ { \\Omega } g ( x ) \\ \\mbox { d i v } \\ T ( x ) \\ d x : T \\in C _ { c } ^ { 1 } ( \\Omega ; \\mathbb { R } ^ { n } ) , \\ | T | \\leq 1 \\right \\} . \\end{align*}"}
+{"id": "2851.png", "formula": "\\begin{align*} \\mathcal { V } _ \\mu ( \\xi ) = \\frac { c } { 2 } \\langle \\xi , \\xi \\rangle - 2 \\pi \\langle \\hat \\rho + \\mu , \\xi \\rangle + \\sum _ { \\alpha \\in R ^ + _ 0 } \\frac { 2 } { \\langle \\alpha , \\hat \\alpha ^ \\vee \\rangle } \\int _ 0 ^ { \\langle \\xi , \\alpha \\rangle } v _ \\alpha ( ) , \\end{align*}"}
+{"id": "727.png", "formula": "\\begin{align*} \\widehat { P _ \\mu f } ( \\xi ) = \\theta \\left ( \\frac { \\xi } { \\mu } \\right ) \\widehat f ( \\xi ) . \\end{align*}"}
+{"id": "2835.png", "formula": "\\begin{align*} \\ell \\left ( y _ i \\right ) \\triangleq \\ell \\left ( c _ { i , k } \\vert y _ i \\right ) = \\ln { \\frac { \\sum _ { \\hat { x } \\in \\mathcal { K } _ { k , 1 } } p ( y _ i \\vert x = \\hat { x } ) } { \\sum _ { \\hat { x } \\in \\mathcal { K } _ { k , 0 } } p ( y _ i \\vert x = \\hat { x } ) } } , \\end{align*}"}
+{"id": "1758.png", "formula": "\\begin{gather*} \\zeta ( s , \\alpha ) = \\sum _ { n = 0 } ^ { \\infty } ( n + \\alpha ) ^ { - s } \\end{gather*}"}
+{"id": "2738.png", "formula": "\\begin{align*} U e _ i = \\left \\{ \\begin{array} { l l } \\varepsilon _ i e _ { \\pi ( i ) } , & 1 \\leq i \\leq \\theta ^ { - 1 } \\\\ \\varepsilon _ i e _ i , & i > \\theta ^ { - 1 } \\end{array} \\right . \\quad ( i \\in \\mathbb N ) , \\end{align*}"}
+{"id": "5322.png", "formula": "\\begin{align*} h ^ { ( n + 2 ) } ( 2 k 2 ^ { - n - 1 } ) = h ^ { ( n + 2 ) } ( ( 2 k + 2 ) 2 ^ { - n - 1 } ) = 1 \\end{align*}"}
+{"id": "8841.png", "formula": "\\begin{align*} \\tau _ { t } \\leq \\tau _ { t _ { 0 } } \\prod _ { i = { t _ { 0 } } } ^ { t - 1 } \\Big ( 1 - \\frac { 2 } { i } \\Big ) \\leq \\tau _ { t _ { 0 } } \\prod _ { i = { t _ { 0 } } } ^ { t - 1 } \\Big ( 1 - \\frac { 1 } { i } \\Big ) \\le \\frac { ( t _ { 0 } - 1 ) \\tau _ { t _ { 0 } } } { t } \\le \\frac { 2 ( t _ { 0 } - 1 ) \\delta _ { t _ { 0 } } } { t } \\enspace . \\end{align*}"}
+{"id": "2200.png", "formula": "\\begin{align*} E ( A , B ) = O \\left ( \\frac { 2 ^ n \\cdot | A | \\cdot | B | } { 2 ^ { 2 n } } \\right ) = O \\left ( \\frac { | A | \\cdot | B | } { 2 ^ { n } } \\right ) . \\end{align*}"}
+{"id": "6862.png", "formula": "\\begin{align*} \\vec { A } x = c \\end{align*}"}
+{"id": "1372.png", "formula": "\\begin{align*} \\varphi _ { 2 - 2 k , - m } ( z ) : = - \\mathcal { M } _ { 2 - 2 k , k } ( - 4 \\pi m y ) e ^ { - 2 \\pi i m x } . \\end{align*}"}
+{"id": "3397.png", "formula": "\\begin{align*} \\alpha _ i = \\dfrac { 1 } { v _ j \\phi _ j } h _ { i j } \\end{align*}"}
+{"id": "3490.png", "formula": "\\begin{align*} & \\psi _ t \\leq \\frac { 1 } { | \\log | t | | } \\varphi - \\frac { 1 } { | \\log | t | | } \\phi _ t - \\frac { 1 } { | \\log | t | | } \\phi _ h \\\\ & \\leq u ( y ) - u ( y ) + \\frac { C } { | \\log | t | | } + C \\epsilon \\leq C \\epsilon . \\end{align*}"}
+{"id": "6645.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty \\Big ( { \\rho } _ { ( 1 ) , \\infty } ^ { ( \\rm c J ) } ( x ; \\beta , p , q ) | _ { \\beta = 2 \\atop q = 0 } - 1 \\Big ) e ^ { i \\tau x } \\ , d x \\mathop { \\sim } \\limits _ { \\tau \\to 0 } \\pi \\sum _ { n = 1 } ^ \\infty { ( - 1 ) ^ n c _ { 2 n } \\over ( 2 n - 1 ) ! } | \\tau | ^ { 2 n - 1 } . \\end{align*}"}
+{"id": "2293.png", "formula": "\\begin{align*} & \\ , \\sum _ { j = 1 } ^ { n } \\theta _ \\alpha \\left ( x - \\frac { j } { n } + \\varepsilon _ j \\right ) = A ( x ) + g _ 1 ( x ) + g _ 2 ( x ) + h ( x ) \\\\ = & \\ , \\sum _ { j = 1 } ^ { n } \\theta _ \\alpha \\left ( x - \\frac { j } { n } \\right ) + \\mathcal { O } \\left ( n e ^ { - \\pi \\alpha n ^ 2 } \\sum _ { j = 1 } ^ { n } \\varepsilon _ j ^ 2 \\right ) + g _ 1 ( x ) + g _ 2 ( x ) + h ( x ) . \\end{align*}"}
+{"id": "2716.png", "formula": "\\begin{align*} ( 1 , 2 , \\dots , n - 1 ) = ( \\underbrace { \\underbrace { s _ { 1 } ^ 1 , \\dots , s _ { p } ^ 1 } _ { p } , \\dots , \\underbrace { s _ { 1 } ^ { d - q } , \\dots , s _ { p } ^ { d - q } } _ { p } } _ { d - q } , \\underbrace { \\underbrace { s _ { 0 } ^ { d - q + 1 } , \\dots , s _ { p } ^ { d - q + 1 } } _ { p + 1 } , \\dots , \\underbrace { s _ { 0 } ^ { d } , \\dots , s _ { p } ^ { d } } _ { p + 1 } } _ { q } ) . \\end{align*}"}
+{"id": "1937.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ t u + u \\ , \\partial _ x u - \\epsilon \\partial ^ 2 _ x u = \\rho V - \\rho u + G ( x , t ) , x \\in [ 0 , L ] , t > 0 , \\end{aligned} \\end{align*}"}
+{"id": "8782.png", "formula": "\\begin{align*} \\gamma = h _ { t } ^ { 2 \\beta } , \\quad \\quad \\gamma = \\frac { h _ { t } } { \\sqrt { T d } } . \\end{align*}"}
+{"id": "8902.png", "formula": "\\begin{align*} \\sigma ^ i \\circ \\varphi = \\varphi \\circ \\begin{pmatrix} 0 & - 1 \\\\ 1 & \\gamma \\end{pmatrix} . \\end{align*}"}
+{"id": "6552.png", "formula": "\\begin{align*} & \\ \\ \\ \\det S ( z ) \\\\ & = \\prod _ { i = ( k , n ) \\in B _ * ^ + } ( z + k \\cdot \\omega + \\mu _ n ) \\prod _ { i = ( k , n ) \\in B _ * ^ - } ( - z - k \\cdot \\omega + \\mu _ n ) \\\\ & \\ \\ \\ + O ( ( \\varepsilon + \\delta ) ^ { 3 / 4 } ) \\\\ & \\sim \\prod _ { l = 1 } ^ { \\# B _ * } ( z - \\sigma _ l ) + O ( ( \\varepsilon + \\delta ) ^ { 3 / 4 } ) . \\end{align*}"}
+{"id": "1971.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { p - 1 } \\omega _ q ^ { \\left ( a ^ t _ { \\sigma ^ j , r } - a ^ t _ { \\sigma ^ j , s } \\right ) - \\left ( a ^ t _ { \\sigma , r } - a ^ t _ { \\sigma , s } \\right ) } = \\sum _ { j = 1 } ^ { p - 1 } \\omega _ p ^ { j \\left ( { r } _ { \\pi _ \\beta ( 1 ) } - { s } _ { \\pi _ \\beta ( 1 ) } \\right ) } = - 1 . \\end{align*}"}
+{"id": "7162.png", "formula": "\\begin{align*} g _ j = f _ j V ^ { - 1 } , ~ \\omega _ j = V \\tau _ j , \\forall 1 \\leq j \\leq n . \\end{align*}"}
+{"id": "5722.png", "formula": "\\begin{align*} i ( 2 \\pi \\ , j + \\xi ) : = \\eta ^ { - 1 } \\ , ( B _ M ) _ { 1 , 1 } + \\ln ( \\gamma _ M ) \\ , . \\end{align*}"}
+{"id": "3133.png", "formula": "\\begin{align*} \\mathbf { H } ^ H _ { l , } \\mathbf { H } _ { l , } ( i , j ) = \\frac { 1 } { N ( \\kappa + 1 ) } \\sum _ { n = 1 } ^ N \\sum _ { n ' = 1 } ^ N g _ { n , i } ^ * g _ { n ' , j } \\sum _ { p = 1 } ^ { L _ l } \\sum _ { p ' = 1 } ^ { L _ l } \\alpha _ { k , p } ^ * \\alpha _ { l , p ' } e ^ { j ( a _ { l , p , n } - a _ { l , p , n ' } ) + j ( \\varphi _ { n ' } - \\varphi _ n ) } . \\end{align*}"}
+{"id": "1036.png", "formula": "\\begin{align*} \\hat { \\lambda } = \\frac { \\| N _ f ( \\overline { u } _ \\eta ) \\| _ \\infty } { m _ \\eta ^ { p _ + - 1 } } + 1 > 0 \\end{align*}"}
+{"id": "7850.png", "formula": "\\begin{align*} \\hat { N } : = \\{ ( \\tau _ 2 , b + \\mathrm { i } t , \\tau _ 1 , \\zeta ^ 1 , \\widetilde { \\zeta } _ 0 , \\widetilde { \\zeta } _ 1 , \\sigma ) \\in \\widetilde { N } \\ ; | \\ ; R ( t , \\tau ) \\geq 0 \\ ; \\} , \\end{align*}"}
+{"id": "7541.png", "formula": "\\begin{align*} ( \\nu _ { 1 } \\times \\nu _ { 2 } ) ^ * ( \\alpha \\boxtimes \\gamma \\boxtimes \\eta ) = s ^ * \\left ( \\left ( ( g _ { 1 } \\times g _ 2 ) ^ * ( s t _ 1 \\times s t _ 2 ) ^ * \\alpha \\right ) \\boxtimes \\gamma \\boxtimes \\eta \\right ) . \\end{align*}"}
+{"id": "3353.png", "formula": "\\begin{align*} S = \\begin{pmatrix} \\alpha _ 3 & \\alpha _ 2 & \\alpha _ 1 \\\\ \\alpha _ 2 & - \\alpha _ 1 & - \\alpha _ 3 \\\\ \\alpha _ 1 & - \\alpha _ 3 & \\alpha _ 2 \\end{pmatrix} , \\alpha _ k = \\sqrt { \\frac { 2 } { 7 } } \\sin \\frac { k \\pi } { 7 } \\end{align*}"}
+{"id": "7876.png", "formula": "\\begin{align*} C _ 0 ^ { t ' , t '' } ( \\mu ' , \\mu '' ) : = \\inf _ \\mu E _ 0 ^ { t ' , t '' } ( \\mu ) , \\end{align*}"}
+{"id": "5752.png", "formula": "\\begin{align*} { \\cal B } _ p : = \\{ X \\subset \\Lambda _ L | X = i + A _ p , i \\in \\Lambda _ L , A _ p \\in { \\cal A } _ p \\} . \\end{align*}"}
+{"id": "674.png", "formula": "\\begin{align*} [ u , v ] ( t ) = \\begin{cases} u ( t ) & t _ 0 < t < t _ 1 \\\\ v ( t ) & t _ 1 < t < t _ 2 \\end{cases} \\end{align*}"}
+{"id": "4633.png", "formula": "\\begin{align*} \\widetilde { \\Phi } _ i = \\sum _ { k _ { i - 1 } \\in \\mathcal { I } _ { e _ { i - 1 } } } \\sum _ { \\ell _ i \\in \\mathcal { I } _ { e _ { i } } } m _ { e _ i , \\ell _ i } ^ * \\otimes m _ { e _ { i - 1 } , k _ { i - 1 } } \\otimes \\bigl ( ( m _ { e _ { i - 1 } , k _ { i - 1 } } ^ * \\otimes \\textup { i d } _ { S _ i } ) \\Phi _ i ( m _ { e _ i , \\ell _ i } ) \\bigr ) \\end{align*}"}
+{"id": "2121.png", "formula": "\\begin{align*} \\rho ( t , x ) = \\sqrt { ( c _ 1 + f _ 0 ( x + t ) ) ( c _ 1 + f _ 0 ( x - t ) ) } . \\end{align*}"}
+{"id": "3550.png", "formula": "\\begin{align*} \\min _ x \\max _ { \\lambda } \\ ; L ( x , y , \\lambda ) = f ( x ) + g ( y ) - \\lambda ^ T ( A x + B y - b ) \\ , \\end{align*}"}
+{"id": "905.png", "formula": "\\begin{align*} t ^ { 1 - \\alpha } \\eta _ { 3 t } - \\eta _ { 3 x x } - \\frac { c } { x } \\eta _ { 3 x } - m x ^ k \\phi _ { 3 x } = 0 , \\end{align*}"}
+{"id": "5593.png", "formula": "\\begin{align*} d _ { p , \\mu } : = \\sup _ { \\underset { \\| u \\| _ { X ^ { 1 , p } _ \\infty } = 1 } { u \\in X ^ { 1 , p } _ \\infty } } \\int _ 0 ^ \\infty \\exp _ p ( \\mu | u | ^ { \\frac { p } { p - 1 } } ) r ^ { \\alpha _ 0 } \\mathrm d r \\end{align*}"}
+{"id": "2940.png", "formula": "\\begin{align*} 0 = \\delta _ { i _ 1 i _ 2 } [ \\sum _ { \\lambda = 1 , \\pi \\in S _ { n } } ^ { n ! } a \\delta _ { k \\pi ( i _ 1 } D _ { i _ 2 \\cdots i _ { n } ) } + \\sum _ { \\lambda = 1 , \\pi \\in S _ { n } } ^ { n ! } b \\delta _ { \\pi ( i _ 1 i _ 2 } D _ { i _ 3 \\cdots i _ n ) k } + \\sum _ { \\lambda = 1 , \\pi \\in S _ { n } } ^ { n ! } d \\epsilon _ { k s \\pi ( i _ 1 } D _ { i _ 2 \\cdots i _ n ) s } + f D _ { k i _ 1 \\dots i _ n } ] . \\end{align*}"}
+{"id": "273.png", "formula": "\\begin{align*} \\Phi ( i ) = \\begin{cases} \\{ j + 1 , i + 1 \\} \\enspace , \\enspace j < i , \\\\ \\{ j + 1 \\} \\enspace , \\enspace j = i , \\\\ \\{ j + 1 , i \\} \\enspace , \\enspace j > i . \\end{cases} \\end{align*}"}
+{"id": "8326.png", "formula": "\\begin{align*} { \\mathbf { s } } _ { \\mathbf { S B } } = { \\mathbf { s } } _ { \\mathbf { S B } } \\mathbf { P } _ { S M } + { \\mathbf { s } } _ { \\mathbf { S B } } { { \\mathbf { w } } } ^ { ' } { \\mathbf { r } } \\end{align*}"}
+{"id": "3949.png", "formula": "\\begin{align*} \\pi = ( t _ 0 , t _ 1 , \\ldots , t _ k ) \\end{align*}"}
+{"id": "6896.png", "formula": "\\begin{align*} k ( K _ Y + B ) \\cdot B \\cdot \\pi ^ * L = \\pi ^ * L \\cdot B \\cdot \\pi ^ * L = \\Delta \\cdot L ^ 2 = 0 , \\end{align*}"}
+{"id": "5916.png", "formula": "\\begin{align*} d [ c a _ { 1 , i } b _ { 1 , j } ] = \\min \\{ d ( c a _ { 1 , i } b _ { 1 , j } ) , \\alpha _ { i } , \\beta _ { j } \\} \\ , , c \\in F ^ { \\times } , \\end{align*}"}
+{"id": "8528.png", "formula": "\\begin{align*} r _ { \\ell } ^ { \\wedge } ( \\bar { z } ) \\geq | w ' - \\tau | - | w - w ' | + \\delta > | w ' - \\tau | - \\frac { \\delta } { 2 } + \\delta = | w ' - \\tau | + \\frac { \\delta } { 2 } , \\end{align*}"}
+{"id": "7795.png", "formula": "\\begin{align*} f = 8 \\pi \\tau _ 2 ^ 2 h ( t ) + 1 6 \\pi \\left ( \\frac { \\chi } { 1 9 2 \\pi } - c _ { \\ell } \\right ) + \\frac { \\tau _ 2 ^ 2 } { ( 2 \\pi ) ^ 2 } \\sum _ { \\hat { \\gamma } \\in \\Lambda ^ + \\cup \\{ 0 \\} } n _ { \\hat { \\gamma } } \\sum _ { ( m , n ) \\in \\mathbb { Z } ^ 2 - \\{ 0 \\} } \\left ( \\frac { 1 } { | m \\tau + n | } + 2 \\pi q _ a t ^ a \\right ) \\frac { e ^ { - S _ { \\hat { \\gamma } , m , n } } } { | m \\tau + n | ^ 2 } \\ , , \\end{align*}"}
+{"id": "3797.png", "formula": "\\begin{align*} { \\rm O T } ( \\beta _ 0 , \\beta _ 1 ) : = \\inf _ { \\alpha \\in \\Gamma ( \\beta _ 0 , \\beta _ 1 ) } ( H _ p , \\alpha ) , \\end{align*}"}
+{"id": "5124.png", "formula": "\\begin{align*} \\ < D ^ * _ { \\rho ( \\cdot ) } ( \\alpha ) , b \\ > = l ^ * ( i _ b d _ A \\alpha ) - i _ b d _ A ( l ^ * ( \\alpha ) ) \\ , \\ , \\ , \\ < \\alpha , D _ { \\rho _ * ( \\cdot ) } ( b ) \\ > = l ( i _ \\alpha d _ { A ^ * } b ) - i _ \\alpha d _ A ( l ( b ) ) . ) \\end{align*}"}
+{"id": "5050.png", "formula": "\\begin{align*} Q _ g \\big ( g _ x \\big ) = 0 , \\end{align*}"}
+{"id": "3672.png", "formula": "\\begin{align*} \\lambda _ 1 ( t ) = \\frac { 1 } { 2 \\ , u ( t ) } \\ , . \\end{align*}"}
+{"id": "5876.png", "formula": "\\begin{align*} k = \\left \\lfloor { ( 1 - \\epsilon ) 2 \\Delta ^ { - 2 } V ^ { - 1 } N ^ { \\zeta } } \\right \\rfloor , \\end{align*}"}
+{"id": "3888.png", "formula": "\\begin{align*} D ( P ) : = \\lim _ { n \\to \\infty } \\frac { 2 \\log n } { - \\log V _ { n } ( P ) } , \\end{align*}"}
+{"id": "289.png", "formula": "\\begin{align*} \\left \\Vert u \\right \\Vert _ { H ^ { 1 } \\left ( \\omega , \\mathbb { B } \\right ) } : = \\left ( \\left \\Vert u \\right \\Vert _ { H ^ { 1 } \\left ( \\omega \\right ) } ^ { 2 } + \\left \\Vert \\operatorname { d i v } \\left ( \\mathbb { B } \\nabla u \\right ) \\right \\Vert _ { L ^ { 2 } \\left ( \\omega \\right ) } ^ { 2 } \\right ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "6581.png", "formula": "\\begin{align*} H = { { H } } ( q ^ { ( r ) } ) & = D ( 0 ) + \\varepsilon \\Delta + \\delta { T } _ { q ^ { ( r ) } } \\end{align*}"}
+{"id": "7793.png", "formula": "\\begin{align*} f ^ { } + f ^ { } = \\frac { \\chi } { 1 2 } + \\frac { \\tau _ 2 ^ 2 } { ( 2 \\pi ) ^ 2 } \\sum _ { \\hat { \\gamma } \\in \\Lambda ^ + \\cup \\{ 0 \\} } n _ { \\hat { \\gamma } } \\sum _ { ( m , n ) \\in \\mathbb { Z } ^ 2 - \\{ 0 \\} } \\left ( \\frac { 1 } { | m \\tau + n | } + 2 \\pi q _ a t ^ a \\right ) \\frac { e ^ { - S _ { \\hat { \\gamma } , m , n } } } { | m \\tau + n | ^ 2 } \\end{align*}"}
+{"id": "8716.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } \\Big ( \\frac { r _ t - r _ { t + 1 } } { \\eta _ t } - \\frac { \\alpha } { 2 } r _ t \\Big ) = \\frac { \\alpha } { 2 } \\sum _ { t = 1 } ^ { T } \\Big ( r _ t ( t - 1 ) - r _ { t + 1 } t \\Big ) \\leq 0 \\enspace . \\end{align*}"}
+{"id": "1347.png", "formula": "\\begin{align*} \\mathcal { C } _ { T } : = \\mathcal { C } \\times _ S T = \\mathbf { P r o j } _ S ( \\bigoplus _ { l \\ge 0 } \\pi _ { \\mathcal { X } * } \\omega _ { \\mathcal { X } / S } ^ { [ l e r ] } ) \\times _ S T \\cong \\mathbf { P r o j } _ T ( \\bigoplus _ { l \\ge 0 } \\pi _ { \\mathcal { X } _ { T } * } \\omega _ { \\mathcal { X } _ T / T } ^ { [ l e r ] } ) . \\end{align*}"}
+{"id": "4135.png", "formula": "\\begin{align*} m _ 1 = & \\min \\left \\{ d - \\sqrt { ( 2 - \\frac { 1 } { r } ) k } - ( 2 - \\frac { 1 } { r } ) ^ { \\frac { 1 } { 4 } } k ^ { \\frac { 1 } { 4 } } - 1 , \\ c ( k ) \\left ( - c ( k ) + d - \\sqrt { ( 2 - \\frac { 1 } { r } ) k } - ( 2 - \\frac { 1 } { r } ) ^ { \\frac { 1 } { 4 } } k ^ { \\frac { 1 } { 4 } } \\right ) \\right \\} , \\\\ m _ 2 = & ( \\frac { 1 } { r } - 1 ) k - c ( k ) \\left ( \\sqrt { ( \\frac { 1 } { r } - 1 ) k } + ( \\frac { 1 } { r } - 1 ) ^ \\frac { 1 } { 4 } k ^ \\frac { 1 } { 4 } \\right ) . \\end{align*}"}
+{"id": "7629.png", "formula": "\\begin{align*} \\mathrm { s u p p } ( B _ k ( \\P ) ) \\supseteq \\bigcup _ { x \\in \\R ^ d } ( \\P _ { x } ) \\supseteq \\bigcup _ { x \\in \\R ^ d } \\{ x \\} = \\R ^ d . \\end{align*}"}
+{"id": "3843.png", "formula": "\\begin{align*} g \\left ( \\frac { \\partial \\varphi } { \\partial t } , \\nu \\right ) = - H , \\end{align*}"}
+{"id": "6578.png", "formula": "\\begin{align*} \\| T ^ { - 1 } ( \\sigma ) \\| \\leq C N _ 2 ^ C e ^ { { N _ 2 } ^ { \\rho _ 2 } } \\leq e ^ { 2 { N _ 2 } ^ { \\rho _ 2 } } = K _ 3 , \\end{align*}"}
+{"id": "5066.png", "formula": "\\begin{align*} \\partial _ m y ( z ) = A y ( z ) , \\end{align*}"}
+{"id": "1664.png", "formula": "\\begin{align*} \\delta ^ { ( m , n ) } _ { \\texttt { a } ; \\mu } ( q ) : = \\prod _ { \\substack { 1 \\leq j < k \\leq n \\\\ \\mu _ j - \\mu _ k = 0 } } \\frac { 1 - q ^ { k - j } } { 1 - q ^ { 1 + k - j } } \\prod _ { \\substack { 1 \\leq j < k \\leq n \\\\ \\mu _ j - \\mu _ k = m } } \\frac { 1 - q ^ { n - k + j } } { 1 - q ^ { n + 1 - k + j } } . \\end{align*}"}
+{"id": "872.png", "formula": "\\begin{align*} u _ t = \\sigma x ^ \\gamma u _ { x x } + f ( x ) u _ x - g ( x ) u , ~ ~ x \\geq 0 , \\sigma > 0 , \\gamma \\neq 2 , \\end{align*}"}
+{"id": "196.png", "formula": "\\begin{align*} & \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T _ C M } - \\widetilde { L _ C } + W _ i ) \\right \\} ^ { ( 1 0 ) } = 4 8 8 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 0 ) } . \\end{align*}"}
+{"id": "2279.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } 1 \\cdot \\sqrt { \\frac { 1 } { n ^ 2 \\pi ^ 2 } \\int _ { \\frac { k + 1 / 2 } { n } } ^ { \\frac { k + 3 / 2 } { n } } f ' ( x ) ^ 2 \\ , d x } & \\leq \\sqrt { n } \\sqrt { \\sum _ { k = 0 } ^ { n - 1 } \\frac { 1 } { n ^ 2 \\pi ^ 2 } \\int _ { \\frac { k + 1 / 2 } { n } } ^ { \\frac { k + 3 / 2 } { n } } f ' ( x ) ^ 2 \\ , d x } \\\\ & = \\sqrt { n } \\sqrt { \\frac { 1 } { n ^ 2 \\pi ^ 2 } \\int _ { 0 } ^ { 1 } f ' ( x ) ^ 2 \\ , d x } \\end{align*}"}
+{"id": "4599.png", "formula": "\\begin{align*} \\langle \\Phi _ { \\lambda _ 1 } ( v _ 1 , p _ 1 ) \\cdots \\Phi _ { \\lambda _ m } ( v _ m , p _ m ) \\rangle = \\beta _ { X , \\vec p ; \\vec \\lambda } ( v _ 1 \\ , { \\otimes } \\cdots { \\otimes } \\ , v _ m ) \\ , . \\end{align*}"}
+{"id": "8340.png", "formula": "\\begin{align*} \\d X ( t ) = a ( X ( t ) , t , \\omega ) \\d t + b ( X ( t ) , t , \\omega ) \\d B ( t ) , X ( 0 ) = x _ 0 \\in ( x _ 1 , x _ 2 ) , \\end{align*}"}
+{"id": "230.png", "formula": "\\begin{align*} \\phi _ \\xi ^ { } ( x ; g ) = \\sum _ { \\nu \\geq 0 } a ^ { } _ \\nu ( \\xi ; g ) e ^ { \\langle \\xi - \\nu , x \\rangle } , \\end{align*}"}
+{"id": "4649.png", "formula": "\\begin{align*} X ^ { ( i ) } : = 1 ^ { \\otimes ( i - 1 ) } \\otimes X \\otimes 1 ^ { \\otimes ( n - i ) } \\in U ( \\mathfrak { g } ) ^ { \\otimes \\# E } \\end{align*}"}
+{"id": "5858.png", "formula": "\\begin{align*} v _ u = \\frac { y _ 1 v _ 1 } { y _ u } = \\frac { y _ 1 v _ 1 } { \\frac { u - 1 } { \\gamma } + y _ 1 } = \\frac { y _ 1 v _ 1 \\gamma } { u - 1 + y _ 1 \\gamma } . \\end{align*}"}
+{"id": "4106.png", "formula": "\\begin{align*} U ( x ) : = \\alpha \\log x , \\quad \\Phi ^ U ( p ) : = \\Phi ( p ) + U ( x ( p ) ) , \\end{align*}"}
+{"id": "4449.png", "formula": "\\begin{align*} \\sum _ { \\langle \\beta \\rangle = 1 \\ , , \\ , \\beta \\leq \\alpha } | | D ^ { \\beta } _ { \\ast } \\mathcal { A } _ 1 D ^ { \\alpha - \\beta } _ { \\ast } ( \\partial _ 1 { \\mathbf V } ) | | ^ 2 _ { L ^ 2 ( \\Omega _ t ) } \\leq C ( K ) | | { \\mathbf V } | | ^ 2 _ { s , \\ast , t } \\ , . \\end{align*}"}
+{"id": "8086.png", "formula": "\\begin{align*} \\begin{aligned} \\uppercase \\expandafter { \\romannumeral 1 } _ { 2 } & \\leq C \\sum \\limits _ { j = 1 } ^ { \\infty } \\mu _ { j } { \\omega ( P _ { j } ) } ^ { - \\frac { 1 } { p } } { \\left ( { \\left ( \\frac { l ( P _ { j } ) } { \\vert x - x _ { P _ { j } } \\vert } \\right ) } ^ { n } \\right ) } ^ { \\gamma } \\chi _ { ( 4 P _ { j } ) ^ { c } } ( x ) \\\\ & \\leq C \\sum \\limits _ { j = 1 } ^ { \\infty } \\mu _ { j } { \\omega ( P _ { j } ) } ^ { - \\frac { 1 } { p } } ( M \\chi _ { P _ { j } } ) ^ { \\gamma } ( x ) \\end{aligned} \\end{align*}"}
+{"id": "6023.png", "formula": "\\begin{align*} D F = \\sum _ { i = 1 } ^ { \\infty } D _ i F \\times e _ i D ^ q F = \\sum _ { i _ 1 , \\cdots , i _ q } D _ { i _ 1 , \\cdots , i _ q } F \\times \\otimes _ { j = 1 } ^ q e _ j . \\end{align*}"}
+{"id": "6.png", "formula": "\\begin{align*} e ^ { \\rm { 2 D } } ( \\rho ) = 4 \\pi \\rho ^ 2 \\delta \\Big ( 1 + \\Big [ 2 \\Gamma + \\frac { 1 } { 2 } + \\log ( \\pi ) \\Big ] \\delta \\Big ) + o ( \\rho ^ 2 \\delta ^ 2 ) , \\end{align*}"}
+{"id": "6258.png", "formula": "\\begin{align*} i \\partial _ t v _ \\lambda + \\partial _ x g _ { [ < \\lambda ] } \\partial _ x v _ \\lambda = f _ \\lambda , v _ \\lambda ( 0 ) = v _ { 0 , \\lambda } , \\end{align*}"}
+{"id": "5057.png", "formula": "\\begin{align*} L ^ 2 _ { g ' , f ' } ( M ; S ^ 2 T ^ * M ) = \\Lambda ( N _ { g , f } ) \\oplus Z ( K _ g ) \\oplus \\Lambda ( P _ { g , f } ) . \\end{align*}"}
+{"id": "7281.png", "formula": "\\begin{align*} \\tilde { B } = \\{ i \\mid \\exists n \\in [ n _ i , n _ { i + 1 } ) \\cap A \\} , \\end{align*}"}
+{"id": "5678.png", "formula": "\\begin{align*} \\ln \\| ( z - T ) ^ { - 1 } e _ 0 \\| _ p ^ p \\geq \\ln \\bigg ( \\frac { \\big ( \\prod \\limits _ { j = 0 } ^ { N - 1 } w _ j \\big ) ^ p } { | z | ^ { p N } } \\bigg ) > 0 . \\end{align*}"}
+{"id": "3931.png", "formula": "\\begin{align*} f ( x ) & = ( x - 1 , x + 1 ) \\\\ g ( x ) & = [ x - 1 , x + 1 ] \\end{align*}"}
+{"id": "7308.png", "formula": "\\begin{align*} V : = S + K \\in G L ( H ) ^ G . \\end{align*}"}
+{"id": "7934.png", "formula": "\\begin{align*} D = \\Big ( D _ 1 \\cup D _ 2 \\cup \\{ u , v \\} \\Big ) \\setminus \\{ u _ 1 , v _ 1 , u _ 2 , v _ 2 \\} . \\end{align*}"}
+{"id": "2895.png", "formula": "\\begin{align*} \\langle \\xi _ \\mu , \\alpha \\rangle - \\langle s _ { \\beta ^ \\vee } \\xi _ \\mu , \\alpha \\rangle = \\langle \\xi _ \\mu , \\beta \\rangle \\langle \\alpha , \\beta ^ \\vee \\rangle . \\end{align*}"}
+{"id": "484.png", "formula": "\\begin{align*} I _ { \\zeta - 1 / 2 } ( \\tau ) = \\left ( \\frac { \\tau } { 2 } \\right ) ^ { \\zeta - 1 / 2 } \\sum _ { k = 0 } ^ \\infty \\frac { ( \\tau ^ 2 / 4 ) ^ k } { k ! \\Gamma ( \\zeta + 1 / 2 + k ) } , \\tau \\in \\C \\setminus ( - \\infty , 0 ] , \\end{align*}"}
+{"id": "8234.png", "formula": "\\begin{align*} W ^ + ( C _ i ) = \\{ m \\in M _ { x _ 1 , 0 , 0 } | \\lim _ { t \\rightarrow + \\infty } \\psi _ t ( m ) \\in C _ i \\} \\end{align*}"}
+{"id": "2524.png", "formula": "\\begin{align*} \\mathcal { R } _ 1 ( \\boldsymbol { \\eta } , \\boldsymbol { \\tau } ) = \\boldsymbol { u } - \\sigma \\partial _ t \\boldsymbol { \\eta } - \\textbf { c u r l } \\ , \\boldsymbol { \\tau } \\mathcal { R } _ 2 ( \\boldsymbol { \\eta } , \\boldsymbol { \\tau } ) = \\boldsymbol { \\tau } - \\nu \\ , \\textbf { c u r l } \\ , \\boldsymbol { \\eta } . \\end{align*}"}
+{"id": "2894.png", "formula": "\\begin{align*} \\langle \\hat \\rho _ v ( \\xi _ \\mu ) , \\beta \\rangle = \\frac { 1 } { 2 } \\sum _ { \\substack { \\alpha \\in R _ 0 \\\\ \\langle \\alpha ^ \\vee , \\beta \\rangle > 0 } } \\bigl ( v _ \\alpha ( \\langle \\xi _ \\mu , \\alpha \\rangle ) - v _ \\alpha ( \\langle s _ { \\beta ^ \\vee } \\xi _ \\mu , \\alpha \\rangle ) \\bigr ) \\langle { \\hat \\alpha } , \\beta \\rangle . \\end{align*}"}
+{"id": "43.png", "formula": "\\begin{align*} \\mathcal { A } _ k : = k ^ 2 + \\rho _ z \\widehat { g } ( k ) , \\mathcal { B } _ k : = \\rho _ z \\widehat g ( k ) . \\end{align*}"}
+{"id": "4467.png", "formula": "\\begin{align*} \\partial _ t \\varphi = \\eta ( { \\mathbf U } ) | _ { x _ 1 = 0 } , \\end{align*}"}
+{"id": "237.png", "formula": "\\begin{align*} \\Phi _ \\xi ^ { } ( x ; g ) : = \\sum _ { \\in W } C ^ { } ( \\xi ; g ) \\phi _ { \\xi } ^ { } ( x ; g ) , \\end{align*}"}
+{"id": "7082.png", "formula": "\\begin{align*} \\mathcal { V } _ 2 \\left ( \\frac { m } { p _ 2 ( p _ 1 q ) ^ 2 } \\right ) = \\frac { p _ 2 ^ { 1 / 4 } ( p _ 1 q ) ^ { 1 / 2 } } { m ^ { 1 / 4 } } \\frac { N ^ { 3 / 4 } } { N ^ { i ( t + \\nu ) } } \\ , \\sum _ { \\pm } \\mathcal { I } ^ { \\pm } ( m , u , q , p _ 1 , p _ 2 ) , \\end{align*}"}
+{"id": "4090.png", "formula": "\\begin{align*} \\varpi _ { 1 , 1 } ( p _ 0 ) + \\varpi _ { 1 , 1 } ( \\sigma ( p _ 0 ) ) = - d \\frac { \\Delta \\varpi _ { \\frac 1 2 , 1 } ( p _ 0 ) } { 2 \\varpi _ { 0 , 1 } ( p _ 0 ) } . \\end{align*}"}
+{"id": "5320.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ L \\ln \\left ( \\frac { L ( C ( N + 1 ) ) ^ { k - 1 } } { \\varepsilon } \\right ) ^ 3 = O ( L \\ln ( \\varepsilon ^ { - 1 } L ) ^ 3 + L ^ 4 \\ln ( C N ) ^ 3 ) \\end{align*}"}
+{"id": "8572.png", "formula": "\\begin{align*} \\sigma ( X , z ) = \\dfrac { \\varepsilon \\zeta ( X ) } { 1 - \\beta b ( X ) } z + \\varepsilon \\zeta ( X ) . \\end{align*}"}
+{"id": "4845.png", "formula": "\\begin{align*} m ( E _ i - I ) = G _ i ^ { - 1 } - G _ i \\end{align*}"}
+{"id": "322.png", "formula": "\\begin{align*} \\left ( \\mathsf { D } _ { j } \\left ( s \\right ) \\psi \\right ) \\left ( \\mathbf { x } \\right ) : = \\int _ { \\Gamma _ { j } } \\left ( \\frac { \\partial } { \\partial \\mathbf { \\tilde { n } } _ { \\mathbf { y } } } G \\left ( \\mathbf { x } , \\mathbf { y } \\right ) \\right ) \\psi \\left ( \\mathbf { y } \\right ) d \\Gamma _ { \\mathbf { y } } \\quad \\mathbf { x } \\in \\mathbb { R } ^ { 3 } \\backslash \\Gamma _ { j } \\end{align*}"}
+{"id": "965.png", "formula": "\\begin{align*} \\int _ { \\Sigma _ i } K \\ , d \\mu _ \\Sigma = 2 \\pi \\chi ( \\Sigma _ i ) - \\int _ { \\partial \\Sigma _ i } \\kappa _ g \\ , d \\mu _ { \\partial \\Sigma } - \\sum _ j \\theta _ { i j } , \\end{align*}"}
+{"id": "6227.png", "formula": "\\begin{align*} D ^ 2 = \\nabla ^ * \\nabla + \\mathcal { R } \\end{align*}"}
+{"id": "7176.png", "formula": "\\begin{align*} \\lambda ( F _ 1 ^ j ) = \\boldsymbol { e _ { \\mathcal { N } _ { j - 1 } + 1 } } , & \\dots , \\lambda ( F _ { n _ j } ^ j ) = \\boldsymbol { e _ { \\mathcal { N } _ j } } , \\end{align*}"}
+{"id": "742.png", "formula": "\\begin{align*} \\tau < \\infty \\implies \\limsup _ { t \\nearrow \\tau } \\left ( \\norm { \\psi _ + ( t ) } _ { L ^ 2 } ^ 2 + \\norm { \\psi _ - ( t ) } _ { L ^ 2 } ^ 2 + \\norm { \\phi _ + ( t ) } _ { H ^ r } ^ 2 \\right ) = \\infty . \\end{align*}"}
+{"id": "215.png", "formula": "\\begin{align*} & \\left \\{ e ^ { \\frac { 1 } { 2 4 } A _ 3 } \\widehat { L } ( T M ) \\right \\} ^ { ( 1 2 ) } = - \\frac { 1 } { 2 } \\left \\{ e ^ { \\frac { 1 } { 2 4 } A _ 3 } \\widehat { A } ( T M ) { \\rm c h } [ 1 7 \\widetilde { T _ C M } + \\wedge ^ 2 \\widetilde { T _ C M } + W _ i + 1 2 8 ] \\right \\} ^ { ( 1 2 ) } \\\\ & + \\frac { 1 } { 6 0 } \\left \\{ c _ 2 ( W _ i ) e ^ { \\frac { 1 } { 2 4 } A _ 3 } \\widehat { A } ( T X ) \\right \\} ^ { ( 1 2 ) } . \\end{align*}"}
+{"id": "2748.png", "formula": "\\begin{align*} \\tilde { S } _ m : = \\big \\{ \\sigma ( j _ 1 ) , \\sigma ( j _ 2 ) , \\sigma ( j _ 3 ) , \\ldots \\sigma ( j _ m ) \\big \\} \\end{align*}"}
+{"id": "6769.png", "formula": "\\begin{align*} \\phi '' _ \\infty ( z ) - c \\phi ' _ \\infty ( z ) + \\phi _ \\infty ( z ) F ( \\phi _ \\infty , \\psi _ \\infty ) ( z ) = 0 . \\end{align*}"}
+{"id": "7300.png", "formula": "\\begin{align*} P _ { S _ { } } ( x , y ) = \\begin{cases} R ( x , y ) - R ( y , x ) & R ( x , y ) > R ( y , x ) , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "4636.png", "formula": "\\begin{align*} \\mathbf { K } = \\prod _ { i = 1 } ^ { n + 1 } K _ { v _ i } = H \\times G ^ { \\times ( n - 1 ) } \\times K \\end{align*}"}
+{"id": "155.png", "formula": "\\begin{align*} X _ n \\longrightarrow \\widetilde { X } ' _ n \\setminus \\big ( \\widetilde { \\{ x _ 2 = 0 \\} } \\cup \\cdots \\cup \\widetilde { \\{ x _ n = 0 \\} } \\big ) \\ , ( t _ 1 , \\dots , t _ n ) \\mapsto ( x _ 1 , \\dots , x _ n ) \\end{align*}"}
+{"id": "6824.png", "formula": "\\begin{align*} T _ 2 ( g ) & : = ( n - 2 ) \\sigma _ 1 ( A _ g ) g - 8 A _ g , \\\\ T _ 4 ( g ) & : = - \\frac { 3 n ^ 2 - 1 2 n - 4 } { 4 } \\sigma _ 1 ( A _ g ) ^ 2 g + 4 ( n - 4 ) \\| A _ g \\| ^ 2 g + 8 ( n - 2 ) \\sigma _ 1 ( A _ g ) A _ g \\\\ & + ( n - 6 ) \\Delta _ g \\sigma _ 1 ( A _ g ) g + 4 8 A _ g ^ 2 - \\frac { 1 6 } { n - 4 } B _ g , \\\\ v _ 6 ( g ) & : = - \\frac { 1 } { 8 } \\sigma _ 3 ( A _ g ) - \\frac { 1 } { 2 4 ( n - 4 ) } \\langle B _ g , A _ g \\rangle _ g , \\end{align*}"}
+{"id": "2998.png", "formula": "\\begin{align*} \\| \\Psi ( x ) \\| ^ 2 = \\sup _ { \\| z \\| = 1 } \\sup _ { v \\in E ^ 0 } \\sum _ { s ( e ) = v } | \\Psi ( x ) z ( e ) | ^ 2 = \\sup _ { \\| z \\| = 1 } \\sup _ { v \\in E ^ 0 } \\sum _ { s ( e ) = v } | x ( e ) z ( e ) | ^ 2 . \\end{align*}"}
+{"id": "5625.png", "formula": "\\begin{align*} D _ X = \\sum D _ i , \\end{align*}"}
+{"id": "5999.png", "formula": "\\begin{align*} ( \\Gamma ) \\sum _ { i = 1 } ^ { \\infty } \\gamma _ { i } = \\lim _ { n \\rightarrow \\infty } \\Gamma _ { n } = \\infty . \\end{align*}"}
+{"id": "7042.png", "formula": "\\begin{align*} w _ { 1 } ( t ) : = v _ { 1 } ( t , q ) , w _ { 2 } ( t ) : = v _ { 2 } ( t , q ) , t \\in [ 0 , T ] . \\end{align*}"}
+{"id": "2597.png", "formula": "\\begin{align*} \\mu ( X , \\{ \\sigma _ n \\} ) = \\varprojlim _ n \\mu _ n ( , \\{ ( \\sigma _ n ) \\} ) \\end{align*}"}
+{"id": "4761.png", "formula": "\\begin{align*} \\theta _ F ( x ) = \\theta _ E ( x ) \\quad \\mathcal H x \\in \\partial ^ e F \\cap \\partial ^ e E . \\end{align*}"}
+{"id": "8595.png", "formula": "\\begin{align*} \\tilde F = \\mu \\big [ h _ b T _ 1 ( z ) [ X , D ] \\frac { 1 } { \\sqrt { \\mu } | D | } , \\Delta \\big ] G + \\mu \\beta b T _ 1 ( z ) [ X , \\mathrm { D } ] \\frac { \\Delta _ X } { \\sqrt { \\mu } | \\mathrm { D } | } G \\end{align*}"}
+{"id": "3008.png", "formula": "\\begin{align*} \\textstyle C _ n \\left ( X _ \\gamma , \\coprod S ^ 1 ; \\mathbb { R } \\right ) = C _ n \\left ( X _ \\gamma ; \\mathbb { R } \\right ) / C _ n \\left ( \\coprod S ^ 1 ; \\mathbb { R } \\right ) \\end{align*}"}
+{"id": "1892.png", "formula": "\\begin{align*} \\partial _ t u + \\frac { 1 } { 2 } \\partial _ x u ^ 2 - \\sqrt { \\epsilon } \\ , \\partial _ x w + \\rho u = \\rho V , \\end{align*}"}
+{"id": "5373.png", "formula": "\\begin{align*} \\frac { p } { q } = \\begin{cases} [ a _ 1 , a _ 2 , \\ldots , a _ { 2 m - 2 } + 1 ] , & ~ \\mbox { i f } ~ a _ { 2 m - 1 } = 1 \\mbox { a n d } ~ m > 1 ; \\\\ [ a _ 1 , a _ 2 , \\ldots , a _ { 2 m - 1 } - 1 , 1 ] , & ~ \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
+{"id": "5133.png", "formula": "\\begin{align*} \\beta ^ * = \\begin{cases} 2 \\alpha , & \\alpha \\leq \\lfloor S / 2 \\rfloor , \\\\ 2 ( S - \\alpha ) , & \\lceil S / 2 \\rceil \\leq \\alpha \\leq S - 1 , \\\\ 1 , & \\alpha = S . \\end{cases} \\end{align*}"}
+{"id": "7828.png", "formula": "\\begin{align*} ( \\tau _ { 0 0 } ) = \\frac { t ^ 3 } { 3 } - b ^ 2 t + \\frac { \\chi \\zeta ( 3 ) } { ( 2 \\pi ) ^ 3 } , ( \\tau _ { 0 1 } ) = b t , ( \\tau _ { 1 1 } ) = - t , \\ , \\end{align*}"}
+{"id": "1061.png", "formula": "\\begin{align*} \\hat { w } _ \\lambda ^ + ( \\overline { u } _ \\lambda ^ * ) = \\inf \\left \\{ \\hat { w } _ \\lambda ^ + ( u ) : \\ : u \\in W ^ { 1 , p ( z ) } ( \\Omega ) \\right \\} . \\end{align*}"}
+{"id": "8917.png", "formula": "\\begin{align*} \\tau ( \\varphi ) = \\sum \\limits _ { \\alpha = 1 } ^ n \\tau ( \\varphi ) ^ \\alpha \\frac { \\partial } { \\partial y _ \\alpha } , \\end{align*}"}
+{"id": "800.png", "formula": "\\begin{align*} - \\frac { N ( p ^ \\flat + p ^ \\sharp ) } { p } + N + \\alpha = - \\frac { N + \\alpha } { N - p s } \\cdot \\frac { p s } { 2 } = - p ^ \\sharp s > - p s . \\end{align*}"}
+{"id": "5622.png", "formula": "\\begin{align*} \\varphi ^ * ( \\omega _ { Y , D _ Y } ) = \\lambda \\omega _ { X , D _ X } . \\end{align*}"}
+{"id": "1331.png", "formula": "\\begin{align*} J ^ { w } ( \\textbf { X } _ { m i n R S S U } ^ { ( n ) } ) & \\geq J ^ { w } ( \\textbf { Z } _ { m i n R S S U } ^ { ( n ) } ) = - \\frac { 1 } { 2 } \\prod _ { i = 1 } ^ { n } [ - 2 J ^ w ( Z _ { i : i } ) ] \\\\ & \\geq - \\frac { 1 } { 2 } [ - 2 J ^ w ( Z _ { 1 : 1 } ) ] ^ n = - \\frac { 1 } { 2 } \\left ( \\frac { \\lambda ^ 2 \\Gamma ( m + 1 ) } { ( 2 \\lambda ) ^ { m + 1 } } \\right ) ^ n . \\end{align*}"}
+{"id": "4546.png", "formula": "\\begin{align*} \\partial _ 1 \\dot H ^ \\pm _ n = - \\partial _ 2 ( \\dot H ^ \\pm _ 2 \\partial _ 1 \\hat \\Phi ^ \\pm ) = - \\partial _ 2 \\dot H ^ \\pm _ 2 \\partial _ 1 \\hat \\Phi ^ \\pm - \\dot H ^ \\pm _ 2 \\partial _ 2 \\partial _ 1 \\hat \\Phi ^ \\pm \\quad \\mbox { i n } \\ , \\ , \\ , \\Omega _ T \\ , . \\end{align*}"}
+{"id": "7111.png", "formula": "\\begin{align*} \\mathcal { G } _ { \\neq 0 } ( X ) = \\int _ { \\mathbb { R } } V ( \\xi ) \\ , \\mathcal { J } ^ + _ 1 ( N _ 0 \\xi , m , q ) \\ , \\ , \\overline { \\mathcal { J } ^ + _ 1 ( N _ 0 \\xi , m _ 1 , q ' ) } \\ , e \\left ( - { X \\xi } \\right ) \\ , d \\xi . \\end{align*}"}
+{"id": "4071.png", "formula": "\\begin{align*} Q = \\Omega \\times ( 0 , T ) , \\Sigma = \\partial \\Omega \\times ( 0 , T ) . \\end{align*}"}
+{"id": "3312.png", "formula": "\\begin{align*} m _ 1 = \\frac { d } { d t } \\phi _ Y ( 0 ) ( - \\mu ) = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\frac { 1 } { \\lambda _ { _ \\Sigma } } , \\end{align*}"}
+{"id": "1484.png", "formula": "\\begin{align*} X _ { 2 j - 1 } = \\frac { \\partial } { \\partial x _ { 2 j - 1 } } - \\frac { 1 } { 2 } x _ { 2 j } \\frac { \\partial } { \\partial z } , \\ ; \\ ; X _ { 2 j } = \\frac { \\partial } { \\partial x _ { 2 j } } + \\frac { 1 } { 2 } x _ { 2 j - 1 } \\frac { \\partial } { \\partial z } , \\ ; \\ ; Z = \\frac { \\partial } { \\partial z } , \\end{align*}"}
+{"id": "2085.png", "formula": "\\begin{align*} \\Lambda = : \\lambda + \\tilde { \\Lambda } , \\hbox { a n d } \\alpha : = 1 + \\tilde { \\alpha } . \\end{align*}"}
+{"id": "7655.png", "formula": "\\begin{align*} Z ( \\mathbf { t } ) = \\int _ { \\mathcal { H } _ N } \\dd \\mathbb { P } ( H ) \\ , e ^ { { \\rm T r } \\ , V _ { \\mathbf { t } } ( H ) } . \\end{align*}"}
+{"id": "2395.png", "formula": "\\begin{align*} R ( \\psi ) : = \\{ x \\in X : d ( T ^ n ( x ) , x ) \\leq \\psi ( n ) \\ , \\textrm { f o r i . m . } n \\in \\N \\} . \\end{align*}"}
+{"id": "3520.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\sim x \\\\ } } \\log ( p ) \\mathrm { t r } ( \\lambda _ p ^ 0 ( \\gamma ) ) = O ( x ^ { \\frac { 1 } { 2 } } \\log ( x ) ^ 2 ) \\end{align*}"}
+{"id": "2443.png", "formula": "\\begin{align*} \\rho ^ { \\leq \\delta } ( x ) : = \\sum _ { n = 1 } ^ N | u ^ { \\leq \\delta } _ n ( x ) | ^ 2 \\rho ^ { \\delta , \\ell } ( x ) : = \\sum _ { n = 1 } ^ N | u ^ { \\delta , \\ell } _ n ( x ) | ^ 2 . \\end{align*}"}
+{"id": "3645.png", "formula": "\\begin{align*} \\omega = \\omega _ { \\epsilon } + x \\wedge \\Omega \\ , \\end{align*}"}
+{"id": "1248.png", "formula": "\\begin{align*} A _ n = & \\int _ { \\R ^ 3 } [ I _ \\alpha \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } ( | ( g _ n ^ 1 ) ^ { - 1 } u _ n | ^ p - | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p ) ] | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n | ^ p d x \\\\ & - \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p ) | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n | ^ p d x . \\end{align*}"}
+{"id": "7186.png", "formula": "\\begin{align*} \\alpha _ 2 & = ( y _ 1 + y _ 2 ) ^ { n _ 2 } y _ 2 \\\\ & = ( y _ 1 ^ { n _ 2 } + y _ 2 ^ { n _ 2 } ) y _ 2 ~ ( \\mbox { a s } ~ \\binom { n _ 2 } { i } ~ \\mbox { i s e v e n f o r } ~ 0 < i < n ) \\\\ & = y _ 2 ^ { n _ 2 + 1 } ~ ( y _ 1 ^ { n _ 2 } = 0 , ~ \\mbox { s i n c e } ~ n _ 2 \\geq n _ 1 + 1 ~ \\mbox { a n d } ~ y _ 1 ^ { n _ 1 + 1 } = 0 ) . \\end{align*}"}
+{"id": "330.png", "formula": "\\begin{align*} \\left \\langle u , \\mathsf { L } _ { j } \\left ( s \\right ) \\overline { v } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } = \\left \\langle \\psi , \\gamma _ { \\operatorname * { N } ; j } ^ { \\operatorname * { e x t } } \\left ( s \\right ) \\overline { v } \\right \\rangle _ { \\Gamma _ { j } } \\quad \\forall v \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } , \\mathbb { A } _ { j } ^ { \\operatorname * { e x t } } \\right ) . \\end{align*}"}
+{"id": "4460.png", "formula": "\\begin{align*} | | \\dot { u } _ n | _ { x _ 1 = 0 } | | ^ 2 _ { H ^ { s - 1 } ( \\Gamma _ t ) } \\lesssim | | { \\mathbf V } | | ^ 2 _ { s , \\ast , t } . \\end{align*}"}
+{"id": "4554.png", "formula": "\\begin{align*} \\tilde { \\mathcal F } _ 2 : = \\partial _ 2 \\tilde { \\mathcal { F } } - \\partial _ 2 \\mathcal B _ 0 \\partial _ t { \\mathbf V } - \\partial _ 2 \\mathcal B _ 1 \\partial _ 1 { \\mathbf V } - \\partial _ 2 \\mathcal B _ 3 { \\mathbf V } \\ , . \\end{align*}"}
+{"id": "2224.png", "formula": "\\begin{align*} A _ 0 : = ( A _ 0 ^ { ( 1 ) } , A _ 0 ^ { ( 2 ) } , A _ 0 ^ { ( 3 ) } ) , \\end{align*}"}
+{"id": "8569.png", "formula": "\\begin{align*} \\mathcal { R } _ 1 ^ { \\mu } [ \\beta b , h , \\overline { V } ] = - \\frac { h ^ 2 } { 2 } ( \\nabla _ X \\cdot \\overline { V } ) ^ 2 - \\frac { 1 } { 3 h } \\big { ( } \\nabla _ X ( h ^ 3 \\nabla _ X \\cdot \\overline { V } ) \\big { ) } \\cdot \\overline { V } - \\frac { 1 } { 2 } h ^ 3 \\Delta _ X ( | \\overline { V } | ^ 2 ) + \\frac { 1 } { 6 h } h ^ 3 \\Delta _ X ( | \\overline { V } | ^ 2 ) . \\end{align*}"}
+{"id": "3516.png", "formula": "\\begin{align*} - \\log \\vert \\zeta _ { \\tau } ( s , \\rho ) \\vert \\leqslant \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { k } \\vert \\mathrm { t r } ( \\mathcal { L } _ { \\tau , s , \\rho } ^ { 2 k } ) \\vert . \\end{align*}"}
+{"id": "2167.png", "formula": "\\begin{align*} d s ^ 2 : = f ^ { ( 0 ) } ( - d t ^ 2 + d r ^ 2 ) + e ^ { u _ 0 } ( r d \\phi ) ^ 2 + e ^ { - u _ 0 } d z ^ 2 , \\end{align*}"}
+{"id": "176.png", "formula": "\\begin{align*} \\theta ' ( v , \\tau + 1 ) = e ^ { \\frac { \\pi \\sqrt { - 1 } } { 4 } } \\theta ' ( v , \\tau ) , ~ ~ \\theta ' ( 0 , - \\frac { 1 } { \\tau } ) = \\frac { 1 } { \\sqrt { - 1 } } \\left ( \\frac { \\tau } { \\sqrt { - 1 } } \\right ) ^ { \\frac { 1 } { 2 } } \\tau \\theta ' ( 0 , \\tau ) . \\end{align*}"}
+{"id": "1207.png", "formula": "\\begin{align*} \\begin{cases} i w _ t + \\Delta w = F ( \\tilde { u } + w ) - F ( \\tilde { u } ) + e , \\\\ w ( 0 ) = u _ 0 - \\tilde { u } ( 0 ) , \\end{cases} \\end{align*}"}
+{"id": "7635.png", "formula": "\\begin{align*} | U ( z ) | = U ^ + ( z ) + U ^ - ( z ) \\le 2 \\tilde { C } ( 1 + | z | ^ { p } ) . \\end{align*}"}
+{"id": "6447.png", "formula": "\\begin{align*} g _ { u , v } = \\left ( x + \\frac { v } { u } w , ( u - v ) y \\right ) \\simeq \\left ( x + \\frac { v } { u } w , c ^ 2 ( u - v ) y \\right ) = g _ { c ^ 2 u , c ^ 2 v } . \\end{align*}"}
+{"id": "2948.png", "formula": "\\begin{align*} G _ { k i _ 1 \\dots i _ n } = \\frac { 2 n - 1 } { n - 1 } \\delta _ { k \\hat { i } _ 1 } D _ { \\hat { i } _ 2 \\dots \\hat { i } _ n } - \\delta _ { \\hat { i } _ 1 \\hat { i } _ 2 } D _ { \\hat { i } _ 3 \\dots \\hat { i } _ n k } + \\epsilon _ { k s \\hat { i } _ 1 } D _ { \\hat { i } _ 2 \\dots \\hat { i } _ n s } + D _ { k i _ 1 \\dots i _ n } \\end{align*}"}
+{"id": "892.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathrm { P r } ^ { ( \\alpha , 2 ) } V \\bigg ( \\mathcal { T } _ t ^ \\alpha u - u _ { x x } - \\frac { c } { x } u _ x - m x ^ k v _ x \\bigg ) | _ { \\eqref { e q u a t i o n 2 } } = 0 , \\\\ & \\mathrm { P r } ^ { ( \\alpha , 2 ) } V \\bigg ( \\mathcal { T } _ t ^ \\alpha v - v _ { x x } - \\frac { c } { x } v _ x - n x ^ k u _ x \\bigg ) | _ { \\eqref { e q u a t i o n 2 } } = 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "6724.png", "formula": "\\begin{align*} \\begin{aligned} \\alpha ( t ) = & C _ 2 t ^ { a + 1 } - C _ 1 \\frac { 1 } { a + 1 } , \\\\ \\beta ( t ) = & C _ 1 \\frac { 2 a + 1 } { a ( a + 1 ) } \\mathfrak { c } _ { p } ^ { - 1 } t ^ a - C _ 2 \\mathfrak { c } _ { p } ^ { - 1 } t ^ { 2 a + 1 } , \\\\ \\gamma ( t ) = & - C _ 2 \\mathfrak { c } _ { p } t - C _ 1 \\mathfrak { c } _ { p } \\frac { 1 } { a ( a + 1 ) } t ^ { - a } . \\end{aligned} \\end{align*}"}
+{"id": "3995.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { M _ s } \\exp \\left ( \\frac { \\alpha _ 0 } { A ^ { \\frac { 1 } { n } } _ { s , k } } ( - \\varphi _ t - s ) ^ { \\frac { n + 1 } { n } } \\right ) \\omega ^ n \\\\ & \\leq \\int _ { M _ s } \\exp \\left ( \\frac { ( n + 1 ) \\alpha _ 0 } { n } \\psi _ { t , k } + \\alpha _ 0 A _ { s , k } \\right ) \\omega ^ n \\\\ & \\leq C \\exp \\left ( \\alpha _ 0 A _ { s , k } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "8223.png", "formula": "\\begin{align*} d ^ c _ { I _ 1 ^ a } K _ 1 ^ a = d ^ c _ { I _ 1 } K _ 1 + a ^ 2 ( x _ 2 d x _ 3 - x _ 3 d x _ 2 ) . \\end{align*}"}
+{"id": "5672.png", "formula": "\\begin{align*} x _ 0 y ^ 2 + B _ 3 ( x _ 0 , x _ 1 , x _ 2 ) y + x _ 1 C _ 4 ( x _ 0 , x _ 1 , x _ 2 ) = 0 , \\end{align*}"}
+{"id": "6668.png", "formula": "\\begin{align*} \\hat { r } ( \\tau ) = \\sum _ { n = 0 } ^ \\infty e _ n \\Big ( { \\tau \\over \\pi } \\Big ) ^ n , \\tau > 0 , \\end{align*}"}
+{"id": "6020.png", "formula": "\\begin{align*} A _ { q , \\kappa , p } ^ { \\delta , \\eta } ( \\gamma _ { N ( t ) } ^ { \\beta } ) = \\Delta ^ { \\frac { p q } { p + \\kappa } } + 2 \\Delta ^ { \\frac { p \\kappa } { p + \\kappa } } . \\end{align*}"}
+{"id": "4525.png", "formula": "\\begin{align*} & | | \\delta \\dot { { \\mathbf V } _ i } | | _ { s , \\ast , T } + | | \\delta \\psi _ i | | _ { H ^ s ( \\Gamma _ T ) } \\\\ & \\leq C \\Big ( | | { f } _ i | | _ { s + 2 , \\ast , T } + | | { g } _ i | | _ { H ^ { s + 2 } ( \\Gamma _ T ) } \\\\ & \\quad + ( | | { f } _ i | | _ { 8 , \\ast , T } + | | { g } _ i | | _ { H ^ { 8 } ( \\Gamma _ T ) } ) | | ( \\tilde { \\mathbf U } ^ a + { \\mathbf V } _ { i + \\frac { 1 } { 2 } } , \\nabla ( \\Psi ^ a + \\Psi _ { i + \\frac { 1 } { 2 } } ) ) | | _ { s + 4 , \\ast , T } \\Big ) . \\end{align*}"}
+{"id": "3074.png", "formula": "\\begin{align*} { \\omega _ { k , n } } = \\left ( { n - 1 } \\right ) \\left ( { \\Theta _ { { k , { \\rm { R } } } , { l _ k } } ^ { \\rm { D } } - \\Theta _ { { \\rm { T } } , k , 0 } ^ { \\rm { A } } } \\right ) , \\end{align*}"}
+{"id": "1667.png", "formula": "\\begin{align*} v _ q ( \\vartheta ) : = \\int _ 0 ^ \\vartheta u _ q ( \\theta ) \\theta u _ q ( \\theta ) : = \\frac { 1 - q ^ 2 } { 1 - 2 q \\cos ( \\theta ) + q ^ 2 } . \\end{align*}"}
+{"id": "1127.png", "formula": "\\begin{align*} [ m _ 1 , m _ 1 ] = 0 , ~ ~ ~ [ m _ 1 , m _ 2 ] = 0 , ~ ~ ~ ~ ~ ~ ~ ~ [ m _ 2 , m _ 2 ] = 0 . \\end{align*}"}
+{"id": "8130.png", "formula": "\\begin{align*} u _ E : = ( 1 - \\chi _ { B _ R } ) u , v _ E : = ( 1 - \\chi _ { B _ R } ) v , \\end{align*}"}
+{"id": "398.png", "formula": "\\begin{align*} \\epsilon \\circ \\psi \\circ \\varphi = p \\circ \\varphi = \\epsilon \\ , . \\end{align*}"}
+{"id": "1912.png", "formula": "\\begin{align*} e _ u = u - u _ h , e _ w = w - w _ h . \\end{align*}"}
+{"id": "4644.png", "formula": "\\begin{align*} D _ { X \\otimes Z Y } = D _ { X \\iota ( Z ) \\otimes Y } \\end{align*}"}
+{"id": "1796.png", "formula": "\\begin{align*} Z _ N ( s , \\alpha ) & = \\sum _ { n = 0 } ^ { \\infty } ( n + \\alpha ) ^ { - s } \\phi \\left ( \\frac { n + \\alpha } { N } \\right ) , \\\\ Z _ N ( s , \\mathbb { X } _ \\alpha ) & = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\mathbb { X } _ \\alpha ( n ) } { ( n + \\alpha ) ^ s } \\phi \\left ( \\frac { n + \\alpha } { N } \\right ) \\end{align*}"}
+{"id": "4478.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { L } ( { \\mathbf V } , \\Psi ) : = \\mathbb { L } ( { \\mathbf U } ^ a + { \\mathbf V } , \\Psi ^ a + \\Psi ) - \\mathbb { L } ( { \\mathbf U } ^ a , \\Psi ^ a ) = \\mathcal { F } ^ a , & \\Omega _ T , \\\\ \\mathcal { B } ( { \\mathbf V } | _ { x _ 1 = 0 } , \\psi ) : = \\mathbb { B } ( { \\mathbf U } ^ a | _ { x _ 1 = 0 } + { \\mathbf V } | _ { x _ 1 = 0 } , \\varphi ^ a + \\psi ) = 0 , & \\Gamma _ T , \\\\ ( { \\mathbf V } , \\psi ) = 0 , & t < 0 . \\end{cases} \\end{align*}"}
+{"id": "1756.png", "formula": "\\begin{align*} \\{ a , b _ 1 \\bullet b _ 2 \\} = \\{ a , b _ 1 \\} \\bullet b _ 2 + ( - 1 ) ^ { \\parallel b _ 1 \\parallel ( \\parallel a \\parallel + 1 ) } b _ 1 \\bullet \\{ a , b _ 2 \\} . \\end{align*}"}
+{"id": "4062.png", "formula": "\\begin{align*} T _ { n } = 1 + 2 \\sum _ { j = 1 } ^ { [ ( k - 1 ) / 2 ] } \\cos 2 j n \\pi a + \\frac { 1 + ( - 1 ) ^ { k } } { 2 } \\cos \\pi n . \\end{align*}"}
+{"id": "5886.png", "formula": "\\begin{align*} r _ { N } = N ^ { - \\beta } \\Phi ( - | \\Delta | + | \\Delta | ) = \\frac { 1 } { 2 } N ^ { - \\beta } . \\end{align*}"}
+{"id": "6209.png", "formula": "\\begin{align*} ( z , t ) ( w , s ) = \\Big ( z + w , t + s + \\frac 1 2 \\mathrm { I m } ( z \\cdot \\overline { w } ) \\Big ) , \\end{align*}"}
+{"id": "5188.png", "formula": "\\begin{align*} \\omega _ i = \\sum _ { j \\leq i } L _ j \\end{align*}"}
+{"id": "5833.png", "formula": "\\begin{align*} & s _ 3 s _ 4 s _ 5 s _ 6 ( s _ { \\alpha _ 7 + \\alpha _ 8 } s _ { \\alpha _ 1 + 2 \\alpha _ 2 + 2 \\alpha _ 3 + 3 \\alpha _ 4 + 2 \\alpha _ 5 + \\alpha _ 6 + \\alpha _ 7 } ) s _ 6 s _ 5 s _ 4 s _ 3 = \\\\ & s _ { \\alpha _ 3 + \\alpha _ 4 + \\alpha _ 5 + \\alpha _ 6 + \\alpha _ 7 + \\alpha _ 8 } s _ { \\alpha _ 1 + 2 \\alpha _ 2 + 3 \\alpha _ 3 + 4 \\alpha _ 4 + 3 \\alpha _ 5 + 2 \\alpha _ 6 + \\alpha _ 7 } . \\\\ \\end{align*}"}
+{"id": "1433.png", "formula": "\\begin{align*} \\sigma _ i = \\frac { 1 } { 2 \\pi } \\lim _ { r \\rightarrow 0 } \\lim _ { k \\rightarrow + \\infty } \\int _ { B _ r ( p ) } e ^ { u _ i ^ k ( x ) } \\mathrm { d } x , \\ \\ i = 1 , 2 , \\ \\ p \\ . \\end{align*}"}
+{"id": "2301.png", "formula": "\\begin{align*} \\mathcal { M } ( t ) : = \\int _ { \\R } | u | ^ { 2 } d x , \\end{align*}"}
+{"id": "5815.png", "formula": "\\begin{align*} & \\alpha ^ { d i } _ { m a x } : = \\alpha _ { n - i + 1 } + 2 \\sum \\limits _ { j = n - i + 2 } ^ { n - 2 } \\alpha _ j + \\alpha _ { n - 1 } + \\alpha _ n \\ ; \\ ; n \\geq 4 , \\\\ & \\alpha ^ { d 3 } _ { m a x } : = \\alpha ^ { a 3 } _ { m a x } : = \\alpha _ { n - 2 } + \\alpha _ { n - 1 } + \\alpha _ n . \\end{align*}"}
+{"id": "6808.png", "formula": "\\begin{align*} d y ' ( z ) & = c y ( z ) - s v ( z ) \\left ( 1 - \\frac { v ( z ) } { q ( u ( z ) ) } \\right ) \\\\ [ 0 . 2 c m ] & \\leq s v ( z ) \\left ( \\frac { v ( z ) } { q ( u ( z ) ) } - \\frac { q ( 1 ) } { q ( 0 ) } - 1 \\right ) < 0 . \\end{align*}"}
+{"id": "5741.png", "formula": "\\begin{align*} \\tilde \\Theta = \\frac { \\int _ { \\mathbb Q _ 4 ^ + } U ( X , t ) ^ 2 x _ { n + 1 } ^ a d X d t } { \\int _ { \\mathbb B _ 1 ^ + } U ( X , 0 ) ^ 2 x _ { n + 1 } ^ a d X } . \\end{align*}"}
+{"id": "3513.png", "formula": "\\begin{align*} \\Vert \\mathcal { L } _ { \\tau , s , \\rho } \\Vert _ { \\mathrm { H S } } ^ 2 = \\sum _ { b \\in \\mathcal { A } } \\sum _ { \\substack { \\textbf { a } , \\textbf { b } \\in Y ( \\tau ) \\\\ \\textbf { a } , \\textbf { b } \\to b } } \\mathrm { t r } \\left ( \\rho ( \\gamma _ { \\textbf { a } } ^ { - 1 } \\gamma _ { \\textbf { b } } ) \\right ) \\mathcal { I } _ { \\textbf { a } , \\textbf { b } } ^ { ( b ) } , \\end{align*}"}
+{"id": "3551.png", "formula": "\\begin{align*} & 0 \\in \\partial f ( x ^ { k + 1 } ) + \\frac { 1 } { \\eta } ( x ^ { k + 1 } - x ^ k - \\eta A ^ T \\lambda ^ k ) \\\\ & 0 \\in \\partial g ^ * ( \\lambda ^ { k + 1 } ) + \\frac { 1 } { \\eta } ( \\lambda ^ { k + 1 } - \\lambda ^ k + \\eta A ( 2 x ^ { k + 1 } - x ^ k ) ) \\ . \\end{align*}"}
+{"id": "4823.png", "formula": "\\begin{align*} \\sum _ { \\alpha , \\beta , \\gamma } S ^ { \\beta q } _ { \\gamma r } ( v ) S ^ { \\alpha p } _ { k \\gamma } ( u + v ) S ^ { i \\alpha } _ { j \\beta } ( u ) = \\sum _ { \\alpha , \\beta , \\gamma } S ^ { \\alpha p } _ { \\beta q } ( u ) S ^ { i \\alpha } _ { \\gamma r } ( u + v ) S ^ { j \\beta } _ { k \\gamma } ( v ) \\end{align*}"}
+{"id": "4081.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\oint _ { C ^ { \\mathfrak { p } } _ { 0 , 3 } } \\phi ( p ) \\cdot \\omega _ { 0 , 3 } ( p , p _ 1 , p _ 2 ) = 0 . \\end{align*}"}
+{"id": "1561.png", "formula": "\\begin{align*} \\partial _ t f + v \\partial _ x f = \\mathcal { L } f = \\frac { 1 } { \\kappa } \\left ( \\rho _ f \\left ( \\alpha \\mathcal { M } _ { u _ f , T _ f } + ( 1 - \\alpha ) \\mathcal { M } _ { \\tau ( x ) } \\right ) - f \\right ) , \\end{align*}"}
+{"id": "5408.png", "formula": "\\begin{align*} \\Delta _ w ( M ( t ) - z I ) & = \\exp \\Big [ N \\Psi ( z , w ; t ) \\Big ] , \\\\ \\Delta ^ 2 ( z , w ; t ) & = ( \\Delta _ w ( M ( t ) - z I ) ) ^ 2 = \\exp \\Big [ 2 N \\Psi ( z , w ; t ) \\Big ] , ( z , w ) \\in \\C \\times \\C ^ { \\times } , \\ , t \\geq 0 , \\end{align*}"}
+{"id": "6218.png", "formula": "\\begin{align*} L _ n ^ \\delta ( y ) e ^ { - y } y ^ \\delta = \\frac { 1 } { n ! } \\frac { \\dd ^ n } { \\dd y ^ n } \\big ( e ^ { - y } y ^ { n + \\delta } \\big ) , y \\ge 0 . \\end{align*}"}
+{"id": "5107.png", "formula": "\\begin{align*} R \\Gamma ^ { g e o } ( G , C ^ { l a } ( G , V ) ) = R \\Gamma ^ { g e o } ( G _ 0 , C ^ { l a } ( G _ 0 , V ) ) . \\end{align*}"}
+{"id": "377.png", "formula": "\\begin{align*} [ n ] = \\{ 0 , x _ 1 , . . . , x _ n , 1 \\} \\ , , \\end{align*}"}
+{"id": "1178.png", "formula": "\\begin{align*} & \\delta _ { 2 } ( m _ { 1 , n + 1 } ) + \\delta _ { 1 } ( m _ { 2 , n + 1 } ) = \\sum _ { \\substack { i + j = n + 1 \\\\ i , j > 0 } } [ m _ { 1 , i } , m _ { 2 , j } ] . \\end{align*}"}
+{"id": "1662.png", "formula": "\\begin{align*} \\Lambda ^ { ( m , n ) } _ { \\texttt { a } } : = \\{ l _ 1 \\omega _ 1 + \\cdots + l _ { n - 1 } \\omega _ { n - 1 } \\mid l _ 1 , \\ldots , l _ { n - 1 } \\in \\mathbb { Z } _ { \\geq 0 } , \\ , l _ 1 + \\cdots + l _ { n - 1 } \\leq m \\} . \\end{align*}"}
+{"id": "7401.png", "formula": "\\begin{align*} \\left ( W _ { n } ^ { M } \\right ) ' ( t ) = \\partial _ { t } W _ { n } ( x _ { t } , t ) \\end{align*}"}
+{"id": "8752.png", "formula": "\\begin{align*} \\int K ( u ) d u = 0 , \\int u K ( u ) d u = 1 , \\int u ^ j K ( u ) d u = 0 , \\ j = 2 , \\dots , \\ell , ~ \\kappa _ \\beta = \\int | u | ^ { \\beta } | K ( u ) | d u < \\infty . \\end{align*}"}
+{"id": "5304.png", "formula": "\\begin{align*} \\hat { x } _ { i + 2 } & \\coloneqq x _ { i - 1 } + \\frac { y _ { i + 2 } - y _ { i - 1 } } { a _ i } \\\\ & = x _ { i - 1 } + \\frac { y _ i - y _ { i - 1 } } { a _ i } + \\frac { y _ { i + 1 } - y _ i } { a _ i } + \\frac { y _ { i + 2 } - y _ { i + 1 } } { a _ i } \\\\ & = x _ { i - 1 } + ( x _ i - x _ { i - 1 } ) + \\frac { a _ { i + 1 } } { a _ i } ( x _ { i + 1 } - x _ i ) + \\frac { a _ { i + 2 } } { a _ i } ( x _ { i + 2 } - x _ { i + 1 } ) \\\\ & = x _ i + \\frac { a _ { i + 1 } } { a _ i } ( x _ { i + 1 } - x _ i ) + \\frac { a _ { i + 2 } } { a _ i } ( x _ { i + 2 } - x _ { i + 1 } ) . \\end{align*}"}
+{"id": "3235.png", "formula": "\\begin{align*} \\Delta _ { \\alpha } ( \\epsilon , R ) = \\Delta _ { \\alpha } ^ { + } ( \\epsilon , R ) + \\Delta _ { \\alpha } ^ { 0 } ( \\epsilon , R ) + \\Delta _ { \\alpha } ^ { - } ( \\epsilon , R ) , \\end{align*}"}
+{"id": "5718.png", "formula": "\\begin{align*} \\Delta ^ { \\xi , \\delta } : = \\frac 1 4 \\ , \\left ( B _ { 1 , 1 } ^ { \\xi , \\delta } - B _ { 2 , 2 } ^ { \\xi , \\delta } + \\delta ^ 2 \\ , \\left ( \\alpha _ 1 ^ { \\xi , \\delta } B _ { 1 , 3 } ^ { \\xi , \\delta } - \\alpha _ 2 ^ { \\xi , \\delta } B _ { 2 , 3 } ^ { \\xi , \\delta } \\right ) \\right ) ^ 2 + \\left ( B _ { 1 , 2 } ^ { \\xi , \\delta } + \\delta ^ 2 \\ , \\alpha _ 2 ^ { \\xi , \\delta } B _ { 1 , 3 } ^ { \\xi , \\delta } \\right ) \\left ( B _ { 2 , 1 } ^ { \\xi , \\delta } + \\delta ^ 2 \\ , \\alpha _ 1 ^ { \\xi , \\delta } B _ { 2 , 3 } ^ { \\xi , \\delta } \\right ) \\ , . \\end{align*}"}
+{"id": "3591.png", "formula": "\\begin{align*} y = { \\bf w } ^ H _ { \\rm B F } { \\bf H } { \\bf f } _ { \\rm B F } s + { \\bf w } ^ H _ { \\rm B F } { \\bf n } . \\end{align*}"}
+{"id": "6065.png", "formula": "\\begin{align*} K _ 1 = \\mathbb { E } \\vert \\int _ { 0 } ^ { t } b ( { X } ^ { { { { \\mathcal { P } } } } , { M _ { { \\ \\mathcal { P } } } } } _ { \\tau ^ { { { \\mathcal { P } } } } ( r ) } ) - b ( { X } ^ { M _ { { \\ \\mathcal { P } } } } _ { r } ) d r \\vert , \\end{align*}"}
+{"id": "850.png", "formula": "\\begin{align*} \\| e v _ { z _ 0 } \\| = 1 = | e v _ { z _ 0 } ( 1 ) | , \\end{align*}"}
+{"id": "6965.png", "formula": "\\begin{align*} T _ 4 : = \\epsilon ^ 2 \\sum _ { k = 1 } ^ d \\sum _ { i = k + 1 } ^ d \\sum _ { j = i + 1 } ^ d \\frac { \\cos ^ 2 ( x _ j / 2 ) - \\cos ^ 2 ( x _ i / 2 ) + \\cos ^ 2 ( x _ k / 2 ) } { Y } , \\end{align*}"}
+{"id": "3096.png", "formula": "\\begin{align*} [ t ^ T A ] _ { i j } = \\left \\{ \\begin{matrix} - \\alpha & i j \\in \\{ 1 2 , 1 3 \\} \\\\ \\alpha - 2 \\beta & i j = 2 3 \\\\ \\gamma - 2 \\beta & i = 1 , j \\ge 4 \\\\ - \\gamma & i \\ge 2 , j \\ge 4 \\end{matrix} \\right . \\end{align*}"}
+{"id": "2128.png", "formula": "\\begin{align*} h _ 2 ( t , x ) = \\partial _ t \\Lambda \\partial _ x \\Lambda + 4 \\sinh ^ 2 ( \\Lambda ) \\partial _ t \\phi \\partial _ x \\phi , \\end{align*}"}
+{"id": "1938.png", "formula": "\\begin{align*} \\frac { { \\rm d } } { { \\rm d } t } \\vec { \\alpha } ( t ) = \\mathcal { L } ( \\vec { \\alpha } , t ) \\end{align*}"}
+{"id": "2699.png", "formula": "\\begin{align*} x _ i ^ 2 + y _ i ^ 2 + z _ i ^ 2 & = x _ i ^ 2 + \\left ( \\frac { b w _ i + e x _ i } { a } \\right ) ^ 2 + w _ i ^ 2 c ^ 2 \\\\ & = \\frac { ( a ^ 2 + e ^ 2 ) } { a ^ 2 } \\left ( x _ i + \\frac { b e w _ i } { a ^ 2 + e ^ 2 } \\right ) ^ 2 + \\frac { w _ i ^ 2 | | \\vec { a } | | ^ 2 } { ( a ^ 2 + e ^ 2 ) } = \\frac { w _ i ^ 2 | | \\vec { a } | | ^ 2 } { ( a ^ 2 + e ^ 2 ) } \\left ( U _ i ^ 2 + 1 \\right ) . \\end{align*}"}
+{"id": "7911.png", "formula": "\\begin{align*} \\mathrm { F P d i m } ( x x ^ \\ast ) = 1 + \\sum _ { y \\in \\Gamma _ 1 } c _ { x , x ^ \\ast } ^ y \\mathrm { F P d i m } ( y ) ^ 2 + 2 \\sum _ { z \\in \\Gamma _ 2 } c _ { x , x ^ \\ast } ^ z \\mathrm { F P d i m } ( z ) ^ 2 , \\end{align*}"}
+{"id": "6226.png", "formula": "\\begin{align*} C _ { p , q } ^ { ( \\nu ) } ( t ) \\coloneqq \\begin{cases} C e ^ { - t d ^ \\nu } & ( t \\ge 1 ) \\\\ C t ^ { - \\mu _ \\nu } & ( 0 < t \\le 1 ) , \\end{cases} \\mu _ \\nu \\coloneqq \\max \\Big \\{ \\frac { d } { \\nu } \\Big ( \\frac { 1 } { \\min \\{ q , q ' \\} } - \\frac { 1 } { \\max \\{ p , p ' \\} } \\Big ) , 0 \\Big \\} , \\end{align*}"}
+{"id": "4746.png", "formula": "\\begin{align*} { \\rm P e r } ( A ; \\R ^ 2 \\setminus \\R _ { + } ^ { 2 } ) = { \\rm P e r } ( A \\setminus \\R _ { + } ^ { 2 } ; \\R ^ 2 \\setminus \\R _ { + } ^ { 2 } ) . \\end{align*}"}
+{"id": "2193.png", "formula": "\\begin{align*} \\C _ M ( x \\mid y ) = \\min \\{ | p | : M ( p , y ) = x \\} . \\end{align*}"}
+{"id": "2953.png", "formula": "\\begin{align*} J _ s ^ n = \\begin{bmatrix} n \\\\ s \\end{bmatrix} - \\begin{bmatrix} n \\\\ s + 1 \\end{bmatrix} \\end{align*}"}
+{"id": "2478.png", "formula": "\\begin{align*} | \\eta ( P _ A ) - \\eta ( P ' _ A ) | & \\leq \\lim _ { n \\rightarrow \\infty } \\log \\left ( \\max _ a | \\mathcal { V } _ a | \\right ) \\cdot \\| T _ { \\overline { a } ^ n } - T _ { \\overline { a } '^ n } \\| _ 1 \\\\ & = \\log \\left ( \\max _ a | \\mathcal { V } _ a | \\right ) \\cdot \\| P _ A - P ' _ A \\| _ 1 . \\end{align*}"}
+{"id": "3120.png", "formula": "\\begin{align*} \\mathbb { E } [ \\rho _ { i , j } ] = \\mathbb { E } \\left [ \\sqrt { \\rho ^ 2 _ { i , j } } \\right ] \\leq \\sqrt { \\mathbb { E } [ \\rho ^ 2 _ { i , j } ] } \\end{align*}"}
+{"id": "4410.png", "formula": "\\begin{align*} \\hat { H } ^ { \\pm } _ 2 \\partial _ 2 \\varphi - \\dot { H } ^ { \\natural \\ , \\pm } _ { N } { \\mp } \\varphi \\partial _ 1 \\hat { H } ^ { \\pm } _ { N } = 0 \\Gamma _ T . \\end{align*}"}
+{"id": "1052.png", "formula": "\\begin{align*} \\hat { u } _ \\eta ^ * \\in [ 0 , u ] , \\ ; \\hat { u } _ \\eta ^ * \\not = 0 . \\end{align*}"}
+{"id": "6115.png", "formula": "\\begin{align*} \\sigma _ A ( T ) \\backslash \\{ 0 \\} = \\Big \\{ g ( T ) : g \\ \\ g ( P ) = 1 \\Big \\} \\backslash \\{ 0 \\} \\end{align*}"}
+{"id": "7834.png", "formula": "\\begin{align*} T = - ( \\tau _ { i j } ) \\mathrm { d } Z ^ i \\mathrm { d } \\overline { Z } ^ j + \\frac { \\chi } { 2 \\pi } \\sum _ { q _ 0 \\in \\mathbb { Z } - \\{ 0 \\} } \\sum _ { n > 0 } e ^ { - 2 \\pi \\mathrm { i } n q _ 0 \\zeta ^ 0 } q _ 0 ^ 2 K _ 0 ( 2 \\pi n \\tau _ 2 | q _ 0 | ) | \\mathrm { d } Z ^ 0 | ^ 2 \\ , . \\end{align*}"}
+{"id": "6274.png", "formula": "\\begin{align*} i u _ t + u _ { x x } = 0 , u ( 0 ) = u _ 0 , \\end{align*}"}
+{"id": "5372.png", "formula": "\\begin{align*} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\cdot \\bigg ( \\cfrac { r } { s } \\bigg ) = \\frac { a r + b s } { c r + d s } , \\end{align*}"}
+{"id": "3174.png", "formula": "\\begin{align*} S = \\begin{bmatrix} 0 & - 1 \\\\ I _ { n - 1 } & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "8093.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } + \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } < \\infty , \\end{align*}"}
+{"id": "4499.png", "formula": "\\begin{align*} e '' _ k : = \\mathcal { L } ' ( { \\mathbf V } _ k , \\Psi _ k ) ( \\delta { \\mathbf V } _ k , \\delta \\Psi _ k ) - \\mathcal { L } ' ( S _ { \\theta _ k } { \\mathbf V } _ k , S _ { \\theta _ k } \\Psi _ k ) ( \\delta { \\mathbf V } _ k , \\delta \\Psi _ k ) , \\end{align*}"}
+{"id": "4536.png", "formula": "\\begin{align*} { \\mathbf V } ^ \\pm _ n = ( \\dot q ^ \\pm , \\dot u ^ \\pm _ n , \\dot H ^ \\pm _ n ) \\ , . \\end{align*}"}
+{"id": "2114.png", "formula": "\\begin{align*} & I _ { 3 } : = \\iint _ { D _ t } \\varphi ( u ) | L \\tilde { \\Lambda } | | Q _ 0 ( \\ln \\alpha , \\tilde { \\Lambda } ) | \\lesssim \\gamma \\varepsilon ^ 2 , \\\\ & I _ 4 : = \\iint _ { D _ t } \\varphi ( u ) | L \\tilde { \\Lambda } | | Q _ 0 ( \\partial _ x ( \\ln \\alpha ) , \\tilde { \\Lambda } ) + Q _ 0 ( \\ln \\alpha , \\partial _ x \\tilde { \\Lambda } ) | \\lesssim \\gamma \\varepsilon ^ 2 . \\end{align*}"}
+{"id": "6459.png", "formula": "\\begin{align*} C ^ \\infty ( A ) : = \\bigcap _ { n \\in \\N } D ( A ^ n ) , \\end{align*}"}
+{"id": "1588.png", "formula": "\\begin{align*} A f = - \\Pi \\mathcal { T } f + \\Pi \\mathcal { T } ^ 2 \\Pi A f , f \\in \\mathcal { H } \\end{align*}"}
+{"id": "6732.png", "formula": "\\begin{align*} \\frac { \\partial g _ { \\alpha \\beta } } { \\partial t } = - 2 \\eta ( t ) f ' ( t ) g _ { \\alpha \\beta } = - 2 t ^ { - 1 } \\left ( 1 + \\frac { m } { 2 t } \\right ) ^ { - 1 } \\left ( 1 - \\frac { m } { 2 t } \\right ) g _ { \\alpha \\beta } . \\end{align*}"}
+{"id": "244.png", "formula": "\\begin{align*} \\Phi _ \\xi ^ { } ( x ; g ) = { \\delta ( x ; g ) } { C ^ { } ( \\rho _ g ; g ) } F _ { B C _ n } ( \\xi , x ; g ) \\end{align*}"}
+{"id": "2510.png", "formula": "\\begin{align*} \\begin{aligned} \\big ( \\sigma \\partial _ t ^ { 1 / 2 } \\boldsymbol { y } , \\partial _ t ^ { 1 / 2 } \\boldsymbol { v } \\big ) = \\big ( \\sigma \\partial _ t \\boldsymbol { y } , \\boldsymbol { v } ^ { \\perp } \\big ) \\mbox { a n d } \\big ( \\sigma \\partial _ t ^ { 1 / 2 } \\boldsymbol { y } , \\partial _ t ^ { 1 / 2 } \\boldsymbol { v } ^ { \\perp } \\big ) = \\big ( \\sigma \\partial _ t \\boldsymbol { y } , \\boldsymbol { v } \\big ) \\end{aligned} \\end{align*}"}
+{"id": "5463.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\exp \\left \\{ \\sum _ { k = 0 } ^ { n - 1 } G ^ * \\left ( \\frac { 1 } { \\pi ( k ) } \\right ) \\pi ( k ) \\right \\} \\pi ( n ) & \\leq e ^ C \\left ( 1 + \\sum _ { n = 2 } ^ \\infty ( n - 1 ) ! \\lambda ^ { - n } \\frac { ( \\log n ) ^ D } { n } \\pi ( n ) \\right ) \\\\ & = e ^ { C } \\left ( 1 + e ^ { - \\lambda } \\sum _ { n = 2 } ^ \\infty \\frac { ( \\log n ) ^ D } { n ^ 2 } \\right ) \\\\ & < \\infty . \\end{align*}"}
+{"id": "811.png", "formula": "\\begin{align*} c + o ( 1 ) & = I [ u _ n ] = \\frac { 1 } { p } [ u _ n ] _ { s , p } ^ p - \\frac { b } { 2 ( p ^ \\sharp ) ^ 2 } \\int _ { \\mathbb { R } ^ N } ( K \\ast | u _ n | ^ { p ^ \\sharp } ) | u _ n | ^ { p ^ \\sharp } d x + \\frac { \\varepsilon _ g } { p _ g } \\| u _ n \\| _ { p _ g } ^ { p _ g } + o ( 1 ) , \\\\ o ( 1 ) & = I ' [ u _ n ] u _ n = [ u _ n ] _ { s , p } ^ p - \\frac { b } { p ^ \\sharp } \\int _ { \\mathbb { R } ^ N } ( K \\ast | u _ n | ^ { p ^ \\sharp } ) | u _ n | ^ { p ^ \\sharp } d x - \\varepsilon _ g \\| u _ n \\| _ { p _ g } ^ { p _ g } + o ( 1 ) . \\end{align*}"}
+{"id": "3038.png", "formula": "\\begin{align*} f ( n ) \\le 4 \\binom { 2 T } { 2 } \\frac { n } { 2 T } + ( 3 + \\varepsilon / 2 ) 2 T \\frac { n ( \\frac { n } { 2 T } - 1 ) } { 2 } \\le ( 3 + \\varepsilon ) \\binom { n } { 2 } , \\end{align*}"}
+{"id": "6671.png", "formula": "\\begin{align*} x ^ 5 R ^ { ( 5 ) } ( x ) & + 1 0 x ^ 4 R ^ { ( 4 ) } ( x ) + x ^ 3 ( 2 0 x ^ 2 - 2 0 q x + 2 2 + 1 0 \\tilde { p } ) R { ''' } ( x ) \\\\ + & x ^ 2 ( 6 4 x ^ 2 - 7 6 q x + 4 + 4 4 \\tilde { p } ) R { '' } ( x ) \\\\ + & 4 x ( 1 6 x ^ 4 - 3 2 q x ^ 3 + 4 ( 4 q ^ 2 + 4 \\tilde { p } + 1 ) x ^ 2 - q ( 6 + 1 6 \\tilde { p } ) x + 4 \\tilde { p } ^ 2 + 7 \\tilde { p } - 1 ) R { ' } ( x ) \\\\ + & 8 ( - 4 q x ^ 3 + 4 ( q ^ 2 + \\tilde { p } ) x ^ 2 - q ( 6 \\tilde { p } - 1 ) x + 2 \\tilde { p } ^ 2 ) R ( x ) = 0 . \\end{align*}"}
+{"id": "2482.png", "formula": "\\begin{align*} \\sum _ { \\mathcal { S } \\in \\Gamma ( G ) } P _ { W ^ * } ( \\mathcal { S } ) = & \\ ; \\sum _ { \\mathcal { S } \\in \\Gamma ( G ) } \\ ; \\ ; \\prod _ { a ' \\in \\mathcal { A } } P _ { W ^ * _ { a ' } } ( \\mathcal { S } _ { a ' } ) \\\\ = & \\ ; \\sum _ { ( \\mathcal { S } _ { a ' } ) _ { a ' \\in \\mathcal { A } } \\in \\prod _ { a ' \\in \\mathcal { A } } \\Gamma ( G _ { a ' } ) } \\ ; \\ ; \\prod _ { a ' \\in \\mathcal { A } } P _ { W ^ * _ { a ' } } ( \\mathcal { S } _ { a ' } ) \\\\ = & \\ ; 1 , \\end{align*}"}
+{"id": "6206.png", "formula": "\\begin{align*} \\mathcal H ^ 2 ( \\varphi ( V ) ) = \\int _ V \\norm { \\varphi _ x \\times \\varphi _ y ( u , v ) } \\d u \\d v , \\end{align*}"}
+{"id": "1956.png", "formula": "\\begin{align*} \\int _ \\R \\int _ I | \\bar { f } | \\ , { \\rm d } x \\ , { \\rm d } v = 0 . \\end{align*}"}
+{"id": "8500.png", "formula": "\\begin{align*} ( \\partial ^ { * } F _ { \\ell } ) _ { \\bar { z } } = _ { \\mathcal { H } ^ { n - 1 } } B ^ { n - 1 } \\left ( 0 , r _ { \\ell } ^ { \\vee } ( \\bar { z } ) \\right ) \\backslash B ^ { n - 1 } \\left ( 0 , r _ { \\ell } ^ { \\wedge } ( \\bar { z } ) \\right ) . \\end{align*}"}
+{"id": "5451.png", "formula": "\\begin{align*} | f ( x ) | & = | \\int _ 0 ^ x f ' ( t ) e ^ { t ^ 2 / 2 } e ^ { - t ^ 2 / 2 } d t | \\\\ & \\leq \\int _ 0 ^ x G ( | f ' ( t ) | ) e ^ { - t ^ 2 / 2 } d t + \\int _ 0 ^ x G ^ * ( e ^ { t ^ 2 / 2 } ) e ^ { - t ^ 2 / 2 } d t \\end{align*}"}
+{"id": "5506.png", "formula": "\\begin{align*} g = a ^ { \\epsilon _ 1 } b ^ { k _ 1 } \\dots a ^ { \\epsilon _ n } b ^ { k _ n } , \\epsilon _ 1 , \\dots , \\epsilon _ n \\in \\{ \\pm 1 \\} , k _ 1 , \\dots , k _ n \\in \\mathbb Z , n \\in \\mathbb Z _ { \\geq 0 } , \\end{align*}"}
+{"id": "8116.png", "formula": "\\begin{align*} v _ \\alpha - q + 1 \\le u _ \\alpha - p \\end{align*}"}
+{"id": "7400.png", "formula": "\\begin{align*} W _ { n } ^ { M } ( t ) : = W _ { n } ( x _ { t } , t ) \\equiv \\max _ { x \\in \\mathbb { T } } W _ { n } ( x , t ) \\end{align*}"}
+{"id": "611.png", "formula": "\\begin{align*} M _ T ( \\phi ) ^ 2 \\le \\left ( \\frac { T } { T _ 0 } \\right ) ^ { 1 - 2 b } M _ { T _ 0 } ( g ) ^ 2 \\le C ^ 2 \\left ( \\frac { T } { T _ 0 } \\right ) ^ { 1 - 2 b } N _ { T _ 0 } ( g ) ^ 2 = C ^ 2 N _ T ( \\phi ) ^ 2 . \\end{align*}"}
+{"id": "345.png", "formula": "\\begin{align*} \\mathbf { u } ^ { \\operatorname * { m u l t } } = \\left ( \\mathbf { u } _ { j } ^ { \\operatorname * { m u l t } } \\right ) _ { j = 1 } ^ { n _ { \\Omega } } = \\left ( \\left ( u _ { \\operatorname * { D } ; j } ^ { \\operatorname * { m u l t } } , u _ { \\operatorname * { N } ; j } ^ { \\operatorname * { m u l t } } \\right ) \\right ) _ { j = 1 } ^ { n _ { \\Omega } } \\in \\mathbb { X } \\left ( \\mathcal { P } _ { \\Omega } \\right ) \\end{align*}"}
+{"id": "4085.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\oint _ { C ^ { \\mathfrak { p } } _ { g , n + 2 } } \\phi ( p ) \\cdot d _ p \\left ( \\frac { \\eta ^ p ( p _ 0 ) } { 2 \\omega _ { 0 , 1 } ( p ) } \\omega _ { g , n + 1 } ( p , J ) \\right ) = - \\frac 1 2 \\omega _ { g , n + 1 } ( p _ 0 , J ) , \\end{align*}"}
+{"id": "1361.png", "formula": "\\begin{align*} R _ { 2 - 2 k } ^ { \\ell - 1 } : = R _ { 2 \\ell - 2 k - 2 } \\circ \\cdots \\circ R _ { 2 - 2 k } \\quad R _ { 2 \\kappa } : = 2 i \\frac { \\partial } { \\partial z } + \\frac { 2 \\kappa } { y } \\end{align*}"}
+{"id": "852.png", "formula": "\\begin{align*} N = N _ 1 + N _ 2 , \\end{align*}"}
+{"id": "5288.png", "formula": "\\begin{align*} \\mathcal { R } ( \\mathbf { n } ) \\leq \\sum _ { ( j _ 1 , \\dots , j _ L ) \\in J } \\prod _ { l = 1 } ^ L { n _ l \\choose j _ l } , \\end{align*}"}
+{"id": "4091.png", "formula": "\\begin{align*} \\frac { d } { d x ( p ) } \\cdot f ( p ) : = \\frac { 1 } { d x ( p ) } d f ( p ) . \\end{align*}"}
+{"id": "4023.png", "formula": "\\begin{align*} L ^ p ( \\mathcal I ; L ^ r ( \\Omega _ \\eta ) ) & : = \\big \\{ v \\in L ^ 1 ( \\mathcal I \\times \\Omega _ \\eta ) : \\ , \\ , v ( t , \\cdot ) \\in L ^ r ( \\Omega _ { \\eta ( t ) } ) \\ , \\ , t , \\ , \\ , \\| v ( t , \\cdot ) \\| _ { L ^ r ( \\Omega _ { \\eta ( t ) } ) } \\in L ^ p ( \\mathcal I ) \\big \\} , \\\\ L ^ p ( \\mathcal I ; W ^ { 1 , r } ( \\Omega _ \\eta ) ) & : = \\big \\{ v \\in L ^ p ( \\mathcal I ; L ^ r ( \\Omega _ \\eta ) ) : \\ , \\ , \\nabla v \\in L ^ p ( \\mathcal I ; L ^ r ( \\Omega _ \\eta ) ) \\big \\} . \\end{align*}"}
+{"id": "7824.png", "formula": "\\begin{align*} \\varphi _ T ( v ^ a , ( \\eta ^ a , \\eta _ i , \\kappa ) ) = ( v ^ a , ( \\eta ^ a - v ^ a , . . . ) ) , \\varphi _ S ( v ^ a , ( \\eta ^ a , \\eta _ i , \\kappa ) ) = ( - \\eta ^ a , ( v ^ a , . . . ) ) \\ , . \\end{align*}"}
+{"id": "2276.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ { n } F _ n ( x _ i - x _ j ) & = \\sum _ { i , j = 1 } ^ { n } \\sum _ { | k | \\leq n } \\left ( 1 - \\frac { | k | } { n } \\right ) e ^ { 2 \\pi i k ( x _ i - x _ j ) } \\\\ & = \\sum _ { | k | \\leq n } \\left ( 1 - \\frac { | k | } { n } \\right ) \\sum _ { i , j = 1 } ^ { n } e ^ { 2 \\pi i k ( x _ i - x _ j ) } \\\\ & = \\sum _ { | k | \\leq n } \\left ( 1 - \\frac { | k | } { n } \\right ) \\left | \\sum _ { j = 1 } ^ { n } e ^ { 2 \\pi i k x _ j } \\right | ^ 2 . \\end{align*}"}
+{"id": "8479.png", "formula": "\\begin{align*} g ^ { \\wedge } ( x ) = \\sup \\left \\{ s \\in \\mathbb { R } : x \\in \\{ g < s \\} ^ { ( 0 ) } \\right \\} = \\sup \\left \\{ s \\in \\mathbb { R } : x \\in \\{ g > s \\} ^ { ( 1 ) } \\right \\} , \\end{align*}"}
+{"id": "5497.png", "formula": "\\begin{align*} c _ - ^ { ( h , g ) } = & \\mathrm { p e x p } \\left ( - \\dfrac { ( Q ^ 8 - Q ^ { - 8 } ) } { ( 1 + s ^ { - \\varphi ( g ) _ { 2 , 2 } } p ^ { \\varphi ( g ) _ { 2 , 1 } } ) ( 1 + s ^ { \\varphi ( h g ) _ { 1 , 2 } } p ^ { - \\varphi ( h g ) _ { 1 , 1 } } ) ( 1 + s ^ { \\varphi ( h g ) _ { 2 , 2 } } p ^ { - \\varphi ( h g ) _ { 2 , 1 } } ) } \\right ) \\qquad \\in \\quad \\mathbf R . \\end{align*}"}
+{"id": "132.png", "formula": "\\begin{align*} x ( t ) = f ( t ) + \\int _ { 0 } ^ { t } k _ 1 ( s , t ) x ( s ) d s + \\int _ { 0 } ^ { t } k _ 2 ( s , t ) x ( s ) d B ( s ) \\end{align*}"}
+{"id": "5042.png", "formula": "\\begin{align*} Y ( \\vec q , \\vec 0 ) = d _ 2 Y ( \\vec q , \\vec 0 ) = \\vec 0 \\vec q \\in B _ { c } . \\end{align*}"}
+{"id": "2446.png", "formula": "\\begin{align*} \\| \\nabla \\phi ^ x \\| _ { L ^ 2 ( \\Omega ) } ^ 2 & = \\langle \\phi ^ x , \\varphi ^ x \\rangle _ { L ^ 2 ( \\Omega ) } \\leq \\| \\phi ^ x \\| _ { L ^ 2 ( \\Omega ) } \\| \\varphi ^ x \\| _ { L ^ 2 ( \\Omega ) } \\leq \\frac { \\| \\nabla \\phi ^ x \\| _ { L ^ 2 ( \\Omega ) } } { \\sqrt { \\lambda _ 1 ( \\Omega ) } } \\| \\varphi ^ x \\| _ { L ^ 2 ( \\Omega ) } , \\end{align*}"}
+{"id": "3437.png", "formula": "\\begin{align*} \\phi _ j = \\frac { 1 } { l } \\max _ { i = 1 } ^ { N } ( \\log | s _ i | + c _ i ) , s _ i \\in H ^ 0 ( X , l L ) , c _ i \\in \\R , \\end{align*}"}
+{"id": "4203.png", "formula": "\\begin{align*} \\sigma ( \\psi ( S ) \\psi ( i T ) - \\psi ( i T ) \\psi ( S ) ^ { \\ast } ) & = \\sigma ( S ( i T ) - ( i T ) S ^ { \\ast } ) = i \\sigma ( S T - T S ^ { \\ast } ) \\\\ & = i \\sigma ( \\psi ( S ) \\psi ( T ) - \\psi ( T ) \\psi ( S ) ^ { \\ast } ) \\\\ & = \\sigma ( \\psi ( S ) ( i \\psi ( T ) ) - ( i \\psi ( T ) ) \\psi ( S ) ^ { \\ast } ) . \\end{align*}"}
+{"id": "3406.png", "formula": "\\begin{align*} \\Theta : = - v _ 1 ^ 2 + v _ 1 ^ 4 + v _ 2 ^ 2 - 1 0 v _ 1 ^ 2 v _ 2 ^ 2 + 9 v _ 1 ^ 4 v _ 2 ^ 2 + v _ 2 ^ 4 - 9 v _ 1 ^ 2 v _ 2 ^ 4 \\neq 0 . \\end{align*}"}
+{"id": "2442.png", "formula": "\\begin{align*} | \\xi | ^ 2 = \\int _ 0 ^ \\infty \\ 1 _ { \\{ | \\xi | ^ 2 > \\ell \\} } \\ , { \\rm d } \\ell , \\end{align*}"}
+{"id": "1592.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ r \\frac { p ^ { \\beta _ j } - 1 } { p - 1 } d _ j \\ , < \\ , N - \\mathcal { M } ( \\beta - 1 ) \\ , . \\end{align*}"}
+{"id": "7100.png", "formula": "\\begin{align*} ( y _ 1 + w ) ^ { 1 / 3 } = y ^ { 1 / 3 } _ 1 \\ , \\left ( 1 + \\frac { 1 } { 3 } \\frac { w } { y } - \\frac { 1 } { 9 } \\left ( \\frac { w } { y } \\right ) ^ 2 + . . . \\right ) = y ^ { 1 / 3 } _ 1 + . \\end{align*}"}
+{"id": "5201.png", "formula": "\\begin{align*} D + \\ell ^ { - 1 } \\sum _ { i > D } | \\tau ( N _ i \\cap Z ) \\cap L _ i ^ 0 | < 3 D = : M , \\end{align*}"}
+{"id": "105.png", "formula": "\\begin{align*} \\ell _ { } : = \\rho ^ { - 1 / 2 } \\widehat { g } ( 0 ) ^ { - 1 / 2 } . \\end{align*}"}
+{"id": "1395.png", "formula": "\\begin{align*} \\sigma ( x ) & = - \\frac { 1 } { \\pi i } \\oint _ { \\Gamma _ N } { \\Bigg ( \\tilde \\varphi ( x , \\mu ) \\varphi ( x , \\mu ) - \\frac { 1 } { 2 } \\Bigg ) \\hat M ( \\mu ) } d \\mu \\\\ & - 2 \\sum _ { k = N + 1 } ^ \\infty \\sum _ { j = 0 } ^ 1 ( - 1 ) ^ j \\alpha _ { k j } \\Bigg ( \\tilde \\varphi _ { k , j } ( x ) \\varphi _ { k , j } ( x ) - \\frac { 1 } { 2 } \\Bigg ) , \\end{align*}"}
+{"id": "7604.png", "formula": "\\begin{align*} a = t - s , & & b = t + s \\end{align*}"}
+{"id": "5022.png", "formula": "\\begin{align*} \\nabla ^ { \\iota ^ * g } _ { r \\partial _ r } \\gamma ' - \\nabla ^ { \\gamma } _ { r \\partial _ r } \\gamma ' = ( \\iota ^ * g ) ^ { - 1 } * \\nabla ^ \\gamma ( \\iota ^ * g ) * ( r \\partial _ r ) * \\gamma ' , \\end{align*}"}
+{"id": "6847.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\nearrow 1 } T ( \\varepsilon ) = \\infty { \\rm a n d } \\lim _ { \\varepsilon \\searrow 1 } T ( \\varepsilon ) = T _ { \\rm c y l } . \\end{align*}"}
+{"id": "8660.png", "formula": "\\begin{align*} y _ 0 = y _ { n + 1 } \\hat { f } \\Big ( \\frac { y _ 1 } { y _ { n + 1 } } , . . . , \\frac { y _ n } { y _ { n + 1 } } \\Big ) \\end{align*}"}
+{"id": "8791.png", "formula": "\\begin{align*} \\sum _ { t = T _ { 1 } + 1 } ^ { T } \\mathbb { E } [ f ( x _ { t } ) - f ^ * ] \\leq \\frac { \\mathcal { B } _ { 3 } } { \\alpha } r _ { T _ { 1 } + 1 } + \\mathcal { B } _ { 4 } \\frac { d } { \\alpha } T ^ { 1 / \\beta } , \\end{align*}"}
+{"id": "4310.png", "formula": "\\begin{align*} \\begin{alignedat} { 3 } s ( q ) & = w \\\\ v ( q ) & = x i + y j + z k \\end{alignedat} \\end{align*}"}
+{"id": "8800.png", "formula": "\\begin{align*} \\forall x \\in \\Theta : \\mathbb { E } \\sum _ { t = 1 } ^ { T } \\big ( f ( x _ t ) - f ( x ) \\big ) & \\le \\sum _ { t = 1 } ^ { T } \\Big ( L h _ t ^ 2 + \\frac { 1 } { 2 \\alpha t } \\mathbb { E } [ \\| { \\hat g } _ t \\| ^ 2 ] \\Big ) . \\end{align*}"}
+{"id": "8135.png", "formula": "\\begin{align*} \\beta ' ( t _ i ) = \\varphi ( \\gamma ( t _ i ) ) = \\varphi ( \\psi ( t _ i ^ * , 1 ) ) = \\beta ( t _ i ^ * ) . \\end{align*}"}
+{"id": "2909.png", "formula": "\\begin{align*} - \\nu ^ \\vee + ( 1 + \\langle \\lambda , \\nu ^ \\vee \\rangle ) = & b _ { \\ell - 1 } \\in R [ { \\lambda + \\nu } ] . \\end{align*}"}
+{"id": "697.png", "formula": "\\begin{align*} \\tau _ R ( \\omega ) = \\sup \\left \\{ t \\in [ 0 , \\tau ( \\omega ) ) \\colon \\right \\} \\end{align*}"}
+{"id": "8430.png", "formula": "\\begin{align*} Y _ { i i } ^ \\star = \\psi ( \\lambda _ i ) = \\max \\Big \\{ \\frac { 1 - \\lambda _ i } { \\lambda _ i } , 0 \\Big \\} , \\end{align*}"}
+{"id": "76.png", "formula": "\\begin{align*} \\tau _ s : = z a _ { s } + \\alpha _ k \\alpha _ { s - k } \\bar z a _ { - s } ^ \\dagger . \\end{align*}"}
+{"id": "6051.png", "formula": "\\begin{align*} \\nabla _ x { X } _ { t } ^ { M _ { { \\mathcal { P } } } } = I _ d + \\int _ { 0 } ^ { t } \\int _ { B _ { M _ { { \\ \\mathcal { P } } } } ( s ) } \\nabla _ x c ( z , { X } _ { s - } ^ { M _ { { \\mathcal { P } } } } ) \\nabla _ x { X } _ { s - } ^ { M _ { { \\mathcal { P } } } } d N ( z , s ) + \\int _ { 0 } ^ { t } \\nabla _ x b ( { X } _ { s } ^ { M _ { { \\mathcal { P } } } } ) \\nabla _ x { X } _ s ^ { M _ { { \\mathcal { P } } } } d s , \\end{align*}"}
+{"id": "5247.png", "formula": "\\begin{align*} \\alpha _ t = \\frac { 1 } { t ^ { c } } , ~ \\gamma _ t = \\frac { \\gamma _ 0 } { \\alpha _ t } , ~ \\forall t \\in \\mathbb { N } _ + , \\end{align*}"}
+{"id": "713.png", "formula": "\\begin{align*} \\mathfrak T ( \\mathbf v ) ( t ) = \\eqref { i n d u c t i o n 1 - t r u n c a t e d } = : \\mathfrak T _ 0 ( t ) + \\mathfrak T _ 1 ( \\mathbf v ) ( t ) + \\mathfrak T _ 2 ( \\mathbf v ) ( t ) , . \\end{align*}"}
+{"id": "4234.png", "formula": "\\begin{align*} d \\omega ^ 1 \\ ! = 0 , \\ \\ \\ d \\omega ^ 2 \\ ! = - \\frac 1 2 \\omega ^ { 1 3 } - \\frac { 1 + 2 x i } { 2 } \\omega ^ { 1 \\bar { 3 } } + x i \\ , \\omega ^ { 3 \\bar { 1 } } , \\ \\ \\ d \\omega ^ 3 \\ ! = \\frac 1 2 \\omega ^ { 1 2 } + \\frac { 2 x - i } { 4 x } \\omega ^ { 1 \\bar { 2 } } + \\frac { i } { 4 x } \\ , \\omega ^ { 2 \\bar { 1 } } , \\end{align*}"}
+{"id": "1841.png", "formula": "\\begin{gather*} \\left \\| f ( s ) - \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + \\alpha ) ^ s } \\right \\| < \\epsilon \\end{gather*}"}
+{"id": "5657.png", "formula": "\\begin{align*} G _ Q ^ 1 = \\left \\{ \\left ( \\begin{array} { c c } \\left ( - \\frac { b ( x , y ) } { a ( x , y ) } \\alpha + \\beta \\right ) & \\left ( - \\frac { c ( x , y ) } { a ( x , y ) } \\alpha \\right ) \\\\ \\alpha & \\beta \\end{array} \\right ) \\right \\} , \\ \\ G _ Q ^ 2 = \\left \\{ \\left ( \\begin{array} { c c } - \\beta & \\left ( \\frac { c ( x , y ) } { a ( x , y ) } \\alpha - \\frac { b ( x , y ) } { a ( x , y ) } \\beta \\right ) \\\\ \\alpha & \\beta \\end{array} \\right ) \\right \\} . \\end{align*}"}
+{"id": "3807.png", "formula": "\\begin{align*} \\gamma = \\varsigma \\nu _ X + \\gamma ^ { \\perp } , \\nu _ X = \\varrho \\gamma + \\nu _ X ^ { \\perp } . \\end{align*}"}
+{"id": "4538.png", "formula": "\\begin{align*} \\tilde { \\mathcal F } = J ^ T \\tilde { \\mathbf F } \\quad \\mbox { a n d } \\quad \\tilde { \\mathbf F } = \\mathcal S ( \\hat { \\mathbf U } ) { \\mathbf F } \\end{align*}"}
+{"id": "437.png", "formula": "\\begin{align*} \\nu _ { \\zeta } ( r , s ) = c _ { \\zeta , \\alpha } \\ , ( r s ) ^ { - \\zeta } \\ , \\left ( \\frac { r \\wedge s } { r \\vee s } \\right ) ^ { \\zeta } \\ , \\frac { 1 } { ( r \\vee s ) ^ { \\alpha + 1 } } \\ , _ 2 \\tilde F _ 1 \\left ( \\zeta + \\frac { \\alpha + 1 } { 2 } , \\frac { \\alpha + 2 } { 2 } ; \\zeta + \\frac 1 2 ; \\frac { s ^ 2 } { r ^ 2 } \\right ) . \\end{align*}"}
+{"id": "6146.png", "formula": "\\begin{align*} ( \\mathbb { Z } , n ) = \\{ \\{ \\ ! \\{ k _ 1 , k _ 2 , \\ldots , k _ n \\} \\ ! \\} \\mid k _ 1 \\leq k _ 2 \\leq \\cdots \\leq k _ n \\} \\simeq \\mathbb { Z } ^ n / \\mathfrak { S } _ n , \\end{align*}"}
+{"id": "2790.png", "formula": "\\begin{align*} I ( T ; x ) = \\ ! \\begin{multlined} [ t ] x ^ { 1 5 } + 6 5 x ^ { 1 4 } + 4 4 3 6 x ^ { 1 3 } + 3 1 5 5 8 x ^ { 1 2 } + 1 0 6 4 0 8 x ^ { 1 1 } + 2 1 8 6 4 9 x ^ { 1 0 } + \\\\ 3 0 3 5 4 0 x ^ 9 + 2 9 9 8 6 0 x ^ 8 + 2 1 6 4 9 3 x ^ 7 + 1 1 5 4 7 5 x ^ 6 + 4 5 4 2 3 x ^ 5 + 1 2 9 8 7 x ^ 4 \\\\ + 2 6 1 7 x ^ 3 + 3 5 1 x ^ 2 + 2 8 x + 1 , \\end{multlined} \\end{align*}"}
+{"id": "8923.png", "formula": "\\begin{align*} d \\varphi ( \\nabla ^ M _ X d \\varphi ^ * ( J Y ) ) = J d \\varphi ( \\nabla ^ M _ X d \\varphi ^ * ( Y ) ) , \\end{align*}"}
+{"id": "6758.png", "formula": "\\begin{align*} f ( P \\overline { \\mathbf { x } } ) = P \\overline { f } ( \\overline { \\mathbf { x } } ) . \\end{align*}"}
+{"id": "7200.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } - \\Delta u _ \\rho = \\sigma _ \\rho & \\textrm { i n } \\Omega , \\\\ u _ \\rho = 0 & \\textrm { i n } \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "7251.png", "formula": "\\begin{align*} K = K _ \\Phi 0 , 1 . \\end{align*}"}
+{"id": "8666.png", "formula": "\\begin{align*} x _ { k + 1 } & = \\phi _ 1 ( x _ 1 , \\dots , x _ k ) \\\\ x _ { k + 2 } & = \\phi _ 2 ( x _ 1 , \\dots , x _ k ) \\\\ \\dots \\\\ x _ { n } & = \\phi _ { n - k } ( x _ 1 , \\dots , x _ k ) \\ , . \\end{align*}"}
+{"id": "1708.png", "formula": "\\begin{align*} \\omega _ { \\texttt { b } ; \\mu } : = e _ 1 + e _ 2 + \\cdots + e _ { _ { \\mu _ 1 } ( \\mu ) } . \\end{align*}"}
+{"id": "7808.png", "formula": "\\begin{align*} \\begin{pmatrix} a & b \\\\ c & d \\\\ \\end{pmatrix} \\cdot \\Big ( - \\frac { 1 } { \\overline { t } } \\Big ) = \\frac { \\overline { t } - t _ { + } ^ { c , d } } { t _ + ^ { c , d } \\overline { t } + 1 } = - \\overline { \\Big [ \\begin{pmatrix} a & b \\\\ c & d \\\\ \\end{pmatrix} \\cdot t \\Big ] } ^ { - 1 } \\ , , \\end{align*}"}
+{"id": "8373.png", "formula": "\\begin{align*} \\lim \\limits _ { T \\rightarrow \\pm \\infty } \\frac { 1 } { | T | } \\bigg \\| \\int _ { 0 } ^ { T } ( - A ) ^ { - \\nu } ( f ( x , Y _ { F } ^ 1 ( \\theta _ r \\omega _ 2 , x ) ) - \\bar f ( x ) ) \\ , d r \\bigg \\| = 0 . \\end{align*}"}
+{"id": "2849.png", "formula": "\\begin{align*} C ( { { \\xi } } ) : = \\prod _ { \\alpha \\in R _ 0 ^ + } \\frac { 1 - t _ { \\alpha } e ^ { - i \\langle \\xi , \\alpha \\rangle } } { 1 - e ^ { - i \\langle \\xi , \\alpha \\rangle } } . \\end{align*}"}
+{"id": "8524.png", "formula": "\\begin{align*} \\mathbb { R } ^ { n } \\backslash E = ( \\mathbb { R } ^ { n } \\backslash E ) \\cap \\{ z < \\bar { z } \\} ) \\cup ( \\mathbb { R } ^ { n } \\backslash E ) \\cap \\{ z \\geq \\bar { z } \\} ) . \\end{align*}"}
+{"id": "8745.png", "formula": "\\begin{align*} \\min \\Big ( \\alpha , \\ , \\frac { d } { \\alpha } T ^ { - \\frac { \\beta - 1 } { \\beta } } \\Big ) = \\min \\Big ( \\max ( \\alpha , T ^ { - 1 / 2 + 1 / \\beta } ) , \\frac { d } { \\sqrt { T } } , \\ , \\frac { d } { \\alpha } T ^ { - \\frac { \\beta - 1 } { \\beta } } \\Big ) \\enspace . \\end{align*}"}
+{"id": "210.png", "formula": "\\begin{align*} { Q _ 1 } ( M , P _ i , \\tau ) = & \\left \\{ e ^ { \\frac { 1 } { 2 4 } E _ 2 ( \\tau ) A _ 3 } \\widehat { L } ( T M , \\nabla ^ { T M } ) { \\rm c h } \\left [ \\bigotimes _ { n = 1 } ^ { \\infty } S _ { q ^ n } ( \\widetilde { T _ C M } ) \\otimes \\bigotimes _ { m = 1 } ^ { \\infty } \\wedge _ { q ^ m } ( \\widetilde { T _ C M } ) \\right ] \\right . \\\\ & \\left . \\cdot \\varphi ( \\tau ) ^ { 8 } { \\rm c h } ( \\mathcal { V } _ i ) \\right \\} ^ { ( 1 2 ) } , \\end{align*}"}
+{"id": "3145.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { r _ i } a _ j ( t ) ^ k = \\frac { 1 } { 2 \\pi i } \\oint _ { \\gamma } z ^ k \\frac { ( f \\circ u _ t ) ' ( z ) } { f ( u _ t ( z ) ) } d z , \\end{align*}"}
+{"id": "3804.png", "formula": "\\begin{align*} \\alpha ( ( Y \\times Y ) \\setminus \\Omega ) = 0 , \\Omega = \\{ ( y _ 0 , y _ 1 ) \\in Y \\times Y \\mid s _ 0 = s _ 1 \\} . \\end{align*}"}
+{"id": "3395.png", "formula": "\\begin{align*} \\Theta : = - v _ 1 ^ 2 + v _ 1 ^ 4 + v _ 2 ^ 2 - 1 0 v _ 1 ^ 2 v _ 2 ^ 2 + 9 v _ 1 ^ 4 v _ 2 ^ 2 + v _ 2 ^ 4 - 9 v _ 1 ^ 2 v _ 2 ^ 4 \\neq 0 . \\end{align*}"}
+{"id": "6069.png", "formula": "\\begin{align*} Y _ { t } = x - y + \\int _ { 0 } ^ { t } \\int _ { \\mathbb { R } ^ { d } } \\Delta _ { s } ^ { c } ( z ) d N ( z , s ) + \\int _ { 0 } ^ { t } \\Delta _ { s } ^ { b } d s . \\end{align*}"}
+{"id": "6194.png", "formula": "\\begin{align*} [ R _ { x \\circ y } , L _ z ] + ( - 1 ) ^ { | x | ( | y | + | z | ) } [ R _ { y \\circ z } , L _ x ] + ( - 1 ) ^ { | z | ( | x | + | y | ) } [ R _ { z \\circ x } , L _ y ] = 0 , \\end{align*}"}
+{"id": "8313.png", "formula": "\\begin{align*} & V _ { f _ 1 } ( v ( \\alpha _ 1 ) , v ( \\alpha _ 2 ) ) > v ( \\alpha ) = \\min \\{ v ( \\alpha _ 1 ) , v ( \\alpha _ 2 ) \\} \\\\ & V _ { f _ 2 } ( v ( \\alpha _ 1 ) , v ( \\alpha _ 2 ) ) > v ( \\alpha ) = \\min \\{ v ( \\alpha _ 1 ) , v ( \\alpha _ 2 ) \\} . \\end{align*}"}
+{"id": "173.png", "formula": "\\begin{align*} \\theta _ 1 ( v , \\tau + 1 ) = e ^ { \\frac { \\pi \\sqrt { - 1 } } { 4 } } \\theta _ 1 ( v , \\tau ) , ~ ~ \\theta _ 1 ( v , - \\frac { 1 } { \\tau } ) = \\left ( \\frac { \\tau } { \\sqrt { - 1 } } \\right ) ^ { \\frac { 1 } { 2 } } e ^ { \\pi \\sqrt { - 1 } \\tau v ^ 2 } \\theta _ 2 ( \\tau v , \\tau ) ; \\end{align*}"}
+{"id": "4138.png", "formula": "\\begin{align*} x \\rhd [ y , z ] & = [ x \\rhd y , z ] + [ y , x \\rhd z ] , \\\\ [ x , y ] \\rhd z & = a _ { \\rhd } ( x , y , z ) - a _ { \\rhd } ( y , x , z ) . \\end{align*}"}
+{"id": "7399.png", "formula": "\\begin{align*} \\partial _ { t } W _ { n } + u _ { n } \\partial _ { x } W _ { n } = 0 . \\end{align*}"}
+{"id": "3308.png", "formula": "\\begin{align*} h _ Z ( x ) = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\lambda _ { _ \\Sigma } e ^ { - \\lambda _ { _ \\Sigma } x } \\mathbb { 1 } _ { [ 0 , + \\infty [ } ( x ) , \\end{align*}"}
+{"id": "6045.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\infty } \\gamma _ { i } = \\lim _ { n \\rightarrow \\infty } \\Gamma _ { n } = \\infty . \\end{align*}"}
+{"id": "2588.png", "formula": "\\begin{align*} v a l ( 1 - f r ) & = v a l ( 1 - f g _ { \\nu } - ( 1 - f r ) ) = v a l ( f ( r - g _ { \\nu } ) ) = \\\\ & = v a l ( f ) + v a l ( r - g _ { \\nu } ) < v a l ( f ) + v a l ( r - g _ { \\mu } ) = \\\\ & = v a l ( f ( r - g _ { \\mu } ) ) = v a l ( 1 - f g _ { \\mu } - ( 1 - f r ) ) = v a l ( 1 - f r ) , \\end{align*}"}
+{"id": "7296.png", "formula": "\\begin{align*} | A _ e \\cap [ 2 ^ { e + n + 3 } , 2 ^ { e + n + 4 } ) | \\leq 2 ^ n = \\frac { 1 } { 2 ^ { e + 3 } } \\cdot | [ 2 ^ { e + n + 3 } , 2 ^ { e + n + 4 } ) | . \\end{align*}"}
+{"id": "243.png", "formula": "\\begin{align*} \\lim _ { x \\to + \\infty } \\Bigl | \\Phi ^ { \\emph { r } } _ { \\xi } ( x ; g ) - \\sum _ { \\emph { w } \\in W } C ^ { \\emph { r } } ( \\emph { w } \\xi ; g ) e ^ { \\langle \\emph { w } \\xi , x \\rangle } \\Bigr | = 0 \\quad \\quad \\emph { R e } ( \\xi ) = 0 . \\end{align*}"}
+{"id": "1519.png", "formula": "\\begin{gather*} \\hat { i } \\ , = \\ , \\lceil \\frac { i } { m } \\rceil , \\ ; \\textrm { w h e r e } \\ ; \\lceil . . \\rceil \\ ; \\textrm { d e n o t e s t h e c e i l f u n c t i o n } \\\\ \\bar { i } = i ( m o d \\ , m ) , \\ ; \\textrm { t h e r e s i d u e m o d u l o } \\ , m \\ , \\textrm { o f } \\ , i \\\\ \\end{gather*}"}
+{"id": "4123.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { k - 2 i + 2 s - a } \\binom { s - \\frac { 1 } { 2 } + k - 2 i - \\ell } { \\ell } \\binom { k + 3 s - 2 a - \\frac { 1 } { 2 } - 2 i - 2 \\ell } { k + 2 s - a - 2 i - \\ell } . \\end{align*}"}
+{"id": "3667.png", "formula": "\\begin{align*} \\sum _ { k , m } \\ , a _ { i k } ( t ) \\ , J _ { k m } ( t ) \\ , a _ { j m } ( t ) = \\delta _ { i j } \\ , . \\end{align*}"}
+{"id": "7149.png", "formula": "\\begin{align*} \\frac { 1 } { r ^ k } \\leq \\varphi _ 0 ' ( r ) & \\leq \\frac { e ^ { k \\| s ^ { - 2 } \\Phi _ 1 ( s ) \\| _ { L ^ 1 ( 1 , \\infty ) } } } { r ^ k } \\\\ & \\le \\begin{cases} \\frac { C } { r ^ { k } } & \\delta = 0 1 \\\\ [ 6 p t ] \\frac { C ( \\log ( h ^ { - 1 } ) ) ^ { \\kappa C } } { r ^ k } & 0 < \\delta < 1 \\end{cases} , h \\in ( 0 , h _ \\delta ] , \\ , 1 \\le r \\le a . \\end{align*}"}
+{"id": "2827.png", "formula": "\\begin{align*} \\ell i _ { 2 } ^ { ( p ) } = \\ell ^ { ( p ) } \\circ \\delta . \\end{align*}"}
+{"id": "2942.png", "formula": "\\begin{align*} 0 = \\delta _ { i _ 1 i _ 2 } [ \\sum _ { \\lambda = 1 , \\pi \\in S _ { n } } ^ { n ! } d \\epsilon _ { k s \\pi ( i _ 1 } D _ { i _ 2 \\cdots i _ n ) s } ] . \\end{align*}"}
+{"id": "3798.png", "formula": "\\begin{align*} \\overline { \\alpha } = ( { \\rm p r d } _ \\theta ) _ { \\# } ( \\theta ^ p \\tilde \\alpha ) , \\end{align*}"}
+{"id": "2406.png", "formula": "\\begin{align*} W ( \\mathcal { S } , ( x _ n ) _ { n = 1 } ^ { \\infty } , h ) \\cap B ( z , r ) \\subseteq \\bigcup _ { \\i \\in \\Omega } S _ { \\i } \\left ( x _ { | \\i | } + Q ( \\i ) \\right ) \\subseteq B ( z , 2 r ) . \\end{align*}"}
+{"id": "1448.png", "formula": "\\begin{align*} \\begin{aligned} \\sum \\limits _ { i = 1 } ^ { n } a ^ { i i } \\left ( - \\frac { 1 } { 2 } \\int _ { \\partial B _ { \\varepsilon } ( 0 ) } | \\nabla u ^ { k } _ { i } | ^ 2 ( x \\cdot \\mathbf { n } ) \\mathrm { d } S + \\int _ { \\partial B _ { \\varepsilon } ( 0 ) } \\big ( x \\cdot \\nabla u ^ k _ i \\big ) \\big ( \\mathbf { n } \\cdot \\nabla u ^ k _ i \\big ) \\mathrm { d } S \\right ) = - 4 \\pi \\sum \\limits _ { i = 1 } ^ { n } a ^ { i i } \\alpha _ { i } ^ { 2 } + o ( 1 ) . \\end{aligned} \\end{align*}"}
+{"id": "4381.png", "formula": "\\begin{align*} | | u | | ^ 2 _ { H ^ m _ { \\ast } ( I \\times \\Omega ) } : = \\sum _ { \\langle \\alpha \\rangle \\leq m } | | D ^ { \\alpha } _ { \\ast } u | | ^ 2 _ { L ^ 2 ( I \\times \\Omega ) } . \\end{align*}"}
+{"id": "2767.png", "formula": "\\begin{align*} \\frac { T _ { \\mu _ 2 } ( w ) } { T _ { \\mu _ 2 } ' ( w ) } + \\frac { w } { 1 + T _ { \\mu _ 2 } ( w ) } = \\frac { T _ { \\mu _ 2 } ( w ) + T _ { \\mu _ 2 } ^ 2 ( w ) + w T _ { \\mu _ 2 } ' ( w ) } { ( 1 + T _ { \\mu _ 2 } ( w ) ) T _ { \\mu _ 2 } ' ( w ) } > 0 . \\end{align*}"}
+{"id": "2493.png", "formula": "\\begin{align*} ( \\pi _ i | _ { S _ i } ) _ * ( K _ { S _ i } + B _ { S _ i } + M _ { S _ i } ) = K _ { S _ { i + 1 } } + B _ { S _ { i + 1 } } + M _ { S _ { i + 1 } } \\quad i . \\end{align*}"}
+{"id": "6003.png", "formula": "\\begin{align*} i i i ) 2 \\overline { b } - \\int _ { \\mathbb { R } ^ { d } } ( 2 \\bar { c } ( z ) + \\bar { c } ^ { 2 } ( z ) ) \\mu ( d z ) : = \\theta > 0 . \\end{align*}"}
+{"id": "2042.png", "formula": "\\begin{align*} T _ \\phi u ( x ' , x _ n ) = u ( x ' , x _ n + \\phi ( x ' ) ) , ( x ' , x _ n ) \\in \\mathbb { R } ^ { n - 1 } \\times [ 0 , + \\infty ) \\end{align*}"}
+{"id": "1233.png", "formula": "\\begin{align*} \\Big | \\sum _ { j = 1 } ^ J v _ n ^ j \\Big | ^ 2 = \\sum _ { j = 1 } ^ J | v _ n ^ j | ^ 2 + \\sum _ { j \\neq k } v _ n ^ j v _ n ^ k , \\end{align*}"}
+{"id": "5560.png", "formula": "\\begin{align*} \\nabla ^ 2 V _ k ( \\zeta ) & = 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\ , \\mbox { i f } \\zeta \\in G _ m , \\\\ V _ k ( \\zeta ) & = 1 - \\Psi _ k ( \\zeta ) \\mbox { i f } \\zeta \\in C _ j , j = 1 , \\ldots , k - 1 , \\\\ V _ k ( \\zeta ) & = 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\ , \\mbox { i f } \\zeta \\in C _ k , \\\\ V _ k ( \\zeta ) & = - \\Psi _ k ( \\zeta ) ~ ~ ~ ~ ~ ~ \\mbox { i f } \\zeta \\in C _ j , j = k + 1 , \\ldots , m . \\end{align*}"}
+{"id": "3125.png", "formula": "\\begin{align*} \\mathbf { E } [ ( \\mathbf { h } _ k ^ T ) ( \\mathbf { h } _ k ^ * ) ] = \\frac { L _ k } { N } \\mathbf { I } _ N \\end{align*}"}
+{"id": "551.png", "formula": "\\begin{align*} \\tau < \\infty \\implies \\limsup _ { t \\nearrow \\tau } \\norm { \\mathbf u } _ { \\mathbf X ^ { \\mathbf s , b } ( 0 , t ) } = \\infty . \\end{align*}"}
+{"id": "4365.png", "formula": "\\begin{align*} \\mathrm { d i v } \\mathbf { H } = 0 \\end{align*}"}
+{"id": "5379.png", "formula": "\\begin{align*} \\phi _ { \\psi ( M ) } = \\psi \\circ \\phi _ M \\circ \\psi ^ { - 1 } \\end{align*}"}
+{"id": "1323.png", "formula": "\\begin{align*} \\Lambda _ V ^ { w } ( u ) & = w ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) \\\\ & = \\alpha ( 1 - u ) ^ { \\frac { \\alpha - m + 1 } { \\alpha } } . \\end{align*}"}
+{"id": "3696.png", "formula": "\\begin{align*} y _ 2 - y ' _ 2 \\approx \\lambda _ \\beta ' ( 2 ^ { - 4 n _ 0 } ) a _ { n _ 0 } = ( 1 - \\beta ) ^ { \\beta / ( 1 - \\beta ) } 2 ^ { - 4 n _ 0 \\beta / ( 1 - \\beta ) } a _ { n _ 0 } \\ll n _ 0 ^ { - 2 } { a _ { n _ 0 } } , \\end{align*}"}
+{"id": "8492.png", "formula": "\\begin{align*} \\int _ { E } \\nabla \\psi ( x ) \\ d x = \\int _ { \\mathbb { R } ^ { n } } \\psi ( x ) \\ d \\mu _ { E } , \\mbox { f o r e v e r y } \\psi \\in C _ { c } ^ { 1 } ( \\mathbb { R } ^ { n } ) . \\end{align*}"}
+{"id": "5272.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ T \\frac { \\gamma _ 0 } { \\gamma _ { t } ^ 2 } \\le \\sum _ { t = 1 } ^ T \\frac { 1 } { \\gamma _ { t } } = \\frac { 1 } { \\gamma _ { 0 } } \\sum _ { t = 1 } ^ T \\frac { 1 } { t ^ c } \\le \\frac { 1 } { \\gamma _ { 0 } ( 1 - c ) } T ^ { 1 - c } . \\end{align*}"}
+{"id": "3605.png", "formula": "\\begin{align*} \\begin{aligned} { \\bf { y } } & = { \\bf { W } } _ { { \\rm { D S } } } ^ H { \\bf { r } } = { \\bf { W } } _ { { \\rm { D S } } } ^ H { { \\bf { H } } _ { { \\rm { D S } } } } { { \\bf { F } } _ { { \\rm { D S } } } } { \\bf { s } } + { \\bf { W } } _ { { \\rm { D S } } } ^ H { \\bf { n } } \\\\ & \\approx \\sqrt { \\frac { E } { { { N _ { \\rm { R } } } } } } \\sum \\limits _ { m = 1 } ^ { { M _ { \\rm { R } } } } { { { \\bf { \\Xi } } _ { { \\rm { D S } } , m } } } { \\bf { s } } + { \\bf { W } } _ { { \\rm { D S } } } ^ H { \\bf { n } } . \\end{aligned} \\end{align*}"}
+{"id": "1524.png", "formula": "\\begin{align*} \\sum _ { V - E \\le k \\le V + E } \\frac { V ^ k } { k ! } \\chi ^ { \\{ a k / b \\} } = \\sum _ { r \\bmod b } \\chi ^ { \\{ a r / b \\} } \\sum _ { \\substack { V - E \\le k \\le V + E \\\\ k \\equiv r \\pmod b } } \\frac { V ^ k } { k ! } . \\end{align*}"}
+{"id": "4787.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 2 \\pi } \\cos j \\theta \\cos 2 k ( \\theta + \\delta ) d \\theta & = ( - 1 ) ^ { 2 k } \\cos 2 k \\delta \\int _ { - \\pi } ^ { \\pi } \\cos ^ 2 2 k \\theta d \\theta \\\\ & = \\cos 2 k \\delta \\frac { 1 } { 2 } \\int _ { - \\pi } ^ { \\pi } [ 1 + \\cos 2 j \\theta ] d \\theta = \\cos 2 k \\delta \\frac { 1 } { 2 } [ 2 \\pi + 0 ] = \\pi \\cos 2 k \\delta \\end{align*}"}
+{"id": "7198.png", "formula": "\\begin{align*} B ^ + _ \\rho ( 0 ) & = \\{ \\lambda ( 0 ) + r \\cos ( \\theta ) \\sin ( \\xi ) { \\bf n } ( 0 ) + r \\sin ( \\theta ) \\sin ( \\xi ) { \\bf b } ( 0 ) , \\ ; ( r , \\theta , \\xi ) \\in [ 0 , \\rho ) \\times [ 0 , 2 \\pi ) \\times ( \\pi , 2 \\pi ) \\} , \\\\ B ^ + _ \\rho ( L ) & = \\{ \\lambda ( L ) + r \\cos ( \\theta ) \\sin ( \\xi ) { \\bf n } ( 0 ) + r \\sin ( \\theta ) \\sin ( \\xi ) { \\bf b } ( 0 ) , \\ ; ( r , \\theta , \\xi ) \\in [ 0 , \\rho ) \\times [ 0 , 2 \\pi ) \\times ( 0 , \\pi ) \\} . \\end{align*}"}
+{"id": "1390.png", "formula": "\\begin{align*} \\left \\Vert E ^ { l } u \\right \\Vert _ { 2 , \\widetilde { P } _ { 0 } ( h ^ { l - 1 } ) _ { \\delta } } = \\left \\Vert E E ^ { l - 1 } u \\right \\Vert _ { 2 , \\widetilde { P } _ { 0 } ( h ^ { l - 1 } ) _ { \\delta } } \\leq C \\left \\Vert E ^ { l - 1 } u \\right \\Vert _ { 2 , ( h ^ { l - 1 } ) _ { \\delta } } , \\end{align*}"}
+{"id": "7537.png", "formula": "\\begin{align*} \\sqrt { ( f _ { 1 } \\times f _ { 2 } ) ^ ! } \\circ \\bar { \\Delta } _ * = i _ { \\bar { \\Delta } * } \\circ \\sqrt { f _ { \\bar { \\Delta } } ^ ! } . \\end{align*}"}
+{"id": "4185.png", "formula": "\\begin{align*} ( \\mathbf { N } ) = ( d - 1 ) \\frac { \\lambda } { | u | } . \\end{align*}"}
+{"id": "8256.png", "formula": "\\begin{align*} f _ { x _ 1 , 0 , 0 } ( \\rho , \\Theta ) = \\rho ^ 2 f _ { \\rho ^ { - 2 } x _ 1 , 0 , 0 } ( 1 , \\Theta ) . \\end{align*}"}
+{"id": "1260.png", "formula": "\\begin{align*} \\| \\nabla e _ { n , T } ^ { n l } \\| _ { L _ t ^ { 2 } L _ x ^ { \\frac 6 5 } } \\lesssim & T ^ { 1 / 2 } \\big | \\frac { x _ n } { \\lambda _ n } \\big | ^ { - 2 b } \\| \\nabla \\Phi _ n \\| _ { L _ t ^ \\infty L _ x ^ r } ^ { 2 ( p - 1 ) } \\sum _ { j = 0 } ^ { 1 } \\big | \\frac { x _ n } { \\lambda _ n } \\big | ^ { - j } \\| \\partial ^ { 1 - j } \\Phi _ n \\| _ { L _ t ^ \\infty L _ x ^ r } \\\\ \\lesssim & T ^ { 1 / 2 } \\big | \\frac { x _ n } { \\lambda _ n } \\big | ^ { - 2 b + \\theta | 5 / 2 - 3 / r | } \\to 0 \\end{align*}"}
+{"id": "1824.png", "formula": "\\begin{gather*} B ( c , M ) = \\left \\{ \\sum _ { M < n \\leq N } \\frac { \\beta _ n } { ( n + c ) ^ s } ~ \\middle | ~ \\right \\} \\end{gather*}"}
+{"id": "7881.png", "formula": "\\begin{align*} \\begin{cases} \\det D ^ 2 u = F ( x , u , \\nabla u ) & \\Omega \\\\ u = 0 & \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "8921.png", "formula": "\\begin{align*} \\sum \\limits _ { i , j = 1 } ^ m g ^ { i j } _ M \\frac { \\partial \\varphi ^ \\alpha } { \\partial x _ i } \\frac { \\partial \\varphi ^ \\beta } { \\partial x _ j } = 0 \\end{align*}"}
+{"id": "6817.png", "formula": "\\begin{align*} w ' ( z ) - c w ( z ) & = f ( u ( z ) ) \\left ( v ( z ) - p ( u ( z ) ) \\right ) \\\\ [ 0 . 2 c m ] & \\geq f ( u ( z ) ) \\left ( \\mu - p ( 0 ) \\right ) > 0 , \\end{align*}"}
+{"id": "7950.png", "formula": "\\begin{align*} D = \\Big ( D _ 1 \\cup D _ 2 \\cup A \\cup \\{ v \\} \\Big ) \\setminus ( A _ 1 \\cup A _ 2 ) . \\end{align*}"}
+{"id": "2475.png", "formula": "\\begin{align*} \\left ( \\bigsqcup _ { a \\in \\mathcal { A } } \\mathcal { V } _ a \\right ) \\times \\left ( \\bigsqcup _ { b \\in \\mathcal { B } } \\mathcal { V } _ b \\right ) = \\bigsqcup _ { ( a , b ) \\in \\mathcal { A } \\times \\mathcal { B } } \\mathcal { V } _ a \\times \\mathcal { V } _ b \\end{align*}"}
+{"id": "6689.png", "formula": "\\begin{align*} h _ j ( \\beta , p , q ) = { i \\over ( 2 \\pi ) ^ j } { \\rm I m } \\ , \\alpha _ j , \\ : \\ : j \\ : { \\rm e v e n } h _ j ( \\beta , p , q ) = { 1 \\over ( 2 \\pi ) ^ j } { \\rm R e } \\ , \\alpha _ j , \\ : \\ : j \\ : { \\rm o d d } \\end{align*}"}
+{"id": "6219.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } L _ n ^ \\delta ( y ) r ^ n = ( 1 - r ) ^ { - \\delta - 1 } e ^ { - \\frac { r } { 1 - r } y } . \\end{align*}"}
+{"id": "373.png", "formula": "\\begin{align*} L _ { \\phi _ 1 , R _ 1 } ( t _ 2 ) = \\int L ( t ) \\phi _ 1 ( R _ 1 \\cdot t _ 1 ) d t _ 1 \\end{align*}"}
+{"id": "5611.png", "formula": "\\begin{align*} g _ { p , \\mu , v } ( t ) & = - t ^ { \\frac { p - 2 } { p - 1 } } \\left ( t \\| v ' \\| _ { L ^ p _ { p - 1 } } ^ p + \\| v \\| _ { L ^ p _ { \\alpha _ 0 } } ^ p \\right ) ^ { \\frac { 1 } { p - 1 } } \\\\ & \\quad + \\dfrac { \\mu } { p ( p - 1 ) } \\dfrac { \\| v \\| _ { L ^ { p p ' } _ { \\alpha _ 0 } } ^ { p p ' } } { \\| v ' \\| ^ p _ { L ^ p _ { p - 1 } } \\| v \\| _ { L ^ { p } _ { \\alpha _ 0 } } ^ { p } } \\left ( - t ( p - 1 ) \\| v ' \\| ^ p _ { L ^ p _ { p - 1 } } + \\| v \\| _ { L ^ { p } _ { \\alpha _ 0 } } ^ { p } \\right ) . \\end{align*}"}
+{"id": "7224.png", "formula": "\\begin{align*} \\{ u < t \\} = \\left \\{ \\begin{array} { l l } \\varnothing , \\\\ F _ i , \\\\ \\Omega , \\end{array} \\quad \\begin{array} { l l } t \\leq y _ 1 , \\\\ y _ i < t \\leq y _ { i + 1 } i = 1 , \\ldots , n - 1 , \\\\ t > y _ n . \\end{array} \\right . \\end{align*}"}
+{"id": "3248.png", "formula": "\\begin{align*} e _ \\ell e _ k + e _ k e _ \\ell = 2 \\delta _ { \\ell , k } \\varepsilon _ { _ \\ell } \\end{align*}"}
+{"id": "5735.png", "formula": "\\begin{align*} H ^ s f ( x , t ) = - \\frac { s } { \\Gamma ( 1 - s ) } \\int _ 0 ^ \\infty \\frac { 1 } { \\tau ^ { 1 + s } } \\big ( P ^ H _ \\tau f ( x , t ) - f ( x , t ) \\big ) d \\tau , \\end{align*}"}
+{"id": "5929.png", "formula": "\\begin{align*} A _ { 2 } \\le \\dfrac { R _ { 3 } - S _ { 2 } } { 2 } + e = \\dfrac { 0 - ( - 2 e ) } { 2 } + e = 2 e \\le d [ a _ { 1 , 2 } b _ { 1 , 2 } ] . \\end{align*}"}
+{"id": "8328.png", "formula": "\\begin{align*} { \\mathbf { s } } = : { \\mathbf { s } } _ { \\mathbf { S B } } \\mathbf { M } = { \\mathbf { s } } _ { \\mathbf { S B } } { { \\mathbf { w } } } ^ { ' } { \\mathbf { r } } ( \\mathbf { I } - \\mathbf { P } _ { S M } ) ^ { - 1 } \\mathbf { M } \\end{align*}"}
+{"id": "864.png", "formula": "\\begin{align*} 1 < p _ 0 : = \\inf _ { s > 0 } \\frac { s g ( s ) } { G ( s ) } \\leq p ^ 0 : = \\sup _ { s > 0 } \\frac { s g ( s ) } { G ( s ) } < + \\infty . \\end{align*}"}
+{"id": "8481.png", "formula": "\\begin{align*} S _ { g } : = \\{ g ^ { \\wedge } \\neq g ^ { \\vee } \\} , \\end{align*}"}
+{"id": "7197.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } - \\Delta u = \\sigma \\delta _ \\Lambda & \\textrm { i n } \\Omega , \\\\ u = 0 & \\textrm { i n } \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
+{"id": "5028.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { C ^ k _ { g , - a , V } } : = \\Vert u \\Vert _ { C ^ k _ { g , ( S - f + 1 ) ^ { - a / 2 } , V } } , \\Vert u \\Vert _ { C ^ { k , \\alpha } _ { g , - a , V } } : = \\Vert u \\Vert _ { C ^ { k , \\alpha } _ { g , ( S - f + 1 ) ^ { - a / 2 } , V } } . \\end{align*}"}
+{"id": "3863.png", "formula": "\\begin{align*} h _ { 1 1 } & = - g ( \\nabla ^ 1 \\nabla _ 1 F , \\nu ) = \\frac { e ^ { - 2 \\lambda ( f + t ) } f '' } { \\sqrt { e ^ { - 2 \\lambda f } ( f ' ) ^ 2 + e ^ { - 4 \\lambda f } } } \\\\ h _ { 1 2 } & = h _ { 2 1 } = 0 \\\\ h _ { 2 2 } & = \\frac { e ^ { - 2 \\lambda ( f + t ) } e ^ { \\lambda t } r f ' } { \\sqrt { e ^ { - 2 \\lambda f } ( f ' ) ^ 2 + e ^ { - 4 \\lambda f } } } \\end{align*}"}
+{"id": "3064.png", "formula": "\\begin{align*} \\left \\{ { \\left ( { { \\alpha _ { { \\rm { T } } , k , l } } , \\Theta _ { { \\rm { T } } , k , l } ^ { \\rm { A } } , \\Theta _ { { \\rm { T } } , k , l } ^ { \\rm { D } } } \\right ) , l \\in { { \\mathcal L } _ { { \\rm { T } } , k } } } \\right \\} _ { k = 1 } ^ K , \\end{align*}"}
+{"id": "7918.png", "formula": "\\begin{align*} & & a _ x & = 0 & & & - b _ x ^ 2 n & = 1 \\\\ \\Rightarrow & & b _ x & = \\pm 1 & & & n & = - 1 , \\\\ & & a _ x & = \\pm 1 / 2 & & & - b _ x ^ 2 n & = 3 / 4 & & \\\\ \\Rightarrow & & b _ x & = \\pm 1 / 2 & & & n & = - 3 , \\\\ & & a _ x & = \\pm 1 & & & b _ x & = 0 & & & . \\end{align*}"}
+{"id": "6580.png", "formula": "\\begin{align*} { { F } } ( q ^ { ( r ) } ) + { { H } } \\Delta _ { r + 1 } q = 0 , \\end{align*}"}
+{"id": "4573.png", "formula": "\\begin{align*} \\mathcal J _ { 2 , 3 } = - \\int _ 0 ^ t \\ ! \\ ! \\ ! \\int _ { \\mathbb R } \\hat c \\ , \\alpha \\ , ( \\dot u ^ + _ N + \\hat D _ 1 \\dot H ^ + _ { N } + \\hat D _ 2 \\dot H ^ - _ N + \\hat D _ 3 \\varphi ) \\ , d x _ 2 d s \\cong \\int _ 0 ^ t \\ ! \\ ! \\ ! \\int _ { \\mathbb R } \\hat c \\ , \\alpha \\ , \\beta \\ , d x _ 2 \\ , d s + \\int _ 0 ^ t \\ ! \\ ! \\ ! \\int _ { \\mathbb R } \\hat c \\ , \\alpha \\ , \\varphi \\ , d x _ 2 \\ , d s \\end{align*}"}
+{"id": "6549.png", "formula": "\\begin{align*} S ( z ) = R _ { B _ * } { H } ( z ) R _ { B _ * } - R _ { B _ * } { H } ( z ) R _ { \\Lambda ^ c } G _ { \\Lambda ^ c } ( z ) R _ { \\Lambda ^ c } { H } ( z ) R _ { B _ * } . \\end{align*}"}
+{"id": "2171.png", "formula": "\\begin{align*} \\sigma \\ast _ { n k } ( t _ 1 \\ast _ { m k } ( s _ 1 , \\ldots , s _ m ) , \\ldots , t _ n \\ast _ { m k } ( s _ 1 , \\ldots , s _ m ) ) = ( \\sigma \\ast _ { n m } ( t _ 1 , \\ldots , t _ n ) ) \\ast _ { m k } ( s _ 1 , \\ldots , s _ m ) . \\end{align*}"}
+{"id": "6159.png", "formula": "\\begin{align*} \\textrm { G M T } ( \\mathbf { k } ) : = \\bigsqcup _ { \\mu ^ { ( n ) } \\in \\textrm { A R } _ n } \\bigsqcup _ { \\mathbf { k } ^ { ( n - 1 ) } \\in \\mu ^ { ( n ) } ( \\mathbf { k } ) } \\bigsqcup _ { \\mu ^ { ( n - 1 ) } \\in \\textrm { A R } _ { n - 1 } } \\bigsqcup _ { \\mathbf { k } ^ { ( n - 2 ) } \\in \\mu ^ { ( n - 1 ) } ( \\mathbf { k } ^ { ( n - 1 ) } ) } \\cdots \\bigsqcup _ { \\mu ^ { ( 2 ) } \\in \\textrm { A R } _ { 2 } } \\bigsqcup _ { \\mathbf { k } ^ { ( 1 ) } \\in \\mu ^ { ( 2 ) } ( \\mathbf { k } ^ { ( 2 ) } ) } \\textrm { A R } _ 1 \\end{align*}"}
+{"id": "4838.png", "formula": "\\begin{align*} \\check { R } ( u ) = \\sum _ { i = 1 } ^ { n } \\Lambda _ i ( u ) P _ i \\end{align*}"}
+{"id": "2734.png", "formula": "\\begin{align*} \\Bigg \\| U e _ i - \\sum _ { k = 1 } ^ { \\lceil \\theta ^ { - 1 } \\rceil - 1 } U e _ { j _ k } \\Bigg \\| = \\Bigg \\| U e _ i - U \\Bigg ( \\sum _ { k = 1 } ^ { \\lceil \\theta ^ { - 1 } \\rceil - 1 } e _ { j _ k } \\Bigg ) \\Bigg \\| = \\Bigg \\| e _ i - \\sum _ { k = 1 } ^ { \\lceil \\theta ^ { - 1 } \\rceil - 1 } e _ { j _ k } \\Bigg \\| = 1 . \\end{align*}"}
+{"id": "7303.png", "formula": "\\begin{align*} D _ { T , S } = \\{ ( a , b ) \\in [ 0 , 1 ] ^ 2 : & \\left [ a \\ge b \\Rightarrow \\exists x \\in [ 0 , 1 ] \\ ; a = S ( x , b ) 0 = T ( x , b ) \\right ] \\\\ & \\left [ b \\ge a \\Rightarrow \\exists x \\in [ 0 , 1 ] \\ ; b = S ( x , a ) 0 = T ( x , a ) \\right ] \\} \\end{align*}"}
+{"id": "433.png", "formula": "\\begin{align*} p _ \\zeta ^ { ( \\alpha ) } ( t , r , s ) & : = \\int _ 0 ^ \\infty p _ \\zeta ^ { ( 2 ) } ( \\tau , r , s ) \\ , \\sigma _ t ^ { ( \\alpha / 2 ) } ( \\tau ) \\ , d \\tau . \\end{align*}"}
+{"id": "5231.png", "formula": "\\begin{align*} \\Pr \\left [ \\max _ { i \\in [ m ] } X _ i \\geq \\mu + t \\right ] \\leq \\sum _ { i = 1 } ^ m \\Pr \\left [ X _ i \\geq \\mu + t \\right ] \\leq m \\exp \\left ( - \\frac { t ^ 2 } { \\sigma ^ 2 } \\right ) . \\end{align*}"}
+{"id": "4755.png", "formula": "\\begin{align*} E _ k = 2 ^ { - 2 k - 1 } T + ( 2 ^ { - 2 k - 1 } , 0 ) \\subset \\Omega _ 2 . \\end{align*}"}
+{"id": "941.png", "formula": "\\begin{align*} u _ x = \\frac { 1 } { \\kappa } [ s _ 1 X _ x \\tilde { u } _ { \\tilde { x } } - r _ 2 X _ x \\tilde { v } _ { \\tilde { x } } + r _ 2 ( s _ { 1 x } v + s _ { 2 x } u + s _ { 3 x } ) - s _ 1 ( r _ { 1 x } u + r _ { 2 x } v + r _ { 3 x } ) ] , \\end{align*}"}
+{"id": "13.png", "formula": "\\begin{align*} \\widetilde { R } = a e ^ { \\frac { 1 } { 2 \\delta } } , \\delta = \\frac { 1 } { 2 } \\log \\Big ( \\frac { \\widetilde { R } } { a } \\Big ) ^ { - 1 } , \\end{align*}"}
+{"id": "4098.png", "formula": "\\begin{align*} & \\omega _ { g , n + 2 } ( p _ 0 , q _ 0 , J ) - \\omega _ { g , n + 2 } ( q _ 0 , p _ 0 , J ) \\\\ & = \\frac { 1 } { 2 \\pi i } \\oint _ { C _ { { \\rm R e c } } ^ p } \\cdot \\frac { \\eta ^ { p } ( p _ 0 ) \\cdot \\Delta \\omega _ { 0 , 2 } ( p , q _ 0 ) - \\eta ^ { p } ( q _ 0 ) \\cdot \\Delta \\omega _ { 0 , 2 } ( p , p _ 0 ) } { 4 \\ , \\omega _ { 0 , 1 } ( p ) } \\cdot \\hat R _ { g , n + 1 } ( p , J ) . \\end{align*}"}
+{"id": "6427.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } { \\frac { \\partial u } { \\partial t } ( x , t ) = d _ { 1 } \\left [ \\left ( J _ { 1 } * u \\right ) ( x , t ) - u ( x , t ) \\right ] + r _ { 1 } u ( x , t ) \\left [ 1 - u ( x , t ) - a v ( x , t ) \\right ] , } \\\\ { \\frac { \\partial v } { \\partial t } ( x , t ) = d _ { 2 } \\left [ \\left ( J _ { 2 } * v \\right ) ( x , t ) - v ( x , t ) \\right ] + r _ 2 v ( x , t ) \\left [ - 1 + b u ( x , t ) - v ( x , t ) \\right ] . } \\end{array} \\right . \\end{align*}"}
+{"id": "3612.png", "formula": "\\begin{align*} f \\left ( x \\right ) = \\left \\{ { \\begin{aligned} & { \\frac { 1 } { 2 } { e ^ { - \\frac { 1 } { 2 } x } } } , x > 0 \\\\ & { 0 , x \\le 0 } \\end{aligned} } \\right . . \\end{align*}"}
+{"id": "2332.png", "formula": "\\begin{align*} f _ { g u } ( \\tau _ { g h } ) = \\nu ( ( g u ) ^ { - 1 } g h ) = \\nu ( u ^ { - 1 } h ) = f _ h ( \\tau _ u ) , \\forall u , g , h \\in G . \\end{align*}"}
+{"id": "1641.png", "formula": "\\begin{align*} f '' ( r ) = \\frac { f ( r ) } { 4 } \\left ( \\bigl ( ( p + q ) \\coth ( r / 2 ) + q \\tanh ( r / 2 ) \\bigr ) ^ 2 - \\frac { p + q } { \\sinh ^ 2 ( r / 2 ) } + \\frac { q } { \\cosh ^ 2 ( r / 2 ) } \\right ) . \\end{align*}"}
+{"id": "6411.png", "formula": "\\begin{align*} \\chi _ \\lambda ( T _ { i _ 1 } ^ { j _ 1 } \\cdots T _ { i _ m } ^ { j _ m } \\otimes T _ { k _ 1 } ^ { \\ell _ 1 } \\cdots T _ { k _ n } ^ { \\ell _ n } ) = \\lambda m n \\varepsilon ( T _ { i _ 1 } ^ { j _ 1 } \\cdots T _ { i _ m } ^ { j _ m } T _ { k _ 1 } ^ { \\ell _ 1 } \\cdots T _ { k _ n } ^ { \\ell _ n } ) \\end{align*}"}
+{"id": "6149.png", "formula": "\\begin{align*} \\# ( \\mathbf { k } ) \\mid _ { \\eta _ = A } = \\begin{cases} ( \\mathbf { k } ) & , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "6908.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\xi } e ^ { \\frac { \\beta ( y - \\xi ) } { c } } e ^ { \\mu | y | } e ^ { - \\mu | \\xi | } d y \\leq \\int _ { - \\infty } ^ { \\xi } e ^ { \\frac { ( \\beta - c \\mu ) ( y - \\xi ) } { c } } d y = \\frac { c } { \\beta - c \\mu } . \\end{align*}"}
+{"id": "7202.png", "formula": "\\begin{align*} \\| \\sigma _ \\rho \\| _ { L _ { \\eta } ^ 2 ( \\Lambda ) } \\le C \\rho ^ { \\eta - 1 - k } \\| \\sigma \\| _ { L ^ 2 ( \\Lambda ) } = C \\rho ^ { \\frac { 1 } { 2 } - \\frac { 1 } { p } + \\varepsilon - k } \\| \\sigma \\| _ { L ^ 2 ( \\Lambda ) } \\end{align*}"}
+{"id": "6628.png", "formula": "\\begin{align*} h _ { j } ( \\beta , p , q ) = ( - 1 ) ^ { j + 1 } h _ { j } ( \\beta , p , - q ) , \\tilde { h } _ { j } ( \\beta , p , q ) = ( - 1 ) ^ j \\tilde { h } _ { j } ( \\beta , p , - q ) . \\end{align*}"}
+{"id": "4060.png", "formula": "\\begin{align*} V _ { 0 , 0 } ( t ) = \\sum _ { n = - \\infty } ^ { \\infty } v _ { n } \\sin n \\pi t , v _ { n } = \\int _ { 0 } ^ { 1 } V _ { 0 , 0 } ( t ) \\sin n \\pi t \\ , d t . \\end{align*}"}
+{"id": "4972.png", "formula": "\\begin{align*} a _ j = 0 \\mbox { f o r e v e r y } s + 1 \\leq j \\leq r - s . \\end{align*}"}
+{"id": "1013.png", "formula": "\\begin{align*} - \\frac { c } { \\mu \\ , \\omega } ( { \\bf E } \\Delta \\varphi + i \\ , | \\omega \\ , k + \\nabla \\varphi | ^ 2 { \\bf E } ) + \\nabla \\times C = \\dfrac { 4 \\pi } { c } { \\bf J } - i \\ , \\omega \\dfrac { \\epsilon } { c } { \\bf E } , \\end{align*}"}
+{"id": "456.png", "formula": "\\begin{align*} \\left ( | D | ^ \\alpha - \\Phi ( \\sigma ) | x | ^ { - \\alpha } \\right ) | x | ^ { - \\sigma } = 0 , x \\neq 0 , \\end{align*}"}
+{"id": "4160.png", "formula": "\\begin{align*} \\bar \\Delta _ k = \\lim _ { T \\rightarrow \\infty } \\frac { 1 } { T } \\int _ { 0 } ^ { T } \\Delta _ k ( \\tau ) d \\tau , \\end{align*}"}
+{"id": "1110.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal J _ + ( u ) & : = \\frac { 1 } { 2 } \\| u \\| ^ 2 - \\frac { 1 } { 2 } \\int _ D { \\alpha ( x ) } | u ( x ) | ^ { 2 } \\ , d \\mu - \\int _ D F _ + ( x , u ( x ) ) d \\mu \\\\ & = \\frac { 1 } { 2 } \\| u \\| ^ 2 - \\frac { 1 } { 2 } \\int _ D { \\alpha ( x ) } | u ( x ) | ^ { 2 } \\ , d \\mu - \\int _ D F ( x , u ^ + ( x ) ) d \\mu , \\end{aligned} \\end{align*}"}
+{"id": "6087.png", "formula": "\\begin{align*} \\begin{aligned} C o v ( \\bar { X } , \\bar { Y } ) & = C o v ( \\frac { 1 } { | S | } \\sum _ { j \\in S _ k } X _ j , \\frac { 1 } { | S | } \\sum _ { j \\in S _ k } Y _ j ) \\\\ & = \\frac { 1 } { | S | ^ 2 } C o v ( \\sum _ { j \\in S _ k } X _ j , \\sum _ { j \\in S _ k } Y _ j ) \\\\ & = \\frac { 1 } { | S | ^ 2 } \\sum _ { j \\in S _ k } C o v ( X _ j , Y _ j ) \\\\ & \\approx \\frac { 1 } { | S | } s _ { X Y } \\\\ \\end{aligned} \\end{align*}"}
+{"id": "1274.png", "formula": "\\begin{align*} P _ { M < \\cdot \\le N } : = P _ { \\le N } - P _ { \\le M } = \\sum _ { M < N ^ \\prime \\le N } P _ { N ^ \\prime } \\end{align*}"}
+{"id": "8318.png", "formula": "\\begin{align*} \\{ N \\} _ { \\beta } \\equiv - \\sum _ { n = 1 } ^ { q } \\omega _ { n } T ^ { n } ( 1 ) \\mod { \\mathbb { Z } } \\end{align*}"}
+{"id": "1428.png", "formula": "\\begin{align*} 2 \\frac { \\partial ^ 2 \\alpha _ i } { \\partial z \\partial \\bar z } - e ^ { 2 \\alpha _ { i - 1 } } + 2 e ^ { 2 \\alpha _ i } - e ^ { 2 \\alpha _ { i + 1 } } = 0 , \\ \\forall \\ i = 0 , 1 , \\cdots , n , \\end{align*}"}
+{"id": "5542.png", "formula": "\\begin{align*} f ( \\eta ( t ) ) = \\gamma ( t ) + \\nu ( t ) + \\i \\mu ( t ) \\end{align*}"}
+{"id": "653.png", "formula": "\\begin{align*} E _ T = E \\cap \\left ( [ 0 , T ] \\times \\Omega \\right ) \\end{align*}"}
+{"id": "7062.png", "formula": "\\begin{align*} S _ r ( N ) & = \\frac { 1 } { K } \\ , \\int _ { \\mathbb { R } } \\ , V _ 1 \\left ( \\frac { \\nu } { K } \\right ) \\mathop { \\mathop { \\sum \\sum } _ { \\substack { n , \\ , m = 1 } } ^ { \\infty } } \\delta _ { n = m } \\ , A _ { \\pi } ( n , r ) m ^ { - i t } \\ , \\lambda _ f ( m ) \\ , \\left ( \\frac { n } { m } \\right ) ^ { i \\nu } \\ , V _ 1 \\left ( \\frac { n } { N } \\right ) \\ , V _ 2 \\left ( \\frac { m } { N } \\right ) \\ , d \\nu , \\end{align*}"}
+{"id": "4280.png", "formula": "\\begin{align*} F _ { n \\times p } { ( j ) } e _ i & = \\begin{cases} e _ { i + j } & \\mbox { i f } 1 \\leq ( i + j ) \\leq n , \\\\ 0 & \\mbox { o t h e r w i s e } , \\end{cases} \\\\ & = \\chi _ { [ 1 , n ] } ( i + j ) e _ { i + j } , \\end{align*}"}
+{"id": "5354.png", "formula": "\\begin{align*} h ^ 1 ( N _ Y ( K _ Y + D - ( k - 1 ) H ) ) = - \\chi ( N _ Y ( K _ Y + D - ( k - 1 ) H ) ) = s . \\end{align*}"}
+{"id": "5419.png", "formula": "\\begin{align*} \\sup _ { k \\leq n \\leq 2 ^ m } | a _ n | \\leq | a _ k | + \\sqrt { 2 } \\sum _ { i = 1 } ^ s \\Big ( \\sum _ { j } { | a _ { u _ { j + 1 } ^ i } - a _ { u _ j ^ i } | ^ 2 } \\Big ) ^ { 1 / 2 } , \\end{align*}"}
+{"id": "1315.png", "formula": "\\begin{align*} l _ 1 ( a ) & = ( \\kappa - 1 ) [ a ^ { 2 ( \\kappa - 1 ) } - a ^ { \\kappa - 2 } - a ^ \\kappa + 1 ] - \\kappa [ a ^ { 2 ( \\kappa - 1 ) } - 2 a ^ { \\kappa - 1 } + 1 ] \\\\ & = - a ^ { 2 ( \\kappa - 1 ) } - ( \\kappa - 1 ) a ^ { \\kappa - 2 } - ( \\kappa - 1 ) a ^ \\kappa + \\kappa - 1 + 2 \\kappa a ^ { \\kappa - 1 } - \\kappa \\\\ & = - a ^ { 2 ( \\kappa - 1 ) } - ( \\kappa - 1 ) a ^ { \\kappa } + 2 \\kappa a ^ { \\kappa - 1 } - ( \\kappa - 1 ) a ^ { \\kappa - 2 } - 1 . \\end{align*}"}
+{"id": "5704.png", "formula": "\\begin{align*} \\big ( ( z - B ' ) ^ { - 1 } f _ 0 \\big ) ( x _ 1 ) = f _ 0 \\big ( ( z - B ) ^ { - 1 } x _ 1 \\big ) = \\sum \\limits _ { k = 0 } ^ { \\infty } \\frac { f _ 0 ( B ^ k x _ 1 ) } { z ^ { k + 1 } } . \\end{align*}"}
+{"id": "761.png", "formula": "\\begin{align*} \\theta ( x ) - \\theta ( y ) - \\theta ( X ) + \\theta ( Y ) = ( I _ 1 - I _ 2 ) ( x - y ) + I _ 2 ( x - y - X + Y ) . \\end{align*}"}
+{"id": "7908.png", "formula": "\\begin{align*} - \\dfrac { 1 } { 2 } & = \\dfrac { - \\mathrm { F P d i m } ( x ) ^ 2 } { 2 } + \\sum _ { y \\in \\Gamma } c _ { x , x ^ \\ast } ^ y \\mathrm { F P d i m } ( y ) . \\end{align*}"}
+{"id": "6829.png", "formula": "\\begin{align*} \\mathcal { Y } _ 4 ( M , g ) = \\mathcal { Y } _ 4 ^ * ( M , g ) \\leqslant \\mathcal { Y } _ 4 ( \\mathbb { S } ^ n , g _ 0 ) . \\end{align*}"}
+{"id": "7538.png", "formula": "\\begin{align*} \\omega _ { \\pi , \\log } = \\omega _ { \\pi } , \\end{align*}"}
+{"id": "7081.png", "formula": "\\begin{align*} \\mathcal { V } _ 2 \\left ( \\frac { m } { p _ 2 ( p _ 1 q ) ^ 2 } \\right ) = 2 \\pi i ^ k \\frac { N } { N ^ { i ( t + \\nu ) } } \\int _ 0 ^ \\infty V _ 2 ( y ) \\ , y ^ { - i ( t + \\nu ) } \\ , e \\left ( - \\frac { N u y } { p _ 1 q Q } \\right ) J _ { k - 1 } \\left ( \\frac { 4 \\pi } { p _ 1 q } \\sqrt { \\frac { m N y } { p _ 2 } } \\right ) \\ { d } y , \\end{align*}"}
+{"id": "8735.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } t ^ { - \\frac { \\beta - 1 } { \\beta } } = 1 + \\int _ { 1 } ^ { T } u ^ { - \\frac { \\beta - 1 } { \\beta } } \\d u \\leq \\beta T ^ { \\frac { 1 } { \\beta } } \\quad \\sum _ { t = 1 } ^ { T } t ^ { - \\frac { 1 } { \\beta } - 1 } \\leq 1 + \\int _ { 1 } ^ { T } u ^ { - \\frac { 1 } { \\beta } - 1 } \\d u \\leq 1 + \\beta \\enspace . \\end{align*}"}
+{"id": "6825.png", "formula": "\\begin{align*} Q _ { 2 m } ( { \\bar { g } } ) = \\frac { 2 } { n - 2 m } u ^ { - \\frac { n + 2 m } { n - 2 m } } P _ { 2 m } ( g ) u , \\end{align*}"}
+{"id": "7434.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\vert y _ 1 \\vert ^ { \\kappa _ 1 + 1 } \\vert y _ 2 \\vert ^ { \\kappa _ 2 + 1 } d z \\leq & C _ 8 \\| y _ 1 \\| _ { m _ 1 } ^ { \\kappa _ 1 + 1 } \\| y _ 2 \\| _ { m _ 2 } ^ { \\kappa _ 2 + 1 } \\\\ \\leq & C _ { 9 } \\| y _ 1 \\| _ { 1 , p } ^ { \\kappa _ 1 + 1 } \\| y _ 2 \\| _ { 1 , p } ^ { \\kappa _ 2 + 1 } , \\end{align*}"}
+{"id": "4093.png", "formula": "\\begin{align*} P ( \\Delta y , x ) = \\Delta y ^ 2 - Q _ 0 ( x ) = 0 . \\end{align*}"}
+{"id": "6509.png", "formula": "\\begin{align*} { T } _ { \\tilde q } ( ( k , n ) ; ( k ' , n ' ) ) = ( p + 1 ) { \\tilde q } _ * ^ p ( k - k ' , n ) \\delta _ { n , n ' } . \\end{align*}"}
+{"id": "5568.png", "formula": "\\begin{align*} \\Phi _ k ( \\zeta ) = \\frac { 1 } { \\pi } \\Im \\log \\phi ( \\psi ( \\zeta , - \\xi , - \\xi _ 1 ) ) , \\end{align*}"}
+{"id": "3239.png", "formula": "\\begin{align*} \\epsilon ^ { - \\frac { 1 } { 1 + 1 / \\eta } } \\ll X = o ( R ) . \\end{align*}"}
+{"id": "1828.png", "formula": "\\begin{gather*} C _ \\beta ( m ) = \\left \\{ n \\in \\mathbb { Z } _ { \\geq 0 } ~ \\middle | ~ m \\beta < \\log ( n + c ) < \\left ( m + \\frac { 1 } { 2 } \\right ) \\beta \\right \\} \\end{gather*}"}
+{"id": "3575.png", "formula": "\\begin{align*} { \\bf a } _ { X } ( Y ) = \\frac { 1 } { \\sqrt { N _ { X } } } { \\left [ 1 , e ^ { j Y } , \\ , \\ldots , \\ , e ^ { j ( N _ { X } - 1 ) Y } \\right ] ^ T } , \\end{align*}"}
+{"id": "8930.png", "formula": "\\begin{align*} \\frac { 1 } { g _ M ( E _ i , E _ i ) } J d \\varphi ( [ E _ i ^ \\prime , E _ i ] ) = \\frac { 1 } { g _ M ( E _ i , E _ i ) } d \\varphi ( \\nabla ^ M _ { E _ i } E _ i ) + \\frac { 1 } { g _ M ( E _ i ^ \\prime , E _ i ^ \\prime ) } d \\varphi ( \\nabla ^ M _ { E _ i ^ \\prime } E _ i ^ \\prime ) \\end{align*}"}
+{"id": "8954.png", "formula": "\\begin{align*} \\Omega _ T : = \\{ ( \\bar { x } , \\bar { y } ) \\in \\R ^ { m + n } : 1 \\leq \\| \\bar { y } \\| < T , | x _ i | < \\vartheta _ i \\| \\bar { y } \\| ^ { - w _ i } , i = 1 , \\ldots , m \\} . \\end{align*}"}
+{"id": "540.png", "formula": "\\begin{align*} \\mathbf X ^ { \\mathbf s , b } ( \\R \\times \\R ^ d ) = X _ { h _ 1 ( \\xi ) } ^ { s _ 1 , b } ( \\R \\times \\R ^ d ) \\times \\dots \\times X _ { h _ n ( \\xi ) } ^ { s _ n , b } ( \\R \\times \\R ^ d ) \\end{align*}"}
+{"id": "1116.png", "formula": "\\begin{align*} \\frac { \\gamma } { \\lambda _ 1 } \\int _ { D } | \\nabla u ^ - _ \\infty | ^ 2 ( x ) d \\mu & \\geq \\gamma \\int _ { D } ( u ^ - _ \\infty ( x ) ) ^ 2 d \\mu \\\\ & = \\int _ { D } | \\nabla u ^ - _ \\infty | ^ 2 ( x ) d \\mu - \\int _ { D } u ^ - _ \\infty ( x ) \\Delta _ \\mu u ^ + _ \\infty ( x ) d \\mu . \\end{align*}"}
+{"id": "1913.png", "formula": "\\begin{align*} \\begin{aligned} e _ u & : = u - u _ h = ( u - Q ^ x _ \\lambda u ) + ( Q ^ x _ \\lambda u - u _ h ) = \\theta _ u - \\eta _ u , \\\\ e _ w & : = w - w _ h = ( w - Q ^ x _ { 1 - \\lambda } w ) + ( Q ^ x _ { 1 - \\lambda } w - w _ h ) = \\theta _ w - \\eta _ w . \\end{aligned} \\end{align*}"}
+{"id": "6536.png", "formula": "\\begin{align*} \\sup _ { \\xi = \\pm 1 , \\ \\sigma \\in \\R } \\# \\left \\{ ( k , n ) \\in \\Lambda _ L : \\ | \\xi ( \\sigma + k \\cdot \\omega ^ { ( 0 ) } ) + \\mu _ n | < \\frac \\eta 2 \\right \\} \\leq b , \\end{align*}"}
+{"id": "5765.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\mathbb E \\langle ( m ^ 1 ) ^ 2 \\rangle _ { \\beta } = 0 , \\end{align*}"}
+{"id": "1778.png", "formula": "\\begin{gather*} \\int _ { S _ 1 } \\omega _ \\lambda ^ q \\ , d \\mathbf { m } ( \\omega _ \\lambda ) = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } e ^ { i q \\theta } \\ , d \\theta = \\begin{cases} 1 & , \\\\ 0 & \\end{cases} \\end{gather*}"}
+{"id": "6257.png", "formula": "\\begin{align*} i \\partial _ t v + \\partial _ x T _ { g ( u ) } \\partial _ x v = f . \\end{align*}"}
+{"id": "7277.png", "formula": "\\begin{align*} [ n ] & = \\{ 0 , 1 , \\ldots , n \\} \\\\ [ n , m ) & = \\{ n , n + 1 , \\ldots , m - 1 \\} \\\\ ( n , m ) & = \\{ n + 1 , n + 2 , \\ldots , m - 1 \\} . \\end{align*}"}
+{"id": "5769.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } { \\mathbb E } \\langle ( m ^ p - \\langle m ^ p \\rangle ) ^ 2 \\rangle = \\lim _ { L \\to \\infty } \\frac { 1 } { | { \\cal B } _ p | ^ 2 } \\sum _ { X , Y \\in { \\cal B } _ p } \\mathbb E ( \\langle \\sigma _ X \\sigma _ { Y } \\rangle - \\langle \\sigma _ X \\rangle \\langle \\sigma _ { Y } \\rangle ) \\leq \\lim _ { L \\to \\infty } \\frac { 1 } { \\beta \\Delta _ p \\sqrt { | { \\cal B } _ p | } } = 0 , \\end{align*}"}
+{"id": "8608.png", "formula": "\\begin{align*} \\mathcal { L } _ { 1 } ^ { \\mu } [ \\beta b ] & = - \\frac { 1 } { \\beta } \\sinh { ( \\beta b ( X ) \\sqrt { \\mu } | \\mathrm { D } | ) } \\mathrm { s e c h } ( \\sqrt { \\mu } | \\mathrm { D } | ) \\dfrac { 1 } { \\sqrt { \\mu } | \\mathrm { D } | } \\\\ \\mathcal { L } _ { 2 } ^ { \\mu } [ \\beta b ] & = - ( \\mathcal { L } _ { 1 } ^ { \\mu } [ \\beta b ] + b ) \\frac { 1 } { \\mu | \\mathrm { D } | ^ 2 } . \\end{align*}"}
+{"id": "8411.png", "formula": "\\begin{align*} U ( T , r , v ) = \\sum _ { P < p \\leq 2 P } ( \\log p ) e \\left ( r p ^ { c } + v \\left ( T - p ^ { c } \\right ) ^ { \\gamma } \\right ) , \\end{align*}"}
+{"id": "6736.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { \\alpha ( t ) } { \\eta ( t ) } \\int _ { \\Sigma _ { t } } \\left ( \\frac { H } { 2 } - \\eta ( u ) | \\nabla u | \\right ) ^ { 2 } = 0 . \\end{align*}"}
+{"id": "5843.png", "formula": "\\begin{align*} w _ 0 ( \\alpha _ 2 ) = s _ { \\alpha _ 2 } ( \\alpha _ 2 ) = - \\alpha _ 2 . \\end{align*}"}
+{"id": "3381.png", "formula": "\\begin{align*} \\begin{cases} \\Upsilon _ 0 = a + \\alpha ^ 0 _ { \\ , 0 \\mu } b ^ \\mu , \\\\ \\Upsilon _ \\mu = 0 . \\end{cases} \\end{align*}"}
+{"id": "409.png", "formula": "\\begin{align*} \\hom ( y b , P \\times [ 0 = 1 ] ) \\cong \\hom ( y b , P ) \\times \\hom ( y b , [ 0 = 1 ] ) \\\\ \\cong \\hom ( y b , P ) \\times 0 \\cong 0 . \\end{align*}"}
+{"id": "317.png", "formula": "\\begin{align*} \\left [ \\left ( \\mathsf { S } _ { j } \\left ( s \\right ) \\varphi \\right ) \\right ] _ { \\operatorname * { D } ; j } \\left ( s \\right ) = 0 \\quad , \\left [ \\left ( \\mathsf { S } _ { j } \\left ( s \\right ) \\varphi \\right ) \\right ] _ { \\operatorname * { N } ; j } ^ { \\operatorname * { e x t } } \\left ( s \\right ) = - \\varphi . \\end{align*}"}
+{"id": "2930.png", "formula": "\\begin{align*} C _ { i j k l } = C _ { j i k l } = C _ { i j l k } \\end{align*}"}
+{"id": "2464.png", "formula": "\\begin{align*} & \\overline { H } \\left ( \\bigsqcup _ { a \\in \\mathcal { A } } ^ { P _ A } G _ a \\right ) = \\sum _ { a \\in \\mathcal { A } } P _ A ( a ) \\overline { H } ( G _ a ) \\\\ \\Longleftrightarrow \\ ; \\ ; & \\overline { H } \\left ( \\bigwedge _ { a \\in \\mathcal { A } } G _ a \\right ) = \\sum _ { a \\in \\mathcal { A } } \\overline { H } ( G _ a ) . \\end{align*}"}
+{"id": "2108.png", "formula": "\\begin{align*} \\mathcal { E } ( 0 ) = C _ 1 \\varepsilon ^ 2 . \\end{align*}"}
+{"id": "4626.png", "formula": "\\begin{align*} ( \\mathcal { R } ( G ^ E ) \\otimes S ) ^ { \\mathbf { K } } = \\bigoplus _ { \\pi \\in ( G ^ E ) ^ \\wedge } ( \\mathcal { R } ^ \\pi ( G ^ E ) \\otimes S ) ^ { \\mathbf { K } } \\end{align*}"}
+{"id": "6233.png", "formula": "\\begin{align*} i u _ t + \\partial ^ 2 _ x u = C ( u , \\bar u , u ) + . \\end{align*}"}
+{"id": "4896.png", "formula": "\\begin{align*} K _ \\xi ^ \\mathfrak { E } ( z ) & = \\frac { E _ + ( z ) E _ + ( \\xi ) ^ * - E _ - ( z ) E _ - ( \\xi ) ^ * } { \\rho _ \\xi ( z ) } \\\\ & = F _ + ( z ) \\left [ \\frac { Q _ + ( z ) Q _ + ( \\xi ) ^ * - Q _ - ( z ) Q _ - ( \\xi ) ^ * } { \\rho _ \\xi ( z ) } \\right ] F _ + ( \\xi ) ^ * \\\\ & = F _ + ( z ) \\Gamma _ \\xi ( z ) F _ + ( \\xi ) ^ * , \\end{align*}"}
+{"id": "594.png", "formula": "\\begin{align*} \\Phi ( t ) = i S _ { \\pm \\xi } ( - t ) P _ \\mu \\left ( P _ \\mu \\psi _ \\mp ^ \\mu ( t ) \\right ) \\mathfrak K _ 1 . \\end{align*}"}
+{"id": "4561.png", "formula": "\\begin{align*} \\vert \\mathcal I _ { 1 , 2 } \\vert & \\le C _ 2 \\Vert \\partial _ 1 \\dot { q } ^ + \\Vert _ { L ^ 2 ( \\Omega _ t ) } \\left ( \\Vert \\dot { H } ^ + _ n \\Vert _ { L ^ 2 ( \\Omega _ t ) } + \\Vert \\partial _ 2 \\dot { H } ^ + _ n \\Vert _ { L ^ 2 ( \\Omega _ t ) } \\right ) \\\\ & \\le C _ 2 \\left \\{ \\int _ 0 ^ t ( I _ { 1 , \\ast } + I _ { 1 , n } ) ( s ) d s \\right \\} \\ , . \\end{align*}"}
+{"id": "6744.png", "formula": "\\begin{align*} { \\widetilde J } = & \\langle x _ { 2 2 } , x _ { 2 1 } , x _ { 1 2 } , x _ { 1 1 } \\rangle \\cap \\langle x _ { 3 3 } , x _ { 2 1 } , x _ { 1 2 } , x _ { 1 1 } \\rangle \\cap \\langle x _ { 3 3 } , x _ { 3 2 } , x _ { 1 2 } , x _ { 1 1 } \\rangle \\\\ \\ & \\cap \\langle x _ { 3 3 } , x _ { 2 3 } , x _ { 2 1 } , x _ { 1 1 } \\rangle \\cap \\langle x _ { 3 3 } , x _ { 3 2 } , x _ { 2 3 } , x _ { 1 1 } \\rangle \\cap \\langle x _ { 3 3 } , x _ { 3 2 } , x _ { 2 3 } , x _ { 2 2 } \\rangle \\end{align*}"}
+{"id": "8283.png", "formula": "\\begin{align*} \\nu ( u ) = \\sum _ { \\beta = 1 } ^ 3 ( u _ { 2 \\beta - 1 } \\bar { u } _ { 2 \\beta } - u _ { 2 \\beta } \\bar { u } _ { 2 \\beta - 1 } ) . \\end{align*}"}
+{"id": "405.png", "formula": "\\begin{align*} [ n ] = \\{ 0 , x _ 1 , . . . , x _ n , 1 \\} , \\end{align*}"}
+{"id": "3529.png", "formula": "\\begin{align*} \\Vert \\gamma \\Vert ^ 2 & = a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 \\\\ & = ( a + d ) ^ 2 + ( b - c ) ^ 2 - 2 \\\\ & \\geqslant ( a + d ) ^ 2 - 2 \\\\ & \\geqslant q ^ { 4 } / 4 - 2 \\\\ & \\geqslant q ^ 4 / 8 . \\end{align*}"}
+{"id": "7174.png", "formula": "\\begin{align*} \\mathcal { F } ( P ) : = \\{ F _ { k _ j } ^ j ~ | ~ 0 \\leq k _ j \\leq n _ j , j = 1 , \\dots , m \\} , \\end{align*}"}
+{"id": "3463.png", "formula": "\\begin{align*} \\det ( D ^ 2 u ) W ( \\nabla u ) = \\frac { C _ 1 } { \\int _ { \\Delta } d x } = \\frac { d _ 0 + \\ldots + d _ m } { d _ 0 \\ldots d _ m } \\frac { ( n - m ) ! m ! } { n ! } . \\end{align*}"}
+{"id": "1373.png", "formula": "\\begin{align*} F = \\sum _ { n = 1 } ^ { n _ 0 } c ( n ) \\mathcal { F } _ { 2 - 2 k , - n } . \\end{align*}"}
+{"id": "8110.png", "formula": "\\begin{align*} \\begin{aligned} \\vert \\Phi _ { j } \\ast T ( a _ { j } ) ( x ) \\vert & = \\left \\vert \\int _ { \\mathbb R ^ { n } } \\Phi _ { t } ( x - y ) T ( a _ { j } ) ( y ) d y \\right \\vert \\\\ & \\leq t ^ { - n } \\int _ { B ( x , t ) } \\vert T ( a _ { j } ) ( y ) \\vert d y \\\\ & \\leq \\sup \\limits _ { y \\in B ( x , t ) } \\vert T ( a _ { j } ) ( y ) \\vert . \\end{aligned} \\end{align*}"}
+{"id": "161.png", "formula": "\\begin{align*} \\nabla ( [ \\omega _ n ^ { ( k ) } ] ) = [ \\omega _ n ^ { ( k - 1 ) } ] \\otimes \\frac { \\d z } { z } \\mbox { f o r } k = 2 , \\ldots , n - 1 . \\end{align*}"}
+{"id": "7145.png", "formula": "\\begin{align*} h \\in ( 0 , h _ \\delta ] \\implies \\eta \\lambda \\in ( 0 , 1 ] , \\ , k \\in [ \\tfrac { 1 } { 3 } , 1 ] , \\eta \\le \\begin{cases} \\min ( \\delta , \\frac { 1 } { 3 } ) & 0 < \\delta \\le 1 \\\\ 1 & \\delta = 0 \\end{cases} . \\end{align*}"}
+{"id": "8612.png", "formula": "\\begin{align*} \\nabla _ X ( \\phi _ 1 ( X , h _ b z ) ) = ( \\nabla _ X \\phi _ 1 ) ( X , h _ b z ) - \\beta \\nabla _ X b ( \\partial _ z \\phi _ 1 ) ( X , h _ b z ) , \\end{align*}"}
+{"id": "3741.png", "formula": "\\begin{align*} ( \\lambda I - \\mathbb { A } ) { \\bf u } = \\varphi \\end{align*}"}
+{"id": "6498.png", "formula": "\\begin{align*} u ^ { ( 0 ) } ( t , n ) = \\sum _ { l = 1 } ^ b a _ l \\cos ( \\omega _ l ^ { ( 0 ) } t ) \\delta _ { n ^ { ( l ) } , n } , \\end{align*}"}
+{"id": "4512.png", "formula": "\\begin{align*} \\mathbb L _ H ( { \\mathbf u } ^ a + \\mathbf { u } _ { i + 1 / 2 } , { \\mathbf H } _ { i + 1 / 2 } - S _ { \\theta _ i } { \\mathbf H } _ i , \\Psi ^ a + \\Psi _ { i + 1 / 2 } ) = F ^ { i + 1 / 2 } _ H \\ , , \\end{align*}"}
+{"id": "6895.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { 1 } { 2 } \\right ) + \\left ( 1 - \\frac { 1 } { 3 } \\right ) + \\left ( 1 - \\frac { 1 } { 6 } \\right ) = 2 . \\end{align*}"}
+{"id": "1333.png", "formula": "\\begin{align*} \\frac { J ^ w ( \\textbf { X } _ { m a x R S S U } ^ { ( n + 1 ) } ) } { J ^ w ( \\textbf { X } _ { m a x R S S U } ^ { ( n ) } ) } & = - 2 J ^ w ( X _ { n + 1 : n + 1 } ) \\\\ & = \\int _ { 0 } ^ { + \\infty } w ( x ) f ^ 2 _ { 1 : n + 1 } ( x ) d x \\\\ & = ( n + 1 ) ^ 2 \\int _ { 0 } ^ { 1 } w ( F ^ { - 1 } ( u ) ) u ^ { 2 n } f ( F ^ { - 1 } ( u ) ) d u \\\\ & \\geq \\frac { ( n + 1 ) ^ 2 } { 2 n + 1 } \\\\ & \\geq 1 . \\end{align*}"}
+{"id": "7560.png", "formula": "\\begin{align*} R _ T = R _ { d , \\beta , N , T } : = \\textrm { m e d } \\left ( \\left \\{ \\sup _ { t \\in [ 0 , T ] } | B ^ { ( k ) } _ t | \\ , : k = 1 , . . . , N \\right \\} \\right ) . \\end{align*}"}
+{"id": "6897.png", "formula": "\\begin{align*} - \\lambda \\Gamma '' - \\mu \\Gamma ' = - f _ * \\Theta - \\frac { \\mu } { a } f _ * D . \\end{align*}"}
+{"id": "6506.png", "formula": "\\begin{align*} q _ * ^ { p + 1 } ( k , n ) = \\sum _ { k ^ { ( 1 ) } + \\cdots + k ^ { ( p + 1 ) } = k } \\prod _ { l = 1 } ^ { p + 1 } q ( k ^ { ( l ) } , n ) , \\end{align*}"}
+{"id": "5035.png", "formula": "\\begin{align*} \\triangle _ f | u | \\geq \\frac { u } { | u | } \\cdot \\triangle _ f u = \\frac { u } { | u | } \\cdot ( L u - b ( u ) ) = - \\lambda | u | - \\frac { u \\cdot b ( u ) } { | u | } \\geq - \\lambda | u | - C ( A ) ( - f ) ^ { - 1 } | u | . \\end{align*}"}
+{"id": "5656.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l } \\alpha ^ 2 A C + \\alpha \\gamma B C + \\gamma ^ 2 C ^ 2 - \\beta ^ 2 A ^ 2 - \\beta \\delta A B - \\delta ^ 2 A C = 0 , \\\\ \\alpha ^ 2 A B + \\alpha \\gamma B ^ 2 + \\gamma ^ 2 B C - 2 \\alpha \\beta A ^ 2 - \\alpha \\delta A B - \\beta \\gamma A B - 2 \\gamma \\delta A C = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "7368.png", "formula": "\\begin{align*} \\partial _ t u = \\Delta u + f \\end{align*}"}
+{"id": "7817.png", "formula": "\\begin{align*} r \\cdot ( \\rho ^ { } , \\zeta ^ i , \\widetilde { \\zeta } _ i ^ { } , \\sigma ^ { } ) = ( r \\rho ^ { } , \\sqrt { r } \\zeta ^ i , \\sqrt { r } \\widetilde { \\zeta } _ i ^ { } , r \\sigma ^ { } ) \\ , . \\end{align*}"}
+{"id": "3854.png", "formula": "\\begin{align*} - H = - \\sin \\theta \\beta e ^ { - \\lambda _ 2 z } - \\sin \\theta \\mu \\lambda _ 2 y e ^ { - \\lambda _ 2 z } + \\cos \\theta \\mu . \\end{align*}"}
+{"id": "8796.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } \\mathbb { E } [ f ( x _ t ) - f ^ * ] \\leq \\frac { 7 2 \\bar { L } ^ 2 \\kappa } { \\alpha } r _ { 1 } + \\mathcal { B } _ { 4 } \\frac { d } { \\alpha } T ^ { \\frac { 1 } { \\beta } } . \\end{align*}"}
+{"id": "2915.png", "formula": "\\begin{align*} \\gamma _ e ^ * K _ { W _ e / X } = F _ X ^ { e - r , * } \\gamma _ r ^ * K _ { W _ r / X } + \\gamma _ e ^ * K _ { W _ e / W _ r } \\leq F _ X ^ { e - r , * } \\gamma _ r ^ * K _ { W _ r / X } + h ^ * ( 1 - p ^ { e - r } ) K _ Y , \\end{align*}"}
+{"id": "8727.png", "formula": "\\begin{align*} \\delta _ { t + 1 } \\leq \\left ( 1 - \\frac { 3 } { t } \\right ) \\delta _ { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { t ^ { p _ { i } + 1 } } \\enspace , \\end{align*}"}
+{"id": "1543.png", "formula": "\\begin{align*} \\overline { \\rho \\ , F ' ( \\rho ) } = \\langle \\lambda _ 1 F ' ( \\lambda _ 1 ) , \\nu \\rangle + m ^ { \\rho F ' ( \\rho ) } , \\end{align*}"}
+{"id": "6193.png", "formula": "\\begin{align*} ( ( x y ) a ) b - ( ( x y ) b ) a - ( ( x a ) b ) y + ( ( x b ) a ) y + ( ( y a ) b ) x - ( ( y b ) a ) x = 0 . \\end{align*}"}
+{"id": "556.png", "formula": "\\begin{align*} f _ R ( t ) = \\norm { \\mathbf u ^ R } _ { \\widetilde { \\mathbf X } ^ { \\mathbf s , b } ( 0 , t ) } ^ 2 , f _ R ( 0 ) = 0 . \\end{align*}"}
+{"id": "6765.png", "formula": "\\begin{align*} d _ X ( c _ { x _ 1 , x } ( T + \\tau + t ) , c _ { x _ 2 , x } & ( T + t ) ) \\\\ \\leq & d _ X ( c _ { x _ 1 , x } ( T + \\tau + t ) , c _ { x _ 1 , x } ( T + \\tau - r ' ) ) \\\\ & + d _ X ( c _ { x _ 1 , x } ( T + \\tau - r ' ) , c _ { x _ 2 , x } ( T - r ' ) ) \\\\ & + d _ X ( c _ { x _ 2 , x } ( T - r ' ) , c _ { x _ 2 , x } ( T + t ) ) \\\\ < & | t + r ' | + \\delta ' + | t + r ' | \\\\ = & 2 | t + r ' | + \\delta ' \\leq 2 | t | + \\delta ' < 2 | t | + 1 . \\end{align*}"}
+{"id": "8903.png", "formula": "\\begin{align*} \\Lambda _ 0 : = \\begin{pmatrix} 2 \\lambda _ { C _ 0 } & \\lambda _ { C _ 0 } \\circ \\beta _ i \\\\ \\lambda _ { C _ 0 } \\circ \\beta _ i & 2 \\lambda _ { C _ 0 } \\end{pmatrix} , \\end{align*}"}
+{"id": "5723.png", "formula": "\\begin{align*} \\Sigma _ { M } ( 1 ) W _ M ^ 0 = e ^ { \\eta ^ { - 1 } \\ , ( B _ M ) _ { 1 , 1 } } \\ , \\gamma _ M \\ , \\Sigma _ { M } ( 0 ) W _ M ^ 0 \\end{align*}"}
+{"id": "6899.png", "formula": "\\begin{align*} \\sigma ^ i _ j ( p ) & = 0 , \\ , \\ , \\textrm { i f } j \\in \\mathcal { I } _ i , \\\\ \\sigma ^ i _ j ( p ) & \\neq 0 , \\ , \\ , \\textrm { i f } j \\notin \\mathcal { I } _ i . \\end{align*}"}
+{"id": "657.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\tau _ R ( \\omega ) = \\tau ( \\omega ) , \\end{align*}"}
+{"id": "414.png", "formula": "\\begin{align*} ( P ^ { ( \\alpha ) } ( t , \\cdot , \\cdot ) f ) ( x ) : = \\int _ { \\R ^ d } P ^ { ( \\alpha ) } ( t , x , y ) f ( y ) \\ , d y , t > 0 , \\ , x \\in \\R ^ d . \\end{align*}"}
+{"id": "3402.png", "formula": "\\begin{align*} P = c R _ 1 - \\tilde { c } R _ 2 , \\end{align*}"}
+{"id": "1510.png", "formula": "\\begin{align*} \\sum \\limits _ { p , q , y , } \\ , c _ { i j } ^ { p q } c _ { q k } ^ { y n } \\ , c _ { p y } ^ { l m } \\ , = \\ , \\sum \\limits _ { y , q , r } \\ , \\ , c _ { j k } ^ { q r } \\ , c _ { i q } ^ { l y } \\ , c _ { y r } ^ { m n } \\end{align*}"}
+{"id": "4861.png", "formula": "\\begin{align*} & ( f ( u ) + f ( - u ) + \\delta f ( u ) f ( - u ) ) E _ i = ( f ( u ) + f ( - u ) + \\delta f ( u ) f ( - u ) ) E _ j \\\\ & ( f ( u ) + f ( - u ) + \\delta f ( u ) f ( - u ) ) ( E _ i - E _ j ) = 0 \\end{align*}"}
+{"id": "5251.png", "formula": "\\begin{align*} \\| z _ { i , t } - \\bar { z } _ { t } \\| & \\le \\tau \\lambda ^ { t - 2 } \\sum _ { j = 1 } ^ n \\| z _ { j , 1 } \\| + \\tau \\sum _ { s = 1 } ^ { t - 2 } \\lambda ^ { t - s - 2 } \\sum _ { j = 1 } ^ n \\| \\epsilon ^ z _ { j , s } \\| + \\| \\epsilon ^ z _ { i , t - 1 } \\| + \\frac { 1 } { n } \\sum _ { j = 1 } ^ n \\| \\epsilon ^ z _ { j , t - 1 } \\| , \\end{align*}"}
+{"id": "7947.png", "formula": "\\begin{align*} \\Psi _ { 1 , 2 } = \\sum _ { i = 1 } ^ 2 ( \\deg _ { G _ i } ( u _ i ) + \\deg _ { G _ i } ( v _ i ) ) \\ge 7 , \\end{align*}"}
+{"id": "108.png", "formula": "\\begin{align*} \\int _ { \\Lambda } \\int _ { B _ u } | \\chi _ { B _ u } | ^ 2 \\dd x \\dd u = | \\Lambda | . \\end{align*}"}
+{"id": "5548.png", "formula": "\\begin{align*} s _ { j , i } = ( i - 1 ) \\frac { 2 \\pi } { n } \\in J _ j = [ 0 , 2 \\pi ] , i = 1 , 2 , \\ldots , n , \\end{align*}"}
+{"id": "7307.png", "formula": "\\begin{align*} \\mu _ - ( T ) = \\dim \\left ( \\oplus _ { \\mu < 0 } \\{ u \\in H : \\ , T u = \\mu u \\} \\right ) , \\end{align*}"}
+{"id": "2071.png", "formula": "\\begin{align*} \\mathbf { I I } & \\leq C _ 1 \\Lambda \\int _ 0 ^ { r _ 0 } \\frac { r ^ n } { t ^ { n / 2 } } \\cdot \\exp \\left ( - \\frac { r ^ 2 } { C _ 1 t } \\right ) \\frac { r } { t } d r \\\\ & = C _ 1 \\Lambda \\int ^ { r _ 0 t ^ { - 1 / 2 } } _ 0 r ^ { n + 1 } d r \\\\ & \\leq C ' ( n , r _ 0 , v _ 0 , g _ 0 , \\delta ) t ^ { - 1 - \\frac { n } 2 } \\end{align*}"}
+{"id": "948.png", "formula": "\\begin{align*} \\big ( D _ t ( C ^ t ) + D _ x ( C ^ x ) \\big ) \\mid _ { \\eqref { c o n s e r v a t i o n s y s t e m } } = 0 . \\end{align*}"}
+{"id": "8346.png", "formula": "\\begin{align*} \\ \\sigma _ { g \\eta } = s ( \\sigma _ \\eta ) \\ \\ \\sigma _ { \\eta } = s ( \\sigma _ { g \\eta } ) \\eta \\in \\partial \\mathbb { F } _ 2 g \\in S . \\end{align*}"}
+{"id": "8837.png", "formula": "\\begin{align*} \\delta _ { t + 1 } \\leq \\left ( 1 - \\frac { 3 } { t } \\right ) \\delta _ { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { t ^ { p _ { i } + 1 } } \\enspace , \\end{align*}"}
+{"id": "2960.png", "formula": "\\begin{align*} \\beta ^ { ( 1 ) } \\circ \\phi _ X ( c ) = \\phi _ Y \\circ \\alpha ( c ) c \\in J _ { \\phi _ X } . \\end{align*}"}
+{"id": "7481.png", "formula": "\\begin{align*} C _ L = \\begin{pmatrix} 0 & 0 & \\ldots & 0 & - \\frac { \\alpha _ 0 } { \\alpha _ d } \\\\ 1 & 0 & \\ldots & 0 & - \\frac { \\alpha _ 1 } { \\alpha _ d } \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\ldots & 1 & - \\frac { \\alpha _ { d - 1 } } { \\alpha _ d } \\end{pmatrix} . \\end{align*}"}
+{"id": "6475.png", "formula": "\\begin{align*} F _ 1 ' \\big | _ { e ^ - } & = b _ e , & F _ 1 ' \\big | _ { e ^ + } & = - b _ e \\\\ F _ 1 \\big | _ { e ^ - } & = - \\frac 1 2 \\ell _ e b _ e + c _ e , & F _ 1 \\big | _ { e ^ + } & = \\frac 1 2 \\ell _ e b _ e + c _ e , \\end{align*}"}
+{"id": "3063.png", "formula": "\\begin{align*} R = \\sum \\limits _ { i = 1 } ^ { N _ { \\rm R } } { \\log _ 2 { \\left ( 1 + \\frac { \\lambda _ i p _ i } { \\sigma ^ 2 } \\right ) } } . \\end{align*}"}
+{"id": "986.png", "formula": "\\begin{align*} x + y \\sqrt d = \\pm \\eta ^ n \\prod _ { p \\in S _ \\pm } \\gamma ' _ p { } ^ { n _ p } p ^ { - l _ p n _ p / 2 } \\end{align*}"}
+{"id": "963.png", "formula": "\\begin{align*} f ( p ) = \\left < \\varphi ( p ) \\wedge \\eta ( p _ 0 ) , \\eta ( p ) \\right > , p \\in \\Sigma , \\end{align*}"}
+{"id": "824.png", "formula": "\\begin{align*} \\tilde { h } ( e ^ { i \\theta _ 0 } ) = e ^ { n _ 0 \\tilde { g _ 1 } ( e ^ { i \\theta _ 0 } ) } = 1 . \\end{align*}"}
+{"id": "4987.png", "formula": "\\begin{align*} d _ B ( a , 0 ) + r _ 0 = d _ B ( b , 1 ) + r _ 0 - \\varepsilon \\Leftrightarrow \\varepsilon = d _ B ( b , 1 ) - d _ B ( a , 0 ) . \\end{align*}"}
+{"id": "4446.png", "formula": "\\begin{align*} | | [ D ^ { \\alpha } _ { \\ast } , \\mathcal { A } _ j ] \\partial _ j { \\mathbf V } | | ^ 2 _ { L ^ 2 ( \\Omega _ t ) } \\leq & \\sum _ { 0 < \\beta \\leq \\alpha } | | D ^ { \\beta } _ { \\ast } \\mathcal { A } _ j D ^ { \\alpha - \\beta } _ { \\ast } \\partial _ j { \\mathbf V } | | ^ 2 _ { L ^ 2 ( \\Omega _ t ) } \\ , . \\end{align*}"}
+{"id": "342.png", "formula": "\\begin{align*} \\mathsf { C } \\left ( s \\right ) : = \\operatorname * { d i a g } \\left [ \\mathsf { C } _ { j } \\left ( s \\right ) : 1 \\leq j \\leq n _ { \\Omega } \\right ] \\quad \\mathsf { C } _ { j } \\left ( s \\right ) : = \\left [ \\begin{array} [ c ] { l l } - \\mathsf { K } _ { j } \\left ( s \\right ) & \\mathsf { V } _ { j } \\left ( s \\right ) \\\\ \\mathsf { W } _ { j } \\left ( s \\right ) & \\mathsf { K } _ { j } ^ { \\prime } \\left ( s \\right ) \\end{array} \\right ] . \\end{align*}"}
+{"id": "1918.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } \\frac { { \\rm d } } { { \\rm d } t } \\| \\theta _ u \\| ^ 2 _ { 0 , I _ h } + \\frac { 1 } { 2 } \\| \\theta _ w \\| ^ 2 _ { 0 , I _ h } \\leq C h ^ { 2 k + 2 } & + C \\| f - f _ h \\| ^ 2 _ { 0 , \\mathcal { T } _ h } + \\| \\theta _ u \\| ^ 2 _ { 0 , I _ h } + h ^ { - 1 } \\| \\theta _ u \\| ^ 4 _ { 0 , I _ h } \\\\ & + \\kappa _ 1 + \\kappa _ 2 + \\kappa _ 3 + \\kappa _ 4 + \\kappa _ 5 . \\end{aligned} \\end{align*}"}
+{"id": "4318.png", "formula": "\\begin{align*} \\begin{alignedat} { 3 } f _ { L } ( t ) = \\frac { 1 } { \\sqrt { M } } \\sum _ { x = 0 } ^ { M - 1 } e ^ { \\mu 2 \\pi \\left ( \\frac { x t } { M } \\right ) } \\ ; f ( x ) \\end{alignedat} \\end{align*}"}
+{"id": "621.png", "formula": "\\begin{align*} I _ 1 - I _ 2 = \\int _ 0 ^ 1 \\left ( \\int _ 0 ^ 1 \\theta '' \\left ( \\rho ( t ) + s \\left [ \\kappa ( t ) - \\rho ( t ) \\right ] \\right ) \\ , d s \\right ) \\left [ \\kappa ( t ) - \\rho ( t ) \\right ] \\ , d t \\end{align*}"}
+{"id": "4337.png", "formula": "\\begin{align*} E _ { \\sf f } = \\cos ( 1 ) { \\bf 1 } + i \\sin ( 1 ) F \\ , , F = \\pi _ \\omega ( B ( { \\sf f } ) ) \\ , . \\end{align*}"}
+{"id": "610.png", "formula": "\\begin{align*} \\norm { u } _ { X ^ { s , b } ( S , T ) } = \\left ( \\int _ { \\R ^ d } \\norm { U ( t , \\xi ) } _ { H ^ b _ t ( S , T ) } ^ 2 \\ , d \\xi \\right ) ^ { 1 / 2 } , \\end{align*}"}
+{"id": "1558.png", "formula": "\\begin{align*} x ^ { A , B , C } _ i = \\lfloor s / 2 \\rfloor + 1 , ~ y ^ { A , B , C } _ i = 0 ~ \\ i \\in A \\end{align*}"}
+{"id": "1102.png", "formula": "\\begin{align*} \\| u \\| : = \\left ( \\int _ { D } | \\nabla u | ^ 2 ( x ) d \\mu \\right ) ^ { 1 / 2 } \\end{align*}"}
+{"id": "6929.png", "formula": "\\begin{align*} & \\overline { \\phi } = \\limsup \\limits _ { \\xi \\rightarrow - \\infty } \\phi ( \\xi ) , \\underline { \\phi } = \\liminf \\limits _ { \\xi \\rightarrow - \\infty } \\phi ( \\xi ) , \\\\ [ 0 . 2 c m ] & \\overline { \\psi } = \\limsup \\limits _ { \\xi \\rightarrow - \\infty } \\psi ( \\xi ) , \\underline { \\psi } = \\liminf \\limits _ { \\xi \\rightarrow - \\infty } \\psi ( \\xi ) , \\end{align*}"}
+{"id": "258.png", "formula": "\\begin{align*} \\hat { a } ^ { } _ \\nu ( \\xi ; g ^ { ( c ) } ) : = \\begin{cases} \\Delta _ ( \\xi ) & \\ \\nu = 0 , \\\\ 0 & \\ \\nu \\not \\geq 0 , \\end{cases} \\end{align*}"}
+{"id": "51.png", "formula": "\\begin{align*} \\mathcal Q ( z ) = \\frac { \\rho _ z ^ 2 } { 2 } \\vert \\Lambda \\vert \\widehat g ( 0 ) + E ^ { \\rm { L H Y } } _ d ( \\rho _ z ) + \\mathcal { K } ^ { \\rm { d i a g } } + \\mathcal R ^ { ( d ) } _ 1 , \\end{align*}"}
+{"id": "7293.png", "formula": "\\begin{align*} B = \\{ 2 n \\mid n \\in A \\} \\cup \\{ 2 n + 1 \\mid n \\notin A \\} . \\end{align*}"}
+{"id": "7103.png", "formula": "\\begin{align*} P ( y ) = - \\frac { t } { 2 \\pi } \\log ( y ^ 2 ) + \\frac { 2 \\sqrt { m N } } { p _ 1 q \\sqrt { p _ 2 } } y + 3 \\lambda \\ , \\xi ^ { 1 / 3 } y ^ { 2 / 3 } . \\end{align*}"}
+{"id": "8502.png", "formula": "\\begin{align*} r _ { \\ell } ^ { \\wedge } ( \\bar { z } ) = | w | + \\delta \\geq | w ' | - | w - w ' | + \\delta > | w ' | + \\frac { \\delta } { 2 } , \\end{align*}"}
+{"id": "1245.png", "formula": "\\begin{align*} \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| e ^ { i t \\Delta } w _ n ^ J \\| _ { S _ 0 ( \\R ) } = 0 . \\end{align*}"}
+{"id": "6171.png", "formula": "\\begin{align*} \\eta _ ( B ) = \\# \\{ ( i , j ) \\mid 1 \\leq j \\leq i \\leq n - 1 , \\ , b _ { i + 1 , j } \\leq b _ { i , j } = b _ { i + 1 , j + 1 } - 1 \\} . \\end{align*}"}
+{"id": "887.png", "formula": "\\begin{align*} & A - \\frac { \\sqrt { a b } } { a } B = \\frac { \\alpha } { t ^ \\alpha } \\mathrm { e } ^ { - \\frac { \\alpha ( x + y ) } { t ^ \\alpha } } \\bigg ( \\frac y x \\bigg ) ^ { - \\frac { 1 + \\sqrt { a b } } { 2 } } I _ { \\sqrt { a b } + 1 } \\bigg ( \\frac { 2 \\alpha \\sqrt { x y } } { t ^ \\alpha } \\bigg ) , \\end{align*}"}
+{"id": "8803.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { 0 \\le | m | \\leq \\ell - 1 } \\frac { 1 } { m ! } D ^ { m } f ( y ) ( x - y ) ^ { m } + \\sum _ { 0 \\le | m | = \\ell } \\frac { 1 } { m ! } D ^ { m } f ( \\zeta _ { x , y } ) ( x - y ) ^ { m } \\enspace . \\end{align*}"}
+{"id": "7233.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ d \\nu ( s _ i ) n _ i = \\nu ( \\pi ) = e ( R ' / R ) \\ , . \\end{align*}"}
+{"id": "2901.png", "formula": "\\begin{align*} ( \\mathrm L _ { \\omega ; 1 } f ) ( \\lambda ) : = \\sum _ { \\nu \\in W _ 0 \\omega } f ( \\lambda + \\nu ) \\end{align*}"}
+{"id": "8753.png", "formula": "\\begin{align*} \\eta _ t = \\min \\left ( \\frac { 1 } { 1 8 L \\kappa } , \\ , ( d T ) ^ { - \\frac { \\beta } { 2 \\beta - 1 } } \\right ) \\qquad h _ { t } = ( d T ) ^ { - \\frac { 1 } { 2 ( 2 \\beta - 1 ) } } \\enspace . \\end{align*}"}
+{"id": "7718.png", "formula": "\\begin{align*} \\{ f , g \\} _ { b a s } : = \\iota ^ \\sharp \\{ \\tilde { f } , \\tilde { g } \\} \\end{align*}"}
+{"id": "2369.png", "formula": "\\begin{align*} I _ { \\mathbb { C } ^ { o ( G ) } } = \\frac { o ( \\Lambda ) } { o ( G ) } \\sum _ { \\mu \\in \\Lambda ^ 0 } f ( \\pi ( \\mu ) ^ { - 1 } \\tau ) \\pi ( \\mu ) = \\frac { o ( \\Lambda ) } { o ( G ) } f ( \\tau ) I _ { \\mathbb { C } ^ { o ( G ) } } + \\frac { o ( \\Lambda ) } { o ( G ) } \\sum _ { \\mu \\in \\Lambda ^ 0 \\setminus \\{ ( 0 , 0 ) \\} } f ( \\pi ( \\mu ) ^ { - 1 } \\tau ) \\pi ( \\mu ) . \\end{align*}"}
+{"id": "7694.png", "formula": "\\begin{align*} ( i _ { B ' , K ' } \\circ \\varphi ) _ { \\sigma } ^ { \\sigma ' } = \\sum _ { y \\in K _ { \\sigma ' } \\backslash G / S _ { \\sigma } } ( i _ { B ' , K ' } \\circ \\varphi ) _ { \\sigma } ^ { \\sigma ' } [ y ] \\end{align*}"}
+{"id": "745.png", "formula": "\\begin{align*} f ( t , \\omega ) = \\int _ 0 ^ t \\norm { \\phi _ + ( s ) } _ { H ^ r } ^ 2 \\ , d s . \\end{align*}"}
+{"id": "6008.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } \\varphi ( z ) \\nu ( d z ) = \\int _ { \\mathbb { R } ^ d } \\int _ { \\mathbb { R } ^ d } P _ { t } ( z , d y ) \\varphi ( y ) \\nu ( d z ) \\end{align*}"}
+{"id": "2330.png", "formula": "\\begin{align*} f _ g ( \\tau _ h ) = ( f \\pi _ { g ^ { - 1 } } ) ( \\pi _ h \\tau ) = f ( \\pi _ { g ^ { - 1 } h } \\tau ) , \\forall g , h \\in G . \\end{align*}"}
+{"id": "8741.png", "formula": "\\begin{align*} \\langle \\hat { g } _ { t } , x _ t - x ^ * \\rangle = \\frac { \\norm { x _ t - x ^ * } ^ 2 - \\norm { x _ { t + 1 } - x ^ * } ^ { 2 } } { \\eta _ t } + \\frac { \\eta _ t } { 2 } \\norm { \\hat { g } _ t } ^ { 2 } . \\end{align*}"}
+{"id": "5254.png", "formula": "\\begin{align*} \\tilde { \\Delta } _ t = \\sum _ { i = 1 } ^ n ( G _ 1 + G _ 2 \\| q _ { i , t + 1 } \\| ) \\| \\epsilon ^ z _ { i , t } \\| - \\sum _ { i = 1 } ^ n \\frac { \\sigma \\| \\epsilon ^ z _ { i , t } \\| ^ 2 } { 2 \\alpha _ { t } } . \\end{align*}"}
+{"id": "1554.png", "formula": "\\begin{align*} \\Delta = y ^ 3 x ^ 2 ( x - \\lambda y ) + z f _ 5 ( x , y , z ) \\end{align*}"}
+{"id": "1280.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left \\{ \\log [ \\frac { \\lambda _ n ^ j } { \\lambda _ n ^ k } ] + \\frac { | x _ n ^ j - x _ n ^ k | ^ 2 } { \\lambda _ n ^ j \\lambda _ n ^ k } + \\frac { | t _ n ^ j ( \\lambda _ n ^ j ) ^ 2 - t _ n ^ k ( \\lambda _ n ^ k ) ^ 2 | } { \\lambda _ n ^ j \\lambda _ n ^ k } \\right \\} = \\infty . \\end{align*}"}
+{"id": "4937.png", "formula": "\\begin{align*} \\Psi ( \\tau ) : = \\sum _ { n \\geq 0 } q ^ { n ( n + 1 ) / 2 } = e ^ { - \\frac { 2 \\pi i \\tau } { 8 } } \\cdot \\dfrac { \\eta ( 2 \\tau ) ^ 2 } { \\eta ( \\tau ) } \\quad \\tilde { \\theta } ( \\tau ) : = \\sum _ { n = - \\infty } ^ \\infty ( - 1 ) ^ n q ^ { n ^ 2 } = \\dfrac { \\eta ( \\tau ) ^ 2 } { \\eta ( 2 \\tau ) } . \\end{align*}"}
+{"id": "4775.png", "formula": "\\begin{align*} \\mu _ { C } ( h C _ { 1 } \\cap C _ { 2 } ) = \\mu _ { C } ( C _ { 1 } ) \\mu _ { C } ( C _ { 2 } ) . \\end{align*}"}
+{"id": "3897.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { | a _ { N _ k } | ^ 2 } { S _ { N _ k } ^ 2 } = 0 . \\end{align*}"}
+{"id": "5835.png", "formula": "\\begin{align*} w _ 0 = & s _ { \\alpha _ 4 } s _ { \\alpha _ 7 } s _ { \\alpha _ 2 + \\alpha _ 4 + \\alpha _ 5 } s _ { \\alpha _ 3 + \\alpha _ 4 + \\alpha _ 5 } \\times \\\\ & s _ { \\scriptscriptstyle { d 6 } , m a x } s _ { \\alpha _ 2 + \\alpha _ 3 + \\alpha _ 4 } s _ { \\alpha _ { m a x } } s _ { \\scriptscriptstyle { e 7 } , m a x } . \\end{align*}"}
+{"id": "65.png", "formula": "\\begin{align*} C \\sum _ { i \\neq j } P _ i \\overline { Q } _ { L , j } g ( P _ i \\overline { Q } _ { L , j } ) ^ { \\dagger } + \\frac { 1 } { 4 } \\mathcal Q _ 4 ^ { } & = C \\widehat g ( 0 ) \\frac { n _ 0 n _ + ^ H } { \\vert \\Lambda \\vert } + \\frac { 1 } { 4 } \\mathcal Q _ 4 ^ { } . \\end{align*}"}
+{"id": "1048.png", "formula": "\\begin{align*} \\Psi ( u _ \\eta ^ * ) = \\inf \\left \\{ \\Psi ( u ) : \\ : u \\in W ^ { 1 , p ( z ) } ( \\Omega ) \\right \\} . \\end{align*}"}
+{"id": "3127.png", "formula": "\\begin{align*} \\mathbf { H } _ { k , } ( m ) = \\sqrt { \\frac { 1 } { \\kappa + 1 } } \\sum _ { n = 1 } ^ N g _ { n , m } \\sum _ { p = 1 } ^ { L _ k } \\alpha _ { k , p } e ^ { j ( \\varphi _ n + b _ { n + 1 } ) } . \\end{align*}"}
+{"id": "7903.png", "formula": "\\begin{align*} c _ { x , y } ^ { z ^ \\ast } = c _ { y , z } ^ { x ^ \\ast } = c _ { z , x } ^ { y ^ \\ast } . \\end{align*}"}
+{"id": "50.png", "formula": "\\begin{align*} \\mathcal Q ( z ) = \\frac { \\rho _ z ^ 2 } { 2 } \\vert \\Lambda \\vert ( \\widehat { g } ( 0 ) + \\widehat { g \\omega } ( 0 ) ) + \\mathcal { K } ^ { \\rm { B o g } } \\end{align*}"}
+{"id": "5975.png", "formula": "\\begin{align*} L _ a f = a \\int _ { \\mathbb { D } } \\frac { f ( s ' ) } { \\left ( ( t _ 1 - s _ 1 ) ^ 2 + a ^ 2 ( t _ 2 - s _ 2 ) ^ 2 \\right ) ^ { 1 / 2 } } d s ' \\end{align*}"}
+{"id": "7289.png", "formula": "\\begin{align*} C ( \\sigma \\mid [ A ] ^ \\omega ) & \\geq C ( \\sigma \\mid [ \\tilde { A } ] ^ \\omega ) - O ( 1 ) \\\\ & \\geq C ^ { 0 ' } ( \\sigma ) - \\log ( 4 / \\delta ) - O ( \\log \\log ( 4 / \\delta ) ) \\\\ & = C ^ { 0 ' } ( \\sigma ) - \\log ( 1 / \\delta ) - O ( \\log \\log ( 1 / \\delta ) ) . \\end{align*}"}
+{"id": "5983.png", "formula": "\\begin{align*} & \\mathcal { N } : H ^ { 1 / 2 } ( \\partial M ) ^ * \\mapsto H ^ { 1 / 2 } ( \\partial M ) , \\\\ & \\mathcal { N } \\psi = u ^ { \\psi } , \\end{align*}"}
+{"id": "3280.png", "formula": "\\begin{align*} \\phi _ X ( t ) & = \\mathcal { F } ^ \\mu ( f _ X ) ( t ) \\\\ & = \\mathcal { F } ^ \\mu \\left ( \\frac { d } { d x } F _ X \\right ) ( t ) \\\\ & = - \\mathcal { F } ^ \\mu \\left ( F _ X \\right ) ( t ) \\mu t . \\end{align*}"}
+{"id": "3283.png", "formula": "\\begin{align*} \\int _ \\mathbb { R } g _ X ( t ) \\phi _ X ( t ) e ^ { - \\mu t y } d t = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } f _ X * ( \\psi _ X ) ^ \\lozenge _ { _ \\Sigma } ( y ) , \\end{align*}"}
+{"id": "1475.png", "formula": "\\begin{align*} \\sigma _ { j } = i \\quad \\sum _ { k = 0 } ^ { i - 1 } d _ { k } + 1 \\leqslant j \\leqslant \\sum _ { k = 1 } ^ { i } d _ { k } , \\end{align*}"}
+{"id": "4366.png", "formula": "\\begin{align*} \\begin{cases} ( \\partial _ t + \\mathbf { u } \\cdot \\nabla ) p + \\rho c ^ 2 \\mathrm { d i v } \\mathbf { u } = 0 , \\\\ \\rho ( \\partial _ t + \\mathbf { u } \\cdot \\nabla ) \\mathbf { u } - ( \\mathbf { H } \\cdot \\nabla ) \\mathbf { H } + \\nabla q = 0 , \\\\ ( \\partial _ t + \\mathbf { u } \\cdot \\nabla ) \\mathbf { H } - ( \\mathbf { H } \\cdot \\nabla ) \\mathbf { u } + \\mathbf { H } \\mathrm { d i v } \\mathbf { u } = 0 , \\\\ ( \\partial _ t + \\mathbf { u } \\cdot \\nabla ) S = 0 , \\\\ \\end{cases} \\end{align*}"}
+{"id": "461.png", "formula": "\\begin{align*} p _ \\zeta ^ { ( 1 ) } ( t , r , s ) & = \\frac { 2 \\Gamma ( \\zeta + 1 ) } { \\sqrt \\pi } \\cdot \\frac { t } { \\left ( r ^ 2 + s ^ 2 + t ^ 2 \\right ) ^ { \\zeta + 1 } } \\\\ & \\times \\ , _ 2 \\tilde { F } _ 1 \\left ( \\zeta + 1 , \\zeta + 2 ; \\zeta + \\frac { 1 } { 2 } ; \\frac { 4 r ^ 2 s ^ 2 } { \\left ( r ^ 2 + s ^ 2 + t ^ 2 \\right ) ^ 2 } \\right ) . \\end{align*}"}
+{"id": "6025.png", "formula": "\\begin{align*} L \\phi ( F ) = \\sum _ { i = 1 } ^ d \\partial _ i \\phi ( F ) L F _ i - \\sum _ { i = 1 } ^ d \\sum _ { j = 1 } ^ d \\partial _ i \\partial _ j \\phi ( F ) \\langle D F _ i , D F _ j \\rangle _ \\mathcal { H } . \\end{align*}"}
+{"id": "3360.png", "formula": "\\begin{align*} \\prod _ { t = 0 } ^ { p - 1 } \\frac { ( x ; \\zeta ) _ { t q } } { ( \\zeta ^ e x ; \\zeta ) _ { t q } } = \\prod _ { t = 0 } ^ { p - 1 } \\frac { ( x ; \\zeta ) _ { e } } { ( \\zeta ^ { t q } x ; \\zeta ) _ { e } } = \\frac { ( x ; \\zeta ) _ k ^ e } { ( x ; \\zeta ^ q ) _ { p e } } , \\end{align*}"}
+{"id": "4703.png", "formula": "\\begin{align*} f ( x y z _ 0 ) = f ( x ) g ( y ) + f ( y ) g ( x ) + \\mu f ( x ) f ( y ) , \\ ; x , y \\in S . \\end{align*}"}
+{"id": "6624.png", "formula": "\\begin{align*} f ( \\tau ; \\beta ) = f \\Big ( - { 2 \\tau \\over \\beta } ; { 4 \\over \\beta } \\Big ) . \\end{align*}"}
+{"id": "8352.png", "formula": "\\begin{align*} d Z ( t ) = B Z ( t ) \\ , d t + d \\omega _ { 2 } ( t ) , Z ( 0 ) = Z _ 0 \\in V , t \\ge 0 \\end{align*}"}
+{"id": "7722.png", "formula": "\\begin{align*} \\{ \\Theta , \\Theta \\} = 0 . \\end{align*}"}
+{"id": "2431.png", "formula": "\\begin{align*} \\frac { \\theta _ \\Gamma ( c _ { \\mathbf { O } _ { 1 m } } ) ^ 2 \\theta _ \\Gamma ( c _ { \\mathbf { O } _ { 2 h } } ) ^ 2 } { \\theta _ \\Gamma ( c _ { \\mathbf { O } _ { 2 m } } ) ^ 2 \\theta _ \\Gamma ( c _ { \\mathbf { O } _ { 1 h } } ) ^ 2 } = \\frac { \\lambda _ h - \\lambda _ m } { \\lambda _ h - \\lambda _ l } \\frac { \\lambda _ k - \\lambda _ m } { \\lambda _ k - \\lambda _ l } \\end{align*}"}
+{"id": "282.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\left ( 1 + \\frac { z } { n } \\right ) ^ { n } = e ^ z . \\end{align*}"}
+{"id": "6022.png", "formula": "\\begin{align*} D _ i F = D _ { e _ i } F = \\langle D F , e _ i \\rangle _ { \\mathcal { H } } . \\end{align*}"}
+{"id": "5328.png", "formula": "\\begin{align*} x = \\max \\{ 0 , x + 1 \\} - 1 \\end{align*}"}
+{"id": "4208.png", "formula": "\\begin{align*} D ( 0 , \\varepsilon ) & = \\sigma _ { \\varepsilon } ( I \\diamond I \\circ _ { \\ast } T ) = \\sigma _ { \\varepsilon } ( \\varphi ( I ) \\diamond \\varphi ( I ) \\circ _ { \\ast } \\varphi ( T ) ) \\\\ & = \\sigma _ { \\varepsilon } ( 2 \\varphi ( I ) ^ { 2 } ( \\varphi ( T ) ^ { \\ast } - \\varphi ( T ) ) ) . \\end{align*}"}
+{"id": "2822.png", "formula": "\\begin{align*} D ( \\langle a \\rangle ) : = a \\otimes a + ( 1 - a ) \\otimes ( 1 - a ) \\end{align*}"}
+{"id": "7236.png", "formula": "\\begin{align*} \\begin{aligned} \\delta _ k & = \\left ( - \\frac { B _ { k k } } { G _ { k k } ^ 2 } \\left ( \\cosh G _ { k k } - 1 \\right ) + \\frac { 1 } { G _ { k k } } \\sinh G _ { k k } \\right ) x _ k , \\\\ \\Delta _ { k k } & = 1 + \\frac { 1 } { 2 } \\left ( \\cosh G _ { k k } - 1 \\right ) + \\frac { 1 } { 2 } \\frac { B _ { k k } ^ 2 } { G _ { k k } ^ 2 } ( \\cosh G _ { k k } - 1 ) - \\frac { B _ { k k } } { G _ { k k } } \\sinh G _ { k k } . \\end{aligned} \\end{align*}"}
+{"id": "2130.png", "formula": "\\begin{align*} \\begin{aligned} & \\partial _ t \\hat p ( t , x ) - \\partial _ x \\hat e ( t , x ) = 0 , \\\\ & \\partial _ t \\hat e ( t , x ) - \\partial _ x \\hat p ( t , x ) = 4 \\sinh ^ 2 ( \\Lambda ) \\left ( \\phi _ x ^ 2 - \\phi _ t ^ 2 \\right ) + \\Lambda _ x ^ 2 - \\Lambda _ t ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "7237.png", "formula": "\\begin{align*} \\Delta = \\frac { 1 } { 2 } \\left ( 1 - R ^ 2 \\right ) + \\frac { 1 } { 2 } \\left ( 1 + R ^ 2 \\right ) \\cosh G - R \\sinh G . \\end{align*}"}
+{"id": "5485.png", "formula": "\\begin{align*} \\mathrm { p e x p } ( 0 : \\Delta ) & = 1 , \\\\ \\mathrm { p e x p } ( - \\mu : \\Delta ) & = \\frac 1 { \\mathrm { p e x p } ( \\mu : \\Delta ) } , \\\\ \\mathrm { p e x p } ( \\mu + \\mu ' : \\Delta ) & = \\mathrm { p e x p } ( \\mu : \\Delta ) \\mathrm { p e x p } ( \\mu ' : \\Delta ) . \\end{align*}"}
+{"id": "8350.png", "formula": "\\begin{align*} \\omega _ 2 ( t ) = \\sum _ { i = 1 } ^ \\infty ( q _ { i i } ^ 2 ) ^ \\frac 1 2 e _ { B , i } \\omega _ 2 ^ i ( t ) , \\quad \\sum _ { i = 1 } ^ \\infty q _ { i i } ^ 2 < \\infty , q _ { i j } ^ 2 = 0 i \\not = j . \\end{align*}"}
+{"id": "2905.png", "formula": "\\begin{align*} \\Phi _ { \\xi } ( 0 ) & = \\sum _ { v \\in W _ 0 } C ( v \\xi ) = \\sum _ { v \\in W _ 0 } \\prod _ { \\alpha \\in R ^ + _ 0 } \\frac { 1 - t _ { \\alpha } e ^ { - i \\langle v \\xi , \\alpha \\rangle } } { 1 - e ^ { - i \\langle v \\xi , \\alpha \\rangle } } \\\\ & \\stackrel { \\star } { = } \\sum _ { v \\in W _ { 0 } } t _ v = \\prod _ { \\alpha \\in R _ 0 ^ + } \\frac { 1 - t _ { \\alpha } e _ t ( \\alpha ) } { 1 - e _ t ( \\alpha ) } > 0 , \\end{align*}"}
+{"id": "79.png", "formula": "\\begin{align*} \\sum _ { k \\in \\mathcal { P } _ H } \\mathcal T _ { \\rm { c o m } } ( k ) = - 2 \\rho _ z \\widehat { g \\omega } ( 0 ) \\sum _ { p \\in \\mathcal P _ L } a _ p ^ \\dagger a _ p + \\mathcal E ' + \\mathcal E , \\end{align*}"}
+{"id": "5309.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n _ 0 } { N \\choose j } \\leq 1 + N ^ { n _ 0 } \\end{align*}"}
+{"id": "7581.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\int _ { 0 } ^ { v } v ^ { - \\frac { 7 } { 8 } } u ^ { - \\frac { 3 } { 8 } } \\exp \\left ( - C K v ^ { - 1 } [ v ^ { \\frac { 3 } { 4 } } - u ^ { \\frac { 3 } { 4 } } ] ^ 2 \\right ) d u d v \\leq K ^ { - 1 / 2 } . \\end{align*}"}
+{"id": "5369.png", "formula": "\\begin{align*} h ^ 1 ( { N _ { X '' } } _ { | _ { Z '' } } ( K _ { Z '' } - 3 H ) ) = 0 . \\end{align*}"}
+{"id": "8929.png", "formula": "\\begin{align*} d \\varphi ( \\nabla ^ M _ { E _ i ^ \\prime } E _ i ^ \\prime ) = - d \\varphi ( \\nabla ^ M _ { E _ i } E _ i ) + J d \\varphi ( [ E _ i ^ \\prime , E _ i ] ) . \\end{align*}"}
+{"id": "5500.png", "formula": "\\begin{align*} g = b ^ { \\epsilon _ 1 } a ^ { k _ 1 } \\dots a ^ { k _ { n - 1 } } b ^ { \\epsilon _ n } a ^ { k _ n } , \\epsilon _ 1 , \\dots , \\epsilon _ n \\in \\{ \\pm 1 \\} , k _ 1 , \\dots , k _ n \\in \\mathbb Z , n \\in \\mathbb Z _ { \\geq 0 } , \\end{align*}"}
+{"id": "3068.png", "formula": "\\begin{align*} { \\xi _ { k , { l _ k } , { j _ k } } } = { \\rho _ k } { \\alpha _ { { k , { \\rm { R } } } , { l _ k } } } { \\alpha _ { { \\rm { T } } , k , { j _ k } } } { \\bf { a } } _ { { \\rm { S , } } k } ^ H \\left ( { \\Theta _ { { k , { \\rm { R } } } , { l _ k } } ^ { \\rm { D } } } \\right ) { { \\bf { a } } _ { { \\rm { S } } , k } } \\left ( { \\Theta _ { { \\rm { T } } , k , { j _ k } } ^ { \\rm { A } } } \\right ) . \\end{align*}"}
+{"id": "1244.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } \\{ \\| \\nabla u _ n \\| _ { L ^ 2 } ^ 2 - \\sum _ { j = 1 } ^ J \\| \\nabla \\phi ^ j \\| _ { L ^ 2 } ^ 2 - \\| \\nabla w _ n ^ J \\| _ { L ^ 2 } ^ 2 \\} = 0 \\\\ & \\lim _ { n \\to \\infty } \\{ P ( u _ n ) - \\sum _ { j = 1 } ^ J P ( g _ n ^ j [ e ^ { i t _ n ^ j \\Delta } \\phi ^ j ] ) - P ( w _ n ^ J ) \\} = 0 . \\end{align*}"}
+{"id": "3763.png", "formula": "\\begin{align*} 2 | E ( G ) | & = \\sum _ { f \\in { F ( G ) } } \\ell ( f ) \\\\ & = \\sum _ { f \\in X } \\ell ( f ) + \\sum _ { f \\in Y } \\ell ( f ) + \\sum _ { f \\in F ( G ) \\setminus { X \\cup Y } } \\ell ( f ) \\\\ & \\geq 3 | X | + 4 | Y | + 5 ( | F ( G ) | - | X | - | Y | ) \\\\ & = 5 | F ( G ) | - 2 | X | - | Y | . \\end{align*}"}
+{"id": "3911.png", "formula": "\\begin{align*} A \\triangle B : = A \\setminus B \\cup B \\setminus A . \\end{align*}"}
+{"id": "2220.png", "formula": "\\begin{align*} \\left ( D _ { a + } ^ \\alpha y \\right ) ( x ) = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\left [ \\frac { y ( a ) } { ( x - a ) ^ \\alpha } + \\underset { a } { \\overset { x } { \\int } } \\frac { y ' ( t ) d t } { ( x - t ) ^ \\alpha } \\right ] , \\end{align*}"}
+{"id": "6849.png", "formula": "\\begin{align*} \\beta _ { \\rm c y l } = \\beta ^ * _ { \\varepsilon _ * , 0 } = \\max _ { \\rho \\in \\mathfrak { p } ^ { - 1 } ( \\{ 0 \\} ) } \\Im ( \\rho ) < + \\infty . \\end{align*}"}
+{"id": "6482.png", "formula": "\\begin{align*} f ' ( 0 ) = \\gamma f ( 0 ) , f ( \\epsilon ) = 0 , \\end{align*}"}
+{"id": "208.png", "formula": "\\begin{align*} & { Q _ 1 } ( M , P _ i , P _ j , \\tau ) = 2 ^ 6 [ h _ 0 ( 8 \\delta _ 1 ) ^ { 7 } + h _ 1 ( 8 \\delta _ 1 ) ^ { 5 } \\varepsilon _ 1 + h _ 2 ( 8 \\delta _ 1 ) ^ { 3 } \\varepsilon _ 1 ^ 2 + h _ 3 ( 8 \\delta _ 1 ) \\varepsilon _ 1 ^ 3 ] , \\end{align*}"}
+{"id": "5595.png", "formula": "\\begin{align*} \\alpha _ { i - j } \\geq \\alpha _ i - j p , \\quad \\forall j = 1 , \\ldots , i . \\end{align*}"}
+{"id": "1550.png", "formula": "\\begin{align*} \\Delta = y ^ 5 ( x - y ) + z f _ 5 ( x , y , z ) , \\end{align*}"}
+{"id": "7788.png", "formula": "\\begin{align*} \\rho = \\frac { \\tau _ 2 ^ 2 } { 1 6 } e ^ { - \\mathcal { K } } - c _ { \\ell } = \\frac { \\tau _ 2 ^ 2 } { 1 6 } K - c _ { \\ell } , K = - 2 ( \\tau _ { i j } ) z ^ i \\overline { z } ^ j \\ , , \\end{align*}"}
+{"id": "4897.png", "formula": "\\begin{align*} \\left [ I - \\frac { z } { x _ 1 } P _ 1 \\right ] ^ { - 1 } = \\left [ I - \\frac { z } { z - x _ 1 } P _ 1 \\right ] \\end{align*}"}
+{"id": "3959.png", "formula": "\\begin{align*} \\frac 1 { f ( u ' , v ' ) } - 1 = - \\frac { f ( u ' , v ' ) - 1 } { f ( u ' , v ' ) } \\end{align*}"}
+{"id": "8304.png", "formula": "\\begin{align*} & ( g _ m + \\Delta _ i ) \\circ u = g _ m \\circ u + \\Delta _ i \\circ u \\equiv g _ m \\circ u + J _ 0 ( u ) ^ { m + 1 } \\Delta _ i ~ ( m + 2 ) , \\\\ & u \\circ ( g _ m + \\Delta _ i ) \\equiv u \\circ g _ m + J ( u ) ( g _ m ) \\Delta _ i \\equiv u \\circ g _ m + J _ 0 ( u ) \\Delta _ i ~ ( m + 2 ) , \\end{align*}"}
+{"id": "3500.png", "formula": "\\begin{align*} & \\max _ { l _ 0 , \\ldots l _ m } \\frac { 1 } { l } \\log \\norm { s _ { l _ 0 , \\ldots l _ m } } _ { V _ { l - \\sum d _ i l _ i } } + \\frac { \\sum l _ i y _ i } { l } \\log | t | \\leq \\frac { 1 } { l } \\log \\norm { s } _ { y , t } + C ( l ) \\\\ \\leq & - ( \\phi _ t + | \\log | t | | \\psi _ t ) ( z ) + | \\log | t | | u ( y ) + \\frac { C } { l } | \\log | t | | + C | \\log | t | | \\epsilon + C ( l ) . \\end{align*}"}
+{"id": "738.png", "formula": "\\begin{align*} \\Phi ( t ) = i S _ { \\pm \\xi } ( - t ) P _ \\mu \\left ( P _ \\mu \\psi _ \\mp ^ \\mu ( t ) \\right ) \\mathfrak K _ 1 . \\end{align*}"}
+{"id": "6608.png", "formula": "\\begin{align*} w ( x ) = { 1 \\over ( 1 - i x ) ^ c ( 1 + i x ) ^ { \\overline { c } } } , c = \\beta ( N + p - 1 ) / 2 + 1 - i q , \\end{align*}"}
+{"id": "6723.png", "formula": "\\begin{align*} F ( t ) : = 4 \\pi \\gamma ( t ) + \\alpha ( t ) \\int _ { \\Sigma _ t } H | \\nabla u | + \\beta ( t ) \\int _ { \\Sigma _ t } | \\nabla u | ^ 2 . \\end{align*}"}
+{"id": "4374.png", "formula": "\\begin{align*} \\begin{cases} \\mathbb { L } ( { \\mathbf U } ^ { \\pm } , \\Psi ^ { \\pm } ) = 0 & [ 0 , T ] \\times \\R ^ 2 _ + , \\\\ \\mathbb { B } ( { \\mathbf U } ^ { + } , { \\mathbf U } ^ { - } , \\varphi ) = 0 & [ 0 , T ] \\times \\Gamma , \\\\ { \\mathbf U } ^ { \\pm } ( 0 , \\mathbf { x } ) = { \\mathbf U } ^ { \\pm } _ 0 & \\R ^ 2 _ + , \\\\ \\varphi ( 0 , x _ 2 ) = \\varphi _ 0 & \\R , \\end{cases} \\end{align*}"}
+{"id": "8185.png", "formula": "\\begin{align*} - x H _ r ( x , N , p ) = A _ r H _ { r + 1 } ( x , N , p ) + B _ r H _ { r } ( x , N , p ) + C _ r H _ { r - 1 } ( x , N , p ) , \\end{align*}"}
+{"id": "7336.png", "formula": "\\begin{align*} E ^ s ( \\lambda , 0 ) & = \\{ u ( 0 ) \\in \\mathbb { R } ^ { 2 n } : \\ , J u ' ( t ) + A ( \\lambda , t ) u ( t ) = 0 , \\ , t \\in \\mathbb { R } ; u ( t ) \\rightarrow 0 , t \\rightarrow \\infty \\} , \\\\ E ^ u ( \\lambda , 0 ) & = \\{ u ( 0 ) \\in \\mathbb { R } ^ { 2 n } : \\ , J u ' ( t ) + A ( \\lambda , t ) u ( t ) = 0 , \\ , t \\in \\mathbb { R } ; u ( t ) \\rightarrow 0 , t \\rightarrow - \\infty \\} , \\end{align*}"}
+{"id": "7191.png", "formula": "\\begin{align*} y _ 1 ^ { n _ 1 } y _ 2 ^ { n _ 2 } = y _ 1 ^ { n _ 1 - 2 } ( y _ 1 ^ 2 y _ 2 ^ { n _ 2 - 1 } ) y _ 2 = y _ 1 ^ { n _ 1 - 2 } y _ 2 ^ { n _ 2 + 1 } y _ 2 = \\cdots = y _ 1 y _ 2 ^ { n _ 1 + n _ 2 - 1 } = y _ 1 y _ 2 ^ { n - 1 } . \\end{align*}"}
+{"id": "5211.png", "formula": "\\begin{align*} \\left ( v \\frac { d B } { d v } \\right ) B ^ { - 1 } \\in \\frac { 1 } { v } \\mathrm { L i e } ( I ) \\iff \\delta _ 0 ( \\omega _ 3 - \\omega _ 1 ) = \\alpha _ 0 \\beta _ 0 ( \\omega _ 2 - \\omega _ 1 ) \\end{align*}"}
+{"id": "5514.png", "formula": "\\begin{align*} c _ + ^ { ( b a ^ { k - 1 } , 1 ) } ( p , s ) = & \\ , \\mathrm { p e x p } \\left ( \\frac { Q ^ 8 - Q ^ { - 8 } } { ( 1 + s ^ { - 1 } ) ( 1 + p ^ { - 1 } s ^ { 2 - k } ) ( 1 + p s ^ { k - 1 } ) } \\right ) = c _ + ^ { ( b a ^ k , 1 ) } ( p / s , s ) , \\\\ c _ - ^ { ( b ^ { - 1 } a ^ { k - 1 } , 1 ) } ( p , s ) = & \\ , \\mathrm { p e x p } \\left ( - \\frac { Q ^ 8 - Q ^ { - 8 } } { ( 1 + s ^ { - 1 } ) ( 1 + p ^ { - 1 } s ^ { 1 - k } ) ( 1 + p s ^ k ) } \\right ) = c _ - ^ { ( b ^ { - 1 } a ^ k , 1 ) } ( p / s , s ) , \\end{align*}"}
+{"id": "5634.png", "formula": "\\begin{align*} X _ 4 = \\Bigl ( y _ 0 y _ 1 + C - L ( x _ 0 y _ 1 - x _ 1 y _ 0 - B ) = 0 \\Bigr ) . \\end{align*}"}
+{"id": "8111.png", "formula": "\\begin{align*} \\begin{aligned} & \\left \\vert \\sum \\limits _ { j } \\mu _ { j } M _ { \\Phi } ( I ^ { l o c } _ { \\alpha } ( b _ { j } ) ) ( x ) \\right \\vert \\\\ & \\leq \\sum \\limits _ { j } \\mu _ { j } \\vert M _ { \\Phi } ( I ^ { l o c } _ { \\alpha } ( b _ { j } ) ) ( x ) \\vert \\chi _ { P _ { j } ^ { * } } ( x ) + \\sum \\limits _ { j } \\mu _ { j } \\vert M _ { \\Phi } ( I ^ { l o c } _ { \\alpha } ( b _ { j } ) ) ( x ) \\vert \\chi _ { ( P _ { j } ^ { * } ) ^ { c } } ( x ) \\\\ & = : \\uppercase \\expandafter { \\romannumeral 1 } + \\uppercase \\expandafter { \\romannumeral 2 } . \\end{aligned} \\end{align*}"}
+{"id": "2235.png", "formula": "\\begin{align*} \\lim \\limits _ { \\delta \\to 0 ^ + } \\frac { 1 } { \\delta ^ { \\sigma + 1 } \\ell ( \\delta ) } \\int _ 0 ^ \\delta t ^ \\sigma \\ell ( t ) \\ , d t = \\frac { 1 } { \\sigma + 1 } . \\end{align*}"}
+{"id": "1421.png", "formula": "\\begin{align*} \\lim _ { K \\to \\infty } \\theta ^ K ( x , \\mu ) = 0 . \\end{align*}"}
+{"id": "6389.png", "formula": "\\begin{align*} \\Delta \\begin{pmatrix} \\alpha & \\beta \\\\ \\gamma & \\delta \\end{pmatrix} = \\begin{pmatrix} \\alpha & \\beta \\\\ \\gamma & \\delta \\end{pmatrix} \\otimes \\begin{pmatrix} \\alpha & \\beta \\\\ \\gamma & \\delta \\end{pmatrix} , \\varepsilon \\begin{pmatrix} \\alpha & \\beta \\\\ \\gamma & \\delta \\end{pmatrix} = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} , \\end{align*}"}
+{"id": "4325.png", "formula": "\\begin{align*} \\omega _ \\Omega ( A ) = ( \\Omega , A \\Omega ) \\ , , \\omega _ \\Psi ( A ) = ( \\Psi , A \\Psi ) \\ \\ ( A \\in \\mathcal { N } ) \\ , , \\end{align*}"}
+{"id": "3146.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Bigg | _ { t = 0 } j _ { 0 , r _ i - 1 } \\left ( f ( u _ t ( z ) ) \\right ) & = j _ { 0 , r _ i - 1 } \\left ( \\frac { d } { d t } \\Bigg | _ { t = 0 } m ^ i _ t ( z ) g _ i ( 0 , z ) \\right ) \\\\ & = j _ { 0 , r _ i - 1 } \\left ( \\frac { d } { d t } \\Bigg | _ { t = 0 } m ^ i _ t ( z ) \\right ) M ( g _ i ) , \\end{align*}"}
+{"id": "3871.png", "formula": "\\begin{align*} \\theta - \\theta _ 0 + \\frac { \\lambda } { \\lambda _ 2 } = e ^ { - \\lambda _ 2 z } \\left ( \\frac { \\lambda } { \\lambda _ 2 } + \\mu y \\right ) \\end{align*}"}
+{"id": "3872.png", "formula": "\\begin{align*} R ^ d = C N \\langle v \\rangle ^ { - n p + 1 } | g ( v ) | ^ { - p } \\end{align*}"}
+{"id": "4388.png", "formula": "\\begin{align*} \\partial _ t \\hat { \\mathbf { H } } ^ { \\pm } + \\frac { 1 } { \\partial _ 1 \\hat { \\Phi } ^ { \\pm } } \\Big ( ( \\hat { \\mathbf { w } } ^ { \\pm } \\cdot \\nabla ) \\hat { \\mathbf { H } } ^ { \\pm } - ( \\hat { \\mathbf { h } } \\cdot \\nabla ) \\hat { \\mathbf { u } } ^ { \\pm } + \\hat { \\mathbf { H } } ^ { \\pm } \\mathrm { d i v } { \\hat { \\mathrm { \\mathbf { v } } } ^ { \\pm } } \\Big ) = 0 , \\end{align*}"}
+{"id": "2916.png", "formula": "\\begin{align*} c d ( K _ U + \\Delta ' ) & = c d ( K _ U + \\Delta | _ U ) + c E + G | _ U \\\\ & \\sim _ { \\mathbb Z } c d ( K _ U + \\Delta | _ U ) + c E + c B | _ U + c d D | _ U - c d ( K _ U + \\Delta | _ U ) \\\\ & = c ( B | _ U + d D | _ U + E ) , \\end{align*}"}
+{"id": "1321.png", "formula": "\\begin{align*} Q _ n & = n ^ n \\prod _ { i = 1 } ^ { n } c _ { i , n } , \\\\ c _ { i , n } & = \\frac { \\binom { 2 i - 2 } { i - 1 } \\binom { 2 n - 2 i } { n - i } } { \\binom { 2 n - 1 } { n - 1 } } , \\end{align*}"}
+{"id": "4256.png", "formula": "\\begin{align*} \\lambda _ { k } = \\min _ { \\stackrel { u \\in \\mathbb { P } _ { k } } { \\| u \\| _ { L ^ { 2 } ( \\Omega ) } = 1 } } \\left \\{ \\int _ { \\Omega } | \\nabla u | ^ 2 \\ , d x + \\alpha \\iint _ { \\mathbb { R } ^ { 2 n } } \\dfrac { | u ( x ) - u ( y ) | ^ 2 } { | x - y | ^ { n + 2 s } } \\ , d x d y \\right \\} , \\end{align*}"}
+{"id": "2254.png", "formula": "\\begin{align*} \\phi _ \\alpha ( x ) = e ^ { - \\pi \\alpha x ^ 2 } , \\ ; \\alpha > 0 , \\widehat { \\phi _ \\alpha } ( \\omega ) = \\frac { 1 } { \\sqrt { \\alpha } } e ^ { - \\frac { \\pi } { \\alpha } \\omega ^ 2 } = \\frac { 1 } { \\sqrt { \\alpha } } \\phi _ { 1 / \\alpha } ( \\omega ) . \\end{align*}"}
+{"id": "4395.png", "formula": "\\begin{align*} L ( \\hat { { \\mathbf U } } ^ { \\pm } , \\hat { \\Psi } ^ { \\pm } ) \\dot { { \\mathbf U } } ^ { \\pm } + C ( \\hat { { \\mathbf U } } ^ { \\pm } , \\hat { \\Psi } ^ { \\pm } ) \\dot { { \\mathbf U } } ^ { \\pm } - \\frac { \\Psi ^ { \\pm } } { \\partial _ 1 \\hat { \\Phi } ^ { \\pm } } \\partial _ 1 \\{ \\mathbb { L } ( \\hat { { \\mathbf U } } ^ { \\pm } , \\hat { \\Psi } ^ { \\pm } ) \\} = \\mathbf { f ^ { \\pm } } . \\end{align*}"}
+{"id": "851.png", "formula": "\\begin{align*} ( N g ) ( z ) : = \\eta ( z ) \\Psi _ 2 ( g ) + ( 1 - \\epsilon ) ( 1 - \\eta ( z ) ) T g ( z ) , \\end{align*}"}
+{"id": "6491.png", "formula": "\\begin{align*} \\Psi ' _ { e _ - } = b _ e , \\Psi ' _ { e _ + } = - b _ e , \\Psi _ { e _ - } = a _ e - \\frac { \\ell _ e } b _ e , \\Psi _ { e _ - } = a _ e + \\frac { \\ell _ e } b _ e . \\end{align*}"}
+{"id": "946.png", "formula": "\\begin{align*} F _ j ( x , t , u _ 1 , \\cdots , u _ s , \\mathcal { T } _ t ^ \\alpha u _ 1 , \\cdots , \\mathcal { T } _ t ^ \\alpha u _ s , u _ { 1 , x } , \\cdots , u _ { s , x } , \\cdots ) = 0 , ~ ~ j = 1 , \\cdots , s , \\end{align*}"}
+{"id": "7048.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } A _ { \\pi } ( 1 , n ) & e \\left ( \\frac { ( a + b q ) n } { p _ 1 q } \\right ) \\ , n ^ { i \\nu } \\ , e \\left ( \\frac { n u } { p _ 1 q Q } \\right ) V _ 1 \\left ( \\frac { n } { N } \\right ) \\\\ = & \\ , \\frac { N ^ { 2 / 3 + i \\nu } } { p _ 1 q } \\sum _ { \\pm } \\sum _ { n _ { 2 } \\sim ( p _ 1 Q K ) ^ 3 / N } \\frac { A _ { \\pi } ( n _ { 2 } , 1 ) } { n ^ { 1 / 3 } _ { 2 } } S \\left ( \\overline { ( a + b q ) } , \\pm n _ { 2 } ; p _ 1 q \\right ) \\eth _ 1 ^ { \\pm } ( . . . ) , \\end{align*}"}
+{"id": "3790.png", "formula": "\\begin{align*} Y : = X \\times \\R _ + , y = ( x , s ) \\in Y \\end{align*}"}
+{"id": "8494.png", "formula": "\\begin{align*} \\nu _ { E } ( x ) = \\lim _ { \\rho \\rightarrow 0 ^ { + } } \\frac { \\mu _ { E } ( B _ { \\rho } ( x ) ) } { | \\mu _ { E } | ( B _ { \\rho } ( x ) ) } \\mbox { e x i s t s a n d b e l o n g s t o } \\partial B _ { 1 } ( 0 ) . \\end{align*}"}
+{"id": "5957.png", "formula": "\\begin{align*} A ^ \\omega _ F ( t , D _ { t ' } ) ^ 2 + i [ D _ { t _ 3 } , A ^ \\omega _ F ( t , D _ { t ' } ) ] - \\widetilde { Q ^ \\omega } ( t , D _ { t ' } ) - \\widetilde { E } ( t ) A ^ \\omega _ F ( t , D _ { t ' } ) = 0 . \\end{align*}"}
+{"id": "6703.png", "formula": "\\begin{align*} H ( X ) = \\sum _ { n > 0 } \\frac { p _ n ( X ) } { n } a _ n , p _ n ( X ) = X _ 1 ^ n + X _ 2 ^ n + \\cdots \\end{align*}"}
+{"id": "4680.png", "formula": "\\begin{align*} | S _ j '' | \\cdot ( \\frac { n } { 2 ( 2 l + 1 ) } - 2 ) \\leq \\sum \\limits _ { v \\in S _ j '' } d _ { G ' } ( v ) \\le e ( S _ j '' , T _ j ) + \\sum \\limits _ { q = 1 } ^ { 2 l + 1 } e ( S _ j '' , S _ q - S _ j '' ) + 2 e ( S _ j '' ) . \\end{align*}"}
+{"id": "2743.png", "formula": "\\begin{align*} U ( e _ i ) = \\varepsilon _ i e _ { \\sigma ( i ) } \\end{align*}"}
+{"id": "4427.png", "formula": "\\begin{align*} \\vert \\lambda ^ \\pm \\vert = \\frac { a ^ \\pm \\vert [ u ] \\vert } { a ^ + \\vert H ^ + \\vert + a ^ - \\vert H ^ - \\vert } \\ , . \\end{align*}"}
+{"id": "4586.png", "formula": "\\begin{align*} Y ( a , z ) = \\sum _ { n \\in \\mathbb { Z } } a _ n \\ , z ^ { - n - 1 } \\ , \\in \\mathrm { E n d } ( \\mathfrak { V } ) [ [ z , z ^ { - 1 } ] ] \\ , , a _ n \\ , { \\in } \\ , \\mathrm { E n d } ( \\mathfrak { V } ) \\ , . \\end{align*}"}
+{"id": "435.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { \\sigma _ t ^ { ( \\alpha / 2 ) } ( \\tau ) } { t } = \\frac { \\Gamma ( \\alpha / 2 + 1 ) \\ , \\sin \\left ( \\pi \\alpha / 2 \\right ) } { \\pi \\ , \\tau ^ { 1 + \\alpha / 2 } } , \\tau > 0 , \\end{align*}"}
+{"id": "8611.png", "formula": "\\begin{align*} \\mu \\beta \\int _ { - h _ b } ^ 0 \\nabla _ X \\phi _ 1 ( X , z ) \\mathrm { d } z = \\mu \\beta h _ b \\int _ { - 1 } ^ 0 ( \\nabla _ X \\phi _ 1 ) ( X , h _ b z ) \\mathrm { d } z . \\end{align*}"}
+{"id": "1464.png", "formula": "\\begin{align*} v _ i = v _ { 2 n + 2 - i } = u _ i , \\ \\beta _ { i } = \\beta _ { 2 n + 2 - i } = \\alpha _ { i } , \\ \\ i \\in I . \\end{align*}"}
+{"id": "8125.png", "formula": "\\begin{align*} C ' _ 2 : = C ' \\setminus C _ 1 = C _ 2 \\setminus \\{ P \\} \\cup \\{ P ' \\} . \\end{align*}"}
+{"id": "7681.png", "formula": "\\begin{align*} \\psi _ \\varepsilon ^ * ( 0 ) = \\frac { \\lambda \\mathbb { E } X ^ * } { c } = 1 - \\frac { \\lambda \\varepsilon \\mathbb { P } ( X > a ) } { c } \\leqslant \\psi ( 0 ) \\leqslant 1 . \\end{align*}"}
+{"id": "8674.png", "formula": "\\begin{align*} \\tilde { \\phi } _ i = \\sum _ { j = 0 } ^ n a _ { i , j } \\phi _ j + b _ i , ~ i = 0 , . . . , n , \\end{align*}"}
+{"id": "5784.png", "formula": "\\begin{align*} s _ { \\delta } s _ { \\tau } s _ { \\delta } = s _ { s _ { \\delta } ( \\tau ) } . \\end{align*}"}
+{"id": "2014.png", "formula": "\\begin{align*} { \\mathrm { E } } : = T _ { \\phi } ^ { - 1 } \\tilde { \\mathrm { E } } T _ { \\phi } \\end{align*}"}
+{"id": "6082.png", "formula": "\\begin{align*} \\bar { X } = \\nabla f _ { S _ k } ( \\omega _ k ) , \\bar { Y } = \\nabla f _ { S _ k } ( \\phi _ k ) , E ( \\bar { Y } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla f _ i ( \\phi _ i ^ k ) \\end{align*}"}
+{"id": "583.png", "formula": "\\begin{align*} \\mathbf u ( t ) = P _ \\mu \\left ( \\mathbf S ( t ) \\mathbf u _ 0 + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf N \\left ( \\Theta _ R ^ { \\mathbf u } ( s ) P _ \\mu \\mathbf u ( s ) \\right ) \\ , d s + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf M \\left ( P _ \\mu \\mathbf u ( s ) \\right ) \\ , d W ( s ) \\right ) , \\end{align*}"}
+{"id": "3345.png", "formula": "\\begin{align*} \\biggl ( \\prod _ { i = 1 } ^ { N } \\frac { \\theta _ i ^ { m ( e ^ \\mathsf { T } A ) _ i } } { ( \\theta _ i ; \\zeta _ i ) _ { e _ i } ^ m } \\biggr ) \\cdot \\tau ( a _ \\zeta ( \\theta ) ^ m ) = \\biggl ( \\sum _ { k \\in ( \\mathbb { Z } / m \\mathbb { Z } ) ^ N } \\zeta ^ { \\overline { Q ( k ) } } \\prod _ { i = 1 } ^ { N } \\frac { \\theta _ i ^ { ( ( k + e ) ^ \\mathsf { T } A ) _ i } } { ( \\theta _ i ; \\zeta _ i ) _ { k _ i + e _ i + 1 } } \\biggr ) ^ m = a _ \\zeta ( \\theta ) ^ m . \\end{align*}"}
+{"id": "7241.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ^ 2 & = \\frac { \\Bigl ( \\Delta \\pm \\sqrt { \\Delta } \\sqrt { \\Delta + 1 - R ^ 2 } - ( 1 - R ) \\Bigr ) ^ 2 \\Bigl ( \\Delta \\pm \\sqrt { \\Delta } \\sqrt { \\Delta + 1 - R ^ 2 } \\Bigr ) ^ 2 } { 2 \\Bigl ( \\Delta \\pm \\sqrt { \\Delta } \\sqrt { \\Delta - 1 + R ^ 2 } + \\frac { 1 } { 2 } \\bigl ( 1 - R ^ 2 \\bigr ) \\Bigr ) ^ 2 } \\frac { 1 + R } { 1 - R } . \\end{aligned} \\end{align*}"}
+{"id": "6157.png", "formula": "\\begin{align*} ( T , s ) \\in \\bigsqcup _ { s \\in S } ( \\mathbf { k } _ s ) , \\ , \\ , \\eta _ ( ( T , s ) ) = \\eta _ ( T ) . \\end{align*}"}
+{"id": "7073.png", "formula": "\\begin{align*} G _ { 0 } ( y ) = d { \\pi } ^ { 3 } y ^ { 2 / 3 } \\ , \\int _ { 0 } ^ \\infty V _ 1 \\left ( \\frac { z } { N } \\right ) e \\left ( \\frac { z u } { p _ 1 q Q } \\right ) \\ , z ^ { - 1 / 3 + i \\nu } \\ , & \\left ( e ( 3 z ^ { 1 / 3 } y ^ { 1 / 3 } ) - e ( - 3 z ^ { 1 / 3 } y ^ { 1 / 3 } ) \\right ) d z \\\\ & + . \\end{align*}"}
+{"id": "3115.png", "formula": "\\begin{align*} \\mathbb { E } [ \\alpha _ { k , p } \\alpha ^ * _ { k , p ' } ] = \\begin{cases} 1 , & p = p ' \\\\ 0 , & p \\neq p ' . \\end{cases} \\end{align*}"}
+{"id": "1067.png", "formula": "\\begin{align*} \\max _ { \\left ( u , m , d \\right ) \\in \\overline { K _ { 1 } ^ { 2 } \\left ( M _ { 2 } \\right ) } \\times \\overline { K _ { 2 } \\left ( M _ { 3 } \\right ) } } \\left \\{ \\max _ { \\overline { Q } _ { T } } \\left [ \\max _ { \\left \\vert \\beta \\right \\vert \\leq 4 } \\left \\vert D ^ { \\beta } F _ { i } \\left ( X _ { i } \\right ) \\right \\vert \\right ] \\leq M _ { 4 } , i = 1 , 2 \\right \\} , \\end{align*}"}
+{"id": "3459.png", "formula": "\\begin{align*} W ( p ) = \\begin{cases} ( 1 + d _ 0 p _ 0 ) ^ { n - 1 } , \\Delta ^ \\vee _ 1 , \\\\ ( 1 + d _ 1 p _ 1 ) ^ { n - 1 } , \\Delta ^ \\vee _ 0 . \\end{cases} \\end{align*}"}
+{"id": "8863.png", "formula": "\\begin{align*} \\delta _ { t } \\leq \\frac { 2 ( t _ { 0 } - 1 ) \\delta _ { t _ { 0 } } } { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { ( c - p _ { i } ) t ^ { p _ { i } } } \\enspace . \\end{align*}"}
+{"id": "5857.png", "formula": "\\begin{align*} u + v x _ 0 = y _ u ( v _ u + \\gamma ) \\end{align*}"}
+{"id": "4145.png", "formula": "\\begin{align*} D _ x z = \\nabla _ x z , D _ x \\alpha = \\nabla _ x \\alpha , D _ E z = E z , D _ { E } \\alpha = E ^ * \\alpha . \\end{align*}"}
+{"id": "940.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u = \\frac { 1 } { \\kappa } ( s _ 1 ( x , t ) \\tilde { u } - r _ 2 ( x , t ) \\tilde { v } + \\delta ) , \\\\ & v = \\frac { 1 } { \\kappa } ( - s _ 2 ( x , t ) \\tilde { u } + r _ 1 ( x , t ) \\tilde { v } + \\varrho ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "6461.png", "formula": "\\begin{align*} ( a _ j ) ^ { 1 / j } : = ( n _ { j _ k } ^ { ( k ) } ) ^ { 1 / j _ k } . \\end{align*}"}
+{"id": "451.png", "formula": "\\begin{align*} \\int \\limits _ { | y | < 2 | x | } d y \\ , | y | ^ { - \\sigma } | y - x | ^ { 2 - d - \\alpha } = | x | ^ { 2 - \\sigma - \\alpha } \\int \\limits _ { | y | < 2 } d y \\ , | y | ^ { - \\sigma } | y - \\omega _ x | ^ { 2 - d - \\alpha } \\lesssim | x | ^ { - \\sigma } \\end{align*}"}
+{"id": "2115.png", "formula": "\\begin{align*} \\begin{aligned} & ~ { } G : = - \\left ( \\partial _ { t } ^ 2 ( \\ln \\alpha ) - \\partial _ { x } ^ 2 ( \\ln \\alpha ) \\right ) - \\dfrac { 1 } { 2 \\alpha ^ 2 } ( ( \\partial _ t \\alpha ) ^ 2 - ( \\partial _ x \\alpha ) ^ 2 ) \\\\ & ~ { } - \\dfrac { 1 } { 2 } ( ( \\partial _ t \\Lambda ) ^ 2 - ( \\partial _ x \\Lambda ) ^ 2 ) - 2 \\sinh ^ 2 \\Lambda ( ( \\partial _ t \\phi ) ^ 2 - ( \\partial _ x \\phi ) ^ 2 ) . \\end{aligned} \\end{align*}"}
+{"id": "3507.png", "formula": "\\begin{align*} Z _ { \\Gamma ' } ( s ) = Z _ \\Gamma ( s , \\lambda _ { \\Gamma / \\Gamma ' } ) . \\end{align*}"}
+{"id": "8579.png", "formula": "\\begin{align*} \\nabla _ X \\sigma = \\varepsilon \\nabla _ X \\Big ( \\dfrac { \\zeta } { h _ b } \\Big ) z + \\varepsilon \\nabla _ X \\zeta . \\end{align*}"}
+{"id": "253.png", "formula": "\\begin{align*} _ { 1 ; m } \\Phi ^ { } _ \\xi ( x ; g ) & : = \\lim _ { 2 \\xi _ 1 \\to m } ( 2 \\xi _ 1 - m ) \\Phi ^ { } _ \\xi ( x ; g ) , \\\\ _ { 1 2 ; m } \\Phi ^ { } _ \\xi ( x ; g ) & : = \\lim _ { \\xi _ 1 + \\xi _ 2 \\to m } ( \\xi _ 1 + \\xi _ 2 - m ) \\Phi ^ { } _ \\xi ( x ; g ) \\end{align*}"}
+{"id": "4086.png", "formula": "\\begin{align*} F _ g : = \\omega _ { g , 0 } : = \\frac { 1 } { 2 - 2 g } \\frac { 1 } { 2 \\pi i } \\oint _ { C _ { g , 1 } ^ { \\mathfrak { p } } } \\phi ( p ) \\cdot \\omega _ { g , 1 } \\end{align*}"}
+{"id": "4459.png", "formula": "\\begin{align*} | | | \\varphi ( t ) | | | ^ 2 _ { H ^ { s - 1 } ( \\R ) } & \\leq C ( K ) \\Big ( | | \\dot { u } _ n | _ { x _ 1 = 0 } | | ^ 2 _ { H ^ { s - 1 } ( \\Gamma _ t ) } + | | \\varphi | | ^ 2 _ { H ^ { s - 1 } ( \\Gamma _ t ) } + \\sum _ { | \\alpha | \\leq s - 1 } | | \\mathcal { G } _ { \\alpha } | | ^ 2 _ { L ^ 2 ( \\Gamma _ t ) } \\Big ) , \\end{align*}"}
+{"id": "8344.png", "formula": "\\begin{align*} E _ \\rho = \\{ ( x , y ) \\in X \\times X \\mid \\rho ( x , y ) < \\infty \\} . \\end{align*}"}
+{"id": "6507.png", "formula": "\\begin{align*} F ( q ) = D q + \\varepsilon \\Delta q + \\delta q _ * ^ { p + 1 } = 0 , \\end{align*}"}
+{"id": "2461.png", "formula": "\\begin{align*} H _ \\kappa ( G ) = \\min _ { V \\in W \\in \\Gamma ( G ) } I ( W ; V ) , \\end{align*}"}
+{"id": "4484.png", "formula": "\\begin{align*} e _ i : = e ' _ i + e '' _ i + e ''' _ i + D _ { i + \\frac { 1 } { 2 } } \\delta \\Psi _ i , \\tilde { e } _ i : = \\tilde { e } ' _ i + \\tilde { e } '' _ i + \\tilde { e } ''' _ i . \\end{align*}"}
+{"id": "5207.png", "formula": "\\begin{align*} s _ 1 \\cdots s _ { i - 1 } ( v _ i ) - s _ 1 \\cdots s _ { i } ( v _ i ) & = s _ 1 \\cdots s _ { i } ( \\bar \\alpha _ i ^ \\vee ) - s _ 1 \\cdots s _ { i } ( 0 ) . \\end{align*}"}
+{"id": "4733.png", "formula": "\\begin{align*} N \\ge \\lim _ k \\bigg | \\frac { \\partial _ { x _ i } u ( d _ k x + ( x ^ k ) ' ) } { d _ k x _ n } \\bigg | = \\lim _ k \\bigg | \\frac { \\partial _ { x _ i } \\tilde u _ k ( x ) } { x _ n } \\bigg | = \\bigg | \\frac { \\partial _ { x _ i } u _ 0 ( x ) } { x _ n } \\bigg | \\end{align*}"}
+{"id": "6056.png", "formula": "\\begin{align*} e ^ { - { s } \\vert a ^ { M _ { { \\mathcal { P } } } } _ { t } \\vert ^ 2 } \\leq e ^ { - { s } \\mathbb { E } \\bar { \\chi } _ { t } ^ { M _ { { \\mathcal { P } } } } } \\leq \\mathbb { E } ( e ^ { - { s } \\bar { \\chi } _ { t } ^ { M _ { { \\mathcal { P } } } } } ) \\leq \\exp ( - \\sum _ { n = 0 } ^ { N ( t ) } ( ( ( \\Gamma _ { n + 1 } \\wedge t ) - \\Gamma _ n ) \\int _ { B ^ c _ { M ( \\gamma _ { n + 1 } ) } } ( 1 - e ^ { - { s } \\underline { c } ( z ) } ) \\overline { \\mu } ( d z ) ) ) . \\end{align*}"}
+{"id": "7974.png", "formula": "\\begin{align*} \\gamma _ t { \\alpha _ \\delta } _ t = { \\alpha _ \\lambda } _ t \\theta _ t = { \\beta _ \\lambda } _ t { \\eta _ F } _ t \\theta _ t = \\gamma _ t \\varsigma _ t p _ t \\pi _ t , \\end{align*}"}
+{"id": "5355.png", "formula": "\\begin{align*} H ^ 1 ( N _ { X ' } \\otimes \\omega _ { X ' } ( - k + 1 ) ) = 0 . \\end{align*}"}
+{"id": "340.png", "formula": "\\begin{align*} \\left \\langle \\mbox { \\boldmath $ \\phi $ } _ { j } , \\mbox { \\boldmath $ \\psi $ } _ { j } \\right \\rangle _ { \\mathbf { X } _ { j } } : = \\left \\langle \\phi _ { \\operatorname * { D } ; j } , \\psi _ { \\operatorname * { N } ; j } \\right \\rangle _ { \\Gamma _ { j } } + \\left \\langle \\psi _ { \\operatorname * { D } ; j } , \\phi _ { \\operatorname * { N } ; j } \\right \\rangle _ { \\Gamma _ { j } } , \\end{align*}"}
+{"id": "1930.png", "formula": "\\begin{align*} \\theta _ f = \\Pi _ { \\lambda _ 1 , \\lambda _ 2 } f - f _ h \\mbox { a n d } \\eta _ f = \\Pi _ { \\lambda _ 1 , \\lambda _ 2 } f - f . \\end{align*}"}
+{"id": "1512.png", "formula": "\\begin{gather*} \\sigma _ x \\sigma _ y = \\sigma _ { \\sigma _ x ( y ) } \\sigma _ { \\gamma _ y ( x ) } \\\\ \\gamma _ y \\gamma _ x = \\gamma _ { \\gamma _ y ( x ) } \\gamma _ { \\sigma _ x ( y ) } \\\\ \\gamma _ { \\sigma _ { \\gamma _ y ( x ) } ( z ) } ( \\sigma _ x ( y ) ) = \\sigma _ { \\gamma _ { \\sigma _ y ( z ) } ( x ) } ( \\gamma _ z ( y ) ) \\end{gather*}"}
+{"id": "2181.png", "formula": "\\begin{align*} \\sum _ { R y \\in X _ R } b _ R ( R x , R y ) | \\Phi ( f ) ( R y ) | = \\sum _ { R y \\in X _ R } \\sum _ { y ' \\in R y } b ( x , y ' ) | f ( y ' ) | = \\sum _ { y \\in X } b ( x , y ) | f ( y ) | , \\end{align*}"}
+{"id": "1768.png", "formula": "\\begin{gather*} ( y ) = \\mathfrak { p } _ 1 ^ { a _ 1 } \\cdots \\mathfrak { p } _ k ^ { a _ k } , \\end{gather*}"}
+{"id": "6803.png", "formula": "\\begin{align*} 0 & \\leq \\phi ' _ 1 ( z _ 1 ) = w ' ( z _ 1 ) - K f ' ( u ( z _ 1 ) ) w ( z _ 1 ) \\\\ [ 0 . 2 c m ] & = c w ( z _ 1 ) - f ( u ( z _ 1 ) ) p ( u ( z _ 1 ) ) + f ( u ( z _ 1 ) ) v ( z _ 1 ) - K f ' ( u ( z _ 1 ) ) w ( z _ 1 ) \\\\ [ 0 . 2 c m ] & = \\left [ c K - p ( u ( z _ 1 ) ) + v ( z _ 1 ) - K ^ 2 f ' ( u ( z _ 1 ) ) \\right ] f ( u ( z _ 1 ) ) < 0 . \\end{align*}"}
+{"id": "5771.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } ( \\mathbb { E } \\langle f R _ { 1 , n + 1 } ^ p \\rangle - \\frac { 1 } { n } \\mathbb { E } \\langle f \\rangle \\mathbb { E } \\langle R _ { 1 , 2 } ^ p \\rangle - \\frac { 1 } { n } \\sum _ { a = 2 } ^ n \\mathbb { E } \\langle f R _ { 1 , a } ^ p \\rangle ) = 0 , \\end{align*}"}
+{"id": "3857.png", "formula": "\\begin{align*} \\sigma h _ { 1 1 } & = - \\lambda _ 2 e ^ { - ( 2 \\lambda _ 2 + \\lambda _ 1 ) z } f _ z , \\\\ \\sigma h _ { 1 2 } & = \\sigma h _ { 2 1 } = 0 , \\\\ \\sigma h _ { 2 2 } & = [ f _ { z z } - 2 \\lambda _ 1 f _ z - \\lambda _ 1 f _ z ^ 3 e ^ { - 2 \\lambda _ 1 z } ] e ^ { - \\lambda _ 1 z } . \\end{align*}"}
+{"id": "6949.png", "formula": "\\begin{align*} \\int _ { \\Omega _ i } f ( x ) d x = \\int _ { \\Omega _ { i + 1 } } f ( x ) d x + \\sum _ { j = 0 } ^ { i - 1 } \\int _ { \\Omega _ i ^ j } f ( x ) d x . \\end{align*}"}
+{"id": "978.png", "formula": "\\begin{align*} ( a , b , c ) = \\left ( \\frac { \\alpha + \\tilde \\alpha } { 2 } , \\frac { \\beta + \\tilde \\beta } { 2 } , \\frac { \\alpha \\beta - \\tilde \\alpha \\tilde \\beta } { \\alpha + \\beta - \\tilde \\alpha - \\tilde \\beta } \\right ) , \\ \\left ( \\frac { \\alpha + \\tilde \\alpha } { 2 } , \\frac { \\beta + \\tilde \\beta } { 2 } , - \\frac { \\alpha + \\beta - \\tilde \\alpha - \\tilde \\beta } { \\alpha \\beta - \\tilde \\alpha \\tilde \\beta } \\right ) \\end{align*}"}
+{"id": "62.png", "formula": "\\begin{align*} \\mathcal Q ^ { \\rm { l o w } } _ 3 : = \\sum _ { i \\neq j } ( P _ i Q _ { L , j } g ( x _ i - x _ j ) Q _ i Q _ j + h . c . ) . \\end{align*}"}
+{"id": "7425.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\partial _ { x } ^ { k + 1 } u _ { n } \\| _ { L ^ { 2 } _ { t , x } } ^ { 2 } & \\le M _ { k } \\| \\partial _ { x } ^ { k } V _ { n } \\| _ { L ^ { 2 } _ { t , x } } + M _ { k } \\| \\partial _ { x } ^ { k } \\rho _ { n } \\| _ { L ^ { 2 } _ { t } L ^ { 2 } _ { x } } + M _ { k } \\| \\partial _ { x } ^ { k - 1 } V _ { n } \\| _ { L ^ { 2 } _ { t , x } } \\\\ [ 1 e x ] & \\le M _ { k } + M _ { k } \\| \\partial _ { x } ^ { k } \\rho _ { n } \\| _ { L ^ { 2 } _ { t } L ^ { 2 } _ { x } } . \\end{aligned} \\end{align*}"}
+{"id": "3107.png", "formula": "\\begin{align*} \\mathbf { h } _ { k } = \\sum _ { p = 1 } ^ { L _ { k } } \\alpha _ { p , k } \\mathbf { b } _ T ^ H ( \\psi _ { 2 , k , p } , \\theta _ { 2 , k , p } ) \\end{align*}"}
+{"id": "2793.png", "formula": "\\begin{align*} ( f * g ) * h ( G ) & = \\sum _ { L \\leq G } ( f * g ) ( L ) h ( G / L ) \\\\ & = \\sum _ { L \\leq G } \\sum _ { H \\leq L } f ( H ) g ( L / H ) h ( G / L ) \\end{align*}"}
+{"id": "4803.png", "formula": "\\begin{align*} \\Phi ^ * [ \\eta _ { i , j , k } ] = \\left \\{ \\begin{array} { l l } \\nu _ i [ \\eta _ { i , j , k } ] + [ \\eta _ { i , j - 1 , k } ] & \\mbox { f o r $ j > 1 $ , } \\\\ \\nu _ i [ \\eta _ { i , j , k } ] & \\mbox { f o r $ j = 1 $ . } \\end{array} \\right . \\end{align*}"}
+{"id": "4725.png", "formula": "\\begin{align*} 0 & = \\partial _ { x _ 1 } \\tilde u _ { k _ m } ( t z _ { l _ m , k _ m } + ( 1 - t ) z _ { k _ m ( m ) } ) | _ { t = 1 } - \\partial _ { x _ 1 } \\tilde u _ { k _ m } ( t z _ { l _ m , k _ m } + ( 1 - t ) z _ { k _ m ( m ) } ) | _ { t = 0 } \\\\ & = \\int _ 0 ^ 1 \\partial _ { e ^ m x _ 1 } \\tilde u _ { k _ m } ( s z _ { l _ m , k _ m } + ( 1 - s ) z _ { k _ m ( m ) } ) | z _ { l _ m , k _ m } - z _ { k _ m ( m ) } | d s . \\end{align*}"}
+{"id": "8678.png", "formula": "\\begin{align*} F _ 1 ( x _ 0 , . . . , x _ { n + 1 } ) = 0 , ~ ~ ~ ~ ~ ~ F _ 2 ( y _ 0 , . . . , y _ { n + 1 } ) = 0 \\end{align*}"}
+{"id": "3921.png", "formula": "\\begin{align*} d _ H ( \\omega _ 1 , \\omega _ 2 ) : = \\max ( \\sup _ { u \\in \\omega _ 1 } d ( u , \\omega _ 2 ) , \\sup _ { u \\in \\omega _ 2 } d ( \\omega _ 1 , u ) ) \\end{align*}"}
+{"id": "5986.png", "formula": "\\begin{align*} \\int _ { \\Gamma _ { \\varepsilon , a } } G _ { \\partial M } ^ { \\sqrt { \\lambda _ { j , \\varepsilon } } } ( x , y ) \\partial _ { \\nu } u _ { j , \\varepsilon } ( y ) d \\mu _ h ( y ) = 0 . \\end{align*}"}
+{"id": "4109.png", "formula": "\\begin{align*} \\left . \\prod _ { j \\ge 0 } \\left ( \\prod _ { i = - 2 k + 4 j + 1 } ^ { - k + 2 j } ( \\ell + i ) \\prod _ { i = k - 2 j } ^ { 2 k - 4 j - 2 } ( \\ell + i ) \\right ) \\middle / \\prod _ { j = 1 } ^ { k - 1 } ( 2 j + 1 ) ^ { k - j } \\right . . \\end{align*}"}
+{"id": "7622.png", "formula": "\\begin{align*} | w | = \\left ( \\sum _ { i = 1 } ^ d w _ i ^ 2 \\right ) ^ { 1 / 2 } \\ge \\frac { 1 } { \\sqrt { d } } \\sum _ { i = 1 } ^ d w _ i \\ge \\frac { a } { \\sqrt { d } } = | w ^ u | , \\end{align*}"}
+{"id": "7407.png", "formula": "\\begin{align*} \\partial _ { x } \\rho _ { n } = - \\rho _ { n } ^ { 2 } \\partial _ { x } \\left ( \\frac { 1 } { \\rho _ { n } } \\right ) , ~ ~ \\partial _ { x } ^ { 2 } \\rho _ { n } = 2 \\rho _ { n } ^ { 3 } \\partial _ { x } \\left ( \\frac { 1 } { \\rho _ { n } } \\right ) ^ { 2 } - \\rho _ { n } ^ { 2 } \\partial _ { x } ^ { 2 } \\left ( \\frac { 1 } { \\rho _ { n } } \\right ) . \\end{align*}"}
+{"id": "2135.png", "formula": "\\begin{align*} \\hat p ( t , r ) = \\tilde e ( t , r ) . \\end{align*}"}
+{"id": "3245.png", "formula": "\\begin{align*} \\Delta _ { \\alpha } ( \\epsilon , R ) & = \\# \\left \\{ ( m , n ) \\in \\Z ^ { 2 } : m ( p / q - \\epsilon ) < n < m ( p / q + \\epsilon ) : 1 \\leq m \\leq \\frac { q R } { \\sqrt { p ^ { 2 } + q ^ { 2 } } } \\right \\} \\\\ & = \\# \\left \\{ ( m , n ) \\in \\Z ^ { 2 } : \\left | n q - m p \\right | < m q \\epsilon : 1 \\leq m \\leq \\frac { q R } { \\sqrt { p ^ { 2 } + q ^ { 2 } } } \\right \\} . \\end{align*}"}
+{"id": "1425.png", "formula": "\\begin{align*} 2 \\frac { \\partial ^ 2 U } { \\partial z \\partial \\bar z } + \\sum _ { j = 0 } ^ n c _ j e ^ { 2 \\alpha _ j ( U ) } \\alpha _ j ^ \\sharp = 0 , \\end{align*}"}
+{"id": "3647.png", "formula": "\\begin{align*} \\Delta _ { \\omega _ 1 } \\phi = \\tilde { k } \\Delta _ { \\omega _ 2 } \\phi \\ , \\end{align*}"}
+{"id": "2441.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ N | u _ n \\rangle \\langle u _ n | \\le \\mathcal { L } _ V ^ { - 1 } L ^ 2 ( \\Omega ) . \\end{align*}"}
+{"id": "2356.png", "formula": "\\begin{align*} \\pi ( \\lambda ) ( x _ g ) _ { g \\in G } = ( \\xi ( g ) x _ { g - k } ) _ { g \\in G } , \\forall ( x _ g ) _ { g \\in G } \\in \\mathbb { C } ^ { o ( G ) } . \\end{align*}"}
+{"id": "2029.png", "formula": "\\begin{align*} ( \\dot { \\mathfrak { b } } ^ { s _ 0 } _ { p , q _ 0 } ( \\Omega ) , \\dot { \\mathfrak { b } } ^ { s _ 1 } _ { p , q _ 1 } ( \\Omega ) ) _ { \\theta , q } = \\dot { \\mathfrak { b } } ^ { s } _ { p , q } ( \\Omega ) \\end{align*}"}
+{"id": "8742.png", "formula": "\\begin{align*} \\frac { \\alpha } { 4 } \\sum _ { t = T _ { 1 } + 1 } ^ { T } [ t ( r _ { t } - r _ { t + 1 } ) - r _ { t } ] \\leq \\frac { T _ 1 \\alpha } { 4 } r _ { T _ { 1 } + 1 } \\leq \\frac { 1 8 \\bar { L } ^ 2 \\kappa } { \\alpha } r _ { T _ { 1 } + 1 } . \\end{align*}"}
+{"id": "10.png", "formula": "\\begin{align*} \\varphi _ { \\mathbb { R } ^ d } = \\begin{dcases} \\frac { 1 } { \\log ( \\tilde R / a ) } \\varphi ^ { 0 } _ { \\mathbb { R } ^ d } , & d = 2 , \\\\ \\frac { 1 } { 1 - a / \\widetilde R } \\varphi ^ { 0 } _ { \\R ^ d } , & d = 3 , \\end{dcases} \\end{align*}"}
+{"id": "8570.png", "formula": "\\begin{align*} \\mathcal { L } _ 1 ^ { \\mu } [ \\beta b ] \\bullet = - b \\mathrm { F } _ 3 \\bullet , \\end{align*}"}
+{"id": "3264.png", "formula": "\\begin{align*} \\mathcal { F } ^ \\mu ( \\tau _ { _ h } f ) ( \\xi ) = \\mathcal { F } ^ \\mu ( f ) ( \\xi ) e ^ { - \\mu h \\xi } \\end{align*}"}
+{"id": "1866.png", "formula": "\\begin{align*} \\left | \\zeta ( s _ 0 , \\mathbb { X } _ \\alpha ) ( \\omega ) - p ( s _ 0 ) \\right | & = \\left | \\frac { 1 } { 2 \\pi i } \\oint _ { | z - s _ 0 | = r } \\frac { \\zeta ( z , \\mathbb { X } _ \\alpha ) ( \\omega ) - p ( z ) } { z - s _ 0 } \\ , d z \\right | \\\\ & \\leq \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } \\left | \\zeta ( r e ^ { i \\theta } , \\mathbb { X } _ \\alpha ) ( \\omega ) - p ( r e ^ { i \\theta } ) \\right | \\ , d \\theta \\end{align*}"}
+{"id": "5726.png", "formula": "\\begin{align*} u ( a , x ) = \\varphi ( a , x ) + \\sum _ { k = 1 } ^ { n - 1 } T ^ { \\circ ( k ) } [ \\varphi ] ( a , x ) + T ^ { \\circ ( n ) } [ u ] ( a , x ) , \\end{align*}"}
+{"id": "6888.png", "formula": "\\begin{align*} q ( \\widetilde { X } , r ) \\cdot r ^ * \\sigma = \\Big ( \\sum _ { i = 1 } ^ r b _ i C _ i \\Big ) \\cdot \\sigma , \\end{align*}"}
+{"id": "4710.png", "formula": "\\begin{align*} g = \\chi ( z _ 0 ) \\chi + \\lambda \\ , f . \\end{align*}"}
+{"id": "8275.png", "formula": "\\begin{align*} ( \\hat \\omega + i \\partial \\bar \\partial u ) ^ { 2 n } = e ^ { \\hat f } \\hat \\omega ^ { 2 n } . \\end{align*}"}
+{"id": "8596.png", "formula": "\\begin{align*} \\Delta _ { X , z } ^ { \\mu } \\phi _ 1 & = \\mu \\beta \\tilde { F } + \\mu ( \\frac { 1 } { h _ b } - 1 ) T _ 1 ( z ) [ X , \\mathrm { D } ] \\frac { \\Delta _ X } { \\sqrt { \\mu } | \\mathrm { D } | } G \\\\ & = \\mu \\beta \\tilde { F } + \\frac { \\mu \\beta b } { h _ b } T _ 1 ( z ) [ X , \\mathrm { D } ] \\frac { \\Delta _ X } { \\sqrt { \\mu } | \\mathrm { D } | } G \\\\ & = \\mu \\beta F . \\end{align*}"}
+{"id": "2170.png", "formula": "\\begin{align*} \\partial _ t p ( t , x ) + \\partial _ x e ( t , x ) = 0 . \\end{align*}"}
+{"id": "3877.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\sup _ { n \\in \\N } I V _ n = 0 . \\end{align*}"}
+{"id": "5020.png", "formula": "\\begin{align*} R = - a ( - f ) ^ { - 1 } - C _ 0 ( - f ) ^ { - 3 } , \\triangle _ f R \\geq \\triangle _ f \\big ( - a ( - f ) ^ { - 1 } - C _ 0 ( - f ) ^ { - 3 } \\big ) \\end{align*}"}
+{"id": "5903.png", "formula": "\\begin{align*} \\| v \\| _ { \\mathcal { X } } = \\left ( \\int _ { 0 } ^ \\infty x _ 2 \\| v ( \\cdot , x _ 2 ) \\| _ { L ^ \\infty ( \\R ) } ^ 2 d x _ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "5861.png", "formula": "\\begin{align*} D _ { N } ( T ) = \\sup _ { 1 \\leq \\nu < \\infty } E _ { \\nu } ( T - \\nu + 1 | T \\geq \\nu ) , \\end{align*}"}
+{"id": "1832.png", "formula": "\\begin{gather*} F _ g ( z ) = \\int _ { \\mathbb { C } } e ^ { z s } \\ , d \\mu _ g ( s ) , \\end{gather*}"}
+{"id": "4001.png", "formula": "\\begin{align*} b _ t f _ l = \\sum _ i \\left ( S _ { n - 1 ; i } ( X ) - \\frac { c } { C ^ m _ n } S _ { m - 1 ; i } ( X ) \\right ) X _ { i \\bar i l } , \\end{align*}"}
+{"id": "6179.png", "formula": "\\begin{align*} m _ i ( \\mu , T , 1 ) = k _ { i + 1 } - \\delta _ \\nwarrow ( \\mu _ { i + 1 } ) + c _ i ( T ) = k _ { i + 1 } + \\delta _ \\nearrow ( \\mu ' _ { i + 1 } ) + c _ { i + 1 } ( T ' ) = m _ { i + 1 } ( \\mu ' , T ' , 0 ) , \\\\ m _ { i + 1 } ( \\mu , T , 0 ) = k _ { i + 1 } + \\delta _ \\nearrow ( \\mu _ { i + 1 } ) + c _ { i + 1 } ( T ) = k _ { i + 1 } - \\delta _ \\nwarrow ( \\mu ' _ { i + 1 } ) + c _ { i } ( T ' ) = m _ { i } ( \\mu ' , T ' , 1 ) , \\end{align*}"}
+{"id": "3712.png", "formula": "\\begin{align*} I : = 2 \\alpha \\int _ { G _ 0 \\cup G ^ + _ 0 } \\frac { x _ 1 } { | x | ^ { 2 + 2 \\alpha } } d x - 2 \\alpha \\int _ { B _ 0 \\cup B ^ + _ 0 } \\frac { x _ 1 } { | x | ^ { 2 + 2 \\alpha } } d x \\end{align*}"}
+{"id": "7519.png", "formula": "\\begin{align*} \\mathbf { D } ( \\rho , r u ) = \\frac { \\rho u } { r ^ 3 } \\begin{pmatrix} u & - 1 \\\\ 0 & 0 \\end{pmatrix} + \\frac { \\theta } { r } \\begin{pmatrix} 1 / r & 0 \\\\ - \\rho u & \\rho \\end{pmatrix} \\end{align*}"}
+{"id": "2857.png", "formula": "\\begin{align*} w = s _ { j _ 1 } \\cdots s _ { j _ \\ell } , \\end{align*}"}
+{"id": "4114.png", "formula": "\\begin{align*} D _ 1 = \\left ( D ( 2 j - i , i + n - k - 1 ) \\right ) _ { 1 \\leq i , j \\leq k } . \\end{align*}"}
+{"id": "1173.png", "formula": "\\begin{align*} & \\bar { m _ { 1 , n } } - m _ { 1 , n } = \\delta ^ 1 \\phi _ n , \\\\ & \\bar { m _ { 2 , n } } - m _ { 2 , n } = \\delta _ 2 \\phi _ n . \\end{align*}"}
+{"id": "1035.png", "formula": "\\begin{align*} - \\Delta _ { p ( z ) } \\overline { u } _ \\eta - \\Delta _ { q ( z ) } \\overline { u } _ \\eta + \\overline { u } _ \\eta ^ { p ( z ) - 1 } = \\eta \\mbox { i n } \\Omega , \\ ; \\frac { \\partial \\overline { u } _ \\eta } { \\partial n } = 0 \\mbox { o n } \\partial \\Omega . \\end{align*}"}
+{"id": "1533.png", "formula": "\\begin{align*} N _ p ( x ) = \\frac x { p \\log x } \\left ( 1 + O \\left ( \\frac 1 { \\sqrt { \\log _ 2 x } } \\right ) \\right ) \\sum _ { k \\le K \\log _ 2 x } \\frac { e ^ { - \\gamma \\xi } } { \\Gamma ( 1 + \\xi ) } \\frac { ( \\log u ) ^ { k - 1 } } { ( k - 1 ) ! } \\sum _ { \\substack { P ^ + ( A ) \\le p \\\\ \\Omega ( A ) = k } } \\frac 1 A , \\end{align*}"}
+{"id": "262.png", "formula": "\\begin{align*} \\lim _ { c \\to + \\infty } \\gamma _ M ( g ^ { ( c ) } ) \\ , e ^ { c \\langle \\xi , \\rho _ M \\rangle } C ^ { } ( \\xi ; g ^ { ( c ) } ) = C ^ { } ( \\xi ; g ) , \\end{align*}"}
+{"id": "6423.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\sup _ { | x | \\leq R , t \\geq t _ n } v _ { n } ( x + c t , t ) = 0 . \\end{align*}"}
+{"id": "5694.png", "formula": "\\begin{align*} \\begin{aligned} \\| ( z - T ) ^ { - 1 } x _ r \\| _ p & = \\Big ( \\sum \\limits _ { n = 0 } ^ { \\infty } \\big | f _ n \\big ( ( z - T ) ^ { - 1 } x _ r \\big ) \\big | ^ p \\Big ) ^ \\frac { 1 } { p } \\\\ & \\leq \\big | f _ 0 \\big ( ( | z | - T ) ^ { - 1 } x _ r \\big ) \\big | \\bigg ( 1 + \\sum \\limits _ { n = 1 } ^ { \\infty } \\Big ( \\frac { r ^ n } { ( n + m ) ! \\prod \\limits _ { j = 1 } ^ n w _ j } \\Big ) ^ p \\bigg ) ^ \\frac { 1 } { p } \\\\ & = \\big | f _ 0 \\big ( ( | z | - T ) ^ { - 1 } x _ r \\big ) \\big | \\| x _ r \\| _ p . \\end{aligned} \\end{align*}"}
+{"id": "86.png", "formula": "\\begin{align*} \\sum _ { m \\in \\Z } \\langle \\Psi ^ m , \\mathcal H \\Psi ^ m \\rangle - \\langle \\Psi , \\mathcal H \\Psi \\rangle = \\sum _ { \\vert k \\vert \\leq 2 } \\delta _ k \\langle \\Psi , \\mathcal H ^ { ( k ) } \\Psi \\rangle , \\end{align*}"}
+{"id": "2252.png", "formula": "\\begin{align*} V ^ 2 ( g ) = \\{ f ( x ) = \\sum _ { k \\in \\Z } c _ k \\ , g ( x - k ) \\mid ( c _ k ) \\in \\ell ^ 2 ( \\Z ) \\} . \\end{align*}"}
+{"id": "6698.png", "formula": "\\begin{align*} G _ { \\lambda / \\mu , f / g } ( x ) = \\det \\left ( \\sum _ { s = 0 } ^ \\infty \\binom { i - j } { s } \\beta ^ s G ^ { [ f _ i / g _ j ] } _ { \\lambda _ i - \\mu _ j - i + j + s } ( x ) \\right ) . \\end{align*}"}
+{"id": "6326.png", "formula": "\\begin{align*} V = \\sqrt { g _ { < \\lambda ^ \\sigma } } ( V _ 1 + i P _ { \\gg \\epsilon ^ 2 } ^ y V _ 2 ) , W = \\frac { i } 2 \\partial _ y P _ { \\lesssim \\epsilon ^ 2 } ^ y V _ 2 . \\end{align*}"}
+{"id": "6205.png", "formula": "\\begin{align*} J _ 2 \\varphi = \\sqrt { \\det \\begin{pmatrix} \\norm { \\varphi _ x } ^ 2 & \\varphi _ x \\bullet \\varphi _ y \\\\ \\varphi _ x \\bullet \\varphi _ y & \\norm { \\varphi _ y } ^ 2 \\end{pmatrix} } = \\norm { \\varphi _ x } \\norm { \\varphi _ y } \\sqrt { 1 - \\cos \\theta } = \\norm { \\varphi _ x \\times \\varphi _ y } . \\end{align*}"}
+{"id": "6059.png", "formula": "\\begin{align*} F _ n ( x ) = { X } _ { t } ^ { M _ { { \\mathcal { P } } _ n } } ( x ) . \\end{align*}"}
+{"id": "4931.png", "formula": "\\begin{align*} I _ 1 ( k ) = \\dfrac { 1 } { 4 8 \\cdot 4 ^ { k - 1 } } . \\end{align*}"}
+{"id": "7611.png", "formula": "\\begin{align*} J _ 1 ( \\beta , N , T , \\mathbf { R _ 1 } ) & \\leq C N \\int \\int _ { \\hat { \\mathbf { R } } _ 1 \\cap \\{ t - s > \\delta \\} } \\frac { 1 } { 2 \\pi ( t - s ) } e ^ { - \\frac { \\beta ^ { 2 / 3 } N ^ { 2 / 3 } ( t - s ) ^ 2 } { 4 ( t - s ) } } d s d t \\\\ & + C N \\int \\int _ { \\hat { \\mathbf { R } } _ 1 \\cap \\{ t - s \\leq \\delta \\} } d s d t . \\end{align*}"}
+{"id": "95.png", "formula": "\\begin{align*} | \\mathcal { E } _ d ^ { } ( \\rho ) | \\leq \\begin{dcases} C \\rho ^ 2 \\widehat { g } ( 0 ) ^ 3 \\rho R ^ 2 \\log ( \\widehat { g } ( 0 ) ) , & d = 2 , \\\\ C \\rho ^ 2 \\widehat { g } ( 0 ) ^ 3 \\rho R ^ 2 \\sqrt { \\rho \\widehat { g } ( 0 ) ^ 3 } , & d = 3 . \\end{dcases} \\end{align*}"}
+{"id": "3267.png", "formula": "\\begin{align*} \\mathcal { F } ^ \\mu \\left ( \\frac { d ^ m } { d x ^ m } f \\right ) ( \\xi ) = \\mathcal { F } ^ \\mu \\left ( f \\right ) ( \\xi ) ( - \\mu \\xi ) ^ m . \\end{align*}"}
+{"id": "6515.png", "formula": "\\begin{align*} \\frac { \\partial ^ l v _ s ( m ) } { \\partial m ^ l } = \\frac 1 2 ( \\frac 1 2 - 1 ) \\cdots ( \\frac 1 2 - l + 1 ) v _ s ^ { - ( 2 l - 1 ) } ( m ) : = \\lambda _ l v _ s ^ { - ( 2 l - 1 ) } ( m ) . \\end{align*}"}
+{"id": "261.png", "formula": "\\begin{align*} \\lim _ { c \\to + \\infty } \\gamma _ L ( g ^ { ( c ) } ) \\Phi _ \\xi ^ { \\emph { b c } } ( x + c \\rho _ L ; g ^ { ( c ) } ) = \\Phi ^ { \\emph { c s } } _ \\xi ( x ; g ) , \\end{align*}"}
+{"id": "5176.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } { \\rm A o I } ( S _ { \\mathrm { z } } , Q _ { \\mathrm { u n i } } , F ^ * ) + \\frac { 3 } { 2 } \\log _ 2 { \\delta } = \\frac { 3 } { 2 } h ( X ) . \\end{align*}"}
+{"id": "7500.png", "formula": "\\begin{align*} \\Gamma : \\xi ^ 2 - 4 \\xi = \\eta ^ 2 \\end{align*}"}
+{"id": "8047.png", "formula": "\\begin{align*} \\left \\| f \\right \\| _ { h _ { \\omega , a t o m } ^ { p , q , s } } = \\inf \\left \\{ \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } + \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } \\right \\} \\end{align*}"}
+{"id": "7117.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\Delta ^ 2 V = p & \\hbox { \\rm i n } \\ D \\\\ V _ { | \\partial D } = 0 & \\hbox { \\rm o n } \\ \\partial D \\\\ ( \\Delta V ) _ { | \\partial D } = 0 & \\hbox { \\rm o n } \\ \\partial D \\\\ ( \\nabla \\Delta V \\cdot n ) _ { | \\partial D } = 0 & \\hbox { \\rm o n } \\ \\partial D \\end{array} \\right . \\end{align*}"}
+{"id": "2257.png", "formula": "\\begin{align*} \\min _ x \\sum _ { j = 1 } ^ n \\sum _ { k \\in \\Z } e ^ { - \\pi \\alpha \\left ( k + \\frac { x _ j - x } { \\delta } \\right ) ^ 2 } , \\end{align*}"}
+{"id": "5357.png", "formula": "\\begin{align*} H ^ 1 ( { N _ { X ' } } _ { | _ Z } ( K _ Z - ( k - 1 ) H ) ) = 0 . \\end{align*}"}
+{"id": "557.png", "formula": "\\begin{align*} \\tau _ R ( \\omega ) = \\sup \\left \\{ t \\in [ 0 , \\infty ) \\colon \\right \\} . \\end{align*}"}
+{"id": "5093.png", "formula": "\\begin{align*} I _ w ( r ) = r ^ { 1 - n } \\ , \\int _ { b = r } | w | ^ 2 \\ , | \\nabla b | \\ , . \\end{align*}"}
+{"id": "7551.png", "formula": "\\begin{align*} & \\ , \\ , \\Phi ^ { ( R _ 1 , R _ 2 ) , \\beta , D _ 1 \\cup _ { p } D _ 2 } _ { p _ 1 , \\ldots , p _ n } ( \\{ \\tau _ i \\} _ { i = 1 } ^ { | Q | } , Q ) ( - ) \\\\ & = \\sum _ { \\beta _ 1 + \\beta _ 2 = \\beta } \\left ( \\Phi ^ { R _ 1 , \\beta _ 1 , D _ 1 } _ { p _ 1 , \\ldots , p _ s , p } ( \\{ \\tau _ i \\} _ { x _ i \\in Q \\cap D _ 1 } , Q \\cap D _ 1 ) \\otimes \\Phi ^ { R _ 2 , \\beta _ 2 , D _ 2 } _ { p _ { s + 1 } , \\ldots , p _ n , p } ( \\{ \\tau _ i \\} _ { x _ i \\in Q \\cap D _ 2 } , Q \\cap D _ 2 ) \\right ) ( - \\boxtimes \\eta ) , \\end{align*}"}
+{"id": "4102.png", "formula": "\\begin{align*} \\omega _ { 0 , 2 } ( p _ 0 , p _ 1 ) : = B ( p _ 0 , p _ 1 ) - \\frac { d x ( p _ 0 ) \\cdot d x ( p _ 1 ) } { ( x ( p _ 0 ) - x ( p _ 1 ) ) ^ 2 } . \\end{align*}"}
+{"id": "4197.png", "formula": "\\begin{align*} D ( 0 , \\varepsilon ) & = \\sigma _ { \\varepsilon } ( [ \\varphi ( \\frac { I } { 2 } ) \\bullet \\varphi ( T ) , \\varphi ( i I ) ] _ { \\ast } ) = \\sigma _ { \\varepsilon } ( [ \\frac { I } { 2 } \\bullet T , i I ] _ { \\ast } ) \\\\ & = \\sigma _ { \\varepsilon } ( i ( T - T ^ { \\ast } ) ) , \\end{align*}"}
+{"id": "2335.png", "formula": "\\begin{align*} \\pi _ g \\tau _ e & = \\sum _ { h \\in G } f _ h ( \\tau _ e ) \\tau _ { g h } = \\sum _ { u \\in G } f _ { g ^ { - 1 } u } ( \\tau _ e ) \\tau _ { u } = \\sum _ { u \\in G } f _ { g ^ { - 1 } u } ( \\tau _ { g ^ { - 1 } g } ) \\tau _ { u } \\\\ & = \\sum _ { u \\in G } f _ { u } ( \\tau _ g ) \\tau _ { u } = \\tau _ g , \\forall g \\in G \\end{align*}"}
+{"id": "6752.png", "formula": "\\begin{align*} ( E ^ { e _ 1 } \\circ O ^ { o _ 1 } \\circ \\dotsb \\circ E ^ { e _ { l } } \\circ O ^ { o _ l } ) ( n ) = ( E ^ { e _ 1 + \\dotsb + e _ { l } } \\circ O ^ { o _ 1 + \\dotsb + o _ l } ) ( n ) + C = W . \\end{align*}"}
+{"id": "8325.png", "formula": "\\begin{align*} \\left ( \\mathbf { P } _ { { S M } } \\right ) _ { i j } = 0 { k \\not = i - 1 \\mod ( K + 1 ) } \\end{align*}"}
+{"id": "5203.png", "formula": "\\begin{align*} \\mbox { $ A _ u = A ' , \\ , B _ v = B ' , \\ , D _ u = S , \\ , $ a n d $ \\ , D _ v = T $ } \\end{align*}"}
+{"id": "3282.png", "formula": "\\begin{align*} \\phi _ X ( t ) = \\int _ \\mathbb { R } f _ X ( x ) e ^ { \\mu t x } d x \\psi _ X ( x ) = \\int _ \\mathbb { R } g _ X ( t ) e ^ { \\mu t x } d t , \\end{align*}"}
+{"id": "5758.png", "formula": "\\begin{align*} \\sum _ { X , Y \\in { \\cal B } _ p } { \\mathbb E } J _ X ^ p \\sigma _ X ^ { a } J _ Y ^ p \\sigma _ Y ^ { b } - \\sum _ { X , Y \\in { \\cal B } _ p } { \\mathbb E } J _ X ^ p \\sigma _ X ^ { a } { \\mathbb E } J _ Y ^ p \\sigma _ Y ^ { b } = | { \\cal B } _ p | \\Delta _ p ^ 2 R ^ p _ { a , b } , \\end{align*}"}
+{"id": "1574.png", "formula": "\\begin{align*} & \\sup _ { x \\in \\mathbb { T } } \\vert \\mathcal { F } ( T _ 1 ) ( x ) - \\mathcal { F } ( T _ 2 ) ( x ) \\vert \\leq C _ { \\alpha , \\underline { T } , \\underline { \\tau } , \\kappa } \\| T _ 1 - T _ 2 \\| _ { L ^ { \\infty } ( \\mathbb { T } ) } \\end{align*}"}
+{"id": "7954.png", "formula": "\\begin{align*} \\frac { \\chi _ i ( \\tau ) } { \\chi _ i ( I _ n ) } = \\sum _ { j = 1 } ^ k \\frac { n _ j ( n _ j - 2 j + 1 ) } { n ( n - 1 ) } , \\end{align*}"}
+{"id": "4847.png", "formula": "\\begin{align*} G _ i ^ 2 = \\alpha + \\beta E _ i + \\gamma G _ i + \\delta G _ i ^ { - 1 } \\end{align*}"}
+{"id": "2403.png", "formula": "\\begin{align*} Z _ { n } : = \\bigcup _ { \\i \\in W _ n } S _ { \\i , \\tt } \\left ( x _ n + \\prod _ { i = 1 } ^ d \\left [ - \\mathcal H ( n ) , \\mathcal H ( n ) \\right ] \\right ) . \\end{align*}"}
+{"id": "901.png", "formula": "\\begin{align*} \\tau = \\tau ( t ) , ~ ~ \\xi = \\xi _ 1 ( x , t ) , \\\\ \\end{align*}"}
+{"id": "3992.png", "formula": "\\begin{align*} \\phi ( s _ 0 + d ) = 0 , \\end{align*}"}
+{"id": "1627.png", "formula": "\\begin{align*} \\tilde { \\Delta } f ( x ) = - 1 + a _ { r } - a _ { r - 1 } + 2 \\sum _ { l = 0 } ^ { r - 1 } a _ l - 2 f ^ { * } + 2 \\epsilon . \\end{align*}"}
+{"id": "1606.png", "formula": "\\begin{align*} \\mathcal G _ T = \\{ P ^ { ( k - 2 ) } _ J \\in \\mathcal P ^ { ( k - 2 ) } : J \\in [ T ] ^ { k - 2 } \\} \\end{align*}"}
+{"id": "5532.png", "formula": "\\begin{align*} P _ i ( \\xi ) = e _ { i } ( \\xi ) + \\sum _ { 0 \\le j < i } c _ j e _ j ( \\xi ) . \\end{align*}"}
+{"id": "6679.png", "formula": "\\begin{align*} \\overline { W } _ 1 ( x ; N , \\beta , p , q ) = { N \\over x } + N \\sum _ { l = 1 } ^ \\infty { W _ 1 ^ { l } ( x ; \\beta , p , q ) \\over N ^ l } . \\end{align*}"}
+{"id": "7574.png", "formula": "\\begin{align*} \\begin{aligned} \\log Q _ T \\big ( A ^ { ( > ) } _ { T , r _ 2 ( T ) } \\big ) & = \\log q _ T ^ { ( > ) } - \\log Z _ { N , T } \\\\ & \\leq - \\frac { C N r _ 2 ( T ) ^ 2 } { T } + f ( \\beta , N , T ) , \\end{aligned} \\end{align*}"}
+{"id": "6961.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } & \\sum _ { k = 1 } ^ d \\sum _ { i = 1 } ^ { k - 1 } \\sum _ { j = i + 1 } ^ { k - 1 } \\frac { \\cos ^ 2 ( x _ k / 2 ) } { ( \\sin ( x _ k / 2 ) - \\sin ( x _ i / 2 ) + \\epsilon ^ 2 ) ( \\sin ( x _ k / 2 ) - \\sin ( x _ j / 2 ) + \\epsilon ^ 2 ) } \\\\ & = \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ d \\sum _ { i = k + 1 } ^ d \\sum _ { j = i + 1 } ^ d \\frac { \\cos ^ 2 ( x _ j / 2 ) } { ( \\sin ( x _ j / 2 ) - \\sin ( x _ k / 2 ) + \\epsilon ^ 2 ) ( \\sin ( x _ j / 2 ) - \\sin ( x _ i / 2 ) + \\epsilon ^ 2 ) } . \\end{align*}"}
+{"id": "5629.png", "formula": "\\begin{align*} \\ \\ \\ D _ { \\binom { 2 } { 2 } } = \\Bigl ( x _ 0 x _ 1 x _ 3 ^ 2 + B x _ 3 x + C x ^ 2 = 0 \\Bigr ) . \\end{align*}"}
+{"id": "2614.png", "formula": "\\begin{align*} I ^ Q ( q ) = \\frac { 1 } { 2 } \\inf \\{ \\int _ 0 ^ 1 \\dot w ^ 0 ( x ) ^ 2 \\ , d x + \\int _ 0 ^ \\infty \\dot w ( t ) ^ 2 \\ , d t + \\int _ 0 ^ \\infty \\int _ 0 ^ 1 \\dot k ( x , t ) ^ 2 \\ , d x \\ , d t \\} \\ , , \\end{align*}"}
+{"id": "6987.png", "formula": "\\begin{align*} \\overline { \\log d e n s } ( S _ 1 \\cap S _ 2 ) & \\geq \\underline { \\log d e n s } ( S _ 1 ) + \\underline { \\log d e n s } ( S _ 2 ) - \\overline { \\log d e n s } ( S _ 1 \\cup S _ 2 ) \\\\ & \\geq 1 - \\zeta + 1 - 1 = 1 - \\zeta . \\end{align*}"}
+{"id": "1251.png", "formula": "\\begin{align*} D _ n = 2 \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p ) | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p d x \\end{align*}"}
+{"id": "8175.png", "formula": "\\begin{align*} \\theta _ { i j } ^ \\star - \\theta _ { a b } ^ \\star = \\frac { n } { k } ( ( r - 2 ) ( b - j ) + ( r - 1 ) ( a - i ) ) . \\end{align*}"}
+{"id": "916.png", "formula": "\\begin{align*} m n x ^ k ( \\eta _ 1 - \\phi _ 1 ) - n \\eta _ { 2 x } + m \\phi _ { 2 x } = 0 , \\end{align*}"}
+{"id": "195.png", "formula": "\\begin{align*} & \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T _ C M } - \\widetilde { L _ C } - 4 8 8 + W _ i ) \\right \\} ^ { ( 1 0 ) } \\\\ & = A _ 1 \\left \\{ e ^ { \\frac { 1 } { 2 4 } A _ 1 } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) - \\frac { e ^ { \\frac { 1 } { 2 4 } A _ 1 } - 1 } { A _ 1 } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T _ C M } - \\widetilde { L _ C } - 4 8 8 + W _ i ) \\right \\} ^ { ( 6 ) } . \\end{align*}"}
+{"id": "8532.png", "formula": "\\begin{align*} g ( z ) = \\begin{cases} 0 , \\quad \\quad \\ \\mbox { i f } z \\in ( - \\infty , a ) \\\\ \\lambda ( r _ { \\ell } ( z ) - r _ { \\ell } ( a ) ) , \\mbox { i f } z \\in ( a , b ) \\\\ \\lambda ( r _ { \\ell } ( b ) - r _ { \\ell } ( a ) ) , \\mbox { i f } z \\in ( b , + \\infty ) . \\\\ \\end{cases} . \\end{align*}"}
+{"id": "6635.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ 2 } ( \\rho _ { ( 1 ) , \\infty } ^ { \\rm O C P } ( \\mathbf r ; Q ) - 1 ) e ^ { i \\mathbf k \\cdot \\mathbf r } \\ , d \\mathbf r = \\sum _ { j = 0 } ^ \\infty { ( - 1 ) ^ j \\over ( j ! ) ^ 2 } \\Big ( { | \\mathbf k | ^ 2 \\over 4 } \\Big ) ^ j \\int _ { \\mathbb R ^ 2 } ( \\rho _ { ( 1 ) , \\infty } ^ { \\rm O C P } ( \\mathbf r ; Q ) - 1 ) | \\mathbf r | ^ { 2 j } \\ , d \\mathbf r , \\end{align*}"}
+{"id": "2161.png", "formula": "\\begin{align*} \\tanh 2 \\phi = \\frac { 2 \\left ( m ^ { - 1 } - m \\right ) } { \\alpha ^ d \\left ( m ^ { d + 1 } + m ^ { - d - 1 } \\right ) - \\alpha ^ { - d } \\left ( m ^ { d - 1 } + m ^ { - d + 1 } \\right ) } . \\end{align*}"}
+{"id": "7960.png", "formula": "\\begin{align*} a ' = \\begin{cases} a & , \\\\ a + n & . \\end{cases} r ' = \\begin{cases} r + n & , \\\\ r & . \\end{cases} \\end{align*}"}
+{"id": "575.png", "formula": "\\begin{align*} S \\le t \\le S ' \\wedge \\tau _ R \\implies \\mathbf \\Phi ( \\mathbf V ) ( t ) = \\mathbf u ( t ) , \\end{align*}"}
+{"id": "6382.png", "formula": "\\begin{align*} ( \\mathrm { I d } \\otimes \\varepsilon ) ( \\chi ) & = 0 ; \\\\ ( \\varepsilon \\otimes \\mathrm { I d } ) ( \\chi ) & = 0 . \\end{align*}"}
+{"id": "2795.png", "formula": "\\begin{align*} \\Phi _ { N , s , p } ( r ) & : = \\mathrm { v o l } ( \\mathbb { S } ^ { N - 2 } ) \\int _ { - 1 } ^ 1 \\frac { \\left ( 1 - t ^ 2 \\right ) ^ { \\frac { N - 3 } { 2 } } } { \\left ( 1 - 2 r t + r ^ 2 \\right ) ^ { \\frac { N + p s } { 2 } } } \\dd t , & & N \\ge 2 , \\\\ \\Phi _ { 1 , s , p } ( r ) & : = \\left ( \\frac { 1 } { ( 1 - r ) ^ { 1 + p s } } + \\frac { 1 } { ( 1 + r ) ^ { 1 + p s } } \\right ) , & & N = 1 . \\end{align*}"}
+{"id": "6631.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ 2 } ( \\rho _ { ( 1 ) , \\infty } ^ { \\rm O C P } ( \\mathbf r ; Q ) - 1 ) \\ , d \\mathbf r = - Q . \\end{align*}"}
+{"id": "2427.png", "formula": "\\begin{align*} c _ { \\gamma } ( s _ 0 ^ { - 1 } \\gamma _ i ^ { - 1 } s _ 0 ^ { - 1 } ) \\frac { u _ \\gamma ( a _ i ) u _ { s _ 0 ^ { - 1 } \\gamma s _ 0 } ( a _ i ) } { u _ { \\gamma } ( a _ 0 ) u _ { s _ 0 ^ { - 1 } \\gamma s _ 0 } ( a _ 0 ) } = c _ { \\gamma } ( \\gamma _ i ) c _ { a _ i , a _ 0 } ( s _ 0 ^ { - 1 } \\gamma s _ 0 ^ { - 1 } \\gamma _ 0 ^ { - 1 } ) \\end{align*}"}
+{"id": "7170.png", "formula": "\\begin{align*} B _ m \\xrightarrow { \\pi _ m } B _ { m - 1 } \\xrightarrow { \\pi _ { m - 1 } } \\cdots \\xrightarrow { \\pi _ 2 } B _ 1 \\xrightarrow { \\pi _ 1 } B _ 0 = \\lbrace \\mbox { p t } \\rbrace \\end{align*}"}
+{"id": "543.png", "formula": "\\begin{align*} [ \\mathbf u , \\mathbf v ] ( t ) = \\begin{cases} \\mathbf u ( t ) & \\\\ \\mathbf v ( t ) & . \\end{cases} \\end{align*}"}
+{"id": "698.png", "formula": "\\begin{align*} \\tau < \\infty \\implies \\limsup _ { t \\nearrow \\tau } \\norm { \\mathbf u } _ { \\mathbf X ^ { \\mathbf s , b } ( 0 , t ) } = \\infty . \\end{align*}"}
+{"id": "6324.png", "formula": "\\begin{align*} V _ 1 = h ( u _ { [ < \\lambda ^ { \\sigma } ] } ) \\partial _ x u _ { [ < \\lambda ^ { \\sigma } ] } , V _ 2 = - \\Re \\int _ { - \\infty } ^ x h _ 1 ( u _ { [ < \\lambda ^ { \\sigma } ] } ) ( \\partial _ x u _ { [ < \\lambda ^ { \\sigma } ] } ) ^ 2 \\ , d x ' . \\end{align*}"}
+{"id": "8920.png", "formula": "\\begin{align*} \\tau _ V ( \\psi \\circ \\varphi ) = d \\psi ( \\tau _ V ( \\varphi ) ) + \\mathrm { t r a c e } \\nabla d \\psi ( d \\varphi , d \\varphi ) . \\end{align*}"}
+{"id": "5049.png", "formula": "\\begin{align*} \\triangle _ f h ' = R ( \\gamma '' , h ) + S ( \\gamma '' , h ' ) , \\end{align*}"}
+{"id": "8332.png", "formula": "\\begin{align*} \\mathbf { s } ^ * = \\mathbf { s } _ { \\mathbf { S B } } ^ * ( 0 ) \\mathbf { M } \\end{align*}"}
+{"id": "799.png", "formula": "\\begin{align*} 0 & = a ( t _ \\infty ^ \\flat ) ^ { p - 1 } \\int _ { \\mathbb { R } ^ N } | u ^ \\flat | ^ { p } d x - \\frac { b ( t _ \\infty ^ \\flat ) ^ { 2 \\cdot p ^ \\flat - 1 } } { p ^ \\flat } \\int _ { \\mathbb { R } ^ N } ( K \\ast | u ^ \\flat | ^ { p ^ \\flat } ) | u ^ \\flat | ^ { p ^ \\flat } d x \\\\ & = \\left ( a ( t _ \\infty ^ \\flat ) ^ { p - 1 } - \\frac { b ( t _ \\infty ^ \\flat ) ^ { 2 \\cdot p ^ \\flat - 1 } } { p ^ \\flat } \\right ) \\int _ { \\mathbb { R } ^ N } | u ^ \\flat | ^ { p } d x \\end{align*}"}
+{"id": "4569.png", "formula": "\\begin{align*} \\mathcal J _ 1 : = \\int _ 0 ^ t \\ ! \\ ! \\ ! \\int _ { \\mathbb R } \\hat c \\ , \\partial _ s \\alpha \\ , \\beta \\ , d x _ 2 d s \\quad \\mbox { a n d } \\mathcal J _ 2 : = \\int _ 0 ^ t \\ ! \\ ! \\ ! \\int _ { \\mathbb R } \\hat c \\ , \\partial _ s \\alpha \\ , \\varphi \\ , d x _ 2 d s \\ , . \\end{align*}"}
+{"id": "6685.png", "formula": "\\begin{align*} \\overline { W } _ { 2 } ^ { 0 } ( x , y ) = \\overline { W } _ { 2 } ^ { 1 } ( x , y ) = 0 , \\overline { W } _ { 3 } ^ { 0 } ( x , y , z ) = 0 . \\end{align*}"}
+{"id": "8080.png", "formula": "\\begin{align*} \\uppercase \\expandafter { \\romannumeral 1 } = : \\sum \\limits _ { i } \\sum \\limits _ { P \\in B _ { i , 0 } } \\mu _ { P } ^ { i } b _ { P } ^ { i } ( x ) \\end{align*}"}
+{"id": "2740.png", "formula": "\\begin{align*} \\big ( U ( e _ k ) \\big ) ( \\sigma ( i ) ) = 0 \\big ( U ^ { - 1 } ( e _ k ) \\big ) ( \\tilde { \\sigma } ( i ) ) = 0 . \\end{align*}"}
+{"id": "5863.png", "formula": "\\begin{align*} T _ { S L } = \\inf \\Big \\{ t : \\max _ { k = t - s + 1 \\in \\mathcal { K } } \\ell ( \\mathbf { p } _ { s t } ) \\geq C _ { \\gamma } \\Big \\} . \\end{align*}"}
+{"id": "4378.png", "formula": "\\begin{align*} H ^ { \\pm } _ { N } | _ { x _ 1 = 0 } = 0 , \\quad \\mbox { o n } \\ , \\ , \\ , \\{ x _ 1 = 0 \\} \\times \\mathbb R \\ , , \\end{align*}"}
+{"id": "1099.png", "formula": "\\begin{align*} - \\int _ D ( \\Delta _ \\mu u ( x ) ) v ( x ) d \\mu = & - \\int _ D \\left ( \\frac { \\sum _ { y \\sim x } w ( x , y ) ( u ( y ) - u ( x ) ) } { \\mu ( x ) } \\right ) v ( x ) d \\mu \\\\ = & \\int _ D \\frac { 1 } { 2 \\mu ( x ) } \\sum _ { y \\sim x } w ( x , y ) ( u ( y ) - u ( x ) ) ( v ( y ) - v ( x ) ) d \\mu , \\end{align*}"}
+{"id": "4992.png", "formula": "\\begin{align*} \\exp \\biggl ( \\sum _ { k = 1 } ^ \\infty \\frac { A ( x ^ k ) } { k } \\biggr ) & = \\exp \\biggl ( \\sum _ { k = 1 } ^ \\infty \\frac 1 k ( x ^ { k } + x ^ { 2 k } + x ^ { 3 k } + x ^ { 4 k } + x ^ { 5 k } + x ^ { 6 k } ) \\biggr ) \\\\ & = 1 + x + 2 x ^ { 2 } + 3 x ^ { 3 } + 5 x ^ { 4 } + 7 x ^ { 5 } + 1 1 x ^ { 6 } + 1 4 x ^ { 7 } + 2 0 x ^ { 8 } + 2 6 x ^ { 9 } + \\cdots \\end{align*}"}
+{"id": "4967.png", "formula": "\\begin{align*} P _ { 3 , 1 } ( n ) P _ { 3 , 5 } ( n ) = \\Phi _ { 6 n } ( 3 ) . \\end{align*}"}
+{"id": "6446.png", "formula": "\\begin{align*} F _ { n + 1 } = \\begin{bmatrix} x \\\\ z \\end{bmatrix} \\cdot F _ n - \\begin{bmatrix} y ^ 2 \\\\ w ^ 2 \\end{bmatrix} \\cdot F _ { n - 1 } . \\end{align*}"}
+{"id": "3028.png", "formula": "\\begin{align*} \\rho _ n > \\rho ( P _ { 2 ; 2 s + 1 } ^ { s + 1 } ) \\geq \\rho ( P _ { 2 ; 1 5 } ^ 8 ) = 2 . 0 9 0 4 + . \\end{align*}"}
+{"id": "8418.png", "formula": "\\begin{align*} L _ { 1 } = \\max \\left ( L , \\frac { P } { k } , \\frac { P } { k + q } \\right ) , L _ { 2 } = \\min \\left ( 2 L , \\frac { 2 P } { k } , \\frac { 2 P } { k + q } \\right ) \\end{align*}"}
+{"id": "1515.png", "formula": "\\begin{gather*} Z = \\ , \\{ T _ { i } ^ { k } \\mid 1 \\leq i \\leq n , \\ , 1 \\leq k \\leq n ' \\} \\\\ t ( T _ { i } ^ { k } , T _ { j } ^ { l } ) \\ , = ( g _ i ^ k ( T _ { j } ^ { l } ) \\ , , \\ , f _ j ^ l ( T _ { i } ^ { k } ) ) \\end{gather*}"}
+{"id": "2970.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { i = 1 } ^ M \\Theta _ { x _ i , x _ i } \\eta - \\eta \\Big \\| < \\frac { \\varepsilon } { 2 } \\end{align*}"}
+{"id": "7352.png", "formula": "\\begin{align*} \\Delta \\xi ( \\tau ) & = \\Big \\langle \\Big ( \\sum _ { a < b } ^ { 0 , u } y ^ a y ^ b \\Big ) ^ 2 \\Big \\rangle _ \\tau - \\Big \\langle \\sum _ { a < b } ^ { 0 , u } y ^ a y ^ b \\Big \\rangle _ \\tau ^ 2 = \\sum _ { a < b } ^ { 0 , u } \\sum _ { c < d } ^ { 0 , u } \\big ( \\langle y ^ a y ^ b y ^ c y ^ d \\rangle _ \\tau - \\langle y ^ a y ^ b \\rangle _ \\tau \\langle y ^ c y ^ d \\rangle _ \\tau \\big ) \\ge 0 , \\end{align*}"}
+{"id": "3004.png", "formula": "\\begin{align*} \\Lambda ( b ) ( v ) = \\sum _ { u \\in \\alpha ^ { - 1 } ( v ) } b ( u ) \\end{align*}"}
+{"id": "3675.png", "formula": "\\begin{align*} \\langle \\nabla u _ i , \\nabla \\ , \\left ( f - \\frac { 1 } { 4 } \\sum _ j u _ j ^ 2 \\right ) \\rangle = \\langle \\nabla u _ i , \\nabla f - \\frac { u _ i } { 2 } \\nabla u _ i \\rangle = 0 \\ , . \\end{align*}"}
+{"id": "3849.png", "formula": "\\begin{align*} e ^ { - \\lambda _ 2 z ( s ) } y ' ( s ) = \\cos \\theta ( s ) , z ' ( s ) = \\sin \\theta ( s ) , \\end{align*}"}
+{"id": "2310.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { q } _ { t } L ^ { r } _ { x } } \\lesssim \\| f \\| _ { X ^ { 0 , \\frac 1 2 , 1 } } , \\frac { 2 } { q } + \\frac { 1 } { r } = \\frac { 1 } { 2 } , 2 \\leq q , r \\leq \\infty , \\end{align*}"}
+{"id": "2477.png", "formula": "\\begin{align*} P _ A = \\beta P ' _ A + ( 1 - \\beta ) P '' _ A . \\end{align*}"}
+{"id": "4022.png", "formula": "\\begin{align*} S _ L = \\{ x \\in \\R ^ 3 : \\ , \\mathrm { d i s t } ( x , \\partial \\Omega ) < L \\} . \\end{align*}"}
+{"id": "7696.png", "formula": "\\begin{align*} { \\varphi ' } _ { \\sigma ' } ^ { \\sigma '' } [ x ' ] \\circ \\varphi _ { \\sigma } ^ { \\sigma ' } [ x ] = \\sum _ { x '' \\in S '' _ { \\sigma '' } \\backslash G / S _ { \\sigma } } \\bigl ( { \\varphi ' } _ { \\sigma ' } ^ { \\sigma '' } [ x ' ] \\circ \\varphi _ { \\sigma } ^ { \\sigma ' } [ x ] \\bigr ) [ x '' ] \\end{align*}"}
+{"id": "1047.png", "formula": "\\begin{align*} \\beta ( z , x ) = \\left \\{ \\begin{array} { l l } C _ 0 ( x ^ + ) ^ { \\tau ( z ) - 1 } - C _ 4 ( x ^ + ) ^ { r ( z ) - 1 } + \\vartheta ( x ^ + ) ^ { p ( z ) - 1 } , & \\hbox { i f } x \\leq u ( z ) \\\\ C _ 0 u ( z ) ^ { \\tau ( z ) - 1 } - C _ 4 u ( z ) ^ { r ( z ) - 1 } + \\vartheta u ( z ) ^ { p ( z ) - 1 } , & \\hbox { i f } u ( z ) < x . \\end{array} \\right . \\end{align*}"}
+{"id": "3485.png", "formula": "\\begin{align*} \\varphi = \\phi _ h + \\phi _ t + | \\log | t | | \\psi _ t , | \\phi _ h | \\leq C . \\end{align*}"}
+{"id": "8876.png", "formula": "\\begin{align*} H ^ * ( B T ) = \\R [ u _ 1 , \\ldots , u _ { \\ell } ] . \\end{align*}"}
+{"id": "8694.png", "formula": "\\begin{align*} e ^ { \\frac { x } { \\hbar } } = \\int \\frac { ( \\frac { x } { \\hbar } ) ^ { \\frac { t } { \\hbar } } } { \\Gamma ( \\frac { t } { \\hbar } + 1 ) } \\frac { d t } { \\hbar } . \\end{align*}"}
+{"id": "2003.png", "formula": "\\begin{align*} \\left ( s , \\frac { 1 } { p _ \\theta } , \\frac { 1 } { q _ \\theta } \\right ) : = ( 1 - \\theta ) \\left ( s _ 0 , \\frac { 1 } { p _ 0 } , \\frac { 1 } { q _ 0 } \\right ) + \\theta \\left ( s _ 1 , \\frac { 1 } { p _ 1 } , \\frac { 1 } { q _ 1 } \\right ) \\end{align*}"}
+{"id": "803.png", "formula": "\\begin{align*} - \\frac { ( N - p s ) ( p ^ \\flat + p ^ \\sharp ) } { p } + N + \\alpha = p ^ \\flat s < p s , \\end{align*}"}
+{"id": "163.png", "formula": "\\begin{align*} \\theta ( v , \\tau ) = 2 q ^ { \\frac { 1 } { 8 } } { \\rm s i n } ( \\pi v ) \\prod _ { j = 1 } ^ { \\infty } [ ( 1 - q ^ j ) ( 1 - e ^ { 2 \\pi \\sqrt { - 1 } v } q ^ j ) ( 1 - e ^ { - 2 \\pi \\sqrt { - 1 } v } q ^ j ) ] , \\end{align*}"}
+{"id": "8063.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { j \\in \\mathbb N } \\sum \\limits _ { Q \\in \\Pi _ { j + N } } \\vert Q \\vert \\psi _ { j } ( x - u _ { Q } ) ( \\psi _ { j } \\ast h ) ( u _ { Q } ) \\end{align*}"}
+{"id": "4914.png", "formula": "\\begin{align*} g ( \\tau ) = \\sum _ { \\ell = 0 } ^ { k - 1 } \\alpha _ k ( \\ell ) \\cdot F ( \\tau ) ^ \\ell \\theta ( 2 \\tau ) ^ { 4 ( k - \\ell ) + 2 } + \\sum _ { \\ell = k } ^ { 2 k } \\beta _ k ( \\ell ) \\cdot F ( \\tau ) ^ { 2 k - \\ell } F ( 2 \\tau ) ^ { \\ell - k } \\theta ( 2 \\tau ) ^ 2 + \\gamma _ k \\cdot \\dfrac { F ( \\tau ) F ( 2 \\tau ) ^ k } { \\theta ( 2 \\tau ) ^ 2 } , \\end{align*}"}
+{"id": "8707.png", "formula": "\\begin{align*} \\eta _ t = \\min \\left ( \\frac { \\mathfrak { y } } { d } , \\ , d ^ { - \\frac { 2 ( \\beta - 1 ) } { 2 \\beta - 1 } } T ^ { - \\frac { \\beta } { 2 \\beta - 1 } } \\right ) \\qquad h _ t = \\mathfrak { h } \\cdot T ^ { - \\frac { 1 } { 2 ( 2 \\beta - 1 ) } } \\enspace . \\end{align*}"}
+{"id": "1903.png", "formula": "\\begin{align*} \\widehat { u _ h ^ 2 } = \\frac { 1 } { 2 } [ ( u _ h ^ + ) ^ 2 + ( u _ h ^ - ) ^ 2 - 2 \\max | u | ( u _ h ^ + - u _ h ^ - ) ] , \\end{align*}"}
+{"id": "5298.png", "formula": "\\begin{align*} \\sum _ { j _ 1 \\leq \\frac { n _ 1 + \\min \\{ n _ 1 , d _ 1 \\} } { 2 } } f _ { j _ 1 , d _ 1 } ( n _ 1 ) \\sum _ { j _ 2 \\leq \\frac { n _ 2 + \\min \\{ n _ 2 , d _ 2 \\} } { 2 } } f _ { j _ 2 , d _ 2 } ( n _ 2 ) \\dots \\sum _ { j _ L = 0 } ^ { \\min \\{ n _ L , d _ L \\} } f _ { j _ L , d _ L } ( n _ L ) . \\end{align*}"}
+{"id": "107.png", "formula": "\\begin{align*} \\chi _ { B _ u } ( x ) : = \\chi \\left ( \\frac { x - u } { \\ell _ { } } \\right ) , \\end{align*}"}
+{"id": "3098.png", "formula": "\\begin{align*} \\alpha ^ 2 + 2 ( n - 2 ) \\beta ^ 2 + \\binom { n - 2 } { 2 } \\gamma ^ 2 \\le ( 1 - \\delta ) ^ 2 . \\end{align*}"}
+{"id": "2696.png", "formula": "\\begin{align*} F ( \\omega ) & = ( c _ 0 + 2 c _ 2 + 4 c _ 4 + \\ldots ) + \\sqrt { 2 } ( c _ 1 + 2 c _ 3 + 4 c _ 5 + \\cdots ) , \\\\ G ( \\omega ) & = ( d _ 0 + 2 d _ 2 + 4 d _ 4 + \\ldots ) + \\sqrt { 2 } ( d _ 1 + 2 d _ 3 + 4 d _ 5 + \\cdots ) , \\end{align*}"}
+{"id": "3989.png", "formula": "\\begin{align*} \\int _ M ( \\chi + \\tilde \\chi ) ^ n = c \\int _ M ( \\chi + \\tilde \\chi ) ^ m \\wedge \\omega ^ { n - m } , \\end{align*}"}
+{"id": "3532.png", "formula": "\\begin{align*} \\begin{aligned} z _ { t } & = \\Delta z - \\chi \\nabla \\cdot ( z \\nabla w ) , & x \\in \\Omega , t > 0 , \\\\ w _ { t } & = \\Delta w - \\lambda w + z , & x \\in \\Omega , t > 0 . \\end{aligned} \\end{align*}"}
+{"id": "5359.png", "formula": "\\begin{align*} H ^ 1 ( \\O _ { L _ 1 \\cup \\ldots \\cup L _ { 2 b } } ( 1 ) ( - B _ b ) ) = 0 , \\ \\hbox { w h e r e } \\ B _ b = ( L _ 1 \\cup \\ldots \\cup L _ { 2 b } ) \\cap ( M _ 1 \\cup \\ldots \\cup M _ b ) . \\end{align*}"}
+{"id": "4550.png", "formula": "\\begin{align*} \\tilde { \\mathcal F } _ \\sigma : = \\sigma \\partial _ 1 \\tilde { \\mathcal { F } } - \\sigma \\partial _ 1 \\mathcal B _ 0 \\partial _ t { \\mathbf V } + \\sigma ^ \\prime \\mathcal B _ 1 \\partial _ 1 { \\mathbf V } - \\sigma \\partial _ 1 \\mathcal B _ 2 \\partial _ 2 { \\mathbf V } - \\sigma \\partial _ 1 \\mathcal B _ 3 { \\mathbf V } \\ , . \\end{align*}"}
+{"id": "339.png", "formula": "\\begin{align*} u = \\mathsf { S } _ { j } \\left ( s \\right ) \\gamma _ { \\operatorname * { N } ; j } ^ { \\operatorname * { e x t } , - } \\left ( s \\right ) u ^ { - } - \\mathsf { D } _ { j } \\left ( s \\right ) \\gamma _ { \\operatorname * { D } ; j } ^ { - } \\left ( s \\right ) u ^ { - } . \\end{align*}"}
+{"id": "4189.png", "formula": "\\begin{align*} h ( \\boldsymbol { N } ) & = \\log \\left ( 2 \\pi e ^ 3 \\right ) + 2 \\log ( \\lambda ) - \\log ( 1 + \\lambda | u | ) \\\\ & ~ ~ - \\lambda | u | e ^ { \\lambda | u | } \\big ( e \\cdot ( - 1 - \\lambda | u | ) - 3 \\cdot ( - \\lambda | u | ) \\big ) , \\end{align*}"}
+{"id": "1834.png", "formula": "\\begin{align*} \\left \\| \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + \\alpha ) ^ s } - \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + c ) ^ s } \\right \\| & \\leq \\sum _ { n = 0 } ^ { N } \\left \\| ( n + \\alpha ) ^ { - s } - ( n + c ) ^ { - s } \\right \\| \\\\ & = \\sum _ { n = 0 } ^ { N } \\left \\| \\int _ { c } ^ { \\alpha } ( - s ) ( n + u ) ^ { - s - 1 } \\ , d u \\right \\| \\end{align*}"}
+{"id": "768.png", "formula": "\\begin{align*} \\norm { \\theta _ R ( \\norm { \\mathbf V } _ { t } ^ 2 ) \\mathbf V } _ { T _ { \\mathbf v } } = \\norm { \\theta _ R ( \\norm { \\mathbf U } _ { t } ^ 2 ) \\mathbf U - \\theta _ R ( \\norm { \\mathbf V } _ { t } ^ 2 ) \\mathbf V } _ { T _ { \\mathbf V } } . \\end{align*}"}
+{"id": "2677.png", "formula": "\\begin{align*} \\Delta _ { g } \\varphi _ t \\ = \\ n - t G + _ { g } ^ { h } , \\end{align*}"}
+{"id": "7929.png", "formula": "\\begin{align*} \\dim _ \\mathbb { C } ( Z _ j , I ( g _ k ) ) = \\dim _ \\mathbb { C } ( F ( Z _ j ) , g _ k ) = 3 , \\end{align*}"}
+{"id": "1665.png", "formula": "\\begin{align*} V ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } ( \\boldsymbol { \\xi } ) : = \\sum _ { 1 \\le j < k \\le n } \\int _ 0 ^ { \\xi _ j - \\xi _ k } v _ q ( \\vartheta ) \\vartheta + \\sum _ { 1 \\leq j \\leq n } \\left ( { \\textstyle \\frac { m } { 2 } } \\xi _ j ^ 2 - 2 \\pi ( \\varrho _ { \\texttt { a } ; j } + \\lambda _ j ) \\xi _ j \\right ) , \\end{align*}"}
+{"id": "2778.png", "formula": "\\begin{align*} I ( G - N [ v _ 0 ] ) & = x \\cdot ( 2 x + 1 ) ^ { 2 k + 5 } \\\\ & = x \\cdot \\bigg [ \\sum _ { i = 0 } ^ { 2 k + 5 } \\binom { 2 k + 5 } { i } ( 2 x ) ^ { i } \\bigg ] \\\\ & = x \\cdot [ ( 2 k ) ^ { 2 k + 5 } + \\dots ] \\\\ & = 2 ^ { 2 k + 5 } x ^ { 2 k + 6 } + \\dots \\end{align*}"}
+{"id": "536.png", "formula": "\\begin{align*} \\norm { M _ 2 ( g ) } _ { \\mathcal L _ 2 ( L ^ 2 , H ^ r ) } = \\norm { g \\mathfrak K _ 2 } _ { \\mathcal L _ 2 ( L ^ 2 , H ^ { r - 1 } ) } . \\end{align*}"}
+{"id": "35.png", "formula": "\\begin{align*} \\mathcal { Q } _ 2 ^ { 1 } : = \\sum _ { i \\neq j } P _ { i } Q _ { j } g ( x _ { i } - x _ { j } ) P _ { j } Q _ { j } . \\end{align*}"}
+{"id": "6655.png", "formula": "\\begin{align*} b _ 0 = \\int _ { - \\infty } ^ \\infty \\Big ( R ( x ) - { 1 \\over \\pi } \\Big ) \\ , d x , b _ 1 = { p ^ 2 \\over 2 \\pi } \\end{align*}"}
+{"id": "6393.png", "formula": "\\begin{align*} a \\triangleleft b & : = \\mathcal { R } ^ { - 1 } ( a _ { 1 } \\otimes b _ { 1 } ) a _ { 2 } \\mathcal { R } ( a _ { 3 } \\otimes b _ { 2 } ) , \\\\ b \\triangleright a & : = \\mathcal { R } ^ { - 1 } ( b _ { 1 } \\otimes a _ { 1 } ) a _ { 2 } \\mathcal { R } ( b _ { 2 } \\otimes a _ { 3 } ) . \\end{align*}"}
+{"id": "2138.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathcal { I } ( t ) = : \\mathcal { J } _ 1 + \\mathcal { J } _ 2 + \\mathcal { J } _ 3 . \\end{align*}"}
+{"id": "1473.png", "formula": "\\begin{align*} A _ { t } = W _ { \\tau ( t ) } , \\end{align*}"}
+{"id": "384.png", "formula": "\\begin{align*} \\alpha ( y u ) = \\alpha ( y ) u . \\end{align*}"}
+{"id": "5937.png", "formula": "\\begin{align*} d [ - a _ { 1 , n + 2 } b _ { 1 , n } ] = d ( - a _ { 1 , n + 2 } b _ { 1 , n } ) \\ge 2 e + R _ { n + 1 } - 1 > 2 e - R _ { n + 2 } + R _ { n + 1 } - 1 + S _ { n } \\end{align*}"}
+{"id": "1621.png", "formula": "\\begin{align*} U ' _ { \\mathcal X _ { t - k } \\to y _ { t - k } } = \\left \\{ u ' \\in \\mathcal P _ { \\mathcal X _ t } : \\big | \\{ e \\in \\mathcal A _ { \\mathcal Y } : \\{ \\beta ^ { 1 } _ { \\mathcal X _ 1 } , \\alpha ^ { 1 } _ { \\mathcal X _ 2 } , \\dots , \\beta ^ { t - k - 2 } _ { \\mathcal X _ { t - k - 1 } } , u ' \\} \\subseteq e \\} \\big | \\ge \\frac { \\rho } { 2 ^ { t - k - 1 } } \\prod _ { j \\in [ k ] \\setminus [ t - k ] } | \\mathcal P _ { \\mathcal X _ j } | \\right \\} . \\end{align*}"}
+{"id": "5423.png", "formula": "\\begin{align*} \\Sigma _ s : = \\begin{cases} \\Sigma _ { \\leq s } & s = 2 ^ u , \\\\ \\Sigma _ { \\leq s } \\setminus \\Sigma _ { \\leq s / 2 ^ u } & s > 2 ^ u , \\end{cases} \\end{align*}"}
+{"id": "4315.png", "formula": "\\begin{align*} \\begin{alignedat} { 3 } F _ { R } ( t ) = \\frac { 1 } { \\sqrt { M } } \\sum _ { x = 0 } ^ { M - 1 } f ( x ) \\ ; e ^ { - \\mu 2 \\pi \\left ( \\frac { x t } { M } \\right ) } \\end{alignedat} \\end{align*}"}
+{"id": "3976.png", "formula": "\\begin{align*} f ( a , \\mu ) ( z ) = \\int _ { C ^ \\alpha } \\int _ { { \\sf T } ^ 2 } F \\big ( a ( z ) , b ( z ' ) \\big ) k ( z , z ' ) \\ , d z ' \\mu ( d b ) . \\end{align*}"}
+{"id": "5527.png", "formula": "\\begin{align*} \\mathcal { P } _ d ( \\C ^ n ) = \\mbox { s p a n } \\{ e _ i : = z ^ { \\alpha ( i ) } = z _ 1 ^ { \\alpha _ 1 ( i ) } \\cdots z _ n ^ { \\alpha _ n ( i ) } ; \\ i \\in \\N \\mbox { a n d } \\deg ( e _ i ) \\le d \\} \\end{align*}"}
+{"id": "2918.png", "formula": "\\begin{align*} K _ { W _ e / U } \\le d _ e ^ * K _ { W _ r \\times _ { V ^ r } V ^ e / U } & = d _ e ^ * K _ { W _ r \\times _ { V ^ r } V ^ e / W _ r } + b _ { e , r } ^ * K _ { W _ r / U } \\\\ & = \\alpha _ e ^ * K _ { V ^ e / V ^ r } + b _ { e , r } ^ * K _ { W _ r / U } = \\alpha _ e ^ * ( 1 - p ^ { e - r } ) K _ V + b _ { e , r } ^ * K _ { W _ r / U } , \\end{align*}"}
+{"id": "7583.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { v } v ^ { - \\frac { 7 } { 8 } } ( v - w ) ^ { - \\frac { 3 } { 8 } } \\exp \\left ( - K v ^ { - \\frac { 3 } { 2 } } w ^ 2 \\right ) d w \\\\ & = \\int _ { 0 } ^ { v / 2 } v ^ { - \\frac { 7 } { 8 } } ( v - w ) ^ { - \\frac { 3 } { 8 } } \\exp \\left ( - K v ^ { - \\frac { 3 } { 2 } } w ^ 2 \\right ) d w \\\\ & + \\int _ { v / 2 } ^ { v } v ^ { - \\frac { 7 } { 8 } } ( v - w ) ^ { - \\frac { 3 } { 8 } } \\exp \\left ( - K v ^ { - \\frac { 3 } { 2 } } w ^ 2 \\right ) d w \\\\ & = : H _ 1 ( v ) + H _ 2 ( v ) . \\end{align*}"}
+{"id": "7836.png", "formula": "\\begin{align*} ( T ) ( p ) < - \\frac { t ^ 4 } { 3 } + \\frac { t \\chi } { 4 ( 2 \\pi ) ^ 3 } \\sum _ { ( m , n ) \\in \\mathbb { Z } ^ 2 - ( 0 , 0 ) } \\frac { 1 } { | m \\tau + n | ^ 3 } = - t \\cdot R _ 1 ( t , \\tau ) < 0 \\ , . \\end{align*}"}
+{"id": "7833.png", "formula": "\\begin{align*} \\frac { \\chi } { 4 ( 2 \\pi ) ^ 3 } \\sum _ { ( m , n ) \\in \\mathbb { Z } ^ 2 - ( 0 , 0 ) } \\frac { 1 } { | m \\tau + n | ^ 3 } > \\frac { \\chi } { 4 ( 2 \\pi ) ^ 3 } \\sum _ { n \\in \\mathbb { Z } - \\{ 0 \\} } \\frac { 1 } { | n | ^ 3 } = \\frac { \\chi \\zeta ( 3 ) } { 2 ( 2 \\pi ) ^ 3 } \\ ; . \\end{align*}"}
+{"id": "4327.png", "formula": "\\begin{align*} S ( \\omega _ \\Psi \\circ \\alpha \\| \\omega _ \\Omega \\circ \\alpha ) = S ( \\omega _ \\Psi \\| \\omega _ \\Omega ) \\ , . \\end{align*}"}
+{"id": "8599.png", "formula": "\\begin{align*} \\mathcal { L } _ { 1 } ^ { \\mu } [ \\beta b ] \\nabla _ X \\psi = - \\frac { 1 } { \\beta } \\sinh { ( \\beta b ( X ) \\sqrt { \\mu } | \\mathrm { D } | ) } \\mathrm { s e c h } ( \\sqrt { \\mu } | \\mathrm { D } | ) \\dfrac { 1 } { \\sqrt { \\mu } | \\mathrm { D } | } \\nabla _ X \\psi . \\end{align*}"}
+{"id": "1713.png", "formula": "\\begin{align*} \\Delta ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } = \\left ( \\det \\left [ H ^ { ( m , n ) } _ { \\texttt { b } ; j , k } ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } ) \\right ] _ { 1 \\leq j , k \\leq n } \\right ) ^ { - 1 } , \\end{align*}"}
+{"id": "6898.png", "formula": "\\begin{align*} \\begin{aligned} \\Lambda : P & \\rightrightarrows [ 0 , 1 ] ^ n \\\\ p & \\mapsto \\tau ( p ) + N \\Gamma ( p ) . \\end{aligned} \\end{align*}"}
+{"id": "1290.png", "formula": "\\begin{align*} C _ 0 \\leq \\lim _ { n \\to \\infty } \\sum _ { j = 1 } ^ { \\infty } P ( g _ n ^ j [ \\phi ^ j ] ) \\leq C _ 0 \\sum _ { j = 1 } ^ { \\infty } \\| \\nabla \\phi ^ j \\| _ { L ^ 2 } ^ { 2 p } . \\end{align*}"}
+{"id": "362.png", "formula": "\\begin{align*} \\phi _ \\ell ( h ) = k . \\end{align*}"}
+{"id": "7029.png", "formula": "\\begin{align*} E [ v ] = \\mathop { \\inf } _ { \\rho \\in \\mathcal I _ N ( \\Omega ) } \\left \\{ F _ L [ \\rho ] - \\int _ { \\Omega } v \\ , d \\rho \\right \\} . \\end{align*}"}
+{"id": "7388.png", "formula": "\\begin{align*} \\partial _ { t } V _ { n } + \\left ( u _ { n } + \\frac { \\lambda _ { n } ( \\rho _ { n } ) } { \\rho _ { n } ^ { 2 } } \\partial _ { x } \\rho _ { n } \\right ) \\partial _ { x } V _ { n } - \\frac { \\lambda _ { n } ( \\rho _ { n } ) } { \\rho _ { n } } \\partial _ { x } ^ { 2 } V _ { n } = - \\frac { ( \\lambda _ { n } ' ( \\rho _ { n } ) \\rho _ { n } + \\lambda _ { n } ( \\rho _ { n } ) ) } { ( \\lambda _ { n } ( \\rho _ { n } ) ) ^ { 2 } } V _ { n } ^ { 2 } . \\end{align*}"}
+{"id": "0.png", "formula": "\\begin{align*} w ( ( x , y ) ) = \\begin{cases} \\frac { n - 1 } { n + m - 2 } & \\textrm { i f } y = 0 ; \\\\ \\frac { m - 1 } { n + m - 2 } & \\textrm { i f } x = 0 ; \\\\ \\frac { 1 } { n + m - 2 } & \\textrm { o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "4059.png", "formula": "\\begin{align*} U _ { 0 , 0 } ( t ) = \\sum _ { n = - \\infty } ^ { \\infty } u _ { n } \\cos n \\pi t , u _ { n } = \\int _ { 0 } ^ { 1 } U _ { 0 , 0 } ( t ) \\cos n \\pi t \\ , d t \\end{align*}"}
+{"id": "6653.png", "formula": "\\begin{align*} c _ { 2 } = - { 1 \\over 2 \\pi ^ 2 } p ^ 2 , c _ { 2 n + 2 } = { 1 \\over \\pi ^ 2 } { 2 n - 1 \\over 2 ( n + 1 ) } ( p ^ 2 - n ^ 2 ) c _ { 2 n } , ( n \\ge 1 ) . \\end{align*}"}
+{"id": "7842.png", "formula": "\\begin{align*} R _ i ( | \\tau | t , - 1 / \\tau ) = | \\tau | ^ 3 R _ i ( t , \\tau ) , i = 1 , 2 \\end{align*}"}
+{"id": "6657.png", "formula": "\\begin{align*} b _ { 2 n - 1 } = \\pi { ( - 1 ) ^ n c _ { 2 n } \\over ( 2 n - 1 ) ! } ( n \\ge 1 ) , b _ { 2 n } = 0 ( n \\ge 1 ) . \\end{align*}"}
+{"id": "8957.png", "formula": "\\begin{align*} \\Delta _ T ( u ) = | \\{ ( \\bar { p } , \\bar { q } ) \\in \\Z ^ m \\times \\Z ^ n : 0 < \\| \\bar { q } \\| < T \\ \\ | p _ i + L _ { u } ^ { ( i ) } ( q _ 1 , \\ldots , q _ n ) | < \\vartheta _ i \\| \\bar { q } \\| ^ { - w _ i } \\\\ \\ i = 1 , \\ldots , m \\ ( \\bar { p } , \\bar { q } ) = v \\mod N \\} | . \\end{align*}"}
+{"id": "1051.png", "formula": "\\begin{align*} \\sigma ( \\hat { u } _ \\eta ^ * ) = \\min \\left \\{ \\sigma ( u ) : \\ : u \\in W ^ { 1 , p ( z ) } ( \\Omega ) \\right \\} . \\end{align*}"}
+{"id": "3749.png", "formula": "\\begin{align*} \\sin \\Big ( ( \\frac { \\alpha + i - j } { 3 } ) 3 \\pi \\Big ) = \\Big ( 2 \\cos \\Big ( ( \\frac { \\alpha + i - j } { 3 } ) \\pi \\Big ) + 1 \\Big ) \\sin \\Big ( ( \\frac { \\alpha + i - j } { 3 } ) \\pi \\Big ) \\end{align*}"}
+{"id": "4574.png", "formula": "\\begin{align*} \\vert \\mathcal J _ { 2 , 3 } \\vert \\le C _ 3 \\left \\{ \\int _ 0 ^ t ( I ( s ) + I _ { 1 , n } ( s ) ) d s + \\int _ 0 ^ t \\Vert \\varphi ( s ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } d s \\right \\} \\ , . \\end{align*}"}
+{"id": "8315.png", "formula": "\\begin{align*} \\mathcal { D } _ { 2 } = \\mathcal { D } _ { 2 1 } \\oplus \\cdots \\oplus \\mathcal { D } _ { 2 d } . \\end{align*}"}
+{"id": "3387.png", "formula": "\\begin{align*} \\partial _ \\nu b ^ \\mu - b ^ \\mu \\alpha ^ 0 _ { \\ , \\nu 0 } - b ^ \\mu \\alpha ^ 0 _ { \\ , \\nu \\tau } b ^ \\tau + \\alpha ^ { \\mu } _ { \\ , \\nu \\tau } b ^ \\tau = \\delta ^ \\mu _ \\nu \\Upsilon _ 0 = \\frac { \\delta ^ \\mu _ \\nu } { n + 1 } \\left ( 2 a + \\partial _ \\mu b ^ \\mu + \\alpha ^ { i } _ { \\ , i \\nu } b ^ \\nu \\right ) . \\end{align*}"}
+{"id": "7190.png", "formula": "\\begin{align*} y _ 1 ^ { n _ 1 } y _ 2 ^ { n _ 2 } = y _ 1 ^ { n _ 1 - 2 } ( y _ 1 ^ 2 y _ 2 ^ { n _ 2 - 1 } ) y _ 2 = y _ 1 ^ { n _ 1 - 2 } y _ 2 ^ { n _ 2 + 1 } y _ 2 = \\cdots = y _ 2 ^ { n _ 1 + n _ 2 } = y _ 2 ^ n . \\end{align*}"}
+{"id": "2056.png", "formula": "\\begin{align*} \\{ \\ , u \\in [ \\dot { \\mathrm { H } } ^ { s _ 0 , p _ 0 } \\cap \\dot { \\mathrm { H } } ^ { s _ 1 , p _ 1 } ] ( \\Omega ) \\ , | \\ , u _ { | _ { \\partial \\Omega } } = 0 \\ , \\} \\simeq [ \\dot { \\mathrm { H } } ^ { s _ 0 , p _ 0 } _ 0 \\cap \\dot { \\mathrm { H } } ^ { s _ 1 , p _ 1 } _ 0 ] ( \\Omega ) \\end{align*}"}
+{"id": "3910.png", "formula": "\\begin{align*} \\frac { h _ K ( v _ 1 ) } { h _ L ( v _ 1 ) } = \\frac { h _ K ( v _ 2 ) } { h _ L ( v _ 2 ) } , \\end{align*}"}
+{"id": "3456.png", "formula": "\\begin{align*} M u ( E ) \\geq \\int _ { \\nabla u ( E \\cap ( D ^ 2 u ) ) } W ( p ) d p = \\int _ { E \\cap ( D ^ 2 u ) } C _ 1 d \\mu _ 0 = \\int _ { E } C _ 1 d \\mu _ 0 , \\forall E \\subset \\Delta . \\end{align*}"}
+{"id": "8814.png", "formula": "\\begin{align*} - 2 \\eta _ { t } \\mathbb { E } \\big [ \\langle x ( t - 1 ) - \\mathbf { 1 } _ { n } \\otimes \\bar { x } ( t - 1 ) , g ( t ) - \\mathbf { 1 } _ { n } \\otimes \\bar { g } ( t ) \\rangle | \\mathcal { F } _ { t - 1 } \\big ] & \\le \\lambda h ( t - 1 ) + \\frac { 2 \\eta _ { t } ^ { 2 } } { \\lambda } \\sum _ { i = 1 } ^ { n } \\norm { \\nabla f _ { i } ( x ^ { i } ( t - 1 ) ) } ^ { 2 } + \\\\ & \\quad + \\frac { 2 \\eta _ { t } ^ { 2 } n } { \\lambda } ( \\kappa _ { \\beta } L ) ^ { 2 } d ^ { 2 } h _ { t } ^ { 2 ( \\beta - 1 ) } \\end{align*}"}
+{"id": "809.png", "formula": "\\begin{align*} c + o ( 1 ) & = I [ u _ n ] = \\frac { 1 } { p } ( [ u _ n ] _ { s , p } ^ p + a \\| u _ n \\| _ p ^ p ) - \\frac { b } { 2 ( p ^ \\flat ) ^ 2 } \\int _ { \\mathbb { R } ^ N } ( K \\ast | u _ n | ^ { p ^ \\flat } ) | u _ n | ^ { p ^ \\flat } d x + o ( 1 ) , \\\\ o ( 1 ) & = I ' [ u _ n ] u _ n = [ u _ n ] _ { s , p } ^ p + a \\| u _ n \\| _ p ^ p - \\frac { b } { p ^ \\flat } \\int _ { \\mathbb { R } ^ N } ( K \\ast | u _ n | ^ { p ^ \\flat } ) | u _ n | ^ { p ^ \\flat } d x + o ( 1 ) . \\end{align*}"}
+{"id": "7659.png", "formula": "\\begin{align*} 0 & = M _ { 1 ^ 6 } + 1 5 M _ { 1 ^ 4 } M _ { 1 , 1 } - 5 M _ { 3 , 1 ^ 3 } - 1 5 M _ { 3 , 1 } M _ { 1 , 1 } - 5 M _ { 3 , 3 } + 9 M _ { 5 , 1 } \\\\ 0 & = M _ { 1 ^ 8 } + 2 8 M _ { 1 ^ 6 } M _ { 1 , 1 } + 3 5 ( M _ { 1 ^ 4 } ) ^ 2 + 7 M _ { 3 , 1 ^ 5 } + 7 0 M _ { 3 , 1 ^ 3 } M _ { 1 , 1 } + 3 5 M _ { 3 , 1 } M _ { 1 ^ 4 } - 3 5 M _ { 3 , 3 } M _ { 1 , 1 } \\\\ & - 7 0 ( M _ { 3 , 1 } ) ^ 2 - 2 1 M _ { 5 , 1 ^ 3 } - 6 3 M _ { 5 , 1 } M _ { 1 , 1 } - 4 2 M _ { 5 , 3 } + 9 0 M _ { 7 , 1 } . \\end{align*}"}
+{"id": "1689.png", "formula": "\\begin{align*} & H ^ { ( n , m ) } _ { \\texttt { b } ; j , k } : = \\partial _ { \\xi _ j } \\partial _ { \\xi _ k } V ^ { ( n , m ) } _ { \\texttt { b } ; \\lambda } ( \\boldsymbol { \\xi } ) \\\\ & = \\begin{cases} 2 ( m + 1 ) + u _ { q _ 0 } ( \\xi _ j ) + u _ { q _ 1 } ( \\xi _ j ) + \\sum _ { \\substack { 1 \\leq l \\leq n \\\\ l \\neq j } } \\bigl ( u _ q ( \\xi _ j + \\xi _ l ) + u _ q ( \\xi _ j - \\xi _ l ) \\bigr ) & \\\\ u _ q ( \\xi _ j + \\xi _ k ) - u _ q ( \\xi _ j - \\xi _ k ) & \\\\ \\end{cases} \\end{align*}"}
