diff --git "a/data_tmp/process_21/tokenized_finally.jsonl" "b/data_tmp/process_21/tokenized_finally.jsonl"
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-{"id": "3361.png", "formula": "\\begin{align*} \\int _ X w \\cdot ( \\phi _ { \\varepsilon } \\cdot a _ t ) d \\mu _ X = \\int _ Y w ( y a _ t ) d \\mu _ Y ( y ) + O ( \\varepsilon ^ { p ' } \\| w \\| _ l ) \\end{align*}"}
-{"id": "956.png", "formula": "\\begin{align*} X _ i = \\beta _ i Y + \\varepsilon _ i , \\ ; \\ ; \\ ; \\ ; \\textrm { C o v } _ B ( Y , \\varepsilon _ i ) = 0 , \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} M ^ d = ( M ^ d _ s ) _ { s \\geq 0 } : = ( \\mathbb E ( \\xi ^ d | \\mathcal F _ { s } ) ) _ { s \\geq 0 } , \\ ; \\ ; \\ ; M ^ c = ( M ^ c _ s ) _ { s \\geq 0 } : = ( \\mathbb E ( \\xi ^ c | \\mathcal F _ { s } ) ) _ { s \\geq 0 } . \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{align*} \\Lambda ^ { N + 1 } _ { N } ( x , d y ) = \\frac { N ! \\Delta _ N ( y ) } { \\Delta _ { N + 1 } ( x ) } \\boldmath { 1 } ( y \\in W ^ { N , N + 1 } ( x ) ) d y , \\end{align*}"}
-{"id": "9920.png", "formula": "\\begin{align*} [ \\kappa ( \\alpha ) ] ( x , y ) = v _ N ( x - y ) \\alpha ( x , y ) . \\end{align*}"}
-{"id": "702.png", "formula": "\\begin{align*} d \\mu ( \\eta , \\bar \\eta , z , \\bar z ) = \\vert { \\cal N } _ 2 \\vert ^ { - 2 } \\sigma _ { 2 j } ( \\eta , \\bar \\eta ) d \\eta d \\bar \\eta ~ d m ( z , \\bar z ) , \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} \\alpha _ n = \\ell ^ 3 [ 4 f ( x _ n ) x _ n ^ \\prime - g ( x _ n ) ] = \\ell ^ 3 \\left ( 1 2 x _ n ^ 2 x _ n ^ \\prime + 1 6 A x _ n ^ \\prime - 3 x _ n ^ 3 + 5 A x _ n + 2 7 B \\right ) , \\end{align*}"}
-{"id": "1317.png", "formula": "\\begin{align*} \\left ( \\lambda _ { 4 } - \\lambda _ { 1 } \\lambda _ { 4 } ^ { 2 } \\right ) g ^ { 5 } + 2 \\left ( \\lambda _ { 5 } - \\lambda _ { 1 } \\lambda _ { 4 } \\lambda _ { 5 } \\right ) g ^ { 3 } - \\left ( \\lambda _ { 1 } \\lambda _ { 5 } ^ { 2 } \\right ) g = 0 . \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , S _ 2 ] = [ [ L _ { - r - 1 } , \\ , S _ { r - 1 } ] , \\ , S _ 2 ] = [ [ L _ { - r - 1 } , \\ , S _ 2 ] , \\ , S _ { r - 1 } ] \\subseteq [ L _ { - r + 1 } , \\ , L _ { r - 1 } ] \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\dim H ^ 2 ( G _ { L , T } , Z _ { i } ( U _ X ) ) \\leq B n ^ { 2 g - 1 } . \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{align*} f _ n = \\sum _ { d \\mid n } \\xi _ 1 ( n / d ) \\xi _ 2 ( d ) d ^ { k - 1 } \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } t _ k ^ i \\ , y _ { i n } = 0 , k = 1 , \\dots , n - 1 . \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{align*} W _ { k ' } : = \\{ u _ k ( r \\mathfrak { e } _ { k ' } ) : r \\in \\R \\} . \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} | \\Pi _ q [ h ] ( x ) | = \\chi _ q ( x ) \\sum _ { j = 1 } ^ L P _ j ( \\frac { x - x _ q } { l ( q ) } ) \\int _ q h ( y ) P _ j ( \\frac { y - x _ q } { l ( q ) } ) \\ , \\frac { d y } { l ( q ) ^ d } . \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} ^ { - 1 } = \\begin{pmatrix} d & c \\\\ b & a \\end{pmatrix} . \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} h _ n = \\frac { 1 } { 2 } \\sum _ { k \\in \\mathbb { Z } + 1 / 2 } : \\chi _ { k } \\chi _ { 2 n - k } : = \\frac { 1 } { 2 } \\sum _ { i \\in \\mathbb { Z } } : \\chi _ { - i - \\frac { 1 } { 2 } } \\chi _ { 2 n + i + \\frac { 1 } { 2 } } : \\end{align*}"}
-{"id": "293.png", "formula": "\\begin{align*} X \\triangleright ( a b ) = ( X _ { ( 1 ) } \\triangleright a ) ( X _ { ( 2 ) } \\triangleright b ) , ( a b ) \\triangleleft X = ( a \\triangleleft X _ { ( 1 ) } ) ( b \\triangleleft X _ { ( 2 ) } ) . \\end{align*}"}
-{"id": "2218.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty h ( t ) h ( t ) ^ \\dag d t + \\int _ { - \\infty } ^ \\infty g ( t ) h ( t ) ^ \\dag d t + \\sum _ { j = 1 } ^ N \\widehat { h } ( - i k _ j ) C _ j ^ 2 \\widehat { h } ( - i k _ j ) ^ \\dag = 0 . \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} D _ { n - 1 } ( x ) : = \\det \\begin{bmatrix} c _ 0 & c _ 1 & \\cdots & c _ { n - 2 } & 1 \\\\ c _ 1 & c _ 2 & \\cdots & c _ { n - 1 } & x \\\\ \\vdots & \\vdots & & \\vdots \\\\ c _ { n - 1 } & c _ { n } & \\cdots & c _ { 2 n - 3 } & x ^ { n - 1 } \\end{bmatrix} \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{align*} X \\triangleright \\mathsf { M } _ { m } ^ { n } = \\pi ( S ( X _ { ( 1 ) } ) ) \\mathsf { M } _ { m } ^ { n } \\pi ( X _ { ( 2 ) } ) , \\quad \\mathsf { N } _ { m } ^ { n } \\triangleleft X = \\pi ( X _ { ( 1 ) } ) \\mathsf { N } _ { m } ^ { n } \\pi ( S ( X _ { ( 2 ) } ) ) . \\end{align*}"}
-{"id": "5820.png", "formula": "\\begin{align*} 0 \\ge & \\Delta v + A ( x ) v \\\\ = & \\triangle ( u _ 1 \\beta ) + A ( x ) u _ 1 \\beta \\\\ = & u _ 1 \\triangle \\beta + \\Delta u _ 1 \\beta + \\nabla u _ 1 \\cdot \\nabla \\beta + A ( x ) u _ 1 \\beta \\\\ = & u _ 1 \\triangle \\beta - \\lambda u _ 1 \\beta + \\nabla u _ 1 \\cdot \\nabla \\beta + A ( x ) u _ 1 \\beta . \\\\ \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} W \\circ \\sigma = \\tau \\circ W \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{align*} \\mu \\Lambda ^ { \\infty } _ N = \\mu _ N , \\end{align*}"}
-{"id": "8662.png", "formula": "\\begin{align*} k _ 0 = \\sup _ { 0 \\le s \\le t _ 0 } K _ 2 ( s ) . \\end{align*}"}
-{"id": "9430.png", "formula": "\\begin{align*} d = \\tau \\nabla \\times b , \\ ; \\ ; \\ ; v = ( b \\cdot \\nabla ) b + \\tau \\nabla \\tau . \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{align*} \\left \\vert x - \\frac { p ' } { q ' } \\right \\vert ~ = ~ \\frac { x } { q ' ( p + p ' y ) } \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} I ( g \\cdot w ) = \\chi ( g ) I ( w ) \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} B ( Q , f , m _ { \\bullet } , c _ { \\bullet } ) = K Q / J ( Q , f , m _ { \\bullet } , c _ { \\bullet } ) , \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} [ \\omega _ i ] ( 1 , h ) & = h \\cdot \\texttt { r i g h t } ( \\omega _ i ( h ) ) = h \\cdot ( h ^ { - 1 } f _ i ( 1 , h ) ) = f _ i ( 1 , h ) \\\\ [ \\omega _ i ] ( - 1 , h ) & = h \\cdot \\texttt { l e f t } ( \\omega _ i ( h ) ) = h \\cdot ( h ^ { - 1 } f _ i ( - 1 , h ) ) = f _ i ( - 1 , h ) \\end{align*}"}
-{"id": "6704.png", "formula": "\\begin{align*} \\int \\limits _ { | x - e | + | x | \\leq 2 \\tau } \\frac { q ( x ) } { | x | | x - e | } d x & = \\frac { 1 } { 2 } \\int \\limits _ { 1 } ^ { 2 \\tau } \\int \\limits _ { 0 } ^ { \\pi } \\int \\limits _ { 0 } ^ { 2 \\pi } q ( \\ln ( r + \\sqrt { r ^ { 2 } - 1 } ) , \\theta , \\phi ) \\sin \\phi d \\theta d \\phi d r . \\end{align*}"}
-{"id": "8230.png", "formula": "\\begin{align*} \\overline { \\Lambda } = \\Lambda _ { 1 } ( \\infty , \\Omega _ { 2 } ) . \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{align*} q _ { i , j } & = \\left ( \\mathbf { A ^ T W A } \\right ) _ { i , j } \\\\ & = u _ { i , j } - u _ { i + 1 , j } \\\\ & = w _ { i , j } + w _ { i + 1 , j + 1 } - w _ { i , j + 1 } - w _ { i + 1 , j } . \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} U _ n ( \\omega ) = T ( \\gamma _ n , \\omega ) \\hat { U } ^ { ( n ) } _ n ( \\omega ) = \\hat { T } ^ { ( n ) } ( \\hat { \\gamma } ^ { ( n ) } _ n , \\omega ) . \\end{align*}"}
-{"id": "7956.png", "formula": "\\begin{align*} Q ( \\cdot , t ) : = \\left ( \\frac { - 1 } { 4 \\pi | \\cdot | } * \\mbox { d i v $ G $ } \\right ) ( \\cdot , t ) \\in L ^ q ( \\mathbb R ^ 3 ) , \\quad \\forall q \\in ( 3 , \\infty ) , \\end{align*}"}
-{"id": "7815.png", "formula": "\\begin{align*} u _ t + b u _ x + u _ { x x x } + u _ { x y y } + ( g ( u ) ) _ x + ( \\psi ( t , x , y ) u ) _ x = f ( t , x , y ) . \\end{align*}"}
-{"id": "6326.png", "formula": "\\begin{align*} \\bar { q } ( \\mu ) = \\min \\{ q _ 1 ( \\mu ) , q _ 2 ( \\mu ) \\} , \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} \\mathbf { X } _ { 0 h } = \\{ \\mathbf { v } _ { h } \\in \\mathbf { X } _ { h } \\ ; | \\ ; ( \\nabla \\cdot \\mathbf { v } _ { h } , \\ , q _ { h } ) = 0 , \\ ; \\forall q _ { h } \\in X _ { h } ^ { 1 } \\} . \\end{align*}"}
-{"id": "8821.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { n - m } \\left ( n - \\sum _ { \\ell = 1 } ^ { j - 1 } c _ i \\right ) \\leq \\left ( \\prod _ { j = 0 } ^ { m - 1 } ( n - 2 j ) \\right ) ( n - 2 m ) ! . \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} L _ { - 4 } = [ L _ { - 2 } , \\ , L _ { - 2 } ] . \\end{align*}"}
-{"id": "4334.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ( E ) = 2 ^ 8 \\cdot 3 ^ 2 \\cdot 5 ^ 2 \\cdot 1 1 ^ 2 \\cdot 1 3 ^ 2 \\cdot 1 7 ^ 2 \\cdot 1 9 ^ 2 \\cdot 2 3 ^ 2 \\cdot 2 9 ^ 2 \\cdot 3 1 ^ 2 \\cdot 3 7 ^ 3 \\cdot p ^ \\prime ; \\\\ j ( E ) = \\frac { 2 ^ 4 \\cdot 7 ^ 3 \\cdot 6 1 ^ 3 \\cdot 3 4 7 ^ 3 \\cdot ( p '' ) ^ 3 } { 3 ^ 2 \\cdot 5 ^ 2 \\cdot 1 1 ^ 2 \\cdot 1 3 ^ 2 \\cdot 1 7 ^ 2 \\cdot 1 9 ^ 2 \\cdot 2 3 ^ 2 \\cdot 2 9 ^ 2 \\cdot 3 1 ^ 2 \\cdot p ^ \\prime } , \\end{cases} \\end{align*}"}
-{"id": "3693.png", "formula": "\\begin{align*} H ^ { \\beta } ( z ) & \\to e ^ { - \\alpha } _ y ( z ) = \\exp ( - \\sum _ { n \\ge 1 } y _ n z ^ n ) \\exp ( \\sum _ { n \\ge 1 } \\frac { \\partial } { n \\partial y _ n } z ^ { - n } ) e ^ { - \\alpha } z ^ { - h ^ y _ 0 } \\\\ H ^ { \\gamma } ( z ) & \\to \\partial _ z e ^ { \\alpha } _ y ( z ) = : h ^ y ( z ) \\exp ( \\sum _ { n \\ge 1 } y _ n z ^ n ) \\exp ( - \\sum _ { n \\ge 1 } \\frac { \\partial } { n \\partial y _ n } z ^ { - n } ) e ^ { \\alpha } z ^ { h ^ y _ 0 } : . \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{align*} \\left [ \\hat { Q } ^ k , \\ ; \\hat { Q } ^ l \\right ] = 0 \\ ; , \\quad \\left [ \\hat { P } ^ k , \\ ; \\hat { P } ^ l \\right ] = 0 \\ ; , \\quad \\left [ \\hat { Q } ^ k , \\ ; \\hat { P } ^ l \\right ] = i \\ , \\delta ^ { k l } \\ , \\hat { \\mathbb { I } } \\ ; , k , \\ , l = 1 , \\ , \\ldots , \\ , n \\ ; . \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{align*} \\dot { \\phi } = \\sqrt { 2 } \\psi ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "103.png", "formula": "\\begin{align*} \\pi _ 1 ( M ) \\geqslant \\pi _ 1 ( \\langle g ^ 2 , b \\rangle ) \\geqslant \\pi _ 1 ( \\langle g ^ 2 , g ^ { - ( n + 1 ) } b g ^ { n + 1 } b \\rangle ) = \\langle \\pi _ 1 ( g ^ 2 ) , \\pi _ 1 ( g ^ { - ( n + 1 ) } b g ^ { n + 1 } b ) \\rangle = G _ 1 \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} & \\mbox { G i v e n a s e n s i n g m a t r i x } A \\in \\mathbb { C } ^ { m \\times n } \\ ; \\mbox { a n d m a g n i t u d e s } | y | \\in \\mathbb { R } _ { \\ge 0 } ^ m \\ ; , \\\\ & \\mbox { o b t a i n a s i g n a l } \\rho \\in \\mathbb { C } ^ n \\ ; \\mbox { s u c h t h a t } \\vert y \\vert = \\vert A \\rho \\vert . \\end{align*}"}