+{"id": "1566.png", "formula": "\\begin{align*} e ^ { t / \\kappa } g ( x + v t , v ) - g ( x , v ) = \\int _ 0 ^ t \\frac { e ^ { s / \\kappa } } { \\kappa } \\rho _ g ( x + v s ) \\left ( \\alpha \\mathcal { M } _ { T ( x + v s ) } ( v ) + ( 1 - \\alpha ) \\mathcal { M } _ { \\tau ( x + v s ) } ( v ) \\right ) \\mathrm { d } s . \\end{align*}"}
+{"id": "5037.png", "formula": "\\begin{align*} | \\nabla ^ { m , \\gamma _ s } ( \\iota ^ * g _ s - \\gamma _ s ) | _ { \\gamma _ s } & \\leq C r ^ { - 1 } m = 0 , 1 , 2 , \\ldots , 1 1 , \\\\ | \\iota ^ * V _ s + \\tfrac 1 2 r \\partial _ r | _ { \\gamma _ s } & \\leq C . \\end{align*}"}
+{"id": "4678.png", "formula": "\\begin{align*} | S _ j ' | \\cdot ( \\frac { n } { 2 ( 2 l + 1 ) } - 2 ) \\leq \\sum \\limits _ { v \\in S _ j ' } d _ { G ' } ( v ) \\le e ( S _ j ' , T _ j ) + \\sum \\limits _ { q = 1 } ^ { 2 l + 1 } e ( S _ j ' , S _ q - S _ j ' ) + 2 e ( S _ j ' ) . \\end{align*}"}
+{"id": "7171.png", "formula": "\\begin{align*} P : = \\prod _ { j = 1 } ^ m \\Delta ^ { n _ j } , \\end{align*}"}
+{"id": "425.png", "formula": "\\begin{align*} \\Phi _ { d , \\ell } ^ { ( 2 ) } ( \\sigma ) : = ( \\ell + \\sigma ) ( d + \\ell - \\sigma - 2 ) , \\sigma \\in \\R , \\end{align*}"}
+{"id": "7516.png", "formula": "\\begin{align*} M _ u ( v ) : = \\frac { 1 } { \\sqrt { 2 \\pi \\theta } } \\exp \\left ( - \\frac { | v - u | ^ 2 } { 2 \\theta } \\right ) . \\end{align*}"}
+{"id": "4515.png", "formula": "\\begin{align*} | | { \\bf v } [ { \\bf u } ^ a + S _ { \\theta _ i } { \\bf u } _ { i } ] | | _ { s , \\ast , T } \\le C \\delta ^ 2 \\begin{cases} \\theta _ i ^ { ( s + 2 - \\alpha ) _ + } & s + 2 \\not = \\alpha , \\\\ \\theta _ i & s + 2 = \\alpha \\end{cases} s \\in \\{ 6 , \\dots , \\tilde { \\alpha } + 6 \\} , \\end{align*}"}
+{"id": "5848.png", "formula": "\\begin{align*} \\footnotesize & w _ 0 ( \\alpha _ 1 ) = - s _ { \\alpha _ 1 + \\alpha _ 2 } ( 2 \\alpha _ 1 + \\alpha _ 2 ) = - ( ( 4 \\alpha _ 1 + 2 \\alpha _ 2 ) - ( 3 \\alpha _ 1 + 2 \\alpha _ 2 ) ) = - \\alpha _ 1 , \\\\ & w _ 0 ( \\alpha _ 2 ) = s _ { \\alpha _ 1 + \\alpha _ 2 } ( 3 \\alpha _ 1 + 2 \\alpha _ 2 ) = 3 ( 2 \\alpha _ 1 + \\alpha _ 2 ) - 2 ( 3 \\alpha _ 1 + 2 \\alpha _ 2 ) = - \\alpha _ 2 , \\end{align*}"}
+{"id": "3325.png", "formula": "\\begin{align*} g ( \\tau ) = \\begin{pmatrix} q ^ { - 1 / 6 0 } \\displaystyle { \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { n ^ 2 } } { ( q ; q ) _ n } } , & q ^ { 1 1 / 6 0 } \\displaystyle { \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { n ^ 2 + n } } { ( q ; q ) _ n } } \\end{pmatrix} ^ \\mathsf { T } \\end{align*}"}
+{"id": "3274.png", "formula": "\\begin{align*} F _ X ( x ) = P ( X \\leq x ) . \\end{align*}"}
+{"id": "592.png", "formula": "\\begin{align*} X _ \\pm ( t ) = X _ \\pm ( 0 ) + \\int _ 0 ^ t \\Psi ( s ) \\ , d s + \\int _ 0 ^ t \\Phi ( s ) \\ , d W ( s ) , \\end{align*}"}
+{"id": "2236.png", "formula": "\\begin{align*} \\lim \\limits _ { \\delta \\to 0 ^ + } \\frac { 1 } { G ( \\delta ) } \\int _ 0 ^ \\delta \\frac { G ( t ) } { t } d t = \\frac { 1 } { p } . \\end{align*}"}
+{"id": "4116.png", "formula": "\\begin{align*} \\frac { 2 \\cdot 6 \\cdots ( 2 k - 2 ) \\cdot ( 2 k + 6 ) ( 2 k + 1 0 ) \\cdots ( 4 k + 2 ) } { ( k + 2 ) ( k + 3 ) \\cdots ( 2 k + 1 ) } & = \\frac { 2 ^ { k / 2 } ( k - 1 ) ! ! \\ , 2 ^ { k / 2 } \\frac { ( 2 k + 1 ) ! ! } { ( k + 1 ) ! ! } } { \\frac { ( 2 k + 1 ) ! } { ( k + 1 ) ! } } \\\\ & = \\frac { 2 ^ k ( k - 1 ) ! ! \\ , ( k + 1 ) ! } { ( k + 1 ) ! ! } \\cdot \\frac { ( 2 k + 1 ) ! ! } { ( 2 k + 1 ) ! } \\\\ & = 2 ^ k k ! \\cdot \\frac { 1 } { ( 2 k ) ! ! } = 1 , \\end{align*}"}
+{"id": "3685.png", "formula": "\\begin{align*} \\sum _ { u = 1 } ^ n \\| \\phi _ { u } \\| ^ 2 = k \\| \\phi \\| ^ 2 . \\end{align*}"}
+{"id": "3819.png", "formula": "\\begin{align*} s _ * : = ( \\mu _ 0 ( X ) + \\mu _ 1 ( X ) + \\nu _ X ( X \\times X ) ) ^ { \\frac 1 p } . \\end{align*}"}
+{"id": "2617.png", "formula": "\\begin{align*} H ( t ) & = Y ( t ) - \\int _ 0 ^ t Y ( t - s ) \\ , d F ( s ) \\ , , \\\\ X ^ { ( 0 ) } ( t ) & = W ^ 0 ( F _ 0 ( t ) ) - ( 1 - F _ 0 ( t ) ) X ( 0 ) ^ - \\ , , \\\\ U ( x , t ) & = K ( F ( x ) , \\mu t ) \\intertext { a n d } \\Theta ( t ) & = - \\int _ { \\R _ + ^ 2 } \\mathbf { 1 } _ { \\{ s + x \\le t \\} } \\ , d U ( x , s ) = - \\int _ { \\R _ + ^ 2 } \\mathbf { 1 } _ { \\{ s + x \\le t \\} } \\ , \\dot K ( F ( x ) , \\mu s ) \\ , d F ( x ) \\ , \\mu d s \\end{align*}"}
+{"id": "2600.png", "formula": "\\begin{align*} \\mathcal D _ r ( X ) = \\{ ( V _ 1 , V _ 2 ) : \\{ 0 \\} \\subset V _ 1 \\subset V _ 2 \\subset \\mathbb C ^ { p + q } , \\dim V _ 1 = q - r , \\ , \\dim V _ 2 = p + r \\} . \\end{align*}"}
+{"id": "2055.png", "formula": "\\begin{align*} \\frac { 1 } { p } = ( 1 - \\theta ) s _ 0 + \\theta s _ 1 \\end{align*}"}
+{"id": "6747.png", "formula": "\\begin{align*} O ^ w ( G ( n ) + H ) & = \\left ( \\frac { 3 } { 2 } \\right ) ^ w ( G ( n ) + H + 1 ) - 1 \\\\ & = \\left ( \\frac { 3 } { 2 } \\right ) ^ w G ( n ) + \\left ( \\frac { 3 } { 2 } \\right ) ^ w H + \\left ( \\frac { 3 } { 2 } \\right ) ^ w - 1 \\\\ & = ( O ^ w \\circ G ) ( n ) + \\left ( \\frac { 3 } { 2 } \\right ) ^ w H \\end{align*}"}
+{"id": "7772.png", "formula": "\\begin{align*} \\mathcal { W } : = R \\sum _ { \\gamma } \\Omega ( \\gamma ) \\widetilde { Z } _ { \\gamma } \\sum _ { n > 0 } \\frac { e ^ { - 2 \\pi \\mathrm { i } n \\zeta _ { \\gamma } } } { n } K _ 0 ( 4 \\pi R n | \\widetilde { Z } _ { \\gamma } | ) \\ , , \\end{align*}"}
+{"id": "5081.png", "formula": "\\begin{align*} y ( z ) = E ( A z ) v ^ c , \\end{align*}"}
+{"id": "3355.png", "formula": "\\begin{align*} \\sum _ { n _ 1 , n _ 2 \\geq 0 } \\frac { q ^ { 2 n _ 1 ^ 2 + 6 n _ 1 n _ 2 + 6 n _ 2 ^ 2 } } { ( q ; q ) _ { n _ 1 } ( q ^ 3 ; q ^ 3 ) _ { n _ 2 } } = ( - q ^ 2 , - q ^ 3 , - q ^ 4 , - q ^ 6 ; q ^ 6 ) _ \\infty . \\end{align*}"}
+{"id": "6512.png", "formula": "\\begin{align*} { H } ( \\sigma ) = { D } ( \\sigma ) + \\varepsilon \\Delta + \\delta { T } _ \\phi , \\ \\sigma \\in \\R , \\end{align*}"}
+{"id": "4276.png", "formula": "\\begin{align*} \\epsilon _ { \\pi } ( r _ i ) = 1 \\epsilon _ { \\pi } ( s _ i ) = ( - 1 ) ^ { 1 + r _ i + s _ i } 1 \\leq i \\leq k _ 1 + k _ 2 . \\end{align*}"}
+{"id": "6885.png", "formula": "\\begin{align*} \\delta \\smallfrown \\pi _ * ( \\ell ( \\widetilde { U } , r ) \\smallfrown [ \\widetilde { U } ] ) = \\pi _ * ( \\ell ( \\widetilde { U } , r ) \\smallfrown [ \\widetilde { T } ] ) \\in H ^ { B M } _ 0 ( \\{ 0 \\} , \\mathbb { R } ) . \\end{align*}"}
+{"id": "1249.png", "formula": "\\begin{align*} B _ n = & \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p ) | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } ( | g _ n ^ { - 1 } u _ n | ^ p - | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p ) d x \\\\ & - \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p ) | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p \\end{align*}"}
+{"id": "6789.png", "formula": "\\begin{align*} & \\underline { v } = \\overline { v } - r q ( 1 ) e ^ { \\left ( \\lambda _ 1 + \\varepsilon \\right ) z } , \\\\ [ 0 . 2 c m ] & \\underline { v } ' = \\overline { v } ' - r q ( 1 ) \\left ( \\lambda _ 1 + \\varepsilon \\right ) e ^ { \\left ( \\lambda _ 1 + \\varepsilon \\right ) z } , \\\\ [ 0 . 2 c m ] & \\underline { v } '' = \\overline { v } '' - r q ( 1 ) \\left ( \\lambda _ 1 + \\varepsilon \\right ) ^ 2 e ^ { \\left ( \\lambda _ 1 + \\varepsilon \\right ) z } . \\end{align*}"}
+{"id": "3112.png", "formula": "\\begin{align*} \\mathbf { H } _ k = \\mathbf { H } _ { k , } + \\mathbf { H } _ { k , } \\end{align*}"}
+{"id": "7766.png", "formula": "\\begin{align*} \\eta _ { \\pm } ^ { } : = \\Big ( \\eta ^ { } - 4 \\rho _ 3 ^ { } \\widetilde { \\eta } \\Big ) \\pm \\Big ( \\sum _ { \\gamma } \\Omega ( \\gamma ) \\eta _ { \\gamma } ^ { } - 4 \\rho ^ { } \\widetilde { \\eta } \\Big ) \\ , , \\widetilde { \\eta } : = \\mathrm { d } ^ c \\log ( r ) \\ , , \\end{align*}"}
+{"id": "7463.png", "formula": "\\begin{align*} \\bar \\Psi _ { a , c } ( \\alpha , x ) = \\overline { \\Psi _ { \\bar a , \\bar c } ( \\alpha , x ) } = \\alpha \\big ( ( 1 - x / \\ddot a ) / \\ddot c \\big ) \\frac { \\| x / \\ddot a \\| ^ { \\dot c } } { \\| 1 - x / \\ddot a \\| ^ { 1 - \\dot a } } . \\end{align*}"}
+{"id": "6327.png", "formula": "\\begin{align*} i \\partial _ { \\tau } v _ { \\lambda } + \\tilde P ^ y _ \\lambda ( i P _ { \\lesssim \\epsilon ^ 2 } ^ y V _ 2 \\partial _ y + \\frac { i } 2 P _ { \\lesssim \\epsilon ^ 2 } ^ y \\partial _ y V _ 2 ) \\tilde P ^ y _ \\lambda v _ { \\lambda } + \\partial ^ 2 _ y v _ { \\lambda } = \\tilde { f } _ \\lambda , \\end{align*}"}
+{"id": "167.png", "formula": "\\begin{align*} \\varphi ( \\tau ) ^ 8 { \\rm c h } ( \\mathcal { V } _ i ) = \\frac { 1 } { 2 } \\left ( \\prod _ { l = 1 } ^ 8 \\theta _ 1 ( y _ l ^ i , \\tau ) + \\prod _ { l = 1 } ^ 8 \\theta _ 2 ( y _ l ^ i , \\tau ) + \\prod _ { l = 1 } ^ 8 \\theta _ 3 ( y _ l ^ i , \\tau ) \\right ) , \\end{align*}"}
+{"id": "3229.png", "formula": "\\begin{align*} S _ { \\alpha } ( \\epsilon , R ) & = \\textnormal { A r e a } \\left ( \\bigcup _ { z \\in \\Z ^ { 2 } \\cap \\textnormal { S e c t } _ { \\alpha , \\epsilon } ( R ) } \\square _ { z } \\right ) \\leq \\textnormal { A r e a } \\left ( \\textnormal { S e c t } ^ { + } _ { \\alpha , \\epsilon } ( R ' ) \\right ) \\\\ & = \\textnormal { A r e a } ( \\textnormal { S e c t } _ { \\alpha , \\epsilon } ( R ) ) + O \\left ( R \\right ) . \\end{align*}"}
+{"id": "6220.png", "formula": "\\begin{align*} A _ 1 = \\begin{bmatrix} - I / 2 & - I / 2 \\\\ O & O \\end{bmatrix} , A _ 2 = \\begin{bmatrix} O & O \\\\ I / 2 & - I \\end{bmatrix} , A _ 3 = \\begin{bmatrix} O & O \\\\ I & - I \\end{bmatrix} , A _ 4 = \\begin{bmatrix} - I / 2 & - I \\\\ O & O \\end{bmatrix} . \\end{align*}"}
+{"id": "8927.png", "formula": "\\begin{align*} \\nabla _ { E _ i ^ \\prime } ^ \\varphi d \\varphi ( E _ i ^ \\prime ) = - \\nabla _ { E _ i } ^ \\varphi d \\varphi ( E _ i ) + J d \\varphi ( [ E _ i ^ \\prime , E _ i ] ) \\end{align*}"}
+{"id": "8098.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\left \\| \\sum \\limits _ { j = N } ^ { \\infty } \\lambda _ { j } a _ { j } \\right \\| _ { h _ { \\omega } ^ { p } } = 0 , \\ \\lim _ { N \\to \\infty } \\left \\| \\sum \\limits _ { j = N } ^ { \\infty } \\mu _ { j } b _ { j } \\right \\| _ { h _ { \\omega } ^ { p } } = 0 \\end{align*}"}
+{"id": "6348.png", "formula": "\\begin{align*} J ^ 4 _ { \\lambda } ( u , u ) = g _ { [ < \\lambda ] } J ^ 4 _ { m a i n } ( u _ \\lambda , u _ \\lambda ) , \\end{align*}"}
+{"id": "2077.png", "formula": "\\begin{align*} d s ^ 2 = f ( t , x ) ( d x ^ 2 - d t ^ 2 ) + g _ { a b } ( t , x ) d x ^ a d x ^ b . \\end{align*}"}
+{"id": "4329.png", "formula": "\\begin{align*} \\omega _ { { \\sf f } _ 1 \\cdots { \\sf f } _ n } ( A ) = ( F _ 1 \\cdots F _ n \\Omega _ \\omega , A F _ 1 \\cdots F _ n \\Omega _ \\omega ) \\ \\ ( A \\in \\mathcal { N } _ \\omega ) \\end{align*}"}
+{"id": "7755.png", "formula": "\\begin{align*} f = 2 \\pi ( r ^ 2 - 8 c _ { \\ell } - \\sum _ { \\gamma } \\Omega ( \\gamma ) \\iota _ V \\eta _ { \\gamma } ^ { } ) , c _ { \\ell } \\in \\mathbb { R } , \\end{align*}"}
+{"id": "1339.png", "formula": "\\begin{align*} K _ { \\mathcal { X } _ T / T } + \\Delta _ T = \\sigma ^ * ( K _ { \\mathcal { X } / S } + \\Delta ) \\end{align*}"}
+{"id": "1346.png", "formula": "\\begin{align*} X = \\boldsymbol { \\mathrm { P r o j } } _ C ( \\bigoplus _ { m \\ge 0 } f _ * L ^ { \\otimes m } ) = \\boldsymbol { \\mathrm { P r o j } } _ C ( \\bigoplus _ { m \\ge 0 } f _ * \\mathcal { O } _ X ( m D ) ) \\cong \\boldsymbol { \\mathrm { P r o j } } _ C ( \\bigoplus _ { m \\ge 0 } f ' _ * L '^ { \\otimes m } ) = X ' . \\end{align*}"}
+{"id": "2498.png", "formula": "\\begin{align*} \\int _ \\Omega \\boldsymbol { y } _ d \\cdot \\nabla v \\ , d \\boldsymbol { x } = 0 \\forall \\ , v \\in H ^ 1 _ 0 ( \\Omega ) . \\end{align*}"}
+{"id": "7333.png", "formula": "\\begin{align*} A _ \\lambda ( + \\infty ) : = \\lim _ { t \\rightarrow \\infty } A _ \\lambda ( t ) , A _ \\lambda ( - \\infty ) : = \\lim _ { t \\rightarrow - \\infty } A _ \\lambda ( t ) , \\quad \\lambda \\in I , \\end{align*}"}
+{"id": "5692.png", "formula": "\\begin{align*} k _ { x } = \\limsup _ { z \\rightarrow 0 } \\frac { \\ln \\| ( z - T ) ^ { - 1 } x \\| _ p } { \\ln \\| ( z - T ) ^ { - 1 } \\| } \\geq \\limsup _ { z \\rightarrow 0 } \\frac { \\ln \\big | f _ 0 \\big ( ( z - T ) ^ { - 1 } x \\big ) \\big | } { \\ln \\| ( z - T ) ^ { - 1 } \\| } . \\end{align*}"}
+{"id": "6577.png", "formula": "\\begin{align*} M = \\# V \\leq N ^ { \\rho _ 2 / 4 } , \\ K _ 2 = e ^ { 2 { N _ 1 } ^ { \\rho _ 2 } } . \\end{align*}"}
+{"id": "58.png", "formula": "\\begin{align*} \\mathcal P _ L = \\lbrace p \\in \\Lambda ^ * , 0 < \\vert p \\vert \\leq K _ L \\ell ^ { - 1 } \\rbrace , \\mathcal P _ H = \\lbrace k \\in \\Lambda ^ * , \\vert k \\vert \\geq K _ H \\ell ^ { - 1 } \\rbrace , \\end{align*}"}
+{"id": "2766.png", "formula": "\\begin{align*} t ( z ) = \\frac { z } { 1 + z } \\frac { 1 } { z ^ 2 } \\mathbb { E } [ X _ { \\mu _ 1 } ] \\mathbb { E } [ X _ { \\mu _ 2 } ] + f ( z ) = \\frac { C } { z } + f ( z ) \\end{align*}"}
+{"id": "5169.png", "formula": "\\begin{align*} & { \\rm e s s } \\inf L ' - \\frac { E [ L '^ 2 ] } { 2 E [ L ' ] } \\\\ = & \\lceil - \\log _ 2 { \\delta M } \\rceil - \\frac { E [ L '^ 2 ] } { 2 E [ L ' ] } \\\\ \\ge & - \\log _ 2 { \\delta M } - 1 - \\frac { E [ L ^ 2 ] } { 2 E [ L ] } - \\frac { 3 } { 2 } \\\\ \\ge & - \\log _ 2 { \\delta M } - 1 - \\frac { 1 } { 2 } h ( X ) + \\frac { 1 } { 2 } \\log _ 2 { \\delta } - \\frac { 3 } { 2 } - \\epsilon \\\\ = & - \\frac { 1 } { 2 } \\log _ 2 { \\delta } - \\log _ 2 { M } - \\frac { 1 } { 2 } h ( X ) - \\frac { 5 } { 2 } - \\epsilon \\\\ > & 0 . \\end{align*}"}
+{"id": "3842.png", "formula": "\\begin{align*} ( c , \\gamma ) : = \\int c ( x , y ) d \\gamma ( x , y ) , c ( x , y ) = | x - y | ^ 2 , \\end{align*}"}
+{"id": "331.png", "formula": "\\begin{align*} \\mathbf { j } _ { \\operatorname * { m } } = \\mathbb { A } _ { j } ^ { \\operatorname * { e x t } } \\nabla _ { \\operatorname * { p w } ; j } u _ { \\operatorname * { m } } = \\mathbb { A } _ { j } ^ { \\operatorname * { e x t } } \\nabla _ { \\operatorname * { p w } ; j } u _ { \\operatorname * { u w } } . \\end{align*}"}
+{"id": "2552.png", "formula": "\\begin{align*} \\mu _ \\mathcal { L } ( \\{ x \\in B _ 1 ( 0 ) \\mid | x | K ( x ) > \\alpha \\} ) & \\leq \\mu _ \\mathcal { L } ( \\{ x \\in B _ 1 ( 0 ) \\mid | x | \\cdot C | x | ^ { - N / r } > \\alpha \\} ) \\\\ & = O ( \\alpha ^ { - \\frac { N r } { N - r } } ) \\end{align*}"}
+{"id": "7208.png", "formula": "\\begin{align*} \\| I _ 2 ( D ^ { \\beta _ \\perp } & \\mathcal { E } ^ { k _ s } _ \\rho ) \\| _ { L ^ 2 _ { \\gamma , \\mu } ( C ( \\Lambda [ 0 , \\frac { L } { 2 } ] , 2 \\rho ) ) } \\\\ & \\le \\| I _ 2 ( D ^ { \\beta _ \\perp } \\mathcal { E } ^ { k _ s } _ \\rho ) \\| _ { L ^ 2 _ { \\gamma , \\mu } ( \\widetilde { \\bf C } _ 0 ^ { 2 \\rho } ) } + \\| I _ 2 ( D ^ { \\beta _ \\perp } \\mathcal { E } ^ { k _ s } _ \\rho ) \\| _ { L ^ 2 _ { \\gamma , \\mu } ( \\cup _ { j = 3 } ^ { J - 1 } { \\bf C } _ j ^ { 2 \\rho } ) } \\\\ & = : I + I I . \\end{align*}"}
+{"id": "6117.png", "formula": "\\begin{align*} n ( t ) = { c '' ( t ) \\over \\kappa ( t ) } \\qquad \\mbox { a n d } b ( t ) = c ' ( t ) \\times n ( t ) , \\end{align*}"}
+{"id": "8435.png", "formula": "\\begin{align*} V _ { p } ( v , t ) = \\begin{cases} \\int _ 0 ^ t \\| \\nabla v ( s ) \\| _ { B ^ { \\frac { d } { p } } _ { p , \\infty } \\cap L ^ \\infty } \\dd s , \\mathrm { i f } \\sigma < 1 + \\frac { d } { p } , \\\\ \\int _ 0 ^ t \\| \\nabla v ( s ) \\| _ { B ^ { \\sigma - 1 } _ { p , r } } \\dd s , \\mathrm { i f } \\sigma > 1 + \\frac { d } { p } \\ \\mathrm { o r } \\ \\{ \\sigma = 1 + \\frac { d } { p } \\mbox { a n d } r = 1 \\} . \\end{cases} \\end{align*}"}
+{"id": "366.png", "formula": "\\begin{align*} \\omega ^ { \\phi } _ { H } ( \\overline { h } _ 1 , \\overline { h } _ 2 ) = \\phi _ { R ( h _ 1 ) } ( h _ 2 ) h _ 2 ^ { - 1 } = \\phi _ { R ( h _ 3 z _ 1 ) } ( h _ 4 z _ 2 ) ( h _ 4 z _ 2 ) ^ { - 1 } . \\end{align*}"}
+{"id": "482.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty t ^ { \\nu + 1 } \\exp ( - p ^ 2 t ^ 2 ) \\cdot I _ { \\nu } ( b t ) \\ , d t = \\frac { b ^ \\nu } { ( 2 p ^ 2 ) ^ { \\nu + 1 } } \\exp \\left ( \\frac { b ^ 2 } { 4 p ^ 2 } \\right ) \\end{align*}"}
+{"id": "3233.png", "formula": "\\begin{align*} \\frac { R } { \\sqrt { 1 + ( \\alpha - \\epsilon ) ^ { 2 } } } - \\frac { R } { \\sqrt { 1 + ( \\alpha + \\epsilon ) ^ { 2 } } } & = \\frac { R } { \\sqrt { 1 + \\alpha ^ { 2 } } } \\left ( 1 + O ( \\epsilon ) \\right ) - \\frac { R } { \\sqrt { 1 + \\alpha ^ { 2 } } } \\left ( 1 + O ( \\epsilon ) \\right ) \\\\ & = O ( R \\epsilon ) . \\end{align*}"}
+{"id": "5996.png", "formula": "\\begin{align*} \\Psi ^ { - \\infty } ( \\mathbb { R } ^ n ) : = \\bigcup _ { m \\in \\mathbb { R } } \\Psi ^ m ( \\mathbb { R } ^ n ) . \\end{align*}"}
+{"id": "4785.png", "formula": "\\begin{align*} \\Delta ^ { u } _ n { = } \\int _ { 0 } ^ { u } \\ ! \\ ! \\ ! x ^ n \\exp { ( { { - } \\beta } x ^ 2 ) } d x { = } { - } \\frac { u ^ { n { - } 1 } \\exp { ( { - } \\beta u ^ 2 ) } } { 2 \\beta } { + } \\frac { n { - } 1 } { 2 \\beta } \\Delta ^ { u } _ { n { - } 2 } . \\end{align*}"}
+{"id": "1109.png", "formula": "\\begin{align*} \\begin{aligned} \\chi ( \\varrho ) & \\leq \\frac 1 \\varrho \\displaystyle \\sup _ { v \\in \\mathbb { B } _ { \\varrho } } \\Psi ( v ) \\\\ & \\leq \\frac 1 \\varrho \\displaystyle \\sup _ { v \\in \\overline { \\mathbb { B } } _ \\varrho } \\left | \\int _ D F ( x , v ( x ) ) d \\mu \\right | \\\\ & \\leq \\frac 1 \\varrho \\displaystyle \\sup _ { v \\in \\overline { \\mathbb { B } } _ \\varrho } \\int _ D \\left | F ( x , v ( x ) ) \\right | d \\mu . \\end{aligned} \\end{align*}"}
+{"id": "184.png", "formula": "\\begin{align*} & \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T _ C M } - \\widetilde { L _ C } - 1 6 + W _ i + W _ j ) \\right \\} ^ { ( 1 4 ) } \\\\ & = - 2 4 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 4 ) } . \\end{align*}"}
+{"id": "1730.png", "formula": "\\begin{align*} \\frac { 1 } { ( 2 \\pi ) ^ { n - 1 } n ^ { 1 / 2 } } \\int _ { \\mathbb { A } ^ { ( n ) } _ { \\texttt { a } } } f ( \\boldsymbol { \\xi } ) \\boldsymbol { \\xi } = \\frac { 1 } { n \\ , m ^ { n - 1 } } \\sum _ { \\lambda \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { a } } } f \\left ( \\frac { 2 \\pi \\lambda } { m } \\right ) \\delta ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } ( 1 ) \\end{align*}"}
+{"id": "2962.png", "formula": "\\begin{align*} ( \\alpha \\times \\beta ) \\circ \\gamma ^ X _ z ( \\iota _ X ( x ) ) = z ( \\alpha \\times \\beta ) ( \\iota _ X ( x ) ) = \\gamma ^ Y _ z \\circ ( \\alpha \\times \\beta ) ( \\iota _ X ( x ) ) , \\end{align*}"}
+{"id": "586.png", "formula": "\\begin{align*} \\mathbf u ^ \\mu ( t ) - \\mathbf u ( t ) = \\Delta _ 1 ^ \\mu ( t ) + \\Delta _ 2 ^ \\mu ( t ) + \\Delta _ 3 ^ \\mu ( t ) , \\end{align*}"}
+{"id": "7465.png", "formula": "\\begin{align*} \\langle x , u | T ( a , c ) | y , v \\rangle : = \\langle x ; v - u \\rangle \\delta _ F ( x - y + u ) \\Psi _ { a , c } ( v - u ) . \\end{align*}"}
+{"id": "3434.png", "formula": "\\begin{align*} s = \\sum _ { l _ 0 , \\ldots l _ m } F _ 0 ^ { l _ 0 } \\ldots F _ m ^ { l _ m } \\sum _ a s _ { l _ 0 \\ldots l _ m , a } t ^ a , \\end{align*}"}
+{"id": "4209.png", "formula": "\\begin{align*} D ( - 4 i , \\varepsilon ) & = \\sigma _ { \\varepsilon } ( - 4 i I ) = \\sigma _ { \\varepsilon } ( I \\diamond I \\circ _ { \\ast } i I ) = \\sigma _ { \\varepsilon } ( \\varphi ( I ) \\diamond \\varphi ( I ) \\circ _ { \\ast } \\varphi ( i I ) ) \\\\ & = \\sigma _ { \\varepsilon } ( - 4 \\varphi ^ { 2 } ( I ) \\varphi ( i I ) ) . \\end{align*}"}
+{"id": "3163.png", "formula": "\\begin{align*} ( \\phi _ 0 ) _ * W _ { \\widehat { L } } = z _ 0 + ( z _ 1 + z _ 2 ) ^ 2 . \\end{align*}"}
+{"id": "8780.png", "formula": "\\begin{align*} \\Big ( \\sum _ { k = 1 } ^ { T } \\eta _ { k } \\Big ) ^ { - 1 } = \\frac { 1 } { \\min \\Big ( \\frac { T } { 1 8 L \\kappa } , T \\gamma \\Big ) } = \\max \\Big ( \\frac { 1 8 L \\kappa } { T } , \\frac { 1 } { T \\gamma } \\Big ) \\leq \\frac { 1 8 L \\kappa } { T } + \\frac { 1 } { T \\gamma } . \\end{align*}"}
+{"id": "6678.png", "formula": "\\begin{align*} \\overline { W } _ 1 ( x ; N , \\beta , p , q ) = { 1 \\over x } \\sum _ { k = 0 } ^ \\infty { c _ { k } ^ { ( \\widetilde { \\rm c J ) } } ( N , \\beta , p , q ) \\over x ^ k } , | x | > 1 . \\end{align*}"}
+{"id": "5679.png", "formula": "\\begin{align*} \\frac { \\sum \\limits _ { j = 0 } ^ { N - 1 } \\ln w _ j } { \\ln | z | } & \\leq \\frac { \\sum \\limits _ { j = 0 } ^ { N - 1 } \\ln w _ j } { \\ln \\delta _ 2 } \\\\ & = \\big ( \\sum \\limits _ { j = 0 } ^ { N - 1 } \\ln w _ j \\big ) \\bigg ( \\frac { \\varepsilon ( \\sum \\limits _ { j = 0 } ^ { N - 1 } \\ln w _ j ) } { \\varepsilon { N } - 1 } \\bigg ) ^ { - 1 } = N - \\frac { 1 } { \\varepsilon } . \\end{align*}"}
+{"id": "5472.png", "formula": "\\begin{align*} \\Lambda ^ 2 V o l ( M ) ^ 2 \\left ( \\int _ { M } f \\big ( \\Delta f \\big ) d V _ { g } \\right ) ^ 2 \\geq \\Big ( \\int _ M \\Delta f d V _ g \\Big ) ^ 4 = | \\nabla f | ^ 4 _ { | _ { \\partial M } } | \\partial M | ^ 4 . \\end{align*}"}
+{"id": "7452.png", "formula": "\\begin{align*} J ( y _ 1 , y _ 2 , \\cdots , y _ m ) = & \\frac { 1 } { p } \\| ( y _ 1 , y _ 2 , \\cdots , y _ m ) \\| _ { Y , 1 } + \\frac { 1 } { q } \\| ( \\nabla y _ 1 , \\nabla y _ 2 , \\cdots , \\nabla y _ m ) \\| _ { Y , 3 } \\\\ & - \\sum _ { i = 1 } ^ { n } \\frac { 1 } { 1 - \\nu } \\int _ { \\Omega } a _ i \\vert y _ i \\vert ^ { 1 - \\nu } d z \\\\ & - \\lambda \\int _ { \\Omega } \\vert y _ 1 \\vert ^ { \\kappa _ 1 + 1 } \\vert y _ 2 \\vert ^ { \\kappa _ 2 + 1 } \\cdots \\vert y _ m \\vert ^ { \\kappa _ m + 1 } d z . \\end{align*}"}
+{"id": "7044.png", "formula": "\\begin{align*} w _ { 2 } ( x , t ) = \\int _ { 0 } ^ { t } \\int _ { \\Omega } h _ { D } ( x , y , t - \\tau ) r ( \\tau ) \\Delta _ X f ( y , \\tau ) \\ , d y + \\int _ { \\Omega } h _ { D } ( x , y , t ) \\Delta _ X \\varphi ( y ) \\ , d y . \\end{align*}"}
+{"id": "1374.png", "formula": "\\begin{align*} n ^ { 1 - 2 k } \\mathcal { F } _ { 2 - 2 k , - n } = \\mathcal { F } _ { 2 - 2 k , - 1 } \\big | _ { 2 - 2 k } T _ n . \\end{align*}"}
+{"id": "1932.png", "formula": "\\begin{align*} \\mathcal { K } ^ 1 ( v , \\eta _ f , \\theta _ f ) = \\sum _ { i , j } \\int _ { T _ { i j } } \\eta _ f v \\ , \\partial _ x \\theta _ f \\ , { \\rm d } x \\ , { \\rm d } v + \\sum _ { i , j } \\int _ { J _ j } \\left ( \\widehat { v \\ , \\eta _ f } [ \\ ! [ \\theta _ f ] \\ ! ] \\right ) _ { i - 1 / 2 , v } \\ , { \\rm d } v \\ , . \\end{align*}"}
+{"id": "1896.png", "formula": "\\begin{align*} \\left ( w _ h , q _ h \\right ) + \\sqrt { \\epsilon } \\ , b _ h ( u _ h , q _ h ) = 0 \\forall \\ , \\ , q _ h \\in X _ h , \\end{align*}"}
+{"id": "6999.png", "formula": "\\begin{align*} \\widetilde { } ( e , e ) = ( e , e ) - 2 \\big ( D ^ 2 \\l ( e , e ) - e ( \\l ) e ( \\l ) \\big ) . \\end{align*}"}
+{"id": "8138.png", "formula": "\\begin{align*} W _ { a , b } \\colon x ^ 6 + y ^ 6 + 1 + a ( x ^ 4 y ^ 2 + x ^ 2 y ^ 4 + x ^ 4 + x ^ 2 + y ^ 4 + y ^ 2 ) + b x ^ 2 y ^ 2 = 0 , \\end{align*}"}
+{"id": "5062.png", "formula": "\\begin{align*} ( d ( \\Pi | _ { P _ 2 } ) _ p ) ^ { - 1 } ( T _ \\gamma Q _ 1 ) = T _ p P _ 1 , \\end{align*}"}
+{"id": "4270.png", "formula": "\\begin{align*} J _ { \\lambda _ k } ( u ) & \\ge \\dfrac { \\lambda _ { k + 1 } - \\lambda _ k } { 2 } \\int _ { \\Omega } | u ( x ) | ^ 2 d x - \\int _ { \\Omega } F ( x , u ( x ) ) d x \\\\ & \\ge - M \\int _ { \\Omega } | u ( x ) | d x \\ge - \\tilde { M } \\| u ( x ) \\| _ { \\mathbb { X } ( \\Omega ) } \\ge - \\tilde { M } R = : - C _ { R } , \\end{align*}"}
+{"id": "1852.png", "formula": "\\begin{gather*} \\mathbf { P } ( \\Omega _ 0 ) = \\delta ^ { N + 1 } + O \\left ( \\frac { N ( \\log { \\Delta } ) ^ { N + 1 } } { \\Delta } \\right ) , \\\\ \\mathbf { E } _ { \\Omega _ 0 } \\left [ \\| \\zeta ( s , \\mathbb { X } _ \\alpha ) - \\zeta _ N ( s , \\mathbb { X } _ \\alpha ) \\| ^ 2 \\right ] \\ll \\mathbf { P } ( \\Omega _ 0 ) N ^ { 1 - 2 \\sigma _ 0 } + L ^ { 1 - 2 \\sigma _ 0 } + \\frac { N L ( \\log { \\Delta } ) ^ { N + 1 } } { \\Delta } \\end{gather*}"}
+{"id": "1199.png", "formula": "\\begin{align*} \\coprod _ { g \\in T } \\beta _ g \\colon \\coprod _ { g \\in T } G / P \\cap P ^ g \\xrightarrow { \\sim } G / P \\times G / P , \\beta _ g ( [ x ] _ { P \\cap P ^ g } ) = ( [ x ] _ P , [ x g ^ { - 1 } ] _ P ) \\end{align*}"}
+{"id": "8477.png", "formula": "\\begin{align*} \\partial ^ { e } E : = \\mathbb { R } ^ { n } \\backslash ( E ^ { ( 0 ) } \\cup E ^ { ( 1 ) } ) \\end{align*}"}
+{"id": "7654.png", "formula": "\\begin{align*} \\dd \\mathbb { P } ( H ) = \\frac { \\sqrt { \\Delta ( \\boldsymbol { \\lambda } , \\boldsymbol { \\lambda } ) } } { 2 ^ { \\frac { N } { 2 } } ( 2 \\pi ) ^ { \\frac { N ^ 2 } { 2 } } } \\ , \\dd H \\ , e ^ { - \\frac { 1 } { 2 } { \\rm T r } ( \\Lambda H ^ 2 ) } . \\end{align*}"}
+{"id": "8828.png", "formula": "\\begin{align*} f _ { \\omega } ( x ) = \\sum _ { i = 1 } ^ { d } \\bigg [ \\Phi _ { \\omega _ { i } } ^ { 1 } ( x _ { i } ) \\mathbb { 1 } _ { ( - \\infty , 0 ) } ( x _ { i } ) + \\Phi _ { \\omega _ { i } } ^ { 2 } ( x _ { i } ) \\mathbb { 1 } _ { [ 0 , a ] } ( x _ { i } ) + \\Phi _ { \\omega _ { i } } ^ { 3 } ( x _ { i } ) \\mathbb { 1 } _ { ( a , \\infty ) } ( x _ { i } ) \\bigg ] , \\end{align*}"}
+{"id": "8202.png", "formula": "\\begin{align*} x = \\frac { 1 } { 4 } d ^ c ( \\rho ^ 2 ) ( T ) = - \\frac { 1 } { 4 } d ( \\rho ^ 2 ) ( I T ) \\end{align*}"}
+{"id": "6768.png", "formula": "\\begin{align*} w '' ( z ) + \\alpha w ' ( z ) + f ( z ) w ( z ) = h ( z ) , \\end{align*}"}
+{"id": "5371.png", "formula": "\\begin{align*} ( x y ) z & = \\left \\{ \\begin{array} { l l } ( x y ) z & \\mbox { i f } x y \\mbox { a n d } ( x y ) z \\mbox { a r e d e f i n e d i n } L \\\\ \\sigma ( \\top _ { I } ) = \\top _ { I } & \\mbox { o t h e r w i s e } \\end{array} \\right . \\\\ x ( y z ) & = \\left \\{ \\begin{array} { l l } x ( y z ) & \\mbox { i f } y z \\mbox { a n d } x ( y z ) \\mbox { a r e d e f i n e d i n } L \\\\ \\sigma ( \\top _ { I } ) = \\top _ { I } & \\mbox { o t h e r w i s e } \\end{array} \\right . \\end{align*}"}
+{"id": "2054.png", "formula": "\\begin{align*} b = a - u \\in ( \\dot { \\mathrm { B } } ^ { \\frac { 1 } { p } } _ { p , 1 } ( \\Omega ) + \\mathrm { B } ^ { s } _ { p , q } ( \\Omega ) ) \\cap \\dot { \\mathrm { B } } ^ { 1 + \\frac { 1 } { p } } _ { p , 1 } ( \\Omega ) \\subset \\dot { \\mathrm { B } } ^ { \\frac { 1 } { p } } _ { p , 1 } ( \\Omega ) \\cap \\dot { \\mathrm { B } } ^ { 1 + \\frac { 1 } { p } } _ { p , 1 } ( \\Omega ) \\end{align*}"}
+{"id": "3349.png", "formula": "\\begin{align*} \\tilde { f } \\Bigl ( \\frac { i \\varepsilon } { 2 \\pi } \\Bigr ) = e ^ { - \\frac { \\Lambda } { \\varepsilon } } ( K + O ( \\varepsilon ) ) \\end{align*}"}
+{"id": "1739.png", "formula": "\\begin{align*} \\mathbf { x } ^ { ( m , n ) } _ { { \\lambda } } : = \\left ( x ^ { ( m + n ) } _ { { \\lambda } _ 1 + n - 1 } , x ^ { ( m + n ) } _ { { \\lambda } _ 2 + n - 2 } , \\ldots , x ^ { ( m + n ) } _ { { \\lambda } _ { n - 1 } + 1 } , x ^ { ( m + n ) } _ { { \\lambda } _ n } \\right ) , \\end{align*}"}
+{"id": "1622.png", "formula": "\\begin{align*} X _ { \\ell } = \\{ x _ r \\in V ( F ^ { ( k ) } _ t ) : r \\equiv t + \\ell \\pmod k , ~ ~ x \\in \\{ a , b , c , d \\} , ~ ~ 0 \\le r \\le t \\} \\ell \\in [ k ] . \\end{align*}"}
+{"id": "3059.png", "formula": "\\begin{align*} R = { \\log _ 2 } \\det \\left ( { { \\bf { I } } _ { { N _ { \\rm R } } \\times { N _ { \\rm R } } } + \\frac { 1 } { { { \\sigma ^ 2 } } } { { \\bf { W } } ^ H } { \\bf { H F } } { { \\bf { F } } ^ H } { { \\bf { H } } ^ H } { \\bf { W } } } \\right ) . \\end{align*}"}
+{"id": "8361.png", "formula": "\\begin{align*} G = \\left ( \\begin{array} { c c } h ( u _ 1 ) & 0 \\\\ 0 & { \\rm i d } \\\\ \\end{array} \\right ) \\end{align*}"}
+{"id": "1176.png", "formula": "\\begin{align*} & \\delta _ { 1 } ( m _ { 1 , n + 1 } ) = \\frac { 1 } { 2 } \\sum _ { \\substack { i + j = n + 1 \\\\ i , j > 0 } } [ m _ { 1 , i } , m _ { 1 , j } ] , \\end{align*}"}
+{"id": "1726.png", "formula": "\\begin{align*} 2 ( m + n ) \\xi + v _ { q _ 0 } ( \\xi ) + v _ { q _ 1 } ( \\xi ) = 2 \\pi ( l + 1 ) \\end{align*}"}
+{"id": "8015.png", "formula": "\\begin{align*} X ( t ) = \\sum _ { h = 0 } ^ H v _ { i _ h } T _ { ( i _ h ) } ( t ) = v _ { i _ k } t + \\sum _ { \\substack { h = 0 \\\\ h \\not = k } } ^ H ( v _ { i _ h } - v _ { i _ k } ) T _ { ( i _ h ) } ( t ) = g _ k ( T _ { ( - i _ k ) } ^ H ( t ) ) , \\end{align*}"}
+{"id": "8406.png", "formula": "\\begin{align*} \\Xi & = \\frac { v N ^ { 2 \\gamma - 2 } } { d \\log ^ 7 N } \\sum _ { P < p \\leq 2 P } ( \\log p ) \\sum _ { z = 1 } ^ { 2 Z - 1 } \\theta _ { z } ( p ^ c ) \\\\ & = \\frac { v N ^ { 2 \\gamma - 2 } } { d \\log ^ 7 N } \\left ( \\sum _ { P < p \\leq 2 P } ( \\log p ) - \\sum _ { P < p \\leq 2 P } ( \\log p ) \\theta _ { 0 } ( p ^ c ) \\right ) \\\\ & \\ll \\frac { v N ^ { 3 \\gamma - 2 } } { d \\log ^ 7 N } , \\end{align*}"}
+{"id": "7143.png", "formula": "\\begin{align*} \\begin{cases} ( \\partial _ t ^ 2 - c ^ 2 ( x ) \\Delta ) u ( x , t ) = 0 , & ( x , t ) \\in \\left ( \\mathbb { R } ^ n \\setminus \\Omega \\right ) \\times ( 0 , \\infty ) , \\ , n \\ge 2 , \\\\ u ( x , 0 ) = u _ 0 ( x ) , \\\\ \\partial _ t u ( x , 0 ) = u _ 1 ( x ) , \\\\ u ( t , x ) = 0 , & ( x , t ) \\in \\partial \\Omega \\times ( 0 , \\infty ) , \\end{cases} \\end{align*}"}
+{"id": "1579.png", "formula": "\\begin{align*} ( - v \\partial _ x ) ^ * h ( x , v ) & = v \\partial _ x h ( x , v ) - v \\frac { \\partial _ x g ( x , v ) } { g ( x , v ) } h ( x , v ) \\\\ & = v \\partial _ x h ( x , v ) - g ( x , v ) ^ { - 1 } \\rho _ g ( x ) ( \\alpha \\mathcal { M } _ { T _ g ( x ) } ( v ) + ( 1 - \\alpha ) \\mathcal { M } _ { \\tau ( x ) } ( v ) ) h ( x , v ) - h ( x , v ) , \\end{align*}"}
+{"id": "2282.png", "formula": "\\begin{align*} A ( x ) & = \\sum _ { k \\in n \\mathbb { Z } } e ^ { - \\pi \\alpha k ^ 2 } \\left ( \\sum _ { j = 1 } ^ { n } e ^ { 2 \\pi i k \\varepsilon _ j } e ^ { - 2 \\pi i k \\frac { j } { n } } \\right ) e ^ { 2 \\pi i k x } \\\\ & = \\sum _ { k \\in n \\mathbb { Z } } e ^ { - \\pi \\alpha k ^ 2 } \\left ( \\sum _ { j = 1 } ^ { n } e ^ { 2 \\pi i k \\varepsilon _ j } \\right ) e ^ { 2 \\pi i k x } , \\end{align*}"}
+{"id": "6280.png", "formula": "\\begin{align*} p _ \\lambda ( \\xi , \\eta ) = ( \\xi + \\eta ) a ( \\xi , \\eta ) , e _ \\lambda ( \\xi , \\eta ) = ( \\xi + \\eta ) ^ 2 a ( \\xi , \\eta ) . \\end{align*}"}
+{"id": "417.png", "formula": "\\begin{align*} \\index { $ [ u ] _ { \\ell , m } $ } [ u ] _ { \\ell , m } ( x ) : = u ( | x | ) | x | ^ { \\ell } Y _ { \\ell , m } ( \\omega _ x ) , \\mbox { f o r } a . e . \\ ; \\ ; x \\in \\R ^ d \\setminus \\{ 0 \\} , \\end{align*}"}
+{"id": "2703.png", "formula": "\\begin{align*} B = \\{ p _ { 1 } , p _ { 2 } , p _ { 1 } p _ { 2 } , p _ { 1 } p _ { 2 } ^ { 2 } \\} \\end{align*}"}
+{"id": "8155.png", "formula": "\\begin{align*} q _ { i j } ( x y ) = v ^ \\star _ { i j } ( \\theta ^ \\star _ { x y } , \\mu ^ \\star _ { x y } ) \\end{align*}"}
+{"id": "5820.png", "formula": "\\begin{align*} & ( \\alpha ^ { d , n - i } _ { m a x } , \\alpha _ i ) = ( \\alpha _ { i + 1 } , \\alpha _ i ) = - 1 , \\\\ & ( \\alpha ^ { d , n - i + 2 } _ { m a x } , \\alpha _ i ) = ( \\alpha _ { i - 1 } + 2 \\alpha _ i + 2 \\alpha _ { i + 1 } , \\alpha _ i ) = - 1 + 4 - 2 = 1 , \\\\ & \\alpha ^ { d , n - i + 2 } _ { m a x } - \\alpha ^ { d , n - i } _ { m a x } = \\alpha _ { i - 1 } + 2 \\alpha _ i + \\alpha _ { i + 1 } . \\end{align*}"}
+{"id": "7268.png", "formula": "\\begin{align*} \\tilde c ( I , \\omega ) = - \\log f ' _ I ( x _ \\omega ) . \\end{align*}"}
+{"id": "468.png", "formula": "\\begin{align*} \\frac { \\Gamma ( \\zeta + ( 1 + \\alpha ) / 2 + 2 k ) } { \\Gamma ( \\zeta + ( 1 + \\alpha ) / 2 ) } = 2 ^ { 2 k } \\frac { \\Gamma ( \\frac { \\zeta + ( 1 + \\alpha ) / 2 } { 2 } + k ) \\Gamma ( \\frac { \\zeta + ( 3 + \\alpha ) / 2 } { 2 } + k ) } { \\Gamma ( \\frac { \\zeta + ( 1 + \\alpha ) / 2 } { 2 } ) \\Gamma ( \\frac { \\zeta + ( 3 + \\alpha ) / 2 } { 2 } ) } , k \\in \\N _ 0 , \\end{align*}"}
+{"id": "2068.png", "formula": "\\begin{align*} e ^ { - 2 F } F _ { i i } = e ^ { - 2 F } \\left ( \\mathfrak { F } ' \\frac { \\rho _ { i i } } { R } + \\mathfrak { F } '' \\frac { \\rho _ i \\rho _ i } { R ^ 2 } \\right ) \\le \\frac { C _ n L } { R } + \\frac { C _ n } { R ^ 2 } , \\end{align*}"}
+{"id": "2831.png", "formula": "\\begin{align*} \\ell ^ { ( p ) } ( r e s _ { s _ { p - 1 } } ( \\mathfrak { q } _ { p - 1 } ) ) - \\ell ^ { ( p ) } ( r e s _ { s _ 1 } ( \\mathfrak { q } _ 1 ) ) = r e s _ { s = 0 } \\Omega ^ { ( p ) } ( \\mathfrak { q } _ { p - 1 } - \\mathfrak { q } _ 1 ) . \\end{align*}"}
+{"id": "755.png", "formula": "\\begin{align*} \\norm { \\mathbf U } _ { T } = \\left ( \\sum _ { i = 1 } ^ n \\norm { U _ i } _ { T } ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "699.png", "formula": "\\begin{align*} \\mathbf u ( t ) = \\mathbf v ( t ) . \\end{align*}"}
+{"id": "109.png", "formula": "\\begin{align*} v ^ { B } ( x ) & : = \\frac { v ( x ) } { \\chi * \\chi ( x / \\ell _ { } ) } , & w _ { B _ u } ( x , y ) : = \\chi _ { B _ u } ( x ) v ^ B ( x - y ) \\chi _ { B _ u } ( y ) , \\\\ v _ 1 ^ { B } ( x ) & : = \\frac { g ( x ) } { \\chi * \\chi ( x / \\ell _ { } ) } , & w _ { 1 , B _ u } ( x , y ) : = \\chi _ { B _ u } ( x ) v _ 1 ^ B ( x - y ) \\chi _ { B _ u } ( y ) , \\\\ v _ 2 ^ { B } ( x ) & : = \\frac { g ( x ) ( 1 + \\omega ( x ) ) } { \\chi * \\chi ( x / \\ell _ { } ) } , & w _ { 2 , B _ u } ( x , y ) : = \\chi _ { B _ u } ( x ) v _ 2 ^ B ( x - y ) \\chi _ { B _ u } ( y ) , \\end{align*}"}
+{"id": "281.png", "formula": "\\begin{align*} d _ 0 ^ n ( y ^ n , z ^ n ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n d _ 0 ( y _ i , z _ i ) , \\end{align*}"}
+{"id": "4617.png", "formula": "\\begin{align*} \\mathfrak { g } = \\bigoplus _ { \\overline { r } \\in \\mathbb { Z } / m \\mathbb { Z } } \\mathfrak { g } _ { \\overline { r } } . \\end{align*}"}
+{"id": "3729.png", "formula": "\\begin{align*} \\begin{cases} \\dfrac { d { \\bf u } } { d t } + \\mathbb { A } { \\bf u } = 0 , \\ t > 0 , \\\\ { \\bf u } ( 0 ) = { \\bf u } _ 0 , \\end{cases} \\end{align*}"}
+{"id": "7051.png", "formula": "\\begin{align*} \\frac { N } { ( p _ 1 Q K ) ^ { 3 / 2 } } \\times \\frac { N } { p _ 1 p ^ { 1 / 2 } _ 2 Q t } \\times \\sqrt { p _ 1 Q } \\times \\sqrt { K } = \\frac { N } { p _ 1 \\ , p _ 2 ^ { 1 / 2 } \\ , t } . \\end{align*}"}
+{"id": "2526.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\| \\cdot \\| } ^ { \\oplus } ( \\boldsymbol { \\eta } , \\boldsymbol { \\tau } ) = \\frac { 1 } { \\underline { c } } \\sqrt { \\| \\mathcal { R } _ 1 ( \\boldsymbol { \\eta } , \\boldsymbol { \\tau } ) \\| ^ 2 + \\| \\mathcal { R } _ 2 ( \\boldsymbol { \\eta } , \\boldsymbol { \\tau } ) \\| ^ 2 } , \\end{align*}"}
+{"id": "3435.png", "formula": "\\begin{align*} S k ( X ) \\simeq \\Delta = \\{ \\sum _ 0 ^ m x _ i = 1 , x _ i \\geq 0 \\} \\subset \\R ^ { n + 1 } , \\end{align*}"}
+{"id": "1514.png", "formula": "\\begin{gather*} t : Z ^ 2 \\rightarrow Z ^ 2 \\\\ t ( \\ , ( x _ i , y _ k ) \\ , , \\ , ( x _ j , y _ l ) \\ , ) \\ , = \\ , ( \\ ; ( x _ { \\sigma _ i ( j ) } , y _ { \\alpha _ k ( l ) } ) \\ , , \\ , ( x _ { \\gamma _ j ( i ) } , y _ { \\beta _ l ( k ) } ) \\ ; ) \\end{gather*}"}
+{"id": "7957.png", "formula": "\\begin{align*} S _ 1 = \\{ u _ { i , j } : i \\equiv j \\ ! \\ ! \\ ! \\ ! \\pmod 2 \\} , ~ S _ 2 = \\{ u _ { i , j } : i \\not \\equiv j \\ ! \\ ! \\ ! \\ ! \\pmod 2 \\} , \\end{align*}"}
+{"id": "6538.png", "formula": "\\begin{align*} { \\rm D C } ( N ) & = \\left \\{ \\omega \\in \\Omega : \\ | k \\cdot \\omega | \\geq N ^ { - 1 0 ^ 5 d b ^ 5 } \\ { \\rm f o r } \\ \\forall \\ 0 < | k | \\leq 1 0 N \\right \\} . \\end{align*}"}
+{"id": "7169.png", "formula": "\\begin{align*} \\| y \\| _ { N ^ c , g } = \\| P _ { N ^ c } V y \\| = \\| V y - P _ N V y \\| \\geq \\| V y \\| - \\| P _ N V y \\| = \\| y \\| - \\| P _ N V y \\| \\geq \\| y \\| - \\| P _ N V P _ M \\| \\| y \\| . \\end{align*}"}
+{"id": "6443.png", "formula": "\\begin{align*} M = \\begin{pmatrix} A & - V \\\\ B & U \\end{pmatrix} , ~ M ' = \\begin{pmatrix} A & - V ' \\\\ B & U ' \\end{pmatrix} \\end{align*}"}
+{"id": "2752.png", "formula": "\\begin{align*} 1 + | b _ { \\hat { \\pi } ( j ) } | = \\| y - z _ j \\| \\geq \\| y - y _ j \\| = \\| U ( x ) - U ( x _ j ) \\| = \\| x - x _ j \\| \\geq 1 + | a _ j | . \\end{align*}"}
+{"id": "3497.png", "formula": "\\begin{align*} l _ 1 - l _ 0 = l _ 1 ' - l _ 0 ' , \\ldots , l _ m - l _ 0 = l _ m ' - l _ 0 ' , \\end{align*}"}
+{"id": "3392.png", "formula": "\\begin{align*} P _ { \\sigma \\nu } & = - \\frac { 1 } { n - 1 } \\left ( \\Omega ^ 0 _ \\nu ( e _ \\sigma , e _ 0 ) + \\Omega ^ \\rho _ { \\mu } ( e _ \\sigma , e _ \\rho ) - \\frac { 1 } { n + 1 } \\Omega ^ 0 _ 0 ( e _ \\sigma , e _ \\nu ) - \\frac { 1 } { n + 1 } \\Omega ^ \\rho _ \\rho ( e _ \\sigma , e _ \\nu ) \\right ) , \\\\ \\tilde { P } _ { \\sigma \\nu } & = - \\frac { 1 } { n - 2 } \\left ( \\Omega ^ \\rho _ \\mu ( e _ \\sigma , e _ \\rho ) - \\frac { 1 } { n } \\Omega ^ { \\rho } _ \\rho ( e _ \\sigma , e _ \\nu ) \\right ) . \\end{align*}"}
+{"id": "6187.png", "formula": "\\begin{align*} f _ t \\circ \\Gamma _ { \\mathbf { k } , x } = \\Gamma _ { f _ t ( \\mathbf { k } ) , f _ t ( x ) } \\circ f _ t . \\end{align*}"}
+{"id": "7546.png", "formula": "\\begin{align*} & \\ , \\big \\langle \\gamma _ 1 , \\gamma _ 2 , \\delta _ A , \\eta _ i \\big \\rangle _ { 0 , \\beta _ 1 } \\big \\langle \\eta ^ i , \\gamma _ 3 , \\delta _ B , \\delta \\big \\rangle _ { 0 , \\beta _ 2 } \\\\ & = \\Phi _ { 0 , | A | + 3 , \\beta _ 1 } \\otimes \\Phi _ { 0 , | B | + 3 , \\beta _ 2 } ( [ \\overline { M } _ { 0 , | A | + 3 } \\times \\overline { M } _ { 0 , | B | + 3 } ] \\boxtimes \\delta _ A \\boxtimes \\gamma _ 1 \\boxtimes \\gamma _ 2 \\boxtimes \\eta \\boxtimes \\delta _ B \\boxtimes \\gamma _ 3 \\boxtimes \\delta ) . \\end{align*}"}
+{"id": "5045.png", "formula": "\\begin{align*} C ^ { k , \\alpha } _ { - 2 } ( M ; S ^ 2 T ^ * M ) = K \\oplus K _ k ^ \\perp , C ^ { k , \\alpha } _ { - 2 , \\nabla f } ( M ; S ^ 2 T ^ * M ) = K \\oplus K _ { k , \\nabla f } ^ \\perp \\end{align*}"}
+{"id": "2168.png", "formula": "\\begin{align*} \\partial ^ { 2 } _ t u _ 0 = \\frac { 1 } { r } \\partial _ { r } ( r \\partial _ r u _ 0 ) . \\end{align*}"}
+{"id": "2328.png", "formula": "\\begin{align*} \\lambda _ g \\theta _ f x & = \\lambda _ g \\left ( \\sum _ { h \\in G } f _ h ( x ) \\delta _ h \\right ) = \\sum _ { h \\in G } f _ h ( x ) \\delta _ { g h } = \\sum _ { u \\in G } f _ { g ^ { - 1 } u } ( x ) \\delta _ u \\\\ & = \\sum _ { u \\in G } f ( \\pi _ { u ^ { - 1 } g } x ) \\delta _ u = \\sum _ { u \\in G } ( f \\pi _ { u ^ { - 1 } } ) ( \\pi _ g x ) \\delta _ u = \\theta _ f ( \\pi _ g x ) \\in \\theta _ f ( \\mathcal { X } ) , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "8913.png", "formula": "\\begin{align*} [ \\mathbb { Q } ( \\xi ) : \\mathbb { Q } ] = [ \\mathbb { Q } ( \\xi ) : \\mathbb { Q } ( \\xi + \\xi ^ { - 1 } ) ] [ \\mathbb { Q } ( \\xi + \\xi ^ { - 1 } ) : \\mathbb { Q } ] = 2 \\cdot 3 = 6 . \\end{align*}"}
+{"id": "7453.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { m } \\kappa _ i + m \\right ) \\int _ { \\Omega } \\prod _ { i = 1 } ^ { m } \\vert y _ i \\vert ^ { \\kappa _ i + 1 } d z \\leq C _ { 2 1 } \\| ( y _ 1 , y _ 2 , \\cdots , y _ m ) \\| _ { Y , 1 } ^ { \\frac { \\sum _ { i = 1 } ^ { m } \\kappa _ i + m } { p } } . \\end{align*}"}
+{"id": "6213.png", "formula": "\\begin{align*} \\pi ( z , w ) ( g _ 1 \\otimes g _ 2 ) = \\pi ( z _ 1 , w _ 1 ) g _ 1 \\otimes \\pi ( z _ 2 , w _ 2 ) g _ 2 . \\end{align*}"}
+{"id": "7518.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t \\rho + \\partial _ x ( \\rho u ) = 0 , \\\\ \\partial _ t ( r u ) + \\partial _ x \\left \\{ r u ^ 2 + p ( n ) + \\theta \\rho \\right \\} = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "8237.png", "formula": "\\begin{align*} ( d ^ c _ { I _ 1 } \\hat { K } ) ( Y + a T ) = - d \\hat { K } ( I _ 1 Y + a I _ 1 T ) = - d \\hat { K } ( I _ Y ) - a d \\hat { K } ( I _ 1 T ) . \\end{align*}"}
+{"id": "1206.png", "formula": "\\begin{align*} P _ { M < \\cdot \\le N } : = P _ { \\le N } - P _ { \\le M } = \\sum _ { M < N ^ \\prime \\le N } P _ { N ^ \\prime } , \\end{align*}"}
+{"id": "6783.png", "formula": "\\begin{align*} P _ 1 ( u , v ) ( z ) & = \\frac { 1 } { \\nu _ 1 ^ + - \\nu _ 1 ^ - } \\left \\{ \\int _ { - \\infty } ^ { z } e ^ { \\nu _ 1 ^ - ( z - y ) } + \\int _ { z } ^ { \\infty } e ^ { \\nu _ 1 ^ + ( z - y ) } \\right \\} F ( u , v ) ( y ) d y , \\\\ [ 0 . 2 c m ] P _ 2 ( u , v ) ( z ) & = \\frac { 1 } { d \\left ( \\nu _ 2 ^ + - \\nu _ 2 ^ - \\right ) } \\left \\{ \\int _ { - \\infty } ^ { z } e ^ { \\nu _ 2 ^ - ( z - y ) } + \\int _ { z } ^ { \\infty } e ^ { \\nu _ 2 ^ + ( z - y ) } \\right \\} G ( u , v ) ( y ) d y , \\end{align*}"}
+{"id": "4711.png", "formula": "\\begin{align*} y _ i = x _ i + \\sum _ { j = 0 } ^ { i - 1 } b _ { i , j } x _ j + \\varepsilon _ i , \\end{align*}"}
+{"id": "1600.png", "formula": "\\begin{align*} V ( \\mathcal A ) = \\bigcup _ { \\mathcal X \\in [ I ] ^ { k - 1 } } \\mathcal { P } _ { \\mathcal X } , ~ ~ ~ E ( \\mathcal A ) = \\bigcup _ { \\mathcal Y \\in [ I ] ^ { k } } E ( \\mathcal A _ { \\mathcal Y } ) \\end{align*}"}
+{"id": "7891.png", "formula": "\\begin{align*} \\begin{cases} u _ t - \\log \\det D ^ 2 u - k \\log ( u ^ { \\star } ) = - g ( x , u ) & Q _ T \\\\ u = \\Phi & \\partial ^ { \\star } Q _ T \\end{cases} \\end{align*}"}
+{"id": "3315.png", "formula": "\\begin{align*} \\frac { \\displaystyle \\prod _ { i = 0 } ^ { t - 1 } ( q ^ m - q ^ { a + i } ) \\prod _ { i = 0 } ^ { b - t - 1 } ( q ^ { a } - q ^ i ) } { \\displaystyle \\prod _ { i = 0 } ^ { b - 1 } ( q ^ m - q ^ i ) } \\begin{bmatrix} b \\\\ t \\end{bmatrix} _ q . \\end{align*}"}
+{"id": "5087.png", "formula": "\\begin{align*} P ^ { - 1 } \\left ( \\begin{array} { c } v _ { 1 } ^ c \\\\ \\vdots \\\\ v _ n ^ c \\end{array} \\right ) = \\widetilde { v } _ { 1 } v _ 1 ^ c + \\cdots + \\widetilde { v } _ n v _ n ^ c , \\end{align*}"}
+{"id": "3840.png", "formula": "\\begin{align*} s = \\frac 1 2 ( \\log \\rho + 1 ) , \\end{align*}"}
+{"id": "5065.png", "formula": "\\begin{align*} \\triangle g ^ { i j } - \\nabla _ V g ^ { i j } = 2 \\nabla _ { k l } ^ 2 x ^ i \\nabla _ { k l } x ^ j - g ^ { i j } . \\end{align*}"}
+{"id": "6415.png", "formula": "\\begin{align*} s ^ * : = \\inf _ { 0 < \\lambda < + \\infty } \\frac { d _ { 1 } \\left [ \\int _ { \\mathbb { R } } J _ { 1 } ( y ) \\mathrm { e } ^ { \\lambda y } \\mathrm { d } y - 1 \\right ] + r _ 1 } { \\lambda } . \\end{align*}"}
+{"id": "593.png", "formula": "\\begin{align*} \\Psi ( t ) = - i M S _ { \\pm \\xi } ( - t ) P _ \\mu ^ 2 \\psi _ { \\mp } ^ \\mu ( t ) + i S _ { \\pm \\xi } ( - t ) P _ \\mu ( \\Theta P _ \\mu \\phi ^ \\mu \\cdot \\Theta P _ \\mu \\psi _ { \\mp } ^ \\mu ) ( t ) - M _ { \\mathfrak K _ 1 } S _ { \\pm \\xi } ( - t ) P _ \\mu ^ 2 \\psi _ \\pm ^ \\mu ( t ) \\end{align*}"}
+{"id": "2931.png", "formula": "\\begin{align*} C _ { i j k l } = C _ { k l i j } \\end{align*}"}
+{"id": "2958.png", "formula": "\\begin{align*} 3 ^ n = \\sum _ { s = 0 } ^ n ( 1 + 2 s ) J _ s ^ n . \\end{align*}"}
+{"id": "2503.png", "formula": "\\begin{align*} \\boldsymbol { v } ( \\boldsymbol { x } , t ) = \\boldsymbol { v } _ 0 ^ c ( \\boldsymbol { x } ) + \\sum _ { k = 1 } ^ { \\infty } \\left ( \\boldsymbol { v } _ k ^ c ( \\boldsymbol { x } ) \\cos ( k \\omega t ) + \\boldsymbol { v } _ k ^ s ( \\boldsymbol { x } ) \\sin ( k \\omega t ) \\right ) \\end{align*}"}
+{"id": "3012.png", "formula": "\\begin{align*} \\Phi \\left ( \\theta _ 1 \\right ) + \\Phi \\left ( \\theta _ 2 \\right ) = \\left ( f _ \\vee \\right ) _ * \\left [ \\Sigma _ \\vee \\right ] . \\end{align*}"}
+{"id": "4738.png", "formula": "\\begin{align*} 4 = { \\rm P e r } ( A ) + { \\rm P e r } ( B \\setminus A ) > { \\rm P e r } ( B ) + 2 { \\rm P e r } _ B ( A ) = 2 . \\end{align*}"}
+{"id": "5377.png", "formula": "\\begin{align*} B _ { \\Psi ( \\alpha ) } = \\Psi \\circ B _ \\alpha \\circ \\Psi ^ { - 1 } \\end{align*}"}
+{"id": "6589.png", "formula": "\\begin{align*} \\Delta _ 1 q = - H _ M ^ { - 1 } { F } ( q ^ { ( 0 ) } ) , \\end{align*}"}
+{"id": "6920.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow + \\infty } \\psi ( \\xi _ n ) = \\psi _ - < \\phi _ - \\psi ' ( \\xi _ n ) = 0 . \\end{align*}"}
+{"id": "8356.png", "formula": "\\begin{align*} \\bigg \\| B \\int _ { - t } ^ 0 S _ B ( - r ) \\omega _ 2 ( r ) d r \\bigg \\| \\le & \\bigg \\| B \\int _ { - 1 } ^ 0 S _ B ( - r ) \\omega _ 2 ( r ) d r \\bigg \\| \\\\ & + \\bigg \\| B \\int _ { - t } ^ { - 1 } S _ B ( - r ) \\omega _ 2 ( r ) d r \\bigg \\| . \\end{align*}"}
+{"id": "4927.png", "formula": "\\begin{align*} \\Omega _ D : = \\dfrac { 1 } { \\sqrt { 2 \\pi | D | } } \\left ( \\prod _ { j = 1 } ^ { | D | - 1 } \\Gamma \\left ( \\dfrac { j } { | D | } \\right ) ^ { \\chi _ D ( j ) } \\right ) ^ { \\frac { 1 } { h ' ( D ) } } , \\end{align*}"}
+{"id": "7126.png", "formula": "\\begin{align*} \\left [ D ^ { 1 0 } , u \\cdot \\nabla \\right ] \\theta = \\sum _ { k = 0 } ^ 9 \\begin{pmatrix} 1 0 \\\\ k \\end{pmatrix} \\nabla ^ \\bot D ^ { 1 0 - k } \\psi \\cdot \\nabla D ^ k \\theta . \\end{align*}"}
+{"id": "2222.png", "formula": "\\begin{align*} { \\bf A C } ( G ) : = \\underset { j = 1 } { \\overset { 3 } { \\oplus } } A C ( [ 0 ; l _ j ] ) . \\end{align*}"}
+{"id": "6071.png", "formula": "\\begin{align*} \\vert \\partial ^ { \\beta _ 0 } _ x \\nabla P _ t \\varphi ( x ) \\vert = \\vert \\mathbb { E } [ \\partial ^ { \\beta _ 0 } _ x ( \\nabla \\varphi ( X _ t ( x ) ) ] \\vert = \\vert \\sum _ { \\vert \\alpha _ 0 \\vert \\leq \\vert \\beta _ 0 \\vert } \\mathbb { E } [ ( \\partial ^ { \\alpha _ 0 } \\nabla \\varphi ) ( X _ t ( x ) ) \\mathbf { P } _ { \\alpha _ 0 } ( x ) ] \\vert \\leq \\Vert \\nabla \\varphi \\Vert _ { k , \\infty } \\sum _ { \\vert \\alpha _ 0 \\vert \\leq \\vert \\beta _ 0 \\vert } \\mathbb { E } \\vert \\mathbf { P } _ { \\alpha _ 0 } ( x ) \\vert , \\end{align*}"}
+{"id": "5109.png", "formula": "\\begin{align*} R \\Gamma ^ { ? } ( G , V ) = [ C ^ { ? } ( G ^ { \\bullet } , V ) , d ^ n ] . \\end{align*}"}
+{"id": "5341.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ L \\ln ( \\varepsilon _ k ^ { - 1 } ) ^ 3 & = \\sum _ { k = 1 } ^ L ( \\ln ( L \\varepsilon ^ { - 1 } ) + ( L - k ) \\ln ( C ( N + 1 ) ) ) ^ 3 \\\\ & = O ( L \\ln ( L \\varepsilon ^ { - 1 } ) ^ 3 + L ^ 4 \\ln ( C N ) ^ 3 ) \\end{align*}"}
+{"id": "5510.png", "formula": "\\begin{align*} O _ B ^ { ( \\sigma ( g ) ) } = \\widehat S ^ { - 1 } O _ A ^ { ( g ) } \\widehat S , O _ A ^ { ( \\sigma ( g ) ) } = \\widehat S ^ { - 1 } O _ B ^ { ( g ) } \\widehat S \\qquad \\textrm { f o r a l l } g \\in F r e e _ 2 . \\end{align*}"}
+{"id": "960.png", "formula": "\\begin{align*} \\left . \\frac { \\partial } { \\partial t } \\mathcal { F } _ { 1 , \\theta } \\left [ \\Sigma _ t \\right ] \\right \\vert _ { t = 0 } = \\int _ \\Sigma \\left < P _ 1 \\nabla f , \\nabla f \\right > - 2 H _ 1 \\left ( H _ 2 + c \\right ) f ^ 2 \\ , d \\mu _ \\Sigma + \\int _ { \\partial \\Sigma } \\left \\vert { P _ 1 \\nu } \\right \\vert \\alpha _ \\theta f ^ 2 \\ , d \\mu _ { \\partial \\Sigma } . \\end{align*}"}
+{"id": "5156.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } { \\rm A o I } ( S _ { \\mathrm { z } } , Q _ { { \\mathrm { u n i } } } , F _ \\mathrm { s } ) - \\frac { 3 } { 2 } { H [ Q _ { { \\mathrm { u n i } } } ( X ) ] } = 0 . \\end{align*}"}
+{"id": "1154.png", "formula": "\\begin{align*} & m _ { 2 , t } ( x , m _ { 1 , t } ( y , z ) ) + m _ { 1 , t } ( x , m _ { 2 , t } ( y , z ) ) \\\\ & = m _ { 2 , t } ( m _ { 1 , t } ( x , y ) , z ) + m _ { 1 , t } ( m _ { 2 , t } ( x , y ) , z ) - m _ { 2 , t } ( m _ { 1 , t } ( x , z ) , y ) - m _ { 1 , t } ( m _ { 2 , t } ( x , z ) , y ) \\end{align*}"}
+{"id": "4000.png", "formula": "\\begin{align*} 0 = \\frac { \\partial _ l | \\nabla \\varphi _ t | ^ 2 } { | \\nabla \\varphi _ t | ^ 2 } - A \\varphi _ { t , l } - \\frac { \\varphi _ { t , l } } { ( \\varphi _ t - \\inf _ M \\varphi _ t + 1 ) ^ 2 } , \\end{align*}"}
+{"id": "8926.png", "formula": "\\begin{align*} g ( E _ i , E _ i ) = g ( E _ i ^ \\prime , E _ i ^ \\prime ) . \\end{align*}"}
+{"id": "5696.png", "formula": "\\begin{align*} \\big ( ( n + m ) ! \\prod \\limits _ { j = 1 } ^ { n } w _ j \\big ) ^ { - 1 } & = \\big ( ( n + m ) ! \\prod \\limits _ { j = 1 } ^ { m _ 0 } w _ j \\prod \\limits _ { j = m _ 0 + 1 } ^ { n } w _ j \\big ) ^ { - 1 } \\\\ & \\leq ( \\prod \\limits _ { j = 1 } ^ { m _ 0 } w _ j ) ^ { - 1 } \\frac { ( n + 2 m _ 0 ) ! } { ( n + m ) ! } \\\\ & \\leq ( \\prod \\limits _ { j = 1 } ^ { m _ 0 } w _ j ) ^ { - 1 } \\frac { 1 } { ( n + m ) ( n + m + 2 ) } . \\end{align*}"}
+{"id": "1717.png", "formula": "\\begin{align*} \\frac { 1 } { 4 \\pi ^ 2 } \\int _ { ^ { ( 2 ) } _ { \\texttt { b } } } f ( X _ 1 , X _ 2 ) & \\frac { \\sqrt { \\rho _ { \\texttt { b } } ( X _ 1 , X _ 2 ) } } { O _ { \\texttt { b } } ( X _ 1 , X _ 2 ; q , q _ 0 ) } X _ 1 X _ 2 \\\\ & = \\sum _ { m \\geq \\lambda _ 1 \\geq \\lambda _ 2 \\geq 0 } f \\bigl ( \\boldsymbol { X } ^ { ( m , 2 ) } _ { \\texttt { b } ; ( \\lambda _ 1 , \\lambda _ 2 ) } \\bigr ) \\hat { \\Delta } ^ { ( m , 2 ) } _ { \\texttt { b } ; ( \\lambda _ 1 , \\lambda _ 2 ) } , \\end{align*}"}
+{"id": "8369.png", "formula": "\\begin{align*} k ( \\omega ) = \\sup _ { x \\not = y \\in C ( \\omega ) } \\log \\bigg ( \\frac { \\| \\phi ( 1 , \\omega , x ) - \\phi ( 1 , \\omega , y ) \\| } { \\| x - y \\| } \\bigg ) \\end{align*}"}
+{"id": "6819.png", "formula": "\\begin{align*} \\partial _ u H = & [ ( u + u ^ * + e _ 2 ) ( u - u ^ * ) ] / [ ( u + e _ 2 ) u ] , \\partial _ v H = \\varrho ( v - v ^ * ) / v , \\\\ [ 0 . 2 c m ] \\partial _ { u u } H = & \\left [ u ^ * ( u ^ * + e _ 2 ) ( 2 u + e _ 2 ) \\right ] / [ ( u + e _ 2 ) u ] ^ 2 , \\ \\ \\partial _ { v v } H = \\varrho v ^ * / v ^ 2 . \\end{align*}"}
+{"id": "2265.png", "formula": "\\begin{align*} E _ { \\Phi } ( \\Gamma ) = \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\sum _ { j = 1 } ^ n \\sum _ { \\ell \\in \\Z } \\frac { 1 } { \\sqrt { \\alpha } } \\phi _ { 1 / \\alpha } ( \\ell + x _ j - x _ k ) = \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\sum _ { j = 1 } ^ n \\theta _ \\alpha ( x _ j - x _ k ) , \\end{align*}"}
+{"id": "4432.png", "formula": "\\begin{align*} | \\hat { H } _ { 2 } ^ + | = | \\hat { H } _ { 2 } ^ - | = : | \\hat { H } _ 2 | , \\hat { c } > \\hat { c } _ A = \\sqrt { \\frac { | \\hat { H } _ 2 | ^ 2 } { \\hat \\rho } } \\end{align*}"}
+{"id": "6083.png", "formula": "\\begin{align*} E ( \\theta _ { \\gamma } ) = E ( \\bar { X } ) = \\nabla f ( \\omega _ k ) \\end{align*}"}
+{"id": "1844.png", "formula": "\\begin{gather*} \\mathbf { E } _ { \\Omega _ 0 } [ \\mathcal { X } ] = \\int _ { \\Omega _ 0 } \\mathcal { X } \\ , d \\mathbf { P } . \\end{gather*}"}
+{"id": "2786.png", "formula": "\\begin{align*} \\begin{multlined} x ^ { k + 4 } \\cdot x ^ { k + 1 } \\cdot 1 7 x ^ 4 + x ^ { k + 4 } \\cdot ( 2 ^ k + k ) x ^ k \\cdot x ^ 5 + ( 2 ^ { k + 3 } + k + 3 ) x ^ { k + 3 } \\cdot x ^ { k + 1 } \\cdot x ^ 5 \\\\ = ( 2 k + 9 \\cdot 2 ^ k + 2 0 ) x ^ { 2 k + 9 } , \\end{multlined} \\end{align*}"}
+{"id": "2668.png", "formula": "\\begin{align*} R _ { \\overline { H } _ { l _ 0 - 1 } ^ { \\perp _ { \\bar { f } } } } & = \\dim ( \\overline { H } _ { l _ 0 - 1 } ^ { \\perp _ { \\bar { f } } } ) - \\dim ( \\overline { H } _ { \\bar { f } | _ { \\overline { H } _ { l _ 0 - 1 } ^ { \\perp _ { \\bar { f } } } } } ^ { \\perp _ { \\bar { f } } } ) \\\\ & = R _ f - ( l _ 0 - 1 ) - \\dim ( \\overline { H } _ { l _ 0 - 1 } ^ { \\perp _ { \\bar { f } } } \\bigcap ( \\overline { H } _ { l _ 0 - 1 } ^ { \\perp _ { \\bar { f } } } ) ^ { \\perp _ { \\bar { f } } } ) \\\\ & = R _ f - 2 ( l _ 0 - 1 ) \\geq 2 . \\end{align*}"}
+{"id": "5302.png", "formula": "\\begin{align*} g ( x ) = \\begin{cases} f ( x ) & x < x _ { i - 1 } , \\\\ y _ { i - 1 } + a _ { i + 1 } ( x - x _ { i - 1 } ) & x _ { i - 1 } \\leq x < \\hat { x } _ { i + 1 } , \\\\ f ( x - \\hat { x } _ { i + 1 } + x _ { i + 1 } ) & \\hat { x } _ { i + 1 } \\leq x . \\end{cases} \\end{align*}"}
+{"id": "988.png", "formula": "\\begin{align*} ( A - C ) ^ 2 ( B ^ 2 + Z ^ 2 ) = ( B - C ) ^ 2 ( A ^ 2 + Z ^ 2 ) . \\end{align*}"}
+{"id": "2945.png", "formula": "\\begin{align*} \\delta _ { i _ 1 i _ 2 } [ \\delta _ { k i _ 1 } D _ { i _ 2 \\pi ( i _ 3 \\cdots i _ { n } ) } ] & = \\delta _ { k 1 } D _ { 1 \\pi ( i _ 3 \\cdots i _ { n } } + \\delta _ { k 2 } D _ { 2 \\pi ( i _ 3 \\cdots i _ { n } ) } + \\delta _ { k 3 } D _ { 3 \\pi ( i _ 3 \\cdots i _ { n } ) } \\\\ & = D _ { k \\pi ( i _ 3 \\cdots i _ { n } ) } \\end{align*}"}
+{"id": "1453.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { Q } _ 1 & = 2 \\sum \\limits _ { i \\in I } \\sigma _ i ^ 2 - 2 \\sum \\limits _ { i = 1 } ^ { n - 1 } \\sigma _ { i } \\sigma _ { i + 1 } - 2 \\sigma _ { n } \\sigma _ { n + 1 } - 2 \\sigma _ { n + 1 } \\sigma _ { 1 } \\\\ & = ( \\sigma _ 1 - \\sigma _ 2 ) ^ { 2 } + ( \\sigma _ 2 - \\sigma _ 3 ) ^ { 2 } + \\cdots + ( \\sigma _ n - \\sigma _ { n + 1 } ) ^ { 2 } + ( \\sigma _ { n + 1 } - \\sigma _ 1 ) ^ { 2 } . \\end{aligned} \\end{align*}"}
+{"id": "2261.png", "formula": "\\begin{align*} \\vartheta ( z ; \\tau ) = \\sum _ { k \\in \\Z } e ^ { \\pi i \\tau k ^ 2 } e ^ { 2 \\pi i k z } . \\end{align*}"}
+{"id": "6947.png", "formula": "\\begin{align*} \\psi ( x _ 1 , . . , x _ { i - 1 } , - \\pi , x _ { i + 1 } , . . , x _ d ) = \\psi ( x _ 1 , . . , x _ { i - 1 } , \\pi , x _ { i + 1 } , . . , x _ d ) , \\end{align*}"}
+{"id": "7751.png", "formula": "\\begin{align*} \\Omega ( \\gamma ) = \\begin{cases} \\Omega ( q _ 0 \\gamma ^ 0 ) = - \\chi , q _ 0 \\neq 0 \\\\ \\Omega ( \\gamma ) = 0 . \\\\ \\end{cases} \\end{align*}"}
+{"id": "4239.png", "formula": "\\begin{align*} d \\omega ^ 1 = - 2 i \\ , \\omega ^ { 1 \\bar { 3 } } , \\ d \\omega ^ 2 = 2 i \\ , \\omega ^ { 2 \\bar { 3 } } , \\ d \\omega ^ 3 = 0 , \\end{align*}"}
+{"id": "8281.png", "formula": "\\begin{align*} f ( t ) = \\left ( \\begin{matrix} A ( t ) & & & \\\\ & A ( t ) & & \\\\ & & A ( t ) & \\\\ & & & 1 \\end{matrix} \\right ) , \\end{align*}"}
+{"id": "1594.png", "formula": "\\begin{align*} \\forall p \\in \\mathcal { P } , \\ \\forall N \\in \\Z ^ + \\ ! , \\ \\forall 1 \\leq \\beta \\leq N , \\ \\sigma ( ( \\Z / p \\Z ) ^ N \\ ! , \\ , \\Z / p ^ \\beta \\Z ) \\ , = \\ , N ( p - 1 ) - 1 . \\end{align*}"}
+{"id": "2547.png", "formula": "\\begin{align*} I ' [ u ] v & = \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\frac { | u ( x ) - u ( y ) | ^ { p - 2 } ( u ( x ) - u ( y ) ) ( v ( x ) - v ( y ) ) } { | x - y | ^ { N + p s } } d x d y \\\\ & \\quad \\quad + \\int _ { \\mathbb { R } ^ N } V ( x ) | u | ^ { p - 2 } u v d x - \\frac { 1 } { p _ { r ; s } ^ { \\uparrow * } } \\int _ { \\mathbb { R } ^ N } ( K \\ast g ( u ) ) g ' ( u ) v d x - \\varepsilon _ W \\int _ { \\mathbb { R } ^ N } W ( x ) f ' ( u ) v d x . \\end{align*}"}
+{"id": "1341.png", "formula": "\\begin{align*} S _ { L } ( F ) : = \\frac { 1 } { L ^ n } \\int _ 0 ^ \\infty \\mathrm { v o l } ( L - t F ) d t , \\end{align*}"}
+{"id": "5753.png", "formula": "\\begin{align*} P _ p ( J _ X ^ p ) : = \\frac { 1 } { \\sqrt { 2 \\pi \\Delta _ p ^ 2 } } \\exp \\Big [ - \\frac { ( J _ X ^ p - \\mu _ p ) ^ 2 } { 2 \\Delta _ p ^ 2 } \\Big ] . \\end{align*}"}
+{"id": "2291.png", "formula": "\\begin{align*} \\| g _ 1 ( x ) \\| _ { L ^ 2 } & \\geq e ^ { - \\pi \\alpha \\left ( \\frac { n - 1 } { 2 } \\right ) ^ 2 } \\left ( \\sum _ { 1 \\leq | k | \\leq \\frac { n - 1 } { 2 } } \\left | \\ \\sum _ { j = 1 } ^ { n } \\varepsilon _ j e ^ { - 2 \\pi i k \\frac { j } { n } } + \\mathcal { O } \\left ( k ^ 2 \\sum _ { j = 1 } ^ { n } \\varepsilon _ j ^ 2 \\right ) \\right | ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "6857.png", "formula": "\\begin{align*} \\mathcal { X } _ { ( j ) } = \\begin{cases} \\mathcal { G } _ 1 C & j = 1 , \\\\ ( \\mathcal { G } _ j ) _ { ( 2 ) } C & \\end{cases} \\end{align*}"}
+{"id": "1241.png", "formula": "\\begin{align*} \\begin{cases} i w _ t + \\Delta w = F ( \\tilde { u } + w ) - F ( \\tilde { u } ) + e , \\\\ w ( 0 ) = u _ 0 - \\tilde { u } ( 0 ) , \\end{cases} \\end{align*}"}
+{"id": "3144.png", "formula": "\\begin{align*} j _ { 0 , \\mathbf { v } } ( u ) = \\left ( j _ { 0 , l _ 1 - 1 } ( f _ 1 ( u ) ) , \\dots , j _ { 0 , l _ N - 1 } ( f _ N ( u ) ) \\right ) , \\end{align*}"}
+{"id": "1381.png", "formula": "\\begin{align*} \\varrho _ { 6 , 1 } ^ { i } \\left ( \\mathcal { F } _ { - 4 , - 1 } \\right ) = \\mathcal { G } _ { \\mathcal { F } _ { - 4 , - 1 } } ( z , i ) = - \\frac { 8 } { \\pi } \\Psi _ { 6 , - 1 } ^ i ( z ) . \\end{align*}"}
+{"id": "6136.png", "formula": "\\begin{align*} ( \\varphi _ 1 \\times \\varphi _ 2 \\times \\cdots \\times \\varphi _ k ) ( s ) = ( \\varphi _ 1 ( s _ 1 ) , \\varphi _ 2 ( s _ 2 ) , \\ldots , \\varphi _ k ( s _ k ) ) . \\end{align*}"}
+{"id": "5991.png", "formula": "\\begin{align*} \\| \\tilde { \\psi } \\| _ { H ^ { 1 / 2 } ( \\mathbb D ) ^ * } = \\frac { 1 } { \\lambda _ j - \\lambda _ { j , \\varepsilon } } O ( \\varepsilon ) . \\end{align*}"}
+{"id": "3834.png", "formula": "\\begin{align*} ( x _ 0 , s _ 0 , \\rho _ 0 ( x _ 0 ) , x _ 0 , s _ 0 , \\rho _ 1 ( x _ 1 ) ) _ { \\# } \\alpha \\in \\mathcal S ^ p _ { = } ( \\mu _ 0 , \\mu _ 1 ) . \\end{align*}"}
+{"id": "3759.png", "formula": "\\begin{gather*} f ( 0 ) = u ' ( 0 ) - a B u ( 0 ) , \\\\ g ( 0 ) = v ' ( 0 ) - b B v ( 0 ) = u '' ( 0 ) - ( a + b ) B u ' ( 0 ) + a b B ^ 2 u ( 0 ) . \\end{gather*}"}
+{"id": "4466.png", "formula": "\\begin{align*} \\partial _ 2 \\varphi = \\mu ( { \\mathbf U } ) | _ { x _ 1 = 0 } = \\frac { H ^ + _ 1 H ^ + _ 2 + H ^ - _ 1 H ^ - _ 2 } { ( H ^ { + } _ 2 ) ^ 2 + ( H ^ - _ 2 ) ^ 2 } \\Big | _ { x _ 1 = 0 } \\ , , \\end{align*}"}
+{"id": "4953.png", "formula": "\\begin{align*} N _ a = \\sum _ { 1 \\leq k \\leq n , ~ k \\equiv a \\pmod * { m } } F ( k ) = \\sum _ { 1 \\leq k \\leq n , ~ k \\equiv a \\pmod * { m } } \\sum _ { d | \\gcd ( k , n ) } \\mu ( d ) = \\sum _ { d | n } \\mu ( d ) N ( a , d ) , \\end{align*}"}
+{"id": "1969.png", "formula": "\\begin{align*} { a ^ t _ { \\sigma ^ j , s } - a ^ t _ { \\sigma , s } } = \\frac { j q } { p } { s } _ { \\pi _ \\beta ( 1 ) } . \\end{align*}"}
+{"id": "3366.png", "formula": "\\begin{align*} \\ll X ^ { \\varepsilon } \\sum _ { \\substack { z ^ { 1 / 2 } \\leq a \\leq z \\\\ P ^ + ( a ) \\leq \\xi \\\\ ( a , q ) = 1 } } S ( a , 1 , 1 , 1 ) \\ll \\frac { X ^ { 1 + \\varepsilon } } { z ^ { 1 / 2 } } + X ^ { \\varepsilon } Y . \\end{align*}"}
+{"id": "1806.png", "formula": "\\begin{gather*} \\frac { 1 } { T } \\int _ { 0 } ^ { T } | \\zeta ( s + i \\tau , \\alpha ) | \\ , d \\tau = \\frac { 1 } { T } \\int _ { t } ^ { T + t } | \\zeta ( \\sigma + i \\tau , \\alpha ) | \\ , d \\tau \\leq \\frac { 1 } { T } \\int _ { - ( T + | t | ) } ^ { T + | t | } | \\zeta ( \\sigma + i \\tau , \\alpha ) | \\ , d \\tau \\end{gather*}"}
+{"id": "1839.png", "formula": "\\begin{gather*} \\left \\| f ( s ) - \\sum _ { n = 0 } ^ { M } \\frac { 1 } { ( n + c ) ^ s } - \\sum _ { M < n \\leq N } \\frac { \\beta _ n } { ( n + c ) ^ s } \\right \\| < \\frac { \\epsilon } { 3 } . \\end{gather*}"}
+{"id": "5167.png", "formula": "\\begin{align*} & \\bigg | \\frac { E [ L ^ 2 ] } { 2 E [ L ] } + \\frac { 1 } { 2 } \\log _ 2 { \\delta } - \\frac { 1 } { 2 } h ( X ) \\bigg | \\\\ < & \\bigg | \\frac { E [ L ^ 2 ] } { 2 E [ L ] } - \\frac { 1 } { 2 } { H [ Q _ { { \\mathrm { u n i } } } ( X ) ] } \\bigg | \\\\ + & \\bigg | \\frac { 1 } { 2 } { H [ Q _ { { \\mathrm { u n i } } } ( X ) ] } + \\frac { 1 } { 2 } \\log _ 2 { \\delta } - \\frac { 1 } { 2 } h ( X ) \\bigg | \\\\ < & \\frac { \\epsilon } { 2 } + \\frac { \\epsilon } { 2 } \\\\ = & \\epsilon . \\end{align*}"}
+{"id": "930.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty ( C L _ 2 ^ u ( y ) + D L _ 2 ^ v ( y ) ) \\mathrm { e } ^ { - \\lambda y ^ 2 } \\mathrm { d } y = \\frac { x ^ { 1 + \\sqrt { m n } + c } } { ( 1 + \\frac { 4 t ^ \\alpha } { \\alpha } \\lambda ) ^ { \\frac { 3 + \\sqrt { m n } - c } { 2 } } } \\mathrm { e } ^ { - \\frac { \\lambda x ^ 2 } { ( 1 + \\frac { 4 t ^ \\alpha } { \\alpha } \\lambda ) } } . \\end{align*}"}
+{"id": "939.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\tilde { t } = Y ( t ) , \\\\ & \\tilde { x } = X ( x , t ) , \\\\ & \\tilde { u } = r _ 1 ( x , t ) u + r _ 2 ( x , t ) v + r _ 3 ( x , t ) , \\\\ & \\tilde { v } = s _ 1 ( x , t ) v + s _ 2 ( x , t ) u + s _ 3 ( x , t ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "6646.png", "formula": "\\begin{align*} J _ \\nu ( z ) \\sim \\Big ( { 2 \\over \\pi z } \\Big ) ^ { 1 / 2 } \\bigg ( \\cos \\omega \\sum _ { k = 0 } ^ \\infty ( - 1 ) ^ k { a _ { 2 k } ( \\nu ) \\over z ^ { 2 k } } - \\sin \\omega \\sum _ { k = 0 } ^ \\infty ( - 1 ) ^ k { a _ { 2 k + 1 } ( \\nu ) \\over z ^ { 2 k + 1 } } \\bigg ) , \\end{align*}"}
+{"id": "3887.png", "formula": "\\begin{align*} \\phi _ 0 ( M ) = \\prod _ G M \\mathrm { a n d } \\phi _ 1 ( M ) = \\prod _ { g \\in G } g . M . \\end{align*}"}
+{"id": "2202.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\# A \\cdot \\# B & = & 2 ^ { k n - \\alpha n - \\beta n + ( 2 k - 1 ) n - \\beta \\pm O ( \\log n ) } = 2 ^ { ( 3 k - 1 ) n - \\alpha n - 2 \\beta n \\pm O ( \\log n ) } \\\\ & = & 2 ^ { ( 3 k - 1 ) n - \\C ( m _ A ) - \\beta n \\pm O ( \\log n ) } . \\end{array} \\end{align*}"}
+{"id": "4841.png", "formula": "\\begin{align*} \\check { R } ( u _ 0 ) = \\sum _ { i } ^ { n } \\left [ \\prod _ { j = 1 } ^ { i - 1 } \\left ( \\frac { \\lambda _ j } { \\lambda _ { j + 1 } } f _ j ( u _ 0 ) + 1 \\right ) \\prod _ { k = i } ^ { n - 1 } \\left ( f _ k ( u _ 0 ) + \\frac { \\lambda _ k } { \\lambda _ { k + 1 } } \\right ) \\lambda _ n \\right ] P _ i \\end{align*}"}
+{"id": "4852.png", "formula": "\\begin{align*} & \\xi E _ i + \\xi ^ 2 \\delta E _ i + \\xi ^ 3 E _ i = \\xi E _ { i + 1 } + \\xi ^ 2 \\delta E _ { i + 1 } + \\xi ^ 3 E _ { i + 1 } \\\\ & ( \\xi + \\xi ^ 2 \\delta + \\xi ^ 3 ) E _ i = ( \\xi + \\xi ^ 2 \\delta + \\xi ^ 3 ) E _ { i + 1 } \\end{align*}"}
+{"id": "5033.png", "formula": "\\begin{align*} b _ 0 : = \\frac { a } 2 \\cdot \\frac { \\frac { n } 2 + W } { - f } - \\Big ( \\frac { a ^ 2 } { 4 } + \\frac { a } 2 \\Big ) \\frac { | \\nabla f | ^ 2 } { ( - f ) ^ 2 } . \\end{align*}"}
+{"id": "3833.png", "formula": "\\begin{align*} \\mathcal S ^ p _ { = } ( \\mu _ 0 , \\mu _ 1 ) : = \\Big \\{ \\eta \\in M ( X ^ 2 \\times \\R _ + ^ 4 ) \\mid \\pi ^ { x _ i } _ { \\# } ( s _ i ^ p w _ i \\eta ) = \\mu _ i \\Big \\} . \\end{align*}"}
+{"id": "8697.png", "formula": "\\begin{align*} 9 f ^ { \\prime \\prime } ( x ) ^ 2 f ^ { ( 5 ) } ( x ) - 4 5 f ^ { \\prime \\prime } ( x ) f ^ { \\prime \\prime \\prime } ( x ) f ^ { ( 4 ) } ( x ) + 4 0 f ^ { \\prime \\prime \\prime } ( x ) ^ 3 = 0 , \\end{align*}"}
+{"id": "8627.png", "formula": "\\begin{align*} \\mathcal { A } ^ { \\mu } [ \\beta b ] \\bullet = \\frac { \\beta } { 2 } \\big ( \\mathrm { F } _ 3 ( b \\bullet ) + b \\mathrm { F } _ 3 \\bullet \\big ) + \\frac { \\mu \\beta } { 2 } \\big ( \\mathrm { F } _ 4 \\Delta _ X ( b \\bullet ) + b \\mathrm { F } _ 4 \\Delta _ X \\bullet \\big ) - \\frac { \\mu \\beta ^ 3 } { 1 2 } \\big ( b ^ 3 \\Delta _ X \\mathrm { F } _ 3 \\bullet + \\Delta _ X \\mathrm { F } _ 3 ( b ^ 3 \\bullet ) \\Big ) , \\end{align*}"}
+{"id": "2762.png", "formula": "\\begin{align*} t ( w ) : = \\frac { w } { 1 + w } T _ 1 ^ { - 1 } ( w ) T _ 2 ^ { - 1 } ( w ) , \\end{align*}"}
+{"id": "2317.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } z + \\gamma \\partial ^ { 3 } _ { x } z - \\delta \\partial ^ { 5 } _ { x } z + z \\partial _ { x } z = 0 , \\\\ z ( x , 0 ) = z _ 0 ( x ) . \\end{cases} \\end{align*}"}
+{"id": "5449.png", "formula": "\\begin{align*} I _ t & = \\frac { e ^ { 2 \\rho t } - 1 } { c _ t } \\int _ { 1 + \\frac { 1 } { c _ t } } ^ { e ^ { 2 \\rho t } + \\frac { 1 } { c _ t } } \\frac { d v } { v ^ 2 - d _ t ^ 2 } = \\frac { e ^ { 2 \\rho t } - 1 } { c _ t } \\left [ \\frac { 1 } { 2 d _ t } \\log \\left ( \\frac { v - d _ t } { v + d _ t } \\right ) \\right ] _ { 1 + \\frac { 1 } { c _ t } } ^ { e ^ { 2 \\rho t } + \\frac { 1 } { c _ t } } . \\end{align*}"}
+{"id": "1096.png", "formula": "\\begin{align*} \\| u \\| _ { W ^ { 1 , 2 } ( D ) } : = \\| u \\| _ { L ^ 2 ( D ) } + \\| | \\nabla u | \\| _ { L ^ 2 ( D ) } , \\end{align*}"}
+{"id": "5128.png", "formula": "\\begin{align*} C = 1 / \\min _ { { \\Delta } \\in \\mathcal { D } } \\sum _ { t \\in [ T ] , s \\in \\mathcal { E } ( t ) } \\Delta _ { t , s } \\end{align*}"}
+{"id": "3582.png", "formula": "\\begin{align*} { \\bf { H } } = { { \\bf { R } } } { \\bf { \\Xi T } } ^ H = { \\bf R } _ { \\rm I } { \\bf \\Xi } _ { \\rm I } { \\bf T } _ { \\rm I } ^ H + { \\bf R } _ { \\rm A } { \\bf \\Xi } _ { \\rm A } { \\bf T } _ { \\rm A } ^ H , \\end{align*}"}
+{"id": "7320.png", "formula": "\\begin{align*} & s L _ \\lambda + ( 1 - s ) ( ( I _ H - P _ n ) L _ \\lambda ( I _ H - P _ n ) + P _ n L _ \\lambda P _ n ) \\\\ & = Q + s K _ \\lambda + ( 1 - s ) ( ( I _ H - P _ n ) K _ \\lambda ( I _ H - P _ n ) + P _ n K _ \\lambda P _ n ) , \\end{align*}"}
+{"id": "3682.png", "formula": "\\begin{align*} ( D _ k ) _ { \\sigma , \\tau } = \\begin{cases} | E _ { \\sigma } | & \\sigma = \\tau , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "2134.png", "formula": "\\begin{align*} \\hat e ( t , r ) = \\tilde p ( t , r ) , \\end{align*}"}
+{"id": "3379.png", "formula": "\\begin{align*} \\hat { \\alpha } ^ i _ { j k } = \\alpha ^ i _ { j k } + 2 \\delta ^ i _ { ( j } \\Upsilon _ { k ) } . \\end{align*}"}
+{"id": "4828.png", "formula": "\\begin{align*} \\sum _ { p , q } S ^ { q l } _ { p k } ( - u ) S ^ { i p } _ { j q } ( u ) = \\rho ( u ) \\rho ( - u ) \\delta _ { i k } \\delta _ { j l } \\ ; . \\end{align*}"}
+{"id": "4765.png", "formula": "\\begin{align*} m _ { \\phi } ( x ) = \\underset { g \\in W _ { a } } { \\sup } ( r _ { \\phi } ( g ^ { - 1 } x ) - | g | ) \\end{align*}"}
+{"id": "2919.png", "formula": "\\begin{align*} 0 \\le E & \\sim _ { \\mathbb Z } h ^ { e , * } ( 1 - p ^ { e - r } ) K _ Z + \\gamma _ e ^ * M _ e \\\\ & = p ^ e ( B | _ U + ( 1 + d _ { \\delta , t } + \\lambda _ t ) D | _ U ) + h ^ { e , * } \\big ( ( t ( 1 - p ^ e ) ( 1 - \\delta ) + 1 - p ^ { e - r } ) K _ V + H \\big ) \\\\ & \\le ( p ^ e + 1 ) B | _ U + \\mu _ e D | _ U + h ^ { e , * } \\big ( ( t ( 1 - p ^ e ) ( 1 - \\delta ) + 1 - p ^ { e - r } ) K _ V \\big ) , \\end{align*}"}
+{"id": "5668.png", "formula": "\\begin{align*} 5 b _ 0 - b _ 1 & = 7 , \\\\ - 7 \\leq b _ 0 - 5 b _ 1 & \\leq 7 , \\end{align*}"}
+{"id": "2580.png", "formula": "\\begin{align*} & ( z , h ) + ( z ' , h ' ) = ( z + z ' - \\beta ( h , h ' ) , h + h ' ) = \\\\ & = ( z ' + z - \\beta ( h ' , h ) , h ' + h ) = ( z ' , h ' ) + ( z , h ) . \\end{align*}"}
+{"id": "2853.png", "formula": "\\begin{align*} \\hat e _ t ( \\eta ) : = t _ { \\vartheta } ^ { \\langle \\hat \\rho ^ \\vee _ s , \\eta \\rangle } t _ { \\varphi } ^ { \\langle \\rho ( \\hat R _ 0 ^ \\vee ) - \\hat \\rho ^ \\vee _ s , \\eta \\rangle } = \\prod _ { \\beta \\in { \\hat R _ 0 ^ + } } t _ { \\beta } ^ { \\langle \\eta , \\beta ^ \\vee \\rangle / 2 } ( \\eta \\in { \\hat Q } ) \\end{align*}"}
+{"id": "2419.png", "formula": "\\begin{align*} \\lim _ { j \\to + \\infty } \\frac { \\abs { \\Omega _ j } } { \\abs { \\partial \\Omega _ j } ^ { \\frac { 3 } { 2 } } } = \\frac { 1 } { 6 \\sqrt { \\pi } } . \\end{align*}"}
+{"id": "5228.png", "formula": "\\begin{align*} \\eta _ t ^ w = 2 \\eta _ t D ^ 2 , \\textrm { a n d } \\eta _ t ^ q = 2 \\eta _ t \\ln m . \\end{align*}"}
+{"id": "4905.png", "formula": "\\begin{align*} \\zeta ( 2 k ) = \\dfrac { ( - 1 ) ^ { k + 1 } B _ { 2 k } 2 ^ { 2 k } } { 2 ( 2 k ) ! } \\pi ^ { 2 k } , \\frac { z } { e ^ { z } - 1 } : = \\sum _ { n = 0 } ^ { \\infty } \\frac { B _ { n } z ^ { n } } { n ! } , \\end{align*}"}
+{"id": "1921.png", "formula": "\\begin{align*} a _ h ^ 0 ( e _ f , \\psi _ h ) : = - \\sum _ { i , j } \\int _ { T _ { i j } } e _ f v \\ , \\partial _ x \\psi _ h \\ , { \\rm d } v \\ , { \\rm d } x - \\sum _ { j = 1 } ^ { N _ v } \\sum _ { i = 0 } ^ { N _ x - 1 } \\int _ { J _ j } \\left ( \\widehat { v e _ f } [ \\ ! [ \\psi _ h ] \\ ! ] \\right ) _ { i + 1 / 2 , v } \\ , { \\rm d } v , \\end{align*}"}
+{"id": "5596.png", "formula": "\\begin{align*} | u ( r ) | ^ p & = \\int _ r ^ \\infty \\left ( | u ( s ) | ^ p \\right ) ' \\mathrm d s = \\int _ r ^ \\infty p | u ( s ) | ^ { p - 2 } u ( s ) u ' ( s ) \\mathrm d s \\\\ & \\leq p \\int _ r ^ \\infty | u ( s ) | ^ { p - 1 } | u ' ( s ) | \\left ( \\dfrac { s } { r } \\right ) ^ { \\frac { \\alpha _ 0 ( p - 1 ) + \\alpha _ 1 } { p } } \\mathrm d s \\\\ & \\leq p \\| u \\| _ { L ^ p _ { \\alpha _ 0 } } ^ { \\frac { p - 1 } { p } } \\| u ' \\| _ { L ^ p _ { \\alpha _ 1 } } ^ { \\frac { 1 } { p } } r ^ { - \\frac { \\alpha _ 0 ( p - 1 ) + \\alpha _ 1 } { p } } . \\end{align*}"}
+{"id": "1273.png", "formula": "\\begin{align*} \\| f \\| _ { L _ { t } ^ q L ^ r _ x ( I \\times \\R ^ 3 ) } = \\bigg ( \\int _ { I } \\| f ( t , x ) \\| _ { L ^ r _ x } ^ q d t \\bigg ) ^ \\frac { 1 } { q } \\end{align*}"}
+{"id": "1236.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta ) u _ { n } ^ J + | x | ^ { - b } F ( u _ n ^ J ) = | x | ^ { - b } [ F ( \\sum _ { j = 1 } ^ J v _ n ^ j ) - \\sum _ { j = 1 } ^ J F ( v _ n ^ j ) ] . \\end{align*}"}
+{"id": "4321.png", "formula": "\\begin{align*} S ( \\omega _ \\ell \\| \\omega ) = \\left . \\frac { 1 } { 2 } \\frac { d } { d t } \\right | _ { t = 0 } \\sigma ( \\Lambda _ t \\ell , \\ell ) \\ , , \\ell \\in L \\ , . \\end{align*}"}
+{"id": "5432.png", "formula": "\\begin{align*} 3 \\sum _ { \\substack { i < j \\\\ r \\neq i , j } } d _ T ( i , j ) x _ i x _ j x _ r & = 3 \\sum _ { i < j < k } [ d _ T ( i , j ) + d _ T ( i , k ) + d _ T ( j , k ) ] x _ i x _ j x _ k \\\\ & = 6 \\sum _ { i < j < k } d _ T ( i , j , k ) x _ i x _ j x _ k \\\\ & = \\sum _ { i , j , k } d _ T ( i , j , k ) x _ i x _ j x _ k \\end{align*}"}
+{"id": "623.png", "formula": "\\begin{align*} M = \\norm { \\mathbf U } _ T + \\norm { \\mathbf V } _ T \\end{align*}"}
+{"id": "3606.png", "formula": "\\begin{align*} { { P } _ { { \\rm { D S } } , k } } = \\left \\{ \\frac { { { \\left ( \\sum \\limits _ { m = 1 } ^ { { M _ { \\rm { R } } } } \\left | [ { { { \\bf { \\Xi } } _ { { \\rm { D S } } , m } } } ] _ { k , k } \\right | \\right ) ^ 2 } E } } { { { N _ { \\rm { R } } } { M _ { \\rm { R } } } { \\sigma ^ 2 } } } < \\gamma _ { \\rm t h } \\right \\} , \\end{align*}"}
+{"id": "2385.png", "formula": "\\begin{align*} J ^ { a f t e r } ( W , t ) = r ( t - T ) + \\log W \\end{align*}"}
+{"id": "7935.png", "formula": "\\begin{align*} D = \\Big ( D _ 1 \\cup D _ 2 \\cup \\{ u , v \\} \\Big ) \\setminus \\{ u _ 1 , u _ 2 , v _ 2 \\} \\ 1 \\end{align*}"}
+{"id": "5484.png", "formula": "\\begin{align*} \\mathrm { p e x p } \\left ( Q ^ { m _ 1 } p ^ { m _ 2 } s ^ { m _ 3 } \\ , \\mathbf { : } \\ , 1 - p ^ { a _ 1 } s ^ { b _ 1 } \\ , \\mathbf { : } \\ , \\dots \\ , \\mathbf { : } \\ , 1 - p ^ { a _ n } s ^ { b _ n } \\right ) : = \\underline { ( Q ^ { m _ 1 } p ^ { m _ 2 } s ^ { m _ 3 } ; p ^ { a _ 1 } s ^ { b _ 1 } , \\dots , p ^ { a _ n } s ^ { b _ n } ) } \\qquad \\in \\quad \\mathbf R ^ \\times \\end{align*}"}
+{"id": "4754.png", "formula": "\\begin{align*} J = \\bigcup _ { k = 0 } ^ \\infty \\left ( \\left ( 2 ^ { - 2 k - 1 } , 2 ^ { - 2 k - 1 } + 2 ^ { - 4 k - 3 } \\right ) \\cup \\left ( 2 ^ { - 2 k } - 2 ^ { - 4 k - 3 } , 2 ^ { - 2 k } \\right ) \\right ) \\times \\left \\{ 0 \\right \\} . \\end{align*}"}
+{"id": "4222.png", "formula": "\\begin{align*} \\rho = \\ , & \\ , a _ 1 e ^ { 1 2 3 } + a _ 2 e ^ { 1 2 4 } + a _ 3 e ^ { 1 3 5 } + a _ 4 ( e ^ { 2 3 4 } - e ^ { 1 2 5 } ) + a _ 5 ( e ^ { 2 3 5 } - e ^ { 1 3 4 } ) + a _ 6 ( e ^ { 2 3 6 } - e ^ { 1 4 5 } ) \\\\ [ 2 p t ] & + a _ 7 ( e ^ { 2 4 5 } - 2 e ^ { 1 2 6 } ) + a _ 8 ( e ^ { 1 4 6 } + e ^ { 2 5 6 } ) + a _ 9 ( 2 e ^ { 1 3 6 } + e ^ { 3 4 5 } ) + a _ { 1 0 } ( e ^ { 3 4 6 } - e ^ { 1 5 6 } ) + a _ { 1 1 } e ^ { 4 5 6 } , \\end{align*}"}
+{"id": "293.png", "formula": "\\begin{align*} \\mathsf { L } _ { j } ^ { - } \\left ( s \\right ) u _ { j } = 0 \\qquad \\Omega _ { j } . \\end{align*}"}
+{"id": "7544.png", "formula": "\\begin{align*} \\rho _ * \\rho ^ * i _ * i ^ * l _ * l ^ * & = \\rho _ * h ^ * i ^ * i _ * i ^ * l _ * l ^ * \\\\ & = \\rho _ * h ^ * i ^ * l _ * l ^ * \\\\ & = \\rho _ * \\rho ^ * l _ * l ^ * . \\end{align*}"}
+{"id": "5643.png", "formula": "\\begin{align*} y ( y + Q ) - C + x _ 3 ( x _ 0 y + B + x _ 1 ( y + Q ) ) = 0 , \\end{align*}"}
+{"id": "4400.png", "formula": "\\begin{align*} \\begin{cases} \\mathbb { L } ' _ e ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) \\dot { { \\mathbf U } } = \\mathbf { f } & \\Omega _ T , \\\\ \\mathbb { B } ' _ e ( \\hat { { \\mathbf U } } , \\hat { \\varphi } ) ( \\dot { { \\mathbf U } } , \\varphi ) = \\mathbf { g } & \\Gamma _ T , \\\\ ( \\dot { { \\mathbf U } } , \\varphi ) = 0 & t < 0 , \\\\ \\end{cases} \\end{align*}"}
+{"id": "8041.png", "formula": "\\begin{align*} g _ { d } ( f ) ( x ) = \\left \\{ \\sum \\limits _ { j \\in \\mathbb N } \\sum \\limits _ { Q \\in \\Pi _ { j } } \\vert \\phi _ { j } \\ast f ( x _ { Q } ) \\vert ^ 2 \\chi _ { Q } ( x ) \\right \\} ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "7098.png", "formula": "\\begin{align*} \\mathcal { G } ( . . . ) = \\int _ { \\mathbb { R } } V ( \\xi ) \\ , \\left | \\mathcal { J } ^ { + } _ 1 ( N _ 0 \\xi , m , q ) \\right | ^ 2 \\ , e \\left ( - \\frac { n _ 2 N _ 0 \\xi } { P n ^ 2 _ 1 } \\right ) d \\xi . \\end{align*}"}
+{"id": "8568.png", "formula": "\\begin{align*} \\mathcal { I } ^ { \\mu } [ h ] \\bullet = \\mathrm { I d } - \\frac { \\mu } { 3 h } \\sqrt { \\mathrm { F } _ 2 } \\nabla _ X \\Big ( h ^ 3 \\sqrt { \\mathrm { F } _ 2 } \\nabla _ X \\cdot \\bullet \\Big ) , \\end{align*}"}
+{"id": "116.png", "formula": "\\begin{align*} \\mathcal { K } _ { } : = \\sum _ { k \\in \\Lambda ^ * } \\tau _ k a _ k ^ { \\dagger } a _ k + \\frac { 1 } { 2 | \\Lambda | } \\sum _ { k \\in \\Lambda ^ * } \\widehat g _ k ( a _ 0 ^ \\dagger a _ 0 ^ \\dagger a _ k a _ { - k } + h . c . ) . \\end{align*}"}
+{"id": "7204.png", "formula": "\\begin{align*} D ^ { \\beta _ s } \\sigma _ \\rho ( r , \\theta , s ) = ( \\sigma ^ { ( k _ s ) } ) _ \\rho + \\mathcal { E } ^ { k _ s } _ \\rho ( r , \\theta , s ) - \\mathcal { E } ^ { k _ s } _ { L - \\rho } ( r , \\theta , s ) . \\end{align*}"}
+{"id": "5125.png", "formula": "\\begin{align*} D _ X ( [ a , b ] ) + D _ X ^ * ( i _ b i _ a \\varphi ) = & [ a , D _ X ( b ) ] + i _ { D _ X ( b ) } i _ a \\varphi - [ b , D _ X ( a ) ] - i _ { D _ X ( a ) } i _ b \\varphi \\\\ & + D _ { [ \\rho ( b ) , X ] } ( a ) - D _ { [ \\rho ( a ) , X ] } ( b ) . \\end{align*}"}
+{"id": "1419.png", "formula": "\\begin{gather*} \\tilde \\varphi _ { n , i } ( x ) = \\varphi _ { n , i } ( x ) + \\sum _ { k = 1 } ^ \\infty ( \\tilde Q _ { n , i ; k , 0 } ( x ) \\varphi _ { k , 0 } ( x ) - \\tilde Q _ { n , i ; k , 1 } ( x ) \\varphi _ { k , 1 } ( x ) ) , \\\\ \\tilde \\varphi _ { n , i } ( x ) = \\varphi ^ K _ { n , i } ( x ) + \\sum _ { k = 1 } ^ K ( \\tilde Q _ { n , i ; k , 0 } ( x ) \\varphi ^ K _ { k , 0 } ( x ) - \\tilde Q _ { n , i ; k , 1 } ( x ) \\varphi ^ K _ { k , 1 } ( x ) ) . \\end{gather*}"}
+{"id": "1369.png", "formula": "\\begin{gather*} \\mathcal { M } _ m : = \\left \\{ M \\in \\textnormal { M a t } _ { 2 \\times 2 } ( \\Z ) \\ : : \\ : \\det ( M ) = m \\right \\} \\end{gather*}"}
+{"id": "7060.png", "formula": "\\begin{align*} S _ r ( N ) : = \\mathop { \\sum } _ { n = 1 } ^ { \\infty } A _ { \\pi } ( n , r ) \\lambda _ f ( n ) n ^ { - i t } V _ 1 \\left ( \\frac { n } { N } \\right ) . \\end{align*}"}
+{"id": "7280.png", "formula": "\\begin{align*} A = \\bigcup _ { i \\in \\tilde { A } } [ n _ i , n _ { i + 1 } ) . \\end{align*}"}
+{"id": "5074.png", "formula": "\\begin{align*} E ( A ) E ( B ) = \\sum _ { p \\geq 0 } \\sum _ { n = 0 } ^ p \\frac { A ^ n B ^ { p - n } } { m ( p - n ) m ( n ) } . \\end{align*}"}
+{"id": "8173.png", "formula": "\\begin{align*} & [ A ^ * _ { 0 1 } , [ A ^ * _ { 0 1 } , [ A ^ * _ { 0 1 } , A _ { 0 1 } ] ] ] = \\left ( \\frac { n ( n - 1 ) } { k ( n - k ) } \\right ) ^ 2 [ A ^ * _ { 0 1 } , A _ { 0 1 } ] , \\\\ & [ A _ { 0 1 } , 2 A _ { 0 1 } A ^ * _ { 0 1 } A _ { 0 1 } - \\{ A _ { 0 1 } ^ 2 , A ^ * _ { 0 1 } \\} + 2 ( r - 1 ) \\{ A _ { 0 1 } , A ^ * _ { 0 1 } \\} + ( c _ 1 + c _ 2 A _ { 1 0 } + A _ { 1 0 } ^ 2 ) A ^ * _ { 0 1 } ] = 0 , \\end{align*}"}
+{"id": "7270.png", "formula": "\\begin{align*} X _ 1 ( \\omega ) : = \\tilde c ( \\omega _ 1 , \\sigma ( \\omega ) ) = - \\log f ' _ { \\omega _ 1 } ( x _ { \\sigma ( \\omega ) } ) . \\end{align*}"}
+{"id": "3711.png", "formula": "\\begin{align*} f ( 1 ) \\ge \\frac 2 5 \\left ( 1 + \\frac { 2 2 } { 2 5 } \\right ) + \\frac 1 5 \\ , \\frac { 2 1 } { 2 5 } = 0 . 9 2 , \\end{align*}"}
+{"id": "3742.png", "formula": "\\begin{align*} ( \\lambda ^ 3 I - A ) u = f \\end{align*}"}
+{"id": "2350.png", "formula": "\\begin{align*} \\sum _ { g \\in G } { f _ 2 } _ g ( x ) { \\tau _ 2 } _ g & = \\sum _ { g \\in G } ( f _ 1 \\pi _ { g ^ { - 1 } } ) ( x ) \\pi _ g \\tau _ 1 = \\sum _ { g \\in G } ( f U ^ { - 1 } ) ( \\pi _ { g ^ { - 1 } } x ) \\pi _ g U \\tau \\\\ & = \\sum _ { g \\in G } ( f \\pi _ { g ^ { - 1 } } ) ( U ^ { - 1 } x ) U \\pi _ g \\tau = U \\left ( \\sum _ { g \\in G } ( f \\pi _ { g ^ { - 1 } } ) ( U ^ { - 1 } x ) \\pi _ g \\tau \\right ) \\\\ & = U \\left ( \\sum _ { g \\in G } f _ g ( U ^ { - 1 } x ) \\tau _ g \\right ) = U U ^ { - 1 } x = x , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "6313.png", "formula": "\\begin{align*} e _ \\lambda = e _ \\lambda ^ 1 + \\partial _ x ( g _ { [ < \\lambda ^ { \\sigma } ] } - g _ { [ < \\lambda ] } ) \\partial _ x w _ \\lambda . \\end{align*}"}
+{"id": "1259.png", "formula": "\\begin{align*} e ^ { l i n } _ { n , T } = & \\Delta [ \\chi _ n ( \\frac { x - x _ n } { \\lambda _ n } ) ] e ^ { i ( t + \\lambda _ n ^ 2 t _ n ) \\Delta } g _ n [ P _ n \\phi ] \\\\ & + 2 \\nabla [ \\chi _ n ( \\frac { x - x _ n } { \\lambda _ n } ) ] e ^ { i ( t + \\lambda _ n ^ 2 t _ n ) \\Delta } \\nabla g _ n [ P _ n \\phi ] \\end{align*}"}
+{"id": "2754.png", "formula": "\\begin{align*} \\mathcal G ( v ) = \\sum _ { n \\ge 1 } g _ n v ^ n \\end{align*}"}
+{"id": "3547.png", "formula": "\\begin{align*} \\Vert w _ { 0 j } - w _ { 0 } \\Vert _ { W ^ { 1 , 1 } ( \\mathbb { R } ^ { N } ) } & = \\Vert w _ { 0 j } - w _ { 0 } \\Vert _ { W ^ { 1 , 1 } ( B _ { r _ { j } } ) } \\\\ & \\leqslant \\Vert w _ { 0 j } \\Vert _ { W ^ { 1 , 1 } ( B _ { r _ { j } } ) } + C \\Vert w _ { 0 } \\Vert _ { W ^ { 1 , \\infty } ( \\mathbb { R } ^ { N } ) } \\vert B _ { r _ { j } } \\vert , \\end{align*}"}
+{"id": "1093.png", "formula": "\\begin{align*} \\Delta _ \\mu u ( x ) : = \\frac { 1 } { \\mu ( x ) } \\sum _ { y \\sim x } w ( x , y ) ( u ( y ) - u ( x ) ) , \\end{align*}"}
+{"id": "432.png", "formula": "\\begin{align*} p _ \\zeta ^ { ( 2 ) } ( t , r , s ) = t ^ { - \\frac { 2 \\zeta + 1 } { 2 } } p _ \\zeta ^ { ( 2 ) } \\left ( 1 , \\frac { r } { t ^ { 1 / 2 } } , \\frac { s } { t ^ { 1 / 2 } } \\right ) , \\end{align*}"}
+{"id": "5750.png", "formula": "\\begin{align*} U _ j ( X , t ) = \\frac { U ( r _ j X , r _ j ^ 2 t ) } { \\bigg ( \\frac { 1 } { r _ j ^ { n + 3 + a } } \\int _ { \\mathbb Q _ { r _ j } ^ + } U ^ 2 x _ { n + 1 } ^ a d X d t \\bigg ) ^ { 1 / 2 } } , \\ , \\ , \\ , \\ \\ \\ \\ j \\geq 1 . \\end{align*}"}
+{"id": "2812.png", "formula": "\\begin{align*} J _ k = - \\int _ { \\Omega } Y _ t \\cdot \\nabla U _ { t , p } ( - \\Delta ) ^ { s } z _ k \\dd x - \\int _ { \\Omega } U _ { t , p } \\mathcal { L } _ { \\mathcal { K } _ { Y _ t } } z _ k \\dd x = : J _ k ^ 1 + J _ k ^ 2 . \\end{align*}"}
+{"id": "4293.png", "formula": "\\begin{align*} ( \\kappa - 1 ) \\frac { 1 } { n ^ { k _ 1 + k _ 2 + 1 } } \\sum _ { i = 1 \\atop i ^ \\prime = 1 } ^ { n } \\left ( \\sum _ { J \\in \\Pi _ 1 ( \\pi ) } g ( i , i ^ \\prime , J _ 1 , J _ 2 , p , n ) + \\sum _ { J \\in \\Pi _ 2 ( \\pi ) } g ( i , i ^ \\prime , J _ 1 , J _ 2 , p , n ) \\right ) + o ( 1 ) , \\end{align*}"}
+{"id": "2753.png", "formula": "\\begin{align*} 1 + | a _ { \\hat { \\pi } ^ { - 1 } ( j ) } | = \\| x - v _ j \\| \\geq \\| x - u _ j \\| = \\| U ( x ) - U ( u _ j ) \\| = \\| y - U ( u _ j ) \\| \\geq 1 + | b _ j | . \\end{align*}"}
+{"id": "5951.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ g u _ f + g ( F , \\nabla _ g u _ f ) + \\omega ^ 2 u _ f = 0 , \\\\ u _ f \\arrowvert _ { \\partial M } = f \\in C ^ \\infty ( \\partial M ) . \\end{cases} \\end{align*}"}
+{"id": "6026.png", "formula": "\\begin{align*} c ) \\vert L \\phi ( F ) \\vert \\leq \\vert \\nabla \\phi ( F ) \\vert \\vert L F \\vert + \\sup _ { \\vert \\beta \\vert = 2 } \\vert \\partial ^ \\beta \\phi ( F ) \\vert \\vert F \\vert _ { 1 , 1 } ^ { 2 } . \\end{align*}"}
+{"id": "5296.png", "formula": "\\begin{align*} 1 + \\sum _ { i = 1 } ^ { L - k } 3 \\cdot 4 ^ { i - 1 } + \\sum _ { i = 1 } ^ { k } 2 ^ i \\cdot 4 ^ { L - k } & = 1 + ( 4 ^ { L - k } - 1 ) + ( 2 ^ { k + 1 } - 2 ) 4 ^ { L - k } \\\\ & = ( 2 ^ { k + 1 } - 1 ) 4 ^ { L - k } \\end{align*}"}
+{"id": "8076.png", "formula": "\\begin{align*} B _ { i , 1 } = \\left \\{ Q \\colon Q \\in \\bigcup _ { j \\geq 1 } \\Pi _ { j + N } , \\vert Q \\cap \\Omega _ { i , 1 } \\vert > \\frac { 1 } { 2 } \\vert Q \\vert , \\vert Q \\cap \\Omega _ { i + 1 , 1 } \\vert \\leq \\frac { 1 } { 2 } \\vert Q \\vert \\right \\} . \\end{align*}"}
+{"id": "5687.png", "formula": "\\begin{align*} \\frac { \\ln \\| ( z - T ) ^ { - 1 } e _ 0 \\| _ p } { \\ln \\| ( z - T ) ^ { - 1 } \\| } \\geq \\frac { \\ln C \\| ( z - T ) ^ { - 1 } \\| } { \\ln \\| ( z - T ) ^ { - 1 } \\| } = \\frac { \\ln C } { \\ln \\| ( z - T ) ^ { - 1 } \\| } + 1 \\end{align*}"}
+{"id": "1431.png", "formula": "\\begin{align*} \\begin{cases} \\Delta u _ i + \\sum \\limits _ { j \\in I } k _ { i j } e ^ { u _ j } = 4 \\pi \\alpha _ i \\delta _ 0 \\ \\ B _ 1 ( 0 ) \\subseteq \\mathbb { R } ^ 2 , \\ \\ \\forall \\ i \\in I , \\\\ \\\\ \\sum \\limits _ { i \\in I } u _ i \\equiv C , \\ C \\ , \\end{cases} \\end{align*}"}
+{"id": "7625.png", "formula": "\\begin{align*} \\langle w , X \\rangle \\ge 0 \\mathcal { Q } \\mathrm { - q . s . } \\Rightarrow \\langle w , X \\rangle = 0 \\mathcal { Q } \\mathrm { - q . s . } \\end{align*}"}
+{"id": "6028.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\infty } = \\mathop { \\bigcap } \\limits _ { l = 1 } ^ { \\infty } \\mathop { \\bigcap } \\limits _ { p = 1 } ^ { \\infty } \\mathcal { D } _ { l , p } , \\mathcal { H } _ { l } = \\mathcal { D } _ { l , 2 } . \\end{align*}"}
+{"id": "8602.png", "formula": "\\begin{align*} \\phi _ 2 = - ( \\dfrac { z ^ 2 } { 2 } + h _ b z ) \\dfrac { \\zeta } { h _ b } \\big ( 1 + \\dfrac { h } { h _ b } \\big ) \\Delta _ { X } \\psi . \\end{align*}"}
+{"id": "3336.png", "formula": "\\begin{align*} G ( Q , \\alpha ) = \\frac { 1 } { \\delta ^ N } \\sum _ { k \\in ( \\mathbb { Z } / \\delta \\mathbb { Z } ) ^ N } \\mathbf { e } ( \\bar { \\alpha } Q ( k ) ) \\end{align*}"}
+{"id": "5011.png", "formula": "\\begin{align*} g = \\gamma + O ( r ^ { - 2 } ) , \\qquad \\nabla ^ g f = - \\tfrac 1 2 r \\partial _ r ( r _ 0 , \\infty ) \\times \\partial X , \\end{align*}"}
+{"id": "6870.png", "formula": "\\begin{align*} \\sqrt { - 1 } ( \\theta _ i \\partial \\overline { \\partial } \\theta _ i - \\partial \\theta _ i \\wedge \\overline { \\partial } \\theta _ i ) \\ > - A _ i \\cdot \\pi _ i ^ * \\Omega _ i = - A _ i \\cdot p r _ X ^ * \\omega . \\end{align*}"}
+{"id": "5928.png", "formula": "\\begin{align*} A _ { 1 } \\le \\dfrac { R _ { 2 } - S _ { 1 } } { 2 } + e = \\dfrac { - 2 e - S _ { 1 } } { 2 } + e = \\dfrac { - S _ { 1 } } { 2 } \\le 0 \\le d [ a _ { 1 } b _ { 1 } ] . \\end{align*}"}
+{"id": "2491.png", "formula": "\\begin{align*} a ( E _ i , S _ i , B _ { S _ i } + M _ { S _ i } ) = a ( E _ { i + 1 } , S _ { i + 1 } , B _ { S _ { i + 1 } } + M _ { S _ { i + 1 } } ) \\quad i \\gg 0 . \\end{align*}"}
+{"id": "4502.png", "formula": "\\begin{align*} p ^ { \\pm } _ { i + \\frac 1 2 } = S _ { \\theta _ i } p ^ \\pm _ i \\ , , u ^ { \\pm } _ { 2 , i + \\frac { 1 } { 2 } } = S _ { \\theta _ i } u ^ { \\pm } _ { 2 , i } \\ , , S ^ { \\pm } _ { i + \\frac { 1 } { 2 } } = S _ { \\theta _ i } S ^ { \\pm } _ i , \\end{align*}"}
+{"id": "6176.png", "formula": "\\begin{align*} \\eta _ ( A ) & = \\# \\{ ( i , j ) \\mid 1 \\leq j \\leq i \\leq n - 1 , \\ , d _ { i + 1 , j } \\leq d _ { i , j } = d _ { i + 1 , j + 1 } \\} \\\\ & = \\# \\{ ( i , j ) \\mid 1 \\leq j \\leq i \\leq n - 1 , \\ , b _ { i + 1 , j } \\leq b _ { i , j } = b _ { i + 1 , j + 1 } - 1 \\} = : \\eta _ ( B ) . \\end{align*}"}
+{"id": "7913.png", "formula": "\\begin{align*} \\alpha = 1 + 2 \\sum _ { x \\in \\Gamma } x x ^ \\ast , \\end{align*}"}
+{"id": "888.png", "formula": "\\begin{align*} & C + \\frac { \\sqrt { a b } } { a } D = \\frac { \\sqrt { a b } } { a } \\frac { \\alpha } { t ^ \\alpha } \\mathrm { e } ^ { - \\frac { \\alpha ( x + y ) } { t ^ \\alpha } } \\bigg ( \\frac y x \\bigg ) ^ { \\frac { \\sqrt { a b } - 1 } { 2 } } I _ { \\sqrt { a b } - 1 } \\bigg ( \\frac { 2 \\alpha \\sqrt { x y } } { t ^ \\alpha } \\bigg ) , \\end{align*}"}
+{"id": "497.png", "formula": "\\begin{align*} X = \\begin{pmatrix} \\psi \\\\ \\phi \\\\ \\dot \\phi \\end{pmatrix} , A = \\begin{pmatrix} - \\alpha \\partial _ x - i M \\beta & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & \\partial _ x ^ 2 - m ^ 2 & 0 \\end{pmatrix} , \\mathcal M ( X ) = \\begin{pmatrix} i \\beta \\psi \\mathfrak K _ 1 \\\\ 0 \\\\ \\phi \\mathfrak K _ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "2643.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\limsup _ { n \\to \\infty } P ( \\abs { \\Theta _ n ^ { ( l ) } ( t ) - \\Theta _ n ( t ) } > \\epsilon ) ^ { 1 / b _ n ^ 2 } = 0 \\ , . \\end{align*}"}
+{"id": "8817.png", "formula": "\\begin{align*} \\kappa _ { \\beta } L d h _ { t } ^ { \\beta - 1 } \\norm { \\bar { x } ( t ) - x ^ { * } } + \\frac { \\bar { L } } { n } \\sum _ { i = 1 } ^ { n } \\norm { x ^ { i } ( t ) - \\bar { x } ( t ) } \\norm { \\bar { x } ( t ) - x ^ { * } } \\leq \\frac { ( \\kappa _ { \\beta } L ) ^ { 2 } } { \\alpha } d ^ { 2 } & h _ { t } ^ { 2 ( \\beta - 1 ) } + \\frac { \\alpha } { 4 } \\norm { \\bar { x } ( t ) - x ^ { * } } ^ { 2 } + \\\\ & + \\frac { \\bar { L } t \\alpha } { n } \\sum _ { i = 1 } ^ { n } \\norm { x ^ { i } ( t ) - \\bar { x } } ^ { 2 } + \\frac { \\bar { L } } { 4 n t \\alpha } \\mathcal { K } ^ { 2 } . \\end{align*}"}
+{"id": "1784.png", "formula": "\\begin{gather*} \\sum _ { n = 1 } ^ { \\infty } | a _ n | ^ 2 ( \\log { n } ) ^ 2 < \\infty . \\end{gather*}"}
+{"id": "4266.png", "formula": "\\begin{align*} \\mathcal { J } _ { \\lambda _ k } ( u ) = & \\dfrac { 1 } { 2 } \\mathcal { B } _ { \\alpha } ( u , u ) - \\dfrac { \\lambda _ k } { 2 } \\int _ { \\Omega } | u ( x ) | ^ 2 d x - \\int _ { \\Omega } ( F ( x , u ^ - ( x ) + u ^ 0 ( x ) ) - F ( x , u ^ 0 ( x ) ) ) d x \\\\ & - \\int _ { \\Omega } F ( x , u ^ 0 ( x ) ) d x \\end{align*}"}
+{"id": "6667.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to 0 ^ \\pm } \\int _ { - \\infty } ^ \\infty \\Big ( R ( x ) - { 1 \\over \\pi } \\Big ) e ^ { i \\tau x } \\ , d x = - p - i q \\ , { \\rm s g n } \\ , \\tau , \\end{align*}"}
+{"id": "1139.png", "formula": "\\begin{align*} \\psi ( h ) = ~ & h , ~ h \\in ( \\mathfrak { g } _ { c o m } ) ^ 0 = \\mathfrak { g } ^ 0 , \\\\ \\psi ( ( h _ 1 , \\ldots , h _ { n + 1 } ) ) = ~ & h _ 1 + \\cdots + h _ { n + 1 } , ~ ( h _ 1 , \\ldots , h _ { n + 1 } ) \\in ( \\mathfrak { g } _ { c o m } ) ^ n \\end{align*}"}
+{"id": "3143.png", "formula": "\\begin{align*} j ^ D _ { 0 , 1 } ( u ) = ( d f ) _ { u ( 0 ) } ^ { - 1 } ( \\lambda ) \\in N _ { D / Z , u ( 0 ) } . \\end{align*}"}
+{"id": "3506.png", "formula": "\\begin{align*} Z _ \\Gamma ( s , \\rho _ 1 \\oplus \\rho _ 2 ) = Z _ \\Gamma ( s , \\rho _ 1 ) Z _ \\Gamma ( s , \\rho _ 2 ) , \\end{align*}"}
+{"id": "4468.png", "formula": "\\begin{align*} \\partial _ t { \\mathbf U } = - ( A _ 0 ( { \\mathbf U } ) ) ^ { - 1 } \\Big ( \\tilde { A } _ 1 ( { \\mathbf U } , \\Psi ) \\partial _ 1 { \\mathbf U } + A _ 2 ( { \\mathbf U } ) \\partial _ 2 { \\mathbf U } \\Big ) , \\end{align*}"}
+{"id": "1418.png", "formula": "\\begin{gather*} \\zeta _ { n i } ^ K ( x ) = \\left \\{ \\begin{array} { l l } - \\sum _ { k = K + 1 } ^ \\infty ( \\tilde Q _ { n , i ; k , 0 } ( x ) \\varphi _ { k , 0 } ( x ) - \\tilde Q _ { n , i ; k , 1 } ( x ) \\varphi _ { k , 1 } ( x ) ) , & n \\le K , \\\\ 0 , & n > K , \\end{array} \\right . \\end{gather*}"}
+{"id": "4065.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathcal { R } _ k q _ 0 ( a - t ) = \\mathcal { K } _ 0 ( \\mathcal { R } _ { k - 1 } q _ 0 ( a - t ) ) , & \\mathcal { R } _ k \\tilde { q } _ 0 ( a - t ) & = \\mathcal { K } _ 0 ( \\mathcal { R } _ { k - 1 } \\tilde { q } _ 0 ( a - t ) ) , \\\\ & \\mathcal { R } _ k p ( a - t ) = \\mathcal { K } _ 1 ( \\mathcal { R } _ { k - 1 } p ( a - t ) ) , & \\mathcal { R } _ k \\tilde { p } ( a - t ) & = \\mathcal { K } _ 1 ( \\mathcal { R } _ { k - 1 } \\tilde { p } ( a - t ) ) \\end{aligned} \\end{align*}"}
+{"id": "2147.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\int _ { | x - v t | \\leq \\omega ( t ) } \\frac 1 { \\alpha ^ 2 } \\left ( \\alpha _ t ^ 2 + \\alpha _ x ^ 2 \\right ) ( t , x ) d x = 0 , \\end{align*}"}
+{"id": "6900.png", "formula": "\\begin{align*} \\liminf _ { ( t _ n , h _ n ) \\to ( 0 ^ + , h ) } t _ n ^ { - 1 } d ( \\lambda ( p ) + t _ n v , \\Lambda ( p + t _ n h _ n ) ) = 0 , \\end{align*}"}
+{"id": "4804.png", "formula": "\\begin{align*} L ( i , j , T ) = \\left \\{ \\begin{array} { l l } ( \\log T ) ^ { j - 1 } & \\mbox { i f $ \\nu _ i $ s a t i s f i e s ( \\ref { e q n : R E S } ) s t r i c t l y } \\\\ ( \\log T ) ^ { j } & \\mbox { i f $ \\nu _ i $ s a t i s f i e s e q u a l i t y i n ( \\ref { e q n : R E S } ) } \\end{array} \\right . \\end{align*}"}
+{"id": "3114.png", "formula": "\\begin{align*} \\mathbf { h } _ { k } = \\frac { 1 } { \\sqrt { N } } \\sum _ { p = 1 } ^ { L _ { k } } \\alpha _ { p , k } \\mathbf { b } ^ H ( \\phi _ { 2 , k , p } , \\vartheta _ { 2 , k , p } ) \\end{align*}"}
+{"id": "1283.png", "formula": "\\begin{align*} B _ n = & \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p ) | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } ( | g _ n ^ { - 1 } u _ n | ^ p - | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p ) d x \\\\ & - \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p ) | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p \\end{align*}"}
+{"id": "7264.png", "formula": "\\begin{align*} f _ { I } ( [ 0 , 1 ] ) \\cap f _ { J } ( [ 0 , 1 ] ) = \\emptyset . \\end{align*}"}
+{"id": "658.png", "formula": "\\begin{align*} \\tau _ R < \\tau ( \\omega ) \\implies f ( \\tau _ R ( \\omega ) , \\omega ) = R . \\end{align*}"}
+{"id": "4272.png", "formula": "\\begin{align*} \\mathcal { A } _ n ( \\phi ) = \\sum _ { i = 1 } ^ { n } \\phi ( \\lambda _ i ) , \\end{align*}"}
+{"id": "3179.png", "formula": "\\begin{align*} u = S y , \\end{align*}"}
+{"id": "2226.png", "formula": "\\begin{align*} \\varphi _ j ( x ) = \\frac { b _ j } { \\Gamma ( \\alpha ) } x ^ { \\alpha - 1 } - \\frac { 1 } { \\Gamma ( \\alpha ) } \\underset { 0 } { \\overset { x } { \\int } } \\frac { - k _ j \\varphi _ j ( s ) + \\left ( D _ { l _ j - } ^ { \\alpha } \\varphi _ j \\right ) ( s ) } { \\left ( x - s \\right ) ^ { 1 - \\alpha } } d s , \\end{align*}"}
+{"id": "4712.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\\\ b _ { 1 , 0 } & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & 0 \\\\ b _ { 4 , 0 } & b _ { 4 , 1 } & b _ { 4 , 2 } & b _ { 4 , 3 } & 1 \\\\ \\end{bmatrix} \\begin{bmatrix} x _ 0 \\\\ x _ 1 \\\\ x _ 2 \\\\ x _ 3 \\\\ x _ 4 \\\\ \\end{bmatrix} + \\begin{bmatrix} 0 \\\\ \\varepsilon _ 1 \\\\ 0 \\\\ 0 \\\\ \\varepsilon _ 4 \\\\ \\end{bmatrix} \\end{align*}"}
+{"id": "5392.png", "formula": "\\begin{align*} \\frac { \\partial \\rho _ N ( t , z ) } { \\partial t } + \\frac { \\partial j _ N ( t , z ) } { \\partial z } = 0 , z \\in \\C , \\ , t \\geq 0 , \\end{align*}"}
+{"id": "5378.png", "formula": "\\begin{align*} \\sigma \\wedge \\tau = B _ { \\tau } ( \\sigma ) = B ^ { - 1 } _ { \\sigma } ( \\tau ) . \\end{align*}"}
+{"id": "3411.png", "formula": "\\begin{align*} X _ t = \\{ F _ 0 F _ 1 \\ldots F _ m + t F = 0 \\} \\subset M . \\end{align*}"}
+{"id": "3631.png", "formula": "\\begin{align*} \\overline { \\nabla } _ X P = \\mu \\ , X + \\omega ( X ) P , \\quad \\forall X \\in \\Gamma ( T N ) , \\end{align*}"}
+{"id": "1133.png", "formula": "\\begin{align*} \\begin{cases} l _ 1 : \\mathfrak { g } \\otimes M \\to M , \\\\ r _ 1 : M \\otimes \\mathfrak { g } \\to M , \\end{cases} ; \\begin{cases} l _ 2 : \\mathfrak { g } \\otimes M \\to M , \\\\ r _ 2 : M \\otimes \\mathfrak { g } \\to M , \\end{cases} \\end{align*}"}
+{"id": "3648.png", "formula": "\\begin{align*} \\Phi _ \\omega [ \\phi ] = \\int _ M \\Delta _ { \\omega } \\phi \\ . \\end{align*}"}
+{"id": "403.png", "formula": "\\begin{align*} ( \\pi _ 0 ) _ Z ( F ^ X ) = \\hom _ { Z } ( X , F ) / \\ ! \\ ! \\sim _ Z \\ , , \\end{align*}"}
+{"id": "2623.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\limsup _ { n \\to \\infty } P ( \\sup _ { x \\in [ 0 , 1 ] } \\abs { B _ n ( x , t ) } > r ) ^ { 1 / b _ n ^ 2 } = 0 \\ , . \\end{align*}"}
+{"id": "3457.png", "formula": "\\begin{align*} M u ( \\Delta ) = \\int _ { \\nabla u ( \\Delta ) } W ( p ) d p \\leq \\int _ { \\Delta ^ \\vee } W ( p ) d p = C _ 1 . \\end{align*}"}
+{"id": "8722.png", "formula": "\\begin{align*} K ( r ) = \\begin{cases} 1 , & r \\geq 0 \\\\ - 1 , & r < 0 \\end{cases} \\enspace . \\end{align*}"}
+{"id": "37.png", "formula": "\\begin{align*} | z \\rangle = e ^ { - \\big ( \\frac { | z | ^ 2 } { 2 } + z a ^ { \\dagger } _ 0 \\big ) } \\ , \\Omega , \\end{align*}"}
+{"id": "5179.png", "formula": "\\begin{align*} f = \\sum _ { \\ell = 1 } ^ L \\sum _ { i = 1 } ^ { R _ \\ell } f _ \\ell [ i ] \\end{align*}"}
+{"id": "7823.png", "formula": "\\begin{align*} ( A , 0 , 0 ) \\cdot ( 1 , v ^ a , ( \\eta ^ a , \\eta _ i , \\kappa ) ) \\cdot ( A ^ { - 1 } , 0 , 0 ) = ( 1 , \\varphi _ A ( v ^ a , ( \\eta ^ a , \\eta _ i , \\kappa ) ) ) \\end{align*}"}
+{"id": "8467.png", "formula": "\\begin{align*} \\mathcal { H } ^ { n - 1 } ( \\{ ( z , w ) \\in \\partial ^ { * } F _ { \\ell } : \\nu _ { w } ^ { F _ { \\ell } } ( z , w ) = 0 \\} \\cap ( \\Omega \\times \\mathbb { R } ^ { n - 1 } ) ) = 0 , \\end{align*}"}
+{"id": "8153.png", "formula": "\\begin{align*} q _ { i j } ( x , y ) = \\frac { \\binom { n } { i } } { \\binom { k } { i } } K _ { i } ( x , k - y , r - 1 ) H _ { j } ( y , n - i , k - i ) , \\end{align*}"}
+{"id": "4389.png", "formula": "\\begin{align*} \\mathrm { d i v } \\hat { \\mathbf { h } } ^ { \\pm } = 0 , \\hat { H } ^ { \\pm } _ N | _ { x _ 1 = 0 } = 0 , \\end{align*}"}
+{"id": "3624.png", "formula": "\\begin{align*} & { \\mathcal Y } ( \\phi , D ) : = _ y \\{ { \\mathcal E } ( \\cdot , \\phi ) \\ : : ( y , \\phi ) \\in { \\mathcal B } , \\ y ( \\Omega ) = D \\} \\end{align*}"}
+{"id": "22.png", "formula": "\\begin{align*} \\ell _ { \\delta } = \\frac { a } { 2 } e ^ { \\frac { 1 } { 2 \\delta } } e ^ { \\Gamma } = \\frac { 1 } { 2 \\sqrt { \\rho \\delta } } e ^ { \\Gamma } ( 1 + o ( 1 ) ) , \\end{align*}"}
+{"id": "1986.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ { p ^ { m } - 1 } \\omega _ q ^ { \\left ( a ^ { t _ 1 } _ { \\sigma , r } - a ^ { t _ 2 } _ { \\sigma , r } \\right ) } = \\sum _ { r = 0 } ^ { p ^ { m } - 1 } \\omega _ p ^ { d _ r } = 0 . \\end{align*}"}
+{"id": "1250.png", "formula": "\\begin{align*} C _ n = & - \\int _ { \\R ^ 3 } [ I _ \\alpha \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } ( | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p + | \\phi ^ 1 | ^ p ) ] | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p d x \\\\ & + \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n | ^ p ) | x - x _ n ^ 1 \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p d x . \\end{align*}"}
+{"id": "2298.png", "formula": "\\begin{align*} 3 3 = n _ { 2 } + 3 n _ { 3 } + 3 t _ { 3 } . \\end{align*}"}
+{"id": "2343.png", "formula": "\\begin{align*} \\pi _ g \\pi _ h & = \\theta _ \\tau \\lambda _ { g } \\theta _ f \\theta _ \\tau \\lambda _ { h } \\theta _ f = \\theta _ \\tau \\theta _ f \\theta _ \\tau \\lambda _ { g } \\lambda _ h \\theta _ f = I _ \\mathcal { X } \\theta _ \\tau \\lambda _ { g h } \\theta _ f = \\pi _ { g h } , \\forall g , h \\in G . \\end{align*}"}
+{"id": "904.png", "formula": "\\begin{align*} t ^ { 1 - \\alpha } \\eta _ { 2 t } - \\eta _ { 2 x x } - \\frac { c } { x } \\eta _ { 2 x } - m x ^ k \\phi _ { 1 x } = 0 , \\end{align*}"}
+{"id": "4210.png", "formula": "\\begin{align*} \\sigma _ { \\varepsilon } ( - 2 i ( T S ^ { \\ast } + S ^ { \\ast } T ) ) & = \\sigma _ { \\varepsilon } ( T \\diamond S \\circ _ { \\ast } i I ) = \\sigma _ { \\varepsilon } ( \\psi ( T ) \\diamond \\psi ( S ) \\circ _ { \\ast } \\psi ( i I ) ) \\\\ & = \\sigma _ { \\varepsilon } ( - 2 i ( \\psi ( T ) \\psi ( S ) ^ { \\ast } + \\psi ( S ) ^ { \\ast } \\psi ( T ) ) ) . \\end{align*}"}
+{"id": "6740.png", "formula": "\\begin{align*} \\mathcal { B } ( p , q ) = \\int _ { 0 } ^ { 1 } t ^ { p - 1 } ( 1 - t ) ^ { q - 1 } d t . \\end{align*}"}
+{"id": "6514.png", "formula": "\\begin{align*} \\omega ^ { ( 0 ) } & = \\omega ^ { ( 0 ) } ( \\alpha , \\theta _ 0 , m ) = \\left ( \\sqrt { \\cos ( \\theta _ 0 + n ^ { ( l ) } \\cdot \\alpha ) + m } \\right ) _ { l = 1 } ^ b , \\\\ \\mu _ n & = \\mu _ n ( \\alpha , \\theta _ 0 , m ) = \\sqrt { \\cos ( \\theta _ 0 + n \\cdot \\alpha ) + m } , \\ n \\in \\Z ^ d . \\end{align*}"}
+{"id": "6130.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n \\left ( u X _ i + v X _ i ^ { - 1 } + w \\right ) \\prod _ { 1 \\leq p < q \\leq n } \\left ( u E _ { k _ p } + v E _ { k _ q } ^ { - 1 } + w E _ { k _ p } E _ { k _ q } ^ { - 1 } \\right ) \\tilde { s } _ { ( k _ n , k _ { n - 1 } , \\ldots , k _ 1 ) } ( X _ 1 , X _ 2 , \\ldots , X _ n ) . \\end{align*}"}
+{"id": "8331.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { K } \\alpha _ k ^ i \\widehat { P _ { i j } ( k ) } = \\sum _ { k = 0 } ^ { K } \\alpha _ k ^ i \\frac { N _ { i j } ( k ) } { N _ i ( k ) } = \\sum _ { k = 0 } ^ { K } \\frac { N _ i ( k ) } { N _ i } \\frac { N _ { i j } ( k ) } { N _ i ( k ) } = \\frac { 1 } { N _ i } \\sum _ { k = 0 } ^ { K } N _ { i j } ( k ) = \\frac { N _ { i j } } { N _ i } = \\left ( \\widehat { \\mathbf { P } _ M ^ S } \\right ) _ { i j } \\end{align*}"}
+{"id": "4309.png", "formula": "\\begin{align*} \\begin{alignedat} { 3 } q & = w + x i + y j + z k \\\\ & = s ( q ) + v ( q ) \\end{alignedat} \\end{align*}"}
+{"id": "7646.png", "formula": "\\begin{align*} Z ^ \\ast = - \\mathrm { s i g n } ( x _ 0 ) | Z ^ \\ast | \\frac { w } { | w | } . \\end{align*}"}
+{"id": "6859.png", "formula": "\\begin{align*} \\mathcal { X } \\times _ 1 A _ 1 + \\mathcal { X } \\times _ 2 A _ 2 + \\dots + \\mathcal { X } \\times _ d A _ d = \\mathcal { C } , \\end{align*}"}
+{"id": "6242.png", "formula": "\\begin{align*} E _ i = \\hat u ^ i \\end{align*}"}
+{"id": "31.png", "formula": "\\begin{align*} n _ 0 = a ^ { \\dagger } _ 0 a _ 0 , n _ + = \\sum _ { k \\in \\Lambda ^ * } a ^ { \\dagger } _ k a _ k . \\end{align*}"}
+{"id": "2995.png", "formula": "\\begin{align*} \\| \\psi ^ * ( g ) x \\| ^ 2 = \\| ( \\psi ^ * ( g ) x \\mid \\psi ^ * ( g ) x ) _ A \\| _ { \\infty } = \\sup _ { v \\in E ^ 0 } \\sum _ { s ( e ) = v } | g ( \\psi ( e ) ) x ( e ) | ^ 2 \\le \\| g \\| _ { \\infty } ^ 2 \\| x \\| ^ 2 , \\end{align*}"}
+{"id": "3626.png", "formula": "\\begin{align*} E ( \\varphi ) = \\frac { 1 } { 2 } \\int _ D | d \\varphi | ^ 2 v ^ g , \\end{align*}"}
+{"id": "3530.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\sim x \\\\ } } \\log ( p ) \\mathrm { t r } \\left ( \\lambda _ p ^ 0 ( \\gamma _ { \\textbf { a } } ^ { - 1 } \\gamma _ { \\textbf { b } } ) \\right ) = \\sum _ { \\substack { p \\sim x \\\\ } } \\log ( p ) p = O ( x ^ 2 ) . \\end{align*}"}
+{"id": "4920.png", "formula": "\\begin{align*} \\eta \\left ( - \\frac { 1 } { \\tau } \\right ) = \\sqrt { - i \\tau } \\eta ( \\tau ) , \\quad \\eta \\left ( \\tau + \\frac { 1 } { 2 } \\right ) = e ^ { \\frac { 2 \\pi i } { 4 8 } } \\dfrac { \\eta ( 2 \\tau ) ^ 3 } { \\eta ( \\tau ) \\eta ( 4 \\tau ) } . \\end{align*}"}
+{"id": "5220.png", "formula": "\\begin{align*} ( n - 1 ) ( n - 2 - \\lambda ) = ( \\frac { n ( n - 3 ) } { 2 } ) \\mu . \\end{align*}"}
+{"id": "8686.png", "formula": "\\begin{align*} \\psi = g \\big ( v _ 1 , . . . , v _ n \\big ) ~ v _ 0 ^ { - \\frac { n } { 2 } } v _ { n + 1 } ^ { - 1 } . \\end{align*}"}
+{"id": "7529.png", "formula": "\\begin{align*} \\tau : = \\frac { n _ \\ast } { r _ \\ast } \\in ( 0 , 1 ) \\ , , \\kappa : = \\frac { r _ \\ast u _ \\ast ^ 2 } { p _ \\ast } \\in ( 0 , \\infty ) \\ , , \\kappa _ \\ast : = \\frac { n _ \\ast p ' _ \\ast } { p _ \\ast } \\in ( 0 , \\infty ) \\ , , \\end{align*}"}
+{"id": "74.png", "formula": "\\begin{align*} \\bar z a _ p ^ \\dagger b _ { p - k } + \\alpha _ k \\alpha _ { p - k } z b _ { p - k } a _ { - p } & = \\bar z a _ p ^ \\dagger b _ { p - k } + \\alpha _ k \\alpha _ { p - k } z a _ { - p } b _ { p - k } + \\alpha _ k \\alpha _ { p - k } z [ b _ { p - k } , a _ { - p } ] , \\end{align*}"}
+{"id": "8692.png", "formula": "\\begin{align*} \\Gamma ( p _ 0 t + r _ 0 ) . . . \\Gamma ( p _ { m - 1 } t + r _ { m - 1 } ) = a e ^ { b t } \\Gamma ( p _ { m } t + r _ { m } ) . . . \\Gamma ( p _ { n + 1 } t + r _ { n + 1 } ) \\end{align*}"}
+{"id": "168.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ 8 ( 2 \\pi \\sqrt { - 1 } y _ l ^ i ) ^ 2 = - \\frac { 1 } { 3 0 } c _ 2 ( W _ i ) , \\end{align*}"}
+{"id": "1675.png", "formula": "\\begin{align*} M _ { \\texttt { a } ; \\mu } ( \\boldsymbol { \\xi } ) : = \\frac { 1 } { N _ { \\texttt { a } ; \\mu } } \\sum _ { \\sigma \\in S _ { n } } \\exp ( i \\xi _ { \\sigma _ 1 } \\mu _ 1 + \\cdots + i \\xi _ { \\sigma _ { n } } \\mu _ { n } ) \\end{align*}"}
+{"id": "571.png", "formula": "\\begin{align*} \\mathbf V ( t ) = \\mathbf S ( t - S ) \\mathbf U ( S ) + i \\int _ { S \\wedge \\tau _ R } ^ { t \\wedge \\tau _ R } \\mathbf S ( t - \\sigma ) \\mathbf N ( \\mathbf u ( s ) ) \\ , d s + i \\int _ { S } ^ t \\mathbf S ( t - s ) \\mathbf M ( \\mathbf V ( s ) ) \\ , d W ( s ) . \\end{align*}"}
+{"id": "539.png", "formula": "\\begin{align*} \\mathbf h ( D _ x ) \\mathbf f = \\begin{pmatrix} h _ 1 ( D _ x ) f _ 1 \\\\ \\vdots \\\\ h _ n ( D _ x ) f _ n \\end{pmatrix} , \\mathbf S ( t ) \\mathbf f = \\begin{pmatrix} S _ { h _ 1 ( \\xi ) } ( t ) f _ 1 \\\\ \\vdots \\\\ S _ { h _ n ( \\xi ) } ( t ) f _ n \\end{pmatrix} , \\end{align*}"}
+{"id": "3966.png", "formula": "\\begin{align*} \\Phi ( G ) = G ' = Z ( G ) = \\Big \\{ 1 + \\sum _ { i < j } a _ { i j } E _ { i j } \\ , \\Big | \\ , a _ { i j } = 0 \\textrm { i f } j \\leq \\lceil n / 2 \\rceil \\textrm { o r } i \\geq \\lceil n / 2 \\rceil \\Big \\} \\end{align*}"}
+{"id": "3026.png", "formula": "\\begin{align*} \\phi ( H _ { 2 k + 1 } ) = \\begin{cases} \\phi ( T _ 6 ) \\ , \\left ( \\phi ( T _ 7 ) - \\phi ( P _ 5 ) \\right ) \\ ; \\ ; \\ ; ; \\\\ [ . 3 e m ] \\phi ( T _ k ) \\left ( \\phi ( T _ { k + 1 } ) - \\phi ( T _ { k - 1 } ) \\right ) . \\\\ \\end{cases} \\end{align*}"}
+{"id": "3525.png", "formula": "\\begin{align*} \\mathrm { t r } ( \\lambda _ p ( \\gamma ) ) = \\textbf { 1 } _ { B _ p } ( s ^ { - 1 } \\gamma s ) + \\sum _ { j = 0 } ^ { p - 1 } \\textbf { 1 } _ { B _ p } ( t ^ { - j } \\gamma t ^ j ) , \\end{align*}"}
+{"id": "8109.png", "formula": "\\begin{align*} \\begin{aligned} \\uppercase \\expandafter { \\romannumeral 2 } = \\sum \\limits _ { j } \\frac { \\mu _ { j } } { \\omega ( P _ { j } ) ^ { 1 / p } } ( M ( T ( b _ { j } ) ) ( x ) \\chi _ { 2 \\sqrt { n } P _ { j } ^ { * } } ( x ) + ( M ( \\chi _ { P _ { j } } ) ( x ) ) ^ { \\gamma } \\chi _ { ( 2 \\sqrt { n } P _ { j } ^ { * } ) ^ { c } } ( x ) ) \\end{aligned} \\end{align*}"}
+{"id": "7726.png", "formula": "\\begin{align*} ( \\tilde { K } + \\tilde { L } | _ N ) \\cap \\wedge ^ 2 E | _ N = ( K + L | _ N ) \\wedge E | _ N , \\end{align*}"}
+{"id": "8649.png", "formula": "\\begin{align*} \\int _ { \\mathcal { S } } \\tilde { P } ( \\Sigma ) \\nabla ^ { \\mu } _ { X , z } \\tilde u \\cdot \\nabla _ { X , z } ^ { \\mu } \\tilde \\varphi \\ : \\mathrm { d } z \\mathrm { d } X & = \\int _ { \\mathcal { S } } \\tilde f \\tilde \\varphi \\ : \\mathrm { d } z \\mathrm { d } X + \\int _ { \\R ^ d } g \\ : \\tilde \\varphi | _ { z = - 1 } \\ : \\mathrm { d } X , \\end{align*}"}
+{"id": "5017.png", "formula": "\\begin{align*} \\gamma = d r ^ 2 + r \\big ( d r \\otimes ( \\pi ^ * \\beta ) + ( \\pi ^ * \\beta ) \\otimes d r \\big ) + r ^ 2 \\pi ^ * h \\end{align*}"}
+{"id": "212.png", "formula": "\\begin{align*} & Q _ 1 ( M , P _ i , \\tau ) = 2 ^ 6 \\left \\{ e ^ { \\frac { 1 } { 2 4 } E _ 2 ( \\tau ) A _ 3 } \\left ( \\prod _ { j = 1 } ^ { 6 } \\frac { x _ j \\theta ' ( 0 , \\tau ) \\theta _ 1 ( x _ j , \\tau ) } { \\theta ( x _ j , \\tau ) \\theta _ 1 ( 0 , \\tau ) } \\right ) \\right . \\\\ & \\left . \\frac { 1 } { 2 } \\left ( \\prod _ { l = 1 } ^ 8 \\theta _ 1 ( y _ l ^ i , \\tau ) + \\prod _ { l = 1 } ^ 8 \\theta _ 2 ( y _ l ^ i , \\tau ) + \\prod _ { l = 1 } ^ 8 \\theta _ 3 ( y _ l ^ i , \\tau ) \\right ) \\right \\} ^ { ( 1 2 ) } , \\end{align*}"}
+{"id": "8499.png", "formula": "\\begin{align*} \\ell ^ { \\wedge } ( z ) = \\ell ^ { \\vee } ( z ) = \\ell ( z ) > 0 \\mbox { f o r a l l } z \\in \\mathring { J } . \\end{align*}"}
+{"id": "114.png", "formula": "\\begin{align*} \\mathcal { H } _ N = \\mathcal { H } _ { \\Lambda } ( \\rho ) _ N + \\rho \\widehat { g } ( 0 ) N . \\end{align*}"}
+{"id": "4766.png", "formula": "\\begin{align*} a _ { \\phi } ( x ) = \\underset { g \\in F _ { \\ell } \\setminus W _ { a } } { \\sup } ( r _ { \\phi } ( g ^ { - 1 } x ) - | g | ) . \\end{align*}"}
+{"id": "7727.png", "formula": "\\begin{align*} \\tilde { K } \\cap \\wedge ^ 2 E | _ N = K \\wedge E | _ N \\ ; \\ ; \\ ; \\tilde { L } | _ N \\cap \\wedge ^ 2 E | _ N = L | _ N \\wedge E | _ N , \\end{align*}"}
+{"id": "4231.png", "formula": "\\begin{align*} 2 \\ , F = i \\ , ( r ^ 2 \\omega ^ { 1 \\bar 1 } + s ^ 2 \\omega ^ { 2 \\bar 2 } + t ^ 2 \\omega ^ { 3 \\bar 3 } ) + u \\omega ^ { 1 \\bar 2 } - \\bar u \\omega ^ { 2 \\bar 1 } + v \\omega ^ { 2 \\bar 3 } - \\bar v \\omega ^ { 3 \\bar 2 } + z \\omega ^ { 1 \\bar 3 } - \\bar z \\omega ^ { 3 \\bar 1 } , \\end{align*}"}
+{"id": "165.png", "formula": "\\begin{align*} \\theta _ 2 ( v , \\tau ) = \\prod _ { j = 1 } ^ { \\infty } [ ( 1 - q ^ j ) ( 1 - e ^ { 2 \\pi \\sqrt { - 1 } v } q ^ { j - \\frac { 1 } { 2 } } ) ( 1 - e ^ { - 2 \\pi \\sqrt { - 1 } v } q ^ { j - \\frac { 1 } { 2 } } ) ] , \\end{align*}"}
+{"id": "436.png", "formula": "\\begin{align*} _ 2 \\tilde F _ 1 \\left ( \\frac a 2 , \\frac { a + 1 } { 2 } ; a - b + 1 ; \\frac { 4 z } { ( 1 + z ) ^ 2 } \\right ) = ( 1 + z ) ^ a \\ , _ 2 \\tilde F _ 1 \\left ( a , b ; a - b + 1 ; z \\right ) , | z | < 1 , \\end{align*}"}
+{"id": "241.png", "formula": "\\begin{align*} \\xi = ( \\xi _ 1 , \\ldots , \\xi _ n ) \\stackrel { } { \\longrightarrow } ( \\epsilon _ 1 \\xi _ { \\sigma ^ { - 1 } ( 1 ) } , \\ldots , \\epsilon _ n \\xi _ { \\sigma ^ { - 1 } ( n ) } ) = : \\xi , \\end{align*}"}
+{"id": "3690.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m j _ i ( n / m - j _ i + 1 ) = \\left ( \\sum _ { i = 1 } ^ m j _ i \\right ) n / m - \\sum _ { i = 1 } ^ m j _ i ( j _ i - 1 ) , \\end{align*}"}
+{"id": "5294.png", "formula": "\\begin{align*} \\mathcal { R } ( g _ { n _ 2 } \\circ g _ { n _ 1 } ) & = \\mathcal { R } ( g _ { n _ 2 } ) \\Tilde { \\mathcal { R } } ( g _ { n _ 1 } ) + \\mathcal { R } ( g _ { n _ 1 } ) - \\Tilde { \\mathcal { R } } ( g _ { n _ 1 } ) \\\\ & = ( \\mathcal { R } ( g _ { n _ 2 } ) - 1 ) \\Tilde { \\mathcal { R } } ( g _ { n _ 1 } ) + \\mathcal { R } ( g _ { n _ 1 } ) \\\\ \\Tilde { \\mathcal { R } } ( g _ { n _ 2 } \\circ g _ { n _ 1 } ) & = \\Tilde { \\mathcal { R } } ( g _ { n _ 2 } ) \\Tilde { \\mathcal { R } } ( g _ { n _ 1 } ) . \\end{align*}"}
+{"id": "8937.png", "formula": "\\begin{align*} b _ t ' : = d i a g ( e ^ { t / m } , \\ldots , e ^ { t / m } , e ^ { - t / n } , \\ldots , e ^ { - t / n } ) , \\ t > 0 . \\end{align*}"}
+{"id": "6356.png", "formula": "\\begin{align*} \\begin{aligned} J ^ 4 _ { \\lambda } ( u , v ) = & \\ g _ { [ < \\lambda ] } ^ { x _ 0 } J ^ 4 _ { m a i n } ( u _ \\lambda , v _ \\lambda ) + ( g _ { [ < \\lambda ] } - g _ { [ < \\lambda ] } ^ { x _ 0 } ) \\left ( M _ \\lambda ( v ) E _ \\lambda ( u ) - 2 P _ \\lambda ( u ) P _ \\lambda ( v ) \\right ) \\\\ : = & \\ J ^ { 4 , a } _ { \\lambda } ( u , v ) + J ^ { 4 , b } _ { \\lambda } ( u , v ) . \\end{aligned} \\end{align*}"}
+{"id": "3918.png", "formula": "\\begin{align*} \\hat \\omega = \\{ t u : \\} . \\end{align*}"}
+{"id": "3209.png", "formula": "\\begin{align*} \\| y - y _ k \\| ^ 2 _ { A A ^ T } = \\| x - x _ k \\| ^ 2 , \\end{align*}"}
+{"id": "2964.png", "formula": "\\begin{align*} U _ z ( x ) = z x , \\alpha _ z ( a ) = a , x \\in X , \\ \\ a \\in A , \\ \\ z \\in \\mathbb { T } \\end{align*}"}
+{"id": "3147.png", "formula": "\\begin{align*} v _ { \\infty } \\cdot D = 0 , \\end{align*}"}
+{"id": "1191.png", "formula": "\\begin{align*} a _ 0 . ( a _ g ) _ { g } . a _ 1 : = ( a _ 0 a _ g g ( a _ 1 ) ) _ { g } a _ 0 , a _ 1 \\in A \\mathrm { a n d } ( a _ g ) _ g \\in \\prod _ G A . \\end{align*}"}
+{"id": "8020.png", "formula": "\\begin{align*} P \\bigg \\{ \\bigcap _ { j \\in I _ { H ^ - } } \\{ N _ { j } ( t ) = 0 \\} \\bigg \\} & = P \\big \\{ V ( 0 ) \\in \\{ v _ { i _ 0 } , \\dots , v _ { i _ M } \\} \\big \\} \\sum _ { n = 0 } ^ \\infty P \\{ N ( t ) = n \\} \\ , \\alpha _ { I _ H } ^ n \\\\ & = e ^ { - \\Lambda ( t ) ( 1 - \\alpha _ { I _ H } ) } \\sum _ { i \\in I _ H } p _ i . \\end{align*}"}
+{"id": "1653.png", "formula": "\\begin{align*} \\mathbf { x } \\leq \\mathbf { y } \\Longleftrightarrow x _ 1 + \\cdots + x _ k \\leq y _ 1 + \\cdots + y _ k ( k = 1 , \\ldots , n ) . \\end{align*}"}
+{"id": "666.png", "formula": "\\begin{align*} \\mathbb E \\left ( \\norm { \\int _ 0 ^ T F ( t ) \\ , d W ( t ) } _ H ^ 2 \\right ) = \\mathbb E \\left ( \\int _ 0 ^ T \\norm { F ( t ) } _ { \\mathcal L _ 2 ( K , H ) } ^ 2 \\ , d t \\right ) . \\end{align*}"}
+{"id": "5395.png", "formula": "\\begin{align*} \\overline { v ( z , w _ { \\ast } ; 0 ) } + \\frac { w - w _ { \\ast } } { 2 t } = 0 . \\end{align*}"}
+{"id": "8495.png", "formula": "\\begin{align*} \\int _ { E } \\nabla \\psi ( x ) \\ d x = \\int _ { \\partial ^ { * } E } \\phi ( x ) \\nu _ { E } ( x ) \\ d \\mathcal { H } ^ { n - 1 } ( x ) \\mbox { f o r e v e r y } \\psi \\in C _ { c } ^ { 1 } ( \\mathbb { R } ^ { n } ) . \\end{align*}"}
+{"id": "8163.png", "formula": "\\begin{align*} & B = \\frac { ( y + x - k ) ( y + x - n - 1 ) ( y + k - n ) } { ( 2 y + x - n ) ( 2 y + x - n - 1 ) } \\ , , & & D = - \\frac { y ( y + x - k - 1 ) ( y + k - n - 1 ) } { ( 2 y + x - n - 1 ) ( 2 y + x - n - 2 ) } . \\end{align*}"}
+{"id": "8624.png", "formula": "\\begin{align*} H ( \\zeta , \\psi ) = \\frac { 1 } { 2 } \\int _ { \\R ^ d } \\zeta ^ 2 \\ : \\mathrm { d } X + \\frac { 1 } { 2 \\mu } \\int _ { \\R ^ d } \\psi \\mathcal { G } ^ { \\mu } \\psi \\ : \\mathrm { d } X , \\end{align*}"}
+{"id": "2963.png", "formula": "\\begin{align*} \\sigma _ g ( \\underline { \\iota } _ A ( a ) ) = \\underline { \\iota } _ A ( \\alpha _ g ( a ) ) \\sigma _ g ( \\underline { \\iota } _ X ( x ) ) = \\underline { \\iota } _ X ( { U _ g x } ) \\end{align*}"}
+{"id": "2154.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\int _ { | x - v t | \\leq \\omega ( t ) } \\frac 1 { \\beta ^ 2 } \\left ( \\beta _ t ^ 2 + \\beta _ x ^ 2 \\right ) ( t , x ) d x = 0 , \\end{align*}"}
+{"id": "5644.png", "formula": "\\begin{align*} y + x _ 1 x _ 3 = 0 , \\end{align*}"}
+{"id": "8834.png", "formula": "\\begin{align*} \\Gamma ( z ) = \\int _ { 0 } ^ \\infty x ^ { z - 1 } \\exp ( - x ) \\d x , z > 0 \\enspace . \\end{align*}"}
+{"id": "5774.png", "formula": "\\begin{align*} 2 \\lim _ { L \\rightarrow \\infty } { \\mathbb E } \\langle ( R _ { 1 , 2 } ^ p - { \\mathbb E } \\langle R _ { 1 , 2 } ^ p \\rangle ) ^ 2 \\rangle = 3 \\lim _ { L \\rightarrow \\infty } { \\mathbb E } \\langle ( R _ { 1 , 2 } ^ p - \\langle R _ { 1 , 2 } ^ p \\rangle ) ^ 2 \\rangle , \\end{align*}"}
+{"id": "123.png", "formula": "\\begin{align*} A \\ll B \\ ; \\Leftrightarrow \\ ; \\begin{cases} A \\leq C ( \\rho a ^ 3 ) ^ { \\zeta } B , & d = 3 , \\\\ A \\leq C \\delta ^ { \\zeta } B , & d = 2 . \\end{cases} \\end{align*}"}
+{"id": "3056.png", "formula": "\\begin{align*} { { \\bf { H } } _ { k , { \\rm { R } } } } = \\sum \\nolimits _ { l \\in { \\mathcal L } _ { k , { \\rm { R } } } } { { \\alpha _ { { k , { \\rm { R } } } , l } } { { \\bf { a } } _ { \\rm { R } } } \\left ( { \\Theta _ { { k , { \\rm { R } } } , l } ^ { \\rm { A } } } \\right ) { \\bf { a } } _ { { \\rm { S , } } k } ^ H \\left ( { \\Theta _ { { k , { \\rm { R } } } , l } ^ { \\rm { D } } } \\right ) } , \\end{align*}"}
+{"id": "4653.png", "formula": "\\begin{align*} \\tau _ v ( \\phi _ v ) : = \\sum _ { t _ v } \\phi _ v ( u _ { t _ v } ^ { ( v ) , * } ) \\otimes u _ { t _ v } ^ { ( v ) } \\end{align*}"}
+{"id": "928.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty ( A L _ 1 ^ u ( y ) + B L _ 1 ^ v ( y ) ) \\mathrm { e } ^ { - \\lambda y ^ 2 } \\mathrm { d } y = \\frac { - m x ^ { 1 + \\sqrt { m n } + c } } { \\sqrt { m n } ( 1 + \\frac { 4 t ^ \\alpha } { \\alpha } \\lambda ) ^ { \\frac { 3 + \\sqrt { m n } - c } { 2 } } } \\mathrm { e } ^ { - \\frac { \\lambda x ^ 2 } { ( 1 + \\frac { 4 t ^ \\alpha } { \\alpha } \\lambda ) } } , \\end{align*}"}
+{"id": "3549.png", "formula": "\\begin{align*} & \\ \\min _ { x , y } \\ ; f ( x ) + g ( y ) \\\\ & \\mathrm { s . t . } \\ ; A x + B y = b \\ , \\end{align*}"}
+{"id": "1996.png", "formula": "\\begin{align*} ( \\dot { \\mathrm { H } } ^ { - s , p ' } ( \\mathbb { R } ^ n ) ) ' = \\dot { \\mathrm { H } } ^ { s , p } ( \\mathbb { R } ^ n ) \\end{align*}"}
+{"id": "2864.png", "formula": "\\begin{align*} d _ { \\lambda , \\nu } : = \\begin{cases} \\theta ( \\lambda + \\nu ) e _ t ( - \\nu ) h _ t ^ { ( \\langle \\lambda , \\hat \\nu \\rangle ) } & \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "8322.png", "formula": "\\begin{align*} d _ { \\beta } ( \\beta ^ { - 2 } x ) \\overset { \\nu } { = } 1 0 0 0 0 ( t - 2 ) ( 2 t - 2 ) ( t - 2 ) ( t - 1 ) ^ { \\infty } . \\end{align*}"}
+{"id": "4880.png", "formula": "\\begin{align*} | | f | | ^ 2 : = \\int _ { - \\infty } ^ \\infty | | E _ + ^ { - 1 } ( t ) f ( t ) | | _ \\mathfrak { X } ^ 2 d t < \\infty \\end{align*}"}
+{"id": "7636.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } u ( k , w ) = - \\infty \\end{align*}"}
+{"id": "6524.png", "formula": "\\begin{align*} k \\cdot \\omega ^ { ( 0 ) } + \\mu _ n = ( k + e _ { l ' } ) \\cdot \\omega ^ { ( 0 ) } , \\end{align*}"}
+{"id": "8704.png", "formula": "\\begin{align*} x _ { n + 1 } = f ( x _ 1 , . . . , x _ n ) . \\end{align*}"}
+{"id": "1540.png", "formula": "\\begin{align*} \\overline { f } = \\langle f ( \\lambda ) , \\nu _ { t , x } \\rangle + m ^ { f } \\end{align*}"}
+{"id": "1720.png", "formula": "\\begin{align*} \\sum _ { 0 \\leq j \\leq n _ { \\texttt { c } } } | C _ { \\texttt { c } } ( \\boldsymbol { \\xi } ^ { ( 1 , n ) } _ { \\texttt { c } ; \\omega _ { \\texttt { c } ; j } } ) | ^ { - 2 } \\Delta ^ { ( 1 , n ) } _ { \\texttt { c } ; \\omega _ { \\texttt { c } ; j } } = \\prod _ { 1 \\leq j \\leq n } \\frac { 1 - q } { 1 - q ^ j } , \\end{align*}"}
+{"id": "5366.png", "formula": "\\begin{align*} h ^ 1 ( N _ { Y _ j } ( K _ { Y _ j } + D _ j - 3 H ) ) = h ^ 1 ( \\O _ { \\P ^ 1 } ( - 2 ) ^ { \\oplus 2 } ) = 2 . \\end{align*}"}
+{"id": "1408.png", "formula": "\\begin{gather*} \\varphi _ { k , 0 } ( x ) = \\cos k x - \\varkappa _ k x \\sin k x + \\delta _ x ( k + \\varkappa _ k ) + \\theta _ k ( x ) , \\varphi _ { k , 1 } ( x ) = \\cos k x + \\delta _ x ( k ) , \\\\ \\tilde \\varphi _ { k , 0 } ( x ) = \\cos k x - \\varkappa _ k x \\sin k x + \\theta _ k ( x ) , \\tilde \\varphi _ { k , 1 } ( x ) = \\cos k x . \\end{gather*}"}
+{"id": "6612.png", "formula": "\\begin{align*} w ^ { ( \\widetilde { \\rm c J } ) } ( \\theta ) = e ^ { q ( \\theta - \\pi ) } | 1 - e ^ { i \\theta } | ^ { \\beta p } . \\end{align*}"}
+{"id": "5314.png", "formula": "\\begin{align*} P ( x ) & = 2 x \\prod _ { k = 1 } ^ { n - 1 } ( x + \\xi ^ k ) + ( - x + \\xi ^ 0 ) \\prod _ { k = 1 } ^ { n - 1 } ( x + \\xi ^ k ) \\\\ & \\geq 2 x \\prod _ { k = 1 } ^ { n - 1 } \\xi ^ k + ( - x + \\xi ^ 0 ) \\prod _ { k = 1 } ^ { n - 1 } | - x + \\xi ^ k | \\\\ & \\geq 2 \\xi ^ n \\prod _ { k = 1 } ^ { n - 1 } \\xi ^ k + \\prod _ { k = 0 } ^ { n - 1 } | - x + \\xi ^ k | \\\\ & = 2 \\xi ^ { n + 1 \\choose 2 } + | P ( - x ) | \\\\ \\Rightarrow P ( x ) + P ( - x ) & \\geq P ( x ) - | P ( - x ) | \\\\ & \\geq 2 \\xi ^ { n + 1 \\choose 2 } . \\end{align*}"}
+{"id": "2603.png", "formula": "\\begin{align*} T _ p ( \\alpha ) = \\tilde { \\alpha } = ( \\tilde { \\alpha } _ n ) _ { n \\in \\N _ 0 } \\alpha \\in \\mathcal { A } , \\end{align*}"}
+{"id": "7154.png", "formula": "\\begin{align*} H _ { 1 } = \\sum _ { i _ 1 , i _ 2 , i _ 3 , i _ 4 } k _ { i _ 1 , i _ 2 , i _ 3 , i _ 4 } q _ 1 ^ { i _ 1 } p _ 1 ^ { i _ 2 } q _ 2 ^ { i _ 3 } p _ 2 ^ { i _ 4 } , \\end{align*}"}
+{"id": "4606.png", "formula": "\\begin{align*} \\phi _ n = 2 \\ ; ( 0 , 1 ) \\times [ 2 ^ { - n } , 1 ) , \\quad \\phi _ n = 1 \\ ; ( 0 , 1 ) \\times ( 0 , 2 ^ { - ( n + 1 ) } ] . \\end{align*}"}
+{"id": "1823.png", "formula": "\\begin{gather*} F ( z ) = \\int _ { \\mathbb { C } } e ^ { z s } \\ , d \\mu ( s ) \\end{gather*}"}
+{"id": "2629.png", "formula": "\\begin{align*} L _ n ( x , t ) = B _ n ( F ( x ) , \\frac { \\hat A _ n ( t ) } { n } ) \\ , . \\end{align*}"}
+{"id": "1777.png", "formula": "\\begin{gather*} y = ( n _ 1 + \\alpha ) ^ { m _ 1 } \\cdots ( n _ k + \\alpha ) ^ { m _ k } . \\end{gather*}"}
+{"id": "5719.png", "formula": "\\begin{align*} R _ M ( j , \\xi , x ) & : = \\phi _ j ( \\xi , x ) - e ^ { 2 i \\pi \\ , j \\ , x } \\left ( 1 + \\sum _ { \\ell = 1 } ^ { M } b _ \\ell ( j , \\xi ) \\ , r _ \\ell ( x ) \\right ) \\ , , \\\\ \\tilde { R } _ M ( j , \\xi , x ) & : = \\tilde { \\phi } _ j ( \\xi , x ) - e ^ { 2 i \\pi \\ , j \\ , x } \\left ( 1 + \\sum _ { \\ell = 1 } ^ { M } \\tilde { b } _ \\ell ( j , \\xi ) \\ , \\tilde { r } _ \\ell ( x ) \\right ) \\ , , \\end{align*}"}
+{"id": "6743.png", "formula": "\\begin{align*} t \\left ( t + \\frac { m } { 2 } \\right ) \\left ( t - \\frac { m } { 2 } \\right ) \\tilde { \\alpha } '' ( t ) + \\left ( a ( \\frac { m } { 2 } ) ^ 2 + 2 ( 1 - a ) \\frac { m } { 2 } t + ( a + 2 ) t ^ 2 \\right ) \\tilde { \\alpha } ' ( t ) = 0 . \\end{align*}"}
+{"id": "8515.png", "formula": "\\begin{align*} B _ { \\rho } ( ( \\bar { z } , w ) ) \\cap \\{ z = \\zeta \\} \\subset B ^ { n - 1 } ( w , \\rho ) \\mbox { f o r e v e r y } \\zeta \\in ( \\bar { z } , \\bar { z } + \\rho ) . \\end{align*}"}
+{"id": "483.png", "formula": "\\begin{align*} & I _ \\zeta ^ { ( \\alpha ) } ( r , s ) : = \\int _ 0 ^ \\infty \\frac { \\Gamma ( \\tfrac \\alpha 2 + 1 ) \\sin ( \\tfrac { \\pi \\alpha } { 2 } ) } { \\pi \\ , \\tau ^ { 1 + \\alpha / 2 } } \\cdot \\frac { ( r s ) ^ { \\frac 1 2 - \\zeta } } { 2 \\tau } \\exp \\left ( - \\frac { r ^ 2 + s ^ 2 } { 4 \\tau } \\right ) I _ { \\zeta - \\frac 1 2 } \\left ( \\frac { r s } { 2 \\tau } \\right ) \\ , d \\tau \\end{align*}"}
+{"id": "2602.png", "formula": "\\begin{align*} H _ m ( \\alpha ) : = \\begin{bmatrix} \\alpha _ { 0 } & \\alpha _ { 1 } & \\cdots & \\alpha _ { m } \\\\ \\alpha _ { 1 } & \\alpha _ { 2 } & \\cdots & \\alpha _ { m + 1 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\alpha _ { m } & \\alpha _ { m + 1 } & \\cdots & \\alpha _ { 2 m } \\\\ \\end{bmatrix} . \\end{align*}"}
+{"id": "8456.png", "formula": "\\begin{align*} \\sum _ { \\substack { w \\\\ A , B \\\\ j A s , i B s } } \\mathrm { a d } _ { w } = \\binom { i + j } { i , j } \\tau _ { i , j } + \\binom { i + j - 1 } { i - 1 , j } \\mathrm { a d } _ { \\sigma _ { i - 1 , j - 1 } } . \\end{align*}"}
+{"id": "4514.png", "formula": "\\begin{align*} | | { \\bf v } [ { \\bf u } ^ a + { \\bf u } _ { i } ] | | _ { s , \\ast , T } \\le C \\delta ^ 2 \\begin{cases} \\theta _ i ^ { ( s + 2 - \\alpha ) _ + } & s + 2 \\not = \\alpha , \\\\ \\theta _ i & s + 2 = \\alpha , \\end{cases} s \\in \\{ 6 , \\dots , \\tilde { \\alpha } - 2 \\} , \\end{align*}"}
+{"id": "5252.png", "formula": "\\begin{align*} z _ { i , t } = \\sum _ { j = 1 } ^ n [ W _ { t - 1 } ] _ { i j } z _ { j , t - 1 } + \\epsilon ^ z _ { i , t - 1 } . \\end{align*}"}
+{"id": "7146.png", "formula": "\\begin{align*} a = h ^ { - M } = \\begin{cases} h ^ { - \\frac { \\sigma } { k } - T \\eta \\lambda } = h ^ { - \\frac { \\sigma } { k } } ( \\log ( h ^ { - 1 } ) ) ^ T \\ge h ^ { - \\frac { 1 } { 3 } } & 0 < \\delta \\le 1 , \\\\ h ^ { - 1 - T \\eta \\lambda - t \\eta } = e ^ { t } h ^ { - 1 } ( \\log ( h ^ { - 1 } ) ) ^ T \\ge h ^ { - 1 } & \\delta = 0 . \\end{cases} \\end{align*}"}
+{"id": "5526.png", "formula": "\\begin{align*} e _ j ( z ) : = z ^ { \\alpha ( j ) } = z _ 1 ^ { \\alpha _ 1 ( j ) } \\cdots z _ 2 ^ { \\alpha _ n ( j ) } , j = 1 , 2 , \\dots . \\end{align*}"}
+{"id": "2000.png", "formula": "\\begin{align*} [ \\dot { \\mathrm { B } } ^ { s _ 0 } _ { p _ 0 , q _ 0 } ( \\mathbb { R } ^ n ) , \\dot { \\mathrm { B } } ^ { s _ 1 } _ { p _ 1 , q _ 1 } ( \\mathbb { R } ^ n ) ] _ { \\theta } = \\dot { \\mathrm { B } } ^ { s } _ { p _ \\theta , q _ \\theta } ( \\mathbb { R } ^ n ) \\end{align*}"}
+{"id": "91.png", "formula": "\\begin{align*} \\alpha = \\mathcal B ^ { - 1 } \\big ( \\mathcal A - \\sqrt { \\mathcal A ^ 2 - \\mathcal B ^ 2 } \\big ) , c _ 0 = \\frac { 2 \\bar \\kappa } { \\mathcal A + \\mathcal B + \\sqrt { \\mathcal A ^ 2 - \\mathcal B ^ 2 } } . \\end{align*}"}
+{"id": "5498.png", "formula": "\\begin{align*} c _ { \\pm } ^ { ( h , g a ) } ( p , s ) = c _ { \\pm } ^ { ( h , g ) } ( p s , s ) , c _ { \\pm } ^ { ( h , g b ) } ( p , s ) = c _ { \\pm } ^ { ( h , g ) } ( p , s / p ) \\qquad \\textrm { f o r a l l } h , g \\in F r e e _ 2 . \\end{align*}"}
+{"id": "1557.png", "formula": "\\begin{align*} \\Delta = y ^ 3 x ^ 3 + z f _ 5 ( x , y , z ) \\end{align*}"}
+{"id": "686.png", "formula": "\\begin{align*} \\norm { \\mathbf f } _ { \\mathbf H ^ { \\mathbf s } ( \\R ^ d ) } = \\left ( \\norm { f _ 1 } _ { H ^ { s _ 1 } ( \\R ^ d ) } ^ 2 + \\dots + \\norm { f _ n } _ { H ^ { s _ n } ( \\R ^ d ) } ^ 2 \\right ) ^ { 1 / 2 } \\mathbf f = \\begin{pmatrix} f _ 1 \\\\ \\vdots \\\\ f _ n \\end{pmatrix} . \\end{align*}"}
+{"id": "7367.png", "formula": "\\begin{align*} \\gamma _ c = 1 + 3 \\sqrt [ 3 ] { p ^ 2 ( 1 - p ) } + 3 \\sqrt [ 3 ] { p ( 1 - p ) ^ 2 } . \\end{align*}"}
+{"id": "7511.png", "formula": "\\begin{align*} \\begin{aligned} A _ { 3 } = & \\frac { 1 } { ( q - 1 ) ^ 2 q ^ 4 } ( q ^ { m - 1 } - 1 ) ( q ^ 5 - q ^ m - q ^ { 3 + m } + q ^ { 4 + m } ) = \\\\ & \\frac { 1 } { ( q - 1 ) ^ 2 q ^ 4 } ( q ^ { m - 1 } - 1 ) ( q ^ 5 + q ^ m ( q ^ 4 - q ^ 3 - 1 ) ) > 0 \\end{aligned} \\end{align*}"}
+{"id": "4660.png", "formula": "\\begin{align*} \\mathbf { X } _ { v | s } = \\bigotimes _ { e \\in \\mathcal { S } ( v | s ) } X _ e , \\qquad \\mathbf { Y } _ { v | t } = \\bigotimes _ { e ^ \\prime \\in \\mathcal { S } ( v | t ) } Y _ { e ^ \\prime } . \\end{align*}"}
+{"id": "8172.png", "formula": "\\begin{align*} & [ A ^ * _ { 1 0 } , [ A ^ * _ { 1 0 } , [ A ^ * _ { 1 0 } , A _ { 1 0 } ] ] ] = \\left ( \\frac { n ( r - 1 ) } { k } \\right ) ^ 2 [ A ^ * _ { 1 0 } , A _ { 1 0 } ] , \\\\ & [ A _ { 1 0 } , [ A _ { 1 0 } , [ A _ { 1 0 } , A ^ * _ { 1 0 } ] ] ] = ( r - 1 ) ^ 2 [ A _ { 1 0 } , A ^ * _ { 1 0 } ] . \\end{align*}"}
+{"id": "1623.png", "formula": "\\begin{align*} & e _ 1 = ( d _ { t - k + 2 } , \\dots , d _ { t - k + i } , b _ t ~ ~ ~ ~ ~ ~ , d _ { t - k + i + 1 } , \\dots , d _ { t - k + j - 1 } , d _ t , d _ { t - k + j } , \\dots , d _ { t - 1 } ) , \\\\ & e _ 2 = ( d _ { t - k + 2 } , \\dots , d _ { t - k + i } , d _ { t - k + 1 } , d _ { t - k + i + 1 } , \\dots , d _ { t - k + j - 1 } , d _ t , d _ { t - k + j } , \\dots , d _ { t - 1 } ) \\end{align*}"}
+{"id": "1044.png", "formula": "\\begin{align*} u _ \\mu \\in [ 0 , \\mu _ \\lambda ] , \\ ; u _ \\mu \\not = 0 , \\ ; u _ \\mu \\not = u _ \\lambda \\mbox { ( s i n c e $ \\lambda < \\mu $ ) . } \\end{align*}"}
+{"id": "1939.png", "formula": "\\begin{align*} \\frac { { \\rm d } } { { \\rm d } t } \\vec { \\beta } ( t ) = \\mathcal { M } ( \\vec { \\beta } , t ) , \\end{align*}"}
+{"id": "5910.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } [ G _ 1 + G _ 2 ] = 0 . \\end{align*}"}
+{"id": "6688.png", "formula": "\\begin{align*} \\alpha _ 2 = & \\frac { \\alpha ( 1 + \\bar { \\alpha } - \\kappa ) ( - 2 + \\alpha - \\bar { \\alpha } + 2 \\kappa ) } { 2 \\kappa ^ 3 } , \\\\ \\alpha _ 3 = & \\frac { \\alpha ( 1 + \\bar { \\alpha } - \\kappa ) \\left ( 6 + \\alpha ^ 2 + \\bar { \\alpha } ^ 2 - 3 \\alpha ( 2 + \\bar { \\alpha } - 2 \\kappa ) - 5 \\bar { \\alpha } ( \\kappa - 1 ) - 1 1 \\kappa + 6 \\kappa ^ 2 \\right ) } { 6 \\kappa ^ 4 } , \\end{align*}"}
+{"id": "3135.png", "formula": "\\begin{align*} \\mathbb { E } [ | \\mathbf { H } _ { k , } \\mathbf { H } _ { l , } ^ H | ^ 2 ] = M \\frac { L _ k L _ l } { ( \\kappa + 1 ) ^ 2 } . \\end{align*}"}
+{"id": "2489.png", "formula": "\\begin{align*} \\overline { H } ( G ) = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } H _ \\chi \\Big ( G ^ { \\wedge n } [ \\mathcal { S } ^ n _ \\epsilon ] \\Big ) , \\end{align*}"}
+{"id": "5174.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } { \\rm A o I } ( S _ { \\mathrm { z } } , Q _ { { \\mathrm { u n i } } } , F ^ * ) - \\frac { 3 } { 2 } H [ Q _ { \\mathrm { u n i } } ( X ) ] = 0 \\end{align*}"}
+{"id": "2360.png", "formula": "\\begin{align*} S _ { f , \\tau } : \\mathbb { C } ^ { o ( G ) } \\ni x \\mapsto S _ { f , \\tau } x \\coloneqq \\sum _ { j = 1 } ^ { n } f _ j ( x ) \\tau _ j \\in \\mathbb { C } ^ { o ( G ) } \\end{align*}"}
+{"id": "4753.png", "formula": "\\begin{align*} \\Omega _ 3 = \\bigcup _ { k = 0 } ^ \\infty \\left ( \\left ( 2 ^ { - 4 k - 3 } T + ( 2 ^ { - 2 k - 1 } , 0 ) \\right ) \\cup \\left ( 2 ^ { - 4 k - 3 } T + ( 2 ^ { - 2 k } - 2 ^ { - 4 k - 3 } , 0 ) \\right ) \\right ) \\cup J . \\end{align*}"}
+{"id": "1800.png", "formula": "\\begin{gather*} Z _ N ( s , \\alpha ) = \\frac { 1 } { 2 \\pi i } \\int _ { c - i \\infty } ^ { c + i \\infty } \\zeta ( s + w , \\alpha ) \\widehat { \\phi } ( w ) N ^ w \\ , d w \\end{gather*}"}
+{"id": "8095.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\left \\| \\sum \\limits _ { j = N } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } = 0 , \\ \\lim _ { N \\to \\infty } \\left \\| \\sum \\limits _ { j = N } ^ { \\infty } \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } = 0 . \\end{align*}"}
+{"id": "1311.png", "formula": "\\begin{align*} \\begin{cases} \\ll _ 1 = 1 , \\\\ \\ll _ k = 1 , \\ , \\Lambda _ k = \\infty , 2 \\le k \\le p - 1 , \\end{cases} \\end{align*}"}
+{"id": "5274.png", "formula": "\\begin{align*} \\epsilon _ s \\sum _ { t = 1 } ^ T \\sum _ { i = 1 } ^ n \\| [ g _ { i , t } ( x _ { i , t } ) ] _ + \\| \\le \\tilde { h } _ T ( x _ s ) . \\end{align*}"}
+{"id": "4020.png", "formula": "\\begin{align*} B ^ s _ { p , q } ( \\mathcal O ) & : = \\{ f | _ { \\mathcal O } : \\ , f \\in B ^ s _ { p , q } ( \\R ^ m ) \\} , \\\\ \\| g \\| _ { B ^ s _ { p , q } ( \\mathcal O ) } & : = \\inf \\{ \\| f \\| _ { B ^ s _ { p , q } ( \\R ^ m ) } : \\ , f | _ { \\mathcal O } = g \\} . \\end{align*}"}
+{"id": "7677.png", "formula": "\\begin{align*} 0 \\leqslant \\varphi ( 0 ) \\leqslant \\frac { \\varepsilon } { h ^ * _ 0 } = \\frac { \\varepsilon } { h _ 0 + \\varepsilon / l } , \\ \\ 0 \\leqslant \\varphi ( 1 ) \\leqslant \\frac { \\varepsilon ( 1 - h _ 1 ) } { ( h _ 0 + \\varepsilon / l ) ^ 2 } , \\end{align*}"}
+{"id": "1894.png", "formula": "\\begin{align*} \\rho _ h = \\sum _ { j = 1 } ^ { N _ v } \\int _ { J _ j } f _ h \\ , { \\rm d } v \\mbox { a n d } \\rho _ h V _ h = \\sum _ { j = 1 } ^ { N _ v } \\int _ { J _ j } v f _ h \\ , { \\rm d } v , \\end{align*}"}
+{"id": "2196.png", "formula": "\\begin{align*} E ( A , B ) = O \\left ( \\frac { D _ L \\cdot \\# A \\cdot \\# B } { \\# R } \\right ) . \\end{align*}"}
+{"id": "8520.png", "formula": "\\begin{align*} E = : E _ { 1 } \\cup ( ( 0 , \\tau ) + E _ { 2 } ) \\in \\mathcal { K } ( \\ell ) \\end{align*}"}
+{"id": "4052.png", "formula": "\\begin{align*} \\rho _ { 0 , 0 , n } = n \\pi + \\frac { q _ 1 ( 0 ) + ( - 1 ) ^ { n + 1 } q _ 1 ( 1 ) } { n \\pi } \\sin n \\pi a + \\frac { \\kappa _ { 0 , 0 , n } } { n } , \\{ \\kappa _ { 0 , 0 , n } \\} \\in l _ 2 , n \\in \\mathbb { Z } \\ ! \\setminus \\ ! \\{ 0 \\} ; \\end{align*}"}
+{"id": "3907.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { C } { \\sigma _ N ^ 2 } = 0 . \\end{align*}"}
+{"id": "6762.png", "formula": "\\begin{align*} Q = \\left [ \\begin{array} { c c } 1 & 1 \\\\ 2 & 1 \\end{array} \\right ] , \\end{align*}"}
+{"id": "8028.png", "formula": "\\begin{align*} \\| { \\mathcal { F } _ { \\alpha } } ( f ) \\| _ { \\alpha , 2 } = \\| f \\| _ { \\alpha , 2 } . \\end{align*}"}
+{"id": "1180.png", "formula": "\\begin{align*} & m ' _ { 1 , t } = m _ { 1 , t } + m _ { 1 , n + 1 } t ^ { n + 1 } , \\\\ & m ' _ { 2 , t } = m _ { 2 , t } + m _ { 2 , n + 1 } t ^ { n + 1 } . \\end{align*}"}
+{"id": "3216.png", "formula": "\\begin{align*} \\underbrace { L ^ { - 1 } A ^ T } _ { { \\hat { A } } ^ T } \\underbrace { A L ^ { - T } } _ { \\hat { A } } \\underbrace { L ^ T x } _ { \\phantom { \\hat { I } } \\hat { x } \\phantom { \\hat { I } } } = \\underbrace { L ^ { - 1 } A ^ T } _ { { \\hat { A } } ^ T } b . \\end{align*}"}
+{"id": "3838.png", "formula": "\\begin{align*} \\mathrm { U O T } ( \\mu _ 0 , \\mu _ 1 ) : = \\inf _ { ( \\rho , v , r ) \\in { \\rm A d m } ( \\mu _ 0 , \\mu _ 1 ) } \\left \\{ \\int \\int _ 0 ^ 1 ( | v | ^ 2 + | r | ^ 2 ) \\rho \\ , d x d t \\right \\} \\end{align*}"}
+{"id": "2002.png", "formula": "\\begin{align*} s : = ( 1 - \\theta ) s _ 0 + \\theta s _ 1 \\end{align*}"}
+{"id": "1114.png", "formula": "\\begin{align*} \\sup \\Big \\{ \\big | \\langle \\ , \\mathcal J ' _ + ( u _ k ) , v \\ , \\rangle \\big | \\ , : \\ ; v \\in W ^ { 1 , 2 } _ 0 ( D ) \\ , , \\| v \\| = 1 \\Big \\} \\to 0 , \\end{align*}"}
+{"id": "77.png", "formula": "\\begin{align*} \\sum _ { k \\in \\mathcal { P } _ H } \\mathcal T _ { \\rm { c o m } } ( k ) = - \\frac { 1 } { \\vert \\Lambda \\vert ^ 2 } \\sum _ { k \\in \\mathcal { P } _ H , p \\in \\mathcal P _ L } \\frac { \\vert z \\vert ^ 2 \\widehat g ( k ) ^ 2 } { ( 1 - \\varepsilon _ K ) \\mathcal D _ k } a _ p ^ \\dagger a _ p + \\mathcal E , \\end{align*}"}
+{"id": "3937.png", "formula": "\\begin{align*} \\lambda ( f ^ { - 1 } ( \\omega ) \\triangle g ^ { - 1 } ( \\omega ) ) = 0 , \\end{align*}"}
+{"id": "2584.png", "formula": "\\begin{align*} a \\cdot b & = \\sum _ { h \\in S } a _ { h } t ^ { h } \\cdot \\sum _ { h \\in S ' } b _ { h } t ^ { h } = \\sum _ { l \\in S + S ' } \\left ( \\sum _ { \\substack { h \\in S \\\\ h ' \\in S ' \\\\ h + h ' = l } } a _ { h } b _ { h ' } p ^ { - \\beta ( h , h ' ) } \\right ) t ^ { l } = \\\\ & = \\sum _ { l \\in S + S ' } \\left ( \\sum _ { \\substack { h \\in S \\\\ h ' \\in S ' \\\\ h + h ' = l } } b _ { h ' } a _ { h } p ^ { - \\beta ( h ' , h ) } \\right ) t ^ { l } = \\\\ & = \\sum _ { h \\in S ' } b _ { h } t ^ { h } \\cdot \\sum _ { h \\in S } a _ { h } t ^ { h } = b \\cdot a . \\end{align*}"}
+{"id": "3159.png", "formula": "\\begin{align*} W _ L = ( z _ 0 + \\dotsi + z _ { d - 1 } ) ^ d + z _ d + \\dotsi + z _ n . \\end{align*}"}
+{"id": "8449.png", "formula": "\\begin{align*} \\exp \\left ( - f - \\sum _ { k = 2 } ^ { \\infty } \\frac { ( - 1 ) ^ k P _ { k } \\mathrm { a d } _ { B } ^ { k } } { k } \\right ) & = \\exp \\left ( - \\sum _ { k = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ k \\mathrm { a d } _ { B } ^ { k } } { k z ^ k } + H ( z ) \\right ) = \\left ( \\frac { \\mathrm { a d } _ { B } } { z } + 1 \\right ) \\exp ( H ( z ) ) , \\end{align*}"}
+{"id": "2086.png", "formula": "\\begin{align*} E [ \\Lambda , \\phi ; \\alpha ] ( t ) : = - \\int [ \\kappa \\partial _ t \\alpha ( h _ 1 - 2 h _ 2 ) ] ( t , x ) d x , \\end{align*}"}
+{"id": "5280.png", "formula": "\\begin{align*} & \\sum _ { t = 1 } ^ T \\alpha _ { t } = \\sum _ { t = 1 } ^ T \\frac { 1 } { \\mu t } = \\sum _ { t = 2 } ^ { T } \\frac { 1 } { \\mu t } + \\frac { 1 } { \\mu } \\le \\int _ { 1 } ^ { T } \\frac { 1 } { \\mu t } d t + \\frac { 1 } { \\mu } \\le \\frac { 1 } { \\mu } ( \\log ( T ) + 1 ) . \\end{align*}"}
+{"id": "3731.png", "formula": "\\begin{align*} \\mathcal { D } ( \\mathbb { A } ) = \\mathcal { H } ^ 1 \\times \\mathcal { H } ^ { \\frac { 2 } { 3 } } \\times \\mathcal { H } ^ { \\frac { 1 } { 3 } } . \\end{align*}"}
+{"id": "8024.png", "formula": "\\begin{align*} \\frac { \\partial f _ i } { \\partial t } = - E _ i f _ i + G _ i \\sum _ { j = 0 } ^ i f _ j + G _ i \\sum _ { j = i + 1 } ^ D f _ j , \\ \\ i = 1 , \\dots D . \\end{align*}"}
+{"id": "5581.png", "formula": "\\begin{align*} | B _ t ( v ) \\setminus B _ { t - 1 } ( v ) | = d ( d - 1 ) ^ { t - 1 } = ( 1 - o ( 1 ) ) d ^ t \\ll n ^ { 1 - \\alpha } . \\end{align*}"}
+{"id": "2923.png", "formula": "\\begin{align*} L _ { i j k l } = \\frac { 3 } { 2 } ( \\delta _ { i j } \\delta _ { k l } + \\delta _ { i k } \\delta _ { j l } ) - \\delta _ { i l } \\delta _ { j k } \\end{align*}"}
+{"id": "8669.png", "formula": "\\begin{align*} \\int G ( x _ 0 , . . . , x _ { n + 1 } ) e ^ { \\frac { 1 } { \\hbar } ( x _ 0 y _ 0 + . . . + x _ { n + 1 } y _ { n + 1 } ) } d x _ 0 . . . d x _ { n + 1 } = ( 2 \\pi \\hbar ) ^ { \\frac { n + 2 } { 2 } } \\hat { G } ( y _ 0 , . . . , y _ { n + 1 } ) \\end{align*}"}
+{"id": "1651.png", "formula": "\\begin{align*} J _ Z ^ 2 ( V ) = - \\Vert Z \\Vert ^ 2 \\ , V . \\end{align*}"}
+{"id": "572.png", "formula": "\\begin{align*} \\mathbf V ( t ) = \\mathbf u ( t ) , \\end{align*}"}
+{"id": "4605.png", "formula": "\\begin{align*} Z _ { i , j } ( A ) = \\mathrm { d i m } _ { \\mathbb { C } } ( \\mathrm { H o m } _ { A \\mathrm { m o d } A } ( i \\ , { \\otimes ^ { - } } A \\ , { \\otimes ^ { + } } j , A ) ) \\end{align*}"}
+{"id": "6240.png", "formula": "\\begin{align*} S ' = L ^ 1 _ t L ^ 2 _ x + L ^ { \\frac 4 3 } _ t L ^ 1 _ x . \\end{align*}"}
+{"id": "8083.png", "formula": "\\begin{align*} \\left \\| \\uppercase \\expandafter { \\romannumeral 1 } _ { 1 } \\right \\| _ { L _ { \\omega } ^ { p } } = \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\mu _ { j } h _ { j } ( x ) \\right \\| _ { L _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\mu _ { j } \\chi _ { 4 P _ { j } } } { \\omega ( P _ { j } ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } . \\end{align*}"}
+{"id": "4379.png", "formula": "\\begin{align*} \\mathbf { h } ^ { \\pm } = ( H ^ { \\pm } _ n , H ^ { \\pm } _ 2 \\partial _ 1 \\Phi ^ { \\pm } ) , H ^ { \\pm } _ n = H ^ { \\pm } _ 1 - H ^ { \\pm } _ 2 \\partial _ 2 \\Psi ^ { \\pm } , H ^ { \\pm } _ N | _ { x _ 1 = 0 } = H ^ { \\pm } _ n | _ { x _ 1 = 0 } . \\end{align*}"}
+{"id": "6046.png", "formula": "\\begin{align*} \\overline { \\mu } \\{ \\underline { c } > \\frac { 1 } { u } \\} = \\overline { \\mu } \\{ \\vert z \\vert < ( \\frac { \\ln u } { a _ 2 } ) ^ { \\frac { 1 } { p } } \\} \\geq \\frac { r _ d } { 2 } ( \\frac { \\ln ( u - 1 ) } { a _ 2 } ) ^ { \\frac { d } { p } } , \\end{align*}"}
+{"id": "4676.png", "formula": "\\begin{align*} e ( \\widetilde { S _ i } , \\widetilde { T _ i } ) & \\geq | \\widetilde { S _ i } | \\cdot ( \\frac { n } { 2 ( 2 l + 1 ) } - 1 ) - ( 4 k - 2 ) k n \\\\ & = | \\widetilde { S _ i } | \\cdot \\frac { 2 n } { 4 ( 2 l + 1 ) + 1 } + | \\widetilde { S _ i } | \\cdot ( \\frac { n } { 3 2 l ^ 2 + 3 6 l + 1 0 } - 1 ) - ( 4 k - 2 ) k n \\\\ & > | \\widetilde { S _ i } | \\cdot \\frac { 2 n } { 4 ( 2 l + 1 ) + 1 } \\end{align*}"}
+{"id": "3489.png", "formula": "\\begin{align*} & \\int _ { | \\frac { - \\log | z _ i | } { | \\log | t | | } - y _ i | \\lesssim \\epsilon } | \\frac { 1 } { | \\log | t | | } \\varphi - u ( y ) | d \\mu _ t \\\\ & \\lesssim \\epsilon \\int _ { | \\frac { - \\log | z _ i | } { | \\log | t | | } - y _ i | \\lesssim \\epsilon } d \\mu _ t + \\epsilon ^ { m + 1 } \\norm { \\psi _ t } _ { L ^ \\infty } \\lesssim \\epsilon ^ { m + 1 } . \\end{align*}"}
+{"id": "357.png", "formula": "\\begin{align*} \\psi _ H \\ ; R = B \\ ; \\psi _ H . \\end{align*}"}
+{"id": "2316.png", "formula": "\\begin{align*} B : = \\{ ( u , v ) \\in X ^ { s , \\frac 1 2 + \\theta } _ { \\lambda } \\times Y ^ { k } _ { \\lambda , \\theta } : \\| u \\| _ { X ^ { s , \\frac 1 2 + \\theta } _ { \\lambda } } \\leq M \\| v \\| _ { Y ^ { k } _ { \\lambda , \\theta } } \\leq \\epsilon _ 0 \\} \\end{align*}"}
+{"id": "8521.png", "formula": "\\begin{align*} r _ { \\ell } ^ { \\vee } ( \\bar { z } ) = \\mbox { a p l i m } ( r _ { \\ell } , ( - \\infty , \\bar { z } ) , \\bar { z } ) \\mbox { a n d } r _ { \\ell } ^ { \\wedge } ( \\bar { z } ) = \\mbox { a p l i m } ( r _ { \\ell } , ( \\bar { z } , + \\infty ) , \\bar { z } ) . \\end{align*}"}
+{"id": "7827.png", "formula": "\\begin{align*} M ^ { } = \\{ ( Z ^ 0 , Z ^ 1 ) = Z ^ 0 ( 1 , z ^ 1 ) \\in \\mathbb { C } ^ { 2 } \\ ; | \\ ; Z ^ 0 \\in \\mathbb { C } ^ { \\times } , \\ ; z ^ 1 \\in \\overline { M } ^ { } \\} , \\mathfrak { F } ^ { } = - \\frac { 1 } { 6 } \\frac { ( Z ^ 1 ) ^ 3 } { Z ^ 0 } \\ , . \\end{align*}"}
+{"id": "2594.png", "formula": "\\begin{align*} d ^ h _ 0 ( p , g _ 1 , \\ldots , g _ m ) & = ( g _ 1 ^ { - 1 } p , g _ 2 , \\ldots , g _ m ) \\\\ d ^ h _ i ( p , g _ 1 , \\ldots , g _ m ) & = ( p , g _ 1 , \\ldots , g _ i \\cdot g _ { i + 1 } , \\ldots , g _ n ) \\\\ s ^ h _ 0 ( p , g _ 1 , \\ldots , g _ m ) & = ( p , \\sigma f ( g _ 1 ) , g _ 1 , \\ldots , g _ n ) \\\\ s ^ h _ i ( p , g _ 1 , \\ldots , g _ m ) & = ( p , g _ 1 , \\ldots , g _ i , \\sigma f ( g _ i ) , g _ { i + 1 } , \\ldots , g _ n ) . \\end{align*}"}
+{"id": "1350.png", "formula": "\\begin{align*} h ^ 0 ( \\mathcal { Y } _ s , g _ s ^ * ( m \\epsilon \\mathcal { H } _ s + k f _ s ^ * \\mathcal { O } ( 1 ) ) ) = \\chi ( \\mathcal { X } _ s , m \\epsilon \\mathcal { H } _ s + k f _ s ^ * \\mathcal { O } ( 1 ) ) \\end{align*}"}
+{"id": "3919.png", "formula": "\\begin{align*} \\omega ^ * & = \\bigcap _ { u \\in \\omega } \\{ v \\in S ^ { n - 1 } : u \\cdot v \\leq 0 \\} . \\end{align*}"}
+{"id": "2534.png", "formula": "\\begin{align*} R _ P = \\{ ( \\alpha _ i , r _ i ) : i \\in S _ M \\cap P \\} , C _ P = \\{ ( \\beta _ i , r _ i ) : i \\in S _ N \\cap P \\} . \\end{align*}"}
+{"id": "8514.png", "formula": "\\begin{align*} & ( ( 0 , \\tau ) + E _ { 2 } ) \\cap B _ { \\bar { \\rho } } ( ( \\bar { z } , w ) ) \\\\ & = \\left ( ( 0 , \\tau ) + ( F _ { \\ell } \\cap \\{ z \\geq \\bar { z } \\} ) \\right ) \\cap B _ { \\bar { \\rho } } ( ( \\bar { z } , w ) ) \\\\ & \\subset \\left \\{ ( z ' , w ' ) \\in \\mathbb { R } \\times \\mathbb { R } ^ { n - 1 } : z ' \\geq \\bar { z } , \\ \\ \\frac { \\epsilon } { 2 } < | w ' - \\tau | < r _ { \\ell } ( z ' ) \\right \\} \\cap B _ { \\bar { \\rho } } ( ( \\bar { z } , w ) ) . \\end{align*}"}
+{"id": "5914.png", "formula": "\\begin{align*} F J _ { 0 , 1 } ( M ) \\cong F J _ { 0 , 1 } ( N _ { j } ^ { n } ( \\Delta \\pi ) ) = F N _ { j } ^ { n } ( \\Delta \\pi ) \\end{align*}"}
+{"id": "2587.png", "formula": "\\begin{align*} O _ { v a l } & = O _ { w _ { 1 } } \\cap K ( t ^ { H } , \\beta ) , \\\\ O _ { v a l } & = O _ { w _ { 2 } } \\cap K ( t ^ { H } , \\beta ) \\end{align*}"}
+{"id": "2194.png", "formula": "\\begin{align*} \\C ( x \\mid y ) : = \\C _ U ( x \\mid y ) . \\end{align*}"}
+{"id": "766.png", "formula": "\\begin{align*} \\kappa ( t ) - \\kappa ( r ) = \\theta _ R ( \\norm { \\mathbf U } _ { t } ^ 2 ) - \\theta _ R ( \\norm { \\mathbf V } _ { t } ^ 2 ) - \\left [ \\theta _ R ( \\norm { \\mathbf U } _ { r } ^ 2 ) - \\theta _ R ( \\norm { \\mathbf V } _ { r } ^ 2 ) \\right ] \\end{align*}"}
+{"id": "3990.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( \\chi + t \\chi + \\tilde \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi _ t \\right ) ^ n = c \\left ( \\chi + t \\chi + \\tilde \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi _ t \\right ) ^ m \\wedge \\omega ^ { n - m } + b _ t f \\omega ^ n , \\end{aligned} \\end{align*}"}
+{"id": "8431.png", "formula": "\\begin{align*} f _ ( X ) = - \\log \\det X + \\sum _ { j = 1 } ^ q \\alpha _ j \\log \\det ( A _ j X A _ j ^ T ) \\end{align*}"}
+{"id": "2088.png", "formula": "\\begin{align*} \\begin{aligned} & \\alpha ( t , x ) > 0 , \\partial _ t \\alpha ( t , x ) > 0 , \\\\ & ( \\partial _ t \\alpha ) ^ 2 ( t , x ) - ( \\partial _ x \\alpha ) ^ 2 ( t , x ) < 0 , \\forall ( t , x ) \\in [ 0 , \\infty ) \\times \\mathbb R . \\end{aligned} \\end{align*}"}
+{"id": "3129.png", "formula": "\\begin{align*} | \\mathbf { H } _ { k } \\cdot \\mathbf { H } _ l ^ H | ^ 2 = \\mathbf { H } _ k \\mathbf { H } _ l ^ H \\mathbf { H } _ l \\mathbf { H } _ k ^ H \\end{align*}"}
+{"id": "7711.png", "formula": "\\begin{align*} \\psi ^ \\sharp \\overline { x } ^ i = \\psi ^ i ( x ) = x ^ i , \\ ; \\ ; \\psi ^ \\sharp \\overline { e } ^ \\mu = e ^ \\mu , \\ ; \\ ; \\psi ^ \\sharp \\overline { p } ^ I = p ^ I . \\end{align*}"}
+{"id": "6098.png", "formula": "\\begin{align*} \\Delta _ { k + 1 } = E ( f ( \\omega _ { k + 1 } ) ) - f ( \\omega ^ * ) , \\xi = 1 - 2 \\mu \\sigma _ 1 \\alpha _ 1 , \\zeta = \\frac { 2 L \\alpha _ 2 \\sigma _ 1 \\sigma _ 2 } { 1 - \\sigma _ 2 } \\end{align*}"}
+{"id": "5267.png", "formula": "\\begin{align*} & \\frac { 1 } { n } \\sum _ { t = 1 } ^ T \\sum _ { j = 1 } ^ n \\| [ g _ { t } ( x _ { j , t } ) ] _ + \\| ^ 2 \\le \\varepsilon _ 3 + \\sum _ { t = 1 } ^ T \\sum _ { i = 1 } ^ n 2 ( \\| [ g _ { i , t } ( x _ { i , t } ) ] _ + \\| ^ 2 + G _ 2 ^ 2 \\tilde { \\varepsilon } _ 4 \\| \\epsilon ^ z _ { i , t } \\| ^ 2 ) . \\end{align*}"}
+{"id": "3855.png", "formula": "\\begin{align*} e ^ { - \\lambda _ 1 z } = \\sigma H , \\end{align*}"}
+{"id": "8621.png", "formula": "\\begin{align*} \\dfrac { 1 } { \\mu } | \\mathcal { G } ^ { \\mu } \\psi - \\mathcal { G } _ 0 \\psi | _ { H ^ { s } } & = | \\nabla _ X \\cdot ( h ( \\overline V - \\overline { V } _ 0 ) ) | _ { H ^ s } \\\\ & \\leq M ( s + 3 ) | \\nabla _ X \\psi | _ { H ^ { s + 5 } } . \\end{align*}"}
+{"id": "81.png", "formula": "\\begin{align*} \\mathcal Q _ 3 ^ { ( 3 ) } = - \\frac { 1 } { \\vert \\Lambda \\vert } \\sum _ { k \\in \\mathcal { P } _ H , p \\in \\mathcal P _ L } \\frac { \\widehat g ( k ) \\alpha _ { p - k } } { \\sqrt { 1 - \\alpha _ k ^ 2 \\vphantom { \\alpha _ { p - k } ^ 2 } } \\sqrt { 1 - \\alpha _ { p - k } ^ 2 } } \\big ( \\bar z b ^ \\dagger _ { k - p } a ^ \\dagger _ p b _ { k } + z b _ { k } ^ \\dagger a _ p b _ { k - p } \\big ) , \\end{align*}"}
+{"id": "2548.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } V | u | ^ p d x & = \\int _ { \\mathbb { R } ^ N } ( ( V - t ) _ { + } + t ) | u | ^ p d x - \\int _ { \\mathbb { R } ^ N } ( V - t ) _ { - } | u | ^ p d x \\\\ & \\geq \\int _ { \\mathbb { R } ^ N } ( ( V - t ) _ { + } + t ) | u | ^ p d x - \\| ( V - t ) _ { - } \\| _ { N / ( p s ) } \\| u \\| _ { p _ s ^ { * } } ^ p \\\\ & \\geq \\int _ { \\mathbb { R } ^ N } ( ( V - t ) _ { + } + t ) | u | ^ p d x - S ^ { - 1 } _ { D ^ { s , p } } \\| ( V - t ) _ { - } \\| _ { N / ( p s ) } \\| u \\| _ { D ^ { s , p } } ^ p . \\end{align*}"}
+{"id": "1949.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { T } & : X \\rightarrow X \\\\ u ^ * & \\longmapsto u = \\mathcal { T } ( u ^ * ) \\end{aligned} \\end{align*}"}
+{"id": "789.png", "formula": "\\begin{align*} o ( 1 ) \\cdot \\| u _ n \\| & = I ' [ u _ n ] u _ n \\\\ & = [ u _ n ] _ { s , p } ^ p + a \\| u _ n \\| _ p ^ p \\\\ & \\phantom { = } - b \\int _ { \\mathbb { R } ^ N } ( K \\ast F ( u _ n ) ) f ( u _ n ) u _ n d x - \\varepsilon _ g \\| u _ n \\| _ { p _ g } ^ { p _ g } , \\end{align*}"}
+{"id": "3040.png", "formula": "\\begin{align*} a ^ 2 = 2 b ^ 2 . \\end{align*}"}
+{"id": "6719.png", "formula": "\\begin{align*} \\begin{aligned} & 4 \\pi - \\frac { ( 1 + k ) ^ { 2 } } { ( 1 - k ) ^ { 2 } } ( \\eta ( r _ { 0 } ) ) ^ { 2 } \\int _ { \\Sigma } | \\nabla u | ^ 2 \\le 8 \\pi a \\left ( \\frac { \\mathfrak { m } _ { A D M } } { m } - 1 \\right ) ( 1 - a \\eta ( r _ { 0 } ) ) . \\end{aligned} \\end{align*}"}
+{"id": "8774.png", "formula": "\\begin{align*} b _ { t + 1 } < \\left ( 1 - \\frac { 1 } { t } \\right ) b _ { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { t ^ { p _ { i } + 1 } } , \\end{align*}"}
+{"id": "4872.png", "formula": "\\begin{align*} B = \\begin{bmatrix} q & 0 & 0 & 0 \\\\ t & 0 & p & 0 \\\\ t & p & 0 & 0 \\\\ \\frac { 2 t ^ 2 } { q - p } & - t & - t & q \\end{bmatrix} \\end{align*}"}
+{"id": "1122.png", "formula": "\\begin{align*} \\lambda _ { m , p } : = \\inf _ { u \\in C _ 0 ^ m ( D ) \\setminus \\{ 0 \\} } \\frac { \\displaystyle \\int _ { D } | \\nabla ^ m u | ^ p ( x ) d \\mu } { \\displaystyle \\int _ { D } | u ( x ) | ^ p d \\mu } . \\end{align*}"}
+{"id": "4716.png", "formula": "\\begin{align*} a _ { j _ 1 , l } - a _ { j 1 , l } ^ { * } & = 0 , \\ ; f o r \\ ; i _ 1 < l < j _ 1 \\\\ a _ { j _ 1 , i _ 1 } - a _ { j 1 , i _ 1 } ^ { * } & = 1 , \\\\ a _ { j _ 1 , l } - a _ { j 1 , l } ^ { * } & = b _ { i _ 1 , l } , \\ ; f o r \\ ; l < i _ 1 \\\\ \\gamma _ { j _ 1 } - \\gamma _ { j _ 1 } ^ { * } = \\varepsilon _ { i _ 1 } . \\end{align*}"}
+{"id": "3344.png", "formula": "\\begin{align*} \\tau \\biggl ( \\prod _ { i = 1 } ^ { N } D _ { \\zeta _ i } ( \\theta _ i ) ^ { - 1 } \\biggr ) = \\prod _ { i = 1 } ^ { N } D _ { \\zeta _ i } ( \\theta _ i ) ^ { - 1 } \\frac { \\theta _ i ^ { m ( e ^ \\mathsf { T } A ) _ i } } { ( \\theta _ i ; \\zeta _ i ) _ { e _ i } ^ m } \\end{align*}"}
+{"id": "5599.png", "formula": "\\begin{align*} \\int _ R ^ \\infty | u | ^ { ( p - 1 + j ) p ' } r ^ \\theta \\mathrm d r & \\leq C ^ { ( p - 1 + j ) p ' } \\| u \\| _ { X ^ { 1 , p } _ \\infty } ^ { ( p - 1 + j ) p ' } R ^ { ( j - 2 ) \\frac { \\alpha _ 0 + 1 } { p } } \\int _ R ^ { \\infty } r ^ { \\theta - ( p + 1 ) \\frac { \\alpha _ 0 + 1 } p } \\mathrm d r \\\\ & = \\dfrac { p C ^ { ( p - 1 + j ) p ' } R ^ { ( j - p - 3 ) \\frac { \\alpha _ 0 + 1 } p + \\theta + 1 } } { ( p + 1 ) ( \\alpha _ 0 + 1 ) - p \\theta - p } \\| u \\| _ { X ^ { 1 , p } _ \\infty } ^ { ( p - 1 + j ) p ' } . \\end{align*}"}
+{"id": "8931.png", "formula": "\\begin{align*} \\tau ( \\varphi ) = - d \\varphi ( \\sum \\limits _ { j = 1 } ^ s \\nabla ^ M _ { u _ j } u _ j ) . \\end{align*}"}
+{"id": "3519.png", "formula": "\\begin{align*} \\lambda _ p = \\textbf { 1 } _ \\Gamma \\oplus \\lambda _ p ^ 0 . \\end{align*}"}
+{"id": "6370.png", "formula": "\\begin{align*} \\chi = a ( 1 \\otimes 1 ) + n ( x \\otimes x ) + o ( x \\otimes x g ) + r ( x g \\otimes x ) + s ( x g \\otimes x g ) . \\end{align*}"}
+{"id": "1791.png", "formula": "\\begin{gather*} g _ { \\alpha , T } ( \\underline { m } ) = \\frac { 1 } { T } \\int _ { 0 } ^ { T } \\ , d \\tau = 1 \\end{gather*}"}
+{"id": "4923.png", "formula": "\\begin{align*} ( G _ { \\ell } ^ { ( \\underline { a } , N ) } | _ \\ell \\gamma ) ( \\tau ) = G _ { \\ell } ^ { ( \\underline { a } \\gamma , N ) } ( \\tau ) . \\end{align*}"}
+{"id": "324.png", "formula": "\\begin{align*} \\left \\langle u , \\mathsf { L } _ { j } \\left ( s \\right ) \\overline { \\mathsf { N } _ { j } \\left ( \\overline { s } \\right ) u } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } = \\left \\langle u , \\mathsf { L } _ { j } \\left ( s \\right ) \\mathsf { N } _ { j } \\left ( s \\right ) \\overline { u } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } = \\left \\Vert u \\right \\Vert _ { L ^ { 2 } \\left ( \\mathbb { R } ^ { 3 } \\right ) } ^ { 2 } . \\end{align*}"}
+{"id": "17.png", "formula": "\\begin{align*} v ( x ) = v _ { \\R ^ { d } } ( x ^ { * } ) , v _ { \\R ^ { d } } : \\R ^ { d } \\to \\R _ { + } . \\end{align*}"}
+{"id": "5936.png", "formula": "\\begin{align*} A _ { n } \\le R _ { n + 1 } - S _ { n } + d [ - a _ { 1 , n + 1 } b _ { 1 , n - 1 } ] \\le R _ { n + 1 } - S _ { n } + ( 1 - R _ { n + 1 } ) = 1 - S _ { n } = d [ a _ { 1 , n } b _ { 1 , n } ] . \\end{align*}"}
+{"id": "1308.png", "formula": "\\begin{align*} \\widehat { P _ t \\mu } ( \\xi ) = \\exp ( - 2 \\pi ^ 2 t | \\xi | ^ 2 ) \\hat { \\mu } ( \\xi ) , \\xi \\in \\Z ^ d , \\ , t > 0 . \\end{align*}"}
+{"id": "300.png", "formula": "\\begin{align*} \\mbox { \\boldmath $ \\gamma $ } _ { \\operatorname * { C } ; j } ^ { \\mathbb { B } , \\sigma } : = \\left ( \\gamma _ { \\operatorname * { D } ; j } ^ { \\sigma } , \\gamma _ { \\operatorname * { N } ; j } ^ { \\mathbb { B } , \\sigma } \\right ) . \\end{align*}"}
+{"id": "2936.png", "formula": "\\begin{align*} \\delta _ { j _ { x _ 1 } j _ { x _ 2 } } D _ { j _ { x _ 1 } j _ { x _ 2 } \\pi ( j _ 3 \\cdots j _ { n } ) k } = 0 \\end{align*}"}
+{"id": "4722.png", "formula": "\\begin{align*} C = \\bigcup _ { ( v _ i ) , \\pi } C _ { ( v _ i ) , \\pi } , \\end{align*}"}
+{"id": "2893.png", "formula": "\\begin{align*} c \\langle \\xi _ \\mu , \\beta \\rangle + \\langle \\hat \\rho _ v ( \\xi _ \\mu ) , \\beta \\rangle = 2 \\pi \\langle \\hat \\rho + \\mu , \\beta \\rangle . \\end{align*}"}
+{"id": "3859.png", "formula": "\\begin{align*} \\frac { d ^ 2 \\gamma ^ 1 } { d t ^ 2 } - 2 \\lambda _ 1 \\frac { d \\gamma ^ 1 } { d t } \\frac { d \\gamma ^ 3 } { d t } & = 0 \\\\ \\frac { d ^ 2 \\gamma ^ 3 } { d t ^ 2 } + 2 \\lambda _ 1 e ^ { - 2 \\lambda _ 1 \\gamma ^ 3 } \\left ( \\frac { d \\gamma ^ 1 } { d t } \\right ) ^ 2 & = 0 . \\end{align*}"}
+{"id": "1324.png", "formula": "\\begin{align*} J ^ { w } ( \\textbf { X } _ { m i n R S S U } ^ { ( n ) } ) & = - \\frac { ( n ! ) ^ 2 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( \\int _ { 0 } ^ { 1 } ( 1 - u ) ^ { 2 i - 2 } w ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) d u \\right ) \\\\ & = - \\frac { ( n ! ) ^ 2 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( \\int _ { 0 } ^ { 1 } ( 1 - u ) ^ { 2 i - 2 } \\alpha ( 1 - u ) ^ { \\frac { \\alpha - m + 1 } { \\alpha } } d u \\right ) \\\\ & = - \\frac { ( n ! ) ^ 2 \\alpha ^ n } { 2 } \\prod _ { i = 1 } ^ { n } \\frac { 1 } { 2 i \\alpha - m + 1 } . \\end{align*}"}
+{"id": "8465.png", "formula": "\\begin{align*} \\mathcal { K } ( \\ell ) = \\{ E \\subset \\mathbb { R } ^ { n } : E \\mbox { i s $ \\ell $ - d i s t r i b u t e d a n d } P ( F _ { \\ell } ) = P ( E ) \\} . \\end{align*}"}
+{"id": "1563.png", "formula": "\\begin{align*} \\partial _ t f + v \\partial _ x f = L f = \\frac { 1 } { \\kappa } \\left ( \\rho _ f \\left ( \\alpha \\mathcal { M } _ { T ( x ) } ( v ) + ( 1 - \\alpha ) \\mathcal { M } _ { \\tau ( x ) } ( v ) \\right ) - f \\right ) \\end{align*}"}
+{"id": "4289.png", "formula": "\\begin{align*} g ( i , i ^ \\prime , J _ 1 , J _ 2 , p , n ) = I ( i , J _ { 1 } , p , n ) I ( i , J _ { 2 } , p , n ) \\delta _ { 0 } ( \\sum _ { q = 1 } ^ { 2 k _ 1 } ( - 1 ) ^ { q } j _ { q } ) \\delta _ { 0 } ( \\sum _ { q = 1 } ^ { 2 k _ 2 } ( - 1 ) ^ { q } j _ { q } ^ \\prime ) . \\end{align*}"}
+{"id": "5714.png", "formula": "\\begin{align*} L _ 0 ^ { \\delta } q _ 1 ^ { \\delta } ( 0 , \\cdot ) & \\equiv 0 \\ , , & L _ 0 ^ { \\delta } q _ 2 ^ { \\delta } ( 0 , \\cdot ) & = E _ { 1 2 } ^ { \\delta } \\delta ^ 2 \\ q _ 1 ^ { \\delta } ( 0 , \\cdot ) \\ , , & L _ 0 ^ { \\delta } q _ 3 ^ { \\delta } ( 0 , \\cdot ) & = E _ { 1 3 } ^ { \\delta } \\delta \\ q _ 1 ^ { \\delta } ( 0 , \\cdot ) \\ , , \\end{align*}"}
+{"id": "1014.png", "formula": "\\begin{align*} \\Delta \\psi = 0 , | \\nabla \\psi | = \\omega / v _ 2 , \\end{align*}"}
+{"id": "3102.png", "formula": "\\begin{align*} y _ k = \\mathbf { h } _ { k } \\boldsymbol { \\Theta } \\mathbf { G } \\mathbf { x } + n _ k \\end{align*}"}
+{"id": "6376.png", "formula": "\\begin{align*} \\chi _ { 1 2 } = a ( 1 \\otimes 1 \\otimes 1 ) + n ( x \\otimes x \\otimes 1 ) - n ( x g \\otimes x g \\otimes 1 ) . \\end{align*}"}
+{"id": "8631.png", "formula": "\\begin{align*} \\mathcal { R } _ 1 ^ { \\mu } [ \\beta b , h , \\overline { V } ] = - \\frac { h ^ 2 } { 2 } ( \\nabla _ X \\cdot \\overline { V } ) ^ 2 - \\frac { 1 } { 3 h } \\big { ( } \\nabla _ X ( h ^ 3 \\nabla _ X \\cdot \\overline { V } ) \\big { ) } \\cdot \\overline { V } - \\frac { 1 } { 2 } h ^ 3 \\Delta _ X ( | \\overline { V } | ^ 2 ) + \\frac { 1 } { 6 h } h ^ 3 \\Delta _ X ( | \\overline { V } | ^ 2 ) . \\end{align*}"}
+{"id": "5971.png", "formula": "\\begin{align*} 1 = \\sigma _ { - 1 } ( N ^ \\omega _ { \\partial M } ) ( t , \\xi ' ) \\sigma _ 1 ( \\Lambda _ { g , F } ^ \\omega ) ( t , \\xi ' ) + S ^ { - 1 } _ { 1 , 0 } \\implies \\sigma _ { - 1 } ( N ^ \\omega _ { \\partial M } ) ( t , \\xi ' ) = \\frac { \\chi ( \\xi ' ) } { \\sigma _ 1 ( \\Lambda _ { g , F } ^ \\omega ) ( t , \\xi ' ) } . \\end{align*}"}
+{"id": "1901.png", "formula": "\\begin{align*} \\widehat { w _ h } : = \\left ( 1 - \\lambda \\right ) w _ h ^ - + \\lambda w _ h ^ + = \\{ w _ h \\} + \\left ( \\frac { 2 \\lambda - 1 } { 2 } \\right ) [ \\ ! [ w _ h ] \\ ! ] , \\end{align*}"}
+{"id": "7665.png", "formula": "\\begin{align*} \\psi ( u ) & = \\mathbb { P } \\left ( \\bigcup _ { t \\geqslant 0 } \\left \\{ W ( t ) < 0 \\right \\} \\right ) \\\\ & = \\mathbb { P } \\left ( \\inf _ { n \\geqslant 1 } \\left \\{ u + c T _ n - \\sum _ { i = 1 } ^ { n } X _ n \\right \\} < 0 \\right ) \\\\ & = \\mathbb { P } \\left ( \\sup _ { n \\geqslant 1 } \\sum _ { k = 1 } ^ n ( X _ k - c \\theta _ k ) > u \\right ) . \\end{align*}"}
+{"id": "513.png", "formula": "\\begin{align*} \\tau _ R < \\tau ( \\omega ) \\implies f ( \\tau _ R ( \\omega ) , \\omega ) = R . \\end{align*}"}
+{"id": "8911.png", "formula": "\\begin{align*} h _ 0 ^ { - 1 } ( h _ 0 ( x ) ) = x + x _ 1 + \\ldots + x _ { d - 1 } . \\end{align*}"}
+{"id": "8730.png", "formula": "\\begin{align*} \\delta _ { t } \\leq \\frac { 2 ( t _ { 0 } - 1 ) \\delta _ { t _ { 0 } } } { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { ( 2 - p _ { i } ) t ^ { p _ { i } } } \\enspace . \\end{align*}"}
+{"id": "4273.png", "formula": "\\begin{align*} T _ { n \\times p } = \\left ( \\begin{array} { c c c c c } a _ { 0 } & a _ { - 1 } & a _ { - 2 } & \\cdots & a _ { 1 - p } \\\\ a _ { 1 } & a _ { 0 } & a _ { - 1 } & \\cdots & a _ { 2 - p } \\\\ a _ { 2 } & a _ { 1 } & a _ { 0 } & \\cdots & a _ { 3 - p } \\\\ \\vdots & \\vdots & \\vdots & \\cdots & \\vdots \\\\ a _ { n - 1 } & a _ { n - 2 } & a _ { n - 3 } & \\cdots & a _ { n - p } \\end{array} \\right ) . \\end{align*}"}
+{"id": "943.png", "formula": "\\begin{align*} h ' ( \\tilde { x } , \\tilde { t } ) = \\frac { h ( Z ( \\tilde { x } , \\tilde { t } ) , Y ^ { - 1 } ( \\tilde { t } ) ) X _ x ^ 2 ( Z ( \\tilde { x } , \\tilde { t } ) , Y ^ { - 1 } ( \\tilde { t } ) ) } { \\tilde { t } ^ { \\alpha - 1 } ( \\mathcal { T } _ t ^ \\alpha Y ) ( Y ^ { - 1 } ( \\tilde { t } ) ) } , \\end{align*}"}
+{"id": "5659.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l } \\alpha \\cdot e _ 0 = \\alpha \\cdot e _ 1 = \\alpha \\cdot e ' _ 0 = \\alpha \\cdot e ' _ 1 = 1 , \\\\ e _ 0 \\cdot e ' _ 1 = e ' _ 0 \\cdot e _ 1 = 0 , \\\\ e _ 0 \\cdot e _ 0 ' = e _ 1 \\cdot e _ 1 ' = 3 . \\end{array} \\right . \\end{align*}"}
+{"id": "717.png", "formula": "\\begin{align*} \\mathbf \\Phi ( \\mathbf V ) ( t ) = \\mathbf S ( t - S ) \\mathbf U ( S ) + i \\int _ { S \\wedge \\tau _ R } ^ { t \\wedge \\tau _ R } \\mathbf S ( t - s ) \\mathbf N ( \\mathbf u ( s ) ) \\ , d s + i \\int _ { S } ^ t \\mathbf S ( t - s ) \\mathbf M ( [ \\mathbf u , \\mathbf V ] ( s ) ) \\ , d W ( s ) , \\end{align*}"}
+{"id": "8146.png", "formula": "\\begin{align*} \\frac { q _ { m n } ( i j ) } { m _ { m n } } = \\frac { p _ { i j } ( m n ) } { k _ { i j } } \\ , , \\end{align*}"}
+{"id": "3777.png", "formula": "\\begin{align*} f ^ * ( x ^ * ) : = \\sup _ { x \\in E } \\ , \\langle x ^ * , x \\rangle - f ( x ) . \\end{align*}"}
+{"id": "8254.png", "formula": "\\begin{align*} x _ 1 = x _ 1 ( e ^ { \\tau ( m , x _ 1 ) } \\cdot m ) & = \\int _ 0 ^ { \\tau ( m , x _ 1 ) } \\frac { d } { d t } ( x _ 1 ( e ^ t \\cdot m ) ) d t = \\int _ 0 ^ { \\tau ( m , x _ 1 ) } | T | ^ 2 ( e ^ t \\cdot m ) ) d t \\\\ & \\geq C ^ { - 1 } \\int _ 0 ^ { \\tau ( m , x _ 1 ) } \\rho ^ 2 ( e ^ t \\cdot m ) d t \\geq C ^ { - 1 } \\int _ 0 ^ { \\tau ( m , x _ 1 ) } \\rho ^ 2 ( m ) d t \\\\ & = C ^ { - 1 } \\rho ^ 2 ( m ) \\tau ( m , x _ 1 ) . \\end{align*}"}
+{"id": "5480.png", "formula": "\\begin{align*} \\underline { f _ 1 } + \\underline { f _ 2 } : = \\underline { f _ 1 \\big | _ { U _ 1 \\cap U _ 2 } + f _ 2 \\big | _ { U _ 1 \\cap U _ 2 } } , \\underline { f _ 1 } * \\underline { f _ 2 } : = \\underline { f _ 1 \\big | _ { U _ 1 \\cap U _ 2 } f _ 2 \\big | _ { U _ 1 \\cap U _ 2 } } . \\end{align*}"}
+{"id": "8956.png", "formula": "\\begin{align*} B = \\{ ( \\bar { x } , \\bar { y } ) \\in \\R ^ { m + n } : \\upsilon _ 1 \\leq \\| \\bar { y } \\| \\leq \\upsilon _ 2 , | x _ i | < \\vartheta _ { i } ( \\epsilon ) \\| \\bar { y } \\| ^ { - w _ i } i = 1 , \\ldots , m \\} . \\end{align*}"}
+{"id": "1783.png", "formula": "\\begin{gather*} \\mathbf { E } [ \\mathcal { X } _ m \\overline { \\mathcal { X } _ n } ] = \\begin{cases} 1 & , \\\\ 0 & . \\end{cases} \\end{gather*}"}
+{"id": "6142.png", "formula": "\\begin{align*} \\eta ( ( u _ 1 , u _ 2 ) ) = ( \\eta _ 1 \\times \\eta _ 2 ) ( u ) = ( \\eta _ 1 ( u _ 1 ) , \\eta _ 2 ( u _ 2 ) ) , \\end{align*}"}
+{"id": "4824.png", "formula": "\\begin{align*} \\check { R } _ i ( u ) = \\sum _ { k l m p } S ^ { k m } _ { l p } ( u ) \\cdot I ^ { ( 1 ) } \\otimes I ^ { ( 2 ) } \\otimes \\cdots \\otimes E ^ { ( i ) } _ { p k } \\otimes E ^ { ( i + 1 ) } _ { m l } \\otimes I ^ { ( i + 2 ) } \\otimes \\cdots \\otimes I ^ { ( n ) } \\end{align*}"}
+{"id": "5331.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n i k _ i = n . \\end{align*}"}
+{"id": "5941.png", "formula": "\\begin{align*} & A = 4 | u _ j ( x ^ * ) | ^ 2 e ^ { \\phi ( x ^ * ) } , \\\\ [ 1 e m ] & B = 4 \\pi | u _ j ( x ^ * ) | ^ 2 e ^ { \\phi ( x ^ * ) } ( H ( x ^ * ) - \\partial _ { \\nu } \\phi ( x ^ * ) ) , \\\\ [ 1 e m ] & C = | u _ j ( x ^ * ) | ^ 2 e ^ { \\phi ( x ^ * ) } \\left ( \\frac { 8 \\log 2 - 6 } { \\pi } ( H ( x ^ * ) - \\partial _ { \\nu } \\phi ( x ^ * ) ) - 1 6 R _ { \\partial M } ^ { \\lambda _ j } ( x ^ * , x ^ * ) \\right ) . \\end{align*}"}
+{"id": "1767.png", "formula": "\\begin{gather*} \\zeta ( s , \\mathbb { X } _ \\alpha ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\mathbb { X } _ \\alpha ( n ) } { ( n + \\alpha ) ^ s } \\quad \\zeta ( s , \\mathbb { Y } _ \\alpha ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\mathbb { Y } _ \\alpha ( n ) } { ( n + \\alpha ) ^ s } \\end{gather*}"}
+{"id": "7554.png", "formula": "\\begin{align*} R : \\mathbb { C } ^ * \\to F = ( \\mathbb { C } ^ * ) ^ 3 , t \\mapsto ( t ^ { - \\sigma _ 1 } , t ^ { - \\sigma _ 2 } , t ^ { - \\sigma _ 3 } ) , \\ , \\ , \\ , \\mathrm { w i t h } \\ , \\ , \\ , \\sigma _ i \\in \\mathbb { Z } . \\end{align*}"}
+{"id": "6695.png", "formula": "\\begin{align*} \\tilde { c } _ k ^ { ( \\rm c J ) } ( N , \\beta , p , q ) = ( 4 / \\beta ^ 2 ) \\tilde { c } _ k ^ { ( \\rm c J ) } ( - \\beta N / 2 , 4 / \\beta , - \\beta p / 2 , - 2 q / \\beta ) , \\end{align*}"}
+{"id": "8756.png", "formula": "\\begin{align*} \\hat f _ t ( x ) = \\mathbb { E } f ( x + h _ t \\tilde \\zeta ) , \\forall x \\in \\mathbb { R } ^ d , \\end{align*}"}
+{"id": "7709.png", "formula": "\\begin{align*} \\psi ^ \\sharp \\overline { x } ^ i = \\psi ^ i ( x ) , \\ ; \\ ; \\psi ^ \\sharp \\overline { e } ^ \\mu = f ^ \\mu _ { \\nu } ( x ) e ^ \\nu , \\ ; \\ ; \\psi ^ \\sharp \\overline { p } ^ I = g _ { \\mu \\nu } ^ I ( x ) e ^ \\mu e ^ \\nu + h ^ I _ J ( x ) p ^ J . \\end{align*}"}
+{"id": "8715.png", "formula": "\\begin{align*} \\Big ( \\sum _ { t = 1 } ^ { T } \\eta _ t \\Big ) ^ { - 1 } = \\frac { 1 } { \\min \\big ( \\frac { T } { 1 6 \\kappa \\bar { L } d } , T \\gamma \\big ) } = \\max \\Big ( \\frac { 1 8 \\kappa \\bar { L } d } { T } , \\frac { 1 } { T \\gamma } \\Big ) \\leq \\frac { 1 6 \\kappa \\bar { L } d } { T } + \\frac { 1 } { T \\gamma } . \\end{align*}"}
+{"id": "7616.png", "formula": "\\begin{align*} \\beta = ( \\beta _ { i , j } ) _ { i \\in I , j \\in J } \\colon \\bigoplus _ { i \\in I _ n } N ( U _ i ) \\to \\bigoplus _ { j \\in I _ { n - 1 } } N ( V _ j ) , \\end{align*}"}
+{"id": "7361.png", "formula": "\\begin{align*} { \\rm S u p p } ( \\rho _ Y ) = \\Big \\{ x \\Big | f ( x ) \\geq 0 \\Big \\} . \\end{align*}"}
+{"id": "7354.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } F _ N ( \\gamma ) = \\frac { \\gamma } { 4 } \\int x ^ 2 \\rho _ { S } ( x ) \\ , d x - \\mathcal { J } [ \\rho _ { \\sqrt { \\gamma } S } , \\ , \\rho _ { \\sqrt { \\gamma } S } \\boxplus \\rho _ { \\rm s c } ] . \\end{align*}"}
+{"id": "2456.png", "formula": "\\begin{align*} & G _ 1 ^ { \\wedge n } = G _ 1 \\wedge . . . \\wedge G _ 1 ( n ) . \\\\ & G _ 1 ^ { \\vee n } = G _ 1 \\vee . . . \\vee G _ 1 ( n ) . \\end{align*}"}
+{"id": "3290.png", "formula": "\\begin{align*} m _ \\ell = E [ X ^ \\ell ] = \\frac { d ^ \\ell } { d t ^ \\ell } \\phi _ X ( 0 ) ( - \\mu ) ^ \\ell . \\end{align*}"}
+{"id": "7120.png", "formula": "\\begin{align*} \\forall \\lambda \\geq 0 , | \\{ \\rho ( t ) > \\lambda \\} | = | \\{ \\rho _ 0 > \\lambda \\} | . \\end{align*}"}
+{"id": "8847.png", "formula": "\\begin{align*} \\Big ( \\sum _ { k = 1 } ^ { T } \\eta _ { k } \\Big ) ^ { - 1 } = \\frac { 1 } { \\min \\Big ( \\frac { T } { 1 8 L \\kappa } , T \\gamma \\Big ) } = \\max \\Big ( \\frac { 1 8 L \\kappa } { T } , \\frac { 1 } { T \\gamma } \\Big ) \\leq \\frac { 1 8 L \\kappa } { T } + \\frac { 1 } { T \\gamma } . \\end{align*}"}
+{"id": "5264.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n G _ 1 \\| x _ { i , t } - x _ { j , t } \\| \\le n \\varepsilon _ 1 G _ 1 + \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ n \\Big ( \\frac { \\tilde { \\varepsilon } _ 2 ^ 2 G _ 1 ^ 2 \\alpha _ t } { \\sigma } + \\frac { \\sigma \\| \\epsilon ^ z _ { i , t } \\| ^ 2 } { 4 \\alpha _ t } \\Big ) . \\end{align*}"}
+{"id": "5576.png", "formula": "\\begin{align*} y _ j \\approx \\log ( C _ \\ell ) + \\beta _ \\ell x _ j , j = 1 , \\ldots , 2 0 , \\end{align*}"}
+{"id": "4577.png", "formula": "\\begin{align*} \\int _ { \\mathbb R } \\vert \\varphi ( t ) \\vert ^ 2 d x _ 2 - \\int _ 0 ^ t \\ ! \\ ! \\ ! \\int _ { \\mathbb R } ( \\partial _ 2 \\hat u ^ + _ 2 ) \\varphi ^ 2 \\ , d x _ 2 \\ , d s - 2 \\int _ 0 ^ t \\ ! \\ ! \\ ! \\int _ { \\mathbb R } \\dot u ^ + _ N \\varphi \\ , d x _ 2 \\ , d s - 2 \\int _ 0 ^ t \\ ! \\ ! \\ ! \\int _ { \\mathbb R } ( \\partial _ 1 \\hat u ^ + _ N ) \\varphi ^ 2 \\ , d x _ 2 \\ , d s = 0 \\ , , \\end{align*}"}
+{"id": "6753.png", "formula": "\\begin{align*} ( E ^ { e _ 1 } \\circ O ^ { o _ 1 } \\circ E ^ { e _ 2 } \\circ \\dotsb \\circ O ^ { o _ l } \\circ E ^ { e _ { l + 1 } } ) ( n ) = ( E ^ { \\sigma _ e } \\circ O ^ { \\sigma _ o } ) ( n ) + C = W . \\end{align*}"}
+{"id": "3433.png", "formula": "\\begin{align*} H ^ 0 ( M , l L ) = \\tilde { W } _ l \\oplus F _ 0 \\ldots F _ m H ^ 0 ( M , ( l - \\sum d _ i ) L ) \\end{align*}"}
+{"id": "777.png", "formula": "\\begin{align*} I [ u ] = \\frac { 1 } { p } [ u ] _ { s , p } ^ p + \\frac { 1 } { p } a \\int _ { \\mathbb { R } ^ N } | u | ^ p d x - \\frac { b } { 2 } \\int _ { \\mathbb { R } ^ N } ( K \\ast F ( u ) ) F ( u ) d x - \\frac { \\varepsilon _ g } { p _ g } \\int _ { \\mathbb { R } ^ N } | u | ^ { p _ g } d x , \\end{align*}"}
+{"id": "6625.png", "formula": "\\begin{align*} f ( \\tau ; \\beta ) = 1 + \\sum _ { j = 1 } ^ \\infty p _ j ( 2 / \\beta ) ( \\tau / 2 \\pi ) ^ j \\end{align*}"}
+{"id": "7041.png", "formula": "\\begin{align*} r ( t ) = \\frac { \\sum _ { n = 1 } ^ { \\infty } \\varphi _ { n } e ^ { - \\lambda _ { n } t } \\phi _ { n } ( q ) } { w ( t ) } + \\frac { 1 } { w ( t ) } \\int _ { 0 } ^ { t } \\left ( \\sum _ { n = 1 } ^ { \\infty } \\phi _ { n } ( q ) f _ { n } ( \\tau ) e ^ { - \\lambda _ { n } ( t - \\tau ) } \\right ) r ( \\tau ) d \\tau \\end{align*}"}
+{"id": "4369.png", "formula": "\\begin{align*} \\begin{cases} & [ j ] = 0 , [ H _ N ] = 0 , \\\\ & j [ u _ N ] + \\vert N \\vert ^ 2 [ q ] = 0 , j [ u _ { \\tau } ] = H _ N [ H _ { \\tau } ] , \\\\ & H _ N [ u _ { \\tau } ] = j [ \\frac { H _ { \\tau } } { \\rho } ] , \\\\ & j [ e + \\frac { 1 } { 2 } \\frac { | \\mathbf { H } | ^ 2 } { \\rho } ] + [ q u _ N - H _ N ( \\mathbf { H } \\cdot \\mathbf { u } ) ] = 0 . \\end{cases} \\end{align*}"}
+{"id": "2661.png", "formula": "\\begin{align*} \\psi ( \\delta ) = \\sup _ { 0 \\le t _ 0 \\le t _ 1 \\le \\ldots \\le t _ l \\le T } \\{ \\sum _ { i = 1 } ^ l \\abs { f ( t _ i ) - f ( t _ { i - 1 } } + M \\sum _ { i = 1 } ^ l \\abs { F ( t _ i ) - F ( t _ { i - 1 } ) } : \\ , \\sum _ { l = 1 } ^ l ( t _ i - t _ { i - 1 } ) \\le \\delta \\} \\end{align*}"}
+{"id": "85.png", "formula": "\\begin{align*} \\langle \\Psi , \\mathcal H \\Psi \\rangle = \\sum _ { \\vert k \\vert \\leq 2 } \\langle \\Psi , \\mathcal H ^ { ( k ) } \\Psi \\rangle = \\sum _ { m \\in \\Z } \\sum _ { \\vert k \\vert \\leq 2 } \\theta _ { \\mathcal M } ( m ) ^ 2 \\langle \\Psi , \\mathcal H ^ { ( k ) } \\Psi \\rangle , \\end{align*}"}
+{"id": "5235.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ T \\| [ g ( x _ { t } ) ] _ + \\| ^ 2 . \\end{align*}"}
+{"id": "492.png", "formula": "\\begin{align*} \\mathfrak K _ j f ( x ) = \\int _ { \\R } \\mathfrak k _ j ( x - y ) f ( y ) d y \\end{align*}"}
+{"id": "8199.png", "formula": "\\begin{align*} g _ 0 = \\bar { g } + \\frac { 1 } { \\rho ^ 2 } \\sum _ { i = 1 } ^ 3 ( d x _ i ) ^ 2 + \\rho ^ 2 \\eta ^ 2 , \\end{align*}"}
+{"id": "3954.png", "formula": "\\begin{align*} f ( x , y ) = \\frac { u v } { x v } \\frac { x y } { u y } \\end{align*}"}
+{"id": "6718.png", "formula": "\\begin{align*} \\begin{aligned} & 4 \\pi - \\frac { ( 1 + k ) ^ { 2 } } { ( 1 - k ) ^ { 2 } } ( \\eta ( r _ { 0 } ) ) ^ { 2 } \\int _ { \\Sigma } | \\nabla u | ^ 2 \\\\ \\ge & \\frac { ( 1 + k ) ^ { 2 } } { 2 k } \\frac { ( 1 - a \\eta ( r _ { 0 } ) ) } { \\eta ( r _ { 0 } ) } \\left \\{ 4 \\pi \\frac { ( 1 - k ) ^ { 2 } } { ( 1 + k ) ^ { 2 } } - \\eta ( r _ { 0 } ) \\int _ { \\Sigma } H | \\nabla u | + ( \\eta ( r _ { 0 } ) ) ^ { 2 } \\int _ { \\Sigma } | \\nabla u | ^ 2 \\right \\} , \\end{aligned} \\end{align*}"}
+{"id": "143.png", "formula": "\\begin{align*} { \\rm K a z } _ { e m } ^ E ( \\sigma . t _ { a _ i \\pi _ \\mu a _ j ^ { - 1 } } ) = { \\rm K a z } _ { e m } ^ E \\big ( t _ { \\sigma ( a _ i ) \\sigma ( \\pi _ \\mu ) \\sigma ( a _ j ) ^ { - 1 } } \\big ) = { \\rm K a z } _ { e m } ^ E \\big ( t _ { c _ i \\pi _ \\mu c _ j ^ { - 1 } } \\big ) = t _ { c _ i ' \\pi _ \\mu ' c _ j '^ { - 1 } } . \\end{align*}"}
+{"id": "7944.png", "formula": "\\begin{align*} S _ i = ( S \\cap V _ i ) \\cup ( X _ i \\cup Y _ i ) \\ 1 \\end{align*}"}
+{"id": "1359.png", "formula": "\\begin{align*} \\mathbb { J } _ { 2 k } ^ { i \\infty } : = D ^ { 2 k - 1 } \\left ( M _ { 2 - 2 k } ^ ! \\right ) \\qquad \\mathbb { J } _ { 2 k , 0 } ^ { i \\infty } : = D ^ { 2 k - 1 } \\left ( S _ { 2 - 2 k } ^ ! \\right ) , \\end{align*}"}
+{"id": "4048.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ 1 U _ { 0 , 0 } ( t ) d t & = 0 , \\int _ 0 ^ 1 V _ { 0 , 0 } ( t ) d t = q _ 1 ( a ) - ( 1 - a ) q _ 1 ( 0 ) - a q _ 1 ( 1 ) , \\\\ \\int _ 0 ^ 1 V _ { 1 , 1 } ( t ) d t & = q _ 1 ( a ) , \\int _ 0 ^ 1 V _ { 0 , 1 } ( t ) d t = q _ 1 ( a ) - q _ 1 ( 0 ) , \\int _ 0 ^ 1 V _ { 1 , 0 } ( t ) d t = q _ 1 ( 1 ) - q _ 1 ( a ) . \\end{aligned} \\end{align*}"}
+{"id": "2013.png", "formula": "\\begin{align*} \\tilde { \\mathrm { E } } u ( x ' , x _ n ) & : = \\begin{cases} u ( x ' , x _ n ) \\ , & ( x ' , x _ n ) \\in \\mathbb { R } ^ { n - 1 } \\times ( 0 , + \\infty ) \\\\ u ( x ' , - x _ n ) \\ , & ( x ' , x _ n ) \\in \\mathbb { R } ^ { n - 1 } \\times ( - \\infty , 0 ) \\end{cases} \\end{align*}"}
+{"id": "8887.png", "formula": "\\begin{align*} \\int _ { G ( k , \\C ^ n ) } c _ 1 ( S ) ^ { m _ 1 } \\cdots c _ k ( S ) ^ { m _ k } = \\sum _ I \\dfrac { \\prod _ { r = 1 } ^ k \\sigma _ r ( u _ { i _ 1 } , \\dots , u _ { i _ k } ) ^ { m _ r } } { \\prod _ { i \\in I } \\prod _ { j \\in J } ( u _ i - u _ j ) } , \\end{align*}"}
+{"id": "2834.png", "formula": "\\begin{align*} y _ i = x _ i \\cdot \\mathrm { e } ^ { j \\phi } + n _ i \\end{align*}"}
+{"id": "303.png", "formula": "\\begin{align*} \\Omega _ { j } ^ { - } & : = \\Omega _ { j } , & \\Omega _ { j } ^ { + } & : = \\mathbb { R } ^ { 3 } \\backslash \\overline { \\Omega _ { j } } , \\\\ \\mathbb { A } _ { j } ^ { + } & : = \\left . \\mathbb { A } _ { j } ^ { \\operatorname * { e x t } } \\right \\vert _ { \\Omega _ { j } ^ { + } } , & p _ { j } ^ { + } & : = \\left . p _ { j } ^ { \\operatorname * { e x t } } \\right \\vert _ { \\Omega _ { j } ^ { + } } . \\end{align*}"}
+{"id": "504.png", "formula": "\\begin{align*} \\mathcal L _ { \\mathrm { n o i s e } } ( \\psi , \\phi , x , t ) = \\mathcal L _ { \\mathrm { D i r a c } } ^ { \\mathrm { n o i s e } } ( \\psi , x , t ) + \\mathcal L _ { \\mathrm { m e s o n } } ^ { \\mathrm { n o i s e } } ( \\phi , x , t ) = \\psi ^ * \\beta \\psi \\xi _ 1 + \\frac 1 2 \\phi ^ 2 \\xi _ 2 . \\end{align*}"}
+{"id": "294.png", "formula": "\\begin{align*} \\gamma _ { \\operatorname * { D } ; j } ^ { \\sigma } u = \\left . u \\right \\vert _ { \\Gamma _ { j } } . \\end{align*}"}
+{"id": "7015.png", "formula": "\\begin{align*} x ^ 2 - 3 y ^ 2 = - 8 , \\end{align*}"}
+{"id": "3895.png", "formula": "\\begin{align*} \\frac { 1 - | f ' ( 0 ) | } { 1 + | f ' ( 0 ) | } \\sum _ { n = 1 } ^ N | a _ n | ^ 2 \\leq \\sigma _ N ^ 2 = \\norm { \\sum _ { n = 1 } ^ N a _ n f ^ n } { 2 } ^ 2 \\leq \\frac { 1 + | f ' ( 0 ) | } { 1 - | f ' ( 0 ) | } \\sum _ { n = 1 } ^ N | a _ n | ^ 2 , N = 1 , 2 , \\ldots \\end{align*}"}
+{"id": "4362.png", "formula": "\\begin{align*} \\int _ \\mathbb { R } \\chi ( t ) \\omega ( A _ 1 { \\tt T } _ t ( A _ 2 ) ) \\ , d t = 0 \\ \\ ( A _ 1 , A _ 2 \\in { \\tt C A R } ( \\mathcal { K } , \\Gamma ) ) \\end{align*}"}
+{"id": "7178.png", "formula": "\\begin{align*} \\Tilde { I } = \\big < \\{ x _ { \\mathcal { N } _ { j - 1 } + 1 } x _ { \\mathcal { N } _ { j - 1 } + 2 } \\dots x _ { \\mathcal { N } _ j } x _ { n + j } ~ | ~ j = 1 , \\dots , m \\} \\big > , \\end{align*}"}
+{"id": "7457.png", "formula": "\\begin{align*} \\langle x \\rangle = e ^ { 2 \\pi i \\dot x \\ddot x } , x = ( \\dot x , \\ddot x ) \\end{align*}"}
+{"id": "2839.png", "formula": "\\begin{align*} \\tau = ( 1 , 0 , 3 , 2 , 5 , 4 , \\dots , N - 1 , N - 2 ) \\end{align*}"}
+{"id": "269.png", "formula": "\\begin{align*} T \\overset { } { = } \\{ \\Delta ^ { \\{ 0 , 2 , 4 \\} } , \\ \\Delta ^ { \\{ 1 , 2 , 3 \\} } , \\ \\Delta ^ { \\{ 0 , 1 , 3 \\} } , \\ \\Delta ^ { \\{ 1 , 3 , 4 \\} } , \\ \\Delta ^ { \\{ 0 , 1 , 2 \\} } \\} ; \\end{align*}"}
+{"id": "7209.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\displaystyle d X _ t ^ { i , N } = b \\bigl ( X _ t ^ { i , N } , \\hat { \\mu } _ t ^ { N } \\bigr ) d t + \\displaystyle \\alpha _ t ^ { i , N } d t + \\sqrt { 2 } d B _ t ^ { i , N } \\\\ \\bigl ( X _ { t _ 0 } ^ { 1 , N } , \\dots , X _ { t _ 0 } ^ { N , N } \\bigr ) = \\mathbf { x } ^ N _ 0 \\end{array} \\right . \\end{align*}"}
+{"id": "523.png", "formula": "\\begin{align*} \\norm { u } _ { X _ { h ( \\xi ) } ^ { s , b } ( S , T ) } ^ 2 = \\int \\langle \\xi \\rangle ^ { 2 s } \\norm { e ^ { i t h ( \\xi ) } \\mathcal F _ x u ( t , \\xi ) } _ { H _ t ^ b ( S , T ) } ^ 2 d \\xi . \\end{align*}"}
+{"id": "1217.png", "formula": "\\begin{align*} D _ n = 2 \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p ) | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p d x . \\end{align*}"}
+{"id": "41.png", "formula": "\\begin{align*} \\mathcal Q ( z ) : = \\frac { 1 } { 2 } \\rho _ { z } ^ 2 \\vert \\Lambda \\vert ( \\widehat g ( 0 ) + \\widehat { g \\omega } ( 0 ) ) + \\mathcal { K } ^ { \\rm { B o g } } , \\end{align*}"}
+{"id": "4744.png", "formula": "\\begin{align*} { \\rm P e r } ( A ) & = { \\rm P e r } ( A ; \\R _ { + } ^ { 2 } ) + { \\rm P e r } ( A ; \\R ^ 2 \\setminus \\R _ { + } ^ { 2 } ) \\\\ & \\ge { \\rm P e r } ( A \\cap \\R _ { + } ^ { 2 } ; \\R _ { + } ^ { 2 } ) + { \\rm P e r } ( A ; \\R ^ 2 \\setminus \\R _ { + } ^ { 2 } ) \\ge { \\rm P e r } ( A ) . \\end{align*}"}
+{"id": "1444.png", "formula": "\\begin{align*} \\sigma ( r , x _ { 0 } ; u ) : = \\frac { 1 } { 2 \\pi } \\int _ { B _ { r } ( x _ { 0 } ) } e ^ { u ( x ) } \\mathrm { d } x . \\end{align*}"}
+{"id": "1477.png", "formula": "\\begin{align*} \\Delta _ { \\mathbb { G } } : = \\sum _ { j = 1 } ^ { d _ { 1 } } X _ { j } ^ { 2 } \\end{align*}"}
+{"id": "7045.png", "formula": "\\begin{align*} \\tilde { w } _ { 2 } ( t ) = \\tilde { w } _ { 1 } ^ { \\prime } ( t ) - r ( t ) f ( q , t ) . \\end{align*}"}
+{"id": "5537.png", "formula": "\\begin{align*} \\eta _ j ( t ) = c _ j + r _ j e ^ { - \\i t } , t \\in J _ j = [ 0 , 2 \\pi ] , j = 1 , \\ldots , m . \\end{align*}"}
+{"id": "4531.png", "formula": "\\begin{align*} \\hat \\Psi ^ \\pm ( t , { \\bf x } ) : = \\chi ( \\pm x _ 1 ) \\hat \\varphi ( t , x _ 2 ) \\ , , \\quad \\forall \\ , { \\bf x } \\in \\mathbb R ^ 2 _ + \\ , \\ , \\ , t \\in ( - \\infty , T ] \\ , , \\end{align*}"}
+{"id": "4039.png", "formula": "\\begin{align*} \\begin{aligned} q _ 0 ( a - t ) & = \\mathcal { K } _ 0 ( q _ 0 ( a + t ) ) , & \\tilde { q } _ 0 ( a - t ) & = \\mathcal { K } _ 0 ( \\tilde { q } _ 0 ( a + t ) ) , \\\\ p ( a - t ) & = \\mathcal { K } _ 1 ( p ( a + t ) ) , & \\tilde { p } ( a - t ) & = \\mathcal { K } _ 1 ( \\tilde { p } ( a + t ) ) , \\end{aligned} \\end{align*}"}
+{"id": "8432.png", "formula": "\\begin{align*} g ( X ) = - \\sum _ { i = 1 } ^ p \\beta _ i \\log \\det X _ i , h ( X ) = - \\sum _ { j = 1 } ^ q \\alpha _ j \\log \\det \\Big ( \\sum _ { i = 1 } ^ p A _ { i j } X _ i A _ { i j } ^ T + \\rho I _ { m _ j } \\Big ) , \\end{align*}"}
+{"id": "677.png", "formula": "\\begin{align*} \\norm { \\theta _ R \\left ( \\sum _ { i = 1 } ^ n \\norm { u _ i } _ { \\widetilde X ^ { s _ i , b } _ { h _ i ( \\xi ) } ( 0 , t ) } ^ 2 \\right ) u _ j ( t ) } _ { X ^ { s _ j , b } _ { h _ j ( \\xi ) } ( 0 , T ) } \\leqslant C \\sqrt { R } , \\end{align*}"}
+{"id": "6881.png", "formula": "\\begin{align*} \\ell ( \\widetilde { X } , r ) \\cdot r ^ * \\sigma = ( \\ell ( \\widetilde { X } , r ) \\smallfrown [ \\widetilde { X } ] ) \\cdot r ^ * \\sigma . \\end{align*}"}
+{"id": "1870.png", "formula": "\\begin{align*} | \\zeta ^ { ( n ) } ( \\sigma _ 0 + i \\tau , \\alpha ) - z _ n | & = \\left | \\frac { n ! } { 2 \\pi i } \\oint _ { | s - \\sigma _ 0 | = r } \\frac { \\zeta ( s + i \\tau , \\alpha ) - f ( s ) } { ( s - \\sigma _ 0 ) ^ { n + 1 } } \\ , d s \\right | \\\\ & \\leq \\frac { n ! } { r ^ n } \\sup _ { s \\in K } | \\zeta ( s + i \\tau , \\alpha ) - f ( s ) | . \\end{align*}"}
+{"id": "6021.png", "formula": "\\begin{align*} D \\phi ( F ) = \\sum _ { i = 1 } ^ d \\partial _ i \\phi ( F ) D F _ i , \\end{align*}"}
+{"id": "70.png", "formula": "\\begin{align*} \\mathcal Q _ 3 ^ { \\rm { s o f t } } ( z ) = \\frac { 1 } { \\vert \\Lambda \\vert } \\sum _ { k \\in \\mathcal P _ H , p \\in \\mathcal P _ L } \\widehat g ( k ) \\big ( \\bar z a _ p ^ \\dagger a _ { p - k } a _ k + h . c . \\big ) . \\end{align*}"}
+{"id": "6618.png", "formula": "\\begin{align*} c _ \\infty ^ { ( \\widetilde { \\rm c J } ) } ( \\tau ; \\beta , p , q ) = \\int _ { - \\infty } ^ \\infty \\Big ( \\rho _ { ( 1 ) , \\infty } ^ { ( \\widetilde { \\rm c J } ) } ( x ; \\beta , p , q ) - 1 \\Big ) e ^ { i \\tau x } \\ , d x . \\end{align*}"}
+{"id": "7848.png", "formula": "\\begin{align*} \\widetilde { N } / ( \\mathrm { S L } ( 2 , \\mathbb { Z } ) \\ltimes \\Lambda ) = F / \\Lambda \\ , . \\end{align*}"}
+{"id": "2977.png", "formula": "\\begin{align*} \\textsf { A } = \\textsf { R S } \\textrm { a n d } \\textsf { B } = \\textsf { S R } . \\end{align*}"}
+{"id": "126.png", "formula": "\\begin{align*} \\begin{matrix} K _ L = K _ \\ell ^ { 4 + 2 \\varepsilon } , & K _ H = K _ \\ell ^ { 4 + 3 \\varepsilon } , & \\frac { \\mathcal M } { N } = K _ \\ell ^ { - 1 2 - 1 0 \\varepsilon } , \\\\ \\varepsilon _ { \\rm { g a p } } = K _ \\ell ^ { - 2 } , & \\varepsilon _ K = K _ \\ell ^ { - 1 8 + 2 d + ( d - 1 6 ) \\varepsilon } , & \\end{matrix} \\end{align*}"}
+{"id": "6537.png", "formula": "\\begin{align*} \\eta = ( \\varepsilon + \\delta ) ^ { \\frac { 1 } { 8 b } } , \\ L = ( \\varepsilon + \\delta ) ^ { - \\frac { 1 } { 1 0 ^ 4 d b ^ 4 } } \\end{align*}"}
+{"id": "5865.png", "formula": "\\begin{align*} \\lambda _ { 1 } > 0 \\mbox { a n d } \\lambda _ { 2 } = \\sqrt { \\frac { \\log \\gamma } { \\log \\log \\gamma } } . \\end{align*}"}
+{"id": "7797.png", "formula": "\\begin{align*} \\left [ \\left ( \\frac { ( m \\tau _ 1 + n ) ( t ^ { - 1 } - t ) - 2 m \\tau _ 2 } { | m \\tau + n | ^ 2 } \\right ) \\frac { 1 + ( t _ { + } ^ { m , n } ) ^ 2 } { ( t - t _ { + } ^ { m , n } ) ^ 2 } - \\frac { ( t ^ { - 2 } + 1 ) } { m \\xi ^ 0 + n } \\frac { 1 + t _ { + } ^ { m , n } t } { t - t _ { + } ^ { m , n } } \\right ] \\Big | _ { m = 0 } = \\frac { 2 } { t | n | } \\ , . \\end{align*}"}
+{"id": "4163.png", "formula": "\\begin{align*} \\bar \\Delta _ { k } = \\frac { \\hat { \\rho } _ { k } ^ { 2 } + 3 \\hat { \\rho } _ { k } + 1 } { \\mu ( 1 + \\hat { \\rho } _ k ) } + \\frac { ( \\hat { \\rho } _ k + 1 ) ^ 2 } { \\mu \\rho _ k } , \\end{align*}"}
+{"id": "1811.png", "formula": "\\begin{gather*} \\lim _ { n \\to \\infty } \\int _ { S } F \\ , d P _ n = \\int _ { S } F \\ , d Q . \\end{gather*}"}
+{"id": "1229.png", "formula": "\\begin{align*} u _ n ( 0 ) = \\sum _ { j = 1 } ^ { J } g _ n ^ j [ e ^ { i t _ n ^ j \\Delta } \\phi ^ j ] + W _ n ^ J \\end{align*}"}
+{"id": "1915.png", "formula": "\\begin{align*} \\left ( \\widehat { \\eta _ u } \\right ) _ { i + 1 / 2 } = \\{ \\eta _ u \\} _ { i + 1 / 2 } + \\left ( \\frac { 1 - 2 \\lambda } { 2 } \\right ) [ \\ ! [ \\eta _ u ] \\ ! ] _ { i + 1 / 2 } = 0 , \\end{align*}"}
+{"id": "7030.png", "formula": "\\begin{align*} E ^ \\Phi [ v ] : = \\mathop { \\inf } _ { \\rho \\in \\mathcal I _ N ( \\Omega ) } \\left \\{ F ^ \\Phi _ L [ \\rho ] - \\int _ { \\Omega } v \\ , d \\rho \\right \\} . \\end{align*}"}
+{"id": "7902.png", "formula": "\\begin{align*} \\begin{cases} \\det D ^ 2 \\varphi = \\lambda \\frac { ( - \\varphi ^ { \\star } ) ^ { n + 2 + k } } { \\varphi ^ { n + 2 + p } } \\\\ \\nabla \\varphi ( \\mathbb R ^ n ) = \\Omega ^ { \\circ } . \\end{cases} \\end{align*}"}
+{"id": "7450.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\vert \\nabla { y _ 1 } _ * \\vert ^ { p - 2 } \\nabla { y _ 1 } _ * \\cdot \\nabla h d z & + \\int _ { \\Omega } \\eta \\vert \\nabla { y _ 1 } _ * \\vert ^ { q - 2 } \\nabla { y _ 1 } _ * \\cdot \\nabla h d z \\\\ \\geq & \\int _ { \\Omega } a _ 1 { y _ 1 } _ * ^ { - \\nu } h d z + \\lambda ( \\kappa _ 1 + 1 ) \\int _ { \\Omega } \\vert { y _ 1 } _ * \\vert ^ { \\kappa _ 1 } \\vert { y _ 2 } _ * \\vert ^ { \\kappa _ 2 + 1 } h d z , \\end{align*}"}
+{"id": "106.png", "formula": "\\begin{align*} B _ u = u + \\Big [ - \\frac { \\ell _ { } } { 2 } , \\frac { \\ell _ { } } { 2 } \\Big ] ^ d . \\end{align*}"}
+{"id": "7410.png", "formula": "\\begin{align*} \\partial _ { t } ( \\rho _ { n } W _ { n } ) + \\partial _ { x } ( \\rho _ { n } W _ { n } u _ { n } ) = \\rho _ { n } ( \\partial _ { t } W _ { n } + u _ { n } \\partial _ { x } W _ { n } ) + W _ { n } ( \\partial _ { t } \\rho _ { n } + \\partial _ { x } ( \\rho _ { n } u _ { n } ) ) = 0 , \\end{align*}"}
+{"id": "2682.png", "formula": "\\begin{align*} ( t \\omega _ { 0 } - _ { \\omega _ { 0 } } + d d ^ c \\varphi _ t ) ^ { n } = e ^ { ( 1 + \\frac { \\alpha } { 2 \\beta } ) \\varphi _ t } \\omega _ { 0 } ^ { n } . \\end{align*}"}
+{"id": "7586.png", "formula": "\\begin{align*} | D _ t ^ { ( 0 ) } + D _ s ^ { ( \\ell ) } | ^ 2 & \\ge \\big ( | D _ t | - | D _ s | \\big ) ^ 2 + 2 | D _ t | \\cdot | D _ s | \\theta ^ 2 \\\\ & \\ge \\beta ^ { 1 / 2 } N ^ { 1 / 2 } \\left [ \\big ( t ^ { 3 / 4 } - s ^ { 3 / 4 } \\big ) ^ 2 + 2 t ^ { 3 / 4 } s ^ { 3 / 4 } \\theta ^ 2 \\right ] . \\end{align*}"}
+{"id": "4323.png", "formula": "\\begin{align*} { \\tt T } _ t ( B ( { \\sf f } ) ) = B ( u _ t { \\sf f } ) \\ \\ ( { \\sf f } \\in \\mathcal { K } \\ , , \\ t \\in \\mathbb { R } ) \\ , . \\end{align*}"}
+{"id": "323.png", "formula": "\\begin{gather*} \\left \\langle u , \\mathsf { L } _ { j } \\left ( s \\right ) \\overline { v } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } = \\left \\langle \\psi , \\gamma _ { \\operatorname * { N } ; j } ^ { \\operatorname * { e x t } } \\left ( s \\right ) \\overline { v } \\right \\rangle _ { \\Gamma j } \\qquad \\forall v \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } , \\mathbb { A } _ { j } ^ { \\operatorname * { e x t } } \\right ) . \\end{gather*}"}
+{"id": "3134.png", "formula": "\\begin{align*} \\mathbb { E } [ \\mathbf { H } ^ H _ { l , } \\mathbf { H } _ { l , } ( i , i ) ] = \\begin{cases} \\frac { L _ l } { N ( \\kappa + 1 ) } , & n = n ' ~ ~ p = p ' , \\\\ 0 , & . \\end{cases} \\end{align*}"}
+{"id": "6368.png", "formula": "\\begin{align*} ( t _ { H , H } \\otimes \\mathrm { I d } _ { H } ) ( 1 _ { H } \\otimes 1 _ { H } \\otimes 1 _ { H } ) = \\chi ^ { i } \\otimes \\chi _ { i } \\otimes 1 _ { H } = \\chi _ { 1 2 } \\end{align*}"}
+{"id": "2530.png", "formula": "\\begin{align*} I _ { } = \\frac { \\mathcal { M } ^ \\oplus _ { | \\cdot | } ( \\beta _ 1 , \\beta _ 2 , \\beta _ 3 ; \\boldsymbol { \\eta } , \\boldsymbol { \\zeta } , \\boldsymbol { \\tau } , \\boldsymbol { \\rho } ) ^ 2 } { \\| ( \\boldsymbol { y } - \\boldsymbol { \\eta } , \\boldsymbol { p } - \\boldsymbol { \\zeta } ) \\| _ { \\boldsymbol { H } ^ { \\textbf { c u r l } , \\frac { 1 } { 2 } } } } \\end{align*}"}
+{"id": "71.png", "formula": "\\begin{align*} a _ k = \\frac { b _ k - \\alpha _ k b _ { - k } ^ \\dagger } { \\sqrt { 1 - \\alpha _ k ^ 2 \\vphantom { \\alpha _ { p - k } ^ 2 } } } , a _ { p - k } = \\frac { b _ { p - k } - \\alpha _ { p - k } b _ { k - p } ^ \\dagger } { \\sqrt { 1 - \\alpha _ { p - k } ^ 2 } } . \\end{align*}"}
+{"id": "7392.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } \\frac { d } { d t } \\| V _ { n } \\| _ { L ^ { 2 } _ { x } } ^ { 2 } & + \\frac { 1 } { 5 } \\gamma _ { n } \\underline { \\rho _ { n } } ^ { \\gamma _ { n } } \\left \\| \\partial _ { x } V _ { n } \\right \\| _ { L ^ { 2 } _ { x } } ^ { 2 } \\le \\| V _ { n } \\| _ { L ^ { 2 } _ { x } } ^ { 2 } \\left \\{ C _ { 1 } + C _ { 2 } \\| V _ { n } \\| _ { L ^ { 2 } _ { x } } ^ { 2 } \\right \\} , \\end{aligned} \\end{align*}"}
+{"id": "124.png", "formula": "\\begin{align*} K _ \\ell \\ll \\begin{cases} \\delta ^ { - \\frac { 1 } { 2 6 } } , \\ ; & \\ ; d = 2 , \\\\ ( \\rho a ^ 3 ) ^ { - \\frac 1 { 2 8 } } , \\ ; & \\ ; d = 3 . \\end{cases} \\end{align*}"}
+{"id": "5563.png", "formula": "\\begin{align*} V _ k ( \\zeta ) = \\sum _ { j = 1 } ^ { k - 1 } ( 1 - P _ { k j } ) \\sigma _ j ( \\zeta ) - \\sum _ { j = k + 1 } ^ { m } P _ { k j } \\sigma _ j ( \\zeta ) . \\end{align*}"}
+{"id": "8443.png", "formula": "\\begin{align*} B _ { 0 } = - 1 , B _ { 1 } = \\frac { 1 } { 2 } , B _ { 2 } = \\frac { 1 } { 6 } , \\dots . \\end{align*}"}
+{"id": "1813.png", "formula": "\\begin{gather*} \\psi _ N ( \\underline { \\gamma } ) = \\sum _ { n = 0 } ^ { k - 1 } \\frac { \\gamma _ n } { ( n + \\alpha ) ^ s } \\phi \\left ( \\frac { n + \\alpha } { N } \\right ) \\end{gather*}"}
+{"id": "3860.png", "formula": "\\begin{align*} \\tau _ 1 & = \\partial _ y f \\partial _ x + \\partial _ y \\\\ \\tau _ 2 & = \\partial _ z f \\partial _ x + \\partial _ z . \\end{align*}"}
+{"id": "3075.png", "formula": "\\begin{align*} \\begin{aligned} { \\bf { F } } & = { { \\bf { A } } _ { { \\rm { T , } } \\bot } } { { \\bf { P } } ^ { 1 / 2 } } , \\\\ { \\bf { W } } & = { { \\bf { A } } _ { { \\rm { R , } } \\bot } } , \\end{aligned} \\end{align*}"}
+{"id": "1316.png", "formula": "\\begin{align*} & l _ 1 ^ { ' } ( a ) = - 2 ( \\kappa - 1 ) a ^ { 2 \\kappa - 3 } - \\kappa ( \\kappa - 1 ) a ^ { \\kappa - 1 } + 2 \\kappa ( \\kappa - 1 ) a ^ { \\kappa - 2 } - ( \\kappa - 1 ) ( \\kappa - 2 ) a ^ { \\kappa - 3 } \\\\ & = ( \\kappa - 1 ) a ^ { \\kappa - 3 } [ - 2 a ^ \\kappa - \\kappa a ^ 2 + 2 \\kappa a - \\kappa + 2 ] = ( \\kappa - 1 ) a ^ { \\kappa - 3 } [ - \\kappa ( a - 1 ) ^ 2 - 2 ( a ^ \\kappa - 1 ) ] \\ge 0 . \\end{align*}"}
+{"id": "8420.png", "formula": "\\begin{align*} Y ^ { ( j ) } ( \\ell ) = \\int _ { k } ^ { k + q } \\left ( j t ^ { j - 1 } h ^ { ( j ) } ( t \\ell ) + \\ell t ^ { j } h ^ { ( j + 1 ) } ( t \\ell ) \\right ) \\mathrm { d } t , j \\geq 1 . \\end{align*}"}
+{"id": "597.png", "formula": "\\begin{align*} \\mathcal H ( \\psi ^ \\mu ( t ) ) - \\mathcal H ( \\psi ^ \\mu ( 0 ) ) = \\int _ 0 ^ t \\left ( \\norm { P _ { \\mu } ( P _ { \\mu } \\psi ^ \\mu ( s ) ) \\mathfrak K _ 1 } _ { \\mathcal L _ 2 } ^ 2 - \\norm { \\left ( P _ { \\mu } \\psi ^ \\mu ( s ) \\right ) \\mathfrak K _ 1 } _ { \\mathcal L _ 2 } ^ 2 \\right ) \\ , d s , \\end{align*}"}
+{"id": "790.png", "formula": "\\begin{align*} c + o ( 1 ) & = I [ u _ n ] \\\\ & = \\frac { 1 } { p } ( [ u _ n ] _ { s , p } ^ p + a \\| u _ n \\| _ p ^ p ) \\\\ & \\phantom { = } - \\frac { b } { 2 } \\int _ { \\mathbb { R } ^ N } ( K \\ast F ( u _ n ) ) F ( u _ n ) d x - \\frac { \\varepsilon _ g } { p _ g } \\| u _ n \\| _ { p _ g } ^ { p _ g } . \\end{align*}"}
+{"id": "7927.png", "formula": "\\begin{align*} F ( I ( g _ j ) ) = 7 g _ j + 3 \\sum _ { k \\neq j } g _ k . \\end{align*}"}
+{"id": "1394.png", "formula": "\\begin{align*} \\tilde \\psi ( x ) = ( E + \\tilde H ( x ) ) \\psi ( x ) , x \\in [ 0 , \\pi ] , \\end{align*}"}
+{"id": "254.png", "formula": "\\begin{align*} \\Delta _ ( \\xi ) : = \\prod _ { \\substack { \\mu > 0 \\\\ _ \\mu \\cap \\overline { } \\neq \\emptyset } } \\langle \\mu - 2 \\xi , \\mu \\rangle , \\end{align*}"}
+{"id": "2868.png", "formula": "\\begin{align*} ^ { \\nu , ( c ) } _ { \\lambda , \\omega } ( t ) = \\begin{cases} W _ { 0 ; \\omega } ( t ) \\bigl ( U _ { \\lambda , \\omega } ( t ) - U _ { 0 , \\omega } ( t ) \\bigr ) & \\\\ W _ { 0 ; \\omega } ( t ) V _ { \\lambda , \\nu - \\lambda } ( t ) & \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "3334.png", "formula": "\\begin{align*} \\Lambda = - \\sum _ { i = 1 } ^ N d _ i ^ { - 1 } \\mathrm { L } ( z _ i ) \\end{align*}"}
+{"id": "4144.png", "formula": "\\begin{align*} \\bar { x } \\rhd \\bar { y } & = H _ { \\bar { x } } \\bar { y } = \\overline { \\nabla _ x y } , & \\bar { x } \\rhd \\bar { E } _ 1 & = H _ { \\bar { x } } \\rhd \\bar { E } _ 1 = \\overline { \\nabla _ x E _ 1 } , \\\\ \\bar { E } _ 1 \\rhd \\bar { x } & = \\xi _ { \\bar { E } _ 1 } \\bar { x } = \\overline { E _ 1 x } & \\bar { E } _ 1 \\rhd \\bar { E } _ 2 & = \\xi _ { \\bar { E } _ 1 } \\bar { E } _ 2 = \\overline { E _ 1 E _ 2 } - \\overline { E _ 2 E _ 1 } . \\end{align*}"}
+{"id": "3418.png", "formula": "\\begin{align*} | f | = \\begin{cases} e ^ { - o r d _ 0 ( f ) } , & z = 0 , \\\\ e ^ { - \\log | f ( z ) | _ \\infty / \\log | z | _ \\infty } , & . \\end{cases} \\end{align*}"}
+{"id": "387.png", "formula": "\\begin{align*} \\alpha ( \\eta x ) = j \\big ( ( \\eta x ) ^ * \\eta , ( \\eta x ) ' \\big ) = j ( 1 _ X , x ) = x , \\end{align*}"}
+{"id": "7640.png", "formula": "\\begin{align*} \\lambda ^ { \\gamma \\rq { } } ( U ( x ) + C ) & \\geq \\lambda ^ { \\gamma \\rq { } } ( U ( - \\underline x ) + C ) = \\lambda ^ { \\gamma \\rq { } } U ^ + ( \\overline x ) \\ge U ^ + ( \\overline x ) . \\end{align*}"}
+{"id": "175.png", "formula": "\\begin{align*} \\theta _ 3 ( v , \\tau + 1 ) = \\theta _ 2 ( v , \\tau ) , ~ ~ \\theta _ 3 ( v , - \\frac { 1 } { \\tau } ) = \\left ( \\frac { \\tau } { \\sqrt { - 1 } } \\right ) ^ { \\frac { 1 } { 2 } } e ^ { \\pi \\sqrt { - 1 } \\tau v ^ 2 } \\theta _ 3 ( \\tau v , \\tau ) , \\end{align*}"}
+{"id": "2474.png", "formula": "\\begin{align*} \\overline { H } ( G \\overset { ( \\alpha , 1 - \\alpha ) } { \\sqcup } G _ 5 ) = \\alpha \\overline { H } ( G ) + ( 1 - \\alpha ) \\overline { H } ( G _ 5 ) . \\end{align*}"}
+{"id": "3471.png", "formula": "\\begin{align*} \\omega _ { E _ J } ^ { n - m } = d _ 0 \\ldots d _ m ( L ^ { n + 1 } \\cdot M ) \\frac { C _ 0 } { m ! } \\Omega _ { E _ J } \\wedge \\overline { \\Omega } _ { E _ J } . \\end{align*}"}
+{"id": "8423.png", "formula": "\\begin{align*} A ( N ) & = \\sum _ { P < p \\leq 2 P } ( \\log p ) \\left ( \\left ( N + 1 - \\left [ p ^ { c } \\right ] \\right ) ^ { \\gamma } - \\left ( N - \\left [ p ^ { c } \\right ] \\right ) ^ { \\gamma } \\right ) \\\\ & = \\gamma \\sum _ { P < p \\leq 2 P } ( \\log p ) \\left ( \\left ( N - [ p ^ c ] \\right ) ^ { \\gamma - 1 } + \\O ( N ^ { \\gamma - 2 } ) \\right ) \\asymp N ^ { 2 \\gamma - 1 } . \\end{align*}"}
+{"id": "7532.png", "formula": "\\begin{align*} P = P _ X \\times _ k M . \\end{align*}"}
+{"id": "6387.png", "formula": "\\begin{align*} \\mathcal { R } * \\chi = \\chi ^ { \\mathrm { o p } } * \\mathcal { R } \\end{align*}"}
+{"id": "98.png", "formula": "\\begin{align*} \\Phi ( z ) : = \\langle z \\vert \\Psi \\rangle , z \\in \\mathbb { C } , \\end{align*}"}
+{"id": "909.png", "formula": "\\begin{align*} \\frac { c } { x } \\bigg ( - \\tau _ t + \\frac { 1 - \\alpha } { t } \\tau \\bigg ) + n x ^ k \\eta _ 2 - t ^ { 1 - \\alpha } \\xi _ { 1 t } - 2 \\eta _ { 1 x } + \\xi _ { 1 x x } + \\frac { c } { x ^ 2 } \\xi _ 1 + \\frac { c } { x } \\xi _ { 1 x } - m x ^ k \\phi _ 2 = 0 , \\end{align*}"}
+{"id": "1902.png", "formula": "\\begin{align*} \\widehat { u _ h } : = \\lambda u _ h ^ - + \\left ( 1 - \\lambda \\right ) u _ h ^ + = \\{ u _ h \\} + \\left ( \\frac { 1 - 2 \\lambda } { 2 } \\right ) [ \\ ! [ u _ h ] \\ ! ] , \\end{align*}"}
+{"id": "7127.png", "formula": "\\begin{align*} \\| \\rho ^ N \\| _ { H ^ m } \\leq \\| \\rho _ 0 \\| _ { H ^ m } \\exp \\left ( C \\| \\rho _ 0 \\| _ { L ^ \\infty } t + \\frac { \\| \\nabla \\rho _ 0 \\| _ { L ^ \\infty } } { \\| \\rho _ 0 \\| _ { L ^ \\infty } } \\left ( \\exp ( C \\| \\rho _ 0 \\| _ { L ^ \\infty } t ) - 1 \\right ) \\right ) = : B _ { \\rho _ 0 } ( t ) . \\end{align*}"}
+{"id": "4672.png", "formula": "\\begin{align*} | S | & > ( 2 m + 1 ) \\cdot ( \\frac { 2 n } { 2 k + 3 } - 2 ) - \\binom { 2 m + 1 } { 2 } \\cdot \\epsilon n \\\\ & = \\frac { 2 n } { 2 k + 3 } \\cdot ( 2 m + 1 ) - 2 ( 2 m + 1 ) - ( 2 m + 1 ) \\cdot m \\cdot \\epsilon n \\\\ & > \\frac { 6 k } { ( 2 k + 3 ) \\cdot 3 k + 1 } \\cdot ( 2 m + 1 ) n \\end{align*}"}
+{"id": "378.png", "formula": "\\begin{align*} X ^ + \\ : = \\ \\Phi _ ! \\ , t _ * ( X ) \\ = \\ \\sum _ { \\varphi : \\Phi } X ^ { [ \\varphi ] } \\ , . \\end{align*}"}
+{"id": "5716.png", "formula": "\\begin{align*} \\widetilde { D } _ { 0 } ^ { 0 } \\ , = \\ , \\begin{pmatrix} k _ 0 ^ 3 \\ , 8 \\pi ^ 2 & 0 & 0 \\\\ - 6 \\pi \\ , k _ 0 ^ 3 & k _ 0 ^ 3 \\ , 8 \\pi ^ 2 & 0 \\\\ 0 & 0 & - k _ 0 ^ 3 \\ , 4 \\pi ^ 2 \\end{pmatrix} \\ , . \\end{align*}"}
+{"id": "1440.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n + 1 } \\sigma _ i ^ 2 - \\sum _ { i = 1 } ^ { n + 1 } \\sigma _ i \\sigma _ { i + 1 } = 2 \\sum _ { i = 1 } ^ { n + 1 } \\mu _ i \\sigma _ i . \\footnote { C o n v e n t i o n a l l y , w e s e t $ \\sigma _ { \\ell } = \\sigma _ i $ i f $ \\ell \\equiv i \\ ( \\mathrm { m o d } \\ n + 1 ) $ . T h e r e f o r e , $ \\sigma _ { n + 2 } = \\sigma _ 1 $ . } \\end{align*}"}
+{"id": "2100.png", "formula": "\\begin{align*} | \\alpha _ 1 ^ { ( n ) } ( 2 u ) | \\leq \\frac { K _ 2 \\gamma } { \\varphi ^ { 3 / 4 } ( u ) } , n = 0 , 1 . \\end{align*}"}
+{"id": "8951.png", "formula": "\\begin{align*} \\hat { f } ( a ^ s \\Lambda _ { u , v } ) = \\sum _ { ( \\bar { p } , \\bar { q } ) \\in \\Z ^ { m + n } } f \\left ( e ^ { w _ 1 s } ( p _ 1 + \\langle \\bar { u } _ 1 , \\bar { q } \\rangle + v _ 1 ) , \\ldots , e ^ { w _ m s } ( p _ m + \\langle \\bar { u } _ m , \\bar { q } \\rangle + v _ m ) , e ^ { - s } \\bar { q } \\right ) . \\end{align*}"}
+{"id": "827.png", "formula": "\\begin{align*} \\eta ( e ^ { i \\theta _ 0 } ) & = ( \\Psi _ { \\epsilon } \\circ \\tilde { h } ) ( e ^ { i \\theta _ 0 } ) \\\\ & = \\Psi _ { \\epsilon } ( 1 ) \\\\ & = 1 , \\end{align*}"}
+{"id": "6432.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow + \\infty } \\{ \\sup _ { t \\geq t _ n , | x - c t | \\leq R } \\tilde { v } _ n ( x , t ) \\} = 0 . \\end{align*}"}
+{"id": "5801.png", "formula": "\\begin{align*} \\begin{array} { l l l } & w _ i = s _ i w _ { i + 1 } s _ i & i = 1 , \\dots , n - 1 , \\\\ & w _ i = s _ { \\alpha _ n + \\alpha _ { n - 1 } + \\dots + \\alpha _ i } & i = 1 , \\dots , n . \\end{array} \\end{align*}"}
+{"id": "1661.png", "formula": "\\begin{align*} C _ { \\texttt { a } } ( \\xi _ 1 , \\ldots , \\xi _ n ; q ) = C _ { \\texttt { a } } ( \\boldsymbol { \\xi } ; q ) : = \\prod _ { 1 \\leq j < k \\leq n } \\left ( \\frac { 1 - q e ^ { - i ( \\xi _ { j } - \\xi _ k ) } } { 1 - e ^ { - i ( \\xi _ { j } - \\xi _ k ) } } \\right ) , \\end{align*}"}
+{"id": "1358.png", "formula": "\\begin{align*} \\frac { E _ 8 ( z ) } { \\Delta ( z ) } = q ^ { - 1 } + 5 0 4 + 7 3 7 6 4 q + 2 6 9 5 0 4 0 q ^ 2 + O \\left ( q ^ 3 \\right ) , \\end{align*}"}
+{"id": "8060.png", "formula": "\\begin{align*} { \\rm s u p p } \\psi _ 0 \\subseteq \\{ x \\in \\mathbb R ^ { n } \\colon \\vert x \\vert \\leq 1 \\} ; \\ \\int { \\psi _ 0 } = 1 , \\end{align*}"}
+{"id": "8948.png", "formula": "\\begin{align*} \\{ ( \\bar { x } , \\bar { y } ) \\in \\R ^ { m + n } : \\upsilon _ 1 \\leq \\| \\bar { y } \\| \\leq \\upsilon _ 2 , | x _ i | \\leq \\vartheta \\| \\bar { y } \\| ^ { - w _ i } , i = 1 , \\ldots , n \\} , \\end{align*}"}
+{"id": "6991.png", "formula": "\\begin{align*} l _ i < u _ i \\quad \\lim _ { i \\to \\infty } l _ i = \\infty . \\end{align*}"}
+{"id": "6757.png", "formula": "\\begin{align*} \\mathcal { A } ( c ) = \\mathcal { A } ( d ) \\implies \\mathcal { B } ( c ) = \\mathcal { B } ( d ) \\end{align*}"}
+{"id": "8829.png", "formula": "\\begin{align*} x ^ { * } = [ \\frac { 1 - \\omega _ { 1 } } { 2 } \\frac { \\pi \\sqrt { 6 } } { 2 4 } \\frac { h _ { T } ^ { \\beta - 1 } } { \\bar { \\alpha } } , \\dots , \\frac { 1 - \\omega _ { d } } { 2 } \\frac { \\pi \\sqrt { 6 } } { 2 4 } \\frac { h _ { T } ^ { \\beta - 1 } } { \\bar { \\alpha } } ] , \\end{align*}"}
+{"id": "7658.png", "formula": "\\begin{align*} 0 & = ( D _ 1 ^ 6 - 5 D _ 1 ^ 3 D _ 3 - 5 D _ 3 ^ 2 + 9 D _ 1 D _ 5 ) ( Z , Z ) ( \\mathbf { 0 } ) \\\\ 0 & = ( D _ 1 ^ 8 + 7 D _ 1 ^ 5 D _ 3 - 3 5 D _ 1 ^ 2 D _ 3 ^ 2 - 2 1 D _ 1 ^ 3 D _ 5 - 4 2 D _ 3 D _ 5 + 9 0 D _ 1 D _ 7 ) ( Z , Z ) ( \\mathbf { 0 } ) \\end{align*}"}
+{"id": "1819.png", "formula": "\\begin{gather*} A ^ 2 ( U ) = \\bigcup _ { N _ 0 \\geq 0 } \\bigcap _ { N > N _ 0 } \\bigcap _ { \\alpha \\in \\mathcal { A } _ \\rho ( c ) } \\Gamma ( \\alpha , N ) ^ { ( \\epsilon ) } , \\end{gather*}"}
+{"id": "8711.png", "formula": "\\begin{align*} \\delta _ { t + 1 } \\leq \\left ( 1 - \\frac { 2 } { t } \\right ) \\delta _ { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { t ^ { p _ { i } + 1 } } \\enspace , \\end{align*}"}
+{"id": "6175.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j ' \\leq k } a _ { i + 1 , j ' } = 1 \\Leftrightarrow d _ { i + 1 , j } \\leq d _ { i , j } = d _ { i + 1 , j + 1 } , \\end{align*}"}
+{"id": "1021.png", "formula": "\\begin{align*} q ( R ) ^ E = - \\tfrac { k } { k + 1 } \\dd _ 0 ^ * \\dd _ 0 + \\tfrac { ( n + k - 2 ) ( n + 2 k - 4 ) } { ( n + 2 k - 2 ) ( n + k - 3 ) } \\dd _ 0 \\dd _ 0 ^ * + ( P _ 3 ^ E ) ^ * P _ 3 ^ E . \\end{align*}"}
+{"id": "2581.png", "formula": "\\begin{align*} f \\left ( ( z , h ) + ( z ' , h ' ) \\right ) & = f ( z + z ' - \\beta ( h , h ' ) , h + h ' ) = \\\\ & = z + z ' - \\beta ( h , h ' ) + \\alpha ( h + h ' ) = \\\\ & = z + z ' + \\alpha ( h ) + \\alpha ( h ' ) = \\\\ & = z + \\alpha ( h ) + z ' + \\alpha ( h ' ) = f ( z , h ) + f ( z ' , h ' ) . \\end{align*}"}
+{"id": "577.png", "formula": "\\begin{align*} \\mathbf U ( t ) & = \\mathbf S ( t ) \\mathbf u _ 0 + i \\int _ 0 ^ { t \\wedge \\mu } \\mathbf S ( t - s ) \\mathbf N ( \\mathbf U ( s ) ) \\ , d s + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf M ( \\mathbf U ( s ) ) \\ , d W ( s ) , \\\\ \\mathbf V ( t ) & = \\mathbf S ( t ) \\mathbf u _ 0 + i \\int _ 0 ^ { t \\wedge \\mu } \\mathbf S ( t - s ) \\mathbf N ( \\mathbf V ( s ) ) \\ , d s + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf M ( \\mathbf V ( s ) ) \\ , d W ( s ) . \\end{align*}"}
+{"id": "7570.png", "formula": "\\begin{align*} I _ 2 ( \\beta , N , T ) & = E ^ { \\mathbb { P } _ T ^ \\lambda } \\left [ \\sum _ { k = 1 } ^ { N } \\left \\{ \\int _ { 0 } ^ { T } \\lambda _ k ( t ) \\cdot d B ^ { ( k ) } _ t - \\frac { 1 } { 2 } \\int _ { 0 } ^ { T } | \\lambda _ k ( t ) | ^ 2 d t \\right \\} \\right ] \\\\ & = - \\frac { N } { 2 } \\kappa ^ 2 ( \\alpha + 1 ) ^ 2 E \\left [ \\int _ { 0 } ^ { T } t ^ { 2 \\alpha } d t \\right ] \\\\ & = - \\frac { \\kappa ^ 2 ( \\alpha + 1 ) ^ 2 } { 2 } N \\cdot \\frac { T ^ { 2 \\alpha + 1 } } { 2 \\alpha + 1 } . \\end{align*}"}
+{"id": "3166.png", "formula": "\\begin{align*} W _ S = \\frac { ( 1 + x ) ^ 2 ( 1 + y ) ^ 2 } { x y } - 4 . \\end{align*}"}
+{"id": "1855.png", "formula": "\\begin{align*} \\widetilde { U } _ { s _ n , t _ n } ( 0 , \\Delta ) & = \\int _ { \\mathbb { R } } U _ { s _ n , t _ n } ( x , \\Delta ) \\ , d x \\\\ & = \\int _ { \\mathbb { R } } \\mathbf { 1 } _ { ( s _ n , t _ n ) } ( x ) \\ , d x + O \\left ( \\int _ { \\mathbb { R } } K _ { s _ n , t _ n } ( x , \\Delta ) \\ , d x \\right ) \\\\ & = 2 \\pi \\delta + O \\left ( \\frac { 1 } { \\Delta } \\right ) , \\end{align*}"}
+{"id": "6155.png", "formula": "\\begin{align*} S = \\begin{cases} ( \\emptyset , \\emptyset ) & ( \\{ p _ { i , j } \\} , \\{ q _ { i , j } \\} ) = 0 , \\\\ ( \\mathbf { k } ) \\mid _ { \\eta _ = A } & ( \\{ p _ { i , j } \\} , \\{ q _ { i , j } \\} ) = + 1 , \\\\ - ( \\mathbf { k } ) \\mid _ { \\eta _ = A } & ( \\{ p _ { i , j } \\} , \\{ q _ { i , j } \\} ) = - 1 . \\end{cases} \\end{align*}"}
+{"id": "1684.png", "formula": "\\begin{align*} P _ { \\texttt { b } ; \\mu } ( \\boldsymbol { \\xi } ; q , q _ 0 ) & : = \\\\ \\sum _ { \\substack { \\sigma \\in S _ n \\\\ \\epsilon \\in \\{ 1 , - 1 \\} ^ n } } & C _ { \\texttt { b } } ( \\epsilon _ 1 \\xi _ { \\sigma _ 1 } , \\ldots , \\epsilon _ n \\xi _ { \\sigma _ n } ; q , q _ 0 ) \\exp ( i \\epsilon _ 1 \\xi _ { \\sigma _ 1 } \\mu _ 1 + \\cdots + i \\epsilon _ n \\xi _ { \\sigma _ n } \\mu _ n ) , \\end{align*}"}
+{"id": "8429.png", "formula": "\\begin{align*} X ^ \\star = ( C + \\Sigma ) ^ { 1 / 2 } Q \\psi ( \\Lambda ) Q ^ T ( C + \\Sigma ) ^ { 1 / 2 } - \\Sigma . \\end{align*}"}
+{"id": "7702.png", "formula": "\\begin{align*} & \\iota ^ \\sharp ( x ^ i ) = \\iota ^ * ( x ^ i ) = 0 , \\ ; \\ ; \\ ; 1 \\leq i \\leq r _ 0 , \\\\ & \\iota ^ \\sharp ( e ^ \\mu ) = 0 , \\ ; \\ ; 1 \\leq \\mu \\leq r _ 1 , \\\\ & \\iota ^ \\sharp ( p ^ I ) = 0 , \\ ; \\qquad \\ ; 1 \\leq I \\leq r _ 2 , \\end{align*}"}
+{"id": "4053.png", "formula": "\\begin{align*} \\rho _ { 1 , 1 , n } = n \\pi + \\frac { \\kappa _ { 1 , 1 , n } } { n } , \\ , \\ , \\{ \\kappa _ { 1 , 1 , n } \\} \\in l _ 2 , n \\in \\mathbb { Z } ; \\end{align*}"}
+{"id": "7756.png", "formula": "\\begin{align*} V ^ P : = \\widetilde { V } + f _ 3 \\partial _ { \\sigma } , f _ 3 : = f - \\frac { 1 } { 2 } g _ { N } ( V , V ) \\end{align*}"}
+{"id": "7789.png", "formula": "\\begin{align*} f = f ^ { } + f ^ { } - 1 6 \\pi c _ { \\ell } + f ^ { } , \\end{align*}"}
+{"id": "1111.png", "formula": "\\begin{align*} \\begin{aligned} \\langle \\mathcal { J } ' _ { + } ( u ) , v \\rangle = { \\langle u , v \\rangle } - \\int _ D \\alpha ( x ) u ( x ) v ( x ) d \\mu - \\int _ D f _ + ( x , u ( x ) ) v ( x ) d \\mu , \\end{aligned} \\end{align*}"}
+{"id": "490.png", "formula": "\\begin{align*} \\alpha = \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} , \\beta = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "242.png", "formula": "\\begin{align*} L ^ { \\emph { r } } _ x \\Phi ^ { \\emph { r } } _ \\xi ( x ; g ) = \\langle \\xi , \\xi \\rangle \\Phi ^ { \\emph { r } } _ \\xi ( x ; g ) \\end{align*}"}
+{"id": "8647.png", "formula": "\\begin{align*} \\int _ { \\mathcal { S } _ b } P ( \\Sigma _ b ) \\nabla ^ { \\mu } _ { X , z } u \\cdot \\nabla _ { X , z } ^ { \\mu } \\varphi \\ : \\mathrm { d } z \\mathrm { d } X & = \\int _ { \\mathcal { S } _ b } f \\varphi \\ : \\mathrm { d } z \\mathrm { d } X + \\int _ { \\R ^ d } g \\ : \\varphi | _ { z = - h _ b } \\ : \\mathrm { d } X , \\end{align*}"}
+{"id": "5404.png", "formula": "\\begin{align*} \\sum _ { j : j \\neq k } S _ { m j } ( t ) S ^ { - 1 } _ { j \\ell } ( t ) = \\delta _ { m \\ell } - S _ { m k } ( t ) S ^ { - 1 } _ { k \\ell } ( t ) , 1 \\leq k , \\ell , m \\leq N , \\ , t \\geq 0 . \\end{align*}"}
+{"id": "6838.png", "formula": "\\begin{align*} & K _ { 0 } = 2 ^ { - 8 } ( n - 6 ) ^ 2 ( n - 2 ) ^ 2 ( n + 2 ) ^ 2 & \\\\ & K _ { 2 } = 2 ^ { - 4 } ( 3 n ^ 4 - 2 4 n ^ 3 + 7 2 n ^ 2 - 9 6 n + 3 0 4 ) & \\\\ & K _ { 4 } = 2 ^ { - 2 } ( 3 n ^ 2 - 1 2 n + 4 4 ) & \\\\ & J _ { 0 } = 2 ^ { - 3 } ( 3 n ^ 4 - 1 8 n ^ 3 - 1 9 2 n ^ 2 + 1 8 6 4 n - 3 9 5 2 ) & \\\\ & J _ { 1 } = 2 ^ { - 1 } ( 3 n ^ 3 + 3 n ^ 2 - 2 4 4 n + 6 2 0 ) & \\\\ & J _ { 2 } = 2 n ^ 2 + 1 3 n - 6 8 & \\\\ & J _ { 3 } = 2 ( n + 1 ) & \\\\ & L _ { 0 } = 2 ^ { - 2 } ( 3 n ^ 2 - 1 2 n - 2 0 ) & \\end{align*}"}
+{"id": "8628.png", "formula": "\\begin{align*} \\mathcal { B } [ \\beta b ] ^ * \\nabla _ X \\psi = b \\nabla _ X \\nabla _ X \\cdot ( b \\nabla _ X \\psi ) + b \\nabla _ X ( b \\nabla _ X \\cdot ( h _ b \\nabla _ X \\psi ) ) + 2 b \\nabla _ X ( h _ b \\nabla _ X b \\cdot \\nabla _ X \\psi ) . \\end{align*}"}
+{"id": "8187.png", "formula": "\\begin{align*} \\hat { P } : = & x ( x , v _ 1 ^ { - \\alpha _ 1 } ) v _ 1 ^ { - \\alpha _ 1 } ( v _ 1 ^ { - \\alpha _ 1 } , v _ 1 ^ { \\alpha _ 1 } ) v _ 1 ^ { \\alpha _ 1 } ( v _ 1 ^ { \\alpha _ 1 } , v _ { 2 } ^ { \\beta _ 1 } ) v _ 2 ^ { \\beta _ 1 } \\dots \\\\ & \\dots v _ { n - 1 } ^ { \\alpha _ { n - 1 } } ( v _ { n - 1 } ^ { \\alpha _ { n - 1 } } , v _ { n } ^ { \\beta _ { n - 1 } } ) v _ n ^ { \\beta _ { n - 1 } } ( v _ n ^ { \\beta _ { n - 1 } } , v _ n ^ { - \\beta _ { n - 1 } } ) v _ n ^ { - \\beta _ { n - 1 } } ( v _ { n } ^ { - \\beta _ { n - 1 } } , y ) y \\end{align*}"}
+{"id": "4274.png", "formula": "\\begin{align*} H _ { n \\times p } = \\left ( \\begin{array} { c c c c c } a _ { n - 1 } & a _ { n - 2 } & a _ { n - 3 } & \\cdots & a _ { n - p } \\\\ a _ { n - 2 } & a _ { n - 3 } & a _ { n - 4 } & \\cdots & a _ { n - p - 1 } \\\\ a _ { n - 3 } & a _ { n - 4 } & a _ { n - 5 } & \\cdots & a _ { n - p - 2 } \\\\ \\vdots & \\vdots & \\vdots & \\cdots & \\vdots \\\\ a _ { 0 } & a _ { - 1 } & a _ { - 2 } & \\cdots & a _ { 1 - p } \\end{array} \\right ) . \\end{align*}"}
+{"id": "550.png", "formula": "\\begin{align*} \\tau _ R ( \\omega ) = \\sup \\left \\{ t \\in [ 0 , \\tau ( \\omega ) ) \\colon \\right \\} \\end{align*}"}
+{"id": "3955.png", "formula": "\\begin{align*} f ( x , v ) = f ( x ' , v ) \\\\ f ( u , y ) = f ( u , y ' ) \\end{align*}"}
+{"id": "7112.png", "formula": "\\begin{align*} S _ { r } ( N ) \\ , \\ll \\ , S _ { r , 1 } ( N ) \\ , = S ^ 0 _ { r , 1 } ( N ) + S ^ { \\neq 0 } _ { r , 1 } ( N ) . \\end{align*}"}
+{"id": "4064.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { R } _ k q _ 0 ( t ) & = \\mathcal { K } _ 0 ( \\mathcal { R } _ { k - 1 } q _ 0 ( t ) ) , & \\mathcal { R } _ k \\tilde { q } _ 0 ( t ) & = \\mathcal { K } _ 0 ( \\mathcal { R } _ { k - 1 } \\tilde { q } _ 0 ( t ) ) , \\\\ \\mathcal { R } _ k p ( t ) & = \\mathcal { K } _ 1 ( \\mathcal { R } _ { k - 1 } p ( t ) ) , & \\mathcal { R } _ k \\tilde { p } ( t ) & = \\mathcal { K } _ 1 ( \\mathcal { R } _ { k - 1 } \\tilde { p } ( t ) ) \\end{aligned} \\end{align*}"}
+{"id": "6692.png", "formula": "\\begin{align*} h _ j ( \\beta , p , q ) = \\Big ( - { 2 \\over \\beta } \\Big ) ^ { j + 1 } h _ j \\Big ( { 4 \\over \\beta } , - { \\beta p \\over 2 } , - { 2 q \\over \\beta } \\Big ) , \\tilde { h } _ j ( \\beta , p , q ) = \\Big ( - { 2 \\over \\beta } \\Big ) ^ { j + 1 } \\tilde { h } _ j \\Big ( { 4 \\over \\beta } , - { \\beta p \\over 2 } , - { 2 q \\over \\beta } \\Big ) . \\end{align*}"}
+{"id": "6418.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { t \\rightarrow \\infty } \\sup _ { | x | > c t } w ( x , t ) = 0 ; \\end{align*}"}
+{"id": "4961.png", "formula": "\\begin{align*} P _ { 2 , 1 } ( n ) P _ { 2 , 3 } ( n ) = \\Phi _ { 4 n } ( 2 ) . \\end{align*}"}
+{"id": "5499.png", "formula": "\\begin{align*} O _ A ^ { ( b a ^ k g ) } = & \\ , c _ + ^ { ( b a ^ k , g ) } O _ A ^ { ( g ) } + \\frac { c _ + ^ { ( b a ^ k , g ) } - \\left ( c _ + ^ { ( b a ^ k , g ) } \\right ) ^ { - 1 } } { \\big ( Q ^ 2 - Q ^ { - 2 } \\big ) ^ 2 } O _ B ^ { ( 1 ) } O _ B ^ { ( 1 ) } O _ A ^ { ( g ) } , \\\\ O _ A ^ { ( b ^ { - 1 } a ^ k g ) } = & \\ , c _ - ^ { ( b ^ { - 1 } a ^ k , g ) } O _ A ^ { ( g ) } + \\frac { c _ - ^ { ( b ^ { - 1 } a ^ k , g ) } - \\left ( c _ - ^ { ( b ^ { - 1 } a ^ k , g ) } \\right ) ^ { - 1 } } { \\big ( Q ^ 2 - Q ^ { - 2 } \\big ) ^ 2 } O _ B ^ { ( 1 ) } O _ B ^ { ( 1 ) } O _ A ^ { ( g ) } , \\end{align*}"}
+{"id": "6684.png", "formula": "\\begin{align*} \\overline { W } _ { 1 } ^ { 0 } ( x ) = \\frac { 1 } { x } , \\overline { W } _ { 1 } ^ { 1 } ( x ) = \\frac { \\kappa p + i q } { \\kappa x ( 1 - x ) } , \\overline { W } _ { 1 } ^ { 2 } ( x ) = \\frac { ( \\kappa p - i q - \\kappa + 1 ) ( \\kappa p + i q ) } { \\kappa ^ 2 ( x - 1 ) ^ 2 } , \\end{align*}"}
+{"id": "3498.png", "formula": "\\begin{align*} \\norm { s } _ { y , t } ^ 2 \\geq & ( 1 - C ( l ) | t | ^ \\epsilon ) \\sum \\norm { F _ 0 ^ { l _ 0 } \\ldots F _ m ^ { l _ m } s _ { l _ 0 , \\ldots l _ m } } _ { y , t } ^ 2 \\\\ \\geq & C ( l ) ^ { - 1 } \\sum \\norm { s _ { l _ 0 , \\ldots l _ m } } _ { V _ { l - \\sum d _ i l _ i } } ^ 2 | t | ^ { 2 \\sum l _ i y _ i } . \\end{align*}"}
+{"id": "6528.png", "formula": "\\begin{align*} | k | \\leq 2 L , \\ n \\neq n ' , \\ | ( n , n ' ) | \\leq L , \\ ( n , n ' ) \\notin \\{ n ^ { ( l ) } \\} _ { l = 1 } ^ b \\times \\{ n ^ { ( l ) } \\} _ { l = 1 } ^ b . \\end{align*}"}
+{"id": "8695.png", "formula": "\\begin{align*} S ( t ) S ( - t ) = 1 . \\end{align*}"}
+{"id": "2125.png", "formula": "\\begin{align*} \\hat e ( t , x ) = - \\tilde e ( t , x ) , \\end{align*}"}
+{"id": "2089.png", "formula": "\\begin{align*} \\lim _ { t \\longrightarrow + \\infty } \\int _ { | x - v t | \\lesssim \\omega ( t ) } \\left [ ( \\partial _ t \\Lambda ) ^ 2 + ( \\partial _ x \\Lambda ) ^ 2 + \\sinh ^ 2 ( \\Lambda ) \\Big ( ( \\partial _ t \\phi ) ^ 2 + ( \\partial _ x \\phi ) ^ 2 \\Big ) \\right ] d x = 0 . \\end{align*}"}
+{"id": "1924.png", "formula": "\\begin{align*} \\left ( \\Pi _ { \\lambda _ 1 , \\lambda _ 2 } g \\right ) ^ { - , - } _ { i + \\frac { 1 } { 2 } , j + \\frac { 1 } { 2 } } = g ^ { - , - } _ { i + \\frac { 1 } { 2 } , j + \\frac { 1 } { 2 } } , \\mbox { i f } \\left ( i , j \\right ) \\in \\left ( \\ , \\gamma _ 2 , \\gamma _ 1 \\ , \\right ) \\end{align*}"}
+{"id": "886.png", "formula": "\\begin{align*} & A + \\frac { \\sqrt { a b } } { a } B = \\frac { \\alpha } { t ^ \\alpha } \\mathrm { e } ^ { - \\frac { \\alpha ( x + y ) } { t ^ \\alpha } } \\bigg ( \\frac y x \\bigg ) ^ { \\frac { \\sqrt { a b } - 1 } { 2 } } I _ { \\sqrt { a b } - 1 } \\bigg ( \\frac { 2 \\alpha \\sqrt { x y } } { t ^ \\alpha } \\bigg ) , \\end{align*}"}
+{"id": "3610.png", "formula": "\\begin{align*} { { P } _ { { \\rm { D B } } } } = \\left \\{ \\frac { { E { \\sum \\limits _ { m = 1 } ^ { { M _ { \\rm { R } } } } { \\sum \\limits _ { k = 1 } ^ K { \\sum \\limits _ { n = 1 } ^ K { { \\left | { \\left [ { { { \\bf { \\Xi } } _ { { \\rm { D B } } , m } } } \\right ] _ { k , k } ^ * { { \\left [ { { { \\bf { \\Xi } } _ { { \\rm { D B } } , m } } } \\right ] } _ { n , n } } } \\right | } } } } } } } { { K { \\sigma ^ 2 } } } < \\gamma _ { \\rm t h } \\right \\} . \\end{align*}"}
+{"id": "757.png", "formula": "\\begin{align*} \\nu ( t ) - \\nu ( r ) = 2 \\int _ { \\R ^ d } \\int _ r ^ t \\int _ 0 ^ s \\frac { \\left [ V ( s , \\zeta ) - V ( \\sigma , \\zeta ) \\right ] \\overline { \\left [ W ( s , \\zeta ) - W ( \\sigma , \\zeta ) \\right ] } } { ( s - \\sigma ) ^ { 1 + 2 b } } \\ , d \\sigma \\ , d s \\ , d \\zeta , \\end{align*}"}
+{"id": "6373.png", "formula": "\\begin{align*} 3 ) \\ r - s - n - o - \\lambda a = - \\lambda a + n + o - r + s ; \\end{align*}"}
+{"id": "5197.png", "formula": "\\begin{align*} M = D ^ 2 \\ell / ( D - \\ell ) \\geq \\ell \\sum _ { w \\not \\sim v } | N _ w \\cap N _ v | / ( D - \\ell ) \\geq \\ell \\sum _ { w \\not \\sim v } | \\tau ( N _ w \\cap N _ v ) | / ( D - \\ell ) \\end{align*}"}
+{"id": "1300.png", "formula": "\\begin{align*} & \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| u _ n ^ J \\| _ { S ( \\R ) } \\lesssim 1 , \\\\ & \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| e ^ { i t \\Delta } ( u _ { 0 , n } - u _ { n } ^ J ( 0 ) ) \\| _ { S ( \\R ) } = 0 \\\\ & \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| \\nabla [ ( i \\partial _ t + \\Delta ) u _ n ^ J + | x | ^ { - b } ( I _ { \\alpha } \\ast | u _ n ^ J | ^ p ) | u _ n ^ J | ^ { p - 2 } u _ n ^ J ] \\| _ { N ( \\R ) } = 0 . \\end{align*}"}
+{"id": "2706.png", "formula": "\\begin{align*} \\begin{array} { r c l } ( a _ 1 , { \\dots } , a _ { 2 k + 1 } ) \\cdot ( a ' _ 1 , a ' _ 2 , { \\dots } , a ' _ { 2 k ' + 1 } ) & : = & ( a _ 1 , { \\dots } , a _ { 2 k } , a _ { 2 k + 1 } \\cdot a ' _ 1 , a ' _ 2 , { \\dots } , a ' _ { 2 k ' + 1 } ) , \\end{array} \\end{align*}"}
+{"id": "7175.png", "formula": "\\begin{align*} v _ { 0 \\dots 0 } = F _ 1 ^ 1 \\cap \\dots \\cap F _ { n _ 1 } ^ 1 \\cap \\dots \\cap F _ 1 ^ m \\cap \\dots \\cap F _ { n _ m } ^ m . \\end{align*}"}
+{"id": "8105.png", "formula": "\\begin{align*} \\uppercase \\expandafter { \\romannumeral 2 } \\leq \\sum \\limits _ { j = 1 } ^ { \\infty } \\vert \\mu _ { j } \\vert \\vert T ( b _ { j } ) ( x ) \\vert \\chi _ { P _ { j } ^ { * } } ( x ) + \\sum \\limits _ { j = 1 } ^ { \\infty } \\vert \\mu _ { j } \\vert \\vert T ( b _ { j } ) ( x ) \\vert \\chi _ { ( P _ { j } ^ { * } ) ^ { c } } ( x ) = : \\uppercase \\expandafter { \\romannumeral 1 } _ { 1 } + \\uppercase \\expandafter { \\romannumeral 1 } _ { 2 } \\end{align*}"}
+{"id": "2437.png", "formula": "\\begin{align*} u _ n = \\lambda _ n ^ { - \\frac { 1 } { 2 } } \\varphi _ n \\in H _ 0 ^ 1 ( \\Omega ) \\end{align*}"}
+{"id": "6837.png", "formula": "\\begin{align*} P _ { \\rm a n g } : = 2 \\partial ^ { ( 4 ) } _ t \\Delta _ { \\theta } - J _ 3 \\partial ^ { ( 3 ) } _ t \\Delta _ { \\theta } + J _ 2 \\partial ^ { ( 2 ) } _ t \\Delta _ { \\theta } - J _ 1 \\partial _ t \\Delta _ { \\theta } + J _ 0 \\Delta _ { \\theta } + 3 \\partial ^ { ( 2 ) } _ t \\Delta ^ 2 _ { \\theta } - L _ 0 \\Delta ^ 2 _ { \\theta } + \\Delta ^ 3 _ { \\theta } \\end{align*}"}
+{"id": "7210.png", "formula": "\\begin{align*} \\displaystyle \\hat { \\mu } _ t ^ { N } : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\delta _ { X _ t ^ { i , N } } \\end{align*}"}
+{"id": "4372.png", "formula": "\\begin{align*} \\mathbf { U } ^ { \\pm } ( 0 , \\mathbf { x } ) = \\mathbf { U } ^ { \\pm } _ 0 ( \\mathbf { x } ) , \\mathbf { x } \\in \\Omega ^ { \\pm } ( 0 ) , \\varphi ( 0 , x _ 2 ) = \\varphi _ 0 ( x _ 2 ) , x _ 2 \\in \\R . \\end{align*}"}
+{"id": "4443.png", "formula": "\\begin{align*} | | | ( \\tilde { \\mathcal { A } } \\mathcal { A } _ 3 \\mathbf V ) ( t ) | | | ^ 2 _ { s - 1 , \\ast } & \\lesssim | | \\tilde { \\mathcal { A } } \\mathcal { A } _ 3 \\mathbf V | | ^ 2 _ { s , \\ast , t } \\\\ & \\leq C ( K ) \\Big ( | | \\mathbf V | | ^ 2 _ { s , \\ast , t } + | | \\dot { { \\mathbf U } } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ t ) } | | \\hat { W } | | ^ 2 _ { s + 2 , \\ast , t } \\Big ) \\\\ & \\leq C ( K ) \\mathcal { M } ( t ) . \\end{align*}"}
+{"id": "4523.png", "formula": "\\begin{align*} R _ k = \\partial _ 1 ( \\mathcal { L } ( { \\mathbf V } _ k , \\Psi _ k ) - \\mathcal { F } ^ a + \\mathcal { I } ) , t > 0 , \\end{align*}"}
+{"id": "2322.png", "formula": "\\begin{align*} \\pi _ g \\pi _ h = \\pi _ { g h } , \\forall g , h \\in G . \\end{align*}"}
+{"id": "6571.png", "formula": "\\begin{align*} \\mathbb { D } = \\left \\{ z \\in \\mathbb { C } : \\ | \\Re z | \\leq \\xi , \\ | \\Im z | \\leq \\xi _ 1 \\right \\} \\end{align*}"}
+{"id": "3109.png", "formula": "\\begin{align*} \\mathbf { b } _ R ( \\psi _ 1 , \\theta _ 1 ) & = \\frac { 1 } { \\sqrt { N } } \\mathbf { b } ( \\phi _ 1 , \\vartheta _ 1 ) , \\\\ \\mathbf { b } _ T ( \\psi _ { 2 , k , p } , \\theta _ { 2 , k , p } ) & = \\frac { 1 } { \\sqrt { N } } \\mathbf { b } ( \\phi _ { 2 , k , p } , \\vartheta _ { 2 , k , p } ) \\end{align*}"}
+{"id": "4352.png", "formula": "\\begin{align*} S ( \\tilde { \\omega } \\| \\omega ) = { \\rm T r } ( \\varrho _ \\omega ( \\log ( \\varrho _ \\omega ) - \\log ( \\varrho _ { \\tilde { \\omega } } ) ) ) \\ , . \\end{align*}"}
+{"id": "6463.png", "formula": "\\begin{align*} & P _ D F = 0 , \\\\ & P _ N F ' = 0 , \\\\ & P _ R F ' = \\Lambda P _ R F . \\end{align*}"}
+{"id": "1447.png", "formula": "\\begin{align*} \\nabla u ^ k _ i ( x ) = - 2 \\alpha _ i x / | x | ^ 2 + o ( 1 ) \\ . \\end{align*}"}
+{"id": "4295.png", "formula": "\\begin{align*} \\epsilon _ \\pi ( u _ 1 ) j _ { u _ 1 } = \\epsilon _ \\pi ( u _ 2 ) j _ { u _ 2 } = x _ \\ell , \\end{align*}"}
+{"id": "8792.png", "formula": "\\begin{align*} \\frac { \\alpha } { 4 } \\sum _ { t = T _ { 1 } + 1 } ^ { T } [ t ( r _ { t } - r _ { t + 1 } ) - r _ { t } ] \\leq \\frac { T _ 1 \\alpha } { 4 } r _ { T _ { 1 } + 1 } \\leq \\frac { 1 8 \\bar { L } ^ 2 \\kappa } { \\alpha } r _ { T _ { 1 } + 1 } . \\end{align*}"}
+{"id": "526.png", "formula": "\\begin{align*} [ u , v ] ( t ) = \\begin{cases} u ( t ) & t _ 0 < t < t _ 1 \\\\ v ( t ) & t _ 1 < t < t _ 2 \\end{cases} \\end{align*}"}
+{"id": "3391.png", "formula": "\\begin{align*} \\omega _ { U _ j } = h ^ * _ { i j } \\omega _ { H } + \\textrm { A d } _ { h _ { i j } ^ { - 1 } } \\omega _ { U _ i } . \\end{align*}"}
+{"id": "3108.png", "formula": "\\begin{align*} \\mathbf { b } _ T ( \\psi _ { 2 , k , p } , \\theta _ { 2 , k , p } ) = \\frac { 1 } { \\sqrt { N } } \\left [ e ^ { j 2 \\pi n _ 1 \\frac { d _ R } { \\lambda } \\cos ( \\theta _ { 2 , k , p } ) \\sin ( \\psi _ { 2 , k , p } ) } \\right ] ^ T _ { n _ 1 \\in \\mathcal { I } ( N _ 1 ) } \\otimes \\frac { 1 } { \\sqrt { N } } \\left [ e ^ { j 2 \\pi n _ 2 \\frac { d _ R } { \\lambda } \\sin ( \\theta _ { 2 , k , p } ) } \\right ] ^ T _ { n _ 2 \\in \\mathcal { I } ( N _ 2 ) } . \\end{align*}"}
+{"id": "8312.png", "formula": "\\begin{align*} v _ p ( X ) = v _ p ( ( x _ 1 , \\cdots , x _ d ) ) = \\min \\{ v _ p ( x _ 1 ) , \\cdots , v _ p ( x _ d ) \\} , \\end{align*}"}
+{"id": "1450.png", "formula": "\\begin{align*} \\mathcal { P } _ 1 = 2 \\pi \\sum \\limits _ { 1 \\leq i < j \\leq n } a ^ { i j } \\Big ( \\sum \\limits _ { t \\in I } k _ { i t } \\sigma ^ k _ t ( r _ k ) - 2 \\alpha _ i \\Big ) \\Big ( \\sum \\limits _ { t \\in I } k _ { j t } \\sigma ^ k _ t ( r _ k ) - 2 \\alpha _ { j } \\Big ) - 8 \\pi \\sum \\limits _ { 1 \\leq i < j \\leq n } a ^ { i j } \\alpha _ i \\alpha _ j + o ( 1 ) \\end{align*}"}
+{"id": "3800.png", "formula": "\\begin{align*} \\overline F ( s ) : = \\begin{cases} 0 , & s = 1 , \\\\ + \\infty , & \\end{cases} \\end{align*}"}
+{"id": "4964.png", "formula": "\\begin{align*} \\Phi _ { 6 n } ( 3 ) = \\Phi _ { 1 2 n } ( \\sqrt { 3 } ) . \\end{align*}"}
+{"id": "776.png", "formula": "\\begin{align*} F ( u ) = \\displaystyle \\frac { 1 } { p ^ \\flat } | u | ^ { p ^ \\flat } + \\frac { 1 } { p ^ \\sharp } | u | ^ { p ^ \\sharp } , \\end{align*}"}
+{"id": "2453.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ N | v _ n \\rangle \\langle v _ n | \\le 1 L ^ 2 ( \\R ^ d ) . \\end{align*}"}
+{"id": "5786.png", "formula": "\\begin{align*} s _ { \\alpha } s _ { \\beta } s _ { \\alpha } = \\begin{cases} s _ { \\alpha + \\beta } ( \\alpha , \\beta ) = - 1 , \\\\ s _ { \\alpha - \\beta } ( \\alpha , \\beta ) = 1 . \\\\ \\end{cases} \\end{align*}"}
+{"id": "7933.png", "formula": "\\begin{align*} \\varPsi ( j \\pm ) \\ , = \\ , ( m + 1 - j ) \\mp \\end{align*}"}
+{"id": "2319.png", "formula": "\\begin{align*} h = \\sum _ { { g \\in G } } \\langle h , \\tau _ g \\rangle \\tau _ g , \\forall h \\in \\mathcal { H } . \\end{align*}"}
+{"id": "7123.png", "formula": "\\begin{align*} \\widehat { \\Psi ^ 0 _ { k } } = | k | ^ { - 2 } \\widehat { \\gamma ^ 0 _ k } \\ ; \\chi _ 0 ( | k | ^ { 1 / 2 } Z ) \\end{align*}"}
+{"id": "4070.png", "formula": "\\begin{align*} \\mathcal { K } _ 0 ( f ( x ) ) = p _ 0 ( x ) , \\mathcal { K } _ 1 ( g ( x ) ) = - p ( x ) \\end{align*}"}
+{"id": "6750.png", "formula": "\\begin{align*} C = W - ( E ^ { \\sigma _ e } \\circ O ^ { \\sigma _ o } ) ( n ) \\end{align*}"}
+{"id": "7663.png", "formula": "\\begin{align*} \\tau _ u = \\begin{cases} & \\inf \\{ t \\geqslant 0 : W ( t ) < 0 \\} , \\\\ & \\infty , \\mbox { \\ i f \\ } \\ W ( t ) \\geqslant 0 \\ \\mbox { \\ f o r a l l \\ } \\ t \\geqslant 0 \\end{cases} \\end{align*}"}
+{"id": "8367.png", "formula": "\\begin{align*} X ( \\omega ) = \\sup _ { x \\in C ( \\omega ) } \\| x \\| _ B \\end{align*}"}
+{"id": "7105.png", "formula": "\\begin{align*} | n _ 2 | \\ll \\ , \\frac { p _ 1 C ^ 2 \\lambda r n _ 1 } { q _ 1 N _ 0 } : = \\ , N _ 2 . \\end{align*}"}
+{"id": "4251.png", "formula": "\\begin{align*} \\| u \\| _ { \\mathbb { X } } : = \\left ( \\int _ { \\Omega } | \\nabla u | ^ 2 \\ , d x \\right ) ^ { 1 / 2 } , u \\in \\mathbb { X } ( \\Omega ) \\ , , \\end{align*}"}
+{"id": "5571.png", "formula": "\\begin{align*} r = F ( \\xi ) . \\end{align*}"}
+{"id": "2939.png", "formula": "\\begin{align*} G _ { k i _ 1 \\dots i _ n } = \\sum _ { \\lambda = 1 , \\pi \\in S _ { n } } ^ { n ! } a \\delta _ { k \\pi ( i _ 1 } D _ { i _ 2 \\cdots i _ { n } ) } + \\sum _ { \\lambda = 1 , \\pi \\in S _ { n } } ^ { n ! } b \\delta _ { \\pi ( i _ 1 i _ 2 } D _ { i _ 3 \\cdots i _ n ) k } + \\sum _ { \\lambda = 1 , \\pi \\in S _ { n } } ^ { n ! } d \\epsilon _ { k s \\pi ( i _ 1 } D _ { i _ 2 \\cdots i _ n ) s } + f D _ { k i _ 1 \\dots i _ n } \\end{align*}"}
+{"id": "4188.png", "formula": "\\begin{align*} h ( W ) : = - \\int _ { - \\infty } ^ \\infty f _ W ( w ) \\log f _ W ( w ) \\mathrm { d } w \\end{align*}"}
+{"id": "8088.png", "formula": "\\begin{align*} \\begin{aligned} \\uppercase \\expandafter { \\romannumeral 1 } & = \\sum \\limits _ { j = 1 } ^ { \\infty } \\lambda _ { j } \\vert g ( a _ { j } ) ( x ) \\vert \\chi _ { 2 Q _ { j } } ( x ) + \\sum \\limits _ { j = 1 } ^ { \\infty } \\lambda _ { j } \\vert g ( a _ { j } ) ( x ) \\vert \\chi _ { ( 2 Q _ { j } ) ^ { c } } ( x ) \\\\ & : = \\uppercase \\expandafter { \\romannumeral 1 } _ { 1 } + \\uppercase \\expandafter { \\romannumeral 1 } _ { 2 } \\end{aligned} \\end{align*}"}
+{"id": "6886.png", "formula": "\\begin{align*} h ^ * \\ell ( \\widetilde { U } , r ) = \\ell ( h ^ * T _ { \\widetilde { U } } , \\mu ) = \\ell ( \\widetilde { S } , \\mu ) . \\end{align*}"}
+{"id": "1647.png", "formula": "\\begin{align*} L _ P \\phi ( r ) = ( P - 1 ) \\phi '' ( r ) + \\frac { f ' ( r ) } { f ( r ) } \\phi ' ( r ) . \\end{align*}"}
+{"id": "8112.png", "formula": "\\begin{align*} 1 + \\frac { \\tau q } { ( \\tau p ) ^ { ' } } = \\tau p \\left ( 1 - \\frac { \\alpha } { \\tau n } \\right ) = \\left ( \\frac { n - \\alpha + N + 1 } { n } \\right ) q . \\end{align*}"}
+{"id": "3874.png", "formula": "\\begin{align*} \\alpha _ 1 & = \\frac { 2 d } { d + 2 s } , \\\\ \\alpha _ 2 & = \\frac { 4 s } { d + 2 s } , \\\\ m & = \\frac 1 2 \\left ( ( q - \\gamma - 2 s ) \\frac { d + 2 s } { 2 s } + \\gamma + 2 s + 1 \\right ) . \\end{align*}"}
+{"id": "8426.png", "formula": "\\begin{align*} f ( X ) = - \\log \\det \\Sigma + \\sum _ { i = 1 } ^ n ( \\lambda _ i Y _ { i i } - \\log ( 1 + Y _ { i i } ) ) . \\end{align*}"}
+{"id": "8601.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ z ^ 2 \\phi _ 2 = - \\dfrac { \\zeta } { h _ b } \\big ( 1 + \\dfrac { h } { h _ b } \\big ) \\Delta _ X \\psi \\ \\ \\mathrm { i n } \\ \\ \\mathcal { S } _ b , \\\\ \\phi _ 2 | _ { z = 0 } = 0 , \\ \\ \\partial _ z \\phi _ 2 | _ { z = - 1 + \\beta b } = 0 . \\end{cases} \\end{align*}"}
+{"id": "6865.png", "formula": "\\begin{align*} ( \\mathcal { R } _ i ) _ { ( i ) } = ( \\overline { \\mathcal { C } } _ i ) _ { ( i ) } - A _ i ( \\overline { \\mathcal { Y } } _ { \\vec k } ^ i ) _ { ( i ) } - ( \\overline { \\mathcal { Y } } _ { \\vec k } ^ i ) _ { ( i ) } B _ i , \\end{align*}"}
+{"id": "4879.png", "formula": "\\begin{align*} \\mathcal { B } ( \\mathfrak { E } ) = \\{ f : \\mathbb { C } \\to \\mathfrak { X } ~ \\mbox { e n t i r e f u n c t i o n } ~ | E _ + ^ { - 1 } f \\in H ^ 2 _ \\mathfrak { X } ( \\mathbb { C } _ + ) , ~ E _ - ^ { - 1 } f \\in H ^ 2 _ \\mathfrak { X } ( \\mathbb { C } _ + ) ^ \\perp \\} . \\end{align*}"}
+{"id": "8442.png", "formula": "\\begin{align*} \\iota _ { K Z B } ( x ) = \\log \\left ( \\exp \\left ( - \\pi _ { \\mathrm { S p a n } _ { K } \\{ B \\} } ( \\log ( x ) ) \\right ) x ) \\right ) . \\end{align*}"}
+{"id": "7521.png", "formula": "\\begin{align*} 0 \\leq | \\sigma ( x ) | \\leq \\min \\left \\{ 1 , | x | \\right \\} , \\sigma ' ( x ) = ( 1 + x ^ 2 ) ^ { - 3 / 2 } \\ , . \\end{align*}"}
+{"id": "8872.png", "formula": "\\begin{align*} T \\times G / T & \\to G / T , \\\\ t \\cdot x T & = t x T . \\end{align*}"}
+{"id": "3230.png", "formula": "\\begin{align*} S _ { \\alpha } ( \\epsilon , R ) \\geq \\textnormal { A r e a } \\left ( \\textnormal { S e c t } ^ { - } _ { \\alpha , \\epsilon } ( R '' ) \\right ) = \\textnormal { A r e a } ( \\textnormal { S e c t } _ { \\alpha , \\epsilon } ( R ) ) + O \\left ( R \\right ) . \\end{align*}"}
+{"id": "5099.png", "formula": "\\begin{align*} V = V ^ { R l a } = I ^ { \\bullet , R l a } = I ^ { \\bullet , l a } , \\end{align*}"}
+{"id": "3799.png", "formula": "\\begin{align*} \\alpha = ( x _ 0 , \\rho ^ { 1 / p } _ 0 ( x _ 0 ) , x _ 1 , \\rho ^ { 1 / p } _ 1 ( x _ 1 ) ) _ { \\# } \\gamma , \\end{align*}"}
+{"id": "7091.png", "formula": "\\begin{align*} \\Omega = \\sum _ { n _ { 2 } \\ll N _ 0 / n ^ 2 _ 1 } ^ { \\infty } \\ , \\left | \\displaystyle \\sum _ { q _ 2 \\sim C / q _ 1 } \\ , \\ , \\sum _ { m \\sim M _ 1 } \\frac { \\lambda _ f ( m ) } { m ^ { 1 / 4 } } \\ , \\mathcal { C } ^ { + } ( n ^ 2 _ 1 n _ 2 , m ; q ) \\ , \\mathcal { J } ^ { + } _ 1 ( n ^ 2 _ 1 n _ 2 , m , q ) \\right | ^ 2 . \\end{align*}"}
+{"id": "929.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty ( C L _ 2 ^ u ( y ) + D L _ 2 ^ v ( y ) ) \\mathrm { e } ^ { - \\lambda y ^ 2 } \\mathrm { d } y = \\frac { \\sqrt { m n } } { m ( 1 + \\frac { 4 t ^ \\alpha } { \\alpha } \\lambda ) ^ { \\frac { c + 1 + \\sqrt { m n } } { 2 } } } \\mathrm { e } ^ { - \\frac { \\lambda x ^ 2 } { ( 1 + \\frac { 4 t ^ \\alpha } { \\alpha } \\lambda ) } } , \\end{align*}"}
+{"id": "558.png", "formula": "\\begin{align*} \\tau = \\sup _ { R } \\tau _ R , \\end{align*}"}
+{"id": "5487.png", "formula": "\\begin{align*} \\widehat \\delta _ { ( p , s ) \\mapsto ( p ^ a s ^ b , p ^ c s ^ d ) } : \\mathbf S \\rightarrow \\mathbf S , \\widehat \\delta _ { ( p , s ) } ( f ) : = f \\circ \\delta _ { ( p , s ) \\mapsto ( p ^ a s ^ b , p ^ c s ^ d ) } . \\end{align*}"}
+{"id": "1325.png", "formula": "\\begin{align*} \\Lambda _ V ^ { w } ( u ) & = w ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) = u ^ m . \\end{align*}"}
+{"id": "8279.png", "formula": "\\begin{align*} N ( \\mathbb { H } ) = \\{ u \\in \\mathbb { H } ^ l | | u ^ 0 | ^ 2 = | u ^ 1 | ^ 2 = | u ^ 2 | ^ 2 = | u ^ 3 | ^ 2 = \\frac { 1 } { 4 } , g _ 0 ( u ^ p , u ^ q ) = \\delta _ { p q } \\} . \\end{align*}"}
+{"id": "7651.png", "formula": "\\begin{align*} \\langle w , X ^ \\mu \\rangle + \\langle w , Z ^ \\mu \\rangle = | w | \\Big ( X ^ w + \\langle \\frac { w } { | w | } , Z ^ \\mu \\rangle \\Big ) , \\end{align*}"}
+{"id": "1437.png", "formula": "\\begin{align*} \\begin{cases} \\Delta u _ i + \\sum \\limits _ { j \\in I } k _ { i j } ^ { \\mathbf { c } ^ t } e ^ { u _ j } = 4 \\pi \\alpha _ i \\delta _ 0 \\ \\ B _ 1 ( 0 ) \\subseteq \\mathbb { R } ^ 2 , \\ \\ \\forall \\ i \\in I , \\\\ \\\\ u _ 1 + 2 \\sum \\limits _ { i \\in I \\setminus \\{ 1 , n + 1 \\} } u _ i + u _ { n + 1 } \\equiv 0 , \\end{cases} \\end{align*}"}
+{"id": "1666.png", "formula": "\\begin{align*} \\varrho _ { \\texttt { a } ; j } : = \\frac { 1 } { 2 } \\bigl ( n + 1 - 2 j \\bigr ) ( j = 1 , \\ldots , n ) \\end{align*}"}
+{"id": "8452.png", "formula": "\\begin{align*} d g = g \\omega _ 0 - \\omega g , \\end{align*}"}
+{"id": "814.png", "formula": "\\begin{align*} c + o ( 1 ) & = \\left ( \\frac { 1 } { p } - \\frac { 1 } { 2 \\cdot p ^ \\sharp } \\right ) [ u _ n ] _ { s , p } ^ p + \\left ( \\frac { 1 } { p _ s ^ * } - \\frac { 1 } { 2 \\cdot 2 ^ \\sharp } \\right ) \\int _ { \\mathbb { R } ^ N } ( K \\ast | u _ n | ^ { p ^ \\sharp } ) | u _ n | ^ { p ^ \\sharp } d x \\\\ & \\geq \\left ( \\frac { 1 } { p } - \\frac { 1 } { 2 \\cdot p ^ \\sharp } \\right ) [ u _ n ] _ { s , p } ^ p . \\end{align*}"}
+{"id": "3786.png", "formula": "\\begin{align*} F ^ * ( \\phi ) = \\exp ( \\phi ) - 1 , R ^ * ( \\psi ) = - \\log ( 1 - \\psi ) , \\end{align*}"}
+{"id": "3227.png", "formula": "\\begin{align*} S _ { \\alpha } ( \\epsilon , R ) = \\textnormal { A r e a } ( \\textnormal { S e c t } _ { \\alpha , \\epsilon } ( R ) ) + \\beta / \\epsilon + O ( \\epsilon ^ { 2 } R ^ { 2 } ) , \\end{align*}"}
+{"id": "2250.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\Gamma } \\left ( \\int _ { 0 } ^ \\infty e ^ { - \\alpha \\gamma ^ 2 } \\ , d \\mu ( \\alpha ) \\right ) = \\int _ 0 ^ \\infty \\left ( \\sum _ { \\gamma \\in \\Gamma } e ^ { - \\alpha \\gamma ^ 2 } \\right ) \\ , d \\mu ( \\alpha ) . \\end{align*}"}
+{"id": "381.png", "formula": "\\begin{align*} j ( u ^ * c , x \\circ c ^ * u ) = j ( c , x ) \\circ u . \\end{align*}"}
+{"id": "6614.png", "formula": "\\begin{align*} c _ \\infty ^ { ( \\widetilde { \\rm c J } ) } ( \\tau ; \\beta , p , q ) = \\lim _ { N , k \\to \\infty } c _ k ^ { ( \\widetilde { \\rm c J } ) } ( N , \\beta , p , q ) \\Big | _ { \\tau = 2 \\pi k / N } , ( \\tau \\ne 0 ) , \\end{align*}"}
+{"id": "6343.png", "formula": "\\begin{align*} u _ 0 ^ h = P _ { < h } u _ 0 . \\end{align*}"}
+{"id": "2775.png", "formula": "\\begin{align*} I ( G - N [ v _ 0 ] ) & = x \\cdot ( 2 x + 1 ) ^ { 2 k + 4 } \\\\ & = x \\cdot \\bigg [ \\sum _ { i = 0 } ^ { 2 k + 4 } \\binom { 2 k + 4 } { i } ( 2 x ) ^ { i } \\bigg ] \\\\ & = x \\cdot [ ( 2 k ) ^ { 2 k + 4 } + \\dots ] \\\\ & = 2 ^ { 2 k + 4 } x ^ { 2 k + 5 } + \\dots \\end{align*}"}
+{"id": "8390.png", "formula": "\\begin{align*} D = N ^ { \\delta } , z = N ^ { \\frac { \\delta } { 2 } - \\varepsilon } , P ( z ) = \\prod _ { 2 < p < z } p . \\end{align*}"}
+{"id": "5385.png", "formula": "\\begin{align*} k & = r a n k ( G G ^ T ) + n u l ( G G ^ T ) \\\\ & = r a n k ( G G ^ T ) + d i m ( H u l l ( C ) ) . \\end{align*}"}
+{"id": "6844.png", "formula": "\\begin{align*} \\Delta _ { \\theta } \\chi _ j ( \\theta ) = - \\lambda _ { j } \\chi _ j ( \\theta ) . \\end{align*}"}
+{"id": "8343.png", "formula": "\\begin{align*} V = \\bigcup _ { \\bar y \\in C } V _ { \\bar y } . \\end{align*}"}
+{"id": "6932.png", "formula": "\\begin{align*} \\liminf \\limits _ { t \\rightarrow + \\infty } \\inf \\limits _ { | x | < c t } \\varphi ( x , t ) = \\frac { 1 + b } { 2 } . \\end{align*}"}
+{"id": "8296.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde W ( \\alpha , \\alpha _ t , \\beta ) & \\leq O ( 1 ) e ^ { - \\frac { \\mu } { 4 } t } \\left ( W ( \\alpha ( x , 0 ) , \\alpha _ t ( x , 0 ) , \\beta ( x , 0 ) ) + \\norm { p _ 0 } ^ 2 _ { H ^ 1 ( 0 , 1 ) } + \\norm { ( \\bar u _ 0 , \\bar u _ 1 ) } ^ 2 _ { H ^ 1 ( 0 , 1 ) \\times L ^ 2 ( 0 , 1 ) } \\right ) , \\\\ & \\leq O ( \\varepsilon ^ { 3 } ) e ^ { - \\frac { \\mu } { 4 } t } . \\end{aligned} \\end{align*}"}
+{"id": "8151.png", "formula": "\\begin{align*} p _ { i j } ( x y ) = v _ { i j } ( \\theta _ { x y } , \\mu _ { x y } ) \\end{align*}"}
+{"id": "5494.png", "formula": "\\begin{align*} O _ A ^ { ( 1 ) } = \\widehat S ^ { - 2 } O _ A ^ { ( 1 ) } \\widehat S ^ 2 . \\end{align*}"}
+{"id": "2509.png", "formula": "\\begin{align*} \\| \\boldsymbol { v } \\| _ { \\boldsymbol { H } ^ { \\emph { \\textbf { c u r l } } , \\frac { 1 } { 2 } } } ^ 2 : = T ( \\| \\boldsymbol { v } _ 0 ^ c \\| _ { \\Omega } ^ 2 + \\| \\emph { \\textbf { c u r l } } \\ , \\boldsymbol { v } _ 0 ^ c \\| _ { \\Omega } ^ 2 ) + \\frac { T } { 2 } \\sum _ { k = 1 } ^ { \\infty } \\Big ( ( 1 + k \\omega ) \\| \\boldsymbol { v } _ k \\| _ { \\Omega } ^ 2 + \\| \\emph { \\textbf { c u r l } } \\ , \\boldsymbol { v } _ k \\| _ { \\Omega } ^ 2 ] \\Big ) . \\end{align*}"}
+{"id": "7301.png", "formula": "\\begin{align*} P _ { S _ P } ( x , y ) = \\begin{cases} \\frac { R ( x , y ) - R ( y , x ) } { 1 - R ( y , x ) } & R ( x , y ) > R ( y , x ) , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "4271.png", "formula": "\\begin{align*} \\sup _ { \\substack { u \\in H _ { k } \\\\ \\| u \\| _ { \\mathbb { X } ( \\Omega ) } = T } } J _ { \\lambda _ k } ( u ) < - C _ R \\le \\inf _ { u \\in \\mathbb { P } _ { k + 1 } } J _ { \\lambda _ k } ( u ) , \\end{align*}"}
+{"id": "7875.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ M \\phi \\rho e ^ { - f } \\ , d V = & \\int _ M \\left [ \\left ( - \\frac 1 2 | \\N \\phi | ^ 2 + \\frac 1 2 R ^ { H , f } \\right ) \\rho + | \\N \\phi | ^ 2 \\rho + ( R ^ { H , f } - R ^ { H , f } ) \\phi \\rho \\right ] e ^ { - f } \\ , d V \\\\ = & \\int _ M \\left [ \\frac 1 2 | \\N \\phi | ^ 2 + \\frac 1 2 R ^ { H , f } \\right ] \\rho e ^ { - f } \\ , d V , \\end{align*}"}
+{"id": "1082.png", "formula": "\\begin{align*} \\int _ D | \\alpha ( x ) | d \\mu < \\mu _ 0 ^ 2 \\lambda _ 1 \\mbox { w i t h } \\ , \\ , \\mu _ 0 : = \\min _ { x \\in D } \\mu ( x ) > 0 \\end{align*}"}
+{"id": "8839.png", "formula": "\\begin{align*} \\delta _ { t + 1 } \\leq \\left ( 1 - \\frac { 2 } { t } \\right ) \\delta _ { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { t ^ { p _ { i } + 1 } } \\enspace , \\end{align*}"}
+{"id": "801.png", "formula": "\\begin{align*} & \\phantom { = } \\left ( a ( t _ \\lambda ^ \\flat ) ^ { p - 1 } - \\frac { b ( t _ \\lambda ^ \\flat ) ^ { 2 \\cdot p ^ \\flat - 1 } } { p ^ \\flat } \\right ) \\int _ { \\mathbb { R } ^ N } | u ^ \\flat | ^ { p } d x \\\\ & \\geq ( t _ \\lambda ^ \\flat ) ^ { p - 1 } \\int _ { \\mathbb { R } ^ N } \\tilde { a } ( \\lambda x ) | u ^ \\flat | ^ { p } d x - \\frac { b ( t _ \\lambda ^ \\flat ) ^ { 2 \\cdot p ^ \\flat - 1 } } { p ^ \\flat } \\int _ { \\mathbb { R } ^ N } | u ^ \\flat | ^ { p } d x > 0 \\end{align*}"}
+{"id": "1940.png", "formula": "\\begin{align*} F ( t , x , v ) = \\left ( v - 1 \\right ) \\cos ( x - t ) e ^ { - v ^ 2 } - \\left ( 2 v \\left ( \\sin ( x - t ) - v \\right ) + 1 \\right ) \\left ( 1 + \\sin ( x - t ) \\right ) e ^ { - v ^ 2 } \\end{align*}"}
+{"id": "3461.png", "formula": "\\begin{align*} u ' = \\begin{cases} \\frac { 1 } { d _ 0 } \\left ( - 1 + ( \\frac { d _ 0 + d _ 1 } { d _ 1 } x ) ^ { 1 / n } \\right ) , & 0 \\leq x \\leq \\frac { d _ 1 } { d _ 0 + d _ 1 } , \\\\ \\frac { 1 } { d _ 1 } \\left ( 1 - ( \\frac { d _ 0 + d _ 1 } { d _ 0 } ( 1 - x ) ) ^ { 1 / n } \\right ) , & \\frac { d _ 1 } { d _ 0 + d _ 1 } \\leq x \\leq 1 . \\end{cases} \\end{align*}"}
+{"id": "1645.png", "formula": "\\begin{align*} g ( r ) = \\frac { 1 } { 4 r ^ 2 W ( r ) } \\end{align*}"}
+{"id": "4137.png", "formula": "\\begin{align*} x \\rhd y & = \\nabla _ x y , & [ x , y ] & = - T ( x , y ) + R ( x , y ) , \\\\ x \\rhd E & = \\nabla _ x E , & [ E , x ] & = - E x , \\\\ E \\rhd y & = E y , & [ E _ 1 , E _ 2 ] & = - E _ 1 E _ 2 + E _ 2 E _ 1 . \\\\ E _ 1 \\rhd E _ 2 & = E _ 1 E _ 2 - E _ 2 E _ 1 , \\end{align*}"}
+{"id": "2025.png", "formula": "\\begin{align*} b = u - a \\in ( \\mathrm { C } _ c ^ \\infty ( \\Omega ) + \\dot { \\mathrm { H } } ^ { s _ 0 , p } ( \\mathbb { R } ^ n ) ) \\cap \\dot { \\mathrm { H } } ^ { s _ 1 , p } ( \\mathbb { R } ^ n ) \\end{align*}"}
+{"id": "4825.png", "formula": "\\begin{align*} \\check { R } ( 0 ) = C \\cdot I \\end{align*}"}
+{"id": "3744.png", "formula": "\\begin{align*} \\sigma _ p ( - \\mathbb { A } ) = \\{ \\lambda \\in \\mathbb { C } : \\lambda ^ 3 \\in \\sigma _ p ( - A ) \\} , \\end{align*}"}
+{"id": "2714.png", "formula": "\\begin{align*} B = ( ( 1 , 1 , 1 , 3 , 2 ) , ( 0 , 2 , 1 , 3 , 2 ) , ( 0 , 2 , 1 , 2 , 3 ) , ( 0 , 1 , 2 , 2 , 3 ) , ( 0 , 1 , 1 , 3 , 3 ) ) . \\end{align*}"}
+{"id": "5450.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda ^ 2 } \\log \\left ( \\int e ^ { - t \\lambda L f } d \\mu \\right ) & = \\frac { 1 } { \\lambda ^ 2 } \\log \\left ( 1 + \\sum _ { k = 2 } ^ \\infty \\frac { ( - t \\lambda ) ^ k } { k ! } \\int ( L f ) ^ k d \\mu \\right ) & ( \\mbox { s i n c e } \\int L f d \\mu = 0 ) \\\\ & \\leq \\sum _ { k = 2 } ^ \\infty \\frac { t ^ k \\lambda ^ { k - 2 } } { k ! } \\int | L f | ^ k d \\mu & ( \\mbox { s i n c e } \\log ( 1 + x ) \\leq x ) \\\\ & \\leq \\int e ^ { t | L f | } d \\mu . \\end{align*}"}
+{"id": "1119.png", "formula": "\\begin{align*} \\| u \\| : = \\sum _ { k = 0 } ^ { m } \\| | \\nabla ^ { k } u | \\| _ { L ^ p ( D ) } , \\end{align*}"}
+{"id": "8148.png", "formula": "\\begin{align*} q _ { i j } ( m n ) = v ^ \\star _ { i j } ( \\theta ^ \\star _ { m n } , \\mu ^ \\star _ { m n } ) \\ , , \\end{align*}"}
+{"id": "5413.png", "formula": "\\begin{align*} f ( \\tau ) = - \\Delta u ( \\sigma _ y ( g ^ { - 1 } ( \\tau ) ) ) \\ \\ \\ \\ \\ \\tau \\in ( 0 , u ( z ) ) . \\end{align*}"}
+{"id": "3570.png", "formula": "\\begin{align*} \\Delta ^ h _ { R _ { 0 2 } } = \\left [ \\begin{matrix} 3 & 5 & - 3 \\\\ 5 & 1 1 & - 5 \\\\ - 3 & - 5 & 3 \\end{matrix} \\right ] , \\quad \\Delta ^ h _ P = \\left [ \\begin{matrix} 4 & 8 & 0 \\\\ 8 & 1 6 & 0 \\\\ 0 & 0 & 8 \\end{matrix} \\right ] , \\Delta ^ 0 _ { \\mathcal { R } } = \\left [ \\begin{matrix} 0 & - 2 & - 6 \\\\ - 2 & 0 & - 1 2 \\\\ - 6 & - 1 2 & 0 \\end{matrix} \\right ] \\end{align*}"}
+{"id": "5826.png", "formula": "\\begin{align*} w ^ { ( 4 ) } _ 1 = & s _ 2 s _ 4 s _ 5 s _ 3 s _ 4 s _ 2 = ( s _ 2 s _ 4 s _ 5 s _ 4 s _ 2 ) ( s _ 2 s _ 4 s _ 3 s _ 4 s _ 2 ) = \\\\ & s _ { \\alpha _ 2 + \\alpha _ 4 + \\alpha _ 5 } s _ { \\alpha _ 2 + \\alpha _ 4 + \\alpha _ 3 } . \\end{align*}"}
+{"id": "4982.png", "formula": "\\begin{align*} ( j _ * \\circ i _ * ) ( f ) \\circ ( j \\circ i ) = j _ * ( i _ * ( f ) ) \\circ j \\circ i = j \\circ i _ * ( f ) \\circ i = j \\circ ( i \\circ f ) , \\end{align*}"}
+{"id": "646.png", "formula": "\\begin{align*} \\mathcal L ( \\psi , \\phi ) = \\mathcal L _ { \\mathrm { D i r a c } } ( \\psi ) + \\mathcal L _ { \\mathrm { m e s o n } } ( \\phi ) + \\mathcal L _ { \\mathrm { Y u k a w a } } ( \\psi , \\phi ) . \\end{align*}"}
+{"id": "274.png", "formula": "\\begin{align*} D _ k = ( 0 , 0 ) < ( 0 , 1 ) < \\cdots < ( 0 , k ) < ( 1 , n ) . \\end{align*}"}
+{"id": "788.png", "formula": "\\begin{align*} f _ 2 ( x ) = \\frac { C _ 2 } { ( C _ 3 + | x - x _ 0 | ^ { 2 } ) ^ { \\frac { N + \\alpha } { 2 } } } . \\end{align*}"}
+{"id": "474.png", "formula": "\\begin{align*} q _ { \\beta , \\mu } ( r ) : = \\frac { \\beta } { h _ { \\beta , \\mu } ( r ) } \\int _ 0 ^ \\infty t ^ { \\beta - 1 } ( p _ t \\mu ) ( r ) \\ , d t = \\frac { \\beta h _ { \\beta - 1 , \\mu } ( r ) } { h _ { \\beta , \\mu } ( r ) } . \\end{align*}"}
+{"id": "305.png", "formula": "\\begin{align*} \\ell _ { j } \\left ( s \\right ) \\left ( u , v \\right ) : = \\left \\langle \\mathbb { A } _ { j } ^ { \\operatorname * { e x t } } \\nabla u , \\overline { \\nabla v } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } + s ^ { 2 } \\left \\langle p _ { j } ^ { \\operatorname * { e x t } } u , \\overline { v } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } \\qquad \\forall u , v \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) , \\end{align*}"}
+{"id": "7063.png", "formula": "\\begin{align*} n = m \\ , \\ , \\iff \\ , \\ , n \\equiv m \\ , \\mathrm { m o d } \\ , p _ 1 \\ , \\ , \\iff \\ , \\ , \\frac { n - m } { p _ 1 } = 0 . \\end{align*}"}
+{"id": "7643.png", "formula": "\\begin{align*} u ( k , w ) & = \\inf _ { \\mathbb { Q } \\in B _ { k } ( \\mathbb { P } ) } \\mathbb { E } _ { \\mathbb { Q } } \\left [ U ( x _ 0 + x _ 0 \\langle w , X \\rangle ) \\right ] \\\\ & \\leq \\inf _ { \\mathbb { Q } \\in B _ { k } ( \\mathbb { P } ) } \\mathbb { E } _ { \\mathbb { Q } } \\left [ U ( x _ 0 + x _ 0 \\langle w ^ u , X \\rangle ) \\right ] = u ( k , w ^ u ) \\end{align*}"}
+{"id": "2284.png", "formula": "\\begin{align*} g _ 1 ( x ) & = \\sum _ { 1 \\leq | k | \\leq \\frac { n - 1 } { 2 } } e ^ { - \\pi \\alpha k ^ 2 } \\left ( \\sum _ { j = 1 } ^ { n } e ^ { 2 \\pi i k \\varepsilon _ j } e ^ { - 2 \\pi i k \\frac { j } { n } } \\right ) e ^ { 2 \\pi i k x } \\end{align*}"}
+{"id": "2244.png", "formula": "\\begin{align*} p _ \\alpha ( x ) = \\sum _ { \\gamma \\in \\Gamma } e ^ { - \\alpha ( x - \\gamma ) ^ 2 } \\end{align*}"}
+{"id": "2237.png", "formula": "\\begin{align*} \\rho _ r \\in C _ c ^ { \\infty } ( \\mathbb { R } ) , s u p p ( \\rho _ r ) = B ( 0 , r ) \\qquad \\int _ { \\mathbb { R } ^ N } \\rho _ r \\ , d x = 1 . \\end{align*}"}
+{"id": "7028.png", "formula": "\\begin{align*} \\mathop { \\lim } _ { n \\to + \\infty } F _ { L , \\Omega } ^ { \\Phi ^ n } [ \\rho ] = F _ { L , \\Omega } [ \\rho ] . \\end{align*}"}
+{"id": "4900.png", "formula": "\\begin{align*} S _ 2 ( z ) : = \\left \\{ \\begin{array} { l l } \\exp [ - \\frac { z } { x _ 1 } P _ 1 ] [ ( z - x _ 1 - z P _ 1 ) [ \\frac { E _ + ( z ) - E _ + ( x _ 1 ) } { z - x _ 1 } ] + E _ + ( x _ 1 ) ] , z \\neq x _ 1 \\\\ \\exp ( - P _ 1 ) [ E _ + ( x _ 1 ) - x _ 1 P _ 1 E _ + ' ( x _ 1 ) ] , z = x _ 1 \\end{array} \\right . \\end{align*}"}
+{"id": "283.png", "formula": "\\begin{align*} \\operatorname { d i v } \\left ( \\mathbb { A } \\nabla w \\right ) + s ^ { 2 } p \\ , w = 0 \\quad \\Omega \\subset \\mathbb { R } ^ { 3 } . \\end{align*}"}
+{"id": "1854.png", "formula": "\\begin{gather*} \\mathbf { E } \\left [ \\prod _ { n = 0 } ^ { N } \\mathbb { X } _ \\alpha ( n ) ^ { m _ n } \\right ] = \\begin{cases} 1 & \\\\ 0 & \\end{cases} \\end{gather*}"}
+{"id": "4661.png", "formula": "\\begin{align*} ( \\mathbf { X } \\otimes \\mathbf { Y } ) \\cdot ( f \\otimes u ) : = D _ { \\mathbf { X } \\otimes \\mathbf { Y } } ( f ) \\otimes u \\end{align*}"}
+{"id": "429.png", "formula": "\\begin{align*} L ^ 2 ( \\R _ + , r ^ { d - 1 + 2 \\ell } d r ) \\ni u \\mapsto ( U u ) ( r ) = r ^ { ( d - 1 + 2 \\ell ) / 2 } u ( r ) \\in L ^ 2 ( \\R _ + , d r ) . \\end{align*}"}
+{"id": "3268.png", "formula": "\\begin{align*} f * g ( y ) = \\int _ \\mathbb { R } f ( x ) g ( y - x ) d x . \\end{align*}"}
+{"id": "3269.png", "formula": "\\begin{align*} \\mathcal { F } ^ \\mu ( f * g ) ( \\xi ) = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } \\mathcal { F } ^ \\mu ( f e _ { _ \\Sigma } ) ( \\xi ) \\mathcal { F } ^ \\mu ( g _ { _ \\Sigma } ) ( \\xi ) . \\end{align*}"}
+{"id": "1535.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { - X } e ^ { - u ^ 2 / 2 } \\ , \\mathrm d u = \\int _ X ^ \\infty e ^ { - u ^ 2 / 2 } \\ , \\mathrm d u \\le \\frac 1 X \\int _ X ^ \\infty u e ^ { - u ^ 2 / 2 } \\ , \\mathrm d u \\ll \\frac 1 X e ^ { - X ^ 2 / 2 } . \\end{align*}"}
+{"id": "1676.png", "formula": "\\begin{align*} N _ { \\texttt { a } ; \\mu } : = \\prod _ { \\substack { 1 \\leq j < k \\leq n \\\\ \\mu _ j - \\mu _ k = 0 } } \\frac { 1 + k - j } { k - j } . \\end{align*}"}
+{"id": "2601.png", "formula": "\\begin{align*} \\alpha _ { n } = \\int _ { K } x ^ { n } d \\mu n \\in \\N _ 0 . \\end{align*}"}
+{"id": "326.png", "formula": "\\begin{align*} \\forall \\left ( \\mathbf { j } , u \\right ) \\in \\left ( \\mathbf { H } \\left ( \\mathbb { R } ^ { 3 } , \\operatorname * { d i v } \\right ) , L ^ { 2 } \\left ( \\mathbb { R } ^ { 3 } \\right ) \\right ) b \\left ( \\left ( \\mathbf { j } , u \\right ) , \\left ( \\mathbf { m } , v \\right ) \\right ) = 0 . \\end{align*}"}
+{"id": "4588.png", "formula": "\\begin{align*} Y ( \\omega , z ) = \\sum _ { n \\in \\mathbb { Z } } L _ n \\ , z ^ { - n - 2 } \\end{align*}"}
+{"id": "8900.png", "formula": "\\begin{align*} \\sigma ^ i \\circ \\psi _ i = \\psi _ i \\circ \\begin{pmatrix} 0 & - 1 \\\\ 1 & \\beta _ i \\end{pmatrix} . \\end{align*}"}
+{"id": "3945.png", "formula": "\\begin{align*} { x _ 0 v _ 1 } = h _ K ( v _ 1 ) \\geq r _ K ( w ) v _ 1 = x ' v _ 1 + c v v _ 1 = x ' v _ 1 + c \\frac { ( 1 - t ) v _ 1 + t v _ 2 } { \\lVert ( 1 - t ) v _ 1 + t v _ 2 \\rVert } v _ 1 > x ' v _ 1 \\end{align*}"}
+{"id": "7412.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } } \\frac { ( \\partial _ { x } w _ { n } ) ^ { 2 } } { \\rho _ { n } } ( x , t ) ~ d x = \\int _ { \\mathbb { T } } \\frac { ( \\partial _ { x } w _ { n } ^ { 0 } ) ^ { 2 } } { \\rho _ { n } ^ { 0 } } ~ d x . \\end{align*}"}
+{"id": "792.png", "formula": "\\begin{align*} I _ { \\tilde { a } } [ t u ] & \\leq C _ 1 ' t ^ p \\| u \\| ^ p - C _ 2 ' t ^ { 2 \\cdot p ^ \\sharp } \\| u \\| ^ { 2 \\cdot p ^ \\sharp } - C _ 3 ' t ^ { 2 \\cdot p ^ \\flat } \\| u \\| ^ { 2 \\cdot p ^ \\flat } \\\\ & \\phantom { = } - C _ 4 ' t ^ { p ^ \\flat + p ^ \\sharp } \\| u \\| ^ { p ^ \\flat + p ^ \\sharp } - C _ 5 ' t ^ { p _ g } \\| u \\| ^ { p _ g } . \\end{align*}"}
+{"id": "5300.png", "formula": "\\begin{align*} \\sum _ { j _ 1 \\leq \\frac { n _ 1 + \\min \\{ n _ 1 , d _ 1 \\} } { 2 } } f _ { j _ 1 , d _ 1 } ( n _ 1 ) \\sum _ { j _ 2 \\leq \\frac { n _ 2 + \\min \\{ n _ 2 , d _ 2 \\} } { 2 } } f _ { j _ 2 , d _ 2 } ( n _ 2 ) \\dots \\sum _ { j _ L = 0 } ^ { \\min \\{ n _ L , d _ L \\} } f _ { j _ L , d _ L } ( n _ L ) . \\end{align*}"}
+{"id": "487.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\frac { \\Gamma ( \\frac { \\gamma + 1 + 2 k } { 2 } ) } { k ! \\Gamma ( \\zeta + \\frac 1 2 + k ) } \\left ( \\frac { 1 } { 4 t } \\right ) ^ k , \\end{align*}"}
+{"id": "1493.png", "formula": "\\begin{align*} g _ { t } = \\left ( B _ { t } , \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } \\langle U ^ { ( 1 ) } B _ { s } , d B _ { s } \\rangle , \\ldots , \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } \\langle U ^ { ( n ) } B _ { s } , d B _ { s } \\rangle \\right ) , \\end{align*}"}
+{"id": "3531.png", "formula": "\\begin{align*} \\begin{aligned} u _ { t } & = \\Delta u - \\chi \\nabla \\cdot ( u \\nabla w ) , & x \\in \\Omega , t > 0 , \\\\ v _ { t } & = \\Delta v - \\xi \\nabla \\cdot ( v \\nabla w ) , & x \\in \\Omega , t > 0 , \\\\ w _ { t } & = \\Delta w - \\lambda w + \\alpha u + \\beta v , & x \\in \\Omega , t > 0 , \\end{aligned} \\end{align*}"}
+{"id": "3482.png", "formula": "\\begin{align*} \\int _ Y e ^ { - \\alpha v } \\frac { \\omega ^ n } { ( Y ) } \\leq A , \\forall v \\in P S H ( Y , \\omega ) \\sup _ Y v = 0 . \\end{align*}"}
+{"id": "6346.png", "formula": "\\begin{align*} u = u ^ 0 + \\sum _ { k \\geq 0 } u ^ { k + 1 } - u ^ k . \\end{align*}"}
+{"id": "5306.png", "formula": "\\begin{align*} \\min \\{ a _ i \\cdot x : i = 1 , \\dots , k \\} . \\end{align*}"}
+{"id": "5666.png", "formula": "\\begin{align*} e _ 1 = e _ 0 + \\lambda f \\ ; \\ ; e _ 1 ^ \\prime = e _ 0 + \\mu f , \\ ; \\ ; \\lambda , \\mu \\geq 0 . \\end{align*}"}
+{"id": "1907.png", "formula": "\\begin{align*} \\int _ { I _ i } \\left ( ( Q ^ x _ \\lambda \\Upsilon ) - \\Upsilon \\right ) z \\ , { \\rm d } x = 0 , \\forall \\ , \\ , z \\in \\mathbb { P } ^ { k - 1 } ( I _ i ) , \\ , \\ , 1 \\leq i \\leq N _ x , \\end{align*}"}
+{"id": "5041.png", "formula": "\\begin{align*} | \\partial ^ m ( g _ { i j } - \\delta _ { i j } ) | \\leq c ^ { - 1 } \\rho _ p ^ { - m } \\leq c ^ { - 1 } m = 0 , \\ldots , k + 1 0 . \\end{align*}"}
+{"id": "613.png", "formula": "\\begin{align*} \\lim _ { t \\searrow 0 } f ( t ) = 0 , \\end{align*}"}
+{"id": "5516.png", "formula": "\\begin{align*} \\widetilde c _ { \\pm } ^ { ( h , g a ^ { - 1 } ) } ( p , s ) \\overset { ( \\ref { e q : c t i l d e D e f } ) } { = } c _ { \\pm } ^ { ( \\sigma ( h ) , \\sigma ( g a ^ { - 1 } ) ) } ( s , p ^ { - 1 } ) = c _ { \\pm } ^ { ( \\sigma ( h ) , \\sigma ( g ) b ^ { - 1 } ) } ( s , p ^ { - 1 } ) \\overset { ( \\ref { e q : c p c m R i g h t M u l t i p l i c a t i o n } ) } { = } c _ { \\pm } ^ { ( \\sigma ( h ) , \\sigma ( g ) ) } ( s , s / p ) \\overset { ( \\ref { e q : c t i l d e D e f } ) } { = } \\widetilde c _ { \\pm } ^ { ( h , g ) } ( p / s , s ) . \\end{align*}"}
+{"id": "2107.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) = \\mathcal { E } _ 0 ( t ) + \\mathcal { E } _ 1 ( t ) . \\end{align*}"}
+{"id": "8113.png", "formula": "\\begin{align*} \\textrm { $ x _ { i j } ^ { ( r ) } > x _ { k l } ^ { ( s ) } $ i f : ` ` $ r < s $ '' , o r ` ` $ r = s , i < k $ '' , o r ` ` $ r = s , i = k , j < l $ '' . } \\end{align*}"}
+{"id": "3950.png", "formula": "\\begin{align*} l ( \\pi , \\gamma ) = \\sum _ { i = 1 } ^ { k } d \\big ( \\gamma ( t _ i ) , \\gamma ( t _ { i + 1 } ) \\big ) . \\end{align*}"}
+{"id": "6413.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } { \\frac { \\partial u } { \\partial t } ( x , t ) = d _ { 1 } [ ( J _ { 1 } * u ) ( x , t ) - u ( x , t ) ] + r _ { 1 } u ( x , t ) [ \\alpha ( x - s t ) - u ( x , t ) - a v ( x , t ) ] , } \\\\ { \\frac { \\partial v } { \\partial t } ( x , t ) = d _ { 2 } [ ( J _ { 2 } * v ) ( x , t ) - v ( x , t ) ] + r _ 2 v ( x , t ) [ - 1 + b u ( x , t ) - v ( x , t ) ] , } \\end{array} \\right . \\end{align*}"}
+{"id": "518.png", "formula": "\\begin{align*} \\int _ 0 ^ T F ( t ) \\ , d W ( t ) = \\lim _ { n \\to \\infty } \\sum _ { j , k = 1 } ^ n \\left ( \\int _ 0 ^ T F _ { j k } ( t ) \\ , d B _ k ( t ) \\right ) f _ j , \\end{align*}"}
+{"id": "2377.png", "formula": "\\begin{align*} \\min _ { u } \\left \\{ { \\mathcal { A } _ 2 } V ^ { a f t e r } ( x , t ) + C ( t , x , u ) \\right \\} = 0 \\end{align*}"}
+{"id": "4124.png", "formula": "\\begin{align*} \\sum _ { i = a } ^ { k - 1 } c _ 1 ( s - a , i - a ) \\cdot D ( k - 2 i + j , - k + 2 s - \\tfrac { 3 } { 2 } - j ) = 0 . \\end{align*}"}
+{"id": "5658.png", "formula": "\\begin{align*} \\alpha \\ = \\ ( \\pi _ { | D _ X } ) ^ * \\mathcal { O } _ { \\mathbb { P } ^ 2 } ( 1 ) \\ \\sim \\ h - e _ 0 - e _ 1 \\ \\sim \\ e _ 0 + e ' _ 0 \\ \\sim \\ e _ 1 + e ' _ 1 \\ . \\end{align*}"}
+{"id": "2745.png", "formula": "\\begin{align*} x _ k : = k ^ { - 1 } \\cdot \\theta ^ { - 1 } \\cdot ( e _ k , e _ { k + 1 } , \\ldots , e _ { 2 k - 1 } ) \\end{align*}"}
+{"id": "223.png", "formula": "\\begin{align*} L _ x ^ { } = \\sum _ { 1 \\leq j \\leq n } \\Bigl ( \\frac { \\partial ^ 2 } { \\partial x _ j ^ 2 } - g _ S e ^ { - x _ j } - a _ n e ^ { - 2 x _ j } \\Bigr ) - \\sum _ { 1 \\leq j < k \\leq n } \\frac { \\frac { 1 } { 2 } g _ M ( g _ M - 1 ) } { \\sinh ^ { 2 } { \\frac { 1 } { 2 } } ( x _ j - x _ k ) } . \\end{align*}"}
+{"id": "5282.png", "formula": "\\begin{align*} \\epsilon _ s \\sum _ { t = 1 } ^ T \\sum _ { i = 1 } ^ n \\| [ g _ { i , t } ( x _ { i , t } ) ] _ + \\| \\le \\tilde { h } _ T ( x _ s ) + \\hat { h } _ T ( x _ s ) . \\end{align*}"}
+{"id": "3202.png", "formula": "\\begin{align*} \\left \\Vert x - x _ { k - 1 } \\right \\Vert _ { A ^ { T } A } ^ { 2 } - \\left \\Vert x - x _ { k } \\right \\Vert _ { A ^ { T } A } ^ { 2 } = \\phi _ { k } ^ { 2 } \\left ( w _ { k } ^ { T } v _ { k } - \\frac { \\theta _ { k + 1 } } { \\rho _ { k } } v _ { k + 1 } ^ { T } w _ { k } \\right ) . \\end{align*}"}
+{"id": "8203.png", "formula": "\\begin{align*} x _ j = \\frac { 1 } { 4 } d ^ c _ { I _ j } ( \\rho ^ 2 ) ( T ) = - \\frac { 1 } { 4 } d ( \\rho ^ 2 ) ( I _ j T ) \\end{align*}"}
+{"id": "1134.png", "formula": "\\begin{align*} & [ ( x , m ) , ( y , n ) ] ^ { 1 } = ( [ x , y ] _ 1 , l _ 1 ( x , n ) + r _ 1 ( m , y ) ) ; \\\\ & [ ( x , m ) , ( y , n ) ] ^ { 2 } = ( [ x , y ] _ 2 , l _ 2 ( x , n ) + r _ 2 ( m , y ) ) , \\end{align*}"}
+{"id": "1007.png", "formula": "\\begin{align*} \\begin{cases} \\nabla \\times { \\bf H } = \\dfrac { 4 \\pi } { c } { \\bf J } + \\dfrac { 1 } { c } \\dfrac { \\partial { \\bf D } } { \\partial t } \\\\ \\nabla \\cdot { \\bf D } = 4 \\pi \\rho \\\\ \\nabla \\times { \\bf E } = - \\dfrac { 1 } { c } \\dfrac { \\partial { \\bf B } } { \\partial t } \\\\ \\nabla \\cdot { \\bf B } = 0 , \\end{cases} \\end{align*}"}
+{"id": "3775.png", "formula": "\\begin{align*} R ( s ) : = \\begin{cases} s F ( 1 / s ) & s > 0 , \\\\ F _ { \\infty } ' & s = 0 . \\end{cases} \\end{align*}"}
+{"id": "6586.png", "formula": "\\begin{align*} { F } \\big | _ { \\mathcal { S } } \\left ( q ^ { ( 0 ) } \\right ) = 0 \\end{align*}"}
+{"id": "3652.png", "formula": "\\begin{align*} \\omega _ { \\epsilon } = \\omega - \\frac { 1 } { n - 2 } \\bot \\omega \\wedge \\Omega + \\frac { \\bot ^ 2 \\omega } { 2 ( n - 1 ) ( n - 2 ) } \\Omega \\wedge \\Omega \\ . \\end{align*}"}
+{"id": "7072.png", "formula": "\\begin{align*} G _ { 0 } ( y ) = 2 d { \\pi } ^ { 3 } y ^ { 2 / 3 } i \\int _ { 0 } ^ \\infty { v _ 1 } ( z ) \\ , z ^ { - 1 / 3 } \\ , s i n ( 6 \\pi z ^ { 1 / 3 } y ^ { 1 / 3 } ) d z + , \\end{align*}"}
+{"id": "1925.png", "formula": "\\begin{align*} \\left ( \\Pi _ { \\lambda _ 1 , \\lambda _ 2 } g \\right ) ^ { \\lambda _ 1 , \\lambda _ 2 } _ { i + \\frac { 1 } { 2 } , j + \\frac { 1 } { 2 } } = g ^ { \\lambda _ 1 , \\lambda _ 2 } _ { i + \\frac { 1 } { 2 } , j + \\frac { 1 } { 2 } } , \\mbox { i f } \\left ( i , j \\right ) \\in ( \\Bbbk _ 2 ^ + , \\Bbbk _ 1 ^ + ) \\end{align*}"}
+{"id": "4858.png", "formula": "\\begin{align*} E = ( G + G ^ { - 1 } ) / m - I \\end{align*}"}
+{"id": "6717.png", "formula": "\\begin{align*} u = 1 - \\frac { \\mathfrak { c } _ { p } } { a } r ^ { - a } + O _ 2 \\left ( r ^ { - a - \\tilde { \\tau } } \\right ) , \\ r = | x | \\rightarrow \\infty , \\end{align*}"}
+{"id": "382.png", "formula": "\\begin{align*} j \\circ c = \\alpha \\circ \\chi ( c , x ) \\circ c = \\alpha \\circ \\eta \\circ x = x . \\end{align*}"}
+{"id": "7358.png", "formula": "\\begin{align*} \\rho _ Y ( x ) = \\begin{cases} \\frac { \\sqrt [ 3 ] { 2 A ^ 2 } - 2 \\big ( q \\gamma ( x ^ 2 - 3 ) - q \\sqrt { \\gamma } x + q + 3 \\gamma \\big ) } { \\pi 2 ^ { \\frac { 5 } { 3 } } \\sqrt { 3 q \\gamma } \\sqrt [ 3 ] { A } } & x \\in { \\rm S u p p } ( \\rho _ Y ) \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "5917.png", "formula": "\\begin{align*} d [ c a _ { i , j } ] : = d [ c a _ { 1 , i - 1 } a _ { 1 , j } ] = \\min \\{ d ( c a _ { i , j } ) , \\alpha _ { i - 1 } , \\alpha _ { j } \\} , \\end{align*}"}
+{"id": "3082.png", "formula": "\\begin{align*} { C _ k } = { \\left | { \\xi _ k ^ \\star } \\right | ^ 2 } \\left ( { \\frac { E } { { { N _ { \\rm { R } } } { \\sigma ^ 2 } } } + \\sum \\nolimits _ { i = 1 } ^ { { N _ { \\rm { R } } } } { \\frac { 1 } { { { N _ { \\rm { R } } } { { \\left | { \\xi _ i ^ \\star } \\right | } ^ 2 } } } } - \\frac { 1 } { { { { \\left | { \\xi _ k ^ \\star } \\right | } ^ 2 } } } } \\right ) , \\end{align*}"}
+{"id": "2646.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\limsup _ { n \\to \\infty } \\sup _ { t \\in [ 0 , L ] } P ( \\sup _ { s \\in [ 0 , \\delta ] } d ' ( X _ n ( t + s ) , X _ n ( t ) ) > \\epsilon ) ^ { 1 / r _ n } = 0 \\ , . \\end{align*}"}
+{"id": "5186.png", "formula": "\\begin{align*} \\omega _ i = \\sum _ { j \\leq i } L _ j \\end{align*}"}
+{"id": "270.png", "formula": "\\begin{align*} T = \\{ \\Delta ^ { \\{ 0 , 2 , 4 \\} } , \\ \\Delta ^ { \\{ 1 , 2 , 3 \\} } , \\ \\Delta ^ { \\{ 0 , 1 , 3 \\} } , \\ \\Delta ^ { \\{ 1 , 3 , 4 \\} } , \\ \\Delta ^ { \\{ 0 , 1 , 2 \\} } \\} \\end{align*}"}
+{"id": "5592.png", "formula": "\\begin{align*} d _ { p , \\mu , \\theta } : = \\sup _ { \\underset { \\| u \\| _ { X ^ { 1 , p } _ \\infty } = 1 } { u \\in X ^ { 1 , p } _ \\infty } } \\int _ 0 ^ \\infty \\exp _ p ( \\mu | u | ^ { \\frac { p } { p - 1 } } ) r ^ { \\theta } \\mathrm d r \\end{align*}"}
+{"id": "2329.png", "formula": "\\begin{align*} a _ { g , h } = \\nu ( g ^ { - 1 } h ) , \\forall g , h \\in G . \\end{align*}"}
+{"id": "1355.png", "formula": "\\begin{align*} \\delta _ { ( \\mathcal { X } _ s , \\mathcal { D } _ s ) } ( \\epsilon \\mathcal { H } _ s + f _ s ^ * \\mathcal { O } ( 1 ) ) = \\lim _ { l \\to \\infty } \\delta _ { l m _ \\epsilon , ( \\mathcal { X } _ s , \\mathcal { D } _ s ) } ( \\epsilon \\mathcal { H } _ s + f _ s ^ * \\mathcal { O } ( 1 ) ) \\ge u - \\delta _ 0 \\end{align*}"}
+{"id": "408.png", "formula": "\\begin{align*} \\hom ( y b , P \\times [ 0 = 1 ] ) \\cong \\hom ( y b , P ) \\times \\hom ( y b , [ 0 = 1 ] ) \\to \\hom ( y b , [ 0 = 1 ] ) . \\end{align*}"}
+{"id": "8525.png", "formula": "\\begin{align*} \\lim _ { \\rho \\rightarrow 0 ^ { + } } \\frac { \\mathcal { H } ^ { n } ( ( \\mathbb { R } ^ { n } \\backslash E ) \\cap B _ { \\rho } ( ( \\bar { z } , w ) \\cap \\{ z < \\bar { z } \\} ) } { \\omega _ { n } \\rho ^ { n } } = 0 . \\end{align*}"}
+{"id": "8451.png", "formula": "\\begin{gather*} p _ n = \\sum _ { 2 a _ 2 + 3 a _ 3 + \\dots + n a _ n = n } \\frac { 1 } { a _ 2 ! a _ 3 ! \\dots a _ n ! } \\prod _ { k = 2 } ^ { n } \\left ( \\frac { ( - 1 ) ^ { k + 1 } P _ k } { k } \\right ) ^ { a _ k } \\\\ q _ n = \\sum _ { a _ 1 + 2 a _ 2 + \\dots + n a _ n = n } \\frac { 1 } { a _ 1 ! a _ 2 ! \\dots a _ n ! } \\prod _ { k = 1 } ^ { n } \\left ( \\frac { ( - 1 ) ^ { k + 1 } P _ k } { k } \\right ) ^ { a _ k } , \\end{gather*}"}
+{"id": "7539.png", "formula": "\\begin{align*} \\Phi _ { g , n , \\beta } ( ( \\iota _ * \\alpha ) \\boxtimes \\gamma ) = \\sum _ { \\beta _ 1 + \\beta _ 2 = \\beta } \\left ( \\Phi _ { g _ 1 , n _ 1 + 1 , \\beta _ 1 } \\otimes \\Phi _ { g _ 2 , n _ 2 + 1 , \\beta _ 2 } \\right ) ( \\alpha \\boxtimes ( \\gamma \\boxtimes \\eta ) ) . \\end{align*}"}
+{"id": "8748.png", "formula": "\\begin{align*} H ^ 2 ( \\mathbf { P } _ { \\omega , T } , \\mathbf { P } _ { \\omega ' , T } ) & \\leq 2 \\Big ( 1 - \\big ( 1 - T ^ { - 1 } \\big ) ^ T \\Big ) \\leq 3 / 2 \\enspace . \\end{align*}"}
+{"id": "2634.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\limsup _ { n \\to \\infty } P ( \\sup _ { s \\in [ 0 , t ] } \\abs { \\Theta _ n ( s ) } > r ) ^ { 1 / b _ n ^ 2 } = 0 \\ , . \\end{align*}"}
+{"id": "8642.png", "formula": "\\begin{align*} \\| \\mathrm { F } _ 0 u - u \\| _ { H ^ { k , 0 } ( \\mathcal { S } _ b ) } \\leq \\| \\mathrm { F } _ 0 u - u \\| _ { H ^ { k , 0 } ( \\mathcal { S } ) } & = \\Big { \\| } \\dfrac { \\cosh { ( ( z + 1 ) \\sqrt { \\mu } | \\mathrm { D } | ) } } { \\cosh { ( \\sqrt { \\mu } | \\mathrm { D } | ) } } u - u \\Big { \\| } _ { H ^ { k , 0 } ( \\mathcal { S } ) } \\lesssim \\mu | \\nabla _ X u | _ { H ^ { k + 1 } } . \\end{align*}"}
+{"id": "1276.png", "formula": "\\begin{align*} e ^ { i ( t - a _ { j + 1 } ) \\Delta } w ( a _ { j + 1 } ) = & e ^ { i t \\Delta } w ( 0 ) + i \\int _ { 0 } ^ { a _ { j + 1 } } e ^ { i ( t - s ) \\Delta } ( F ( \\tilde { u } + w ) - F ( u ) ) d s \\\\ & - i \\int _ { 0 } ^ { a _ { j + 1 } } e ^ { i ( t - s ) \\Delta } e d s . \\end{align*}"}
+{"id": "4367.png", "formula": "\\begin{align*} A _ 0 ( { \\bf { U } } ) \\partial _ t { \\bf { U } } + A _ 1 ( { \\bf { U } } ) \\partial _ 1 { \\bf { U } } + A _ 2 ( { \\bf { U } } ) \\partial _ 2 { \\bf { U } } = 0 , \\end{align*}"}
+{"id": "5650.png", "formula": "\\begin{align*} \\Gamma _ 0 \\cup \\Gamma _ 1 = ( x _ 0 = x _ 2 = 0 ) \\cup ( x _ 0 = x _ 3 = 0 ) \\end{align*}"}
+{"id": "3226.png", "formula": "\\begin{align*} S _ { \\alpha } ( \\epsilon , R ) = \\frac { \\epsilon q ^ { 2 } R ^ { 2 } } { p ^ { 2 } + q ^ { 2 } } + \\frac { 1 } { q ^ { 2 } \\epsilon } \\left \\{ \\frac { \\epsilon q ^ 2 R } { \\sqrt { p ^ { 2 } + q ^ { 2 } } } \\right \\} \\left ( 1 - \\left \\{ \\frac { \\epsilon q ^ 2 R } { \\sqrt { p ^ { 2 } + q ^ { 2 } } } \\right \\} \\right ) + O \\left ( 1 + ( R \\epsilon ) ^ { 2 } \\right ) , \\end{align*}"}
+{"id": "78.png", "formula": "\\begin{align*} \\sum _ { k \\in \\mathcal { P } _ H } \\mathcal T _ { \\rm { c o m } } ( k ) = - \\big ( 1 + \\mathcal O ( \\varepsilon _ K + \\ell ^ 2 \\rho \\widehat g ( 0 ) K _ H ^ { - 2 } ) \\big ) \\frac { \\rho _ z } { \\vert \\Lambda \\vert } \\sum _ { k \\in \\mathcal { P } _ H } \\frac { \\widehat g ( k ) ^ 2 } { k ^ 2 } \\sum _ { p \\in \\mathcal P _ L } a _ p ^ \\dagger a _ p + \\mathcal E , \\end{align*}"}
+{"id": "3695.png", "formula": "\\begin{align*} y : = ( \\lambda _ \\beta ( 2 ^ { - 4 n _ 0 } ) , \\lambda _ \\beta ( 2 ^ { - 4 n _ 0 } ) ) y ' : = \\left ( \\lambda _ \\beta ( 2 ^ { - 4 n _ 0 } ) + 2 n _ 0 ^ { - 2 } { a _ { n _ 0 } } , \\lambda _ \\beta ( 2 ^ { - 4 n _ 0 } - a _ { n _ 0 } ) \\right ) , \\end{align*}"}
+{"id": "8265.png", "formula": "\\begin{align*} \\mu _ { \\lambda ( M \\times \\mathbb { S } ^ 1 \\times \\mathbb { R } ^ 3 ) } = \\lambda ^ 2 \\mu _ { M \\times \\mathbb { S } ^ 1 \\times \\mathbb { R } ^ 3 } . \\end{align*}"}
+{"id": "6058.png", "formula": "\\begin{align*} \\underline { \\lim } _ { u \\rightarrow \\infty } \\frac { 1 } { \\ln u } \\overline { \\mu } ( \\underline { c } \\geq \\frac { 1 } { u } ) = \\infty . \\end{align*}"}
+{"id": "816.png", "formula": "\\begin{align*} | I ' [ \\tilde { u } _ n ] \\varphi | & = | I ' [ u _ n ] ( \\varphi ( \\cdot - y _ n ) ) | \\\\ & \\leq \\| I ' [ u _ n ] \\| _ { W ^ { - s , p ' } } \\| \\varphi ( \\cdot - y _ n ) \\| _ { W ^ { s , p } } = o ( 1 ) \\cdot \\| \\varphi \\| _ { W ^ { s , p } } , \\end{align*}"}
+{"id": "5202.png", "formula": "\\begin{align*} \\mbox { $ N _ { J ( Z ) } ( v ) \\cap Z = N _ { J } ( u ) \\cap Z ~ $ , } \\end{align*}"}
+{"id": "6029.png", "formula": "\\begin{align*} D ^ { q } F = \\lim _ { n \\rightarrow \\infty } D ^ { q } F _ { n } i n L ^ p ( \\Omega ; \\mathcal { H } ^ { \\otimes q } ) , L F = \\lim _ { n \\rightarrow \\infty } L F _ { n } i n L ^ p ( \\Omega ) . \\end{align*}"}
+{"id": "8757.png", "formula": "\\begin{align*} \\forall x \\in \\Theta : \\ \\ \\mathbb { E } \\sum _ { t = 1 } ^ { T } \\big ( f ( x _ t ) - f ( x ) \\big ) \\le \\min \\left ( G B T , 2 \\sqrt { 3 L } \\sigma \\frac { d } { \\sqrt { \\alpha } } \\sqrt { T } + A _ 4 \\frac { d ^ 2 } { \\alpha } \\log T \\right ) , \\end{align*}"}
+{"id": "8271.png", "formula": "\\begin{align*} \\Delta _ { \\omega _ 1 ^ a } w = \\frac { 1 } { 2 n } \\frac { ( \\omega _ 1 ^ a ) ^ { 2 n - 1 } \\wedge d d ^ c _ { I _ 1 ^ a } w } { ( \\omega _ 1 ^ a ) ^ { 2 n } } . \\end{align*}"}
+{"id": "4117.png", "formula": "\\begin{align*} G ( n ) & : = \\left . \\prod _ { i \\ge 0 } \\left ( \\prod _ { s = - 2 n + 4 i + 1 } ^ { - n + 2 i } ( 2 n + s ) \\prod _ { s = n - 2 i } ^ { 2 n - 4 i - 2 } ( 2 n + s ) \\right ) \\middle / \\prod _ { i = 1 } ^ { n - 1 } ( 2 i + 1 ) ^ { n - i } \\right . \\\\ & = \\left . \\prod _ { i \\ge 0 } \\left ( \\prod _ { s = 4 i + 1 } ^ { n + 2 i } s \\prod _ { s = 3 n - 2 i } ^ { 4 n - 4 i - 2 } s \\right ) \\middle / \\prod _ { i = 1 } ^ { n - 1 } ( 2 i + 1 ) ^ { n - i } \\right . . \\end{align*}"}
+{"id": "8152.png", "formula": "\\begin{align*} & \\theta _ { x y } = p _ { 1 0 } ( x y ) = k ( r - 2 ) - x ( r - 1 ) \\ , , \\\\ & \\mu _ { x y } = p _ { 0 1 } ( x y ) = ( r - 1 ) ( ( k - x - y ) ( n - k - y ) - y ) \\ , . \\end{align*}"}
+{"id": "754.png", "formula": "\\begin{align*} \\lim _ { t \\searrow 0 } f ( t ) = 0 , \\end{align*}"}
+{"id": "1267.png", "formula": "\\begin{align*} \\Big | \\sum _ { j = 1 } ^ J v _ n ^ j \\Big | ^ 2 = \\sum _ { j = 1 } ^ J | v _ n ^ j | ^ 2 + \\sum _ { j \\neq k } v _ n ^ j v _ n ^ k , \\end{align*}"}
+{"id": "7647.png", "formula": "\\begin{align*} Z ^ \\ast = - \\mathrm { s i g n } ( x _ 0 ) | Z ^ \\ast | \\frac { w } { | w | } . \\end{align*}"}
+{"id": "983.png", "formula": "\\begin{align*} ( k ) = ( x + y \\sqrt d ) ( x - y \\sqrt d ) , \\end{align*}"}
+{"id": "5148.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } H [ Q _ { { \\mathrm { u n i } } } ( X ) ] + \\log _ 2 \\delta = h ( X ) . \\end{align*}"}
+{"id": "6950.png", "formula": "\\begin{align*} 0 \\leq \\int _ { \\Omega } | \\nabla u + F _ \\epsilon u | ^ 2 d x & = \\int _ { \\Omega } | \\nabla u | ^ 2 d x + \\int _ { \\Omega } | F _ \\epsilon | ^ 2 | u | ^ 2 d x + \\int _ { \\Omega } F _ \\epsilon \\cdot \\nabla | u | ^ 2 d x \\\\ & = \\int _ { \\Omega } | \\nabla u | ^ 2 d x - \\int _ { \\Omega } \\Big ( F _ \\epsilon - | F _ \\epsilon | ^ 2 \\Big ) | u | ^ 2 d x . \\end{align*}"}
+{"id": "5238.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g ( x _ { i , t } ) ] _ + \\| . \\end{align*}"}
+{"id": "2873.png", "formula": "\\begin{align*} \\underbrace { T _ j T _ k T _ j \\cdots } _ { m _ { j k } \\ { \\rm f a c t o r s } } = \\underbrace { T _ k T _ j T _ k \\cdots } _ { m _ { j k } \\ { \\rm f a c t o r s } } ( 0 \\leq j \\neq k \\leq n ) , \\end{align*}"}
+{"id": "6547.png", "formula": "\\begin{align*} \\inf _ { \\xi = \\pm 1 , i = ( k , n ) \\in \\Lambda \\setminus B _ * } | \\xi ( \\sigma ^ * + k \\cdot \\omega ^ { ( 0 ) } ) + \\mu _ n | \\geq \\frac { ( \\varepsilon + \\delta ) ^ { \\frac { 1 } { 8 b } } } { 2 } . \\end{align*}"}
+{"id": "6879.png", "formula": "\\begin{align*} \\ell \\smallfrown \\sigma : = ( \\ell \\smallsmile u ) \\smallfrown [ X ] . \\end{align*}"}
+{"id": "1338.png", "formula": "\\begin{align*} B _ { C } : = \\sum _ { P } ( 1 - b _ { P } ) P , \\end{align*}"}
+{"id": "4387.png", "formula": "\\begin{align*} & \\hat { { \\mathbf U } } ^ { \\pm } \\in W ^ { 3 , \\infty } ( \\Omega _ T ) , \\quad \\hat { \\varphi } \\in W ^ { 4 , \\infty } ( \\Gamma _ T ) , \\\\ & | | \\hat { { \\mathbf U } } ^ { \\pm } | | _ { W ^ { 3 , \\infty } ( \\Omega _ T ) } + | | \\hat { \\varphi } | | _ { W ^ { 4 , \\infty } ( \\Gamma _ T ) } \\leq K , \\end{align*}"}
+{"id": "6594.png", "formula": "\\begin{align*} \\min _ { \\xi = \\pm 1 , 1 \\leq l \\leq N _ 1 ^ C } | k \\cdot \\omega + \\xi \\sqrt { \\zeta _ l } | > e ^ { - \\frac 1 4 N _ 1 ^ { \\rho _ 2 / 2 } } , \\end{align*}"}
+{"id": "2341.png", "formula": "\\begin{align*} \\pi _ g \\tau _ e = \\theta _ \\tau \\lambda _ g \\theta _ f \\tau _ e = \\theta _ \\tau \\lambda _ g \\theta _ f \\theta _ \\tau \\delta _ e = \\theta _ \\tau \\theta _ f \\theta _ \\tau \\lambda _ g \\delta _ e = I _ \\mathcal { X } \\theta _ \\tau \\lambda _ g \\delta _ e = \\theta _ \\tau \\delta _ g = \\tau _ g , \\forall g \\in G \\end{align*}"}
+{"id": "4046.png", "formula": "\\begin{align*} \\Delta _ { \\alpha , \\beta } ( \\rho ) = \\rho ^ { - 1 } \\left ( ( - 1 ) ^ { \\alpha } \\left ( \\rho \\cos \\rho - W _ { \\alpha , \\beta } ( a , \\rho ) \\right ) + \\int \\limits _ { 0 } ^ { 1 } ( U _ { \\alpha , \\beta } ( t ) \\sin \\rho t + V _ { \\alpha , \\beta } ( t ) \\cos \\rho t ) \\ , d t \\right ) , \\end{align*}"}
+{"id": "740.png", "formula": "\\begin{align*} I I I _ + + I I I _ - = \\int _ 0 ^ t \\norm { P _ { \\mu } ( P _ { \\mu } \\psi ^ \\mu ( s ) ) \\mathfrak K _ 1 } _ { \\mathcal L _ 2 } ^ 2 d s = 2 M _ { \\mathfrak K _ 1 } \\int _ 0 ^ t \\norm { P _ { \\mu } ( P _ { \\mu } \\psi ^ \\mu ( s ) ) } _ { L _ 2 } ^ 2 d s , \\end{align*}"}
+{"id": "7864.png", "formula": "\\begin{align*} & \\left ( 1 + \\psi ( r _ n ) \\right ) ^ 3 - \\left ( 1 - \\psi ( r _ n ) \\right ) ^ 2 \\leq \\left ( 1 + \\psi ( r _ n ) \\right ) ^ 3 - \\left ( 1 - \\psi ( r _ n ) \\right ) ^ 3 \\\\ & = 2 \\ , \\psi ( r _ n ) \\left ( 2 + 2 \\psi ( r _ n ) ^ 2 + 1 - \\psi ( r _ n ) ^ 2 \\right ) \\leq 2 r _ n ^ { - 1 } ( 3 + r _ n ^ { - 2 } ) \\leq 8 r _ n ^ { - 1 } \\leq \\frac { 1 } { \\log { n } } \\end{align*}"}
+{"id": "2606.png", "formula": "\\begin{align*} T _ p ( \\alpha ) = \\left ( \\det \\begin{bmatrix} \\alpha _ n & \\alpha _ { n + 1 } & \\alpha _ { n + 2 } \\\\ \\alpha _ { n + 1 } & \\alpha _ { n + 2 } & \\alpha _ { n + 3 } \\\\ \\alpha _ { n + 2 } & \\alpha _ { n + 3 } & \\alpha _ { n + 4 } \\end{bmatrix} \\right ) _ { n \\in \\N _ 0 } . \\end{align*}"}
+{"id": "1041.png", "formula": "\\begin{align*} k ( z , x ) = \\left \\{ \\begin{array} { l l } f ( z , x ^ + ) + \\vartheta ( x ^ + ) ^ { p ( z ) - 1 } , & \\hbox { i f } x \\leq u _ \\lambda ( z ) \\\\ f ( z , u _ \\lambda ( z ) ) + \\vartheta u _ \\lambda ( z ) ^ { p ( z ) - 1 } , & \\hbox { i f } u _ \\lambda ( z ) < x . \\end{array} \\right . \\end{align*}"}
+{"id": "3983.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) u ^ \\# = \\Phi ^ \\# \\end{align*}"}
+{"id": "7416.png", "formula": "\\begin{align*} \\Psi _ { n } ( 1 , t ) & = \\int _ { \\mathbb { T } } \\rho _ { n } ^ { 0 } ( y ) - \\langle \\rho _ { n } \\rangle ~ d y - \\int _ { 0 } ^ { t } \\rho _ { n } u _ { n } ( 1 , s ) ~ d s \\\\ [ 1 e x ] & = - \\int _ { 0 } ^ { t } \\rho _ { n } u _ { n } ( 1 , s ) ~ d s = - \\int _ { 0 } ^ { t } \\rho _ { n } u _ { n } ( 0 , s ) ~ d s = \\Psi _ { n } ( 0 , t ) . \\end{align*}"}
+{"id": "2423.png", "formula": "\\begin{align*} \\theta _ H ( c ) = \\sum _ { c _ \\gamma \\in \\Lambda _ H } \\xi _ { H , c _ \\gamma } ( c ) . \\end{align*}"}
+{"id": "4505.png", "formula": "\\begin{align*} \\varepsilon ^ i : = & \\partial _ t \\psi _ i - u _ { 1 , i } \\vert _ { x _ 1 = 0 } + ( \\partial _ t \\varphi ^ a - u _ 1 ^ a \\vert _ { x _ 1 = 0 } + u _ 2 ^ a \\vert _ { x _ 1 = 0 } \\partial _ 2 \\varphi ^ a ) + ( u ^ a _ 2 + u _ { 2 , i } ) \\vert _ { x _ 1 = 0 } \\partial _ 2 \\psi _ i \\\\ & + u _ { 2 , i } \\vert _ { x _ 1 = 0 } \\partial _ 2 \\varphi ^ a = \\mathcal B ( { \\mathbf V } _ i \\vert _ { x _ 1 = 0 } , \\psi _ i ) _ 1 \\ , . \\end{align*}"}
+{"id": "4044.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { R } f : & = ( \\mathcal { R } _ { 1 } f , \\mathcal { R } _ { 2 } f , \\ldots , \\mathcal { R } _ { k } f ) ^ { T } , \\\\ \\mathcal { Q } f : & = ( \\mathcal { Q } _ { 1 } f , \\mathcal { Q } _ { 2 } f , \\ldots , \\mathcal { Q } _ { k } f ) ^ { T } , \\end{aligned} \\end{align*}"}
+{"id": "6704.png", "formula": "\\begin{align*} H ^ \\ast ( X ) = \\sum _ { n > 0 } \\frac { p _ n ( X ) } { n } a _ { - n } . \\end{align*}"}
+{"id": "7594.png", "formula": "\\begin{align*} \\gamma _ 1 = \\frac { 5 } { 4 } , \\gamma _ 2 = \\frac { 3 } { 4 } , \\end{align*}"}
+{"id": "3724.png", "formula": "\\begin{align*} \\| \\cdot \\| _ { \\mathcal { H } ^ \\alpha } : = \\| A ^ { \\alpha } \\cdot \\| _ \\mathcal { H } \\end{align*}"}
+{"id": "2331.png", "formula": "\\begin{align*} \\nu ( g ^ { - 1 } h ) = f _ g ( \\tau _ h ) , \\forall g , h \\in G . \\end{align*}"}
+{"id": "8498.png", "formula": "\\begin{align*} P ( F _ { \\ell } ; B \\times \\mathbb { R } ^ { n - 1 } ) = \\int _ { B } \\sqrt { ( \\mathcal { H } ^ { n - 2 } ( \\partial ^ { * } ( F _ { \\ell } ) _ { z } ) ) ^ 2 + | \\nabla \\ell ( z ) | ^ { 2 } } \\ d z + | D ^ { s } \\ell | ( B ) , \\end{align*}"}
+{"id": "1609.png", "formula": "\\begin{align*} | \\mathcal { K } _ k ( \\hat { P } ^ { ( k - 2 ) } _ K ) | = ( 1 \\pm \\eta ) \\prod \\limits _ { \\ell = 1 } ^ { k - 2 } ( \\frac { 1 } { a _ { \\ell } } ) ^ { \\binom { k } { \\ell } } n ^ k . \\end{align*}"}
+{"id": "5794.png", "formula": "\\begin{align*} ( \\alpha + 3 \\beta , \\alpha + 3 \\beta ) = ( \\alpha , \\alpha ) + 6 ( \\alpha , \\beta ) + 9 ( \\beta , \\beta ) = 3 - 6 ( \\frac { 3 } { 2 } ) + 9 = 3 . \\end{align*}"}
+{"id": "1312.png", "formula": "\\begin{align*} - \\Delta u = P _ \\varepsilon ( \\mu _ t ^ { B , H } - \\mathfrak { m } ) . \\end{align*}"}
+{"id": "7878.png", "formula": "\\begin{align*} T ( x ) = \\exp _ x ( \\N \\phi ( x ) ) , \\end{align*}"}
+{"id": "8324.png", "formula": "\\begin{align*} f _ i ( x ) : = a _ { i , 0 } x ^ d + a _ { i , 1 } x ^ { d - 1 } + \\cdots + a _ { i , d } = 0 \\end{align*}"}
+{"id": "1955.png", "formula": "\\begin{align*} \\begin{aligned} 4 \\epsilon ^ { - 1 } \\ , T \\left ( \\| m _ 0 f _ 1 \\| ^ 2 _ { L ^ \\infty ( 0 , T ; L ^ 2 ( I ) ) } + \\| u _ 1 \\| ^ 2 _ { X } + \\| u _ 2 \\| ^ 2 _ X + \\frac { \\epsilon ^ 2 } { 4 } \\ , \\right ) \\leq \\frac { 1 } { 2 } . \\end{aligned} \\end{align*}"}
+{"id": "3980.png", "formula": "\\begin{align*} f ( u , \\mu ) ( z ) = \\int F \\big ( u ( z ) , v _ 1 ( z ) , \\dots , v _ m ( z ) \\big ) \\ , \\mu ^ { \\otimes m } ( d v _ 1 \\dots d v _ m ) \\end{align*}"}
+{"id": "5708.png", "formula": "\\begin{align*} \\big | \\big ( ( z - B ' ) ^ { - 1 } f _ 0 \\big ) ( \\xi _ r ) \\big | & = \\big | f _ 0 \\big ( ( z - B ) ^ { - 1 } \\xi _ r \\big ) \\big | = \\big | \\frac { 1 } { z } + \\frac { z ^ { m - 1 } } { r ^ { m } } \\big ( e ^ { \\frac { r } { z } } - \\sum \\limits _ { n = 0 } ^ { m } \\frac { 1 } { n ! } \\frac { r ^ n } { z ^ n } \\big ) \\big | . \\end{align*}"}
+{"id": "7645.png", "formula": "\\begin{align*} | w ^ \\ast | = | \\lim _ { k \\to \\infty } w _ k | = \\lim _ { k \\to \\infty } | w _ k | = | w ^ u | . \\end{align*}"}
+{"id": "2906.png", "formula": "\\begin{align*} L _ \\omega \\Phi _ { \\xi _ \\mu } = \\mathrm m _ \\omega ( e ^ { i \\xi _ \\mu } ) \\Phi _ { \\xi _ \\mu } ( \\omega \\in P ^ + , \\mu \\in { \\hat P } _ { c } ) . \\end{align*}"}
+{"id": "2529.png", "formula": "\\begin{align*} I _ { } = \\frac { \\mathcal { M } _ { | \\cdot | } ^ { \\oplus } ( \\beta ; \\boldsymbol { \\eta } , \\boldsymbol { \\tau } ) ^ 2 } { \\| \\boldsymbol { y } - \\boldsymbol { \\eta } \\| _ { \\boldsymbol { H } ^ { \\textbf { c u r l } , \\frac { 1 } { 2 } } } } \\end{align*}"}
+{"id": "8043.png", "formula": "\\begin{align*} \\left \\| f \\right \\| _ { h _ { \\omega } ^ { p } ( \\mathbb { R } ^ { n } ) } = \\left \\| g _ { d } ( f ) \\right \\| _ { L _ { \\omega } ^ { p } ( \\mathbb { R } ^ { n } ) } . \\end{align*}"}
+{"id": "2645.png", "formula": "\\begin{align*} \\overline { \\dot w ^ 0 } ( x ) - \\overline p ( F _ 0 ^ { - 1 } ( x ) ) = 0 \\ , , \\\\ \\overline { \\dot w } ( t ) - \\sigma \\overline p ( t ) + \\sigma \\int _ 0 ^ \\infty \\overline p ( t + s ) F ' ( s ) \\ , d s = 0 \\ , , \\\\ \\overline { \\dot k } ( x , t ) - \\overline p ( \\frac { t } { \\mu } + F ^ { - 1 } ( x ) ) = 0 \\ , . \\end{align*}"}
+{"id": "6400.png", "formula": "\\begin{align*} c _ { 1 2 } c _ { 2 3 } c _ { 1 2 } = c _ { 2 3 } c _ { 1 2 } c _ { 2 3 } \\end{align*}"}
+{"id": "7340.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} J \\begin{pmatrix} u ' _ 2 \\\\ u ' _ 4 \\end{pmatrix} + \\begin{pmatrix} b _ \\lambda ( t ) & d _ \\lambda ( t ) \\\\ d _ \\lambda ( t ) & h _ \\lambda ( t ) \\end{pmatrix} \\begin{pmatrix} u _ 2 \\\\ u _ 4 \\end{pmatrix} & = 0 , t \\in \\mathbb { R } \\\\ \\lim _ { t \\rightarrow \\pm \\infty } u ( t ) & = 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7232.png", "formula": "\\begin{align*} R ^ h \\{ t _ 1 , \\dots , t _ d \\} / ( \\sigma ( F ) ) = R ^ h \\{ t _ 2 , \\dots , t _ d \\} \\simeq R ^ h \\{ t _ 1 , \\dots , t _ d \\} / ( t _ 1 ^ { n _ 1 } \\cdots t _ d ^ { n _ d } - u \\pi ) \\ , . \\ , \\end{align*}"}
+{"id": "7870.png", "formula": "\\begin{align*} E ( \\mu ) = \\frac 1 2 \\int _ 0 ^ 1 \\int _ M | \\N \\phi | ^ 2 \\rho e ^ { - f } \\ , d V \\ , d s . \\end{align*}"}
+{"id": "4029.png", "formula": "\\begin{align*} \\partial _ i 1 = 0 , \\partial _ i x _ j = \\delta _ { i j } 1 \\otimes 1 ( j = 1 , \\ldots , m ) , \\end{align*}"}
+{"id": "1897.png", "formula": "\\begin{align*} \\mathcal { B } _ h ( u _ h ; f _ h , \\psi _ h ) : = \\sum _ { i = 1 } ^ { N _ x } \\sum _ { j = 1 } ^ { N _ v } \\mathcal { B } _ { i j } ^ h ( u _ h ; f _ h , \\psi _ h ) , \\end{align*}"}
+{"id": "8229.png", "formula": "\\begin{align*} W ^ + _ Z ( D _ i ) & = \\{ z \\in Z | \\lim _ { t \\rightarrow + \\infty } \\phi _ t ( z ) \\in D _ i \\} , \\\\ W ^ - _ Z ( D _ i ) & = \\{ z \\in Z | \\lim _ { t \\rightarrow - \\infty } \\phi _ t ( z ) \\in D _ i \\} . \\end{align*}"}
+{"id": "2777.png", "formula": "\\begin{align*} \\ ! \\begin{multlined} [ t ] x ^ { k + 2 } \\cdot x ^ { k + 1 } \\cdot 1 1 x ^ 3 + x ^ { k + 2 } \\cdot ( 2 ^ k + k ) x ^ k \\cdot x ^ 4 + ( 2 ^ { k + 1 } + k + 1 ) x ^ { k + 1 } \\cdot x ^ { k + 1 } \\cdot x ^ 4 = \\\\ ( 3 \\cdot 2 ^ k + 2 k + 1 2 ) x ^ { 2 k + 6 } , \\end{multlined} \\end{align*}"}
+{"id": "6424.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial v _ { \\infty } } { \\partial t } ( x , t ) & = d _ { 2 } \\left [ \\left ( J _ { 2 } * v _ { \\infty } \\right ) ( x , t ) - v _ { \\infty } ( x , t ) \\right ] + r _ 2 v _ { \\infty } ( x , t ) \\left ( - 1 - v _ { \\infty } ( x , t ) \\right ) \\\\ & \\leq d _ { 2 } \\left [ \\left ( J _ { 2 } * v _ { \\infty } \\right ) ( x , t ) - v _ { \\infty } ( x , t ) \\right ] - r _ 2 v _ { \\infty } ( x , t ) , \\end{aligned} \\end{align*}"}
+{"id": "1987.png", "formula": "\\begin{align*} \\sum _ { \\sigma = 0 } ^ { p ^ k - 1 } \\mathcal { C } ( \\mathbf { a } _ \\sigma ^ { t _ 1 } , \\mathbf { a } _ \\sigma ^ { t _ 2 } ) ( 0 ) = 0 . \\end{align*}"}
+{"id": "2691.png", "formula": "\\begin{align*} U : = & f ( i ) f ( - i ) - g ( i ) g ( - i ) - h ( i ) h ( - i ) + t ( i ) t ( - i ) , \\\\ V : = & f ( i ) h ( - i ) + f ( - i ) h ( i ) - g ( i ) t ( - i ) - g ( - i ) t ( i ) . \\end{align*}"}
+{"id": "7374.png", "formula": "\\begin{align*} l o g ( R ^ k _ { j } ) ~ | \\{ Y ^ { ( k ) } _ { i j } < a _ 1 ~ o r ~ Y ^ { ( k ) } _ { i j } > a _ 2 \\} = { V _ { j k } ^ { ( 2 ) } } ' \\beta _ j ^ { ( 2 ) } + { U _ { j k } ^ { ( 2 ) } } ' b _ k ^ { ( 2 ) } + \\varepsilon _ { i j k } ^ { ( 2 ) } , \\end{align*}"}
+{"id": "1420.png", "formula": "\\begin{align*} \\lim _ { K \\to \\infty } \\| \\theta ^ K ( x ) \\| _ { l _ 2 } = 0 , \\lim _ { K \\to \\infty } \\| \\{ \\theta _ { n 0 } ^ K ( x ) - \\theta _ { n 1 } ^ K ( x ) \\} \\| _ { l _ 1 } = 0 , \\end{align*}"}
+{"id": "6854.png", "formula": "\\begin{align*} \\mathcal { X } = \\mathcal G \\times _ 1 B _ 1 \\times _ 2 B _ 2 \\times _ 3 \\dots \\times _ d B _ d . \\end{align*}"}
+{"id": "5380.png", "formula": "\\begin{align*} R _ H ( t , s ) = \\frac { 1 } { 2 } \\left ( t ^ { 2 H } + s ^ { 2 H } - | t - s | ^ { 2 H } \\right ) , \\ \\ t , s \\in [ 0 , T ] . \\end{align*}"}
+{"id": "6364.png", "formula": "\\begin{align*} \\mathcal { R } \\chi = \\chi ^ \\mathrm { o p } \\mathcal { R } \\end{align*}"}
+{"id": "4747.png", "formula": "\\begin{align*} { \\rm P e r } ( E ; U ) = { \\rm P e r } ( F ; U ) . \\end{align*}"}
+{"id": "869.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\widetilde { u } ( z ) - \\widetilde { u } ( w ) \\right | & \\leq 2 \\left \\| \\widetilde { u } \\right \\| _ { C ( X , d ) } \\\\ & = 2 . 3 ^ { \\alpha - \\frac { s } { p _ { 0 } } } \\left \\| \\widetilde { u } \\right \\| _ { C ( X , d ) } \\frac { 1 } { 3 ^ { \\alpha - \\frac { s } { p _ { 0 } } } } \\\\ & \\leq 2 . 3 ^ { \\alpha - \\frac { s } { p _ { 0 } } } C _ 3 \\| u \\| _ { W _ s ^ { \\alpha , G } ( X , d , \\mu ) } d ( z , w ) ^ { \\alpha - \\frac { s } { p _ { 0 } } } . \\end{aligned} \\end{align*}"}
+{"id": "6964.png", "formula": "\\begin{align*} & T _ 3 : = \\\\ & \\sum _ { k = 1 } ^ d \\sum _ { i = k + 1 } ^ d \\sum _ { j = i + 1 } ^ d \\frac { \\cos ^ 2 ( x _ j / 2 ) ( \\sin ( x _ i / 2 ) - \\sin ( x _ k / 2 ) ) - \\cos ^ 2 ( x _ i / 2 ) ( \\sin ( x _ j / 2 ) - \\sin ( x _ k / 2 ) ) } { Y } \\\\ & + \\sum _ { k = 1 } ^ d \\sum _ { i = k + 1 } ^ d \\sum _ { j = i + 1 } ^ d \\frac { \\cos ^ 2 ( x _ k / 2 ) ( \\sin ( x _ j / 2 ) - \\sin ( x _ i / 2 ) ) } { Y } , \\end{align*}"}
+{"id": "7462.png", "formula": "\\begin{align*} a = \\frac { x y } { x + y - 1 } , g ' ( a ) = - \\frac { ( x + y - 1 ) ^ 3 } { x y ( x - 1 ) ( y - 1 ) } \\end{align*}"}
+{"id": "1019.png", "formula": "\\begin{align*} q ( R ) K = - k \\ , P _ 1 ^ * P _ 1 K \\ , + \\ , ( n + k - 2 ) \\ , P _ 2 ^ * P _ 2 K \\ , + \\ , P _ 3 ^ * P _ 3 \\ , K . \\end{align*}"}
+{"id": "1115.png", "formula": "\\begin{align*} - \\Delta _ \\mu u _ \\infty ( x ) - \\gamma u _ \\infty ( x ) = ( u ^ + _ \\infty ( x ) ) ^ { p - 1 } { \\rm i n } \\mathop D \\limits ^ \\circ . \\end{align*}"}
+{"id": "5793.png", "formula": "\\begin{align*} ( \\alpha + 2 \\beta , \\alpha + 2 \\beta ) = ( \\alpha , \\alpha ) + 4 ( \\alpha , \\beta ) + 4 ( \\beta , \\beta ) = 4 - 8 + 8 = 4 . \\end{align*}"}
+{"id": "1070.png", "formula": "\\begin{align*} \\Omega _ { \\gamma } = \\Omega \\cap \\left \\{ x _ { 1 } \\in \\left ( - A _ { 1 } + \\gamma , A _ { 1 } \\right ) \\right \\} , Q _ { \\gamma , T } = \\Omega _ { \\gamma } \\times \\left ( 0 , T \\right ) \\subset Q _ { T } . \\end{align*}"}
+{"id": "4715.png", "formula": "\\begin{align*} P _ 1 + P _ 2 & = ( x _ { i _ 2 } + z _ { i _ 2 } ) ( z _ { i _ 1 } + v _ { i _ 1 } ) + ( z _ { i _ 2 } + v _ { i _ 2 } ) ( x _ { i _ 1 } + v _ { i _ 1 } ) . \\end{align*}"}
+{"id": "8137.png", "formula": "\\begin{align*} d _ T ( g _ t ( s ) , g _ t ( s ' ) ) = d _ T ( \\psi ( t , 0 ) , \\gamma ( t ) ) \\left | s - s ' \\right | . \\end{align*}"}
+{"id": "1376.png", "formula": "\\begin{align*} g = \\sum _ { n = 1 } ^ { n _ 0 } c ( n ) n ^ { 2 k - 1 } R _ { 2 - 2 k } ^ { \\ell - 1 } \\left ( \\mathcal { F } _ { 2 - 2 k , - 1 } \\big | _ { 2 - 2 k } T _ n \\right ) . \\end{align*}"}
+{"id": "8870.png", "formula": "\\begin{align*} \\chi ( M ) = \\frac { 1 } { 2 \\pi } \\int _ M K { \\rm v o l } . \\end{align*}"}
+{"id": "8750.png", "formula": "\\begin{align*} \\eta _ t = \\min \\left ( \\frac { 1 } { 8 \\kappa \\bar { L } d } , \\ , d ^ { - \\frac { 2 ( \\beta - 1 ) } { 2 \\beta - 1 } } T ^ { - \\frac { \\beta } { 2 \\beta - 1 } } \\right ) \\qquad h _ t = \\left ( \\frac { d ^ 2 } { T } \\right ) ^ { \\frac { 1 } { 2 ( 2 \\beta - 1 ) } } \\enspace , \\end{align*}"}
+{"id": "4333.png", "formula": "\\begin{align*} S ( \\omega _ { { \\sf f } _ 1 \\cdots { \\sf f } _ n } \\| \\omega _ { { \\sf g } _ 1 \\cdots { \\sf g } _ m } ) = S ( \\omega _ { { \\sf g } _ m \\cdots { \\sf g } _ 1 { \\sf f } _ 1 \\cdots { \\sf f } _ n } \\| \\omega ) . \\end{align*}"}
+{"id": "3770.png", "formula": "\\begin{align*} { F ' _ { \\infty } = \\lim _ { s \\to \\infty } \\frac { F ( s ) } { s } } \\end{align*}"}
+{"id": "5315.png", "formula": "\\begin{align*} P ( x ) + P ( - x ) & \\geq P ( x ) \\\\ & \\geq \\prod _ { k = 0 } ^ { n - 1 } \\xi ^ k \\\\ & \\geq \\xi ^ { n \\choose 2 } \\\\ & > 2 \\xi ^ { n + 1 \\choose 2 } , \\end{align*}"}
+{"id": "2756.png", "formula": "\\begin{align*} R ( w ) = \\frac { r _ A } { 2 \\pi i } \\int _ { \\partial \\mathbb { D } } \\frac { J _ { [ a , b ] } ( r _ A v ) - 1 / \\mathcal G ( r _ A v ) } { \\mathcal G ( r _ A v ) - w } G ' ( J _ { [ a , b ] } ( r _ A v ) ) J _ { [ a , b ] } ' ( r _ A v ) \\mathrm { d } v = : \\int _ { \\partial \\mathbb D } u ( v ) \\mathrm { d } v . \\end{align*}"}
+{"id": "1038.png", "formula": "\\begin{align*} \\hat { \\varphi } ( \\hat { u } ) = \\inf \\left \\{ \\hat { \\varphi } ( u ) : \\ : u \\in W ^ { 1 , p ( z ) } ( \\Omega ) \\right \\} . \\end{align*}"}
+{"id": "104.png", "formula": "\\begin{align*} \\langle n _ + ^ { H } \\rangle _ { \\Psi } \\leq C \\begin{dcases} C _ B N \\ , K _ L ^ { - 2 } K _ { \\ell } ^ 2 \\widehat { g } ( 0 ) , & d = 2 , \\\\ C _ B N \\ , K _ L ^ { - 2 } K _ { \\ell } ^ 2 \\sqrt { \\rho a ^ 3 } , & d = 3 . \\end{dcases} \\end{align*}"}
+{"id": "1660.png", "formula": "\\begin{align*} P _ { \\texttt { a } ; \\mu } ( \\boldsymbol { \\xi } ; q ) : = \\sum _ { \\sigma \\in S _ { n } } C _ { \\texttt { a } } ( \\xi _ { \\sigma _ 1 } , \\ldots , \\xi _ { \\sigma _ { n } } ; q ) \\exp ( i \\xi _ { \\sigma _ 1 } \\mu _ 1 + \\cdots + i \\xi _ { \\sigma _ { n } } \\mu _ { n } ) , \\end{align*}"}
+{"id": "4493.png", "formula": "\\begin{align*} \\mathbb { L } '' ( \\hat { \\mathbf U } , \\hat { \\Psi } ) ( ( { \\mathbf V } , \\Psi ) , ( \\tilde { { \\mathbf V } } , \\tilde { \\Psi } ) ) : = \\frac { d } { d \\varepsilon } \\mathbb { L } ' ( \\hat { \\mathbf U } + \\varepsilon \\tilde { { \\mathbf V } } , \\hat { \\Psi } + \\varepsilon \\tilde { \\Psi } ) ( { \\mathbf V } , \\Psi ) \\Big | _ { \\varepsilon = 0 } , \\end{align*}"}
+{"id": "4970.png", "formula": "\\begin{align*} f ( x ) = f _ 1 ( x ^ { t ( q ) } ) \\end{align*}"}
+{"id": "2838.png", "formula": "\\begin{align*} \\hat { \\phi } _ \\mathrm { f } = \\frac { 1 } { M } \\operatorname { a r c t a n } \\left ( \\frac { \\sum _ { i = 0 } ^ { N _ \\mathrm { c } - 1 } \\operatorname { I m } \\left ( \\frac { y _ i ^ { M } } { \\lvert y _ i \\rvert ^ { M - 1 } } \\right ) } { \\sum _ { i = 0 } ^ { N _ \\mathrm { c } - 1 } \\operatorname { R e } \\left ( \\frac { y _ i ^ { M } } { \\lvert y _ i \\rvert ^ { M - 1 } } \\right ) } \\right ) . \\end{align*}"}
+{"id": "428.png", "formula": "\\begin{align*} \\int _ { \\R _ + } \\frac { | h ( r ) \\cdot u ( r ) | ^ 2 } { r ^ \\alpha } \\ , r ^ { d - 1 + 2 \\ell } \\ , d r = \\int _ { \\R ^ d } \\frac { | [ h \\cdot u ] _ { \\ell , m } ( x ) | ^ 2 } { | x | ^ \\alpha } \\ , d x . \\end{align*}"}
+{"id": "1993.png", "formula": "\\begin{align*} [ X , Y ] _ { \\theta } & : = \\left \\{ \\ , f ( \\theta ) \\ , \\big { | } \\ , f \\in \\mathrm { F } ( X , Y ) \\ , \\right \\} \\\\ \\left \\lVert { x } \\right \\rVert _ { [ X , Y ] _ { \\theta } } & : = \\underset { \\substack { f \\in \\mathrm { F } ( X , Y ) , \\\\ f ( \\theta ) = x } } { \\inf } \\left \\lVert { f } \\right \\rVert _ { \\mathrm { F } ( X , Y ) } \\end{align*}"}
+{"id": "4255.png", "formula": "\\begin{align*} \\mathcal { J } _ { \\lambda } ' ( u ) ( \\varphi ) & = \\int _ { \\Omega } \\langle \\nabla u , \\nabla \\varphi \\rangle \\ , d x + \\alpha \\iint _ { \\mathbb { R } ^ { 2 n } } \\dfrac { ( u ( x ) - u ( y ) ) ( \\varphi ( x ) - \\varphi ( y ) ) } { | x - y | ^ { n + 2 s } } \\ , d x d y \\\\ & - \\lambda \\int _ { \\Omega } u ( x ) \\varphi ( x ) \\ , d x - \\int _ { \\Omega } f ( x , u ( x ) ) \\varphi ( x ) \\ , d x \\textrm { f o r e v e r y } \\varphi \\in \\mathbb { X } ( \\Omega ) \\ , . \\end{align*}"}
+{"id": "3603.png", "formula": "\\begin{align*} { \\bf { r } } _ m = { \\bf { H } } _ m { \\bf F } _ { \\rm D S } { \\bf s } + { \\bf { n } } _ m . \\end{align*}"}
+{"id": "5664.png", "formula": "\\begin{align*} c _ 1 + 3 = r ( L ' ) \\cdot r ( \\xi ) = 2 a + b _ 0 + b _ 1 \\end{align*}"}
+{"id": "8744.png", "formula": "\\begin{align*} \\| { \\hat g } \\| ^ 2 & = \\frac { d ^ 2 } { 4 h ^ 2 } \\| ( f ( x + h r \\zeta ) - f ( x - h r \\zeta ) + \\xi - \\xi ' ) \\zeta K ( r ) \\| ^ 2 \\\\ & = \\frac { d ^ 2 } { 4 h ^ 2 } ( f ( x + h r \\zeta ) - f ( x - h r \\zeta ) + \\xi - \\xi ' ) ^ 2 K ^ 2 ( r ) . \\end{align*}"}
+{"id": "5346.png", "formula": "\\begin{align*} c K _ S ^ 2 + ( c + 1 2 ) c _ 2 ( S ) = \\frac { 6 d } { c + 2 } [ c ( c + 2 ) k ^ 2 - c ( 5 c + 1 2 ) k + 6 c ^ 2 + 2 0 c + 1 2 ] . \\end{align*}"}
+{"id": "5795.png", "formula": "\\begin{align*} w _ 0 = s _ 1 ( s _ 2 s _ 1 ) ( s _ 3 s _ 2 s _ 1 ) \\dots ( s _ n s _ { n - 1 } { \\dots } s _ 2 s _ 1 ) \\end{align*}"}
+{"id": "4757.png", "formula": "\\begin{align*} E _ k \\subset B _ { r _ k } ( y ) = B _ { r / C } ( y ) , \\end{align*}"}
+{"id": "8084.png", "formula": "\\begin{align*} \\left \\| \\uppercase \\expandafter { \\romannumeral 1 } _ { 1 } \\right \\| _ { L _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\mu _ { j } \\chi _ { P _ { j } } } { \\omega ( P _ { j } ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } . \\end{align*}"}
+{"id": "1384.png", "formula": "\\begin{align*} G ( z ) | T _ 2 - ( - 2 4 ) G ( z ) = q + 1 6 8 6 8 4 0 9 q ^ 2 + 2 7 9 6 8 7 5 1 4 9 1 4 3 3 3 q ^ 3 + O \\left ( q ^ 4 \\right ) , \\end{align*}"}
+{"id": "1053.png", "formula": "\\begin{align*} \\hat { g } ( z , x ) = \\left \\{ \\begin{array} { l l } f ( z , u _ \\mu ( z ) ) + \\vartheta u _ \\mu ( z ) ^ { p ( z ) - 1 } , & \\\\ f ( z , x ) + \\vartheta x ^ { p ( z ) - 1 } , & \\\\ f ( z , u _ \\eta ( z ) ) + \\vartheta u _ \\eta ( z ) ^ { p ( z ) - 1 } , & \\end{array} \\right . \\end{align*}"}
+{"id": "4684.png", "formula": "\\begin{align*} \\min _ { \\pi \\in X } \\sup _ { \\lambda \\geq 0 } L ( \\pi , \\lambda ) = \\sup _ { \\lambda \\geq 0 } \\min _ { \\pi \\in X } L ( \\pi , \\lambda ) . \\end{align*}"}
+{"id": "898.png", "formula": "\\begin{align*} m x ^ k \\bigg ( \\eta _ u - \\tau _ t + \\frac { 1 - \\alpha } { t } \\tau \\bigg ) - 2 \\eta _ { x v } - m k x ^ { k - 1 } \\xi - m x ^ k ( \\phi _ v - \\xi _ x ) = 0 , \\end{align*}"}
+{"id": "6163.png", "formula": "\\begin{align*} A _ 2 = \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\end{pmatrix} , & & A _ 3 = \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\end{pmatrix} , & & A _ 6 = \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\end{pmatrix} , & & O = \\begin{pmatrix} 0 \\\\ 0 \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\end{pmatrix} , & & \\end{align*}"}
+{"id": "8675.png", "formula": "\\begin{align*} \\int e ^ { \\frac { \\phi _ 0 y _ 0 + \\phi _ 1 y _ 1 + . . . + \\phi _ n y _ n } { \\hbar } } ~ \\psi ~ d \\phi _ 1 \\wedge . . . d \\phi _ n = ( 2 \\pi \\hbar ) ^ { \\frac { n } { 2 } } \\cdot y _ 0 ^ { - \\frac { n } { 2 } } ~ \\hat { g } \\Bigg ( \\frac { y _ 1 } { y _ 0 } , . . . , \\frac { y _ n } { y _ 0 } \\Bigg ) \\cdot e ^ { \\frac { y _ 0 } { \\hbar } \\hat { f } \\big ( \\frac { y _ 1 } { y _ 0 } , . . . , \\frac { y _ n } { y _ 0 } \\big ) } . \\end{align*}"}
+{"id": "3909.png", "formula": "\\begin{align*} \\mu = \\lambda ( K , \\cdot ) ? \\end{align*}"}
+{"id": "7440.png", "formula": "\\begin{align*} \\| ( y _ 1 , y _ 2 ) \\| _ { 1 , p } & \\leq C _ { 1 2 } ( \\| y _ 1 \\| _ p ^ { 1 - \\nu } + \\| y _ 2 \\| _ p ^ { 1 - \\nu } ) \\\\ & \\leq C _ { 1 3 } ( \\| y _ 1 \\| _ { 1 , p } ^ { 1 - \\nu } + \\| y _ 2 \\| _ { 1 , p } ^ { 1 - \\nu } ) \\\\ & \\leq C _ { 1 4 } \\| ( y _ 1 , y _ 2 ) \\| _ { 1 , p } ^ { \\frac { 1 - \\nu } { p } } , \\end{align*}"}
+{"id": "4301.png", "formula": "\\begin{align*} M ' _ { r } = \\sum _ { \\pi \\in \\mathcal { D P } _ 2 ( 2 r ) } \\int _ { [ 0 , 1 ] \\times [ - \\lambda , 1 ] ^ { r } } \\prod _ { s = 1 } ^ { r } \\chi _ { [ 0 , \\lambda ] } \\left ( x _ 0 - \\sum _ { \\ell = i } ^ { 2 s - 1 } \\epsilon _ \\pi ( i ) x _ { \\pi ( i ) } \\right ) \\chi _ { [ 0 , 1 ] } \\left ( x _ 0 - \\sum _ { i = 1 } ^ { 2 s } \\epsilon _ \\pi ( i ) x _ { \\pi ( i ) } \\right ) \\prod _ { l = 0 } ^ { r } \\mathrm { ~ d } x _ l , \\end{align*}"}
+{"id": "8597.png", "formula": "\\begin{align*} \\phi _ 1 = - h _ b \\frac { \\sinh ( \\frac { z } { h _ b } \\sqrt { \\mu } | \\mathrm { D } | ) } { \\cosh ( \\sqrt { \\mu } | \\mathrm { D } | ) } \\frac { 1 } { \\sqrt { \\mu } | \\mathrm { D } | } \\nabla _ X \\cdot \\big ( \\mathcal { L } _ { 1 } ^ { \\mu } [ \\beta b ] \\nabla _ X \\psi \\big { ) } , \\end{align*}"}
+{"id": "958.png", "formula": "\\begin{align*} \\mathcal { F } _ { 1 , \\theta } [ \\Sigma _ t ] = - \\int _ \\Sigma H _ 2 ( t ) \\left < \\xi _ t , \\eta _ t \\right > \\ , d \\mu _ { \\Sigma _ t } + \\int _ { \\partial \\Sigma } \\left < \\xi _ t , ( P _ 1 \\nu - \\vert { P _ 1 \\nu } \\vert \\cos \\theta \\ , \\overline \\nu ) _ t \\right > \\ , d \\mu _ { \\partial \\Sigma _ t } , \\end{align*}"}
+{"id": "6798.png", "formula": "\\begin{align*} u ( z ) & = 1 - \\alpha e ^ { \\lambda _ 1 z } - \\beta e ^ { \\lambda _ 2 z } - \\gamma e ^ { \\lambda _ 3 z } + h . o . t . , \\\\ [ 0 . 2 c m ] w ( z ) & = 0 - \\alpha \\lambda _ 1 e ^ { \\lambda _ 1 z } - \\beta \\lambda _ 2 e ^ { \\lambda _ 2 z } - \\gamma \\lambda _ 3 e ^ { \\lambda _ 3 z } + h . o . t . , \\\\ [ 0 . 2 c m ] v ( z ) & = 0 - \\alpha \\psi \\left ( \\lambda _ { 1 } \\right ) e ^ { \\lambda _ 1 z } - \\beta \\psi \\left ( \\lambda _ { 2 } \\right ) e ^ { \\lambda _ 2 z } + h . o . t . . \\end{align*}"}
+{"id": "7517.png", "formula": "\\begin{align*} L _ u ( f ) = \\theta \\partial _ v \\left \\{ M _ u \\ , \\partial _ v ( M _ { u } ^ { - 1 } f ) \\right \\} , \\end{align*}"}
+{"id": "875.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathcal { T } _ t ^ \\alpha u = u _ { x x } + \\frac { c } { x } u _ x + m x ^ k v _ x , \\\\ & \\mathcal { T } _ t ^ \\alpha v = v _ { x x } + \\frac { c } { x } v _ x + n x ^ k u _ x , ~ ~ x > 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4642.png", "formula": "\\begin{align*} \\widetilde { \\Theta } : = \\sum _ { \\ell _ 1 \\in \\mathcal { I } _ { e _ 1 } } m _ { e _ 1 , \\ell _ 1 } ^ * \\otimes \\Theta ( m _ { e _ 1 , \\ell _ 1 } ) \\in \\bigl ( \\mathbf { M } ^ { \\pi _ { \\mathcal { S } ( v _ 1 ) } } \\otimes L \\bigr ) ^ { K _ { v _ 1 } } \\end{align*}"}
+{"id": "2040.png", "formula": "\\begin{align*} u = S _ { \\phi } ^ { - 1 } A + S _ { \\phi } ^ { - 1 } B \\in \\mathrm { L } ^ { p } ( \\partial \\Omega ) + \\dot { \\mathrm { H } } ^ { 1 , p } ( \\partial \\Omega ) \\end{align*}"}
+{"id": "1113.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - \\Delta _ { \\mu } u ( x ) = \\alpha ( x ) u ( x ) + f _ + ( x , u ( x ) ) x \\in \\mathop D \\limits ^ \\circ \\\\ \\medskip \\ , \\ , u | _ { \\partial D } = 0 . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "5218.png", "formula": "\\begin{align*} \\mu - \\lambda = \\frac { 2 k } { v - 1 } \\end{align*}"}
+{"id": "5316.png", "formula": "\\begin{align*} \\xi ^ n = e ^ { - \\sqrt { n } } \\leq e ^ { - 1 } < \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "3278.png", "formula": "\\begin{align*} \\phi _ X ( t ) = E [ e ^ { \\mu t X } ] = \\int _ \\mathbb { R } f _ X ( x ) e ^ { \\mu t x } d x = \\mathcal { F } ^ \\mu ( f _ X ) ( t ) . \\end{align*}"}
+{"id": "1914.png", "formula": "\\begin{align*} \\left ( \\theta _ w , q _ h \\right ) + \\sqrt { \\epsilon } b _ h ( \\theta _ u , q _ h ) = \\left ( \\eta _ w , q _ h \\right ) , \\forall \\ , \\ , q _ h \\in X _ h . \\end{align*}"}
+{"id": "2043.png", "formula": "\\begin{align*} \\mathcal { K } ^ { s , p } ( \\mathbb { R } ^ { n } _ + ) : = \\mathrm { H } ^ { s , p } ( \\mathbb { R } _ + , \\mathrm { L } ^ p ( \\mathbb { R } ^ { n - 1 } ) ) \\cap \\mathrm { H } ^ { s - 1 , p } ( \\mathbb { R } _ + , \\mathrm { H } ^ { 1 , p } ( \\mathbb { R } ^ { n - 1 } ) ) \\end{align*}"}
+{"id": "5391.png", "formula": "\\begin{align*} \\Xi ( t , \\cdot ) : = \\frac { 1 } { N } \\sum _ { j = 1 } ^ N \\delta _ { \\Lambda _ j ( t ) } ( \\cdot ) , t \\geq 0 , \\end{align*}"}
+{"id": "2654.png", "formula": "\\begin{align*} K ( x , t ) = - \\int _ 0 ^ x \\frac { K ( y , t ) } { 1 - y } \\ , d y + B ( x , t ) \\ , , \\end{align*}"}
+{"id": "7477.png", "formula": "\\begin{align*} \\Gamma _ n ( z ) \\Gamma _ n ( 1 - z ) = 1 , n \\in \\{ 0 , 1 \\} . \\end{align*}"}
+{"id": "8368.png", "formula": "\\begin{align*} C ( \\omega ) = \\overline { \\bigcup _ { n \\in \\mathbb { N } } \\{ x _ n ( \\omega ) \\} } . \\end{align*}"}
+{"id": "295.png", "formula": "\\begin{align*} \\gamma _ { \\operatorname * { D } ; j } ^ { - } \\left ( \\left . u \\right \\vert _ { \\Omega _ { j } } \\right ) = \\gamma _ { \\operatorname * { D } ; j } ^ { + } \\left ( \\left . u \\right \\vert _ { \\Omega _ { j } ^ { + } } \\right ) \\end{align*}"}
+{"id": "4319.png", "formula": "\\begin{align*} W ( \\ell ) W ( \\ell ' ) = { \\rm e } ^ { - i \\sigma ( \\ell , \\ell ' ) / 2 } W ( \\ell + \\ell ' ) \\ , , W ( \\ell ) ^ * = W ( - \\ell ) W ( 0 ) = { \\bf 1 } \\ , . \\end{align*}"}
+{"id": "8776.png", "formula": "\\begin{align*} b _ { t + 1 } < \\left ( 1 - \\frac { 2 } { t } \\right ) b _ { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { t ^ { p _ { i } + 1 } } , \\end{align*}"}
+{"id": "632.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\left ( - i \\partial _ t - i \\alpha \\partial _ x + M \\beta \\right ) \\psi & = \\phi \\beta \\psi + \\beta \\psi \\xi _ 1 , \\\\ \\left ( \\partial _ t ^ 2 - \\partial _ x ^ 2 + m ^ 2 \\right ) \\phi & = \\psi ^ * \\beta \\psi + \\phi \\xi _ 2 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "8289.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( A _ \\varepsilon ( u , v , p ) , ( u , v , p ) \\right ) _ { \\mathcal { H } } & \\leq \\left ( \\frac { d ^ 2 e ^ \\mu } { 4 \\mu } - 1 \\right ) \\int _ 0 ^ 1 \\abs { p _ x } ^ 2 d x + \\left ( b ^ 2 ( 2 \\sinh { \\mu } + 4 \\cosh { \\mu } ) - c + \\frac { d ^ 2 e ^ \\mu } { 4 \\mu } \\right ) \\abs { p ( 0 ) } ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "3636.png", "formula": "\\begin{align*} \\theta ( \\eta ) \\theta ( P - \\phi \\eta ) = 0 . \\end{align*}"}
+{"id": "1185.png", "formula": "\\begin{align*} F ( X ) = \\{ F _ k ( X ) \\} _ k \\mathrm { a n d } \\quad \\overline { F } ( M ) = \\{ \\overline { F } _ k ( M ) \\} _ k \\end{align*}"}
+{"id": "8734.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } \\Big ( \\frac { r _ t - r _ { t + 1 } } { \\eta _ t } - \\frac { \\alpha } { 2 } r _ t \\Big ) = \\frac { \\alpha } { 2 } \\sum _ { t = 1 } ^ { T } \\Big ( r _ t ( t - 1 ) - r _ { t + 1 } t \\Big ) \\leq 0 \\enspace . \\end{align*}"}
+{"id": "5388.png", "formula": "\\begin{align*} d \\Lambda _ j ( t ) = ( S ^ { - 1 } ( t ) d M ( t ) S ( t ) ) _ { j j } , 1 \\leq j \\leq N , \\ , t \\geq 0 , \\end{align*}"}
+{"id": "6037.png", "formula": "\\begin{align*} ( \\mathbf { B } _ q ^ \\prime ) \\quad \\vert \\mathbb { E } ( \\partial ^ \\beta f ( F ) \\Upsilon _ \\eta ( \\det \\sigma _ F ) G ) \\vert \\leq C _ q ^ \\prime \\Vert f \\Vert _ \\infty \\times \\frac { 1 } { \\eta ^ { 2 q } } , \\quad \\forall \\vert \\beta \\vert = q . \\end{align*}"}
+{"id": "5348.png", "formula": "\\begin{align*} \\begin{aligned} \\binom { k + c } { c + 2 } = h ^ 0 ( \\O _ { \\P ^ { c + 2 } } ( k - 2 ) ) = h ^ 0 ( \\O _ S ( k - 2 ) ) = \\chi ( \\O _ S ( k - 2 ) ) + h ^ 1 ( \\O _ S ( k - 2 ) ) \\ge \\\\ \\ge \\chi ( \\O _ S ) + \\frac { d ( k - 2 ) } { 2 } \\big ( 3 - k + \\frac { 4 ( k - 1 ) } { c + 2 } \\big ) . \\end{aligned} \\end{align*}"}
+{"id": "6928.png", "formula": "\\begin{align*} \\liminf \\limits _ { n \\rightarrow + \\infty } f \\left ( \\phi , \\psi \\right ) ( \\xi _ n ) & \\geq f \\left ( \\gamma _ { 2 n _ 0 - 1 } , \\gamma _ { 2 n _ 0 - 2 } \\right ) \\\\ [ 0 . 2 c m ] & > ( 1 - \\gamma _ { 2 n _ 0 - 1 } ) \\left ( \\frac { \\gamma _ { 2 n _ 0 - 1 } } { b } - 1 \\right ) - \\displaystyle \\frac { m \\gamma _ { 2 n _ 0 - 2 } } { \\gamma _ { 2 n _ 0 - 3 } + a \\gamma _ { 2 n _ 0 - 2 } } = 0 , \\end{align*}"}
+{"id": "2674.png", "formula": "\\begin{align*} \\omega _ t \\ = \\ t \\omega _ { 0 } - _ { \\omega _ { 0 } } + d d ^ c \\varphi _ t \\ > \\ 0 , \\end{align*}"}
+{"id": "4807.png", "formula": "\\begin{align*} \\Phi \\circ \\varphi _ t = \\varphi _ { A t } \\circ \\Phi , \\end{align*}"}
+{"id": "7981.png", "formula": "\\begin{align*} | w _ { d _ { 1 } - 1 } | \\ge \\frac { c _ { d _ { 1 } , 1 } } { ( d _ 1 - 1 ) ! } , \\ \\ w _ { d _ { 1 } - 2 } = 0 , \\ \\ | w _ { d _ { 1 } - 3 } | \\leq \\frac { ( s _ { d _ { 1 } } ) ^ 2 } { ( d _ 1 - 3 ) ! } , | w _ { d _ { 1 } - 4 } | \\leq \\frac { ( s _ { d _ { 1 } } ) ^ 3 } { ( d _ 1 - 4 ) ! } , \\dots , | w _ 0 | \\leq ( s _ { d _ { 1 } } ) ^ { d _ { 1 } - 1 } . \\end{align*}"}
+{"id": "5540.png", "formula": "\\begin{align*} F ( \\infty ) = \\infty , \\lim _ { \\zeta \\to \\infty } ( F ( \\zeta ) - \\zeta ) = 0 , \\end{align*}"}
+{"id": "7228.png", "formula": "\\begin{align*} P _ M ^ R ( t ) = \\dfrac { P _ M ^ S ( t ) } { 1 - t ( P _ R ^ S ( t ) - 1 ) } . \\end{align*}"}
+{"id": "309.png", "formula": "\\begin{align*} \\ell _ { j } \\left ( s \\right ) \\left ( v , w \\right ) = \\left \\langle - \\left [ v \\right ] _ { \\operatorname * { N } ; j } ^ { \\operatorname * { e x t } } \\left ( s \\right ) , \\gamma _ { \\operatorname * { D } ; j } \\left ( s \\right ) \\overline { w } \\right \\rangle _ { \\Gamma _ { j } } , \\qquad \\forall w \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) . \\end{align*}"}
+{"id": "400.png", "formula": "\\begin{align*} \\epsilon ( c , f ) = f ( 1 _ c ) . \\end{align*}"}
+{"id": "8119.png", "formula": "\\begin{align*} p _ 0 - q _ 1 = u _ \\alpha - u _ \\beta . \\end{align*}"}
+{"id": "8503.png", "formula": "\\begin{align*} B _ { \\rho } ( ( \\bar { z } , w ) ) \\cap \\{ z = \\zeta \\} \\subset B ^ { n - 1 } ( w , \\rho ) \\mbox { f o r e v e r y } \\zeta \\in ( \\bar { z } - \\rho , \\bar { z } + \\rho ) . \\end{align*}"}
+{"id": "7445.png", "formula": "\\begin{align*} F _ { ( y _ 1 , y _ 2 ) } ( t _ { 0 } ) = \\max \\limits _ { t \\in ( 0 , \\infty ) } F _ { ( y _ 1 , y _ 2 ) } ( t ) , \\end{align*}"}
+{"id": "685.png", "formula": "\\begin{align*} \\mathbf H ^ { \\mathbf s } ( \\R ^ d ) = H ^ { s _ 1 } ( \\R ^ d ) \\times \\dots \\times H ^ { s _ n } ( \\R ^ d ) \\end{align*}"}
+{"id": "4130.png", "formula": "\\begin{align*} G ( k , s , S _ k ) \\sum _ { \\ell = 0 } ^ { k - 2 r - 2 s + 2 } V ( k , s , r , \\ell ) + ( S _ r - 1 ) H ( k , s , r , S _ k , S _ r ) \\sum _ { \\ell = 0 } ^ { k - 2 r - 2 s + 2 } V ( k , s , r , \\ell ) = 0 . \\end{align*}"}
+{"id": "2008.png", "formula": "\\begin{align*} \\Psi ( \\mathbb { R } ^ n _ + ) = \\Omega \\Psi ^ { - 1 } ( \\Omega ) = \\mathbb { R } ^ n _ + \\mathrm { d e t } ( \\nabla \\Psi ) = \\mathrm { d e t } ( \\nabla ( \\Psi ^ { - 1 } ) ) = 1 \\end{align*}"}
+{"id": "2363.png", "formula": "\\begin{align*} S _ { f , \\tau , \\Lambda } x = \\sum _ { \\lambda \\in \\Lambda } f ( \\pi ( \\lambda ) ^ { - 1 } x ) \\pi ( \\lambda ) \\tau = \\frac { o ( \\Lambda ) } { o ( G ) } \\sum _ { \\mu \\in \\Lambda ^ 0 } f ( \\pi ( \\mu ) ^ { - 1 } \\tau ) \\pi ( \\mu ) x = \\frac { o ( \\Lambda ) } { o ( G ) } S _ { f , x , \\Lambda ^ 0 } \\tau , \\forall x \\in \\mathbb { C } ^ { o ( G ) } . \\end{align*}"}
+{"id": "6791.png", "formula": "\\begin{align*} & \\underline { v } ' = \\overline { v } ' + r q ( 1 ) e ^ { \\lambda z } \\left [ \\frac { 1 } { 2 } ( - z ) ^ { - 1 / 2 } - \\lambda ( - z ) ^ { 1 / 2 } \\right ] , \\\\ [ 0 . 2 c m ] & \\underline { v } '' = \\overline { v } '' + r q ( 1 ) e ^ { \\lambda z } \\left [ \\lambda ( - z ) ^ { - 1 / 2 } + \\frac { 1 } { 4 } ( - z ) ^ { - 3 / 2 } - \\lambda ^ 2 ( - z ) ^ { 1 / 2 } \\right ] . \\end{align*}"}
+{"id": "1286.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\liminf _ { n \\to \\infty } \\big [ P ( f _ n ) - P ( g _ n ^ j \\phi ^ j ) - P ( r _ n ^ J ) \\big ] = 0 . \\end{align*}"}
+{"id": "3439.png", "formula": "\\begin{align*} - \\log | s _ i | = \\min \\{ l _ 0 x _ 0 + \\ldots l _ m x _ m + a | s ^ { ( i ) } _ { l _ 0 \\ldots l _ m , a } \\neq 0 \\} . \\end{align*}"}
+{"id": "4243.png", "formula": "\\begin{align*} d T = - \\frac { 1 6 } { t ^ 2 } e ^ { 1 2 5 6 } - \\frac { 1 6 } { t ^ 2 } e ^ { 3 4 5 6 } . \\end{align*}"}
+{"id": "7414.png", "formula": "\\begin{align*} \\| \\partial _ { x } \\pi _ { n } \\| _ { L ^ { \\infty } _ { t } L ^ { 2 } _ { x } } & = \\| \\sqrt { \\rho _ { n } } ( \\sqrt { \\rho _ { n } } w _ { n } - \\sqrt { \\rho _ { n } } u _ { n } ) \\| _ { L ^ { \\infty } _ { t } L ^ { 2 } _ { x } } \\le C , \\end{align*}"}
+{"id": "779.png", "formula": "\\begin{align*} - \\Delta u + u = ( I _ \\alpha \\ast | u | ^ q ) | u | ^ { q - 2 } u + g ( u ) \\quad \\mathbb { R } ^ N \\end{align*}"}
+{"id": "489.png", "formula": "\\begin{align*} \\psi ( 0 , x , \\omega ) = \\psi _ 0 ( x , \\omega ) , \\phi ( 0 , x , \\omega ) = \\phi _ 0 ( x , \\omega ) , \\partial _ t \\phi ( 0 , x , \\omega ) = \\phi _ 1 ( x , \\omega ) . \\end{align*}"}
+{"id": "8259.png", "formula": "\\begin{align*} f _ { x _ 1 , 0 , 0 } ( \\rho , \\Theta ) = \\frac { 1 } { 2 } \\rho ^ 2 + \\sum _ { \\nu \\geq 2 } k _ \\nu ( \\Theta ) \\frac { x _ 1 ^ \\nu } { \\rho ^ { 2 \\nu - 2 } } , \\end{align*}"}
+{"id": "3594.png", "formula": "\\begin{align*} { { \\bf { D } } _ { { N _ { \\rm { R } } } } } = { \\rm { d i a g } } \\left ( { N _ { { \\rm { S } } , 1 } ^ 2 \\rho _ 1 ^ 2 \\ , \\ldots , \\ , N _ { { \\rm { S } } , { N _ { \\rm { R } } } } ^ 2 \\rho _ { { N _ { \\rm { R } } } } ^ 2 } \\right ) , \\end{align*}"}
+{"id": "4265.png", "formula": "\\begin{align*} \\lim _ { \\substack { u \\in H ^ { - } _ { k } \\\\ \\| u \\| _ { \\mathbb { X } ( \\Omega ) } \\to + \\infty } } J _ { \\lambda _ k } ( u ) = - \\infty , \\end{align*}"}
+{"id": "3745.png", "formula": "\\begin{align*} A ( \\lambda ^ 3 I + A ) ^ { - 1 } = I - \\lambda ^ 3 ( \\lambda ^ 3 I + A ) ^ { - 1 } . \\end{align*}"}
+{"id": "7263.png", "formula": "\\begin{align*} \\left ( P _ s ^ { n N } \\left ( f \\right ) \\right ) ' ( x ) = \\sum _ { \\omega \\in \\Omega ^ { n N } } \\mathbb { Q } \\left ( [ \\omega ] \\right ) \\left ( P _ { s , \\omega , n N } \\left ( f \\right ) \\right ) ' ( x ) . \\end{align*}"}
+{"id": "8626.png", "formula": "\\begin{align*} \\mathrm { R H S } _ 2 : & = \\frac { \\mu \\beta } { 4 } \\int _ { \\R ^ d } \\mathrm { F } _ 4 \\Delta _ X \\mathcal { L } ^ { \\mu } _ 1 [ \\beta b ] \\nabla _ X \\psi \\cdot \\nabla _ X \\psi \\ : \\mathrm { d } X \\\\ & = - \\frac { \\mu \\beta } { 4 } \\int _ { \\R ^ d } \\mathrm { F } _ 4 \\Delta _ X ( b \\nabla _ X \\psi ) \\cdot \\nabla _ X \\psi \\ : \\mathrm { d } X + O ( \\mu ^ 2 \\beta ^ 3 ) . \\end{align*}"}
+{"id": "2985.png", "formula": "\\begin{align*} E ^ { \\infty } = \\Big \\{ e _ 1 e _ 2 \\ldots \\in \\prod _ { i = 1 } ^ { \\infty } E ^ 1 \\mid s ( e _ i ) = r ( e _ { i + 1 } ) \\Big \\} \\end{align*}"}
+{"id": "7953.png", "formula": "\\begin{align*} \\lambda _ i = \\sum _ { g \\in S } \\frac { \\chi _ i ( g ) } { \\chi _ i ( 1 _ G ) } , \\end{align*}"}
+{"id": "6971.png", "formula": "\\begin{align*} \\omega ( x ) : = \\sum _ { j = 1 } ^ d \\sin ^ 2 ( x _ j / 2 ) . \\end{align*}"}
+{"id": "3390.png", "formula": "\\begin{align*} i ^ * \\omega _ U = \\begin{pmatrix} i ^ * \\alpha _ 0 ^ 0 - \\frac { 1 } { n + 1 } i ^ * \\alpha ^ l _ l & 0 & 0 \\\\ i ^ * \\alpha ^ \\mu _ 0 & i ^ * \\alpha ^ \\mu _ \\nu - \\frac { 1 } { n + 1 } i ^ * \\alpha ^ l _ l \\delta ^ \\mu _ \\nu & \\dd y ^ \\mu \\\\ - P _ { \\sigma 0 } \\dd y ^ \\sigma & - P _ { \\sigma \\nu } \\dd y ^ \\sigma & - \\frac { 1 } { n + 1 } i ^ * \\alpha ^ l _ l \\end{pmatrix} . \\end{align*}"}
+{"id": "2337.png", "formula": "\\begin{align*} \\tau _ g = \\pi _ g \\tau _ e , f _ g = f _ e \\pi _ { g ^ { - 1 } } , \\forall g \\in G \\end{align*}"}
+{"id": "8691.png", "formula": "\\begin{align*} x _ 0 ^ { p _ 0 } . . . x _ { m - 1 } ^ { p _ { m - 1 } } = x _ { m } ^ { p _ { m } } . . . x _ { n + 1 } ^ { p _ { n + 1 } } \\end{align*}"}
+{"id": "1209.png", "formula": "\\begin{align*} \\| e ^ { i ( t - a _ { j + 1 } ) \\Delta } w ( a _ { j + 1 } ) \\| _ { S ( I _ { j + 1 } ) } \\leq & 2 C \\epsilon + \\frac 1 3 \\sum _ { k = 1 } ^ { j } \\| w \\| _ { S ( I _ k ) } \\\\ & + C \\sum _ { k = 1 } ^ { j } ( \\eta ^ { p + 1 } \\| w \\| _ { S ( I _ k ) } ^ { p - 2 } + \\eta \\| w \\| _ { S ( I _ k ) } ^ { 2 ( p - 1 ) } + \\| w \\| _ { S ( I _ k ) } ^ { 2 p - 1 } ) . \\\\ \\leq & 2 C \\epsilon + 4 \\sum _ { k = 1 } ^ { j } \\gamma _ j . \\end{align*}"}
+{"id": "8329.png", "formula": "\\begin{align*} M _ { i j } = \\begin{cases} 1 & ( j - 1 ) ( K + 1 ) \\leq i \\leq j ( K + 1 ) \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "7436.png", "formula": "\\begin{align*} \\lambda ( \\kappa _ 1 + \\kappa _ 2 + 2 ) & ( \\kappa _ 1 + \\kappa _ 2 + 1 ) \\int _ { \\Omega } \\vert y _ 1 \\vert ^ { \\kappa _ 1 + 1 } \\vert y _ 2 \\vert ^ { \\kappa _ 2 + 1 } d z \\\\ \\leq & ( p - 1 ) \\| ( y _ 1 , y _ 2 ) \\| _ { 1 , p } + ( q - 1 ) \\| ( \\nabla y _ 1 , \\nabla y _ 2 ) \\| _ { q , \\eta } \\\\ & + \\nu \\int _ { \\Omega } \\left [ a _ 1 \\vert y _ 1 \\vert ^ { 1 - \\nu } + a _ 2 \\vert y _ 2 \\vert ^ { 1 - \\nu } \\right ] d z . \\end{align*}"}
+{"id": "1950.png", "formula": "\\begin{align*} \\partial _ t f + v \\ , \\partial _ x f + \\partial _ v \\left ( \\left ( u ^ * - v \\right ) f \\right ) = 0 , \\end{align*}"}
+{"id": "7588.png", "formula": "\\begin{align*} t ^ { 3 / 4 } - s ^ { 3 / 4 } = C r ^ { - \\frac { 1 } { 4 } } ( t - s ) \\ge C t ^ { - \\frac { 1 } { 4 } } ( t - s ) . \\end{align*}"}
+{"id": "3649.png", "formula": "\\begin{align*} \\delta \\Phi _ \\omega [ \\phi ] = 0 \\iff \\Delta _ { \\mathcal { E } ( \\omega ) } \\phi = 0 \\ . \\end{align*}"}
+{"id": "2984.png", "formula": "\\begin{align*} E ^ n _ I = \\{ ( e _ 1 , v _ 1 , \\ldots , e _ n , v _ n ) \\mid e _ i \\in E ^ 1 , \\ , v _ i \\in E ^ 0 _ I , \\ , s ( e _ i ) = \\alpha ( v _ i ) , v _ i = \\psi ( e _ { i + 1 } ) i \\ge 1 \\} . \\end{align*}"}
+{"id": "5193.png", "formula": "\\begin{align*} \\psi _ { K , U } ^ { e q } ( x ) - \\psi _ { K , U } ( t , x ) = \\int _ { U \\setminus K } p _ { U \\setminus K } ( t , x , y ) \\psi _ { K , U } ^ { e q } ( y ) d y \\ . \\end{align*}"}
+{"id": "5856.png", "formula": "\\begin{align*} m = | \\mathcal { D } ( X ) | \\geq | D ( \\pi ( C ) ) | \\geq q / | K | + 1 \\geq p ^ { n - k } + 1 , \\end{align*}"}
+{"id": "8518.png", "formula": "\\begin{align*} ( \\partial ^ { * } E ) _ { \\bar { z } } & \\subset \\mathbb { R } ^ { n - 1 } \\bigg \\backslash \\left ( \\bigg \\{ w \\in \\mathbb { R } ^ { n - 1 } : | w | > r _ { \\ell } ^ { \\vee } ( \\bar { z } ) \\bigg \\} \\bigg \\backslash \\{ \\tau \\} \\right ) \\\\ & = \\overline { B ^ { n - 1 } \\left ( 0 , r _ { \\ell } ^ { \\vee } ( \\bar { z } ) \\right ) } \\cup \\{ \\tau \\} , \\end{align*}"}
+{"id": "1752.png", "formula": "\\begin{align*} N ^ { ( m , n ) } _ \\epsilon : = \\bigl ( 2 ( m + n - 1 ) + \\epsilon _ + + \\epsilon _ - + \\tilde { \\epsilon } _ + + \\tilde { \\epsilon } _ - \\bigr ) ^ n . \\end{align*}"}
+{"id": "218.png", "formula": "\\begin{align*} & h ' _ 0 = - \\left \\{ e ^ { \\frac { 1 } { 2 4 } A _ 3 } \\widehat { A } ( T M ) \\right \\} ^ { ( 1 2 ) } \\\\ & h ' _ 1 = \\left \\{ e ^ { \\frac { 1 } { 2 4 } A _ 3 } \\widehat { A } ( T M ) { \\rm c h } ( \\widetilde { T _ C M } + 1 2 0 ) \\right \\} ^ { ( 1 2 ) } \\\\ & h ' _ 2 = \\left \\{ - e ^ { \\frac { 1 } { 2 4 } A _ 3 } \\widehat { A } ( T M ) { \\rm c h } ( 3 7 1 2 + 8 1 \\widetilde { T _ C M } + \\wedge ^ 2 \\widetilde { T _ C M } + W _ i ) + e ^ { \\frac { 1 } { 2 4 } A _ 3 } A _ 3 \\widehat { A } ( T M ) \\right \\} ^ { ( 1 2 ) } \\end{align*}"}
+{"id": "3881.png", "formula": "\\begin{align*} f ( u ( i _ 1 ( y _ 1 ) ) ) = v ( \\alpha ( i _ 1 ( y _ 1 ) ) ) = v ( \\beta ( i _ 1 ( y _ 1 ) ) ) = i _ 1 ( y _ 1 ) . \\end{align*}"}
+{"id": "8637.png", "formula": "\\begin{align*} L _ 1 ^ { \\mu } ( \\beta b ( X ) , \\xi ) & = - b ( X ) \\mathrm { s e c h } ( \\sqrt { \\mu } | \\xi | ) \\\\ & - \\frac { 1 } { 6 \\beta } \\Big ( \\beta ^ 3 b ( X ) ^ 3 \\int _ 0 ^ 1 \\cosh ( t \\beta b ( X ) \\sqrt { \\mu } | \\xi | ) ( 1 - t ) ^ 2 \\ : \\mathrm { d } t \\Big ) ( \\sqrt { \\mu } | \\xi | ) ^ 2 \\mathrm { s e c h } ( \\sqrt { \\mu } | \\xi | ) . \\end{align*}"}
+{"id": "2572.png", "formula": "\\begin{align*} v _ 0 & : = ( - \\sqrt { N } , \\dots , - \\sqrt { N } ) , \\\\ v _ i & : = v _ 0 + ( 0 , \\dots , \\underbrace { ( n + \\sqrt { n } ) \\sqrt { N } } _ { i - { \\rm t h \\ c o o r d . } } , \\dots , 0 ) \\hbox { f o r } i = 1 , \\dots , n , \\end{align*}"}
+{"id": "8470.png", "formula": "\\begin{align*} H _ { x , \\nu } ^ { + } = \\left \\{ y \\in \\mathbb { R } ^ { n } : \\langle ( y - x ) , \\nu \\rangle \\geq 0 \\right \\} \\end{align*}"}
+{"id": "4503.png", "formula": "\\begin{align*} u _ { 1 , i + 1 / 2 } : = S _ { \\theta _ i } u _ { 1 , i } + \\mathcal R _ T \\mathcal G \\ , , \\end{align*}"}
+{"id": "6148.png", "formula": "\\begin{align*} \\# ( \\boldsymbol { k } ) = \\prod _ { 1 \\leq i < j \\leq n } \\frac { k _ j - k _ i } { j - i } , \\end{align*}"}
+{"id": "2620.png", "formula": "\\begin{align*} J _ n ( t ) & = \\int _ 0 ^ t \\frac { U _ n ( t - x , x ) } { 1 - F ( x ) } \\ , d F ( x ) \\end{align*}"}
+{"id": "3956.png", "formula": "\\begin{align*} \\nabla _ x f ( x ' , y ' ) = \\frac { u \\cdot v } { u \\cdot y ' } \\frac { ( x ' \\cdot v ) y ' - ( x ' \\cdot y ' ) v } { ( x ' \\cdot v ) ^ 2 } \\end{align*}"}
+{"id": "8878.png", "formula": "\\begin{align*} ( x T ) w = x w T . \\end{align*}"}
+{"id": "2306.png", "formula": "\\begin{align*} \\hat { f } ( \\xi , \\tau ) = \\int _ { \\R \\times \\R } e ^ { - i ( x \\xi + t \\tau ) } u ( x , t ) d x d t . \\end{align*}"}
+{"id": "2198.png", "formula": "\\begin{align*} A : = \\{ x ' \\ : \\ \\C ( x ' \\mid m _ A ) \\le \\C ( x \\mid m _ A ) \\} B : = \\{ y ' \\ : \\ \\C ( y ' \\mid m _ A ) \\le \\C ( y \\mid m _ A ) \\} . \\end{align*}"}
+{"id": "2340.png", "formula": "\\begin{align*} \\lambda _ g \\theta _ f \\theta _ \\tau = \\theta _ f \\theta _ \\tau \\lambda _ g , \\forall g \\in G . \\end{align*}"}
+{"id": "7637.png", "formula": "\\begin{align*} U ( \\lambda x ) & \\leq U ( \\overline x ) = U ^ + ( \\overline x ) . \\end{align*}"}
+{"id": "8149.png", "formula": "\\begin{align*} A _ { 1 0 } A _ { i j } = & ( k - i - j + 1 ) ( r - 2 ) A _ { i - 1 , j } + \\left ( i ( r - 3 ) + j ( r - 2 ) \\right ) A _ { i j } + ( i + 1 ) A _ { i + 1 , j } , \\\\ A _ { 0 1 } A _ { i j } = & ( k - i - j + 1 ) ( r - 2 ) j A _ { i - 1 , j } + ( k - i - j + 1 ) ( n - k - j + 1 ) ( r - 1 ) A _ { i , j - 1 } \\\\ & + ( i + 1 ) j A _ { i + 1 , j } + ( j + 1 ) ^ 2 A _ { i , j + 1 } + ( i + 1 ) ( n - k - j + 1 ) ( r - 1 ) A _ { i + 1 , j - 1 } \\\\ & + ( j + 1 ) ^ 2 ( r - 2 ) A _ { i - 1 , j + 1 } + j \\left ( k - i - j + ( r - 2 ) i + ( n - k - j ) ( r - 1 ) \\right ) A _ { i , j } . \\end{align*}"}
+{"id": "3062.png", "formula": "\\begin{align*} { p _ i } = \\max { \\left ( 0 , { \\mu - \\frac { { \\sigma ^ 2 } } { { \\lambda _ i } } } \\right ) } , \\end{align*}"}
+{"id": "8239.png", "formula": "\\begin{align*} p _ 0 ^ * ( d ^ c _ { I _ 1 } ( \\frac { 1 } { 2 } f _ { x _ 1 } ) ) ( Y + a T ) = d ^ c _ { I _ 1 } ( \\frac { 1 } { 2 } f _ { x _ 1 } ) ( Y ) = d ^ c _ { I _ 1 } ( \\hat { K } | _ { \\mu ^ { - 1 } ( 0 ) } ) ( Y ) = - d \\hat { K } ( I _ 1 Y ) , \\end{align*}"}
+{"id": "2996.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ j a _ i \\cdot ( a _ i \\mid a ) _ A = \\sum _ { i = 1 } ^ j a _ i a _ i ^ * a = u _ j a \\to a \\end{align*}"}
+{"id": "7897.png", "formula": "\\begin{align*} \\begin{cases} ( u ^ { \\star } ) ^ k \\det D ^ 2 u = ( 1 - s u ) ^ { n + k } & \\Omega \\\\ u = 0 & \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "4681.png", "formula": "\\begin{align*} e ( S _ j '' , T _ j ) & \\geq | S _ j '' | \\cdot ( \\frac { n } { 2 ( 2 l + 1 ) } - 2 ) - \\left ( \\sum \\limits _ { q = 1 } ^ { 2 l + 1 } e ( S _ j '' , S _ q - S _ j '' ) + 2 e ( S _ j '' ) \\right ) \\\\ & \\geq | S _ j '' | \\cdot ( \\frac { n } { 2 ( 2 l + 1 ) } - 2 ) - ( 2 l + 3 ) k n \\\\ & > | S _ j '' | \\cdot \\frac { n } { 4 ( 2 l + 1 ) } \\end{align*}"}
+{"id": "839.png", "formula": "\\begin{align*} ( T f ) ( z ) = \\sum _ { n = 0 } ^ \\infty \\left ( 1 - \\dfrac { 1 } { n + 1 } \\right ) a _ n z ^ n . \\end{align*}"}
+{"id": "4506.png", "formula": "\\begin{align*} \\varepsilon ^ i & = \\left ( \\mathcal B ( { \\mathbf V } _ i , \\psi _ i ) _ 1 - \\mathcal B ( { \\mathbf V } _ { i - 1 } , \\psi _ { i - 1 } ) _ 1 \\right ) + \\mathcal B ( { \\mathbf V } _ { i - 1 } , \\psi _ { i - 1 } ) _ 1 \\\\ & = \\partial _ t \\delta \\psi _ { i - 1 } - \\delta u _ { 1 , i - 1 } + ( u _ 2 ^ a + u _ { 2 , i - 1 } ) \\partial _ 2 \\delta \\psi _ { i - 1 } + \\delta u _ { 2 , i - 1 } \\partial _ 2 ( \\varphi ^ a + \\delta \\psi _ { i - 1 } + \\psi _ { i - 1 } ) \\\\ & \\quad + \\mathcal B ( { \\mathbf V } _ { i - 1 } , \\psi _ { i - 1 } ) _ 1 \\ , . \\end{align*}"}
+{"id": "4801.png", "formula": "\\begin{align*} \\Pi _ k f ( x ) : = \\hat \\nu _ { k , x } ^ { - 1 } \\int _ { C _ k ^ \\perp ( x ) } f ( z ) \\ , d \\nu _ { k , x } , \\ ; \\ ; \\ ; \\ ; \\Delta _ k f : = f - \\Pi _ k f , \\ ; \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\ ; \\delta _ k f : = \\Pi _ k f - \\Pi _ { k - 1 } f . \\end{align*}"}
+{"id": "3367.png", "formula": "\\begin{align*} \\ll \\frac { X } { ( \\log z ) ^ 2 } \\exp \\Big ( \\sum _ { p \\leq z } \\frac { \\lambda _ 1 ( p ) + \\lambda _ 2 ( p ) } { p } \\Big ) + X ^ { \\varepsilon } Y z ^ 3 . \\end{align*}"}
+{"id": "6402.png", "formula": "\\begin{align*} c _ { 1 2 } c _ { 2 3 } t _ { 1 2 } & = t _ { 2 3 } c _ { 1 2 } c _ { 2 3 } c _ { 2 3 } t _ { 1 2 } c _ { 2 3 } ^ { - 1 } = c _ { 1 2 } ^ { - 1 } t _ { 2 3 } c _ { 1 2 } \\\\ t _ { 1 2 } c _ { 2 3 } c _ { 1 2 } & = c _ { 2 3 } c _ { 1 2 } t _ { 2 3 } c _ { 2 3 } ^ { - 1 } t _ { 1 2 } c _ { 2 3 } = c _ { 1 2 } t _ { 2 3 } c _ { 1 2 } ^ { - 1 } \\end{align*}"}
+{"id": "6426.png", "formula": "\\begin{align*} A _ { m } ( \\beta ) > A _ { m } ( \\beta ^ * ) = B ( \\beta ^ * ) > B ( \\beta ) 0 < \\beta < \\beta ^ * . \\end{align*}"}
+{"id": "1749.png", "formula": "\\begin{align*} \\Delta _ \\lambda ^ { ( m , n ) } : = \\prod _ { 1 \\leq j \\leq n } \\Delta _ { \\lambda _ j + n - j } ^ { ( m + n ) } , \\end{align*}"}
+{"id": "3199.png", "formula": "\\begin{align*} A ^ { T } A ( x - x _ { k } ) = - \\theta _ { k + 1 } \\phi _ { k } v _ { k + 1 } , x _ { k } - x _ { k - 1 } = \\frac { \\phi _ { k } } { \\rho _ { k } } w _ { k } \\end{align*}"}
+{"id": "6177.png", "formula": "\\begin{align*} \\begin{tabular} { c c c } \\hline $ \\eta _ = 0 $ & $ \\eta _ = 1 $ & $ \\eta _ = 2 $ \\\\ \\hline $ 1 $ & $ 3 $ & $ 2 $ \\\\ \\hline \\end{tabular} . \\end{align*}"}
+{"id": "6279.png", "formula": "\\begin{align*} M _ \\lambda ( u ) = A ( u , \\bar u ) . \\end{align*}"}
+{"id": "7669.png", "formula": "\\begin{align*} & \\tau _ u = \\begin{cases} & \\min \\{ t \\in \\mathbb { N } : W ( t ) < 0 \\} , \\\\ & \\infty , \\mbox { \\ i f \\ } \\ W ( t ) \\geqslant 0 \\ \\mbox { \\ f o r a l l \\ } \\ t \\in \\mathbb { N } , \\end{cases} \\\\ & \\psi ( u ) = \\mathbb { P } \\left ( \\tau _ { u } < \\infty \\right ) = \\mathbb { P } \\left ( \\sup _ { k \\geqslant 1 } \\sum _ { i = 1 } ^ k ( X _ i - c ) > u \\right ) , \\ , u \\in \\mathbb { N } _ 0 . \\end{align*}"}
+{"id": "1890.png", "formula": "\\begin{align*} | w | _ { m , \\mathcal { T } _ h } = \\left ( \\sum _ { R \\in \\mathcal { T } _ h } | w | ^ 2 _ { m , R } \\right ) ^ \\frac { 1 } { 2 } , \\ , \\| w \\| _ { m , \\mathcal { T } _ h } = \\left ( \\sum _ { R \\in \\mathcal { T } _ h } \\| w \\| ^ 2 _ { m , R } \\right ) ^ \\frac { 1 } { 2 } \\forall \\ , w \\in H ^ m ( \\mathcal { T } _ h ) , \\ , \\ , m \\geq 0 , \\end{align*}"}
+{"id": "6321.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial y } { \\partial t } = & \\int _ { - \\infty } ^ x \\partial _ t \\frac { 1 } { \\sqrt { g _ { [ < \\lambda ^ { \\sigma } ] } } } \\ , d x ' \\\\ = & \\ \\Re \\int _ { - \\infty } ^ x h ( u _ { [ < \\lambda ^ { \\sigma } ] } ) \\partial _ x ^ 2 u _ { [ < \\lambda ^ { \\sigma } ] } + h _ 1 ( u _ { [ < \\lambda ^ { \\sigma } ] } ) ( \\partial _ x u _ { [ < \\lambda ^ { \\sigma } ] } ) ^ 2 \\ , d x ' . \\end{aligned} \\end{align*}"}
+{"id": "5902.png", "formula": "\\begin{align*} v _ 2 ( s , x _ 2 ) = \\int _ 0 ^ { x _ 2 } \\partial _ 2 v _ 2 ( s , t ) d t . \\end{align*}"}
+{"id": "5180.png", "formula": "\\begin{align*} m _ d ( f ) ( g _ 1 , \\ldots , g _ { d - 1 } ) = \\int _ G f ^ * ( g ) f ( g g _ 1 ) \\ldots f ( g g _ { d - 1 } ) \\ ; d g \\end{align*}"}
+{"id": "4598.png", "formula": "\\begin{align*} \\vec { \\mathcal H } _ { \\vec \\lambda } : = { \\mathcal H } _ { \\lambda _ 1 } { \\otimes _ { \\mathbb C } ^ { } } \\cdots { \\otimes _ { \\mathbb C } ^ { } } \\ , { \\mathcal H } _ { \\lambda _ m } \\end{align*}"}
+{"id": "3540.png", "formula": "\\begin{align*} \\chi u w _ { r } & = u _ { r } - \\chi w _ { r } - \\sqrt { \\chi ( u + 1 ) } g _ { 1 } , \\\\ \\xi v w _ { r } & = v _ { r } - \\xi w _ { r } - \\sqrt { \\xi ( v + 1 ) } g _ { 2 } , \\end{align*}"}
+{"id": "1790.png", "formula": "\\begin{align*} g _ { \\alpha , T } ( \\underline { m } ) & = \\int _ { \\mathcal { T } ^ k } \\prod _ { j = 1 } ^ { k } \\gamma _ j ^ { m _ j } \\ , d \\mu _ { \\alpha , T } ( \\underline { \\gamma } ) \\\\ & = \\frac { 1 } { T } \\int _ { 0 } ^ { T } \\left \\{ ( n _ 1 + \\alpha ) ^ { m _ 1 } \\cdots ( n _ k + \\alpha ) ^ { m _ k } \\right \\} ^ { - i \\tau } \\ , d \\tau \\end{align*}"}
+{"id": "2255.png", "formula": "\\begin{align*} \\Gamma = \\bigcup _ { k = 1 } ^ n \\L + x _ k , \\L = \\delta \\Z , \\ , \\delta > 0 , x _ k \\neq x _ j , \\ , k \\neq j , x _ k \\in [ 0 , \\delta ) . \\end{align*}"}
+{"id": "1046.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\Delta _ { p ( z ) } u - \\Delta _ { q ( z ) } u + \\hat { \\lambda } _ \\eta | u | ^ { p ( z ) - 1 } = C _ 0 u ^ { \\tau ( z ) - 1 } - C _ 4 u ^ { r ( z ) - 1 } \\Omega , \\\\ \\displaystyle \\frac { \\partial u } { \\partial n } = 0 \\mbox { o n } \\partial \\Omega , \\ u \\geq 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "6120.png", "formula": "\\begin{align*} \\frac { \\zeta '' } { \\zeta ' } ( s ) = \\frac { \\zeta '' } { \\zeta ' } ( 0 ) - 2 - \\frac { 2 } { s - 1 } + \\sum _ { n \\geq 1 } \\left ( \\frac { 1 } { s + a _ n } - \\frac { 1 } { a _ n } \\right ) + \\sum _ { \\rho _ 1 } \\left ( \\frac { 1 } { s - \\rho _ 1 } + \\frac { 1 } { \\rho _ 1 } \\right ) , \\end{align*}"}
+{"id": "7034.png", "formula": "\\begin{align*} \\boxed { D _ { L , \\Omega } ^ { \\Phi } [ \\rho ] = \\sup _ { \\begin{array} { c } v \\in { \\rm S p a n } \\{ \\Phi \\} , \\\\ \\forall \\Psi \\in \\mathcal H _ 1 ^ N , \\ ; \\langle \\Psi | H _ { N , \\Omega } ^ v | \\Psi \\rangle \\geq 0 \\\\ \\end{array} } \\int v d \\rho , } \\end{align*}"}
+{"id": "5544.png", "formula": "\\begin{align*} \\eta ^ { ( i ) } _ j ( t ) = c ^ { ( i ) } _ j + r ^ { ( i ) } _ j e ^ { \\i t } , 0 \\le t \\le 2 \\pi , j = 1 , \\ldots , m . \\end{align*}"}
+{"id": "8604.png", "formula": "\\begin{align*} \\begin{cases} \\frac { h } { h _ b } \\nabla ^ { \\mu } _ { X , z } \\cdot P ( \\Sigma _ b ) \\nabla _ { X , z } ^ { \\mu } u = f \\mathcal { S } _ b \\\\ u | _ { z = 0 } = 0 , \\frac { h } { h _ b } | n _ b | \\partial _ { n _ { b } } ^ { P _ b } u | _ { z = - h _ b } = g , \\end{cases} \\end{align*}"}
+{"id": "5961.png", "formula": "\\begin{align*} \\partial _ { t _ 3 } v + A ^ \\omega _ F v - \\widetilde { E } v = - i w , \\end{align*}"}
+{"id": "7183.png", "formula": "\\begin{align*} \\Tilde { A } : = & \\begin{pmatrix} \\textbf { 1 } & \\boldsymbol { 0 } & \\dots & \\boldsymbol { 0 } \\\\ \\boldsymbol { \\beta _ 1 ^ 2 } & \\textbf { 1 } & \\dots & \\boldsymbol { 0 } \\\\ \\vdots & \\vdots & \\dots & \\vdots \\\\ \\boldsymbol { \\beta _ 1 ^ m } & \\boldsymbol { \\beta _ 2 ^ m } & \\dots & \\textbf { 1 } \\\\ \\end{pmatrix} _ { m \\times m } , \\end{align*}"}
+{"id": "4291.png", "formula": "\\begin{align*} \\tau _ \\pi ( r ) j _ { r } = \\tau _ \\pi ( s ) j _ { s } B _ { \\ell } = \\{ r , s \\} , \\end{align*}"}
+{"id": "7909.png", "formula": "\\begin{align*} \\tilde { \\varphi } ( x x ^ \\ast ) = \\sum _ { g \\in G } \\tilde { \\varphi } ( g ) + \\sum _ { y \\in B \\setminus G } \\tilde { \\varphi } ( y ) = \\sum _ { g \\in G } \\varphi ( g ) , \\end{align*}"}
+{"id": "7227.png", "formula": "\\begin{align*} b _ { i , j } ( [ x _ 1 | \\cdots | x _ n ] \\otimes g \\otimes r ) & = [ \\underline { x } _ 1 | \\cdots | \\underline { x } _ j | m _ i ( [ x _ { j + 1 } | \\cdots | x _ { j + i } ] ) | x _ { j + i + 1 } | \\cdots | x _ n ] \\otimes g \\otimes r , & \\\\ b ' _ i ( [ x _ 1 | \\cdots | x _ n ] \\otimes g \\otimes r ) & = [ \\underline { x } _ 1 | \\cdots | \\underline { x } _ { n - i + 1 } ] \\otimes m _ i ^ G ( [ x _ { n - i + 2 } | \\cdots | x _ n ] \\otimes g ) \\otimes r . \\end{align*}"}
+{"id": "2090.png", "formula": "\\begin{align*} \\lim _ { t \\longrightarrow + \\infty } \\int _ { | x - v t | \\le \\omega ( t ) } \\left ( ( \\partial _ t \\det g ) ^ 2 + ( \\partial _ x \\det g ) ^ 2 \\right ) ( t , x ) d x = 0 . \\end{align*}"}
+{"id": "2718.png", "formula": "\\begin{align*} G \\coloneqq \\sum _ { i , j \\ge 0 } ( - 1 ) ^ { i + j } \\sum _ { \\substack { P \\subseteq \\{ 2 , 3 , \\dots , n - 1 \\} , \\\\ \\# P = i } } \\sum _ { \\substack { Q \\subseteq \\{ 1 , n \\} , \\\\ \\# Q = j } } \\binom { d - \\sum _ { p \\in P } ( \\alpha _ { p } + 1 ) - \\sum _ { q \\in Q } \\alpha _ { q } + n - 2 } { n - 1 } \\end{align*}"}
+{"id": "5160.png", "formula": "\\begin{align*} E [ L ^ 2 ] = \\sum _ i p _ i \\log _ 2 ^ 2 { p _ i } . \\end{align*}"}
+{"id": "2704.png", "formula": "\\begin{align*} M \\Big ( ( c _ { j } ) _ { j = 1 } ^ { l } \\Big ) \\times F = 0 \\end{align*}"}
+{"id": "5978.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta _ g ^ F + 1 ) u = 0 & M , \\\\ \\partial _ \\nu u = \\psi & \\partial M . \\end{cases} \\end{align*}"}
+{"id": "6919.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } } J _ 2 ( y ) ( - y ) \\int _ { 0 } ^ { 1 } [ \\psi ( \\xi _ n - \\tau y ) - \\psi ( - \\tau y ) ] d \\tau d y \\\\ [ 0 . 2 c m ] \\leq & 2 \\int _ { \\mathbb { R } } J _ 2 ( y ) | y | d y \\leq 4 \\int _ { \\mathbb { R } } J _ 2 ( y ) e ^ { y } d y < \\infty . \\end{align*}"}
+{"id": "6874.png", "formula": "\\begin{align*} \\sigma ( f ( u ) ) ) = ( \\iota ( g . ( f . u ) ) ) _ { g \\in G _ i } = ( \\iota ( ( g f ) . u ) ) _ { g \\in G _ i } = f . ( \\sigma ( u ) ) . \\end{align*}"}
+{"id": "3405.png", "formula": "\\begin{align*} ( - v _ 1 ^ 2 + v _ 1 ^ 4 + v _ 2 ^ 2 - 1 0 v _ 1 ^ 2 v _ 2 ^ 2 + 9 v _ 1 ^ 4 v _ 2 ^ 2 + v _ 2 ^ 4 - 9 v _ 1 ^ 2 v _ 2 ^ 4 ) V _ 1 ^ 2 = ( \\tilde { c } - c ) v _ 1 ^ 2 ( v _ 2 ^ 2 + v _ 3 ^ 2 ) ^ 2 . \\end{align*}"}
+{"id": "7129.png", "formula": "\\begin{align*} I _ k : = \\int _ 0 ^ \\infty | \\Psi ^ { ( k ) } ( Z ) | ^ 2 \\omega _ k ( r Z ) \\ : d Z . \\end{align*}"}
+{"id": "7021.png", "formula": "\\begin{align*} F _ { L } [ \\rho ] = \\sup _ { \\substack { v \\in L ^ { \\frac { 3 } { 2 } } ( \\R ^ 3 ) + L ^ { \\infty } ( \\R ^ 3 ) \\\\ H ^ { v } _ N \\geq 0 } } \\left \\{ \\int _ { \\R ^ 3 } v ( x ) \\rho ( x ) \\dd x \\right \\} , \\end{align*}"}
+{"id": "1092.png", "formula": "\\begin{align*} d ( x , y ) : = \\left \\{ \\begin{array} { l l } \\displaystyle \\min _ { P ( x , y ) \\in \\wp ( x , y ) } \\{ \\ell ( P ( x , y ) ) \\} & \\mbox { i f } \\ , \\ \\ \\ x \\neq y \\\\ 0 & \\mbox { i f } \\ , \\ \\ \\ x = y . \\end{array} \\right . \\end{align*}"}
+{"id": "5096.png", "formula": "\\begin{align*} C ^ { s m } ( G , V ) = \\prod _ { g \\in G / H } C ^ { s m } ( g H , V ) . \\end{align*}"}
+{"id": "1253.png", "formula": "\\begin{align*} \\Delta W + ( I _ \\alpha \\ast | \\cdot | ^ { - b } | W | ^ p ) | x | ^ { - b } | W | ^ { p - 2 } W = 0 . \\end{align*}"}
+{"id": "813.png", "formula": "\\begin{align*} [ u _ n ] _ { s , p } ^ p & \\geq S ^ \\sharp \\left ( \\int _ { \\mathbb { R } ^ N } ( K \\ast | u _ n | ^ { p ^ \\sharp } ) | u _ n | ^ { p ^ \\sharp } d x \\right ) ^ { \\frac { p } { 2 \\cdot p ^ \\sharp } } + o ( 1 ) \\\\ & = S ^ \\sharp ( p ^ \\sharp / b ) ^ { \\frac { p } { 2 \\cdot p ^ \\sharp } } \\left ( [ u _ n ] _ { s , p } ^ p \\right ) ^ { \\frac { p } { 2 \\cdot p ^ \\sharp } } + o ( 1 ) , \\end{align*}"}
+{"id": "8286.png", "formula": "\\begin{align*} \\norm { u _ 0 - \\bar u _ 0 } _ { H ^ 1 ( 0 , 1 ) } + \\norm { u _ 1 - \\bar u _ 1 } _ { L ^ 2 ( 0 , 1 ) } + \\norm { p _ 0 - \\bar p _ 0 - d \\tfrac { 1 + c x } { c } u _ 0 ( 1 ) } _ { L ^ 2 ( 0 , 1 ) } = O ( \\varepsilon ^ { 3 / 2 } ) , \\end{align*}"}
+{"id": "6832.png", "formula": "\\begin{align*} A _ n & : = \\frac { 3 n ^ 2 - 6 n - 3 2 } { 4 n ( n - 1 ) } \\\\ B _ n & : = \\frac { 3 n ^ 4 - 1 2 n ^ 3 - 5 2 n ^ 2 + 1 2 8 n + 1 9 2 } { 1 6 n ^ 2 ( n - 1 ) ^ 2 } \\\\ C _ n & : = \\frac { ( n ^ 4 - 2 0 n ^ 2 + 6 4 ) ( n - 6 ) } { 6 4 n ^ 2 ( n - 1 ) ^ 3 } \\end{align*}"}
+{"id": "3422.png", "formula": "\\begin{align*} \\mu _ t = \\frac { \\Omega _ t \\wedge \\overline { \\Omega } _ t } { \\int _ { X _ t } \\Omega _ t \\wedge \\overline { \\Omega } _ t } , \\end{align*}"}
+{"id": "7599.png", "formula": "\\begin{align*} \\begin{aligned} & J _ 2 ( \\beta , N , T ) \\\\ & \\leq C N ^ 2 \\int _ 0 ^ T \\int _ 0 ^ T \\left ( \\left ( \\frac { 1 } { ( t + s ) ^ { 1 / 2 } } \\int _ { | z _ 1 | < 2 } e ^ { - \\frac { ( z _ 1 - \\beta ^ { 1 / 3 } N ^ { 1 / 3 } ( t - s ) ) ^ 2 } { 2 ( t + s ) } } d z _ 1 \\right ) \\wedge 1 \\right ) d s d t . \\end{aligned} \\end{align*}"}
+{"id": "2345.png", "formula": "\\begin{align*} \\theta _ f \\theta _ \\tau \\{ a _ h \\} _ { h \\in G } = \\sum _ { g \\in G } \\eta ( g ) \\rho _ g \\{ a _ h \\} _ { h \\in G } , \\forall \\{ a _ h \\} _ { h \\in G } \\in \\ell ^ p ( G ) , \\end{align*}"}
+{"id": "6384.png", "formula": "\\begin{align*} 0 & = \\varepsilon ( \\chi ^ { i } ) 1 _ { H } \\otimes \\chi _ { i } = ( m \\otimes \\mathrm { I d } ) ( S \\otimes \\mathrm { I d } \\otimes \\mathrm { I d } ) ( \\Delta \\otimes \\mathrm { I d } ) ( \\chi ) \\\\ & \\overset { \\eqref { e q : C C 4 ' 2 } } = ( m \\otimes \\mathrm { I d } ) ( S \\otimes \\mathrm { I d } \\otimes \\mathrm { I d } ) ( 1 _ { H } \\otimes \\chi + ( \\mathrm { I d } \\otimes \\rho ^ { l } ) ( \\chi ) ) \\\\ & = \\chi + ( m \\otimes \\mathrm { I d } ) ( S \\otimes \\rho ^ { l } ) ( \\chi ) . \\end{align*}"}
+{"id": "582.png", "formula": "\\begin{align*} \\widehat { P _ \\mu f } ( \\xi ) = \\theta \\left ( \\frac { \\xi } { \\mu } \\right ) \\widehat f ( \\xi ) . \\end{align*}"}
+{"id": "7018.png", "formula": "\\begin{align*} \\boxed { F _ L [ \\rho ] : = \\mathop { \\inf } _ { \\Gamma \\in \\mathfrak { S } _ 1 ^ + ( \\mathcal H _ 0 ^ N ) , \\ ; \\rho _ \\Gamma = \\rho } { \\rm T r } \\left [ \\bigg ( - \\frac { 1 } { 2 } \\Delta + V ) \\Gamma \\bigg ) \\right ] , } \\end{align*}"}
+{"id": "4477.png", "formula": "\\begin{align*} \\mathcal { F } ^ a : = \\begin{cases} - \\mathbb { L } ( { \\mathbf U } ^ a , \\Psi ^ a ) , \\ ; & t > 0 , \\\\ ~ ~ ~ ~ 0 , \\ ; & t < 0 . \\\\ \\end{cases} \\end{align*}"}
+{"id": "3993.png", "formula": "\\begin{align*} d = C ^ { \\frac { 1 } { \\alpha } } \\phi ^ { \\frac { \\beta - 1 } { \\alpha } } ( s _ 0 ) 2 ^ { \\frac { \\beta } { \\beta - 1 } } . \\end{align*}"}
+{"id": "3105.png", "formula": "\\begin{align*} \\mathbf { a } _ T ( \\psi ^ { } ) = \\frac { 1 } { \\sqrt { M } } \\left [ e ^ { j 2 \\pi n _ 1 \\frac { d _ B } { \\lambda } \\sin ( \\psi ^ { } ) } \\right ] ^ T _ { m \\in \\mathcal { I } ( M ) } \\end{align*}"}
+{"id": "2570.png", "formula": "\\begin{align*} c _ 1 ( \\xi ) : = & \\frac { 1 } { 9 6 } \\left ( 6 4 e ^ { i ( \\xi _ 1 + 2 \\xi _ 2 ) } - 2 0 8 e ^ { i ( \\xi _ 1 + \\xi _ 2 ) } + 2 0 3 e ^ { i \\xi _ 1 } - 8 3 e ^ { i ( \\xi _ 1 - \\xi _ 2 ) } - 6 2 e ^ { 2 i \\xi _ 2 } + 1 9 9 e ^ { i \\xi _ 2 } - 5 8 + 1 0 9 e ^ { - 2 i \\xi _ 2 } \\right . \\\\ & \\left . - 1 7 e ^ { i ( \\xi _ 2 - \\xi _ 1 ) } + 4 9 e ^ { - i \\xi _ 1 } - 4 e ^ { _ 2 i \\xi _ 1 } \\right ) , \\\\ c _ 2 ( \\xi ) : = & ( 1 - c _ 1 ( \\xi ) / c _ 1 ( 0 ) ) ^ 2 , c _ 3 ( \\xi ) : = \\frac { c _ 2 ( \\xi ) ^ 2 - 1 } { c _ 1 ( \\xi ) } . \\end{align*}"}
+{"id": "6627.png", "formula": "\\begin{align*} c _ \\infty ^ { ( \\widetilde { \\rm c J } ) } ( \\tau ; \\beta , p , q ) = \\sum _ { j = 0 } ^ \\infty h _ j ( \\beta , p , q ) \\tau ^ { j } { \\rm s g n } \\ , \\tau + \\sum _ { j = 0 } ^ \\infty \\tilde { h } _ j ( \\beta , p , q ) \\tau ^ { j } , | \\tau | < | \\tau ^ * | , \\end{align*}"}
+{"id": "5393.png", "formula": "\\begin{align*} \\phi ( z , w ; t ) & : = \\lim _ { N \\to \\infty } \\phi _ N ( z , w ; t ) \\\\ & = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\left [ \\frac { n } { 2 \\pi t } \\int _ { \\C } \\exp \\left \\{ n \\left ( \\phi ( z , w ' ; 0 ) - \\frac { | w - w ' | ^ 2 } { 2 t } \\right ) \\right \\} m ( d w ' ) \\right ] \\\\ & = \\max _ { w ' } \\left ( \\phi ( z , w ' ; 0 ) - \\frac { | w - w ' | ^ 2 } { 2 t } \\right ) , ( z , w ) \\in \\C \\times \\C ^ { \\times } , \\ , t \\geq 0 , \\end{align*}"}
+{"id": "2208.png", "formula": "\\begin{align*} \\lambda _ 1 = D _ L \\cdot D _ R = \\Theta ( q ^ { k - 2 } \\cdot q ^ { k - 3 } \\cdot q ^ { k } ) \\lambda _ 2 = \\lambda _ 3 = \\ldots \\lambda _ { \\# L } \\le D _ L . \\end{align*}"}
+{"id": "1682.png", "formula": "\\begin{align*} \\mathbb { A } ^ { ( n ) } _ { \\texttt { b } } : = \\{ \\boldsymbol { \\xi } = ( \\xi _ 1 , \\ldots , \\xi _ n ) \\in \\mathbb { R } ^ n \\mid \\pi > \\xi _ 1 > \\xi _ 2 > \\cdots > \\xi _ n > 0 \\} . \\end{align*}"}
+{"id": "7464.png", "formula": "\\begin{align*} a _ 3 = a _ 2 + a _ 4 , c _ 3 = a _ 0 + c _ 4 , c _ 1 = c _ 0 + a _ 4 , a _ 1 = a _ 0 + a _ 2 , c _ 2 = c _ 1 + c _ 3 . \\end{align*}"}
+{"id": "8815.png", "formula": "\\begin{align*} \\kappa _ { \\beta } L d h _ { t } ^ { \\beta - 1 } \\norm { \\bar { x } ( t ) - x ^ { * } } + \\frac { \\bar { L } } { n } \\sum _ { i = 1 } ^ { n } \\norm { x ^ { i } ( t ) - \\bar { x } ( t ) } \\norm { \\bar { x } ( t ) - x ^ { * } } . \\end{align*}"}
+{"id": "3087.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta R & \\approx \\sum \\limits _ { k = 1 } ^ { { N _ { \\rm { R } } } } { { { \\log } _ 2 } \\left ( { \\frac { { 1 + { { { \\bar C } _ k } } } } { { 1 + { { \\bar C } _ k } \\left ( { 1 - { 2 ^ { - 1 - { x _ k } } } } \\right ) } } } \\right ) } \\\\ & < \\sum \\limits _ { k = 1 } ^ { { N _ { \\rm { R } } } } { { { \\log } _ 2 } \\left ( { \\frac { { 1 } } { { { 1 - { 2 ^ { - 1 - { x _ k } } } } } } } \\right ) } . \\end{aligned} \\end{align*}"}
+{"id": "5908.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\big ( G _ 3 + G _ 4 \\big ) = 0 . \\end{align*}"}
+{"id": "5242.png", "formula": "\\begin{align*} x _ { i , t } = \\sum _ { j = 1 } ^ n [ W _ { t } ] _ { i j } z _ { j , t } . \\end{align*}"}
+{"id": "7861.png", "formula": "\\begin{align*} S _ 2 \\approx & \\ , e ^ { - 2 \\alpha r _ n } \\sum _ { k = r _ n + 1 } ^ { n - 1 } ( n - k ) ( 1 + \\psi ( k - r _ n ) ) \\leq ( 1 + \\psi ( 1 ) ) e ^ { - 2 \\alpha r _ n } \\sum _ { k = r _ n + 1 } ^ { n - 1 } ( n - k ) \\\\ \\leq & ( 1 + \\psi ( 1 ) ) e ^ { - 2 \\alpha r _ n } n ^ 2 \\leq \\frac { 1 + \\psi ( 1 ) } { ( \\log { n } ) ^ 2 } \\leq \\frac { 1 + \\psi ( 1 ) } { \\log { n } } \\end{align*}"}
+{"id": "7199.png", "formula": "\\begin{align*} \\textrm { F i n d } u \\in H ^ 1 _ { 0 , \\gamma } ( \\Omega ) : \\int _ \\Omega \\nabla u \\nabla v = \\int _ \\Lambda \\sigma v , \\forall v \\in H ^ 1 _ { 0 , - \\gamma } ( \\Omega ) . \\end{align*}"}
+{"id": "3614.png", "formula": "\\begin{align*} \\begin{aligned} & { \\mathbb E } \\left \\{ { \\left | { \\beta _ { n , { \\rm { R } } , { l _ n } } ^ * { \\beta _ { m , { \\rm { R } } , { l _ m } } } } \\right | } \\right \\} \\\\ = & \\frac { 1 } { 2 } { \\mathbb E } \\left \\{ \\sqrt { { \\left | \\sqrt { 2 } { \\beta _ { n , { \\rm { R } } , { l _ n } } } \\right | ^ 2 } } \\right \\} { \\mathbb E } \\left \\{ \\sqrt { { \\left | \\sqrt { 2 } { { \\beta _ { m , { \\rm { R } } , { l _ m } } } } \\right | ^ 2 } } \\right \\} . \\end{aligned} \\end{align*}"}
+{"id": "179.png", "formula": "\\begin{align*} E _ 4 ( \\tau ) = 1 + 2 4 0 q + 2 1 6 0 q ^ 2 + 6 7 2 0 q ^ 3 + \\cdots , \\end{align*}"}
+{"id": "3577.png", "formula": "\\begin{align*} { \\bf { y } } = { { \\bf { W } } ^ H } { \\bf { H F s } } + { { \\bf { W } } ^ H } { \\bf { n } } , \\end{align*}"}
+{"id": "6211.png", "formula": "\\begin{align*} q _ j \\mapsto V _ j = - i \\partial _ { x _ j } - \\frac { 1 } { 2 } y _ j , p _ j \\mapsto W _ j = - i \\partial _ { y _ j } + \\frac { 1 } { 2 } x _ j . \\end{align*}"}
+{"id": "8760.png", "formula": "\\begin{align*} d \\mathbf { P } _ f = d F ( y _ 1 - f ( z _ 1 ) ) \\prod _ { t = 2 } ^ T d F \\bigg ( y _ t - f \\Big ( \\Phi _ { t } ( z _ 1 , y _ 1 , \\ldots , y _ { t - 1 } ) \\Big ) \\bigg ) \\enspace . \\end{align*}"}
+{"id": "278.png", "formula": "\\begin{align*} r : \\Delta ^ 1 \\times \\Delta ^ n \\xlongrightarrow { } \\Delta ^ { n + 1 } , \\enspace r ( \\epsilon , i ) = \\begin{cases} i , \\enspace i = 0 , \\\\ n , \\enspace i = 1 . \\end{cases} \\end{align*}"}
+{"id": "5085.png", "formula": "\\begin{align*} y ( z ) = E ( A z ) v ^ c , \\end{align*}"}
+{"id": "2035.png", "formula": "\\begin{align*} Q _ { ( a , b ) } : = \\prod _ { j = 1 } ^ { n - 1 } [ a _ j , b _ j ] K _ { ( a , b ) } : = \\{ \\ , ( x ' , \\phi ( x ' ) ) x ' \\in Q _ { ( a , b ) } \\ , \\} \\end{align*}"}
+{"id": "2137.png", "formula": "\\begin{align*} \\omega ( t ) : = \\frac { t } { \\log ^ 2 t } , \\frac { \\omega ' ( t ) } { \\omega ( t ) } = \\frac { 1 } { t } \\left ( 1 - \\frac { 2 } { \\log t } \\right ) . \\end{align*}"}
+{"id": "3856.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\sigma h _ { 1 1 } & = & f _ { y y } e ^ { - \\lambda _ 1 z } , \\\\ \\sigma h _ { 1 2 } & = & \\sigma h _ { 2 1 } = ( \\lambda _ 2 - \\lambda _ 1 ) f _ y e ^ { - \\lambda _ 1 z } , \\\\ \\sigma h _ { 2 2 } & = & 0 . \\end{array} \\end{align*}"}
+{"id": "6742.png", "formula": "\\begin{align*} 0 = t \\left ( t + \\frac { m } { 2 } \\right ) ^ { 2 } \\bar { \\alpha } '' ( t ) + \\left ( t + \\frac { m } { 2 } \\right ) \\left ( ( a - 2 ) t - a \\frac { m } { 2 } \\right ) \\bar { \\alpha } ' ( t ) + 2 ( 1 - a ) t \\bar { \\alpha } ( t ) . \\end{align*}"}
+{"id": "53.png", "formula": "\\begin{align*} b _ k = \\frac { 1 } { \\sqrt { 1 - \\alpha _ k ^ 2 } } \\Big ( a _ k + \\alpha _ k a _ { - k } ^ \\dagger \\Big ) , \\alpha _ k = \\frac { k ^ 2 + \\rho _ z \\widehat g ( k ) - \\sqrt { k ^ 4 + 2 k ^ 2 \\rho _ z \\widehat g ( k ) } } { \\rho _ z \\widehat g ( k ) } , \\end{align*}"}
+{"id": "430.png", "formula": "\\begin{align*} \\langle [ u ] _ { \\ell , m } , P ^ { ( \\alpha ) } ( t , \\cdot , \\cdot ) [ u ] _ { \\ell , m } \\rangle _ { L ^ 2 ( \\R ^ d ) } = \\langle u , p _ { ( d - 1 + 2 \\ell ) / 2 } ^ { ( \\alpha ) } ( t , \\cdot , \\cdot ) u \\rangle _ { L ^ 2 ( \\R _ + , r ^ { d - 1 + 2 \\ell } d r ) } , \\end{align*}"}
+{"id": "5638.png", "formula": "\\begin{align*} 0 = ( y _ 1 + x _ 1 L ) ( y _ 0 - x _ 0 L ) + x _ 0 x _ 1 L ^ 2 + B L + C = y _ 0 y _ 1 + C - L ( x _ 0 y _ 1 - x _ 1 y _ 0 - B ) , \\end{align*}"}
+{"id": "2513.png", "formula": "\\begin{align*} \\int _ { Q } \\left ( \\sigma \\partial _ t ^ { 1 / 2 } \\boldsymbol { y } \\cdot \\partial _ t ^ { 1 / 2 } \\boldsymbol { v } ^ { \\perp } + \\nu \\ , \\textbf { c u r l } \\ , \\boldsymbol { y } \\cdot \\textbf { c u r l } \\ , \\boldsymbol { v } \\right ) \\ , d \\boldsymbol { x } \\ , d t = \\int _ { Q } \\boldsymbol { u } \\cdot \\boldsymbol { v } \\ , d \\boldsymbol { x } \\ , d t \\end{align*}"}
+{"id": "7343.png", "formula": "\\begin{align*} \\dot { \\widetilde { A } } ( 0 , t ) = \\begin{cases} ( \\arctan t ) J \\dot { S } _ { 0 } , & t \\geq 0 \\\\ 0 , & t < 0 , \\end{cases} \\end{align*}"}
+{"id": "786.png", "formula": "\\begin{align*} S ^ \\sharp = S _ { H , L } \\coloneqq \\inf _ { u \\in D ^ { s , p } \\setminus \\{ 0 \\} } \\frac { [ u ] _ { s , p } ^ p } { \\left ( \\int _ { \\mathbb { R } ^ N } ( K \\ast | u | ^ { p ^ \\sharp } ) | u | ^ { p ^ \\sharp } d x \\right ) ^ { p / ( 2 \\cdot p ^ \\sharp ) } } , \\end{align*}"}
+{"id": "3417.png", "formula": "\\begin{align*} A _ { e ^ { - 1 } } = \\{ f = \\sum _ { \\alpha \\in \\Z } c _ \\alpha t ^ \\alpha \\in \\C ( \\ ! ( t ) \\ ! ) | \\norm { f } _ { h y b } : = \\sum \\norm { c _ \\alpha } _ { h y b } e ^ { - \\alpha } \\} . \\end{align*}"}
+{"id": "974.png", "formula": "\\begin{align*} ( a , b , c ) = & \\pm \\left ( x _ { ( 2 m - 1 ) ( 2 n - 1 ) } ^ { ( d ) } , x _ { ( 2 m - 1 ) ( 2 n + 1 ) } ^ { ( d ) } , \\frac { y _ { ( 2 m - 1 ) \\cdot 2 n } ^ { ( d ) } } { y _ { 2 m - 1 } ^ { ( d ) } } \\right ) , \\\\ & \\pm ( x _ { 2 n - 1 } ^ { ( 2 ) } , - x _ { 2 n + 1 } ^ { ( 2 ) } , x _ { 2 n } ^ { ( 2 ) } ) \\end{align*}"}
+{"id": "7771.png", "formula": "\\begin{align*} \\xi ^ { i } & = \\zeta ^ { i } - \\mathrm { i } R ( t ^ { - 1 } z ^ i + t \\overline { z } ^ i ) \\\\ \\widetilde { \\xi } _ i & = \\widetilde { \\zeta } _ { i } - \\mathrm { i } R ( t ^ { - 1 } F _ { i } + t \\overline { F } _ i ) \\\\ \\alpha & = \\sigma - \\mathrm { i } R ( t ^ { - 1 } \\langle \\widetilde { Z } , \\zeta \\rangle + t \\langle \\overline { \\widetilde { Z } } , \\zeta \\rangle ) - 8 \\mathrm { i } c _ { \\ell } \\log ( t ) \\ , , \\end{align*}"}
+{"id": "5828.png", "formula": "\\begin{align*} w _ 0 = w _ 1 w _ 2 w _ 3 w _ 4 \\end{align*}"}
+{"id": "8924.png", "formula": "\\begin{align*} 0 = d f ( \\tau _ V ( \\varphi ) ) + \\mathrm { t r a c e } \\nabla d f ( d \\varphi , d \\varphi ) \\end{align*}"}
+{"id": "8400.png", "formula": "\\begin{align*} \\sum _ { z = 0 } ^ { 2 Z - 1 } \\theta _ { z } ( x ) = 1 x \\in \\mathbb { R } . \\end{align*}"}
+{"id": "5479.png", "formula": "\\begin{align*} \\mathbf S : = \\left \\{ f : U \\rightarrow \\mathbb C \\ ; \\Big | \\ ; f \\ ; \\textrm { i s h o l o m o r p h i c o n o p e n d e n s e } \\ ; U \\subset \\big ( \\mathbb C ^ \\times \\big ) ^ 3 \\right \\} . \\end{align*}"}
+{"id": "6720.png", "formula": "\\begin{align*} \\begin{aligned} & 4 \\pi + 2 \\int _ { \\Sigma } H | \\nabla u | - 2 ^ { 4 a } ( I _ { a } ( 1 ) ) ^ { 2 } \\int _ { \\Sigma } | \\nabla u | ^ 2 \\ge 0 , \\end{aligned} \\end{align*}"}
+{"id": "1424.png", "formula": "\\begin{align*} \\Delta u _ i + \\sum \\limits _ { j \\in I } k _ { i j } e ^ { u _ j } = 0 , \\ \\ i \\in I , \\end{align*}"}
+{"id": "6527.png", "formula": "\\begin{align*} \\tilde k = ( k + e _ { l ' } - e _ { l '' } ) \\in \\Z ^ { b } \\setminus \\{ 0 \\} , \\ v ( m ) = \\omega ^ { ( 0 ) } \\in \\R ^ { b } , \\end{align*}"}
+{"id": "186.png", "formula": "\\begin{align*} & Q ( M , P _ i , \\tau ) = \\left \\{ e ^ { \\frac { 1 } { 2 4 } E _ 2 ( \\tau ) A _ 1 } \\left ( \\prod _ { j = 1 } ^ { 7 } \\frac { x _ j \\theta ' ( 0 , \\tau ) } { \\theta ( x _ j , \\tau ) } \\right ) \\frac { \\sqrt { - 1 } \\theta ( u , \\tau ) } { \\theta _ 1 ( 0 , \\tau ) \\theta _ 2 ( 0 , \\tau ) \\theta _ 3 ( 0 , \\tau ) } \\right . \\\\ & \\left . \\frac { 1 } { 2 } \\left ( \\prod _ { l = 1 } ^ 8 \\theta _ 1 ( y _ l ^ i , \\tau ) + \\prod _ { l = 1 } ^ 8 \\theta _ 2 ( y _ l ^ i , \\tau ) + \\prod _ { l = 1 } ^ 8 \\theta _ 3 ( y _ l ^ i , \\tau ) \\right ) \\right \\} ^ { ( 1 4 ) } . \\end{align*}"}
+{"id": "8179.png", "formula": "\\begin{align*} \\theta _ { i j } - \\theta _ { a b } = ( r - 1 ) ( a - i ) . \\end{align*}"}
+{"id": "7365.png", "formula": "\\begin{align*} { \\rm M M S E } '' ( \\gamma ) = { \\rm M M S E } '' ( 1 ) + a ( \\gamma - 1 ) \\big ( \\log | \\gamma - 1 | + b \\big ) + o \\big ( ( \\gamma - 1 ) \\big ) . \\end{align*}"}
+{"id": "4149.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Phi _ t = f ( \\Phi _ t ) , \\Phi _ 0 = \\phi \\end{align*}"}
+{"id": "4296.png", "formula": "\\begin{align*} A _ k = \\big \\{ ( j _ 1 , \\ldots , j _ { 2 k } ) \\in \\{ - ( p - 1 ) , \\ldots , - 1 , 1 , \\ldots , ( n - 1 ) \\} ^ { 2 k } : \\sum _ { q = 1 } ^ { 2 k } ( - 1 ) ^ { q } j _ { q } = 0 \\big \\} . \\end{align*}"}
+{"id": "434.png", "formula": "\\begin{align*} & \\int _ 0 ^ \\infty p _ \\zeta ^ { ( \\alpha ) } ( t , r , s ) s ^ { 2 \\zeta } \\ , d s = 1 , \\\\ & \\int _ 0 ^ \\infty p _ \\zeta ^ { ( \\alpha ) } ( t , r , z ) p _ \\zeta ^ { ( \\alpha ) } ( t ' , z , s ) z ^ { 2 \\zeta } \\ , d z = p _ \\zeta ^ { ( \\alpha ) } ( t + t ' , r , s ) , \\\\ & p _ \\zeta ^ { ( \\alpha ) } ( t , r , s ) = t ^ { - \\frac { 2 \\zeta + 1 } { \\alpha } } p _ \\zeta ^ { ( \\alpha ) } \\left ( 1 , \\frac { r } { t ^ { 1 / \\alpha } } , \\frac { s } { t ^ { 1 / \\alpha } } \\right ) . \\end{align*}"}
+{"id": "7829.png", "formula": "\\begin{align*} ( ( \\tau ) ) = - \\left ( \\frac { t ^ 4 } { 3 } + \\frac { t \\chi \\zeta ( 3 ) } { ( 2 \\pi ) ^ 3 } \\right ) , ( \\tau _ { i j } ) Z ^ i \\overline { Z } ^ j = | Z ^ 0 | ^ 2 \\left ( - \\frac { 2 t ^ 3 } { 3 } + \\frac { \\chi \\zeta ( 3 ) } { ( 2 \\pi ) ^ 3 } \\right ) \\end{align*}"}
+{"id": "5426.png", "formula": "\\begin{align*} v _ j ^ s ( \\xi ) : = \\sum _ { a / q \\in \\Sigma _ s } G ( a / q ) \\big ( \\Theta _ { N _ j } - \\Theta _ { N _ { j - 1 } } \\big ) ( \\xi - a / q ) \\eta _ { j ^ \\tau } ( \\xi - a / q ) \\end{align*}"}
+{"id": "4545.png", "formula": "\\begin{align*} \\mathcal { A } \\partial _ 1 { \\mathbf V } = \\mathcal { F } - \\mathcal { A } _ 0 \\partial _ t { \\mathbf V } - \\mathcal H _ { ( 0 ) } \\sigma \\partial _ 1 { \\mathbf V } - \\mathcal { A } _ 2 \\partial _ 2 { \\mathbf V } - \\mathcal { A } _ 3 { \\mathbf V } \\quad \\mbox { i n } \\ , \\ , \\ , \\Omega _ T \\ , , \\end{align*}"}
+{"id": "6701.png", "formula": "\\begin{align*} \\langle \\psi _ { m _ 1 } \\cdots \\psi _ { m _ { r } } \\psi ^ \\ast _ { n _ r } \\cdots \\psi ^ \\ast _ { n _ { 1 } } \\rangle = \\det ( \\langle \\psi _ { m _ i } \\psi ^ \\ast _ { n _ j } \\rangle ) _ { 1 \\leq i , j \\leq r } . \\end{align*}"}
+{"id": "5539.png", "formula": "\\begin{align*} M ( s , t ) = \\frac { 1 } { \\pi } \\Re \\left ( \\frac { \\eta ' ( t ) } { \\eta ( t ) - \\eta ( s ) } \\right ) , ( s , t ) \\in J \\times J . \\end{align*}"}
+{"id": "6477.png", "formula": "\\begin{align*} \\left < P _ D \\Psi , G \\right > = - \\left < \\Psi , \\Lambda P _ R F \\right > \\end{align*}"}
+{"id": "7317.png", "formula": "\\begin{align*} M ^ \\ast _ \\lambda L _ \\lambda M _ \\lambda = Q + K _ \\lambda , \\lambda \\in I , \\end{align*}"}
+{"id": "6109.png", "formula": "\\begin{align*} \\sigma _ A ( T ) = \\Big \\{ g ( A T ) : g \\in \\mathcal { P } _ A ( T ) \\ \\Big \\} , \\end{align*}"}
+{"id": "6751.png", "formula": "\\begin{align*} C = W - \\frac { 1 } { 2 ^ { \\sigma _ e } } \\left [ \\left ( \\frac { 3 } { 2 } \\right ) ^ { \\sigma _ o } ( n + 1 ) - 1 \\right ] = W - \\frac { 3 ^ { \\sigma _ o } ( n + 1 ) - 2 ^ { \\sigma _ o } } { 2 ^ { \\sigma _ o + \\sigma _ e } } . \\end{align*}"}
+{"id": "6529.png", "formula": "\\begin{align*} | k | \\leq 2 L , \\ \\exists \\ 1 \\leq l ' \\neq l '' \\leq b , \\ { \\rm s . t . , } \\ n = n ^ { ( l ' ) } , \\ n ' = n ^ { ( l '' ) } , \\ k + e _ { l ' } - e _ { l '' } \\neq 0 . \\end{align*}"}
+{"id": "4031.png", "formula": "\\begin{align*} \\frac { n } { 2 } \\cdot \\epsilon _ { 0 } ^ 4 \\ln n \\le \\sum _ { x \\in V } \\sum _ { i = 1 } ^ { r } p ( x , S _ i ) = \\sum _ { i = 1 } ^ { r } \\sum _ { x \\in V } p ( x , S _ i ) \\ ; . \\end{align*}"}
+{"id": "778.png", "formula": "\\begin{align*} - \\Delta u + V ( x ) u = ( I _ \\alpha \\ast | u | ^ q ) | u | ^ { q - 2 } u \\quad \\mathbb { R } ^ N \\end{align*}"}
+{"id": "560.png", "formula": "\\begin{align*} \\mathbb E \\left ( \\int _ 0 ^ t \\norm { \\mathbf S ( t - s ) \\mathbf M ( \\mathbf u ( s ) ) } _ { \\mathcal L _ 2 ( K , \\mathbf H ^ { \\mathbf s } ) } ^ 2 \\ , d s \\right ) = \\mathbb E \\left ( \\int _ 0 ^ t \\norm { \\mathbf M ( \\mathbf u ( s ) ) } _ { \\mathcal L _ 2 ( K , \\mathbf H ^ { \\mathbf s } ) } ^ 2 \\ , d s \\right ) \\le C \\mathbb E \\left ( \\int _ 0 ^ t \\norm { \\mathbf u ( s ) } _ { \\mathbf H ^ { \\mathbf s } } ^ 2 \\ , d s \\right ) , \\end{align*}"}
+{"id": "1166.png", "formula": "\\begin{align*} & \\sum _ { i , j \\geq 0 } \\phi _ i ( m \\rq _ { 1 , j } ( x , y ) ) t ^ { i + j } = \\sum _ { i , j , k \\geq 0 } m _ { 1 , i } ( \\phi _ j ( x ) , \\phi _ k ( y ) ) t ^ { i + j + k } , \\\\ & \\sum _ { i , j \\geq 0 } \\phi _ i ( m \\rq _ { 2 , j } ( x , y ) ) t ^ { i + j } = \\sum _ { i , j , k \\geq 0 } m _ { 2 , i } ( \\phi _ j ( x ) , \\phi _ k ( y ) ) t ^ { i + j + k } . \\end{align*}"}
+{"id": "6884.png", "formula": "\\begin{align*} \\ell ( \\widetilde { U } , r ) \\smallfrown [ \\widetilde { T } ] = ( \\ell ( \\widetilde { U } , r ) \\smallsmile \\pi ^ * \\delta ) \\smallfrown [ \\widetilde { U } ] = \\pi ^ * \\delta \\smallfrown ( \\ell ( \\widetilde { U } , r ) \\smallfrown [ \\widetilde { U } ] ) . \\end{align*}"}
+{"id": "6442.png", "formula": "\\begin{align*} M = \\begin{pmatrix} a _ 0 + y ^ n c + w ^ n c ' & a _ 1 - x ^ n c - z ^ n c ' \\\\ b _ 0 - y ^ n d - w ^ n d ' & b _ 1 + x ^ n d + z ^ n d ' \\end{pmatrix} . \\end{align*}"}
+{"id": "6367.png", "formula": "\\begin{align*} t _ { H , H \\otimes H } ( 1 _ { H } \\otimes 1 _ { H } \\otimes 1 _ { H } ) = \\chi ^ { i } \\otimes ( \\chi _ { i } \\cdot ( 1 _ { H } \\otimes 1 _ { H } ) ) = \\chi ^ { i } \\otimes \\chi _ { i 1 } \\otimes \\chi _ { i 2 } = ( \\mathrm { I d } _ { H } \\otimes \\Delta ) ( \\chi ) , \\end{align*}"}
+{"id": "162.png", "formula": "\\begin{align*} \\nabla ( [ \\omega _ 1 ^ { ( 1 ) } ] ) = [ \\omega _ 1 ^ { ( 0 ) } ] \\otimes \\frac { \\d z } { 1 - z } . \\end{align*}"}
+{"id": "2870.png", "formula": "\\begin{align*} \\delta ( { { \\xi } } ) = \\sum _ { w \\in W _ 0 } ( - 1 ) ^ { \\ell ( w ) } e ^ { i \\langle w \\xi , \\rho \\rangle } = \\prod _ { \\alpha \\in R _ 0 ^ + } ( e ^ { i \\langle \\xi , \\alpha \\rangle / 2 } - e ^ { - i \\langle \\xi , \\alpha \\rangle / 2 } ) . \\end{align*}"}
+{"id": "6564.png", "formula": "\\begin{align*} C N _ 1 ^ { C } N _ 1 ^ d \\leq N _ 1 ^ { C } = N ^ { \\frac { C } { \\widetilde C ^ 2 } } \\leq N ^ { \\rho _ 2 / 4 } / 2 , \\end{align*}"}
+{"id": "5297.png", "formula": "\\begin{align*} j \\leq \\min \\left \\{ \\frac { n _ l + d } { 2 } , n _ l \\right \\} = \\frac { n _ l + \\min \\{ n _ l , d _ l \\} } { 2 } . \\end{align*}"}
+{"id": "5431.png", "formula": "\\begin{align*} \\frac { 1 } { k } D _ z p _ T ( \\mathbf { y } ) & = \\sum _ { a + b + c = k - 1 } x _ u ^ a x _ v ^ b x _ w ^ c \\binom { k - 1 } { a , b , c } d _ T ( u , v , w ) \\\\ & + \\sum _ { a + c = k - 1 } x _ u ^ a x _ w ^ c \\binom { k - 1 } { a , c } ( d _ T ( u , w ) - d _ T ( u , v , w ) ) \\\\ & + x _ u ^ { k - 1 } ( d _ T ( u ) - d _ T ( u , w ) ) \\\\ & = 2 ( x _ u + x _ v + x _ w ) ^ { k - 1 } - ( x _ u + x _ w ) ^ { k - 1 } - x _ u ^ { k - 1 } \\\\ & = 0 - ( - \\zeta ) ^ { k - 1 } - 1 ^ { k - 1 } = 0 . \\end{align*}"}
+{"id": "6306.png", "formula": "\\begin{align*} \\partial _ t w _ \\lambda + \\partial _ x g _ { [ < \\lambda ] } \\partial _ x w _ \\lambda = e _ \\lambda , w _ \\lambda ( T ) = w _ { T , \\lambda } , \\end{align*}"}
+{"id": "3885.png", "formula": "\\begin{align*} \\gamma ( r _ 1 ( f ( x _ 0 ) ) ) = r _ 2 ( i _ 2 ( \\gamma ( r _ 1 ( f ( x _ 0 ) ) ) ) ) & = r _ 2 ( \\alpha ( f ( x _ 0 ) ) ) \\\\ & = r _ 2 ( \\beta ( f ( x _ 0 ) ) ) \\\\ & = r _ 2 ( i _ 2 ( \\delta ( r _ 1 ( f ( x _ 0 ) ) ) ) ) = \\delta ( r _ 1 ( f ( x _ 0 ) ) ) \\end{align*}"}
+{"id": "3831.png", "formula": "\\begin{align*} \\tilde H ( x _ 0 , x _ 1 , s _ 0 , s _ 1 , w _ 0 , w _ 1 ) = \\begin{cases} a H ( x _ 0 , s _ 0 , x _ 1 , s _ 1 ) , \\quad & w _ 0 = w _ 1 = a , \\\\ + \\infty , \\quad & \\end{cases} \\end{align*}"}
+{"id": "2052.png", "formula": "\\begin{align*} \\mathrm { E } T _ \\phi u = \\mathrm { E } T _ \\phi a + \\mathrm { E } T _ \\phi b \\in { \\mathrm { L } } ^ { p } ( \\mathbb { R } ^ n ) + { \\mathrm { H } } ^ { 1 , p } ( \\mathbb { R } ^ n ) \\subset \\mathrm { L } ^ p ( \\mathbb { R } ^ n ) + \\dot { \\mathrm { H } } ^ { 1 , p } ( \\mathbb { R } ^ n ) \\end{align*}"}
+{"id": "7831.png", "formula": "\\begin{align*} \\begin{cases} \\Omega ( q _ 0 \\gamma ^ 0 ) = - \\chi , q _ 0 \\neq 0 \\\\ \\Omega ( \\gamma ) = 0 , \\\\ \\end{cases} \\end{align*}"}
+{"id": "4887.png", "formula": "\\begin{align*} f _ n ( z ) = 1 - ( 1 - z ) \\exp ( z + \\frac { 1 } { 2 } z ^ 2 + \\ldots + \\frac { 1 } { n } z ^ n ) \\end{align*}"}
+{"id": "1630.png", "formula": "\\begin{align*} \\begin{cases} \\Delta { } ^ { \\mathcal D } \\ ! E _ r = 1 & \\Omega _ r \\\\ { } ^ { \\mathcal D } \\ ! E _ r = 0 & \\partial \\Omega _ r \\end{cases} \\end{align*}"}
+{"id": "3476.png", "formula": "\\begin{align*} x = ( - \\frac { \\log | F _ 0 | _ { h ^ { \\otimes d _ 0 } } } { | \\log | t | | } , \\ldots - \\frac { \\log | F _ m | _ { h ^ { \\otimes d _ m } } } { | \\log | t | | } ) \\in U _ k , k = 0 , 1 , \\ldots m , \\end{align*}"}
+{"id": "3412.png", "formula": "\\begin{align*} | f + g | \\leq \\max \\{ | f | + | g | \\} , | f g | = | f | | g | . \\end{align*}"}
+{"id": "3113.png", "formula": "\\begin{align*} \\mathbf { H } _ { k , } = \\frac { 1 } { N } \\sqrt { \\frac { \\kappa } { \\kappa + 1 } } \\sum _ { p = 1 } ^ { L _ k } \\alpha _ { k , p } \\sum _ { n _ 1 = 0 } ^ { N _ 1 - 1 } \\sum _ { n _ 2 = 0 } ^ { N _ 2 - 1 } \\chi _ { k , p , n _ 1 \\cdot N _ 2 + n _ 2 } \\mathbf { a } ^ H ( \\phi ^ ) . \\end{align*}"}
+{"id": "2993.png", "formula": "\\begin{align*} E _ { I _ n } ^ { n } \\simeq E _ { I _ { n - 1 } } ^ n \\times _ { s , \\alpha _ n } E ^ 0 _ { I _ { n } } & \\simeq ( \\cdots ( ( E ^ n \\times _ { s , \\alpha _ 1 } E ^ 0 _ { I _ 1 } ) \\times _ { s , \\alpha _ 2 } E ^ 0 _ { I _ 2 } ) \\times _ { s , \\alpha _ 3 } \\cdots ) \\times _ { s , \\alpha _ n } E _ { I _ n } ^ 0 \\\\ & \\simeq E ^ n \\times _ { s , \\alpha _ 1 \\circ \\cdots \\circ \\alpha _ n } E _ { I _ n } ^ 0 = E ^ n \\times _ { s , \\alpha } E _ { I _ n } ^ 0 . \\end{align*}"}
+{"id": "1150.png", "formula": "\\begin{align*} \\delta _ c ( m ) ( a ) : = ~ & x \\cdot _ 1 m - m \\cdot _ 1 x = x \\cdot _ 2 m - m \\cdot _ 2 x , m \\in C ^ 0 _ { c o m } ( \\mathfrak { g } , M ) x \\in \\mathfrak { g } , \\\\ \\delta _ c ( h _ 1 , \\ldots , h _ n ) : = ~ & ( \\delta _ 1 h _ 1 , \\ldots , \\underbrace { \\delta _ 1 h _ i + \\delta _ 2 h _ { i - 1 } } _ { i - } , \\ldots , \\delta _ 2 h _ n ) , \\end{align*}"}
+{"id": "982.png", "formula": "\\begin{align*} O _ F = \\begin{cases} \\mathbb Z \\left [ \\dfrac { 1 + \\sqrt d } { 2 } \\right ] & d \\equiv 1 \\ ( \\mathrm { m o d } \\ 4 ) , \\\\ \\mathbb Z [ \\sqrt d ] & d \\equiv 2 , \\ 3 \\ ( \\mathrm { m o d } \\ 4 ) . \\end{cases} \\end{align*}"}
+{"id": "8089.png", "formula": "\\begin{align*} \\left \\| \\uppercase \\expandafter { \\romannumeral 1 } _ { 1 } \\right \\| _ { L _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } . \\end{align*}"}
+{"id": "1740.png", "formula": "\\begin{align*} \\emph { W } ^ { ( m , n ) } _ { { \\lambda } } : = \\prod _ { 1 \\leq j < k \\leq n } \\Bigl ( x ^ { ( m + n ) } _ { \\lambda _ j + n - j } - x ^ { ( m + n ) } _ { \\lambda _ k + n - k } \\Bigr ) ^ 2 \\prod _ { 1 \\leq j \\leq n } \\emph { w } ^ { ( m + n ) } _ { { \\lambda } _ j + n - j } , \\end{align*}"}
+{"id": "984.png", "formula": "\\begin{align*} ( a + b \\sqrt d ) ( a - b \\sqrt d ) & = \\prod _ { p \\in S _ \\pm } ( p ) ^ { l _ p n _ p + r _ p } \\prod _ { q \\notin S _ \\pm } ( q ) ^ { r _ q } \\\\ & = \\prod _ { p \\in S _ \\pm } ( \\gamma _ p ) ^ { n _ p } ( \\widetilde { \\gamma _ p } ) ^ { n _ p } ( p ) ^ { r _ p } \\prod _ { q \\notin S _ \\pm } ( q ) ^ { r _ q } , \\end{align*}"}
+{"id": "5580.png", "formula": "\\begin{align*} \\Phi ( G , k ) : = \\min _ { S \\subset V ( G ) ; | S | \\leq k } \\frac { | N ( S ) \\setminus S | } { | S | } . \\end{align*}"}
+{"id": "6066.png", "formula": "\\begin{align*} K _ 2 = \\mathbb { E } \\vert \\int _ { 0 } ^ { t } \\int _ { B _ { M _ { { \\mathcal { P } } } ( r ) } } { c } ( z , { X } ^ { { { { \\mathcal { P } } } } , { M _ { { \\mathcal { P } } } } } _ { \\tau ^ { { \\ { \\ \\mathcal { P } } } } ( r ) - } ) - { c } ( z , { X } ^ { M _ { { \\mathcal { P } } } } _ { r - } ) { N } ( d z , d r ) \\vert . \\end{align*}"}
+{"id": "5602.png", "formula": "\\begin{align*} g ' _ { N , q } ( t ) \\exp _ q ( t ) ^ 2 & = \\dfrac { p - 1 } { \\Gamma ( p ) } t ^ { p - 2 } \\exp _ q ( t ) - \\dfrac { q - 1 } { \\Gamma ( q ) } t ^ { q - 2 } \\exp _ p ( t ) \\\\ & = \\sum _ { j = 0 } ^ \\infty \\left ( \\dfrac { p - 1 } { \\Gamma ( p ) \\Gamma ( q + j ) } - \\dfrac { q - 1 } { \\Gamma ( q ) \\Gamma ( p + j ) } \\right ) t ^ { p + q - 3 } \\\\ & = \\sum _ { j = 0 } ^ \\infty \\dfrac { ( p - 1 ) \\prod _ { i = 0 } ^ { j - 1 } ( p + i ) - ( q - 1 ) \\prod _ { i = 0 } ^ { j - 1 } ( q + j ) } { \\Gamma ( q + j ) \\Gamma ( p + j ) } t ^ { p + q - 3 } \\\\ & > 0 , \\end{align*}"}
+{"id": "4341.png", "formula": "\\begin{align*} { \\rm P f } ( { \\bf A } ) ^ 2 = { \\rm d e t } ( { \\bf A } ) \\end{align*}"}
+{"id": "7773.png", "formula": "\\begin{align*} \\mathrm { L } ( \\exp ( { 2 \\pi \\mathrm { i } \\xi _ { \\gamma } } ) ) = \\mathrm { L i } _ 2 ( \\exp ( 2 \\pi \\mathrm { i } \\xi _ { \\gamma } ) ) + \\pi \\mathrm { i } \\xi _ { \\gamma } \\log ( 1 - \\exp ( 2 \\pi \\mathrm { i } \\xi _ { \\gamma } ) ) ; \\end{align*}"}
+{"id": "4148.png", "formula": "\\begin{align*} u \\rhd [ v , w ] & = [ u \\rhd v , w ] + [ v , u \\rhd w ] , \\\\ [ u , v ] \\rhd w & = u \\rhd ( v \\rhd w ) - ( u \\rhd v ) \\rhd w - v \\rhd ( u \\rhd w ) + ( v \\rhd u ) \\rhd w . \\end{align*}"}
+{"id": "3917.png", "formula": "\\begin{align*} \\ , \\omega = \\{ t u : \\} , \\end{align*}"}
+{"id": "3771.png", "formula": "\\begin{align*} \\gamma _ i = \\sigma _ i \\mu _ i + \\gamma _ i ^ \\perp , \\mu _ i = \\rho _ i \\gamma _ i + \\mu _ i ^ \\perp . \\end{align*}"}
+{"id": "7819.png", "formula": "\\begin{align*} H _ { 2 n + 2 } : = \\{ ( \\eta ^ i , \\widetilde { \\eta } _ i , \\kappa ) \\in _ { 2 n + 3 } ( \\mathbb { R } ) \\ ; | \\ ; \\eta ^ 0 = 0 \\} \\ , . \\end{align*}"}
+{"id": "4907.png", "formula": "\\begin{align*} \\beta ( 2 k + 1 ) = \\dfrac { ( - 1 ) ^ k E _ { 2 k } } { 4 ^ { k + 1 } ( 2 k ) ! } \\pi ^ { 2 k + 1 } , \\frac { 2 } { e ^ { z } + e ^ { - z } } : = \\sum _ { n = 0 } ^ { \\infty } \\frac { E _ { n } z ^ { n } } { n ! } . \\end{align*}"}
+{"id": "2381.png", "formula": "\\begin{align*} \\min _ { u } \\left \\{ { \\mathcal { A } _ 2 } V ^ { a f t e r } ( x , t ) + C ( t , x , u ) \\right \\} = 0 \\end{align*}"}
+{"id": "225.png", "formula": "\\begin{align*} x \\to x + c \\rho _ M , \\rho _ M = \\sum _ { 1 \\leq j \\leq n } ( n - j ) e _ j \\end{align*}"}
+{"id": "8357.png", "formula": "\\begin{align*} Z ( \\omega _ 2 ) = & B \\int _ { - \\infty } ^ 0 S _ B ( - q ) \\omega _ 2 ( q ) \\ , d q \\\\ & = \\lim _ { t \\to \\infty } \\Big ( B \\int _ { - t } ^ 0 S _ B ( - q ) \\omega _ 2 ( q ) \\ , d q - S _ B ( t ) \\omega _ 2 ( - t ) + S _ B ( t ) Z _ 0 \\Big ) \\end{align*}"}
+{"id": "7621.png", "formula": "\\begin{align*} U ( x ) : = x 1 _ { \\{ x > 0 \\} } - \\frac { 1 } { q } [ ( 1 - x ) ^ { q } - 1 ] 1 _ { \\{ x \\leq 0 \\} } . \\end{align*}"}
+{"id": "5782.png", "formula": "\\begin{align*} \\lim _ { J _ 1 ^ 0 \\rightarrow 0 } \\lim _ { L \\rightarrow \\infty } { \\mathbb E } \\langle ( { R _ { 1 , 2 } ^ c } - \\langle R _ { 1 , 2 } ^ c \\rangle ) ^ 2 \\rangle = 0 . \\end{align*}"}
+{"id": "1171.png", "formula": "\\begin{align*} & m \\rq _ { 2 , n } ( x , y ) + \\phi _ n ( m \\rq _ { 2 , 0 } ( x , y ) ) = m _ { 2 , n } ( x , y ) + m _ { 2 , 0 } ( \\phi _ n ( x ) , y ) + m _ { 2 , 0 } ( x , \\phi _ n ( y ) ) , \\\\ & m \\rq _ { 2 , n } ( x , y ) - m _ { 2 , n } ( x , y ) = m _ { 2 } ( \\phi _ n ( x ) , y ) + m _ 2 ( x , \\phi _ n ( y ) ) - \\phi _ n ( m \\rq _ 2 ( x , y ) ) = \\delta _ 2 \\phi _ n ( x , y ) . \\end{align*}"}
+{"id": "5888.png", "formula": "\\begin{align*} p ^ { n } \\leq 2 N ^ { - 4 \\beta + 2 - 2 \\zeta } \\mbox { a n d } p ^ { n } = o ( q ^ n ) . \\end{align*}"}
+{"id": "8928.png", "formula": "\\begin{align*} \\frac { 1 } { g ( E _ i , E _ i ) } J d \\varphi ( [ E _ i ^ \\prime , E _ i ] ) = \\frac { 1 } { g ( E _ i ^ \\prime , E _ i ^ \\prime ) } \\nabla _ { E _ i ^ \\prime } ^ \\varphi d \\varphi ( E _ i ^ \\prime ) + \\frac { 1 } { g ( E _ i , E _ i ) } \\nabla _ { E _ i } ^ \\varphi d \\varphi ( E _ i ) \\\\ \\end{align*}"}
+{"id": "5414.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\xi } _ x ( t ) = \\mathbf { v } ( \\xi _ x ( t ) ) , & \\\\ \\xi _ x ( 0 ) = x . \\ \\ & \\end{cases} \\end{align*}"}
+{"id": "7469.png", "formula": "\\begin{align*} g _ { a , c } ( x , z ) = g _ { b , a } ( y , x ) = g _ { c , b } ( z , y ) , a + b + c = \\varpi , x y z = 1 . \\end{align*}"}
+{"id": "2880.png", "formula": "\\begin{align*} \\mathcal { C } ( P ) ^ { W } : = \\{ f \\in \\mathcal { C } ( P ) \\mid w f = f , \\ w \\in W \\} \\end{align*}"}
+{"id": "8282.png", "formula": "\\begin{align*} A ( t ) = \\left ( \\begin{matrix} \\cos t & \\sin t \\\\ - \\sin t & \\cos t \\end{matrix} \\right ) . \\end{align*}"}
+{"id": "6300.png", "formula": "\\begin{align*} J ^ 4 _ { m a i n } ( u , v ) = 4 | \\partial _ x ( u \\bar v ) | ^ 2 + \\partial _ x ^ 2 ( | u | ^ 2 | v | ^ 2 ) , \\end{align*}"}
+{"id": "6595.png", "formula": "\\begin{align*} J _ { j , \\xi } = \\{ \\sigma \\in \\R : \\ | \\sigma + k \\cdot \\omega + \\xi \\mu _ n | \\leq C ( \\varepsilon + \\delta ) \\} , \\end{align*}"}
+{"id": "5478.png", "formula": "\\begin{align*} & ( z ; p _ 1 , \\dots , p _ n ) = ( z ; p _ { \\nu ( 1 ) , \\dots , \\nu ( n ) } ) \\qquad \\textrm { f o r a l l } \\quad \\nu \\in S _ n , \\\\ & ( z ; p _ 1 , \\dots , p _ n ) = ( z ; p _ 2 , \\dots , p _ n ) ( z p _ 1 ; p _ 1 , \\dots , p _ n ) , \\end{align*}"}
+{"id": "4007.png", "formula": "\\begin{align*} 0 = \\sum _ l X _ { l \\bar l i } - w A \\varphi _ { t , i } , \\end{align*}"}
+{"id": "6940.png", "formula": "\\begin{align*} \\dot x _ 1 = x _ 2 , \\ ; \\ ; \\ ; \\dot x _ 2 = ( 1 - x _ 1 ^ 2 ) x _ 2 - x _ 1 + u . \\end{align*}"}
+{"id": "1577.png", "formula": "\\begin{align*} \\exp \\left ( - \\frac { | v - u _ f | ^ 2 } { 2 T _ f } \\right ) = \\exp \\left ( - \\frac { v ^ 2 } { 2 T _ g } \\right ) \\left ( 1 + \\frac { \\varepsilon v ^ 2 } { 2 T _ g \\rho _ g } \\int _ { - \\infty } ^ { \\infty } \\left [ \\frac { u ^ 2 } { T _ g } - 1 \\right ] h ( x , u ) \\mathrm { d } u + \\frac { \\varepsilon v m _ h } { T _ g \\rho _ g } \\right ) . \\end{align*}"}
+{"id": "6664.png", "formula": "\\begin{align*} R ( x ) - { 1 \\over \\pi } \\mathop { \\sim } \\limits _ { x \\to \\infty } { 1 \\over \\pi } \\sum _ { n = 1 } ^ \\infty { \\pi ^ { n } d _ { n } \\over x ^ { n } } . \\end{align*}"}
+{"id": "7794.png", "formula": "\\begin{align*} f ^ { } = \\frac { \\tau _ 2 ^ 2 } { ( 2 \\pi ) ^ 2 } \\sum _ { \\hat { \\gamma } \\in \\Lambda ^ + \\cup \\{ 0 \\} } n _ { \\hat { \\gamma } } \\sum _ { n \\in \\mathbb { Z } - \\{ 0 \\} } \\left ( \\frac { 1 } { | n | } + 2 \\pi q _ a t ^ a \\right ) \\frac { e ^ { - S _ { \\hat { \\gamma } , 0 , n } } } { n ^ 2 } \\\\ \\end{align*}"}
+{"id": "8268.png", "formula": "\\begin{align*} H ( m , | x | ) = \\sqrt { | x | } H ( \\frac { 1 } { \\sqrt { | x | } } m , 1 ) , \\end{align*}"}
+{"id": "1528.png", "formula": "\\begin{align*} \\sum _ { \\substack { P ^ + ( A ) \\le y \\\\ \\Omega ( A ) = J } } \\frac 1 A < z ^ { - J } \\sum _ { \\substack { P ^ + ( A ) \\leq y } } \\frac { z ^ { \\Omega ( A ) } } { A } & = z ^ { - J } \\left ( 1 - \\frac { z } { 2 } \\right ) ^ { - 1 } \\prod _ { 3 \\le p \\leq y } \\left ( 1 - \\frac { z } { p } \\right ) ^ { - 1 } \\\\ & \\ll \\frac { z ^ { - J } } { 2 - z } \\exp \\left ( z \\sum _ { 3 \\le p \\le y } \\frac 1 p \\right ) \\ll \\frac { z ^ { - J } } { 2 - z } ( \\log y ) ^ z . \\end{align*}"}
+{"id": "8510.png", "formula": "\\begin{align*} r _ { \\ell } ^ { \\vee } ( \\bar { z } ) = \\mbox { a p l i m } ( r _ { \\ell } , ( - \\infty , \\bar { z } ) , \\bar { z } ) \\mbox { a n d } r _ { \\ell } ^ { \\wedge } ( \\bar { z } ) = \\mbox { a p l i m } ( r _ { \\ell } , ( \\bar { z } , + \\infty ) , \\bar { z } ) = 0 . \\end{align*}"}
+{"id": "3984.png", "formula": "\\begin{align*} f ( u ^ { i , n } , \\mu ^ n ) = \\frac { 1 } { n } \\sum _ { j = 1 } ^ n F \\big ( u ^ { i , n } , u ^ { j , n } \\big ) . \\end{align*}"}
+{"id": "235.png", "formula": "\\begin{align*} \\lim _ { x \\to + \\infty } | \\phi ^ { \\emph { r } } _ { \\xi } ( x ; g ) - e ^ { \\langle \\xi , x \\rangle } | = 0 \\quad \\emph { R e } ( \\xi ) = 0 , \\end{align*}"}
+{"id": "3088.png", "formula": "\\begin{align*} \\Delta R \\approx \\sum \\limits _ { k = 1 } ^ { { N _ { \\rm { R } } } } { { { \\log } _ 2 } \\left ( { { { 1 + { { { \\bar C } _ k } } } } } \\right ) } - \\sum \\limits _ { k = 1 } ^ { { N _ { \\rm { R } } } } { { { \\log } _ 2 } \\left ( { { { 1 + { { \\bar C } _ k } \\left ( { 1 - { 2 ^ { - 1 - { x _ k } } } } \\right ) } } } \\right ) } . \\end{align*}"}
+{"id": "2914.png", "formula": "\\begin{align*} K _ { X ^ e / X } = K _ { X ^ e } - F ^ { e , * } K _ X = ( 1 - p ^ e ) K _ X , \\end{align*}"}
+{"id": "6781.png", "formula": "\\begin{align*} ( u ( - \\infty ) , v ( - \\infty ) ) = ( 1 , 0 ) , ( u ( + \\infty ) , v ( + \\infty ) ) = E , \\end{align*}"}
+{"id": "5429.png", "formula": "\\begin{align*} \\omega _ n ^ s ( \\xi ) : = \\sum _ { 2 ^ { \\kappa _ s / \\tau } \\leq j \\leq n } ( \\Theta _ { N _ j } - \\Theta _ { N _ { j - 1 } } ) ( \\xi ) \\eta _ { j ^ \\tau } ( \\xi ) . \\end{align*}"}
+{"id": "4513.png", "formula": "\\begin{align*} F _ H ^ { i + 1 / 2 } : = \\Delta _ 1 + \\Delta _ 2 - S _ { \\theta _ i } \\mathbb L _ H ( { \\mathbf u } ^ a + \\mathbf { u } _ { i } , { \\mathbf H } ^ a + { \\mathbf H } _ i , \\Psi ^ a + \\Psi _ { i } ) \\ , ; \\end{align*}"}
+{"id": "7020.png", "formula": "\\begin{align*} \\Gamma = \\sum _ { i = 1 } ^ { + \\infty } \\alpha _ i | \\Psi _ i \\rangle \\langle \\Psi _ i | , \\end{align*}"}
+{"id": "3899.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } Q _ N \\varphi ( N ) ^ { 1 / 8 } = \\frac { 1 } { 2 } , \\end{align*}"}
+{"id": "5343.png", "formula": "\\begin{align*} A ^ h = \\{ m \\in \\mathbb { N } : ( \\exists p \\in A ) p ^ { h ( p ) } \\mid m \\} \\in { \\cal U } . \\end{align*}"}
+{"id": "6399.png", "formula": "\\begin{align*} \\chi \\tau _ { H , H } ( a \\otimes b ) = \\chi ^ i ( b ) \\chi _ i ( a ) = \\tau _ { H ^ \\circ , H ^ \\circ } ( \\chi ^ i \\otimes \\chi _ i ) ( a \\otimes b ) = \\tau _ { H ^ \\circ , H ^ \\circ } ( \\chi ) ( a \\otimes b ) \\end{align*}"}
+{"id": "8915.png", "formula": "\\begin{align*} ' d _ 2 ^ { 1 , 0 } ( \\overline { z ^ 1 } ) & = \\overline { d ^ { ( 0 , 1 ) , 1 } ( d ^ { \\underline { 0 } , 2 } ( z ^ 1 _ 1 ) ) + d ^ { ( 0 , 1 ) , 2 } ( d ^ { \\underline { 0 } , 2 } ( z ^ 1 _ 1 ) ) + d ^ { ( 1 , 0 ) , 1 } ( d ^ { \\underline { 0 } , 1 } ( z ^ 1 _ 2 ) ) - d ^ { ( 1 , 0 ) , 2 } ( d ^ { \\underline { 0 } , 1 } ( z ^ 1 _ 2 ) ) } \\\\ & = \\overline { d ^ { ( 1 , 0 ) , 2 } d ^ { \\underline { 0 } , 1 } ( z ^ 1 _ 1 - z ^ 1 _ 2 ) } . \\end{align*}"}
+{"id": "3247.png", "formula": "\\begin{align*} & \\frac { R } { \\sqrt { p ^ { 2 } + q ^ { 2 } } } \\left ( 1 + 2 \\left \\lfloor \\frac { \\epsilon q ^ { 2 } R } { \\sqrt { p ^ { 2 } + q ^ 2 } } \\right \\rfloor \\right ) = \\frac { 2 q ^ 2 } { p ^ { 2 } + q ^ { 2 } } \\epsilon R ^ { 2 } + \\left ( 1 - 2 \\left \\{ \\frac { \\epsilon q ^ { 2 } R } { \\sqrt { p ^ { 2 } + q ^ { 2 } } } \\right \\} \\right ) \\frac { R } { \\sqrt { p ^ { 2 } + q ^ { 2 } } } \\end{align*}"}
+{"id": "8248.png", "formula": "\\begin{align*} g _ 0 ( \\frac { \\partial \\Psi } { \\partial x _ 1 } , \\frac { \\partial \\Psi } { \\partial x _ 2 } ) = - V g _ 0 ( I _ 1 T , \\frac { \\partial \\Psi } { \\partial x _ 2 } ) = - V \\omega _ 1 ( T , \\frac { \\partial \\Psi } { \\partial x _ 2 } ) = V d x _ 1 ( \\frac { \\partial \\Psi } { \\partial x _ 2 } ) = 0 . \\end{align*}"}
+{"id": "5993.png", "formula": "\\begin{align*} p ( x , \\xi ) \\sim \\sum _ { j = 0 } ^ \\infty p _ { m - j } ( x , \\xi ) . \\end{align*}"}
+{"id": "8888.png", "formula": "\\begin{align*} \\int _ { \\C P ^ 2 } & x ^ 2 = \\int _ { G ( 1 , \\C ^ 3 ) } c _ 1 ( S ) ^ 2 = \\sum _ { i = 1 } ^ 3 \\frac { u _ i ^ 2 } { \\prod _ { j \\ne i } ( u _ i - u _ j ) } \\\\ = & \\frac { u _ 1 ^ 2 } { ( u _ 1 - u _ 2 ) ( u _ 1 - u _ 3 ) } + \\frac { u _ 2 ^ 2 } { ( u _ 2 - u _ 1 ) ( u _ 2 - u _ 3 ) } + \\frac { u _ 3 ^ 2 } { ( u _ 3 - u _ 1 ) ( u _ 3 - u _ 2 ) } , \\end{align*}"}
+{"id": "8853.png", "formula": "\\begin{align*} \\frac { \\alpha } { 4 } \\sum _ { t = T _ { 1 } + 1 } ^ { T } [ t ( r _ { t } - r _ { t + 1 } ) - r _ { t } ] \\leq \\frac { T _ 1 \\alpha } { 4 } r _ { T _ { 1 } + 1 } \\leq \\frac { 1 8 \\bar { L } ^ 2 \\kappa } { \\alpha } r _ { T _ { 1 } + 1 } . \\end{align*}"}
+{"id": "7550.png", "formula": "\\begin{align*} \\Phi ^ { R _ 1 \\cdot R _ 2 , \\beta , D } _ { p _ 1 , \\ldots , p _ n } ( \\{ \\tau _ i \\} _ { i = 1 } ^ { | Q | } , Q ) = \\Phi ^ { ( R _ 1 , R _ 2 ) , \\beta _ 1 + \\beta _ 2 , D _ 1 \\cup _ { p } D _ 2 } _ { p _ 1 , \\ldots , p _ n } ( \\{ \\tau _ i \\} _ { i = 1 } ^ { | Q | } , Q ' ) . \\end{align*}"}
+{"id": "5617.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } I _ 3 = 0 \\mbox { u n i f o r m l y i n } n . \\end{align*}"}
+{"id": "455.png", "formula": "\\begin{align*} \\left ( - \\Delta - \\Phi ( \\sigma ) | x | ^ { - 2 } \\right ) | x | ^ { - \\sigma } = 0 , x \\neq 0 . \\end{align*}"}
+{"id": "2022.png", "formula": "\\begin{align*} \\left ( s , \\frac { 1 } { p _ \\theta } , \\frac { 1 } { q _ \\theta } \\right ) : = ( 1 - \\theta ) \\left ( s _ 0 , \\frac { 1 } { p _ 0 } , \\frac { 1 } { q _ 0 } \\right ) + \\theta \\left ( s _ 1 , \\frac { 1 } { p _ 1 } , \\frac { 1 } { q _ 1 } \\right ) \\end{align*}"}
+{"id": "1272.png", "formula": "\\begin{align*} \\frac { d } { d t } M ( t ) = & 8 \\int _ { \\R ^ 3 } | \\nabla u | ^ 2 - ( I _ \\alpha \\ast | \\cdot | ^ b | u | ^ p ) | x | ^ { - b } | u | ^ p d x \\\\ & + \\mathcal { O } \\left ( \\int _ { | x | > R } | \\nabla u | ^ 2 + | x | ^ { - 2 } | u | ^ 2 + ( I _ \\alpha \\ast | \\cdot | ^ b | u | ^ p ) | x | ^ { - b } | u | ^ p d x \\right ) , \\end{align*}"}
+{"id": "5243.png", "formula": "\\begin{align*} \\frac { q _ { i , t + 1 } ^ \\top g _ { i , t } ( x _ { i , t } ) } { \\gamma _ t } = [ g _ { i , t } ( x _ { i , t } ) ] _ + ^ \\top g _ { i , t } ( x _ { i , t } ) = \\| g _ { i , t } ( x _ { i , t } ) \\| ^ 2 \\end{align*}"}
+{"id": "752.png", "formula": "\\begin{align*} M _ T ( \\phi ) ^ 2 \\le \\left ( \\frac { T } { T _ 0 } \\right ) ^ { 1 - 2 b } M _ { T _ 0 } ( g ) ^ 2 \\le C ^ 2 \\left ( \\frac { T } { T _ 0 } \\right ) ^ { 1 - 2 b } N _ { T _ 0 } ( g ) ^ 2 = C ^ 2 N _ T ( \\phi ) ^ 2 . \\end{align*}"}
+{"id": "1508.png", "formula": "\\begin{align*} ( c \\otimes I d _ V ) ( I d _ V \\otimes c ) ( c \\otimes I d _ V ) \\ , = \\ , ( I d _ V \\otimes c ) ( c \\otimes I d _ V ) ( I d _ V \\otimes c ) \\end{align*}"}
+{"id": "2741.png", "formula": "\\begin{align*} 1 = \\| e _ i \\pm e _ k \\| = \\| U ( e _ i ) \\pm U ( e _ k ) \\| \\geq \\big | 1 \\pm \\big ( U ( e _ k ) \\big ) ( \\sigma ( i ) ) \\big | , \\end{align*}"}
+{"id": "3030.png", "formula": "\\begin{align*} \\phi ( H _ { 2 k + 1 } ) & = \\phi ( T _ k ) \\left ( \\phi ( T _ { k + 1 } ) - \\phi ( T _ { k - 1 } ) \\right ) \\\\ [ . 2 e m ] & = \\phi ( T _ k ) \\left ( \\phi ( P _ 2 ) \\phi ( P _ { k - 1 } ) - \\lambda \\phi ( P _ 2 ) \\phi ( P _ { k - 4 } ) - \\phi ( P _ 2 ) \\phi ( P _ { k - 3 } ) + \\lambda \\phi ( P _ 2 ) \\phi ( P _ { k - 6 } ) \\right ) \\\\ [ . 2 e m ] & = \\phi ( P _ 2 ) \\phi ( T _ k ) q ( \\lambda ) . \\end{align*}"}
+{"id": "5555.png", "formula": "\\begin{align*} \\omega ( r ) = \\sum _ { j = 1 } ^ { k } \\left ( \\sigma _ { \\frac { m } { 2 } - j + 1 } ( \\zeta _ 0 ) + \\sigma _ { \\frac { m } { 2 } + j } ( \\zeta _ 0 ) \\right ) \\end{align*}"}
+{"id": "4259.png", "formula": "\\begin{align*} H _ k ^ { 0 } = \\textrm { s p a n } _ { \\R } \\{ u _ j \\ : \\ \\lambda _ j = \\lambda _ k \\} , \\end{align*}"}
+{"id": "2528.png", "formula": "\\begin{align*} \\mathcal { Q } _ N = \\frac { T } { 2 } \\sum _ { k = N + 1 } ^ \\infty \\| \\boldsymbol { y _ d } _ k \\| _ { \\Omega } ^ 2 = \\frac { T } { 2 } \\sum _ { k = N + 1 } ^ \\infty \\left ( \\| \\boldsymbol { y _ d } _ k ^ c \\| _ { \\Omega } ^ 2 + \\| \\boldsymbol { y _ d } _ k ^ s \\| _ { \\Omega } ^ 2 \\right ) = \\| \\boldsymbol { y _ d } - \\boldsymbol { y _ d } _ N \\| \\end{align*}"}
+{"id": "4977.png", "formula": "\\begin{align*} \\bigl [ x , \\sum _ { i = 1 } ^ n x ^ { 1 + 2 ^ i } \\bigr ] = [ x , x R ( x ) ] \\ { \\rm \\ w i t h \\ } R ( x ) : = \\sum _ { i = 1 } ^ n x ^ { 2 ^ i } . \\end{align*}"}
+{"id": "1754.png", "formula": "\\begin{align*} \\frac { 1 } { n ! } \\int _ a ^ { b } \\cdots \\int _ a ^ b & \\det \\left [ f _ j ( x _ k ) \\right ] _ { 1 \\leq j , k \\leq n } \\det \\left [ g _ j ( x _ k ) \\right ] _ { 1 \\leq j , k \\leq n } \\prod _ { 1 \\leq j \\leq n } ( x _ j ) \\ , x _ 1 \\cdots x _ n \\\\ & = \\det \\left [ \\int _ a ^ b f _ j ( x ) g _ k ( x ) ( x ) x \\right ] _ { 1 \\leq j , k \\leq n } . \\end{align*}"}
+{"id": "6739.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { ( 1 + k ) ^ { 2 } } { 2 k } \\frac { 1 } { \\eta ( r _ { 0 } ) } \\left \\{ 4 \\pi \\frac { ( 1 - k ) ^ { 2 } } { ( 1 + k ) ^ { 2 } } - \\frac { 1 } { 4 } \\int _ { \\Sigma } H ^ { 2 } + \\int _ { \\Sigma } \\left ( \\frac { H } { 2 } - \\eta ( r _ { 0 } ) | \\nabla u | \\right ) ^ { 2 } \\right \\} \\le 8 \\pi \\left ( \\frac { \\mathfrak { m } _ { A D M } } { m } - 1 \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "5206.png", "formula": "\\begin{align*} \\widetilde { \\psi _ x } ( a , t ) & = \\left ( \\dots , s _ i u _ i ( a _ i ) s _ i , u _ { i + 1 } ( a _ { i + 1 } ) s _ { i + 1 } , \\dots \\right ) \\\\ & = \\left ( \\dots , u _ i ( - a _ i ^ { - 1 } ) s _ i u _ i ( - a _ i ) a _ i ^ { \\bar \\alpha _ { i } ^ \\vee } , u _ { i + 1 } ( a _ { i + 1 } ) s _ { i + 1 } , \\dots \\right ) \\\\ & \\simeq \\left ( \\dots , u _ i ( - a _ i ^ { - 1 } ) s _ i u _ i ( - a _ i ) , a _ i ^ { \\bar \\alpha _ { i } ^ \\vee } u _ { i + 1 } ( a _ { i + 1 } ) s _ { i + 1 } , \\dots \\right ) . \\end{align*}"}
+{"id": "2433.png", "formula": "\\begin{align*} \\rho ( x ) : = \\sum _ { n = 1 } ^ N | u _ n ( x ) | ^ 2 . \\end{align*}"}
+{"id": "8482.png", "formula": "\\begin{align*} [ \\ , g \\ , ] ( x ) : = \\begin{cases} g ^ { \\vee } ( x ) - g ^ { \\wedge } ( x ) , & \\mbox { i f } x \\in S _ { g } \\\\ 0 , & \\mbox { e l s e w h e r e . } \\end{cases} \\end{align*}"}
+{"id": "726.png", "formula": "\\begin{align*} \\mathbb E \\left ( \\norm { \\mathbf U - \\mathbf V } _ { \\mathbf X ^ { \\mathbf s , b } ( T , T + \\delta ) } ^ 2 \\right ) = 0 , \\end{align*}"}
+{"id": "3254.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { C } + \\mathcal { D } } = \\widetilde { \\mathcal { C } } + \\widetilde { \\mathcal { D } } , \\widetilde { \\widetilde { \\mathcal { C } } } = \\mathcal { C } , \\widetilde { \\mathcal { C } \\mathcal { D } } = \\widetilde { \\mathcal { D } } \\widetilde { \\mathcal { C } } . \\end{align*}"}
+{"id": "2555.png", "formula": "\\begin{align*} & \\phantom { = } \\left \\| | \\varphi | ^ { p _ { r ; s } ^ { \\uparrow * } } \\int _ { \\mathbb { R } ^ N \\setminus B _ R ( 0 ) } K ( x - y ) | v _ n ( y ) | ^ { p _ { r ; s } ^ { \\uparrow * } } d y \\right \\| _ { L ^ \\sigma _ x ( \\operatorname { s u p p } \\varphi ) } \\\\ & \\leq \\| K \\wedge C _ { \\varphi , R } \\| _ { \\tilde { r } } \\| | v _ n | ^ { p _ { r ; s } ^ { \\uparrow * } } \\| _ { \\ell _ r } \\| | \\varphi | ^ { p _ { r ; s } ^ { \\uparrow * } } \\| _ { q _ \\varphi } \\leq C _ 3 \\end{align*}"}
+{"id": "207.png", "formula": "\\begin{align*} & { Q _ 2 } ( M , P _ i , P _ j , \\tau ) = h _ 0 ( 8 \\delta _ 2 ) ^ { 7 } + h _ 1 ( 8 \\delta _ 2 ) ^ { 5 } \\varepsilon _ 2 + h _ 2 ( 8 \\delta _ 2 ) ^ { 3 } \\varepsilon _ 2 ^ 2 + h _ 3 ( 8 \\delta _ 2 ) \\varepsilon _ 2 ^ 3 , \\end{align*}"}
+{"id": "6686.png", "formula": "\\begin{align*} \\int _ 0 ^ { 2 \\pi } \\rho _ { ( 1 ) , N } ^ { ( \\widetilde { \\rm c J } ) } ( \\theta ; \\beta , p , q ) \\ , d \\theta = N = { N \\over 2 \\pi } \\delta _ { k = 0 } \\int _ 0 ^ { 2 \\pi } e ^ { i k \\theta } \\ , d \\theta , k \\in \\mathbb Z \\backslash \\{ 0 \\} . \\end{align*}"}
+{"id": "8784.png", "formula": "\\begin{align*} \\frac { \\eta _ { t } } { 2 } ( - 1 + 9 L \\kappa \\eta _ { t } ) 2 \\alpha = - \\frac { \\alpha } { 3 6 L \\kappa } \\leq - \\frac { 1 } { T _ { 1 } + 1 } . \\end{align*}"}
+{"id": "3643.png", "formula": "\\begin{align*} \\omega \\in \\mathcal { I } ^ n \\iff \\Delta _ { \\omega } \\phi = 0 \\ \\forall \\phi \\ . \\end{align*}"}
+{"id": "3281.png", "formula": "\\begin{align*} \\mathcal { F } ^ \\mu \\left ( F _ X \\right ) ( t ) = \\frac { 1 } { t } \\phi _ X ( t ) \\mu . \\end{align*}"}
+{"id": "1020.png", "formula": "\\begin{align*} q ( R ) ^ E = - k \\ , ( P _ 1 ^ E ) ^ * P _ 1 ^ E \\ , + \\ , ( n + k - 2 ) \\ , ( P _ 2 ^ E ) ^ * P _ 2 ^ E \\ , + \\ , ( P _ 3 ^ E ) ^ * P _ 3 ^ E , \\end{align*}"}
+{"id": "6195.png", "formula": "\\begin{align*} [ R _ x , L _ y ] = [ L _ x , R _ y ] . \\end{align*}"}
+{"id": "7623.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ d w ^ i = 1 . \\end{align*}"}
+{"id": "7173.png", "formula": "\\begin{align*} V ( P ) : = \\{ v _ { \\ell _ 1 \\ell _ 2 \\dots \\ell _ m } : = ( v _ { \\ell _ 1 } ^ 1 , v _ { \\ell _ 2 } ^ 2 , \\dots , v _ { \\ell _ m } ^ m ) ~ | ~ 0 \\leq \\ell _ j \\leq n _ j \\} . \\end{align*}"}
+{"id": "5234.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { t = 1 } ^ T [ g ( x _ { t } ) ] _ + \\Big \\| , \\end{align*}"}
+{"id": "4014.png", "formula": "\\begin{align*} ( \\chi + t ' _ j \\chi + \\tilde \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi _ { t ' _ j } ) ^ n = c ( \\chi + t ' _ j \\chi + \\tilde \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi _ { t ' _ j } ) ^ m \\wedge \\omega ^ { n - m } + b _ { t ' _ j } f _ 2 \\omega ^ n . \\end{align*}"}
+{"id": "8762.png", "formula": "\\begin{align*} \\mathbf { E } _ { \\omega , T } \\big [ \\norm { z _ { T } - x ^ { * } _ { \\omega } } ^ { 2 } \\big ] & \\geq \\frac { 1 } { 4 } \\mathbf { E } _ { \\omega , T } \\big [ \\norm { x _ { \\omega } ^ { * } - x _ { \\hat { \\omega } } ^ { * } } ^ { 2 } \\big ] \\\\ & = \\alpha ^ { - 2 } r ^ { 2 } h ^ { 2 \\beta - 2 } \\mathbf { E } _ { \\omega , T } \\rho ( \\hat { \\omega } , \\omega ) , \\end{align*}"}
+{"id": "1830.png", "formula": "\\begin{gather*} \\sum _ { n \\in C _ \\beta ( m ) } \\frac { 1 } { n + c } \\geq \\sum _ { e ^ { m \\beta } < n < e ^ { ( m + 1 / 2 ) \\beta } - 1 } \\frac { 1 } { n + 1 } = \\frac { \\beta } { 2 } + o ( 1 ) \\end{gather*}"}
+{"id": "5509.png", "formula": "\\begin{align*} O _ B ^ { ( a ^ { - 1 } ) } = \\widetilde \\rho O _ B ^ { ( 1 ) } + \\frac { \\widetilde \\rho - \\widetilde \\rho ^ { - 1 } } { \\big ( Q ^ 2 - Q ^ { - 2 } \\big ) ^ 2 } O _ A ^ { ( 1 ) } O _ A ^ { ( 1 ) } O _ B ^ { ( 1 ) } , \\qquad \\textrm { w h e r e } \\quad \\widetilde \\rho = \\mathrm { p e x p } \\left ( - \\frac { Q ^ 8 - Q ^ { - 8 } } { ( 1 + p ^ { - 1 } ) ( 1 + s ) ( 1 + p s ^ { - 1 } ) } \\right ) . \\end{align*}"}
+{"id": "4013.png", "formula": "\\begin{align*} ( \\chi + t _ i \\chi + \\tilde \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi _ { t _ i } ) ^ n = c ( \\chi + t _ i \\chi + \\tilde \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi _ { t _ i } ) ^ m \\wedge \\omega ^ { n - m } + b _ { t _ i } f _ 1 \\omega ^ n \\end{align*}"}
+{"id": "8738.png", "formula": "\\begin{align*} \\frac { \\eta _ { t } } { 2 } ( - 1 + 9 L \\kappa \\eta _ { t } ) 2 \\alpha = - \\frac { \\alpha } { 3 6 L \\kappa } \\leq - \\frac { 1 } { T _ { 1 } + 1 } . \\end{align*}"}
+{"id": "3237.png", "formula": "\\begin{align*} \\Delta _ { \\alpha } ( \\epsilon , R ) = \\frac { \\epsilon R ^ { 2 } } { ( 1 + \\alpha ^ { 2 } ) } + O \\left ( \\frac { R } { q } + \\frac { 1 } { \\epsilon q ^ 2 } + \\epsilon q R + 1 \\right ) . \\end{align*}"}
+{"id": "817.png", "formula": "\\begin{align*} \\frac { 4 \\log N } { 3 N p _ 2 } = \\frac { 4 ( \\log 2 + \\log \\log n + \\log 1 / p _ 1 ) p _ 1 } { 3 p _ 2 \\log n } \\lesssim \\frac 2 3 . \\end{align*}"}
+{"id": "4096.png", "formula": "\\begin{align*} Q _ 0 ( x ) = \\sum _ { i = 0 } ^ { 2 \\tilde g + 1 } C _ i ^ { ( 0 ) } x ^ i . \\end{align*}"}
+{"id": "6186.png", "formula": "\\begin{align*} f _ t \\circ \\Gamma _ { \\mathbf { k } , \\pm \\infty } = \\Gamma _ { f _ t ( \\mathbf { k } ) , \\pm \\infty } \\circ f _ t . \\end{align*}"}
+{"id": "5400.png", "formula": "\\begin{align*} o _ { j k } : = a ^ { - 1 } _ { j k } a _ { k j } , 1 \\leq j , k \\leq N . \\end{align*}"}
+{"id": "2366.png", "formula": "\\begin{align*} S _ { f , \\tau , \\Lambda } = \\sum _ { \\mu \\in \\Lambda ^ 0 } c _ \\mu \\pi ( \\mu ) . \\end{align*}"}
+{"id": "6922.png", "formula": "\\begin{align*} \\limsup \\limits _ { n \\rightarrow + \\infty } f \\left ( \\phi , \\psi \\right ) ( \\xi _ n ) \\leq f \\left ( \\phi _ + , \\psi _ - \\right ) = f \\left ( 1 , \\psi _ - \\right ) = - \\displaystyle \\frac { m \\psi _ - } { 1 + a \\psi _ - } < 0 . \\end{align*}"}
+{"id": "7115.png", "formula": "\\begin{align*} \\int _ D f \\ , \\eta \\ , d x + \\langle g , \\nabla \\eta \\cdot { n } \\rangle _ { - \\frac 1 2 } - \\langle h , \\eta \\rangle _ { - \\frac 3 2 } = 0 \\end{align*}"}
+{"id": "1792.png", "formula": "\\begin{gather*} g _ { \\alpha , T } ( \\underline { m } ) = \\frac { 1 } { - i \\Delta } \\frac { \\left \\{ ( n _ 1 + \\alpha ) ^ { m _ 1 } \\cdots ( n _ k + \\alpha ) ^ { m _ k } \\right \\} ^ { - i T } - 1 } { T } , \\end{gather*}"}
+{"id": "4576.png", "formula": "\\begin{align*} I _ 0 ( t ) \\le & \\frac { C _ 3 } { \\varepsilon } \\left \\{ \\Vert { \\mathbf F } \\Vert ^ 2 _ { L ^ 2 ( \\Omega _ t ) } + \\Vert \\partial _ s { \\mathbf F } \\Vert _ { L ^ 2 ( \\Omega _ t ) } ^ 2 + \\int _ 0 ^ t ( I _ { 1 , \\ast } + I _ { 1 , n } ) ( s ) d s + \\int _ 0 ^ t \\Vert \\varphi ( s ) \\Vert _ { L ^ 2 ( \\mathbb R ) } ^ 2 d s \\right \\} \\\\ & + \\varepsilon \\left \\{ I ( t ) + I _ { 1 , n } ( t ) \\right \\} + \\frac { C _ 2 } { \\varepsilon } \\Vert \\varphi ( t ) \\Vert _ { L ^ 2 ( \\mathbb R ) } ^ 2 \\ , . \\end{align*}"}
+{"id": "1586.png", "formula": "\\begin{align*} \\Pi \\mathcal { T } \\Pi h = 0 \\end{align*}"}
+{"id": "6843.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\rm c y l } ( v _ { \\varepsilon } + t \\psi ) - \\mathcal { N } _ { \\rm c y l } ( v _ { \\varepsilon } ) = t P _ { \\rm c y l } \\psi - t c _ n \\left ( - P _ { \\rm c y l } v + \\frac { n + 6 } { n - 6 } c _ n v _ { \\varepsilon } ^ { \\frac { 1 2 } { n - 6 } } \\psi \\right ) + \\mathcal { O } ( t ^ 2 ) { \\rm a s } t \\rightarrow 0 \\end{align*}"}
+{"id": "4411.png", "formula": "\\begin{align*} B _ 0 ( \\hat { { \\mathbf U } } ) \\partial _ t \\dot { { \\mathbf U } } + \\tilde { B } _ 1 ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) \\partial _ 1 \\dot { { \\mathbf U } } + B _ 2 ( \\hat { { \\mathbf U } } ) \\partial _ 2 \\dot { { \\mathbf U } } + \\tilde { C } ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) \\dot { { \\mathbf U } } = \\tilde { \\mathbf F } ( \\hat { { \\mathbf U } } ) , \\end{align*}"}
+{"id": "1002.png", "formula": "\\begin{align*} [ [ { \\bf G } ( x ) ] ] = { \\bf G _ + } ( x ) - { \\bf G _ - } ( x ) , . \\end{align*}"}
+{"id": "4695.png", "formula": "\\begin{align*} \\lambda _ { \\min } ( n - i , q ) > \\frac { i } { i + 1 } = 1 - \\frac { 1 } { i + 1 } . \\end{align*}"}
+{"id": "1343.png", "formula": "\\begin{align*} S _ L ( P ) = \\frac { 1 } { \\mathrm { d e g } \\ , L } \\int ^ { \\mathrm { d e g } \\ , L } _ 0 ( \\mathrm { d e g } \\ , L - t ) d t = \\frac { \\mathrm { d e g } \\ , L } { 2 } . \\end{align*}"}
+{"id": "8719.png", "formula": "\\begin{align*} \\min \\Big ( \\alpha _ 0 , \\ , \\frac { d } { \\alpha _ 0 } T ^ { - \\frac { \\beta - 1 } { \\beta } } \\Big ) = \\min \\Big ( T ^ { - 1 / 2 + 1 / \\beta } , \\frac { d } { \\sqrt { T } } \\Big ) \\enspace , \\end{align*}"}
+{"id": "4497.png", "formula": "\\begin{align*} | | e ' _ k | | _ { s , \\ast , T } & \\lesssim \\delta ^ 2 \\Delta ^ 2 _ k ( \\theta ^ { ( s + 2 - \\alpha ) _ + + 1 0 - 2 \\alpha } _ k + \\theta ^ { s + 6 - 2 \\alpha } _ k ) \\\\ & \\lesssim \\delta ^ 2 \\theta ^ { L _ 1 ( s ) - 1 } _ k \\Delta _ k . \\end{align*}"}
+{"id": "1530.png", "formula": "\\begin{align*} \\sum _ { \\substack { P ^ + ( A ) \\le y \\\\ \\Omega ( A ) = k } } \\frac 1 A = I + O \\left ( \\exp ( - K \\sqrt { \\log y } ) \\int _ { | z | = 2 - \\epsilon } \\frac { ( \\log y ) ^ { \\Re z } } { | z | ^ { k + 1 } } \\ , | d z | \\right ) , \\end{align*}"}
+{"id": "1990.png", "formula": "\\begin{align*} K ( t , z , X , Y ) : = \\underset { \\substack { ( x , y ) \\in X \\times Y , \\\\ z = x + y } } { \\inf } \\left ( { \\left \\lVert { x } \\right \\rVert _ { X } + t \\left \\lVert { y } \\right \\rVert _ { Y } } \\right ) \\end{align*}"}
+{"id": "4648.png", "formula": "\\begin{align*} ( x _ e ) _ { e \\in E } \\mapsto \\sum _ { i = 1 } ^ n 1 ^ { \\otimes ( i - 1 ) } \\otimes x _ { e _ i } \\otimes 1 ^ { \\otimes ( n - i ) } \\end{align*}"}
+{"id": "1307.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } ^ d } | f | ^ { 2 n } ( x ) \\d x = \\sum _ { \\xi _ 1 , \\xi _ 2 , \\cdots , \\xi _ { 2 n } \\in \\Z ^ d } \\prod _ { i = 1 } ^ { 2 n } \\hat { f } ( \\xi _ i ) \\dd _ 0 \\Big ( \\sum _ { i = 1 } ^ { 2 n } \\xi _ i \\Big ) , \\end{align*}"}
+{"id": "5517.png", "formula": "\\begin{align*} \\Psi ( e _ A ) = - \\frac 1 { \\big ( Q ^ 2 - Q ^ { - 2 } \\big ) ^ 2 } O _ A ^ { ( 1 ) } O _ A ^ { ( 1 ) } = & \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) , \\\\ [ 0 . 5 e m ] \\Psi ( e _ B ) = - \\frac 1 { \\big ( Q ^ 2 - Q ^ { - 2 } \\big ) ^ 2 } O _ B ^ { ( 1 ) } O _ B ^ { ( 1 ) } = & \\left ( \\begin{array} { c c c } \\frac { 1 } { 2 } & 0 & - \\big ( Q ^ 2 - Q ^ { - 2 } \\big ) ^ { - 2 } \\\\ 0 & 1 & 0 \\\\ - \\frac 1 4 \\big ( Q ^ 2 - Q ^ { - 2 } \\big ) ^ 2 & 0 & \\frac { 1 } { 2 } \\end{array} \\right ) . \\end{align*}"}
+{"id": "5475.png", "formula": "\\begin{align*} ( z ; p _ 1 , \\dots , p _ { j - 1 } , p _ j ^ { - 1 } , p _ { j + 1 } , \\dots , p _ n ) = \\frac 1 { ( p _ j z , p _ 1 , \\dots , p _ { j - 1 } , p _ j , p _ { j + 1 } , \\dots , p _ n ) } . \\end{align*}"}
+{"id": "7067.png", "formula": "\\begin{align*} S _ 1 ( . . . ) = & \\frac { N ^ { 2 / 3 + i \\nu } } { p _ 1 q r ^ { 2 / 3 } } \\sum _ { \\pm } \\sum _ { n _ { 1 } | p _ 1 q r } \\ , n ^ { 1 / 3 } _ 1 \\sum _ { n _ { 2 } \\ll N _ 0 / n ^ 2 _ 1 } \\frac { A _ { \\pi } ( n _ { 2 } , n _ { 1 } ) } { n ^ { 1 / 3 } _ { 2 } } S \\left ( r \\overline { ( a + b q ) } , \\pm n _ { 2 } ; p _ 1 q r / n _ { 1 } \\right ) \\\\ & \\times \\eth _ 1 ^ { \\pm } ( n ^ 2 _ 1 n _ 2 , u , q , p _ 1 ) , \\end{align*}"}
+{"id": "3747.png", "formula": "\\begin{align*} \\varUpsilon _ j ^ \\alpha = 2 \\cos \\dfrac { 2 \\pi ( \\alpha + j ) } { 3 } + 1 , j = 0 , 1 , 2 . \\end{align*}"}
+{"id": "6989.png", "formula": "\\begin{align*} f '' + A ( z ) f ' + B ( z ) f = H ( z ) , \\end{align*}"}
+{"id": "533.png", "formula": "\\begin{align*} B = - b + 1 - \\varepsilon . \\end{align*}"}
+{"id": "8213.png", "formula": "\\begin{align*} \\frac { d } { d \\tau } ( \\rho ^ 2 ( m _ \\tau ) ) & = d ( \\rho ^ 2 ) ( - I _ 1 T ( m _ \\tau ) ) = 4 x _ 1 ( m _ \\tau ) , \\\\ \\frac { d } { d \\tau } ( x _ 1 ( m _ \\tau ) ) & = d x _ 1 ( - I _ 1 T ( m _ \\tau ) ) = - g _ 0 ( I _ 1 T , - I _ 1 T ) = | T | ^ 2 _ { g _ 0 } ( m _ \\tau ) . \\end{align*}"}
+{"id": "5995.png", "formula": "\\begin{align*} p ( x , D ) v : = \\int _ { \\mathbb { R } ^ n } e ^ { i \\xi \\cdot x } p ( x , \\xi ) \\widehat { v } ( \\xi ) d \\xi . \\end{align*}"}
+{"id": "2824.png", "formula": "\\begin{align*} \\ell i _ { 2 } = ( \\ell _ { 2 } \\wedge \\ell _ 1 ) \\circ \\delta . \\end{align*}"}
+{"id": "1483.png", "formula": "\\begin{align*} X = \\frac { \\partial } { \\partial x } - \\frac { y } { 2 } \\frac { \\partial } { \\partial z } , \\ ; \\ ; Y = \\frac { \\partial } { \\partial y } + \\frac { x } { 2 } \\frac { \\partial } { \\partial z } , \\ ; \\ ; Z = \\frac { \\partial } { \\partial z } . \\end{align*}"}
+{"id": "4311.png", "formula": "\\begin{align*} f ( t ) = x ( t ) i + y ( t ) j + z ( t ) k \\end{align*}"}
+{"id": "8591.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ { X , z } ^ { \\mu } \\phi _ 1 = \\mu \\beta F \\ \\ \\mathrm { i n } \\ \\ \\mathcal { S } _ b , \\\\ \\phi _ 1 | _ { z = 0 } = 0 , \\ \\ \\partial _ z \\phi _ 1 | _ { z = - 1 + \\beta b } = - \\nabla _ X \\cdot \\big ( \\mathcal { L } _ { 1 } ^ { \\mu } [ \\beta b ] \\nabla _ X \\psi \\big ) , \\end{cases} \\end{align*}"}
+{"id": "7678.png", "formula": "\\begin{align*} \\varphi ( u ) = \\frac { 1 } { h _ 0 } \\left ( \\varphi ( u - 1 ) - \\sum _ { k = 1 } ^ { u } \\varphi ( u - k ) h _ k \\right ) , \\ , u \\in \\mathbb { N } . \\end{align*}"}
+{"id": "4667.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } S _ i & = \\{ { N _ { G ' } ( v _ i ) , i \\in [ 2 m + 1 ] } \\} \\mbox { ~ a n d ~ } S = \\bigcup \\limits _ { i = 1 } ^ { 2 m + 1 } S _ i ; \\\\ T _ i & = \\{ { N _ { G ' - S } ( S _ i ) , i \\in [ 2 m + 1 ] } \\} \\mbox { ~ a n d ~ } T = \\bigcup \\limits _ { i = 1 } ^ { 2 m + 1 } T _ i ; \\\\ H & = G ' - S - T . \\end{array} \\right . \\end{align*}"}
+{"id": "5531.png", "formula": "\\begin{align*} \\max _ { z \\in K } | D ^ { \\alpha ( k ) } P _ { k } ( z ) | & \\le M ^ { \\abs { \\alpha ( k _ 0 ) } } ( \\alpha ( k _ 0 ) ! ) ^ r \\ \\left ( M \\alpha _ 1 ( k ) ^ r \\max _ { z \\in K } | P _ { k } ( z ) | \\right ) \\\\ & = M ^ { | \\alpha ( k ) | } ( \\alpha ( k ) ! ) ^ r \\ \\max _ { z \\in K } | P _ k ( z ) | . \\end{align*}"}
+{"id": "8754.png", "formula": "\\begin{align*} \\mathbb { P } ( S = t ) = \\frac { \\eta _ t } { \\sum _ { s = 1 } ^ { T } \\eta _ { s } } . \\end{align*}"}
+{"id": "6425.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } { \\frac { \\partial \\bar { v } _ { \\infty } } { \\partial t } ( x , t ) = d _ { 2 } \\left [ \\left ( J _ { 2 } * \\bar { v } _ { \\infty } \\right ) ( x , t ) - \\bar { v } _ { \\infty } ( x , t ) \\right ] - r _ 2 \\bar { v } _ { \\infty } ( x , t ) , x \\in \\R , t > - t _ { 0 } , t _ 0 \\in \\R , } \\\\ { \\bar { v } _ { \\infty } ( x , - t _ 0 ) = v _ { \\infty } ( x , - t _ 0 ) , x \\in \\R , t _ 0 \\in \\R . } \\end{array} \\right . \\end{align*}"}
+{"id": "1170.png", "formula": "\\begin{align*} & m \\rq _ { 1 , n } ( x , y ) + \\phi _ n ( m \\rq _ { 1 , 0 } ( x , y ) ) = m _ { 1 , n } ( x , y ) + m _ { 1 , 0 } ( \\phi _ n ( x ) , y ) + m _ { 1 , 0 } ( x , \\phi _ n ( y ) ) , \\\\ & m \\rq _ { 1 , n } ( x , y ) - m _ { 1 , n } ( x , y ) = m _ { 1 } ( \\phi _ n ( x ) , y ) + m _ 1 ( x , \\phi _ n ( y ) ) - \\phi _ n ( m \\rq _ 1 ( x , y ) ) = \\delta _ 1 \\phi _ n ( x , y ) , \\end{align*}"}
+{"id": "8603.png", "formula": "\\begin{align*} \\phi _ 2 = \\int _ { z } ^ 0 \\int _ { - 1 + \\beta b } ^ { z ' } \\dfrac { \\zeta } { h _ b } \\big ( 1 + \\dfrac { h } { h _ b } \\big ) \\Delta _ X \\psi \\ : \\mathrm { d } z '' \\mathrm { d } z ' = - ( \\dfrac { z ^ 2 } { 2 } + h _ b z ) \\dfrac { \\zeta } { h _ b } \\big ( 1 + \\dfrac { h } { h _ b } \\big ) \\Delta _ X \\psi . \\end{align*}"}
+{"id": "1314.png", "formula": "\\begin{align*} \\begin{cases} \\ll _ 1 = 1 , \\\\ \\ll _ k = 1 , \\ , \\Lambda _ k = \\infty , 2 \\le k \\le p - 1 , \\end{cases} \\end{align*}"}
+{"id": "8358.png", "formula": "\\begin{align*} Z ( t , \\omega _ 2 ) = & S _ B ( t - r ) Z _ 0 + \\int _ r ^ t S _ B ( t - q ) \\ , d \\omega _ 2 ( q ) \\\\ = & S _ B ( t - r ) Z _ 0 + \\int _ 0 ^ { t - r } S _ B ( t - r - q ) \\ , d \\theta _ r \\omega _ 2 ( q ) \\\\ = & S _ B ( t - r ) Z _ 0 + B \\int _ 0 ^ { t - r } S _ B ( t - r - q ) \\theta _ r \\omega _ 2 ( q ) \\ , d q + \\theta _ r \\omega _ 2 ( t - r ) . \\end{align*}"}
+{"id": "153.png", "formula": "\\begin{align*} { \\rm K a z } _ m ^ F \\circ { \\rm \\overline { B r } } = { \\rm \\overline { B r } ' } \\circ { \\rm K a z } _ { e m } ^ E , \\end{align*}"}
+{"id": "1467.png", "formula": "\\begin{align*} f _ i ( j ) = \\left \\{ \\begin{aligned} & j + 1 , \\ & & \\ j = i , 2 l - i , \\\\ & j - 1 , \\ & & \\ j = i + 1 , 2 l + 1 - i , \\\\ & j , \\ & & \\ j \\neq i , i + 1 , 2 l - i , 2 l + 1 - i , \\end{aligned} \\right . \\ \\ \\ \\ 0 \\leq i \\leq l . \\end{align*}"}
+{"id": "6801.png", "formula": "\\begin{align*} u ( z ) & = 1 - \\alpha e ^ { \\lambda _ 1 z } - \\beta z e ^ { \\lambda _ 2 z } + h . o . t . , \\\\ [ 0 . 2 c m ] w ( z ) & = 0 - \\alpha \\lambda _ 1 e ^ { \\lambda _ 1 z } - \\beta \\left [ \\lambda _ 2 z + f ( 1 ) / ( 2 \\lambda _ 2 - c ) \\right ] e ^ { \\lambda _ 2 z } + h . o . t . , \\\\ [ 0 . 2 c m ] v ( z ) & = 0 - \\alpha \\psi \\left ( \\lambda _ { 1 } \\right ) e ^ { \\lambda _ 1 z } - \\beta e ^ { \\lambda _ 2 z } + h . o . t . . \\end{align*}"}
+{"id": "202.png", "formula": "\\begin{align*} { Q _ 2 } ( M , P _ i , P _ j , \\tau ) = & \\left \\{ e ^ { \\frac { 1 } { 2 4 } E _ 2 ( \\tau ) A _ 2 } \\widehat { A } ( T M , \\nabla ^ { T M } ) { \\rm c h } \\left [ \\bigotimes _ { n = 1 } ^ { \\infty } S _ { q ^ n } ( \\widetilde { T _ C M } ) \\otimes \\bigotimes _ { m = 1 } ^ { \\infty } \\wedge _ { - q ^ { m - \\frac { 1 } { 2 } } } ( \\widetilde { T _ C M } ) \\right ] \\right . \\\\ & \\left . \\cdot \\varphi ( \\tau ) ^ { 1 6 } { \\rm c h } ( \\mathcal { V } _ i ) { \\rm c h } ( \\mathcal { V } _ j ) \\right \\} ^ { ( 1 2 ) } , \\end{align*}"}
+{"id": "3328.png", "formula": "\\begin{align*} \\xi _ A \\coloneqq \\sum _ { i = 1 } ^ N [ z _ i ] , \\end{align*}"}
+{"id": "7507.png", "formula": "\\begin{align*} \\begin{cases} & A _ { 2 } + A _ { 3 } + A _ { 4 } = q ^ { 2 m } + q ^ m + 1 \\\\ & ( q ^ { 2 } - 1 ) A _ { 2 } + ( q ^ { 3 } - 1 ) A _ { 3 } + ( q ^ { 4 } - 1 ) A _ { 4 } = ( q ^ { m + 2 } - 1 ) ( q ^ m + 1 ) \\\\ & ( q ^ { 2 } - 1 ) ( q - 1 ) A _ { 2 } + ( q ^ { 3 } - 1 ) ( q ^ { 2 } - 1 ) A _ { 3 } + ( q ^ { 4 } - 1 ) ( q ^ { 3 } - 1 ) A _ { 4 } = ( q ^ { m + 2 } - 1 ) ( q ^ { m + 1 } - 1 ) \\end{cases} \\end{align*}"}
+{"id": "6039.png", "formula": "\\begin{align*} R _ { q } ( \\delta , x ) = \\frac { 1 } { q ! } \\sum _ { \\left \\vert \\alpha \\right \\vert = q } \\int _ { 0 } ^ { 1 } d \\lambda ( 1 - \\lambda ) ^ { q } \\int _ { \\mathbb { R } ^ { d } } d y \\phi _ { \\delta } ( y ) y ^ { \\alpha } \\partial ^ { \\alpha } f ( x + \\lambda y ) \\end{align*}"}
+{"id": "1817.png", "formula": "\\begin{gather*} X ^ { ( \\epsilon ) } = \\left \\{ f \\in A ^ 2 ( U ) ~ \\middle | ~ \\right \\} . \\end{gather*}"}
+{"id": "5632.png", "formula": "\\begin{align*} D _ 5 ^ b = \\Bigl ( x _ 1 y ^ 2 + B y + x _ 0 C = 0 \\Bigr ) . \\end{align*}"}
+{"id": "4011.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { M _ s } \\exp \\left ( \\frac { \\beta } { A ^ { \\frac { 1 } { n } } _ { s , k } } \\left ( 2 v _ t + \\frac { 1 - r } { r } \\varphi _ 2 - \\frac { 1 } { r } \\varphi _ 1 \\right ) ^ { \\frac { n + 1 } { n } } \\right ) \\omega ^ n \\\\ & \\leq \\int _ { M _ s } \\exp \\left ( - \\frac { n + 1 } { n } \\beta \\psi _ k + \\beta A _ { s , k } \\right ) \\omega ^ n \\\\ & \\leq C e ^ { \\beta A _ { s , k } } . \\end{aligned} \\end{align*}"}
+{"id": "7964.png", "formula": "\\begin{align*} \\sigma \\rho ^ { i } = \\rho ^ { r i } \\sigma ~ ~ i \\in \\Z _ n , ~ i \\equiv 0 \\ ! \\ ! \\ ! \\ ! \\pmod 2 . \\end{align*}"}
+{"id": "8519.png", "formula": "\\begin{align*} P ( E ; \\{ z = \\bar { z } \\} ) & = \\mathcal { H } ^ { n - 1 } ( \\partial ^ { * } E \\cap \\{ z = \\bar { z } \\} ) = \\mathcal { H } ^ { n - 1 } ( ( \\partial ^ { * } E ) _ { \\bar { z } } ) \\\\ & \\leq \\mathcal { H } ^ { n - 1 } \\left ( B ^ { n - 1 } \\left ( ( 0 , r _ { \\ell } ^ { \\vee } ( \\bar { z } ) \\right ) \\right ) \\\\ & = \\mathcal { H } ^ { n - 1 } ( \\partial ^ { * } F _ { \\ell } \\cap \\{ z = \\bar { z } \\} ) \\\\ & = P ( F _ { \\ell } ; \\{ z = \\bar { z } \\} ) \\leq P ( E ; \\{ z = \\bar { z } \\} ) , \\end{align*}"}
+{"id": "7033.png", "formula": "\\begin{align*} \\boxed { D : = \\mathop { \\sup } _ { \\begin{array} { c } ( y , S ) \\in \\mathbb { R } ^ m \\times \\mathcal S ^ n \\\\ A ^ * ( y ) + S = C \\\\ S \\succcurlyeq 0 \\\\ \\end{array} } \\langle b , y \\rangle } \\end{align*}"}
+{"id": "3639.png", "formula": "\\begin{align*} p ( \\alpha ) = \\alpha - \\mathfrak { c } \\wedge ( \\chi \\mathbin { \\lrcorner } \\alpha ) \\ . \\end{align*}"}
+{"id": "8161.png", "formula": "\\begin{align*} G & = K _ { i - 1 } ( x , k - y , r - 1 ) \\frac { \\binom { n } { i - 1 } } { \\binom { k } { i - 1 } } \\Big ( \\frac { ( n - j - k ) ( j + 1 ) } { n - i - 2 j } H _ { j + 1 } ( y , n - ( i - 1 ) , k - ( i - 1 ) ) \\\\ & \\quad + \\frac { ( k - i - j + 1 ) ( n - i - j + 2 ) } { n - i - 2 j + 2 } H _ { j } ( y , n - ( i - 1 ) , k - ( i - 1 ) ) \\Big ) . \\end{align*}"}
+{"id": "3172.png", "formula": "\\begin{gather*} \\widehat { \\mathfrak { m } } ^ b _ 0 = q \\mathbf { m } _ 0 ( L , \\xi _ b ) = q W ( \\mathbf { m } _ b ) . \\\\ \\widehat { \\mathfrak { m } } ^ b _ 1 ( x ) = d _ b ( \\widehat { \\mathfrak { m } } ^ b _ 0 ) ( x ) , \\ \\ \\ x \\in H ^ 1 ( L , \\mathbb { C } ) . \\end{gather*}"}
+{"id": "7423.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\partial _ { x } ^ { k } u _ { n } \\| _ { L ^ { \\infty } _ { t } L ^ { 2 } _ { x } } & \\le M _ { k } + M _ { k } \\| \\partial _ { x } ^ { k - 1 } \\rho _ { n } \\| _ { L ^ { \\infty } _ { t } L ^ { 2 } _ { x } } \\le M _ { k } . \\end{aligned} \\end{align*}"}
+{"id": "5123.png", "formula": "\\begin{align*} \\pi ^ \\sharp = \\rho _ * \\circ \\rho ^ * : T ^ * M \\to T M \\end{align*}"}
+{"id": "3042.png", "formula": "\\begin{align*} \\int _ { ( 0 , a ) } ^ { ( b , 0 ) } 1 d s = \\sqrt { a ^ 2 + b ^ 2 } . \\end{align*}"}
+{"id": "4334.png", "formula": "\\begin{align*} S ( \\omega _ { \\sf f } \\| \\omega ) & = i \\left . \\frac { d } { d t } \\right | _ { t = 0 } ( \\Omega _ \\omega , F \\Delta ^ { i t } F \\Omega _ \\omega ) \\end{align*}"}
+{"id": "748.png", "formula": "\\begin{align*} \\mathbb E \\left ( \\int _ 0 ^ t \\norm { \\mathbb 1 _ { \\sigma \\le \\tau _ R } \\phi _ + ( \\sigma ) } _ { H ^ r } ^ 2 \\ , d \\sigma \\right ) = \\int _ 0 ^ t \\mathbb E \\left ( \\norm { \\mathbb 1 _ { \\sigma \\le \\tau _ R } \\phi _ + ( \\sigma ) } _ { H ^ r } ^ 2 \\right ) \\ , d \\sigma \\le \\int _ 0 ^ t \\mathbb E \\left ( A _ R ( \\sigma ) \\right ) \\ , d \\sigma , \\end{align*}"}
+{"id": "7719.png", "formula": "\\begin{align*} ( C _ N ^ \\infty ) _ { b a s } ( V ) = \\{ f \\in C ^ \\infty _ N ( V ) \\ ; | \\ ; X ( f ) = 0 \\ , \\ , \\forall X \\in \\Gamma _ F ( V ) \\} . \\end{align*}"}
+{"id": "8594.png", "formula": "\\begin{align*} \\partial _ z ^ 2 \\phi _ 1 & = \\frac { \\mu } { h _ b } T _ 1 ( z ) [ X , \\mathrm { D } ] \\frac { \\Delta _ X } { \\sqrt { \\mu } | \\mathrm { D } | } G . \\end{align*}"}
+{"id": "877.png", "formula": "\\begin{align*} V = \\xi ( x , t , u , v ) \\partial _ x + \\tau ( x , t , u , v ) \\partial _ t + \\eta ( x , t , u , v ) \\partial _ u + \\phi ( x , t , u , v ) \\partial _ v , \\end{align*}"}
+{"id": "3539.png", "formula": "\\begin{align*} f & \\equiv f ( r , t ) : = - \\Delta w + \\lambda w - \\alpha u - \\beta v , \\\\ g _ { 1 } & \\equiv g _ { 1 } ( r , t ) : = \\frac { u _ { r } } { \\sqrt { \\chi ( u + 1 ) } } - \\sqrt { \\chi ( u + 1 ) } w _ { r } , \\\\ g _ { 2 } & \\equiv g _ { 2 } ( r , t ) : = \\frac { v _ { r } } { \\sqrt { \\xi ( v + 1 ) } } - \\sqrt { \\xi ( v + 1 ) } w _ { r } . \\end{align*}"}
+{"id": "1808.png", "formula": "\\begin{gather*} \\mathbf { E } \\left [ | \\zeta ( s , \\mathbb { X } _ \\alpha ) | \\right ] \\leq \\left \\{ \\sum _ { n = 0 } ^ { \\infty } \\frac { 1 } { ( n + \\alpha ) ^ { 2 \\sigma } } \\right \\} ^ { 1 / 2 } \\ll \\frac { 1 } { \\sqrt { 2 \\sigma - 1 } } , \\end{gather*}"}
+{"id": "2527.png", "formula": "\\begin{align*} \\mathcal { P } _ N = \\frac { T } { 2 } \\sum _ { k = N + 1 } ^ \\infty \\| \\boldsymbol { u } _ k \\| _ { \\Omega } ^ 2 = \\frac { T } { 2 } \\sum _ { k = N + 1 } ^ \\infty \\left ( \\| \\boldsymbol { u } _ k ^ c \\| _ { \\Omega } ^ 2 + \\| \\boldsymbol { u } _ k ^ s \\| _ { \\Omega } ^ 2 \\right ) = \\| \\boldsymbol { u } - \\boldsymbol { u } _ N \\| \\end{align*}"}
+{"id": "7559.png", "formula": "\\begin{align*} Q _ T ( S ) : = \\frac { 1 } { Z _ T } E ^ { P _ T } \\big [ \\mathbf { 1 } _ S \\mathcal { E } _ T \\big ] , & & Z _ T : = E ^ { P _ T } \\big [ \\mathcal { E } _ T \\big ] . \\end{align*}"}
+{"id": "7921.png", "formula": "\\begin{align*} - 1 = \\epsilon _ x \\alpha \\mathrm { F P d i m } ( x ) + \\epsilon _ y \\alpha ^ { - 1 } \\mathrm { F P d i m } ( y ) . \\end{align*}"}
+{"id": "7108.png", "formula": "\\begin{align*} \\mathcal { I } _ { y _ 1 } = \\int _ 0 ^ \\infty \\ , 4 y ^ 2 _ 1 \\ , U ( y ^ 2 _ 1 ) \\ , U ( y ^ 2 _ 1 + y _ 3 ) \\ , e \\left ( t \\left ( \\frac { y _ 3 } { y _ 1 } \\right ) + \\frac { 2 \\sqrt { N } } { p _ 1 q \\sqrt { p _ 2 } } \\left ( { \\sqrt { m } } - { \\sqrt { m _ 1 } } \\right ) y _ 1 \\right ) d y _ 1 \\end{align*}"}
+{"id": "4192.png", "formula": "\\begin{align*} \\sigma _ { \\varepsilon } ( [ T _ { 1 } \\bullet T _ { 2 } , T _ { 3 } ] _ { \\ast } ) = \\sigma _ { \\varepsilon } ( [ \\varphi ( T _ { 1 } ) \\bullet \\varphi ( T _ { 2 } ) , \\varphi ( T _ { 3 } ) ] _ { \\ast } ) , \\end{align*}"}
+{"id": "993.png", "formula": "\\begin{align*} \\mathbb { P } ( X \\ge ( 1 - \\epsilon ) \\ell \\alpha \\ \\mathrm { a n d } \\ Y \\ge ( 1 - \\epsilon ) \\ell \\alpha ) & = 1 - \\mathbb { P } ( X < ( 1 - \\epsilon ) \\ell \\alpha \\ \\mathrm { o r } \\ Y < ( 1 - \\epsilon ) \\ell \\alpha ) \\\\ & \\ge 1 - \\mathbb { P } ( X < ( 1 - \\epsilon ) \\ell \\alpha ) - \\mathbb { P } ( X < ( 1 - \\epsilon ) \\ell \\alpha ) \\\\ & \\ge 1 - 4 e ^ { - \\frac { \\epsilon ^ 2 \\ell \\alpha } 3 } \\end{align*}"}
+{"id": "3252.png", "formula": "\\begin{align*} \\mathcal { C } = \\sum _ { \\ell = 0 } ^ n \\langle \\mathcal { C } \\rangle _ { _ \\ell } = \\langle \\mathcal { C } \\rangle _ { _ 0 } + \\langle \\mathcal { C } \\rangle _ { _ 1 } + \\cdots + \\langle \\mathcal { C } \\rangle _ { _ n } \\end{align*}"}
+{"id": "7721.png", "formula": "\\begin{align*} p ^ \\sharp _ 1 ( e ) . p ^ \\sharp _ 1 ( e ' ) = p ^ \\sharp _ 2 ( e . e ' ) . \\end{align*}"}
+{"id": "3719.png", "formula": "\\begin{align*} \\phi _ \\infty ( x ) : = \\sum _ { n = 1 } ^ \\infty a _ n \\phi _ 1 \\left ( \\frac { x - ( 4 - a _ n , 2 ^ { - 4 n } ) } { a _ n } \\right ) . \\end{align*}"}
+{"id": "5572.png", "formula": "\\begin{align*} h ( r ) = U _ k ( \\zeta _ 0 ) . \\end{align*}"}
+{"id": "1553.png", "formula": "\\begin{align*} \\Delta = y ^ 4 x ^ 2 + z f _ 5 ( x , y , z ) , \\end{align*}"}
+{"id": "4650.png", "formula": "\\begin{align*} \\mathbf { X } = \\bigotimes _ { e \\in E } X _ e , \\qquad \\quad \\mathbf { Y } = \\bigotimes _ { e ^ \\prime \\in E } Y _ { e ^ \\prime } . \\end{align*}"}
+{"id": "4349.png", "formula": "\\begin{align*} \\check { { \\bf I } } = \\left ( \\begin{array} { c c c c c } 0 & 0 & \\ldots & 0 & 1 \\\\ 0 & 0 & \\ldots & 1 & 0 \\\\ \\cdot & & & & \\cdot \\\\ \\cdot & & & & \\cdot \\\\ 1 & 0 & \\ldots & 0 & 0 \\end{array} \\right ) \\end{align*}"}
+{"id": "8309.png", "formula": "\\begin{align*} \\sigma ( Z ) = u ( Z ) ~ Z \\in ( F ) . \\end{align*}"}
+{"id": "2989.png", "formula": "\\begin{align*} \\pi _ { \\lambda ^ c } ( z ^ { \\gcd ( n , b ) } ) & = ( ( z ^ { \\gcd ( n , b ) } ) ^ { q _ b } , \\lambda ^ c ( z ^ { \\gcd ( n , b ) } ) ^ { q _ n } ) = ( z ^ b , \\lambda ^ { k n + d } z ^ n ) \\\\ & = ( ( \\lambda ^ { k \\gcd ( n , b ) } z ) ^ { q _ b } , \\lambda ^ d ( \\lambda ^ { k \\gcd ( n , b ) } z ) ^ { q _ n } ) = \\pi _ { \\lambda ^ d } ( ( \\lambda ^ k z ) ^ { \\gcd ( n , b ) } ) . \\end{align*}"}
+{"id": "2806.png", "formula": "\\begin{align*} \\mathcal { E } _ Y ( u , u ) : = \\iint _ { \\R ^ N \\times \\R ^ N } ( u ( x ) - u ( y ) ) ^ 2 \\mathcal { K } _ Y ( x , y ) \\dd x \\dd y , \\end{align*}"}
+{"id": "5535.png", "formula": "\\begin{align*} \\mathcal { C } = \\bigcap _ { \\ell = 0 } ^ { \\infty } E _ \\ell \\end{align*}"}
+{"id": "112.png", "formula": "\\begin{align*} \\mathcal { H } _ { B _ u } ( \\rho _ { \\mu } ) _ n : = \\sum _ { j = 1 } ^ n \\Big ( \\mathcal { T } _ { j , u } - \\rho _ { \\mu } \\int _ { \\mathbb { R } ^ d } w _ { 1 , B _ u } ( x _ j , y ) \\mathrm { d } y \\Big ) + \\sum _ { i < j } ^ n w _ { B _ u } ( x _ i , x _ j ) . \\end{align*}"}
+{"id": "1277.png", "formula": "\\begin{align*} \\| e ^ { i ( t - a _ { j + 1 } ) \\Delta } w ( a _ { j + 1 } ) \\| _ { S ( I _ { j + 1 } ) } \\leq & 2 C \\epsilon + \\frac 1 3 \\sum _ { k = 1 } ^ { j } \\| w \\| _ { S ( I _ k ) } \\\\ & + C \\sum _ { k = 1 } ^ { j } ( \\eta ^ { p + 1 } \\| w \\| _ { S ( I _ k ) } ^ { p - 2 } + \\eta \\| w \\| _ { S ( I _ k ) } ^ { 2 ( p - 1 ) } + \\| w \\| _ { S ( I _ k ) } ^ { 2 p - 1 } ) \\\\ \\leq & 2 C \\epsilon + 4 \\sum _ { k = 1 } ^ { j } \\gamma _ j \\end{align*}"}
+{"id": "370.png", "formula": "\\begin{align*} r \\cdot t = r \\cdot ( t _ 1 , \\ldots , t _ { \\nu } ) = ( r _ 1 \\cdot t _ 1 , \\ldots , r _ { \\nu } \\cdot t _ { \\nu } ) . \\end{align*}"}
+{"id": "16.png", "formula": "\\begin{align*} x ^ { * } = \\min _ { z \\in \\mathbb { Z } ^ { d } } \\vert x - z \\ell \\vert , \\end{align*}"}
+{"id": "8779.png", "formula": "\\begin{align*} \\mathbb { E } [ \\norm { \\nabla f ( x _ S ) } ^ 2 ] & \\leq \\frac { 4 \\delta ( 1 ) } { \\sum _ { k = 1 } ^ { T } \\eta _ k } + 4 \\sum _ { t = 1 } ^ { T } \\frac { \\eta _ t B _ { t } ^ { 2 } + \\eta _ { t } ^ 2 V _ t } { \\sum _ { k = 1 } ^ { T } \\eta _ k } , \\end{align*}"}
+{"id": "4567.png", "formula": "\\begin{align*} ( \\mathcal { B } _ 1 { \\mathbf V } _ { t } , { \\mathbf V } _ { t } ) | _ { x _ 1 = 0 } = 2 [ \\hat { u } _ 2 - \\hat \\lambda \\hat { H } _ 2 ] \\partial _ t \\dot { q } ^ + \\partial _ { t } \\partial _ 2 \\varphi + \\mbox { l . o . t } \\ , , \\end{align*}"}
+{"id": "7565.png", "formula": "\\begin{align*} e ^ { \\alpha t } f ' ( 0 ) = - \\alpha q + ( 1 - \\alpha ) p = p - \\alpha ( p + q ) = p - \\alpha < 0 . \\end{align*}"}
+{"id": "1104.png", "formula": "\\begin{align*} \\frac { \\langle u , v \\rangle } { \\lambda } - \\frac { 1 } { \\lambda } \\int _ D \\alpha ( x ) u ( x ) v ( x ) d \\mu - \\int _ D f ( x , u ( x ) ) v ( x ) d \\mu = 0 , \\end{align*}"}
+{"id": "7262.png", "formula": "\\begin{align*} | | \\varphi | | _ { ( b ) } = | | \\varphi | | _ \\infty + \\frac { | | \\varphi ' | | _ { \\infty } } { | b | } . \\end{align*}"}
+{"id": "1716.png", "formula": "\\begin{align*} G _ { \\texttt { b } } ( \\xi _ 1 , \\xi _ 2 ) = \\sum _ { m \\geq \\mu _ 1 > \\mu _ 2 \\geq 0 } e ^ { i \\mu _ 1 \\xi _ 1 + i \\mu _ 2 \\xi _ 2 } + \\frac { 1 } { 1 + q } \\sum _ { m \\geq \\mu _ 1 = \\mu _ 2 \\geq 0 } e ^ { i \\mu _ 1 \\xi _ 1 + i \\mu _ 2 \\xi _ 2 } . \\end{align*}"}
+{"id": "8901.png", "formula": "\\begin{align*} \\varphi ^ { \\vee } \\circ \\lambda _ { \\tau } \\circ \\varphi = \\begin{pmatrix} 2 \\lambda _ N & \\lambda _ N \\circ \\gamma \\\\ \\lambda _ N \\circ \\gamma & 2 \\lambda _ N \\end{pmatrix} , \\end{align*}"}
+{"id": "128.png", "formula": "\\begin{align*} f ( t ) = F ^ T T _ W W ( t ) \\end{align*}"}
+{"id": "1469.png", "formula": "\\begin{align*} \\sigma _ { u _ i } = \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { R } ^ 2 } e ^ { u _ i ( y ) } \\mathrm { d } y , \\ \\ \\ i \\in J _ 0 . \\end{align*}"}
+{"id": "4996.png", "formula": "\\begin{align*} S _ { \\Sigma } = S - 2 \\mathrm { R i c } ( \\nu , \\nu ) + 4 H ^ 2 - | h | ^ 2 . \\end{align*}"}
+{"id": "8746.png", "formula": "\\begin{align*} \\min \\Big ( \\alpha _ 0 , \\ , \\frac { d } { \\alpha _ 0 } T ^ { - \\frac { \\beta - 1 } { \\beta } } \\Big ) = \\min \\Big ( T ^ { - 1 / 2 + 1 / \\beta } , \\frac { d } { \\sqrt { T } } \\Big ) \\enspace , \\end{align*}"}
+{"id": "5336.png", "formula": "\\begin{align*} \\tanh ( x ) = \\frac { 2 } { 1 + e ^ { - 2 x } } - 1 . \\end{align*}"}
+{"id": "3772.png", "formula": "\\begin{align*} ( c , \\gamma ) : = \\int _ { X \\times X } c ( x _ 0 , x _ 1 ) d \\gamma ( x _ 0 , x _ 1 ) \\end{align*}"}
+{"id": "2240.png", "formula": "\\begin{align*} u _ k = \\eta _ k u . \\end{align*}"}
+{"id": "6962.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } & \\sum _ { k = 1 } ^ d \\sum _ { i = 1 } ^ { k - 1 } \\sum _ { j = k + 1 } ^ d \\frac { \\cos ^ 2 ( x _ k / 2 ) } { ( \\sin ( x _ k / 2 ) - \\sin ( x _ i / 2 ) + \\epsilon ^ 2 ) ( \\sin ( x _ j / 2 ) - \\sin ( x _ k / 2 ) + \\epsilon ^ 2 ) } \\\\ & = - \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ d \\sum _ { i = k + 1 } ^ { d } \\sum _ { j = i + 1 } ^ d \\frac { \\cos ^ 2 ( x _ i / 2 ) } { ( \\sin ( x _ i / 2 ) - \\sin ( x _ k / 2 ) + \\epsilon ^ 2 ) ( \\sin ( x _ j / 2 ) - \\sin ( x _ i / 2 ) + \\epsilon ^ 2 ) } . \\end{align*}"}
+{"id": "4346.png", "formula": "\\begin{align*} W ^ { ( 2 ) } _ \\omega ( { \\sf f } _ j , { \\sf f } _ k ) = \\delta _ { j k } \\ , , \\ \\ ( j , k = 1 , \\ldots , m ) \\ , . \\end{align*}"}
+{"id": "4833.png", "formula": "\\begin{align*} & b _ i b _ j = b _ j b _ i ; \\ | i - j | \\geq 2 \\\\ & b _ i b _ { i + 1 } b _ i = b _ { i + 1 } b _ i b _ { i + 1 } \\end{align*}"}
+{"id": "2988.png", "formula": "\\begin{align*} \\pi _ { \\omega } ( z ) = ( z ^ { q _ b } , \\omega z ^ { q _ n } ) . \\end{align*}"}
+{"id": "5654.png", "formula": "\\begin{align*} \\varphi ( x _ 0 : x _ 1 : x _ 2 : x _ 3 ) = ( A x _ 0 : A x _ 1 : A x _ 2 : - A x _ 3 - B ) , \\end{align*}"}
+{"id": "5549.png", "formula": "\\begin{align*} \\nabla ^ 2 \\sigma _ k ( \\zeta ) & = 0 \\quad \\mbox { i f } \\zeta \\in G _ m , \\\\ \\sigma _ k ( \\zeta ) & = \\delta _ { k , j } \\mbox { i f } \\zeta \\in C _ j , j = 1 , \\ldots , m , \\end{align*}"}
+{"id": "102.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { A } } _ k : = ( 1 - \\varepsilon ) k ^ 2 + \\rho _ z \\widehat { g } _ k . \\end{align*}"}
+{"id": "7605.png", "formula": "\\begin{align*} H _ 1 ( \\beta , N , T ) & \\le C N ^ 2 \\int _ { 0 } ^ { 2 T } \\frac { 1 } { ( 2 \\pi b ) ^ { 1 / 2 } } \\left ( \\int _ { \\delta } ^ { \\infty } e ^ { - \\beta ^ { 2 / 3 } N ^ { 2 / 3 } \\frac { a ^ 2 } { 4 b } } d a \\right ) d b . \\end{align*}"}
+{"id": "8171.png", "formula": "\\begin{align*} & [ A ^ * _ { 0 1 } , A _ { 1 0 } ] = 0 . \\end{align*}"}
+{"id": "4662.png", "formula": "\\begin{align*} e ( v ) : = \\# \\mathcal { S } ( v | s ) + \\# \\mathcal { S } ( v | t ) \\end{align*}"}
+{"id": "2708.png", "formula": "\\begin{align*} \\begin{array} { r c l } [ u \\cdot v ] ^ { \\boxtimes a } & = & [ u ] ^ { \\boxtimes a } \\cdot [ v ] ^ { \\boxtimes a } . \\end{array} \\end{align*}"}
+{"id": "264.png", "formula": "\\begin{align*} \\lim _ { c \\to + \\infty } e ^ { - c | J | } V ^ { } _ { \\varepsilon J } ( \\xi ; g ^ { ( c ) } ) & = V ^ { } _ { \\varepsilon J } ( \\xi ; g ) , \\\\ \\lim _ { c \\to + \\infty } e ^ { - c p } U ^ { } _ { K , p } ( \\xi ; g ^ { ( c ) } ) & = U ^ { } _ { K , p } ( \\xi ; g ) , \\\\ \\lim _ { c \\to + \\infty } e ^ { - c \\ell } E ^ { } _ \\ell ( x + c \\rho _ L ) & = E ^ { } _ \\ell ( x ) , \\end{align*}"}
+{"id": "559.png", "formula": "\\begin{align*} \\tau _ R < \\infty \\implies \\norm { \\mathbf u ^ R } _ { \\widetilde { \\mathbf X } ^ { \\mathbf s , b } ( 0 , \\tau _ R ) } ^ 2 = R , \\end{align*}"}
+{"id": "7796.png", "formula": "\\begin{align*} \\alpha ^ { } : = \\alpha + \\frac { 1 } { 2 } ( \\alpha ^ { } - \\xi ^ { i } \\widetilde { \\xi } _ i ^ { } ) , \\widetilde { \\xi } _ i ^ { } : = \\widetilde { \\xi } _ i - \\widetilde { \\xi } _ i ^ { } \\ , . \\end{align*}"}
+{"id": "1454.png", "formula": "\\begin{align*} \\sum \\limits _ { i \\in I } \\left ( n _ { i , { j } } - n _ { i + 1 , { j } } \\right ) ^ { 2 } = 2 n _ { { j } , { j } } . \\footnote { H e r e $ \\sigma _ { \\ell } = \\sigma _ i $ i f $ \\ell \\equiv i \\ ( \\mathrm { m o d } \\ n + 1 ) $ . T h e r e f o r e , $ \\sigma _ { n + 2 } = \\sigma _ 1 $ a n d $ \\sigma _ { 0 } = \\sigma _ { n + 1 } $ , $ n _ { n + 2 , j } = n _ { 1 , j } $ a n d $ n _ { 0 , j } = n _ { n + 1 , j } $ f o r a n y $ j \\in I $ . } \\end{align*}"}
+{"id": "3188.png", "formula": "\\begin{align*} S = \\max _ { m \\leq j < k } \\Delta _ { j : k } / \\Delta _ j \\end{align*}"}
+{"id": "4947.png", "formula": "\\begin{align*} \\Psi ( \\tau ) = e ^ { - \\frac { 2 \\pi i \\tau } { 8 } } \\cdot \\dfrac { \\eta ( 2 \\tau ) ^ 2 } { \\eta ( \\tau ) } , \\end{align*}"}
+{"id": "8846.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } t ^ { - \\frac { 1 } { \\beta } - 1 } \\leq 1 + \\int _ { 1 } ^ { T } u ^ { - \\frac { 1 } { \\beta } - 1 } \\d u = 1 - \\beta u ^ { - \\frac { 1 } { \\beta } } \\Big | _ { 1 } ^ { T } = 1 + \\beta - \\beta T ^ { - \\frac { 1 } { \\beta } } \\leq 1 + \\beta \\enspace . \\end{align*}"}
+{"id": "1167.png", "formula": "\\begin{align*} & m \\rq _ { 1 , 0 } ( x , y ) = m _ 1 ( x , y ) , \\\\ & m \\rq _ { 2 , 0 } ( x , y y ) = m _ 2 ( x , y ) . \\end{align*}"}
+{"id": "5025.png", "formula": "\\begin{align*} \\int _ { M ^ * } ( 4 \\pi ) ^ { - 1 } e ^ { - H } d g ^ * = \\int _ 0 ^ \\infty ( 4 \\pi ) ^ { - 1 } e ^ { - H ( r ) } \\ , 2 \\pi q ( r ) d r = 1 \\end{align*}"}
+{"id": "3752.png", "formula": "\\begin{align*} \\sigma ( \\mathbb { A } ^ \\alpha ) = \\{ \\lambda ^ \\alpha : \\lambda \\in \\sigma ( \\mathbb { A } ) \\} = \\{ \\lambda ^ \\alpha : \\lambda ^ 3 \\in \\sigma ( A ) \\} \\end{align*}"}
+{"id": "8091.png", "formula": "\\begin{align*} \\left \\| \\uppercase \\expandafter { \\romannumeral 1 } _ { 2 } \\right \\| _ { L _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } . \\end{align*}"}
+{"id": "7526.png", "formula": "\\begin{align*} \\lambda _ + ^ 2 = \\nu p ' + \\theta ( 1 - \\nu ) \\quad \\textrm { w i t h } \\nu = n / r \\in ( 0 , 1 ) \\ , . \\end{align*}"}
+{"id": "8385.png", "formula": "\\begin{align*} N = m _ { 1 } ^ { k } + m _ { 2 } ^ { k } + \\cdots + m _ { s } ^ { k } \\end{align*}"}
+{"id": "7291.png", "formula": "\\begin{align*} C ^ { 0 ' } ( \\sigma ) \\leq k + n + O ( \\log ( n ) ) = k + \\log ( 1 / \\delta ) + O ( \\log \\log ( 1 / \\delta ) ) . \\end{align*}"}
+{"id": "5960.png", "formula": "\\begin{align*} ( D _ { t _ 3 } + i A ^ \\omega _ F ) u _ f & = v , \\ ; \\ \\left . u _ f \\right \\vert _ { t _ 3 = 0 } = f , \\\\ ( D _ { t _ 3 } + i \\widetilde { E } - i A ^ \\omega _ F ) v & = w \\in C ^ \\infty ( [ 0 , T ] ; \\mathcal { D } ' ( \\mathbb { R } ^ 2 ) ) , \\end{align*}"}
+{"id": "6158.png", "formula": "\\begin{align*} ( \\mathbf { k } ) ^ + = \\{ \\mathbf { k } \\} , & & ( \\mathbf { k } ) ^ - = \\emptyset . \\end{align*}"}
+{"id": "3444.png", "formula": "\\begin{align*} u ^ { * * } ( x ) = \\max _ { p \\in \\Delta ^ \\vee } \\langle p , x \\rangle - u ^ * ( p ) . \\end{align*}"}
+{"id": "830.png", "formula": "\\begin{align*} | \\eta ( z ) | + ( 1 - \\epsilon ) | 1 - \\eta ( z ) | & = | \\Psi _ { \\epsilon } ( \\tilde { h } ( z ) ) | + ( 1 - \\epsilon ) | 1 - \\Psi _ { \\epsilon } ( \\tilde { h } ( z ) ) | \\\\ & \\leq 1 . \\end{align*}"}
+{"id": "1634.png", "formula": "\\begin{align*} Q '^ { X + 2 a } A '^ { X + a } T '^ { C - X } = Q '^ { 2 a } A '^ a Q '^ X A '^ X T '^ { C - X } = a ^ a Q ^ X T ^ { C - X } . \\end{align*}"}
+{"id": "2991.png", "formula": "\\begin{align*} r _ I ( k , z ) = \\psi ( z ^ { q _ b } ) = z ^ { q _ b a } = z ^ { m / \\gcd ( n , b ) } s _ I ( k , z ) = \\lambda ^ k z ^ { q _ n } = \\lambda ^ k z ^ { n / \\gcd ( n , b ) } . \\end{align*}"}
+{"id": "2223.png", "formula": "\\begin{align*} \\langle \\varphi , \\psi \\rangle : = \\underset { j = 1 } { \\overset { 3 } { \\sum } } \\underset { e _ j } { \\int } \\varphi _ j ( x ) \\cdot \\psi _ j ^ * ( x ) d x . \\end{align*}"}
+{"id": "8389.png", "formula": "\\begin{align*} 1 < c < \\frac { 2 4 7 } { 2 3 8 } , \\gamma = \\frac { 1 } { c } , P = 1 0 ^ { - 9 } N ^ { \\gamma } , \\delta = \\frac { 2 4 7 \\gamma - 2 3 8 } { 2 2 5 } - \\varepsilon , \\end{align*}"}
+{"id": "8285.png", "formula": "\\begin{align*} \\begin{cases} \\bar p _ \\tau = \\bar p _ { x x } , & ( x , \\tau ) \\in ( 0 , 1 ) \\times ( 0 , \\infty ) , \\\\ \\bar p _ x ( 0 , \\tau ) = c \\bar p ( 0 , \\tau ) , & \\tau \\in ( 0 , \\infty ) , \\\\ \\bar p _ x ( 1 , \\tau ) = 0 , & \\tau \\in ( 0 , \\infty ) , \\\\ \\bar p ( x , 0 ) = \\bar p _ 0 ( x ) , & x \\in ( 0 , 1 ) , \\end{cases} \\end{align*}"}
+{"id": "3426.png", "formula": "\\begin{align*} \\norm { s } _ { F S , k } ( x ) = \\frac { | s | ( x ) } { ( \\max _ { 1 \\leq i \\leq N } { | s _ i | ( x ) e ^ { c _ i } ) } ^ { 1 / k } } . \\end{align*}"}
+{"id": "4687.png", "formula": "\\begin{align*} \\mu _ j ^ n ( S , W ) = \\begin{cases} 0 & j \\neq 4 , \\\\ d _ n & j = 4 . \\end{cases} \\end{align*}"}
+{"id": "3214.png", "formula": "\\begin{align*} \\Delta ^ { \\mathrm { C R A I G } } _ { { \\ell : k } } \\equiv \\sum _ { j = { \\ell } } ^ { { k } } \\zeta ^ 2 _ { j + 1 } \\approx \\| x - x _ { { \\ell } } \\| ^ 2 . \\end{align*}"}
+{"id": "8394.png", "formula": "\\begin{align*} \\Sigma _ { j } = \\sum _ { d \\leq D } \\lambda ( d ) \\sum _ { P < p \\leq 2 P } ( \\log p ) \\psi \\left ( - \\frac { 1 } { d } \\left ( N + j - \\left [ p ^ { c } \\right ] \\right ) ^ { \\gamma } \\right ) , j = 0 , 1 . \\end{align*}"}
+{"id": "2854.png", "formula": "\\begin{align*} { \\mathrm m } _ \\omega ( e ^ { i \\xi } ) M _ \\lambda ^ { ( c ) } ( \\xi ) = U _ { \\lambda , \\omega } ( t ) M _ \\lambda ^ { ( c ) } ( \\xi ) + \\sum _ { \\substack { \\nu \\in W _ 0 \\omega \\\\ \\lambda + \\nu \\in { P _ c } } } V _ { \\lambda , \\nu } ( t ) M _ { \\lambda + \\nu } ^ { ( c ) } ( \\xi ) . \\end{align*}"}
+{"id": "7792.png", "formula": "\\begin{align*} \\lim _ { z \\to 0 } z K _ 1 ( 2 \\pi \\tau _ 2 n z ) = \\frac { 1 } { 2 \\pi n \\tau _ 2 } , \\lim _ { z \\to 0 } z ^ 2 K _ 1 ( 2 \\pi \\tau _ 2 n z ) = 0 , \\lim _ { z \\to 0 } z ^ 2 K _ 0 ( 2 \\pi \\tau _ 2 n z ) = 0 \\ , . \\end{align*}"}
+{"id": "980.png", "formula": "\\begin{align*} x + y \\sqrt d = \\pm \\prod _ { i = 1 } ^ s ( a _ i + b _ i \\sqrt d ) \\end{align*}"}
+{"id": "3053.png", "formula": "\\begin{align*} \\C \\delta : \\delta : = \\frac { 4 } { \\sqrt { 3 } } \\Big [ \\frac { 1 } { 2 } \\alpha _ 1 \\Big ( | e _ 1 ^ * \\delta e _ 1 | ^ 2 + | \\nu ^ * \\delta \\nu | ^ 2 + | \\eta ^ * \\delta \\eta | ^ 2 \\Big ) + 3 \\alpha _ 2 | \\mathrm { t r } \\ , \\delta | ^ 2 \\Big ] \\ , , \\qquad \\textrm { f o r e v e r y } \\delta \\in \\R ^ { 2 \\times 2 } \\ , . \\end{align*}"}
+{"id": "6559.png", "formula": "\\begin{align*} \\frac { \\det S ( z ) } { \\prod _ { l \\notin K } ( z - \\sigma _ l ) } = g ( z ) \\prod _ { l \\in K } ( z - z _ l ) , \\end{align*}"}
+{"id": "8491.png", "formula": "\\begin{align*} | D g | ( a , b ) = \\sup \\left \\{ \\sum _ { i = 1 } ^ { M } | g ( x _ { i + 1 } ) - g ( x _ { i } ) | : \\ a < x _ { 1 } < x _ { 2 } < \\cdots < x _ { M } < b \\right \\} \\end{align*}"}
+{"id": "2104.png", "formula": "\\begin{align*} C _ { t _ 0 , u _ 0 } : = \\left \\{ ( t , x ) : \\ ; u = \\frac { t + x } { 2 } = u _ 0 , \\ , 0 \\leq t \\leq t _ 0 \\right \\} . \\end{align*}"}
+{"id": "2536.png", "formula": "\\begin{align*} r _ i < r _ { \\sigma ( i ) } \\textrm { o r } ( r _ i = r _ { \\sigma ( i ) } \\beta _ { \\tau ( i ) } \\leq \\beta _ { \\sigma ( i ) } ) \\end{align*}"}
+{"id": "5897.png", "formula": "\\begin{align*} \\alpha _ n ( T , \\phi ) = \\Delta _ n ( \\phi ) \\circ u _ n \\bigl ( \\alpha ( T ) \\bigr ) \\circ \\lambda . \\end{align*}"}
+{"id": "6310.png", "formula": "\\begin{align*} i \\partial _ t v _ \\lambda + \\partial _ x g _ { [ < \\lambda ^ \\sigma ] } \\partial _ x v _ \\lambda = f _ \\lambda , v _ \\lambda ( 0 ) = v _ { 0 , \\lambda } , \\end{align*}"}
+{"id": "2277.png", "formula": "\\begin{align*} 0 & = \\sum _ { 0 \\leq k \\leq n - 1 } f \\left ( \\frac { k + 1 / 2 } { n } \\right ) \\geq n X + \\max _ { 0 \\leq k \\leq n - 1 } f \\left ( \\frac { k + 1 / 2 } { n } \\right ) \\end{align*}"}
+{"id": "1258.png", "formula": "\\begin{align*} \\| v _ { n , T } ( 0 ) - \\phi _ n \\| _ { \\dot { H } ^ 1 } & = \\| [ e ^ { - i t T \\Delta } \\chi _ n P _ n e ^ { i T \\Delta } - 1 ] \\phi \\| _ { \\dot { H } ^ 1 } \\\\ & = \\| [ \\chi _ n P _ n - 1 ] e ^ { i T \\Delta } \\phi \\| _ { \\dot { H } ^ 1 } \\to 0 \\end{align*}"}
+{"id": "565.png", "formula": "\\begin{align*} B : = b ' + 1 - \\varepsilon > \\frac 1 2 . \\end{align*}"}
+{"id": "1945.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( u u _ x , u _ { x x } \\right ) & \\leq \\| u \\| _ { L ^ 2 ( I ) } \\| u _ x \\| _ { L ^ \\infty ( I ) } \\| u _ { x x } \\| _ { L ^ 2 ( I ) } \\\\ & \\leq \\sqrt { 2 } \\ , \\| u \\| _ { L ^ 2 ( I ) } \\| u _ x \\| _ { L ^ 2 ( I ) } ^ \\frac { 1 } { 2 } \\| u _ { x x } \\| ^ \\frac { 3 } { 2 } _ { L ^ 2 ( I ) } \\\\ & \\leq \\epsilon ^ { - 3 } \\ , \\| u \\| ^ 4 _ { L ^ 2 ( I ) } \\| u _ x \\| ^ 2 _ { L ^ 2 ( I ) } + \\frac { 3 \\epsilon } { 4 } \\| u _ { x x } \\| ^ 2 _ { L ^ 2 ( I ) } . \\end{aligned} \\end{align*}"}
+{"id": "8006.png", "formula": "\\begin{align*} \\Bigl ( ( v _ 0 , \\dots , v _ M ) \\Bigr ) = \\Bigl ( v _ 1 - v _ 0 \\ \\cdots \\ v _ M - v _ 0 \\Bigr ) = \\begin{pmatrix} \\begin{array} { l } 1 ^ T \\\\ \\mathrm { V } \\end{array} \\end{pmatrix} - 1 , \\end{align*}"}
+{"id": "6352.png", "formula": "\\begin{align*} j ^ 6 _ { \\lambda } ( \\xi _ 1 , \\xi _ 2 , \\xi _ 3 , \\xi _ 4 , \\xi _ 5 , \\xi _ 6 ) = b _ \\lambda ( \\xi _ 1 ) b _ \\lambda ( \\xi _ 2 ) b _ \\lambda ( \\xi _ 3 ) b _ \\lambda ( \\xi _ 4 ) b _ \\lambda ( \\xi _ 5 ) b _ \\lambda ( \\xi _ 6 ) + \\tilde j ^ 6 _ \\lambda ( \\xi _ 1 , \\xi _ 2 , \\xi _ 3 , \\xi _ 4 , \\xi _ 5 , \\xi _ 6 ) , \\end{align*}"}
+{"id": "7774.png", "formula": "\\begin{align*} h ( t ) = \\frac { 1 } { 6 } k _ { a b c } t ^ a t ^ b t ^ c , \\end{align*}"}
+{"id": "5067.png", "formula": "\\begin{align*} m ( p ) = \\int _ { 0 } ^ { \\infty } t ^ { p - 1 } e ( t ) d t , p \\ge 0 . \\end{align*}"}
+{"id": "8017.png", "formula": "\\begin{align*} T ^ H _ { ( - i _ k ) } ( t ) = \\bigl ( g _ k ^ H \\bigr ) ^ { - 1 } \\Bigl ( X ^ H ( t ) \\Bigr ) = \\Biggl ( \\bigl ( g _ k ^ H \\bigr ) _ h ^ { - 1 } \\Bigl ( X ^ H ( t ) \\Bigr ) \\Biggr ) _ { \\substack { h = 0 , \\dots , H \\\\ h \\not = k } } = \\Biggl [ v _ { i } ^ H - v _ { i _ k } ^ H \\Biggr ] ^ { - 1 } _ { \\substack { i \\in I _ H \\\\ i \\not = i _ k } } \\ , \\Bigl ( X ^ H ( t ) - v _ { i _ k } ^ H t \\Bigr ) . \\end{align*}"}
+{"id": "1269.png", "formula": "\\begin{align*} \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| e ^ { i t \\Delta } ( u _ { 0 , n } - u _ { n } ^ J ( 0 ) ) \\| _ { S ( \\R ) } \\leq \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| e ^ { i t \\Delta } w _ n ^ J \\| _ { S ( \\R ) } = 0 . \\end{align*}"}
+{"id": "2139.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d t } \\mathcal { I } ( t ) \\geq & ~ { } \\frac { 1 - | v | - \\delta } { \\omega ( t ) } \\int \\rho ' \\left ( \\frac { x - v t } { \\omega ( t ) } \\right ) \\hat e ( t , x ) - \\frac { C _ { \\delta } } { t \\log ^ 2 t } . \\end{aligned} \\end{align*}"}
+{"id": "2435.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\int _ { \\R ^ d } \\ 1 _ { \\{ | \\xi | ^ 2 - V ( x ) < 0 \\} } \\ , { \\rm d } \\xi \\ , { \\rm d } x = C _ { { \\rm c l } , d } \\int _ { \\R ^ d } [ V ( x ) ] _ + ^ { \\frac { d } { 2 } } \\ , { \\rm d } x . \\end{align*}"}
+{"id": "2697.png", "formula": "\\begin{align*} X + \\sqrt { 2 } Y & = \\left ( \\frac { \\alpha _ 1 + \\alpha _ 2 } { 2 } + \\frac { \\sqrt { 2 } } { 2 } ( \\beta _ 1 + \\beta _ 2 ) \\right ) ^ 2 + \\left ( \\frac { \\alpha _ 1 - \\alpha _ 2 } { 2 } + \\frac { \\sqrt { 2 } } { 2 } ( \\beta _ 1 - \\beta _ 2 ) \\right ) ^ 2 \\\\ & + \\left ( \\frac { \\alpha _ 3 + \\alpha _ 4 } { 2 } + \\frac { \\sqrt { 2 } } { 2 } ( \\beta _ 3 + \\beta _ 4 ) \\right ) ^ 2 + \\left ( \\frac { \\alpha _ 3 - \\alpha _ 4 } { 2 } + \\frac { \\sqrt { 2 } } { 2 } ( \\beta _ 3 - \\beta _ 4 ) \\right ) ^ 2 . \\end{align*}"}
+{"id": "3546.png", "formula": "\\begin{align*} \\varphi ( \\epsilon ) : = \\int _ { 0 } ^ { 1 } \\rho ^ { N - 1 } ( \\rho ^ { 2 } + \\epsilon ) ^ { - \\frac { N } { 2 } } d \\rho \\epsilon > 0 . \\end{align*}"}
+{"id": "236.png", "formula": "\\begin{align*} \\mathbb { C } _ { } ^ n : = \\{ \\xi \\in \\mathbb { C } ^ n \\mid 2 \\xi _ j \\not \\in \\mathbb { Z } , \\ , \\xi _ j \\pm \\xi _ k \\not \\in \\mathbb { Z } \\} , \\end{align*}"}
+{"id": "6924.png", "formula": "\\begin{align*} \\gamma _ { n + 1 } < \\gamma _ { n + 3 } < u _ 2 ^ * < \\gamma _ { n + 4 } < \\gamma _ { n + 2 } n = 2 k , \\ k \\in \\mathbb N . \\end{align*}"}
+{"id": "787.png", "formula": "\\begin{align*} U ( x ) = C ( 1 + | x | ^ { \\frac { p } { p - 1 } } ) ^ { s - \\frac { N } { p } } \\end{align*}"}
+{"id": "6043.png", "formula": "\\begin{align*} i i i ) 2 \\overline { b } - \\int _ { \\mathbb { R } ^ d } ( 2 \\bar { c } ( z ) + \\bar { c } ^ { 2 } ( z ) ) \\mu ( d z ) : = \\theta > 0 . \\end{align*}"}
+{"id": "1449.png", "formula": "\\begin{align*} \\mathcal { P } _ 3 = - 4 \\pi \\sum \\limits _ { i \\in I } \\sigma ^ k _ { i } ( r _ k ) + o ( 1 ) . \\end{align*}"}
+{"id": "689.png", "formula": "\\begin{align*} Z ^ { \\mathbf s , b } ( S , T ) = \\mathbb L ^ 2 \\left ( \\Omega , \\mathbf X ^ { \\mathbf s , b } ( S , T ) \\cap C \\left ( [ S , T ] , \\mathbf H ^ { \\mathbf s } \\right ) \\right ) \\end{align*}"}
+{"id": "5140.png", "formula": "\\begin{align*} \\Delta _ t = t - U _ t . \\end{align*}"}
+{"id": "2760.png", "formula": "\\begin{align*} G _ { \\mu } ( z ) \\approx \\frac { r _ B ^ 2 } { N } \\sum _ { j = 0 } ^ { N - 1 } \\frac { g ' ( r _ B \\xi _ N ^ j ) \\xi _ N ^ { 2 j } } { g ( r _ B \\xi _ N ^ j ) - z } . \\end{align*}"}
+{"id": "6488.png", "formula": "\\begin{align*} ( L \\Phi ) _ { e _ \\pm } = \\ell _ e \\Phi _ { e _ \\pm } . \\end{align*}"}
+{"id": "1345.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\delta _ { ( X , \\Delta ) } ( \\epsilon A - ( K _ X + \\Delta ) ) = 2 \\frac { ( \\max _ { p \\in \\mathbb { P } ^ 1 } \\mathrm { o r d } _ p ( B _ { \\mathbb { P } ^ 1 } ) - 1 ) } { u } = \\delta _ { ( \\mathbb { P } ^ 1 , B _ { \\mathbb { P } ^ 1 } ) } ( - K _ { \\mathbb { P } ^ 1 } - B _ { \\mathbb { P } ^ 1 } - M _ { \\mathbb { P } ^ 1 } ) \\end{align*}"}
+{"id": "7101.png", "formula": "\\begin{align*} \\mathcal { J } ^ { + } _ 1 ( N _ 0 \\xi , m , q ) = \\int _ 0 ^ \\infty S ( y ) \\ , e \\left ( P ( y ) \\right ) , \\end{align*}"}
+{"id": "6941.png", "formula": "\\begin{align*} \\dot { x } = u \\end{align*}"}
+{"id": "7735.png", "formula": "\\begin{align*} g ( e ) = \\left \\lbrace \\begin{array} { l l l l l l l l l } ( 1 , 0 ) & & & \\rm i f ~ \\it e \\in E ( S _ { \\rm 1 } ) - E ( S _ { \\rm 2 } ) ; \\\\ ( 0 , 1 ) & & & \\rm i f ~ \\it e \\in E ( S _ { \\rm 2 } ) - E ( S _ { \\rm 1 } ) ; \\\\ ( 1 , 1 ) & & & \\rm i f ~ \\it e \\in E ( S _ { \\rm 1 } ) \\cap E ( S _ { \\rm 2 } ) . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "5276.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { j = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { t } ( x _ { j , t } ) ] _ + \\| \\le n \\varepsilon _ 1 G _ 2 + \\Big ( \\frac { \\varepsilon _ { 5 } } { 1 - c } + n \\varepsilon _ { 6 } \\varepsilon _ { 7 } \\Big ) T ^ { 1 - c } , \\end{align*}"}
+{"id": "6632.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ 2 } ( \\rho _ { ( 1 ) , \\infty } ^ { \\rm O C P } ( \\mathbf r ; Q ) - 1 ) | \\mathbf r | ^ 2 \\ , d \\mathbf r = - { 2 \\over \\pi \\beta } \\Big ( ( 1 - \\beta / 4 ) + ( \\beta / 4 ) Q \\Big ) Q . \\end{align*}"}
+{"id": "5729.png", "formula": "\\begin{align*} H ^ s u ( x , t ) = - V ( x , t ) u ( x , t ) . \\end{align*}"}
+{"id": "8588.png", "formula": "\\begin{align*} \\mathcal { L } _ { 1 } ^ { \\mu } [ \\beta b ] \\nabla _ X \\psi = - \\frac { 1 } { \\beta } \\sinh { ( \\beta b ( X ) \\sqrt { \\mu } | \\mathrm { D } | ) } \\mathrm { s e c h } ( \\sqrt { \\mu } | \\mathrm { D } | ) \\dfrac { 1 } { \\sqrt { \\mu } | \\mathrm { D } | } \\nabla _ X \\psi , \\end{align*}"}
+{"id": "4593.png", "formula": "\\begin{align*} c _ { M _ 2 , M _ 1 } ^ { } ( \\mathcal { Y } ^ { M _ 2 , M _ 1 } ( m _ 2 , z ) \\ , m _ 1 ) = \\mathrm e ^ { z L _ { - 1 } } \\mathcal { Y } ^ { M _ 1 , M _ 2 } ( m _ 1 , - z ) \\ , m _ 2 \\ , . \\end{align*}"}
+{"id": "2712.png", "formula": "\\begin{align*} A = ( ( 1 , 1 , 1 , 3 , 2 ) , ( 0 , 2 , 1 , 3 , 2 ) , ( 0 , 1 , 2 , 3 , 2 ) , ( 0 , 1 , 2 , 2 , 3 ) , ( 0 , 1 , 1 , 3 , 3 ) ) . \\end{align*}"}
+{"id": "1518.png", "formula": "\\begin{align*} g _ i ^ k g _ j ^ l ( T _ s ^ m ) \\ , = \\ , g _ i ^ k ( T _ { \\sigma _ j ( s ) } ^ { \\alpha _ l ( m ) } ) = \\ , T _ { \\sigma _ i \\sigma _ j ( s ) } ^ { \\alpha _ k \\alpha _ l ( m ) } \\\\ g _ { \\sigma _ i ( j ) } ^ { \\alpha _ k ( l ) } \\ , g _ { \\gamma _ j ( i ) } ^ { \\beta _ l ( k ) } ( T _ s ^ m ) = g _ { \\sigma _ i ( j ) } ^ { \\alpha _ k ( l ) } ( T _ { \\sigma _ { \\gamma _ j ( i ) } ( s ) } ^ { \\alpha _ { \\beta _ l ( k ) } ( m ) } ) \\ , = \\ , T _ { \\sigma _ { \\sigma _ i ( j ) } \\sigma _ { \\gamma _ j ( i ) } ( s ) } ^ { \\alpha _ { \\alpha _ k ( l ) } \\alpha _ { \\beta _ l ( k ) } ( m ) } \\end{align*}"}
+{"id": "5321.png", "formula": "\\begin{align*} g ^ { ( n + 1 ) } ( x ) = g ^ { ( n ) } ( x ) + 2 ^ { - 2 n - 3 } ( h ^ { ( n + 2 ) } ( x ) - 1 ) . \\end{align*}"}
+{"id": "8862.png", "formula": "\\begin{align*} \\delta _ { t + 1 } \\leq \\left ( 1 - \\frac { c } { t } \\right ) \\delta _ { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { t ^ { p _ { i } + 1 } } \\enspace , \\end{align*}"}
+{"id": "5984.png", "formula": "\\begin{align*} a _ { \\varepsilon _ 1 } ( u , u ) = a _ { \\varepsilon _ 2 } ( u , u ) < ( \\lambda + \\delta ) ( u , u ) _ { L ^ 2 ( M , e ^ \\phi d \\mu _ g ) } , u \\in L . \\end{align*}"}
+{"id": "4402.png", "formula": "\\begin{align*} \\hat { H } _ 2 ^ { \\pm } \\partial _ 2 \\varphi - \\dot { H } ^ { \\pm } _ N \\mp \\varphi \\partial _ 1 \\hat { H } ^ { \\pm } _ N = g ^ { \\pm } _ 3 \\\\ \\Gamma _ { T } . \\end{align*}"}
+{"id": "376.png", "formula": "\\begin{align*} L _ { \\phi _ { \\mu } , R _ { \\mu } } ( t _ { \\mu ' } ) = \\int L ( t ) \\phi _ { \\mu } ( R _ { \\mu } \\cdot t _ { \\mu } ) d t _ { \\mu } \\end{align*}"}
+{"id": "7919.png", "formula": "\\begin{align*} - 1 / 2 = a _ x \\mathrm { F P d i m } ( x ) + a _ y \\mathrm { F P d i m } ( y ) . \\end{align*}"}
+{"id": "2543.png", "formula": "\\begin{align*} A ^ + = R ^ + C ^ + = R ^ \\mathrm { T } ( R R ^ \\mathrm { T } ) ^ { - 1 } ( C ^ \\mathrm { T } C ) ^ { - 1 } C ^ \\mathrm { T } = R ^ \\mathrm { T } ( C ^ \\mathrm { T } A R ^ \\mathrm { T } ) ^ { - 1 } C ^ \\mathrm { T } \\end{align*}"}
+{"id": "3162.png", "formula": "\\begin{align*} d ^ d z ^ { d - 1 } & = \\lambda / z _ k \\quad \\hbox { i f } 0 \\leq k \\leq d - 1 , \\\\ z _ k & = \\lambda \\quad \\quad \\ \\hbox { i f } d \\leq k \\leq n . \\end{align*}"}
+{"id": "7985.png", "formula": "\\begin{align*} \\Big ( ( \\mathfrak { s } _ 1 ) ^ { d - \\mathfrak { d } _ 1 } , \\dots , ( \\mathfrak { s } _ 1 ) ^ { \\mathfrak { d } _ 1 - \\mathfrak { d } _ 1 } , 1 , \\dots , 1 \\Big ) : = \\mathfrak { D } ( \\mathfrak { d } _ 1 , \\mathfrak { s } _ 1 ) \\end{align*}"}
+{"id": "5430.png", "formula": "\\begin{align*} \\frac { 1 } { k } D _ z p _ T ( \\mathbf { y } ) & = \\sum _ { a + b + c = k - 1 } x _ u ^ a x _ v ^ b x _ w ^ c \\binom { k - 1 } { a , b , c } d _ T ( z , u , v , w ) \\\\ & + \\sum _ { b + c = k - 1 } x _ v ^ b x _ w ^ c \\binom { k - 1 } { b , c } ( d _ T ( z , v , w ) - d _ T ( z , u , v , w ) ) \\\\ & + x _ v ^ { k - 1 } ( d _ T ( z , v ) - d _ T ( z , v , w ) ) \\\\ & = \\alpha ( x _ u + x _ v + x _ w ) ^ { k - 1 } - ( x _ v + x _ w ) ^ { k - 1 } - x _ v ^ { k - 1 } \\\\ & = 0 - ( - 1 ) ^ { k - 1 } - ( - \\zeta ) ^ { k - 1 } = 0 . \\end{align*}"}
+{"id": "5780.png", "formula": "\\begin{align*} m _ + = \\lim _ { \\mu _ 1 \\searrow 0 } \\lim _ { L \\to \\infty } \\mathbb E \\langle m ^ 1 \\rangle _ \\beta = 0 , \\end{align*}"}
+{"id": "8053.png", "formula": "\\begin{align*} \\left < T ( f ) , g \\right > = \\int \\mathcal { K } ( x , y ) f ( y ) g ( x ) d x d y \\end{align*}"}
+{"id": "5244.png", "formula": "\\begin{align*} \\Big ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { t } ( x _ { i , t } ) ] _ + \\| \\Big ) ^ 2 \\le \\frac { T } { n } \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { t } ( x _ { i , t } ) ] _ + \\| ^ 2 , \\end{align*}"}
+{"id": "6296.png", "formula": "\\begin{align*} d _ \\lambda ^ 2 = \\| v _ \\lambda ( 0 ) \\| _ { L ^ 2 } ^ 2 + \\sup _ { x _ 0 } \\| v _ \\lambda f _ \\lambda ^ { x _ 0 } \\| _ { L ^ 1 } . \\end{align*}"}
+{"id": "8162.png", "formula": "\\begin{align*} G & = \\frac { ( n - j - k ) ( j + 1 ) } { n - i - 2 j } q _ { i - 1 , j + 1 } ( x , y ) + \\frac { ( k - i - j + 1 ) ( n - i - j + 2 ) } { n - i - 2 j + 2 } q _ { i - 1 , j } ( x , y ) . \\end{align*}"}
+{"id": "1880.png", "formula": "\\begin{align*} \\rho ( t , x ) = \\int _ { \\R } f ( t , x , v ) \\ , { \\rm d } v \\mbox { a n d } V ( t , x ) = \\frac { 1 } { \\rho } \\int _ { \\R } f ( t , x , v ) v \\ , { \\rm d } v . \\end{align*}"}
+{"id": "3195.png", "formula": "\\begin{align*} \\left \\Vert x - x _ { k - 1 } \\right \\Vert _ { A ^ { T } A } ^ { 2 } - \\left \\Vert x - x _ { k } \\right \\Vert _ { A ^ { T } A } ^ { 2 } = \\phi _ { k } ^ { 2 } , \\end{align*}"}
+{"id": "3733.png", "formula": "\\begin{align*} \\mathbf { u } = \\left [ \\begin{smallmatrix} u \\\\ 0 \\\\ - u \\end{smallmatrix} \\right ] \\end{align*}"}
+{"id": "3295.png", "formula": "\\begin{align*} \\| x g _ { _ \\Sigma } \\| _ 2 ^ 2 = \\int _ { \\mathbb { R } } x ^ 2 g ^ 2 _ { _ \\Sigma } ( x ) d x = \\int _ { \\mathbb { R } } x ^ 2 f _ { _ \\Sigma } ( x ) d x = ( m _ 2 ) _ { _ \\Sigma } . \\end{align*}"}
+{"id": "947.png", "formula": "\\begin{align*} V _ i = \\xi _ i \\partial _ x + \\tau _ i \\partial _ t + \\sum _ { j = 1 } ^ s \\eta ^ j _ i \\partial _ { u _ j } , ~ ~ i = 1 , \\cdots , n . \\end{align*}"}
+{"id": "7156.png", "formula": "\\begin{align*} r _ { 1 } ( H _ { 1 } ) = ( k _ { 0 , 2 , 1 , 0 } - k _ { 1 , 1 , 1 , 0 } + ( h - \\alpha _ { 1 } ) k _ { 1 , 2 , 1 , 1 } - 2 ( h - \\alpha _ { 1 } ) k _ { 2 , 1 , 2 , 0 } ) \\dfrac { 1 } { x _ { 1 } } x _ { 2 } ^ 2 y _ { 2 } + \\cdots . \\end{align*}"}
+{"id": "7787.png", "formula": "\\begin{align*} \\theta _ { + } ^ P | _ { \\overline { N } } = \\theta _ { + } ^ { P , } + \\theta _ { + } ^ { P , } + \\theta _ { + } ^ { P , } , \\quad \\theta _ { 3 } ^ P | _ { \\overline { N } } = \\theta _ { 3 } ^ { P , } + \\theta _ { 3 } ^ { P , } + \\theta _ { 3 } ^ { P , } , \\end{align*}"}
+{"id": "6916.png", "formula": "\\begin{align*} & \\phi _ + = \\limsup \\limits _ { \\xi \\rightarrow + \\infty } \\phi ( \\xi ) , \\phi _ - = \\liminf \\limits _ { \\xi \\rightarrow + \\infty } \\phi ( \\xi ) , \\\\ [ 0 . 2 c m ] & \\psi _ { + } = \\limsup \\limits _ { \\xi \\rightarrow + \\infty } \\psi ( \\xi ) , \\psi _ - = \\liminf \\limits _ { \\xi \\rightarrow + \\infty } \\psi ( \\xi ) . \\end{align*}"}
+{"id": "820.png", "formula": "\\begin{align*} \\tilde { g _ 1 } ( e ^ { i \\theta _ 0 } ) = 0 , \\end{align*}"}
+{"id": "7487.png", "formula": "\\begin{align*} \\mathrm { T r } ( A _ L ) = \\mathrm { N } _ { q ^ m / q } ( \\alpha _ 1 / \\alpha _ 2 ) ( G _ { m } - u ^ { \\sigma ^ { - 1 } } G ^ \\sigma _ { m - 2 } ) = \\mathrm { N } _ { q ^ m / q } ( \\alpha _ 1 / \\alpha _ 2 ) ( G _ m + G ^ \\sigma _ m + G ^ \\sigma _ { m - 1 } ) . \\end{align*}"}
+{"id": "3504.png", "formula": "\\begin{align*} X _ t = \\{ F _ 0 F _ 1 + t F = 0 \\} \\subset M . \\end{align*}"}
+{"id": "8396.png", "formula": "\\begin{align*} W ( v ) = \\sum _ { P < p \\leq 2 P } ( \\log p ) e \\left ( v \\left ( N + j - \\left [ p ^ { c } \\right ] \\right ) ^ { \\gamma } \\right ) . \\end{align*}"}
+{"id": "3022.png", "formula": "\\begin{align*} \\phi ( P _ n ) = \\lambda \\phi ( P _ { n - 1 } ) - \\phi ( P _ { n - 2 } ) . \\end{align*}"}
+{"id": "6189.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ n \\left ( u X _ i + v X _ i ^ { - 1 } + w \\right ) \\prod _ { 1 \\leq p < q \\leq n } \\left ( u E _ { k _ p } + v E _ { k _ q } ^ { - 1 } + w E _ { k _ p } E _ { k _ q } ^ { - 1 } \\right ) \\tilde { s } _ { ( k _ n , k _ { n - 1 } , \\ldots , k _ 1 ) } ( X _ 1 , X _ 2 , \\ldots , X _ n ) , \\end{align*}"}
+{"id": "808.png", "formula": "\\begin{align*} \\| u _ n \\| _ { p _ g } \\leq C \\| u _ n \\| _ { p _ s ^ * } ^ \\theta \\| u _ n \\| _ p ^ { 1 - \\theta } = o ( 1 ) . \\end{align*}"}
+{"id": "1024.png", "formula": "\\begin{align*} \\pi _ k ^ * ( K ) \\cdot \\pi _ l ^ * ( K ' \\otimes \\xi ) = \\pi _ { k + l } ^ * ( ( K \\cdot K ' ) \\otimes \\xi ) \\end{align*}"}
+{"id": "5736.png", "formula": "\\begin{align*} U _ i , \\ U _ t , \\ x _ { n + 1 } ^ a U _ { x _ { n + 1 } } \\ \\in \\ H ^ { \\alpha } ( \\mathbb B _ 4 ^ + \\times ( - 1 6 , 0 ] ) , \\ \\ \\ \\ i = 1 , 2 , . . , n . \\end{align*}"}
+{"id": "3657.png", "formula": "\\begin{align*} L [ \\phi , \\psi ] = \\psi \\phi _ 1 \\phi _ { 2 2 } + \\frac { 1 } { 2 } \\phi _ 1 \\phi _ 3 - \\frac { 1 } { 2 } \\psi ^ 2 \\phi _ { 2 2 } + \\frac { 1 } { 2 } \\phi _ 2 \\phi _ 4 \\ . \\end{align*}"}
+{"id": "5764.png", "formula": "\\begin{align*} \\lim _ { \\mu _ 1 \\searrow 0 } \\lim _ { L \\to \\infty } \\mathbb E \\langle m ^ 1 \\rangle _ \\beta = \\lim _ { \\mu _ 1 \\searrow 0 } \\lim _ { L \\to \\infty } \\Big [ \\frac { 1 } { \\beta } \\frac { \\partial } { \\partial \\mu _ 1 } p _ L ( \\beta , \\Delta _ 1 , \\Delta _ 2 , \\mu _ 1 , \\mu _ 2 ) \\Big ] _ { \\Delta _ 1 = \\sqrt { \\frac { \\mu _ 1 } { \\beta _ { \\rm N } } } } = 0 , \\end{align*}"}
+{"id": "3579.png", "formula": "\\begin{align*} { \\xi _ { k , { l } , { j } } } = { \\rho _ k } { \\alpha _ { { k , { \\rm { R } } } , { l } } } { \\alpha _ { { \\rm { T } } , k , { j } } } { \\bf { a } } _ { { \\rm { S , } } k } ^ H \\left ( { \\Phi _ { { k } , { l } } ^ { \\rm { D } } } \\right ) { { \\bf { a } } _ { { \\rm { S } } , k } } \\left ( { \\Theta _ { k , { j } } ^ { \\rm { A } } } \\right ) . \\end{align*}"}
+{"id": "3299.png", "formula": "\\begin{align*} m _ 2 = \\frac { d ^ 2 } { d t ^ 2 } \\phi _ X ( 0 ) ( - \\mu ) ^ 2 = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\frac { \\beta _ { _ \\Sigma } ^ 3 - \\alpha _ { _ \\Sigma } ^ 3 } { 3 } . \\end{align*}"}
+{"id": "5327.png", "formula": "\\begin{align*} k ( x ) = \\begin{cases} x & x \\in ( - \\infty , 1 ] , \\\\ 1 & \\end{cases} \\end{align*}"}
+{"id": "464.png", "formula": "\\begin{align*} I _ { \\zeta - 1 / 2 } ( \\tau ) = \\left ( \\frac { \\tau } { 2 } \\right ) ^ { \\zeta - 1 / 2 } \\sum _ { k = 0 } ^ \\infty \\frac { ( \\tau ^ 2 / 4 ) ^ k } { k ! \\Gamma ( \\zeta + 1 / 2 + k ) } , \\tau \\in \\C \\setminus ( - \\infty , 0 ] \\end{align*}"}
+{"id": "280.png", "formula": "\\begin{align*} d ( x , z ) = \\log \\frac { 1 } { z ( x ) } \\end{align*}"}
+{"id": "7267.png", "formula": "\\begin{align*} = \\sum _ { n \\geq 0 } \\int e ^ { - s \\sigma ( \\eta , \\ , z ) } \\frac { 1 } { 2 \\pi } \\int _ \\mathbb { R } e ^ { i \\theta ( \\sigma ( \\eta , z ) + t ) } \\hat { f } ( \\eta . z , \\ , \\theta ) d \\theta d \\mathbf { p } ^ n ( \\eta ) . \\end{align*}"}
+{"id": "7994.png", "formula": "\\begin{align*} b _ { 1 } g _ { 1 } | _ { { \\textstyle \\bigcup _ { i = 1 } ^ { m } } \\Omega _ { i } } + \\cdots + b _ { n } g _ { n } | _ { { \\textstyle \\bigcup _ { i = 1 } ^ { m } } \\Omega _ { i } } = - b _ { n + 1 } f _ { 1 } | _ { { \\textstyle \\bigcup _ { i = 1 } ^ { m } } \\Omega _ { i } } - \\cdots - b _ { n + m } f _ { m } | _ { { \\textstyle \\bigcup _ { i = 1 } ^ { m } } \\Omega _ { i } } \\end{align*}"}
+{"id": "508.png", "formula": "\\begin{align*} E _ T = E \\cap \\left ( [ 0 , T ] \\times \\Omega \\right ) \\end{align*}"}
+{"id": "4105.png", "formula": "\\begin{align*} P ( x , y ) = 4 x ^ 2 y ^ 2 - ( x ^ 2 + 4 \\lambda _ { \\infty } x + 4 \\lambda _ { 0 } ^ 2 ) \\end{align*}"}
+{"id": "6680.png", "formula": "\\begin{align*} c _ \\infty ^ { \\widetilde { \\rm ( c J ) } } ( \\tau ; \\beta , p , q ) = \\sum _ { l = 0 } ^ \\infty \\Big ( { \\tau \\over 2 \\pi } \\Big ) ^ l \\alpha _ l ( \\beta , p , q ) . \\end{align*}"}
+{"id": "3883.png", "formula": "\\begin{align*} i _ 1 ( g ( y _ 0 ) ) = f ( i _ 0 ( y _ 0 ) ) = f ( u ( i _ 1 ( y _ 1 ) ) ) \\overset { \\eqref { e q q q } } { = } i _ 1 ( y _ 1 ) \\end{align*}"}
+{"id": "6381.png", "formula": "\\begin{align*} ( \\mathrm { I d } \\otimes \\Delta ) ( \\chi ) & = \\chi \\otimes 1 _ { H } + ( \\rho ^ r \\otimes \\mathrm { I d } ) ( \\chi ) ; \\\\ ( \\Delta \\otimes \\mathrm { I d } ) ( \\chi ) & = 1 _ { H } \\otimes \\chi + ( \\mathrm { I d } \\otimes \\rho ^ l ) ( \\chi ) . \\end{align*}"}
+{"id": "5733.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u = V u , \\end{align*}"}
+{"id": "1567.png", "formula": "\\begin{align*} ( e ^ { 1 / \\kappa | v | } - 1 ) g ( x , v ) & = \\int _ 0 ^ 1 \\frac { e ^ { y / \\kappa | v | } } { \\kappa | v | } \\rho _ g ( x + y ) \\left ( \\alpha \\mathcal { M } _ { T ( x + y ) } ( v ) + ( 1 - \\alpha ) \\mathcal { M } _ { \\tau ( x + y ) } ( v ) \\right ) \\mathrm { d } y , \\end{align*}"}
+{"id": "4010.png", "formula": "\\begin{align*} \\tilde X : = \\chi + \\frac { \\delta } { 2 } \\chi + \\sqrt { - 1 } X _ { 1 \\bar 1 } d z ^ 1 \\wedge d \\bar z ^ 1 . \\end{align*}"}
+{"id": "4635.png", "formula": "\\begin{align*} K _ { v _ i } = \\begin{cases} H & \\hbox { f o r } \\ , i = 1 , \\\\ G & \\hbox { f o r } \\ , i = 2 , \\ldots , n , \\\\ K & \\hbox { f o r } \\ , i = n + 1 , \\end{cases} \\end{align*}"}
+{"id": "2044.png", "formula": "\\begin{align*} \\lVert u \\rVert _ { \\dot { \\mathcal { K } } ^ { s , p } ( \\mathbb { R } ^ { n } _ + ) } : = \\lVert ( \\partial _ { x _ n } u , \\nabla ' u ) \\rVert _ { \\dot { \\mathrm { H } } ^ { s - 1 , p } ( \\mathbb { R } _ + , \\mathrm { L } ^ p ( \\mathbb { R } ^ { n - 1 } ) ) } \\end{align*}"}
+{"id": "7311.png", "formula": "\\begin{align*} - \\langle u , v \\rangle = \\langle S _ - ( T ) P _ - ( T ) T P _ - ( T ) S _ - ( T ) u , v \\rangle , u , v \\in H _ - , \\end{align*}"}
+{"id": "4986.png", "formula": "\\begin{align*} d _ Y ( a , b ) \\leqslant \\min \\{ \\rho ( a , y ) , \\rho ' ( a , y ) \\} + \\min \\{ \\rho ( y , b ) , \\rho ' ( y , b ) \\} = d _ Y ( a , y ) + d _ Y ( y , b ) . \\end{align*}"}
+{"id": "2990.png", "formula": "\\begin{align*} \\lambda ^ c z ^ { q _ n } = \\lambda ^ d v ^ { q _ n } = \\lambda ^ d ( \\lambda ^ { k \\gcd ( n , b ) } z ) ^ { q _ n } = \\lambda ^ { k n + d } z ^ { q _ n } . \\end{align*}"}
+{"id": "6390.png", "formula": "\\begin{align*} \\chi ( \\xi \\otimes \\eta ) = | \\xi | \\cdot | \\eta | \\varepsilon ( \\xi \\eta ) . \\end{align*}"}
+{"id": "2262.png", "formula": "\\begin{align*} \\theta ( x ; \\alpha ) = \\sum _ { k \\in \\Z } e ^ { - \\pi \\alpha k ^ 2 } e ^ { 2 \\pi i k x } = \\sum _ { k \\in \\Z } e ^ { - \\pi \\alpha k ^ 2 } \\cos ( 2 \\pi k x ) = 1 + 2 \\sum _ { k \\geq 1 } e ^ { - \\pi \\alpha k ^ 2 } \\cos ( 2 \\pi k x ) . \\end{align*}"}
+{"id": "7631.png", "formula": "\\begin{align*} ( \\P ^ \\ast ) \\supseteq \\{ \\pm e _ i : i = 1 , \\dots , d \\} , \\end{align*}"}
+{"id": "5518.png", "formula": "\\begin{align*} O _ A ^ { ( g a ) } ( p , s ) = O _ A ^ { ( g ) } ( p s , s ) , O _ B ^ { ( g a ) } ( p , s ) = O _ B ^ { ( g ) } ( p s , s ) . \\end{align*}"}
+{"id": "1826.png", "formula": "\\begin{gather*} \\sum _ { n = 0 } ^ { \\infty } \\left | \\left \\langle ( n + c ) ^ { - s } , g ( s ) \\right \\rangle \\right | < \\infty \\end{gather*}"}
+{"id": "650.png", "formula": "\\begin{align*} \\mathcal L ( \\psi , \\phi , x , t ) = \\mathcal L _ { \\mathrm { D i r a c } } ( \\psi ) + \\mathcal L _ { \\mathrm { m e s o n } } ( \\phi ) + \\mathcal L _ { \\mathrm { Y u k a w a } } ( \\psi , \\phi ) + \\mathcal L _ { \\mathrm { n o i s e } } ( \\psi , \\phi , x , t ) \\end{align*}"}
+{"id": "3666.png", "formula": "\\begin{align*} I _ i ' ( 0 ) = 1 - 2 \\ , \\lambda _ i \\ , , J _ { i j } ' ( 0 ) = ( 1 - 2 \\ , \\lambda _ i ) \\ , \\delta _ { i j } { } E _ i ' ( 0 ) \\leq 0 \\ , . \\end{align*}"}
+{"id": "5311.png", "formula": "\\begin{align*} \\sigma _ h ( x ) = \\frac { \\rho ( \\alpha + h x ) - 2 \\rho ( \\alpha ) + \\rho ( \\alpha - h x ) } { h ^ 2 \\rho '' ( \\alpha ) } . \\end{align*}"}
+{"id": "5579.png", "formula": "\\begin{align*} c ( G ) = o ( g ( \\delta - 1 ) ^ { ( 1 + o ( 1 ) ) \\frac { g } { 4 } } ) . \\end{align*}"}
+{"id": "140.png", "formula": "\\begin{align*} G _ F = \\coprod _ { \\mu \\in X _ * ( \\textbf { T } ) ^ { - } } \\textbf { G } ( \\mathfrak { o } _ F ) \\varpi _ { \\mu } \\textbf { G } ( \\mathfrak { o } _ F ) , \\end{align*}"}
+{"id": "8096.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j = N } ^ { \\infty } \\lambda _ { j } a _ { j } \\right \\| _ { h _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = N } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } ; \\end{align*}"}
+{"id": "6942.png", "formula": "\\begin{align*} \\dot { x } _ 1 = x _ 2 ; \\ ; \\ ; \\dot { x } _ 2 = - x _ 1 + x _ 2 ( 1 - x _ 1 ^ 2 ) + u . \\end{align*}"}
+{"id": "4422.png", "formula": "\\begin{align*} [ \\hat { u } _ 2 - \\lambda \\hat { H } _ 2 ] = 0 , \\end{align*}"}
+{"id": "5898.png", "formula": "\\begin{align*} L ^ q ( \\mathcal { D } , \\mu ) = \\{ f : \\mathcal { D } \\rightarrow \\R : \\| f \\| _ { L ^ q ( \\mathcal { D } , \\mu ) } < \\infty \\} \\end{align*}"}
+{"id": "7017.png", "formula": "\\begin{align*} \\frac { 2 n ' + 1 } { 2 m ' + 1 } - \\sqrt { \\frac { a } { b } } = \\frac { \\pm 4 r ( n ' - m ' ) + b - a } { \\sqrt { b } ( 2 m ' + 1 ) ( \\sqrt { b } ( 2 n ' + 1 ) + \\sqrt { a } ( 2 m ' + 1 ) ) } . \\end{align*}"}
+{"id": "8578.png", "formula": "\\begin{align*} P ( \\Sigma _ b ) = \\begin{pmatrix} \\dfrac { h } { h _ b } \\mathrm { I d } & - \\sqrt { \\mu } \\nabla _ X \\sigma \\\\ - \\sqrt { \\mu } ( \\nabla _ X \\sigma ) ^ T & \\dfrac { h _ b + \\mu h _ b | \\nabla _ X \\sigma | ^ 2 } { h } \\end{pmatrix} , \\end{align*}"}
+{"id": "1701.png", "formula": "\\begin{align*} X _ j : = M _ { \\texttt { b } ; e _ 1 + \\cdots + e _ j } ( \\boldsymbol { \\xi } ) , j = 1 , \\ldots , n \\end{align*}"}
+{"id": "2103.png", "formula": "\\begin{align*} C _ { t _ 0 , \\underline { u } _ 0 } : = \\left \\{ ( t , x ) : \\ ; u = \\frac { t - x } { 2 } = \\underline { u } _ 0 , \\ , 0 \\leq t \\leq t _ 0 \\right \\} , \\end{align*}"}
+{"id": "7883.png", "formula": "\\begin{align*} \\begin{cases} \\det D ^ 2 u = \\lambda ( u ^ { \\star } ) ^ { - k } ( - u ) ^ p & \\Omega \\\\ u = 0 & \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "448.png", "formula": "\\begin{align*} \\langle [ u ] _ { \\ell , m } , P ^ { ( \\alpha ) } ( t , \\cdot , \\cdot ) [ u ] _ { \\ell , m } \\rangle _ { L ^ 2 ( \\R ^ d ) } & = \\langle u , p _ \\zeta ^ { ( \\alpha ) } ( t , \\cdot , \\cdot ) u \\rangle _ { L ^ 2 ( \\R _ + , r ^ { 2 \\zeta } d r ) } , t > 0 \\end{align*}"}
+{"id": "7164.png", "formula": "\\begin{align*} f _ j ( V ^ { - 1 } x ) = f _ j \\left ( \\sum _ { k = 1 } ^ { n } g _ k ( x ) \\tau _ k \\right ) = \\sum _ { k = 1 } ^ { n } g _ k ( x ) f _ j ( \\tau _ k ) = g _ j ( x ) , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "4911.png", "formula": "\\begin{align*} M _ { k , \\chi , \\psi } ( \\tau ) = c _ k + \\mathcal { A } _ k \\sum _ { n \\geq 1 } a ( n ) q ^ { n / R } , c _ k = \\begin{cases} 2 L ( k , \\psi ) , & \\chi : \\ ; , \\\\ 0 , & , \\end{cases} \\end{align*}"}
+{"id": "6150.png", "formula": "\\begin{align*} \\# ( \\mathbf { k } ) = ( \\mathbf { k } ) \\cdot \\# ( \\mathbf { k } ' ) = \\prod _ { 1 \\leq i < j \\leq n } \\frac { k _ j - k _ i } { j - i } , \\end{align*}"}
+{"id": "4840.png", "formula": "\\begin{align*} \\Lambda _ i ( u ) = \\prod _ { j = 1 } ^ { i - 1 } \\left ( \\frac { \\lambda _ j } { \\lambda _ { j + 1 } } f _ j ( u ) + 1 \\right ) \\prod _ { k = i } ^ { n - 1 } \\left ( f _ k ( u ) + \\frac { \\lambda _ k } { \\lambda _ { k + 1 } } \\right ) \\lambda _ n \\end{align*}"}
+{"id": "411.png", "formula": "\\begin{align*} P _ n = \\{ p ( x _ 1 , . . . , x _ n ) \\ | \\ \\ x _ 1 , . . . , x _ n \\} \\end{align*}"}
+{"id": "4471.png", "formula": "\\begin{align*} ( H ^ { \\pm } _ N ) _ j | _ { x _ 1 = 0 } = 0 , j \\geq 2 , \\end{align*}"}
+{"id": "1631.png", "formula": "\\begin{align*} Q ' = A ^ { - 1 } Q , A ' = A , T ' = T . \\end{align*}"}
+{"id": "1470.png", "formula": "\\begin{align*} \\mathbf { f } _ { m , j } = \\sqrt { \\alpha _ { m , j } } \\mathbf { a } ( \\theta _ { m , j } ) , \\end{align*}"}
+{"id": "870.png", "formula": "\\begin{align*} u _ t = x u _ { x x } + f ( x ) u _ x , ~ ~ x \\geq 0 , \\end{align*}"}
+{"id": "5424.png", "formula": "\\begin{align*} \\Xi _ { \\leq j ^ { \\tau u } } ( \\xi ) : = \\sum _ { a / q \\in \\Sigma _ { \\leq F ( j ^ { \\tau u } ) } } \\eta _ { j ^ { \\tau } } ( \\xi - a / q ) \\end{align*}"}
+{"id": "1050.png", "formula": "\\begin{align*} k ( z , x ) = \\left \\{ \\begin{array} { l l } C _ 0 ( x ^ + ) ^ { \\tau ( z ) - 1 } - C _ 4 ( x ^ + ) ^ { r ( z ) - 1 } + \\vartheta ( x ^ + ) ^ { p ( z ) - 1 } , & \\hbox { i f } x \\leq u ( z ) \\\\ C _ 0 u ( z ) ^ { \\tau ( z ) - 1 } - C _ 4 u ( z ) ^ { r ( z ) - 1 } + \\vartheta u ( z ) ^ { p ( z ) - 1 } , & \\hbox { i f } u ( z ) < x . \\end{array} \\right . \\end{align*}"}
+{"id": "3762.png", "formula": "\\begin{align*} \\begin{aligned} & u ( t ) = e ^ { t a B } \\varphi + \\dfrac { 1 } { b - a } B ^ { - 1 } \\Big [ e ^ { t b B } - e ^ { t a B } \\Big ] ( \\psi - a B \\varphi ) \\\\ & \\dfrac { 1 } { 1 + b } B ^ { - 2 } \\Big [ \\dfrac { 1 } { b - a } e ^ { t b B } - \\dfrac { 1 + b } { ( 1 + a ) ( b - a ) } e ^ { t a B } + \\dfrac { 1 } { 1 + a } e ^ { - t B } \\Big ] ( \\xi - ( a + b ) B \\psi + a b B \\varphi ) , \\end{aligned} \\end{align*}"}
+{"id": "2072.png", "formula": "\\begin{align*} \\mathbf { I } & \\leq C _ 1 \\delta \\int ^ \\infty _ { r _ 0 } \\frac { r ^ n } { t ^ { n / 2 } } \\cdot \\exp \\left ( - \\frac { r ^ 2 } { C _ 1 t } \\right ) \\frac { 1 } { r t } d r \\\\ & \\leq t ^ { - 1 } C _ 1 \\delta \\int ^ \\infty _ { 0 } r ^ { n - 1 } \\exp \\left ( - \\frac { r ^ 2 } { C _ 1 } \\right ) d r \\\\ & \\leq C _ 2 t ^ { - 1 } \\delta . \\end{align*}"}
+{"id": "8799.png", "formula": "\\begin{align*} \\nabla \\hat f _ t ( x ) & = \\frac { 1 } { V ( \\mathbb { B } _ { d } ) h _ t ^ d } \\int _ { \\norm { v } = h _ t } f ( x + v ) \\frac { v } { \\norm { v } } { \\rm d } s _ { h _ t } ( v ) = \\frac { d } { V ( S _ d ) h _ t } \\int _ { \\norm { u } = 1 } f ( x + h _ t u ) u \\ , { \\rm d } s _ 1 ( u ) \\\\ & = \\frac { d } { V ( S _ d ) h _ t } \\int _ { \\norm { u } = 1 } f ( x + h _ t u ) u \\ , { \\rm d } s _ 1 ( u ) = \\mathbb { E } \\Big [ \\frac { d } { h _ t } f ( x + h _ t \\zeta _ t ) \\zeta _ t \\Big ] \\end{align*}"}
+{"id": "2983.png", "formula": "\\begin{align*} \\alpha _ 1 ^ { - 1 } ( K ) = K \\times _ { s , \\alpha } \\alpha ^ { - 1 } ( s ( K ) ) = ( K \\times \\alpha ^ { - 1 } ( s ( K ) ) ) \\cap E ^ 1 _ I . \\end{align*}"}
+{"id": "7498.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta _ \\gamma = & \\mathrm { T r } ( A _ \\gamma ) ^ 2 - 4 \\det ( A _ \\gamma ) \\\\ & = \\gamma ^ { 2 m } \\Bigl [ ( 2 G _ { m } ( \\gamma ) + G _ { m - 1 } ( \\gamma ) ) ^ 2 - 4 \\Bigl ( G ^ 2 _ m ( \\gamma ) + G _ { m - 1 } ( \\gamma ) G _ { m } ( \\gamma ) + \\frac { G _ { m - 1 } ^ 2 ( \\gamma ) } { \\gamma } \\Bigr ) \\Bigr ] \\\\ & = \\gamma ^ { 2 m - 1 } ( \\gamma - 4 ) G ^ 2 _ { m - 1 } ( \\gamma ) \\end{aligned} \\end{align*}"}
+{"id": "364.png", "formula": "\\begin{align*} x ^ { - 1 } \\phi _ { g } ( h ) h ^ { - 1 } x = & \\ ; \\phi _ { g } ( h _ 1 ^ { - 1 } ) \\phi _ { g } ( h ) h ^ { - 1 } \\phi _ { g } ( h _ 1 ) \\\\ = & \\ ; \\big ( \\phi _ { g } ( h _ 1 ^ { - 1 } h ) \\big ) \\big ( h ^ { - 1 } h _ 1 \\big ) \\big ( h ^ { - 1 } _ 1 \\phi _ { g } ( h _ 1 ) \\big ) \\\\ = & \\ ; \\big ( \\phi _ { g } ( h _ 1 ^ { - 1 } h ) \\ ; ( h ^ { - 1 } _ 1 h ) ^ { - 1 } \\big ) \\ ; \\big ( h ^ { - 1 } _ 1 \\phi _ { g } ( h _ 1 ) \\big ) \\in H ^ { ( 2 ) } , \\\\ \\end{align*}"}
+{"id": "1679.png", "formula": "\\begin{align*} f ( \\boldsymbol { \\xi } ) = P _ { \\texttt { a } ; \\mu } ( \\boldsymbol { \\xi } ; q ) P _ { \\texttt { a } ; \\nu } ( \\boldsymbol { \\xi } ; q ) \\quad \\mu \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { a } } , \\ \\nu \\in \\Lambda ^ { ( m - 1 , n ) } _ { \\texttt { a } } . \\end{align*}"}
+{"id": "5291.png", "formula": "\\begin{align*} - \\frac { 3 } { 2 } & \\max \\{ 0 , ( 2 n + 1 ) x - 1 \\} \\\\ & \\max \\{ 0 , ( 2 n + 1 ) x - 3 \\} \\\\ - & \\max \\{ 0 , - ( 2 n + 1 ) x + 5 \\} \\\\ ( - 1 ) ^ i & \\max \\{ 0 , ( 2 n + 1 ) x - ( 2 i - 1 ) \\} i = 4 , \\dots , n \\end{align*}"}
+{"id": "8781.png", "formula": "\\begin{align*} \\mathbb { E } [ \\norm { \\nabla f ( x _ S ) } ^ 2 ] & \\leq \\frac { 7 2 L \\kappa \\delta ( 1 ) } { T } + \\mathcal { C } _ { 5 } \\sum _ { t = 1 } ^ { T } \\Big [ \\gamma d h _ { t } ^ { 2 ( \\beta - 1 ) } + \\gamma ^ 2 d \\Big ( \\frac { \\sigma ^ 2 } { h _ { t } ^ 2 } + h _ { t } ^ 2 \\Big ) \\Big ] \\frac { 1 } { T } + \\\\ & \\quad \\quad + \\frac { 4 \\delta ( 1 ) } { T \\gamma } + \\mathcal { C } _ { 5 } \\sum _ { t = 1 } ^ { T } \\Big [ d h _ { t } ^ { 2 ( \\beta - 1 ) } + \\gamma d \\Big ( \\frac { \\sigma ^ 2 } { h _ { t } ^ 2 } + h _ { t } ^ 2 \\Big ) \\Big ] \\frac { 1 } { T } , \\end{align*}"}
+{"id": "6934.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow + \\infty } \\phi ( x _ n ) = \\underline { \\phi } \\phi ' ( x _ n ) = 0 , \\\\ [ 0 . 2 c m ] \\lim \\limits _ { n \\rightarrow + \\infty } \\phi ( y _ n ) = \\overline { \\phi } \\phi ' ( y _ n ) = 0 . \\end{align*}"}
+{"id": "7475.png", "formula": "\\begin{align*} \\operatorname { B } ( z , w ) : = \\int _ { 0 } ^ 1 t ^ { z - 1 } ( 1 - t ) ^ { w - 1 } \\operatorname { d } \\ ! t . \\end{align*}"}
+{"id": "2912.png", "formula": "\\begin{align*} M _ \\lambda ( \\xi ) = \\sum _ { \\mu \\in P ^ + , \\ , \\mu \\leq \\lambda } n _ { \\lambda , \\mu } ( t ) m _ \\mu ( e ^ { i \\xi } ) , \\end{align*}"}
+{"id": "5871.png", "formula": "\\begin{align*} T _ { S } ( \\epsilon _ { 0 } ) = \\inf \\Big \\{ t : \\max _ { k = t - s + 1 \\in \\mathcal { K } } \\sum _ { n = 1 } ^ { N } g ( Z ^ { n + } _ { s t } ) \\geq C _ { \\gamma } \\Big \\} , \\end{align*}"}
+{"id": "5088.png", "formula": "\\begin{align*} [ \\R ( h ) ] _ { i j } = \\sum _ { m , n } \\R _ { i m j n } \\ , h _ { m n } \\ , . \\end{align*}"}
+{"id": "2885.png", "formula": "\\begin{align*} T _ j \\mathbf { e } ^ { i { \\xi } } = _ j ( s _ j \\boldsymbol { \\xi } ) \\mathbf { e } ^ { i \\boldsymbol { \\xi } } + _ j ( s _ j \\boldsymbol { \\xi } ) \\mathbf { e } ^ { i s _ j \\boldsymbol { \\xi } } = _ j ( - \\boldsymbol { \\xi } ) \\mathbf { e } ^ { i \\boldsymbol { \\xi } } + _ j ( - \\boldsymbol { \\xi } ) \\mathbf { e } ^ { i s _ j \\boldsymbol { \\xi } } , \\end{align*}"}
+{"id": "1218.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\liminf _ { n \\to \\infty } \\Big [ P ( f _ n ) - \\sum _ { j = 1 } ^ J P ( g _ n ^ j \\phi ^ j ) - P ( r _ n ^ J ) \\Big ] = 0 . \\end{align*}"}
+{"id": "2739.png", "formula": "\\begin{align*} \\big | \\big ( U ( e _ i ) \\big ) ( \\sigma ( i ) ) \\big | = 1 \\big | \\big ( U ^ { - 1 } ( e _ i ) \\big ) ( \\tilde { \\sigma } ( i ) ) \\big | = 1 , \\end{align*}"}
+{"id": "1284.png", "formula": "\\begin{align*} C _ n = & - \\int _ { \\R ^ 3 } [ I _ \\alpha \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } ( | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p + | \\phi ^ 1 | ^ p ) ] | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p d x \\\\ & + \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n | ^ p ) | x - x _ n ^ 1 \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p d x . \\end{align*}"}
+{"id": "5428.png", "formula": "\\begin{align*} w ^ s ( \\xi ) : = \\sum _ { a / q \\in \\Sigma _ s } G ( a / q ) \\tilde { \\eta } _ { 2 ^ { \\kappa _ s } } ( \\xi - a / q ) , \\Pi ^ s ( \\xi ) : = \\sum _ { a / q \\in \\Sigma _ s } \\tilde { \\eta } _ { 2 ^ { \\kappa _ s } } ( \\xi - a / q ) , \\end{align*}"}
+{"id": "8103.png", "formula": "\\begin{align*} \\uppercase \\expandafter { \\romannumeral 1 } \\leq \\sum \\limits _ { j = 1 } ^ { \\infty } \\vert \\lambda _ { j } \\vert \\vert T ( a _ { j } ) ( x ) \\vert \\chi _ { Q _ { j } ^ { * } } ( x ) + \\sum \\limits _ { j = 1 } ^ { \\infty } \\vert \\lambda _ { j } \\vert \\vert T ( a _ { j } ) ( x ) \\vert \\chi _ { ( Q _ { j } ^ { * } ) ^ { c } } ( x ) = : \\uppercase \\expandafter { \\romannumeral 1 } _ { 1 } + \\uppercase \\expandafter { \\romannumeral 1 } _ { 2 } . \\end{align*}"}
+{"id": "7690.png", "formula": "\\begin{align*} K = \\bigcap _ { i = 1 } ^ n K _ i \\cap \\bigcap _ { i ' = 1 } ^ n K ' _ { i ' } . \\end{align*}"}
+{"id": "5002.png", "formula": "\\begin{align*} \\int _ { B _ p ( T + \\frac { \\eta } { 8 } ) } e ^ { - b r } - \\int _ { B _ p ( T - \\frac { \\eta } { 8 } ) } e ^ { - b r } = f ' ( \\xi ) \\left ( T + \\frac { \\eta } { 8 } - T + \\frac { \\eta } { 8 } \\right ) , \\end{align*}"}
+{"id": "2866.png", "formula": "\\begin{align*} \\theta ( \\lambda ) : = \\left | { \\{ a \\in R ^ + \\mid a ( \\lambda ) = - 2 \\} } \\right | . \\end{align*}"}
+{"id": "2162.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { \\alpha ^ d \\left ( m ^ { d + 1 } + m ^ { - d - 1 } \\right ) + \\alpha ^ { - d } \\left ( m ^ { d - 1 } + m ^ { - d + 1 } \\right ) } { m ^ { d } + m ^ { - d } } \\\\ & ~ { } = \\frac { ( \\alpha ^ { d } m + \\alpha ^ { - d } m ^ { - 1 } ) m ^ d + ( \\alpha ^ { d } m ^ { - 1 } + \\alpha ^ { - d } m ) m ^ { - d } } { m ^ { d } + m ^ { - d } } \\geq 1 . \\end{aligned} \\end{align*}"}
+{"id": "8192.png", "formula": "\\begin{align*} \\omega _ 1 = \\bar { \\omega } _ 1 + d x _ 1 \\wedge \\eta + V d x _ 2 \\wedge d x _ 3 , \\end{align*}"}
+{"id": "2450.png", "formula": "\\begin{align*} \\hat { u _ n ^ { \\leq \\delta } } ( \\xi ) : = \\ 1 _ { \\{ | 2 \\pi \\xi | ^ { 2 s } \\leq \\delta \\} } ( \\xi ) \\hat { u } _ n ( \\xi ) , \\hat { u _ n ^ { \\delta , \\ell } } ( \\xi ) : = \\ 1 _ { \\{ \\delta < | 2 \\pi \\xi | ^ { 2 s } \\leq \\ell \\} } ( \\xi ) \\hat { u } _ n ( \\xi ) , \\end{align*}"}
+{"id": "7014.png", "formula": "\\begin{align*} c > ( 0 . 0 1 6 8 5 a _ 1 ( a _ 1 ' ) ^ { - 1 } a _ 2 ^ { - 1 } ( a _ 2 - a _ 1 ) ^ { - 2 } ) ^ { - 1 } = 0 . 0 1 6 8 5 ^ { - 1 } a _ 1 ^ { - 1 } a _ 1 ' a _ 2 ( a _ 2 - a _ 1 ) ^ 2 \\end{align*}"}
+{"id": "854.png", "formula": "\\begin{align*} ( N _ 2 g ) ( z ) = ( 1 - \\epsilon ) ( 1 - \\eta ( z ) ) T g ( z ) , \\end{align*}"}
+{"id": "7765.png", "formula": "\\begin{align*} f ^ { } = - 2 \\pi \\sum _ { \\gamma } \\Omega ( \\gamma ) \\iota _ V \\eta _ { \\gamma } ^ { } \\ , . \\end{align*}"}
+{"id": "8255.png", "formula": "\\begin{align*} \\rho ^ 2 ( e ^ { \\tau ( m , x _ 1 ) } \\cdot m ) - \\rho ^ 2 ( m ) & = \\int _ 0 ^ { \\tau ( m , x _ 1 ) } \\frac { d } { d t } \\rho ^ 2 ( e ^ t \\cdot m ) d t = 4 \\int _ 0 ^ { \\tau ( m , x _ 1 ) } x _ 1 ( e ^ t \\cdot m ) d t \\\\ & \\leq 4 \\int _ 0 ^ { \\tau ( m , x _ 1 ) } x _ 1 d t = 4 x _ 1 \\tau ( m , x _ 1 ) . \\end{align*}"}
+{"id": "3399.png", "formula": "\\begin{align*} v _ j \\phi _ j h _ { i k } = v _ k \\phi _ k h _ { i j } , 1 \\leq i \\neq j \\neq k \\neq i \\leq 3 . \\end{align*}"}
+{"id": "4455.png", "formula": "\\begin{align*} \\mathcal { J } ( t ) \\leq \\sum _ { \\pm } \\sum ^ 4 _ { i = 1 } \\mathcal { J } ^ { \\pm } _ i ( t ) , \\end{align*}"}
+{"id": "3385.png", "formula": "\\begin{align*} \\alpha ^ 0 _ { \\mu \\nu } b ^ \\nu = 0 . \\end{align*}"}
+{"id": "3683.png", "formula": "\\begin{align*} \\sum _ { u \\in \\sigma } d _ { k - 1 } ( \\sigma \\setminus \\{ u \\} ) = ( k - 1 ) d _ k ( \\sigma ) + 2 | E _ { \\sigma } | . \\end{align*}"}
+{"id": "6727.png", "formula": "\\begin{align*} \\begin{cases} \\Psi : \\Sigma \\times ( r _ 0 , + \\infty ) \\rightarrow \\ M , \\\\ \\partial _ t \\Psi ( p , t ) = f ' ( t ) \\frac { \\nabla u } { | \\nabla u | ^ 2 } = f ' ( t ) \\frac { 1 } { | \\nabla u | } \\nu . \\end{cases} \\end{align*}"}
+{"id": "4693.png", "formula": "\\begin{align*} \\begin{cases} l ^ n _ { r } = \\mu \\cdot l ^ n _ { r - 1 } = \\mu ^ { r } \\cdot l _ 0 , & \\forall r \\geq 1 , \\\\ ( A ^ r ) _ { ( S , W ) } = l ^ n _ { i , r } = ( \\mu ^ r \\cdot l _ 0 ) ^ t _ i . \\end{cases} \\end{align*}"}
+{"id": "3453.png", "formula": "\\begin{align*} M u = C _ 1 \\mu _ 0 , C _ 1 = \\int _ { \\Delta ^ \\vee } W d p = \\frac { d _ 0 + \\ldots + d _ m } { d _ 0 \\ldots d _ m } \\frac { ( n - m ) ! } { n ! } . \\end{align*}"}
+{"id": "4621.png", "formula": "\\begin{align*} U ( \\mathfrak { g } ) ^ { \\mathfrak { k } } : = C _ { U ( \\mathfrak { g } ) } \\bigl ( U ( \\mathfrak { k } ) \\bigr ) \\end{align*}"}
+{"id": "5588.png", "formula": "\\begin{align*} \\Theta ^ { n - 1 } ( \\mu , x ) & = \\lim _ { r \\to 0 } \\frac { 1 } { r ^ { n - 1 } } \\int _ { B _ r ( x ) } | \\nabla u _ * | ^ 2 \\ , d x \\\\ & \\leq C _ { n + 1 } \\lim _ { r \\to 0 } r ^ { 2 } u _ * | u _ * | \\leq 1 \\\\ & = 0 \\end{align*}"}
+{"id": "5223.png", "formula": "\\begin{align*} p _ i = \\frac { 1 / \\sqrt { n _ m } + 1 } { 1 / \\sqrt { n _ m } + \\sqrt { n _ m / n _ i } } \\end{align*}"}
+{"id": "5080.png", "formula": "\\begin{align*} J _ { \\lambda , i } ^ p = ( \\lambda I _ i + N _ i ) ^ p = \\sum _ { h = 0 } ^ { \\min \\{ i - 1 , p \\} } \\binom { p } { h } \\lambda ^ { p - h } I _ i N _ i ^ h = \\sum _ { h = 0 } ^ { \\min \\{ i - 1 , p \\} } \\binom { p } { h } \\lambda ^ { p - h } N _ i ^ h . \\end{align*}"}
+{"id": "8648.png", "formula": "\\begin{align*} ( u \\circ \\Sigma _ b ^ { - 1 } ) \\circ \\Sigma ( X , z ) = u ( X , z h _ b ) : = \\tilde u ( X , z ) , \\end{align*}"}
+{"id": "3138.png", "formula": "\\begin{align*} \\mathbb { E } [ x _ i \\cdot x ^ * _ { i ' } ] = \\begin{cases} \\frac { L _ k L _ l } { ( \\kappa + 1 ) ^ 2 } , & n = m , p = q , n ' = m ' , p ' = q ' , \\\\ 0 , & ~ , \\end{cases} \\end{align*}"}
+{"id": "3824.png", "formula": "\\begin{align*} z _ W : = \\int _ { Y \\times Y } \\exp ( - W ) d \\nu _ Y < + \\infty . \\end{align*}"}
+{"id": "121.png", "formula": "\\begin{align*} \\langle \\mathcal { H } \\rangle _ { \\Psi } \\geq b \\frac { K _ L ^ 2 } { \\ell ^ 2 } \\langle n _ + ^ H \\rangle _ { \\Psi } + \\frac { 1 } { 2 } \\langle \\mathcal { Q } _ 4 ^ { } \\rangle _ { \\Psi } + \\frac { 1 } { 2 } \\rho N \\widehat { g } ( 0 ) \\times \\begin{dcases} \\Big ( 1 - C \\sqrt { \\rho a ^ 3 } \\frac { K _ { L } } { K _ { \\ell } } \\Big ) , & d = 3 , \\\\ \\Big ( 1 - C \\widehat { g } ( 0 ) \\Big ) , & d = 2 , \\end{dcases} \\end{align*}"}
+{"id": "1301.png", "formula": "\\begin{align*} \\Big | \\sum _ { j = 1 } ^ J v _ n ^ j \\Big | ^ 2 = \\sum _ { j = 1 } ^ J | v _ n ^ j | ^ 2 + \\sum _ { j \\neq k } v _ n ^ j v _ n ^ k , \\end{align*}"}
+{"id": "4558.png", "formula": "\\begin{align*} \\hat { H } ^ { + } _ 2 \\partial _ 2 \\varphi = \\dot { H } ^ { + } _ { N } + \\varphi \\partial _ 1 \\hat { H } ^ { + } _ { N } \\quad \\mbox { a n d } \\hat { H } ^ { - } _ 2 \\partial _ 2 \\varphi = \\dot { H } ^ { - } _ { N } - \\varphi \\partial _ 1 \\hat { H } ^ { - } _ { N } \\qquad \\mbox { o n } \\ , \\ , \\ , \\Gamma _ T \\ , . \\end{align*}"}
+{"id": "6118.png", "formula": "\\begin{align*} \\tilde { c } ( t ) = - \\int _ { 0 } ^ t \\frac { n ' ( u ) + \\mathfrak { t } ( u ) b ( u ) } { \\kappa ( u ) } \\d u , \\end{align*}"}
+{"id": "8160.png", "formula": "\\begin{align*} G & = \\frac { \\binom { n } { i } } { \\binom { k } { i } } ( k - i + 1 ) K _ { i - 1 } ( x , k - y , r - 1 ) H _ j ( y , n - i , k - i ) + \\frac { \\binom { n } { i } } { \\binom { k } { i } } K _ { i - 1 } ( x , k - y , r - 1 ) \\\\ & \\quad \\Big ( A _ j ( n - i , k - i ) H _ { j + 1 } ( y , n - i , k - i ) + B _ j ( n - i , k - i ) H _ j ( y , n - i , k - i ) \\\\ & + C _ j ( n - i , k - i ) H _ { j - 1 } ( y , n - i , k - i ) \\Big ) . \\end{align*}"}
+{"id": "4068.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k ( - 1 ) ^ j y _ j = 0 . \\end{align*}"}
+{"id": "3898.png", "formula": "\\begin{align*} \\max \\{ n _ k , j _ k \\} < \\min \\{ n _ { k + 1 } , j _ { k + 1 } \\} , k = 1 , \\ldots , p - 1 . \\end{align*}"}
+{"id": "8409.png", "formula": "\\begin{align*} W ( v ) = \\sum _ { z = 1 } ^ { 2 Z - 1 } V _ { z } ( v ) + \\O \\left ( \\frac { N ^ { 2 \\gamma - 1 } } { d \\log ^ 4 N } \\right ) . \\end{align*}"}
+{"id": "3821.png", "formula": "\\begin{align*} \\hat \\eta : = ( { \\rm p r d } _ \\theta ) _ { \\# } ( \\theta ^ p \\tilde \\eta ) . \\end{align*}"}
+{"id": "3722.png", "formula": "\\begin{align*} \\| S \\| _ { \\mathcal { B } ( E ) } : = \\sup _ { x \\in E , \\ x \\neq 0 } \\dfrac { \\| S x \\| _ { E } } { \\| x \\| _ { E } } , \\ \\forall S \\in \\mathcal { B } ( E ) . \\end{align*}"}
+{"id": "1601.png", "formula": "\\begin{align*} d ( H ^ { ( i ) } | H ^ { ( i - 1 ) } ) : = \\begin{cases} \\frac { | E ( H ^ { ( i ) } ) \\cap \\mathcal { K } _ i ( H ^ { ( i - 1 ) } ) | } { | \\mathcal { K } _ i ( H ^ { ( i - 1 ) } ) | } & | \\mathcal { K } _ i ( H ^ { ( i - 1 ) } ) | > 0 , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "5705.png", "formula": "\\begin{align*} \\limsup \\limits _ { z \\rightarrow 0 } \\big | \\big ( ( z - B ' ) ^ { - 1 } f _ 0 \\big ) ( x _ 1 ) \\big | = + \\infty . \\end{align*}"}
+{"id": "8629.png", "formula": "\\begin{align*} \\mathcal { B } [ \\beta b ] \\bullet & = b \\nabla _ X ( \\nabla _ X \\cdot ( b \\bullet ) ) + h _ b \\nabla _ X \\big { ( } b \\nabla _ X \\cdot ( b \\bullet ) \\big { ) } + 2 h _ b ( \\nabla _ X b ) \\nabla _ X \\cdot ( b \\bullet ) , \\end{align*}"}
+{"id": "895.png", "formula": "\\begin{align*} \\frac { 1 - \\alpha } { t } \\tau - \\tau _ t + 2 \\xi _ x = 0 , \\end{align*}"}
+{"id": "97.png", "formula": "\\begin{align*} \\mathcal { K } ^ { \\mathrm { d i a g } } _ { \\varepsilon } = \\frac { \\varepsilon } { 2 } \\sum _ { k \\in \\mathcal P _ H } k ^ { 2 } a _ { k } ^ { \\dagger } a _ { k } \\end{align*}"}
+{"id": "453.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 + z ) ^ a } \\ , _ 2 \\tilde F _ 1 \\left ( \\frac a 2 , \\frac { a + 1 } { 2 } ; a - b + 1 ; \\frac { 4 z } { ( 1 + z ) ^ 2 } \\right ) = \\ , _ 2 \\tilde F _ 1 \\left ( a , b ; a - b + 1 ; z \\right ) , \\end{align*}"}
+{"id": "4985.png", "formula": "\\begin{align*} d _ Y ( a , y ) < \\rho ' ( a , y ) \\leqslant \\rho ' ( a , b ) + \\rho ' ( b , y ) = d _ Y ( a , b ) + d _ Y ( b , y ) . \\end{align*}"}
+{"id": "1746.png", "formula": "\\begin{align*} R ( \\xi ) = \\frac { f \\bigl ( \\cos ( \\xi ) \\bigr ) } { \\prod _ { 1 \\leq r \\leq d } \\bigl ( 1 - 2 a _ r \\cos ( \\xi ) + a _ r ^ 2 \\bigr ) } , \\end{align*}"}
+{"id": "2837.png", "formula": "\\begin{align*} \\hat { \\phi } = \\arg \\max _ { \\phi \\in [ 0 , 2 \\pi ) } \\ell ( \\phi ) . \\end{align*}"}
+{"id": "224.png", "formula": "\\begin{align*} a _ j \\equiv 2 \\ \\ ( j = 1 , \\ldots , n - 1 ) a _ n \\equiv { \\textstyle \\frac { 1 } { 4 } } \\end{align*}"}
+{"id": "7322.png", "formula": "\\begin{align*} \\| L ^ n _ \\lambda u \\| = \\| P _ n L _ \\lambda u \\| \\geq C \\| u \\| , u \\in H _ n . \\end{align*}"}
+{"id": "8533.png", "formula": "\\begin{align*} E : = \\left \\{ ( z , w ) \\in \\mathbb { R } \\times \\mathbb { R } ^ { n - 1 } : | w - g ( z ) e | < r _ { \\ell } ( z ) \\right \\} . \\end{align*}"}
+{"id": "3811.png", "formula": "\\begin{align*} f ( x ) & = \\sup _ { x ^ * \\in \\R ^ 2 _ + } x \\cdot x ^ * - f ^ * ( x ^ * ) = \\left ( \\sup _ { x _ 0 ^ * > 0 } x _ 0 ^ * x _ 0 - s _ 0 F ( x _ 0 ^ * ) \\right ) + \\left ( \\sup _ { x _ 1 ^ * > 0 } x _ 1 ^ * x _ 1 - s _ 1 F ( x _ 1 ^ * ) \\right ) \\\\ & = ( s _ 0 F ) ^ * ( x _ 0 ) + ( s _ 1 F ) ^ * ( x _ 1 ) \\equiv s _ 0 F ^ * \\left ( \\frac { \\phi _ 0 } { s _ 0 } \\right ) + s _ 1 F ^ * \\left ( \\frac { \\phi _ 1 } { s _ 1 } \\right ) \\end{align*}"}
+{"id": "1927.png", "formula": "\\begin{align*} \\left ( \\Pi _ { \\lambda _ 1 , \\lambda _ 2 } g \\right ) ^ { 1 - \\lambda _ 1 , \\lambda _ 2 } _ { i - \\frac { 1 } { 2 } , j + \\frac { 1 } { 2 } } = g ^ { 1 - \\lambda _ 1 , \\lambda _ 2 } _ { i - \\frac { 1 } { 2 } , j + \\frac { 1 } { 2 } } , \\mbox { i f } \\left ( i , j \\right ) \\in ( \\Bbbk _ 2 ^ + , \\Bbbk _ 1 ^ - ) \\end{align*}"}
+{"id": "1262.png", "formula": "\\begin{align*} E _ c = \\inf \\{ c : v _ 0 \\in L ( c ) , \\| v \\| _ { S ( I _ { \\max } ) } = \\infty \\} \\end{align*}"}
+{"id": "8668.png", "formula": "\\begin{align*} ( f , g ) ~ ~ ~ \\leftrightsquigarrow ~ ~ ~ G ( x _ 0 , . . . , x _ { n + 1 } ) = \\delta \\Big ( \\frac { x _ { n + 1 } } { x _ 0 } - f \\Big ( \\frac { x _ 1 } { x _ 0 } , . . . , \\frac { x _ n } { x _ 0 } \\Big ) \\Big ) ~ g \\Big ( \\frac { x _ 1 } { x _ 0 } , . . . , \\frac { x _ n } { x _ 0 } \\Big ) ~ x _ 0 ^ { - \\frac { n + 2 } { 2 } } \\end{align*}"}
+{"id": "6873.png", "formula": "\\begin{align*} w = \\log \\ , \\left ( \\sum _ i \\theta _ i ^ 2 \\exp ( w _ i ) \\right ) . \\end{align*}"}
+{"id": "798.png", "formula": "\\begin{align*} [ u ^ \\sharp ] _ { s , p } ^ p = \\int _ { \\mathbb { R } ^ N } ( K \\ast | u ^ \\sharp | ^ { p ^ \\sharp } ) | u ^ \\sharp | ^ { p ^ \\sharp } d x . \\end{align*}"}
+{"id": "6100.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta _ { k + 1 } & = \\xi ^ { k + 1 } \\Delta _ 0 + \\Delta _ 0 \\zeta \\sum _ { i = 0 } ^ { k } \\beta ^ i \\xi ^ { k - i } \\\\ & \\leq \\xi ^ { k + 1 } \\Delta _ 0 ( 1 + \\zeta \\frac { 1 - ( \\frac { \\beta } { \\xi } ) ^ { k + 1 } } { \\xi - \\beta } ) \\\\ & \\leq \\xi ^ { k + 1 } \\Delta _ 0 ( 1 + \\frac { \\zeta } { \\xi - \\beta } ) \\end{aligned} \\end{align*}"}
+{"id": "3319.png", "formula": "\\begin{align*} q ^ { b ( t - ( m - a ) ) } \\begin{bmatrix} m - a \\\\ t \\end{bmatrix} _ q \\alpha \\beta \\end{align*}"}
+{"id": "3228.png", "formula": "\\begin{align*} \\textnormal { A r e a } \\left ( \\textnormal { S e c t } ^ { + } _ { \\alpha , \\epsilon } ( R ' ) \\right ) & = \\theta \\cdot ( R ' ) ^ { 2 } = \\theta \\cdot \\left ( R + \\frac { \\sqrt { 2 } } { 2 \\theta } + O ( 1 ) \\right ) ^ { 2 } \\\\ & = \\textnormal { A r e a } ( \\textnormal { S e c t } _ { \\alpha , \\epsilon } ( R ) ) + O \\left ( R \\right ) , \\end{align*}"}
+{"id": "2370.png", "formula": "\\begin{align*} f ( \\tau ) = \\frac { o ( G ) } { o ( \\Lambda ) } , f ( \\pi ( \\mu ) ^ { - 1 } \\tau ) = 0 , \\forall \\mu \\in \\Lambda ^ 0 \\setminus \\{ ( 0 , 0 ) \\} . \\end{align*}"}
+{"id": "8320.png", "formula": "\\begin{align*} 1 = 2 t \\beta ^ { - 1 } - 2 t \\beta ^ { - 2 } + t \\beta ^ { - 3 } . \\end{align*}"}
+{"id": "3130.png", "formula": "\\begin{align*} \\mathbb { E } [ | \\mathbf { H } _ { k , } \\cdot \\mathbf { H } _ { l , } ^ H | ^ 2 ] = \\frac { L _ k L _ l } { N ^ 2 } \\left ( \\frac { \\kappa } { \\kappa + 1 } \\right ) ^ 2 . \\end{align*}"}
+{"id": "7203.png", "formula": "\\begin{align*} { \\bf C } ^ \\delta _ 0 : = C ( \\Lambda [ 0 , \\rho ] , \\delta ) , \\quad { \\bf C } ^ \\delta _ j : = C ( \\Lambda [ 2 ^ { j - 1 } \\rho , 2 ^ { j } \\rho ] , \\delta ) , \\textrm { f o r } j = 1 , \\dots , J . \\end{align*}"}
+{"id": "3584.png", "formula": "\\begin{align*} { \\bf F } _ { \\rm S M } = \\sqrt { \\frac { E } { N _ { \\rm R } } } { \\bf T } _ { \\rm S M } , { \\bf W } _ { \\rm S M } = { \\bf R } _ { \\rm S M } . \\end{align*}"}
+{"id": "2671.png", "formula": "\\begin{align*} N ( H _ r ) = p ^ { s - ( r + 1 ) } \\Big ( 1 - p ^ { - \\frac { R _ f } { 2 } } \\sum \\limits _ { ( u , - \\alpha ) \\in H _ { r } , u \\in \\Im L _ f } v ( f ( x _ { u } ) \\Big ) . \\end{align*}"}
+{"id": "268.png", "formula": "\\begin{align*} T \\overset { } { = } \\{ \\Delta ^ { \\{ 0 , 2 , 4 \\} } , \\ \\Delta ^ { \\{ 1 , 2 , 3 \\} } , \\ \\Delta ^ { \\{ 0 , 1 , 3 \\} } , \\ \\Delta ^ { \\{ 1 , 3 , 4 \\} } , \\ \\Delta ^ { \\{ 0 , 1 , 2 \\} } \\} ; \\end{align*}"}
+{"id": "1751.png", "formula": "\\begin{align*} ( x ) & = \\frac { ( 1 - \\epsilon _ + x ) ( 1 + \\epsilon _ - x ) } { 2 \\pi \\sqrt { 1 - x ^ 2 } \\prod _ { 1 \\leq r \\leq d } ( 1 + 2 a _ r x + a _ r ^ 2 ) } , \\\\ ^ { ( m + 1 ) } _ l & = \\Delta _ l ^ { ( m + 1 ) } \\frac { ( 1 - \\epsilon _ + x ^ { ( m + 1 ) } _ l ) ( 1 + \\epsilon _ - x ^ { ( m + 1 ) } _ l ) } { \\prod _ { 1 \\leq r \\leq d } ( 1 + 2 a _ r x ^ { ( m + 1 ) } _ l + a _ r ^ 2 ) } . \\end{align*}"}
+{"id": "1934.png", "formula": "\\begin{align*} \\begin{aligned} | \\mathcal { K } ^ 2 ( u _ h - v , f , \\theta _ f ) | \\leq & C \\left ( h ^ k \\| u _ h - u \\| _ { \\infty , I _ h } + h ^ { k + 1 } \\| u \\| _ { W ^ { 1 , \\infty } ( I ) } \\right ) \\| f \\| _ { k + 1 , \\Omega } \\| \\theta _ f \\| _ { 0 , \\mathcal { T } _ h } . \\end{aligned} \\end{align*}"}
+{"id": "8844.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } \\Big ( \\frac { r _ t - r _ { t + 1 } } { \\eta _ t } - \\frac { \\alpha } { 2 } r _ t \\Big ) = \\frac { \\alpha } { 2 } \\sum _ { t = 1 } ^ { T } \\Big ( r _ t ( t - 1 ) - r _ { t + 1 } t \\Big ) \\leq 0 \\enspace . \\end{align*}"}
+{"id": "380.png", "formula": "\\begin{align*} \\mathsf { T F i b } = \\mathcal { C } ^ \\pitchfork \\ , . \\end{align*}"}
+{"id": "4217.png", "formula": "\\begin{align*} 0 = N _ F ( e _ 1 , e _ 2 ) & = 2 ( A C - B \\mu ) e _ 1 + 2 ( A B - \\lambda C ) e _ 2 - ( 1 + A ^ 2 + B ^ 2 + C ^ 2 - \\lambda \\mu + \\lambda \\tau + \\mu \\tau ) e _ 3 , \\\\ [ 5 p t ] 0 = N _ F ( e _ 1 , e _ 3 ) & = - 2 ( A \\tau + B C ) e _ 1 - ( 1 - A ^ 2 - B ^ 2 + C ^ 2 + \\lambda \\mu - \\lambda \\tau + \\mu \\tau ) e _ 2 - 2 ( A B - \\lambda C ) e _ 3 , \\\\ [ 5 p t ] 0 = N _ F ( e _ 2 , e _ 3 ) & = ( 1 - A ^ 2 + B ^ 2 - C ^ 2 + \\lambda \\mu + \\lambda \\tau - \\mu \\tau ) e _ 1 + 2 ( A \\tau + B C ) e _ 2 + 2 ( A C - B \\mu ) e _ 3 . \\end{align*}"}
+{"id": "6420.png", "formula": "\\begin{align*} | \\alpha _ n ( x - s t ) | \\leq \\max \\{ - \\alpha ( - \\infty ) , 1 \\} = \\bar { \\alpha } . \\end{align*}"}
+{"id": "2840.png", "formula": "\\begin{align*} b \\ge \\frac { 1 } { 1 6 k } \\frac { \\binom { n - a - 2 } { k } } { \\binom { n - a - 2 } { k - 2 } } \\ge \\frac { 1 } { 3 2 k ^ 3 } ( n - a ) ^ { 2 } , \\end{align*}"}
+{"id": "3023.png", "formula": "\\begin{align*} g _ { n + 2 } ( \\lambda ) = \\lambda g _ { n + 1 } ( \\lambda ) - g _ { n } ( \\lambda ) \\end{align*}"}
+{"id": "5698.png", "formula": "\\begin{align*} T = \\begin{bmatrix} B & 0 \\\\ e _ 1 \\otimes f _ { 0 } & A \\end{bmatrix} \\end{align*}"}
+{"id": "2110.png", "formula": "\\begin{align*} \\begin{cases*} \\rho _ 1 : = Q _ 0 ( \\partial _ x ( \\ln \\alpha ) , \\tilde { \\Lambda } ) + Q _ 0 ( \\ln \\alpha , \\partial _ x \\tilde { \\Lambda } ) , \\\\ \\rho _ 2 : = - 2 \\Big [ \\sinh ( 2 \\lambda + 2 \\tilde { \\Lambda } ) \\left ( Q _ 0 ( \\partial _ x \\phi , \\phi ) + Q _ 0 ( \\phi , \\partial _ x \\phi ) \\right ) + 2 \\partial _ x \\tilde { \\Lambda } \\cosh ( 2 \\lambda + 2 \\tilde { \\Lambda } ) Q _ 0 ( \\phi , \\phi ) \\Big ] . \\end{cases*} \\end{align*}"}
+{"id": "8404.png", "formula": "\\begin{align*} V _ { z } ( v ) = \\sum _ { P < p \\leq 2 P } ( \\log p ) \\theta _ { z } ( p ^ c ) e \\left ( v \\left ( N + j - \\left [ p ^ { c } \\right ] + \\frac { z } { 2 Z } \\right ) ^ { \\gamma } \\right ) . \\end{align*}"}
+{"id": "2720.png", "formula": "\\begin{align*} \\binom { r + t - n } { t - n } - \\binom { r - 2 + t - n } { t - n } = \\binom { r + 1 } { 1 } - \\binom { r - 2 + 1 } { 1 } = 2 , \\end{align*}"}
+{"id": "6760.png", "formula": "\\begin{align*} P = \\left [ \\begin{array} { c c } 1 & 0 \\\\ 1 & 0 \\\\ 0 & 1 \\end{array} \\right ] , \\end{align*}"}
+{"id": "5637.png", "formula": "\\begin{align*} x _ 0 y _ 1 = x _ 1 y _ 0 + x _ 0 x _ 1 L + B . \\end{align*}"}
+{"id": "6140.png", "formula": "\\begin{align*} ( ( _ T \\times \\varphi ) \\circ ( \\varphi \\times _ S ) ) ( ( B , B ) ) = ( B , B ^ \\dag ) \\ne ( B ^ \\dag , B ) = ( ( \\varphi \\times _ T ) \\circ ( _ S \\times \\varphi ) ) ( ( B , B ) ) , \\end{align*}"}
+{"id": "7558.png", "formula": "\\begin{align*} L _ T ( x ) = L _ { T , d , N } ( x ) : = \\sum _ { k = 1 } ^ { N } | \\{ t \\in [ 0 , T ] : B ^ { ( k ) } _ t \\in \\mathbf { B } _ 1 ( x ) \\} | \\end{align*}"}
+{"id": "2607.png", "formula": "\\begin{align*} \\left ( \\det \\begin{bmatrix} \\alpha _ n & \\alpha _ { n + 1 } \\\\ \\alpha _ { n + 1 } & \\alpha _ { n + 2 } \\end{bmatrix} \\right ) _ { n \\in \\N _ 0 } . \\end{align*}"}
+{"id": "1812.png", "formula": "\\begin{gather*} \\int _ { H ( D ) } F \\ , d P _ { \\alpha , T } = \\frac { 1 } { T } \\int _ { 0 } ^ { T } F ( \\zeta ( s + i \\tau , \\alpha ) ) \\ , d \\tau , \\\\ \\int _ { H ( D ) } F \\ , d Q _ { \\alpha } = \\mathbf { E } \\left [ F ( \\zeta ( s , \\mathbb { X } _ \\alpha ) ) \\right ] \\end{gather*}"}
+{"id": "1388.png", "formula": "\\begin{align*} E P \\left ( x , D \\right ) \\upsilon = \\upsilon \\Omega \\upsilon \\in \\mathcal { E } ^ { ^ { \\prime } } \\left ( \\Omega \\right ) , \\end{align*}"}
+{"id": "3258.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ 2 ( \\mathbb { R } , C \\ell _ { p , q } ) } ^ 2 = \\int _ \\mathbb { R } | f ( x ) | ^ 2 d x . \\end{align*}"}
+{"id": "415.png", "formula": "\\begin{align*} H ^ s ( \\R ^ d ) : = \\left \\{ f \\in L ^ 2 ( \\R ^ d ) : \\ , \\int _ { \\R ^ d } | \\xi | ^ { 2 s } | \\hat f ( \\xi ) | ^ 2 \\ , d \\xi < \\infty \\right \\} , s \\geq 0 . \\end{align*}"}
+{"id": "3364.png", "formula": "\\begin{align*} d _ D ( \\mu , \\nu ) \\geq | \\mu ( R a ) - \\nu ( R a ) | = R | \\mu ( a ) - \\nu ( a ) | \\end{align*}"}
+{"id": "3936.png", "formula": "\\begin{align*} \\lambda ( f ^ { - 1 } ( \\omega ) \\triangle g ^ { - 1 } ( \\omega ) ) = 0 , \\end{align*}"}
+{"id": "2877.png", "formula": "\\begin{align*} ( \\mathcal J f ) ( \\lambda ) : = t [ \\lambda ] ^ { - 1 } ( T _ { w _ \\lambda } f ) ( \\lambda _ + ) . \\end{align*}"}