-{"id": "593.png", "formula": "\\begin{align*} Q \\log | f _ t ' ( z ) | & = ( h _ 0 ' - h _ t ) ( z ) + b _ t \\\\ & = ( h ' - h \\circ f _ t ) ( z ) + \\frac { 2 } { \\sqrt \\kappa } \\log | z | - \\frac { 2 } { \\sqrt \\kappa } \\log | f _ t ( z ) | + b _ t \\\\ & = ( h ' - h \\circ f _ t ) ( z ) + \\frac { 2 } { \\sqrt \\kappa } \\log \\frac { | z | } { | f _ t ( z ) | } + b _ t . \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} u _ b ( \\xi , y ) = u _ { c _ * } ( \\xi ) + 2 b \\cos ( y ) \\psi _ * ( \\xi ) + \\tilde { u } _ b ( \\xi , y ) , \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} M z + N \\bar { z } = p \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 2 } { x ^ 2 } + \\frac { A _ 1 } { x } + A _ 0 \\right ) Y , \\frac { \\partial Y } { \\partial t _ 1 } = ( E _ 3 x + B _ 1 ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = \\left ( \\frac { 1 } { t _ 2 } N x + B _ 2 \\right ) Y . \\end{gather*}"}
-{"id": "2844.png", "formula": "\\begin{align*} M ( \\vec { \\nu } , \\vec \\gamma ) : = \\bigl \\{ f \\in R \\ ; \\big | \\ ; T _ { ( \\alpha , 2 ( \\mu _ \\alpha - \\nu _ \\alpha ) ) } ^ - ( f ) = \\gamma _ \\alpha \\cdot T _ { ( \\alpha , 2 ( \\mu _ \\alpha - \\nu _ \\alpha ) ) } ^ + ( f ) \\mbox { \\rm f o r a l l } \\alpha \\in \\Pi \\bigr \\} . \\end{align*}"}
-{"id": "9511.png", "formula": "\\begin{align*} w ^ * { } \\lim _ \\beta \\sum _ { j \\in J } \\phi ( x m _ j ^ * ) m _ j = 0 , \\ \\ \\ x \\in M . \\end{align*}"}
-{"id": "526.png", "formula": "\\begin{align*} \\rho \\tilde J \\phi ( x , v ) = \\rho ( x ) e ^ { - \\int _ 0 ^ { \\tau _ + ( x , v ) } \\sigma ( x + s v , v ) d s } \\phi ( x + \\tau _ + ( x , v ) v , v ) , \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} K ( s , t ) = \\begin{cases} w ( s ) - \\left ( w ( a ) + \\psi ( \\lambda ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ a , t ) , \\\\ w ( s ) - \\left ( w ( a ) + ( 1 + \\psi ( 1 - \\lambda ) ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ t , b ] , \\end{cases} \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{align*} s _ { q r , k } ^ { ( - 1 ) } = \\delta _ { 1 , k } - p ( q - k ) - p ( r - k ) + p ( q r - k ) , \\end{align*}"}
-{"id": "9791.png", "formula": "\\begin{align*} Q _ k ( M , c ^ 2 g ) = c ^ { 2 - n } Q _ k ( M , g ) , \\end{align*}"}
-{"id": "9728.png", "formula": "\\begin{align*} | | | x _ 0 ^ * + \\lambda y ^ * | | | + | | | x _ 0 ^ * - \\lambda y ^ * | | | & \\geq \\langle x _ 0 + \\rho \\ , e _ N , x _ 0 ^ * + \\lambda y ^ * \\rangle + \\langle x _ 0 - \\rho \\ , e _ N , x _ 0 ^ * - \\lambda y ^ * \\rangle \\\\ & > 2 - 6 \\lambda \\delta + 2 \\lambda \\rho = 2 + \\lambda \\left ( \\frac { 2 } { 3 } - ( 6 + \\lambda ) \\delta \\right ) . \\end{align*}"}
-{"id": "9455.png", "formula": "\\begin{align*} h ( t , x ) = h _ 0 \\big ( \\Phi ( 0 , t , x ) \\big ) \\exp \\left \\{ - \\int ^ t _ 0 \\nabla \\cdot v \\big ( s , \\Phi ( s , t , x ) \\big ) { \\rm d } s \\right \\} \\end{align*}"}
-{"id": "5346.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } & = - y + a _ { 0 2 } y ^ 2 , \\\\ \\dot { y } & = ( b _ { 0 2 } + b _ { 1 2 } x ) y ^ 2 + ( b _ { 1 1 } x + b _ { 2 1 } x ^ 2 ) y + ( b _ { 2 0 } x ^ 2 + b _ { 3 0 } x ^ 3 + x ) \\\\ & = x + b _ { 2 0 } x ^ 2 + b _ { 1 1 } x y + b _ { 0 2 } y ^ 2 + b _ { 3 0 } x ^ 3 + b _ { 2 1 } x ^ 2 y + b _ { 1 2 } x y ^ 2 , \\end{aligned} \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} : \\chi ( z ) \\chi ( w ) : = \\sum _ { m , n \\in \\mathbb { Z } + 1 / 2 } : \\chi _ m \\chi _ n : z ^ { - m - 1 / 2 } w ^ { - n - 1 / 2 } = \\sum _ { m , n \\in \\mathbb { Z } } : \\chi _ { - m - \\frac { 1 } { 2 } } \\chi _ { - n - \\frac { 1 } { 2 } } : z ^ { m } w ^ { n } , \\end{align*}"}
-{"id": "4235.png", "formula": "\\begin{align*} w _ i = u _ i + \\log M , i = 1 , 2 , \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{align*} \\langle A ^ { - 1 } _ { \\varepsilon , \\delta } \\overline { \\sigma _ { l } } , \\overline { \\sigma _ s } \\rangle _ { \\omega _ { \\varepsilon } } = \\langle \\biggl ( \\sum _ { 1 \\leq m \\leq k } \\frac { 1 + \\delta \\lambda _ { m } } { \\varepsilon + \\lambda _ { m } } + \\frac { n - k - 1 } { \\varepsilon } \\biggl ) ^ { - 1 } \\overline { \\sigma _ { l } } , \\overline { \\sigma _ { s } } \\rangle _ { \\omega _ { \\varepsilon } } \\leq \\frac { \\varepsilon } { n - k - 1 } \\delta _ { l s } \\end{align*}"}
-{"id": "4850.png", "formula": "\\begin{align*} \\left \\langle \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} D _ { H _ i } \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} ^ { - 1 } , I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle . \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} { \\displaystyle \\mathcal { U } _ h ( \\Omega ) = \\{ u _ h \\in C ( \\overline { \\Omega } ) : \\ , \\ , \\ , u _ h | _ { e } \\in P _ 1 , \\ , \\ , \\ , u _ h | _ { \\partial \\Omega } = 0 \\} , } \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} d e g \\big ( | 0 \\rangle \\big ) = 0 ; d e g \\chi _ { - j } | 0 \\rangle = j ; d e g \\Big ( \\left ( \\chi _ { - j _ k } \\right ) ^ { m _ k } \\dots \\left ( \\chi _ { - j _ 2 } \\right ) ^ { m _ 2 } \\left ( \\chi _ { - j _ 1 } \\right ) ^ { m _ 1 } | 0 \\rangle \\Big ) = m _ k \\cdot j _ k + \\dots m _ 2 \\cdot j _ 2 + m _ 1 \\cdot j _ 1 , \\end{align*}"}
-{"id": "2972.png", "formula": "\\begin{align*} \\sup _ { j \\ge 0 } \\ \\| P _ { n } T ^ { \\ , j } P _ { l } \\ , x \\| & \\le \\Bigl ( \\prod _ { r = 1 } ^ { M _ { 0 } - 1 } C _ { \\varphi ^ { r - 1 } ( l ) } \\Bigr ) \\ , \\cdot \\ , C _ { \\varphi ^ { M _ { 0 } - 1 } ( l ) } \\ , \\cdot \\ , \\Bigl \\| \\sum _ { k = b _ l } ^ { b _ { l + 1 } - 1 } \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Bigr ) \\ , x _ { k } e _ { k } \\ \\Bigr \\| _ { 1 } \\\\ & \\le C _ { l } \\ , \\cdot \\ , \\Bigl \\| \\sum _ { k = b _ { l } } ^ { b _ { l + 1 } - 1 } \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Bigr ) \\ , x _ { k } e _ { k } \\Bigr \\| _ { 1 } , \\end{align*}"}
-{"id": "9539.png", "formula": "\\begin{align*} \\| X \\| = \\sup \\| R X C \\| \\end{align*}"}
-{"id": "7557.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\mu ( \\pi ( N ( L , W ) ) ) > 0 & & \\mu ( \\pi ( S ( L , W ) ) ) = 0 . \\end{array} \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} c _ 1 ( X ) = \\big ( ( 1 + \\frac 2 k ) [ D _ \\infty ] + ( 1 - \\frac 2 k ) [ D _ 0 ] \\big ) . \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 1 e _ 4 + \\alpha _ 2 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 3 e _ 4 + \\alpha _ 4 e _ 5 , [ e _ 2 , e _ 3 ] = \\gamma _ 2 e _ 5 . \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{align*} \\underset { t \\to 0 } { \\lim } \\left [ P ^ { s , N } _ { H P } ( t ) f \\right ] ( x ) = \\underset { t \\to 0 } { \\lim } \\left [ \\mathcal { S } ^ N ( t ) f \\circ \\mathsf { e v a l } _ N \\right ] ( U ^ * x U ) = \\left [ f \\circ \\mathsf { e v a l } _ N \\right ] ( U ^ * x U ) = f ( x ) . \\end{align*}"}
-{"id": "7046.png", "formula": "\\begin{align*} \\frac { d u } { d t } + L _ { 0 } u \\left ( t \\right ) = f \\left ( t \\right ) , u \\left ( 0 \\right ) = 0 , t \\in \\left ( 0 , \\infty \\right ) . \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} L = K ( I - K ) ^ { - 1 } \\ , , K = L ( I + L ) ^ { - 1 } . \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ 0 ^ t \\langle ( h u _ s + \\widetilde U ) \\otimes w , \\nabla w _ k - \\nabla w \\rangle d \\tau = 0 . \\end{align*}"}
-{"id": "2735.png", "formula": "\\begin{align*} | u ^ 2 - v ^ 2 D | & = v ^ 2 \\left | \\left ( \\frac { u } { v } + \\sqrt { D } \\right ) \\left ( \\frac { u } { v } - \\sqrt { D } \\right ) \\right | \\\\ & \\geqslant v ^ 2 \\left ( 2 \\sqrt { D } - \\frac { \\varepsilon } { B ^ 2 } \\right ) \\frac { C ( D ) } { v ^ 2 } \\\\ & = 2 C ( D ) \\sqrt { D } - \\frac { \\varepsilon C ( D ) } { B ^ 2 } = 2 C ( D ) \\sqrt { D } - o ( 1 ) . \\end{align*}"}
-{"id": "6763.png", "formula": "\\begin{align*} \\| 1 _ { \\tilde \\Lambda } \\| _ { B ^ t _ { p , q } ( \\real ^ d ) } < \\infty \\ , , \\mbox { i f } 0 < t < 1 / p \\mbox { a n d } 0 < q < \\infty \\mbox { o r } t = 1 / p \\mbox { a n d } q = \\infty \\ , . \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} \\underset { l \\rightarrow \\infty } { \\lim } \\mathbf { X } _ { l } ^ { ( \\omega ) } \\left ( s _ { 1 } , s _ { 2 } \\right ) = \\mathbf { X } _ { \\infty } \\left ( s _ { 1 } , s _ { 2 } \\right ) \\ . \\end{align*}"}
-{"id": "2651.png", "formula": "\\begin{align*} S _ n ( U _ { p , \\varepsilon } ) : = \\{ L _ G ( e _ 1 ) L _ G ( e _ 2 ) \\cdots L _ G ( e _ n ) : e _ 1 e _ 2 \\cdots e _ n \\in \\Gamma _ n ( G ; f , U _ { p , \\varepsilon } ) \\} , \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{align*} { } _ B M _ A = \\bigoplus _ { ( g , e ) \\in F \\times E } { } _ g m _ e \\left ( B g \\otimes e A \\right ) \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { k \\geq 0 } { \\mathfrak p } _ k ( a , b , \\gamma , \\delta , \\epsilon ) x ^ k , Q ( x ) = \\sum _ { k \\geq 0 } { \\mathfrak q } _ k ( a , b , \\gamma , \\delta , \\epsilon ) x ^ k , \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} J = \\mathrm { d i a g } \\{ J _ 1 , J _ 2 , \\ldots , J _ s \\} \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} s _ { i + 1 } = \\sum _ { k = 1 } ^ m a _ k \\cdot s _ { k + 1 } . \\end{align*}"}
-{"id": "7004.png", "formula": "\\begin{align*} \\ - \\varepsilon u ^ { \\left ( 2 \\right ) } \\left ( x , \\varepsilon \\right ) + B u ^ { \\left ( 1 \\right ) } \\left ( x , \\varepsilon \\right ) + \\left ( A + \\lambda \\right ) u \\left ( x , \\varepsilon \\right ) = 0 , x \\in \\left ( 0 , T \\right ) , \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} f ^ { - 1 } \\circ \\psi _ { p _ n } ( \\zeta ) = \\psi _ { f ^ { - 1 } ( p _ n ) } ( \\lambda _ { p _ n , - 1 } \\cdot \\zeta ) . \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} \\sigma \\left ( A \\right ) = \\sigma \\left ( S \\right ) \\cup \\sigma \\left ( C \\right ) , \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} p _ u = \\frac { C _ u } { n } \\leq \\frac { C + \\alpha _ n } { n } \\leq \\frac { C + 1 } { n } \\leq \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{align*} \\mu _ 1 \\left ( A _ 1 \\right ) = 1 \\ \\textnormal { a n d } \\ \\mu _ 2 \\left ( A _ 1 \\right ) = 0 . \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} & ( - 1 ) ^ k s ^ k \\partial ^ k ( s ^ { \\lambda _ 1 - \\lambda _ 2 } \\mu ) \\\\ & = ( - 1 ) ^ k s ^ k \\sum _ { j = 0 } ^ k \\binom { k } { j } ( - 1 ) ^ { k - j } \\frac { \\Gamma ( k - j + \\lambda _ 2 - \\lambda _ 1 ) } { \\Gamma ( \\lambda _ 2 - \\lambda _ 1 ) } s ^ { \\lambda _ 1 - \\lambda _ 2 + j - k } \\partial ^ j \\mu \\\\ & = s ^ { \\lambda _ 1 - \\lambda _ 2 } \\sum _ { j = 0 } ^ k \\binom { k } { j } \\frac { \\Gamma ( k - j + \\lambda _ 2 - \\lambda _ 1 ) } { \\Gamma ( \\lambda _ 2 - \\lambda _ 1 ) } ( - 1 ) ^ j s ^ { j } \\partial ^ j \\mu \\geq 0 . \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{align*} k \\alpha = - \\frac { \\pi } { 2 } + ( - 1 ) ^ { ( n + 1 ) } \\arcsin p \\left ( x , y , k \\right ) + n \\pi . \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{align*} 2 H _ { 0 } \\left [ \\left ( \\frac { f } { p } \\right ) ^ { \\frac { 4 } { 3 } } \\left ( \\frac { r } { g } \\right ) ^ { \\frac { 2 } { 3 } } - \\left ( \\frac { p } { f } \\right ) ^ { \\frac { 2 } { 3 } } \\left ( \\frac { g } { r } \\right ) ^ { \\frac { 4 } { 3 } } \\right ] ^ { \\frac { 3 } { 2 } } = \\frac { f \\dot { p } } { p } + \\frac { g \\dot { r } } { r } - 2 . \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} C ( d , \\delta ) \\ge \\begin{cases} h ( w ) - { \\displaystyle \\max _ { 0 \\le x \\le \\delta / 2 } } \\Big [ w \\ , h \\Big ( \\frac { x } { w } \\Big ) + ( 1 - w ) h \\Big ( \\frac { x } { 1 - w } \\Big ) \\Big ] & \\textup { i f } w < \\frac { 1 } { 2 } \\\\ 1 - h ( \\delta ) & \\textup { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} a ^ { 2 } c + a ^ { 2 } + b ^ { 2 } + c ^ { 2 } + 1 = 0 , \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{gather*} \\displaystyle f '' ( r ) + \\Big ( \\frac { N - 1 } { r } + \\frac { r } { 2 } \\Big ) f ' ( r ) + \\frac { 1 } { \\alpha } f ( r ) + | f ( r ) | ^ { \\alpha } f ( r ) = 0 \\\\ f ( 0 ) = a , f ' ( 0 ) = 0 \\end{gather*}"}