+{"id": "8336.png", "formula": "\\begin{align*} \\mathbf { s } _ { \\mathbf { S B } } ^ * ( 1 ) : = \\mathbf { s } _ { \\mathbf { S B } } ^ * ( 0 ) \\mathbf { P } _ { S M } + \\mathbf { r } _ { \\mathbf { S B } } ^ * ( 0 ) \\end{align*}"}
+{"id": "3168.png", "formula": "\\begin{gather*} \\partial _ s u + J _ t ( \\partial _ t u - X _ H ( u ) ) = 0 , \\hbox { a n d } \\ \\int _ Z \\abs { d u } ^ 2 < + \\infty . \\end{gather*}"}
+{"id": "3460.png", "formula": "\\begin{align*} \\begin{cases} ( 1 + d _ 0 u ' ) ^ { n - 1 } u '' = \\frac { 1 } { n d _ 0 } + \\frac { 1 } { n d _ 1 } , \\Delta _ 1 , \\\\ ( 1 - d _ 1 u ' ) ^ { n - 1 } u '' = \\frac { 1 } { n d _ 0 } + \\frac { 1 } { n d _ 1 } , \\Delta _ 0 , \\end{cases} \\end{align*}"}
+{"id": "668.png", "formula": "\\begin{align*} \\norm { f } _ { H ^ b ( \\R ) } ^ 2 = \\norm { f } _ { L ^ 2 ( \\R ) } ^ 2 + c _ b \\norm { f } _ { S ^ b ( \\R ) } ^ 2 , \\end{align*}"}
+{"id": "5110.png", "formula": "\\begin{align*} \\N _ K = 0 \\Longleftrightarrow \\left \\{ \\begin{array} { l } \\N _ r = 0 , \\\\ D _ X ( l ( \\sigma ) ) - l ( D _ X ( \\sigma ) ) = 0 , \\\\ l ( D _ { [ X , Y ] } ( \\sigma ) ) - [ D _ X , D _ Y ] ( \\sigma ) - D _ { [ X , Y ] _ r } ( \\sigma ) = 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "6287.png", "formula": "\\begin{align*} c ^ 4 _ { \\lambda , m } + i \\Delta ^ 4 \\xi ^ 2 \\ , b ^ 4 _ { \\lambda , m } = i \\Delta ^ 4 \\xi \\ , r ^ 4 _ { \\lambda , m } . \\end{align*}"}
+{"id": "862.png", "formula": "\\begin{align*} \\| u \\| _ { W _ s ^ { \\alpha , G } ( X , d , \\mu ) } : = \\| u \\| _ { L ^ G ( X , \\mu ) } + [ u ] _ { W _ s ^ { \\alpha , G } ( X , d , \\mu ) } , \\end{align*}"}
+{"id": "7536.png", "formula": "\\begin{align*} Z ^ s : = Z \\cap ( W ^ s ) ^ n . \\end{align*}"}
+{"id": "69.png", "formula": "\\begin{align*} \\mathcal Q _ 3 ^ { \\rm { s o f t } } = \\int _ { \\mathbb { C } } \\mathcal Q _ 3 ^ { \\rm { s o f t } } ( z ) \\vert z \\rangle \\langle z \\rvert \\dd z , \\end{align*}"}
+{"id": "222.png", "formula": "\\begin{align*} L _ x ^ { } = \\sum _ { 1 \\leq j \\leq n } \\frac { \\partial ^ 2 } { \\partial x _ j ^ 2 } & - \\sum _ { 1 \\leq j < n } a _ j e ^ { - x _ j + x _ { j + 1 } } - a _ { n - 1 } e ^ { - x _ { n - 1 } - x _ n } \\\\ & - \\frac { \\frac { 1 } { 4 } g _ S ( g _ S + 2 g _ L - 1 ) } { \\sinh ^ 2 \\textstyle { \\frac { 1 } { 2 } } ( x _ n ) } - \\frac { g _ L ( g _ L - 1 ) } { \\sinh ^ 2 ( x _ n ) } \\end{align*}"}
+{"id": "368.png", "formula": "\\begin{align*} \\overline { S } \\psi _ 1 = \\eta _ 1 \\overline { R } \\textrm { a n d } \\psi _ 1 \\overline { \\phi } _ { \\overline { g } } = \\overline { \\varphi } _ { \\eta _ 1 ( \\overline { g } ) } \\psi _ 1 . \\end{align*}"}
+{"id": "291.png", "formula": "\\begin{align*} \\mathsf { L } _ { j } ^ { \\sigma } \\left ( s \\right ) w : = - \\operatorname { d i v } \\left ( \\mathbb { A } _ { j } ^ { \\sigma } \\nabla w \\right ) + s ^ { 2 } p _ { j } ^ { \\sigma } w \\quad \\Omega _ { j } ^ { \\sigma } , \\end{align*}"}
+{"id": "7390.png", "formula": "\\begin{align*} \\frac { \\lambda _ { n } ' ( \\rho _ { n } ) \\rho _ { n } + \\lambda _ { n } ( \\rho _ { n } ) } { ( \\lambda _ { n } ( \\rho _ { n } ) ) ^ { 2 } } = \\frac { \\gamma _ { n } + 2 } { \\gamma _ { n } \\rho _ { n } ^ { \\gamma _ { n } + 1 } } . \\end{align*}"}
+{"id": "6433.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow + \\infty } \\{ \\sup _ { t \\geq t _ n , | x - c t | \\leq R } \\tilde { u } _ n ( x , t ) \\} = 1 , R > 0 . \\end{align*}"}
+{"id": "773.png", "formula": "\\begin{align*} ~ ~ \\Phi ( g _ 1 ) | V = \\Phi ( g _ 2 ) | V , ~ ~ ~ \\Phi ( g _ 1 ) | U = \\Phi ( g _ 2 ) | U . \\end{align*}"}
+{"id": "3193.png", "formula": "\\begin{align*} B _ { k } = \\left [ \\begin{array} { c c c c c } \\alpha _ { 1 } \\\\ \\beta _ { 2 } & \\ddots \\\\ & \\ddots & \\ddots \\\\ & & \\beta _ { k - 1 } & \\alpha _ { k - 1 } \\\\ & & & \\beta _ { k } & \\alpha _ { k } \\\\ & & & & \\beta _ { k + 1 } \\end{array} \\right ] . \\end{align*}"}
+{"id": "5783.png", "formula": "\\begin{align*} \\lim _ { J _ 1 ^ 0 \\rightarrow 0 } \\lim _ { L \\rightarrow \\infty } \\langle ( { R _ { 1 , 2 } ^ c } - { \\mathbb E } \\langle R _ { 1 , 2 } ^ c \\rangle ) ^ 2 \\rangle = 0 , \\end{align*}"}
+{"id": "6792.png", "formula": "\\begin{align*} \\mathcal { V } \\left ( \\underline { u } , \\underline { v } \\right ) = & d \\overline { v } '' + d r q ( 1 ) e ^ { \\lambda z } \\left [ \\lambda ( - z ) ^ { - 1 / 2 } + \\frac { 1 } { 4 } ( - z ) ^ { - 3 / 2 } - \\lambda ^ 2 ( - z ) ^ { 1 / 2 } \\right ] \\\\ [ 0 . 2 c m ] & - c \\overline { v } ' - c r q ( 1 ) e ^ { \\lambda z } \\left [ \\frac { 1 } { 2 } ( - z ) ^ { - 1 / 2 } - \\lambda ( - z ) ^ { 1 / 2 } \\right ] + s \\overline { v } - s r q ( 1 ) ( - z ) ^ { 1 / 2 } e ^ { \\lambda z } - \\frac { s \\underline { v } ^ 2 } { q ( \\underline { u } ) } . \\end{align*}"}
+{"id": "7278.png", "formula": "\\begin{align*} D _ i = \\{ n \\in D \\mid i \\} . \\end{align*}"}
+{"id": "4609.png", "formula": "\\begin{align*} \\mathcal { H } _ \\chi : = \\{ h \\in \\mathcal { H } \\ , \\ , | \\ , \\ , y \\cdot h = \\chi ( y ) h \\forall \\ , y \\in I \\} \\end{align*}"}
+{"id": "8933.png", "formula": "\\begin{align*} ( ( A , v ) , \\Lambda ) \\mapsto A . \\Lambda + v = \\{ A . w + v : w \\in \\Lambda \\} , \\end{align*}"}
+{"id": "5401.png", "formula": "\\begin{align*} \\widetilde { a } _ { k \\ell } = \\overline { c _ k } c _ { \\ell } a _ { k \\ell } . \\end{align*}"}
+{"id": "4873.png", "formula": "\\begin{align*} T = \\begin{bmatrix} 0 & 0 & 0 & 0 \\\\ 0 & - 1 & 1 & 0 \\\\ 0 & 1 & - 1 & 0 \\\\ 0 & 0 & 0 & 0 \\end{bmatrix} \\end{align*}"}
+{"id": "1303.png", "formula": "\\begin{align*} \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| e ^ { i t \\Delta } ( u _ { 0 , n } - u _ { n } ^ J ( 0 ) ) \\| _ { S ( \\R ) } \\leq \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| e ^ { i t \\Delta } w _ n ^ J \\| _ { S ( \\R ) } = 0 . \\end{align*}"}
+{"id": "1317.png", "formula": "\\begin{align*} J ( X ) = - \\frac { 1 } { 2 } \\int _ { 0 } ^ { \\infty } f ^ 2 ( x ) d x = - \\frac { 1 } { 2 } E \\left ( f ( X ) \\right ) . \\end{align*}"}
+{"id": "6160.png", "formula": "\\begin{gather*} \\forall s \\in \\textrm { M T } ( \\mathbf { k } ) , \\ , \\ , \\eta _ ( s ) = s , \\\\ \\forall s = \\left ( \\mathbf { k } ^ { ( n ) } , \\mu ^ { ( n ) } , \\mathbf { k } ^ { ( n - 1 ) } , \\mu ^ { ( n - 1 ) } , \\ldots , \\mathbf { k } ^ { ( 1 ) } , \\mu ^ { ( 1 ) } \\right ) \\in \\textrm { G M T } ( \\mathbf { k } ) , \\ , \\ , \\eta _ ( s ) = ( \\mathbf { k } _ 1 , \\mathbf { k } _ 2 , \\ldots , \\mathbf { k } _ n ) . \\end{gather*}"}
+{"id": "2152.png", "formula": "\\begin{align*} \\beta _ x - \\beta _ t = - 2 \\tilde \\alpha _ 0 ' ( - 2 \\underline u ) + 2 \\alpha _ 1 ( - 2 \\underline u ) > 0 , \\end{align*}"}
+{"id": "2081.png", "formula": "\\begin{align*} \\det g = \\alpha ^ 2 . \\end{align*}"}
+{"id": "1023.png", "formula": "\\begin{align*} ( \\pi ^ * a ) ( \\pi ^ * b ) & = \\tfrac 1 { k ! } g ( a , v ^ k ) \\tfrac 1 { l ! } g ( b , v ^ l ) = g ( a _ 1 , v ) \\dotsm g ( a _ k , v ) g ( b _ 1 , v ) \\dotsm g ( b _ l , v ) \\\\ & = \\tfrac 1 { ( k + l ) ! } g ( a \\cdot b , v ^ { k + l } ) = \\pi ^ * ( a \\cdot b ) , \\end{align*}"}
+{"id": "6469.png", "formula": "\\begin{align*} \\lambda _ n \\big ( R _ \\epsilon \\big ) = 0 + \\nu _ n \\epsilon ^ 2 + O ( \\epsilon ^ 3 ) , \\end{align*}"}
+{"id": "1456.png", "formula": "\\begin{align*} 2 x ^ 2 - 2 \\big ( n _ { { i _ 0 } - 1 , { j } } + n _ { { i _ 0 } + 1 , { j } } \\big ) x + n _ { { i _ 0 } - 1 , { j } } ^ 2 + n _ { { i _ 0 } + 1 , { j } } ^ 2 + \\sum \\limits _ { i \\in I \\setminus \\{ { i _ 0 } - 1 , { i _ 0 } \\} } \\big ( n _ { i , { j } } - n _ { i + 1 , { j } } \\big ) ^ { 2 } - 2 n _ { j j } = 0 . \\end{align*}"}
+{"id": "4952.png", "formula": "\\begin{align*} t _ k ( n ) = - \\dfrac { 1 } { E _ { 2 k } } \\cdot \\sigma _ { \\chi _ { - 4 } ; 2 k } ( 2 n + k + 1 ) + a _ k ( 2 n + k + 1 ) . \\end{align*}"}
+{"id": "7571.png", "formula": "\\begin{align*} \\kappa = \\beta ^ { \\frac { 1 } { d + 2 } } N ^ { \\frac { 1 } { d + 2 } } , \\alpha = - \\frac { d - 1 } { d + 2 } , d = 1 , 2 , 3 . \\end{align*}"}
+{"id": "1503.png", "formula": "\\begin{align*} & X _ { t } = W _ { 0 } ( t ) , & A _ { 1 } ( t ) = W _ { 1 } ( \\tau ( t ) ) , & & \\ldots \\ldots & & A _ { n } ( t ) = W _ { n } ( \\tau ( t ) ) , \\end{align*}"}
+{"id": "3572.png", "formula": "\\begin{align*} { \\bf { H } } = \\sum \\nolimits _ { k \\in { \\mathcal K } } { { \\rho _ k } { { \\bf { H } } _ { k , { \\rm { R } } } } { { \\bf { \\Gamma } } _ k } { { \\bf { H } } _ { { \\rm { T } } , k } } } , \\end{align*}"}
+{"id": "6305.png", "formula": "\\begin{align*} i \\partial _ t v _ \\lambda + \\partial _ x g _ { [ < \\lambda ] } \\partial _ x v _ \\lambda = f _ \\lambda , v _ \\lambda ( 0 ) = v _ { 0 , \\lambda } . \\end{align*}"}
+{"id": "994.png", "formula": "\\begin{align*} \\mathbb { P } ( Z ' = 0 ) & = \\sum _ { s = 0 } ^ \\ell \\mathbb { P } ( Z ' = 0 , Z = s ) = \\sum _ { s = 0 } ^ \\ell \\mathbb { P } ( Z ' = 0 \\ \\big | \\ Z = s ) \\mathbb { P } ( Z = s ) \\\\ & \\le \\sum _ { s = 0 } ^ \\ell ( 1 - \\beta _ k ) ^ s \\mathbb { P } ( Z = s ) = \\sum _ { s < c \\ell } ( 1 - \\beta _ k ) ^ s \\mathbb { P } ( Z = s ) + \\sum _ { s \\ge c \\ell } ( 1 - \\beta _ k ) ^ s \\mathbb { P } ( Z = s ) \\\\ & \\le \\mathbb { P } ( Z < c \\ell ) + ( 1 - \\beta _ k ) ^ { c \\ell } \\mathbb { P } ( Z \\ge c \\ell ) . \\end{align*}"}
+{"id": "3213.png", "formula": "\\begin{align*} \\| x - x _ { { \\ell } } \\| ^ 2 = \\sum _ { j = { \\ell } } ^ { { k } } \\zeta ^ 2 _ { j + 1 } + \\| x - x _ { { k + 1 } } \\| ^ 2 \\end{align*}"}
+{"id": "6599.png", "formula": "\\begin{align*} b _ f ( t , x , v ) = ( v , E _ f ( t , x ) ) \\ t \\in [ 0 , T ] , \\ x , v \\in \\R ^ d . \\end{align*}"}
+{"id": "7316.png", "formula": "\\begin{align*} ( \\pi _ { Q _ S } \\circ \\sigma _ V ) ( T ) & = \\pi _ { Q _ S } ( h \\cdot \\sigma ( \\tau _ { h ^ { - 1 } } ( T ) ) ) = \\tau _ h ( \\pi _ { Q _ S } ( \\sigma ( \\tau _ { h ^ { - 1 } } ( T ) ) ) ) \\\\ & = \\tau _ h ( \\tau _ { h ^ { - 1 } } ( T ) ) = T , T \\in \\mathcal { U } _ V , \\end{align*}"}
+{"id": "6562.png", "formula": "\\begin{align*} \\widetilde T _ Q ( E ; \\theta ) = R _ Q ( \\cos ( \\theta + n \\cdot \\alpha ) \\delta _ { n , n ' } + m - E + \\varepsilon \\Delta ) R _ Q . \\end{align*}"}
+{"id": "5459.png", "formula": "\\begin{align*} | f ( x ) | & = | \\int _ 0 ^ x f ' ( t ) e ^ { t ^ 2 / 2 } e ^ { - t ^ 2 / 2 } d t | \\\\ & \\leq \\int _ 0 ^ x G ( | f ' ( t ) | ) e ^ { - t ^ 2 / 2 } d t + \\int _ 0 ^ x G ^ * ( e ^ { t ^ 2 / 2 } ) e ^ { - t ^ 2 / 2 } d t \\end{align*}"}
+{"id": "7607.png", "formula": "\\begin{align*} H _ 1 ( \\beta , N , T ) & \\le C N ^ 2 \\int _ { 0 } ^ { 2 T } C N ^ { - 1 / 3 } \\beta ^ { - 1 / 3 } d b \\\\ & \\le C \\beta ^ { - 1 / 3 } N ^ { 5 / 3 } T ^ { } . \\end{align*}"}
+{"id": "4147.png", "formula": "\\begin{align*} \\eta \\rhd c & = t ( c ; \\eta ) , \\\\ \\eta \\rhd t ( c ; \\tau _ 1 , \\ldots , \\tau _ r ) & = t ( c ; \\eta , \\tau _ 1 , \\ldots , \\tau _ r ) + t ( c ; \\eta \\rhd \\tau _ 1 , \\ldots , \\tau _ r ) + \\ldots + t ( c ; \\tau _ 1 , \\ldots , \\eta \\rhd \\tau _ r ) , \\end{align*}"}
+{"id": "3659.png", "formula": "\\begin{align*} p _ \\mu & = \\eta ^ { \\mu \\nu } \\phi _ { \\nu } , \\mu = 1 , \\ldots , n \\ , \\\\ e & = - \\frac { 1 } { 2 } \\eta ^ { \\mu \\nu } \\phi _ \\mu \\phi _ \\nu - \\frac { 1 } { 2 } m ^ 2 \\phi ^ 2 \\ , \\end{align*}"}
+{"id": "567.png", "formula": "\\begin{align*} \\mathbf u ( t ) = \\mathbf S ( t - S ) \\mathbf u ( S ) + i \\int _ S ^ t \\mathbf S ( t - s ) \\mathbf N \\left ( \\Theta _ R ^ { \\mathbf u } ( s ) \\mathbf u ( s ) \\right ) \\ , d s + i \\int _ S ^ t \\mathbf S ( t - s ) \\mathbf M \\left ( \\mathbf u ( s ) \\right ) \\ , d W ( s ) \\end{align*}"}
+{"id": "7809.png", "formula": "\\begin{align*} \\widetilde { N } _ S = \\{ p \\in \\widetilde { N } \\ ; | \\ ; \\epsilon < \\tau _ 2 , \\ ; \\ ; \\epsilon < \\frac { \\tau _ 2 } { | \\tau _ 1 | ^ 2 + | \\tau _ 2 | ^ 2 } , \\ ; \\ ; t ^ a > K , \\ ; \\ ; | \\tau | t ^ a > K \\} \\end{align*}"}
+{"id": "3944.png", "formula": "\\begin{align*} x ' v + c & = ( x ' + c v ) v = r _ K ( w ) v = h _ K ( v ) > r _ K ( u ) v = x _ 0 v \\Leftrightarrow \\\\ c & > x ' v - x _ 0 v = 0 \\end{align*}"}
+{"id": "6456.png", "formula": "\\begin{align*} \\| F ^ { ( n ) } ( x ) \\| \\le A n ! \\frac { ( 2 k r ) ^ n } { r ^ n M _ n ^ * } = A ( 2 k ) ^ n M _ n . \\end{align*}"}
+{"id": "4570.png", "formula": "\\begin{align*} \\vert \\mathcal J _ { 1 , 1 } + \\mathcal J _ { 1 , 3 } \\vert & \\le C _ 3 \\left \\{ \\Vert { \\mathbf V } \\Vert ^ 2 _ { L ^ 2 ( \\Omega _ t ) } + \\Vert \\partial _ s { \\mathbf V } \\Vert ^ 2 _ { L ^ 2 ( \\Omega _ t ) } + \\Vert \\partial _ 1 { \\mathbf V } _ n \\Vert ^ 2 _ { L ^ 2 ( \\Omega _ t ) } \\right \\} \\\\ & \\le C _ 3 \\int _ 0 ^ t ( I _ { 1 , \\ast } + I _ { 1 , n } ) ( s ) d s \\ , . \\end{align*}"}
+{"id": "7503.png", "formula": "\\begin{align*} G _ 4 ( \\gamma ) = \\frac { \\gamma ^ 2 - 3 \\gamma + 1 } { \\gamma ^ 2 } . \\end{align*}"}
+{"id": "6992.png", "formula": "\\begin{align*} C _ 1 = \\frac { 1 } { 2 } ( h + 1 ) C , X _ 1 = \\frac { 1 } { 2 } ( h + 1 ) X , h = X ^ { - n } C ^ { - m } . \\end{align*}"}
+{"id": "6265.png", "formula": "\\begin{align*} \\begin{aligned} \\| P _ { \\mu } ( u _ { \\mu _ 1 } u _ { \\mu _ 2 } g ( u _ { < \\mu _ 2 } ) \\| _ { L ^ 2 } \\lesssim & \\ \\| u _ { \\mu _ 1 } \\| _ { L ^ 6 } \\| u _ { \\mu _ 2 } \\| _ { L ^ 6 } \\| P _ { \\approx \\mu } g '' ( u _ { < \\mu _ 2 } ) \\| _ { L ^ 6 } \\\\ \\lesssim & \\ C ^ 3 \\epsilon ^ 3 c _ { \\mu _ 1 } c _ { \\mu _ 2 } ^ 2 \\mu _ 1 ^ { - s } \\mu _ 2 ^ { - s } \\left ( \\frac { \\mu _ 2 } { \\mu } \\right ) ^ N . \\end{aligned} \\end{align*}"}
+{"id": "1636.png", "formula": "\\begin{align*} T = Q Q ^ { - 1 } T = \\sqrt { q t } . \\end{align*}"}
+{"id": "7524.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t n + \\partial _ x ( n u ) = 0 , \\\\ & \\partial _ t \\rho + \\partial _ x ( \\rho u ) = 0 , \\\\ & \\partial _ t ( r u ) + \\partial _ x \\left \\{ r u ^ 2 + p ( n ) \\right \\} = 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "8042.png", "formula": "\\begin{align*} h _ { \\omega } ^ { p } ( \\mathbb { R } ^ { n } ) = \\{ f \\in \\mathcal { S } ^ { \\prime } ( \\mathbb { R } ^ { n } ) \\colon \\left \\| f \\right \\| _ { h _ { \\omega } ^ { p } ( \\mathbb { R } ^ { n } ) } < \\infty \\} , \\end{align*}"}
+{"id": "1426.png", "formula": "\\begin{align*} 2 \\frac { \\partial ^ 2 w _ i } { \\partial z \\partial \\bar z } + e ^ { 2 ( w _ i - w _ { i - 1 } ) } - e ^ { 2 ( w _ { i + 1 } - w _ i ) } = 0 , \\ \\forall \\ i \\in I , \\footnote { H e r e w e s e t $ w _ { i } = w _ { n + 1 + i } $ f o r a n y $ i \\in \\mathbb { N } \\cup \\{ 0 \\} $ . } \\end{align*}"}
+{"id": "8559.png", "formula": "\\begin{align*} \\widehat { \\mathrm { F } ( \\mathrm { D } ) u } ( \\xi ) = F ( \\xi ) \\widehat { u } ( \\xi ) . \\end{align*}"}
+{"id": "4721.png", "formula": "\\begin{align*} B = \\bigcup _ { ( x , y , s , t ) } B _ { ( x , y , s , t ) } , \\end{align*}"}
+{"id": "7249.png", "formula": "\\begin{align*} K = \\bigcup _ { i = 1 } ^ n f _ i ( K ) . \\end{align*}"}
+{"id": "2545.png", "formula": "\\begin{align*} L ^ { q , \\infty } ( \\Omega ) & \\coloneqq \\{ f \\mid , \\sup _ { t > 0 } t \\lambda _ f ( t ) ^ { 1 / q } < \\infty \\} \\\\ & = \\{ f \\mid , \\| f \\| _ { L ^ { q , \\infty } ( \\Omega ) } \\coloneqq \\sup _ { s > 0 } s ^ { 1 / q } f ^ * ( s ) < \\infty \\} , \\end{align*}"}
+{"id": "4017.png", "formula": "\\begin{align*} \\chi ^ n = g _ 1 \\chi ^ m \\wedge \\omega ^ { n - m } . \\end{align*}"}
+{"id": "7489.png", "formula": "\\begin{align*} y ^ { \\sigma ^ 2 } - ( 1 + \\lambda ) y ^ { \\sigma } + \\lambda y = 0 . \\end{align*}"}
+{"id": "3996.png", "formula": "\\begin{align*} \\omega _ { i \\bar j } = \\omega \\left ( \\frac { \\partial } { \\partial z ^ i } , \\frac { \\partial } { \\partial \\bar z ^ j } \\right ) , \\left [ \\omega ^ { i \\bar j } \\right ] = \\left [ \\omega _ { i \\bar j } \\right ] ^ { - 1 } , \\end{align*}"}
+{"id": "6912.png", "formula": "\\begin{align*} \\mathcal { N } _ 1 \\left [ \\underline { \\phi } \\right ] ( \\xi ) = & \\int _ { \\mathbb R } J _ i ( \\xi - y ) \\underline { \\phi } ( y ) d y - \\underline { \\phi } ( \\xi ) \\\\ [ 0 . 2 c m ] \\geq & 1 - \\frac { 1 - b } { 2 } e ^ { \\eta \\xi } \\int _ { \\mathbb R } J _ 1 ( y ) e ^ { - \\eta y } d y - \\left ( 1 - \\displaystyle \\frac { 1 - b } { 2 } e ^ { \\eta \\xi } \\right ) \\\\ [ 0 . 2 c m ] = & \\frac { 1 - b } { 2 } e ^ { \\eta \\xi } \\left ( 1 - \\int _ { \\mathbb R } J _ 1 ( y ) e ^ { - \\eta y } d y \\right ) . \\end{align*}"}
+{"id": "4373.png", "formula": "\\begin{align*} \\bar { \\mathbf { U } } ( \\mathbf { x } ) : = \\begin{cases} \\bar { \\mathbf { U } } ^ + : = ( \\bar { p } ^ + , 0 , \\bar { u } ^ + _ 2 , 0 , \\bar { H } ^ + _ 2 , \\bar { S } ^ + ) , & x _ 1 > 0 , \\\\ \\bar { \\mathbf { U } } ^ - : = ( \\bar { p } ^ - , 0 , \\bar { u } ^ - _ 2 , 0 , \\bar { H } ^ - _ 2 , \\bar { S } ^ - ) , & x _ 1 < 0 , \\\\ \\end{cases} \\end{align*}"}
+{"id": "4637.png", "formula": "\\begin{align*} \\mathbf { S } : = L \\otimes S _ 2 \\otimes \\cdots \\otimes S _ n \\otimes N \\end{align*}"}
+{"id": "6806.png", "formula": "\\begin{align*} & \\psi _ 1 ( z ) = y ( z ) - \\frac { c } { 2 d } v ( z ) , \\\\ [ 0 . 2 c m ] & \\psi _ 2 ( z ) = y ( z ) + \\frac { s q ( 1 ) } { c q ( 0 ) } v ( z ) . \\end{align*}"}
+{"id": "1149.png", "formula": "\\begin{align*} C ^ 0 _ { c o m } ( \\mathfrak { g } , M ) : = ~ & \\{ m \\in M ~ | ~ x \\cdot _ 1 m - m \\cdot _ 1 x = x \\cdot _ 2 m - m \\cdot _ 2 x , ~ \\forall x \\in \\mathfrak { g } \\} , \\\\ C ^ n _ { c o m } ( \\mathfrak { g } , M ) : = ~ & \\underbrace { C ^ n ( \\mathfrak { g } , M ) \\oplus \\cdots \\oplus C ^ n ( \\mathfrak { g } , M ) } _ { n } , ~ n \\geq 1 . \\end{align*}"}
+{"id": "5761.png", "formula": "\\begin{align*} \\mathbb E \\langle H \\rangle = - \\sum _ { p \\in Q } | { \\cal B } _ p | \\mu _ p . \\end{align*}"}
+{"id": "8054.png", "formula": "\\begin{align*} [ \\omega ] _ { A _ { p } } = \\sup \\limits _ { Q } \\left ( \\frac { 1 } { Q } \\int _ { Q } \\omega ( x ) d x \\right ) \\left ( \\frac { 1 } { Q } \\int _ { Q } \\omega ( x ) ^ { - \\frac { 1 } { p - 1 } } d x \\right ) ^ { p - 1 } < \\infty , \\end{align*}"}
+{"id": "3055.png", "formula": "\\begin{align*} { { \\bf { H } } _ { { \\rm { T } } , k } } = \\sum \\nolimits _ { l \\in { \\mathcal L } _ { { \\rm T } , k } } { { \\alpha _ { { \\rm { T } } , k , l } } { { \\bf { a } } _ { { \\rm { S } } , k } } \\left ( { \\Theta _ { { \\rm { T } } , k , l } ^ { \\rm { A } } } \\right ) { \\bf { a } } _ { \\rm { T } } ^ H \\left ( { \\Theta _ { { \\rm { T } } , k , l } ^ { \\rm { D } } } \\right ) } , \\end{align*}"}
+{"id": "2796.png", "formula": "\\begin{align*} & \\int _ { \\Omega } \\Big ( ( x \\cdot \\nabla v ) \\Delta u + ( x \\cdot \\nabla u ) \\Delta v \\Big ) \\dd x \\\\ & = ( N - 2 ) \\int _ { \\Omega } \\nabla u \\cdot \\nabla v \\dd x + \\int _ { \\partial \\Omega } \\Big ( \\partial _ \\nu u ( x \\cdot \\nabla v ) + \\partial _ \\nu v ( x \\cdot \\nabla u ) - ( \\nabla u \\cdot \\nabla v ) ( x \\cdot \\nu ) \\Big ) \\dd \\sigma , \\end{align*}"}
+{"id": "7244.png", "formula": "\\begin{align*} \\gamma = \\sqrt { \\alpha ^ 2 + 8 \\Delta ^ 2 \\delta ^ 2 } . \\end{align*}"}
+{"id": "2094.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } ^ 2 \\alpha - \\partial _ { x } ^ 2 \\alpha = 0 \\\\ ( \\alpha , \\partial _ t \\alpha ) | _ { t = 0 } = ( 1 + \\tilde \\alpha _ 0 , \\alpha _ 1 ) \\\\ \\mbox { w i t h } ( \\tilde \\alpha _ 0 , \\alpha _ 1 ) \\in C _ c ^ { \\infty } ( \\mathbb { R } ) \\times \\mathcal { S } ( \\mathbb { R } ) . \\end{cases} \\end{align*}"}
+{"id": "8123.png", "formula": "\\begin{align*} | C _ 2 | = | C _ { \\ell , S } | . \\end{align*}"}
+{"id": "4587.png", "formula": "\\begin{align*} Y ( | 0 \\rangle , z ) \\ , a = a Y ( a , z ) | 0 \\rangle \\in a \\ , { + } \\ , z \\ , \\mathfrak { V } [ [ z ] ] ~ a \\ , { \\in } \\ , \\mathfrak { V } \\ , , \\end{align*}"}
+{"id": "1090.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - \\Delta _ { \\mu } u ( x ) = f ( x , u ( x ) ) x \\in \\mathop D \\limits ^ \\circ \\\\ \\medskip \\ , \\ , u | _ { \\partial D } = 0 , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "8698.png", "formula": "\\begin{align*} A _ 1 = 0 , ~ A _ 2 = 0 , \\dots \\end{align*}"}
+{"id": "4682.png", "formula": "\\begin{align*} | H | & = n - \\sum \\limits _ { i = 1 } ^ { 2 l + 1 } ( | S _ i | + | T _ i | ) - | V ( C _ { 2 l + 1 } ) | \\\\ & \\leq n - ( 2 l + 1 ) \\left ( \\frac { n } { 2 ( 2 l + 1 ) } - 2 + \\frac { n } { 2 ( 2 l + 1 ) } - 1 \\right ) - ( 2 l + 1 ) \\\\ & \\leq 2 ( 2 l + 1 ) . \\end{align*}"}
+{"id": "3717.png", "formula": "\\begin{align*} | u ^ \\infty ( t , x ) - u ^ \\infty ( t , x ' ) | \\le C _ \\alpha ' \\sum _ { n = 1 } ^ { N - 1 } a _ n ^ 3 \\lambda _ \\beta ( 2 ^ { - 4 n - 3 } ) ^ { - 2 - 2 \\alpha } | x - x ' | + C _ \\alpha ' \\sum _ { n = N } ^ { \\infty } a _ n ^ { 2 - 2 \\alpha } \\end{align*}"}
+{"id": "1228.png", "formula": "\\begin{align*} E _ c = \\sup \\{ c : v _ 0 \\in L ( c ) , \\| v \\| _ { S ( I _ { \\max } ) } < \\infty \\} , \\end{align*}"}
+{"id": "4701.png", "formula": "\\begin{align*} f ( x y z _ 0 ) = \\chi ( z _ 0 ) f ( x ) \\chi ( y ) + \\chi ( z _ 0 ) f ( y ) \\chi ( x ) , \\ ; x , y \\in S , \\end{align*}"}
+{"id": "3177.png", "formula": "\\begin{align*} \\begin{bmatrix} X & Y \\end{bmatrix} = \\Lambda ^ { - 1 } \\begin{bmatrix} \\Delta & I _ n \\end{bmatrix} \\begin{bmatrix} a I _ n & c I _ n \\\\ b I _ n & d I _ n \\end{bmatrix} \\begin{bmatrix} \\Gamma { } & 0 \\\\ 0 & I _ n \\end{bmatrix} . \\end{align*}"}
+{"id": "8932.png", "formula": "\\begin{align*} | p _ i + L _ { u } ^ i ( q _ 1 , q _ 2 , \\ldots , q _ n ) | \\leq \\vartheta _ i \\| \\bar { q } \\| ^ { - w _ i } , \\ \\ ( \\bar { p } , \\bar { q } ) = v \\ ( m o d \\ N ) \\ \\ \\ i = 1 , \\ldots , m , \\end{align*}"}
+{"id": "7979.png", "formula": "\\begin{align*} s _ { d _ { 1 } , 0 } : = 2 ^ { - \\mathfrak { k } _ 0 / d _ { 1 } } . \\end{align*}"}
+{"id": "2120.png", "formula": "\\begin{align*} f ( t , x ) = \\rho ( t , x ) \\exp \\left ( \\frac { 1 } { 2 } \\int _ { x - t } ^ { x + t } \\frac { f _ 1 ( s ) d s } { c _ 1 + f _ 0 ( s ) } + \\frac { 1 } { 2 } \\int _ { 0 } ^ t \\left [ \\int _ { x + s - t } ^ { x + t - s } G ( s , y ) d y \\right ] d s \\right ) , \\end{align*}"}
+{"id": "371.png", "formula": "\\begin{align*} K _ { \\phi _ { \\mu } , R _ { \\mu } } ( \\ldots , t _ { \\mu - 1 } , t _ { \\mu + 1 } , \\ldots ) : = \\int K ( t ) \\phi _ { \\mu } ( R _ { \\mu } \\cdot t _ { \\mu } ) d t _ { \\mu } \\end{align*}"}
+{"id": "267.png", "formula": "\\begin{align*} T \\overset { } { = } \\{ \\Delta ^ { \\{ 0 , 2 , 4 \\} } , \\ \\Delta ^ { \\{ 1 , 2 , 3 \\} } , \\ \\Delta ^ { \\{ 0 , 1 , 3 \\} } , \\ \\Delta ^ { \\{ 1 , 3 , 4 \\} } , \\ \\Delta ^ { \\{ 0 , 1 , 2 \\} } \\} ; \\end{align*}"}
+{"id": "2816.png", "formula": "\\begin{align*} b _ { N , s , \\theta } \\int _ { \\Omega } \\frac { | u | } { | x | ^ { \\theta + 2 s } } \\dd x & \\leq b _ { N , s , \\theta } \\liminf _ { t \\to 0 ^ + } \\int _ { \\Omega } \\frac { U _ { t , 1 } } { | x | ^ { \\theta + 2 s } } \\dd x \\\\ & \\leq \\lim _ { t \\to 0 ^ + } \\int _ { \\Omega } \\frac { u } { ( t ^ 2 + u ^ 2 ) ^ { 1 / 2 } } \\frac { ( - \\Delta ) ^ { s } u } { | x | ^ { \\theta } } \\dd x \\\\ & = \\int _ { \\Omega } \\frac { \\mathrm { s i g n } ( u ) ( - \\Delta ) ^ { s } u } { | x | ^ { \\theta } } \\dd x . \\end{align*}"}
+{"id": "1727.png", "formula": "\\begin{align*} p _ l ( \\xi ; q _ 0 ) : = c ( \\xi ; q _ 0 ) e ^ { i l \\xi } + c ( - \\xi ; q _ 0 ) e ^ { - i l \\xi } , c ( \\xi ; q _ 0 ) : = \\frac { 1 - q _ 0 e ^ { - i \\xi } } { 1 - e ^ { - 2 i \\xi } } . \\end{align*}"}
+{"id": "519.png", "formula": "\\begin{align*} \\mathbb E \\left ( \\norm { \\int _ 0 ^ T F ( t ) \\ , d W ( t ) } _ H ^ 2 \\right ) = \\mathbb E \\left ( \\int _ 0 ^ T \\norm { F ( t ) } _ { \\mathcal L _ 2 ( K , H ) } ^ 2 \\ , d t \\right ) . \\end{align*}"}
+{"id": "4890.png", "formula": "\\begin{align*} | | G _ n ( z ) ^ { - 1 } P ( z ) - I | | & = | | G _ n ( z ) ^ { - 1 } \\lim _ { k \\to \\infty } G _ k ( z ) - I | | \\\\ & = \\lim _ { k \\to \\infty } | | G _ n ( z ) ^ { - 1 } G _ k ( z ) - I | | \\\\ & = \\lim _ { k \\to \\infty } | | B _ { n + 1 } \\exp ( \\tilde { B } _ { n + 1 } ) \\ldots B _ k \\exp ( \\tilde { B } _ k ) - I | | \\\\ & \\leq \\lim _ { k \\to \\infty } \\exp \\left [ \\sum _ { s = n + 1 } ^ k \\frac { | \\frac { z - z _ 0 } { z _ s - z _ 0 } | ^ { s + 1 } } { 1 - | \\frac { z - z _ 0 } { z _ s - z _ 0 } | } \\right ] - 1 , \\end{align*}"}
+{"id": "8188.png", "formula": "\\begin{align*} g | _ L = V \\left ( \\sum _ { i = 1 } ^ 3 \\theta _ i ^ 2 + \\theta ^ 2 \\right ) = V \\sum _ { i = 1 } ^ 3 \\theta _ i ^ 2 + \\frac { 1 } { V } \\eta ^ 2 . \\end{align*}"}
+{"id": "5455.png", "formula": "\\begin{align*} x - \\frac { 1 } { V ' ( x ) } - \\sqrt { 2 W ( x ) } & = - \\frac { 1 } { V ' ( x ) } + \\frac { x ^ 2 - 2 W ( x ) } { x + \\sqrt { 2 W ( x ) } } \\\\ & = - \\frac { 1 } { V ' ( x ) } - \\frac { 2 \\log H ( x ) } { x + \\sqrt { 2 W ( x ) } } . \\end{align*}"}
+{"id": "8307.png", "formula": "\\begin{align*} \\sigma ( h ( Z ) ) = u ( h ( Z ) ) ~ Z \\in ( F ) . \\end{align*}"}
+{"id": "5742.png", "formula": "\\begin{align*} \\Theta _ { \\rho } \\overset { d e f } { = } \\frac { \\int _ { \\mathbb Q ^ { + } _ { 4 } } U ( X , t ) ^ 2 x _ { n + 1 } ^ a d X d t } { \\rho ^ 2 \\int _ { \\mathbb B ^ { + } _ \\rho } U ( X , 0 ) ^ 2 x _ { n + 1 } ^ a d X } . \\end{align*}"}
+{"id": "5488.png", "formula": "\\begin{align*} D _ A : = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & - i Q & 0 \\\\ 0 & 0 & - 1 \\end{array} \\right ) \\end{align*}"}
+{"id": "6746.png", "formula": "\\begin{align*} [ O ^ a , E ^ b ] ( n ) & \\equiv ( O ^ a \\circ E ^ b ) ( n ) - ( E ^ b \\circ O ^ a ) ( n ) \\\\ & = \\left ( \\frac { 3 } { 2 } \\right ) ^ a \\left ( \\frac { n } { 2 ^ b } + 1 \\right ) - 1 - \\frac { 1 } { 2 ^ b } \\left ( \\frac { 3 } { 2 } \\right ) ^ a ( n + 1 ) + \\frac { 1 } { 2 ^ b } \\\\ & = \\left ( \\frac { 3 } { 2 } \\right ) ^ a \\left ( 1 - \\frac { 1 } { 2 ^ b } \\right ) + \\frac { 1 } { 2 ^ b } - 1 \\end{align*}"}
+{"id": "3427.png", "formula": "\\begin{align*} \\phi _ { F S , k } = \\frac { 1 } { k } \\max _ { 1 \\leq i \\leq N } ( \\log | s _ i | + c _ i ) , \\end{align*}"}
+{"id": "1152.png", "formula": "\\begin{align*} H ^ n _ { c o m } ( \\mathfrak { g } , M ) : = \\frac { Z ^ n _ { c o m } ( \\mathfrak { g } , M ) } { B ^ n _ { c o m } ( \\mathfrak { g } , M ) } , n \\geq 0 \\end{align*}"}
+{"id": "992.png", "formula": "\\begin{align*} \\prod _ { u = 1 } ^ { \\beta _ j } ( q ^ { n - k } - q ^ { n _ { i - 1 } + p _ { j - 1 } + u - 1 - s _ { j - 1 } } ) & = \\prod _ { u = 1 } ^ { \\beta _ j } ( q ^ { n - k } - q ^ { ( n _ { i - 1 } - k ) + k + p _ { j - 1 } + u - 1 - s _ { j - 1 } } ) \\\\ & = \\prod _ { u = 1 } ^ { \\beta _ j } ( q ^ { n - k } - q ^ { ( n _ { i - 1 } - k ) + k - \\beta _ j - \\cdots - \\beta _ { k _ 0 + 1 } + ( u - 1 ) } ) . \\end{align*}"}
+{"id": "4956.png", "formula": "\\begin{align*} | A _ 1 | = | A _ 3 | = | B _ 5 | = | B _ 7 | , ~ | A _ 5 | = | A _ 7 | = | B _ 1 | = | B _ 3 | . \\end{align*}"}
+{"id": "4152.png", "formula": "\\begin{align*} \\frac { d } { d t } \\exp ^ * ( t \\alpha ) & = \\exp ^ * ( t \\alpha ) * \\alpha = \\exp ^ { \\cdot } ( t \\alpha ) \\cdot ( \\exp ^ { \\cdot } ( t \\alpha ) \\rhd \\alpha ) , \\\\ \\frac { d } { d t } \\exp ^ { \\cdot } ( t \\alpha ) & = \\exp ^ { \\cdot } ( t \\alpha ) \\cdot \\alpha . \\end{align*}"}
+{"id": "1871.png", "formula": "\\begin{gather*} \\epsilon ' = \\epsilon \\cdot \\left ( 1 + \\frac { 1 } { r } + \\cdots + \\frac { N ! } { r ^ N } \\right ) ^ { - 1 } . \\end{gather*}"}
+{"id": "5922.png", "formula": "\\begin{align*} d [ - a _ { 1 , n + 1 } b _ { 1 , n - 1 } ] + d [ - a _ { 1 , n + 2 } b _ { 1 , n } ] > 0 + ( 1 - R _ { n + 2 } ) & = 2 e + ( 1 - 2 e ) - R _ { n + 2 } \\\\ & = 2 e + S _ { n } - R _ { n + 2 } . \\end{align*}"}
+{"id": "5987.png", "formula": "\\begin{align*} d \\mu _ h ( y ) = a \\varepsilon ^ 2 ( 1 + \\varepsilon ^ 2 Q _ \\varepsilon ( s ' ) ) d s _ 1 \\wedge d s _ 2 , \\end{align*}"}
+{"id": "1973.png", "formula": "\\begin{align*} \\sum _ { \\sigma = 0 } ^ { p ^ k - 1 } \\mathcal { A } ( \\mathbf { a } _ \\sigma ^ t ) ( \\tau ) = 0 . \\end{align*}"}
+{"id": "4683.png", "formula": "\\begin{align*} L ( t \\pi _ 1 + ( 1 - t ) \\pi _ 2 , \\lambda ) = t L ( \\pi _ 1 , \\lambda ) + ( 1 - t ) L ( \\pi _ 2 , \\lambda ) . \\end{align*}"}
+{"id": "6607.png", "formula": "\\begin{align*} x = i { 1 - e ^ { i \\theta } \\over 1 + e ^ { i \\theta } } = \\tan ( \\theta / 2 ) , \\end{align*}"}
+{"id": "6986.png", "formula": "\\begin{align*} \\left | \\frac { f ' ( z ) } { f ( z ) } \\right | = O ( | z | ^ { - \\alpha } ) \\end{align*}"}
+{"id": "4575.png", "formula": "\\begin{align*} \\vert \\mathcal J _ { 2 } \\vert \\le & \\varepsilon \\left \\{ I ( t ) + I _ { 1 , n } ( t ) \\right \\} + \\frac { C _ 2 } { \\varepsilon } \\Vert \\varphi ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } \\\\ & + C _ 3 \\left \\{ \\int _ 0 ^ t ( I ( s ) + I _ { 1 , n } ( s ) ) d s + \\int _ 0 ^ t \\Vert \\varphi ( s ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } d s \\right \\} \\ , . \\end{align*}"}
+{"id": "1011.png", "formula": "\\begin{align*} I V = ( { \\bf E } \\cdot \\nabla ) ( \\omega \\ , k + \\nabla \\varphi ) & = E _ 1 ( \\varphi _ { x _ 1 x _ 1 } , \\varphi _ { x _ 1 x _ 2 } , \\varphi _ { x _ 1 x _ 3 } ) + E _ 2 ( \\varphi _ { x _ 2 x _ 1 } , \\varphi _ { x _ 2 x _ 2 } , \\varphi _ { x _ 2 x _ 3 } ) + E _ 3 ( \\varphi _ { x _ 3 x _ 1 } , \\varphi _ { x _ 3 x _ 2 } , \\varphi _ { x _ 3 x _ 3 } ) \\\\ & = ( { \\bf E } \\cdot \\nabla \\varphi _ { x _ 1 } , { \\bf E } \\cdot \\nabla \\varphi _ { x _ 2 } , { \\bf E } \\cdot \\nabla \\varphi _ { x _ 3 } ) . \\end{align*}"}
+{"id": "2961.png", "formula": "\\begin{align*} ( \\alpha \\times \\beta ) \\circ \\gamma ^ X _ z ( \\iota _ A ( a ) ) = \\iota _ B \\circ \\alpha ( a ) = \\gamma ^ Y _ z \\circ ( \\alpha \\times \\beta ) ( \\iota _ A ( a ) ) , \\end{align*}"}
+{"id": "4651.png", "formula": "\\begin{align*} D ( h \\otimes u ) : = D ( h ) \\otimes u \\end{align*}"}
+{"id": "8504.png", "formula": "\\begin{align*} ( \\partial ^ { * } F _ { \\ell } ) _ { \\bar { z } } & = _ { \\mathcal { H } ^ { n - 1 } } \\overline { B ^ { n - 1 } \\left ( 0 , r _ { \\ell } ^ { \\vee } ( \\bar { z } ) \\right ) } \\backslash B ^ { n - 1 } \\left ( 0 , r _ { \\ell } ^ { \\wedge } ( \\bar { z } ) \\right ) \\\\ & = _ { \\mathcal { H } ^ { n - 1 } } B ^ { n - 1 } \\left ( 0 , r _ { \\ell } ^ { \\vee } ( \\bar { z } ) \\right ) \\backslash B ^ { n - 1 } \\left ( 0 , r _ { \\ell } ^ { \\wedge } ( \\bar { z } ) \\right ) , \\end{align*}"}
+{"id": "8099.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } + \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } \\leq C \\left \\| f \\right \\| _ { h _ { \\omega } ^ { p } } \\end{align*}"}
+{"id": "5417.png", "formula": "\\begin{align*} A _ { M _ 1 , \\ldots , M _ d ; S _ 1 , \\ldots , S _ d } ^ { n _ 1 , \\ldots , n _ d } f ( x ) \\coloneqq \\frac { 1 } { M _ 1 \\cdots M _ d } \\sum _ { n _ 1 = 1 } ^ { M _ 1 } \\ldots \\sum _ { n _ d = 1 } ^ { M _ d } f ( S _ 1 ^ { n _ 1 } \\cdots S _ d ^ { n _ d } x ) , x \\in X , \\end{align*}"}
+{"id": "5844.png", "formula": "\\begin{align*} w _ 0 ( \\alpha _ 3 ) = s _ { \\alpha _ 2 + 2 \\alpha _ 3 + 2 \\alpha _ 4 } ( - \\alpha _ 3 ) = - \\alpha _ 3 . \\end{align*}"}
+{"id": "5523.png", "formula": "\\begin{align*} \\psi _ { i j } = f ^ { { \\widehat g l } _ 2 } \\big ( X _ i p ^ { - 1 / 2 } , X _ i ^ { - 1 } , p ^ { 1 / 2 } \\big \\vert X _ j s ^ { - 1 / 2 } , X _ j ^ { - 1 } , s ^ { 1 / 2 } \\big | Q ^ 4 , - Q ^ { - 4 } \\big ) \\end{align*}"}
+{"id": "20.png", "formula": "\\begin{align*} \\widehat g ( 0 ) = \\begin{cases} 8 \\pi \\delta , & d = 2 , \\\\ 8 \\pi a , & d = 3 . \\end{cases} \\end{align*}"}
+{"id": "6339.png", "formula": "\\begin{align*} \\begin{aligned} N _ \\lambda ^ { l , 2 } v = & \\ L ( P _ { \\lambda } g ( u _ { < \\lambda } ) , \\partial ^ 2 v _ { < \\lambda } ) + L ( P _ \\lambda g ( u _ { < \\lambda } ) , \\partial ^ 2 v _ { \\lambda } ) + \\sum _ { \\mu \\gg \\lambda } L ( P _ \\mu g ( u _ { < \\lambda } ) , \\partial ^ 2 v _ \\mu ) \\\\ : = & \\ N _ \\lambda ^ { l , 2 , l o } v + N _ \\lambda ^ { l , 2 , m e d } v + N _ \\lambda ^ { l , 2 , h i } v , \\end{aligned} \\end{align*}"}
+{"id": "6499.png", "formula": "\\begin{align*} u ( t , n ) = \\sum _ { k \\in \\Z ^ b } q ( k , n ) \\cos ( k \\cdot \\omega t ) \\end{align*}"}
+{"id": "8434.png", "formula": "\\begin{align*} & u v = { T } _ { u } v + { T } _ { v } u + { R } ( u , v ) \\quad { T } _ { u } v = \\sum _ { j \\in \\mathbb { Z } } { S } _ { j - 1 } u { \\Delta } _ j v \\quad \\mbox { a n d } \\quad { R } ( u , v ) = \\sum _ { j \\in \\mathbb { Z } } { \\Delta } _ j u \\tilde { \\Delta } _ j v . \\end{align*}"}
+{"id": "3678.png", "formula": "\\begin{align*} L ( G ) _ { u , v } = \\begin{cases} d _ G ( v ) & u = v , \\\\ - 1 & \\{ u , v \\} \\in E , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "7939.png", "formula": "\\begin{align*} S _ i = ( S \\cap V _ i ) \\cup \\{ u _ i , v _ i \\} \\ 1 \\end{align*}"}
+{"id": "7810.png", "formula": "\\begin{align*} \\widetilde { N } _ { K , \\epsilon } : = \\{ ( \\tau _ 2 , b ^ a + \\mathrm { i } t ^ a , \\zeta ^ i , \\widetilde { \\zeta } _ i , \\sigma ) \\in \\widetilde { N } \\ ; | \\ ; \\tau _ 2 > \\epsilon , \\ ; \\ ; t ^ a > K , \\ ; \\ ; a = 1 , . . . , n \\} \\ , . \\end{align*}"}
+{"id": "783.png", "formula": "\\begin{align*} \\varepsilon ^ { 2 s } ( - \\Delta ) ^ s u + V ( x ) u = \\varepsilon ^ { - \\alpha } ( K \\ast F ( u ) ) F ' ( u ) \\mathbb { R } ^ N \\end{align*}"}
+{"id": "8709.png", "formula": "\\begin{align*} \\Gamma ( z ) = \\int _ { 0 } ^ \\infty x ^ { z - 1 } \\exp ( - x ) \\d x , z > 0 \\enspace . \\end{align*}"}
+{"id": "3167.png", "formula": "\\begin{align*} W = x + y + \\frac { 1 } { x } + \\frac { 1 } { x ^ 2 y } . \\end{align*}"}
+{"id": "3576.png", "formula": "\\begin{align*} { \\bf { H } } = \\sum \\limits _ { k = 1 } ^ K { \\sum \\limits _ { { l } = 1 } ^ { { L _ { { \\rm R } } } } { \\sum \\limits _ { { j } = 0 } ^ { { L _ { { \\rm T } } } } { { \\xi _ { k , { l } , { j } } } { { \\bf { a } } _ { \\rm { R } } } \\left ( { \\Phi _ { { k } , { l } } ^ { \\rm { A } } } \\right ) { \\bf { a } } _ { \\rm { T } } ^ H \\left ( { \\Theta _ { k , { j } } ^ { \\rm { D } } } \\right ) } } } = { { \\bf { R } } } { \\bf { \\Xi T } } ^ H , \\end{align*}"}
+{"id": "2612.png", "formula": "\\begin{align*} Q _ n ( t ) = Q _ n ^ { ( 0 ) } ( t ) + ( Q _ n ( 0 ) - n ) ^ + + A _ n ( t ) - \\int _ 0 ^ t \\int _ 0 ^ t \\mathbf { 1 } _ { \\{ s + x \\le t \\} } \\ , d \\ , \\sum _ { i = 1 } ^ { { { \\hat A } _ n } ( s ) } \\mathbf { 1 } _ { \\{ \\eta _ i \\le x \\} } \\ , , \\end{align*}"}
+{"id": "6995.png", "formula": "\\begin{align*} e \\triangleq e _ 1 + i e _ 2 , ~ ~ ~ n \\triangleq e _ 3 + i e _ 4 , ~ ~ ~ \\o \\triangleq \\o _ 1 + i \\o _ 2 , ~ ~ ~ \\O \\triangleq \\o _ 3 + i \\o _ 4 , ~ ~ ~ ~ ~ ~ \\\\ \\mathrm { I I } _ 1 \\triangleq ( \\o _ { 1 3 } + i \\o _ { 2 3 } ) + i ( \\o _ { 1 4 } + i \\o _ { 2 4 } ) , ~ ~ ~ \\mathrm { I I } _ 2 \\triangleq ( \\o _ { 1 3 } - i \\o _ { 2 3 } ) + i ( \\o _ { 1 4 } - i \\o _ { 2 4 } ) . \\end{align*}"}
+{"id": "6380.png", "formula": "\\begin{align*} \\delta ^ i _ n ( x _ 1 \\otimes \\cdots \\otimes x _ n ) = \\begin{cases} 1 \\otimes x _ 1 \\otimes \\cdots \\otimes x _ n , i = 0 \\\\ x _ 1 \\otimes \\cdots \\otimes x _ { i - 1 } \\otimes \\Delta ( x _ i ) \\otimes x _ { i + 1 } \\otimes \\cdots \\otimes x _ n , 1 \\leq i \\leq n \\\\ x _ 1 \\otimes \\cdots \\otimes x _ { n } \\otimes 1 , i = n + 1 . \\end{cases} \\end{align*}"}
+{"id": "7925.png", "formula": "\\begin{align*} 3 - ( a + b ) = c _ { x , x } ^ { \\phi ^ 2 ( x ) } = c _ { \\phi ( x ) , \\phi ( x ) } ^ x = c _ { x , \\phi ( x ) } ^ { \\phi ( x ) } . \\end{align*}"}
+{"id": "1004.png", "formula": "\\begin{align*} I _ 3 = \\int _ { \\Omega _ - ^ { \\varepsilon } } G _ 1 ^ - \\varphi _ { x _ 2 } - G _ 2 ^ - \\varphi _ { x _ 1 } \\ , d x = \\int _ { \\partial \\Omega ^ { \\varepsilon } _ - } \\varphi ( G _ 1 ^ - n _ 2 ^ { \\varepsilon } - G _ 2 ^ - n _ 1 ^ { \\varepsilon } ) \\ , d \\sigma + \\int _ { \\Omega ^ { \\varepsilon } _ { - } } \\varphi ( ( G _ 2 ^ - ) _ { x _ 1 } - ( G _ 1 ^ - ) _ { x _ 2 } ) \\ , d x . \\end{align*}"}
+{"id": "1935.png", "formula": "\\begin{align*} T _ 1 = \\sum _ { i , j } \\int _ { T _ { i j } } ( u - u _ h ) \\ , \\partial _ v f \\ , \\theta _ f \\ , { \\rm d } v \\ , { \\rm d } x . \\end{align*}"}
+{"id": "7724.png", "formula": "\\begin{align*} \\{ \\{ \\{ \\Theta , \\tilde { e } \\} , e \\} , f \\} & = \\pounds _ { \\rho ( \\{ \\tilde { e } , e \\} ) } f + \\{ \\{ \\Theta , e \\} , \\{ \\tilde { e } , f \\} \\} - \\{ \\tilde { e } , \\{ \\{ \\Theta , e \\} , f \\} \\} \\\\ & = \\pounds _ { \\rho ( \\{ \\tilde { e } , e \\} ) } f + \\pounds _ { \\rho ( e ) } ( \\pounds _ X f ) - \\pounds _ { X } ( \\pounds _ { \\rho ( e ) } f ) . \\end{align*}"}
+{"id": "4456.png", "formula": "\\begin{align*} l = \\begin{cases} 2 , & \\alpha _ 0 \\neq s , \\\\ 0 , & \\alpha _ 0 = s . \\end{cases} \\end{align*}"}
+{"id": "6283.png", "formula": "\\begin{align*} \\frac { d } { d t } M ( v _ \\lambda ) = \\partial _ x [ g _ { [ < \\lambda ] } P ( v _ \\lambda ) ] + F ^ { p a r a } _ { \\lambda , m } , F ^ { p a r a } _ { \\lambda , m } = 2 \\Im ( f _ \\lambda \\bar v _ \\lambda ) . \\end{align*}"}
+{"id": "8871.png", "formula": "\\begin{align*} \\int _ M c _ 1 ^ { i _ 1 } \\cdots c _ r ^ { i _ r } , \\sum _ { k = 1 } ^ r k \\cdot i _ k = n , \\end{align*}"}
+{"id": "6496.png", "formula": "\\begin{align*} u ( t , x ) = \\sum _ { ( k , n ) \\in \\Z ^ b \\times \\Z ^ d } q ( k , n ) \\cos ( k \\cdot \\omega t ) \\delta _ n ( x ) , \\end{align*}"}
+{"id": "1745.png", "formula": "\\begin{align*} u _ a ( \\theta ) : = & \\frac { 1 - a ^ 2 } { 1 - 2 a \\cos ( \\theta ) + a ^ 2 } ( | a | < 1 , \\ , \\theta \\in \\mathbb { R } ) , \\end{align*}"}
+{"id": "4269.png", "formula": "\\begin{align*} \\begin{aligned} & \\left | \\int _ { \\Omega } F ( x , u _ j ^ 0 ( x ) ) d x \\right | \\le | J _ { \\lambda _ k } ( u _ j ) | + \\bigg | \\dfrac { 1 } { 2 } \\mathcal { B } _ { \\alpha } ( u _ j ^ + , u _ j ^ + ) + \\dfrac { 1 } { 2 } \\mathcal { B } _ { \\alpha } ( u _ j ^ - , u _ j ^ - ) \\\\ & - \\dfrac { \\lambda _ k } { 2 } \\int _ { \\Omega } \\left ( | u _ j ^ + ( x ) | ^ 2 + | u _ j ^ - ( x ) | ^ 2 \\right ) d x \\ - \\int _ { \\Omega } \\left ( F ( x , u _ j ( x ) ) - F ( x , u _ j ^ 0 ( x ) ) \\right ) d x \\bigg | \\end{aligned} \\end{align*}"}
+{"id": "865.png", "formula": "\\begin{align*} \\begin{aligned} & G \\left ( \\left | m _ u \\left ( B \\left ( z , r _ 1 \\right ) \\right ) - m _ u \\left ( B \\left ( z , r _ 2 \\right ) \\right ) \\right | \\right ) \\\\ & \\leq 4 b ^ 2 2 ^ { s + \\alpha p _ { 0 } } [ u ] _ { W ^ { \\alpha , G } _ { s } ( X , d , \\mu ) } ^ { p _ { 0 } } \\left ( r _ 1 ^ { \\alpha p _ { 0 } - s } + \\frac { r _ 2 ^ { \\alpha p _ { 0 } } } { r _ 1 ^ s } \\right ) \\end{aligned} \\end{align*}"}
+{"id": "1077.png", "formula": "\\begin{align*} \\overline { v } \\left ( x , t \\right ) = \\widetilde { v } _ { t } \\left ( x , t \\right ) , \\overline { w } \\left ( x , t \\right ) = \\widetilde { w } _ { t } \\left ( x , t \\right ) . \\end{align*}"}
+{"id": "8064.png", "formula": "\\begin{align*} \\begin{aligned} f ( x ) & = \\sum \\limits _ { j \\in \\mathbb N } \\psi _ { j } \\ast \\psi _ { j } \\ast f ( x ) \\\\ & = \\sum \\limits _ { j \\in \\mathbb N } \\sum \\limits _ { Q \\in \\Pi _ { j + N } } \\int _ { Q } \\psi _ { j } ( x - u ) ( \\psi _ { j } \\ast f ) ( u ) d u \\\\ & = : T _ { N } ( f ) ( x ) + R _ { n } ( f ) ( x ) \\end{aligned} \\end{align*}"}
+{"id": "5590.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } | \\nabla u _ i | ^ 2 \\ , d x \\rightharpoonup \\mu = \\frac { 1 } { 2 } | \\nabla u _ i | ^ 2 \\ , d x + \\mu _ { \\Sigma } \\end{align*}"}
+{"id": "3734.png", "formula": "\\begin{align*} { \\rm R e } \\left < \\mathbb { A } \\mathbf { u } , \\mathbf { u } \\right > _ X = \\left < \\left [ \\begin{smallmatrix} 0 \\\\ - u \\\\ A u \\end{smallmatrix} \\right ] , \\left [ \\begin{smallmatrix} u \\\\ 0 \\\\ - u \\end{smallmatrix} \\right ] \\right > _ X = - { \\rm R e } \\langle A u , u \\rangle _ \\mathcal { H } < 0 . \\end{align*}"}
+{"id": "1015.png", "formula": "\\begin{align*} \\frac 1 2 \\Delta \\ ( | \\nabla u | ^ 2 \\ ) = | D ^ 2 u | ^ 2 \\end{align*}"}
+{"id": "8478.png", "formula": "\\begin{align*} g ^ { \\vee } ( x ) = \\inf \\left \\{ s \\in \\mathbb { R } : x \\in \\{ g > s \\} ^ { ( 0 ) } \\right \\} = \\inf \\left \\{ s \\in \\mathbb { R } : x \\in \\{ g < s \\} ^ { ( 1 ) } \\right \\} \\end{align*}"}
+{"id": "5486.png", "formula": "\\begin{align*} \\frac { \\mu } { \\prod _ { i = 1 } ^ n ( 1 - p ^ { a _ i } s ^ { b _ i } ) } = \\frac { \\mu ' } { \\prod _ { i = 1 } ^ { n ' } ( 1 - p ^ { a _ i ' } s ^ { b _ i ' } ) } \\qquad \\in \\quad \\mathbb S _ \\times ^ { - 1 } \\mathbf M . \\end{align*}"}
+{"id": "563.png", "formula": "\\begin{align*} \\widehat { \\mathbf I } ( t ) = \\mathcal F \\mathbf I ( t ) = \\int _ 0 ^ t \\mathcal F \\mathbf \\Lambda ^ { \\mathbf s } \\mathbf S ( - s ) \\mathbf M ( \\mathbf u ( s ) ) \\ , d W ( s ) \\in L ^ 2 \\left ( \\Omega , L ^ 2 _ \\xi \\right ) \\end{align*}"}
+{"id": "6911.png", "formula": "\\begin{align*} \\int _ { \\mathbb R } J _ 1 ( \\xi - y ) \\underline { \\phi } ( y ) d y \\geq \\int _ { \\mathbb R } J _ 1 ( \\xi - y ) \\left ( 1 - \\displaystyle \\frac { 1 - b } { 2 } e ^ { \\eta y } \\right ) d y = 1 - \\frac { 1 - b } { 2 } e ^ { \\eta \\xi } \\int _ { \\mathbb R } J _ 1 ( y ) e ^ { - \\eta y } d y . \\end{align*}"}
+{"id": "3061.png", "formula": "\\begin{align*} \\begin{aligned} { \\bf { F } } & = { \\bf { V } } { { \\bf { P } } ^ { 1 / 2 } } , \\\\ { \\bf { W } } & = { \\bf { U } } , \\end{aligned} \\end{align*}"}
+{"id": "6827.png", "formula": "\\begin{align*} Q _ n = Q _ 6 ( g _ 0 ) = \\frac { n ( n ^ 4 - 2 0 n ^ 2 + 6 4 ) } { 2 ^ 5 } , \\end{align*}"}
+{"id": "664.png", "formula": "\\begin{align*} W ( t ) = \\sum _ { k = 1 } ^ \\infty B _ k ( t ) e _ k , \\end{align*}"}
+{"id": "4786.png", "formula": "\\begin{align*} { } _ 1 F _ 1 \\left ( - \\frac { m } 2 , \\frac 1 2 , - \\frac { t ^ 2 } { 4 \\beta } \\right ) = \\Gamma \\left ( \\frac m 2 + 1 \\right ) L _ { \\frac m 2 } ^ { - \\frac 1 2 } \\left ( - \\frac { t ^ 2 } { 4 \\beta } \\right ) , \\end{align*}"}
+{"id": "6228.png", "formula": "\\begin{align*} D _ A ^ * & D _ A \\phi = ( D + \\i A \\cdot ) ( D + \\i A \\cdot ) \\phi \\\\ & = D ^ * D \\phi + \\i \\ , e _ i \\cdot \\nabla ^ S _ { e _ i } ( A _ j e _ j \\cdot \\phi ) + \\i A \\cdot D \\phi - \\sum _ { i , j } A _ i A _ j e _ i \\cdot e _ j \\cdot \\phi \\\\ & = ( \\nabla ^ S ) ^ * \\nabla ^ S \\phi + \\mathcal { R } \\phi + \\i \\ , d A \\cdot \\phi - \\i \\ , \\div A \\ , \\phi - 2 \\i \\sum _ { i } A _ i \\nabla _ { e _ i } \\phi + | A | ^ 2 \\phi . \\end{align*}"}
+{"id": "145.png", "formula": "\\begin{align*} \\overline { \\sigma ' ( \\pi ' ) } = \\sigma ' ( \\overline { \\pi } ' ) = \\overline { \\pi ' } \\overline { \\sigma ' ( s ' ) } . \\end{align*}"}
+{"id": "1780.png", "formula": "\\begin{gather*} \\zeta _ N ( s , \\mathbb { X } _ \\alpha ) = \\sum _ { n = 0 } ^ { N } \\frac { \\mathbb { X } _ \\alpha ( n ) } { ( n + \\alpha ) ^ s } \\quad \\zeta _ N ( s , \\mathbb { Y } _ \\alpha ) = \\sum _ { n = 0 } ^ { N } \\frac { \\mathbb { Y } _ \\alpha ( n ) } { ( n + \\alpha ) ^ s } \\end{gather*}"}
+{"id": "7053.png", "formula": "\\begin{align*} & \\psi ( q , x ) = 1 + h ( q , x ) , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , h ( q , x ) = O \\left ( \\frac { Q } { q } \\left ( \\frac { q } { Q } + | x | \\right ) ^ A \\right ) \\\\ & \\psi ( q , x ) \\ll | x | ^ { - A } \\\\ & x ^ j \\frac { \\partial ^ j } { \\partial x ^ j } \\psi ( q , x ) \\ll \\ \\min \\left \\lbrace \\frac { Q } { q } , \\frac { 1 } { | x | } \\right \\rbrace \\ \\log Q \\end{align*}"}
+{"id": "917.png", "formula": "\\begin{align*} 2 ( m \\phi _ { 2 x } + n \\eta _ { 2 x } ) + m n ( k + 1 ) x ^ k \\bigg ( - \\frac { 1 - \\alpha } { t } \\tau + \\tau _ t \\bigg ) + 2 m n k x ^ { k - 1 } \\sigma _ 1 = 0 , \\end{align*}"}
+{"id": "2216.png", "formula": "\\begin{align*} \\left ( D _ { a + } ^ \\alpha y \\right ) ( x ) : = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\left ( \\frac { d } { d x } \\right ) ^ n \\underset { a } { \\overset { x } { \\int } } \\frac { y ( t ) d t } { ( x - t ) ^ { \\alpha - n + 1 } } , \\ , ( n = \\left [ R e ( \\alpha ) \\right ] + 1 , \\ , x > a ) , \\end{align*}"}
+{"id": "1235.png", "formula": "\\begin{align*} \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| e ^ { i t \\Delta } ( u _ { 0 , n } - u _ { n } ^ J ( 0 ) ) \\| _ { S ( \\R ) } \\leq \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| e ^ { i t \\Delta } w _ n ^ J \\| _ { S ( \\R ) } = 0 . \\end{align*}"}
+{"id": "4354.png", "formula": "\\begin{align*} \\varrho _ \\beta = \\frac { 1 } { { \\rm T r } ( { \\rm e } ^ { - \\beta \\boldsymbol { H } } ) } { \\rm e } ^ { - \\beta \\boldsymbol { H } } \\end{align*}"}
+{"id": "6662.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty \\Big ( { \\rho } _ { ( 1 ) , \\infty } ^ { ( \\rm c J ) } ( x ; \\beta , p , q ) | _ { \\beta = 2 } - 1 \\Big ) e ^ { i \\tau x } \\ , d x \\mathop { \\sim } \\limits _ { \\tau \\to 0 } \\pi \\sum _ { n = 1 } ^ \\infty d _ n { i ^ n \\over ( n - 1 ) ! } \\tau ^ { n - 1 } { \\rm s g n } \\ , \\tau . \\end{align*}"}
+{"id": "3302.png", "formula": "\\begin{align*} \\phi _ Y ( t ) & = \\mathcal { F } ^ \\mu \\left ( g _ Y \\right ) ( t ) = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\sqrt { \\frac { \\lambda _ { _ \\Sigma } } { \\pi } } \\int _ \\mathbb { R } e ^ { - \\lambda _ { _ \\Sigma } x ^ 2 } e ^ { \\mu t x } d x = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } e ^ { - \\frac { t ^ 2 } { 4 \\lambda _ { _ \\Sigma } } } . \\end{align*}"}
+{"id": "3699.png", "formula": "\\begin{align*} A ^ \\pm _ x : = \\{ y \\in D ^ + \\ , | \\ , \\pm K _ 1 ( x , y ) > 0 \\} . \\end{align*}"}
+{"id": "7225.png", "formula": "\\begin{align*} & w = u \\qquad , \\\\ & w = v \\qquad , \\end{align*}"}
+{"id": "4596.png", "formula": "\\begin{align*} S ( \\chi _ M ( \\tau ) ^ { } ) = \\chi _ M ^ { } ( { - } \\frac { 1 } { \\tau } ) T ( \\chi _ M ( \\tau ) ^ { } ) = \\chi _ M ^ { } ( \\tau { + } 1 ) \\ , , \\end{align*}"}
+{"id": "4854.png", "formula": "\\begin{align*} \\check { R } ( u ) = L ( u ) B + M ( u ) I + N ( u ) B ^ { - 1 } \\end{align*}"}
+{"id": "4930.png", "formula": "\\begin{align*} g _ k : = \\prod _ { \\ell = 1 } ^ \\infty ( 1 + e ^ { - 2 ^ { k } \\pi \\ell } ) . \\end{align*}"}
+{"id": "8300.png", "formula": "\\begin{align*} S _ J & = \\prod _ { i \\in J } S _ i & S _ { { - } J } & = \\prod _ { i \\in J ^ { \\text c } } S _ i \\end{align*}"}
+{"id": "1239.png", "formula": "\\begin{align*} \\| f \\| _ { L _ { t } ^ q L ^ r _ x ( I \\times \\R ^ 3 ) } = \\bigg ( \\int _ { I } \\| f ( t , x ) \\| _ { L ^ r _ x } ^ q d t \\bigg ) ^ \\frac { 1 } { q } \\end{align*}"}
+{"id": "8036.png", "formula": "\\begin{align*} { \\rm s u p p } \\widehat \\phi _ { 0 } \\subseteq \\{ \\xi \\in \\mathbb { R } ^ { n } \\colon \\vert \\xi \\vert \\leq 2 \\} ; \\ \\widehat \\phi _ { 0 } \\{ \\xi \\} = 1 , \\ { \\rm i f } \\ \\vert \\xi \\vert \\leq 1 , \\ \\end{align*}"}
+{"id": "4420.png", "formula": "\\begin{align*} \\mbox { l . o . t . } : = & - 2 [ \\partial _ 1 \\hat u _ N - \\hat \\lambda \\partial _ 1 \\hat H _ N ] \\dot q ^ + \\varphi - 2 [ \\partial _ 1 \\hat q ] \\varphi \\partial _ t \\varphi - 2 [ \\partial _ 1 \\hat q ] ( \\hat u ^ - _ 2 - \\hat \\lambda \\hat H ^ - _ 2 ) \\varphi \\partial _ 2 \\varphi \\\\ & - 2 [ \\partial _ 1 \\hat q ] ( \\partial _ 1 \\hat u ^ - _ N - \\hat \\lambda ^ - \\partial _ 1 \\hat H ^ - _ N ) \\varphi ^ 2 \\ , . \\end{align*}"}
+{"id": "1733.png", "formula": "\\begin{align*} \\delta _ { \\texttt { b } ; \\lambda } ^ { ( m , n ) } ( 1 ) = \\prod _ { \\substack { 1 \\leq j < k \\leq n \\\\ \\lambda _ j - \\lambda _ k = 0 } } \\frac { k - j } { 1 + k - j } . \\end{align*}"}
+{"id": "5860.png", "formula": "\\begin{align*} \\mu ^ { n } _ { t } = \\Delta \\mathbf { 1 } _ { \\{ t \\geq \\nu , n \\in \\mathcal { N } \\} } , \\mbox { f o r s o m e } \\Delta \\neq 0 , \\end{align*}"}
+{"id": "2546.png", "formula": "\\begin{align*} I [ u ] & = \\frac { 1 } { p } \\| u \\| _ { D ^ { s , p } } ^ p + \\frac { 1 } { p } \\int _ { \\mathbb { R } ^ N } V ( x ) | u | ^ p d x \\\\ & \\quad - \\frac { 1 } { 2 \\cdot p _ { r ; s } ^ { \\uparrow * } } \\int _ { \\mathbb { R } ^ N } ( K \\ast g ( u ) ) g ( u ) d x - \\varepsilon _ W \\int _ { \\mathbb { R } ^ N } W ( x ) f ( u ) d x . \\end{align*}"}
+{"id": "7680.png", "formula": "\\begin{align*} \\mathbb { E } X ^ * & = \\mathbb { E } X - \\varepsilon \\mathbb { P } ( X > a ) < \\mathbb { E } X , \\\\ \\mathbb { P } ( X ^ * \\leqslant X ) & = \\mathbb { P } ( X ^ * \\leqslant X , X > a ) + \\mathbb { P } ( X ^ * \\leqslant X , X \\leqslant a ) = 1 . \\end{align*}"}
+{"id": "500.png", "formula": "\\begin{align*} \\psi _ \\pm ( 0 ) = f _ \\pm \\in L ^ 2 \\left ( \\Omega , H ^ s ( \\R , \\C ) \\right ) , \\phi _ \\pm ( 0 ) = g _ \\pm \\in L ^ 2 \\left ( \\Omega , H ^ r ( \\R , \\C ) \\right ) , \\overline { g _ + } = g _ - . \\end{align*}"}
+{"id": "6333.png", "formula": "\\begin{align*} i \\partial _ t v _ \\lambda + \\partial _ x g _ { [ < \\lambda ^ \\sigma ] } \\partial _ x v _ \\lambda + V _ { < \\lambda ^ { \\sigma } } \\partial _ x v _ \\lambda = f _ \\lambda + f _ \\lambda ^ { e r r } , v _ \\lambda ( 0 ) = v _ { 0 , \\lambda } + v _ { 0 , \\lambda } ^ { e r r } , \\end{align*}"}
+{"id": "6113.png", "formula": "\\begin{align*} \\sigma ^ r _ A ( T ) \\backslash \\{ 0 \\} = \\Big \\{ g ( T ) : g \\in \\mathcal { P } _ I ^ r ( T ) \\ \\ g ( P ) = 1 \\Big \\} \\backslash \\{ 0 \\} \\end{align*}"}
+{"id": "1192.png", "formula": "\\begin{align*} F _ B ( A ) = A \\otimes B \\simeq \\prod _ { i = 1 } ^ n B [ e _ { i } ^ { - 1 } ] = \\prod _ { j = 1 } ^ s ( B [ e _ j ^ { - 1 } ] ) ^ { \\times n _ j } \\end{align*}"}
+{"id": "7579.png", "formula": "\\begin{align*} \\begin{aligned} J _ 2 ( \\beta , N , T ) & \\leq C N ^ 2 \\int _ { 0 } ^ { \\pi } \\iint _ { 0 \\le s \\le t \\le T } \\frac { 1 } { ( t + s ) ^ { 3 / 2 } } \\\\ & \\times \\int _ { \\mathbf { B } _ 2 ( 0 ) } \\exp \\left ( - \\frac { \\big | z - D _ t ^ { ( k ) } + D _ s ^ { ( \\ell ) } \\big | ^ 2 } { 2 ( t + s ) } \\right ) d z d s d t d \\theta . \\end{aligned} \\end{align*}"}
+{"id": "23.png", "formula": "\\begin{align*} \\vert \\mathcal { E } \\vert \\leq \\begin{cases} C \\rho ^ { 2 } | \\Lambda | \\delta ^ { 2 + \\eta } , & d = 2 , \\\\ C \\rho ^ { 2 } | \\Lambda | a \\big ( \\rho a ^ { 3 } \\big ) ^ { \\frac 1 2 + \\eta } , & d = 3 . \\end{cases} \\end{align*}"}
+{"id": "8183.png", "formula": "\\begin{align*} - x Q _ i ( x ; \\alpha , \\beta , n ) = A _ i Q _ { i + 1 } ( x ; \\alpha , \\beta , n ) - ( A _ i + C _ i ) Q _ i ( x ; \\alpha , \\beta , n ) + C _ i Q _ { i - 1 } ( x ; \\alpha , \\beta , n ) , \\end{align*}"}
+{"id": "4083.png", "formula": "\\begin{align*} \\oint _ { p \\in C ^ { \\mathfrak { p } } _ { \\frac 1 2 , 2 } } \\oint _ { q \\in \\check C _ + } = \\oint _ { q \\in \\check C _ + } \\oint _ { p \\in C ^ { \\mathfrak { p } } _ { \\frac 1 2 , 2 } } . \\end{align*}"}
+{"id": "6222.png", "formula": "\\begin{align*} p _ t ( z ) & = ( 8 \\pi ) ^ { - d } e ^ { - d t } \\sum _ { k = 0 } ^ \\infty ( e ^ { - 2 t } ) ^ k L _ k ^ { d - 1 } \\Big ( \\frac { | z | ^ 2 } { 2 } \\Big ) e ^ { - \\frac { | z | ^ 2 } { 4 } } \\\\ & = ( 8 \\pi ) ^ { - d } e ^ { - d t } ( 1 - e ^ { - 2 t } ) ^ { - d } \\exp \\Big ( - \\frac { 1 + e ^ { - 2 t } } { 1 - e ^ { - 2 t } } \\frac { | z | ^ 2 } { 4 } \\Big ) \\\\ & = ( 1 6 \\pi \\sinh t ) ^ { - d } \\exp \\Big ( - \\frac 1 4 \\coth ( t ) | z | ^ 2 \\Big ) . \\end{align*}"}
+{"id": "4140.png", "formula": "\\begin{align*} [ \\xi _ A , H _ a ] _ J = - H _ { A a } , [ H _ a , H _ b ] _ J = - H _ { \\overline { T } ( a , b ) } + \\xi _ { \\overline { R } ( a , b ) } . \\end{align*}"}
+{"id": "3537.png", "formula": "\\begin{align*} w ( r , t ) & = w ( r _ { 0 } ( t ) , t ) - \\int _ { r } ^ { r _ { 0 } ( t ) } w _ { r } ( \\rho , t ) d \\rho \\\\ & \\leqslant K + \\int _ { r } ^ { r _ { 0 } ( t ) } \\vert w _ { r } ( \\rho , t ) \\vert d \\rho \\\\ & \\leqslant K + r ^ { 1 - N } \\int _ { r } ^ { r _ { 0 } ( t ) } \\rho ^ { N - 1 } \\vert w _ { r } ( \\rho , t ) \\vert d \\rho \\\\ & \\leqslant K + C K r ^ { 1 - N } . \\end{align*}"}
+{"id": "8398.png", "formula": "\\begin{align*} H = d N ^ { 1 - \\gamma } ( \\log N ) ^ { 3 } . \\end{align*}"}
+{"id": "5229.png", "formula": "\\begin{align*} \\eta _ j = \\frac { 1 } { M \\sqrt { j } } , \\end{align*}"}
+{"id": "8049.png", "formula": "\\begin{align*} \\left \\| \\left ( \\sum \\limits _ { j = 1 } ^ { \\infty } \\left ( \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right ) ^ { \\eta } \\right ) ^ { 1 / \\eta } \\right \\| _ { L _ { \\omega } ^ { p } } + \\left \\| \\left ( \\sum \\limits _ { j = 1 } ^ { \\infty } \\left ( \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right ) ^ { \\eta } \\right ) ^ { 1 / \\eta } \\right \\| _ { L _ { \\omega } ^ { p } } \\leq C _ { \\eta } \\left \\| f \\right \\| _ { h _ { \\omega } ^ { p } } \\end{align*}"}
+{"id": "2409.png", "formula": "\\begin{align*} A _ { \\overline { \\i } } ( B ( 0 , c r ) ) \\subseteq \\sum _ { k = 1 } ^ { \\infty } A _ { \\overline { \\i } } ^ k ( B ( 0 , r ) ) \\end{align*}"}
+{"id": "622.png", "formula": "\\begin{align*} T _ { \\mathbf U } = \\sup \\left \\{ t \\in [ 0 , T ] \\colon \\norm { \\mathbf U } _ t ^ 2 < 2 R \\right \\} . \\end{align*}"}
+{"id": "476.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty m ( d z ) \\ , p _ t ^ { ( \\beta , \\mu ) } ( r , z ) p _ { t ' } ^ { ( \\beta , \\mu ) } ( z , s ) = p _ { t + t ' } ^ { ( \\beta , \\mu ) } ( r , s ) , r , s , t , t ' > 0 . \\end{align*}"}
+{"id": "6638.png", "formula": "\\begin{align*} { } _ 1 F _ 1 ( p ; 2 p ; 2 i X ) = \\Gamma ( p + 1 / 2 ) \\left ( { X \\over 2 } \\right ) ^ { - p + 1 / 2 } e ^ { i X } J _ { p - 1 / 2 } ( X ) , \\end{align*}"}
+{"id": "1692.png", "formula": "\\begin{align*} \\kappa _ { \\texttt { b } ; \\pm } ( q , q _ 0 , q _ 1 ) : = & \\frac { 1 } { 2 } \\left ( \\frac { 1 - | q _ 0 | } { 1 + | q _ 0 | } \\right ) ^ { \\pm 1 } + \\frac { 1 } { 2 } \\left ( \\frac { 1 - | q _ 1 | } { 1 + | q _ 1 | } \\right ) ^ { \\pm 1 } \\\\ & + ( n - 1 ) \\left ( \\frac { 1 - | q | } { 1 + | q | } \\right ) ^ { \\pm 1 } . \\end{align*}"}
+{"id": "4.png", "formula": "\\begin{align*} \\mathbf { 1 } _ { A } = \\begin{cases} 1 & \\textrm { i f $ A $ i s t r u e } ; \\\\ 0 & \\textrm { i f $ A $ i s f a l s e } . \\\\ \\end{cases} \\end{align*}"}
+{"id": "3640.png", "formula": "\\begin{align*} \\bot \\alpha : = X _ { \\Omega } \\mathbin { \\lrcorner } \\alpha \\ . \\end{align*}"}
+{"id": "7608.png", "formula": "\\begin{align*} \\begin{aligned} J _ 2 ( \\beta , N , T , \\hat { \\mathbf { R } } _ 2 ) & \\leq N ^ 2 | \\hat { \\mathbf { R } } _ 2 | \\\\ & \\leq C N ^ 2 \\beta ^ { - 1 / 3 } N ^ { - 1 / 3 } T \\\\ & \\leq C \\beta ^ { - 1 / 3 } N ^ { 5 / 3 } T . \\end{aligned} \\end{align*}"}
+{"id": "7782.png", "formula": "\\begin{align*} t _ { \\pm } ^ { c , d } = \\frac { c \\tau _ 1 + d \\pm | c \\tau + d | } { c \\tau _ 2 } \\ , . \\end{align*}"}
+{"id": "1810.png", "formula": "\\begin{gather*} \\zeta ( s + i \\tau , \\alpha ) - Z _ N ( s + i \\tau , \\alpha ) = \\frac { 1 } { 2 \\pi i } \\oint _ { \\partial K _ { M + 1 } } \\frac { \\zeta ( z + i \\tau , \\alpha ) - Z _ N ( z + i \\tau , \\alpha ) } { z - s } \\ , d z . \\end{gather*}"}
+{"id": "6353.png", "formula": "\\begin{align*} \\tilde j ^ 6 _ \\lambda = q ( \\xi ) ( \\xi _ { o d d } - \\xi _ { e v e n } ) . \\end{align*}"}
+{"id": "7342.png", "formula": "\\begin{align*} u _ - ( t ) & = \\sqrt { t ^ 2 + 1 } \\ , e ^ { - t \\arctan ( t ) } \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} , \\ , t \\leq 0 , \\\\ u _ + ( t ) & = \\sqrt { t ^ 2 + 1 } \\ , e ^ { - t \\arctan ( t ) } \\begin{pmatrix} \\cos \\left ( \\frac { \\lambda } { 2 } \\right ) \\\\ \\sin \\left ( \\frac { \\lambda } { 2 } \\right ) \\end{pmatrix} , \\ , t \\geq 0 , \\end{align*}"}
+{"id": "3086.png", "formula": "\\begin{align*} \\Delta R \\approx \\sum \\limits _ { k = 1 } ^ { { N _ { \\rm { R } } } } { { { \\log } _ 2 } \\left ( { \\frac { { 1 + { { { \\bar C } _ k } } } } { { 1 + { { \\bar C } _ k } \\left ( { 1 - { 2 ^ { - 1 - { x _ k } } } } \\right ) } } } \\right ) } . \\end{align*}"}
+{"id": "5757.png", "formula": "\\begin{align*} m ^ p : = \\frac { 1 } { | { \\cal B } _ p | } \\sum _ { X \\in { \\cal B } _ p } \\sigma _ X \\end{align*}"}
+{"id": "2879.png", "formula": "\\begin{align*} ( \\mathcal { J } f ) ( \\lambda ) = & \\ , t [ \\lambda ] ^ { - 1 } ( T _ { w _ { \\lambda } } f ) ( \\lambda _ + ) = t _ j ^ { - 1 } t [ s _ j \\lambda ] ^ { - 1 } ( T _ { w _ { s _ j \\lambda } } T _ { j } f ) ( ( s _ j \\lambda ) _ + ) \\\\ \\stackrel { } { = } & \\ , t _ j ^ { - 1 } \\sum _ { \\mu \\in P , \\ , \\mu \\preceq s _ j \\lambda } J _ { s _ j \\lambda , \\mu } ( T _ { j } f ) ( \\mu ) = \\sum _ { \\mu \\in P , \\ , \\mu \\preceq \\lambda } J _ { \\lambda , \\mu } f ( \\mu ) , \\end{align*}"}
+{"id": "3853.png", "formula": "\\begin{align*} \\tau _ 1 & = f _ y e ^ { - \\lambda _ 1 z } E _ 1 + e ^ { - \\lambda _ 2 z } E _ 2 , \\\\ \\tau _ 2 & = f _ z e ^ { - \\lambda _ 1 z } E _ 1 + E _ 3 , \\end{align*}"}
+{"id": "4892.png", "formula": "\\begin{align*} \\left [ I - \\frac { z - z _ 0 } { z _ 1 - z _ 0 } P _ 1 \\right ] ^ { - 1 } = \\left [ I - \\frac { z - z _ 0 } { z - z _ 1 } P _ 1 \\right ] \\end{align*}"}
+{"id": "5976.png", "formula": "\\begin{align*} L _ a \\left ( { K _ a } ^ { - 1 } { ( 1 - | t ' | ^ 2 ) ^ { - 1 / 2 } } \\right ) = 1 , \\ ; \\ K _ a = \\frac { \\pi } { 2 } \\int _ { 0 } ^ { 2 \\pi } \\left ( \\cos ^ 2 \\theta + \\frac { \\sin ^ 2 \\theta } { a ^ 2 } \\right ) ^ { - 1 / 2 } d \\theta . \\end{align*}"}
+{"id": "5155.png", "formula": "\\begin{align*} { \\rm A o I } ( S _ { \\mathrm { z } } , Q ^ * , F ^ * ) = & \\min _ { \\{ Q \\in \\mathcal { Q } \\} } \\min _ { \\{ L \\in \\mathcal { L } \\} } { \\frac { E [ L ^ 2 ] } { 2 E [ L ] } + E [ L ] } \\\\ & \\mbox { s . t . } \\sum _ i 2 ^ { - l _ i } \\le 1 \\\\ & \\sum _ i \\int _ { a _ { i - 1 } } ^ { a _ i } ( x - c _ i ) ^ 2 f ( x ) d x \\le D \\\\ & \\ l _ i \\in R ^ + . \\end{align*}"}
+{"id": "2574.png", "formula": "\\begin{align*} \\beta ( h , h ' ) = \\alpha ( h + h ' ) - \\alpha ( h ) - \\alpha ( h ' ) , \\end{align*}"}
+{"id": "8896.png", "formula": "\\begin{align*} h ^ * \\big ( b ( u ) \\big ) \\coloneqq h ^ * \\big ( b ( u _ 1 , \\ldots , u _ { \\ell } ) \\big ) = b ( h ^ * u _ 1 , \\ldots , h ^ * u _ { \\ell } ) = b ( a _ 1 , \\ldots , a _ { \\ell } ) , \\end{align*}"}
+{"id": "1379.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { \\infty } \\lambda _ r c _ f ( r ) = 0 . \\end{align*}"}
+{"id": "8338.png", "formula": "\\begin{align*} \\d X ( t ) = a ( X ( t ) ) \\d t + b ( X ( t ) ) \\d W ( t ) . \\end{align*}"}
+{"id": "8066.png", "formula": "\\begin{align*} R _ { N } ( f ) ( x ) = \\sum \\limits _ { j \\in \\mathbb N } \\sum \\limits _ { Q \\in \\Pi _ { j + N } } \\int _ { Q } [ \\psi _ { j } ( x - u ) ( \\psi _ { j } \\ast f ) ( u ) - \\psi _ { j } ( x - u _ { Q } ) ( \\psi _ { j } \\ast f ) ( u _ { Q } ) ] d u , \\end{align*}"}
+{"id": "1610.png", "formula": "\\begin{align*} | E _ R | \\le \\frac { 2 \\delta _ k | \\mathcal K _ k ( \\mathcal G _ T ) | } { 2 \\sqrt { \\delta _ k } | \\mathcal K _ k ( \\mathcal G _ T ) \\cap \\mathrm { C r o s s } _ { \\mathcal Y } | } \\le 2 \\sqrt { \\delta _ k } \\binom { a _ 1 } { k } . \\end{align*}"}
+{"id": "935.png", "formula": "\\begin{align*} E ( x , t , u , v ) = 0 , ~ x \\in \\Omega , ~ t > 0 , \\end{align*}"}
+{"id": "8070.png", "formula": "\\begin{align*} \\begin{aligned} \\left \\| f \\right \\| _ { h _ { \\omega } ^ { p } } & \\approx \\left \\| \\left \\{ \\sum \\limits _ { j \\in \\mathbb N } \\sum \\limits _ { Q \\in \\Pi _ { j + N } } \\sup \\limits _ { u \\in Q } \\vert ( \\psi _ { j } \\ast f ) ( u ) \\vert ^ { 2 } \\chi _ { Q } \\right \\} ^ { \\frac { 1 } { 2 } } \\right \\| _ { L _ { \\omega } ^ { p } } \\\\ \\end{aligned} \\end{align*}"}
+{"id": "3750.png", "formula": "\\begin{align*} [ \\mathbb { A } ( \\lambda I + \\mathbb { A } ) ^ { - 1 } ] _ { i j } = \\dfrac { ( - 1 ) ^ { i - j } } { 3 } \\Big ( 2 \\cos \\Big ( ( \\frac { \\alpha + i - j } { 3 } ) \\pi \\Big ) + 1 \\Big ) A ^ { \\frac { \\alpha + i - j } { 3 } } . \\end{align*}"}
+{"id": "6055.png", "formula": "\\begin{align*} a ^ { { \\mathcal { P } } } _ t = ( \\sum \\limits _ { \\substack { 1 \\leq i \\leq n } } \\gamma _ i \\int _ { \\{ \\vert z \\vert \\geq M ( \\gamma _ i ) \\} } \\underline { c } ( z ) \\mu ( d z ) + ( t - \\Gamma _ n ) \\int _ { \\{ \\vert z \\vert \\geq M ( \\gamma _ { n + 1 } ) \\} } \\underline { c } ( z ) \\mu ( d z ) ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "6713.png", "formula": "\\begin{align*} \\tau ^ { \\prime } ( x ) = \\tau ^ { \\prime } \\circ \\pi _ { l } ( x ) = \\pi _ r ( x ) = y _ 0 . \\end{align*}"}
+{"id": "6914.png", "formula": "\\begin{align*} \\underline { \\psi } ( \\xi ) = e ^ { \\lambda _ 1 \\xi } \\left ( 1 - r e ^ { \\varepsilon \\xi } \\right ) \\underline { \\psi } ' ( \\xi ) = e ^ { \\lambda _ 1 \\xi } \\left ( \\lambda _ 1 - r ( \\lambda _ 1 + \\varepsilon ) e ^ { \\varepsilon \\xi } \\right ) . \\end{align*}"}
+{"id": "4566.png", "formula": "\\begin{align*} ( \\mathcal { B } _ 1 { \\mathbf V } _ { t } , { \\mathbf V } _ { t } ) | _ { x _ 1 = 0 } = 2 [ \\partial _ t \\dot { q } ( \\partial _ t \\dot { u } _ { N } - \\hat { \\lambda } \\partial _ t \\dot { H } _ { N } ) ] , \\end{align*}"}
+{"id": "5846.png", "formula": "\\begin{align*} w _ 0 = s _ 2 s _ 1 s _ 2 s _ 1 s _ 2 s _ 1 . \\end{align*}"}
+{"id": "8424.png", "formula": "\\begin{align*} d _ { h ^ \\ast } ( \\nabla g ( X ) + Y , \\nabla h ( X ) ) & \\geq \\tfrac { \\gamma } { 2 } \\| \\nabla g ( X ) + Y - \\nabla h ( X ) \\| _ F ^ 2 \\\\ & = \\tfrac { \\gamma } { 2 } \\| \\nabla g ( X ) + Y - \\nabla h ( X ) - ( \\nabla g ( X ^ \\star ) + Y ^ \\star - \\nabla h ( X ^ \\star ) ) \\| _ F ^ 2 \\\\ & \\geq \\tfrac { \\gamma } { 2 } \\big | \\| \\nabla f ( X ) - \\nabla f ( X ^ \\star ) + ( Y - Y ^ \\star ) \\| _ F ^ 2 , \\end{align*}"}
+{"id": "4943.png", "formula": "\\begin{align*} F ( \\tau ) = \\dfrac { \\eta ( 4 \\tau ) ^ 8 } { \\eta ( 2 \\tau ) ^ 4 } , F ( 2 \\tau ) = \\dfrac { \\eta ( 8 \\tau ) ^ 8 } { \\eta ( 4 \\tau ) ^ 4 } , \\theta ( 2 \\tau ) = \\dfrac { \\eta ( 4 \\tau ) ^ 5 } { \\eta ( 2 \\tau ) ^ 2 \\eta ( 8 \\tau ) ^ 2 } , \\end{align*}"}
+{"id": "2057.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\left ( \\partial _ s + \\Delta _ { y , g ( s ) } \\right ) G ( x , t ; y , s ) = 0 , \\quad \\Omega \\times \\Omega \\times [ 0 , t ) ; \\\\ \\lim _ { s \\to t ^ - } G ( x , t ; y , s ) = \\delta _ x ( y ) , \\quad \\ ; x \\in \\Omega ; \\\\ G ( x , t ; y , s ) = 0 , \\quad \\ ; x \\in \\Omega \\ ; \\ ; y \\in \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "5290.png", "formula": "\\begin{align*} h ( x ) = 1 - \\max \\{ 0 , 1 - 3 x \\} - \\max \\{ 0 , 3 x - 1 \\} + \\max \\{ 0 , 6 x - 4 \\} . \\end{align*}"}
+{"id": "1801.png", "formula": "\\begin{gather*} Z _ N ( s , \\alpha ) = \\frac { 1 } { 2 \\pi i } \\int _ { - \\delta - i \\infty } ^ { - \\delta + i \\infty } \\zeta ( s + w , \\alpha ) \\widehat { \\phi } ( w ) N ^ w \\ , d w + \\zeta ( s , \\alpha ) + \\widehat { \\phi } ( 1 - s ) N ^ { 1 - s } . \\end{gather*}"}
+{"id": "3670.png", "formula": "\\begin{align*} f ( x , t ) = \\frac { | x | ^ 2 } { 4 } - \\frac { n } { 2 } \\ , \\log u ( t ) { } g ( t ) = u ( t ) \\ , \\delta _ { i j } \\ , , \\end{align*}"}
+{"id": "6397.png", "formula": "\\begin{align*} \\chi ( S \\left ( b \\right ) \\otimes S ( a ) ) = \\chi ( a \\otimes b ) a , b \\in H , \\end{align*}"}
+{"id": "3753.png", "formula": "\\begin{align*} \\begin{cases} \\dfrac { d { \\bf u } } { d t } + \\mathbb { A } ^ \\alpha { \\bf u } = 0 , \\ t > 0 , 0 < \\alpha < \\alpha ^ * , \\\\ { \\bf u } ( 0 ) = { \\bf u } _ 0 . \\end{cases} \\end{align*}"}
+{"id": "4024.png", "formula": "\\begin{align*} \\begin{aligned} & \\eta _ m \\rightharpoonup ^ \\ast \\eta \\quad L ^ \\infty ( \\mathcal I _ 1 ; B ^ { \\theta } _ { p , p } \\cap W ^ { 2 , 2 } ( \\omega ) ) \\sup _ { \\mathcal I _ 1 } \\mathrm { L i p } ( \\partial \\Omega _ { \\eta ( t ) } ) \\leq \\delta , \\\\ & \\qquad \\qquad \\partial _ t \\eta _ m \\rightharpoonup \\partial _ t \\eta \\quad L ^ { 3 } ( \\mathcal I _ 1 ; W ^ { 1 , q _ 0 } ( \\omega ) ) , \\end{aligned} \\end{align*}"}
+{"id": "912.png", "formula": "\\begin{align*} n x ^ k \\bigg ( \\phi _ 1 - \\tau _ t + \\frac { 1 - \\alpha } { t } \\tau \\bigg ) - 2 \\phi _ { 2 x } - n k x ^ { k - 1 } \\xi - n x ^ k ( \\eta _ 1 - \\xi _ { 1 x } ) = 0 , \\end{align*}"}
+{"id": "1922.png", "formula": "\\begin{align*} \\mathcal { N } ( u ; f , \\psi _ h ) : = - \\sum _ { i , j } \\int _ { T _ { i j } } f ( u - v ) \\ , \\partial _ v \\psi _ h \\ , { \\rm d } v \\ , { \\rm d } x \\ , \\ , - \\sum _ { i = 1 } ^ { N _ x } \\sum _ { j = 0 } ^ { N _ v - 1 } \\int _ { I _ i } \\left ( ( u - v ) f [ \\ ! [ \\psi _ h ] \\ ! ] \\right ) _ { x , j + 1 / 2 } \\ , { \\rm d } x \\ , , \\end{align*}"}
+{"id": "7630.png", "formula": "\\begin{align*} \\P ^ \\ast : = \\frac { 1 } { 2 d } ( \\P _ { - e _ 1 } + \\P _ { e _ 1 } + \\P _ { - e _ 2 } + \\dots + \\P _ { e _ d } ) , \\end{align*}"}
+{"id": "7844.png", "formula": "\\begin{align*} s ( \\tau ) : = \\frac { \\chi } { 4 ( 2 \\pi ) ^ 3 } \\sum _ { ( m , n ) \\in \\mathbb { Z } ^ 2 - ( 0 , 0 ) } \\frac { 1 } { | m \\tau + n | ^ 3 } , \\ , \\end{align*}"}
+{"id": "3784.png", "formula": "\\begin{align*} \\phi _ i : = R ^ * ( \\psi _ i ) \\iff \\psi _ i = - F ^ * ( - \\phi _ i ) , \\end{align*}"}
+{"id": "3837.png", "formula": "\\begin{align*} { \\rm U O T } ( \\mu _ 0 , \\mu _ 1 ) = \\inf _ { \\xi \\in \\widetilde { \\mathcal S } ^ p _ = ( \\mu _ 0 , \\mu _ 1 ) } \\int _ { X ^ 2 \\times \\R ^ 3 _ + } w \\ , H ( x _ 0 , s _ 0 , x _ 1 , s _ 1 ) d \\xi , \\end{align*}"}
+{"id": "4949.png", "formula": "\\begin{align*} t _ k ( n ) \\sim \\begin{cases} \\dfrac { 1 } { 2 ^ { 2 k } E _ { 2 k } } \\cdot \\sigma _ { \\chi _ { - 4 } ; 2 k } ( 2 n + k + 1 ) , & k \\equiv 0 \\pmod { 2 } , \\\\ - \\dfrac { 1 } { E _ { 2 k } } \\cdot \\sigma _ { \\chi _ { - 4 } ; 2 k } ( 2 n + k + 1 ) , & k \\equiv 1 \\pmod { 2 } , \\end{cases} \\end{align*}"}
+{"id": "3324.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { n ^ 2 } } { ( q ; q ) _ n } = \\frac { 1 } { ( q , q ^ 4 ; q ^ 5 ) _ \\infty } , \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { n ^ 2 + n } } { ( q ; q ) _ n } = \\frac { 1 } { ( q ^ 2 , q ^ 3 ; q ^ 5 ) _ \\infty } , \\end{align*}"}
+{"id": "2113.png", "formula": "\\begin{align*} K : = \\max \\{ K _ 1 , K _ 2 \\} . \\end{align*}"}
+{"id": "1910.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( \\frac { \\partial e _ u } { \\partial t } , \\phi _ h \\right ) + a _ h ( u , \\phi _ h ) & - a _ h ( u _ h , \\phi _ h ) + \\sqrt { \\epsilon } \\ , b _ h ( e _ w , \\phi _ h ) \\\\ & + \\left ( \\rho u - \\rho _ h u _ h , \\phi _ h \\right ) = \\left ( \\rho V - \\rho _ h V _ h , \\phi _ h \\right ) \\forall \\ , \\ , \\phi _ h \\in X _ h , \\end{aligned} \\end{align*}"}
+{"id": "1984.png", "formula": "\\begin{align*} \\omega _ q ^ { \\left ( a ^ { t _ 1 } _ { \\sigma , r } - a ^ { t _ 2 } _ { \\sigma , s } \\right ) } + \\sum _ { j = 1 } ^ { p - 1 } \\omega _ q ^ { \\left ( a ^ { t _ 1 } _ { \\sigma , r ^ j } - a ^ { t _ 2 } _ { \\sigma , s ^ j } \\right ) } = 0 . \\end{align*}"}
+{"id": "341.png", "formula": "\\begin{align*} \\left \\langle \\mbox { \\boldmath $ \\phi $ } , \\mbox { \\boldmath $ \\psi $ } \\right \\rangle _ { \\mathbb { X } } : = \\sum _ { 1 \\leq j \\leq n _ { \\Omega } } \\left \\langle \\mbox { \\boldmath $ \\phi $ } _ { j } , \\mbox { \\boldmath $ \\psi $ } _ { j } \\right \\rangle _ { \\mathbf { X } _ { j } } . \\end{align*}"}
+{"id": "1336.png", "formula": "\\begin{align*} J ^ w ( \\textbf { X } _ { m i n R S S U } ^ { ( n ) } ) & = - \\frac { ( n ! ) ^ 2 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( \\int _ { 0 } ^ { 1 } w ( F ^ { - 1 } ( u ) ) ( 1 - u ) ^ { 2 i - 2 } f ( F ^ { - 1 } ( u ) ) d u \\right ) \\\\ & \\geq - \\frac { ( n ! ) ^ 2 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( \\int _ { 0 } ^ { 1 } k M ( 1 - u ) ^ { 2 i - 2 } d u \\right ) \\\\ & = - \\frac { ( n ! ) ^ 2 k ^ n M ^ n } { 2 ( 2 n - 1 ) ! ! } . \\end{align*}"}
+{"id": "878.png", "formula": "\\begin{align*} \\mathrm { P r } ^ { ( \\alpha , n ) } V = V + \\eta ^ { \\alpha , t } \\partial _ { \\mathcal { T } _ t ^ \\alpha u } + \\phi ^ { \\alpha , t } \\partial _ { \\mathcal { T } _ t ^ \\alpha v } + \\eta ^ x \\partial _ { u _ x } + \\phi ^ x \\partial _ { v _ x } + \\eta ^ { x x } \\partial _ { u _ { x x } } + \\phi ^ { x x } \\partial _ { v _ { x x } } + \\cdots , \\end{align*}"}
+{"id": "6190.png", "formula": "\\begin{align*} \\tilde { s } _ { ( k _ n , k _ { n - 1 } , \\ldots , k _ 1 ) } = s _ { ( k _ n - ( n - 1 ) , k _ { n - 1 } - ( n - 2 ) , \\ldots , k _ 1 - 0 ) } . \\end{align*}"}
+{"id": "1917.png", "formula": "\\begin{align*} \\| \\partial _ x \\theta _ u \\| _ { 0 , I _ h } + \\sum _ { i = 0 } ^ { N _ x - 1 } h ^ { - \\frac { 1 } { 2 } } [ \\ ! [ \\theta _ u ] \\ ! ] _ { i + 1 / 2 } \\leq C \\epsilon ^ { - \\frac { 1 } { 2 } } \\| \\theta _ w \\| _ { 0 , I _ h } . \\end{align*}"}
+{"id": "1465.png", "formula": "\\begin{align*} \\begin{cases} \\Delta v _ i + \\sum \\limits _ { j = 1 } ^ { 2 n } \\hat { k } _ { i j } e ^ { v _ j } = 4 \\pi \\beta _ i \\delta _ 0 \\ \\ B _ 1 ( 0 ) , \\ \\forall \\ i = 1 , \\cdots , 2 n , \\\\ v _ 1 + v _ 2 + \\cdots + v _ { 2 n } \\equiv 0 , \\end{cases} \\end{align*}"}
+{"id": "6260.png", "formula": "\\begin{align*} i \\partial _ t u _ \\lambda + \\partial _ x g _ { [ < \\lambda ] } \\partial _ x u _ \\lambda = N _ \\lambda ( u ) , v _ \\lambda ( 0 ) = u _ { 0 , \\lambda } , \\end{align*}"}
+{"id": "8475.png", "formula": "\\begin{align*} \\theta ( E , x ) = \\lim _ { \\rho \\rightarrow 0 ^ { + } } \\frac { \\mathcal { H } ^ { n } ( E \\cap B _ { \\rho } ( x ) ) } { \\omega _ { n } \\rho ^ { n } } , \\mbox { f o r $ \\mathcal { H } ^ { n } $ - a . e . } x \\in \\mathbb { R } ^ { n } . \\end{align*}"}
+{"id": "3224.png", "formula": "\\begin{align*} S _ { \\alpha } ( \\epsilon , R ) & : = \\# \\{ \\Z ^ { 2 } \\cap \\textnormal { S e c t } _ { \\alpha , \\epsilon } ( R ) \\} \\\\ & \\phantom { : } = \\# \\{ ( m , n ) \\in \\mathbb { Z } _ { > 0 } \\times \\Z : m ^ { 2 } + n ^ { 2 } \\leq R ^ { 2 } , | n - \\alpha m | < m \\epsilon \\} \\end{align*}"}
+{"id": "6031.png", "formula": "\\begin{align*} \\lambda ( F ) = \\inf _ { \\vert \\zeta \\vert = 1 } \\langle \\sigma _ F \\zeta , \\zeta \\rangle \\end{align*}"}
+{"id": "1458.png", "formula": "\\begin{align*} \\sigma _ { s } ^ { * } = 2 \\sum \\limits _ { q = j - 1 } ^ { s - 1 } \\sum \\limits _ { k = j } ^ { 2 j + l - q - 1 } \\mu _ { k } - 2 \\sum \\limits _ { q = j - 1 } ^ { s - 1 } \\sum \\limits _ { k = j } ^ { q } \\mu _ { k } - \\big ( \\sigma _ { 2 j + l - s } - \\sigma _ { j + l + 1 } \\big ) + \\sigma _ { j - 1 } , \\ \\ s \\in J . \\end{align*}"}
+{"id": "7613.png", "formula": "\\begin{align*} \\overline { \\gamma _ { i , j } } = \\begin{cases} \\pm [ ( q _ i ^ 1 ) _ { \\pm 1 } ] & \\ ; j = j _ { \\pm } ( i ) \\ ; \\ ; j _ { - } ( i ) \\not = j _ { + } ( i ) ; \\\\ [ ( q _ i ^ 1 ) _ { + 1 } ] - [ ( q _ i ^ 1 ) _ { - 1 } ] & \\ ; j = j _ { - } ( i ) = j _ { + } ( i ) ; \\\\ 0 & \\ ; j \\notin \\{ j _ { - } ( i ) , j _ { + } ( i ) \\} . \\end{cases} \\end{align*}"}
+{"id": "307.png", "formula": "\\begin{align*} \\lambda _ { j } : = \\min \\left \\{ \\lambda _ { j } \\left ( p _ { j } ^ { \\operatorname * { e x t } } \\right ) , \\lambda _ { j } \\left ( \\mathbb { A } _ { j } ^ { \\operatorname * { e x t } } \\right ) \\right \\} \\quad \\Lambda _ { j } : = \\max \\left \\{ \\Lambda _ { j } \\left ( p _ { j } ^ { \\operatorname * { e x t } } \\right ) , \\Lambda _ { j } \\left ( \\mathbb { A } _ { j } ^ { \\operatorname * { e x t } } \\right ) \\right \\} . \\end{align*}"}
+{"id": "953.png", "formula": "\\begin{align*} L = p ( x , t ) ( \\mathcal { T } _ t ^ \\alpha u - x u _ { x x } - a v _ x ) + q ( x , t ) ( \\mathcal { T } _ t ^ \\alpha v - x v _ { x x } - b u _ x ) , \\end{align*}"}
+{"id": "4080.png", "formula": "\\begin{align*} & \\omega _ { \\frac 1 2 , 2 } ( p _ 0 , p _ 1 ) + \\omega _ { \\frac 1 2 , 2 } ( \\sigma ( p _ 0 ) , p _ 1 ) \\\\ & = - \\frac { 1 } { 2 \\omega _ { 0 , 1 } ( p _ 0 ) } \\left ( - \\frac { d \\Delta y ( p _ 0 ) } { \\Delta y ( p _ 0 ) } \\cdot \\Delta _ 0 \\omega _ { 0 , 2 } ( p _ 0 , p _ 1 ) + d x ( p _ 0 ) \\cdot d _ 0 \\left ( \\frac { \\Delta _ 0 \\omega _ { 0 , 2 } ( p _ 0 , p _ 1 ) } { d x ( p _ 0 ) } \\right ) \\right ) . \\end{align*}"}
+{"id": "3555.png", "formula": "\\begin{align*} 0 & \\in \\partial f ( x ^ { k + 1 } ) + \\eta A ^ T ( A x ^ { k + 1 } + B y ^ { k + 1 } - b - \\frac { 1 } { \\eta } \\lambda ^ { k + 1 } ) = \\partial f ( x ^ { k + 1 } ) + \\eta A ^ T ( A x ^ { k + 1 } - A x ^ k + \\frac { 1 } { \\eta } \\lambda ^ { k } - \\frac { 2 } { \\eta } \\lambda ^ { k + 1 } ) \\ . \\end{align*}"}
+{"id": "7444.png", "formula": "\\begin{align*} F _ 1 ( t ) = & ( p - \\kappa _ 1 - \\kappa _ 2 - 2 ) t ^ { p + \\nu - 1 } \\| ( y _ 1 , y _ 2 ) \\| _ { 1 , p } + ( q - \\kappa _ 1 - \\kappa _ 2 - 2 ) t ^ { q + \\nu - 1 } \\| ( \\nabla y _ 1 , \\nabla y _ 2 ) \\| _ { q , \\eta } \\end{align*}"}
+{"id": "46.png", "formula": "\\begin{align*} \\langle f | a _ { 0 } ^ { \\dagger } a _ { 0 } g \\rangle = \\langle f | a _ { 0 } a _ { 0 } ^ { \\dagger } g \\rangle - \\langle f | g \\rangle = \\frac { 1 } { \\pi } \\int _ { \\mathbb { C } } | z | ^ { 2 } \\langle f | z \\rangle \\langle z | g \\rangle \\dd z - \\frac { 1 } { \\pi } \\int _ { \\mathbb { C } } \\langle f | z \\rangle \\langle z | g \\rangle \\dd z , \\end{align*}"}
+{"id": "4641.png", "formula": "\\begin{align*} \\bigl ( \\mathbf { M } ^ { \\pi _ { \\mathcal { S } ( v _ { n + 1 } ) } } \\otimes N \\bigr ) ^ { K _ { v _ { n + 1 } } } = \\bigl ( M ^ { \\pi _ { n } } \\otimes N \\bigr ) ^ { K } \\simeq \\textup { H o m } _ K ( N ^ * , M ^ { \\pi _ { n } } ) . \\end{align*}"}
+{"id": "1146.png", "formula": "\\begin{align*} \\widetilde { ( \\delta _ 1 h ) } = ( - 1 ) ^ { n - 1 } ~ [ \\mu _ 1 , \\widetilde { h } ] \\widetilde { ( \\delta _ 2 h ) } = ( - 1 ) ^ { n - 1 } ~ [ \\mu _ 2 , \\widetilde { h } ] , \\end{align*}"}
+{"id": "1430.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial z \\partial \\bar z } \\log | f _ i | ^ 2 = \\frac { | f _ { i + 1 } | ^ 2 } { | f _ i | ^ 2 } - \\frac { | f _ i | ^ 2 } { | f _ { i - 1 } | ^ 2 } . \\end{align*}"}
+{"id": "612.png", "formula": "\\begin{align*} f ( t ) = \\norm { u } _ { \\widetilde X ^ { s , b } ( 0 , t ) } ^ 2 . \\end{align*}"}
+{"id": "828.png", "formula": "\\begin{align*} \\eta ( \\overline { \\mathbb { D } } \\setminus U ) & = \\Psi _ { \\epsilon } ( \\tilde { h } ( \\overline { \\mathbb { D } } \\setminus U ) ) \\\\ & \\subseteq \\Psi _ { \\epsilon } ( \\delta _ 1 \\mathbb { D } ) \\\\ & \\subseteq \\epsilon ^ 2 \\mathbb { D } . \\end{align*}"}
+{"id": "3654.png", "formula": "\\begin{align*} \\phi _ { 1 1 } - c \\phi _ { 2 2 } = 0 \\ , \\end{align*}"}
+{"id": "4863.png", "formula": "\\begin{align*} f ( u ) = \\frac { 2 } { \\delta } \\frac { y ( u ) } { 1 - y ( u ) } \\end{align*}"}
+{"id": "7397.png", "formula": "\\begin{align*} \\rho _ { n } ^ { 2 } \\partial _ { t } W _ { n } = \\rho _ { n } \\partial _ { t } \\partial _ { x } w _ { n } - \\partial _ { x } w _ { n } \\partial _ { t } \\rho _ { n } . \\end{align*}"}
+{"id": "4829.png", "formula": "\\begin{align*} S ^ { i k } _ { j l } ( u ) = \\left [ \\frac { r ( i ) r ( l ) } { r ( j ) r ( k ) } \\right ] ^ { \\frac { 1 } { 2 } } F ( u ) \\cdot S ^ { j l } _ { \\bar { k } \\bar { i } } ( \\lambda - u ) \\end{align*}"}
+{"id": "1572.png", "formula": "\\begin{align*} J _ + ( x ) : = \\int _ 0 ^ 1 \\rho _ { g ^ T } ( x + y ) \\int _ 0 ^ { \\infty } | v | \\frac { e ^ { - ( 1 - y ) / \\kappa | v | } } { \\kappa ( 1 - e ^ { - 1 / \\kappa | v | } ) } \\left ( \\alpha \\mathcal { M } _ { T ( x + y ) } ( v ) + ( 1 - \\alpha ) \\mathcal { M } _ { \\tau ( x + y ) } ( v ) \\right ) \\mathrm { d } v . \\end{align*}"}
+{"id": "3713.png", "formula": "\\begin{align*} 2 \\alpha \\int _ { B _ 0 } \\frac { x _ 1 } { | x | ^ { 2 + 2 \\alpha } } d x = \\frac { 2 ^ { 1 - 2 \\alpha } } { 1 - 2 \\alpha } - 2 ^ { - \\alpha } f ( 1 ) - 2 ^ { - \\alpha } \\mu _ \\alpha \\end{align*}"}
+{"id": "564.png", "formula": "\\begin{align*} \\widetilde { \\mathbf I } ( \\tau , \\xi , \\omega ) = \\int _ 0 ^ T e ^ { - i t \\tau } \\ , \\widehat { \\mathbf I } ( t , \\xi , \\omega ) \\ , d t , \\end{align*}"}
+{"id": "6440.png", "formula": "\\begin{align*} \\tilde { f } _ t = \\begin{pmatrix} A & - ( T V _ 1 + ( 1 - T ) V _ 2 ) \\\\ B & T U _ 1 + ( 1 - T ) U _ 2 \\end{pmatrix} . \\end{align*}"}
+{"id": "3641.png", "formula": "\\begin{align*} \\Delta _ { \\omega } \\phi : = ( j ^ 1 \\phi ) ^ * \\omega \\ . \\end{align*}"}
+{"id": "5894.png", "formula": "\\begin{align*} ( { S _ n ' } ^ { \\textup { s a t } } , { N _ n ' } ^ { \\textup { s a t } } ) = ( S _ n ' \\otimes _ { \\Z _ p [ N _ n ' ] } \\Z _ p [ { N _ n ' } ^ { \\textup { s a t } } ] , { N _ n ' } ^ { \\textup { s a t } } ) \\end{align*}"}
+{"id": "8045.png", "formula": "\\begin{align*} h _ { \\omega , a t o m } ^ { p , q , s } ( \\mathbb { R } ^ { n } ) = \\left \\{ f \\in \\mathcal { S } ^ { \\prime } ( \\mathbb { R } ^ { n } ) \\colon f = \\sum \\limits _ { j } \\lambda _ { j } a _ { j } + \\sum \\limits _ { j } \\mu _ { j } b _ { j } \\right \\} , \\end{align*}"}
+{"id": "3782.png", "formula": "\\begin{align*} - g ( A x ) = 0 \\iff \\phi _ 0 \\oplus \\phi _ 1 \\leq c - g ( A x ) = - \\infty \\ , \\end{align*}"}
+{"id": "7109.png", "formula": "\\begin{align*} T ( X , 3 \\lambda Z ) = \\int _ { \\mathbb { R } } \\xi ^ 2 \\ , V ( \\xi ^ 3 ) \\ , e \\left ( 3 \\lambda Y \\xi - X \\xi ^ 3 \\right ) \\ , d \\xi . \\end{align*}"}
+{"id": "4184.png", "formula": "\\begin{align*} f _ \\mathbf { Y } ( \\mathbf { y } ) & = \\int _ { \\mathbb { R } ^ { d - 1 } } f _ { \\mathbf { Y } | \\mathbf { X } } ( \\mathbf { y } | \\mathbf { x } ) f _ \\mathbf { X } ( \\mathbf { x } ) \\mathrm { d } \\mathbf { x } \\\\ & = \\int _ { \\mathbb { R } ^ { d - 1 } } f _ \\mathbf { N } ( \\mathbf { n } ) f _ \\mathbf { X } ( \\mathbf { y } - \\mathbf { n } ) \\mathrm { d } \\mathbf { n } . \\end{align*}"}
+{"id": "4139.png", "formula": "\\begin{align*} [ F _ 1 , F _ 2 ] = [ F _ 1 , F _ 2 ] _ { \\mathfrak { g } } , F _ 1 \\rhd F _ 2 = Y _ { F _ 1 } F _ 2 . \\end{align*}"}
+{"id": "8419.png", "formula": "\\begin{align*} Y ( \\ell ) = Y _ { k , q } ( \\ell ) = h ( ( k + q ) \\ell ) - h ( k \\ell ) . \\end{align*}"}
+{"id": "8327.png", "formula": "\\begin{align*} { \\mathbf { s } } _ { \\mathbf { S B } } = { \\mathbf { s } } _ { \\mathbf { S B } } { { \\mathbf { w } } } ^ { ' } { \\mathbf { r } } ( \\mathbf { I } - \\mathbf { P } _ { S M } ) ^ { - 1 } \\end{align*}"}
+{"id": "3988.png", "formula": "\\begin{align*} ( \\chi + \\tilde \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ) ^ n = c ( \\chi + \\tilde \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ) ^ m \\wedge \\omega ^ { n - m } , \\sup _ M \\varphi = 0 , \\end{align*}"}
+{"id": "5130.png", "formula": "\\begin{align*} C _ { \\alpha , \\beta } = \\frac { 1 } { \\beta T } \\sum _ { \\gamma = ( \\alpha + \\beta - S ) ^ + } ^ { \\min ( \\alpha , \\beta ) } \\min ( \\beta , 2 \\gamma ) \\cdot \\binom { \\alpha } { \\gamma } \\cdot \\binom { S - \\alpha } { \\beta - \\gamma } . \\end{align*}"}
+{"id": "4758.png", "formula": "\\begin{align*} \\frac { { \\rm d i s t } ( w , \\partial \\Omega ) } { \\ell ( \\gamma _ { y , w } ) } \\le \\frac { 2 ^ { - 4 k - 4 } } { 2 ^ { - 2 k - 3 } } = 2 ^ { - 2 k - 1 } < \\frac { 1 } { C } , \\end{align*}"}
+{"id": "3448.png", "formula": "\\begin{align*} \\bar { \\Delta } ^ \\vee = \\cup _ k \\bar { \\Delta } ^ \\vee _ k , \\bar { \\Delta } ^ \\vee _ k = ( \\Delta ^ \\vee _ k ) . \\end{align*}"}
+{"id": "8816.png", "formula": "\\begin{align*} \\frac { \\bar { L } } { n } \\sum _ { i = 1 } ^ { n } \\norm { x ^ { i } ( t ) - \\bar { x } ( t ) } \\norm { \\bar { x } ( t ) - x ^ { * } } \\leq \\frac { \\bar { L } t \\alpha } { n } \\sum _ { i = 1 } ^ { n } \\norm { x ^ { i } ( t ) - \\bar { x } } ^ { 2 } + \\frac { \\bar { L } } { 4 n t \\alpha } \\mathcal { K } ^ { 2 } . \\end{align*}"}
+{"id": "1570.png", "formula": "\\begin{align*} & \\int _ 0 ^ { \\infty } g ^ T ( x , v ) \\mathrm { d } v = \\int _ 0 ^ { \\infty } \\int _ 0 ^ 1 \\frac { e ^ { - ( 1 - y ) / \\kappa | v | } } { \\kappa | v | ( 1 - e ^ { - 1 / \\kappa | v | } ) } \\rho _ g ( x + y ) \\left ( \\alpha \\mathcal { M } _ { T ( y + x ) } + ( 1 - \\alpha ) \\mathcal { M } _ { \\tau ( y + x ) } \\right ) \\mathrm { d } y \\mathrm { d } v \\\\ & = \\int _ 0 ^ 1 [ \\alpha \\mathcal { K } _ 1 ( T , y ) + ( 1 - \\alpha ) \\mathcal { K } _ 1 ( \\tau , y ) ] \\rho _ g ( x + y ) \\mathrm { d } y . \\end{align*}"}
+{"id": "2398.png", "formula": "\\begin{align*} T _ { \\i } ( x ) - x \\in E _ n \\iff & A _ { \\i } ^ { - 1 } ( x ) - x \\in \\sum _ { k = 1 } ^ { n } A _ { i _ 1 \\ldots i _ { k } } ^ { - 1 } t _ { i _ { k } } + E _ n \\\\ \\iff & x \\in ( A _ { \\i } ^ { - 1 } - I ) ^ { - 1 } \\left ( \\sum _ { k = 1 } ^ { n } A _ { i _ 1 \\ldots i _ { k } } ^ { - 1 } t _ { i _ { k } } + E _ n \\right ) . \\end{align*}"}
+{"id": "3947.png", "formula": "\\begin{align*} r _ K ( w ) v _ 2 = x ' v _ 2 + c v v _ 2 > x _ 0 v _ 2 = h _ L ( v _ 2 ) . \\end{align*}"}
+{"id": "2664.png", "formula": "\\begin{align*} d _ { r } ( C _ { D } ) = n - \\max \\big \\{ | D \\cap H | : H \\in [ \\mathbb { F } , s - r ] _ { p } \\big \\} . \\end{align*}"}
+{"id": "1882.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\R } \\int _ I \\ , v ^ 2 \\ , f ( t , x , v ) \\ , { \\rm d } x \\ , { \\rm d } v + \\int _ { I } u ^ 2 ( t , x ) \\ , { \\rm d } x \\leq \\int _ { \\R } \\int _ I v ^ 2 f _ 0 ( x , v ) \\ , { \\rm d } x \\ , { \\rm d } v + \\int _ { I } u _ 0 ^ 2 ( x ) \\ , { \\rm d } x , t \\in [ 0 , T ] \\end{aligned} \\end{align*}"}
+{"id": "6813.png", "formula": "\\begin{align*} G ( u , v ) & = ( v - v ^ * ) \\left [ ( q ( u ) - q ( u ^ * ) ) - ( u - u ^ * ) \\right ] / q ( u ) \\\\ [ 0 . 2 c m ] & = ( v - v ^ * ) \\left [ ( u - u ^ * ) - ( u - u ^ * ) \\right ] / q ( u ) = 0 , \\end{align*}"}
+{"id": "2145.png", "formula": "\\begin{align*} \\begin{aligned} \\hat e _ 0 & = \\frac { d ^ 2 \\alpha \\partial _ t \\alpha } { ( \\partial _ x \\alpha ) ^ 2 - ( \\partial _ t \\alpha ) ^ 2 } \\left ( \\frac { ( \\partial _ x \\alpha ) ^ 2 - ( \\partial _ t \\alpha ) ^ 2 } { \\alpha ^ 2 } \\right ) = d ^ 2 \\partial _ t ( \\ln \\alpha ) . \\end{aligned} \\end{align*}"}
+{"id": "4752.png", "formula": "\\begin{align*} \\Omega _ 2 \\coloneqq \\bigcup _ { k = 0 } ^ \\infty \\left ( 2 ^ { - 2 k - 1 } T + ( 2 ^ { - 2 k - 1 } , 0 ) \\right ) \\cup J , \\end{align*}"}
+{"id": "3011.png", "formula": "\\begin{align*} \\Sigma _ \\vee = \\Sigma _ { k , 1 } \\vee \\Sigma _ { \\ell , 1 } . \\end{align*}"}
+{"id": "866.png", "formula": "\\begin{align*} \\begin{aligned} & G \\left ( \\left | m _ u \\left ( B \\left ( z , r _ 1 \\right ) \\right ) - m _ u \\left ( B \\left ( z , r _ 2 \\right ) \\right ) \\right | \\right ) \\\\ & \\leq 4 b ^ 2 2 ^ { s + \\alpha p _ { 0 } } [ u ] _ { W ^ { \\alpha , G } _ { s } ( X , d , \\mu ) } ^ { p ^ { 0 } } \\left ( r _ 1 ^ { \\alpha p _ { 0 } - s } + \\frac { r _ 2 ^ { \\alpha p _ { 0 } } } { r _ 1 ^ s } \\right ) \\end{aligned} \\end{align*}"}
+{"id": "7327.png", "formula": "\\begin{align*} f _ \\lambda ( w ) & = \\frac { 1 } { 2 } \\int _ \\Omega { \\langle \\nabla u ( x ) , \\nabla u ( x ) \\rangle \\ , d x } - \\frac { 1 } { 2 } \\int _ \\Omega { \\langle \\nabla v ( x ) , \\nabla v ( x ) \\rangle \\ , d x } + \\int _ \\Omega { F ( \\lambda , x , u ( x ) , v ( x ) ) \\ , d x } , \\end{align*}"}
+{"id": "404.png", "formula": "\\begin{align*} ( A = B ) \\simeq ( A \\simeq B ) \\ , . \\end{align*}"}
+{"id": "8445.png", "formula": "\\begin{align*} \\iota _ { \\nabla } \\circ j ^ { d R } _ { \\nabla } ( x ) = \\log \\left [ \\exp \\left ( - p _ { \\nabla } ^ { H } ( x ) \\right ) p ^ { c r } _ { \\nabla } ( x ) \\right ] . \\end{align*}"}
+{"id": "6406.png", "formula": "\\begin{align*} c _ { 2 3 } t _ { 1 2 } c _ { 2 3 } ^ { - 1 } = c _ { 1 2 } ^ { - 1 } t _ { 2 3 } c _ { 1 2 } = c _ { 2 3 } ^ { - 1 } t _ { 1 2 } c _ { 2 3 } = c _ { 1 2 } t _ { 2 3 } c _ { 1 2 } ^ { - 1 } \\end{align*}"}
+{"id": "6101.png", "formula": "\\begin{align*} \\begin{aligned} E ( f ( x _ { l + 1 } ) - f ( \\omega _ * ) ) \\leq \\xi E ( f ( x _ l ) - f ( \\omega ^ * ) ) \\\\ \\xi = \\frac { ( 1 - \\sigma _ 2 ) ( 1 - \\beta ) + 2 L \\alpha _ 2 \\sigma _ 1 \\sigma _ 2 \\beta ( 1 - \\beta ^ m ) } { 2 \\mu \\sigma _ 1 \\alpha _ 1 m ( 1 - \\sigma _ 2 ) ( 1 - \\beta ) } \\\\ \\end{aligned} \\end{align*}"}
+{"id": "3428.png", "formula": "\\begin{align*} \\phi _ { k , t } = \\frac { 1 } { k } \\max _ { 1 \\leq i \\leq N } ( \\log | s _ i | _ { h ^ { \\otimes k } } + c _ i | \\log | t | | ) , \\end{align*}"}
+{"id": "93.png", "formula": "\\begin{align*} I ( d ) = \\int _ { \\mathbb { R } ^ d } \\Big ( \\mathcal { A } ( k ) - \\sqrt { \\mathcal { A } ( k ) ^ 2 - \\mathcal { B } ( k ) ^ 2 } \\Big ) \\dd k - \\frac { K _ 2 ^ 2 } { \\kappa } \\int _ { \\mathbb { R } ^ d } G _ d ( k ) \\dd k , \\end{align*}"}
+{"id": "7786.png", "formula": "\\begin{align*} \\widetilde { \\xi } _ i ^ { } : = \\widetilde { \\xi } _ i - \\widetilde { \\xi } _ i ^ { } , \\alpha ^ { } : = \\alpha + \\frac { 1 } { 2 } ( \\alpha ^ { } - \\xi ^ { i } \\widetilde { \\xi } _ i ^ { } ) \\ , . \\end{align*}"}
+{"id": "5249.png", "formula": "\\begin{align*} \\alpha _ t = \\frac { 1 } { \\mu t } , ~ \\gamma _ t = \\frac { \\gamma _ 0 } { \\alpha _ t } , ~ \\forall t \\in \\mathbb { N } _ + , \\end{align*}"}
+{"id": "2736.png", "formula": "\\begin{align*} \\| x \\| = \\theta \\cdot \\sum _ { i = 1 } ^ { d } \\big \\| E _ i x \\big \\| . \\end{align*}"}
+{"id": "4944.png", "formula": "\\begin{align*} f _ k ( \\tau _ r ) = \\dfrac { \\eta ( 2 ^ { r + 2 } i ) ^ { 8 k - 2 } \\eta ( 2 ^ { r + 3 } i ) ^ 4 } { \\eta ( 2 ^ { r + 1 } i ) ^ { 4 k } } = \\dfrac { a ^ { 4 k + 2 } } { 2 ^ { 2 k ( r + 4 ) + r + 5 } } \\cdot e ^ { \\frac { \\pi ( 2 k + 1 ) } { 3 } - 2 ^ { r + 1 } ( k + 1 ) \\pi - \\frac { 2 k + 1 } { \\pi } \\sum _ { m = 1 } ^ { r + 1 } 2 ^ m L _ m - \\frac { 2 ^ { r + 2 } ( 4 k + 1 ) } { \\pi } L _ { r + 2 } - \\frac { 2 ^ { r + 4 } } { \\pi } L _ { r + 3 } } . \\end{align*}"}
+{"id": "7873.png", "formula": "\\begin{align*} E _ 0 ( \\mu ) : = \\frac 1 2 \\int _ { t ' } ^ { t '' } \\int _ M \\left [ | \\N \\phi | ^ 2 + R ^ { H , f } \\right ] \\rho e ^ { - f } \\ , d V d t . \\end{align*}"}
+{"id": "5069.png", "formula": "\\begin{align*} E ( M ) = \\left ( \\begin{array} { c c c c c } E ( m _ { 1 1 } ) & 0 & 0 & \\cdots & 0 \\\\ 0 & E ( m _ { 2 2 } ) & 0 & \\cdots & 0 \\\\ \\vdots & & \\ddots & & \\vdots \\\\ 0 & \\cdots & 0 & E ( m _ { n - 1 n - 1 } ) & 0 \\\\ 0 & \\cdots & 0 & 0 & E ( m _ { n n } ) \\end{array} \\right ) , \\end{align*}"}
+{"id": "5398.png", "formula": "\\begin{align*} \\frac { \\partial v ( z , w ; t ) } { \\partial t } = 2 \\frac { \\partial | v ( z , w ; t ) | ^ 2 } { \\partial w } , ( z , w ) \\in \\C \\times \\C ^ { \\times } , \\ , t \\geq 0 , \\end{align*}"}
+{"id": "7223.png", "formula": "\\begin{align*} \\substack { \\mathcal H ^ h ( \\partial ^ * E \\setminus E ^ { ( 1 / 2 ) } ) = 0 \\\\ E } \\quad \\Longrightarrow \\quad \\quad \\Longrightarrow \\quad \\end{align*}"}
+{"id": "3029.png", "formula": "\\begin{align*} \\phi ( P _ { k - 5 } ) \\phi ( P _ { k - 2 } ) = \\phi ( P _ { k - 4 } ) \\phi ( P _ { k - 3 } ) - \\lambda \\mbox { a n d } \\phi ( P _ { k - 6 } ) \\phi ( P _ { k - 4 } ) = \\phi ^ 2 ( P _ { k - 5 } ) - 1 . \\end{align*}"}
+{"id": "7567.png", "formula": "\\begin{align*} p _ N & = P _ T \\Big ( \\sup _ { t \\in [ 0 , T ] } | B ^ { ( k ) } _ t | > r _ 2 ( T ) \\Big ) \\\\ & \\leq C \\frac { 1 } { r _ 2 ( T ) } \\exp \\left ( - \\frac { r _ 2 ( T ) ^ 2 } { 2 T } \\right ) , \\end{align*}"}
+{"id": "6430.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\limsup _ { t \\rightarrow \\infty } \\tilde { v } _ n ( c t , t ) = 0 . \\end{align*}"}
+{"id": "2732.png", "formula": "\\begin{align*} U e _ i = \\left \\{ \\begin{array} { l l } \\varepsilon _ i e _ { \\pi ( i ) } , & 1 \\leq i \\leq \\lceil \\theta ^ { - 1 } \\rceil \\\\ \\varepsilon _ i e _ i , & i > \\lceil \\theta ^ { - 1 } \\rceil \\end{array} \\right . \\quad ( i \\in \\mathbb N ) \\end{align*}"}
+{"id": "952.png", "formula": "\\begin{align*} \\begin{aligned} C ^ x _ i = & \\xi _ i L + \\sum _ { j = 1 } ^ { s } \\left ( W ^ j _ i \\frac { \\delta L } { \\delta u _ { j , x } } + D _ x ( W ^ j _ i ) \\frac { \\delta L } { \\delta u _ { j , x x } } + D ^ 2 _ x ( W ^ j _ i ) \\frac { \\delta L } { \\delta u _ { j , x x x } } + \\cdots \\right ) , \\\\ C ^ t _ i = & \\tau _ i L + \\sum _ { j = 1 } ^ { s } \\left ( W ^ j _ i \\frac { \\delta L } { \\delta u _ { j , t } } \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "8022.png", "formula": "\\begin{align*} \\sum _ { h = 1 } ^ H c _ h e ^ { \\beta c _ h } \\prod _ { \\substack { j = 1 \\\\ j \\not = h } } ^ { H } \\bigl ( e ^ { \\beta c _ j } - 1 \\bigr ) = \\sum _ { h = 1 } ^ { H } ( - 1 ) ^ { H - h } \\sum _ { i \\in \\mathcal { C } _ h ^ { \\{ 1 , \\dots , H \\} } } ( c _ { i _ 1 } + \\dots + c _ { i _ h } ) \\ , e ^ { \\beta ( c _ { i _ 1 } + \\dots + c _ { i _ h } ) } . \\end{align*}"}
+{"id": "6125.png", "formula": "\\begin{align*} 2 \\pi H _ f ( \\sigma _ 1 , \\sigma _ 2 ) = \\phi _ f ' ( \\sigma _ 2 ) - \\phi _ f ' ( \\sigma _ 1 ) . \\end{align*}"}
+{"id": "4809.png", "formula": "\\begin{align*} | \\varphi | ^ \\perp _ { \\alpha , m } = \\sup _ { x \\in S _ \\gamma } \\sup _ { x \\neq y \\in C ^ \\perp _ 0 ( x ) } \\frac { \\left \\| \\varphi ( x ) - \\varphi ( y ) \\right \\| } { d ( x , y ) ^ \\alpha } . \\end{align*}"}
+{"id": "3021.png", "formula": "\\begin{align*} \\phi ( G _ { n + 2 } ) = \\lambda \\phi ( G _ { n + 1 } ) - \\phi ( G _ { n } ) . \\end{align*}"}
+{"id": "823.png", "formula": "\\begin{align*} | \\tilde { h } ( z ) | & = e ^ { n _ 0 \\mbox { R e } \\tilde { g _ 1 } ( z ) } \\\\ & \\leq e ^ 0 \\\\ & = 1 , \\end{align*}"}
+{"id": "8440.png", "formula": "\\begin{align*} & x t ^ n \\cdot ( a \\otimes f _ { ( m ) } ) = a \\otimes ( x _ L f ) _ { ( n + m ) } , \\\\ & x t ^ { - n - 1 } \\cdot ( a \\otimes f _ { ( m ) } ) = ( x t ^ { - n - 1 } ) a \\otimes f _ { ( m ) } , \\end{align*}"}
+{"id": "6598.png", "formula": "\\begin{align*} \\mathcal { S } = \\mathcal { S } _ 1 \\cup \\mathcal { S } _ 2 \\end{align*}"}
+{"id": "4233.png", "formula": "\\begin{align*} \\tau ^ { 1 } = \\sqrt { 1 - | u | ^ 2 } \\ , \\omega ^ { 1 } , \\tau ^ { 2 } = u \\ , \\omega ^ { 1 } + i \\ , \\omega ^ { 2 } , \\tau ^ { 3 } = t \\ , \\omega ^ { 3 } . \\end{align*}"}
+{"id": "5970.png", "formula": "\\begin{align*} 1 = \\sum _ { \\gamma } \\frac { 1 } { \\gamma ! } \\partial _ { \\xi ' } ^ \\gamma \\sigma ( N ^ \\omega _ { \\partial M } ) D _ t ^ \\gamma \\sigma ( \\Lambda _ { g , F } ^ \\omega ) + S ^ { - \\infty } . \\end{align*}"}
+{"id": "6606.png", "formula": "\\begin{align*} \\prod _ { l = 1 } ^ N w ( x _ l ) \\prod _ { 1 \\le j < k \\le N } | x _ k - x _ j | ^ \\beta . \\end{align*}"}
+{"id": "7501.png", "formula": "\\begin{align*} X ^ { \\sigma + 1 } - \\overline { \\gamma } X + \\overline { \\gamma } = 0 \\end{align*}"}
+{"id": "6541.png", "formula": "\\begin{align*} \\Sigma _ N = \\bigcup _ { \\sqrt { N } \\leq N ' \\leq N , \\ \\Lambda \\in ( 0 , n ) + \\mathcal { E R } _ 0 ( N ' ) , \\ | n | \\leq 2 N ' } \\Sigma _ \\Lambda , \\end{align*}"}
+{"id": "457.png", "formula": "\\begin{align*} ( | D | ^ \\alpha f ) ( x ) = \\Phi _ \\ell ( \\sigma ) | x | ^ { - \\alpha } f ( x ) \\end{align*}"}
+{"id": "4128.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k - 1 } \\sum _ { \\ell = 0 } ^ { k - 2 i } U ( k , s , i , \\ell ) + \\sum _ { r = 1 } ^ s \\sum _ { \\ell = 0 } ^ { k - 2 r - 2 s + 2 } V ( k , s , r , \\ell ) = 0 . \\end{align*}"}
+{"id": "1833.png", "formula": "\\begin{gather*} \\sup _ { \\alpha \\in \\mathcal { A } _ \\rho ( c ) } \\left \\| \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + \\alpha ) ^ s } - \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + c ) ^ s } \\right \\| < \\epsilon \\end{gather*}"}
+{"id": "7749.png", "formula": "\\begin{align*} \\chi _ { i } ( G \\circ H ) \\leq | \\mathbb { P } | = \\chi _ { i } ( G ) i _ { H } + \\chi _ { 2 } ( G ) \\big { ( } | V ( H ) | - i _ { H } \\big { ) } = \\chi _ { 2 } ( G ) | V ( H ) | - i _ { H } \\big { ( } \\chi _ { 2 } ( G ) - \\chi _ { i } ( G ) \\big { ) } . \\end{align*}"}
+{"id": "4262.png", "formula": "\\begin{align*} \\sup _ { u ^ - \\in H _ k ^ - \\setminus \\{ 0 \\} } \\left \\{ 1 + \\dfrac { \\alpha [ u ^ - ] ^ 2 - \\lambda _ k \\| u ^ - \\| ^ 2 _ { L ^ 2 ( \\Omega ) } } { \\| u ^ - \\| _ { \\mathbb { X } ( \\Omega ) } ^ 2 } \\right \\} = - \\gamma . \\end{align*}"}
+{"id": "221.png", "formula": "\\begin{align*} & \\left \\{ \\widehat { L } ( T M ) [ 2 { \\rm c h } ( \\widetilde { T _ C M } ) - 8 + { \\rm c h } ( W _ i ) ] \\right \\} ^ { ( 1 2 ) } \\\\ & = \\left \\{ \\widehat { A } ( T M ) { \\rm c h } [ 2 1 1 6 \\widetilde { T _ C M } + 4 \\wedge ^ 2 \\widetilde { T _ C M } + 4 W _ i - 1 5 8 7 2 ] \\right \\} ^ { ( 1 2 ) } . \\end{align*}"}
+{"id": "6250.png", "formula": "\\begin{align*} D = \\{ \\xi = \\xi _ 1 - \\xi _ 2 + \\cdots \\} . \\end{align*}"}
+{"id": "6147.png", "formula": "\\begin{align*} A = ( \\{ \\ ! \\{ a _ { 1 , 1 } \\} \\ ! \\} , \\{ \\ ! \\{ a _ { 2 , 1 } , a _ { 2 , 2 } \\} \\ ! \\} , \\ldots , \\{ \\ ! \\{ a _ { n , 1 } , a _ { n , 2 } , \\ldots , a _ { n , n } \\} \\ ! \\} ) , a _ { i , j } \\leq a _ { i , j + 1 } ( \\forall 1 \\leq j < i \\leq n ) , \\end{align*}"}
+{"id": "6035.png", "formula": "\\begin{align*} \\Vert D G _ n - D F _ n \\Vert _ { L ^ 2 ( \\Omega ; \\mathcal { H } ) } \\leq \\sum _ { i = n } ^ { m _ { n } } \\gamma _ { i } ^ { n } \\Vert D F _ { i } - D F _ n \\Vert _ { L ^ 2 ( \\Omega ; \\mathcal { H } ) } \\rightarrow 0 . \\end{align*}"}
+{"id": "6693.png", "formula": "\\begin{align*} \\lim _ { p \\to 0 } { 1 \\over p } c _ \\infty ^ { ( \\widetilde { \\rm c J ) } } ( \\tau ; \\beta , p , 0 ) = - { \\pi \\beta \\over | \\tau | } \\Big ( 1 + c _ \\infty ^ { ( \\widetilde { \\rm c J ) } } ( \\tau ; \\beta , 1 , 0 ) \\Big ) . \\end{align*}"}
+{"id": "5308.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = 0 } ^ { n _ 0 } { N \\choose j } \\right ) ^ L . \\end{align*}"}
+{"id": "257.png", "formula": "\\begin{align*} e ^ { - c \\langle \\xi , \\rho _ K \\rangle } \\Delta _ ( \\xi ) \\phi _ \\xi ^ { } ( x + c \\rho _ K ; g ^ { ( c ) } ) = \\sum _ { \\nu \\geq 0 } \\hat { a } ^ { } _ { \\nu } ( \\xi ; g ^ { ( c ) } ) e ^ { \\langle \\xi - \\nu , x \\rangle } , \\end{align*}"}
+{"id": "2197.png", "formula": "\\begin{align*} \\# L = \\frac { q ^ { k + 1 } - 1 } { q - 1 } , \\ \\# R = \\frac { ( q ^ { k + 1 } - 1 ) ( q ^ { k } - 1 ) } { ( q - 1 ) ^ 2 } , \\ D _ L = \\frac { ( q ^ { k } - 1 ) ( q ^ { k - 1 } - 1 ) } { ( q - 1 ) ^ 2 } , \\ D _ R = \\frac { q ^ { k - 1 } - 1 } { q - 1 } \\end{align*}"}
+{"id": "200.png", "formula": "\\begin{align*} & \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) ( 2 0 + { \\rm c h } ( \\overline { W _ i } ) - 8 { \\rm c h } ( { W _ i } ) + B _ 1 \\wedge [ { \\rm c h } ( { W _ i } ) - 8 ] + B _ 2 ) \\right \\} ^ { ( 1 0 ) } \\\\ & = 6 1 9 2 0 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 0 ) } . \\end{align*}"}
+{"id": "1252.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\liminf _ { n \\to \\infty } \\big [ P ( f _ n ) - P ( g _ n ^ j \\phi ^ j ) - P ( r _ n ^ J ) \\big ] = 0 . \\end{align*}"}
+{"id": "488.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\left ( - i \\partial _ t - i \\alpha \\partial _ x + M \\beta \\right ) \\psi & = \\phi \\beta \\psi + \\beta \\psi \\xi _ 1 , \\\\ \\left ( \\partial _ t ^ 2 - \\partial _ x ^ 2 + m ^ 2 \\right ) \\phi & = \\psi ^ * \\beta \\psi + \\phi \\xi _ 2 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "7504.png", "formula": "\\begin{align*} G _ 6 ( \\gamma ) = \\frac { \\gamma ^ 3 - 5 \\gamma ^ 2 + 6 \\gamma - 1 } { \\gamma ^ 3 } \\end{align*}"}
+{"id": "1509.png", "formula": "\\begin{align*} c ( e _ i \\otimes e _ j ) \\ , = \\ , \\sum \\limits _ { k , l } \\ , c _ { i j } ^ { k l } \\ , e _ k \\otimes e _ l \\end{align*}"}
+{"id": "3890.png", "formula": "\\begin{align*} \\begin{gathered} s _ \\alpha ^ j \\colon C ^ n ( \\mathcal { C } ) \\to C ^ { n + 1 } ( \\mathcal { C } ) \\ , ; j = 1 , \\dots , n \\ , , \\ ; \\alpha = - 1 , 1 \\ , , \\\\ s _ \\alpha ^ j ( S ) ( a _ 1 , \\dots , a _ { n + 1 } ) = \\begin{cases} S ( a _ 1 , \\dots , a _ { j - 1 } , a _ { j + 1 } \\dots , a _ { n + 1 } ) & a _ j \\neq \\alpha \\\\ 0 & a _ j = \\alpha \\ , . \\end{cases} \\end{gathered} \\end{align*}"}
+{"id": "4990.png", "formula": "\\begin{align*} g ( x _ j ) = g ( f _ { \\ell _ j } \\circ \\dotsc \\circ f _ 1 ( x _ 0 ) ) = f _ { \\ell _ j } \\circ \\dotsb \\circ f _ 1 ( x _ i ) = \\begin{cases} x _ { i + j } , & i \\\\ x _ { i - j } , & i \\end{cases} \\end{align*}"}
+{"id": "3058.png", "formula": "\\begin{align*} { \\bf { y } } = { { \\bf { W } } ^ H } { \\bf { H F s } } + { { \\bf { W } } ^ H } { \\bf { n } } , \\end{align*}"}
+{"id": "4494.png", "formula": "\\begin{align*} \\mathbb { B } '' ( ( { \\mathbf V } , \\psi ) , ( \\tilde { { \\mathbf V } } , \\tilde { \\psi } ) ) : = \\frac { d } { d \\varepsilon } \\mathbb { B } ' ( \\hat { \\mathbf U } + \\varepsilon \\tilde { { \\mathbf V } } , \\hat { \\varphi } + \\varepsilon \\tilde { \\psi } ) ( { \\mathbf V } , \\psi ) \\Big | _ { \\varepsilon = 0 } , \\end{align*}"}
+{"id": "3517.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\leqslant x \\\\ } } N _ p ( \\sigma ) = \\sum _ { \\nu \\in \\mathbb { N } } S ( \\frac { x } { 2 ^ \\nu } , \\sigma ) \\end{align*}"}
+{"id": "6650.png", "formula": "\\begin{align*} x ^ 2 R ''' ( x ) + 4 x R '' ( x ) + ( 2 - 4 p ^ 2 + 4 x ^ 2 ) R ' ( x ) - { 4 p ^ 2 \\over x } R ( x ) = 0 . \\end{align*}"}
+{"id": "3018.png", "formula": "\\begin{align*} \\phi ( H _ { 1 0 } ) = \\phi ( P _ 2 \\cup C _ { 1 , 1 ; 6 } ^ { 0 , 3 } ) ; \\phi ( H _ { 1 3 } ) = \\phi ( P _ { 2 ; 4 } ^ 1 \\cup C _ { 1 ; 6 } ^ 0 ) ; \\phi ( H _ { 1 5 } ) = \\phi ( P _ { 1 , 2 , 6 } ^ { 1 , 3 } \\cup C _ 6 ) , \\end{align*}"}
+{"id": "3348.png", "formula": "\\begin{align*} \\xi _ { A , d } = \\sum _ { i = 1 } ^ N d _ i ^ { - 1 } [ z _ i ] . \\end{align*}"}
+{"id": "3198.png", "formula": "\\begin{align*} \\left \\Vert x - x _ { k - 1 } \\right \\Vert _ { A ^ { T } A } ^ { 2 } - \\left \\Vert x - x _ { k } \\right \\Vert _ { A ^ { T } A } ^ { 2 } = \\phi _ { k } ^ { 2 } \\left ( { 1 + \\ell _ { k - 1 } ^ { ( w ) } } - \\ell _ { k } ^ { ( w ) } \\right ) . \\end{align*}"}
+{"id": "6945.png", "formula": "\\begin{align*} u ( \\dots , x _ j , \\dots , x _ i , \\dots ) = - u ( \\dots , x _ i , \\dots , x _ j , \\dots ) , \\end{align*}"}
+{"id": "7926.png", "formula": "\\begin{align*} \\dfrac { 1 } { p \\mathrm { F P d i m } ( R ) } + \\sum _ { j = 1 } ^ n \\dfrac { 1 } { f _ j } = \\dfrac { 1 } { p } . \\end{align*}"}
+{"id": "2064.png", "formula": "\\begin{align*} 2 \\sum _ { j = 2 } ^ { n - 1 } R _ { i j j i } & = \\left ( \\sum _ { j = 2 } ^ { n - 2 } ( R _ { 1 j j 1 } + R _ { 1 ( j + 1 ) ( j + 1 ) 1 } ) \\right ) + R _ { 1 ( n - 1 ) ( n - 1 ) 1 } + R _ { 1 2 2 1 } \\\\ & \\geq 0 . \\end{align*}"}
+{"id": "8809.png", "formula": "\\begin{align*} D ^ { m } f ( x ) \\nu ^ { m } = \\frac { \\partial ^ { | m | } f ( x ) } { \\partial ^ { m _ { 1 } } x _ { 1 } \\cdots \\partial ^ { m _ { d } } x _ { d } } \\nu _ { 1 } ^ { m _ { 1 } } \\cdots \\nu _ { d } ^ { m _ { d } } . \\end{align*}"}
+{"id": "5651.png", "formula": "\\begin{align*} K _ X + D _ X \\ = \\ \\sigma ^ * ( K _ { \\mathbb { P } ^ 3 } + D ) . \\end{align*}"}
+{"id": "2823.png", "formula": "\\begin{align*} \\ell i _ { 2 } ( [ s + a t ] ) = - \\frac { a ^ 3 } { 2 s ^ 2 ( 1 - s ) ^ 2 } . \\end{align*}"}
+{"id": "4969.png", "formula": "\\begin{align*} f _ n ^ { ( \\alpha \\bmod { \\delta } ) } ( 1 ) & = \\prod _ { a \\in ( \\Z / n _ 0 ) ^ \\times : a \\equiv \\pm \\alpha \\pmod * { \\delta } } \\frac { ( 1 - \\zeta _ { n _ 0 } ^ a ) } { ( 1 - \\zeta _ { n _ 0 } ^ { a p ' } ) } \\\\ & = \\frac { f _ { \\frac { n } { p } } ^ { ( \\alpha \\bmod { \\delta } ) } ( 1 ) } { f _ { \\frac { n } { p } } ^ { ( \\alpha p ' \\bmod { \\delta } ) } ( 1 ) } . \\end{align*}"}
+{"id": "3787.png", "formula": "\\begin{align*} H ( x _ 0 , s _ 0 , x _ 1 , s _ 1 ) : = \\inf _ { t > 0 } \\Bigg \\{ t \\left ( R \\left ( \\frac { s _ 0 } { t } \\right ) + R \\left ( \\frac { s _ 1 } { t } \\right ) + c ( x _ 0 , x _ 1 ) \\right ) \\Bigg \\} . \\end{align*}"}
+{"id": "5477.png", "formula": "\\begin{align*} ( z ; p _ 1 , \\dots , p _ n ) : = \\Big ( \\big ( p _ 1 ^ { ( \\epsilon _ 1 - 1 ) / 2 } \\dots p _ n ^ { ( \\epsilon _ n - 1 ) / 2 } z ; p _ 1 ^ { \\epsilon _ 1 } , \\dots , p _ n ^ { \\epsilon _ n } \\big ) \\Big ) ^ { \\epsilon _ 1 \\dots \\epsilon _ n } \\qquad \\textrm { f o r } ( z , p _ 1 , \\dots , p _ n ) \\in \\Omega _ { \\boldsymbol \\epsilon } . \\end{align*}"}
+{"id": "8288.png", "formula": "\\begin{align*} A _ \\varepsilon ( u , v , p ) = \\left ( v , u _ { x x } , \\varepsilon ^ { - 1 } p _ { x x } \\right ) , \\end{align*}"}
+{"id": "3845.png", "formula": "\\begin{align*} g = e ^ { - 2 \\lambda _ 1 z } d x ^ 2 + e ^ { - 2 \\lambda _ 2 z } d y ^ 2 + d z ^ 2 . \\end{align*}"}
+{"id": "6930.png", "formula": "\\begin{align*} & \\left ( \\phi ( - \\infty ) , \\psi ( - \\infty ) \\right ) = ( b , 0 ) , \\\\ [ 0 . 2 c m ] & \\left ( \\phi ( - \\infty ) , \\psi ( - \\infty ) \\right ) = ( 1 , 0 ) , \\\\ [ 0 . 2 c m ] & \\left ( \\phi ( - \\infty ) , \\psi ( - \\infty ) \\right ) = ( u _ 1 ^ * , u _ 1 ^ * ) , \\\\ [ 0 . 2 c m ] & \\left ( \\phi ( - \\infty ) , \\psi ( - \\infty ) \\right ) = ( u _ 2 ^ * , u _ 2 ^ * ) . \\end{align*}"}
+{"id": "5949.png", "formula": "\\begin{align*} G ^ \\omega _ M ( x , y ) & = \\sum _ { k \\neq j } \\frac { u _ k ( x ) u _ k ( y ) } { \\lambda _ k - \\omega ^ 2 } e ^ { \\phi ( y ) } + \\frac { u _ j ( x ) u _ j ( y ) } { \\lambda _ j - \\omega ^ 2 } e ^ { \\phi ( y ) } , \\\\ & : = N ^ \\omega _ M ( x , y ) + \\frac { u _ j ( x ) u _ j ( y ) } { \\lambda _ j - \\omega ^ 2 } e ^ { \\phi ( y ) } , \\end{align*}"}
+{"id": "3586.png", "formula": "\\begin{align*} { \\bar R _ { { \\rm { S M } } , n } } = { \\mathbb E } { \\left \\{ { { { \\log } _ 2 } \\left ( { 1 + \\frac { x } { 2 c _ n } } \\right ) } \\right \\} } , \\end{align*}"}
+{"id": "1859.png", "formula": "\\begin{gather*} \\mathcal { E } _ 2 = \\bigcup _ { \\substack { ( m _ 0 , \\ldots , m _ N ) \\in \\ , \\mathbb { Z } ^ { N + 1 } \\\\ \\forall n , ~ | m _ n | \\leq \\Delta } } \\bigcup _ { \\substack { N < n _ 1 , n _ 2 < L \\\\ n _ 1 \\neq n _ 2 } } \\left \\{ \\alpha \\in \\mathcal { A } ~ \\middle | ~ \\prod _ { n = 0 } ^ { N } ( n + \\alpha ) ^ { m _ n } = \\frac { n _ 2 + \\alpha } { n _ 1 + \\alpha } \\right \\} . \\end{gather*}"}
+{"id": "1396.png", "formula": "\\begin{gather*} \\sigma ( x ) = - 2 \\sum _ { i = 0 } ^ 1 \\sum _ { k \\in I _ i } ( - 1 ) ^ { i } \\left ( \\sum _ { j = 0 } ^ { m _ { k i } - 1 } \\alpha _ { k + j , i } \\sum _ { p = 0 } ^ { j } \\tilde \\varphi _ { k + p , i } ( x ) \\varphi _ { k + ( j - p ) , i } ( x ) - \\frac { 1 } { 2 } \\alpha _ { k i } \\right ) . \\end{gather*}"}
+{"id": "6229.png", "formula": "\\begin{align*} & \\bigl ( ( \\nabla ^ S ) ^ * \\nabla ^ S - 2 \\i \\ , \\nabla ^ S _ A + | A | ^ 2 + \\mathcal { R } - \\i \\ , \\div A + \\i \\ , d A \\cdot \\\\ & + 1 + \\i \\ , d V \\cdot \\nu \\cdot + V ^ 2 - A _ 0 ^ 2 + d A _ 0 \\cdot \\bigr ) \\psi _ n = ( D _ { A , V , A _ 0 } + \\i ) \\phi _ n . \\end{align*}"}
+{"id": "2059.png", "formula": "\\begin{align*} a _ k : = \\begin{cases} \\displaystyle \\sup _ { \\Omega \\setminus B _ { g _ 0 } ( x , r _ k ) } G _ { \\Omega } ( x , 1 ; \\cdot , t _ k ) \\quad & \\Omega \\setminus B _ { g _ 0 } ( x , r _ k ) \\neq \\phi ; \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "7874.png", "formula": "\\begin{align*} \\frac { d } { d s } E _ 0 ( \\mu ( \\cdot , s ) ) = \\left . \\int _ M \\phi \\partial _ s \\rho e ^ { - f } \\ , d V \\right | _ { t = t ' } ^ { t '' } - \\int _ { t ' } ^ { t '' } \\int _ M \\left [ \\partial _ t \\phi + \\frac 1 2 | \\nabla \\phi | ^ 2 - \\frac 1 2 R ^ { H , f } \\right ] \\partial _ s \\rho e ^ { - f } \\ , d V d t . \\end{align*}"}
+{"id": "525.png", "formula": "\\begin{align*} \\norm { \\theta ( t ) S _ { h ( \\xi ) } ( t ) f } _ { X ^ { s , b } _ { h ( \\xi ) } } = \\norm { \\theta } _ { H ^ b ( \\R ) } \\norm { f } _ { H ^ s ( \\R ^ d ) } \\end{align*}"}
+{"id": "859.png", "formula": "\\begin{align*} \\xi = \\prod _ { p \\in \\Pi } ~ p ^ { \\chi _ { \\xi } ( p ) } \\ , \\end{align*}"}
+{"id": "1087.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - \\Delta _ { \\mu } u ( x ) = \\alpha ( x ) u ( x ) + f ( x , u ( x ) ) x \\in \\mathop D \\limits ^ \\circ \\\\ \\medskip \\ , \\ , u | _ { \\partial D } = 0 , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "442.png", "formula": "\\begin{align*} \\beta & = \\frac 1 \\alpha \\left ( 2 \\zeta - \\gamma - \\eta \\right ) , \\end{align*}"}
+{"id": "604.png", "formula": "\\begin{align*} \\norm { \\Theta \\psi _ - ^ R } _ { X ^ { 0 , 1 / 2 + \\varepsilon - r } _ { - \\xi } ( 0 , T ) } = \\norm { \\Theta \\psi _ - ^ R } _ { X ^ { 0 , b } _ { - \\xi } ( 0 , T ) } ^ { \\mu } \\norm { \\Theta \\psi _ - ^ R } _ { X ^ { 0 , 0 } _ { - \\xi } ( 0 , T ) } ^ { 1 - \\mu } , \\end{align*}"}
+{"id": "1793.png", "formula": "\\begin{gather*} \\int _ { \\mathcal { T } ^ k } \\prod _ { j = 1 } ^ { k } \\gamma _ j ^ { m _ j } \\ , d \\nu _ \\alpha ( \\underline { \\gamma } ) = \\mathbf { E } \\left [ \\mathbb { X } _ \\alpha ( n _ 1 ) ^ { m _ 1 } \\cdots \\mathbb { X } _ \\alpha ( n _ k ) ^ { m _ k } \\right ] = g _ \\alpha ( \\underline { m } ) \\end{gather*}"}
+{"id": "7085.png", "formula": "\\begin{align*} \\mathcal { J } ^ { \\pm } ( n ^ 2 _ 1 n _ 2 , m , q ) = \\displaystyle \\int _ { \\mathbb { R } } \\ , V _ 1 \\left ( \\frac { \\nu } { K } \\right ) \\ , \\displaystyle \\int _ { \\mathbb { R } } W ( u ) \\ , g ( u , q ) \\ , \\eth _ 1 ^ { \\pm } ( n ^ 2 _ 1 n _ 2 , u , q , p _ 1 ) \\ , \\mathcal { I } ^ { \\pm } ( m , u , q , p _ 1 , p _ 2 ) \\ , d u \\ , d \\nu . \\end{align*}"}
+{"id": "3478.png", "formula": "\\begin{align*} - d d ^ c \\log | F _ i | _ { h ^ { \\otimes d _ i } } = - d d ^ c ( | F _ i | e ^ { - d _ i \\phi _ h } ) = d _ i d d ^ c \\phi _ h = d _ i \\omega _ M . \\end{align*}"}
+{"id": "2288.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } e ^ { 2 \\pi i k \\varepsilon _ j } e ^ { - 2 \\pi i k \\frac { j } { n } } = 2 \\pi i k \\sum _ { j = 1 } ^ { n } \\varepsilon _ j e ^ { - 2 \\pi i k \\frac { j } { n } } + \\mathcal { O } \\left ( k ^ 2 \\sum _ { j = 1 } ^ { n } \\varepsilon _ j ^ 2 \\right ) . \\end{align*}"}
+{"id": "7945.png", "formula": "\\begin{align*} S _ i = ( S \\cap V _ i ) \\cup X _ i \\end{align*}"}
+{"id": "2379.png", "formula": "\\begin{align*} \\min _ { u } \\left \\{ { \\mathcal { A } _ 1 } V ( x , t ) + h \\left [ V ^ { a f t e r } ( x _ t ^ { a f t e r } , t ) - V ^ { p r e } ( x _ t , t ) \\right ] + C ( t , x , u ) \\right \\} = 0 \\end{align*}"}
+{"id": "6471.png", "formula": "\\begin{align*} f _ 1 \\big | _ e = b _ e x + c _ e . \\end{align*}"}
+{"id": "263.png", "formula": "\\begin{align*} \\lim _ { c \\to + \\infty } \\gamma _ L ( g ^ { ( c ) } ) \\ , e ^ { c \\langle \\xi , \\rho _ L \\rangle } C ^ { } ( \\xi ; g ^ { ( c ) } ) = C ^ { } ( \\xi ; g ) . \\end{align*}"}
+{"id": "5778.png", "formula": "\\begin{align*} m _ + : = \\lim _ { \\mu _ 1 \\searrow 0 } \\lim _ { L \\to \\infty } \\mathbb E \\langle m ^ 1 \\rangle _ \\beta , \\ \\ \\ q _ + : = \\lim _ { \\mu _ 1 \\searrow 0 } \\lim _ { L \\to \\infty } \\mathbb E \\langle R _ { 1 , 2 } \\rangle _ \\beta . \\end{align*}"}
+{"id": "3037.png", "formula": "\\begin{align*} f ( n ) \\le 4 \\binom { 2 T } { 2 } + [ 3 \\cdot ( 2 T ) + 2 ( t - 1 ) ] ( n - 2 T ) + f ( n - 2 T ) \\le 4 \\binom { 2 T } { 2 } + ( 3 + \\varepsilon / 2 ) 2 T ( n - 2 T ) + f ( n - 2 T ) . \\end{align*}"}
+{"id": "6111.png", "formula": "\\begin{align*} \\psi \\left ( A ( T - \\lambda ) S \\right ) = 0 ; \\end{align*}"}
+{"id": "2589.png", "formula": "\\begin{align*} O _ { v a l _ { K } } & = \\left \\{ x \\in K ( t ^ { H } , \\beta ) \\ , \\vert \\ , v a l _ { K } ( x ) \\geq 0 \\right \\} = \\\\ & = \\left \\{ x \\in K ( t ^ { H } , \\beta ) _ { \\delta } \\ , \\vert \\ , h _ { 0 } > 0 \\ , \\ , \\ , \\ , h _ { 0 } = 0 \\ , \\wedge \\ , v ( a _ { h _ { 0 } } ) \\geq 0 , \\ , h _ { 0 } = m i n \\ , s u p p ( x ) \\right \\} = \\\\ & = \\left \\{ x \\in K [ t ^ { H } , \\beta ] _ { \\delta } \\ , \\vert \\ , x \\in ( t ) \\ , \\ , \\ , \\ , v ( a _ { 0 } ) \\geq 0 \\right \\} = \\\\ & = O _ { K } + ( t ) , \\end{align*}"}
+{"id": "29.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { i \\neq j } v ( x _ i - x _ j ) = \\sum _ { j = 0 } ^ 4 { \\mathcal Q } _ j ^ { \\rm r e n } , \\end{align*}"}
+{"id": "2221.png", "formula": "\\begin{align*} \\left ( D _ { b - } ^ \\alpha y \\right ) ( x ) = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\left [ \\frac { y ( b ) } { ( b - x ) ^ \\alpha } - \\underset { x } { \\overset { b } { \\int } } \\frac { y ' ( t ) d t } { ( t - x ) ^ \\alpha } \\right ] . \\end{align*}"}
+{"id": "5021.png", "formula": "\\begin{align*} \\triangle _ f ( - f ) ^ { - b } & = b ( \\triangle _ f f ) ( - f ) ^ { - b - 1 } + b ( b + 1 ) | \\nabla f | ^ 2 ( - f ) ^ { - b - 2 } \\\\ & = - b ( - f ) ^ { - b } - 1 2 b ( - f ) ^ { - b - 1 } + b ( b + 1 ) ( - f - R + 1 0 ) ( - f ) ^ { b - 2 } \\\\ & \\leq - b ( - f ) ^ { - b } + b ( b - 1 1 ) ( - f ) ^ { - b - 1 } + 1 1 b ( b + 1 ) ( - f ) ^ { b - 2 } < - b ( - f ) ^ { - b } , \\end{align*}"}
+{"id": "328.png", "formula": "\\begin{align*} \\left [ u \\right ] _ { \\operatorname * { D } ; j } \\left ( s \\right ) = \\psi . \\end{align*}"}
+{"id": "6264.png", "formula": "\\begin{align*} g ( u _ { < 1 } ) = u _ { < 1 } ^ 2 h ( u _ { < 1 } ) , \\end{align*}"}
+{"id": "5268.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { t } ( x _ { i , t } ) ] _ + \\| ^ 2 \\le \\varepsilon _ 3 + \\varepsilon _ 4 \\tilde { h } _ T , ~ \\forall y \\in \\mathcal { X } _ T . \\end{align*}"}
+{"id": "8773.png", "formula": "\\begin{align*} \\mathbb { E } [ \\norm { \\nabla f ( z _ S ) } ^ 2 ] & \\geq 2 \\sum _ { t = 1 } ^ { T } p _ t \\mathbb { E } [ f ( z _ t ) - f ^ * ] \\\\ & \\geq 2 \\mathbb { E } [ f ( \\sum _ { t = 1 } ^ { T } p _ t z _ t ) - f ^ * ] . \\end{align*}"}
+{"id": "3240.png", "formula": "\\begin{align*} \\frac { 1 } { \\epsilon q ^ { 2 } } = o \\left ( \\epsilon X R \\right ) . \\end{align*}"}
+{"id": "5545.png", "formula": "\\begin{align*} c ^ { ( 0 ) } _ j = w _ j , r ^ { ( 0 ) } _ j = \\frac { \\ , L \\ , } { 2 } , j = 1 , \\ldots , m . \\end{align*}"}
+{"id": "7822.png", "formula": "\\begin{align*} v \\cdot \\begin{pmatrix} \\widetilde { \\xi } _ 0 ^ { } \\\\ \\widetilde { \\xi } _ a ^ { } \\\\ \\alpha ^ { } \\\\ \\end{pmatrix} = \\begin{pmatrix} \\widetilde { \\xi } _ 0 ^ { } - \\widetilde { \\xi } _ a ^ { } v ^ a \\\\ \\widetilde { \\xi } _ a ^ { } \\\\ \\alpha ^ { } \\\\ \\end{pmatrix} \\ , . \\end{align*}"}
+{"id": "1781.png", "formula": "\\begin{gather*} ( \\gamma _ 0 , \\ldots , \\gamma _ N ) \\mapsto \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + \\alpha ) ^ s } \\end{gather*}"}
+{"id": "8868.png", "formula": "\\begin{align*} \\min \\Big ( \\alpha , \\ , \\frac { d } { \\alpha } T ^ { - \\frac { \\beta - 1 } { \\beta } } \\Big ) = \\min \\Big ( \\max ( \\alpha , T ^ { - \\frac { \\beta + 2 } { 2 \\beta } } ) , \\frac { d } { \\sqrt { T } } , \\ , \\frac { d } { \\alpha } T ^ { - \\frac { \\beta - 1 } { \\beta } } \\Big ) \\enspace . \\end{align*}"}
+{"id": "4881.png", "formula": "\\begin{align*} \\langle f , g \\rangle : = \\int _ { - \\infty } ^ \\infty \\langle E _ + ^ { - 1 } ( t ) f ( t ) , E _ + ^ { - 1 } ( t ) g ( t ) \\rangle _ \\mathfrak { X } d t , \\end{align*}"}
+{"id": "5406.png", "formula": "\\begin{align*} d S ^ { - 1 } _ { m k } ( t ) = S ^ { - 1 } _ { m k } ( t ) d V _ { m m } ( t ) + \\sum _ { j : j \\not = m } S ^ { - 1 } _ { j k } ( t ) d V _ { m j } = - S ^ { - 1 } _ { m k } ( t ) d U _ { m m } ( t ) - \\sum _ { j : j \\not = m } S ^ { - 1 } _ { j k } ( t ) d U _ { m j } , \\end{align*}"}
+{"id": "1908.png", "formula": "\\begin{align*} \\left ( \\widehat { Q ^ x _ \\lambda \\Upsilon } \\right ) _ { i + 1 / 2 } = \\lambda \\Upsilon ^ - _ { i + 1 / 2 } + \\left ( 1 - \\lambda \\right ) \\Upsilon ^ + _ { i + 1 / 2 } , \\ , \\ , 1 \\leq i \\leq N _ x - 1 . \\end{align*}"}
+{"id": "4717.png", "formula": "\\begin{align*} \\Gamma _ { D _ 1 } ( w ) = H . \\end{align*}"}
+{"id": "6079.png", "formula": "\\begin{align*} \\min \\limits _ { \\omega \\in R ^ { d } } f ( \\omega ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f _ i ( \\omega ) \\end{align*}"}
+{"id": "951.png", "formula": "\\begin{align*} \\frac { \\delta } { \\delta u _ j } = \\frac { \\partial } { \\partial u _ j } + \\sum _ { l = 1 } ^ \\infty ( - 1 ) ^ l D _ { i 1 } D _ { i 2 } \\cdots D _ { i l } \\frac { \\partial } { \\partial _ { u _ j , i _ 1 i _ 2 \\cdots i _ l } } . \\end{align*}"}
+{"id": "8272.png", "formula": "\\begin{align*} \\frac { d } { d \\tau } ( \\frac { x _ 1 } { \\rho } ( e ^ \\tau \\cdot m ) ) = \\frac { \\rho ^ 2 | T | ^ 2 - 2 x _ 1 ^ 2 } { \\rho ^ 3 } ( e ^ \\tau \\cdot m ) . \\end{align*}"}
+{"id": "4030.png", "formula": "\\begin{align*} \\partial _ i ( x _ { i ( 1 ) } \\cdots x _ { i ( \\ell ) } ) = \\sum _ { k = 1 } ^ \\ell \\delta _ { i , i ( k ) } x _ { i ( 1 ) } \\cdots x _ { i ( k - 1 ) } \\otimes x _ { i ( k + 1 ) } \\cdots x _ { i ( \\ell ) } . \\end{align*}"}
+{"id": "89.png", "formula": "\\begin{align*} \\mathcal { A } & ( a ^ { \\dagger } _ + a _ + + a ^ { \\dagger } _ - a _ - ) + \\mathcal { B } ( a _ + ^ { \\dagger } a ^ { \\dagger } _ - + a _ + a _ - ) + \\kappa ( a ^ { \\dagger } _ + + a _ - ) + \\overline { \\kappa } ( a _ + + a _ - ^ { \\dagger } ) \\\\ & = \\mathcal D ( b ^ \\dagger _ + b _ + + b ^ \\dagger _ - b _ - ) - \\frac { 1 } { 2 } \\alpha \\mathcal { B } ( [ a _ + , a ^ { \\dagger } _ + ] + [ a _ - , a ^ { \\dagger } _ - ] ) - \\frac { 2 | \\kappa | ^ 2 } { \\mathcal { A } + \\mathcal { B } } , \\end{align*}"}
+{"id": "8204.png", "formula": "\\begin{align*} \\mu ( q \\cdot m ) = \\phi ( q ) \\mu ( m ) , \\end{align*}"}
+{"id": "247.png", "formula": "\\begin{align*} \\sum _ { \\substack { J \\subset \\{ 1 , \\ldots , n \\} , \\ , 0 \\leq | J | \\leq \\ell \\\\ \\epsilon _ j \\in \\{ 1 , - 1 \\} , \\ ; j \\in J } } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! U ^ { \\emph { r } } _ { J ^ c , \\ , \\ell - | J | } ( \\xi ; g ) V ^ { \\emph { r } } _ { \\epsilon J } ( \\xi ; g ) \\Phi ^ { \\emph { r } } _ { \\xi + e _ { \\epsilon J } } ( x ; g ) = E ^ { \\emph { r } } _ \\ell ( x ) \\Phi ^ { \\emph { r } } _ { \\xi } ( x ; g ) , \\end{align*}"}
+{"id": "4980.png", "formula": "\\begin{align*} x = a \\sigma ( a ) ^ { - 1 } . \\end{align*}"}
+{"id": "5884.png", "formula": "\\begin{align*} D _ { N } ( T _ { S L } ) \\leq k \\sum _ { j = 0 } ^ { \\infty } ( 1 - \\eta ) ^ { j } + \\gamma P _ { 1 } ( \\# \\mathcal { N } \\notin J _ { N } ) \\sim k . \\end{align*}"}
+{"id": "8144.png", "formula": "\\begin{align*} A _ { i j } = v _ { i j } ( A _ { 1 0 } , A _ { 0 1 } ) \\ , , \\end{align*}"}
+{"id": "5460.png", "formula": "\\begin{align*} \\int _ { x _ o } ^ x W ^ { - 1 } ( t ^ 2 / 2 ) d t & \\leq \\frac { 1 } { \\kappa } \\int _ { \\sqrt { 2 / 3 } } ^ x \\left ( \\frac { t ^ 2 } { 2 } - \\frac { \\beta - 1 } { \\beta } \\log \\left ( \\frac { t ^ 2 } { 2 } \\right ) + 1 \\right ) ^ \\frac { 1 } { \\beta } d t \\\\ & = \\frac { 1 } { \\kappa } \\int _ { \\frac { 1 } { 3 } } ^ { \\frac { x ^ 2 } { 2 } } \\left ( u - \\frac { \\beta - 1 } { \\beta } \\log u + 1 \\right ) ^ \\frac { 1 } { \\beta } \\frac { d u } { \\sqrt { 2 u } } . \\end{align*}"}
+{"id": "2011.png", "formula": "\\begin{align*} T _ \\phi u : = u \\circ \\Psi \\quad T _ \\phi ^ { - 1 } v = v \\circ \\Psi ^ { - 1 } \\end{align*}"}
+{"id": "2922.png", "formula": "\\begin{align*} L _ { k j _ 1 \\cdots j _ s l _ 1 \\cdots l _ { s - 1 } } D _ { l _ 1 \\cdots l _ { s - 1 } } = \\frac { 2 s - 1 } { s - 1 } \\delta _ { \\hat { k j _ 1 } } D _ { \\hat { j } _ 2 \\cdots \\hat { j } _ s } - \\delta _ { \\hat { j } _ 1 \\hat { j } _ 2 } D _ { \\hat { j } _ 3 \\cdots \\hat { j } _ s k } \\end{align*}"}
+{"id": "3169.png", "formula": "\\begin{gather*} u ( - \\infty , - ) = y _ 0 \\quad \\hbox { a n d } u ( + \\infty , - ) = y _ 1 . \\end{gather*}"}
+{"id": "5565.png", "formula": "\\begin{align*} r = F ( \\xi ) - z _ 0 . \\end{align*}"}
+{"id": "4782.png", "formula": "\\begin{align*} S _ k ( a ) & = - \\frac { u ^ { i - 1 } \\exp { ( - u ^ 2 ) } } { 2 } \\Big | _ { - a } ^ { \\infty } + \\frac { 1 } { 2 } ( i - 1 ) \\int _ { - a } ^ { \\infty } u ^ { k - 2 } \\exp { ( - u ^ 2 ) } d u \\\\ & = \\frac { a ^ { k - 1 } \\exp { ( - a ^ 2 ) } } { 2 } + \\frac { 1 } { 2 } ( k - 1 ) S _ { k - 2 } ( a ) . \\end{align*}"}
+{"id": "8222.png", "formula": "\\begin{align*} d ^ c _ { I _ 1 ^ a } K _ 1 & = d ^ c _ { I _ 1 } K _ 1 - \\frac { a ^ 2 V } { V + a ^ 2 } 2 x _ 1 \\theta , \\\\ d ^ c _ { I _ 1 ^ a } x _ 1 & = \\frac { V } { V + a ^ 2 } \\theta , \\end{align*}"}
+{"id": "5900.png", "formula": "\\begin{align*} v ( x _ 1 , x _ 2 ) = \\int _ 0 ^ { x _ 2 } \\partial _ 2 v ( x _ 1 , s ) d s . \\end{align*}"}
+{"id": "6275.png", "formula": "\\begin{align*} M ( u ) = | u | ^ 2 , P ( u ) = i ( \\bar u \\partial _ x u - u \\partial _ x \\bar u ) , E ( u ) = - \\bar u \\partial _ x ^ 2 u + 2 | \\partial _ x u | ^ 2 - u \\partial _ x ^ 2 \\bar u . \\end{align*}"}
+{"id": "8867.png", "formula": "\\begin{align*} r _ { T _ 0 + 1 } \\leq r _ 1 + \\sum _ { t = 1 } ^ { T _ 0 } \\left ( \\frac { 1 6 } { \\alpha ^ 2 T _ 0 } \\left ( b L h _ t ^ { \\beta - 1 } \\right ) ^ 2 + \\frac { 6 4 } { \\alpha ^ 2 T _ 0 ^ 2 } \\left ( { \\sf V } _ 2 \\bar { L } ^ 2 h _ t ^ { 2 } + { \\sf V } _ 3 \\sigma ^ 2 h _ t ^ { - 2 } \\right ) \\right ) \\ , . \\end{align*}"}
+{"id": "475.png", "formula": "\\begin{align*} p _ t ^ { ( \\beta , \\mu ) } ( t , r , s ) & : = \\sum _ { n \\geq 0 } p _ t ^ { ( n , \\beta , \\mu ) } ( r , s ) , p _ t ^ { ( 0 , \\beta , \\mu ) } ( r , s ) : = p _ t ( r , s ) , \\\\ p _ t ^ { ( n , \\beta , \\mu ) } ( r , s ) & : = \\int _ 0 ^ t d \\tau \\int _ 0 ^ \\infty m ( d z ) \\ , p _ \\tau ( r , z ) q _ { \\beta , \\mu } ( z ) p _ { t - \\tau } ^ { ( n - 1 , \\beta , \\mu ) } ( z , s ) , n \\in \\N . \\end{align*}"}
+{"id": "360.png", "formula": "\\begin{align*} \\eta ( g _ 1 g _ 2 ) = S ( \\psi ( h _ { g _ 1 g _ 2 } ) ) , \\end{align*}"}
+{"id": "3470.png", "formula": "\\begin{align*} \\omega _ { E _ J } ^ { n - m } = ( L ^ { n - m } \\cdot E _ J ) \\frac { \\Omega _ { E _ J } \\wedge \\overline { \\Omega } _ { E _ J } } { \\int _ { E _ J } \\Omega _ { E _ J } \\wedge \\overline { \\Omega } _ { E _ J } } = d _ 0 \\ldots d _ m ( L ^ { n + 1 } \\cdot M ) \\frac { \\Omega _ { E _ J } \\wedge \\overline { \\Omega } _ { E _ J } } { \\int _ { E _ J } \\Omega _ { E _ J } \\wedge \\overline { \\Omega } _ { E _ J } } , \\end{align*}"}
+{"id": "8808.png", "formula": "\\begin{align*} \\int u ^ j K ( u ) d u = 0 , \\ j = 2 , \\dots , \\ell , ~ \\int | u | ^ { \\beta } | K ( u ) | d u < \\infty , \\end{align*}"}
+{"id": "219.png", "formula": "\\begin{align*} & - 2 \\left \\{ \\widehat { L } ( T M ) \\right \\} ^ { ( 1 2 ) } = \\left \\{ \\widehat { A } ( T M ) { \\rm c h } [ 1 7 \\widetilde { T _ C M } + \\wedge ^ 2 \\widetilde { T _ C M } + W _ i + 1 2 8 ] \\right \\} ^ { ( 1 2 ) } . \\end{align*}"}
+{"id": "6161.png", "formula": "\\begin{align*} \\begin{pmatrix} b _ { j + 1 , 0 } \\\\ b _ { j + 1 , 1 } \\end{pmatrix} = T _ j \\begin{pmatrix} b _ { j , 0 } \\\\ b _ { j , 1 } \\end{pmatrix} , \\end{align*}"}
+{"id": "6818.png", "formula": "\\begin{align*} f ( u ) = a u , p ( u ) = \\frac { 1 - u } { a } , q ( u ) = u + e _ 2 , h ( u ) = 1 , \\mu = e _ 2 , \\end{align*}"}
+{"id": "7522.png", "formula": "\\begin{align*} \\lambda = \\lambda _ 0 = u \\qquad \\textrm { a n d } \\lambda _ \\pm = u \\pm \\sqrt { ( n p ' + \\theta \\rho ) / r } \\ , . \\end{align*}"}
+{"id": "5652.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } L '^ 2 \\cdot D _ { X ^ \\prime } & \\xi \\cdot L ' \\cdot D _ { X ^ \\prime } \\\\ \\xi \\cdot L ' \\cdot D _ { X ^ \\prime } & \\xi ^ 2 \\cdot D _ { X ^ \\prime } \\end{array} \\right ) = \\left ( \\begin{array} { c c } 2 & c _ 1 + 3 \\\\ c _ 1 + 3 & c _ 1 ^ 2 + 3 c _ 1 - 2 c _ 2 \\end{array} \\right ) . \\end{align*}"}
+{"id": "7707.png", "formula": "\\begin{align*} a ^ j X _ j + b ^ \\nu Y _ \\nu + c ^ J Z _ J = 0 , \\end{align*}"}
+{"id": "4347.png", "formula": "\\begin{align*} S ( \\omega _ { { \\sf f } _ 1 \\cdots { \\sf f } _ m } \\| \\omega ) = \\sum _ { j = 1 } ^ m S ( \\omega _ { { \\sf f } _ j } \\| \\omega ) \\ , . \\end{align*}"}
+{"id": "8273.png", "formula": "\\begin{align*} \\rho ( \\Psi ( m , x ) ) & \\leq \\rho ( m ) + C \\int _ 0 ^ { \\tau ( m , x _ 1 ) } \\frac { d } { d t } ( \\frac { x _ 1 } { \\rho } ( e ^ t \\cdot m ) ) d t \\\\ & = \\rho ( m ) + C \\frac { x _ 1 } { \\rho ( \\Psi ( m , x ) ) } . \\end{align*}"}
+{"id": "4933.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\sin ( 2 \\pi n t ) e ^ { - 2 ^ k \\pi ( m + 1 ) t } d t & = \\dfrac { 1 } { 2 i } \\left ( \\int _ 0 ^ \\infty e ^ { 2 \\pi i n t } e ^ { - 2 ^ k \\pi ( m + 1 ) t } d t - \\int _ 0 ^ \\infty e ^ { - 2 \\pi i n t } e ^ { - 2 ^ k \\pi ( m + 1 ) t } d t \\right ) \\\\ & = - \\dfrac { 1 } { 2 i } \\left ( \\dfrac { 1 } { 2 \\pi ( i n - 2 ^ { k - 1 } ( m + 1 ) ) } + \\dfrac { 1 } { 2 \\pi ( i n + 2 ^ { k - 1 } ( m + 1 ) ) } \\right ) \\\\ & = \\dfrac { 1 } { 2 \\pi } \\left ( \\dfrac { n } { n ^ 2 + 2 ^ k ( m + 1 ) ^ 2 } \\right ) . \\end{align*}"}
+{"id": "1858.png", "formula": "\\begin{align*} \\mathbf { E } \\left [ \\| \\zeta ( s , \\mathbb { X } _ \\alpha ) - \\zeta _ L ( s , \\mathbb { X } _ \\alpha ) \\| ^ 2 \\right ] & = \\sum _ { m , n > L } \\mathbf { E } [ \\mathbb { X } _ \\alpha ( m ) \\overline { \\mathbb { X } _ \\alpha ( n ) } ] \\left \\langle ( m + \\alpha ) ^ { - s } , ( n + \\alpha ) ^ { - s } \\right \\rangle \\\\ & = \\sum _ { n > L } \\| ( n + \\alpha ) ^ { - s } \\| ^ 2 \\end{align*}"}
+{"id": "617.png", "formula": "\\begin{align*} \\mu ( t ) = t ^ { 1 - 2 b } f ( t ) , f ( t ) = \\frac { 1 } { t } \\int _ 0 ^ t g ( s ) \\ , d s , g ( s ) = \\int _ { \\R ^ m } V ( s , \\zeta ) \\overline { W ( s , \\zeta ) } \\ , d \\zeta , \\end{align*}"}
+{"id": "7384.png", "formula": "\\begin{align*} H _ { n } ( \\rho _ { n } ) : = \\frac { 1 } { \\gamma _ { n } + 1 } \\rho _ { n } ^ { \\gamma _ { n } + 1 } . \\end{align*}"}
+{"id": "7980.png", "formula": "\\begin{align*} \\mathfrak { d } _ 1 : = d _ 1 . \\end{align*}"}
+{"id": "7703.png", "formula": "\\begin{align*} & \\{ x ^ 1 , \\ldots , x ^ { r _ 0 } \\} , \\\\ & \\{ e ^ 1 , \\ldots , e ^ { r _ 1 } , x ^ i e ^ \\mu \\} , \\ ; \\ ; i = 1 , \\ldots , r _ 0 , \\\\ & \\{ p ^ 1 , \\ldots , p ^ { r _ 2 } , e ^ \\mu e ^ \\gamma , x ^ i p ^ I , x ^ i e ^ \\mu e ^ \\nu \\} , \\ ; \\ ; i = 1 , \\ldots , r _ 0 , \\ ; \\ ; \\gamma = 1 , \\ldots , r _ 1 , \\end{align*}"}
+{"id": "6665.png", "formula": "\\begin{align*} d _ 3 = - { q ( p ^ 2 + q ^ 2 ) \\over 2 \\pi ^ 3 } , \\ : \\ : d _ 4 = { ( p ^ 2 + q ^ 2 ) ( 1 - p ^ 2 - 5 q ^ 2 ) \\over 8 \\pi ^ 4 } , \\ : \\ : d _ 5 = { q ( p ^ 2 + q ^ 2 ) ( 5 - 3 p ^ 2 - 7 q ^ 2 ) \\over 8 \\pi ^ 5 } . \\end{align*}"}
+{"id": "4009.png", "formula": "\\begin{align*} \\begin{aligned} \\left | X _ { i \\bar j l } - X _ { j \\bar l } \\frac { w _ i } { w } \\right | ^ 2 & = X _ { i \\bar j l } X _ { j \\bar i \\bar l } - 2 \\mathfrak { R e } \\left \\{ X _ { i \\bar j l } X _ { l \\bar j } \\frac { w _ { \\bar i } } { w } \\right \\} + X _ { j \\bar l } X _ { l \\bar j } \\frac { w _ i w _ { \\bar i } } { w ^ 2 } , \\end{aligned} \\end{align*}"}
+{"id": "1190.png", "formula": "\\begin{align*} i _ e = A \\to \\prod _ G A . \\end{align*}"}
+{"id": "8634.png", "formula": "\\begin{align*} \\mathcal { Q } ^ { \\mu } [ b ] \\bullet = \\Big ( \\mathrm { F } _ 3 \\nabla _ X ( b \\nabla _ X \\cdot \\bullet ) + \\nabla _ X ( b \\mathrm { F } _ 3 \\nabla _ X \\cdot \\bullet ) \\Big ) + \\Big ( \\mathrm { F } _ 4 \\nabla _ X \\nabla _ X \\cdot \\big ( b \\bullet \\big ) + b \\mathrm { F } _ 4 \\Delta _ X \\bullet \\Big ) , \\end{align*}"}
+{"id": "4928.png", "formula": "\\begin{align*} \\eta ( 2 ^ k i ) = \\dfrac { \\pi ^ { 1 / 4 } } { \\Gamma ( 3 / 4 ) \\cdot 2 ^ { \\frac { k + 1 } { 2 } } } \\cdot e ^ { - \\frac { \\pi ( 2 ^ { k } - 1 ) ^ 2 } { 1 2 \\cdot 2 ^ k } - \\frac { 1 } { 2 \\pi } \\sum _ { \\ell = 1 } ^ k 2 ^ \\ell L _ \\ell } , \\end{align*}"}
+{"id": "4999.png", "formula": "\\begin{align*} - 2 \\int _ { 0 } ^ { R } \\phi ( r ) \\phi ' ( r ) f ( r ) d r = \\int _ { 0 } ^ { R } f ' ( r ) \\phi ^ 2 ( r ) d r . \\end{align*}"}
+{"id": "2217.png", "formula": "\\begin{align*} \\left ( D _ { b - } ^ \\alpha y \\right ) ( x ) : = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\left ( - \\frac { d } { d x } \\right ) ^ n \\underset { x } { \\overset { b } { \\int } } \\frac { y ( t ) d t } { ( t - x ) ^ { \\alpha - n + 1 } } , \\ , ( n = \\left [ R e ( \\alpha ) \\right ] + 1 , \\ , x < b ) , \\end{align*}"}
+{"id": "402.png", "formula": "\\begin{align*} c \\otimes _ Z Z ^ * \\delta _ 0 = c \\otimes \\delta _ 0 \\ , , \\end{align*}"}
+{"id": "1436.png", "formula": "\\begin{align*} \\sigma _ i = \\frac { 1 } { 2 \\pi } \\lim _ { r \\rightarrow 0 } \\lim _ { k \\rightarrow + \\infty } \\int _ { B _ r ( 0 ) } e ^ { u _ i ^ k ( x ) } \\mathrm { d } x , \\ \\ i \\in I . \\end{align*}"}
+{"id": "4884.png", "formula": "\\begin{align*} E _ - ( z ) F _ - ( z ) & = E _ + ( z ) F _ - ( z ) \\\\ E _ + ( z ) F _ + ( z ) & = F _ - ( z ) E _ + ( z ) \\end{align*}"}
+{"id": "8238.png", "formula": "\\begin{align*} ( d ^ c _ { I _ 1 } \\hat { K } ) ( Y + a T ) = - d \\hat { K } ( I _ 1 Y ) + x _ 1 \\eta ( Y + a T ) . \\end{align*}"}
+{"id": "5027.png", "formula": "\\begin{align*} \\square ^ * \\big ( ( 4 \\pi \\tau _ t ) ^ { - n / 2 } e ^ { - f _ t } \\big ) = ( - \\partial _ t - \\triangle + R ) \\big ( ( 4 \\pi \\tau _ t ) ^ { - n / 2 } e ^ { - f _ t } \\big ) = 0 \\end{align*}"}
+{"id": "7840.png", "formula": "\\begin{align*} g _ { N } ( V , V ) = & 8 \\pi T _ { i \\overline { j } } Z ^ i \\overline { Z } ^ j + 2 \\pi | W _ 0 ^ { } ( V ) | ^ 2 T ^ { 0 \\overline { 0 } } \\ , \\\\ & = 8 \\pi \\tau _ 2 ^ 2 \\left ( \\frac { 2 t ^ 3 } { 3 } - \\frac { \\chi } { 4 ( 2 \\pi ) ^ 3 } \\sum _ { ( m , n ) \\in \\mathbb { Z } ^ 2 - ( 0 , 0 ) } \\left ( \\frac { 3 ( m \\tau _ 1 + n ) ^ 2 } { | m \\tau + n | ^ 5 } - \\frac { 1 } { | m \\tau + n | ^ 3 } \\right ) \\right ) + 2 \\pi | W _ 0 ^ { } ( V ) | ^ 2 T ^ { 0 \\overline { 0 } } \\end{align*}"}
+{"id": "5920.png", "formula": "\\begin{align*} A _ { n } & = \\min \\{ \\dfrac { R _ { n + 1 } - S _ { n } } { 2 } + e , R _ { n + 1 } - S _ { n } + d [ - a _ { 1 , n + 1 } b _ { 1 , n - 1 } ] \\} \\ge \\min \\{ \\dfrac { 1 } { 2 } + e , 1 \\} = 1 > d [ a _ { 1 , n } b _ { 1 , n } ] . \\end{align*}"}
+{"id": "472.png", "formula": "\\begin{align*} ( p _ t \\mu ) ( r ) : = \\int _ 0 ^ \\infty p _ t ( r , s ) \\mu ( d s ) . \\end{align*}"}
+{"id": "3785.png", "formula": "\\begin{align*} \\Psi : = \\{ \\psi = ( \\psi _ 0 , \\psi _ 1 ) \\in C _ b ( X ) \\times C _ b ( X ) \\ ; \\mid \\ ; R ^ * ( \\psi _ i ) \\in C _ b ( X ) , \\ ; R ^ * ( \\psi _ 0 ) \\oplus R ^ * ( \\psi _ 1 ) \\leq c \\} . \\end{align*}"}
+{"id": "5187.png", "formula": "\\begin{align*} \\alpha _ 1 = L _ 1 - L _ 2 , \\dots , \\alpha _ { n - 1 } = L _ { n - 1 } - L _ n , \\alpha _ n = L _ n \\end{align*}"}
+{"id": "3876.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sup _ { n \\in \\N } \\P \\big ( \\tau _ n ^ N < T \\big ) = 0 . \\end{align*}"}
+{"id": "7439.png", "formula": "\\begin{align*} ( p - 1 ) \\| ( y _ 1 , y _ 2 ) \\| _ { 1 , p } < & \\lambda ( \\kappa _ 1 + \\kappa _ 2 + 2 ) ( \\kappa _ 1 + \\kappa _ 2 + 1 ) \\int _ { \\Omega } \\vert y _ 1 \\vert ^ { \\kappa _ 1 + 1 } \\vert y _ 2 \\vert ^ { \\kappa _ 2 + 1 } d z \\\\ & - ( q - 1 ) \\| ( \\nabla y _ 1 , \\nabla y _ 2 ) \\| _ { q , \\eta } \\\\ & - \\nu \\int _ { \\Omega } \\left [ a _ 1 \\vert y _ 1 \\vert ^ { 1 - \\nu } + a _ 2 \\vert y _ 2 \\vert ^ { 1 - \\nu } \\right ] d z . \\end{align*}"}
+{"id": "1862.png", "formula": "\\begin{align*} \\mathbf { E } _ { \\Omega ' _ 0 } \\left [ \\| \\zeta ( s , \\mathbb { Y } _ \\alpha ) - \\zeta _ N ( s , \\mathbb { Y } _ \\alpha ) \\| ^ 2 \\right ] & = \\mathbf { E } [ \\mathbf { 1 } _ { \\Omega ' _ 0 } ] \\cdot \\mathbf { E } \\left [ \\| \\zeta ( s , \\mathbb { Y } _ \\alpha ) - \\zeta _ N ( s , \\mathbb { Y } _ \\alpha ) \\| ^ 2 \\right ] \\\\ & \\ll \\mathbf { P } ( \\Omega ' _ 0 ) N ^ { 1 - 2 \\sigma _ 0 } \\end{align*}"}
+{"id": "706.png", "formula": "\\begin{align*} \\mathbf I ( t ) = \\mathbf \\Lambda ^ { \\mathbf s } \\int _ 0 ^ t \\mathbf S ( - s ) \\mathbf M ( \\mathbf u ( s ) ) \\ , d W ( s ) = \\int _ 0 ^ t \\mathbf \\Lambda ^ { \\mathbf s } \\mathbf S ( - s ) \\mathbf M ( \\mathbf u ( s ) ) \\ , d W ( s ) , \\end{align*}"}
+{"id": "5938.png", "formula": "\\begin{align*} d [ - a _ { 1 , n + 1 } b _ { 1 , n - 1 } ] + d [ - a _ { 1 , n + 2 } b _ { 1 , n } ] & > ( 1 - R _ { n + 1 } ) + ( 2 e - R _ { n + 2 } + R _ { n + 1 } - 1 + S _ { n } ) \\\\ & = 2 e + S _ { n } - R _ { n + 2 } . \\end{align*}"}
+{"id": "998.png", "formula": "\\begin{align*} \\langle \\nabla \\cdot { \\bf G } , \\varphi \\rangle = - \\sum _ { i = 1 } ^ 3 \\langle G _ i , \\partial _ { x _ i } \\varphi \\rangle , \\end{align*}"}
+{"id": "6138.png", "formula": "\\begin{align*} ( \\psi _ 1 \\times \\psi _ 2 ) \\circ ( \\varphi _ 1 \\times \\varphi _ 2 ) = ( \\psi _ 1 \\circ \\varphi _ 1 ) \\times ( \\psi _ 2 \\circ \\varphi _ 2 ) . \\end{align*}"}
+{"id": "3251.png", "formula": "\\begin{align*} \\{ e _ { _ \\Sigma } = e _ { \\sigma _ 1 } e _ { \\sigma _ 2 } \\cdots e _ { \\sigma _ \\ell } , \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] , 1 \\leq \\sigma _ 1 < \\sigma _ 2 < \\cdots < \\sigma _ \\ell \\leq n \\} \\cup \\{ e _ \\emptyset = 1 \\} \\end{align*}"}
+{"id": "2624.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\limsup _ { n \\to \\infty } P ( \\sup _ { x \\in [ 0 , 1 ] } B _ n ( x , t ) \\ge r ) ^ { 1 / b _ n ^ 2 } = 0 \\ , . \\end{align*}"}
+{"id": "2679.png", "formula": "\\begin{align*} 0 \\ < \\ g \\ = \\ \\omega _ t \\mid _ { x _ { 1 } } \\ \\leq \\ t \\omega _ { 0 } - _ { \\omega _ { 0 } } ( x _ { 1 } ) , \\end{align*}"}
+{"id": "2725.png", "formula": "\\begin{align*} U \\bigg ( \\sum _ { i = 1 } ^ { \\infty } a _ i e _ i \\bigg ) = \\sum _ { i = 1 } ^ { \\theta ^ { - 1 } } \\varepsilon _ i a _ i e _ { \\pi ( i ) } + \\sum _ { i = \\theta ^ { - 1 } + 1 } ^ { \\infty } \\varepsilon _ i a _ i e _ i \\end{align*}"}
+{"id": "5335.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\left | \\frac { e ^ { x + r e ^ { i t } } } { 1 + e ^ { x + r e ^ { i t } } } \\right | r ^ { - n } \\ , \\mathrm { d } t & \\leq \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\frac { 1 } { \\sin ( r ) } r ^ { - n } \\ , \\mathrm { d } t \\\\ & = \\frac { 1 } { \\sin ( r ) r ^ n } . \\end{align*}"}
+{"id": "4818.png", "formula": "\\begin{align*} R ( u ) & = \\sum R ^ { k l } _ { i j } ( u ) E _ { i k } \\otimes E _ { j l } \\ ; , \\end{align*}"}
+{"id": "5740.png", "formula": "\\begin{align*} \\Theta \\overset { d e f } { = } \\frac { \\int _ { \\mathbb Q _ 5 ^ + } U ( X , t ) ^ 2 x _ { n + 1 } ^ a d X d t } { \\int _ { \\mathbb B _ 1 ^ + } U ( X , 0 ) ^ 2 x _ { n + 1 } ^ a d X } . \\end{align*}"}
+{"id": "4175.png", "formula": "\\begin{align*} \\frac { K _ { \\nu / 2 } ( s | \\tilde { \\omega } | ) } { ( s | \\tilde { \\omega } | ) ^ { \\frac { \\nu } { 2 } } } & = \\frac { 2 ^ { \\frac { \\nu - 2 } { 2 } } \\Gamma \\left ( \\frac { \\nu + 1 } { 2 } \\right ) } { \\sqrt { \\pi } | \\tilde { \\omega } | ^ { \\nu } } \\int _ { \\mathbb { R } } \\left ( s ^ 2 + \\tilde { n } ^ 2 \\right ) ^ { - \\tfrac { \\nu + 1 } { 2 } } e ^ { i \\tilde { \\omega } \\tilde { n } } ~ \\mathrm { d } \\tilde { n } \\end{align*}"}
+{"id": "8170.png", "formula": "\\begin{align*} & A _ { 1 0 } E _ { i j } = E _ { i j } A _ { 1 0 } = \\theta _ { i j } E _ { i j } , A _ { 0 1 } E _ { i j } = E _ { i j } A _ { 0 1 } = \\mu _ { i j } E _ { i j } , \\\\ & A _ { 1 0 } ^ \\star E _ { i j } ^ \\star = E _ { i j } ^ \\star A _ { 1 0 } ^ \\star = \\theta _ { i j } ^ \\star E _ { i j } ^ \\star , A _ { 0 1 } ^ \\star E _ { i j } ^ \\star = E _ { i j } ^ \\star A _ { 0 1 } ^ \\star = \\mu _ { i j } ^ \\star E _ { i j } ^ \\star . \\end{align*}"}
+{"id": "4398.png", "formula": "\\begin{align*} [ \\partial _ 1 \\hat q ] : = \\partial _ 1 \\hat { q } ^ + \\vert _ { x _ 1 = 0 } + \\partial _ 1 \\hat { q } ^ - \\vert _ { x _ 1 = 0 } \\ , . \\end{align*}"}
+{"id": "6748.png", "formula": "\\begin{align*} E ^ w ( G ( n ) + H ) = \\frac { 1 } { 2 ^ w } ( G ( n ) + H ) = \\frac { 1 } { 2 ^ w } G ( n ) + \\frac { 1 } { 2 ^ w } H = ( E ^ w \\circ G ) ( n ) + \\left ( \\frac { 1 } { 2 } \\right ) ^ w H . \\end{align*}"}
+{"id": "1513.png", "formula": "\\begin{align*} r ( i , \\ , \\sigma _ i ^ { - 1 } ( j ) ) \\ , = \\ , ( j , \\ , \\sigma _ j ^ { - 1 } ( i ) ) \\end{align*}"}
+{"id": "1750.png", "formula": "\\begin{align*} R ( \\boldsymbol { \\xi } ) = \\frac { f \\bigl ( \\cos ( \\xi _ 1 ) , \\ldots , \\cos ( \\xi _ n ) \\bigr ) } { \\prod _ { \\substack { 1 \\leq r \\leq d \\\\ 1 \\leq j \\leq n } } \\bigl ( 1 - 2 a _ r \\cos ( \\xi _ j ) + a _ r ^ 2 \\bigr ) } , \\end{align*}"}
+{"id": "8316.png", "formula": "\\begin{align*} \\mu _ { 0 i _ 1 } = \\min \\{ \\mu _ { 0 j } \\ | \\ 0 < j \\leq n , \\ b _ { n - j } ( x ) \\ne 0 \\} . \\end{align*}"}
+{"id": "2554.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } ( K \\ast | u _ n | ^ { p _ { r ; s } ^ { \\uparrow * } } ) | u _ n | ^ { p _ { r ; s } ^ { \\uparrow * } } d x & = \\int _ { \\mathbb { R } ^ N } d \\xi + \\xi _ \\infty ; \\\\ \\limsup _ { n \\to \\infty } \\| u _ n \\| _ { D ^ { s , p } } ^ p & = \\int _ { \\mathbb { R } ^ N } d \\mu + \\mu _ \\infty ; \\\\ \\limsup _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } | u _ n | ^ { p _ s ^ * } d x & = \\int _ { \\mathbb { R } ^ N } d \\nu + \\nu _ \\infty ; \\end{align*}"}
+{"id": "1318.png", "formula": "\\begin{align*} J ^ { w } ( X ) & = - \\frac { 1 } { 2 } \\int _ { 0 } ^ { \\infty } w ( x ) f ^ 2 ( x ) d x = - \\frac { 1 } { 2 } \\int _ { 0 } ^ { \\infty } w ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) d u = - \\frac { 1 } { 2 } E ( \\Lambda _ X ^ { w } ( U ) ) , \\end{align*}"}
+{"id": "5278.png", "formula": "\\begin{align*} & \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { t } ( x _ { i , t } ) ] _ + \\| \\le \\sqrt { \\varepsilon _ 3 T + \\varepsilon _ 4 T ( \\tilde { h } _ T ( y ) + \\hat { h } _ T ( y ) ) } , ~ \\forall y \\in \\mathcal { X } _ T . \\end{align*}"}
+{"id": "6398.png", "formula": "\\begin{align*} \\chi _ { 1 t } \\left ( a \\otimes b \\right ) \\varepsilon \\left ( c \\right ) + \\chi _ { 1 \\left ( n - t \\right ) } \\left ( a \\triangleleft b \\otimes c \\right ) = \\chi _ { 1 n } \\left ( a \\otimes b c \\right ) \\end{align*}"}
+{"id": "2683.png", "formula": "\\begin{align*} \\int _ { X } e ^ { ( 1 + \\frac { \\alpha } { 2 \\beta } ) \\varphi _ t ^ { \\ast } } \\omega _ { 0 } ^ { n } \\leq \\int _ { X } \\omega _ { 0 } ^ { n } \\ = : \\ c _ { 2 } , \\end{align*}"}
+{"id": "956.png", "formula": "\\begin{align*} V _ 3 = & t ^ { 1 + \\alpha } \\partial _ t + \\alpha x t ^ \\alpha \\partial _ x - \\bigg ( \\bigg ( \\frac { \\alpha ( c + 1 ) t ^ \\alpha } { 2 } + \\frac { \\alpha ^ 2 x ^ 2 } { 4 } \\bigg ) u + \\frac { m \\alpha t ^ \\alpha } { 2 } v \\bigg ) \\partial _ u - \\\\ & \\bigg ( \\bigg ( \\frac { \\alpha ( c + 1 ) t ^ \\alpha } { 2 } + \\frac { \\alpha ^ 2 x ^ 2 } { 4 } \\bigg ) v + \\frac { n \\alpha t ^ \\alpha } { 2 } u \\bigg ) \\partial _ v , \\end{align*}"}
+{"id": "7324.png", "formula": "\\begin{align*} u ' ( t ) = - ( \\nabla f _ \\lambda ) ( u ( t ) ) \\end{align*}"}
+{"id": "1997.png", "formula": "\\begin{align*} \\left ( s , \\frac { 1 } { p _ \\theta } , \\frac { 1 } { q _ \\theta } \\right ) : = ( 1 - \\theta ) \\left ( s _ 0 , \\frac { 1 } { p _ 0 } , \\frac { 1 } { q _ 0 } \\right ) + \\theta \\left ( s _ 1 , \\frac { 1 } { p _ 1 } , \\frac { 1 } { q _ 1 } \\right ) \\end{align*}"}
+{"id": "7692.png", "formula": "\\begin{align*} \\bigl ( ( \\rho ' \\circ \\rho ) \\otimes _ 0 ( \\varphi ' \\circ \\varphi ) \\bigr ) _ { \\sigma } ^ { \\sigma '' } = \\sum _ { \\sigma ' \\in \\Sigma ' } ( \\rho ' \\otimes _ 0 \\varphi ' ) _ { \\sigma ' } ^ { \\sigma '' } \\circ ( \\rho \\otimes _ 0 \\varphi ) _ { \\sigma } ^ { \\sigma ' } . \\end{align*}"}
+{"id": "4436.png", "formula": "\\begin{align*} \\mathcal { K } : = \\tilde { \\mathcal { A } } \\Big ( \\mathcal { F } - \\mathcal { A } _ 0 \\partial _ t { \\mathbf V } - \\mathcal { A } _ 2 \\partial _ 2 { \\mathbf V } - \\mathcal { A } _ 3 { \\mathbf V } - \\mathcal { A } _ { ( 0 ) } \\partial _ 1 { \\mathbf V } \\Big ) , \\end{align*}"}
+{"id": "3986.png", "formula": "\\begin{align*} ( \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ) ^ n = c ( \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ) ^ m \\wedge \\omega ^ { n - m } , \\sup _ M \\varphi = 0 \\end{align*}"}
+{"id": "1549.png", "formula": "\\begin{align*} \\Delta = y ^ 3 ( x - y ) ^ 3 + z f _ 5 ( x , y , z ) , \\end{align*}"}
+{"id": "633.png", "formula": "\\begin{align*} \\psi ( 0 , x , \\omega ) = \\psi _ 0 ( x , \\omega ) , \\phi ( 0 , x , \\omega ) = \\phi _ 0 ( x , \\omega ) , \\partial _ t \\phi ( 0 , x , \\omega ) = \\phi _ 1 ( x , \\omega ) . \\end{align*}"}
+{"id": "3743.png", "formula": "\\begin{align*} \\rho ( \\mathbb { A } ) = \\{ \\lambda \\in \\mathbb { C } : \\lambda ^ 3 \\in \\rho ( A ) \\} , \\end{align*}"}
+{"id": "6284.png", "formula": "\\begin{align*} \\frac { d } { d t } P ( v _ \\lambda ) = \\partial _ x [ g _ { [ < \\lambda ] } E ( v _ \\lambda ) ] + F ^ { p a r a } _ { \\lambda , p } , F ^ { p a r a } _ { \\lambda , p } = 2 \\Re ( \\partial _ x f _ \\lambda \\bar v _ \\lambda - f _ \\lambda \\partial _ x \\bar v _ \\lambda ) . \\end{align*}"}
+{"id": "144.png", "formula": "\\begin{align*} K _ E c _ i \\pi _ \\mu c _ j ^ { - 1 } K _ E & = K _ E \\sigma ( a _ i ) \\sigma ( \\pi _ \\mu ) \\sigma ( a _ j ^ { - 1 } ) K _ E \\\\ & = K _ E \\sigma ( a _ i ) \\pi _ \\mu \\sigma ( s _ \\mu a _ j ^ { - 1 } ) K _ E \\\\ & = K _ E \\sigma ( a _ i ) \\pi _ \\mu \\sigma ( d _ j ^ { - 1 } ) K _ E , \\end{align*}"}
+{"id": "5153.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } { \\rm A o I } ( S _ { \\mathrm { z } } , Q _ { \\mathrm { u n i } } , F ^ { * } ) + \\frac { 3 } { 2 } \\log _ 2 { \\delta } = \\frac { 3 } { 2 } h ( X ) . \\end{align*}"}
+{"id": "8953.png", "formula": "\\begin{align*} \\Theta _ { \\infty } ( s ) = \\Theta _ { \\infty } ( | s | ) = \\int _ { \\R ^ { m + n } } f ( a ^ { | s | } \\bar { z } ) f ( \\bar { z } ) ) \\ , d \\bar { z } = \\underset { \\| \\bar { y } \\| \\in [ \\upsilon _ 1 , \\upsilon _ 2 ] \\cap [ e ^ { | s | } \\upsilon _ 1 , e ^ { | s | } \\upsilon _ 2 ] } { \\int } \\prod _ { i = 1 } ^ { m } \\left ( \\frac { 2 \\vartheta } { \\| \\bar { y } \\| ^ { - w _ i } } \\right ) \\ , d \\bar { y } . \\end{align*}"}
+{"id": "8493.png", "formula": "\\begin{align*} P ( E ; G ) : = | \\mu _ { E } | ( G ) . \\end{align*}"}
+{"id": "6166.png", "formula": "\\begin{align*} c _ i ( T ) & = \\sum _ { j = i + 1 } ^ { n } \\delta _ { \\swarrow } ( t _ { i , j } ) - \\sum _ { j = 1 } ^ { i - 1 } \\delta _ { \\searrow } ( t _ { i , j } ) \\\\ T ( \\mathbf { k } ) & = ( k _ 1 + c _ 1 ( T ) , k _ 2 + c _ 2 ( T ) , \\ldots , k _ n + c _ n ( T ) ) \\in \\mathbb { Z } ^ n , \\end{align*}"}
+{"id": "4770.png", "formula": "\\begin{align*} \\mu ( F ) & = \\int _ { X _ { \\alpha } } \\mu ^ { \\alpha } ( F | A ) d \\mu _ { \\alpha } ( A ) \\\\ & = \\int _ { X } \\mu ^ { \\alpha } ( F | N _ { \\alpha } ( x ) ) d \\mu ( x ) . \\end{align*}"}
+{"id": "3553.png", "formula": "\\begin{align*} 0 & \\in \\partial g ( y ^ { k + 1 } ) + \\eta B ^ T ( A x ^ k + B y ^ { k + 1 } - b - \\frac { 1 } { \\eta } \\lambda ^ k ) = \\partial g ( y ^ { k + 1 } ) + \\eta B ^ T \\frac { 1 } { \\eta } ( \\lambda ^ k - \\lambda ^ { k + 1 } - \\lambda ^ k ) = \\partial g ( y ^ { k + 1 } ) - B ^ T \\lambda ^ { k + 1 } \\ , \\end{align*}"}
+{"id": "7887.png", "formula": "\\begin{align*} \\underline \\lambda _ { n + k + 1 } ( \\Omega ) : = \\inf _ { u \\in \\mathcal C _ 0 } \\left \\{ \\frac { ( n + k + 1 ) H _ { n + k } ( u ) } { \\int _ \\Omega | u | ^ { n + k + 1 } } \\right \\} . \\end{align*}"}
+{"id": "6269.png", "formula": "\\begin{align*} N _ \\lambda ^ { 2 , m e d } u _ \\lambda ^ { x _ 0 } = \\lambda L ( P _ { \\lambda } \\partial g ( u _ { < \\lambda } ) , \\partial u _ { \\lambda } , u _ \\lambda ) \\end{align*}"}
+{"id": "4875.png", "formula": "\\begin{align*} \\check { R } ( u ) = I - \\frac { u } { k - u } T = \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & \\frac { k } { k - u } & \\frac { - u } { k - u } & 0 \\\\ 0 & \\frac { - u } { k - u } & \\frac { k } { k - u } & 0 \\\\ 0 & 0 & 0 & 1 \\end{bmatrix} \\end{align*}"}
+{"id": "277.png", "formula": "\\begin{align*} C _ j = ( 0 , 0 ) < ( 0 , 1 ) \\cdots < ( 0 , j ) < ( 1 , i ) < \\cdots < ( 1 , n ) \\end{align*}"}
+{"id": "3679.png", "formula": "\\begin{align*} ( B _ { n , k , \\ell } ) _ { \\sigma , \\eta } = \\begin{cases} 1 & \\eta \\subset \\sigma , \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "4252.png", "formula": "\\begin{align*} H ^ s _ 0 ( \\Omega ) : = \\left \\{ H ^ { s } ( \\R ^ n ) : \\ , \\right \\} \\end{align*}"}
+{"id": "4771.png", "formula": "\\begin{align*} h ^ { + } ( x ) _ { f } = \\begin{cases} x _ { h _ { f } } & f \\in ( B ' \\setminus B _ { N } ) \\cap W _ { a } \\\\ x _ { g } & f = h _ { g } g \\in ( B ' \\setminus B _ { N } ) \\cap W _ { a } \\\\ x _ { f } & \\end{cases} \\end{align*}"}
+{"id": "8038.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { j = 0 } ^ { \\infty } \\phi _ { j } \\ast \\phi _ { j } \\ast f ( x ) \\end{align*}"}
+{"id": "1899.png", "formula": "\\begin{align*} b _ h ( w _ h , \\phi _ h ) : = \\sum _ { i = 1 } ^ { N _ x } \\int _ { I _ i } w _ h \\partial _ x \\phi _ h \\ , { \\rm d } x + \\sum _ { i = 0 } ^ { N _ x - 1 } \\left ( \\widehat { w _ h } [ \\ ! [ \\phi _ h ] \\ ! ] \\right ) _ { i + 1 / 2 } \\end{align*}"}
+{"id": "2455.png", "formula": "\\begin{align*} H _ { \\chi } ( G ) = \\inf \\lbrace H ( c ( V ) ) \\ : | \\ : c G \\rbrace . \\end{align*}"}
+{"id": "6182.png", "formula": "\\begin{align*} \\eta _ ( ( T , \\mathbf { l } , \\mu ) ) & = \\eta _ ( T ) , \\\\ \\eta _ ( ( T , \\mathbf { l } , \\mu ) ) & = \\eta _ ( T ) + \\eta _ ( \\mu ) . \\end{align*}"}
+{"id": "3317.png", "formula": "\\begin{align*} P _ { i + 1 } ^ { ( a + v ) } = \\frac { x _ { a + v , i } } { z _ i } , 1 - P _ { i + 1 } ^ { ( a + v ) } = \\frac { y _ { a + v } } { z _ i } \\end{align*}"}
+{"id": "3867.png", "formula": "\\begin{align*} - d N _ { \\varphi ( s , t ) } \\varphi _ s ( s , t ) & = - \\frac { d } { d s } N ( \\varphi ( s , t ) ) = - \\frac { d } { d s } \\left ( - \\sin \\theta ( s ) E _ 2 + \\cos \\theta ( s ) E _ 3 \\right ) \\\\ & = \\cos \\theta ( s ) \\theta ' ( s ) E _ 2 + \\sin \\theta ( s ) \\theta ' ( s ) E _ 3 = \\theta ' ( s ) \\varphi _ s ( s , t ) . \\\\ - d N _ { \\varphi ( s , t ) } \\varphi _ t ( s , t ) & = - \\frac { d } { d t } N ( \\varphi ( s , t ) ) = 0 . \\end{align*}"}
+{"id": "7338.png", "formula": "\\begin{align*} A ( \\lambda , t ) = \\begin{pmatrix} a _ \\lambda ( t ) & 0 & c _ \\lambda ( t ) & 0 \\\\ 0 & b _ \\lambda ( t ) & 0 & d _ \\lambda ( t ) \\\\ c _ \\lambda ( t ) & 0 & e _ \\lambda ( t ) & 0 \\\\ 0 & d _ \\lambda ( t ) & 0 & h _ \\lambda ( t ) \\end{pmatrix} \\end{align*}"}
+{"id": "921.png", "formula": "\\begin{align*} & t ^ { 1 - \\alpha } \\eta _ { 2 x t } - \\frac { k + c } { x } \\eta _ { 2 x x } + \\frac { k + c } { x ^ 2 } \\eta _ { 2 x } - \\frac 1 4 m ( k + 1 ) x ^ k ( - ( 1 - \\alpha ) t ^ { - 1 - \\alpha } \\tau \\\\ & + ( 1 - \\alpha ) t ^ { - \\alpha } \\tau _ t - t ^ { 1 - \\alpha } \\tau _ { t t } ) + \\frac 1 2 m k x ^ { k - 1 } t ^ { 1 - \\alpha } \\sigma _ { 1 t } - \\frac 1 2 m ( k + c ) ( k - 2 ) x ^ { k - 3 } \\sigma _ 1 = 0 . \\end{align*}"}
+{"id": "111.png", "formula": "\\begin{align*} \\mathcal { H } _ { \\Lambda } ( \\rho _ { \\mu } ) _ n : = \\sum _ { j = 1 } ^ n \\Big ( - \\Delta _ j - \\rho _ { \\mu } \\int _ { \\mathbb { R } ^ d } g ( x _ j - y ) \\mathrm { d } y \\Big ) + \\sum _ { i < j } ^ n v ( x _ i - x _ j ) . \\end{align*}"}
+{"id": "7404.png", "formula": "\\begin{align*} \\partial _ { t } W _ { n } ^ { M } ( t ) = - u _ { n } ( x _ { t } , t ) \\underbrace { \\partial _ { x } W _ { n } ( x _ { t } , t ) } _ { = 0 } = 0 , \\end{align*}"}
+{"id": "7868.png", "formula": "\\begin{align*} W ( \\mu _ 1 , \\mu _ 2 ) ^ 2 : = \\inf \\int _ { M \\times M } d ^ 2 ( x , y ) \\ , d \\gamma ( x , y ) , \\end{align*}"}
+{"id": "1401.png", "formula": "\\begin{gather*} \\alpha _ n = \\dfrac { 2 } { \\pi } + \\varkappa _ n , n \\ge 1 . \\end{gather*}"}
+{"id": "5814.png", "formula": "\\begin{align*} \\varepsilon ' _ i = \\alpha ^ { c , n - i + 1 } _ { m a x } \\ ; \\ ; i \\geq 2 , \\varepsilon ' _ n = \\alpha ^ { c 1 } _ { m a x } = \\alpha ' _ n . \\end{align*}"}
+{"id": "5059.png", "formula": "\\begin{align*} \\deg ( M , N , \\iota , Q , X ) : = \\deg ( \\Pi | _ P , X ) \\end{align*}"}
+{"id": "2861.png", "formula": "\\begin{align*} R [ \\lambda ] : = \\{ a \\in R ^ + \\mid a ( \\lambda ) < 0 \\} . \\end{align*}"}
+{"id": "8541.png", "formula": "\\begin{align*} | D \\ell | ( J ) - \\frac { 1 } { k } \\leq \\sum _ { i = 1 } ^ { N _ { k } - 1 } | \\ell ( z _ { i + 1 } ^ { k } ) - \\ell ( z _ { i } ^ { k } ) | \\leq | D \\ell | ( J ) . \\end{align*}"}
+{"id": "5150.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } H [ Q _ { \\mathrm { u n i } } ( X ) ] - H [ Q ^ * ( X ) ] = 0 . \\end{align*}"}
+{"id": "2830.png", "formula": "\\begin{align*} \\ell ^ { ( p ) } ( r e s _ { s ' } ( \\mathfrak { q } ' ) ) - \\ell ^ { ( p ) } ( r e s _ { s } ( \\mathfrak { q } ) ) = r e s _ { s = 0 } \\Omega ^ { ( p ) } ( \\mathfrak { q } ' - \\mathfrak { q } ) . \\end{align*}"}
+{"id": "8682.png", "formula": "\\begin{align*} & F _ 1 ( x _ 0 , . . . , x _ { n + 1 } ) \\cdot ( \\psi ) = 0 , \\\\ & F _ 2 ( \\partial _ { x _ 0 } , . . . , \\partial _ { x _ { n + 1 } } ) \\cdot ( \\psi ) = 0 , \\\\ & \\Big ( \\sum _ { i = 0 } ^ { n + 1 } x _ i \\partial _ { x _ i } + \\frac { n + 2 } { 2 } \\Big ) \\psi = 0 . \\end{align*}"}
+{"id": "1602.png", "formula": "\\begin{align*} d ( H ^ { ( i ) } | \\mathbf { Q } ) : = \\begin{cases} \\frac { | E ( H ^ { ( i ) } ) \\cap \\mathcal { K } _ i ( \\mathbf { Q } ) | } { | \\mathcal { K } _ i ( \\mathbf { Q } ) | } & | \\mathcal { K } _ i ( \\mathbf { Q } ) | > 0 , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "5940.png", "formula": "\\begin{align*} - \\Delta _ g u - g ( F , \\nabla _ g u ) = \\lambda u , \\left . u \\right | _ { \\Gamma _ \\varepsilon } = 0 , \\left . \\partial _ \\nu u \\right | _ { \\partial M \\setminus \\Gamma _ \\varepsilon } = 0 , \\end{align*}"}
+{"id": "8724.png", "formula": "\\begin{align*} \\Gamma ( z ) = \\int _ { 0 } ^ \\infty x ^ { z - 1 } \\exp ( - x ) \\d x , z > 0 \\enspace . \\end{align*}"}
+{"id": "527.png", "formula": "\\begin{align*} \\norm { S _ { h ( \\xi ) } ( t ) f } _ { X _ { h ( \\xi ) } ^ { s , b } ( 0 , T ) } = \\norm { 1 } _ { H ^ b ( 0 , T ) } \\norm { f } _ { H ^ s } \\end{align*}"}
+{"id": "3544.png", "formula": "\\begin{align*} \\mathcal { F } ( t ) < - 2 C _ { 1 } K ^ { 2 } t \\in ( 0 , t _ { 0 } ) , \\mathcal { F } ( t _ { 0 } ) = - 2 C _ { 1 } K ^ { 2 } . \\end{align*}"}
+{"id": "5467.png", "formula": "\\begin{align*} \\log \\left [ y - \\log y + \\log \\left ( 1 + \\frac { 1 } { y } \\right ) \\right ] \\leq \\log y - \\frac { \\log y } { y } + \\frac { 1 } { y ^ 2 } . \\end{align*}"}
+{"id": "7220.png", "formula": "\\begin{align*} \\inf _ { ( \\mu , \\omega ) \\in \\mathbb { B } } \\sup _ { \\nu \\in \\mathbb { A } } \\mathcal { L } \\bigl ( ( \\mu , \\omega ) , \\nu \\bigr ) = \\max _ { \\nu \\in \\mathbb { A } } \\inf _ { ( \\mu , \\omega ) \\in \\mathbb { B } } \\mathcal { L } \\bigl ( ( \\mu , \\omega ) , \\nu \\bigr ) . \\end{align*}"}
+{"id": "8200.png", "formula": "\\begin{align*} d ( \\alpha ( X ) ) ( Y ) = ( L _ X \\alpha ) ( Y ) - ( d \\alpha ) ( X , Y ) . \\end{align*}"}
+{"id": "6291.png", "formula": "\\begin{align*} h ( \\xi , \\eta ) = h _ 1 ( \\xi , \\eta ) ( \\xi - \\eta ) ^ 2 . \\end{align*}"}
+{"id": "7172.png", "formula": "\\begin{align*} \\mathcal { N } _ s : = \\sum _ { j = 1 } ^ s n _ j , \\end{align*}"}
+{"id": "1964.png", "formula": "\\begin{align*} \\begin{aligned} & P 1 : \\sum _ { i = 0 } ^ { M - 1 } \\mathcal { A } ( \\mathbf { c } _ i ^ e ) ( \\tau ) = 0 , | \\tau | \\in \\left ( \\mathcal { T } _ { 1 } \\cup \\mathcal { T } _ { 2 } \\right ) \\cap \\mathcal { T } ; \\\\ & P 2 : \\sum _ { i = 0 } ^ { M - 1 } \\mathcal { C } ( \\mathbf { c } _ i ^ e , \\mathbf { c } _ i ^ { e ' } ) ( \\tau ) = 0 , | \\tau | \\in \\{ 0 \\} \\cup \\mathcal { T } _ { 1 } \\cup \\mathcal { T } _ { 2 } ; \\\\ & \\end{aligned} \\end{align*}"}
+{"id": "8267.png", "formula": "\\begin{align*} \\lambda ( M , g _ 1 ) = \\mu _ \\lambda ^ { - 1 } ( 0 ) / \\mathbb { S } ^ 1 . \\end{align*}"}
+{"id": "5671.png", "formula": "\\begin{align*} D _ { X ^ \\prime } \\cdot e _ 0 & = - 5 b _ 0 + b _ 1 + 6 \\geq 0 , \\\\ D _ { X ^ \\prime } \\cdot e _ 0 ^ \\prime & = 5 b _ 0 - b _ 1 + 6 \\geq 0 , \\\\ D _ { X ^ \\prime } \\cdot e _ 1 & = b _ 0 - 5 b _ 1 + 6 \\geq 0 , \\\\ D _ { X ^ \\prime } \\cdot e _ 1 ^ \\prime & = - b _ 0 + 5 b _ 1 + 6 \\geq 0 . \\end{align*}"}
+{"id": "7024.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ K \\omega _ k = 1 \\quad \\Gamma _ { \\rm o p t } ^ \\Phi = \\sum _ { k = 1 } ^ K \\omega _ k | \\Psi _ k \\rangle \\langle \\Psi _ k | . \\end{align*}"}
+{"id": "88.png", "formula": "\\begin{align*} \\sum _ { m \\in \\Z } \\langle \\Psi ^ m , \\mathcal H \\Psi ^ m \\rangle - \\langle \\Psi , \\mathcal H \\Psi \\rangle = \\delta _ 1 \\langle d _ 1 ^ L \\rangle _ \\Psi + \\delta _ 2 \\langle d _ 2 ^ L \\rangle _ \\Psi , \\end{align*}"}
+{"id": "3370.png", "formula": "\\begin{align*} \\mathcal { R } _ { f _ 1 , f _ 2 } ( y , T ) & \\ll \\frac { 1 } { q ^ { 1 - \\varepsilon } } + \\frac { T ^ { \\varepsilon } } { T } ( q T ) ^ { 1 / 2 + \\theta } \\Big ( q + \\frac { T } { Q } \\Big ) ^ { 1 / 2 } \\Big ( 1 + \\frac { T } { q Q } \\Big ) ^ { 1 / 2 } \\\\ & \\ll \\frac { 1 } { q ^ { 1 - \\varepsilon } } + T ^ { \\varepsilon } ( q T ) ^ { \\theta } \\Big ( \\frac { q } { T ^ { 1 / 2 } } + \\frac { T ^ { 1 / 2 } } { Q } \\Big ) . \\end{align*}"}
+{"id": "4657.png", "formula": "\\begin{align*} \\bigl ( ( \\mathbf { X } _ { v | s } \\otimes \\mathbf { Y } _ { v | t } ) \\cdot T \\bigr ) ( \\xi ) : = ( \\mathbf { X } _ { v | s } \\otimes \\mathbf { Y } _ { v | t } ) \\cdot ( T ( \\xi ) ) \\end{align*}"}
+{"id": "5078.png", "formula": "\\begin{align*} E ( A ) E ( - A ) = \\frac { 4 } { 3 } I . \\end{align*}"}
+{"id": "893.png", "formula": "\\begin{align*} t ^ { 1 - \\alpha } \\eta _ t - \\eta _ { x x } - \\frac { c } { x } \\eta _ x - m x ^ k \\phi _ x = 0 , \\end{align*}"}
+{"id": "7540.png", "formula": "\\begin{align*} \\gamma _ { 1 2 } ^ * ( \\alpha \\boxtimes \\gamma ) & = s _ { 1 2 } ^ * \\left ( \\left ( g _ { 1 2 } ^ * ( s t _ 1 \\times s t _ 2 ) ^ * \\alpha \\right ) \\boxtimes \\gamma \\right ) \\\\ & = s _ { 1 2 } ^ * s ^ * _ 3 \\left ( \\left ( ( g _ { 1 } \\times g _ 2 ) ^ * ( s t _ 1 \\times s t _ 2 ) ^ * \\alpha \\right ) \\boxtimes \\gamma \\right ) , \\end{align*}"}
+{"id": "7438.png", "formula": "\\begin{align*} \\lambda \\int _ { \\Omega } \\vert y _ 1 \\vert ^ { \\kappa _ 1 + 1 } \\vert y _ 2 \\vert ^ { \\kappa _ 2 + 1 } d z \\leq & \\frac { p - 1 + \\nu } { ( \\kappa _ 1 + \\kappa _ 2 + 2 ) ( \\kappa _ 1 + \\kappa _ 2 + 1 + \\nu ) } \\| ( y _ 1 , y _ 2 ) \\| _ { 1 , p } \\\\ & + \\frac { q - 1 + \\nu } { ( \\kappa _ 1 + \\kappa _ 2 + 2 ) ( \\kappa _ 1 + \\kappa _ 2 + 1 + \\nu ) } \\| ( \\nabla y _ 1 , \\nabla y _ 2 ) \\| _ { q , \\eta } . \\end{align*}"}
+{"id": "125.png", "formula": "\\begin{align*} \\begin{cases} K _ \\ell ^ { - 2 6 - 1 9 \\varepsilon } \\gg \\delta , \\ ; & \\ ; d = 2 , \\\\ K _ { \\ell } ^ { - 2 8 - 1 6 \\varepsilon } \\gg \\rho a ^ 3 , \\ ; & \\ ; d = 3 . \\end{cases} \\end{align*}"}
+{"id": "3650.png", "formula": "\\begin{align*} \\bot ( \\omega \\wedge \\Omega ) = ( \\bot \\omega ) \\Omega + ( n - 2 ) \\omega \\ , \\end{align*}"}
+{"id": "8836.png", "formula": "\\begin{align*} \\frac { \\Gamma ( \\beta ) } { ( \\ell - 1 ) ! } = \\frac { \\Gamma ( \\beta ) } { \\Gamma ( \\ell ) } = \\frac { \\ell \\Gamma ( \\ell + ( \\beta - \\ell ) ) } { \\Gamma ( \\ell + 1 ) } \\leq \\ell ^ { \\beta - \\ell } \\leq \\ell \\enspace , \\end{align*}"}
+{"id": "8118.png", "formula": "\\begin{align*} q _ 0 \\ge p _ 0 + v _ \\alpha - u _ \\alpha , p _ 1 \\ge q _ 1 + v _ \\beta - u _ \\beta . \\end{align*}"}
+{"id": "5044.png", "formula": "\\begin{align*} | \\partial ^ m \\Gamma | \\leq C ( A ) \\rho ^ { - 1 - m } , m = 0 , \\ldots , k + 9 , \\end{align*}"}
+{"id": "4199.png", "formula": "\\begin{align*} D ( 4 i , \\varepsilon ) & = \\sigma _ { \\varepsilon } ( 4 i I ) = \\sigma _ { \\varepsilon } ( [ I \\bullet i I , I ] _ { \\ast } ) = \\sigma _ { \\varepsilon } ( [ \\varphi ( I ) \\bullet \\varphi ( i I ) , \\varphi ( I ) ] _ { \\ast } ) \\\\ & = \\sigma _ { \\varepsilon } ( [ \\varphi ( I ) \\varphi ( i I ) + \\varphi ( i I ) \\varphi ( I ) ^ { \\ast } , \\varphi ( I ) ] _ { \\ast } ) = \\sigma _ { \\varepsilon } ( 4 \\varphi ^ { 2 } ( I ) \\varphi ( i I ) ) . \\end{align*}"}
+{"id": "6591.png", "formula": "\\begin{align*} \\omega _ l ^ 2 & = ( \\omega _ l ^ { ( 0 ) } ) ^ 2 + \\varepsilon [ \\frac { 2 } { a _ l } ( \\Delta q ^ { ( 1 ) } ) ( \\pm e _ l , n ^ { ( l ) } ) ] ( \\omega , a ) \\\\ & \\ \\ + \\delta [ \\frac { 2 } { a _ l } ( q ^ { ( 1 ) } ) _ { * } ^ { p + 1 } ( \\pm e _ l , n ^ { ( l ) } ) ] ( \\omega , a ) , \\ , \\l = 1 , 2 , . . . , b . \\end{align*}"}
+{"id": "2715.png", "formula": "\\begin{align*} [ n - 1 ] \\setminus ( \\{ n - 1 , n - 2 , \\dots , \\kappa _ 2 - 1 \\} \\sqcup \\{ 1 , 2 , \\dots , \\kappa _ 1 \\} ) = \\{ \\kappa _ 1 + 1 , \\dots , \\kappa _ 2 - 2 \\} . \\end{align*}"}
+{"id": "4302.png", "formula": "\\begin{align*} \\widehat { \\sigma } _ { j _ 1 , j _ 2 } = \\sum _ { \\pi \\in \\mathcal { D P } _ 2 ( 2 j _ 1 , 2 j _ 2 ) } f _ I ^ - ( \\pi ) + ( \\kappa - 1 ) \\sum _ { \\pi \\in \\mathcal { D P } _ { 2 , 4 } ( 2 j _ 1 , 2 j _ 2 ) } f _ { I I } ^ - ( \\pi ) , \\end{align*}"}
+{"id": "1460.png", "formula": "\\begin{align*} ( \\sigma _ { f ( 1 ) } - \\sigma _ { f ( 2 ) } ) ^ { 2 } + ( \\sigma _ { f ( 2 ) } - \\sigma _ { f ( 3 ) } ) ^ { 2 } + \\cdots + ( \\sigma _ { f ( n ) } - \\sigma _ { f ( n + 1 ) } ) ^ { 2 } + ( \\sigma _ { f ( n + 1 ) } - \\sigma _ 1 ) ^ { 2 } = 4 \\sum \\limits _ { i \\in I } \\mu _ { f ( i ) } \\sigma _ { f ( i ) } . \\end{align*}"}
+{"id": "5600.png", "formula": "\\begin{align*} \\exp _ p ( t ) = \\mathrm { e } ^ t - \\mathrm { e } ^ t \\dfrac { p - 1 } { \\Gamma ( p ) } \\int _ t ^ \\infty s ^ { p - 2 } \\mathrm { e } ^ { - s } \\mathrm d s , \\quad \\forall t \\in ( 0 , \\infty ) . \\end{align*}"}
+{"id": "4074.png", "formula": "\\begin{align*} y ^ 2 - ( x - a ) ( x - b ) ( x - c ) = 0 . \\end{align*}"}
+{"id": "3892.png", "formula": "\\begin{align*} \\alpha _ z & = g \\circ ( f \\circ \\alpha _ x \\circ ( \\overline { f } \\ast \\det ) ) \\circ ( \\overline { g } \\ast \\det ) \\\\ & = ( g \\circ f ) \\circ \\alpha _ x \\circ ( ( \\overline { f } \\circ \\overline { g } ) \\ast \\det ) . \\end{align*}"}
+{"id": "1160.png", "formula": "\\begin{align*} & [ m _ { 1 , 1 } , m _ { 1 , 0 } ] + [ m _ { 1 , 0 } , m _ { 1 , 1 } ] + [ m _ { 1 , 0 } , m _ { 1 , 0 } ] = 0 , \\\\ & [ m _ { 2 , 1 } , m _ { 2 , 0 } ] + [ m _ { 2 , 0 } , m _ { 2 , 1 } ] + [ m _ { 2 , 0 } , m _ { 2 , 0 } ] = 0 , \\\\ & [ m _ { 1 , 1 } , m _ { 2 , 0 } ] + [ m _ { 1 , 0 } , m _ { 2 , 1 } ] + [ m _ { 1 , 0 } , m _ { 2 , 0 } ] = 0 . \\end{align*}"}
+{"id": "6254.png", "formula": "\\begin{align*} c _ { \\leq k } = ( \\sum _ { j \\leq k } c _ j ^ 2 ) ^ \\frac 1 2 , c _ { \\geq k } = ( \\sum _ { j \\geq k } c _ j ^ 2 ) ^ \\frac 1 2 . \\end{align*}"}
+{"id": "5127.png", "formula": "\\begin{align*} R \\leq \\frac { L } { \\sum _ { t \\in [ T ] , s \\in \\mathcal { E } ( t ) } \\log _ d | \\mathcal { Q } _ { t , s } | } = \\frac { L } { \\sum _ { t \\in [ T ] , s \\in \\mathcal { E } ( t ) } \\log _ d \\delta _ { t , s } } . \\end{align*}"}
+{"id": "6839.png", "formula": "\\begin{align*} u _ { x _ 0 , \\varepsilon } ( x ) = \\left ( \\frac { 2 \\varepsilon } { 1 + \\varepsilon ^ { 2 } | x - x _ 0 | ^ { 2 } } \\right ) ^ { \\frac { n - 6 } { 2 } } . \\end{align*}"}
+{"id": "7035.png", "formula": "\\begin{align*} D _ { L , \\Omega } ^ { \\Phi } [ \\rho ] = \\sup _ { v \\in { \\rm S p a n } \\{ \\Phi \\} } \\left \\{ \\int v \\dd \\rho - E [ v ] \\right \\} . \\end{align*}"}
+{"id": "619.png", "formula": "\\begin{align*} \\theta ( X ) - \\theta ( Y ) = \\left ( \\int _ 0 ^ 1 \\theta ' \\left ( \\rho ( t ) \\right ) \\ , d t \\right ) ( X - Y ) = : I _ 2 ( X - Y ) . \\end{align*}"}
+{"id": "6737.png", "formula": "\\begin{align*} \\begin{aligned} \\lim \\limits _ { t \\rightarrow + \\infty } F ( t ) \\ge - 8 \\pi C _ { 2 } \\mathfrak { c } _ { p } ( \\mathfrak { m } _ { A D M } - m ) . \\end{aligned} \\end{align*}"}
+{"id": "3758.png", "formula": "\\begin{align*} \\begin{gathered} g ( t ) = e ^ { - t B } g ( 0 ) , \\\\ f ( t ) = e ^ { t b B } f ( 0 ) + \\int _ { 0 } ^ t e ^ { ( t - s ) b B } g ( s ) d s , \\\\ u ( t ) = e ^ { t a B } u ( 0 ) + \\int _ { 0 } ^ t e ^ { ( t - s ) a B } f ( s ) d s . \\end{gathered} \\end{align*}"}
+{"id": "1887.png", "formula": "\\begin{align*} \\Gamma _ x : = \\bigcup \\limits _ { i , j } \\left \\{ \\{ x _ { i - \\frac { 1 } { 2 } } \\} \\times J _ j \\right \\} \\mbox { a n d } \\Gamma _ v : = \\bigcup \\limits _ { i , j } \\left \\{ I _ i \\times \\{ v _ { j - \\frac { 1 } { 2 } } \\} \\right \\} . \\end{align*}"}
+{"id": "2438.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\rho ( x ) { \\rm d } x & = \\sum _ { n = 1 } ^ N \\int _ { \\Omega } | u _ n ( x ) | ^ 2 \\ , { \\rm d } x = \\sum _ { n = 1 } ^ N \\frac { 1 } { \\lambda _ n } \\\\ & = \\sum _ { n = 1 } ^ N \\frac { 1 } { n } \\cdot \\frac { | \\Omega | } { 4 \\pi } \\left ( 1 + o ( 1 ) _ { n \\to \\infty } \\right ) = \\ln ( N ) \\cdot \\frac { | \\Omega | } { 4 \\pi } \\left ( 1 + o ( 1 ) _ { N \\to \\infty } \\right ) . \\end{align*}"}
+{"id": "7005.png", "formula": "\\begin{align*} \\int _ { M ^ 2 } \\left | v _ { 1 1 } \\right | ^ 2 d A _ x = \\int _ { M ^ 2 } \\left ( | v _ { 1 \\bar 1 } | ^ 2 - 3 | v _ { 1 } | ^ 2 \\right ) d A _ x , \\end{align*}"}
+{"id": "3618.png", "formula": "\\begin{align*} W _ i ( F ) \\begin{cases} \\geq c _ { \\textrm { b u l k } } \\left ( | F | ^ { r } + \\left ( \\displaystyle \\frac { | F | ^ 3 } { \\det F } \\right ) ^ { r - 1 } + ( \\det F ) ^ { - s } \\right ) & \\det F > 0 \\\\ [ 2 m m ] = + \\infty & \\det F \\le 0 , \\end{cases} \\end{align*}"}
+{"id": "2413.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } \\frac { \\mu _ { \\lambda } ( B ( x , r ) ) } { 2 r } = d _ { \\lambda } ( x ) \\end{align*}"}
+{"id": "3941.png", "formula": "\\begin{align*} \\bigcup _ { 0 < t < 1 } P ( ( 1 - t ) v _ 1 + t v _ 2 ) = \\bigcup _ { 0 < t < 1 } P ( ( 1 - t ) c v _ 1 + t v _ 2 ) , \\end{align*}"}
+{"id": "137.png", "formula": "\\begin{align*} \\| x ( t ) - x _ m ( t ) \\| _ 2 ^ 2 = E \\big ( | x ( t ) - x _ m ( t ) | ^ 2 \\big ) \\le R _ 1 e ^ { R _ 2 } = O ( h ^ 2 ) \\end{align*}"}
+{"id": "2585.png", "formula": "\\begin{align*} \\beta _ { e } ( h , h ' ) = e \\beta ( h , h ' ) & = e ( \\alpha ( h + h ' ) - \\alpha ( h ) - \\alpha ( h ' ) ) = \\\\ & = e \\alpha ( h + h ' ) - e \\alpha ( h ) - e \\alpha ( h ' ) = \\\\ & = \\alpha _ { e } ( h + h ' ) - \\alpha _ { e } ( h ) - \\alpha _ { e } ( h ' ) . \\end{align*}"}
+{"id": "3735.png", "formula": "\\begin{align*} \\sigma ( \\mathbb { A } ) = \\{ \\lambda \\in \\mathbb { C } : \\lambda ^ 3 \\in \\sigma ( A ) \\} . \\end{align*}"}
+{"id": "2742.png", "formula": "\\begin{align*} 1 > \\| U ( e _ { \\theta ^ { - 1 } + 1 } ) - \\varepsilon e _ m \\| = \\| e _ { \\theta ^ { - 1 } + 1 } - \\varepsilon \\cdot U ^ { - 1 } ( e _ m ) \\| . \\end{align*}"}
+{"id": "1471.png", "formula": "\\begin{align*} \\mathbf { h } _ { M , j } = \\frac { \\sqrt { \\gamma _ j \\mu } } { d _ { M , j } } \\mathbf { a } ( \\theta _ { M , j } ) , \\end{align*}"}
+{"id": "8573.png", "formula": "\\begin{align*} J _ { \\Sigma _ b } = \\begin{pmatrix} \\mathrm { I d } & \\mathbf { 0 } \\\\ ( \\nabla _ X \\sigma ) ^ T & 1 + \\partial _ z \\sigma \\end{pmatrix} , \\end{align*}"}
+{"id": "2205.png", "formula": "\\begin{align*} H = \\left ( \\begin{array} { c c } 0 & J \\\\ J ^ \\top & 0 \\end{array} \\right ) \\end{align*}"}
+{"id": "2647.png", "formula": "\\begin{align*} w _ L ( X , \\delta ) = \\sup _ { t , s \\in [ 0 , L ] : \\ , \\abs { t - s } \\le \\delta } d ' ( X ( t ) , X ( s ) ) \\ , . \\end{align*}"}
+{"id": "4623.png", "formula": "\\begin{align*} c _ { \\phi , m } ^ \\pi ( g ) : = \\phi \\bigl ( \\pi ( g ) m \\bigr ) = ( \\pi ^ * ( g ^ { - 1 } ) \\phi ) ( m ) \\end{align*}"}
+{"id": "5068.png", "formula": "\\begin{align*} E _ { m } ( M ) = \\sum _ { p \\geq 0 } \\frac { M ^ p } { m ( p ) } . \\end{align*}"}
+{"id": "2049.png", "formula": "\\begin{align*} v ( t ) = e ^ { - t ( \\mathrm { I } - \\Delta ' ) ^ \\frac { 1 } { 2 } } v ( 0 ) + \\int _ { 0 } ^ { t } e ^ { - ( t - s ) ( \\mathrm { I } - \\Delta ' ) ^ \\frac { 1 } { 2 } } F ( s ) \\ , \\mathrm { d } s \\end{align*}"}
+{"id": "2860.png", "formula": "\\begin{align*} t [ \\lambda ] : = \\prod _ { a \\in R [ \\lambda ] } t _ { a ' } \\end{align*}"}
+{"id": "6074.png", "formula": "\\begin{align*} \\frac { \\gamma _ { i } } { \\gamma _ n } = \\prod _ { k = i } ^ { n - 1 } \\frac { \\gamma _ k } { \\gamma _ { k + 1 } } \\leq \\prod _ { k = i } ^ { n - 1 } e ^ { 2 \\overline { \\omega } ( \\gamma _ { k + 1 } ) } \\leq e ^ { 2 \\overline { \\omega } ( \\Gamma _ n - \\Gamma _ i ) } . \\end{align*}"}
+{"id": "8550.png", "formula": "\\begin{align*} \\widetilde { E } : = \\left \\{ ( z , w ) \\in J \\times \\mathbb { R } ^ { n - 1 } : | w - \\lambda ( r _ { \\ell } ( z ) - r _ { \\ell } ( a ) ) e | < r _ { \\ell } ( z ) \\right \\} , \\end{align*}"}
+{"id": "829.png", "formula": "\\begin{align*} | \\eta ( z ) | & = | \\Psi _ { \\epsilon } ( \\tilde { h } ( z ) ) | \\\\ & \\leq | \\tilde { h } ( z ) | \\\\ & \\leq 1 , \\end{align*}"}
+{"id": "8613.png", "formula": "\\begin{align*} \\partial _ z ( \\phi _ 1 ( X , h _ b z ) ) = h _ b ( \\partial _ z \\phi _ 1 ) ( X , h _ b z ) , \\end{align*}"}
+{"id": "7097.png", "formula": "\\begin{align*} \\mathcal { H } ( . . . ) = & \\sum _ { n _ { 2 } \\in \\mathbb { Z } } ^ { \\infty } \\ , V \\left ( \\frac { n _ 2 } { N _ 0 / n ^ 2 _ 1 } \\right ) \\ , \\mathcal { C } ^ { + } ( n ^ 2 _ 1 n _ 2 , m ; q ) \\ , \\overline { \\mathcal { C } ^ { + } ( n ^ 2 _ 1 n _ 2 , m ; q ) } \\\\ & \\times \\ , \\ , \\ , \\mathcal { J } ^ { + } _ 1 ( n ^ 2 _ 1 n _ 2 , m , q ) \\ , \\overline { \\mathcal { J } ^ { + } _ 1 ( n ^ 2 _ 1 n _ 2 , m , q ) } . \\end{align*}"}
+{"id": "4811.png", "formula": "\\begin{align*} \\Pi _ k \\varphi ( x ) : = \\hat \\nu _ { k , x } ^ { - 1 } \\int _ { C _ k ^ \\perp ( x ) } \\varphi \\ , d \\nu _ { k , x } , \\ ; \\ ; \\ ; \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\ ; \\ ; \\ ; \\Delta _ k : = \\varphi - \\Pi _ k \\varphi , \\end{align*}"}
+{"id": "5825.png", "formula": "\\begin{align*} w ^ { ( 1 ) } _ 1 w ^ { ( 2 ) } _ 1 w ^ { ( 3 ) } _ 1 = s _ { \\alpha _ 7 } s _ { \\alpha _ 4 } s _ { \\alpha _ 3 + \\alpha _ 4 + \\alpha _ 5 } s _ { \\alpha ^ { d 6 } _ { m a x } } \\end{align*}"}
+{"id": "769.png", "formula": "\\begin{align*} u ( t ) = \\phi ( t ) S _ { h ( \\xi ) } ( t ) f \\end{align*}"}
+{"id": "2010.png", "formula": "\\begin{align*} { \\mathcal { E } u } _ { | _ { \\Omega } } = u \\end{align*}"}
+{"id": "3455.png", "formula": "\\begin{align*} | \\det ( \\nabla T ) | = \\frac { C _ 1 } { W ( T ( x ) ) \\int _ { \\Delta } d x } , . \\end{align*}"}
+{"id": "4125.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k - 1 } c _ 1 ( s , i ) \\cdot D ( k - 2 i , - k + 2 s - \\tfrac { 3 } { 2 } ) = 0 . \\end{align*}"}
+{"id": "1247.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( P ( u _ n ) - P ( u _ n - \\phi _ n ^ 1 ) - P ( \\phi _ n ^ 1 ) ) = 0 \\end{align*}"}
+{"id": "7838.png", "formula": "\\begin{align*} g _ N = 2 \\pi \\left ( T _ { i \\overline { j } } \\mathrm { d } Z ^ i \\mathrm { d } \\overline { Z } ^ j + ( W _ i + W _ i ^ { } ) T ^ { i \\overline { j } } ( \\overline { W } _ j + \\overline { W } _ j ^ { } ) \\right ) , \\end{align*}"}
+{"id": "2649.png", "formula": "\\begin{align*} w ' _ L ( X , \\delta ) = \\inf _ { \\substack { 0 = t _ 0 < t _ 1 < \\ldots < t _ k = L : \\\\ t _ j - t _ { j - 1 } > \\delta } } \\ , \\ , \\max _ { j = 1 , \\ldots , k } \\sup _ { u , v \\in [ t _ { j - 1 } , t _ j ) } d ' ( X ( u ) , X ( v ) ) \\ , . \\end{align*}"}
+{"id": "7895.png", "formula": "\\begin{align*} ( \\det D ^ 2 \\tilde u _ 1 ) ^ { 1 / n } L ( u _ 2 - \\tilde u _ 1 + ( x _ n ^ 2 - A | x ' | ^ 3 ) ) & \\geq o _ { | x | } ( 1 ) + ( c + O ( | x ' | ) ) ( \\det D ^ 1 \\tilde u _ 1 ) ^ { \\frac { 1 } { n } } \\sum _ { i = 1 } ^ n u ^ { i i } \\\\ & \\geq c + o _ { | x | } ( 1 ) + O ( | x ' | ) \\\\ & \\geq 0 \\end{align*}"}
+{"id": "3541.png", "formula": "\\begin{align*} f & = - \\Delta w + \\lambda w - \\alpha u - \\beta v , \\\\ g _ { 1 } & = \\frac { \\nabla u } { \\sqrt { \\chi ( u + 1 ) } } - \\sqrt { \\chi ( u + 1 ) } \\nabla w , \\\\ g _ { 2 } & = \\frac { \\nabla v } { \\sqrt { \\xi ( v + 1 ) } } - \\sqrt { \\xi ( v + 1 ) } \\nabla w . \\end{align*}"}
+{"id": "3273.png", "formula": "\\begin{align*} f _ X ( x ) = \\frac { d } { d x } F _ X ( x ) , \\end{align*}"}
+{"id": "6544.png", "formula": "\\begin{align*} I _ { \\xi , i } = \\left \\{ \\sigma \\in \\R : \\ | \\xi ( \\sigma + k \\cdot \\omega ^ { ( 0 ) } ) + \\mu _ n | \\leq \\frac { 1 } { 1 0 0 } ( \\varepsilon + \\delta ) ^ { \\frac { 1 } { 8 b } } \\right \\} . \\end{align*}"}
+{"id": "4298.png", "formula": "\\begin{align*} ( X _ n ( t ) ) ^ { k } - ( X _ n ( s ) ) ^ { k } & = ( \\widetilde { X } _ n ( t - s ) ) ^ { k } + \\sum _ { d = 1 } ^ { k - 1 } \\binom { k } { d } ( \\widetilde { X } _ n ( t - s ) ) ^ d ( X _ n ( s ) ) ^ { k - d } , \\end{align*}"}
+{"id": "5215.png", "formula": "\\begin{align*} 1 + m _ 1 + m _ 2 = v . \\end{align*}"}
+{"id": "5525.png", "formula": "\\begin{align*} \\max _ { z \\in K } \\left | \\frac { \\partial P } { \\partial z _ m } ( z ) \\right | \\le M d ^ r \\max _ { z \\in K } | P ( z ) | \\mbox { f o r } z = ( z _ 1 , \\dots , z _ n ) \\end{align*}"}
+{"id": "3072.png", "formula": "\\begin{align*} \\sum \\limits _ { n = 1 } ^ { { Z _ { K , { N _ { \\rm { R } } } } } } { \\prod \\limits _ { k = 1 } ^ K { L _ { { k , { \\rm { R } } } } ^ { { I _ { n , k } } } } } , \\end{align*}"}
+{"id": "2558.png", "formula": "\\begin{align*} \\| \\eta _ R u _ n \\| _ { D ^ { s , p } } ^ { p } & \\leq ( 1 + \\delta ) \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\frac { | \\eta _ R ( x ) | ^ p | u _ n ( x ) - u _ n ( y ) | ^ p } { | x - y | ^ { N + s p } } d x d y \\\\ & \\phantom { = } + C ( \\delta ) \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\frac { | u _ n ( y ) | ^ p | \\eta _ R ( x ) - \\eta _ R ( y ) | ^ p } { | x - y | ^ { N + s p } } d x d y . \\end{align*}"}
+{"id": "3187.png", "formula": "\\begin{align*} \\min _ { z \\in \\mathbb { R } ^ n } \\| z \\| \\mbox { s u b j e c t t o } A z = b , A \\in \\mathbb { R } ^ { m \\times n } , \\ b \\in \\mathbb { R } ^ { m } , \\ b \\in \\mathcal { R } ( A ) , \\end{align*}"}
+{"id": "4817.png", "formula": "\\begin{align*} R _ { 1 2 } ( u ) R _ { 1 3 } ( u + v ) R _ { 2 3 } ( v ) = R _ { 2 3 } ( v ) R _ { 1 3 } ( u + v ) R _ { 1 2 } ( u ) \\end{align*}"}
+{"id": "6711.png", "formula": "\\begin{align*} \\frac { \\prod _ { k = 1 } ^ { f _ i } ( 1 + \\beta x _ k ) } { \\prod _ { k = 1 } ^ { g _ j - 1 } ( 1 + \\beta x _ k ) } \\frac { ( t + \\beta ) ^ { r - j } } { ( 1 + \\beta z ^ { - 1 } ) ^ { r - i + 1 } } \\frac { \\prod _ { k = f _ i + 1 } ^ { g _ r - 1 } ( 1 - x _ k z ) } { \\prod _ { k = g _ j } ^ { g _ r - 1 } ( 1 - x _ k t ) } \\frac { z _ i ^ { - r } } { t - z } \\cdot t ^ { \\mu _ j } d t , \\end{align*}"}
+{"id": "876.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathcal { T } _ t ^ \\alpha u = M ( x , t , u , v , u ^ { ( 1 ) } , v ^ { ( 1 ) } , \\cdots , u ^ { ( n ) } , v ^ { ( n ) } ) , \\\\ & \\mathcal { T } _ t ^ \\alpha v = N ( x , t , u , v , u ^ { ( 1 ) } , v ^ { ( 1 ) } , \\cdots , u ^ { ( n ) } , v ^ { ( n ) } ) , \\end{aligned} \\right . \\end{align*}"}
+{"id": "1417.png", "formula": "\\begin{gather*} \\theta _ { n i } ^ K ( x ) = \\left \\{ \\begin{array} { l l } \\varphi _ { n , i } ( x ) - \\varphi _ { n , i } ^ K ( x ) , & n \\le K , \\\\ 0 , & n > K , \\end{array} \\right . \\end{gather*}"}
+{"id": "6677.png", "formula": "\\begin{align*} \\overline { W } _ 1 ( x ; N , \\beta , p , q ) : = \\int _ { 0 } ^ { 2 \\pi } { \\rho _ { ( 1 ) , N } ^ { \\widetilde { \\rm ( c J ) } } ( \\theta ; \\beta , p , q ) \\over x - e ^ { i \\theta } } \\ , d \\theta , x \\notin { \\mathcal C } _ 1 , \\end{align*}"}
+{"id": "7617.png", "formula": "\\begin{align*} \\beta _ { i , j } = \\begin{cases} \\pm N ( [ c ( g _ i ) _ { \\pm } ] ) & \\ ; j = j _ { \\pm } ( i ) \\ ; \\ ; j _ { - } ( i ) \\not = j _ { + } ( i ) ; \\\\ N ( [ c ( g _ i ) _ { + } ] ) - N ( [ c ( g _ i ) _ { - } ] ) & \\ ; j = j _ { - } ( i ) = j _ { + } ( i ) ; \\\\ 0 & \\ ; j \\notin \\{ j _ { - } ( i ) , j _ { + } ( i ) \\} . \\end{cases} \\end{align*}"}
+{"id": "7495.png", "formula": "\\begin{align*} ( \\lambda + 1 ) ^ { \\sigma ^ 2 - 1 } = \\lambda ^ { \\sigma - 1 } , \\end{align*}"}
+{"id": "3275.png", "formula": "\\begin{align*} m _ \\ell = E [ X ^ \\ell ] = \\int _ \\mathbb { R } x ^ \\ell f _ X ( x ) d x . \\end{align*}"}
+{"id": "5411.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\sigma } _ y ( t ) = \\nabla u ( \\sigma _ y ( t ) ) , & \\\\ \\sigma _ y ( 0 ) = y . \\ \\ & \\end{cases} \\end{align*}"}
+{"id": "2781.png", "formula": "\\begin{align*} x \\cdot I ( G - N [ v _ 0 ] ) & = x \\cdot ( 2 x + 1 ) ^ { 2 k + 4 } \\cdot ( 3 x ^ 2 + 4 x + 1 ) \\\\ & = x ( 3 x ^ 2 + 4 x + 1 ) \\cdot \\bigg [ \\sum _ { i = 0 } ^ { 2 k + 4 } \\binom { 2 k + 4 } { i } ( 2 x ) ^ { i } ] \\\\ & = x ( 3 x ^ 2 + 4 x + 1 ) \\cdot [ ( 2 x ) ^ { 2 k + 4 } + \\dots ] \\\\ & = 3 \\cdot 2 ^ { 2 k + 4 } x ^ { 2 k + 7 } + \\dots \\end{align*}"}
+{"id": "7693.png", "formula": "\\begin{align*} \\sum _ { \\sigma ' \\in \\Sigma ' } ( \\rho ' \\otimes _ 0 \\varphi ' ) _ { \\sigma ' } ^ { \\sigma '' } \\circ ( \\rho \\otimes _ 0 \\varphi ) _ { \\sigma } ^ { \\sigma ' } = \\sum _ { \\sigma ' \\in \\widehat { \\Sigma ' } } ( \\rho ' \\otimes _ 0 \\varphi ' ) _ { \\sigma ' } ^ { \\sigma '' } \\circ ( \\rho \\otimes _ 0 \\varphi ) _ { \\sigma } ^ { \\sigma ' } . \\end{align*}"}
+{"id": "4340.png", "formula": "\\begin{align*} a _ { j , k } = W ^ { ( 2 ) } _ \\omega ( { \\sf f } _ j , { \\sf f } _ k ) \\ \\ ( 1 \\le j < k \\le 2 m ) \\ , . \\end{align*}"}
+{"id": "1864.png", "formula": "\\begin{gather*} \\sup _ { s \\in K } | g ( s ) - p ( s ) | = \\lim _ { n \\to \\infty } \\sup _ { s \\in K } | g _ n ( s ) - p ( s ) | \\geq \\frac { \\epsilon } { 2 } , \\end{gather*}"}
+{"id": "6172.png", "formula": "\\begin{align*} \\eta _ ( A ) & = \\sum _ { 1 \\leq i < i ' \\leq n , \\ , 1 \\leq j ' \\leq j \\leq n } a _ { i j } a _ { i ' j ' } \\\\ & = \\sum _ { 1 \\leq i ' \\leq n } \\sum _ { 1 \\leq j \\leq n } \\sum _ { 1 \\leq i < i ' } \\sum _ { 1 \\leq j ' \\leq j } a _ { i j } a _ { i ' j ' } \\\\ & = \\sum _ { 1 \\leq i ' \\leq n } \\sum _ { 1 \\leq j \\leq n } \\left ( \\sum _ { 1 \\leq i < i ' } a _ { i j } \\right ) \\left ( \\sum _ { 1 \\leq j ' \\leq j } a _ { i ' j ' } \\right ) . \\end{align*}"}
+{"id": "731.png", "formula": "\\begin{align*} \\Delta _ 2 ^ \\mu ( t ) = \\Delta _ { 2 , 1 } ^ \\mu ( t ) + \\Delta _ { 2 , 2 } ^ \\mu ( t ) , \\end{align*}"}
+{"id": "7527.png", "formula": "\\begin{align*} Z : = \\big \\{ \\mathbf { v } \\in \\R ^ 3 \\ , : \\ , ( \\mathbf { A } - \\lambda _ + \\mathbf { I } ) \\mathbf { v } \\in \\mathrm { R a n } \\ , \\mathbf D \\big \\} \\ , ; \\end{align*}"}
+{"id": "2451.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ N | ( - \\Delta _ { \\Omega } ) ^ { \\frac { s } { 2 } } u _ n \\rangle \\langle ( - \\Delta _ { \\Omega } ) ^ { \\frac { s } { 2 } } u _ n | \\le 1 L ^ 2 ( \\Omega ) . \\end{align*}"}
+{"id": "7779.png", "formula": "\\begin{align*} Z _ { \\gamma ^ i } = Z ^ i , Z _ { \\widetilde { \\gamma } _ i } = - Z _ i = - \\frac { \\partial \\mathfrak { F } } { \\partial Z ^ i } \\ , . \\end{align*}"}
+{"id": "3677.png", "formula": "\\begin{align*} \\bar { f } _ t & = f _ t = \\frac { n } { 2 } - S - \\Delta \\ , f = \\frac { n } { 2 } - S _ N - \\left ( \\Delta _ N \\ , \\bar { f } + \\frac { k } { 2 } \\right ) \\\\ & = \\frac { n - k } { 2 } - S _ N - \\Delta _ N \\ , \\bar { f } \\ , . \\end{align*}"}
+{"id": "8886.png", "formula": "\\begin{align*} \\int _ { G / T } f ( y _ 1 , \\ldots , y _ { \\ell } ) = \\sum _ { w \\in W } \\frac { w \\cdot f ( u ) } { w \\cdot \\big ( \\prod _ { \\alpha \\in \\Delta ^ + } c _ 1 ( S _ { \\alpha } ) \\big ) } . \\end{align*}"}
+{"id": "1057.png", "formula": "\\begin{align*} \\gamma _ \\lambda ( u _ 0 ) < \\inf \\left \\{ \\gamma _ \\lambda ( u ) : \\ : \\| u - u _ 0 \\| = \\rho \\right \\} = m _ \\lambda . \\end{align*}"}
+{"id": "5098.png", "formula": "\\begin{align*} V ^ { R l a } = \\varinjlim _ { b \\to + \\infty } \\varprojlim _ { a \\to - \\infty } ( \\tau ^ { [ a , b ] } V ) ^ { R l a } , \\end{align*}"}
+{"id": "1608.png", "formula": "\\begin{align*} \\mathcal G _ T = \\{ P ^ { ( k - 2 ) } _ J \\in \\mathcal P ^ { ( k - 2 ) } : J \\in [ T ] ^ { k - 2 } \\} . \\end{align*}"}
+{"id": "4439.png", "formula": "\\begin{align*} | | \\tilde { \\mathcal { A } } \\mathcal { A } _ j \\partial _ j { \\mathbf V } | | ^ 2 _ { s - 1 , \\ast , t } & \\lesssim | | \\tilde { \\mathcal { A } } \\mathcal { A } _ j { \\mathbf V } | | ^ 2 _ { s , \\ast , t } \\\\ & \\leq C ( K ) \\Big ( | | { \\mathbf V } | | ^ 2 _ { s , \\ast , t } + | | \\dot { { \\mathbf U } } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ t ) } | | \\hat { W } | | ^ 2 _ { s , \\ast , t } \\Big ) , \\end{align*}"}
+{"id": "190.png", "formula": "\\begin{align*} & \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T _ C M } - \\widetilde { L _ C } - 8 + W _ i ) \\right \\} ^ { ( 1 4 ) } \\\\ & = - 2 6 4 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 4 ) } . \\end{align*}"}
+{"id": "5746.png", "formula": "\\begin{align*} \\tilde N : = \\textstyle { ( \\int _ { \\mathbb B _ { 1 } ^ + } U ( X , 0 ) ^ 2 x _ { n + 1 } ^ a d X ) ^ { - 1 } } + M \\log ( M \\Theta ) + M \\| V \\| _ { 1 } ^ { 1 / 2 s } , \\end{align*}"}
+{"id": "3563.png", "formula": "\\begin{align*} \\delta ( \\alpha P + \\beta Q ) = \\alpha ( \\alpha - \\beta ) \\cdot \\delta ( P ) + \\beta ( \\beta - \\alpha ) \\cdot \\delta ( Q ) + 4 \\alpha \\beta \\cdot \\delta \\left ( \\frac { P + Q } { 2 } \\right ) . \\end{align*}"}
+{"id": "7877.png", "formula": "\\begin{align*} E _ 0 ^ { t ' + u , t '' + u } ( \\mu _ u ) = \\frac 1 2 \\int _ { t ' } ^ { t '' } \\int _ M ( | \\N \\phi ( t , u ) | ^ 2 + R ^ { H , f } ) \\rho ( t , u ) e ^ { - f } \\ , d V d t , \\end{align*}"}
+{"id": "239.png", "formula": "\\begin{align*} c _ v ( z ) = \\begin{cases} \\frac { \\Gamma ( z ) } { \\Gamma ( g _ M + z ) } & \\ \\ \\ , \\\\ \\Gamma ( z ) & \\ , \\end{cases} \\end{align*}"}
+{"id": "1340.png", "formula": "\\begin{align*} \\delta _ { m r , ( X , \\Delta ) } ( L ) & : = \\inf _ { D _ { m r } } \\mathrm { l c t } ( X , \\Delta ; D _ { m r } ) , \\\\ \\delta _ { ( X , \\Delta ) } ( L ) & : = \\lim _ { m \\to \\infty } \\delta _ { m r , ( X , \\Delta ) } ( L ) , \\end{align*}"}
+{"id": "3210.png", "formula": "\\begin{align*} A ^ T y _ 0 + A ^ T \\mathcal { K } _ k ( A A ^ T , r _ 0 ) = x _ 0 + \\mathcal { K } _ k ( A ^ T A , A ^ T r _ 0 ) . \\end{align*}"}
+{"id": "964.png", "formula": "\\begin{align*} \\begin{dcases*} L _ 1 f + 2 H _ 1 H _ 2 f = 0 & i n $ \\Sigma $ \\\\ \\frac { \\partial f } { \\partial \\nu } - f = 0 & o n $ \\partial \\Sigma $ \\end{dcases*} . \\end{align*}"}
+{"id": "4094.png", "formula": "\\begin{align*} \\hat P ^ { } ( \\Delta \\hat y , \\hat x ) \\cdot \\Psi ^ { } : = \\left ( \\Delta \\hat y ^ 2 - Q _ 0 ( \\hat x ) \\right ) \\cdot \\Psi ^ { } = 0 . \\end{align*}"}
+{"id": "6860.png", "formula": "\\begin{align*} \\mathcal { X } \\times _ 1 A _ 1 + \\mathcal { X } \\times _ 2 A _ 2 + \\dots + \\mathcal { X } \\times _ d A _ d = \\mathcal { C } , \\end{align*}"}
+{"id": "3658.png", "formula": "\\begin{align*} \\phi _ { 1 1 } - \\phi _ { 2 2 } - \\phi _ { 3 3 } - \\phi _ { 4 4 } + m ^ 2 \\phi ^ 2 = 0 \\ , \\end{align*}"}
+{"id": "548.png", "formula": "\\begin{align*} \\mathbf u ( t ) = \\mathbf S ( t ) \\mathbf u _ 0 + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf N ( \\mathbf u ( s ) ) \\ , d s \\\\ + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf M ( \\mathbf u ( s ) ) \\ , d W ( s ) \\end{align*}"}
+{"id": "3868.png", "formula": "\\begin{align*} V = \\lambda F _ 2 + \\mu F _ 3 = \\lambda e ^ { - \\lambda _ 2 z } E _ 2 + \\mu ( \\lambda _ 1 x e ^ { - \\lambda _ 1 z } E _ 1 + \\lambda _ 2 y e ^ { - \\lambda _ 2 z } E _ 2 + E _ 3 ) . \\end{align*}"}
+{"id": "4026.png", "formula": "\\begin{align*} h ( \\mu ) : = \\int _ { \\R ^ m } - \\rho \\log \\rho , \\end{align*}"}
+{"id": "3562.png", "formula": "\\begin{align*} L ( \\bar x ^ k , \\lambda ) - L ( x , \\bar \\lambda ^ k ) \\leq \\frac 1 k \\sum _ { i = 0 } ^ { k - 1 } L ( x ^ { i + 1 } , \\lambda ) - L ( x , \\lambda ^ { i + 1 } ) \\leq \\frac { 1 } { 2 k } \\| z ^ { 0 } - z \\| _ { P } ^ 2 + \\frac { D \\sum _ { i = 1 } ^ k \\| \\epsilon ^ { i } \\| _ 2 } { k } \\ , \\end{align*}"}
+{"id": "7076.png", "formula": "\\begin{align*} S _ 1 ( . . . ) = & \\frac { N ^ { 2 / 3 + i \\nu } } { p _ 1 q r ^ { 2 / 3 } } \\sum _ { \\pm } \\sum _ { n _ { 1 } | p _ 1 q r } \\ , n ^ { 1 / 3 } _ 1 \\sum _ { n _ { 2 } \\ll N _ 0 / n ^ 2 _ 1 } \\frac { A _ { \\pi } ( n _ { 2 } , n _ { 1 } ) } { n ^ { 1 / 3 } _ { 2 } } S \\left ( r \\overline { ( a + b q ) } , \\pm n _ { 2 } ; p _ 1 q r / n _ { 1 } \\right ) \\\\ & \\times \\eth _ 1 ^ { \\pm } ( n ^ 2 _ 1 n _ 2 , u , q , p _ 1 ) , \\end{align*}"}
+{"id": "2408.png", "formula": "\\begin{align*} \\frac { | D e t ( A _ { \\overline { \\i } } ) | } { C } \\leq \\left | D e t \\left ( \\sum _ { k = 1 } ^ { \\infty } A _ { \\overline { \\i } } ^ k \\right ) \\right | \\leq C \\left | D e t ( A _ { \\overline { \\i } } ) \\right | \\end{align*}"}
+{"id": "5891.png", "formula": "\\begin{align*} \\phi ( m ) = m ^ p ( 1 + p \\delta _ { \\log } ( m ) ) \\end{align*}"}
+{"id": "5089.png", "formula": "\\begin{align*} \\Delta \\ , f + S & = \\frac { n } { 2 } \\ , , \\end{align*}"}
+{"id": "3006.png", "formula": "\\begin{align*} \\textstyle H _ * \\left ( X , \\gamma ; R \\right ) = H _ * \\left ( X _ \\gamma , \\coprod S ^ 1 ; R \\right ) . \\end{align*}"}
+{"id": "7715.png", "formula": "\\begin{align*} D = \\{ \\widetilde { e } , \\cdot \\} : \\Gamma ( E _ 1 ^ * ) \\to \\Gamma ( E _ 1 ^ * ) . \\end{align*}"}
+{"id": "4353.png", "formula": "\\begin{align*} U _ t \\pi ^ { P } ( B ( { \\sf f } ) ) U _ t ^ * = \\pi ^ { P } ( B ( u _ t { \\sf f } ) ) U _ t \\Omega ^ { P } = \\Omega ^ { P } \\ \\ ( { \\sf f } \\in \\mathcal { K } \\ , , \\ t \\in \\mathbb { R } ) \\end{align*}"}
+{"id": "63.png", "formula": "\\begin{align*} \\mathcal Q _ 3 ^ { } - \\mathcal Q _ 3 ^ { } = \\sum _ { i \\neq j } ( P _ i \\overline { Q } _ { L , j } g ( x _ i - x _ j ) Q _ i Q _ j + h . c . ) . \\end{align*}"}
+{"id": "4805.png", "formula": "\\begin{align*} B _ T = g _ { \\sigma \\log T } B _ 1 , \\end{align*}"}
+{"id": "6241.png", "formula": "\\begin{align*} S ^ { s } = \\langle D \\rangle ^ { - s } S . \\end{align*}"}
+{"id": "4336.png", "formula": "\\begin{align*} S ( \\omega _ { { \\sf f } _ 1 \\cdots { \\sf f } _ n } \\| \\omega _ { { \\sf g } _ 1 \\cdots { \\sf g } _ m } ) & = S ( \\omega \\circ \\alpha _ U \\| \\omega \\circ \\alpha _ V ) \\\\ & = S ( \\omega \\circ \\alpha _ { V ^ * U } \\| \\omega ) = S ( \\omega _ { { \\sf g } _ m \\cdots { \\sf g } _ 1 { \\sf f } _ 1 \\cdots { \\sf f } _ n } \\| \\omega ) \\ , . \\end{align*}"}
+{"id": "2484.png", "formula": "\\begin{align*} \\chi \\left ( \\left ( \\bigsqcup _ { a \\in \\mathcal { A } } ^ { P _ A } G _ a \\right ) [ \\mathcal { S } ] \\right ) & = \\chi \\left ( \\bigsqcup _ { a \\in \\mathcal { A } } ^ { P _ A } G _ a [ \\mathcal { S } _ a ] \\right ) \\\\ & = \\max _ { a \\in \\mathcal { A } } \\chi \\left ( G _ a [ \\mathcal { S } _ a ] \\right ) \\\\ & = \\max _ { a \\in \\mathcal { A } } \\omega \\left ( G _ a [ \\mathcal { S } _ a ] \\right ) , \\end{align*}"}
+{"id": "2027.png", "formula": "\\begin{align*} { \\mathrm { L } } ^ { p } ( \\Omega ) \\cap \\dot { \\mathrm { B } } ^ { s } _ { p , q } ( \\Omega ) = { \\mathrm { B } } ^ { s } _ { p , q } ( \\Omega ) \\end{align*}"}
+{"id": "6728.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { \\Sigma _ t } | \\nabla u | ^ 2 = & \\int _ { \\Sigma _ t } f ' ( t ) \\frac { 1 } { | \\nabla u | } \\nu ( | \\nabla u | ^ 2 ) + | \\nabla u | ^ 2 H f ' ( t ) \\frac { 1 } { | \\nabla u | } \\\\ = & \\int _ { \\Sigma _ t } f ' ( t ) \\frac { 2 } { 1 - p } | \\nabla u | H + f ' ( t ) H | \\nabla u | \\\\ = & - a f ' ( t ) \\int _ { \\Sigma _ t } H | \\nabla u | . \\end{align*}"}
+{"id": "4799.png", "formula": "\\begin{align*} C ^ r _ \\alpha ( \\Omega ) = \\left \\{ f \\in C ^ r ( \\Omega ) \\cap H ^ \\perp _ \\alpha ( \\Omega ) : \\sum _ { 0 \\leq p \\leq r } \\sum _ { | i | = p } \\| \\partial ^ i f \\| _ { C ^ 0 } + | \\partial ^ i f | ^ \\perp _ \\alpha < \\infty \\right \\} \\end{align*}"}
+{"id": "885.png", "formula": "\\begin{align*} ( u _ 1 , v _ 1 ) = \\bigg ( 1 , \\frac { \\sqrt { a b } } { a } \\bigg ) , ~ ~ ( u _ 2 , v _ 2 ) = x ^ { 1 + \\sqrt { a b } } \\bigg ( 1 , - \\frac { \\sqrt { a b } } { a } \\bigg ) , \\end{align*}"}
+{"id": "5854.png", "formula": "\\begin{align*} s _ { \\alpha _ k + \\dots \\alpha _ { n - 2 } + \\alpha _ { n - 1 } } \\prod _ { n \\geq j \\geq k } s _ j = \\big ( \\prod _ { n - 1 \\geq j \\geq { k + 1 } } s _ j \\big ) s _ { \\alpha _ k + \\alpha _ { k + 1 } + \\dots { \\alpha _ { n - 1 } + \\alpha _ n } } \\end{align*}"}
+{"id": "5227.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ \\infty \\frac { 1 } { t ^ 2 } = \\frac { \\pi ^ 2 } { 6 } \\leq 2 . \\end{align*}"}
+{"id": "7447.png", "formula": "\\begin{align*} \\left ( \\frac { C _ { 1 8 } } { \\lambda C _ { 1 9 } C _ { 2 0 } ^ { \\kappa _ 1 + \\kappa _ 2 + 2 } } \\right ) ^ { \\frac { p } { \\kappa _ 1 + \\kappa _ 2 + 2 } } \\leq \\left ( \\| { y _ 1 } _ n \\| _ { m _ 1 } ^ { p } + \\| { y _ 2 } _ n \\| _ { m _ 2 } ^ { p } \\right ) . \\end{align*}"}
+{"id": "6503.png", "formula": "\\begin{align*} D ( k , n ) = \\mu _ n ^ 2 - ( k \\cdot \\omega ) ^ 2 = ( \\mu _ n + k \\cdot \\omega ) ( \\mu _ n - k \\cdot \\omega ) . \\end{align*}"}
+{"id": "8823.png", "formula": "\\begin{align*} \\langle a , b \\rangle = \\norm { a } ^ 2 + \\langle a , b - c \\rangle + \\langle a , c - a \\rangle . \\end{align*}"}
+{"id": "6674.png", "formula": "\\begin{align*} h _ { n - 1 } ( \\beta , p , q ) | _ { \\beta = 4 } = \\pi g _ n { i ^ n \\over ( n - 1 ) ! } . \\end{align*}"}
+{"id": "2069.png", "formula": "\\begin{align*} \\mathcal { R } ( x _ 0 , t ) & \\leq \\int _ M G ( x _ 0 , t ; y , 0 ) \\ , \\mathcal { R } ( y , 0 ) \\ , d \\mathrm { v o l } _ { g _ 0 } ( y ) \\\\ & \\leq \\int ^ \\infty _ 0 \\frac { r ^ n } { t ^ { n / 2 } } \\cdot \\exp \\left ( - \\frac { r ^ 2 } { C _ 1 t } \\right ) \\frac { C _ 1 } { r t } k ( x _ 0 , r ) d r . \\end{align*}"}
+{"id": "2136.png", "formula": "\\begin{align*} E [ \\Lambda , \\phi ; \\alpha ] : = \\int _ { \\mathbb { R } } \\hat e ( t , x ) d x \\end{align*}"}
+{"id": "6181.png", "formula": "\\begin{align*} \\eta _ ( ( T , \\mathbf { l } , \\mu ) ) & = \\eta _ ( T ) , \\\\ \\eta _ ( ( T , \\mathbf { l } , \\mu ) ) & = \\eta _ ( T ) + \\eta _ ( \\mu ) , \\end{align*}"}
+{"id": "6234.png", "formula": "\\begin{align*} \\begin{aligned} i u _ t + ( 1 + a | u | ^ 2 ) \\partial _ x ^ 2 u = b u | u | ^ 2 , u ( 0 ) = u _ 0 , \\end{aligned} \\end{align*}"}
+{"id": "7662.png", "formula": "\\begin{align*} W ( t ) = u + c t - \\sum _ { i = 1 } ^ { \\Theta ( t ) } X _ i , \\ t \\geqslant 0 , \\end{align*}"}
+{"id": "5121.png", "formula": "\\begin{align*} \\sigma = \\sum _ { k = 1 } ^ n f _ k \\sigma _ k + g _ k \\ , l ( \\sigma _ k ) , \\end{align*}"}
+{"id": "2436.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\lambda _ n } { n } = \\frac { 4 \\pi } { | \\Omega | } . \\end{align*}"}
+{"id": "5913.png", "formula": "\\begin{align*} W _ { 1 } ^ { n } ( c ) : = \\begin{cases} \\mathbb { H } ^ { \\frac { n - 2 } { 2 } } \\perp [ 1 , - c ] & , \\\\ \\mathbb { H } ^ { \\frac { n - 1 } { 2 } } \\perp [ c ] & , \\end{cases} \\end{align*}"}
+{"id": "6745.png", "formula": "\\begin{align*} ( E ^ { e _ 1 } \\circ O ^ { o _ 1 } \\circ E ^ { e _ 2 } \\circ \\dotsb \\circ O ^ { o _ l } \\circ E ^ { e _ { l + 1 } } ) ( n ) = \\lceil ( E ^ { e _ 1 + \\dotsb + e _ { l + 1 } } \\circ O ^ { o _ 1 + \\dotsb + o _ { l } } ) ( n ) \\rceil . \\end{align*}"}
+{"id": "7936.png", "formula": "\\begin{align*} \\Psi _ { 1 , 2 } = \\sum _ { i = 1 } ^ 2 ( \\deg _ { G _ i } ( u _ i ) + \\deg _ { G _ i } ( v _ i ) ) . \\end{align*}"}
+{"id": "2733.png", "formula": "\\begin{align*} \\Bigg \\| U e _ i - \\sum _ { k = 1 } ^ { \\lceil \\theta ^ { - 1 } \\rceil - 1 } U e _ { j _ k } \\Bigg \\| = \\bigg \\| \\varepsilon _ i e _ { \\lceil \\theta ^ { - 1 } \\rceil } - \\sum _ { k = 1 } ^ { \\lceil \\theta ^ { - 1 } \\rceil - 1 } \\varepsilon _ { j _ k } e _ { j _ k } \\bigg \\| = \\theta \\cdot \\lceil \\theta ^ { - 1 } \\rceil > 1 . \\end{align*}"}
+{"id": "4339.png", "formula": "\\begin{align*} W _ \\omega ^ { ( n ) } ( { \\sf f } _ 1 , \\ldots , { \\sf f } _ n ) = \\omega ( B ( { \\sf f } _ 1 ) \\cdots B ( { \\sf f } _ n ) ) \\ , . \\end{align*}"}
+{"id": "3197.png", "formula": "\\begin{align*} \\beta _ { 1 } e _ { 1 } - B _ { k } y _ { k } = Q _ { k } ^ { T } \\begin{bmatrix} 0 \\\\ \\bar { \\phi } _ { k + 1 } \\end{bmatrix} ; \\end{align*}"}
+{"id": "520.png", "formula": "\\begin{align*} \\widehat f ( \\xi ) = \\mathcal F f ( \\xi ) = \\int _ { \\R ^ d } e ^ { - i x \\xi } f ( x ) \\ , d x ( \\xi \\in \\R ^ d ) . \\end{align*}"}
+{"id": "2763.png", "formula": "\\begin{align*} T _ { \\mu } ( z ) \\approx \\frac { r _ B ^ 2 } { N } \\sum _ { j = 0 } ^ { N - 1 } \\frac { ( J _ { [ a , b ] } ^ { - } ) ' ( t ( r _ B \\xi _ N ^ j ) ) t ' ( r _ B \\xi _ N ^ j ) \\xi _ N ^ { 2 j } } { J _ { [ a , b ] } ^ { - } ( t ( r _ B \\xi _ N ^ j ) ) - z } . \\end{align*}"}
+{"id": "8136.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n + 1 } d \\left ( \\beta ' ( t _ i ) , \\beta ' ( t _ { i + 1 } ) \\right ) = \\sum _ { i = 0 } ^ { n + 1 } d \\left ( \\beta ( t _ i ^ * ) , \\beta ( t _ { i + 1 } ^ * ) \\right ) \\end{align*}"}
+{"id": "6086.png", "formula": "\\begin{align*} V a r ( \\theta _ { \\gamma = \\gamma ^ * } ) = V a r ( \\bar { X } ) ( 1 - \\rho _ { \\bar { X } \\bar { Y } } ^ 2 ) \\end{align*}"}
+{"id": "6682.png", "formula": "\\begin{align*} \\overline { W } _ 2 ( x _ 1 , x _ 2 ) = \\Big \\langle ( G ( x _ 1 ) - \\langle G ( x _ 1 ) \\rangle ) ( G ( x _ 2 ) - \\langle G ( x _ 2 ) \\rangle ) \\Big \\rangle , \\end{align*}"}
+{"id": "8011.png", "formula": "\\begin{align*} X ' ( t ) \\stackrel { d } { = } f \\bigl ( X ( t ) \\bigr ) = \\mathrm { V } ' \\begin{pmatrix} \\begin{array} { l } 1 ^ T \\\\ \\mathrm { V } \\end{array} \\end{pmatrix} ^ { - 1 } \\begin{pmatrix} \\begin{array} { c } t \\\\ X ( t ) \\end{array} \\end{pmatrix} , \\end{align*}"}
+{"id": "3588.png", "formula": "\\begin{align*} { \\rm { E i } } \\left ( x \\right ) = \\int \\nolimits _ { - \\infty } ^ x { \\frac { { { e ^ t } } } { t } d t } , { \\rm { } } x < 0 . \\end{align*}"}
+{"id": "3136.png", "formula": "\\begin{align*} \\mathbf { H } _ { k , } \\cdot \\mathbf { H } ^ H _ { l , } = \\frac { 1 } { N ( \\kappa + 1 ) } ( x _ 1 + \\cdots + x _ M ) \\end{align*}"}
+{"id": "8728.png", "formula": "\\begin{align*} \\delta _ { t } \\leq \\frac { ( t _ { 0 } - 1 ) \\delta _ { t _ { 0 } } } { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { ( 3 - p _ { i } ) t ^ { p _ { i } } } \\enspace . \\end{align*}"}
+{"id": "5528.png", "formula": "\\begin{align*} \\begin{array} { r c c l } \\partial _ i : & \\N ^ n & \\longrightarrow & \\N ^ n \\\\ & \\beta & \\longmapsto & \\partial _ i \\beta = \\begin{cases} ( 0 , \\dots , 0 ) & \\beta _ i = 0 \\\\ ( \\beta _ 1 , \\dots , \\beta _ { i - 1 } , \\beta _ i - 1 , \\beta _ { i + 1 } , \\dots , \\beta _ n ) & \\beta _ i \\ge 1 \\end{cases} , \\end{array} \\end{align*}"}
+{"id": "4452.png", "formula": "\\begin{align*} \\Sigma _ 3 ( t ) & \\lesssim | | \\partial _ 1 \\mathcal { A } _ { ( 0 ) } \\partial _ 1 { \\mathbf V } | | ^ 2 _ { s - 4 , \\ast , t } + | | \\partial _ 1 ( \\partial _ 1 \\mathcal { A } _ { ( 0 ) } \\partial _ 1 { \\mathbf V } ) | | ^ 2 _ { s - 4 , \\ast , t } \\\\ & \\leq C ( K ) \\Big ( | | { \\mathbf V } | | ^ 2 _ { s , \\ast , t } + | | \\dot { { \\mathbf U } } | | ^ 2 _ { W ^ { 2 , \\infty } _ { \\ast } ( \\Omega _ t ) } | | \\hat { W } | | ^ 2 _ { s , \\ast , t } \\Big ) . \\end{align*}"}
+{"id": "3739.png", "formula": "\\begin{align*} ( \\lambda I - \\mathbb { A } ) { \\bf u } = \\varphi . \\end{align*}"}
+{"id": "3915.png", "formula": "\\begin{align*} H ^ - ( \\alpha , v ) : = \\{ x : x \\cdot v \\leq \\alpha \\} \\\\ H ^ + ( \\alpha , v ) : = \\{ x : x \\cdot v \\geq \\alpha \\} \\\\ H ( \\alpha , v ) : = \\{ x : x \\cdot v = \\alpha \\} . \\end{align*}"}
+{"id": "4248.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u ( x ) : = C ( n , s ) \\ , \\mbox { P . V . } \\int _ { \\R ^ n } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { n + 2 s } } \\ , d y \\ , , \\end{align*}"}
+{"id": "6403.png", "formula": "\\begin{align*} \\chi ( T _ i ^ k \\otimes T _ j ^ \\ell ) = t _ { i j } ^ { k \\ell } , \\qquad \\chi ( 1 \\otimes T _ i ^ j ) = 0 = \\chi ( T _ i ^ j \\otimes 1 ) , \\end{align*}"}
+{"id": "1728.png", "formula": "\\begin{align*} \\Delta ^ { ( m + n ) } _ { l } = & \\left | c \\bigl ( \\xi ^ { ( m + n ) } _ { l } ; q _ 0 \\bigr ) \\right | ^ 2 \\left ( \\sum _ { 0 \\leq \\nu < m + n } \\left | p _ { \\nu } ( \\xi ^ { ( m + n ) } _ { l } ; q _ 0 ) \\right | ^ 2 \\right ) ^ { - 1 } \\\\ = & \\Delta _ { \\texttt { b } ; l } ^ { ( m + n - 1 , 1 ) } = \\left ( 2 ( m + n ) + u _ { q _ 0 } \\bigl ( \\xi ^ { ( m + n ) } _ l \\bigr ) + u _ { q _ 1 } \\bigl ( \\xi ^ { ( m + n ) } _ l \\bigr ) \\right ) ^ { - 1 } , \\end{align*}"}
+{"id": "5789.png", "formula": "\\begin{align*} s _ { \\beta } s _ { \\alpha } s _ { \\beta } = \\begin{cases} s _ { \\alpha + 2 \\beta } ( \\alpha , \\beta ) = - 1 , \\\\ s _ { \\alpha - 2 \\beta } ( \\alpha , \\beta ) = 1 . \\\\ \\end{cases} \\end{align*}"}
+{"id": "7685.png", "formula": "\\begin{align*} \\varphi _ \\varepsilon ^ * ( u ) & = e ^ { - A } \\left ( 1 + \\sum _ { n = 1 } ^ { \\infty } \\left ( 1 - e ^ { - A } \\right ) ^ n H ^ { * n } ( u ) \\right ) \\\\ & = \\varphi _ \\varepsilon ^ * ( 0 ) \\left ( 1 + \\sum _ { n = 1 } ^ { \\infty } \\left ( \\psi _ \\varepsilon ^ * ( 0 ) \\right ) ^ n H ^ { * n } ( u ) \\right ) , \\ , u \\geqslant 0 , \\end{align*}"}
+{"id": "5010.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 2 } ^ { n } K ( e _ 1 , e _ 1 ) \\geq - C _ 0 ( n - 1 ) . \\end{align*}"}
+{"id": "630.png", "formula": "\\begin{align*} \\norm { \\theta _ R \\left ( \\sum _ { i = 1 } ^ n \\norm { \\phi _ i } _ { \\widetilde H ^ { b } ( 0 , t ) } ^ 2 \\right ) \\phi _ j ( t ) } _ { H ^ { b } ( 0 , T ) } \\leqslant C \\sqrt { R } , \\end{align*}"}
+{"id": "6437.png", "formula": "\\begin{align*} g _ { u , v } \\oplus \\pi _ { \\mathrm { N } } ^ * \\left ( X / v \\right ) = \\pi _ { \\mathrm { N } } ^ * \\left ( X / u \\right ) \\end{align*}"}
+{"id": "2662.png", "formula": "\\begin{align*} \\phi _ k ( \\delta ) = \\sup _ { 0 \\le t _ 0 \\le t _ 1 \\le \\ldots \\le t _ l \\le T } \\{ \\sum _ { i = 1 } ^ l \\abs { g _ k ( t _ i ) - g _ k ( t _ { i - 1 } ) } : \\ , \\sum _ { l = 1 } ^ l ( t _ i - t _ { i - 1 } ) \\le \\delta \\} \\ , . \\end{align*}"}
+{"id": "6548.png", "formula": "\\begin{align*} \\inf _ { \\xi = \\pm 1 , i = ( k , n ) \\in \\Lambda \\setminus B _ * } | \\xi ( z + k \\cdot \\omega ) + \\mu _ n | \\geq \\frac { ( \\varepsilon + \\delta ) ^ { \\frac { 1 } { 8 b } } } { 5 } . \\end{align*}"}
+{"id": "3589.png", "formula": "\\begin{align*} \\begin{aligned} { { \\bar R } _ { { \\rm { S M , u p p e r } } } } & = \\sum \\nolimits _ { n \\in { \\mathcal K } _ { \\rm S M } } { \\log _ 2 } \\left ( 1 + \\frac { 1 } { c _ n } \\right ) \\\\ & = \\sum \\nolimits _ { n \\in { \\mathcal K } _ { \\rm S M } } { \\log _ 2 } \\left ( 1 + \\frac { { { E { N _ { \\rm { T } } } N _ { { \\rm { S } } , n } ^ 2 \\rho _ n ^ 2 \\kappa } } } { { { { \\sigma ^ 2 } L _ { \\rm R } \\left ( { \\kappa + 1 } \\right ) } } } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "4100.png", "formula": "\\begin{align*} y ^ r - P ( x ) = 0 , \\end{align*}"}
+{"id": "5972.png", "formula": "\\begin{align*} 1 & = \\sigma _ { - 1 } ( N ^ \\omega _ { \\partial M } ) ( t , \\xi ' ) \\sigma _ 1 ( \\Lambda _ { g , F } ^ \\omega ) ( t , \\xi ' ) + \\sigma _ { - 1 } ( N ^ \\omega _ { \\partial M } ) ( t , \\xi ' ) \\sigma _ 0 ( \\Lambda _ { g , F } ^ \\omega ) ( t , \\xi ' ) , \\\\ & + \\sigma _ { - 2 } ( N ^ \\omega _ { \\partial M } ) ( t , \\xi ' ) \\sigma _ 1 ( \\Lambda _ { g , F } ^ \\omega ) ( t , \\xi ' ) + \\nabla _ { \\xi ' } \\sigma _ { - 1 } ( N ^ \\omega _ { \\partial M } ) \\cdot D _ { t ' } \\sigma _ 1 ( \\Lambda _ { g , F } ^ \\omega ) + S ^ { - 2 } _ { 1 , 0 } . \\end{align*}"}
+{"id": "5628.png", "formula": "\\begin{align*} D _ 4 = \\Bigl ( x _ 0 x _ 1 x _ 3 ^ 2 + B x _ 3 + C = 0 \\Bigr ) . \\end{align*}"}
+{"id": "4936.png", "formula": "\\begin{align*} g _ k = 2 ^ { - 1 / 2 } \\cdot e ^ { \\frac { \\pi } { 3 \\cdot 2 ^ { k + 2 } } - \\frac { 2 ^ { k - 1 } } { \\pi } \\cdot L _ k } . \\end{align*}"}
+{"id": "7389.png", "formula": "\\begin{align*} \\| V _ { n } \\| ^ { 2 } _ { L ^ { \\infty } _ { t } L ^ { 2 } _ { x } } & + C _ { 3 } \\left \\| \\partial _ { x } V _ { n } \\right \\| ^ { 2 } _ { L ^ { 2 } _ { t , x } } \\\\ [ 1 e x ] & \\le T \\| V _ { n } ( 0 ) \\| _ { L ^ { 2 } _ { x } } ^ { 2 } \\exp \\left ( 2 C _ { 1 } T + 2 C _ { 2 } \\| V _ { n } \\| ^ { 2 } _ { L ^ { 2 } _ { t , x } } \\right ) \\left \\{ C _ { 1 } T + C _ { 2 } \\| V _ { n } ( 0 ) \\| _ { L ^ { 2 } _ { x } } ^ { 2 } \\right ) = : \\mathcal { V } _ { 1 } \\end{align*}"}
+{"id": "8334.png", "formula": "\\begin{align*} \\mathbf { s } ^ * \\ , \\cdot \\ , _ { \\alpha } \\mathbf { P } _ { M } ^ I ( 0 ) = \\mathbf { s } _ { \\mathbf { S B } } ^ * ( 0 ) \\mathbf { P } _ { S M } \\mathbf { M } \\end{align*}"}
+{"id": "4520.png", "formula": "\\begin{align*} \\tilde { e } ''' _ k : = \\left [ \\begin{array} { c } 0 \\\\ 0 \\\\ \\delta H ^ + _ k \\cdot ( S _ { \\theta _ k } H ^ + _ k - H ^ + _ { k + \\frac { 1 } { 2 } } ) - \\delta H ^ - _ k \\cdot ( S _ { \\theta _ k } H ^ - _ k - H ^ - _ { k + \\frac { 1 } { 2 } } ) \\end{array} \\right ] \\ , . \\end{align*}"}
+{"id": "5677.png", "formula": "\\begin{align*} \\| ( z - T ) ^ { - 1 } e _ 0 \\| ^ { p } _ { p } & = \\| \\sum \\limits _ { n = 0 } ^ { \\infty } \\frac { T ^ n } { z ^ { n + 1 } } e _ 0 \\| ^ { p } _ { p } = \\| \\frac { e _ 0 } { z } + \\sum \\limits _ { n = 1 } ^ { \\infty } \\frac { \\prod \\limits _ { j = 0 } ^ { n - 1 } w _ j } { z ^ { n + 1 } } e _ n \\| ^ { p } _ { p } \\\\ & = \\frac { 1 } { | z | ^ p } + \\frac { 1 } { | z | ^ p } \\sum \\limits _ { n = 1 } ^ { \\infty } \\frac { \\big ( \\prod \\limits _ { j = 0 } ^ { n - 1 } w _ j \\big ) ^ p } { | z | ^ { p n } } . \\end{align*}"}
+{"id": "918.png", "formula": "\\begin{align*} \\phi _ 2 = \\frac n m \\eta _ 2 + \\frac { 1 } { m x ^ k } ( \\phi _ { 1 x } - \\eta _ { 1 x } ) , \\end{align*}"}
+{"id": "3481.png", "formula": "\\begin{align*} \\omega _ { C Y , t } = \\omega _ t + | \\log | t | | d d ^ c \\psi _ t . \\end{align*}"}
+{"id": "1140.png", "formula": "\\begin{align*} \\psi [ ( h _ 1 , \\ldots , h _ { m + 1 } ) , ( k _ 1 , \\ldots , k _ { n + 1 } ) ] _ c = ~ & \\sum _ { i = 1 } ^ { m + n + 1 } \\sum _ { q + r = i + 1 } [ h _ q , k _ r ] \\\\ = ~ & [ h _ 1 + \\cdots + h _ { m + 1 } , ~ k _ 1 + \\cdots + k _ { n + 1 } ] \\\\ = ~ & [ \\psi ( h _ 1 , \\ldots , h _ { m + 1 } ) , \\psi ( k _ 1 , \\ldots , k _ { n + 1 } ) ] , \\end{align*}"}
+{"id": "211.png", "formula": "\\begin{align*} { Q _ 2 } ( M , P _ i , \\tau ) = & \\left \\{ e ^ { \\frac { 1 } { 2 4 } E _ 2 ( \\tau ) A _ 3 } \\widehat { A } ( T M , \\nabla ^ { T M } ) { \\rm c h } \\left [ \\bigotimes _ { n = 1 } ^ { \\infty } S _ { q ^ n } ( \\widetilde { T _ C M } ) \\otimes \\bigotimes _ { m = 1 } ^ { \\infty } \\wedge _ { - q ^ { m - \\frac { 1 } { 2 } } } ( \\widetilde { T _ C M } ) \\right ] \\right . \\\\ & \\left . \\cdot \\varphi ( \\tau ) ^ { 8 } { \\rm c h } ( \\mathcal { V } _ i ) \\right \\} ^ { ( 1 2 ) } . \\end{align*}"}
+{"id": "6734.png", "formula": "\\begin{align*} u = 1 - \\frac { \\mathfrak { c } _ { p } } { a } r ^ { - a } \\left ( 1 + O _ { 2 } ( r ^ { - \\tilde { \\tau } } ) \\right ) , \\hbox { a s } r \\to \\infty . \\end{align*}"}
+{"id": "5394.png", "formula": "\\begin{align*} v ( z , w ; t ) : = \\frac { \\partial \\phi ( z , w ; t ) } { \\partial w } , ( z , w ) \\in \\C \\times \\C ^ { \\times } , \\ , t \\geq 0 . \\end{align*}"}
+{"id": "2158.png", "formula": "\\begin{align*} m : = \\frac { \\mu } { \\alpha } = \\frac 1 { \\alpha } ( w - \\beta - \\sqrt { ( w - \\beta ) ^ 2 - \\alpha ^ 2 } ) , \\end{align*}"}
+{"id": "1062.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\hat { w } _ \\lambda ( \\hat { v } _ \\lambda ^ * ) \\leq \\hat { w } _ \\lambda ( \\hat { u } _ \\lambda ^ * ) < \\inf \\left \\{ \\hat { w } ( u ) : \\ : \\| u - \\hat { u } _ \\lambda ^ * \\| = \\rho \\right \\} = \\hat { m } _ \\lambda , & \\\\ \\| \\hat { v } _ \\lambda ^ * - \\hat { u } _ \\lambda ^ * \\| > \\rho . & \\end{array} \\right . \\end{align*}"}
+{"id": "5557.png", "formula": "\\begin{align*} \\psi ( \\zeta , \\xi , \\xi _ 1 ) = \\frac { ( \\zeta - \\xi ) ( \\xi _ 1 - \\overline { \\xi } ) + \\i ( \\zeta - \\overline { \\xi } ) ( \\xi _ 1 - \\xi ) } { ( \\zeta - \\overline { \\xi } ) ( \\xi _ 1 - \\xi ) + \\i ( \\zeta - \\xi ) ( \\xi _ 1 - \\overline { \\xi } ) } , \\phi ( w ) = \\frac { w - \\i } { \\i w - 1 } . \\end{align*}"}
+{"id": "7975.png", "formula": "\\begin{align*} \\gamma _ d { \\beta _ \\tau } _ d = { \\beta _ \\lambda } _ d \\varphi _ d = { \\beta ' _ \\lambda } _ d \\varphi _ d = \\gamma _ d { \\beta ' _ \\tau } _ d . \\end{align*}"}
+{"id": "3930.png", "formula": "\\begin{align*} \\forall \\omega \\lambda ( f ^ { - 1 } ( \\omega ) \\triangle g ^ { - 1 } ( \\omega ) ) = 0 , \\end{align*}"}
+{"id": "2212.png", "formula": "\\begin{align*} \\left \\Vert \\sum _ { k = 1 } ^ { \\infty } \\alpha _ { k } c _ { k } \\right \\Vert _ { \\mathrm { o p } } \\leq \\sqrt { \\sum _ { k = 1 } ^ { \\infty } \\left | \\alpha _ { k } \\right | ^ { 2 } } \\end{align*}"}
+{"id": "7075.png", "formula": "\\begin{align*} n _ 1 ^ 2 n _ 2 \\ll \\ , { \\max } \\left \\{ \\frac { ( p _ 1 q K ) ^ 3 r } { N } , N ^ { 1 / 2 } ( p _ 1 K ) ^ { 3 / 2 } r \\ , u ^ 3 \\right \\} : = { N _ 0 } . \\end{align*}"}
+{"id": "722.png", "formula": "\\begin{align*} \\mathbf U ( t ) = \\mathbf V ( t ) \\end{align*}"}
+{"id": "1575.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } v ^ 3 g ^ T ( x , v ) \\mathrm { d } v & = \\int _ 0 ^ { \\infty } \\int _ 0 ^ 1 v ^ 3 \\frac { e ^ { - ( 1 - y ) / \\kappa | v | } } { \\kappa | v | ( 1 - e ^ { - 1 / \\kappa | v | } ) } \\rho _ g ( x + y ) \\left ( \\alpha \\mathcal { M } _ { T ( y + x ) } + ( 1 - \\alpha ) \\mathcal { M } _ { \\tau ( y + x ) } \\right ) \\mathrm { d } y \\mathrm { d } v . \\end{align*}"}
+{"id": "4842.png", "formula": "\\begin{align*} \\check { R } ( 0 ) = \\sum _ { i } ^ { n } \\left [ \\prod _ { j = 1 } ^ { i - 1 } \\left ( \\frac { \\lambda _ j } { \\lambda _ { j + 1 } } f _ j ( 0 ) + 1 \\right ) \\prod _ { k = i } ^ { n - 1 } \\left ( f _ k ( 0 ) + \\frac { \\lambda _ k } { \\lambda _ { k + 1 } } \\right ) \\lambda _ n \\right ] P _ i = C \\cdot I \\end{align*}"}
+{"id": "2007.png", "formula": "\\begin{align*} \\Omega : = \\{ \\ , ( x ' , x _ n ) \\in \\mathbb { R } ^ { n - 1 } \\times \\mathbb { R } \\ , | \\ , x _ n > \\phi ( x ' ) \\ , \\} \\end{align*}"}
+{"id": "1994.png", "formula": "\\begin{align*} \\varphi _ j ( \\xi ) : = \\varphi ( 2 ^ { - j } \\xi ) \\psi _ j ( \\xi ) : = \\varphi _ { j } ( \\xi / 2 ) - \\varphi _ { j } ( \\xi ) \\end{align*}"}
+{"id": "6795.png", "formula": "\\begin{align*} & 1 = \\lim _ { z \\rightarrow - \\infty } \\underline { u } ( z ) \\leq \\liminf _ { z \\rightarrow - \\infty } u ( z ) \\leq \\limsup _ { z \\rightarrow - \\infty } u ( z ) \\leq \\lim _ { z \\rightarrow - \\infty } \\overline { u } ( z ) = 1 , \\\\ [ 0 . 2 c m ] & 0 = \\lim _ { z \\rightarrow - \\infty } \\underline { v } ( z ) \\leq \\liminf _ { z \\rightarrow - \\infty } v ( z ) \\leq \\limsup _ { z \\rightarrow - \\infty } v ( z ) \\leq \\lim _ { z \\rightarrow - \\infty } \\overline { v } ( z ) = 0 , \\end{align*}"}
+{"id": "7710.png", "formula": "\\begin{align*} ( d \\Psi ) _ { x _ 0 } = \\left ( \\begin{matrix} \\left ( \\frac { \\partial \\psi ^ i } { \\partial x ^ j } ( x _ 0 ) \\right ) & 0 & 0 \\\\ 0 & \\left ( f ^ \\mu _ \\nu ( x _ 0 ) \\right ) ^ t & 0 \\\\ 0 & 0 & \\left ( h ^ I _ J ( x _ 0 ) \\right ) ^ t \\end{matrix} \\right ) . \\end{align*}"}
+{"id": "579.png", "formula": "\\begin{align*} \\Delta _ 1 ( t ) & = i \\int _ { S \\wedge \\mu } ^ { t \\wedge \\mu } \\mathbf S ( t - s ) \\left [ \\mathbf N ( \\mathbf U ( s ) ) - \\mathbf N ( \\mathbf V ( s ) ) \\right ] \\ , d s , \\\\ \\Delta _ 2 ( t ) & = i \\int _ S ^ t \\mathbf S ( t - s ) \\left [ \\mathbf M ( \\mathbf U ( s ) ) - \\mathbf M ( \\mathbf V ( s ) ) \\right ] \\ , d W ( s ) . \\end{align*}"}
+{"id": "1626.png", "formula": "\\begin{align*} g ( x _ 0 , x ) = g ( r ) = \\sum _ { k = r } ^ { \\infty } \\frac { 1 } { \\partial B ( k ) } , \\end{align*}"}
+{"id": "5184.png", "formula": "\\begin{align*} F ( f ) ( V \\otimes W ) ^ * = [ F ( f ) ( V ) \\otimes F ( f ) ( W ) ] a _ 2 ( f ) ( V \\otimes W ) ^ { - 1 } . \\end{align*}"}
+{"id": "8342.png", "formula": "\\begin{align*} L _ t [ u ] ( x ) : = a ( x , t ) u ' ( x ) + \\tfrac 1 2 b ^ 2 ( x , t ) u '' ( x ) , x \\in [ x _ 1 , x _ 2 ] , \\ t \\geqslant 0 , \\end{align*}"}
+{"id": "8376.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| x - x _ n \\| = 0 . \\end{align*}"}
+{"id": "4051.png", "formula": "\\begin{align*} \\begin{gathered} \\sum _ { j = 0 } ^ { ( k - 1 ) / 2 } ( - 1 ) ^ { j } U _ { 1 , 0 } ( 2 j a + t ) = \\sum _ { j = 1 } ^ { ( k - 1 ) / 2 } ( - 1 ) ^ { j } U _ { 1 , 0 } ( 2 j a - t ) \\ ; ( 0 , a ) , \\\\ \\sum _ { j = 0 } ^ { ( k - 1 ) / 2 } ( - 1 ) ^ { j } V _ { 1 , 0 } ( 2 j a + t ) + \\sum _ { j = 1 } ^ { ( k - 1 ) / 2 } ( - 1 ) ^ { j } V _ { 1 , 0 } ( 2 j a - t ) = 0 \\ ; ( 0 , a ) . \\end{gathered} \\end{align*}"}
+{"id": "2615.png", "formula": "\\begin{align*} 1 - F _ 0 ( t ) + \\mu t - \\mu \\int _ 0 ^ t ( t - s ) d F ( s ) = 1 \\ , , \\end{align*}"}
+{"id": "3358.png", "formula": "\\begin{align*} D _ \\zeta ( x ) = \\prod _ { t = 1 } ^ { m - 1 } ( 1 - \\zeta ^ t x ) ^ t \\end{align*}"}
+{"id": "44.png", "formula": "\\begin{align*} \\mathcal Q _ 2 ^ { \\rm { e x } } ( z ) & = \\rho _ z \\sum _ { k \\neq 0 } \\big ( \\widehat { g \\omega } ( k ) + \\widehat { g \\omega } ( 0 ) \\big ) a _ k ^ \\dagger a _ k . \\end{align*}"}
+{"id": "100.png", "formula": "\\begin{align*} \\varepsilon = \\frac { K _ { \\ell } ^ 4 } { K _ L ^ 2 } \\ll 1 , \\end{align*}"}
+{"id": "6812.png", "formula": "\\begin{align*} \\mathcal { L } ' ( \\chi ( z ) ) = & \\left [ \\left ( u - u ^ * \\right ) ( p ( u ) - p ( u ^ * ) ) - \\left ( v - v ^ * \\right ) ^ 2 \\right ] / q ( u ) - w ^ 2 \\partial _ { u u } H - d y ^ 2 \\partial _ { v v } H + G ( u , v ) , \\end{align*}"}
+{"id": "5151.png", "formula": "\\begin{align*} \\lim _ { D \\to 0 } { \\rm A o I } ( S _ { \\mathrm { z } } , Q _ { \\mathrm { u n i } } , F ^ { * } ) - { \\rm A o I } ( S ^ * , Q ^ * , F ^ * ) = 0 . \\end{align*}"}
+{"id": "5212.png", "formula": "\\begin{align*} D \\cdot c _ { j , i } & = D _ { j , i } D _ { j - 1 , i } ^ { - 1 } c _ { j , i } = \\chi _ { j , i , i } ( D ) c _ { j , i } ; \\\\ D \\cdot a _ { j , k , l , m } & = D _ { j , k } D _ { j - 1 , l } ^ { - 1 } a _ { j , k , l , m } = \\chi _ { j , k , l } ( D ) a _ { j , k , l , m } , \\end{align*}"}
+{"id": "4250.png", "formula": "\\begin{align*} \\displaystyle F ( x , t ) = \\int _ 0 ^ t f ( x , \\tau ) d \\tau \\to \\pm \\infty \\textrm { a s } | t | \\to + \\infty , \\end{align*}"}
+{"id": "6332.png", "formula": "\\begin{align*} \\partial _ t | u _ \\mu | ^ 2 = \\partial _ x ( g _ { [ < \\mu ] } P ( u _ \\mu ) + 2 \\Im ( N _ \\mu ( u ) \\cdot \\bar u _ \\mu ) , \\end{align*}"}
+{"id": "6994.png", "formula": "\\begin{align*} & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ d \\o _ i = \\sum _ { j = 1 } ^ 4 \\o _ { i j } \\wedge \\o _ j , ~ ~ ~ 1 \\leq i \\leq 4 ; \\\\ & d \\o _ { i j } - \\sum _ { k = 1 } ^ { 4 } \\o _ { i k } \\wedge \\o _ { k j } + \\sum _ { 1 \\leq k < l \\leq 4 } K _ { i j k l } \\o _ k \\wedge \\o _ l = 0 , ~ ~ ~ 1 \\leq i , j \\leq 4 , \\end{align*}"}
+{"id": "1702.png", "formula": "\\begin{align*} \\frac { 1 } { ( 2 \\pi ) ^ { n } } \\int _ { ^ { ( n ) } _ { \\texttt { b } } } f ( X _ 1 , \\ldots X _ n ) & \\frac { \\sqrt { \\rho _ { \\texttt { b } } ( X _ 1 , \\ldots , X _ n ) } } { O _ { \\texttt { b } } ( X _ 1 , \\ldots , X _ n ; q , q _ 0 ) } X _ 1 \\cdots X _ n \\\\ & = \\sum _ { \\lambda \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { b } } } f \\bigl ( \\boldsymbol { X } ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } \\bigr ) \\hat { \\Delta } ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } , \\end{align*}"}
+{"id": "7867.png", "formula": "\\begin{align*} \\left ( \\frac { n } { 3 } \\right ) ^ 4 \\sum _ p \\sum _ q R _ p ^ 2 R _ q ^ 2 \\leq \\left ( \\frac { n } { 3 } \\right ) ^ 4 \\sum _ p R _ p ^ 2 \\sum _ q R _ q ^ 2 = \\left ( \\frac { n } { 3 } \\right ) ^ 4 \\left ( \\sum _ { p } R _ p ^ 2 \\right ) ^ 2 , \\end{align*}"}
+{"id": "2975.png", "formula": "\\begin{align*} E ^ n & \\coloneqq E ^ 1 \\times _ { s , r } \\cdots \\times _ { s , r } E ^ 1 = \\Big \\{ e _ 1 e _ 2 \\cdots e _ n \\in \\prod ^ n E ^ 1 \\mid s ( e _ i ) = r ( e _ { i + 1 } ) \\Big \\} . \\end{align*}"}
+{"id": "8699.png", "formula": "\\begin{align*} \\tilde { f } ( x _ 1 , . . . , x _ n ) = \\lambda ~ f \\Big ( \\sum _ { j = 1 } ^ n a _ { 1 , j } x _ j + b _ 1 , . . . , \\sum _ { j = 1 } ^ n a _ { n , j } x _ j + b _ n \\Big ) + \\sum _ { j = 1 } ^ n c _ j x _ j + p , \\end{align*}"}
+{"id": "3043.png", "formula": "\\begin{align*} \\C : = \\frac { \\partial ^ 2 W } { \\partial M ^ 2 } ( \\mathrm { I d } ) \\ , . \\end{align*}"}
+{"id": "3103.png", "formula": "\\begin{align*} \\mathbf { G } = \\sqrt { \\frac { \\kappa } { \\kappa + 1 } } \\mathbf { G } _ { } + \\sqrt { \\frac { 1 } { \\kappa + 1 } } \\mathbf { G } _ { } \\end{align*}"}
+{"id": "8620.png", "formula": "\\begin{align*} R _ 2 = \\frac { 1 } { \\mu } \\nabla _ X \\cdot ( b ( \\mathrm { F } _ 4 - 1 ) \\nabla _ X G ) . \\end{align*}"}
+{"id": "2544.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s _ p u + V ( x ) | u | ^ { p - 2 } u = \\frac { 1 } { p _ { r ; s } ^ { \\uparrow * } } ( K \\ast g ( u ) ) g ' ( u ) + \\varepsilon _ W W ( x ) f ' ( u ) \\quad \\mathbb { R } ^ N , \\end{align*}"}
+{"id": "3774.png", "formula": "\\begin{align*} c ( x _ 0 , x _ 1 ) : = \\begin{cases} - \\log ( \\cos ^ 2 ( \\d ( x _ 0 , x _ 1 ) ) ) & \\d ( x _ 0 , x _ 1 ) < \\pi / 2 \\\\ + \\infty & . \\end{cases} \\end{align*}"}
+{"id": "4658.png", "formula": "\\begin{align*} k _ v \\bullet _ v \\bigl ( \\mathbf { X } _ { v | s } \\otimes \\mathbf { Y } _ { v | t } \\bigr ) : = \\Bigl ( \\bigotimes _ { e \\in \\mathcal { S } ( v | s ) } \\textup { A d } ( k _ v ) X _ e \\Bigr ) \\otimes \\Bigl ( \\bigotimes _ { e ^ \\prime \\in \\mathcal { S } ( v | t ) } \\textup { A d } ( k _ v ) Y _ { e ^ \\prime } \\Bigr ) . \\end{align*}"}
+{"id": "3957.png", "formula": "\\begin{align*} f ( u ' , v ' ) - 1 = f ( u ' , v ' ) - f ( u , v ' ) = \\nabla _ x f ( u + t ( u ' - u ) , v ' ) ( u ' - u ) \\end{align*}"}
+{"id": "1278.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } \\{ \\| \\nabla u _ n \\| _ { L ^ 2 } ^ 2 - \\sum _ { j = 1 } ^ J \\| \\nabla \\phi ^ j \\| _ { L ^ 2 } ^ 2 - \\| \\nabla w _ n ^ J \\| _ { L ^ 2 } ^ 2 \\} = 0 \\\\ & \\lim _ { n \\to \\infty } \\{ P ( u _ n ) - \\sum _ { j = 1 } ^ J P ( g _ n ^ j [ e ^ { i t _ n ^ j \\Delta } \\phi ^ j ] ) - P ( w _ n ^ J ) \\} = 0 . \\end{align*}"}
+{"id": "2083.png", "formula": "\\begin{align*} \\begin{aligned} & ~ { } G : = - \\left ( \\partial _ { t } ^ 2 ( \\ln \\alpha ) - \\partial _ { x } ^ 2 ( \\ln \\alpha ) \\right ) - \\dfrac { 1 } { 2 \\alpha ^ 2 } ( ( \\partial _ t \\alpha ) ^ 2 - ( \\partial _ x \\alpha ) ^ 2 ) \\\\ & ~ { } - \\dfrac { 1 } { 2 } ( ( \\partial _ t \\Lambda ) ^ 2 - ( \\partial _ x \\Lambda ) ^ 2 ) - 2 \\sinh ^ 2 \\Lambda ( ( \\partial _ t \\phi ) ^ 2 - ( \\partial _ x \\phi ) ^ 2 ) . \\end{aligned} \\end{align*}"}
+{"id": "920.png", "formula": "\\begin{align*} \\eta _ 1 = & \\frac 1 8 ( - ( 1 - \\alpha ) t ^ { - 1 - \\alpha } \\tau + ( 1 - \\alpha ) t ^ { - \\alpha } \\tau _ t - t ^ { 1 - \\alpha } \\tau _ { t t } ) x ^ 2 - \\frac 1 2 t ^ { 1 - \\alpha } \\sigma _ { 1 t } x \\\\ & + \\frac { k + 1 } { 4 } \\bigg ( \\tau _ t - \\frac { 1 - \\alpha } { t } \\tau \\bigg ) + \\frac { k - c } { 2 x } \\sigma _ 1 + \\frac { 1 } { m x ^ k } \\eta _ { 2 x } + \\frac 1 2 \\sigma _ 2 . \\end{align*}"}
+{"id": "4418.png", "formula": "\\begin{align*} ( \\mathcal { B } _ 1 { \\mathbf V } , { \\mathbf V } ) | _ { x _ 1 = 0 } = 2 [ \\dot { q } ( \\dot { u } _ N - \\hat { \\lambda } \\dot { H } _ N ) ] , \\end{align*}"}
+{"id": "4795.png", "formula": "\\begin{align*} \\varphi _ t ( z ) : = ( z _ 0 - t , z _ 1 - D _ \\gamma ^ { - 1 } t , z _ 2 - D _ \\gamma ^ { - 2 } t , \\dots ) , \\end{align*}"}
+{"id": "4171.png", "formula": "\\begin{align*} y _ i = & \\frac { \\sqrt { w _ i ( 1 + \\gamma _ i ) | h _ { i , i } | ^ 2 p _ i } } { \\sum _ { j = 1 } ^ { L } | h _ { i , j } | ^ 2 p _ j + \\sigma ^ 2 _ j } , \\\\ \\tilde { y } _ k = & \\frac { \\sqrt { \\sum _ { j \\ne k } | \\tilde { h } _ { k , j } | ^ 2 p _ j + \\tilde { \\sigma } ^ 2 _ k } } { w _ k ( 1 - \\tilde { \\gamma } _ k ) | \\tilde { h } _ { k , k } | ^ 2 p _ k + \\varepsilon } . \\end{align*}"}
+{"id": "5878.png", "formula": "\\begin{align*} E _ { s , V } L ( \\epsilon _ { 1 } ) \\geq P ( W \\geq V ) E _ { s , V } L _ { V } = [ 1 + o ( 1 ) ] E _ { s , V } L _ { V } . \\end{align*}"}
+{"id": "7956.png", "formula": "\\begin{align*} \\ell _ h = \\frac { 2 ^ \\varepsilon n } { \\gcd ( r , n ) } \\ell _ s = \\frac { 2 n } { \\gcd ( a , n ) } , \\end{align*}"}
+{"id": "2871.png", "formula": "\\begin{align*} \\xi _ \\mu ( 0 ) : = \\frac { 2 \\pi } { h + c } ( \\hat \\rho + \\mu ) ( \\mu \\in \\hat P _ c ) , \\end{align*}"}
+{"id": "3124.png", "formula": "\\begin{align*} \\mathbb { E } [ \\mathbf { G } _ { } \\mathbf { G } ^ H _ { } ] = M \\mathbf { I } _ N . \\end{align*}"}
+{"id": "703.png", "formula": "\\begin{align*} f _ R ( t ) = \\norm { \\mathbf u ^ R } _ { \\widetilde { \\mathbf X } ^ { \\mathbf s , b } ( 0 , t ) } ^ 2 , f _ R ( 0 ) = 0 . \\end{align*}"}
+{"id": "2992.png", "formula": "\\begin{align*} r _ I ( k , z ) = \\psi ( \\gamma ^ k z ^ { q _ b } ) = \\gamma ^ { k a } z ^ { q _ b a } = \\gamma ^ { k a } z ^ { m / \\gcd ( n , b ) } s _ I ( k , z ) = z ^ { q _ n } = z ^ { n / \\gcd ( n , b ) } . \\end{align*}"}
+{"id": "8303.png", "formula": "\\begin{align*} & f \\circ g = ( f _ 1 ( g _ 1 , \\cdots , g _ n ) , \\cdots , f _ n ( g _ 1 , \\cdots , g _ n ) ) \\\\ & g \\circ f = ( g _ 1 ( f _ 1 , \\cdots , f _ n ) , \\cdots , g _ n ( f _ 1 , \\cdots , f _ n ) ) . \\end{align*}"}
+{"id": "5402.png", "formula": "\\begin{align*} \\lim _ { w \\to 0 } \\int _ { B _ { | w | } ( \\lambda _ 1 ) } \\phi ( z ) r ^ { ( 1 ) } _ { \\ell } ( z , w ) m ( d z ) = 0 , \\ell = 1 , 2 . \\end{align*}"}
+{"id": "2680.png", "formula": "\\begin{align*} e ^ { ( 1 + \\frac { \\alpha } { 2 \\beta } ) \\sup _ { X } \\varphi _ t ( x ) } \\ = \\ e ^ { ( 1 + \\frac { \\alpha } { 2 \\beta } ) \\varphi _ t ( x _ { 1 } ) } \\ \\leq \\ \\frac { ( t \\omega _ { 0 } - _ { \\omega _ { 0 } } ) ) ^ { n } } { \\omega _ { 0 } ^ { n } } \\ \\leq \\ C , \\end{align*}"}
+{"id": "600.png", "formula": "\\begin{align*} \\norm { \\psi _ + ^ R ( t ) } _ { L ^ 2 } ^ 2 + \\norm { \\psi _ - ^ R ( t ) } _ { L ^ 2 } ^ 2 = \\norm { \\psi _ 0 } _ { L ^ 2 } ^ 2 \\end{align*}"}
+{"id": "6307.png", "formula": "\\begin{align*} \\| w _ \\lambda \\| _ { X _ \\lambda } ^ 2 = \\| w _ \\lambda ( T ) \\| _ { L ^ 2 } ^ 2 + \\sup _ { x _ 0 \\in \\R } \\| w ^ { x _ 0 } _ \\lambda e _ \\lambda \\| _ { L ^ 1 } , \\end{align*}"}
+{"id": "3046.png", "formula": "\\begin{align*} \\psi ^ { \\C } ( \\zeta ) = \\lim _ { \\frac { r _ 2 } { r _ 1 } \\to + \\infty } \\psi _ { r _ 1 , r _ 2 } ^ { \\C } ( \\zeta ) \\ , , \\end{align*}"}
+{"id": "5635.png", "formula": "\\begin{align*} \\Gamma = \\left \\lbrace \\begin{array} { l } Q = y - x _ 0 L = 0 , \\\\ F = x _ 0 x _ 1 L ^ 2 + B L + C = 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "2886.png", "formula": "\\begin{align*} _ j ( \\boldsymbol { \\xi } ) & = \\frac { 1 - t _ { j } e ^ { - i \\langle \\xi , \\alpha _ j \\rangle } } { 1 - e ^ { - i \\langle \\xi , \\alpha _ j \\rangle } } , & _ j ( \\boldsymbol { \\xi } ) & = t _ j - _ j ( \\boldsymbol { \\xi } ) = _ j ( - \\boldsymbol { \\xi } ) - 1 . \\end{align*}"}
+{"id": "5802.png", "formula": "\\begin{align*} w _ 0 = \\prod \\limits _ { i = 1 } ^ n s _ { \\varepsilon _ i } . \\end{align*}"}
+{"id": "4283.png", "formula": "\\begin{align*} ( T ^ { ( 1 ) } _ { n \\times p } ) ' e _ i = \\sum _ { j = 0 } ^ { p - 1 } a ^ { ( 1 ) } _ { - j } \\chi _ { [ 1 , p ] } ( i + j ) e _ { i + j } + \\sum _ { j = 1 } ^ { n - 1 } a ^ { ( 1 ) } _ { j } \\chi _ { [ 1 , p ] } ( i - j ) e _ { i - j } = \\sum _ { j = - ( p - 1 ) } ^ { n - 1 } a ^ { ( 1 ) } _ { j } \\chi _ { [ 1 , p ] } ( i - j ) e _ { i - j } . \\end{align*}"}
+{"id": "3858.png", "formula": "\\begin{align*} \\Gamma _ { 1 1 } ^ 3 & = \\lambda _ 1 e ^ { - 2 \\lambda _ 1 z } \\\\ \\Gamma _ { 2 2 } ^ 3 & = \\lambda _ 2 e ^ { - 2 \\lambda _ 2 z } \\\\ \\Gamma _ { 1 3 } ^ 1 & = \\Gamma _ { 3 1 } ^ 1 = - \\lambda _ 1 \\\\ \\Gamma _ { 2 3 } ^ 2 & = \\Gamma _ { 3 2 } ^ 2 = - \\lambda _ 2 \\\\ \\Gamma _ { i j } ^ k & = 0 \\hbox { e n t o d o s l o s o t r o s c a s o s } \\end{align*}"}
+{"id": "8875.png", "formula": "\\begin{align*} B T & = B S ^ 1 \\times \\cdots \\times B S ^ 1 \\\\ & = \\C P ^ { \\infty } \\times \\cdots \\times \\C P ^ { \\infty } . \\end{align*}"}
+{"id": "7576.png", "formula": "\\begin{align*} | D ^ { ( k ) } _ t - D ^ { ( \\ell ) } _ s | ^ 2 & = | D ^ { ( k ) } _ t | ^ 2 + | D ^ { ( \\ell ) } _ s | ^ 2 - 2 | D ^ { ( k ) } _ t | \\cdot | D ^ { ( \\ell ) } _ s | \\cos \\theta \\\\ & = \\left ( | D ^ { ( k ) } _ t | - | D ^ { ( \\ell ) } _ s | \\right ) ^ 2 + 2 | D ^ { ( k ) } _ t | \\cdot | D ^ { ( \\ell ) } _ s | \\left [ 1 - \\cos \\theta \\right ] . \\end{align*}"}
+{"id": "1759.png", "formula": "\\begin{gather*} D = \\{ s \\in \\mathbb { C } \\mid 1 / 2 < \\sigma < 1 \\} , \\\\ \\mathcal { A } = \\{ 0 < \\alpha < 1 \\mid \\} , \\end{gather*}"}
+{"id": "6811.png", "formula": "\\begin{align*} & \\left ( u - u ^ * \\right ) ( p ( u ) - v ) = \\left ( u - u ^ * \\right ) ( p ( u ) - p ( u ^ * ) ) + \\left ( u - u ^ * \\right ) ( v ^ * - v ) , \\\\ [ 0 . 2 c m ] & \\left ( v - v ^ * \\right ) \\left ( q ( u ) - v \\right ) = \\left ( v - v ^ * \\right ) \\left ( q ( u ) - q ( u ^ * ) \\right ) - \\left ( v - v ^ * \\right ) ^ 2 . \\end{align*}"}
+{"id": "5221.png", "formula": "\\begin{align*} 2 ( n - 1 ) ( n - 2 ) - n ( n - 3 ) \\mu = 2 ( n - 1 ) \\lambda . \\end{align*}"}
+{"id": "7585.png", "formula": "\\begin{align*} \\begin{aligned} | D _ t ^ { ( 0 ) } - D _ s ^ { ( \\ell ( \\theta ) ) } | ^ 2 & = \\beta ^ { \\frac { 1 } { 2 } } N ^ { \\frac { 1 } { 2 } } \\left ( t ^ { \\frac { 3 } { 4 } } - s ^ { \\frac { 3 } { 4 } } \\cos \\theta \\right ) ^ 2 \\\\ & + \\beta ^ { \\frac { 1 } { 2 } } N ^ { \\frac { 1 } { 2 } } \\left ( s ^ { \\frac { 3 } { 4 } } \\sin \\theta \\right ) ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "8513.png", "formula": "\\begin{align*} \\{ z = \\bar { z } \\} \\backslash \\{ ( \\bar { z } , \\tau ) \\} \\subset \\left ( ( 0 , \\tau ) + E _ { 2 } \\right ) ^ { ( 0 ) } . \\end{align*}"}
+{"id": "3380.png", "formula": "\\begin{align*} \\Upsilon _ k = \\frac { 1 } { n + 1 } \\left ( \\hat { \\alpha } ^ i _ { i k } - \\alpha ^ i _ { i k } \\right ) . \\end{align*}"}
+{"id": "4408.png", "formula": "\\begin{align*} \\begin{cases} \\mathbb { L } ' _ e ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) \\dot { { \\mathbf U } ^ \\natural } = \\mathbf { F } & \\Omega _ T , \\\\ \\mathbb { B } ' _ e ( \\hat { { \\mathbf U } } , \\hat { \\varphi } ) ( \\dot { { \\mathbf U } ^ \\natural } , \\varphi ) = 0 & \\Gamma _ T , \\\\ ( \\dot { { \\mathbf U } ^ \\natural } , \\varphi ) = 0 & t < 0 , \\end{cases} \\end{align*}"}
+{"id": "3438.png", "formula": "\\begin{align*} s _ i = \\sum F _ 0 ^ { l _ 0 } \\ldots F _ m ^ { l _ m } \\sum s ^ { ( i ) } _ { l _ 0 \\ldots l _ m , a } t ^ a . \\end{align*}"}
+{"id": "1680.png", "formula": "\\begin{align*} ^ { ( n ) } _ { \\texttt { a } } & : = \\left \\{ \\bigl ( X _ 1 ( \\boldsymbol { \\xi } ) , \\ldots , X _ { n - 1 } ( \\boldsymbol { \\xi } ) \\bigr ) \\mid \\boldsymbol { \\xi } \\in \\mathbb { A } ^ { ( n ) } _ { \\texttt { a } } \\right \\} , \\\\ \\boldsymbol { X } ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } & : = \\bigl ( X _ 1 ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } ) , \\ldots , X _ { n - 1 } ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } ) \\bigr ) . \\end{align*}"}
+{"id": "654.png", "formula": "\\begin{align*} u _ n ( s , \\omega ) = u ( 0 , \\omega ) \\mathbb 1 _ { \\{ 0 \\} } ( s ) + \\sum _ { i = 1 } ^ { 2 ^ n } u ( t _ i , \\omega ) \\mathbb 1 _ { ( t _ { i - 1 } , t _ i ] } ( s ) . \\end{align*}"}
+{"id": "7431.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\lambda } = \\left \\{ ( y _ 1 , y _ 2 ) \\in W _ { 0 } ^ { 1 , \\mathcal { H } } ( \\Omega ) \\times W _ { 0 } ^ { 1 , \\mathcal { H } } ( \\Omega ) \\backslash \\{ ( 0 , 0 ) \\} : \\psi _ { ( y _ 1 , y _ 2 ) } ^ { \\prime } ( t ) \\bigg \\vert _ { t = 1 } = 0 \\right \\} . \\end{align*}"}
+{"id": "6191.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ n \\eta _ { X _ i } - \\eta _ u + \\eta _ v \\right ) ( T ) = { \\sum } _ n T = \\sum \\mathbf { k } , \\end{align*}"}
+{"id": "32.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n P _ j g ( x _ i - x _ j ) P _ j = \\frac { 1 } { \\vert \\Lambda \\vert } \\sum _ { j = 1 } ^ n P _ j \\int _ { \\Lambda } g ( x _ i - y ) d y = \\frac { n _ 0 } { | \\Lambda | } \\widehat g ( 0 ) . \\end{align*}"}
+{"id": "6342.png", "formula": "\\begin{align*} N _ \\lambda ^ { l , 2 , m e d } v v _ \\lambda ^ { x _ 0 } = \\mu ^ 2 L ( P _ { \\mu } g ( u _ { < \\lambda } ) , v _ { \\mu } , v _ \\lambda ) . \\end{align*}"}
+{"id": "1668.png", "formula": "\\begin{align*} & H ^ { ( n , m ) } _ { \\texttt { a } ; j , k } ( \\boldsymbol { \\xi ) } : = \\partial _ { \\xi _ j } \\partial _ { \\xi _ k } V ^ { ( n , m ) } _ { \\texttt { a } ; \\lambda } ( \\boldsymbol { \\xi } ) \\\\ & = \\begin{cases} m + \\sum _ { \\substack { 1 \\leq l \\leq n \\\\ l \\neq j } } u _ q ( \\xi _ j - \\xi _ l ) & \\\\ - u _ q ( \\xi _ j - \\xi _ k ) & \\\\ \\end{cases} , \\end{align*}"}
+{"id": "4756.png", "formula": "\\begin{align*} r _ k : = \\sqrt { 2 } \\cdot 2 ^ { - 4 k - 1 } < \\delta / C 2 ^ { 2 k + 1 } > C . \\end{align*}"}
+{"id": "3440.png", "formula": "\\begin{align*} \\tilde { \\phi } _ j = \\frac { 1 } { l } \\max _ { i = 1 } ^ N \\max \\{ l _ 0 \\log | F _ 0 | + \\ldots + l _ m \\log | F _ m | + c _ i - a : s ^ { ( i ) } _ { l _ 0 \\ldots l _ m , a } \\neq 0 \\} . \\end{align*}"}
+{"id": "313.png", "formula": "\\begin{align*} \\ell _ { j } \\left ( s \\right ) \\left ( \\mathsf { N } _ { j } \\left ( s \\right ) \\circ \\mathsf { L } _ { j } \\left ( s \\right ) v , w \\right ) = \\left \\langle \\mathsf { L } _ { j } \\left ( s \\right ) v , \\overline { w } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } = \\ell _ { j } \\left ( s \\right ) \\left ( v , w \\right ) \\qquad \\forall w \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) . \\end{align*}"}
+{"id": "3246.png", "formula": "\\begin{align*} \\Delta _ { \\alpha } ( \\epsilon , R ) & = \\sum _ { | d | \\leq \\frac { \\epsilon q ^ { 2 } R } { \\sqrt { p ^ { 2 } + q ^ { 2 } } } } \\# \\left \\{ ( m , n ) \\in \\Z ^ { 2 } : n q - m p = d , \\frac { | d | } { q \\epsilon } < m \\leq \\frac { q R } { \\sqrt { p ^ { 2 } + q ^ { 2 } } } \\right \\} \\\\ & = \\sum _ { | d | \\leq \\frac { \\epsilon q ^ { 2 } R } { \\sqrt { p ^ { 2 } + q ^ { 2 } } } } \\sum _ { \\substack { \\frac { | d | } { q \\epsilon } < m \\leq \\frac { R q } { \\sqrt { p ^ { 2 } + q ^ { 2 } } } \\\\ m \\equiv - d \\bar { p } \\ ; ( q ) } } 1 , \\end{align*}"}
+{"id": "6188.png", "formula": "\\begin{align*} f _ t \\circ \\Gamma _ { \\mathbf { k } , + \\infty } = \\Gamma _ { f _ t ( \\mathbf { k } ) , + \\infty } \\circ f _ t . \\end{align*}"}
+{"id": "473.png", "formula": "\\begin{align*} h _ { \\beta , \\mu } ( r ) : = \\int _ 0 ^ \\infty t ^ \\beta ( p _ t \\mu ) ( r ) \\ , d t . \\end{align*}"}
+{"id": "2568.png", "formula": "\\begin{align*} I _ { \\mathcal { M } } [ u ] & = \\frac { 1 } { p } \\mathcal { M } ( \\| u \\| _ { D ^ { s , p } } ^ p ) + \\frac { 1 } { p } \\int _ { \\mathbb { R } ^ N } V ( x ) | u | ^ p d x \\\\ & \\phantom { = } - \\frac { 1 } { 2 \\cdot p _ { r ; s } ^ { \\uparrow * } } \\varepsilon _ K \\int _ { \\mathbb { R } ^ N } ( K \\ast g ( u ) ) g ( u ) d x - \\varepsilon _ W \\int _ { \\mathbb { R } ^ N } W ( x ) f ( u ) d x . \\end{align*}"}
+{"id": "2384.png", "formula": "\\begin{align*} \\pi _ 1 = \\frac { \\mu - r - h } { \\sigma ^ 2 } \\end{align*}"}
+{"id": "2269.png", "formula": "\\begin{align*} \\max _ { x } f _ \\alpha ( x ) = \\max _ { x } \\sum _ { j = 1 } ^ n \\theta _ \\alpha \\left ( x + x _ j \\right ) \\end{align*}"}
+{"id": "6863.png", "formula": "\\begin{align*} \\vec { A } = \\sum _ { i = 1 } ^ d I _ { n _ d } \\otimes \\cdots \\otimes I _ { n _ { i + 1 } } \\otimes A _ i \\otimes I _ { n _ { i - 1 } } \\otimes \\cdots \\otimes I _ { n _ 1 } , A _ i \\in \\C ^ { n _ i \\times n _ i } . \\end{align*}"}
+{"id": "7816.png", "formula": "\\begin{align*} \\mathfrak { g } = \\mathbb { R } \\ltimes ( \\mathfrak { s l } ( 2 , \\mathbb { R } ) \\ltimes ( \\mathbb { R } ^ n \\ltimes \\mathfrak { h } _ { 2 n + 2 } ) ) \\ , . \\end{align*}"}
+{"id": "3253.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { C } } = \\sum _ { \\ell = 0 } ^ n ( - 1 ) ^ { \\frac { \\ell ( \\ell - 1 ) } { 2 } } \\overline { \\langle \\mathcal { C } \\rangle _ { _ \\ell } } \\end{align*}"}
+{"id": "7739.png", "formula": "\\begin{align*} \\begin{array} { l l l l l l l l l l l } 0 = \\omega ( C ) - \\omega ( C ) & = | E _ { g ' = ( 0 , 0 ) } \\cap E ( C ) | + | E ( C ) - E _ { g ' = ( 0 , 0 ) } | - \\omega ( C ) \\\\ & \\le \\frac { \\omega ( C ) } { 4 } + \\Omega + 1 + 1 + \\Omega ( 7 - 1 - 2 ) - \\omega ( C ) \\\\ & = - 2 < 0 , \\end{array} \\end{align*}"}
+{"id": "6243.png", "formula": "\\begin{align*} \\xi _ 4 = \\xi _ 1 - \\xi _ 2 + \\xi _ 3 . \\end{align*}"}
+{"id": "5352.png", "formula": "\\begin{align*} H ^ 0 ( N _ Y ( K _ Y + D - ( k - 1 ) H ) ) = 0 . \\end{align*}"}
+{"id": "1145.png", "formula": "\\begin{align*} \\widetilde { h } \\big ( ( x _ 1 , m _ 1 ) , \\ldots , ( x _ n , m _ n ) \\big ) = \\big ( 0 , h ( x _ 1 , \\ldots , x _ n ) \\big ) , \\end{align*}"}
+{"id": "5869.png", "formula": "\\begin{align*} g _ { M } ( x ) = \\log [ 1 + \\epsilon _ { 0 } ( \\lambda _ { M } e ^ { x / 2 } - 1 ) ] , \\lambda _ { M } > 0 . \\end{align*}"}
+{"id": "8939.png", "formula": "\\begin{align*} \\int _ { U ^ - } \\theta ^ - = \\int _ { U ^ 0 } \\theta ^ 0 = 1 \\end{align*}"}
+{"id": "2805.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s | \\cdot | ^ { - \\theta } ( x ) = 2 ^ { 2 s } \\frac { \\Gamma \\left ( \\frac { N - \\theta } { 2 } \\right ) \\Gamma ( \\frac { 2 s + \\theta } { 2 } ) } { \\Gamma \\left ( \\frac { N - \\theta - 2 s } { 2 } \\right ) \\Gamma \\left ( \\tfrac { \\theta } { 2 } \\right ) } \\ | x | ^ { - ( \\theta + 2 s ) } , x \\neq 0 , N > \\theta > - 2 s , \\end{align*}"}
+{"id": "6997.png", "formula": "\\begin{align*} R _ { 1 2 3 4 } = K _ { 1 2 3 4 } + 2 ( | \\phi | ^ 2 - | \\psi | ^ 2 ) . \\end{align*}"}
+{"id": "5038.png", "formula": "\\begin{align*} \\triangle _ { V _ s + 2 \\frac { \\nabla w } w } ( w Y ) - \\left ( 1 + \\left ( \\frac { \\triangle _ { V _ s } w } { w } - \\frac 1 2 \\right ) - 2 \\frac { | \\nabla w | ^ 2 } { w ^ 2 } \\right ) ( w Y ) + \\frac 1 2 \\big ( d V ^ \\flat ( w Y , \\cdot ) \\big ) ^ { \\sharp } = w U \\end{align*}"}
+{"id": "2635.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } P ( \\sup _ { s \\in [ 0 , t ] } \\abs { \\frac { 1 } { n } \\ , Q _ n ^ { ( 0 ) } ( s ) - ( 1 - F _ 0 ( s ) ) } > \\epsilon ) ^ { 1 / b _ n ^ 2 } = 0 \\ , . \\end{align*}"}
+{"id": "8166.png", "formula": "\\begin{align*} [ X , Y ^ \\star ] = 0 \\ , . \\end{align*}"}
+{"id": "7667.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\mathbb { P } ( \\eta _ 1 + \\ldots + \\eta _ n \\leqslant x ) = \\sum _ { n = 1 } ^ { \\infty } \\mathbb { P } \\left ( e ^ { - t ( \\eta _ 1 + \\ldots + \\eta _ n ) } \\geqslant e ^ { - t x } \\right ) \\leqslant e ^ { t x } \\sum _ { n = 1 } ^ { \\infty } \\left ( \\mathbb { E } e ^ { - t \\eta } \\right ) ^ n < \\infty , \\end{align*}"}
+{"id": "7422.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { \\mathbb { T } } \\rho \\phi ~ d x = \\int _ { \\mathbb { T } } \\rho u \\phi ' ~ d x \\in L ^ { 1 } ( 0 , T ) . \\end{align*}"}
+{"id": "3886.png", "formula": "\\begin{align*} a \\wedge \\bigvee _ { s \\in S } s = \\bigvee _ { s \\in S } ( a \\wedge s ) \\end{align*}"}
+{"id": "1748.png", "formula": "\\begin{align*} \\frac { 1 } { ( 2 \\pi ) ^ n \\ , n ! } & \\int _ 0 ^ \\pi \\cdots \\int _ 0 ^ \\pi R ( \\boldsymbol { \\xi } ) \\rho _ \\epsilon ( \\boldsymbol { \\xi } ) \\xi _ 1 \\cdots \\xi _ n = \\\\ & \\sum _ { { \\lambda } \\in \\Lambda ^ { ( m , n ) } } R \\bigl ( \\boldsymbol { \\xi } _ { { \\lambda } } ^ { ( m , n ) } \\bigr ) \\rho _ \\epsilon \\bigl ( \\boldsymbol { \\xi } _ { { \\lambda } } ^ { ( m , n ) } \\bigr ) \\Delta _ \\lambda ^ { ( m , n ) } . \\end{align*}"}
+{"id": "2263.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\sum _ { k \\in \\Z } e ^ { - \\pi \\alpha k ^ 2 } e ^ { 2 \\pi i k x } \\ , d x = \\sum _ { k \\in \\Z } e ^ { - \\pi \\alpha k ^ 2 } \\int _ 0 ^ 1 e ^ { 2 \\pi i k x } \\ , d x = \\sum _ { k \\in \\Z } e ^ { - \\pi \\alpha k ^ 2 } \\delta _ { k , 0 } = 1 , \\end{align*}"}
+{"id": "8767.png", "formula": "\\begin{align*} \\frac { \\partial ^ { 2 } f _ { \\omega } ( x _ { i } ) } { \\partial ^ { 2 } x _ { i } } = 2 \\alpha ( 1 + \\delta ) + \\omega _ { i } r h ^ { \\beta } \\frac { \\partial ^ { 2 } \\eta ( x _ { i } h ^ { - 1 } ) } { \\partial ^ { 2 } x _ { i } } \\geq 2 \\alpha - \\frac { \\alpha } { \\sqrt { d } } \\geq \\alpha . \\end{align*}"}
+{"id": "5163.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } { \\rm A o I } ( S _ { \\mathrm { z } } , Q _ { { \\mathrm { u n i } } } , F _ \\mathrm { s } ) - \\frac { 3 } { 2 } { H [ Q _ { { \\mathrm { u n i } } } ( X ) ] } = 0 . \\end{align*}"}
+{"id": "1083.png", "formula": "\\begin{align*} \\begin{aligned} & \\qquad \\ , \\ , \\mbox { t h e r e a r e } \\ , \\ , \\beta > 2 \\ , \\ , \\mbox { a n d } \\ , \\ , r _ 0 > 0 \\ , \\ , \\mbox { s u c h t h a t } \\\\ & t f ( x , t ) \\geq \\beta F ( x , t ) > 0 \\ , \\ , \\mbox { f o r a n y } \\ , \\ , | t | \\geq r _ 0 \\ , \\ , \\mbox { a n d e v e r y } \\ , \\ , x \\in D , \\end{aligned} \\end{align*}"}
+{"id": "5101.png", "formula": "\\begin{align*} j _ * ( V _ { h } ) _ { h } = R \\varprojlim _ { h } V _ { h } . \\end{align*}"}
+{"id": "3051.png", "formula": "\\begin{align*} \\nabla u ^ { R , \\zeta } = R \\beta _ { \\R ^ 2 } ^ { \\zeta , \\C } \\ , , \\end{align*}"}
+{"id": "8793.png", "formula": "\\begin{align*} \\sum _ { t = T _ { 1 } + 1 } ^ { T } \\mathbb { E } [ f ( x _ { t } ) - f ^ * ] \\leq \\frac { \\mathcal { B } _ { 3 } } { \\alpha } r _ { 1 } + \\frac { \\mathcal { B } _ { 3 } \\mathcal { B } _ { 4 } } { 1 8 \\bar { L } ^ 2 \\kappa } \\beta \\frac { d } { \\alpha } T _ { 1 } ^ { \\frac { 1 } { \\beta } } + \\mathcal { B } _ { 4 } \\frac { d } { \\alpha } T ^ { \\frac { 1 } { \\beta } } . \\end{align*}"}
+{"id": "4516.png", "formula": "\\begin{align*} | | S _ { \\theta _ i } \\Delta _ 3 | | _ { s , \\ast , T } \\le C \\theta _ i ^ { s - 6 } | | \\Delta _ 3 | | _ { 6 , \\ast , T } \\le C \\theta _ i ^ { s - 6 } \\cdot \\delta \\theta _ i ^ { 8 - \\alpha } = C \\delta \\theta _ i ^ { s + 2 - \\alpha } , \\quad \\mbox { f o r } \\ , \\ , \\ , s \\ge 6 , \\end{align*}"}
+{"id": "4495.png", "formula": "\\begin{align*} \\mathbb { B } '' ( ( { \\mathbf V } , \\psi ) , ( \\tilde { { \\mathbf V } } , \\tilde { \\psi } ) ) : = \\left [ \\begin{array} { c } \\tilde { u } ^ + _ 2 \\partial _ 2 \\psi + \\partial _ 2 \\tilde { \\psi } u ^ + _ 2 \\\\ \\tilde { u } ^ - _ 2 \\partial _ 2 \\psi + \\partial _ 2 \\tilde { \\psi } u ^ - _ 2 \\\\ \\mathbf { H } ^ + \\cdot \\tilde { \\mathbf { H } } ^ + - \\mathbf { H } ^ - \\cdot \\tilde { \\mathbf { H } } ^ - \\end{array} \\right ] \\ , . \\end{align*}"}
+{"id": "2348.png", "formula": "\\begin{align*} f _ { u g } ( \\tau _ { u h } ) = f _ g ( \\tau _ h ) , \\forall u , g , h \\in G . \\end{align*}"}
+{"id": "4095.png", "formula": "\\begin{align*} V _ \\gamma : = \\frac 1 2 \\int _ { \\gamma } \\Delta d \\log \\Psi . \\end{align*}"}
+{"id": "4016.png", "formula": "\\begin{align*} \\left ( \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi \\right ) ^ n = e ^ b g \\left ( \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi \\right ) ^ m \\wedge \\omega ^ { n - m } . \\end{align*}"}
+{"id": "6820.png", "formula": "\\begin{align*} & 1 - u - a v = - ( u - u ^ * ) - a ( v - v ^ * ) , \\\\ [ 0 . 2 c m ] & u + e _ 2 - v = ( u - u ^ * ) - ( v - v ^ * ) , \\end{align*}"}
+{"id": "3789.png", "formula": "\\begin{align*} H ( x _ 0 , s _ 0 , x _ 1 , s _ 1 ) = s _ 0 + s _ 1 - 2 \\sqrt { s _ 0 s _ 1 } \\exp \\left ( \\frac { - c ( x _ 0 , x _ 1 ) } { 2 } \\right ) . \\end{align*}"}
+{"id": "299.png", "formula": "\\begin{align*} \\gamma _ { \\operatorname * { N } ; j } ^ { \\mathbb { B } , - } \\left ( \\left . u \\right \\vert _ { \\Omega _ { j } ^ { - } } \\right ) = - \\gamma _ { \\operatorname * { N } , j } ^ { \\mathbb { B } , + } \\left ( \\left . u \\right \\vert _ { \\Omega _ { j } ^ { + } } \\right ) \\end{align*}"}
+{"id": "4040.png", "formula": "\\begin{align*} C ( x , \\rho ) = \\cos \\rho ( x \\ ! - \\ ! a ) \\ ! + \\ ! \\int \\limits _ { a } ^ { x } q _ { 1 } ( t ) \\sin \\rho ( x \\ ! - \\ ! t ) d t \\ ! + \\ ! \\int \\limits _ { a } ^ { x } q _ { 0 } ( t ) \\frac { \\sin \\rho ( x \\ ! - \\ ! t ) } { \\rho } d t , \\end{align*}"}
+{"id": "4622.png", "formula": "\\begin{align*} \\bigl ( ( g _ 1 , g _ 2 ) \\cdot f \\bigr ) ( g ) : = f ( g _ 1 ^ { - 1 } g g _ 2 ) . \\end{align*}"}
+{"id": "4207.png", "formula": "\\begin{align*} D ( 0 , \\varepsilon ) & = \\sigma _ { \\varepsilon } ( T \\diamond S \\circ _ { \\ast } I ) = \\sigma _ { \\varepsilon } ( \\varphi ( T ) \\diamond \\varphi ( S ) \\circ _ { \\ast } \\varphi ( I ) ) \\\\ & = \\sigma _ { \\varepsilon } ( 2 \\varphi ( T ) \\varphi ( I ) ^ { \\ast } - 2 \\varphi ( I ) \\varphi ( T ) ) . \\end{align*}"}
+{"id": "4101.png", "formula": "\\begin{align*} \\omega _ { 0 , 1 } ( p _ 0 ) : = y ( p _ 0 ) d x ( p _ 0 ) . \\end{align*}"}
+{"id": "8910.png", "formula": "\\begin{align*} \\beta _ i ( [ p - x ] ) = [ p _ i + p _ { d - i } - x _ i - x _ { d - i } ] , \\end{align*}"}
+{"id": "3014.png", "formula": "\\begin{align*} \\psi \\left ( g _ 2 , g _ 3 \\right ) - \\psi \\left ( g _ 1 g _ 2 , g _ 3 \\right ) + \\psi \\left ( g _ 1 , g _ 2 g _ 3 \\right ) - \\psi \\left ( g _ 1 , g _ 2 \\right ) = 0 \\end{align*}"}
+{"id": "8761.png", "formula": "\\begin{align*} f _ { \\omega } ( u ) = \\alpha ( 1 + \\delta ) \\norm { u } ^ { 2 } / 2 + \\sum _ { i = 1 } ^ { d } \\omega _ { i } r h ^ { \\beta } \\eta ( u _ { i } h ^ { - 1 } ) , u = ( u _ 1 , \\dots , u _ d ) , \\end{align*}"}
+{"id": "3270.png", "formula": "\\begin{align*} g = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } g _ { _ \\Sigma } , \\end{align*}"}
+{"id": "1364.png", "formula": "\\begin{align*} \\Delta _ { 2 \\kappa } : = - y ^ 2 \\left ( \\frac { \\partial ^ 2 } { \\partial x ^ 2 } + \\frac { \\partial ^ 2 } { \\partial y ^ 2 } \\right ) + 2 i \\kappa y \\left ( \\frac { \\partial } { \\partial x } + i \\frac { \\partial } { \\partial y } \\right ) . \\end{align*}"}
+{"id": "797.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } | u ^ \\flat | ^ { p } d x = \\int _ { \\mathbb { R } ^ N } ( K \\ast | u ^ \\flat | ^ { p ^ \\flat } ) | u ^ \\flat | ^ { p ^ \\flat } d x \\end{align*}"}
+{"id": "2944.png", "formula": "\\begin{align*} \\delta _ { i _ 1 i _ 2 } [ \\delta _ { i _ 1 i _ 2 } D _ { \\pi ( i _ 3 \\cdots i _ n ) k } ] & = \\delta _ { 1 1 } D _ { \\pi ( i _ 3 \\cdots i _ n ) k } + \\delta _ { 2 2 } D _ { \\pi ( i _ 3 \\cdots i _ n ) k } + \\delta _ { 3 3 } D _ { \\pi ( i _ 3 \\cdots i _ n ) k } \\\\ & = 3 D _ { \\pi ( i _ 3 \\cdots i _ n ) k } . \\end{align*}"}
+{"id": "7329.png", "formula": "\\begin{align*} F ( \\lambda , x , u , - v ) = F ( \\lambda , x , u , v ) \\lambda \\in I , x \\in \\Omega , ( u , v ) \\in \\mathbb { R } ^ 2 . \\end{align*}"}
+{"id": "2575.png", "formula": "\\begin{align*} \\beta ( h , h ' ) + \\beta ( h + h ' , h '' ) = \\beta ( h ' , h '' ) + \\beta ( h , h ' + h '' ) . \\end{align*}"}