-{"id": "370.png", "formula": "\\begin{align*} < P _ i , P _ j > = \\int ^ { b } _ { a } P _ i \\ P _ j \\ w d x = | | P _ i | | ^ 2 \\delta _ { i j } , \\ i , j \\in \\mathbb { N } _ 0 \\end{align*}"}
-{"id": "4681.png", "formula": "\\begin{align*} U ^ \\mu ( x ) = \\int \\log \\frac { 1 } { | x - t | } d \\mu ( t ) , \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} \\widetilde { G } ( \\mathbf { x } ) = \\sum \\limits _ { \\substack { \\mathbf { k } \\in \\mathbb { Z } ^ m \\\\ \\Vert \\mathbf { k } \\Vert _ \\infty \\leqslant X } } b _ X ( \\mathbf { k } ) e ( \\mathbf { k } \\cdot \\mathbf { x } ) + O _ { c , C , \\varepsilon } \\Big ( \\frac { \\log X } { \\sigma _ G X } \\Big ) , \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{align*} R ^ e _ n ( s _ 0 ) & = ( - e + 1 , 1 ) ( - e + 2 , 2 ) \\cdots ( 0 , e ) \\\\ R ^ e _ n ( s _ 1 ) & = ( 1 , e + 1 ) ( 2 , e + 2 ) \\cdots ( e , 2 e ) \\\\ R ^ e _ n ( s _ m ) & = ( ( m - 1 ) e + 1 , m e + 1 ) ( ( m - 1 ) e + 2 , m e + 2 ) \\cdots ( m e , ( m + 1 ) e ) , \\end{align*}"}
-{"id": "8258.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial x _ { j + 1 } } - \\frac { \\partial } { \\partial x _ j } \\right ) \\psi | _ { x _ { j + 1 } = x _ j ^ + } = \\alpha \\psi | _ { x _ { j + 1 } = x _ j ^ + } , \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} { \\displaystyle - 2 \\mathrm { i } \\Delta t ( \\partial \\theta ^ { k } _ { \\Psi } , \\partial { \\theta _ { \\Psi } ^ { k } } ) + \\Delta t B \\left ( \\overline { \\mathbf { A } } ^ { k } _ { h } ; \\overline { \\theta } _ { \\Psi } ^ { k } , \\partial { \\theta _ { \\Psi } ^ { k } } \\right ) = J _ 1 ^ { ( k ) } + J _ 2 ^ { ( k ) } + J _ 3 ^ { ( k ) } + J _ 4 ^ { ( k ) } , } \\end{align*}"}
-{"id": "4378.png", "formula": "\\begin{align*} \\sum _ { k \\ge 0 } \\binom { 2 k + e } { k } x ^ { k } = \\frac { 1 } { \\sqrt { 1 - 4 x } } \\left ( \\frac { 1 - \\sqrt { 1 - 4 x } } { 2 x } \\right ) ^ { e } , \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} L _ { 2 m k + m } + L _ { 2 m k - m } = L _ m L _ { 2 m k } \\ , , \\quad \\mbox { $ m $ e v e n } \\ , . \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{align*} \\phi _ \\lambda = 0 , \\qquad \\lambda \\in ( - 1 , 0 ) \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{align*} M _ 1 S ( 0 ) M _ 1 ^ { - 1 } = \\begin{bmatrix} I _ \\mu & 0 \\\\ 0 & - I _ { n - \\mu } \\end{bmatrix} , \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) ^ { p } = \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} + \\begin{pmatrix} 0 & b \\\\ c & 0 \\end{pmatrix} p \\end{align*}"}
-{"id": "3833.png", "formula": "\\begin{align*} b _ 1 ( 2 n ) ^ 2 - b _ 1 ( 2 n - 1 ) b _ 1 ( 2 n + 1 ) & = b _ 1 ( 2 n ) ^ 2 - b _ 1 ( 2 n - 2 ) b _ 1 ( 2 n ) \\\\ & = b _ 1 ( 2 n ) \\left ( b _ 1 ( 2 n ) - b _ 1 ( 2 n - 2 ) \\right ) = b _ { 1 } ( 2 n ) b _ { 1 } ( n ) . \\\\ \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} P _ 0 = \\xi ( P ) + [ \\ell ] Q ^ \\prime , \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} V ^ - ( z ) = \\exp \\Big ( - \\sum _ { n > 0 } \\frac { 1 } { n } h _ n z ^ { - 2 n } \\Big ) ; V ^ + ( z ) = \\exp \\Big ( \\sum _ { n > 0 } \\frac { 1 } { n } h _ { - n } z ^ { 2 n } \\Big ) \\end{align*}"}
-{"id": "8165.png", "formula": "\\begin{align*} { \\bf A } _ 1 = { \\bf F } \\left [ \\begin{array} { c c } { \\bf I } _ r & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf 0 } \\end{array} \\right ] { \\bf F } ' , \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ { n } A _ { i j } y _ { j } = \\left ( \\sum \\limits _ { j = 1 } ^ { n } c _ { i j } x _ { j } \\right ) \\mathbf { f = } \\left ( \\lambda x _ { i } \\right ) \\mathbf { f = } \\lambda \\left ( x _ { i } \\mathbf { f } \\right ) = \\lambda y _ { i } \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} p _ { j } ^ { X } ( A ) : = \\sup _ { p _ { j } ( x ) = 1 } p _ { j } ( A x ) , \\ , j \\in \\N , \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} \\left ( \\sum _ { i \\in I } f _ { i } \\right ) \\circ x = \\sum _ { i \\in I } ( f _ { i } \\circ x ) ; \\end{align*}"}
-{"id": "6029.png", "formula": "\\begin{align*} \\frac { ( \\alpha _ 2 - \\lambda _ i ) } { ( \\alpha _ 2 - \\lambda _ 1 ) } \\cdot \\mathbf M _ { i j } = \\frac { ( \\lambda _ { n + 2 } - \\lambda _ i ) ( \\lambda _ { n + 3 } - \\lambda _ i ) } { ( \\lambda _ { n + 2 } - \\lambda _ 1 ) ( \\lambda _ { n + 3 } - \\lambda _ 1 ) } \\cdot \\mathbf M _ { 1 j } \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} w _ R ( t ) : = \\rho _ R ( R t ) , \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} \\omega ( E ) _ j = \\frac { 1 } { \\pi } \\sum _ { k \\leq j } \\int _ { E _ { k - 1 } ^ + } ^ { E _ k ^ - } \\frac { g _ k ( E , t ) } { \\sqrt { ( t - E _ { k - 1 } ^ + ) ( E _ k ^ - - t ) } } d t . \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} F _ { t } ( x ^ 2 ) = \\frac { 1 } { ( 1 - x ) ^ { t } } F _ { t } ( x ) & = \\left ( \\sum _ { n = 0 } ^ { \\infty } { t + n - 1 \\choose n } x ^ n \\right ) \\left ( \\sum _ { n = 0 } ^ { \\infty } f _ { n } ( t ) x ^ n \\right ) \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\left ( \\sum _ { k = 0 } ^ { n } { t + n - k - 1 \\choose n - k } f _ { k } ( t ) \\right ) x ^ { n } . \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} \\frac { H _ u ( x _ 0 , r _ 1 ) } { r _ 1 ^ { n + a } } = \\frac { H _ u ( x _ 0 , r _ 0 ) } { r _ 0 ^ { n + a } } \\ , e ^ { 2 \\int _ { r _ 0 } ^ { r _ 1 } \\frac { I _ u ( x _ 0 , t ) } { t } \\d t } . \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} \\bigl ( y ( P ) \\bigr ) ^ 2 \\bigl [ 4 f ( x ( P ) ) x ( [ 2 ] P ) - g ( x ( P ) ) \\bigr ] = \\Delta ^ \\prime , \\end{align*}"}
-{"id": "9722.png", "formula": "\\begin{align*} G = \\left ( \\underbrace { \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} } _ { I } \\underbrace { \\begin{array} { c c c } 0 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ 1 & 0 & 0 \\end{array} } _ { C } \\underbrace { \\begin{array} { c c c } 0 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ 1 & 0 & 0 \\end{array} } _ { C } \\underbrace { \\begin{array} { c c c } 1 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ 1 & 0 & 1 \\end{array} } _ { I + C - C _ 1 } \\right ) \\end{align*}"}
-{"id": "3271.png", "formula": "\\begin{align*} \\mathfrak { T } _ { \\mathbb { G T } _ c ( N ) } = \\inf \\{ t > 0 : \\exists \\ 1 \\le i < j \\le n \\le N \\textnormal { s . t } X _ i ^ { ( n ) } ( t ) = X _ j ^ { ( n ) } ( t ) \\} , \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { ( t A ) ^ { n } } { n ! } \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} D _ { n , k } ( 1 , x ) & = k \\ , x \\ , D _ { n - 1 , 2 } ( 1 , x ) + D _ n ( 1 , x ) , \\ , \\ , \\ , \\ , n \\geq 1 . \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} \\mathbf { \\Xi } _ { \\mathrm { p } } \\left ( - t \\right ) = \\mathbf { \\Xi } _ { \\mathrm { p } } \\left ( t \\right ) ^ { \\mathrm { t } } , t \\in \\mathbb { R } \\ , \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} K ( s , t ) = \\begin{cases} w ( s ) - \\left ( w ( a ) + \\psi ( \\lambda ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ a , t ) , \\\\ w ( s ) - \\left ( w ( a ) + ( 1 + \\psi ( 1 - \\lambda ) ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ t , b ] . \\end{cases} \\end{align*}"}
-{"id": "2937.png", "formula": "\\begin{align*} \\| T ^ { \\ , n } z - x _ { l } \\| \\le \\Bigl | \\Bigl | \\ , T ^ { \\ , n } \\Bigl ( \\sum _ { s = 1 } ^ { j _ { m } + j } z _ { s } \\Bigr ) - x _ { l } \\ , \\Bigr | \\Bigr | + \\sum _ { s > j _ { m } + j } \\| T ^ { \\ , n } z _ { s } \\| . \\end{align*}"}
-{"id": "9904.png", "formula": "\\begin{align*} \\lambda ( \\rho ) = \\min _ { \\Phi } ~ ~ - \\textrm { d i v } ( \\rho \\nabla \\Phi ) ^ T \\cdot \\textrm { H e s s } \\mathcal { \\bar F } ( \\rho ) \\cdot \\textrm { d i v } ( \\rho \\nabla \\Phi ) \\ , \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{align*} B = \\left [ \\begin{array} { c c c c } 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & p \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} D _ 1 ^ a D _ 2 ^ b f = \\delta _ 1 ^ { * a } * \\delta _ 2 ^ { * b } * f . \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{align*} \\left [ \\phi _ 0 ^ * , \\phi _ 0 \\right ] = \\rho . \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{align*} J _ 1 ^ 0 ( k ) ^ { - 1 } J _ 1 ( k ) = I _ n + o ( 1 ) , | k | \\to \\infty , \\ ; \\ ; k \\in \\overline { \\mathbb { C } } ^ { + } . \\end{align*}"}
-{"id": "9717.png", "formula": "\\begin{align*} P = \\begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{align*} c _ k = \\sum _ { j = 1 } ^ { n - 1 } \\sigma _ j t _ j ^ k , ( k = 0 , \\dots , 2 n - 3 ) , c _ { 2 n - 2 } = \\sum _ { j = 1 } ^ { n - 1 } \\sigma _ j t _ j ^ { 2 n - 2 } + \\lambda . \\end{align*}"}
-{"id": "9958.png", "formula": "\\begin{align*} x ( t _ * ^ + ) = \\Pi _ j x ( t _ * ^ - ) \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} y ( [ n ] P ) = \\frac { \\Psi ^ \\prime _ { n + 2 } ( x , y ) \\left ( \\Psi ^ \\prime _ { n - 1 } ( x , y ) \\right ) ^ 2 - \\Psi ^ \\prime _ { n - 2 } ( x , y ) \\left ( \\Psi ^ \\prime _ { n + 1 } ( x , y ) \\right ) ^ 2 } { 4 y \\left ( \\Psi ^ \\prime _ { n } ( x , y ) \\right ) ^ 3 } , \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} \\hat { \\gamma } _ { C A ' B ' } ^ { ( n ) } : = \\left ( \\mathcal { B } _ \\eta \\otimes \\mathbb { I } _ { A ' B ' } \\right ) \\left ( \\hat { \\gamma } _ { A A ' } ^ { ( n ) } \\otimes \\hat { \\gamma } _ { B B ' } ^ { ( n ) } \\right ) \\ ; , \\end{align*}"}