+{"id": "2259.png", "formula": "\\begin{align*} \\min _ x \\sum _ { j = 1 } ^ n \\sum _ { k \\in \\Z } e ^ { - \\pi \\alpha k ^ 2 } e ^ { 2 \\pi i k ( x _ j - x ) } = \\min _ x \\sum _ { j = 1 } ^ n \\theta _ \\alpha ( x _ j - x ) . \\end{align*}"}
+{"id": "6805.png", "formula": "\\begin{align*} 0 & \\leq \\phi ' _ 2 ( z _ 2 ) = w ' ( z _ 2 ) + K f ' ( u ( z _ 2 ) ) w ( z _ 2 ) \\\\ [ 0 . 2 c m ] & = c w ( z _ 2 ) - f ( u ( z _ 2 ) ) p ( u ( z _ 2 ) ) + f ( u ( z _ 2 ) ) v ( z _ 2 ) + K f ' ( u ( z _ 2 ) ) w ( z _ 2 ) \\\\ [ 0 . 2 c m ] & = \\left [ - ( K ^ 2 f ' ( u ( z _ 2 ) ) + c K ) + v ( z _ 2 ) - p ( u ( z _ 2 ) ) \\right ] f ( u ( z _ 2 ) ) < 0 , \\end{align*}"}
+{"id": "7309.png", "formula": "\\begin{align*} Q _ S = 2 P _ + ( V ) - I _ H \\in \\mathcal { F S } ^ i ( H ) ^ G \\end{align*}"}
+{"id": "805.png", "formula": "\\begin{align*} ( S ^ * ) ^ { N / ( p s ) } ( t _ 0 ^ * ) ^ { p - 1 } - \\frac { b } { p ^ \\sharp } C _ U ( S ^ * ) ^ { \\frac { N + \\alpha } { p s } } ( t _ 0 ^ * ) ^ { 2 \\cdot p ^ \\sharp - 1 } - \\varepsilon _ g ( S ^ * ) ^ { N / ( p s ) } ( t _ 0 ^ * ) ^ { p _ s ^ * - 1 } = 0 . \\end{align*}"}
+{"id": "4405.png", "formula": "\\begin{align*} \\begin{cases} \\mathbb { L } ' _ e ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) \\dot { { \\mathbf U } } ^ { \\natural } = \\mathbf { F } = \\mathbf { f } - \\mathbb { L } ' _ e ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) \\tilde { { \\mathbf U } } & \\Omega _ T , \\\\ \\mathbb { B } ' _ e ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) ( \\dot { { \\mathbf U } } ^ { \\natural } , \\varphi ) = 0 & \\Gamma _ T . \\end{cases} \\end{align*}"}
+{"id": "8339.png", "formula": "\\begin{align*} \\d X ( t ) = a ( X ( t ) ) \\d t + b ( X ( t ) ) \\d W ( t ) , \\ X ( 0 ) = x _ 0 \\in \\R ^ n , \\end{align*}"}
+{"id": "3376.png", "formula": "\\begin{align*} \\begin{cases} \\phi ^ 0 ( r , y ) = r + a ( y ) r ^ 2 + o ( r ^ 2 ) , \\\\ \\phi ^ \\mu ( r , y ) = y ^ \\mu + b ^ { \\mu } ( y ) r + c ^ { \\mu } ( y ) r ^ 2 + o ( r ^ 2 ) . \\end{cases} \\end{align*}"}
+{"id": "3512.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\tau , s , \\rho } f ( z ) = \\sum _ { \\substack { \\textbf { a } \\in Y ( \\tau ) \\\\ \\textbf { a } \\to b } } \\gamma _ a ' ( z ) ^ s \\rho ( \\gamma _ a ) ^ { - 1 } f ( \\gamma _ a ( z ) ) \\ \\mathrm { i f } \\ z \\in D _ b , \\end{align*}"}
+{"id": "1977.png", "formula": "\\begin{align*} a ^ t _ { \\sigma , s } - a ^ t _ { \\sigma , s ^ j } = j \\left ( \\frac { q } { p } s _ { \\pi _ { \\hat { \\beta } } ( \\hat { \\gamma } - 2 ) } + \\frac { q } { p } s _ { \\pi _ { \\hat { \\beta } } ( \\hat { \\gamma } ) } + \\lambda _ { \\pi _ { \\hat { \\beta } } ( \\hat { \\gamma } - 1 ) } \\right ) . \\end{align*}"}
+{"id": "3480.png", "formula": "\\begin{align*} \\int _ { X _ t } e ^ { - \\alpha v } d \\mu _ t \\leq A , \\forall v \\in P S H ( X _ t , \\frac { 1 } { | \\log | t | | } \\omega _ M | _ { X _ t } ) \\sup _ { X _ t } v = 0 . \\end{align*}"}
+{"id": "5149.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\frac { D } { \\delta ^ 2 } = \\frac { 1 } { 1 2 } . \\end{align*}"}
+{"id": "8008.png", "formula": "\\begin{align*} X ( t ) = g _ k \\Bigl ( T _ { ( - k ) } ( t ) \\Bigr ) = v _ k t + \\Biggl [ v _ h - v _ k \\Biggr ] _ { \\substack { h = 0 , \\dots , D \\\\ h \\not = k } } T _ { ( - k ) } ( t ) , \\ \\ t \\ge 0 , \\end{align*}"}
+{"id": "7260.png", "formula": "\\begin{align*} \\mu _ \\omega ( C _ u ) = \\mu _ \\omega ( f ^ { ( \\omega _ 1 ) } _ { u _ 1 } \\circ \\circ \\circ f ^ { ( \\omega _ n ) } _ { u _ n } ( [ 0 , 1 ] ) ) = \\eta ^ { ( \\omega ) } ( [ u _ 1 , . . . , u _ n ] ) = \\prod _ { i = 1 } ^ n \\mathbf { p } ^ { ( \\omega _ i ) } _ { u _ i } . \\end{align*}"}
+{"id": "5963.png", "formula": "\\begin{align*} I = N ^ \\omega _ { \\partial M } \\Lambda ^ \\omega _ { g , F } + \\Psi ^ { - \\infty } , \\end{align*}"}
+{"id": "7255.png", "formula": "\\begin{align*} \\mu _ \\omega = \\sum _ { u \\in X _ 1 ^ { ( \\omega ) } } p _ u ^ { ( \\omega _ 1 ) } f _ u ^ { ( \\omega _ 1 ) } \\mu _ { \\sigma ( \\omega ) } . \\end{align*}"}
+{"id": "1814.png", "formula": "\\begin{gather*} \\frac { 1 } { T } \\int _ { 0 } ^ { T } F ( Z _ N ( s + i \\tau , \\alpha ) ) \\ , d \\tau = \\int _ { \\mathcal { T } ^ k } ( F \\circ \\psi _ N ) \\ , d \\mu _ { \\alpha , T } , \\\\ \\mathbf { E } \\left [ F ( Z _ N ( s , \\mathbb { X } _ \\alpha ) ) \\right ] = \\int _ { \\mathcal { T } ^ k } ( F \\circ \\psi _ N ) \\ , d \\nu _ { \\alpha } . \\end{gather*}"}
+{"id": "388.png", "formula": "\\begin{align*} j ( u ^ * c , x \\circ c ^ * u ) = j ( c , x ) \\circ u . \\end{align*}"}
+{"id": "4941.png", "formula": "\\begin{align*} \\eta ( 8 i ) = \\dfrac { \\pi ^ { 1 / 4 } \\cdot ( \\sqrt { 2 } - 1 ) ^ { 1 / 8 } \\cdot ( 1 - 2 ^ { - 1 / 4 } ) ^ { 1 / 2 } } { \\Gamma ( 3 / 4 ) \\cdot 2 ^ { 5 3 / 3 2 } } . \\end{align*}"}
+{"id": "3330.png", "formula": "\\begin{align*} g ( \\tau + 1 ) = \\begin{pmatrix} \\zeta _ { 1 8 } ^ { - 1 } & 0 & 0 \\\\ 0 & \\zeta _ { 1 8 } ^ 5 & 0 \\\\ 0 & 0 & \\zeta _ { 1 8 } ^ { 1 1 } \\end{pmatrix} g ( \\tau ) , g ( - \\frac { 1 } { \\tau } ) = \\begin{pmatrix} \\alpha _ 1 & \\alpha _ 2 & \\alpha _ 4 \\\\ \\alpha _ 2 & - \\alpha _ 4 & - \\alpha _ 1 \\\\ \\alpha _ 4 & - \\alpha _ 1 & \\alpha _ 2 \\end{pmatrix} g ( \\frac { \\tau } { 3 } ) \\end{align*}"}
+{"id": "4193.png", "formula": "\\begin{align*} \\sigma _ { \\varepsilon } ( T _ { 1 } \\diamond T _ { 2 } \\circ _ { \\ast } T _ { 3 } ) = \\sigma _ { \\varepsilon } ( \\varphi ( T _ { 1 } ) \\diamond \\varphi ( T _ { 2 } ) \\circ _ { \\ast } \\varphi ( T _ { 3 } ) ) , \\end{align*}"}
+{"id": "8087.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\mu _ { j } b _ { j } \\right \\| _ { h _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\mu _ { j } g ( b _ { j } ) \\right \\| _ { L _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\mu _ { j } \\chi _ { P _ { j } } } { \\omega ( P _ { j } ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } . \\end{align*}"}
+{"id": "7379.png", "formula": "\\begin{align*} { \\rm H C } ( ( A , M ) / ( R , P ) ) : = { \\rm H H } ( ( A , M ) / ( R , P ) ) _ { h S ^ 1 } , \\\\ { \\rm H C } ^ - ( ( A , M ) / ( R , P ) ) : = { \\rm H H } ( ( A , M ) / ( R , P ) ) ^ { h S ^ 1 } , \\\\ { \\rm H P } ( ( A , M ) / ( R , P ) ) : = { \\rm H H } ( ( A , M ) / ( R , P ) ) ^ { t S ^ 1 } , \\end{align*}"}
+{"id": "6957.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\nabla u | ^ 2 d x \\geq & ( \\alpha ( d - 2 ) - \\alpha ^ 2 ) \\int _ { \\Omega } \\frac { | u | ^ 2 } { | x | ^ 2 } d x + \\alpha ( d ^ 2 - d ) \\int _ { \\Omega } \\frac { | u | ^ 2 } { | x | ^ 2 } d x \\\\ & = ( - \\alpha ^ 2 + \\alpha ( d - 2 ) ) \\int _ { \\Omega } \\frac { | u | ^ 2 } { | x | ^ 2 } d x . \\end{align*}"}
+{"id": "2648.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\limsup _ { n \\to \\infty } P ( w _ L ( X _ n , \\delta ) > \\epsilon ) ^ { 1 / r _ n } = 0 \\ , . \\end{align*}"}
+{"id": "2817.png", "formula": "\\begin{align*} \\left [ \\frac { b _ { N , s / 2 , s } } { 2 } \\right ] ^ { 2 } \\int _ { \\Omega } \\frac { u ( x ) ^ { 2 } } { | x | ^ { 2 s } } \\dd x \\leq \\left [ \\frac { b _ { N , s / 2 , s } } { 2 } \\right ] ^ { 2 } \\liminf _ { n \\to \\infty } \\int _ { \\Omega } \\frac { u _ n ( x ) ^ { 2 } } { | x | ^ { 2 s } } \\dd x & \\leq \\lim _ { n \\to \\infty } \\int _ \\Omega | ( - \\Delta ) ^ { s / 2 } u _ n | ^ 2 \\dd x \\\\ & = \\int _ \\Omega | ( - \\Delta ) ^ { s / 2 } u | ^ 2 \\dd x . \\end{align*}"}
+{"id": "7941.png", "formula": "\\begin{align*} S _ 2 = ( S \\cap V _ 2 ) \\cup ( X _ 2 \\cup Y _ 2 ) \\ 1 \\end{align*}"}
+{"id": "3359.png", "formula": "\\begin{align*} \\frac { D _ \\zeta ( x ) ^ p } { D _ { \\zeta ^ q } ( x ) } \\equiv \\left ( \\prod _ { t = 1 } ^ { p - 1 } \\frac { 1 } { ( x ; \\zeta ) _ { t q } } \\right ) ^ m \\quad \\mod ( 1 - x ^ m ) ^ m . \\end{align*}"}
+{"id": "8549.png", "formula": "\\begin{align*} P ( \\widetilde { E } ; J \\times \\mathbb { R } ^ { n - 1 } ) = P ( F _ { \\ell } ; J \\times \\mathbb { R } ^ { n - 1 } ) . \\end{align*}"}
+{"id": "683.png", "formula": "\\begin{align*} \\mathcal F ( v \\mathfrak K e _ j ) ( \\xi ) = \\frac { 1 } { ( 2 \\pi ) ^ d } \\int \\widehat v ( \\xi - \\eta ) \\widehat { \\mathfrak k * e _ j } ( \\eta ) \\ , d \\eta = \\frac { 1 } { ( 2 \\pi ) ^ d } \\int \\widehat v ( \\xi - \\eta ) \\widehat { \\mathfrak k } ( \\eta ) \\widehat { e _ j } ( \\eta ) \\ , d \\eta . \\end{align*}"}
+{"id": "1625.png", "formula": "\\begin{align*} \\Delta _ y g ( x , y ) = \\frac { \\delta _ x ( y ) } { m ( x ) } . \\end{align*}"}
+{"id": "3220.png", "formula": "\\begin{align*} \\| x - x _ { k } \\| _ { A ^ { T } A } ^ { 2 } = \\left \\Vert r _ { k } \\right \\Vert ^ { 2 } - \\left \\Vert b - b _ { | \\mathcal { R } ( A ) } \\right \\Vert ^ { 2 } \\end{align*}"}
+{"id": "2899.png", "formula": "\\begin{align*} T _ 0 \\phi _ \\xi = \\sum _ { v \\in W _ 0 } \\frac { t _ 0 - 1 } { 1 - e ^ { i \\langle v \\xi , { \\alpha _ 0 } \\rangle } } C ( v \\xi ) e ^ { i v \\xi } + \\sum _ { v \\in W _ 0 } \\frac { 1 - t _ 0 e ^ { i \\langle v \\xi , - { \\alpha _ 0 } \\rangle } } { 1 - e ^ { i \\langle v \\xi , - { \\alpha _ 0 } \\rangle } } C ( s ' _ 0 v \\xi ) e ^ { i c \\langle v \\xi , { \\alpha _ 0 } \\rangle } e ^ { i v \\xi } . \\end{align*}"}
+{"id": "1126.png", "formula": "\\begin{align*} [ x , [ y , z ] _ 1 ] _ 2 + [ x , [ y , z ] _ 2 ] _ 1 = [ [ x , y ] _ 1 , z ] _ 2 + [ [ x , y ] _ 2 , z ] _ 1 - [ [ x , z ] _ 1 , y ] _ 2 - [ [ x , z ] _ 2 , y ] _ 1 . \\end{align*}"}
+{"id": "4194.png", "formula": "\\begin{align*} \\sigma _ { \\varepsilon } ( [ T _ { 1 } \\bullet T _ { 2 } , T _ { 3 } ] _ { \\ast } ) = \\sigma _ { \\varepsilon } ( [ \\varphi ( T _ { 1 } ) \\bullet \\varphi ( T _ { 2 } ) , \\varphi ( T _ { 3 } ) ] _ { \\ast } ) , ~ ~ ( T _ { 1 } , T _ { 2 } , T _ { 3 } \\in B ( H ) ) . \\end{align*}"}
+{"id": "6225.png", "formula": "\\begin{align*} C _ \\nu ( t ) \\coloneqq \\begin{cases} C e ^ { - t d ^ \\nu } & ( t \\ge 1 ) \\\\ C t ^ { - \\lambda _ \\nu } & ( 0 < t \\le 1 ) , \\end{cases} \\lambda _ \\nu \\coloneqq \\max \\Big \\{ \\frac { d } { \\nu } \\Big ( \\frac { 1 } { p _ 2 } - \\frac { 1 } { p _ 1 } \\Big ) , 0 \\Big \\} , \\end{align*}"}
+{"id": "2952.png", "formula": "\\begin{align*} J _ s ^ { ( n + 1 ) } = \\begin{cases} J _ { s - 1 } ^ n + J _ s ^ n + J _ { s + 1 } ^ n & , 1 \\le s \\le n \\\\ J _ 1 ^ n & , s = 0 \\\\ J _ n ^ n & , s = n + 1 \\end{cases} \\end{align*}"}
+{"id": "4300.png", "formula": "\\begin{align*} \\tilde { N } _ { k } ( t ) = N _ k ( t ) + k t ^ { k / 2 } M _ { k - 1 } b _ 0 ( t ) , \\end{align*}"}
+{"id": "5985.png", "formula": "\\begin{align*} \\mathrm { d i m } \\left ( \\mathrm { K e r } ( - \\Delta _ { M i x , \\varepsilon _ 1 } ^ F - \\lambda ) \\oplus L \\right ) = \\mathrm { d i m } \\left ( \\mathrm { K e r } \\left ( - \\Delta _ { M i x , \\varepsilon _ 1 } ^ F - \\lambda \\right ) \\right ) + \\mathrm { d i m } L . \\end{align*}"}
+{"id": "8224.png", "formula": "\\begin{align*} d K _ 1 ^ a ( X ) = \\frac { 1 } { 2 } d ( \\rho ^ 2 ) ( \\bar { X } ) + 2 ( V + a ^ 2 ) x _ 1 a _ 1 + ( 2 V + a ^ 2 ) ( x _ 2 a _ 2 + x _ 3 a _ 3 ) . \\end{align*}"}
+{"id": "5048.png", "formula": "\\begin{align*} \\triangle _ f h = R ( \\gamma '' , h ) + S ( \\gamma '' , h ) , \\end{align*}"}
+{"id": "6410.png", "formula": "\\begin{align*} \\chi ( T _ i ^ k \\otimes T _ j ^ \\ell ) T _ k ^ m T _ \\ell ^ n = t _ { i j } ^ { k \\ell } T _ k ^ m T _ \\ell ^ n = T _ i ^ k T _ j ^ \\ell t _ { k \\ell } ^ { m n } = T _ i ^ k T _ j ^ \\ell \\chi ( T _ k ^ m \\otimes T _ \\ell ^ n ) , \\end{align*}"}
+{"id": "603.png", "formula": "\\begin{align*} \\frac 1 2 + \\varepsilon - r = \\mu b . \\end{align*}"}
+{"id": "863.png", "formula": "\\begin{align*} [ u ] _ { W _ s ^ { \\alpha , G } ( X , d , \\mu ) } : = \\inf \\left \\lbrace \\lambda > 0 ; \\Phi ^ { \\alpha , G } _ { s } \\left ( \\frac { u } { \\lambda } \\right ) \\leq 1 \\right \\rbrace . \\end{align*}"}
+{"id": "1089.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - \\Delta _ { \\mu } u ( x ) = \\gamma u ( x ) + | u ( x ) | ^ { p - 2 } u ( x ) x \\in \\mathop D \\limits ^ \\circ \\\\ \\medskip \\ , \\ , u | _ { \\partial D } = 0 , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "529.png", "formula": "\\begin{align*} \\norm { \\theta _ R \\left ( \\sum _ { i = 1 } ^ n \\norm { u _ i } _ { \\widetilde X ^ { s _ i , b } _ { h _ i ( \\xi ) } ( 0 , t ) } ^ 2 \\right ) u _ j ( t ) } _ { X ^ { s _ j , b } _ { h _ j ( \\xi ) } ( 0 , T ) } \\leqslant C \\sqrt { R } , \\end{align*}"}
+{"id": "4889.png", "formula": "\\begin{align*} | f _ n ( z ) | \\leq \\sum _ { k = 0 } ^ \\infty \\frac { f _ n ^ k ( 0 ) } { k ! } | z | ^ k = f _ n ( | z | ) . \\end{align*}"}
+{"id": "5104.png", "formula": "\\begin{align*} C = K \\{ T \\} [ T ^ { - 1 } ] \\oplus K \\{ T ^ { - 1 } \\} [ T ] \\end{align*}"}
+{"id": "5204.png", "formula": "\\begin{align*} N _ 1 ( u ) = A ' \\cup S , \\ , N _ 1 ( v ) = B ' \\cup T , \\ , N _ 2 ( u ) = A ' \\cup T , \\ , N _ 2 ( v ) = B ' \\cup S \\end{align*}"}
+{"id": "6312.png", "formula": "\\begin{align*} \\partial _ t w _ \\lambda + \\partial _ x g _ { [ < \\lambda ^ { \\sigma } ] } \\partial _ x w _ \\lambda = e _ \\lambda ^ 1 , w _ \\lambda ( T ) = w _ { T , \\lambda } , \\end{align*}"}
+{"id": "8546.png", "formula": "\\begin{align*} P ( E _ { 1 } ^ { k } ; J \\times \\mathbb { R } ^ { n - 1 } ) = P ( E _ { 2 } ^ { k } ; J \\times \\mathbb { R } ^ { n - 1 } ) = \\cdots & = P ( E _ { N _ { k } } ^ { k } ; J \\times \\mathbb { R } ^ { n - 1 } ) \\\\ & = P ( F _ { \\ell ^ { k } } ; J \\times \\mathbb { R } ^ { n - 1 } ) . \\end{align*}"}
+{"id": "3045.png", "formula": "\\begin{align*} \\Gamma ^ { \\zeta , \\C } ( \\cdot ) = \\hat \\Gamma ^ { \\zeta , \\C } ( \\cdot ) = - \\Gamma ^ { \\zeta , \\C } ( \\cdot + \\pi ) \\ , ; \\end{align*}"}
+{"id": "2659.png", "formula": "\\begin{align*} g ( t ) = f ( t ) + \\int _ 0 ^ t g ( t - s ) ^ + \\ , d F ( s ) \\ , , t \\in \\R _ + \\ , . \\end{align*}"}
+{"id": "6872.png", "formula": "\\begin{align*} K = 2 \\sum _ i A _ i \\exp ( C _ i ) \\end{align*}"}
+{"id": "1928.png", "formula": "\\begin{align*} \\left ( \\Pi _ { \\lambda _ 1 , \\lambda _ 2 } g \\right ) ^ { 1 - \\lambda _ 1 , 1 - \\lambda _ 2 } _ { i - \\frac { 1 } { 2 } , j - \\frac { 1 } { 2 } } = g ^ { 1 - \\lambda _ 1 , 1 - \\lambda _ 2 } _ { i - \\frac { 1 } { 2 } , j - \\frac { 1 } { 2 } } , \\mbox { i f } \\left ( i , j \\right ) \\in ( \\Bbbk _ 2 ^ - , \\Bbbk _ 1 ^ - ) . \\end{align*}"}
+{"id": "2425.png", "formula": "\\begin{align*} u _ { \\alpha a , \\gamma \\alpha a } ( \\alpha z ) = c u _ { a , \\alpha ^ { - 1 } \\gamma \\alpha a } ( z ) \\end{align*}"}
+{"id": "4464.png", "formula": "\\begin{align*} & | | \\dot { { \\mathbf U } } | | _ { s , \\ast , T } + | | \\varphi | | _ { H ^ s ( \\Gamma _ T ) } \\\\ & \\quad \\leq C ( K _ 0 ) \\Big ( | | \\mathbf { f } | | _ { s + 2 , \\ast , T } + | | \\mathbf { g } | | _ { H ^ { s + 2 } ( \\Gamma _ T ) } + ( | | \\mathbf { f } | | _ { 8 , \\ast , T } + | | \\mathbf { g } | | _ { H ^ { 8 } ( \\Gamma _ T ) } ) | | \\hat { W } | | _ { s + 4 , \\ast , T } \\Big ) , \\end{align*}"}
+{"id": "673.png", "formula": "\\begin{align*} \\norm { \\theta ( t ) S _ { h ( \\xi ) } ( t ) f } _ { X ^ { s , b } _ { h ( \\xi ) } } = \\norm { \\theta } _ { H ^ b ( \\R ) } \\norm { f } _ { H ^ s ( \\R ^ d ) } \\end{align*}"}
+{"id": "2917.png", "formula": "\\begin{align*} T : = \\left \\{ t \\in \\mathbb Z _ { ( p ) } \\ , \\middle | \\ , \\begin{tabular} { c } \\textup { t h e r e i s a $ \\lambda _ t \\in \\mathbb Z _ { > 0 } $ a n d a $ \\mathbb Z _ { ( p ) } $ - W e i l d i v i s o r $ E _ t \\ge 0 $ o n $ U $ } \\\\ \\textup { s u c h t h a t $ B | _ U + \\lambda _ t D | _ U - t h ^ * K _ V \\sim _ { \\mathbb Z _ { ( p ) } } E _ t $ } \\end{tabular} \\right \\} . \\end{align*}"}
+{"id": "8702.png", "formula": "\\begin{align*} \\tilde { \\phi } _ i = \\sum _ { j = 0 } ^ n a _ { i , j } \\phi _ j + b _ i , ~ i = 0 , . . . , n , \\end{align*}"}
+{"id": "3601.png", "formula": "\\begin{align*} \\begin{aligned} & { E _ { { \\rm { t h } } } } \\Big { | } _ { N _ { \\rm R } = 3 } - { E _ { { \\rm { t h } } } } \\Big { | } _ { N _ { \\rm R } = 2 } \\\\ = & \\frac { { { \\sigma ^ 2 } { L _ { \\rm { R } } } \\left ( { \\kappa + 1 } \\right ) } } { { 2 { N _ { \\rm { T } } } \\kappa { C ^ 2 } } } \\left ( { \\sqrt { 9 + 3 \\pi \\left ( { K - 1 } \\right ) } - { K \\pi } + \\pi - 3 } \\right ) \\\\ \\mathop \\le \\limits ^ { \\left ( a \\right ) } & 0 , \\end{aligned} \\end{align*}"}
+{"id": "3313.png", "formula": "\\begin{align*} m _ 2 = \\frac { d ^ 2 } { d t ^ 2 } \\phi _ Y ( 0 ) ( - \\mu ) ^ 2 = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\frac { 2 } { \\lambda _ { _ \\Sigma } ^ 2 } . \\end{align*}"}
+{"id": "5776.png", "formula": "\\begin{align*} P _ p ( J ^ p _ X ) : = \\sqrt { \\frac { L ^ { p - 1 } } { 2 \\pi \\Delta _ p ^ 2 } } \\exp \\Big [ - \\frac { L ^ { p - 1 } } { 2 \\Delta _ p ^ 2 } ( J ^ p _ X - L ^ { 1 - p } \\mu _ p ) ^ 2 \\Big ] . \\end{align*}"}
+{"id": "365.png", "formula": "\\begin{align*} \\overline { \\phi } _ { \\overline { g } } ( \\overline { h } ) = \\overline { \\phi _ g ( h ) } = \\overline { \\phi _ g ( h ) h ^ { - 1 } h } = \\overline { h } , \\end{align*}"}
+{"id": "410.png", "formula": "\\begin{align*} \\hom ( y ( k ) , P \\times [ \\bot = \\top ] ) \\cong \\hom ( y ( k ) , P ) \\times \\hom ( y ( k ) , [ 0 = 1 ] ) \\\\ \\cong \\hom ( y ( k ) , P ) \\times 1 \\cong \\hom ( y ( k ) , P ) \\cong P ( k ) . \\end{align*}"}
+{"id": "3374.png", "formula": "\\begin{align*} \\hat { \\alpha } ^ i _ j = \\alpha ^ i _ j + \\Upsilon _ k \\theta ^ k \\delta ^ i _ j + \\theta ^ i \\Upsilon _ j , \\end{align*}"}
+{"id": "3486.png", "formula": "\\begin{align*} y = ( y _ 0 , \\ldots y _ m ) , \\sum _ 0 ^ m y _ i = 1 , y _ i \\gtrsim \\epsilon , \\end{align*}"}
+{"id": "7753.png", "formula": "\\begin{align*} N \\cong \\{ \\zeta : \\Gamma \\to \\mathbb { R } / \\mathbb { Z } \\ ; | \\ ; \\zeta _ { \\gamma + \\gamma ' } = \\zeta _ { \\gamma } + \\zeta _ { \\gamma ' } \\} \\ , . \\end{align*}"}
+{"id": "4674.png", "formula": "\\begin{align*} | S | & \\ge ( 2 m + 1 ) \\cdot ( \\frac { n } { 2 ( 2 l + 1 ) } - 2 ) - \\binom { 2 m + 1 } { 2 } \\cdot \\epsilon n \\\\ & = \\frac { n } { 2 ( 2 l + 1 ) } \\cdot ( 2 m + 1 ) - 2 ( 2 m + 1 ) - ( 2 m + 1 ) \\cdot m \\cdot \\epsilon n \\\\ & > \\frac { 2 } { 4 ( 2 l + 1 ) + 1 } \\cdot ( 2 m + 1 ) n \\end{align*}"}
+{"id": "7535.png", "formula": "\\begin{align*} \\sqrt { ( g \\circ f ) ^ ! } = f ^ ! \\circ \\sqrt { g ^ ! } , \\end{align*}"}
+{"id": "7245.png", "formula": "\\begin{align*} B = R G = \\frac { \\alpha } { \\gamma } G , \\end{align*}"}
+{"id": "6910.png", "formula": "\\begin{align*} \\int _ { \\mathbb R } J _ 1 ( \\xi - y ) \\underline { \\phi } ( y ) d y \\geq & \\frac { 1 + b } { 2 } \\int _ { \\mathbb R } J _ 1 ( \\xi - y ) d y = \\frac { 1 + b } { 2 } , \\end{align*}"}
+{"id": "6180.png", "formula": "\\begin{gather*} t ' _ { p , q } = \\begin{cases} t _ { p , q } & p < q < i \\\\ t _ { p , q - 1 } & p < i < q \\\\ t _ { p - 1 , q - 1 } & i < p < q \\\\ r ( \\mu _ p ) & p < i = q \\\\ r ( \\mu _ q ) & p = i < q \\end{cases} \\\\ \\mu ' _ 1 = \\mu _ i . \\end{gather*}"}
+{"id": "4238.png", "formula": "\\begin{align*} d \\sigma ^ 1 = 2 i \\ , \\sigma ^ { 1 3 } , d \\sigma ^ 2 = \\frac { 4 u } { s ^ 2 } \\ , \\sigma ^ { 1 3 } - 2 i \\ , \\sigma ^ { 2 3 } , d \\sigma ^ 3 = 0 , \\end{align*}"}
+{"id": "2847.png", "formula": "\\begin{align*} x \\mapsto x ^ \\vee : = 2 x / \\langle x , x \\rangle ( x \\in V \\setminus \\{ 0 \\} ) . \\end{align*}"}
+{"id": "3552.png", "formula": "\\begin{align*} & \\frac { 1 } { \\eta } ( x ^ k - x ^ { k + 1 } ) + A ^ T ( \\lambda ^ k - \\lambda ^ { k + 1 } ) \\in \\partial f ( x ^ { k + 1 } ) - A ^ T \\lambda ^ { k + 1 } \\\\ & \\frac { 1 } { \\eta } ( \\lambda ^ k - \\lambda ^ { k + 1 } ) + A ( x ^ k - x ^ { k + 1 } ) \\in \\partial g ^ * ( \\lambda ^ { k + 1 } ) + A x ^ { k + 1 } \\end{align*}"}
+{"id": "2551.png", "formula": "\\begin{align*} & \\phantom { = } \\int _ { \\mathbb { R } ^ N } ( K \\ast g ( u _ n ) ) g ( u _ n ) d x - \\int _ { \\mathbb { R } ^ N } ( K \\ast g ( u _ n - u ) ) g ( u _ n - u ) d x \\\\ & = \\int _ { \\mathbb { R } ^ N } ( K \\ast ( g ( u _ n ) - g ( u _ n - u ) ) ) ( g ( u _ n ) - g ( u _ n - u ) ) d x \\\\ & \\phantom { = } + 2 \\int _ { \\mathbb { R } ^ N } ( K \\ast ( g ( u _ n ) - g ( u _ n - u ) ) ) g ( u _ n - u ) d x \\\\ & \\to \\int _ { \\mathbb { R } ^ N } ( K \\ast g ( u ) ) g ( u ) d x . \\end{align*}"}
+{"id": "959.png", "formula": "\\begin{align*} \\begin{dcases*} T _ 1 f = - L _ 1 f - q _ r f = \\lambda f , & $ ~ \\Sigma $ \\\\ \\frac { \\partial f } { \\partial \\nu } + \\alpha _ \\theta f = 0 , & $ ~ \\partial \\Sigma $ \\end{dcases*} , \\end{align*}"}
+{"id": "4227.png", "formula": "\\begin{align*} \\Omega \\wedge F ^ 2 = 0 , \\Omega ^ { 0 , 2 } = \\Omega ^ { 2 , 0 } = 0 . \\end{align*}"}
+{"id": "1097.png", "formula": "\\begin{align*} \\langle u , v \\rangle : = \\int _ { D } \\Gamma ( u , v ) ( x ) d \\mu + \\int _ { D } u ( x ) v ( x ) d \\mu , \\ \\hbox { f o r a l l } \\ u , v \\in W ^ { 1 , 2 } ( D ) . \\end{align*}"}
+{"id": "7542.png", "formula": "\\begin{align*} 1 : = \\frac { [ X ^ { F _ 0 } ] } { e ^ { F _ 0 } ( N _ { X ^ { F _ 0 } } X ) } \\in A _ * ^ { F _ 0 } ( Z ( \\phi ) ) _ { l o c } . \\end{align*}"}
+{"id": "1929.png", "formula": "\\begin{align*} e _ f : = f - f _ h : = \\left ( \\Pi _ { \\lambda _ 1 , \\lambda _ 2 } f - f _ h \\right ) - \\left ( \\Pi _ { \\lambda _ 1 , \\lambda _ 2 } f - f \\right ) = : \\theta _ f - \\eta _ f , \\end{align*}"}
+{"id": "334.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l } \\left [ \\left ( \\mathsf { D } _ { j } \\left ( s \\right ) \\psi \\right ) \\right ] _ { \\operatorname * { D } ; j } \\left ( s \\right ) = \\psi , & \\left [ \\left ( \\mathsf { D } _ { j } \\left ( s \\right ) \\psi \\right ) \\right ] _ { \\operatorname * { N } ; j } ^ { \\operatorname * { e x t } } \\left ( s \\right ) = 0 . \\end{array} \\end{align*}"}
+{"id": "8825.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\mathbb { E } \\big [ \\norm { g ^ { i } ( t ) - \\bar { g } ( t ) } ^ { 2 } | \\mathcal { F } _ { t - 1 } \\big ] & \\leq \\sum _ { i = 1 } ^ { n } \\mathbb { E } \\big [ \\norm { g ^ { i } ( t ) } ^ { 2 } | \\mathcal { F } _ { t - 1 } \\big ] \\\\ & \\leq 9 \\kappa d \\sum _ { i = 1 } ^ { n } \\norm { \\nabla f _ { i } ( x ^ { i } { ( t - 1 ) } ) } ^ { 2 } + n d ^ { 2 } \\big ( \\frac { 9 h _ { t } ^ { 2 } \\kappa \\bar { L } ^ { 2 } } { 8 } + \\frac { 3 \\kappa \\sigma ^ { 2 } } { 2 h _ { t } ^ { 2 } } \\big ) , \\end{align*}"}
+{"id": "3408.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle \\sum _ { i = 1 } ^ 3 \\delta _ i v _ i ^ 2 = 1 , \\ \\sum _ { i = 1 } ^ 3 \\delta _ i v _ i V _ i = 0 , \\ \\sum _ { i = 1 } ^ 3 \\delta _ i V _ i ^ 2 = ( c - \\tilde { c } ) \\ \\mbox { a n d } \\ \\sum _ { i = 1 } ^ 3 { \\frac { V _ i } { v _ i } } = 0 \\end{array} \\end{align*}"}
+{"id": "4719.png", "formula": "\\begin{align*} D _ x & = \\{ ( \\ell _ i , \\lambda _ j ) : \\} , \\\\ D _ y & = \\{ ( \\lambda _ j , \\ell _ i ) : \\} . \\end{align*}"}
+{"id": "3632.png", "formula": "\\begin{align*} \\overline { \\nabla } _ X V = \\mu \\ , X + \\omega ( X ) P + \\phi \\ , A ( X ) - X ( \\phi ) \\eta . \\end{align*}"}
+{"id": "1607.png", "formula": "\\begin{align*} \\mathcal K _ k ( \\mathcal G _ T ) = \\{ K \\in \\mathrm { C r o s s } _ k : P ^ { ( k - 2 ) } _ J \\in \\mathcal G _ T ~ J \\in [ K ] ^ { k - 2 } \\} \\end{align*}"}
+{"id": "1202.png", "formula": "\\begin{align*} \\mathrm { p r } _ g h ( [ x ] _ P \\otimes [ y _ P ] ) = [ x ] _ P \\cdot [ y g ] _ P = \\begin{cases} [ x ] _ P & \\mathrm { i f } \\ ; [ x ] _ P = [ y g ] _ P \\\\ 0 & \\mathrm { o t h e r w i s e } \\end{cases} \\end{align*}"}
+{"id": "2458.png", "formula": "\\begin{align*} H _ \\kappa ( G ) = \\min _ { V \\in W \\in \\Gamma ( G ) } I ( W ; V ) , \\end{align*}"}
+{"id": "970.png", "formula": "\\begin{align*} \\Pr [ D _ i = x _ i ] \\geq ( 1 - \\exp ( - \\mu \\cdot ( q - 1 / 2 ) ^ 2 / ( 2 q ) ) ) \\cdot ( 1 - \\rho ^ * ) . \\end{align*}"}
+{"id": "2684.png", "formula": "\\begin{align*} c _ 3 \\ : = \\ \\inf \\left \\{ \\int _ X e ^ v \\omega _ 0 ^ n \\ | \\ v \\in ( X , c _ 0 b _ 0 \\omega _ 0 ) , \\ n \\log \\frac { ( n + 1 ) \\delta c _ { 1 } } { ( 2 \\alpha n + 2 \\beta ) c _ { 2 } } \\le \\sup _ X v \\le \\log ( c _ 0 + b _ 0 ) ^ n \\right \\} , \\end{align*}"}
+{"id": "4103.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { r - 1 } B ( \\sigma ^ k ( p _ 0 ) , p _ 1 ) = \\frac { d x ( p _ 0 ) \\cdot d x ( p _ 1 ) } { ( x ( p _ 0 ) - x ( p _ 1 ) ^ 2 } . \\end{align*}"}
+{"id": "6545.png", "formula": "\\begin{align*} \\Sigma _ \\Lambda = \\bigcup _ { \\xi = \\pm 1 , i \\in \\Lambda } ( \\Sigma _ \\Lambda \\cap I _ { \\xi , i } ) . \\end{align*}"}
+{"id": "318.png", "formula": "\\begin{align*} \\ell _ { j } \\left ( s \\right ) \\left ( \\mathsf { S } _ { j } \\left ( s \\right ) \\varphi , w \\right ) = \\left \\langle \\left ( \\gamma _ { \\operatorname * { D } ; j } \\left ( s \\right ) \\right ) ^ { \\prime } \\varphi , \\overline { w } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } \\quad \\forall w \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) , \\end{align*}"}
+{"id": "6244.png", "formula": "\\begin{align*} \\Delta ^ 4 \\xi = 0 , \\Delta ^ 4 \\xi = \\xi _ 1 - \\xi _ 2 + \\xi _ 3 - \\xi _ 4 . \\end{align*}"}
+{"id": "2935.png", "formula": "\\begin{align*} \\delta _ { j _ { x _ 1 } j _ { x _ 2 } } D _ { j _ { x _ 1 } j _ { x _ 2 } \\pi ( j _ 3 \\cdots j _ { n } ) } = 0 . \\end{align*}"}
+{"id": "889.png", "formula": "\\begin{align*} & C - \\frac { \\sqrt { a b } } { a } D = - \\frac { \\sqrt { a b } } { a } \\frac { \\alpha } { t ^ \\alpha } \\mathrm { e } ^ { - \\frac { \\alpha ( x + y ) } { t ^ \\alpha } } \\bigg ( \\frac y x \\bigg ) ^ { - \\frac { 1 + \\sqrt { a b } } { 2 } } I _ { \\sqrt { a b } + 1 } \\bigg ( \\frac { 2 \\alpha \\sqrt { x y } } { t ^ \\alpha } \\bigg ) . \\end{align*}"}
+{"id": "899.png", "formula": "\\begin{align*} n x ^ k \\bigg ( \\phi _ v - \\tau _ t + \\frac { 1 - \\alpha } { t } \\tau \\bigg ) - 2 \\phi _ { x u } - n k x ^ { k - 1 } \\xi - n x ^ k ( \\eta _ u - \\xi _ x ) = 0 , \\end{align*}"}
+{"id": "8778.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } \\eta _ t \\mathbb { E } [ \\norm { \\nabla f ( x _ t ) } ^ 2 ] & \\leq 4 \\sum _ { t = 1 } ^ { T } \\Big ( \\delta ( t ) - \\delta ( t + 1 ) \\Big ) + 4 \\sum _ { t = 1 } ^ { T } \\eta _ t B _ { t } ^ { 2 } + 4 \\sum _ { t = 1 } ^ { T } \\eta _ { t } ^ 2 V _ t \\\\ & = 4 \\delta ( 1 ) + 4 \\sum _ { t = 1 } ^ { T } \\eta _ t B _ { t } ^ { 2 } + 4 \\sum _ { t = 1 } ^ { T } \\eta _ { t } ^ 2 V _ t . \\end{align*}"}
+{"id": "873.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ t = u _ { x x } + \\frac { c } { x } u _ x + m x ^ k v _ x , \\\\ & v _ t = v _ { x x } + \\frac { c } { x } v _ x + n x ^ k u _ x , ~ ~ x > 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4976.png", "formula": "\\begin{align*} s ^ m = 2 ^ { m a } \\geq 2 ^ { 2 ( 2 n + 1 ) } = q ^ 2 \\geq 2 ^ { 1 4 } \\end{align*}"}
+{"id": "2021.png", "formula": "\\begin{align*} ( \\dot { \\mathrm { H } } ^ { s _ 0 , p } _ 0 ( \\Omega ) , \\dot { \\mathrm { H } } ^ { s _ 1 , p } _ 0 ( \\Omega ) ) _ { \\frac { s - s _ 0 } { s _ 1 - s _ 0 } , q } = \\dot { \\mathrm { B } } ^ { s } _ { p , q , 0 } ( \\Omega ) = \\dot { \\mathrm { B } } ^ { s } _ { p , q } ( \\Omega ) , \\end{align*}"}
+{"id": "4673.png", "formula": "\\begin{align*} | \\widetilde { S _ i } | = \\left | S _ i - \\bigcup _ { \\substack { j \\in [ 2 m + 1 ] \\\\ i \\neq j } } ( S _ i \\cap S _ j ) \\right | > \\frac { 2 n } { 2 k + 3 } - 2 - 2 m \\cdot \\epsilon n \\ge \\frac { n } { k + 2 } \\end{align*}"}
+{"id": "3331.png", "formula": "\\begin{align*} f _ Q ( q ) = \\sum _ { n \\in \\mathbb { N } ^ N } \\frac { q ^ { Q ( n ) } } { ( q ^ { d _ 1 } ; q ^ { d _ 1 } ) _ { n _ 1 } \\cdots ( q ^ { d _ N } ; q ^ { d _ N } ) _ { n _ N } } \\end{align*}"}
+{"id": "6106.png", "formula": "\\begin{align*} \\left ( P - P L P \\right ) ^ { - 1 } = \\sum _ { n \\ge 0 } ( P L P ) ^ n = \\sum _ { n \\ge 0 } P L ^ n P . \\end{align*}"}
+{"id": "6532.png", "formula": "\\begin{align*} \\inf _ { x \\in I _ i } \\left | \\frac { d ^ { r _ 1 } f } { d x ^ { r _ 1 } } ( x ) \\right | & \\geq \\left | \\frac { d ^ { r _ 1 } f } { d x ^ { r _ 1 } } ( x _ 0 ) \\right | - \\sup _ { x \\in I _ i } \\left | \\frac { d ^ { r _ 1 } f } { d x ^ { r _ 1 } } ( x ) - \\frac { d ^ { r _ 1 } f } { d x ^ { r _ 1 } } ( x _ 0 ) \\right | \\\\ & \\geq \\tau - \\sup _ { x \\in I _ i } \\left | \\frac { d ^ { r _ 1 + 1 } f } { d x ^ { r _ 1 + 1 } } ( x ) \\right | \\cdot | I _ i | \\\\ & \\geq \\tau - A \\cdot \\frac { \\tau } { 2 A } = \\frac \\tau 2 . \\end{align*}"}
+{"id": "154.png", "formula": "\\begin{align*} ( x _ 1 , \\dots , x _ n ) = ( t _ 1 t _ 2 \\cdots t _ n , t _ 2 \\cdots t _ n , \\dots , t _ { n - 1 } t _ n , t _ n ) , \\end{align*}"}
+{"id": "5442.png", "formula": "\\begin{align*} F _ 2 ( x ) : = \\sup \\big \\{ F ( x _ 1 ) + F ( x _ 2 ) : x _ 1 , x _ 2 \\geq 0 , \\ x _ 1 + x _ 2 = x \\big \\} , x \\geq 0 . \\end{align*}"}
+{"id": "38.png", "formula": "\\begin{align*} a _ 0 | z \\rangle = z \\ , | z \\rangle \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; 1 = \\frac { 1 } { \\pi } \\int _ { \\mathbb { C } } | z \\rangle \\langle z | \\ , \\dd z . \\end{align*}"}
+{"id": "5569.png", "formula": "\\begin{align*} V _ k ( \\zeta ) = U _ k ( \\zeta ) - \\Psi _ k ( \\zeta ) - \\Phi _ k ( \\zeta ) \\end{align*}"}
+{"id": "2338.png", "formula": "\\begin{align*} f _ { u g } ( \\tau _ { u h } ) = f _ g ( \\tau _ h ) , \\forall u , g , h \\in G . \\end{align*}"}
+{"id": "3912.png", "formula": "\\begin{align*} \\rho _ K ( u ) = \\max \\{ a : a u \\in K \\} . \\end{align*}"}
+{"id": "6983.png", "formula": "\\begin{align*} y '' + \\left ( Q ( z ) - \\frac { 1 } { 4 } ( h e ^ { p ( z ) } ) ^ 2 - \\frac { 1 } { 2 } h ' ( z ) e ^ { p ( z ) } - \\frac { 1 } { 2 } h ( z ) p ' ( z ) e ^ { p ( z ) } \\right ) y = 0 . \\end{align*}"}
+{"id": "7321.png", "formula": "\\begin{align*} F _ 2 ( \\lambda , x , \\eta ( \\lambda , x ) ) = 0 , ( \\lambda , x ) \\in I \\times B _ X . \\end{align*}"}
+{"id": "2326.png", "formula": "\\begin{align*} \\sum _ { { g \\in G } } f _ g ( x ) \\tau _ g = \\sum _ { { g \\in G } } \\zeta _ g ( U x ) V \\delta _ g = V \\left ( \\sum _ { { g \\in G } } \\zeta _ g ( U x ) \\delta _ g \\right ) = V U x = x . \\end{align*}"}
+{"id": "6602.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\norm * { b _ n - b } _ { L ^ \\infty } = 0 , \\end{align*}"}
+{"id": "1463.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 , n + 1 } \\sigma _ { i } \\Big ( \\sum \\limits _ { j \\in I } k _ { i j } ^ { \\mathbf { c } ^ t } \\sigma _ { j } - 2 \\mu _ { i } \\Big ) + 2 \\sum \\limits _ { i \\in I \\setminus \\{ 1 , n + 1 \\} } \\sigma _ { i } \\Big ( \\sum \\limits _ { j \\in I } k _ { i j } ^ { \\mathbf { c } ^ t } \\sigma _ { j } - 2 \\mu _ { i } \\Big ) = 2 \\Big ( \\sum \\limits _ { i = 1 , n + 1 } \\mu _ i \\sigma _ i + 2 \\sum \\limits _ { i \\in I \\setminus \\{ 1 , n + 1 \\} } \\mu _ i \\sigma _ i \\Big ) , \\end{align*}"}
+{"id": "546.png", "formula": "\\begin{align*} \\mathbf N ( \\mathbf 0 ) = \\mathbf 0 \\end{align*}"}
+{"id": "6470.png", "formula": "\\begin{align*} f ( x ; \\epsilon ) = f _ 0 ( x ) + \\epsilon f _ 1 ( x ) + O ( \\epsilon ^ 2 ) \\end{align*}"}
+{"id": "3173.png", "formula": "\\begin{align*} \\bar { \\gamma } = { } \\sqrt [ n ] { \\gamma _ 1 \\gamma _ 2 \\cdots { } \\gamma _ n } \\end{align*}"}
+{"id": "7915.png", "formula": "\\begin{align*} 0 = 1 + ( \\zeta + \\zeta ^ { - 1 } ) \\mathrm { F P d i m } ( x ) , \\end{align*}"}
+{"id": "7948.png", "formula": "\\begin{align*} A = \\{ v _ j \\in Q \\colon v _ { 1 , j } \\in D _ 1 \\mbox { o r } v _ { 2 , j } \\in D _ 1 \\mbox { w h e r e } j \\in [ r ] \\} . \\end{align*}"}
+{"id": "3080.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta R & = { R _ { { \\rm { P L , u p p e r } } } } - { R _ { { \\rm { Q L , u p p e r } } } } = \\sum \\limits _ { k = 1 } ^ { { N _ { \\rm { R } } } } { { { \\log } _ 2 } \\frac { { { \\sigma ^ 2 } + { \\mathbb E } \\left \\{ { { { \\left | { \\xi _ k ^ \\star } \\right | } ^ 2 { p _ k ^ \\star } } } \\right \\} } } { { { \\sigma ^ 2 } + { \\mathbb E } \\left \\{ { { { \\left | { { \\xi _ k } } \\right | } ^ 2 { p _ k ^ \\star } } } \\right \\} } } } . \\end{aligned} \\end{align*}"}
+{"id": "3587.png", "formula": "\\begin{align*} c _ n = { { { \\frac { { { \\sigma ^ 2 } L _ { \\rm R } \\left ( { \\kappa + 1 } \\right ) } } { { E { N _ { \\rm { T } } } N _ { { \\rm { S } } , n } ^ 2 \\rho _ n ^ 2 \\kappa } } } } } . \\end{align*}"}
+{"id": "7351.png", "formula": "\\begin{align*} & g _ M = \\sum _ { k \\le M } \\ln \\int \\prod _ { a = 0 } ^ u d p _ X ( y ^ a ) \\exp \\Big ( \\tau _ { \\pi _ k } \\sum _ { a < b } ^ { 0 , u } y ^ a y ^ b \\Big ) = \\sum _ { k \\le M } \\ln \\int \\prod _ { a = 0 } ^ u d p _ X ( y ^ a ) \\exp \\Big ( \\tau _ { k } \\sum _ { a < b } ^ { 0 , u } y ^ a y ^ b \\Big ) \\end{align*}"}
+{"id": "1442.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & s _ 1 = r _ 2 \\\\ & s _ 2 = r _ 2 + 1 \\\\ & \\quad \\quad \\vdots \\\\ & s _ { n - r _ 2 + 2 } = n + 1 \\end{aligned} \\right . \\left \\{ \\begin{aligned} & s _ { n - r _ 2 + 3 } = 1 \\\\ & s _ { n - r _ 2 + 4 } = 2 \\\\ & \\qquad \\vdots \\\\ & s _ { | J _ 0 | } = r _ 1 \\end{aligned} \\right . \\ \\ ; \\end{align*}"}
+{"id": "8653.png", "formula": "\\begin{align*} y _ i = - \\frac { \\partial f ( t _ 1 , . . . , t _ n ) } { \\partial t _ i } , ~ ~ ~ i = 1 , . . . , n \\end{align*}"}
+{"id": "7006.png", "formula": "\\begin{align*} W \\left ( e , \\bar { e } , \\bar { e } , n \\right ) = - W \\left ( n , \\bar { n } , \\bar { e } , n \\right ) , ~ ~ ~ W \\left ( e , \\bar { e } , e , n \\right ) = W \\left ( n , \\bar { n } , e , n \\right ) , \\end{align*}"}
+{"id": "8775.png", "formula": "\\begin{align*} b _ { t } < \\frac { 2 ( t _ { 0 } - 1 ) b _ { t _ { 0 } } } { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { ( 1 - p _ { i } ) t ^ { p _ { i } } } . \\end{align*}"}
+{"id": "6295.png", "formula": "\\begin{align*} q ^ 4 _ { \\lambda , m } : = h _ 3 \\left ( \\frac 1 2 ( \\xi _ 1 + \\xi _ 2 + \\xi _ 3 + \\xi _ 4 ) , \\frac 1 2 ( ( \\xi _ 1 - \\xi _ 3 ) ^ 2 + ( \\xi _ 2 - \\xi _ 4 ) ^ 2 ) \\right ) . \\end{align*}"}
+{"id": "5626.png", "formula": "\\begin{align*} K _ W + D _ W = p ^ * ( K _ X + D _ X ) + \\sum a _ j E _ j , \\end{align*}"}
+{"id": "6880.png", "formula": "\\begin{align*} \\ell ( \\widetilde { X } , r ) \\cdot r ^ * \\sigma = \\Big ( \\sum _ { i = 1 } ^ r a _ i C _ i \\Big ) \\cdot \\sigma . \\end{align*}"}
+{"id": "5032.png", "formula": "\\begin{align*} \\triangle _ f u + ( b _ 0 - \\lambda ) u = v , \\end{align*}"}
+{"id": "7769.png", "formula": "\\begin{align*} | Z ^ 0 | ^ 2 = 1 6 ( \\rho + c _ { \\ell } ) e ^ { \\mathcal { K } } = 4 R ^ 2 \\ , , \\end{align*}"}
+{"id": "5139.png", "formula": "\\begin{align*} { \\bf y } _ t = { \\bf M } _ t { \\bf x } _ t = \\sum _ { k \\in [ K ] } { \\bf M } _ { t , k } \\underbrace { \\begin{bmatrix} V _ { t , s _ 1 } ^ k \\\\ V _ { t , s _ 2 } ^ k \\\\ \\vdots \\\\ V _ { t , s _ n } ^ k \\end{bmatrix} } _ { { \\bf V } _ { t , k } } { \\bf W } _ k , \\end{align*}"}
+{"id": "8689.png", "formula": "\\begin{align*} f = p _ 3 ( u _ 2 ) + q _ 3 ( u _ 2 ) u _ 1 , ~ ~ ~ x _ 1 = p _ 1 ( u _ 2 ) + q _ 1 ( u _ 2 ) u _ 1 , ~ ~ ~ x _ 2 = p _ 2 ( u _ 2 ) + q _ 2 ( u _ 2 ) u _ 1 , \\end{align*}"}
+{"id": "7996.png", "formula": "\\begin{align*} | f _ { \\lambda , \\alpha } ( x ) - f _ { \\lambda , \\alpha } ( y ) | & = ( x - \\lambda ) ^ { \\alpha } - ( y - \\lambda ) ^ { \\alpha } \\\\ & = \\int _ { y - \\lambda } ^ { x - \\lambda } \\alpha t ^ { \\alpha - 1 } d t \\\\ & \\leq \\int _ { y - \\lambda } ^ { x - \\lambda } \\alpha ( t - y + \\lambda ) ^ { \\alpha - 1 } d t = ( x - y ) ^ { \\alpha } = | x - y | ^ { \\alpha } \\end{align*}"}
+{"id": "6132.png", "formula": "\\begin{align*} R & = \\{ \\ ! \\{ \\ , \\{ v , \\varphi ( v ) \\} \\mid v \\in S ^ { + } \\sqcup T ^ { - } \\ , \\} \\ ! \\} \\\\ B & = \\{ \\ ! \\{ \\ , \\{ v , \\psi ( v ) \\} \\mid v \\in T ^ { + } \\sqcup U ^ { - } \\ , \\} \\ ! \\} . \\end{align*}"}
+{"id": "6938.png", "formula": "\\begin{align*} K ^ { \\star } \\ ! = \\ ! ( \\frac { 1 } { M } \\ ! \\sum _ { k = 1 } ^ { M } \\Psi ( y _ k ) \\Psi ( x _ k ) ^ T ) ( \\frac { 1 } { M } \\ ! \\sum _ { k = 1 } ^ { M } \\Psi ( x _ k ) \\Psi ( x _ k ) ^ T ) ^ { \\dagger } . \\end{align*}"}
+{"id": "2695.png", "formula": "\\begin{align*} A = & f ( 1 ) ^ 2 - g ( 1 ) ^ 2 \\\\ B = & f ( - 1 ) ^ 2 - g ( - 1 ) ^ 2 \\\\ C = & | f ( i ) | ^ 2 - | g ( i ) | ^ 2 \\\\ D = & \\left ( | f ( \\omega ) | ^ 2 + | g ( \\omega ) | ^ 2 \\right ) \\left ( | f ( \\omega ^ 3 ) | ^ 2 + | g ( \\omega ^ 3 ) | ^ 2 \\right ) . \\end{align*}"}
+{"id": "1254.png", "formula": "\\begin{align*} J ( f ) = \\frac { P ( f ) } { \\| \\nabla f \\| _ { L ^ 2 } ^ { 2 p } } . \\end{align*}"}
+{"id": "178.png", "formula": "\\begin{align*} { \\rm c h } ( \\mathcal { V } _ i ) = 1 + ( 2 4 8 - c _ 2 ( W _ i ) + \\cdots ) q + \\cdots , \\end{align*}"}
+{"id": "4568.png", "formula": "\\begin{align*} \\partial _ t \\varphi = \\dot u ^ + _ N + \\hat D _ 1 \\dot H ^ + _ { N } + \\hat D _ 2 \\dot H ^ - _ N + \\hat D _ 3 \\varphi \\ , , \\end{align*}"}
+{"id": "6038.png", "formula": "\\begin{align*} f _ { \\delta } ( x ) = f \\ast \\phi _ { \\delta } ( x ) = \\int _ { \\mathbb { R } ^ { d } } f ( y ) \\phi _ { \\delta } ( x - y ) d y . \\end{align*}"}
+{"id": "589.png", "formula": "\\begin{align*} \\rho ( \\mu , S ) = \\norm { \\mathbf u ^ \\mu - \\mathbf u } _ { L ^ 2 ( \\Omega , \\mathbf X ^ { \\mathbf s , b } ( 0 , S ) ) } + \\norm { ( 1 - P _ \\mu ) \\mathbf u } _ { L ^ 2 ( \\Omega , \\mathbf X ^ { \\mathbf s , b } ( 0 , T ) ) } \\to 0 , \\end{align*}"}
+{"id": "1699.png", "formula": "\\begin{align*} f ( \\boldsymbol { \\xi } ) = P _ { \\texttt { b } ; \\mu } ( \\boldsymbol { \\xi } ; q , q _ 0 ) P _ { \\texttt { b } ; \\nu } ( \\boldsymbol { \\xi } ; q , q _ 0 ) \\quad \\mu , \\nu \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { b } } \\end{align*}"}
+{"id": "6953.png", "formula": "\\begin{align*} - 2 \\sum _ { k = 1 } ^ d \\sum _ { i = 1 } ^ { k - 1 } \\sum _ { j = k + 1 } ^ { d } & \\frac { 1 } { ( ( x _ k - x _ i ) + \\epsilon ^ 2 ) ( ( x _ j - x _ k ) + \\epsilon ^ 2 ) } \\\\ & = - 2 \\sum _ { k = 1 } ^ d \\sum _ { i = k + 1 } ^ { d } \\sum _ { j = i + 1 } ^ { d } \\frac { 1 } { ( ( x _ i - x _ k ) + \\epsilon ^ 2 ) ( ( x _ j - x _ i ) + \\epsilon ^ 2 ) } . \\end{align*}"}
+{"id": "701.png", "formula": "\\begin{align*} \\mathbf u ( t ) = \\mathbf S ( t ) \\mathbf u _ 0 + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf N \\left ( \\Theta _ R ^ { \\mathbf u } ( s ) \\mathbf u ( s ) \\right ) \\ , d s + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf M \\left ( \\mathbf u ( s ) \\right ) \\ , d W ( s ) , \\end{align*}"}
+{"id": "7606.png", "formula": "\\begin{align*} \\frac { 1 } { ( 2 \\pi b ) ^ { 1 / 2 } } \\int _ { \\delta } ^ { \\infty } e ^ { - \\beta ^ { 2 / 3 } N ^ { 2 / 3 } \\frac { a ^ 2 } { 4 b } } d a & \\leq \\frac { N ^ { - 1 / 3 } \\beta ^ { - 1 / 3 } } { ( 2 \\pi b N ^ { - 2 / 3 } \\beta ^ { - 2 / 3 } ) ^ { 1 / 2 } } \\int _ { - \\infty } ^ { \\infty } e ^ { - \\frac { a ^ 2 } { 4 b N ^ { - 2 / 3 } \\beta ^ { - 2 / 3 } } } d a \\\\ & \\leq C N ^ { - 1 / 3 } \\beta ^ { - 1 / 3 } . \\end{align*}"}
+{"id": "6192.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ n \\eta _ { X _ i } - \\eta _ u + \\eta _ v \\right ) ( ( \\mu , ( A , T ) ) ) = { \\sum } _ n A - ( ) + ( ) . \\end{align*}"}
+{"id": "1415.png", "formula": "\\begin{align*} \\tilde \\varphi _ { n , i } ( x ) = \\varphi ^ K _ { n , i } ( x ) + \\sum _ { k = 1 } ^ K ( \\tilde Q _ { n , i ; k , 0 } ( x ) \\varphi ^ K _ { k , 0 } ( x ) - \\tilde Q _ { n , i ; k , 1 } ( x ) \\varphi ^ K _ { k , 1 } ( x ) ) , n = \\overline { 1 , K } . \\end{align*}"}
+{"id": "3070.png", "formula": "\\begin{align*} { \\bf { H } } = { { \\bf { A } } _ { \\rm { R } } } { { \\bf { \\Xi } } _ \\parallel } { \\bf { A } } _ { \\rm { T } } ^ H + { { \\bf { A } } _ { { \\rm { R , } } \\bot } } { { \\bf { \\Xi } } _ \\bot } { \\bf { A } } _ { { \\rm { T , } } \\bot } ^ H , \\end{align*}"}
+{"id": "5420.png", "formula": "\\begin{align*} ( A v ) _ \\gamma = \\abs { \\gamma } v _ \\gamma . \\end{align*}"}
+{"id": "6648.png", "formula": "\\begin{align*} c _ { 2 n } ( p ) = - { \\alpha _ n \\over \\pi ^ { 2 n } } \\prod _ { l = 0 } ^ { n - 1 } ( p ^ 2 - l ^ 2 ) , \\end{align*}"}
+{"id": "4779.png", "formula": "\\begin{align*} \\beta _ { p , a } ( t ) = P _ { 1 } ^ { t } + \\dots + P _ { m } ^ { t } \\end{align*}"}
+{"id": "4073.png", "formula": "\\begin{align*} P ( x , y ) = a ( x ) y ^ 2 + b ( x ) y + c ( x ) = 0 , \\end{align*}"}
+{"id": "7315.png", "formula": "\\begin{align*} ( M , K ) \\cdot ( \\tilde { M } , \\tilde { K } ) = ( M \\tilde { M } , K + M \\tilde { K } M ^ \\ast ) , \\end{align*}"}
+{"id": "4849.png", "formula": "\\begin{align*} B & = \\lambda _ 1 P _ 1 + \\lambda _ 2 P _ 2 \\\\ & = \\lambda _ 1 ( P _ 1 + \\frac { \\lambda _ 2 } { \\lambda _ 1 } P _ 2 ) \\\\ & = \\lambda _ 1 ( I + ( \\frac { \\lambda _ 2 } { \\lambda _ 1 } - 1 ) P _ 2 ) \\end{align*}"}
+{"id": "18.png", "formula": "\\begin{align*} \\omega _ { \\R ^ { d } } : = 1 - \\varphi _ { \\R ^ { d } } , g _ { \\R ^ { d } } : = v _ { \\R ^ { d } } ( 1 - \\omega _ { \\R ^ { d } } ) = v _ { \\R ^ { d } } \\varphi _ { \\R ^ { d } } , \\end{align*}"}
+{"id": "4058.png", "formula": "\\begin{align*} \\rho _ { k \\left ( n - \\frac { 1 } { 2 } \\right ) + \\frac { 1 } { 2 } } = k \\left ( n - \\frac { 1 } { 2 } \\right ) \\pi , \\ , n \\in \\mathbb { Z } . \\end{align*}"}
+{"id": "2836.png", "formula": "\\begin{align*} \\ell ( \\phi ) = \\sum _ { i = 0 } ^ { N _ \\mathrm { c } - 1 } \\ln { \\sum _ { x \\in \\mathcal { K } } \\exp { \\left ( - \\frac { 1 } { 2 \\sigma ^ 2 } { \\vert y _ i - x \\mathrm { e } ^ { j \\phi } \\vert } ^ 2 \\right ) } } . \\end{align*}"}
+{"id": "6438.png", "formula": "\\begin{align*} M _ n = \\begin{pmatrix} x ^ n A & y ^ n B \\\\ z ^ n A & w ^ n B \\end{pmatrix} M _ n ' = \\begin{pmatrix} x ^ n A & z ^ n B \\\\ y ^ n A & w ^ n B \\end{pmatrix} . \\end{align*}"}
+{"id": "6080.png", "formula": "\\begin{align*} g _ k = \\nabla f _ { S _ k } ( \\omega _ k ) - \\nabla f _ { S _ k } ( \\phi _ k ) + \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla f _ i ( \\phi _ i ^ k ) \\end{align*}"}
+{"id": "7955.png", "formula": "\\begin{align*} \\lambda _ { \\bf i } = \\sum _ { j = 1 } ^ k \\frac { n _ j ( n _ j - 2 j + 1 ) } { 2 } . \\end{align*}"}
+{"id": "5324.png", "formula": "\\begin{align*} g ^ { ( n ) } ( ( 2 k + 1 ) 2 ^ { - n - 1 } ) & = \\frac { 1 } { 2 } ( g ^ { ( n ) } ( k 2 ^ { - n } ) + g ^ { ( n ) } ( ( k + 1 ) 2 ^ { - n } ) ) \\\\ & = \\left ( k ^ 2 + k + \\frac { 1 } { 2 } \\right ) 4 ^ { - n } \\\\ h ^ { ( n + 2 ) } ( ( 2 k + 1 ) 2 ^ { - n - 1 } ) & = - 1 \\\\ \\Rightarrow g ^ { ( n + 1 ) } ( ( 2 k + 1 ) 2 ^ { - n - 1 } ) & = \\left ( k ^ 2 + k + \\frac { 1 } { 2 } \\right ) 4 ^ { - n } + 2 ^ { - 2 n - 3 } ( - 2 ) \\\\ & = ( ( 2 k + 1 ) 2 ^ { - n - 1 } ) ^ 2 , \\end{align*}"}
+{"id": "4161.png", "formula": "\\begin{align*} \\bar \\Delta _ { k } = \\frac { 1 + \\rho _ { k } + 3 \\hat { \\rho } _ { k } + 3 \\hat { \\rho } _ { k } \\rho _ { k } + 3 \\hat { \\rho } _ { k } ^ { 2 } + \\hat { \\rho } _ { k } ^ { 2 } \\rho _ { k } + \\hat { \\rho } _ { k } ^ { 3 } } { \\mu \\rho _ { k } \\left ( 1 + \\hat { \\rho } _ { k } \\right ) } , \\end{align*}"}
+{"id": "1365.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\geq - n _ 0 } c _ f ^ + ( n ) q ^ { n } + c _ f ^ - ( 0 ) y ^ { 1 - 2 \\kappa } + \\sum _ { \\substack { n \\leq n _ 0 \\\\ n \\neq 0 } } c _ f ^ - ( n ) \\Gamma ( 1 - 2 \\kappa , - 4 \\pi n y ) q ^ { n } , \\end{align*}"}
+{"id": "3323.png", "formula": "\\begin{align*} q ^ { - r d ( m - a - t ) } \\begin{bmatrix} m - a \\\\ t \\end{bmatrix} _ q \\alpha \\beta \\end{align*}"}
+{"id": "2386.png", "formula": "\\begin{align*} d W & = W \\left ( \\pi _ 1 \\mu d t + \\pi _ 1 \\sigma d B + ( 1 - \\pi _ 1 ) r d t \\right ) \\\\ d W ^ 2 & = W ^ 2 \\pi _ 1 ^ 2 \\sigma ^ 2 d t \\end{align*}"}
+{"id": "7275.png", "formula": "\\begin{align*} Y [ n ] = \\mathop { \\sum } \\limits _ { m = - \\infty } ^ { \\infty } X [ m ] \\cdot h [ n - m ] + W [ n ] , \\end{align*}"}
+{"id": "1073.png", "formula": "\\begin{align*} \\left . \\begin{array} { c } \\left ( u _ { t } - L _ { 0 } u \\right ) ^ { 2 } \\varphi _ { \\lambda , \\nu } \\psi ^ { - \\nu + 1 } = \\left [ \\left ( z _ { 1 } + z _ { 3 } \\right ) + z _ { 2 } + z _ { 4 } \\right ] ^ { 2 } \\psi ^ { - \\nu + 1 } \\geq \\\\ \\geq \\left ( z _ { 1 } + z _ { 3 } \\right ) ^ { 2 } \\psi ^ { - \\nu + 1 } + 2 z _ { 1 } z _ { 2 } \\psi ^ { - \\nu + 1 } + 2 z _ { 1 } z _ { 4 } \\psi ^ { - \\nu + 1 } + 2 z _ { 2 } z _ { 3 } \\psi ^ { - \\nu + 1 } + 2 z _ { 3 } z _ { 4 } \\psi ^ { - \\nu + 1 } . \\end{array} \\right . \\end{align*}"}
+{"id": "3622.png", "formula": "\\begin{align*} \\int _ { y ( \\Omega ) } \\phi \\ , { \\rm d } X = \\eta \\ , { \\mathcal L } ^ 3 ( y ( \\Omega ) ) \\end{align*}"}
+{"id": "3700.png", "formula": "\\begin{align*} L ' _ x & : = \\{ y \\in D ^ + \\ , | \\ , y _ 1 \\in ( x _ 1 , 1 ) \\ , \\ , \\& \\ , \\ , y _ 2 \\in ( 0 , y _ 1 - x _ 1 + x _ 2 ) \\} \\subseteq L _ t , \\\\ L '' _ x & : = \\big [ ( ( 0 , x _ 1 ] \\cup [ 1 , \\infty ) ) \\times ( 0 , x _ 2 ) \\big ] \\setminus L ''' _ x , \\end{align*}"}
+{"id": "7247.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta & = \\cosh B - \\sinh B \\\\ & = e ^ { - B } \\\\ \\log ( \\Delta ^ { - 1 } ) & = B \\\\ \\log ( \\Sigma _ t ) & = B , \\end{aligned} \\end{align*}"}
+{"id": "2290.png", "formula": "\\begin{align*} \\| g _ 1 ( x ) \\| _ { L ^ 2 } \\geq e ^ { - \\pi \\alpha \\left ( \\frac { n - 1 } { 2 } \\right ) ^ 2 } \\left ( \\sum _ { 1 \\leq | k | \\leq \\frac { n - 1 } { 2 } } \\left | \\sum _ { j = 1 } ^ { n } e ^ { 2 \\pi i k \\varepsilon _ j } e ^ { - 2 \\pi i k \\frac { j } { n } } \\right | ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "4535.png", "formula": "\\begin{align*} I _ { 1 , \\ast } ( t ) : = I ( t ) + I _ 0 ( t ) + I _ { \\sigma } ( t ) + I _ { 2 } ( t ) \\ , . \\end{align*}"}
+{"id": "2404.png", "formula": "\\begin{align*} m _ { k + 1 } : = \\inf \\left \\{ n \\in \\N : ( y _ { n } + E _ { n } ) \\cap \\bigcup _ { i = 1 } ^ { k } ( y _ { m _ i } + E _ { m _ i } ) = \\emptyset \\right \\} . \\end{align*}"}
+{"id": "6015.png", "formula": "\\begin{align*} d _ { T V } ( \\mu , \\nu ) = \\sup \\limits _ { \\Vert f \\Vert _ { \\infty } \\leq 1 } \\big \\vert \\int _ { \\mathbb { R } ^ { d } } f ( x ) \\mu ( d x ) - \\int _ { \\mathbb { R } ^ { d } } f ( x ) \\nu ( d x ) \\big \\vert . \\end{align*}"}
+{"id": "4177.png", "formula": "\\begin{align*} \\Phi ^ { ( d ) } _ { \\mathbf { N } } ( \\boldsymbol { \\omega } ) & = e ^ { \\lambda | u | } \\exp ( - \\lambda \\| \\mathbf { w } \\| ) \\\\ & = \\exp \\bigg ( - \\lambda \\Big ( \\sqrt { \\norm { \\boldsymbol { \\omega } } ^ 2 + \\vert u \\vert ^ 2 } - \\vert u \\vert \\Big ) \\bigg ) , \\end{align*}"}
+{"id": "325.png", "formula": "\\begin{align*} f \\left ( \\left ( \\mathbf { m } , v \\right ) \\right ) : = \\left \\langle \\psi , \\gamma _ { \\mathbf { n } ; j } \\left ( s \\right ) \\overline { \\mathbf { m } } \\right \\rangle _ { \\Gamma _ { j } } . \\end{align*}"}
+{"id": "8580.png", "formula": "\\begin{align*} \\nabla _ { X , z } ^ { \\mu } \\phi _ b = I ^ { \\mu } ( J _ { \\Sigma _ b } ) ^ T ( I ^ { \\mu } ) ^ { - 1 } ( \\nabla ^ { \\mu } _ { X , z } \\Phi ) \\circ \\Sigma _ b . \\end{align*}"}
+{"id": "5190.png", "formula": "\\begin{align*} \\omega _ i = \\sum _ { j \\leq i } L _ j \\end{align*}"}
+{"id": "4518.png", "formula": "\\begin{align*} e ''' _ k : = \\mathcal { L } ' ( S _ { \\theta _ k } { \\mathbf V } _ k , S _ { \\theta _ k } \\Psi _ k ) ( \\delta { \\mathbf V } _ k , \\delta \\Psi _ k ) - \\mathcal { L } ' ( { \\mathbf V } _ { k + \\frac { 1 } { 2 } } , \\Psi _ { k + \\frac { 1 } { 2 } } ) ( \\delta { \\mathbf V } _ k , \\delta \\Psi _ k ) , \\end{align*}"}
+{"id": "8574.png", "formula": "\\begin{align*} \\begin{cases} \\nabla ^ { \\mu } _ { X , z } \\cdot P ( \\Sigma _ b ) \\nabla _ { X , z } ^ { \\mu } \\phi _ b = 0 S _ b \\\\ \\phi _ b | _ { z = 0 } = \\psi , \\partial _ { n _ { b } } ^ { P _ b } \\phi _ b | _ { z = - h _ b } = 0 , \\end{cases} \\end{align*}"}
+{"id": "3025.png", "formula": "\\begin{align*} \\phi ( H _ n ) = \\lambda \\phi ( H _ { n - 1 } ) - \\phi ( H _ { n - 2 } ) . \\end{align*}"}
+{"id": "3200.png", "formula": "\\begin{align*} \\left \\Vert x - x _ { k - 1 } \\right \\Vert _ { A ^ { T } A } ^ { 2 } - \\left \\Vert x - x _ { k } \\right \\Vert _ { A ^ { T } A } ^ { 2 } = \\phi _ { k } ^ { 2 } \\left ( \\left [ - \\frac { \\theta _ k \\phi _ { k - 1 } } { \\rho _ k \\phi _ k } \\right ] w _ { k } ^ { T } v _ { k } - \\frac { \\theta _ { k + 1 } } { \\rho _ { k } } v _ { k + 1 } ^ { T } w _ { k } \\right ) . \\end{align*}"}
+{"id": "1649.png", "formula": "\\begin{align*} f ( t ) = t ^ { n - 1 } \\left ( 1 - \\frac { s ( o ) } { 6 n } t ^ 2 + O ( t ^ 4 ) \\right ) \\end{align*}"}
+{"id": "2689.png", "formula": "\\begin{align*} \\alpha _ 0 = a _ 0 - a _ 4 , \\alpha _ 1 = a _ 1 - a _ 5 , & \\alpha _ 2 = a _ 2 - a _ 6 , \\alpha _ 3 = a _ 3 - a _ 7 , \\\\ \\gamma _ 0 = a _ 0 + a _ 4 , \\gamma _ 1 = a _ 1 + a _ 5 , & \\gamma _ 2 = a _ 2 + a _ 6 , \\gamma _ 3 = a _ 3 + a _ 7 , \\end{align*}"}
+{"id": "5259.png", "formula": "\\begin{align*} & \\sum _ { t = 1 } ^ T \\sum _ { i = 1 } ^ n \\frac { q _ { i , t + 1 } ^ \\top g _ { i , t } ( x _ { i , t } ) } { \\gamma _ t } \\le h _ T + \\sum _ { t = 1 } ^ T \\frac { n F } { \\gamma _ { t } } + \\sum _ { t = 1 } ^ T \\frac { \\tilde { \\Delta } _ t } { \\gamma _ { t } } + \\sum _ { t = 1 } ^ T \\sum _ { i = 1 } ^ n \\frac { \\Delta _ { i , t } ( y ) } { \\gamma _ { t } } . \\end{align*}"}
+{"id": "3518.png", "formula": "\\begin{align*} Z _ { \\Gamma _ 0 ( p ) } = Z _ { \\Gamma } ( s , \\lambda _ p ) , \\end{align*}"}
+{"id": "1025.png", "formula": "\\begin{align*} w ( \\pi _ k ^ * ( K ) ) & = \\tfrac { d } { d t } \\big | _ { t = 0 } \\pi _ k ^ * ( K ) ( v _ t ) = \\tfrac 1 { k ! } \\tfrac { d } { d t } \\big | _ { t = 0 } g ( K , v _ t ^ k ) = \\tfrac 1 { ( k - 1 ) ! } g ( K , w \\cdot v ^ { k - 1 } ) \\\\ & = \\tfrac 1 { ( k - 1 ) ! } g ( w \\lrcorner K , v ^ { k - 1 } ) , \\end{align*}"}
+{"id": "4572.png", "formula": "\\begin{align*} \\vert \\mathcal J _ { 1 } \\vert \\le & \\varepsilon I _ { 1 , n } ( t ) + \\frac { C _ 2 } { \\varepsilon } \\left \\{ \\Vert { \\mathbf F } \\Vert _ { L ^ 2 ( \\Omega _ t ) } ^ 2 + \\Vert \\varphi ( t ) \\Vert _ { L ^ 2 ( \\mathbb R ) } ^ 2 + \\int _ 0 ^ t \\Vert \\varphi ( s ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } d s \\right \\} \\\\ & + \\frac { C _ 3 } { \\varepsilon } \\int _ 0 ^ t ( I _ { 1 , \\ast } + I _ { 1 , n } ) ( s ) d s \\ , . \\end{align*}"}
+{"id": "8765.png", "formula": "\\begin{align*} \\norm { \\nabla f _ { \\omega } ( z ) - \\nabla f _ { \\omega } ( x ) } ^ 2 \\leq \\sum _ { i = 1 } ^ { d } \\Big [ 8 \\alpha ^ 2 ( 1 + \\delta ) ^ 2 ( z _ { i } - x _ i ) ^ 2 + 2 r ^ 2 h ^ { 2 ( \\beta - 1 ) } \\big ( \\eta _ { 0 } ( z _ i h ^ { - 1 } ) - \\eta _ { 0 } ( x _ { i } h ^ { - 1 } ) \\big ) ^ 2 \\Big ] . \\end{align*}"}
+{"id": "6395.png", "formula": "\\begin{align*} a \\triangleleft 1 _ { H } = \\mathcal { R } ^ { - 1 } ( a _ { 1 } \\otimes 1 _ { H } ) a _ { 2 } \\mathcal { R } ( a _ { 3 } \\otimes 1 _ { H } ) = \\varepsilon ( a _ { 1 } ) a _ { 2 } \\varepsilon ( a _ { 3 } ) = a \\end{align*}"}
+{"id": "6575.png", "formula": "\\begin{align*} T ( \\sigma ) = { { H } } _ { \\Lambda ( \\tilde N ) } ( \\sigma ) , \\ \\xi = \\xi _ 1 = e ^ { - 1 0 N _ 1 ^ { \\rho _ 2 } } . \\end{align*}"}
+{"id": "1958.png", "formula": "\\begin{align*} J ( G ) \\ = \\ { \\bf z } _ { N ( x _ 1 ) } J ( G _ 1 ) + x _ 1 J ( G \\setminus \\{ x _ 1 , y _ 1 \\} ) , \\end{align*}"}
+{"id": "7912.png", "formula": "\\begin{align*} \\dfrac { 1 } { 2 } = \\dfrac { 1 } { 2 } \\left ( \\mathrm { F P d i m } ( x ) ^ 2 - \\sum _ { y \\in \\Gamma _ 1 } c _ { x , x ^ \\ast } ^ y \\mathrm { F P d i m } ( y ) ^ 2 \\right ) - \\sum _ { z \\in \\Gamma _ 2 } c _ { x , x ^ \\ast } ^ z \\mathrm { F P d i m } ( z ) ^ 2 \\end{align*}"}
+{"id": "2302.png", "formula": "\\begin{align*} \\mathcal { Q } ( t ) : = \\int _ { \\R } \\alpha v ^ { 2 } + 2 \\varepsilon \\operatorname { I m } ( u \\partial _ { x } \\overline { u } ) d x , \\end{align*}"}
+{"id": "189.png", "formula": "\\begin{align*} & \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T _ C M } - \\widetilde { L _ C } + 2 5 6 + W _ i ) \\right \\} ^ { ( 1 4 ) } \\\\ & = A _ 1 \\left \\{ e ^ { \\frac { 1 } { 2 4 } A _ 1 } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) - \\frac { e ^ { \\frac { 1 } { 2 4 } A _ 1 } - 1 } { A _ 1 } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T _ C M } - \\widetilde { L _ C } + 2 5 6 + W _ i ) \\right \\} ^ { ( 1 0 ) } . \\end{align*}"}
+{"id": "1582.png", "formula": "\\begin{align*} \\int \\left ( \\frac { v ^ 2 } { T _ g } - \\frac { d _ 3 v } { \\sqrt { T _ g } } - 1 \\right ) ^ 2 g \\mathrm { d } v & = \\frac { \\rho _ g T _ g ^ 2 ( 3 + d _ 4 ) } { T _ g } - \\frac { 2 \\rho _ g d _ 3 ^ 2 T _ g ^ { 3 / 2 } } { T _ g ^ { 3 / 2 } } + \\frac { \\rho _ g d _ 3 ^ 2 T _ g } { T _ g } - 2 \\frac { \\rho _ g T _ g } { T _ g } + \\rho _ g \\\\ & = \\rho _ g \\left ( 2 + d _ 4 - d _ 3 ^ 2 \\right ) . \\end{align*}"}
+{"id": "4706.png", "formula": "\\begin{align*} f ( x y z _ 0 ) = f ( x ) g ( y ) + f ( y ) g ( x ) + f ( x ) \\psi ( y ) , \\end{align*}"}
+{"id": "1242.png", "formula": "\\begin{align*} e ^ { i ( t - a _ { j + 1 } ) \\Delta } w ( a _ { j + 1 } ) = & e ^ { i t \\Delta } w ( 0 ) + i \\int _ { 0 } ^ { a _ { j + 1 } } e ^ { i ( t - s ) \\Delta } ( F ( \\tilde { u } + w ) - F ( u ) ) d s \\\\ & - i \\int _ { 0 } ^ { a _ { j + 1 } } e ^ { i ( t - s ) \\Delta } e d s . \\end{align*}"}
+{"id": "1883.png", "formula": "\\begin{align*} \\mathcal { T } _ { h } = \\left \\{ T _ { i j } = I _ i \\times J _ j : 1 \\leq i \\leq N _ x , 1 \\leq j \\leq N _ v \\right \\} , \\end{align*}"}
+{"id": "6460.png", "formula": "\\begin{align*} \\| A ^ n y ( t ) \\| = \\| y ^ { ( n ) } ( t ) \\| \\le C h ^ n M _ n , \\end{align*}"}
+{"id": "6755.png", "formula": "\\begin{align*} ( E ^ { e _ 1 } \\circ O ^ { o _ 1 } \\circ E ^ { e _ 2 } \\circ \\dotsb \\circ O ^ { o _ l } \\circ E ^ { e _ { l + 1 } } ) ( n ) = \\lceil ( E ^ { \\sigma _ e } \\circ O ^ { \\sigma _ o } ) ( n ) \\rceil . \\end{align*}"}
+{"id": "7920.png", "formula": "\\begin{align*} 0 = 1 + 2 \\mathrm { R e } ( \\varphi ( x ) ^ 2 ) + 2 \\mathrm { R e } ( \\varphi ( y ) ^ 2 ) = 1 - 2 + 2 = 1 . \\end{align*}"}
+{"id": "2678.png", "formula": "\\begin{align*} \\frac { 1 } { G } \\langle - g + t h , h \\rangle _ { g } \\ = \\ - 1 + \\frac { t } { G } | h | _ { g } ^ { 2 } \\ \\geq \\ - 1 . \\end{align*}"}
+{"id": "4759.png", "formula": "\\begin{align*} P ( E , \\overline { F } ) = P ( F , \\partial E ) . \\end{align*}"}
+{"id": "7386.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\sqrt { \\rho _ { n } } \\partial _ { x } p _ { n } ( \\rho _ { n } ) \\| _ { L ^ { \\infty } _ { t } L ^ { 2 } _ { x } } ^ { 2 } & = \\| \\sqrt { \\rho _ { n } } ( w _ { n } - u _ { n } ) \\| _ { L ^ { \\infty } _ { t } L ^ { 2 } _ { x } } ^ { 2 } \\le \\left ( \\| \\sqrt { \\rho _ { n } } w _ { n } \\| _ { L ^ { \\infty } _ { t } L ^ { 2 } _ { x } } ^ { 2 } + \\| \\sqrt { \\rho _ { n } } u _ { n } \\| _ { L ^ { \\infty } _ { t } L ^ { 2 } _ { x } } ^ { 2 } \\right ) \\\\ [ 1 e x ] & \\le ( T + 3 ) E _ { n } , \\end{aligned} \\end{align*}"}
+{"id": "2314.png", "formula": "\\begin{align*} \\| u \\| _ { Y ^ { k } _ { \\lambda , \\theta } } : = \\| w _ { k , \\lambda , \\theta } ( \\xi , \\tau ) \\hat { u } ( \\xi , \\tau ) \\| _ { L ^ 2 _ { \\xi , \\tau } } + \\| u \\| _ { L ^ { \\infty } _ t H ^ k _ x } . \\end{align*}"}
+{"id": "4004.png", "formula": "\\begin{align*} \\begin{aligned} & 2 n \\left ( A + 1 \\right ) S _ n ( X ) \\\\ & \\geq \\frac { A \\delta _ 0 t } { 4 } \\sum _ l \\left ( S _ { n - 1 ; l } ( X ) - \\frac { c } { C ^ m _ n } S _ { m - 1 ; l } ( X ) \\right ) + \\frac { ( n - m ) A t } { 2 } S _ n ( X ) \\\\ & \\geq \\frac { A ( n - m ) \\delta _ 0 t } { 4 n } S _ { n - 1 } ( X ) + \\frac { ( n - m ) A t } { 2 } S _ n ( X ) , \\end{aligned} \\end{align*}"}
+{"id": "4089.png", "formula": "\\begin{align*} \\lim _ { \\epsilon _ 2 = 0 } ^ { \\epsilon _ 1 = { \\rm f i x e d } } \\omega _ n = \\sum _ { g \\geq 0 } \\epsilon _ 1 ^ { 2 g } \\cdot \\omega _ { g , n } ^ { ( 2 g ) } = \\sum _ { g \\geq 0 } \\epsilon _ 1 ^ { 2 g } \\cdot \\varpi _ { g , n } \\end{align*}"}
+{"id": "8952.png", "formula": "\\begin{align*} \\sum _ { s = - \\infty } ^ { \\infty } \\Theta _ { \\infty } ( s ) = \\sum _ { s = - \\infty } ^ { \\infty } \\int _ { \\R ^ { d } } f ( a ^ s x ) f ( x ) ) \\ , d x . \\end{align*}"}
+{"id": "5076.png", "formula": "\\begin{align*} E ( A ) = \\sum _ { p \\geq 0 } \\frac { I ^ p } { 2 ^ p } = 2 I , \\end{align*}"}
+{"id": "3307.png", "formula": "\\begin{align*} \\sigma ^ 2 = m _ 2 - m _ 1 ^ 2 = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\frac { 1 } { 2 \\lambda _ { _ \\Sigma } } . \\end{align*}"}
+{"id": "7568.png", "formula": "\\begin{align*} | \\lambda _ k ( t ) | = \\kappa ( \\alpha + 1 ) t ^ \\alpha , \\end{align*}"}
+{"id": "8207.png", "formula": "\\begin{align*} g _ a \\leq g _ 0 + a ^ 2 \\sum _ { i = 1 } ^ 3 ( d x _ i ) ^ 2 . \\end{align*}"}
+{"id": "3565.png", "formula": "\\begin{align*} \\langle \\omega _ { P + \\vect { y } } , \\vect { x } \\rangle - \\langle \\omega _ { P } , \\vect { x } \\rangle = \\langle \\omega _ { P + \\vect { x } } , \\vect { y } \\rangle - \\langle \\omega _ { P } , \\vect { y } \\rangle , \\vect { x } , \\vect { y } \\in \\vect { A } . \\end{align*}"}
+{"id": "3940.png", "formula": "\\begin{align*} \\gamma _ n : = \\bigcup _ { k = v } ^ n \\omega _ { \\frac 1 { k } } . \\end{align*}"}
+{"id": "353.png", "formula": "\\begin{align*} h _ 1 \\circ _ R h _ 2 : = h _ 1 \\phi _ { R ( h _ 1 ) } ( h _ 2 ) \\end{align*}"}
+{"id": "5893.png", "formula": "\\begin{align*} G \\coloneqq Q ^ { \\textup { g p } } / M ^ { \\textup { g p } } = \\widetilde { N } ^ { \\textup { g p } } / M _ A ^ { \\textup { g p } } \\cong N ^ { \\textup { g p } } \\end{align*}"}
+{"id": "2295.png", "formula": "\\begin{align*} ( d - 1 ) ^ { 2 } - d _ { 1 } ( d - d _ { 1 } - 1 ) = \\tau ( C ) , \\end{align*}"}
+{"id": "1187.png", "formula": "\\begin{align*} \\mu \\circ \\gamma = \\eta \\mathrm { a n d } ( \\mu \\otimes 1 _ A ) \\circ ( 1 _ A \\otimes \\gamma ) = ( 1 _ A \\otimes \\mu ) \\circ ( \\gamma \\otimes 1 _ A ) . \\end{align*}"}
+{"id": "5639.png", "formula": "\\begin{align*} \\begin{cases} y _ 0 y _ 1 + C - L ( x _ 0 y _ 1 - x _ 1 y _ 0 - B ) & = 0 , \\\\ x _ 0 ( y _ 1 + x _ 1 L ) - x _ 1 ( y _ 0 + x _ 0 L ) - B & = x _ 0 y _ 1 - x _ 1 y _ 0 - B = 0 , \\end{cases} \\end{align*}"}
+{"id": "6603.png", "formula": "\\begin{align*} f ( t , \\cdot ) = \\Gamma ( t , \\cdot ) _ \\# f _ 0 \\ t \\in [ 0 , T ] , \\end{align*}"}
+{"id": "8583.png", "formula": "\\begin{align*} \\dfrac { h } { h _ b } \\nabla ^ { \\mu } _ { X , z } \\cdot P ( \\Sigma _ b ) \\nabla ^ { \\mu } _ { X , z } \\phi _ b = \\Delta ^ { \\mu } _ { X , z } \\phi _ b + \\mu \\varepsilon A [ \\nabla _ X , \\partial _ z ] \\phi _ b , \\end{align*}"}
+{"id": "6834.png", "formula": "\\begin{align*} \\mathfrak { F } ( u ) ( t , \\theta ) = e ^ { \\frac { 6 - n } { 2 } t } u ( e ^ { - t } \\theta ) = v ( t , \\theta ) { \\rm w i t h } \\theta = x / | x | . \\end{align*}"}
+{"id": "3662.png", "formula": "\\begin{align*} \\partial _ t \\ , \\langle \\nabla u , \\nabla f \\rangle = - g _ t ( \\nabla u , \\nabla f ) + \\langle \\nabla u _ t , \\nabla f \\rangle + \\langle \\nabla u , \\nabla f _ t \\rangle \\ , . \\end{align*}"}
+{"id": "2470.png", "formula": "\\begin{align*} P _ A = t P ' _ A + ( 1 - t ) P '' _ A . \\end{align*}"}
+{"id": "4945.png", "formula": "\\begin{align*} J ( k , r ) : = \\frac { \\pi ( 2 k + 1 ) } { 3 } - \\frac { 2 k + 1 } { \\pi } \\sum _ { m = 1 } ^ { r + 1 } 2 ^ m L _ m - \\frac { 2 ^ { r + 2 } ( 4 k + 1 ) } { \\pi } L _ { r + 2 } - \\frac { 2 ^ { r + 4 } } { \\pi } L _ { r + 3 } . \\end{align*}"}
+{"id": "8556.png", "formula": "\\begin{align*} & M _ 0 = C ( \\frac { 1 } { h _ { \\min } } , \\frac { 1 } { h _ { b , \\min } } , | \\zeta | _ { H ^ { t _ 0 } } , | b | _ { H ^ { t _ 0 } } ) \\\\ & M ( s ) = C ( M _ 0 , | \\zeta | _ { H ^ { \\max \\{ t _ 0 + 2 , s \\} } } , | b | _ { H ^ { \\max \\{ t _ 0 + 2 , s \\} } } ) \\\\ & N ( s ) = C ( M ( s ) , | \\nabla _ X \\psi | _ { H ^ { s } } ) . \\end{align*}"}
+{"id": "2808.png", "formula": "\\begin{align*} \\mathcal { E } _ Y ( u , u ) = - 2 \\int _ { \\Omega } Y \\cdot \\nabla u ( - \\Delta ) ^ s u \\dd x . \\end{align*}"}
+{"id": "8085.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 0 } ^ { \\infty } C \\frac { 2 ^ { i n } } { ( 1 + 2 ^ { i } \\vert x - x _ { P _ { j } } \\vert ) ^ { M } } \\leq \\frac { C } { \\vert x - x _ { P _ { j } } \\vert ^ { M } } . \\end{align*}"}
+{"id": "7371.png", "formula": "\\begin{align*} \\partial _ t u = \\Delta u + f \\end{align*}"}
+{"id": "383.png", "formula": "\\begin{align*} j ( c ' , x u ' ) = \\alpha \\circ \\chi ( c ' , x u ' ) = \\alpha \\circ \\chi ( c , x ) \\circ u = j ( c , x ) u , \\end{align*}"}
+{"id": "8805.png", "formula": "\\begin{align*} K ( r ) = \\begin{cases} 1 , & r \\geq 0 \\\\ - 1 , & r < 0 \\end{cases} \\enspace . \\end{align*}"}
+{"id": "8511.png", "formula": "\\begin{align*} | w | = r _ { \\ell } ^ { \\vee } ( \\bar { z } ) + \\delta . \\end{align*}"}
+{"id": "1581.png", "formula": "\\begin{align*} \\langle \\mathcal { C } h , h \\rangle = \\frac { 1 } { \\kappa } \\left [ \\alpha \\mathcal { I } + ( 1 - \\alpha ) \\mathcal { J } \\right ] \\end{align*}"}
+{"id": "4299.png", "formula": "\\begin{align*} | T _ 2 | \\leq \\frac { ( t - s ) ^ 2 } { n ^ { 4 } } \\sum _ { i _ 1 , i _ 2 , i _ 3 , i _ 4 = 1 } ^ n \\alpha ^ 2 O ( 1 ) = ( t - s ) ^ 2 \\alpha ^ 2 O ( 1 ) . \\end{align*}"}
+{"id": "8771.png", "formula": "\\begin{align*} H ^ 2 ( \\mathbf { P } _ { \\omega , T } , \\mathbf { P } _ { \\omega ' , T } ) & \\leq 2 \\Big ( 1 - \\big ( 1 - T ^ { - 1 } \\big ) ^ T \\Big ) \\\\ & \\leq 2 ( 1 - \\frac { 1 } { 4 } ) = 3 / 2 . \\end{align*}"}
+{"id": "5435.png", "formula": "\\begin{align*} c _ r & = \\sum _ j \\left ( \\frac { ( 2 - d _ r ) ( 2 - d _ j ) } { 2 ( n - 1 ) } + \\left \\{ \\begin{array} { l l } - d _ r / 2 & r = j \\\\ a _ { r j } / 2 & r \\neq j \\end{array} \\right . \\right ) \\\\ & = \\frac { 2 - d _ r } { 2 ( n - 1 ) } \\sum _ j ( 2 - d _ j ) - \\frac { d _ r } { 2 } + \\frac { d _ r } { 2 } \\\\ & = \\frac { 2 - d _ r } { 2 ( n - 1 ) } ( 2 n - 2 ( n - 1 ) ) = \\frac { 2 - d _ r } { n - 1 } . \\end{align*}"}
+{"id": "352.png", "formula": "\\begin{align*} \\operatorname { R e } \\left ( c \\left ( s \\right ) \\left ( \\mbox { \\boldmath $ \\alpha $ } , \\mbox { \\boldmath $ \\alpha $ } \\right ) - \\frac { 1 } { 2 } \\left \\langle \\mbox { \\boldmath $ \\alpha $ } , \\overline { \\mbox { \\boldmath $ \\alpha $ } } \\right \\rangle _ { \\mathbb { X } } \\right ) = \\operatorname { R e } c \\left ( s \\right ) \\left ( \\mbox { \\boldmath $ \\alpha $ } , \\mbox { \\boldmath $ \\alpha $ } \\right ) \\end{align*}"}
+{"id": "3600.png", "formula": "\\begin{align*} { E _ { { \\rm { t h } } } } = \\frac { { { \\sigma ^ 2 } { L _ { \\rm { R } } } \\left ( { \\kappa + 1 } \\right ) } } { { { N _ { \\rm { T } } } \\kappa } } \\frac { { \\sqrt { 9 + 3 \\pi \\left ( { K - 1 } \\right ) } - 3 } } { { 2 { C ^ 2 } } } \\end{align*}"}
+{"id": "493.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} d \\psi = ( - \\alpha \\partial _ x - i M \\beta ) \\psi \\ , d t + i \\phi \\beta \\psi \\ , d t + i \\beta \\psi \\mathfrak K _ 1 \\circ d W , \\\\ d \\phi = \\dot \\phi \\ , d t , \\\\ d \\dot \\phi = ( \\partial _ x ^ 2 - m ^ 2 ) \\phi \\ , d t + \\psi ^ * \\beta \\psi \\ , d t + \\phi \\mathfrak K _ 2 \\circ d W , \\end{gathered} \\right . \\end{align*}"}
+{"id": "3194.png", "formula": "\\begin{align*} x _ { k } = V _ { k } y _ { k } , y _ k = \\arg \\min \\| \\beta _ { 1 } e _ { 1 } - B _ { k } y \\| \\end{align*}"}
+{"id": "6969.png", "formula": "\\begin{align*} \\sum _ { n \\in \\Z ^ d } \\frac { | u ( n ) | ^ 2 } { | n | ^ { 2 } } = \\int _ { Q _ d } | \\nabla \\psi ( x ) | ^ 2 d x , \\end{align*}"}
+{"id": "8127.png", "formula": "\\begin{align*} \\| u \\| _ { \\dot { W } ^ { s , 2 } ( \\R ^ N ) } ^ 2 \\sim \\| u \\| _ { \\dot { H } ^ s ( \\R ^ N ) } ^ 2 = \\int _ { \\R ^ N } | \\xi | ^ { 2 s } | \\hat { u } ( \\xi ) | ^ 2 d \\xi \\end{align*}"}
+{"id": "2845.png", "formula": "\\begin{align*} \\mathcal { V } ^ { ( n , c ) } _ { \\mu } ( \\xi ) = \\sum _ { 1 \\le j < k \\le n } \\int _ 0 ^ { \\xi _ j - \\xi _ k } v ( ) + \\sum _ { 1 \\leq j \\leq n } \\left ( { \\textstyle \\frac { c } { 2 } } \\xi _ j ^ 2 - 2 \\pi ( \\rho _ j + \\mu _ j ) \\xi _ j \\right ) , \\end{align*}"}
+{"id": "8117.png", "formula": "\\begin{align*} \\bar { s } _ { i j k } ( C ' ) = \\bar { s } _ { i j k } ( C ) . \\end{align*}"}
+{"id": "2859.png", "formula": "\\begin{align*} x _ + : = w _ x x \\in A _ c . \\end{align*}"}
+{"id": "152.png", "formula": "\\begin{align*} ( { \\rm K a z } _ m ^ F \\circ { \\rm \\overline { B r } } ) ( h ) = ( { \\rm \\overline { B r } } ' \\circ { \\rm K a z } _ m ^ E ) ( h ) . \\end{align*}"}
+{"id": "2124.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde p = \\tilde p [ \\Lambda , \\phi , \\alpha ] : = & ~ { } \\kappa \\partial _ x \\alpha \\left [ ( \\partial _ t \\Lambda ) ^ 2 + ( \\partial _ x \\Lambda ) ^ 2 + 4 \\sinh ^ 2 ( \\Lambda ) \\Big ( ( \\partial _ t \\phi ) ^ 2 + ( \\partial _ x \\phi ) ^ 2 \\Big ) \\right ] \\\\ & ~ { } - 2 \\kappa \\partial _ t \\alpha \\left ( \\partial _ x \\Lambda \\partial _ t \\Lambda + 4 \\partial _ x \\phi \\partial _ t \\phi \\sinh ^ 2 ( \\Lambda ) \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "1802.png", "formula": "\\begin{gather*} Z _ N ( s , \\mathbb { X } _ \\alpha ) ( \\omega ) = \\frac { 1 } { 2 \\pi i } \\int _ { c - i \\infty } ^ { c + i \\infty } \\zeta ( s + w , \\mathbb { X } _ \\alpha ) ( \\omega ) \\widehat { \\phi } ( w ) N ^ w \\ , d w \\end{gather*}"}
+{"id": "4469.png", "formula": "\\begin{align*} [ u _ { 1 , 0 } ] - [ u _ { 2 , 0 } ] \\partial _ 2 \\varphi _ 0 = 0 , [ p _ 0 + \\frac { | H _ 0 | ^ 2 } { 2 } ] = 0 . \\end{align*}"}
+{"id": "1724.png", "formula": "\\begin{align*} \\Delta ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } = \\prod _ { 1 \\leq j \\leq n } \\Delta _ { \\lambda _ j + n - j } ^ { ( m + n ) } , \\end{align*}"}
+{"id": "7380.png", "formula": "\\begin{align*} B ( 1 , ( \\gamma ( m ) ^ { - 1 } , \\gamma ( m ) ) ) & = ( 1 , ( 1 , \\gamma ( m ) ^ { - 1 } , \\gamma ( m ) ) ) - ( 1 , ( 1 , \\gamma ( m ) , \\gamma ( m ) ^ { - 1 } ) ) \\\\ & + ( 1 , ( \\gamma ( m ) ^ { - 1 } , 1 , \\gamma ( m ) ) ) - ( 1 , ( \\gamma ( m ) , 1 , \\gamma ( m ) ^ { - 1 } ) ) . \\end{align*}"}
+{"id": "2061.png", "formula": "\\begin{align*} & \\int _ { B _ k \\cap \\Omega } G _ { \\Omega } ( x , 1 ; z , t _ k ) G _ { \\Omega } ( z , t _ k ; y , t _ { k + 1 } ) \\ , d \\mathrm { v o l } _ { g ( t _ k ) } ( z ) \\\\ & \\le \\int _ { \\Omega \\setminus B _ 0 ( x , r _ k ) } G _ { \\Omega } ( x , 1 ; z , t _ k ) G _ { \\Omega } ( z , t _ k ; y , t _ { k + 1 } ) \\ , d \\mathrm { v o l } _ { g ( t _ k ) } ( z ) \\leq a _ k . \\end{align*}"}
+{"id": "718.png", "formula": "\\begin{align*} [ \\mathbf u , \\mathbf V ] ( t ) = \\begin{cases} \\mathbf u ( t ) & \\\\ \\mathbf V ( t ) & \\end{cases} \\end{align*}"}
+{"id": "8305.png", "formula": "\\begin{align*} ( F \\circ h ) \\circ u & = F \\circ u \\circ h , ~ \\\\ & = u \\circ ( F \\circ h ) , ~ , \\\\ ( h \\circ F ) \\circ u & = h \\circ u \\circ F , ~ \\\\ & = u \\circ ( h \\circ F ) , ~ . \\end{align*}"}
+{"id": "7009.png", "formula": "\\begin{align*} & m \\equiv n \\equiv 0 \\ ( \\bmod \\ 2 ) \\ \\ x _ { ( 0 ) } = y _ { ( 1 ) } = | z _ { ( 0 ) } | = | z _ { ( 1 ) } | = 2 \\ \\ z _ { ( 0 ) } z _ { ( 1 ) } > 0 ; \\\\ & m \\equiv n \\equiv 1 \\ ( \\bmod \\ 2 ) \\ \\ x _ { ( 0 ) } = y _ { ( 1 ) } = r _ 2 , \\ | z _ { ( 0 ) } | = t , \\ | z _ { ( 1 ) } | = s _ 2 \\ \\ z _ { ( 0 ) } z _ { ( 1 ) } > 0 . \\end{align*}"}
+{"id": "6129.png", "formula": "\\begin{align*} \\forall s \\in S ^ + \\sqcup T ^ - , \\ , \\eta ( \\varphi ( s ) ) = \\eta ( s ) . \\end{align*}"}
+{"id": "954.png", "formula": "\\begin{align*} \\left \\{ \\ \\begin{aligned} & \\frac { \\delta L } { \\delta u } = F ^ * _ 1 = - t ^ { 1 - \\alpha } p _ t - ( 1 - \\alpha ) t ^ { - \\alpha } p - 2 p _ x + b q _ x - x p _ { x x } = 0 , \\\\ & \\frac { \\delta L } { \\delta v } = F ^ * _ 2 = - t ^ { 1 - \\alpha } q _ t - ( 1 - \\alpha ) t ^ { - \\alpha } q - 2 q _ x + b p _ x - x q _ { x x } = 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "2599.png", "formula": "\\begin{align*} \\mathcal F _ { a , b } ( X ) = \\{ ( V _ 1 , V _ 2 ) : \\{ 0 \\} \\subset V _ 1 \\subset V _ 2 \\subset \\mathbb C ^ { p + q } , \\dim V _ 1 = a , \\ , \\dim V _ 2 = b \\} . \\end{align*}"}
+{"id": "2271.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } e ^ { 2 \\pi i k \\varepsilon _ j } e ^ { - 2 \\pi i k \\frac { j } { n } } & = 2 \\pi i k \\sum _ { j = 1 } ^ { n } \\varepsilon _ j e ^ { - 2 \\pi i k \\frac { j } { n } } + \\mathcal { O } \\left ( k ^ 2 \\sum _ { j = 1 } ^ { n } \\varepsilon _ j ^ 2 \\right ) , \\end{align*}"}
+{"id": "7624.png", "formula": "\\begin{align*} \\langle w , X \\rangle \\ge 0 \\P \\mathrm { a . s . } \\Rightarrow \\langle w , X \\rangle = 0 \\P \\mathrm { a . s . } \\end{align*}"}
+{"id": "2018.png", "formula": "\\begin{align*} \\left ( s , \\frac { 1 } { p } \\right ) : = ( 1 - \\theta ) \\left ( s _ 0 , \\frac { 1 } { p _ 0 } \\right ) + \\theta \\left ( s _ 1 , \\frac { 1 } { p _ 1 } \\right ) \\end{align*}"}
+{"id": "793.png", "formula": "\\begin{align*} \\Gamma \\coloneqq \\{ \\gamma \\in C ( [ 0 , 1 ] ; H ^ s ( \\mathbb { R } ^ N ) ) \\mid \\gamma ( 0 ) = 0 , I _ { \\tilde { a } } [ \\gamma ( 1 ) ] < 0 \\} . \\end{align*}"}
+{"id": "2184.png", "formula": "\\begin{align*} ( Q _ \\alpha ^ { n - 1 } ) _ { v , w } \\geq \\prod _ { i = 1 } ^ { n - 1 } ( Q _ \\alpha ) _ { w _ i , w _ { i + 1 } } & \\geq \\prod _ { i = 1 } ^ { n - 1 } b ( w _ i , ( z _ { i + 1 } - z _ i ) w _ { i + 1 } ) \\exp ( \\langle \\alpha , z _ { i + 1 } - z _ i \\rangle ) \\\\ & = \\exp ( \\langle \\alpha , z _ n - z _ 1 \\rangle ) \\prod _ { i = 1 } ^ { n - 1 } b ( z _ i w _ i , z _ { i + 1 } w _ { i + 1 } ) > 0 . \\end{align*}"}
+{"id": "1328.png", "formula": "\\begin{align*} \\Lambda _ Z ^ { w } ( u ) = \\frac { w ( \\phi ( F ^ { - 1 } ( u ) ) ) } { \\phi ^ \\prime ( F ^ { - 1 } ( u ) ) } f ( F ^ { - 1 } ( u ) ) \\leq w ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) = \\Lambda _ X ^ { w } ( u ) . \\end{align*}"}
+{"id": "12.png", "formula": "\\begin{align*} \\varphi ^ { 0 } _ { \\mathbb { R } ^ 2 } ( x ) = \\log \\Big ( \\frac { \\lvert x \\rvert } { a } \\Big ) , \\varphi ^ { 0 } _ { \\mathbb { R } ^ 3 } ( x ) : = 1 - \\frac { a } { \\lvert x \\rvert } . \\end{align*}"}
+{"id": "5216.png", "formula": "\\begin{align*} k + m _ 1 \\theta _ 1 + m _ 2 \\theta _ 2 = 0 . \\end{align*}"}
+{"id": "3141.png", "formula": "\\begin{align*} \\langle \\omega , \\pi _ 2 ( Z \\backslash D , L ) \\rangle = 0 \\quad \\hbox { a n d } \\eta _ L = 0 . \\end{align*}"}
+{"id": "7888.png", "formula": "\\begin{align*} \\mathcal J ( u ) : = H _ { n + k } ( u ) - \\int _ { \\Omega } \\left ( \\int _ u ^ 0 F ( x , s ) d s \\right ) . \\end{align*}"}
+{"id": "6468.png", "formula": "\\begin{align*} \\lambda _ n \\big ( R _ \\epsilon \\big ) = 0 + \\mu _ n \\epsilon + O ( \\epsilon ^ 2 ) , n \\leq m _ 0 . \\end{align*}"}
+{"id": "7965.png", "formula": "\\begin{align*} \\sigma ( u _ { 0 , 1 } ) = v _ { 0 , 0 } , ~ \\sigma ( v _ { 0 , 0 } ) = u _ { 0 , 1 } ~ ~ \\sigma ( v _ { 1 , 0 } ) = u _ { r , 1 } . \\end{align*}"}
+{"id": "8859.png", "formula": "\\begin{align*} H ^ 2 ( \\mathbf { P } _ { \\omega , T } , \\mathbf { P } _ { \\omega ' , T } ) & \\leq 2 \\Big ( 1 - \\big ( 1 - T ^ { - 1 } \\big ) ^ T \\Big ) \\leq 3 / 2 \\enspace . \\end{align*}"}
+{"id": "1060.png", "formula": "\\begin{align*} K _ { \\hat { w } _ \\lambda } \\subseteq [ \\hat { v } _ \\lambda ^ * , \\hat { u } _ \\lambda ^ * ] \\cap C ^ 1 ( \\overline { \\Omega } ) , \\ ; K _ { \\hat { w } _ \\lambda ^ + } = \\{ 0 , \\hat { u } _ \\lambda ^ * \\} , \\ ; K _ { \\hat { w } _ \\lambda ^ - } = \\{ 0 , \\hat { v } _ \\lambda ^ * \\} . \\end{align*}"}
+{"id": "4963.png", "formula": "\\begin{align*} | A _ j | + | B _ j | = \\phi ( 1 2 n ) / 4 = \\phi ( 3 n ) / 2 \\end{align*}"}
+{"id": "3477.png", "formula": "\\begin{align*} & \\omega _ t = \\omega _ M + d d ^ c \\phi _ t \\\\ = & \\omega _ M + \\frac { 1 } { | \\log | t | | } \\sum _ { 1 \\leq i , j \\leq m } \\frac { \\partial ^ 2 u } { \\partial x _ i \\partial x _ j } d \\log | F _ i | _ { h ^ { \\otimes d _ i } } \\wedge d ^ c \\log | F _ j | _ { h ^ { \\otimes d _ j } } \\\\ & - \\sum _ 1 ^ m \\frac { \\partial u } { \\partial x _ i } d d ^ c \\log | F _ i | _ { h ^ { \\otimes d _ i } } . \\end{align*}"}
+{"id": "2178.png", "formula": "\\begin{align*} T _ g T _ l ^ n h _ 1 & = T _ l ^ n T _ g T _ { [ g , l ^ n ] } h _ 1 \\\\ & = T _ l ^ n T _ g T _ q ^ n T _ { r _ n } h _ 1 \\\\ & = T _ l ^ n T _ g \\gamma ( q ^ { - 1 } ) ^ n h _ 1 \\\\ & = \\gamma ( q ) ^ { - n } h _ 1 ( g ^ { - 1 } x _ 0 ) T _ l ^ n h _ 1 , \\end{align*}"}
+{"id": "1663.png", "formula": "\\begin{align*} \\sum _ { \\mu \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { a } } } P _ { \\texttt { a } ; \\mu } \\bigl ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } ; q \\bigr ) \\overline { P _ { \\texttt { a } ; \\mu } \\bigl ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { a } ; \\kappa } ; q \\bigr ) } \\delta ^ { ( m , n ) } _ { \\texttt { a } ; \\mu } ( q ) = 0 \\quad \\ \\lambda \\neq \\kappa \\end{align*}"}
+{"id": "4281.png", "formula": "\\begin{align*} G _ { n \\times p } { ( j ) } e _ i & = \\begin{cases} e _ { i - j } & \\mbox { i f } 1 \\leq ( i - j ) \\leq n , \\\\ 0 & \\mbox { o t h e r w i s e } , \\end{cases} \\\\ & = \\chi _ { [ 1 , n ] } ( i - j ) e _ { i - j } . \\end{align*}"}
+{"id": "5791.png", "formula": "\\begin{align*} s _ { \\beta } s _ { \\alpha } s _ { \\beta } = \\begin{cases} s _ { \\alpha + 3 \\beta } ( \\alpha , \\beta ) = - \\frac { 3 } { 2 } , \\\\ s _ { \\alpha - 3 \\beta } ( \\alpha , \\beta ) = \\frac { 3 } { 2 } . \\\\ \\end{cases} \\end{align*}"}
+{"id": "4529.png", "formula": "\\begin{align*} \\partial _ t \\varphi + \\hat { u } ^ \\pm _ 2 \\partial _ 2 \\varphi - \\dot { u } ^ \\pm _ N \\mp \\varphi \\partial _ 1 \\hat { u } ^ \\pm _ N = 0 \\ , , \\dot { q } ^ + - \\dot { q } ^ - + \\varphi [ \\partial _ 1 \\hat { q } ] = 0 \\quad \\mbox { o n } \\ , \\ , \\ , \\Gamma _ T \\ , , \\end{align*}"}
+{"id": "8102.png", "formula": "\\begin{align*} \\left \\| \\uppercase \\expandafter { \\romannumeral 1 } \\right \\| _ { L _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } . \\end{align*}"}
+{"id": "160.png", "formula": "\\begin{align*} \\nabla ( [ \\omega _ n ^ { ( 0 ) } ] ) = 0 . \\end{align*}"}
+{"id": "4179.png", "formula": "\\begin{align*} \\rho : = | u | \\sqrt { r ^ 2 + \\lambda ^ 2 } ; ~ ~ \\mathrm { d } \\rho = | u | \\frac { r } { \\sqrt { r ^ 2 + \\lambda ^ 2 } } \\mathrm { d } r = | u | ^ 2 \\frac { r } { \\rho } \\mathrm { d } r \\end{align*}"}
+{"id": "3020.png", "formula": "\\begin{align*} \\vartheta ( H _ n ) = \\begin{cases} \\phantom { - } 1 , & \\mbox { i f { \\ ; } $ n = 4 s $ ; } \\\\ [ . 5 m m ] \\phantom { - } ( 2 s + 1 ) \\lambda , & \\mbox { i f { \\ ; } $ n = 4 s + 1 $ ; } \\\\ [ . 5 m m ] - 1 , & \\mbox { i f { \\ ; } $ n = 4 s + 2 $ ; } \\\\ [ . 5 m m ] - ( 2 s + 2 ) \\lambda , & \\mbox { i f { \\ ; } $ n = 4 s + 3 $ . } \\\\ [ . 3 m m ] \\end{cases} \\end{align*}"}
+{"id": "3033.png", "formula": "\\begin{align*} \\phi ( H _ { 1 1 } ) = \\lambda ( \\lambda - 1 ) ^ 2 ( \\lambda + 1 ) ^ 2 ( \\lambda ^ 2 - 3 ) ( \\lambda ^ 4 - 5 \\lambda ^ 2 + 2 ) . \\end{align*}"}
+{"id": "7226.png", "formula": "\\begin{align*} w \\coloneqq \\begin{cases} \\ u & \\Omega \\setminus E _ s , \\\\ \\ v & \\Omega \\cap E _ s . \\end{cases} \\end{align*}"}
+{"id": "2355.png", "formula": "\\begin{align*} \\pi ( \\lambda ) = \\pi ( k , \\xi ) \\coloneqq M _ \\xi T _ k . \\end{align*}"}
+{"id": "7299.png", "formula": "\\begin{align*} P _ { \\max } ( x , y ) = \\begin{cases} R ( x , y ) & R ( x , y ) > R ( y , x ) , \\\\ 0 & . \\end{cases} \\end{align*}"}
+{"id": "2540.png", "formula": "\\begin{align*} 0 \\le P _ n ( x ) & \\le Q ( x ) + a _ r x ^ r + | x | ^ { r + 1 } \\sum _ { i = r + 1 } ^ n | a _ i | | x | ^ { i - r - 1 } \\\\ & \\le Q ( x ) + a _ r x ^ r + B x ^ { r + 1 } , \\end{align*}"}
+{"id": "3607.png", "formula": "\\begin{align*} { { R } _ { { \\rm { D S } } } } \\approx \\frac { 1 } { M _ { \\rm R } } \\sum \\limits _ { n = 1 } ^ { { N _ { \\rm { R } } } } { { { \\log } _ 2 } \\left ( 1 + \\frac { { { \\left ( \\sum \\limits _ { m = 1 } ^ { { M _ { \\rm { R } } } } \\left | [ { { { \\bf { \\Xi } } _ { { \\rm { D S } } , m } } } ] _ { n , n } \\right | \\right ) ^ 2 } E } } { { { N _ { \\rm { R } } } { M _ { \\rm { R } } } { \\sigma ^ 2 } } } \\right ) } . \\end{align*}"}
+{"id": "822.png", "formula": "\\begin{align*} \\tilde { h } ( z ) : = e ^ { n _ 0 \\tilde { g _ 1 } ( z ) } ( z \\in \\overline { \\mathbb { D } } ) . \\end{align*}"}
+{"id": "8618.png", "formula": "\\begin{align*} \\mathcal { L } _ { 1 } ^ { \\mu } [ \\beta b ] \\nabla _ X \\psi = - \\beta b \\nabla _ X \\psi + \\mu R . \\end{align*}"}
+{"id": "8378.png", "formula": "\\begin{align*} \\bar { b } ( t , x , \\mu ) = \\int _ { \\mathbb { R } ^ { m } } b ( t , x , \\mu , z ) \\nu ^ { t , x , \\mu } ( d z ) , \\end{align*}"}
+{"id": "2878.png", "formula": "\\begin{align*} ( \\mathcal { J } f ) ( \\lambda ) = \\sum _ { \\mu \\in P , \\ , \\mu \\preceq \\lambda } J _ { \\lambda , \\mu } f ( \\mu ) , ( f \\in \\mathcal C ( P ) , \\lambda \\in P ) \\end{align*}"}
+{"id": "8820.png", "formula": "\\begin{align*} h _ { t } \\leq \\rho ^ { 2 } h ( t - 1 ) + \\rho ^ { 2 } \\eta _ { t } ^ { 2 } \\sum _ { i = 1 } ^ { n } \\norm { g ^ { i } ( t ) - \\bar { g } ( t ) } ^ { 2 } - 2 \\rho ^ { 2 } \\eta _ { t } \\sum _ { i = 1 } ^ { n } \\langle x ^ { i } ( t ) - \\bar { x } ( t ) , g ^ { i } ( t ) - \\bar { g } ( t ) \\rangle . \\end{align*}"}
+{"id": "3140.png", "formula": "\\begin{align*} \\mu _ L ( u ) = 2 u \\cdot D + \\eta _ L ( \\partial u ) , \\end{align*}"}
+{"id": "1168.png", "formula": "\\begin{align*} & m \\rq _ { 1 , 1 } ( x , y ) + \\phi _ 1 ( m \\rq _ { 1 , 0 } ( x , y ) ) = m _ { 1 , 1 } ( x , y ) + m _ { 1 , 0 } ( \\phi _ 1 ( x ) , y ) + m _ { 1 , 0 } ( x , \\phi _ 1 ( y ) ) , \\\\ & m \\rq _ { 2 , 1 } ( x , y ) + \\phi _ 1 ( m \\rq _ { 2 , 0 } ( x , y ) ) = m _ { 2 , 1 } ( x , y ) + m _ { 2 , 0 } ( \\phi _ 1 ( x ) , y ) + m _ { 2 , 0 } ( x , \\phi _ 1 ( y ) ) . \\end{align*}"}
+{"id": "4401.png", "formula": "\\begin{align*} \\mathrm { d i v } \\dot { \\mathbf h } ^ { \\pm } = r ^ { \\pm } \\Omega _ T , \\end{align*}"}
+{"id": "1906.png", "formula": "\\begin{align*} b _ h ( u _ h , w _ h ) + b _ h ( w _ h , u _ h ) = 0 . \\end{align*}"}
+{"id": "8617.png", "formula": "\\begin{align*} R _ 5 & = - \\zeta h \\dfrac { \\nabla _ X b } { h _ b ^ 3 } \\Big ( \\cosh { ( \\beta b ( X ) \\sqrt { \\mu } | \\mathrm { D } | ) } \\mathrm { s e c h } { ( \\sqrt { \\mu } | \\mathrm { D } | ) } - 1 \\Big ) \\frac { 1 } { \\mu | \\mathrm { D } | ^ 2 } \\Delta _ X \\psi . \\end{align*}"}
+{"id": "3896.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { | a _ N | ^ 2 } { \\sum _ { n \\geq N } | a _ n | ^ 2 } = 0 . \\end{align*}"}
+{"id": "4191.png", "formula": "\\begin{align*} h ( \\mathbf { Z } ) = h _ 0 ( | u | \\lambda '' ) + \\log ( 2 \\pi e ^ 3 ) - 2 \\log ( | u | ) . \\end{align*}"}
+{"id": "8635.png", "formula": "\\begin{align*} \\mathcal { B } [ \\beta b ] \\bullet & = b \\nabla _ X ( \\nabla _ X \\cdot ( b \\bullet ) ) + h _ b \\nabla _ X \\big { ( } b \\nabla _ X \\cdot ( b \\bullet ) \\big { ) } + 2 h _ b ( \\nabla _ X b ) \\nabla _ X \\cdot ( b \\bullet ) \\end{align*}"}
+{"id": "1203.png", "formula": "\\begin{align*} A _ P ^ G \\otimes ( A _ P ^ G ) ^ { h W } \\simeq ( A _ P ^ G \\otimes A _ P ^ G ) ^ { h W } \\simeq ( \\prod _ W A _ P ^ G ) ^ { h W } = A ^ G _ P \\end{align*}"}
+{"id": "2640.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\limsup _ { n \\to \\infty } P ( \\sup _ { s \\in [ 0 , t ] } \\abs { M _ n ( s ) } > r ) ^ { 1 / b _ n ^ 2 } = 0 \\intertext { a n d } \\lim _ { \\delta \\to 0 } \\limsup _ { n \\to \\infty } \\sup _ { s \\in [ 0 , t ] } P ( \\sup _ { s ' \\in [ 0 , \\delta ] } \\abs { M _ n ( s ' + s ) - M _ n ( s ) } > \\epsilon ) ^ { 1 / b _ n ^ 2 } = 0 \\ , , \\end{align*}"}
+{"id": "185.png", "formula": "\\begin{align*} Q ( M , P _ i , \\tau ) = & \\left \\{ e ^ { \\frac { 1 } { 2 4 } E _ 2 ( \\tau ) A _ 1 } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } \\left [ \\bigotimes _ { n = 1 } ^ { \\infty } S _ { q ^ n } ( \\widetilde { T _ C M } ) \\otimes \\bigotimes _ { m = 1 } ^ { \\infty } \\wedge _ { - q ^ m } ( \\widetilde { L _ C } ) \\right ] \\right . \\\\ & \\left . \\cdot \\varphi ( \\tau ) ^ { 8 } { \\rm c h } ( \\mathcal { V } _ i ) \\right \\} ^ { ( 1 4 ) } . \\end{align*}"}
+{"id": "4006.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = - \\frac { c } { C ^ m _ n } \\sum _ i S _ { n - m - 1 ; i } ( X ^ { - 1 } ) ( X ^ { i \\bar i } ) ^ 2 X _ { i \\bar i l } + b _ t f _ l S _ n ( X ^ { - 1 } ) - b _ t S _ n ( X ^ { - 1 } ) f \\sum _ i X ^ { i \\bar i } X _ { i \\bar i l } , \\end{aligned} \\end{align*}"}
+{"id": "1877.png", "formula": "\\begin{gather*} N _ \\alpha ( \\sigma _ 1 , \\sigma _ 2 , T ) \\geq n ( T ) : = \\max \\left \\{ n ~ \\middle | ~ \\tau _ n + r \\leq T \\right \\} \\end{gather*}"}
+{"id": "2362.png", "formula": "\\begin{align*} x = \\sum _ { \\lambda \\in \\Lambda } \\phi ( \\pi ( \\lambda ) ^ { - 1 } x ) \\pi ( \\lambda ) \\tau = \\sum _ { \\lambda \\in \\Lambda } f ( \\pi ( \\lambda ) ^ { - 1 } x ) \\pi ( \\lambda ) \\omega , \\forall x \\in \\mathbb { C } ^ { o ( G ) } . \\end{align*}"}
+{"id": "2285.png", "formula": "\\begin{align*} g _ 2 ( x ) & = \\sum _ { \\frac { n - 1 } { 2 } < | k | \\leq n - 1 } e ^ { - \\pi \\alpha k ^ 2 } \\left ( \\sum _ { j = 1 } ^ { n } e ^ { 2 \\pi i k \\varepsilon _ j } e ^ { - 2 \\pi i k \\frac { j } { n } } \\right ) e ^ { 2 \\pi i k x } , \\end{align*}"}
+{"id": "5988.png", "formula": "\\begin{align*} 0 = \\frac { 1 } { 2 \\pi } \\varepsilon L _ a \\tilde { v } _ \\varepsilon + a \\varepsilon ^ 2 \\frac { \\tilde { u } _ j ( t ' ) } { \\lambda _ j - \\lambda _ { j , \\varepsilon } } \\int _ { \\mathbb { D } } \\tilde { u } _ j ( s ' ) \\tilde { v } _ \\varepsilon ( s ' ) e ^ { \\tilde { \\phi } ( s ' ) } d s ' + \\mathcal { R } _ \\varepsilon \\tilde { v } _ \\varepsilon + \\mathcal { R } ^ { \\lambda _ j , \\lambda _ { j , \\varepsilon } } \\tilde { v } . \\end{align*}"}
+{"id": "515.png", "formula": "\\begin{align*} \\left \\{ \\tau _ R > t \\right \\} \\cap \\left \\{ \\tau \\leq t \\right \\} = \\emptyset . \\end{align*}"}
+{"id": "7562.png", "formula": "\\begin{align*} q _ T ^ { ( < ) } & = Q _ T ( A ^ { ( < ) } _ { T , r _ 1 ( T ) } ) = E ^ { P _ T } \\Big [ \\mathbf { 1 } _ { A ^ { ( < ) } _ { T , r _ 1 ( T ) } } \\mathcal { E } _ T \\Big ] , \\\\ q _ T ^ { ( > ) } & = Q _ T ( A ^ { ( > ) } _ { T , r _ 2 ( T ) } ) = E ^ { P _ T } \\Big [ \\mathbf { 1 } _ { A ^ { ( > ) } _ { T , r _ 2 ( T ) } } \\mathcal { E } _ T \\Big ] . \\end{align*}"}
+{"id": "288.png", "formula": "\\begin{align*} H ^ { 1 } \\left ( \\omega , \\mathbb { B } \\right ) : = \\left \\{ u \\in H ^ { 1 } \\left ( \\omega \\right ) \\mid \\operatorname { d i v } \\left ( \\mathbb { B } \\nabla u \\right ) \\in L ^ { 2 } \\left ( \\omega \\right ) \\right \\} \\end{align*}"}
+{"id": "945.png", "formula": "\\begin{align*} g ' _ 2 ( \\tilde { x } , \\tilde { t } ) = & \\frac { X _ x ( Z ( \\tilde { x } , \\tilde { t } ) , Y ^ { - 1 } ( \\tilde { t } ) ) } { \\kappa \\tilde { t } ^ { \\alpha - 1 } ( \\mathcal { T } _ t ^ \\alpha Y ) ( Y ^ { - 1 } ( \\tilde { t } ) ) } \\bigg [ 2 h ( Z ( \\tilde { x } , \\tilde { t } ) , Y ^ { - 1 } ( \\tilde { t } ) ) ( s _ 2 s _ { 1 x } - s _ 1 s _ { 2 x } ) + f _ 1 s _ 1 s _ 2 \\\\ & - g _ 1 s _ 2 ^ 2 - f _ 2 s _ 1 s _ 2 + g _ 2 s _ 1 ^ 2 \\bigg ] , \\end{align*}"}
+{"id": "8345.png", "formula": "\\begin{align*} x E _ G ^ X y \\iff \\exists g \\in G g x = y , x , y \\in X . \\end{align*}"}
+{"id": "8156.png", "formula": "\\begin{align*} & \\theta ^ \\star _ { x y } = q _ { 1 0 } ( x y ) = \\frac { n } { k } \\Big ( ( r - 2 ) ( k - y ) - x ( r - 1 ) \\Big ) \\ , , \\\\ & \\mu ^ \\star _ { x y } = q _ { 0 1 } ( x y ) = ( n - 1 ) \\left ( 1 - \\frac { n } { k ( n - k ) } y \\right ) \\ , . \\end{align*}"}
+{"id": "5005.png", "formula": "\\begin{align*} A ( R ) \\leq \\sum _ { k = 0 } ^ { n } \\int _ { B _ p ( R - 1 4 k \\Lambda ) \\setminus B _ p ( R - 1 4 ( k + 1 ) \\Lambda ) } d A + A ( 1 4 \\Lambda ) . \\end{align*}"}
+{"id": "691.png", "formula": "\\begin{align*} [ \\mathbf u , \\mathbf v ] ( t ) = \\begin{cases} \\mathbf u ( t ) & \\\\ \\mathbf v ( t ) & . \\end{cases} \\end{align*}"}
+{"id": "4886.png", "formula": "\\begin{align*} K _ w ( z ) & = \\frac { E _ + ( z ) F _ + ( z ) F _ + ( w ) ^ * E _ + ( w ) ^ * - E _ - ( z ) F _ - ( z ) F _ - ( w ) ^ * E _ - ( w ) ^ * } { \\rho _ w ( z ) } \\\\ & = E _ + ( z ) K _ w ^ \\mathfrak { F } ( z ) E _ + ( w ) ^ * + F _ - ( z ) K _ w ^ \\mathfrak { E } ( z ) F _ - ( w ) ^ * . \\end{align*}"}
+{"id": "6790.png", "formula": "\\begin{align*} \\mathcal { V } \\left ( \\overline { u } , \\overline { v } \\right ) \\leq & d \\overline { v } '' - c \\overline { v } ' + s \\overline { v } \\\\ [ 0 . 2 c m ] = & - d h q ( 1 ) \\lambda \\left ( 2 + \\lambda z \\right ) e ^ { \\lambda z } + c h q ( 1 ) \\left ( 1 + \\lambda z \\right ) e ^ { \\lambda z } - s h q ( 1 ) z e ^ { \\lambda z } \\\\ [ 0 . 2 c m ] = & - h q ( 1 ) e ^ { \\lambda z } \\left [ P ( c , \\lambda ) z + ( 2 d \\lambda - c ) \\right ] = 0 . \\end{align*}"}
+{"id": "4479.png", "formula": "\\begin{align*} ( { \\mathbf V } _ i , \\Psi _ i , \\psi _ i ) | _ { t < 0 } = 0 , \\Psi ^ + _ { i } | _ { x _ 1 = 0 } = \\Psi ^ - _ { i } | _ { x _ 1 = 0 } = \\psi _ i . \\end{align*}"}
+{"id": "1243.png", "formula": "\\begin{align*} \\| e ^ { i ( t - a _ { j + 1 } ) \\Delta } w ( a _ { j + 1 } ) \\| _ { S ( I _ { j + 1 } ) } \\leq & 2 C \\epsilon + \\frac 1 3 \\sum _ { k = 1 } ^ { j } \\| w \\| _ { S ( I _ k ) } \\\\ & + C \\sum _ { k = 1 } ^ { j } ( \\eta ^ { p + 1 } \\| w \\| _ { S ( I _ k ) } ^ { p - 2 } + \\eta \\| w \\| _ { S ( I _ k ) } ^ { 2 ( p - 1 ) } + \\| w \\| _ { S ( I _ k ) } ^ { 2 p - 1 } ) \\\\ \\leq & 2 C \\epsilon + 4 \\sum _ { k = 1 } ^ { j } \\gamma _ j \\end{align*}"}
+{"id": "101.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { K } } ^ { } = \\frac { 1 } { 2 } \\sum _ { k \\neq 0 } \\widetilde { \\mathcal { A } } _ k \\big ( a _ k ^ \\dagger a _ k + a _ { - k } ^ \\dagger a _ { - k } \\big ) + \\frac { 1 } { 2 } \\sum _ { k \\neq 0 } \\mathcal { B } _ k \\big ( a _ k ^ \\dagger a _ { - k } ^ \\dagger + a _ k a _ { - k } \\big ) , \\end{align*}"}
+{"id": "874.png", "formula": "\\begin{align*} \\mathcal { T } _ t ^ \\alpha u = x u _ { x x } + f ( x ) u _ x , ~ ~ x \\geq 0 , \\end{align*}"}
+{"id": "6404.png", "formula": "\\begin{align*} t _ { 1 2 } c _ { 2 3 } + c _ { 1 2 } t _ { 2 3 } c ^ { - 1 } _ { 1 2 } c _ { 2 3 } & = c _ { 2 3 } t _ { 1 2 } + c _ { 2 3 } c _ { 1 2 } t _ { 2 3 } c ^ { - 1 } _ { 1 2 } \\\\ t _ { 2 3 } c _ { 1 2 } + c _ { 2 3 } t _ { 1 2 } c ^ { - 1 } _ { 2 3 } c _ { 1 2 } & = c _ { 1 2 } t _ { 2 3 } + c _ { 1 2 } c _ { 2 3 } t _ { 1 2 } c ^ { - 1 } _ { 2 3 } \\end{align*}"}
+{"id": "973.png", "formula": "\\begin{align*} ( a , b , c ) = \\left ( a , b , \\frac { a b ' + a ' b } { b ' + a ' } \\right ) , \\ \\left ( a , b , \\frac { a b ' - a ' b } { b ' - a ' } \\right ) \\end{align*}"}
+{"id": "5829.png", "formula": "\\begin{align*} w _ 1 = & ( s _ { \\alpha _ 3 + \\alpha _ 4 } s _ { \\alpha _ 2 + \\alpha _ 3 + \\alpha _ 4 + \\alpha _ 5 } s _ { \\alpha _ 1 + \\alpha _ 2 + \\alpha _ 3 + \\alpha _ 4 + \\alpha _ 5 + \\alpha _ 6 } ) _ { i \\rightarrow i + 2 } \\\\ & s _ { \\alpha _ 5 + \\alpha _ 6 } s _ { \\alpha _ 4 + \\alpha _ 5 + \\alpha _ 6 + \\alpha _ 7 } s _ { \\alpha _ 3 + \\alpha _ 4 + \\alpha _ 5 + \\alpha _ 6 + \\alpha _ 7 + \\alpha _ 8 } . \\end{align*}"}
+{"id": "1837.png", "formula": "\\begin{gather*} g ( s ) = f ( s ) - \\sum _ { n = 0 } ^ { M } \\frac { 1 } { ( n + c ) ^ s } \\end{gather*}"}
+{"id": "8721.png", "formula": "\\begin{align*} \\eta _ t = \\min \\left ( \\frac { \\mathfrak { y } } { d } , \\ , d ^ { - \\frac { 2 ( \\beta - 1 ) } { 2 \\beta - 1 } } T ^ { - \\frac { \\beta } { 2 \\beta - 1 } } \\right ) \\qquad h _ t = \\mathfrak { h } \\cdot T ^ { - \\frac { 1 } { 2 ( 2 \\beta - 1 ) } } \\enspace . \\end{align*}"}
+{"id": "2144.png", "formula": "\\begin{align*} \\Lambda ^ { ( 0 ) } ( t , x ) = u _ 0 , \\hbox { a n d } \\phi ^ { ( 0 ) } = n \\pi , n \\in \\mathbb Z . \\end{align*}"}
+{"id": "7269.png", "formula": "\\begin{align*} S _ n ( \\omega ) = - \\log f ' _ { \\omega | _ n } ( x _ { \\sigma ^ n ( \\omega ) } ) . \\end{align*}"}
+{"id": "7860.png", "formula": "\\begin{align*} S _ 0 \\leq B ^ 6 \\sum _ { k = 1 } ^ { \\lfloor r _ n / 2 \\rfloor } k Z _ { \\ell _ k } ( \\omega _ { \\ell _ k } ) \\preceq B ^ 6 r _ n ^ 2 e ^ { - \\alpha r _ n } \\leq \\frac { B ^ 6 r _ n ^ 2 } { n \\log { n } } \\preceq \\frac { 1 } { \\log { n } } . \\end{align*}"}
+{"id": "2180.png", "formula": "\\begin{align*} & ( G / R ) V _ R = \\{ R g R v \\mid R g \\in G / R , v \\in V \\} \\\\ & \\ : \\ : = \\{ R g v \\mid g \\in G , v \\in V \\} = \\{ R x \\mid x \\in G V \\} = X _ R , \\end{align*}"}
+{"id": "1762.png", "formula": "\\begin{gather*} \\mathcal { A } ( c , d ) = \\{ \\alpha \\in \\mathcal { A } \\mid \\} , \\end{gather*}"}
+{"id": "5213.png", "formula": "\\begin{align*} m _ { j , k + 1 } & = m _ { j + 1 , k } \\\\ & = m _ { j + 2 , k + 1 } + n _ { j + 1 } - n _ j \\\\ & = m _ { j + 3 , k } + n _ { j + 1 } - n _ j \\\\ & = m _ { j + 4 , k + 1 } + 2 ( n _ { j + 1 } - n _ j ) = \\cdots \\\\ & = m _ { j + 2 N , k + 1 } + N ( n _ { j + 1 } - n _ j ) . \\end{align*}"}
+{"id": "4748.png", "formula": "\\begin{align*} { \\rm P e r } ( A \\cap \\R _ { + } ^ { 2 } ; \\R _ { + } ^ { 2 } ) = \\sum _ { n = 1 } ^ \\infty { \\rm P e r } ( A \\cap \\overline { T } _ n ; \\overline { T } _ n ) . \\end{align*}"}
+{"id": "2997.png", "formula": "\\begin{align*} \\Psi ( x ) z ( e ) = \\sum _ { i = 1 } ^ \\infty ( \\rho _ i x ) ^ { 1 / 2 } ( e ) \\sum _ { s ( f ) = s ( e ) } ( \\rho _ i x ) ^ { 1 / 2 } ( f ) z ( f ) = \\sum _ { i = 1 } ^ { \\infty } \\rho _ i ( e ) x ( e ) z ( e ) = x ( e ) z ( e ) . \\end{align*}"}
+{"id": "496.png", "formula": "\\begin{align*} d X = A X d t + \\mathcal N ( X ) d t + \\mathcal M ( X ) \\circ d W , \\end{align*}"}
+{"id": "2189.png", "formula": "\\begin{align*} ( \\deg ( x ) - \\lambda ) f _ 1 ( x ) & = \\sum _ { y \\in X } b ( x , y ) f _ 1 ( y ) \\\\ & = \\sum _ { y \\in X } b ( x , y ) \\mu _ x f _ 2 ( y ) \\\\ & = ( \\deg ( x ) - \\lambda ) \\mu _ x f _ 2 ( x ) \\end{align*}"}
+{"id": "4338.png", "formula": "\\begin{align*} \\omega _ { E _ { \\sf f } \\Omega _ \\omega } ( A ) = ( E _ { \\sf f } \\Omega _ \\omega , A E _ { \\sf f } \\Omega _ \\omega ) \\ \\ ( A \\in \\mathcal { N } _ \\omega ) \\ , . \\end{align*}"}
+{"id": "5709.png", "formula": "\\begin{align*} \\| ( z - A ) ^ { - 1 } \\| & = \\| ( z - B ) ^ { - 1 } \\| \\geq \\frac { 1 } { \\| \\xi _ 1 \\| _ p } \\| ( z - B ) ^ { - 1 } \\xi _ 1 \\| _ p \\\\ & \\geq \\frac { 1 } { \\| \\xi _ 1 \\| _ p } | f _ 0 ( ( z - B ) ^ { - 1 } \\xi _ 1 ) | \\\\ & = \\frac { 1 } { \\| \\xi _ 1 \\| _ p } \\big | \\frac { 1 } { z } + z ^ { m - 1 } \\big ( e ^ { \\frac { 1 } { z } } - \\sum \\limits _ { n = 0 } ^ { m } \\frac { 1 } { n ! } \\frac { 1 } { z ^ n } \\big ) \\big | \\end{align*}"}
+{"id": "861.png", "formula": "\\begin{align*} \\langle \\gamma ' \\rangle \\supset \\Big ( \\langle \\gamma ' \\rangle \\cap C ^ { H ' } _ { H ' _ j } \\Big ) \\supset \\Big ( \\langle \\gamma ' \\rangle \\cap C ^ { H ' } _ { j } \\Big ) = C ^ { H ' } _ { \\gamma ' , j } . \\end{align*}"}
+{"id": "6761.png", "formula": "\\begin{align*} M P = \\left [ \\begin{array} { c c } 1 & 1 \\\\ 1 & 1 \\\\ 2 & 1 \\end{array} \\right ] . \\end{align*}"}
+{"id": "5313.png", "formula": "\\begin{align*} R ( x ) = x \\frac { P ( x ) } { P ( x ) + P ( - x ) } \\end{align*}"}
+{"id": "842.png", "formula": "\\begin{align*} \\mathcal { F } = \\{ \\widetilde { T f } : f \\in S _ { A ( \\mathbb { D } ) } \\} . \\end{align*}"}
+{"id": "7216.png", "formula": "\\begin{align*} v ^ N ( t , \\mathbf { x } ^ N ) : = \\mathbb { P } \\left [ \\forall s \\in [ 0 , t ] , \\hat { \\mu } _ s ^ N \\in \\Omega _ { \\infty } \\right ] , \\end{align*}"}
+{"id": "8751.png", "formula": "\\begin{align*} K ( r ) = \\begin{cases} 1 , & r \\geq 0 \\\\ - 1 , & r < 0 \\end{cases} \\enspace . \\end{align*}"}
+{"id": "8645.png", "formula": "\\begin{align*} \\Big { | } \\int _ { - 1 + \\beta b ( \\cdot ) } ^ 0 u ( \\cdot , z ) \\ : \\mathrm { d } z \\Big { | } _ { H ^ k } ^ 2 \\leq M ( k ) \\big { ( } \\| \\nabla _ { X , z } ^ { \\mu } u \\| ^ 2 _ { H ^ { k , 0 } ( \\mathcal { S } _ b ) } + \\sum \\limits _ { j = 1 } ^ k \\| \\partial _ z ^ { j } u \\| _ { H ^ { k - j , 0 } ( \\mathcal { S } _ b ) } ^ 2 \\big { ) } . \\end{align*}"}
+{"id": "4614.png", "formula": "\\begin{align*} J _ \\chi = \\textup { E n d } ( \\mathcal { H } _ \\chi ) . \\end{align*}"}
+{"id": "2258.png", "formula": "\\begin{align*} \\min _ x p _ \\alpha ( x ) = \\min _ x \\sum _ { j = 1 } ^ n \\sum _ { k \\in \\Z } e ^ { - \\pi \\alpha ( k + x _ j - x ) ^ 2 } . \\end{align*}"}
+{"id": "5317.png", "formula": "\\begin{align*} P ( x ) + P ( - x ) & \\leq | P ( x ) | + | P ( - x ) | \\\\ & = ( x + 1 ) \\prod _ { k = 1 } ^ { n - 1 } | x + \\xi ^ k | + ( - x + 1 ) \\prod _ { k = 1 } ^ { n - 1 } | - x + \\xi ^ k | \\\\ & \\leq ( x + 1 ) \\prod _ { k = 1 } ^ { n - 1 } 2 + ( - x + 1 ) \\prod _ { k = 1 } ^ { n - 1 } 2 \\\\ & = 2 ^ n . \\end{align*}"}
+{"id": "4728.png", "formula": "\\begin{align*} | \\tilde u _ j ( x ) | & = | \\frac { u _ j ( ( x ^ j ) ' + d _ j x ) - u _ j ( ( x ^ j ) ' ) } { d _ j ^ 2 } | \\\\ & \\le \\frac { | \\nabla u _ j ( r _ { x ^ j } ) | | x | } { d _ j } = \\frac { | \\nabla u _ j ( r _ { x ^ j } ) - \\nabla u _ j ( x ^ j ) | | x | } { d _ j } \\le T _ 1 , \\end{align*}"}
+{"id": "6875.png", "formula": "\\begin{align*} c _ i ( \\widetilde { X } , r ) : = c _ i ( T _ { \\widetilde { X } } , r ) . \\end{align*}"}
+{"id": "5821.png", "formula": "\\begin{align*} & \\alpha ^ { a 3 } _ { m a x } : = \\alpha _ 3 + \\alpha _ 4 + \\alpha _ 5 , \\\\ & \\alpha ^ { a 5 } _ { m a x } : = \\alpha _ 1 + \\alpha _ 3 + \\alpha _ 4 + \\alpha _ 5 + \\alpha _ 6 , \\end{align*}"}
+{"id": "5108.png", "formula": "\\begin{align*} d ^ n ( f ) ( g _ 1 , \\ldots , g _ { n + 1 } ) = g _ 1 f ( g _ 2 , \\ldots , g _ { n + 1 } ) + \\sum _ { i = 1 } ^ { n } ( - 1 ) ^ { i } f ( g _ 1 , \\ldots , g _ { i - 1 } , g _ i g _ { i + 1 } , \\ldots , g _ { n + 1 } ) + ( - 1 ) ^ { n + 1 } f ( g _ 1 , \\ldots , g _ n ) . \\end{align*}"}
+{"id": "4430.png", "formula": "\\begin{align*} \\hat a ^ + \\vert \\hat { H } _ 2 ^ + \\vert + \\hat a ^ - \\vert \\hat { H } _ 2 ^ - \\vert = \\frac { \\hat c ^ + \\hat c ^ + _ A } { ( ( \\hat c ^ + ) ^ 2 + ( \\hat c ^ + _ A ) ^ 2 ) ^ { 1 / 2 } } + \\frac { \\hat c ^ - \\hat c ^ - _ A } { ( ( \\hat c ^ - ) ^ 2 + ( \\hat c ^ - _ A ) ^ 2 ) ^ { 1 / 2 } } \\end{align*}"}
+{"id": "7862.png", "formula": "\\begin{align*} \\mu \\left ( \\{ C _ { r _ n } ( \\sigma ^ i \\underline { x } ) = C _ { r _ n } ( \\sigma ^ { i + k } \\underline { x } ) \\} \\right ) = & \\mu \\left ( \\sigma ^ { - i } S _ k ( r _ n ) \\right ) = \\sum _ { C \\in \\mathcal { C } _ { r _ n } } \\mu ( C \\cap \\sigma ^ { - k } C ) \\\\ \\geq & ( 1 - \\psi ( k - r _ n ) ) \\sum _ { C \\in \\mathcal { C } _ { r _ n } } \\mu ( C ) ^ 2 \\geq ( 1 - \\psi ( r _ n ) ) Z _ { r _ n } ( 1 ) , \\end{align*}"}
+{"id": "5466.png", "formula": "\\begin{align*} x G ^ * ( 1 / x ) & \\leq \\psi ^ { - 1 } ( \\log _ 2 \\Phi ^ { - 1 } ( ( 1 / x ) ) ) \\\\ & \\leq \\psi ^ { - 1 } \\left ( \\log _ 2 \\left ( \\frac { 1 } { x \\log ( 1 / x ) } \\left [ 1 + \\frac { 1 } { \\log ( 1 / x ) } \\right ] \\right ) \\right ) \\\\ & = \\psi ^ { - 1 } \\left ( \\log \\left [ y - \\log y + \\log \\left ( 1 + \\frac { 1 } { y } \\right ) \\right ] \\right ) . \\end{align*}"}
+{"id": "7285.png", "formula": "\\begin{align*} C ( \\sigma \\mid \\ ; \\geq N ) = \\max _ { n \\geq N } C ( \\sigma \\mid n ) . \\end{align*}"}
+{"id": "7893.png", "formula": "\\begin{align*} \\begin{cases} ( u ^ { \\star } ) ^ k \\det D ^ 2 u = | u | ^ { p } & \\Omega \\\\ u = 0 & \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "5870.png", "formula": "\\begin{align*} D _ { N , k } ( T ) = \\sup _ { k \\leq \\nu < \\infty } E _ { \\nu } ( T - \\nu + 1 | T \\geq \\nu ) . \\end{align*}"}
+{"id": "4212.png", "formula": "\\begin{align*} ( J _ \\rho \\alpha ) ( v ) \\ , \\phi ( \\rho ) = \\alpha \\wedge \\iota _ v \\rho \\wedge \\rho , \\alpha \\in V ^ * , v \\in V . \\end{align*}"}
+{"id": "3817.png", "formula": "\\begin{align*} \\nu _ X = \\frac { 1 } { \\mu _ 0 ( X ) \\mu _ 1 ( X ) } \\mu _ 0 \\otimes \\mu _ 1 , \\end{align*}"}
+{"id": "1030.png", "formula": "\\begin{align*} [ x ] _ F = [ x ] _ F ^ { * G } = \\bigcap _ n \\Big [ \\bigcup _ m A _ { n , m } \\Big ] ^ { * G } = \\bigcap _ n \\bigcap _ V \\Big [ \\bigcup _ m A _ { n , m } \\Big ] ^ { \\Delta V } = \\bigcap _ n \\bigcap _ V \\bigcup _ m A _ { n , m } ^ { \\Delta V } \\end{align*}"}
+{"id": "5367.png", "formula": "\\begin{align*} h ^ 1 ( { N _ { X ' } } _ { | _ { Y _ j } } ( K _ { Y _ j } + D _ j - 3 H ) ) = h ^ 0 ( { N ^ * _ { X ' } } _ { | _ { Y _ j } } ( 3 H - D _ j ) ) = h ^ 0 ( N ^ * _ { Y _ j } ( 3 H - D _ j ) ) - 2 = h ^ 1 ( N _ { Y _ j } ( K _ { Y _ j } + D _ j - 3 H ) ) - 2 = 0 . \\end{align*}"}
+{"id": "4827.png", "formula": "\\begin{align*} \\check { R } ( 0 ) \\check { R } ( 0 ) & = \\rho ( 0 ) \\rho ( 0 ) I \\\\ C \\cdot C & = \\rho ( 0 ) ^ 2 \\end{align*}"}
+{"id": "5192.png", "formula": "\\begin{align*} f [ r ] = \\sum _ { \\ell = 0 } ^ L \\sum _ { m = - \\ell } ^ \\ell A _ { \\ell } ^ m [ r ] Y _ { \\ell } ^ m . \\end{align*}"}
+{"id": "4813.png", "formula": "\\begin{align*} \\| \\Pi _ k \\varphi \\| ^ \\perp _ { \\alpha + r , m } = \\| \\Pi _ k \\varphi \\| _ { C ^ 0 } + | \\Pi _ k \\varphi | ^ \\perp _ { \\alpha + r , m } \\leq \\| \\varphi \\| _ { C ^ 0 } + 1 + 2 C _ \\mu \\lambda _ 0 ^ { \\Lambda k } \\leq 2 + 2 C _ \\mu \\lambda _ 0 ^ { \\Lambda k } \\leq 4 C _ \\mu \\lambda _ 0 ^ { \\Lambda k } . \\end{align*}"}
+{"id": "1966.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf { a } ^ t _ \\sigma = \\boldsymbol { g } & + \\frac { q } { p } \\left ( \\sum _ { \\beta = 1 } ^ { k } \\sigma _ \\beta \\boldsymbol { x } _ { \\pi _ \\beta ( 1 ) } + \\sum _ { \\beta = 1 } ^ { k } t _ \\beta \\boldsymbol { x } _ { \\pi _ \\beta ( m _ { \\beta } ) } \\right . \\\\ & \\left . + \\sum _ { \\beta = 1 } ^ { \\delta } t _ { k + \\beta } \\boldsymbol { x } _ { m - \\delta + \\beta } \\right ) , \\end{aligned} \\end{align*}"}
+{"id": "2347.png", "formula": "\\begin{align*} \\lambda _ g \\theta _ f \\theta _ \\tau \\delta _ h & = \\lambda _ g \\left ( \\sum _ { u \\in G } \\eta ( u ) \\rho _ u \\delta _ h \\right ) = \\sum _ { u \\in G } \\eta ( u ) \\lambda _ g \\rho _ u \\delta _ h \\\\ & = \\left ( \\sum _ { u \\in G } \\eta ( u ) \\rho _ u \\right ) \\lambda _ g \\delta _ h = \\theta _ f \\theta _ \\tau \\lambda _ g \\delta _ h , \\forall g \\in G . \\end{align*}"}
+{"id": "6145.png", "formula": "\\begin{align*} \\textrm { G T } ( \\mathbf { k } ) = \\textrm { G T } ( k _ 1 , k _ 2 , \\ldots , k _ n ) = \\bigsqcup _ { \\mathbf { l } \\in \\underline { [ k _ 1 , k _ 2 ) } \\times \\cdots \\times \\underline { [ k _ { n - 1 } , k _ n ) } } \\textrm { G T } ( \\mathbf { l } ) . \\end{align*}"}
+{"id": "5263.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } l _ t ( x _ { i , t } ) & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { n } l _ { j , t } ( x _ { i , t } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { n } l _ { j , t } ( x _ { j , t } ) + \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { n } ( l _ { j , t } ( x _ { i , t } ) - l _ { j , t } ( x _ { j , t } ) ) \\\\ & \\le \\sum _ { i = 1 } ^ { n } l _ { i , t } ( x _ { i , t } ) + \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { n } G _ 1 \\| x _ { i , t } - x _ { j , t } \\| , \\end{align*}"}
+{"id": "6993.png", "formula": "\\begin{align*} \\gamma _ k = \\left ( Y _ { k + 1 } ^ { \\frac { m } { n } } M _ { k } \\right ) ^ { \\frac { n } { m + n } } \\end{align*}"}
+{"id": "5000.png", "formula": "\\begin{align*} - 6 ( \\alpha - H ^ 2 ) \\int _ { 0 } ^ { R } \\phi ( r ) \\phi ' ( r ) A ( r ) = 3 ( \\alpha - H ^ 2 ) \\int _ { B _ { p } ( R ) } \\phi ^ 2 . \\end{align*}"}
+{"id": "1584.png", "formula": "\\begin{align*} \\sqrt { \\rho _ g ( x ) } \\left \\langle \\tilde { h } , \\frac { 1 } { \\sqrt { 2 } } \\left [ \\frac { v ^ 2 } { T _ g } - 1 \\right ] \\mathcal { M } _ { T _ g } \\right \\rangle \\leq \\left ( \\int _ { - \\infty } ^ { \\infty } \\frac { \\tilde { h } ^ 2 } { g ^ 2 } \\rho _ g ( x ) \\mathcal { M } _ { T _ g } ( v ) \\mathrm { d } v \\right ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "5183.png", "formula": "\\begin{align*} a _ 2 ( f ) ( V \\otimes W ) = [ F ( f ) ( V ) \\otimes F ( f ) ( W ) ] C _ { V , W } [ F ( f ) ( V _ 1 ) ^ * \\oplus \\ldots \\oplus F ( f ) ( V _ r ) ^ * ] C _ { V , W } ^ * \\end{align*}"}
+{"id": "3503.png", "formula": "\\begin{align*} u ( x ) = \\max _ { p \\in \\Delta ^ \\vee } \\langle p , x \\rangle - u ^ * ( p ) \\geq - u ^ * ( - \\frac { l _ i } { l } ) + \\sum _ 0 ^ m \\frac { - l _ i } { l } x _ i , \\end{align*}"}
+{"id": "2466.png", "formula": "\\begin{align*} H _ \\kappa ( G _ 6 ) & = H ( V _ 6 ) - \\max _ { V _ 6 \\in W _ 6 \\in \\Gamma ( G _ 6 ) } H ( V _ 6 | W _ 6 ) \\\\ & = 1 + \\log 3 - \\log 3 = 1 \\end{align*}"}
+{"id": "3034.png", "formula": "\\begin{align*} | H | \\le 2 + 2 ( n - 2 ) + 2 \\cdot \\lfloor \\frac { ( n - 2 ) ^ 2 } { 4 } \\rfloor = 2 \\cdot \\lfloor \\frac { n ^ 2 } { 4 } \\rfloor . \\end{align*}"}
+{"id": "4008.png", "formula": "\\begin{align*} \\begin{aligned} 0 & \\geq \\sum _ l X _ { l \\bar l i \\bar i } - \\frac { w _ i w _ { \\bar i } } { w } - w A \\varphi _ { t , i \\bar i } \\\\ & = \\sum _ l \\left ( X _ { i \\bar i l \\bar l } - R _ { l \\bar l i \\bar i } X _ { i \\bar i } + R _ { i \\bar i l \\bar l } X _ { l \\bar l } + G _ { i \\bar i l \\bar l } \\right ) - \\frac { w _ i w _ { \\bar i } } { w } - w A \\varphi _ { t , i \\bar i } . \\end{aligned} \\end{align*}"}
+{"id": "3985.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) u ^ { i , n } ( \\omega ) = f \\Big ( u ^ { i , n } ( \\omega ) , u ^ { U _ n ( \\cdot ) , n } ( \\omega ) \\Big ) \\xi ^ i ( \\omega ) + g \\Big ( u ^ { i , n } ( \\omega ) , u ^ { U _ n ( \\cdot ) , n } ( \\omega ) \\Big ) \\end{align*}"}
+{"id": "5899.png", "formula": "\\begin{align*} \\| v ( x _ 1 , \\cdot ) \\| _ { L ^ 2 ( ( 0 , 1 ) ) } & \\leq N \\Big ( \\| v \\| _ { L ^ 2 ( ( x _ 1 - 1 , x _ 1 ) \\times ( 0 , 1 ) ) } + \\| \\nabla v \\| _ { L ^ 2 ( ( x _ 1 - 1 , x _ 1 ) \\times ( 0 , 1 ) ) } \\Big ) \\\\ & \\leq N \\| \\nabla v \\| _ { L ^ 2 ( ( x _ 1 - 1 , x _ 1 ) \\times ( 0 , 1 ) ) } , \\end{align*}"}
+{"id": "7084.png", "formula": "\\begin{align*} \\mathcal { C } ^ { \\pm } ( n ^ 2 _ 1 n _ 2 , m ; q ) = \\sideset { } { ^ \\star } { \\displaystyle \\sum } _ { a \\ , \\mathrm { m o d } \\ , q } \\ , \\ , \\ , \\sum _ { b \\ , \\mathrm { m o d } \\ , p _ 1 } \\ , S \\left ( r \\overline { ( a + b q ) } , \\pm n _ { 2 } ; p _ 1 q r / n _ { 1 } \\right ) \\ , e \\left ( - m \\frac { \\overline { ( a + b q ) p _ 2 } } { p _ 1 q } \\right ) , \\end{align*}"}
+{"id": "8413.png", "formula": "\\begin{align*} U ( T , r , v ) = U _ { 1 } - U _ { 2 } - U _ { 3 } + \\mathcal { O } \\left ( N ^ { \\frac { \\gamma } { 2 } } \\right ) , \\end{align*}"}
+{"id": "6217.png", "formula": "\\begin{align*} \\frac { 1 } { p } = \\frac { 1 - \\theta } { p _ 1 } + \\frac { \\theta } { p _ 2 } , \\frac { 1 } { q } = \\frac { 1 - \\theta } { q _ 1 } + \\frac { \\theta } { q _ 2 } , r = r _ 1 ( 1 - \\theta ) + r _ 2 \\theta , s = s _ 1 ( 1 - \\theta ) + s _ 2 \\theta , \\end{align*}"}
+{"id": "1889.png", "formula": "\\begin{align*} [ \\ ! [ \\phi _ h ] \\ ! ] _ { i + \\frac { 1 } { 2 } , v } & : = \\left ( \\phi _ h \\right ) ^ + _ { i + \\frac { 1 } { 2 } , v } - \\left ( \\phi _ h \\right ) ^ - _ { i + \\frac { 1 } { 2 } , v } \\quad \\forall \\ , \\phi _ h \\in \\mathcal { Z } _ h , \\\\ \\{ \\phi _ h \\} _ { i + \\frac { 1 } { 2 } , v } & : = \\frac { 1 } { 2 } \\left ( \\left ( \\phi _ h \\right ) ^ + _ { i + \\frac { 1 } { 2 } , v } + \\left ( \\phi _ h \\right ) ^ - _ { i + \\frac { 1 } { 2 } , v } \\right ) \\quad \\forall \\ , \\phi _ h \\in \\mathcal { Z } _ h . \\end{align*}"}
+{"id": "8048.png", "formula": "\\begin{align*} h _ { \\omega } ^ { p } ( \\mathbb { R } ^ { n } ) = h _ { \\omega , a t o m } ^ { p , q , s } ( \\mathbb { R } ^ { n } ) \\end{align*}"}
+{"id": "7355.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } | F _ N ( \\gamma ) - \\tilde { F } _ N ( \\gamma ) | = 0 . \\end{align*}"}
+{"id": "8010.png", "formula": "\\begin{align*} T _ { ( - k ) } ( t ) = g ^ { - 1 } _ k \\bigl ( X ( t ) \\bigr ) = \\Bigl ( g _ { k , h } ^ { - 1 } \\bigl ( X ( t ) \\bigr ) \\Bigr ) _ { h \\not = k } = \\Bigl ( g _ { \\cdot , h } ^ { - 1 } \\bigl ( t , X ( t ) \\bigr ) \\Bigr ) _ { h \\not = k } = \\Bigl [ v _ h - v _ k \\Bigr ] _ { h \\not = k } ^ { - 1 } \\bigl ( X ( t ) - v _ k t \\bigr ) . \\end{align*}"}
+{"id": "8277.png", "formula": "\\begin{align*} e ^ { i t } \\cdot ( u _ 1 , \\dots , u _ n ) = ( u _ 1 e ^ { i a _ 1 t } , \\dots , u _ n e ^ { i a _ n t } ) \\end{align*}"}
+{"id": "4957.png", "formula": "\\begin{align*} | A _ j | + | B _ j | = \\phi ( 8 n ) / 4 = \\phi ( n ) \\end{align*}"}
+{"id": "6006.png", "formula": "\\begin{align*} X _ { t } ^ { \\mathcal { P } , { M _ { \\mathcal { P } } } } = x + \\int _ { 0 } ^ { t } b ( X _ { \\tau ( r ) } ^ { \\mathcal { P } , { M _ { \\mathcal { P } } } } ) d r + \\int _ { 0 } ^ { t } \\int _ { B _ { M _ { { \\mathcal { P } } } ( r ) } } { c } ( z , X _ { \\tau ( r ) - } ^ { \\mathcal { P } , { M _ { \\mathcal { P } } } } ) N ( d z , d r ) . \\end{align*}"}
+{"id": "4790.png", "formula": "\\begin{align*} C ( k _ 1 , k _ 2 , j ; \\psi , \\eta ) { = } \\frac { 1 } { 4 ^ { k _ 1 { + } k _ 2 } k _ 1 ! ( k _ 1 { + } 2 j ) ! k _ 2 ! ( k _ 2 { + } j ) ! } \\left ( \\frac { \\mu ^ 2 _ 1 \\sigma ^ 4 _ { 2 } { + } \\mu ^ 2 _ 2 \\sigma ^ 4 _ { 1 } } { \\sigma ^ 4 _ 1 \\sigma ^ 4 _ { 2 } } \\right ) ^ { k _ { 1 } } \\left ( \\frac { ( \\sigma ^ 2 _ 2 { - } \\sigma ^ 2 _ 1 ) ^ 2 } { 1 6 \\sigma ^ 4 _ 1 \\sigma ^ 4 _ { 2 } } \\right ) ^ { k _ 2 } . \\end{align*}"}
+{"id": "4526.png", "formula": "\\begin{align*} & | | \\delta { { \\mathbf V } _ i } | | _ { s , \\ast , T } + | | \\delta \\psi _ i | | _ { H ^ s ( \\Gamma _ T ) } \\\\ & \\leq C \\Big ( | | { f } _ i | | _ { s + 2 , \\ast , T } + | | { g } _ i | | _ { H ^ { s + 2 } ( \\Gamma _ T ) } \\\\ & \\quad + ( | | { f } _ i | | _ { 8 , \\ast , T } + | | { g } _ i | | _ { H ^ { 8 } ( \\Gamma _ T ) } ) | | ( \\tilde { \\mathbf U } ^ a + { \\mathbf V } _ { i + \\frac { 1 } { 2 } } , \\nabla ( \\Psi ^ a + \\Psi _ { i + \\frac { 1 } { 2 } } ) ) | | _ { s + 4 , \\ast , T } \\Big ) . \\end{align*}"}
+{"id": "4195.png", "formula": "\\begin{align*} D ( 0 , \\varepsilon ) & = \\sigma _ { \\varepsilon } ( [ i I \\bullet \\varphi ^ { - 1 } ( \\frac { i I } { 2 } ) , S ] _ { \\ast } ) = \\sigma _ { \\varepsilon } ( [ \\varphi ( i I ) \\bullet \\frac { i I } { 2 } , \\frac { i I } { 2 } ] _ { \\ast } ) \\\\ & = \\sigma _ { \\varepsilon } ( \\frac { - 1 } { 2 } ( \\varphi ( i I ) + \\varphi ( i I ) ^ { \\ast } ) ) . \\end{align*}"}
+{"id": "8893.png", "formula": "\\begin{align*} H _ G ^ * ( X ) = H _ T ^ * ( X ) ^ W . \\end{align*}"}
+{"id": "4768.png", "formula": "\\begin{align*} \\sum _ { n = N } ^ { \\infty } \\mu _ { p } ( \\{ x \\mid v _ { \\phi } ( g _ { n } ^ { - 1 } x ) > n \\} ) = \\sum _ { n = N } ^ { \\infty } \\mu _ { p } ( \\{ x \\mid v _ { \\phi } ( x ) > n \\} ) < \\epsilon . \\end{align*}"}
+{"id": "3039.png", "formula": "\\begin{align*} k \\ge x + \\sum _ { i = 1 } ^ m y _ i + \\sum _ { i = 1 } ^ m ( \\frac { k } { 4 } + 2 y _ i - x ) = \\frac { m k } { 4 } + 3 \\sum _ { i = 1 } ^ m y _ i - ( m - 1 ) x > \\frac { ( m + 3 ) k } { 4 } - ( m - 7 ) x . \\end{align*}"}
+{"id": "2779.png", "formula": "\\begin{align*} x ^ { k + 3 } \\cdot x ^ { k + 1 } \\cdot x ^ 4 = x ^ { 2 k + 8 } , \\end{align*}"}
+{"id": "7124.png", "formula": "\\begin{align*} \\Delta ^ 2 \\left ( \\left ( ( 1 + t ) ^ { - 1 } f _ 1 ( x , Z ) + ( 1 + t ) ^ { - 3 / 2 } f _ 2 ( x , Z ) \\right ) \\chi ( z ) \\right ) = f ( x , Z ) \\chi ( z ) + O ( ( 1 + t ) ^ { - 1 } ) H ^ { - 2 } . \\end{align*}"}
+{"id": "8824.png", "formula": "\\begin{align*} h ( t ) & \\leq \\rho ^ { 2 } \\sum _ { i = 1 } ^ { n } \\norm { x ^ { i } ( t - 1 ) - \\bar { x } ( t - 1 ) - \\eta _ { t } \\big ( g ^ { i } ( t ) - \\bar { g } ( t ) \\big ) } ^ { 2 } \\\\ & \\leq \\rho ^ { 2 } h ( t - 1 ) + \\rho ^ { 2 } \\eta _ { t } ^ { 2 } \\sum _ { i = 1 } ^ { n } \\norm { g ^ { i } ( t ) - \\bar { g } ( t ) } ^ { 2 } - 2 \\rho ^ { 2 } \\eta _ { t } \\sum _ { i = 1 } ^ { n } \\langle x ^ { i } ( t - 1 ) - \\bar { x } ( t - 1 ) , g ^ { i } ( t ) - \\bar { g } ( t ) \\rangle , \\end{align*}"}
+{"id": "7433.png", "formula": "\\begin{align*} \\| ( y _ 1 , y _ 2 ) \\| _ { 1 , p } \\leq \\left ( \\frac { ( \\kappa _ 1 + \\kappa _ 2 + \\nu + 1 ) C _ 4 } { \\kappa _ 1 + \\kappa _ 2 + 2 - p } \\right ) ^ { \\frac { p } { p + \\gamma - 1 } } : = C _ 5 . \\end{align*}"}
+{"id": "1330.png", "formula": "\\begin{align*} J ^ { w _ 1 } ( \\textbf { X } _ { m i n R S S U } ^ { ( n ) } ) & = - \\frac { ( n ! ) ^ 2 } { 2 ( n - 1 ) ! ! } \\prod _ { i = 1 } ^ { n } E ( \\Lambda _ X ^ { w _ 1 } ( B _ { 1 , 2 i - 1 } ) ) \\\\ & \\le - \\frac { ( n ! ) ^ 2 } { 2 ( n - 1 ) ! ! } \\prod _ { i = 1 } ^ { n } E ( \\Lambda _ Y ^ { w _ 2 } ( B _ { 1 , 2 i - 1 } ) ) \\\\ & = J ^ { w _ 2 } ( \\textbf { Y } _ { m i n R S S U } ^ { ( n ) } ) . \\\\ \\end{align*}"}
+{"id": "7778.png", "formula": "\\begin{align*} \\sum _ { \\hat { \\gamma } = q _ a \\gamma ^ a \\in \\Lambda ^ + } | n _ { \\hat { \\gamma } } | e ^ { - \\epsilon q _ a t ^ a } \\end{align*}"}
+{"id": "2596.png", "formula": "\\begin{align*} \\mu _ 0 ( X , \\{ \\sigma _ n \\} ) : = \\pi _ 0 X = \\ast \\end{align*}"}
+{"id": "7484.png", "formula": "\\begin{align*} A _ L = \\mathrm { N } _ { q ^ m / q } ( \\alpha _ 1 / \\alpha _ 2 ) \\begin{pmatrix} - u ^ { \\sigma ^ { - 1 } } G _ { m - 2 } ^ { \\sigma } & - ( \\alpha _ 0 / \\alpha _ 1 ) G ^ \\sigma _ { m - 1 } \\\\ ( \\alpha _ 2 / \\alpha _ 1 ) ^ { \\sigma ^ { - 1 } } G _ { m - 1 } & G _ m \\end{pmatrix} , \\end{align*}"}
+{"id": "7040.png", "formula": "\\begin{align*} w ( t ) = u ( q , t ) , t \\in \\lbrack 0 , T ] . \\end{align*}"}
+{"id": "4517.png", "formula": "\\begin{align*} \\partial _ t Y + \\sum _ { j = 1 } ^ 2 \\mathcal D _ j ( b ) \\partial _ j Y + \\mathcal Q ( b ) Y = F _ H ^ { i + 1 / 2 } , \\end{align*}"}
+{"id": "1257.png", "formula": "\\begin{align*} v _ { n , T } ( t ) & = g _ n [ \\chi _ n P _ n e ^ { i ( \\lambda _ n ^ { - 2 } t + t _ n ) \\Delta } \\phi ] \\\\ & = \\chi ( \\frac { x - x _ n } { \\lambda _ n } ) e ^ { i ( t + \\lambda _ n ^ 2 t _ n ) \\Delta } g _ n [ P _ n \\phi ] . \\end{align*}"}
+{"id": "3421.png", "formula": "\\begin{align*} \\Omega _ t = b _ 0 ^ { - 1 } f _ 1 f _ 2 ^ { - 1 } z _ 0 ^ { a _ 0 } \\ldots z _ k ^ { a _ k } d \\log z _ 1 \\wedge \\ldots d \\log z _ k \\wedge d z _ { k + 1 } \\wedge \\ldots d z _ n . \\end{align*}"}
+{"id": "8194.png", "formula": "\\begin{align*} g _ \\mathbb { H } = \\sum _ { \\alpha = 0 } ^ 3 ( d q _ \\alpha ) ^ 2 . \\end{align*}"}
+{"id": "3568.png", "formula": "\\begin{align*} G ^ 0 _ { i j } = - \\frac 1 2 \\widehat { g } ( z _ { R _ j } - z _ { R _ i } , z _ { R _ j } - z _ { R _ i } ) = - \\frac 1 2 g ( \\vect { R _ i R _ j } , \\vect { R _ i R _ j } ) . \\end{align*}"}
+{"id": "7597.png", "formula": "\\begin{align*} \\kappa = \\beta ^ { 1 / 3 } N ^ { 1 / 3 } , \\alpha = 0 , \\end{align*}"}
+{"id": "8353.png", "formula": "\\begin{align*} Z _ p ( t , \\omega ) & = S _ B ( t ) \\pi _ p Z _ 0 + \\pi _ p \\omega _ 2 ( t ) + B \\int _ 0 ^ t S _ B ( t - r ) \\pi _ p \\omega _ 2 ( r ) d r \\end{align*}"}
+{"id": "4423.png", "formula": "\\begin{align*} \\lambda ^ + = s g n ( \\hat H ^ + _ 2 ) \\frac { \\hat a ^ + [ \\hat u _ 2 ] } { \\hat a ^ + | \\hat H ^ + _ 2 | + \\hat a ^ - | \\hat H ^ - _ 2 | } , \\\\ \\lambda ^ - = - s g n ( \\hat H ^ - _ 2 ) \\frac { \\hat a ^ - [ \\hat u _ 2 ] } { \\hat a ^ + | \\hat H ^ + _ 2 | + \\hat a ^ - | \\hat H ^ - _ 2 | } . \\end{align*}"}
+{"id": "480.png", "formula": "\\begin{align*} t ^ { 2 / \\alpha } \\sigma _ t ^ { ( \\alpha / 2 ) } ( t ^ { 2 / \\alpha } \\tau ) = \\sigma _ 1 ^ { ( \\alpha / 2 ) } ( \\tau ) . \\end{align*}"}
+{"id": "5120.png", "formula": "\\begin{align*} g ^ \\flat = B ^ \\flat \\circ r - r ^ * \\circ B ^ \\flat , \\Sigma _ X = B ^ \\flat \\circ D ^ { r } _ X - D ^ { r , * } _ X \\circ B ^ \\flat . \\end{align*}"}
+{"id": "7601.png", "formula": "\\begin{align*} J _ 2 ( \\beta , N , T , \\hat { \\mathbf { R } } _ i ) : = N ^ 2 \\iint _ { \\hat { \\mathbf { R } } _ i } \\int _ { \\mathbf { B } _ 2 ( 0 ) } f ^ { ( 1 , 1 , \\alpha ) } _ { t , s } ( z ) d z d s d t , i = 1 , 2 , \\end{align*}"}
+{"id": "2495.png", "formula": "\\begin{align*} N _ { 1 , t _ 1 } : = N _ \\sigma \\big ( K _ { X _ 1 } + ( B _ 1 + t _ 1 N _ 1 ) + ( M _ 1 + t _ 1 P _ 1 ) \\big ) = ( 1 + t _ 1 ) N _ 1 \\end{align*}"}
+{"id": "6318.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } = \\frac { \\partial } { \\partial { \\tau } } \\cdot \\frac { \\partial { \\tau } } { \\partial t } + \\frac { \\partial } { \\partial y } \\cdot \\frac { \\partial y } { \\partial t } . \\end{align*}"}
+{"id": "1344.png", "formula": "\\begin{align*} \\delta _ { ( X , \\Delta ) } ( L ) = \\frac { 2 } { \\mathrm { d e g } \\ , L } \\inf _ P A _ { ( X , \\Delta ) } ( P ) = 2 \\frac { ( 1 - \\max _ { P \\in X } \\mathrm { o r d } _ P ( \\Delta ) ) } { \\mathrm { d e g } \\ , L } . \\end{align*}"}
+{"id": "5701.png", "formula": "\\begin{align*} \\begin{aligned} \\ln \\| ( z - T ) ^ { - 1 } \\| & \\approx \\ln \\| ( z - A ) ^ { - 1 } e _ 1 \\| _ p \\| ( z - B ' ) ^ { - 1 } f _ 0 \\| _ q \\\\ & \\approx \\ln \\| ( z - A ) ^ { - 1 } \\| ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "5577.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { 8 } \\left ( C _ \\ell - 0 . 9 0 0 6 1 3 e ^ { 0 . 0 4 1 0 6 9 \\ell } \\right ) ^ 2 \\approx 1 . 7 8 \\times 1 0 ^ { - 6 } . \\end{align*}"}
+{"id": "8032.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } | m _ { \\sigma } ( x ) | ^ { 2 } \\frac { d \\sigma } { \\sigma } = 1 \\ , ; \\ ; x \\in \\R _ { + } ^ { n } . \\end{align*}"}
+{"id": "1789.png", "formula": "\\begin{gather*} g ( \\underline { m } ) = \\lim _ { n \\to \\infty } g _ n ( \\underline { m } ) \\end{gather*}"}
+{"id": "8501.png", "formula": "\\begin{align*} r _ { \\ell } ^ { \\wedge } ( \\bar { z } ) = | w | + \\delta . \\end{align*}"}
+{"id": "5946.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ g G _ M ^ \\omega ( x , y ) - _ g ( F ( y ) G _ M ^ \\omega ( x , y ) ) + \\omega ^ 2 G _ M ^ \\omega ( x , y ) = - \\delta _ x ( y ) , \\\\ \\left . \\partial _ \\nu G _ M ^ \\omega ( x , y ) - g _ y ( F ( y ) , \\nu ) G _ M ^ \\omega ( x , y ) \\right \\vert _ { y \\in \\partial M } = 0 , \\end{cases} \\end{align*}"}
+{"id": "2780.png", "formula": "\\begin{align*} \\ ! \\begin{multlined} [ t ] x ^ { k + 3 } \\cdot x ^ { k + 1 } \\cdot 1 1 x ^ 3 + x ^ { k + 3 } \\cdot ( 2 ^ k + k ) x ^ k \\cdot x ^ 4 + ( 2 ^ { k + 2 } + k + 2 ) x ^ { k + 2 } \\cdot x ^ { k + 1 } \\cdot x ^ 4 = \\\\ ( 5 \\cdot 2 ^ k + 2 k + 1 3 ) x ^ { 2 k + 7 } , \\end{multlined} \\end{align*}"}
+{"id": "7331.png", "formula": "\\begin{align*} J = \\begin{pmatrix} 0 & - I _ n \\\\ I _ n & 0 \\end{pmatrix} \\end{align*}"}
+{"id": "2311.png", "formula": "\\begin{align*} | J | = | 5 \\delta \\xi ^ { 4 } + 3 \\gamma \\xi ^ { 2 } + 2 \\xi _ 1 | . \\end{align*}"}
+{"id": "4312.png", "formula": "\\begin{align*} e ^ q = c o s ( | q | ) + s i n ( | q | ) \\frac { q } { | q | } \\end{align*}"}
+{"id": "5601.png", "formula": "\\begin{align*} f _ p ' ( t ) & = \\dfrac { p - 1 } { \\Gamma ( p ) } t ^ { p - 2 } + \\exp _ p ( t ) - \\mathrm { e } ^ t + \\dfrac { p - 1 } { \\Gamma ( p ) } \\left [ \\mathrm { e } ^ t \\int _ t ^ \\infty s ^ { p - 2 } \\mathrm { e } ^ { - s } \\mathrm d s - \\mathrm { e } ^ t t ^ { p - 2 } \\mathrm { e } ^ { - t } \\right ] \\\\ & = \\exp _ p ( t ) - \\mathrm { e } ^ t + \\mathrm { e } ^ t \\dfrac { p - 1 } { \\Gamma ( p ) } \\int _ t ^ \\infty s ^ { p - 2 } \\mathrm { e } ^ { - s } \\mathrm d s \\\\ & = f _ p ( t ) . \\end{align*}"}
+{"id": "8854.png", "formula": "\\begin{align*} \\norm { x _ { t + 1 } - x _ { p } } ^ { 2 } & \\leq \\norm { x _ { t } - \\eta _ { t } \\hat { g } _ { t } - x _ { p } } ^ 2 \\\\ & = \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 \\\\ & \\leq ( 1 - 2 \\eta _ { t } \\alpha ) \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } - \\nabla f ( x _ t ) , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 , \\end{align*}"}
+{"id": "2841.png", "formula": "\\begin{align*} | F _ { x y } \\cup F _ { y x } \\cup F _ { y y _ m } \\cup F _ { y y _ n } | & \\le \\Delta - 1 + 2 + 2 \\\\ & = \\Delta + 3 \\end{align*}"}
+{"id": "3.png", "formula": "\\begin{align*} \\alpha _ { n - 1 } = \\sum \\limits _ { i = n } ^ { n + t - 1 } \\frac { 1 } { 2 n - 1 - i } + z - \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "6967.png", "formula": "\\begin{align*} W _ \\epsilon ( x ) : = F _ \\epsilon - | F _ \\epsilon | ^ 2 \\geq & d \\alpha \\omega _ \\epsilon ^ { - 1 } - \\Big ( 2 \\alpha + 4 \\alpha ^ 2 \\Big ) \\omega \\omega _ \\epsilon ^ { - 2 } + \\Big ( 2 \\alpha + 4 \\alpha ^ 2 \\Big ) \\omega _ \\epsilon ^ { - 2 } \\sum _ j \\sin ^ 4 ( x _ j / 2 ) \\\\ & - 2 \\alpha \\omega \\omega _ \\epsilon ^ { - 1 } + T _ 2 ^ \\epsilon ( x ) + T _ 1 ^ \\epsilon ( x ) - \\frac { 1 } { 2 } T _ 3 ^ \\epsilon ( x ) - \\frac { 1 } { 2 } T _ 4 ^ \\epsilon ( x ) . \\end{align*}"}
+{"id": "1638.png", "formula": "\\begin{align*} f ( r ) = 2 ^ { p + q } ( \\sinh ( r / 2 ) ) ^ { p + q } ( \\cosh ( r / 2 ) ) ^ q = 2 ^ p ( \\sinh ( r / 2 ) ) ^ p ( \\sinh r ) ^ q . \\end{align*}"}
+{"id": "479.png", "formula": "\\begin{align*} \\lim _ { t \\searrow 0 } \\frac { \\sigma _ t ^ { ( \\frac \\alpha 2 ) } ( \\tau ) } { t } = \\frac { \\Gamma ( \\alpha / 2 + 1 ) \\ , \\sin ( \\pi \\alpha / 2 ) } { \\pi \\ , \\tau ^ { 1 + \\alpha / 2 } } , \\tau > 0 . \\end{align*}"}
+{"id": "1055.png", "formula": "\\begin{align*} \\gamma _ \\lambda \\Big | _ { [ u _ \\mu , u _ \\eta ] } = \\hat { \\gamma } _ \\lambda \\Big | _ { [ u _ \\mu , u _ \\eta ] } . \\end{align*}"}
+{"id": "693.png", "formula": "\\begin{align*} \\mathbf M ( \\mathbf f ) = \\begin{pmatrix} M _ 1 ( \\mathbf f ) \\\\ \\vdots \\\\ M _ n ( \\mathbf f ) \\end{pmatrix} \\mathbf N ( \\mathbf f ) = \\begin{pmatrix} N _ 1 ( \\mathbf f ) \\\\ \\vdots \\\\ N _ n ( \\mathbf f ) \\end{pmatrix} , \\end{align*}"}
+{"id": "7039.png", "formula": "\\begin{align*} \\tilde { \\nu } : = \\max _ { x \\in \\bar { \\Omega } } \\nu ( x ) . \\end{align*}"}
+{"id": "3110.png", "formula": "\\begin{align*} \\mathbf { b } ( \\phi , \\vartheta ) = [ e ^ { j b _ { n _ 1 \\cdot N _ 2 + n _ 2 } ( \\phi , \\vartheta ) } ] ^ T _ { n _ 1 \\in \\mathcal { I } ( N _ 1 ) , n _ 2 \\in \\mathcal { I } ( N _ 2 ) } \\end{align*}"}
+{"id": "6052.png", "formula": "\\begin{align*} { \\Phi } ( s , z , \\omega ) = \\bar { c } ( z ) [ \\vert { Y } ^ { M _ { { \\mathcal { \\ P } } } } _ { s - } \\vert _ { 1 , l } + \\vert { X } ^ { M _ { { \\mathcal { P } } } } _ { s - } \\vert _ { 1 , l } ^ { 2 l } + \\vert { Y } ^ { M _ { { \\mathcal { P } } } } _ { s - } \\vert _ { l - 1 } ^ { 2 l } ] , \\end{align*}"}
+{"id": "2566.png", "formula": "\\begin{align*} & \\phantom { = } \\sup _ n \\int _ { \\Omega } | W ( x ) f ' ( u _ n ) u _ n | d x \\\\ & \\leq \\sum _ { j = 1 , 2 } \\| W \\| _ { L ^ { \\frac { p _ s ^ * } { p _ s ^ * - q _ j } } ( \\Omega ) } \\sup _ n ( \\| u _ n \\| _ { p _ s ^ * } ^ { q _ j - 1 } \\| u _ n \\| _ { p _ s ^ * } ) \\\\ & \\leq C \\sum _ { j = 1 , 2 } \\| W \\| _ { L ^ { \\frac { p _ s ^ * } { p _ s ^ * - q _ j } } ( \\Omega ) } \\end{align*}"}
+{"id": "3090.png", "formula": "\\begin{align*} \\begin{aligned} \\mathop { \\max } \\limits _ { \\left \\{ { { n _ k } } \\right \\} _ { k = 1 } ^ { { N _ { \\rm { R } } } } } & \\sum \\limits _ { k = 1 } ^ { { N _ { \\rm { R } } } } { { { \\log } _ 2 } \\left ( { { { 1 + { { \\bar C } _ k } \\left ( { 1 - { 2 ^ { - 1 - ( { x _ k } + n _ k ) } } } \\right ) } } } \\right ) } \\\\ { \\rm { s . t . } } \\quad & { n _ k } \\in { { \\mathbb Z } ^ + } \\\\ & \\sum \\limits _ { k = 1 } ^ { { N _ { \\rm { R } } } } { { n _ k } } = 1 . \\end{aligned} \\end{align*}"}
+{"id": "2924.png", "formula": "\\begin{align*} T _ { i j } = \\alpha \\delta _ { i j } + \\epsilon _ { i j s } v _ s + D _ { i j } . \\end{align*}"}
+{"id": "4836.png", "formula": "\\begin{align*} B = \\sum _ { i = 1 } ^ { n } \\lambda _ i P _ i \\end{align*}"}
+{"id": "1760.png", "formula": "\\begin{gather*} \\sup _ { x \\in [ - 1 , 1 ] } \\left | \\sum _ { n = 1 } ^ { m _ k } a _ n x ^ n - f ( x ) \\right | \\to 0 \\end{gather*}"}
+{"id": "4698.png", "formula": "\\begin{align*} g ( x y ) = g ( x ) g ( y ) - f ( x ) f ( y ) , \\ , x , y \\in S \\end{align*}"}
+{"id": "5265.png", "formula": "\\begin{align*} \\| [ g _ { i , t } ( x _ { i , t } ) ] _ + \\| ^ 2 & = \\| [ g _ { i , t } ( x _ { i , t } ) ] _ + - [ g _ { i , t } ( x _ { j , t } ) ] _ + + [ g _ { i , t } ( x _ { j , t } ) ] _ + \\| ^ 2 \\\\ & \\ge \\frac { 1 } { 2 } \\| [ g _ { i , t } ( x _ { j , t } ) ] _ + \\| ^ 2 - \\| [ g _ { i , t } ( x _ { i , t } ) ] _ + - [ g _ { i , t } ( x _ { j , t } ) ] _ + \\| ^ 2 \\\\ & \\ge \\frac { 1 } { 2 } \\| [ g _ { i , t } ( x _ { j , t } ) ] _ + \\| ^ 2 - \\| g _ { i , t } ( x _ { i , t } ) - g _ { i , t } ( x _ { j , t } ) \\| ^ 2 \\\\ & \\ge \\frac { 1 } { 2 } \\| [ g _ { i , t } ( x _ { j , t } ) ] _ + \\| ^ 2 - G _ 2 ^ 2 \\| x _ { i , t } - x _ { j , t } \\| ^ 2 , \\end{align*}"}
+{"id": "5850.png", "formula": "\\begin{align*} s _ { \\alpha _ 2 } ( 3 \\alpha _ 1 + \\alpha _ 2 ) = 3 \\alpha _ 1 + \\alpha _ 2 - \\frac { 2 ( - \\frac { 3 } { 2 } ) } { 3 } \\alpha _ 2 = 3 \\alpha _ 1 + 2 \\alpha _ 2 . \\end{align*}"}
+{"id": "7885.png", "formula": "\\begin{align*} \\delta ^ 2 H _ { n + k } ( u ) [ \\phi , \\psi ] & = \\int _ \\Omega u ^ { i j } \\phi _ i \\psi _ j ( u ^ { \\star } ) ^ k \\det D ^ 2 u + k \\int _ \\Omega \\phi \\psi ( u ^ { \\star } ) ^ { k - 1 } \\det D ^ 2 u . \\end{align*}"}
+{"id": "7139.png", "formula": "\\begin{align*} - \\int _ 0 ^ \\infty \\frac { \\Psi ( Z ) } { Z } \\left ( \\frac { d } { d Z } \\frac { \\Psi ( Z ) } { Z } w _ 2 \\right ) ' \\ : d Z = \\int _ 0 ^ \\infty \\left ( \\frac { d } { d Z } \\frac { \\Psi ( Z ) } { Z } \\right ) ^ 2 w _ 2 ( Z ) \\ : d Z . \\end{align*}"}
+{"id": "3972.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) { \\sf u } = { \\sf f } ( { \\sf u } ) \\ , \\xi ^ { [ 1 , n ] } + { \\sf g } ( { \\sf u } ) , \\end{align*}"}
+{"id": "5409.png", "formula": "\\begin{align*} d \\Lambda _ j ^ { \\rm H } ( t ) = d B _ j ( t ) + \\frac { \\beta } { 2 } \\sum _ { 1 \\leq k \\leq N , k \\not = j } \\frac { d t } { \\Lambda _ j ^ { \\rm H } ( t ) - \\Lambda _ k ^ { \\rm H } ( t ) } , 1 \\leq j \\leq N , \\ , t \\geq 0 . \\end{align*}"}
+{"id": "3779.png", "formula": "\\begin{align*} x \\equiv ( \\phi _ 0 , \\phi _ 1 ) , A x \\equiv \\phi _ 0 \\oplus \\phi _ 1 , - f ( - x ) = \\sum _ i - \\mathcal { F } ^ * ( - \\phi _ i \\mid \\mu _ i ) . \\end{align*}"}
+{"id": "910.png", "formula": "\\begin{align*} \\frac { c } { x } \\bigg ( - \\tau _ t + \\frac { 1 - \\alpha } { t } \\tau \\bigg ) + m x ^ k \\phi _ 2 - t ^ { 1 - \\alpha } \\xi _ { 1 t } - 2 \\phi _ { 1 x } + \\xi _ { 1 x x } + \\frac { c } { x ^ 2 } \\xi _ 1 + \\frac { c } { x } \\xi _ { 1 x } - n x ^ k \\eta _ 2 = 0 , \\end{align*}"}
+{"id": "3009.png", "formula": "\\begin{align*} \\bar { g } = \\left [ \\bar { a } _ 1 , \\bar { b } _ 1 \\right ] \\cdots \\left [ \\bar { a } _ k , \\bar { b } _ k \\right ] , \\end{align*}"}
+{"id": "355.png", "formula": "\\begin{align*} ( h _ 1 \\circ _ R h _ 2 ) \\cdot h _ 1 ^ { - 1 } \\cdot ( h _ 1 \\circ _ R h _ 3 ) = & h _ 1 \\cdot \\phi _ { R ( h _ 1 ) } ( h _ 2 ) \\cdot ( h _ 1 ^ { - 1 } \\cdot h _ 1 ) \\cdot \\phi _ { R ( h _ 1 ) } ( h _ 3 ) \\\\ = & h _ 1 \\cdot ( \\phi _ { R ( h _ 1 ) } ( h _ 2 \\cdot h _ 3 ) ) \\\\ = & h _ 1 \\circ _ R ( h _ 2 \\cdot h _ 3 ) , \\end{align*}"}
+{"id": "2428.png", "formula": "\\begin{align*} \\theta _ \\Gamma ( c ) = \\sum _ { c _ \\gamma \\in \\Lambda _ { \\Gamma } } \\xi _ { \\Gamma , c _ \\gamma } ( c ) . \\end{align*}"}
+{"id": "8040.png", "formula": "\\begin{align*} g ( f ) ( x ) = \\left \\{ \\sum \\limits _ { i \\in \\mathbb N } \\vert \\phi _ { i } \\ast f ( x ) \\vert ^ { 2 } \\right \\} ^ { \\frac { 1 } { 2 } } \\end{align*}"}
+{"id": "696.png", "formula": "\\begin{align*} f ( t ) = \\norm { \\mathbf u } _ { \\widetilde { \\mathbf X } ^ { \\mathbf s , b } ( 0 , t ) } ^ 2 , f ( 0 ) = 0 . \\end{align*}"}
+{"id": "7488.png", "formula": "\\begin{align*} \\ell _ { \\lambda } : x _ 1 = \\lambda x _ 0 \\textnormal { o r } \\ell _ { \\infty } : x _ 0 = 0 \\end{align*}"}
+{"id": "4383.png", "formula": "\\begin{align*} | | u | | ^ 2 _ { m , \\ast , t } = \\int ^ t _ { - \\infty } | | | u ( s ) | | | ^ 2 _ { m , \\ast } d s \\ , . \\end{align*}"}
+{"id": "1461.png", "formula": "\\begin{align*} ( \\sigma _ 1 - \\sigma _ 2 ) ^ { 2 } + ( \\sigma _ 2 - \\sigma _ 3 ) ^ { 2 } + \\cdots + ( \\sigma _ n - \\sigma _ { n + 1 } ) ^ { 2 } = 2 \\left ( \\mu _ 1 \\sigma _ 1 + 2 \\sum \\limits _ { i \\in I \\setminus \\{ 1 , n + 1 \\} } \\mu _ i \\sigma _ i + \\mu _ { n + 1 } \\sigma _ { n + 1 } \\right ) . \\end{align*}"}
+{"id": "6522.png", "formula": "\\begin{align*} \\sup _ { 1 \\leq l \\leq \\beta } \\left | \\frac { \\partial ^ l k \\cdot \\tilde \\omega } { \\partial m ^ l } ( m ) \\right | & = \\sup _ { 1 \\leq l \\leq \\beta } | k \\cdot { \\bf M } _ l ( m ) | \\\\ & \\geq \\xi = \\tilde c | k | _ 2 c _ \\star ^ { \\beta ( \\beta - 1 ) } , \\end{align*}"}
+{"id": "2560.png", "formula": "\\begin{align*} & \\phantom { = } \\lim _ { n \\to \\infty } I _ { 1 , n } ' = \\int _ { \\mathbb { R } ^ N } \\varphi _ { x _ i , \\varepsilon } d \\mu = \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\frac { | u _ 0 ( x ) - u _ 0 ( y ) | ^ { p } \\varphi _ { x _ i , \\varepsilon } ( x ) } { | x - y | ^ { N + p s } } d x d y + \\mu _ i , \\end{align*}"}
+{"id": "344.png", "formula": "\\begin{align*} \\left ( \\mathsf { C } _ { j } \\left ( s \\right ) - \\frac { 1 } { 2 } \\operatorname * { I d } \\nolimits _ { j } \\right ) \\mbox { \\boldmath $ \\gamma $ } _ { \\operatorname * { C } ; j } ^ { \\operatorname * { e x t } , - } \\left ( s \\right ) u ^ { - } = 0 , \\end{align*}"}
+{"id": "7531.png", "formula": "\\begin{align*} \\Phi _ { g , n , \\beta } ( ( \\iota _ * \\alpha ) \\boxtimes \\gamma ) = \\sum _ { \\beta _ 1 + \\beta _ 2 = \\beta } \\left ( \\Phi _ { g _ 1 , n _ 1 + 1 , \\beta _ 1 } \\otimes \\Phi _ { g _ 2 , n _ 2 + 1 , \\beta _ 2 } \\right ) ( \\alpha \\boxtimes ( \\gamma \\boxtimes \\eta ) ) , \\end{align*}"}
+{"id": "2688.png", "formula": "\\begin{align*} \\mathcal { D } _ { S D _ { 1 6 } } ( F ( X , Y ) ) & = \\prod _ { j = 0 } ^ 7 f ( \\omega _ 8 ^ j ) f ( \\omega _ 8 ^ { 3 j } ) - g ( \\omega _ 8 ^ j ) g ( \\omega _ 8 ^ { 3 j } ) \\\\ & = M A _ 2 ^ 2 A _ 3 ^ 2 \\end{align*}"}
+{"id": "7279.png", "formula": "\\begin{align*} E = \\{ n \\in D _ k \\mid B \\cap ( B _ 0 \\cup \\ldots \\cup B _ k ) n \\} . \\end{align*}"}
+{"id": "7598.png", "formula": "\\begin{align*} \\begin{aligned} & J _ 2 ( \\beta , N , T ) \\\\ & : = C N ^ 2 \\int _ 0 ^ T \\int _ 0 ^ T \\int _ { \\mathbf { B } _ 2 ( 0 ) } f ^ { ( k , \\ell , \\alpha ) } _ { t , s } ( z ) d z d s d t \\\\ & \\leq C N ^ 2 \\int _ 0 ^ T \\int _ 0 ^ T \\left ( \\left ( \\frac { 1 } { ( t + s ) ^ { 1 / 2 } } \\int _ { | z _ 1 | < 2 } e ^ { - \\frac { ( z _ 1 - \\beta ^ { 1 / 3 } N ^ { 1 / 3 } ( t - s ) ) ^ 2 } { 2 ( t + s ) } } d z _ 1 \\right ) \\wedge 1 \\right ) d s d t . \\end{aligned} \\end{align*}"}
+{"id": "2670.png", "formula": "\\begin{align*} N ( H _ r ) = p ^ { s - ( r + 1 ) } \\Big ( 1 + ( - 1 ) ^ { \\frac { ( R _ f - 1 ) ( p - 1 ) } { 4 } } p ^ { - \\frac { R _ f - 1 } { 2 } } \\varepsilon _ { f } \\sum _ { ( u , - \\alpha ) \\in H _ { r } , y _ 1 \\in \\Im L _ f } \\bar { \\eta } ( f ( x _ { u } ) \\big ) \\Big ) . \\end{align*}"}
+{"id": "5959.png", "formula": "\\begin{align*} \\sigma _ 1 ( A ^ \\omega _ F ) & = - \\sqrt { \\widetilde { q _ 2 } } , \\\\ \\sigma _ 0 ( A ^ \\omega _ F ) & = \\frac { 1 } { 2 \\sqrt { \\widetilde { q _ 2 } } } ( \\nabla _ { \\xi ' } \\sqrt { \\widetilde { q _ 2 } } \\cdot D _ { t ' } \\sqrt { \\widetilde { q _ 2 } } - \\widetilde { q _ 1 } - \\partial _ { t _ 3 } \\sqrt { \\widetilde { q _ 2 } } + \\widetilde { E } \\sqrt { \\widetilde { q _ 2 } } ) . \\end{align*}"}
+{"id": "4480.png", "formula": "\\begin{align*} { \\mathbf V } _ { i + 1 } = { \\mathbf V } _ i + \\delta { \\mathbf V } _ i , \\ ; \\Psi _ { i + 1 } = \\Psi _ i + \\delta \\Psi _ i , \\ ; \\psi _ { i + 1 } = \\psi _ i + \\delta \\psi _ i , \\end{align*}"}
+{"id": "7742.png", "formula": "\\begin{align*} x ^ * ( x ) = \\sum _ { i = 1 } ^ n \\lambda _ i f _ { S _ i } ( x ) \\leq \\sum _ { i = 1 } ^ n \\vert \\lambda _ i \\vert \\vert f _ { S _ i } ( x ) \\vert & \\leq \\left ( \\sum _ { i = 1 } ^ n \\vert \\lambda _ i \\vert ^ q \\right ) ^ \\frac { 1 } { q } \\left ( \\sum _ { i = 1 } ^ n \\vert f _ { S _ i } ( x ) \\vert ^ p \\right ) ^ \\frac { 1 } { p } \\\\ & \\leq \\left ( \\sum _ { i = 1 } ^ n \\left \\vert \\sum _ { t \\in S _ i } x ( t ) \\right \\vert ^ p \\right ) ^ \\frac { 1 } { p } \\leq 1 \\end{align*}"}
+{"id": "5663.png", "formula": "\\begin{align*} r ( \\xi ) = a \\alpha + b _ 0 e _ 0 + b _ 1 e _ 1 \\end{align*}"}
+{"id": "2844.png", "formula": "\\begin{align*} R _ \\lambda R _ { \\omega _ r } = c _ { \\omega _ r } ( t ) \\sum _ { \\substack { J \\subseteq \\{ 1 , \\ldots , n \\} , \\ , | J | = r \\\\ \\lambda + \\bar { e } _ J \\in \\Lambda ^ { ( n ) } } } R _ { \\lambda + \\bar { e } _ J } \\prod _ { \\substack { 1 \\leq j < k \\leq n \\\\ j \\in J , \\ , k \\not \\in J \\\\ \\lambda _ j = \\lambda _ k } } \\frac { 1 - t ^ { k - j + 1 } } { 1 - t ^ { k - j } } . \\end{align*}"}
+{"id": "7618.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } w _ k = w ^ u , \\end{align*}"}
+{"id": "5501.png", "formula": "\\begin{align*} O _ A ^ { ( a ^ k g ) } = O _ A ^ { ( g ) } , \\qquad \\textrm { f o r a l l } k \\in \\mathbb Z , g \\in F r e e _ 2 . \\end{align*}"}
+{"id": "5724.png", "formula": "\\begin{align*} \\alpha _ 2 ^ { 0 , 0 } \\ , = \\ , \\frac { B _ { 3 , 2 } ^ { 0 , 0 } } { B _ { 2 , 2 } ^ { 0 , 0 } - B _ { 3 , 3 } ^ { 0 , 0 } } \\ , = \\ , \\frac { B _ { 3 , 2 } ^ { 0 , 0 } } { 1 2 \\pi \\ , k _ 0 ^ 3 } \\ , . \\end{align*}"}
+{"id": "3599.png", "formula": "\\begin{align*} { E _ { { \\rm { t h } } } } = \\frac { { { \\sigma ^ 2 } { L _ { \\rm { R } } } \\left ( { \\kappa + 1 } \\right ) \\pi \\left ( { K - 1 } \\right ) } } { { 2 { N _ { \\rm { T } } } \\kappa { C ^ 2 } } } . \\end{align*}"}
+{"id": "5100.png", "formula": "\\begin{align*} m : \\Gamma _ { \\underline { \\iota } } : = \\Gamma _ { \\iota _ 1 } \\times \\cdots \\times \\Gamma _ { \\iota _ { r } } \\to G \\end{align*}"}
+{"id": "1757.png", "formula": "\\begin{align*} k \\cdot \\varphi _ { k * } \\left ( \\Delta a \\right ) & = \\varphi _ { k * } \\circ r _ * \\left ( p _ { k * } [ S ^ 1 ] \\times a \\right ) \\\\ & = r _ { * } \\circ ( 1 \\times \\varphi _ k ) _ * \\left ( [ S ^ 1 ] \\times a \\right ) \\\\ & = r _ * ( [ S ^ 1 ] \\times k ^ i a ) \\\\ & = k ^ i \\cdot \\Delta ( a ) , \\end{align*}"}
+{"id": "6525.png", "formula": "\\begin{align*} \\tilde k = ( k , 1 , - 1 ) \\in \\Z ^ { b + 2 } \\setminus \\{ 0 \\} , \\ v ( m ) = ( \\omega ^ { 0 } , \\mu _ n , \\mu _ { n ' } ) \\in \\R ^ { b + 2 } . \\end{align*}"}
+{"id": "521.png", "formula": "\\begin{align*} \\norm { u } _ { X ^ { s , b } _ { h ( \\xi ) } ( S , T ) } = \\inf \\left \\{ \\norm { v } _ { X ^ { s , b } _ { h ( \\xi ) } } \\colon \\right \\} . \\end{align*}"}
+{"id": "2769.png", "formula": "\\begin{align*} I ( G ; x ) = \\sum _ { k = 0 } ^ { \\alpha ( G ) } { s _ k } x ^ { k } = { s _ 0 } + { s _ 1 } x + { s _ 2 } x ^ { 2 } + . . . + { s _ { \\alpha ( G ) } } x ^ { \\alpha ( G ) } , \\end{align*}"}
+{"id": "3181.png", "formula": "\\begin{align*} \\begin{bmatrix} A ^ { \\mathsf { T } } P + P A + C ^ { \\mathsf { T } } C & C ^ { \\mathsf { T } } D + P B \\\\ D ^ { \\mathsf { T } } C + B ^ { \\mathsf { T } } P & D ^ { \\mathsf { T } } D - 1 \\end{bmatrix} \\end{align*}"}
+{"id": "5689.png", "formula": "\\begin{align*} \\| ( z - T ) ^ { - 1 } \\| = \\big \\| \\sum \\limits _ { n = 0 } ^ { \\infty } \\frac { T ^ n } { z ^ { n + 1 } } \\big \\| & \\leq \\sum \\limits _ { n = 0 } ^ { \\infty } \\frac { \\| T ^ n \\| } { | z | ^ { n + 1 } } = \\sum \\limits _ { n = 0 } ^ { \\infty } \\frac { \\beta _ n } { | z | ^ { n + 1 } } \\\\ & = \\| ( z - T ) ^ { - 1 } e _ { 0 } \\| _ 1 \\leq \\| ( z - T ) ^ { - 1 } \\| . \\end{align*}"}
+{"id": "6144.png", "formula": "\\begin{gather*} \\forall s \\in S ^ { + } \\sqcup T ^ { - } , \\ , \\eta ( s ) = s , \\\\ \\forall s \\in S ^ { - } \\sqcup T ^ { + } , \\ , \\eta ( s ) = \\Gamma ( s ) . \\end{gather*}"}
+{"id": "6968.png", "formula": "\\begin{align*} 2 \\sum _ { k = 1 } ^ d \\sum _ { i = 1 } ^ { k - 1 } & \\sin ^ 2 ( x _ i / 2 ) + \\sin ^ 2 ( x _ k / 2 ) + \\sin ( x _ i / 2 ) \\sin ( x _ k / 2 ) \\\\ & = \\sum _ { k = 1 } ^ d \\sum _ { i = 1 } ^ d \\sin ^ 2 ( x _ i / 2 ) + \\sin ^ 2 ( x _ k / 2 ) + \\sin ( x _ i / 2 ) \\sin ( x _ k / 2 ) - 3 \\omega ( x ) \\\\ & = ( 2 d - 3 ) \\omega + \\Big ( \\sum _ { i = 1 } ^ d \\sin ( x _ i / 2 ) \\Big ) ^ 2 \\leq ( 3 d - 3 ) \\omega . \\end{align*}"}
+{"id": "3150.png", "formula": "\\begin{align*} [ \\mathbf { z } _ t ] = t \\times [ \\mathbf { z } _ 1 ] \\end{align*}"}
+{"id": "919.png", "formula": "\\begin{align*} \\phi _ 1 = \\frac 1 4 ( - ( 1 - \\alpha ) t ^ { - 1 - \\alpha } \\tau + ( 1 - \\alpha ) t ^ { - \\alpha } \\tau _ t - t ^ { 1 - \\alpha } \\tau _ { t t } ) x ^ 2 - t ^ { 1 - \\alpha } \\sigma _ { 1 t } x - \\eta _ 1 - \\frac { c } { x } \\sigma _ 1 + \\sigma _ 2 , \\end{align*}"}
+{"id": "1120.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\mathcal { L } _ { m , p } u ( x ) = \\lambda f ( x , u ( x ) ) x \\in \\mathop D \\limits ^ \\circ \\\\ \\smallskip \\smallskip \\ , \\ , | \\nabla ^ { j } u | = 0 \\ , \\ , \\ \\mbox { o n } \\partial D , \\mbox { w i t h } \\ , j = 0 , . . . , m - 1 , \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "8348.png", "formula": "\\begin{align*} \\rho _ s ( \\sigma _ { \\eta _ { k _ i } } , \\sigma _ { \\eta _ { k _ j } } ) \\le T , i , j = 0 , 1 , 2 . \\end{align*}"}
+{"id": "3535.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { N } } u ( \\cdot , t ) d x = \\int _ { \\mathbb { R } ^ { N } } u _ { 0 } d x ~ ~ \\int _ { \\mathbb { R } ^ { N } } v ( \\cdot , t ) d x = \\int _ { \\mathbb { R } ^ { N } } v _ { 0 } d x t \\in ( 0 , T _ { m a x } ) . \\end{align*}"}
+{"id": "5707.png", "formula": "\\begin{align*} \\limsup \\limits _ { z \\rightarrow 0 } \\big | \\big ( \\alpha _ 1 - \\big ( ( z - B ' ) ^ { - 1 } f _ 0 \\big ) ( x _ 1 ) \\big ) \\big | = + \\infty . \\end{align*}"}
+{"id": "704.png", "formula": "\\begin{align*} \\tau _ R ( \\omega ) = \\sup \\left \\{ t \\in [ 0 , \\infty ) \\colon \\right \\} . \\end{align*}"}
+{"id": "2828.png", "formula": "\\begin{align*} \\Omega ^ { ( p ) } \\big ( \\sigma ( \\underline { e } ^ { \\alpha t ^ a } \\wedge \\underline { e } ^ { \\beta t ^ b } \\wedge \\underline { e } ^ { \\gamma t ^ c } ) \\big ) - \\Omega ^ { ( p ) } ( \\underline { e } ^ { \\alpha t ^ a } \\wedge \\underline { e } ^ { \\beta t ^ b } \\wedge \\underline { e } ^ { \\gamma t ^ c } ) = d ( \\sum _ { i + j + k = q \\atop { 0 \\leq i , j , k } } \\frac { x ^ q } { q } S ( a , b , c ; i , j , k ) \\alpha ^ { ( i ) } \\beta ^ { ( j ) } \\gamma ^ { ( k ) } ) , \\end{align*}"}
+{"id": "990.png", "formula": "\\begin{align*} a & = \\frac { l - n ^ 2 / l } { 2 n } = \\frac { l ^ 2 - n ^ 2 } { 2 l n } , & a ' & = \\frac { l + n ^ 2 / l } { 2 n } = \\frac { l ^ 2 + n ^ 2 } { 2 l n } , \\\\ b & = \\frac { m - n ^ 2 / m } { 2 n } = \\frac { m ^ 2 - n ^ 2 } { 2 m n } , & b ' & = \\frac { m + n ^ 2 / m } { 2 n } = \\frac { m ^ 2 + n ^ 2 } { 2 m n } , \\end{align*}"}
+{"id": "6449.png", "formula": "\\begin{align*} \\widetilde { \\mu } ( [ g _ { u , v } ] ) : = \\mu ( [ X / u ] ) - \\mu ( [ X / v ] ) . \\end{align*}"}
+{"id": "3430.png", "formula": "\\begin{align*} W _ l = \\bigoplus _ { \\sum d _ i l _ i \\leq l } F _ 0 ^ { l _ 0 } F _ 1 ^ { l _ 1 } \\ldots F _ m ^ { l _ m } V _ { l - \\sum d _ i l _ i } \\to H ^ 0 ( M , l L ) \\end{align*}"}
+{"id": "6889.png", "formula": "\\begin{align*} \\frac { 1 } { \\alpha _ \\epsilon ^ 3 } \\cdot ( \\alpha - \\beta ) \\cdot \\alpha _ \\epsilon ^ { 2 } = \\frac { 2 \\epsilon \\cdot \\alpha ^ { 2 } \\omega + \\epsilon ^ 2 \\cdot ( \\alpha - \\beta ) \\omega ^ 2 } { 3 \\epsilon \\cdot \\alpha ^ { 2 } \\omega + 3 \\epsilon ^ 2 \\cdot \\alpha \\omega ^ 2 + \\epsilon ^ 3 \\cdot \\omega ^ 3 } \\sim _ { \\epsilon \\to 0 } \\frac { 2 \\epsilon \\cdot \\alpha ^ { 2 } \\omega } { 3 \\epsilon \\cdot \\alpha ^ { 2 } \\omega } = \\frac { 2 } { 3 } . \\end{align*}"}
+{"id": "6408.png", "formula": "\\begin{align*} [ t _ { 2 3 } , c _ { 1 2 } t _ { 2 3 } c ^ { - 1 } _ { 1 2 } ] = [ t _ { 1 2 } , t _ { 2 3 } ] = [ c _ { 2 3 } t _ { 1 2 } c ^ { - 1 } _ { 2 3 } , t _ { 1 2 } ] . \\end{align*}"}
+{"id": "7250.png", "formula": "\\begin{align*} \\omega \\in \\mathcal { A } ^ { \\mathbb N } \\mapsto x _ \\omega : = \\lim _ { m \\rightarrow \\infty } f _ { \\omega | _ m } ( x _ 0 ) , \\end{align*}"}
+{"id": "1504.png", "formula": "\\begin{align*} \\vert B _ { t } \\vert ^ { 2 } = 2 \\int _ { 0 } ^ { t } \\vert B _ { s } \\vert d X _ { s } + m t , \\end{align*}"}
+{"id": "198.png", "formula": "\\begin{align*} & \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) ( { \\rm c h } ( \\overline { W _ j } ) + { \\rm c h } ( \\overline { W _ i } ) + { \\rm c h } ( { W _ i } ) { \\rm c h } ( { W _ j } ) - 1 6 { \\rm c h } ( { W _ j } ) - 1 6 { \\rm c h } ( { W _ i } ) + 1 0 4 \\right . \\\\ & \\left . + B _ 1 \\wedge [ { \\rm c h } ( { W _ j } ) + { \\rm c h } ( { W _ i } ) - 1 6 ] + B _ 2 ) \\right \\} ^ { ( 1 4 ) } \\\\ & = - 1 9 6 6 3 2 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 4 ) } . \\end{align*}"}
+{"id": "8856.png", "formula": "\\begin{align*} \\min \\Big ( \\alpha , \\ , \\frac { d } { \\alpha } T ^ { - \\frac { \\beta - 1 } { \\beta } } \\Big ) = \\min \\Big ( \\max ( \\alpha , T ^ { - 1 / 2 + 1 / \\beta } ) , \\frac { d } { \\sqrt { T } } , \\ , \\frac { d } { \\alpha } T ^ { - \\frac { \\beta - 1 } { \\beta } } \\Big ) \\enspace . \\end{align*}"}
+{"id": "7736.png", "formula": "\\begin{align*} \\begin{array} { l l l l l l l l l l l } 0 = \\omega ( C ) - \\omega ( C ) & \\le | E _ { f = ( 0 , 0 ) } \\cap E ( C ) | + \\Omega | E ( C ) - E _ { f = ( 0 , 0 ) } | - \\omega ( C ) \\\\ & \\le \\frac { \\omega ( C ) } { 4 } + \\Omega ( \\frac { \\omega ( C ) } { 4 } + 1 - \\Omega ) - \\omega ( C ) \\\\ & \\le \\frac { 1 } { 6 4 } \\left ( \\omega ( C ) - 2 0 \\right ) ^ 2 - 6 < 0 , \\end{array} \\end{align*}"}
+{"id": "5241.png", "formula": "\\begin{align*} \\omega _ { t + 1 } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\nabla f _ { i , t } ( x _ { t } ) + ( \\nabla g _ { t } ( x _ { t } ) ) ^ \\top q _ { t + 1 } . \\end{align*}"}
+{"id": "4356.png", "formula": "\\begin{align*} \\omega ( A ) = { \\rm T r } ( \\varrho \\pi ^ P ( A ) ) \\tilde { \\omega } ( A ) = { \\rm T r } ( \\tilde { \\varrho } \\pi ^ P ( A ) ) \\ \\ ( A \\in { \\tt C A R } ( \\mathcal { K } , \\Gamma ) ) \\ , , \\end{align*}"}
+{"id": "6636.png", "formula": "\\begin{align*} \\rho _ { ( k ) , N } ^ { ( \\rm c J ) } ( \\theta _ 1 , \\dots , \\theta _ k ) = N ( N - 1 ) \\cdots ( N - k + 1 ) \\int _ { ( - \\pi , \\pi ] ^ { N - k } } d \\theta _ { k + 1 } \\cdots d \\theta _ N \\ , p _ { N , \\beta } ^ { ( \\rm c J ) } ( \\theta _ 1 , \\dots , \\theta _ N ) . \\end{align*}"}
+{"id": "6763.png", "formula": "\\begin{align*} \\mathbf { x } ^ { + } / \\dot { \\mathbf { x } } = f ( \\mathbf { x } ) . \\end{align*}"}
+{"id": "7855.png", "formula": "\\begin{align*} y ^ 2 = \\Pi _ { i = 1 } ^ { g + 1 } ( x - a _ i ) ( x - \\tfrac { 1 } { a _ i } ) = ( x ^ { g + 1 } - a ) ( x ^ { g + 1 } - \\tfrac { 1 } { a } ) \\end{align*}"}
+{"id": "4592.png", "formula": "\\begin{align*} \\ell _ M ( \\mathcal { Y } ^ { \\mathfrak { V } , M } ( a , z ) \\ , m ) = Y ^ M ( a , z ) m ~ ~ a \\ , { \\in } \\ , \\mathfrak { V } \\ , , \\ m \\ , { \\in } \\ , M \\ , . \\end{align*}"}
+{"id": "8295.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d t } V _ 1 ( u , u _ t ) \\leq - \\frac { \\mu } { 2 } V _ 1 ( u , u _ t ) - q ( a , \\mu ) \\abs { u _ t ( 1 , t ) } ^ 2 & + \\left ( F ( \\mu ) \\left ( \\tfrac { b d } { c } \\right ) ^ 2 - \\mu e ^ { - \\mu } \\right ) \\abs { u ( 1 , t ) } ^ 2 + F ( \\mu ) b ^ 2 \\abs { \\tilde p ( 0 , t ) } ^ 2 , \\end{aligned} \\end{align*}"}
+{"id": "5653.png", "formula": "\\begin{align*} c _ 1 \\left ( r ( \\widetilde { \\alpha } ) \\right ) = c _ 1 ( \\alpha ) . \\end{align*}"}
+{"id": "3560.png", "formula": "\\begin{align*} L ( x ^ { k + 1 } , \\lambda ^ { k + 1 } ) - L ( x , \\lambda ^ { k + 1 } ) & = f ( x ^ { k + 1 } ) - f ( x ) - ( \\lambda ^ { k + 1 } ) ^ T A ( x ^ { k + 1 } - x ) \\\\ & \\leq ( \\nabla f ( x ^ k ) - A ^ T \\lambda ^ { k + 1 } ) ^ T ( x ^ { k + 1 } - x ) + \\frac { L } { 2 } \\| x ^ { k } - x ^ { k + 1 } \\| _ 2 ^ 2 \\ . \\end{align*}"}
+{"id": "5205.png", "formula": "\\begin{align*} N ' _ 1 ( u ) = A ' \\cup B ' \\cup S , \\ , N ' _ 1 ( v ) = T , \\ , N ' _ 2 ( u ) = A ' \\cup B ' \\cup T , \\ , N ' _ 2 ( v ) = S . \\end{align*}"}
+{"id": "5872.png", "formula": "\\begin{align*} \\mu ^ { n } _ { t } = \\Delta \\mathbf { 1 } _ { \\{ t \\geq \\nu , n \\in \\mathcal { N } \\} } , \\mbox { f o r s o m e } \\Delta > 0 , \\end{align*}"}
+{"id": "7334.png", "formula": "\\begin{align*} D ^ 2 _ 0 f _ \\lambda ( u , v ) = b ( u , v ) + \\int ^ \\infty _ { - \\infty } { \\langle A ( \\lambda , t ) u ( t ) , v ( t ) \\rangle \\ , d t } \\end{align*}"}
+{"id": "6178.png", "formula": "\\begin{align*} m _ i ( \\mu , T , 0 ) = k _ i + \\delta _ \\nearrow ( \\mu _ i ) + c _ i ( T ) , & & m _ i ( \\mu , T , 1 ) = k _ { i + 1 } - \\delta _ \\nwarrow ( \\mu _ { i + 1 } ) + c _ i ( T ) . \\end{align*}"}
+{"id": "8439.png", "formula": "\\begin{align*} & \\rho ' _ R ( g ) ( h , s ) = ( g h , s ) , \\\\ & \\mu ' _ R ( h , s ) = \\mathrm { A d } ^ * _ h ( \\chi _ s ) , \\end{align*}"}
+{"id": "8384.png", "formula": "\\begin{align*} & ( t - s ) ^ { 2 \\beta } \\int _ { s } ^ { t } e ^ { - \\lambda ( t - r ) } ( r - s ) ^ { - \\alpha } ( t - r ) ^ { \\alpha - 1 } \\mathrm { d } r \\cr & = ( t - s ) ^ { 2 \\gamma } ( t - s ) ^ { 2 ( \\beta - \\gamma ) } \\int _ { 0 } ^ { 1 } e ^ { - \\lambda ( t - s ) ( 1 - v ) } v ^ { - \\alpha } ( 1 - v ) ^ { \\alpha - 1 } \\mathrm { d } v \\leq ( t - s ) ^ { 2 \\gamma } K ( \\lambda ) \\end{align*}"}
+{"id": "7548.png", "formula": "\\begin{align*} P _ { 0 , F } = P _ 0 \\times _ { \\mathbb { C } ^ * } R . \\end{align*}"}
+{"id": "5338.png", "formula": "\\begin{align*} 0 & \\leq \\ln ( 1 + e ^ x ) - x \\\\ & = \\ln ( 1 + e ^ { - x } ) \\\\ & \\leq e ^ { - x } \\end{align*}"}
+{"id": "1482.png", "formula": "\\begin{align*} \\rho : \\mathbb { N } \\rightarrow \\mathbb { N } , \\ ; \\rho ( n ) = 8 p + q , \\ ; \\ ; n = \\cdot 2 ^ { 4 p + q } , \\ ; 0 \\leqslant q \\leqslant 3 . \\end{align*}"}
+{"id": "4831.png", "formula": "\\begin{align*} & \\ ; \\ ; = S ^ { \\bar { i } \\bar { k } } _ { \\bar { j } \\bar { l } } ( u ) \\\\ S ^ { i k } _ { j l } ( u ) \\ ; \\ ; & \\ ; \\ ; = S ^ { j l } _ { i k } ( u ) \\\\ & \\ ; \\ ; = S ^ { k i } _ { l j } ( u ) \\end{align*}"}
+{"id": "5271.png", "formula": "\\begin{align*} & \\sum _ { t = 1 } ^ T \\alpha _ { t } = \\sum _ { t = 2 } ^ { T } \\frac { 1 } { t ^ c } + 1 \\le \\int _ { 1 } ^ { T } \\frac { 1 } { t ^ c } d t + 1 \\le \\frac { T ^ { 1 - c } } { 1 - c } . \\end{align*}"}
+{"id": "3487.png", "formula": "\\begin{align*} | \\frac { - \\log | z _ i | } { | \\log | t | | } - y _ i | \\lesssim \\epsilon , i = 0 , \\ldots m . \\end{align*}"}
+{"id": "503.png", "formula": "\\begin{align*} \\mathcal L _ { \\mathrm { Y u k a w a } } ( \\psi , \\phi ) = \\phi \\psi ^ * \\beta \\psi . \\end{align*}"}
+{"id": "5881.png", "formula": "\\begin{align*} P _ { s , V } ( L _ { V } \\geq J ) \\rightarrow 0 , J = \\exp ( 2 N ^ { \\zeta } / 3 ) . \\end{align*}"}
+{"id": "1351.png", "formula": "\\begin{align*} h ^ 0 ( \\mathcal { Y } _ s , g _ s ^ * ( m \\epsilon \\mathcal { H } _ s + m f _ s ^ * \\mathcal { O } ( 1 ) ) ) = \\frac { m ^ { d } \\epsilon ^ { d - 1 } } { ( d - 1 ) ! } ( ( \\mathcal { H } _ s ^ { d - 1 } \\cdot f _ s ^ * \\mathcal { O } ( 1 ) ) + O ( \\epsilon ) + O ( ( m \\epsilon ) ^ { - 1 } ) ) . \\end{align*}"}
+{"id": "8633.png", "formula": "\\begin{align*} \\mathcal { L } _ 1 ^ { \\mu } [ \\beta b ] = - b \\mathrm { F _ 3 } + O ( \\mu \\beta ^ 2 ) , \\mathcal { L } _ 2 ^ { \\mu } [ \\beta b ] = - \\frac { 1 } { 2 } b \\mathrm { F } _ 4 + \\frac { \\beta ^ 2 } { 6 } b ^ 3 \\mathrm { F } _ 3 + O ( \\mu \\beta ^ 4 ) , \\end{align*}"}
+{"id": "6609.png", "formula": "\\begin{align*} w ( x ) = x ^ { \\lambda _ 1 } ( 1 - x ) ^ { \\lambda _ 2 } \\chi _ { 0 < x < 1 } , \\end{align*}"}
+{"id": "3340.png", "formula": "\\begin{align*} u = \\biggl ( \\prod _ { i = 1 } ^ { N } \\theta _ i ^ { b _ i / d _ i } ( 1 - z _ i ) ^ { - \\frac { 1 } { m } } D _ { \\zeta _ i } ( \\zeta _ i \\theta _ i ) ^ { - \\frac { 1 } { m } } \\biggr ) \\sum _ { k \\in ( \\mathbb { Z } / m \\mathbb { Z } ) ^ N } \\zeta ^ { \\overline { Q ( k ) } } \\prod _ { i = 1 } ^ { N } \\frac { \\theta _ i ^ { ( k ^ \\mathsf { T } A ) _ i } } { ( \\zeta _ i \\theta _ i ; \\zeta _ i ) _ { k _ i } } \\end{align*}"}
+{"id": "2982.png", "formula": "\\begin{align*} r _ I ^ { - 1 } ( \\overline { V } ) & = \\psi ^ { - 1 } ( \\overline { V } ) \\times _ { s , \\alpha } \\alpha ^ { - 1 } ( s ( \\psi ^ { - 1 } ( \\overline { V } ) ) ) = ( \\psi ^ { - 1 } ( \\overline { V } ) \\times \\alpha ^ { - 1 } ( s ( \\psi ^ { - 1 } ( \\overline { V } ) ) ) ) \\cap E ^ 1 _ I \\end{align*}"}
+{"id": "7310.png", "formula": "\\begin{align*} S _ + ( T ) P _ + ( T ) T P _ + ( T ) S _ + ( T ) = I _ { H _ + } . \\end{align*}"}
+{"id": "687.png", "formula": "\\begin{align*} \\mathbf h ( D _ x ) \\mathbf f = \\begin{pmatrix} h _ 1 ( D _ x ) f _ 1 \\\\ \\vdots \\\\ h _ n ( D _ x ) f _ n \\end{pmatrix} , \\mathbf S ( t ) \\mathbf f = \\begin{pmatrix} S _ { h _ 1 ( \\xi ) } ( t ) f _ 1 \\\\ \\vdots \\\\ S _ { h _ n ( \\xi ) } ( t ) f _ n \\end{pmatrix} , \\end{align*}"}
+{"id": "1835.png", "formula": "\\begin{align*} \\left \\| \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + \\alpha ) ^ s } - \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + c ) ^ s } \\right \\| \\leq \\rho \\sqrt { K } \\sum _ { n = 0 } ^ { N } ( n + c / 2 ) ^ { - 3 / 2 } \\leq \\rho \\sqrt { K } \\zeta ( 3 / 2 , c / 2 ) , \\end{align*}"}
+{"id": "1534.png", "formula": "\\begin{align*} N _ p ( x ) = \\frac x { p \\log x } \\left ( 1 + O \\left ( \\frac { ( \\log _ 2 p ) ^ 2 } { \\log _ 2 x } \\right ) \\right ) \\sum _ { k \\le K \\log _ 2 x } \\frac { g ( p , \\mu ) } { \\Gamma ( 1 + \\mu ) } \\frac { ( \\log _ 2 x ) ^ { k - 1 } } { ( k - 1 ) ! } \\sum _ { \\substack { P ^ + ( A ) \\le p \\\\ \\Omega ( A ) = k } } \\frac 1 A \\end{align*}"}
+{"id": "6016.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } \\phi ( x ) d x = 1 , \\int _ { \\mathbb { R } ^ d } y ^ { \\beta _ { 1 } } \\phi ( y ) d y = 0 | \\beta _ { 1 } | \\geq 1 , \\int _ { \\mathbb { R } ^ d } | y | ^ { m } | \\partial _ { \\beta _ { 2 } } \\phi ( y ) | d y < \\infty m \\in \\mathbb { N } . \\end{align*}"}
+{"id": "7966.png", "formula": "\\begin{align*} \\sigma ( u _ { 0 , 0 } ) = v _ { 0 , 1 } , ~ \\sigma ( u _ { 1 , 1 } ) = v _ { r , 1 } ~ ~ \\sigma ( v _ { 0 , 1 } ) = u _ { 0 , 0 } . \\end{align*}"}
+{"id": "4409.png", "formula": "\\begin{align*} \\mathrm { d i v } \\dot { \\mathbf h ^ \\natural } ^ { \\pm } = 0 \\Omega _ T , \\end{align*}"}
+{"id": "3027.png", "formula": "\\begin{align*} \\phi ( H _ { 2 k + 1 } ) = \\phi ( T _ h ) f ( \\lambda ) \\end{align*}"}
+{"id": "7054.png", "formula": "\\begin{align*} G _ { \\ell } ( y ) : = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\sigma ) } ( \\pi ^ 3 y ) ^ { - s } \\ , \\prod _ { i = 1 } ^ { 3 } \\frac { \\Gamma \\left ( \\frac { 1 + s + { \\bf \\alpha } _ { i } + \\ell } { 2 } \\right ) } { \\Gamma \\left ( \\frac { - s - { \\bf \\alpha } _ { i } + \\ell } { 2 } \\right ) } \\tilde \\psi ( - s ) d s . \\end{align*}"}
+{"id": "1389.png", "formula": "\\begin{align*} \\exists A > 0 , \\ \\left \\Vert E ^ { l } u \\right \\Vert _ { 2 , \\left ( \\tilde { P } _ { 0 } \\right ) ^ { l } } \\leq A ^ { l } \\left \\Vert u \\right \\Vert _ { L ^ { 2 } } , \\forall u \\in C _ { 0 } ^ { \\infty } \\left ( \\Omega \\right ) , \\ l = 0 , 1 , . . . , \\end{align*}"}
+{"id": "2367.png", "formula": "\\begin{align*} S _ { f , \\tau , \\Lambda } = \\frac { o ( \\Lambda ) } { o ( G ) } \\sum _ { \\mu \\in \\Lambda ^ 0 } f ( \\pi ( \\mu ) ^ { - 1 } \\tau ) \\pi ( \\mu ) . \\end{align*}"}
+{"id": "4419.png", "formula": "\\begin{align*} ( \\mathcal B _ 1 { \\mathbf V } \\cdot { \\mathbf V } ) = 2 [ \\hat u _ 2 - \\hat \\lambda \\hat H _ 2 ] \\dot q ^ + \\partial _ 2 \\varphi + \\mbox { l . o . t } \\ , , \\quad \\mbox { o n } \\ , \\ , \\ , \\{ x _ 1 = 0 \\} \\ , , \\end{align*}"}
+{"id": "3836.png", "formula": "\\begin{align*} \\int _ X f ( x ) d \\pi ^ { x _ i } _ { \\# } ( s ^ p w _ i \\xi ) = \\int _ { X ^ 2 \\times \\R _ + ^ 3 } s ^ p w _ i f ( x _ i ) d \\xi . \\end{align*}"}
+{"id": "4984.png", "formula": "\\begin{align*} d _ Y ( a , y ) = \\min \\{ \\rho ( a , y ) , \\rho ' ( a , y ) \\} & \\leqslant \\min \\{ \\rho ( a , b ) + \\rho ( b , y ) , \\rho ' ( a , b ) + \\rho ' ( b , y ) \\} \\\\ & = d _ B ( a , b ) + \\min \\{ \\rho ( b , y ) , \\rho ' ( b , y ) \\} . \\end{align*}"}
+{"id": "1005.png", "formula": "\\begin{align*} \\langle g ( \\cdot , t ) , \\varphi \\rangle = \\int _ { \\Omega } g ( x , t ) \\ , \\varphi ( x ) \\ , d x = \\int _ { \\Omega _ - } g ( x , t ) \\ , \\varphi ( x ) \\ , d x + \\int _ { \\Omega _ + } g ( x , t ) \\ , \\varphi ( x ) \\ , d x , \\end{align*}"}
+{"id": "5.png", "formula": "\\begin{align*} e ^ { \\rm { 3 D } } ( \\rho ) = 4 \\pi \\rho ^ 2 a \\Big ( 1 + \\frac { 1 2 8 } { 1 5 \\sqrt { \\pi } } \\sqrt { \\rho a ^ 3 } \\Big ) + o \\big ( \\rho ^ 2 a \\sqrt { \\rho a ^ 3 } \\big ) . \\end{align*}"}
+{"id": "2497.png", "formula": "\\begin{align*} \\min _ { \\boldsymbol { y } , \\boldsymbol { u } } \\mathcal { J } ( \\boldsymbol { y } , \\boldsymbol { u } ) = \\min _ { \\boldsymbol { y } , \\boldsymbol { u } } \\frac { 1 } { 2 } \\| \\boldsymbol { y } - \\boldsymbol { y _ d } \\| _ { L ^ 2 ( Q ) } ^ 2 + \\frac { \\alpha } { 2 } \\| \\boldsymbol { u } \\| _ { L ^ 2 ( Q ) } ^ 2 \\end{align*}"}
+{"id": "7136.png", "formula": "\\begin{align*} - \\int _ 0 ^ \\infty \\Psi ^ { ( 5 ) } ( Z ) \\Psi ^ { ( 3 ) } ( Z ) w _ 1 ( Z ) \\ : d Z = \\int _ 0 ^ \\infty ( \\Psi ^ { ( 4 ) } ( Z ) ) ^ 2 w _ 1 ( Z ) + \\int _ 0 ^ \\infty \\Psi ^ { ( 4 ) } ( Z ) \\Psi ^ { ( 3 ) } ( Z ) w _ 1 ' ( Z ) \\ : d Z . \\end{align*}"}
+{"id": "1393.png", "formula": "\\begin{align*} \\tilde \\varphi _ { n , i } ( x ) = \\varphi _ { n , i } ( x ) + \\sum _ { k = 1 } ^ { \\infty } ( \\tilde Q _ { n , i ; k , 0 } ( x ) \\varphi _ { k , 0 } ( x ) - \\tilde Q _ { n , i ; k , 1 } ( x ) \\varphi _ { k , 1 } ( x ) ) , n \\ge 1 , \\ , i = 0 , 1 , \\end{align*}"}
+{"id": "1714.png", "formula": "\\begin{align*} G _ { \\texttt { b } } ( \\xi _ 1 , \\ldots , \\xi _ n ) : = \\sum _ { \\mu \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { b } } } \\delta ^ { ( m , n ) } _ { \\texttt { b } ; \\mu } ( q ) e ^ { i \\mu _ 1 \\xi _ 1 + \\cdots + i \\mu _ n \\xi _ n } . \\end{align*}"}
+{"id": "5927.png", "formula": "\\begin{align*} d [ - a _ { 1 , 3 } b _ { 1 } ] + d [ - a _ { 1 , 4 } b _ { 1 , 2 } ] = 1 + ( 2 e - 1 ) > 2 e + ( 2 - 2 e ) - R _ { 4 } = 2 e + S _ { 2 } - R _ { 4 } . \\end{align*}"}
+{"id": "4521.png", "formula": "\\begin{align*} D _ { k + \\frac { 1 } { 2 } } \\delta \\Psi _ k = \\frac { \\delta \\Psi _ k } { \\partial _ 1 ( \\Phi ^ a + \\Psi _ { k + \\frac { 1 } { 2 } } ) } R _ k , \\end{align*}"}
+{"id": "8764.png", "formula": "\\begin{align*} \\big ( \\nabla f _ { \\omega } ( u ) \\big ) _ { i } = 2 \\alpha ( 1 + \\delta ) u _ { i } + \\omega _ { i } r h ^ { \\beta - 1 } \\eta _ { 0 } ( u _ { i } h ^ { - 1 } ) . \\end{align*}"}
+{"id": "2608.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { m } & \\sum _ { k = 0 } ^ { m } \\left [ ( a + b ) \\tilde { \\alpha } _ { j + k + 1 } - \\tilde { \\alpha } _ { j + k + 2 } - a b \\tilde { \\alpha } _ { j + k } \\right ] \\xi _ { j } \\xi _ { k } \\\\ = & \\int _ a ^ b \\cdots \\int _ a ^ b \\left ( \\sum _ { j = 0 } ^ { m } ( x _ 1 \\cdots x _ d ) ^ { j } \\xi _ { j } \\right ) ^ 2 ( x _ 1 \\cdots x _ d ) f ( x _ 1 \\cdots x _ d ) p ( \\pmb x ) d \\mu \\cdots d \\mu , \\end{align*}"}
+{"id": "8759.png", "formula": "\\begin{align*} \\norm { \\mathbb { E } [ \\hat { g } _ { t } | x _ t ] - \\nabla f ( x _ { t } ) } = 0 , \\end{align*}"}
+{"id": "6054.png", "formula": "\\begin{align*} \\frac { 1 } { \\vert \\chi _ { t } ^ { M _ { { \\mathcal { P } } } } + \\vert a ^ { { { \\mathcal { P } } } } _ { t } \\vert ^ 2 \\vert ^ { 2 d p } } = \\frac { 1 } { \\Gamma ( 2 d p ) } \\int _ { 0 } ^ { \\infty } { s } ^ { 2 d p - 1 } e ^ { - { s } ( \\chi _ { t } ^ { M _ { { \\mathcal { P } } } } + \\vert a ^ { { { \\mathcal { P } } } } _ { t } \\vert ^ 2 ) } d { s } , \\end{align*}"}
+{"id": "5669.png", "formula": "\\begin{align*} r ( \\xi ) = \\ a \\alpha + b _ 0 e _ 0 + b _ 1 e _ 1 , \\end{align*}"}
+{"id": "4485.png", "formula": "\\begin{align*} E _ i : = \\sum ^ { i - 1 } _ { k = 0 } e _ k , \\tilde { E } _ i : = \\sum ^ { i - 1 } _ { k = 0 } \\tilde { e } _ k . \\end{align*}"}
+{"id": "784.png", "formula": "\\begin{align*} - \\Delta u - \\lambda u = \\alpha | u | ^ { 2 ^ * - 2 } u + \\beta ( K \\ast | u | ^ { 2 ^ \\sharp } ) | u | ^ { 2 ^ \\sharp - 2 } u \\Omega , \\end{align*}"}
+{"id": "2445.png", "formula": "\\begin{align*} \\rho ^ { \\leq \\delta } ( x ) = \\sum _ { n = 1 } ^ N \\left | \\langle \\nabla \\phi ^ x , \\nabla u _ n \\rangle _ { L ^ 2 ( \\Omega ) } \\right | ^ 2 \\leq \\| \\nabla \\phi ^ x \\| _ { L ^ 2 ( \\Omega ) } ^ 2 . \\end{align*}"}
+{"id": "3970.png", "formula": "\\begin{align*} \\tau _ x ( k [ A ] ) ( a ) = k [ A ( x \\oplus a ) ] \\simeq \\bigoplus _ { \\xi \\in A ( x ) } k [ A ( a ) ] \\end{align*}"}
+{"id": "2066.png", "formula": "\\begin{align*} e ^ { - 2 n ( n - 1 ) t } \\tilde { \\mathcal { R } } ( x , t ) & = \\varphi ( x , t ) \\leq u ( x , t ) + v \\\\ & \\leq t ^ { - 1 } + ( \\frac 1 4 r _ 0 ) ^ { - 2 } \\end{align*}"}
+{"id": "1172.png", "formula": "\\begin{align*} & \\bar { m _ { 1 , t } } = \\Phi ^ { - 1 } _ t \\circ m _ { 1 , t } \\circ ( \\Phi _ t \\otimes \\Phi _ t ) , \\\\ & \\bar { m _ { 2 , t } } = \\Phi ^ { - 1 } _ t \\circ m _ { 2 , t } \\circ ( \\Phi _ t \\otimes \\Phi _ t ) . \\end{align*}"}
+{"id": "7148.png", "formula": "\\begin{align*} \\mathcal { W } = \\frac { r } { 2 } , q = 0 , \\Phi = - \\frac { k } { r + \\Phi _ 1 ( r ) } , \\varphi ' = \\tau h ^ { - \\sigma } e ^ { - \\int _ 1 ^ r \\frac { k } { s + \\Phi _ 1 ( s ) } d s } . \\end{align*}"}
+{"id": "8855.png", "formula": "\\begin{align*} \\| { \\hat g } \\| ^ 2 & = \\frac { d ^ 2 } { 4 h ^ 2 } \\| ( f ( x + h r \\zeta ) - f ( x - h r \\zeta ) + \\xi - \\xi ' ) \\zeta K ( r ) \\| ^ 2 \\\\ & = \\frac { d ^ 2 } { 4 h ^ 2 } ( f ( x + h r \\zeta ) - f ( x - h r \\zeta ) + \\xi - \\xi ' ) ^ 2 K ^ 2 ( r ) . \\end{align*}"}
+{"id": "2855.png", "formula": "\\begin{align*} V _ { \\lambda , \\nu } ( t ) & : = \\prod _ { \\substack { \\beta \\in { \\hat R _ 0 ^ + } \\\\ \\langle \\lambda , \\beta \\rangle = 0 \\\\ \\langle \\nu , \\beta \\rangle > 0 } } \\frac { 1 - t _ { \\beta } \\hat e _ t ( \\beta ) } { 1 - \\hat e _ t ( \\beta ) } \\prod _ { \\substack { \\beta \\in { \\hat R _ 0 ^ + } \\\\ \\langle \\lambda , \\beta \\rangle = c \\\\ \\langle \\nu , \\beta \\rangle < 0 } } \\frac { 1 - t _ { \\beta } \\hat h _ t \\hat e _ t ( - \\beta ) } { 1 - \\hat h _ t \\hat e _ t ( - \\beta ) } \\end{align*}"}
+{"id": "67.png", "formula": "\\begin{align*} \\mathcal Q _ 3 ^ { \\rm { l o w } } - \\mathcal Q _ 3 ^ { \\rm { s o f t } } = \\frac { 1 } { \\vert \\Lambda \\vert } \\sum _ { \\substack { k \\in \\mathcal P _ H ^ c , k \\neq 0 \\\\ p \\in \\mathcal P _ L } } \\widehat g ( k ) \\big ( a _ 0 ^ \\dagger a _ p ^ \\dagger a _ { p - k } a _ k + h . c . \\big ) . \\end{align*}"}
+{"id": "3165.png", "formula": "\\begin{align*} W _ { \\mathbb { P } ^ 1 \\times \\mathbb { P } ^ 1 } = x + y + \\frac { 1 } { x } + \\frac { 1 } { y } . \\end{align*}"}
+{"id": "392.png", "formula": "\\begin{align*} ( \\mathcal { C } \\otimes \\delta ) ^ { \\pitchfork } = \\mathcal { F } . \\end{align*}"}
+{"id": "3389.png", "formula": "\\begin{align*} \\omega _ U = \\begin{pmatrix} \\alpha _ 0 ^ 0 - \\frac { 1 } { n + 1 } \\alpha ^ l _ l & \\alpha ^ 0 _ \\mu & \\dd r \\\\ \\alpha ^ \\mu _ 0 & \\alpha ^ \\mu _ \\nu - \\frac { 1 } { n + 1 } \\alpha ^ l _ l \\delta ^ \\mu _ \\nu & \\dd y ^ \\mu \\\\ - P _ { i 0 } \\dd x ^ i & - P _ { i \\nu } \\dd x ^ i & - \\frac { 1 } { n + 1 } \\alpha ^ l _ l \\end{pmatrix} , \\end{align*}"}
+{"id": "4172.png", "formula": "\\begin{align*} f ^ { ( \\nu ) } _ N ( n ) & = \\frac { \\Gamma \\left ( \\frac { \\nu + 1 } { 2 } \\right ) } { \\sqrt { \\nu \\pi } \\Gamma \\left ( \\frac { \\nu } { 2 } \\right ) } \\left ( 1 + \\tfrac { n ^ 2 } { \\nu } \\right ) ^ { - \\tfrac { \\nu + 1 } { 2 } } , n \\in \\mathbb { R } , \\end{align*}"}
+{"id": "8236.png", "formula": "\\begin{align*} \\omega _ { 1 , x _ 1 , 0 , 0 } = i \\partial \\bar \\partial _ { I _ 1 } f _ { x _ 1 , 0 , 0 } + x _ 1 \\Omega _ 0 , \\end{align*}"}
+{"id": "7370.png", "formula": "\\begin{align*} T ( g ) ( t , x ) = \\int _ \\mathbb { R } \\int _ { \\mathbb { R } ^ d } \\Omega ( t - s , x - y ) g ( s , y ) \\ , d s \\ , d y . \\end{align*}"}
+{"id": "8950.png", "formula": "\\begin{align*} \\{ ( \\bar { x } , \\bar { y } ) \\in \\R ^ { m + n } : \\upsilon _ 1 \\leq \\| \\bar { y } \\| \\leq \\upsilon _ 2 , | x _ i | \\leq \\vartheta \\| \\bar { y } \\| ^ { - w _ i } , i = 1 , \\ldots , n \\} , \\end{align*}"}
+{"id": "6926.png", "formula": "\\begin{align*} ( 1 - \\gamma _ { n + 3 } ) \\left ( \\displaystyle \\frac { \\gamma _ { n + 3 } } { b } - 1 \\right ) = \\displaystyle \\frac { m \\gamma _ { n + 2 } } { \\gamma _ { n + 1 } + a \\gamma _ { n + 2 } } < \\displaystyle \\frac { m \\gamma _ { n } } { \\gamma _ { n - 1 } + a \\gamma _ { n } } = ( 1 - \\gamma _ { n + 1 } ) \\left ( \\displaystyle \\frac { \\gamma _ { n + 1 } } { b } - 1 \\right ) , \\end{align*}"}
+{"id": "1605.png", "formula": "\\begin{align*} | \\mathcal { K } _ k ( \\hat { P } ^ { ( j ) } _ K ) | = ( 1 \\pm \\eta ) \\prod \\limits _ { \\ell = 1 } ^ j ( \\frac { 1 } { a _ { \\ell } } ) ^ { \\binom { k } { \\ell } } n ^ k . \\end{align*}"}
+{"id": "6412.png", "formula": "\\begin{align*} 0 = ( 2 + k ) b _ 1 ^ 2 - ( 2 + k ) D b _ 2 ^ 2 - 2 z _ 1 b _ 1 + 2 z _ 2 D b _ 2 - 2 \\end{align*}"}
+{"id": "8732.png", "formula": "\\begin{align*} \\eta _ t = \\min \\left ( \\frac { \\mathfrak { y } } { d } , \\ , \\Xi _ T \\right ) \\qquad h _ t = \\mathfrak { h } \\cdot T ^ { - \\frac { 1 } { 2 ( 2 \\beta - 1 ) } } \\enspace , \\end{align*}"}
+{"id": "213.png", "formula": "\\begin{align*} & Q _ 2 ( M , P _ i , \\tau ) = \\left \\{ e ^ { \\frac { 1 } { 2 4 } E _ 2 ( \\tau ) A _ 3 } \\left ( \\prod _ { j = 1 } ^ { 6 } \\frac { x _ j \\theta ' ( 0 , \\tau ) \\theta _ 2 ( x _ j , \\tau ) } { \\theta ( x _ j , \\tau ) \\theta _ 2 ( 0 , \\tau ) } \\right ) \\right . \\\\ & \\left . \\frac { 1 } { 2 } \\left ( \\prod _ { l = 1 } ^ 8 \\theta _ 1 ( y _ l ^ i , \\tau ) + \\prod _ { l = 1 } ^ 8 \\theta _ 2 ( y _ l ^ i , \\tau ) + \\prod _ { l = 1 } ^ 8 \\theta _ 3 ( y _ l ^ i , \\tau ) \\right ) \\right \\} ^ { ( 1 2 ) } . \\end{align*}"}
+{"id": "8671.png", "formula": "\\begin{align*} f = \\phi _ 0 ( u _ 1 , . . . , u _ n ) , ~ ~ ~ x _ i = \\phi _ i ( u _ 1 , . . . , u _ n ) , ~ i = 1 , . . . , n , \\end{align*}"}
+{"id": "1954.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\bar { u } \\partial _ x u _ 1 + u _ 2 \\bar { u } _ x \\| _ { L ^ 2 ( [ 0 , T ] \\times I ) } & \\leq \\| \\bar { u } \\| _ { L ^ \\infty ( [ 0 , T ] \\times I ) } \\| \\partial _ x u _ 1 \\| _ { L ^ 2 ( [ 0 , T ] \\times I ) } \\\\ & + \\| u _ 2 \\| _ { L ^ 2 ( [ 0 , T ] ; L ^ \\infty ( I ) ) } \\| \\bar { u } \\| _ { L ^ \\infty ( [ 0 , T ] ; H ^ 1 ( I ) ) } \\\\ & \\leq T ^ \\frac { 1 } { 2 } \\left ( \\| u _ 1 \\| _ { X } + \\| u _ 2 \\| _ X \\right ) \\| \\bar { u } \\| _ { X } . \\end{aligned} \\end{align*}"}
+{"id": "2399.png", "formula": "\\begin{align*} \\L _ { d } \\left ( \\sum _ { l = 1 } ^ { \\infty } A _ { \\i } ^ { l } E \\right ) \\ll | D e t ( A _ { \\i } ) | \\L _ { d } ( E ) \\end{align*}"}
+{"id": "2287.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } \\theta _ \\alpha \\left ( x - \\frac { j } { n } + \\varepsilon _ j \\right ) = \\sum _ { k \\in \\mathbb { Z } } e ^ { - \\pi \\alpha k ^ 2 } \\left ( \\sum _ { j = 1 } ^ { n } e ^ { 2 \\pi i k \\varepsilon _ j } e ^ { - 2 \\pi i k \\frac { j } { n } } \\right ) e ^ { 2 \\pi i k x } . \\end{align*}"}
+{"id": "4898.png", "formula": "\\begin{align*} \\left [ I - \\frac { z } { x _ 1 } P _ 1 \\right ] ^ { - 1 } A ( z ) = ( z - x _ 1 - z P _ 1 ) \\left [ \\frac { A ( z ) - A ( x _ 1 ) } { z - x _ 1 } \\right ] + A ( x _ 1 ) \\end{align*}"}
+{"id": "7856.png", "formula": "\\begin{align*} \\theta = ( b , b c , a b , b c a ^ { - g } ) . \\end{align*}"}
+{"id": "4797.png", "formula": "\\begin{align*} | f | ^ \\perp _ \\alpha = \\sup _ { x \\in S _ \\gamma } \\sup _ { x \\neq y \\in C ^ \\perp _ 0 ( x ) } \\frac { \\left | f ( x ) - f ( y ) \\right | } { d ( x , y ) ^ \\alpha } , \\end{align*}"}
+{"id": "6464.png", "formula": "\\begin{align*} & P _ D F = 0 , \\\\ & P _ N F ' = 0 , \\\\ & P _ R F ' = \\epsilon \\Lambda P _ R F . \\end{align*}"}
+{"id": "7151.png", "formula": "\\begin{align*} w ' = 2 r , 0 < r < a . \\end{align*}"}
+{"id": "2904.png", "formula": "\\begin{align*} \\Phi _ { \\xi } ( \\lambda ) = M _ \\lambda ( \\xi ) = M ^ { ( c ) } _ \\lambda ( \\xi ) \\quad \\ \\xi \\in \\mathcal { P } _ c \\ \\ \\lambda \\in P _ c . \\end{align*}"}
+{"id": "3296.png", "formula": "\\begin{align*} f _ X ( x ) = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\mathbb { 1 } _ { \\left [ \\alpha _ { _ \\Sigma } , \\beta _ { _ \\Sigma } \\right ] } . \\end{align*}"}
+{"id": "6615.png", "formula": "\\begin{align*} \\rho _ { ( 1 ) , \\infty } ^ { ( \\rm c J ) } ( x ; \\beta , p , q ) : = \\lim _ { N \\to \\infty } { 2 \\pi \\over N } \\rho _ { ( 1 ) , N } ^ { ( \\rm c J ) } ( - \\pi \\ , { \\rm s g n } ( x ) + 2 \\pi x / N ; \\beta , p , q ) , \\end{align*}"}
+{"id": "7468.png", "formula": "\\begin{align*} - x _ 0 + x _ 1 + z _ 0 & = 0 , & - x _ 2 + x _ 3 + z _ 1 & = 0 , \\\\ x _ 1 - x _ 2 + x _ 3 & = 0 , & - x _ 0 + x _ 1 + x _ 3 & = 0 , \\end{align*}"}
+{"id": "5137.png", "formula": "\\begin{align*} \\mbox { r a n k } ( { \\bf M } ) & = N , & & { \\bf M } { \\bf J } _ { 2 N } { \\bf M } ^ T = { \\bf 0 } _ { N \\times N } \\end{align*}"}
+{"id": "561.png", "formula": "\\begin{align*} \\mathbb E \\left ( \\norm { \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf M ( \\mathbf u ( s ) ) \\ , d W ( s ) } _ { \\mathbf H ^ { \\mathbf s } } ^ 2 \\right ) = \\mathbb E \\left ( \\int _ 0 ^ t \\norm { \\mathbf M ( \\mathbf u ( s ) ) } _ { \\mathcal L _ 2 ( K , \\mathbf H ^ { \\mathbf s } ) } ^ 2 \\ , d s \\right ) . \\end{align*}"}
+{"id": "4015.png", "formula": "\\begin{align*} \\left ( \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi \\right ) ^ n = g \\left ( \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi \\right ) ^ m \\wedge \\omega ^ { n - m } , \\end{align*}"}
+{"id": "1496.png", "formula": "\\begin{align*} X _ { t } = \\sum _ { k = 1 } ^ { m } \\int _ { 0 } ^ { t } \\frac { B _ { k } ( s ) } { \\vert B _ { s } \\vert } d B _ { s } , \\end{align*}"}
+{"id": "6183.png", "formula": "\\begin{align*} \\forall x \\geq X _ + , \\ , \\Gamma _ { \\mathbf { k } , x } = \\Gamma _ { \\mathbf { k } , X _ + } , \\\\ \\forall x \\leq X _ - , \\ , \\Gamma _ { \\mathbf { k } , x } = \\Gamma _ { \\mathbf { k } , X _ - } , \\end{align*}"}
+{"id": "756.png", "formula": "\\begin{align*} \\mu ( t ) & = t ^ { - 2 b } \\int _ { \\R ^ d } \\int _ 0 ^ t V ( s , \\zeta ) \\overline { W ( s , \\zeta ) } \\ , d s \\ , d \\zeta , \\\\ \\nu ( t ) & = 2 \\int _ { \\R ^ d } \\int _ 0 ^ t \\int _ 0 ^ s \\frac { \\left [ V ( s , \\zeta ) - V ( \\sigma , \\zeta ) \\right ] \\overline { \\left [ W ( s , \\zeta ) - W ( \\sigma , \\zeta ) \\right ] } } { ( s - \\sigma ) ^ { 1 + 2 b } } \\ , d \\sigma \\ , d s \\ , d \\zeta . \\end{align*}"}
+{"id": "1613.png", "formula": "\\begin{align*} | \\mathcal K _ k ( \\hat P ^ { ( k - 1 ) } ) | = ( 1 \\pm \\eta ) \\prod \\limits _ { \\ell = 1 } ^ { k - 1 } ( \\frac { 1 } { a _ { \\ell } } ) ^ { \\binom { k } { \\ell } } n ^ k , \\end{align*}"}
+{"id": "8547.png", "formula": "\\begin{align*} E ^ { k } : = E _ { N _ { k } } ^ { k } = \\left \\{ ( z , w ) \\in J \\times \\mathbb { R } ^ { n - 1 } : | w - \\lambda ( r _ { \\ell ^ { k } } ( z ) - r _ { \\ell ^ { k } } ( a ) ) e | < r _ { \\ell } ( z ) \\right \\} , \\end{align*}"}
+{"id": "902.png", "formula": "\\begin{align*} \\eta = \\eta _ 1 ( x , t ) u + \\eta _ 2 ( x , t ) v + \\eta _ 3 ( x , t ) , ~ ~ \\phi = \\phi _ 1 ( x , t ) v + \\phi _ 2 ( x , t ) u + \\phi _ 3 ( x , t ) . \\\\ \\end{align*}"}
+{"id": "7349.png", "formula": "\\begin{align*} \\langle y ^ a _ k y ^ b _ \\ell \\rangle = \\langle y ^ a _ k \\rangle \\langle y ^ b _ \\ell \\rangle \\ \\forall \\ a < b \\ \\mbox { a n d } \\ k \\neq \\ell . \\end{align*}"}
+{"id": "4247.png", "formula": "\\begin{align*} \\norm { \\hat { g } } _ 1 & = \\langle \\hat { g } _ S , \\hat h _ { S } \\rangle + \\| \\hat { g } _ { { S ^ { c } } } \\| _ { 1 } \\\\ & \\geq \\langle \\hat { g } _ S , h _ { S } \\rangle + \\norm { \\hat { g } _ { S ^ c } } _ 1 + \\langle g _ S , h _ S \\rangle - \\langle g _ S , h _ { S } \\rangle \\\\ & = \\norm { g } _ 1 - \\langle \\eta _ S , h _ S \\rangle + \\norm { \\hat { g } _ { S ^ c } } _ 1 . \\end{align*}"}
+{"id": "2557.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } f _ n d x & = \\limsup _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } f _ n \\eta _ R d x + \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } ( 1 - \\eta _ R ) ( f _ n d x ) . \\end{align*}"}
+{"id": "1123.png", "formula": "\\begin{align*} \\begin{aligned} & \\qquad \\ , \\ , \\mbox { t h e r e a r e } \\ , \\ , \\beta > p \\ , \\ , \\mbox { a n d } \\ , \\ , r _ 0 > 0 \\ , \\ , \\mbox { s u c h t h a t } \\\\ & t f ( x , t ) \\geq \\beta F ( x , t ) > 0 \\ , \\ , \\mbox { f o r a n y } \\ , \\ , | t | \\geq r _ 0 \\ , \\ , \\mbox { a n d e v e r y } \\ , \\ , x \\in D , \\end{aligned} \\end{align*}"}
+{"id": "2387.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\pi _ 1 } E _ t \\left ( d J ^ { p r e } \\right ) = \\left ( \\mu - r \\right ) d t - \\pi _ 1 \\sigma ^ 2 d t - h d t = 0 \\end{align*}"}
+{"id": "5717.png", "formula": "\\begin{align*} Q _ { \\xi , \\delta } \\ , : = \\ , \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ - \\delta \\ , \\alpha _ 1 ^ { \\xi , \\delta } & - \\delta \\ , \\alpha _ 2 ^ { \\xi , \\delta } & 0 \\end{pmatrix} \\ , . \\end{align*}"}
+{"id": "2318.png", "formula": "\\begin{align*} \\| h \\| ^ 2 = \\sum _ { g \\in G } | \\langle h , \\tau _ g \\rangle | ^ 2 , \\forall h \\in \\mathcal { H } . \\end{align*}"}
+{"id": "7070.png", "formula": "\\begin{align*} \\mathcal { V } ^ { \\pm } _ 1 \\left ( \\frac { n _ { 1 } ^ 2 n _ { 2 } } { ( p _ 1 q ) ^ 3 r } \\right ) = \\frac { 1 } { 2 \\pi ^ { 3 / 2 } } \\bigg \\{ G _ { 0 } \\left ( \\frac { n _ { 1 } ^ 2 n _ { 2 } } { ( p _ 1 q ) ^ 3 r } \\right ) \\mp i G _ { 1 } \\left ( \\frac { n _ { 1 } ^ 2 n _ { 2 } } { ( p _ 1 q ) ^ 3 r } \\right ) \\bigg \\} , \\end{align*}"}
+{"id": "316.png", "formula": "\\begin{align*} \\mathsf { L } _ { j } ^ { \\sigma } \\left ( s \\right ) u ^ { \\sigma } = 0 \\quad \\Omega _ { j } ^ { \\sigma } \\end{align*}"}
+{"id": "2452.png", "formula": "\\begin{align*} \\rho ^ { \\leq \\delta } ( x ) = \\sum _ { n = 1 } ^ N \\left | \\langle \\phi ^ x , ( - \\Delta _ { \\Omega } ) ^ { \\frac { s } { 2 } } u _ n \\rangle _ { L ^ 2 ( \\Omega ) } \\right | ^ 2 \\leq \\| \\phi ^ x \\| _ { L ^ 2 ( \\Omega ) } ^ 2 . \\end{align*}"}
+{"id": "2031.png", "formula": "\\begin{align*} \\nu : = \\frac { 1 } { { \\sqrt { \\lvert \\nabla ' \\phi \\rvert ^ 2 + 1 } } } ( \\nabla ' \\phi , - 1 ) \\end{align*}"}
+{"id": "6519.png", "formula": "\\begin{align*} \\sup _ { 1 \\leq l \\leq \\beta } | { \\bf M } _ l ( m ) | _ 1 & \\leq \\sup _ { 1 \\leq l \\leq \\beta } \\sum _ { s = 1 } ^ \\beta | \\lambda _ l v _ s ^ { - l } | \\leq \\sup _ { 1 \\leq l \\leq \\beta } \\sum _ { s = 1 } ^ \\beta | \\lambda _ l | : = K _ 1 ( \\beta ) . \\end{align*}"}
+{"id": "1479.png", "formula": "\\begin{align*} & \\theta ^ { \\ell } _ { g } ( d g _ { t } ) = \\left ( d B _ { t } , 0 \\right ) , \\\\ & g _ { 0 } = e , \\end{align*}"}
+{"id": "6990.png", "formula": "\\begin{align*} h ^ \\ast _ 2 ( T ) = \\sum _ { P _ i \\in \\mathcal { P } \\setminus \\mathcal { I } } \\left ( | E ( P _ i ) | - 1 \\right ) + \\sum _ { P _ i \\in \\mathcal { I } } \\left ( | E ( P _ i ) | - 2 \\right ) . \\end{align*}"}
+{"id": "1138.png", "formula": "\\begin{align*} & [ ( h _ 1 , \\ldots , h _ { m + 1 } ) , ( k _ 1 , \\ldots , k _ { n + 1 } ) ] _ c : = \\\\ & = \\big ( [ h _ 1 , k _ 1 ] , ~ [ h _ 1 , k _ 2 ] + [ h _ 2 , k _ 1 ] , ~ \\ldots , \\underbrace { [ h _ 1 , k _ i ] + [ h _ 2 , k _ { i - 1 } ] + \\cdots + [ h _ i , k _ 1 ] } _ { i } , ~ \\ldots , [ h _ { m + 1 } , k _ { n + 1 } ] \\big ) , \\end{align*}"}
+{"id": "6905.png", "formula": "\\begin{align*} ( \\phi , \\psi ) ( - \\infty ) = ( 1 , 0 ) \\ \\ ( \\phi , \\psi ) ( + \\infty ) = ( u _ 2 ^ * , u _ 2 ^ * ) , \\end{align*}"}
+{"id": "5636.png", "formula": "\\begin{align*} x _ 1 y ^ 2 + B y + x _ 0 C = Q \\left ( x _ 1 ( y + x _ 0 L ) + B \\right ) + F x _ 0 . \\end{align*}"}
+{"id": "725.png", "formula": "\\begin{align*} \\Delta _ 1 ( t ) = i \\int _ T ^ { t } \\mathbf S ( t - s ) \\left [ \\mathbf N ( \\mathbb 1 _ { s \\le \\mu } \\mathbf U ( s ) ) - \\mathbf N ( \\mathbb 1 _ { s \\le \\mu } \\mathbf V ( s ) ) \\right ] \\ , d s . \\end{align*}"}
+{"id": "1080.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - \\Delta _ { \\mu } u ( x ) = \\alpha ( x ) u ( x ) + \\lambda f ( x , u ( x ) ) x \\in \\mathop D \\limits ^ \\circ \\\\ \\medskip \\ , \\ , u | _ { \\partial D } = 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "4049.png", "formula": "\\begin{align*} \\begin{gathered} \\sum _ { j = 0 } ^ { [ ( k - 1 ) / 2 ] } U _ { 0 , 0 } ( 2 j a + t ) + \\sum _ { j = 1 } ^ { [ k / 2 ] } U _ { 0 , 0 } ( 2 j a - t ) = 0 \\ ; ( 0 , a ) , \\\\ \\sum _ { j = 0 } ^ { [ ( k - 1 ) / 2 ] } V _ { 0 , 0 } ( 2 j a + t ) = \\sum _ { j = 1 } ^ { [ k / 2 ] } V _ { 0 , 0 } ( 2 j a - t ) \\ ; ( 0 , a ) ; \\end{gathered} \\end{align*}"}
+{"id": "3321.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n x _ { k , j } \\mu _ { i , j , l } = \\eta _ { i , k , l } , ( i , k , l ) \\in [ n - k ] \\times [ r ] \\times [ d ] . \\end{align*}"}
+{"id": "5507.png", "formula": "\\begin{align*} \\widetilde c _ { \\pm } ^ { ( h , g ) } : = \\delta _ { ( p , s ) \\mapsto ( s , p ^ { - 1 } ) } \\left ( c _ { \\pm } ^ { ( \\sigma ( h ) , \\sigma ( g ) ) } \\right ) \\qquad \\in \\quad \\mathbf R , h , g \\in F r e e _ 2 . \\end{align*}"}
+{"id": "6262.png", "formula": "\\begin{align*} c _ \\lambda ( \\xi _ 1 , \\xi _ 2 , \\xi _ 3 ) = p _ \\lambda ( \\xi _ 1 - \\xi _ 2 + \\xi _ 3 ) c _ { d i a g } ( \\xi _ 1 , \\xi _ 2 , \\xi _ 3 ) c ( \\xi _ 1 , \\xi _ 2 , \\xi _ 3 ) . \\end{align*}"}
+{"id": "5702.png", "formula": "\\begin{align*} ( z - T ) ^ { - 1 } x = \\begin{bmatrix} ( z - B ) ^ { - 1 } x _ 1 \\\\ - \\big ( ( z - B ' ) ^ { - 1 } f _ 0 \\big ) ( x _ 1 ) ( z - A ) ^ { - 1 } e _ 1 + ( z - A ) ^ { - 1 } x _ 2 \\end{bmatrix} . \\end{align*}"}
+{"id": "1165.png", "formula": "\\begin{align*} & \\sum _ { i \\geq 0 } \\phi _ i \\bigg ( \\sum _ { j \\geq 0 } m \\rq _ { 1 , j } ( x , y ) t ^ j \\bigg ) t ^ i = \\sum _ { i \\geq 0 } m _ { 1 , i } \\bigg ( \\sum _ { j \\geq 0 } \\phi _ j ( x ) t ^ j , \\sum _ { k \\geq 0 } \\phi _ k ( y ) t ^ k \\bigg ) t ^ i , \\\\ & \\sum _ { i \\geq 0 } \\phi _ i \\bigg ( \\sum _ { j \\geq 0 } m \\rq _ { 2 , j } ( x , y ) t ^ j \\bigg ) t ^ i = \\sum _ { i \\geq 0 } m _ { 2 , i } \\bigg ( \\sum _ { j \\geq 0 } \\phi _ j ( x ) t ^ j , \\sum _ { k \\geq 0 } \\phi _ k ( y ) t ^ k \\bigg ) t ^ i . \\\\ \\end{align*}"}
+{"id": "8918.png", "formula": "\\begin{align*} \\tau ( \\varphi ) ^ { \\alpha } = \\sum \\limits _ { i , j = 1 } ^ m g ^ { i j } \\left ( \\frac { \\partial ^ 2 \\varphi ^ { \\alpha } } { \\partial x _ i \\partial x _ j } - \\sum \\limits _ { k = 1 } ^ m { ^ M \\Gamma } _ { i j } ^ k \\frac { \\partial \\varphi ^ { \\alpha } } { \\partial x _ k } + \\sum \\limits _ { \\beta , \\gamma = 1 } ^ n { ^ N L } _ { \\beta \\gamma } ^ \\alpha \\frac { \\partial \\varphi ^ \\beta } { \\partial x _ i } \\frac { \\partial \\varphi ^ \\gamma } { \\partial x _ j } \\right ) . \\end{align*}"}
+{"id": "6156.png", "formula": "\\begin{align*} \\eta _ { , 1 } \\left ( \\begin{array} { c c c c c } & & 4 & & \\\\ & 2 & & 5 & \\\\ 2 & & 5 & & 7 \\end{array} \\right ) = \\{ 4 \\} , & & \\eta _ { , 3 } \\left ( \\begin{array} { c c c c c } & & 4 & & \\\\ & 2 & & 5 & \\\\ 2 & & 5 & & 7 \\end{array} \\right ) = \\{ 2 , 5 , 7 \\} , \\end{align*}"}
+{"id": "908.png", "formula": "\\begin{align*} t ^ { 1 - \\alpha } \\phi _ { 3 t } - \\phi _ { 3 x x } - \\frac { c } { x } \\phi _ { 3 x } - n x ^ k \\eta _ { 3 x } = 0 , \\end{align*}"}
+{"id": "2920.png", "formula": "\\begin{align*} 0 \\ne H ^ 0 ( T , \\mathcal E ( - t ) ) & \\subseteq H ^ 0 \\big ( T , ( f _ * \\mathcal O _ X ( m L ) ) ( - t ) \\big ) \\\\ & = H ^ 0 ( T , f _ * \\mathcal O _ X ( m L - G ) ) = H ^ 0 ( X , \\mathcal O _ X ( m L - G ) ) , \\end{align*}"}
+{"id": "6943.png", "formula": "\\begin{align*} P ^ T = \\Lambda ^ { - 1 } K \\Lambda . \\end{align*}"}
+{"id": "2294.png", "formula": "\\begin{align*} 9 \\binom { k } { 2 } + 3 k d + \\binom { d } { 2 } = n _ { 2 } + 3 t _ { 5 } + 3 n _ { 3 } . \\end{align*}"}
+{"id": "6535.png", "formula": "\\begin{align*} A = \\sup _ { m \\in [ 2 , 3 ] } \\sup _ { 1 \\leq l \\leq b + 1 } \\left | \\frac { d ^ l f } { d m ^ l } ( m ) \\right | \\leq C ( b ) | k | , \\end{align*}"}
+{"id": "6821.png", "formula": "\\begin{align*} \\mathcal { H } ( u , v ) = & - ( u + u ^ * + e _ 2 ) ( u - u ^ * ) ^ 2 + [ \\varrho s - a ( u + u ^ * + e _ 2 ) ] ( u - u ^ * ) ( v - v ^ * ) - \\varrho s ( v - v ^ * ) ^ 2 \\\\ [ 0 . 2 c m ] = & - \\left [ \\sqrt { ( u + u ^ * + e _ 2 ) } ( u - u ^ * ) - \\frac { \\varrho s - a ( u + u ^ * + e _ 2 ) } { 2 \\sqrt { ( u + u ^ * + e _ 2 ) } } ( v - v ^ * ) \\right ] ^ 2 \\\\ [ 0 . 2 c m ] & \\quad - \\left ( \\varrho s - \\frac { [ \\varrho s - a ( u + u ^ * + e _ 2 ) ] ^ 2 } { 4 ( u + u ^ * + e _ 2 ) } \\right ) ( v - v ^ * ) ^ 2 . \\end{align*}"}
+{"id": "1287.png", "formula": "\\begin{align*} \\Delta W + ( I _ \\alpha \\ast | \\cdot | ^ { - b } | W | ^ p ) | x | ^ { - b } | W | ^ { p - 2 } W = 0 . \\end{align*}"}
+{"id": "3916.png", "formula": "\\begin{align*} F ( K , v ) : = H _ { K } ( v ) \\cap K . \\end{align*}"}
+{"id": "3813.png", "formula": "\\begin{align*} \\eta = ( x _ 0 , \\rho _ 0 ( x _ 0 ) , x _ 1 , \\rho _ 1 ( x _ 1 ) , \\rho ( x _ 0 , x _ 1 ) ) _ { \\# } \\gamma \\end{align*}"}
+{"id": "1884.png", "formula": "\\begin{align*} I _ i = [ x _ { i - \\frac { 1 } { 2 } } , x _ { i + \\frac { 1 } { 2 } } ] \\forall \\ , \\ , 1 \\leq i \\leq N _ x ; J _ j = [ v _ { j - \\frac { 1 } { 2 } } , v _ { j + \\frac { 1 } { 2 } } ] \\ , \\quad \\forall \\ , \\ , 1 \\leq j \\leq N _ v . \\end{align*}"}
+{"id": "47.png", "formula": "\\begin{align*} a ^ { \\dagger } _ 0 a _ 0 = \\frac { 1 } { \\pi } \\int _ { \\mathbb { C } } ( | z | ^ 2 - 1 ) \\vert z \\rangle \\langle z \\vert \\dd z , \\end{align*}"}
+{"id": "3534.png", "formula": "\\begin{align*} \\mathcal { F } ( u , v ) ( t ) = \\frac { 1 } { 2 } \\int _ { \\Omega } \\vert \\nabla w \\vert ^ { 2 } d x + \\frac { 1 } { 2 } \\int _ { \\Omega } w ^ { 2 } d x - \\int _ { \\Omega } z w d x + \\int _ { \\Omega } z \\log z d x . \\end{align*}"}
+{"id": "8662.png", "formula": "\\begin{align*} x _ i \\to x _ i , x _ { n + i } \\to x _ { n + i } + \\sum _ { j = 1 } ^ n b _ { i j } x _ j + c _ i , i = 1 , \\dots , n , b _ { i j } = b _ { j i } \\in \\mathbf k , c _ i \\in \\mathbf k \\ , . \\end{align*}"}
+{"id": "5822.png", "formula": "\\begin{align*} w _ 1 = s _ { \\alpha _ { m a x } } . \\end{align*}"}
+{"id": "369.png", "formula": "\\begin{align*} r _ { \\mu } \\cdot t _ { \\mu } = r _ { \\mu } \\cdot ( t _ 1 ^ { \\mu } , \\ldots , t _ { q _ { \\mu } } ^ { \\mu } ) = ( r _ { \\mu } ^ { d ^ { \\mu } _ 1 } t _ 1 ^ { \\mu } , \\ldots , r _ { \\mu } ^ { d ^ { \\mu } _ { q _ { \\mu } } } t _ { q _ { \\mu } } ^ { \\mu } ) . \\end{align*}"}
+{"id": "4760.png", "formula": "\\begin{align*} P ( E , \\overline { E _ 1 } ) = 2 \\ne 1 = P ( E _ 1 , \\partial E ) . \\end{align*}"}
+{"id": "1565.png", "formula": "\\begin{align*} P _ c S _ r \\delta _ { x _ 0 , v _ 0 } & = c \\delta _ { x _ 0 + r v _ 0 } 1 _ { | v | \\leq 1 } , \\\\ S _ { s - r } P _ c S _ r \\delta _ { ( x _ 0 , v _ 0 ) } & = c \\delta _ { x _ 0 + r v _ 0 } ( x - ( s - r ) v ) 1 _ { | v | \\leq 1 } , \\\\ P _ c S _ { s - r } P _ c S _ r \\delta _ { ( x _ 0 , v _ 0 ) } & = \\frac { c ^ 2 } { ( s - r ) ^ d } 1 _ { | v | \\leq 1 } 1 _ { | x _ 0 + r v _ 0 - x | \\leq ( s - r ) 1 } . \\end{align*}"}
+{"id": "6133.png", "formula": "\\begin{align*} \\xi \\circ ( \\psi \\circ \\varphi ) = ( \\xi \\circ \\psi ) \\circ \\varphi . \\end{align*}"}
+{"id": "8250.png", "formula": "\\begin{align*} g _ 0 ( \\frac { \\partial \\Psi } { \\partial x _ 2 } , d \\Psi ( Y ) ) = \\frac { 1 } { 2 x _ 1 } ( g _ { 0 , 0 , 0 } - g _ { x _ 1 , 0 , 0 } ) ( \\xi _ 3 ( m _ 0 ) , Y ) . \\end{align*}"}
+{"id": "4214.png", "formula": "\\begin{align*} N _ { F } ( X , Y ) = [ F X , F Y ] - F [ X , F Y ] - F [ F X , Y ] - [ X , Y ] . \\end{align*}"}
+{"id": "6848.png", "formula": "\\begin{align*} T _ { \\rm c y l } = \\frac { 2 \\pi } { \\beta _ { \\rm c y l } } \\end{align*}"}
+{"id": "7962.png", "formula": "\\begin{align*} \\gcd ( r , n ) \\in \\{ 1 , 2 \\} , ~ ~ \\gcd ( n , r ) = 2 , ~ ~ m ~ . \\end{align*}"}
+{"id": "4510.png", "formula": "\\begin{align*} | | u _ 2 ^ a \\partial _ 2 ( S _ { \\theta _ i } - I ) & \\psi _ i | | _ { H ^ { s - 1 } ( \\Gamma _ T ) } \\lesssim | | u ^ a _ 2 | | _ { L ^ \\infty ( \\Gamma _ T ) } | | ( I - S _ { \\theta _ i } ) \\psi _ i | | _ { H ^ s ( \\Gamma _ T ) } \\\\ & + | | u ^ a | | _ { s , \\ast , T } | | ( I - S _ { \\theta _ i } ) \\psi _ i | | _ { H ^ 2 ( \\Gamma _ T ) } \\leq C \\delta \\theta _ i ^ { s - \\alpha } \\ , , \\quad \\mbox { f o r } \\ , \\ , \\ , s \\in \\{ 6 , \\dots , \\tilde \\alpha \\} \\ , . \\end{align*}"}
+{"id": "166.png", "formula": "\\begin{align*} \\theta _ 3 ( v , \\tau ) = \\prod _ { j = 1 } ^ { \\infty } [ ( 1 - q ^ j ) ( 1 + e ^ { 2 \\pi \\sqrt { - 1 } v } q ^ { j - \\frac { 1 } { 2 } } ) ( 1 + e ^ { - 2 \\pi \\sqrt { - 1 } v } q ^ { j - \\frac { 1 } { 2 } } ) ] , \\end{align*}"}
+{"id": "4335.png", "formula": "\\begin{align*} \\omega _ { { \\sf f } _ 1 \\cdots { \\sf f } _ n } = \\omega \\circ \\alpha _ U \\omega _ { { \\sf g } _ 1 \\cdots { \\sf g } _ m } = \\omega \\circ \\alpha _ V \\ , . \\end{align*}"}
+{"id": "284.png", "formula": "\\begin{align*} \\left \\Vert v \\right \\Vert _ { H ^ { 1 } \\left ( \\omega \\right ) ; s } : = \\left ( \\left \\Vert \\nabla v \\right \\Vert _ { \\mathbf { L } ^ { 2 } \\left ( \\omega \\right ) } ^ { 2 } + \\left \\vert s \\right \\vert ^ { 2 } \\left \\Vert v \\right \\Vert _ { L ^ { 2 } \\left ( \\omega \\right ) } ^ { 2 } \\right ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "4348.png", "formula": "\\begin{align*} { \\bf A } = \\left ( \\begin{array} { c c } 0 & \\check { { \\bf I } } \\\\ - \\check { { \\bf I } } & 0 \\end{array} \\right ) \\end{align*}"}
+{"id": "6985.png", "formula": "\\begin{align*} \\log ^ + | f ( r e ^ { \\iota \\theta } | = O ( r ^ \\frac { m + 2 } { 2 } ) \\end{align*}"}
+{"id": "465.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ \\infty \\frac { ( 2 \\beta ) ^ { - 2 k } \\Gamma ( \\zeta + ( 1 + \\alpha ) / 2 + 2 k ) } { k ! \\Gamma ( \\zeta + \\frac 1 2 + k ) } \\\\ & = \\Gamma ( \\zeta + ( 1 + \\alpha ) / 2 ) \\ , _ 2 \\tilde { F } _ 1 \\left ( \\frac { \\zeta + ( 1 + \\alpha ) / 2 } { 2 } , \\frac { \\zeta + ( 3 + \\alpha ) / 2 } { 2 } ; \\zeta + \\frac 1 2 ; \\beta ^ { - 2 } \\right ) . \\end{align*}"}
+{"id": "4483.png", "formula": "\\begin{align*} D _ { i + \\frac { 1 } { 2 } } : = \\frac { 1 } { \\partial _ 1 ( \\Phi ^ a + \\Psi _ { i + \\frac { 1 } { 2 } } ) } \\partial _ 1 \\mathbb { L } ( { \\mathbf U } ^ a + { \\mathbf V } _ { i + \\frac { 1 } { 2 } } , \\Psi ^ a + \\Psi _ { i + \\frac { 1 } { 2 } } ) , \\end{align*}"}
+{"id": "1755.png", "formula": "\\begin{align*} \\sum _ { \\substack { L \\subset \\{ 0 , \\ldots , m + n - 1 \\} \\\\ | L | = n } } \\det F _ L \\det G _ L = \\det \\left ( F G ^ T \\right ) , \\end{align*}"}
+{"id": "7839.png", "formula": "\\begin{align*} W _ i = \\mathrm { d } \\zeta _ { \\widetilde { \\gamma } _ i } + \\tau _ { i j } \\mathrm { d } \\zeta _ { \\gamma ^ j } , W _ i ^ { } = - \\sum _ { \\gamma } \\Omega ( \\gamma ) q _ i ( \\gamma ) \\left ( A _ { \\gamma } ^ { } - \\mathrm { i } V _ { \\gamma } ^ { } \\mathrm { d } \\zeta _ { \\gamma } \\right ) \\ , . \\end{align*}"}
+{"id": "7969.png", "formula": "\\begin{align*} B ( f ) ( F ( d ) ( x ) ) = F ( c ) ( A ( f ) ( x ) ) . \\end{align*}"}
+{"id": "8269.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow 0 + } \\mu _ \\lambda ^ { - 1 } ( 0 ) / \\mathbb { S } ^ 1 = ( \\mu ^ { - 1 } ( 0 ) \\times \\mathbb { R } ^ 3 ) / \\mathbb { S } ^ 1 = M _ 0 \\times \\mathbb { R } ^ 3 . \\end{align*}"}
+{"id": "5813.png", "formula": "\\begin{align*} w _ 0 ( \\varepsilon _ i ) = s _ { \\varepsilon ' _ i } ( \\varepsilon ' _ i ) = - \\varepsilon ' _ i . \\end{align*}"}
+{"id": "5743.png", "formula": "\\begin{align*} \\begin{cases} \\operatorname { s u p p } ( \\nabla \\phi ) \\cap \\{ x _ { n + 1 } > 0 \\} \\subset \\mathbb B _ 3 ^ + \\setminus \\mathbb B _ { 2 } ^ + \\\\ | \\operatorname { d i v } ( x _ { n + 1 } ^ a \\nabla \\phi ) | \\leq C x _ { n + 1 } ^ a \\ \\mathbf 1 _ { \\mathbb B _ 3 ^ + \\setminus \\mathbb B _ { 2 } ^ + } . \\end{cases} \\end{align*}"}
+{"id": "6048.png", "formula": "\\begin{align*} \\frac { 1 } { \\ln u } \\overline { \\mu } \\{ \\underline { c } > \\frac { 1 } { u } \\} \\geq \\frac { 1 } { \\ln u } \\int _ 1 ^ { ( \\underline { \\sigma } ( u - 1 ) ) ^ { \\frac { 1 } { 4 } } } \\frac { 1 } { \\vert z \\vert ^ { 1 - \\alpha } } d z = \\frac { ( \\underline { \\sigma } ( u - 1 ) ) ^ { \\frac { \\alpha } { 4 } } - 1 } { \\alpha \\ln u } , \\end{align*}"}
+{"id": "672.png", "formula": "\\begin{align*} \\norm { \\overline u } _ { X _ { h ( \\xi ) } ^ { s , b } ( S , T ) } = \\norm { u } _ { X _ { - h ( - \\xi ) } ^ { s , b } ( S , T ) } . \\end{align*}"}
+{"id": "4173.png", "formula": "\\begin{align*} \\mathbb { E } [ e ^ { i \\omega N } ] = \\frac { K _ { \\nu / 2 } ( \\sqrt { \\nu } | \\omega | ) \\cdot ( \\sqrt { \\nu } | \\omega | ) ^ { \\tfrac { \\nu } { 2 } } } { \\Gamma \\left ( \\frac { \\nu } { 2 } \\right ) 2 ^ { \\frac { \\nu - 2 } { 2 } } } , \\omega \\in \\mathbb { R } . \\end{align*}"}
+{"id": "2159.png", "formula": "\\begin{align*} \\begin{aligned} \\cosh { \\Lambda } + \\cos 2 \\phi \\sinh { \\Lambda } = & ~ { } \\frac { 2 \\alpha ^ d \\left ( m ^ { d + 1 } + m ^ { - d - 1 } \\right ) } { m ^ { d } + m ^ { - d } } \\\\ \\cosh { \\Lambda } - \\cos 2 \\phi \\sinh { \\Lambda } = & ~ { } \\frac { 2 \\alpha ^ { - d } \\left ( m ^ { d - 1 } + m ^ { - d + 1 } \\right ) } { m ^ { d } + m ^ { - d } } \\\\ \\sin 2 \\phi \\sinh { \\Lambda } = & ~ { } \\frac { 2 \\left ( m ^ { - 1 } - m \\right ) } { m ^ { d } + m ^ { - d } } . \\end{aligned} \\end{align*}"}
+{"id": "4595.png", "formula": "\\begin{align*} \\chi _ M ^ { } ( \\tau ) = \\mathrm { t r } _ M ^ { } q ^ { L _ 0 - c / 2 4 } q = \\exp ( 2 \\pi i \\tau ) \\ , , \\ \\tau \\ , { \\in } \\ , \\mathbb { H } \\ , , \\end{align*}"}
+{"id": "8230.png", "formula": "\\begin{align*} \\frac { d } { d t } ( e ^ { i t } \\cdot m ) & = T ( e ^ { i t } \\cdot m ) , \\\\ \\frac { d } { d t } ( e ^ \\tau \\cdot m ) & = - I _ 1 T ( e ^ \\tau \\cdot m ) . \\end{align*}"}
+{"id": "8258.png", "formula": "\\begin{align*} f _ { x _ 1 , 0 , 0 } ( \\rho , \\Theta ) = \\rho ^ 2 f _ { \\rho ^ { - 2 } x _ 1 , 0 , 0 } ( 1 , \\Theta ) = \\sum _ { \\nu \\geq 0 } k _ \\nu ( \\Theta ) \\frac { x _ 1 ^ \\nu } { \\rho ^ { 2 \\nu - 2 } } . \\end{align*}"}
+{"id": "4983.png", "formula": "\\begin{align*} r _ 0 & : = \\max \\{ d _ B ( a , b ) \\colon a , b \\in B \\} + 1 , \\\\ \\varepsilon & : = \\frac 1 2 \\min \\{ | d _ B ( a , 0 ) - d _ B ( b , 1 ) | > 0 \\colon a , b \\in B \\} , \\\\ r _ 1 & : = r _ 0 - \\varepsilon . \\end{align*}"}
+{"id": "19.png", "formula": "\\begin{align*} \\omega ( x ) : = \\omega _ { \\R ^ { d } } ( x ^ { * } ) , g ( x ) : = g ( x ^ { * } ) , x \\in \\Lambda . \\end{align*}"}
+{"id": "9.png", "formula": "\\begin{align*} \\mathrm { E } _ { 2 } ( v , \\widetilde { R } ) = \\frac { 2 \\pi } { \\log ( \\frac { \\widetilde R } { a } ) } , \\mathrm { E } _ { 3 } ( v , \\widetilde { R } ) = \\frac { 4 \\pi a } { 1 - a / \\widetilde R } . \\end{align*}"}
+{"id": "4317.png", "formula": "\\begin{align*} \\begin{alignedat} { 3 } f _ { R } ( t ) = \\frac { 1 } { \\sqrt { M } } \\sum _ { x = 0 } ^ { M - 1 } f ( x ) \\ ; e ^ { \\mu 2 \\pi \\left ( \\frac { x t } { M } \\right ) } \\end{alignedat} \\end{align*}"}
+{"id": "7743.png", "formula": "\\begin{align*} f ( z ) = \\alpha f ( x ) + \\beta f ( ( 1 + \\delta ) x _ 0 ) - \\gamma f ( ( 1 + \\delta ) x _ 0 ) & \\leq \\alpha + ( \\beta - \\gamma ) ( 1 + \\delta ) \\\\ & \\leq \\alpha + \\beta ( 1 + \\delta ) \\leq 1 + \\delta \\\\ & = f ( ( 1 + \\delta ) x _ 0 ) . \\end{align*}"}
+{"id": "3369.png", "formula": "\\begin{align*} \\frac { 1 } { q } \\Big | \\sum _ { \\substack { q \\mid n _ 1 \\pm b n _ 2 \\\\ n _ 1 , n _ 2 \\ll q ^ { 1 + \\varepsilon } } } \\lambda _ 1 ( n _ 1 ) \\lambda _ 2 ( n _ 2 ) \\chi ( n _ 2 ) G \\Big ( \\frac { n _ 1 } { q } , \\frac { n _ 2 } { q } \\Big ) \\Big | \\end{align*}"}
+{"id": "5631.png", "formula": "\\begin{align*} D _ 5 ^ a = \\Bigl ( x _ 0 y ^ 2 + B y + x _ 1 C = 0 \\Bigr ) . \\end{align*}"}
+{"id": "5552.png", "formula": "\\begin{align*} f _ k ( \\eta ( t ) ) = \\sum _ { j = 1 } ^ m a _ { k j } \\left ( \\gamma _ j ( t ) + \\nu _ j ( t ) + \\i \\mu _ j ( t ) \\right ) \\end{align*}"}
+{"id": "1596.png", "formula": "\\begin{align*} \\left | \\binom { U } { k } \\cap E \\right | \\ge d \\binom { | U | } { k } - \\mu | V | ^ k , \\end{align*}"}
+{"id": "8362.png", "formula": "\\begin{align*} \\tilde V _ \\alpha = \\bigg \\{ u \\in \\tilde V : \\| u \\| _ \\alpha ^ 2 = \\sum _ { i = 1 } ^ \\infty \\tilde \\lambda _ { i } ^ { 2 \\alpha } | \\hat u _ i | ^ 2 \\} < \\infty , u = \\sum _ { i = 1 } ^ \\infty \\hat u _ i e _ { J , i } \\bigg \\} \\tilde V = \\tilde V _ 0 . \\end{align*}"}
+{"id": "2020.png", "formula": "\\begin{align*} \\dot { \\mathrm { B } } ^ { s } _ { p , q , 0 } ( \\Omega ) = \\dot { \\mathrm { B } } ^ { s } _ { p , q } ( \\Omega ) \\end{align*}"}
+{"id": "7592.png", "formula": "\\begin{align*} J _ 1 ( \\beta , N , T ) \\leq C \\sum _ { i = 1 } ^ 2 \\hat J ( \\beta , N , T , \\hat { \\mathbf { R } } _ i ) . \\end{align*}"}
+{"id": "7238.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { \\frac { 1 } { 2 } G } & = \\frac { \\sqrt { \\Delta } \\pm \\sqrt { \\Delta - 1 + R ^ 2 } } { ( 1 - R ) } . \\end{aligned} \\end{align*}"}
+{"id": "7890.png", "formula": "\\begin{align*} \\begin{cases} u _ t - \\log \\det D ^ 2 u - k \\log ( u ^ { \\star } ) = - g ( x , u ) & Q _ T \\\\ u = u _ 0 & \\overline \\Omega \\times \\{ 0 \\} \\\\ u = 0 & \\partial \\Omega \\times ( 0 , T ) . \\end{cases} \\end{align*}"}
+{"id": "7591.png", "formula": "\\begin{align*} \\hat J ( \\beta , N , T , \\hat { \\mathbf { R } } _ i ) = N \\iint _ { \\hat { \\textbf { R } } _ i } \\int _ { \\mathbf { B } _ 2 ( 0 ) } f ^ { ( 1 , 1 , \\alpha ) } _ { t , s } ( z ) d z d s d t . \\end{align*}"}
+{"id": "8407.png", "formula": "\\begin{align*} W ( v ) = \\sum _ { z = 1 } ^ { 2 Z - 1 } V _ { z } ( v ) + \\O \\left ( \\frac { v N ^ { 3 \\gamma - 2 } } { d \\log ^ 7 N } \\right ) + \\O \\left ( \\frac { N ^ { 2 \\gamma - 1 } } { d \\log ^ 6 N } \\right ) . \\end{align*}"}
+{"id": "4200.png", "formula": "\\begin{align*} D ( - 4 , \\varepsilon ) & = \\sigma _ { \\varepsilon } ( - 4 I ) = \\sigma _ { \\varepsilon } ( [ I \\bullet i I , i I ] _ { \\ast } ) = \\sigma _ { \\varepsilon } ( [ \\varphi ( I ) \\bullet \\varphi ( i I ) , \\varphi ( i I ) ] _ { \\ast } ) \\\\ & = \\sigma _ { \\varepsilon } ( [ \\varphi ( I ) \\varphi ( i I ) + \\varphi ( i I ) \\varphi ( I ) ^ { \\ast } , \\varphi ( i I ) ] _ { \\ast } ) = \\sigma _ { \\varepsilon } ( 4 \\varphi ^ { 2 } ( I ) \\varphi ( i I ) ) . \\end{align*}"}
+{"id": "2209.png", "formula": "\\begin{align*} ( v , i ) ^ { j \\hat { \\rho } _ 1 } = ( v ^ j , i ) ^ { \\hat { \\rho } _ 1 } = ( v ^ { j \\rho _ 1 } , - i ) = ( v ^ { \\rho _ 1 j } , - i ) = ( v ^ { \\rho _ 1 } , - i ) ^ j = ( v , i ) ^ { \\hat { \\rho } _ 1 j } , \\end{align*}"}
+{"id": "530.png", "formula": "\\begin{align*} \\mathbf H ^ { s , r } = H ^ s ( \\R , \\C ) \\times H ^ s ( \\R , \\C ) \\times H ^ r ( \\R , \\C ) \\end{align*}"}
+{"id": "2856.png", "formula": "\\begin{align*} ( s _ j s _ k ) ^ { m _ { j k } } = 1 , j , k \\in \\{ 0 , \\ldots , n \\} \\end{align*}"}
+{"id": "3333.png", "formula": "\\begin{align*} 1 - z _ i = \\prod _ { j = 1 } ^ N z _ j ^ { A _ { i j } } \\end{align*}"}
+{"id": "3542.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { B _ { 2 } } \\vert \\nabla w \\vert ^ { 2 } d x & \\leqslant \\frac { 1 } { 1 2 } \\int _ { \\mathbb { R } ^ { N } } ( \\alpha u + \\beta v ) w d x + C K ^ { 2 } r _ { 0 } ^ { 1 - N } + C K r _ { 0 } ^ { \\frac { 1 - N } { 2 } } \\Vert f \\Vert _ { 2 } \\\\ & + C K + C r _ { 0 } \\Vert f \\Vert _ { 2 } ^ { 2 } + C \\sqrt { K } \\Vert g _ { 1 } \\Vert _ { 2 } + C \\sqrt { K } \\Vert g _ { 2 } \\Vert _ { 2 } + \\int _ { B _ { 2 } } w ^ { 2 } d x , \\end{aligned} \\end{align*}"}
+{"id": "3510.png", "formula": "\\begin{align*} \\mathbb { L } ( z ) = \\log \\vert z \\vert + \\mathrm { a r g } ( z ) , \\end{align*}"}
+{"id": "7132.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\Psi ^ { ( 5 ) } ( Z ) ( \\Psi w ) ' ( Z ) \\ : d Z = \\int _ 0 ^ \\infty ( \\Psi ^ { ( 3 ) } ( Z ) ^ 2 w ( Z ) + \\sum _ { k = 1 } ^ 3 \\begin{pmatrix} 3 k \\end{pmatrix} \\int _ 0 ^ \\infty \\Psi ^ { ( 3 ) } ( Z ) \\Psi ^ { ( 3 - k ) } ( Z ) w ^ { ( k ) } ( Z ) \\ : d Z . \\end{align*}"}
+{"id": "6033.png", "formula": "\\begin{align*} \\mathbb { E } \\left \\vert G _ { n } - F \\right \\vert \\leq \\sum _ { i = n } ^ { m _ { n } } \\gamma _ { i } ^ { n } \\times \\mathbb { E } \\left \\vert F _ { i } - F \\right \\vert \\rightarrow 0 . \\end{align*}"}
+{"id": "1227.png", "formula": "\\begin{align*} \\| v _ n \\| _ { S ( \\R ) } \\leq C \\| \\phi \\| _ { \\dot { H } ^ 1 } \\ \\textnormal { a n d } \\ \\limsup _ { T \\to \\infty } \\lim _ { n \\to \\infty } \\| v _ n - v _ { n , T } \\| _ { S ( \\R ) } = 0 . \\end{align*}"}
+{"id": "5375.png", "formula": "\\begin{align*} l = l ( \\frac { r } { s } ) = \\begin{cases} 0 , & ~ \\mbox { i f } ~ a _ { 2 m } \\geq 2 ; \\\\ a _ { 2 m - 1 } , & ~ \\mbox { i f } ~ a _ { 2 m } = 1 . \\end{cases} \\end{align*}"}
+{"id": "5458.png", "formula": "\\begin{align*} x - \\frac { 1 } { V ' ( x ) } - \\sqrt { 2 W ( x ) } & \\leq - \\frac { 1 } { x } - \\frac { 2 \\log H ( x ) } { x + \\sqrt { 2 W ( x ) } } \\\\ & = - \\frac { 1 + I ( x ) } { x } \\end{align*}"}
+{"id": "6315.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x } = \\frac { 1 } { \\sqrt { g _ { [ < \\lambda ^ { \\sigma } ] } } } \\frac { \\partial } { \\partial y } , \\end{align*}"}
+{"id": "5119.png", "formula": "\\begin{align*} g ( R ( Z , \\cdot ) ( r ( X ) ) - r ( R ( Z , \\cdot ) ( X ) ) , Y ) = 0 , \\end{align*}"}
+{"id": "1148.png", "formula": "\\begin{align*} & \\widetilde { ( \\delta _ 1 \\circ \\delta _ 2 + \\delta _ 2 \\circ \\delta _ 1 ) ( h ) } \\\\ & = ( - 1 ) ^ { n } ~ [ \\mu _ 1 , \\widetilde { \\delta _ 2 h } ] ~ + ~ ( - 1 ) ^ { n } ~ [ \\mu _ 2 , \\widetilde { \\delta _ 1 h } ] \\\\ & = - [ \\mu _ 1 , [ \\mu _ 2 , \\widetilde { h } ] ] - [ \\mu _ 2 , [ \\mu _ 1 , \\widetilde { h } ] \\\\ & = - [ [ \\mu _ 1 , \\mu _ 2 ] , \\widetilde { h } ] = 0 ~ ~ ~ ~ ( ~ [ \\mu _ 1 , \\mu _ 2 ] = 0 ) . \\end{align*}"}
+{"id": "5495.png", "formula": "\\begin{align*} \\varphi : \\mathrm { F r e e } _ 2 \\rightarrow S L ( 2 , { \\mathbb Z } ) , \\ \\ \\ \\ \\ \\varphi ( a ) = \\left ( \\begin{array} { c c } 1 & - 1 \\\\ 0 & 1 \\end{array} \\right ) , \\ \\ \\ \\ \\ \\varphi ( b ) = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 1 & 1 \\end{array} \\right ) . \\end{align*}"}
+{"id": "4306.png", "formula": "\\begin{align*} q ^ * = w - x i - y j - z k \\end{align*}"}
+{"id": "3580.png", "formula": "\\begin{align*} { \\omega _ { k , n } } = \\left ( { n - 1 } \\right ) \\left ( { { \\Phi _ { { k } , { l } } ^ { \\rm { D } } } - { \\Theta _ { k , { j } } ^ { \\rm { A } } } } \\right ) , \\end{align*}"}
+{"id": "5667.png", "formula": "\\begin{align*} D _ { X ^ \\prime } \\cdot e _ 1 & = 6 + b _ 0 - 5 b _ 1 \\geq - 1 , \\\\ D _ { X ^ \\prime } \\cdot e _ 1 ^ \\prime & = 6 - b _ 0 + 5 b _ 1 \\geq - 1 . \\end{align*}"}
+{"id": "7147.png", "formula": "\\begin{align*} \\log ( r ) - \\int _ 1 ^ r \\frac { 1 } { s + \\Phi _ 1 ( s ) } d s & = \\int _ 1 ^ r \\frac { 1 } { s } - \\frac { 1 } { s + \\Phi _ 1 ( s ) } d s \\\\ & = \\int _ 1 ^ r \\frac { \\Phi _ 1 ( s ) } { s ( s + \\Phi _ 1 ( s ) ) } d s \\end{align*}"}
+{"id": "4794.png", "formula": "\\begin{align*} d ( y , z ) = d _ x ( y , z ) : = \\lambda _ 0 ^ { - k ( y , z ) } , \\end{align*}"}
+{"id": "6759.png", "formula": "\\begin{align*} M P = P Q . \\end{align*}"}
+{"id": "4547.png", "formula": "\\begin{align*} I _ { 1 , n } ( t ) & \\le C _ 0 \\Vert { \\mathbf F } ( t ) \\Vert _ { L ^ 2 ( \\mathbb R ^ 2 _ + ) } ^ 2 + C _ 1 \\left \\{ I ( t ) + I _ 0 ( t ) + I _ { \\sigma } ( t ) + I _ 2 ( t ) \\right \\} \\\\ & = C _ 0 \\Vert { \\mathbf F } ( t ) \\Vert _ { L ^ 2 ( \\mathbb R ^ 2 _ + ) } ^ 2 + C _ 1 I _ { 1 , \\ast } ( t ) \\ , . \\end{align*}"}
+{"id": "7137.png", "formula": "\\begin{align*} - \\int _ 0 ^ \\infty \\frac { \\Psi ( Z ) } { Z } \\Psi ^ { ( 3 ) } ( Z ) w _ 1 ( Z ) \\ : d Z = \\int _ 0 ^ \\infty \\Psi '' ( Z ) \\frac { d } { d Z } \\left ( \\frac { \\Psi ( Z ) } { Z } w _ 1 ( Z ) \\right ) \\ : d Z . \\end{align*}"}
+{"id": "8052.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j } \\mu _ { j } b _ { j } \\right \\| _ { h _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } . \\end{align*}"}
+{"id": "7596.png", "formula": "\\begin{align*} \\begin{aligned} s ^ { \\frac { 3 } { d + 2 } } & \\leq t ^ { \\frac { 3 } { d + 2 } } \\leq C \\beta ^ { - \\frac { 1 } { d + 2 } } N ^ { - \\frac { 1 } { d + 2 } } + s ^ { \\frac { 3 } { d + 2 } } , \\\\ s & \\leq C \\beta ^ { - \\frac { 1 } { 3 } } N ^ { - \\frac { 1 } { 3 } } . \\end{aligned} \\end{align*}"}
+{"id": "4946.png", "formula": "\\begin{align*} \\Psi ( \\tau ) = \\Psi ( q ) : = \\sum _ { n \\geq 0 } q ^ { \\frac { n ( n + 1 ) } { 2 } } = \\sum _ { n \\geq 0 } q ^ { T _ n } . \\end{align*}"}
+{"id": "502.png", "formula": "\\begin{align*} \\mathcal L _ { \\mathrm { D i r a c } } ( \\psi ) = \\psi ^ * \\left ( i \\partial _ t + i \\alpha \\partial _ x - M \\beta \\right ) \\psi , \\mathcal L _ { \\mathrm { m e s o n } } ( \\phi ) = \\frac 1 2 ( \\partial _ t \\phi ) ^ 2 - \\frac 1 2 ( \\partial _ x \\phi ) ^ 2 - \\frac 1 2 m ^ 2 \\phi ^ 2 , \\end{align*}"}
+{"id": "615.png", "formula": "\\begin{align*} \\mu ( t ) & = t ^ { - 2 b } \\int _ { \\R ^ d } \\int _ 0 ^ t V ( s , \\zeta ) \\overline { W ( s , \\zeta ) } \\ , d s \\ , d \\zeta , \\\\ \\nu ( t ) & = 2 \\int _ { \\R ^ d } \\int _ 0 ^ t \\int _ 0 ^ s \\frac { \\left [ V ( s , \\zeta ) - V ( \\sigma , \\zeta ) \\right ] \\overline { \\left [ W ( s , \\zeta ) - W ( \\sigma , \\zeta ) \\right ] } } { ( s - \\sigma ) ^ { 1 + 2 b } } \\ , d \\sigma \\ , d s \\ , d \\zeta . \\end{align*}"}
+{"id": "8403.png", "formula": "\\begin{align*} W _ { z } ( v ) = V _ { z } ( v ) + \\O \\left ( \\frac { v N ^ { 2 \\gamma - 2 } } { d \\log ^ 7 N } \\sum _ { P < p \\leq 2 P } ( \\log p ) \\theta _ { z } ( p ^ c ) \\right ) , \\end{align*}"}
+{"id": "1200.png", "formula": "\\begin{align*} \\mathrm { p r } _ g \\tau ( [ x ] _ P \\otimes [ y ] _ P ) = \\begin{cases} [ x ] _ P & \\mathrm { i f } \\ ; \\exists \\ , p \\in P \\ ; \\mathrm { s . t } \\ ; y ^ { - 1 } x p g ^ { - 1 } \\in P \\\\ 0 & \\mathrm { o t h e r w i s e } \\end{cases} \\end{align*}"}
+{"id": "4973.png", "formula": "\\begin{align*} \\sum ^ s _ { i = 1 } i a _ i \\leq s , ~ \\sum ^ s _ { i = 1 } i a _ { r + 1 - i } \\leq s . \\end{align*}"}
+{"id": "1327.png", "formula": "\\begin{align*} J ^ { w } ( \\textbf { X } _ { m a x R S S U } ^ { ( n ) } ) & = - \\frac { ( n ! ) ^ 2 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( \\int _ { 0 } ^ { 1 } u ^ { 2 i - 2 } w ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) d u \\right ) \\\\ & = - \\frac { ( n ! ) ^ 2 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( \\int _ { 0 } ^ { 1 } u ^ { 2 i - 2 } u ^ m d u \\right ) \\\\ & = - \\frac { ( n ! ) ^ 2 } { 2 } \\prod _ { i = 1 } ^ { n } \\frac { 1 } { m + 2 i - 1 } . \\end{align*}"}
+{"id": "1188.png", "formula": "\\begin{align*} \\xi _ M \\colon M \\xrightarrow { \\gamma \\otimes 1 _ M } A \\otimes A \\otimes M \\xrightarrow { 1 _ A \\otimes \\mathrm { a c t } } A \\otimes M = A \\otimes U _ A M . \\end{align*}"}
+{"id": "8247.png", "formula": "\\begin{align*} \\frac { \\partial \\Psi } { \\partial x _ 2 } = \\frac { 1 } { 2 x _ 1 } \\xi _ 3 - \\frac { 1 } { 2 x _ 1 } d \\psi _ { x _ 1 , 0 , 0 } ( \\xi _ 3 ( m ) ) . \\end{align*}"}
+{"id": "275.png", "formula": "\\begin{align*} \\Lambda ^ n _ 0 K = A _ { - 1 } \\xlongrightarrow { } A _ 0 \\xlongrightarrow { } \\cdots \\xlongrightarrow { } A _ { n - 1 } \\xlongrightarrow { } A _ n = K _ n \\end{align*}"}
+{"id": "3242.png", "formula": "\\begin{align*} \\epsilon q R = O \\left ( \\epsilon X R \\right ) , \\end{align*}"}
+{"id": "772.png", "formula": "\\begin{align*} \\norm { \\theta _ R \\left ( \\sum _ { i = 1 } ^ n \\norm { \\phi _ i } _ { \\widetilde H ^ { b } ( 0 , t ) } ^ 2 \\right ) \\phi _ j ( t ) - \\theta _ R \\left ( \\sum _ { i = 1 } ^ n \\norm { \\Phi _ i } _ { \\widetilde H ^ { b } ( 0 , t ) } ^ 2 \\right ) \\Phi _ j ( t ) } _ { H ^ { b } ( 0 , T ) } \\leqslant C \\sum _ { i = 1 } ^ n \\norm { \\phi _ i - \\Phi _ i } _ { H ^ { b } ( 0 , T ) } , \\end{align*}"}
+{"id": "2201.png", "formula": "\\begin{align*} A : = \\{ x ' \\ : \\ \\C ( x ' \\mid m _ A ) \\le \\C ( x \\mid m _ A ) \\} B : = \\{ \\langle y ' , z ' \\rangle \\ : \\ \\C ( \\langle y ' , z ' \\rangle \\mid m _ A ) \\le \\C ( \\langle y , z \\rangle \\mid m _ A ) \\} . \\end{align*}"}
+{"id": "8652.png", "formula": "\\begin{align*} \\| \\nabla _ { X , z } ^ { \\mu } u \\| _ { H ^ { k , 0 } ( \\mathcal { S } _ b ) } & \\leq M ( k + 1 ) ( \\| \\nabla _ { X , z } ^ { \\mu } \\tilde u \\| _ { H ^ { k , 0 } ( \\mathcal { S } ) } + \\sum \\limits _ { j = 0 } ^ { k } \\| \\partial _ z ^ j \\tilde u \\| _ { H ^ { k , 0 } ( \\mathcal { S } ) } ) \\\\ & \\leq M ( k + 1 ) ( | g | _ { H ^ k } + \\sum \\limits _ { j = 0 } ^ { k } \\| \\partial _ z ^ j f \\| _ { H ^ { k - j , 0 } ( \\mathcal { S } _ b ) } ) . \\end{align*}"}
+{"id": "6032.png", "formula": "\\begin{align*} G _ { n } = \\sum _ { i = n } ^ { m _ { n } } \\gamma _ { i } ^ { n } \\times F _ { i } \\in \\mathcal { S } \\end{align*}"}
+{"id": "535.png", "formula": "\\begin{align*} \\mathcal F ( v \\mathfrak K e _ j ) ( \\xi ) = \\frac { 1 } { ( 2 \\pi ) ^ d } \\int \\widehat v ( \\xi - \\eta ) \\widehat { \\mathfrak k * e _ j } ( \\eta ) \\ , d \\eta = \\frac { 1 } { ( 2 \\pi ) ^ d } \\int \\widehat v ( \\xi - \\eta ) \\widehat { \\mathfrak k } ( \\eta ) \\widehat { e _ j } ( \\eta ) \\ , d \\eta . \\end{align*}"}
+{"id": "1174.png", "formula": "\\begin{align*} & \\bar { m _ { 1 , t } } = m _ { 1 , t } + m _ { 1 , n + 1 } t ^ { n + 1 } , \\\\ & \\bar { m _ { 2 , t } } = m _ { 2 , t } + m _ { 2 , n + 1 } t ^ { n + 1 } , \\end{align*}"}
+{"id": "5751.png", "formula": "\\begin{align*} \\int _ { Q _ 1 } U _ j ( ( x , 0 ) , t ) ^ 2 d x d t & \\leq r _ j ^ { a + 1 - N _ 1 } \\int _ { Q _ { r _ j } } U ( ( x , 0 ) , t ) ^ 2 d x d t \\\\ & = r _ j ^ { a + 1 - N _ 1 } \\int _ { Q _ { r _ j } } u ( x , t ) ^ 2 d x d t \\\\ & \\stackrel { \\eqref { c o n t r a } } \\leq C r _ j ^ { a + 1 + N _ 1 } . \\end{align*}"}
+{"id": "5685.png", "formula": "\\begin{align*} k _ { e _ { n _ 0 } } = \\limsup \\limits _ { | z | \\rightarrow 0 } \\frac { \\ln \\| ( | z | - T ) ^ { - 1 } e _ { n _ 0 } \\| _ p } { \\ln \\| ( | z | - T ) ^ { - 1 } \\| } . \\end{align*}"}
+{"id": "8801.png", "formula": "\\begin{align*} \\forall x \\in \\Theta : \\mathbb { E } \\sum _ { t = 1 } ^ { T } \\big ( f ( x _ t ) - f ( x ) \\big ) & \\le \\sum _ { t = 1 } ^ { T } \\Big [ \\Big ( L + \\frac { 9 L ^ 2 d ^ 2 } { 4 \\alpha t } \\Big ) h _ t ^ 2 + \\frac { 3 d ^ 2 \\sigma ^ 2 } { 4 h _ t ^ 2 \\alpha t } + \\frac { 9 G ^ 2 d } { 2 \\alpha t } \\Big ] . \\end{align*}"}
+{"id": "987.png", "formula": "\\begin{align*} \\begin{cases} n + \\displaystyle \\sum _ { p \\in S _ - } n _ p \\equiv r \\ ( \\mathrm { m o d } \\ 2 ) & x ^ 2 - d y ^ 2 = - 1 \\ , \\\\ \\displaystyle \\sum _ { p \\in S _ - } n _ p \\equiv r \\ ( \\mathrm { m o d } \\ 2 ) & . \\end{cases} \\end{align*}"}
+{"id": "5323.png", "formula": "\\begin{align*} g ^ { ( n + 1 ) } ( 2 k 2 ^ { - n - 1 } ) & = g ^ { ( n ) } ( k 2 ^ { - n } ) = ( k 2 ^ { - n } ) ^ 2 \\\\ g ^ { ( n + 1 ) } ( ( 2 k + 2 ) 2 ^ { - n - 1 } ) & = g ^ { ( n ) } ( ( k + 1 ) 2 ^ { - n } ) = ( ( k + 1 ) 2 ^ { - n } ) ^ 2 . \\end{align*}"}
+{"id": "688.png", "formula": "\\begin{align*} \\mathbf X ^ { \\mathbf s , b } ( \\R \\times \\R ^ d ) = X _ { h _ 1 ( \\xi ) } ^ { s _ 1 , b } ( \\R \\times \\R ^ d ) \\times \\dots \\times X _ { h _ n ( \\xi ) } ^ { s _ n , b } ( \\R \\times \\R ^ d ) \\end{align*}"}
+{"id": "1690.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j , k \\leq n } H ^ { ( m , n ) } _ { \\texttt { b } ; j , k } x _ j x _ k = & \\sum _ { 1 \\leq j \\leq n } \\Bigl ( 2 ( m + 1 ) + u _ { q _ 0 } ( \\xi _ j ) + u _ { q _ 1 } ( \\xi _ j ) \\Bigr ) x _ j ^ 2 \\\\ & + \\sum _ { 1 \\leq j < k \\leq n } \\Bigl ( u _ q ( \\xi _ j + \\xi _ k ) ( x _ j + x _ k ) ^ 2 + u _ q ( \\xi _ j - \\xi _ k ) ( x _ j - x _ k ) ^ 2 \\Bigr ) \\\\ \\ge & 2 ( m + 1 ) \\sum _ { 1 \\leq j \\leq n } x _ j ^ 2 . \\end{align*}"}
+{"id": "8798.png", "formula": "\\begin{align*} \\| { \\hat g } \\| ^ 2 & = \\frac { d ^ 2 } { 4 h ^ 2 } \\| ( f ( x + h r \\zeta ) - f ( x - h r \\zeta ) + \\xi - \\xi ' ) \\zeta K ( r ) \\| ^ 2 \\\\ & = \\frac { d ^ 2 } { 4 h ^ 2 } ( f ( x + h r \\zeta ) - f ( x - h r \\zeta ) + \\xi - \\xi ' ) ^ 2 K ^ 2 ( r ) . \\end{align*}"}
+{"id": "3479.png", "formula": "\\begin{align*} ( L ^ n ) = L ^ n \\cdot X _ t = L ^ n \\cdot ( - K _ M ) = ( \\sum d _ i ) ( L ^ { n + 1 } \\cdot M ) , \\end{align*}"}
+{"id": "2241.png", "formula": "\\begin{align*} \\varphi ( a , s , \\delta ) = \\frac { a ^ p } { p ( 1 - s ) } \\delta ^ { p ( 1 - s ) } \\left ( K _ { N , p } + K _ { N , p } | \\ln ( a ) | + K _ { N , p , \\ln } + \\frac { K _ { N , p } } { p } \\Big ( 1 - p ( 1 - s ) \\ln ( \\delta ) \\Big ) \\right ) , \\end{align*}"}
+{"id": "6304.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\iint ( g _ { [ < \\mu ] } - g _ { [ < \\mu ] } ^ { x _ 0 } ) \\left ( M ( v _ \\mu ^ { x _ 0 } ) E ( v _ \\lambda ) - P ( v _ \\lambda ) P ( v _ \\mu ^ { x _ 0 } ) \\right ) d x d t \\right | \\lesssim & \\ \\epsilon ^ 2 C ^ 2 \\lambda ^ 2 | x _ 0 | d _ { \\lambda } ^ 2 d _ { \\mu } ^ 2 . \\end{aligned} \\end{align*}"}
+{"id": "8866.png", "formula": "\\begin{align*} \\eta _ t = \\min \\left ( \\frac { \\mathfrak { y } } { d } , \\ , \\Xi _ T \\right ) \\qquad h _ t = \\mathfrak { h } T ^ { - \\frac { 1 } { 2 ( 2 \\beta - 1 ) } } \\enspace , \\end{align*}"}
+{"id": "1909.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\| Q ^ x _ \\lambda \\Upsilon - \\Upsilon \\| _ { 0 , I _ h } & \\leq C h _ x ^ { k + 1 } | \\Upsilon | _ { k + 1 , I _ h } , \\\\ \\| Q ^ x _ \\lambda \\Upsilon - \\Upsilon \\| _ { 0 , \\Gamma _ x } & \\leq C h _ x ^ { k + 1 / 2 } | \\Upsilon | _ { k + 1 , I _ h } . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7837.png", "formula": "\\begin{align*} f & = 2 \\pi \\tau _ 2 ^ 2 \\left ( \\frac { 2 t ^ 3 } { 3 } \\right ) - \\frac { \\tau _ 2 ^ 2 \\chi } { 2 ( 2 \\pi ) ^ 2 } \\sum _ { ( m , n ) \\in \\mathbb { Z } ^ 2 - \\{ 0 \\} } \\frac { 1 } { | m \\tau + n | ^ 3 } \\\\ & = 4 \\pi \\tau _ 2 ^ 2 \\left ( \\frac { t ^ 3 } { 3 } - \\frac { \\chi } { 8 ( 2 \\pi ) ^ 3 } \\sum _ { ( m , n ) \\in \\mathbb { Z } ^ 2 - ( 0 , 0 ) } \\frac { 1 } { | m \\tau + n | ^ 3 } \\right ) \\ , \\\\ & > 4 \\pi \\tau _ 2 ^ 2 R _ 1 ( t , \\tau ) . \\end{align*}"}
+{"id": "5646.png", "formula": "\\begin{align*} x _ 3 ^ 2 Q + x _ 3 B + C = 0 \\ , \\end{align*}"}
+{"id": "3870.png", "formula": "\\begin{align*} \\theta ' = - \\sin \\theta \\lambda e ^ { - \\lambda _ 2 z } - \\sin \\theta \\mu \\lambda _ 2 y e ^ { - \\lambda _ 2 z } + \\cos \\theta \\mu = \\frac { d } { d s } \\left [ e ^ { - \\lambda _ 2 z } \\left ( \\frac { \\lambda } { \\lambda _ 2 } + \\mu y \\right ) \\right ] . \\end{align*}"}
+{"id": "5841.png", "formula": "\\begin{align*} w _ 0 = s _ { \\alpha _ 2 } s _ { \\alpha ^ { c 2 } _ { m a x } } s _ { \\alpha ^ { c 3 } _ { m a x } } s _ { \\alpha _ { m a x } } . \\end{align*}"}
+{"id": "6706.png", "formula": "\\begin{align*} \\prod _ { k = q } ^ p X _ k = \\begin{cases} X _ q X _ { q + 1 } \\cdots X _ p & ( p \\geq q ) \\\\ 1 & ( p = q - 1 ) \\\\ X ^ { - 1 } _ { p + 1 } X ^ { - 1 } _ { p + 2 } \\cdots X ^ { - 1 } _ { q - 1 } & ( p < q - 1 ) \\end{cases} \\end{align*}"}
+{"id": "4119.png", "formula": "\\begin{align*} \\sum _ { j = a } ^ { 2 s - a + 1 } \\binom { 2 s - 2 a + 1 } { j - a } \\cdot \\big ( \\big ) = 0 . \\end{align*}"}
+{"id": "8571.png", "formula": "\\begin{align*} \\mathcal { L } _ 2 ^ { \\mu } [ \\beta b ] \\bullet = - \\frac { 1 } { 2 } b \\mathrm { F } _ 4 \\bullet + \\frac { \\beta ^ 2 } { 6 } b ^ 3 \\mathrm { F } _ 3 \\bullet , \\end{align*}"}
+{"id": "2188.png", "formula": "\\begin{align*} \\sum _ { y \\in X } b ( x , y ) f _ t ( y ) & \\leq \\left ( \\sum _ { y \\in X } b ( x , y ) f _ 1 ( y ) \\right ) ^ t \\left ( \\sum _ { y \\in X } b ( x , y ) f _ 2 ( y ) \\right ) ^ { ( 1 - t ) } \\\\ & = ( \\deg ( x ) - \\lambda _ 1 ) ^ t f _ { \\alpha _ 1 } ( x ) ^ t ( \\deg ( x ) - \\lambda _ 2 ) ^ { ( 1 - t ) } f _ { \\alpha _ 2 } ( x ) ^ { ( 1 - t ) } \\\\ & \\leq ( \\deg ( x ) - \\lambda _ t ) f _ t ( x ) \\end{align*}"}
+{"id": "7695.png", "formula": "\\begin{align*} \\bigl ( i _ { B ' | _ { S ' _ { \\sigma ' } , K ' _ { \\sigma ' } } } \\circ \\varphi _ { \\sigma } ^ { \\sigma ' } [ x ] \\bigr ) [ y ] = ( i _ { B ' , K ' } \\circ \\varphi ) _ { \\sigma } ^ { \\sigma ' } [ y ] . \\end{align*}"}
+{"id": "5058.png", "formula": "\\begin{align*} ( \\Pi , \\Xi _ g ) : U _ g \\longrightarrow \\{ G = 0 \\} \\subset U _ 1 \\times U _ 2 . \\end{align*}"}
+{"id": "3256.png", "formula": "\\begin{align*} | \\mathcal { C } | = \\sqrt { \\langle \\mathcal { C } \\widetilde { \\mathcal { C } } \\rangle _ 0 } = \\sqrt { \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } \\mathcal { C } _ { _ \\Sigma } ^ 2 } . \\end{align*}"}
+{"id": "129.png", "formula": "\\begin{align*} k ( s , t ) = W ^ T ( s ) T _ W K T _ W W ( t ) = W ^ T ( t ) T _ W K ^ T T _ W W ( s ) \\end{align*}"}
+{"id": "259.png", "formula": "\\begin{align*} \\lim _ { c \\to + \\infty } \\hat { a } ^ { } _ \\nu ( \\xi ; g ^ { ( c ) } ) = \\begin{cases} \\Delta _ ( \\xi ) a ^ { } _ \\nu ( \\xi ; g ) & \\ K = M , \\\\ \\Delta _ ( \\xi ) a ^ { } _ \\nu ( \\xi ; g ) & \\ K = L . \\end{cases} \\end{align*}"}
+{"id": "3044.png", "formula": "\\begin{align*} \\beta ^ { \\zeta , \\C } _ { \\R ^ 2 } ( \\rho , \\theta ) : = \\frac { 1 } { \\rho } \\Gamma ^ { \\zeta , \\C } ( \\theta ) \\ , , \\end{align*}"}
+{"id": "7454.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { n } \\kappa _ i + m \\right ) \\int _ { \\Omega } \\prod _ { i = 1 } ^ { m } \\vert y _ i \\vert ^ { \\kappa _ i + 1 } d z \\leq & C _ { 2 2 } \\prod _ { i = 1 } ^ { m } \\| y _ i \\| _ { 1 , p } ^ { \\kappa _ i + 1 } . \\end{align*}"}
+{"id": "4249.png", "formula": "\\begin{align*} \\mbox { P . V . } \\int _ { \\R ^ n } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { n + 2 s } } \\ , d y \\\\ = \\lim _ { \\epsilon \\to 0 } \\int _ { \\{ y \\in \\R ^ n \\ , : \\ , | y - x | \\geq \\epsilon \\} } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { n + 2 s } } \\ , d y , \\end{align*}"}
+{"id": "5836.png", "formula": "\\begin{align*} \\alpha _ 2 , \\ ; \\alpha _ 3 , \\ ; \\alpha _ 5 , \\ ; \\alpha _ 7 , \\ ; \\alpha ^ { d 4 } _ { m a x } , \\ ; \\alpha ^ { d 6 } _ { m a x } , \\ ; \\alpha ^ { e 7 } _ { m a x } , \\ ; \\alpha _ { m a x } . \\end{align*}"}
+{"id": "5699.png", "formula": "\\begin{align*} ( z - T ) ^ { - 1 } = \\begin{bmatrix} ( z - B ) ^ { - 1 } & 0 \\\\ - ( z - A ) ^ { - 1 } e _ 1 \\otimes ( z - B ' ) ^ { - 1 } f _ 0 & ( z - A ) ^ { - 1 } \\end{bmatrix} , \\end{align*}"}
+{"id": "1979.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { p - 1 } \\omega _ q ^ { \\left ( a ^ t _ { \\sigma , r ^ j } - a ^ t _ { \\sigma , s ^ j } \\right ) - \\left ( a ^ t _ { \\sigma , r } - a ^ t _ { \\sigma , s } \\right ) } = \\sum _ { j = 1 } ^ { p - 1 } \\omega _ p ^ { j \\left ( s _ { \\pi _ { \\hat { \\beta } } ( \\hat { \\gamma } ) } - r _ { \\pi _ { \\hat { \\beta } } ( \\hat { \\gamma } ) } \\right ) } = - 1 . \\end{align*}"}
+{"id": "7248.png", "formula": "\\begin{align*} \\Sigma ( t ) = e ^ { t \\log \\Sigma _ t } = \\left ( \\Sigma _ t \\right ) ^ t , \\end{align*}"}
+{"id": "1532.png", "formula": "\\begin{align*} N _ p ( x ) = 1 + \\sum _ { k \\le K \\log _ 2 x } \\sum _ { \\substack { A \\le x / p \\\\ P ^ + ( A ) \\le p \\\\ \\Omega ( A ) = k } } \\sum _ { \\substack { B \\le x / A p \\\\ P ^ - ( B ) \\ge p \\\\ \\Omega ( B ) = k } } 1 = \\sum _ { 0 \\le k \\le K \\log _ 2 x } \\sum _ { \\substack { A \\le x / p \\\\ P ^ + ( A ) \\le p \\\\ \\Omega ( A ) = k } } \\Phi _ k \\left ( \\frac x { A p } , p \\right ) . \\end{align*}"}
+{"id": "5334.png", "formula": "\\begin{align*} \\frac { 1 } { n ! } \\frac { \\mathrm { d } ^ n } { \\mathrm { d } x ^ n } \\frac { e ^ x } { 1 + e ^ x } & = \\frac { 1 } { 2 \\pi i } \\oint _ { \\partial B ( x , r ) } \\frac { e ^ z } { ( 1 + e ^ z ) ( z - x ) ^ { n + 1 } } \\ , \\mathrm { d } z \\\\ & = \\frac { 1 } { 2 \\pi i } \\int _ 0 ^ { 2 \\pi } \\frac { e ^ { x + r e ^ { i t } } } { ( 1 + e ^ { x + r e ^ { i t } } ) ( r e ^ { i t } ) ^ { n + 1 } } \\cdot r i e ^ { i t } \\ , \\mathrm { d } t \\end{align*}"}
+{"id": "6485.png", "formula": "\\begin{align*} P _ D = 0 , P _ N = 0 , P _ R = I _ 2 , \\Lambda = \\begin{pmatrix} \\gamma & 0 \\\\ 0 & - \\gamma \\end{pmatrix} . \\end{align*}"}
+{"id": "8463.png", "formula": "\\begin{align*} r _ { \\ell } ( z ) = \\left ( \\frac { \\ell ( z ) } { \\omega _ { n - 1 } } \\right ) ^ { \\frac { 1 } { n - 1 } } \\mbox { f o r } \\mathcal { H } ^ { 1 } \\mbox { - a . e . } z \\in \\mathbb { R } , \\end{align*}"}
+{"id": "7858.png", "formula": "\\begin{align*} M _ n ( \\underline { x } ) = \\max \\{ k : x _ i , \\dots , x _ { i + k - 1 } = x _ j , \\dots , x _ { j + k - 1 } \\} \\end{align*}"}
+{"id": "8565.png", "formula": "\\begin{align*} \\mathcal { B } [ \\beta b ] \\bullet & = b \\nabla _ X ( \\nabla _ X \\cdot ( b \\bullet ) ) + h _ b \\nabla _ X \\big { ( } b \\nabla _ X \\cdot ( b \\bullet ) \\big { ) } + 2 h _ b ( \\nabla _ X b ) \\nabla _ X \\cdot ( b \\bullet ) , \\end{align*}"}
+{"id": "5284.png", "formula": "\\begin{align*} \\mathbf { n } = ( n _ 0 , n _ 1 , \\dots , n _ { L + 1 } ) \\in \\mathbb { N } ^ { L + 2 } . \\end{align*}"}
+{"id": "5105.png", "formula": "\\begin{align*} D = K \\{ T \\} \\oplus K \\{ T ^ { - 1 } \\} \\end{align*}"}
+{"id": "7557.png", "formula": "\\begin{align*} & \\ , \\frac { s _ { \\square } } { 2 \\hbar } ( \\ln ( z ) - \\ln ( s _ { \\square } + m ) + \\ln ( s _ { \\square } ) - \\sum _ { \\square ' \\neq \\square } \\ln ( s _ { \\square } - s _ { \\square ' } ) + \\sum _ { \\square ' \\neq \\square } \\ln ( s _ { \\square ' } - s _ { \\square } ) \\\\ & - \\sum _ { s = 1 } ^ 3 \\sum _ { \\square ' \\neq \\square } \\ln ( s _ { \\square ' } - s _ { \\square } + \\hbar _ s ) + \\sum _ { s = 1 } ^ 3 \\sum _ { \\square ' \\neq \\square } \\ln ( s _ { \\square } - s _ { \\square ' } + \\hbar _ s ) + o ( \\hbar ) ) . \\end{align*}"}
+{"id": "1000.png", "formula": "\\begin{align*} \\nabla \\cdot ( \\nabla \\times { \\bf G } ) = 0 , \\end{align*}"}
+{"id": "6355.png", "formula": "\\begin{align*} \\begin{aligned} J ^ 4 _ { \\lambda } ( u , v ) = \\int & g _ { [ < \\lambda ] } ^ { x _ 0 } \\left ( M _ \\lambda ( u ) E _ \\lambda ( v ) - 2 P _ \\lambda ( u ) P _ \\lambda ( v ) \\right ) \\\\ & \\ + g _ { [ < \\lambda ] } \\left ( M _ \\lambda ( v ) E _ \\lambda ( u ) - 2 P _ \\lambda ( u ) P _ \\lambda ( v ) \\right ) \\ , d x . \\end{aligned} \\end{align*}"}
+{"id": "894.png", "formula": "\\begin{align*} t ^ { 1 - \\alpha } \\phi _ t - \\phi _ { x x } - \\frac { c } { x } \\phi _ x - n x ^ k \\eta _ x = 0 , \\end{align*}"}
+{"id": "6637.png", "formula": "\\begin{align*} { \\rho } _ { ( k ) , \\infty } ^ { ( \\rm c J ) } ( x _ 1 , \\dots , x _ k ; \\beta , p , q ) \\Big | _ { \\beta = 2 } = \\det \\Big [ { K } _ \\infty ^ { ( p , q ) } ( x _ j , x _ l ) \\Big ] _ { j , l = 1 , \\dots , k } , \\end{align*}"}
+{"id": "7644.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } u ( k , w _ n ) = - \\infty , \\end{align*}"}
+{"id": "3889.png", "formula": "\\begin{align*} \\partial ^ \\alpha _ j \\colon C ^ n ( \\mathcal { C } ) \\to C ^ { n - 1 } ( \\mathcal { C } ) \\ , ; j = 1 , \\dots , n \\ , , \\ ; \\alpha = - 1 , 0 , 1 \\ , . \\end{align*}"}
+{"id": "4235.png", "formula": "\\begin{align*} \\tau ^ { 1 } = r \\ , \\omega ^ { 1 } , \\tau ^ { 2 } = \\frac { \\Delta } { t } \\ , \\omega ^ { 2 } , \\tau ^ { 3 } = \\frac { v } { t } \\ , \\omega ^ { 2 } + i t \\ , \\omega ^ { 3 } , \\end{align*}"}
+{"id": "5283.png", "formula": "\\begin{align*} & \\frac { 1 } { n } \\sum _ { j = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { t } ( x _ { j , t } ) ] _ + \\| \\le n \\varepsilon _ 1 G _ 2 + \\frac { \\varepsilon _ 5 } { \\mu } ( \\log ( T ) + 1 ) + \\frac { \\varepsilon _ 6 } { \\epsilon _ s } \\Big ( \\varepsilon _ { 9 } + \\frac { n F \\log ( T ) } { \\gamma _ { 0 } \\mu } \\Big ) . \\end{align*}"}
+{"id": "180.png", "formula": "\\begin{align*} E _ 6 ( \\tau ) = 1 - 5 0 4 q - 1 6 6 3 2 q ^ 2 - 1 2 2 9 7 6 q ^ 3 + \\cdots . \\end{align*}"}
+{"id": "1334.png", "formula": "\\begin{align*} J ^ w ( \\textbf { X } _ { m i n R S S U } ^ { ( n ) } ) & = - \\frac { ( n ! ) ^ 2 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( \\int _ { 0 } ^ { 1 } w ( F ^ { - 1 } ( u ) ) ( 1 - u ) ^ { 2 i - 2 } f ( F ^ { - 1 } ( u ) ) d u \\right ) \\\\ & \\geq - \\frac { ( n ! ) ^ 2 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( \\int _ { 0 } ^ { 1 } w ( F ^ { - 1 } ( u ) ) f ( F ^ { - 1 } ( u ) ) d u \\right ) \\\\ & = ( n ! ) ^ 2 J ^ w ( \\textbf { X } _ { S R S } ^ { ( n ) } ) . \\end{align*}"}
+{"id": "6597.png", "formula": "\\begin{align*} \\mathcal { K } = \\Pi _ { \\omega , a } \\left ( X \\cap ( ( \\Gamma _ { r _ 1 } \\cap I _ 0 ) \\times \\R ) \\right ) \\end{align*}"}
+{"id": "7619.png", "formula": "\\begin{align*} U ( x ) : = - \\frac { 1 } { \\delta } [ ( x + 1 ) ^ { - \\delta } - 1 ] 1 _ { \\{ x > 0 \\} } - \\frac { 1 } { p } [ ( 1 - x ) ^ { p } - 1 ] 1 _ { \\{ x \\leq 0 \\} } \\end{align*}"}
+{"id": "2383.png", "formula": "\\begin{align*} S _ t = \\tilde { S _ t } 1 _ { t < \\tau } \\end{align*}"}
+{"id": "2559.png", "formula": "\\begin{align*} & \\phantom { = } \\lim _ { R \\to \\infty } \\limsup _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\frac { | u _ n ( y ) | ^ p | \\eta _ R ( x ) - \\eta _ R ( y ) | ^ p } { | x - y | ^ { N + s p } } d x d y \\\\ & = \\lim _ { R \\to \\infty } \\limsup _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } \\int _ { \\mathbb { R } ^ N } \\frac { | ( 1 - \\eta _ R ( x ) ) - ( 1 - \\eta _ R ( y ) ) | ^ p } { | x - y | ^ { N + s p } } | u _ n ( y ) | ^ p d x d y = 0 , \\end{align*}"}
+{"id": "2392.png", "formula": "\\begin{align*} f ' ( t ) - h f ( t ) + h r ( t - T ) = g - r \\end{align*}"}
+{"id": "2673.png", "formula": "\\begin{align*} ( t \\omega _ { 0 } - _ { \\omega _ { 0 } } + d d ^ c \\varphi _ t ) ^ { n } = e ^ { ( 1 + \\frac { \\alpha } { 2 \\beta } ) \\varphi _ t } \\omega _ { 0 } ^ { n } . \\end{align*}"}
+{"id": "5465.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty | f ( n ) - f ( 0 ) | \\pi ( n ) & = \\sum _ { n = 0 } ^ \\infty | \\sum _ { k = 0 } ^ { n - 1 } \\nabla f ( k ) | \\pi ( n ) \\\\ & \\leq \\sum _ { k = 0 } ^ \\infty | \\nabla f ( k ) | \\pi ( k ) \\sum _ { n = k + 1 } ^ \\infty \\frac { \\pi ( n ) } { \\pi ( k ) } \\\\ & \\leq A \\sum _ { k = 0 } ^ \\infty | \\nabla f ( k ) | \\pi ( k ) \\end{align*}"}
+{"id": "8555.png", "formula": "\\begin{align*} P ( E ) & = P ( F _ { \\ell } ; \\{ z < a \\} ) + P ( F _ { \\ell } ; \\{ z < b \\} \\backslash \\overline { \\{ z < a \\} } ) + P ( F _ { \\ell } ; \\{ z > b \\} ) \\\\ & = P ( F _ { \\ell } ) . \\end{align*}"}
+{"id": "11.png", "formula": "\\begin{align*} - \\Delta \\varphi ^ { 0 } _ { \\mathbb { R } ^ d } + \\frac { 1 } { 2 } v \\varphi _ { \\mathbb { R } ^ d } ^ { 0 } = 0 , \\end{align*}"}
+{"id": "528.png", "formula": "\\begin{align*} \\norm { S _ { h ( \\xi ) } ( t ) f } _ { \\widetilde X _ { h ( \\xi ) } ^ { s , b } ( 0 , T ) } = T ^ { 1 / 2 - b } \\norm { f } _ { H ^ s } . \\end{align*}"}
+{"id": "7733.png", "formula": "\\begin{align*} \\omega ^ * ( e ) = \\left \\lbrace \\begin{array} { l l l l l l l l l } \\omega ( x y ) + \\omega ( y z ) & & & \\rm i f ~ \\it e = e ^ * ; \\\\ \\omega ( e ) & & & \\rm i f ~ \\it e \\in E ( G ^ * ) - \\left \\lbrace e ^ * \\right \\rbrace . \\end{array} \\right . \\end{align*}"}
+{"id": "3960.png", "formula": "\\begin{align*} d ( H ) \\leq d ( H / K ) + d ( K ) \\leq { k ^ 2 / 4 + 1 } + { k ( n - k ) } = n k - \\frac { 3 } 4 k ^ 2 + 1 \\leq \\frac { 1 } 3 n ^ 2 + 1 . \\end{align*}"}
+{"id": "4870.png", "formula": "\\begin{align*} f ( u ) = \\pm \\frac { u } { k - u } \\ ; , \\end{align*}"}
+{"id": "1391.png", "formula": "\\begin{gather*} \\mathit { l } y = - ( y ^ { [ 1 ] } ) ' - \\sigma ( x ) y ^ { [ 1 ] } - \\sigma ^ 2 ( x ) y , \\end{gather*}"}
+{"id": "1162.png", "formula": "\\begin{align*} & [ m _ { 1 , 1 } , m _ { 1 } ] = 0 , \\\\ & [ m _ { 2 , 1 } , m _ { 2 } ] = 0 , \\\\ & [ m _ { 1 , 1 } , m _ { 2 } ] + [ m _ { 1 } , m _ { 2 , 1 } ] = 0 . \\end{align*}"}
+{"id": "4528.png", "formula": "\\begin{align*} \\mathcal { L } ( { \\mathbf V } _ i , \\Psi _ i ) - \\mathcal { F } ^ a = ( S _ { \\theta _ { i - 1 } } - I ) \\mathcal { F } ^ a + ( I - S _ { \\theta _ { i - 1 } } ) E _ { i - 1 } + e _ { i - 1 } . \\end{align*}"}
+{"id": "4835.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { n } ( B - \\lambda _ i I ) = 0 \\end{align*}"}
+{"id": "5798.png", "formula": "\\begin{align*} & \\alpha _ i = \\varepsilon _ i - \\varepsilon _ { i + 1 } i = 1 , \\dots , n - 1 , \\alpha _ n = \\varepsilon _ n , \\\\ & \\norm { \\alpha _ i } = \\sqrt { 2 } \\ ; \\ ; i = 1 , \\dots , n - 1 , \\norm { \\alpha _ n } = 1 , \\end{align*}"}
+{"id": "7638.png", "formula": "\\begin{align*} \\lambda ( U ( x ) + C ) & \\geq \\lambda ( U ( - \\underline x ) + C ) = \\lambda U ^ + ( \\overline x ) \\ge U ^ + ( \\overline x ) \\end{align*}"}
+{"id": "3371.png", "formula": "\\begin{align*} \\mathcal { R } _ { f _ 1 , f _ 2 } ( y , T ) & \\ll T ^ { - 1 0 } + \\frac { 1 } { T } \\Bigg ( \\frac { ( q + T / Q ) T } { q ( \\log T ) ^ { 1 / 9 } } \\Big ) + T ^ { \\varepsilon } \\Big ( \\frac { ( q + T / Q ) T } { q T ^ { \\gamma / 4 } } \\Big ) + \\Big ( q + \\frac { T } { Q } \\Big ) T ^ { 3 \\gamma } \\Bigg ) \\\\ & \\ll \\frac { 1 } { ( \\log T ) ^ { 1 / 9 } } + \\frac { T } { q Q } + \\frac { q } { T ^ { 1 - 3 \\gamma - \\varepsilon } } . \\end{align*}"}
+{"id": "4232.png", "formula": "\\begin{align*} d \\omega ^ 1 = K \\ , \\omega ^ { 1 3 } + L \\ , \\omega ^ { 1 \\bar { 3 } } , \\ d \\omega ^ 2 = - K \\ , \\omega ^ { 2 3 } - L \\ , \\omega ^ { 2 \\bar { 3 } } , \\ d \\omega ^ 3 = 0 , \\end{align*}"}
+{"id": "6867.png", "formula": "\\begin{align*} \\rho _ h = A V _ { h } Y + V _ { h } Y B - C Z ^ H , \\end{align*}"}
+{"id": "4543.png", "formula": "\\begin{align*} I ( t ) \\le C _ 0 \\Vert { \\mathbf F } \\Vert _ { L ^ 2 ( \\Omega _ t ) } ^ 2 + C _ 1 \\left \\{ \\Vert \\varphi ( t ) \\Vert _ { L ^ 2 ( \\mathbb R ) } ^ 2 + \\int _ 0 ^ t ( I + I _ { 1 , n } ) ( s ) d s \\right \\} + C _ 2 \\int _ 0 ^ t \\Vert \\varphi ( s ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } d s \\end{align*}"}
+{"id": "3131.png", "formula": "\\begin{align*} \\mathbb { E } [ | \\mathbf { H } _ { k , } \\cdot \\mathbf { H } _ { l , } ^ H | ^ 2 ] = \\mathbb { E } [ | \\mathbf { H } _ { k , } \\cdot \\mathbf { H } _ { l , } ^ H | ^ 2 ] = \\frac { L _ k L _ l } { N } \\frac { \\kappa } { ( \\kappa + 1 ) ^ 2 } . \\end{align*}"}
+{"id": "5923.png", "formula": "\\begin{align*} d ( ( - 1 ) ^ { ( n + 2 ) / 2 } a _ { 1 , n + 2 } ) = d [ ( - 1 ) ^ { ( n + 2 ) / 2 } a _ { 1 , n + 2 } ] = 1 - R _ { n + 2 } < 2 e \\end{align*}"}
+{"id": "2229.png", "formula": "\\begin{align*} k [ X _ { \\sigma _ i ^ c } ] = \\bigoplus _ { m \\in \\sigma _ i ^ \\vee \\cap N _ i ^ \\vee } k [ G ] ^ { ( B ) } _ { - m } \\end{align*}"}
+{"id": "5642.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l } F _ 3 = x _ 0 y + B + x _ 1 ( y + Q ) = 0 , \\\\ G _ 4 = y ( y + Q ) - C = 0 , \\end{array} \\right . \\end{align*}"}
+{"id": "1500.png", "formula": "\\begin{align*} & d ( A _ { i } ( t ) + A _ { j } ( t ) ) = \\frac { 1 } { 2 } \\langle U ^ { ( i ) } B _ { t } + U ^ { ( j ) } B _ { t } , d B _ { t } \\rangle , \\end{align*}"}
+{"id": "5921.png", "formula": "\\begin{align*} A _ { n - 1 } & = \\min \\{ \\dfrac { R _ { n } - S _ { n - 1 } } { 2 } + e , R _ { n } - S _ { n - 1 } + d [ - a _ { 1 , n } b _ { 1 , n - 2 } ] , \\\\ & R _ { n } - S _ { n - 1 } + R _ { n + 1 } - S _ { n - 2 } + d [ a _ { 1 , n + 1 } b _ { 1 , n - 3 } ] \\} \\\\ & \\ge \\min \\{ \\dfrac { 1 - 2 e } { 2 } + e , ( 1 - 2 e ) + 2 e , ( 1 - 2 e ) + ( 2 e + 1 ) \\} = \\dfrac { 1 } { 2 } > d [ a _ { 1 , n - 1 } b _ { 1 , n - 1 } ] . \\end{align*}"}
+{"id": "5047.png", "formula": "\\begin{align*} Q _ g ( g + h + T ( \\gamma ' - \\gamma ) ) = 0 . \\end{align*}"}
+{"id": "5965.png", "formula": "\\begin{align*} - u _ f ( x ) & = \\int _ M G ^ \\omega _ M ( x , z ) \\Delta _ g u _ f d \\mu _ g ( z ) + \\int _ { \\partial M } u _ f ( z ) \\partial _ \\nu G ^ \\omega _ M ( x , z ) - G ^ \\omega _ M ( x , z ) \\partial _ \\nu u _ f ( z ) d \\mu _ h ( z ) \\\\ & + \\int _ M g _ z ( F ( z ) , \\nabla _ g u _ f ) G ^ \\omega _ M ( x , z ) d \\mu _ g ( z ) - \\int _ { \\partial M } u _ f ( z ) G ^ \\omega _ M ( x , z ) F ( z ) \\cdot \\nu d \\mu _ h ( z ) \\\\ & + \\omega ^ 2 \\int _ M u _ f ( z ) G ^ \\omega _ M ( x , z ) d \\mu _ g ( z ) . \\end{align*}"}
+{"id": "6714.png", "formula": "\\begin{align*} \\partial _ { J } \\sigma = O \\left ( | x | ^ { - \\tau - | J | } \\right ) \\ \\ \\ x \\rightarrow \\infty \\ \\ J \\ \\ | J | \\le 2 . \\end{align*}"}
+{"id": "4368.png", "formula": "\\begin{align*} A _ 0 ( { \\bf { U } } ) : = \\mathrm { d i a g } \\Big \\{ \\frac { 1 } { \\rho c ^ 2 } , \\rho , \\rho , 1 , 1 , 1 \\Big \\} , \\end{align*}"}
+{"id": "3661.png", "formula": "\\begin{align*} I _ { \\epsilon } = \\{ t _ 0 \\in [ a , b ] \\ , | \\ , h ( t ) \\leq h ( a ) + ( c + \\epsilon ) \\ , ( t - a ) { } t \\leq t _ 0 \\} \\ , . \\end{align*}"}
+{"id": "3250.png", "formula": "\\begin{align*} C \\ell _ { p , q } = \\bigoplus _ { \\ell = 0 } ^ n C \\ell _ { p , q } ^ \\ell \\end{align*}"}
+{"id": "3243.png", "formula": "\\begin{align*} 1 = o \\left ( \\frac { R } { q } \\right ) = o \\left ( \\epsilon X R \\right ) . \\end{align*}"}
+{"id": "7679.png", "formula": "\\begin{align*} \\varphi _ \\varepsilon ^ * ( 0 ) = \\frac { m ^ { * ( 1 ) } _ 0 } { \\mathbb { P } ( S ^ * _ N = 0 ) } \\end{align*}"}
+{"id": "4740.png", "formula": "\\begin{align*} A ^ \\lambda = \\bigcap _ { i = 1 } ^ \\infty A _ i ^ \\lambda \\end{align*}"}
+{"id": "6237.png", "formula": "\\begin{align*} u = u _ 0 + \\int _ 0 ^ \\infty u _ k \\ , d k . \\end{align*}"}
+{"id": "6107.png", "formula": "\\begin{align*} \\left ( P - P T P \\right ) ^ { - 1 } = \\sum _ { n \\ge 0 } ( P T P ) ^ n = \\sum _ { n \\ge 0 } P T ^ n P . \\end{align*}"}
+{"id": "5981.png", "formula": "\\begin{align*} 0 & = \\int _ M ( - \\Delta _ g ^ F u ^ \\psi ( x ) + u ^ \\psi ( x ) ) u ^ \\psi ( x ) e ^ { \\phi ( x ) } d \\mu _ g ( x ) \\\\ & = \\| \\nabla _ g u ^ \\psi \\| _ { L ^ 2 ( M , e ^ \\phi d \\mu _ g ) } ^ 2 + \\| u ^ \\psi \\| _ { L ^ 2 ( M , e ^ \\phi d \\mu _ g ) } ^ 2 - \\langle e ^ \\phi \\mathcal { N } \\psi , \\psi \\rangle , \\end{align*}"}
+{"id": "7007.png", "formula": "\\begin{align*} a _ 1 z ^ 2 - c x _ 1 ^ 2 & = 4 ( a _ 1 - c ) \\\\ a _ 2 z ^ 2 - c x _ 2 ^ 2 & = 4 ( a _ 2 - c ) \\\\ b z ^ 2 - c y ^ 2 & = 4 ( b - c ) \\end{align*}"}
+{"id": "6775.png", "formula": "\\begin{align*} \\omega _ \\infty '' ( z ) - c \\omega _ \\infty ' ( z ) + F ( 0 , \\mu ) \\omega _ \\infty ( z ) = 0 \\end{align*}"}
+{"id": "4307.png", "formula": "\\begin{align*} | q | = \\sqrt { w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 } \\end{align*}"}
+{"id": "6510.png", "formula": "\\begin{align*} { F } \\big | _ { \\mathcal { S } ^ c } \\left ( q \\right ) = 0 , \\ \\mathcal { S } ^ c = \\Z ^ b \\times \\Z ^ d \\setminus \\mathcal { S } , \\end{align*}"}
+{"id": "6861.png", "formula": "\\begin{align*} \\mathcal { Y } _ { \\vec k } \\times _ 1 A _ 1 ^ { ( k _ 1 ) } + \\dots + \\mathcal { Y } _ { \\vec k } \\times _ d A _ d ^ { ( k _ d ) } = \\mathcal { C } _ { \\vec k } , \\end{align*}"}
+{"id": "1552.png", "formula": "\\begin{align*} \\Delta = y ^ 6 + z f _ 5 ( x , y , z ) \\end{align*}"}
+{"id": "5522.png", "formula": "\\begin{align*} { \\mathcal N } _ { \\lambda , \\mu } ^ { ( k ) } \\big ( u \\vert q , s \\big ) = \\prod \\limits _ { b \\geq a \\geq 1 } \\ \\big ( 1 - u q ^ { - \\mu _ a + \\lambda _ { b + 1 } } s ^ { - a + b } \\big ) ^ { \\delta _ { a - b - k \\vert N } } \\ \\big ( 1 - u q ^ { \\lambda _ { a } - \\mu _ b } s ^ { a - b - 1 } \\big ) ^ { \\delta _ { a - b + k + 1 \\vert N } } \\end{align*}"}
+{"id": "450.png", "formula": "\\begin{align*} & - \\int _ 0 ^ \\infty d r \\ , r ^ { d - 1 - 2 \\sigma - 1 } \\overline { u ( r ) } u ' ( r ) \\\\ & = \\int _ 0 ^ \\infty d r \\ , r ^ { d - 2 - 2 \\sigma } \\overline { u ' ( r ) } u ( r ) + ( d - 2 - 2 \\sigma ) \\int _ 0 ^ \\infty d r \\ , r ^ { d - 3 - 2 \\sigma } | u ( r ) | ^ 2 \\end{align*}"}
+{"id": "151.png", "formula": "\\begin{align*} K _ { E ' } a _ i ' \\pi _ \\mu ' a _ j '^ { - 1 } K _ { E ' } \\cap G _ { F ' } = \\coprod _ { \\nu \\in X _ * ( \\textbf { T } ) ^ { - } } \\coprod _ { ( d _ i ' , d _ j ' ) \\in T _ \\nu ( F ' ) } K _ { E ' } a _ i ' \\pi ' _ \\mu a _ j '^ { - 1 } K _ { E ' } \\cap K _ { F ' } d _ i ' \\varpi ' _ \\nu d _ j '^ { - 1 } K _ { F ' } . \\end{align*}"}
+{"id": "6626.png", "formula": "\\begin{align*} p _ j ( u ) = \\left \\{ \\begin{array} { l l } ( 1 - u ) ^ 2 \\sum _ { l = 0 } ^ { j - 2 } b _ { j , l } u ^ l ( b _ { j , 0 } = 1 , \\ , b _ { j , l } = b _ { j , j - 2 - l } ) \\ : j \\ , { \\rm e v e n } \\\\ ( 1 - u ) \\sum _ { l = 0 } ^ { j - 1 } b _ { j , l } u ^ l ( b _ { j , 0 } = 1 , \\ , b _ { j , l } = b _ { j , j - 1 - l } ) \\ : j \\ , { \\rm o d d . } \\end{array} \\right . \\end{align*}"}
+{"id": "2801.png", "formula": "\\begin{align*} c _ { N , s } : = \\frac { s 2 ^ { 2 s } \\Gamma ( \\frac { N + 2 s } { 2 } ) } { \\pi ^ { N / 2 } \\Gamma ( 1 - s ) } \\end{align*}"}
+{"id": "1069.png", "formula": "\\begin{align*} \\widetilde { f } _ { 0 } \\left ( x , t \\right ) = \\widetilde { g } _ { 0 } \\left ( x , t \\right ) = 0 \\left ( x , t \\right ) \\in S _ { T } \\diagdown \\left ( \\Gamma _ { T } ^ { - } \\cup \\Gamma _ { T } ^ { + } \\right ) . \\end{align*}"}
+{"id": "1983.png", "formula": "\\begin{align*} \\sum _ { \\sigma = 0 } ^ { p ^ { k } - 1 } \\omega _ q ^ { \\left ( a ^ { t _ 1 } _ { \\sigma , r } - a ^ { t _ 2 } _ { \\sigma , s } \\right ) } = 0 . \\end{align*}"}
+{"id": "2215.png", "formula": "\\begin{align*} \\left ( I _ { b - } ^ \\alpha f \\right ) ( x ) : = \\frac { 1 } { \\Gamma ( \\alpha ) } \\underset { x } { \\overset { b } { \\int } } \\frac { f ( t ) d t } { ( t - x ) ^ { 1 - \\alpha } } , \\ , ( x < b ; \\ , R e ( \\alpha > 0 ) ) \\end{align*}"}
+{"id": "966.png", "formula": "\\begin{align*} \\operatorname { s u p p } ( q ) & = \\{ q \\} & & , \\\\ \\operatorname { s u p p } ( a ) & = \\textstyle \\bigcup \\{ \\operatorname { s u p p } ( x ) \\ , | \\ , x \\in a \\} & & , \\end{align*}"}
+{"id": "3664.png", "formula": "\\begin{align*} | \\nabla u | ^ 2 = g ( \\nabla u , \\nabla u ) = g ^ { - 1 } ( d u , d u ) \\ , , \\end{align*}"}
+{"id": "3180.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{bmatrix} N & M \\\\ I _ n & S \\end{bmatrix} & = \\begin{bmatrix} N & M \\\\ I _ n & S \\end{bmatrix} \\begin{bmatrix} - S & 0 \\\\ I _ n & I _ n \\end{bmatrix} \\begin{bmatrix} - S ^ { \\mathsf { T } } & 0 \\\\ S ^ { \\mathsf { T } } & I _ n \\end{bmatrix} , \\\\ & = \\begin{bmatrix} M - N S & M \\\\ 0 & S \\end{bmatrix} \\begin{bmatrix} - S ^ { \\mathsf { T } } & 0 \\\\ S ^ { \\mathsf { T } } & I _ n \\end{bmatrix} , \\\\ \\end{aligned} \\end{align*}"}
+{"id": "649.png", "formula": "\\begin{align*} \\mathcal L _ { \\mathrm { n o i s e } } ( \\psi , \\phi , x , t ) = \\mathcal L _ { \\mathrm { D i r a c } } ^ { \\mathrm { n o i s e } } ( \\psi , x , t ) + \\mathcal L _ { \\mathrm { m e s o n } } ^ { \\mathrm { n o i s e } } ( \\phi , x , t ) = \\psi ^ * \\beta \\psi \\xi _ 1 + \\frac 1 2 \\phi ^ 2 \\xi _ 2 . \\end{align*}"}
+{"id": "4132.png", "formula": "\\begin{align*} I ( k , s , S _ k ) G ( k , s , S _ k ) \\sum _ { r = 1 } ^ { s } \\sum _ { \\ell = 0 } ^ { k - 2 r - 2 s + 2 } V ( k , s , r , \\ell ) = 0 . \\end{align*}"}
+{"id": "3994.png", "formula": "\\begin{align*} \\int _ { M } \\exp \\left ( \\frac { ( n + 1 ) \\alpha _ 0 } { n } \\psi _ { t , k } \\right ) \\omega ^ n \\leq C , \\end{align*}"}
+{"id": "2897.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in R _ 0 ^ + } t _ { \\alpha } \\langle \\beta , { \\alpha ^ \\vee } \\rangle \\langle \\alpha , \\beta ^ \\vee \\rangle = \\frac { 2 } { n } \\sum _ { \\alpha \\in R _ 0 } t _ { \\alpha } \\end{align*}"}
+{"id": "6169.png", "formula": "\\begin{align*} ( \\mathbf { k } ) = \\prod _ { 1 \\leq p < q \\leq n } ( E _ { k _ p } + E _ { k _ q } ^ { - 1 } - E _ { k _ p } E _ { k _ q } ^ { - 1 } ) \\prod _ { 1 \\leq i < j \\leq n } \\frac { k _ j - k _ i } { j - i } , \\end{align*}"}
+{"id": "8244.png", "formula": "\\begin{align*} 0 = L _ { \\xi _ i } ( g _ 0 ( \\tilde { X } , T ) ) = ( L _ { \\xi _ i } g _ 0 ) ( \\tilde { X } , T ) + g _ 0 ( [ \\xi _ i , \\tilde { X } ] , T ) + g _ 0 ( \\tilde { X } , [ \\xi _ i , T ] ) . \\end{align*}"}
+{"id": "8891.png", "formula": "\\begin{align*} E = P \\times _ G F \\simeq { \\underline { h } } ^ * ( E G ) \\times _ V F \\simeq \\underline { h } ^ * ( E G \\times _ G F ) = \\underline { h } ^ * F _ G . \\end{align*}"}
+{"id": "7429.png", "formula": "\\begin{align*} \\rho _ { \\mathcal { H } } ( y ) : = \\int _ { \\Omega } \\mathcal { H } ( z , \\vert y ( z ) \\vert ) d z = \\int _ { \\Omega } ( \\vert y ( z ) \\vert ^ p + \\eta ( z ) \\vert y ( z ) \\vert ^ q ) d z . \\end{align*}"}
+{"id": "8713.png", "formula": "\\begin{align*} \\tau _ { t } \\leq \\tau _ { t _ { 0 } } \\prod _ { i = { t _ { 0 } } } ^ { t - 1 } \\Big ( 1 - \\frac { 2 } { i } \\Big ) \\leq \\tau _ { t _ { 0 } } \\prod _ { i = { t _ { 0 } } } ^ { t - 1 } \\Big ( 1 - \\frac { 1 } { i } \\Big ) \\le \\frac { ( t _ { 0 } - 1 ) \\tau _ { t _ { 0 } } } { t } \\le \\frac { 2 ( t _ { 0 } - 1 ) \\delta _ { t _ { 0 } } } { t } \\enspace . \\end{align*}"}
+{"id": "8205.png", "formula": "\\begin{align*} g _ a & = \\bar { g } + ( V + a ^ 2 ) \\sum _ { i = 1 } ^ 3 ( d x _ i ) ^ 2 + \\frac { 1 } { V + a ^ 2 } \\eta ^ 2 , \\\\ I _ i ^ a & = I _ i - \\frac { a ^ 2 V } { a ^ 2 + V } \\theta \\otimes I _ i T + a ^ 2 d x _ i \\otimes T , \\\\ \\omega _ i ^ a & = \\omega _ i + a ^ 2 d x _ j \\wedge d x _ k , \\end{align*}"}
+{"id": "7553.png", "formula": "\\begin{align*} \\Gamma _ { 2 \\hbar } ( z + 2 \\hbar ) = \\frac { z } { 2 \\hbar } \\Gamma _ { 2 \\hbar } ( z ) , \\end{align*}"}
+{"id": "3769.png", "formula": "\\begin{align*} H ( x _ 0 , s _ 0 , x _ 1 , s _ 1 ) = s _ 0 + s _ 1 - 2 \\sqrt { s _ 0 s _ 1 } \\exp \\left ( \\frac { - c ( x _ 0 , x _ 1 ) } { 2 } \\right ) , \\end{align*}"}
+{"id": "7615.png", "formula": "\\begin{align*} \\alpha _ { i , j } = \\begin{cases} \\pm M ( ( q _ i ^ 1 ) _ { \\pm } ) & \\ ; j = j _ { \\pm } ( i ) \\ ; \\ ; j _ { - } ( i ) \\not = j _ { + } ( i ) ; \\\\ M ( ( q _ i ^ 1 ) _ { + 1 } ) - M ( ( q _ i ^ 1 ) _ { - 1 } ) & \\ ; j = j _ { - } ( i ) = j _ { + } ( i ) ; \\\\ 0 & \\ ; j \\notin \\{ j _ { - } ( i ) , j _ { + } ( i ) \\} . \\end{cases} \\end{align*}"}
+{"id": "6794.png", "formula": "\\begin{align*} & \\underline { u } ' ( z _ 1 ^ - ) = - \\beta < 0 = \\underline { u } ' ( z _ 1 ^ + ) , \\\\ [ 0 . 2 c m ] & \\underline { v } ' ( z _ 2 ^ - ) = - q ( 1 ) h e ^ { \\lambda z _ 2 } / 2 < 0 = \\underline { v } ' ( z _ 2 ^ + ) , \\\\ [ 0 . 2 c m ] & \\overline { v } ' ( ( - 2 / \\lambda ) ^ + ) = 0 < q ( 1 ) h e ^ { - 2 } = \\overline { v } ' ( ( - 2 / \\lambda ) ^ - ) . \\end{align*}"}