-{"id": "4584.png", "formula": "\\begin{align*} { \\rm R e } \\langle g ( \\vec { e } ) , \\ , \\vec { e } \\rangle = \\Vert \\vec { e } \\Vert _ { H } ^ { 2 } - { \\rm R e } \\langle \\vec { c } , \\ , \\vec { e } \\rangle \\geq \\Vert \\vec { e } \\Vert _ { H } \\big ( \\Vert \\vec { e } \\Vert _ { H } - \\Vert \\vec { c } \\Vert _ { H } \\big ) , \\forall \\ , \\vec { e } \\in H , \\end{align*}"}
-{"id": "9527.png", "formula": "\\begin{align*} E _ { n , j } = \\left \\{ \\omega \\in \\Omega : \\left | \\sum _ { i = 1 } ^ r \\langle C _ { n , j } ( \\omega ) ^ * X ( \\omega ) C _ { n , j } ( \\omega ) \\xi _ i ( \\omega ) , \\eta _ i ( \\omega ) \\rangle \\right | < \\delta r K ^ 2 \\right \\} . \\end{align*}"}
-{"id": "2966.png", "formula": "\\begin{align*} X _ l : = \\Bigl \\| \\sum _ { k = b _ l } ^ { b _ { l + 1 } - 1 } \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ s \\Bigr ) \\ , x _ k e _ k \\Bigr \\| , l \\ge 0 . \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} t = t _ { \\rm T U R N I N G } = l n \\left ( \\frac { \\mu K _ 1 h _ M } { \\Gamma } \\right ) . \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{align*} \\varphi = \\begin{cases} 1 , { \\rm i f } \\ \\frac { 3 } { 4 } \\leq r \\leq \\frac { 6 } { 5 } \\\\ 0 , { \\rm i f } \\ r < \\frac { 2 } { 3 } \\ { \\rm o r } \\ r > \\frac { 5 } { 4 } \\end{cases} , | \\nabla \\varphi | \\lesssim 1 . \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} h ^ { 2 } ( P , Q ) = 1 - \\rho ( P , Q ) \\qquad \\mbox { e t } \\qquad \\rho \\left ( P ^ { \\otimes n } , Q ^ { \\otimes n } \\right ) = \\rho ^ { n } ( P , Q ) . \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} ( x _ { n } ) _ { \\omega } ( \\xi _ n ) _ { \\omega } : = ( x _ n \\xi _ n ) _ { \\omega } . \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} { N ^ { w \\ \\ , t '' } _ { t , t ' } } = \\begin{cases} 1 & \\ | t - t ' | + 1 \\leq t '' \\leq \\{ t + t ' - 1 , 2 w - t - t ' \\} , t + t ' + t '' \\ \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1160.png", "formula": "\\begin{align*} \\begin{cases} i \\frac { d \\rho } { d t } ( t ) = [ H ( t ) , \\rho ( t ) ] , \\ \\ \\ \\ \\ \\ \\ \\ & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ t \\in ( 0 , T ) , \\\\ \\rho ( 0 ) = \\rho ^ 0 , \\ \\ \\ \\ & ( [ H , \\rho ] = H \\rho - \\rho H ) , \\\\ \\end{cases} \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} \\hat { A } _ { n _ { \\epsilon } } ( n ) = \\left \\{ \\sum _ { i = 1 } ^ { n } t ^ { ( n ) } ( f _ i ) \\leq 2 \\mu n ^ { 1 + \\epsilon } \\right \\} \\end{align*}"}
-{"id": "9945.png", "formula": "\\begin{align*} E \\dot { x } = A x , \\end{align*}"}
-{"id": "9470.png", "formula": "\\begin{align*} ( P , Q ) = \\left ( \\begin{bmatrix} I _ { k - 1 } & T & . \\\\ . & 1 & . \\\\ . & . & I _ { n - k } \\end{bmatrix} , \\begin{bmatrix} I _ { k - 1 } & U & . \\\\ . & 1 & . \\\\ . & . & I _ { n - k } \\end{bmatrix} \\right ) \\end{align*}"}
-{"id": "7096.png", "formula": "\\begin{align*} U ^ * = \\{ t ^ * _ i : i \\le d \\} \\cup \\{ u ^ * : u \\in B ^ + \\setminus A _ e \\} \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} f \\circ W = W \\circ g . \\end{align*}"}
-{"id": "8062.png", "formula": "\\begin{align*} \\Delta _ { r , l } ( x ) : = \\{ ( y , s ) : s \\in ( \\widehat { \\gamma } ( l , r ) , 0 ) , y \\in ( \\widehat { \\pi } ^ { ( x , 0 ) } _ l ( s ) , \\widehat { \\pi } ^ { ( x , 0 ) } _ r ( s ) ) \\} . \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} \\dot \\phi _ t = F ( \\phi _ t ) , \\phi _ T = c > 0 , \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{align*} c o n s t \\times \\int _ { W ^ N } ^ { } \\det \\left ( \\phi _ { \\omega } ^ { ( j - 1 ) } ( x _ { N + 1 - i } ) \\right ) ^ N _ { i , j = 1 } \\Delta _ N ( x ) f ( x ) d x = \\int _ { \\mathbb { R } ^ N } ^ { } \\Delta ^ 2 _ N ( x ) F _ { \\omega } ( x _ 1 ) \\cdots F _ { \\omega } ( x _ N ) \\bar { \\hat { f } } ( x ) d x , \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} \\tau _ { k } = \\int \\limits _ { 0 } ^ { x _ { k } } \\gamma _ { k } ^ { - 1 } \\left ( y \\right ) d y \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{align*} \\mathfrak { L } _ s ^ { ( N ) } = h ^ { - 1 } _ N ( \\theta ) \\circ \\sum _ { i = 1 } ^ { N } L ^ { s , ( N ) } _ { \\theta _ i } \\circ h _ N ( \\theta ) - c o n s t _ { N , s } , \\end{align*}"}
-{"id": "4365.png", "formula": "\\begin{align*} \\Psi _ 1 ^ { - 1 } H \\Psi _ 1 = \\overline { \\Psi _ 1 ^ { - 1 } H \\Psi _ 1 } . \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} \\begin{pmatrix} M & N \\\\ [ . 1 c m ] \\overline { N } & \\overline { M } \\end{pmatrix} \\begin{pmatrix} z \\\\ [ . 2 c m ] \\overline { w } \\end{pmatrix} = \\begin{pmatrix} p \\\\ [ . 2 c m ] \\overline { p } \\end{pmatrix} \\end{align*}"}
-{"id": "8974.png", "formula": "\\begin{align*} B ^ N _ { \\tau _ t } = \\sup _ { 0 \\le \\tau \\le t } B ^ N _ { \\tau } . \\end{align*}"}
-{"id": "7894.png", "formula": "\\begin{align*} \\phi ( - \\nu ( x ) ) = \\phi ^ { \\circ \\circ } ( - \\nu ( x ) ) = \\sup _ { \\phi ^ \\circ ( y ) \\le 1 } - \\nu ( x ) \\cdot y = \\sup _ { y \\in B _ { \\phi ^ \\circ } = W _ \\phi } \\nu ( x ) \\cdot y = x \\cdot y . \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} Q = \\frac { v _ \\epsilon ''' } { v _ \\epsilon '' } + ( 1 - \\alpha ) ( \\log \\theta ) ' + A t \\end{align*}"}
-{"id": "9878.png", "formula": "\\begin{align*} \\sum _ { | \\gamma | = L } Q ( \\gamma ) \\leq \\sum _ { s = 1 } ^ { L / 2 + 1 } \\sum _ { | \\sigma _ 1 | + \\dots + | \\sigma _ s | = L } \\ \\frac { 1 } { s ! } \\prod _ { i = 1 } ^ { s } Q ( \\sigma _ i ) \\ : , \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{align*} ( 1 + J ^ s ) / ( 1 + J ^ { s + 1 } ) = J ^ s / J ^ { s + 1 } \\end{align*}"}
-{"id": "606.png", "formula": "\\begin{align*} \\begin{aligned} { \\xi } _ { j _ n } \\left ( \\bigcup _ { l \\neq n } K _ { j _ l } ^ { \\varepsilon / 3 } \\right ) < \\kappa / 4 \\end{aligned} \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} T _ i = [ g _ i ' ; m _ { i , 1 } , \\ldots , m _ { i , r _ i } ] \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} Q _ { r , a } ( X _ 1 , \\cdots , X _ r ) = \\sum _ { f _ 1 , f _ 2 , \\cdots , f _ r \\ge 0 } \\sigma _ a ( [ p ; f _ 1 , \\cdots , f _ r ] ) X _ 1 ^ { f _ 1 } \\cdots X _ r ^ { f _ r } \\end{align*}"}
-{"id": "7256.png", "formula": "\\begin{align*} [ y _ 1 , y _ 2 ] = y _ 3 , [ y _ 2 , y _ 1 ] = \\frac { \\beta _ 3 } { \\alpha _ 4 \\gamma _ 1 \\gamma _ 5 } y _ 5 , [ y _ 2 , y _ 2 ] = y _ 4 + \\theta y _ 5 , [ y _ 1 , y _ 3 ] = y _ 4 , [ y _ 2 , y _ 3 ] = y _ 5 , [ y _ 1 , y _ 4 ] = y _ 5 . \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{align*} F _ { \\omega } ( x ) = e ^ { i \\gamma _ 1 x - \\frac { \\gamma _ 2 } { 2 } x ^ 2 } \\prod _ { k = 1 } ^ { \\infty } \\frac { e ^ { - i \\alpha _ k ^ + x } } { 1 - i \\alpha _ k ^ + x } \\prod _ { k = 1 } ^ { \\infty } \\frac { e ^ { i \\alpha _ k ^ - x } } { 1 + i \\alpha _ k ^ - x } \\ . \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\mathcal M ( X ( T ) ; \\mathcal P _ X ) = \\bigcup _ { a \\in R _ 0 ( M ; \\mathcal P _ M ) } \\mathcal M ( X _ 1 ; a ; \\mathcal P _ { X _ 1 } ) \\times \\mathcal M ( X _ 2 ; a ; \\mathcal P _ { X _ 2 } ) . \\end{align*}"}
-{"id": "356.png", "formula": "\\begin{align*} w = \\sum _ { m , n } c ^ m _ n v _ m \\otimes f ^ n \\in V ( \\Lambda ) \\otimes V ( \\Lambda ) ^ * . \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} D _ { n , k } ( 1 , x ) & = k \\ , x \\ , D _ { n - 2 , 1 } ( 1 , x ) + D _ n ( 1 , x ) , \\ , \\ , \\ , \\ , n \\geq 2 \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} d \\varphi ( t ) = \\frac { \\sqrt { 6 } } { 3 \\ , y ( t ) ^ 5 } \\ , ( f ^ { 1 2 3 4 } + f ^ { 1 2 5 6 } + f ^ { 3 4 5 6 } ) = \\tau _ 0 ( t ) \\star _ t \\varphi ( t ) + \\star _ t \\tau _ 3 ( t ) , \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} S = \\begin{pmatrix} 4 & 3 & 5 \\\\ 5 & 4 & 3 \\\\ 3 & 5 & 4 \\end{pmatrix} C = \\begin{pmatrix} 4 & 3 \\\\ - 3 & 4 \\end{pmatrix} \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{gather*} \\left ( Z ( X _ { i , \\lambda } , T ) \\prod _ { j = 0 } ^ { n - 1 } \\big ( 1 - q ^ j T \\big ) \\right ) ^ { ( - 1 ) ^ n } = \\det \\big ( I - T \\mathrm { F r o b } ^ * \\colon H ^ { n } _ c ( U _ { i , \\lambda } ) \\big ) . \\end{gather*}"}
-{"id": "2001.png", "formula": "\\begin{align*} G _ { \\mu } ( \\{ z \\} _ N | \\{ \\overline { \\alpha } \\} ) = \\mathrm { d e t } _ N \\Bigg ( \\prod _ { j = 0 } ^ \\mu ( z _ k + \\alpha _ j ) - \\prod _ { j = 0 } ^ \\mu ( z _ k ^ { - 1 } + \\alpha _ j ) \\Bigg ) . \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} \\partial _ { t } u _ { m } = i \\left [ \\Delta u _ { m } + \\sum \\limits _ { j = 1 } ^ { N } \\left ( a _ { m j } + b _ { m j } \\left ( x , t \\right ) \\right ) u _ { j } \\right ] , x \\in R ^ { n } , t \\in \\left ( 0 , T \\right ) . \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} \\partial _ t f = \\mathcal { L } \\cdot f + \\mathcal { N } ( f ) , \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{align*} c ( T { \\widetilde { X } } ) = ( 1 + E ) \\left ( \\prod _ { i = 1 } ^ d \\frac { 1 + f ^ * Z _ i - E } { 1 + f ^ * Z _ i } \\right ) f ^ * c ( T X ) . \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} H _ n = \\left \\{ \\pi _ n \\subseteq B _ { 8 \\mu \\beta _ 1 ^ { - 1 } n ^ { 1 + \\epsilon } } \\right \\} \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{align*} \\dim F = \\sum _ { i = 1 } ^ \\infty ( 2 i + 1 ) f _ { 2 i + 1 } - \\sum _ { i = 1 } ^ \\infty ( 2 i - 1 ) f _ { 2 i } . \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{align*} w _ 0 = \\begin{cases} ( \\overline { 2 } , \\overline { 3 } , \\ldots , \\overline { n + 1 } , 1 ) & \\textrm { i f } k < n + 1 \\ / ; \\\\ ( 1 , \\overline { 2 } , \\overline { 3 } , \\ldots , \\overline { n + 1 } ) & \\textrm { i f } k = n + 1 \\ / ; \\end{cases} \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} P _ { n } T ^ { \\ , j } P _ { l } \\ , x = \\smash [ t ] { P _ { n } T ^ { \\ , j } \\ , \\Bigl ( \\sum _ { k = b _ { l + 1 } - N } ^ { b _ { l + 1 } - 1 } x _ { k } e _ { k } \\ \\Bigr ) , } \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} \\textbf { y } _ { 0 } & = \\textbf { H } _ { 0 } \\textbf { x } _ { 0 } + \\textbf { v } _ { 0 } + \\textbf { z } _ { 0 } \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} p _ v = \\sum _ { \\lambda \\in v \\Lambda ^ { e } } t _ \\lambda t _ \\lambda ^ * . \\end{align*}"}
-{"id": "153.png", "formula": "\\begin{align*} - 2 c _ k \\sum _ { j = k + 1 } ^ { 2 k } \\binom { 2 k } { j } ( - 1 ) ^ { j - k } ( 1 - \\cos ( ( j - k ) \\theta ) ) \\ , \\frac { d \\theta } { 2 \\pi } . \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{gather*} \\mathcal { B } ^ * : = \\left \\{ \\omega _ { \\mathbf { m } } \\colon 0 \\leq m _ i < d \\mbox { f o r } i = 0 , \\dots , n \\ \\ \\sum _ { i = 0 } ^ n m _ i \\equiv 0 \\bmod d \\right \\} \\end{gather*}"}
-{"id": "8730.png", "formula": "\\begin{align*} a _ n = \\sum _ { k = 0 } ^ { n } { n \\choose k } ( - 1 ) ^ k b _ k \\ ; \\ ; \\ ; \\mbox { \\rm i f a n d o n l y i f } \\ ; \\ ; \\ ; b _ n = \\sum _ { k = 0 } ^ { n } { n \\choose k } ( - 1 ) ^ k a _ k , \\end{align*}"}
-{"id": "1736.png", "formula": "\\begin{align*} \\Delta \\phi _ 1 = \\cos ^ { - 1 } \\left ( \\frac { r _ s - \\Delta T _ s ( v _ { o , m a x } + v _ c ) } { r _ s } \\right ) + \\sin ^ { - 1 } \\frac { r _ c } { r _ s } \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} N ( R ) : = \\# \\{ p _ n \\mid | p _ n | \\leq R \\} . \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{align*} \\det \\left ( \\hat { p } ^ { ( N + 1 ) , s } _ t ( x _ i , y _ j ) \\right ) ^ { N } _ { i , j = 1 } d y , \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{align*} u _ 0 = \\sum _ { k + 1 \\leq | \\alpha | \\leq m - 1 } a _ { \\alpha } x ^ { \\alpha } . \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} \\xi _ \\varepsilon ^ z ( y ) = - \\log ( | z - y | \\vee \\varepsilon ) + \\phi ^ z ( y ) , \\end{align*}"}
-{"id": "3845.png", "formula": "\\begin{align*} \\hat { P } _ { i , k + 1 } ( \\sqrt { x } , T ) & = P _ { i , k + 1 } ( x , T ) , \\\\ \\hat { P } _ { i + 2 ^ k , k + 1 } ( \\sqrt { x } , T ) & = P _ { i + 2 ^ k , k + 1 } ( x , T ) , \\\\ \\hat { Q } _ { i , k + 1 } ( \\sqrt { x } , T ) & = Q _ { i , k + 1 } ( x , T ) , \\\\ \\hat { Q } _ { i + 2 ^ k , k + 1 } ( \\sqrt { x } , T ) & = Q _ { i + 2 ^ k , k + 1 } ( x , T ) \\end{align*}"}
-{"id": "2900.png", "formula": "\\begin{align*} R _ { l } : = \\liminf _ { p \\to \\infty } \\bigl ( \\omega _ { p r + l } \\cdots \\omega _ { r + l } \\omega _ { l } \\bigr ) ^ { 1 / p } > \\max ( 1 , \\Vert A \\Vert ) . \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} \\psi = e ^ { 1 2 3 4 } + e ^ { 1 2 5 6 } + e ^ { 1 3 6 7 } + e ^ { 1 4 5 7 } + e ^ { 2 3 5 7 } - e ^ { 2 4 6 7 } + e ^ { 3 4 5 6 } . \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} \\{ \\lambda ^ n _ 0 ( B _ 1 , B _ 2 ) = t \\} = & \\{ | B _ 1 ( s _ 1 ) - B _ 2 ( s _ 1 ) | > 0 s _ 1 \\in [ t , t + 1 ] \\} \\\\ & \\cap \\{ | B _ 1 ( t ) - B _ 2 ( t ) | = 1 / n \\} \\cap ( E ^ n _ t ) ^ c , \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} \\begin{array} { l r l l l } H _ 0 \\left ( A , ( N \\otimes _ B M ) \\right ) = & ( e A f \\otimes h B g ) & \\oplus & ( e ' A f \\otimes h B g ' ) \\ \\ \\oplus \\\\ & ( e A f ' \\otimes h ' B g ) & \\oplus & ( e ' A f ' \\otimes h ' B g ' ) . \\end{array} \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ s ^ t \\langle \\widetilde U \\otimes ( w _ k - w ) , \\nabla \\varphi \\rangle d \\tau = 0 . \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\big ( \\deg _ { \\widetilde { X } } ( v _ { i } ) - 2 \\big ) \\leq N \\cdot \\big ( \\deg _ { \\widetilde { Y } } ( a ) - 2 \\big ) \\cdot ( 3 - 2 ) \\leq N \\cdot \\big ( \\deg _ { \\widetilde { Y } } ( a ) - 2 \\big ) \\cdot \\big ( \\deg _ { \\widetilde { Z } } ( b ) - 2 \\big ) . \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} L ' = \\{ ( \\emptyset , 2 4 ) < ( 2 , 1 2 4 ) < ( 2 4 , P ) \\} \\end{align*}"}
-{"id": "10021.png", "formula": "\\begin{align*} | S _ t h _ v ( x ) | \\lesssim v ^ { ( n + 1 ) / 2 } ( \\log 1 / v ) ^ { n - 1 } | t | ^ { - n / 2 } \\lesssim \\frac { v } { | t | ^ { n / 2 } } = \\frac { v } { | t | ^ \\gamma } \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} C _ { n } = \\sum _ { m = 1 } ^ { n } \\sum _ { \\underset { k _ { i } \\ge 1 } { k _ { 1 } + \\dots + k _ { m } = n } } C _ { k _ { 1 } - 1 } \\dots C _ { k _ { m } - 1 } . \\end{align*}"}
-{"id": "9989.png", "formula": "\\begin{align*} \\tilde { E } _ 2 = \\tilde { E } _ 1 - g h ^ T \\end{align*}"}
-{"id": "821.png", "formula": "\\begin{align*} \\bigl [ e ^ { t \\Delta } \\P \\theta _ 0 e _ 3 \\bigr ] _ { j } = \\Bigl ( \\int \\theta _ 0 \\Bigr ) K _ { j , 3 } ( \\cdot , t ) + \\sum _ { h = 1 } ^ 3 F _ { j , h , 3 } ( \\cdot , t ) * V _ h . \\end{align*}"}
-{"id": "707.png", "formula": "\\begin{align*} [ e _ i , f _ j ] = \\delta _ { i j } h _ j , [ h _ i , e _ j ] = a _ { i j } e _ j , [ h _ i , f _ j ] = - a _ { i j } f _ j \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} \\rho _ { \\alpha , \\varepsilon } ( x ) & : = 2 \\pi \\sqrt { 1 - x ^ 2 } \\psi _ { \\alpha , \\varepsilon } ( x ) \\\\ & = \\varepsilon _ { \\alpha } - \\varepsilon x + \\frac { 2 } { \\alpha } \\sqrt { 1 - x ^ 2 } \\arctan \\left ( \\frac { \\sqrt { 1 - x ^ 2 } } { \\sqrt { \\alpha ^ 2 - 1 } } \\right ) , x \\in [ - 1 , 1 ] \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} D _ { p ^ l + 1 , k } ( 1 , x ) & = \\Big ( \\frac { 1 } { 2 } - \\frac { k } { 4 } \\Big ) \\ , ( 1 - 4 x ) ^ { \\frac { p ^ l + 1 } { 2 } } + \\frac { k } { 4 } \\ , ( 1 - 4 x ) ^ { \\frac { p ^ l - 1 } { 2 } } + \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} Y _ i = \\frac { 1 - \\alpha _ i } { \\beta _ i } X _ i . \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{align*} H x _ i - x _ i ^ 2 = ( n - 1 ) \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{align*} \\| g \\| _ { q _ * , s } \\leq C \\| \\nabla g \\| _ { q , s } , \\frac { 1 } { q _ * } = \\frac { 1 } { q } - \\frac { 1 } { 3 } , q \\in ( 1 , 3 ) , \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{align*} d \\vec { L } = \\partial \\vec { L } + \\bar { \\partial } \\vec { L } \\in \\Omega ^ { ( 1 , 0 ) } ( D ^ 2 _ { \\ast } , \\C ^ n ) \\oplus \\Omega ^ { ( 0 , 1 ) } ( D ^ 2 _ { \\ast } , \\C ^ n ) \\end{align*}"}
-{"id": "2054.png", "formula": "\\begin{gather*} A _ 0 = \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} \\begin{pmatrix} 1 - p _ 1 - p _ 2 & \\theta ^ 0 \\end{pmatrix} , A _ { t _ i } = \\begin{pmatrix} q _ i \\\\ 1 \\end{pmatrix} \\begin{pmatrix} p _ i & \\theta ^ { t _ i } - p _ i q _ i \\end{pmatrix} , \\\\ N = \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} , N _ i = \\frac { q _ i ( p _ i q _ i - \\theta ^ { t _ i } ) } { t _ i } N , i = 1 , 2 . \\end{gather*}"}
-{"id": "7759.png", "formula": "\\begin{align*} X = ( I _ { 1 } , \\ldots , I _ { r - 1 } , I _ { r } , I _ { r + 1 } , \\ldots I _ { n } ) , \\\\ Y = ( I _ { 1 } , \\ldots , I _ { r - 1 } , I _ { r + 1 } , I _ { r } , \\ldots I _ { n } ) . \\\\ \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} \\lim _ { k ^ { \\prime } \\rightarrow \\infty } \\sup _ { k \\in ( k ^ { \\prime } , \\infty ) } \\left [ - 2 A ( x , y ) A _ { 0 } ( x , y ) \\cos \\left ( k \\alpha \\right ) + \\hat { f } \\left ( x , y , k \\right ) \\right ] = 2 A ( x , y ) A _ { 0 } ( x , y ) . \\end{align*}"}
-{"id": "1930.png", "formula": "\\begin{align*} \\int _ { P _ { k } } ( \\pi \\circ \\tau _ { k } \\circ v \\circ \\phi ) \\eta _ { k } d v _ { g _ { k } } = 0 , \\end{align*}"}
-{"id": "5386.png", "formula": "\\begin{align*} P _ { \\perp } \\begin{pmatrix} M z \\\\ [ . 1 c m ] \\overline { N } z \\end{pmatrix} = P _ { \\perp } \\begin{pmatrix} p \\\\ [ . 1 5 c m ] \\overline { p } \\end{pmatrix} \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{align*} \\mathfrak { h } _ 0 & = \\mathfrak { h } _ 1 , \\textrm { o n } \\{ | x | = a _ 1 , | y | \\le b _ 1 \\} ; \\\\ \\mathfrak { h } _ 0 & = \\mathfrak { h } _ 2 , \\textrm { o n } \\{ | x | \\le a _ 2 , | y | = b _ 1 \\} ; \\\\ \\mathfrak { h } _ 0 & = h _ s , \\textrm { o n } T _ s , \\ , s \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "8913.png", "formula": "\\begin{align*} y ( 0 ) = y _ 0 , \\quad { } y ' ( 0 ) = y ' _ 0 . \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\lambda } ^ d ( x \\in L _ n , y \\in L _ n ) & = E P _ { \\lambda } ^ { \\omega } ( \\forall 0 \\leq i \\leq n - 1 , U _ { x _ i , x _ { i + 1 } } < H _ { x _ i } , U _ { y _ i , y _ { i + 1 } } < H _ { y _ i } ) \\\\ & = E [ \\prod _ { i = 0 } ^ { n - 1 } F ( x _ i , y _ i ; x _ { i + 1 } , y _ { i + 1 } ) ] . \\end{align*}"}
-{"id": "8305.png", "formula": "\\begin{align*} f _ { 1 / n } ( \\tau ) & : = t _ n ( \\tau ) = \\left ( \\frac { \\eta ( \\tau ) } { \\eta ( n \\tau ) } \\right ) ^ { 2 4 / ( n - 1 ) } , \\\\ f _ { 1 / 1 } ( \\tau ) & : = \\frac { 1 } { t _ n ( \\tau ) } = \\left ( \\frac { \\eta ( n \\tau ) } { \\eta ( \\tau ) } \\right ) ^ { 2 4 / ( n - 1 ) } \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } \\frac { 1 } { m - n + 1 + i } < \\int _ { m - n } ^ { m } \\frac { d t } { t } = \\log \\frac { m } { m - n } < \\frac { n } { m - n } \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} u _ { \\beta } u _ { \\gamma } = \\sum _ { \\alpha } c _ { \\beta , \\gamma } ^ { \\alpha } u _ { \\alpha } \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\# { \\cal E } _ i = r \\right ) \\leq { n - 1 \\choose r - 1 } T _ r p _ u ^ { r - 1 } ( 1 - p _ d ) ^ { r ( n - r ) } . \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{align*} \\phi = \\phi _ 0 + \\phi _ { - 1 } + \\phi _ { < - 1 } \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} \\frac { \\eta ^ { i ' i '' } - \\sum _ { m ' , m '' = 1 } ^ \\infty ( R ^ { - 1 } _ { m ' } ) _ { j ' } ^ { i ' } \\eta ^ { j ' j '' } ( R ^ { - 1 } _ { m '' } ) _ { j '' } ^ { i '' } ( \\psi ' ) ^ { m ' } ( \\psi '' ) ^ { m '' } } { \\psi ' + \\psi '' } e _ { i ' } \\otimes e _ { i '' } \\end{align*}"}
-{"id": "4264.png", "formula": "\\begin{align*} \\Lambda ( E , s ) = N _ E ^ { s / 2 } ( 2 \\pi ) ^ { - s } \\Gamma ( s ) L ( E , s ) , \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\frac 1 2 \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] = \\bigg ( 1 - \\frac z 2 \\bigg ) ^ { - \\alpha } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 \\alpha & \\frac 1 2 + \\frac 1 2 \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac { z ^ 2 } { ( z - 2 ) ^ 2 } \\bigg ] . \\end{align*}"}
-{"id": "9376.png", "formula": "\\begin{align*} \\phi = \\sum _ { i , j , k } m _ { i j k } \\ , \\chi _ { F _ i ^ 1 } \\otimes \\chi _ { F _ j ^ 2 } \\otimes \\chi _ { F _ k ^ 3 } . \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{gather*} A ^ { S _ 1 G _ 1 S _ 2 } ( z ) = \\frac { 1 } { z ^ { 4 / 3 } } \\begin{pmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ - t & 0 & 0 \\end{pmatrix} \\\\ \\hphantom { A ^ { S _ 1 G _ 1 S _ 2 } ( z ) = } { } + \\frac { 1 } { z } \\begin{pmatrix} - p _ 1 q _ 1 - \\theta ^ 0 _ 1 - \\frac { 2 } { 3 } & 0 & 0 \\\\ 0 & p _ 2 q _ 2 - \\frac { 1 } { 3 } & 0 \\\\ 0 & 0 & p _ 1 q _ 1 - p _ 2 q _ 2 - \\theta ^ 0 _ 2 - 1 \\end{pmatrix} + \\cdots . \\end{gather*}"}