+{"id": "3151.png", "formula": "\\begin{align*} \\frac { r ! } { r _ 1 ! \\dotsi r _ N ! } \\tau ^ D _ { \\alpha } ( L ) + r ! \\langle \\psi _ { r - 2 } \\emph { } \\rangle _ { \\alpha } = W _ L ^ r [ \\alpha ] . \\end{align*}"}
+{"id": "66.png", "formula": "\\begin{align*} \\mathcal Q _ 3 ^ { \\rm { l o w } } = \\frac { 1 } { \\vert \\Lambda \\vert } \\sum _ { p \\in \\mathcal P _ L , k \\neq 0 } \\widehat g ( k ) \\big ( a _ 0 ^ \\dagger a _ p ^ \\dagger a _ { p - k } a _ k + h . c . \\big ) . \\end{align*}"}
+{"id": "6047.png", "formula": "\\begin{align*} \\overline { \\mu } \\{ \\underline { c } > \\frac { 1 } { u } \\} = \\overline { \\mu } \\{ \\vert z \\vert < ( a _ 2 u - 1 ) ^ { \\frac { 1 } { p } } \\} \\geq \\frac { r _ d } { 2 } ( a _ 2 ( u - 1 ) - 1 ) ^ { \\frac { d } { p } } , \\end{align*}"}
+{"id": "7446.png", "formula": "\\begin{align*} C _ { 1 8 } \\leq \\lambda C _ { 1 9 } \\frac { \\| { y _ 1 } _ n \\| _ { m _ 1 } ^ { \\kappa _ 1 + 1 } \\| { y _ 2 } _ n \\| _ { m _ 2 } ^ { \\kappa _ 2 + 1 } } { \\| { y _ 1 } _ n \\| _ { m _ 1 } ^ p + \\| { y _ 2 } _ n \\| _ { m _ 2 } ^ p } . \\end{align*}"}
+{"id": "8658.png", "formula": "\\begin{align*} \\hat { g } ( \\vec { y } ) = g ( \\vec { x } ) \\cdot \\det \\Bigg ( - \\frac { \\partial ^ 2 f } { \\partial x _ i \\partial x _ j } \\Bigg ) _ { i , j } ^ { - \\frac { 1 } { 2 } } \\Bigg \\rvert _ { \\vec { x } = \\vec { t } } . \\end{align*}"}
+{"id": "5347.png", "formula": "\\begin{align*} \\binom { k + c } { c + 1 } = h ^ 0 ( \\O _ { \\P ^ { c + 1 } } ( k - 1 ) ) \\le h ^ 0 ( \\O _ C ( k - 1 ) ) = d ( k - 1 ) - g + 1 . \\end{align*}"}
+{"id": "1423.png", "formula": "\\begin{align*} q ( x ) & = - \\frac { 1 } { \\pi i } \\oint _ { \\Gamma _ N } { \\Bigg ( \\tilde \\varphi ' ( x , \\mu ) \\varphi ( x , \\mu ) + \\tilde \\varphi ( x , \\mu ) \\varphi ' ( x , \\mu ) \\Bigg ) \\hat M ( \\mu ) } d \\mu \\\\ & - 2 \\sum _ { k = N + 1 } ^ \\infty \\sum _ { j = 0 } ^ 1 ( - 1 ) ^ j \\alpha _ { k j } ( \\tilde \\varphi ' _ { k , j } ( x ) \\varphi _ { k , j } ( x ) + \\tilde \\varphi _ { k , j } ( x ) \\varphi ' _ { k , j } ( x ) ) . \\end{align*}"}
+{"id": "6565.png", "formula": "\\begin{align*} A _ k = R _ { \\Lambda ' } ( { D } ( \\sigma ) + \\varepsilon \\Delta ) R _ { \\Lambda ' } . \\end{align*}"}
+{"id": "8725.png", "formula": "\\begin{align*} \\frac { \\Gamma ( d + 1 ) } { \\Gamma ( d + \\beta ) } & = \\frac { \\Gamma ( d + 1 ) } { \\Gamma \\big ( d + \\underbrace { ( \\beta - \\ell ) } _ { \\in ( 0 , 1 ] } \\big ) \\prod _ { i = 1 } ^ { \\ell } \\big ( d + \\beta - i \\big ) } \\leq \\frac { ( d + \\beta - \\ell ) ^ { 1 - ( \\beta - \\ell ) } } { \\prod _ { i = 1 } ^ { \\ell } \\big ( d + \\beta - i \\big ) } \\leq \\frac { 1 } { d ^ { \\beta - 1 } } \\enspace , \\end{align*}"}
+{"id": "989.png", "formula": "\\begin{align*} A ^ 2 + Z ^ 2 = d A ' { } ^ 2 , B ^ 2 + Z ^ 2 = d B ' { } ^ 2 , \\end{align*}"}
+{"id": "5019.png", "formula": "\\begin{align*} R + | \\nabla f | ^ 2 = - f + 1 0 . \\end{align*}"}
+{"id": "5337.png", "formula": "\\begin{align*} \\frac { 2 ^ { n + 1 } } { \\sin ( r ) r ^ n } & = \\frac { 2 ^ { n + 1 } } { \\frac { 1 } { 2 } \\left ( \\frac { 5 \\pi } { 6 } \\right ) ^ n } \\\\ & = 4 \\left ( \\frac { 1 2 } { 5 \\pi } \\right ) ^ n . \\end{align*}"}
+{"id": "2542.png", "formula": "\\begin{align*} | & f ( x ) - P _ n ( x ) | \\\\ & \\le c ( k , r ) A \\begin{cases} | x - x _ 0 | ^ { m + 1 } \\rho _ n ^ { r - m - 1 } ( x ) \\omega _ \\ell ( f ^ { ( r ) } , \\rho _ n ( x ) ) , & \\ ; m \\le r - 1 , \\\\ | x - x _ 0 | ^ { r } \\omega _ \\ell ( f ^ { ( r ) } , | x - x _ 0 | ^ { 1 / \\ell } \\rho _ n ^ { 1 - 1 / \\ell } ( x ) ) , & \\ ; m = r . \\end{cases} \\end{align*}"}
+{"id": "2429.png", "formula": "\\begin{align*} \\theta _ { \\Gamma } ( c ) = \\theta _ { \\Gamma } ( c ^ { - 1 } c _ \\gamma ) = \\xi ^ { - 1 } _ { \\Gamma , c _ \\gamma } ( c ^ { - 1 } ) \\theta _ { \\Gamma } ( c ^ { - 1 } ) \\end{align*}"}
+{"id": "2650.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\limsup _ { n \\to \\infty } P ( w _ L ' ( X _ n , \\delta ) > \\epsilon ) ^ { 1 / r _ n } = 0 \\ , . \\end{align*}"}
+{"id": "4776.png", "formula": "\\begin{align*} D = C \\cap \\{ x \\mid a ^ { * } ( x ) \\leq M m ^ { * } ( x ) \\leq M \\} \\end{align*}"}
+{"id": "5630.png", "formula": "\\begin{align*} \\ \\ \\ D _ { \\binom { 1 } { 2 } } ^ a = \\Bigl ( x _ 0 x _ 3 ^ 2 + B x _ 3 x + x _ 1 C x ^ 2 = 0 \\Bigr ) . \\end{align*}"}
+{"id": "7717.png", "formula": "\\begin{align*} \\widetilde { K } : = ( \\pi '' ) ^ { - 1 } ( \\phi ( K _ 2 ) ) \\subseteq \\mathbb { A } _ E | _ N . \\end{align*}"}
+{"id": "7761.png", "formula": "\\begin{align*} \\overline { N } : = \\{ p \\in P | _ { N ' } \\quad | ( Z ^ 0 ) = 0 , \\} \\ , . \\end{align*}"}
+{"id": "4507.png", "formula": "\\begin{align*} [ \\partial _ t , S _ { \\theta _ i } ] \\psi _ i = \\partial _ t ( S _ { \\theta _ i } - I ) \\psi _ i + ( I - S _ { \\theta _ i } ) \\partial _ t \\psi _ i \\ , , \\end{align*}"}
+{"id": "8308.png", "formula": "\\begin{align*} & u ( h ( Z ) ) = h ( u ( Z ) ) ~ Z \\in \\mathcal { Z } \\\\ \\Rightarrow & u \\circ h - h \\circ u = 0 ~ ~ \\mathcal { Z } . \\end{align*}"}
+{"id": "6196.png", "formula": "\\begin{align*} \\{ a \\cdot b , c \\} = ( - 1 ) ^ { | b | | c | } \\{ a , c \\} \\cdot b + a \\cdot \\{ b , c \\} . \\end{align*}"}
+{"id": "7879.png", "formula": "\\begin{align*} u \\mapsto C _ 0 ^ { t ' + u , t '' + u } ( \\mu _ u ( t ' + u ) , \\mu _ u ( t '' + u ) ) = C _ 0 ^ { t ' + u , t '' + u } ( \\mu ' ( t ' + u ) , \\mu '' ( t '' + u ) ) \\end{align*}"}
+{"id": "3527.png", "formula": "\\begin{align*} \\mathrm { t r } ( \\gamma ) = a + d = ( a ' + d ' ) q \\pm 2 = \\pm 2 \\mod q ^ 2 \\end{align*}"}
+{"id": "5541.png", "formula": "\\begin{align*} \\gamma ( t ) = \\Im \\eta ( t ) , t \\in J . \\end{align*}"}
+{"id": "8874.png", "formula": "\\begin{align*} y _ i = c _ 1 ( L _ { { \\chi } _ i } ) \\in H ^ 2 ( G / T ) , u _ i = c _ 1 ( S _ { \\chi _ i } ) \\in H ^ 2 ( B T ) . \\end{align*}"}
+{"id": "6659.png", "formula": "\\begin{align*} h _ { 2 j } ( \\beta , p , q ) \\Big | _ { \\beta = 2 \\atop q = 0 } = c _ { 2 j } \\ : \\ : ( j \\ge 0 ) , h _ { 2 j - 1 } ( \\beta , p , q ) \\Big | _ { \\beta = 2 \\atop q = 0 } = \\tilde { h } _ { j } ( \\beta , p , q ) \\Big | _ { \\beta = 2 \\atop q = 0 } = 0 \\ : \\ : ( j \\ge 1 ) . \\end{align*}"}
+{"id": "2375.png", "formula": "\\begin{align*} \\min _ { u } \\left \\{ { \\mathcal { A } } V ( x , t ) + C ( t , x , u ) \\right \\} = 0 \\end{align*}"}
+{"id": "2598.png", "formula": "\\begin{align*} \\alpha = \\varprojlim _ n \\alpha _ n \\end{align*}"}
+{"id": "2219.png", "formula": "\\begin{align*} \\left ( D _ { b - } ^ \\alpha y \\right ) ( x ) = - \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\frac { d } { d x } \\underset { x } { \\overset { b } { \\int } } \\frac { y ( t ) d t } { ( t - x ) ^ \\alpha } , \\ , ( x < b ) \\end{align*}"}
+{"id": "5543.png", "formula": "\\begin{align*} F ( \\zeta ) = \\zeta - \\i f ( \\zeta ) , \\zeta \\in G _ m \\cup \\Gamma . \\end{align*}"}
+{"id": "172.png", "formula": "\\begin{align*} \\theta ( v , \\tau + 1 ) = e ^ { \\frac { \\pi \\sqrt { - 1 } } { 4 } } \\theta ( v , \\tau ) , ~ ~ \\theta ( v , - \\frac { 1 } { \\tau } ) = \\frac { 1 } { \\sqrt { - 1 } } \\left ( \\frac { \\tau } { \\sqrt { - 1 } } \\right ) ^ { \\frac { 1 } { 2 } } e ^ { \\pi \\sqrt { - 1 } \\tau v ^ 2 } \\theta ( \\tau v , \\tau ) ; \\end{align*}"}
+{"id": "724.png", "formula": "\\begin{align*} \\Delta _ 1 ( t ) & = i \\int _ { T \\wedge \\mu } ^ { t \\wedge \\mu } \\mathbf S ( t - s ) \\left [ \\mathbf N ( \\mathbf U ( s ) ) - \\mathbf N ( \\mathbf V ( s ) ) \\right ] \\ , d s , \\\\ \\Delta _ 2 ( t ) & = i \\int _ T ^ t \\mathbf S ( t - s ) \\left [ \\mathbf M ( \\mathbf U ( s ) ) - \\mathbf M ( \\mathbf V ( s ) ) \\right ] \\ , d W ( s ) . \\end{align*}"}
+{"id": "7922.png", "formula": "\\begin{align*} 2 & = a _ x + a _ y \\pm ( b _ x + b _ y ) \\sqrt { n } . \\end{align*}"}
+{"id": "301.png", "formula": "\\begin{align*} \\mbox { \\boldmath $ \\gamma $ } _ { \\operatorname * { C } ; j } ^ { \\mathbb { B } } : H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } , \\mathbb { B } \\right ) \\rightarrow H ^ { 1 / 2 } \\left ( \\Gamma _ { j } \\right ) \\times H ^ { - 1 / 2 } \\left ( \\Gamma _ { j } \\right ) , \\quad \\mbox { \\boldmath $ \\gamma $ } _ { \\operatorname * { C } ; j } ^ { \\mathbb { B } } u : = \\left ( \\gamma _ { \\operatorname * { D } ; j } u , \\gamma _ { \\operatorname * { N } ; j } ^ { \\mathbb { B } } u \\right ) . \\end{align*}"}
+{"id": "3999.png", "formula": "\\begin{align*} 1 = \\frac { c } { C ^ m _ n } S _ { n - m } ( X ^ { - 1 } ) + b _ t f S _ n ( X ^ { - 1 } ) , \\end{align*}"}
+{"id": "5310.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n _ 0 } { N \\choose j } & \\leq 1 + N ^ { n _ 0 - 1 } + { N \\choose n _ 0 } \\\\ & \\leq 1 + N ^ { n _ 0 - 1 } + N ^ { n _ 0 - 1 } ( N - 1 ) \\\\ & = 1 + N ^ { n _ 0 } . \\end{align*}"}
+{"id": "7652.png", "formula": "\\begin{align*} | w | ( X ^ w + Y ^ \\mu ) = \\langle w , X ^ \\mu \\rangle + \\langle w , \\frac { w } { | w | } Y ^ \\mu \\rangle | \\frac { w } { | w | } Y ^ \\mu | = | Y ^ \\mu | . \\end{align*}"}
+{"id": "346.png", "formula": "\\begin{align*} u _ { j } : = \\left . \\left ( \\mathsf { S } _ { j } \\left ( s \\right ) u _ { \\operatorname * { N } ; j } ^ { \\operatorname * { m u l t } } - \\mathsf { D } _ { j } \\left ( s \\right ) u _ { \\operatorname * { D } ; j } ^ { \\operatorname * { m u l t } } \\right ) \\right \\vert _ { \\Omega _ { j } ^ { - } } , \\quad 1 \\leq j \\leq n _ { \\Omega } . \\end{align*}"}
+{"id": "1367.png", "formula": "\\begin{align*} f ( z ) \\big | V _ m : = f ( m z ) = m ^ { - \\kappa } f ( z ) \\Big | _ { 2 \\kappa } \\begin{pmatrix} m & 0 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "7899.png", "formula": "\\begin{align*} \\begin{cases} \\det D ^ 2 v _ { s } = ( \\| u _ { s } \\| _ { L ^ { \\infty } } ^ { - 1 } - s v _ { s } ) ^ { n + k } ( u _ { s } ^ { \\star } ) ^ { - k } & \\Omega \\\\ v _ { s } = 0 & \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "3222.png", "formula": "\\begin{align*} \\textnormal { S e c t } _ { \\alpha , \\epsilon } ( R ) & : = \\{ ( x , y ) \\in \\R _ { > 0 } \\times \\R : x ^ { 2 } + y ^ { 2 } \\leq R ^ { 2 } , y / x \\in I _ { \\epsilon } ( \\alpha ) \\} , \\end{align*}"}
+{"id": "4216.png", "formula": "\\begin{align*} F = \\begin{pmatrix} \\lambda & A & B \\\\ A & \\mu & C \\\\ B & C & \\tau \\end{pmatrix} , \\end{align*}"}
+{"id": "7657.png", "formula": "\\begin{align*} Z ( \\mathbf { t } ) Z ( \\mathbf { s } ) = \\mathop { { \\rm R e s } } _ { z = 0 } \\ , \\frac { \\dd z } { z } \\ , e ^ { \\sum _ { k \\geq 0 } z ^ { 2 k + 1 } ( t _ { 2 k + 1 } - s _ { 2 k + 1 } ) } Z \\big ( \\mathbf { t } - 2 [ z ^ { - 1 } ] \\big ) \\ , Z \\big ( \\mathbf { s } + 2 [ z ^ { - 1 } ] \\big ) , \\end{align*}"}
+{"id": "7725.png", "formula": "\\begin{align*} K \\cap T ^ * M = \\mathrm { A n n } ( \\rho ( K ^ \\perp ) ) = \\mathrm { A n n } ( T N ) , \\end{align*}"}
+{"id": "1972.png", "formula": "\\begin{align*} \\omega _ q ^ { \\left ( a ^ t _ { \\sigma , r } - a ^ t _ { \\sigma , s } \\right ) } + \\sum _ { j = 1 } ^ { p - 1 } \\omega _ q ^ { \\left ( a ^ t _ { \\sigma ^ j , r } - a ^ t _ { \\sigma ^ j , s } \\right ) } = 0 . \\end{align*}"}
+{"id": "3342.png", "formula": "\\begin{align*} a _ \\zeta ( \\theta ) = \\sum _ { k \\in ( \\mathbb { Z } / m \\mathbb { Z } ) ^ N } \\zeta ^ { \\overline { Q ( k ) } } \\prod _ { i = 1 } ^ { N } \\frac { \\theta _ i ^ { ( k ^ \\mathsf { T } A ) _ i } } { ( \\theta _ i ; \\zeta _ i ) _ { k _ i + 1 } } . \\end{align*}"}
+{"id": "618.png", "formula": "\\begin{align*} \\theta ( x ) - \\theta ( y ) = \\left ( \\int _ 0 ^ 1 \\theta ' \\left ( \\kappa ( t ) \\right ) \\ , d t \\right ) ( x - y ) = : I _ 1 ( x - y ) \\end{align*}"}
+{"id": "3024.png", "formula": "\\begin{align*} g _ n & = \\lambda g _ { n - 1 } - g _ { n - 2 } \\\\ [ . 2 e m ] & = \\lambda \\left ( \\phi ( P _ { l - 1 } ) g _ { n - l } - \\phi ( P _ { l - 2 } ) g _ { n - l - 1 } \\right ) - \\left ( \\phi ( P _ { l - 2 } ) g _ { n - l } - \\phi ( P _ { l - 3 } ) g _ { n - l - 1 } \\right ) \\\\ [ . 2 e m ] & = \\left ( \\lambda \\phi ( P _ { l - 1 } ) - \\phi ( P _ { l - 2 } ) \\right ) g _ { n - l } - \\left ( \\lambda \\phi ( P _ { l - 2 } ) - \\phi ( P _ { l - 3 } ) \\right ) g _ { n - l - 1 } \\\\ [ . 2 e m ] & = \\phi ( P _ l ) g _ { n - l } - \\phi ( P _ { l - 1 } ) g _ { n - l - 1 } , \\end{align*}"}
+{"id": "7369.png", "formula": "\\begin{align*} \\partial _ t u = \\Delta u + \\vec b \\cdot \\vec \\nabla u + c u + f \\end{align*}"}
+{"id": "2361.png", "formula": "\\begin{align*} x = \\sum _ { j = 1 } ^ { n } f _ j ( x ) \\omega _ j = \\sum _ { j = 1 } ^ { n } g _ j ( x ) \\tau _ j , \\forall x \\in \\mathbb { C } ^ { o ( G ) } . \\end{align*}"}
+{"id": "7096.png", "formula": "\\begin{align*} \\Omega = \\ , \\displaystyle \\sum _ { q _ 2 \\sim C / q _ 1 } \\ , \\displaystyle \\sum _ { q ' _ 2 \\sim C / q _ 1 } \\ , \\ , \\sum _ { m \\sim M _ 1 } \\ , \\sum _ { m _ 1 \\sim M _ 1 } \\frac { \\lambda _ f ( m ) \\ , \\lambda _ f ( m _ 1 ) } { ( m m _ 1 ) ^ { 1 / 4 } } \\ , \\mathcal { H } ( . . . ) , \\end{align*}"}
+{"id": "8693.png", "formula": "\\begin{align*} I ( \\vec { y } ) = \\int x _ 1 ^ { b _ 1 } . . . x _ n ^ { b _ n } e ^ { \\frac { 1 } { \\hbar } ( x _ 1 ^ { a _ 1 } . . . x _ n ^ { a _ n } + x _ 1 y _ 1 + . . . + x _ n y _ n ) } . \\end{align*}"}
+{"id": "7366.png", "formula": "\\begin{align*} g ( u ) = u ^ 4 - 2 \\sqrt { \\gamma } u ^ 3 + ( \\gamma - 1 ) u ^ 2 + 2 p \\sqrt { \\gamma } u - p \\gamma . \\end{align*}"}
+{"id": "3946.png", "formula": "\\begin{align*} x ' v & = x _ 0 v \\Leftrightarrow \\\\ x ' \\frac { ( 1 - t ) v _ 1 + t v _ 2 } { \\lVert ( 1 - t ) v _ 1 + t v _ 2 \\rVert } & = x _ 0 \\frac { ( 1 - t ) v _ 1 + t v _ 2 } { \\lVert ( 1 - t ) v _ 1 + t v _ 2 \\rVert } \\end{align*}"}
+{"id": "7880.png", "formula": "\\begin{align*} f ^ \\prime ( t ) = \\frac { ( \\tan t ) ^ \\prime \\cdot t - \\tan t } { t ^ 2 } = \\frac { \\frac { t } { \\cos ^ 2 t } - \\tan t } { t ^ 2 } = \\frac { t - \\sin t \\cos t } { t ^ 2 \\cos ^ 2 t } = \\frac { t - \\sin ( 2 t ) / 2 } { t ^ 2 \\cos ^ 2 t } . \\end{align*}"}
+{"id": "3041.png", "formula": "\\begin{align*} b ^ 2 = 2 A ^ 2 \\end{align*}"}
+{"id": "8616.png", "formula": "\\begin{align*} \\overline { V } _ { \\mathrm { a p p } } = \\int _ { - 1 + \\beta b } ^ { 0 } \\big { [ } \\frac { 1 } { h _ b } \\nabla _ X \\phi _ { \\mathrm { a p p } } - \\frac { 1 } { h } ( \\varepsilon \\nabla _ X \\Big ( \\dfrac { \\zeta } { h _ b } \\Big ) z + \\varepsilon \\nabla _ X \\zeta ) \\partial _ z \\phi _ { \\mathrm { a p p } } \\big { ] } \\ : \\mathrm { d } z . \\end{align*}"}
+{"id": "5273.png", "formula": "\\begin{align*} & \\Big ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { t } ( x _ { i , t } ) ] _ + \\| \\Big ) ^ 2 \\le \\varepsilon _ 3 T + \\frac { n \\varepsilon _ 4 K T } { \\gamma _ 0 } + \\frac { n \\varepsilon _ 4 ( F \\sigma + 2 G _ 1 ^ 2 ) } { \\sigma \\gamma _ { 0 } ( 1 - c ) } T ^ { 2 - c } . \\end{align*}"}
+{"id": "8044.png", "formula": "\\begin{align*} \\left \\| a \\right \\| _ { L ^ { q } } \\leq \\vert Q \\vert ^ { \\frac { 1 } { q } } \\omega ( Q ) ^ { - \\frac { 1 } { p } } { \\rm a n d } \\int _ { Q } { a ( x ) x ^ { \\alpha } } d x = 0 , \\ { \\rm f o r \\ a l l } \\ \\vert \\alpha \\vert \\leq s . \\end{align*}"}
+{"id": "1438.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ g \\tilde { u } _ i ^ k + \\sum \\limits _ { j \\in I } k _ { i j } \\rho ^ k _ { j } \\left ( \\frac { \\tilde h ^ k _ { j } e ^ { \\tilde { u } ^ k _ j } } { \\int _ { M } \\tilde { h } ^ k _ { j } e ^ { \\tilde { u } ^ k _ j } \\mathrm { d } V _ { g } } - \\dfrac { 1 } { | M | } \\right ) = 0 \\ \\ M , \\ \\ \\forall \\ i \\in I , \\\\ \\sum \\limits _ { i \\in I } \\tilde { u } ^ k _ i \\equiv 0 , \\end{cases} \\end{align*}"}
+{"id": "6523.png", "formula": "\\begin{align*} C _ { k , n , n ' } & = \\left \\{ \\begin{array} { l l } | ( k , 1 , - 1 ) | _ 2 & { \\rm i f } \\ n , n ' \\notin \\{ n ^ { ( l ) } \\} _ { 1 \\leq l \\leq b } , \\\\ | ( k + e _ { l ' } , - 1 ) | _ 2 & { \\rm i f } \\ n = n ^ { ( l ' ) } , \\ n ' \\notin \\{ n ^ { ( l ) } \\} _ { 1 \\leq l \\leq b } , \\ 1 \\leq l ' \\leq b , \\\\ | ( k - e _ { l ' } , 1 ) | _ 2 & { \\rm i f } \\ n \\notin \\{ n ^ { ( l ) } \\} _ { 1 \\leq l \\leq b } , \\ n ' = n ^ { ( l ' ) } , \\ 1 \\leq l ' \\leq b . \\end{array} \\right . \\end{align*}"}
+{"id": "7266.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } B ^ n ( f ) ( z , t ) = \\lim _ { s \\rightarrow 0 ^ + } \\sum _ { n \\geq 0 } B _ s ^ n ( f ) ( z , t ) . \\end{align*}"}
+{"id": "2481.png", "formula": "\\begin{align*} P _ { W ^ * | A = a } ( \\mathcal { S } ) = & \\sum _ { v _ a \\in \\mathcal { V } _ a } P _ { V _ a } ( v _ a ) P _ { W ^ * | A = a , V = v _ a } ( \\mathcal { S } ) \\\\ = & \\sum _ { v _ a \\in \\mathcal { V } _ a } P _ { V _ a } ( v _ a ) P _ { W ^ * _ a | V _ a = v _ a } ( \\mathcal { S } _ a ) \\prod _ { \\substack { a ' \\in \\mathcal { A } \\\\ a ' \\neq a } } P _ { W ^ * _ { a ' } } ( \\mathcal { S } _ { a ' } ) \\\\ = & \\prod _ { a ' \\in \\mathcal { A } } P _ { W ^ * _ { a ' } } ( \\mathcal { S } _ { a ' } ) . \\end{align*}"}
+{"id": "2516.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } \\boldsymbol { M _ h } & - \\boldsymbol { K _ h } \\\\ - \\boldsymbol { K _ h } & - \\alpha ^ { - 1 } \\boldsymbol { M _ h } \\end{array} \\right ) \\left ( \\begin{array} { c } \\underline { \\boldsymbol { y } } _ 0 ^ c \\\\ \\underline { \\boldsymbol { p } } _ 0 ^ c \\end{array} \\right ) = \\left ( \\begin{array} { c } { \\underline { \\boldsymbol { y } } _ d ^ c } _ 0 \\\\ 0 \\end{array} \\right ) . \\end{align*}"}
+{"id": "1028.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { i = 1 } ^ n \\lambda _ i x _ i \\mid n \\in \\omega , \\ ; \\lambda _ i \\in \\mathbb { Q } , \\ ; x _ i \\in D \\right \\} , \\end{align*}"}
+{"id": "6966.png", "formula": "\\begin{align*} T _ 3 = - \\sum _ { k = 1 } ^ d \\sum _ { i = k + 1 } ^ d \\sum _ { j = i + 1 } ^ d \\frac { ( X _ i - X _ k ) ( X _ j - X _ k ) ( X _ j - X _ i ) } { ( X _ i - X _ k + \\epsilon ^ 2 ) ( X _ j - X _ k + \\epsilon ^ 2 ) ( X _ j - X _ i + \\epsilon ^ 2 ) } . \\end{align*}"}
+{"id": "5018.png", "formula": "\\begin{align*} \\Phi _ { \\iota _ 1 , \\iota _ 2 } \\big ( [ ( g , V , \\gamma ) ] \\big ) = [ ( \\phi ^ * g , \\phi ^ * V , \\gamma ) ] \\end{align*}"}
+{"id": "1523.png", "formula": "\\begin{align*} \\frac 1 x \\sum _ { n \\le x } \\log P ^ { \\left ( \\frac 1 2 \\right ) } ( n ) = A ( \\log x ) ^ { \\varphi ' } \\left ( 1 + O \\left ( \\frac { ( \\log _ 3 x ) ^ { 3 / 2 } } { \\sqrt { \\log _ 2 x } } \\right ) \\right ) \\end{align*}"}
+{"id": "1147.png", "formula": "\\begin{align*} \\delta _ 1 \\circ \\delta _ 2 + \\delta _ 2 \\circ \\delta _ 1 = 0 . \\end{align*}"}
+{"id": "2422.png", "formula": "\\begin{align*} u _ { a , b } ( z ) = \\prod _ { h \\in H } \\frac { z - h a } { z - h b } \\end{align*}"}
+{"id": "4763.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n _ k } { f ( \\hat { T } ^ i ( b , x ) ) } \\geq - 1 . \\end{align*}"}
+{"id": "3329.png", "formula": "\\begin{align*} g ( \\tau ) = \\begin{pmatrix} q ^ { - 1 / 1 8 } f _ 1 ( q ) \\\\ q ^ { 5 / 1 8 } f _ 2 ( q ) \\\\ q ^ { 1 1 / 1 8 } f _ 3 ( q ) \\\\ \\end{pmatrix} \\end{align*}"}
+{"id": "1313.png", "formula": "\\begin{align*} h ( t ) \\le G ^ { - 1 } ( G ( 1 ) - C _ 1 t ) = G ^ { - 1 } ( - C _ 1 t ) , t \\ge 0 , \\end{align*}"}
+{"id": "1153.png", "formula": "\\begin{align*} & m _ { 1 , t } ( x , ( m _ { 1 , t } ( y , z ) ) = m _ { 1 , t } ( m _ { 1 , t } ( x , y ) , z ) - m _ { 1 , t } ( m _ { 1 , t } ( x , z ) , y ) , \\\\ & m _ { 2 , t } ( x , ( m _ { 2 , t } ( y , z ) ) = m _ { 2 , t } ( m _ { 2 , t } ( x , y ) , z ) - m _ { 2 , t } ( m _ { 2 , t } ( x , z ) , y ) . \\end{align*}"}
+{"id": "2636.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } P ( \\sup _ { s \\in [ 0 , t ] } \\abs { \\frac { 1 } { n } \\ , Q _ n ^ { ( 0 ) } ( s ) + \\frac { 1 } { n } \\ , A _ n ( s ) - \\frac { 1 } { n } \\ , \\int _ { 0 } ^ s A _ n ( s - x ) \\ , d F ( x ) - 1 } > \\epsilon ) ^ { 1 / b _ n ^ 2 } = 0 \\ , . \\end{align*}"}
+{"id": "248.png", "formula": "\\begin{align*} V ^ { \\emph { r } } _ { \\epsilon J } ( \\xi ; g ) & = \\prod _ { j \\in J } w ( \\epsilon _ j \\xi _ j ) \\prod _ { \\substack { j \\in J \\\\ k \\not \\in J } } v ( \\epsilon _ j \\xi _ j + \\xi _ k ) v ( \\epsilon _ j \\xi _ j - \\xi _ k ) \\\\ & \\times \\prod _ { \\substack { j , j ^ \\prime \\in J \\\\ j < j ^ \\prime } } v ( \\epsilon _ j \\xi _ j + \\epsilon _ { j ^ \\prime } \\xi _ { j ^ \\prime } ) v ( \\epsilon _ j \\xi _ j + \\epsilon _ { j ^ \\prime } \\xi _ { j ^ \\prime } + 1 ) , \\end{align*}"}
+{"id": "6266.png", "formula": "\\begin{align*} N ^ 1 _ \\lambda ( u ) = L ( \\partial _ x g ( u _ { < \\lambda } ) , u _ \\lambda ) = L ( h ( u _ { < \\lambda } ) , u _ { < \\lambda } , \\partial _ x u _ { < \\lambda } , u _ \\lambda ) , \\end{align*}"}
+{"id": "1747.png", "formula": "\\begin{align*} \\texttt { D } = 2 m + \\tilde { \\epsilon } _ + + \\tilde { \\epsilon } _ - - \\tilde { d } - 1 . \\end{align*}"}
+{"id": "147.png", "formula": "\\begin{align*} t _ { c _ i ' \\pi _ \\mu ' c _ j '^ { - 1 } } = t _ { \\sigma ' ( a _ i ' ) \\sigma ' ( \\pi _ \\mu ' ) \\sigma ' ( a _ j '^ { - 1 } ) } = \\sigma ' . t _ { a _ i ' \\pi _ \\mu ' a _ j '^ { - 1 } } . \\end{align*}"}
+{"id": "7573.png", "formula": "\\begin{align*} \\begin{aligned} \\log Q _ T \\big ( A ^ { ( < ) } _ { T , r _ 1 ( T ) } \\big ) & = \\log q _ T ^ { ( < ) } - \\log Z _ { N , T } \\\\ & \\leq - \\beta C \\frac { N ^ 2 T ^ 2 } { ( 1 + r _ 1 ( T ) ) ^ d } + f ( \\beta , N , T ) , \\end{aligned} \\end{align*}"}
+{"id": "2729.png", "formula": "\\begin{align*} \\mathcal { S } _ { \\alpha } : = \\Bigg \\{ \\bigcup _ { i = 1 } ^ { d } S ^ { i } _ { \\beta } \\colon d \\leq S ^ { 1 } _ { \\beta } < S ^ { 2 } _ { \\beta } < \\cdots < S ^ { d } _ { \\beta } , \\ , \\ , \\big \\{ S ^ { i } _ { \\beta } \\big \\} _ { i = 1 } ^ d \\subset \\mathcal { S } _ { \\beta } d \\in \\mathbb { N } \\Bigg \\} \\cup \\big \\{ \\emptyset \\big \\} ; \\end{align*}"}
+{"id": "4762.png", "formula": "\\begin{align*} \\pi _ b g \\nu = \\nu _ b . \\end{align*}"}
+{"id": "110.png", "formula": "\\begin{align*} \\mathcal T _ u : = Q _ { B _ u } \\chi _ { B _ u } \\Big ( - \\Delta _ { \\R ^ d } - s ^ { - 2 } \\ell _ { \\rm { G P } } ^ { - 2 } \\Big ) _ + \\chi _ { B _ u } Q _ { B _ u } + b \\ell ^ { - 2 } _ { \\rm { G P } } Q _ { B _ u } . \\end{align*}"}
+{"id": "782.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u + u = ( K \\ast F ( u ) ) F ' ( u ) \\mathbb { R } ^ N \\end{align*}"}
+{"id": "1536.png", "formula": "\\begin{align*} \\ll \\sum _ { \\substack { n \\le x : \\ , p \\parallel n \\\\ \\lambda \\le R _ p ( n ) < \\lambda + 1 / \\Omega ( n ) } } \\frac 1 { \\Omega ( n ) } \\ll \\sum _ { \\substack { n \\le x : \\ , p | n \\\\ \\Omega ( n ) \\le \\frac 1 3 \\log _ 2 x } } 1 + \\frac 1 { \\log _ 2 x } \\sum _ { \\substack { n \\le x : \\ , p \\parallel n \\\\ \\lambda \\le R _ p ( n ) < \\lambda + 1 / \\Omega ( n ) } } 1 . \\end{align*}"}
+{"id": "2523.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\boldsymbol { \\eta } } ( \\boldsymbol { v } ) = \\int _ { Q } \\Big ( ( \\boldsymbol { u } - \\sigma \\partial _ t \\boldsymbol { \\eta } - \\textbf { c u r l } \\ , \\boldsymbol { \\tau } ) \\cdot \\boldsymbol { v } + ( \\boldsymbol { \\tau } - \\nu \\ , \\textbf { c u r l } \\ , \\boldsymbol { \\eta } ) \\cdot \\textbf { c u r l } \\ , \\boldsymbol { v } \\Big ) \\ , d \\boldsymbol { x } \\ , d t . \\end{align*}"}
+{"id": "8335.png", "formula": "\\begin{align*} \\mathbf { r } ^ * ( 0 ) = \\mathbf { r } _ { \\mathbf { S B } } ^ * ( 0 ) \\mathbf { M } \\end{align*}"}
+{"id": "6392.png", "formula": "\\begin{align*} \\chi ( \\alpha \\otimes \\alpha \\delta - q \\beta \\gamma ) = 2 \\varepsilon ( \\alpha ^ 2 \\delta ) - 2 q \\varepsilon ( \\alpha \\beta \\gamma ) = 2 \\neq 0 = \\chi ( \\alpha \\otimes 1 ) . \\end{align*}"}
+{"id": "8178.png", "formula": "\\begin{align*} ( \\mu _ { i j } ^ \\star - \\mu _ { a , j \\pm 1 } ^ \\star ) ^ 2 = \\left ( \\frac { n ( n - 1 ) } { k ( n - k ) } \\right ) ^ 2 . \\end{align*}"}
+{"id": "1295.png", "formula": "\\begin{align*} \\| v _ n \\| _ { S ( \\R ) } \\leq C \\| \\phi \\| _ { \\dot { H } ^ 1 } \\ a n d \\ \\limsup _ { T \\to \\infty } \\lim _ { n \\to \\infty } \\| v _ n - v _ { n , T } \\| _ { S ( \\R ) } = 0 . \\end{align*}"}
+{"id": "3019.png", "formula": "\\begin{align*} \\vartheta ( T _ n ) = \\begin{cases} \\phantom { - } 1 , & \\mbox { i f { \\ ; } $ \\epsilon = 0 $ ; } \\\\ [ . 5 m m ] \\phantom { - } ( 2 s + 1 ) \\lambda , & \\mbox { i f { \\ ; } $ \\epsilon = 1 $ ; } \\\\ [ . 5 m m ] - 1 , \\phantom { n u l l n u l l } & \\mbox { i f { \\ ; } $ \\epsilon = 2 $ ; } \\\\ [ . 5 m m ] - 2 ( s + 1 ) \\lambda , & \\mbox { i f { \\ ; } $ \\epsilon = 3 $ ; } \\end{cases} \\end{align*}"}
+{"id": "7353.png", "formula": "\\begin{align*} { \\rm M M S E } ( \\gamma ) = \\frac { 1 } { \\gamma } \\Big ( 1 - \\Phi ( \\sqrt { \\gamma } S + Z ) \\Big ) \\end{align*}"}
+{"id": "2522.png", "formula": "\\begin{align*} \\int _ { Q } \\textbf { c u r l } \\ , \\boldsymbol { \\tau } \\cdot \\boldsymbol { v } \\ , d \\boldsymbol { x } \\ , d t = \\int _ { Q } \\boldsymbol { \\tau } \\cdot \\textbf { c u r l } \\ , \\boldsymbol { v } \\ , d \\boldsymbol { x } \\ , d t \\forall \\ , \\boldsymbol { v } \\in \\boldsymbol { H } ^ { \\textbf { c u r l } , 0 } _ 0 ( Q ) \\forall \\ , \\boldsymbol { \\tau } \\in \\boldsymbol { H } ^ { \\textbf { c u r l } , 0 } ( Q ) , \\end{align*}"}
+{"id": "6494.png", "formula": "\\begin{align*} 0 = \\left < \\Psi ' , P _ D \\Psi \\right > = \\left < P _ D \\Psi ' , \\Psi \\right > = \\left < \\Psi ' , \\Psi \\right > = \\left < \\Psi ' , A \\right > - \\frac 1 2 \\left < \\Psi ' , L \\Psi ' \\right > . \\end{align*}"}
+{"id": "1735.png", "formula": "\\begin{align*} \\rho _ \\epsilon ( \\boldsymbol { \\xi } ) : = \\prod _ { 1 \\leq j \\leq n } 2 ^ { \\epsilon _ + + \\epsilon _ - } & \\bigl ( 1 + \\epsilon _ + \\cos ( \\xi _ j ) \\bigr ) \\bigl ( 1 - \\epsilon _ - \\cos ( \\xi _ j ) \\bigr ) \\\\ & \\times \\prod _ { 1 \\leq j < k \\leq n } \\bigl ( \\cos ( \\xi _ j ) - \\cos ( \\xi _ k ) \\bigr ) ^ 2 , \\end{align*}"}
+{"id": "5073.png", "formula": "\\begin{align*} E ( A + B ) = \\sum _ { p \\geq 0 } \\sum _ { n = 0 } ^ p \\frac { p ! } { n ! ( p - n ) ! m ( p ) } A ^ n B ^ { p - n } \\end{align*}"}
+{"id": "2151.png", "formula": "\\begin{align*} \\beta _ x = \\tilde \\alpha _ 0 ' ( 2 u ) - \\tilde \\alpha _ 0 ' ( - 2 \\underline { u } ) + \\alpha _ 1 ( 2 u ) + \\alpha _ 1 ( - 2 \\underline { u } ) > 0 . \\end{align*}"}
+{"id": "6845.png", "formula": "\\begin{align*} \\Delta ^ 2 _ { \\theta } \\chi _ j ( \\theta ) = \\lambda _ j ^ 2 \\chi _ j ( \\theta ) { \\rm a n d } \\Delta ^ 3 _ { \\theta } \\chi _ j ( \\theta ) = - \\lambda _ j ^ 3 \\chi _ j ( \\theta ) . \\end{align*}"}
+{"id": "6199.png", "formula": "\\begin{align*} [ r , k ] _ 0 & = i d _ R , \\\\ [ r , k ] _ n ( x ) & = \\begin{cases} 0 , & k \\nmid n , \\\\ r ^ l x - r ^ { l - 1 } x r , & n = k l , \\end{cases} \\end{align*}"}
+{"id": "1786.png", "formula": "\\begin{gather*} \\zeta ( s , \\mathbb { X } _ \\alpha ) ( \\omega ) = \\lim _ { N \\to \\infty } \\zeta _ N ( s , \\mathbb { X } _ \\alpha ) ( \\omega ) \\end{gather*}"}
+{"id": "3365.png", "formula": "\\begin{align*} & \\ll X ^ { \\varepsilon } \\sum _ { \\substack { p \\leq z ^ { 1 / 2 } \\\\ p \\nmid q } } S ( p ^ { e _ 0 } , 1 , 1 , 1 ) \\ll X ^ { 1 + \\varepsilon } \\Big ( \\sum _ { p \\leq z ^ { 1 / 4 } } \\frac { 1 } { z ^ { 1 / 2 } } + \\sum _ { z ^ { 1 / 4 } \\leq p \\leq z ^ { 1 / 2 } } \\frac { 1 } { p ^ 2 } \\Big ) + z ^ { 1 / 2 } X ^ { \\varepsilon } Y \\\\ & \\ll \\frac { X ^ { 1 + \\varepsilon } } { z ^ { 1 / 4 } } + z ^ { 1 / 2 } X ^ { \\varepsilon } Y . \\end{align*}"}
+{"id": "401.png", "formula": "\\begin{align*} A ^ + = \\sum _ { \\varphi : \\Phi } A ^ { [ \\varphi ] } \\quad . \\end{align*}"}
+{"id": "7421.png", "formula": "\\begin{align*} ( 1 - \\rho ) \\pi = 0 \\mathbb { T } \\times [ 0 , T ] . \\end{align*}"}
+{"id": "7140.png", "formula": "\\begin{align*} V = V _ L + V _ S . \\end{align*}"}
+{"id": "7363.png", "formula": "\\begin{align*} { \\rm M M S E } ( \\gamma ) = \\frac { 1 } { \\gamma } \\Big ( 1 - \\frac { 8 \\pi ^ 2 } { 3 } \\int _ { L ( \\gamma ) } ^ { U ( \\gamma ) } \\rho _ Y ^ 3 ( x ) \\ , d x \\Big ) . \\end{align*}"}
+{"id": "5075.png", "formula": "\\begin{align*} \\frac { p ! } { n ! ( p - n ) ! } = \\frac { m ( p ) } { m ( n ) m ( p - n ) } , 0 \\le n \\le p , \\end{align*}"}
+{"id": "7807.png", "formula": "\\begin{align*} \\mathcal { Z } _ D : = \\mathcal { Z } - \\widetilde { N } \\times S ^ 1 , \\end{align*}"}
+{"id": "3416.png", "formula": "\\begin{align*} \\norm { \\cdot } _ { h y b } = \\max \\{ | \\cdot | _ 0 , | \\cdot | _ \\infty \\} , \\end{align*}"}
+{"id": "1978.png", "formula": "\\begin{align*} a ^ t _ { \\sigma , r ^ j } - a ^ t _ { \\sigma , s ^ j } - \\left ( a ^ t _ { \\sigma , r } - a ^ t _ { \\sigma , s } \\right ) = j \\frac { q } { p } \\left ( s _ { \\pi _ { \\hat { \\beta } } ( \\hat { \\gamma } ) } - r _ { \\pi _ { \\hat { \\beta } } ( \\hat { \\gamma } ) } \\right ) . \\end{align*}"}
+{"id": "4665.png", "formula": "\\begin{align*} s ^ { ( k + 1 ) } = ( M _ 1 + \\Omega _ { 1 } + I - L _ 1 ) ^ { - 1 } [ ( N _ { 1 } + I - L _ 1 ) s ^ { ( k ) } + ( \\Omega _ 1 - A _ { 1 } ) | s ^ { ( k ) } | - r q ] \\end{align*}"}
+{"id": "8593.png", "formula": "\\begin{align*} \\partial _ z \\phi _ 1 | _ { z = - h _ b } & = - \\mathcal { F } ^ { - 1 } \\Big { ( } \\frac { \\cosh ( \\frac { z } { h _ b ( X ) } \\sqrt { \\mu } | \\xi | ) } { \\cosh ( \\sqrt { \\mu } | \\xi | ) } \\hat { G } ( \\xi ) \\Big { ) } ( X ) | _ { z = - h _ b ( X ) } \\\\ & = - G ( X ) . \\end{align*}"}
+{"id": "7019.png", "formula": "\\begin{align*} V ( x _ 1 , \\ldots , x _ N ) = \\sum _ { 1 \\leq i < j \\leq N } \\frac { 1 } { | x _ i - x _ j | } . \\end{align*}"}
+{"id": "747.png", "formula": "\\begin{align*} A _ R ( t , \\omega ) = \\sup _ { 0 \\le s \\le t } \\norm { \\mathbb 1 _ { s \\le \\tau _ R ( \\omega ) } \\phi _ + ( s ) } _ { H ^ r } ^ 2 \\end{align*}"}
+{"id": "7276.png", "formula": "\\begin{align*} \\Pr \\Big ( \\frac { 1 } { l \\cdot k } \\sum _ { i = 0 } ^ { l \\cdot k - 1 } \\big ( X _ u [ i ] \\big ) ^ 2 > P \\Big ) \\le \\delta . \\end{align*}"}
+{"id": "4718.png", "formula": "\\begin{align*} \\Gamma _ { D _ 2 } ( \\lambda _ 1 \\lambda _ 2 \\dots \\lambda _ { n - 2 k } ) = G - H . \\end{align*}"}
+{"id": "7781.png", "formula": "\\begin{align*} \\widetilde { \\zeta } _ i ^ { } : = \\widetilde { \\zeta } _ i - \\widetilde { \\zeta } _ i ^ { } , \\sigma ^ { } : = \\sigma - \\sigma ^ { } \\ , , \\end{align*}"}
+{"id": "1591.png", "formula": "\\begin{align*} & \\int _ v ( 1 + v ^ 2 ) [ M _ T ( v ) - M _ { T ( x ) } ( v ) ] ^ 2 M _ T ^ { - 1 } d v = \\int _ v ( 1 + v ^ 2 ) \\left [ 1 - \\frac { M _ { T ( x ) } ( v ) } { M _ T ( v ) } \\right ] ^ 2 M _ T ( v ) \\mathrm { d } v \\\\ & \\sim \\int _ v ( 1 + v ^ 2 ) M _ T ( v ) \\left [ 1 - \\sqrt { \\frac { T } { T ( x ) } } e ^ { - \\frac { | v | ^ 2 } { 2 } ( T ( x ) ^ { - 1 } - T ^ { - 1 } ) } \\right ] ^ 2 \\mathrm { d } v . \\end{align*}"}
+{"id": "2771.png", "formula": "\\begin{align*} I ( G ; x ) = I ( G - v ; x ) + x \\cdot I ( G - N [ v ] ; x ) . \\end{align*}"}
+{"id": "5177.png", "formula": "\\begin{align*} f \\mapsto m _ d ( f ) \\colon = \\int _ G \\overbrace { g \\cdot f \\otimes \\ldots \\otimes g \\cdot f } ^ { } \\otimes \\overline { g \\cdot f } \\ ; d g \\end{align*}"}
+{"id": "4061.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { j = 0 } ^ { [ ( k - 1 ) / 2 ] } U _ { 0 , 0 } ( 2 j a + t ) + \\sum _ { j = 1 } ^ { [ k / 2 ] } U _ { 0 , 0 } ( 2 j a - t ) = \\sum _ { n = - \\infty } ^ { \\infty } u _ { n } T _ { n } \\cos n \\pi t = 0 , t \\in ( 0 , a ) , \\\\ \\sum _ { j = 0 } ^ { [ ( k - 1 ) / 2 ] } V _ { 0 , 0 } ( 2 j a + t ) - \\sum _ { j = 1 } ^ { [ k / 2 ] } V _ { 0 , 0 } ( 2 j a - t ) = \\sum _ { n = - \\infty } ^ { \\infty } v _ { n } T _ { n } \\sin n \\pi t = 0 , t \\in ( 0 , a ) , \\end{aligned} \\end{align*}"}
+{"id": "6749.png", "formula": "\\begin{align*} ( E ^ { e _ 1 } \\circ O ^ { o _ 1 } \\circ E ^ { e _ 2 } \\circ \\dotsb \\circ O ^ { o _ l } \\circ E ^ { e _ { l + 1 } } ) ( n ) = ( E ^ { \\sigma _ e } \\circ O ^ { \\sigma _ o } ) ( n ) + C = W . \\end{align*}"}
+{"id": "4611.png", "formula": "\\begin{align*} J _ \\chi : = \\{ x \\vert _ { \\mathcal { H } _ \\chi } \\ , \\ , | \\ , \\ , x \\in J \\} \\end{align*}"}
+{"id": "6766.png", "formula": "\\begin{align*} \\bigl | F ( \\Phi _ s ( z ) , t ) | = ( 1 - \\frac { | t | } { \\widehat \\alpha } ) \\varphi ( \\Phi _ { t } ( z ) ) \\le \\frac { \\widehat \\alpha - | t | } { \\widehat \\alpha } \\le \\frac { | s | } { \\widehat \\alpha } . \\end{align*}"}
+{"id": "5262.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n \\| x _ { i , t } - x _ { j , t } \\| ^ 2 \\le \\sum _ { i = 1 } ^ n 4 \\| z _ { i , t } - \\bar { z } _ t \\| ^ 2 . \\end{align*}"}
+{"id": "7515.png", "formula": "\\begin{align*} p ( n ) : = C n ^ \\gamma \\quad \\textrm { w i t h } C > 0 , \\gamma > 1 . \\end{align*}"}
+{"id": "3673.png", "formula": "\\begin{align*} \\lambda _ i ( t ) = \\frac { 1 } { 2 } { } t \\in [ t _ 0 , t _ 1 ] { } 1 \\leq i \\leq k \\ , . \\end{align*}"}
+{"id": "1516.png", "formula": "\\begin{gather*} g _ i ^ k \\ , , \\ , f _ j ^ l \\ , : Z \\rightarrow Z \\\\ g _ i ^ k ( T _ { j } ^ { l } ) \\ , = \\ , T _ { \\sigma _ i ( j ) } ^ { \\alpha _ k ( l ) } \\\\ f _ j ^ l ( T _ { i } ^ { k } ) \\ , = \\ , T _ { \\gamma _ j ( i ) } ^ { \\beta _ l ( k ) } \\end{gather*}"}
+{"id": "1595.png", "formula": "\\begin{align*} \\pi ( F ) = \\lim _ { n \\to \\infty } \\frac { e x ( n , F ) } { \\binom { n } { k } } . \\end{align*}"}
+{"id": "7417.png", "formula": "\\begin{align*} \\partial _ { x } \\Psi _ { n } & = \\rho _ { n } ^ { 0 } ( x ) - \\langle \\rho _ { n } \\rangle - \\int _ { 0 } ^ { t } \\partial _ { x } ( \\rho _ { n } u _ { n } ) ( x , s ) ~ d s \\\\ [ 1 e x ] & = \\rho _ { n } ^ { 0 } ( x ) - \\langle \\rho _ { n } \\rangle + \\rho _ { n } ( x , t ) - \\rho _ { n } ( x , 0 ) = \\rho _ { n } - \\langle \\rho _ { n } \\rangle , \\end{align*}"}
+{"id": "5700.png", "formula": "\\begin{align*} k _ { ( A , e _ 1 ) } = \\lim \\limits _ { z \\rightarrow 0 } \\frac { \\ln \\| ( z - A ) ^ { - 1 } e _ 1 \\| _ p } { \\ln \\| ( z - A ) ^ { - 1 } \\| } = 1 . \\end{align*}"}
+{"id": "4201.png", "formula": "\\begin{align*} \\sigma _ { \\varepsilon } ( 2 ( T S - S T ^ { \\ast } ) ) & = \\sigma _ { \\varepsilon } ( [ I \\bullet T , S ] _ { \\ast } ) = \\sigma _ { \\varepsilon } ( [ \\psi ( I ) \\bullet \\psi ( T ) , \\psi ( S ) ] _ { \\ast } ) \\\\ & = \\sigma _ { \\varepsilon } ( 2 ( \\psi ( T ) \\psi ( S ) - \\psi ( S ) \\psi ( T ) ^ { \\ast } ) ) . \\end{align*}"}
+{"id": "7603.png", "formula": "\\begin{align*} \\delta = \\beta ^ { - 1 / 3 } N ^ { - 1 / 3 } . \\end{align*}"}
+{"id": "4267.png", "formula": "\\begin{align*} \\mathcal { B } _ { \\alpha } ( u _ j ^ 0 , u _ j ^ 0 ) = \\lambda _ { k } \\int _ { \\Omega } | u _ j ^ 0 ( x ) | ^ 2 d x . \\end{align*}"}
+{"id": "407.png", "formula": "\\begin{align*} P \\times [ 0 = 1 ] \\cong [ 0 = 1 ] \\quad P ( n ) = 1 \\ . \\end{align*}"}
+{"id": "6802.png", "formula": "\\begin{align*} \\phi _ 1 ( z ) = w ( z ) - K f ( u ( z ) ) , \\\\ [ 0 . 2 c m ] \\phi _ 2 ( z ) = w ( z ) + K f ( u ( z ) ) . \\end{align*}"}
+{"id": "6972.png", "formula": "\\begin{align*} \\sum _ { n \\in \\Z ^ d } \\frac { | u ( n ) | ^ 2 } { | n | ^ 2 } = \\sum _ { j = 1 } ^ d \\sum _ { n \\in \\Z ^ d } | u _ j | ^ 2 = \\int _ { Q _ d } \\sum _ { j = 1 } ^ d | \\widehat { u _ j } | ^ 2 d x . \\end{align*}"}
+{"id": "5275.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { i , t } ( x _ { i , t } ) ] _ + \\| \\le n \\varepsilon _ { 7 } T ^ { 1 - c } , \\end{align*}"}
+{"id": "4602.png", "formula": "\\begin{align*} B _ { \\vec \\lambda } ( X , \\vec p ) : = { ( \\vec { \\mathcal H } _ { \\vec \\lambda } ^ * ) } _ 0 \\ , . \\end{align*}"}
+{"id": "2641.png", "formula": "\\begin{align*} \\Theta ^ { ( l ) } _ n ( t ) = - \\int _ { \\R _ + ^ 2 } I _ { t } ^ { ( l ) } ( x , s ) d U _ n ( x , s ) \\ , , \\end{align*}"}
+{"id": "4320.png", "formula": "\\begin{align*} \\omega _ { \\ell } ( A ) = \\omega ( W ( \\ell ) A W ( \\ell ) ^ * ) ( A \\in \\mathcal { W } ( L , \\sigma ) ) \\end{align*}"}
+{"id": "3848.png", "formula": "\\begin{align*} \\begin{array} { r c l } h ^ 1 _ { 1 } + h ^ 2 _ { 2 } & = & \\frac 1 { E G - F ^ 2 } [ - G g ( \\nabla _ { \\tau _ 1 } \\tau _ 1 , \\nu ) + F g ( \\nabla _ { \\tau _ 2 } \\tau _ 1 , \\nu ) + F g ( \\nabla _ { \\tau _ 1 } \\tau _ 2 , \\nu ) - E g ( \\nabla _ { \\tau _ 2 } \\tau _ 2 , \\nu ) ] , \\\\ \\end{array} \\end{align*}"}
+{"id": "6480.png", "formula": "\\begin{align*} f ' ( 0 ) = 0 , - f ' ( \\epsilon ) = 0 , \\end{align*}"}
+{"id": "5990.png", "formula": "\\begin{align*} 0 = \\tilde { \\psi } _ { \\varepsilon } + 2 \\pi a \\varepsilon \\frac { L _ { a } ^ { - 1 } \\tilde { u } _ j } { \\lambda _ j - \\lambda _ { j , \\varepsilon } } + \\frac { 2 \\pi } { \\varepsilon } L _ { a } ^ { - 1 } \\left ( \\mathcal { R } _ \\varepsilon + \\mathcal { R } ^ { \\lambda _ j , \\lambda _ { j , \\varepsilon } } \\right ) \\tilde { \\psi } _ \\varepsilon . \\end{align*}"}
+{"id": "8736.png", "formula": "\\begin{align*} \\Big ( \\sum _ { k = 1 } ^ { T } \\eta _ { k } \\Big ) ^ { - 1 } = \\frac { 1 } { \\min \\Big ( \\frac { T } { 1 8 L \\kappa } , T \\gamma \\Big ) } = \\max \\Big ( \\frac { 1 8 L \\kappa } { T } , \\frac { 1 } { T \\gamma } \\Big ) \\leq \\frac { 1 8 L \\kappa } { T } + \\frac { 1 } { T \\gamma } . \\end{align*}"}
+{"id": "1712.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi ^ 2 } \\int _ { ^ { ( 3 ) } _ { \\texttt { a } } } f ( X _ 1 , X _ 2 ) & \\frac { \\sqrt { \\rho _ { \\texttt { a } } ( X _ 1 , X _ 2 ) } } { O _ { \\texttt { a } } ( X _ 1 , X _ 2 ; q ) } X _ 1 X _ 2 \\\\ & = \\sum _ { \\substack { l _ 1 , l _ 2 \\geq 0 \\\\ l _ 1 + l _ 2 \\leq m } } f \\bigl ( \\boldsymbol { X } ^ { ( m , 3 ) } _ { \\texttt { a } ; l _ 1 \\omega _ 1 + l _ 2 \\omega _ 2 } \\bigr ) \\hat { \\Delta } ^ { ( m , 3 ) } _ { \\texttt { a } ; l _ 1 \\omega _ 1 + l _ 2 \\omega _ 2 } , \\end{align*}"}
+{"id": "462.png", "formula": "\\begin{align*} p _ 0 ^ { ( 1 ) } ( t , r , s ) & = \\frac { 2 t } { \\pi \\left ( r ^ 2 + s ^ 2 + t ^ 2 \\right ) \\left ( 1 - 4 r ^ 2 s ^ 2 \\cdot \\left ( r ^ 2 + s ^ 2 + t ^ 2 \\right ) ^ { - 2 } \\right ) } , \\\\ p _ 1 ^ { ( 1 ) } ( t , r , s ) & = \\frac { 4 } { \\pi } \\ , \\frac { t } { ( r ^ 2 - s ^ 2 ) ^ 2 + t ^ 2 ( t ^ 2 + 2 r ^ 2 + 2 s ^ 2 ) } . \\end{align*}"}
+{"id": "1848.png", "formula": "\\begin{gather*} \\widetilde { U } _ { s , t } ( 0 , \\Delta ) = \\int _ { \\mathbb { R } } U _ { s , t } ( x , \\Delta ) \\ , d x \\leq \\int _ { \\mathbb { R } } \\mathbf { 1 } _ { ( s , t ) } ( x ) \\ , d x + \\int _ { \\mathbb { R } } K _ { s , t } ( x , \\Delta ) \\ , d x \\end{gather*}"}
+{"id": "2666.png", "formula": "\\begin{align*} \\dim ( H ) + \\dim ( H ^ { \\perp _ f } ) = s . \\end{align*}"}
+{"id": "2511.png", "formula": "\\begin{align*} \\int _ { Q } \\boldsymbol { p } \\cdot \\nabla v \\ , d \\boldsymbol { x } \\ , d t = 0 \\forall \\ , v \\in H ^ { 1 , 0 } _ 0 ( Q ) . \\end{align*}"}
+{"id": "2308.png", "formula": "\\begin{align*} \\left | \\int f _ 1 \\cdots f _ n d \\xi d \\tau \\right | \\leq \\| f _ 1 \\| _ { L ^ { p _ 1 } _ { \\xi , \\tau } } \\cdots \\| f _ 1 \\| _ { L ^ { p _ 1 } _ { \\xi , \\tau } } , \\frac { 1 } { p _ 1 } + \\cdots + \\frac { 1 } { p _ n } = 1 \\end{align*}"}
+{"id": "4407.png", "formula": "\\begin{align*} | | ( g ^ { + } _ 3 , g ^ { - } _ 3 ) | | _ { H ^ { s + 1 } ( \\Gamma _ T ) } \\leq C | | ( \\mathcal { G } ^ { + } , \\mathcal { G } ^ { - } ) | | _ { H ^ { s + 1 } ( \\Gamma _ T ) } \\leq C \\Big ( | | \\mathbf { f } | | _ { s + 2 , \\ast , T } + | | ( g ^ { + } _ 1 , g ^ { - } _ 1 ) | | _ { H ^ { s + 2 } ( \\Gamma _ T ) } \\Big ) . \\end{align*}"}
+{"id": "5258.png", "formula": "\\begin{align*} & \\frac { 2 G _ 2 ^ 2 \\alpha _ { t } \\| q _ { i , t + 1 } \\| ^ 2 } { \\sigma } - q _ { i , t + 1 } ^ \\top g _ { i , t } ( x _ { i , t } ) = \\Big ( \\frac { 2 G _ 2 ^ 2 \\gamma _ 0 } { \\sigma } - 1 \\Big ) \\gamma _ { t } \\| [ g _ { i , t } ( x _ { i , t } ) ] _ + \\| ^ 2 \\le 0 , \\end{align*}"}
+{"id": "3536.png", "formula": "\\begin{align*} \\zeta _ { R } ( x ) : = \\eta ( \\vert x \\vert - R ) R > 0 , \\end{align*}"}
+{"id": "8729.png", "formula": "\\begin{align*} \\delta _ { t + 1 } \\leq \\left ( 1 - \\frac { 2 } { t } \\right ) \\delta _ { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { t ^ { p _ { i } + 1 } } \\enspace , \\end{align*}"}
+{"id": "4544.png", "formula": "\\begin{align*} \\mathcal A _ { ( 0 ) } \\partial _ 1 { \\mathbf V } = \\mathcal H _ { ( 0 ) } \\sigma \\partial _ 1 { \\mathbf V } \\end{align*}"}
+{"id": "6034.png", "formula": "\\begin{align*} \\Vert G _ n \\Vert _ { L , l , p } \\leq \\sum _ { i = n } ^ { m _ n } \\gamma _ i ^ n \\Vert F _ i \\Vert _ { L , l , p } \\leq K _ { l , p } . \\end{align*}"}
+{"id": "5968.png", "formula": "\\begin{align*} I = N ^ \\omega _ { \\partial M } \\Lambda ^ \\omega _ { g , F } + \\Psi ^ { - \\infty } \\end{align*}"}
+{"id": "5024.png", "formula": "\\begin{align*} \\int _ N ( 4 \\pi ) ^ { - ( n - 1 ) / 2 } e ^ { - ( f ( r , \\cdot ) - F ( r ) ) } d h = 1 , \\int _ 0 ^ \\infty ( 4 \\pi ) ^ { - 1 / 2 } e ^ { - F ( r ) } r ^ { n - 1 } d r = 1 . \\end{align*}"}
+{"id": "3680.png", "formula": "\\begin{align*} B _ { n , k , k - 1 } L _ { k - 1 } ( G ) = L _ k ( G ) B _ { n , k , k - 1 } . \\end{align*}"}
+{"id": "975.png", "formula": "\\begin{align*} l _ p & = \\min \\{ l \\in \\mathbb Z \\mid , \\ l > 0 \\} , \\\\ \\gamma _ p & = s _ p + t _ p \\sqrt d \\end{align*}"}
+{"id": "3948.png", "formula": "\\begin{align*} \\nabla \\frac { \\rho _ { K ^ * } } { \\rho _ { L ^ * } } ( u ) = 0 . \\end{align*}"}
+{"id": "72.png", "formula": "\\begin{align*} \\mathcal Q _ 3 ^ { ( 1 ) } + ( 1 - \\varepsilon _ K ) \\mathcal K ^ { \\rm { d i a g } } _ H = ( 1 - \\varepsilon _ K ) \\sum _ { k \\in \\mathcal { P } _ H } \\mathcal D _ k c _ k ^ \\dagger c _ k + \\sum _ { k \\in \\mathcal { P } _ H } \\mathcal T ( k ) , \\end{align*}"}
+{"id": "7555.png", "formula": "\\begin{align*} \\Gamma _ { 2 \\hbar } ( - d _ { ( 0 , 0 , 0 ) } 2 \\hbar ) & = \\frac { ( - 1 ) ^ { d _ { ( 0 , 0 , 0 ) } } } { d _ { ( 0 , 0 , 0 ) } ! } \\cdot \\Gamma _ { 2 \\hbar } ( 0 ) = \\frac { ( - 1 ) ^ { d _ { ( 0 , 0 , 0 ) } } } { d _ { ( 0 , 0 , 0 ) } ! } \\cdot \\frac { 1 } { e ^ { T _ 0 } ( 1 ) } , \\end{align*}"}
+{"id": "5680.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow 0 } \\frac { \\ln | \\varphi ( | z | ) | } { \\ln \\| ( z - T ) ^ { - 1 } e _ 0 \\| _ p } & = \\lim \\limits _ { z \\rightarrow 0 } \\frac { k \\ln | z | + \\ln \\big | \\sum \\limits _ { j = 0 } ^ { n } c _ j | z | ^ j \\big | } { \\ln \\| ( z - T ) ^ { - 1 } e _ 0 \\| _ p } \\\\ & = \\lim \\limits _ { z \\rightarrow 0 } \\Big ( \\frac { k \\ln | z | } { \\ln \\| ( z - T ) ^ { - 1 } e _ 0 \\| _ p } + \\frac { \\ln \\big | \\sum \\limits _ { j = 0 } ^ { n } c _ j | z | ^ j \\big | } { \\ln \\| ( z - T ) ^ { - 1 } e _ 0 \\| _ p } \\Big ) . \\end{align*}"}
+{"id": "8884.png", "formula": "\\begin{align*} i _ w ^ * \\tilde { y _ i } = w \\cdot u _ i . \\end{align*}"}
+{"id": "2191.png", "formula": "\\begin{align*} | A - B | = O ( \\log n ) , \\ A \\le B + O ( \\log n ) , B \\le A + O ( \\log n ) \\end{align*}"}
+{"id": "1319.png", "formula": "\\begin{align*} J ^ { w } ( \\textbf { X } _ { S R S } ^ { ( n ) } ) = \\frac { - 1 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( \\int _ { 0 } ^ { \\infty } w ( x _ i ) f ^ 2 ( x _ i ) d x _ i \\right ) = \\frac { - 1 } { 2 } \\left ( - 2 J ^ { w } ( X ) \\right ) ^ n = \\frac { - 1 } { 2 } \\left ( E ( \\Lambda _ X ^ { w } ( U ) ) \\right ) ^ n . \\end{align*}"}
+{"id": "3101.png", "formula": "\\begin{align*} [ q ] _ { i j } = \\left \\{ \\begin{matrix} 1 & i j = 1 2 \\\\ \\frac { 3 - \\sqrt { 3 } } { 2 } + o ( 1 ) \\approx 0 . 6 3 + o ( 1 ) & \\end{matrix} \\right . \\end{align*}"}
+{"id": "4358.png", "formula": "\\begin{align*} \\tilde { \\omega } ( A ) & = { \\rm T r } ( F \\varrho F ^ * \\pi ^ P ( A ) ) = { \\rm T r } ( \\varrho F ^ * \\pi ^ P ( A ) F ) \\\\ & = { \\rm T r } ( \\varrho \\pi ^ P ( B ^ * A B ) ) = \\omega ( B ^ * A B ) \\ \\ ( A \\in { \\tt C A R } ( \\mathcal { K } , \\Gamma ) ) \\end{align*}"}
+{"id": "4376.png", "formula": "\\begin{align*} \\tilde { A } _ 1 ( { \\mathbf U } ^ { \\pm } , \\Psi ^ { \\pm } ) = \\frac { 1 } { \\partial _ 1 \\Phi ^ { \\pm } } \\Big ( A _ 1 ( { \\mathbf U } ^ { \\pm } ) - A _ 0 ( { \\mathbf U } ^ { \\pm } ) \\partial _ t \\Psi ^ { \\pm } - A _ 2 ( { \\mathbf U } ^ { \\pm } ) \\partial _ 2 \\Psi ^ { \\pm } \\Big ) , \\end{align*}"}
+{"id": "8341.png", "formula": "\\begin{align*} \\d X ( t ) = a ( X ( t ) , t , \\omega ) \\d t + b ( X ( t ) , t , \\omega ) \\d W ( t ) , X ( 0 ) = x _ 0 \\in ( x _ 1 , x _ 2 ) , \\end{align*}"}
+{"id": "8145.png", "formula": "\\begin{align*} p _ { i j } ( m n ) = v _ { i j } ( \\theta _ { m n } , \\mu _ { m n } ) \\ , , \\end{align*}"}
+{"id": "700.png", "formula": "\\begin{align*} \\mathbf U ( t ) = \\mathbf S ( t ) \\mathbf u _ 0 + i \\int _ 0 ^ { t \\wedge \\tau _ R } \\mathbf S ( t - s ) \\mathbf N ( \\mathbf u ( s ) ) \\ , d s \\\\ + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf M ( \\mathbf U ( s ) ) \\ , d W ( s ) \\end{align*}"}
+{"id": "968.png", "formula": "\\begin{align*} \\Pr [ X _ \\kappa = 1 ] \\geq ( 1 - \\gamma ) \\cdot \\left ( \\frac { \\beta - \\alpha \\cdot \\gamma } { ( 1 + \\rho ) \\cdot \\beta } \\right ) \\cdot \\left ( 1 - \\frac { 2 \\cdot \\beta \\cdot \\rho } { \\alpha \\cdot \\gamma } \\right ) \\end{align*}"}
+{"id": "7382.png", "formula": "\\begin{align*} p _ { n } ( \\rho _ { n } ) = \\rho _ { n } ^ { \\gamma _ { n } } . \\end{align*}"}
+{"id": "6064.png", "formula": "\\begin{align*} { \\Phi } ( r , z , \\omega ) = \\vert \\nabla _ x { c } ( z , { X } ^ { M _ { { \\mathcal { P } } _ n } } _ { r - } ) \\vert \\vert D ^ Z { X } ^ { M _ { { \\ \\mathcal { P } } _ n } } _ { r - } - D ^ Z { X } ^ { M _ { { \\ \\mathcal { P } } _ m } } _ { r - } \\vert _ { l _ 2 } , \\end{align*}"}
+{"id": "4603.png", "formula": "\\begin{align*} \\mathbb { D } ^ { X , Y } = \\underline { \\mathrm { N a t } } ( - \\ , { \\otimes _ { A } } \\ , X , - \\ , { \\otimes _ { A } } \\ , Y ) : = \\underline { \\mathrm { H o m } } _ { \\mathcal F u n _ { A , B } } ( - \\ , { \\otimes _ { A } } \\ , X , - \\ , { \\otimes _ { A } } \\ , Y ) \\ , \\in \\mathcal { Z } ( \\mathcal { C } ) \\ , . \\end{align*}"}
+{"id": "4115.png", "formula": "\\begin{align*} D ( i , - \\tfrac { 1 } { 2 } ) = \\sum _ { m = 0 } ^ { i } \\binom { i } { m } \\binom { - \\tfrac { 1 } { 2 } } { m } 2 ^ m = \\begin{cases} 0 , & i , \\\\ \\left | \\dbinom { - \\tfrac { 1 } { 2 } } { \\tfrac { i } { 2 } } \\right | , & i . \\end{cases} \\end{align*}"}
+{"id": "570.png", "formula": "\\begin{align*} \\mathfrak T _ 1 ( \\mathbf v ) - \\mathfrak T _ 1 ( \\mathbf w ) = i \\int _ S ^ t \\mathbf S ( t - \\sigma ) \\left [ \\mathbf N \\left ( \\Theta _ R ^ { [ \\mathbf u , \\mathbf v ] } ( \\sigma ) \\mathbf v ( \\sigma ) \\right ) - \\mathbf N \\left ( \\Theta _ R ^ { [ \\mathbf u , \\mathbf w ] } ( \\sigma ) \\mathbf w ( \\sigma ) \\right ) \\right ] \\ , d \\sigma \\end{align*}"}
+{"id": "8251.png", "formula": "\\begin{align*} g _ 0 ( \\frac { \\partial \\Psi } { \\partial x _ 2 } , \\frac { \\partial \\Psi } { \\partial x _ 2 } ) & = \\frac { 1 } { 4 x _ 1 ^ 2 } g _ 0 ( \\xi _ 3 - d \\psi _ { x _ 1 , 0 , 0 } ( \\xi _ 3 ( m ) ) , \\xi _ 3 - d \\psi _ { x _ 1 , 0 , 0 } ( \\xi _ 3 ( m ) ) ) \\\\ & = \\frac { 1 } { 4 x _ 1 ^ 2 } [ \\rho ^ 2 - 2 \\rho ^ 2 ( m ) + g _ { x _ 1 , 0 , 0 } ( \\xi _ 3 ( m ) , \\xi _ 3 ( m ) ) ] . \\end{align*}"}
+{"id": "8433.png", "formula": "\\begin{align*} \\mathbf { S } ^ { 0 } _ t ( u _ 0 ) = v _ 1 + v _ 2 + , \\end{align*}"}
+{"id": "470.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty p _ t ( r , z ) p _ { t ' } ( z , s ) m ( d z ) = p _ { t + t ' } ( r , s ) , r , s , t , t ' > 0 , \\end{align*}"}
+{"id": "8842.png", "formula": "\\begin{align*} \\left ( \\sum _ { t = 1 } ^ { T } \\eta _ t \\right ) ^ { - 1 } = \\frac { 1 } { \\min \\left ( \\frac { T } { 8 \\kappa \\bar { L } d } , T a _ T \\right ) } = \\max \\left ( \\frac { 8 \\kappa \\bar { L } d } { T } , \\frac { 1 } { T a _ T } \\right ) \\leq \\frac { 8 \\kappa \\bar { L } d } { T } + \\frac { 1 } { T a _ T } \\enspace . \\end{align*}"}
+{"id": "1886.png", "formula": "\\begin{align*} 0 < h _ v = \\max _ { 1 \\leq j \\leq N _ v } h _ j ^ v , \\mbox { w h e r e } h _ j ^ v = v _ { j + \\frac { 1 } { 2 } } - v _ { j - \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "7418.png", "formula": "\\begin{align*} \\partial _ { t } ( \\rho _ { n } p _ { n } ) + \\partial _ { x } ( \\rho _ { n } p _ { n } u _ { n } ) = - \\lambda _ { n } ( \\rho _ { n } ) \\partial _ { x } u _ { n } . \\end{align*}"}
+{"id": "6979.png", "formula": "\\begin{align*} f '' + A ( z ) f ' + B ( z ) f = 0 , \\end{align*}"}
+{"id": "8906.png", "formula": "\\begin{align*} T ( X ) = \\{ a \\in A \\mid X \\subset \\Theta + a \\} . \\end{align*}"}
+{"id": "7865.png", "formula": "\\begin{align*} \\sigma ^ i \\underline { x } = & \\left ( x _ i ^ { r _ n - r } , x _ j ^ { r _ n } , \\dots , x _ i ^ { r _ n } , \\dots , x _ j ^ { r _ n } , \\dots \\right ) = \\left ( x _ i ^ { r } , x _ { j } ^ { r _ n - r } , x _ { j + r _ n } ^ r , \\dots , x _ i ^ { r } , x _ { j } ^ { r _ n - r } , \\dots , x _ { j } ^ { r _ n - r } , x _ { j + r _ n } ^ r , \\dots \\right ) . \\end{align*}"}
+{"id": "4814.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\langle \\varphi , \\partial _ { i _ 1 } \\cdots \\partial _ { i _ p } \\mathcal { L } \\eta \\rangle \\ , d \\mu = \\prod _ { j = 1 } ^ p \\lambda _ { i _ j } ^ { - 1 } \\int _ { \\Omega } \\langle \\varphi \\circ \\Phi , \\partial _ { i _ 1 } \\cdots \\partial _ { i _ p } \\eta \\rangle \\ , d \\mu . \\end{align*}"}
+{"id": "1474.png", "formula": "\\begin{align*} N : = \\sum _ { i = 1 } ^ { r } d _ { i } , \\end{align*}"}
+{"id": "2658.png", "formula": "\\begin{align*} k ( x , t ) = ( 1 - x ) \\int _ 0 ^ x \\frac { b _ y ( y , t ) } { 1 - y } \\ , d y \\ , , \\end{align*}"}
+{"id": "3901.png", "formula": "\\begin{align*} \\sum _ { n = J _ { P _ N } + 1 } ^ { N } | a _ n | ^ 2 < \\varphi ( N ) ^ { 1 / 8 } S _ N ^ 2 \\end{align*}"}
+{"id": "3232.png", "formula": "\\begin{align*} \\sqrt { 1 + ( \\alpha \\pm \\epsilon ) ^ { 2 } } & = \\sqrt { 1 + \\alpha ^ 2 } ( 1 + O ( \\epsilon ) ) . \\end{align*}"}
+{"id": "6320.png", "formula": "\\begin{align*} y = \\int _ { 0 } ^ { x } \\frac { 1 } { \\sqrt { g _ { [ < \\lambda ^ { \\sigma } ] } } } \\ , d x ' , \\end{align*}"}
+{"id": "8317.png", "formula": "\\begin{align*} \\mu _ { i _ 1 i _ 2 } = \\min \\{ \\mu _ { i _ 1 j } \\ | \\ i _ 1 < j \\leq n , \\ b _ { n - j } ( x ) \\ne 0 \\} \\end{align*}"}
+{"id": "3559.png", "formula": "\\begin{align*} L ( x ^ { k + 1 } , \\lambda ) - L ( x ^ { k + 1 } , \\lambda ^ { k + 1 } ) & = g ^ * ( \\lambda ^ { k + 1 } ) - g ^ * ( \\lambda ) - ( x ^ { k + 1 } ) ^ T A ^ T ( \\lambda - \\lambda ^ { k + 1 } ) \\\\ & \\leq ( \\nabla g ^ * ( \\lambda ^ { k + 1 } ) + A x ^ { k + 1 } ) ^ T ( \\lambda ^ { k + 1 } - \\lambda ) + \\frac { L } { 2 } \\| \\lambda ^ k - \\lambda ^ { k + 1 } \\| _ 2 ^ 2 \\ , \\end{align*}"}
+{"id": "3122.png", "formula": "\\begin{align*} \\mathbb { E } [ \\| \\mathbf { H } _ { k , } \\| ^ 2 ] = \\frac { L _ k } { N } \\frac { \\kappa } { \\kappa + 1 } . \\end{align*}"}
+{"id": "7467.png", "formula": "\\begin{align*} w _ e = \\sum _ { i = 1 } ^ n a _ i \\end{align*}"}
+{"id": "2476.png", "formula": "\\begin{align*} & \\left ( \\sum _ { a \\in \\mathcal { A } } P _ A ( a ) P _ { V _ a } \\right ) \\left ( \\sum _ { b \\in \\mathcal { B } } P _ B ( b ) P _ { V _ b } \\right ) \\\\ & = \\sum _ { ( a , b ) \\in \\mathcal { A } \\times \\mathcal { B } } P _ A ( a ) P _ B ( b ) P _ { V _ a } P _ { V _ b } . \\end{align*}"}
+{"id": "7656.png", "formula": "\\begin{align*} Z ( \\mathbf { t } ) = \\frac { \\sqrt { \\Delta ( \\boldsymbol { \\lambda } , \\boldsymbol { \\lambda } ) } } { 2 ^ { \\frac { N ^ 2 } { 2 } } \\ , ( 2 \\pi ) ^ { \\frac { N } { 2 } } \\ , \\prod _ { n = 1 } ^ { N - 1 } n ! } \\ , \\mathop { { \\rm P f } } _ { 0 \\leq m , n \\leq N - 1 } \\big ( K _ { m , n } ( \\mathbf { t } ) \\big ) , \\end{align*}"}
+{"id": "7319.png", "formula": "\\begin{align*} \\| ( I _ H - P _ n ) L _ \\lambda u \\| = \\| Q u + ( I _ H - P _ n ) K _ \\lambda u \\| \\geq \\frac { 1 } { 2 } \\| u \\| , u \\in H ^ \\perp _ n , \\end{align*}"}
+{"id": "6416.png", "formula": "\\begin{align*} s _ { * } : = \\inf _ { 0 < \\lambda < + \\infty } \\frac { d _ { 2 } \\left [ \\int _ { \\mathbb { R } } J _ { 2 } ( y ) \\mathrm { e } ^ { \\lambda y } \\mathrm { d } y - 1 \\right ] + r _ 2 ( b - 1 ) } { \\lambda } . \\end{align*}"}
+{"id": "1095.png", "formula": "\\begin{align*} | \\nabla u | ( x ) : = \\sqrt { \\Gamma ( u ) ( x ) } = \\left ( \\frac { 1 } { 2 \\mu ( x ) } \\sum _ { y \\sim x } w ( x , y ) ( u ( y ) - u ( x ) ) ^ 2 \\right ) ^ { 1 / 2 } , \\end{align*}"}
+{"id": "3751.png", "formula": "\\begin{align*} [ \\mathbb { A } ( \\lambda I + \\mathbb { A } ) ^ { - 1 } ] _ { i j } = \\dfrac { ( - 1 ) ^ { i - j } } { 3 } \\Big ( 2 \\cos \\Big ( ( \\frac { \\alpha + i - j } { 3 } ) \\pi \\Big ) + 1 \\Big ) A ^ { \\frac { \\alpha + i - j } { n } } . \\end{align*}"}
+{"id": "3377.png", "formula": "\\begin{align*} g = \\begin{pmatrix} 1 & 0 \\\\ b ^ \\alpha & \\delta ^ \\alpha _ \\beta \\end{pmatrix} , g ^ { - 1 } = \\begin{pmatrix} 1 & 0 \\\\ - b ^ \\alpha & \\delta ^ \\alpha _ \\beta \\end{pmatrix} \\end{align*}"}
+{"id": "8584.png", "formula": "\\begin{align*} \\mathcal { L } ^ { \\mu } [ \\beta b ] = - \\cosh { ( ( - 1 + \\beta b ( X ) ) \\sqrt { \\mu } D ) } ^ { - 1 } \\sinh { ( \\beta b ( X ) \\sqrt { \\mu } D ) } \\mathrm { s e c h } ( \\sqrt { \\mu } D ) . \\end{align*}"}
+{"id": "8355.png", "formula": "\\begin{align*} B \\int _ 0 ^ t S _ B ( t - r ) \\omega _ 2 ( - t ) d r = S _ B ( t ) \\omega _ 2 ( - t ) - \\omega _ 2 ( - t ) . \\end{align*}"}
+{"id": "6251.png", "formula": "\\begin{align*} K ( x _ 1 , x _ 2 , \\cdots , x _ { k } ) = ( 2 \\pi ) ^ { - \\frac { k } 2 } \\hat q ( - x _ 1 , x _ 2 , \\cdots , ( - 1 ) ^ k x _ { k } ) . \\end{align*}"}
+{"id": "4131.png", "formula": "\\begin{align*} G ( k , s , S _ k ) \\sum _ { r = 1 } ^ { s } \\sum _ { \\ell = 0 } ^ { k - 2 r - 2 s + 2 } V ( k , s , r , \\ell ) - \\sum _ { \\ell = 0 } ^ { k - 2 s } H ( k , s , 1 , S _ k , S _ r ) V ( k , s , 1 , \\ell ) = 0 . \\end{align*}"}
+{"id": "6906.png", "formula": "\\begin{align*} F _ { 1 } ( \\phi , \\psi ) ( \\xi ) : & = \\beta \\phi ( \\xi ) + d _ { 1 } \\mathcal { N } _ { 1 } [ \\phi ] ( \\xi ) + \\phi ( \\xi ) f \\left ( \\phi , \\psi \\right ) ( \\xi ) , \\\\ [ 0 . 2 c m ] F _ { 2 } ( \\phi , \\psi ) ( \\xi ) : & = \\beta \\psi ( \\xi ) + d _ { 2 } \\mathcal { N } _ { 2 } [ \\psi ] ( \\xi ) + \\psi ( \\xi ) g \\left ( \\phi , \\psi \\right ) ( \\xi ) , \\end{align*}"}
+{"id": "8167.png", "formula": "\\begin{align*} & [ X ^ \\star , [ X ^ \\star , X ] ] = X - ( r - 3 ) X ^ \\star - ( r - 2 ) ( Y ^ \\star - k ) , \\\\ & [ X , [ X ^ \\star , X ] ] = ( r - 3 ) X - ( r - 1 ) ^ 2 X ^ \\star - ( r - 1 ) ( r - 2 ) ( Y ^ \\star - k ) . \\end{align*}"}
+{"id": "6248.png", "formula": "\\begin{align*} L u ( x ) = \\int K ( y ) u ^ y ( x ) d y \\end{align*}"}
+{"id": "5084.png", "formula": "\\begin{align*} E ( A z ) v ^ c = \\left ( \\begin{array} { c c c } E ( \\lambda _ { 1 } z ) v _ { 1 } & \\cdots & E ( \\lambda _ { n } z ) v _ n \\end{array} \\right ) \\left ( \\begin{array} { c } c _ 1 \\\\ \\vdots \\\\ c _ n \\end{array} \\right ) = \\sum _ { i = 1 } ^ n c _ i E ( \\lambda _ i z ) v _ i . \\end{align*}"}
+{"id": "3117.png", "formula": "\\begin{align*} \\mathbb { E } [ \\chi _ { k , p , n } \\chi ^ * _ { k , p , n ' } ] = \\begin{cases} 1 , & n = n ' \\\\ 0 , & n \\neq n ' . \\end{cases} \\end{align*}"}
+{"id": "2941.png", "formula": "\\begin{align*} 0 = \\delta _ { i _ 1 i _ 2 } [ f D _ { k i _ 1 \\dots i _ n } ] \\end{align*}"}
+{"id": "8807.png", "formula": "\\begin{align*} \\int K ( u ) d u = 0 , ~ \\int u K ( u ) d u = 1 \\end{align*}"}
+{"id": "3425.png", "formula": "\\begin{align*} C _ 0 \\int _ { E _ J } \\Omega _ { E _ J } \\wedge \\overline { \\Omega } _ { E _ J } = \\frac { 1 } { \\int _ { S k ( X ) } d x _ 1 \\ldots d x _ m } = m ! , \\end{align*}"}
+{"id": "7273.png", "formula": "\\begin{align*} \\frac { h '' \\circ g ( y ) \\cdot g ' ( y ) } { h ' \\circ g ( y ) } = \\frac { h '' ( y ) } { h ' ( y ) } . \\end{align*}"}
+{"id": "8159.png", "formula": "\\begin{align*} \\theta ^ \\star _ { x , y } q _ { i j } ( x , y ) & = \\frac { n ( r - 3 ) i } { k } q _ { i j } ( x , y ) + \\frac { n ( i + 1 ) } { k } \\frac { \\binom { n } { i } } { \\binom { k } { i } } K _ { i + 1 } ( x , k - y , r - 1 ) H _ j ( y , n - i , k - i ) \\\\ & \\quad + \\frac { n ( r - 2 ) } { k } \\frac { \\binom { n } { i } } { \\binom { k } { i } } ( k - y - i + 1 ) K _ { i - 1 } ( x , k - y , r - 1 ) H _ j ( y , n - i , k - i ) \\ , . \\end{align*}"}
+{"id": "1578.png", "formula": "\\begin{align*} ( 2 \\pi T _ f ) ^ { - 1 / 2 } & = \\frac { 1 } { \\sqrt { 2 \\pi T _ g } } \\left ( 1 - \\frac { \\varepsilon } { 2 } \\left ( \\frac { P _ h } { P _ g } - \\frac { \\rho _ h } { \\rho _ g } \\right ) \\right ) + o ( \\varepsilon ) \\\\ & = \\frac { 1 } { \\sqrt { 2 \\pi T _ g } } \\left ( 1 - \\frac { \\varepsilon } { 2 \\rho _ g } \\int _ { - \\infty } ^ { \\infty } \\left [ \\frac { u ^ 2 } { T _ g } - 1 \\right ] h ( x , u ) \\mathrm { d } u \\right ) + o ( \\varepsilon ) . \\end{align*}"}
+{"id": "2512.png", "formula": "\\begin{align*} 0 = ( \\boldsymbol { u } ( t ) , \\nabla \\psi ) _ { \\Omega } = ( \\boldsymbol { u } _ 0 ^ c , \\nabla \\psi ) _ { \\Omega } + \\sum _ { k = 0 } ^ { \\infty } [ ( \\boldsymbol { u } _ k ^ c , \\nabla \\psi ) _ { \\Omega } \\cos ( k \\omega t ) + ( \\boldsymbol { u } _ k ^ s , \\nabla \\psi ) _ { \\Omega } \\sin ( k \\omega t ) ] \\end{align*}"}
+{"id": "3671.png", "formula": "\\begin{align*} \\langle \\nabla v , \\nabla f \\rangle = g _ { i k } ( g ^ { i j } v _ j ) ( g ^ { k m } f _ m ) = g ^ { i j } v _ i f _ j = a ^ { - 1 } \\ , v _ i \\frac { x _ i } { 2 } \\ , . \\end{align*}"}
+{"id": "6699.png", "formula": "\\begin{align*} G _ { \\lambda / \\mu , [ [ f / g ] ] } ( x ) = \\det \\left ( \\sum _ { s = 0 } ^ \\infty \\binom { i - j } { s } \\beta ^ s G ^ { [ [ f _ i / g _ j ] ] } _ { \\lambda _ i - \\mu _ j - i + j + s } ( x ) \\right ) \\end{align*}"}
+{"id": "4542.png", "formula": "\\begin{align*} & \\int _ { \\mathbb R } \\vert f ( t ) \\vert _ { x _ 1 = 0 } \\vert ^ 2 d x _ 2 = - \\int _ { \\mathbb R } \\int _ 0 ^ { + \\infty } \\partial _ 1 \\left ( \\vert f ( t ) \\vert _ { x _ 1 = 0 } \\vert ^ 2 \\right ) d x _ 1 d x _ 2 \\\\ & = - \\int _ { \\mathbb R ^ 2 _ + } 2 f \\partial _ 1 f ( t ) \\ , d x \\le \\Vert f ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ^ 2 _ + ) } + \\Vert \\partial _ 1 f ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ^ 2 _ + ) } \\ , , \\end{align*}"}
+{"id": "6476.png", "formula": "\\begin{align*} F _ 1 = - \\frac 1 2 L F _ 1 ' + C , \\end{align*}"}
+{"id": "8790.png", "formula": "\\begin{align*} \\sum _ { t = T _ { 1 } + 1 } ^ { T } \\mathbb { E } [ f ( x _ { t } ) - f ^ * ] \\leq \\sum _ { t = T _ { 1 } + 1 } ^ { T } \\Big [ \\frac { r _ { t } - r _ { t + 1 } } { 2 \\eta _ t } - \\frac { \\alpha } { 4 } r _ t \\Big ] + \\sum _ { t = T _ { 1 } + 1 } ^ { T } \\Big [ 2 ( \\kappa _ { \\beta } L ) ^ 2 \\frac { d } { \\alpha } h _ { t } ^ { 2 ( \\beta - 1 ) } + \\frac { 3 \\eta _ t } { 4 } \\kappa \\left ( \\frac { \\sigma ^ 2 } { h _ { t } ^ 2 } + \\frac { 3 { \\bar L } ^ 2 } { 2 } h _ { t } ^ 2 \\right ) d \\Big ] , \\end{align*}"}
+{"id": "3208.png", "formula": "\\begin{align*} y _ k \\in y _ 0 + \\mathcal { K } _ k ( A A ^ T , r _ 0 ) , \\| y - y _ k \\| _ { A A ^ T } = \\min _ { z \\in y _ 0 + \\mathcal { K } _ k ( A A ^ T , r _ 0 ) } \\| y - z \\| _ { A A ^ T } . \\end{align*}"}
+{"id": "4988.png", "formula": "\\begin{align*} \\delta ( D _ n ) & = 3 \\ell + 1 = 2 ^ { m - 1 } ( 6 k + 3 ) + 1 , \\\\ \\beta _ { 2 n } & = 2 ^ { m + 1 } ( k + 1 ) = 2 ^ { m - 1 } ( 4 k + 4 ) , \\end{align*}"}
+{"id": "2561.png", "formula": "\\begin{align*} \\mu _ \\infty + V _ \\infty = \\xi _ \\infty + \\xi _ \\infty ' + \\varepsilon _ W W _ \\infty \\end{align*}"}
+{"id": "569.png", "formula": "\\begin{align*} \\mathfrak T ( \\mathbf v ) ( t ) = \\eqref { i n d u c t i o n 1 - t r u n c a t e d } = : \\mathfrak T _ 0 ( t ) + \\mathfrak T _ 1 ( \\mathbf v ) ( t ) + \\mathfrak T _ 2 ( \\mathbf v ) ( t ) , . \\end{align*}"}
+{"id": "3303.png", "formula": "\\begin{align*} \\frac { d } { d t } ( \\phi _ Y ) _ { _ \\Sigma } ( t ) = - \\frac { t } { 2 \\lambda _ { _ \\Sigma } } e ^ { - \\frac { t ^ 2 } { 4 \\lambda _ { _ \\Sigma } } } , \\end{align*}"}
+{"id": "220.png", "formula": "\\begin{align*} & \\left \\{ e ^ { \\frac { 1 } { 2 4 } A _ 3 } \\widehat { L } ( T M ) [ - A _ 3 + 2 { \\rm c h } ( \\widetilde { T _ C M } ) - 8 + { \\rm c h } ( W _ i ) ] \\right \\} ^ { ( 1 2 ) } \\\\ & = \\left \\{ e ^ { \\frac { 1 } { 2 4 } A _ 3 } \\widehat { A } ( T M ) { \\rm c h } [ 2 1 1 6 \\widetilde { T _ C M } + 4 \\wedge ^ 2 \\widetilde { T _ C M } + 4 W _ i - 1 5 8 7 2 ] \\right \\} ^ { ( 1 2 ) } \\\\ & - \\frac { 2 } { 1 5 } \\left \\{ c _ 2 ( W _ i ) e ^ { \\frac { 1 } { 2 4 } A _ 3 } \\widehat { A } ( T X ) \\right \\} ^ { ( 1 2 ) } . \\end{align*}"}
+{"id": "8197.png", "formula": "\\begin{align*} e ^ { i t } \\cdot ( z _ 1 , w _ 1 , \\dots , z _ n , w _ n ) = ( e ^ { i t } z _ 1 , e ^ { - i t } w _ 1 , \\dots , e ^ { i t } z _ n , e ^ { - i t } w _ n ) . \\end{align*}"}
+{"id": "7408.png", "formula": "\\begin{align*} \\partial _ { t } P _ { n } ^ { M } = - u _ { n } \\partial _ { x } P _ { n } ^ { M } + \\rho _ { n } ^ { - 1 } \\partial _ { x } w _ { n } - P _ { n } ^ { M } p _ { n } '' ( \\rho _ { n } ) | \\rho _ { n } ^ { 2 } \\partial _ { x } P _ { n } ^ { M } | ^ { 2 } - 2 p _ { n } ' ( \\rho _ { n } ) \\rho _ { n } ^ { 2 } \\left ( \\partial _ { x } P _ { n } ^ { M } \\right ) ^ { 2 } + \\rho _ { n } p _ { n } ' ( \\rho _ { n } ) \\partial _ { x } ^ { 2 } P _ { n } ^ { M } . \\end{align*}"}
+{"id": "6202.png", "formula": "\\begin{gather*} [ x _ 1 , x _ 2 , x _ 3 ] = - \\epsilon ( x _ 1 , x _ 2 ) [ x _ 2 , x _ 1 , x _ 3 ] , \\\\ \\epsilon ( x _ 3 , x _ 1 ) [ x _ 1 , x _ 2 , x _ 3 ] + \\epsilon ( x _ 1 , x _ 2 ) [ x _ 2 , x _ 3 , x _ 1 ] + \\epsilon ( x _ 2 , x _ 3 ) [ x _ 3 , x _ 1 , x _ 2 ] = 0 \\end{gather*}"}
+{"id": "272.png", "formula": "\\begin{align*} \\Lambda ^ n _ i K = A _ { - 1 } \\xlongrightarrow { } A _ 0 \\xlongrightarrow { } \\cdots \\xlongrightarrow { } A _ { n - 1 } \\xlongrightarrow { } A _ n = K _ n \\end{align*}"}
+{"id": "7849.png", "formula": "\\begin{align*} \\frac { t ^ 3 } { 3 } > \\frac { \\chi } { 4 ( 2 \\pi ) ^ 3 } \\sum _ { ( m , n ) \\in \\mathbb { Z } ^ 2 - ( 0 , 0 ) } \\frac { 1 } { | m \\tau + n | ^ 3 } > \\frac { \\chi } { 4 ( 2 \\pi ) ^ 3 } \\sum _ { n \\in \\mathbb { Z } - \\{ 0 \\} } \\frac { 1 } { | n | ^ 3 } = \\frac { \\chi \\zeta ( 3 ) } { 2 ( 2 \\pi ) ^ 3 } > 0 \\ ; . \\end{align*}"}
+{"id": "1066.png", "formula": "\\begin{align*} B = \\max \\left ( 1 , \\max _ { i , j } \\left \\Vert a _ { i j } \\right \\Vert _ { C ^ { 2 } \\left ( \\overline { \\Omega } \\right ) } \\right ) , \\end{align*}"}
+{"id": "3191.png", "formula": "\\begin{align*} \\| x - x _ k \\| ^ 2 _ { A ^ T A } = \\| A ( x - x _ k ) \\| ^ 2 = \\| b - r - A x _ k \\| ^ 2 = \\| r _ k - r \\| ^ 2 = \\| r _ k \\| ^ 2 - \\| r \\| ^ 2 , \\end{align*}"}
+{"id": "7803.png", "formula": "\\begin{align*} \\frac { t + t _ { + } ^ { 0 , n } } { t - t _ { + } ^ { 0 , n } } : = \\begin{dcases} - 1 n > 0 \\\\ 1 n < 0 \\\\ \\end{dcases} \\ , . \\end{align*}"}
+{"id": "3373.png", "formula": "\\begin{align*} \\sum _ { i = 3 } ^ n \\alpha _ i x _ i = 0 \\end{align*}"}
+{"id": "4997.png", "formula": "\\begin{align*} K _ { \\Sigma } = \\frac { S } { 2 } - \\mathrm { R i c } ( \\nu , \\nu ) + 2 H ^ 2 - \\frac { 1 } { 2 } | h | ^ 2 . \\end{align*}"}
+{"id": "5905.png", "formula": "\\begin{align*} \\varphi = 0 [ - 1 , 1 ] , \\varphi = 1 \\R \\setminus ( - 2 , 2 ) , \\| \\varphi \\| _ { C ^ 2 ( \\R ) } \\leq C _ 0 . \\end{align*}"}
+{"id": "1738.png", "formula": "\\begin{align*} \\emph { W } ^ { ( n ) } ( \\mathbf { x } ) : = \\prod _ { 1 \\leq j < k \\leq n } ( x _ j - x _ k ) ^ 2 \\prod _ { 1 \\leq j \\leq n } \\emph { w } ( x _ j ) , \\end{align*}"}
+{"id": "8554.png", "formula": "\\begin{align*} P ( E ; \\{ z = a \\} ) = P ( E ; \\{ z = b \\} ) = 0 . \\end{align*}"}
+{"id": "7704.png", "formula": "\\begin{align*} X _ x ( f g | _ x ) = X _ x ( f | _ x ) g _ 0 ( x ) + ( - 1 ) ^ { | X _ x | | f | } f _ 0 ( x ) X _ x ( g | _ x ) , \\end{align*}"}
+{"id": "7357.png", "formula": "\\begin{align*} G _ { \\rho _ Y } ( z ) + \\frac { \\sqrt { \\gamma } } { q } \\frac { 1 } { 1 - \\sqrt { \\gamma } G _ { \\rho _ Y } ( z ) } + \\frac { 1 } { G _ { \\rho _ Y } ( z ) } = z . \\end{align*}"}
+{"id": "5721.png", "formula": "\\begin{align*} \\det \\left ( \\Sigma _ { M } ( 1 ) - e ^ { \\eta ^ { - 1 } \\ , ( B _ M ) _ { 1 , 1 } } \\ , \\gamma _ M \\ , \\Sigma _ { M } ( 0 ) \\right ) \\ , = \\ , 0 \\end{align*}"}
+{"id": "2313.png", "formula": "\\begin{align*} & P _ 1 : = \\{ ( \\xi , \\tau ) \\in \\R ^ 2 : | \\tau - p _ { \\lambda } ( \\xi ) | \\leq \\frac { 3 1 } { 3 2 } \\delta | \\xi | ^ 5 + \\frac { 7 } { 8 } \\lambda ^ 2 \\gamma | \\xi | ^ 3 \\} , \\\\ & P _ 2 : = \\R ^ 2 \\setminus P _ 1 , \\end{align*}"}
+{"id": "8417.png", "formula": "\\begin{align*} h ( t ) = r t ^ { c } + v \\left ( T - t ^ { c } \\right ) ^ { \\gamma } . \\end{align*}"}
+{"id": "6331.png", "formula": "\\begin{align*} I _ { \\mu } = \\mu ^ 2 \\int _ 0 ^ T | u _ \\mu ( t , x _ t ) | ^ 2 d s . \\end{align*}"}
+{"id": "8215.png", "formula": "\\begin{align*} d \\Phi _ a | _ m = ( d L _ { a ^ 2 x _ 1 ( m ) } ) | _ m - a ^ 2 I _ 1 T | _ { \\Phi _ a ( m ) } \\otimes ( d x _ 1 ) | _ m . \\end{align*}"}
+{"id": "4392.png", "formula": "\\begin{align*} C ( \\hat { { \\mathbf U } } ^ { \\pm } , \\hat { \\Psi } ^ { \\pm } ) Y = ( Y , \\nabla _ y A _ 0 ( \\hat { { \\mathbf U } } ^ { \\pm } ) ) \\partial _ t \\hat { { \\mathbf U } } ^ { \\pm } + ( Y , \\nabla _ y \\tilde { A } _ 1 ( \\hat { { \\mathbf U } } ^ { \\pm } , \\hat { \\Psi } ^ { \\pm } ) ) \\partial _ 1 \\hat { { \\mathbf U } } ^ { \\pm } + ( Y , \\nabla _ y A _ 2 ( \\hat { { \\mathbf U } } ^ { \\pm } ) ) \\partial _ 2 \\hat { { \\mathbf U } } ^ { \\pm } \\ , , \\end{align*}"}
+{"id": "8731.png", "formula": "\\begin{align*} \\tau _ { t } \\leq \\tau _ { t _ { 0 } } \\prod _ { i = { t _ { 0 } } } ^ { t - 1 } \\Big ( 1 - \\frac { 2 } { i } \\Big ) \\leq \\tau _ { t _ { 0 } } \\prod _ { i = { t _ { 0 } } } ^ { t - 1 } \\Big ( 1 - \\frac { 1 } { i } \\Big ) \\le \\frac { ( t _ { 0 } - 1 ) \\tau _ { t _ { 0 } } } { t } \\le \\frac { 2 ( t _ { 0 } - 1 ) \\delta _ { t _ { 0 } } } { t } \\enspace . \\end{align*}"}
+{"id": "8216.png", "formula": "\\begin{align*} ( \\iota _ { I _ 1 T } \\omega _ 3 ) ( Y ) = \\omega _ 3 ( I _ 1 T , Y ) = g _ 0 ( I _ 3 I _ 1 T , Y ) = g _ 0 ( I _ 2 T , Y ) = \\omega _ 2 ( T , Y ) = - d x _ 2 ( Y ) , \\end{align*}"}
+{"id": "6044.png", "formula": "\\begin{align*} \\overline { \\omega } = \\overline { \\lim _ { n \\rightarrow \\infty } } \\frac { \\gamma _ { n } - \\gamma _ { n + 1 } } { \\gamma _ { n + 1 } ^ { 2 } } , \\end{align*}"}
+{"id": "4871.png", "formula": "\\begin{align*} \\check { R } ( u ) = I \\pm \\frac { u } { k - u } E \\end{align*}"}
+{"id": "1068.png", "formula": "\\begin{align*} M = \\max \\left ( M _ { 1 } , M _ { 2 } , M _ { 3 } , M _ { 4 } , M _ { 1 } / c , M _ { 2 } / c , M _ { 3 } / c , M _ { 4 } / c \\right ) . \\end{align*}"}
+{"id": "1981.png", "formula": "\\begin{align*} \\sum _ { \\sigma = 0 } ^ { p ^ k - 1 } \\mathcal { C } ( \\mathbf { a } _ \\sigma ^ { t _ 1 } , \\mathbf { a } _ \\sigma ^ { t _ 2 } ) ( \\tau ) = \\sum _ { r = 0 } ^ { p ^ { m } - \\tau - 1 } \\sum _ { \\sigma = 0 } ^ { p ^ { k } - 1 } \\omega _ q ^ { \\left ( a ^ { t _ 1 } _ { \\sigma , r } - a ^ { t _ 2 } _ { \\sigma , r + \\tau } \\right ) } = 0 . \\end{align*}"}
+{"id": "1541.png", "formula": "\\begin{align*} \\overline { \\rho ^ 2 } = \\langle \\lambda _ 1 ^ 2 , \\nu \\rangle + m ^ { \\rho ^ { 2 } } , \\end{align*}"}
+{"id": "3514.png", "formula": "\\begin{align*} \\det ( 1 - A ) = \\exp \\left ( - \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { k } \\mathrm { t r } ( A ^ k ) \\right ) , \\end{align*}"}
+{"id": "933.png", "formula": "\\begin{align*} \\frac { 1 } { 2 y } \\bigg ( C + \\frac { \\sqrt { m n } } { m } D \\bigg ) = \\frac { \\sqrt { m n } } { m } \\frac { \\alpha } { 4 t ^ \\alpha } \\mathrm { e } ^ { - \\frac { \\alpha ( x ^ 2 + y ^ 2 ) } { 4 t ^ \\alpha } } \\bigg ( \\frac y x \\bigg ) ^ { \\frac { c + \\sqrt { m n } - 1 } { 2 } } I _ { \\frac { c + \\sqrt { m n } - 1 } { 2 } } \\bigg ( \\frac { \\alpha \\sqrt { x y } } { 2 t ^ \\alpha } \\bigg ) , \\end{align*}"}
+{"id": "5838.png", "formula": "\\begin{align*} & w _ 1 = s _ 1 s _ 2 ( s _ 3 s _ 2 s _ 4 ( s _ 3 s _ 2 s _ 1 s _ 2 s _ 3 ) s _ 4 s _ 2 s _ 3 ) s _ 2 s _ 1 \\\\ & w _ 2 = s _ 4 s _ 3 s _ 2 s _ 3 s _ 4 s _ 3 s _ 2 s _ 3 s _ 2 \\\\ \\end{align*}"}
+{"id": "7102.png", "formula": "\\begin{align*} S ( y ) = 2 y \\ , U ( y ^ 2 - w ) , \\end{align*}"}
+{"id": "2015.png", "formula": "\\begin{align*} \\dot { \\mathrm { H } } ^ { s _ 0 , p _ 0 } ( \\Omega ) \\cap \\dot { \\mathrm { H } } ^ { s _ 1 , p _ 1 } ( \\Omega ) = [ \\dot { \\mathrm { H } } ^ { s _ 0 , p _ 0 } \\cap \\dot { \\mathrm { H } } ^ { s _ 1 , p _ 1 } ] ( \\Omega ) \\end{align*}"}
+{"id": "1432.png", "formula": "\\begin{align*} \\Delta u _ i + \\sum \\limits _ { j = 1 } ^ { n } a _ { i j } e ^ { u _ j } = 4 \\pi \\alpha _ i \\delta _ 0 \\ \\ B _ 1 ( 0 ) \\subseteq \\mathbb { R } ^ 2 , \\ \\ \\forall \\ i = 1 , \\cdots , n , \\end{align*}"}
+{"id": "6068.png", "formula": "\\begin{align*} \\mathbb { E } \\vert { X } ^ { M _ \\mathcal { P } } _ { \\gamma _ 1 } - X _ { \\gamma _ 1 } \\vert \\leq \\bar { C } { \\gamma _ 1 } e ^ { \\bar { C } { \\gamma _ 1 } } \\sqrt { \\varepsilon _ { M ( { \\gamma _ 1 } ) } } = \\bar { C } { \\gamma _ 1 } ^ 2 . \\end{align*}"}
+{"id": "4968.png", "formula": "\\begin{align*} f _ n ^ { ( 1 \\bmod { 8 } ) } ( \\sqrt { 2 } ) & = P _ { 2 , 1 } ( n ) = \\Phi _ { 8 n , 1 } ( \\sqrt { 2 } ) , \\\\ f _ n ^ { ( 3 \\bmod { 8 } ) } ( \\sqrt { 2 } ) & = P _ { 2 , 3 } ( n ) = \\Phi _ { 8 n , 3 } ( \\sqrt { 2 } ) , \\\\ f _ n ^ { ( 1 \\bmod { 1 2 } ) } ( \\sqrt { 3 } ) & = P _ { 3 , 1 } ( n ) = \\Phi _ { 1 2 n , 1 } ( \\sqrt { 3 } ) , \\\\ f _ n ^ { ( 5 \\bmod { 1 2 } ) } ( \\sqrt { 3 } ) & = P _ { 3 , 5 } ( n ) = \\Phi _ { 1 2 n , 5 } ( \\sqrt { 3 } ) \\end{align*}"}
+{"id": "979.png", "formula": "\\begin{align*} x + y \\sqrt d = \\pm \\eta ^ n \\prod _ { p \\in S _ \\pm } \\gamma ' _ p { } ^ { n _ p } p ^ { k _ p - l _ p n _ p / 2 } \\prod _ { q \\notin S _ \\pm } q ^ { k _ q } \\end{align*}"}
+{"id": "6434.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } v _ { \\infty } ( c t , t ) = b - 1 > 0 . \\end{align*}"}
+{"id": "4940.png", "formula": "\\begin{align*} \\eta ( 4 \\tau ) ^ 2 = e ^ { 3 \\pi i \\tau / 4 } \\cdot \\dfrac { \\eta ( \\tau ) \\cdot \\Psi ( \\tau ) \\cdot \\Psi ( 2 \\tau ) } { \\eta ( 2 \\tau ) } . \\end{align*}"}
+{"id": "5255.png", "formula": "\\begin{align*} & \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\sum _ { t = 1 } ^ { T } ( l _ { i , t } ( x _ { i , t } ) - l _ { i , t } ( y _ { t } ) ) \\le \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\Big ( - q _ { i , t + 1 } ^ \\top g _ { i , t } ( x _ { i , t } ) + \\frac { 1 } { n } \\tilde { \\Delta } _ t + \\Delta _ { i , t } ( y ) \\Big ) . \\end{align*}"}
+{"id": "3126.png", "formula": "\\begin{align*} \\mathbf { H } _ { k , } ( m ) = \\frac { 1 } { N } \\sqrt { \\frac { \\kappa } { \\kappa + 1 } } \\sum _ { p = 1 } ^ { L _ k } \\sum _ { n _ 1 = 0 } ^ { N _ 1 - 1 } \\sum _ { n _ 2 = 0 } ^ { N _ 2 - 1 } \\alpha _ { k , p } \\chi _ { k , p , n _ 1 \\cdot N _ 2 + n _ 2 } e ^ { - j 2 \\pi ( m - 1 ) \\phi ^ { } } \\end{align*}"}
+{"id": "2012.png", "formula": "\\begin{align*} { \\mathrm { E } u } _ { | _ { \\Omega } } = u \\end{align*}"}
+{"id": "590.png", "formula": "\\begin{align*} \\Delta _ 3 ^ \\mu ( t ) = \\Delta _ { 3 , 1 } ^ \\mu ( t ) + \\Delta _ { 3 , 2 } ^ \\mu ( t ) + \\Delta _ { 3 , 3 } ^ \\mu ( t ) , \\end{align*}"}
+{"id": "6360.png", "formula": "\\begin{align*} u = \\sum _ { \\nu \\in J ^ S _ k } e ^ { i t \\lambda _ \\nu } u _ \\nu e _ \\nu . \\end{align*}"}
+{"id": "2986.png", "formula": "\\begin{align*} E _ I ^ 1 = E ^ 1 \\times _ { s , r } E ^ 1 = \\{ ( e ' , e ) \\in E ^ 1 \\times E ^ 1 : s ( e ' ) = r ( e ) \\} \\end{align*}"}
+{"id": "3754.png", "formula": "\\begin{align*} u ''' ( t ) + 3 \\varUpsilon _ 0 ^ \\alpha A ^ { \\frac { \\alpha } { 3 } } u '' ( t ) + 3 \\varUpsilon _ 0 ^ \\alpha A ^ { \\frac { 2 \\alpha } { 3 } } u '' ( t ) + A ^ { \\alpha } u ( t ) = 0 , t > 0 , \\end{align*}"}
+{"id": "5950.png", "formula": "\\begin{align*} G ^ \\omega _ { \\partial M } ( x , y ) = N ^ \\omega _ { \\partial M } ( x , y ) + \\frac { u _ j ( x ) u _ j ( y ) } { \\lambda _ j - \\omega ^ 2 } e ^ { \\phi ( y ) } . \\end{align*}"}
+{"id": "4142.png", "formula": "\\begin{align*} & \\nabla T ( v _ 1 , v _ 2 ; v _ 3 ) = ( \\nabla _ { v _ 3 } T ) ( v _ 1 , v _ 2 ) , \\end{align*}"}
+{"id": "2047.png", "formula": "\\begin{align*} \\lVert u \\rVert _ { \\dot { \\mathrm { H } } ^ { \\alpha , p } ( \\mathbb { R } _ + , \\mathrm { L } ^ p ( \\mathbb { R } ^ { n - 1 } ) ) } : = \\inf _ { \\substack { U _ { | _ { \\mathbb { R } _ + } } = u \\\\ U \\in \\dot { \\mathrm { H } } ^ { \\alpha , p } ( \\mathbb { R } , \\mathrm { L } ^ p ( \\mathbb { R } ^ { n - 1 } ) ) } } \\lVert U \\rVert _ { \\dot { \\mathrm { H } } ^ { \\alpha , p } ( \\mathbb { R } , \\mathrm { L } ^ p ( \\mathbb { R } ^ { n - 1 } ) ) } \\end{align*}"}
+{"id": "2045.png", "formula": "\\begin{align*} \\dot { \\mathrm { H } } ^ { \\alpha , p } ( \\mathbb { R } , \\mathrm { L } ^ p ( \\mathbb { R } ^ { n - 1 } ) ) : = \\left \\{ \\ , u \\in \\mathrm { L } ^ { r } ( \\mathbb { R } , \\mathrm { L } ^ p ( \\mathbb { R } ^ { n - 1 } ) ) \\ , \\big | \\ , ( - \\partial _ { x _ n } ^ 2 ) ^ { \\frac { \\alpha } { 2 } } u \\in \\mathrm { L } ^ { p } ( \\mathbb { R } , \\mathrm { L } ^ p ( \\mathbb { R } ^ { n - 1 } ) ) = \\mathrm { L } ^ p ( \\mathbb { R } ^ { n } ) \\ , \\right \\} \\end{align*}"}
+{"id": "8050.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } + \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } < \\infty . \\end{align*}"}
+{"id": "3361.png", "formula": "\\begin{align*} \\prod _ { t = 0 } ^ { p - 1 } ( \\zeta ^ { t q } x ; \\zeta ) _ { e } = \\prod _ { i = 0 } ^ { e - 1 } \\prod _ { t = 0 } ^ { p - 1 } ( 1 - \\zeta ^ { t q + i } x ) = \\prod _ { t = 0 } ^ { p e - 1 } ( 1 - \\zeta ^ { t q } x ) = ( x ; \\zeta ^ q ) _ { p e } . \\end{align*}"}
+{"id": "5770.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\mathbb E \\langle ( R _ { 1 , 2 } ^ p - \\langle R _ { 1 , 2 } ^ p \\rangle ) ^ 2 \\rangle = 0 . \\end{align*}"}
+{"id": "7639.png", "formula": "\\begin{align*} U ( \\lambda x ) & \\leq U ( \\overline x ) = U ^ + ( \\overline x ) . \\end{align*}"}
+{"id": "4910.png", "formula": "\\begin{align*} W ( \\psi ^ 0 ) = \\displaystyle \\sum _ { a = 0 } ^ { r _ \\psi - 1 } \\psi ^ 0 ( a ) e ^ { 2 \\pi i a / M } \\mathcal { A } _ k = \\dfrac { 2 ( - 2 \\pi i ) ^ { k } W ( \\psi ^ 0 ) } { R ^ k ( k - 1 ) ! } , \\end{align*}"}
+{"id": "1297.png", "formula": "\\begin{align*} u _ n ( 0 ) = \\sum _ { j = 1 } ^ { J } g _ n ^ j [ e ^ { t _ n ^ j \\Delta } \\phi ^ j ] + W _ n ^ J \\end{align*}"}
+{"id": "7466.png", "formula": "\\begin{align*} c _ i \\mapsto c _ i ' = c _ i + \\lambda , \\forall i \\in \\{ 1 , 2 , \\dots , n \\} . \\end{align*}"}
+{"id": "1201.png", "formula": "\\begin{align*} \\mathrm { p r } _ g \\tau ( [ x ] _ P \\otimes [ y ] _ P ) = \\begin{cases} [ x _ P ] & \\mathrm { i f } \\ ; y ^ { - 1 } x g ^ { - 1 } \\in P \\\\ 0 & \\mathrm { o t h e r w i s e } \\end{cases} \\end{align*}"}
+{"id": "2333.png", "formula": "\\begin{align*} \\Delta _ g \\pi _ g x & = \\Delta _ g \\left ( \\sum _ { h \\in G } f _ h ( x ) \\tau _ { g h } \\right ) = \\sum _ { h \\in G } f _ h ( x ) \\Delta _ g \\tau _ { g h } = \\sum _ { h \\in G } f _ h ( x ) \\sum _ { u \\in G } f _ { g u } ( \\tau _ { g h } ) \\tau _ u \\\\ & = \\sum _ { h \\in G } f _ h ( x ) \\sum _ { u \\in G } f _ { u } ( \\tau _ { h } ) \\tau _ u = \\sum _ { h \\in G } f _ h ( x ) \\tau _ h = x , \\forall x \\in \\mathcal { X } \\end{align*}"}
+{"id": "6072.png", "formula": "\\begin{align*} \\sup _ x \\mathbb { E } \\vert \\det \\sigma _ { X _ { t , r } \\circ X _ { t - 1 , t } ^ { \\mathcal { P } , { M _ { \\mathcal { P } } } } ( x ) } - \\det \\sigma _ { X _ { t , r } \\circ X _ { t - 1 , t } ( x ) } \\vert ^ { \\frac { 2 } { 1 + \\varepsilon _ 0 } } = \\sup _ x \\mathbb { E } \\vert \\det \\sigma _ { F _ { r - t + 1 } ^ { \\mathcal { P } , { M _ \\mathcal { P } } } ( x ) } - \\det \\sigma _ { F _ { r - t + 1 } ( x ) } \\vert ^ { \\frac { 2 } { 1 + \\varepsilon _ 0 } } \\leq C \\vert { \\mathcal { P } ^ { t - 1 , t } } \\vert ^ { \\frac { 2 } { ( 2 + \\varepsilon _ 0 ) ( 1 + \\varepsilon _ 0 ) } } . \\end{align*}"}
+{"id": "4707.png", "formula": "\\begin{align*} g ( z _ 0 ^ 2 ) g ( x y ) = g ( z _ 0 ) g ( x ) g ( y ) - g ( z _ 0 ) f ( x ) f ( y ) + f ( z _ 0 ^ 2 ) f ( x y ) , \\ ; x , y \\in S . \\end{align*}"}
+{"id": "1085.png", "formula": "\\begin{align*} \\lambda ^ * : = \\frac 1 2 \\sup _ { \\varrho > 0 } \\frac { \\varrho } { \\displaystyle \\max _ { { \\footnotesize \\begin{array} { c } x \\in D \\\\ | s | \\leq \\kappa \\sqrt { \\varrho } \\end{array} } } \\left | \\int _ 0 ^ { s } f ( x , t ) d t \\right | } \\in ( 0 , + \\infty ] \\end{align*}"}
+{"id": "5415.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = f ( | x | , u ) , \\ \\ u > 0 & \\ \\ B _ R , \\\\ u = 0 , \\ \\ & \\ \\ \\partial B _ R . \\end{cases} \\end{align*}"}
+{"id": "3738.png", "formula": "\\begin{align*} ( \\lambda ^ 3 I - A ) u = 0 . \\end{align*}"}
+{"id": "3932.png", "formula": "\\begin{align*} f ^ { - 1 } ( \\omega ) \\triangle g ^ { - 1 } ( \\omega ) = \\Big ( \\bigcup _ { x \\in \\omega } [ x - 1 , x + 1 ] \\Big ) \\setminus \\Big ( \\bigcup _ { x \\in \\omega } ( x - 1 , x + 1 ) \\Big ) \\end{align*}"}
+{"id": "3407.png", "formula": "\\begin{align*} \\begin{array} { c } \\displaystyle v _ i ( \\delta _ j v _ j ^ 2 - \\delta _ k v _ k ^ 2 ) V _ j + v _ j ( \\delta _ i v _ i ^ 2 - \\delta _ k v _ k ^ 2 ) V _ i = 0 . \\\\ \\end{array} \\end{align*}"}
+{"id": "4878.png", "formula": "\\begin{align*} K _ w ( z ) : = \\left \\{ \\begin{array} { l l } \\frac { E _ + ( z ) E _ + ( w ) ^ * - E _ - ( z ) E _ - ( w ) ^ * } { \\rho _ w ( z ) } & \\mbox { i f } z \\neq \\overline { w } \\\\ \\frac { E _ + ^ { ' } ( \\overline { w } ) E _ + ( w ) ^ * - E _ - ^ { ' } ( \\overline { w } ) E _ - ( w ) ^ * } { - 2 \\pi i } & \\mbox { i f } z = \\overline { w } \\end{array} \\right . \\end{align*}"}
+{"id": "1951.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( u _ t , \\phi \\right ) + \\left ( u u _ x , \\phi \\right ) + \\epsilon \\left ( u _ x , \\phi _ x \\right ) = \\left ( \\rho V - \\rho u , \\phi \\right ) , \\forall \\ , \\ , \\phi \\in H ^ 1 ( I ) \\end{aligned} \\end{align*}"}
+{"id": "3597.png", "formula": "\\begin{align*} { \\rm { t } } { { \\rm { r } } _ 2 } \\left ( { { \\bf { D } } _ { K } ^ { 1 / 2 } } \\right ) = \\sum \\limits _ { n = 1 } ^ { K } { \\sum \\limits _ { m > n } ^ { K } { { \\rho _ n } { \\rho _ m } { N _ { { \\rm { S , } } n } } { N _ { { \\rm { S , } } m } } } } . \\end{align*}"}
+{"id": "4213.png", "formula": "\\begin{align*} d \\omega ^ 1 = \\omega ^ { 2 3 } , d \\omega ^ 2 = - \\omega ^ { 1 3 } , d \\omega ^ 3 = \\omega ^ { 1 2 } . \\end{align*}"}
+{"id": "4012.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ M \\left | \\varphi _ { i _ j } - \\varphi _ { i _ k } \\right | ^ { q ^ * } \\omega ^ n & \\leq \\Vert \\varphi _ { i _ j } - \\varphi _ { i _ k } \\Vert ^ { q ^ * - 1 } _ { L ^ \\infty } \\int _ M \\left | \\varphi _ { i _ j } - \\varphi _ { i _ k } \\right | \\omega ^ n \\\\ & \\leq C \\int _ M \\left | \\varphi _ { i _ j } - \\varphi _ { i _ k } \\right | \\omega ^ n \\\\ & \\to 0 ( j , k \\to \\infty ) , \\end{aligned} \\end{align*}"}
+{"id": "2416.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ { 3 } \\sum _ { \\abs { \\beta } = 0 } ^ k \\abs { x } ^ { \\abs { \\beta } + \\tau } \\abs { \\partial _ \\beta \\eta _ { i j } } = O ( 1 ) & & \\abs { x } \\to + \\infty . \\end{align*}"}
+{"id": "1947.png", "formula": "\\begin{align*} I _ 3 = - \\frac { 1 } { 2 } \\int _ { \\R } \\int _ I \\left ( 1 + | v | ^ k \\right ) \\left ( u - v \\right ) \\partial _ v \\left \\vert \\partial _ x f \\right \\vert ^ 2 \\ , { \\rm d } x \\ , { \\rm d } v . \\end{align*}"}
+{"id": "7575.png", "formula": "\\begin{align*} | D ^ { ( k ) } _ t | = \\kappa t ^ { \\alpha + 1 } , & & | D ^ { ( \\ell ) } _ s | = \\kappa s ^ { \\alpha + 1 } . \\end{align*}"}
+{"id": "4632.png", "formula": "\\begin{align*} \\widetilde { \\Phi } _ i : = \\sum _ { k \\in \\mathcal { I } _ { e _ i } } m _ { e _ i , k } ^ { * } \\otimes \\Phi _ i ( m _ { e _ i , k } ) \\in \\bigl ( \\mathbf { M } ^ { \\pi _ { \\mathcal { S } ( v _ i ) } } \\otimes S _ i \\bigr ) ^ { K _ { v _ i } } . \\end{align*}"}
+{"id": "7614.png", "formula": "\\begin{align*} \\alpha = ( \\alpha _ { i , j } ) _ { i \\in I , j \\in J } \\colon \\bigoplus _ { i \\in I } M ( G / U _ i ) \\to \\bigoplus _ { j \\in J } M ( G / V _ j ) , \\end{align*}"}
+{"id": "5436.png", "formula": "\\begin{align*} a _ 1 ^ 2 d _ T ( 1 , 2 ) & + a _ 1 \\left [ \\sum _ { j \\geq 3 } ( d _ T ( 1 , 2 ) - d _ T ( 1 , j ) + d _ T ( 2 , j ) ) a _ j \\right ] \\\\ & + \\sum _ { j , k \\geq 3 } ( d _ T ( 2 , j ) - 2 d _ T ( j , k ) ) a _ j a _ k = 0 , \\end{align*}"}
+{"id": "2925.png", "formula": "\\begin{align*} J _ 0 ^ 2 = 1 , J _ 1 ^ 2 = 1 , J _ 2 ^ 2 = 1 . \\end{align*}"}
+{"id": "2176.png", "formula": "\\begin{align*} b ( x , y ) & = - \\left ( \\sum _ { z \\in X } b ( x , z ) ( 1 _ { g y } ( g x ) - 1 _ { g y } ( g z ) ) + c ( x ) 1 _ { g y } ( g x ) \\right ) \\\\ & = - H T _ { g ^ { - 1 } } 1 _ { g y } ( x ) = - T _ { g ^ { - 1 } } H 1 _ { g y } ( x ) \\\\ & = - \\left ( \\sum _ { z \\in X } b ( g x , z ) ( 1 _ { g y } ( g x ) - 1 _ { g y } ( z ) ) + c ( g x ) 1 _ { g y } ( g x ) \\right ) \\\\ & = b ( g x , g y ) , \\end{align*}"}
+{"id": "194.png", "formula": "\\begin{align*} & \\left \\{ e ^ { \\frac { 1 } { 2 4 } A _ 1 } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } ( \\widetilde { T _ C M } - \\widetilde { L _ C } - 8 + W _ i ) - e ^ { \\frac { 1 } { 2 4 } A _ 1 } A _ 1 \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 0 ) } \\\\ & = 4 8 0 \\left \\{ e ^ { \\frac { 1 } { 2 4 } A _ 1 } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 0 ) } . \\end{align*}"}
+{"id": "4610.png", "formula": "\\begin{align*} C _ A ( I ) : = \\{ x \\in A \\ , \\ , | \\ , \\ , x y = y x \\forall y \\in I \\} \\end{align*}"}
+{"id": "5438.png", "formula": "\\begin{align*} \\log \\int e ^ f d \\gamma _ n \\leq 1 0 \\int \\frac { e ^ { \\frac { 1 } { 2 } | \\nabla f | ^ 2 } } { 1 + | \\nabla f | } d \\gamma _ n \\mbox { f o r a l l s m o o t h } \\ f \\ \\mbox { w i t h } \\int f d \\mu = 0 . \\end{align*}"}
+{"id": "6605.png", "formula": "\\begin{align*} \\prod _ { l = 1 } ^ N w ^ { ( \\rm c J ) } ( \\theta _ l ) \\prod _ { 1 \\le j < k \\le N } | e ^ { i \\theta _ k } - e ^ { i \\theta _ j } | ^ \\beta , w ^ { ( \\rm c J ) } ( \\theta ) = e ^ { q \\theta } | 1 + e ^ { i \\theta } | ^ { \\beta p } \\ : \\ : ( - \\pi < \\theta \\le \\pi ) , \\end{align*}"}
+{"id": "7058.png", "formula": "\\begin{align*} \\mathfrak { I } = \\int _ a ^ b g ( t ) e ( f ( t ) ) d t , \\end{align*}"}
+{"id": "4791.png", "formula": "\\begin{align*} F _ R ( u ; \\sigma , \\rho ) = \\frac { u ^ 2 } { 2 \\sigma ^ 2 \\sqrt { 1 { - } \\rho ^ 2 } } \\sum _ { k { = } 0 } ^ \\infty \\frac 1 { 4 ^ k ( k ! ) ^ 2 } \\frac { u ^ { 4 k } } { 2 k + 1 } \\rho ^ { 2 k } { } _ 1 F _ 1 \\left ( 2 k + 1 , 2 k + 2 , - \\frac { u ^ 2 } { 2 \\sigma ^ 2 ( 1 - \\rho ^ 2 ) } \\right ) , \\end{align*}"}
+{"id": "8789.png", "formula": "\\begin{align*} \\langle \\hat { g } _ { t } , x _ t - x ^ * \\rangle = \\frac { \\norm { x _ t - x ^ * } ^ 2 - \\norm { x _ { t + 1 } - x ^ * } ^ { 2 } } { \\eta _ t } + \\frac { \\eta _ t } { 2 } \\norm { \\hat { g } _ t } ^ { 2 } . \\end{align*}"}
+{"id": "5444.png", "formula": "\\begin{align*} \\Gamma ( \\Phi ( f ) , g ) = \\sum _ { i = 1 } ^ n \\partial _ i \\Phi ( f ) \\Gamma ( f _ i , g ) , f , g \\in \\mathcal { A } . \\end{align*}"}
+{"id": "2946.png", "formula": "\\begin{align*} \\delta _ { i _ 1 i _ 2 } [ \\delta _ { \\pi ( i _ n i _ 1 } D _ { i _ 2 \\cdots i _ { n - 1 } ) k } ] & = \\delta _ { \\pi ( i _ n 1 } D _ { 1 \\cdots i _ { n - 1 } ) k } + \\delta _ { \\pi ( i _ n 2 } D _ { 2 \\cdots i _ { n - 1 } ) k } + \\delta _ { \\pi ( i _ n 3 } D _ { 3 \\cdots i _ { n - 1 } ) k } \\\\ & = D _ { \\pi ( i _ n i _ 3 \\cdots i _ { n - 1 } ) k } . \\end{align*}"}
+{"id": "7556.png", "formula": "\\begin{align*} \\overline { z } = \\frac { 1 } { s _ i } \\prod _ { s = 1 } ^ 3 \\prod _ { j \\neq i } \\frac { s _ i - s _ j - \\hbar _ s } { s _ i - s _ j + \\hbar _ s } , \\end{align*}"}
+{"id": "3723.png", "formula": "\\begin{align*} A ^ \\alpha = \\dfrac { \\sin ( \\pi \\alpha ) } { \\pi } \\int _ { 0 } ^ { \\infty } \\lambda ^ { \\alpha - 1 } A ( I \\lambda + A ) ^ { - 1 } d t , \\end{align*}"}
+{"id": "6644.png", "formula": "\\begin{align*} { \\rho } _ { ( 1 ) , \\infty } ^ { ( \\rm c J ) } ( x ; \\beta , p , q ) | _ { \\beta = 2 \\atop q = 0 } - 1 \\mathop { \\sim } \\limits _ { x \\to \\infty } \\sum _ { n = 1 } ^ \\infty { c _ { 2 n } \\over x ^ { 2 n } } , \\end{align*}"}
+{"id": "8830.png", "formula": "\\begin{align*} f ^ { * } _ { \\omega } = \\min _ { x \\mathbb { R } ^ { d } } f _ { \\omega } ( x ) = - \\sum _ { i = 1 } ^ { n } \\frac { 1 - \\omega _ { i } } { 4 \\tilde { \\alpha } } h _ { T } ^ { 2 ( \\beta - 2 ) } . \\end{align*}"}
+{"id": "8360.png", "formula": "\\begin{align*} d u ( t ) = J u ( t ) \\ , d t + F ( u ( t ) ) \\ , d t + G ( u ( t ) ) \\ , ( d \\omega _ 1 ( t ) , d \\omega _ 2 ( t ) ) \\end{align*}"}
+{"id": "7785.png", "formula": "\\begin{align*} \\lim _ { c \\to 0 } \\frac { t _ - ^ { c , n } } { t _ - ^ { c , n } } \\left ( \\frac { 1 + t _ { + } ^ { c , n } t } { t - t _ { + } ^ { c , n } } \\right ) = \\lim _ { c \\to 0 } \\frac { t _ - ^ { c , n } - t } { t _ - ^ { c , n } t + 1 } = \\frac { 1 } { t } \\ , , \\quad \\lim _ { c \\to 0 } \\frac { t _ - ^ { c , n } } { t _ - ^ { c , n } } \\left ( \\frac { 1 - t _ { + } ^ { c , n } t } { t - t _ { + } ^ { c , n } } \\right ) = \\lim _ { c \\to 0 } \\frac { t _ - ^ { c , n } + t } { t _ - ^ { c , n } t + 1 } = \\frac { 1 } { t } . \\end{align*}"}
+{"id": "8934.png", "formula": "\\begin{align*} U : = \\left \\{ \\begin{pmatrix} I _ m & u \\\\ 0 & I _ { n + 1 } \\end{pmatrix} : u \\in M _ { m \\times ( n + 1 ) } ( \\R ) \\right \\} < G . \\end{align*}"}
+{"id": "5958.png", "formula": "\\begin{align*} \\sum _ { \\gamma } \\frac { 1 } { \\gamma ! } \\partial _ { \\xi ' } ^ \\gamma \\sigma ( A ^ \\omega _ F ) D _ t ^ \\gamma \\sigma ( A ^ \\omega _ F ) - \\partial _ { t _ 3 } \\sigma ( A ^ \\omega _ F ) - \\sigma ( \\widetilde { Q ^ \\omega } ) - \\widetilde { E } ( t ) \\sigma ( A ^ \\omega _ F ) = 0 . \\end{align*}"}
+{"id": "7229.png", "formula": "\\begin{align*} B _ n = \\bigoplus _ { k \\ge 0 } \\bigoplus _ { \\substack { \\ell _ 1 , \\dots , \\ell _ k \\ge 1 , m \\ge 0 , \\\\ \\ell _ 1 + \\cdots + \\ell _ k + m + k = n } } A _ { \\ell _ 1 } \\otimes _ S \\cdots \\otimes _ S A _ { \\ell _ k } \\otimes _ S G _ m \\otimes _ S R . \\end{align*}"}
+{"id": "723.png", "formula": "\\begin{align*} \\mathbf U ( t ) - \\mathbf V ( t ) = \\Delta _ 1 ( t ) + \\Delta _ 2 ( t ) , \\end{align*}"}
+{"id": "5550.png", "formula": "\\begin{align*} g _ k ( \\zeta ) = b _ k + f _ k ( \\zeta ) - \\sum _ { j = 1 } ^ { m } a _ { k j } \\log ( \\zeta - c _ j ) \\end{align*}"}
+{"id": "246.png", "formula": "\\begin{align*} \\rho _ g : = { \\textstyle ( ( n - 1 ) g _ M + \\frac { 1 } { 2 } g _ S + g _ L , ( n - 2 ) g _ M + \\frac { 1 } { 2 } g _ S + g _ L , \\ldots , \\frac { 1 } { 2 } g _ S + g _ L ) } . \\end{align*}"}
+{"id": "2890.png", "formula": "\\begin{align*} C _ { s _ j w } ( \\boldsymbol { \\xi } ) _ j ( w \\boldsymbol { \\xi } ) = C _ { w } ( \\boldsymbol { \\xi } ) _ j ( - w \\boldsymbol { \\xi } ) \\quad \\ w \\in W _ 0 , \\ j \\in \\{ 1 , \\dots , n \\} . \\end{align*}"}
+{"id": "356.png", "formula": "\\begin{align*} \\psi _ H \\ ; \\phi _ { R ( h ) } = \\phi _ { B ( \\psi _ H ( h ) ) } \\ ; \\psi _ H \\end{align*}"}
+{"id": "4169.png", "formula": "\\begin{align*} \\gamma _ i & = \\frac { | h _ { i , i } | ^ 2 p _ i } { \\sum _ { j \\ne i } | h _ { i , j } | ^ 2 p _ j + \\sigma _ i ^ 2 } , \\\\ \\tilde { \\gamma } _ k & = \\frac { | \\tilde { h } _ { k , k } | ^ 2 p _ k } { \\sum _ { j = 1 } ^ { L } | \\tilde { h } _ { k , j } | ^ 2 p _ j + \\tilde { \\sigma } ^ 2 _ k } . \\end{align*}"}
+{"id": "1492.png", "formula": "\\begin{align*} & ( d B _ { t } , 0 ) = \\theta ^ { \\ell } _ { g _ { t } } ( d g _ { t } ) \\\\ & = \\left ( d \\overline { g } _ { t } , d \\widehat { g } _ { 1 } ( t ) - \\frac { 1 } { 2 } \\langle U ^ { ( 1 ) } \\overline { g } _ { t } , d \\overline { g } _ { t } \\rangle , \\ldots , d \\widehat { g } _ { n } ( t ) - \\frac { 1 } { 2 } \\langle U ^ { ( n ) } \\overline { g } _ { t } , d \\overline { g } _ { t } \\rangle \\right ) , \\end{align*}"}
+{"id": "7963.png", "formula": "\\begin{align*} \\sigma ( u _ { i , j } ) = v _ { r i , j + 1 } ~ ~ u _ { i , j } \\in S _ 1 . \\end{align*}"}
+{"id": "99.png", "formula": "\\begin{align*} \\langle \\Psi , \\mathcal { H } \\Psi \\rangle = \\frac { 1 } { \\pi } \\int _ { \\mathbb { C } } \\langle \\Phi ( z ) , ( \\mathcal { K } ( z ) + \\mathcal { Q } _ 3 ^ { } + \\mathcal { Q } _ 4 ^ { } + \\mathcal { R } _ 0 ) \\Phi ( z ) \\rangle _ { } \\mathrm { d } z . \\end{align*}"}
+{"id": "7728.png", "formula": "\\begin{align*} \\tilde { \\mathfrak { g } } = \\mathfrak { h } [ 2 ] \\oplus \\mathfrak { a } [ 1 ] \\oplus \\mathfrak { g } . \\end{align*}"}
+{"id": "584.png", "formula": "\\begin{align*} \\lim _ { \\mu \\to \\infty } \\norm { ( 1 - P _ \\mu ) \\mathbf M ( \\mathbf f ) } _ { \\mathcal L _ 2 ( K , \\mathbf H ^ \\mathbf s ) } = 0 \\end{align*}"}
+{"id": "6643.png", "formula": "\\begin{align*} { \\rho } _ { ( 1 ) , \\infty } ^ { ( \\rm c J ) } ( x ; \\beta , p , q ) | _ { \\beta = 2 \\atop p = 2 , q = 0 } = 1 - { 2 \\over \\pi ^ 2 x ^ 2 } - { 3 \\over 2 \\pi ^ 4 x ^ 4 } + { 3 - 2 \\pi ^ 2 x ^ 2 \\over 2 \\pi ^ 4 x ^ 4 } \\cos 2 \\pi x + { 3 \\over \\pi ^ 3 x ^ 3 } \\sin 2 \\pi x . \\end{align*}"}
+{"id": "4428.png", "formula": "\\begin{align*} \\lambda ^ + = s g n ( H ^ + ) \\frac { a ^ + [ u ] } { a ^ + \\vert H ^ + \\vert + a ^ - \\vert H ^ - \\vert } \\quad \\mbox { a n d } \\lambda ^ - = - s g n ( H ^ - ) \\frac { a ^ - [ u ] } { a ^ + \\vert H ^ + \\vert + a ^ - \\vert H ^ - \\vert } \\end{align*}"}
+{"id": "1885.png", "formula": "\\begin{align*} 0 < h _ x = \\max _ { 1 \\leq i \\leq N _ x } h _ i ^ x , \\mbox { w h e r e } h _ i ^ x = x _ { i + \\frac { 1 } { 2 } } - x _ { i - \\frac { 1 } { 2 } } , \\end{align*}"}
+{"id": "3825.png", "formula": "\\begin{align*} { \\rm U O T } ( \\mu _ 0 , \\mu _ 1 ) = ( H _ p , \\overline { \\alpha } ) = { \\rm O T } ( \\overline { \\alpha } _ 0 , \\overline { \\alpha } _ 1 ) , \\end{align*}"}
+{"id": "5239.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { t } ( x _ { i , t } ) ] _ + \\| \\end{align*}"}
+{"id": "321.png", "formula": "\\begin{align*} \\mathsf { D } _ { j } \\left ( s \\right ) : = \\mathsf { N } _ { j } \\left ( s \\right ) \\circ \\left ( \\gamma _ { \\operatorname * { N } ; j } ^ { \\operatorname * { e x t } } \\right ) ^ { \\prime } \\left ( s \\right ) . \\end{align*}"}
+{"id": "6835.png", "formula": "\\begin{align*} P _ { \\rm c y l } : = P _ { \\rm r a d } + P _ { \\rm a n g } , \\end{align*}"}
+{"id": "5711.png", "formula": "\\begin{align*} \\ln \\| ( z - T ) ^ { - 1 } x _ r \\| _ p & \\approx \\ln e ^ { \\frac { 1 } { | z | } } + \\ln \\big | \\frac { 1 } { z } + \\frac { z ^ { m - 1 } } { r ^ { m } } \\big ( e ^ { \\frac { r } { z } } - \\sum \\limits _ { n = 0 } ^ { m } \\frac { 1 } { n ! } \\frac { r ^ n } { z ^ n } \\big ) \\big | . \\end{align*}"}
+{"id": "1989.png", "formula": "\\begin{align*} \\Omega = \\{ \\ , ( x ' , x _ n ) \\in \\mathbb { R } ^ { n - 1 } \\times \\mathbb { R } \\ , | \\ , x _ n > \\phi ( x ' ) \\ , \\} \\end{align*}"}
+{"id": "4108.png", "formula": "\\begin{align*} Z _ n & = \\det _ { 0 \\le i , j \\le n - 1 } \\left ( [ u ^ i v ^ j ] \\frac { 1 + u } { 1 - v - 4 u v - u ^ 2 v + u ^ 2 v ^ 2 } \\right ) \\\\ & = \\det _ { 0 \\le i , j \\le n - 1 } \\left ( 2 ^ i \\binom { i + 2 j - 1 } { 2 j + 1 } - \\binom { i - 1 } { 2 j + 1 } \\right ) . \\end{align*}"}
+{"id": "4696.png", "formula": "\\begin{align*} P _ j ( q ) : = \\frac { ( q ^ { 2 j - 2 } - 1 ) } { q ^ { j - 2 } ( q ^ 2 - 1 ) } + j - 1 > n . \\end{align*}"}
+{"id": "3318.png", "formula": "\\begin{align*} P ( p ) = \\frac { \\displaystyle \\prod _ { v = 0 } ^ { t - 1 } y _ { a + v } \\prod _ { i = 0 } ^ { b - t - 1 } x _ { a , i } } { \\prod _ { i = 0 } ^ { b - 1 } z _ i } q ^ { f _ p } = Q ( a , b , t ) q ^ { f _ p } . \\end{align*}"}
+{"id": "944.png", "formula": "\\begin{align*} g ' _ 1 ( \\tilde { x } , \\tilde { t } ) = & \\frac { X _ x ( Z ( \\tilde { x } , \\tilde { t } ) , Y ^ { - 1 } ( \\tilde { t } ) ) } { \\kappa \\tilde { t } ^ { \\alpha - 1 } ( \\mathcal { T } _ t ^ \\alpha Y ) ( Y ^ { - 1 } ( \\tilde { t } ) ) } \\big [ 2 h ( Z ( \\tilde { x } , \\tilde { t } ) , Y ^ { - 1 } ( \\tilde { t } ) ) ( r _ 2 r _ { 1 x } - r _ 1 r _ { 2 x } ) - f _ 1 r _ 1 r _ 2 \\\\ & + g _ 1 r _ 1 ^ 2 + f _ 2 r _ 1 r _ 2 - g _ 2 r _ 2 ^ 2 \\big ] , \\end{align*}"}
+{"id": "5926.png", "formula": "\\begin{align*} d [ - a _ { 1 , 3 } b _ { 1 } ] = \\min \\{ d ( - a _ { 1 , 3 } b _ { 1 } ) , \\alpha _ { 3 } , \\beta _ { 1 } \\} = 1 . \\end{align*}"}
+{"id": "991.png", "formula": "\\begin{align*} \\frac { a b ' + a ' b } { b ' + a ' } & = \\frac { 1 } { 2 } \\cdot \\frac { ( l ^ 2 - n ^ 2 ) ( m ^ 2 + n ^ 2 ) + ( l ^ 2 + n ^ 2 ) ( m ^ 2 - n ^ 2 ) } { l n ( m ^ 2 + n ^ 2 ) + m n ( l ^ 2 + n ^ 2 ) } = \\frac { 1 } { 2 } \\cdot \\frac { 2 ( l ^ 2 m ^ 2 - n ^ 4 ) } { n ( l m + n ^ 2 ) ( l + m ) } = \\frac { l m - n ^ 2 } { ( l + m ) n } , \\\\ \\frac { a b ' - a ' b } { b ' - a ' } & = - \\left ( \\frac { a b ' + a ' b } { b ' + a ' } \\right ) ^ { - 1 } = - \\frac { ( l + m ) n } { l m - n ^ 2 } . \\end{align*}"}
+{"id": "486.png", "formula": "\\begin{align*} _ 1 \\tilde F _ 1 ( a ; b ; z ) = \\frac { 1 } { \\Gamma ( a ) } \\sum _ { k = 0 } ^ \\infty \\frac { \\Gamma ( a + k ) } { k ! \\Gamma ( b + k ) } z ^ k , z \\in \\C . \\end{align*}"}
+{"id": "1959.png", "formula": "\\begin{align*} \\deg _ { x _ p } ( u _ 1 \\cdots u _ N ) + \\deg _ { y _ p } ( u _ 1 \\cdots u _ N ) = \\deg _ { x _ p } ( v _ 1 \\cdots v _ N ) + \\deg _ { y _ p } ( v _ 1 \\cdots v _ N ) = N . \\end{align*}"}
+{"id": "4951.png", "formula": "\\begin{align*} t _ k ( n ) = \\dfrac { 1 } { 2 ^ { 2 k } E _ { 2 k } } \\cdot \\sigma _ { \\chi _ { - 4 } ; 2 k } ( 2 n + k + 1 ) + a _ k ( 2 n + k + 1 ) . \\end{align*}"}
+{"id": "591.png", "formula": "\\begin{align*} \\norm { \\psi ( t ) } _ { L ^ 2 } ^ 2 = \\norm { \\psi _ 0 } _ { L ^ 2 } ^ 2 \\end{align*}"}
+{"id": "6640.png", "formula": "\\begin{align*} { \\rho } _ { ( 1 ) , \\infty } ^ { ( \\rm c J ) } ( x ; \\beta , p , q ) | _ { \\beta = 2 \\atop q = 0 } = { \\pi ^ 2 | x | \\over 2 } \\Big ( ( J _ { p - 1 / 2 } ( \\pi x ) ) ^ 2 + ( J _ { p + 1 / 2 } ( \\pi x ) ) ^ 2 - { 2 p \\over \\pi x } J _ { p - 1 / 2 } ( \\pi x ) J _ { p + 1 / 2 } ( \\pi x ) \\Big ) . \\end{align*}"}
+{"id": "6374.png", "formula": "\\begin{align*} \\chi = a ( 1 \\otimes 1 ) + n ( x \\otimes x ) - n ( x g \\otimes x g ) \\end{align*}"}
+{"id": "1197.png", "formula": "\\begin{align*} ( A _ { N _ \\alpha } ^ G ) ^ { h H _ \\alpha } \\simeq ( \\prod _ { G / N _ \\alpha } k ) ^ { h H _ \\alpha } \\simeq \\prod _ { G / N _ \\alpha H _ \\alpha } k \\simeq A ^ G _ { N _ \\alpha H _ \\alpha } = A ^ G _ { H _ \\alpha } \\end{align*}"}
+{"id": "8943.png", "formula": "\\begin{align*} \\alpha _ s ( \\Lambda ) = s u p \\{ d _ { \\Lambda } ( V ) ^ { - 1 } : \\} . \\end{align*}"}
+{"id": "3839.png", "formula": "\\begin{align*} { \\rm A d m } ( \\mu _ 0 , \\mu _ 1 ) : = \\{ \\rho , v , r \\ , \\mid \\ , \\partial _ t \\rho + \\div ( \\rho v ) = \\rho r , \\rho | _ { t = i } = \\mu _ i \\} . \\end{align*}"}
+{"id": "6918.png", "formula": "\\begin{align*} c [ \\psi ( \\xi _ n ) - \\psi ( 0 ) ] - d _ 2 \\int _ { 0 } ^ { \\xi _ n } \\mathcal { N } _ 2 \\left [ \\psi \\right ] ( \\xi ) d \\xi = \\int _ { 0 } ^ { \\xi _ n } \\psi ( \\xi ) g ( \\phi , \\psi ) ( \\xi ) d \\xi . \\end{align*}"}
+{"id": "4386.png", "formula": "\\begin{align*} \\hat a ^ \\pm : = \\sqrt { \\frac { 1 } { \\hat { \\rho } ^ { \\pm } ( 1 + \\frac { ( \\hat { c } ^ { \\pm } _ A ) ^ 2 } { ( \\hat { c } ^ { \\pm } ) ^ 2 } ) } } \\ , . \\end{align*}"}
+{"id": "7290.png", "formula": "\\begin{align*} \\tilde { A } = \\{ n \\mid n n \\leq 2 N \\} \\cup \\{ 2 n + 1 \\mid n \\in A \\} . \\end{align*}"}
+{"id": "775.png", "formula": "\\begin{align*} U _ { \\ell } = \\prod _ { i = 1 } ^ { \\ell } \\ \\{ x _ i \\} ~ \\times ~ \\prod _ { i = \\ell + 1 } ^ { \\infty } \\ X _ { q _ i , 2 , 1 } \\end{align*}"}
+{"id": "5952.png", "formula": "\\begin{align*} \\sigma _ 1 ( \\Lambda _ { g , F } ^ \\omega ) & = - \\sqrt { \\widetilde { q _ 2 } } , \\\\ \\sigma _ 0 ( \\Lambda _ { g , F } ^ \\omega ) & = \\frac { 1 } { 2 \\sqrt { \\widetilde { q _ 2 } } } ( \\nabla _ { \\xi ' } \\sqrt { \\widetilde { q _ 2 } } \\cdot D _ { t ' } \\sqrt { \\widetilde { q _ 2 } } - \\widetilde { q _ 1 } - \\partial _ { t _ 3 } \\sqrt { \\widetilde { q _ 2 } } + \\widetilde { E } \\sqrt { \\widetilde { q _ 2 } } ) , \\end{align*}"}
+{"id": "4076.png", "formula": "\\begin{align*} \\omega ( p _ 0 , . . , p _ n ) = f ( z _ 0 , . . , z _ n ) d z _ 0 \\otimes \\cdots \\otimes d z _ n , \\end{align*}"}
+{"id": "290.png", "formula": "\\begin{align*} \\partial \\Omega = \\Gamma _ { \\operatorname * { D } } \\cup \\Gamma _ { \\operatorname * { N } } \\end{align*}"}
+{"id": "1490.png", "formula": "\\begin{align*} & X _ { j } = \\frac { \\partial } { \\partial \\overline { x } _ { j } } - \\frac { 1 } { 2 } \\sum _ { s = 1 } ^ { n } \\left ( \\sum _ { i = 1 } ^ { m } U ^ { ( s ) } _ { j i } \\overline { x } _ { i } \\right ) \\frac { \\partial } { \\partial \\widehat { x } _ { s } } , & j = 1 , \\ldots , m , \\\\ & Z _ { i } = \\frac { \\partial } { \\partial \\widehat { x } _ { i } } , & i = 1 , \\ldots , n . \\end{align*}"}
+{"id": "3016.png", "formula": "\\begin{align*} \\left < \\left [ \\delta \\phi \\right ] , \\alpha \\right > = n \\cdot \\phi \\left ( w \\right ) . \\end{align*}"}
+{"id": "3621.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Vert \\phi _ n - \\phi \\Vert _ { L ^ 1 ( y _ n ( \\Omega ) \\cap y ( \\Omega ) ) } = 0 . \\end{align*}"}
+{"id": "6078.png", "formula": "\\begin{align*} T ^ { - 1 } \\left ( T ^ { k + 1 } a _ { + } \\right ) \\cap \\bigcup _ { j = 0 } ^ { \\infty } T ^ j \\left ( \\{ a _ { + } \\} \\cup \\{ a _ { - } \\} \\right ) = T ^ { k } a _ { + } , k \\ge 1 . \\end{align*}"}
+{"id": "5117.png", "formula": "\\begin{align*} g ^ \\flat ( D ^ r _ Z ( X ) ) - D ^ { r , * } _ Z ( g ^ \\flat ( X ) ) = \\ , & g ^ \\flat ( \\nabla _ Z \\ , r ( X ) ) - r ^ * ( g ^ \\flat ( \\nabla _ Z X ) , \\\\ D ^ { r , * } _ Y ( g ^ \\flat ( \\nabla _ X Z ) ) - D ^ { r , * } _ X ( g ^ \\flat ( \\nabla _ Y Z ) ) + r ^ * g ^ \\flat ( \\nabla _ { [ X , Y ] } Z ) = \\ , & g ^ \\flat \\left ( \\nabla _ X D ^ r _ Y ( Z ) - \\nabla _ Y D ^ r _ X ( Z ) \\vphantom { D ^ r _ { [ X , Y ] } ( Z ) } \\right . \\\\ & \\left . - D ^ r _ { [ X , Y ] } ( Z ) - \\nabla _ { [ X , Y ] _ r } Z \\right ) . \\end{align*}"}
+{"id": "7215.png", "formula": "\\begin{align*} \\lim _ { \\delta \\rightarrow 0 } \\mathcal { U } ^ { \\delta } ( t _ 0 , \\mu _ 0 ) = \\inf _ { \\delta > 0 } \\mathcal { U } ^ { \\delta } ( t _ 0 , \\mu _ 0 ) = \\mathcal { U } ( t _ 0 , \\mu _ 0 ) + \\gamma \\end{align*}"}
+{"id": "924.png", "formula": "\\begin{align*} V _ 3 = & t ^ { 1 + \\alpha } \\partial _ t + \\alpha x t ^ \\alpha \\partial _ x - \\bigg ( \\bigg ( \\frac { \\alpha ( c + 1 ) t ^ \\alpha } { 2 } + \\frac { \\alpha ^ 2 x ^ 2 } { 4 } \\bigg ) u + \\frac { m \\alpha t ^ \\alpha } { 2 } v \\bigg ) \\partial _ u - \\\\ & \\bigg ( \\bigg ( \\frac { \\alpha ( c + 1 ) t ^ \\alpha } { 2 } + \\frac { \\alpha ^ 2 x ^ 2 } { 4 } \\bigg ) v + \\frac { n \\alpha t ^ \\alpha } { 2 } u \\bigg ) \\partial _ v , \\end{align*}"}
+{"id": "3894.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { | a _ N | ^ 2 } { \\sum _ { n = 1 } ^ N | a _ n | ^ 2 } = 0 . \\end{align*}"}
+{"id": "4867.png", "formula": "\\begin{align*} y ( u ) + y ( v ) - y ( u + v ) \\left [ \\left ( 1 - \\frac { 4 } { \\delta ^ 2 } \\right ) y ( u ) y ( v ) \\right ] = 0 \\end{align*}"}
+{"id": "136.png", "formula": "\\begin{align*} \\Big ( X - F - m \\hat { H _ 1 } - m \\hat { H _ 2 } \\Big ) = [ 0 ] _ { m \\times 1 } \\end{align*}"}
+{"id": "6151.png", "formula": "\\begin{align*} & ( \\cdots , f _ i - f _ j , f _ j - f _ k , f _ k - f _ l , \\cdots ) \\\\ & \\qquad = ( \\cdots , ( f _ i - f _ j ) + ( f _ j - f _ k ) , f _ j - f _ k , ( f _ k - f _ l ) + ( f _ j - f _ k ) , \\cdots ) \\\\ & \\qquad = ( \\cdots , f _ i - f _ k , f _ j - f _ k , f _ j - f _ l , \\cdots ) \\\\ & \\qquad = - ( \\cdots , f _ i - f _ k , f _ k - f _ j , f _ j - f _ l , \\cdots ) , \\end{align*}"}
+{"id": "4874.png", "formula": "\\begin{align*} T ^ 2 = \\begin{bmatrix} 0 & 0 & 0 & 0 \\\\ 0 & - 1 & 1 & 0 \\\\ 0 & 1 & - 1 & 0 \\\\ 0 & 0 & 0 & 0 \\end{bmatrix} ^ 2 = \\begin{bmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 2 & - 2 & 0 \\\\ 0 & - 2 & 2 & 0 \\\\ 0 & 0 & 0 & 0 \\end{bmatrix} = - 2 T \\end{align*}"}
+{"id": "6357.png", "formula": "\\begin{align*} g _ { [ < \\lambda ] } - g _ { [ < \\lambda ] } ^ { x _ 0 } = \\int _ { 0 } ^ { x _ 0 } \\partial _ x g _ { [ < \\lambda ] } ^ y d y . \\end{align*}"}
+{"id": "3528.png", "formula": "\\begin{align*} \\vert a + d \\vert = \\vert \\mathrm { t r } ( \\gamma ) \\vert \\geqslant q ^ { 2 } - 2 \\geqslant q ^ 2 / 2 . \\end{align*}"}
+{"id": "6726.png", "formula": "\\begin{align*} \\frac { H } { 2 | \\nabla u | _ { g } } = \\eta ( r ) = \\mathfrak { c } _ { p } ^ { - 1 } r ^ { a } \\left ( 1 + \\frac { m } { 2 r } \\right ) ^ { 2 a - 1 } \\left ( ( 1 - \\frac { m } { 2 r } \\right ) . \\end{align*}"}
+{"id": "4120.png", "formula": "\\begin{align*} \\sum _ { j = a } ^ { a + b } \\binom b { j - a } & \\sum _ { \\ell = 0 } ^ { k - 2 i + j } \\binom { { n - 1 } + k - 2 i - \\ell } { { n - 1 } - j - \\ell , k - 2 i + j - \\ell , \\ell } \\\\ & = \\sum _ { \\ell = 0 } ^ { k - 2 i + a + b } \\binom { { n - 1 } + k - 2 i - \\ell } { \\ell } \\sum _ { j = a } ^ { a + b } \\binom b { a + b - j } \\binom { { n - 1 } + k - 2 i - 2 \\ell } { k - 2 i + j - \\ell } \\\\ & = \\sum _ { \\ell = 0 } ^ { k - 2 i + a + b } \\binom { { n - 1 } + k - 2 i - \\ell } { \\ell } \\binom { { n - 1 } + k + b - 2 i - 2 \\ell } { k + a + b - 2 i - \\ell } . \\end{align*}"}
+{"id": "157.png", "formula": "\\begin{align*} \\gamma _ n = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n ( - 1 ) ^ { n - i } \\gamma ^ i _ n . \\end{align*}"}
+{"id": "6709.png", "formula": "\\begin{align*} \\prod _ { i : 1 \\to N } P _ i : = P _ 1 P _ 2 \\cdots P _ N , \\prod _ { i : N \\to 1 } P _ i : = P _ N \\cdots P _ 2 P _ 1 . \\end{align*}"}
+{"id": "3585.png", "formula": "\\begin{align*} { \\bf { y } } = { \\bf { W } } _ { { \\rm { S M } } } ^ H { \\bf { H } } { \\bf F } _ { \\rm S M } { \\bf s } + { \\bf { W } } _ { { \\rm { S M } } } ^ H { \\bf { n } } \\end{align*}"}
+{"id": "8397.png", "formula": "\\begin{align*} \\Sigma _ { j } ^ { ( 1 ) } = \\sum _ { d \\leq D } \\lambda ( d ) \\sum _ { 0 < | h | \\leq H } a _ { h } W \\left ( - \\frac { h } { d } \\right ) \\ll \\sum _ { d \\leq D } \\sum _ { 1 \\leq h \\leq H } \\frac { 1 } { h } \\left | W \\left ( \\frac { h } { d } \\right ) \\right | . \\end{align*}"}
+{"id": "6394.png", "formula": "\\begin{align*} \\chi ( a \\otimes b c ) & = \\chi ( a \\otimes b ) \\varepsilon ( c ) + \\chi ( a \\triangleleft b \\otimes c ) , \\\\ \\chi ( a b \\otimes c ) & = \\varepsilon ( a ) \\chi ( b \\otimes c ) + \\chi ( a \\otimes b \\triangleright c ) \\end{align*}"}
+{"id": "595.png", "formula": "\\begin{align*} I _ + + I _ - = - 2 M _ { \\mathfrak K _ 1 } \\int _ 0 ^ t \\norm { P _ { \\mu } \\psi ^ \\mu ( s ) } _ { L ^ 2 } ^ 2 d s = - \\int _ 0 ^ t \\norm { \\left ( P _ { \\mu } \\psi ^ \\mu ( s ) \\right ) \\mathfrak K _ 1 } _ { \\mathcal L _ 2 } ^ 2 d s , \\end{align*}"}
+{"id": "6518.png", "formula": "\\begin{align*} | v _ { l _ 1 } ^ { - 2 } ( m ) - v _ { l _ 2 } ^ { - 2 } ( m ) | & = | v _ { l _ 1 } ^ { - 2 } ( m ) v _ { l _ 2 } ^ { - 2 } ( m ) | \\cdot | \\mu _ { n ^ { ( l _ 1 ) } } ^ 2 - \\mu _ { n ^ { ( l _ 2 ) } } ^ 2 | \\\\ & \\geq \\frac { 1 } { 2 \\pi ^ 2 } c _ \\star ^ 2 , \\end{align*}"}
+{"id": "5803.png", "formula": "\\begin{align*} & \\varepsilon _ i + \\varepsilon _ { i + 1 } = \\alpha _ i + 2 \\alpha _ { i + 1 } + \\dots + 2 \\alpha _ { n - 1 } + 2 \\alpha _ n , \\\\ & p _ i = s _ { \\varepsilon _ i + \\varepsilon _ { i + 1 } } s _ { \\alpha _ i } . \\end{align*}"}
+{"id": "1729.png", "formula": "\\begin{align*} \\det \\left [ H ^ { ( m , n ) } _ { \\texttt { b } ; j , k } ( \\boldsymbol { \\xi } ) \\right ] _ { 1 \\leq j , k \\leq n } & = \\det \\Bigl ( \\bigl [ 2 ( m + n ) + u _ { q _ 0 } ( \\xi _ j ) + u _ { q _ 1 } ( \\xi _ j ) \\bigr ] _ { 1 \\leq j \\leq n } \\Bigr ) \\\\ & = \\prod _ { 1 \\leq j \\leq n } \\bigl ( 2 ( m + n ) + u _ { q _ 0 } ( \\xi _ j ) + u _ { q _ 1 } ( \\xi _ j ) \\bigr ) . \\end{align*}"}
+{"id": "5236.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\Big ( \\sum _ { t = 1 } ^ { T } l _ { t } ( x _ { i , t } ) - \\min _ { x \\in \\mathcal { X } } \\sum _ { t = 1 } ^ { T } l _ { t } ( x ) \\Big ) . \\end{align*}"}
+{"id": "3288.png", "formula": "\\begin{align*} \\frac { d ^ \\ell } { d t ^ \\ell } \\phi _ X ( t ) & = \\frac { d } { d t } \\left ( \\frac { d ^ { \\ell - 1 } } { d x ^ { \\ell - 1 } } \\phi _ X ( t ) \\right ) \\\\ & = \\frac { d } { d t } \\left ( \\int _ \\mathbb { R } x ^ { \\ell - 1 } f _ X ( x ) e ^ { \\mu t x } d x \\mu ^ { \\ell - 1 } \\right ) \\\\ & = \\int _ \\mathbb { R } x ^ { \\ell - 1 } f _ X ( x ) \\frac { d } { d t } \\left ( e ^ { \\mu t x } \\right ) d x \\mu ^ { \\ell - 1 } \\\\ & = \\int _ \\mathbb { R } x ^ \\ell f _ X ( x ) e ^ { \\mu t x } d x \\mu ^ \\ell . \\end{align*}"}
+{"id": "7972.png", "formula": "\\begin{align*} = 1 . \\end{align*}"}
+{"id": "995.png", "formula": "\\begin{align*} ( 1 - \\beta _ k ) ^ { c \\ell } & = ( 1 - ( 1 - \\frac 1 q ) ^ k ) ^ { c \\ell } = ( 1 - e ^ { \\delta k } ) ^ { c \\ell } \\\\ & = ( 1 - e ^ { \\delta \\ln ( n ) } ) ^ { c \\ell } = ( 1 - n ^ { \\delta } ) ^ { c \\ell } = \\left ( ( 1 - \\frac 1 { n ^ { - \\delta } } ) ^ { n ^ { - \\delta } } \\right ) ^ { \\frac { c n ^ { 1 + \\delta } } { \\ln ( n ) } } \\\\ \\end{align*}"}
+{"id": "1163.png", "formula": "\\begin{align*} & \\Phi _ t \\circ m _ { 1 , t } \\rq = m _ { 1 , t } \\circ ( \\Phi _ t \\otimes \\Phi _ t ) , \\\\ & \\Phi _ t \\circ m _ { 2 , t } \\rq = m _ { 2 , t } \\circ ( \\Phi _ t \\otimes \\Phi _ t ) . \\end{align*}"}
+{"id": "8925.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\mathrm { t r a c e } ( A ) & = & \\sum _ { i = 1 } ^ { n } ( A ( E _ i , E _ i ) + A ( J E _ i , J E _ i ) ) \\\\ \\\\ & = & - \\sum _ { i = 1 } ^ { n } g ( E _ i , E _ i ) B ^ \\perp \\\\ \\\\ & = & - n B ^ \\perp . \\end{array} \\end{align*}"}
+{"id": "1074.png", "formula": "\\begin{align*} \\left . \\begin{array} { c } 2 z _ { 1 } z _ { 4 } \\psi ^ { - \\nu + 1 } = - 2 \\lambda ^ { 2 } \\nu ^ { 2 } \\psi ^ { \\nu - 1 } \\left [ 1 - 2 \\psi ^ { - \\nu } \\left ( \\nu - 1 \\right ) / \\left ( \\lambda \\nu \\right ) \\right ] v _ { t } v = \\partial _ { t } V _ { 2 } , \\\\ V _ { 2 } = - \\lambda ^ { 2 } \\nu ^ { 2 } \\psi ^ { \\nu - 1 } \\left [ 1 - 2 \\psi ^ { - \\nu } \\left ( \\nu - 1 \\right ) / \\left ( \\lambda \\nu \\right ) \\right ] v ^ { 2 } . \\end{array} \\right . \\end{align*}"}
+{"id": "8195.png", "formula": "\\begin{align*} \\kappa _ \\alpha & = d q _ 0 \\wedge d q _ \\alpha + d q _ \\beta \\wedge d q _ \\gamma , \\end{align*}"}
+{"id": "2628.png", "formula": "\\begin{align*} U _ n ( x , t ) = K _ n ( F ( x ) , \\frac { \\hat A _ n ( t ) } { n } ) \\ , . \\end{align*}"}
+{"id": "6505.png", "formula": "\\begin{align*} ( \\mu _ n ^ 2 - ( k \\cdot \\omega ) ^ 2 ) q ( k , n ) + \\varepsilon \\Delta q ( k , n ) + \\delta q _ * ^ { p + 1 } ( k , n ) = 0 , \\end{align*}"}
+{"id": "8126.png", "formula": "\\begin{align*} \\| u \\| ^ p _ { \\dot { W } ^ { s , p } ( \\R ^ N ) } = \\iint _ { \\R ^ N \\times \\R ^ N } \\frac { | u ( x ) - u ( y ) | ^ p } { | x - y | ^ { N + s p } } d x d y \\end{align*}"}
+{"id": "7508.png", "formula": "\\begin{align*} \\binom { \\frac { q ^ { \\ell } - 1 } { q - 1 } } { 2 } = \\frac { q } { 2 } \\frac { q ^ \\ell - 1 } { q - 1 } \\frac { q ^ { \\ell - 1 } - 1 } { q - 1 } , \\ell \\geq 2 . \\end{align*}"}
+{"id": "599.png", "formula": "\\begin{align*} s = 0 , \\frac 1 4 < r < \\frac 1 2 , \\max ( r , 1 - 2 r ) < b < 1 / 2 . \\end{align*}"}
+{"id": "5912.png", "formula": "\\begin{align*} c ^ { \\# } : = \\begin{cases} \\Delta & , \\\\ 1 + 4 \\rho r ^ { - 1 } \\pi ^ { - d ( c ) } & . \\end{cases} \\end{align*}"}
+{"id": "5071.png", "formula": "\\begin{align*} \\partial _ m E ( A z ) = \\partial _ m \\sum _ { p \\geq 0 } \\frac { A ^ p } { m ( p ) } z ^ p = \\sum _ { p \\geq 0 } \\frac { A ^ { p + 1 } } { m ( p ) } z ^ p = A \\sum _ { p \\geq 0 } \\frac { A ^ p } { m ( p ) } z ^ p = A E ( A z ) . \\end{align*}"}
+{"id": "4604.png", "formula": "\\begin{align*} \\mathrm { B l } _ { \\mathcal { C } } ( \\mathcal { S } ) \\ , { : = } \\ , \\mathrm { B l } _ { \\mathcal { C } } ( \\varSigma _ { \\mathcal { S } } ; \\mathbb { F } _ { \\ell _ { \\mathrm { i n } } } ( \\mathsf { b } _ { \\mathrm { i n } } ) , \\mathbb { F } _ { \\ell _ { \\mathrm { o u t } } } ( \\mathsf { b } _ { \\mathrm { o u t } } ) ) \\ , , \\end{align*}"}
+{"id": "605.png", "formula": "\\begin{align*} \\norm { \\Theta \\psi _ - ^ R } _ { X ^ { 0 , 0 } _ { - \\xi } ( 0 , T ) } = \\norm { \\Theta \\psi _ - ^ R } _ { L ^ 2 ( ( 0 , T ) \\times \\R ) } \\le \\sqrt { T } \\norm { \\psi _ 0 } _ { L ^ 2 } . \\end{align*}"}
+{"id": "7243.png", "formula": "\\begin{align*} R = \\frac { \\alpha } { \\gamma } , \\end{align*}"}
+{"id": "7564.png", "formula": "\\begin{align*} P \\left ( S _ { n } > \\alpha n \\right ) & = P \\big ( \\exp \\left ( t S _ { n } \\right ) > \\exp \\left ( \\alpha t n \\right ) \\big ) \\\\ & \\le \\frac { E \\left [ \\exp \\left ( t S _ { n } \\right ) \\right ] } { \\exp \\left ( \\alpha t n \\right ) } \\\\ & = \\exp \\left ( - \\alpha t n \\right ) \\left [ q + p e ^ t \\right ] ^ { n } \\\\ & = \\left ( e ^ { - \\alpha t } \\left [ q + p e ^ t \\right ] \\right ) ^ { n } . \\end{align*}"}
+{"id": "1671.png", "formula": "\\begin{align*} m ( \\xi _ j - \\xi _ k ) + \\sum _ { 1 \\leq l \\leq n } \\Bigl ( v _ q ( \\xi _ j - \\xi _ l ) - v _ q ( \\xi _ k - \\xi _ l ) \\Bigr ) = 2 \\pi ( \\lambda _ j - \\lambda _ k + k - j ) , \\end{align*}"}
+{"id": "1001.png", "formula": "\\begin{align*} \\Omega _ - \\cap B ( x _ 0 , r ) & = \\{ ( x _ 1 , x _ 2 , x _ 3 ) \\in B ( x _ 0 , r ) : x _ 3 < \\psi ( x _ 1 , x _ 2 ) \\} \\\\ \\Omega _ + \\cap B ( x _ 0 , r ) & = \\{ ( x _ 1 , x _ 2 , x _ 3 ) \\in B ( x _ 0 , r ) : x _ 3 > \\psi ( x _ 1 , x _ 2 ) \\} . \\end{align*}"}
+{"id": "3462.png", "formula": "\\begin{align*} E = \\{ x \\in \\Delta : \\exists p \\in \\nabla u ( x ) \\cap \\cup _ k \\partial \\bar { \\Delta } ^ \\vee _ k \\} \\end{align*}"}
+{"id": "3475.png", "formula": "\\begin{align*} \\phi _ { k , t } = \\max \\{ \\frac { 1 } { k } \\log \\Pi _ { i = 0 } ^ m | F _ i | _ { h ^ { \\otimes d _ i } } ^ { - k p _ i } + \\frac { 1 + \\sum d _ i p _ i } { N _ 0 } \\max \\log | \\tau _ l | _ { h ^ { \\otimes N _ 0 } } - u ^ * ( p ) | \\log | t | | : p \\in \\Delta ^ \\vee \\cap \\frac { 1 } { k } \\Z ^ { m + 1 } \\} . \\end{align*}"}
+{"id": "103.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\rho _ z ^ 2 + \\rho ^ 2 - \\rho \\rho _ z = \\frac { 1 } { 2 } ( \\rho - \\rho _ z ) ^ 2 + \\frac { 1 } { 2 } \\rho ^ 2 , \\end{align*}"}
+{"id": "7195.png", "formula": "\\begin{align*} \\mathfrak { a } _ 2 ^ 2 \\mathfrak { a } _ 3 ^ 3 & = ( 1 \\otimes y _ 2 ^ 2 + y _ 2 \\otimes 1 ) ( 1 \\otimes y _ 3 ^ 3 - y _ 3 \\otimes y _ 3 ^ 2 + y _ 3 ^ 2 \\otimes y _ 3 - y _ 3 ^ 3 \\otimes 1 ) \\\\ & = ( y _ 1 y _ 2 + y _ 2 y _ 3 ) \\otimes y _ 1 y _ 2 y _ 3 - y _ 1 y _ 2 y _ 3 \\otimes ( y _ 1 y _ 2 + y _ 2 y _ 3 ) . \\end{align*}"}
+{"id": "8037.png", "formula": "\\begin{align*} \\vert \\widehat { \\phi _ 0 } ( \\xi ) \\vert ^ { 2 } + \\sum \\limits _ { j = 1 } ^ { \\infty } \\vert \\widehat { \\phi } ( 2 ^ { - j } \\xi ) \\vert ^ { 2 } = 1 . \\end{align*}"}
+{"id": "969.png", "formula": "\\begin{align*} K = O ( \\max \\{ k ( 1 + \\log ( k ) / r ( \\lambda ) ) , r ( \\lambda ) \\} ) . \\end{align*}"}
+{"id": "2142.png", "formula": "\\begin{align*} u _ 0 ( t , x ) = d \\ln \\alpha , \\end{align*}"}
+{"id": "7620.png", "formula": "\\begin{align*} U ( x ) : = \\log ( x ) 1 _ { \\{ x > 1 \\} } + [ x - 1 ] 1 _ { \\{ x \\leq 1 \\} } . \\end{align*}"}
+{"id": "8832.png", "formula": "\\begin{align*} d \\mathbf { P } _ { \\omega , T } ( z _ 1 , y _ { 1 } , \\dots , y _ { T } ) = d F \\big ( y _ { 1 } - f _ \\omega ( z _ { 1 } ) \\big ) \\prod _ { i = 2 } ^ { T } d F \\Big ( y _ { i } - f _ \\omega \\big ( \\Phi _ i ( z _ 1 , y _ 1 , \\dots , y _ { i - 1 } ) \\big ) \\Big ) . \\end{align*}"}
+{"id": "1204.png", "formula": "\\begin{align*} ( A _ P ^ G ) ^ { h U } \\simeq ( \\prod _ { G / P } k ) ^ { V / P } \\simeq \\prod _ { G / V } k = A ^ G _ V . \\end{align*}"}
+{"id": "3106.png", "formula": "\\begin{align*} \\mathbf { G } _ { } = \\mathbf { b } _ R ( \\psi _ 1 , \\theta _ 1 ) \\mathbf { a } ^ H _ T ( \\psi ^ { } ) . \\end{align*}"}
+{"id": "1352.png", "formula": "\\begin{align*} & h ^ 0 ( \\mathcal { Y } _ s , m \\epsilon F '' _ s + g _ s ^ * ( m \\epsilon \\mathcal { H } _ s + m f _ s ^ * \\mathcal { O } ( 1 ) ) - k a _ { 0 j } G _ s ) \\\\ & = h ^ 0 ( \\mathcal { Y } _ s , m \\epsilon F '' _ s + g _ s ^ * ( m \\epsilon \\mathcal { H } _ s + ( m - k ) f _ s ^ * \\mathcal { O } ( 1 ) ) ) . \\end{align*}"}
+{"id": "1827.png", "formula": "\\begin{gather*} F _ g ( 0 ) = \\left \\langle 1 , g ( s ) \\right \\rangle = 0 \\quad \\frac { d ^ k } { d z ^ k } F _ g ( z ) \\bigg | _ { z = 0 } = ( - 1 ) ^ k \\langle s ^ k , g ( s ) \\rangle = 0 \\end{gather*}"}
+{"id": "4564.png", "formula": "\\begin{align*} \\tilde { \\mathcal F } _ t : = \\partial _ t \\tilde { \\mathcal { F } } - \\partial _ t \\mathcal B _ 1 \\partial _ 1 { \\mathbf V } - \\partial _ t \\mathcal B _ 2 \\partial _ 2 { \\mathbf V } - \\partial _ t \\mathcal B _ 3 { \\mathbf V } \\ , . \\end{align*}"}
+{"id": "296.png", "formula": "\\begin{align*} \\gamma _ { \\mathbf { n } ; j } ^ { - } \\left ( \\mbox { \\boldmath $ \\psi $ } ^ { - } \\right ) = \\left \\langle \\left . \\mbox { \\boldmath $ \\psi $ } ^ { - } \\right \\vert _ { \\Gamma _ { j } } , \\mathbf { n } _ { j } \\right \\rangle \\quad \\gamma _ { \\mathbf { n } ; j } ^ { + } \\left ( \\mbox { \\boldmath $ \\psi $ } ^ { + } \\right ) = \\left \\langle \\left . \\mbox { \\boldmath $ \\psi $ } ^ { + } \\right \\vert _ { \\Gamma _ { j } } , - \\mathbf { n } _ { j } \\right \\rangle , \\end{align*}"}
+{"id": "1587.png", "formula": "\\begin{align*} A : = ( 1 + ( \\mathcal { T } \\Pi ) ^ * \\mathcal { T } \\Pi ) ^ { - 1 } ( \\mathcal { T } \\Pi ) ^ * . \\end{align*}"}
+{"id": "5113.png", "formula": "\\begin{align*} D _ { \\rho ( a ) } ^ * ( \\mu ) & = i _ a d _ A ( \\mu _ l ) - i _ { l ( a ) } d _ A \\mu , \\\\ i _ a d _ A D _ X ^ * ( \\mu ) & = D _ X ^ * ( d _ A \\mu ) ( a ; \\ , \\cdot ) + \\mathcal { R } _ X ( \\mu ) ( a ; \\ , \\cdot ) , \\end{align*}"}
+{"id": "6007.png", "formula": "\\begin{align*} W _ { 1 } ( \\nu , \\mu ) = \\sup \\{ \\left \\vert \\int _ { \\mathbb { R } ^ d } \\varphi d ( \\nu - \\mu ) \\right \\vert : \\left \\Vert \\nabla \\varphi \\right \\Vert _ { \\infty } \\leq 1 \\} . \\end{align*}"}
+{"id": "4003.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( S _ { n - 1 ; i } ( X ) - \\frac { c } { C ^ m _ n } S _ { m - 1 ; i } ( X ) \\right ) X _ { i \\bar i } & \\leq \\sum _ l \\left ( S _ { n - 1 ; l } ( X ) - \\frac { c } { C ^ m _ n } S _ { m - 1 ; l } ( X ) \\right ) X _ { l \\bar l } \\\\ & \\leq n S _ n ( X ) , \\end{aligned} \\end{align*}"}
+{"id": "2037.png", "formula": "\\begin{align*} ( \\mathrm { L } ^ p ( \\partial \\Omega ) , \\dot { \\mathrm { H } } ^ { 1 , p } ( \\partial \\Omega ) ) _ { s , p } = \\dot { \\mathrm { B } } ^ { s } _ { p , p } ( \\partial \\Omega ) \\end{align*}"}
+{"id": "4866.png", "formula": "\\begin{align*} & f ( u ) + f ( v ) + f ( u ) f ( v ) \\delta + f ( u ) f ( u + v ) f ( v ) - f ( u + v ) = 0 \\ ; . \\end{align*}"}
+{"id": "3595.png", "formula": "\\begin{align*} { \\rm { t } } { { \\rm { r } } _ k } \\left ( { \\bf { A } } \\right ) = \\sum \\limits _ { \\left \\{ { \\vec { a } } \\right \\} } { \\prod \\limits _ { i = 1 } ^ { k } { \\lambda _ { x , a _ i } } } = \\sum \\limits _ { \\left \\{ { \\vec { a } } \\right \\} } { \\left | { { { \\left [ { \\bf { A } } \\right ] } _ { { \\vec { a } } , { \\vec { a } } } } } \\right | } \\end{align*}"}
+{"id": "8890.png", "formula": "\\begin{align*} f _ * \\bigg ( \\frac { 1 } { 1 - x } \\bigg ) = \\frac { 1 } { c ( E ) } . \\end{align*}"}
+{"id": "7791.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\hat { \\gamma } } ^ { ( \\nu ) } = \\sum _ { q _ 0 \\in \\mathbb { Z } } f ( x + q _ 0 , y , \\zeta ^ i , R ) \\end{align*}"}
+{"id": "4994.png", "formula": "\\begin{align*} ( n + 1 ) g _ { n + 1 } = \\sum _ { k = 0 } ^ n ( k + 1 ) f _ { k + 1 } g _ { n - k } \\end{align*}"}
+{"id": "4548.png", "formula": "\\begin{align*} \\partial _ 1 { \\mathbf V } _ \\sigma = \\partial _ 1 ( \\sigma \\partial _ 1 { \\mathbf V } ) = \\sigma \\partial _ 1 \\partial _ 1 { \\mathbf V } + \\sigma ^ \\prime \\partial _ 1 { \\mathbf V } \\ , ; \\end{align*}"}
+{"id": "6481.png", "formula": "\\begin{align*} f ' ( 0 ) = \\gamma f ( 0 ) , - f ' ( \\epsilon ) = 0 , \\end{align*}"}
+{"id": "5055.png", "formula": "\\begin{align*} \\langle \\Lambda ( u ) , v \\rangle _ { g ' , f ' } = \\langle u , v \\rangle _ { g , f } , \\end{align*}"}
+{"id": "3338.png", "formula": "\\begin{align*} S _ { Q , \\zeta } ( \\varepsilon ) = \\biggl ( \\prod _ { i = 1 } ^ { N } D _ { \\zeta _ i } ( \\zeta _ i \\theta _ i ) ^ { - \\frac { 1 } { m } } \\biggr ) \\sum _ { k \\in ( \\mathbb { Z } / m \\mathbb { Z } ) ^ N } \\zeta ^ { \\overline { Q ( k ) } } \\biggl ( \\prod _ { i = 1 } ^ { N } \\frac { \\theta _ i ^ { ( k ^ \\mathsf { T } A ) _ i } } { ( \\zeta _ i \\theta _ i ; \\zeta _ i ) _ { k _ i } } \\biggr ) I _ { Q , \\zeta } ( k , \\varepsilon ) , \\end{align*}"}
+{"id": "1923.png", "formula": "\\begin{align*} \\Pi _ { \\lambda _ 1 , \\lambda _ 2 } g = \\left ( \\Pi _ { \\lambda _ 1 } \\otimes \\Pi _ { \\lambda _ 2 } \\right ) g , \\forall g \\in \\mathcal { C } ^ 0 ( \\overline { T _ { i j } } ) , 1 \\leq i \\leq N _ x , 1 \\leq j \\leq N _ v . \\end{align*}"}
+{"id": "1222.png", "formula": "\\begin{align*} C _ 0 \\leq \\lim _ { n \\to \\infty } \\sum _ { j = 1 } ^ { \\infty } P ( g _ n ^ j [ \\phi ^ j ] ) \\leq C _ 0 \\sum _ { j = 1 } ^ { \\infty } \\| \\nabla \\phi ^ j \\| _ { L ^ 2 } ^ { 2 p } . \\end{align*}"}
+{"id": "2672.png", "formula": "\\begin{align*} ( t \\omega _ { 0 } - _ { \\omega _ { 0 } } + d d ^ c \\varphi _ t ) ^ { n } = e ^ { ( 1 + \\frac { \\alpha } { 2 \\beta } ) \\varphi _ t } \\omega _ { 0 } ^ { n } \\end{align*}"}
+{"id": "3596.png", "formula": "\\begin{align*} { \\rm { t } } { { \\rm { r } } _ 1 } \\left ( { { { \\bf { D } } _ { K } } } \\right ) = \\sum \\limits _ { n = 1 } ^ { K } { N _ { { \\rm { S , } } n } ^ 2 \\rho _ n ^ 2 } , \\end{align*}"}
+{"id": "5043.png", "formula": "\\begin{align*} ( d \\phi _ Z ) _ i ^ j = \\delta _ i ^ j + \\partial _ i Z ^ j + \\partial _ i Y _ p ^ j ( \\rho _ p ^ { - 1 } \\vec q , \\rho _ p ^ { - 1 } Z ( \\vec q ) ) + \\partial _ { l + n } Y _ p ^ j ( \\rho _ p ^ { - 1 } \\vec q , \\rho _ p ^ { - 1 } Z ( \\vec q ) ) \\partial _ { i } Z ^ l . \\end{align*}"}
+{"id": "3084.png", "formula": "\\begin{align*} { b _ { \\min , k } } = \\left \\lceil { { { \\log } _ 2 } \\frac { { { N _ { { \\rm { S } } , k } } } } { 2 } } \\right \\rceil . \\end{align*}"}
+{"id": "3142.png", "formula": "\\begin{align*} \\partial _ s u + J _ { s , t } ( u ) \\partial _ t u = 0 , \\end{align*}"}
+{"id": "1616.png", "formula": "\\begin{align*} \\mathcal S ^ { \\rho } _ { \\mathcal Y \\setminus \\{ y _ \\ell \\} \\to y _ \\ell } : = \\{ v \\in \\mathcal P _ { \\mathcal Y \\setminus \\{ y _ { \\ell } \\} } : \\deg _ { \\mathcal Y \\to y _ { \\ell } } ( v ) \\ge \\rho \\} . \\end{align*}"}
+{"id": "3524.png", "formula": "\\begin{align*} I = t ^ 0 , \\ , t , \\ , t ^ 2 , \\ , \\dots , \\ , t ^ { p - 1 } , \\ , s \\end{align*}"}
+{"id": "8605.png", "formula": "\\begin{align*} \\cosh ( x ) & = 1 + \\frac { x ^ 2 } { 2 } \\int _ 0 ^ 1 \\cosh ( t x ) ( 1 - t ) \\ : d t \\\\ \\frac { 1 } { \\cosh ( x ) } & = 1 + \\frac { x ^ 2 } { 2 } \\int _ 0 ^ 1 \\Big ( \\frac { \\tanh ( t x ) ^ 2 } { \\cosh ( t x ) } - \\frac { 1 } { \\cosh ( t x ) ^ 3 } \\Big ) ( 1 - t ) \\ : d t , \\end{align*}"}
+{"id": "1520.png", "formula": "\\begin{align*} \\frac 1 x \\# \\left \\{ n \\le x : \\frac { \\log _ 2 P ^ { ( \\alpha ) } ( n ) - \\alpha \\log _ 2 x } { \\sqrt { \\log _ 2 x } } < t \\right \\} = \\Phi \\left ( \\frac { t } { \\sqrt { \\alpha ( 1 - \\alpha ) } } \\right ) + O _ \\alpha \\left ( \\frac { 1 } { \\sqrt { \\log _ 3 x } } \\right ) \\end{align*}"}
+{"id": "7857.png", "formula": "\\begin{align*} f ( \\{ q _ { j } : j \\in J _ { 1 } \\} ) = \\{ q _ { l } : l \\in J _ { 2 } \\} . \\end{align*}"}
+{"id": "2119.png", "formula": "\\begin{align*} \\begin{aligned} v ( t , x ) : = & \\frac { 1 } { 2 } \\left [ \\ln ( c _ 1 + f _ 0 ( x + t ) ) + \\ln ( c _ 1 + f _ 0 ( x - t ) ) \\right ] \\\\ & + \\frac { 1 } { 2 } \\int _ { x - t } ^ { x + t } \\frac { f _ 1 ( s ) d s } { c _ 1 + f _ 0 ( s ) } + \\frac { 1 } { 2 } \\int _ { 0 } ^ t \\left [ \\int _ { x + s - t } ^ { x + t - s } G ( s , y ) d y \\right ] d s = : v _ 1 + v _ 2 + v _ 3 . \\end{aligned} \\end{align*}"}
+{"id": "5564.png", "formula": "\\begin{align*} U _ k ( \\zeta ) = \\Psi _ k ( \\zeta ) + \\sum _ { j = 1 } ^ { k - 1 } \\sigma _ j ( \\zeta ) - \\sum _ { \\begin{subarray} { c } j = 1 \\\\ j \\ne k \\end{subarray} } ^ { m } P _ { k j } \\sigma _ j ( \\zeta ) . \\end{align*}"}
+{"id": "1443.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & a ^ { i 1 } + a ^ { i n } = 1 , \\ a ^ { 1 j } + a ^ { n j } = 1 , \\\\ & a ^ { i j } = a ^ { j i } , \\ \\ i \\neq j , \\end{aligned} \\right . \\ \\ \\ i , j = 1 , 2 , \\cdots , n . \\end{align*}"}
+{"id": "3289.png", "formula": "\\begin{align*} \\frac { d ^ \\ell } { d t ^ \\ell } \\phi _ X ( t ) ( - \\mu ) ^ \\ell = \\int _ \\mathbb { R } x ^ \\ell f _ X ( x ) e ^ { \\mu t x } d x . \\end{align*}"}
+{"id": "7676.png", "formula": "\\begin{align*} \\varphi ^ * ( 0 ) + \\varphi ^ * ( 1 ) s + \\varphi ^ * ( 2 ) s ^ 2 + \\ldots = \\frac { 1 - \\mathbb { E } X ^ * } { G _ { X ^ * } ( s ) - s } = \\frac { \\varepsilon } { G _ { X ^ * } ( s ) - s } , \\ , | s | < 1 , \\end{align*}"}
+{"id": "7900.png", "formula": "\\begin{align*} \\frac { \\det D ^ 2 u ( x ) } { ( u ^ { \\star } ) ^ { n + 2 } ( x ) } = \\varphi ^ { n + 2 } \\left ( \\frac { x } { - u ( x ) } \\right ) \\det D ^ 2 \\varphi \\left ( \\frac { x } { - u ( x ) } \\right ) . \\end{align*}"}
+{"id": "501.png", "formula": "\\begin{align*} \\mathcal L ( \\psi , \\phi ) = \\mathcal L _ { \\mathrm { D i r a c } } ( \\psi ) + \\mathcal L _ { \\mathrm { m e s o n } } ( \\phi ) + \\mathcal L _ { \\mathrm { Y u k a w a } } ( \\psi , \\phi ) . \\end{align*}"}
+{"id": "2169.png", "formula": "\\begin{align*} \\Lambda ^ { ( 0 ) } = J _ 0 ( r ) \\sin ( t ) , \\end{align*}"}
+{"id": "7800.png", "formula": "\\begin{align*} T = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\\\ \\end{pmatrix} , S = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\\\ \\end{pmatrix} \\end{align*}"}
+{"id": "8768.png", "formula": "\\begin{align*} H ^ 2 ( \\mathbf { P } _ { \\omega , T } , \\mathbf { P } _ { \\omega ' , T } ) & \\leq 2 \\Big ( 1 - \\big ( 1 - T ^ { - 1 } \\big ) ^ T \\Big ) \\\\ & \\leq 2 ( 1 - \\frac { 1 } { 4 } ) = 3 / 2 . \\end{align*}"}
+{"id": "719.png", "formula": "\\begin{align*} S \\le t \\le S ' \\wedge \\tau _ R \\implies \\mathbf \\Phi ( \\mathbf V ) ( t ) = \\mathbf u ( t ) , \\end{align*}"}
+{"id": "4308.png", "formula": "\\begin{align*} q = \\frac { i + j + k } { \\sqrt { 3 } } \\end{align*}"}
+{"id": "8196.png", "formula": "\\begin{align*} g _ 0 = \\sum _ { a = 1 } ^ n | d z _ a | ^ 2 + \\sum _ { a = 1 } ^ n | d w _ a | ^ 2 . \\end{align*}"}
+{"id": "1363.png", "formula": "\\begin{align*} f ( z ) \\big | _ { 2 \\kappa } \\gamma : = \\det ( \\gamma ) ^ { \\kappa } ( c z + d ) ^ { - 2 \\kappa } f ( \\gamma z ) . \\end{align*}"}
+{"id": "3089.png", "formula": "\\begin{align*} \\begin{aligned} \\mathop { \\max } \\limits _ { \\left \\{ { { x _ k } } \\right \\} _ { k = 1 } ^ { { N _ { \\rm { R } } } } } & \\sum \\limits _ { k = 1 } ^ { { N _ { \\rm { R } } } } { { { \\log } _ 2 } \\left ( { { { 1 + { { \\bar C } _ k } \\left ( { 1 - { 2 ^ { - 1 - { x _ k } } } } \\right ) } } } \\right ) } \\\\ { \\rm { s . t . } } \\quad & { x _ k } \\in { { \\mathbb Z } ^ + } \\\\ & \\sum \\limits _ { k = 1 } ^ { { N _ { \\rm { R } } } } { { x _ k } } = { B } - { B _ { { \\rm { m i n } } } } . \\end{aligned} \\end{align*}"}
+{"id": "931.png", "formula": "\\begin{align*} \\frac { 1 } { 2 y } \\bigg ( A + \\frac { \\sqrt { m n } } { m } B \\bigg ) = \\frac { \\alpha } { 4 t ^ \\alpha } \\mathrm { e } ^ { - \\frac { \\alpha ( x ^ 2 + y ^ 2 ) } { 4 t ^ \\alpha } } \\bigg ( \\frac y x \\bigg ) ^ { \\frac { c + \\sqrt { m n } - 1 } { 2 } } I _ { \\frac { c + \\sqrt { m n } - 1 } { 2 } } \\bigg ( \\frac { \\alpha \\sqrt { x y } } { 2 t ^ \\alpha } \\bigg ) , \\end{align*}"}
+{"id": "4033.png", "formula": "\\begin{align*} \\frac { n } { 2 } \\cdot \\epsilon _ { 0 } ^ 4 \\ln n \\leq \\sum _ { i = 1 } ^ { r } \\frac { n } { \\ln n } \\sum _ { j = 1 } ^ { \\infty } z _ j e ^ { - z _ j + 1 } \\leq r \\frac { n } { \\ln n } \\bigg ( \\frac { e } { \\epsilon _ 0 } + \\frac { 1 } { \\epsilon _ 0 } + \\frac { 2 } { \\epsilon _ 0 } \\bigg ) \\leq \\frac { 6 r n } { \\epsilon _ 0 \\ln n } . \\end{align*}"}
+{"id": "2858.png", "formula": "\\begin{align*} A _ c = \\{ x \\in V \\mid 0 \\leq \\langle x , \\beta \\rangle \\leq c , \\ \\forall \\beta \\in { \\hat R _ 0 ^ + } \\} . \\end{align*}"}
+{"id": "728.png", "formula": "\\begin{align*} \\mathbf u ( t ) = P _ \\mu \\left ( \\mathbf S ( t ) \\mathbf u _ 0 + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf N \\left ( \\Theta _ R ^ { \\mathbf u } ( s ) P _ \\mu \\mathbf u ( s ) \\right ) \\ , d s + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf M \\left ( P _ \\mu \\mathbf u ( s ) \\right ) \\ , d W ( s ) \\right ) , \\end{align*}"}
+{"id": "5781.png", "formula": "\\begin{align*} \\lim _ { J _ 1 ^ 0 \\rightarrow 0 } \\lim _ { L \\rightarrow \\infty } \\langle ( { R _ { 1 , 2 } ^ c } - \\langle R _ { 1 , 2 } ^ c \\rangle ) ^ 2 \\rangle = 0 , \\end{align*}"}
+{"id": "5162.png", "formula": "\\begin{align*} E [ L ] & = - \\sum _ i f ( x _ i ) \\delta \\log _ 2 { \\big ( f ( x _ i ) \\delta \\big ) } \\\\ & = - \\sum _ i f ( x _ i ) \\delta \\log _ 2 { f ( x _ i ) } - \\log _ 2 { \\delta } . \\end{align*}"}
+{"id": "6543.png", "formula": "\\begin{align*} \\Sigma _ \\Lambda \\subset \\bigcup _ { \\xi = \\pm 1 , i = ( k , n ) \\in \\Lambda } I _ { \\xi , i } , \\end{align*}"}
+{"id": "7886.png", "formula": "\\begin{align*} \\| u \\| _ { H _ { n + k } } : = [ H _ { n + k } ( u ) ] ^ { \\frac { 1 } { n + k + 1 } } \\end{align*}"}
+{"id": "1106.png", "formula": "\\begin{align*} v _ y ( x ) : = \\left \\{ \\begin{array} { l l } \\displaystyle ( - \\Delta _ { \\mu } u ( y ) - \\alpha ( y ) u ( y ) - \\lambda f ( y , u ( y ) ) ) \\delta _ y ^ { x } & \\mbox { i f } \\ , \\ \\ \\ x \\not \\in \\partial D \\\\ 0 & \\mbox { i f } \\ , \\ \\ \\ x \\in \\partial D , \\end{array} \\right . \\end{align*}"}
+{"id": "7314.png", "formula": "\\begin{align*} ( \\pi _ { Q _ S } \\circ \\sigma ) ( T ) = \\pi _ { Q _ S } ( ( S _ 0 ( T ) ^ { - 1 } ) ^ \\ast , 0 ) = ( S _ 0 ( T ) ^ { - 1 } ) ^ \\ast Q _ S S _ 0 ( T ) ^ { - 1 } = T , T \\in \\mathcal { U } . \\end{align*}"}
+{"id": "4230.png", "formula": "\\begin{align*} \\begin{cases} \\ d e ^ 1 = d e ^ 2 = 0 , d e ^ 3 = \\frac { 2 s } { r } \\ , e ^ { 1 5 } , d e ^ 4 = \\frac { 2 s } { r } \\ , e ^ { 2 5 } , d e ^ 5 = 0 , d e ^ 6 = \\pm \\frac { 2 } { r s } \\ , ( e ^ { 1 3 } + e ^ { 2 4 } ) ; \\end{cases} \\end{align*}"}
+{"id": "3943.png", "formula": "\\begin{align*} r _ K ( w ) = x ' + c v \\end{align*}"}
+{"id": "3878.png", "formula": "\\begin{align*} \\tau _ { M , m } : = \\tau _ M ^ 1 \\wedge \\tau _ { M , m } ^ 2 \\wedge \\tau _ { m } ^ 3 . \\end{align*}"}
+{"id": "601.png", "formula": "\\begin{align*} \\psi _ { \\pm } ^ R ( t ) = S _ { \\pm \\xi } ( t ) f _ \\pm + \\sum _ { j = 1 } ^ 4 \\Psi _ { j , \\pm } ( t ) \\end{align*}"}
+{"id": "2591.png", "formula": "\\begin{align*} S = F _ { - 1 } T \\subset F _ 0 T \\subset \\cdots \\subset F _ r T = T \\end{align*}"}
+{"id": "412.png", "formula": "\\begin{align*} s ^ * p ( x _ 1 , . . . , x _ n ) = p \\big ( s ( x _ 1 ) , \\dots , s ( x _ n ) \\big ) \\ , , \\end{align*}"}
+{"id": "2260.png", "formula": "\\begin{align*} f _ \\alpha ( x ) = \\frac { 1 } { \\sqrt { \\alpha } } \\ , p _ { 1 / \\alpha } ( x ) . \\end{align*}"}
+{"id": "2883.png", "formula": "\\begin{align*} \\phi _ { { \\xi } } = \\sum _ { w \\in W _ 0 } C ( w { \\xi } ) \\mathbf { e } ^ { i w { \\xi } } , \\end{align*}"}
+{"id": "6388.png", "formula": "\\begin{align*} \\mathcal { R } * \\chi = \\chi \\tau * \\mathcal { R } \\end{align*}"}
+{"id": "8857.png", "formula": "\\begin{align*} \\min \\Big ( \\alpha _ 0 , \\ , \\frac { d } { \\alpha _ 0 } T ^ { - \\frac { \\beta - 1 } { \\beta } } \\Big ) = \\min \\Big ( T ^ { - 1 / 2 + 1 / \\beta } , \\frac { d } { \\sqrt { T } } \\Big ) \\enspace , \\end{align*}"}
+{"id": "84.png", "formula": "\\begin{align*} \\sum _ { m \\in \\Z } \\langle \\Psi ^ m , \\mathcal H \\Psi ^ m \\rangle = \\sum _ { m \\in \\Z } \\sum _ { \\vert k \\vert \\leq 2 } \\theta _ { \\mathcal M } ( m ) \\theta _ { \\mathcal M } ( m + k ) \\langle \\Psi , \\mathcal H ^ { ( k ) } \\Psi \\rangle . \\end{align*}"}
+{"id": "5196.png", "formula": "\\begin{align*} A ^ v _ { w z } = \\{ \\tau _ w = \\tau _ z \\neq \\tau _ x \\ , \\forall x \\in J ^ v _ { w z } \\} , \\end{align*}"}
+{"id": "5437.png", "formula": "\\begin{align*} \\int f e ^ f d \\mu \\leq c \\int | \\nabla f | ^ 2 e ^ f d \\mu \\mbox { f o r a l l s m o o t h } \\ f \\ \\mbox { o n } \\ \\mathbb { R } ^ n \\ \\mbox { s u c h t h a t } \\ , \\int e ^ f d \\mu = 1 \\end{align*}"}
+{"id": "8833.png", "formula": "\\begin{align*} \\mathbf { E } _ { \\omega , T } | Z _ { T } - f _ { \\omega } ^ { * } | & \\geq \\frac { 1 } { 2 } \\mathbf { E } _ { \\omega , T } | f _ { \\omega } ^ { * } - f _ { \\hat { \\omega } } ^ { * } | \\\\ & = \\frac { 1 } { 2 } \\big [ \\frac { h _ { T } ^ { 2 \\beta - 2 } } { \\tilde { \\alpha } } \\big ] \\mathbf { E } _ { \\omega , T } \\rho ( \\hat { \\omega } , \\omega ) , \\end{align*}"}
+{"id": "6010.png", "formula": "\\begin{align*} ( \\Gamma ) \\sum _ { i = 1 } ^ { \\infty } \\gamma _ { i } = \\lim _ { n \\rightarrow \\infty } \\Gamma _ { n } = \\infty . \\end{align*}"}
+{"id": "8883.png", "formula": "\\begin{align*} \\int _ { G / T } f ( \\tilde { y } _ 1 , \\ldots , \\tilde { y } _ { \\ell } ) = \\sum _ { w \\in W } \\frac { i _ w ^ * f ( \\tilde { y } ) } { e ^ T ( \\nu _ w ) } , \\end{align*}"}
+{"id": "5967.png", "formula": "\\begin{align*} f ( x ) & = \\int _ { \\partial M } G ^ \\omega _ { \\partial M } ( x , z ) \\Lambda ^ \\omega _ { g , F } f ( z ) d \\mu _ h ( y ) , \\\\ & = \\int _ { \\partial M } N ^ \\omega _ { \\partial M } ( x , z ) \\Lambda ^ \\omega _ { g , F } f ( z ) d \\mu _ h ( z ) + \\int _ { \\partial M } e ^ { \\phi ( z ) } \\frac { u _ j ( x ) u _ j ( z ) } { \\lambda _ j - \\omega ^ 2 } \\Lambda _ { g , F } ^ \\omega f ( z ) d \\mu _ h ( z ) . \\end{align*}"}
+{"id": "2382.png", "formula": "\\begin{align*} \\min _ { u } \\left \\{ { \\mathcal { A } _ 1 } V ( x , t ) + h \\left [ V ^ { a f t e r } ( x _ t ^ { a f t e r } , t ) - V ^ { p r e } ( x _ t , t ) \\right ] + C ( t , x , u ) \\right \\} = 0 \\end{align*}"}
+{"id": "5281.png", "formula": "\\begin{align*} & \\Big ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { t } ( x _ { i , t } ) ] _ + \\| \\Big ) ^ 2 \\le \\varepsilon _ 3 T + \\varepsilon _ 4 T \\Big ( \\varepsilon _ { 9 } + \\frac { n F \\log ( T ) } { \\gamma _ { 0 } \\mu } \\Big ) , \\end{align*}"}
+{"id": "5356.png", "formula": "\\begin{align*} H ^ 1 ( { N _ { X ' } } _ { | _ Y } ( K _ Y + D - ( k - 1 ) H ) ) = 0 \\end{align*}"}
+{"id": "764.png", "formula": "\\begin{align*} M = \\norm { \\mathbf U } _ T + \\norm { \\mathbf V } _ T \\end{align*}"}
+{"id": "7043.png", "formula": "\\begin{align*} w _ { 1 } ( x , t ) = \\int _ { 0 } ^ { t } \\int _ { \\Omega } h _ { D } ( x , y , t - \\tau ) r ( \\tau ) f ( y , \\tau ) \\ , d y + \\int _ { \\Omega } h _ { D } ( x , y , t ) \\varphi ( y ) \\ , d y \\end{align*}"}
+{"id": "1542.png", "formula": "\\begin{align*} \\overline { F ( \\rho ) } = \\langle F ( \\lambda _ 1 ) , \\nu \\rangle + m ^ { F ( \\rho ) } \\end{align*}"}
+{"id": "5513.png", "formula": "\\begin{align*} - \\frac { O _ A ^ { ( 1 ) } O _ A ^ { ( g a ^ { - 1 } ) } O _ A ^ { ( 1 ) } } { \\big ( Q ^ 2 - Q ^ { - 2 } \\big ) ^ 2 } & = D _ A ^ { - 1 } O _ A ^ { ( g ) } ( p / s , s ) D _ A , \\\\ \\frac { Q O _ A ^ { ( 1 ) } O _ B ^ { ( g a ^ { - 1 } ) } - Q ^ { - 1 } O _ B ^ { ( g a ^ { - 1 } ) } O _ A ^ { ( 1 ) } } { Q ^ 2 - Q ^ { - 2 } } & = D _ A ^ { - 1 } O _ B ^ { ( g ) } ( p / s , s ) D _ A . \\end{align*}"}
+{"id": "454.png", "formula": "\\begin{align*} \\frac { 2 ^ { \\alpha + 1 } \\Gamma ( \\zeta + \\frac { \\alpha + 1 } { 2 } ) } { | \\Gamma ( - \\alpha / 2 ) | } = \\frac { 2 ^ { 1 + \\alpha } \\Gamma ( \\frac \\alpha 2 + 1 ) \\sin ( \\frac { \\pi \\alpha } { 2 } ) \\Gamma ( \\zeta + \\frac { \\alpha + 1 } { 2 } ) } { \\pi } , \\end{align*}"}
+{"id": "3192.png", "formula": "\\begin{align*} \\left \\Vert x - x _ { k } \\right \\Vert _ { A ^ { T } A } ^ { 2 } - \\left \\Vert x - x _ { k + 1 } \\right \\Vert _ { A ^ { T } A } ^ { 2 } = \\Vert x _ { k + 1 } - x _ { k } \\Vert _ { A ^ { T } A } ^ 2 + 2 \\left ( x _ { k + 1 } - x _ { k } \\right ) ^ { T } A ^ { T } A \\left ( x - x _ { k + 1 } \\right ) . \\end{align*}"}
+{"id": "460.png", "formula": "\\begin{align*} p _ \\zeta ^ { ( \\alpha ) } ( t , r , s ) & \\sim _ { \\zeta , \\alpha } \\frac { t } { | r - s | ^ { 1 + \\alpha } ( r + s ) ^ { 2 \\zeta } + t ^ { \\frac { 1 + \\alpha } { \\alpha } } ( t ^ { \\frac 1 \\alpha } + r + s ) ^ { 2 \\zeta } } . \\end{align*}"}
+{"id": "5481.png", "formula": "\\begin{align*} W = \\bigcup _ { \\alpha \\in I } W \\cap U _ \\alpha \\ ; \\subset \\ ; U \\ ; \\subset \\ ; \\big ( \\mathbb C ^ \\times \\big ) ^ 3 \\end{align*}"}
+{"id": "1407.png", "formula": "\\begin{gather*} \\tilde \\psi _ { n i } = \\psi _ { n i } + \\sum _ { k , j } \\tilde H _ { n i , k j } ( x ) \\psi _ { k j } , ( n , i ) , ( k , j ) \\in V , \\end{gather*}"}
+{"id": "1685.png", "formula": "\\begin{align*} \\Lambda ^ { ( m , n ) } _ { \\texttt { b } } : = \\{ ( \\mu _ 1 , \\ldots , \\mu _ n ) \\in \\mathbb { Z } ^ n \\mid m \\geq \\mu _ 1 \\geq \\cdots \\geq \\mu _ n \\geq 0 \\} \\end{align*}"}
+{"id": "2749.png", "formula": "\\begin{align*} x _ k : = \\theta ^ { - 1 } \\cdot m ^ { - 1 } \\cdot \\sum _ { i = 1 } ^ m e _ { j _ i } . \\end{align*}"}
+{"id": "8299.png", "formula": "\\begin{align*} S _ { - i } & = \\prod _ { j \\not = i } S _ j & s _ { - i } & \\in S _ { - i } \\end{align*}"}
+{"id": "1770.png", "formula": "\\begin{gather*} y ^ h = \\varpi _ { \\mathfrak { p } _ 1 } ^ { a _ 1 } \\cdots \\varpi _ { \\mathfrak { p } _ k } ^ { a _ k } u _ 1 ^ { b _ 1 } \\cdots u _ d ^ { b _ d } , \\end{gather*}"}
+{"id": "8421.png", "formula": "\\begin{align*} \\Omega _ { 3 } = \\Omega _ { 3 , 1 } + \\Omega _ { 3 , 2 } + \\Omega _ { 3 , 3 } + \\Omega _ { 3 , 4 } , \\end{align*}"}
+{"id": "2267.png", "formula": "\\begin{align*} E _ \\Phi ( \\Gamma _ 0 ) \\leq E _ \\Phi ( \\Gamma ) \\Gamma = \\Gamma _ 0 . \\end{align*}"}
+{"id": "6452.png", "formula": "\\begin{align*} [ v _ 1 v _ 2 ] & = [ v _ 1 ] + [ v _ 2 ] + \\eta . [ v _ 1 ] . [ v _ 2 ] \\\\ & = [ v _ 1 ] + [ v _ 2 ] + \\eta . [ c _ 1 ^ 2 s ] . [ c _ 2 ^ 2 ( 1 - s ) ] \\\\ & = [ v _ 1 ] + [ v _ 2 ] + \\eta . ( [ c _ 1 ^ 2 ] + [ s ] ) . ( [ c _ 2 ^ 2 ] + [ 1 - s ] ) \\\\ & = [ v _ 1 ] + [ v _ 2 ] , \\end{align*}"}
+{"id": "1698.png", "formula": "\\begin{align*} \\hat { \\Delta } ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } : = \\Biggl ( \\sum _ { \\mu \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { b } } } \\left | P _ { \\texttt { b } ; \\mu } \\bigl ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } ; q , q _ 0 \\bigr ) \\right | ^ 2 \\delta ^ { ( m , n ) } _ { \\texttt { b } ; \\mu } ( q ) \\Biggr ) ^ { - 1 } . \\end{align*}"}
+{"id": "61.png", "formula": "\\begin{align*} \\mathcal Q _ 3 ^ { \\rm { s o f t } } = \\frac { 1 } { \\vert \\Lambda \\vert } \\sum _ { \\substack { k \\in \\mathcal P _ H , \\\\ p \\in \\mathcal P _ L } } \\widehat g ( k ) \\big ( a _ 0 ^ \\dagger a _ p ^ \\dagger a _ { p - k } a _ k + h . c . \\big ) . \\end{align*}"}
+{"id": "7628.png", "formula": "\\begin{align*} \\mathrm { s u p p } ( B _ k ( \\P ) ) = \\R ^ d . \\end{align*}"}
+{"id": "573.png", "formula": "\\begin{align*} \\mathbf \\Phi ( \\mathbf V ) ( t ) = \\mathbf S ( t - S ) \\mathbf U ( S ) + i \\int _ { S \\wedge \\tau _ R } ^ { t \\wedge \\tau _ R } \\mathbf S ( t - s ) \\mathbf N ( \\mathbf u ( s ) ) \\ , d s + i \\int _ { S } ^ t \\mathbf S ( t - s ) \\mathbf M ( [ \\mathbf u , \\mathbf V ] ( s ) ) \\ , d W ( s ) , \\end{align*}"}
+{"id": "7825.png", "formula": "\\begin{align*} \\widetilde { N } _ S = \\{ ( \\rho , z ^ a , \\zeta ^ i , \\widetilde { \\zeta } _ i , \\sigma ) \\in \\widetilde { N } \\ ; | \\ ; \\epsilon < \\tau _ 2 , \\ ; \\ ; \\epsilon < \\frac { \\tau _ 2 } { | \\tau _ 1 | ^ 2 + | \\tau _ 2 ^ 2 | } , \\ ; \\ ; t ^ a > K , \\ ; \\ ; | \\tau | t ^ a > K \\} \\ , . \\end{align*}"}
+{"id": "4445.png", "formula": "\\begin{align*} | | D ^ { \\alpha } _ { \\ast } \\mathcal { F } | | ^ 2 _ { L ^ 2 ( \\Omega _ t ) } & \\lesssim | | J ^ T \\mathbf { F } | | ^ 2 _ { s , \\ast , t } \\\\ & \\leq C ( K ) \\Big ( | | \\mathbf { F } | | ^ 2 _ { s , \\ast , t } + | | \\mathbf { F } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ t ) } | | \\hat { W } | | ^ 2 _ { s , \\ast , t } \\Big ) . \\end{align*}"}
+{"id": "6778.png", "formula": "\\begin{align*} 0 \\geq d \\psi '' _ \\infty ( 0 ) = - v _ 0 G ( \\phi _ \\infty ( 0 ) , v _ 0 ) . \\end{align*}"}
+{"id": "8539.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\mathcal { H } ^ { n - 2 } \\left ( ( \\partial ^ { * } F _ { \\ell ^ { k } } ) _ { z } \\right ) = \\mathcal { H } ^ { n - 2 } \\left ( ( \\partial ^ { * } F _ { \\ell } ) _ { z } \\right ) \\mbox { f o r } \\mathcal { H } ^ { 1 } \\mbox { - a . e } z \\in J . \\end{align*}"}
+{"id": "2079.png", "formula": "\\begin{align*} \\partial _ { t } ^ 2 \\alpha - \\partial _ { x } ^ 2 \\alpha = 0 . \\end{align*}"}
+{"id": "7089.png", "formula": "\\begin{align*} \\mathcal { J } ^ + _ 1 ( n _ 1 ^ 2 n _ 2 , m , q ) = \\int _ 0 ^ \\infty U ( y ) \\ , y ^ { - i t } \\ , e \\left ( \\frac { 2 \\sqrt { m N y } } { p _ 1 q \\sqrt { p _ 2 } } + \\frac { 3 ( N ( y + w ) n _ 1 ^ 2 n _ 2 ) ^ { 1 / 3 } } { p _ 1 q r ^ { 1 / 3 } } \\right ) d y . \\end{align*}"}
+{"id": "5842.png", "formula": "\\begin{align*} & ( s _ 4 s _ 3 s _ 2 s _ 3 s _ 4 ) s _ 3 s _ 2 s _ 3 s _ 2 = ( s _ 4 s _ { \\alpha _ 2 + 2 \\alpha _ 3 } s _ 4 ) s _ 3 s _ 2 s _ 3 s _ 2 = \\\\ & s _ { \\alpha _ 2 + 2 \\alpha _ 3 + 2 \\alpha _ 4 } ( s _ 3 s _ 2 s _ 3 ) s _ 2 = s _ { \\alpha _ 2 + 2 \\alpha _ 3 + 2 \\alpha _ 4 } s _ { \\alpha _ 2 + 2 \\alpha _ 3 } s _ 2 . \\end{align*}"}
+{"id": "1709.png", "formula": "\\begin{align*} e ^ { i m \\xi _ j } = ( - 1 ) ^ { n - 1 } \\prod _ { \\substack { 1 \\leq k \\leq n \\\\ k \\neq j } } \\left ( \\frac { 1 - q e ^ { i ( \\xi _ j - \\xi _ k ) } } { e ^ { i ( \\xi _ j - \\xi _ k ) } - q } \\right ) ( j = 1 , \\ldots , n ) . \\end{align*}"}
+{"id": "7201.png", "formula": "\\begin{align*} \\eta ^ * = \\eta + \\frac { n } { p } - \\frac { n } { q } - \\alpha . \\end{align*}"}
+{"id": "2067.png", "formula": "\\begin{align*} e ^ { - 2 F } | \\nabla F | ^ 2 = e ^ { - 2 F } \\left ( \\mathfrak { F } ' \\right ) ^ 2 \\frac { | \\nabla \\rho | ^ 2 } { R ^ 2 } \\le C \\frac { | \\nabla \\rho | ^ 2 } { R ^ 2 } \\le \\frac { 2 C } { R ^ 2 } . \\end{align*}"}
+{"id": "6497.png", "formula": "\\begin{align*} u ( t , n ) = \\sum _ { k \\in \\Z ^ b } q ( k , n ) \\cos ( k \\cdot \\omega t ) , \\end{align*}"}
+{"id": "1135.png", "formula": "\\begin{align*} [ \\alpha , \\alpha ] = 0 . \\end{align*}"}
+{"id": "6405.png", "formula": "\\begin{align*} c _ { 2 3 } t _ { 1 2 } c ^ { - 1 } _ { 2 3 } = c _ { 1 2 } t _ { 2 3 } c _ { 1 2 } ^ { - 1 } \\end{align*}"}
+{"id": "7896.png", "formula": "\\begin{align*} \\begin{cases} ( u ^ { \\star } ) ^ k \\det D ^ 2 u = \\lambda | u | ^ { n + k } & \\Omega \\\\ u = 0 & \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "5064.png", "formula": "\\begin{align*} \\triangle x ^ i - V \\cdot \\nabla x ^ i = 0 \\end{align*}"}
+{"id": "2875.png", "formula": "\\begin{align*} ( J _ j f ) ( \\lambda ) : = \\begin{cases} - \\sum _ { k = 1 } ^ { a _ j ( \\lambda ) } f ( \\lambda - k \\alpha _ j ) & \\ a _ j ( \\lambda ) > 0 , \\\\ 0 & \\ a _ j ( \\lambda ) = 0 , \\\\ \\sum _ { k = 0 } ^ { - a _ j ( \\lambda ) - 1 } f ( \\lambda + k \\alpha _ j ) & \\ a _ j ( \\lambda ) < 0 , \\end{cases} \\end{align*}"}
+{"id": "2660.png", "formula": "\\begin{align*} g _ { k } ( t ) = f ( t ) + \\int _ 0 ^ t g _ { k - 1 } ( t - s ) ^ + \\ , d F ( s ) \\ , . \\end{align*}"}
+{"id": "6697.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ m \\binom m j ( 1 - q ) ^ { m - j } & q ^ j ( 1 - | H _ 1 | ^ { - 1 } ) ^ { j } + | H _ 1 | ^ m \\exp ( - r ' e ( H _ 1 ) ^ { - 2 } ) \\\\ & = ( 1 - q | H _ 1 | ) ^ m + | H _ 1 | ^ m \\exp ( - r ' e ( H _ 1 ) ^ { - 2 } ) \\\\ & \\leq \\exp ( - m q | H _ 1 | ) + | H _ 1 | ^ m \\exp ( - r ' e ( H _ 1 ) ^ { - 2 } ) , \\\\ \\end{align*}"}
+{"id": "6502.png", "formula": "\\begin{align*} i u _ { t } + \\varepsilon \\Delta + V u = 0 , \\ , \\mathbb Z ^ d , \\end{align*}"}
+{"id": "7714.png", "formula": "\\begin{align*} E _ 1 = E , \\quad \\ ; \\ ; \\widetilde { E } = \\mathbb { A } _ E ^ * , \\ ; \\ ; \\phi _ E \\colon \\widetilde { E } \\twoheadrightarrow \\wedge ^ 2 E _ 1 , \\end{align*}"}
+{"id": "3554.png", "formula": "\\begin{align*} \\frac { 1 } { \\eta } ( \\lambda ^ { k } - \\lambda ^ { k + 1 } ) - A ( x ^ k - x ^ { k + 1 } ) = B y ^ { k + 1 } + A x ^ { k + 1 } - b \\ , \\end{align*}"}
+{"id": "8274.png", "formula": "\\begin{align*} \\hat { f } = \\log \\frac { \\Omega _ Y \\wedge \\bar \\Omega _ Y } { ( \\hat \\omega / c ) ^ { 2 n } } , \\end{align*}"}
+{"id": "3066.png", "formula": "\\begin{align*} { \\xi _ { k , { l _ k } , { j _ k } } } \\buildrel \\Delta \\over = { \\rho _ k } { \\alpha _ { { k , { \\rm { R } } } , { l _ k } } } { \\alpha _ { { \\rm { T } } , k , { j _ k } } } { \\bf { a } } _ { { \\rm { S , } } k } ^ H \\left ( { \\Theta _ { { k , { \\rm { R } } } , { l _ k } } ^ { \\rm { D } } } \\right ) { { \\bf { \\Gamma } } _ k } { { \\bf { a } } _ { { \\rm { S } } , k } } \\left ( { \\Theta _ { { \\rm { T } } , k , { j _ k } } ^ { \\rm { A } } } \\right ) \\end{align*}"}
+{"id": "4843.png", "formula": "\\begin{align*} \\Lambda _ i ( u ) = \\prod _ { j = 1 } ^ { i - 1 } \\left ( \\frac { \\lambda _ j } { \\lambda _ { j + 1 } } y ( u ) + 1 \\right ) \\prod _ { k = i } ^ { n - 1 } \\left ( y ( u ) + \\frac { \\lambda _ k } { \\lambda _ { k + 1 } } \\right ) \\lambda _ n \\end{align*}"}
+{"id": "94.png", "formula": "\\begin{align*} \\frac { 1 } { 2 ( 2 \\pi ) ^ d } \\int _ { \\mathbb { R } ^ d } \\Big ( \\sqrt { k ^ 4 - 2 k ^ 2 \\rho \\widehat { g } ( k ) } - k ^ 2 \\rho \\widehat { g } ( k ) - \\rho ^ 2 G _ d ( k ) \\Big ) \\dd k = \\frac { \\rho ^ 2 } { 2 } I ^ { } _ d \\widehat { g } ( 0 ) \\lambda _ d ^ { } + \\mathcal { E } _ d ^ { } ( \\rho ) , \\end{align*}"}
+{"id": "5469.png", "formula": "\\begin{align*} v = ( 1 + \\alpha f ) ^ { - \\frac { n - 2 } { 2 } } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ ; \\mathrm { a n d } \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ ; \\alpha ^ { - 1 } = \\max _ M \\left ( f ^ 2 + \\frac { n ( n - 1 ) } { R _ g } | \\nabla f | ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
+{"id": "8101.png", "formula": "\\begin{align*} \\left \\vert T ( f ) ( x ) \\right \\vert \\leq \\sum \\limits _ { j } \\vert \\lambda _ { j } \\vert \\vert T ( a _ { j } ) ( x ) \\vert + \\sum \\limits _ { j } \\vert \\mu _ { j } \\vert \\vert T ( b _ { j } ) ( x ) \\vert = : \\uppercase \\expandafter { \\romannumeral 1 } + \\uppercase \\expandafter { \\romannumeral 2 } . \\end{align*}"}
+{"id": "8710.png", "formula": "\\begin{align*} \\frac { \\Gamma ( d + 1 ) } { \\Gamma ( d + \\beta ) } & = \\frac { \\Gamma ( d + 1 ) } { \\Gamma \\big ( d + \\underbrace { ( \\beta - \\ell ) } _ { \\in ( 0 , 1 ] } \\big ) \\prod _ { i = 1 } ^ { \\ell } \\big ( d + \\beta - i \\big ) } \\leq \\frac { ( d + \\beta - \\ell ) ^ { 1 - ( \\beta - \\ell ) } } { \\prod _ { i = 1 } ^ { \\ell } \\big ( d + \\beta - i \\big ) } \\leq \\frac { 1 } { d ^ { \\beta - 1 } } \\enspace , \\end{align*}"}
+{"id": "8005.png", "formula": "\\begin{align*} \\bigl ( X ( t ) \\bigr ) = ( v _ 0 t , \\dots , v _ M t ) , \\ \\ t \\ge 0 , \\end{align*}"}
+{"id": "3737.png", "formula": "\\begin{align*} \\begin{cases} \\lambda u + v = 0 , \\\\ \\lambda v + w = 0 , \\\\ - A u + \\lambda w = 0 . \\end{cases} \\end{align*}"}
+{"id": "2076.png", "formula": "\\begin{align*} _ { \\mu \\nu } ( \\widetilde g ) = 0 . \\end{align*}"}
+{"id": "2926.png", "formula": "\\begin{align*} J _ 0 ^ 3 = 1 , J _ 1 ^ 3 = 3 , J _ 2 ^ 3 = 2 , J _ 3 ^ 3 = 1 \\end{align*}"}
+{"id": "6672.png", "formula": "\\begin{align*} g _ 1 = - \\frac { q } { 2 \\pi } , g _ 2 = \\frac { 1 } { 8 \\pi ^ 2 } ( 2 \\tilde { p } - q ^ 2 ) , g _ 3 = - \\frac { 1 } { 1 6 \\pi ^ 3 } q \\left ( - 1 - 2 \\tilde { p } + q ^ 2 \\right ) \\\\ g _ 4 = \\frac { 1 } { 1 2 8 \\pi ^ 4 } \\left ( - 1 6 \\tilde { p } - 4 \\tilde { p } ^ 2 + 1 9 q ^ 2 + 1 2 \\tilde { p } q ^ 2 - 5 q ^ 4 \\right ) , \\end{align*}"}
+{"id": "3446.png", "formula": "\\begin{align*} u ^ { * * } ( x ) = \\max _ { p \\in \\Delta ^ \\vee } \\langle p , x \\rangle - u ^ * ( p ) \\leq u ( x ) , x \\in \\Delta . \\end{align*}"}
+{"id": "4780.png", "formula": "\\begin{align*} P _ { 1 } ^ { t } + \\cdots + P _ { m } ^ { t } = Q _ { 1 } ^ { t } + \\cdots + Q _ { n } ^ { t } \\end{align*}"}
+{"id": "1208.png", "formula": "\\begin{align*} e ^ { i ( t - a _ { j + 1 } ) \\Delta } w ( a _ { j + 1 } ) = & e ^ { i t \\Delta } w ( 0 ) + i \\int _ { 0 } ^ { a _ { j + 1 } } e ^ { i ( t - s ) \\Delta } ( F ( \\tilde { u } + w ) - F ( u ) ) d s \\\\ & - i \\int _ { 0 } ^ { a _ { j + 1 } } e ^ { i ( t - s ) \\Delta } e d s , \\end{align*}"}
+{"id": "8811.png", "formula": "\\begin{align*} \\langle a , b \\rangle = \\norm { a } ^ 2 + \\langle a , b - c \\rangle + \\langle a , c - a \\rangle . \\end{align*}"}
+{"id": "2772.png", "formula": "\\begin{align*} I ( G _ 1 \\cup G _ 2 ; x ) = I ( G _ 1 ; x ) \\cdot I ( G _ 2 ; x ) . \\end{align*}"}
+{"id": "2122.png", "formula": "\\begin{align*} \\kappa ( t , x ) = \\dfrac { \\alpha } { ( \\partial _ x \\alpha ) ^ 2 - ( \\partial _ t \\alpha ) ^ 2 } , \\end{align*}"}
+{"id": "2111.png", "formula": "\\begin{align*} \\begin{aligned} I _ 1 : = & ~ { } \\iint _ { D _ t } \\varphi ( \\underline { u } ) | \\underline { L } \\tilde { \\Lambda } | | Q _ 0 ( \\ln \\alpha , \\tilde { \\Lambda } ) | \\\\ \\lesssim & ~ { } \\iint _ { D _ t } \\varphi ( \\underline { u } ) | \\underline { L } \\tilde { \\Lambda } | [ | L ( \\ln \\alpha ) | | \\underline { L } \\tilde { \\Lambda } | + | \\underline { L } ( \\ln \\alpha ) | | L \\tilde { \\Lambda } | ] = : I _ { 1 , 1 } + I _ { 1 , 2 } . \\end{aligned} \\end{align*}"}
+{"id": "3796.png", "formula": "\\begin{align*} { \\rm U O T } ( \\mu _ 0 , \\mu _ 1 ) = \\inf _ { ( \\beta _ 0 , \\beta _ 1 ) } \\big \\{ { \\rm O T } ( \\beta _ 0 , \\beta _ 1 ) \\ , \\mid \\ , \\beta _ i \\in \\overline { S } ^ p _ { = } ( \\mu _ i ) \\big \\} , \\end{align*}"}
+{"id": "7961.png", "formula": "\\begin{align*} \\sigma ( v _ { i , j } ) = v _ { i , j } , ~ \\sigma ( v _ { i + m , j } ) = v _ { i + m , j } , ~ \\sigma ( v _ { i , j + 1 } ) = v _ { i + m , j + 1 } ~ ~ \\sigma ( v _ { i + m , j + 1 } ) = v _ { i , j + 1 } . \\end{align*}"}
+{"id": "2129.png", "formula": "\\begin{align*} \\hat e = - \\kappa ( \\partial _ t \\alpha h _ 1 - 2 \\partial _ x \\alpha h _ 2 ) , \\hat p = \\kappa ( \\partial _ x \\alpha h _ 1 - 2 \\partial _ t \\alpha h _ 2 ) . \\end{align*}"}
+{"id": "6422.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\sup _ { | x | \\leq R , t \\geq t _ n } u _ { n } ( x + c t , t ) = 0 . \\end{align*}"}
+{"id": "1517.png", "formula": "\\begin{gather*} g _ i ^ k g _ j ^ l \\ , = \\ , g _ { \\sigma _ i ( j ) } ^ { \\alpha _ k ( l ) } \\ , g _ { \\gamma _ j ( i ) } ^ { \\beta _ l ( k ) } \\\\ f _ j ^ l f _ i ^ k \\ , = \\ , f _ { \\gamma _ j ( i ) } ^ { \\beta _ l ( k ) } \\ , f _ { \\sigma _ i ( j ) } ^ { \\alpha _ k ( l ) } \\\\ f _ { \\sigma _ { \\gamma _ j ( i ) } ( s ) } ^ { \\alpha _ { \\beta _ l ( k ) } ( m ) } \\ , g _ i ^ k ( T _ j ^ l ) \\ ; = \\ ; g _ { \\gamma _ { \\sigma _ j ( s ) } ( i ) } ^ { \\beta _ { \\alpha _ l ( m ) } ( k ) } \\ , \\ , f _ s ^ m ( T _ j ^ l ) \\end{gather*}"}
+{"id": "6956.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ d E T ( k ) = 2 \\epsilon ^ 2 \\sum _ { k = 1 } ^ d \\sum _ { i = k + 1 } ^ { d } \\sum _ { j = i + 1 } ^ { d } \\frac { 1 } { ( ( x _ i - x _ k ) + \\epsilon ^ 2 ) ( ( x _ j - x _ k ) + \\epsilon ^ 2 ) ( ( x _ j - x _ i ) + \\epsilon ^ 2 ) } . \\end{align*}"}
+{"id": "6349.png", "formula": "\\begin{align*} E r r = \\int _ 0 ^ T \\int _ \\R \\partial _ x ^ 2 g _ { [ < \\lambda ] } | u _ \\lambda | ^ 2 | u _ \\lambda | ^ 2 d x d t . \\end{align*}"}
+{"id": "7931.png", "formula": "\\begin{align*} F ( I ( e ) ) = 4 e \\oplus 2 g \\oplus 3 X \\oplus 5 Y . \\end{align*}"}
+{"id": "6982.png", "formula": "\\begin{align*} f = y \\exp \\{ - \\frac { 1 } { 2 } \\int _ 0 ^ z h ( z ) e ^ { p ( z ) } d z \\} . \\end{align*}"}
+{"id": "6909.png", "formula": "\\begin{align*} c ^ * = \\inf \\limits _ { \\lambda > 0 } \\left \\{ \\frac { 1 } { \\lambda } \\left [ d _ 2 \\left ( \\int _ { \\mathbb R } J _ 2 ( y ) e ^ { - \\lambda y } d y - 1 \\right ) + s \\right ] \\right \\} \\end{align*}"}
+{"id": "4749.png", "formula": "\\begin{align*} { \\rm P e r } ( A ) & = { \\rm P e r } ( A ; \\R ^ 2 \\setminus \\R _ { + } ^ { 2 } ) + { \\rm P e r } ( A ; \\R _ { + } ^ { 2 } ) \\\\ & = { \\rm P e r } ( A \\setminus \\R _ { + } ^ { 2 } ; \\R ^ 2 \\setminus \\R _ { + } ^ { 2 } ) + { \\rm P e r } ( A \\cap \\R _ { + } ^ { 2 } ; \\R _ { + } ^ { 2 } ) \\\\ & = { \\rm P e r } ( A \\setminus \\R _ { + } ^ { 2 } ; \\R ^ 2 \\setminus \\R _ { + } ^ { 2 } ) + \\sum _ { n = 1 } ^ \\infty { \\rm P e r } ( A \\cap \\overline { T } _ n ; \\overline { T } _ n ) . \\end{align*}"}
+{"id": "5586.png", "formula": "\\begin{align*} \\int _ { B _ r ( x ) } | \\nabla u _ * | ^ 2 \\ , d x = \\lim _ { i \\to \\infty } \\int _ { B _ r ( x ) } | \\nabla u _ i | ^ 2 \\ , d x \\end{align*}"}
+{"id": "602.png", "formula": "\\begin{align*} \\norm { \\Psi _ { 1 , \\pm } } _ { L ^ 1 ( \\Omega , X ^ { 0 , b } _ { + \\xi } ( 0 , T ) ) } \\le C M \\norm { \\psi _ { \\mp } ^ R } _ { L ^ 1 ( \\Omega , X ^ { 0 , 0 } _ { + \\xi } ( 0 , T ) ) } = C M \\norm { \\psi _ { \\mp } ^ R } _ { L ^ 1 ( \\Omega , L ^ 2 ( ( 0 , T ) \\times \\R ) ) } \\le C M \\sqrt { T } \\norm { \\psi _ 0 } _ { L ^ 2 ( \\Omega , L ^ 2 ) } . \\end{align*}"}
+{"id": "8262.png", "formula": "\\begin{align*} g _ { x _ 1 , 0 , 0 } ( \\rho , \\Theta ) = g _ { 0 , 0 , 0 } + x _ 1 H _ { x _ 1 , 0 , 0 } + \\sum _ { \\nu \\geq 2 } h _ \\nu ( \\Theta ) ( \\frac { x _ 1 } { \\rho ^ 2 } ) ^ \\nu . \\end{align*}"}
+{"id": "1719.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { c } } } | C _ { \\texttt { c } } ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { c } ; \\lambda } ) | ^ { - 2 } \\Delta ^ { ( m , n ) } _ { \\texttt { c } ; \\lambda } = \\prod _ { 1 \\leq j \\leq n } \\frac { 1 - q } { 1 - q ^ j } \\end{align*}"}
+{"id": "1843.png", "formula": "\\begin{gather*} ( \\gamma _ 0 , \\ldots , \\gamma _ N ) \\mapsto \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + \\alpha ) ^ s } \\end{gather*}"}
+{"id": "6458.png", "formula": "\\begin{align*} y ( t ) = e ^ { t A } y _ 0 , \\end{align*}"}
+{"id": "3991.png", "formula": "\\begin{align*} \\int _ M f \\omega ^ n = \\int _ M \\omega ^ n , \\int _ M ( \\chi + t \\chi + \\tilde \\chi ) ^ n = c \\int _ M ( \\chi + t \\chi + \\tilde \\chi ) ^ m \\wedge \\omega ^ { n - m } + b _ t \\int _ M \\omega ^ n . \\end{align*}"}
+{"id": "538.png", "formula": "\\begin{align*} \\norm { \\mathbf f } _ { \\mathbf H ^ { \\mathbf s } ( \\R ^ d ) } = \\left ( \\norm { f _ 1 } _ { H ^ { s _ 1 } ( \\R ^ d ) } ^ 2 + \\dots + \\norm { f _ n } _ { H ^ { s _ n } ( \\R ^ d ) } ^ 2 \\right ) ^ { 1 / 2 } \\mathbf f = \\begin{pmatrix} f _ 1 \\\\ \\vdots \\\\ f _ n \\end{pmatrix} . \\end{align*}"}
+{"id": "4860.png", "formula": "\\begin{align*} & ( I _ i + f ( u ) E _ i ) ( I _ i + f ( - u ) E _ i ) = ( I _ j + f ( u ) E _ j ) ( I _ j + f ( - u ) E _ j ) \\\\ & = I _ i + ( f ( u ) + f ( - u ) ) E _ i + f ( u ) f ( - u ) E _ i ^ 2 \\\\ & = I _ j + ( f ( u ) + f ( - u ) ) E _ j + f ( u ) f ( - u ) E _ j ^ 2 \\\\ & = I _ i + ( f ( u ) + f ( - u ) ) E _ i + f ( u ) f ( - u ) \\delta E _ i \\\\ & = I _ j + ( f ( u ) + f ( - u ) ) E _ j + f ( u ) f ( - u ) \\delta E _ j \\end{align*}"}
+{"id": "804.png", "formula": "\\begin{align*} & \\phantom { = } \\int _ { \\mathbb { R } ^ N } ( K \\ast | u _ \\varepsilon ^ * | ^ { p ^ \\sharp } ) | u _ \\varepsilon ^ * | ^ { p ^ \\sharp } d x \\\\ & \\geq C _ U ( S ^ * ) ^ { \\frac { N + \\alpha } { p s } } - | O ( \\varepsilon ^ { \\frac { N + \\alpha } { 2 ( p - 1 ) } } ) | - | O ( \\varepsilon ^ { \\frac { N + \\alpha } { p - 1 } } ) | \\\\ & = C _ U ( S ^ * ) ^ { \\frac { N + \\alpha } { p s } } - | O ( \\varepsilon ^ { \\frac { N + \\alpha } { 2 ( p - 1 ) } } ) | , \\end{align*}"}
+{"id": "690.png", "formula": "\\begin{align*} \\norm { \\mathbf u } _ { Z ^ { \\mathbf s , b } ( S , T ) } = \\norm { \\mathbf u } _ { L ^ 2 \\left ( \\Omega , \\mathbf X ^ { \\mathbf s , b } ( S , T ) \\right ) } + \\norm { \\mathbf u } _ { L ^ 2 \\left ( \\Omega , C \\left ( [ S , T ] , \\mathbf H ^ { \\mathbf s } \\right ) \\right ) } . \\end{align*}"}
+{"id": "6533.png", "formula": "\\begin{align*} { \\rm m e a s } ( \\{ x \\in I : \\ | f ( x ) | \\leq \\eta \\} ) & \\leq \\sum _ { i = 1 } ^ N \\frac { r ( r + 3 ) } { \\tau } \\eta ^ { \\frac 1 r } \\leq \\frac { C ( r ) A | I | } { \\tau ^ 2 } \\eta ^ { \\frac 1 r } , \\end{align*}"}
+{"id": "4819.png", "formula": "\\begin{align*} P x \\otimes y = y \\otimes x \\end{align*}"}
+{"id": "6453.png", "formula": "\\begin{align*} [ ( A , B ) ] \\boxplus [ s _ 0 , s _ 1 ] : = [ M ^ T \\cdot ( s _ 0 , s _ 1 ) ^ T ] . \\end{align*}"}
+{"id": "4264.png", "formula": "\\begin{align*} \\lim _ { \\substack { u \\in H _ { k } \\\\ \\| u \\| _ { \\mathbb { X } ( \\Omega ) } \\to + \\infty } } J _ { \\lambda _ k } ( u ) = - \\infty , \\end{align*}"}
+{"id": "1646.png", "formula": "\\begin{align*} \\Delta _ P \\phi ( r ) = \\vert \\phi ' ( r ) \\vert ^ { P - 2 } L _ P \\phi ( r ) , \\end{align*}"}
+{"id": "6238.png", "formula": "\\begin{align*} ( i \\partial _ t + \\partial ^ 2 _ x ) u = f , u ( 0 ) = u _ 0 . \\end{align*}"}
+{"id": "1088.png", "formula": "\\begin{align*} \\begin{aligned} & \\mbox { t h e r e a r e } \\ , \\ , \\beta > 2 \\ , \\ , \\mbox { a n d } \\ , \\ , r _ 0 > 0 \\ , \\ , \\mbox { s u c h t h a t } \\\\ & t f ( x , t ) \\geq \\beta F ( x , t ) > 0 \\ , \\ , \\mbox { f o r a n y } \\ , \\ , t \\geq r _ 0 \\ , \\ , \\mbox { a n d e v e r y } \\ , \\ , \\ , x \\in D . \\end{aligned} \\end{align*}"}
+{"id": "4826.png", "formula": "\\begin{align*} \\check { R } ( - u ) \\check { R } ( u ) = \\rho ( u ) \\rho ( - u ) I \\ ; . \\end{align*}"}
+{"id": "6774.png", "formula": "\\begin{align*} \\varpi _ - ( z ) = \\frac { \\pi _ { + } - \\pi _ { - } e ^ { - \\sqrt { c ^ { 2 } + 4 m } \\left ( z - z _ { 2 } \\right ) } } { 1 - e ^ { - \\sqrt { c ^ { 2 } + 4 m } \\left ( z - z _ { 2 } \\right ) } } , \\end{align*}"}
+{"id": "5720.png", "formula": "\\begin{align*} \\frac { \\d V _ \\lambda } { \\d x } & \\ , = \\ , A _ \\lambda \\ , V _ \\lambda \\ , , & V _ \\lambda ( 1 ) & = e ^ { i \\ , \\xi } \\ , V _ \\lambda ( 0 ) \\ , , \\end{align*}"}
+{"id": "1427.png", "formula": "\\begin{align*} w _ { n + 1 + i } = w _ i , \\ i \\in I , \\ \\ \\sum _ { i \\in I } w _ i = 0 . \\end{align*}"}
+{"id": "6322.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial y } { \\partial t } = & \\Re h ( u _ { [ < \\lambda ^ { \\sigma } ] } ) \\partial _ x u _ { [ < \\lambda ^ { \\sigma } ] } - \\Re \\int _ { - \\infty } ^ x h _ 1 ( u _ { [ < \\lambda ^ { \\sigma } ] } ) ( \\partial _ x u _ { [ < \\lambda ^ { \\sigma } ] } ) ^ 2 \\ , d x ' . \\end{aligned} \\end{align*}"}
+{"id": "1952.png", "formula": "\\begin{align*} \\bar { f } _ t + v \\bar { f } _ x + \\partial _ v \\left ( \\bar { u } ^ * f _ 1 + u _ 2 ^ * \\bar { f } - v \\bar { f } \\right ) = 0 , \\mbox { w i t h } \\bar { f } ( 0 , x , v ) = 0 , \\end{align*}"}
+{"id": "1382.png", "formula": "\\begin{align*} \\frac { 1 } { 2 ^ 7 \\pi \\alpha } \\Psi _ { 6 , - 1 } ^ i ( z ) = \\frac { \\Delta ( z ) } { E _ 6 ( z ) } = f _ { 6 , i } ( z ) . \\end{align*}"}
+{"id": "7569.png", "formula": "\\begin{align*} \\kappa ( \\alpha + 1 ) \\int _ { 0 } ^ { t } s ^ { \\alpha } d s = \\kappa t ^ { \\alpha + 1 } , \\end{align*}"}
+{"id": "5713.png", "formula": "\\begin{align*} \\sigma ( L ) = \\bigcup _ { \\xi \\in ( - \\pi , \\pi ] } \\sigma _ { p e r } ( L _ { \\xi } ) \\ , , \\end{align*}"}
+{"id": "6521.png", "formula": "\\begin{align*} \\beta K _ 1 ( \\beta ) ^ { \\beta - 1 } \\xi & = \\frac 1 2 c ' c _ \\star ^ { \\beta ( \\beta - 1 ) } | k | _ 2 \\\\ & \\leq \\frac 1 2 | k | _ 2 \\cdot | \\det ( { \\bf M } _ l ( m ) ) _ { 1 \\leq l \\leq \\beta } | . \\end{align*}"}
+{"id": "3746.png", "formula": "\\begin{align*} \\| A ^ { \\alpha } ( \\lambda ^ 3 I + A ) ^ { - 1 } x \\| _ \\mathcal { H } & \\leqslant C \\| ( \\lambda ^ 3 I + A ) ^ { - 1 } x \\| _ \\mathcal { H } ^ { 1 - \\alpha } \\| A ( \\lambda ^ 3 I + A ) ^ { - 1 } x \\| _ \\mathcal { H } ^ { \\alpha } \\\\ & \\leqslant \\dfrac { 2 ^ { \\alpha } C } { \\lambda ^ { 3 ( 1 - \\alpha ) } } \\| x \\| _ \\mathcal { H } \\end{align*}"}
+{"id": "8104.png", "formula": "\\begin{align*} \\left \\| \\uppercase \\expandafter { \\romannumeral 2 } \\right \\| _ { L _ { \\omega } ^ { p } } \\leq C \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } . \\end{align*}"}
+{"id": "8314.png", "formula": "\\begin{align*} _ U ( F _ { 1 } ) & = _ U ( F _ { 1 1 } ) \\oplus \\cdots + _ U ( F _ { 1 d } ) \\\\ \\Rightarrow \\mathcal { D } _ { 1 } & = \\mathcal { D } _ { 1 1 } \\oplus \\cdots \\oplus \\mathcal { D } _ { 1 d } , \\end{align*}"}
+{"id": "707.png", "formula": "\\begin{align*} \\widehat { \\mathbf I } ( t ) = \\mathcal F \\mathbf I ( t ) = \\int _ 0 ^ t \\mathcal F \\mathbf \\Lambda ^ { \\mathbf s } \\mathbf S ( - s ) \\mathbf M ( \\mathbf u ( s ) ) \\ , d W ( s ) \\in L ^ 2 \\left ( \\Omega , L ^ 2 _ \\xi \\right ) \\end{align*}"}
+{"id": "3285.png", "formula": "\\begin{align*} m _ \\ell = E [ X ^ \\ell ] = \\frac { d ^ \\ell } { d t ^ \\ell } \\phi _ X ( 0 ) ( - \\mu ) ^ \\ell . \\end{align*}"}
+{"id": "7634.png", "formula": "\\begin{align*} U ^ + ( z ) = U ^ + ( y + x ) & \\le U ^ - ( y - x ) \\\\ & \\le C ( 1 + | y - x | ^ p ) \\\\ & \\le C ( 1 + 2 ^ { p - 1 } ( | 2 y | ^ p + | y + x | ^ p ) ) \\\\ & \\le C ( 1 + 2 ^ { p - 1 } | 2 y | ^ p + 2 ^ { p - 1 } ) ( 1 + | y + x | ^ p ) \\\\ & = \\tilde { C } ( 1 + | z | ^ { p } ) , \\end{align*}"}
+{"id": "5318.png", "formula": "\\begin{align*} \\frac { z ^ { 2 ^ { m + 1 } } } { 1 - z } & \\leq \\frac { 2 ^ { n - 1 } } { \\xi ^ { n + 1 \\choose 2 } } \\left ( 1 - \\frac { \\xi ^ { n + 1 \\choose 2 } } { 2 ^ { n - 1 } } \\right ) ^ { 2 ^ { m + 1 } } \\\\ & \\leq \\frac { 2 ^ { n - 1 } } { \\xi ^ { n + 1 \\choose 2 } } \\exp \\left ( - \\frac { \\xi ^ { n + 1 \\choose 2 } } { 2 ^ { n - 1 } } \\right ) ^ { 2 ^ { m + 1 } } \\\\ & = \\frac { 2 ^ { n - 1 } } { \\xi ^ { n + 1 \\choose 2 } } \\exp \\left ( - 2 ^ { m - n + 2 } \\xi ^ { n + 1 \\choose 2 } \\right ) \\\\ & \\leq e ^ { - \\sqrt { n } } , \\end{align*}"}
+{"id": "5867.png", "formula": "\\begin{align*} R ^ { n } _ { 0 } = 0 , R ^ { n } _ { t } = ( R ^ { n } _ { t - 1 } + \\Delta _ { 0 } X ^ { n } _ { t } - \\Delta _ { 0 } ^ { 2 } / 2 ) ^ { + } , t \\geq 1 . \\end{align*}"}
+{"id": "3203.png", "formula": "\\begin{align*} v _ { k } ^ { T } w _ { k } = v _ { k } ^ { T } \\left ( v _ { k } + \\frac { \\theta _ { k } } { \\rho _ { k - 1 } } w _ { k - 1 } \\right ) = 1 + \\frac { \\theta _ { k } } { \\rho _ { k - 1 } } v _ { k } ^ { T } w _ { k - 1 } . \\end{align*}"}
+{"id": "4734.png", "formula": "\\begin{align*} J ( X ) = \\sum \\limits _ { i = 1 } ^ n \\frac { w _ i } { \\sum \\limits _ { k = 1 } ^ D ( x _ k - s _ { k i } ) ^ 2 } . \\end{align*}"}
+{"id": "7506.png", "formula": "\\begin{align*} A = \\begin{pmatrix} \\alpha & 0 & 0 \\\\ 0 & \\alpha ^ { q ^ s } & 0 \\\\ 0 & 0 & \\alpha ^ { q ^ { 2 s } } \\end{pmatrix} , \\end{align*}"}
+{"id": "6139.png", "formula": "\\begin{align*} ( _ T \\times \\varphi ) \\circ ( \\varphi \\times _ S ) = \\varphi \\times \\varphi = ( \\varphi \\times _ T ) \\circ ( _ S \\times \\varphi ) . \\end{align*}"}
+{"id": "8035.png", "formula": "\\begin{align*} \\left \\| f \\right \\| _ { h _ { \\omega } ^ { p } ( \\mathbb { R } ^ { n } ) } = \\left \\| M _ { \\Phi } ( f ) \\right \\| _ { L _ { \\omega } ^ { p } ( \\mathbb { R } ^ { n } ) } . \\end{align*}"}
+{"id": "5799.png", "formula": "\\begin{align*} \\varepsilon _ i = \\alpha _ n + \\alpha _ { n - 1 } + \\dots + \\alpha _ i , \\ ; \\ ; i < n , \\ ; \\ ; \\varepsilon _ n = \\alpha _ n . \\end{align*}"}
+{"id": "2376.png", "formula": "\\begin{align*} X _ t = 1 _ { t \\leq \\tau } ( X _ 1 ) _ { t } + 1 _ { t \\geq \\tau } ( X _ 2 ) _ { t } + h d t 1 _ { d \\tau } \\left [ ( X _ 1 ) _ { \\tau } + ( X _ 2 ) _ { \\tau } - ( X _ 1 ) _ { \\tau } \\right ] \\end{align*}"}
+{"id": "2974.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { i = 1 } ^ { M _ K } \\sum _ { j = 1 } ^ { N _ K } \\Theta _ { x _ i \\otimes y _ j , x _ i \\otimes y _ j } T - T \\Big \\| < \\varepsilon . \\end{align*}"}
+{"id": "7984.png", "formula": "\\begin{align*} \\mathfrak { s } _ 1 \\in \\begin{cases} \\{ 1 \\} , & \\mathfrak { d } _ 1 = 2 ; \\\\ [ 2 ^ { - \\frac { \\mathfrak { k } _ 0 } { \\mathfrak { d } _ 1 } } , \\frac { 1 } { 2 } ] & \\mathfrak { d } _ 1 > 2 , \\end{cases} \\ \\ 2 ^ { \\mathfrak { k } _ 1 } = 2 ^ { \\mathfrak { k } _ 0 } ( \\mathfrak { s } _ 1 ) ^ { \\mathfrak { d } _ 1 } . \\end{align*}"}
+{"id": "2286.png", "formula": "\\begin{align*} h ( x ) & = \\sum _ { \\substack { | k | \\geq n + 1 \\\\ n \\ , \\nmid \\ , k } } e ^ { - \\pi \\alpha k ^ 2 } \\left ( \\sum _ { j = 1 } ^ { n } e ^ { 2 \\pi i k \\varepsilon _ j } e ^ { - 2 \\pi i k \\frac { j } { n } } \\right ) e ^ { 2 \\pi i k x } . \\end{align*}"}
+{"id": "3816.png", "formula": "\\begin{align*} m _ i > 0 , \\ ; n _ i = \\frac { 1 } { m _ i } , \\ ; \\left ( \\nu _ X ^ i \\right ) ^ { \\perp } . \\end{align*}"}
+{"id": "1031.png", "formula": "\\begin{align*} \\overline { \\{ g \\in G \\mid H _ { \\le n } g \\hat { h } \\cdot y \\cap U _ k \\neq \\emptyset \\} } = \\overline { \\{ g \\in G \\mid H _ { \\le n } g \\cdot x \\cap U _ k \\neq \\emptyset \\} } . \\end{align*}"}
+{"id": "2001.png", "formula": "\\begin{align*} \\mathrm { X } ( \\Omega ) : = \\mathrm { X } ( \\mathbb { R } ^ n ) _ { | _ { \\Omega } } \\end{align*}"}
+{"id": "4929.png", "formula": "\\begin{align*} \\eta ( 2 ^ k i ) & = e ^ { - 2 ^ { k + 1 } \\pi / 2 4 } \\prod _ { \\ell = 1 } ^ \\infty ( 1 - e ^ { - 2 ^ { k + 1 } \\pi \\ell } ) = e ^ { - 2 ^ { k + 1 } \\pi / 2 4 } \\prod _ { \\ell = 1 } ^ \\infty ( 1 - e ^ { - 2 ^ { k } \\pi \\ell } ) \\prod _ { \\ell = 1 } ^ \\infty ( 1 + e ^ { - 2 ^ { k } \\pi \\ell } ) \\\\ & = : e ^ { - 2 ^ k \\pi / 2 4 } \\cdot g _ k \\cdot \\eta ( 2 ^ { k - 1 } i ) , \\end{align*}"}
+{"id": "2472.png", "formula": "\\begin{align*} \\overline { H } ( G ) = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } H _ \\chi \\Big ( G ^ { \\wedge n } [ \\mathcal { S } ^ n _ \\epsilon ] \\Big ) , \\end{align*}"}
+{"id": "385.png", "formula": "\\begin{align*} \\alpha ( y ) = j ( y ^ * \\eta , y ' ) , \\end{align*}"}
+{"id": "5597.png", "formula": "\\begin{align*} \\int _ R ^ \\infty \\exp _ p ( \\mu | u | ^ { p ' } ) r ^ \\theta \\mathrm d r & = \\sum _ { j = 0 } ^ \\infty \\dfrac { \\mu ^ { p - 1 + j } } { \\Gamma ( p + j ) } \\int _ R ^ { \\infty } | u | ^ { ( p - 1 + j ) p ' } r ^ \\theta \\mathrm d r \\\\ & = \\dfrac { \\mu ^ { p - 1 } } { \\Gamma ( p ) } \\| u \\| _ { L ^ { p } _ \\theta } ^ { p } + \\dfrac { \\mu ^ { p } } { \\Gamma ( p + 1 ) } \\| u \\| _ { L ^ { p p ' } _ \\theta } ^ { p p ' } + \\sum _ { j = 2 } ^ \\infty \\dfrac { \\mu ^ { p - 1 + j } } { \\Gamma ( p + j ) } \\int _ R ^ { \\infty } | u | ^ { ( p - 1 + j ) p ' } r ^ \\theta \\mathrm d r . \\end{align*}"}
+{"id": "8297.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } = u _ { x x } , & ( x , t ) \\in ( 0 , 1 ) \\times ( 0 , \\infty ) , \\\\ u ( 0 , t ) = 0 , & t \\in ( 0 , \\infty ) , \\\\ u _ x ( 1 , t ) = - a u _ t ( 1 , t ) + b p ( 0 , t ) , & t \\in ( 0 , \\infty ) , \\\\ \\varepsilon p _ t = p _ { x x } , & ( x , t ) \\in ( 0 , 1 ) \\times ( 0 , \\infty ) , \\\\ p _ x ( 0 , t ) = c p ( 0 , t ) , & t \\in ( 0 , \\infty ) , \\\\ p _ x ( 1 , t ) = d u _ t ( 1 , t ) , & t \\in ( 0 , \\infty ) , \\end{cases} \\end{align*}"}
+{"id": "7682.png", "formula": "\\begin{align*} \\varphi _ \\varepsilon ^ * ( u ) & = \\left ( 1 - \\frac { \\lambda \\mathbb { E } X ^ * } { c } \\right ) \\left ( 1 + \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { \\lambda \\mathbb { E } X ^ * } { c } \\right ) ^ n F _ I ^ { * n } ( u ) \\right ) \\\\ & = \\varphi _ \\varepsilon ^ * ( 0 ) \\left ( 1 + \\sum _ { n = 1 } ^ { \\infty } \\left ( \\psi _ \\varepsilon ^ * ( 0 ) \\right ) ^ n F _ I ^ { * n } ( u ) \\right ) , \\ , u \\geqslant 0 , \\end{align*}"}
+{"id": "5006.png", "formula": "\\begin{align*} K _ { \\Sigma } = - v ^ 2 - \\dfrac { 1 } { 2 } | h | ^ 2 + 2 H ^ 2 , \\end{align*}"}
+{"id": "7688.png", "formula": "\\begin{align*} v ( x _ 0 ) & \\leq v ( z _ \\epsilon ) - P _ \\epsilon ( z _ \\epsilon ) + P _ \\epsilon ( x _ 0 ) \\\\ & \\leq g ( z _ \\epsilon ) - P _ \\epsilon ( z _ \\epsilon ) + P _ \\epsilon ( x _ 0 ) \\\\ & = g ( z _ \\epsilon ) + \\gamma _ \\epsilon n ( x _ 0 ) \\cdot ( z _ \\epsilon - x _ 0 ) \\\\ & \\leq g ( z _ \\epsilon ) - \\gamma _ \\epsilon \\alpha \\abs { z _ \\epsilon - x _ 0 } ^ 2 \\\\ & \\leq g ( z _ \\epsilon ) . \\end{align*}"}
+{"id": "56.png", "formula": "\\begin{align*} \\mathcal Q ( z ) & = \\frac 1 2 \\rho _ z ^ 2 \\vert \\Lambda \\vert \\widehat g ( 0 ) + \\frac { 1 } { 2 } \\sum _ { k \\neq 0 } \\Big ( \\sqrt { k ^ 4 + 2 k ^ 2 \\rho _ z \\widehat g ( k ) } - k ^ 2 - \\rho _ z \\widehat g ( k ) \\Big ) \\\\ & \\quad + \\frac 1 2 \\rho _ z ^ 2 \\vert \\Lambda \\vert \\widehat { g \\omega } ( 0 ) + \\mathcal K ^ { \\rm { d i a g } } . \\end{align*}"}
+{"id": "8566.png", "formula": "\\begin{align*} \\mathcal { L } _ 1 ^ { \\mu } [ \\beta b ] \\bullet = - b \\mathrm { F } _ 3 \\bullet , \\end{align*}"}
+{"id": "6840.png", "formula": "\\begin{align*} v _ { \\rm s p h } ( t , \\theta ) = \\cosh ( t ) ^ { \\frac { 6 - n } { 2 } } . \\end{align*}"}
+{"id": "3111.png", "formula": "\\begin{align*} \\mathbf { a } ( \\phi ^ { } ) = \\frac { 1 } { \\sqrt { M } } [ e ^ { j 2 \\pi m \\phi ^ { } } ] ^ T _ { m \\in \\mathcal { I } ( M ) } . \\end{align*}"}
+{"id": "1842.png", "formula": "\\begin{align*} & \\mathbf { E } [ \\phi _ 1 ( \\mathbb { Y } _ \\alpha ( n _ 1 ) ) \\cdots \\phi _ k ( \\mathbb { Y } _ \\alpha ( n _ k ) ) ] \\\\ & = \\int _ { \\Omega _ 2 } \\prod _ { j = 1 } ^ { k } \\phi _ j ( \\mathbb { Y } _ \\alpha ( n _ j ) ) \\ , d \\mathbf { P } _ 2 = \\prod _ { j = 1 } ^ { k } \\int _ { S ^ 1 } \\phi _ j ( \\omega _ { n _ j } ) \\ , d \\mathbf { m } ( \\omega _ { n _ j } ) = \\prod _ { j = 1 } ^ { k } \\mathbf { E } [ \\phi _ j ( \\mathbb { Y } _ \\alpha ( n _ j ) ) ] \\end{align*}"}
+{"id": "4925.png", "formula": "\\begin{align*} \\dfrac { \\pi } { 2 } = \\lim _ { q \\rightarrow 1 ^ - } ( 1 - q ) \\prod _ { n = 1 } ^ \\infty \\dfrac { ( 1 - q ^ { 2 n } ) ^ 4 } { ( 1 - q ^ n ) ^ 2 } = \\lim _ { q \\rightarrow 1 ^ - } ( 1 - q ) \\dfrac { ( q ^ 2 ; q ^ 2 ) _ \\infty ^ 4 } { ( q ; q ) _ \\infty ^ 2 } . \\end{align*}"}
+{"id": "7510.png", "formula": "\\begin{align*} \\begin{aligned} A _ { 2 } & = \\frac { 1 } { q ^ { 4 } ( q + 1 ) ( q - 1 ) ^ 2 } ( q ^ 7 + q ^ { 2 m } - q ^ { 1 + m } - q ^ { 5 + m } - q ^ { 6 + m } + q ^ { 7 + m } + q ^ { 2 + 2 m } - q ^ { 5 + 2 m } - q ^ { 6 + 2 m } + q ^ { 7 + 2 m } ) \\\\ & = \\frac { 1 } { q ^ { 4 } ( q + 1 ) ( q - 1 ) ^ 2 } ( q ^ 7 + q ^ { m + 1 } ( q ^ { 6 } - q ^ 5 - q ^ 4 - 1 ) + q ^ { 2 m } ( q ^ { 7 } - q ^ 6 - q ^ 5 + q ^ 2 + 1 ) ) > 0 \\end{aligned} \\end{align*}"}
+{"id": "7802.png", "formula": "\\begin{align*} A = \\begin{pmatrix} a & b \\\\ c & d \\\\ \\end{pmatrix} \\in \\mathrm { S L } ( 2 , \\mathbb { Z } ) \\ , . \\end{align*}"}
+{"id": "8007.png", "formula": "\\begin{align*} \\bigl ( X ( t ) \\bigr ) = \\bigcup _ { H = 0 } ^ D \\ \\bigcup _ { i \\in \\mathcal { C } _ { H + 1 } ^ { \\{ 0 , \\dots , D \\} } } \\overset { \\circ } { } ( v _ { i _ 0 } t , \\dots , v _ { i _ H } t ) , \\end{align*}"}
+{"id": "431.png", "formula": "\\begin{align*} p _ \\zeta ^ { ( 2 ) } ( t , r , s ) & : = \\frac { ( r s ) ^ { 1 / 2 - \\zeta } } { 2 t } \\exp \\left ( - \\frac { r ^ 2 + s ^ 2 } { 4 t } \\right ) I _ { \\zeta - 1 / 2 } \\left ( \\frac { r s } { 2 t } \\right ) , \\end{align*}"}
+{"id": "1224.png", "formula": "\\begin{align*} \\| v _ { n , T } ( 0 ) - \\phi _ n \\| _ { \\dot { H } ^ 1 } & = \\left \\| [ e ^ { - i t T \\Delta } \\chi _ n P _ n e ^ { i T \\Delta } - 1 ] \\phi \\right \\| _ { \\dot { H } ^ 1 } \\\\ & = \\| [ \\chi _ n P _ n - 1 ] e ^ { i T \\Delta } \\phi \\| _ { \\dot { H } ^ 1 } \\to 0 \\end{align*}"}
+{"id": "6223.png", "formula": "\\begin{align*} \\| p _ t \\| _ { W ^ { p , q } } = C e ^ { - t d } ( 1 + \\coth t ) ^ { d / p } ( 1 + \\tanh t ) ^ { d / q } , \\end{align*}"}
+{"id": "6203.png", "formula": "\\begin{align*} T = T _ { 0 } \\oplus \\left ( \\bigoplus _ { \\alpha \\in \\Lambda ^ T } T _ { \\alpha } \\right ) . \\end{align*}"}
+{"id": "3702.png", "formula": "\\begin{align*} \\frac { 2 \\alpha } { 4 - 2 ^ { - \\alpha } } V _ { \\alpha , 1 } = \\int _ 0 ^ 1 \\frac { d q } { ( q ^ 2 + 1 ) ^ { \\alpha } } - \\frac { 2 + 2 ^ { - \\alpha } - 2 ^ { 1 - 2 \\alpha } } { ( 1 - 2 \\alpha ) ( 4 - 2 ^ { - \\alpha } ) } . \\end{align*}"}
+{"id": "3238.png", "formula": "\\begin{align*} | \\delta | = \\big | \\alpha - \\frac { p _ { i } } { q _ { i } } \\big | < \\frac { 1 } { q _ { i } q _ { i + 1 } } < \\frac { 1 } { q _ { i } X } < C \\cdot \\frac { 1 } { X ^ { 1 + 1 / \\eta } } . \\end{align*}"}
+{"id": "8276.png", "formula": "\\begin{align*} ( e ^ { \\hat f } - 1 ) \\hat \\omega ^ { 2 n } = i \\partial \\bar \\partial u \\wedge \\sum _ { k = 1 } ^ { 2 n - 1 } \\omega ^ k \\wedge \\hat \\omega ^ { 2 n - 1 - k } , \\end{align*}"}
+{"id": "1255.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } J ( f _ n ) = \\sup _ { f \\in \\dot { H } ^ 1 \\setminus \\{ 0 \\} } J ( f ) = C _ 0 . \\end{align*}"}
+{"id": "5194.png", "formula": "\\begin{align*} \\psi _ { K , U } ^ { e q } ( x ) = - \\int _ { \\partial K } G _ U ( x , y ) \\partial _ \\nu \\psi _ { K , U } ^ { e q } ( y ) \\ , d \\sigma ( y ) \\ . \\end{align*}"}
+{"id": "7377.png", "formula": "\\begin{align*} i \\big ( \\gamma ( g + q h ) \\big ) & = ( g + q h ) \\otimes ( g + q h ) \\\\ & = g \\otimes g + q ( g \\otimes h ) + q ( h \\otimes g ) + q ^ 2 ( h \\otimes h ) \\\\ & = g \\otimes g + \\{ [ g , h ] \\} + \\{ [ h , g ] \\} + q \\{ [ h , h ] \\} \\\\ & = g \\otimes g + \\{ [ g , h ] + [ h , g ] + q [ h , h ] \\} = g \\otimes g = i \\big ( \\gamma ( g ) \\big ) . \\end{align*}"}
+{"id": "5195.png", "formula": "\\begin{align*} \\sum _ { x \\in X } f _ x = | X | D - s \\equiv n D - s = 2 | E ( G ) | . \\end{align*}"}
+{"id": "7231.png", "formula": "\\begin{align*} { \\dim X - 1 = \\dim X _ s < \\dim \\overline { \\mathfrak { m } } _ x / \\overline { \\mathfrak { m } } _ x ^ 2 \\le \\dim X = \\dim \\mathfrak { m } _ x / \\mathfrak { m } _ x ^ 2 } \\end{align*}"}
+{"id": "8062.png", "formula": "\\begin{align*} \\vert \\widehat { \\psi _ 0 } ( \\xi ) \\vert ^ { 2 } + \\sum \\limits _ { j = 1 } ^ { \\infty } \\vert \\widehat { \\psi } ( 2 ^ { - j } \\xi ) \\vert ^ { 2 } = 1 , \\ { \\rm f o r \\ a l l } \\ \\xi \\in \\mathbb R ^ { n } , \\end{align*}"}
+{"id": "3092.png", "formula": "\\begin{align*} { f _ k } \\left ( { { x _ k } } \\right ) < { \\log _ 2 } \\left ( { 1 + \\frac { 1 } { { { 2 ^ { 2 + x _ k } } - 2 } } } \\right ) = { f _ { k , { \\rm u p p e r } } } \\left ( { { x _ k } } \\right ) . \\end{align*}"}
+{"id": "1125.png", "formula": "\\begin{align*} [ x , y ] = \\lambda _ 1 [ x , y ] _ 1 + \\lambda _ 2 [ x , y ] _ 2 , \\end{align*}"}
+{"id": "2981.png", "formula": "\\begin{align*} r _ I ^ { - 1 } ( \\overline { V } ) = \\{ ( e , v ) \\in E ^ 1 _ I \\mid \\psi ( e ) \\in \\overline { V } \\} . \\end{align*}"}
+{"id": "7530.png", "formula": "\\begin{align*} | u _ \\ast | = \\sqrt { p _ \\ast \\tau \\kappa / n _ \\ast } \\ , . \\end{align*}"}
+{"id": "6907.png", "formula": "\\begin{align*} P _ { 1 } ( \\phi , \\psi ) ( \\xi ) & = \\frac { 1 } { c } \\int _ { - \\infty } ^ { \\xi } e ^ { \\frac { \\beta ( y - \\xi ) } { c } } F _ { 1 } ( \\phi , \\psi ) ( y ) d y , \\ \\xi \\in \\mathbb R , \\\\ [ 0 . 2 c m ] P _ { 2 } ( \\phi , \\psi ) ( \\xi ) & = \\frac { 1 } { c } \\int _ { - \\infty } ^ { \\xi } e ^ { \\frac { \\beta ( y - \\xi ) } { c } } F _ { 2 } ( \\phi , \\psi ) ( y ) d y , \\ \\xi \\in \\mathbb R . \\end{align*}"}
+{"id": "42.png", "formula": "\\begin{align*} \\mathcal { K } ^ { \\rm { B o g } } = \\frac { 1 } { 2 } \\sum _ { k \\neq 0 } \\mathcal { A } _ k \\big ( a _ k ^ \\dagger a _ k + a _ { - k } ^ \\dagger a _ { - k } \\big ) + \\frac 1 2 \\sum _ { k \\neq 0 } \\mathcal { B } _ k \\big ( a _ k ^ \\dagger a _ { - k } ^ \\dagger + a _ k a _ { - k } \\big ) , \\end{align*}"}
+{"id": "1405.png", "formula": "\\begin{gather*} \\tilde \\psi ( x ) = ( E + \\tilde H ( x ) ) \\psi ( x ) \\end{gather*}"}
+{"id": "4104.png", "formula": "\\begin{align*} H _ 0 : = \\frac { 1 } { \\sqrt { 3 } } ( J _ 1 + J _ 2 + J _ 3 ) , H _ 1 : = \\frac { 1 } { \\sqrt { 6 } } ( 2 J _ 1 - J _ 2 - J _ 3 ) , H _ 2 : = \\frac { 1 } { \\sqrt { 2 } } ( J _ 2 - J _ 3 ) . \\end{align*}"}
+{"id": "2950.png", "formula": "\\begin{align*} L _ { k j _ 1 \\cdots j _ s l _ 1 \\cdots l _ { s - 1 } } D _ { l _ 1 \\cdots l _ { s - 1 } } = \\frac { 2 s - 1 } { s - 1 } \\delta _ { \\hat { k j _ 1 } } D _ { \\hat { j } _ 2 \\cdots \\hat { j } _ s } - \\delta _ { \\hat { j } _ 1 \\hat { j } _ 2 } D _ { \\hat { j } _ 3 \\cdots \\hat { j } _ s k } \\end{align*}"}
+{"id": "4612.png", "formula": "\\begin{align*} C _ A ( J ) _ \\chi = \\mathbb { C } \\ , \\textup { i d } _ { \\mathcal { H } _ \\chi } = I _ \\chi \\end{align*}"}
+{"id": "399.png", "formula": "\\begin{align*} E ( c ) \\ = \\ { \\textstyle \\coprod _ { A \\in U ( c ) } { \\mathsf { E } l } ( c , A ) } . \\end{align*}"}
+{"id": "3384.png", "formula": "\\begin{align*} 2 ( n - 1 ) a + ( n + 1 ) \\alpha ^ 0 _ { \\mu 0 } b ^ \\mu + ( n - 1 ) \\alpha ^ 0 _ { \\ , 0 \\mu } b ^ \\mu + ( n + 1 ) \\alpha ^ { 0 } _ { \\mu \\nu } b ^ \\mu b ^ \\nu = 2 \\partial _ \\mu b ^ \\mu + 2 \\alpha ^ \\mu _ { \\ , \\mu \\nu } b ^ \\nu . \\end{align*}"}
+{"id": "6974.png", "formula": "\\begin{align*} ( 2 \\pi ) ^ { \\frac { d } { 2 } } n _ k u _ j ( n ) = \\int _ { Q _ d } n _ k \\widehat { u _ j } ( x ) e ^ { i n \\cdot x } = i \\int _ { Q _ d } \\partial _ { x _ k } \\widehat { u _ j } ( x ) e ^ { i n \\cdot x } , \\end{align*}"}
+{"id": "2412.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ { n } a _ i \\lambda ^ { i - 1 } - \\sum _ { i = 1 } ^ { n } a _ i ' \\lambda ^ { i - 1 } \\right | \\geq \\frac { C } { 2 ^ n } . \\end{align*}"}
+{"id": "6973.png", "formula": "\\begin{align*} \\sum _ { n \\in \\Z ^ d } \\sum _ { j = 1 } ^ d | D _ j u ( n ) | ^ 2 = \\sum _ { j = 1 } ^ d \\int _ { Q _ d } | \\widehat { D _ j u } | ^ 2 d x & = 4 \\int _ { Q _ d } | \\widehat { u } ( x ) | ^ 2 \\sum _ { j = 1 } ^ d \\sin ^ 2 ( x _ j / 2 ) \\\\ & = 4 \\int _ { Q _ d } \\Big | \\sum _ { j = 1 } ^ d \\partial _ { x _ j } \\widehat { u _ j } ( x ) \\Big | ^ 2 \\sum _ { j = 1 } ^ d \\sin ^ 2 ( x _ j / 2 ) d x , \\end{align*}"}
+{"id": "1414.png", "formula": "\\begin{gather*} \\varphi ^ K _ { n , 0 } ( x ) = \\xi _ n \\psi ^ K _ { n 0 } ( x ) + \\psi ^ K _ { n 1 } ( x ) , \\varphi ^ K _ { n , 1 } ( x ) = \\psi ^ K _ { n 1 } ( x ) , n = \\overline { 1 , K } . \\end{gather*}"}
+{"id": "3728.png", "formula": "\\begin{align*} \\| \\cdot \\| _ X ^ 2 = \\| \\cdot \\| ^ 2 _ { \\mathcal { H } ^ { \\frac { 2 } { 3 } } } + \\| \\cdot \\| ^ 2 _ { \\mathcal { H } ^ { \\frac { 1 } { 3 } } } + \\| \\cdot \\| ^ 2 _ { \\mathcal { H } } . \\end{align*}"}
+{"id": "2726.png", "formula": "\\begin{align*} U \\bigg ( \\sum _ { i = 1 } ^ { \\infty } a _ i e _ i \\bigg ) = \\sum _ { i = 1 } ^ { \\infty } \\varepsilon _ i a _ i e _ i , \\end{align*}"}
+{"id": "8696.png", "formula": "\\begin{align*} I ( \\vec { y } ) = \\int \\prod _ { i = 1 } ^ n x _ i ^ { b _ i + \\frac { a _ i t } { \\hbar } } e ^ { \\frac { x _ i y _ i } { \\hbar } } ~ \\frac { 1 } { \\Gamma ( \\frac { t } { \\hbar } + 1 ) \\hbar ^ { \\frac { t } { \\hbar } } } ~ \\frac { d t } { \\hbar } ~ d x _ 1 . . . d x _ n . \\end{align*}"}
+{"id": "2256.png", "formula": "\\begin{align*} \\min _ x \\frac { 1 } { \\sqrt { \\alpha } } \\sum _ { \\gamma \\in \\Gamma } \\phi _ { 1 / \\alpha } ( x - \\gamma ) & = \\min _ x \\frac { 1 } { \\sqrt { \\alpha } } \\sum _ { j = 1 } ^ n \\sum _ { k \\in \\Z } e ^ { - \\pi \\frac { \\delta ^ 2 } { \\alpha } \\left ( k + \\frac { x _ j - x } { \\delta } \\right ) ^ 2 } \\\\ & = \\min _ x \\frac { 1 } { \\delta } \\sum _ { j = 1 } ^ n \\sum _ { k \\in \\Z } e ^ { - \\pi \\frac { \\alpha } { \\delta ^ 2 } k ^ 2 } e ^ { 2 \\pi i k \\frac { ( x _ j - x ) } { \\delta } } , \\end{align*}"}
+{"id": "8496.png", "formula": "\\begin{align*} P ( E ; G ) = | \\mu _ { E } | ( G ) = \\mathcal { H } ^ { n - 1 } ( G \\cap \\partial ^ { * } E ) . \\end{align*}"}
+{"id": "5738.png", "formula": "\\begin{align*} a = 1 - 2 s . \\end{align*}"}
+{"id": "7381.png", "formula": "\\begin{align*} w _ { n } = u _ { n } + C _ { n } ( \\rho _ { n } ) . \\end{align*}"}
+{"id": "6335.png", "formula": "\\begin{align*} K ( \\xi , \\eta ) = - \\int e ^ { i x \\xi - y \\eta } \\frac { d } { d x } \\left ( \\frac { 1 } { \\xi - \\eta ( 1 + h ' ) } \\right ) \\ , d x = - \\int e ^ { i x \\xi - y \\eta } \\frac { \\eta h '' } { ( \\xi - \\eta ( 1 + h ' ) ) ^ 2 } \\ , d x , \\end{align*}"}
+{"id": "2373.png", "formula": "\\begin{align*} 0 = f \\left ( \\pi ( \\lambda ) ^ { - 1 } \\left ( \\sum _ { \\mu \\in \\Lambda ^ 0 } c _ \\mu \\pi ( \\mu ) ^ { - 1 } \\tau \\right ) \\right ) = \\sum _ { \\mu \\in \\Lambda ^ 0 } c _ \\mu f ( \\pi ( \\lambda ) ^ { - 1 } \\pi ( \\mu ) ^ { - 1 } \\tau ) = c _ \\lambda \\frac { o ( G ) } { o ( \\Lambda ) } . \\end{align*}"}
+{"id": "5853.png", "formula": "\\begin{align*} & s _ k s _ { k + 1 } \\dots s _ { n - 1 } s _ n s _ { n - 1 } \\dots s _ { k + 1 } s _ k = s _ { \\alpha _ k + \\alpha _ { k + 1 } + \\dots + \\alpha _ { n - 1 } + \\alpha _ n } \\\\ & s _ n s _ { n - 1 } \\dots s _ { k + 1 } s _ k s _ { k + 1 } \\dots s _ { n - 1 } s _ n = s _ { \\alpha _ k + \\alpha _ { k + 1 } + \\dots + \\alpha _ { n - 1 } + \\alpha _ n } \\\\ \\end{align*}"}
+{"id": "6012.png", "formula": "\\begin{align*} N ( t ) = i a n d \\tau ( t ) = \\Gamma _ { i } . \\end{align*}"}
+{"id": "3615.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty { x ^ { n - \\frac { 1 } { 2 } } { e ^ { - \\mu x } } d x } = \\sqrt { \\pi } 2 ^ { - n } \\mu ^ { - n - \\frac { 1 } { 2 } } ( 2 n - 1 ) ! ! , \\end{align*}"}
+{"id": "6114.png", "formula": "\\begin{align*} \\ \\sigma ^ l _ A ( T ) \\backslash \\{ 0 \\} = \\Big \\{ \\overline { g ( L ) } : g \\in \\mathcal { P } _ I ^ l ( L ) \\ \\ g ( P ) = 1 \\Big \\} \\backslash \\{ 0 \\} . \\end{align*}"}
+{"id": "8150.png", "formula": "\\begin{align*} p _ { i j } ( x , y ) = ( r - 1 ) ^ j K _ { i } ( x , k - j , r - 1 ) E _ { j } ( y , n - x , k - x ) , \\end{align*}"}
+{"id": "1998.png", "formula": "\\begin{align*} ( \\dot { \\mathrm { H } } ^ { s _ 0 , p } ( \\mathbb { R } ^ n ) , \\dot { \\mathrm { H } } ^ { s _ 1 , p } ( \\mathbb { R } ^ n ) ) _ { \\theta , q } = ( \\dot { \\mathrm { B } } ^ { s _ 0 } _ { p , q _ 0 } ( \\mathbb { R } ^ n ) , \\dot { \\mathrm { B } } ^ { s _ 1 } _ { p , q _ 1 } ( \\mathbb { R } ^ n ) ) _ { \\theta , q } = \\dot { \\mathrm { B } } ^ { s } _ { p , q } ( \\mathbb { R } ^ n ) \\end{align*}"}
+{"id": "3567.png", "formula": "\\begin{align*} \\begin{aligned} \\widehat { g } _ { \\mathcal { R } } ^ m ( z _ { R _ i } , z _ { R _ j } ) & = \\left ( 4 \\delta \\left ( \\frac { R _ i + R _ j } { 2 } \\right ) - \\delta ( R _ i ) - \\delta ( R _ j ) \\right ) - \\left ( \\delta ( R _ i ) + \\delta ( R _ j ) \\right ) = \\\\ & = 4 \\delta \\left ( \\frac { R _ i + R _ j } { 2 } \\right ) - 2 \\delta ( R _ i ) - 2 \\delta ( R _ j ) , \\end{aligned} \\end{align*}"}
+{"id": "8143.png", "formula": "\\begin{align*} x ^ m y ^ n \\preceq _ { ( \\alpha , \\beta ) } x ^ i y ^ j \\Leftrightarrow \\begin{cases} m + \\alpha n \\leq i + \\alpha j \\\\ \\\\ \\beta m + n \\leq \\beta i + j \\ , , \\end{cases} \\end{align*}"}
+{"id": "8021.png", "formula": "\\begin{align*} \\sum _ { h = 1 } ^ H ( - 1 ) ^ { H - h } \\sum _ { i \\in \\mathcal { C } _ h ^ { \\{ 1 , \\dots , H \\} } } ( c _ { i _ 1 } + \\dots c _ { i _ h } ) ^ m = \\begin{cases} \\begin{array} { l l } 0 , & \\ m < H , \\\\ \\displaystyle \\sum _ { \\substack { n _ 1 , \\dots , n _ H \\ge 1 \\\\ n _ 1 + \\dots + n _ H = m } } c _ 1 ^ { n _ 1 } \\cdots c _ H ^ { n _ H } \\binom { m } { n _ 1 , \\dots , n _ H } , & \\ m \\ge H . \\end{array} \\end{cases} \\end{align*}"}
+{"id": "5443.png", "formula": "\\begin{align*} \\int f L g d \\mu = - \\int \\Gamma ( f , g ) d \\mu . \\end{align*}"}
+{"id": "7230.png", "formula": "\\begin{align*} C : y ^ 2 = \\pi ( x ^ 6 + \\alpha \\pi x ^ 3 + \\pi ^ 2 ) \\ , , \\end{align*}"}
+{"id": "8850.png", "formula": "\\begin{align*} \\norm { x _ { t + 1 } - x _ { p } } ^ { 2 } & = \\norm { x _ { t } - \\eta _ { t } \\hat { g } _ { t } - x _ { p } } ^ 2 \\\\ & = \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 \\\\ & \\leq ( 1 - 2 \\eta _ { t } \\alpha ) \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } - \\nabla f ( x _ t ) , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 , \\end{align*}"}
+{"id": "7871.png", "formula": "\\begin{align*} \\frac { d } { d t } e ^ { - f } d V = - R ^ { H , f } d V . \\end{align*}"}
+{"id": "6288.png", "formula": "\\begin{align*} c ^ 4 _ { \\lambda , m } + i \\Delta ^ 4 \\xi ^ 2 \\ , b ^ 4 _ { \\lambda , m } = i \\Delta ^ 4 \\xi \\ , r ^ 4 _ { \\lambda , m } + i f ^ { 4 , b a l } _ { \\lambda , m } , \\end{align*}"}
+{"id": "620.png", "formula": "\\begin{align*} \\theta ( x ) - \\theta ( y ) - \\theta ( X ) + \\theta ( Y ) = ( I _ 1 - I _ 2 ) ( x - y ) + I _ 2 ( x - y - X + Y ) . \\end{align*}"}
+{"id": "6610.png", "formula": "\\begin{align*} \\langle f ( 1 - i x _ 1 , \\dots , 1 - i x _ N ) \\rangle ^ { ( \\rm C y ) } = \\langle f ( 2 x _ 1 , \\dots , 2 x _ N ) \\rangle ^ { ( \\rm J ) } \\Big | _ { \\lambda _ 1 = - \\beta ( N + p - 1 ) / 2 - 1 + i q \\atop \\lambda _ 2 = - \\beta ( N + p - 1 ) / 2 - 1 - i q } , \\end{align*}"}
+{"id": "2794.png", "formula": "\\begin{align*} \\mathcal { C } _ { N , s , p } : = 2 \\int _ 0 ^ 1 r ^ { p s - 1 } \\left | 1 - r ^ { ( N - p s ) / p } \\right | ^ p \\Phi _ { N , s , p } ( r ) \\dd r , \\end{align*}"}
+{"id": "7894.png", "formula": "\\begin{align*} \\begin{cases} ( u ^ { \\star } ) ^ k \\det D ^ 2 u = F ( x , u ) & \\Omega \\\\ u = 0 & \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "6702.png", "formula": "\\begin{align*} [ a _ m , a _ n ] = m \\delta _ { m + n , 0 } , [ a _ m , \\psi _ n ] = \\psi _ { n - m } , [ a _ m , \\psi ^ \\ast _ n ] = - \\psi ^ \\ast _ { n + m } , \\end{align*}"}
+{"id": "942.png", "formula": "\\begin{align*} v _ x = - \\frac { 1 } { \\kappa } [ s _ 2 X _ x \\tilde { u } _ { \\tilde { x } } - r _ 1 X _ x \\tilde { v } _ { \\tilde { x } } + r _ 1 ( s _ { 1 x } v + s _ { 2 x } u + s _ { 3 x } ) - s _ 2 ( r _ { 1 x } u + r _ { 2 x } v + r _ { 3 x } ) ] , \\end{align*}"}
+{"id": "2449.png", "formula": "\\begin{align*} \\Lambda _ d : = \\frac { 1 } { ( 4 \\pi ) ^ { \\frac { d } { 8 } } } \\left ( \\frac { \\Gamma ( \\frac { d } { 4 } ) } { \\Gamma ( \\frac { 3 d } { 4 } ) } \\right ) ^ { \\frac { 1 } { 2 } } \\left ( \\frac { \\Gamma ( d ) } { \\Gamma ( \\frac { d } { 2 } ) } \\right ) ^ { \\frac { 1 } { 4 } } , \\end{align*}"}
+{"id": "3260.png", "formula": "\\begin{align*} q = | q | e ^ { \\theta \\mu } = | q | \\left ( \\cos \\theta + \\mu \\sin \\theta \\right ) . \\end{align*}"}
+{"id": "8012.png", "formula": "\\begin{align*} T _ { ( \\cdot ) } ( t ) = g ^ { - 1 } \\bigl ( t , X ( t ) \\bigr ) = \\begin{pmatrix} \\begin{array} { l l } 1 & - 1 ^ T \\\\ 0 & I _ D \\end{array} \\end{pmatrix} \\begin{pmatrix} \\begin{array} { c } t \\\\ X ( t ) \\end{array} \\end{pmatrix} = \\begin{pmatrix} \\begin{array} { c } t - \\sum _ { j = 1 } ^ D X _ j ( t ) \\\\ X _ 1 ( t ) \\\\ \\cdot \\\\ X _ D ( t ) \\end{array} \\end{pmatrix} . \\end{align*}"}
+{"id": "6001.png", "formula": "\\begin{align*} \\overline { P } _ { s , t } = \\prod _ { i = N ( s ) } ^ { N ( t ) - 1 } \\overline { P } _ { \\gamma _ { i } } , \\end{align*}"}
+{"id": "7341.png", "formula": "\\begin{align*} \\widetilde { A } ( \\lambda , t ) = \\begin{pmatrix} a _ \\lambda ( t ) & c _ \\lambda ( t ) \\\\ c _ \\lambda ( t ) & e _ \\lambda ( t ) \\end{pmatrix} = \\begin{cases} ( \\arctan t ) J S _ { \\lambda } , t \\geq 0 \\\\ ( \\arctan t ) J S _ 0 , t < 0 , \\end{cases} , \\end{align*}"}
+{"id": "4965.png", "formula": "\\begin{align*} \\Phi _ { 1 2 n , a } ( X ) : = \\prod _ { 1 \\leq k < 1 2 n , ~ \\gcd ( k , 1 2 n ) = 1 , ~ k \\equiv \\pm a \\pmod * { 1 2 } } \\bigl ( X - \\zeta ^ k \\bigr ) , \\end{align*}"}
+{"id": "1231.png", "formula": "\\begin{align*} \\tilde { u } _ n ( t ) = \\sum _ { j = 1 } ^ { J } v _ n ^ j ( t ) \\end{align*}"}
+{"id": "6198.png", "formula": "\\begin{align*} \\alpha \\beta = \\sum _ { x \\le y } \\left ( \\sum _ { x \\le z \\le y } \\alpha _ { x z } \\beta _ { z y } \\right ) e _ { x y } . \\end{align*}"}
+{"id": "4223.png", "formula": "\\begin{align*} d \\big ( \\tilde J _ \\rho \\rho \\big ) = \\ , & \\ , q _ 1 e ^ { 1 2 3 4 } + q _ 2 e ^ { 1 2 3 5 } + q _ 3 e ^ { 1 2 4 5 } + q _ 4 e ^ { 1 2 4 6 } + q _ 5 ( e ^ { 1 2 5 6 } - e ^ { 2 3 4 6 } ) + q _ 6 e ^ { 1 3 4 5 } \\\\ [ 2 p t ] & + q _ 7 ( e ^ { 1 3 4 6 } - e ^ { 2 3 5 6 } ) + q _ 8 e ^ { 1 3 5 6 } + q _ 9 e ^ { 2 3 4 5 } , \\end{align*}"}
+{"id": "576.png", "formula": "\\begin{align*} \\mathbf u ( t ) = \\mathbf S ( t - S ) \\mathbf u ( S ) + i \\int _ { S } ^ t \\mathbf S ( t - s ) \\mathbf N ( \\mathbf u ( s ) ) \\ , d s + i \\int _ { S } ^ t \\mathbf S ( t - s ) \\mathbf M ( \\mathbf u ( s ) ) \\ , d W ( s ) , \\end{align*}"}
+{"id": "7435.png", "formula": "\\begin{align*} \\lambda \\geq ( p - 1 ) C _ 7 ^ { - 1 } \\| ( y _ 1 , y _ 2 ) \\| _ { 1 , p } ^ { 1 - \\frac { \\kappa _ 1 + \\kappa _ 2 + 2 } { p } } = ( p - 1 ) C _ 7 ^ { - 1 } \\| ( y _ 1 , y _ 2 ) \\| _ { 1 , p } ^ { \\frac { p - ( \\kappa _ 1 + \\kappa _ 2 + 2 ) } { p } } . \\end{align*}"}
+{"id": "2060.png", "formula": "\\begin{align*} G _ { \\Omega } ( x , 1 ; y , t _ { k + 1 } ) = \\int _ { \\Omega } G _ { \\Omega } ( x , 1 ; z , t _ k ) G _ { \\Omega } ( z , t _ k ; y , t _ { k + 1 } ) \\ , d \\mathrm { v o l } _ { g ( t _ k ) } ( z ) . \\end{align*}"}
+{"id": "5670.png", "formula": "\\begin{align*} r ( D _ { X ^ \\prime } ) = ( 6 - b _ 0 - b _ 1 ) \\alpha + 2 b _ 0 e _ 0 + 2 b _ 1 e _ 1 . \\end{align*}"}
+{"id": "1265.png", "formula": "\\begin{align*} \\tilde { u } _ n ( t ) = \\sum _ { j = 1 } ^ { J } v _ n ^ j ( t ) \\end{align*}"}
+{"id": "511.png", "formula": "\\begin{align*} \\tau _ R ( \\omega ) = \\sup \\left \\{ t \\in [ 0 , \\tau ( \\omega ) ) \\colon \\right \\} . \\end{align*}"}
+{"id": "1881.png", "formula": "\\begin{align*} m _ k f ( t , x ) = \\int _ { \\R } | v | ^ k f ( t , x , v ) \\ , { \\rm d } v , \\mbox { f o r } k \\in \\N \\cup \\{ 0 \\} . \\end{align*}"}
+{"id": "922.png", "formula": "\\begin{align*} & 4 x ^ { 3 - k } t ^ { 1 - \\alpha } \\eta _ { 2 x t } - 4 ( k + c ) x ^ { 2 - k } \\eta _ { 2 x x } + 4 k ( k + c ) x ^ { 1 - k } \\eta _ { 2 x } - m ( k + 1 ) x ^ 3 ( - ( 1 - \\alpha ) t ^ { - 1 - \\alpha } \\tau \\\\ & + ( 1 - \\alpha ) t ^ { - \\alpha } \\tau _ t - t ^ { 1 - \\alpha } \\tau _ { t t } ) + 2 m k x ^ 2 t ^ { 1 - \\alpha } \\sigma _ { 1 t } + 2 m k ( k + c ) \\sigma _ 1 = 0 . \\end{align*}"}
+{"id": "1294.png", "formula": "\\begin{align*} \\| \\nabla e _ { n , T } ^ { n l } \\| _ { L _ t ^ { 2 } L _ x ^ { \\frac 6 5 } } \\lesssim & T ^ { 1 / 2 } \\big | \\frac { x _ n } { \\lambda _ n } \\big | ^ { - 2 b } \\| \\nabla \\Phi _ n \\| _ { L _ t ^ \\infty L _ x ^ r } ^ { 2 ( p - 1 ) } \\sum _ { j = 0 } ^ { 1 } \\big | \\frac { x _ n } { \\lambda _ n } \\big | ^ { - j } \\| \\partial ^ { 1 - j } \\Phi _ n \\| _ { L _ t ^ \\infty L _ x ^ r } \\\\ \\lesssim & T ^ { 1 / 2 } \\big | \\frac { x _ n } { \\lambda _ n } \\big | ^ { - 2 b + \\theta | 5 / 2 - 3 / r | } \\to 0 \\end{align*}"}
+{"id": "6673.png", "formula": "\\begin{align*} 6 4 ( n + & 4 ) \\pi ^ { 4 } g _ { n + 4 } - 3 2 q ( 1 1 + 4 n ) \\pi ^ { 3 } g _ { n + 3 } \\\\ + & 4 \\left ( 3 2 + 5 4 n + 2 9 n ^ 2 + 5 n ^ 3 + 8 \\tilde { p } ( 3 + 2 n ) + 2 4 q ^ 2 + 1 6 n q ^ 2 \\right ) \\pi ^ { 2 } g _ { n + 2 } \\\\ - & 4 q \\left ( 4 \\tilde { p } ( 1 + 4 n ) + n \\left ( 4 + 1 1 n + 5 n ^ 2 \\right ) \\right ) \\pi g _ { n + 1 } \\\\ + & \\left ( ( n - 1 ) \\left ( 1 6 \\tilde { p } ^ 2 + 2 \\tilde { p } n ( 5 n - 2 ) + n ^ 2 \\left ( - 2 + n + n ^ 2 \\right ) \\right ) \\right ) g _ n = 0 . \\end{align*}"}
+{"id": "2969.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { i = 1 } ^ M \\sum _ { j = 1 } ^ N \\Theta _ { x _ i \\otimes y _ j , x _ i \\otimes y _ j } \\xi - \\xi \\Big \\| < \\varepsilon . \\end{align*}"}
+{"id": "4286.png", "formula": "\\begin{align*} & \\Pi _ 1 ( \\pi ) = \\{ J \\in \\Pi ( \\pi ) : u \\sim _ { \\pi } v , j _ u = \\epsilon _ { \\pi } ( v ) j _ v \\} \\\\ & \\Pi _ 2 ( \\pi ) = \\Pi ( \\pi ) \\setminus \\Pi _ 1 ( \\pi ) . \\end{align*}"}
+{"id": "7133.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\Psi ( Z ) ( \\Psi w ) ' ( Z ) \\frac { d Z } { Z } = \\int _ 0 ^ \\infty ( \\Psi w ) ( Z ) ( \\Psi w ) ' ( Z ) \\frac { d Z } { Z w ( Z ) } = - \\frac { 1 } { 2 } \\int _ 0 ^ \\infty ( \\Psi w ) ^ 2 ( Z ) \\frac { d } { d Z } \\left ( \\frac { 1 } { Z w ( Z ) } \\right ) d Z . \\end{align*}"}
+{"id": "119.png", "formula": "\\begin{align*} \\mathcal { K } _ { } + A _ 0 & \\geq \\frac { 1 } { 2 \\lvert \\Lambda \\rvert } \\sum _ { k \\in \\Lambda ^ * } \\Big ( \\mathcal { A } _ k ( b _ k ^ { \\dagger } b _ k + b _ { - k } ^ { \\dagger } b _ { - k } ) + \\widehat { g } _ k ( b _ k ^ { \\dagger } b _ { - k } ^ { \\dagger } + b _ { - k } b _ k ) \\Big ) \\end{align*}"}
+{"id": "3154.png", "formula": "\\begin{align*} \\omega _ X = \\phi ^ * \\omega _ Y + d d ^ c \\rho \\end{align*}"}
+{"id": "8560.png", "formula": "\\begin{align*} \\mathrm { F } _ 1 = \\dfrac { \\tanh { ( \\sqrt { \\mu } | \\mathrm { D } | ) } } { \\sqrt { \\mu } | \\mathrm { D } | } , \\mathrm { F } _ 2 = \\frac { 3 } { \\mu | \\mathrm { D } | ^ 2 } ( 1 - \\mathrm { F } _ 1 ) , \\mathrm { F } _ 3 = \\mathrm { s e c h } ( \\sqrt { \\mu } | D | ) , \\mathrm { F } _ 4 = \\frac { 2 } { \\mu | \\mathrm { D } | ^ 2 } ( 1 - \\mathrm { F } _ 3 ) . \\end{align*}"}
+{"id": "2980.png", "formula": "\\begin{align*} \\textsf { R } = \\begin{pmatrix} 1 & 0 \\\\ 1 & 0 \\\\ 1 & 1 \\end{pmatrix} \\textrm { a n d } \\textsf { S } = \\begin{pmatrix} 1 & 0 & 1 \\\\ 0 & 1 & 0 \\end{pmatrix} \\end{align*}"}
+{"id": "2938.png", "formula": "\\begin{align*} H _ { k i _ 1 \\dots i _ { n + 1 } j _ 1 \\dots j _ n } D _ { k j _ 1 \\dots j _ n } = \\sum _ { \\lambda = 1 } ^ { ( n + 1 ) ! } c _ { \\lambda } D _ { k i _ 1 \\dots i _ n } = f D _ { k i _ 1 \\dots i _ n } . \\end{align*}"}
+{"id": "8074.png", "formula": "\\begin{align*} \\widetilde \\Omega _ { i , k } = \\left \\{ x \\in \\mathbb { R } ^ { n } \\colon M ( \\chi _ { \\Omega _ { i , k } } ) ( x ) > \\frac { 1 } { 1 0 ^ { n } } \\right \\} . \\end{align*}"}
+{"id": "5697.png", "formula": "\\begin{align*} \\begin{aligned} \\sum \\limits _ { n = 1 } ^ { \\infty } \\big ( & ( n + m ) ! \\prod \\limits _ { j = 1 } ^ { n } w _ j \\big ) ^ { - 1 } \\\\ & \\leq \\sum \\limits _ { n = 1 } ^ { m _ 0 } \\big ( ( n + m ) ! \\prod \\limits _ { j = 1 } ^ { n } w _ j \\big ) ^ { - 1 } + ( \\prod \\limits _ { j = 1 } ^ { m _ 0 } w _ j ) ^ { - 1 } \\sum \\limits _ { n = m _ 0 + 1 } ^ { \\infty } \\frac { 1 } { ( n + m ) ( n + m + 2 ) } < \\infty . \\end{aligned} \\end{align*}"}
+{"id": "660.png", "formula": "\\begin{align*} \\left \\{ \\tau _ R < \\tau \\right \\} \\cap \\left \\{ \\tau _ R \\le t \\right \\} = \\bigcup _ { 0 \\le s \\le t } f _ s ^ { - 1 } \\left ( [ R , \\infty ) \\right ) , \\end{align*}"}
+{"id": "5614.png", "formula": "\\begin{align*} v _ * & = \\lim _ { R \\to \\infty } \\lim _ { n \\to \\infty } \\sum _ { j = 0 } ^ \\infty \\dfrac { \\mu ^ { p - 1 + j } } { \\Gamma ( p + j ) } \\| u ^ * _ { n , R } \\| _ { L _ { \\alpha _ 0 } ^ { p ' ( p - 1 + j ) } } ^ { p ' ( p - 1 + j ) } \\\\ & \\geq \\lim _ { R \\to \\infty } \\lim _ { n \\to \\infty } \\dfrac { \\mu ^ { p - 1 } } { \\Gamma ( p ) } \\| u ^ * _ { n , R } \\| ^ p _ { L ^ p _ { \\alpha _ 0 } } = \\dfrac { \\mu ^ { p - 1 } } { \\Gamma ( p ) } \\eta _ * \\end{align*}"}
+{"id": "8264.png", "formula": "\\begin{align*} \\lambda ( M , g _ 1 ) = \\lambda \\left ( ( M \\times \\mathbb { S } ^ 1 \\times \\mathbb { R } ^ 3 ) / \\ ! \\ ! / \\ ! \\ ! / \\mathbb { S } ^ 1 \\right ) = \\lambda ( M \\times \\mathbb { S } ^ 1 \\times \\mathbb { R } ^ 3 ) / \\ ! \\ ! / \\ ! \\ ! / \\mathbb { S } ^ 1 , \\end{align*}"}
+{"id": "702.png", "formula": "\\begin{align*} \\Theta _ R ^ { \\mathbf u } ( t ) = \\theta _ R \\left ( \\sum _ { i = 1 } ^ n \\norm { u _ i } _ { \\widetilde X ^ { s _ i , b } _ { h _ i ( \\xi ) } ( 0 , t ) } ^ 2 \\right ) . \\end{align*}"}
+{"id": "1980.png", "formula": "\\begin{align*} \\omega _ q ^ { \\left ( a ^ t _ { \\sigma , r } - a ^ t _ { \\sigma , s } \\right ) } + \\sum _ { j = 1 } ^ { p - 1 } \\omega _ q ^ { \\left ( a ^ t _ { \\sigma , r ^ j } - a ^ t _ { \\sigma , s ^ j } \\right ) } = 0 . \\end{align*}"}
+{"id": "7304.png", "formula": "\\begin{align*} \\langle L _ \\lambda u , v \\rangle _ H = ( D ^ 2 _ 0 f _ \\lambda ) ( u , v ) , u , v \\in H , \\end{align*}"}
+{"id": "3569.png", "formula": "\\begin{align*} \\langle v _ P , u _ m \\rangle = \\langle \\dd _ m v _ P , u \\rangle = v _ P ( \\delta + u ) - v _ P ( \\delta ) = \\delta ( P ) + u ( P ) - \\delta ( P ) = 1 . \\end{align*}"}
+{"id": "1657.png", "formula": "\\begin{align*} \\rho _ { \\texttt { a } } ( \\boldsymbol { \\xi } ) : = \\prod _ { 1 \\leq j < k \\leq n } | e ^ { i \\xi _ j } - e ^ { i \\xi _ k } | ^ 2 = 2 ^ { n ( n - 1 ) } \\prod _ { 1 \\leq j < k \\leq n } \\sin ^ 2 \\left ( \\frac { \\xi _ j - \\xi _ k } { 2 } \\right ) \\end{align*}"}
+{"id": "1156.png", "formula": "\\begin{align*} & \\sum _ { i + j = n } [ m _ { 1 , i } , m _ { 1 , j } ] = 0 , \\\\ & \\sum _ { i + j = n } [ m _ { 2 , i } , m _ { 2 , j } ] = 0 . \\end{align*}"}
+{"id": "3850.png", "formula": "\\begin{align*} \\tau _ 1 = \\varphi _ s ( s , r ) & = ( 0 , y ' ( s ) , z ' ( s ) ) = \\cos ( \\theta ) E _ 2 + \\sin ( \\theta ) E _ 3 , \\\\ \\tau _ 2 = \\varphi _ r ( s , r ) & = ( 1 , 0 , 0 ) = e ^ { - \\lambda _ 1 z } E _ 1 , \\end{align*}"}
+{"id": "8739.png", "formula": "\\begin{align*} \\norm { x _ { t + 1 } - x _ { p } } ^ { 2 } & = \\norm { x _ { t } - \\eta _ { t } \\hat { g } _ { t } - x _ { p } } ^ 2 \\\\ & = \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 \\\\ & \\leq ( 1 - 2 \\eta _ { t } \\alpha ) \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } - \\nabla f ( x _ t ) , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 , \\end{align*}"}
+{"id": "7723.png", "formula": "\\begin{align*} T = s _ I p ^ I + A _ { \\mu \\nu \\eta } ( x ) e ^ \\mu e ^ \\nu e ^ \\eta , \\end{align*}"}
+{"id": "4182.png", "formula": "\\begin{align*} g ( s ) & < 3 \\cdot \\frac { s + 1 } { s + 2 } - \\frac { s } { s + 1 } \\cdot \\frac { s + 1 } { ( s + 1 ) + 1 } = \\frac { 2 s + 3 } { s + 2 } < 2 , \\end{align*}"}
+{"id": "245.png", "formula": "\\begin{align*} \\delta ( x ; g ) : = & \\prod _ { 1 \\leq j \\leq n } ( e ^ { \\frac { 1 } { 2 } x _ j } - e ^ { - \\frac { 1 } { 2 } x _ j } ) ^ { g _ S } ( e ^ { x _ j } - e ^ { - x _ j } ) ^ { g _ L } \\\\ & \\times \\prod _ { 1 \\leq j < k \\leq n } ( e ^ { \\frac { 1 } { 2 } ( x _ j + x _ k ) } - e ^ { - \\frac { 1 } { 2 } ( x _ j + x _ k ) } ) ^ { g _ M } ( e ^ { \\frac { 1 } { 2 } ( x _ j - x _ k ) } - e ^ { - \\frac { 1 } { 2 } ( x _ j - x _ k ) } ) ^ { g _ M } \\end{align*}"}
+{"id": "3904.png", "formula": "\\begin{align*} \\norm { \\sum _ { n = 1 } ^ N a _ n f ^ n - \\sum _ { k = 1 } ^ { Q _ N } \\xi _ k } { 2 } \\leq & \\sum _ { k = 1 } ^ { Q _ N } \\| \\eta _ k \\| _ 2 + \\| \\rho _ N \\| _ 2 \\lesssim \\\\ & \\lesssim Q _ N \\varphi ( N ) ^ { 1 / 4 } \\sigma _ N + \\varphi ( N ) ^ { 1 / 1 6 } \\sigma _ N \\lesssim \\varphi ( N ) ^ { 1 / 1 6 } \\sigma _ N . \\end{align*}"}
+{"id": "2965.png", "formula": "\\begin{align*} \\| a h - h \\| ^ 2 & = \\| ( a - 1 ) h \\| ^ 2 = \\| h ^ * ( a - 1 ) ^ 2 h \\| = \\sup \\{ \\phi ( h ^ * ( a - 1 ) ^ 2 h ) \\} \\\\ & \\le \\sup \\{ \\phi ( h ^ * ( b - 1 ) ^ 2 h ) \\} = \\| b h - h \\| ^ 2 , \\end{align*}"}
+{"id": "6573.png", "formula": "\\begin{align*} \\widetilde \\Omega _ N = \\bigcap _ { 1 \\leq l , l ' \\leq N ^ C , \\xi = \\pm 1 , \\xi ' = \\pm 1 , 0 < | k | \\leq 2 N } \\left \\{ \\omega \\in \\Omega : \\ | k \\cdot \\omega + \\xi \\sqrt { \\zeta _ l } - \\xi ' \\sqrt { \\zeta _ { l ' } } | > 2 e ^ { - \\frac 1 4 N _ 1 ^ { \\rho _ 2 / 2 } } \\right \\} . \\end{align*}"}
+{"id": "6556.png", "formula": "\\begin{align*} \\mathbb { D } _ 1 = \\mathbb { D } _ { ( \\frac { 1 } { 1 0 } + \\frac 1 { 1 0 0 b } + \\frac { l _ * - 1 } { 5 0 b } ) ( \\varepsilon + \\delta ) ^ { \\frac { 1 } { 8 b } } } ( \\sigma ^ * ) \\end{align*}"}
+{"id": "3783.png", "formula": "\\begin{align*} \\Phi : = \\{ \\phi = ( \\phi _ 0 , \\phi _ 1 ) \\in C _ b ( X ) \\times C _ b ( X ) \\ ; \\mid \\ ; - F ^ * ( - \\phi _ i ) \\in C _ b ( X ) , \\ ; \\phi _ 0 \\oplus \\phi _ 1 \\leq c \\} . \\end{align*}"}
+{"id": "4802.png", "formula": "\\begin{align*} \\check { H } ^ * ( \\Omega ; \\mathbb { R } ) = \\lim _ { n \\rightarrow \\infty } \\left ( \\check H ^ * ( \\Gamma ; \\mathbb { R } ) , \\gamma ^ * \\right ) . \\end{align*}"}
+{"id": "2156.png", "formula": "\\begin{align*} g ^ { ( 1 ) } = \\frac { 1 } { \\mu \\cosh ( \\rho ) } \\left [ \\begin{array} { c c } e ^ { u _ 0 } ( \\mu ^ 2 e ^ { \\rho } + \\alpha ^ 2 e ^ { - \\rho } ) & \\alpha ^ 2 - \\mu ^ 2 \\\\ \\alpha ^ 2 - \\mu ^ 2 & e ^ { - u _ 0 } ( \\alpha ^ 2 e ^ { \\rho } + \\mu ^ 2 e ^ { - \\rho } ) \\end{array} \\right ] , \\end{align*}"}
+{"id": "1911.png", "formula": "\\begin{align*} \\left ( e _ w , q _ h \\right ) + \\sqrt { \\epsilon } \\ , b _ h ( e _ u , q _ h ) = 0 \\forall \\ , \\ , q _ h \\in X _ h , \\end{align*}"}
+{"id": "5766.png", "formula": "\\begin{align*} \\lim _ { \\mu _ 1 \\searrow 0 } \\lim _ { L \\to \\infty } \\mathbb E \\langle m ^ 1 \\rangle _ { \\beta } \\leq \\lim _ { \\mu _ 1 \\searrow 0 } \\lim _ { L \\to \\infty } \\mathbb E \\langle m ^ 1 \\rangle _ { \\beta _ { \\rm N } } = 0 . \\end{align*}"}
+{"id": "60.png", "formula": "\\begin{align*} n _ + ^ L : = \\sum _ { j = 1 } ^ n \\overline { Q } _ { H , j } . \\end{align*}"}
+{"id": "2153.png", "formula": "\\begin{align*} \\beta _ x + \\beta _ t = 2 \\tilde \\alpha _ 0 ' ( 2 u ) + 2 \\alpha _ 1 ( 2 u ) > 0 . \\end{align*}"}
+{"id": "6061.png", "formula": "\\begin{align*} O _ { 3 , 3 } = \\mathbb { E } ( \\int _ 0 ^ t \\int _ { B _ { M _ { { \\mathcal { P } } _ n } ( r ) } } \\vert \\nabla _ x { c } ( z , { X } ^ { M _ { { \\mathcal { P } } _ n } } _ { { T } _ { i } ^ { k } - } ) \\vert \\vert D ^ Z { X } ^ { M _ { { \\mathcal { P } } _ n } } _ { { T } _ { i } ^ { k } - } - D ^ Z { X } ^ { M _ { { \\mathcal { P } } _ m } } _ { { T } _ { i } ^ { k } - } \\vert _ { l _ 2 } N ( d z , d r ) ) ^ 2 . \\end{align*}"}
+{"id": "6330.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\dot x = & \\ a _ \\xi ( x , \\xi ) \\\\ \\dot \\xi = & \\ - a _ x ( x , \\xi ) . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5030.png", "formula": "\\begin{align*} \\triangle u ' + \\nabla _ V u ' + b ' _ 1 * \\nabla u ' + b ' _ 0 * u ' = v ' , \\end{align*}"}
+{"id": "8717.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } t ^ { - \\frac { \\beta - 1 } { \\beta } } = 1 + \\int _ { 1 } ^ { T } u ^ { - \\frac { \\beta - 1 } { \\beta } } \\d u \\leq \\beta T ^ { \\frac { 1 } { \\beta } } \\quad \\sum _ { t = 1 } ^ { T } t ^ { - \\frac { 1 } { \\beta } - 1 } \\leq 1 + \\int _ { 1 } ^ { T } u ^ { - \\frac { 1 } { \\beta } - 1 } \\d u \\leq 1 + \\beta \\enspace . \\end{align*}"}
+{"id": "2933.png", "formula": "\\begin{align*} G _ { k i _ 1 \\cdots i _ n } = D ^ k _ { i _ 1 \\cdots i _ n } . \\end{align*}"}
+{"id": "6807.png", "formula": "\\begin{align*} d \\psi _ 1 ' ( z ) - \\frac { c } { 2 } \\psi _ 1 ( z ) & = d y ' ( z ) - c y ( z ) + \\frac { c ^ 2 } { 4 d } v ( z ) \\\\ [ 0 . 2 c m ] & = \\left ( \\frac { s v ( z ) } { q ( u ( z ) ) } + \\frac { c ^ 2 } { 4 d } - s \\right ) v ( z ) \\\\ [ 0 . 2 c m ] & \\geq \\frac { s v ^ 2 ( z ) } { q ( u ( z ) ) } > 0 . \\end{align*}"}
+{"id": "2321.png", "formula": "\\begin{align*} x = \\sum _ { g \\in G } f _ g ( x ) \\tau _ g , \\forall x \\in \\mathcal { X } , \\end{align*}"}
+{"id": "6000.png", "formula": "\\begin{align*} \\overline { \\omega } = \\overline { \\omega } ( ( \\gamma _ { n } ) _ { n \\in N } ) = \\overline { \\lim _ { n \\rightarrow \\infty } } \\frac { \\gamma _ { n } - \\gamma _ { n + 1 } } { \\gamma _ { n + 1 } ^ { 2 } } < \\infty . \\end{align*}"}
+{"id": "7144.png", "formula": "\\begin{align*} A ( r ) - ( 1 + \\gamma ) B ( r ) & \\geq w ' \\Big [ E + ( \\varphi ' ) ^ 2 \\big ( 1 + 2 \\mathcal { W } \\Phi - 2 ( 1 + \\gamma ) \\mathcal { W } | \\Phi | ^ 2 \\min \\big ( \\mathcal { W } , \\frac { h } { 4 \\varphi ' } \\big ) \\big ) \\\\ & - 2 ( 1 + \\gamma ) h ^ { - 2 } \\mathcal { W } | V _ S | ^ 2 \\min \\big ( \\mathcal { W } , \\frac { h } { 4 \\varphi ' } \\big ) - V _ L - \\mathcal { W } \\big ( V ' _ L + \\frac { h ^ 2 q } { 4 r ^ 2 } \\big ) \\Big ] . \\end{align*}"}
+{"id": "1275.png", "formula": "\\begin{align*} \\begin{cases} i w _ t + \\Delta w = F ( \\tilde { u } + w ) - F ( \\tilde { u } ) + e , \\\\ w ( 0 ) = u _ 0 - \\tilde { u } ( 0 ) , \\end{cases} \\end{align*}"}
+{"id": "2819.png", "formula": "\\begin{align*} \\frac { 1 } { \\mathrm { v o l } ( \\mathbb { S } ^ { N - 2 } ) } \\psi ( r ) : & = \\int _ { 0 } ^ \\pi \\frac { \\sin ^ { N - 2 } ( \\alpha _ 1 ) } { \\big ( 1 + r ^ 2 - 2 r \\cos ( \\alpha _ 1 ) \\big ) ^ { ( N + 2 s ) / 2 } } \\dd \\alpha _ 1 \\\\ & = 2 \\int _ { - 1 } ^ 1 \\frac { ( 1 - h ^ 2 ) ^ { \\frac { N - 3 } { 2 } } } { \\big ( 1 + r ^ 2 - 2 r h \\big ) ^ { ( N + 2 s ) / 2 } } \\dd h . \\end{align*}"}
+{"id": "6951.png", "formula": "\\begin{align*} | F _ \\epsilon ^ k | ^ 2 = & \\alpha ^ 2 \\frac { x _ k ^ 2 } { ( | x | ^ 2 + \\epsilon ^ 2 ) ^ 2 } + \\sum _ { i = 1 } ^ { k - 1 } \\frac { 1 } { ( ( x _ k - x _ i ) + \\epsilon ^ 2 ) ^ 2 } + \\sum _ { i = k + 1 } ^ { d } \\frac { 1 } { ( ( x _ i - x _ k ) + \\epsilon ^ 2 ) ^ 2 } \\\\ & - 2 \\frac { \\alpha } { | x | ^ 2 + \\epsilon ^ 2 } \\sum _ { i = 1 } ^ { k - 1 } \\frac { x _ k } { ( ( x _ k - x _ i ) + \\epsilon ^ 2 ) } + 2 \\frac { \\alpha } { | x | ^ 2 + \\epsilon ^ 2 } \\sum _ { i = k + 1 } ^ d \\frac { x _ k } { ( ( x _ i - x _ k ) + \\epsilon ^ 2 ) } + E T ( k ) , \\\\ \\end{align*}"}
+{"id": "4511.png", "formula": "\\begin{align*} \\mathbb L _ H ( { \\mathbf u } ^ a + \\mathbf { u } _ { i + 1 / 2 } , \\mathbf { H } ^ a + \\mathbf { H } _ { i + 1 / 2 } , \\Psi ^ a + \\Psi _ { i + 1 / 2 } ) = 0 \\ , , \\quad \\mbox { i n } \\ , \\ , \\ , \\Omega _ T \\ , . \\end{align*}"}
+{"id": "7424.png", "formula": "\\begin{align*} \\partial _ { x } ^ { k + 1 } u _ { n } = \\lambda _ { n } ^ { - 1 } \\partial _ { x } ^ { k } V _ { n } + \\left [ \\partial _ { x } ^ { k } , \\lambda _ { n } ^ { - 1 } \\right ] V _ { n } \\end{align*}"}
+{"id": "6319.png", "formula": "\\begin{align*} i \\partial _ { \\tau } v _ { \\lambda } + \\left ( i \\frac { \\partial y } { \\partial t } + \\partial _ x \\sqrt { _ { \\lambda } g _ { [ < \\lambda ^ { \\sigma } ] } } \\right ) \\partial _ y v _ { \\lambda } + \\partial ^ 2 _ y v _ { \\lambda } = \\tilde { f } _ \\lambda . \\end{align*}"}
+{"id": "6998.png", "formula": "\\begin{align*} e ^ { - 2 \\lambda } \\widetilde { K } ( X , Y , Z , W ) = & K ( X , Y , Z , W ) - | \\nabla \\l | ^ 2 \\Big ( h ( X , Z ) h ( Y , W ) - h ( X , W ) h ( Y , Z ) \\Big ) \\\\ & - h ( Y , W ) \\Big ( D ^ 2 \\lambda ( X , Z ) - X ( \\lambda ) Z ( \\lambda ) \\Big ) - h ( X , Z ) \\Big ( D ^ 2 \\lambda ( Y , W ) - Y ( \\l ) W ( \\l ) \\Big ) \\\\ & + h ( X , W ) \\Big ( D ^ 2 \\l ( Y , Z ) - Y ( \\l ) Z ( \\l ) \\Big ) + h ( Y , Z ) \\Big ( D ^ 2 \\l ( X , W ) - X ( \\l ) W ( \\l ) \\Big ) , \\end{align*}"}
+{"id": "818.png", "formula": "\\begin{align*} \\mathcal { F } = \\{ \\widetilde { T x } ~ ~ | ~ ~ x \\in S _ { X } \\} , \\end{align*}"}
+{"id": "8919.png", "formula": "\\begin{align*} \\tau _ V ( \\varphi ) = \\sum \\limits _ { \\alpha = 1 } ^ n \\tau ( \\varphi ) ^ \\alpha \\frac { \\partial } { \\partial y _ \\alpha } + \\sum \\limits _ { \\alpha = 1 } ^ n \\sum \\limits _ { i = 1 } ^ m V _ i \\frac { \\partial \\varphi ^ \\alpha } { \\partial x _ i } \\frac { \\partial } { \\partial y _ \\alpha } . \\end{align*}"}
+{"id": "7027.png", "formula": "\\begin{align*} \\mathop { \\lim } _ { n \\to + \\infty } F _ { L , \\theta } ^ { \\Phi ^ n , C } [ \\rho ] = F _ { L } [ \\rho ] . \\end{align*}"}
+{"id": "7764.png", "formula": "\\begin{align*} \\eta ^ { } : = - \\sum _ { \\gamma } \\Omega ( \\gamma ) \\eta _ { \\gamma } ^ { } - \\frac { 1 } { 2 } \\iota _ V \\left ( \\frac { g _ N } { 2 \\pi } - \\pi ^ * _ M g _ { M } \\right ) \\ , . \\end{align*}"}
+{"id": "4328.png", "formula": "\\begin{align*} S ( \\omega _ \\Psi \\| \\omega _ \\Omega ) = S ( \\omega _ { U \\Omega } \\| \\omega _ \\Omega ) = i \\left . \\frac { d } { d t } \\right | _ { t = 0 } ( U ^ * \\Omega , \\Delta ^ { i t } _ { \\Omega , \\Omega } U ^ * \\Omega ) \\ , , \\end{align*}"}
+{"id": "2430.png", "formula": "\\begin{align*} \\frac { \\theta _ { \\Gamma , \\mathbf { P _ 1 } } ^ 2 ( t _ \\Gamma ( P _ m ) ) ^ 2 } { \\theta _ { \\Gamma , \\mathbf { P _ 2 } } ^ 2 ( t _ \\Gamma ( P _ m ) ) ^ 2 } \\frac { \\theta _ { \\Gamma , \\mathbf { P _ 2 } } ^ 2 ( t _ \\Gamma ( P _ h ) ) } { \\theta _ { \\Gamma , \\mathbf { P _ 1 } } ^ 2 ( t _ \\Gamma ( P _ h ) ) ^ 2 } = \\frac { \\lambda _ h - \\lambda _ m } { \\lambda _ h - \\lambda _ l } \\frac { \\lambda _ k - \\lambda _ m } { \\lambda _ k - \\lambda _ l } \\end{align*}"}
+{"id": "7750.png", "formula": "\\begin{align*} \\mathfrak { F } = - \\frac { 1 } { 6 } \\frac { ( Z ^ 1 ) ^ 3 } { Z ^ 0 } + \\chi \\frac { ( Z ^ 0 ) ^ 2 \\zeta ( 3 ) } { 2 ( 2 \\pi \\mathrm { i } ) ^ 3 } , \\chi > 0 \\ , , \\end{align*}"}
+{"id": "1459.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & f ( 1 ) = r \\\\ & f ( 2 ) = r + 1 \\\\ & \\quad \\quad \\quad \\vdots \\\\ & f ( n - r + 2 ) = n + 1 \\end{aligned} \\right . \\left \\{ \\begin{aligned} & f ( n - r + 3 ) = 1 \\\\ & f ( n - r + 4 ) = 2 \\\\ & \\quad \\quad \\quad \\vdots \\\\ & f ( n + 1 ) = r - 1 \\end{aligned} \\right . \\quad . \\end{align*}"}
+{"id": "4993.png", "formula": "\\begin{align*} r ( x ) = x h [ r ( x ) ] = x \\exp \\biggl ( \\sum _ { k = 1 } ^ \\infty \\frac { r ( x ^ k ) } { k } \\biggr ) . \\end{align*}"}
+{"id": "3236.png", "formula": "\\begin{align*} \\Delta _ { \\alpha } ^ { 0 } ( \\epsilon , R ) = \\frac { R } { q \\sqrt { 1 + \\alpha ^ { 2 } } } + O ( 1 ) . \\end{align*}"}
+{"id": "4555.png", "formula": "\\begin{align*} \\int _ { \\Omega _ t } ( \\partial _ t \\mathcal B _ 0 + \\partial _ 1 \\mathcal B _ 1 - 2 \\partial _ 2 \\mathcal B _ 2 - \\mathcal B _ 3 ) { \\mathbf V } _ { x _ 2 } \\cdot { \\mathbf V } _ { x _ 2 } d { \\bf x } d s \\le C _ 1 \\Vert { \\mathbf V } _ { x _ 2 } \\Vert _ { L ^ 2 ( \\Omega _ t ) } ^ 2 = C _ 1 \\int _ 0 ^ t I _ 2 ( s ) d s \\ , , \\end{align*}"}
+{"id": "3211.png", "formula": "\\begin{align*} \\left \\Vert x - x _ { k } \\right \\Vert ^ 2 - \\left \\Vert x - x _ { k + 1 } \\right \\Vert ^ 2 = \\Vert x _ { k + 1 } - x _ { k } \\Vert ^ 2 + 2 \\left ( x _ { k + 1 } - x _ { k } \\right ) ^ { T } \\left ( x - x _ { k + 1 } \\right ) . \\end{align*}"}
+{"id": "8211.png", "formula": "\\begin{align*} ( \\alpha - i a ^ 2 \\alpha ( T ) d x _ j ) ( I _ j ^ a X ) = i ( \\alpha - i a ^ 2 \\alpha ( T ) d x _ j ) X . \\end{align*}"}
+{"id": "5682.png", "formula": "\\begin{align*} k _ { e _ { n } } & = \\limsup _ { z \\rightarrow 0 } \\frac { \\ln \\| ( z - T ) ^ { - 1 } e _ { n } \\| _ { p } } { \\ln \\| ( z - T ) ^ { - 1 } \\| } \\\\ & = \\lim _ { z \\rightarrow 0 } \\frac { \\ln \\| ( z - T ) ^ { - 1 } e _ { n } \\| _ { p } } { \\ln \\| ( z - T ) ^ { - 1 } e _ { 0 } \\| _ p } \\limsup _ { z \\rightarrow 0 } \\frac { \\ln \\| ( z - T ) ^ { - 1 } e _ { 0 } \\| _ { p } } { \\ln \\| ( z - T ) ^ { - 1 } \\| } \\\\ & = k _ { e _ { 0 } } . \\end{align*}"}
+{"id": "5796.png", "formula": "\\begin{align*} s _ 1 ( s _ 2 s _ 1 ) ( s _ 3 s _ 2 s _ 1 ) \\dots & ( s _ n s _ { n - 1 } { \\dots } s _ 2 s _ 1 ) = \\\\ & s _ n ( s _ { n - 1 } s _ n ) ( s _ { n - 2 } s _ { n - 1 } s _ n ) \\dots ( s _ 1 s _ 2 { \\dots } s _ { n - 1 } s _ n ) . \\end{align*}"}
+{"id": "721.png", "formula": "\\begin{align*} \\mathbf U ( t ) & = \\mathbf S ( t ) \\mathbf u _ 0 + i \\int _ 0 ^ { t \\wedge \\mu } \\mathbf S ( t - s ) \\mathbf N ( \\mathbf U ( s ) ) \\ , d s + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf M ( \\mathbf U ( s ) ) \\ , d W ( s ) , \\\\ \\mathbf V ( t ) & = \\mathbf S ( t ) \\mathbf u _ 0 + i \\int _ 0 ^ { t \\wedge \\mu } \\mathbf S ( t - s ) \\mathbf N ( \\mathbf V ( s ) ) \\ , d s + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf M ( \\mathbf V ( s ) ) \\ , d W ( s ) . \\end{align*}"}
+{"id": "607.png", "formula": "\\begin{align*} f ( T ) = \\norm { \\psi _ + ^ R } _ { L ^ 1 ( \\Omega , X ^ { 0 , b } _ { + \\xi } ( 0 , T ) ) } + \\norm { \\psi _ - ^ R } _ { L ^ 1 ( \\Omega , X ^ { 0 , b } _ { - \\xi } ( 0 , T ) ) } , \\end{align*}"}
+{"id": "2143.png", "formula": "\\begin{align*} \\det g = \\alpha ^ 2 , g = \\alpha \\ , \\hbox { d i a g } \\left ( e ^ { u _ 0 } , e ^ { - u _ 0 } \\right ) . \\end{align*}"}
+{"id": "7552.png", "formula": "\\begin{align*} z ^ \\beta = z _ 1 ^ { \\beta _ 1 } \\cdots z _ N ^ { \\beta _ N } . \\end{align*}"}
+{"id": "1875.png", "formula": "\\begin{gather*} \\sup _ { | s - \\sigma _ 0 | = r } | \\zeta ( s + i \\tau _ n , \\alpha ) - f ( s ) | < r = \\inf _ { | s - \\sigma _ 0 | = r } | f ( s ) | \\end{gather*}"}
+{"id": "524.png", "formula": "\\begin{align*} \\norm { \\overline u } _ { X _ { h ( \\xi ) } ^ { s , b } ( S , T ) } = \\norm { u } _ { X _ { - h ( - \\xi ) } ^ { s , b } ( S , T ) } . \\end{align*}"}
+{"id": "5368.png", "formula": "\\begin{align*} h ^ 1 ( { N _ { X ' } } _ { | _ Z } ( K _ Z - 3 H ) ) = 0 . \\end{align*}"}
+{"id": "3508.png", "formula": "\\begin{align*} Z _ { \\Gamma ' } ( s ) = Z _ \\Gamma ( s ) Z _ \\Gamma ( s , \\lambda _ { \\Gamma / \\Gamma ' } ^ 0 ) . \\end{align*}"}
+{"id": "8122.png", "formula": "\\begin{align*} V ^ \\alpha _ { p + v _ \\alpha - u _ \\alpha } = \\phi _ \\alpha ^ { - 1 } ( V _ q ^ \\beta | _ k ) \\textrm { w i t h } q = p - u _ \\alpha + \\sum _ { \\beta ' \\stackrel { \\ge k } { \\to } \\alpha } u _ { \\beta ' } . \\end{align*}"}
+{"id": "3691.png", "formula": "\\begin{align*} \\lambda = \\left ( \\sum _ { i = 1 } ^ m k _ i \\right ) n / m - \\sum _ { i = 1 } ^ m k _ i ( k _ i - 1 ) = k n / m = k \\lambda _ n ( L ( G ) ) . \\end{align*}"}
+{"id": "1256.png", "formula": "\\begin{align*} C _ 0 \\leq \\lim _ { n \\to \\infty } \\sum _ { j = 1 } ^ { \\infty } P ( g _ n ^ j [ \\phi ^ j ] ) \\leq C _ 0 \\sum _ { j = 1 } ^ { \\infty } \\| \\nabla \\phi ^ j \\| _ { L ^ 2 } ^ { 2 p } . \\end{align*}"}
+{"id": "7951.png", "formula": "\\begin{align*} \\Psi _ { i } ^ r = \\sum _ { u \\in U _ i } \\deg _ { G _ i } ( u _ i ) \\end{align*}"}
+{"id": "3015.png", "formula": "\\begin{align*} \\left < \\psi , \\sigma _ { g _ 1 , g _ 2 } \\right > = \\psi \\left ( g _ 1 , g _ 2 \\right ) . \\end{align*}"}
+{"id": "3608.png", "formula": "\\begin{align*} { \\bf f } _ { \\rm D B } = \\sqrt { \\frac { E } { K } } { \\bf T } _ { \\rm D B } { \\bf 1 } _ { K \\times 1 } . \\end{align*}"}
+{"id": "142.png", "formula": "\\begin{align*} \\overline { \\sigma ( a ) } ' = \\sigma ( \\overline { a } ) ' = \\sigma ' ( \\overline { a } ' ) = \\overline { \\sigma ' ( a ' ) } , \\end{align*}"}
+{"id": "8023.png", "formula": "\\begin{align*} \\frac { \\partial w _ 1 } { \\partial t } & = \\lambda ( p _ 0 + p _ 1 - 1 ) w _ 1 - \\frac { \\partial f _ 1 } { \\partial x _ 1 } + \\lambda ( p _ 0 + p _ 1 ) \\sum _ { j = 2 } ^ D f _ j \\\\ & = A w _ 1 + B f _ 1 + C \\sum _ { j = 2 } ^ D f _ j , \\end{align*}"}
+{"id": "174.png", "formula": "\\begin{align*} \\theta _ 2 ( v , \\tau + 1 ) = \\theta _ 3 ( v , \\tau ) , ~ ~ \\theta _ 2 ( v , - \\frac { 1 } { \\tau } ) = \\left ( \\frac { \\tau } { \\sqrt { - 1 } } \\right ) ^ { \\frac { 1 } { 2 } } e ^ { \\pi \\sqrt { - 1 } \\tau v ^ 2 } \\theta _ 1 ( \\tau v , \\tau ) ; \\end{align*}"}
+{"id": "4487.png", "formula": "\\begin{align*} | | ( { \\mathbf V } _ k , \\Psi _ k ) | | _ { s , \\ast , T } + | | \\psi _ k | | _ { H ^ { s } ( \\Gamma _ T ) } \\lesssim \\begin{cases} \\delta \\theta ^ { ( s - \\alpha ) _ + } _ k , & s \\neq \\alpha , \\\\ \\delta \\log \\theta _ k , & s = \\alpha , \\\\ \\end{cases} \\end{align*}"}
+{"id": "5464.png", "formula": "\\begin{align*} \\log \\left ( \\sum _ { n = 0 } ^ \\infty e ^ { | f ( n ) | } \\pi ( n ) \\right ) & = \\log \\left ( \\sum _ { n = 0 } ^ \\infty e ^ { | f ( n ) - f ( 0 ) + f ( 0 ) - \\pi ( f ) | } \\pi ( n ) \\right ) \\\\ & \\leq \\log \\left ( \\sum _ { n = 0 } ^ \\infty e ^ { | f ( n ) - f ( 0 ) | } \\pi ( n ) \\right ) + | f ( 0 ) - \\pi ( f ) | . \\end{align*}"}
+{"id": "3161.png", "formula": "\\begin{align*} \\nabla _ \\mathbf { z } W _ L = \\lambda \\nabla _ \\mathbf { z } f \\hbox { f o r s o m e } \\lambda \\in \\mathbb { C } . \\end{align*}"}
+{"id": "21.png", "formula": "\\begin{align*} \\ell : = \\frac { K _ { \\ell } } { \\sqrt { \\rho \\widehat { g } ( 0 ) } } \\end{align*}"}
+{"id": "6842.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\rm c y l } ( v _ { \\varepsilon } + t \\psi ) - \\mathcal { N } _ { \\rm c y l } ( v _ { \\varepsilon } ) & = - P _ { \\rm c y l } v _ { \\varepsilon } + t P _ { \\rm c y l } \\psi - ( v _ { \\varepsilon } + t \\psi ) ^ { \\frac { n + 6 } { n - 6 } } + P _ { \\rm c y l } v _ { \\varepsilon } + v _ { \\varepsilon } ^ { \\frac { n + 6 } { n - 6 } } & \\\\ & = t P _ { \\rm c y l } \\psi - ( v _ { \\varepsilon } + t \\psi ) ^ { \\frac { n + 6 } { n - 6 } } + v _ { \\varepsilon } ^ { \\frac { n + 6 } { n - 6 } } , \\end{align*}"}
+{"id": "3100.png", "formula": "\\begin{align*} [ p ] _ { i j } = \\left \\{ \\begin{matrix} 1 - \\frac { \\sqrt 3 } { 3 } + o ( 1 ) & \\approx 0 . 4 2 + o ( 1 ) & i j \\in \\{ 1 2 , 1 3 \\} \\\\ 2 \\left ( 1 - \\frac { \\sqrt 3 } { 3 } \\right ) + o ( 1 ) & \\approx 0 . 8 5 + o ( 1 ) & i j = 2 3 \\\\ \\frac { 3 - \\sqrt { 3 } } { 2 } + o ( 1 ) & \\approx 0 . 6 3 + o ( 1 ) & \\end{matrix} \\right . \\end{align*}"}
+{"id": "7409.png", "formula": "\\begin{align*} \\partial _ { t } W _ { n } + u _ { n } \\partial _ { x } W _ { n } = 0 . \\end{align*}"}
+{"id": "4156.png", "formula": "\\begin{align*} P _ { \\hat p , \\exp _ { \\hat p } ^ { \\nabla } ( t w ) } ( v ) & = v - \\langle v , w \\rangle w + \\langle v , w \\rangle p _ 2 ( t ) , \\\\ p _ 2 ( t ) & = - \\sin ( t ) \\hat p + \\cos ( t ) w . \\end{align*}"}
+{"id": "7799.png", "formula": "\\begin{align*} \\begin{pmatrix} a & b \\\\ c & d \\\\ \\end{pmatrix} \\cdot t : = \\begin{dcases} t \\\\ - 1 / t \\\\ \\frac { 1 + t ^ { c , d } _ { + } t } { t ^ { c , d } _ { + } - t } \\end{dcases} \\ , . \\end{align*}"}
+{"id": "2731.png", "formula": "\\begin{align*} \\| x \\| _ { \\theta , \\mathcal { F } } = \\max \\big \\{ \\| x \\| _ { \\infty } , \\| x \\| _ T \\big \\} , \\end{align*}"}
+{"id": "5933.png", "formula": "\\begin{align*} 2 e - 1 = ( 2 e - 2 ) + 1 = S _ { n - 2 } - S _ { n - 1 } + \\beta _ { n - 2 } \\le d [ - b _ { n - 2 } b _ { n - 1 } ] \\le \\beta _ { n - 1 } = 2 e - 1 \\end{align*}"}
+{"id": "2538.png", "formula": "\\begin{align*} & a _ { j - 1 } < a ' _ k \\\\ & s _ { j - 1 } < t _ k \\textrm { o r } ( s _ { j - 1 } = t _ k \\textrm { a n d } b _ { j - 1 } < b ' _ k ) \\end{align*}"}
+{"id": "8726.png", "formula": "\\begin{align*} \\frac { \\Gamma ( \\beta ) } { ( \\ell - 1 ) ! } = \\frac { \\Gamma ( \\beta ) } { \\Gamma ( \\ell ) } = \\frac { \\ell \\Gamma ( \\ell + ( \\beta - \\ell ) ) } { \\Gamma ( \\ell + 1 ) } \\leq \\ell ^ { \\beta - \\ell } \\leq \\ell \\enspace , \\end{align*}"}
+{"id": "3350.png", "formula": "\\begin{align*} \\tilde { f } \\Bigl ( \\frac { i \\varepsilon } { 2 \\pi } \\Bigr ) = \\tilde { f } \\Bigl ( \\frac { p i \\varepsilon / 2 \\pi + q } { r i \\varepsilon / 2 \\pi + s } \\Bigr ) = \\tilde { f } \\Bigl ( \\alpha + \\frac { i h } { 2 \\pi m } \\Bigr ) \\end{align*}"}
+{"id": "4186.png", "formula": "\\begin{align*} C : = \\sup _ { f ( X ) : ~ \\mathbb { E } [ X ^ 2 ] \\leq P } I ( X ; Y ) \\end{align*}"}
+{"id": "2872.png", "formula": "\\begin{align*} ( T _ j - t _ j ) ( T _ j + 1 ) = 0 ( 0 \\leq j \\leq n ) , \\end{align*}"}
+{"id": "1371.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\kappa , s } ( - w ) : = w ^ { - \\frac { \\kappa } { 2 } } M _ { - \\frac { \\kappa } { 2 } , s - \\frac { 1 } { 2 } } ( w ) , \\end{align*}"}
+{"id": "3511.png", "formula": "\\begin{align*} Z _ { \\Gamma } ( s , \\rho ) = \\det ( 1 - \\mathcal { L } _ { s , \\rho } ) . \\end{align*}"}
+{"id": "5001.png", "formula": "\\begin{align*} \\psi ( r ) = \\left \\{ \\begin{array} { r c l } 1 , & \\mathrm { i f } & r \\leq t - \\eta \\\\ \\frac { t - r } { \\eta } , & \\mathrm { i f } & t - \\eta < r < t \\\\ 0 , & \\mathrm { i f } & r \\geq t . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "1279.png", "formula": "\\begin{align*} \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| e ^ { i t \\Delta } w _ n ^ J \\| _ { S _ 0 ( \\R ) } = 0 . \\end{align*}"}
+{"id": "6297.png", "formula": "\\begin{align*} \\begin{aligned} J ^ 4 ( v _ \\lambda , v _ \\mu ^ { x _ 0 } ) = & \\ g _ { [ < \\lambda ] } J ^ 4 _ { m a i n } ( v _ \\lambda , v _ \\mu ^ { x _ 0 } ) + ( g _ { [ < \\mu ] } ^ { x _ 0 } - g _ { [ < \\lambda ] } ) \\left ( M ( v _ \\mu ^ { x _ 0 } ) E ( v _ \\lambda ) - P ( v _ \\lambda ) P ( v _ \\mu ^ { x _ 0 } ) \\right ) \\\\ : = & \\ J ^ { 4 , a } ( v _ \\lambda , v _ \\mu ^ { x _ 0 } ) + J ^ { 4 , b } ( v _ \\lambda , v _ \\mu ^ { x _ 0 } ) , \\end{aligned} \\end{align*}"}
+{"id": "2882.png", "formula": "\\begin{align*} \\xi \\in V _ { } : = \\{ \\xi \\in V \\mid \\langle \\xi , \\alpha \\rangle \\not \\in 2 \\pi \\mathbb { Z } , \\ , \\forall \\alpha \\in R _ 0 ^ + \\} , \\end{align*}"}
+{"id": "4630.png", "formula": "\\begin{align*} \\bigsqcup _ { v \\in V } \\mathcal { S } ( v | s ) = E = \\bigsqcup _ { v \\in V } \\mathcal { S } ( v | t ) \\end{align*}"}
+{"id": "1164.png", "formula": "\\begin{align*} & \\Phi _ t ( m \\rq _ { 1 , t } ( x , y ) ) = m _ { 1 , t } ( \\Phi _ t ( x ) , \\Phi _ t ( y ) ) , \\\\ & \\Phi _ t ( m \\rq _ { 2 , t } ( x , y ) ) = m _ { 2 , t } ( \\Phi _ t ( x ) , \\Phi _ t ( y ) ) , \\ , \\ , \\ , ~ x , y \\in \\mathfrak { g } . \\end{align*}"}
+{"id": "4560.png", "formula": "\\begin{align*} \\vert \\mathcal I _ { 1 , 1 } + \\mathcal I _ { 1 , 3 } \\vert & \\le C _ 2 \\Vert \\partial _ 2 \\dot { q } ^ + \\Vert _ { L ^ 2 ( \\Omega _ t ) } \\Vert \\dot { H } ^ + _ n \\Vert _ { L ^ 2 ( \\Omega _ t ) } + C _ 1 \\Vert \\partial _ 2 \\dot { q } ^ + \\Vert _ { L ^ 2 ( \\Omega _ t ) } \\Vert \\partial _ 1 \\dot { H } ^ + _ n \\Vert _ { L ^ 2 ( \\Omega _ t ) } \\\\ & \\le C _ 2 \\left \\{ \\int _ 0 ^ t ( I _ { 1 , \\ast } + I _ { 1 , n } ) ( s ) d s \\right \\} \\end{align*}"}
+{"id": "4027.png", "formula": "\\begin{align*} h ( \\mu * \\gamma _ t ) - h ( \\mu ) = \\frac { 1 } { 2 } \\int _ 0 ^ t \\mathcal { I } ( \\mu * \\gamma _ s ) d s , \\end{align*}"}
+{"id": "3942.png", "formula": "\\begin{align*} h _ K ( v ) \\geq v r _ K ( w ) > v r _ L ( w ) = h _ L ( v ) . \\end{align*}"}
+{"id": "6489.png", "formula": "\\begin{align*} a _ e = F _ { e _ - } , b _ e = F ' _ { e _ - } , c _ e = \\frac 1 2 C _ { e _ - } . \\end{align*}"}
+{"id": "1831.png", "formula": "\\begin{gather*} \\limsup _ { r \\to \\infty } \\frac { \\log | F _ g ( r ) | } { r } = \\limsup _ { m \\to \\infty } \\frac { \\log | F _ g ( \\lambda _ m ) | } { | \\lambda _ m | } . \\end{gather*}"}
+{"id": "2519.png", "formula": "\\begin{align*} & \\int _ { \\Omega } \\big ( \\nu \\ , \\textbf { c u r l } \\ , \\boldsymbol { y } _ 0 ^ c \\cdot \\textbf { c u r l } \\ , \\boldsymbol { v } _ 0 ^ c \\big ) \\ , d \\boldsymbol { x } = \\int _ { \\Omega } \\boldsymbol { u } ^ c _ 0 \\cdot \\boldsymbol { v } _ 0 ^ c \\ , d \\boldsymbol { x } \\end{align*}"}
+{"id": "8646.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\int _ { - 1 + \\beta b ( X ) } ^ 0 | u ( X , z ) | ^ 2 \\ : \\mathrm { d } z \\mathrm { d } X & = \\int _ { \\R ^ d } \\int _ { - 1 + \\beta b ( X ) } ^ 0 | \\int _ { z } ^ 0 ( \\partial _ z u ) ( X , z ' ) \\ : \\mathrm { d } z ' | ^ 2 \\ : \\mathrm { d } z \\mathrm { d } X \\\\ & \\leq ( 1 + \\beta | b | _ { L ^ { \\infty } } ) ^ 2 \\int _ { \\R ^ d } \\int _ { - 1 + \\beta b ( X ) } ^ 0 | ( \\partial _ z u ) ( X , z ' ) | ^ 2 \\ : \\mathrm { d } z ' \\mathrm { d } X . \\end{align*}"}
+{"id": "1043.png", "formula": "\\begin{align*} \\langle \\hat { \\varphi } ' _ \\mu ( u _ \\mu ) , h \\rangle = 0 \\mbox { f o r a l l } h \\in W ^ { 1 , p ( z ) } ( \\Omega ) . \\end{align*}"}
+{"id": "6852.png", "formula": "\\begin{align*} { } \\mathcal { X } \\times _ 1 A _ 1 + \\mathcal { X } \\times _ 2 A _ 2 + \\dots + \\mathcal { X } \\times _ d A _ d = \\mathcal { C } , \\end{align*}"}
+{"id": "5445.png", "formula": "\\begin{align*} P _ { t - s } ( h \\Gamma ( f _ s ) ) & \\leq \\frac { 2 \\rho q ( s ) } { 2 \\rho \\theta - 1 + e ^ { - 2 \\rho ( t - s ) } } \\log \\left ( P _ { t } \\left ( e ^ { q ( 0 ) ( \\Gamma ( f ) ) } \\right ) \\right ) ^ { 1 / q ( 0 ) } \\\\ & = \\frac { 2 \\rho q ( s ) / q ( 0 ) } { 2 \\rho \\theta - 1 + e ^ { - 2 \\rho ( t - s ) } } \\log P _ { t } \\left ( e ^ { \\alpha ( \\Gamma ( f ) ) } \\right ) . \\end{align*}"}
+{"id": "7712.png", "formula": "\\begin{align*} \\widetilde { K } : = K ' + K _ 1 \\wedge E _ 1 ^ * | _ S \\end{align*}"}
+{"id": "7194.png", "formula": "\\begin{align*} \\mathfrak { a } _ 2 ^ 3 \\mathfrak { a } _ 3 & = ( 1 \\otimes y _ 2 ^ 3 - y _ 2 \\otimes y _ 2 ^ 2 + y _ 2 ^ 2 \\otimes y _ 2 - y _ 2 ^ 3 \\otimes 1 ) ( 1 \\otimes y _ 3 - y _ 3 \\otimes 1 ) \\\\ & = y _ 1 y _ 2 \\otimes y _ 2 y _ 3 + y _ 2 y _ 3 \\otimes y _ 1 y _ 2 - y _ 1 y _ 2 y _ 3 \\otimes y _ 2 - y _ 2 \\otimes y _ 1 y _ 2 y _ 3 . \\end{align*}"}
+{"id": "3429.png", "formula": "\\begin{align*} H ^ 1 ( M _ K , ( l - \\sum d _ i ) L ) \\simeq H ^ 1 ( M _ K , l L + K _ M ) = 0 , \\end{align*}"}
+{"id": "8749.png", "formula": "\\begin{align*} \\int K ( r ) \\d r = 0 , \\int r K ( r ) \\d r = 1 , \\int r ^ j K ( r ) \\d r = 0 , \\ j = 2 , \\dots , \\ell , ~ \\kappa _ \\beta \\triangleq \\int | r | ^ { \\beta } | K ( r ) | \\d r < \\infty \\end{align*}"}
+{"id": "329.png", "formula": "\\begin{align*} \\left \\langle \\mathbf { j } , \\nabla \\overline { q } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } + s ^ { 2 } \\left \\langle p _ { j } ^ { \\operatorname * { e x t } } u , \\overline { q } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } = 0 \\quad \\forall q \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } . \\mathbb { A } _ { j } ^ { \\operatorname * [ e x t ] } \\right ) . \\end{align*}"}
+{"id": "631.png", "formula": "\\begin{align*} \\norm { \\theta _ R \\left ( \\sum _ { i = 1 } ^ n \\norm { \\phi _ i } _ { \\widetilde H ^ { b } ( 0 , t ) } ^ 2 \\right ) \\phi _ j ( t ) - \\theta _ R \\left ( \\sum _ { i = 1 } ^ n \\norm { \\Phi _ i } _ { \\widetilde H ^ { b } ( 0 , t ) } ^ 2 \\right ) \\Phi _ j ( t ) } _ { H ^ { b } ( 0 , T ) } \\leqslant C \\sum _ { i = 1 } ^ n \\norm { \\phi _ i - \\Phi _ i } _ { H ^ { b } ( 0 , T ) } , \\end{align*}"}
+{"id": "1650.png", "formula": "\\begin{align*} \\widetilde W ( r ) \\sim \\begin{cases} \\left ( \\frac { n - P } { P } \\right ) ^ P r ^ { - P } & 1 < P < n \\\\ \\left ( \\frac { P - 1 } { P } \\right ) ^ P | r \\log ( r ) | ^ { - P } & P = n \\\\ C _ { P , f } \\ , \\ , r ^ { - \\frac { P ( n - 1 ) } { P - 1 } } & n < P < \\infty \\end{cases} r \\to 0 , \\end{align*}"}
+{"id": "5673.png", "formula": "\\begin{align*} \\widehat { D } _ 5 ^ a \\cdot e _ 1 = \\frac { 1 5 } { 2 } + 3 b - 3 a = - \\frac { k } { 2 } \\ , \\end{align*}"}
+{"id": "4087.png", "formula": "\\begin{align*} \\check \\omega _ { 0 , n + 1 } = \\check \\omega _ { 0 , n + 1 } ^ { ( 0 ) } = \\check \\varpi _ { 0 , n + 1 } , \\check \\omega _ { \\frac 1 2 , n + 1 } = \\check \\omega _ { \\frac 1 2 , n + 1 } ^ { ( 1 ) } = \\check \\varpi _ { \\frac 1 2 , n + 1 } . \\end{align*}"}
+{"id": "8241.png", "formula": "\\begin{align*} i ^ * _ { \\mu ^ { - 1 } ( 0 ) } \\hat { \\omega } _ 1 = p _ 0 ^ * i \\partial \\bar \\partial _ { I _ 1 } f _ { x _ 1 } + x _ 1 i ^ * _ { \\mu ^ { - 1 } ( 0 ) } d \\eta = p _ 0 ^ * ( i \\partial \\bar \\partial _ { I _ 1 } f _ { x _ 1 } + x _ 1 \\Omega _ 0 ) . \\end{align*}"}
+{"id": "6041.png", "formula": "\\begin{align*} R = \\frac { \\det \\sigma _ F - \\det \\sigma _ Q } { \\det \\sigma _ Q } . \\end{align*}"}
+{"id": "2959.png", "formula": "\\begin{align*} d = \\frac { 1 } { 6 } \\left ( T _ { 1 2 3 } - T _ { 1 3 2 } - T _ { 2 1 3 } + T _ { 2 3 1 } + T _ { 3 1 2 } - T _ { 3 2 1 } \\right ) , \\end{align*}"}
+{"id": "7059.png", "formula": "\\begin{align*} S _ r ( N ) : = \\mathop { \\sum } _ { n = 1 } ^ { \\infty } A _ { \\pi } ( n , r ) \\lambda _ f ( n ) n ^ { - i t } V _ 1 \\left ( \\frac { n } { N } \\right ) , \\end{align*}"}
+{"id": "7846.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\mathbb { H } } : = \\{ \\tau \\in \\mathbb { H } \\ ; | \\ ; | \\tau | > 1 , \\ ; \\ ; | \\tau _ 1 | < 1 / 2 \\} \\cup \\{ \\tau \\in \\mathbb { H } \\ ; | \\ ; | \\tau | \\geq 1 , \\ ; \\ ; \\tau _ 1 = - \\frac { 1 } { 2 } \\} \\cup \\{ \\tau \\in \\mathbb { H } \\ ; | \\ ; | \\tau | = 1 , \\ ; \\ ; - \\frac { 1 } { 2 } < \\tau _ 1 \\leq 0 \\} \\ , . \\end{align*}"}
+{"id": "8706.png", "formula": "\\begin{align*} \\tilde { \\phi } _ i = \\frac { \\sum _ { j = 0 } ^ n a _ { i , j } \\phi _ j + a _ { i , n + 1 } } { \\sum _ { j = 0 } ^ n a _ { n + 1 , j } \\phi _ j + a _ { n + 1 , n + 1 } } , ~ i = 0 , . . . , n , \\end{align*}"}
+{"id": "8873.png", "formula": "\\begin{align*} c ( V ) = c ( j ^ * V _ G ) = j ^ * c ( V _ G ) = j ^ * c ^ G ( V ) . \\end{align*}"}
+{"id": "6683.png", "formula": "\\begin{align*} \\overline { W } _ n ( x _ 1 , \\dots , x _ n ; N , \\beta , p , q ) = N ^ { 2 - n } \\sum _ { l = 0 } ^ \\infty { \\overline { W } _ n ^ { l } ( x ; \\beta , p , q ) \\over N ^ l } ; \\end{align*}"}
+{"id": "5225.png", "formula": "\\begin{align*} \\Pr [ S _ n > t ] \\leq \\exp \\left ( - \\frac { 2 t ^ 2 } { \\sum _ { i = 1 } ^ n c _ i ^ 2 } \\right ) . \\end{align*}"}
+{"id": "5919.png", "formula": "\\begin{align*} - 2 e + R _ { n - 1 } = R _ { n - 2 } + R _ { n - 1 } & \\le S _ { n - 2 } + S _ { n - 1 } = 1 - 2 e , \\\\ R _ { n - 1 } + R _ { n } & \\le S _ { n - 1 } + S _ { n } = 2 - 2 e . \\end{align*}"}
+{"id": "4178.png", "formula": "\\begin{align*} c & : = \\dfrac { \\lambda } { \\sqrt { 2 \\pi ^ 3 } } | u | ^ { 3 } e ^ { \\lambda | u | } ; ~ f ( r ) : = \\dfrac { K _ { 3 / 2 } \\left ( | u | \\sqrt { r ^ 2 + \\lambda ^ 2 } \\right ) } { \\left ( | u | \\sqrt { r ^ 2 + \\lambda ^ 2 } \\right ) ^ { 3 / 2 } } . \\end{align*}"}
+{"id": "6249.png", "formula": "\\begin{align*} L ( u _ 1 , \\cdots , u _ k ) ( x ) = \\int K ( y _ 1 , \\cdots , y _ k ) u _ 1 ^ { y _ 1 } ( x ) \\cdots u _ k ^ { y _ k } \\ , d y \\end{align*}"}
+{"id": "6419.png", "formula": "\\begin{align*} \\displaystyle \\liminf _ { t \\rightarrow \\infty } \\inf _ { | x | < c t } w ( x , t ) = k . \\end{align*}"}
+{"id": "238.png", "formula": "\\begin{align*} C ^ { } ( \\xi ; g ) : = \\prod _ { 1 \\leq j \\leq n } c _ w ( \\xi _ j ) \\prod _ { 1 \\leq j < k \\leq n } c _ v ( \\xi _ j + \\xi _ k ) c _ v ( \\xi _ j - \\xi _ k ) , \\end{align*}"}
+{"id": "4196.png", "formula": "\\begin{align*} D ( 0 , \\varepsilon ) & = \\sigma _ { \\varepsilon } ( [ S \\bullet T , \\varphi ^ { - 1 } ( i I ) ] _ { \\ast } ) = \\sigma _ { \\varepsilon } ( [ \\frac { I } { 2 } \\bullet \\varphi ( T ) , i I ] _ { \\ast } ) \\\\ & = \\sigma _ { \\varepsilon } ( i ( \\varphi ( T ) - \\varphi ( T ) ^ { \\ast } ) ) . \\end{align*}"}
+{"id": "8787.png", "formula": "\\begin{align*} \\norm { x _ { t + 1 } - x _ { p } } ^ { 2 } & = \\norm { x _ { t } - \\eta _ { t } \\hat { g } _ { t } - x _ { p } } ^ 2 \\\\ & = \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 \\\\ & \\leq ( 1 - 2 \\eta _ { t } \\alpha ) \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } - \\nabla f ( x _ t ) , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 , \\end{align*}"}
+{"id": "6277.png", "formula": "\\begin{align*} m ( \\xi , \\eta ) = 1 , p ( \\xi , \\eta ) = \\xi + \\eta , e ( \\xi , \\eta ) = ( \\xi + \\eta ) ^ 2 . \\end{align*}"}
+{"id": "6484.png", "formula": "\\begin{align*} f ' ( 0 ) = \\gamma f ( 0 ) , - f ' ( \\epsilon ) = - \\gamma f ( \\epsilon ) . \\end{align*}"}
+{"id": "3118.png", "formula": "\\begin{align*} \\rho _ { k , l } = \\frac { | \\mathbf { H } _ { k } \\cdot \\mathbf { H } _ { l } ^ H | } { \\Vert \\mathbf { H } _ { k } \\Vert \\norm { \\mathbf { H } _ { l } } } . \\end{align*}"}
+{"id": "5763.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\mathbb E \\langle ( m ^ 1 ) ^ 2 \\rangle _ { \\beta } = 0 , \\end{align*}"}
+{"id": "2846.png", "formula": "\\begin{align*} s _ \\lambda ^ { ( c ) } s _ { \\omega _ r } ^ { ( c ) } = \\sum _ { \\substack { J \\subseteq \\{ 1 , \\ldots , n \\} , \\ , | J | = r \\\\ \\lambda + \\bar { e } _ J \\in \\Lambda ^ { ( n , c ) } } } s _ { \\lambda + \\bar { e } _ J } ^ { ( c ) } \\end{align*}"}
+{"id": "7069.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } & A _ { \\pi } ( r , n ) \\ , e \\left ( \\frac { ( a + b q ) n } { p _ 1 q } \\right ) \\ , v _ 1 ( n ) \\\\ & = p _ 1 q \\ , \\sum _ { \\pm } \\sum _ { n _ { 1 } | p _ 1 q r } \\ , \\ , \\sum _ { n _ { 2 } = 1 } ^ { \\infty } \\frac { A _ { \\pi } ( n _ { 2 } , n _ { 1 } ) } { n _ { 1 } n _ { 2 } } S \\left ( r \\overline { ( a + b q ) } , \\pm n _ { 2 } ; p _ 1 q r / n _ { 1 } \\right ) \\ , \\mathcal { V } ^ { \\pm } _ 1 \\left ( \\frac { n _ { 1 } ^ 2 n _ { 2 } } { ( p _ 1 q ) ^ 3 r } \\right ) . \\end{align*}"}
+{"id": "1332.png", "formula": "\\begin{align*} \\frac { J ^ w ( \\textbf { X } _ { m i n R S S U } ^ { ( n + 1 ) } ) } { J ^ w ( \\textbf { X } _ { m i n R S S U } ^ { ( n ) } ) } & = - 2 J ^ w ( X _ { 1 : n + 1 } ) \\\\ & = \\int _ { 0 } ^ { + \\infty } w ( x ) f ^ 2 _ { 1 : n + 1 } ( x ) d x \\\\ & = ( n + 1 ) ^ 2 \\int _ { 0 } ^ { 1 } w ( F ^ { - 1 } ( u ) ) ( 1 - u ) ^ { 2 n } f ( F ^ { - 1 } ( u ) ) d u \\\\ & \\geq \\frac { ( n + 1 ) ^ 2 } { 2 n + 1 } \\\\ & \\geq 1 . \\end{align*}"}
+{"id": "949.png", "formula": "\\begin{align*} L = \\sum _ { j = 1 } ^ { s } p _ j ( x , t ) ( F _ j ) , \\end{align*}"}
+{"id": "692.png", "formula": "\\begin{align*} - i d \\mathbf u ( t ) = \\mathbf h ( D ) \\mathbf u ( t ) \\ , d t + \\mathbf N ( \\mathbf u ( t ) ) \\ , d t + \\mathbf M ( \\mathbf u ( t ) ) \\ , d W ( t ) , \\mathbf u ( 0 ) = \\mathbf u _ 0 , \\end{align*}"}
+{"id": "7798.png", "formula": "\\begin{align*} A = \\begin{pmatrix} a & b \\\\ c & d \\\\ \\end{pmatrix} \\in \\mathrm { S L } ( 2 , \\mathbb { Z } ) \\end{align*}"}
+{"id": "8538.png", "formula": "\\begin{align*} r _ { \\ell } ^ { k } ( z ) : = \\left ( \\frac { \\ell ^ { k } ( z ) } { \\omega _ { n - 1 } } \\right ) ^ { \\frac { 1 } { n - 1 } } \\mbox { f o r e v e r y $ z \\in J $ a n d f o r e v e r y $ k \\in \\mathbb { N } $ } . \\end{align*}"}
+{"id": "3668.png", "formula": "\\begin{align*} 0 & = \\sum _ { p , m } \\ , \\left ( a _ { i p } ' ( 0 ) \\ , \\delta _ { p m } \\ , \\delta _ { j m } + \\delta _ { i p } \\ , J _ { p m } ' ( 0 ) \\ , \\delta _ { j m } + \\delta _ { i p } \\ , \\delta _ { p m } \\ , a _ { j m } ' ( 0 ) \\right ) \\\\ & = a _ { i j } ' ( 0 ) + ( 1 - 2 \\ , \\lambda _ i ) \\ , \\delta _ { i j } + a _ { j i } ' ( 0 ) \\ , . \\end{align*}"}
+{"id": "4743.png", "formula": "\\begin{align*} w ( x , y ) = \\begin{cases} \\min \\big \\{ 1 , { \\rm d i s t } \\big ( ( x , y ) , \\R \\times \\{ - 1 , 0 \\} \\big ) \\big \\} , & ( x , y ) \\ { \\rm w i t h } \\ y \\in [ - 1 , 0 ] \\\\ 1 , & { \\rm o t h e r w i s e } . \\end{cases} \\end{align*}"}