-{"id": "7526.png", "formula": "\\begin{align*} \\left ( 1 + \\frac { 1 } { 2 } G ( y ^ { - 1 } k y ) \\right ) ^ { - 1 } = \\left ( 1 - \\frac { 1 } { 2 } G ( y ^ { - 1 } k y ) \\right ) + O ( t ^ { - 2 } \\mathcal L ) \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} E ( k ) = \\mathbb { I } _ { n ! | \\mathcal { E } | ^ { n - 1 } } \\otimes \\begin{psmallmatrix} 0 & 1 \\\\ 1 & 0 \\end{psmallmatrix} \\otimes e ^ { i k \\boldsymbol { l } } . \\end{align*}"}
-{"id": "7378.png", "formula": "\\begin{align*} A { \\frac { { \\rm d } ^ { 2 } } { { \\rm d } { u } ^ { 2 } } } { \\it T _ 1 } \\left ( u \\right ) + B { \\frac { \\rm d } { { \\rm d } u } } { \\it T _ 1 } \\left ( u \\right ) + \\left ( C + D + E ) \\right ) { \\it T _ 1 } \\left ( u \\right ) = 0 , \\end{align*}"}
-{"id": "6684.png", "formula": "\\begin{align*} \\begin{aligned} ( \\Box - q ( x ) ) v ( x , t ) & = 0 , \\ \\ \\ \\ t > \\lvert x \\rvert \\\\ v ( x , \\lvert x \\rvert ) & = \\frac { 1 } { 8 \\pi } \\int \\limits _ { 0 } ^ { 1 } q ( s x ) d s . \\end{aligned} \\end{align*}"}
-{"id": "7261.png", "formula": "\\begin{align*} \\begin{cases} \\gamma _ 3 = - \\beta _ 5 & \\gamma _ 4 = - \\beta _ 6 \\\\ \\gamma _ 5 = - \\gamma _ 1 & \\gamma _ 6 = - \\gamma _ 2 \\\\ \\gamma _ 7 = 0 = \\gamma _ 8 \\end{cases} \\end{align*}"}
-{"id": "10031.png", "formula": "\\begin{align*} g ( J X , J Y ) = g ( X , Y ) , \\forall X , Y \\in { \\mathfrak X } ( M ) . \\end{align*}"}
-{"id": "3332.png", "formula": "\\begin{align*} p _ N ( a ) \\sim \\frac { 1 } { \\sqrt { 2 \\pi } a } N ^ { \\frac { \\alpha } { 2 } - 1 } \\ , { \\rm e } ^ { \\zeta _ 0 N ^ \\alpha } \\exp \\left ( \\sum _ { k = 1 } ^ { k _ - } \\zeta _ k N ^ { ( \\alpha - 1 ) k + \\alpha } \\right ) \\ , , \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\dot { y } = K - \\alpha ^ 2 y ^ 2 & \\hbox { ; } \\ ; \\ ; \\ ; ( a ) \\\\ y ( t _ { 0 } ) \\ ; \\hbox { g i v e n } & \\ ; \\ ; \\ ; ( b ) \\end{array} \\right . \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{align*} \\begin{array} { l } E ( X ) = \\sum _ { v \\in V } E ( X _ v ) = \\sum _ { v \\in V } P ( X _ v = 1 ) = n / 2 . \\end{array} \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} \\frac { \\lambda ^ { r + 1 } } { ( r + 1 ) ! } \\leq \\left ( \\frac { e \\cdot 2 d ^ 2 } { r + 1 } \\right ) ^ { r + 1 } \\leq 2 ^ { - r } = o ( 1 / n ) . \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} \\rho _ \\theta ( T ) = 1 - \\sum _ { \\emptyset \\neq S _ 1 \\subset T } \\pi _ { S _ 1 } + \\sum _ { \\substack { \\emptyset \\neq S _ 1 , S _ 2 \\subset T \\\\ S _ 1 < S _ 2 } } \\pi _ { S _ 1 } \\pi _ { S _ 2 } - \\sum _ { \\substack { \\emptyset \\neq S _ 1 , S _ 2 , S _ 3 \\subset T \\\\ S _ 1 < S _ 2 < S _ 3 } } \\pi _ { S _ 1 } \\pi _ { S _ 2 } \\pi _ { S _ 3 } + . . . . \\end{align*}"}
-{"id": "9906.png", "formula": "\\begin{align*} x _ { k + 1 } = f ( x _ { k } , u _ { k } ) , \\ ; k = 0 , 1 , \\hdots , \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} V _ k = \\{ ( x , y ) \\in \\R ^ 2 : k ^ 2 | x - k | \\le y \\le 2 k ^ 2 | x - k | \\le 1 \\} . \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} S = \\left ( \\begin{tabular} [ c ] { c | c } $ A + B $ & $ \\mathbf { x } $ \\\\ \\hline $ 2 \\mathbf { y } $ & $ u $ \\end{tabular} \\right ) C = A - B \\end{align*}"}
-{"id": "9359.png", "formula": "\\begin{align*} B _ { k , l , c , q } = F ^ { Q - q } B _ { k , l , c , Q } \\setminus F ^ { Q - q - 1 } B _ { k , l , c , Q } . \\end{align*}"}
-{"id": "7974.png", "formula": "\\begin{align*} \\| v _ { 0 \\varepsilon } - \\bar v _ 0 \\| _ { 3 , \\infty , \\mathbb R ^ 3 } = \\| v _ { 0 \\varepsilon } - v _ 0 \\| _ { 3 , \\infty } \\leq \\varepsilon . \\end{align*}"}
-{"id": "195.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c | c | c } \\alpha _ { t _ 1 } ^ { [ \\sigma ] } & \\dots & \\alpha _ { t _ m } ^ { [ \\sigma ] } \\end{array} \\right ) = \\begin{pmatrix} \\alpha _ { t _ 1 } & \\cdots & \\alpha _ { t _ { m } } \\\\ \\sigma ( \\alpha _ { t _ 1 } ) & \\cdots & \\sigma ( \\alpha _ { t _ { m } } ) \\\\ \\vdots & \\ddots & \\vdots \\\\ \\sigma ^ { n - 1 } ( \\alpha _ { t _ 1 } ) & \\cdots & \\sigma ^ { n - 1 } ( \\alpha _ { t _ { m } } ) \\end{pmatrix} . \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} & \\dim \\overline { N } = \\binom { c + a - e + 3 } { 3 } + \\binom { c + b - e + 3 } { 3 } + \\binom { c - a + 3 } { 3 } \\\\ & + \\binom { c - b + 3 } { 3 } - \\binom { a + b - e + 3 } { 3 } - \\binom { b - a + 3 } { 3 } - \\binom { 2 a - e + 3 } { 3 } \\\\ & - \\binom { 2 b - e + 3 } { 3 } - 3 - t ( e , a , b ) , \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} g ( r ) \\ ; = \\ ; g _ 0 \\ , r ^ { - B } + g _ 1 r ^ B + o ( r ^ { 1 / 2 } ) \\qquad \\textrm { a s } \\ ; r \\downarrow 0 \\ , . \\end{align*}"}
-{"id": "2162.png", "formula": "\\begin{align*} J ( k ) ^ { - 1 } = O \\left ( 1 \\right ) , J ( k ) = - i k A + B + Q ( 0 ) A + o ( 1 ) . \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { q - 1 } \\binom { 2 k } { k } a _ k x ^ k \\equiv ( 1 - 4 x ) ^ { ( q - 1 ) / 2 } \\sum _ { k = 0 } ^ { q - 1 } \\binom { 2 k } { k } b _ k \\left ( \\frac { - x } { 1 - 4 x } \\right ) ^ k \\pmod { p } . \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{align*} \\theta ( E ) = \\sum _ { i \\in Q _ 0 } \\theta _ i \\dim E _ i \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} \\tilde { F } _ { \\varepsilon } \\left ( x , t \\right ) = \\left [ \\sqrt { \\alpha _ { \\varepsilon } \\beta _ { \\varepsilon } } \\mu _ { \\varepsilon } \\left ( t \\right ) \\right ] ^ { \\frac { n } { 2 } + 2 } V _ { 1 } \\left ( \\sqrt { \\alpha _ { \\varepsilon } \\beta _ { \\varepsilon } } \\mu _ { \\varepsilon } \\left ( t \\right ) x , \\beta _ { \\varepsilon } t \\mu _ { \\varepsilon } \\left ( t \\right ) \\right ) , \\end{align*}"}
-{"id": "2854.png", "formula": "\\begin{align*} H : = \\left ( \\frac { \\partial ^ 2 } { \\partial x _ 1 ^ 2 } + \\frac { \\partial ^ 2 } { \\partial x _ 2 ^ 2 } \\right ) - \\sum \\limits _ { \\alpha \\in \\Pi } \\dfrac { \\mu _ \\alpha ( \\mu _ \\alpha + 1 ) } { l _ \\alpha ^ 2 ( x _ 1 , x _ 2 ) } . \\end{align*}"}
-{"id": "2615.png", "formula": "\\begin{align*} \\lambda _ { * } : = \\inf _ { k \\geq 1 } { \\lambda _ { k } \\over \\lambda _ { k + 1 } } > 0 . \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { i + 1 } c _ { i + 2 } ( j ) = P _ { i + 2 } ( 1 ) = F ( 2 ^ { - i - 1 } ) - ( - 1 ) ^ { i + 2 } F ( 0 ) = F ( 2 ^ { - i - 1 } ) \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} ( a - 1 ) ^ 2 ( a + 2 \\frac { b } { c } ) - 4 a \\frac { b } { c } = 9 \\frac { b ^ 2 } { c ^ 2 } . \\end{align*}"}
-{"id": "9314.png", "formula": "\\begin{align*} s ( \\mu ) \\mu + 1 = q \\left ( 1 - \\frac { \\mu } { L } \\right ) = \\left ( 1 - \\frac { \\mu } { L } \\right ) ^ { k + 1 } , \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , S _ 2 ] \\subseteq [ [ L _ { - r - 2 } , \\ , S _ r ] , \\ , S _ 2 ] = [ [ L _ { - r - 2 } , \\ , S _ 2 ] , \\ , S _ r ] \\subseteq [ L _ { - r } , \\ , S _ r ] . \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{align*} u ( x ) = \\sum _ { k = 1 } ^ \\infty g _ k \\Big ( \\frac r { R _ 0 } \\Big ) ^ k \\sin k \\theta \\ , . \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{align*} { } _ { \\mu , \\sigma \\ast } D _ { v , q ; z } ^ { \\alpha , \\eta , p } \\left ( { f ( z ) } \\right ) = e ^ { \\frac { \\sqrt { \\frac { 2 } { \\pi } } } { \\Gamma ( \\alpha ) } \\int _ { 0 } ^ { z } ( \\ln f ( t ) ) ( z - t ) ^ { \\alpha - 1 } t ^ { \\eta } { } I _ { v + \\frac { 1 } { 2 } } ( q ; \\frac { - p z ^ { ^ { \\mu + \\sigma } } } { t ^ { \\mu } \\left ( z - t \\right ) ^ { \\sigma } } ) d t } , \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{align*} B _ { \\varphi _ 1 - \\tau _ i \\varphi _ 2 } ( \\mathbf w _ i ' , \\mathbf w _ j ' ) = B _ { \\varphi _ 1 - \\tau _ j \\varphi _ 2 } ( \\mathbf w _ i ' , \\mathbf w _ j ' ) = 0 . \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{align*} { \\rm I m } \\ \\left ( F _ k ( \\Psi , \\overline { \\Psi } ) \\ \\ \\overline { \\psi _ k } \\right ) = 0 , \\ k = 1 , \\cdots , N . \\end{align*}"}
-{"id": "9718.png", "formula": "\\begin{align*} C _ 1 C & = C _ 1 ^ T C ^ T \\\\ & = ( C C _ 1 ) ^ T \\\\ & = C _ 1 ^ T = C _ 1 , \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} \\xi = b [ D _ \\infty ] - a [ D _ 0 ] \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} \\mathrm { T r } ( P _ C ( g _ i \\gamma ) ) = \\sum _ { j = 1 } ^ 4 a _ j \\mathrm { T r } ( P _ C ( g _ i g _ j ) ) , \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} B ( x , y ) = q ( x + y ) - q ( x ) - q ( y ) \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { m } \\left ( s _ o ( m , j ) - s _ e ( m , j ) \\right ) s _ { j , k } ^ { ( - 1 ) } & = \\delta _ { m , j } . \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} \\lambda [ 0 , T _ { k + 1 } ] = ( 2 ^ k + 1 ) \\| \\lambda ^ { ( k - 1 ) } \\| = ( 2 ^ k + 1 ) ( 2 ^ { k - 1 } + 1 ) \\cdots ( 2 ^ 2 + 1 ) \\cdot 2 > 2 ^ { \\frac { k ( k + 1 ) } { 2 } } . \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{align*} \\Lambda _ { N + 1 } ^ { \\infty } \\Lambda _ N ^ { N + 1 } = \\Lambda _ N ^ { \\infty } , \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} f & = \\max { ( f _ 1 , f _ 2 ) } = \\max { ( p _ 1 - q _ 1 , p _ 2 - q _ 2 ) } \\\\ & = \\max { \\left ( p _ 1 + q _ 2 - ( q _ 1 + q _ 2 ) , p _ 2 + q _ 1 - ( q _ 1 + q _ 2 ) \\right ) } \\\\ & = \\max { ( p _ 1 + q _ 2 , p _ 2 + q _ 1 ) } - ( q _ 1 + q _ 2 ) \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} \\int _ { \\Gamma _ + } A f \\phi d \\xi = \\int _ { \\Gamma _ - } f A _ { \\rm b a c k } \\phi d \\xi . \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} \\varphi ( x ^ \\ast ) = - \\frac 1 2 n + \\frac 1 2 n \\ln n - n \\ln y - \\lambda \\sqrt { n } - \\frac 1 3 \\frac { n ^ { 3 / 2 } } { y ^ 3 } + \\rm { s m a l l e r ~ t e r m s . } \\end{align*}"}
-{"id": "5208.png", "formula": "\\begin{align*} \\hat \\phi ( x , y , z , t ) & = \\Phi ( x , y , z ) \\cos ( \\omega t + \\delta ) , \\\\ \\hat \\eta ( x , y , t ) & = \\xi ( x , y ) \\sin ( \\omega t + \\delta ) , \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} u ( x , 0 ) = u _ 0 ( x ) \\end{align*}"}
-{"id": "7649.png", "formula": "\\begin{align*} K ( s , t ) = \\begin{cases} w ( s ) - \\left ( w ( a ) + \\psi ( \\lambda ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ a , t ) , \\\\ w ( s ) - \\left ( w ( a ) + ( 1 + \\psi ( 1 - \\lambda ) ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ t , b ] , \\end{cases} \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c l } T _ { ( \\alpha , 2 \\mu _ \\alpha ) } ^ + ( f ) & = & \\sum \\limits _ { j = 0 } ^ { \\mu _ \\alpha - 1 } f ^ { ( 2 j ) } _ \\alpha \\frac { \\displaystyle \\sigma ^ j } { \\displaystyle ( 2 j ) ! } \\\\ T _ { ( \\alpha , 2 \\mu _ \\alpha ) } ^ - ( f ) & = & \\sum \\limits _ { j = 0 } ^ { \\mu _ \\alpha - 1 } f ^ { ( 2 j + 1 ) } _ \\alpha \\frac { \\displaystyle \\sigma ^ j } { \\displaystyle ( 2 j + 1 ) ! } . \\end{array} \\right . \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} \\partial _ t F + v \\partial _ x F - \\partial _ x \\phi \\partial _ v F = \\gamma \\partial _ v \\left ( v F + \\partial _ v F \\right ) ~ , ~ \\Delta \\phi = c \\left ( \\int F d v - 1 \\right ) ~ . \\end{align*}"}
-{"id": "9696.png", "formula": "\\begin{align*} C _ { S ^ { \\ast } } \\ = \\ \\bigsqcup _ { u \\in \\mathbb { Z } ^ { d + 1 } } \\Pi _ { S ^ { \\ast } } + u _ 1 \\bar { v } _ 1 + \\cdots + u _ { d + 1 } \\bar { v } _ { d + 1 } \\ , . \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} \\pi _ { \\textnormal { n } ( \\theta ) } \\epsilon E = f \\iff \\epsilon ^ { \\textnormal { h o m } } ( \\theta ) E = f . \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} s _ 3 \\leq ( 2 5 n ) ^ { \\frac { 1 } { 2 } ( u + 1 ) } \\left ( 1 + \\sum _ { i = 1 } ^ { 3 u + 3 } p _ i ( \\kappa ' ) / n ^ i \\right ) e ^ { - 2 \\kappa ' } + ( 2 5 n ) ^ { \\frac { 1 } { 2 } ( u + 1 ) } ( q ( \\kappa ' ) / n ^ { 3 u + 4 } ) e ^ { 2 \\kappa ' } . \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{align*} S ( v _ 0 , k ) & = T ( v _ 0 , k - 1 ) S ( v _ 0 , k - 1 ) \\\\ & = ( 0 , - 1 , \\cdots , - k + 1 , 1 , \\cdots , k ) ( - k + 2 , 1 ) ( - k + 3 , 2 ) \\cdots ( 0 , k - 1 ) \\\\ & = ( - k + 1 , 1 ) ( - k + 2 , 2 ) \\cdots ( 0 , k ) . \\end{align*}"}
-{"id": "6512.png", "formula": "\\begin{align*} \\sum _ { g \\in G } b ( x g ) \\mu ( g ) = b ( x ) \\textrm { f o r a l l } x \\in G , \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} u _ 1 ( z ) = \\sum _ { n = 0 } ^ { a } c _ n z ^ n + \\varphi _ 1 ( z ) \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{align*} \\partial \\bar \\partial \\dot w \\wedge ( \\pi ^ * \\omega _ u ) ^ n & = 0 \\qquad \\bar D \\times X \\\\ \\dot w & = \\psi \\qquad \\partial D \\times X . \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} c _ { i , j - 1 } = c _ { i j } + a _ { i j } - a _ { i - 1 , j } \\end{align*}"}
-{"id": "9891.png", "formula": "\\begin{align*} F ^ C _ w ( \\mathbf { x } ) = \\sum _ \\lambda g _ { w \\lambda } s _ \\lambda ( \\mathbf { x } ) , g _ { w \\lambda } = \\sum _ \\mu 2 ^ { \\ell ( \\mu ) } \\big | \\mathcal { U T } _ w ( \\mu ) \\big | \\ h _ { \\mu \\lambda } \\ . \\end{align*}"}
-{"id": "2407.png", "formula": "\\begin{align*} \\frac d { d t } \\| u ( t ) \\| _ 2 ^ 2 & = \\| u ( t ) \\| _ { p + 1 } ^ { p + 1 } \\leq C \\| u ( t ) \\| _ { H ^ 1 ( \\mathbb R ) } ^ { p + 1 } , \\\\ \\frac d { d t } \\| \\nabla u ( t ) \\| _ 2 ^ 2 & = \\mathrm { R e } \\int _ { \\mathbb R } \\nabla ( | u ( t , x ) | ^ { p - 1 } u ( t , x ) ) \\cdot \\overline { \\nabla u ( t , x ) } d x \\leq C \\| u ( t ) \\| _ { H ^ 1 ( \\mathbb R ) } ^ { p + 1 } , \\end{align*}"}
-{"id": "9463.png", "formula": "\\begin{align*} h _ { 0 } ^ { \\varepsilon } = h _ { 0 } \\ast \\rho _ { \\varepsilon } = \\int _ { \\mathbb { T } ^ 3 } \\rho _ { \\varepsilon } ( x - y ) { \\rm d } \\ : h _ { 0 } ( y ) , \\ ; \\ ; \\ ; B _ { 0 } ^ { \\varepsilon } = B _ { 0 } \\ast \\rho _ { \\varepsilon } \\end{align*}"}
-{"id": "1227.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { N - 1 } \\big | \\mathcal { D } _ { N , \\ell } ^ { h , 2 , 2 } \\big | \\le \\frac { C ( 1 + | x | _ { L ^ p } ) ^ { K } } { \\tau ^ { 2 \\kappa } } \\bigl ( 1 + t _ { N - 1 } ^ { \\frac { 1 } { 2 } - \\frac { 1 } { 2 q } - \\kappa - \\beta } \\bigr ) \\Bigl ( | ( - A ) ^ { - \\beta } h | _ { L ^ { 2 q } } + \\frac { \\Delta t } { t _ { N - 1 } } | h | _ { L ^ { 2 q } } \\Bigr ) . \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} \\dot \\phi _ \\tau = - F ( \\phi _ \\tau ) , \\phi _ 0 = c > 0 . \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{align*} ( R _ j ( x ) ) _ { i _ 1 , \\dots , i _ j , \\dots , i _ n } \\colon = \\begin{cases} 0 , & i _ j = 1 \\\\ x _ { i _ 1 , \\dots , i _ j - 1 , \\dots , i _ n } , & i _ j > 1 , \\end{cases} \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} \\wp ''' ( z ) = 1 2 \\wp \\wp ' , \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} \\frac 1 2 \\big ( ( \\nabla _ { \\bar \\alpha } T ) _ \\gamma ^ { \\enskip \\bar \\alpha \\gamma } + ( \\nabla _ { \\alpha } T ) _ \\gamma ^ { \\enskip \\alpha \\bar \\gamma } \\big ) = - \\bar { z } _ { \\bar \\alpha } \\theta _ \\alpha - z _ \\alpha \\theta _ { \\bar \\alpha } \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} \\smash { \\Bigl | \\Bigl | T ^ { \\ , n } \\Bigl ( \\sum _ { s = 1 } ^ { j _ { m } + j } z _ { s } \\Bigr ) - x _ { l } \\Bigr | \\Bigr | } & \\smash [ t ] { < \\sum _ { u = 0 } ^ { j - 1 } 2 ^ { - ( j _ { m } + u ) } + \\| T ^ { \\ , i } z _ { j _ { m } + j } \\| } \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{align*} \\det \\left ( p ^ { ( N + 1 ) , s } _ t ( x _ i , y _ j ) \\right ) ^ { N + 1 } _ { i , j = 1 } d y . \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} \\sup _ { B _ { s } ( R - d ) } \\| X _ { ( \\Phi ^ t _ \\chi ) ^ \\ast F - F } ( \\psi , \\bar \\psi ) \\| _ { s } & = \\sup _ { B _ { s } ( R - d ) } \\| X _ { F \\circ \\Phi ^ t _ \\chi - F } ( \\psi , \\bar \\psi ) \\| _ { s } \\\\ & \\stackrel { \\eqref { l i e b r e s t s } } { \\leq } \\frac { 5 } { d } \\ , \\sup _ { B _ { s } ( R ) } \\| X _ \\chi ( \\psi , \\bar \\psi ) \\| _ { s } \\ , \\sup _ { B _ { s } ( R ) } \\| X _ F ( \\psi , \\bar \\psi ) \\| _ { s } \\\\ & < 2 \\sup _ { B _ { s } ( R ) } \\| X _ F ( \\psi , \\bar \\psi ) \\| _ { s } . \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} f ( t r _ x + ( 1 - t ) r _ y ) \\le t f ( r _ x ) + ( 1 - t ) f ( r _ y ) = t h ( x ) + ( 1 - t ) h ( y ) \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} \\ , ^ { A B C } \\ , _ { 0 } D ^ { \\alpha } _ { t } u _ k ( t ) + k ^ 2 \\pi ^ 2 u _ k ( t ) = f _ { k } ( t ) , \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{align*} \\Xi ^ { \\lambda } ( K _ { a } \\otimes F _ { a } \\otimes E _ { a } K _ { a } ^ { - 1 } \\otimes K _ { a } ) = \\sum _ { i , j , n , o } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) c _ { n } ^ { j } \\pi ( K _ { a } F _ { a } E _ { a } K _ { \\lambda } ) _ { o } ^ { n } c _ { i } ^ { o } . \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} \\deg ( f _ { m + 2 } ( X ) f ^ 3 _ m ( X ) = \\frac { 1 } { 2 } ( ( m + 2 ) ^ 2 - 1 ) + \\frac { 3 } { 2 } ( m ^ 2 - 1 ) = 2 m ^ 2 + 2 m , \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} T ( z ) = \\mathrm { i } \\ , \\frac { 1 + z } { 1 - z } , \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} \\sigma ( h ( \\tau ) ) = h ( \\sigma \\tau \\sigma ^ { - 1 } ) \\in X _ { ( \\chi ) } , \\end{align*}"}
-{"id": "9736.png", "formula": "\\begin{align*} k _ { i j } ( C , \\tilde C ) = \\min \\{ \\dfrac { m _ { j } . q _ { 1 } \\ldots q _ { i } + l _ { i } . n _ { 1 } . \\ldots n _ { j } } { 2 . l _ { i } . n _ { 1 } . \\ldots n _ { j } } , \\dfrac { m _ { j } . q _ { 1 } \\ldots q _ { i } + l _ { i } . n _ { 1 } . \\ldots n _ { j } } { 2 . m _ { j } . q _ { 1 } \\ldots q _ { i } } \\} . \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} u ( \\mathsf { x , } t ) = \\mathcal { N } \\left ( 2 \\pi \\sinh ( t ) \\right ) ^ { - \\frac { d } { 2 } } \\exp \\left [ - \\frac { \\coth ( t ) \\left \\vert \\mathsf { x } \\right \\vert ^ { 2 } } { 2 } \\right ] \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} m _ 1 = \\prod _ { \\substack { p \\in \\operatorname { D e } \\\\ p | m } } p ^ { v _ p ( m ) } , m _ 2 = \\prod _ { \\substack { p \\notin \\operatorname { D e } \\\\ p | m } } p ^ { v _ p ( m ) } . \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} \\iota _ { R ( \\theta ) } = \\begin{pmatrix} \\iota _ { R _ 1 ( \\theta ) } & 0 \\\\ 0 & \\iota _ { R _ 2 ( \\theta ) } \\end{pmatrix} , \\end{align*}"}
-{"id": "3985.png", "formula": "\\begin{align*} \\Psi _ i ( p , p ) - \\Psi _ i ( 0 , 0 ) \\equiv p \\cdot \\frac { \\partial \\Psi _ i ( x , 0 ) } { \\partial x } \\bigg | _ { x = 0 } + p \\cdot \\frac { \\partial \\Psi _ i ( 0 , y ) } { \\partial y } \\bigg | _ { y = 0 } \\pmod { p ^ 2 } , i = 1 , 2 . \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} \\mathcal { A } ^ { ( P , Q ) } = \\left ( \\mathcal { A } ^ { ( P , Q ) } _ { j _ 1 \\dots j _ n } \\right ) _ { j _ 1 , \\dots , j _ n = 1 } ^ { | \\mathcal { E } | } \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 - 1 } { s _ 2 } \\leq \\frac { \\lambda _ 2 \\cdot c _ 3 - c _ 3 } { s _ 3 } \\leq \\frac { \\lambda _ 3 - c _ 3 } { s _ 3 } = \\frac { \\alpha \\cdot c _ 3 + \\beta - c _ 3 } { s _ 3 } \\leq \\frac { \\alpha \\cdot c _ 4 + \\beta \\cdot u _ 3 - c _ 4 } { s _ 4 } \\leq \\frac { \\lambda _ 4 - c _ 4 } { s _ 4 } . \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{align*} f _ q ^ { \\gamma _ N } = f _ q \\chi _ { E _ q ^ { \\gamma _ N } } , \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} [ q ^ - _ i , q ^ + _ j ] = ( \\mathbb { I } _ 2 - 2 N _ { q _ i } ) ~ \\delta _ { i j } , [ N _ { q _ i } , q ^ + _ j ] = - \\delta _ { i j } q ^ + _ j , [ N _ { q _ i } , q ^ - _ j ] = + \\delta _ { i j } q ^ - _ j [ q _ i ^ - , q _ j ^ - ] = [ q _ i ^ + , q _ j ^ + ] = 0 . \\end{align*}"}
-{"id": "9884.png", "formula": "\\begin{align*} \\Pr { Y _ i \\leq t } & \\leq 2 ( 1 - e ^ { - t } ) + k ( 1 - e ^ { - t } - t e ^ { - t } ) \\leq 2 t + k t ^ 2 / 2 . \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} D = \\sum _ { i = 0 } ^ \\nu p _ i \\frac { d ^ i } { d x ^ i } , \\ p _ i \\in \\mathbb { P } _ { n _ i } , \\end{align*}"}