+{"id": "3164.png", "formula": "\\begin{align*} \\phi _ * W _ { L ^ + } = ( z _ 0 + ( z _ 1 + z _ 2 ) ^ 2 ) ^ 2 - 4 . \\end{align*}"}
+{"id": "4834.png", "formula": "\\begin{align*} u = u + v = v \\end{align*}"}
+{"id": "8615.png", "formula": "\\begin{align*} f \\sim g \\iff f ( X , z ) = r ( X ) g ( X , z ) , \\end{align*}"}
+{"id": "7730.png", "formula": "\\begin{align*} \\nu ( [ u , w ] ) & = \\pounds _ { [ Q , \\eta ( u [ 1 ] ) ] } \\nu ( w ) - \\pounds _ { \\eta ( w ) } ( \\pounds _ Q \\nu ( u [ 1 ] ) ) , \\\\ \\pounds _ { \\eta ( w _ 1 ) } \\nu ( w _ 2 ) & = - \\pounds _ { \\eta ( w _ 2 ) } \\nu ( w _ 1 ) , \\end{align*}"}
+{"id": "4724.png", "formula": "\\begin{align*} \\begin{cases} F ( D ^ 2 u ) = \\chi _ \\Omega & B _ { 1 } ^ { + } \\\\ u = 0 & B ' _ { 1 } \\end{cases} \\end{align*}"}
+{"id": "6707.png", "formula": "\\begin{align*} H ( x ^ { [ f / g ] } ) = H ( x ^ { [ f ] } ) - H ( x ^ { [ g - 1 ] } ) . \\end{align*}"}
+{"id": "6590.png", "formula": "\\begin{align*} q ^ { ( 1 ) } = q ^ { ( 0 ) } + \\Delta _ 1 q = q ^ { ( 0 ) } - H _ M ^ { - 1 } { F } ( q ^ { ( 0 ) } ) . \\end{align*}"}
+{"id": "427.png", "formula": "\\begin{align*} H ( x ) : = [ h ] _ { \\ell , m } ( x ) = | x | ^ { - \\sigma } Y _ { \\ell , m } ( \\omega _ x ) \\end{align*}"}
+{"id": "6566.png", "formula": "\\begin{align*} \\| A _ k ^ { - 1 } \\| & \\leq \\max _ { 1 \\leq l \\leq \\# \\Lambda ' } | - ( \\sigma + k \\cdot \\omega ) ^ 2 + \\zeta _ l ( k ) | ^ { - 1 } \\\\ & = \\max _ { 1 \\leq l \\leq \\# \\Lambda ' } | \\sigma + k \\cdot \\omega - \\sqrt { \\zeta _ l ( k ) } | ^ { - 1 } \\cdot | \\sigma + k \\cdot \\omega + \\sqrt { \\zeta _ l ( k ) } | ^ { - 1 } . \\end{align*}"}
+{"id": "2480.png", "formula": "\\begin{align*} P _ { W ^ * | A = a , V = v _ a } ( \\mathcal { S } ) \\doteq P _ { W ^ * _ a | V _ a = v _ a } ( \\mathcal { S } _ a ) \\prod _ { \\substack { a ' \\in \\mathcal { A } \\\\ a ' \\neq a } } P _ { W ^ * _ { a ' } } ( \\mathcal { S } _ { a ' } ) , \\end{align*}"}
+{"id": "2665.png", "formula": "\\begin{align*} f ( x ) = f ( X ) = f ( x _ { 1 } , x _ { 2 } , \\cdots , x _ { s } ) & = \\sum _ { 1 \\leq i , j \\leq s } F ( v _ i , v _ j ) x _ { i } x _ { j } = X A X ^ T , \\end{align*}"}
+{"id": "3709.png", "formula": "\\begin{align*} \\inf _ { c \\in [ 1 , \\infty ) } J ( c ) \\ge J ( 1 ) + \\int _ 1 ^ \\infty ( 1 - 2 \\alpha ) c ^ { 2 \\alpha - 2 } g ( 1 ) d c = J ( 1 ) + g ( 1 ) , \\end{align*}"}
+{"id": "6378.png", "formula": "\\begin{align*} \\chi = a ( 1 \\otimes 1 ) + n ( x \\otimes x ) + o ( x \\otimes x g ) + r ( x g \\otimes x ) + s ( x g \\otimes x g ) , \\end{align*}"}
+{"id": "1744.png", "formula": "\\begin{align*} 2 \\bigl ( m - d _ \\epsilon - \\tilde { d } _ { \\tilde { \\epsilon } } \\bigr ) \\xi + \\sum _ { 1 \\leq r \\leq d } \\int _ 0 ^ \\xi u _ { a _ r } ( \\theta ) \\theta + \\sum _ { 1 \\leq r \\leq \\tilde { d } } \\int _ 0 ^ \\xi u _ { \\tilde { a } _ r } ( \\theta ) \\theta = \\pi ( 2 l + \\epsilon _ - + \\tilde { \\epsilon } _ - ) , \\end{align*}"}
+{"id": "8935.png", "formula": "\\begin{align*} \\Lambda _ { u , v } = \\left \\{ \\left ( p _ 1 + \\sum _ { j = 1 } ^ n u _ { 1 j } q _ j + v _ 1 , \\ldots , p _ m + \\sum _ { j = 1 } ^ { n } u _ { m j } q _ j + v _ m , \\bar { q } \\right ) : ( \\bar { p } , \\bar { q } ) \\in \\Z ^ m \\times \\Z ^ n \\right \\} \\end{align*}"}
+{"id": "1529.png", "formula": "\\begin{align*} \\sum _ { \\substack { P ^ + ( A ) \\le y \\\\ \\Omega ( A ) = k } } \\frac 1 A = \\eta _ o ( 2 ) \\frac { \\log ^ 2 y } { 2 ^ k } \\left ( 1 + O \\left ( \\frac 1 { ( \\log y ) ^ \\epsilon \\sqrt { \\log _ 2 y } } \\right ) \\right ) . \\end{align*}"}
+{"id": "2565.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } ( K \\ast g ( u _ n ) ) g ' ( u _ n ) u _ n d x = \\int _ { \\mathbb { R } ^ N } ( K \\ast g ( u _ 0 ) ) g ' ( u _ 0 ) u _ 0 d x . \\end{align*}"}
+{"id": "8201.png", "formula": "\\begin{align*} d ( d ^ c K ( T ) ) ( Y ) = - ( d d ^ c K ) ( T , Y ) = - 2 \\omega ( T , Y ) = - 2 ( \\iota _ T \\omega ) ( Y ) . \\end{align*}"}
+{"id": "8415.png", "formula": "\\begin{align*} \\Omega _ { i } = \\sum _ { d \\leq D } \\sum _ { h \\leq H } \\frac { 1 } { h } \\sum _ { | r | \\leq R } \\sup _ { T \\in [ N , N + 2 ] } \\left | U _ { i } \\right | . \\end{align*}"}
+{"id": "2030.png", "formula": "\\begin{align*} ( \\dot { \\mathrm { B } } ^ { - s } _ { p ' , q ' , 0 } ( \\Omega ) ) ' = \\dot { \\mathrm { B } } ^ { s } _ { p , q } ( \\Omega ) \\ , \\ , ( \\dot { \\mathrm { B } } ^ { - s } _ { p ' , q ' } ( \\Omega ) ) ' = \\dot { \\mathrm { B } } ^ { s } _ { p , q , 0 } ( \\Omega ) \\end{align*}"}
+{"id": "6771.png", "formula": "\\begin{align*} \\omega _ n ( z ) = \\frac { \\phi _ n ( z ) } { \\phi ( z _ n ) } = \\frac { \\phi ( z + z _ n ) } { \\phi ( z _ n ) } = \\exp \\left \\{ \\int _ { z _ n } ^ { z _ n + z } \\frac { \\phi ' ( z ) } { \\phi ( z ) } d z \\right \\} . \\end{align*}"}
+{"id": "2150.png", "formula": "\\begin{align*} \\beta _ t = \\tilde \\alpha _ 0 ' ( 2 u ) + \\tilde \\alpha _ 0 ' ( - 2 \\underline { u } ) + \\alpha _ 1 ( 2 u ) - \\alpha _ 1 ( - 2 \\underline { u } ) , \\end{align*}"}
+{"id": "5344.png", "formula": "\\begin{align*} d \\ge \\frac { c + 2 } { 2 k + c } \\binom { k + c } { c + 1 } . \\end{align*}"}
+{"id": "5610.png", "formula": "\\begin{align*} \\| \\psi ' \\| _ { L ^ p _ { p - 1 } } = \\gamma _ n , \\ \\| \\psi \\| _ { L ^ p _ { \\alpha _ 0 } } = 1 , \\ \\| \\psi _ n \\| _ { X ^ { 1 , p } _ { \\infty } } \\to 1 , \\ \\lim _ { R \\to \\infty } \\lim _ { n \\to \\infty } \\int _ R ^ \\infty | \\psi _ n | ^ p r ^ { \\alpha _ 0 } \\mathrm d r = 1 . \\end{align*}"}
+{"id": "5598.png", "formula": "\\begin{align*} \\int _ R ^ \\infty \\exp _ p ( \\mu | u | ^ { p ' } ) r ^ \\theta \\mathrm d r & \\leq C _ p ^ p \\dfrac { \\mu ^ { p - 1 } } { \\Gamma ( p ) } \\| u \\| _ { X ^ { 1 , p } _ \\infty } ^ { p } + C _ { p p ' } ^ { p p ' } \\dfrac { \\mu ^ { p } } { \\Gamma ( p + 1 ) } \\| u \\| _ { X ^ { 1 , p } _ \\infty } ^ { p p ' } \\\\ & \\quad + \\sum _ { j = 2 } ^ \\infty \\dfrac { \\mu ^ { p - 1 + j } } { \\Gamma ( p + j ) } \\int _ R ^ { \\infty } | u | ^ { ( p - 1 + j ) p ' } r ^ \\theta \\mathrm d r . \\end{align*}"}
+{"id": "2183.png", "formula": "\\begin{align*} H _ { b , c } ( \\lim _ { n \\to \\infty } f _ n ) ( x ) & = \\sum _ { y \\sim x } b ( x , y ) \\lim _ { n \\to \\infty } ( f _ n ( x ) - f _ n ( y ) ) + c ( x ) \\lim _ { n \\to \\infty } f _ n ( x ) \\\\ & = \\lim _ { n \\to \\infty } \\left ( \\sum _ { y \\sim x } b ( x , y ) ( f _ n ( x ) - ( f _ n ( y ) ) + c ( x ) f _ n ( x ) \\right ) = \\lim _ { n \\to \\infty } H _ { b , c } f _ n ( x ) = 0 , \\end{align*}"}
+{"id": "643.png", "formula": "\\begin{align*} \\mathcal M _ k [ \\mathcal M _ k ( X ) ] = \\begin{pmatrix} ( i \\mathfrak K _ 1 e _ k \\beta ) ^ 2 \\psi \\\\ 0 \\\\ 0 \\end{pmatrix} = \\begin{pmatrix} - ( \\mathfrak K _ 1 e _ k ) ^ 2 \\psi \\\\ 0 \\\\ 0 \\end{pmatrix} \\end{align*}"}
+{"id": "8763.png", "formula": "\\begin{align*} H ^ 2 ( \\mathbf { P } _ { \\omega , T } , \\mathbf { P } _ { \\omega ' , T } ) & \\leq 2 \\Big ( 1 - \\big ( 1 - T ^ { - 1 } \\big ) ^ T \\Big ) \\\\ & \\leq 2 ( 1 - \\frac { 1 } { 4 } ) = 3 / 2 . \\end{align*}"}
+{"id": "6557.png", "formula": "\\begin{align*} K = \\{ 1 \\leq l \\leq \\# B _ * : \\ \\sigma _ l \\in \\mathbb { D } _ { l _ * - 1 } ( \\sigma ^ * ) \\} . \\end{align*}"}
+{"id": "2346.png", "formula": "\\begin{align*} \\lambda _ g \\rho _ h \\delta _ u = \\lambda _ g \\delta _ { u h ^ { - 1 } } = \\delta _ { g u h ^ { - 1 } } = \\rho _ h \\delta _ { g u } = \\rho _ h \\lambda _ g \\delta _ { u } . \\end{align*}"}
+{"id": "581.png", "formula": "\\begin{align*} \\mathbb E \\left ( \\norm { \\mathbf U - \\mathbf V } _ { \\mathbf X ^ { \\mathbf s , b } ( S , S ' ) } ^ 2 \\right ) = 0 , \\end{align*}"}
+{"id": "1722.png", "formula": "\\begin{align*} R _ { \\texttt { b } } ( \\boldsymbol { \\xi } ) = \\frac { f ( \\boldsymbol { \\xi } ) } { \\prod _ { 1 \\leq j \\leq n } ( 1 - 2 q _ 0 \\cos ( \\xi _ j ) + q _ 0 ^ 2 ) } , \\end{align*}"}
+{"id": "4583.png", "formula": "\\begin{align*} I _ { 1 , \\ast } ( t ) + \\Vert \\varphi ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } \\le & C _ 3 \\int _ 0 ^ t ( I _ { 1 , \\ast } ( s ) + \\Vert \\varphi ( s ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } ) d s + C _ 3 \\Vert { \\mathbf F } \\Vert ^ 2 _ { H ^ 1 _ \\ast ( \\Omega _ t ) } \\ , . \\end{align*}"}
+{"id": "4342.png", "formula": "\\begin{align*} \\frac { d } { d t } { \\rm P f } ( { \\bf A } ( t ) ) = \\frac { 1 } { 2 } { \\rm P f } ( { \\bf A } ( t ) ) \\ , { \\rm T r } \\left ( { \\bf A } ( t ) ^ { - 1 } \\frac { d } { d t } { \\bf A } ( t ) \\right ) \\end{align*}"}
+{"id": "7977.png", "formula": "\\begin{align*} \\mathfrak { k } _ 0 : = k , \\ \\ \\mathfrak { m } _ 0 : = d _ 0 - 1 . \\end{align*}"}
+{"id": "3556.png", "formula": "\\begin{align*} L ( \\bar x ^ k , \\lambda ) - L ( x , \\bar \\lambda ^ k ) \\leq \\frac 1 k \\sum _ { i = 0 } ^ { k - 1 } L ( x ^ { i + 1 } , \\lambda ) - L ( x , \\lambda ^ { i + 1 } ) \\leq \\frac { 1 } { 2 k } \\| z ^ { 0 } - z \\| _ { P } ^ 2 \\ , \\end{align*}"}
+{"id": "2098.png", "formula": "\\begin{align*} \\varphi ( u ) : = ( 1 + | u | ^ 2 ) ^ { 1 + \\delta } \\mbox { w i t h } 0 < \\delta < 1 / 3 . \\end{align*}"}
+{"id": "2004.png", "formula": "\\begin{align*} \\mathrm { \\mathrm { H } } ^ { s , p } _ 0 ( \\Omega ) = \\overline { \\mathrm { C } _ c ^ \\infty ( \\Omega ) } ^ { \\lVert \\cdot \\rVert _ { \\mathrm { \\mathrm { H } } ^ { s , p } ( \\mathbb { R } ^ n ) } } \\quad \\mathrm { B } ^ { s } _ { p , q , 0 } ( \\Omega ) = \\overline { \\mathrm { C } _ c ^ \\infty ( \\Omega ) } ^ { \\lVert \\cdot \\rVert _ { \\mathrm { B } ^ { s } _ { p , q } ( \\mathbb { R } ^ n ) } } \\end{align*}"}
+{"id": "2737.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ d \\big \\| E _ i v \\big \\| > \\theta ^ { - 1 } - \\varepsilon . \\end{align*}"}
+{"id": "6247.png", "formula": "\\begin{align*} u ^ y ( x ) = u ( x - y ) \\end{align*}"}
+{"id": "8480.png", "formula": "\\begin{align*} g _ { 1 } ^ { \\wedge } ( x ) = g _ { 2 } ^ { \\wedge } ( x ) \\mbox { a n d } g _ { 1 } ^ { \\vee } ( x ) = g _ { 2 } ^ { \\vee } ( x ) \\mbox { f o r e v e r y } x \\in \\mathbb { R } ^ { n } . \\end{align*}"}
+{"id": "5013.png", "formula": "\\begin{align*} h = r ^ { - n } e ^ { - r ^ 2 / 4 } \\dot \\gamma + o ( r ^ { - n } e ^ { - r ^ 2 / 4 } ) , \\end{align*}"}
+{"id": "276.png", "formula": "\\begin{align*} A _ { - 1 } = K ^ 0 _ n \\to A _ 1 \\to A _ 2 \\to \\cdots \\to A _ { n - 1 } \\to A _ n = K _ n \\end{align*}"}
+{"id": "7512.png", "formula": "\\begin{align*} A _ { 4 } = \\frac { 1 } { ( q + 1 ) ( q - 1 ) ^ 2 } ( q ^ { m - 1 } - 1 ) ( q ^ { m - 4 } - 1 ) > 0 \\end{align*}"}
+{"id": "7987.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { J - 1 } \\mathfrak { D } ( \\mathfrak { d } _ j , \\mathfrak { s } _ j ) \\end{align*}"}
+{"id": "7971.png", "formula": "\\begin{align*} { \\overline { G } _ { c , d _ 1 , d } } _ { ( h , k , x ) } = { ( B ( h ) \\cdot { F } _ k ) _ { c , d _ 1 , d } } _ { ( h , k , x ) } \\colon B ( h k ) ( F _ d ( x ) ) \\to B ( h ) ( F _ { d _ 1 } ( A ( k ) ( x ) ) ) . \\end{align*}"}
+{"id": "2698.png", "formula": "\\begin{align*} a _ 0 = & \\frac { 1 } { 2 } ( \\alpha _ 1 - \\alpha _ 2 ) , a _ 1 = \\beta _ 1 , a _ 2 = \\frac { 1 } { 2 } ( \\alpha _ 1 + \\alpha _ 2 ) , a _ 3 = \\beta _ 2 , \\\\ b _ 0 = & \\frac { 1 } { 2 } ( \\alpha _ 3 - \\alpha _ 4 ) , b _ 1 = \\beta _ 3 , b _ 2 = \\frac { 1 } { 2 } ( \\alpha _ 3 + \\alpha _ 4 ) , b _ 3 = \\beta _ 4 . \\end{align*}"}
+{"id": "5574.png", "formula": "\\begin{align*} h ( r ) \\sim C _ 0 ( r - 1 ) ^ { \\beta _ 0 } , C _ 0 = \\frac { 2 \\sqrt { 2 } } { \\pi } , \\beta _ 0 = \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "8861.png", "formula": "\\begin{align*} \\frac { \\Gamma ( d + 1 ) } { \\Gamma ( d + \\beta ) } & = \\frac { \\Gamma ( d + 1 ) } { \\Gamma \\big ( d + \\underbrace { ( \\beta - \\ell ) } _ { \\in ( 0 , 1 ] } \\big ) \\prod _ { i = 1 } ^ { \\ell } \\big ( d + \\beta - i \\big ) } \\leq \\frac { ( d + \\beta - \\ell ) ^ { 1 - ( \\beta - \\ell ) } } { \\prod _ { i = 1 } ^ { \\ell } \\big ( d + \\beta - i \\big ) } \\leq \\frac { 1 } { d ^ { \\beta - 1 } } \\enspace , \\end{align*}"}
+{"id": "8810.png", "formula": "\\begin{align*} \\eta _ { t } = \\frac { 2 } { \\alpha t } , h _ { t } = t ^ { - \\frac { 1 } { 2 \\beta } } . \\end{align*}"}
+{"id": "3190.png", "formula": "\\begin{align*} r = b - b _ { | \\mathcal { R } ( A ) } . \\end{align*}"}
+{"id": "4709.png", "formula": "\\begin{align*} ( g - \\lambda f ) ( x y z _ 0 ) = ( g - \\lambda f ) ( x ) ( g - \\lambda f ) ( y ) - ( \\lambda ^ 2 + \\mu \\lambda + 1 ) f ( x ) f ( y ) , \\ , x , y \\in S . \\end{align*}"}
+{"id": "6903.png", "formula": "\\begin{align*} & u ^ * = \\frac { 1 + b _ 1 } { 2 } , \\\\ & u _ { 1 } ^ { * } = \\frac { 1 } { 2 } \\left ( b + 1 - \\sqrt { b ^ { 2 } - 2 \\left ( 1 + \\frac { 2 m } { 1 + a } \\right ) b + 1 } \\right ) , \\\\ & u _ { 2 } ^ { * } = \\frac { 1 } { 2 } \\left ( b + 1 + \\sqrt { b ^ { 2 } - 2 \\left ( 1 + \\frac { 2 m } { 1 + a } \\right ) b + 1 } \\right ) . \\end{align*}"}
+{"id": "4304.png", "formula": "\\begin{align*} q = w + x i + y j + z k \\end{align*}"}
+{"id": "4181.png", "formula": "\\begin{align*} g ( s ) : = s e ^ { s + 1 } ( - ( s + 1 ) ) - 3 s e ^ s ( - s ) , s > 0 . \\end{align*}"}
+{"id": "2768.png", "formula": "\\begin{align*} x f ( x ) & = \\frac { 1 } { \\pi } \\lim _ { \\varepsilon \\to 0 ^ + } \\mathcal { I } ( T _ \\mu ( x - i \\varepsilon ) ) \\\\ & = \\frac { 1 } { \\pi } \\lim _ { \\varepsilon \\to 0 ^ + } \\mathcal { I } ( \\sqrt { a - x + i \\varepsilon } ( c _ { 1 / 2 } + c _ { 3 / 2 } ( a - w ) + c _ { 5 / 2 } ( a - w ) ^ 2 + \\ldots ) ) \\\\ & = \\frac { 1 } { \\pi } \\lim _ { \\varepsilon \\to 0 ^ + } \\sqrt { x - a } \\cdot h ( x ) , \\end{align*}"}
+{"id": "3298.png", "formula": "\\begin{align*} m _ 1 = \\frac { d } { d t } \\phi _ X ( 0 ) ( - \\mu ) = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\frac { \\beta _ { _ \\Sigma } ^ 2 - \\alpha _ { _ \\Sigma } ^ 2 } { 2 } , \\end{align*}"}
+{"id": "4686.png", "formula": "\\begin{align*} f ( x , \\phi _ { \\epsilon } ( x ) ) & = \\inf F _ x , & x \\in I , \\\\ & \\leq \\epsilon + \\inf F _ x , & x \\notin I , \\inf f _ x \\notin - \\infty \\\\ & \\leq - \\epsilon ^ { - 1 } , & x \\notin I , \\inf f _ x = - \\infty . \\end{align*}"}
+{"id": "4444.png", "formula": "\\begin{align*} | | | \\partial _ 2 ( \\dot { H } ^ + _ 2 \\partial _ 1 \\hat { \\Phi } ^ { + } ) ( t ) | | | ^ 2 _ { s - 1 , \\ast , t } & \\leq | | \\dot { H } ^ + _ 2 \\partial _ 1 \\hat { \\Phi } ^ { + } | | ^ 2 _ { s , \\ast , t } \\\\ & \\leq C ( K ) \\Big ( | | { \\mathbf V } | | ^ 2 _ { s , \\ast , t } + | | \\dot { { \\mathbf U } } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ t ) } | | \\hat { W } | | ^ 2 _ { s + 2 , \\ast , t } \\Big ) \\\\ & \\leq C ( K ) \\mathcal { M } ( t ) . \\end{align*}"}
+{"id": "1108.png", "formula": "\\begin{align*} \\begin{aligned} \\| u \\| _ { \\alpha } ^ 2 = \\displaystyle \\langle u , u \\rangle - \\int _ D { \\alpha ( x ) } | u ( x ) | ^ { 2 } d \\mu \\geq \\langle u , u \\rangle = \\| u \\| ^ 2 , \\end{aligned} \\end{align*}"}
+{"id": "4037.png", "formula": "\\begin{align*} y ^ { ( \\alpha ) } ( 0 ) = y ^ { ( \\beta ) } ( 1 ) = 0 , \\end{align*}"}
+{"id": "2424.png", "formula": "\\begin{align*} u _ { a , b } ( \\alpha ( z ) ) = \\prod _ { \\gamma \\in \\Gamma } \\frac { \\alpha ( z ) - \\gamma a } { \\alpha ( z ) - \\gamma b } = \\prod _ { \\gamma \\in \\Gamma } = \\prod _ { \\gamma \\in \\Gamma } \\frac { \\frac { p z + q } { r z + s } - \\gamma a } { \\frac { p z + q } { r z + s } - \\gamma b } = \\prod _ { \\gamma \\in \\Gamma } \\frac { p z + q - r z \\gamma a - s \\gamma a } { p z + q - r z \\gamma b - s \\gamma b } \\end{align*}"}
+{"id": "4579.png", "formula": "\\begin{align*} \\Vert \\partial _ t \\varphi ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } \\le C _ 2 \\{ I ( t ) + I _ { 1 , n } ( t ) + \\Vert \\varphi ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } \\} \\ , . \\end{align*}"}
+{"id": "6210.png", "formula": "\\begin{align*} q _ j \\mapsto X _ j = x _ j , p _ j \\mapsto D _ j = - i \\partial _ { x _ j } , \\end{align*}"}
+{"id": "1404.png", "formula": "\\begin{gather*} \\begin{pmatrix} \\psi _ { n 0 } ( x ) \\\\ \\psi _ { n 1 } ( x ) \\end{pmatrix} = \\begin{pmatrix} \\chi _ { n } & - \\chi _ { n } \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} \\varphi _ { n , 0 } ( x ) \\\\ \\varphi _ { n , 1 } ( x ) \\end{pmatrix} . \\end{gather*}"}
+{"id": "2863.png", "formula": "\\begin{align*} U _ { \\lambda , \\omega } ( t ) = \\sum _ { \\substack { \\nu \\in W _ 0 \\omega \\\\ ( \\lambda + \\nu ) _ + = \\lambda } } t [ \\lambda + \\nu ] + ( 1 - t _ { \\vartheta } ^ { - 1 } ) \\sum _ { \\substack { \\nu \\in W _ 0 \\omega \\\\ w _ { \\lambda + \\nu } \\lambda = \\lambda } } d _ { \\lambda , \\nu } , \\end{align*}"}
+{"id": "1027.png", "formula": "\\begin{align*} \\pi _ k ^ * ( P ^ E _ 3 ) ^ * P ^ E _ 3 = Z _ k ^ * \\pi _ { k } ^ * P ^ E _ 3 = Z _ k ^ * Z _ k \\pi _ { k } ^ * . \\end{align*}"}
+{"id": "4670.png", "formula": "\\begin{align*} e ( S _ { i j } , T _ { i j } ) & \\geq | S _ { i j } | \\cdot ( \\delta ( G ) - 2 ) - \\left ( \\sum \\limits _ { q = 1 } ^ { 2 m + 1 } e ( S _ { i j } , S _ q - S _ { i j } ) + 2 e ( S _ { i j } ) \\right ) \\\\ & \\geq | S _ { i j } | \\cdot ( \\delta ( G ) - 2 ) - ( 2 k + 1 ) k n \\\\ & \\geq \\epsilon n \\cdot ( \\delta ( G ) - 2 ) - ( 2 k + 1 ) k n \\\\ & > \\frac { \\epsilon n } { 2 } \\delta ( G ) \\end{align*}"}
+{"id": "4638.png", "formula": "\\begin{align*} \\underline { S } : = S _ 2 \\otimes \\cdots \\otimes S _ n , \\end{align*}"}
+{"id": "6134.png", "formula": "\\begin{align*} ( \\varphi _ 1 \\times \\varphi _ 2 ) \\times \\varphi _ 3 = \\varphi _ 1 \\times ( \\varphi _ 2 \\times \\varphi _ 3 ) . \\end{align*}"}
+{"id": "1320.png", "formula": "\\begin{align*} J ^ { w } ( \\textbf { X } _ { R S S } ^ { ( n ) } ) = - \\frac { 1 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( - 2 J ^ { w } ( X _ { ( i : n ) i } ) \\right ) = - \\frac { Q _ n } { 2 } \\prod _ { i = 1 } ^ { n } E \\left ( \\Lambda _ X ^ { w } ( B _ { 2 i - 1 : 2 n - 1 } ) \\right ) , \\end{align*}"}
+{"id": "1446.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ { n } a ^ { i j } \\Delta u ^ k _ j + e ^ { u ^ k _ i } - e ^ { u ^ k _ { n + 1 } } = 4 \\pi \\sum \\limits _ { j = 1 } ^ { n } a ^ { i j } \\alpha _ { j } \\delta _ 0 , \\ \\ i = 1 , \\cdots , n . \\end{align*}"}
+{"id": "4730.png", "formula": "\\begin{align*} N \\ge \\lim _ k \\bigg | \\frac { \\partial _ { x _ i } u _ k ( d _ k x + ( x ^ k ) ' ) } { d _ k x _ n } \\bigg | = \\lim _ k \\bigg | \\frac { \\partial _ { x _ i } \\tilde u _ k ( x ) } { x _ n } \\bigg | = \\bigg | \\frac { \\partial _ { x _ i } u _ 0 ( x ) } { x _ n } \\bigg | \\end{align*}"}
+{"id": "3000.png", "formula": "\\begin{align*} \\Phi ^ { - 1 } ( \\xi ) x ( e ) = \\sum _ { s ( f ) = s ( e ) } \\xi ( e , f ) x ( f ) \\end{align*}"}
+{"id": "1611.png", "formula": "\\begin{align*} \\mathcal { K } _ k ( G _ S ) = \\{ \\vec { v } \\in V ^ { [ k ] } : ( \\vec { v } \\mid S ) \\in G _ S \\} . \\end{align*}"}
+{"id": "3509.png", "formula": "\\begin{align*} I _ \\textbf { a } : = D _ \\textbf { a } \\cap \\mathbb { R } \\end{align*}"}
+{"id": "4475.png", "formula": "\\begin{align*} \\mathbb { L } ( { \\mathbf U } , \\Psi ) : = \\left [ \\begin{array} { c } \\mathbb { L } ( { \\mathbf U } ^ + , \\Psi ^ + ) \\\\ \\mathbb { L } ( { \\mathbf U } ^ - , \\Psi ^ - ) \\end{array} \\right ] . \\end{align*}"}
+{"id": "6542.png", "formula": "\\begin{align*} \\Sigma _ \\Lambda = \\left \\{ \\sigma \\in \\R : \\ \\| G _ \\Lambda ( \\sigma ) \\| \\geq e ^ { ( { \\rm d i a m } \\ \\Lambda ) ^ { \\rho _ 2 } } \\right \\} . \\end{align*}"}
+{"id": "1455.png", "formula": "\\begin{align*} 2 x ^ 2 - 2 \\big ( 1 + n _ { { i _ 0 } - 1 , { i _ 0 } } + n _ { { i _ 0 } + 1 , { i _ 0 } } \\big ) x + n _ { { i _ 0 } - 1 , { i _ 0 } } ^ 2 + n _ { { i _ 0 } + 1 , { i _ 0 } } ^ 2 + \\sum \\limits _ { i \\in I \\setminus \\{ { i _ 0 } - 1 , { i _ 0 } \\} } \\big ( n _ { i , { i _ 0 } } - n _ { i + 1 , { i _ 0 } } \\big ) ^ { 2 } = 0 . \\end{align*}"}
+{"id": "3832.png", "formula": "\\begin{align*} { \\rm U O T } ( \\mu _ 0 , \\mu _ 1 ) = \\inf _ { \\eta \\in \\mathcal S ^ p _ = ( \\mu _ 0 , \\mu _ 1 ) } \\int _ { X ^ 2 \\times \\R _ + ^ 4 } \\tilde H ( x _ 0 , x _ 1 , s _ 0 , s _ 1 , w _ 0 , w _ 1 ) d \\eta , \\end{align*}"}
+{"id": "6516.png", "formula": "\\begin{align*} | \\det { \\bf M } ( m ) | = \\prod _ { s = 1 } ^ \\beta | \\lambda _ s v _ s ^ { - 1 } ( m ) | ^ \\beta \\cdot | \\det { \\bf M } ' ( m ) | , \\end{align*}"}
+{"id": "445.png", "formula": "\\begin{align*} h _ { \\beta , \\gamma } ( r ) & = r ^ { 2 \\beta + \\gamma - 2 \\zeta } \\int _ 0 ^ \\infty \\frac { d t } { t } \\ , t ^ \\beta \\int _ 0 ^ \\infty d s \\ , s ^ \\gamma p _ \\zeta ^ { ( 2 ) } ( t , 1 , s ) = C ( \\beta , \\gamma , \\zeta ) r ^ { 2 \\beta + \\gamma - 2 \\zeta } , \\end{align*}"}
+{"id": "7350.png", "formula": "\\begin{align*} \\boldsymbol { \\pi } ^ { a b } = \\boldsymbol { \\pi } \\ \\forall \\ a < b . \\end{align*}"}
+{"id": "4810.png", "formula": "\\begin{align*} \\| \\varphi \\| ^ \\perp _ { \\alpha , m } : = \\| \\varphi \\| _ { C ^ 0 } + | \\varphi | ^ \\perp _ { \\alpha , m } < \\infty . \\end{align*}"}
+{"id": "4692.png", "formula": "\\begin{align*} l _ r ( S , W ) = \\sum _ { T \\perp S } l _ { r - 1 } ( T , W ) = \\sum _ { j = 1 } ^ 6 \\sum _ { T \\in \\C _ j ( S , W ) } l _ { r - 1 } ( T , W ) = \\sum _ { j = 1 } ^ 6 \\mu ^ n _ { i , j } \\cdot l _ { j , r - 1 } ^ n . \\end{align*}"}
+{"id": "7221.png", "formula": "\\begin{align*} \\theta _ E = \\theta _ F \\quad \\mathcal H ^ h \\partial ^ * E \\cap \\partial ^ * F \\end{align*}"}
+{"id": "7022.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 3 } \\varphi _ m \\rho _ \\Gamma = \\int _ { \\mathbb { R } ^ 3 } \\varphi _ m \\rho , \\quad \\forall m = 1 , \\cdots , M . \\end{align*}"}
+{"id": "1299.png", "formula": "\\begin{align*} \\tilde { u } _ n ( t ) = \\sum _ { j = 1 } ^ { J } v _ n ^ j ( t ) \\end{align*}"}
+{"id": "7609.png", "formula": "\\begin{align*} J _ 1 ( \\beta , N , T ) & : = N \\int _ 0 ^ T \\int _ 0 ^ T \\int _ { \\mathbf { B } _ 2 ( 0 ) } f ^ { ( 1 , 1 , \\alpha ) } _ { t , s } ( z ) d z d s d t \\\\ & = N \\int _ 0 ^ T \\int _ 0 ^ T \\frac { 1 } { ( 2 \\pi ( t - s ) ) ^ { 1 / 2 } } \\int _ { | z | < 2 } e ^ { - \\frac { ( z - \\beta ^ { 1 / 3 } N ^ { 1 / 3 } ( t - s ) ) ^ 2 } { 2 ( t - s ) } } d z d s d t . \\end{align*}"}
+{"id": "150.png", "formula": "\\begin{align*} K _ E a _ i \\pi _ \\mu a _ j ^ { - 1 } K _ E \\cap G _ F = \\coprod _ { \\lambda \\in X _ * ( \\textbf { T } ) ^ { - } } \\coprod _ { ( b _ i , b _ j ) \\in T _ \\lambda ( F ) } K _ E a _ i \\pi _ \\mu a _ j ^ { - 1 } K _ E \\cap K _ F b _ i \\varpi _ \\lambda b _ j ^ { - 1 } K _ F . \\end{align*}"}
+{"id": "3469.png", "formula": "\\begin{align*} \\max \\{ \\frac { 1 } { k } \\log \\prod _ { i = 0 } ^ m | F _ i | ^ { - k p _ i } + \\frac { 1 + \\sum d _ i p _ i } { N _ 0 } \\max \\log | \\tau _ l | - u ^ * ( p ) | p \\in \\Delta ^ \\vee \\cap \\frac { 1 } { k } \\Z ^ { m + 1 } \\} . \\end{align*}"}
+{"id": "5214.png", "formula": "\\begin{align*} \\theta _ 1 ^ 2 + \\theta _ 2 ^ 2 = ( \\lambda - \\mu ) ^ 2 + 2 ( k - \\mu ) . \\end{align*}"}
+{"id": "1037.png", "formula": "\\begin{align*} \\hat { f } ( z , x ) = \\left \\{ \\begin{array} { l l } f ( z , x ^ + ) , & \\hbox { i f } x \\leq \\overline { u } _ \\eta ( z ) \\\\ f ( z , \\overline { u } _ \\eta ( z ) ) , & \\hbox { i f } \\overline { u } _ \\eta ( z ) < x . \\end{array} \\right . \\end{align*}"}
+{"id": "4960.png", "formula": "\\begin{align*} P _ { 2 , a } ( n ) = \\Phi _ { 8 n , a } ( \\sqrt { 2 } ) , \\end{align*}"}
+{"id": "6085.png", "formula": "\\begin{align*} \\gamma ^ * = \\frac { C o v ( \\bar X , \\bar Y ) } { V a r ( \\bar Y ) } \\end{align*}"}
+{"id": "1559.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m \\frac { a _ j ! \\ , b _ j ! \\ , c _ j ! } { ( a _ j + b _ j + c _ j ) ! } = \\sum _ { j = 1 } ^ m \\frac { 1 } { \\binom { a _ j + b _ j + c _ j } { a _ j + b _ j } \\binom { a _ j + b _ j } { a _ j } } \\leq \\min ( a , b ) . \\end{align*}"}
+{"id": "2273.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\theta _ { \\alpha } \\left ( x - \\frac { j } { n } \\right ) = n \\sum _ { k \\in \\mathbb { Z } } e ^ { - \\pi \\alpha k ^ 2 n ^ 2 } e ^ { 2 \\pi i k n x } = n \\cdot \\theta ( n x ; n ^ 2 \\alpha ) \\end{align*}"}
+{"id": "5633.png", "formula": "\\begin{align*} D _ { 3 , 4 } = \\begin{cases} y _ 0 y _ 1 + C & = 0 , \\\\ x _ 0 y _ 1 - x _ 1 y _ 0 - B & = 0 . \\end{cases} \\end{align*}"}
+{"id": "737.png", "formula": "\\begin{align*} \\Psi ( t ) = - i M S _ { \\pm \\xi } ( - t ) P _ \\mu \\psi _ { \\mp } ^ \\mu ( t ) + i S _ { \\pm \\xi } ( - t ) P _ \\mu ( \\Theta P _ \\mu \\phi ^ \\mu \\cdot \\Theta P _ \\mu \\psi _ { \\mp } ^ \\mu ) ( t ) - M _ { \\mathfrak K _ 1 } S _ { \\pm \\xi } ( - t ) P _ \\mu ^ 2 \\psi _ \\pm ^ \\mu ( t ) \\end{align*}"}
+{"id": "5816.png", "formula": "\\begin{align*} w _ 0 = s _ n s _ { n - 1 } ( s _ { n - 2 } s _ n s _ { n - 1 } s _ { n - 2 } ) \\dots ( s _ 1 s _ 2 \\dots s _ { n - 2 } s _ n s _ { n - 1 } s _ { n - 2 } \\dots s _ 2 s _ 1 ) . \\end{align*}"}
+{"id": "2937.png", "formula": "\\begin{align*} \\delta _ { j _ { x _ 1 } k } D _ { j _ { x _ 1 } k \\pi ( j _ 2 \\cdots j _ { n } ) } = 0 . \\end{align*}"}
+{"id": "8680.png", "formula": "\\begin{align*} \\psi = x _ 0 ^ { - \\frac { n } { 2 } } e ^ { \\frac { x _ 0 y _ { n + 1 } } { \\hbar } f \\big ( \\frac { x _ 1 } { x _ 0 } , . . . , \\frac { x _ n } { x _ 0 } \\big ) } g \\Big ( \\frac { x _ 1 } { x _ 0 } , . . . , \\frac { x _ n } { x _ 0 } \\Big ) \\end{align*}"}
+{"id": "3160.png", "formula": "\\begin{align*} y _ k & = z _ k / z _ 0 \\ \\quad \\hbox { i f } 1 \\leq k \\leq d - 1 , \\\\ x _ { k - d } & = z _ { k } \\quad \\ \\ \\hbox { i f } d \\leq k \\leq n . \\end{align*}"}
+{"id": "7642.png", "formula": "\\begin{align*} L ^ * & : = \\mathbb { E } _ { \\P ^ \\ast } \\left [ U ^ + \\left ( 1 + | X | \\right ) \\right ] = \\frac { 1 } { 2 d } \\sum _ { i = 1 } ^ d 2 \\left ( \\left ( 1 - \\alpha \\right ) \\mathbb { E } _ { \\P } \\left [ U ^ + \\left ( 1 + | X | \\right ) \\right ] + \\alpha U ^ + ( 2 ) \\right ) \\\\ & = \\left ( 1 - \\alpha \\right ) \\mathbb { E } _ { \\P } \\left [ U ^ + \\left ( 1 + | X | \\right ) \\right ] + \\alpha U ^ + ( 2 ) < \\infty . \\end{align*}"}
+{"id": "3548.png", "formula": "\\begin{align*} \\min _ x \\max _ { \\lambda } \\ ; L ( x , \\lambda ) = f ( x ) - \\lambda ^ T A x - g ^ * ( \\lambda ) \\ , \\end{align*}"}
+{"id": "7928.png", "formula": "\\begin{align*} \\dim _ \\mathbb { C } \\mathrm { H o m } ( I ( g _ 0 ) , Z ) = \\dim _ \\mathbb { C } \\mathrm { H o m } ( g _ j , F ( Z ) ) = 1 \\end{align*}"}
+{"id": "6359.png", "formula": "\\begin{align*} J ^ S _ k = \\{ \\nu ; \\ \\alpha \\varrho ^ { | k | } \\leq \\lambda _ \\nu < \\varrho ^ { | k | } / \\alpha \\} , k \\in \\N ^ * . \\end{align*}"}
+{"id": "4781.png", "formula": "\\begin{align*} \\mu _ { p } ( D \\cap a ^ { n } D ) = \\mu _ { p } ( D ) ^ { 2 } \\geq ( \\prod _ { k = - M } ^ { M } P _ { i _ { a ^ { k } } } ) ^ { t } \\end{align*}"}
+{"id": "1403.png", "formula": "\\begin{gather*} \\xi _ n = | \\rho _ n - \\tilde \\rho _ n | + | \\alpha _ n - \\tilde \\alpha _ n | , \\chi _ { n } = \\left \\{ \\begin{array} { l l } \\xi _ n ^ { - 1 } , & \\xi _ n \\neq 0 , \\\\ 0 , & \\xi _ n = 0 , \\end{array} \\right . n \\ge 1 . \\end{gather*}"}
+{"id": "2410.png", "formula": "\\begin{align*} \\left \\| \\sum _ { k = 1 } ^ { \\infty } A _ { \\overline { \\i } } ^ { k - 1 } x \\right \\| \\geq \\| x \\| - \\left \\| \\sum _ { k = 1 } ^ { \\infty } A _ { \\overline { \\i } } ^ { k } x \\right \\| \\geq \\| x \\| - \\sum _ { k = 1 } ^ { \\infty } \\lambda ^ k \\| x \\| \\geq \\| x \\| \\left ( 1 - \\frac { \\lambda } { 1 - \\lambda } \\right ) \\end{align*}"}
+{"id": "4287.png", "formula": "\\begin{align*} \\int _ { [ 0 , 1 ] \\times [ - m , m ] ^ { | S P ( \\pi ) | } \\times [ - \\lambda , 1 ] ^ { | D P ( \\pi ) | } } \\prod _ { s = 1 } ^ { r } \\chi _ { [ 0 , \\lambda ] } \\left ( x _ 0 + \\sum _ { i = 1 } ^ { 2 s - 1 } \\epsilon _ \\pi ( i ) x _ { \\pi ( i ) } \\right ) \\chi _ { [ 0 , 1 ] } \\left ( x _ 0 + \\sum _ { i = 1 } ^ { 2 s } \\epsilon _ \\pi ( i ) x _ { \\pi ( i ) } \\right ) \\prod _ { l = 0 } ^ { r } \\mathrm { ~ d } x _ l , \\end{align*}"}
+{"id": "7734.png", "formula": "\\begin{align*} f ^ * ( e ) = \\left \\lbrace \\begin{array} { l l l l l l l l l } g ( x y ) & & & \\rm i f ~ \\it e = e ^ * ; \\\\ g ( e ) & & & \\rm i f ~ \\it e \\in E ( G ^ * ) - \\left \\lbrace e ^ * \\right \\rbrace . \\end{array} \\right . \\end{align*}"}
+{"id": "6303.png", "formula": "\\begin{align*} g _ { [ < \\mu ] } - g _ { [ < \\mu ] } ^ { x _ 0 } = \\int _ 0 ^ { x _ 0 } \\partial _ x g _ { [ < \\mu ] } ^ y \\ , d y , \\end{align*}"}
+{"id": "5106.png", "formula": "\\begin{align*} f _ * V = R \\Gamma ^ { g e o } ( G , C ^ { l a } ( G , V ) _ { \\star _ { 1 , 3 } } ) . \\end{align*}"}
+{"id": "5092.png", "formula": "\\begin{align*} \\sum _ i a _ i ^ 2 = \\| V \\| _ { L ^ 2 } ^ 2 \\ , . \\end{align*}"}
+{"id": "2874.png", "formula": "\\begin{align*} T _ j f = ( t _ j s _ j + ( t _ j - 1 ) J _ j ) f ( f \\in \\mathcal C ( P ) , j = 0 , \\ldots , n ) , \\end{align*}"}
+{"id": "8822.png", "formula": "\\begin{align*} - 2 \\eta _ { t } \\sum _ { i = 1 } ^ { n } \\mathbb { E } [ \\langle x ^ { i } ( t ) - \\bar { x } ( t ) , g ^ { i } ( t ) - \\bar { g } ( t ) \\rangle | \\mathcal { F } _ { t - 1 } ] & \\leq \\lambda h ( t - 1 ) + \\frac { \\eta _ { t } ^ { 2 } } { \\lambda } \\kappa n [ C ^ { * } G ^ { 2 } d + \\frac { 3 d ^ { 2 } \\sigma ^ { 2 } } { 2 h _ { t } ^ { 2 } } ] . \\end{align*}"}
+{"id": "8226.png", "formula": "\\begin{align*} g _ a ( X , X ) = g _ 0 ( \\bar { X } , \\bar { X } ) + ( V + a ^ 2 ) ( a _ 1 ^ 2 + a _ 2 ^ 2 + a _ 3 ^ 2 ) + \\frac { V ^ 2 } { V + a ^ 2 } a _ 0 ^ 2 . \\end{align*}"}
+{"id": "6587.png", "formula": "\\begin{align*} \\omega _ l ^ { ( 1 ) } & = \\sqrt { ( \\omega _ l ^ { ( 0 ) } ) ^ 2 + C _ { p + 1 } ^ { p / 2 } 2 ^ { - p } a _ l ^ p \\delta } \\\\ & = \\omega _ l ^ { ( 0 ) } + \\frac { C _ { p + 1 } ^ { p / 2 } 2 ^ { - p } a _ l ^ p \\delta } { \\sqrt { ( \\omega _ l ^ { ( 0 ) } ) ^ 2 + C _ { p + 1 } ^ { p / 2 } 2 ^ { - p } a _ l ^ p \\delta } + \\omega _ l ^ { ( 0 ) } } , \\end{align*}"}
+{"id": "6439.png", "formula": "\\begin{align*} s _ 0 = \\sum _ { i = 0 } ^ n a _ i \\begin{bmatrix} x ^ i y ^ { n - i } \\\\ z ^ i w ^ { n - i } \\end{bmatrix} ~ ~ s _ 1 & = \\sum _ { i = 0 } ^ n b _ i \\begin{bmatrix} x ^ n y ^ { n - i } \\\\ z ^ i w ^ { n - i } \\end{bmatrix} . \\end{align*}"}
+{"id": "5405.png", "formula": "\\begin{align*} \\sum _ { j : j \\neq k } S _ { m j } ( t ) \\sum _ { \\ell = 1 } ^ N S ^ { - 1 } _ { j \\ell } ( t ) d S _ { \\ell k } ( t ) & = d S _ { m k } ( t ) - S _ { m k } ( t ) \\sum _ { \\ell = 1 } ^ N S ^ { - 1 } _ { k \\ell } ( t ) d S _ { \\ell k } ( t ) \\\\ & = d S _ { m k } ( t ) - S _ { m k } ( t ) d U _ { k k } ( t ) , 1 \\leq k , m \\leq N , \\ , t \\geq 0 . \\end{align*}"}
+{"id": "2761.png", "formula": "\\begin{align*} \\begin{bmatrix} \\mathcal G _ { \\mu } ( r _ C \\xi _ M ^ 0 ) \\\\ \\mathcal G _ { \\mu } ( r _ C \\xi _ M ^ 1 ) \\\\ \\vdots \\\\ \\mathcal G _ { \\mu } ( r _ C \\xi _ M ^ { M - 1 } ) \\end{bmatrix} = \\begin{bmatrix} \\xi _ M ^ 0 & & & \\\\ & \\xi _ M ^ 1 & & \\\\ & & \\ddots & \\\\ & & & \\xi _ M ^ { M - 1 } \\end{bmatrix} \\cdot F \\cdot \\begin{bmatrix} g _ 1 r _ C \\\\ g _ 2 r _ C ^ 2 \\\\ \\vdots \\\\ g _ { M } r _ C ^ { M } \\end{bmatrix} , \\end{align*}"}
+{"id": "3969.png", "formula": "\\begin{align*} t ^ { k + r } \\otimes d = j ^ * ( A _ 1 t ^ { r + m - 4 } \\alpha + \\cdots + A _ n t ^ { r + m - 4 n } \\alpha ^ n + A _ m t ^ { r } \\beta + \\cdots + A _ { m + n } t ^ { r - 4 n } \\alpha ^ { n } \\beta ) , \\end{align*}"}
+{"id": "3578.png", "formula": "\\begin{align*} R = { \\log _ 2 } \\det \\left ( { { \\bf { I } } + \\frac { 1 } { { { \\sigma ^ 2 } } } { { \\bf { W } } ^ H } { \\bf { H F } } { { \\bf { F } } ^ H } { { \\bf { H } } ^ H } { \\bf { W } } } \\right ) , \\end{align*}"}
+{"id": "2344.png", "formula": "\\begin{align*} \\tau _ g = \\pi _ g \\tau _ e , f _ g = f _ e \\pi _ { g ^ { - 1 } } , \\forall g \\in G \\end{align*}"}
+{"id": "7910.png", "formula": "\\begin{align*} \\mathrm { F P d i m } ( x ) ^ 2 = \\mathrm { F P d i m } ( x \\phi ( x ) ) = 2 \\sum _ { y \\in \\Gamma _ 0 } c _ { x , \\phi ( x ) } ^ y \\mathrm { F P d i m } ( y ) , \\end{align*}"}
+{"id": "177.png", "formula": "\\begin{align*} f ( g \\tau ) : = f \\left ( \\frac { a \\tau + b } { c \\tau + d } \\right ) = \\chi ( g ) ( c \\tau + d ) ^ k f ( \\tau ) , ~ ~ \\forall g = \\left ( \\begin{array} { c c } \\ a & b \\\\ c & d \\end{array} \\right ) \\in \\Gamma , \\end{align*}"}
+{"id": "1683.png", "formula": "\\begin{align*} \\Lambda ^ { ( n ) } _ { \\texttt { b } } : = \\{ ( \\mu _ 1 , \\ldots , \\mu _ n ) \\in \\mathbb { Z } ^ n \\mid \\mu _ 1 \\geq \\cdots \\geq \\mu _ n \\geq 0 \\} \\end{align*}"}
+{"id": "7056.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ \\infty \\lambda _ f ( m ) e \\left ( \\frac { a m } { c } \\right ) F ( n ) = \\frac { 1 } { c } \\frac { \\eta _ f ( D _ 2 ) } { \\sqrt { D _ 2 } } \\sum _ { m = 1 } ^ \\infty \\lambda _ { f } ( m ) e \\left ( \\frac { - m \\overline { a D _ 2 } } { c } \\right ) H \\left ( \\frac { m } { D _ 2 c ^ 2 } \\right ) , \\end{align*}"}
+{"id": "7653.png", "formula": "\\begin{align*} \\bar { \\Q } : = \\frac { 1 } { 2 } ( \\Q ^ 1 + \\Q ^ 2 ) \\in ( B _ k ( \\P ^ w ) ) . \\end{align*}"}
+{"id": "6255.png", "formula": "\\begin{align*} u _ 0 = \\sum _ { \\lambda \\geq 1 } u _ { 0 , \\lambda } . \\end{align*}"}
+{"id": "249.png", "formula": "\\begin{align*} v ( z ) = \\begin{cases} 1 + g _ M z ^ { - 1 } & \\ \\emph { r = b c } \\ \\ \\emph { r = c s } , \\\\ z ^ { - 1 } & \\ \\emph { r = t } , \\end{cases} \\end{align*}"}
+{"id": "7189.png", "formula": "\\begin{align*} y _ 1 ^ { n _ 1 } y _ 2 ^ { n _ 2 } = y _ 1 ^ { n _ 1 - 1 } ( y _ 1 y _ 2 ^ { n _ 2 } ) = y _ 1 ^ { n _ 1 - 1 } y _ 2 ^ { n _ 2 + 1 } = \\cdots = y _ 2 ^ { n _ 1 + n _ 2 } = y _ 2 ^ n . \\end{align*}"}
+{"id": "3974.png", "formula": "\\begin{align*} u _ t = P _ t ( u _ 0 ) + Z _ t + \\int _ 0 ^ t P _ { t - s } g ( u _ s , \\mu _ s ) d s . \\end{align*}"}
+{"id": "5823.png", "formula": "\\begin{align*} w _ 2 = s _ { \\alpha _ 4 } s _ { \\alpha ^ { a 3 } _ { m a x } } s _ { \\alpha ^ { a 5 } _ { m a x } } . \\end{align*}"}
+{"id": "3928.png", "formula": "\\begin{align*} \\frac { \\lVert u _ 1 - v ( t ) \\rVert } { \\sin ( \\gamma ( t ) ) } & = \\frac { \\lVert u _ 1 \\rVert } { \\sin ( \\pi - \\theta - \\gamma ( t ) ) } \\Leftrightarrow \\\\ { t \\lVert u _ 1 - u _ 2 \\rVert } & = \\lVert u _ 1 \\rVert \\frac { { \\sin ( \\gamma ( t ) ) } } { \\sin ( \\pi - \\theta - \\gamma ( t ) ) } \\Leftrightarrow \\\\ { t \\lVert u _ 1 - u _ 2 \\rVert } & = \\lVert u _ 1 \\rVert \\frac { { \\sin ( \\gamma ( t ) ) } } { \\sin ( \\theta + \\gamma ( t ) ) } \\end{align*}"}
+{"id": "2420.png", "formula": "\\begin{align*} \\limsup _ { j \\to + \\infty } \\frac { 2 } { \\abs { \\partial \\Omega _ j } } \\left ( \\abs { \\Omega _ j } - \\frac { \\abs { \\partial \\Omega _ j } ^ { \\frac { 3 } { 2 } } } { 6 \\sqrt { \\pi } } \\right ) = \\limsup _ { j \\to + \\infty } \\frac { \\abs { \\partial \\Omega _ j } ^ { \\frac { 1 - 3 \\alpha } { 2 } } } { 3 \\alpha \\sqrt { \\pi } } \\left ( ( 6 \\sqrt { \\pi } \\abs { \\Omega _ j } ) ^ { \\alpha } - \\abs { \\partial \\Omega _ j } ^ { \\frac { 3 \\alpha } { 2 } } \\right ) \\end{align*}"}
+{"id": "1548.png", "formula": "\\begin{align*} \\Delta = y ^ 3 ( \\lambda x - y ) ^ 2 ( x - y ) + z f _ 5 ( x , y , z ) \\end{align*}"}
+{"id": "4917.png", "formula": "\\begin{align*} J ( k , r ) & : = \\frac { \\pi ( 2 k + 1 ) } { 3 } - \\frac { 2 k + 1 } { \\pi } \\sum _ { m = 1 } ^ { r + 1 } 2 ^ m L _ m - \\frac { 2 ^ { r + 2 } ( 4 k + 1 ) } { \\pi } L _ { r + 2 } - \\frac { 2 ^ { r + 4 } } { \\pi } L _ { r + 3 } , \\\\ M ( \\ell , r ) & : = - \\frac { \\ell \\cdot 2 ^ { r + 1 } } { \\pi } \\sum _ { m = 1 } ^ { r + 1 } 2 ^ m L _ m - \\frac { 2 ^ { r + 3 } ( k - \\ell + 1 ) } { \\pi } L _ { r + 2 } + \\frac { 2 ^ { r + 5 } ( k - \\ell + 1 ) } { \\pi } L _ { r + 3 } , \\end{align*}"}
+{"id": "7746.png", "formula": "\\begin{align*} \\chi _ { 2 } ( G ) = \\chi ( G ^ { 2 } ) = \\chi \\big { ( } \\mathcal { N } _ { c } ( G ) \\big { ) } = p ( G ) . \\end{align*}"}
+{"id": "1670.png", "formula": "\\begin{align*} m \\xi _ j + \\sum _ { \\substack { 1 \\leq k \\leq n \\\\ k \\neq j } } v _ q ( \\xi _ j - \\xi _ k ) = 2 \\pi ( \\lambda _ j + \\varrho _ { \\texttt { a } , j } ) ( j = 1 , \\ldots , n ) , \\end{align*}"}
+{"id": "6634.png", "formula": "\\begin{align*} b _ 0 ( \\beta ) = - ( \\beta - 6 ) ( \\beta - 8 / 3 ) , b _ 1 ( \\beta ) = { 2 \\over 3 } \\beta ( 2 \\beta - 7 ) , b _ 2 ( \\beta ) = - { 1 \\over 3 } \\beta ^ 2 . \\end{align*}"}
+{"id": "2070.png", "formula": "\\begin{align*} & \\int ^ \\infty _ 0 \\frac { r ^ n } { t ^ { n / 2 } } \\cdot \\exp \\left ( - \\frac { r ^ 2 } { C _ 1 t } \\right ) \\frac { C _ 1 } { r t } k ( x _ 0 , r ) d r \\\\ & = \\left ( \\int ^ \\infty _ { r _ 0 } + \\int ^ { r _ 0 } _ 0 \\right ) \\frac { r ^ n } { t ^ { n / 2 } } \\cdot \\exp \\left ( - \\frac { r ^ 2 } { C _ 1 t } \\right ) \\frac { C _ 1 } { r t } k ( x _ 0 , r ) d r \\\\ & = \\mathbf { I } + \\mathbf { I I } . \\end{align*}"}
+{"id": "2976.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { i } ( x _ i \\cdot ( x _ i \\mid x ) _ { C _ 0 ( E ^ 0 ) } ) - x \\Big \\| ^ 2 = \\sup _ { v \\in E ^ 0 } \\sum _ { s ( e ) = v } \\Big | \\sum _ { i } ( x _ i \\cdot ( x _ i \\mid x ) _ { C _ 0 ( E ^ 0 ) } ) ( e ) - x ( e ) \\Big | ^ 2 \\to 0 . \\end{align*}"}
+{"id": "310.png", "formula": "\\begin{gather*} \\ell _ { j } \\left ( s \\right ) \\left ( \\mathsf { N } _ { j } \\left ( s \\right ) f , w \\right ) = \\left \\langle f , \\overline { w } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } \\qquad \\forall f \\in H ^ { - 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) , \\quad \\forall w \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) . \\end{gather*}"}
+{"id": "443.png", "formula": "\\begin{align*} h _ { \\beta , \\gamma } ( r ) = C ^ { ( \\alpha ) } \\left ( \\beta , \\gamma , \\zeta \\right ) r ^ { - \\eta } q _ { \\beta , \\gamma } ( r ) = \\Psi _ \\zeta ( \\eta ) r ^ { - \\alpha } . \\end{align*}"}
+{"id": "5439.png", "formula": "\\begin{align*} \\kappa \\leq \\kappa _ \\beta : = \\frac { 1 } { \\sqrt { 2 } } + \\frac { \\sqrt { 2 } } { \\beta } . \\end{align*}"}
+{"id": "694.png", "formula": "\\begin{align*} \\mathbf N ( \\mathbf 0 ) = \\mathbf 0 \\end{align*}"}
+{"id": "2583.png", "formula": "\\begin{align*} & = \\sum _ { r \\in S + S ' + S '' } \\left ( \\sum _ { \\substack { h \\in S \\\\ h ' \\in S ' \\\\ h '' \\in S '' \\\\ h + h ' + h '' = r } } a _ { h } b _ { h ' } c _ { h '' } p ^ { - ( \\beta ( h ' , h '' ) + \\beta ( h , h ' + h '' ) ) } \\right ) t ^ { r } = \\end{align*}"}
+{"id": "8489.png", "formula": "\\begin{align*} D ^ { c } g = D ^ { s } g - D ^ { j } g , \\end{align*}"}
+{"id": "2579.png", "formula": "\\begin{align*} & ( ( z , h ) + ( z ' , h ' ) ) + ( z '' , h '' ) = ( z + z ' - \\beta ( h , h ' ) , h + h ' ) + ( z '' , h '' ) = \\\\ & = ( z + z ' + z '' - ( \\beta ( h , h ' ) + \\beta ( h + h ' , h '' ) ) , h + h ' + h '' ) = \\\\ & = ( z + z ' + z '' - ( \\beta ( h ' , h '' ) + \\beta ( h , h ' + h '' ) ) , h + h ' + h '' ) = \\\\ & = ( z , h ) + ( z ' + z '' - \\beta ( h ' , h '' ) , h ' + h '' ) = ( z , h ) + ( ( z ' , h ' ) + ( z '' , h '' ) ) , \\end{align*}"}
+{"id": "7990.png", "formula": "\\begin{align*} 2 ^ { J _ i } = 2 ^ { \\mathfrak { k } _ J } \\prod _ { j : \\mathfrak { d } _ j \\leq i } ( \\mathfrak { s } _ j ) ^ { i - \\mathfrak { d } _ j } = 2 ^ { k } \\prod _ { j = 1 } ^ { J } ( \\mathfrak { s } _ j ) ^ { \\mathfrak { d } _ j } \\prod _ { j : \\mathfrak { d } _ j \\leq i } ( \\mathfrak { s } _ j ) ^ { i - \\mathfrak { d } _ j } . \\end{align*}"}
+{"id": "26.png", "formula": "\\begin{align*} \\sum _ { i < j } v ( x _ i - x _ j ) = \\sum _ { k = 0 } ^ 4 \\mathcal Q _ k , \\end{align*}"}
+{"id": "2802.png", "formula": "\\begin{align*} \\mathcal { K } _ { Y } ( x , y ) = \\frac { c _ { N , s } } { 2 } \\left [ \\operatorname { d i v } Y ( x ) + \\operatorname { d i v } Y ( y ) - ( N + 2 s ) \\frac { ( Y ( x ) - Y ( y ) ) \\cdot ( x - y ) } { | x - y | ^ { 2 } } \\right ] | x - y | ^ { - N - 2 s } . \\end{align*}"}
+{"id": "2323.png", "formula": "\\begin{align*} f _ g = f \\pi _ { g ^ { - 1 } } , \\tau _ g = \\pi _ g \\tau , \\forall g \\in G . \\end{align*}"}
+{"id": "541.png", "formula": "\\begin{align*} Z ^ { \\mathbf s , b } ( S , T ) = \\mathbb L ^ 2 \\left ( \\Omega , \\mathbf X ^ { \\mathbf s , b } ( S , T ) \\cap C \\left ( [ S , T ] , \\mathbf H ^ { \\mathbf s } \\right ) \\right ) \\end{align*}"}
+{"id": "6428.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } u _ { n } ( c t _ n , t _ n ) = 0 . \\end{align*}"}
+{"id": "27.png", "formula": "\\begin{align*} \\mathcal Q _ 0 = \\sum _ { i < j } P _ i P _ j v ( x _ i - x _ j ) P _ j P _ i . \\end{align*}"}
+{"id": "8182.png", "formula": "\\begin{align*} p x K _ i ( x , N , p ) = - ( i + 1 ) K _ { i + 1 } ( x , N , p ) + \\big ( i + ( p - 1 ) ( N - i ) \\big ) K _ i ( x , N , p ) - ( p - 1 ) ( N - i + 1 ) K _ { i - 1 } ( x , N , p ) \\ , . \\end{align*}"}
+{"id": "6741.png", "formula": "\\begin{align*} t ^ { 2 } \\left ( 1 + \\frac { m } { 2 t } \\right ) ^ { 2 } \\alpha '' ( t ) + t \\left ( 1 + \\frac { m } { 2 t } \\right ) \\left ( - a + ( a + 2 ) \\frac { m } { 2 t } \\right ) \\alpha ' ( t ) - 2 ( 2 a + 1 ) \\frac { m } { 2 t } \\alpha ( t ) = 0 \\end{align*}"}
+{"id": "2494.png", "formula": "\\begin{align*} a ( T _ 1 , X _ i ' , B _ i ' + M _ i ' ) = { - } 1 \\quad i , \\end{align*}"}
+{"id": "1672.png", "formula": "\\begin{align*} \\kappa _ { \\texttt { a } ; \\pm } ( q ) : = n \\left ( \\frac { 1 - | q | } { 1 + | q | } \\right ) ^ { \\pm 1 } . \\end{align*}"}
+{"id": "3703.png", "formula": "\\begin{align*} \\Omega _ t \\supseteq L _ t \\supseteq L _ x : = \\{ y \\in L _ t \\ , | \\ , y _ 1 > x _ 1 \\} , \\end{align*}"}
+{"id": "3905.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\| g _ N \\| _ 2 = 0 , \\end{align*}"}
+{"id": "1221.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } J ( f _ n ) = \\sup _ { f \\in \\dot { H } ^ 1 \\setminus \\{ 0 \\} } J ( f ) = C _ 0 . \\end{align*}"}
+{"id": "4343.png", "formula": "\\begin{align*} S ( \\omega _ { { \\sf f } _ 1 \\cdots { \\sf f } _ m } \\| \\omega ) & = i \\left . \\frac { d } { d t } \\right | _ { t = 0 } ( \\Omega _ \\omega , F _ 1 \\cdots F _ m \\Delta ^ { i t } F _ m \\cdots F _ 1 \\Omega _ \\omega ) \\\\ & = i \\left . \\frac { d } { d t } \\right | _ { t = 0 } ( \\Omega _ \\omega , F _ 1 \\cdots F _ m F _ m ( t ) \\cdots F _ 1 ( t ) \\Omega _ \\omega ) \\ , . \\end{align*}"}
+{"id": "1016.png", "formula": "\\begin{align*} v _ 1 \\cdot \\ldots \\cdot v _ k : = \\sum _ { \\sigma \\in S _ k } \\ , v _ { \\sigma ( 1 ) } \\otimes \\ldots \\otimes v _ { \\sigma ( k ) } , \\end{align*}"}
+{"id": "6347.png", "formula": "\\begin{align*} f ^ { 4 , b a l } _ { \\lambda , m } ( \\xi _ 1 , \\xi _ 2 , \\xi _ 3 , \\xi _ 4 ) = q ^ { 4 , b a l } _ { \\lambda , m } [ ( \\xi _ 1 - \\xi _ 2 ) ( \\xi _ 3 - \\xi _ 4 ) + ( \\xi _ 1 - \\xi _ 4 ) ( \\xi _ 2 - \\xi _ 3 ) ] . \\end{align*}"}
+{"id": "5468.png", "formula": "\\begin{align*} R i c _ { \\widehat { g } } - \\frac { R _ { \\widehat { g } } } { 2 } \\widehat { g } = T , \\end{align*}"}
+{"id": "6869.png", "formula": "\\begin{align*} R = V _ { h + 1 } \\left ( V _ { h + 1 } ^ H A - \\begin{bmatrix} V _ h ^ H \\\\ 0 \\end{bmatrix} A \\right ) V _ { h } Y = V _ { h + 1 } \\begin{bmatrix} 0 \\\\ e _ { h + 1 } ^ H V _ { h + 1 } ^ H \\end{bmatrix} A V _ h Y , \\end{align*}"}
+{"id": "5365.png", "formula": "\\begin{align*} H ^ 1 ( \\O _ { L _ 1 \\cup \\ldots \\cup L _ { 2 b - 2 } } ( 1 ) ( - B _ { b - 1 } ) ) = 0 \\ \\hbox { a n d } \\ \\deg ( { B _ { b - 1 } } _ { | L _ { 2 b - 2 } } ) = 1 . \\end{align*}"}
+{"id": "7666.png", "formula": "\\begin{align*} \\mathbb { E } X - c \\ , \\mathbb { E } \\theta = 0 \\Rightarrow \\psi ( u ) = 1 \\end{align*}"}
+{"id": "4668.png", "formula": "\\begin{align*} | S _ { i j } | \\cdot ( \\delta ( G ) - 2 ) \\leq \\sum \\limits _ { v \\in S _ { i j } } d _ { G ' } ( v ) \\le e ( S _ { i j } , T _ { i j } ) + \\sum \\limits _ { q = 1 } ^ { 2 m + 1 } e ( S _ { i j } , S _ q - S _ { i j } ) + 2 e ( S _ { i j } ) . \\end{align*}"}
+{"id": "2028.png", "formula": "\\begin{align*} s : = ( 1 - \\theta ) s _ 0 + \\theta s _ 1 \\end{align*}"}
+{"id": "638.png", "formula": "\\begin{align*} ( \\psi \\mathfrak K _ 1 ) f ( x ) = \\psi ( x ) \\int _ { \\R } \\mathfrak k _ 1 ( x - y ) f ( y ) d y \\end{align*}"}
+{"id": "856.png", "formula": "\\begin{align*} \\Lambda ^ { * - 1 } ( y ) = \\inf _ { \\lambda \\in [ 0 , b ) } \\left ( \\frac { y + \\Lambda ( \\lambda ) } { \\lambda } \\right ) . \\end{align*}"}
+{"id": "2694.png", "formula": "\\begin{align*} m _ 1 & = F ( 1 , 1 , 1 ) F ( - 1 , 1 , 1 ) \\equiv ( \\alpha _ 1 + \\alpha _ 2 + \\gamma _ 1 + \\gamma _ 2 ) ^ 2 , \\\\ m _ 2 & = F ( 1 , - 1 , 1 ) F ( - 1 , - 1 , 1 ) \\equiv ( \\alpha _ 1 + \\alpha _ 2 - \\gamma _ 1 - \\gamma _ 2 ) ^ 2 , \\\\ m _ 3 & = F ( 1 , 1 , - 1 ) F ( - 1 , 1 , - 1 ) \\equiv ( \\alpha _ 1 - \\alpha _ 2 + \\gamma _ 1 - \\gamma _ 2 ) ^ 2 , \\\\ m _ 4 & = F ( 1 , - 1 , - 1 ) F ( - 1 , - 1 , - 1 ) \\equiv ( \\alpha _ 1 - \\alpha _ 2 - \\gamma _ 1 + \\gamma _ 2 ) ^ 2 . \\end{align*}"}
+{"id": "8459.png", "formula": "\\begin{align*} \\log \\left ( e ^ X e ^ Y \\right ) = X + Y + \\frac { 1 } { \\mathrm { a d } _ { Y } } \\left ( 1 - \\frac { \\exp \\left ( \\mathrm { a d } _ { X } \\right ) - 1 } { \\mathrm { a d } _ { X } } \\frac { \\mathrm { a d } _ { X } + \\mathrm { a d } _ { Y } } { \\exp \\left ( \\mathrm { a d } _ { X } + \\mathrm { a d } _ { Y } \\right ) - 1 } \\right ) [ X , Y ] . \\end{align*}"}
+{"id": "1734.png", "formula": "\\begin{align*} R _ { \\texttt { b } } ( \\xi _ 1 , \\ldots , \\xi _ n ) = \\frac { f ( \\xi _ 1 , \\ldots , \\xi _ n ) } { \\prod _ { 1 \\leq j \\leq n } ( 1 - 2 q _ 0 \\cos ( \\xi _ j ) + q _ 0 ^ 2 ) } , \\end{align*}"}
+{"id": "3760.png", "formula": "\\begin{align*} g ( t ) = e ^ { - t B } [ u '' ( 0 ) - ( a + b ) B u ' ( 0 ) + a b B ^ 2 u ( 0 ) ] . \\end{align*}"}
+{"id": "5880.png", "formula": "\\begin{align*} ( 1 - \\epsilon _ { 1 } + \\epsilon _ { 1 } e ^ { k \\Delta ^ { 2 } } ) ^ { V } = o ( \\exp ( 2 N ^ { \\zeta } / 3 ) ) . \\end{align*}"}
+{"id": "6585.png", "formula": "\\begin{align*} q ^ { ( 0 ) } ( e _ l , n ^ { ( l ) } ) = q ^ { ( 0 ) } ( - e _ l , n ^ { ( l ) } ) = a _ l / 2 \\ { \\rm f o r } \\ 1 \\leq l \\leq b . \\end{align*}"}
+{"id": "7995.png", "formula": "\\begin{align*} \\bigcup _ { 1 \\leq q < p } \\left ( \\sum _ { n } X _ { n } \\right ) _ { q } = \\bigcup _ { n \\in \\mathbb { N } } \\left ( \\sum _ { n } X _ { n } \\right ) _ { q _ { n } } . \\end{align*}"}
+{"id": "6572.png", "formula": "\\begin{align*} N _ 2 = N _ 1 ^ { \\widetilde C } , \\ N = N _ 2 ^ { \\widetilde C } . \\end{align*}"}
+{"id": "1612.png", "formula": "\\begin{align*} | \\mathcal K _ k ( \\mathcal G _ T ) \\cap \\mathrm { C r o s s } _ { \\mathcal Y } | = ( 1 \\pm \\eta ) \\prod \\limits _ { \\ell = 1 } ^ { k - 2 } ( \\frac { 1 } { a _ { \\ell } } ) ^ { \\binom { k } { \\ell } } n ^ k , \\end{align*}"}
+{"id": "2722.png", "formula": "\\begin{align*} \\phi ( \\vee _ { k \\in S } u _ k ) = \\wedge _ { k \\in S } H ' _ k , \\phi ( \\vee _ { k \\in S } u ' _ k ) = \\wedge _ { k \\in S } H _ k . \\end{align*}"}
+{"id": "8448.png", "formula": "\\begin{align*} \\nabla _ { K Z B } ^ { \\prime } = d - \\omega _ { K Z B } ^ { \\prime } , \\omega _ { K Z B } ^ { \\prime } = ( \\beta - d f ) B + \\alpha \\exp \\left ( - f - \\sum _ { k = 2 } ^ { \\infty } \\frac { ( - 1 ) ^ k P _ { k } \\mathrm { a d } _ { B } ^ { k } } { k } \\right ) A \\end{align*}"}
+{"id": "7092.png", "formula": "\\begin{align*} L ( n _ 1 , n _ 2 , k ) = \\sum _ { n _ 1 \\ll p _ 1 C r } \\ , \\frac { \\Theta ^ { 1 / 2 } } { n ^ k _ 1 } \\ll _ { \\epsilon } \\ , N ^ { 1 / 6 } _ 0 . \\end{align*}"}
+{"id": "549.png", "formula": "\\begin{align*} f ( t ) = \\norm { \\mathbf u } _ { \\widetilde { \\mathbf X } ^ { \\mathbf s , b } ( 0 , t ) } ^ 2 , f ( 0 ) = 0 . \\end{align*}"}
+{"id": "7832.png", "formula": "\\begin{align*} N : = \\{ ( Z ^ 0 , Z ^ 1 , \\zeta ^ 0 , \\zeta ^ 1 , \\widetilde { \\zeta } _ 0 , \\widetilde { \\zeta } _ 1 ) \\in M ^ { } \\times \\mathbb { R } ^ 4 \\ ; | \\ ; R _ 1 ( t , \\tau ) > 0 , \\ ; \\ ; R _ 2 ( t , \\tau ) > 0 \\ ; \\} \\end{align*}"}
+{"id": "2520.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\boldsymbol { \\eta } } ( \\boldsymbol { v } ) = \\int _ { Q } \\Big ( \\boldsymbol { u } \\cdot \\boldsymbol { v } - \\sigma \\partial _ t ^ { 1 / 2 } \\boldsymbol { \\eta } \\cdot \\partial _ t ^ { 1 / 2 } \\boldsymbol { v } ^ { \\perp } - \\nu \\ , \\textbf { c u r l } \\ , \\boldsymbol { \\eta } \\cdot \\ , \\textbf { c u r l } \\ , \\boldsymbol { v } \\Big ) \\ , d \\boldsymbol { x } \\ , d t . \\end{align*}"}
+{"id": "8363.png", "formula": "\\begin{align*} \\| S _ J ( t - r ) - S _ J ( s - r ) - & S _ J ( t - q ) + S _ J ( s - q ) \\| _ { L ( \\tilde V ) } \\\\ & \\le c ( t - s ) ^ { \\varrho } ( r - q ) ^ { \\nu } ( s - r ) ^ { - ( \\varrho + \\nu ) } . \\end{align*}"}
+{"id": "6988.png", "formula": "\\begin{align*} \\left | \\frac { f ' ( r e ^ { \\iota \\theta _ 0 } ) } { f ( r e ^ { \\iota \\theta _ { 0 } } ) } \\right | = O \\left ( \\frac { 1 } { r ^ 2 } \\right ) , \\end{align*}"}
+{"id": "774.png", "formula": "\\begin{align*} \\xi = \\prod _ { p \\in \\Pi } ~ p ^ { \\chi _ { \\xi } ( p ) } \\end{align*}"}
+{"id": "5864.png", "formula": "\\begin{align*} \\mathcal { K } = \\{ 1 , \\cdots , k _ { 1 } \\} \\cup \\{ \\left \\lfloor { r ^ { j } k _ { 1 } } \\right \\rfloor : j \\geq 1 \\} , k _ { 1 } \\geq 1 , r > 1 . \\end{align*}"}
+{"id": "3823.png", "formula": "\\begin{align*} \\alpha ( ( Y \\times Y \\times \\R _ + ) \\setminus \\Omega ) = 0 , \\Omega = \\{ ( y _ 0 , y _ 1 , S ) \\in Y \\times Y \\times \\R _ + \\mid s _ 0 = s _ 1 \\} . \\end{align*}"}
+{"id": "5222.png", "formula": "\\begin{align*} \\eta _ t ^ { ( i ) } = \\frac { D } { G \\sqrt { t } } , \\ \\eta _ t ^ w = \\frac { 2 D ^ 2 } { \\sqrt { ( 2 D ^ 2 G ^ 2 + 2 \\ln m ) t } } , \\textrm { a n d } \\eta _ t ^ q = \\frac { 2 \\ln m } { \\sqrt { ( 2 D ^ 2 G ^ 2 + 2 \\ln m ) t } } \\end{align*}"}
+{"id": "52.png", "formula": "\\begin{align*} \\mathcal K ^ { \\rm { d i a g } } = \\sum _ { k \\neq 0 } \\mathcal D _ k b _ k ^ \\dagger b _ k , \\mathcal D _ k = \\sqrt { k ^ 4 + 2 k ^ 2 \\rho _ z \\widehat g ( k ) } , \\end{align*}"}
+{"id": "6975.png", "formula": "\\begin{align*} \\widehat { u _ j } = ( 2 \\pi ) ^ { - \\frac { d } { 2 } } \\sum _ { n } \\frac { n _ j } { | n | ^ 2 } u ( n ) e ^ { - i n \\cdot x } = ( 2 \\pi ) ^ { - \\frac { d } { 2 } } \\sum _ { n } - i n _ j a ( n ) e ^ { - i n \\cdot x } = \\partial _ { x _ j } \\psi , \\end{align*}"}
+{"id": "1234.png", "formula": "\\begin{align*} \\Big \\| ( \\sum _ { j = 1 } ^ { J } v _ n ^ j ) ^ 2 \\Big \\| _ { L _ { t } ^ { \\frac { q _ 0 } { 2 } } L _ x ^ { \\frac { r _ 1 } { 2 } } } ^ 2 \\leq \\sum _ { j = 1 } ^ { J } \\| v _ n ^ j \\| _ { S _ 0 ( \\R ) } ^ 2 + \\sum _ { j \\neq k } \\| v _ n ^ j v _ n ^ k \\| _ { L _ { t } ^ { \\frac { q _ 0 } { 2 } } L _ x ^ { \\frac { r _ 1 } { 2 } } } . \\end{align*}"}
+{"id": "5286.png", "formula": "\\begin{align*} \\mathbb { R } ^ { n _ 0 } = \\bigcup _ { R \\in M } \\overline { R } . \\end{align*}"}
+{"id": "927.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty ( A L _ 1 ^ u ( y ) + B L _ 1 ^ v ( y ) ) \\mathrm { e } ^ { - \\lambda y ^ 2 } \\mathrm { d } y = \\frac { 1 } { ( 1 + \\frac { 4 t ^ \\alpha } { \\alpha } \\lambda ) ^ { \\frac { c + 1 + \\sqrt { m n } } { 2 } } } \\mathrm { e } ^ { - \\frac { \\lambda x ^ 2 } { ( 1 + \\frac { 4 t ^ \\alpha } { \\alpha } \\lambda ) } } , \\end{align*}"}
+{"id": "3047.png", "formula": "\\begin{align*} \\psi ^ \\C ( s \\zeta ) = s ^ 2 \\psi ^ { \\C } ( \\zeta ) \\ , . \\end{align*}"}
+{"id": "1480.png", "formula": "\\begin{align*} [ \\mathfrak { z } ^ { \\perp } , \\mathfrak { z } ^ { \\perp } ] = \\mathfrak { z } , \\end{align*}"}
+{"id": "4664.png", "formula": "\\begin{align*} \\Omega _ 1 ( | s | - s ) & = A _ { 1 } \\tau ( | s | + s ) + q \\\\ ( A _ { 1 } \\tau + \\Omega _ 1 ) s & = ( \\Omega _ { 1 } - A _ { 1 } \\tau ) | s | - q \\\\ ( ( M _ { 1 } + I - L _ 1 ) \\tau + \\Omega _ 1 ) s & = ( N _ 1 + I - L _ 1 ) \\tau s + ( \\Omega _ 1 - A _ { 1 } \\tau ) | s | - q . \\end{align*}"}
+{"id": "7318.png", "formula": "\\begin{align*} P _ n Q - Q P _ n & = ( P ^ + _ n + P ^ - _ n ) ( P - ( I _ H - P ) ) - ( P - ( I _ H - P ) ) ( P ^ + _ n + P ^ - _ n ) \\\\ & = P ^ + _ n P - P ^ - _ n ( I _ H - P ) - P P ^ + _ n + ( I _ H - P ) P ^ - _ n = 0 , \\end{align*}"}
+{"id": "2663.png", "formula": "\\begin{align*} \\phi ( \\delta ) = \\sup _ { 0 \\le t _ 0 \\le t _ 1 \\le \\ldots \\le t _ l \\le T } \\{ \\abs { g ( t _ i ) - g ( t _ { i - 1 } ) } : \\ , \\sum _ { l = 1 } ^ l ( t _ i - t _ { i - 1 } ) \\le \\delta \\} \\ , . \\end{align*}"}
+{"id": "1741.png", "formula": "\\begin{align*} \\Lambda ^ { ( m , n ) } : = \\{ \\lambda = ( \\lambda _ 1 , \\ldots , \\lambda _ n ) \\in \\mathbb { Z } ^ n \\mid m \\geq \\lambda _ 1 \\geq \\lambda _ 2 \\geq \\cdots \\geq \\lambda _ n \\geq 0 \\} . \\end{align*}"}
+{"id": "7863.png", "formula": "\\begin{align*} \\sigma ^ i \\underline { x } = & \\left ( x _ i ^ { r _ n } , \\dots , x _ { i + k } ^ { r _ n } , x _ { i + k + r _ n } , x _ { i + k + r _ n + 1 } , \\dots , x _ j ^ { r _ n } , \\dots , x _ { j + l } ^ { r _ n } \\dots \\right ) = \\left ( x _ i ^ { r _ n } , \\dots , x _ i ^ { r _ n } , \\dots , x _ j ^ { r _ n } , \\dots , x _ j ^ { r _ n } , \\dots \\right ) . \\end{align*}"}
+{"id": "7770.png", "formula": "\\begin{align*} V = 2 \\mathrm { i } Z ^ i \\partial _ { Z ^ i } - 2 \\mathrm { i } \\overline { Z } ^ i \\partial _ { \\overline { Z } ^ i } \\ , . \\end{align*}"}
+{"id": "1302.png", "formula": "\\begin{align*} \\Big \\| ( \\sum _ { j = 1 } ^ { J } v _ n ^ j ) ^ 2 \\Big \\| _ { L _ { t } ^ { \\frac { q _ 0 } { 2 } } L _ x ^ { \\frac { r _ 1 } { 2 } } } ^ 2 \\leq \\sum _ { j = 1 } ^ { J } \\| v _ n ^ j \\| _ { S _ 0 ( \\R ) } ^ 2 + \\sum _ { j \\neq k } \\| v _ n ^ j v _ n ^ k \\| _ { L _ { t } ^ { \\frac { q _ 0 } { 2 } } L _ x ^ { \\frac { r _ 1 } { 2 } } } . \\end{align*}"}
+{"id": "3826.png", "formula": "\\begin{align*} { \\rm U O T } ( \\mu _ 0 , \\mu _ 1 ) = \\inf _ { \\alpha \\in M ( Y \\times Y ) } \\int _ { Y \\times Y } \\left ( \\sum _ i s _ i ^ p \\overline R ( \\rho _ i ( x _ i ) ) + H _ p \\right ) d \\alpha , \\end{align*}"}
+{"id": "8955.png", "formula": "\\begin{align*} \\{ ( \\bar { x } , \\bar { y } ) \\in \\R ^ { m + n } : 1 - \\epsilon \\leq \\| \\bar { y } \\| \\leq e + \\epsilon , | x _ i | < \\vartheta _ { i } ( \\epsilon ) \\| \\bar { y } \\| ^ { - w _ i } i = 1 , \\ldots , m \\} . \\end{align*}"}
+{"id": "4669.png", "formula": "\\begin{align*} \\sum \\limits _ { q = 1 } ^ { 2 m + 1 } e ( S _ { i j } , S _ q - S _ { i j } ) + 2 e ( S _ { i j } ) \\leq ( 2 m + 3 ) k \\le ( 2 k + 1 ) k n . \\end{align*}"}
+{"id": "1869.png", "formula": "\\begin{gather*} f ( s ) = \\sum _ { n = 0 } ^ { N } \\frac { z _ n } { n ! } ( s - \\sigma _ 0 ) ^ n \\end{gather*}"}
+{"id": "141.png", "formula": "\\begin{align*} G _ F = \\coprod _ { \\mu \\in X _ * ( \\textbf { T } ) ^ { - } } \\coprod _ { ( b _ i , b _ j ) \\in T _ \\mu ( F ) } K _ F b _ i \\varpi _ \\mu b _ j ^ { - 1 } K _ F . \\end{align*}"}
+{"id": "7914.png", "formula": "\\begin{align*} 3 = \\sum _ { y \\in B } | \\varphi ( y ) | ^ 2 = 1 + \\sum _ { y \\in B \\setminus \\{ 1 _ R \\} } | \\varphi ( y ) | ^ 2 . \\end{align*}"}
+{"id": "2490.png", "formula": "\\begin{align*} \\eta ( P _ A ) = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } H _ \\chi \\Big ( \\big ( { \\sqcup } ^ { P _ A } _ { a \\in \\mathcal { A } } G _ a \\big ) ^ { \\wedge n } [ \\mathcal { S } ^ n _ \\epsilon ] \\Big ) . \\end{align*}"}
+{"id": "7584.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } v ^ { - \\frac { 7 } { 8 } } H _ 1 ( v ) d v & = C K ^ { - 1 / 2 } \\int _ { 0 } ^ { 1 } v ^ { - \\frac { 7 } { 8 } } v ^ { \\frac { 3 } { 8 } } d v \\\\ & = C K ^ { - 1 / 2 } . \\end{align*}"}
+{"id": "1555.png", "formula": "\\begin{align*} \\Delta = y ^ 3 x ^ 3 + z f _ 5 ( x , y , z ) \\end{align*}"}
+{"id": "8530.png", "formula": "\\begin{align*} \\ell = \\ell ^ { a } + \\ell ^ { j } + \\ell ^ { c } , \\end{align*}"}
+{"id": "3322.png", "formula": "\\begin{align*} \\frac { \\displaystyle \\prod _ { i = 0 } ^ { t - 1 } ( q ^ m - q ^ { a + i } ) \\prod _ { i = 0 } ^ { r d - t - 1 } ( q ^ { a } - q ^ i ) } { \\displaystyle \\prod _ { i = 0 } ^ { r d - 1 } ( q ^ m - q ^ i ) } \\begin{bmatrix} r d \\\\ t \\end{bmatrix} _ q . \\end{align*}"}
+{"id": "3185.png", "formula": "\\begin{align*} - \\alpha _ { i + 1 } = \\frac { d \\beta _ i + \\beta _ i d } { i + 1 } , i \\geq 0 . \\end{align*}"}
+{"id": "8231.png", "formula": "\\begin{align*} \\frac { d } { d \\tau } ( \\rho ^ 2 ( e ^ \\tau \\cdot m ) ) & = 4 x _ 1 ( e ^ \\tau \\cdot m ) , \\\\ \\frac { d } { d \\tau } ( x _ 1 ( e ^ \\tau \\cdot m ) ) & = | T | ^ 2 _ { g _ 0 } ( e ^ \\tau \\cdot m ) . \\end{align*}"}
+{"id": "7150.png", "formula": "\\begin{align*} \\begin{gathered} h ^ { - 2 \\sigma } a ^ { - 2 k + \\eta } = h ^ { - 2 \\sigma } ( h ^ { \\frac { \\sigma } { k } } ( \\log ( h ^ { - 1 } ) ) ^ { - T } ) ^ { 2 k - \\eta } \\le e ^ { \\frac { \\sigma } { k } } \\log ( h ^ { - 1 } ) ^ { - \\frac { T } { 3 } } \\le C \\log ( h ^ { - 1 } ) ^ { - \\frac { T } { 3 } } , \\\\ h ^ { - 2 \\sigma } a ^ { - 1 - 2 \\delta + k + \\eta } \\le h ^ { - 2 \\sigma } a ^ { - 2 k + \\eta } \\le C \\log ( h ^ { - 1 } ) ^ { - \\frac { T } { 3 } } . \\end{gathered} \\end{align*}"}
+{"id": "1878.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t f + v \\partial _ x f + \\partial _ v \\left ( \\left ( u - v \\right ) f \\right ) & = 0 \\quad \\mbox { i n } ( 0 , T ] \\times I \\times \\R , \\\\ f ( 0 , x , v ) & = f _ 0 ( x , v ) \\quad \\mbox { i n } I \\times \\R , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4876.png", "formula": "\\begin{align*} \\mbox { i n d } ( F ) = \\dim ( \\ker F ) - \\dim ( ( \\mbox { r n g } ~ F ) ^ \\perp ) . \\end{align*}"}
+{"id": "1804.png", "formula": "\\begin{gather*} \\zeta ( \\sigma + i v , \\alpha ) = \\sum _ { 0 \\leq n \\leq V } \\frac { 1 } { ( n + \\alpha ) ^ { \\sigma + i v } } + \\frac { V ^ { 1 - ( \\sigma + i v ) } } { \\sigma + i v - 1 } + O \\left ( V ^ { - \\sigma } \\right ) \\end{gather*}"}
+{"id": "2800.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u ( x ) : = c _ { N , s } \\ , \\mathrm { p . v . } \\int _ { \\R ^ N } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { N + 2 s } } \\dd y , \\end{align*}"}
+{"id": "8483.png", "formula": "\\begin{align*} \\theta \\left ( \\{ | g - s | > \\epsilon \\} \\cap E ; x \\right ) = 0 , \\mbox { f o r e v e r y } \\epsilon > 0 ( s \\in \\mathbb { R } ) , \\end{align*}"}
+{"id": "4629.png", "formula": "\\begin{align*} \\pi _ { \\mathcal { S } ( v ) } ( k _ v ) : = \\Bigl ( \\bigotimes _ { e \\in \\mathcal { S } ( v | s ) } \\pi _ e ^ * ( k _ v ) \\Bigr ) \\otimes \\Bigl ( \\bigotimes _ { e \\in \\mathcal { S } ( v | t ) } \\pi _ e ( k _ v ) \\Bigr ) . \\end{align*}"}
+{"id": "542.png", "formula": "\\begin{align*} \\norm { \\mathbf u } _ { Z ^ { \\mathbf s , b } ( S , T ) } = \\norm { \\mathbf u } _ { L ^ 2 \\left ( \\Omega , \\mathbf X ^ { \\mathbf s , b } ( S , T ) \\right ) } + \\norm { \\mathbf u } _ { L ^ 2 \\left ( \\Omega , C \\left ( [ S , T ] , \\mathbf H ^ { \\mathbf s } \\right ) \\right ) } . \\end{align*}"}
+{"id": "2809.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow + \\infty } I _ { k } ^ { 1 } = \\int _ { \\Omega } Y \\cdot \\nabla U _ { t , p } ( - \\Delta ) ^ { s } \\left ( \\frac { 1 } { ( t ^ 2 + | \\cdot | ^ 2 ) ^ \\frac { \\theta } { 2 } } \\right ) \\dd x . \\end{align*}"}
+{"id": "8212.png", "formula": "\\begin{align*} \\frac { d } { d t } ( ( i t ) \\cdot m ) & = T ( ( i t ) \\cdot m ) , \\\\ \\frac { d } { d \\tau } ( \\tau \\cdot m ) & = - I _ 1 T ( \\tau \\cdot m ) . \\end{align*}"}
+{"id": "7459.png", "formula": "\\begin{align*} \\phi ( x ) = \\frac { \\Phi _ { \\mathsf { b } } ( \\dot x + \\frac { \\ddot x } 2 ) } { \\Phi _ { \\mathsf { b } } ( \\dot x - \\frac { \\ddot x } 2 ) } , x = ( \\dot x , \\ddot x ) , \\ y = ( \\dot y , \\ddot y ) \\ , . \\end{align*}"}
+{"id": "8446.png", "formula": "\\begin{align*} F ^ { - j } \\left ( V ^ { \\otimes n } _ { d R } \\otimes \\mathcal { O } _ { C } \\right ) = \\begin{cases} \\mathrm { S p a n } _ { \\mathcal { O } _ { C } } \\left \\{ w A _ i B _ j \\sum _ { i = 1 } ^ { g } \\mathrm { d e g } _ { A _ i } w \\leq | j | \\right \\} & j \\leq 0 \\\\ 0 & j > 0 . \\end{cases} \\end{align*}"}
+{"id": "6925.png", "formula": "\\begin{align*} & ( 1 - \\gamma _ { n + 3 } ) \\left ( \\displaystyle \\frac { \\gamma _ { n + 3 } } { b } - 1 \\right ) = \\displaystyle \\frac { m \\gamma _ { n + 2 } } { \\gamma _ { n + 1 } + a \\gamma _ { n + 2 } } > \\displaystyle \\frac { m } { 1 + a } = ( 1 - u _ 2 ^ * ) \\left ( \\displaystyle \\frac { u _ 2 ^ * } { b } - 1 \\right ) , \\end{align*}"}
+{"id": "3663.png", "formula": "\\begin{align*} ( \\det \\ , g ) ' = ( \\det \\ , g ) \\ , g ^ { i j } \\ , ( g _ t ) _ { i j } = 2 \\ , ( \\det \\ , g ) \\ , f _ t \\ , , \\end{align*}"}
+{"id": "8252.png", "formula": "\\begin{align*} \\Psi ^ * g _ 0 = & V ( d x _ 1 ) ^ 2 + \\frac { 1 } { 4 x _ 1 ^ 2 } [ \\rho ^ 2 - 2 \\rho ^ 2 ( m ) + g _ { x _ 1 , 0 , 0 } ( \\xi _ 3 ( m ) , \\xi _ 3 ( m ) ) ] ( ( d x _ 2 ) ^ 2 + ( d x _ 3 ) ^ 2 ) + \\\\ & + \\beta _ 3 d x _ 2 - \\beta _ 2 d x _ 3 + g _ { x _ 1 , 0 , 0 } + \\frac { 1 } { V } \\eta ^ 2 . \\end{align*}"}
+{"id": "495.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} d \\psi = ( - \\alpha \\partial _ x - i M \\beta ) \\psi \\ , d t + i \\phi \\beta \\psi \\ , d t - M _ { \\mathfrak K _ 1 } \\psi \\ , d t + i \\beta \\psi \\mathfrak K _ 1 \\ , d W , \\\\ d \\phi = \\dot \\phi \\ , d t , \\\\ d \\dot \\phi = ( \\partial _ x ^ 2 - m ^ 2 ) \\phi \\ , d t + \\psi ^ * \\beta \\psi \\ , d t + \\phi \\mathfrak K _ 2 \\ , d W , \\end{gathered} \\right . \\end{align*}"}
+{"id": "6579.png", "formula": "\\begin{align*} K _ 2 = e ^ { \\frac 4 3 N _ 1 ^ { \\rho _ 2 } } . \\end{align*}"}
+{"id": "5866.png", "formula": "\\begin{align*} T _ { M e i } = \\inf \\Big \\{ t : \\sum _ { n = 1 } ^ { N } \\max _ { 0 < s \\leq t } ( \\Delta _ { 0 } S ^ { n } _ { s t } - k \\Delta _ { 0 } ^ { 2 } / 2 ) ^ { + } \\geq C _ { \\gamma } \\Big \\} , \\end{align*}"}
+{"id": "6656.png", "formula": "\\begin{align*} \\hat { r } ( \\tau ) = \\sum _ { n = 0 } ^ \\infty b _ n ( { \\pi \\tau } ) ^ n , \\tau > 0 , \\end{align*}"}
+{"id": "5551.png", "formula": "\\begin{align*} \\gamma _ j ( t ) = \\log | \\eta ( t ) - c _ j | , \\end{align*}"}
+{"id": "5166.png", "formula": "\\begin{align*} E [ { \\rm e s s } \\inf L _ j + L _ { j + 1 } ] = { \\rm e s s } \\inf L _ j + E [ L ] . \\end{align*}"}
+{"id": "6421.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\limsup _ { t \\rightarrow + \\infty } u _ { n } ( c t , t ) = 0 , \\end{align*}"}
+{"id": "3483.png", "formula": "\\begin{align*} \\omega = \\frac { 1 } { | \\log | t | | } \\omega _ { C Y , t } , \\omega _ \\phi = \\frac { 1 } { | \\log | t | | } \\omega _ t , \\phi = - \\psi _ t . \\end{align*}"}
+{"id": "3449.png", "formula": "\\begin{align*} \\tilde { \\phi } _ j = u _ j ( - \\log | F _ 0 | , \\ldots - \\log | F _ m | ) , \\end{align*}"}
+{"id": "2469.png", "formula": "\\begin{align*} \\left ( \\bigsqcup _ { a \\in \\mathcal { A } } ^ { P _ A } G _ a \\right ) \\wedge \\left ( \\bigsqcup _ { b \\in \\mathcal { B } } ^ { P _ B } G _ b \\right ) = \\bigsqcup _ { ( a , b ) \\in \\mathcal { A } \\times \\mathcal { B } } ^ { P _ A P _ B } G _ a \\wedge G _ b . \\end{align*}"}
+{"id": "3356.png", "formula": "\\begin{align*} \\sum _ { n _ 1 , n _ 2 \\geq 0 } \\frac { q ^ { \\frac { 3 } { 2 } n _ 1 ^ 2 + 4 n _ 1 n _ 2 + 4 n _ 2 ^ 2 - \\frac { 1 } { 2 } n _ 1 } } { ( q ; q ) _ { n _ 1 } ( q ^ 4 ; q ^ 4 ) _ { n _ 2 } } & = \\frac { 1 } { ( q , q ^ 4 , q ^ 7 ; q ^ 8 ) _ \\infty } , \\\\ \\sum _ { n _ 1 , n _ 2 \\geq 0 } \\frac { q ^ { \\frac { 3 } { 2 } n _ 1 ^ 2 + 4 n _ 1 n _ 2 + 4 n _ 2 ^ 2 + \\frac { 3 } { 2 } n _ 1 + 4 n _ 2 } } { ( q ; q ) _ { n _ 1 } ( q ^ 4 ; q ^ 4 ) _ { n _ 2 } } & = \\frac { 1 } { ( q ^ 3 , q ^ 4 , q ^ 5 ; q ^ 8 ) _ \\infty } . \\end{align*}"}
+{"id": "311.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { H ^ { - 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) ; s } : = \\sup _ { g \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) \\backslash \\left \\{ 0 \\right \\} } \\frac { \\left \\vert \\left \\langle f , \\overline { g } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } \\right \\vert } { \\left \\Vert g \\right \\Vert _ { H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) ; s } } . \\end{align*}"}
+{"id": "1522.png", "formula": "\\begin{align*} \\frac 1 { x / p } \\# \\left \\{ n \\le x : \\ , p | n , \\frac { R _ p ( n ) - \\beta } { ( \\log _ 2 x ) ^ { - 1 / 2 } } < t \\right \\} = \\Phi \\left ( \\frac { t } { \\sqrt { \\beta ( 1 - \\beta ) } } \\right ) + O _ { \\epsilon } \\left ( \\frac 1 { ( \\log _ 2 x ) ^ { 1 / 3 } } \\right ) , \\end{align*}"}
+{"id": "3629.png", "formula": "\\begin{align*} \\tau ( \\mathbf { i } ) = m H , \\tau _ 2 ( \\mathbf { i } ) = - m \\sum _ { i = 1 } ^ m \\left \\{ \\overline { R } ( H , e _ i ) e _ i + \\overline { \\nabla } _ { e _ i } \\overline { \\nabla } _ { e _ i } H - \\overline { \\nabla } _ { \\nabla _ { e _ i } e _ i } H \\right \\} , \\end{align*}"}
+{"id": "7055.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { \\infty } A _ { \\pi } ( m , n ) e \\left ( \\frac { a n } { q } \\right ) \\psi ( n ) \\\\ & = q \\sum _ { \\pm } \\sum _ { n _ { 1 } | q m } \\sum _ { n _ { 2 } = 1 } ^ { \\infty } \\frac { A _ { \\pi } ( n _ { 2 } , n _ { 1 } ) } { n _ { 1 } n _ { 2 } } S \\left ( m \\bar { a } , \\pm n _ { 2 } ; m q / n _ { 1 } \\right ) \\ , G _ { \\pm } \\left ( \\frac { n _ { 1 } ^ 2 n _ { 2 } } { q ^ 3 m } \\right ) \\end{align*}"}
+{"id": "7843.png", "formula": "\\begin{align*} R ( t , \\tau ) : = \\frac { t ^ 3 } { 3 } - \\frac { 9 \\chi } { 8 ( 2 \\pi ) ^ 3 } \\sum _ { ( m , n ) \\in \\mathbb { Z } ^ 2 - ( 0 , 0 ) } \\frac { 1 } { | m \\tau + n | ^ 3 } \\end{align*}"}
+{"id": "1058.png", "formula": "\\begin{align*} \\gamma _ \\lambda ( t _ n u _ n ^ + ) = \\max \\left \\{ \\gamma _ \\lambda ( t u _ n ^ + ) : \\ : 0 \\leq t \\leq 1 \\right \\} . \\end{align*}"}
+{"id": "6690.png", "formula": "\\begin{align*} \\tilde { h } _ j ( \\beta , p , q ) = { 1 \\over ( 2 \\pi ) ^ j } { \\rm R e } \\ , \\alpha _ j , \\ : \\ : j \\ : { \\rm e v e n } \\tilde { h } _ j ( \\beta , p , q ) = { i \\over ( 2 \\pi ) ^ j } { \\rm I m } \\ , \\alpha _ j , \\ : \\ : j \\ : { \\rm o d d } . \\end{align*}"}
+{"id": "4869.png", "formula": "\\begin{align*} y ( u ) = \\frac { 1 } { k } u \\ ; . \\end{align*}"}
+{"id": "4134.png", "formula": "\\begin{align*} c ( k ) & \\leq m _ 1 : = \\min _ { 1 \\leq w \\leq c ( k ) } w \\left ( d - \\sqrt { ( 2 - \\frac { 1 } { r } ) k } - ( 2 - \\frac { 1 } { r } ) ^ { \\frac { 1 } { 4 } } k ^ { \\frac { 1 } { 4 } } \\right ) - w ^ 2 , \\\\ c ( k ) & \\leq m _ 2 : = \\min _ { 1 \\leq w \\leq c ( k ) } ( \\frac { 1 } { r } - 1 ) k - w \\left ( \\sqrt { ( \\frac { 1 } { r } - 1 ) k } + ( \\frac { 1 } { r } - 1 ) ^ \\frac { 1 } { 4 } k ^ \\frac { 1 } { 4 } \\right ) . \\end{align*}"}
+{"id": "2978.png", "formula": "\\begin{align*} j _ X ^ { ( 1 ) } \\circ \\phi _ X ( c ) & = \\sum _ i \\Theta _ { j _ X ( \\phi _ X ( c ) x _ i ) , j _ X ( x _ i ) } = \\phi _ { X \\oplus Y } ( j _ A ( c ) ) \\sum _ i \\Theta _ { j _ X ( x _ i ) , j _ X ( x _ i ) } \\\\ & = \\phi _ { X \\oplus Y } ( j _ A ( c ) ) P _ X = \\phi _ { X \\oplus Y } \\circ j _ A ( c ) , \\end{align*}"}
+{"id": "8157.png", "formula": "\\begin{align*} & \\theta ^ \\star _ { x y } \\ , q _ { i j } ( x , y ) = \\frac { n ( r - 3 ) i } { k } q _ { i j } ( x , y ) + \\frac { n ( i + 1 ) ( k - i - j ) } { k ( n - i - 2 j ) } q _ { i + 1 , j } ( x , y ) + \\frac { n ( i + 1 ) ( n - k - j + 1 ) } { k ( n - i - 2 j + 2 ) } q _ { i + 1 , j - 1 } ( x , y ) \\\\ & + \\frac { n ( n - j - k ) ( j + 1 ) ( r - 2 ) } { k ( n - i - 2 j ) } q _ { i - 1 , j + 1 } ( x , y ) + \\frac { n ( k - i - j + 1 ) ( n - i - j + 2 ) ( r - 2 ) } { k ( n - i - 2 j + 2 ) } q _ { i - 1 , j } ( x , y ) , \\end{align*}"}
+{"id": "413.png", "formula": "\\begin{align*} \\nu ( z ) : = \\mathcal { A } _ { d , - \\alpha } | z | ^ { - d - \\alpha } \\mathcal { A } _ { d , - \\alpha } : = \\frac { 2 ^ { \\alpha } \\Gamma \\big ( ( d + \\alpha ) / 2 \\big ) } { \\pi ^ { d / 2 } | \\Gamma ( - \\alpha / 2 ) | } . \\end{align*}"}
+{"id": "6613.png", "formula": "\\begin{align*} \\rho _ { ( 1 ) , N } ^ { ( \\widetilde { \\rm c J } ) } ( \\theta ; \\beta , p , q ) = { 1 \\over 2 \\pi } \\sum _ { k = - \\infty } ^ \\infty c _ k ^ { ( \\widetilde { \\rm c J ) } } ( N , \\beta , p , q ) e ^ { - i k \\theta } . \\end{align*}"}
+{"id": "6276.png", "formula": "\\begin{align*} \\partial _ t M ( u ) = \\partial _ x P ( u ) , \\partial _ t P ( u ) = \\partial _ x E ( u ) . \\end{align*}"}
+{"id": "7393.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\sqrt { \\rho _ { n } } \\partial _ { x } p _ { n } ( \\rho _ { n } ) \\| _ { L ^ { \\infty } _ { t } L ^ { 2 } _ { x } } = \\| \\gamma _ { n } \\rho _ { n } ^ { \\gamma _ { n } - \\frac { 1 } { 2 } } \\partial _ { x } \\rho _ { n } \\| _ { L ^ { \\infty } _ { t } L ^ { 2 } _ { x } } \\ge \\gamma _ { n } \\underline { \\rho _ { n } } ^ { \\gamma _ { n } - \\frac { 1 } { 2 } } \\| \\partial _ { x } \\rho _ { n } \\| _ { L ^ { \\infty } _ { t } L ^ { 2 } _ { x } } . \\end{aligned} \\end{align*}"}
+{"id": "5613.png", "formula": "\\begin{align*} \\nu _ * \\geq \\dfrac { \\mu ^ { p - 1 } } { \\Gamma ( p ) } \\eta _ * \\ ( * = 0 , \\infty ) \\end{align*}"}
+{"id": "2550.png", "formula": "\\begin{align*} & \\phantom { = } C _ 3 \\| u _ n \\| ^ p - C _ 4 \\varepsilon _ W ( \\| u _ n \\| ^ { q _ 1 } + \\| u _ n \\| ^ { q _ 2 } ) - C _ 2 \\\\ & \\geq C _ 3 \\| u _ n \\| ^ p - C _ 4 \\varepsilon ( \\| u _ n \\| ^ { q _ 1 } + \\| u _ n \\| ^ { q _ 2 } ) - C _ 2 . \\end{align*}"}
+{"id": "8517.png", "formula": "\\begin{align*} & \\bigg \\{ ( \\bar { z } , w ) \\in \\mathbb { R } \\times \\mathbb { R } ^ { n - 1 } : | w | > r _ { \\ell } ^ { \\vee } ( \\bar { z } ) \\bigg \\} \\bigg \\backslash \\{ ( \\bar { z } , \\tau ) \\} \\subset E _ { 1 } ^ { ( 0 ) } \\cap ( ( 0 , \\tau ) + E _ { 2 } ) ^ { ( 0 ) } = E ^ { ( 0 ) } . \\end{align*}"}
+{"id": "5909.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\| \\psi \\nabla ^ 2 \\eta _ n \\| _ { L ^ 2 ( \\R ^ 2 _ + ) } ^ 2 = 0 . \\end{align*}"}
+{"id": "4084.png", "formula": "\\begin{align*} ( 2 - 2 g - n - 1 ) \\cdot \\omega _ { g , n + 1 } ( p _ 0 , J ) = \\frac { 1 } { 2 \\pi i } \\oint _ { C ^ { \\mathfrak { p } } _ { g , n + 2 } } \\phi ( p ) \\cdot \\omega _ { g , n + 2 } ( p , p _ 0 , J ) . \\end{align*}"}
+{"id": "6782.png", "formula": "\\begin{align*} & F ( u , v ) = \\tau u + f ( u ) ( p ( u ) - v ) , \\\\ [ 0 . 2 c m ] & G ( u , v ) = \\tau v + s v \\left ( 1 - \\frac { v } { q ( u ) } \\right ) , \\end{align*}"}
+{"id": "2966.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { i = 1 } ^ M \\sum _ { j = 1 } ^ N \\Theta _ { x _ i \\otimes y _ j , x _ i \\otimes y _ j } \\Big \\| \\le 1 . \\end{align*}"}
+{"id": "2770.png", "formula": "\\begin{align*} I ( G ; x ) = \\sum _ { k = 0 } ^ { \\alpha ( G ) } { s _ k } x ^ { k } = { s _ 0 } + { s _ 1 } x + { s _ 2 } x ^ { 2 } + . . . + { s _ { \\alpha ( G ) } } x ^ { \\alpha ( G ) } , \\end{align*}"}
+{"id": "2627.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } P ( \\abs { b _ n ^ 2 \\langle B _ n \\rangle ( x , t ) - t x } > \\epsilon ) ^ { 1 / b _ n ^ 2 } = 0 \\ , . \\end{align*}"}
+{"id": "8535.png", "formula": "\\begin{align*} | D \\ell | ( J ) \\leq \\sum _ { i = 1 } ^ { N _ { k } - 1 } | \\ell ( z _ { i + 1 } ^ { k } ) - \\ell ( z _ { i } ^ { k } ) | + \\frac { 1 } { k } \\end{align*}"}
+{"id": "7595.png", "formula": "\\begin{align*} \\gamma : = \\frac { 1 } { 4 } ( N \\beta ) ^ { - \\frac { 1 } { 2 } } T ^ { \\frac { 1 } { 2 } } \\log ( T ^ { \\gamma _ 1 } \\beta ^ { \\gamma _ 2 } ) . \\end{align*}"}
+{"id": "2913.png", "formula": "\\begin{align*} M _ \\lambda ( L ) = \\sum _ { \\mu \\in P ^ + , \\ , \\mu \\leq \\lambda } n _ { \\lambda , \\mu } ( t ) L _ \\mu \\end{align*}"}
+{"id": "2270.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } \\theta \\left ( x - \\frac { j } { n } + \\varepsilon _ j \\right ) = \\sum _ { k \\in \\mathbb { Z } } e ^ { - \\pi \\alpha k ^ 2 } \\left ( \\sum _ { j = 1 } ^ { n } e ^ { 2 \\pi i k \\varepsilon _ j } e ^ { - 2 \\pi i k \\frac { j } { n } } \\right ) e ^ { 2 \\pi i k x } . \\end{align*}"}
+{"id": "2797.png", "formula": "\\begin{align*} & \\int _ { \\Omega } \\frac { u ^ { p - 1 } \\Delta u } { | x | ^ \\theta } \\dd x + ( p - 1 ) \\int _ { \\Omega } \\frac { u ^ { p - 2 } | \\nabla u | ^ 2 } { | x | ^ { \\theta } } \\dd x = \\theta \\int _ { \\Omega } \\frac { u ^ { p - 1 } x \\cdot \\nabla u } { | x | ^ { \\theta + 2 } } \\dd x . \\end{align*}"}
+{"id": "1130.png", "formula": "\\begin{align*} & [ e _ 1 , e _ 2 ] _ 1 = e _ 3 , ~ [ e _ 2 , e _ 1 ] _ 1 = - e _ 3 ; \\\\ & [ e _ 1 , e _ 1 ] _ 2 = e _ 2 , ~ [ e _ 2 , e _ 1 ] _ 2 = e _ 3 . \\end{align*}"}
+{"id": "8705.png", "formula": "\\begin{align*} \\int g ( \\vec { x } ) e ^ { \\frac { f ( \\vec { x } ) - \\sum _ { i = 1 } ^ n x _ i \\frac { \\partial f ( \\vec { t } ) } { \\partial t _ i } } { \\hbar } } d \\vec { x } = ( 2 \\pi \\hbar ) ^ { \\frac { n } { 2 } } g ( \\vec { t } ) \\det \\Bigg ( \\frac { \\partial ^ 2 f ( \\vec { t } ) } { \\partial t _ i \\partial t _ j } \\Bigg ) _ { i , j } ^ { - \\frac { 1 } { 2 } } \\cdot e ^ { \\frac { 1 } { \\hbar } \\big ( f ( \\vec { t } ) - \\sum _ { i = 1 } ^ { \\infty } t _ i \\frac { \\partial f ( \\vec { t } ) } { \\partial t _ i } \\big ) } . \\end{align*}"}
+{"id": "1298.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } E ( u _ n ) = \\sum _ { j = 1 } ^ J \\lim _ { n \\to \\infty } E ( e ^ { t _ n ^ j \\Delta } \\phi ^ j ) + \\lim _ { n \\to \\infty } E ( W _ n ^ J ) . \\end{align*}"}
+{"id": "1071.png", "formula": "\\begin{align*} W _ { 1 } = \\left \\Vert \\widetilde { p } _ { 0 t } \\right \\Vert _ { H ^ { 2 , 1 } \\left ( S _ { T } \\right ) } + \\left \\Vert \\widetilde { p } _ { 1 t } \\right \\Vert _ { H ^ { 1 , 0 } \\left ( S _ { T } \\right ) } + \\left \\Vert \\widetilde { q } _ { 0 t } \\right \\Vert _ { H ^ { 2 , 1 } \\left ( S _ { T } \\right ) } + \\left \\Vert \\widetilde { q } _ { 1 t } \\right \\Vert _ { H ^ { 1 , 0 } \\left ( S _ { T } \\right ) } . \\end{align*}"}
+{"id": "4297.png", "formula": "\\begin{align*} X _ n ( t ) & = X _ n ( t ) - X _ n ( s ) + X _ n ( s ) = \\widetilde { X } _ n ( t - s ) + X _ n ( s ) , \\end{align*}"}
+{"id": "24.png", "formula": "\\begin{align*} & \\mathcal { H } _ N = \\sum _ { j = 1 } ^ N - \\Delta _ j + \\sum _ { 1 \\leq i < j \\leq N } v ( x _ i - x _ j ) , \\\\ & \\Lambda = \\Big [ - \\frac { \\ell } { 2 } , \\frac { \\ell } { 2 } \\Big ] ^ d , \\ell = \\frac { K _ { \\ell } } { \\sqrt { \\rho \\widehat { g } ( 0 ) } } . \\end{align*}"}
+{"id": "5031.png", "formula": "\\begin{align*} \\partial _ t u _ t = ( L + \\lambda ) u _ t , u _ t \\xrightarrow [ t \\searrow 0 ] { } u C ^ 0 ( M ; E ) \\end{align*}"}
+{"id": "6883.png", "formula": "\\begin{align*} \\ell ( \\widetilde { U } , r ) \\smallfrown [ \\widetilde { T } ] = ( \\ell ( \\widetilde { U } , r ) \\smallsmile c l ^ { \\widetilde { U } } ( \\widetilde { T } ) ) \\smallfrown [ \\widetilde { U } ] = h ^ * \\ell ( \\widetilde { U } , r ) \\smallfrown [ \\widetilde { S } ] \\in H _ 0 ^ { B M } ( \\Delta , \\mathbb { R } ) . \\end{align*}"}
+{"id": "3505.png", "formula": "\\begin{align*} Z _ { \\Gamma } ( s , \\lambda _ p ^ 0 ) = \\det ( 1 - \\mathcal { L } _ { s , \\lambda _ p ^ 0 } ) , \\end{align*}"}
+{"id": "4384.png", "formula": "\\begin{align*} | | u | | _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ T ) } : = \\sum _ { \\langle \\alpha \\rangle \\leq 1 } | | D ^ { \\alpha } _ { \\ast } u | | _ { L ^ { \\infty } ( \\Omega _ T ) } \\ , , \\end{align*}"}
+{"id": "4360.png", "formula": "\\begin{align*} S ( \\omega _ { { \\sf f _ 1 } \\cdots { \\sf f } _ n } \\| \\omega ) & = S ( \\tilde { \\omega } \\| \\omega ) = i \\left . \\frac { d } { d t } \\right | _ { t = 0 } ( \\Omega _ \\omega , F _ \\omega \\Delta ^ { i t } F _ \\omega ^ * \\Omega _ \\omega ) \\\\ & = i \\left . \\frac { d } { d t } \\right | _ { t = 0 } \\omega ( B { \\tt T } _ t ( B ^ * ) ) = i \\left . \\frac { d } { d t } \\right | _ { t = 0 } { \\rm T r } ( \\varrho F U _ t F ^ * U _ t ^ * ) \\\\ & = { \\rm T r } ( \\varrho F [ \\boldsymbol { H } , F ^ * ] ) \\end{align*}"}
+{"id": "3590.png", "formula": "\\begin{align*} \\begin{aligned} { \\bar R _ { { \\rm { S M } } , n } } & \\le { { { { \\log } _ 2 } \\left ( { 1 + \\frac { { \\mathbb E } \\left \\{ | \\beta _ { n , { \\rm R } , l _ n } | ^ 2 \\right \\} } { c _ n } } \\right ) } } \\\\ & = { { { { \\log } _ 2 } \\left ( { 1 + \\frac { 1 } { c _ n } } \\right ) } } = { \\bar R _ { { \\rm { S M } } , n , { \\rm u p p e r } } } , \\end{aligned} \\end{align*}"}
+{"id": "8607.png", "formula": "\\begin{align*} \\phi _ { \\mathrm { a p p } } = \\psi + \\Big { ( } \\frac { h } { h _ b } \\Big { ) } ^ 2 ( \\phi _ 0 - \\psi ) + \\mu \\beta \\phi _ 1 , \\end{align*}"}
+{"id": "5747.png", "formula": "\\begin{align*} U _ j ( X , t ) = \\frac { U ( r _ j X , r _ j ^ 2 \\ , t ) } { \\bigg ( \\frac { 1 } { r _ j ^ { n + 1 + a } } \\int _ { \\mathbb B _ { r _ j / 2 } ^ + } U ^ 2 ( X , 0 ) x _ { n + 1 } ^ a d X \\bigg ) ^ { 1 / 2 } } , \\ , \\ , \\ , \\ \\ \\ \\ j \\geq 1 . \\end{align*}"}
+{"id": "770.png", "formula": "\\begin{align*} \\norm { u } _ { X ^ { s , b } _ { h ( \\xi ) } ( S , T ) } = \\norm { \\phi } _ { H ^ b ( S , T ) } \\norm { f } _ { H ^ s } \\norm { u } _ { \\widetilde X ^ { s , b } _ { h ( \\xi ) } ( S , T ) } = \\norm { \\phi } _ { \\widetilde H ^ b ( S , T ) } \\norm { f } _ { H ^ s } , \\end{align*}"}
+{"id": "8428.png", "formula": "\\begin{align*} g ( X , Y ) & = - \\beta \\log \\det ( X + Y + \\Sigma _ 2 ) - \\log \\det ( X + \\Sigma _ 1 ) , \\\\ h ( X , Y ) & = - \\alpha \\log \\det ( X + Y + \\Sigma _ 1 ) - \\lambda \\log \\det ( X + \\Sigma _ 2 ) , \\end{align*}"}
+{"id": "5754.png", "formula": "\\begin{align*} J _ X ^ p = \\Delta _ p g _ X ^ p + \\mu _ p . \\end{align*}"}
+{"id": "8687.png", "formula": "\\begin{align*} F _ 2 ( D _ 0 , . . . , D _ { n + 1 } ) \\cdot \\Big ( g ( v _ 1 , . . . , v _ n ) v _ 0 ^ { - \\frac { n } { 2 } } v _ { n + 1 } ^ { - 1 } \\Big ) = 0 . \\end{align*}"}
+{"id": "7448.png", "formula": "\\begin{align*} \\tilde { H } ( x _ 1 , x _ 2 , \\xi ( x _ 1 , x _ 2 ) ) = 0 , \\ \\ \\ \\forall \\ ( x _ 1 , x _ 2 ) \\in B _ { \\epsilon } ( 0 ) . \\end{align*}"}
+{"id": "6014.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\gamma _ { i } ^ { 1 + \\alpha } e ^ { - \\rho ( \\Gamma _ { n } - \\Gamma _ { i } ) } \\leq C _ { \\rho } \\gamma _ { n } ^ { \\alpha } . \\end{align*}"}
+{"id": "4412.png", "formula": "\\begin{align*} \\dot { { \\mathbf U } } = J { \\mathbf V } \\ , , \\end{align*}"}
+{"id": "2504.png", "formula": "\\begin{align*} \\boldsymbol { v } _ k ^ c ( \\boldsymbol { x } ) = \\frac { 2 } { T } \\int _ 0 ^ T \\boldsymbol { v } ( \\boldsymbol { x } , t ) \\cos ( k \\omega t ) \\ , d t , \\boldsymbol { v } _ k ^ s ( \\boldsymbol { x } ) = \\frac { 2 } { T } \\int _ 0 ^ T \\boldsymbol { v } ( \\boldsymbol { x } , t ) \\sin ( k \\omega t ) \\ , d t . \\end{align*}"}
+{"id": "1118.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - \\Delta _ { \\mu } u _ \\infty ( x ) - \\gamma u _ \\infty ( x ) = u _ \\infty ( x ) ^ { p - 1 } x \\in \\mathop D \\limits ^ \\circ \\\\ \\medskip \\ , \\ , u _ \\infty ( x ) \\geq 0 x \\in \\mathop D \\limits ^ \\circ \\\\ \\medskip \\ , \\ , u _ \\infty | _ { \\partial D } = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "1721.png", "formula": "\\begin{align*} \\det \\left [ H ^ { ( m , n ) } _ { \\texttt { a } ; j , k } ( \\boldsymbol { \\xi } ) \\right ] _ { 1 \\leq j , k \\leq n } & = \\det \\bigl [ ( m + n ) \\delta _ { j , k } - 1 \\bigr ] _ { 1 \\leq j , k \\leq n } \\\\ & = { m ( m + n ) ^ { n - 1 } } \\end{align*}"}
+{"id": "4072.png", "formula": "\\begin{align*} F ' ( \\overline { \\varphi } ) - F ' ( \\hat { \\varphi } ) - F '' ( \\hat { \\varphi } ) \\psi = \\frac { 1 } { 2 } F ''' ( \\theta \\overline { \\varphi } + ( 1 - \\theta ) \\hat { \\varphi } ) \\xi ^ 2 + F '' ( \\hat { \\varphi } ) \\rho , \\end{align*}"}
+{"id": "574.png", "formula": "\\begin{align*} [ \\mathbf u , \\mathbf V ] ( t ) = \\begin{cases} \\mathbf u ( t ) & \\\\ \\mathbf V ( t ) & . \\end{cases} \\end{align*}"}
+{"id": "260.png", "formula": "\\begin{align*} \\lim _ { c \\to + \\infty } \\gamma _ M ( g ^ { ( c ) } ) \\Phi _ \\xi ^ { \\emph { b c } } ( x + c \\rho _ M ; g ^ { ( c ) } ) = \\Phi ^ { \\emph { t } } _ \\xi ( x ; g ) , \\end{align*}"}
+{"id": "4245.png", "formula": "\\begin{align*} \\norm { g } _ 1 & \\geq \\norm { \\tilde { g } } _ 1 = \\norm { g + w } _ 1 \\\\ & = \\norm { g + w _ S + w _ { S ^ c } } _ 1 \\\\ & = \\norm { g + w _ S } _ 1 + \\norm { w _ { S ^ c } } _ 1 \\\\ & \\geq \\norm { g } _ 1 - \\norm { w _ S } _ 1 + \\norm { w _ { S ^ c } } _ 1 \\\\ & > \\norm { g } _ 1 \\\\ \\end{align*}"}
+{"id": "6062.png", "formula": "\\begin{align*} { \\Phi } ( r , z , \\omega ) = \\vert \\nabla _ x { c } ( z , { X } ^ { M _ { { \\mathcal { P } } _ m } } _ { r - } ) \\vert \\vert D ^ Z { X } ^ { M _ { { \\ \\mathcal { P } } _ m } } _ { r - } \\vert _ { l _ 2 } , \\end{align*}"}
+{"id": "1335.png", "formula": "\\begin{align*} J ^ { w } ( \\textbf { X } _ { R S S } ^ { ( n ) } ) = - \\frac { 1 } { 2 } \\prod _ { i = 1 } ^ { n } \\left ( - 2 J ^ { w } ( X _ { ( i : n ) } ) \\right ) . \\end{align*}"}
+{"id": "8057.png", "formula": "\\begin{align*} L _ { \\omega } ^ { p } ( \\mathbb { R } ^ { n } ) = \\left \\{ f \\colon \\int \\vert f ( x ) \\vert ^ { p } \\omega ( x ) d x < \\infty \\right \\} \\end{align*}"}
+{"id": "8233.png", "formula": "\\begin{align*} d P ( X ) = \\frac { 1 } { \\rho ( m ) } X - \\frac { 1 } { \\rho ^ 2 ( m ) } \\frac { 1 } { 2 } d ( \\rho ^ 2 ) ( X ) \\frac { \\partial } { \\partial \\rho } | _ { P ( m ) } . \\end{align*}"}
+{"id": "4844.png", "formula": "\\begin{align*} & E _ i E _ j = E _ j E _ i ; \\ | i - j | \\geq 2 \\\\ & E _ i E _ { i \\pm 1 } E _ i = E _ { i \\pm 1 } \\\\ & E _ i ^ 2 = \\delta E _ i \\end{align*}"}
+{"id": "3829.png", "formula": "\\begin{align*} \\Phi _ Y : = \\{ \\phi = ( \\phi _ 0 , \\phi _ 1 ) \\in C _ b ( X ) \\times C _ b ( X ) \\mid ( s _ 0 ^ p \\phi _ 0 ) \\oplus ( s _ 1 ^ p \\phi _ 1 ) \\leq H \\} . \\end{align*}"}
+{"id": "6876.png", "formula": "\\begin{align*} H ^ { B M } _ k ( X , \\mathbb { R } ) = H _ k ( \\overline { X } , \\{ \\infty _ X \\} , \\mathbb { R } ) , \\ H ^ { B M } _ k ( U , \\mathbb { R } ) = H _ k ( \\overline { X } , S \\cup \\{ \\infty _ X \\} , \\mathbb { R } ) , \\end{align*}"}
+{"id": "2190.png", "formula": "\\begin{align*} ( \\deg ( x ) - \\lambda ) f _ 1 ( x ) & = \\sum _ { y \\in S _ 2 } b ( x , y ) f _ 1 ( y ) = \\sum _ { y \\in S _ 2 } b ( x , y ) s _ 2 f _ 2 ( y ) \\\\ & = s _ 2 ( \\deg ( x ) - \\lambda ) f _ 2 ( x ) = \\frac { s _ 2 } { s _ 1 } ( \\deg ( x ) - \\lambda ) f _ 1 ( x ) \\end{align*}"}
+{"id": "5453.png", "formula": "\\begin{align*} \\int _ { t _ o } ^ x \\left [ G ^ * ( e ^ { t ^ 2 / 2 } ) e ^ { - t ^ 2 / 2 } - t \\right ] d t & \\leq - \\int _ { t _ o } ^ x \\frac { 1 . 2 2 8 } { W ^ { - 1 } ( t ^ 2 / 2 ) } d t \\\\ & \\leq - \\int _ { t _ o } ^ x \\frac { 1 . 2 2 8 } { t } d t \\ , = \\ , - 1 . 2 2 8 \\log \\left ( \\frac { x } { t _ o } \\right ) . \\end{align*}"}
+{"id": "8630.png", "formula": "\\begin{align*} \\mathcal { I } ^ { \\mu } [ h ] \\bullet = \\mathrm { I d } - \\frac { \\mu } { 3 h } \\sqrt { \\mathrm { F } _ 2 } \\nabla _ X \\Big ( h ^ 3 \\sqrt { \\mathrm { F } _ 2 } \\nabla _ X \\cdot \\bullet \\Big ) , \\end{align*}"}
+{"id": "2934.png", "formula": "\\begin{align*} \\delta _ { j _ { x _ 1 } j _ { x _ 2 } } D _ { j _ { x _ 1 } j _ { x _ 2 } \\pi ( j _ 3 \\cdots j _ { n - 1 } ) } = 0 \\end{align*}"}
+{"id": "4613.png", "formula": "\\begin{align*} J _ \\chi & = \\textup { E n d } ( \\mathcal { H } _ \\chi ) = C _ A ( I ) _ \\chi , \\\\ I _ \\chi & = \\mathbb { C } \\textup { i d } _ { \\mathcal { H } _ \\chi } = C _ A ( J ) _ \\chi \\end{align*}"}
+{"id": "4126.png", "formula": "\\begin{align*} \\sum _ { i = a } ^ { k - 1 } c _ 2 ( s - a , i - a ) \\cdot D ( k - 2 i + j , - k + 2 s - \\tfrac { 1 } { 2 } - j ) = 0 . \\end{align*}"}
+{"id": "6851.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\nearrow + \\infty } \\| v _ \\varepsilon - v _ { \\rm s p h } \\| _ { \\mathcal { C } ^ \\infty _ { \\rm l o c } ( \\mathbb R ) } = 0 \\end{align*}"}
+{"id": "855.png", "formula": "\\begin{align*} L _ { Z _ { [ n ] } } ( w ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\ell ( w , Z _ i ) . \\end{align*}"}
+{"id": "1186.png", "formula": "\\begin{align*} G ( \\{ X _ k \\} _ k ) = \\lim _ k G _ k ( X _ k ) \\mathrm { a n d } \\overline { G } ( \\{ M _ k \\} _ k ) = \\lim _ k \\overline { G } _ k ( M _ k ) \\end{align*}"}
+{"id": "6270.png", "formula": "\\begin{align*} N _ \\lambda ^ { 2 , h i } u _ \\lambda ^ { x _ 0 } = \\mu ^ 2 L ( P _ { \\mu } g ( u _ { < \\lambda } ) , u _ { \\mu } , u _ \\lambda ) , \\end{align*}"}
+{"id": "5008.png", "formula": "\\begin{align*} \\Delta g + \\mathrm { R i c } ( \\nu , \\nu ) g + | h | ^ 2 g = 0 , \\end{align*}"}
+{"id": "1599.png", "formula": "\\begin{align*} \\pi ( F ) = \\pi _ 0 ( F ) \\ge \\pi _ { 1 } ( F ) \\ge \\dots \\ge \\pi _ { k - 2 } ( F ) \\ge \\pi _ { k - 1 } ( F ) = 0 , \\end{align*}"}
+{"id": "7801.png", "formula": "\\begin{align*} \\begin{pmatrix} a & b \\\\ c & d \\\\ \\end{pmatrix} \\cdot t = \\frac { 1 + t ^ { c , d } _ { + } t } { t ^ { c , d } _ { + } - t } = - \\frac { t _ { - } ^ { c , d } - t } { 1 + t _ { - } ^ { c , d } t } \\ , . \\end{align*}"}
+{"id": "7050.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\infty } \\lambda _ f ( m ) \\ , & e \\left ( - \\frac { ( a + b q ) m } { p _ 1 q } \\right ) \\ , m ^ { - i ( t + \\nu ) } \\ , e \\left ( \\frac { - m u } { p _ 1 q Q } \\right ) V _ 2 \\left ( \\frac { m } { N } \\right ) \\\\ = & \\ , \\frac { N ^ { 3 / 4 - i ( t + \\nu ) } } { ( p _ 1 q ) ^ { 1 / 2 } \\ , p _ 2 ^ { 1 / 4 } } \\sum _ { \\pm } \\sum _ { m \\sim ( p _ 1 Q t ) ^ 2 p _ 2 / N } \\frac { \\lambda _ g ( m ) } { m ^ { 1 / 4 } } e \\left ( - m \\frac { \\overline { ( a + b q ) p _ 2 } } { q } \\right ) \\mathcal { I } ^ { \\pm } ( . . . ) , \\end{align*}"}
+{"id": "537.png", "formula": "\\begin{align*} \\mathbf H ^ { \\mathbf s } ( \\R ^ d ) = H ^ { s _ 1 } ( \\R ^ d ) \\times \\dots \\times H ^ { s _ n } ( \\R ^ d ) \\end{align*}"}
+{"id": "1105.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - \\Delta _ { \\mu } u ( x ) = \\alpha ( x ) u ( x ) + \\lambda f ( x , u ( x ) ) x \\in \\mathop D \\limits ^ \\circ \\\\ \\medskip \\ , \\ , u | _ { \\partial D } = 0 . \\\\ \\end{array} \\right . \\end{align*}"}
+{"id": "1946.png", "formula": "\\begin{align*} I _ 1 = - \\int _ { \\R } \\int _ I \\left ( 1 + | v | ^ k \\right ) \\partial _ x u \\ , \\partial _ v f \\ , \\partial _ x f \\ , { \\rm d } x \\ , { \\rm d } v , I _ 2 = \\int _ { \\R } \\int _ I \\left ( 1 + | v | ^ k \\right ) \\left \\vert \\partial _ x f \\right \\vert ^ 2 \\ , { \\rm d } x \\ , { \\rm d } v \\end{align*}"}
+{"id": "3681.png", "formula": "\\begin{align*} \\phi _ { u } ( \\tau ) = \\begin{cases} \\phi ( \\tau \\cup \\{ u \\} ) & u \\notin \\tau , \\\\ 0 & u \\in \\tau , \\end{cases} \\end{align*}"}
+{"id": "4656.png", "formula": "\\begin{align*} \\mathbf { M } ^ { \\pi _ { \\mathcal { S } ( v ) } } = \\Bigl ( \\bigotimes _ { e \\in \\mathcal { S } ( v | s ) } M ^ { \\pi _ e ^ * } \\Bigr ) \\otimes \\Bigl ( \\bigotimes _ { e ^ \\prime \\in \\mathcal { S } ( v | t ) } M ^ { \\pi _ { e ^ \\prime } } \\Bigr ) , \\end{align*}"}
+{"id": "5040.png", "formula": "\\begin{align*} \\partial _ s d \\xi _ s = - d \\beta _ s . \\end{align*}"}
+{"id": "7066.png", "formula": "\\begin{align*} S _ 2 ( . . . ) = \\sum _ { m = 1 } ^ { \\infty } \\lambda _ f ( m ) \\ , e \\left ( - \\frac { ( a + b q ) m } { p _ 1 q } \\right ) \\ , v _ 2 ( n ) . \\end{align*}"}
+{"id": "7257.png", "formula": "\\begin{align*} \\int g ( x ) \\ , d \\mu _ { \\omega } ( x ) = \\int P _ { 0 , \\omega , N } \\left ( g \\right ) ( x ) \\ , d \\mu _ { \\sigma ^ N \\omega } ( x ) . \\end{align*}"}
+{"id": "8794.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T _ 1 } \\mathbb { E } [ f ( x _ t ) - f ^ * ] \\leq \\frac { 7 2 \\bar { L } ^ 2 \\kappa } { \\alpha } r _ { 1 } + \\mathcal { B } _ { 4 } \\frac { d } { \\alpha } T _ { 1 } ^ { \\frac { 1 } { \\beta } } . \\end{align*}"}
+{"id": "8107.png", "formula": "\\begin{align*} \\left \\| T ( f ) \\right \\| _ { h _ { \\omega } ^ { p } } = \\left \\| M _ { \\Phi } ( T ( f ) ) \\right \\| _ { L _ { \\omega } ^ { p } } \\leq C _ { 1 } \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } + C _ { 2 } \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } . \\end{align*}"}
+{"id": "6162.png", "formula": "\\begin{align*} A _ 1 & = \\begin{pmatrix} 0 & 1 \\\\ 0 & 1 \\end{pmatrix} , & A _ 2 & = \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} , & A _ 3 & = \\begin{pmatrix} 1 & 0 \\\\ 1 & 0 \\end{pmatrix} , \\\\ A _ 4 & = \\begin{pmatrix} 0 & 1 \\\\ - 1 & 1 \\end{pmatrix} , & A _ 5 & = \\begin{pmatrix} 1 & - 1 \\\\ 1 & - 1 \\end{pmatrix} , & A _ 6 & = \\begin{pmatrix} 0 & 0 \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "1101.png", "formula": "\\begin{align*} \\left ( \\int _ { D } | u ( x ) | ^ 2 d \\mu \\right ) ^ { 1 / 2 } = \\left ( \\sum _ { x \\in D } \\mu ( x ) | u ( x ) | ^ 2 \\right ) ^ { 1 / 2 } \\leq \\frac { 1 } { \\sqrt { \\lambda _ 1 } } \\left ( \\int _ { D } | \\nabla u | ^ 2 ( x ) d \\mu \\right ) ^ { 1 / 2 } , \\end{align*}"}
+{"id": "5790.png", "formula": "\\begin{align*} s _ { \\beta } s _ { \\alpha } s _ { \\beta } = \\begin{cases} s _ { \\alpha + 2 \\beta } ( \\alpha , \\beta ) = - 2 , \\\\ s _ { \\alpha - 2 \\beta } ( \\alpha , \\beta ) = 2 . \\\\ \\end{cases} \\end{align*}"}
+{"id": "2625.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\limsup _ { n \\to \\infty } P ( \\sup _ { x \\in [ 0 , 1 ] } \\abs { K _ n ( x , t ) } > r ) ^ { 1 / b _ n ^ 2 } = 0 \\ , . \\end{align*}"}
+{"id": "4597.png", "formula": "\\begin{align*} S ( \\chi _ { M _ i } ^ { } ) = \\sum _ { j \\in \\mathrm I } S _ { i , j } \\ , \\chi _ { M _ j } ^ { } T ( \\chi _ { M _ i } ^ { } ) = T _ { i , i } \\ , \\chi _ { M _ i } ^ { } \\end{align*}"}
+{"id": "3623.png", "formula": "\\begin{align*} C ( y , \\phi ) & = \\lim _ { n \\to \\infty } C ( y _ n , \\phi _ n ) . \\end{align*}"}
+{"id": "4473.png", "formula": "\\begin{align*} M _ 0 : = | | \\tilde { { \\mathbf U } } _ 0 | | _ { { H ^ { \\mu + 1 . 5 } } ( \\mathbb R ^ 2 _ + ) } + | | \\varphi _ 0 | | _ { H ^ { \\mu + 1 . 5 } ( \\R ) } \\ , . \\end{align*}"}
+{"id": "4166.png", "formula": "\\begin{align*} \\underset { p _ i , y _ i , \\tilde y _ k } { } & \\sum ^ { L } _ { i = 1 } \\log ( 1 + Q ^ + _ i ) + \\sum ^ K _ { k = 1 } \\log \\left ( 1 - \\frac { 1 } { Q ^ - _ k } \\right ) \\\\ & \\quad \\ ; 0 \\le p _ i \\le P , \\ ; \\ ; i = 1 , \\ldots , L \\\\ & \\quad \\ ; Q ^ - _ k > 0 , \\ ; \\ ; k = 1 , \\ldots , K , \\end{align*}"}
+{"id": "2814.png", "formula": "\\begin{align*} - b _ { N , s , \\theta } ( N - 2 s - \\theta ) \\int _ { \\Omega } \\frac { U _ { t , p } } { | x | ^ { \\theta + 2 s } } \\dd x - \\theta \\int _ { \\Omega } \\frac { ( - \\Delta ) ^ { s } U _ { t , p } } { | x | ^ { \\theta } } \\dd x = - ( N - 2 s ) \\int _ { \\Omega } \\frac { ( - \\Delta ) ^ { s } U _ { t , p } } { | x | ^ { \\theta } } \\dd x . \\end{align*}"}
+{"id": "8225.png", "formula": "\\begin{align*} d ^ c _ { I _ 1 ^ a } K _ 1 ^ a ( X ) = \\frac { 1 } { 2 } d ^ c _ { I _ 1 } ( \\rho ^ 2 ) ( \\bar { X } ) + 2 V x _ 1 a _ 0 + ( 2 V + a ^ 2 ) ( x _ 2 a _ 3 - x _ 3 a _ 2 ) . \\end{align*}"}
+{"id": "6168.png", "formula": "\\begin{align*} c _ 1 ( T ) = 2 - 0 = 2 , & & c _ 2 ( T ) = 2 - 1 = 1 , & & c _ 3 ( T ) = 1 - 2 = - 1 , & & c _ 4 ( T ) = 0 - 0 = 0 . \\end{align*}"}
+{"id": "1648.png", "formula": "\\begin{align*} G ( r ) = \\beta \\int _ r ^ \\infty \\frac { d t } { ( f ( t ) ) ^ { 1 / ( P - 1 ) } } . \\end{align*}"}
+{"id": "2757.png", "formula": "\\begin{align*} G ( z ) = \\pi \\frac { b - a } { 2 } \\int _ 1 ^ 0 \\sin ( 2 \\pi \\theta ) \\frac { f \\left ( \\frac { b - a } { 2 } \\cos ( 2 \\pi \\theta ) + \\frac { b + a } { 2 } \\right ) } { z - \\left ( \\frac { b - a } { 2 } \\cos ( 2 \\pi \\theta ) + \\frac { b + a } { 2 } \\right ) } \\mathrm { d } \\theta . \\end{align*}"}
+{"id": "4830.png", "formula": "\\begin{align*} \\sum _ { p , q } S ^ { k p } _ { q l } ( \\lambda + u ) S ^ { j q } _ { p i } ( \\lambda - u ) \\frac { r ( q ) r ( p ) } { r ( j ) r ( k ) } = \\rho ( u ) \\rho ( - u ) \\delta _ { i k } \\delta _ { j l } \\end{align*}"}
+{"id": "7566.png", "formula": "\\begin{align*} q _ T ^ { ( > ) } & = P ( S _ N > N / 2 ) \\le \\left ( \\frac { p _ N } { q _ N } \\right ) ^ { N / 2 } \\left [ 2 q _ N \\right ] ^ N \\\\ & = 2 ^ N [ p q ] ^ { N / 2 } \\le ( 4 p _ N ) ^ { N / 2 } \\end{align*}"}
+{"id": "785.png", "formula": "\\begin{align*} - \\Delta u + u = ( K \\ast | u | ^ { 2 ^ \\flat } ) | u | ^ { 2 ^ \\flat - 2 } u + | u | ^ { 2 ^ * - 2 } u + g ( u ) \\mathbb { R } ^ N \\end{align*}"}
+{"id": "7395.png", "formula": "\\begin{align*} \\inf _ { t \\in [ 0 , T ^ { * } ) } \\min _ { x \\in \\mathbb { T } } \\rho _ { n } ( t , x ) = 0 . \\end{align*}"}
+{"id": "1982.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { p - 1 } \\omega _ q ^ { \\left ( a ^ { t _ 1 } _ { \\sigma ^ j , r } - a ^ { t _ 2 } _ { \\sigma ^ j , s } \\right ) - \\left ( a ^ { t _ 1 } _ { \\sigma , r } - a ^ { t _ 2 } _ { \\sigma , s } \\right ) } = \\sum _ { j = 1 } ^ { p - 1 } \\omega _ p ^ { j \\left ( { r } _ { \\pi _ \\beta ( 1 ) } - { s } _ { \\pi _ \\beta ( 1 ) } \\right ) } = - 1 . \\end{align*}"}
+{"id": "1560.png", "formula": "\\begin{align*} \\left [ \\frac { 1 } { p _ 0 ^ { 2 p _ 0 } ( p _ 0 ^ 2 ) ^ { ( q - 1 ) p _ 0 ^ 2 } } \\right ] ^ N = ( p _ 0 ^ { - 2 } ) ^ { ( p _ 0 + ( q - 1 ) p _ 0 ^ 2 ) N } = ( p _ 0 ^ { - 2 } ) ^ k = \\left ( \\frac { q - 1 } { \\sqrt { q } - 1 } \\right ) ^ { 2 k } = ( \\sqrt { q } + 1 ) ^ { 2 k } . \\end{align*}"}
+{"id": "720.png", "formula": "\\begin{align*} \\mathbf u ( t ) = \\mathbf S ( t - S ) \\mathbf u ( S ) + i \\int _ { S } ^ t \\mathbf S ( t - s ) \\mathbf N ( \\mathbf u ( s ) ) \\ , d s + i \\int _ { S } ^ t \\mathbf S ( t - s ) \\mathbf M ( \\mathbf u ( s ) ) \\ , d W ( s ) , \\end{align*}"}
+{"id": "2485.png", "formula": "\\begin{align*} \\Pr ( B _ n = 0 ) = \\frac { \\beta P ' _ A ( \\overline { a } _ n ) } { P _ A ( \\overline { a } _ n ) } . \\end{align*}"}
+{"id": "3017.png", "formula": "\\begin{align*} f _ * \\left ( \\partial _ 1 \\right ) = w ^ k = f _ * ( \\beta ) w ^ \\ell f _ * ( \\beta ) ^ { - 1 } . \\end{align*}"}
+{"id": "7905.png", "formula": "\\begin{align*} 0 = \\sum _ { x \\in B } \\chi _ \\varphi ( x ) \\mathrm { F P d i m } ( x ^ \\ast ) = 1 + p \\sum _ { x \\in \\Gamma } \\chi _ \\varphi ( x ) \\mathrm { F P d i m } ( x ) , \\end{align*}"}
+{"id": "3454.png", "formula": "\\begin{align*} \\int _ A c ( x , T ( x ) ) d \\mu ( x ) , c ( x , y ) = \\frac { 1 } { 2 } | x - y | ^ 2 . \\end{align*}"}
+{"id": "4954.png", "formula": "\\begin{align*} N _ a = \\sum _ { d | n , ~ 3 \\nmid d } \\mu ( d ) N ( a , d ) . \\end{align*}"}
+{"id": "6124.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\limsup _ { \\delta \\to \\infty } \\frac { 1 } { \\delta - \\gamma } \\int _ { \\gamma } ^ \\delta \\int _ { \\alpha _ 1 } ^ { \\beta _ 1 } | f ( \\sigma + i t ) - f _ n ( \\sigma + i t ) | ^ p \\mathrm { d } \\sigma \\mathrm { d } t = 0 , \\end{align*}"}
+{"id": "1564.png", "formula": "\\begin{align*} f ( t , x , v ) \\geq e ^ { - t / \\kappa } f ( 0 , x - v t , v ) + & \\int _ 0 ^ t e ^ { - ( t - s ) / \\kappa } \\int f ( s , x - v ( t - s ) , u ) \\mathrm { d } u \\times \\\\ & \\times \\left ( \\alpha \\mathcal { M } _ { T ( x - v ( t - s ) ) } ( v ) + ( 1 - \\alpha ) \\mathcal { M } _ { \\tau ( x - v ( t - s ) ) } ( v ) \\right ) \\mathrm { d } s . \\end{align*}"}
+{"id": "2971.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { j = 1 } ^ N \\Theta _ { y _ j , y _ j } \\phi ( ( x _ i \\mid \\eta ) _ A ) \\zeta - \\phi ( ( x _ i \\mid \\eta ) _ A ) \\zeta \\Big \\| < \\frac { \\varepsilon } { 2 M } \\end{align*}"}
+{"id": "2051.png", "formula": "\\begin{align*} T _ \\phi u = T _ \\phi a + T _ \\phi b \\in { \\mathrm { L } } ^ { p } ( \\mathbb { R } ^ n _ + ) + { \\mathrm { H } } ^ { 1 , p } ( \\mathbb { R } ^ n _ + ) \\subset \\mathrm { L } ^ p ( \\mathbb { R } ^ n _ + ) + \\dot { \\mathrm { H } } ^ { 1 , p } ( \\mathbb { R } ^ n _ + ) \\end{align*}"}
+{"id": "148.png", "formula": "\\begin{align*} ( { \\rm K a z } _ m ^ F \\circ { \\rm \\overline { B r } } ) ( h ) = \\sum _ { ( a _ i , a _ j ) \\in T _ \\mu ( E ) } \\alpha _ \\mu ( a _ i , a _ j ) ( { \\rm K a z } _ m ^ F \\circ { \\rm \\overline { B r } } ) ( t _ { a _ i \\pi _ \\mu a _ j ^ { - 1 } } ) \\end{align*}"}
+{"id": "8947.png", "formula": "\\begin{align*} a ' = { d i a g } ( e ^ { w _ 1 } , \\ldots , e ^ { w _ m } , e ^ { - 1 } , \\ldots , e ^ { - 1 } ) . \\end{align*}"}
+{"id": "8860.png", "formula": "\\begin{align*} \\int K ( r ) \\d r { = } 0 , \\int r K ( r ) \\d r { = } 1 , \\int r ^ j K ( r ) \\d r { = } 0 , \\ j { = } 2 , \\dots , \\ell , ~ \\kappa _ \\beta { \\triangleq } \\int | r | ^ { \\beta } | K ( r ) | \\d r < \\infty \\enspace . \\end{align*}"}
+{"id": "8337.png", "formula": "\\begin{align*} \\mathbf { s } ^ * = \\mathbf { s } _ { \\mathbf { S B } } ^ * ( 1 ) \\mathbf { M } \\end{align*}"}
+{"id": "7989.png", "formula": "\\begin{align*} 2 ^ { J _ i } = 2 ^ { \\mathfrak { k } _ J } \\prod _ { j : \\mathfrak { d } _ j \\leq i } ( s _ j ) ^ { i - \\mathfrak { d } _ j } , \\end{align*}"}
+{"id": "2711.png", "formula": "\\begin{align*} \\begin{array} { r c c c l c l } \\mu _ M & : = & [ [ \\mu ] ] \\gamma & : & M ^ { \\times 2 } & \\to & M ^ { \\times 1 } \\\\ \\eta _ M & : = & [ [ \\eta ] ] \\gamma & : & M ^ { \\times 0 } & \\to & M ^ { \\times 1 } \\\\ \\omega _ M & : = & [ [ \\omega ] ] \\gamma & : & M ^ { \\times 1 } & \\to & M ^ { \\times 1 } . \\end{array} \\end{align*}"}
+{"id": "6555.png", "formula": "\\begin{align*} \\mathbb { D } _ { 5 b } ( \\sigma ^ * ) \\setminus \\mathbb { D } _ { 0 } ( \\sigma ^ * ) = \\bigcup _ { l = 1 } ^ { 5 b } \\left ( \\mathbb { D } _ l ( \\sigma ^ * ) \\setminus \\mathbb { D } _ { l - 1 } ( \\sigma ^ * ) \\right ) . \\end{align*}"}
+{"id": "2185.png", "formula": "\\begin{align*} H _ { h _ \\alpha } T _ g f & = \\frac { 1 } { h _ \\alpha } H ( h _ \\alpha T _ g f ) \\\\ & = \\frac { 1 } { h _ \\alpha } H ( \\exp ( - \\alpha ( z ) \\rangle ) T _ g ( h _ \\alpha f ) ) \\\\ & = \\frac { 1 } { \\exp ( \\alpha ( z ) ) h _ \\alpha } T _ g H ( h _ \\alpha f ) \\\\ & = T _ g H _ { h _ \\alpha } , \\end{align*}"}
+{"id": "7673.png", "formula": "\\begin{align*} \\mathbb { P } ( X ^ * \\leqslant X ) & = \\sum _ { k = 0 } ^ { \\infty } \\sum _ { m = k } ^ { \\infty } \\mathbb { P } ( X ^ * = k , \\ , X = m ) \\\\ & = \\mathbb { P } ( X ^ * = 0 , \\ , X = l ) + \\sum _ { k = 0 } ^ { \\infty } \\mathbb { P } ( X ^ * = k , \\ , X = k ) = 1 . \\end{align*}"}
+{"id": "3184.png", "formula": "\\begin{align*} \\alpha _ 0 = f \\sum _ { i \\geq 0 } \\alpha _ i = g . \\end{align*}"}
+{"id": "3971.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) u ^ i = f ( u ^ i , \\mu _ t ^ n ) \\ , \\xi _ t ^ i + g ( u ^ i , \\mu _ t ^ n ) , ( 1 \\leq i \\leq n ) , \\end{align*}"}
+{"id": "2092.png", "formula": "\\begin{align*} \\Lambda ( t , x ) : = \\lambda + \\tilde { \\Lambda } ( t , x ) , \\lambda \\neq 0 . \\end{align*}"}
+{"id": "7497.png", "formula": "\\begin{align*} \\Lambda ( \\gamma ) = \\frac { \\det ( A _ \\gamma ) } { \\mathrm { T r } ( A _ \\gamma ) ^ 2 } = \\frac { 1 } { \\gamma ^ m G ^ 2 _ { m - 1 } ( \\gamma ) } . \\end{align*}"}
+{"id": "5457.png", "formula": "\\begin{align*} x - \\frac { 1 } { V ' ( x ) } - \\sqrt { 2 W ( x ) } & = - \\frac { 1 } { V ' ( x ) } - \\frac { 2 \\log H ( x ) } { x + \\sqrt { 2 W ( x ) } } \\\\ & \\leq - \\frac { 1 } { x } - \\frac { 2 \\log H ( x ) } { x } \\\\ & = - \\frac { 2 } { x } \\log ( \\sqrt { e } H ( x ) ) . \\end{align*}"}
+{"id": "7166.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ n a _ j \\omega _ j \\right \\| = \\left \\| \\sum _ { j = 1 } ^ n a _ j V \\tau _ j \\right \\| = \\left \\| V \\left ( \\sum _ { j = 1 } ^ n a _ j \\tau _ j \\right ) \\right \\| = \\left \\| \\sum _ { j = 1 } ^ n a _ j \\tau _ j \\right \\| = \\left ( \\sum _ { j = 1 } ^ n | a _ j | ^ p \\right ) ^ \\frac { 1 } { p } . \\end{align*}"}
+{"id": "4608.png", "formula": "\\begin{align*} \\bigotimes _ { i \\in \\mathcal { I } } M _ i : = M _ { i _ 1 } \\otimes \\cdots \\otimes M _ { i _ s } . \\end{align*}"}
+{"id": "1710.png", "formula": "\\begin{align*} e ^ { 2 i ( m + 1 ) \\xi _ j } = & \\left ( \\frac { 1 - q _ 0 e ^ { i \\xi _ j } } { e ^ { i \\xi _ j } - q _ 0 } \\right ) \\left ( \\frac { 1 - q _ 1 e ^ { i \\xi _ j } } { e ^ { i \\xi _ j } - q _ 1 } \\right ) \\\\ & \\times \\prod _ { \\substack { 1 \\leq k \\leq n \\\\ k \\neq j } } \\left ( \\frac { 1 - q e ^ { i ( \\xi _ j + \\xi _ k ) } } { e ^ { i ( \\xi _ j + \\xi _ k ) } - q } \\right ) \\left ( \\frac { 1 - q e ^ { i ( \\xi _ j - \\xi _ k ) } } { e ^ { i ( \\xi _ j - \\xi _ k ) } - q } \\right ) ( j = 1 , \\ldots , n ) . \\end{align*}"}
+{"id": "6708.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { H ( x ^ { [ f / g ] } ) } \\psi ( z ) e ^ { - H ^ \\ast ( - \\beta ) } & = \\left ( \\frac { 1 } { 1 + \\beta z ^ { - 1 } } \\prod _ { j = g } ^ f \\frac { 1 + \\beta x _ i } { 1 - x _ i z } \\right ) e ^ { - H ^ \\ast ( - \\beta ) } \\psi ( z ) e ^ { H ( x ^ { [ f / g ] } ) } , \\end{aligned} \\end{align*}"}
+{"id": "7274.png", "formula": "\\begin{align*} \\mathop { \\sum } \\limits _ { m = - \\infty } ^ { \\infty } X [ m ] \\delta ( t - m T _ s ) * h ( t ) = \\mathop { \\sum } \\limits _ { m = - \\infty } ^ { \\infty } X [ m ] \\cdot h ( t - m T _ s ) . \\end{align*}"}
+{"id": "4488.png", "formula": "\\begin{align*} | | ( S _ { \\theta _ k } { \\mathbf V } _ k , S _ { \\theta _ k } \\Psi _ k ) | | _ { s , \\ast , T } + | | S _ { \\theta _ k } \\psi _ k | | _ { H ^ s ( \\Gamma _ T ) } \\lesssim \\begin{cases} \\delta \\theta ^ { ( s - \\alpha ) _ + } _ k , & s \\neq \\alpha , \\\\ \\delta \\log \\theta _ k , & s = \\alpha . \\\\ \\end{cases} \\end{align*}"}
+{"id": "6350.png", "formula": "\\begin{align*} j ^ 6 _ { \\lambda } ( \\xi ) = c ( \\xi , \\xi , \\xi ) \\phi _ \\lambda ^ 4 ( \\xi ) . \\end{align*}"}
+{"id": "5868.png", "formula": "\\begin{align*} T _ { M e i } ( \\epsilon _ { 0 } ) = \\inf \\Big \\{ t : \\sum _ { n = 1 } ^ { N } g _ { M } ( R ^ { n } _ { t } ) \\geq C _ { \\gamma } \\Big \\} , \\end{align*}"}
+{"id": "3073.png", "formula": "\\begin{align*} { { \\xi _ { k , { l _ k } , 0 } } = { \\rho _ k } { \\alpha _ { { k , { \\rm { R } } } , { l _ k } } } { \\alpha _ { { \\rm { T } } , k , 0 } } { \\bf { a } } _ { { \\rm { S , } } k } ^ H \\left ( { \\Theta _ { { k , { \\rm { R } } } , { l _ k } } ^ { \\rm { D } } } \\right ) { { \\bf { \\Gamma } } _ k } { { \\bf { a } } _ { { \\rm { S } } , k } } \\left ( { \\Theta _ { { \\rm { T } } , k , 0 } ^ { \\rm { A } } } \\right ) } , \\end{align*}"}
+{"id": "6005.png", "formula": "\\begin{align*} \\varepsilon _ { m } : = \\int _ { \\{ | z | > m \\} } | \\bar { c } ( z ) | ^ { 2 } \\mu ( d z ) + | \\int _ { \\{ | z | > m \\} } \\bar { c } ( z ) \\mu ( d z ) | ^ { 2 } . \\end{align*}"}
+{"id": "5982.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta _ g ^ F + 1 ) u = 0 , & M ; \\\\ \\partial _ \\nu u = \\psi , & \\partial M . \\end{cases} \\end{align*}"}
+{"id": "4284.png", "formula": "\\begin{align*} ( X ^ { ( k ) } \\cdots X ^ { ( 1 ) } ) e _ i & = \\sum _ { j _ { 1 } , \\ldots , j _ { 2 k } = - ( p - 1 ) } ^ { n - 1 } \\prod _ { r = 1 } ^ { k } a ^ { ( r ) } _ { j _ { 2 r - 1 } } a ^ { ( r ) } _ { j _ { 2 r } } \\prod _ { s = 1 } ^ { k } \\chi _ { [ 1 , n ] } ( i + \\sum _ { \\ell = 1 } ^ { 2 s - 1 } ( - 1 ) ^ \\ell j _ \\ell ) \\\\ & \\times \\chi _ { [ 1 , p ] } ( i + \\sum _ { \\ell = 1 } ^ { 2 s } ( - 1 ) ^ \\ell j _ \\ell ) e _ { i - \\sum _ { q = 1 } ^ { 2 k } ( - 1 ) ^ { q } j _ { q } } . \\end{align*}"}
+{"id": "8380.png", "formula": "\\begin{align*} \\mathcal { G } ^ { 0 } \\Big ( \\int _ { 0 } ^ { t } \\dot { u } _ { s } d s \\Big ) = \\bar { X } ^ { h } _ { t } , \\end{align*}"}
+{"id": "4447.png", "formula": "\\begin{align*} \\sum _ { \\langle \\beta \\rangle = 1 \\ , , \\ , \\beta \\leq \\alpha } | | D ^ { \\beta } _ { \\ast } \\mathcal { A } _ j D ^ { \\alpha - \\beta } _ { \\ast } \\partial _ j { \\mathbf V } | | ^ 2 _ { L ^ 2 ( \\Omega _ t ) } \\le C ( K ) | | { \\mathbf V } | | ^ 2 _ { s , \\ast , t } \\ , . \\end{align*}"}
+{"id": "6855.png", "formula": "\\begin{align*} \\mathcal X _ { ( i ) } = B _ i \\mathcal G _ { ( i ) } ( B _ d \\otimes \\dots \\otimes B _ { i + 1 } \\otimes B _ { i - 1 } \\otimes \\dots \\otimes B _ 1 ) . \\end{align*}"}
+{"id": "8.png", "formula": "\\begin{align*} \\mathrm { E } _ { d } ( v , \\widetilde { R } ) = \\inf _ { \\varphi } \\int _ { B ^ { d } ( 0 , \\widetilde { R } ) } \\Big ( \\vert \\nabla \\varphi \\vert ^ 2 + \\frac { 1 } { 2 } v \\varphi ^ 2 \\Big ) \\dd x , \\end{align*}"}
+{"id": "8261.png", "formula": "\\begin{align*} g _ { x _ 1 , 0 , 0 } ( \\rho , \\Theta ) = g _ { \\rho ^ { - 2 } x _ 1 , 0 , 0 } ( 1 , \\Theta ) = \\sum _ { \\nu \\geq 0 } h _ \\nu ( \\Theta ) ( \\frac { x _ 1 } { \\rho ^ 2 } ) ^ \\nu . \\end{align*}"}
+{"id": "4628.png", "formula": "\\begin{align*} \\mathbf { M } ^ { \\pi _ { \\mathcal { S } ( v ) } } : = \\Bigl ( \\bigotimes _ { e \\in \\mathcal { S } ( v | s ) } M ^ { \\pi _ e ^ * } \\Bigr ) \\otimes \\Bigl ( \\bigotimes _ { e \\in \\mathcal { S } ( v | t ) } M ^ { \\pi _ e } \\Bigr ) \\end{align*}"}
+{"id": "6647.png", "formula": "\\begin{align*} \\omega = z - { 1 \\over 2 } \\nu \\pi - { 1 \\over 4 } \\pi , a _ k ( \\nu ) = { 1 \\over ( - 2 ) ^ k k ! } \\Big ( { 1 \\over 2 } - \\nu \\Big ) _ k \\Big ( { 1 \\over 2 } + \\nu \\Big ) _ k . \\end{align*}"}
+{"id": "1873.png", "formula": "\\begin{gather*} f ( s ) = s - \\sigma _ 0 . \\end{gather*}"}
+{"id": "348.png", "formula": "\\begin{align*} \\left \\Vert u \\right \\Vert _ { H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\backslash \\Gamma _ { j } \\right ) ; s } : = \\left ( \\sum _ { \\sigma \\in \\left \\{ + , - \\right \\} } \\left \\Vert u ^ { \\sigma } \\right \\Vert _ { H ^ { 1 } \\left ( \\Omega _ { j } ^ { \\sigma } \\right ) ; s } ^ { 2 } \\right ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "3812.png", "formula": "\\begin{align*} ( a F ) ^ * ( \\phi ) = a F ^ * \\left ( \\frac { \\phi } { a } \\right ) , \\end{align*}"}
+{"id": "2401.png", "formula": "\\begin{align*} S _ { \\i , \\tt } ( x _ n ) - S _ { \\i , \\tt } ( \\pi _ { \\tt } ( ( i _ { n , m } ' ) _ { m = 1 } ^ { \\infty } ) ) = A ^ n x _ n - A ^ n \\pi _ { \\tt } ( ( i _ { n , m } ' ) _ { m = 1 } ^ { \\infty } ) . \\end{align*}"}
+{"id": "2626.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\limsup _ { n \\to \\infty } P ( \\sup _ { x \\in [ 0 , 1 / 2 ] } \\abs { K _ n ( x , t ) } > r ) ^ { 1 / b _ n ^ 2 } = 0 \\ , . \\end{align*}"}
+{"id": "6733.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow + \\infty } \\frac { \\xi ( t ) } { t ^ { - a - 1 } } = \\mathfrak { c } _ { p } \\frac { 1 } { a + 1 } . \\end{align*}"}
+{"id": "421.png", "formula": "\\begin{align*} f = \\sum _ { \\ell \\in L _ d } \\sum _ { m \\in M _ \\ell } f _ { \\ell , m } , \\| f \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 = \\sum _ { \\ell \\in L _ d } \\sum _ { m \\in M _ \\ell } \\| f _ { \\ell , m } \\| _ { L ^ 2 ( \\R ^ d ) } ^ 2 . \\end{align*}"}
+{"id": "2195.png", "formula": "\\begin{align*} H = \\left ( \\begin{array} { c c } 0 & J \\\\ J ^ \\top & 0 \\end{array} \\right ) \\end{align*}"}
+{"id": "5178.png", "formula": "\\begin{align*} m _ d ( f ) ( x _ 1 , \\ldots , x _ d ) = \\int _ G ( g \\cdot f ) ( x _ 1 ) \\ldots ( g \\cdot f ) ( x _ { d - 1 } ) \\overline { ( g \\cdot f ) ( x _ d ) } \\ ; d g \\end{align*}"}
+{"id": "3048.png", "formula": "\\begin{align*} [ v _ { \\delta _ l } ] = \\bar R _ { \\delta _ l } ^ { \\mathrm { T } } \\frac { \\zeta _ { \\delta _ l } } { \\delta _ l } \\qquad \\textrm { o n } \\mathsf { C } \\ , , \\end{align*}"}
+{"id": "7763.png", "formula": "\\begin{align*} N _ { i j } ^ { } : = - 2 \\sum _ { \\gamma } \\Omega ( \\gamma ) V ^ { } _ { \\gamma } q _ i ( \\gamma ) q _ j ( \\gamma ) \\ , . \\end{align*}"}
+{"id": "7732.png", "formula": "\\begin{align*} f _ C ( e ) = \\left \\lbrace \\begin{array} { l l l l l l l l l } a & & & \\rm i f ~ \\it e \\in E ( C ) ; \\\\ ( 0 , 0 ) & & & \\rm i f ~ \\it e \\in E ( G ) \\backslash E ( C ) . \\end{array} \\right . \\end{align*}"}
+{"id": "5029.png", "formula": "\\begin{align*} L u = \\triangle u + \\nabla _ V u + b _ 1 * \\nabla u + b _ 0 * u = v , \\end{align*}"}
+{"id": "3609.png", "formula": "\\begin{align*} { \\bf { r } } = \\left [ { \\bf { r } } _ 1 , \\ , \\ldots , \\ , { \\bf { r } } _ { M _ { \\rm R } } \\right ] ^ T = { \\bf { H } } _ { \\rm D B } { \\bf f } _ { \\rm D B } { s } + { \\bf { n } } , \\end{align*}"}
+{"id": "3963.png", "formula": "\\begin{align*} d ( \\Phi ( A ) ) \\leq d ( \\Phi ( A / A \\cap C ) ) + d ( A \\cap C ) \\leq k + 2 k = 3 k . \\end{align*}"}
+{"id": "4433.png", "formula": "\\begin{align*} | | \\breve { \\mathbf U } ^ { \\pm } | | _ { 9 , \\ast , T } + | | \\hat { \\varphi } | | _ { H ^ { 1 0 } ( \\Gamma _ T ) } \\leq \\hat { K } , \\breve { \\mathbf U } ^ { \\pm } : = \\hat { \\mathbf { U } } ^ { \\pm } - \\bar { \\mathbf { U } } . \\end{align*}"}
+{"id": "7213.png", "formula": "\\begin{align*} J ( t _ 0 , \\mu _ 0 ; ( \\mu , \\alpha ) ) : = \\int _ { t _ 0 } ^ T \\int _ { \\R ^ d } L \\bigl ( x , \\alpha ( t , x ) \\bigr ) d \\mu ( t ) ( x ) d t + \\int _ { t _ 0 } ^ T \\mathcal { F } ( \\mu ( t ) ) d t + \\mathcal { G } ( \\mu ( T ) ) \\end{align*}"}
+{"id": "5806.png", "formula": "\\begin{align*} \\alpha ^ { c i } _ { m a x } : = 2 \\sum \\limits _ { j = n - i + 1 } ^ { n - 1 } \\alpha _ j + \\alpha ' _ n , i \\ge 2 , \\alpha ^ { c 1 } _ { m a x } : = \\alpha ' _ n . \\\\ \\end{align*}"}
+{"id": "8679.png", "formula": "\\begin{align*} \\psi = g \\Big ( \\frac { x _ 1 } { x _ 0 } , . . . , \\frac { x _ n } { x _ 0 } \\Big ) ~ x _ 0 ^ { - \\frac { n + 2 } { 2 } } \\delta \\Big ( \\frac { x _ { n + 1 } } { x _ 0 } - f \\Big ( \\frac { x _ 1 } { x _ 0 } , . . . , \\frac { x _ n } { x _ 0 } \\Big ) \\Big ) \\end{align*}"}
+{"id": "8537.png", "formula": "\\begin{align*} \\ell ^ { k } ( z ) : = \\sum _ { i = 0 } ^ { N _ { k } } \\ell ( z _ { i } ^ { k } ) \\chi _ { [ z _ { i } ^ { k } , z _ { i + 1 } ^ { k } ) } ( z ) , \\end{align*}"}
+{"id": "40.png", "formula": "\\begin{align*} \\mathcal K ( z ) & : = \\mathcal Q ( z ) + \\mathcal Q _ 2 ^ { \\rm { e x } } ( z ) + ( \\rho _ z - \\rho ) n _ + \\widehat { g } ( 0 ) - \\rho \\rho _ z | \\Lambda | \\widehat { g } ( 0 ) + \\rho ^ 2 | \\Lambda | \\widehat { g } ( 0 ) , \\end{align*}"}
+{"id": "2994.png", "formula": "\\begin{align*} \\alpha ^ * ( f ) = f \\circ \\alpha \\psi ^ * ( f ) x ( e ) = f ( \\psi ( e ) ) x ( e ) \\end{align*}"}
+{"id": "4877.png", "formula": "\\begin{align*} ( R _ \\alpha f ) ( z ) : = \\left \\{ \\begin{array} { l l } \\frac { f ( z ) - f ( \\alpha ) } { z - \\alpha } & \\mbox { i f } z \\neq \\alpha \\\\ f ' ( \\alpha ) & \\mbox { i f } z = \\alpha . \\end{array} \\right . \\end{align*}"}
+{"id": "8039.png", "formula": "\\begin{align*} f ( x ) = \\sum \\limits _ { j = 0 } ^ { \\infty } \\sum \\limits _ { Q \\in \\Pi _ { j } } \\vert Q \\vert ( \\phi _ { j } \\ast f ) ( x _ { Q } ) \\phi _ { j } ( x - x _ { Q } ) . \\end{align*}"}
+{"id": "6337.png", "formula": "\\begin{align*} d _ \\lambda = \\epsilon \\lambda ^ { - s } c _ \\lambda . \\end{align*}"}
+{"id": "8542.png", "formula": "\\begin{align*} | D ^ { c } \\ell | ( J ) = | D \\ell | ( J ) = \\lim _ { k \\rightarrow \\infty } \\sum _ { i = 1 } ^ { N _ { k } - 1 } | \\ell ( z _ { i + 1 } ^ { k } ) - \\ell ( z _ { i } ^ { k } ) | = \\lim _ { k \\rightarrow \\infty } | D \\ell ^ { k } | ( J ) = \\lim _ { k \\rightarrow \\infty } | D ^ { c } \\ell ^ { k } | ( J ) . \\end{align*}"}
+{"id": "6864.png", "formula": "\\begin{align*} \\vec { V } _ i & = I _ { n _ d } \\otimes \\cdots \\otimes I _ { n _ { i + 1 } } \\otimes V _ i \\otimes I _ { n _ { i - 1 } } \\otimes \\cdots \\otimes I _ { n _ 1 } , \\\\ \\overline { \\vec { V } } _ i & = { V } _ d \\otimes \\cdots \\otimes V _ { i + 1 } \\otimes I _ { n _ i } \\otimes V _ { i - 1 } \\otimes \\cdots \\otimes V _ 1 . \\end{align*}"}
+{"id": "3076.png", "formula": "\\begin{align*} { \\mathbb E } \\left \\{ | { \\xi _ { k , { l _ k } , 0 } ^ { \\rm r } } | ^ 2 \\right \\} \\buildrel \\Delta \\over = \\xi _ { 0 } ^ \\star , \\forall k . \\end{align*}"}
+{"id": "7768.png", "formula": "\\begin{align*} \\mathcal { Z } _ p : = \\{ J \\in Q _ p \\ ; | \\ ; J ^ { 2 } = - 1 \\} , p \\in \\overline { N } ; \\end{align*}"}
+{"id": "4620.png", "formula": "\\begin{align*} \\theta _ m ( x _ 1 , \\ldots , x _ m ) : = ( x _ m , x _ 1 , \\ldots , x _ { m - 1 } ) . \\end{align*}"}
+{"id": "5966.png", "formula": "\\begin{align*} u _ f ( x ) & = \\int _ { \\partial M } G ^ \\omega _ M ( x , z ) \\partial _ \\nu u _ f ( z ) d \\mu _ h ( z ) . \\end{align*}"}
+{"id": "25.png", "formula": "\\begin{align*} I ^ { \\rm { B o g } } _ { d } = \\begin{dcases} 2 \\Gamma + \\frac { 1 } { 2 } + \\log \\pi , & d = 2 , \\\\ \\frac { 1 2 8 } { 1 5 \\sqrt { \\pi } } , & d = 3 . \\end{dcases} \\end{align*}"}
+{"id": "3309.png", "formula": "\\begin{align*} \\phi _ Z ( t ) & = \\mathcal { F } ^ \\mu \\left ( h _ Z \\right ) ( t ) = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\int _ \\mathbb { R } \\lambda _ { _ \\Sigma } e ^ { - \\lambda _ { _ \\Sigma } x } \\mathbb { 1 } _ { [ 0 , + \\infty [ } ( x ) e ^ { \\mu t x } d x = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\frac { \\lambda _ { _ \\Sigma } } { \\lambda _ { _ \\Sigma } - \\mu t } . \\end{align*}"}
+{"id": "5416.png", "formula": "\\begin{align*} M _ N f ( x ) : = \\frac { 1 } { N } \\sum _ { n = 1 } ^ N f ( S ^ n x ) \\end{align*}"}
+{"id": "314.png", "formula": "\\begin{align*} \\ell _ { j } \\left ( s \\right ) \\left ( \\mathsf { S } _ { j } \\left ( s \\right ) \\varphi , w \\right ) = \\left \\langle \\varphi , \\gamma _ { \\operatorname * { D } ; j } \\left ( s \\right ) \\overline { w } \\right \\rangle _ { \\Gamma _ { j } } \\quad \\forall w \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) . \\end{align*}"}
+{"id": "7641.png", "formula": "\\begin{align*} u ( k , w ) & = \\inf _ { \\mathbb { Q } \\in B _ { k } ( \\mathbb { P } ) } \\mathbb { E } _ { \\mathbb { Q } } \\left [ U ( x _ 0 + x _ 0 \\langle w , X \\rangle ) \\right ] \\\\ & \\leq \\inf _ { \\mathbb { Q } \\in B _ { k } ( \\mathbb { P } ) } \\mathbb { E } _ { \\mathbb { Q } } \\left [ U ( x _ 0 + x _ 0 \\langle w ^ u , X \\rangle ) \\right ] = u ( k , w ^ u ) . \\end{align*}"}
+{"id": "682.png", "formula": "\\begin{align*} \\mathfrak K f ( x ) = \\int _ { \\R ^ d } \\mathfrak k ( x - y ) f ( y ) d y . \\end{align*}"}
+{"id": "2351.png", "formula": "\\begin{align*} \\lambda ( G ) ' = \\rho ( G ) '' \\rho ( G ) ' = \\lambda ( G ) '' . \\end{align*}"}
+{"id": "8033.png", "formula": "\\begin{align*} M _ { \\Phi } ( f ) ( x ) = \\sup \\limits _ { 0 < t < 1 } \\vert \\Phi _ { t } \\ast f ( x ) \\vert . \\end{align*}"}
+{"id": "8474.png", "formula": "\\begin{align*} \\theta _ { * } ( E , x ) = \\liminf _ { \\rho \\rightarrow 0 ^ { + } } \\frac { \\mathcal { H } ^ { n } ( E \\cap B _ { \\rho } ( x ) ) } { \\omega _ { n } \\rho ^ { n } } , \\mbox { a n d } \\theta ^ { * } ( E , x ) = \\limsup _ { \\rho \\rightarrow 0 ^ { + } } \\frac { \\mathcal { H } ^ { n } ( E \\cap B _ { \\rho } ( x ) ) } { \\omega _ { n } \\rho ^ { n } } , \\end{align*}"}
+{"id": "8374.png", "formula": "\\begin{align*} \\lim \\limits _ { T \\rightarrow \\pm \\infty } \\frac { 1 } { | T | } \\bigg \\| \\int _ { 0 } ^ { T } ( - A ) ^ { - \\nu } ( f ( x , Y _ { F } ^ 1 ( \\theta _ r \\omega _ 2 , x ) ) - \\bar f ( x ) ) \\ , d r \\bigg \\| = 0 \\end{align*}"}
+{"id": "3866.png", "formula": "\\begin{align*} \\varphi _ s ( s , t ) & = ( 0 , y ' ( s ) , z ' ( s ) ) = \\cos \\theta E _ 2 + \\sin \\theta E _ 3 . \\\\ \\varphi _ t ( s , t ) & = ( 1 , 0 , 0 ) = e ^ { - \\lambda _ 1 z } E _ 1 , \\end{align*}"}
+{"id": "7155.png", "formula": "\\begin{align*} r _ { 1 } : q _ { 1 } = \\dfrac { 1 } { x _ 1 } , p _ { 1 } = - x _ { 1 } ^ 2 y _ { 1 } - x _ { 1 } x _ { 2 } y _ { 2 } - \\alpha _ { 1 } x _ { 1 } , q _ { 2 } = \\dfrac { x _ 2 } { x _ 1 } , p _ { 2 } = x _ { 1 } y _ { 2 } , \\end{align*}"}
+{"id": "6256.png", "formula": "\\begin{align*} i \\partial _ t v + \\partial _ x g ( u ) \\partial _ x v = N ^ { l i n } _ u ( v , \\partial _ x v ) . \\end{align*}"}
+{"id": "5063.png", "formula": "\\begin{align*} \\deg ( f ) = \\deg ( f | _ { M ' } ) . \\end{align*}"}
+{"id": "5812.png", "formula": "\\begin{align*} w _ 0 = \\prod \\limits _ { i = 1 } ^ n s _ { \\varepsilon _ i } = \\prod \\limits _ { i = 1 } ^ n s _ { \\varepsilon ' _ i } , \\end{align*}"}
+{"id": "2024.png", "formula": "\\begin{align*} [ \\dot { \\mathfrak { b } } ^ { s _ 0 } _ { p _ 0 , q _ 0 } ( \\Omega ) , \\dot { \\mathfrak { b } } ^ { s _ 1 } _ { p _ 1 , q _ 1 } ( \\Omega ) ] _ { \\theta } = \\dot { \\mathfrak { b } } ^ { s } _ { p _ \\theta , q _ \\theta } ( \\Omega ) \\end{align*}"}
+{"id": "4038.png", "formula": "\\begin{align*} - y '' ( x ) + q ( x ) y ( a ) = \\lambda y ( x ) , x \\in ( 0 , 1 ) \\end{align*}"}
+{"id": "884.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & [ \\eta ^ { \\alpha , t } - \\xi u _ { x x } - x \\eta ^ { x x } - a \\phi ^ x ] | _ { \\eqref { s y s t e m E x a m p l e } } = 0 , \\\\ & [ \\phi ^ { \\alpha , t } - \\xi v _ { x x } - x \\phi ^ { x x } - b \\eta ^ x ] | _ { \\eqref { s y s t e m E x a m p l e } } = 0 . \\end{aligned} \\right . \\end{align*}"}
+{"id": "5727.png", "formula": "\\begin{align*} T ^ { \\circ ( n ) } [ u ] ( a , x ) & \\leq C \\frac { ( 2 \\beta ) ^ { n - 1 } } { ( n - 1 ) ! } \\left ( 1 + \\int _ 0 ^ { + \\infty } \\int _ 0 ^ s z ^ { n - 1 } e ^ { - \\beta z } d z \\ , e ^ { - \\rho ( x ) s } d s \\right ) \\\\ & = C \\frac { ( 2 \\beta ) ^ { n - 1 } } { ( n - 1 ) ! } \\left ( 1 + \\frac { ( n - 1 ) ! } { \\rho ( x ) ( \\beta + \\rho ( x ) ) ^ { n } } \\right ) \\xrightarrow [ n \\rightarrow + \\infty ] { } 0 \\end{align*}"}
+{"id": "4303.png", "formula": "\\begin{align*} \\sigma _ { j _ 1 , j _ 2 } = \\sum _ { \\pi \\in \\mathcal { P } _ 2 ( 2 k _ 1 , 2 k _ 2 ) } f ^ { - } _ I ( \\pi ) + ( \\kappa - 1 ) \\sum _ { \\pi \\in \\mathcal { P } _ { 2 , 4 } ( 2 k _ 1 , 2 k _ 2 ) } ( f ^ - _ { I I } ( \\pi ) + f ^ + _ { I I } ( \\pi ) ) , \\end{align*}"}
+{"id": "5456.png", "formula": "\\begin{align*} x - \\frac { 1 } { V ' ( x ) } - \\sqrt { 2 W ( x ) } & \\leq x - \\frac { 1 } { V ' ( x ) } = \\frac { x ^ 4 - 2 } { x ( 1 + x ^ 2 ) } \\leq 0 . \\end{align*}"}
+{"id": "4043.png", "formula": "\\begin{align*} \\Delta _ { \\alpha , \\beta } ( \\rho ) = \\begin{vmatrix} C ^ { ( \\alpha ) } ( 0 , \\rho ) & S ^ { ( \\alpha ) } ( 0 , \\rho ) \\\\ C ^ { ( \\beta ) } ( 1 , \\rho ) & S ^ { ( \\beta ) } ( 1 , \\rho ) \\end{vmatrix} , \\end{align*}"}
+{"id": "7683.png", "formula": "\\begin{align*} \\varphi ( 0 ) \\leqslant \\varphi ^ * _ { \\varepsilon } ( 0 ) \\leqslant \\exp \\left \\{ - \\sum _ { n = 1 } ^ { N } \\frac { \\mathbb { P } ( S ^ * _ n > 0 ) } { n } \\right \\} \\end{align*}"}
+{"id": "3431.png", "formula": "\\begin{align*} \\sum _ { l \\geq 0 } \\dim W _ l y ^ l = \\sum _ { l \\geq 0 } h ^ 0 ( M , l L ) y ^ l . \\end{align*}"}
+{"id": "2075.png", "formula": "\\begin{align*} \\mathbf { I I I } & \\leq \\frac { C _ 6 \\Lambda } { t } \\int ^ { r _ 0 } _ 0 \\frac { \\mathrm { V o l } _ { g _ 0 } \\left ( B _ { g _ 0 } ( x _ 0 , r ) \\right ) } { \\mathrm { V o l } _ { g _ 0 } \\left ( B _ { g _ 0 } ( x _ 0 , \\sqrt { t } ) \\right ) } \\exp \\left ( - \\frac { r ^ 2 } { C _ 1 t } \\right ) r \\ , d r \\\\ & \\leq \\frac { C _ 7 r _ 0 ^ { n + 2 } \\Lambda t ^ { - 1 - \\frac 1 2 } } { { \\mathrm { V o l } _ { g _ 0 } \\left ( B _ { g _ 0 } ( x _ 0 , 1 ) \\right ) } } . \\end{align*}"}
+{"id": "7684.png", "formula": "\\begin{align*} \\varphi ( 0 ) \\leqslant \\exp \\left \\{ - \\sum _ { n = 1 } ^ { N } \\frac { \\mathbb { P } ( S _ n > 0 ) } { n } \\right \\} \\end{align*}"}
+{"id": "3727.png", "formula": "\\begin{align*} X = \\mathcal { H } ^ { \\frac { 2 } { 3 } } \\times \\mathcal { H } ^ { \\frac { 1 } { 3 } } \\times \\mathcal { H } \\end{align*}"}
+{"id": "8077.png", "formula": "\\begin{align*} \\begin{aligned} \\uppercase \\expandafter { \\romannumeral 2 } & = \\sum _ { i = - \\infty } ^ { + \\infty } \\sum _ { \\widetilde { Q } \\in B _ { i , 1 } } \\sum _ { Q \\subset \\widetilde { Q } , Q \\in B _ { i , 1 } } \\vert Q \\vert ( \\psi _ { Q } \\ast h ) ( u _ Q ) \\psi _ { Q } ( x - u _ Q ) \\\\ & = : \\sum _ { i = - \\infty } ^ { + \\infty } \\sum _ { \\widetilde { Q } \\in B _ { i , 1 } } \\lambda _ { \\widetilde { Q } } ^ { i } a _ { \\widetilde { Q } } ^ { i } ( x ) , \\end{aligned} \\end{align*}"}
+{"id": "7032.png", "formula": "\\begin{align*} \\boxed { P : = \\mathop { \\inf } _ { \\begin{array} { c } X \\in \\mathcal S ^ n \\\\ A ( X ) = b \\\\ X \\succcurlyeq 0 \\\\ \\end{array} } \\langle C , X \\rangle . } \\end{align*}"}
+{"id": "6582.png", "formula": "\\begin{align*} { { F } } ( q ^ { ( r ) } ) + R _ { [ - M ^ { r + 1 } , M ^ { r + 1 } ] ^ { b + d } \\setminus \\mathcal { S } } { { H } } R _ { [ - M ^ { r + 1 } , M ^ { r + 1 } ] ^ { b + d } \\setminus \\mathcal { S } } \\Delta _ { r + 1 } q = 0 , \\end{align*}"}
+{"id": "6351.png", "formula": "\\begin{align*} b _ \\lambda ( \\xi ) = c ( \\xi , \\xi , \\xi ) ^ \\frac 1 6 \\phi _ \\lambda ^ \\frac 2 3 . \\end{align*}"}
+{"id": "7818.png", "formula": "\\begin{align*} ( \\eta ^ i , \\widetilde { \\eta } _ i , \\kappa ) \\cdot ( \\zeta ^ i , \\widetilde { \\zeta } _ i ^ { } , \\sigma ^ { } ) = ( \\zeta ^ i + \\eta ^ i , \\widetilde { \\zeta } _ i ^ { } + \\widetilde { \\eta } _ i , \\sigma ^ { } + \\kappa + \\widetilde { \\zeta } _ i ^ { } \\eta ^ i - \\zeta ^ i \\widetilde { \\eta } _ i ) \\ , . \\end{align*}"}
+{"id": "1900.png", "formula": "\\begin{align*} \\widehat { u _ h ^ 2 } : = \\frac { 1 } { 3 } \\left ( \\left ( u _ h ^ + \\right ) ^ 2 + u _ h ^ + u _ h ^ - + \\left ( u _ h ^ - \\right ) ^ 2 \\right ) , \\end{align*}"}
+{"id": "2275.png", "formula": "\\begin{align*} \\max _ { 1 \\leq | k | \\leq n - 1 } \\left | \\sum _ { j = 1 } ^ { n } e ^ { - 2 \\pi i k x _ j } \\right | \\leq 2 0 0 0 n ^ 2 \\cdot e ^ { - \\pi \\alpha ( 2 n - 1 ) } . \\end{align*}"}
+{"id": "4645.png", "formula": "\\begin{align*} X \\cdot Z : = X \\iota ( Z ) \\qquad ( X \\in U ( \\mathfrak { g } ) , \\ , \\ , Z \\in Z ( \\mathfrak { g } ) ) \\end{align*}"}
+{"id": "6329.png", "formula": "\\begin{align*} a ( x , \\xi ) = ( P ^ y _ { \\ll \\epsilon ^ 2 } V _ 2 ) \\ , \\xi \\ , \\tilde p _ \\lambda ^ 2 ( \\xi ) + \\xi ^ 2 . \\end{align*}"}
+{"id": "7523.png", "formula": "\\begin{align*} \\mathbf { D } _ 0 : = \\nu ( 1 - \\nu ) p ' \\begin{pmatrix} 0 & 0 & 0 \\\\ - 1 & 1 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} \\quad \\textrm { a n d } \\mathbf { D } _ 1 : = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & \\nu ^ 2 & 0 \\\\ - ( 1 - \\nu ) u & 0 & 1 - \\nu \\end{pmatrix} \\ , . \\end{align*}"}
+{"id": "1232.png", "formula": "\\begin{align*} & \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| u _ n ^ J \\| _ { S ( \\R ) } \\lesssim 1 , \\\\ & \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| e ^ { i t \\Delta } ( u _ { 0 , n } - u _ { n } ^ J ( 0 ) ) \\| _ { S ( \\R ) } = 0 \\\\ & \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| \\nabla [ ( i \\partial _ t + \\Delta ) u _ n ^ J + | x | ^ { - b } ( I _ { \\alpha } \\ast | u _ n ^ J | ^ p ) | u _ n ^ J | ^ { p - 2 } u _ n ^ J ] \\| _ { N ( \\R ) } = 0 . \\end{align*}"}
+{"id": "8909.png", "formula": "\\begin{align*} & [ \\sigma ^ i ( p ' ) + \\sigma ^ i ( j ( p ' ) ) + \\sigma ^ { - i } ( p ' ) + \\sigma ^ { - i } ( j ( p ' ) ) - \\sigma ^ i ( x ' ) - \\sigma ^ i ( j ( x ' ) ) - \\sigma ^ { - i } ( x ' ) - \\sigma ^ { - i } ( j ( x ' ) ) ] = \\\\ & [ p ' _ i + j ( p ' _ { d - i } ) + p ' _ { d - i } + j ( p ' _ { i } ) - x ' _ i - j ( x ' _ { d - i } ) - x ' _ { d - i } - j ( x ' _ { i } ) ] = \\\\ & ( 1 + j ) ( [ p ' _ i + p ' _ { d - i } - x ' _ i - x ' _ { d - i } ] ) . \\end{align*}"}
+{"id": "6011.png", "formula": "\\begin{align*} \\overline { \\omega } = \\overline { \\omega } ( ( \\gamma _ { n } ) _ { n \\in N } ) = \\overline { \\lim _ { n \\rightarrow \\infty } } \\frac { \\gamma _ { n } - \\gamma _ { n + 1 } } { \\gamma _ { n + 1 } ^ { 2 } } < \\infty . \\end{align*}"}
+{"id": "5584.png", "formula": "\\begin{align*} \\mathbb { E } ( | C | ) \\leq ( 2 \\epsilon + o ( 1 ) ) g ( d - 1 ) ^ { ( 3 + \\log _ { p } ( 4 + o ( 1 ) ) - 2 + o ( 1 ) ) \\frac { g } { 4 } } \\\\ = ( 2 \\epsilon + o ( 1 ) ) g ( d - 1 ) ^ { ( 1 + o ( 1 ) ) \\frac { g } { 4 } } = o ( g ( d - 1 ) ^ { ( 1 + o ( 1 ) ) \\frac { g } { 4 } } ) \\end{align*}"}
+{"id": "3495.png", "formula": "\\begin{align*} \\norm { s } _ { y , t } ^ 2 = \\int _ { U _ { y , t } } | s | _ { h ^ { \\otimes l } } ^ 2 \\sqrt { - 1 } ^ { n ^ 2 } \\Omega _ t \\wedge \\overline { \\Omega } _ t . \\end{align*}"}
+{"id": "1383.png", "formula": "\\begin{align*} \\mathcal F _ { - 1 0 , - 1 } | T _ 2 - 2 ^ { - 1 1 } \\tau ( 2 ) \\mathcal F _ { - 1 0 , - 1 } & = 2 ^ { - 1 1 } g , \\\\ \\mathcal F _ { - 1 0 , - 1 } | T _ 3 - 3 ^ { - 1 1 } \\tau ( 3 ) \\mathcal F _ { - 1 0 , - 1 } & = 3 ^ { - 1 1 } ( j - 7 6 8 ) g . \\end{align*}"}
+{"id": "7997.png", "formula": "\\begin{align*} \\left ( \\sum _ { j _ { 1 } = 1 } ^ { \\infty } \\cdots \\sum _ { j _ { m } = 1 } ^ { \\infty } \\left \\Vert T \\left ( x _ { j _ { 1 } } ^ { \\left ( 1 \\right ) } , \\ldots x _ { j _ { m } } ^ { \\left ( m \\right ) } \\right ) \\right \\Vert \\right ) ^ { 1 / r } \\leq C \\prod _ { k = 1 } ^ { m } \\left \\Vert \\left ( x _ { j } ^ { \\left ( k \\right ) } \\right ) _ { j = 1 } ^ { \\infty } \\right \\Vert _ { s , w } \\end{align*}"}
+{"id": "4034.png", "formula": "\\begin{align*} \\binom { n } { n ^ { 0 . 7 } } \\binom { n } { n ^ { 0 . 5 + 2 \\epsilon _ 0 } } ^ r & < n ^ { n ^ { 0 . 7 } } \\Big ( n ^ { n ^ { 0 . 5 + 2 \\epsilon _ 0 } } \\Big ) ^ r \\\\ & = e ^ { n ^ { 0 . 7 } \\ln n } \\Big ( e ^ { ( n ^ { 0 . 5 + 4 \\epsilon _ 0 } ) r \\ln n } \\Big ) \\\\ & = e ^ { n ^ { 0 . 7 } \\ln n + n ^ { ( 0 . 5 + 4 \\epsilon _ 0 ) } r \\ln n } \\\\ & = e ^ { O ( n ^ { 0 . 7 } \\ln n ) } . \\end{align*}"}
+{"id": "8826.png", "formula": "\\begin{align*} - 2 \\eta _ { t } \\mathbb { E } \\big [ \\langle x ( t - 1 ) - \\mathbf { 1 } _ { n } \\otimes \\bar { x } ( t - 1 ) , g ( t ) - \\mathbf { 1 } _ { n } \\otimes \\bar { g } ( t ) \\rangle | \\mathcal { F } _ { t - 1 } \\big ] & \\le \\lambda h ( t - 1 ) + \\frac { 2 \\eta _ { t } ^ { 2 } } { \\lambda } \\sum _ { i = 1 } ^ { n } \\norm { \\nabla f _ { i } ( x ^ { i } ( t - 1 ) ) } ^ { 2 } + \\\\ & \\quad + \\frac { 2 \\eta _ { t } ^ { 2 } n } { \\lambda } ( \\kappa _ { \\beta } L ) ^ { 2 } d ^ { 2 } h _ { t } ^ { 2 ( \\beta - 1 ) } \\end{align*}"}
+{"id": "555.png", "formula": "\\begin{align*} \\Theta _ R ^ { \\mathbf u } ( t ) = \\theta _ R \\left ( \\sum _ { i = 1 } ^ n \\norm { u _ i } _ { \\widetilde X ^ { s _ i , b } _ { h _ i ( \\xi ) } ( 0 , t ) } ^ 2 \\right ) . \\end{align*}"}
+{"id": "7978.png", "formula": "\\begin{align*} \\sum _ { I _ { d _ { 1 } } } a _ { 1 , I _ { d _ { 1 } } } ( \\theta ) = 1 , \\ \\ \\forall \\theta , \\end{align*}"}
+{"id": "7494.png", "formula": "\\begin{align*} \\lambda ( \\lambda + 1 ) ^ { \\sigma ^ 2 } - \\lambda ^ \\sigma ( \\lambda + 1 ) = 0 . \\end{align*}"}
+{"id": "714.png", "formula": "\\begin{align*} \\mathfrak T _ 1 ( \\mathbf v ) - \\mathfrak T _ 1 ( \\mathbf w ) = i \\int _ S ^ t \\mathbf S ( t - \\sigma ) \\left [ \\mathbf N \\left ( \\Theta _ R ^ { [ \\mathbf u , \\mathbf v ] } ( \\sigma ) \\mathbf v ( \\sigma ) \\right ) - \\mathbf N \\left ( \\Theta _ R ^ { [ \\mathbf u , \\mathbf w ] } ( \\sigma ) \\mathbf w ( \\sigma ) \\right ) \\right ] \\ , d \\sigma \\end{align*}"}
+{"id": "5407.png", "formula": "\\begin{align*} d S ^ { - 1 } _ { j k } ( t ) = - S ^ { - 1 } _ { j k } ( t ) d U _ { j j } ( t ) - \\sum _ { 1 \\leq \\ell \\leq N : \\ell \\not = j } S ^ { - 1 } _ { \\ell k } ( t ) \\frac { ( S ^ { - 1 } ( t ) d M ( t ) S ( t ) ) _ { j \\ell } } { \\Lambda _ { \\ell } ( t ) - \\Lambda _ j ( t ) } , 1 \\leq j , k \\leq N , \\ , t \\geq 0 . \\end{align*}"}
+{"id": "5583.png", "formula": "\\begin{align*} \\mathbb { E } ( | C | ) = n 4 \\epsilon t ( \\frac { d } { 4 + o ( 1 ) } ) ^ { 1 - t } \\\\ \\leq ( 2 \\epsilon + o ( 1 ) ) g ( d - 1 ) ^ { ( \\frac { 3 } { 4 } + o ( 1 ) ) g } ( \\frac { d } { 4 + o ( 1 ) } ) ^ { 1 - t } \\end{align*}"}
+{"id": "5605.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\left | ( u _ { n , R } ^ 0 ) ' \\right | ^ p r ^ { p - 1 } \\mathrm d r & = \\int _ 0 ^ \\infty \\left | \\varphi ^ 0 _ R u _ n ' + ( \\varphi _ R ^ 0 ) ' u _ n \\right | ^ p r ^ { p - 1 } \\mathrm d r \\\\ & = \\int _ 0 ^ \\infty | u _ n ' | ^ p | \\varphi _ R ^ 0 | ^ p r ^ { p - 1 } \\mathrm d r + \\mathcal R _ { n , R } , \\end{align*}"}
+{"id": "2467.png", "formula": "\\begin{align*} \\overline { H } ( G _ 6 \\wedge G _ 8 ) = H _ \\kappa ( G _ 6 ) + H _ \\kappa ( G _ 8 ) = 2 . \\end{align*}"}
+{"id": "1537.png", "formula": "\\begin{align*} 1 \\ge \\frac p x \\sum _ { \\substack { n \\le x : \\ , p | n \\\\ R _ p ( n ) < \\beta + \\frac { t } { \\sqrt { \\log _ 2 x } } } } 1 & \\ge \\frac p x \\sum _ { \\substack { n \\le x : \\ , p | n \\\\ R _ p ( n ) < \\beta + \\frac { \\sqrt { \\log _ 3 x } } { \\sqrt { \\log _ 2 x } } } } 1 \\\\ & = \\Phi \\left ( \\sqrt { \\frac { \\log _ 3 x } { \\beta ( 1 - \\beta ) } } \\right ) + O \\left ( \\frac 1 { ( \\log _ 2 x ) ^ { 1 / 3 } } \\right ) = 1 + O \\left ( \\frac 1 { ( \\log _ 2 x ) ^ { 1 / 3 } } \\right ) \\end{align*}"}
+{"id": "134.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } W ( s ) W ^ T ( s ) T _ W X d B ( s ) = m T _ W \\tilde { X } P _ S \\frac { 1 } { m } T _ W W ( t ) = T _ W \\tilde { X } P _ S T _ W W ( t ) \\end{align*}"}
+{"id": "8540.png", "formula": "\\begin{align*} | D \\ell ^ { k } | ( J ) & = \\sum _ { i = 0 } ^ { N _ { k } } | \\ell ( z _ { i + 1 } ^ { k } ) - \\ell ( z _ { i } ^ { k } ) | \\\\ & = | \\ell ( z _ { 1 } ^ { k } ) - \\ell ( a ) | + \\sum _ { i = 1 } ^ { N _ { k } - 1 } | \\ell ( z _ { i + 1 } ^ { k } ) - \\ell ( z _ { i } ^ { k } ) | + | \\ell ( b ) - \\ell ( z _ { N _ { k } } ^ { k } ) | . \\end{align*}"}
+{"id": "8681.png", "formula": "\\begin{align*} F _ 2 \\Big ( \\hbar \\partial _ { x _ 0 } - \\frac { n \\hbar } { 2 x _ 0 } + Q _ 0 , \\hbar \\partial _ { x _ 1 } + Q _ 1 , . . . , \\hbar \\partial _ { x _ n } + Q _ n , y _ { n + 1 } \\Big ) \\cdot g \\Big ( \\frac { x _ 1 } { x _ 0 } , . . . , \\frac { x _ n } { x _ 0 } \\Big ) = 0 \\end{align*}"}
+{"id": "3543.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { N } } ( \\alpha u + \\beta v ) w d x & \\leqslant 6 C K ^ { 2 } r _ { 0 } ^ { 1 - N } + 6 C K + 6 C K ^ { 2 } + C K ^ { 2 } \\\\ & + 6 C K r _ { 0 } ^ { \\frac { 1 - N } { 2 } } \\Vert f \\Vert _ { 2 } + 6 C r _ { 0 } \\Vert f \\Vert _ { 2 } ^ { 2 } + C K ^ { \\frac { 4 } { N + 4 } } \\Vert f \\Vert _ { 2 } ^ { \\frac { 2 N + 4 } { N + 4 } } \\\\ & + 6 C \\sqrt { K } \\Vert g _ { 1 } \\Vert _ { 2 } + 6 C \\sqrt { K } \\Vert g _ { 2 } \\Vert _ { 2 } . \\end{aligned} \\end{align*}"}
+{"id": "4729.png", "formula": "\\begin{align*} \\begin{cases} F ( D ^ { 2 } u _ 0 ) = 1 & \\mathbb { R } _ + ^ n \\cap \\Omega _ 0 , \\\\ | \\nabla u _ 0 | = 0 & \\mathbb { R } _ + ^ n \\backslash \\Omega _ 0 , \\\\ u _ 0 = 0 & \\mathbb { R } _ + ^ { n - 1 } . \\end{cases} \\end{align*}"}
+{"id": "6361.png", "formula": "\\begin{align*} U _ n ( t ) = \\sum _ { \\lambda _ \\nu \\leq n } \\big ( e ^ { i t \\sqrt { \\lambda _ \\nu } } u _ \\nu ^ + + e ^ { - i t \\sqrt { \\lambda _ \\nu } } u _ \\nu ^ - \\big ) e _ \\nu . \\end{align*}"}
+{"id": "5974.png", "formula": "\\begin{align*} \\sigma _ { - 2 } ( N ^ \\omega _ { \\partial M } ) ( t , \\xi ' ) = - \\frac { \\chi ( \\xi ' ) } { \\sigma _ 1 ( \\Lambda _ { g , F } ^ \\omega ) ( t , \\xi ' ) } \\left ( p _ { - 1 } ( t , \\xi ' ) \\sigma _ 0 ( \\Lambda _ { g , F } ^ \\omega ) ( t , \\xi ' ) + \\nabla _ { \\xi ' } p _ { - 1 } \\cdot D _ { t ' } \\sigma _ 1 ( \\Lambda _ { g , F } ^ \\omega ) \\right ) . \\end{align*}"}
+{"id": "3447.png", "formula": "\\begin{align*} \\bar { \\Delta } ^ \\vee = ( ( 0 , \\ldots - \\frac { 1 } { d _ i } , \\ldots 0 ) , i = 0 , 1 , \\ldots m ) \\subset \\R ^ { m + 1 } / \\R ( 1 , 1 , \\ldots 1 ) . \\end{align*}"}
+{"id": "1098.png", "formula": "\\begin{align*} \\mu _ 0 : = \\min _ { x \\in D } \\mu ( x ) > 0 , \\end{align*}"}
+{"id": "3801.png", "formula": "\\begin{align*} H ( x _ 0 , s _ 0 , x _ 1 , s _ 1 ) = \\begin{cases} a c ( x _ 0 , x _ 1 ) , \\quad & s _ 0 = s _ 1 = a , \\\\ + \\infty , \\quad & \\end{cases} \\end{align*}"}
+{"id": "4055.png", "formula": "\\begin{align*} \\rho _ { 1 , 0 , n } = \\left ( n - \\frac { 1 } { 2 } \\right ) \\pi + \\frac { ( - 1 ) ^ { n + 1 } q _ 1 ( 1 ) } { n \\pi } \\cos \\left ( n - \\frac { 1 } { 2 } \\right ) \\pi a + \\frac { \\kappa _ { 1 , 0 , n } } { n } , \\ , \\ , \\{ \\kappa _ { 1 , 0 , n } \\} \\in l _ 2 , n \\in \\mathbb { Z } . \\end{align*}"}
+{"id": "3404.png", "formula": "\\begin{align*} \\displaystyle V _ j = - \\frac { v _ j ( \\delta _ i v _ i ^ 2 - \\delta _ k v _ k ^ 2 ) } { v _ i ( \\delta _ j v _ j ^ 2 - \\delta _ k v _ k ^ 2 ) } V _ i . \\end{align*}"}
+{"id": "2050.png", "formula": "\\begin{align*} b & = u - a \\in ( { \\mathrm { B } } ^ { s } _ { p , q } ( \\Omega ) + \\mathrm { L } ^ p ( \\Omega ) ) \\cap \\dot { \\mathrm { H } } ^ { 1 , p } ( \\Omega ) \\subset { \\mathrm { H } } ^ { 1 , p } ( \\Omega ) \\end{align*}"}
+{"id": "6846.png", "formula": "\\begin{align*} B _ j & = - K _ 4 + 2 \\lambda _ j \\\\ C _ j & = \\lambda _ j J _ 3 \\\\ D _ j & = K _ 2 + 2 J _ 2 \\lambda _ j \\\\ E _ j & = \\lambda _ j J _ 1 \\\\ F _ { \\varepsilon , j } ( t ) & = - K _ 0 + J _ 0 \\lambda _ j - L _ 0 \\lambda _ j ^ 2 + \\lambda _ j ^ 3 - \\hat { c } _ n v _ { \\varepsilon } ^ { \\frac { 1 2 } { n - 6 } } ( t ) \\end{align*}"}
+{"id": "8073.png", "formula": "\\begin{align*} \\Omega _ { i , k } = \\left \\{ x \\in \\mathbb { R } ^ { n } \\colon S ^ { k } ( h ) ( x ) > 2 ^ i \\right \\} \\end{align*}"}
+{"id": "2973.png", "formula": "\\begin{align*} & \\Big \\| \\sum _ { i = 1 } ^ M \\sum _ { j = 1 } ^ N \\Theta _ { x _ i \\otimes y _ j , x _ i \\otimes y _ j } T - T \\Big \\| \\\\ & \\le \\Big \\| \\sum _ { i = 1 } ^ M \\sum _ { j = 1 } ^ N \\Theta _ { x _ i \\otimes y _ j , x _ i \\otimes y _ j } \\Big \\| \\| T - S \\| + \\Big \\| \\sum _ { i = 1 } ^ M \\sum _ { j = 1 } ^ N \\Theta _ { x _ i \\otimes y _ j , x _ i \\otimes y _ j } S - S \\Big \\| + \\| T - S \\| < \\varepsilon . \\end{align*}"}
+{"id": "8330.png", "formula": "\\begin{align*} \\alpha _ k ^ i = \\frac { N _ i ( k ) } { N _ i } \\end{align*}"}
+{"id": "5325.png", "formula": "\\begin{align*} | g ( x ) - x ^ 2 | & = | 2 ^ { - n } ( 2 k + 1 ) ( x - k 2 ^ { - n } ) + k ^ 2 4 ^ { - n } - x ^ 2 | \\\\ & = | 2 ^ { - n } ( 2 k + 1 ) \\delta + k ^ 2 4 ^ { - n } - ( k 2 ^ { - n } + \\delta ) ^ 2 | \\\\ & = | \\delta ( 2 ^ { - n } - \\delta ) | \\\\ & \\leq 4 ^ { - n } . \\end{align*}"}
+{"id": "6705.png", "formula": "\\begin{align*} e ^ { H ( X ) } = \\sum _ { n = 0 } ^ \\infty \\frac { H ( X ) ^ n } { n ! } , e ^ { H ^ \\ast ( X ) } = \\sum _ { n = 0 } ^ \\infty \\frac { H ^ \\ast ( X ) ^ n } { n ! } \\end{align*}"}
+{"id": "1039.png", "formula": "\\begin{align*} \\Rightarrow \\langle A _ p ( \\hat { u } ) , h \\rangle + \\langle A _ q ( \\hat { u } ) , h \\rangle + \\hat { \\lambda } \\int _ \\Omega | \\hat { u } | ^ { p ( z ) - 2 } \\hat { u } h d z = \\int _ \\Omega \\hat { f } ( z , \\hat { u } ) h d z \\end{align*}"}
+{"id": "8894.png", "formula": "\\begin{align*} F _ T = ( E T \\times F ) / T = ( E G \\times F ) / T \\end{align*}"}
+{"id": "3533.png", "formula": "\\begin{align*} \\begin{aligned} u _ { t } & = \\Delta u - \\chi \\nabla \\cdot ( u \\nabla w ) , & x \\in \\mathbb { R } ^ { N } , t > 0 , \\\\ v _ { t } & = \\Delta v - \\xi \\nabla \\cdot ( v \\nabla w ) , & x \\in \\mathbb { R } ^ { N } , t > 0 , \\\\ w _ { t } & = \\Delta w - \\lambda w + \\alpha u + \\beta v , & x \\in \\mathbb { R } ^ { N } , t > 0 , \\end{aligned} \\end{align*}"}
+{"id": "1995.png", "formula": "\\begin{align*} \\left \\lVert { u } \\right \\rVert _ { \\dot { \\mathrm { H } } ^ { s , p } ( \\mathbb { R } ^ n ) } = \\sup \\limits _ { \\substack { v \\in \\mathcal { V } ^ { - s , p ' } } } \\big \\lvert \\big \\langle u , v \\big \\rangle _ { \\mathbb { R } ^ n } \\big \\rvert \\end{align*}"}
+{"id": "7003.png", "formula": "\\begin{align*} K \\left ( e , \\bar { e } , e , n \\right ) + K \\left ( n , \\bar { n } , e , n \\right ) = W \\left ( e , \\bar { e } , e , n \\right ) + W \\left ( n , \\bar { n } , e , n \\right ) = 2 W \\left ( e , \\bar { e } , e , n \\right ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\end{align*}"}
+{"id": "5172.png", "formula": "\\begin{align*} & \\lim _ { \\delta \\to 0 } \\frac { { \\rm A o I } ( S _ { \\mathrm { z } } , Q _ { \\mathrm { u n i } } , F _ \\mathrm { s } ) } { \\log _ 2 D } \\\\ = & \\lim _ { \\delta \\to 0 } \\frac { { \\rm A o I } ( S _ { \\mathrm { z } } , Q _ { \\mathrm { u n i } } , F _ \\mathrm { s } ) } { H [ Q _ { \\mathrm { u n i } } ( X ) ] } \\frac { H [ Q _ { \\mathrm { u n i } } ( X ) ] } { \\log _ 2 D } \\\\ = & - \\frac { 3 } { 4 } . \\end{align*}"}
+{"id": "1615.png", "formula": "\\begin{align*} \\deg _ { { \\mathcal Y } \\to y _ { \\ell } } ( v ) : = \\frac { | \\{ e \\in E ( \\mathcal A _ { \\mathcal Y } ) : v \\in e \\} | } { \\prod _ { j \\in [ k ] \\setminus \\{ \\ell \\} } | \\mathcal P _ { \\mathcal Y \\setminus \\{ y _ { j } \\} } | } \\end{align*}"}
+{"id": "191.png", "formula": "\\begin{align*} R ( M , P _ i , \\tau ) = & \\left \\{ e ^ { \\frac { 1 } { 2 4 } E _ 2 ( \\tau ) A _ 1 } \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) { \\rm c h } \\left [ \\bigotimes _ { n = 1 } ^ { \\infty } S _ { q ^ n } ( \\widetilde { T _ C M } ) \\otimes \\bigotimes _ { m = 1 } ^ { \\infty } \\wedge _ { - q ^ m } ( \\widetilde { L _ C } ) \\right ] \\right . \\\\ & \\left . \\cdot \\varphi ( \\tau ) ^ { 8 } { \\rm c h } ( \\mathcal { V } _ i ) \\right \\} ^ { ( 1 0 ) } . \\end{align*}"}
+{"id": "3820.png", "formula": "\\begin{align*} \\theta ( x _ 0 , s _ 0 , x _ 1 , s _ 1 , S ) = \\frac { 1 } { s _ * } ( s _ 0 ^ p + s _ 1 ^ p + S ^ p ) ^ { \\frac 1 p } , { \\rm p r d } _ { \\theta } ( x _ 0 , s _ 0 , x _ 1 , s _ 1 , S ) : = \\left ( x _ 0 , \\frac { s _ 0 } { \\theta } , x _ 1 , \\frac { s _ 1 } { \\theta } , \\frac { S } { \\theta } \\right ) . \\end{align*}"}
+{"id": "2118.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t ^ 2 v - \\partial _ x ^ 2 v = G ( t , x ) \\\\ v ( 0 , x ) = \\ln ( f ( 0 , x ) ) = \\ln ( f _ 0 + c _ 1 ) \\\\ \\partial _ t v ( t , x ) | _ { \\{ t = 0 \\} } = \\frac { f _ 1 } { c _ 1 + f _ 0 } . \\end{cases} \\end{align*}"}
+{"id": "2141.png", "formula": "\\begin{align*} d s ^ 2 = f _ 0 ( t , x ) ( d x ^ 2 - d t ^ 2 ) + \\alpha e ^ { u _ 0 } d y ^ 2 + \\alpha e ^ { - u _ 0 } d z ^ 2 , \\end{align*}"}
+{"id": "3320.png", "formula": "\\begin{align*} u _ { i , j , k } = \\sum _ { u = 1 } ^ m \\sum _ { v = 1 } ^ m \\sum _ { w = 1 } ^ m a _ { i , u } b _ { j , v } c _ { k , w } t _ { u , v , w } . \\end{align*}"}
+{"id": "1210.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } \\{ \\| \\nabla u _ n \\| _ { L ^ 2 } ^ 2 - \\sum _ { j = 1 } ^ J \\| \\nabla \\phi ^ j \\| _ { L ^ 2 } ^ 2 - \\| \\nabla w _ n ^ J \\| _ { L ^ 2 } ^ 2 \\} = 0 \\\\ & \\lim _ { n \\to \\infty } \\{ P ( u _ n ) - \\sum _ { j = 1 } ^ J P ( g _ n ^ j [ e ^ { i t _ n ^ j \\Delta } \\phi ^ j ] ) - P ( w _ n ^ J ) \\} = 0 . \\end{align*}"}
+{"id": "4045.png", "formula": "\\begin{align*} \\Delta _ { \\alpha , \\alpha } ( \\rho ) = \\rho ^ { 2 \\alpha - 2 } \\left ( \\rho \\sin \\rho - W _ { \\alpha , \\alpha } ( a , \\rho ) + \\int \\limits _ { 0 } ^ { 1 } ( U _ { \\alpha , \\alpha } ( t ) \\cos \\rho t + V _ { \\alpha , \\alpha } ( t ) \\sin \\rho t ) \\ , d t \\right ) , \\end{align*}"}
+{"id": "2238.png", "formula": "\\begin{align*} u _ r = u ( x ) \\ast \\rho _ r ( x ) . \\end{align*}"}
+{"id": "4899.png", "formula": "\\begin{align*} A _ 0 ^ { ( 2 ) } ( z ) : = \\left \\{ \\begin{array} { l l } \\exp [ - \\frac { z } { x _ 1 } P _ 1 ] [ ( z - x _ 1 - z P _ 1 ) [ \\frac { A ( z ) - A ( x _ 1 ) } { z - x _ 1 } ] + A ( x _ 1 ) ] , z \\neq x _ 1 \\\\ \\exp ( - P _ 1 ) [ A ( x _ 1 ) - x _ 1 P _ 1 A ' ( x _ 1 ) ] , z = x _ 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "8260.png", "formula": "\\begin{align*} g _ { \\zeta , 0 , 0 } ( 1 , \\Theta ) = \\sum _ { \\nu \\geq 0 } h _ \\nu ( \\Theta ) \\zeta ^ \\nu , \\end{align*}"}
+{"id": "5427.png", "formula": "\\begin{align*} \\Lambda _ j ^ s ( \\xi ) : = \\sum _ { a / q \\in \\Sigma _ s } \\big ( \\Theta _ { N _ j } - \\Theta _ { N _ { j - 1 } } \\big ) ( \\xi - a / q ) \\eta _ { j ^ \\tau } ( \\xi - a / q ) . \\end{align*}"}
+{"id": "8821.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\mathbb { E } [ \\norm { g ^ { i } ( t ) - \\bar { g } ( t ) } ^ { 2 } | \\mathcal { F } _ { t - 1 } ] & \\leq \\sum _ { i = 1 } ^ { n } \\mathbb { E } [ \\norm { g ^ { i } } ^ { 2 } | \\mathcal { F } _ { t - 1 } ] \\\\ & \\leq \\kappa n ( C ^ { * } G ^ { 2 } d + \\frac { 3 d ^ { 2 } \\sigma ^ { 2 } } { 2 h _ { t } ^ { 2 } } ) , \\end{align*}"}
+{"id": "2710.png", "formula": "\\begin{align*} \\begin{array} { r c c l } X _ { \\operatorname { G } } ( m , n ) & : = & \\begin{cases} \\{ \\mu \\} & m = 2 , n = 1 \\\\ \\{ \\eta \\} & m = 0 , n = 1 \\\\ \\{ \\omega \\} & m = 1 , n = 1 \\\\ \\emptyset & . \\end{cases} \\end{array} \\end{align*}"}
+{"id": "7525.png", "formula": "\\begin{align*} \\lambda = \\lambda _ 0 = u , \\lambda _ \\pm = u \\pm \\sqrt { n p ' / r } . \\end{align*}"}
+{"id": "4865.png", "formula": "\\begin{align*} & f ( u ) E _ { i } + f ( u + v ) E _ { i + 1 } + f ( v ) E _ { i } + f ( u ) f ( v ) \\delta E _ { i } + f ( u ) f ( u + v ) f ( v ) E _ { i } \\\\ = & f ( v ) E _ { i + 1 } + f ( u + v ) E _ { i } + f ( u ) E _ { i + 1 } + f ( v ) f ( u ) \\delta E _ { i + 1 } + f ( v ) f ( u + v ) f ( u ) E _ { i + 1 } \\end{align*}"}
+{"id": "1538.png", "formula": "\\begin{align*} \\frac 1 x \\sum _ { n \\le x } \\log P ^ { \\left ( \\frac 1 2 \\right ) } ( n ) = \\frac 1 x \\sum _ { p \\le x } M ^ { \\left ( \\frac 1 2 \\right ) } _ p ( x ) \\log p . \\end{align*}"}
+{"id": "5350.png", "formula": "\\begin{align*} \\chi ( N _ Z ( K _ Z - ( k - 1 ) H ) ) = \\chi ( N _ Y ( K _ Y + D - ( k - 1 ) H ) ) = - s . \\end{align*}"}
+{"id": "7578.png", "formula": "\\begin{align*} \\begin{aligned} J _ 2 ( \\beta , N , T ) & \\leq C N ^ 2 \\int _ { 0 } ^ { \\pi } \\iint _ { 0 \\le s \\le t \\le T } \\frac { 1 } { t + s } \\\\ & \\times \\int _ { \\mathbf { B } _ 2 ( 0 ) } \\exp \\left ( - \\frac { \\big | z - D _ t ^ { ( k ) } + D _ s ^ { ( \\ell ) } \\big | ^ 2 } { 2 ( t + s ) } \\right ) d z d s d t d \\theta . \\end{aligned} \\end{align*}"}
+{"id": "5934.png", "formula": "\\begin{align*} A _ { i } \\le \\dfrac { R _ { i + 1 } - S _ { i } } { 2 } + e = \\dfrac { - 2 e - 0 } { 2 } + e = 0 \\le d [ a _ { 1 , i } b _ { 1 , i } ] . \\end{align*}"}
+{"id": "588.png", "formula": "\\begin{align*} \\Delta _ { 2 , 1 } ^ \\mu ( t ) & = i P _ \\mu \\int _ 0 ^ t \\mathbf S ( t - s ) \\left [ \\mathbf N \\left ( \\Theta _ R ^ { \\mathbf u ^ \\mu } ( s ) P _ \\mu \\mathbf u ^ \\mu ( s ) \\right ) - \\mathbf N \\left ( \\Theta _ R ^ { \\mathbf u } ( s ) \\mathbf u ( s ) \\right ) \\right ] \\ , d s , \\\\ \\Delta _ { 2 , 2 } ^ \\mu ( t ) & = i ( P _ \\mu - 1 ) \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf N \\left ( \\Theta _ R ^ { \\mathbf u } ( s ) \\mathbf u ( s ) \\right ) \\ , d s . \\end{align*}"}
+{"id": "1658.png", "formula": "\\begin{align*} \\Lambda ^ { ( n ) } _ { \\texttt { a } } : = \\{ l _ 1 \\omega _ 1 + \\cdots + l _ { n - 1 } \\omega _ { n - 1 } \\mid l _ 1 , \\ldots , l _ { n - 1 } \\in \\mathbb { Z } _ { \\geq 0 } \\} \\end{align*}"}
+{"id": "8942.png", "formula": "\\begin{align*} D ( t _ 1 , \\ldots , t _ r ) : = \\min \\{ t _ i , | t _ i - t _ j | : 1 \\leq i \\neq j \\leq r \\} . \\end{align*}"}
+{"id": "1633.png", "formula": "\\begin{align*} \\tilde { \\iota } _ a ( ( V _ X ) _ { X \\in \\mathbb { Z } } ) _ { Q ' , A ' } & = \\begin{cases} V _ X \\{ X \\} _ { T ' } , & ( Q ' , A ' ) = \\tilde { \\varphi } _ a ( X ) , \\\\ 0 , & , \\end{cases} \\\\ \\iota _ a ( ( V _ X ) _ { X \\in \\mathbb { Z } } ) _ { Q ' , A ' } & = \\begin{cases} V _ X \\{ X \\} _ { T ' } , & ( Q ' , A ' ) = \\varphi ( X ) , \\\\ 0 , & . \\end{cases} \\end{align*}"}
+{"id": "8880.png", "formula": "\\begin{align*} H ^ * ( G / T ) = \\frac { R } { ( R _ + ^ W ) } = \\frac { \\R [ y _ 1 , \\ldots , y _ { \\ell } ] } { ( \\R [ y _ 1 , \\ldots , y _ { \\ell } ] _ + ^ W ) } . \\end{align*}"}
+{"id": "4127.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { k - 1 } c _ 2 ( s , i ) \\cdot D ( k - 2 i , - k + 2 s - \\tfrac { 1 } { 2 } ) = 0 . \\end{align*}"}
+{"id": "3708.png", "formula": "\\begin{align*} g ' ( c ) = \\frac 1 { ( 1 - 2 \\alpha ) ( c ^ 2 + 1 ) ^ \\alpha } \\left ( 1 - \\frac { 2 \\alpha c ^ 2 } { c ^ 2 + 1 } - ( 1 - 2 \\alpha ) \\right ) = \\frac { 2 \\alpha } { ( 1 - 2 \\alpha ) ( c ^ 2 + 1 ) ^ { 1 + \\alpha } } \\ge 0 , \\end{align*}"}
+{"id": "5014.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ M R u \\ , d g _ t \\geq \\int _ M \\big ( ( \\triangle R ) u + \\tfrac 2 n R ^ 2 u - R ( \\triangle u ) + ( 1 - \\tfrac 2 n ) R ^ 2 u - R ^ 2 u \\big ) d g _ t = 0 . \\end{align*}"}
+{"id": "819.png", "formula": "\\begin{align*} U = \\{ z \\in \\overline { \\mathbb { D } } : | z - e ^ { i \\theta _ 0 } | < \\delta _ 2 \\} . \\end{align*}"}
+{"id": "4918.png", "formula": "\\begin{align*} L _ \\ell : = \\sum _ { m , n \\geq 1 } \\dfrac { ( - 1 ) ^ m } { n ^ 2 + 2 ^ \\ell m ^ 2 } . \\end{align*}"}
+{"id": "7118.png", "formula": "\\begin{align*} \\rho = \\Theta + \\theta \\end{align*}"}
+{"id": "2709.png", "formula": "\\begin{align*} \\begin{array} { r c l } [ x ] \\boxtimes [ x ' ] & : = & [ x \\boxtimes x ' ] \\end{array} \\end{align*}"}
+{"id": "2465.png", "formula": "\\begin{align*} \\overline { H } \\big ( G \\wedge G _ 5 ) = \\overline { H } ( G ) + \\overline { H } ( G _ 5 ) = H _ \\kappa ( G ) + \\textstyle \\frac { 1 } { 2 } \\log 5 . \\end{align*}"}
+{"id": "5553.png", "formula": "\\begin{align*} \\sigma _ k ( \\zeta ) = \\Re g _ k ( \\zeta ) \\end{align*}"}
+{"id": "5079.png", "formula": "\\begin{align*} \\liminf _ { p \\to \\infty } m ( p ) ^ { 1 / p } = + \\infty \\end{align*}"}
+{"id": "6568.png", "formula": "\\begin{align*} \\tilde G _ { \\Lambda } ( \\sigma ) = \\bigoplus _ { k \\in \\Pi _ b \\Lambda } B _ k ^ { - 1 } , \\end{align*}"}
+{"id": "8186.png", "formula": "\\begin{align*} & A _ r ( N , p ) = \\frac { ( r - N + p ) ( p - r ) ( r + 1 ) } { ( 2 r - N ) ( N - 2 r - 1 ) } , \\\\ & B _ r ( N , p ) = - \\frac { r ^ 2 N - r N ( N + 1 ) + ( 2 + N ) ( N - p ) p } { ( 2 r - N - 2 ) ( 2 r - N ) } , \\\\ & C _ r ( N , p ) = - \\frac { ( r - N - 1 + p ) ( r - p - 1 ) ( N - r + 2 ) } { ( 2 r - N - 2 ) ( N - 2 r + 3 ) } . \\end{align*}"}
+{"id": "4851.png", "formula": "\\begin{align*} \\xi E _ i + \\xi ^ 2 E _ i ^ 2 + \\xi ^ 3 E _ i E _ { i + 1 } E _ i = \\xi E _ { i + 1 } + \\xi ^ 2 E _ { i + 1 } ^ 2 + \\xi ^ 3 E _ { i + 1 } E _ i E _ { i + 1 } \\end{align*}"}
+{"id": "4226.png", "formula": "\\begin{align*} d T = 2 \\pi ^ 2 \\alpha ' \\big ( p _ 1 ( \\nabla ) - p _ 1 ( A ) \\big ) . \\end{align*}"}
+{"id": "4414.png", "formula": "\\begin{align*} \\mathcal { A } _ 0 ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) \\partial _ t { \\mathbf V } + \\mathcal { A } _ 1 ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) \\partial _ 1 { \\mathbf V } + \\mathcal { A } _ 2 ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) \\partial _ 2 { \\mathbf V } + \\mathcal { A } _ 3 ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) { \\mathbf V } = \\mathcal { F } ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) , \\end{align*}"}
+{"id": "5473.png", "formula": "\\begin{align*} p _ 1 , \\dots , p _ n \\ ; \\in \\ ; \\mathbb D ^ \\times : = \\mathbb D \\backslash \\{ 0 \\} = \\big \\{ w \\in \\mathbb C \\ ; \\big | \\ ; 0 < | w | < 1 \\big \\} \\end{align*}"}
+{"id": "6629.png", "formula": "\\begin{align*} \\overline { h _ { j } ( \\beta , p , q ) } = ( - 1 ) ^ { j + 1 } h _ { j } ( \\beta , p , q ) \\overline { \\tilde { h } _ { j } ( \\beta , p , q ) } = ( - 1 ) ^ j \\tilde { h } _ { j } ( \\beta , p , q ) . \\end{align*}"}
+{"id": "5493.png", "formula": "\\begin{align*} O _ A ^ { ( 1 ) } S ( 1 / s , p ) = S ( 1 / s , p ) O _ B ^ { ( 1 ) } ( p , s ) , \\end{align*}"}
+{"id": "6800.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow - \\infty } \\frac { w ( z ) } { u ( z ) - 1 } = \\lambda _ 1 . \\end{align*}"}
+{"id": "5839.png", "formula": "\\begin{align*} w _ 1 = s _ { \\alpha _ { m a x } } . \\end{align*}"}
+{"id": "6200.png", "formula": "\\begin{align*} e ^ i = ( - 1 ) ^ { n + i + 1 } [ e _ 1 , \\dots , \\hat { e } _ i , \\dots , e _ n ] , i = 1 , \\dots , n + 1 . \\end{align*}"}
+{"id": "7383.png", "formula": "\\begin{align*} \\pi _ { n } ' ( \\rho _ { n } ) : = \\rho _ { n } p ' _ { n } ( \\rho _ { n } ) = \\gamma _ { n } \\rho _ { n } ^ { \\gamma _ { n } } , \\end{align*}"}
+{"id": "7071.png", "formula": "\\begin{align*} G _ { 0 } ( y ) : = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\sigma ) } ( \\pi ^ 3 y ) ^ { - s } \\ , \\prod _ { i = 1 } ^ { 3 } \\frac { \\Gamma \\left ( \\frac { 1 + s + { \\bf \\alpha } _ { i } } { 2 } \\right ) } { \\Gamma \\left ( \\frac { - s - { \\bf \\alpha } _ { i } } { 2 } \\right ) } \\tilde { v _ 1 } ( - s ) d s . \\end{align*}"}
+{"id": "6776.png", "formula": "\\begin{align*} & \\phi '' _ \\infty ( z ) - c \\phi ' _ \\infty ( z ) + \\phi _ \\infty ( z ) F ( \\phi _ \\infty , \\psi _ \\infty ) ( z ) = 0 , \\\\ [ 0 . 2 c m ] & d \\psi '' _ \\infty ( z ) - c \\psi ' _ \\infty ( z ) + \\psi _ \\infty ( z ) G ( \\phi _ \\infty , \\psi _ \\infty ) ( z ) = 0 . \\end{align*}"}
+{"id": "3466.png", "formula": "\\begin{align*} u ( x ) = \\sup _ { p \\in \\Delta ^ \\vee _ k } \\langle p , x \\rangle - u ^ * ( p ) . \\end{align*}"}
+{"id": "8078.png", "formula": "\\begin{align*} a _ { \\widetilde { Q } } ^ { i } ( x ) : = \\frac { 1 } { \\lambda _ { \\widetilde { Q } } ^ { i } } \\sum _ { Q \\subset \\widetilde { Q } } \\vert Q \\vert ( \\psi _ { Q } \\ast h ) ( u _ Q ) \\psi _ { Q } ( x - u _ Q ) \\end{align*}"}
+{"id": "7744.png", "formula": "\\begin{align*} 1 + \\delta - r < f ( z ) = \\alpha f ( x ) + ( \\beta - \\gamma ) ( 1 + \\delta ) \\leq 1 + \\beta \\delta . \\end{align*}"}
+{"id": "3697.png", "formula": "\\begin{align*} T _ { n _ 0 } = O \\left ( 2 ^ { - 4 n _ 0 \\beta / ( 1 - \\beta ) } a _ { n _ 0 } \\left ( 2 ^ { - 4 n _ 0 ( 1 - 2 \\alpha ) / ( 1 - \\beta ) } n _ 0 ^ { - 2 } { a _ { n _ 0 } } \\right ) ^ { - 1 } \\right ) = O \\big ( { n _ 0 ^ 2 } 2 ^ { - 4 n _ 0 ( \\beta + 2 \\alpha - 1 ) / ( 1 - \\beta ) } \\big ) \\to 0 \\end{align*}"}
+{"id": "4220.png", "formula": "\\begin{align*} \\rho = \\ , & \\ , a _ 1 e ^ { 1 2 3 } + a _ 2 e ^ { 1 2 5 } + a _ 3 e ^ { 1 2 6 } + a _ 4 ( e ^ { 1 3 5 } - e ^ { 1 2 4 } ) + a _ 5 e ^ { 1 3 6 } + a _ 6 e ^ { 2 3 4 } + a _ 7 ( e ^ { 1 3 4 } + e ^ { 2 3 5 } ) + a _ 8 e ^ { 2 3 6 } \\\\ [ 2 p t ] & + a _ 9 ( e ^ { 2 5 6 } - e ^ { 1 4 6 } ) + a _ { 1 0 } ( e ^ { 2 4 6 } + e ^ { 3 5 6 } ) + a _ { 1 1 } e ^ { 4 5 6 } , \\end{align*}"}
+{"id": "531.png", "formula": "\\begin{align*} \\tau < \\infty \\implies \\limsup _ { t \\nearrow \\tau } \\norm { ( \\psi _ + , \\psi _ - , \\phi _ + ) } _ { \\mathbf X ^ { s , r , b } ( 0 , t ) } = \\infty , \\end{align*}"}
+{"id": "6302.png", "formula": "\\begin{align*} g _ { [ < \\lambda ] } - g _ { [ < \\mu ] } ^ { x _ 0 } = g _ { [ < \\lambda ] } - g _ { [ < \\mu ] } + g _ { [ < \\mu ] } - g _ { [ < \\mu ] } ^ { x _ 0 } . \\end{align*}"}
+{"id": "1132.png", "formula": "\\begin{align*} & f \\circ [ ~ , ~ ] _ 1 = [ ~ , ~ ] ^ { ' } _ 1 \\circ ( f \\otimes f ) ; \\\\ & f \\circ [ ~ , ~ ] _ 2 = [ ~ , ~ ] ^ { ' } _ 2 \\circ ( f \\otimes f ) . \\end{align*}"}
+{"id": "8094.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j = N } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } + \\left \\| \\sum \\limits _ { j = N } ^ { \\infty } \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } \\to 0 , \\ a s \\ N \\to \\infty . \\end{align*}"}
+{"id": "3566.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ( P ) & = \\frac 1 2 \\widehat { g } ( z _ P , z _ P ) = \\frac 1 2 p ^ t \\cdot G \\cdot p , \\\\ \\widehat { g } ( z _ P , z _ Q ) & = 4 \\delta \\left ( \\frac { P + Q } { 2 } \\right ) - \\delta ( P ) - \\delta ( Q ) = p ^ t \\cdot G \\cdot q , \\\\ G _ { i j } & = \\widehat { g } ( z _ { R _ i } , z _ { R _ j } ) = 4 \\delta \\left ( \\frac { R _ i + R _ j } { 2 } \\right ) - \\delta ( R _ i ) - \\delta ( R _ j ) , \\end{aligned} \\end{align*}"}
+{"id": "8737.png", "formula": "\\begin{align*} \\gamma = h _ { t } ^ { 2 \\beta } , \\quad \\quad \\gamma = \\frac { h _ { t } } { \\sqrt { T d } } . \\end{align*}"}
+{"id": "6057.png", "formula": "\\begin{align*} \\beta _ f ( { s } ) = \\int _ B ( 1 - e ^ { - { s } f ( x ) } ) \\mathcal { M } ( d x ) a n d I _ { t } ^ p ( f ) = \\int _ 0 ^ \\infty { s } ^ { p - 1 } e ^ { - t \\beta _ f ( { s } ) } d { s } . \\end{align*}"}
+{"id": "8386.png", "formula": "\\begin{align*} N = \\left [ m _ { 1 } ^ { c } \\right ] + \\left [ m _ { 2 } ^ { c } \\right ] + \\cdots + \\left [ m _ { s } ^ { c } \\right ] \\end{align*}"}
+{"id": "1604.png", "formula": "\\begin{align*} | \\{ P ^ { ( i ) } \\in \\mathcal { P } ^ { ( i ) } : P ^ { ( i ) } \\subseteq \\mathcal { K } _ i ( \\hat { P } ^ { ( i - 1 ) } ) \\} | = a _ i \\end{align*}"}
+{"id": "1155.png", "formula": "\\begin{align*} & \\sum _ { i + j = n } \\big ( m _ { 1 , i } ( x , ( m _ { 1 , j } ( y , z ) ) - m _ { 1 , i } ( m _ { 1 , j } ( x , y ) , z ) + m _ { 1 , i } ( m _ { 1 , j } ( x , z ) , y ) \\big ) = 0 \\\\ & \\sum _ { i + j = n } \\big ( m _ { 2 , i } ( x , ( m _ { 2 , j } ( y , z ) ) - m _ { 2 , i } ( m _ { 2 , j } ( x , y ) , z ) + m _ { 2 , i } ( m _ { 2 , j } ( x , z ) , y ) \\big ) = 0 . \\end{align*}"}
+{"id": "6513.png", "formula": "\\begin{align*} { D } ( \\sigma ) & = \\mu _ n ^ 2 - ( \\sigma + k \\cdot \\omega ) ^ 2 , \\\\ \\mu _ n & = \\sqrt { \\cos ( n \\cdot \\alpha + \\theta _ 0 ) + m } , \\\\ { T } _ \\phi ( ( k , n ) ; ( k ' , n ' ) ) & = \\phi ( k - k ' , n ) \\delta _ { n , n ' } , \\end{align*}"}
+{"id": "2172.png", "formula": "\\begin{align*} \\sigma \\bullet _ { n } ( t _ 1 \\bullet _ { m } ( a _ 1 , \\ldots , a _ m ) , \\ldots , t _ n \\bullet _ { m } ( a _ 1 , \\ldots , a _ m ) ) = ( \\sigma \\ast _ { n m } ( t _ 1 , \\ldots , t _ n ) ) \\bullet _ { m } ( a _ 1 , \\ldots , a _ m ) . \\end{align*}"}
+{"id": "1580.png", "formula": "\\begin{align*} \\langle f _ 1 , f _ 2 \\rangle = \\langle f _ 1 , f _ 2 \\rangle _ { L _ { x , v } ^ 2 ( g ^ { - 1 } ) } = \\iint f _ 1 ( x , u ) f _ 2 ( x , u ) / g ( x , u ) \\mathrm { d } u \\mathrm { d } x . \\end{align*}"}
+{"id": "7252.png", "formula": "\\begin{align*} \\lim _ n \\frac { d } { d x } \\log f _ { \\xi | _ n } ' ( x ) = \\lim _ n \\frac { d } { d x } \\log f _ { \\zeta | _ n } ' ( x ) , \\end{align*}"}
+{"id": "5364.png", "formula": "\\begin{align*} H ^ 1 ( \\O _ { L _ 1 \\cup \\ldots \\cup L _ { 2 b } } ( 1 ) ( - B _ b ) ) = 0 , \\deg ( { B _ b } _ { | L _ { 2 b } } ) = 1 \\ \\hbox { w h e r e } \\ B _ b = ( L _ 1 \\cup \\ldots \\cup L _ { 2 b } ) \\cap ( M _ 1 \\cup \\ldots \\cup M _ b ) . \\end{align*}"}
+{"id": "8422.png", "formula": "\\begin{align*} Q = \\left [ N ^ { \\frac { 3 5 \\gamma } { 5 3 } - \\frac { 1 3 \\delta } { 5 3 } - \\frac { 2 6 } { 5 3 } } \\right ] , \\end{align*}"}
+{"id": "3884.png", "formula": "\\begin{align*} g ( y _ 0 ) = g ( r _ 0 ( x _ 0 ) ) \\overset { ( b ) } { = } r _ 1 f ( x _ 0 ) = r _ 1 i _ 1 ( y _ 1 ) \\overset { ( i i ) } { = } y _ 1 . \\end{align*}"}
+{"id": "903.png", "formula": "\\begin{align*} t ^ { 1 - \\alpha } \\eta _ { 1 t } - \\eta _ { 1 x x } - \\frac { c } { x } \\eta _ { 1 x } - m x ^ k \\phi _ { 2 x } = 0 , \\end{align*}"}
+{"id": "6495.png", "formula": "\\begin{align*} u _ { t t } + ( \\varepsilon \\Delta + \\cos ( n \\cdot \\alpha + \\theta _ 0 ) \\delta _ { n , n ' } + m ) u + \\delta u ^ { p + 1 } = 0 , \\ , \\mathbb Z ^ d , \\end{align*}"}
+{"id": "8640.png", "formula": "\\begin{align*} L _ 1 ^ { \\mu } ( \\beta b ( X ) , \\xi ) & = - \\Big { ( } b ( X ) + \\frac { \\mu \\beta ^ 2 } { 6 } b ( X ) ^ 3 | \\xi | ^ 2 \\Big { ) } \\mathrm { s e c h } ( \\sqrt { \\mu } | \\xi | ) - \\frac { \\mu ^ 2 \\beta ^ 4 } { 1 2 0 } b ( X ) ^ 5 | \\xi | ^ 4 r _ 2 ( X , \\xi ) , \\end{align*}"}
+{"id": "1370.png", "formula": "\\begin{align*} R _ { 2 \\kappa } \\left ( f \\big | _ { 2 \\kappa } T _ m \\right ) = \\frac 1 m R _ { 2 \\kappa } ( f ) \\big | _ { 2 \\kappa + 2 } T _ m . \\end{align*}"}
+{"id": "6963.png", "formula": "\\begin{align*} 2 \\sum _ { k = 1 } ^ d E T ( k ) = T _ 3 ^ \\epsilon ( x ) + T _ 4 ^ \\epsilon ( x ) , \\end{align*}"}
+{"id": "7449.png", "formula": "\\begin{align*} \\psi _ { ( { y _ 1 } _ * , { y _ 2 } _ * ) } & ( \\xi _ 1 ( t h , t w ) ) \\leq \\psi _ { ( { y _ 1 } _ * , { y _ 2 } _ * ) } ( 1 ) = J ( { y _ 1 } _ * , { y _ 2 } _ * ) \\\\ & = \\inf _ { \\mathcal { N } _ { \\lambda } ^ { - } } J ( y _ 1 , y _ 2 ) \\leq J ( \\xi _ 1 ( t h , t w ) ( { y _ 1 } _ * + t h , { y _ 2 } _ * + t w ) ) . \\\\ \\end{align*}"}
+{"id": "7907.png", "formula": "\\begin{align*} \\mathrm { F P d i m } ( x ) ^ 2 & = \\mathrm { F P d i m } ( x x ^ \\ast ) \\\\ & = 1 + 2 \\sum _ { y \\in \\Gamma } c _ { x , x ^ \\ast } ^ y \\mathrm { F P d i m } ( y ) \\end{align*}"}
+{"id": "7775.png", "formula": "\\begin{align*} g _ { \\overline { M } ^ { } } : = \\frac { \\partial ^ 2 \\mathcal { K } } { \\partial z ^ a \\partial \\overline { z } ^ b } \\mathrm { d } z ^ a \\mathrm { d } \\overline { z } ^ b \\ , \\ ; \\mathcal { K } = - \\log ( 8 h ( t ) ) \\ , . \\end{align*}"}
+{"id": "4788.png", "formula": "\\begin{align*} { } _ { 1 } F _ { 1 } \\left ( 1 , s , z \\right ) = 1 + \\frac { z ^ 2 } s \\ , { } _ { 1 } F _ { 1 } \\left ( 1 , s + 1 , z \\right ) . \\end{align*}"}
+{"id": "985.png", "formula": "\\begin{align*} \\{ ( \\gamma _ p ) , ( \\widetilde { \\gamma _ p } ) \\} = \\{ \\mathfrak p _ p { } ^ { l _ p } , \\widetilde { \\mathfrak p _ p } ^ { l _ p } \\} . \\end{align*}"}
+{"id": "206.png", "formula": "\\begin{align*} { Q _ 1 } ( M , P _ i , P _ j , - \\frac { 1 } { \\tau } ) = 2 ^ { 6 } \\tau ^ { 1 4 } { Q _ 2 } ( M , P _ i , P _ j , \\tau ) . \\end{align*}"}
+{"id": "5246.png", "formula": "\\begin{align*} & \\varepsilon _ 2 = \\frac { ( \\tilde { \\varepsilon } _ 2 ^ 2 + 2 ) G _ 1 ^ 2 } { \\sigma } , ~ \\varepsilon _ 3 = 2 G _ 2 ^ 2 \\tilde { \\varepsilon } _ 3 , ~ \\varepsilon _ 4 = \\frac { 4 \\max \\{ 1 , G _ 2 ^ 2 \\tilde { \\varepsilon } _ 4 \\} } { \\min \\{ 1 , \\frac { \\sigma } { 2 \\gamma _ 0 } \\} } , ~ \\varepsilon _ { 5 } = \\frac { n \\tilde { \\varepsilon } _ 2 G _ 1 G _ 2 } { \\sigma } , ~ \\varepsilon _ { 6 } = \\frac { \\tilde { \\varepsilon } _ 2 G _ 2 ^ 2 \\gamma _ { 0 } + \\sigma } { \\sigma } . \\end{align*}"}
+{"id": "6323.png", "formula": "\\begin{align*} i \\partial _ { \\tau } v _ { \\lambda } + \\left ( V _ 1 + i V _ 2 \\right ) \\partial _ y v _ { \\lambda } + \\partial ^ 2 _ y v _ { \\lambda } = \\tilde { f } _ \\lambda , \\end{align*}"}
+{"id": "2058.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\left ( \\partial _ t - \\Delta _ { x , g ( t ) } - \\mathcal { R } ( g ( x , t ) ) \\right ) G ( x , t ; y , s ) = 0 , \\quad \\Omega \\times \\Omega \\times ( s , T ] ; \\\\ \\lim _ { t \\to s ^ + } G ( x , t ; y , s ) = \\delta _ y ( x ) , \\quad \\ ; y \\in \\Omega ; \\\\ G ( x , t ; y , s ) = 0 , \\quad \\ ; y \\in \\Omega \\ ; \\ ; x \\in \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
+{"id": "4028.png", "formula": "\\begin{align*} h ( \\mu ) = \\frac { 1 } { 2 } \\int _ 0 ^ \\infty \\left ( \\frac { m } { 1 + t } - \\mathcal { I } ( \\mu \\ast \\gamma _ t ) \\right ) \\ d t + \\frac { m } { 2 } \\log ( 2 \\pi e ) . \\end{align*}"}
+{"id": "7036.png", "formula": "\\begin{align*} \\mathop { \\sup } _ { \\begin{array} { c } ( y , S ) \\in \\mathbb { R } ^ J \\times \\mathcal S ^ K \\\\ A ^ * ( y ) + S = C \\\\ S \\succcurlyeq 0 \\\\ \\end{array} } \\langle b , y \\rangle , \\end{align*}"}
+{"id": "4092.png", "formula": "\\begin{align*} \\left ( \\left ( \\hbar \\frac { d } { d x ( p ) } \\right ) ^ 2 - \\sum _ { k \\geq 0 } \\hbar ^ k \\cdot Q _ k ( p ) \\right ) \\cdot \\psi _ \\gamma ( p ; \\hbar ) = 0 , \\end{align*}"}
+{"id": "4205.png", "formula": "\\begin{align*} 2 \\left \\langle T x , x \\right \\rangle & = T r ( T ( x \\otimes x ) + ( x \\otimes x ) T ) \\\\ & = T r ( \\psi ( T ) U ( x \\otimes x ) U ^ { \\ast } + U ( x \\otimes x ) U ^ { \\ast } \\psi ( T ) ) \\\\ & = 2 \\left \\langle U ^ { \\ast } \\psi ( T ) U , x \\right \\rangle . \\end{align*}"}
+{"id": "6119.png", "formula": "\\begin{align*} \\left | 1 - \\frac { ( - 1 ) ^ k \\zeta ^ { ( k ) } ( s ) 2 ^ s } { ( \\log 2 ) ^ k } \\right | \\leq \\frac { \\sqrt { 2 } } { 2 } , \\qquad \\sigma \\geq A , t \\in \\mathbb { R } , k = 1 , 2 . \\end{align*}"}
+{"id": "3764.png", "formula": "\\begin{align*} 2 | E ( G ) | & = \\sum _ { f \\in { F ( G ) } } \\ell ( f ) \\\\ & = \\sum _ { f \\in X } \\ell ( f ) + \\sum _ { f \\in Y } \\ell ( f ) + \\sum _ { f \\in Z } \\ell ( f ) + \\sum _ { f \\in F ( G ) \\setminus { X \\cup Y \\cup Z } } \\ell ( f ) \\\\ & \\geq 3 | X | + 4 | Y | + 5 | Z | + 6 ( | F ( G ) | - | X | - | Y | - | Z | ) \\\\ & = 6 | F ( G ) | - 3 | X | - 2 | Y | - | Z | . \\end{align*}"}
+{"id": "8114.png", "formula": "\\begin{align*} S _ q : = \\bigcup _ p [ c _ { p q } , d _ { p q } ] _ \\mathbb { Z } = \\bigcup _ p [ c ' _ { p q } , d ' _ { p q } ] _ \\mathbb { Z } . \\end{align*}"}
+{"id": "7687.png", "formula": "\\begin{align*} v ( x ) - P _ \\epsilon ( x ) & = v ( \\lambda _ 1 x _ 1 + \\lambda _ 2 x _ 2 ) - P _ \\epsilon ( \\lambda _ 1 x _ 1 + \\lambda _ 2 x _ 2 ) \\\\ & \\leq \\lambda _ 1 ( v ( x _ 1 ) - P _ \\epsilon ( x _ 1 ) ) + \\lambda _ 2 ( v ( x _ 2 ) - P _ \\epsilon ( x _ 2 ) ) \\\\ & < \\lambda _ 2 ( v ( x _ 2 ) - P _ \\epsilon ( x _ 2 ) ) \\\\ & \\leq v ( z _ \\epsilon ) - P _ \\epsilon ( z _ \\epsilon ) . \\end{align*}"}
+{"id": "3765.png", "formula": "\\begin{align*} { \\rm O T } ( \\mu _ 0 , \\mu _ 1 ) = \\inf _ { \\gamma \\in \\Gamma ( \\mu _ 0 , \\mu _ 1 ) } ( c , \\gamma ) , ( c , \\gamma ) : = \\int _ { X \\times X } c ( x _ 0 , x _ 1 ) d \\gamma ( x _ 0 , x _ 1 ) , \\end{align*}"}
+{"id": "825.png", "formula": "\\begin{align*} | \\tilde { h } ( z ) | & = e ^ { n _ 0 \\mbox { R e } \\tilde { g _ 1 } ( z ) } \\\\ & = e ^ { { - n _ 0 } ^ { - \\mbox { R e } \\tilde { g _ 1 } ( z ) } } \\\\ & < \\epsilon _ 1 ^ { - \\mbox { R e } \\tilde { g _ 1 } ( z ) } . \\end{align*}"}
+{"id": "7328.png", "formula": "\\begin{align*} D ^ 2 _ 0 f _ \\lambda ( w _ 1 , w _ 2 ) & = \\int _ \\Omega { \\langle \\nabla u _ 1 ( x ) , \\nabla u _ 2 ( x ) \\rangle \\ , d x } - \\int _ \\Omega { \\langle \\nabla v _ 1 ( x ) , \\nabla v _ 2 ( x ) \\rangle \\ , d x } \\\\ & + \\int _ \\Omega { \\langle S _ \\lambda ( x ) w _ 1 , w _ 2 \\rangle \\ , d x } , w _ 1 = ( u _ 1 , v _ 1 ) , w _ 2 = ( u _ 2 , v _ 2 ) , \\end{align*}"}
+{"id": "8270.png", "formula": "\\begin{align*} \\hat { \\omega } = \\omega _ Y + i \\partial \\bar \\partial ( \\zeta v ) + C i \\partial \\bar \\partial ( ( 1 - \\zeta _ S ) h _ \\alpha ) + c i \\partial \\bar \\partial h _ 1 , \\end{align*}"}
+{"id": "5295.png", "formula": "\\begin{align*} \\Tilde { \\mathbf { n } } = ( 1 , n _ 1 , \\dots , n _ { l - 1 } , n _ { l + 1 } , \\dots , n _ L , n _ l , 1 ) . \\end{align*}"}
+{"id": "6725.png", "formula": "\\begin{align*} m = 2 \\left ( \\mathfrak { c } _ { p } I _ { a } ( \\frac { m } { 2 r _ { 0 } } ) \\right ) ^ { \\frac { 1 } { a } } . \\end{align*}"}
+{"id": "4862.png", "formula": "\\begin{align*} \\frac { f ( u ) } { 1 + \\frac { \\delta } { 2 } f ( u ) } = \\frac { f ( - u ) } { 1 + \\frac { \\delta } { 2 } f ( - u ) } \\end{align*}"}
+{"id": "4916.png", "formula": "\\begin{align*} H _ k ( \\tau _ r ) & = \\dfrac { a ^ { 4 k + 2 } } { 2 ^ { ( 2 k + 1 ) ( r + 3 ) } } \\cdot \\left ( \\dfrac { e ^ { - 2 ^ { r + 1 } ( k + 1 ) \\pi } } { 4 ^ { k + 1 } } + \\sum _ { \\ell = 1 } ^ { k - 1 } \\dfrac { \\alpha _ k ( \\ell ) } { 4 ^ \\ell } \\cdot e ^ { M ( \\ell , r ) } + \\sum _ { \\ell = k } ^ { 2 k } \\dfrac { \\beta _ k ( \\ell ) } { 4 ^ \\ell } \\cdot e ^ { M ( \\ell , r ) } \\right ) \\cdot e ^ { J ( k , r ) } , \\end{align*}"}
+{"id": "4551.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ 2 _ + } ( \\mathcal B _ 0 { \\mathbf V } _ \\sigma \\cdot { \\mathbf V } _ \\sigma ) ( t ) d { \\bf x } = 2 \\int _ { \\Omega _ t } \\tilde { \\mathcal F } _ \\sigma \\cdot { \\mathbf V } _ \\sigma d { \\bf x } d s + \\int _ { \\Omega _ t } ( \\partial _ t \\mathcal B _ 0 + \\partial _ 2 \\mathcal B _ 2 - \\partial _ 1 \\mathcal B _ 1 - \\mathcal B _ 3 ) { \\mathbf V _ { \\sigma } } \\cdot { \\mathbf V _ { \\sigma } } d { \\bf x } d s \\ , . \\end{align*}"}
+{"id": "6214.png", "formula": "\\begin{align*} \\sigma ( z , w ) \\coloneqq J z \\cdot w , J \\coloneqq \\begin{bmatrix} O & I \\\\ - I & O \\end{bmatrix} , \\end{align*}"}
+{"id": "2246.png", "formula": "\\begin{align*} \\theta _ \\alpha ( x ) = \\theta ( x ; \\alpha ) = \\sum _ { k \\in \\Z } e ^ { - \\pi \\alpha k ^ 2 } e ^ { 2 \\pi i k x } . \\end{align*}"}
+{"id": "3035.png", "formula": "\\begin{align*} | H _ { a , a } | + | H _ { \\overline { a } , \\overline { a } } | \\le 2 + 2 ( n - 2 ) + 2 \\cdot \\lfloor \\frac { ( n - 2 ) ^ 2 } { 4 } \\rfloor = 2 \\cdot \\lfloor \\frac { n ^ 2 } { 4 } \\rfloor . \\end{align*}"}
+{"id": "8877.png", "formula": "\\begin{align*} W : = W _ G ( T ) = N _ G ( T ) / T . \\end{align*}"}
+{"id": "2921.png", "formula": "\\begin{align*} { \\pi ^ l } ^ * \\mathcal E & = { \\pi ^ l } ^ * S ^ 0 f _ * ( \\sigma ( X , \\Delta ) \\otimes \\mathcal O _ X ( m L ) ) \\\\ & \\cong S ^ 0 { f _ l } _ * ( \\sigma ( X _ l , \\Delta _ l ) \\otimes \\mathcal O _ { X _ l } ( m L _ l ) ) , \\end{align*}"}
+{"id": "3975.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) u = w ' < \\pi + w ^ \\# , u ( 0 ) = u _ 0 , \\end{align*}"}
+{"id": "2133.png", "formula": "\\begin{align*} g = \\left [ \\begin{matrix} \\alpha ^ 2 e ^ { - u ^ { ( 0 ) } } & 0 \\\\ 0 & e ^ { u ^ { ( 0 ) } } \\end{matrix} \\right ] . \\end{align*}"}
+{"id": "926.png", "formula": "\\begin{align*} ( u _ 1 , v _ 1 ) = \\bigg ( 1 , \\frac { \\sqrt { m n } } { m } \\bigg ) , ~ ~ ( u _ 2 , v _ 2 ) = x ^ { 1 + \\sqrt { m n } - c } \\bigg ( - \\frac { \\sqrt { m n } } { n } , 1 \\bigg ) , \\end{align*}"}
+{"id": "478.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\sigma _ t ^ { ( \\alpha / 2 ) } ( \\tau ) \\ , d \\tau = 1 , t > 0 . \\end{align*}"}
+{"id": "6933.png", "formula": "\\begin{align*} \\liminf \\limits _ { t \\rightarrow + \\infty } \\inf \\limits _ { | x | < c t } w ( x , t ) \\geq \\liminf \\limits _ { t \\rightarrow + \\infty } \\inf \\limits _ { | x | < c t } \\varphi ( x , t ) = ( 1 + b ) / 2 . \\end{align*}"}
+{"id": "1466.png", "formula": "\\begin{align*} f ( j ) + f ( 2 l + 1 - j ) = 2 l + 1 , \\ 0 \\leq j \\leq 2 l + 1 . \\end{align*}"}
+{"id": "4921.png", "formula": "\\begin{align*} S = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} , \\gamma _ 2 = \\begin{pmatrix} 1 & 0 \\\\ 2 & 1 \\end{pmatrix} \\gamma _ 4 = \\begin{pmatrix} 1 & 0 \\\\ 4 & 1 \\end{pmatrix} . \\end{align*}"}
+{"id": "1240.png", "formula": "\\begin{align*} P _ { M < \\cdot \\le N } : = P _ { \\le N } - P _ { \\le M } = \\sum _ { M < N ^ \\prime \\le N } P _ { N ^ \\prime } \\end{align*}"}
+{"id": "667.png", "formula": "\\begin{align*} \\widehat f ( \\xi ) = \\mathcal F f ( \\xi ) = \\int _ { \\R ^ d } e ^ { - i x \\xi } f ( x ) \\ , d x ( \\xi \\in \\R ^ d ) . \\end{align*}"}
+{"id": "372.png", "formula": "\\begin{align*} \\sum _ { j \\in \\Z ^ 2 } \\eta [ D _ j , \\psi _ 1 ] \\eta ' = \\sum _ { \\substack { j \\in \\Z ^ 2 ; \\\\ j _ 1 , j _ 2 \\leq 0 } } \\eta [ D _ j , \\psi _ 1 ] \\eta ' + \\sum _ { \\substack { j \\in \\Z ^ 2 ; \\\\ j _ 1 \\leq 0 < j _ 2 } } \\eta [ D _ j , \\psi _ 1 ] \\eta ' + \\sum _ { \\substack { j \\in \\Z ^ 2 ; \\\\ j _ 2 \\leq 0 < j _ 1 } } \\eta [ D _ j , \\psi _ 1 ] \\eta ' + \\sum _ { \\substack { j \\in \\Z ^ 2 ; \\\\ j _ 1 , j _ 2 > 0 } } \\eta [ D _ j , \\psi _ 1 ] \\eta ' . \\end{align*}"}
+{"id": "6104.png", "formula": "\\begin{align*} \\begin{aligned} V a r ( g _ { S _ l } ^ k ( \\gamma ) ) = \\frac { 1 } { 1 0 0 } \\sum _ { l = 1 } ^ { 1 0 0 } ( g _ { S _ l } ^ k ( \\gamma ) - \\frac { 1 } { 1 0 0 } \\sum _ { l = 1 } ^ { 1 0 0 } g _ { S _ l } ^ k ( \\gamma ) ) ^ 2 , k = 0 , 1 , 2 . . . 1 0 0 \\end{aligned} \\end{align*}"}
+{"id": "204.png", "formula": "\\begin{align*} & \\delta _ 1 ( \\tau ) = \\frac { 1 } { 4 } + 6 q + 6 q ^ 2 + \\cdots , ~ ~ \\varepsilon _ 1 ( \\tau ) = \\frac { 1 } { 1 6 } - q + 7 q ^ 2 + \\cdots , \\\\ & 8 \\delta _ 2 ( \\tau ) = - 1 - 2 4 q ^ { \\frac { 1 } { 2 } } - 2 4 q - 9 6 q ^ { \\frac { 3 } { 2 } } + \\cdots , ~ ~ \\varepsilon _ 2 ( \\tau ) = q ^ { \\frac { 1 } { 2 } } + 8 q + 2 8 q ^ { \\frac { 3 } { 2 } } + \\cdots . \\end{align*}"}
+{"id": "7094.png", "formula": "\\begin{align*} \\sum _ { u \\ , \\mathrm { m o d } \\ , p _ 1 } \\ , e \\left ( \\frac { ( \\alpha n _ 1 - m \\bar { p _ 2 } ) \\overline { q } \\ , \\ , \\overline { u } } { p _ 1 } \\right ) = \\ , \\ , \\begin{cases} p _ 1 & \\ , \\ , \\ , \\ , \\ , p _ 1 \\ , | \\ , ( \\alpha n _ 1 - m \\bar { p _ 2 } ) \\\\ 0 & \\ , \\ , \\ , . \\end{cases} \\end{align*}"}
+{"id": "1962.png", "formula": "\\begin{align*} \\mathcal { C } \\left ( { \\mathbf { u } , \\mathbf { v } } \\right ) ( \\tau ) = \\begin{cases} \\sum _ { i = 0 } ^ { N - 1 - \\tau } \\omega _ q ^ { u _ { i } - v _ { i + \\tau } } , & 0 \\leq \\tau \\leq N - 1 , \\\\ \\sum _ { i = 0 } ^ { N - 1 + \\tau } \\omega _ q ^ { u _ { i - \\tau } - v _ { i } } , & - N + 1 \\leq \\tau \\leq - 1 , \\\\ 0 , & | \\tau | \\geq N , \\end{cases} \\end{align*}"}
+{"id": "6622.png", "formula": "\\begin{align*} { S } _ \\infty ( \\tau ; \\beta ) - 1 = c _ \\infty ^ { ( \\widetilde { \\rm c J } ) } ( \\tau ; \\beta , 1 , 0 ) . \\end{align*}"}
+{"id": "1063.png", "formula": "\\begin{align*} C _ 1 ( \\hat { w } _ \\lambda , y _ 0 ) \\not = 0 . \\end{align*}"}
+{"id": "4290.png", "formula": "\\begin{align*} \\frac { 1 } { n ^ { k _ 1 + k _ 2 + 1 } } \\sum _ { J \\in \\Pi ( \\pi ) } \\sum _ { i = 1 \\atop i ^ \\prime = 1 } ^ { n } | \\Pi ( \\pi ) | V ( \\pi ) & = ( \\kappa - 1 ) \\frac { 1 } { n ^ { k _ 1 + k _ 2 + 1 } } \\sum _ { i = 1 \\atop i ^ \\prime = 1 } ^ { n } \\sum _ { J \\in \\Pi ( \\pi ) } g ( i , i ^ \\prime , J _ 1 , J _ 2 , p , n ) , \\end{align*}"}
+{"id": "6441.png", "formula": "\\begin{align*} \\begin{pmatrix} A & - V \\\\ B & U \\end{pmatrix} \\simeq \\begin{pmatrix} A & - \\frac { 1 } { u ^ 2 } V \\\\ u ^ 2 B & U \\end{pmatrix} . \\end{align*}"}
+{"id": "2225.png", "formula": "\\begin{align*} \\Omega ( \\varphi , \\psi ) = \\langle A \\varphi , \\psi \\rangle - \\langle \\varphi , A \\psi \\rangle \\end{align*}"}
+{"id": "5615.png", "formula": "\\begin{align*} d _ { p , \\mu } \\| u _ { n , R } ^ * \\| ^ p _ { X ^ { 1 , p } _ \\infty } & \\geq \\int _ 0 ^ \\infty \\exp _ p \\left ( \\mu | u _ { n , R } ^ * | ^ { p ' } \\right ) r ^ { \\alpha _ 0 } \\mathrm d r \\\\ & \\quad + \\sum _ { j = 1 } ^ \\infty \\left ( \\dfrac { 1 } { \\| u ^ * _ { n , R } \\| ^ { p ' j } _ { X ^ { 1 , p } _ \\infty } } - 1 \\right ) \\dfrac { \\mu ^ { p - 1 + j } } { \\Gamma ( p + j ) } \\| u _ { n , R } ^ * \\| ^ { p ' ( p - 1 + j ) } _ { L _ { \\alpha _ 0 } ^ { p ' ( p - 1 + j ) } } . \\end{align*}"}
+{"id": "8575.png", "formula": "\\begin{align*} P ( \\Sigma _ b ) = | J _ { \\Sigma _ b } | ( I ^ { \\mu } ) ^ { - 1 } J _ { \\Sigma _ b } ^ { - 1 } ( I ^ { \\mu } ) ^ 2 ( J _ { \\Sigma _ b } ^ { - 1 } ) ^ T ( I ^ { \\mu } ) ^ { - 1 } , \\end{align*}"}
+{"id": "8548.png", "formula": "\\begin{align*} P ( E ^ { k } ; J \\times \\mathbb { R } ^ { n - 1 } ) = P ( F _ { \\ell ^ { k } } ; J \\times \\mathbb { R } ^ { n - 1 } ) , \\mbox { f o r e v e r y } k \\in \\mathbb { N } . \\end{align*}"}
+{"id": "4624.png", "formula": "\\begin{align*} \\mathcal { R } ( G ) = \\bigoplus _ { \\pi \\in G ^ \\wedge } \\mathcal { R } ^ { \\pi } ( G ) \\end{align*}"}
+{"id": "7193.png", "formula": "\\begin{align*} \\mathfrak { a } _ n ^ { 2 ^ r - 1 } & = ( 1 \\otimes y _ n - y _ n \\otimes 1 ) ^ { 2 ^ r - 1 } \\\\ & = \\sum _ { k = 0 } ^ { 2 ^ r - 1 } ( - 1 ) ^ { 2 ^ r - 1 - k } \\binom { 2 ^ r - 1 } { k } ( 1 \\otimes y _ n ) ^ k ( y _ n \\otimes 1 ) ^ { 2 ^ r - 1 - k } \\\\ & = \\sum _ { k = 0 } ^ { 2 ^ r - 1 } ( - 1 ) ^ { 2 ^ r - 1 - k } \\binom { 2 ^ r - 1 } { k } ( y _ n ^ { 2 ^ r - 1 - k } \\otimes y _ n ^ k ) . \\end{align*}"}
+{"id": "6871.png", "formula": "\\begin{align*} \\gamma = \\sqrt { - 1 } \\sum _ i \\theta _ i \\partial \\overline { \\partial } w _ i . \\end{align*}"}
+{"id": "5989.png", "formula": "\\begin{align*} \\int _ { \\mathbb { D } } \\tilde { u } _ j ( s ' ) \\tilde { v } _ \\varepsilon ( s ' ) e ^ { \\tilde { \\phi } ( s ' ) } d s ' = 0 . \\end{align*}"}
+{"id": "1795.png", "formula": "\\begin{gather*} \\phi ( x ) = \\frac { 1 } { 2 \\pi i } \\int _ { c - i \\infty } ^ { c + i \\infty } \\widehat { \\phi } ( w ) x ^ { - w } \\ , d w \\end{gather*}"}
+{"id": "8067.png", "formula": "\\begin{align*} f ( x ) = T _ { N } ( T _ { N } ^ { - 1 } ( f ) ) ( x ) = \\sum \\limits _ { j \\in \\mathbb N } \\sum \\limits _ { Q \\in \\Pi _ { j + N } } \\vert Q \\vert \\psi _ { j } ( x - u _ { Q } ) ( \\psi _ { j } \\ast h ) ( u _ { Q } ) . \\end{align*}"}
+{"id": "1694.png", "formula": "\\begin{align*} \\mathbb { P } ^ { ( m , n ) } _ { \\texttt { b } } : = _ { \\mu \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { b } } } \\{ M _ { \\texttt { b } ; \\mu } ( \\boldsymbol { \\xi } ) \\} , \\end{align*}"}
+{"id": "3001.png", "formula": "\\begin{align*} \\psi ^ * ( j ) = \\psi ^ * ( j ) ^ { 1 / 2 } \\sum _ { i = 1 } ^ { \\infty } \\Theta _ { x _ i , x _ i } \\psi ^ * ( j ) ^ { 1 / 2 } = \\sum _ { i = 1 } ^ { \\infty } \\Theta _ { \\psi ^ * ( j ) ^ { 1 / 2 } x _ i , \\psi ^ * ( j ) ^ { 1 / 2 } x _ i } . \\end{align*}"}
+{"id": "3638.png", "formula": "\\begin{align*} \\Omega ^ k ( \\mathcal { C } ) : = \\{ \\alpha \\in \\Omega ^ k ( J ^ 1 U ) \\ | \\ \\chi \\mathbin { \\lrcorner } \\alpha = 0 \\} \\ . \\end{align*}"}
+{"id": "5237.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\Big \\| \\Big [ \\sum _ { t = 1 } ^ T g ( x _ { i , t } ) \\Big ] _ + \\Big \\| \\end{align*}"}
+{"id": "6691.png", "formula": "\\begin{align*} & \\tilde { h } _ 0 ( \\beta , p , q ) = - p , \\ : \\ : \\tilde { h } _ 1 ( \\beta , p , q ) = { i q ( 1 - \\kappa ) \\over 2 \\pi \\kappa ^ 2 } , \\ : \\ : \\tilde { h } _ 2 ( \\beta , p , q ) = - { ( 1 - \\kappa ) \\over 4 \\pi ^ 2 \\kappa ^ 3 } ( 2 q ^ 2 + \\kappa p + \\kappa ^ 2 ( - 1 + p ) p ) , \\\\ & \\tilde { h } _ 3 ( \\beta , p , q ) = - { i q ( 1 - \\kappa ) \\over 4 8 \\pi ^ 3 \\kappa ^ 4 } ( - 6 + 1 6 q ^ 2 + \\kappa ( 1 1 + 1 2 p ) + 6 \\kappa ^ 2 ( - 1 - 2 p + 2 p ^ 2 ) ) . \\end{align*}"}
+{"id": "8158.png", "formula": "\\begin{align*} \\mu ^ \\star _ { x y } q _ { i j } ( x , y ) & = ( n - 1 ) q _ { i j } ( x , y ) + \\frac { n ( n - 1 ) } { k ( n - k ) } \\frac { \\binom { n } { i } } { \\binom { k } { i } } K _ i ( x , k - y , r - 1 ) ( - y ) H _ j ( y , n - i , k - i ) \\ , . \\end{align*}"}
+{"id": "4154.png", "formula": "\\begin{align*} & \\phi ( \\exp ^ { \\nabla } ( t f ^ { \\hat { p } } | _ p ) ) | _ { \\hat p = p } = ( \\tilde \\phi \\circ e ^ { t H _ { \\bar { f } ( \\hat u ) } } ) ( P _ { \\hat p , p } \\hat u ) | _ { \\hat p = p } \\\\ & = ( \\exp ^ { \\cdot } ( t H _ { \\bar { f } } ) \\rhd \\tilde \\phi ) ( P _ { \\hat p , p } \\hat u ) = ( \\exp ^ { \\cdot } ( t f ) \\rhd \\phi ) ( p ) \\end{align*}"}
+{"id": "5226.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ t \\delta _ j ^ { ( i ) } \\leq 4 D G \\sqrt { 2 \\sum _ { j = 1 } ^ t ( \\eta _ j ^ { ( i ) } ) ^ 2 \\left ( \\ln \\frac { 2 m t ^ 2 } { \\delta } \\right ) } . \\end{align*}"}
+{"id": "5330.png", "formula": "\\begin{align*} D ^ n ( f \\circ g ) = \\sum \\frac { n ! } { k _ 1 ! \\cdots k _ n ! } ( D ^ { k _ 1 + \\dots + k _ n } f \\circ g ) \\prod _ { m = 1 } ^ n \\left ( \\frac { D ^ m g } { m ! } \\right ) ^ { k _ m } , \\end{align*}"}
+{"id": "2791.png", "formula": "\\begin{align*} M ( \\widetilde \\varphi ' ) _ * K ( \\widetilde \\varphi ' ) ^ * & = ( M ( \\widetilde \\varphi ) \\circ M ( k _ y ) ) _ * ( c _ { k _ y } \\circ K ( \\widetilde \\varphi ) ) ^ * = M ( \\widetilde \\varphi ) _ * M ( k _ y ) _ * K ( \\widetilde \\varphi ) ^ * c _ { k _ y } ^ * \\\\ & = M ( \\widetilde \\varphi ) _ * K ( \\widetilde \\varphi ) ^ * M ( k _ y ) _ * c _ { k _ y } ^ * . \\end{align*}"}
+{"id": "4584.png", "formula": "\\begin{align*} \\Vert \\nabla _ { t , x _ 2 } \\varphi \\Vert ^ 2 _ { L ^ 2 ( \\Gamma _ T ) } \\le C \\Vert { \\mathbf F } \\Vert ^ 2 _ { H ^ 1 _ \\ast ( \\Omega _ T ) } \\ , . \\end{align*}"}
+{"id": "2368.png", "formula": "\\begin{align*} x = \\sum _ { \\lambda \\in \\Lambda } f ( \\pi ( \\lambda ) ^ { - 1 } x ) \\pi ( \\lambda ) \\tau , \\forall x \\in \\mathbb { C } ^ { o ( G ) } \\iff f ( \\pi ( \\mu ) ^ { - 1 } \\tau ) = \\frac { o ( G ) } { o ( \\Lambda ) } \\delta _ { \\mu , 0 } , \\forall \\mu \\in \\Lambda ^ 0 . \\end{align*}"}
+{"id": "5461.png", "formula": "\\begin{align*} | f ( n ) | & = | \\sum _ { k = 0 } ^ { n - 1 } \\nabla f ( k ) | \\\\ & \\leq \\sum _ { k = 0 } ^ { n - 1 } | \\nabla f ( k ) | \\frac { 1 } { \\pi ( k ) } \\pi ( k ) \\\\ & \\leq \\sum _ { k = 0 } ^ { n - 1 } G ( | \\nabla f ( k ) | ) \\pi ( k ) + \\sum _ { k = 0 } ^ { n - 1 } G ^ * \\left ( \\frac { 1 } { \\pi ( k ) } \\right ) \\pi ( k ) \\end{align*}"}
+{"id": "3436.png", "formula": "\\begin{align*} - \\log | s | = \\min \\{ l _ 0 x _ 0 + \\ldots l _ m x _ m + a : s _ { l _ 0 \\ldots l _ m , a } \\neq 0 \\} . \\end{align*}"}
+{"id": "652.png", "formula": "\\begin{align*} E = \\left \\{ ( s , \\omega ) \\in [ 0 , \\infty ) \\times \\Omega \\colon 0 \\le s < \\tau ( \\omega ) \\right \\} \\end{align*}"}
+{"id": "3492.png", "formula": "\\begin{align*} s = \\sum F _ 0 ^ { l _ 0 } \\ldots F _ m ^ { l _ m } s _ { l _ 0 , \\ldots l _ m } , \\end{align*}"}
+{"id": "8450.png", "formula": "\\begin{gather*} \\nabla _ { K Z B } = d - \\beta B - \\alpha A - \\alpha \\sum _ { k = 2 } ^ { \\infty } p _ { k } \\mathrm { a d } _ { B } ^ { k } ( A ) , \\nabla _ { K Z B } ^ { \\prime } = d - \\left ( \\beta - d f \\right ) B - \\alpha A - \\alpha \\sum _ { k = 1 } ^ { \\infty } q _ { k } \\mathrm { a d } _ { B } ^ { k } ( A ) , \\end{gather*}"}
+{"id": "6654.png", "formula": "\\begin{align*} - { d ^ 3 \\over d \\tau ^ 3 } \\Big ( ( \\tau ^ 3 - 4 \\tau ) \\hat { r } ( \\tau ) \\Big ) + 4 { d ^ 2 \\over d \\tau ^ 2 } ( \\tau ^ 2 \\hat { r } ( \\tau ) ) - 2 ( 1 - 2 p ^ 2 ) { d \\over d \\tau } ( \\tau \\hat { r } ( \\tau ) ) - 4 p ^ 2 \\hat { r } ( \\tau ) = 0 , \\end{align*}"}
+{"id": "4966.png", "formula": "\\begin{align*} P _ { 3 , a } ( n ) = \\Phi _ { 1 2 n , a } ( \\sqrt { 3 } ) , \\end{align*}"}
+{"id": "5086.png", "formula": "\\begin{align*} & E ( A z ) \\\\ & = \\left ( \\begin{array} { c c c c c c c } \\Delta _ 0 E ( \\lambda _ { 1 } , z ) v _ { 1 , l _ { 1 , 1 } } ^ 1 & \\ ! \\ ! \\ ! \\cdots \\ ! \\ ! \\ ! & \\sum _ { h = 1 } ^ { l _ { 1 , 1 } } \\Delta _ { l _ { 1 , 1 } - h } E ( \\lambda _ { 1 } , z ) v _ { 1 , l _ { 1 , 1 } } ^ { l _ { 1 , 1 } } & \\ ! \\ ! \\ ! \\cdots \\ ! \\ ! \\ ! & \\Delta _ 0 E ( \\lambda _ { k } , z ) v _ { k , l _ { k , q _ k } } ^ 1 & \\ ! \\ ! \\ ! \\cdots \\ ! \\ ! \\ ! & \\sum _ { h = 1 } ^ { l _ { k , q _ k } } \\Delta _ { l _ { k , q _ k } - h } E ( \\lambda _ { k } , z ) v _ { k , l _ { k , q _ k } } ^ { l _ { k , q _ k } } \\end{array} \\right ) P ^ { - 1 } , \\end{align*}"}
+{"id": "7396.png", "formula": "\\begin{align*} W _ { n } = \\frac { \\partial _ { x } w _ { n } } { \\rho _ { n } } . \\end{align*}"}
+{"id": "6336.png", "formula": "\\begin{align*} \\iint v _ \\lambda g _ \\lambda \\ , d x d t - \\langle v _ \\lambda ( T ) , w _ \\lambda ( T ) \\rangle = \\iint w _ \\lambda f _ \\lambda d x d t - \\langle v _ \\lambda ( 0 ) , w _ \\lambda ( 0 ) \\rangle . \\end{align*}"}
+{"id": "75.png", "formula": "\\begin{align*} \\mathcal T ( k ) = - & \\frac { \\widehat g ( k ) ^ 2 } { ( 1 - \\varepsilon _ K ) \\mathcal D _ k ( 1 - \\alpha _ k ^ 2 ) \\vert \\Lambda \\vert ^ 2 } \\sum _ { p , s \\in \\mathcal P _ L } \\frac { 1 } { \\sqrt { 1 - \\alpha _ { p - k } ^ 2 } \\sqrt { 1 - \\alpha _ { s - k } ^ 2 } } \\\\ & \\times ( \\bar z a _ p ^ \\dagger + \\alpha _ k \\alpha _ { p - k } z a _ { - p } ) b _ { p - k } b _ { s - k } ^ \\dagger ( z a _ s + \\alpha _ k \\alpha _ { s - k } \\bar z a _ { - s } ^ \\dagger ) . \\end{align*}"}
+{"id": "1075.png", "formula": "\\begin{align*} \\left ( u _ { t } - L _ { 0 } u \\right ) ^ { 2 } \\varphi _ { \\lambda , \\nu _ { 0 } } = \\left ( u _ { t } ^ { 2 } + \\left ( L _ { 0 } u \\right ) ^ { 2 } \\right ) \\varphi _ { \\lambda , \\nu _ { 0 } } - 2 u _ { t } L _ { 0 } u \\varphi _ { \\lambda , \\nu _ { 0 } } . \\end{align*}"}
+{"id": "4938.png", "formula": "\\begin{align*} \\eta ( 2 i ) = \\dfrac { \\pi ^ { 1 / 4 } } { \\Gamma ( 3 / 4 ) \\cdot 2 ^ { 7 / 8 } } , \\eta ( 4 i ) = \\dfrac { \\pi ^ { 1 / 4 } \\cdot ( \\sqrt { 2 } - 1 ) ^ { 1 / 4 } } { \\Gamma ( 3 / 4 ) \\cdot 2 ^ { 2 1 / 1 6 } } , \\end{align*}"}
+{"id": "8944.png", "formula": "\\begin{align*} \\alpha ( x ) & = \\alpha _ { d + 1 } ( h ( x ) ) = \\alpha _ { d } ( \\pi ( x ) ) , \\end{align*}"}
+{"id": "8455.png", "formula": "\\begin{align*} - \\left ( 1 - \\alpha Y - \\beta X \\right ) ^ { - 1 } \\left ( 1 + ( 1 - \\beta X ) \\sum _ { i \\geq 0 } \\sum _ { k = 1 } ^ { i } \\beta ^ { i - k + 1 } \\alpha p _ k X ^ i \\right ) , \\end{align*}"}
+{"id": "5641.png", "formula": "\\begin{align*} \\widetilde { y } = - y - x _ 3 ( x _ 0 + x _ 1 ) \\end{align*}"}
+{"id": "4380.png", "formula": "\\begin{align*} | | u | | ^ 2 _ { H ^ m _ { \\ast } ( \\Omega ) } : = \\sum _ { \\langle \\alpha \\rangle \\leq m , \\alpha _ 0 = 0 } | | D ^ { \\alpha } _ { \\ast } u | | ^ 2 _ { L ^ 2 ( \\Omega ) } , \\end{align*}"}
+{"id": "4099.png", "formula": "\\begin{align*} \\omega _ { g , 1 } ( p _ 0 ) + \\omega _ { g , 1 } ( \\sigma ( p _ 0 ) = d f _ { g , 1 } ( p _ 0 ) . \\end{align*}"}
+{"id": "2535.png", "formula": "\\begin{align*} r _ i < r _ { \\tau ( i ) } \\textrm { o r } ( r _ i = r _ { \\tau ( i ) } \\alpha _ { \\sigma ( i ) } \\leq \\alpha _ { \\tau ( i ) } ) \\end{align*}"}
+{"id": "5804.png", "formula": "\\begin{align*} w _ 0 = \\prod \\limits _ { i = 1 , 3 , . . . , n - 1 } s _ { \\varepsilon _ i + \\varepsilon _ { i + 1 } } s _ { \\alpha _ i } , \\end{align*}"}
+{"id": "8003.png", "formula": "\\begin{align*} \\nabla ^ 2 K ( x ) = \\nabla ^ 2 H ( \\pi _ { p _ 0 } ^ { - 1 } ( x ) ) [ \\nabla \\pi _ { p _ 0 } ^ { - 1 } ( x ) , \\nabla \\pi _ { p _ 0 } ^ { - 1 } ( x ) ] + \\nabla H ( \\pi _ { p _ 0 } ^ { - 1 } ) [ \\nabla ^ 2 \\pi _ { p _ 0 } ^ { - 1 } ( x ) ] \\end{align*}"}
+{"id": "5945.png", "formula": "\\begin{align*} \\Gamma _ { \\varepsilon , a } : = \\{ \\mathrm { e x p } _ { x ^ * ; h } ( \\varepsilon t _ 1 E _ 1 + \\varepsilon t _ 2 E _ 2 ) \\mid t _ 1 ^ 2 + a ^ { - 2 } t _ 2 ^ 2 \\leq 1 \\} . \\end{align*}"}
+{"id": "7993.png", "formula": "\\begin{align*} g _ { j _ { r } } | _ { { \\textstyle \\bigcup _ { i = 1 } ^ { \\tilde { m } } } \\Omega _ { i } } = \\tilde { b } _ { 1 } g _ { j _ { 1 } } | _ { { \\textstyle \\bigcup _ { i = 1 } ^ { \\tilde { m } } } \\Omega _ { i } } + \\cdots + \\tilde { b } _ { r - 1 } g _ { j _ { r - 1 } } | _ { { \\textstyle \\bigcup _ { i = 1 } ^ { \\tilde { m } } } \\Omega _ { i } } \\end{align*}"}
+{"id": "5129.png", "formula": "\\begin{align*} C ^ * & = \\min ( 1 , 2 C ^ o ) \\end{align*}"}
+{"id": "4769.png", "formula": "\\begin{align*} \\mu _ { p } ( A ) \\leq \\mu _ { p } \\bigg ( \\bigcup _ { n = N } ^ { \\infty } \\{ x \\mid v _ { \\phi } ( g _ { n } ^ { - 1 } x ) > n \\} \\bigg ) \\leq \\sum _ { n = N } ^ { \\infty } \\mu _ { p } ( \\{ x \\mid v _ { \\phi } ( g _ { n } ^ { - 1 } x ) > n \\} ) < \\epsilon . \\end{align*}"}
+{"id": "471.png", "formula": "\\begin{align*} ( p _ t f ) ( r ) : = \\int _ 0 ^ \\infty p _ t ( r , s ) f ( s ) m ( d s ) . \\end{align*}"}
+{"id": "8370.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| \\phi ( t , \\theta _ { - t } \\omega , y ( \\theta _ { - t } \\omega ) ) - Y ( \\omega ) \\| = 0 , \\ , \\lim _ { t \\to \\infty } \\| \\phi ( t , \\omega , y ( \\omega ) ) - Y ( \\theta _ t \\omega ) \\| = 0 \\end{align*}"}
+{"id": "2951.png", "formula": "\\begin{align*} L _ { k j _ 1 \\cdots j _ s l _ 1 \\cdots l _ { s - 1 } } D _ { l _ 1 \\cdots l _ { s - 1 } } = \\frac { 2 s - 1 } { s - 1 } \\delta _ { \\hat { k j _ 1 } } D _ { \\hat { j } _ 2 \\cdots \\hat { j } _ s } - \\delta _ { \\hat { j } _ 1 \\hat { j } _ 2 } D _ { \\hat { j } _ 3 \\cdots \\hat { j } _ s k } . \\end{align*}"}
+{"id": "744.png", "formula": "\\begin{align*} \\tau < \\infty \\implies \\limsup _ { t \\nearrow \\tau } \\norm { \\phi _ + ( t ) } _ { H ^ r } ^ 2 = \\infty . \\end{align*}"}
+{"id": "8544.png", "formula": "\\begin{align*} P ( E ^ { k } ; J \\times \\mathbb { R } ^ { n - 1 } ) = P ( F _ { \\ell ^ { k } } ; J \\times \\mathbb { R } ^ { n - 1 } ) . \\end{align*}"}
+{"id": "736.png", "formula": "\\begin{align*} X _ \\pm ( t ) = X _ \\pm ( 0 ) + \\int _ 0 ^ t \\Psi ( s ) \\ , d s + \\int _ 0 ^ t \\Phi ( s ) \\ , d W ( s ) , \\end{align*}"}
+{"id": "6501.png", "formula": "\\begin{align*} i u _ { t } + \\varepsilon \\Delta + V u + \\delta | u | ^ { 2 p } u = 0 , \\ , \\mathbb Z ^ d , \\end{align*}"}
+{"id": "446.png", "formula": "\\begin{align*} & \\int _ 0 ^ \\infty \\frac { d t } { t } \\ , t ^ \\beta \\cdot 2 ^ { \\gamma - 2 \\zeta } \\Gamma \\left ( \\frac { \\gamma + 1 } { 2 } \\right ) t ^ { \\frac { \\gamma - 2 \\zeta } { 2 } } \\ , _ 1 \\tilde { F } _ 1 \\left ( \\frac { 2 \\zeta - \\gamma } { 2 } ; \\zeta + \\frac 1 2 ; - \\frac { 1 } { 4 t } \\right ) = C ( \\beta , \\gamma , \\zeta ) , \\end{align*}"}
+{"id": "8651.png", "formula": "\\begin{align*} \\frac { 1 + | \\nabla _ X \\theta | ^ 2 } { h } \\partial _ z ^ 2 \\tilde u = \\tilde { f } - \\mu \\nabla _ X \\cdot ( h \\nabla _ X \\tilde u ) + \\mu \\nabla _ X \\cdot ( \\nabla _ X \\theta \\partial _ z \\tilde u ) + \\mu \\partial _ z ( \\nabla _ X \\theta \\cdot \\nabla _ X \\tilde u ) - \\frac { ( \\partial _ z | \\nabla _ X \\theta | ^ 2 ) } { h } \\partial _ z \\tilde u , \\end{align*}"}
+{"id": "3170.png", "formula": "\\begin{align*} ( C ( \\mathbf { L _ 0 } , \\mathbf { L _ 1 } ) , \\mathfrak { n } _ { 0 , 0 } ) = ( C F ^ * ( \\mathbf { L _ 0 } , \\mathbf { L _ 1 } ) , \\mu ^ 1 ) , \\end{align*}"}
+{"id": "6710.png", "formula": "\\begin{align*} \\begin{gathered} \\left ( \\prod _ { j : r \\to 1 } P _ j \\right ) e ^ { - r X } = \\prod _ { j : r \\to 1 } ( e ^ { ( r - j ) X } P _ j e ^ { - ( r - j + 1 ) X } ) = \\prod _ { j : r \\to 1 } \\mathrm { A d } _ { e ^ { ( r - j ) X } } ( P _ j e ^ { - X } ) , \\\\ e ^ { r X } \\prod _ { i : 1 \\to r } P _ i = \\prod _ { i : 1 \\to r } ( e ^ { ( r - i + 1 ) X } P _ i e ^ { - ( r - i ) X } ) = \\prod _ { i : 1 \\to r } \\mathrm { A d } _ { e ^ { ( r - i ) X } } ( e ^ X P _ i ) , \\end{gathered} \\end{align*}"}
+{"id": "8639.png", "formula": "\\begin{align*} | \\mathcal { L } _ 1 ^ { \\mu } [ \\beta b ] u + b u | _ { H ^ s } = \\mu | \\mathcal { R } [ X , \\mathrm { D } ] u | _ { H ^ s } \\lesssim \\mu | u | _ { H ^ { s + 2 } } . \\end{align*}"}
+{"id": "4557.png", "formula": "\\begin{align*} ( \\mathcal { B } _ 1 { \\mathbf V } _ { x _ 2 } , { \\mathbf V } _ { x _ 2 } ) | _ { x _ 1 = 0 } = 2 [ \\hat { u } _ 2 - \\hat \\lambda \\hat { H } _ 2 ] \\partial _ 2 \\dot { q } ^ + \\partial _ { 2 } \\partial _ 2 \\varphi + \\mbox { l . o . t } \\ , , \\end{align*}"}
+{"id": "4018.png", "formula": "\\begin{align*} \\chi ^ n _ { \\varphi _ t } = e ^ { b _ t + t b ' } g ^ t g ^ { 1 - t } _ 2 \\chi ^ m _ { \\varphi _ t } \\wedge \\omega ^ { n - m } , t \\in [ 0 , 1 ] , \\end{align*}"}
+{"id": "57.png", "formula": "\\begin{align*} \\mathcal Q _ 3 ^ { \\rm { r e n } } = \\sum _ { i \\neq j } P _ i Q _ j g ( x _ i - x _ j ) Q _ i Q _ j + h . c . \\end{align*}"}
+{"id": "6122.png", "formula": "\\begin{align*} \\phi _ f ' ( \\alpha ' ) = \\lim _ { ( \\delta - \\gamma ) \\to \\infty } \\frac { 1 } { \\delta - \\gamma } \\int _ { \\gamma } ^ { \\delta } \\mathrm { R e } \\frac { f ' } { f } ( \\alpha ' + i t ) \\mathrm { d } t . \\end{align*}"}
+{"id": "1353.png", "formula": "\\begin{align*} K _ { \\mathcal { Y } _ s } + \\Gamma = g _ s ^ * ( K _ { \\mathcal { X } _ s } + \\mathcal { D } _ s ) , \\end{align*}"}
+{"id": "358.png", "formula": "\\begin{align*} \\eta \\ ; R = S \\ ; \\psi \\textrm { a n d } \\psi \\ ; \\phi _ g = \\varphi _ { \\eta ( g ) } \\ ; \\psi \\textrm { f o r a l l } g \\in G . \\end{align*}"}
+{"id": "4741.png", "formula": "\\begin{align*} T = \\big \\{ ( x , y ) \\in \\mathbb R ^ 2 \\ , : \\ , y \\ge 0 , \\ , y < x < 1 - y \\big \\} . \\end{align*}"}
+{"id": "485.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ \\infty \\frac { \\Gamma \\left ( \\frac { \\gamma + 1 + 2 k } { 2 } \\right ) } { k ! \\Gamma \\left ( \\zeta + \\frac 1 2 + k \\right ) } \\left ( \\frac { 1 } { 4 t } \\right ) ^ k = \\Gamma \\left ( \\frac { 1 } { 2 } ( \\gamma + 1 ) \\right ) \\cdot \\ , _ 1 { \\tilde F } _ 1 \\left ( \\frac { 1 } { 2 } ( \\gamma + 1 ) ; \\zeta + \\frac 1 2 ; \\frac { 1 } { 4 t } \\right ) . \\end{align*}"}
+{"id": "8848.png", "formula": "\\begin{align*} \\gamma = h _ { t } ^ { 2 \\beta } , \\quad \\quad \\gamma = \\frac { h _ { t } } { \\sqrt { T d } } . \\end{align*}"}
+{"id": "3352.png", "formula": "\\begin{align*} g ( - \\frac { 1 } { \\tau } ) = \\begin{pmatrix} S & S \\\\ S & - S \\end{pmatrix} g ( \\frac { \\tau } { 2 } ) , \\end{align*}"}
+{"id": "3494.png", "formula": "\\begin{align*} U _ { y , t } = \\{ y _ i | \\log | t | | \\leq - \\log | z _ i | \\leq y _ i | \\log | t | | + 1 , \\forall 1 \\leq i \\leq m \\} \\subset B ( 1 ) ^ { n - m } \\times ( \\C ^ * ) ^ m , \\end{align*}"}
+{"id": "115.png", "formula": "\\begin{align*} - \\Delta = \\sum _ { k \\in \\Lambda ^ * } \\tau _ k a _ k ^ { \\dagger } a _ k + b \\frac { K _ L ^ 2 } { \\ell ^ 2 } n _ + ^ H , \\end{align*}"}
+{"id": "336.png", "formula": "\\begin{align*} w : = \\mathsf { D } _ { j } \\left ( s \\right ) \\psi - \\mathsf { S } _ { j } \\left ( s \\right ) \\varphi . \\end{align*}"}
+{"id": "836.png", "formula": "\\begin{align*} \\| ( N - T ) x \\| & = \\| \\eta \\Psi _ 2 ( x ) + ( 1 - \\epsilon ) ( 1 - \\eta ) T x - T x \\| \\\\ & = \\| \\eta \\Psi _ 2 ( x ) - \\epsilon ( 1 - \\eta ) T x + T x - \\eta T x - T x \\| \\\\ & = \\| \\eta ( \\Psi _ 2 ( x ) - T x ) - \\epsilon ( 1 - \\eta ) T x \\| \\\\ & \\leq \\| \\eta ( \\Psi _ 2 ( x ) - T x ) \\| + \\epsilon \\| 1 - \\eta \\| \\| T x \\| \\\\ & \\leq \\| \\eta ( \\Psi _ 2 ( x ) - T x ) \\| + 2 \\epsilon . \\end{align*}"}
+{"id": "6623.png", "formula": "\\begin{align*} f ( \\tau ; \\beta ) : = { \\pi \\beta \\over | \\tau | } S _ \\infty ( \\tau ; \\beta ) , 0 < \\tau < { \\rm m i n } \\ , ( 2 \\pi , \\pi \\beta ) , \\end{align*}"}
+{"id": "7737.png", "formula": "\\begin{align*} \\begin{array} { l l l l l l l l l l l } 0 = \\omega ( C ) - \\omega ( C ) & \\le | E _ { f = ( 0 , 0 ) } \\cap E ( C ) | + \\Omega | E ( C ) - E _ { f = ( 0 , 0 ) } | - \\omega ( C ) \\\\ & \\le \\frac { \\omega ( C ) } { 4 } + \\Omega ( \\frac { \\omega ( C ) } { 4 } + 2 - \\Omega ) - \\omega ( C ) \\\\ & = - 3 < 0 , \\end{array} \\end{align*}"}
+{"id": "695.png", "formula": "\\begin{align*} \\mathbf u ( t ) = \\mathbf S ( t ) \\mathbf u _ 0 + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf N ( \\mathbf u ( s ) ) \\ , d s \\\\ + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf M ( \\mathbf u ( s ) ) \\ , d W ( s ) \\end{align*}"}
+{"id": "3424.png", "formula": "\\begin{align*} \\mu _ 0 = ( C _ 0 \\int _ { E _ J } \\Omega _ { E _ J } \\wedge \\overline { \\Omega } _ { E _ J } ) | d x _ 1 \\ldots d x _ m | , \\end{align*}"}
+{"id": "34.png", "formula": "\\begin{align*} \\sum _ { i \\neq j } P _ i Q _ j ( g + g \\omega ) ( x _ i - x _ j ) Q _ j P _ i = \\Big ( \\widehat g ( 0 ) + \\widehat { g \\omega } ( 0 ) \\Big ) \\frac { n _ 0 n _ + } { \\vert \\Lambda \\vert } . \\end{align*}"}
+{"id": "5907.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } G _ 1 = 0 . \\end{align*}"}
+{"id": "914.png", "formula": "\\begin{align*} x ^ k ( n \\eta _ 2 - m \\phi _ 2 ) + \\phi _ { 1 x } - \\eta _ { 1 x } = 0 , \\end{align*}"}
+{"id": "6676.png", "formula": "\\begin{align*} \\tilde { g } _ 2 = - { q \\over \\pi ^ 2 } , \\tilde { g } _ 3 = { 1 \\over 8 \\pi ^ 3 } ( - \\tilde { p } + q ^ 2 ) , \\tilde { g } _ 3 = { 1 \\over 1 6 \\pi ^ 4 } q ( - 1 - 3 \\tilde { p } + 2 q ^ 2 ) . \\end{align*}"}
+{"id": "8198.png", "formula": "\\begin{align*} x _ 1 & = \\frac { 1 } { 2 } \\sum _ { a = 1 } ^ n | z _ a | ^ 2 - \\frac { 1 } { 2 } \\sum _ { a = 1 } ^ n | w _ a | ^ 2 , \\\\ x _ 2 + i x _ 3 & = - i \\sum _ { a = 1 } ^ n z _ a w _ a . \\end{align*}"}
+{"id": "6165.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 \\end{pmatrix} A _ 4 = \\begin{pmatrix} 0 & 1 \\end{pmatrix} , A _ 4 \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\end{align*}"}
+{"id": "2653.png", "formula": "\\begin{align*} P ( X _ n \\notin A ) ^ { 1 / r _ n } \\le \\sum _ { i = 1 } ^ \\infty P ( w _ { i } ( X _ n , \\delta _ i ) > \\frac { 1 } { 2 ^ i } ) ^ { 1 / r _ n } + \\sum _ { i = 1 } ^ \\infty P ( X _ n ( t _ i ) \\notin K _ i ) ^ { 1 / r _ n } < 2 \\epsilon \\ , . \\end{align*}"}
+{"id": "5520.png", "formula": "\\begin{align*} O _ A ^ { ( g b ) } ( p , s ) = \\left ( M ( p , s ) \\right ) ^ { - 1 } O _ A ^ { ( g ) } ( p , s / p ) M ( p , s ) , O _ B ^ { ( g b ) } ( p , s ) = \\left ( M ( p , s ) \\right ) ^ { - 1 } O _ B ^ { ( g ) } ( p , s / p ) M ( p , s ) , \\end{align*}"}
+{"id": "5003.png", "formula": "\\begin{align*} \\int _ { B _ p ( T + \\frac { \\eta } { 8 } ) \\setminus B _ p ( T - \\frac { \\eta } { 8 } ) } e ^ { - b r } = \\frac { \\eta } { 4 } e ^ { - b \\xi } L ( \\xi ) . \\end{align*}"}
+{"id": "7804.png", "formula": "\\begin{align*} \\widetilde { \\xi } _ a ^ { } = \\frac { \\tau _ 2 } { 8 \\pi ^ 2 } \\sum _ { \\hat { \\gamma } \\in \\Lambda ^ { + } \\cup \\{ 0 \\} } n _ { \\hat { \\gamma } } q _ a \\sum _ { ( m , n ) \\in \\mathbb { Z } ^ 2 - \\{ 0 \\} } \\frac { e ^ { - S _ { \\hat { \\gamma } , m , n } } } { | m \\tau + n | ^ 2 } \\frac { 1 + t _ { + } ^ { m , n } t } { t - t _ { + } ^ { m , n } } \\ , . \\end{align*}"}
+{"id": "1545.png", "formula": "\\begin{align*} \\frac 1 2 m ^ 2 K _ S ^ 2 + = P _ m ( S ) \\leq P _ m ( W ) = \\frac 1 2 m ^ 2 K _ W ^ 2 + \\end{align*}"}
+{"id": "422.png", "formula": "\\begin{align*} f _ { \\ell , m } = [ u _ { \\ell , m } ] _ { \\ell , m } , \\end{align*}"}
+{"id": "3398.png", "formula": "\\begin{align*} \\begin{array} { l l l l } \\alpha _ 1 F _ 1 = 0 , \\alpha _ 2 F _ 2 = 0 \\mbox { a n d } \\alpha _ 3 F _ 3 = 0 , \\end{array} \\end{align*}"}
+{"id": "4679.png", "formula": "\\begin{align*} e ( S _ j ' , T _ j ) & \\geq | S _ j ' | \\cdot ( \\frac { n } { 2 ( 2 l + 1 ) } - 2 ) - \\left ( \\sum \\limits _ { q = 1 } ^ { 2 l + 1 } e ( S _ j ' , S _ q - S _ j ' ) + 2 e ( S _ j ' ) \\right ) \\\\ & \\geq | S _ j ' | \\cdot ( \\frac { n } { 2 ( 2 l + 1 ) } - 2 ) - ( 2 l + 3 ) k n \\\\ & \\geq | S _ j ' | \\cdot \\frac { n } { 2 ( 2 l + 1 ) } - ( 2 | S _ j ' | + ( 2 l + 3 ) k n ) \\\\ & = | S _ j ' | \\cdot \\frac { n } { 4 ( 2 l + 1 ) } + | S _ j ' | \\cdot \\frac { n } { 4 ( 2 l + 1 ) } - ( 2 | S _ j ' | + ( 2 l + 3 ) k n ) \\\\ & > | S _ j ' | \\cdot \\frac { n } { 4 ( 2 l + 1 ) } \\end{align*}"}
+{"id": "8701.png", "formula": "\\begin{align*} \\tilde { f } = \\tilde { \\phi } _ 0 ( u _ 1 , . . . , u _ n ) , ~ ~ ~ x _ i = \\tilde { \\phi } _ i ( u _ 1 , . . . , u _ n ) , ~ i = 1 , . . . , n , \\end{align*}"}
+{"id": "3822.png", "formula": "\\begin{align*} \\overline { \\mathcal { F } } ( \\gamma _ i \\mid \\mu _ i ) & = \\int _ X \\overline F ( \\sigma _ i ) d \\mu _ i + F ' _ { \\infty } \\gamma _ i ^ { \\perp } ( X ) , \\\\ { \\mathcal { F } } ( \\gamma \\mid \\nu _ X ) & = \\int _ { X \\times X } F ( \\sigma ) d \\nu _ X + F ' _ { \\infty } \\gamma ^ { \\perp } ( X \\times X ) . \\end{align*}"}
+{"id": "4736.png", "formula": "\\begin{align*} J ( \\Delta X ) = 2 \\sum \\limits _ { i = 1 } ^ D \\frac { \\left [ \\frac 1 4 + r ^ 2 \\right ] \\left [ ( \\frac 1 4 + r ^ 2 ) ^ 2 + \\Delta x _ i ^ 2 \\right ] } { ( \\frac 1 4 + r ^ 2 ) ^ 4 - \\Delta x _ i ^ 4 } \\end{align*}"}
+{"id": "2471.png", "formula": "\\begin{align*} & \\forall k \\in \\mathbb { N } ^ \\star , \\ , T _ { a ^ k } = \\frac { 1 } { k } \\left ( \\right ) \\\\ & T _ { a '^ k } \\underset { k \\rightarrow \\infty } { \\rightarrow } P ' _ { A } & T _ { a ''^ k } \\underset { k \\rightarrow \\infty } { \\rightarrow } P '' _ { A } \\\\ & \\gamma ' ( k ) \\underset { k \\rightarrow \\infty } { \\sim } \\frac { k } { t } & \\gamma '' ( k ) \\underset { k \\rightarrow \\infty } { \\sim } \\frac { k } { 1 - t } . \\end{align*}"}
+{"id": "5490.png", "formula": "\\begin{align*} ( \\mathbf a \\mathbf s ) ^ 2 = \\mathbf s \\mathbf a ^ { - 1 } \\end{align*}"}
+{"id": "5530.png", "formula": "\\begin{align*} \\frac { \\partial P _ { k } } { \\partial z _ m } = \\alpha _ { m } ( k ) z ^ { \\alpha ( k _ 0 ) } + \\sum _ { j = 0 } ^ { k _ 0 } b _ j e _ j = \\alpha _ { m } ( k ) \\left ( e _ { k _ 0 } + \\sum _ { j = 0 } ^ { k _ 0 } \\tilde { c } _ j e _ j \\right ) . \\end{align*}"}
+{"id": "2505.png", "formula": "\\begin{align*} \\boldsymbol { v } _ k = ( \\boldsymbol { v } _ k ^ c , \\boldsymbol { v } _ k ^ s ) ^ T , \\boldsymbol { v } _ k ^ { \\perp } = ( - \\boldsymbol { v } _ k ^ s , \\boldsymbol { v } _ k ^ c ) ^ T , \\textbf { c u r l } \\ , \\boldsymbol { v } _ k = ( \\textbf { c u r l } \\ , \\boldsymbol { v } _ k ^ c , \\textbf { c u r l } \\ , \\boldsymbol { v } _ k ^ s ) ^ T . \\end{align*}"}
+{"id": "509.png", "formula": "\\begin{align*} u _ n ( s , \\omega ) = u ( 0 , \\omega ) \\mathbb 1 _ { \\{ 0 \\} } ( s ) + \\sum _ { i = 1 } ^ { 2 ^ n } u ( t _ i , \\omega ) \\mathbb 1 _ { ( t _ { i - 1 } , t _ i ] } ( s ) . \\end{align*}"}
+{"id": "7406.png", "formula": "\\begin{align*} \\partial _ { t } \\left ( \\frac { 1 } { \\rho _ { n } } \\right ) + u _ { n } \\partial _ { x } \\left ( \\frac { 1 } { \\rho _ { n } } \\right ) = \\frac { 1 } { \\rho _ { n } } \\partial _ { x } u _ { n } . \\end{align*}"}
+{"id": "7917.png", "formula": "\\begin{align*} 2 = | \\varphi ( x ) | ^ 2 = ( 1 / 2 ) ( - 1 + b \\sqrt { n } ) ( 1 / 2 ) ( - 1 - b \\sqrt { n } ) = ( 1 / 4 ) ( 1 - b ^ 2 n ) . \\end{align*}"}
+{"id": "2214.png", "formula": "\\begin{align*} \\left ( I _ { a + } ^ \\alpha f \\right ) ( x ) : = \\frac { 1 } { \\Gamma ( \\alpha ) } \\underset { a } { \\overset { x } { \\int } } \\frac { f ( t ) d t } { ( x - t ) ^ { 1 - \\alpha } } , \\ , ( x > a ; \\ , R e ( \\alpha > 0 ) ) \\end{align*}"}
+{"id": "8181.png", "formula": "\\begin{align*} & K _ i ( x , N , p ) = \\binom { N } { i } ( p - 1 ) ^ i \\hat K _ i ( x ; ( p - 1 ) / p , N ) \\\\ & H _ { i } ( x , N , p ) = \\left ( \\ , \\binom { N } { i } - \\binom { N } { i - 1 } \\right ) Q _ i ( x ; \\mu - N - 1 , - \\mu - 1 , \\mu ) , \\\\ & E _ i ( x , N , p ) = \\binom { p } { i } \\binom { N - p } { i } R _ i ( \\lambda ( x ) ; \\mu - N - 1 , - \\mu - 1 , \\mu ) , \\end{align*}"}
+{"id": "4919.png", "formula": "\\begin{align*} \\sum _ { m , n \\geq 1 } \\dfrac { ( - 1 ) ^ m } { n ^ 2 + 2 ^ \\ell m ^ 2 } = \\begin{cases} - \\dfrac { \\pi ^ 2 } { 2 4 } - \\dfrac { \\pi \\cdot \\log 2 } { 8 } , & \\ell = 1 , \\\\ - \\dfrac { 7 \\pi ^ 2 } { 9 6 } - \\dfrac { \\pi \\cdot \\log 2 } { 3 2 } - \\dfrac { \\pi \\cdot \\log ( \\sqrt { 2 } - 1 ) } { 8 } , & \\ell = 2 , \\\\ - \\dfrac { 3 1 \\pi ^ 2 } { 3 8 4 } - \\dfrac { 5 \\pi \\log 2 } { 1 2 8 } + \\dfrac { \\pi \\cdot \\log ( \\sqrt { 2 } - 1 ) } { 3 2 } - \\dfrac { \\pi \\cdot \\log ( 1 - 2 ^ { - 1 / 4 } ) } { 8 } , & \\ell = 3 . \\end{cases} \\end{align*}"}
+{"id": "5877.png", "formula": "\\begin{align*} E _ { s , V } L ( \\epsilon _ { 1 } ) = ( 1 - \\epsilon _ { 1 } + \\epsilon _ { 1 } e ^ { k \\Delta ^ { 2 } } ) ^ { V } . \\end{align*}"}
+{"id": "4237.png", "formula": "\\begin{align*} d \\omega ^ 1 = 2 i \\ , \\omega ^ { 1 3 } , \\ \\ d \\omega ^ 2 = - 2 i \\ , \\omega ^ { 2 3 } , \\ \\ d \\omega ^ 3 = 0 , \\end{align*}"}
+{"id": "8940.png", "formula": "\\begin{align*} \\| \\Tilde { \\theta } \\| _ { \\C ^ k } \\ll \\rho ^ { - ( 2 k + 2 N ) } , N = n ^ 2 + m ^ 2 + n m + n - 1 \\end{align*}"}
+{"id": "8284.png", "formula": "\\begin{align*} \\begin{cases} \\bar u _ { t t } = \\bar u _ { x x } , & ( x , t ) \\in ( 0 , 1 ) \\times ( 0 , \\infty ) , \\\\ \\bar u ( 0 , t ) = 0 , & t \\in ( 0 , \\infty ) \\\\ \\bar u _ x ( 1 , t ) = - a \\bar u _ t ( 1 , t ) + \\frac { b d } { c } \\bar u ( 1 , t ) , & t \\in ( 0 , \\infty ) \\\\ \\bar u ( x , 0 ) = \\bar u _ 0 ( x ) , \\ : \\bar u _ t ( x , 0 ) = \\bar u _ 1 ( x ) , & x \\in ( 0 , 1 ) , \\end{cases} \\end{align*}"}
+{"id": "1895.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( \\frac { \\partial u _ h } { \\partial t } , \\phi _ h \\right ) + a _ h ( u _ h , \\phi _ h ) + \\sqrt { \\epsilon } \\ , b _ h ( w _ h , \\phi _ h ) & + \\left ( \\rho _ h u _ h , \\phi _ h \\right ) \\\\ & = \\left ( \\rho _ h V _ h , \\phi _ h \\right ) \\forall \\ , \\ , \\phi _ h \\in X _ h , \\end{aligned} \\end{align*}"}
+{"id": "4079.png", "formula": "\\begin{align*} \\omega _ { g , n + 1 } ^ { ( 0 ) } ( p _ 0 , J ) + \\omega _ { g , n + 1 } ^ { ( 0 ) } ( \\sigma ( p _ 0 ) , J ) = 0 . \\end{align*}"}
+{"id": "5620.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\exp _ p ( \\mu | u _ n | ^ { p ' } ) r ^ { \\alpha _ 0 } \\mathrm d r & = \\int _ 0 ^ \\infty \\left [ \\exp _ p ( \\mu | u _ n | ^ { p ' } ) - \\dfrac { \\mu ^ { p - 1 } | u _ n | ^ p } { \\Gamma ( p ) } \\right ] r ^ { \\alpha _ 0 } \\mathrm d r + \\dfrac { \\mu ^ { p - 1 } } { \\Gamma ( p ) } \\int _ 0 ^ \\infty | u _ n | ^ p r ^ { \\alpha _ 0 } \\mathrm d r \\\\ & \\overset { n \\to \\infty } \\longrightarrow \\int _ 0 ^ \\infty \\exp _ p ( \\mu | u | ^ { p ' } ) r ^ { \\alpha _ 0 } \\mathrm d r , \\end{align*}"}
+{"id": "6569.png", "formula": "\\begin{align*} \\tilde G _ \\Lambda ( \\sigma ) = ( R _ \\Lambda ( { D } ( \\sigma ) + \\varepsilon \\Delta ) R _ \\Lambda ) ^ { - 1 } . \\end{align*}"}
+{"id": "2292.png", "formula": "\\begin{align*} \\| g _ 1 ( x ) \\| _ { L ^ 2 } \\geq \\frac { e ^ { - \\pi \\alpha \\left ( \\frac { n - 1 } { 2 } \\right ) ^ 2 } } { 2 } \\left ( \\sum _ { j = 1 } ^ { n } \\varepsilon _ j ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "5462.png", "formula": "\\begin{align*} \\log \\left ( \\int e ^ f d \\pi \\right ) & \\leq \\sum _ { n = 0 } ^ \\infty \\exp \\{ \\lambda ( | \\nabla f ( n ) | + \\log | \\nabla f ( n ) | + \\log \\lambda ) e ^ { | \\nabla f ( n ) | } \\} \\pi ( n ) \\\\ & + \\log \\left ( \\sum _ { n = 0 } ^ \\infty \\exp \\left \\{ \\sum _ { k = 0 } ^ { n - 1 } G ^ * \\left ( \\frac { 1 } { \\pi ( k ) } \\right ) \\pi ( k ) \\right \\} \\pi ( n ) \\right ) \\end{align*}"}
+{"id": "1078.png", "formula": "\\begin{align*} \\widetilde { h } _ { 0 t } \\mid _ { S _ { T } \\diagdown \\left ( \\Gamma _ { T } ^ { - } \\cup \\Gamma _ { T } ^ { + } \\right ) } = \\widetilde { r } _ { 0 t } \\mid _ { S _ { T } \\diagdown \\left ( \\Gamma _ { T } ^ { - } \\cup \\Gamma _ { T } ^ { + } \\right ) } = 0 . \\end{align*}"}
+{"id": "2525.png", "formula": "\\begin{align*} \\mathcal { M } _ { | \\cdot | } ^ { \\oplus } ( \\boldsymbol { \\eta } , \\boldsymbol { \\tau } ) = \\frac { 1 } { \\underline { c } } \\left ( { C _ F ^ { } } \\ , \\| \\mathcal { R } _ 1 ( \\boldsymbol { \\eta } , \\boldsymbol { \\tau } ) \\| + \\| \\mathcal { R } _ 2 ( \\boldsymbol { \\eta } , \\boldsymbol { \\tau } ) \\| \\right ) \\end{align*}"}
+{"id": "7451.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\vert \\nabla { y _ 2 } _ * \\vert ^ { p - 2 } \\nabla { y _ 2 } _ * \\cdot \\nabla w d z & + \\int _ { \\Omega } \\eta \\vert \\nabla { y _ 2 } _ * \\vert ^ { q - 2 } \\nabla { y _ 2 } _ * \\cdot \\nabla w d z \\\\ \\geq & \\int _ { \\Omega } a _ 2 { y _ 2 } _ * ^ { - \\nu } w d z + \\lambda ( \\kappa _ 2 + 1 ) \\int _ { \\Omega } \\vert { y _ 1 } _ * \\vert ^ { \\kappa _ 1 + 1 } \\vert { y _ 2 } _ * \\vert ^ { \\kappa _ 2 } w d z . \\end{align*}"}
+{"id": "2394.png", "formula": "\\begin{align*} W ( \\mathbf { x } , ( E _ n ) ) : = \\{ x \\in X : T ^ n ( x ) \\in x _ { n } + E _ n \\textrm { f o r i . m . } n \\in \\mathbb { N } \\} . \\end{align*}"}
+{"id": "2764.png", "formula": "\\begin{align*} G ( z ) = - \\frac { 1 } { 2 } \\int _ { \\partial \\mathbb { D } } \\frac { ( b - a ) ^ 2 } { 4 } \\frac { 1 } { v } \\left ( v - \\frac { 1 } { v } \\right ) ^ 2 \\frac { \\psi ( J _ { [ a , b ] } ( v ) ) } { z - J _ { [ a , b ] } ( v ) } \\mathrm { d } v , \\end{align*}"}
+{"id": "6358.png", "formula": "\\begin{align*} J ^ { 4 , a } _ { \\lambda \\mu } = M _ \\lambda ( u ) E _ \\mu ( v ) + M _ \\mu ( v ) E _ \\lambda ( u ) - 2 P _ \\lambda ( u ) P _ \\mu ( v ) , \\end{align*}"}
+{"id": "5648.png", "formula": "\\begin{align*} K _ { Z _ 1 } + D _ 1 = \\sigma _ 1 ^ * ( K _ { Z _ 0 } + D _ 0 ) . \\end{align*}"}
+{"id": "826.png", "formula": "\\begin{align*} \\eta : = \\Psi _ { \\epsilon } \\circ \\tilde { h } . \\end{align*}"}
+{"id": "4646.png", "formula": "\\begin{align*} ( \\mathbf { k } \\cdot f ) ( \\mathbf { g } ) : = f ( \\mathbf { k } ^ { - 1 } \\cdot \\mathbf { g } ) \\end{align*}"}
+{"id": "3688.png", "formula": "\\begin{align*} B ^ { T } _ { n , k - 1 , k - 2 } \\phi _ u ( \\eta ) = \\sum _ { \\substack { \\tau \\in \\binom { [ n ] } { k - 1 } , \\\\ \\eta \\subset \\tau } } \\phi _ u ( \\tau ) = \\sum _ { \\substack { \\tau \\in \\binom { [ n ] } { k - 1 } , \\\\ \\eta \\subset \\tau , \\ , u \\notin \\tau } } \\phi ( \\tau \\cup \\{ u \\} ) = \\sum _ { \\substack { \\sigma \\in \\binom { [ n ] } { k } , \\\\ \\eta \\cup \\{ u \\} \\subset \\sigma } } \\phi ( \\sigma ) = B ^ T _ { n , k , k - 1 } \\phi ( \\eta \\cup \\{ u \\} ) = 0 . \\end{align*}"}
+{"id": "8536.png", "formula": "\\begin{align*} | z _ { i + 1 } ^ { k } - z _ { i } ^ { k } | < \\frac { 1 } { k } , \\mbox { f o r e v e r y } i = 1 , \\dots , N _ { k } - 1 . \\end{align*}"}
+{"id": "4600.png", "formula": "\\begin{align*} f ^ { ( i ) } ( \\xi _ i ) = \\sum _ { n \\gg - \\infty } a _ n ^ { ( i ) } \\ , \\xi _ i ^ n . \\end{align*}"}
+{"id": "2889.png", "formula": "\\begin{align*} { \\bf \\imath _ 0 } : = \\sum _ { v \\in W _ 0 } T _ v , \\end{align*}"}
+{"id": "7010.png", "formula": "\\begin{align*} \\frac { 1 } { 2 - \\lambda } & = \\frac { \\log ( 0 . 0 4 2 1 6 a _ 1 a _ 2 ( a _ 2 - a _ 1 ) ^ { - 2 } c ^ 2 ) } { \\log ( 0 . 0 1 6 8 5 8 a _ 1 ( a _ 1 ' ) ^ { - 1 } a _ 2 ^ { - 1 } ( a _ 2 - a _ 1 ) ^ { - 2 } c ) } \\\\ & < \\frac { 2 \\log ( 0 . 2 0 5 3 3 a _ 1 ^ { 1 / 2 } a _ 2 ^ { 1 / 2 } ( a _ 2 - a _ 1 ) ^ { - 1 } c ) } { \\log ( 0 . 0 1 6 8 5 a _ 1 ( a _ 1 ' ) ^ { - 1 } a _ 2 ^ { - 1 } ( a _ 2 - a _ 1 ) ^ { - 2 } c ) } . \\end{align*}"}
+{"id": "2062.png", "formula": "\\begin{align*} k ( x _ 0 , r ) & \\leq C _ n \\int ^ { 2 r } _ r s ^ { - 1 } k ( x _ 0 , s ) d s \\\\ & = C _ n \\left ( f ( x _ 0 , 2 r ) - f ( x _ 0 , r ) \\right ) . \\end{align*}"}
+{"id": "2825.png", "formula": "\\begin{align*} \\ell i _ { 2 } = \\ell \\circ \\delta . \\end{align*}"}
+{"id": "4508.png", "formula": "\\begin{align*} | | ( I - S _ { \\theta _ i } ) & ( \\partial _ t \\varphi ^ a - u ^ a _ 1 + u _ 2 ^ a \\partial _ 2 \\varphi ^ a ) | | _ { H ^ { s - 1 } ( \\Gamma _ T ) } \\le C \\theta _ i ^ { s - \\alpha } | | \\partial _ t \\varphi ^ a - u ^ a _ 1 + u _ 2 ^ a \\partial _ 2 \\varphi ^ a | | _ { H ^ { \\alpha - 1 } ( \\Gamma _ T ) } \\\\ & \\le C \\theta _ i ^ { s - \\alpha } ( | | \\varphi ^ a | | _ { H ^ { \\alpha - 1 } ( \\Gamma _ T ) } + | | u ^ a | | _ { \\alpha , \\ast , T } ) \\le C \\delta \\theta _ i ^ { s - \\alpha } \\ , , \\end{align*}"}
+{"id": "8577.png", "formula": "\\begin{align*} P ( \\Sigma _ b ) = \\begin{pmatrix} ( 1 + \\partial _ z \\sigma ) \\mathrm { I d } & - \\sqrt { \\mu } \\nabla _ X \\sigma \\\\ - \\sqrt { \\mu } ( \\nabla _ X \\sigma ) ^ T & \\dfrac { 1 + \\mu h _ b | \\nabla _ X \\sigma | ^ 2 } { 1 + \\partial _ z \\sigma } \\end{pmatrix} . \\end{align*}"}
+{"id": "1879.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ t u + u \\partial _ x u - \\epsilon \\partial _ x ^ 2 u & = \\rho V - \\rho u \\mbox { i n } ( 0 , T ] \\times I , \\\\ u ( 0 , x ) & = u _ 0 ( x ) \\qquad \\mbox { i n } I , \\end{aligned} \\right . \\end{align*}"}
+{"id": "8081.png", "formula": "\\begin{align*} \\vert g ( f ) ( x ) \\vert \\leq \\sum \\limits _ { j = 1 } ^ { \\infty } \\lambda _ { j } \\vert g ( a _ { j } ) ( x ) \\vert + \\sum \\limits _ { j = 1 } ^ { \\infty } \\mu _ { j } \\vert g ( b _ { j } ) ( x ) \\vert = \\uppercase \\expandafter { \\romannumeral 1 } + \\uppercase \\expandafter { \\romannumeral 2 } . \\end{align*}"}
+{"id": "5016.png", "formula": "\\begin{align*} \\phi _ { t _ 1 } \\circ \\phi _ { t _ 2 } = \\phi _ { t _ 1 t _ 2 } t _ 1 , t _ 2 > 0 . \\end{align*}"}
+{"id": "514.png", "formula": "\\begin{align*} \\left \\{ \\tau _ R > t \\right \\} = \\left \\{ \\tau _ R > t \\right \\} \\cap \\left \\{ \\tau > t \\right \\} , \\end{align*}"}
+{"id": "5389.png", "formula": "\\begin{align*} d S _ { j k } ( t ) = S _ { j k } ( t ) d U _ { k k } ( t ) + \\sum _ { 1 \\leq \\ell \\leq N : \\ell \\neq k } S _ { j \\ell } ( t ) \\dfrac { ( S ^ { - 1 } ( t ) d M ( t ) S ( t ) ) _ { \\ell k } } { \\Lambda _ k ( t ) - \\Lambda _ { \\ell } ( t ) } , 1 \\leq j , k \\leq N , \\ , t \\geq 0 , \\end{align*}"}
+{"id": "3157.png", "formula": "\\begin{align*} J _ Y = I _ Y \\hbox { n e a r } D _ Y . \\end{align*}"}
+{"id": "2448.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ N | u _ n \\rangle \\langle u _ n | \\leq ( - \\Delta _ { \\Omega } ) ^ { - s } L ^ 2 ( \\Omega ) . \\end{align*}"}
+{"id": "8818.png", "formula": "\\begin{align*} b _ { t + 1 } < \\left ( 1 - \\frac { 2 } { t } \\right ) b _ { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { t ^ { p _ { i } + 1 } } , \\end{align*}"}
+{"id": "6923.png", "formula": "\\begin{align*} \\gamma _ { n - 1 } < \\gamma _ { n + 1 } < u _ 2 ^ * < \\gamma _ { n + 2 } < \\gamma _ { n } n = 2 k , \\ k \\in \\mathbb N , \\end{align*}"}
+{"id": "8813.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\mathbb { E } \\big [ \\norm { g ^ { i } ( t ) - \\bar { g } ( t ) } ^ { 2 } | \\mathcal { F } _ { t - 1 } \\big ] & \\leq \\sum _ { i = 1 } ^ { n } \\mathbb { E } \\big [ \\norm { g ^ { i } ( t ) } ^ { 2 } | \\mathcal { F } _ { t - 1 } \\big ] \\\\ & \\leq 9 \\kappa d \\bar { L } ^ { 2 } n \\mathcal { K } ^ { 2 } + n d ^ { 2 } \\big ( \\frac { 9 h _ { t } ^ { 2 } \\kappa \\bar { L } ^ { 2 } } { 8 } + \\frac { 3 \\kappa \\sigma ^ { 2 } } { 2 h _ { t } ^ { 2 } } \\big ) , \\end{align*}"}
+{"id": "8892.png", "formula": "\\begin{align*} f _ * \\big ( ( f ^ * a ) h ^ * b \\big ) = a f _ * ( h ^ * b ) = a \\underline { h } ^ * ( \\pi _ { G * } b ) . \\end{align*}"}
+{"id": "6772.png", "formula": "\\begin{align*} \\omega _ n '' ( z ) - c \\omega _ n ' ( z ) + \\omega _ n ( z ) F ( \\phi _ n , \\psi _ n ) ( z ) = 0 . \\end{align*}"}
+{"id": "1003.png", "formula": "\\begin{align*} I _ 2 & = \\int _ { \\Omega _ - ^ { \\varepsilon } } - G _ 1 ^ - \\varphi _ { x _ 3 } + G _ 3 ^ - \\varphi _ { x _ 1 } \\ , d x = \\int _ { \\partial \\Omega ^ { \\varepsilon } _ - } \\varphi ( G _ 3 ^ - n _ 1 ^ { \\varepsilon } - G _ 1 ^ - n _ 3 ^ { \\varepsilon } ) \\ , d \\sigma + \\int _ { \\Omega ^ { \\varepsilon } _ { - } } \\varphi ( ( G _ 1 ^ - ) _ { x _ 3 } - ( G _ 3 ^ - ) _ { x _ 1 } ) \\ , d x . \\end{align*}"}
+{"id": "6127.png", "formula": "\\begin{align*} \\phi _ f ' ( \\alpha ' - 0 ; \\gamma , \\delta ) = \\phi _ f ' ( \\alpha ' + 0 ; \\gamma , \\delta ) + o ( 1 ) = \\phi _ f ' ( \\alpha ' ) + o ( 1 ) \\end{align*}"}
+{"id": "7359.png", "formula": "\\begin{align*} A & = q ^ { \\frac { 3 } { 2 } } \\big ( \\sqrt { \\gamma } x - 2 \\big ) \\big ( \\gamma ( 2 x ^ 2 - 9 ) + \\sqrt { \\gamma } x - 1 ) + 9 q ^ 2 \\gamma \\big ( \\sqrt { \\gamma } x + 1 \\big ) + \\sqrt { f ( x ) } \\end{align*}"}
+{"id": "3521.png", "formula": "\\begin{align*} \\sum _ { \\substack { 2 \\leqslant p \\leqslant x \\\\ } } \\log ( p ) \\left ( \\frac { d } { p } \\right ) = O ( x ^ { \\frac { 1 } { 2 } } \\log ( d x ) ^ 2 ) \\end{align*}"}
+{"id": "2173.png", "formula": "\\begin{align*} \\sigma \\bullet _ { n } ( t _ 1 \\bullet _ { m } ( a _ 1 , \\ldots , a _ m ) , \\ldots , t _ n \\bullet _ { m } ( a _ 1 , \\ldots , a _ m ) ) = ( \\sigma \\ast _ { n m } ( t _ 1 , \\ldots , t _ n ) ) \\bullet _ { m } ( a _ 1 , \\ldots , a _ m ) . \\end{align*}"}
+{"id": "2005.png", "formula": "\\begin{align*} ( \\mathrm { \\mathrm { H } } ^ { s , p } ( \\Omega ) ) ' = & \\mathrm { H } ^ { - s , p ' } _ 0 ( \\Omega ) \\ , ( \\mathrm { B } ^ { s } _ { p , q } ( \\Omega ) ) ' = \\mathrm { B } ^ { - s } _ { p ' , q ' , 0 } ( \\Omega ) \\\\ & ( \\mathrm { B } ^ { s } _ { p , q , 0 } ( \\Omega ) ) ' = \\mathrm { B } ^ { - s } _ { p ' , q ' } ( \\Omega ) \\end{align*}"}
+{"id": "5131.png", "formula": "\\begin{align*} { \\bf M } = \\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\\\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\\\ 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "3394.png", "formula": "\\begin{align*} V _ 2 = \\dfrac { v _ 2 ( v _ 1 ^ 2 - v _ 3 ^ 2 ) } { v _ 1 ( v _ 2 ^ 2 + v _ 3 ^ 2 ) } V _ 1 , \\ , V _ 3 = - \\dfrac { v _ 3 ( v _ 1 ^ 2 + v _ 2 ^ 2 ) } { v _ 1 ( v _ 2 ^ 2 + v _ 3 ^ 2 ) } V _ 1 \\end{align*}"}
+{"id": "628.png", "formula": "\\begin{align*} u ( t ) = \\phi ( t ) S _ { h ( \\xi ) } ( t ) f \\end{align*}"}
+{"id": "2131.png", "formula": "\\begin{align*} \\partial _ t ^ 2 u ^ { ( 0 ) } = \\frac 1 r \\partial _ r \\left ( r \\partial _ r u ^ { ( 0 ) } \\right ) , ( t , r ) \\in \\mathbb R _ t \\times ( 0 , \\infty ) . \\end{align*}"}
+{"id": "5077.png", "formula": "\\begin{align*} E ( - A ) = \\sum _ { p \\geq 0 } \\frac { ( - I ) ^ p } { 2 ^ p } = I \\sum _ { p \\geq 0 } \\frac { 1 } { ( - 2 ) ^ p } = \\frac { 2 } { 3 } I , \\end{align*}"}
+{"id": "5534.png", "formula": "\\begin{align*} \\nabla ^ 2 u ( z ) & = 0 \\mbox { i f } z \\in \\Omega , \\\\ u ( z ) & = 1 \\mbox { i f } z \\in \\partial \\Omega \\cap \\overline { B ( z _ 0 , r ) } , \\\\ u ( z ) & = 0 \\mbox { i f } z \\in \\partial \\Omega \\backslash \\overline { B ( z _ 0 , r ) } . \\end{align*}"}
+{"id": "3908.png", "formula": "\\begin{align*} 1 & = | L _ { N , 1 } | + | L _ { N , 2 } | + \\cdots + | L _ { N , 2 N + 1 } | \\\\ & \\leq ( 1 + 1 + \\cdots + 1 ) ^ { 1 / q } \\cdot ( | L _ { N , 1 } | ^ p + | L _ { N , 2 } | ^ p + \\cdots + | L _ { N , 2 N + 1 } | ^ p ) ^ { 1 / p } \\\\ & \\leq ( 2 N + 1 ) ^ { \\frac { p - 1 } { p } } \\cdot ( | L _ { N , 1 } | ^ p + | L _ { N , 2 } | ^ p + \\cdots + | L _ { N , 2 N + 1 } | ^ p ) ^ { 1 / p } \\end{align*}"}
+{"id": "7360.png", "formula": "\\begin{align*} f ( x ) = q \\Big ( q \\big ( \\sqrt { \\gamma } x - 2 \\big ) \\big ( 2 \\gamma x ^ 2 + \\sqrt { \\gamma } x - 9 \\gamma - 1 \\big ) + 9 ( \\gamma ^ { \\frac { 3 } { 2 } } x + \\gamma ) \\Big ) ^ 2 - 4 \\Big ( q \\gamma ( x ^ 2 - 3 ) - q \\sqrt { \\gamma } x + q + 3 \\gamma \\big ) ^ 3 \\Big ) \\end{align*}"}
+{"id": "8636.png", "formula": "\\begin{align*} \\mathcal { M } _ m ( L ) = \\sup \\limits _ { | \\alpha | \\leq \\lceil \\frac { d } { 2 } \\rceil + 1 } \\sup \\limits _ { | \\gamma | \\leq \\lceil \\frac { d } { 2 } \\rceil + 1 } \\sup \\limits _ { ( X , \\xi ) \\in \\R ^ d \\times \\R ^ d } \\Big { \\{ } \\langle \\xi \\rangle ^ { - ( m - | \\gamma | ) } | \\partial _ X ^ { \\alpha } \\partial _ { \\xi } ^ { \\gamma } L ( X , \\xi ) | \\Big { \\} } . \\end{align*}"}
+{"id": "5944.png", "formula": "\\begin{align*} \\lambda _ { j , \\varepsilon } - \\lambda _ j & = 4 u _ j ^ 2 ( x ^ * ) e ^ { \\phi ( x ^ * ) } \\varepsilon + \\frac { 4 } { \\pi } ( H ( x ^ * ) - F ^ \\perp ( x ^ * ) ) u _ j ^ 2 ( x ^ * ) e ^ { \\phi ( x ^ * ) } \\varepsilon ^ 2 \\log \\varepsilon + O ( \\varepsilon ^ 2 ) . \\end{align*}"}
+{"id": "5834.png", "formula": "\\begin{align*} \\tau = \\alpha _ 1 + 2 \\alpha _ 2 + 3 \\alpha _ 3 + 4 \\alpha _ 4 + 3 \\alpha _ 5 + 3 \\alpha _ 6 + 2 \\alpha _ 7 + \\alpha _ 8 . \\end{align*}"}
+{"id": "2339.png", "formula": "\\begin{align*} f _ { u g } ( \\tau _ { u h } ) & = ( f _ e \\pi _ { ( u g ) ^ { - 1 } } ) ( \\pi _ { u h } \\tau _ e ) = ( f _ e \\pi _ { g ^ { - 1 } u ^ { - 1 } } ) ( \\pi _ { u h } \\tau _ e ) = ( f _ e \\pi _ { g ^ { - 1 } } ) ( \\pi _ { h } \\tau _ e ) = f _ g ( \\tau _ h ) . \\end{align*}"}
+{"id": "6466.png", "formula": "\\begin{align*} & P _ D F = 0 , \\\\ & ( I - P _ D ) F ' = 0 , \\end{align*}"}
+{"id": "5810.png", "formula": "\\begin{align*} & \\varepsilon _ i = \\alpha _ n + \\alpha _ { n - 1 } + \\dots + \\alpha _ i , \\ ; \\ ; i < n , \\\\ & \\varepsilon ' _ i = \\alpha ' _ n + 2 \\alpha _ { n - 1 } + \\dots + 2 \\alpha _ i \\ ; \\ ; i < n , \\\\ & \\varepsilon _ n = \\alpha _ n , \\ ; \\ ; \\varepsilon ' _ i = 2 \\varepsilon _ n = 2 \\alpha _ n = \\alpha ' _ n . \\end{align*}"}
+{"id": "2174.png", "formula": "\\begin{align*} \\sigma \\bullet _ n ^ A ( x _ 1 , \\ldots , x _ n ) & = p _ 1 ( \\sigma \\bullet _ n ^ R ( y _ 1 , \\ldots , y _ n ) ) \\\\ \\sigma \\bullet _ n ^ A ( x ' _ 1 , \\ldots , x ' _ n ) & = p _ 2 ( \\sigma \\bullet _ n ^ R ( y _ 1 , \\ldots , y _ n ) ) \\end{align*}"}
+{"id": "2149.png", "formula": "\\begin{align*} \\beta ( t , x ) : = C + \\tilde \\alpha _ 0 ( 2 u ) - \\tilde \\alpha _ 0 ( - 2 \\underline { u } ) + \\int _ { 0 } ^ { 2 u } \\alpha _ 1 ( s ) d s + \\int _ { 0 } ^ { - 2 \\underline { u } } \\alpha _ 1 ( s ) d s , C \\in \\mathbb R . \\end{align*}"}
+{"id": "6902.png", "formula": "\\begin{align*} \\mathcal { N } _ i [ w ] ( x , t ) = \\int _ { \\mathbb R } J _ i ( x - y ) w ( y , t ) d y - w ( x , t ) , \\end{align*}"}
+{"id": "4509.png", "formula": "\\begin{align*} u _ 2 ^ a \\partial _ 2 S _ { \\theta _ i } \\psi _ i - S _ { \\theta _ i } ( u ^ a _ 2 \\partial _ 2 \\psi _ i ) = u _ 2 ^ a \\partial _ 2 ( S _ { \\theta _ i } - I ) \\psi _ i + ( I - S _ { \\theta _ i } ) ( u _ 2 ^ a \\partial _ 2 \\psi _ i ) \\ , , \\end{align*}"}
+{"id": "6520.png", "formula": "\\begin{align*} \\xi = \\frac 1 2 \\beta ^ { - 1 } K _ 1 ( \\beta ) ^ { 1 - \\beta } c ' c _ \\star ^ { \\beta ( \\beta - 1 ) } | k | _ 2 , \\end{align*}"}
+{"id": "6858.png", "formula": "\\begin{align*} \\mathcal { X } \\times _ 1 A _ 1 + \\mathcal { X } \\times _ 2 A _ 2 + \\dots + \\mathcal { X } \\times _ d A _ d = c _ 1 \\times _ 2 c _ 2 \\times _ 3 \\dots \\times _ d c _ d , c _ i \\in \\C ^ { n _ i } , \\end{align*}"}
+{"id": "265.png", "formula": "\\begin{align*} T ( x , v ) = \\sum \\limits _ \\lambda m _ \\lambda ( x ) e ^ G _ \\lambda ( v ) = \\sum \\limits _ \\lambda s _ \\lambda ( x ) s ^ G _ { \\lambda ^ * } ( v ) = \\sum \\limits _ \\lambda e _ \\lambda ( x ) m ^ G _ \\lambda ( v ) . \\end{align*}"}
+{"id": "2228.png", "formula": "\\begin{align*} \\beta ( e _ i ) = \\begin{cases} u _ { \\alpha _ i } 1 \\leq i \\leq \\ell \\\\ u _ { \\rho _ i } \\ell + 1 \\leq i \\leq n \\end{cases} \\end{align*}"}
+{"id": "2908.png", "formula": "\\begin{align*} \\theta ( \\mu ) = \\begin{cases} \\theta ( s _ j \\mu ) + 1 & , \\\\ \\theta ( s _ j \\mu ) & . \\end{cases} \\end{align*}"}
+{"id": "2036.png", "formula": "\\begin{align*} \\lVert w \\rVert _ { \\dot { \\mathrm { B } } ^ { s } _ { \\infty , \\infty } ( \\partial \\Omega ) } = \\sup _ { \\substack { ( x , y ) \\in \\partial \\Omega ^ 2 \\\\ x \\neq y } } \\frac { \\lvert w ( x ) - w ( y ) \\rvert } { \\lvert x - y \\rvert ^ { s } } \\end{align*}"}
+{"id": "7869.png", "formula": "\\begin{align*} P ^ \\infty ( M ) : = \\{ \\mu \\in P ( M ) : d \\mu = \\rho \\ , e ^ { - f } \\ , d V , \\ , \\rho \\in C ^ \\infty ( M ) , \\ \\rho > 0 \\} . \\end{align*}"}
+{"id": "3450.png", "formula": "\\begin{align*} \\nabla u ( x ) = \\{ p \\in \\Delta ^ \\vee \\simeq \\bar { \\Delta } ^ \\vee | u ( x ' ) \\geq u ( x ) + \\langle x - x ' , p \\rangle x ' \\in \\Delta \\} \\end{align*}"}
+{"id": "6207.png", "formula": "\\begin{align*} & [ \\delta '' , L ] = i D ' & [ \\delta ' , L ] & - i D '' \\\\ & [ \\Lambda , D '' ] = - i \\delta ' & [ \\Lambda , D ' ] & = i \\delta '' . \\end{align*}"}
+{"id": "5855.png", "formula": "\\begin{align*} \\theta \\cdot f ( \\gamma ) = \\sum _ { \\substack { \\delta \\in { X ( \\gamma ) } } } f ( \\delta ) \\end{align*}"}
+{"id": "6077.png", "formula": "\\begin{align*} \\ell _ { 1 } ( h _ { 0 } ) = 1 \\ell _ k ( h _ { 0 } ) = 0 . \\end{align*}"}
+{"id": "2502.png", "formula": "\\begin{align*} \\boldsymbol { H } ^ { 0 , 1 } ( Q ) & : = \\{ \\boldsymbol { v } \\in \\boldsymbol { L ^ 2 } ( Q ) : \\partial _ t \\boldsymbol { v } \\in \\boldsymbol { L ^ 2 } ( Q ) \\} , \\\\ \\boldsymbol { H } ^ { 0 , 1 } _ { p e r } ( Q ) & : = \\{ \\boldsymbol { v } \\in \\boldsymbol { H } ^ { 0 , 1 } ( Q ) : \\boldsymbol { v } ( \\boldsymbol { x } , 0 ) = \\boldsymbol { v } ( \\boldsymbol { x } , T ) \\ \\mbox { f o r a l m o s t a l l } \\boldsymbol { x } \\in \\Omega \\} . \\end{align*}"}
+{"id": "8388.png", "formula": "\\begin{align*} N = \\left [ p ^ { c } \\right ] + \\left [ m ^ { c } \\right ] , \\end{align*}"}
+{"id": "3810.png", "formula": "\\begin{align*} \\tilde \\Phi : = \\{ \\phi = ( \\phi _ 0 , \\phi _ 1 ) \\in C _ b ( X ) \\times C _ b ( X ) \\ ; \\mid \\ ; - F ^ * ( - \\phi _ i ) \\in C _ b ( X ) \\} \\end{align*}"}
+{"id": "997.png", "formula": "\\begin{align*} \\mathbb { P } ( Y _ j \\not \\in U \\ \\big | \\ Y _ 1 = y _ 1 , \\dots , Y _ { j - 1 } = y _ { j - 1 } ) & = \\sum _ { y \\in V - U } \\mathbb { P } ( Y _ j = y \\ \\big | \\ Y _ 1 = y _ 1 , \\dots , Y _ { j - 1 } = y _ { j - 1 } ) \\\\ & \\ge | V - U | \\alpha \\beta = ( q ^ n - q ^ m ) \\alpha \\beta . \\end{align*}"}
+{"id": "566.png", "formula": "\\begin{align*} I _ m ( t ) = \\int _ S ^ t \\mathbf S ( t - s ) \\mathbf N ( \\mathbf u _ m ( s ) ) \\ , d s \\end{align*}"}
+{"id": "7207.png", "formula": "\\begin{align*} I I & \\le C \\bigg ( \\sum _ { j = 1 } ^ { J - 1 } ( 2 ^ j \\rho ) ^ { 2 \\mu } ( I I _ A + I I _ B + I I _ C ) ^ 2 \\bigg ) ^ \\frac { 1 } { 2 } \\\\ & \\le C \\rho ^ { \\gamma + \\frac { 3 } { 2 } - k _ \\perp - \\frac { 1 } { p } } | \\log ( \\rho ) | \\| \\sigma \\| _ { H ^ 1 ( \\Lambda ) } \\bigg ( \\sum _ { j = 1 } ^ { J - 1 } ( 2 ^ j \\rho ) ^ { 2 \\mu + \\frac { 2 } { p } } \\bigg ) ^ \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "1310.png", "formula": "\\begin{align*} h ( t ) \\le G ^ { - 1 } ( G ( 1 ) - C _ 1 t ) = G ^ { - 1 } ( - C _ 1 t ) , t \\ge 0 , \\end{align*}"}
+{"id": "1511.png", "formula": "\\begin{gather*} \\sigma _ { \\sigma _ x ( y ) } \\gamma _ { y } ( x ) = x \\\\ \\gamma _ { \\gamma _ y ( x ) } \\sigma _ x ( y ) = y \\end{gather*}"}
+{"id": "5691.png", "formula": "\\begin{align*} k _ { e _ m } & = \\limsup _ { z \\rightarrow 0 } \\frac { \\ln \\| ( z - T ) ^ { - 1 } e _ m \\| _ p } { \\ln \\| ( z - T ) ^ { - 1 } \\| } \\\\ & \\leq \\limsup _ { z \\rightarrow 0 } \\frac { \\ln \\| ( z - T ) ^ { - 1 } e _ m \\| _ p } { \\ln \\| ( z - T ) ^ { - 1 } e _ N \\| _ p } = \\frac { m + 1 } { N + 1 } . \\end{align*}"}
+{"id": "3809.png", "formula": "\\begin{align*} G ( s ) : = s \\log s + ( c _ 0 - 1 ) s + \\frac 1 { c _ 1 } , c _ 0 = \\frac 1 2 \\log ( \\mu _ 0 ( X ) \\mu _ 1 ( X ) ) , \\ , c _ 1 = 2 \\frac { \\mu _ 0 ( X ) \\mu _ 1 ( X ) } { \\mu _ 0 ( X ) + \\mu _ 1 ( X ) } . \\end{align*}"}
+{"id": "2182.png", "formula": "\\begin{align*} \\Phi ^ { - 1 } ( H _ { b _ R , c _ R } \\Phi ( f ) ) ( x ) & = \\sum _ { R y \\in X _ R } b _ R ( R x , R y ) ( \\Phi ( f ) ( R x ) - \\Phi ( f ) ( R y ) ) + c ( R x ) \\Phi ( f ) ( R x ) \\\\ & = \\sum _ { R y \\in X _ R } \\sum _ { y ' \\in R y } b ( x , y ' ) ( f ( x ) - f ( y ' ) ) + c ( x ) f ( x ) \\\\ & = \\sum _ { y \\in X } b ( x , y ) ( f ( x ) - f ( y ) ) + c ( x ) f ( x ) \\\\ & = H _ { b , c } f ( x ) , \\end{align*}"}
+{"id": "5492.png", "formula": "\\begin{align*} \\widehat S ^ { - 1 } O _ A ^ { ( 1 ) } \\widehat S = O _ B ^ { ( 1 ) } , \\widehat S ^ { - 1 } O _ B ^ { ( 1 ) } \\widehat S = O _ A ^ { ( 1 ) } . \\end{align*}"}
+{"id": "3846.png", "formula": "\\begin{align*} \\widetilde { E } _ 1 ( x , y , z ) & = \\frac { d } { d s } | _ { s = 0 } ( s , 0 , 0 ) ( x , y , z ) = \\partial _ x = e ^ { - \\lambda _ 1 z } E _ 1 , \\\\ \\widetilde { E } _ 2 ( x , y , z ) & = \\partial _ y = e ^ { - \\lambda _ 2 z } E _ 2 , \\\\ \\widetilde { E } _ 3 ( x , y , z ) & = \\lambda _ 1 x \\ , \\partial _ x + \\lambda _ 2 y \\ , \\partial _ y + \\partial _ z = \\lambda _ 1 x e ^ { - \\lambda _ 1 z } E _ 1 + \\lambda _ 2 y e ^ { - \\lambda _ 2 z } E _ 2 + E _ 3 . \\end{align*}"}
+{"id": "2541.png", "formula": "\\begin{align*} Q _ 2 ( x ) : = P _ 2 ( x ) / x & = f ' ( 0 ) + \\frac { a } { 1 2 } x ^ 3 + \\frac { b } { 6 } x ^ 2 + \\frac { c _ 2 } { 2 } x \\ge 0 , x \\in [ 0 , 1 ] , \\\\ Q _ 1 ( x ) : = P _ 1 ( x ) / x & = f ' ( 0 ) + \\frac { a } { 1 2 } x ^ 3 + \\frac { b } { 6 } x ^ 2 + \\frac { c _ 1 } { 2 } x \\ge 0 , x \\in [ - 1 , 0 ] . \\end{align*}"}
+{"id": "8387.png", "formula": "\\begin{align*} N = \\left [ m _ { 1 } ^ { c } \\right ] + \\left [ m _ { 2 } ^ { c } \\right ] \\end{align*}"}
+{"id": "4492.png", "formula": "\\begin{align*} \\tilde { e } ' _ k = \\frac { 1 } { 2 } \\mathbb { B } '' ( ( \\delta { \\mathbf V } _ k , \\delta \\psi _ k ) , ( \\delta { \\mathbf V } _ k , \\delta \\psi _ k ) ) , \\end{align*}"}
+{"id": "216.png", "formula": "\\begin{align*} & { Q _ 2 } ( M , P _ i , \\tau ) = h _ 0 ' ( 8 \\delta _ 2 ) ^ { 5 } + h _ 1 ' ( 8 \\delta _ 2 ) ^ { 3 } \\varepsilon _ 2 + h _ 2 ' ( 8 \\delta _ 2 ) \\varepsilon _ 2 ^ 2 , \\end{align*}"}
+{"id": "4702.png", "formula": "\\begin{align*} g ( z _ 0 ^ 2 ) g ( x y ) = g ( z _ 0 ) [ g ( x ) g ( y ) - f ( x ) f ( y ) ] + f ( z _ 0 ^ 2 ) f ( x y ) , \\ ; x , y \\in S . \\end{align*}"}
+{"id": "8664.png", "formula": "\\begin{align*} \\int \\Bigg ( e ^ { - { \\vec y \\cdot \\vec y \\over 2 \\hbar } } \\Big ( \\int g ( \\vec x ) e ^ { f ( \\vec x ) + \\vec x \\cdot \\vec y \\over \\hbar } d ^ n \\vec x \\Big ) \\Bigg ) e ^ { \\vec y \\cdot \\vec z \\over \\hbar } d ^ n \\vec y = ( 2 \\pi \\hbar ) ^ { n / 2 } e ^ { { \\vec z \\cdot \\vec z \\over 2 \\hbar } } \\int e ^ { { 1 \\over \\hbar } ( f ( \\vec x ) + { \\vec x \\cdot \\vec x \\over 2 } + \\vec x \\cdot \\vec z ) } d ^ n \\vec x \\end{align*}"}
+{"id": "1184.png", "formula": "\\begin{align*} U _ k \\circ \\overline { F } _ k \\simeq F _ k \\circ U \\mathrm { a n d } U \\circ \\overline { G } _ k = G _ k \\circ U _ k . \\end{align*}"}
+{"id": "4846.png", "formula": "\\begin{align*} & G _ { i + 1 } G _ i E _ { i + 1 } = E _ i G _ { i + 1 } G _ i = E _ i E _ { i + 1 } \\\\ & G _ { i + 1 } E _ i G _ { i + 1 } = G _ i ^ { - 1 } E _ { i + 1 } G _ i ^ { - 1 } \\\\ & G _ { i + 1 } E _ i E _ { i + 1 } = G _ i ^ { - 1 } E _ { i + 1 } \\\\ & G _ i E _ i = E _ i G _ i = l ^ { - 1 } E _ i \\\\ & E _ i G _ { i + 1 } E _ i = l E _ i \\end{align*}"}
+{"id": "8772.png", "formula": "\\begin{align*} \\max _ { \\omega \\in \\Omega } \\mathbf { E } _ { \\omega , T } \\big [ \\norm { z _ { T } - x ^ { * } _ { \\omega } } ^ { 2 } \\big ] \\geq 0 . 0 1 \\times d \\alpha ^ { - 2 } r ^ { 2 } h ^ { 2 } = 0 . 0 1 \\times r ^ { 2 } \\min \\Big ( 1 , \\ , \\frac { d } { \\alpha ^ { 2 } } T ^ { - 1 } \\Big ) . \\end{align*}"}
+{"id": "6095.png", "formula": "\\begin{align*} E ( g _ k ) = \\nabla f ( \\omega _ k ) \\end{align*}"}
+{"id": "8469.png", "formula": "\\begin{align*} P ( F _ { \\ell } ; \\Omega \\times \\mathbb { R } ) = P ( E ; \\Omega \\times \\mathbb { R } ) , \\end{align*}"}
+{"id": "2175.png", "formula": "\\begin{align*} H T _ { g ^ { - 1 } } f ( x ) & = \\sum _ { y \\in X } b ( x , y ) ( f ( g x ) - f ( g y ) ) + c ( x ) f ( g x ) \\\\ & = \\sum _ { y \\in X } b ( g x , g y ) ( f ( g x ) - f ( g y ) ) + c ( g x ) f ( g x ) \\\\ & = \\sum _ { y \\in X } b ( g x , y ) ( f ( g x ) - f ( y ) ) + c ( g x ) f ( g x ) \\\\ & = H f ( g x ) = T _ { g ^ { - 1 } } H f ( x ) \\end{align*}"}
+{"id": "4385.png", "formula": "\\begin{align*} \\partial _ t \\hat { \\varphi } = \\hat { u } ^ { \\pm } _ N | _ { x _ 1 = 0 } \\ , , \\end{align*}"}
+{"id": "1818.png", "formula": "\\begin{gather*} \\Gamma ( \\alpha , N ) = \\left \\{ \\sum _ { n = 0 } ^ { N } \\frac { \\gamma _ n } { ( n + \\alpha ) ^ s } ~ \\middle | ~ \\right \\} \\end{gather*}"}
+{"id": "2324.png", "formula": "\\begin{align*} f _ g ( h ) = f ( \\pi _ { g ^ { - 1 } } h ) = \\langle \\pi _ { g ^ { - 1 } } h , \\tau \\rangle = \\langle h , \\pi _ g \\tau \\rangle = \\langle h , \\tau _ g \\rangle , \\forall h \\in \\mathcal { H } . \\end{align*}"}
+{"id": "1788.png", "formula": "\\begin{gather*} g ( \\underline { m } ) = \\int _ { \\mathcal { T } ^ k } \\prod _ { j = 1 } ^ { k } \\gamma _ j ^ { m _ j } \\ , d \\mu ( \\underline { \\gamma } ) \\end{gather*}"}
+{"id": "2987.png", "formula": "\\begin{align*} F ^ 1 = E _ I ^ 1 \\times _ { s _ I , \\psi } E ^ 1 = \\{ ( e ' , x , e ) \\in E ^ 1 \\times E ^ 0 _ I \\times E ^ 1 \\mid s ( e ' ) = \\alpha ( x ) x = \\psi ( e ) \\} \\end{align*}"}
+{"id": "2815.png", "formula": "\\begin{align*} b _ { N , s , \\theta } \\int _ { \\Omega } \\frac { U _ { t , p } } { | x | ^ { \\theta + 2 s } } \\dd x & = \\int _ { \\Omega } \\frac { ( - \\Delta ) ^ { s } U _ { t , p } } { | x | ^ { \\theta } } \\dd x \\\\ & \\leq \\int _ { \\Omega } \\frac { \\phi ' _ t ( u ) ( - \\Delta ) ^ { s } u } { | x | ^ { \\theta } } \\dd x = p \\int _ { \\Omega } \\frac { u } { ( t ^ 2 + u ^ 2 ) ^ { 1 / 2 } } \\frac { ( t ^ 2 + u ^ 2 ) ^ { ( p - 1 ) / 2 } ( - \\Delta ) ^ { s } u } { | x | ^ { \\theta } } \\dd x . \\end{align*}"}
+{"id": "8858.png", "formula": "\\begin{align*} H ^ 2 ( \\mathbf { P } _ { \\omega , T } , \\mathbf { P } _ { \\omega ' , T } ) \\leq 2 \\left ( 1 - \\left ( 1 - \\frac { I _ { 0 } r ^ 2 T ^ { - 1 } } { 2 } \\right ) ^ T \\right ) \\enspace , \\end{align*}"}
+{"id": "4739.png", "formula": "\\begin{align*} | D { \\rm d i s t } _ r ( \\cdot , \\partial \\Omega ) | ( \\Omega ) = \\int _ 0 ^ r { \\rm P e r } ( \\{ { \\rm d i s t } _ r ( \\cdot , \\partial \\Omega ) > s \\} ; \\Omega ) \\ , { \\rm d } s = \\int _ 0 ^ r { \\rm P e r } ( \\Omega \\setminus B ( \\partial \\Omega , s ) ) \\ , { \\rm d } s . \\end{align*}"}
+{"id": "1342.png", "formula": "\\begin{align*} \\delta _ { ( X , \\Delta ) } ( L ) = \\inf _ F \\frac { A _ { ( X , \\Delta ) } ( F ) } { S _ L ( F ) } \\end{align*}"}
+{"id": "2867.png", "formula": "\\begin{align*} G ( t ) = \\sum _ { w \\in G } t _ w \\end{align*}"}
+{"id": "3189.png", "formula": "\\begin{align*} A ^ T A x = A ^ T b . \\end{align*}"}
+{"id": "3522.png", "formula": "\\begin{align*} \\sum _ { n \\leqslant x } \\Lambda ( n ) \\chi ( n ) = O ( x ^ { \\frac { 1 } { 2 } } \\log ( d x ) ^ 2 ) , \\end{align*}"}
+{"id": "2655.png", "formula": "\\begin{align*} W ^ 0 ( x ) = - \\int _ 0 ^ x \\frac { W ^ 0 ( y ) } { 1 - y } \\ , d y + W ' ( x ) \\ , , \\end{align*}"}
+{"id": "883.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathcal { T } _ t ^ \\alpha u = x u _ { x x } + a v _ x , \\\\ & \\mathcal { T } _ t ^ \\alpha v = x v _ { x x } + b u _ x , ~ ~ x > 0 , \\end{aligned} \\right . \\end{align*}"}
+{"id": "4800.png", "formula": "\\begin{align*} \\| f \\| _ { r , \\alpha } : = \\sum _ { 0 \\leq p \\leq r } \\sum _ { | i | = p } \\| \\partial ^ i f \\| _ { C ^ 0 } + | \\partial ^ i f | ^ \\perp _ \\alpha . \\end{align*}"}
+{"id": "3396.png", "formula": "\\begin{align*} \\displaystyle v _ i ( \\delta _ j v _ j ^ 2 - \\delta _ k v _ k ^ 2 ) V _ j + v _ j ( \\delta _ i v _ i ^ 2 - \\delta _ k v _ k ^ 2 ) V _ i = 0 . \\end{align*}"}
+{"id": "5759.png", "formula": "\\begin{align*} \\beta \\Delta _ p ^ 2 = \\mu _ p , \\end{align*}"}
+{"id": "1198.png", "formula": "\\begin{align*} \\pi _ 0 ( A _ P ^ G ) = \\widehat { H } ^ 0 ( P ; k ) = k / | P | = k \\end{align*}"}
+{"id": "1851.png", "formula": "\\begin{gather*} \\mathbf { 1 } _ { \\Omega _ 0 } = \\prod _ { n = 0 } ^ { N } \\mathbf { 1 } _ { A ( s _ n , t _ n ) } ( \\mathbb { X } _ \\alpha ( n ) ) , \\end{gather*}"}
+{"id": "7185.png", "formula": "\\begin{align*} y _ 1 ^ { n _ 1 } ( y _ 1 + y _ 2 ) ^ { n _ 2 } & = y _ 1 ^ { n _ 1 } \\{ y _ 2 ^ { n _ 2 } + y _ 1 \\cdot f ( y _ 1 , y _ 2 ) \\} ~ ( \\mbox { w h e r e } ~ f ( y _ 1 , y _ 2 ) ~ \\mbox { i s a f u n c t i o n o f } ~ y _ 1 ~ \\mbox { a n d } ~ y _ 2 ) \\\\ & = y _ 1 ^ { n _ 1 } y _ 2 ^ { n _ 2 } + y _ 1 ^ { n _ 1 + 1 } \\cdot f ( y _ 1 , y _ 2 ) \\\\ & = y _ 1 ^ { n _ 1 } y _ 2 ^ { n _ 2 } ~ ( \\mbox { a s } ~ y _ 1 ^ { n _ 1 + 1 } = 0 ) . \\end{align*}"}
+{"id": "1100.png", "formula": "\\begin{align*} \\| u \\| _ { W _ 0 ^ { 1 , 2 } ( D ) } = \\left ( \\int _ { D } | \\nabla u | ^ 2 ( x ) d \\mu \\right ) ^ { 1 / 2 } \\end{align*}"}
+{"id": "3768.png", "formula": "\\begin{align*} S ( \\mu _ 0 , \\mu _ 1 ) = \\Big \\{ \\alpha \\in M ( Y \\times Y ) \\mid \\pi ^ { x _ i } _ { \\# } ( s _ i \\alpha ) = \\mu _ i . \\Big \\} \\end{align*}"}
+{"id": "8476.png", "formula": "\\begin{align*} E ^ { ( s ) } : = \\{ x \\in \\mathbb { R } ^ { n } \\ : \\ \\theta ( E , x ) = s \\} \\end{align*}"}
+{"id": "8922.png", "formula": "\\begin{align*} d \\varphi _ x \\left ( ( \\nabla ^ M _ v d \\varphi ^ * ( J Y ) ) _ x \\right ) = J _ { \\varphi ( x ) } d \\varphi _ x \\left ( ( \\nabla ^ M _ v d \\varphi ^ * ( Y ) ) _ x \\right ) , \\end{align*}"}
+{"id": "8720.png", "formula": "\\begin{align*} r _ { T _ 0 + 1 } \\leq r _ 1 + \\sum _ { t = 1 } ^ { T _ 0 } \\left ( \\frac { 8 } { \\alpha ^ 2 T _ 0 } \\left ( { \\sf b } L h _ t ^ { \\beta - 1 } \\right ) ^ 2 + \\frac { 1 6 } { \\alpha ^ 2 T _ 0 ^ 2 } \\left ( { \\sf v _ 2 } \\bar { L } ^ 2 h _ t ^ { 2 } + { \\sf v _ 3 } \\sigma ^ 2 h _ t ^ { - 2 } \\right ) \\right ) \\enspace . \\end{align*}"}
+{"id": "4218.png", "formula": "\\begin{align*} \\rho = \\ , & \\ , a _ 1 e ^ { 1 2 3 } + a _ 2 e ^ { 1 2 4 } + a _ 3 e ^ { 1 2 5 } + a _ 4 e ^ { 1 2 6 } + a _ 5 e ^ { 1 3 4 } + a _ 6 e ^ { 1 3 5 } + a _ 7 e ^ { 1 3 6 } + a _ 8 e ^ { 2 3 4 } + a _ 9 e ^ { 2 3 5 } \\\\ & + a _ { 1 0 } e ^ { 2 3 6 } + a _ { 1 1 } e ^ { 4 5 6 } , \\end{align*}"}
+{"id": "5181.png", "formula": "\\begin{align*} a _ d ( f ) ( V _ 1 \\otimes \\ldots \\otimes V _ { d - 1 } ) = \\int _ { G ^ { d - 1 } } \\left ( \\int _ G f ^ * ( g ) f ( g g _ 1 ) \\ldots f ( g g _ { d - 1 } ) \\ ; d g \\right ) D _ { V _ 1 } ( g _ 1 ) ^ * \\otimes \\ldots D _ { V _ { d - 1 } } ( g _ { d - 1 } ^ * ) \\ ; d g _ 1 \\ldots d g _ { d - 1 } \\end{align*}"}
+{"id": "1131.png", "formula": "\\begin{align*} & [ e _ 1 , e _ 1 ] = e _ 2 , ~ [ e _ 2 , e _ 2 ] = e _ 3 , ~ [ e _ 1 , e _ 2 ] = e _ 3 . \\end{align*}"}
+{"id": "343.png", "formula": "\\begin{align*} c \\left ( s \\right ) \\left ( \\mbox { \\boldmath $ \\phi $ } , \\mbox { \\boldmath $ \\psi $ } \\right ) : = \\left \\langle \\mathsf { C } \\left ( s \\right ) \\mbox { \\boldmath $ \\phi $ } , \\overline { \\mbox { \\boldmath $ \\psi $ } } \\right \\rangle _ { \\mathbb { X } } . \\end{align*}"}
+{"id": "6853.png", "formula": "\\begin{align*} A V _ { k + 1 } \\underline { K _ k } = V _ { k + 1 } \\underline { H _ k } . \\end{align*}"}
+{"id": "5116.png", "formula": "\\begin{align*} \\Upsilon & = g ( R ( X , Z ) ( Y ) , W ) - g ( R ( Y , Z ) ( X ) , W ) - g ( R ( Z , W ) ( X ) , Y ) + g ( \\nabla _ Z \\nabla _ X Y - \\nabla _ Z \\nabla _ Y X , W ) \\\\ & = \\underbrace { g ( R ( X , Z ) ( Y ) , W ) + g ( R ( Z , Y ) ( X ) , W ) + g ( R ( Y , X ) ( Z ) , W ) } _ { = 0 \\ ; } + g ( \\nabla _ Z [ X , Y ] , W ) , \\end{align*}"}
+{"id": "1502.png", "formula": "\\begin{align*} & X _ { t } = W _ { 0 } ( \\langle X \\rangle _ { t } ) , & A _ { 1 } ( t ) = W _ { 1 } ( \\langle A _ { 1 } \\rangle _ { t } ) , & & \\ldots \\ldots & & A _ { n } ( t ) = W _ { n } ( \\langle A _ { n } \\rangle _ { t } ) , \\end{align*}"}
+{"id": "2821.png", "formula": "\\begin{align*} H ( x ) + ( 1 - x ) H ( \\frac { y } { 1 - x } ) = H ( y ) + ( 1 - y ) H ( \\frac { x } { 1 - y } ) . \\end{align*}"}
+{"id": "5293.png", "formula": "\\begin{align*} g = g _ { n _ l } \\circ g _ { n _ { l - 1 } } \\circ \\dots \\circ g _ { n _ 1 } \\end{align*}"}
+{"id": "1486.png", "formula": "\\begin{align*} \\vert x \\vert : = \\left ( \\vert \\overline { x } \\vert _ { \\R ^ { m } } ^ { 4 } + \\vert \\widehat { x } \\vert _ { \\R ^ { n } } ^ { 2 } \\right ) ^ { \\frac { 1 } { 4 } } \\end{align*}"}
+{"id": "1291.png", "formula": "\\begin{align*} v _ { n , T } ( t ) & = g _ n [ \\chi _ n P _ n e ^ { i ( \\lambda _ n ^ { - 2 } t + t _ n ) \\Delta } \\phi ] \\\\ & = \\chi ( \\frac { x - x _ n } { \\lambda _ n } ) e ^ { i ( t + \\lambda _ n ^ 2 t _ n ) \\Delta } g _ n [ P _ n \\phi ] . \\end{align*}"}
+{"id": "6212.png", "formula": "\\begin{align*} C _ \\nu ( t ) = \\begin{cases} C e ^ { - t d ^ \\nu } & ( t \\ge 1 ) \\\\ C t ^ { - \\lambda _ \\nu } & ( 0 < t \\le 1 ) , \\end{cases} \\lambda _ \\nu \\coloneqq \\max \\Big \\{ \\frac { d } { \\nu } \\Big ( \\frac { 1 } { p _ 2 } - \\frac { 1 } { p _ 1 } \\Big ) , 0 \\Big \\} , \\end{align*}"}
+{"id": "2676.png", "formula": "\\begin{align*} \\frac { 1 } { G } _ { i \\overline { j } } ^ { g } g ^ { i \\overline { q } } g ^ { p \\overline { j } } h _ { p \\overline { q } } & \\ = \\ \\frac { 1 } { G } \\langle ^ { g } , h \\rangle _ { g } \\ = \\ \\frac { 1 } { G } \\langle - g + t h - \\frac { \\alpha } { 2 \\beta } d d ^ c \\varphi _ t , h \\rangle _ { g } \\\\ & \\ = \\ \\frac { 1 } { G } \\langle - g + t h , h \\rangle _ { g } - \\frac { 1 } { G } \\frac { \\alpha } { 2 \\beta } \\langle \\varphi _ t , h \\rangle _ { g } . \\end{align*}"}
+{"id": "8632.png", "formula": "\\begin{align*} \\begin{cases} \\nabla _ X \\psi = \\overline { V } + O ( \\mu ) \\\\ \\nabla _ X \\psi = \\overline { V } - \\frac { \\mu } { 3 h } \\nabla _ X ( h ^ 3 \\nabla _ X \\cdot \\overline { V } ) + \\mu ^ 2 R . \\end{cases} \\end{align*}"}
+{"id": "4981.png", "formula": "\\begin{align*} \\bigl ( x \\sigma ^ { - 1 } \\bigr ) ^ { 2 n + 1 } = x \\cdot \\sigma ( x ) \\cdot \\sigma ^ 2 ( x ) \\cdot \\ldots \\cdot \\sigma ^ { 2 n } ( x ) . \\end{align*}"}
+{"id": "193.png", "formula": "\\begin{align*} E _ 4 ( \\tau ) ^ 2 = 1 + 4 8 0 q + 6 1 9 2 0 q ^ 2 + \\cdots . \\end{align*}"}
+{"id": "7372.png", "formula": "\\begin{align*} d X ^ { ( k ) } ( t ) = \\mu ^ { ( k ) } d t + \\Sigma ^ { ( k ) \\frac { 1 } { 2 } } d W ^ { ( k ) } ( t ) \\end{align*}"}
+{"id": "7650.png", "formula": "\\begin{align*} X ^ w : = \\langle \\frac { w } { | w | } , X ^ \\mu \\rangle \\end{align*}"}
+{"id": "333.png", "formula": "\\begin{align*} \\mathsf { L } _ { j } ^ { \\sigma } \\left ( s \\right ) w ^ { \\sigma } = 0 \\quad \\Omega _ { j } ^ { \\sigma } \\sigma \\in \\left \\{ + , - \\right \\} , \\end{align*}"}
+{"id": "7235.png", "formula": "\\begin{align*} \\lambda ( t ) = ( \\lambda _ t ) ^ t . \\end{align*}"}
+{"id": "416.png", "formula": "\\begin{align*} \\kappa _ { \\rm c } ^ { ( \\alpha ) } : = \\frac { 2 ^ { \\alpha } \\Gamma \\left ( ( d + \\alpha ) / 4 \\right ) ^ 2 } { \\Gamma \\left ( ( d - \\alpha ) / 4 \\right ) ^ 2 } . \\end{align*}"}
+{"id": "3802.png", "formula": "\\begin{align*} { \\rm O T } ( \\mu _ 0 , \\mu _ 1 ) = \\inf _ { \\gamma \\in \\Gamma ( \\mu _ 0 , \\mu _ 1 ) } \\Big \\{ \\inf _ { a > 0 } \\int _ { X \\times X } \\frac { 1 } { a } c ( x _ 0 , x _ 1 ) d ( a \\gamma ) \\Big \\} = \\inf _ { \\gamma \\in \\Gamma ( \\mu _ 0 , \\mu _ 1 ) } \\int _ { X \\times X } c ( x _ 0 , x _ 1 ) d \\gamma , \\end{align*}"}
+{"id": "8685.png", "formula": "\\begin{align*} F _ 2 ( \\partial _ { x _ 0 } , . . . , \\partial _ { x _ { n + 1 } } ) \\cdot ( \\psi ) = 0 \\end{align*}"}
+{"id": "6184.png", "formula": "\\begin{align*} f _ t \\circ \\Gamma _ { \\mathbf { k } , \\pm \\infty } = \\Gamma _ { f _ t ( \\mathbf { k } ) , \\pm \\infty } \\circ f _ t . \\end{align*}"}
+{"id": "3925.png", "formula": "\\begin{align*} K \\hat { + } _ t L : = ( ( 1 - t ) K ^ * + t L ^ * ) ^ * . \\end{align*}"}
+{"id": "6261.png", "formula": "\\begin{align*} i \\partial _ t v _ \\lambda + \\partial _ x g _ { [ < \\lambda ] } \\partial _ x v _ \\lambda = N ^ { l i n } _ \\lambda v , v _ \\lambda ( 0 ) = v _ { 0 , \\lambda } , \\end{align*}"}
+{"id": "4704.png", "formula": "\\begin{align*} g ( x ) g ( y z _ 0 ) = f ( x ) f ( y z _ 0 ) , \\ ; x , y \\in S . \\end{align*}"}
+{"id": "4413.png", "formula": "\\begin{align*} J ^ { \\pm } = \\left [ \\begin{array} { c c c c c c } 1 & 0 & 0 & - \\hat { H } ^ { \\pm } _ 1 & - \\hat { H } ^ { \\pm } _ { \\tau } & 0 \\\\ 0 & 1 & \\partial _ 2 \\hat { \\Psi } ^ { \\pm } & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & \\partial _ 2 \\hat { \\Psi } ^ { \\pm } & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 \\end{array} \\right ] , \\end{align*}"}
+{"id": "4424.png", "formula": "\\begin{align*} [ u ] = \\lambda ^ + H ^ + - \\lambda ^ - H ^ - . \\end{align*}"}
+{"id": "2514.png", "formula": "\\begin{align*} a ( \\boldsymbol { y } , \\boldsymbol { v } ) = \\int _ { Q } \\left ( \\sigma \\partial _ t ^ { 1 / 2 } \\boldsymbol { y } \\cdot \\partial _ t ^ { 1 / 2 } \\boldsymbol { v } ^ { \\perp } + \\nu \\ , \\emph { \\textbf { c u r l } } \\ , \\boldsymbol { y } \\cdot \\emph { \\textbf { c u r l } } \\ , \\boldsymbol { v } \\right ) \\ , d \\boldsymbol { x } \\ , d t . \\end{align*}"}
+{"id": "751.png", "formula": "\\begin{align*} \\norm { u } _ { X ^ { s , b } ( S , T ) } = \\left ( \\int _ { \\R ^ d } \\norm { U ( t , \\xi ) } _ { H ^ b _ t ( S , T ) } ^ 2 \\ , d \\xi \\right ) ^ { 1 / 2 } , \\end{align*}"}
+{"id": "1953.png", "formula": "\\begin{align*} \\left ( \\bar { u } _ t , \\phi \\right ) _ I - \\epsilon \\ , \\left ( \\bar { u } _ { x x } , \\phi \\right ) _ I = \\left ( \\int _ { \\R } \\left ( v \\bar { f } + u _ 2 \\bar { f } - \\bar { u } f _ 1 \\right ) \\ , { \\rm d } v , \\phi \\right ) _ I - \\left ( \\left ( \\bar { u } \\partial _ x u _ 1 + u _ 2 \\bar { u } _ x \\right ) , \\phi \\right ) _ I , \\end{align*}"}
+{"id": "5979.png", "formula": "\\begin{align*} & \\mathcal { N } : H ^ { 1 / 2 } ( \\partial M ) ^ * \\mapsto H ^ { 1 / 2 } ( \\partial M ) , \\\\ & \\mathcal { N } \\psi = u ^ { \\psi } \\arrowvert _ { \\partial M } , \\end{align*}"}
+{"id": "2746.png", "formula": "\\begin{align*} \\| U ( x _ k ) + U ( e _ i ) \\| = \\| x _ k + e _ i \\| = \\| x _ k - e _ i \\| = \\| U ( x _ k ) - U ( e _ i ) \\| \\end{align*}"}
+{"id": "4619.png", "formula": "\\begin{align*} \\mathfrak { g } ^ { ( m ) } = ( \\mathfrak { g } ^ { \\times m } ) ^ { \\theta _ m } \\end{align*}"}
+{"id": "3263.png", "formula": "\\begin{align*} \\mathcal { F } ^ \\mu ( \\mathcal { C } f + \\mathcal { D } g ) ( \\xi ) = \\mathcal { C } \\mathcal { F } ^ \\mu ( f ) ( \\xi ) + \\mathcal { D } \\mathcal { F } ^ \\mu ( g ) ( \\xi ) . \\end{align*}"}
+{"id": "5102.png", "formula": "\\begin{align*} \\varinjlim _ { h } V _ { h } ^ { \\vee } = \\varinjlim _ { h } W _ { h } \\end{align*}"}
+{"id": "4726.png", "formula": "\\begin{align*} 0 & = \\int _ 0 ^ 1 \\partial _ { e ^ m x _ 1 } \\tilde u _ { k _ m } ( s z _ { l _ m , k _ m } + ( 1 - s ) z _ { k _ m ( m ) } ) d s \\\\ & = \\int _ 0 ^ { t ( \\alpha ) } \\partial _ { e ^ m x _ 1 } \\tilde u _ { k _ m } ( s z _ { l _ m , k _ m } + ( 1 - s ) z _ { k _ m ( m ) } ) d s \\\\ & + \\int _ { t ( \\alpha ) } ^ { t ( R ) } \\partial _ { e ^ m x _ 1 } \\tilde u _ { k _ m } ( s z _ { l _ m , k _ m } + ( 1 - s ) z _ { k _ m ( m ) } ) d s \\\\ & + \\int _ { t ( R ) } ^ { 1 } \\partial _ { e ^ m x _ 1 } \\tilde u _ { k _ m } ( s z _ { l _ m , k _ m } + ( 1 - s ) z _ { k _ m ( m ) } ) d s . \\end{align*}"}
+{"id": "4539.png", "formula": "\\begin{align*} \\left \\vert \\int _ { \\Gamma _ t } ( \\mathcal B _ 1 { \\mathbf V } \\cdot { \\mathbf V } ) \\vert _ { x _ 1 = 0 } \\ , d x _ 2 d s \\right \\vert \\le C _ 1 \\left \\{ \\Vert \\dot q ^ + \\vert _ { x _ 1 = 0 } \\Vert _ { L ^ 2 ( \\Gamma _ t ) } ^ 2 + \\Vert \\varphi ( t ) \\Vert _ { L ^ 2 ( \\mathbb R ) } ^ 2 \\right \\} + C _ 2 \\Vert \\varphi \\Vert ^ 2 _ { L ^ 2 ( \\Gamma _ t ) } \\ , . \\end{align*}"}
+{"id": "1042.png", "formula": "\\begin{align*} \\hat { \\varphi } _ \\mu ( u _ \\mu ) = \\inf \\left \\{ \\hat { \\varphi } _ \\mu ( u ) : \\ : u \\in W ^ { 1 , p ( z ) } ( \\Omega ) \\right \\} . \\end{align*}"}
+{"id": "5939.png", "formula": "\\begin{align*} L \\cong \\begin{pmatrix} a & f _ { 1 } & f _ { 2 } & f _ { 4 } \\\\ f _ { 1 } & b & f _ { 3 } & f _ { 5 } \\\\ f _ { 2 } & f _ { 3 } & c & f _ { 6 } \\\\ f _ { 4 } & f _ { 5 } & f _ { 6 } & d \\end{pmatrix} . \\end{align*}"}
+{"id": "3780.png", "formula": "\\begin{align*} f ^ * ( \\phi ) = \\sup _ { s > 0 } \\left ( s \\phi - f ( s ) \\right ) . \\end{align*}"}
+{"id": "6553.png", "formula": "\\begin{align*} \\left ( \\mathbb { D } _ { l _ * } ( \\sigma ^ * ) \\setminus \\mathbb { D } _ { l _ * - 1 } ( \\sigma ^ * ) \\right ) \\cap \\{ \\sigma _ l \\} _ { 1 \\leq l \\leq \\# B _ * } = \\emptyset . \\end{align*}"}
+{"id": "6137.png", "formula": "\\begin{align*} ( \\varphi _ 1 \\times \\varphi _ 2 \\times \\cdots \\times \\varphi _ k ) ( s ) = ( s _ 1 , s _ 2 , \\ldots , s _ { j - 1 } , \\varphi _ j ( s _ j ) , s _ { j + 1 } , \\ldots , s _ k ) , \\end{align*}"}
+{"id": "7297.png", "formula": "\\begin{align*} | A _ e \\cap [ k ] | & \\leq | A _ e \\cap [ 2 ^ { m + 1 } - 1 ] | \\\\ & \\leq \\frac { 2 ^ { m + 1 } } { 2 ^ { e + 3 } } = \\frac { 2 ^ m } { 2 ^ { e + 2 } } \\leq \\frac { k + 1 } { 2 ^ { e + 2 } } \\end{align*}"}
+{"id": "6108.png", "formula": "\\begin{align*} \\sigma _ A ( T ) \\subseteq \\overline { D } ( 0 , \\alpha \\} \\ \\ \\alpha = \\max ( \\| T \\| _ A , \\| L \\| _ A ) . \\end{align*}"}
+{"id": "59.png", "formula": "\\begin{align*} n _ + ^ H : = \\sum _ { j = 1 } ^ n \\overline Q _ { L , j } , \\end{align*}"}
+{"id": "3272.png", "formula": "\\begin{align*} \\int _ \\mathbb { R } ( f _ X ) _ { _ \\Sigma } ( x ) d x = 1 \\{ ( f _ X ) _ { _ \\Sigma } < 0 \\} = \\emptyset . \\end{align*}"}
+{"id": "1705.png", "formula": "\\begin{align*} Q _ { \\texttt { a } ; \\mu } ( \\boldsymbol { \\xi } ) : = P _ { \\texttt { a } ; \\mu } ( \\boldsymbol { \\xi } ; q ) - q ^ { _ { \\mu _ 1 } ( \\mu ) _ { \\mu _ n } ( \\mu ) } P _ { \\texttt { a } ; \\mu - \\omega _ { \\texttt { a } ; \\mu } } ( \\boldsymbol { \\xi } ; q ) \\end{align*}"}
+{"id": "5269.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { t = 1 } ^ T \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n \\| x _ { i , t } - x _ { j , t } \\| & \\le n \\varepsilon _ 1 + \\tilde { \\varepsilon } _ 2 \\sum _ { t = 1 } ^ { T } \\sum _ { j = 1 } ^ n \\| \\epsilon ^ z _ { j , t } \\| \\\\ & \\le n \\varepsilon _ 1 + \\frac { \\tilde { \\varepsilon } _ 2 } { \\sigma } \\sum _ { t = 1 } ^ { T } \\sum _ { j = 1 } ^ n ( G _ 2 \\gamma _ { 0 } \\| [ g _ { j , t } ( x _ { j , t } ) ] _ + \\| + G _ 1 \\alpha _ { t } ) . \\end{align*}"}
+{"id": "7119.png", "formula": "\\begin{align*} \\int _ \\Omega S _ \\perp \\theta ' = 0 , \\| S _ \\parallel \\| _ { L ^ 2 } \\lesssim \\frac { \\sigma } { ( 1 + t ) ^ { 1 + \\delta } } . \\end{align*}"}
+{"id": "3902.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\varphi ( N ) ^ { 1 / 8 } P _ N = 1 . \\end{align*}"}
+{"id": "5755.png", "formula": "\\begin{align*} | i - j | : = \\sum _ { s = 1 } ^ d | i _ s - j _ s | . \\end{align*}"}
+{"id": "3354.png", "formula": "\\begin{align*} \\sum _ { n _ 1 , n _ 2 \\geq 0 } \\frac { q ^ { \\frac { 3 } { 2 } n _ 1 ^ 2 + 4 n _ 1 n _ 2 + 4 n _ 2 ^ 2 - \\frac { 1 } { 2 } n _ 1 } } { ( q ; q ) _ { n _ 1 } ( q ^ 2 ; q ^ 2 ) _ { n _ 2 } } x ^ { n _ 1 + 2 n _ 2 } = \\sum _ { n \\geq 0 } \\frac { q ^ { n ^ 2 } } { ( q ; q ) _ n } x ^ n \\end{align*}"}
+{"id": "7502.png", "formula": "\\begin{align*} X ^ { \\sigma + 1 } + X ^ \\sigma + ( 1 - \\overline { \\gamma } ) X + 1 = 0 \\end{align*}"}
+{"id": "5410.png", "formula": "\\begin{align*} \\begin{cases} \\mathbf { v } \\cdot \\nabla \\mathbf { v } = - \\nabla P & \\ \\ \\Omega , \\\\ \\nabla \\cdot \\mathbf { v } = 0 \\ , \\ \\ , \\ \\ \\ \\ \\ \\ \\ \\ \\ , & \\ \\ \\Omega , \\end{cases} \\end{align*}"}
+{"id": "5785.png", "formula": "\\begin{align*} s _ { \\alpha } s _ { \\beta } s _ { \\alpha } = \\begin{cases} s _ { \\alpha + \\beta } ( \\alpha , \\beta ) = - 1 , \\\\ s _ { \\alpha - \\beta } ( \\alpha , \\beta ) = 1 . \\\\ \\end{cases} \\end{align*}"}
+{"id": "2388.png", "formula": "\\begin{align*} \\pi _ 1 = \\frac { \\mu - r } { \\sigma ^ 2 } \\end{align*}"}
+{"id": "760.png", "formula": "\\begin{align*} \\theta ( X ) - \\theta ( Y ) = \\left ( \\int _ 0 ^ 1 \\theta ' \\left ( \\rho ( t ) \\right ) \\ , d t \\right ) ( X - Y ) = : I _ 2 ( X - Y ) . \\end{align*}"}
+{"id": "3628.png", "formula": "\\begin{align*} E _ 2 ( \\varphi ) = \\frac { 1 } { 2 } \\int _ D | \\tau ( \\varphi ) | ^ 2 v ^ g . \\end{align*}"}
+{"id": "8797.png", "formula": "\\begin{align*} \\norm { x _ { t + 1 } - x _ { p } } ^ { 2 } & \\leq \\norm { x _ { t } - \\eta _ { t } \\hat { g } _ { t } - x _ { p } } ^ 2 \\\\ & = \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 \\\\ & \\leq ( 1 - 2 \\eta _ { t } \\alpha ) \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } - \\nabla f ( x _ t ) , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 , \\end{align*}"}
+{"id": "8177.png", "formula": "\\begin{align*} \\mu _ { i j } ^ \\star - \\mu _ { a b } ^ \\star = \\frac { n ( n - 1 ) } { k ( n - k ) } ( b - j ) . \\end{align*}"}
+{"id": "6253.png", "formula": "\\begin{align*} \\begin{aligned} g ( u ) = & \\ g ( u _ 0 ) + \\int _ { 0 } ^ \\infty g ' ( u _ { < k } ) u _ k \\ , d k \\\\ = & \\ g ( u _ 0 ) + \\int _ { 0 } ^ \\infty g ' ( u _ { 0 } ) u _ k \\ , d k + \\iint _ { 0 < k _ 2 < k _ 1 } g '' ( u _ { < k _ 2 } ) u _ { k _ 2 } u _ { k _ 1 } \\ , d k _ 1 d k _ 2 . \\end{aligned} \\end{align*}"}
+{"id": "3619.png", "formula": "\\begin{align*} \\langle V , u \\rangle = \\langle V ^ { \\ell } , u \\rangle = \\lim _ { n \\to \\infty } \\langle V _ n ^ \\ell , u \\rangle \\end{align*}"}
+{"id": "4032.png", "formula": "\\begin{align*} \\sum _ { x \\in V } p ( x , S _ i ) = \\sum _ { j = 1 } ^ { \\ln n } \\sum _ { x \\in L _ j } p ( x , S _ i ) \\leq \\frac { n } { \\ln n } \\sum _ { j = 1 } ^ { \\ln n } z _ j e ^ { - z _ j + 1 } \\leq \\frac { n } { \\ln n } \\sum _ { j = 1 } ^ { \\infty } z _ j e ^ { - z _ j + 1 } . \\end{align*}"}
+{"id": "5660.png", "formula": "\\begin{align*} x _ 0 x _ 1 x _ 3 ^ 2 + B x _ 3 x + C x ^ 2 = 0 \\end{align*}"}
+{"id": "4344.png", "formula": "\\begin{align*} \\frac { 1 } { i } \\left . \\frac { d } { d t } \\right | _ { t = 0 } { \\bf A } ( t ) = \\left ( \\begin{array} { c c } 0 & { \\bf a } \\\\ - { \\bf a } ^ T & 0 \\end{array} \\right ) \\end{align*}"}
+{"id": "3725.png", "formula": "\\begin{align*} u ''' ( t ) + 3 \\varUpsilon _ 0 ^ \\alpha A ^ { \\frac { \\alpha } { 3 } } u '' ( t ) + 3 \\varUpsilon _ 0 ^ \\alpha A ^ { \\frac { 2 \\alpha } { 3 } } u '' ( t ) + A ^ { \\alpha } u ( t ) = 0 , t > 0 , \\end{align*}"}
+{"id": "7161.png", "formula": "\\begin{align*} \\| x \\| = \\left \\| \\sum _ { j = 1 } ^ n f _ j ( x ) \\tau _ j \\right \\| = \\left ( \\sum _ { j = 1 } ^ n | f _ j ( x ) | ^ p \\right ) ^ \\frac { 1 } { p } , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "7906.png", "formula": "\\begin{align*} \\sum _ { \\varphi \\in \\mathrm { I r r } ( R ) } \\dfrac { \\dim ( \\varphi ) } { f _ \\varphi } = 1 , \\end{align*}"}
+{"id": "4926.png", "formula": "\\begin{align*} g ( \\tau ) = \\sum _ { \\ell = 0 } ^ { k - 1 } \\alpha _ k ( \\ell ) \\cdot F ( \\tau ) ^ \\ell \\theta ( 2 \\tau ) ^ { 4 ( k - \\ell ) + 2 } + \\sum _ { \\ell = k } ^ { 2 k } \\beta _ k ( \\ell ) \\cdot F ( \\tau ) ^ { 2 k - \\ell } F ( 2 \\tau ) ^ { \\ell - k } \\theta ( 2 \\tau ) ^ 2 + \\gamma _ k \\cdot \\dfrac { F ( \\tau ) F ( 2 \\tau ) ^ k } { \\theta ( 2 \\tau ) ^ 2 } , \\end{align*}"}
+{"id": "2765.png", "formula": "\\begin{align*} T _ { \\mu } \\left ( \\frac { 1 } { z } \\right ) & = \\int _ a ^ b \\frac { x } { 1 / z - x } \\mathrm { d } \\mu ( x ) = \\int _ a ^ b \\frac { z x } { 1 - z x } \\mathrm { d } \\mu ( x ) \\\\ & = \\int _ a ^ b \\left ( \\sum _ { n = 1 } ^ { + \\infty } ( z x ) ^ n \\mathrm { d } \\mu ( x ) \\right ) = \\sum _ { n = 1 } ^ { + \\infty } z ^ n \\int _ a ^ b x ^ n \\mathrm { d } \\mu ( x ) = z \\mathbb { E } [ X _ \\mu ] + z ^ 2 f ( z ) , \\end{align*}"}
+{"id": "4837.png", "formula": "\\begin{align*} P _ i = \\prod _ { i \\neq j } \\frac { S - \\lambda _ j } { \\lambda _ i - \\lambda _ j } \\end{align*}"}
+{"id": "7514.png", "formula": "\\begin{align*} p ' , \\ , p '' > 0 \\quad \\textrm { i n $ ( 0 , \\infty ) $ } \\quad \\textrm { a n d } \\lim _ { n \\to + \\infty } \\displaystyle \\frac { p ( n ) } { n } = + \\infty . \\end{align*}"}
+{"id": "1487.png", "formula": "\\begin{align*} \\langle U ^ { ( i ) } \\overline { x } , U ^ { ( j ) } \\overline { x } \\rangle = 0 , \\end{align*}"}
+{"id": "6450.png", "formula": "\\begin{align*} [ g _ { u _ 1 , v _ 1 } ] \\oplus \\ldots \\oplus [ g _ { u _ s , v _ s } ] = 0 . \\end{align*}"}
+{"id": "8393.png", "formula": "\\begin{align*} \\left [ p ^ { c } \\right ] + \\left [ m ^ { c } \\right ] = N , ( m , P ( z ) ) = 1 . \\end{align*}"}
+{"id": "5911.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } [ G _ 3 + G _ 4 ] = 0 . \\end{align*}"}
+{"id": "1562.png", "formula": "\\begin{align*} \\rho _ f ( x ) & = \\int _ { - \\infty } ^ \\infty f ( x , v ) \\mathrm { d } v , \\rho _ f ( x ) u _ f ( x ) = \\int _ { - \\infty } ^ \\infty v f ( x , v ) \\mathrm { d } v , \\quad \\\\ & P _ f ( x ) = \\int _ { - \\infty } ^ \\infty v ^ 2 f ( x , v ) \\mathrm { d } v = \\rho _ f ( x ) [ T _ f ( x ) + u _ f ( x ) ^ 2 ] \\end{align*}"}
+{"id": "1590.png", "formula": "\\begin{align*} H ( f ) : = \\frac { 1 } { 2 } \\| f \\| ^ 2 + \\varepsilon \\langle A f , f \\rangle \\end{align*}"}
+{"id": "8302.png", "formula": "\\begin{align*} L ( x _ 1 , x _ 2 ) = ( L _ 1 ( x _ 1 , x _ 2 ) , L _ 2 ( x _ 1 , x _ 2 ) ) = \\left ( x _ 1 + \\frac { 1 } { p } L _ 2 ( x _ 1 ^ { p ^ { h _ 1 } } , x _ 2 ^ { p ^ { h _ 1 } } ) , x _ 2 + \\frac { 1 } { p } L _ 1 ( x _ 1 ^ { p ^ { h _ 2 } } , x _ 2 ^ { p ^ { h _ 2 } } ) \\right ) , \\end{align*}"}
+{"id": "4157.png", "formula": "\\begin{align*} P _ { \\hat p , p } ( v ) & = v + \\langle v , w \\rangle \\left ( - \\sqrt { 1 - \\langle \\hat p , p \\rangle ^ 2 } \\hat p + ( \\langle \\hat p , p \\rangle - 1 ) w \\right ) , \\\\ w & = \\frac { p - \\langle \\hat p , p \\rangle \\hat p } { | p - \\langle \\hat p , p \\rangle \\hat p | } . \\end{align*}"}
+{"id": "659.png", "formula": "\\begin{align*} \\left \\{ \\tau _ R \\le t \\right \\} = \\left \\{ \\tau _ R = \\tau \\le t \\right \\} \\cup \\left ( \\left \\{ \\tau _ R < \\tau \\right \\} \\cap \\left \\{ \\tau _ R \\le t \\right \\} \\right ) . \\end{align*}"}
+{"id": "439.png", "formula": "\\begin{align*} ( r \\vee s ) ^ { - 1 - \\alpha } | 1 - s ^ 2 / r ^ 2 | ^ { - 1 - \\alpha } & = r ^ { 1 + \\alpha } | r ^ 2 - s ^ 2 | ^ { - 1 - \\alpha } \\sim | r - s | ^ { - 1 - \\alpha } \\ 0 < s < r , \\end{align*}"}
+{"id": "250.png", "formula": "\\begin{align*} w ( z ) = \\begin{cases} \\left ( 1 + ( \\frac { 1 } { 2 } g _ S + g _ L ) z ^ { - 1 } \\right ) \\left ( 1 + g _ S ( 1 + 2 z ) ^ { - 1 } \\right ) & \\ \\emph { r = b c } \\ \\ \\emph { r = t } , \\\\ \\frac { 1 } { 2 } z ^ { - 1 } \\left ( \\frac { 1 } { 2 } + g _ S ( 1 + 2 z ) ^ { - 1 } \\right ) & \\ \\emph { r = c s } , \\end{cases} \\end{align*}"}
+{"id": "3862.png", "formula": "\\begin{align*} \\bar { g } _ { 1 1 } & = e ^ { - 2 \\lambda ( f + t ) } + e ^ { - 2 \\lambda t } ( f ' ) ^ 2 \\\\ \\bar { g } _ { 1 2 } & = g _ { 2 1 } = 0 \\\\ \\bar { g } _ { 2 2 } & = e ^ { - 2 \\lambda ( f + t ) } r ^ 2 . \\end{align*}"}
+{"id": "8582.png", "formula": "\\begin{align*} ( \\nabla _ X \\Phi ) \\circ \\Sigma _ b = \\begin{pmatrix} \\mathrm { I d } & \\mathbf { 0 } \\\\ \\mathbf { 0 } ^ T & 0 \\end{pmatrix} ( J _ { \\Sigma _ b } ^ { - 1 } ) ^ T \\nabla _ { X , z } \\phi _ b , \\end{align*}"}
+{"id": "8382.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } \\Big ( \\int _ { \\Delta } ^ { T } | X _ { t ( \\Delta ) } ^ { \\delta , \\epsilon , h ^ { \\delta } } - X _ { t - \\Delta } ^ { \\delta , \\epsilon , h ^ { \\delta } } | ^ { 2 } d t \\Big ) \\leq & C _ { T , H , a , M , | x | , | y | , \\delta , \\beta _ { 2 } } ( \\Delta ^ { 2 } \\vee \\Delta ^ { 2 H } ) . \\end{aligned} \\end{align*}"}
+{"id": "5034.png", "formula": "\\begin{align*} - \\triangle _ { g ' _ p } u + \\nabla _ { r _ p ^ { 2 } \\nabla ^ g f } u - r _ p ^ 2 b ( u ) = r _ p ^ 2 \\lambda u . \\end{align*}"}
+{"id": "4562.png", "formula": "\\begin{align*} \\left \\vert \\int _ { \\Gamma _ t } ( \\mathcal B _ 1 { \\mathbf V } _ { x _ 2 } \\cdot { \\mathbf V } _ { x _ 2 } ) \\vert _ { x _ 1 = 0 } \\ , d x _ 2 d s \\right \\vert \\le C _ 2 \\left \\{ \\int _ 0 ^ t ( I _ { 1 , \\ast } + I _ { 1 , n } ) ( s ) d s + \\int _ 0 ^ t \\Vert \\varphi ( s ) \\Vert _ { L ^ 2 ( \\mathbb R ) } ^ 2 d s \\right \\} \\ , . \\end{align*}"}
+{"id": "374.png", "formula": "\\begin{align*} K _ { \\phi _ { \\mu } , R _ { \\mu } } ( \\ldots , t _ { \\mu - 1 } , t _ { \\mu + 1 } , \\ldots ) : = \\int K ( t ) \\phi _ { \\mu } ( R _ { \\mu } \\cdot t _ { \\mu } ) d t _ { \\mu } \\end{align*}"}
+{"id": "7302.png", "formula": "\\begin{align*} D _ S = \\{ ( a , b ) \\in [ 0 , 1 ] ^ 2 : & \\left [ a \\ge b \\Rightarrow \\exists x \\in [ 0 , 1 ] \\ ; a = S ( x , b ) b = 1 \\Rightarrow x = 0 \\right ] \\\\ a n d & \\left [ b \\ge a \\Rightarrow \\exists x \\in [ 0 , 1 ] \\ ; b = S ( x , a ) a = 1 \\Rightarrow x = 0 \\right ] \\} \\end{align*}"}
+{"id": "7131.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\Psi ^ { ( 5 ) } ( Z ) ( \\Psi w ) ' ( Z ) \\ : d Z = \\int _ 0 ^ \\infty \\Psi ^ { ( 3 ) } ( Z ) ( \\Psi w ) ^ { ( 3 ) } ( Z ) \\ : d Z - \\Psi ^ { ( 4 ) } ( 0 ) ( \\Psi w ) ' ( 0 ) + \\Psi ^ { ( 3 ) } ( 0 ) ( \\Psi w ) '' ( 0 ) . \\end{align*}"}
+{"id": "5052.png", "formula": "\\begin{align*} L _ { g ' } ( Z ( \\kappa ) ) = D _ 2 G _ { ( \\gamma ' , \\kappa ' ) } ( \\kappa ) , \\end{align*}"}
+{"id": "3378.png", "formula": "\\begin{align*} \\hat { \\alpha } ^ i _ j = \\alpha ^ i _ j + \\Upsilon \\delta ^ i _ j + \\Upsilon _ i \\dd x ^ j . \\end{align*}"}
+{"id": "7513.png", "formula": "\\begin{align*} L _ { u } ( f ) : = \\partial _ v \\bigl \\{ ( v - u ) f + \\theta \\partial _ v f \\bigr \\} \\end{align*}"}
+{"id": "2630.png", "formula": "\\begin{align*} U _ n ( x , t ) = - \\int _ 0 ^ x \\frac { U _ n ( y , t ) } { 1 - F ( y ) } \\ , d F ( y ) + L _ n ( x , t ) \\end{align*}"}
+{"id": "4225.png", "formula": "\\begin{align*} a _ 2 & = \\frac { - a _ { 1 1 } a _ 4 a _ 6 + 4 a _ { 1 1 } a _ 5 a _ 7 - 3 a _ { 1 0 } a _ 6 a _ 7 - a _ { 1 0 } a _ 4 a _ 8 + a _ 6 ^ 2 a _ 8 + 2 a _ 5 a _ 8 ^ 2 - 2 a _ 7 a _ 8 a _ 9 } { 2 a _ { 1 1 } a _ 9 - a _ { 1 0 } ^ 2 } , \\\\ [ 5 p t ] a _ 4 & = \\frac { - a _ { 1 1 } a _ 5 a _ 6 + a _ { 1 0 } a _ 6 ^ 2 + 2 a _ { 1 1 } a _ 3 a _ 7 - a _ { 1 0 } a _ 5 a _ 8 + a _ 3 a _ 8 ^ 2 - 2 a _ { 1 0 } a _ 7 a _ 9 + 3 a _ 6 a _ 8 a _ 9 } { 2 ( 2 a _ { 1 1 } a _ 9 - a _ { 1 0 } ^ 2 ) } , \\\\ [ 5 p t ] a _ 5 & = \\frac { a _ { 1 1 } a _ 3 a _ 6 + a _ { 1 0 } a _ 3 a _ 8 - a _ { 1 0 } a _ 6 a _ 9 - 2 a _ 8 a _ 9 ^ 2 } { 2 a _ { 1 1 } a _ 9 - a _ { 1 0 } ^ 2 } . \\end{align*}"}
+{"id": "4594.png", "formula": "\\begin{align*} \\theta _ { M _ 1 \\otimes _ { \\mathfrak { V } } M _ 2 } = c _ { M _ 2 , M _ 1 } ^ { } \\ , { \\circ } \\ , c _ { M _ 1 , M _ 2 } ^ { } \\ , { \\circ } \\ , ( \\theta _ { M _ 1 } \\ , { \\otimes _ { \\mathfrak { V } } } \\ , \\theta _ { M _ 2 } ) \\ , . \\end{align*}"}
+{"id": "8046.png", "formula": "\\begin{align*} \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\lambda _ j \\chi _ { Q _ j } } { \\omega ( Q _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } + \\left \\| \\sum \\limits _ { j = 1 } ^ { \\infty } \\frac { \\mu _ j \\chi _ { P _ j } } { \\omega ( P _ j ) ^ { \\frac { 1 } { p } } } \\right \\| _ { L _ { \\omega } ^ { p } } < \\infty . \\end{align*}"}
+{"id": "8831.png", "formula": "\\begin{align*} \\norm { \\nabla f _ { \\omega } ( x ) } ^ { 2 } = \\frac { 8 } { 3 } \\frac { \\bar { \\alpha } ^ { 2 } } { \\tilde { \\alpha } ^ { 2 } } h _ { T } ^ { 2 \\beta - 2 } \\sum _ { i = 1 } ^ { d } \\cos ^ { 2 } ( \\frac { 2 \\sqrt { 6 } } { 3 } \\bar { \\alpha } x _ { i } h _ { T } ^ { 1 - \\beta } ) . \\end{align*}"}
+{"id": "1957.png", "formula": "\\begin{align*} \\begin{aligned} \\left \\Vert \\int _ { \\R } v \\bar { f } \\ , { \\rm d } v \\right \\Vert _ { L ^ 2 ( [ 0 , T ] \\times I ) } & \\lesssim T \\| \\bar { u } ^ * \\| _ { X } \\left ( \\| m _ 2 f _ 1 \\| _ { L ^ \\infty ( 0 , T ; L ^ 1 ( I ) ) } + \\| u _ 2 ^ * \\| _ { X } \\| m _ 1 f _ 1 \\| _ { L ^ \\infty ( 0 , T ; L ^ 1 ( I ) ) } \\right . \\\\ & \\left . + \\| u _ 2 ^ * \\| ^ 2 _ { X } \\| m _ 0 f _ 1 \\| _ { L ^ \\infty ( 0 , T ; L ^ 1 ( I ) ) } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "149.png", "formula": "\\begin{align*} ( { \\rm \\overline { B r } } ' \\circ { \\rm K a z } _ { e m } ^ E ) ( h ) = \\sum _ { \\mu \\in X _ * ( \\textbf { T } ) ^ { - } } \\sum _ { ( a _ i , a _ j ) \\in T _ \\mu ( E ' ) } \\alpha _ \\mu ( a _ i , a _ j ) { \\rm \\overline { B r } } ' ( t _ { a _ i ' \\pi _ \\mu ' a _ j '^ { - 1 } } ) . \\end{align*}"}
+{"id": "302.png", "formula": "\\begin{align*} \\left [ \\mbox { \\boldmath $ \\beta $ } \\right ] _ { j , k } : = \\left ( \\left . \\beta _ { \\operatorname * { D } ; j } \\right \\vert _ { \\Gamma _ { j , k } } - \\left . \\beta _ { \\operatorname * { D } ; k } \\right \\vert _ { \\Gamma _ { j , k } } , - \\left . \\beta _ { \\operatorname * { N } ; j } \\right \\vert _ { \\Gamma _ { j , k } } - \\left . \\beta _ { \\operatorname * { N } ; k } \\right \\vert _ { \\Gamma _ { j , k } } \\right ) \\end{align*}"}
+{"id": "5070.png", "formula": "\\begin{align*} E ( A ) E ( B ) & = \\sum _ { p \\geq 0 } \\frac { A ^ p } { m ( p ) } \\sum _ { p \\geq 0 } \\frac { B ^ p } { m ( p ) } \\\\ & = \\sum _ { p \\geq 0 } \\sum _ { n = 0 } ^ p \\frac { A ^ n } { m ( n ) } \\frac { B ^ { p - n } } { m ( p - n ) } = \\sum _ { p \\geq 0 } \\sum _ { n = 0 } ^ p \\frac { B ^ { p - n } } { m ( n ) } \\frac { A ^ n } { m ( p - n ) } \\\\ & = \\sum _ { p \\geq 0 } \\sum _ { l = 0 } ^ p \\frac { B ^ { l } } { m ( l ) } \\frac { A ^ { p - l } } { m ( p - l ) } = \\sum _ { p \\geq 0 } \\frac { B ^ p } { m ( p ) } \\sum _ { p \\geq 0 } \\frac { A ^ p } { m ( p ) } = E ( B ) E ( A ) . \\end{align*}"}
+{"id": "5594.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty | u _ n ^ { ( i ) } - u ^ { ( i ) } | ^ p r ^ { \\alpha _ i } \\mathrm d r \\overset { n \\to \\infty } \\longrightarrow 0 , \\quad \\forall i = 0 , \\ldots , k . \\end{align*}"}
+{"id": "3362.png", "formula": "\\begin{align*} d ( p , q ) = \\sup \\big \\{ | f ( p ) - f ( q ) | \\mid f \\in C ^ \\infty ( M ) , \\| [ D , f ] \\| \\leq 1 \\big \\} , \\end{align*}"}
+{"id": "2820.png", "formula": "\\begin{align*} & ( - \\Delta ) ^ { s } ( t ^ 2 + | \\cdot | ^ 2 ) ^ { - \\frac { \\theta } { 2 } } ( x ) \\\\ & = \\frac { c _ { N , s } } { | x | ^ { 2 s } } \\int _ { 0 } ^ 1 \\psi ( r ) \\Bigg ( r ^ { N - 1 } \\left ( \\frac { 1 } { ( t ^ 2 + | x | ^ 2 ) ^ { \\frac { \\theta } { 2 } } } - \\frac { 1 } { ( t ^ 2 + r ^ 2 | x | ^ 2 ) ^ { \\frac { \\theta } { 2 } } } \\right ) + r ^ { 2 s - 1 } \\left ( \\frac { 1 } { ( t ^ 2 + | x | ^ 2 ) ^ { \\frac { \\theta } { 2 } } } - \\frac { 1 } { ( t ^ 2 + \\frac { | x | ^ 2 } { r ^ 2 } ) ^ { \\frac { \\theta } { 2 } } } \\right ) \\Bigg ) \\dd r . \\end{align*}"}
+{"id": "506.png", "formula": "\\begin{align*} \\mathbb L ^ 2 ( \\Omega , Z ) = \\left \\{ u \\in L ^ 2 ( \\Omega , Z ) \\colon \\right \\} . \\end{align*}"}
+{"id": "3653.png", "formula": "\\begin{align*} k \\omega _ \\epsilon = \\mathcal { E } ( \\tilde { L } \\beta ) \\ , \\end{align*}"}
+{"id": "7698.png", "formula": "\\begin{align*} { \\rho ' } _ { x ' \\sigma ' } ^ { \\sigma '' } \\cdot \\rho _ { x \\sigma } ^ { \\sigma ' } = { \\rho ' } _ { g ' \\sigma ' } ^ { \\sigma '' } \\cdot \\rho _ { g \\sigma } ^ { \\sigma ' } \\\\ = { \\rho ' } _ { u '' g ' \\sigma ' } ^ { u '' \\sigma '' } \\cdot \\rho _ { u '' g ' u ' g \\sigma } ^ { u '' g ' u ' \\sigma ' } \\\\ = { \\rho ' } _ { u '' g ' \\sigma ' } ^ { \\sigma '' } \\cdot \\rho _ { u '' g ' u ' g \\sigma } ^ { u '' g ' \\sigma ' } \\end{align*}"}
+{"id": "1888.png", "formula": "\\begin{align*} \\left ( \\phi \\right ) ^ + _ { i + \\frac { 1 } { 2 } , v } = \\lim _ { \\eth \\to 0 ^ + } \\phi _ h \\left ( x _ { i + \\frac { 1 } { 2 } } + \\eth , v \\right ) \\mbox { a n d } \\left ( \\phi \\right ) ^ - _ { i + \\frac { 1 } { 2 } , v } = \\lim _ { \\eth \\to 0 ^ + } \\phi _ h \\left ( x _ { i + \\frac { 1 } { 2 } } - \\eth , v \\right ) . \\end{align*}"}
+{"id": "5925.png", "formula": "\\begin{align*} 1 - R _ { n + 2 } = d ( - a _ { n + 1 } a _ { n + 2 } ) = d ( ( - 1 ) ^ { ( n + 2 ) / 2 } a _ { 1 , n + 2 } ) = d ( c ) \\end{align*}"}
+{"id": "2813.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } J _ k = \\lim _ { k \\to \\infty } \\int _ { \\Omega } \\frac { Y \\cdot \\nabla z _ k } { ( t ^ 2 + | x | ^ 2 ) ^ \\frac { \\theta } { 2 } } ( - \\Delta ) ^ { s } U _ { t , p } \\dd x = 0 . \\end{align*}"}
+{"id": "4663.png", "formula": "\\begin{align*} ( \\Omega _ { 1 } + M _ { 1 } ) s = N _ { 1 } s + ( \\Omega _ { 1 } - A _ { 1 } ) | s | - r q , \\end{align*}"}
+{"id": "6583.png", "formula": "\\begin{align*} { { G } } _ { r + 1 } = \\left ( R _ { [ - M ^ { r + 1 } , M ^ { r + 1 } ] ^ { b + d } \\setminus \\mathcal { S } } { { H } } R _ { [ - M ^ { r + 1 } , M ^ { r + 1 } ] ^ { b + d } \\setminus \\mathcal { S } } \\right ) ^ { - 1 } . \\end{align*}"}
+{"id": "6539.png", "formula": "\\begin{align*} \\Omega _ N & = \\left \\{ \\begin{array} { l l } \\Omega & { \\rm i f } \\ N _ 0 \\leq N \\leq ( \\varepsilon + \\delta ) ^ { - \\frac { 1 } { 1 0 ^ 4 d b ^ 4 } } , \\\\ { \\rm D C } ( N ) \\cap \\widetilde \\Omega _ N & { \\rm i f } N > ( \\varepsilon + \\delta ) ^ { - \\frac { 1 } { 1 0 ^ 4 d b ^ 4 } } , \\end{array} \\right . \\end{align*}"}
+{"id": "3814.png", "formula": "\\begin{align*} \\nu _ X = \\nu _ X ^ 0 \\otimes v _ X ^ 1 \\end{align*}"}
+{"id": "7218.png", "formula": "\\begin{align*} Y _ t ^ { i , N } & : = x _ 0 ^ { 1 , N } + \\int _ { t _ 0 } ^ t \\alpha ^ { i , N } _ s d s + \\int _ { t _ 0 } ^ t b ^ { i , N } ( \\textbf { Y } _ s ^ N ) d s + \\sqrt { 2 } \\int _ { t _ 0 } ^ t d B ^ { i , N } _ s . \\end{align*}"}
+{"id": "3279.png", "formula": "\\begin{align*} f _ X ( x ) = \\int _ \\mathbb { R } \\phi _ X ( x ) e ^ { - \\mu t x } d x = \\mathcal { F } ^ { - \\mu } ( \\phi _ X ) ( t ) . \\end{align*}"}
+{"id": "1975.png", "formula": "\\begin{align*} \\begin{aligned} \\tau = & s - r = \\sum _ { i = m - \\delta + 1 } ^ m \\left ( s _ i - r _ i \\right ) p ^ { i - 1 } \\\\ & \\leq p ^ { m - \\delta } + p ^ { m - \\delta + 1 } + \\hdots + p ^ { m - 1 } = p ^ { m } - p ^ { m - \\delta } . \\end{aligned} \\end{align*}"}
+{"id": "3091.png", "formula": "\\begin{align*} \\begin{aligned} { f _ k } \\left ( { { x _ k } } \\right ) = & { \\log _ 2 } \\left ( { { 1 + { { \\bar C } _ k } \\left ( { 1 - { 2 ^ { - 1 - \\left ( { { x _ k } + 1 } \\right ) } } } \\right ) } } \\right ) \\\\ & - { \\log _ 2 } \\left ( { { 1 + { { \\bar C } _ k } \\left ( { 1 - { 2 ^ { - 1 - { x _ k } } } } \\right ) } } \\right ) \\\\ = & { \\log _ 2 } \\left ( { 1 + \\frac { 1 } { { { 2 ^ { 2 + { x _ k } } } \\left ( { 1 / { { \\bar C } _ k } + 1 } \\right ) - 2 } } } \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "7086.png", "formula": "\\begin{align*} \\mathcal { J } ^ { + } ( n ^ 2 _ 1 n _ 2 , m , q ) = \\displaystyle \\int _ { \\mathbb { R } } \\ , V _ 1 \\left ( \\frac { \\nu } { K } \\right ) \\ , \\displaystyle \\int _ { \\mathbb { R } } W ( u ) \\ , g ( u , q ) \\ , \\eth _ 1 ^ { + } ( n ^ 2 _ 1 n _ 2 , u , q , p _ 1 ) \\ , \\mathcal { I } ^ { + } ( m , u , q , p _ 1 , p _ 2 ) \\ , d u \\ , d \\nu . \\end{align*}"}
+{"id": "7492.png", "formula": "\\begin{align*} \\lambda ^ { \\sigma ^ 2 + 1 } - \\lambda ^ { \\sigma + 1 } - \\lambda ^ { \\sigma } + \\lambda = 0 . \\end{align*}"}
+{"id": "6112.png", "formula": "\\begin{align*} \\sigma _ A ( T ) = \\Big \\{ \\varphi ( T ) : \\varphi \\ \\ \\varphi ( A ) \\neq 0 \\Big \\} \\subseteq \\sigma ( T ) . \\end{align*}"}
+{"id": "3732.png", "formula": "\\begin{align*} { \\bf u } = \\begin{bmatrix} u \\\\ v \\\\ w \\end{bmatrix} { \\bf u } _ 0 = \\begin{bmatrix} u _ 0 \\\\ u _ 1 \\\\ u _ 2 \\end{bmatrix} . \\end{align*}"}
+{"id": "4498.png", "formula": "\\begin{align*} | | e ' _ k | | _ { s , \\ast , T } & \\lesssim \\delta ^ 2 \\Delta ^ 2 _ k ( \\theta ^ { 1 1 - 2 \\alpha } _ k + \\theta ^ { 4 - \\alpha } _ k ) \\\\ & \\lesssim \\delta ^ 2 \\theta ^ { L _ 1 ( \\alpha - 2 ) - 1 } _ k \\Delta _ k . \\end{align*}"}
+{"id": "2896.png", "formula": "\\begin{align*} c \\langle \\xi _ \\mu , \\varphi \\rangle + \\langle \\hat \\rho _ v ( \\xi _ \\mu ) , \\varphi \\rangle \\geq 2 \\pi c + \\pi \\sum _ { \\alpha \\in R _ 0 ^ + } \\langle \\alpha , \\varphi ^ \\vee \\rangle \\langle \\varphi , \\frac { \\alpha ^ \\vee } { m _ \\alpha } \\rangle { = } 2 \\pi ( 1 + c + \\langle \\hat \\rho , \\varphi \\rangle ) , \\end{align*}"}
+{"id": "7295.png", "formula": "\\begin{align*} \\sum _ { e \\in \\N } \\frac { 1 } { 2 ^ { e + 2 } } = \\frac { 1 } { 2 } . \\end{align*}"}
+{"id": "4697.png", "formula": "\\begin{align*} P _ i ( q ) - P _ j ( q ) - ( i - j ) & = \\frac { ( q ^ { 2 i - 2 } - 1 ) q ^ { j - 2 } - ( q ^ { 2 j - 2 } - 1 ) q ^ { i - 2 } } { q ^ { j + i - 4 } ( q ^ 2 - 1 ) } = \\frac { ( q ^ { j + i - 2 } + 1 ) ( q ^ { i - 2 } - q ^ { j - 2 } ) } { q ^ { j + i - 4 } ( q ^ 2 - 1 ) } . \\end{align*}"}
+{"id": "2374.png", "formula": "\\begin{align*} 0 = \\sum _ { \\mu \\in \\Lambda ^ 0 } d _ \\mu f ( \\pi ( \\mu ) ^ { - 1 } \\pi ( \\lambda ) ^ { - 1 } \\tau ) = d _ \\lambda \\frac { o ( G ) } { o ( \\Lambda ) } . \\end{align*}"}
+{"id": "4294.png", "formula": "\\begin{align*} & \\Pi _ 1 ( \\pi ) = \\{ J = ( J _ 1 , J _ 2 ) \\in \\Pi ( \\pi ) : ( J _ 1 , J _ 2 ) ( \\ref { e q n : t a u c o n d i t i o n 1 } ) ( \\ref { e q n : t a u c o n d i t i o n 2 } ) \\} , \\\\ & \\Pi _ 2 ( \\pi ) = \\{ J = ( J _ 1 , J _ 2 ) \\in \\Pi ( \\pi ) : ( J _ 1 , - J _ 2 ) ( \\ref { e q n : t a u c o n d i t i o n 1 } ) ( \\ref { e q n : t a u c o n d i t i o n 2 } ) \\} . \\end{align*}"}
+{"id": "7332.png", "formula": "\\begin{align*} \\mathcal { H } _ \\lambda ( t , u ) = \\frac { 1 } { 2 } \\langle A ( \\lambda , t ) u , u \\rangle + R ( \\lambda , t , u ) , \\end{align*}"}
+{"id": "3148.png", "formula": "\\begin{align*} \\mu _ L ( \\beta ) = 2 \\quad \\beta \\cdot D _ i = 1 . \\end{align*}"}
+{"id": "7958.png", "formula": "\\begin{align*} S _ 3 = \\{ v _ { i , j } : i \\equiv j \\ ! \\ ! \\ ! \\ ! \\pmod 2 \\} , ~ S _ 4 = \\{ v _ { i , j } : i \\not \\equiv j \\ ! \\ ! \\ ! \\ ! \\pmod 2 \\} . \\end{align*}"}
+{"id": "4694.png", "formula": "\\begin{align*} d _ n , q ^ { 2 n - 5 } , \\pm q ^ { 2 n - 4 } , \\pm q ^ { 3 n - 6 } . \\end{align*}"}
+{"id": "516.png", "formula": "\\begin{align*} \\left \\{ \\tau _ R > t \\right \\} \\cap \\left \\{ \\tau > t \\right \\} = \\bigcup _ { n = 1 } ^ \\infty \\bigcap _ { s \\in \\mathbb Q \\cap [ 0 , t ) } \\left ( \\left \\{ f _ s < R - \\frac 1 n \\right \\} \\cap \\left \\{ \\tau > t \\right \\} \\right ) , \\end{align*}"}
+{"id": "4457.png", "formula": "\\begin{align*} \\nabla _ { t , x _ 2 } \\varphi = \\mathbf { H } ( \\hat { { \\mathbf U } } , \\hat { \\varphi } ) { \\mathbf V } _ n | _ { x _ 1 = 0 } + \\mathbf { G } ( \\hat { { \\mathbf U } } , \\hat { \\varphi } ) \\varphi . \\end{align*}"}
+{"id": "4848.png", "formula": "\\begin{align*} \\check { R } ( u ) = B + \\lambda _ 1 \\lambda _ 2 y ( u ) B ^ { - 1 } \\end{align*}"}
+{"id": "915.png", "formula": "\\begin{align*} ( - ( 1 - \\alpha ) t ^ { - 1 - \\alpha } \\tau + ( 1 - \\alpha ) t ^ { - \\alpha } \\tau _ { t } - t ^ { 1 - \\alpha } \\tau _ { t t } ) x - 2 t ^ { 1 - \\alpha } \\sigma _ { 1 t } - 2 \\eta _ { 1 x } - 2 \\phi _ { 1 x } + \\frac { 2 c } { x ^ 2 } \\sigma _ 1 = 0 , \\end{align*}"}
+{"id": "8412.png", "formula": "\\begin{align*} R = d N ^ { 1 - \\gamma } ( \\log N ) ^ { 1 2 } . \\end{align*}"}
+{"id": "2755.png", "formula": "\\begin{align*} g ( w ) : = G _ { \\mu _ 1 } ^ { - 1 } ( w ) + G _ { \\mu _ 2 } ^ { - 1 } ( w ) - \\frac { 1 } { w } , \\end{align*}"}
+{"id": "5339.png", "formula": "\\begin{align*} | \\rho ( x ) - P _ m ( x ) | & = \\left | \\frac { \\rho ^ { ( m + 1 ) } ( \\xi ) } { m ! } ( x - a _ J ) ^ { m + 1 } \\right | \\\\ & \\leq \\left | \\frac { \\rho ^ { ( m + 1 ) } ( \\xi ) } { m ! } \\right | \\end{align*}"}
+{"id": "8349.png", "formula": "\\begin{align*} \\| u \\| _ { \\infty } = \\| u \\| _ { \\infty , T _ 1 , T _ 2 } = \\sup _ { s \\in [ T _ 1 , T _ 2 ] } \\| u ( s ) \\| . \\end{align*}"}
+{"id": "4036.png", "formula": "\\begin{align*} \\ell y : = - y ^ { \\prime \\prime } ( x ) + Q ( x , \\rho ) y ( a ) = \\rho ^ { 2 } y ( x ) , 0 < x < 1 , \\end{align*}"}
+{"id": "3997.png", "formula": "\\begin{align*} X = \\chi + t \\chi + \\tilde \\chi + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi _ t . \\end{align*}"}
+{"id": "8804.png", "formula": "\\begin{align*} \\bigg | f ( x ) - \\sum _ { 0 \\le | m | \\leq \\ell } \\frac { 1 } { m ! } D ^ { m } f ( x ) ( x - y ) ^ { m } \\bigg | & = \\abs { \\sum _ { 0 \\le | m | = \\ell } \\frac { 1 } { m ! } \\left ( D ^ { m } f ( \\zeta _ { x , y } ) - D ^ m f ( y ) \\right ) ( x - y ) ^ { m } } \\\\ & \\leq \\frac { 1 } { \\ell ! } \\abs { f ^ { ( \\ell ) } ( \\zeta _ { x , y } ) [ ( x - y ) , \\ldots , ( x - y ) ] - f ^ { ( \\ell ) } ( y ) [ ( x - y ) , \\ldots , ( x - y ) ] } \\enspace . \\end{align*}"}
+{"id": "4322.png", "formula": "\\begin{align*} \\Gamma u _ t = u _ t \\Gamma \\ \\ ( t \\in \\mathbb { R } ) \\ , . \\end{align*}"}
+{"id": "2078.png", "formula": "\\begin{align*} \\partial _ t \\left ( \\alpha \\partial _ t g g ^ { - 1 } \\right ) - \\partial _ x \\left ( \\alpha \\partial _ x g g ^ { - 1 } \\right ) = 0 , \\det g = \\alpha ^ 2 . \\end{align*}"}
+{"id": "3689.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { i = 1 } ^ m \\lambda _ { t _ i } ^ { i } : \\ , 1 \\leq t _ i \\leq n _ i \\ , \\ , \\forall i \\in [ m ] \\right \\} . \\end{align*}"}
+{"id": "2807.png", "formula": "\\begin{align*} \\mathcal { K } _ Y ( x , y ) = \\frac { c _ { N , s } } { 2 } \\left [ \\operatorname { d i v } Y ( x ) + \\operatorname { d i v } Y ( y ) - ( N + 2 s ) \\frac { ( Y ( x ) - Y ( y ) ) \\cdot ( x - y ) } { | x - y | ^ { 2 } } \\right ] | x - y | ^ { - N - 2 s } , \\end{align*}"}
+{"id": "3684.png", "formula": "\\begin{align*} \\sum _ { u \\in \\sigma } d _ { k - 1 } ( \\sigma \\setminus \\{ u \\} ) = ( k - 1 ) d _ k ( \\sigma ) + 2 | E _ { \\sigma } | . \\end{align*}"}
+{"id": "1439.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ g \\hat { u } _ i ^ k + \\sum \\limits _ { j \\in I } k _ { i j } ^ { \\mathbf { c } ^ { t } } \\rho ^ k _ { j } \\left ( \\frac { \\hat h ^ k _ { j } e ^ { \\hat { u } ^ k _ j } } { \\int _ { M } \\hat { h } ^ k _ { j } e ^ { \\hat { u } ^ k _ j } \\mathrm { d } V _ { g } } - \\dfrac { 1 } { | M | } \\right ) = 0 \\ \\ M , \\ \\ \\forall \\ i \\in I , \\\\ \\hat { u } _ 1 ^ k + 2 \\sum \\limits _ { i \\in I \\setminus \\{ 1 , n + 1 \\} } \\hat { u } _ i ^ k + \\hat { u } _ { n + 1 } ^ k \\equiv 0 , \\end{cases} \\end{align*}"}
+{"id": "1366.png", "formula": "\\begin{align*} \\Delta _ { 2 \\kappa } = - \\xi _ { 2 - 2 \\kappa } \\circ \\xi _ { 2 \\kappa } , \\end{align*}"}
+{"id": "5818.png", "formula": "\\begin{align*} \\footnotesize & \\alpha ^ { d 3 } _ { m a x } , \\ ; \\alpha ^ { d 5 } _ { m a x } , \\ ; \\alpha ^ { d 7 } _ { m a x } , \\ ; \\dots , \\ ; \\alpha ^ { d , n - 2 } _ { m a x } , \\ ; \\alpha _ { m a x } , \\\\ & \\alpha _ 1 , \\alpha _ 3 , \\dots , \\alpha _ { n - 4 } , \\alpha _ { n - 2 } . \\end{align*}"}
+{"id": "1220.png", "formula": "\\begin{align*} J ( f ) = \\frac { P ( f ) } { \\| \\nabla f \\| _ { L ^ 2 } ^ { 2 p } } . \\end{align*}"}
+{"id": "5219.png", "formula": "\\begin{align*} \\tau \\sigma \\tau ^ { - 1 } ( \\{ a , b \\} ) & = \\tau \\sigma ( \\{ m ^ { - 1 } a , m ^ { - 1 } b \\} ) = \\tau ( \\{ m ^ { - 1 } a + 1 , m ^ { - 1 } b + 1 \\} ) \\\\ & = \\{ a + m , b + m \\} = \\sigma ^ m ( \\{ a , b \\} ) . \\end{align*}"}
+{"id": "844.png", "formula": "\\begin{align*} \\| \\widetilde { T f } ( z ) - \\widetilde { T f } ( e ^ { i \\theta _ 0 } ) \\| & = \\| x ^ * ( f ) \\tilde { h } ( z ) - x ^ * ( f ) \\tilde { h } ( e ^ { i \\theta _ 0 } ) \\| \\\\ & \\leq \\| x ^ * ( f ) \\| \\| \\tilde { h } ( z ) - \\tilde { h } ( e ^ { i \\theta _ 0 } ) \\| \\\\ & \\leq \\| x ^ * \\| \\| \\tilde { h } ( z ) - \\tilde { h } ( e ^ { i \\theta _ 0 } ) \\| \\\\ & < \\| x ^ * \\| \\dfrac { \\epsilon } { \\| x ^ * \\| } \\\\ & = \\epsilon , \\end{align*}"}
+{"id": "8553.png", "formula": "\\begin{align*} E = _ { \\mathcal { H } ^ { n } } \\left ( F _ { \\ell } \\cap \\{ z < a \\} \\right ) \\cup \\left [ \\widetilde { E } \\cap \\left ( \\{ z < b \\} \\backslash \\{ z < a \\} \\right ) \\right ] \\cup \\left [ \\lambda ( r _ { \\ell } ( b ) - r _ { \\ell } ( a ) ) e + \\left ( F _ { \\ell } \\backslash \\{ z < b \\} \\right ) \\right ] . \\end{align*}"}
+{"id": "6436.png", "formula": "\\begin{align*} [ f ] \\oplus [ g ] & : = ( [ f _ 0 ] \\oplus [ n \\pi ] ) \\oplus ( [ g _ 0 ] \\oplus [ m \\pi ] ) \\\\ & = ( [ f _ 0 ] \\oplus [ g _ 0 ] ) \\oplus [ ( n + m ) \\pi ] . \\end{align*}"}
+{"id": "3468.png", "formula": "\\begin{align*} \\phi _ k ^ { N A } = \\max \\{ - \\sum _ 0 ^ m p _ i \\log | F _ i | - u ^ * ( p ) | p = ( p _ 0 , \\ldots p _ m ) \\in \\Delta ^ \\vee \\cap \\frac { 1 } { k } \\Z ^ { m + 1 } \\} . \\end{align*}"}
+{"id": "7258.png", "formula": "\\begin{align*} P _ { s , \\omega , n N } \\left ( g \\right ) ( x ) = P _ { s , \\sigma ^ { \\tilde { n } N } \\omega , N n - N \\tilde { n } } \\left ( P _ { s , \\omega , \\tilde { n } N } \\left ( g \\right ) \\right ) \\left ( x \\right ) . \\end{align*}"}
+{"id": "1771.png", "formula": "\\begin{gather*} \\mathbf { m } ( A ( s , t ) ) = \\frac { t - s } { 2 \\pi } , \\end{gather*}"}
+{"id": "1531.png", "formula": "\\begin{align*} I \\coloneqq \\frac 1 { 2 \\pi i } \\oint \\limits _ { | z | = 2 - \\epsilon } \\frac { \\eta ( z ) ( \\log y ) ^ z } { z ^ { k + 1 } } \\ , \\mathrm { d } z = \\eta _ o ( 2 ) \\frac { \\log ^ 2 y } { 2 ^ k } - \\frac 1 { 2 \\pi i } \\oint \\limits _ { | z | = 2 + \\epsilon _ 1 } \\frac { \\eta ( z ) ( \\log y ) ^ z } { z ^ { k + 1 } } \\ , \\mathrm { d } z \\end{align*}"}
+{"id": "8945.png", "formula": "\\begin{align*} \\alpha _ { d + 1 } \\left ( \\begin{pmatrix} A & v \\\\ 0 & 1 \\end{pmatrix} \\Z ^ { d + 1 } \\right ) = \\alpha _ d ( A \\Z ^ d ) A \\in S L _ { d } ( \\R ) , v \\in \\R ^ d . \\end{align*}"}
+{"id": "1742.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ 0 ^ \\pi R ( \\xi ) \\rho _ \\epsilon ( \\xi ) \\xi = \\sum _ { 0 \\leq { l } \\leq m } R \\bigl ( \\xi _ { { l } } ^ { ( m + 1 ) } \\bigr ) \\rho _ \\epsilon \\bigl ( \\xi _ { l } ^ { ( m + 1 ) } \\bigr ) { \\Delta } ^ { ( m + 1 ) } _ { { l } } , \\end{align*}"}
+{"id": "6293.png", "formula": "\\begin{align*} h _ 1 ( \\xi , \\eta ) = h _ 2 ( \\xi + \\eta , \\xi - \\eta ) , \\end{align*}"}
+{"id": "4454.png", "formula": "\\begin{align*} J _ 0 ( t ) = - 2 \\int _ { \\mathbb R ^ 2 _ + } \\Big ( \\mathcal { A } ^ 2 _ 0 D ^ { \\alpha } \\partial ^ { \\alpha _ 3 - 1 } _ 1 \\partial _ t { \\mathbf V } \\cdot D ^ { \\alpha } _ { \\ast } { \\mathbf V } \\Big ) ( t ) d \\mathbf { x } . \\end{align*}"}
+{"id": "4416.png", "formula": "\\begin{align*} \\mathcal { A } _ 1 = \\mathcal { A } + \\mathcal { A } _ { ( 0 ) } , \\mathcal { A } = \\mathrm { d i a g } \\Big ( \\frac { 1 } { \\partial _ 1 \\hat { \\Phi } ^ + } \\mathcal { E } _ { 1 2 } , \\frac { 1 } { \\partial _ 1 \\hat { \\Phi } ^ - } \\mathcal { E } _ { 1 2 } \\Big ) , \\mathcal { A } _ { ( 0 ) } | _ { x _ 1 = 0 } = 0 . \\end{align*}"}
+{"id": "5421.png", "formula": "\\begin{align*} \\eta ( x ) = \\begin{cases} 1 & | x | _ \\infty \\leq \\tfrac { 1 } { 3 2 | \\Gamma | } , \\\\ 0 & | x | _ \\infty \\geq \\tfrac { 1 } { 1 6 | \\Gamma | } . \\end{cases} \\end{align*}"}
+{"id": "2411.png", "formula": "\\begin{align*} \\left \\| \\sum _ { k = 1 } ^ { \\infty } A _ { \\overline { \\i } } ^ { k - 1 } x \\right \\| \\leq \\| x \\| + \\left \\| \\sum _ { k = 1 } ^ { \\infty } A _ { \\overline { \\i } } ^ { k } x \\right \\| \\leq \\| x \\| + \\sum _ { k = 1 } ^ { \\infty } \\lambda ^ k \\| x \\| \\leq \\| x \\| \\left ( 1 + \\frac { \\lambda } { 1 - \\lambda } \\right ) \\end{align*}"}
+{"id": "6856.png", "formula": "\\begin{align*} \\mathcal { X } _ { ( i _ 1 , \\dots , i _ d ) } = \\left ( B _ { ( i _ { j + 1 } , \\dots , i _ d ) } ^ T \\otimes A _ { ( i _ 1 , \\dots , i _ { j - 1 } ) } \\right ) ( G _ j ( i _ j ) ) , \\end{align*}"}
+{"id": "1961.png", "formula": "\\begin{align*} \\mathcal { G } \\ = \\ \\{ & x _ { 6 } t _ { u _ 2 } - y _ { 6 } t _ { u _ 1 } , \\ , \\ , x _ { 5 } t _ { u _ 3 } - y _ { 5 } t _ { u _ 1 } , \\ , \\ , x _ { 6 } t _ { u _ 4 } - y _ { 6 } t _ { u _ 3 } , \\ , \\ , x _ { 5 } t _ { u _ 3 } - y _ { 5 } t _ { u _ 1 } , \\ , \\ , x _ { 4 } t _ { u _ 5 } - y _ { 4 } t _ { u _ 4 } , \\ , \\ , \\\\ & x _ { 3 } t _ { u _ 6 } - y _ { 3 } t _ { u _ 5 } , \\ , \\ , x _ { 2 } t _ { u _ 7 } - y _ { 2 } t _ { u _ 1 } , \\ , \\ , x _ { 1 } t _ { u _ 8 } - y _ { 1 } t _ { u _ 1 } , \\ , \\ , t _ { u _ 1 } t _ { u _ 4 } - t _ { u _ 2 } t _ { u _ 3 } \\} . \\end{align*}"}
+{"id": "2091.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\int _ { | x - v t | \\leq \\omega ( t ) } \\left [ ( \\partial _ t \\Lambda ) ^ 2 + ( \\partial _ x \\Lambda ) ^ 2 + \\sinh ^ 2 ( \\Lambda ) \\Big ( ( \\partial _ t \\phi ) ^ 2 + ( \\partial _ x \\phi ) ^ 2 \\Big ) \\right ] ( t , x ) d x = 0 . \\end{align*}"}
+{"id": "6467.png", "formula": "\\begin{align*} \\lambda _ n \\big ( R _ \\epsilon \\big ) = \\lambda ^ 0 _ n + O ( \\epsilon ) , \\lambda ^ 0 _ n : = \\lambda _ n \\big ( R _ 0 \\big ) > 0 . \\end{align*}"}
+{"id": "5558.png", "formula": "\\begin{align*} \\Psi _ k ( \\zeta ) = \\frac { 1 } { \\pi } \\Im \\log \\phi ( \\psi ( \\zeta , \\xi , \\xi _ 1 ) ) , \\zeta \\in G _ m . \\end{align*}"}
+{"id": "6341.png", "formula": "\\begin{align*} N _ \\lambda ^ { l , 2 , m e d } v v _ \\lambda ^ { x _ 0 } = L ( P _ { \\lambda } \\partial ^ 2 g ( u _ { < \\lambda } ) , v _ { \\lambda } , v _ \\lambda ) . \\end{align*}"}
+{"id": "1289.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } J ( f _ n ) = \\sup _ { f \\in \\dot { H } ^ 1 \\setminus \\{ 0 \\} } J ( f ) = C _ 0 . \\end{align*}"}
+{"id": "8827.png", "formula": "\\begin{align*} x _ { i + 1 } = \\Phi _ { i + 1 } ( y _ { 1 } , . . . , y _ { i } ) , i = 1 , 2 , . . . , \\end{align*}"}
+{"id": "2578.png", "formula": "\\begin{align*} \\beta ' ( h , h ' ) & = \\alpha ' ( h + h ' ) - \\alpha ' ( h ) - \\alpha ' ( h ' ) = \\\\ & = \\alpha ( h + h ' ) - \\psi ( h + h ' ) - ( \\alpha ( h ) - \\psi ( h ) ) - ( \\alpha ( h ' ) - \\psi ( h ' ) ) = \\\\ & = \\beta ( h , h ' ) - ( \\psi ( h + h ' ) - \\psi ( h ) - \\psi ( h ' ) ) = \\\\ & = \\beta ( h , h ' ) - \\psi ^ { * } ( h , h ' ) \\end{align*}"}
+{"id": "2127.png", "formula": "\\begin{align*} h _ 1 ( t , x ) = ( \\partial _ t \\Lambda ) ^ 2 + ( \\partial _ x \\Lambda ) ^ 2 + 4 \\sinh ^ 2 ( \\Lambda ) ( ( \\partial _ x \\phi ) ^ 2 + ( \\partial _ t \\phi ) ^ 2 ) \\geq 0 , \\end{align*}"}
+{"id": "2531.png", "formula": "\\begin{align*} \\boldsymbol { u } ( \\boldsymbol { x } , t ) = ( 0 , 0 , e ^ t ( \\cos { t } + ( 2 \\pi ^ 2 + 1 ) \\sin { t } ) \\sin { \\pi x _ 1 } \\sin { \\pi x _ 2 } ) ^ T \\end{align*}"}
+{"id": "6301.png", "formula": "\\begin{align*} E r r = \\int _ 0 ^ T \\int _ \\R \\partial _ x ^ 2 g _ { [ < \\lambda ] } | v _ \\lambda | ^ 2 | v _ \\mu ^ { x _ 0 } | ^ 2 d x d t . \\end{align*}"}
+{"id": "2312.png", "formula": "\\begin{align*} \\int _ { \\R } \\| \\varphi _ { j } ( \\lambda ) \\hat { f } ( \\xi , \\lambda + p ( \\xi ) ) \\| _ { L ^ { 2 } _ { \\xi } } d \\lambda & = \\int _ { | \\lambda | \\sim 2 ^ j } \\| \\varphi _ { j } ( \\lambda ) \\hat { f } ( \\xi , \\lambda + p ( \\xi ) ) \\| _ { L ^ { 2 } _ { \\xi } } d \\lambda \\\\ & \\lesssim 2 ^ { \\frac { j } { 2 } } \\| \\varphi _ { j } ( \\tau - p ( \\xi ) ) \\hat { f } ( \\xi , \\tau ) \\| _ { L ^ { 2 } _ { \\xi , \\tau } } , \\end{align*}"}
+{"id": "1725.png", "formula": "\\begin{align*} \\Delta _ { l } ^ { ( m + n ) } : = \\Delta ^ { ( m + n - 1 , 1 ) } _ { \\texttt { b } ; l } = \\left ( 2 ( m + n ) + u _ { q _ 0 } \\bigl ( \\xi ^ { ( m + n ) } _ l \\bigr ) + u _ { q _ 1 } \\bigl ( \\xi ^ { ( m + n ) } _ l \\bigr ) \\right ) ^ { - 1 } \\end{align*}"}
+{"id": "6936.png", "formula": "\\begin{align*} \\int _ { X } [ \\mathbb { K } _ t \\varphi ] \\psi d x = \\int _ { X } [ \\mathbb { P } _ t \\psi ] \\varphi d x , ~ \\forall \\psi , \\varphi . \\end{align*}"}
+{"id": "2468.png", "formula": "\\begin{align*} \\overline { H } \\left ( \\bigsqcup _ { a \\in \\mathcal { A } } ^ { P _ A } G _ a ^ { \\wedge 3 } \\right ) = 3 \\overline { H } \\left ( \\bigsqcup _ { a \\in \\mathcal { A } } ^ { P _ A } G _ a \\right ) = 3 \\sum _ { a \\in \\mathcal { A } } P _ A ( a ) \\overline { H } ( G _ a ) = 3 ; \\end{align*}"}
+{"id": "8755.png", "formula": "\\begin{align*} \\hat g _ t = \\frac { d } { 2 h _ t } ( y _ t - y ' _ t ) \\zeta _ t , \\end{align*}"}
+{"id": "3851.png", "formula": "\\begin{align*} h _ { 1 1 } & = - \\theta ' - \\lambda _ 2 \\cos \\theta , \\\\ h _ { 1 2 } & = 0 , \\\\ h _ { 2 1 } & = 0 , \\\\ h _ { 2 2 } & = - \\lambda _ 1 e ^ { - 2 \\lambda _ 1 z } \\cos \\theta . \\end{align*}"}
+{"id": "4655.png", "formula": "\\begin{align*} \\mathbf { X } _ { v | s } = \\bigotimes _ { e \\in \\mathcal { S } ( v | s ) } X _ e , \\mathbf { Y } _ { v | t } = \\bigotimes _ { e ^ \\prime \\in \\mathcal { S } ( v | t ) } Y _ { e ^ \\prime } , \\end{align*}"}
+{"id": "8240.png", "formula": "\\begin{align*} i _ { \\mu ^ { - 1 } ( 0 ) } ^ * d _ { I _ 1 } ^ c \\hat { K } = p _ 0 ^ * d ^ c _ { I _ 1 } ( \\frac { 1 } { 2 } f _ { x _ 1 } ) + x _ 1 i _ { \\mu ^ { - 1 } ( 0 ) } ^ * \\eta . \\end{align*}"}
+{"id": "4519.png", "formula": "\\begin{align*} \\tilde { e } ''' _ k : = & \\mathcal { B } ' ( S _ { \\theta _ k } { \\mathbf V } _ k | _ { x _ 1 = 0 } , S _ { \\theta _ k } \\psi _ k ) ( ( \\delta { \\mathbf V } _ k ) | _ { x _ 1 = 0 } , \\delta \\psi _ k ) \\\\ & - \\mathcal { B } ' ( { \\mathbf V } _ { k + \\frac { 1 } { 2 } } | _ { x _ 1 = 0 } , \\psi _ { k + \\frac { 1 } { 2 } } ) ( ( \\delta { \\mathbf V } _ k ) | _ { x _ 1 = 0 } , \\delta \\psi _ k ) . \\\\ \\end{align*}"}
+{"id": "7480.png", "formula": "\\begin{align*} \\kappa ( X , K _ X + \\Delta + f ^ * L ) & = \\kappa ( F , K _ F + \\Delta | _ F ) + \\kappa ( A , \\widehat { \\det } f _ * \\mathcal { O } _ X ( l D ) \\otimes \\mathcal { O } _ A ( L ) ) \\\\ & = \\kappa ( F , K _ F + \\Delta | _ F ) + \\dim V ^ 0 ( A , f _ * \\mathcal { O } _ X ( l D ) \\otimes \\mathcal { O } _ A ( L ) ) . \\end{align*}"}
+{"id": "6921.png", "formula": "\\begin{align*} c [ \\phi ( \\xi _ n ) - \\phi ( 0 ) ] - & d _ 1 \\int _ { 0 } ^ { \\xi _ n } \\mathcal { N } _ 1 \\left [ \\phi \\right ] ( \\xi ) d \\xi = \\int _ { 0 } ^ { \\xi _ n } \\phi ( \\xi ) f \\left ( \\phi , \\psi \\right ) ( \\xi ) d \\xi . \\end{align*}"}
+{"id": "2865.png", "formula": "\\begin{align*} h _ t : = t _ { \\vartheta } \\ , e _ t ( - \\alpha _ 0 ) \\quad e _ t ( \\nu ) : = \\prod _ { \\alpha \\in R ^ + _ 0 } t _ { \\alpha } ^ { \\langle \\nu , \\alpha ^ \\vee \\rangle / 2 } . \\end{align*}"}
+{"id": "73.png", "formula": "\\begin{align*} c _ k = b _ k + \\frac { 1 } { \\mathcal D _ k ( 1 - \\varepsilon _ K ) \\vert \\Lambda \\vert } \\sum _ { p \\in \\mathcal P _ L } \\frac { \\widehat g ( k ) } { \\sqrt { 1 - \\alpha _ { k } ^ { 2 } \\vphantom { \\alpha _ { p - k } ^ 2 } } \\sqrt { 1 - \\alpha _ { p - k } ^ 2 } } \\Big ( z b _ { p - k } ^ \\dagger a _ p + \\alpha _ k \\alpha _ { p - k } \\bar z a _ { - p } ^ \\dagger b _ { p - k } ^ \\dagger \\Big ) , \\end{align*}"}
+{"id": "547.png", "formula": "\\begin{align*} \\norm { M _ i ( \\mathbf f ) } _ { \\mathcal L _ 2 ( L ^ 2 , H ^ { s _ i } ) } \\le \\sum _ { j = 1 } ^ n \\norm { f _ j \\mathfrak K _ { i , j } } _ { \\mathcal L _ 2 ( L ^ 2 , H ^ { s _ i - \\sigma _ { i , j } } ) } \\le C \\sum _ { j = 1 } ^ n \\norm { f _ j } _ { H ^ { s _ j } } \\norm { \\mathfrak k _ { i , j } } _ { H ^ { \\abs { s _ j } } } \\end{align*}"}
+{"id": "205.png", "formula": "\\begin{align*} \\delta _ 2 ( - \\frac { 1 } { \\tau } ) = \\tau ^ 2 \\delta _ 1 ( \\tau ) , ~ ~ ~ ~ ~ ~ \\varepsilon _ 2 ( - \\frac { 1 } { \\tau } ) = \\tau ^ 4 \\varepsilon _ 1 ( \\tau ) , \\end{align*}"}
+{"id": "6866.png", "formula": "\\begin{align*} ( \\mathcal R _ i ) _ { ( i ) } = \\left ( R _ { i } ^ G ( A _ i ) \\circ C _ i \\right ) ( { R _ { i } ^ G } ^ H ( - B _ i ) \\circ ^ { - 1 } Z _ i ) ^ H , R _ { i } ^ G ( z ) = \\chi _ { i } ( z ) / Q _ { i } ( z ) , \\end{align*}"}
+{"id": "3564.png", "formula": "\\begin{align*} \\delta ^ h _ P ( Q ) = 2 p ^ t \\Delta p + 2 q ^ t \\Delta q - 4 \\left ( \\frac { p + q } { 2 } \\right ) ^ t \\Delta \\left ( \\frac { p + q } { 2 } \\right ) = ( q - p ) ^ t \\Delta ( q - p ) . \\end{align*}"}
+{"id": "8864.png", "formula": "\\begin{align*} \\tau _ { t } \\leq \\tau _ { t _ { 0 } } \\prod _ { i = { t _ { 0 } } } ^ { t - 1 } \\Big ( 1 - \\frac { c } { i } \\Big ) \\leq \\tau _ { t _ { 0 } } \\prod _ { i = { t _ { 0 } } } ^ { t - 1 } \\Big ( 1 - \\frac { 1 } { i } \\Big ) \\le \\frac { ( t _ { 0 } - 1 ) \\tau _ { t _ { 0 } } } { t } \\le \\frac { 2 ( t _ { 0 } - 1 ) \\delta _ { t _ { 0 } } } { t } \\enspace . \\end{align*}"}
+{"id": "2967.png", "formula": "\\begin{align*} ( \\psi _ N ( x \\otimes y ) ) _ i = \\begin{cases} \\phi ( ( x _ i \\mid x ) _ A ) y & i \\le N ; \\\\ 0 & \\end{cases} \\end{align*}"}
+{"id": "8898.png", "formula": "\\begin{align*} f ^ * f _ * b ( a ) = \\sum _ { w \\in S _ n } w \\cdot \\left ( \\dfrac { b ( a ) } { \\prod _ { i < j } ( a _ i - a _ j ) } \\right ) , \\end{align*}"}
+{"id": "1495.png", "formula": "\\begin{align*} A _ { i } ( t ) = \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } \\langle U ^ { ( i ) } B _ { s } , d B _ { s } \\rangle , \\ ; i = 1 , \\ldots , n . \\end{align*}"}
+{"id": "648.png", "formula": "\\begin{align*} \\mathcal L _ { \\mathrm { Y u k a w a } } ( \\psi , \\phi ) = \\phi \\psi ^ * \\beta \\psi . \\end{align*}"}
+{"id": "7898.png", "formula": "\\begin{align*} \\begin{cases} ( u ^ { \\star } ) ^ k \\det D ^ 2 u = ( 1 - s u ) ^ { p } & \\Omega \\\\ u = 0 & \\partial \\Omega . \\end{cases} \\end{align*}"}
+{"id": "8708.png", "formula": "\\begin{align*} \\int K ( r ) \\d r { = } 0 , \\int r K ( r ) \\d r { = } 1 , \\int r ^ j K ( r ) \\d r { = } 0 , \\ j { = } 2 , \\dots , \\ell , ~ \\kappa _ \\beta { \\triangleq } \\int | r | ^ { \\beta } | K ( r ) | \\d r < \\infty \\end{align*}"}
+{"id": "8029.png", "formula": "\\begin{align*} { \\cal { T } } _ { x } ^ { \\alpha } f ( y ) = \\int _ { \\R _ { + } ^ { n } } \\omega _ { \\alpha } ( x , y , z ) f ( z ) d \\mu _ { \\alpha } ( z ) , \\end{align*}"}
+{"id": "8485.png", "formula": "\\begin{align*} \\theta \\left ( \\{ g > - M \\} \\cap E ; x \\right ) = 0 , \\mbox { f o r e v e r y } M > 0 ( s = - \\infty ) . \\end{align*}"}
+{"id": "5623.png", "formula": "\\begin{align*} \\begin{pmatrix} - 2 & 0 \\\\ 0 & 4 \\end{pmatrix} \\end{align*}"}
+{"id": "1412.png", "formula": "\\begin{gather*} \\tilde \\psi ( x ) = ( E + \\tilde H ^ K ( x ) ) \\psi ^ K ( x ) \\end{gather*}"}
+{"id": "2402.png", "formula": "\\begin{align*} \\sum _ { n \\in \\mathcal { N } } \\tilde { h } ( n ) = \\infty . \\end{align*}"}
+{"id": "8310.png", "formula": "\\begin{align*} u \\circ F & = F \\circ u \\\\ u \\circ G & = G \\circ u . \\end{align*}"}
+{"id": "2911.png", "formula": "\\begin{align*} ( T _ { w _ { \\lambda + \\nu } } f ) ( ( \\lambda + \\nu ) _ + ) & = ( T _ { w _ { \\lambda + s ' _ j \\nu } } T _ j f ) ( ( \\lambda + s ' _ j \\nu ) _ + ) \\\\ & = ( T _ j f ) ( \\lambda + s ' _ j \\nu ) - d _ { \\lambda , s ' _ j \\nu } ( 1 - t _ { \\vartheta } ^ { - 1 } ) ( T _ j f ) ( \\lambda ) \\end{align*}"}
+{"id": "1794.png", "formula": "\\begin{gather*} \\widehat { \\phi } ( w ) = \\int _ { 0 } ^ { \\infty } \\phi ( x ) x ^ w \\ , \\frac { d x } { x } \\end{gather*}"}
+{"id": "2457.png", "formula": "\\begin{align*} \\overline { H } ( G ) = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } H _ { \\chi } ( G ^ { \\wedge n } ) . \\end{align*}"}
+{"id": "3523.png", "formula": "\\begin{align*} \\mathrm { t r } ( \\lambda _ p ^ 0 ( \\gamma ) ) = \\begin{cases} p & \\\\ \\left ( \\frac { d ( \\gamma ) } { p } \\right ) & . \\end{cases} \\end{align*}"}
+{"id": "4901.png", "formula": "\\begin{align*} T _ 2 ( z ) : = \\left \\{ \\begin{array} { l l } \\exp [ - \\frac { z } { x _ 1 } P _ 1 ] [ ( z - x _ 1 - z P _ 1 ) [ \\frac { E _ - ( z ) - E _ - ( x _ 1 ) } { z - x _ 1 } ] + E _ - ( x _ 1 ) ] , z \\neq x _ 1 \\\\ \\exp ( - P _ 1 ) [ E _ - ( x _ 1 ) - x _ 1 P _ 1 E _ - ' ( x _ 1 ) ] , z = x _ 1 . \\end{array} \\right . \\end{align*}"}
+{"id": "7533.png", "formula": "\\begin{align*} r \\circ \\ , s \\ , \\circ \\ , S _ 0 \\ , \\circ \\ , \\overline { \\Omega } & = S _ 2 \\circ R ^ { N C } _ { 2 } \\circ p \\circ \\Omega _ Y \\\\ & = R ^ { P C } _ { 2 } \\circ T _ 2 \\circ p \\circ \\Omega _ Y \\stackrel { R ^ { P C } _ { 2 } \\circ T _ 2 ( \\gamma ) } { \\leadsto } 0 . \\end{align*}"}
+{"id": "3726.png", "formula": "\\begin{align*} u ( 0 ) = \\varphi \\in \\mathcal { H } ^ { \\frac { 2 } { 3 } } , \\ u ' ( 0 ) = \\psi \\in \\mathcal { H } ^ { \\frac { 2 - \\alpha } { 3 } } , \\ u '' ( 0 ) = \\xi \\in \\mathcal { H } ^ { \\frac { 2 - \\alpha } { 3 } } , \\end{align*}"}
+{"id": "8758.png", "formula": "\\begin{align*} \\hat { g } _ { j } ( t ) & = \\big ( e _ { j } ^ { \\top } A x _ t + x _ { t } ^ { \\top } A e _ { j } + b ^ { \\top } e _ { j } \\Big ) r K ( r ) + \\left ( \\frac { \\xi _ { j } ( t ) - \\xi ^ { ' } _ { j } ( t ) } { 2 h _ { t } } \\right ) K ( r ) \\\\ & = \\langle \\nabla f ( x _ t ) , e _ { j } \\rangle r K ( r ) + \\left ( \\frac { \\xi _ { j } ( t ) - \\xi ^ { ' } _ { j } ( t ) } { 2 h _ { t } } \\right ) K ( r ) , \\end{align*}"}
+{"id": "663.png", "formula": "\\begin{align*} \\bigcap _ { 0 \\le s < t _ k } f _ { s } ^ { - 1 } \\left ( \\left [ 0 , R \\right ] \\right ) = \\bigcap _ { 0 \\le s _ i < t _ k } f _ { s _ i } ^ { - 1 } \\left ( \\left [ 0 , R \\right ] \\right ) . \\end{align*}"}
+{"id": "4530.png", "formula": "\\begin{align*} ( { \\mathbf V } , \\varphi ) = ( 0 , 0 ) \\quad \\mbox { f o r } \\ , \\ , \\ , t < 0 \\ , . \\end{align*}"}
+{"id": "2539.png", "formula": "\\begin{align*} D ( x \\odot \\phi _ \\nu \\ , x ^ { L ( \\nu ) } ) & = \\mathrm { d i a g } ( \\phi _ \\nu ) x ^ { L ( \\nu ) } + \\mathrm { d i a g } ( x ) \\phi _ \\nu ( L ( \\nu ) ^ T \\odot x ^ { - \\bar 1 } ) ^ T x ^ { L ( \\nu ) } \\\\ & = \\mathrm { d i a g } ( \\phi _ \\nu ) x ^ { L ( \\nu ) } + ( \\phi _ \\nu \\odot x ) ( L ( \\nu ) ^ T \\odot x ^ { - \\bar 1 } ) ^ T x ^ { L ( \\nu ) } \\end{align*}"}
+{"id": "5396.png", "formula": "\\begin{align*} w _ { \\ast } = w + 2 t \\overline { v ( z , w _ { \\ast } ; 0 ) } , \\end{align*}"}
+{"id": "1963.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { C } \\left ( C ^ { e } , C ^ { e ^ { \\prime } } \\right ) ( \\tau ) & = \\sum _ { l = 0 } ^ { M - 1 } \\mathcal { C } \\left ( \\mathbf { c } _ { l } ^ { e } , \\mathbf { c } _ { l } ^ { e ^ { \\prime } } \\right ) ( \\tau ) \\\\ & = \\begin{cases} M N , & \\tau = 0 , e = e ^ { \\prime } , \\\\ 0 , & 0 < | \\tau | < Z , e = e ' , \\\\ 0 , & | \\tau | < Z , e \\neq e ' , \\end{cases} \\end{aligned} \\end{align*}"}
+{"id": "3491.png", "formula": "\\begin{align*} h ^ { \\otimes N _ 0 } ( h e ^ { - 2 ( \\phi _ t + | \\log | t | | \\psi _ t ) } ) ^ { \\otimes ( l - N _ 0 ) } = h ^ { \\otimes l } e ^ { - 2 ( l - N _ 0 ) ( \\phi _ t + | \\log | t | | \\psi _ t ) } , \\end{align*}"}
+{"id": "229.png", "formula": "\\begin{align*} \\nu \\geq 0 \\Leftrightarrow \\nu _ 1 + \\cdots + \\nu _ k \\geq 0 \\ \\ k = 1 , \\ldots , n , \\end{align*}"}
+{"id": "8454.png", "formula": "\\begin{align*} G ^ { * } _ { 1 , 0 } = \\int \\beta , G ^ { * } _ { 0 , 1 } = \\int \\alpha , \\end{align*}"}
+{"id": "5138.png", "formula": "\\begin{align*} { \\bf M } _ 1 = \\begin{bmatrix} 1 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 1 \\end{bmatrix} , ~ ~ { \\bf M } _ 2 = \\begin{bmatrix} 1 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0 \\end{bmatrix} . \\end{align*}"}
+{"id": "4153.png", "formula": "\\begin{align*} \\exp ^ * ( t f ) \\rhd \\phi = \\phi \\circ e ^ { t f } , \\end{align*}"}
+{"id": "7437.png", "formula": "\\begin{align*} \\nu \\int _ { \\Omega } \\left [ a _ 1 \\vert y _ 1 \\vert ^ { 1 - \\nu } + a _ 2 \\vert y _ 2 \\vert ^ { 1 - \\nu } \\right ] d z = & \\nu \\| ( y _ 1 , y _ 2 ) \\| _ { 1 , p } + \\nu \\| ( \\nabla y _ 1 , \\nabla y _ 2 ) \\| _ { q , \\eta } \\\\ & - \\lambda \\nu ( \\kappa _ 1 + \\kappa _ 2 + 2 ) \\int _ { \\Omega } \\vert y _ 1 \\vert ^ { \\kappa _ 1 + 1 } \\vert y _ 2 \\vert ^ { \\kappa _ 2 + 1 } d z . \\end{align*}"}
+{"id": "2850.png", "formula": "\\begin{align*} { P _ c } : = \\{ \\lambda \\in P \\mid 0 \\leq \\langle \\lambda , \\beta \\rangle \\leq c , \\ \\forall \\beta \\in { \\hat R _ 0 ^ + } \\} , \\\\ \\hat P _ c : = { \\{ \\mu \\in \\hat P \\mid 0 \\leq \\langle \\mu , \\alpha \\rangle \\leq c , \\ , \\forall \\alpha \\in R _ 0 ^ + \\} } , \\end{align*}"}
+{"id": "7884.png", "formula": "\\begin{align*} H _ { n + k } ( u ) : = \\frac { 1 } { n + k + 1 } \\int _ { \\Omega } ( u ^ { \\star } ) ^ { k } ( - u ) \\det D ^ 2 u . \\end{align*}"}
+{"id": "6687.png", "formula": "\\begin{align*} \\alpha _ 0 = - p - \\frac { i q } { \\kappa } , \\alpha _ 1 = \\left ( \\frac { 1 } { \\kappa } - 1 \\right ) p + p ^ 2 + \\frac { q ( - q + ( \\kappa - 1 ) i ) } { \\kappa ^ 2 } , \\end{align*}"}
+{"id": "5403.png", "formula": "\\begin{align*} d U _ { j k } ( t ) = \\sum _ { \\ell = 1 } ^ N S ^ { - 1 } _ { j \\ell } ( t ) d S _ { \\ell k } ( t ) = \\dfrac { ( S ^ { - 1 } ( t ) d M ( t ) S ( t ) ) _ { j k } } { \\Lambda _ k ( t ) - \\Lambda _ j ( t ) } , 1 \\leq j \\not = k \\leq N , \\ , t \\geq 0 . \\end{align*}"}
+{"id": "2097.png", "formula": "\\begin{align*} L = \\partial _ t + \\partial _ x , \\underline { L } = \\partial _ t - \\partial _ x . \\end{align*}"}
+{"id": "7760.png", "formula": "\\begin{align*} \\left ( \\tau _ { i j } \\right ) , \\tau _ { i j } : = \\frac { \\partial ^ 2 \\mathfrak { F } } { \\partial Z ^ i \\partial Z ^ j } \\end{align*}"}
+{"id": "55.png", "formula": "\\begin{align*} \\mathcal A _ k ( a _ k ^ \\dagger a _ k & + a _ { - k } ^ \\dagger a _ { - k } ) + \\mathcal B _ k ( a _ k ^ \\dagger a _ { - k } ^ \\dagger + a _ k a _ { - k } ) = \\\\ & \\mathcal D _ k ( b _ k ^ \\dagger b _ k + b _ { - k } ^ \\dagger b _ { - k } ) + \\sqrt { k ^ 4 + 2 k ^ 2 \\rho _ z \\widehat g ( k ) } - k ^ 2 - \\rho _ z \\widehat g ( k ) . \\end{align*}"}
+{"id": "8558.png", "formula": "\\begin{align*} \\mathcal { G } ^ { \\mu } [ \\varepsilon \\zeta , \\beta b ] \\psi = ( \\partial _ z \\Phi - \\mu \\varepsilon \\nabla _ X \\zeta \\cdot \\nabla _ X \\Phi ) _ { | _ { z = \\varepsilon \\zeta } } . \\end{align*}"}
+{"id": "4723.png", "formula": "\\begin{align*} \\begin{cases} \\Delta u = \\chi _ { u > 0 } & B _ 1 , \\\\ u \\ge 0 & B _ 1 \\\\ u = g & \\partial B _ 1 , \\ \\end{cases} \\end{align*}"}
+{"id": "5649.png", "formula": "\\begin{align*} K _ { Z _ 2 } + D _ 2 = \\sigma _ 2 ^ * ( K _ { Z _ 1 } + D _ 1 ) , \\end{align*}"}
+{"id": "5189.png", "formula": "\\begin{align*} \\alpha _ 1 = L _ 1 - L _ 2 , \\dots , \\alpha _ { n - 1 } = L _ { n - 1 } - L _ n , \\alpha _ n = L _ { n - 1 } + L _ n \\end{align*}"}
+{"id": "4832.png", "formula": "\\begin{align*} S ^ { i k } _ { j l } ( u ) = 0 \\textrm { u n l e s s } i + j = k + l \\end{align*}"}
+{"id": "5830.png", "formula": "\\begin{align*} w _ 3 = & s _ 1 s _ { \\alpha _ 2 + \\alpha _ 4 } s _ { \\alpha _ 3 + \\alpha _ 4 + \\alpha _ 5 } s _ { \\alpha _ 2 + \\alpha _ 3 + 2 \\alpha _ 4 + \\alpha _ 5 + \\alpha _ 6 } \\\\ & s _ { \\alpha _ 2 + \\alpha _ 3 + \\alpha _ 4 + \\alpha _ 5 + \\alpha _ 6 + \\alpha _ 7 } s _ { \\alpha _ 7 + \\alpha _ 8 } s _ 6 s _ { \\alpha _ 2 + \\alpha _ 4 + \\alpha _ 5 } . \\end{align*}"}
+{"id": "1861.png", "formula": "\\begin{gather*} \\mathbf { P } ( \\Omega ' _ 0 ) = \\prod _ { n = 0 } ^ { N } \\mathbf { P } \\left ( \\mathbb { Y } _ \\alpha ( n ) \\in A ( \\theta _ n - \\pi \\delta , \\theta _ n + \\pi \\delta ) \\right ) = \\delta ^ { N + 1 } \\end{gather*}"}
+{"id": "2787.png", "formula": "\\begin{align*} x \\cdot I ( G - N [ v _ 0 ] ) & = x \\cdot ( 2 x + 1 ) ^ { 2 k + 4 } \\cdot ( 3 x ^ 2 + 4 x + 1 ) \\\\ & = x ( 3 x ^ 2 + 4 x + 1 ) \\cdot \\bigg [ \\sum _ { i = 0 } ^ { 2 k + 4 } \\binom { 2 k + 4 } { i } ( 2 x ) ^ { i } ] \\\\ & = x ( 3 x ^ 2 + 4 x + 1 ) \\cdot [ ( 2 x ) ^ { 2 k + 1 } + \\dots ] \\\\ & = 3 \\cdot 2 ^ { 2 k + 2 } x ^ { 2 k + 5 } + \\dots \\end{align*}"}
+{"id": "7852.png", "formula": "\\begin{align*} l _ { \\gamma } = \\{ t \\ ; | \\ ; \\widetilde { Z } _ { \\gamma } / t \\in \\mathbb { R } _ { < 0 } \\} \\ , . \\end{align*}"}
+{"id": "2632.png", "formula": "\\begin{align*} G _ n ( \\alpha , t ) = ( e ^ { \\alpha / ( b _ n \\sqrt { n } ) } - 1 - \\frac { \\alpha } { b _ n \\sqrt { n } } ) \\sum _ { i = 1 } ^ { { { \\hat A } _ n } ( t ) } \\int _ 0 ^ { \\eta _ i \\wedge ( t - \\hat \\tau _ { n , i } ) } \\frac { d F ( u ) } { 1 - F ( u ) } \\ , . \\end{align*}"}
+{"id": "8723.png", "formula": "\\begin{align*} \\int K ( r ) \\d r { = } 0 , \\int r K ( r ) \\d r { = } 1 , \\int r ^ j K ( r ) \\d r { = } 0 , \\ j { = } 2 , \\dots , \\ell , ~ \\kappa _ \\beta { \\triangleq } \\int | r | ^ { \\beta } | K ( r ) | \\d r < \\infty \\end{align*}"}
+{"id": "7882.png", "formula": "\\begin{align*} \\begin{cases} \\det D ^ 2 u = ( - u ) ^ p & \\Omega \\\\ u = 0 & \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "8457.png", "formula": "\\begin{align*} \\sum _ { \\substack { w \\\\ A , B \\\\ j A s , i B s } } \\mathrm { a d } _ { w } & = \\sum _ { \\substack { w \\\\ A , B \\\\ j - 1 A s , i B s } } \\mathrm { a d } _ { w } \\mathrm { a d } _ { A } + \\sum _ { \\substack { w \\\\ A , B \\\\ j A s , i - 1 B s } } \\mathrm { a d } _ { w } \\mathrm { a d } _ { B } , \\end{align*}"}
+{"id": "1491.png", "formula": "\\begin{align*} A _ { i } ( t ) = \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } \\langle U ^ { ( i ) } B _ { s } , d B _ { s } \\rangle , \\ ; i = 1 , \\ldots , n . \\end{align*}"}
+{"id": "5962.png", "formula": "\\begin{align*} \\Lambda _ { g , F } ^ \\omega f = \\left . \\partial _ { t _ 3 } u \\right \\vert _ { t _ 3 = 0 } = \\left . A _ F ^ \\omega ( t , D _ { t ' } ) u \\right \\vert _ { t _ 3 = 0 } . \\end{align*}"}
+{"id": "4647.png", "formula": "\\begin{align*} \\mathbf { k } \\cdot ( D f ) = ( \\mathbf { k } \\bullet D ) ( \\mathbf { k } \\cdot f ) \\end{align*}"}
+{"id": "1576.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } v ^ 4 g ^ T ( x , v ) \\mathrm { d } v & = \\int _ 0 ^ { \\infty } \\int _ 0 ^ 1 v ^ 3 \\frac { e ^ { - ( 1 - y ) / \\kappa } } { \\kappa | v | ( 1 - e ^ { - 1 / \\kappa | v | } ) } \\rho _ g ( x + y ) \\left ( \\alpha \\mathcal { M } _ { T ( y + x ) } + ( 1 - \\alpha ) \\mathcal { M } _ { \\tau ( y + x ) } \\right ) \\mathrm { d } y \\mathrm { d } v . \\end{align*}"}
+{"id": "491.png", "formula": "\\begin{align*} \\xi _ j = \\frac { d \\mathcal W _ j } { d t } , \\mathcal W _ j = \\mathfrak K _ j W ( j = 1 , 2 ) , \\end{align*}"}
+{"id": "6822.png", "formula": "\\begin{align*} R ( \\varrho s ) : = ( \\varrho s ) ^ 2 - 2 ( a + 2 ) ( u + u ^ * + e _ 2 ) \\varrho s + a ^ 2 ( u + u ^ * + e _ 2 ) ^ 2 < 0 , \\end{align*}"}
+{"id": "6617.png", "formula": "\\begin{align*} \\rho _ { ( 1 ) , \\infty } ^ { ( \\widetilde { \\rm c J } ) } ( x ; \\beta , p , q ) - 1 = { 1 \\over 2 \\pi } \\int _ { - \\infty } ^ \\infty c _ \\infty ^ { ( \\widetilde { \\rm c J } ) } ( \\tau ; \\beta , p , q ) e ^ { - i \\tau x } \\ , d \\tau , \\end{align*}"}
+{"id": "5060.png", "formula": "\\begin{align*} \\deg ( M , N , \\iota , Q _ 1 , \\{ \\gamma \\} ) = \\deg ( M , N , \\iota , Q _ 2 , \\{ \\gamma \\} ) \\end{align*}"}
+{"id": "1737.png", "formula": "\\begin{align*} \\frac { 1 } { n ! } \\int _ a ^ { b } \\cdots \\int _ a ^ b f ( \\mathbf { x } ) \\emph { W } ^ { ( n ) } ( \\mathbf { x } ) \\emph { d } x _ 1 \\cdots \\emph { d } x _ n = \\sum _ { { \\lambda } \\in \\Lambda ^ { ( m , n ) } } f \\bigl ( \\mathbf { x } ^ { ( m , n ) } _ { { \\lambda } } \\bigr ) \\emph { W } ^ { ( m , n ) } _ { { \\lambda } } , \\end{align*}"}
+{"id": "5307.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 | f - g | \\ , \\mathrm { d } x & \\geq \\frac { 1 } { 8 } \\frac { 3 ^ { L ^ 2 } - ( 3 ^ { L - 1 } - 1 ) ^ L - 2 } { 3 ^ { L ^ 2 } } \\\\ & \\geq \\frac { 1 } { 8 } \\frac { 3 ^ { L ^ 2 } - 3 ^ { ( L - 1 ) L } } { 3 ^ { L ^ 2 } } \\\\ & = \\frac { 1 } { 8 } ( 1 - 3 ^ { - L } ) \\\\ & \\geq \\frac { 1 } { 9 } . \\end{align*}"}
+{"id": "8301.png", "formula": "\\begin{align*} [ p ] _ F ( X ) = ( p x _ 1 , p x _ 2 ) ~ \\ 2 ) \\ \\ [ p ] _ F ( X ) \\equiv ( x _ 2 ^ { h _ 1 } , x _ 1 ^ { h _ 2 } ) \\ ( \\ p ) , \\end{align*}"}
+{"id": "2390.png", "formula": "\\begin{align*} \\frac { \\left ( \\mu - r - h \\right ) ^ 2 ( 2 - \\sigma ^ 2 ) } { 2 \\sigma ^ 2 } + h r ( t - T ) + r + f ' ( t ) - h f ( t ) = 0 \\end{align*}"}
+{"id": "5454.png", "formula": "\\begin{align*} \\int _ { t _ o } ^ \\infty e ^ { \\int _ { t _ o } ^ x \\left [ G ^ * ( e ^ { t ^ 2 / 2 } ) e ^ { - t ^ 2 / 2 } - t \\right ] d t } d x & \\leq \\int _ { t _ o } ^ \\infty \\left ( \\frac { x } { t _ o } \\right ) ^ { - 1 . 2 2 8 } d x \\ , = \\ , \\frac { 1 } { 0 . 2 2 8 t _ o } . \\end{align*}"}
+{"id": "522.png", "formula": "\\begin{align*} S _ { h ( \\xi ) } ( t ) = e ^ { - i t h ( D _ x ) } , \\end{align*}"}
+{"id": "6978.png", "formula": "\\begin{align*} x _ 1 ( E , y ) : = \\sqrt { - J ( E , \\hat y ) + y _ 1 ^ 2 } \\end{align*}"}
+{"id": "4731.png", "formula": "\\begin{align*} | \\tilde u _ k ( x ) | & = | \\frac { u ( ( x ^ k ) ' + d _ k x ) - u ( ( x ^ k ) ' ) } { d _ k ^ 2 } | \\\\ & \\le \\frac { | \\nabla u ( r _ { x ^ k } ) | | x | } { d _ k } = \\frac { | \\nabla u ( r _ { x ^ k } ) - \\nabla u ( x ^ k ) | | x | } { d _ k } \\le T _ 1 , \\end{align*}"}
+{"id": "1629.png", "formula": "\\begin{align*} \\begin{cases} \\Delta { } ^ { \\mathcal D } \\ ! E = 1 & N \\\\ { } ^ { \\mathcal D } \\ ! E = 0 & \\partial N . \\end{cases} \\end{align*}"}
+{"id": "4893.png", "formula": "\\begin{align*} \\left [ I - \\frac { z - z _ 0 } { z _ 1 - z _ 0 } P _ 1 \\right ] ^ { - 1 } A ( z ) = [ ( z - z _ 1 ) - ( z - z _ 0 ) P _ 1 ] \\left [ \\frac { A ( z ) - A ( z _ 1 ) } { z - z _ 1 } \\right ] + A ( z _ 1 ) \\end{align*}"}
+{"id": "2887.png", "formula": "\\begin{align*} \\phi _ { \\boldsymbol { \\xi } } = \\sum _ { w \\in W _ 0 } C _ w ( \\boldsymbol { \\xi } ) \\mathbf { e } ^ { i w { \\xi } } , \\end{align*}"}
+{"id": "406.png", "formula": "\\begin{align*} u _ * ( P ) [ n ] & = P [ n ] , \\\\ u _ * ( P ) ( n ) & = \\{ * \\} , \\end{align*}"}
+{"id": "4133.png", "formula": "\\begin{align*} c ( k ) \\leq & \\min _ { 1 \\leq w \\leq c ( k ) } w d - w ( w - 1 ) + w = \\min \\{ d + 1 , ( d - c ( k ) + 2 ) c ( k ) \\} , \\\\ c ( k ) \\leq & \\min _ { 1 \\leq w \\leq c ( k ) } k - w d + w = k - ( d - 1 ) c ( k ) . \\end{align*}"}
+{"id": "5892.png", "formula": "\\begin{align*} ( A / I \\langle M \\rangle , M _ A \\oplus M ) = ( A / I \\langle \\N ^ S \\rangle , M _ A \\oplus \\N ^ S ) \\end{align*}"}
+{"id": "6894.png", "formula": "\\begin{align*} L \\cdot C = L \\cdot C _ 1 = L \\cdot C _ 2 . \\end{align*}"}
+{"id": "8655.png", "formula": "\\begin{align*} A _ k = 0 k \\ge 1 . \\end{align*}"}
+{"id": "8462.png", "formula": "\\begin{align*} \\ell ( z ) = \\mathcal { H } ^ { n - 1 } \\left ( B ^ { n - 1 } ( 0 , r _ { \\ell } ( z ) \\right ) , \\mbox { f o r } \\mathcal { H } ^ { 1 } \\mbox { - a . e . } z \\in \\mathbb { R } , \\end{align*}"}
+{"id": "710.png", "formula": "\\begin{align*} I _ m ( t ) = \\int _ S ^ t \\mathbf S ( t - s ) \\mathbf N ( \\mathbf u _ m ( s ) ) \\ , d s \\end{align*}"}
+{"id": "1238.png", "formula": "\\begin{align*} \\frac { d } { d t } M ( t ) = & 8 \\int _ { \\R ^ 3 } | \\nabla u | ^ 2 - ( I _ \\alpha \\ast | \\cdot | ^ b | u | ^ p ) | x | ^ { - b } | u | ^ p d x \\\\ & + \\mathcal { O } \\left ( \\int _ { | x | > R } | \\nabla u | ^ 2 + | x | ^ { - 2 } | u | ^ 2 + ( I _ \\alpha \\ast | \\cdot | ^ b | u | ^ p ) | x | ^ { - b } | u | ^ p d x \\right ) . \\end{align*}"}
+{"id": "5448.png", "formula": "\\begin{align*} I _ t & \\coloneqq \\int _ 0 ^ t \\frac { 2 \\rho ( e ^ { 2 \\rho t } - 1 ) } { 2 \\rho \\alpha e ^ { 2 \\rho s } ( e ^ { 2 \\rho t } - 1 ) + ( e ^ { 2 \\rho ( t - s ) } - 1 ) ( e ^ { - 2 \\rho ( t - s ) } - 1 ) } d s \\\\ & = \\int _ 1 ^ { e ^ { 2 \\rho t } } \\frac { e ^ { 2 \\rho t } - 1 } { u \\left [ 2 \\rho \\alpha u ( e ^ { 2 \\rho t } - 1 ) + 2 - \\frac { e ^ { 2 \\rho t } } { u } - e ^ { - 2 \\rho t } u \\right ] } d u \\\\ & = \\int _ 1 ^ { e ^ { 2 \\rho t } } \\frac { e ^ { 2 \\rho t } - 1 } { u ^ 2 c _ t + 2 u - e ^ { 2 \\rho t } } d u \\end{align*}"}
+{"id": "5567.png", "formula": "\\begin{align*} f ( \\xi ) = f ( \\eta _ k ( s ) ) = \\gamma _ k ( s ) + h _ k ( s ) + \\i \\mu _ k ( s ) \\end{align*}"}
+{"id": "1822.png", "formula": "\\begin{gather*} \\limsup _ { m \\to \\infty } \\frac { \\log | F ( \\lambda _ m ) | } { | \\lambda _ m | } = \\limsup _ { r \\to \\infty } \\frac { \\log | F ( r ) | } { r } . \\end{gather*}"}
+{"id": "1196.png", "formula": "\\begin{align*} \\pi _ 0 ( k ) = \\widehat { H } ^ 0 ( G ; k ) = k / | G | , \\end{align*}"}
+{"id": "6977.png", "formula": "\\begin{align*} p _ 1 = \\mathcal P ( z , q _ 1 , \\hat p ) : = z + \\tilde { \\mathcal P } ( z , q _ 1 , \\hat p ) \\end{align*}"}
+{"id": "4399.png", "formula": "\\begin{align*} \\mathbb { L } ' _ e ( \\hat { { \\mathbf U } } , \\hat { \\Psi } ) \\dot { { \\mathbf U } } : = \\left [ \\begin{array} { c } \\mathbb { L } ' _ e ( \\hat { { \\mathbf U } } ^ + , \\hat { \\Psi } ^ + ) \\dot { { \\mathbf U } } ^ + \\\\ \\mathbb { L } ' _ e ( \\hat { { \\mathbf U } } ^ - , \\hat { \\Psi } ^ - ) \\dot { { \\mathbf U } } ^ - \\end{array} \\right ] \\ , . \\end{align*}"}
+{"id": "2268.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { \\alpha } } \\ , p _ { 1 / \\alpha } ( x ) & = f _ { \\alpha } ( x ) = \\sum _ { k \\in \\Z } e ^ { - \\pi \\alpha k ^ 2 } e ^ { 2 \\pi i k x } \\\\ & \\leq \\sum _ { k \\in \\Z } e ^ { - \\pi \\alpha k ^ 2 } \\left | e ^ { 2 \\pi i k x } \\right | = \\sum _ { k \\in \\Z } e ^ { - \\pi \\alpha k ^ 2 } \\\\ & = f _ { \\alpha } ( 0 ) = \\frac { 1 } { \\sqrt { \\alpha } } \\ , p _ { 1 / \\alpha } ( 0 ) , \\forall \\alpha > 0 . \\end{align*}"}
+{"id": "6385.png", "formula": "\\begin{align*} \\chi ^ { i } \\otimes \\chi _ { i 1 } \\otimes \\chi _ { i 2 } = \\chi ^ { i } \\otimes \\chi _ { i } \\otimes 1 _ { H } - 1 _ { H } \\otimes \\chi ^ { i } \\otimes \\chi _ { i } + \\chi ^ { i } _ { 1 } \\otimes \\chi ^ { i } _ { 2 } \\otimes \\chi _ { i } . \\end{align*}"}
+{"id": "6639.png", "formula": "\\begin{align*} { K } _ \\infty ^ { ( p , q ) } ( x , y ) \\Big | _ { q = 0 } = \\pi ( x y ) ^ { 1 / 2 } { J _ { p + 1 / 2 } ( \\pi x ) J _ { p - 1 / 2 } ( \\pi y ) - J _ { p - 1 / 2 } ( \\pi x ) J _ { p + 1 / 2 } ( \\pi y ) \\over 2 ( x - y ) } . \\end{align*}"}
+{"id": "8193.png", "formula": "\\begin{align*} \\omega _ i = \\bar { \\omega } _ i + d x _ i \\wedge \\eta + V d x _ j \\wedge d x _ k . \\end{align*}"}
+{"id": "7163.png", "formula": "\\begin{align*} \\| V x \\| = \\left \\| \\sum _ { j = 1 } ^ { n } f _ j ( x ) \\omega _ j \\right \\| = \\left ( \\sum _ { j = 1 } ^ n | f _ j ( x ) | ^ p \\right ) ^ \\frac { 1 } { p } = \\left \\| \\sum _ { j = 1 } ^ { n } f _ j ( x ) \\tau _ j \\right \\| = \\| x \\| . \\end{align*}"}
+{"id": "3981.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) u = f ( { \\bf u } , { \\bf v } ) \\ , \\xi + g ( u , v ) , \\end{align*}"}
+{"id": "6221.png", "formula": "\\begin{align*} m ( x ) = x ^ { - \\delta / 2 } e ^ { i x ^ \\gamma / 2 } , \\gamma , \\delta > 0 , \\end{align*}"}
+{"id": "7196.png", "formula": "\\begin{align*} \\mathfrak { a } _ 2 \\mathfrak { a } _ 3 ^ 3 \\mathfrak { a } _ 4 ^ 3 & = ( 1 \\otimes y _ 2 - y _ 2 \\otimes 1 ) ( 1 \\otimes y _ 3 ^ 3 - y _ 3 \\otimes y _ 3 ^ 2 + y _ 3 ^ 2 \\otimes y _ 3 - y _ 3 ^ 3 \\otimes 1 ) ( 1 \\otimes y _ 4 ^ 3 - y _ 4 \\otimes y _ 4 ^ 2 \\\\ & + y _ 4 ^ 2 \\otimes y _ 4 - y _ 4 ^ 3 \\otimes 1 ) \\\\ & = y _ 1 y _ 2 y _ 3 y _ 4 \\otimes y _ 2 y _ 3 y _ 4 - y _ 2 y _ 3 y _ 4 \\otimes y _ 1 y _ 2 y _ 3 y _ 4 . \\end{align*}"}
+{"id": "6814.png", "formula": "\\begin{align*} \\mathcal { L } ( \\chi ( z ) ) & = \\varrho ( c u - w ) + c \\int _ { q ( 0 ) } ^ v \\frac { \\eta - q ( 0 ) } { \\eta } d \\eta - d y + \\frac { d q ( 0 ) y } { v } . \\end{align*}"}
+{"id": "6298.png", "formula": "\\begin{align*} J ^ 4 _ { m a i n } ( u , v ) = M ( u ) E ( v ) + M ( v ) E ( u ) - 2 P ( u ) P ( v ) . \\end{align*}"}
+{"id": "5511.png", "formula": "\\begin{align*} O _ A ^ { ( g ) } = \\widehat S ^ { - 2 } O _ A ^ { ( g ) } \\widehat S ^ 2 \\qquad \\textrm { f o r a l l } g \\in F r e e _ 2 . \\end{align*}"}
+{"id": "6633.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ 2 } ( \\rho _ { ( 1 ) , \\infty } ^ { \\rm O C P } ( \\mathbf r ; Q ) - 1 ) | \\mathbf r | ^ 4 \\ , d \\mathbf r = { 1 \\over ( \\pi \\beta ) ^ 2 } \\Big ( b _ 0 ( \\beta ) + b _ 1 ( \\beta ) Q + b _ 2 ( \\beta ) Q ^ 2 \\Big ) Q , \\end{align*}"}
+{"id": "5882.png", "formula": "\\begin{align*} P _ { s , V } ( L _ { V } \\geq c _ { \\gamma } ) ( = P _ { s , V } ( L _ { V } \\geq c _ { \\gamma } | T \\geq s ) ) \\rightarrow 0 , \\end{align*}"}
+{"id": "3705.png", "formula": "\\begin{align*} I ( b ) = 2 \\alpha \\int _ { G _ b \\cup G _ b ^ + } \\frac { x _ 2 } { | x | ^ { 2 + 2 \\alpha } } d x - 2 \\alpha \\int _ { B _ b ^ + } \\frac { x _ 2 } { | x | ^ { 2 + 2 \\alpha } } d x . \\end{align*}"}
+{"id": "8879.png", "formula": "\\begin{align*} R = \\R [ y _ 1 , \\ldots , y _ { \\ell } ] \\simeq \\R [ u _ 1 , \\ldots , u _ { \\ell } ] . \\end{align*}"}
+{"id": "338.png", "formula": "\\begin{align*} \\ell _ { j } \\left ( s \\right ) \\left ( d , v \\right ) = 0 \\quad \\forall v \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } \\right ) . \\end{align*}"}
+{"id": "7754.png", "formula": "\\begin{align*} T : = \\pi ^ * g _ { M } - \\sum _ { \\gamma } \\Omega ( \\gamma ) V _ { \\gamma } ^ { } \\pi ^ * | \\mathrm { d } Z _ { \\gamma } | ^ 2 \\end{align*}"}
+{"id": "6246.png", "formula": "\\begin{align*} m ( D ) u ( x ) = \\int K ( y ) u ( x - y ) d y , \\hat K = m . \\end{align*}"}
+{"id": "8068.png", "formula": "\\begin{align*} B _ { l } = \\{ Q \\colon l ( Q ) = 2 ^ { - j - N } , \\ Q \\subset B ( 0 , l ) , \\ \\vert j \\vert \\leq l \\} , \\end{align*}"}
+{"id": "3294.png", "formula": "\\begin{align*} \\| f \\| _ 1 = \\| g \\| _ 2 ^ 2 = 1 , \\end{align*}"}
+{"id": "332.png", "formula": "\\begin{align*} \\left \\langle \\mathsf { D } _ { j } \\left ( s \\right ) \\psi , \\mathsf { L } _ { j } \\left ( s \\right ) \\overline { v } \\right \\rangle _ { \\mathbb { R } ^ { 3 } } = \\left \\langle \\psi , \\gamma _ { \\operatorname * { N } ; j } ^ { \\operatorname * { e x t } } \\left ( s \\right ) \\overline { v } \\right \\rangle _ { \\Gamma _ { j } } \\qquad \\forall v \\in H ^ { 1 } \\left ( \\mathbb { R } ^ { 3 } , \\mathbb { A } _ { j } ^ { \\operatorname * { e x t } } \\right ) . \\end{align*}"}
+{"id": "2693.png", "formula": "\\begin{align*} \\ell _ 1 + \\ell _ 3 & = 4 \\left ( ( \\alpha _ 1 + \\gamma _ 1 ) ^ 2 + ( \\alpha _ 2 + \\gamma _ 2 ) ^ 2 + ( \\beta _ 1 + \\delta _ 1 ) ^ 2 + ( \\beta _ 2 + \\delta _ 2 ) ^ 2 \\right ) , \\\\ \\ell _ 2 + \\ell _ 4 & = 4 \\left ( ( \\alpha _ 1 - \\gamma _ 1 ) ^ 2 + ( \\alpha _ 2 - \\gamma _ 2 ) ^ 2 + ( \\beta _ 1 - \\delta _ 1 ) ^ 2 + ( \\beta _ 2 - \\delta _ 2 ) ^ 2 \\right ) . \\end{align*}"}
+{"id": "2179.png", "formula": "\\begin{align*} b _ R ( R g x , R g y ) & = \\sum _ { r \\in R } b ( r g x , r g y ) = \\sum _ { r \\in R } b ( g ^ { - 1 } r g x , g ^ { - 1 } r g y ) \\\\ & = \\sum _ { \\underset { r \\in R } { r ' = g ^ { - 1 } r g } } b ( r ' x , r ' y ) = \\sum _ { r \\in R } b ( r x , r y ) = b _ R ( R x , R y ) , \\end{align*}"}
+{"id": "3914.png", "formula": "\\begin{align*} H _ K ( v ) = \\{ x : x \\cdot v = h _ K ( v ) \\} . \\end{align*}"}
+{"id": "6868.png", "formula": "\\begin{align*} ( V _ h ^ H A V _ { h } ) Y + Y B - V _ h ^ H C Z ^ H = 0 . \\end{align*}"}
+{"id": "8243.png", "formula": "\\begin{align*} g _ x = h _ x + | x | H _ x . \\end{align*}"}
+{"id": "2105.png", "formula": "\\begin{align*} \\begin{cases*} \\square \\tilde { \\Lambda } = Q _ 0 ( \\ln \\alpha , \\tilde { \\Lambda } ) - 2 \\sinh ( 2 \\lambda + 2 \\tilde { \\Lambda } ) Q _ 0 ( \\phi , \\phi ) , \\\\ \\square \\phi = Q _ 0 ( \\ln \\alpha , \\phi ) + \\dfrac { \\sinh ( 2 \\lambda + 2 \\tilde { \\Lambda } ) } { \\sinh ^ 2 ( \\lambda + \\tilde { \\Lambda } ) } Q _ 0 ( \\phi , \\tilde { \\Lambda } ) . \\end{cases*} \\end{align*}"}
+{"id": "2717.png", "formula": "\\begin{align*} A = ( ( 1 , 0 , \\ldots , 0 , d - 1 ) , ( 0 , 1 , 0 , \\ldots , 0 , d - 1 ) , \\ldots , ( 0 , \\ldots , 0 , d ) ) . \\end{align*}"}
+{"id": "1820.png", "formula": "\\begin{gather*} L ^ \\perp = \\{ x \\in H \\mid \\} . \\end{gather*}"}
+{"id": "4363.png", "formula": "\\begin{align*} \\hat { \\chi } ( s ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ \\mathbb { R } \\chi ( t ) { \\rm e } ^ { - i s t } \\ , d t \\ \\ ( s \\in \\mathbb { R } ) \\ , . \\end{align*}"}
+{"id": "5168.png", "formula": "\\begin{align*} & E [ { \\rm e s s } \\inf L _ j + L _ { j + 1 } ] - \\frac { E [ \\int _ { L _ j } ^ { L _ j + L _ { j + 1 } } t d t ] } { E [ L _ { j + 1 } ] } \\\\ = & - \\log _ 2 { \\delta M } - \\frac { E [ L ^ 2 ] } { 2 E [ L ] } \\\\ > & - \\log _ 2 { \\delta M } - \\frac { 1 } { 2 } h ( X ) + \\frac { 1 } { 2 } \\log _ 2 { \\delta } - \\epsilon \\\\ = & - \\frac { 1 } { 2 } \\log _ 2 { \\delta } - \\log _ 2 { M } - \\frac { 1 } { 2 } h ( X ) - \\epsilon \\\\ > & 0 . \\end{align*}"}
+{"id": "6887.png", "formula": "\\begin{align*} \\hat { c } _ 2 ( X ) - \\hat { c _ 2 } ( \\Omega _ X ^ { [ 1 ] } ( \\mathrm { l o g } \\ , B ) ) + ( K _ X + B ) \\cdot B = [ C ] \\in H ^ { 4 } ( X , \\mathbb { R } ) . \\end{align*}"}
+{"id": "4451.png", "formula": "\\begin{align*} \\sum _ { \\langle \\alpha \\rangle \\le s , \\alpha _ 1 = 0 , \\alpha _ 3 \\geq 1 } | | D ^ { \\alpha } _ { \\ast } { \\mathbf V } ( t ) | | ^ 2 _ { L ^ 2 ( \\mathbb R ^ 2 _ + ) } \\leq C ( K ) \\mathcal { M } ( t ) \\end{align*}"}
+{"id": "6804.png", "formula": "\\begin{align*} w ' ( z ) = & c w ( z ) + f ( u ( z ) ) v ( z ) - f ( u ( z ) ) p ( u ( z ) ) \\\\ [ 0 . 2 c m ] < & [ - c K + v ( z ) - p ( u ( z ) ) ] f ( u ( z ) ) \\\\ [ 0 . 2 c m ] < & ( q ( 1 ) - c K ) f ( u ( z ) ) < 0 \\end{align*}"}
+{"id": "2204.png", "formula": "\\begin{align*} \\C ( x , y ) - | m _ A | - | m _ B | - | m _ C | - O ( \\log n ) = 1 . 5 n - O ( \\log n ) . \\end{align*}"}
+{"id": "7713.png", "formula": "\\begin{align*} X _ { \\{ f , g \\} } = [ X _ f , X _ g ] . \\end{align*}"}
+{"id": "2447.png", "formula": "\\begin{align*} \\frac { | \\Omega | } { 4 \\pi ^ 2 z _ { 0 , 1 } ^ 2 } \\delta = \\kappa ^ 2 . \\end{align*}"}
+{"id": "2320.png", "formula": "\\begin{align*} \\tau _ g = \\pi _ g \\tau , \\forall g \\in G . \\end{align*}"}
+{"id": "7128.png", "formula": "\\begin{align*} b _ { n , k } = e ^ { i k x } \\begin{cases} \\cos ( \\omega _ { n , k } z ) - \\frac { \\cos ( \\omega _ { n , k } ) } { \\cosh ( r _ { n , k } ) } \\cosh ( r _ { n , k } z ) , & n \\in 2 \\N , \\\\ \\sin ( \\omega _ { n , k } z ) - \\frac { \\sin ( \\omega _ { n , k } ) } { \\sinh ( r _ { n , k } ) } \\sinh ( r _ { n , k } z ) , & n \\in 2 \\N + 1 , \\end{cases} \\end{align*}"}
+{"id": "5608.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\lim _ { n \\to \\infty } \\int _ R ^ \\infty \\left ( \\exp _ p ( \\mu | u _ n | ^ { p ' } ) - \\dfrac { \\mu ^ { p - 1 } } { \\Gamma ( p ) } | u _ n | ^ { p } \\right ) r ^ { \\alpha _ 0 } \\mathrm d r = 0 . \\end{align*}"}
+{"id": "7889.png", "formula": "\\begin{align*} \\delta \\mathcal J ( u ) [ \\phi ] = - \\int _ \\Omega \\phi \\left ( ( u ^ { \\star } ) ^ k \\det D ^ 2 u - F ( x , u ) \\right ) . \\end{align*}"}
+{"id": "1624.png", "formula": "\\begin{align*} \\begin{cases} ( \\partial _ t + \\Delta _ y ) \\ , p _ t ( x , y ) = 0 & \\forall x , y \\in V \\ , \\ , t > 0 \\\\ p _ 0 ( x , y ) = \\frac { \\delta _ x ( y ) } { m ( x ) } , \\\\ \\end{cases} \\end{align*}"}
+{"id": "1362.png", "formula": "\\begin{align*} R _ { 2 - 2 k } ^ { 2 k - 1 } = - ( 4 \\pi ) ^ { 2 k - 1 } D ^ { 2 k - 1 } , \\end{align*}"}
+{"id": "8838.png", "formula": "\\begin{align*} \\delta _ { t } \\leq \\frac { ( t _ { 0 } - 1 ) \\delta _ { t _ { 0 } } } { t } + \\sum _ { i = 1 } ^ { N } \\frac { a _ { i } } { ( 3 - p _ { i } ) t ^ { p _ { i } } } \\enspace . \\end{align*}"}
+{"id": "802.png", "formula": "\\begin{align*} ( t _ 0 ^ \\sharp ) ^ { p - 1 } - \\frac { b } { p ^ \\sharp } ( t _ 0 ^ \\sharp ) ^ { 2 \\cdot p ^ \\sharp - 1 } = 0 , \\end{align*}"}
+{"id": "614.png", "formula": "\\begin{align*} \\norm { \\mathbf U } _ { T } = \\left ( \\sum _ { i = 1 } ^ n \\norm { U _ i } _ { T } ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "3938.png", "formula": "\\begin{align*} \\gamma _ n : = \\bigcup _ { k = 1 } ^ n \\omega _ { \\frac 1 { k } } . \\end{align*}"}
+{"id": "5749.png", "formula": "\\begin{align*} \\int _ { \\mathbb B _ { 1 / 2 } ^ + } U _ 0 ^ 2 ( X , 0 ) x _ { n + 1 } ^ a d X = 1 . \\end{align*}"}
+{"id": "7415.png", "formula": "\\begin{align*} \\Psi _ { n } ( x , t ) = \\int _ { 0 } ^ { x } \\left ( \\rho _ { n } ^ { 0 } ( y ) - \\langle \\rho _ { n } \\rangle \\right ) ~ d y - \\int _ { 0 } ^ { t } \\rho _ { n } u _ { n } ( x , s ) ~ d s , \\end{align*}"}
+{"id": "4146.png", "formula": "\\begin{align*} \\chi _ 1 \\cdot \\chi _ 2 & = \\chi _ 1 \\otimes \\chi _ 2 , & \\epsilon _ 1 \\cdot \\epsilon _ 2 & = \\epsilon _ 1 \\otimes \\epsilon _ 2 \\\\ \\chi _ 1 \\cdot \\epsilon _ 2 & = \\chi _ 1 \\otimes \\epsilon _ 2 , & E \\cdot \\chi _ 2 & = \\chi _ 2 \\otimes E - D _ { E } \\chi _ 2 . \\end{align*}"}
+{"id": "7473.png", "formula": "\\begin{align*} \\Gamma _ { n } ( z ) : = \\sqrt { \\frac { 2 } { \\pi } } \\Gamma ( z ) \\cos ( \\pi ( n - z ) / 2 ) , n \\in \\{ 0 , 1 \\} \\ , . \\end{align*}"}
+{"id": "8079.png", "formula": "\\begin{align*} \\lambda _ { \\widetilde { Q } } ^ { i } : = \\widetilde C \\frac { \\omega ( \\widetilde Q ) ^ { \\frac { 1 } { p } } } { \\vert \\widetilde Q \\vert ^ { \\frac { 1 } { q } } } \\left \\| \\left \\{ \\sum _ { Q \\subset \\widetilde { Q } } \\vert \\psi _ { Q } \\ast h ( u _ { Q } ) \\vert ^ 2 \\chi _ { Q } \\right \\} ^ { 1 / 2 } \\right \\| _ { L ^ { q } } . \\end{align*}"}
+{"id": "5232.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } l _ t ( x _ t ) - \\min _ { x \\in \\mathcal { X } } \\sum _ { t = 1 } ^ { T } l _ t ( x ) , \\end{align*}"}
+{"id": "1845.png", "formula": "\\begin{align*} U _ { s , t } ( z , \\Delta ) & = \\frac { 1 } { 2 } \\left \\{ H \\left ( \\frac { \\Delta } { 2 \\pi } ( z - s ) \\right ) + H \\left ( \\frac { \\Delta } { 2 \\pi } ( t - z ) \\right ) \\right \\} , \\\\ K _ { s , t } ( z , \\Delta ) & = \\frac { 1 } { 2 } \\left \\{ K \\left ( \\frac { \\Delta } { 2 \\pi } ( z - s ) \\right ) + K \\left ( \\frac { \\Delta } { 2 \\pi } ( t - z ) \\right ) \\right \\} \\end{align*}"}
+{"id": "8895.png", "formula": "\\begin{align*} E G \\times _ G ( G / T ) \\simeq E G / T = E T / T = B T . \\end{align*}"}
+{"id": "7219.png", "formula": "\\begin{align*} \\inf _ { ( \\mu , \\beta ) } J '^ l \\bigl ( t _ 0 , \\mu _ 0 , ( \\mu , \\beta ) \\bigr ) = \\inf _ { ( \\mu , \\omega ) \\in \\mathbb { A } } \\sup _ { \\nu \\in \\mathbb { B } } \\mathcal { L } \\bigl ( ( \\mu , \\omega ) , \\nu \\bigr ) \\end{align*}"}
+{"id": "7179.png", "formula": "\\begin{align*} \\Lambda _ { \\Tilde { J } } = \\begin{pmatrix} \\lambda ( F _ 1 ) ^ t & \\lambda ( F _ 2 ) ^ t & \\dots & \\lambda ( F _ { n + m } ) ^ t \\end{pmatrix} _ { ( n \\times ( n + m ) ) } \\cdot \\begin{pmatrix} x _ 1 & x _ 2 & \\dots & x _ { n + m } \\end{pmatrix} ^ t _ { ( n + m ) \\times 1 } . \\end{align*}"}
+{"id": "5955.png", "formula": "\\begin{align*} \\sigma ( \\widetilde { Q ^ \\omega } ) ( t , \\xi ' ) & = \\widetilde { q _ 1 } ( t , \\xi ' ) + \\widetilde { q _ 2 } ( t , \\xi ' ) - \\omega ^ 2 . \\end{align*}"}
+{"id": "7700.png", "formula": "\\begin{align*} ( \\rho \\otimes \\varphi ) _ { ( \\sigma , s ) } ^ { ( \\sigma ' , s ' ) } : = \\rho \\otimes \\varphi _ { s } ^ { s ' } \\end{align*}"}
+{"id": "2253.png", "formula": "\\begin{align*} \\sum _ { k \\in \\Z } f ( k + x ) = \\sum _ { \\ell \\in \\Z } \\widehat { f } ( \\ell ) e ^ { 2 \\pi i \\ell x } . \\end{align*}"}
+{"id": "6278.png", "formula": "\\begin{align*} a ( \\xi , \\eta ) = \\phi _ \\lambda ( \\xi ) \\phi _ \\lambda ( \\eta ) \\end{align*}"}
+{"id": "7534.png", "formula": "\\begin{align*} r \\circ \\ , s \\ , \\circ \\ , S _ 0 \\ , \\circ \\ , \\overline { \\Omega } & = r \\circ S _ 1 \\circ R ^ { N C } _ { 1 } \\circ \\Omega _ Y \\\\ & = r \\circ \\ , R ^ { P C } _ { 1 } \\circ T _ 1 \\circ \\Omega _ Y \\\\ & = R ^ { P C } _ { 2 } \\circ p ' \\circ T _ 1 \\circ \\Omega _ Y \\stackrel { R ^ { P C } _ { 2 } \\circ p ' ( \\eta ) } { \\leadsto } 0 . \\end{align*}"}
+{"id": "1551.png", "formula": "\\begin{align*} \\Delta = y ^ 4 ( y - \\lambda x ) ^ 2 + z f _ 5 ( x , y , z ) \\end{align*}"}
+{"id": "1398.png", "formula": "\\begin{align*} \\rho _ n = n - M _ 1 - 1 + \\varepsilon _ n , \\varepsilon _ n = o ( 1 ) , n \\to \\infty . \\end{align*}"}
+{"id": "3693.png", "formula": "\\begin{align*} & \\int _ { x + [ 0 , 2 ] \\times [ - 2 , 2 ] } y _ 1 \\left ( \\frac { 1 } { | x + h e _ 2 - y | ^ { 2 + 2 \\alpha } } - \\frac { 1 } { | x - y | ^ { 2 + 2 \\alpha } } \\right ) { \\rm s g n } \\left ( y _ 2 - x _ 2 - \\frac h 2 \\right ) 2 M ( y _ 1 ) d y \\\\ & = \\int _ { x + [ - 2 , 2 ] ^ 2 } y _ 1 \\left ( \\frac { 1 } { | x + h e _ 2 - y | ^ { 2 + 2 \\alpha } } - \\frac { 1 } { | x - y | ^ { 2 + 2 \\alpha } } \\right ) { \\rm s g n } \\left ( y _ 1 \\left ( y _ 2 - x _ 2 - \\frac h 2 \\right ) \\right ) M ( y _ 1 ) d y , \\end{align*}"}
+{"id": "4041.png", "formula": "\\begin{align*} S ( x , \\rho ) = \\frac { \\sin \\rho ( x - a ) } { \\rho } . \\end{align*}"}
+{"id": "5191.png", "formula": "\\begin{align*} \\omega _ { n - 1 } = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ n L _ j , \\quad \\textrm { a n d } \\omega _ n = \\frac { 1 } { 2 } ( L _ 1 + \\dots + L _ { n - 1 } - L _ n ) . \\end{align*}"}
+{"id": "6173.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq i < i ' } a _ { i j } = 1 \\Leftrightarrow c _ { i ' - 1 , j } = 1 \\Leftrightarrow \\exists \\tilde { j } \\in \\{ 1 , 2 , \\ldots , i ' - 1 \\} , d _ { i ' - 1 , \\tilde { j } } = j . \\end{align*}"}
+{"id": "981.png", "formula": "\\begin{align*} x + y \\sqrt d = \\pm ( a + b \\sqrt d ) ^ n \\end{align*}"}
+{"id": "4021.png", "formula": "\\begin{align*} \\sup _ { w : \\ , \\| w \\| _ { W ^ { s , p } ( \\mathcal O ) } = 1 } \\| \\varphi \\ , w \\| _ { W ^ { s , p } ( \\mathcal O ) } < \\infty . \\end{align*}"}
+{"id": "4326.png", "formula": "\\begin{align*} S ( \\omega _ \\Psi \\| \\omega _ \\Omega ) = - ( \\Omega , \\log ( \\Delta _ { \\Psi , \\Omega } ) \\Omega ) = i \\left . \\frac { d } { d t } \\right | _ { t = 0 } ( \\Omega , \\Delta ^ { i t } _ { \\Psi , \\Omega } \\Omega ) \\ , , \\end{align*}"}
+{"id": "4924.png", "formula": "\\begin{align*} \\sum _ { \\ell \\geq 1 } \\ell ^ r t ^ \\ell = \\dfrac { t A _ r ( t ) } { ( 1 - t ) ^ { r + 1 } } \\end{align*}"}
+{"id": "5805.png", "formula": "\\begin{align*} w _ 0 = s _ { \\alpha _ n } \\prod \\limits _ { i = 1 , 3 , . . . , n - 2 } s _ { \\varepsilon _ i + \\varepsilon _ { i + 1 } } s _ { \\alpha _ i } , \\end{align*}"}
+{"id": "6630.png", "formula": "\\begin{align*} \\prod _ { l = 1 } ^ N | z _ l | ^ { \\beta Q } e ^ { - \\beta \\pi | z _ l | ^ 2 / 2 } \\prod _ { 1 \\le j < k \\le N } | z _ k - z _ j | ^ \\beta , z _ l \\in \\mathbb C . \\end{align*}"}
+{"id": "2093.png", "formula": "\\begin{align*} \\begin{cases*} \\partial _ { \\mu } ( m ^ { \\mu \\nu } \\alpha \\partial _ { \\nu } \\Psi ) = F ( \\Psi , \\partial \\Psi ) \\\\ ( \\Psi , \\partial _ t \\Psi ) | _ { \\{ t = 0 \\} } = ( \\Psi _ 0 , \\Psi _ 1 ) \\in \\mathcal { H } . \\end{cases*} \\end{align*}"}
+{"id": "6504.png", "formula": "\\begin{align*} V = { \\rm d i a g } ( \\mu _ n ) _ { n \\in \\Z ^ d } \\ { \\rm w i t h } \\ \\mu _ n = \\sqrt { \\cos ( n \\cdot \\alpha + \\theta _ 0 ) + m } \\end{align*}"}
+{"id": "5994.png", "formula": "\\begin{align*} p ( x , \\xi ) - \\sum _ { j = 0 } ^ N p _ { m - j } ( x , \\xi ) \\in S ^ { m - N - 1 } _ { 1 , 0 } ( T ^ * \\mathbb { R } ^ n ) . \\end{align*}"}
+{"id": "4808.png", "formula": "\\begin{align*} \\langle f , g \\circ \\Phi ^ n \\rangle & = V _ 0 \\int _ { B _ { 1 } } f \\circ \\varphi _ { \\lambda ^ { - n } _ 0 s } ( p ) \\cdot g \\circ \\varphi _ { s } ( p ) \\ , d s + \\sum _ { i = 2 } ^ { d ^ + } \\bar { c } _ { i , n } ( f , g ) \\left ( \\frac { \\lambda _ i } { \\lambda _ 1 } \\right ) ^ n + \\mathcal { O } ( \\lambda ^ { - n } _ 0 ) \\\\ \\end{align*}"}
+{"id": "3973.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) u = \\zeta + g ( u , \\mu ) \\end{align*}"}
+{"id": "5529.png", "formula": "\\begin{align*} \\alpha ( k _ 0 ) = \\partial _ m \\alpha ( k ) = ( \\alpha _ 1 ( k ) , \\dots , \\alpha _ { m - 1 } ( k ) , \\alpha _ m ( k ) - 1 , \\alpha _ { m + 1 } ( k ) , \\dots , \\alpha _ n ( k ) ) . \\end{align*}"}
+{"id": "8644.png", "formula": "\\begin{align*} | u | _ { z = - h _ b } | _ { L ^ 2 } \\lesssim \\| \\nabla _ { X , z } ^ { \\mu } u \\| _ { L ^ 2 ( \\mathcal { S } _ b ) } . \\end{align*}"}
+{"id": "5261.png", "formula": "\\begin{align*} \\sum _ { t = 3 } ^ { T } \\sum _ { s = 1 } ^ { t - 2 } \\lambda ^ { t - s - 2 } \\sum _ { j = 1 } ^ n \\| \\epsilon ^ z _ { j , s } \\| & = \\sum _ { t = 1 } ^ { T - 2 } \\sum _ { j = 1 } ^ n \\| \\epsilon ^ z _ { j , t } \\| \\sum _ { s = 0 } ^ { T - t - 2 } \\lambda ^ s \\le \\frac { 1 } { 1 - \\lambda } \\sum _ { t = 1 } ^ { T - 2 } \\sum _ { j = 1 } ^ n \\| \\epsilon ^ z _ { j , t } \\| . \\end{align*}"}
+{"id": "1674.png", "formula": "\\begin{align*} \\mathbb { P } ^ { ( m , n ) } _ { \\texttt { a } } : = _ { \\mu \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { a } } } \\{ M _ { \\texttt { a } ; \\mu } ( \\boldsymbol { \\xi } ) \\} , \\end{align*}"}
+{"id": "4578.png", "formula": "\\begin{align*} \\Vert \\partial _ 2 \\varphi ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } & \\le C _ 2 \\{ \\Vert { \\mathbf V } _ n ( t ) \\vert _ { x _ 1 = 0 } \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } + \\Vert \\varphi ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } \\} \\\\ & \\le C _ 2 \\{ I ( t ) + I _ { 1 , n } ( t ) + \\Vert \\varphi ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } \\} \\ , , \\end{align*}"}
+{"id": "4524.png", "formula": "\\begin{align*} \\mathcal { I } : = \\int ^ 1 _ 0 \\mathbb { L } ' ( \\mathbf { U } ^ a + { \\mathbf V } _ k + \\tau ( { \\mathbf V } _ { k + \\frac { 1 } { 2 } } - { \\mathbf V } _ k ) , \\Psi ^ a + \\Psi _ k + \\tau ( \\Psi _ { k + \\frac { 1 } { 2 } } - \\Psi _ k ) ) ( { \\mathbf V } _ { k + \\frac { 1 } { 2 } } - { \\mathbf V } _ k , \\Psi _ { k + \\frac { 1 } { 2 } } - \\Psi _ k ) d \\tau . \\end{align*}"}
+{"id": "5342.png", "formula": "\\begin{align*} h _ A ( p ) = \\min \\{ n \\in \\mathbb { N } : p ^ n \\in A \\} . \\end{align*}"}
+{"id": "5422.png", "formula": "\\begin{align*} \\eta _ N ( \\xi ) : = \\eta \\big ( 2 ^ { N \\cdot A - N ^ { \\chi } \\cdot \\rm { I d } } \\xi \\big ) \\end{align*}"}
+{"id": "1215.png", "formula": "\\begin{align*} B _ n = & \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p ) | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } ( | g _ n ^ { - 1 } u _ n | ^ p - | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p ) d x \\\\ & - \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p ) | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p . \\end{align*}"}
+{"id": "287.png", "formula": "\\begin{align*} H _ { \\operatorname * { l o c } } ^ { k } \\left ( \\omega \\right ) : = \\left \\{ v : \\chi v \\in H ^ { k } \\left ( \\omega \\right ) \\chi \\in C _ { 0 } ^ { \\infty } \\left ( \\mathbb { R } ^ { 3 } \\right ) \\right \\} \\end{align*}"}
+{"id": "3473.png", "formula": "\\begin{align*} \\phi _ t = \\max _ { p \\in \\Delta ^ \\vee } \\left ( \\sum _ 0 ^ m - p _ i \\log | F _ i | _ { h ^ { \\otimes d _ i } } - u ^ * ( p ) | \\log | t | | \\right ) , \\end{align*}"}
+{"id": "3305.png", "formula": "\\begin{align*} m _ 1 = \\frac { d } { d t } \\phi _ Y ( 0 ) ( - \\mu ) = 0 , \\end{align*}"}
+{"id": "6154.png", "formula": "\\begin{align*} \\# \\left ( \\mathbf { k } ; \\{ p _ { i , j } \\} , \\{ q _ { i , j } \\} \\right ) & = ( \\mathbf { k } ) \\cdot ( \\{ p _ { i , j } \\} , \\{ q _ { i , j } \\} ) \\cdot \\# ( \\mathbf { k } ' ) \\\\ & = ( \\{ p _ { i , j } \\} , \\{ q _ { i , j } \\} ) \\cdot \\prod _ { 1 \\leq i < j \\leq n } \\frac { k _ j - k _ i } { j - i } , \\end{align*}"}
+{"id": "4888.png", "formula": "\\begin{align*} g _ n ( z ) = I - ( I - z P _ n ) \\exp ( z P _ n + \\frac { 1 } { 2 } z ^ 2 P _ n + \\ldots + \\frac { 1 } { n } z ^ n P _ n ) . \\end{align*}"}
+{"id": "4989.png", "formula": "\\begin{align*} x _ { 2 i + 1 } & : = f _ 1 ( x _ { 2 i } ) , & x _ { 2 i + 2 } & : = f _ 2 ( x _ { 2 i + 1 } ) , & i & \\in \\omega . \\end{align*}"}
+{"id": "1056.png", "formula": "\\begin{align*} K _ { \\hat { \\gamma } _ \\lambda } = \\{ u _ 0 \\} . \\end{align*}"}
+{"id": "1128.png", "formula": "\\begin{align*} & [ m _ 1 , m _ 2 ] \\\\ & = m _ 1 \\circ m _ 2 + m _ 2 \\circ m _ 1 \\\\ & = m _ 1 ( x _ 1 , m _ 2 ( x _ 2 , x _ 3 ) ) + m _ 2 ( x _ 1 , m _ 1 ( x _ 2 , x _ 3 ) ) + m _ 1 ( m _ 2 ( x _ 1 , x _ 3 ) , x _ 2 ) + m _ 2 ( m _ 1 ( x _ 1 , x _ 3 ) , x _ 2 ) \\\\ & - m _ 1 ( m _ 2 ( x _ 1 , x _ 2 ) , x _ 3 ) - m _ 2 ( m _ 1 ( x _ 1 , x _ 2 ) , x _ 3 ) . \\end{align*}"}
+{"id": "6197.png", "formula": "\\begin{align*} ( f g ) ( x , y ) & = \\sum _ { x \\le z \\le y } f ( x , z ) g ( z , y ) \\end{align*}"}
+{"id": "4654.png", "formula": "\\begin{align*} U ( \\mathfrak { g } ) ^ { ( v ) } : = U ( \\mathfrak { g } ) ^ { \\otimes \\# \\mathcal { S } ( v | s ) } \\otimes U ( \\mathfrak { g } ) ^ { \\otimes \\# \\mathcal { S } ( v | t ) } . \\end{align*}"}
+{"id": "7600.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { \\mathbf { R } } _ 1 & = \\{ ( t , s ) \\in [ 0 , T ] ^ 2 : \\beta ^ { 1 / 3 } N ^ { 1 / 3 } ( t - s ) > C _ 0 \\} \\\\ \\hat { \\mathbf { R } } _ 2 & = \\{ ( t , s ) \\in [ 0 , T ] ^ 2 : \\beta ^ { 1 / 3 } N ^ { 1 / 3 } ( t - s ) \\leq C _ 0 \\} . \\end{aligned} \\end{align*}"}
+{"id": "1485.png", "formula": "\\begin{align*} x \\cdot y = \\left ( \\overline { x } + \\overline { y } , \\widehat { x } _ { 1 } + \\widehat { y } _ { 1 } + \\frac { 1 } { 2 } \\langle U ^ { ( 1 ) } \\overline { x } , \\overline { y } \\rangle , \\ldots , \\widehat { x } _ { n } + \\widehat { y } _ { n } + \\frac { 1 } { 2 } \\langle U ^ { ( n ) } \\overline { x } , \\overline { y } \\rangle \\right ) , \\end{align*}"}
+{"id": "1161.png", "formula": "\\begin{align*} & [ m _ { 1 , 1 } , m _ { 1 } ] + [ m _ { 1 } , m _ { 1 , 1 } ] + [ m _ { 1 } , m _ { 1 } ] = 0 , \\\\ & [ m _ { 2 , 1 } , m _ { 2 } ] + [ m _ { 2 } , m _ { 2 , 1 } ] + [ m _ { 2 } , m _ { 2 } ] = 0 , \\\\ & [ m _ { 1 , 1 } , m _ { 2 } ] + [ m _ { 1 } , m _ { 2 , 1 } ] + [ m _ { 1 } , m _ { 2 } ] = 0 . \\end{align*}"}
+{"id": "7254.png", "formula": "\\begin{align*} 0 < D : = \\min \\lbrace - \\log | f ' ( x ) | : f \\in \\Phi , x \\in I \\rbrace , D ' : = \\max \\lbrace - \\log | f ' ( x ) | : f \\in \\Phi , x \\in I \\rbrace < \\infty . \\end{align*}"}
+{"id": "4330.png", "formula": "\\begin{align*} \\Delta ^ { i t } \\pi _ \\omega ( B ( { \\sf f } ) ) \\Delta ^ { - i t } = \\pi _ \\omega ( B ( u _ t { \\sf f } ) ) \\end{align*}"}
+{"id": "2177.png", "formula": "\\begin{align*} k ( g ^ { - 1 } v ) & = T _ { g } k ( v ) > C T _ { g } k ( q ^ { - 1 } v ) \\\\ & = C T _ { g } T _ q k ( v ) = C T _ q T _ g k ( v ) = C T _ q k ( g ^ { - 1 } v ) \\end{align*}"}
+{"id": "729.png", "formula": "\\begin{align*} \\lim _ { \\mu \\to \\infty } \\norm { \\mathbf u ^ \\mu - \\mathbf u } _ { Z ^ { \\mathbf s , b } ( 0 , T ) } = 0 . \\end{align*}"}
+{"id": "8438.png", "formula": "\\begin{align*} & \\mu _ L ( h , \\phi ) = \\phi , \\\\ & \\mu _ R ( h , \\phi ) = \\mathrm { A d } ^ * _ h ( \\phi ) . \\end{align*}"}
+{"id": "1753.png", "formula": "\\begin{align*} \\xi ^ { ( m + n ) } _ { { l } } = \\frac { \\pi \\bigl ( { l } + \\frac { 1 } { 2 } ( \\epsilon _ - + \\tilde { \\epsilon } _ - ) \\bigr ) } { m + n - 1 + \\frac { 1 } { 2 } ( \\epsilon _ + + \\epsilon _ - + \\tilde { \\epsilon } _ + + \\tilde { \\epsilon } _ - ) } ( 0 \\le { l } < m + n ) . \\end{align*}"}
+{"id": "762.png", "formula": "\\begin{align*} I _ 1 - I _ 2 = \\int _ 0 ^ 1 \\left ( \\int _ 0 ^ 1 \\theta '' \\left ( \\rho ( t ) + s \\left [ \\kappa ( t ) - \\rho ( t ) \\right ] \\right ) \\ , d s \\right ) \\left [ \\kappa ( t ) - \\rho ( t ) \\right ] \\ , d t \\end{align*}"}
+{"id": "3156.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\mu _ { L _ X } ( u ) - u \\cdot D _ X = \\frac { 1 } { 2 } \\mu _ { L _ Y } ( v ) - v \\cdot D _ Y . \\end{align*}"}
+{"id": "7689.png", "formula": "\\begin{align*} \\nabla \\phi ( x _ 0 ) & = p + n , \\\\ D ^ 2 \\phi ( x _ 0 ) & = \\alpha I + \\gamma n n ^ T > 0 . \\end{align*}"}
+{"id": "5305.png", "formula": "\\begin{align*} \\min \\{ a , b \\} & = \\frac { 1 } { 2 } ( a + b - | a - b | ) \\\\ & = \\frac { 1 } { 2 } ( a + b - \\max \\{ a - b , 0 \\} - \\max \\{ - a + b , 0 \\} ) \\end{align*}"}
+{"id": "1985.png", "formula": "\\begin{align*} \\sum _ { \\sigma = 0 } ^ { p ^ k - 1 } \\mathcal { C } ( \\mathbf { a } _ \\sigma ^ { t _ 1 } , \\mathbf { a } _ \\sigma ^ { t _ 2 } ) ( 0 ) = \\sum _ { \\sigma = 0 } ^ { p ^ { k } - 1 } \\sum _ { r = 0 } ^ { p ^ { m } - 1 } \\omega _ q ^ { \\left ( a ^ { t _ 1 } _ { \\sigma , r } - a ^ { t _ 2 } _ { \\sigma , r } \\right ) } = 0 . \\end{align*}"}
+{"id": "7272.png", "formula": "\\begin{align*} \\left | f _ { \\omega | _ { \\tau _ k { ( \\omega ) } + 1 } ^ { \\beta _ k ( \\omega ) } } ' ( x ) \\right | = \\Theta _ { C ' } \\left ( e ^ { - \\frac { \\epsilon k } { 8 } } \\right ) . \\end{align*}"}
+{"id": "1285.png", "formula": "\\begin{align*} D _ n = 2 \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p ) | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p d x \\end{align*}"}
+{"id": "5277.png", "formula": "\\begin{align*} & \\sum _ { t = 1 } ^ T \\sum _ { i = 1 } ^ n \\frac { 1 } { 2 } \\Big ( \\frac { q _ { i , t + 1 } ^ \\top g _ { i , t } ( x _ { i , t } ) } { \\gamma _ t } + \\frac { \\sigma \\| \\epsilon ^ z _ { i , t } \\| ^ 2 } { 2 \\gamma _ 0 } \\Big ) \\le h _ T ( y ) + \\tilde { h } _ T ( y ) + \\hat { h } _ T ( y ) , ~ \\forall y \\in \\mathcal { X } _ T , \\end{align*}"}
+{"id": "83.png", "formula": "\\begin{align*} d _ { 1 } ^ L & : = \\sum _ { i \\neq j } ( P _ i + Q _ { H , i } ) \\overline { Q } _ { H , j } v ( x _ i - x _ j ) \\overline { Q } _ { H , i } \\overline { Q } _ { H , j } + h . c . \\\\ & + \\sum _ { i \\neq j } \\overline { Q } _ { H , i } ( P _ j + Q _ { H , j } ) v ( x _ i - x _ j ) ( P _ i + Q _ { H , i } ) ( P _ j + Q _ { H , j } ) + h . c . \\intertext { a n d } d _ 2 ^ L & : = \\sum _ { i \\neq j } ( P _ i + Q _ { H , i } ) ( P _ j + Q _ { H , j } ) v ( x _ i - x _ j ) \\overline { Q } _ { H , j } \\overline { Q } _ { H , i } + h . c . \\end{align*}"}
+{"id": "6185.png", "formula": "\\begin{align*} \\psi _ 2 \\circ \\varphi _ 2 = ( \\psi _ 1 \\sqcup ( \\psi _ 2 \\setminus \\psi _ 1 ) ) \\circ ( \\varphi _ 1 \\sqcup ( \\varphi _ 2 \\setminus \\varphi _ 1 ) ) = ( \\psi _ 1 \\circ \\varphi _ 1 ) \\sqcup ( ( \\psi _ 2 \\setminus \\psi _ 1 ) \\circ ( \\varphi _ 2 \\setminus \\varphi _ 1 ) ) . \\end{align*}"}
+{"id": "6236.png", "formula": "\\begin{align*} P _ k u : = \\frac { d } { d k } P _ { < k } u . \\end{align*}"}
+{"id": "3304.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } ( \\phi _ Y ) _ { _ \\Sigma } ( t ) = \\left ( \\frac { t ^ 2 } { 4 \\lambda _ { _ \\Sigma } ^ 2 } - \\frac { 1 } { 2 \\lambda _ { _ \\Sigma } } \\right ) e ^ { - \\frac { t ^ 2 } { 2 \\lambda _ { _ \\Sigma } } } . \\end{align*}"}
+{"id": "8656.png", "formula": "\\begin{align*} \\int g ( \\vec { x } ) e ^ { \\frac { f ( \\vec { x } ) + \\vec { x } \\cdot \\vec { y } } { \\hbar } } d ^ n \\vec { x } = ( 2 \\pi \\hbar ) ^ { \\frac { n } { 2 } } \\hat { g } ( \\vec { y } ) e ^ { \\frac { \\hat { f } ( \\vec { y } ) } { \\hbar } } . \\end{align*}"}
+{"id": "4571.png", "formula": "\\begin{align*} \\vert \\mathcal J _ { 1 , 2 } \\vert \\le & \\varepsilon I _ { 1 , n } ( t ) + \\frac { C _ 2 } { \\varepsilon } \\left \\{ \\Vert { \\mathbf F } \\Vert _ { L ^ 2 ( \\Omega _ t ) } ^ 2 + \\Vert \\varphi ( t ) \\Vert _ { L ^ 2 ( \\mathbb R ) } ^ 2 + \\int _ 0 ^ t \\Vert \\varphi ( s ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } d s \\right \\} \\\\ & + \\frac { C _ 3 } { \\varepsilon } \\int _ 0 ^ t ( I _ { 1 , \\ast } + I _ { 1 , n } ) ( s ) d s \\end{align*}"}
+{"id": "857.png", "formula": "\\begin{align*} \\lambda _ i ^ * = \\frac { \\alpha _ i } { 2 \\sum _ { j \\neq i } \\alpha _ j ^ 2 \\sigma ^ 2 + 2 \\sigma _ N ^ 2 } . \\end{align*}"}
+{"id": "8881.png", "formula": "\\begin{align*} \\int _ M \\tilde { \\omega } = \\sum _ { p \\in M ^ T } \\frac { i _ p ^ * \\tilde { \\omega } } { e ^ T ( \\nu _ p ) } , \\end{align*}"}
+{"id": "8606.png", "formula": "\\begin{align*} \\phi _ 0 - \\psi = \\mu ( \\frac { z ^ 2 } { 2 } + z ) | \\mathrm { D } | ^ 2 \\psi + \\mu ^ 2 z ^ 2 R , \\end{align*}"}
+{"id": "3961.png", "formula": "\\begin{align*} \\Omega _ t ( G ) = \\langle x \\in G \\ , | \\ , x ^ { p ^ t } = 1 \\rangle \\textrm { \\ a n d \\ } \\mho _ t ( G ) = \\langle x ^ { p ^ t } \\ , | \\ , x \\in G \\rangle . \\end{align*}"}
+{"id": "3363.png", "formula": "\\begin{align*} d _ L ( \\mu , \\nu ) : = \\sup \\{ | \\mu ( x ) - \\nu ( x ) | \\mid x \\in X , L ( x ) \\leq 1 \\} \\end{align*}"}
+{"id": "8206.png", "formula": "\\begin{align*} \\mu ^ { - 1 } ( \\lambda ) = \\{ ( m , q ) \\in M \\times \\mathbb { H } | q = ( q _ 0 , a x _ 1 ( m ) - \\lambda _ 1 , a x _ 2 ( m ) - \\lambda _ 2 , a x _ 3 ( m ) - \\lambda _ 3 ) \\} . \\end{align*}"}
+{"id": "1081.png", "formula": "\\begin{align*} \\lambda _ 1 : = \\inf _ { u \\in C _ 0 ( D ) \\setminus \\{ 0 \\} } \\frac { \\displaystyle \\int _ { D } | \\nabla u | ^ 2 ( x ) d \\mu } { \\displaystyle \\int _ { D } | u ( x ) | ^ 2 d \\mu } . \\end{align*}"}
+{"id": "3815.png", "formula": "\\begin{align*} \\mu _ i = m _ i \\nu _ X ^ i , \\nu _ X ^ i = n _ i \\mu _ i + \\left ( \\nu _ X ^ i \\right ) ^ { \\perp } , \\end{align*}"}
+{"id": "4591.png", "formula": "\\begin{align*} Y ( b , z ) = \\sum _ { n \\in \\mathbb { Z } } b _ { ( n ) } \\ , z ^ { - n - h } . \\end{align*}"}
+{"id": "3756.png", "formula": "\\begin{align*} \\dfrac { d } { d t } \\left [ \\begin{smallmatrix} u \\\\ v \\\\ w \\end{smallmatrix} \\right ] + \\mathbb { A } ^ \\alpha \\left [ \\begin{smallmatrix} u \\\\ v \\\\ w \\end{smallmatrix} \\right ] = 0 , \\ t > 0 , 0 < \\alpha < \\alpha ^ * , \\\\ \\end{align*}"}
+{"id": "709.png", "formula": "\\begin{align*} B : = b ' + 1 - \\varepsilon > \\frac 1 2 . \\end{align*}"}
+{"id": "3583.png", "formula": "\\begin{align*} { \\bf { a } } _ { \\rm { T } } ^ H \\left ( { \\Theta _ { { m } , 0 } ^ { \\rm { D } } } \\right ) { { \\bf { a } } _ { \\rm { T } } } \\left ( { \\Theta _ { { n } , 0 } ^ { \\rm { D } } } \\right ) = 0 , m \\ne n . \\end{align*}"}
+{"id": "7282.png", "formula": "\\begin{align*} d _ A ( n ) = \\frac { | A \\cap [ n ] | } { n + 1 } . \\end{align*}"}
+{"id": "5374.png", "formula": "\\begin{align*} \\frac { u } { v } = \\begin{cases} [ a _ 1 , a _ 2 , \\ldots , a _ { 2 m - 1 } , a _ { 2 m } - 1 ] , & ~ \\mbox { i f } ~ a _ { 2 m } \\geq 2 ; \\\\ [ a _ 1 , a _ 2 , \\ldots , a _ { 2 m - 2 } ] , & ~ \\mbox { i f } ~ a _ { 2 m } = 1 . \\end{cases} \\end{align*}"}
+{"id": "5954.png", "formula": "\\begin{align*} \\widetilde { Q ^ \\omega } ( t , D _ { t ' } ) & : = \\sum _ { \\alpha , \\beta } h ^ { \\alpha \\beta } ( t ) D _ { t _ \\alpha } D _ { t _ \\beta } \\\\ & - i \\sum _ { \\alpha , \\beta } \\left ( \\frac { 1 } { 2 } h ^ { \\alpha \\beta } ( t ) \\partial _ { t _ \\alpha } \\log \\delta ( t ) + \\partial _ { t _ \\alpha } h ^ { \\alpha \\beta } ( t ) - F ^ \\alpha ( t ) h ^ \\beta _ \\alpha ( t ) \\right ) D _ { t _ \\beta } - \\omega ^ 2 . \\end{align*}"}
+{"id": "1211.png", "formula": "\\begin{align*} \\limsup _ { J \\to J ^ * } \\limsup _ { n \\to \\infty } \\| e ^ { i t \\Delta } w _ n ^ J \\| _ { S ^ 1 ( \\R ) } = 0 . \\end{align*}"}
+{"id": "843.png", "formula": "\\begin{align*} \\mathcal { F } = \\{ x ^ * ( f ) \\tilde { h } : f \\in S _ { A ( \\mathbb { D } ) } \\} . \\end{align*}"}
+{"id": "4491.png", "formula": "\\begin{align*} e ' _ k = \\int ^ 1 _ 0 \\mathbb { L } '' ( { \\mathbf U } ^ a + { \\mathbf V } _ k + \\tau \\delta { \\mathbf V } _ k , \\Psi ^ a + \\Psi _ k + \\tau \\delta \\Psi _ k ) ( ( \\delta { \\mathbf V } _ k , \\delta \\Psi _ k ) , ( \\delta { \\mathbf V } _ k , \\delta \\Psi _ k ) ) ( 1 - \\tau ) d \\tau , \\end{align*}"}
+{"id": "7460.png", "formula": "\\begin{align*} \\delta _ F ( f ( x ) ) = \\sum _ { a \\ , : \\ , f ( a ) = 0 } \\frac { 1 } { \\| f ' ( a ) \\| } \\delta _ F ( x - a ) \\ , , \\end{align*}"}
+{"id": "6208.png", "formula": "\\begin{align*} F _ t ( V ^ \\leq ( \\epsilon ) ) & = F _ { t } ( \\bigcup _ { t _ i \\geq 0 } F _ { t _ i } ( S _ 1 ( \\epsilon ) ) ) = \\bigcup _ { t _ i \\geq 0 } F _ { t + t _ i } ( S _ 1 ( \\epsilon ) ) \\\\ & = \\bigcup _ { t _ i ' \\geq t } F _ { t _ i ' } ( S _ 1 ( \\epsilon ) ) \\subset \\bigcup _ { t ' \\geq 0 } F _ { t ' } ( S _ 1 ( \\epsilon ) ) = V ^ \\leq ( \\epsilon ) , \\end{align*}"}
+{"id": "4110.png", "formula": "\\begin{align*} D ( i , j ) & = [ u ^ n v ^ m ] \\frac { 1 } { 1 - u - v - u v } \\\\ & = \\sum _ { \\ell = 0 } ^ n ( - 1 ) ^ { n - \\ell } \\binom n \\ell \\binom { m + \\ell } \\ell 2 ^ \\ell \\\\ & = \\sum _ { \\ell = 0 } ^ i \\binom i \\ell \\binom { j } \\ell 2 ^ \\ell \\\\ & = \\sum _ { \\ell = 0 } ^ i \\binom { i + j - \\ell } { i - \\ell , j - \\ell , \\ell } . \\end{align*}"}
+{"id": "6981.png", "formula": "\\begin{align*} f '' + h ( z ) e ^ { p ( z ) } f ' + Q ( z ) f = 0 \\end{align*}"}
+{"id": "327.png", "formula": "\\begin{align*} u \\leftarrow \\frac { \\bar { s } } { \\left \\vert s \\right \\vert ^ { 3 } } v - \\frac { \\bar { s } } { \\left \\vert s \\right \\vert ^ { 3 } } \\frac { 1 } { p _ { j } ^ { \\operatorname * { e x t } } } \\operatorname * { d i v } \\mathbf { m \\quad } \\mathbf { j } = - \\frac { s } { \\left \\vert s \\right \\vert } \\left ( 1 + \\frac { 1 } { \\left \\vert s \\right \\vert ^ { 2 } } \\right ) \\mathbf { m } \\end{align*}"}
+{"id": "2928.png", "formula": "\\begin{align*} H _ { i j k } = H _ { j i k } . \\end{align*}"}
+{"id": "6375.png", "formula": "\\begin{align*} ( \\mathrm { I d } \\otimes \\Delta ) ( \\chi ) = a ( 1 \\otimes 1 \\otimes 1 ) + n ( x \\otimes x \\otimes 1 ) + n ( x \\otimes g \\otimes x ) - n ( x g \\otimes x g \\otimes g ) - n ( x g \\otimes 1 \\otimes x g ) \\end{align*}"}
+{"id": "6317.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & y = y ( t , x ) \\\\ & \\tau = t . \\end{aligned} \\right . \\end{align*}"}
+{"id": "7192.png", "formula": "\\begin{align*} x _ { \\mathcal { N } _ { j - 1 } + 1 } = x _ { \\mathcal { N } _ { j - 1 } + 2 } = \\cdots = x _ { \\mathcal { N } _ j } = y _ j . \\end{align*}"}
+{"id": "6308.png", "formula": "\\begin{align*} \\| w _ \\lambda \\| _ { X _ \\lambda ^ \\sharp } = \\inf _ { w _ \\lambda = \\sum w _ \\lambda ^ j } \\sum \\| w _ \\lambda ^ j \\| _ { X _ \\lambda } . \\end{align*}"}
+{"id": "6053.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ d \\langle \\partial _ { z _ j } { c } ( z , x ) , \\zeta \\rangle ^ { 2 } \\geq \\underline { c } ( z ) \\vert \\zeta \\vert ^ 2 . \\end{align*}"}
+{"id": "3182.png", "formula": "\\begin{align*} \\bar { \\gamma } = 8 / 3 \\sqrt { K } . \\end{align*}"}
+{"id": "961.png", "formula": "\\begin{align*} \\mathcal { I } _ { 1 , \\theta } ( f _ 1 , f _ 2 ) = \\int _ \\Sigma \\left < P _ 1 \\nabla f _ 1 , \\nabla f _ 2 \\right > - 2 H _ 1 \\left ( H _ 2 + c \\right ) f _ 1 f _ 2 \\ , d \\mu _ \\Sigma + \\int _ { \\partial \\Sigma } \\left \\vert { P _ 1 \\nu } \\right \\vert \\alpha _ \\theta f _ 1 f _ 2 \\ , d \\mu _ { \\partial \\Sigma } , \\end{align*}"}
+{"id": "6338.png", "formula": "\\begin{align*} i \\partial _ t v + g ( u ) \\partial _ x ^ 2 v = - g ' ( u ) \\partial _ x ^ 2 u v + N _ 1 ( u , \\partial u ) v + N _ 2 ( u , \\partial u ) \\partial v , \\end{align*}"}
+{"id": "2618.png", "formula": "\\begin{align*} X ( t ) = ( 1 - F ( t ) ) X ( 0 ) ^ + + X ^ { ( 0 ) } ( t ) + \\int _ 0 ^ t X ( t - s ) ^ + \\ , d F ( s ) + H ( t ) + \\Theta ( t ) \\ , . \\end{align*}"}
+{"id": "4211.png", "formula": "\\begin{align*} \\sigma ( T ( x \\otimes x ) + ( x \\otimes x ) T ) & = \\bigcap _ { \\alpha > 0 } \\sigma _ { \\frac { \\delta } { \\alpha } } ( T ( x \\otimes x ) + ( x \\otimes x ) T ) \\\\ & = \\bigcap _ { \\alpha > 0 } \\sigma _ { \\frac { \\delta } { \\alpha } } ( \\psi ( T ) ( x \\otimes x ) + ( x \\otimes x ) \\psi ( T ) ) \\\\ & = \\sigma ( \\psi ( T ) ( x \\otimes x ) + ( x \\otimes x ) \\psi ( T ) ) . \\end{align*}"}
+{"id": "4474.png", "formula": "\\begin{align*} \\sum ^ j _ { l = 0 } ( [ u _ { 1 , j - l } ] - [ u _ { 2 , j - l } ] \\partial _ 2 \\varphi _ l ) = 0 , [ p _ j ] + \\sum ^ { j - 1 } _ { l = 0 } C _ { l , j - 1 } [ ( H _ l , H _ { j - l } ) ] = 0 , \\quad \\mbox { o n } \\ , \\ , \\{ x _ 1 = 0 \\} \\ , , \\end{align*}"}
+{"id": "5547.png", "formula": "\\begin{align*} \\frac { 1 } { 2 m } \\sum _ { j = 1 } ^ { m } \\left ( | w ^ { ( i ) } _ j - w _ j | + | L ^ { ( i ) } _ j - L | \\right ) < \\varepsilon \\quad { \\rm o r } i > { \\tt M a x } \\end{align*}"}
+{"id": "2724.png", "formula": "\\begin{align*} U e _ i = \\left \\{ \\begin{array} { l l } \\varepsilon _ i e _ { \\pi ( i ) } , & 1 \\leq i \\leq \\lfloor \\theta ^ { - 1 } \\rfloor \\\\ \\varepsilon _ i e _ i , & i > \\lfloor \\theta ^ { - 1 } \\rfloor \\end{array} \\right . \\quad ( i \\in \\mathbb N ) \\end{align*}"}
+{"id": "8014.png", "formula": "\\begin{align*} P \\{ W _ m ^ { ( h ) } \\in A , \\ , V ( T _ n ) = v _ h , \\ , C _ { n , h } = m \\} = P \\{ W _ m ^ { ( h ) } \\in A \\} P \\{ V ( T _ n ) = v _ h , \\ , C _ { n , h } = m \\} \\end{align*}"}
+{"id": "5706.png", "formula": "\\begin{align*} & - \\big ( ( z - B ' ) ^ { - 1 } f _ 0 \\big ) ( x _ 1 ) ( z - A ) ^ { - 1 } e _ 1 + ( z - A ) ^ { - 1 } x _ 2 \\\\ & = \\big ( - \\big ( ( z - B ' ) ^ { - 1 } f _ 0 \\big ) ( x _ 1 ) + \\alpha _ 1 \\big ) ( z - A ) ^ { - 1 } e _ 1 + ( z - A ) ^ { - 1 } \\widetilde { x } _ 2 \\end{align*}"}
+{"id": "7217.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\displaystyle \\int _ { \\R ^ d } | D _ m \\Psi ( m , x ) | ^ 2 d m ( x ) \\neq 0 \\mbox { , w h e n e v e r } \\Psi ( m ) = 0 . \\end{array} \\end{align*}"}
+{"id": "8936.png", "formula": "\\begin{align*} ( A , u ) \\rightarrow \\begin{pmatrix} A & u \\\\ 0 & 1 \\end{pmatrix} A \\in S L _ { m + n } ( \\R ) , \\ u \\in \\R ^ { m + n } . \\end{align*}"}
+{"id": "4292.png", "formula": "\\begin{align*} & \\tau _ \\pi ( r _ 1 ) j _ { r _ 1 } = \\tau _ \\pi ( s _ 1 ) j _ { s _ 1 } = \\tau _ \\pi ( r _ 2 ) j _ { r _ 2 } = \\tau _ \\pi ( s _ 2 ) j _ { s _ 2 } \\\\ & \\tau _ \\pi ( r _ 1 ) j _ { r _ 1 } = \\tau _ \\pi ( s _ 1 ) j _ { s _ 1 } = - \\tau _ \\pi ( r _ 2 ) j _ { r _ 2 } = - \\tau _ \\pi ( s _ 2 ) j _ { s _ 2 } . \\end{align*}"}
+{"id": "6508.png", "formula": "\\begin{align*} D = { \\rm d i a g } _ { ( k , n ) \\in \\Z ^ { b + d } } ( \\mu _ n ^ 2 - ( k \\cdot \\omega ) ^ 2 ) , \\end{align*}"}
+{"id": "2533.png", "formula": "\\begin{align*} \\boldsymbol { y _ d } ( \\boldsymbol { x } , t ) = ( 0 , 0 , e ^ t ( \\sin { t } + ( 2 \\pi ^ 2 + 1 ) ( ( 2 \\pi ^ 2 + 1 ) \\sin { t } - \\cos { t } ) ) \\sin { \\pi x _ 1 } \\sin { \\pi x _ 2 } ) ^ T \\end{align*}"}
+{"id": "3158.png", "formula": "\\begin{align*} ( \\phi ^ { \\vee } ) ^ * W _ { L _ X } = W ^ { Y \\backslash D _ Y } _ { L _ Y } + ( W _ { L _ Y } ^ { D _ Y } ) ^ r - r ! \\langle \\psi _ { r - 2 } \\emph { } \\rangle _ r . \\end{align*}"}
+{"id": "1829.png", "formula": "\\begin{gather*} \\sum _ { n = 0 } ^ { \\infty } | F _ g ( \\log ( n + c ) ) | \\geq \\sum _ { m \\notin A _ \\beta } \\sum _ { n \\in C _ \\beta ( m ) } | F _ g ( \\log ( n + c ) ) | \\geq \\sum _ { m \\notin A _ \\beta } \\sum _ { n \\in C _ \\beta ( m ) } \\frac { 1 } { n + c } . \\end{gather*}"}
+{"id": "7561.png", "formula": "\\begin{align*} A ^ { ( < ) } _ { T , r } & = \\{ R _ T \\le r \\} , \\\\ A ^ { ( > ) } _ { T , r } & = \\{ R _ T \\ge r \\} . \\end{align*}"}
+{"id": "2521.png", "formula": "\\begin{align*} \\big ( \\sigma \\partial _ t ^ { 1 / 2 } \\boldsymbol { \\eta } , \\partial _ t ^ { 1 / 2 } \\boldsymbol { v } ^ { \\perp } \\big ) = \\big ( \\sigma \\partial _ t \\boldsymbol { \\eta } , \\boldsymbol { v } \\big ) \\end{align*}"}
+{"id": "6722.png", "formula": "\\begin{align*} \\begin{cases} \\alpha ' ( t ) - ( 2 a + 1 ) \\eta ( t ) f ' ( t ) \\alpha ( t ) - a f ' ( t ) \\beta ( t ) = 0 , \\\\ \\beta ' ( t ) + ( 2 a + 1 ) ( \\eta ( t ) ) ^ 2 f ' ( t ) \\alpha ( t ) = 0 , \\\\ \\gamma ' ( t ) = - f ' ( t ) \\alpha ( t ) , \\end{cases} \\end{align*}"}
+{"id": "3178.png", "formula": "\\begin{align*} S \\begin{bmatrix} d _ 1 & 0 & \\cdots { } & 0 \\\\ 0 & d _ 2 & \\ddots { } & \\vdots { } \\\\ \\vdots { } & \\ddots { } & \\ddots { } & 0 \\\\ 0 & \\cdots { } & 0 & d _ n \\end{bmatrix} S ^ \\mathsf { T } = \\begin{bmatrix} d _ n & 0 & \\cdots { } & 0 \\\\ 0 & d _ 1 & \\ddots { } & \\vdots { } \\\\ \\vdots { } & \\ddots { } & \\ddots { } & 0 \\\\ 0 & \\cdots { } & 0 & d _ { n - 1 } \\end{bmatrix} , \\end{align*}"}
+{"id": "3736.png", "formula": "\\begin{align*} \\lambda I - \\mathbb { A } = \\begin{bmatrix} \\lambda I & I & 0 \\\\ 0 & \\lambda I & I \\\\ - A & 0 & \\lambda I \\end{bmatrix} , \\end{align*}"}
+{"id": "2264.png", "formula": "\\begin{align*} \\vartheta ( z ; \\tau ) & = \\prod _ { k \\geq 1 } \\left ( 1 - e ^ { 2 k \\pi i \\tau } \\right ) \\left ( 1 + e ^ { ( 2 k - 1 ) \\pi i \\tau } e ^ { 2 \\pi i z } \\right ) \\left ( 1 + e ^ { ( 2 k - 1 ) \\pi i \\tau } e ^ { - 2 \\pi i z } \\right ) \\\\ & = \\prod _ { k \\geq 1 } \\left ( 1 - e ^ { 2 k \\pi i \\tau } \\right ) \\left ( 1 + 2 \\cos ( 2 \\pi z ) e ^ { ( 2 k - 1 ) \\pi i \\tau } + e ^ { 2 ( 2 k - 1 ) \\pi i \\tau } \\right ) . \\end{align*}"}
+{"id": "1497.png", "formula": "\\begin{align*} d A _ { i } ( t ) = \\frac { 1 } { 2 } \\langle U ^ { ( i ) } B _ { t } , d B _ { t } \\rangle , \\end{align*}"}
+{"id": "4421.png", "formula": "\\begin{align*} \\hat { \\lambda } ^ \\pm : = \\lambda ( \\hat { { \\mathbf U } } ^ { \\pm } ) : = \\eta ( x _ 1 ) \\lambda ^ { \\pm } ( t , x _ 2 ) , \\end{align*}"}
+{"id": "5303.png", "formula": "\\begin{align*} g ( x ) \\coloneqq \\begin{cases} f ( x ) & x < x _ { i - 1 } , \\\\ y _ { i - 1 } + a _ i ( x - x _ { i - 1 } ) & x _ { i - 1 } \\leq x < \\hat { x } _ { i + 2 } , \\\\ f ( \\frac { a _ i } { a _ { i + 2 } } ( x - \\hat { x } _ { i + 2 } ) + x _ { i + 2 } ) & \\hat { x } _ { i + 2 } \\leq x \\end{cases} \\end{align*}"}
+{"id": "5562.png", "formula": "\\begin{align*} \\nabla ^ 2 V _ k ( \\zeta ) & = 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\ : \\ : \\mbox { i f } \\zeta \\in G _ m , \\\\ V _ k ( \\zeta ) & = 1 - P _ { k j } \\mbox { i f } \\zeta \\in C _ j , j = 1 , \\ldots , k - 1 , \\\\ V _ k ( \\zeta ) & = 0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\ : \\ : \\mbox { i f } \\zeta \\in C _ k , \\\\ V _ k ( \\zeta ) & = - P _ { k j } ~ ~ ~ ~ \\ : \\ : \\ : \\mbox { i f } \\zeta \\in C _ j , j = k + 1 , \\ldots , m . \\end{align*}"}
+{"id": "7378.png", "formula": "\\begin{align*} { \\rm F i l } ^ * { \\rm T H H } ( ( A , M ) ; { \\Bbb Z } _ p ) : = R \\Gamma _ { \\rm l q s y n } ( ( A , M ) , \\tau _ { \\ge 2 * } { \\rm T H H } ( ( - , - ) ; { \\Bbb Z } _ p ) ) \\\\ { \\rm F i l } ^ * { \\rm T C } ^ - ( ( A , M ) ; { \\Bbb Z } _ p ) : = R \\Gamma _ { \\rm l q s y n } ( ( A , M ) , \\tau _ { \\ge 2 * } { \\rm T C } ^ - ( ( - , - ) ; { \\Bbb Z } _ p ) ) \\\\ { \\rm F i l } ^ * { \\rm T P } ( ( A , M ) ; { \\Bbb Z } _ p ) : = R \\Gamma _ { \\rm l q s y n } ( ( A , M ) , \\tau _ { \\ge 2 * } { \\rm T P } ( ( - , - ) ; { \\Bbb Z } _ p ) ) \\end{align*}"}
+{"id": "2459.png", "formula": "\\begin{align*} \\overline { H } \\left ( \\bigsqcup _ { a \\in \\mathcal { A } } ^ { P _ A } G _ a \\right ) = \\frac { 1 } { k } \\overline { H } \\left ( \\bigwedge _ { a \\in \\mathcal { A } } G _ a ^ { \\wedge k P _ A ( a ) } \\right ) . \\end{align*}"}
+{"id": "4882.png", "formula": "\\begin{align*} | | P f | | _ { \\mathcal { B } ( \\mathfrak { E } ) } & = \\int _ { - \\infty } ^ \\infty | | E _ + ^ { - 1 } ( t ) P ( t ) f ( t ) | | ^ 2 d t \\\\ & = \\int _ { - \\infty } ^ \\infty | | ( E _ + ^ o ) ^ { - 1 } ( t ) P ( t ) ^ { - 1 } P ( t ) f ( t ) | | ^ 2 d t \\\\ & = \\int _ { - \\infty } ^ \\infty | | ( E _ + ^ o ) ^ { - 1 } ( t ) f ( t ) | | ^ 2 d t \\\\ & = | | f | | _ { \\mathcal { B } ( \\mathfrak { E } ^ 0 ) } . \\end{align*}"}
+{"id": "1815.png", "formula": "\\begin{gather*} \\lim _ { T \\to \\infty } \\frac { 1 } { T } \\int _ { 0 } ^ { T } F ( Z _ N ( s + i \\tau , \\alpha ) ) \\ , d \\tau = \\mathbf { E } \\left [ F ( Z _ N ( s , \\mathbb { X } _ \\alpha ) ) \\right ] \\end{gather*}"}
+{"id": "5973.png", "formula": "\\begin{align*} 0 & = \\sigma _ { - 1 } ( N ^ \\omega _ { \\partial M } ) ( t , \\xi ' ) \\sigma _ 0 ( \\Lambda _ { g , F } ^ \\omega ) ( t , \\xi ' ) + \\sigma _ { - 2 } ( N ^ \\omega _ { \\partial M } ) ( t , \\xi ' ) \\sigma _ 1 ( \\Lambda _ { g , F } ^ \\omega ) ( t , \\xi ' ) \\\\ & + \\nabla _ { \\xi ' } \\sigma _ { - 1 } ( N ^ \\omega _ { \\partial M } ) \\cdot D _ { t ' } \\sigma _ 1 ( \\Lambda _ { g , F } ^ \\omega ) . \\end{align*}"}
+{"id": "8121.png", "formula": "\\begin{align*} ( S _ k ^ \\alpha = ) \\sum _ { \\beta ' \\stackrel { > k } { \\to } \\alpha } u _ { \\beta ' } < u _ \\alpha - p + 1 \\le \\sum _ { \\beta ' \\stackrel { \\ge k } { \\to } \\alpha } u _ { \\beta ' } . \\end{align*}"}
+{"id": "8461.png", "formula": "\\begin{align*} \\ell ( z ) = \\mathcal { H } ^ { n - 1 } ( E _ { z } ) \\mbox { f o r } \\mathcal { H } ^ { 1 } \\mbox { - a . e . } z \\in \\mathbb { R } , \\end{align*}"}
+{"id": "6893.png", "formula": "\\begin{align*} \\sum _ i \\left ( 1 - \\frac { 1 } { m _ i } \\right ) b _ i = 2 . \\end{align*}"}
+{"id": "5686.png", "formula": "\\begin{align*} k _ { e _ { n _ 0 } } = \\lim \\limits _ { j \\rightarrow \\infty } \\frac { \\ln \\| ( r _ j - T ) ^ { - 1 } e _ { n _ 0 } \\| _ p } { \\ln \\| ( r _ j - T ) ^ { - 1 } \\| } . \\end{align*}"}
+{"id": "3386.png", "formula": "\\begin{align*} - 2 b ^ \\mu a + 2 c ^ \\mu - b ^ \\mu \\alpha ^ 0 _ { \\ , 0 0 } - b ^ \\mu \\alpha ^ 0 _ { \\ , 0 \\tau } b ^ \\tau - b ^ \\mu \\alpha ^ 0 _ { \\ , \\sigma 0 } b ^ \\sigma + \\alpha ^ \\mu _ { \\ , 0 \\tau } b ^ \\tau + \\alpha ^ \\mu _ { \\ , \\sigma 0 } b ^ \\sigma + \\alpha ^ \\mu _ { \\ , \\sigma \\tau } b ^ \\sigma b ^ \\tau = 0 . \\end{align*}"}
+{"id": "2157.png", "formula": "\\begin{align*} e ^ { u _ 0 } = \\alpha ^ d , e ^ \\rho = \\left ( \\frac { \\mu } { \\alpha } \\right ) ^ d , e ^ { - \\rho } = \\left ( \\frac { \\mu } { \\alpha } \\right ) ^ { - d } , \\cosh \\rho = \\frac 1 2 \\left ( \\left ( \\frac { \\mu } { \\alpha } \\right ) ^ d + \\left ( \\frac { \\mu } { \\alpha } \\right ) ^ { - d } \\right ) , \\end{align*}"}
+{"id": "8688.png", "formula": "\\begin{align*} \\int u _ 0 e ^ { \\frac { 1 } { 2 \\hbar } \\sum _ { i = 0 } ^ { n + 1 } \\sum _ { j , k = 0 } ^ n a _ { i , j , k } y _ i u _ j u _ k } d u _ 0 . . . d u _ n \\end{align*}"}
+{"id": "4939.png", "formula": "\\begin{align*} \\Psi ( 2 i ) = \\dfrac { \\pi ^ { 1 / 4 } ( 2 - 2 ^ { 1 / 2 } ) ^ { 1 / 2 } \\cdot e ^ { \\pi / 2 } } { \\Gamma ( 3 / 4 ) \\cdot 2 ^ 2 } , \\Psi ( 4 i ) = \\dfrac { \\pi ^ { 1 / 4 } ( 1 - 2 ^ { - 1 / 4 } ) \\cdot e ^ { \\pi } } { \\Gamma ( 3 / 4 ) \\cdot 2 ^ 2 } . \\end{align*}"}
+{"id": "2164.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\int _ { | x - v t | \\leq \\omega ( t ) } \\left ( \\Lambda _ t ^ 2 + \\Lambda _ x ^ 2 + \\sinh ^ 2 ( \\Lambda ) ( \\phi _ t ^ 2 + \\phi _ x ^ 2 ) \\right ) ( t , x ) d x = 0 . \\end{align*}"}
+{"id": "6584.png", "formula": "\\begin{align*} u ^ { ( 0 ) } ( t , n ) = \\sum _ { ( k , n ) \\in \\mathcal { S } } q ^ { ( 0 ) } ( k , n ) \\cos ( e _ l \\cdot \\omega ^ { ( 0 ) } t ) , \\end{align*}"}
+{"id": "1809.png", "formula": "\\begin{gather*} d ( f , g ) \\leq \\sum _ { \\nu = 1 } ^ { M } 2 ^ { - \\nu } d _ \\nu ( f , g ) + \\sum _ { \\nu = M + 1 } ^ { \\infty } 2 ^ { - \\nu } \\leq d _ M ( f , g ) + 2 ^ { - M } \\end{gather*}"}
+{"id": "8905.png", "formula": "\\begin{align*} \\sigma ^ i \\circ \\varphi = \\varphi \\circ \\begin{pmatrix} 0 & - 1 \\\\ 1 & \\gamma \\end{pmatrix} , \\end{align*}"}
+{"id": "3794.png", "formula": "\\begin{align*} \\theta ( y _ 0 , y _ 1 ) = \\frac { 1 } { s _ * } ( s _ 0 ^ p + s _ 1 ^ p ) ^ { \\frac 1 p } , { \\rm p r d } _ { \\theta } ( x _ 0 , s _ 0 , x _ 1 , s _ 1 ) : = \\left ( x _ 0 , \\frac { s _ 0 } { \\theta } , x _ 1 , \\frac { s _ 1 } { \\theta } \\right ) . \\end{align*}"}
+{"id": "6996.png", "formula": "\\begin{align*} R _ { 1 2 1 2 } = K _ { 1 2 1 2 } + | H | ^ 2 - 2 ( | \\phi | ^ 2 + | \\psi | ^ 2 ) ; \\end{align*}"}
+{"id": "4677.png", "formula": "\\begin{align*} | H | & = n - | S | - | T | - | V ( C _ { 2 l + 1 } ) | \\\\ & < n - \\frac { 4 n } { 4 ( 2 l + 1 ) + 1 } \\cdot ( 2 l + 1 ) n = \\frac { n } { 4 ( 2 l + 1 ) + 1 } . \\end{align*}"}
+{"id": "2727.png", "formula": "\\begin{align*} \\chi _ F = \\sum _ { i \\in F } e _ i \\in \\{ 0 , 1 \\} ^ { \\mathbb { N } } . \\end{align*}"}
+{"id": "8941.png", "formula": "\\begin{align*} \\Tilde { \\theta } ( u ^ - u ^ 0 ) : = \\theta ^ - ( u ^ - ) \\theta ^ 0 ( u ^ 0 ) \\Delta ( u ^ 0 ) ^ { - 1 } . \\end{align*}"}
+{"id": "2073.png", "formula": "\\begin{align*} \\mathbf { I } & \\leq \\frac { C _ 2 } { t } \\int ^ \\infty _ { \\sqrt { t } } \\left ( \\frac { r } { \\sqrt { t } } \\right ) ^ n \\exp \\left ( - \\frac { r ^ 2 } { C _ 1 t } \\right ) \\cdot \\frac { k ( x _ 0 , r ) } { r } \\ , d r \\leq \\frac { C _ 3 } { t } \\int ^ \\infty _ { \\sqrt { t } } \\frac { k ( x _ 0 , r ) } { r } \\ , d r . \\end{align*}"}
+{"id": "5587.png", "formula": "\\begin{align*} \\mu ( B _ r ( x ) ) = \\int _ { B _ r ( x ) } | \\nabla u _ * | ^ 2 \\ , d x \\end{align*}"}
+{"id": "5647.png", "formula": "\\begin{align*} \\cdots \\to Z _ { i } \\to Z _ { i - 1 } \\to \\dots \\to Z _ 1 \\to Z _ 0 = Z , \\end{align*}"}
+{"id": "252.png", "formula": "\\begin{align*} V ^ { } _ { \\{ \\epsilon j , \\epsilon ^ \\prime j ^ \\prime \\} } ( \\xi ; g ) = & w ( \\epsilon \\xi _ j ) w ( \\epsilon ^ \\prime \\xi _ { j ^ \\prime } ) v ( \\epsilon \\xi _ j + \\epsilon ^ \\prime \\xi _ { j ^ \\prime } ) v ( 1 + \\epsilon \\xi _ j + \\epsilon ^ \\prime \\xi _ { j ^ \\prime } ) \\\\ & \\times \\prod _ { \\substack { 1 \\leq k \\leq n \\\\ k \\neq j , j ^ \\prime } } v ( \\epsilon \\xi _ j + \\xi _ k ) v ( \\epsilon \\xi _ j - \\xi _ k ) , \\end{align*}"}
+{"id": "1697.png", "formula": "\\begin{align*} \\frac { 1 } { ( 2 \\pi ) ^ { n } } \\int _ { \\mathbb { A } ^ { ( n ) } _ { \\texttt { b } } } f ( \\boldsymbol { \\xi } ) | C _ { \\texttt { b } } ( \\boldsymbol { \\xi } ; q , q _ 0 ) | ^ { - 2 } \\boldsymbol { \\xi } = \\sum _ { \\lambda \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { b } } } f \\bigl ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } \\bigr ) \\hat { \\Delta } ^ { ( m , n ) } _ { \\texttt { b } ; \\lambda } , \\end{align*}"}
+{"id": "684.png", "formula": "\\begin{align*} \\norm { M _ 2 ( g ) } _ { \\mathcal L _ 2 ( L ^ 2 , H ^ r ) } = \\norm { g \\mathfrak K _ 2 } _ { \\mathcal L _ 2 ( L ^ 2 , H ^ { r - 1 } ) } . \\end{align*}"}
+{"id": "6040.png", "formula": "\\begin{align*} \\left \\vert \\mathbb { E } ( R _ { q } ( \\delta , F ) \\Upsilon _ \\eta ( \\det \\sigma _ F ) ) \\right \\vert \\leq C ^ \\prime _ { q } \\frac { \\left \\Vert f \\right \\Vert _ { \\infty } } { \\eta ^ { 2 q } } \\int _ { \\mathbb { R } ^ { d } } d y \\phi _ { \\delta } ( y ) \\left \\vert y \\right \\vert ^ { q } = C ^ \\prime _ { q } \\int _ { \\mathbb { R } ^ { d } } \\phi ( y ) \\left \\vert y \\right \\vert ^ { q } d y \\left \\Vert f \\right \\Vert _ { \\infty } \\frac { \\delta ^ { q } } { \\eta ^ { 2 q } } . \\end{align*}"}
+{"id": "3458.png", "formula": "\\begin{align*} \\Delta ^ \\vee _ 0 = \\{ ( 0 , p _ 1 ) : 0 \\geq p _ 1 \\geq - \\frac { 1 } { d _ 1 } \\} , \\Delta ^ \\vee _ 1 = \\{ ( p _ 0 , 0 ) : 0 \\geq p _ 0 \\geq - \\frac { 1 } { d _ 0 } \\} , \\end{align*}"}
+{"id": "7079.png", "formula": "\\begin{align*} \\mathcal { V } _ 2 \\left ( \\frac { m } { p _ 2 ( p _ 1 q ) ^ 2 } \\right ) = 2 \\pi i ^ k \\int _ 0 ^ \\infty v _ 2 ( y ) \\ , J _ { k - 1 } \\left ( \\frac { 4 \\pi } { p _ 1 q } \\sqrt { \\frac { m y } { p _ 2 } } \\right ) \\ { d } y . \\end{align*}"}
+{"id": "5792.png", "formula": "\\begin{align*} ( \\alpha + 2 \\beta , \\alpha + 2 \\beta ) = ( \\alpha , \\alpha ) + 4 ( \\alpha , \\beta ) + 4 ( \\beta , \\beta ) = 2 - 4 + 4 = 2 . \\end{align*}"}
+{"id": "2199.png", "formula": "\\begin{align*} \\# A \\cdot \\# B = 2 ^ { 2 n - \\alpha n - \\beta n + 2 n - \\beta \\pm O ( \\log n ) } = 2 ^ { 4 n - \\alpha n - 2 \\beta n \\pm O ( \\log n ) } = 2 ^ { 4 n - \\C ( m _ A ) - \\beta n \\pm O ( \\log n ) } . \\end{align*}"}
+{"id": "3561.png", "formula": "\\begin{align*} L ( x ^ { k + 1 } , \\lambda ) - L ( x , \\lambda ^ { k + 1 } ) & = L ( x ^ { k + 1 } , \\lambda ) - L ( x ^ { k + 1 } , \\lambda ^ { k + 1 } ) + L ( x ^ { k + 1 } , \\lambda ^ { k + 1 } ) - L ( x , \\lambda ^ { k + 1 } ) \\\\ & \\leq \\langle F ^ { k + 1 } , z ^ { k + 1 } - z \\rangle + \\frac { L } { 2 } \\| z ^ { k } - z ^ { k + 1 } \\| _ 2 ^ 2 \\ , \\end{align*}"}
+{"id": "7356.png", "formula": "\\begin{align*} \\rho _ S ( x ) = \\big ( 1 - \\frac { 1 } { q } \\big ) ^ + \\delta ( x ) + \\frac { \\sqrt { \\Big ( x - \\big ( \\frac { 1 } { \\sqrt { q } } - 1 \\big ) ^ 2 \\Big ) \\Big ( \\big ( \\frac { 1 } { \\sqrt { q } } + 1 \\big ) ^ 2 - x \\Big ) } } { 2 \\pi x } . \\end{align*}"}
+{"id": "2852.png", "formula": "\\begin{align*} \\hat h _ t : = t _ { \\alpha _ 0 } \\ , \\hat e _ t ( - \\alpha _ 0 ^ \\vee ) \\end{align*}"}
+{"id": "2365.png", "formula": "\\begin{align*} \\sum _ { \\mu \\in G \\times \\widehat { G } } c _ \\mu ( 1 - d _ { \\mu , \\lambda } ) \\pi ( \\mu ) = 0 , \\forall \\lambda \\in \\Lambda . \\end{align*}"}
+{"id": "7214.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\partial _ t u + H ( x , D u ) - \\Delta u = \\psi _ 1 \\nu + \\varphi _ 1 & \\mbox { i n } [ t _ 0 , T ] \\times \\R ^ d , \\\\ u ( T , x ) = \\psi _ 2 & \\mbox { i n } \\R ^ d , \\end{array} \\right . \\end{align*}"}
+{"id": "4688.png", "formula": "\\begin{align*} \\begin{cases} 1 = \\Psi ( x , y ) , \\\\ \\Psi ( x , w ) = 1 = \\Psi ( y , u ) , \\\\ \\Psi ( x , u ) = 0 = \\Psi ( y , w ) . \\end{cases} \\end{align*}"}
+{"id": "598.png", "formula": "\\begin{align*} \\mathcal H ( \\psi ( t ) ) - \\mathcal H ( \\psi ( 0 ) ) = 0 . \\end{align*}"}
+{"id": "6799.png", "formula": "\\begin{align*} u ( z ) & = 1 - \\alpha z e ^ { \\lambda _ 1 z } - \\beta e ^ { \\lambda _ 2 z } + h . o . t . , \\\\ [ 0 . 2 c m ] w ( z ) & = 0 - \\alpha \\left [ \\lambda _ 1 z + f ( 1 ) / ( 2 \\lambda _ { 1 } - c ) \\right ] e ^ { \\lambda _ 1 z } - \\beta \\lambda _ 2 e ^ { \\lambda _ 2 z } + h . o . t . , \\\\ [ 0 . 2 c m ] v ( z ) & = 0 - \\alpha e ^ { \\lambda _ 1 z } - \\beta \\psi \\left ( \\lambda _ { 2 } \\right ) e ^ { \\lambda _ 2 z } + h . o . t . . \\end{align*}"}
+{"id": "4438.png", "formula": "\\begin{align*} | | \\tilde { \\mathcal { A } } \\mathcal { F } | | ^ 2 _ { s - 1 , \\ast , t } \\leq C ( K ) \\Big ( | | \\mathbf { F } | | ^ 2 _ { s - 1 , \\ast , t } + | | \\mathbf { F } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ t ) } | | \\hat { W } | | ^ 2 _ { s - 1 , \\ast , t } \\Big ) . \\end{align*}"}
+{"id": "6290.png", "formula": "\\begin{align*} \\{ \\xi _ 1 , \\xi _ 3 \\} = \\{ \\xi _ 2 , \\xi _ 4 \\} = \\{ \\xi , \\eta \\} . \\end{align*}"}
+{"id": "2999.png", "formula": "\\begin{align*} \\Theta _ { x , x } z ( e ) = x ^ 2 ( e ) z ( e ) = \\Psi ( x ^ 2 ) z ( e ) \\end{align*}"}
+{"id": "6036.png", "formula": "\\begin{align*} \\overline { \\lim } _ { n \\rightarrow \\infty } \\Vert \\sum \\limits _ { i = n } ^ { m _ { n } } \\gamma _ { i } ^ { n } ( D F _ { i } , D \\bar { F } _ { i } ) - ( D F , D \\bar { F } ) \\Vert _ { L ^ 2 ( \\Omega ; \\mathcal { H } ) } = 0 . \\end{align*}"}
+{"id": "7731.png", "formula": "\\begin{align*} \\overline { \\nabla } _ Z ( \\overline { ( Y , \\Delta ) } ) = \\overline { ( [ Z , Y ] , [ \\nabla _ Z , \\Delta ] ) } , \\end{align*}"}
+{"id": "6473.png", "formula": "\\begin{align*} P _ R F _ 1 & = ( I - P _ D - P _ N ) F _ 1 = ( I - P _ N ) F _ 1 , \\\\ P _ R F _ 1 ' & = ( I - P _ D - P _ N ) F _ 1 ' = ( I - P _ D ) F _ 1 ' , \\end{align*}"}
+{"id": "5904.png", "formula": "\\begin{align*} \\phi = 1 ( - 1 / 2 , 1 / 2 ) , \\phi = 0 \\R \\setminus ( - 1 , 1 ) , \\| \\phi \\| _ { C ^ 2 ( \\R ) } \\leq C _ 0 . \\end{align*}"}
+{"id": "1635.png", "formula": "\\begin{align*} Q ^ { 2 k - l } T ^ { 2 k - 2 k + l } = Q ^ { 2 k - l } T ^ l = Q ^ { 2 k } Q ^ { - l } T ^ l = q ^ k ( \\sqrt { t } ) ^ l . \\end{align*}"}
+{"id": "1034.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\Delta _ { p ( z ) } u - \\Delta _ { q ( z ) } u + u ^ { p ( z ) - 1 } = \\eta \\Omega , \\\\ \\displaystyle \\frac { \\partial u } { \\partial n } = 0 \\mbox { o n } \\partial \\Omega , \\ u \\geq 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "1472.png", "formula": "\\begin{align*} A _ { t } : = \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } B _ { 1 } ( s ) d B _ { 2 } ( s ) - B _ { 2 } ( s ) d B _ { 1 } ( s ) \\end{align*}"}
+{"id": "758.png", "formula": "\\begin{align*} \\mu ( t ) = t ^ { 1 - 2 b } f ( t ) , f ( t ) = \\frac { 1 } { t } \\int _ 0 ^ t g ( s ) \\ , d s , g ( s ) = \\int _ { \\R ^ m } V ( s , \\zeta ) \\overline { W ( s , \\zeta ) } \\ , d \\zeta , \\end{align*}"}
+{"id": "8347.png", "formula": "\\begin{align*} \\sigma _ \\eta = ( s _ 1 , \\dots , s _ { n _ 0 } ) ^ \\smallfrown \\bar s \\sigma _ \\zeta = ( t _ 1 , \\dots , t _ { n _ 1 } ) ^ \\smallfrown \\bar s . \\end{align*}"}
+{"id": "4783.png", "formula": "\\begin{align*} { } _ { 1 } F _ { 1 } \\left ( 1 , s , z \\right ) = 1 + \\frac { z ^ 2 } s \\ , { } _ { 1 } F _ { 1 } \\left ( 1 , s + 1 , z \\right ) . \\end{align*}"}
+{"id": "4742.png", "formula": "\\begin{align*} T _ n = \\big ( 2 ^ { - 2 n + 1 } T + ( 2 ^ { - 2 n + 1 } , 0 ) \\big ) S = \\bigcup _ { n = 1 } ^ { \\infty } T _ n . \\end{align*}"}
+{"id": "7182.png", "formula": "\\begin{align*} A : = & \\begin{pmatrix} \\boldsymbol { \\alpha _ 1 ^ 1 } & \\boldsymbol { \\alpha _ 2 ^ 1 } & \\dots & \\boldsymbol { \\alpha _ m ^ 1 } \\\\ \\boldsymbol { \\alpha _ 1 ^ 2 } & \\boldsymbol { \\alpha _ 2 ^ 2 } & \\dots & \\boldsymbol { \\alpha _ m ^ 2 } \\\\ \\vdots & \\vdots & \\dots & \\vdots \\\\ \\boldsymbol { \\alpha _ 1 ^ m } & \\boldsymbol { \\alpha _ 2 ^ m } & \\dots & \\boldsymbol { \\alpha _ m ^ m } \\\\ \\end{pmatrix} _ { m \\times m } . \\end{align*}"}
+{"id": "3383.png", "formula": "\\begin{align*} \\Upsilon _ \\mu = 0 . \\end{align*}"}
+{"id": "3300.png", "formula": "\\begin{align*} \\sigma ^ 2 = m _ 2 - m _ 1 ^ 2 = \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\frac { \\beta _ { _ \\Sigma } ^ 3 - \\alpha _ { _ \\Sigma } ^ 3 } { 3 } - \\left ( \\sum _ { \\Sigma \\subseteq [ \\ ! [ 1 , n ] \\ ! ] } e _ { _ \\Sigma } \\frac { \\beta _ { _ \\Sigma } ^ 2 - \\alpha _ { _ \\Sigma } ^ 2 } { 2 } \\right ) ^ 2 . \\end{align*}"}
+{"id": "4615.png", "formula": "\\begin{align*} x \\cdot m : = \\frac { d } { d t } \\bigg | _ { t = 0 } \\pi ( \\exp ( t x ) ) m \\end{align*}"}
+{"id": "834.png", "formula": "\\begin{align*} ( N x ) ( z ) : = \\eta ( z ) \\Psi _ 2 ( x ) + ( 1 - \\epsilon ) ( 1 - \\eta ( z ) ) T x ( z ) , \\end{align*}"}
+{"id": "3419.png", "formula": "\\begin{align*} \\Omega = f _ 1 z _ 0 ^ { a _ 0 + b _ 0 - 1 } \\ldots z _ k ^ { a _ k + b _ k - 1 } d z _ 0 \\wedge \\ldots d z _ n , \\end{align*}"}
+{"id": "4922.png", "formula": "\\begin{align*} G _ { \\ell } ^ { ( \\underline { a } , N ) } ( \\tau ) : = \\sideset { } { ' } \\sum _ { \\substack { m _ 1 , m _ 2 \\in \\mathbb { Z } \\\\ m _ 1 \\equiv a _ 1 \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\pmod { N } \\\\ m _ 2 \\equiv a _ 2 \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\pmod { N } } } \\dfrac { 1 } { ( m _ 1 \\tau + m _ 2 ) ^ { \\ell } } \\end{align*}"}
+{"id": "7344.png", "formula": "\\begin{align*} \\Gamma ( L \\mid _ { H _ G } , 0 ) [ u _ \\ast ] & = \\int ^ \\infty _ { 0 } { \\left \\langle \\dot { \\widetilde { A } } ( 0 , t ) u _ \\ast ( t ) , u _ { \\ast } ( t ) \\right \\rangle \\ , d t } + \\int ^ 0 _ { - \\infty } { \\left \\langle \\dot { \\widetilde { A } } ( 0 , t ) u _ \\ast ( t ) , u _ { \\ast } ( t ) \\right \\rangle \\ , d t } \\\\ & = \\int ^ \\infty _ { 0 } { \\arctan ( t ) \\langle J \\dot { S } _ 0 u _ { \\ast } ( t ) , u _ { \\ast } ( t ) \\rangle \\ , d t } \\\\ & = - \\int ^ \\infty _ { 0 } { \\arctan ( t ) ( t ^ 2 + 1 ) e ^ { - 2 t \\arctan ( t ) } \\ , d t } < 0 , \\end{align*}"}
+{"id": "54.png", "formula": "\\begin{align*} \\vert \\mathcal R _ 1 ^ { ( d ) } ( \\rho _ z ) \\vert \\leq \\begin{cases} C | \\Lambda | \\rho _ z ^ { 2 } \\delta ^ { 2 } K _ { \\ell } ^ { - 1 } , & d = 2 , \\\\ C | \\Lambda | \\rho _ z ^ { 2 } a \\big ( \\rho _ z a ^ { 3 } \\big ) ^ { \\frac 1 2 } \\log ( \\rho _ z ) K _ { \\ell } ^ { - 1 } , & d = 3 . \\end{cases} \\end{align*}"}
+{"id": "2590.png", "formula": "\\begin{align*} F _ { \\sigma } = & [ D _ { \\sigma } \\ , \\vert \\ , K ] , \\\\ F _ { \\sigma } ^ { ' } = & [ D _ { \\sigma } ^ { ' } \\ , \\vert \\ , K ' ] \\end{align*}"}
+{"id": "8232.png", "formula": "\\begin{align*} \\bar { \\xi } _ 1 = d p _ { x _ 1 , 0 , 0 } ( \\xi _ 1 - \\frac { 2 x _ 1 } { | T | ^ 2 } T ) . \\end{align*}"}
+{"id": "2009.png", "formula": "\\begin{align*} \\frac { 1 } { q } = \\frac { 1 } { p } - \\frac { s } { n } \\end{align*}"}
+{"id": "656.png", "formula": "\\begin{align*} \\tau _ R ( \\omega ) = \\sup \\left \\{ t \\in [ 0 , \\tau ( \\omega ) ) \\colon \\right \\} . \\end{align*}"}
+{"id": "662.png", "formula": "\\begin{align*} \\left \\{ \\tau _ R = \\tau \\le t \\right \\} = \\bigcap _ P \\left ( \\bigcup _ { \\substack { t _ k \\in P \\\\ t _ k < t } } \\bigcap _ { 0 \\le s < t _ k } f _ { s } ^ { - 1 } \\left ( \\left [ 0 , R \\right ] \\right ) \\cap \\{ t _ k \\le \\tau \\le t _ { k + 1 } \\} \\right ) , \\end{align*}"}
+{"id": "3065.png", "formula": "\\begin{align*} \\left \\{ { \\left ( { { \\alpha _ { { k , { \\rm { R } } } , i } } , \\Theta _ { { k , { \\rm { R } } } , i } ^ { \\rm { A } } , \\Theta _ { { k , { \\rm { R } } } , i } ^ { \\rm { D } } } \\right ) , i \\in { { \\mathcal L } _ { { k , { \\rm { R } } } } } } \\right \\} _ { k = 1 } ^ K . \\end{align*}"}
+{"id": "6600.png", "formula": "\\begin{align*} f ( t , \\cdot ) = ( \\Gamma _ f ( t , \\cdot ) ) _ \\# f _ 0 \\ t \\ge 0 , \\end{align*}"}
+{"id": "609.png", "formula": "\\begin{align*} \\norm { u } _ { X ^ { s , b } } : = \\left ( \\int _ { \\R ^ d } \\left ( \\norm { U ( t , \\xi ) } _ { L ^ 2 _ t ( \\R ) } ^ 2 + \\norm { U ( t , \\xi ) } _ { S ^ b _ t ( \\R ) } ^ 2 \\right ) \\ , d \\xi \\right ) ^ { 1 / 2 } . \\end{align*}"}
+{"id": "5390.png", "formula": "\\begin{align*} S ^ { - 1 } _ { j j } ( t ) d S _ { j j } ( t ) = - \\sum _ { 1 \\leq k \\leq N : k \\neq j } S ^ { - 1 } _ { j k } ( t ) d S _ { k j } ( t ) , 1 \\leq j \\leq N . \\end{align*}"}
+{"id": "418.png", "formula": "\\begin{align*} \\| [ u ] _ { \\ell , m } \\| _ { L ^ 2 ( \\R ^ d ) } = \\| u \\| _ { L ^ 2 ( \\R _ + , r ^ { d - 1 + 2 \\ell } d r ) } , \\end{align*}"}
+{"id": "4285.png", "formula": "\\begin{align*} M _ { r } & = \\sum _ { \\pi \\in \\mathcal { P } _ 2 ( 2 r ) } \\int _ { [ 0 , 1 ] \\times [ - m , m ] ^ { | S P ( \\pi ) | } \\times [ - \\lambda , 1 ] ^ { | D P ( \\pi ) | } } \\prod _ { s = 1 } ^ { r } \\chi _ { [ 0 , \\lambda ] } \\left ( x _ 0 + \\sum _ { i = 1 } ^ { 2 s - 1 } \\epsilon _ \\pi ( i ) x _ { \\pi ( i ) } \\right ) \\\\ & \\times \\chi _ { [ 0 , 1 ] } \\left ( x _ 0 + \\sum _ { i = 1 } ^ { 2 s } \\epsilon _ \\pi ( i ) x _ { \\pi ( i ) } \\right ) \\prod _ { l = 0 } ^ { r } \\mathrm { ~ d } x _ l , \\end{align*}"}
+{"id": "8505.png", "formula": "\\begin{align*} E : = E _ { 1 } \\cup ( ( 0 , \\tau ) + E _ { 2 } ) \\in \\mathcal { K } ( \\ell ) \\mbox { f o r e v e r y } \\tau \\in \\mathbb { R } ^ { n - 1 } . \\end{align*}"}
+{"id": "2619.png", "formula": "\\begin{align*} \\Theta _ n ( t ) = J _ n ( t ) - M _ n ( t ) \\ , , \\end{align*}"}
+{"id": "1006.png", "formula": "\\begin{align*} \\dfrac { d } { d t } \\langle g ( \\cdot , t ) , \\varphi \\rangle = \\int _ { \\Omega _ - } \\dfrac { \\partial g } { \\partial t } ( x , t ) \\ , \\varphi ( x ) \\ , d x + \\int _ { \\Omega _ + } \\dfrac { \\partial g } { \\partial t } ( x , t ) \\ , \\varphi ( x ) \\ , d x . \\end{align*}"}
+{"id": "1452.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum \\limits _ { 1 \\leq i , j \\leq n } a ^ { i j } \\Big ( \\sum \\limits _ { t \\in I } k _ { i t } \\sigma _ { t } - 2 \\alpha _ i \\Big ) \\Big ( \\sum \\limits _ { s \\in I } k _ { j s } \\sigma _ s - 2 \\alpha _ { j } \\Big ) - 4 \\sum \\limits _ { 1 \\leq i , j \\leq n } a ^ { i j } \\alpha _ i \\alpha _ j - 4 \\sum \\limits _ { i \\in I } \\sigma _ { i } = 0 . \\end{aligned} \\end{align*}"}
+{"id": "8364.png", "formula": "\\begin{align*} J = \\left ( \\begin{array} { c c } A & 0 \\\\ 0 & B \\\\ \\end{array} \\right ) . \\end{align*}"}
+{"id": "7141.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } m _ L ( r ) = 0 , r ^ { - 1 } m _ L ( r ) \\in L ^ 1 [ 1 , \\infty ) . \\end{align*}"}
+{"id": "1499.png", "formula": "\\begin{align*} & \\langle A _ { i } , A _ { j } \\rangle _ { t } = 0 , \\ ; \\ , i \\neq j \\ , \\ , \\{ 1 , \\ldots , n \\} \\\\ & \\langle A _ { i } , X \\rangle _ { t } = 0 , \\ ; \\ , i = 1 , \\ldots n . . \\end{align*}"}
+{"id": "5270.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { j = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { t } ( x _ { j , t } ) ] _ + \\| & \\le \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { i , t } ( x _ { j , t } ) ] _ + \\| \\\\ & = \\frac { 1 } { n } \\sum _ { t = 1 } ^ T \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n \\| [ g _ { i , t } ( x _ { i , t } ) ] _ + + [ g _ { i , t } ( x _ { j , t } ) ] _ + - [ g _ { i , t } ( x _ { i , t } ) ] _ + \\| \\\\ & \\le \\frac { 1 } { n } \\sum _ { t = 1 } ^ T \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ n ( \\| [ g _ { i , t } ( x _ { i , t } ) ] _ + \\| + G _ 2 \\| x _ { i , t } - x _ { j , t } \\| ) , \\end{align*}"}
+{"id": "5845.png", "formula": "\\begin{align*} w _ 0 ( \\alpha _ 4 ) & = s _ { \\alpha _ 2 + 2 \\alpha _ 3 + 2 \\alpha _ 4 } ( \\alpha _ 2 + 2 \\alpha _ 3 + \\alpha _ 4 ) = \\\\ & ( \\alpha _ 2 + 2 \\alpha _ 3 + \\alpha _ 4 ) - ( \\alpha _ 2 + 2 \\alpha _ 3 + 2 \\alpha _ 4 ) = - \\alpha _ 4 , \\\\ \\end{align*}"}
+{"id": "3467.png", "formula": "\\begin{align*} \\phi ^ { N A } = u ( - \\log | F _ 0 | , \\ldots - \\log | F _ m | ) , \\end{align*}"}
+{"id": "1547.png", "formula": "\\begin{align*} K _ X \\equiv \\sum _ { i = 1 } ^ k ( m _ i - 1 ) F _ i . \\end{align*}"}
+{"id": "5885.png", "formula": "\\begin{align*} | E _ { 1 } Z ^ { n } | = | \\Delta | , \\end{align*}"}
+{"id": "8562.png", "formula": "\\begin{align*} \\mathcal { M } _ m ( L ) = \\sup \\limits _ { | \\alpha | \\leq \\lceil \\frac { d } { 2 } \\rceil + 1 } \\sup \\limits _ { | \\gamma | \\leq \\lceil \\frac { d } { 2 } \\rceil + 1 } \\sup \\limits _ { ( X , \\xi ) \\in \\R ^ d \\times \\R ^ d } \\Big { \\{ } \\langle \\xi \\rangle ^ { - ( m - | \\gamma | ) } | \\partial _ X ^ { \\alpha } \\partial _ { \\xi } ^ { \\gamma } L ( X , \\xi ) | \\Big { \\} } . \\end{align*}"}
+{"id": "4453.png", "formula": "\\begin{align*} \\Sigma _ 1 ( t ) = \\tilde { \\Sigma } _ 1 ( t ) + J _ 0 ( t ) \\ , , \\end{align*}"}
+{"id": "441.png", "formula": "\\begin{align*} \\Psi _ \\zeta ( \\eta ) : = \\frac { 2 ^ \\alpha \\Gamma \\left ( \\frac { 2 \\zeta + 1 - \\eta } { 2 } \\right ) \\Gamma \\left ( \\frac { \\alpha + \\eta } { 2 } \\right ) } { \\Gamma \\left ( \\frac { \\eta } { 2 } \\right ) \\Gamma \\left ( \\frac { 2 \\zeta + 1 - \\eta - \\alpha } { 2 } \\right ) } . \\end{align*}"}
+{"id": "8552.png", "formula": "\\begin{align*} P ( \\widetilde { E } ; J \\times \\mathbb { R } ^ { n - 1 } ) = P ( F _ { \\ell } ; J \\times \\mathbb { R } ^ { n - 1 } ) . \\end{align*}"}
+{"id": "4915.png", "formula": "\\begin{align*} & ( 1 ) \\alpha _ k ( 0 ) = 0 = \\gamma _ { k } , ( 2 ) \\displaystyle \\sum _ { \\ell = 0 } ^ { k - 1 } \\dfrac { \\alpha _ k ( \\ell ) } { 4 ^ \\ell } + \\sum _ { \\ell = k } ^ { 2 k } \\dfrac { \\beta _ k ( \\ell ) } { 4 ^ \\ell } + \\dfrac { \\gamma _ k } { 2 ^ { 4 k + 3 } } = 0 , \\\\ & ( 3 ) \\displaystyle \\sum _ { \\ell = 0 } ^ { k - 1 } ( - 1 ) ^ \\ell \\dfrac { \\alpha _ k ( \\ell ) } { 4 ^ { \\ell } } + \\sum _ { \\ell = k } ^ { 2 k } ( - 1 ) ^ \\ell \\dfrac { \\beta _ k ( \\ell ) } { 4 ^ \\ell } - \\dfrac { \\gamma _ k } { 2 ^ { 4 k + 2 } } = 0 . \\end{align*}"}
+{"id": "4465.png", "formula": "\\begin{align*} | | \\mathbf { F } | | _ { s , \\ast , T } \\leq C ( K ) \\Big ( | | \\mathbf { f } | | _ { s + 2 , \\ast , T } + | | \\mathbf { g } | | _ { H ^ { s + 2 } ( \\Gamma _ T ) } + ( | | \\mathbf { f } | | _ { 8 , \\ast , T } + | | \\mathbf { g } | | _ { H ^ 8 ( \\Gamma _ T ) } ) | | \\hat { \\mathbf U } , \\hat { \\Psi } | | _ { s + 2 , \\ast , T } \\Big ) . \\end{align*}"}
+{"id": "6201.png", "formula": "\\begin{align*} B ( f , g ) ( x , y ) = \\begin{cases} 0 , & x = y , \\\\ \\sigma ( x , y ) [ f , g ] ( x , y ) , & x < y , \\end{cases} \\end{align*}"}
+{"id": "4067.png", "formula": "\\begin{align*} Y = B _ { \\alpha , \\beta } X \\end{align*}"}
+{"id": "4500.png", "formula": "\\begin{align*} \\tilde { e } '' _ k : = \\mathcal { B } ' ( { \\mathbf V } _ k | _ { x _ 1 = 0 } , \\psi _ k ) ( \\delta { \\mathbf V } _ k | _ { x _ 1 = 0 } , \\delta \\psi _ k ) - \\mathcal { B } ' ( S _ { \\theta _ k } { \\mathbf V } _ k | _ { x _ 1 = 0 } , S _ { \\theta _ k } \\psi _ k ) ( \\delta { \\mathbf V } _ k | _ { x _ 1 = 0 } , \\delta \\psi _ k ) . \\end{align*}"}
+{"id": "3740.png", "formula": "\\begin{align*} ( \\lambda I - \\mathbb { A } ) { \\bf u } = 0 . \\end{align*}"}
+{"id": "5688.png", "formula": "\\begin{align*} k _ { e _ 0 } = \\lim \\limits _ { z \\rightarrow 0 } \\frac { \\ln \\| ( z - T ) ^ { - 1 } e _ 0 \\| _ p } { \\ln \\| ( z - T ) ^ { - 1 } \\| } = 1 . \\end{align*}"}
+{"id": "932.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 y } \\bigg ( - \\frac { m } { \\sqrt { m n } } A + B \\bigg ) y ^ { 1 + \\sqrt { m n } - c } \\\\ ~ ~ = & - \\frac { m x ^ { 1 + \\sqrt { m n } - c } } { \\sqrt { m n } } \\frac { \\alpha } { 4 t ^ \\alpha } \\mathrm { e } ^ { - \\frac { \\alpha ( x ^ 2 + y ^ 2 ) } { t ^ \\alpha } } \\bigg ( \\frac y x \\bigg ) ^ { \\frac { 1 + \\sqrt { m n } - c } { 2 } } I _ { \\frac { 1 + \\sqrt { m n } - c } { 2 } } \\bigg ( \\frac { \\alpha \\sqrt { x y } } { 2 t ^ \\alpha } \\bigg ) , \\end{align*}"}
+{"id": "3913.png", "formula": "\\begin{align*} h _ K ( x ) = \\max \\{ x \\cdot y : y \\in K \\} . \\end{align*}"}
+{"id": "6831.png", "formula": "\\begin{align*} a _ n : = \\frac { ( n - 6 ) ( n + 4 ) } { 4 n ( n - 1 ) } , b _ n : = \\frac { ( n - 4 ) ( n + 2 ) } { 4 n ( n - 1 ) } , { \\rm a n d } c _ n : = \\frac { n - 2 } { 4 ( n - 1 ) } \\end{align*}"}
+{"id": "2126.png", "formula": "\\begin{align*} \\hat p ( t , x ) = \\tilde p ( t , x ) . \\end{align*}"}
+{"id": "6576.png", "formula": "\\begin{align*} K _ 1 = O ( \\tilde N ) . \\end{align*}"}
+{"id": "3204.png", "formula": "\\begin{align*} s _ k = A ^ T ( b - A x _ k ) = - \\theta _ { k + 1 } \\phi _ k v _ { k + 1 } , \\end{align*}"}
+{"id": "3292.png", "formula": "\\begin{align*} \\sigma ^ 2 = \\frac { d ^ 2 } { d t ^ 2 } \\phi _ X ( 0 ) ( - \\mu ) ^ 2 - \\left [ \\frac { d } { d t } \\phi _ X ( 0 ) ( - \\mu ) \\right ] ^ 2 = \\left [ \\frac { d } { d t } \\phi _ X ( 0 ) \\right ] ^ 2 - \\frac { d ^ 2 } { d t ^ 2 } \\phi _ X ( 0 ) . \\end{align*}"}
+{"id": "5103.png", "formula": "\\begin{align*} d _ { n } ^ i ( g _ n , \\cdots , g _ 1 , x ) = \\begin{cases} ( g _ n , \\ldots , g _ 2 , g _ 1 x ) & \\mbox { i f } i = 0 \\\\ ( g _ n , \\ldots , g _ { i + 1 } g _ i , \\ldots , g _ 1 , x ) & \\mbox { i f } 0 < i < n \\\\ ( g _ { n - 1 } , \\ldots , g _ 1 , x ) & \\mbox { i f } i = n \\end{cases} \\end{align*}"}
+{"id": "2622.png", "formula": "\\begin{align*} K ( x , t ) = - \\int _ 0 ^ x \\frac { K ( y , t ) } { 1 - y } \\ , d y + B ( x , t ) \\ , , \\ ; \\ ; t \\in \\R _ + \\ , , x \\in [ 0 , 1 ] \\ , . \\end{align*}"}
+{"id": "3929.png", "formula": "\\begin{align*} \\lambda ( \\{ x \\mid f ( x ) \\neq g ( x ) \\} ) = 0 \\end{align*}"}
+{"id": "7872.png", "formula": "\\begin{align*} E ( t ) : = E ( \\mu ( \\cdot , t ) ) = \\frac 1 2 \\int _ 0 ^ 1 \\int _ M | \\N \\phi ( s , t ) | ^ 2 \\rho ( s , t ) e ^ { - f } \\ , d V \\ , d s , \\end{align*}"}
+{"id": "7325.png", "formula": "\\begin{align*} \\phi ( [ E ] - [ F ] ) = ( \\dim ( E ) - \\dim ( F ) , \\dim ( E ^ G ) - \\dim ( F ^ G ) ) , \\end{align*}"}
+{"id": "568.png", "formula": "\\begin{align*} \\mathbf v ( t ) = \\mathbf S ( t - S ) \\mathbf u ( S ) + i \\int _ S ^ t \\mathbf S ( t - \\sigma ) \\mathbf N ( \\Theta _ R ^ { [ \\mathbf u , \\mathbf v ] } ( \\sigma ) \\mathbf v ( \\sigma ) ) \\ , d \\sigma \\\\ + i \\int _ S ^ t \\mathbf S ( t - \\sigma ) \\mathbf M ( \\mathbf v ( \\sigma ) ) \\ , d W ( \\sigma ) , \\end{align*}"}
+{"id": "4463.png", "formula": "\\begin{align*} | | \\dot { { \\mathbf U } } | | _ { s , \\ast , T } + | | \\varphi | | _ { H ^ s ( \\Gamma _ T ) } & \\leq C ( K ) T ^ { \\frac { 1 } { 2 } } e ^ { C ( K ) T } \\Big ( | | \\mathbf { F } | | _ { s , \\ast , T } + ( | | \\dot { { \\mathbf U } } | | _ { 6 , \\ast , T } \\\\ & \\quad + | | \\varphi | | _ { H ^ 6 ( \\Gamma _ T ) } + | | \\mathbf { F } | | _ { 4 , \\ast , T } ) | | \\hat { W } | | _ { s + 4 , \\ast , T } \\Big ) . \\end{align*}"}
+{"id": "4112.png", "formula": "\\begin{align*} \\left . \\prod _ { i \\ge 0 } \\left ( \\prod _ { s = - 2 k + 4 i + 1 } ^ { - k + 2 i } ( 2 n + s ) \\prod _ { s = k - 2 i } ^ { 2 k - 4 i - 2 } ( 2 n + s ) \\right ) \\middle / \\prod _ { i = 1 } ^ { k - 1 } ( 2 i + 1 ) ^ { k - i } \\right . . \\end{align*}"}
+{"id": "637.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} d \\psi = ( - \\alpha ^ j \\partial _ j - i M \\beta ) \\psi \\ , d t + i \\phi \\beta \\psi \\ , d t + i \\beta \\psi \\mathfrak K _ 1 \\circ d W , \\\\ d \\phi = \\dot \\phi \\ , d t , \\\\ d \\dot \\phi = ( \\partial _ x ^ 2 - m ^ 2 ) \\phi \\ , d t + \\psi ^ * \\beta \\psi \\ , d t + \\phi \\mathfrak K _ 2 \\circ d W , \\end{gathered} \\right . \\end{align*}"}
+{"id": "4441.png", "formula": "\\begin{align*} | | \\partial _ 2 ( \\dot { H } ^ + _ 2 \\partial _ 1 \\hat { \\Phi } ^ { + } ) | | ^ 2 _ { s - 1 , \\ast , t } & \\leq | | \\dot { H } ^ + _ 2 \\partial _ 1 \\hat { \\Phi } ^ { + } | | ^ 2 _ { s , \\ast , t } \\\\ & \\leq C ( K ) \\Big ( | | { \\mathbf V } | | ^ 2 _ { s , \\ast , t } + | | \\dot { { \\mathbf U } } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ t ) } | | \\hat { W } | | ^ 2 _ { s , \\ast , t } \\Big ) . \\end{align*}"}
+{"id": "7973.png", "formula": "\\begin{align*} \\rho _ c ( A ( f ) ( x ) ) \\circ \\overline { F _ 1 } _ f & = \\overline { F _ 2 } _ f \\circ B ( f ) ( \\rho _ d ( x ) ) , \\end{align*}"}
+{"id": "3271.png", "formula": "\\begin{align*} | f ( x ) | ^ 2 & = \\frac { 1 } { 4 \\pi ^ 2 } \\left | \\int _ { \\mathbb { R } } \\mathcal { F } ^ \\mu ( f ) ( \\xi ) e ^ { \\mu \\xi x } d \\xi \\right | ^ 2 \\\\ & \\leq \\frac { 1 } { 4 \\pi ^ 2 } \\int _ { \\mathbb { R } } ( \\rho + \\xi ^ 2 ) \\left | \\mathcal { F } ^ \\mu ( f ) ( \\xi ) \\right | ^ 2 \\frac { e ^ { 2 \\mu \\xi x } } { \\rho + \\xi ^ 2 } d \\xi . \\end{align*}"}
+{"id": "7184.png", "formula": "\\begin{align*} \\alpha _ j & = x _ { \\mathcal { N } _ { j - 1 } + 1 } x _ { \\mathcal { N } _ { j - 1 } + 2 } \\cdots x _ { \\mathcal { N } _ j } y _ j \\\\ & = ( y _ j + \\beta _ { 1 1 } ^ j y _ 1 + \\beta _ { 2 1 } ^ j y _ 2 + \\cdots + \\beta _ { ( j - 1 ) 1 } ^ j y _ { j - 1 } ) ( y _ j + \\beta _ { 1 2 } ^ j y _ 1 + \\beta _ { 2 2 } ^ j y _ 2 + \\\\ & \\cdots + \\beta _ { ( j - 1 ) 2 } ^ j y _ { j - 1 } ) \\cdots ( y _ j + \\beta _ { 1 n _ j } ^ j y _ 1 + \\beta _ { 2 n _ j } ^ j y _ 2 + \\cdots + \\beta _ { ( j - 1 ) n _ j } ^ j y _ { j - 1 } ) y _ j , \\end{align*}"}
+{"id": "2631.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\limsup _ { n \\to \\infty } P ( \\sup _ { s \\in [ 0 , t ] } \\abs { U _ n ( s , t - s ) } > r ) ^ { 1 / b _ n ^ 2 } = 0 \\\\ \\intertext { a n d } \\lim _ { r \\to \\infty } \\limsup _ { n \\to \\infty } P ( \\sup _ { s \\in [ 0 , t ] } \\abs { M _ n ( s ) } > r ) ^ { 1 / b _ n ^ 2 } = 0 \\ , . \\end{align*}"}
+{"id": "4857.png", "formula": "\\begin{align*} G ^ 2 = ( m + l ^ { - 1 } ) G - ( m l ^ { - 1 } + 1 ) I + l ^ { - 1 } G ^ { - 1 } \\end{align*}"}
+{"id": "4883.png", "formula": "\\begin{align*} F _ + ( z ) E _ + ( z ) & = E _ + ( z ) F _ + ( z ) \\\\ F _ + ( z ) E _ - ( z ) & = E _ - ( z ) F _ + ( z ) \\\\ F _ - ( z ) E _ + ( z ) & = E _ + ( z ) F _ - ( z ) \\\\ F _ - ( z ) E _ - ( z ) & = E _ - ( z ) F _ - ( z ) . \\end{align*}"}
+{"id": "6457.png", "formula": "\\begin{align*} y ' ( t ) = A y ( t ) \\end{align*}"}
+{"id": "7385.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } } H _ { n } ( \\rho _ { n } ( t ) ) - H _ { n } ( \\rho _ { n } ^ { 0 } ) ~ d x + \\| \\sqrt { \\rho _ { n } } \\partial _ { x } p _ { n } ( \\rho _ { n } ) \\| ^ { 2 } _ { L ^ { 2 } _ { t } L ^ { 2 } _ { x } } = \\int _ { 0 } ^ { t } \\int _ { \\mathbb { T } } ( \\partial _ { x } p _ { n } ( \\rho _ { n } ) ) \\rho _ { n } w _ { n } ~ d x d s . \\end{align*}"}
+{"id": "4063.png", "formula": "\\begin{align*} ( q _ 1 ( 0 ) + ( - 1 ) ^ { n + 1 } q _ 1 ( 1 ) ) \\sin n \\pi a = \\eta _ n + o ( 1 ) , \\ \\ | n | \\rightarrow \\infty . \\end{align*}"}
+{"id": "5248.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { t = 1 } ^ T \\| [ g _ { t } ( x _ { i , t } ) ] _ + \\| = \\mathcal { O } ( T ^ { 1 - c } ) . \\end{align*}"}
+{"id": "2357.png", "formula": "\\begin{align*} V _ \\tau W _ f = o ( G ) f ( \\tau ) I _ { \\mathbb { C } ^ { o ( G ) } } . \\end{align*}"}
+{"id": "3121.png", "formula": "\\begin{align*} \\mathbb { E } [ \\| \\mathbf { H } _ k \\| ^ 2 ] = \\mathbb { E } [ \\| \\mathbf { H } _ { k , } \\| ^ 2 ] + \\mathbb { E } [ \\| \\mathbf { H } _ { k , } \\| ^ 2 ] + \\mathbb { E } [ \\mathbf { H } _ { k , } \\cdot \\mathbf { H } ^ H _ { k , } ] + \\mathbb { E } [ \\mathbf { H } _ { k , } \\cdot \\mathbf { H } ^ H _ { k , } ] . \\end{align*}"}
+{"id": "3581.png", "formula": "\\begin{align*} { \\xi ^ { \\star } _ { k , { l } , { j } } } = { \\rho _ k } { \\alpha _ { { k , { \\rm { R } } } , { l } } } { \\alpha _ { { \\rm { T } } , k , { j } } } . \\end{align*}"}
+{"id": "3346.png", "formula": "\\begin{align*} v = \\biggl ( \\prod _ { i = 1 } ^ { N } h _ i \\biggr ) \\cdot \\frac { a _ { \\zeta ^ q } ( \\tilde { \\theta } ) } { a _ \\zeta ( \\theta ) ^ p } \\in H ^ \\times \\end{align*}"}
+{"id": "6917.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow + \\infty } \\psi ( \\xi _ n ) = \\psi _ - = \\psi _ + < \\phi _ - \\lim \\limits _ { n \\rightarrow + \\infty } \\psi ' ( \\xi _ n ) = 0 . \\end{align*}"}
+{"id": "7134.png", "formula": "\\begin{align*} \\| \\Psi \\| _ { \\mathcal H } ^ 2 = \\int _ 0 ^ \\infty ( \\Psi ^ { ( 3 ) } ( Z ) ^ 2 W ( r Z ) \\ : d Z + \\int _ 0 ^ \\infty \\Psi ^ 2 ( Z ) \\left ( \\frac { W ( r Z ) } { Z ^ 2 } + r \\frac { W ' ( r Z ) } { Z } \\right ) d Z , \\end{align*}"}
+{"id": "2943.png", "formula": "\\begin{align*} 0 = \\delta _ { i _ 1 i _ 2 } [ \\sum _ { \\lambda = 1 , \\pi \\in S _ { n } } ^ { n ! } a \\delta _ { k \\pi ( i _ 1 } D _ { i _ 2 \\cdots i _ { n } ) } + \\sum _ { \\lambda = 1 , \\pi \\in S _ { n } } ^ { n ! } b \\delta _ { \\pi ( i _ 1 i _ 2 } D _ { i _ 3 \\cdots i _ n ) k } ] . \\end{align*}"}
+{"id": "8009.png", "formula": "\\begin{align*} \\begin{pmatrix} \\begin{array} { l } t \\\\ X ( t ) \\end{array} \\end{pmatrix} = g \\Bigl ( T _ { ( \\cdot ) } ( t ) \\Bigr ) = \\begin{pmatrix} \\begin{array} { l } 1 ^ T \\\\ \\mathrm { V } \\end{array} \\end{pmatrix} T _ { ( \\cdot ) } ( t ) \\end{align*}"}
+{"id": "7738.png", "formula": "\\begin{align*} \\begin{array} { l l l l l l l l l l l } 3 5 \\ge \\omega ( C ) & = 4 | E _ { g = ( 0 , 0 ) } \\cap E ( C ) | \\\\ & \\ge 4 ( \\Omega + 4 ) \\\\ \\end{array} \\end{align*}"}
+{"id": "5977.png", "formula": "\\begin{align*} \\int _ { \\Gamma _ { \\varepsilon , a } } h _ x \\left ( F ^ { \\parallel } ( x ) , \\frac { \\exp _ { x ; h } ( y ) } { | \\exp _ { x ; h } ( y ) | _ h } \\right ) v ( y ) d \\mu _ h ( y ) = \\varepsilon ^ 2 R _ { F , a } \\tilde v ( t ' ) + \\varepsilon ^ 3 \\mathcal { A } _ \\varepsilon \\tilde v ( t ' ) , \\end{align*}"}
+{"id": "1589.png", "formula": "\\begin{align*} \\| A f \\| ^ 2 = \\langle f , \\mathcal { T } \\Pi A f \\rangle & - \\| \\mathcal { T } \\Pi A f \\| ^ 2 \\leq \\| ( 1 - \\Pi ) f \\| \\| \\mathcal { T } \\Pi A f \\| - \\| \\mathcal { T } \\Pi A f \\| ^ 2 \\\\ & \\| A f \\| ^ 2 + \\| \\mathcal { T } \\Pi A f \\| ^ 2 \\lesssim \\frac { \\| ( 1 - \\Pi ) f \\| ^ 2 } { 2 \\varepsilon } + \\frac { \\varepsilon } { 2 } \\| \\mathcal { T } \\Pi A f \\| ^ 2 \\end{align*}"}
+{"id": "4955.png", "formula": "\\begin{align*} \\frac { \\phi ( n ) } { n } = \\sum _ { d | n } \\frac { \\mu ( d ) } { d } = \\sum _ { d | n , ~ 3 \\nmid d } \\frac { \\mu ( d ) } { d } - \\sum _ { 3 d ' = d | n , 3 \\nmid d ' } \\frac { \\mu ( d ' ) } { 3 d ' } = \\frac { 2 } { 3 } \\sum _ { d | n , ~ 3 \\nmid d } \\frac { \\mu ( d ) } { d } . \\end{align*}"}
+{"id": "5582.png", "formula": "\\begin{align*} | B _ { t + 1 } ( S ) | \\geq | S | ( 1 + ( 1 - o ( 1 ) ) \\frac { d } { 4 } ) ^ { t + 1 } = | S | ( \\frac { d } { 4 + o ( 1 ) } ) ^ { t + 1 } . \\end{align*}"}
+{"id": "5730.png", "formula": "\\begin{align*} \\Delta u = V ( x ) u , \\end{align*}"}
+{"id": "8908.png", "formula": "\\begin{align*} p ' , p ' _ 1 : = \\sigma ( p ' ) , \\ldots , p ' _ { d - 1 } = p ' _ { 2 k } : = \\sigma ^ { d - 1 } ( p ' ) , \\end{align*}"}
+{"id": "7924.png", "formula": "\\begin{align*} d _ 2 ^ 2 = m _ 1 d _ 1 + m _ 2 d _ 2 \\end{align*}"}
+{"id": "3207.png", "formula": "\\begin{align*} A A ^ T y = b , \\end{align*}"}
+{"id": "4590.png", "formula": "\\begin{align*} \\mathfrak { V } = \\bigoplus _ { n \\in \\mathbb { Z } } \\mathfrak { V } _ n \\mathfrak { V } _ n = \\{ v \\ , { \\in } \\ , \\mathfrak { V } \\ , \\vert \\ , L _ 0 v \\ , { = } \\ , n v \\} \\ , . \\end{align*}"}
+{"id": "8292.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d t } V _ 1 ( \\bar u , \\bar u _ t ) & \\leq - \\mu V _ 1 ( \\bar u , \\bar u _ t ) - q ( a , \\mu ) \\abs { \\bar u _ t ( 1 , t ) } ^ 2 + ( 2 \\sinh ( \\mu ) + 4 \\cosh { \\mu } ) \\left ( \\tfrac { b d } { c } \\right ) ^ 2 \\abs { \\bar u ( 1 , t ) } ^ 2 , \\end{aligned} \\end{align*}"}
+{"id": "6435.png", "formula": "\\begin{align*} \\displaystyle \\tilde { u } _ { n } ( x , t ) = u ( x + c \\tau _ { n } , \\tau _ n + t ) , \\quad \\tilde { v } _ { n } ( x , t ) = v ( x + c \\tau _ { n } , \\tau _ n + t ) , \\end{align*}"}
+{"id": "2685.png", "formula": "\\begin{align*} \\mathcal { D } _ G ( F ) & = \\prod _ { x ^ 3 = 1 , y ^ 3 = 1 } f ( x , y ) f ( x ^ { - 1 } , y ^ { - 1 } ) - g ( x , y ) g ( x ^ { - 1 } , y ^ { - 1 } ) . \\end{align*}"}
+{"id": "375.png", "formula": "\\begin{align*} \\eta [ T , \\psi _ 1 ] \\eta ' = \\sum _ { \\substack { j \\in \\Z _ { \\geq 0 } ^ 2 } } 2 ^ { - j _ 2 } E _ j , \\end{align*}"}
+{"id": "891.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\mathcal { T } _ \\tau ^ \\alpha u = u _ { y y } + \\frac { q } { ( q - 2 ) y } u _ y + a ' \\frac { 2 } { 2 - q } y ^ { \\frac { q } { q - 2 } } v _ y , \\\\ & \\mathcal { T } _ \\tau ^ \\alpha v = v _ { y y } + \\frac { q } { ( q - 2 ) y } v _ y + b ' \\frac { 2 } { 2 - q } y ^ { \\frac { q } { q - 2 } } u _ y . \\end{aligned} \\right . \\end{align*}"}
+{"id": "1292.png", "formula": "\\begin{align*} \\| v _ { n , T } ( 0 ) - \\phi _ n \\| _ { \\dot { H } ^ 1 } & = \\| [ e ^ { - i t T \\Delta } \\chi _ n P _ n e ^ { i T \\Delta } - 1 ] \\phi \\| _ { \\dot { H } ^ 1 } \\\\ & = \\| [ \\chi _ n P _ n - 1 ] e ^ { i T \\Delta } \\phi \\| _ { \\dot { H } ^ 1 } \\to 0 \\end{align*}"}
+{"id": "6530.png", "formula": "\\begin{align*} A = \\sup _ { x \\in I } \\sup _ { 1 \\leq l \\leq r + 1 } \\left | \\frac { d ^ l f } { d x ^ l } ( x ) \\right | < \\infty . \\end{align*}"}
+{"id": "5788.png", "formula": "\\begin{align*} s _ { \\alpha } ( \\beta ) = \\alpha - ( \\alpha , \\beta ) \\beta = \\alpha + \\beta \\end{align*}"}
+{"id": "146.png", "formula": "\\begin{align*} K _ { E ' } c _ i ' \\pi _ \\mu ' c _ j '^ { - 1 } K _ { E ' } & = K _ { E ' } \\sigma ' ( a _ i ' ) \\pi _ \\mu ' \\sigma ' ( d _ j '^ { - 1 } ) K _ { E ' } \\\\ & = K _ { E ' } \\sigma ' ( a _ i ' ) \\sigma ' ( \\pi _ \\mu ' ) \\sigma ' ( a _ j '^ { - 1 } ) K _ { E ' } , \\end{align*}"}
+{"id": "5515.png", "formula": "\\begin{align*} D _ A ^ { - 1 } O _ B ^ { ( 1 ) } ( p / s , s ) O _ B ^ { ( 1 ) } ( p / s , s ) D _ A O _ A ^ { ( 1 ) } = O _ A ^ { ( 1 ) } O _ B ^ { ( 1 ) } ( p , s ) O _ B ^ { ( 1 ) } ( p , s ) \\end{align*}"}
+{"id": "214.png", "formula": "\\begin{align*} { Q _ 1 } ( M , P _ i , - \\frac { 1 } { \\tau } ) = 2 ^ { 6 } \\tau ^ { 1 0 } { Q _ 2 } ( M , P _ i , \\tau ) . \\end{align*}"}
+{"id": "7256.png", "formula": "\\begin{align*} P _ { s , \\omega , N } \\left ( g \\right ) ( x ) : = \\sum _ { I \\in X _ N ^ { ( \\omega ) } } \\eta ^ { ( \\omega ) } ( I ) e ^ { 2 \\pi s c ( I , x ) } g \\circ f _ I ( x ) , \\eta ^ { ( \\omega ) } ( I ) : = \\prod _ { n = 1 } ^ { | I | } \\textbf { p } ^ { ( \\omega _ n ) } _ { I _ n } . \\end{align*}"}
+{"id": "2686.png", "formula": "\\begin{align*} A : & = f ( 1 , 1 ) ^ 2 - g ( 1 , 1 ) ^ 2 = \\left ( f ( 1 , 1 ) + g ( 1 , 1 ) \\right ) ( f ( 1 , 1 ) - g ( 1 , 1 ) ) , \\\\ B _ 1 : & = f ( 1 , \\omega ) f ( 1 , \\omega ^ 2 ) - g ( 1 , \\omega ) g ( 1 , \\omega ^ 2 ) , \\\\ B _ 2 : & = f ( \\omega , 1 ) f ( \\omega ^ 2 , 1 ) - g ( \\omega , 1 ) g ( \\omega ^ 2 , 1 ) , \\\\ B _ 3 : & = f ( \\omega , \\omega ) f ( \\omega ^ 2 , \\omega ^ 2 ) - g ( \\omega , \\omega ) g ( \\omega ^ 2 , \\omega ^ 2 ) , \\\\ B _ 4 : & = f ( \\omega , \\omega ^ 2 ) f ( \\omega ^ 2 , \\omega ) - g ( \\omega , \\omega ^ 2 ) g ( \\omega ^ 2 , \\omega ) . \\end{align*}"}
+{"id": "5681.png", "formula": "\\begin{align*} p \\ln \\| & ( z - T ) ^ { - 1 } e _ n \\| _ p = \\ln \\varphi _ n ( | z | ) + p \\ln \\| ( z - T ) ^ { - 1 } e _ 0 \\| _ p + \\ln \\Big ( 1 - \\| ( z - T ) ^ { - 1 } e _ 0 \\| _ p ^ { - p } \\psi _ n ( | z | ) \\Big ) . \\end{align*}"}
+{"id": "6836.png", "formula": "\\begin{align*} P _ { \\rm r a d } : = \\partial _ t ^ { ( 6 ) } - K _ { 4 } \\partial _ t ^ { ( 4 ) } + K _ { 2 } \\partial _ t ^ { ( 2 ) } - K _ { 0 } \\end{align*}"}
+{"id": "6850.png", "formula": "\\begin{align*} \\mathbb S _ { T } \\times \\mathbb S ^ { n - 1 } \\simeq \\overline { \\mathcal { I } } _ T \\times \\mathbb S ^ { n - 1 } : = \\mathcal { C } ^ n _ { T } . \\end{align*}"}
+{"id": "2053.png", "formula": "\\begin{align*} T _ \\phi u ( x ' , 0 ) = u ( x ' , 0 + \\phi ( x ' ) ) = S _ \\phi [ u _ { | _ { \\partial \\Omega } } ] ( x ' ) \\end{align*}"}
+{"id": "3093.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { { N _ { \\rm { R } } } } { \\frac { 1 } { { { N _ { \\rm { R } } } { { \\left | { \\xi _ i ^ \\star } \\right | } ^ 2 } } } } \\approx \\sum \\limits _ { i = 1 } ^ { { N _ { \\rm { R } } } } { \\frac { 1 } { { { N _ { \\rm { R } } } { { \\left | { \\xi _ k ^ \\star } \\right | } ^ 2 } } } } = \\frac { 1 } { { { { \\left | { \\xi _ k ^ \\star } \\right | } ^ 2 } } } . \\end{align*}"}
+{"id": "7064.png", "formula": "\\begin{align*} S _ r ( N ) = \\ , \\frac { 1 } { K Q p _ 1 } \\displaystyle \\int _ { \\mathbb { R } } W ( u ) \\ , \\int _ { \\mathbb { R } } \\ , V _ 1 \\left ( \\frac { \\nu } { K } \\right ) \\ , \\displaystyle \\sum _ { 1 \\leq q \\leq Q } \\frac { g ( q , u ) } { q } \\sideset { } { ^ \\star } { \\displaystyle \\sum } _ { a \\ , \\mathrm { m o d } \\ , q } \\ , \\ , \\sum _ { b \\ , \\mathrm { m o d } \\ , p _ 1 } \\ , S _ 1 ( . . . ) S _ 2 ( . . . ) \\mathrm { d } u \\ , \\mathrm { d } \\nu , \\end{align*}"}
+{"id": "2023.png", "formula": "\\begin{align*} ( \\dot { \\mathfrak { h } } ^ { s _ 0 , p } ( \\Omega ) , \\dot { \\mathfrak { h } } ^ { s _ 1 , p } ( \\Omega ) ) _ { \\theta , q } = \\dot { \\mathfrak { b } } ^ { s } _ { p , q } ( \\Omega ) \\end{align*}"}
+{"id": "6252.png", "formula": "\\begin{align*} D _ 0 = \\{ 0 = \\xi _ 1 - \\xi _ 2 + \\cdots \\} . \\end{align*}"}
+{"id": "2336.png", "formula": "\\begin{align*} ( f _ e \\pi _ { g ^ { - 1 } } ) ( x ) & = f _ e \\left ( \\sum _ { h \\in G } f _ h ( x ) \\tau _ { g ^ { - 1 } h } \\right ) = \\sum _ { h \\in G } f _ h ( x ) f _ e ( \\tau _ { g ^ { - 1 } h } ) = \\sum _ { h \\in G } f _ h ( x ) f _ { g ^ { - 1 } g } ( \\tau _ { g ^ { - 1 } h } ) \\\\ & = \\sum _ { h \\in G } f _ h ( x ) f _ g ( \\tau _ { h } ) = f _ g \\left ( \\sum _ { h \\in G } f _ h ( x ) \\tau _ { h } \\right ) = f _ g ( x ) , \\forall g \\in G , \\forall x \\in \\mathcal { X } . \\end{align*}"}
+{"id": "2187.png", "formula": "\\begin{align*} A _ { v , w } & = H _ f ( 1 _ { G w } ) ( v ) \\\\ & = f ( v ) ^ { - 1 } \\sum _ { y \\in X } b ( v , y ) ( f ( v ) 1 _ { G w } ( v ) - f ( y ) 1 _ { G w } ( y ) ) + c ( v ) 1 _ { G w } ( v ) \\\\ & = - f ( v ) ^ { - 1 } \\sum _ { y \\in G w } f ( y ) b ( v , y ) \\leq 0 . \\end{align*}"}
+{"id": "1893.png", "formula": "\\begin{align*} w - \\sqrt { \\epsilon } \\ , \\partial _ x u = 0 . \\end{align*}"}
+{"id": "1598.png", "formula": "\\begin{align*} | E | \\ge d \\binom { | V | } { k } - \\mu n ^ k . \\end{align*}"}
+{"id": "2038.png", "formula": "\\begin{align*} S _ { \\phi } u = S _ { \\phi } a + S _ { \\phi } b \\in \\mathrm { L } ^ { p } ( \\mathbb { R } ^ { n - 1 } ) + \\dot { \\mathrm { H } } ^ { 1 , p } ( \\mathbb { R } ^ { n - 1 } ) \\end{align*}"}
+{"id": "676.png", "formula": "\\begin{align*} \\norm { S _ { h ( \\xi ) } ( t ) f } _ { \\widetilde X _ { h ( \\xi ) } ^ { s , b } ( 0 , T ) } = T ^ { 1 / 2 - b } \\norm { f } _ { H ^ s } . \\end{align*}"}
+{"id": "8209.png", "formula": "\\begin{align*} d x _ j ( I _ j ^ a X ) & = d x _ j ( I _ j X - \\frac { a ^ 2 V } { a ^ 2 + V } \\theta ( X ) I _ j T + a ^ 2 d x _ j ( X ) T ) \\\\ & = - \\theta ( X ) + \\frac { a ^ 2 } { a ^ 2 + V } \\theta ( X ) + 0 \\\\ & = - \\frac { V } { a ^ 2 + V } \\theta ( X ) , \\end{align*}"}
+{"id": "7135.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\Psi ^ { ( 3 ) } ( \\Phi W ( r \\cdot ) ) ^ { ( 3 ) } + \\int _ 0 ^ \\infty \\Psi ( Z ) \\frac { ( \\Phi ( Z ) W ( r Z ) ) ' } { Z } \\ : d Z = \\int _ 0 ^ \\infty \\frac { S ( Z ) } { Z } ( \\Phi ( Z ) W ( r Z ) ) ' \\ : d Z . \\end{align*}"}
+{"id": "923.png", "formula": "\\begin{align*} ( k - 1 ) ( k + c ) ( m \\sigma _ 1 x ^ { k - 1 } + \\eta _ { 2 x } ) = 0 . \\end{align*}"}
+{"id": "8882.png", "formula": "\\begin{align*} \\int _ { G / T } f ( y _ 1 , \\ldots , y _ { \\ell } ) = \\int _ { G / T } f ( \\tilde { y } _ 1 , \\ldots , \\tilde { y } _ { \\ell } ) , \\end{align*}"}
+{"id": "5932.png", "formula": "\\begin{align*} A _ { 3 } & = \\min \\{ \\dfrac { R _ { 4 } - S _ { 3 } } { 2 } + e , R _ { 4 } - S _ { 3 } + d [ - a _ { 1 , 4 } b _ { 1 , 2 } ] \\} \\\\ & \\ge \\min \\{ \\dfrac { 1 - 2 e } { 2 } + e , ( 1 - 2 e ) + 2 e \\} = \\dfrac { 1 } { 2 } > d [ a _ { 1 , 3 } b _ { 1 , 3 } ] . \\end{align*}"}
+{"id": "8321.png", "formula": "\\begin{align*} Q _ { \\beta } = & \\{ \\pm ( 3 , 2 ) , \\pm ( 1 , 1 ) , \\pm ( 2 , 2 ) , \\pm ( 2 , 1 ) , \\pm ( 1 , 0 ) , \\pm ( 3 , 1 ) , \\pm ( 0 , 1 ) , \\\\ & \\ \\pm ( 2 , 0 ) , \\pm ( 1 , - 1 ) , \\pm ( 3 , 3 ) , \\pm ( 1 , 2 ) , \\pm ( 2 , 3 ) , \\pm ( 0 , 2 ) , ( 0 , 0 ) \\} \\\\ \\end{align*}"}
+{"id": "7668.png", "formula": "\\begin{align*} W ( t ) = u + c t - \\sum _ { i = 1 } ^ { \\lfloor t \\rfloor } X _ i , \\ , t \\geqslant 0 , \\end{align*}"}
+{"id": "1655.png", "formula": "\\begin{align*} \\mathbb { A } ^ { ( n ) } _ { \\texttt { a } } : = \\{ ( \\xi _ 1 , \\ldots , \\xi _ n ) \\in \\mathbb { R } ^ { n } _ 0 \\mid \\xi _ 1 > \\xi _ 2 > \\cdots > \\xi _ { n } > \\xi _ 1 - 2 \\pi \\} \\end{align*}"}
+{"id": "8852.png", "formula": "\\begin{align*} \\langle \\hat { g } _ { t } , x _ t - x ^ * \\rangle = \\frac { \\norm { x _ t - x ^ * } ^ 2 - \\norm { x _ { t + 1 } - x ^ * } ^ { 2 } } { \\eta _ t } + \\frac { \\eta _ t } { 2 } \\norm { \\hat { g } _ t } ^ { 2 } . \\end{align*}"}
+{"id": "1868.png", "formula": "\\begin{gather*} f ^ { ( n ) } ( \\sigma _ 0 ) = \\frac { n ! } { 2 \\pi i } \\oint _ { | s - \\sigma _ 0 | = r } \\frac { f ( s ) } { ( s - \\sigma _ 0 ) ^ { n + 1 } } \\ , d s \\end{gather*}"}
+{"id": "3665.png", "formula": "\\begin{align*} I _ i ( 0 ) = 1 , \\ , J _ { i j } ( 0 ) = \\delta _ { i j } , \\ , E _ i ( 0 ) = \\lambda _ i ( 0 ) { } D _ { i j } ( 0 ) = \\lambda _ i ( 0 ) \\ , \\delta _ { i j } \\ , . \\end{align*}"}
+{"id": "2486.png", "formula": "\\begin{align*} \\Pr \\left ( T _ { B ^ n , \\overline { a } ^ n } \\underset { n \\rightarrow \\infty } { \\rightarrow } ( \\beta P ' _ A , ( 1 - \\beta ) P '' _ A ) \\right ) = 1 . \\end{align*}"}
+{"id": "1619.png", "formula": "\\begin{align*} U _ { \\mathcal X _ t \\to y _ t } = \\{ u \\in \\mathcal P _ { \\mathcal X _ t } : \\{ \\alpha ^ { 1 } _ { \\mathcal X _ 1 } , \\dots , \\alpha ^ { t - 1 } _ { \\mathcal X _ { t - 1 } } , u \\} \\frac { \\rho } { 2 ^ { t - 1 } } \\prod _ { j \\in [ k ] \\setminus [ t ] } | \\mathcal P _ { \\mathcal X _ j } | \\mathcal A _ { \\mathcal Y } \\} . \\end{align*}"}
+{"id": "5807.png", "formula": "\\begin{align*} \\{ \\varepsilon _ i \\mid i = 1 , 2 , \\dots n \\} . \\end{align*}"}
+{"id": "8471.png", "formula": "\\begin{align*} H _ { x , \\nu } ^ { - } = \\left \\{ y \\in \\mathbb { R } ^ { n } : \\langle ( y - x ) , \\nu \\rangle \\leq 0 \\right \\} . \\end{align*}"}
+{"id": "2082.png", "formula": "\\begin{align*} g = \\alpha \\left ( \\begin{array} { c c } \\cosh { \\Lambda } + \\cos 2 \\phi \\sinh { \\Lambda } & \\sin 2 \\phi \\sinh { \\Lambda } \\\\ \\sin 2 \\phi \\sinh { \\Lambda } & \\cosh { \\Lambda } - \\cos 2 \\phi \\sinh { \\Lambda } \\end{array} \\right ) . \\end{align*}"}
+{"id": "5056.png", "formula": "\\begin{align*} L ^ 2 _ { g , f } ( M ; S ^ 2 T ^ * M ) = N _ { g , f } \\oplus K _ { g , f } \\oplus P _ { g , f } , \\end{align*}"}
+{"id": "7784.png", "formula": "\\begin{align*} \\lim _ { c \\to 0 } \\frac { 1 + t _ { + } ^ { c , n } t } { t - t _ { + } ^ { c , n } } = - t \\ , , \\lim _ { c \\to 0 } \\frac { 1 - t _ { + } ^ { c , n } t } { t - t _ { + } ^ { c , n } } = t . \\end{align*}"}
+{"id": "4113.png", "formula": "\\begin{align*} \\left . \\prod _ { i \\ge 0 } \\left ( \\prod _ { s = - 2 k + 4 i + 1 } ^ { - k + 2 i } ( 2 n + s + 1 ) \\prod _ { s = k - 2 i } ^ { 2 k - 4 i - 2 } ( 2 n + s + 1 ) \\right ) \\middle / \\prod _ { i = 1 } ^ { k - 1 } ( 2 i + 1 ) ^ { k - i } \\right . . \\end{align*}"}
+{"id": "4461.png", "formula": "\\begin{align*} \\mathcal { N } ( T ) & = | | \\mathbf { F } | | ^ 2 _ { s , \\ast , T } + ( | | \\dot { { \\mathbf U } } | | ^ 2 _ { W ^ { 2 , \\infty } _ { \\ast } ( \\Omega _ T ) } + | | \\varphi | | ^ 2 _ { W ^ { 2 , \\infty } ( \\Gamma _ T ) } + | | \\mathbf { F } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ T ) } ) | | \\hat { W } | | ^ 2 _ { s + 4 , \\ast , T } . \\end{align*}"}
+{"id": "5591.png", "formula": "\\begin{align*} \\langle \\delta V , X \\rangle & = \\lim _ { i \\to \\infty } \\langle \\delta V _ i , X \\rangle \\\\ & = 0 \\end{align*}"}
+{"id": "5200.png", "formula": "\\begin{align*} X = X ( \\tau ) = | T \\cap Z | = \\sum _ { z \\in N _ v } X _ z , \\end{align*}"}
+{"id": "5690.png", "formula": "\\begin{align*} \\| ( z - T ) ^ { - 1 } e _ m \\| _ p ^ p = \\| \\sum \\limits _ { i = 0 } ^ { m } \\frac { 1 } { z ^ { i + 1 } } T ^ i e _ m \\| _ p ^ p & = \\| \\frac { e _ m } { z } + \\sum \\limits _ { i = 1 } ^ { m } \\big ( \\frac { 1 } { z ^ { i + 1 } } \\prod \\limits _ { j = 0 } ^ { i - 1 } w _ { m - j } \\big ) e _ { m - i } \\| _ p ^ p \\\\ & = \\frac { 1 } { | z | ^ p } + \\sum \\limits _ { i = 1 } ^ { m } \\Big | \\frac { 1 } { z ^ { i + 1 } } \\prod \\limits _ { j = 0 } ^ { i - 1 } w _ { m - j } \\Big | ^ p . \\end{align*}"}
+{"id": "7472.png", "formula": "\\begin{align*} h _ { a , c } ( \\alpha , \\gamma ) = h _ { b , a } ( \\beta , \\alpha ) = h _ { c , b } ( \\gamma , \\beta ) , a + b + c = \\varpi , \\alpha + \\beta + \\gamma = 0 . \\end{align*}"}
+{"id": "3967.png", "formula": "\\begin{align*} B S = S B = 0 , \\ e _ i e _ j = \\left \\{ \\begin{array} { l l } \\varphi ( e _ i , e _ j ) & \\textrm { i f } i < j , \\\\ 0 & \\textrm { i f } i \\geq j \\end{array} \\right . \\end{align*}"}
+{"id": "2414.png", "formula": "\\begin{align*} \\L _ { 1 } ( E _ \\infty ) & \\geq \\limsup _ { Q \\to \\infty } \\frac { ( \\sum _ { n = N } ^ { Q } \\L _ { 1 } ( E _ n ) ) ^ 2 } { \\sum _ { n , m = N } ^ { Q } \\L _ { 1 } ( E _ n \\cap E _ m ) } \\\\ & \\geq \\limsup _ { Q \\to \\infty } \\frac { ( \\frac { r d _ { \\lambda } ( x ' ) } { 4 } ) ^ 2 \\left ( \\sum _ { n = N } ^ { Q } h ( n ) \\right ) ^ 2 } { r d _ { \\lambda } ( x ' ) \\left ( \\sum _ { n = N } ^ { Q } h ( n ) + \\left ( \\sum _ { n = N } ^ { Q } h ( n ) \\right ) ^ 2 \\right ) } \\\\ & \\gg r d _ { \\lambda } ( x ' ) . \\end{align*}"}
+{"id": "5333.png", "formula": "\\begin{align*} H _ { 2 m } ( x ) & = ( 2 m ) ! \\sum _ { l = 0 } ^ m \\frac { ( - 1 ) ^ { m - l } } { ( 2 l ) ! ( m - l ) ! } ( 2 x ) ^ { 2 l } \\\\ H _ { 2 m + 1 } ( x ) & = ( 2 m + 1 ) ! \\sum _ { l = 0 } ^ m \\frac { ( - 1 ) ^ { m - l } } { ( 2 l + 1 ) ! ( m - l ) ! } ( 2 x ) ^ { 2 l + 1 } . \\end{align*}"}
+{"id": "2700.png", "formula": "\\begin{align*} x _ 1 x _ 2 & + y _ 1 y _ 2 + z _ 1 z _ 2 = x _ 1 x _ 2 + \\frac { ( b w _ 1 + e x _ 1 ) } { a } \\frac { ( b w _ 2 + e x _ 2 ) } { a } + w _ 1 w _ 2 c ^ 2 \\\\ & = \\frac { ( a ^ 2 + e ^ 2 ) } { a ^ 2 } \\left ( x _ 1 + \\frac { b e w _ 1 } { a ^ 2 + e ^ 2 } \\right ) \\left ( x _ 2 + \\frac { b e w _ 2 } { a ^ 2 + e ^ 2 } \\right ) + \\frac { | | \\vec { a } | | ^ 2 } { a ^ 2 + e ^ 2 } \\\\ & = \\frac { w _ 1 w _ 2 | | \\vec { a } | | ^ 2 } { a ^ 2 + e ^ 2 } \\left ( U _ 1 U _ 2 + 1 \\right ) . \\end{align*}"}
+{"id": "3558.png", "formula": "\\begin{align*} & f ( x ^ { k + 1 } ) - f ( x ) \\ = f ( x ^ { k + 1 } ) - f ( x ^ k ) + f ( x ^ k ) - f ( x ) \\\\ \\ \\leq & \\langle \\nabla f ( x ^ k ) , x ^ { k + 1 } - x ^ k \\rangle + \\frac { L } { 2 } \\| x ^ k - x ^ { k + 1 } \\| _ 2 ^ 2 + \\langle \\nabla f ( x ^ k ) , x ^ { k } - x \\rangle = \\langle \\nabla f ( x ^ k ) , x ^ { k + 1 } - x \\rangle + \\frac { L } { 2 } \\| x ^ k - x ^ { k + 1 } \\| _ 2 ^ 2 \\ , \\end{align*}"}
+{"id": "4978.png", "formula": "\\begin{align*} V ( x ) : = [ x , x R ( x ) ] . \\end{align*}"}
+{"id": "6383.png", "formula": "\\begin{align*} \\chi ^ { i } \\otimes 1 _ { \\Bbbk } \\otimes \\chi _ { i } = 1 _ { H } \\otimes \\varepsilon ( \\chi ^ { i } ) \\otimes \\chi _ { i } + \\chi ^ { i } \\otimes 1 _ { \\Bbbk } \\otimes \\chi _ { i } \\end{align*}"}
+{"id": "3616.png", "formula": "\\begin{align*} { \\mathbb E } \\left \\{ { \\left | { \\beta _ { n , { \\rm { R } } , { l _ n } } ^ * { \\beta _ { m , { \\rm { R } } , { l _ m } } } } \\right | } \\right \\} = \\frac { \\pi } { 4 } . \\end{align*}"}
+{"id": "8561.png", "formula": "\\begin{align*} \\mathcal { L } [ X , D ] u ( X ) : = \\mathcal { F } ^ { - 1 } \\big { ( } L ( X , \\xi ) \\hat { u } ( \\xi ) \\big { ) } ( X ) , \\end{align*}"}
+{"id": "7078.png", "formula": "\\begin{align*} S _ 2 ( . . . ) = & \\sum _ { m = 1 } ^ \\infty \\lambda _ f ( m ) e \\left ( - \\frac { ( a + b q ) m } { p _ 1 q } \\right ) v _ 2 ( n ) \\\\ & = \\frac { 1 } { p _ 1 q } \\frac { \\eta _ f ( p _ 2 ) } { \\sqrt { p _ 2 } } \\sum _ { m = 1 } ^ \\infty \\lambda _ { f } ( m ) e \\left ( - m \\frac { \\overline { ( a + b q ) p _ 2 } } { p _ 1 q } \\right ) \\mathcal { V } _ 2 \\left ( \\frac { m } { p _ 2 ( p _ 1 q ) ^ 2 } \\right ) , \\\\ \\end{align*}"}
+{"id": "8700.png", "formula": "\\begin{align*} f = \\phi _ 0 ( u _ 1 , . . . , u _ n ) , ~ ~ ~ x _ i = \\phi _ i ( u _ 1 , . . . , u _ n ) , ~ i = 1 , . . . , n , \\end{align*}"}
+{"id": "6097.png", "formula": "\\begin{align*} \\begin{aligned} E ( \\| g _ { k - 1 } \\| ^ 2 ) & \\leq \\beta ^ { k - 1 } E ( \\| g _ 0 \\| ^ 2 ) \\\\ & \\leq \\beta ^ { k - 1 } \\| E ( g _ 0 ) \\| ^ 2 \\\\ & \\leq \\beta ^ { k - 1 } \\| \\nabla f ( \\omega _ 0 ) \\| ^ 2 \\\\ & \\leq 2 \\beta ^ { k - 1 } L ( f ( \\omega _ 0 ) - f ( \\omega ^ * ) ) \\end{aligned} \\end{align*}"}
+{"id": "3128.png", "formula": "\\begin{align*} \\mathbf { H } _ { k , } ( m ) \\mathbf { H } ^ * _ { k , } ( m ) = \\frac { 1 } { N } \\sqrt { \\frac { \\kappa } { ( \\kappa + 1 ) ^ 2 } } \\sum _ { p = 1 } ^ { L _ k } \\sum _ { n _ 1 = 0 } ^ { N _ 1 - 1 } \\sum _ { n _ 2 = 0 } ^ { N _ 2 - 1 } \\sum _ { n ' = 1 } ^ N \\sum _ { p ' = 1 } ^ { L _ k } \\alpha _ { k , p } \\alpha _ { k , p ' } g ^ * _ { n ' , m } \\chi _ { k , p , n _ 1 \\cdot N _ 2 + n _ 2 } e ^ { - j [ 2 \\pi ( m - 1 ) \\phi ^ { } + \\varphi _ { n ' } + b _ { n ' + 1 } ] } . \\end{align*}"}
+{"id": "7671.png", "formula": "\\begin{align*} \\hat { \\varphi } ( u ) = \\mathbb { P } \\left ( \\sup _ { k \\geqslant 1 } \\sum _ { i = 1 } ^ k ( X _ i - c ) < u \\right ) . \\end{align*}"}
+{"id": "3924.png", "formula": "\\begin{align*} \\omega _ 1 \\tilde + _ t \\omega _ 2 : = \\{ P ( ( 1 - t ) u + t v ) \\mid u \\in \\omega _ 1 , v \\in \\omega _ 2 \\} . \\end{align*}"}
+{"id": "634.png", "formula": "\\begin{align*} \\alpha = \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} , \\beta = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
+{"id": "158.png", "formula": "\\begin{align*} \\sum _ { j \\geq 0 } ( j + 1 ) ^ r x ^ j = \\frac { E _ r ( x ) } { ( 1 - x ) ^ { r + 1 } } . \\end{align*}"}
+{"id": "8843.png", "formula": "\\begin{align*} \\Big ( \\sum _ { t = 1 } ^ { T } \\eta _ t \\Big ) ^ { - 1 } = \\frac { 1 } { \\min \\big ( \\frac { T } { 1 6 \\kappa \\bar { L } d } , T \\gamma \\big ) } = \\max \\Big ( \\frac { 1 8 \\kappa \\bar { L } d } { T } , \\frac { 1 } { T \\gamma } \\Big ) \\leq \\frac { 1 6 \\kappa \\bar { L } d } { T } + \\frac { 1 } { T \\gamma } . \\end{align*}"}
+{"id": "5767.png", "formula": "\\begin{align*} \\lim _ { L \\rightarrow \\infty } { \\mathbb E } \\langle ( m ^ p - \\langle m ^ p \\rangle ) ^ 2 \\rangle = 0 . \\end{align*}"}
+{"id": "624.png", "formula": "\\begin{align*} \\kappa = \\psi - \\chi \\end{align*}"}
+{"id": "580.png", "formula": "\\begin{align*} \\Delta _ 1 ( t ) = i \\int _ S ^ { t } \\mathbf S ( t - s ) \\left [ \\mathbf N ( \\mathbb 1 _ { s \\le \\mu } \\mathbf U ( s ) ) - \\mathbf N ( \\mathbb 1 _ { s \\le \\mu } \\mathbf V ( s ) ) \\right ] \\ , d s . \\end{align*}"}
+{"id": "2750.png", "formula": "\\begin{align*} \\tilde { S } _ m : = \\big \\{ \\sigma ( j _ 1 ) , \\sigma ( j _ 2 ) , \\ldots \\sigma ( j _ m ) \\big \\} \\end{align*}"}
+{"id": "4198.png", "formula": "\\begin{align*} D ( 0 , \\varepsilon ) & = \\sigma _ { \\varepsilon } ( [ i T \\bullet \\varphi ^ { - 1 } ( i I ) , S ] _ { \\ast } ) = \\sigma _ { \\varepsilon } ( [ \\varphi ( i T ) \\bullet i I , \\varphi ( S ) ] _ { \\ast } ) \\\\ & = \\sigma _ { \\varepsilon } ( 2 i ( \\varphi ( i T ) + \\varphi ( i T ) ^ { \\ast } ) ) . \\end{align*}"}
+{"id": "1213.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left ( P ( u _ n ) - P ( u _ n - \\phi _ n ^ 1 ) - P ( \\phi _ n ^ 1 ) \\right ) = 0 , \\end{align*}"}
+{"id": "4440.png", "formula": "\\begin{align*} | | \\tilde { \\mathcal { A } } \\mathcal { A } _ 3 { \\mathbf V } | | ^ 2 _ { s - 1 , \\ast , t } \\leq C ( K ) \\Big ( | | { \\mathbf V } | | ^ 2 _ { s - 1 , \\ast , t } + | | \\dot { { \\mathbf U } } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ t ) } | | \\hat { W } | | ^ 2 _ { s + 1 , \\ast , t } \\Big ) , \\end{align*}"}
+{"id": "2095.png", "formula": "\\begin{align*} \\begin{aligned} & \\sup _ { t \\in [ 0 , T ] } \\norm { ( \\psi , \\partial _ t \\psi ) } _ { \\mathcal { H } } \\\\ & \\leq C \\left ( \\norm { ( \\psi _ 0 , \\psi _ 1 ) } _ { H ^ { 1 } ( \\mathbb { R } ) \\times L ^ { 2 } ( \\mathbb { R } ) } + \\int _ { 0 } ^ { T } \\norm { f } _ { H ^ { k - 1 } ( \\mathbb { R } ) } ( t ) d t \\right ) \\exp \\left ( C \\int _ 0 ^ T \\norm { \\partial a } _ { L ^ { \\infty } ( \\mathbb { R } ) } ( t ) \\right ) . \\end{aligned} \\end{align*}"}
+{"id": "1656.png", "formula": "\\begin{align*} \\mathbb { R } ^ { n } _ 0 : = \\{ ( \\xi _ 1 , \\ldots \\xi _ n ) \\in \\mathbb { R } ^ { n } \\mid \\xi _ 1 + \\cdots + \\xi _ { n } = 0 \\} . \\end{align*}"}
+{"id": "3982.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) u _ n = f ( u _ n , v ) \\ , \\xi _ n + g ( u _ n , v ) - c _ n ( t ) ( f \\partial _ 1 f ) ( u _ n , v ) , \\end{align*}"}
+{"id": "3692.png", "formula": "\\begin{align*} \\mu & = k ( n - k + 1 ) - \\lambda = k ( n - k + 1 ) - k n / m \\\\ & = k ( ( m - 1 ) n / m - k + 1 ) = k ( \\lambda _ 2 ( L ( \\bar { G } ) ) - k + 1 ) . \\end{align*}"}
+{"id": "8154.png", "formula": "\\begin{align*} & H _ i ( x , N , p ) = \\frac { \\binom { N } { i } - \\binom { N } { i - 1 } } { \\binom { p } { x } \\binom { N - p } { x } } E _ x ( i , N , p ) . \\end{align*}"}
+{"id": "5072.png", "formula": "\\begin{align*} E ( A ) = \\sum _ { p \\ge 0 } \\frac { A ^ p } { m ( p ) } = \\sum _ { p \\geq 0 } \\frac { ( C B C ^ { - 1 } ) ^ p } { m ( p ) } = \\sum _ { p \\geq 0 } \\frac { C B ^ p C ^ { - 1 } } { m ( p ) } = C \\sum _ { p \\geq 0 } \\frac { B ^ p } { m ( p ) } C ^ { - 1 } = C E ( B ) C ^ { - 1 } . \\end{align*}"}
+{"id": "3906.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { \\sum _ { k = 1 } ^ { Q _ N } \\norm { \\xi _ k } { 2 } ^ 2 } { \\sigma _ N ^ 2 } = 1 . \\end{align*}"}
+{"id": "6391.png", "formula": "\\begin{align*} \\chi ( \\xi \\otimes \\alpha \\delta \\eta ) & = | \\xi | ( 2 + | \\eta | ) \\varepsilon ( \\xi \\alpha \\delta \\eta ) \\\\ & = | \\xi | ( 2 + | \\eta | ) \\varepsilon ( \\xi \\eta ) \\\\ & = | \\xi | ( 2 + | \\eta | ) \\varepsilon ( \\xi \\delta \\alpha \\eta ) + ( q - q ^ { - 1 } ) | \\xi | ( 2 + | \\eta | ) \\underbrace { \\varepsilon ( \\xi \\beta \\gamma ) } _ { = 0 } \\\\ & = \\chi ( \\xi \\otimes ( \\delta \\alpha + ( q - q ^ { - 1 } ) \\beta \\gamma ) \\eta ) \\end{align*}"}
+{"id": "734.png", "formula": "\\begin{align*} R _ 2 ( \\mu ) = \\norm { ( 1 - P _ \\mu ) \\mathbf u } _ { \\mathbf X ^ { \\mathbf s , b } ( 0 , T ) } \\to 0 . \\end{align*}"}
+{"id": "3287.png", "formula": "\\begin{align*} \\frac { d ^ { \\ell - 1 } } { d t ^ { \\ell - 1 } } \\phi _ X ( t ) = \\int _ \\mathbb { R } x ^ { \\ell - 1 } f _ X ( x ) e ^ { \\mu t x } d x \\mu ^ { \\ell - 1 } . \\end{align*}"}
+{"id": "3776.png", "formula": "\\begin{align*} R ( s ) = s - \\log s - 1 . \\end{align*}"}
+{"id": "3206.png", "formula": "\\begin{align*} \\Delta ^ { \\mathrm { L S Q R } } _ { \\ell : k } \\equiv \\sum _ { j = \\ell } ^ { k } { \\phi } _ { j + 1 } ^ 2 \\approx \\| x - x _ { \\ell } \\| ^ 2 _ { A ^ T A } = \\| r _ { \\ell } \\| ^ 2 - \\| r \\| ^ 2 . \\end{align*}"}
+{"id": "5873.png", "formula": "\\begin{align*} T _ { S L } ^ { 1 } = \\inf \\Big \\{ t : \\max _ { k = t - s + 1 \\in \\mathcal { K } } \\ell ( \\mathbf { p } _ { s t } ) \\geq C _ { \\gamma } \\Big \\} , \\end{align*}"}
+{"id": "847.png", "formula": "\\begin{align*} T f = x ^ * ( f ) h , \\end{align*}"}
+{"id": "7866.png", "formula": "\\begin{align*} \\mu ( \\sigma ^ { - i } S _ k ( r _ n ) \\cap \\sigma ^ { - j } S _ l ( r _ n ) ) & \\leq B ^ { 1 0 } \\sum _ { \\substack { A , B \\in \\mathcal { C } _ r \\\\ C \\in \\mathcal { C } _ { r _ n - r } } } \\mu ( A ) ^ 2 \\mu ( B ) ^ 2 \\mu ( C ) ^ 3 = C ^ { 1 0 } Z _ { r } ( 1 ) ^ 2 Z _ { r _ n - r } ( 2 ) \\\\ & \\preceq B ^ { 1 0 } e ^ { - 4 \\alpha r _ n } e ^ { - 3 \\alpha ( r _ n - r ) } \\leq B ^ { 1 0 } e ^ { - 3 \\alpha r _ n } . \\end{align*}"}
+{"id": "675.png", "formula": "\\begin{align*} \\norm { S _ { h ( \\xi ) } ( t ) f } _ { X _ { h ( \\xi ) } ^ { s , b } ( 0 , T ) } = \\norm { 1 } _ { H ^ b ( 0 , T ) } \\norm { f } _ { H ^ s } \\end{align*}"}
+{"id": "2639.png", "formula": "\\begin{align*} X ^ { ( 0 ) } _ n ( t ) = X '^ { ( 0 ) } _ n ( t , \\frac { Q _ n ( 0 ) } { n } \\wedge 1 ) - ( 1 - F _ 0 ( t ) ) X _ n ( 0 ) ^ - \\ , . \\end{align*}"}
+{"id": "1640.png", "formula": "\\begin{align*} f ' ( r ) = \\frac { f ( r ) } { 2 } \\bigl ( ( p + q ) \\coth ( r / 2 ) + q \\tanh ( r / 2 ) \\bigr ) . \\end{align*}"}
+{"id": "1860.png", "formula": "\\begin{gather*} ( n _ 1 + \\alpha ) ( n _ 2 + \\alpha ) ^ { - 1 } \\prod _ { n = 0 } ^ { N } ( n + \\alpha ) ^ { m _ n } = 1 \\end{gather*}"}
+{"id": "3081.png", "formula": "\\begin{align*} { \\mu ^ \\star } = \\frac { E } { { { N _ { \\rm { R } } } } } + \\sum \\nolimits _ { i = 1 } ^ { { N _ { \\rm { R } } } } { \\frac { { { \\sigma ^ 2 } } } { { { N _ { \\rm { R } } } { { \\left | { \\xi _ i ^ \\star } \\right | } ^ 2 } } } } . \\end{align*}"}
+{"id": "6070.png", "formula": "\\begin{align*} \\partial ^ { \\beta _ 0 } _ x P _ t \\varphi ( x ) = \\mathbb { E } [ \\partial ^ { \\beta _ 0 } _ x ( \\varphi ( X _ t ( x ) ) ] = \\sum _ { \\vert \\alpha _ 0 \\vert \\leq \\vert \\beta _ 0 \\vert } \\mathbb { E } [ ( \\partial ^ { \\alpha _ 0 } \\varphi ) ( X _ t ( x ) ) \\mathbf { P } _ { \\alpha _ 0 } ( x ) ] , \\end{align*}"}
+{"id": "4995.png", "formula": "\\begin{align*} s ( x ) = x \\sum _ { n \\ne 1 } h _ n [ s ( x ) ] = x \\bigl ( h [ s ( x ) ] - s ( x ) \\bigr ) . \\end{align*}"}
+{"id": "6230.png", "formula": "\\begin{align*} \\partial _ \\kappa A _ \\mu ^ n ( x ) = & - \\sum _ { \\nu > \\{ \\mu , \\kappa \\} } \\int _ 0 ^ { x ^ \\nu } \\partial _ \\kappa B _ { \\mu \\nu } ( x ^ 1 , \\ldots , x ^ { \\nu - 1 } , t , 0 , \\ldots , 0 ) d t \\\\ & + \\delta _ { \\mu > \\kappa } B _ { \\mu \\kappa } ( x ^ 1 , \\ldots , x ^ { \\kappa } , 0 , \\ldots , 0 ) \\end{align*}"}
+{"id": "8522.png", "formula": "\\begin{align*} P ( E ) = \\mathcal { H } ^ { n - 1 } ( \\partial ^ { * } F _ { \\ell } \\cap \\{ z < \\bar { z } \\} ) + \\mathcal { H } ^ { n - 1 } ( \\partial ^ { * } E \\cap \\{ z = \\bar { z } \\} ) + \\mathcal { H } ^ { n - 1 } ( \\partial ^ { * } F _ { \\ell } \\cap \\{ z > \\bar { z } \\} ) . \\end{align*}"}
+{"id": "1825.png", "formula": "\\begin{gather*} h _ N ( s ) = \\sum _ { M < n \\leq N } \\frac { | \\left \\langle ( n + c ) ^ { - s } , g ( s ) \\right \\rangle | } { \\left \\langle ( n + c ) ^ { - s } , g ( s ) \\right \\rangle } \\frac { 1 } { ( n + c ) ^ s } \\end{gather*}"}
+{"id": "4002.png", "formula": "\\begin{align*} \\partial _ l | \\nabla \\varphi _ t | ^ 2 = \\sum _ i ( \\varphi _ { t , i } \\varphi _ { t , l \\bar i } + \\varphi _ { t , i l } \\varphi _ { t , \\bar i } ) , \\end{align*}"}
+{"id": "3674.png", "formula": "\\begin{align*} g ( t ) = g _ N ( t ) + \\sum _ i d u _ i ^ 2 \\ , . \\end{align*}"}
+{"id": "4406.png", "formula": "\\begin{align*} \\mathrm { d i v } \\dot { \\mathbf h } ^ { \\natural \\pm } = 0 , ( \\hat { H } ^ { \\pm } _ 2 \\partial _ 2 \\varphi - \\dot { H } ^ { \\natural \\pm } _ N { \\mp } \\varphi \\partial _ 1 \\hat { H } ^ { \\pm } _ N ) | _ { x _ 1 = 0 } = 0 \\end{align*}"}
+{"id": "1261.png", "formula": "\\begin{align*} \\| v _ n \\| _ { S ( \\R ) } \\leq C \\| \\phi \\| _ { \\dot { H } ^ 1 } \\ a n d \\ \\limsup _ { T \\to \\infty } \\lim _ { n \\to \\infty } \\| v _ n - v _ { n , T } \\| _ { S ( \\R ) } = 0 . \\end{align*}"}
+{"id": "5684.png", "formula": "\\begin{align*} { ( z - T ) ^ { - 1 } x } & = \\sum \\limits _ { n = 0 } ^ { \\infty } \\hat { x } _ { n } ( z - T ) ^ { - 1 } e _ n = \\sum \\limits _ { n = n _ 0 } ^ { \\infty } \\hat { x } _ { n } ( z - T ) ^ { - 1 } e _ n \\\\ & = \\sum \\limits _ { n = n _ 0 } ^ { \\infty } \\hat { x } _ { n } \\beta _ { n } ^ { - 1 } z ^ { n - 1 } \\sum \\limits _ { k = n } ^ { \\infty } \\frac { \\beta _ k } { z ^ k } e _ k = \\sum \\limits _ { k = n _ 0 } ^ { \\infty } \\frac { \\beta _ k } { z ^ k } ( \\sum \\limits _ { n = n _ 0 } ^ { k } \\hat { x } _ { n } \\beta _ { n } ^ { - 1 } z ^ { n - 1 } ) e _ k . \\end{align*}"}
+{"id": "6093.png", "formula": "\\begin{align*} \\beta _ k ^ { P R P } = \\frac { g _ k ^ T ( g _ k - g _ { k - 1 } ) } { \\| g _ { k - 1 } \\| ^ 2 } , \\beta _ k ^ { F R } = \\frac { \\| g _ k \\| ^ 2 } { \\| g _ { k - 1 } \\| ^ 2 } \\end{align*}"}
+{"id": "871.png", "formula": "\\begin{align*} u _ t = \\sigma x ^ \\gamma u _ { x x } + f ( x ) u _ x - \\mu x ^ r u , ~ ~ \\sigma > 0 , \\end{align*}"}
+{"id": "8897.png", "formula": "\\begin{align*} f _ * \\big ( ( f ^ * c ) b ( a ) \\big ) = c f _ * \\big ( b ( a ) \\big ) c \\in H ^ * ( M ) . \\end{align*}"}
+{"id": "4282.png", "formula": "\\begin{align*} ( T ^ { ( 1 ) } _ { n \\times p } ) e _ i = \\sum _ { j = 0 } ^ { n - 1 } a ^ { ( 1 ) } _ { j } \\chi _ { [ 1 , n ] } ( i + j ) e _ { i + j } + \\sum _ { j = 1 } ^ { p - 1 } a ^ { ( 1 ) } _ { - j } \\chi _ { [ 1 , n ] } ( i - j ) e _ { i - j } = \\sum _ { j = - ( p - 1 ) } ^ { n - 1 } a ^ { ( 1 ) } _ { j } \\chi _ { [ 1 , n ] } ( i + j ) e _ { i + j } . \\end{align*}"}
+{"id": "1183.png", "formula": "\\begin{align*} & \\phi \\circ m ^ E _ 1 ( ( 0 , x ) , ( 0 , y ) ) = m '^ { E } _ 1 ( \\phi ( 0 , x ) , \\phi ( 0 , y ) ) , \\\\ & \\phi \\circ m ^ E _ 2 ( ( 0 , x ) , ( 0 , y ) ) = m '^ { E } _ 2 ( \\phi ( 0 , x ) , \\phi ( 0 , y ) ) , \\end{align*}"}
+{"id": "8747.png", "formula": "\\begin{align*} H ^ 2 ( \\mathbf { P } _ { \\omega , T } , \\mathbf { P } _ { \\omega ' , T } ) \\leq 2 \\left ( 1 - \\left ( 1 - \\frac { I _ { 0 } r ^ 2 T ^ { - 1 } } { 2 } \\right ) ^ T \\right ) \\enspace . \\end{align*}"}
+{"id": "5797.png", "formula": "\\begin{align*} w _ 1 = s _ 1 , \\ ; w _ 2 = s _ 2 s _ 1 , \\ ; \\dots , \\ ; w _ { n - 1 } = s _ { n - 1 } { \\dots } s _ 2 s _ 1 , \\ ; w _ { n } = s _ { n } { \\dots } s _ 2 s _ 1 . \\end{align*}"}
+{"id": "6309.png", "formula": "\\begin{align*} \\| f _ \\lambda \\| _ { X _ \\lambda ^ * } = \\sup _ { \\mu \\approx \\lambda } \\sup _ { \\| w _ \\mu \\| _ { X _ \\mu } \\leq 1 } \\| f _ \\lambda w _ \\mu \\| _ { L ^ 1 } . \\end{align*}"}
+{"id": "96.png", "formula": "\\begin{align*} b _ k : = a _ k + \\frac { 2 } { \\varepsilon \\vert \\Lambda \\vert } \\sum _ { p \\in \\mathcal { P } _ L } \\ , \\frac { \\widehat { g } ( k ) } { k ^ 2 } z a _ { p - k } ^ { \\dagger } a _ p , \\end{align*}"}
+{"id": "3083.png", "formula": "\\begin{align*} { { { { \\left | { \\tilde \\Theta _ { { k , { \\rm { R } } } } ^ { \\rm { D } } - \\Theta _ { { k , { \\rm { R } } } } ^ { \\rm { D } } } \\right | } _ { \\max } } } } = \\frac { { 2 \\pi } } { { { 2 ^ { { b _ { \\min , k } } + 1 } } } } . \\end{align*}"}
+{"id": "680.png", "formula": "\\begin{align*} ( \\psi _ + , \\psi _ - , \\phi _ + ) ( t ) = ( \\Psi _ + , \\Psi _ - , \\Phi _ + ) ( t ) . \\end{align*}"}
+{"id": "8371.png", "formula": "\\begin{align*} \\tilde Y _ F ^ { 1 } ( \\omega _ 2 , x ) = \\int _ { - \\infty } ^ 0 S _ { B } ( - r ^ \\prime ) \\tilde g _ 1 ( x , \\tilde Y ^ 1 _ F ( \\theta _ { r ^ \\prime } \\omega _ 2 , x ) , \\theta _ { r ^ \\prime } \\omega _ 2 ) d r ^ \\prime . \\end{align*}"}
+{"id": "7780.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } A _ { i j } Z ^ i Z ^ j , A _ { i j } = A _ { j i } \\in \\mathbb { R } \\end{align*}"}
+{"id": "4229.png", "formula": "\\begin{align*} d e ^ 1 = d e ^ 2 = d e ^ 3 = d e ^ 4 = 0 , d e ^ 5 = t \\ , e ^ { 1 3 } - t \\ , e ^ { 2 4 } , d e ^ 6 = t \\ , e ^ { 1 4 } + t \\ , e ^ { 2 3 } , \\end{align*}"}
+{"id": "7456.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ { p } y - { \\rm d i v } ( \\eta \\lvert \\nabla { y } \\rvert ^ { q - 2 } \\nabla y ) = f ( z , y ) , & z \\in \\Omega , \\\\ y = 0 , & z \\in \\partial \\Omega , \\end{cases} \\end{align*}"}
+{"id": "7441.png", "formula": "\\begin{align*} \\psi _ { ( y _ 1 , y _ 2 ) } ^ { \\prime } ( t ) = t ^ { \\kappa _ 1 + \\kappa _ 2 + 1 } \\left [ F _ { ( y _ 1 , y _ 2 ) } ( t ) - \\lambda ( \\kappa _ 1 + \\kappa _ 2 + 2 ) \\int _ { \\Omega } \\vert y _ 1 \\vert ^ { \\kappa _ 1 + 1 } \\vert y _ 2 \\vert ^ { \\kappa _ 2 + 1 } d z \\right ] , \\end{align*}"}
+{"id": "3900.png", "formula": "\\begin{align*} \\frac { 1 } { { \\sigma } _ N } \\norm { \\sum _ { n = 1 } ^ N a _ n f ^ n - \\sum _ { k = 1 } ^ { Q _ N } \\xi _ k } { 2 } \\leq \\varphi ( N ) ^ { 1 / 1 6 } \\end{align*}"}
+{"id": "1687.png", "formula": "\\begin{align*} \\delta ^ { ( m , n ) } _ { \\texttt { b } ; \\mu } ( q ) : = \\prod _ { \\substack { 1 \\leq j < k \\leq n \\\\ \\mu _ j - \\mu _ k = 0 } } \\frac { 1 - q ^ { k - j } } { 1 - q ^ { 1 + k - j } } . \\end{align*}"}
+{"id": "2571.png", "formula": "\\begin{align*} \\Delta = \\{ \\bar { x } \\in \\R ^ n \\mid x _ j \\ge - \\sqrt { N } \\hbox { f o r } j = 1 , \\dots , n , \\sum _ { 1 \\le j \\le n } x _ j \\le \\sqrt { n N } \\} . \\end{align*}"}
+{"id": "2719.png", "formula": "\\begin{align*} H \\coloneqq \\sum _ { i , j \\ge 0 } ( - 1 ) ^ { i + j } \\sum _ { \\substack { P \\subseteq \\{ 2 , 3 , \\dots , n - 1 \\} , \\\\ \\# P = i } } \\sum _ { \\substack { Q \\subseteq \\{ 1 , n \\} , \\\\ \\# Q = j } } \\binom { d - \\sum _ { p \\in P } ( \\alpha _ { p } - 1 ) - \\sum _ { q \\in Q } \\alpha _ { q } } { n - 1 } \\end{align*}"}
+{"id": "7545.png", "formula": "\\begin{align*} 1 : = [ Z ^ s / G ] \\in A _ * ^ { F _ 0 } ( Z ^ s / G ) . \\end{align*}"}
+{"id": "169.png", "formula": "\\begin{align*} E _ 2 ( \\tau + 1 ) = E _ 2 ( \\tau ) , ~ ~ E _ 2 ( - \\frac { 1 } { \\tau } ) = \\tau ^ 2 E _ 2 ( \\tau ) - \\frac { 6 \\sqrt { - 1 } \\tau } { \\pi } , \\end{align*}"}
+{"id": "6271.png", "formula": "\\begin{align*} g ( u ) - g ( u _ { < \\lambda } ) = \\sum _ { \\mu \\gtrsim \\lambda } u _ { \\mu } g ' ( u _ { < \\lambda } ) + \\sum _ { \\mu _ 1 \\geq \\mu _ 2 \\gtrsim \\lambda } u _ { \\mu _ 1 } u _ { \\mu _ 2 } g '' ( u _ { < \\mu _ 2 } ) . \\end{align*}"}
+{"id": "6913.png", "formula": "\\begin{align*} K _ 1 & = \\frac { 1 - b } { 2 } \\left ( \\frac { 1 } { b } - \\frac { 1 - b } { 2 b } e ^ { \\eta \\xi } - 1 \\right ) - \\frac { m e ^ { ( \\lambda _ 1 - \\eta ) \\xi } } { \\underline { \\phi } ( \\xi ) + a e ^ { \\lambda _ 1 \\xi } } . \\end{align*}"}
+{"id": "298.png", "formula": "\\begin{align*} \\gamma _ { \\operatorname * { N } ; j } ^ { \\mathbb { B } , - } u ^ { - } = \\left \\langle \\mathbb { B } _ { j } ^ { - } \\nabla u ^ { - } , \\mathbf { n } _ { j } \\right \\rangle \\quad \\gamma _ { \\operatorname * { N } ; j } ^ { \\mathbb { B } , + } u ^ { + } = \\left \\langle \\mathbb { B } _ { j } ^ { + } \\nabla u ^ { + } , - \\mathbf { n } _ { j } \\right \\rangle . \\end{align*}"}
+{"id": "7482.png", "formula": "\\begin{align*} A _ L = C _ L C _ L ^ \\sigma \\ldots C _ { L } ^ { \\sigma ^ { m - 1 } } . \\end{align*}"}
+{"id": "7610.png", "formula": "\\begin{align*} J _ 1 ( \\beta , N , T , \\hat { \\mathbf { R } } _ i ) : = N \\iint _ { \\hat { \\mathbf { R } } _ i } \\int _ { \\mathbf { B } _ 2 ( 0 ) } f ^ { ( 1 , 1 , \\alpha ) } _ { t , s } ( z ) d z d s d t , i = 1 , 2 , \\end{align*}"}
+{"id": "4204.png", "formula": "\\begin{align*} D ( 0 , \\frac { \\varepsilon } { 2 } ) & = \\sigma _ { \\frac { \\varepsilon } { 2 } } ( [ i P , Q ] _ { \\ast } ) = \\sigma _ { \\frac { \\varepsilon } { 2 } } ( [ \\psi ( i P ) , \\psi ( Q ) ] _ { \\ast } ) \\\\ & = \\sigma _ { \\frac { \\varepsilon } { 2 } } ( i ( \\psi ( P ) \\psi ( Q ) + \\psi ( Q ) \\psi ( P ) ) ) , \\end{align*}"}
+{"id": "5158.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\frac { E [ L ^ 2 ] } { E [ L ] ^ 2 } = 1 . \\end{align*}"}
+{"id": "199.png", "formula": "\\begin{align*} & \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) ( 2 0 + { \\rm c h } ( \\overline { W _ i } ) - 8 { \\rm c h } ( { W _ i } ) + B _ 1 \\wedge [ { \\rm c h } ( { W _ i } ) - 8 ] + B _ 2 ) \\right \\} ^ { ( 1 4 ) } \\\\ & = - 1 3 5 4 3 2 \\left \\{ \\widehat { A } ( T X ) { \\rm e x p } ( \\frac { c } { 2 } ) \\right \\} ^ { ( 1 4 ) } . \\end{align*}"}
+{"id": "4959.png", "formula": "\\begin{align*} \\Phi _ { 8 n , a } ( X ) : = \\prod _ { 1 \\leq k < 8 n , ~ \\gcd ( k , 2 n ) = 1 , ~ k \\equiv \\pm a \\pmod * { 8 } } \\bigl ( X - \\zeta ^ k \\bigr ) , \\end{align*}"}
+{"id": "6944.png", "formula": "\\begin{align*} \\nabla \\cdot ( { f \\rho + g \\bar \\rho } ) & = \\nabla \\cdot ( f \\rho ) + \\sum _ { i = 1 } ^ { m } \\nabla \\cdot ( g _ i \\rho u _ i ) \\\\ & = - \\mathcal { P } _ { f } \\rho - \\sum _ { i = 1 } ^ m \\mathcal { P } _ { g _ i } \\bar \\rho _ i \\\\ & \\approx \\Psi ^ T ( \\frac { I - P _ 0 ^ T } { \\Delta t } v ) + \\Psi ^ T ( \\frac { I - ( P _ i - P _ 0 ) ^ T } { \\Delta t } w _ i ) . \\end{align*}"}
+{"id": "6891.png", "formula": "\\begin{align*} \\hat { c } _ 1 ( \\Omega _ { Y } ^ { [ 1 ] } ( \\mathrm { l o g } \\ , B ) ) = K _ Y + B = \\pi ^ * ( K _ X + \\Delta ) - E , \\end{align*}"}
+{"id": "7998.png", "formula": "\\begin{align*} B ( x , x + 1 ) = \\frac 1 2 B ( x , x ) , x > 0 \\end{align*}"}
+{"id": "315.png", "formula": "\\begin{align*} \\mathsf { S } _ { j } \\left ( s \\right ) = \\mathsf { N } _ { j } \\left ( s \\right ) \\circ \\left ( \\gamma _ { \\operatorname * { D } ; j } \\left ( s \\right ) \\right ) ^ { \\prime } . \\end{align*}"}
+{"id": "6365.png", "formula": "\\begin{align*} t _ { M , N } \\colon M \\otimes N \\rightarrow M \\otimes N , m \\otimes n \\mapsto \\chi \\cdot ( m \\otimes n ) = ( \\chi ^ i \\cdot m ) \\otimes ( \\chi _ i \\cdot n ) , \\end{align*}"}
+{"id": "8529.png", "formula": "\\begin{align*} B _ { \\rho } ( ( \\bar { z } , w ) ) \\cap \\{ z = \\zeta \\} \\subset B ^ { n - 1 } ( w , \\rho ) \\mbox { f o r e v e r y } \\zeta \\in ( \\bar { z } - \\rho , \\bar { z } + \\rho ) . \\end{align*}"}
+{"id": "957.png", "formula": "\\begin{align*} \\begin{dcases*} \\nu = \\cos \\theta \\ , \\overline \\nu + \\sin \\theta \\ , \\overline \\eta \\\\ \\eta = - \\sin \\theta \\ , \\overline \\nu + \\cos \\theta \\ , \\overline \\eta \\end{dcases*} , \\begin{dcases*} \\overline \\nu = \\cos \\theta \\ , \\nu - \\sin \\theta \\ , \\eta \\\\ \\overline \\eta = \\sin \\theta \\ , \\nu + \\cos \\theta \\ , \\eta \\end{dcases*} . \\end{align*}"}
+{"id": "6216.png", "formula": "\\begin{align*} \\| f \\| _ { M ^ { p , q } } \\coloneqq \\begin{cases} \\| f \\| _ { M ^ { p , q } _ { } } & ( n ) \\\\ \\| f \\| _ { M ^ { p , q } _ { } } & ( n ) , \\end{cases} \\| f \\| _ { W ^ { p , q } } \\coloneqq \\begin{cases} \\| f \\| _ { W ^ { p , q } _ { } } & ( n ) \\\\ \\| f \\| _ { W ^ { p , q } _ { } } & ( n ) . \\end{cases} \\end{align*}"}
+{"id": "5036.png", "formula": "\\begin{align*} \\triangle _ f ( - f ) ^ { - a } = a ( \\triangle _ f f ) ( - f ) ^ { - a - 1 } - a ( a + 1 ) | \\nabla f | ^ 2 ( - f ) ^ { - a - 2 } \\leq - a ( - f ) ^ { - a } + C ( A ) ( - f ) ^ { - a - 1 } . \\end{align*}"}
+{"id": "6102.png", "formula": "\\begin{align*} \\begin{aligned} \\min \\limits _ { \\omega } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } ( y _ i - x _ i \\omega ) ^ 2 + \\lambda \\| \\omega \\| ^ 2 \\end{aligned} \\end{align*}"}
+{"id": "4450.png", "formula": "\\begin{align*} | | D ^ { \\alpha } _ { \\ast } ( \\mathcal { A } _ 3 { \\mathbf V } ) | | ^ 2 _ { L ^ 2 ( \\Omega _ t ) } & \\lesssim | | \\mathcal { A } _ 3 { \\mathbf V } | | ^ 2 _ { s , \\ast , t } \\\\ & \\leq C ( K ) \\Big ( | | { \\mathbf V } | | ^ 2 _ { s , \\ast , t } + | | \\dot { { \\mathbf U } } | | ^ 2 _ { W ^ { 1 , \\infty } _ { \\ast } ( \\Omega _ t ) } | | \\hat { W } | | ^ 2 _ { s + 2 , \\ast , t } \\Big ) . \\end{align*}"}
+{"id": "5645.png", "formula": "\\begin{align*} 0 \\ = \\ - \\lambda \\iota + \\lambda a \\ & = \\ ( \\pi ^ { * } K _ X \\ + \\ E ) \\ \\cdot \\ [ \\ell - ( a \\lambda ) e ] \\\\ & = \\ ( \\chi ^ * \\pi '^ { * } K _ W \\ + \\ t \\widetilde { H } _ b ) \\ \\cdot \\ [ \\ell - ( a \\lambda ) e ] \\\\ & = \\ K _ W \\cdot ( \\pi ' \\circ \\chi ) _ * [ \\ell - ( a \\lambda ) e ] \\ + \\ t \\lambda ( b - a ) < 0 . \\end{align*}"}
+{"id": "6153.png", "formula": "\\begin{align*} ( \\cdots , f _ i - f _ j , f _ j - f _ k ) & = ( \\cdots , ( f _ i - f _ j ) + ( f _ j - f _ k ) , f _ j - f _ k ) \\\\ & = - ( \\cdots , f _ i - f _ k , f _ k - f _ j ) . \\end{align*}"}
+{"id": "6841.png", "formula": "\\begin{align*} \\varepsilon _ { * } : = K _ 0 ^ { \\frac { n - 6 } { 1 2 } } = \\left ( \\frac { ( n - 6 ) ( n - 2 ) ( n + 2 ) } { 8 } \\right ) ^ { \\frac { n - 6 } { 6 } } . \\end{align*}"}
+{"id": "5895.png", "formula": "\\begin{align*} S _ \\infty ^ i = ( \\N _ \\infty \\oplus _ { \\N } \\cdots \\oplus _ { \\N } \\N _ \\infty ) ^ { } \\longrightarrow \\N _ \\infty \\oplus ( \\N _ \\infty / \\N ) ^ { \\oplus i } \\end{align*}"}
+{"id": "3005.png", "formula": "\\begin{align*} \\textstyle X _ \\gamma = \\left ( X \\amalg \\left ( \\coprod S ^ 1 \\times [ 0 , 1 ] \\right ) \\right ) / \\sim , \\end{align*}"}
+{"id": "2400.png", "formula": "\\begin{align*} S _ { \\j , \\tt } \\left ( B _ { \\tt } ( s , \\i , n ) \\right ) = \\pi _ { \\tt } ( \\j \\i ) + E _ n . \\end{align*}"}
+{"id": "6073.png", "formula": "\\begin{align*} u _ { n } : = \\sum _ { i = 1 } ^ { n } \\gamma _ { i } ^ { 1 + \\alpha } e ^ { - \\rho ( \\Gamma _ { n } - \\Gamma _ { i } ) } \\leq C \\gamma _ { n } ^ { \\alpha } . \\end{align*}"}
+{"id": "1539.png", "formula": "\\begin{align*} \\beta + 2 \\sqrt { \\beta ( 1 - \\beta ) } - 1 = \\varphi ' - \\frac { 5 \\sqrt { 5 } } 4 \\beta '^ 2 + O \\left ( \\beta '^ 3 \\right ) . \\end{align*}"}
+{"id": "2903.png", "formula": "\\begin{align*} W _ { \\lambda } ( t ) & = \\sum _ { w \\in W _ { \\lambda } } t _ w = \\prod _ { \\substack { \\alpha \\in { \\hat R _ 0 ^ + } \\\\ \\langle \\lambda , \\alpha \\rangle = 0 } } \\frac { 1 - t _ { \\alpha } \\hat e _ t ( \\alpha ) } { 1 - \\hat e _ t ( \\alpha ) } \\prod _ { \\substack { \\alpha \\in { \\hat R _ 0 ^ + } \\\\ \\langle \\lambda , \\alpha \\rangle = c } } \\frac { 1 - t _ { \\alpha } \\hat h _ t \\hat e _ t ( - \\alpha ) } { 1 - \\hat h _ t \\hat e _ t ( - \\alpha ) } , \\end{align*}"}
+{"id": "1931.png", "formula": "\\begin{align*} \\mathcal { N } ( u ; f , \\theta _ f ) & - \\mathcal { N } ^ h ( u _ h ; f _ h , \\theta _ f ) = \\sum _ { i , j } \\int _ { I _ i } \\left ( \\frac { 2 \\lambda _ 2 - 1 } { 2 } \\right ) \\left \\vert ( u _ h - v ) \\right \\vert [ \\ ! [ \\theta _ f ] \\ ! ] ^ 2 _ { x , j - 1 / 2 } \\ , { \\rm d } x \\\\ & + \\sum _ { i , j } \\int _ { T _ { i j } } \\left ( ( u - u _ h ) \\ , \\partial _ v f \\ , \\theta _ f - \\frac { 1 } { 2 } \\theta ^ 2 _ f \\right ) \\ , { \\rm d } v \\ , { \\rm d } x + \\mathcal { K } ^ 2 ( u _ h - v , f , \\theta _ f ) , \\end{align*}"}
+{"id": "3781.png", "formula": "\\begin{align*} \\eta \\ , \\colon \\ , \\R \\to \\R , \\eta ( s ) : = c s , \\eta ^ * ( \\phi ) : = \\sup _ { s > 0 } \\left ( s \\phi - \\eta ( s ) \\right ) \\begin{cases} + \\infty , & \\phi > c \\\\ 0 , & \\phi \\leq c \\end{cases} \\end{align*}"}
+{"id": "7806.png", "formula": "\\begin{align*} \\begin{pmatrix} p & q \\\\ m & n \\\\ \\end{pmatrix} \\in \\mathrm { S L } ( 2 , \\mathbb { Z } ) \\ , . \\end{align*}"}
+{"id": "5053.png", "formula": "\\begin{align*} \\big \\langle \\psi ^ * ( Z ( \\kappa ) ) , L _ { \\psi ^ * g ' } ( T ( \\gamma ^ * ) ) \\big \\rangle _ { \\psi ^ * g ' , \\psi ^ * f ' } = \\big \\langle Z ( \\kappa ) , L _ { g ' } ( T ( \\gamma ^ * ) ) \\big \\rangle _ { g ' , f ' } . \\end{align*}"}
+{"id": "2160.png", "formula": "\\begin{align*} \\cosh \\Lambda = \\frac { \\alpha ^ d \\left ( m ^ { d + 1 } + m ^ { - d - 1 } \\right ) + \\alpha ^ { - d } \\left ( m ^ { d - 1 } + m ^ { - d + 1 } \\right ) } { m ^ { d } + m ^ { - d } } , \\end{align*}"}
+{"id": "1468.png", "formula": "\\begin{align*} \\Delta u _ i + \\sum \\limits _ { j \\in J _ 0 } k _ { i j } ^ { \\prime } e ^ { u _ j } = 4 \\pi \\sum \\limits _ { \\ell = 1 } ^ { N } \\alpha _ { \\ell , i } \\delta _ { p _ { \\ell } } \\ \\ \\mathbb { R } ^ 2 , \\ \\int _ { \\mathbb { R } ^ 2 } e ^ { u _ i ( y ) } \\mathrm { d } y < + \\infty , \\ \\ \\forall \\ i \\in J _ 0 , \\end{align*}"}
+{"id": "4821.png", "formula": "\\begin{align*} \\check { R } _ i ( u ) = I ^ { ( 1 ) } \\otimes I ^ { ( 2 ) } \\otimes \\cdots \\otimes \\check { R } ( u ) ^ { i \\otimes i + 1 } \\otimes I ^ { ( i + 2 ) } \\otimes \\cdots \\otimes I ^ { ( m ) } \\end{align*}"}
+{"id": "7362.png", "formula": "\\begin{align*} A = 1 6 x + \\frac { 3 2 \\times 2 ^ { 1 / 3 } ( - 3 + 3 \\gamma + x ^ 2 ) } { B } + 2 ^ { 2 / 3 } B \\end{align*}"}
+{"id": "1129.png", "formula": "\\begin{align*} & [ e _ 1 , e _ 1 ] _ 1 = e _ 3 ; \\\\ & [ e _ 1 , e _ 1 ] _ 2 = e _ 2 , ~ [ e _ 2 , e _ 1 ] _ 2 = e _ 3 . \\end{align*}"}
+{"id": "1905.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } \\frac { { \\rm d } } { { \\rm d } t } \\| u _ h \\| ^ 2 _ { 0 , I _ h } + \\| w _ h \\| ^ 2 _ { 0 , I _ h } & + a _ h ( u _ h , u _ h ) + \\sqrt { \\epsilon } \\ , b _ h ( w _ h , u _ h ) \\\\ & + \\sqrt { \\epsilon } \\ , b _ h ( u _ h , w _ h ) + \\left ( \\left ( \\rho _ h u _ h - \\rho _ h V _ h \\right ) , u _ h \\right ) = 0 . \\end{aligned} \\end{align*}"}
+{"id": "1216.png", "formula": "\\begin{align*} C _ n = & - \\int _ { \\R ^ 3 } [ I _ \\alpha \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } ( | ( g _ n ^ 1 ) ^ { - 1 } u _ n - \\phi ^ 1 | ^ p + | \\phi ^ 1 | ^ p ) ] | x - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p d x \\\\ & + \\int _ { \\R ^ 3 } ( I _ { \\alpha } \\ast | \\cdot - x _ n ^ 1 / \\lambda _ n ^ 1 | ^ { - b } | ( g _ n ^ 1 ) ^ { - 1 } u _ n | ^ p ) | x - x _ n ^ 1 \\lambda _ n ^ 1 | ^ { - b } | \\phi ^ 1 | ^ p d x . \\end{align*}"}
+{"id": "4167.png", "formula": "\\begin{align*} Q ^ + _ i & = 2 y _ i \\sqrt { | h _ { i i } | ^ 2 p _ i } - y ^ 2 _ i \\left ( \\sum \\nolimits _ { j \\ne i } | h _ { i j } | ^ 2 p _ j + \\sigma ^ 2 _ i \\right ) , \\\\ Q ^ - _ k & = 2 \\tilde y _ k \\sqrt { \\sum \\nolimits _ { j } | \\tilde h _ { k j } | ^ 2 p _ j + \\tilde \\sigma ^ 2 _ k } - \\tilde y ^ 2 _ k | \\tilde h _ { k k } | ^ 2 p _ k . \\end{align*}"}
+{"id": "4155.png", "formula": "\\begin{align*} \\exp _ { \\hat p } ^ { \\nabla } ( v ) & = \\cos ( | v | ) \\hat { p } + \\sin ( | v | ) \\frac { v } { | v | } , \\\\ \\\\ \\exp _ { \\hat p } ^ { \\nabla , - 1 } ( p ) & = \\cos ^ { - 1 } ( \\langle \\hat p , p \\rangle ) \\frac { p - \\langle \\hat p , p \\rangle \\hat p } { | p - \\langle \\hat p , p \\rangle \\hat p | } . \\end{align*}"}
+{"id": "113.png", "formula": "\\begin{align*} \\mathcal { H } _ { \\Lambda } ( \\rho _ { \\mu } ) _ n \\geq \\sum _ { j = 1 } ^ n \\frac { b } { \\ell ^ 2 } Q _ { \\Lambda , j } + \\frac { 1 } { \\lvert B \\rvert } \\int _ { \\Lambda } \\ , \\mathcal { H } _ { B _ u } ( \\rho _ { \\mu } ) _ n \\mathrm { d } u . \\end{align*}"}
+{"id": "8543.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } P ( F _ { \\ell ^ { k } } ; J \\times \\mathbb { R } ^ { n - 1 } ) & = \\lim _ { k \\rightarrow \\infty } \\left ( \\int _ { J } \\mathcal { H } ^ { n - 2 } \\left ( ( \\partial ^ { * } F _ { \\ell ^ { k } } ) _ { z } \\right ) \\ d z + | D ^ { c } \\ell ^ { k } | ( J ) \\right ) \\\\ & = \\int _ { J } \\mathcal { H } ^ { n - 2 } ( ( \\partial ^ { * } F _ { \\ell } ) _ { z } \\ d z + | D ^ { c } \\ell | ( J ) \\\\ & = P ( F _ \\ell ; J \\times \\mathbb { R } ^ { n - 1 } ) , \\end{align*}"}
+{"id": "8004.png", "formula": "\\begin{align*} | \\nabla H ( \\pi _ { p _ 0 } ^ { - 1 } ( x ) ) | = | \\nabla H ( \\pi _ { p _ 0 } ^ { - 1 } ( x ) ) - \\nabla H ( p _ 0 ) | \\leq C \\max _ { S ^ m } | \\nabla ^ 2 H | \\cdot | \\pi _ { p _ 0 } ^ { - 1 } ( x ) - p _ 0 | \\leq C ( 1 + | x | ^ 2 ) ^ { - \\frac 1 2 } . \\end{align*}"}
+{"id": "8381.png", "formula": "\\begin{align*} \\begin{aligned} J _ { 2 } ( t ) & \\leq \\int _ { 0 } ^ { t } | \\bar { X } _ { s } ^ { h } | ^ { 2 } d s + C _ { H , T } K ( t ^ { 2 } ) { ( \\kappa ( | \\bar { X } _ { t } ^ { 0 } | ) ^ { 2 } + 1 ) } \\cdot \\| u _ { t } \\| ^ { 2 } _ { \\mathcal { H } _ { H } } . \\end{aligned} \\end{align*}"}
+{"id": "6245.png", "formula": "\\begin{align*} \\Delta ^ 4 \\xi ^ 2 = 0 , \\Delta ^ 4 \\xi ^ 2 = \\xi _ 1 ^ 2 - \\xi _ 2 ^ 2 + \\xi _ 3 ^ 2 - \\xi _ 4 ^ 2 . \\end{align*}"}
+{"id": "4364.png", "formula": "\\begin{align*} \\int _ \\mathbb { R } \\chi ( t ) \\omega ( A _ 1 { \\tt T } _ t ( A _ 2 ) ) \\ , d t = \\int _ { \\mathbb { R } } \\chi ( t + i \\beta ) \\omega ( { \\tt T } _ t ( A _ 2 ) A _ 1 ) \\ , d t \\ \\ ( A _ 1 , A _ 2 \\in { \\tt C A R } ( \\mathcal { K } , \\Gamma ) ) \\end{align*}"}
+{"id": "228.png", "formula": "\\begin{align*} g _ S ^ { ( c ) } = 2 g _ S \\quad g _ L ^ { ( c ) } ( g _ L ^ { ( c ) } - 1 ) = { \\textstyle \\frac { e ^ { 2 c } } { 1 6 } } ( g _ L ^ { ( c ) } > 0 ) , \\end{align*}"}
+{"id": "2701.png", "formula": "\\begin{align*} ( x _ 1 ^ 2 + y _ 1 ^ 2 + z _ 1 ^ 2 ) ( x _ 2 ^ 2 + y _ 2 ^ 2 + z _ 2 ^ 2 ) & = \\frac { w _ 1 ^ 2 w _ 2 ^ 2 | | \\vec { a } | | ^ 4 } { ( a ^ 2 + e ^ 2 ) ^ 2 } ( U _ 1 ^ 2 + 1 ) \\ ; \\left ( \\frac { 1 } { U _ 1 ^ 2 } + 1 \\right ) \\\\ & = \\frac { w _ 1 ^ 2 w _ 2 ^ 2 | | \\vec { a } | | ^ 4 } { ( a ^ 2 + e ^ 2 ) ^ 2 } \\left ( 2 + U _ 1 ^ 2 + \\frac { 1 } { U _ 1 ^ 2 } \\right ) \\geq \\frac { 4 w _ 1 ^ 2 w _ 2 ^ 2 | | \\vec { a } | | ^ 4 } { ( a ^ 2 + e ^ 2 ) ^ 2 } . \\end{align*}"}
+{"id": "8306.png", "formula": "\\begin{align*} \\sigma ( Z ) = u ( Z ) ~ Z \\in ( F ) . \\end{align*}"}
+{"id": "1385.png", "formula": "\\begin{align*} M _ { 0 } = 1 M _ { p } ^ { 2 } \\leq M _ { p - 1 } M _ { p + 1 } , \\forall p \\in \\mathbb { N } ^ { \\ast } , \\end{align*}"}
+{"id": "8885.png", "formula": "\\begin{align*} e ^ T ( \\nu _ w ) = w \\cdot \\bigg ( \\prod _ { \\alpha \\in \\Delta ^ + } c _ 1 ( S _ { \\alpha } ) \\bigg ) , \\end{align*}"}
+{"id": "8287.png", "formula": "\\begin{align*} \\left ( ( u , v ) , ( \\tilde u , \\tilde v ) \\right ) _ { H } = 2 \\int _ 0 ^ 1 e ^ { \\mu x } ( u _ x + v ) ( \\tilde u _ x + \\tilde v ) + e ^ { - \\mu x } ( u _ x - v ) ( \\tilde u _ x - \\tilde v ) d x , \\end{align*}"}
+{"id": "1677.png", "formula": "\\begin{align*} \\frac { 1 } { ( 2 \\pi ) ^ { n - 1 } n ^ { 1 / 2 } } \\int _ { \\mathbb { A } ^ { ( n ) } _ { \\texttt { a } } } f ( \\boldsymbol { \\xi } ) | C _ { \\texttt { a } } ( \\boldsymbol { \\xi } ; q ) | ^ { - 2 } \\boldsymbol { \\xi } = \\sum _ { \\lambda \\in \\Lambda ^ { ( m , n ) } _ { \\texttt { a } } } f \\bigl ( \\boldsymbol { \\xi } ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } \\bigr ) \\hat { \\Delta } ^ { ( m , n ) } _ { \\texttt { a } ; \\lambda } , \\end{align*}"}
+{"id": "7455.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { m } \\| y _ i \\| _ { 1 , p } ^ { \\kappa _ i + 1 } \\leq C _ { 2 6 } \\| ( y _ 1 , y _ 2 , \\cdots , y _ m ) \\| _ { Y , 1 } ^ { \\frac { \\sum _ { i = 1 } ^ { m } \\kappa _ i + m } { p } } , \\end{align*}"}
+{"id": "661.png", "formula": "\\begin{align*} \\left \\{ \\tau _ R < \\tau \\right \\} \\cap \\left \\{ \\tau _ R \\le t \\right \\} = \\bigcap _ { k = 1 } ^ \\infty \\left ( \\bigcup _ { 0 \\le s _ i \\le t } f _ { s _ i } ^ { - 1 } \\left ( \\left [ R - \\frac { 1 } { k } , \\infty \\right ) \\right ) \\right ) , \\end{align*}"}
+{"id": "3701.png", "formula": "\\begin{align*} L ''' _ x : = \\{ y \\in D ^ + \\ , | \\ , y _ 1 \\in ( x _ 1 - x _ 2 , x _ 1 ) \\ , \\ , \\& \\ , \\ , y _ 2 \\in ( 0 , y _ 1 - x _ 1 + x _ 2 ) \\} \\end{align*}"}
+{"id": "4868.png", "formula": "\\begin{align*} y ( u ) + y ( v ) - y ( u + v ) = 0 \\ ; . \\end{align*}"}
+{"id": "6050.png", "formula": "\\begin{align*} \\mathcal { S } = \\{ F = f ( \\omega , ( { Z } _ { i } ^ { k } ) _ { \\substack { 1 \\leq k \\leq m ^ { \\prime } \\\\ 1 \\leq i \\leq m } } , \\Delta ) : f \\in C _ { \\mathcal { G } , p } , m , m ^ { \\prime } \\in \\mathbb { N } \\} . \\end{align*}"}
+{"id": "197.png", "formula": "\\begin{align*} & B _ 1 = \\widetilde { T _ C M } - \\widetilde { L _ C } , \\\\ & B _ 2 = \\wedge ^ 2 \\widetilde { L _ C } - \\widetilde { L _ C } - \\widetilde { T _ C M } \\otimes \\widetilde { L _ C } + S ^ 2 \\widetilde { T _ C M } + \\widetilde { T _ C M } . \\end{align*}"}
+{"id": "739.png", "formula": "\\begin{align*} I _ + + I _ - = - 2 M _ { \\mathfrak K _ 1 } \\int _ 0 ^ t \\norm { P _ { \\mu } \\psi ^ \\mu ( s ) } _ { L ^ 2 } ^ 2 d s . \\end{align*}"}
+{"id": "4784.png", "formula": "\\begin{align*} { } _ { 1 } F _ { 1 } \\left ( 1 , s , z \\right ) = & \\sum _ { k = 0 } ^ { \\infty } \\frac { ( 1 ) _ { k } } { ( s ) _ { k } } \\frac { z ^ { k } } { k ! } \\\\ = & 1 + \\sum _ { k = 1 } ^ { \\infty } \\frac { k ! } { s ( s + 1 ) \\hdots ( s + k - 1 ) } \\frac { z ^ { k } } { k ! } \\\\ = & 1 + \\frac { z } s \\sum _ { k = 1 } ^ { \\infty } \\frac { ( k - 1 ) ! } { ( s + 1 ) \\hdots ( s + k - 1 ) } \\frac { z ^ { ( k - 1 ) } } { ( k - 1 ) ! } = 1 + \\frac { z } s \\ , { } _ { 1 } F _ { 1 } \\left ( 1 , s + 1 , z \\right ) \\end{align*}"}
+{"id": "4490.png", "formula": "\\begin{align*} \\tilde { e } ' _ k : = \\mathcal { B } ( { \\mathbf V } _ { k + 1 } | _ { x _ 1 = 0 } , \\psi _ { k + 1 } ) - \\mathcal { B } ( { \\mathbf V } _ { k } | _ { x _ 1 = 0 } , \\psi _ { k } ) - \\mathcal { B } ' ( { \\mathbf V } _ { k } | _ { x _ 1 = 0 } , \\psi _ { k } ) ( \\delta { \\mathbf V } _ k | _ { x _ 1 = 0 } , \\delta \\psi _ k ) . \\end{align*}"}
+{"id": "1225.png", "formula": "\\begin{align*} e ^ { l i n } _ { n , T } = & \\Delta [ \\chi _ n ( \\frac { x - x _ n } { \\lambda _ n } ) ] e ^ { i ( t + \\lambda _ n ^ 2 t _ n ) \\Delta } g _ n [ P _ n \\phi ] \\\\ & + 2 \\nabla [ \\chi _ n ( \\frac { x - x _ n } { \\lambda _ n } ) ] e ^ { i ( t + \\lambda _ n ^ 2 t _ n ) \\Delta } \\nabla g _ n [ P _ n \\phi ] \\end{align*}"}
+{"id": "1386.png", "formula": "\\begin{align*} \\left \\Vert D ^ { \\alpha } u \\right \\Vert _ { L ^ { 2 } \\left ( \\omega _ { \\delta } \\right ) } \\leq C ^ { k } \\sum _ { i = 0 } ^ { k } \\binom { k } { i } \\left ( \\frac { k } { \\delta } \\right ) ^ { \\left ( k - i \\right ) d m } \\left \\Vert Q ^ { i } u \\right \\Vert _ { L ^ { 2 } \\left ( \\omega \\right ) } \\end{align*}"}
+{"id": "2353.png", "formula": "\\begin{align*} \\rho _ g J A J \\delta _ h & = \\rho _ g J A \\delta _ { h ^ { - 1 } } = \\rho _ g J \\left ( \\sum _ { u \\in G } \\zeta _ u ( A \\delta _ { h ^ { - 1 } } ) \\delta _ u \\right ) = \\sum _ { u \\in G } \\zeta _ u ( A \\delta _ { h ^ { - 1 } } ) \\rho _ g J \\delta _ u \\\\ & = \\sum _ { u \\in G } \\zeta _ u ( A \\delta _ { h ^ { - 1 } } ) \\rho _ g \\delta _ { u ^ { - 1 } } = \\sum _ { u \\in G } \\zeta _ u ( A \\delta _ { h ^ { - 1 } } ) \\delta _ { u ^ { - 1 } g ^ { - 1 } } . \\end{align*}"}
+{"id": "8134.png", "formula": "\\begin{align*} \\psi ( t _ i ^ * , 1 ) = \\gamma ( t _ i ) \\psi ( t , 1 ) \\neq \\gamma ( t _ i ) t > t _ i ^ * , \\end{align*}"}
+{"id": "5695.png", "formula": "\\begin{align*} k _ { x _ r } = \\limsup \\limits _ { | z | \\rightarrow 0 } \\frac { \\ln \\big | f _ 0 \\big ( ( | z | - T ) ^ { - 1 } x _ r \\big ) \\big | } { \\ln \\| ( | z | - T ) ^ { - 1 } \\| } , ~ ~ \\forall r \\in ( 0 , 1 ] . \\end{align*}"}
+{"id": "2898.png", "formula": "\\begin{align*} \\mathcal J T _ j \\mathcal J ^ { - 1 } \\Phi _ \\xi = \\mathcal J T _ j \\phi _ \\xi = t _ j \\mathcal J \\phi _ \\xi = t _ j \\Phi _ \\xi . \\end{align*}"}
+{"id": "3484.png", "formula": "\\begin{align*} \\frac { - \\log | z _ i | } { | \\log | t | | } \\gtrsim \\epsilon , i = 0 , 1 , \\ldots m . \\end{align*}"}
+{"id": "7528.png", "formula": "\\begin{align*} \\frac { r } { r _ \\ast } = \\frac { r _ \\ast u _ \\ast ^ 2 } { r _ \\ast u _ \\ast ^ 2 + p _ \\ast - p ( n ) } \\ , . \\end{align*}"}
+{"id": "2065.png", "formula": "\\begin{align*} e ^ { - c _ n t } \\phi ( x , t ) & \\leq \\int _ M G ( x , t ; y , 0 ) \\phi ( y , 0 ) \\ , d \\mathrm { v o l } _ { h _ 0 } ( y ) . \\end{align*}"}
+{"id": "28.png", "formula": "\\begin{align*} v ( x _ { i } - x _ { j } ) = ( P _ { i } + Q _ { i } ) ( P _ { j } + Q _ { j } ) v ( x _ { i } - x _ { j } ) ( P _ { j } + Q _ { j } ) ( P _ { i } + Q _ { i } ) \\end{align*}"}
+{"id": "4257.png", "formula": "\\begin{align*} \\mathbb { P } _ { k } : = \\left \\{ u \\in \\mathbb { X } ( \\Omega ) : \\mathcal { B } _ { \\alpha } ( u , u _ j ) = 0 \\ , \\textrm { f o r e v e r y } j = 1 , \\ldots , k - 1 \\right \\} ; \\end{align*}"}
+{"id": "3707.png", "formula": "\\begin{align*} J ' ( c ) = ( 1 - 2 \\alpha ) c ^ { 2 \\alpha - 2 } \\left ( \\frac c { ( 1 - 2 \\alpha ) ( c ^ 2 + 1 ) ^ \\alpha } - f ( c ) - f ( 1 ) + \\frac { 2 ^ { - 1 - \\alpha } } { 1 - 2 \\alpha } - 2 ^ { - 1 - \\alpha } \\mu _ \\alpha \\right ) = : ( 1 - 2 \\alpha ) c ^ { 2 \\alpha - 2 } g ( c ) . \\end{align*}"}
+{"id": "2046.png", "formula": "\\begin{align*} \\dot { \\mathrm { H } } ^ { \\alpha , p } ( \\mathbb { R } _ + , \\mathrm { L } ^ p ( \\mathbb { R } ^ { n - 1 } ) ) : = \\dot { \\mathrm { H } } ^ { \\alpha , p } ( \\mathbb { R } , \\mathrm { L } ^ p ( \\mathbb { R } ^ { n - 1 } ) ) _ { | _ { \\mathbb { R } _ + } } \\end{align*}"}
+{"id": "5827.png", "formula": "\\begin{align*} & \\alpha ^ { d 4 } _ { m a x } : = \\alpha _ 2 + \\alpha _ 3 + 2 \\alpha _ 4 + \\alpha _ 5 , \\\\ & \\alpha ^ { d 6 } _ { m a x } : = \\alpha _ 2 + \\alpha _ 3 + 2 \\alpha _ 4 + 2 \\alpha _ 5 + 2 \\alpha _ 6 + \\alpha _ 7 , \\\\ & \\alpha ^ { e 7 } _ { m a x } : = 2 \\alpha _ 1 + 2 \\alpha _ 2 + 3 \\alpha _ 3 + 4 \\alpha _ 4 + 3 \\alpha _ 5 + 2 \\alpha _ 6 + \\alpha _ 7 , \\end{align*}"}
+{"id": "5470.png", "formula": "\\begin{align*} \\int _ M f \\Delta | \\nabla f | ^ 2 d V _ g = \\int _ M | \\nabla f | ^ 2 \\Delta f d V _ g + | \\nabla f | ^ 3 _ { | _ { \\partial M } } | \\partial M | . \\end{align*}"}
+{"id": "3501.png", "formula": "\\begin{align*} & \\sum _ 0 ^ m \\frac { l _ i } { l } \\log | F _ i | _ { h ^ { \\otimes d _ i } } ( z ) \\geq - C ( l ) - \\frac { 1 } { l } \\log \\norm { s _ { l _ 0 , \\ldots l _ m } } _ { V _ { l - \\sum d _ i l _ i } } \\\\ & \\geq ( \\phi _ t + | \\log | t | | \\psi _ t ) ( z ) - | \\log | t | | u ( y ) + \\frac { \\sum l _ i y _ i } { l } \\log | t | - ( \\frac { C } { l } | \\log | t | | + C | \\log | t | | \\epsilon + C ( l ) ) . \\end{align*}"}
+{"id": "554.png", "formula": "\\begin{align*} \\mathbf u ( t ) = \\mathbf S ( t ) \\mathbf u _ 0 + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf N \\left ( \\Theta _ R ^ { \\mathbf u } ( s ) \\mathbf u ( s ) \\right ) \\ , d s + i \\int _ 0 ^ t \\mathbf S ( t - s ) \\mathbf M \\left ( \\mathbf u ( s ) \\right ) \\ , d W ( s ) , \\end{align*}"}
+{"id": "3400.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } A \\alpha _ 3 ^ 2 + B + C = 0 , \\\\ \\\\ \\widetilde { A } \\alpha _ 3 ^ 2 + \\widetilde { B } + \\widetilde { C } = 0 , \\\\ \\\\ F _ { 3 3 } \\alpha _ 3 ^ 2 + F _ { 3 4 } + F _ { 3 5 } = 0 . \\end{array} \\right . \\end{align*}"}
+{"id": "796.png", "formula": "\\begin{align*} ( S ^ * ) ^ { \\frac { 2 \\cdot p ^ \\sharp } { p } } A ( \\varepsilon _ g ) = \\frac { b } { p ^ \\sharp } C _ U A ( \\varepsilon _ g ) ^ { \\frac { 2 \\cdot p ^ \\sharp } { p } } + \\varepsilon _ g ( S ^ * ) ^ { \\frac { p _ s ^ * } { p } } A ( \\varepsilon _ g ) ^ { \\frac { p _ s ^ * } { p } } , \\end{align*}"}
+{"id": "5824.png", "formula": "\\begin{align*} & \\alpha ^ { d 4 } _ { m a x } : = \\alpha _ 2 + \\alpha _ 3 + 2 \\alpha _ 4 + \\alpha _ 5 , \\\\ & \\alpha ^ { d 6 } _ { m a x } : = \\alpha _ 2 + \\alpha _ 3 + 2 \\alpha _ 4 + 2 \\alpha _ 5 + 2 \\alpha _ 6 + \\alpha _ 7 , \\end{align*}"}
+{"id": "2949.png", "formula": "\\begin{align*} T _ { i _ 1 \\cdots i _ n } = \\sum _ { J = 1 } ^ { J _ 0 ^ n } \\alpha _ J H _ { i _ 1 \\cdots i _ n } ^ { J } + \\sum _ { J = 1 } ^ { J _ 1 ^ n } H _ { i _ 1 \\cdots i _ n j } ^ { J } [ v _ j ^ J ] + \\sum _ { s = 2 } ^ { J _ s ^ n } \\sum _ { J = 1 } ^ { J _ 0 ^ n } H _ { i _ 1 \\cdots i _ n j _ 1 \\cdots j _ s } ^ { J } [ D _ { j _ 1 \\cdots j _ n } ^ { J } ] \\end{align*}"}
+{"id": "7394.png", "formula": "\\begin{align*} \\partial _ { x } ^ { 2 } u _ { n } = \\lambda _ { n } ^ { - 1 } \\partial _ { x } V _ { n } + V _ { n } \\partial _ { x } ( \\lambda _ { n } ^ { - 1 } ) \\end{align*}"}
+{"id": "6369.png", "formula": "\\begin{align*} \\chi = a ( 1 \\otimes 1 ) + b ( 1 \\otimes g ) + e ( g \\otimes 1 ) + f ( g \\otimes g ) + n ( x \\otimes x ) + o ( x \\otimes x g ) + r ( x g \\otimes x ) + s ( x g \\otimes x g ) . \\end{align*}"}
+{"id": "6773.png", "formula": "\\begin{align*} \\varpi ' ( z ) & = c \\varpi ( z ) - \\varpi ^ 2 ( z ) - F ( \\phi , \\psi ) ( z ) \\\\ [ 0 . 2 c m ] & \\leq c \\varpi ( z ) - \\varpi ^ 2 ( z ) + m = - ( \\varpi ( z ) - \\pi _ + ) ( \\varpi ( z ) - \\pi _ - ) , \\end{align*}"}
+{"id": "4793.png", "formula": "\\begin{align*} \\Omega _ \\gamma = \\lim _ { \\leftarrow } ( \\Gamma , \\gamma ) \\end{align*}"}
+{"id": "2278.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 f ( x ) ^ 2 \\ , d x & \\leq n ^ 2 X ^ 2 + \\frac { 1 } { n ^ 2 \\pi ^ 2 } \\int _ 0 ^ 1 f ' ( x ) ^ 2 \\ , d x \\\\ & + 2 | X | \\sqrt { n } \\sum _ { k = 0 } ^ { n - 1 } \\sqrt { \\frac { 1 } { n ^ 2 \\pi ^ 2 } \\int _ { \\frac { k + 1 / 2 } { n } } ^ { \\frac { k + 3 / 2 } { n } } f ' ( x ) ^ 2 \\ , d x } \\end{align*}"}
+{"id": "2389.png", "formula": "\\begin{align*} J ^ { p r e } ( t , W ) = \\frac { - g + \\exp ( h ( t - T ) ( g - 2 r ) + r ( 2 + h ( t - T ) } { h } + \\log W \\end{align*}"}
+{"id": "8845.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { T } t ^ { - \\frac { \\beta - 1 } { \\beta } } = 1 + \\int _ { 1 } ^ { T } u ^ { - \\frac { \\beta - 1 } { \\beta } } \\d u = \\beta T ^ { \\frac { 1 } { \\beta } } + 1 - \\beta \\leq \\beta T ^ { \\frac { 1 } { \\beta } } \\enspace , \\end{align*}"}
+{"id": "1849.png", "formula": "\\begin{gather*} \\int _ { \\mathbb { R } } K _ { s , t } ( x , \\Delta ) \\ , d x = \\int _ { \\mathbb { R } } K \\left ( \\frac { \\Delta x } { 2 \\pi } \\right ) \\ , d x = \\frac { 2 \\pi } { \\Delta } \\int _ { \\mathbb { R } } \\left ( \\frac { \\sin \\pi x } { \\pi x } \\right ) ^ 2 \\ , d x \\ll \\frac { 1 } { \\Delta } \\end{gather*}"}
+{"id": "6135.png", "formula": "\\begin{align*} \\varphi _ 1 \\times \\varphi _ 2 \\times \\cdots \\times \\varphi _ k = ( \\cdots ( ( \\varphi _ 1 \\times \\varphi _ 2 ) \\times \\varphi _ 3 ) \\times \\cdots ) \\times \\varphi _ k . \\end{align*}"}
+{"id": "2669.png", "formula": "\\begin{align*} | N ( u , v ) | & = p ^ { s - 2 } + p ^ { s _ 2 - 2 } \\sum _ { z _ { 1 } \\in \\mathbb { F } _ { p } ^ { \\ast } } \\sigma _ { z _ { 1 } } \\Big ( \\varepsilon _ { f } ( p ^ { \\ast } ) ^ { \\frac { R _ { f } } { 2 } } p ^ { s _ 1 - R _ f } \\zeta _ { p } ^ { - f ( x _ { c } ) } \\Big ) \\\\ & = p ^ { s - 2 } + p ^ { s - 2 - R _ f } \\varepsilon _ { f } \\sum _ { z _ { 1 } \\in \\mathbb { F } _ { p } ^ { \\ast } } \\sigma _ { z _ { 1 } } \\Big ( ( p ^ { \\ast } ) ^ { \\frac { R _ { f } } { 2 } } \\zeta _ { p } ^ { - f ( x _ { c } ) } \\Big ) . \\end{align*}"}
+{"id": "7107.png", "formula": "\\begin{align*} \\mathcal { I } _ { y _ 1 } = \\int _ 0 ^ \\infty \\ , U ( y _ 1 ) \\ , U ( y _ 1 + y _ 3 ) \\ , e \\left ( t \\log \\left ( 1 + \\frac { y _ 3 } { y _ 1 } \\right ) + \\frac { 2 \\sqrt { N } } { p _ 1 q \\sqrt { p _ 2 } } \\left ( { \\sqrt { m y _ 1 } } - { \\sqrt { m _ 1 ( y _ 1 + y _ 3 ) } } \\right ) \\right ) d y _ 1 \\end{align*}"}
+{"id": "8176.png", "formula": "\\begin{align*} ( \\theta _ { i j } ^ \\star - \\theta _ { i \\pm 1 , j } ^ \\star ) ^ 2 = \\left ( \\frac { n } { k } ( r - 1 ) \\right ) ^ 2 . \\end{align*}"}
+{"id": "6152.png", "formula": "\\begin{align*} ( f _ i - f _ j , f _ j - f _ k , \\cdots ) & = ( f _ i - f _ j , ( f _ i - f _ j ) + ( f _ j - f _ k ) , \\cdots ) \\\\ & = - ( f _ j - f _ i , f _ i - f _ k , \\cdots ) , \\end{align*}"}
+{"id": "3413.png", "formula": "\\begin{align*} f = \\sum _ { \\alpha \\in \\N ^ { | J | } } f _ \\alpha z _ 1 ^ { \\alpha _ 0 } \\ldots z _ { | J | } ^ { \\alpha _ { | J | } } , \\end{align*}"}
+{"id": "1865.png", "formula": "\\begin{gather*} \\liminf _ { T \\to \\infty } P _ { \\alpha , T } ( A ) \\geq Q _ \\alpha ( A ) = \\mathbf { P } \\left ( \\sup _ { s \\in K } | \\zeta ( s , \\mathbb { X } _ \\alpha ) - p ( s ) | < \\frac { \\epsilon } { 2 } \\right ) . \\end{gather*}"}
+{"id": "3978.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) u ^ \\# = \\Phi ^ \\# \\end{align*}"}
+{"id": "2792.png", "formula": "\\begin{align*} f * ( g * h ) ( G ) & = \\sum _ { H \\leq G } f ( H ) ( g * h ) ( G / H ) \\\\ & = \\sum _ { H \\leq G } \\sum _ { K \\leq G / H } f ( H ) g ( K ) h ( G / H / K ) \\\\ & = \\sum _ { H \\leq G } \\sum _ { H \\leq L \\leq G } f ( H ) g ( L / H ) h \\left ( \\frac { G / H } { L / H } \\right ) \\\\ & = \\sum _ { H \\leq G } \\sum _ { H \\leq L \\leq G } f ( H ) g ( L / H ) h ( G / L ) \\end{align*}"}
+{"id": "7088.png", "formula": "\\begin{align*} \\mathcal { J } ^ + ( n _ 1 ^ 2 n _ 2 , m , q ) = \\frac { q } { Q } \\ , \\mathcal { J } ^ + _ 1 ( n _ 1 ^ 2 n _ 2 , m , q ) , \\end{align*}"}
+{"id": "3069.png", "formula": "\\begin{align*} { \\bf { a } } _ { \\rm { T } } ^ H \\left ( { \\Theta _ { { \\rm { T } } , { m } , i _ m } ^ { \\rm { D } } } \\right ) { { \\bf { a } } _ { \\rm { T } } } \\left ( { \\Theta _ { { \\rm { T } } , { n } , i _ n } ^ { \\rm { D } } } \\right ) = 0 , \\end{align*}"}
+{"id": "1637.png", "formula": "\\begin{align*} \\Delta h ( r ) = h '' ( r ) + \\frac { f ' ( r ) } { f ( r ) } h ' ( r ) , \\end{align*}"}
+{"id": "950.png", "formula": "\\begin{align*} F _ j ^ * = \\frac { \\delta L } { \\delta u _ j } = 0 , ~ ~ j = 1 , \\cdots , s , \\end{align*}"}
+{"id": "4580.png", "formula": "\\begin{align*} I _ { 1 , \\ast } ( t ) + \\Vert \\varphi ( t ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } \\le & \\frac { C _ 3 } { \\varepsilon } \\int _ 0 ^ t ( I _ { 1 , \\ast } ( s ) + I _ { 1 , n } ( s ) + \\Vert \\varphi ( s ) \\Vert ^ 2 _ { L ^ 2 ( \\mathbb R ) } ) d s \\\\ & + \\varepsilon \\left \\{ I ( t ) + I _ { 1 , n } ( t ) \\right \\} + \\frac { C _ 3 } { \\varepsilon } \\Vert { \\mathbf F } \\Vert ^ 2 _ { H ^ 1 _ \\ast ( \\Omega _ t ) } \\ , . \\end{align*}"}
+{"id": "8788.png", "formula": "\\begin{align*} \\norm { x _ { t + 1 } - x _ { p } } ^ { 2 } & \\leq \\norm { x _ { t } - \\eta _ { t } \\hat { g } _ { t } - x _ { p } } ^ 2 \\\\ & = \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 \\\\ & \\leq ( 1 - 2 \\eta _ { t } \\alpha ) \\norm { x _ { t } - x _ { p } } ^ 2 - 2 \\eta _ { t } \\langle \\hat { g } _ { t } - \\nabla f ( x _ t ) , x _ { t } - x _ { p } \\rangle + \\eta _ { t } ^ 2 \\norm { \\hat { g } _ { t } } ^ 2 , \\end{align*}"}
+{"id": "2372.png", "formula": "\\begin{align*} f ( \\pi ( \\mu ) ^ { - 1 } \\tau ) = \\frac { o ( G ) } { o ( \\Lambda ) } \\delta _ { \\mu , 0 } , \\forall \\mu \\in \\Lambda ^ 0 . \\end{align*}"}
+{"id": "8785.png", "formula": "\\begin{align*} \\delta ( T _ { 1 } + 1 ) & \\leq ( 1 - a ) ^ { T _ 1 } \\delta ( 1 ) + \\sum _ { s = 0 } ^ { T _ 1 } U ( T _ 1 - s + 1 ) \\\\ & \\leq \\delta ( 1 ) + \\mathcal { A } _ { 1 } \\frac { d } { \\min \\{ \\alpha , \\alpha ^ 2 \\} } \\sum _ { s = 0 } ^ { T _ { 1 } } ( 1 - a ) ^ s ( T _ { 1 } - s + 1 ) ^ { - \\frac { 2 \\beta - 1 } { \\beta } } . \\end{align*}"}
+{"id": "3791.png", "formula": "\\begin{align*} \\beta : = ( x , \\rho _ i ( x ) ) _ { \\# } \\gamma _ i . \\end{align*}"}
+{"id": "4777.png", "formula": "\\begin{align*} H _ { - } ^ { n } = \\{ h \\in H _ { C } ^ { - } \\mid ( h x ) _ { g } = x _ { g } g \\neq a ^ { k } - M - n < k < - M x \\in X _ { p } \\} \\end{align*}"}
+{"id": "2848.png", "formula": "\\begin{align*} M _ \\lambda ( \\xi ) = \\sum _ { v \\in W _ 0 } C ( v \\xi ) e ^ { i \\langle v \\xi , \\lambda \\rangle } \\end{align*}"}
+{"id": "2638.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } P ( \\sup _ { s \\in [ 0 , t ] } \\abs { \\frac { 1 } { n } \\ , Q _ n ( s ) - 1 } > \\epsilon ) ^ { 1 / b _ n ^ 2 } = 0 \\ , . \\end{align*}"}
+{"id": "5279.png", "formula": "\\begin{align*} \\frac { 1 } { \\alpha _ { t + 1 } } - \\frac { 1 } { \\alpha _ { t } } - \\mu = \\frac { t + 1 } { ( t + 1 ) ^ { 1 - c } } - \\frac { t } { t ^ { 1 - c } } - \\mu < \\frac { 1 } { t ^ { 1 - c } } - \\mu \\le 0 , ~ \\forall t \\ge \\varepsilon _ { 8 } . \\end{align*}"}
+{"id": "2642.png", "formula": "\\begin{align*} \\Theta ^ { ( l ) } ( t ) = - \\int _ { \\R _ + ^ 2 } I _ { t } ^ { ( l ) } ( x , s ) \\dot K ( F ( x ) , \\mu s ) \\ , d F ( x ) \\ , \\mu d s \\ , . \\end{align*}"}
+{"id": "6492.png", "formula": "\\begin{align*} \\Psi = A - \\frac 1 2 L \\Psi ' , \\end{align*}"}
+{"id": "6588.png", "formula": "\\begin{align*} H _ M ^ { - 1 } = ( R _ { \\Lambda _ M \\setminus \\mathcal { S } } ( { D } ( 0 ) + \\varepsilon \\Delta + \\delta { T } _ { q ^ { ( 0 ) } } ) R _ { \\Lambda _ M \\setminus \\mathcal { S } } ) ^ { - 1 } . \\end{align*}"}
+{"id": "4288.png", "formula": "\\begin{align*} S _ { J } = \\{ | j _ 1 | , | j _ 2 | , \\ldots , | j _ p | \\} . \\end{align*}"}
+{"id": "4537.png", "formula": "\\begin{align*} C _ m = C _ m ( \\Vert \\hat { \\mathbf U } ^ \\pm \\Vert _ { W ^ { m , \\infty } ( \\Omega _ T ) } , \\Vert \\nabla _ { t , x _ 2 } \\hat \\varphi \\Vert _ { W ^ { m , \\infty } ( \\Gamma _ T ) } , k ) \\ , , \\end{align*}"}
+{"id": "1033.png", "formula": "\\begin{align*} f ( z , x ) = 0 \\mbox { f o r a . a . } z \\in \\Omega , \\mbox { a l l } x \\leq 0 . \\end{align*}"}
+{"id": "8115.png", "formula": "\\begin{align*} \\bar { r } _ { i j k } + \\bar { s } _ { i j k } = \\min ( i - 1 , u _ { \\alpha } - 1 - \\min ( a _ \\alpha - i , b _ \\alpha - j ' ) ) + \\min ( a _ \\alpha - i , u _ { \\alpha } - \\min ( i , j ' ) ) \\end{align*}"}
+{"id": "7946.png", "formula": "\\begin{align*} \\Psi _ { 1 , 2 } = \\sum _ { i = 1 } ^ 2 ( \\deg _ { G _ i } ( u _ i ) + \\deg _ { G _ i } ( v _ i ) ) = 6 . \\end{align*}"}
+{"id": "2782.png", "formula": "\\begin{align*} x ^ { n + 3 } \\cdot x ^ { n + 1 } \\cdot x ^ 5 = x ^ { 2 k + 9 } , \\end{align*}"}
+{"id": "6935.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { [ \\mathbb { P } _ t \\psi ] ( x ) - \\psi ( x ) } { t } = - \\nabla \\cdot ( f ( x ) \\psi ( x ) ) = : { \\cal P } _ f \\psi . \\end{align*}"}
+{"id": "5397.png", "formula": "\\begin{align*} v ( z , w ; t ) = v ( z , w + 2 t \\overline { v ( z , w ; t ) } ; 0 ) . \\end{align*}"}
+{"id": "4170.png", "formula": "\\begin{align*} Q ^ + _ i & = 2 y _ i \\sqrt { w _ i ( 1 + \\gamma _ i ) | h _ { i , i } | ^ 2 p _ i } - y ^ 2 _ i \\left ( \\sum _ { j = 1 } ^ { L } | h _ { i , j } | ^ 2 p _ j + \\sigma ^ 2 _ i \\right ) , \\\\ Q _ k ^ - & = { 2 \\tilde { y } _ k \\sqrt { \\sum _ { j \\ne k } | \\tilde { h } _ { k , j } | ^ 2 p _ j + \\tilde { \\sigma } ^ 2 _ k } - \\tilde { y } _ k ^ 2 w _ k ( 1 - \\tilde { \\gamma } _ k ) | \\tilde { h } _ { k , k } | ^ 2 p _ k } . \\end{align*}"}
+{"id": "7485.png", "formula": "\\begin{align*} G _ k + G ^ { \\sigma } _ { k - 1 } + u G _ { k - 2 } ^ { \\sigma ^ 2 } = 0 . \\end{align*}"}
+{"id": "2499.png", "formula": "\\begin{align*} \\mathcal { L } ( \\boldsymbol { y } , \\boldsymbol { u } , \\boldsymbol { p } ) : = \\mathcal { J } ( \\boldsymbol { y } , \\boldsymbol { u } ) + \\int _ 0 ^ T \\int _ { \\Omega } \\big ( \\sigma \\frac { \\partial \\boldsymbol { y } } { \\partial t } + \\textbf { c u r l } ( \\nu \\ , \\textbf { c u r l } \\ , \\boldsymbol { y } ) - \\boldsymbol { u } \\big ) \\boldsymbol { p } \\ , d \\boldsymbol { x } d t , \\end{align*}"}
+{"id": "2758.png", "formula": "\\begin{align*} G ( z ) \\approx - \\frac { \\pi } { N } \\sum _ { k = 0 } ^ { N } { } ^ { '' } \\frac { b - a } { 2 } \\sin \\left ( \\frac { 2 \\pi k } { N } \\right ) \\frac { f \\left ( \\frac { b - a } { 2 } \\cos \\left ( \\frac { 2 \\pi k } { N } \\right ) + \\frac { b + a } { 2 } \\right ) } { z - \\left ( \\frac { b - a } { 2 } \\cos \\left ( \\frac { 2 \\pi k } { N } \\right ) + \\frac { b + a } { 2 } \\right ) } , \\end{align*}"}
+{"id": "7916.png", "formula": "\\begin{align*} 0 = 1 + ( \\varphi ( x ) + \\varphi ( x ^ \\ast ) ) \\mathrm { F P d i m } ( x ) . \\end{align*}"}
+{"id": "3368.png", "formula": "\\begin{align*} G ( x _ 1 , x _ 2 ) = \\frac { 4 e ( x _ 0 x _ 2 ) } { L _ 1 L _ 2 } K ^ { \\ast } _ { i t _ 1 } ( x _ 1 ) K ^ { \\ast } _ { i t _ 2 } ( | y _ 0 | x _ 2 ) \\end{align*}"}
+{"id": "2230.png", "formula": "\\begin{align*} k [ X _ { \\sigma _ 2 ^ c } ] = \\bigoplus _ { m \\in \\sigma _ 2 ^ \\vee \\cap N _ 2 ^ \\vee } k [ G ] ^ { ( B ) } _ { - m } \\to k [ X _ { \\sigma _ 1 ^ c } ] = \\bigoplus _ { m \\in \\sigma _ 1 ^ \\vee \\cap N _ 1 ^ \\vee } k [ G ] ^ { ( B ) } _ { - m } \\end{align*}"}
+{"id": "8473.png", "formula": "\\begin{align*} E _ { 1 } = _ { \\mathcal { H } ^ { n } } E _ { 2 } \\mbox { i f } \\mathcal { H } ^ { n } ( E _ { 1 } \\triangle E _ { 2 } ) = 0 . \\end{align*}"}
+{"id": "7167.png", "formula": "\\begin{align*} \\| P _ S x \\| = \\left \\| \\sum _ { j \\in S } f _ j ( x ) \\tau _ j \\right \\| = \\left ( \\sum _ { j \\in S } | f _ j ( x ) | ^ p \\right ) ^ \\frac { 1 } { p } = \\| x \\| _ { S , f } , \\forall x \\in \\mathcal { X } \\end{align*}"}
+{"id": "1296.png", "formula": "\\begin{align*} E _ c = \\inf \\{ c : v _ 0 \\in L ( c ) , \\| v \\| _ { S ( I _ { \\max } ) } = \\infty \\} \\end{align*}"}
+{"id": "4713.png", "formula": "\\begin{align*} P & = ( x _ 4 + b _ { 4 , 3 } x _ 3 + b _ { 4 , 1 } x _ 1 + \\varepsilon _ 4 ) ( x _ 2 + b _ { 2 , 1 } x _ 1 + \\varepsilon _ 2 ) ( x _ 0 + \\varepsilon _ 0 ) \\\\ Q & = ( x _ 3 + a _ { 3 , 0 } x _ 0 + \\gamma _ 3 ) ( x _ 2 + a _ { 2 , 0 } x _ 0 + \\gamma _ 2 ) ( x _ 1 + a _ { 1 , 0 } x _ 0 + \\gamma _ 0 ) \\end{align*}"}
+{"id": "6526.png", "formula": "\\begin{align*} \\tilde k = ( k - e _ { l ' } , 1 ) \\in \\Z ^ { b + 1 } \\setminus \\{ 0 \\} , \\ v ( m ) = ( \\omega ^ { ( 0 ) } , \\mu _ n ) \\in \\R ^ { b + 1 } , \\end{align*}"}