-{"id": "4088.png", "formula": "\\begin{align*} \\Omega = \\left \\{ B _ 1 , B _ 2 , B _ 3 , B _ 4 , B _ 5 \\right \\} , \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} \\varrho ^ { ( \\beta , \\chi _ { x } ^ { ( \\Omega ) } ( \\omega ) , \\vartheta , \\lambda ) } = \\varrho ^ { ( \\beta , \\omega , \\vartheta , \\lambda ) } \\circ \\chi _ { x } \\ , x \\in \\mathfrak { L } = \\mathbb { Z } ^ { d } \\ . \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} E = \\sum _ { j = 1 } ^ n k _ j ^ 2 . \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} \\sigma _ { t } = \\left ( \\lambda _ { 1 } + 2 t , \\lambda _ { 2 } \\pm t \\exp \\left ( i \\theta \\right ) , \\lambda _ { 3 } , \\ldots , \\overline { \\lambda } _ { 3 } , \\overline { \\lambda } _ { 2 } \\pm t \\exp \\left ( - i \\theta \\right ) \\right ) \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{align*} & \\left ( \\left | \\frac { e ^ { \\lambda y } \\sin ( \\theta - \\lambda \\xi ) - \\sin \\theta } { \\lambda } \\right | ^ 2 + \\left | \\frac { e ^ { \\lambda y } \\cos ( \\theta - \\lambda \\xi ) - \\cos \\theta } { \\lambda } \\right | ^ 2 \\right ) ^ 2 \\\\ & + 4 \\left ( \\frac { e ^ { 2 \\lambda y } - 1 } { 2 \\lambda ^ 2 y } ( - \\xi + \\lambda z ) + \\frac 1 { \\lambda ^ 2 } e ^ { \\lambda y } \\sin ( \\lambda \\xi ) \\right ) ^ 2 \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{align*} E _ i \\circ f ( x _ 1 , \\dots , x _ { i - 1 } , x _ i , x _ { i + 1 } , \\dots , x _ n ) = f \\bigl ( x _ 1 , \\dots , x _ { i - 1 } , 0 , x _ { i + 1 } , \\dots , x _ n \\bigr ) . \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} \\left | \\dfrac { d ^ { n + m } } { d x ^ { n + m } } \\phi ' ( x ) \\right | = M ( \\phi , m + 1 , j ) ^ n \\left | M ( \\phi , m + 1 , j ) ^ { - n } \\phi ^ { n + m + 1 } ( x ) \\right | \\leqslant M ^ n \\ , \\sup _ { n } p _ { ( m + 1 , j ) } \\big ( M ^ { - n } \\phi ^ { ( n ) } \\big ) , \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} a \\Big ( \\sum _ { i \\in Q _ 0 } \\xi _ i \\Big ) = \\theta ( a ) = \\Big ( \\sum _ { i \\in Q _ 0 } \\xi _ i \\Big ) a \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} \\chi _ { k , n } ^ { ( t _ { 1 } , . . . , t _ { k + n } ) } ( x _ { 1 } , . . . , x _ { n + k } | \\rho ) = \\sum _ { j \\geq 0 } \\rho ^ { j } \\prod _ { m = 1 } ^ { k } T _ { j + t _ { m } } ( x _ { m } ) \\prod _ { m = k + 1 } ^ { n + k } U _ { j + t _ { m } } ( x _ { m } ) , \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} y ^ { 3 } + z ^ { 3 } + y z ^ { 2 } + \\epsilon ( y ^ { 2 } + z ^ { 2 } ) + b y z = 0 . \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 0 } ^ { 2 } \\varepsilon ^ { \\frac { j } { 2 } } \\left \\vert \\lambda \\right \\vert ^ { 1 - \\frac { j } { 2 } } \\left \\Vert u ^ { \\left ( j \\right ) } \\right \\Vert _ { L _ { p } \\left ( R ; E \\right ) } + \\left \\Vert A u \\right \\Vert _ { L _ { p } \\left ( R ; E \\right ) } \\leq C \\left \\Vert \\bar { f } \\right \\Vert _ { L _ { p } \\left ( R ; E \\right ) } . \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} \\int \\limits _ { 0 } ^ { 1 } \\left \\Vert \\sum \\limits _ { j = 1 } ^ { m } r _ { j } \\left ( y \\right ) \\Psi \\left ( \\eta _ { j } , h \\right ) u _ { j } \\right \\Vert _ { E } d y \\leq C \\int \\limits _ { 0 } ^ { 1 } \\sum \\limits _ { i = 0 } ^ { 1 } \\left \\Vert \\sum \\limits _ { j = 1 } ^ { m } B _ { i } \\left ( \\eta _ { j } , h \\right ) r _ { j } \\left ( y \\right ) u _ { j } \\right \\Vert _ { E } d y \\leq \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{align*} \\nabla _ f U = 1 - R . \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} L _ r \\supseteq [ S _ 1 , \\ , S _ { r - 1 } ] \\supseteq [ S _ 1 , \\ , [ L _ { - 1 } , \\ , S _ r ] ] = [ [ L _ { - 1 } , \\ , S _ 1 ] , \\ , S _ r ] = S _ r \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} [ L _ { - r + 1 } , \\ , [ L _ { - 2 } , \\ , S _ r ] ] = 0 \\end{align*}"}
-{"id": "9973.png", "formula": "\\begin{align*} x ^ T Q _ 2 x = \\left [ \\begin{array} { c } x _ 1 \\\\ x _ 2 \\end{array} \\right ] ^ T \\tilde { Q } _ 2 \\left [ \\begin{array} { c } x _ 1 \\\\ x _ 2 \\end{array} \\right ] ^ T \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} \\sigma _ a ( F _ r ) = \\sigma _ a ( G _ r ) + p ^ { a \\ell } \\ , | F _ { r - 1 } | \\ , \\sigma _ { a - 1 } ( F _ { r - 1 } ) . \\end{align*}"}
-{"id": "7525.png", "formula": "\\begin{align*} y \\phi _ { - 1 } y ^ { - 1 } = O ( t ^ { - 1 / 2 } \\mathcal L ) , k = O ( t ^ { - 1 } \\mathcal L ) , G ( y ^ { - 1 } k y ) = O ( t ^ { - 1 } \\mathcal L ) . \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} \\alpha ^ \\prime & = \\alpha ( R - 2 ) ^ 2 \\\\ & = z ^ 3 - 4 z ^ 2 + 6 z - 3 \\end{align*}"}
-{"id": "6751.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { n } \\sum \\limits _ { i = 0 } ^ { m + 2 } \\varepsilon _ { k } ^ { \\frac { i } { m + 2 } } \\left \\vert \\lambda \\right \\vert ^ { 1 - \\frac { i } { m + 2 } } \\left \\Vert \\frac { \\partial ^ { i } u } { \\partial x _ { k } ^ { i } } \\right \\Vert _ { L ^ { q } \\left ( R _ { + } ^ { n } ; E \\right ) } + \\left \\Vert A u \\right \\Vert _ { L ^ { q } \\left ( R _ { + } ^ { n } ; E \\right ) } \\leq C \\left \\Vert f \\right \\Vert _ { W ^ { m , q } \\left ( R _ { + } ^ { n } ; E \\right ) } \\end{align*}"}
-{"id": "3562.png", "formula": "\\begin{align*} \\ 1 _ V \\ 1 _ W \\tau = \\ 1 _ { V \\cap W } \\tau . \\end{align*}"}
-{"id": "5829.png", "formula": "\\begin{align*} L _ m \\sum _ { k = 1 } ^ n { L _ { 2 m k } } = L _ { m + 2 m n } - L _ m \\ , , \\quad \\mbox { $ m $ o d d } \\ , . \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { M } | K _ 1 ^ { ( k ) } | \\leq \\frac { C } { \\Delta t } \\left ( h ^ { 2 r + 2 } + ( \\Delta t ) ^ { 4 } \\right ) + C \\sum _ { k = 1 } ^ { M } \\| \\mathbf { v } \\| _ { \\mathbf { L } ^ 2 } ^ { 2 } . \\end{align*}"}
-{"id": "4237.png", "formula": "\\begin{align*} \\psi ^ c ( s , \\theta _ { 1 } , \\theta _ { 2 } , \\dots , \\theta _ { d - 1 } ) \\ , = \\ , ( c _ { 1 } s \\theta ^ { a _ { 1 } } , c _ { 2 } s \\theta ^ { a _ { 2 } } , \\dots , c _ { n } s \\theta ^ { a _ { n } } ) . \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} | \\Delta _ k ( E _ 1 , \\ldots , E _ k ) | \\ge \\frac { \\left ( \\prod \\limits _ { j = 1 } ^ k | E _ j | - \\nu _ k ( 0 ) \\right ) ^ 2 } { \\sum \\limits _ { t \\in \\mathbb F _ q ^ * } \\nu _ k ^ 2 ( t ) } . \\end{align*}"}
-{"id": "885.png", "formula": "\\begin{align*} \\underset { \\kappa \\rightarrow 0 } { l i m } \\quad \\underset { T \\geq 1 } { s u p } \\mathbb { E } \\left | \\eta _ \\psi ^ T - \\eta _ { \\psi _ \\kappa } ^ T \\right | ^ 2 = 0 , \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} \\frac { d \\hat c _ 0 } { d t } = ( \\mu h _ F - \\Lambda T _ s ) e ^ { - t } - \\frac { \\Gamma } { K _ 1 } + \\Lambda T _ s + \\Lambda \\theta _ F . \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} x _ { 1 } & = \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\cosh \\rho \\cos \\phi \\\\ x _ { 2 } & = \\frac { 1 } { 2 } \\sinh \\rho \\sin \\theta \\sin \\phi \\\\ x _ { 3 } & = \\frac { 1 } { 2 } \\sinh \\rho \\cos \\theta \\sin \\phi \\\\ \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} { p _ { o u t , K } } = \\Pr \\left ( { G _ K \\triangleq \\prod \\nolimits _ { k = 1 } ^ K { \\left ( { 1 + { \\gamma _ k } } \\right ) } < { 2 ^ { \\mathcal R } } } \\right ) = { F _ { G _ K } } \\left ( { { 2 ^ { \\mathcal R } } } \\right ) , \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 , [ y _ 1 , y _ 2 ] = y _ 3 , [ y _ 2 , y _ 1 ] = \\theta _ 1 y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 2 ] = y _ 4 + \\theta _ 3 y _ 5 , [ y _ 1 , y _ 3 ] = y _ 5 . \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} \\begin{aligned} \\int \\limits _ { | x - e | + | x | \\leq 2 \\tau } \\frac { q ( x ) } { | x | | x - e | } d x & = \\frac { 1 } { 2 } \\int \\limits _ { \\cosh \\rho \\leq 2 \\tau } \\int \\limits _ { 0 } ^ { \\pi } \\int \\limits _ { 0 } ^ { 2 \\pi } q ( \\rho , \\theta , \\phi ) \\sinh \\rho \\sin \\phi d \\theta d \\phi d \\rho . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} \\xi _ { 1 } z + \\xi _ { 2 } = 0 . \\end{align*}"}
-{"id": "3656.png", "formula": "\\begin{align*} \\beta _ \\chi ( z ^ 2 ) = \\frac { \\chi ( z ) - \\chi ( - z ) } { 2 z } ; \\gamma _ \\chi ( z ^ 2 ) = \\frac { \\chi ( z ) + \\chi ( - z ) } { 2 } . \\end{align*}"}
-{"id": "7681.png", "formula": "\\begin{align*} 2 - 2 g ( C ) = C \\cdot C \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} \\| f \\| _ { A ^ { 2 \\alpha } _ \\alpha } = \\left ( \\int _ { \\mathbb { D } } | f ( z ) | ^ { 2 \\alpha } \\ , ( \\alpha - 1 ) ( 1 - | z | ^ 2 ) ^ \\alpha \\ , d \\mu ( z ) \\right ) ^ \\frac { 1 } { 2 \\alpha } \\leq \\| f \\| _ { H ^ 2 } . \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} \\begin{pmatrix} M z \\\\ [ . 1 c m ] \\overline { N } z \\end{pmatrix} + \\begin{pmatrix} N \\bar { z } \\\\ [ . 1 c m ] \\overline { M } \\bar { z } \\end{pmatrix} = \\begin{pmatrix} p \\\\ [ . 1 c m ] \\overline { p } \\end{pmatrix} \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} \\Psi ( 0 ) = \\frac { ( 1 + \\alpha ) _ b ( 1 + \\alpha - \\gamma - \\delta ) _ b } { ( 1 + \\alpha - \\gamma ) _ b ( 1 + \\alpha - \\delta ) _ b } \\cdot { } _ 4 F _ 3 \\bigg [ \\begin{matrix} 1 - \\epsilon & - b & \\gamma & \\delta \\\\ & 1 & \\alpha - \\epsilon + 1 & \\gamma + \\delta - b - \\alpha \\end{matrix} \\bigg | \\ , 1 \\bigg ] . \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} P ( P g - ( 1 + c ) P _ { n } g > 0 ) = P \\left ( P g - P _ { n } g > \\frac { c P g } { 1 + c } \\right ) . \\end{align*}"}
-{"id": "4687.png", "formula": "\\begin{align*} \\psi _ { \\alpha , 0 } ( x ) = \\frac { 1 } { 2 \\alpha } + \\frac { 1 } { \\alpha \\pi \\sqrt { 1 - x ^ 2 } } \\int _ 0 ^ { \\sqrt { \\alpha ^ 2 - 1 } } \\frac { s ^ 2 } { s ^ 2 + 1 - x ^ 2 } d s , x \\in [ - 1 , 1 ] . \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} \\begin{bmatrix} m _ 1 + m _ 3 & m _ 1 & m _ 3 \\\\ m _ 1 & m _ 1 + m _ 2 & m _ 2 \\\\ m _ 3 & m _ 2 & m _ 2 + m _ 3 \\end{bmatrix} , \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} \\partial _ { t } u = \\Delta u + A u + V \\left ( x , t \\right ) u , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} M _ L ( f x _ { n + 1 } ^ d , e ) = \\left [ \\begin{array} { c c c } C _ { ( 0 , 0 ) } & \\ldots & C _ { ( 0 , q - 1 ) } \\\\ \\vdots & & \\vdots \\\\ C _ { ( q - 1 , 0 ) } & \\ldots & C _ { ( q - 1 , q - 1 ) } \\\\ \\end{array} \\right ] \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} K _ { I , X } ( x , y ) = \\sqrt { \\rho _ { I , X } ( x ) \\rho _ { I , X } ( y ) } \\sum _ { j = 0 } ^ { N - 1 } \\varphi _ j ( x ) \\varphi _ j ( y ) . \\end{align*}"}
-{"id": "7258.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 1 e _ 3 + \\alpha _ 2 e _ 4 + \\alpha _ 3 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 4 e _ 3 + \\alpha _ 5 e _ 4 + \\alpha _ 6 e _ 5 , [ e _ 2 , e _ 1 ] = \\beta _ 1 e _ 3 + \\beta _ 2 e _ 4 + \\beta _ 3 e _ 5 , \\\\ [ e _ 2 , e _ 2 ] = \\beta _ 4 e _ 3 + \\beta _ 5 e _ 4 + \\beta _ 6 e _ 5 , [ e _ 1 , e _ 3 ] = \\gamma _ 1 e _ 4 + \\gamma _ 2 e _ 4 , [ e _ 2 , e _ 3 ] = \\gamma _ 3 e _ 4 + \\gamma _ 4 e _ 5 . \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = t _ 1 & x = t _ 2 & x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\begin{matrix} 0 \\\\ \\theta ^ 0 \\end{matrix} & \\begin{matrix} 0 \\\\ \\theta ^ { t _ 1 } \\end{matrix} & \\begin{matrix} 0 \\\\ \\theta ^ { t _ 2 } \\end{matrix} & \\overbrace { \\begin{matrix} 1 & \\theta ^ \\infty _ 1 / 2 \\\\ - 1 & \\theta ^ \\infty _ 1 / 2 \\end{matrix} } \\ \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "4453.png", "formula": "\\begin{align*} \\mathbf { E } _ { t } ^ { \\mathbf { A } } \\left ( \\mathbf { x } \\right ) : = \\int \\nolimits _ { 0 } ^ { 1 } \\left [ E _ { \\mathbf { A } } ( t , \\alpha x ^ { ( 2 ) } + ( 1 - \\alpha ) x ^ { ( 1 ) } ) \\right ] ( x ^ { ( 2 ) } - x ^ { ( 1 ) } ) \\mathrm { d } \\alpha \\ . \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{align*} \\| \\pi _ j ( [ x ] _ { j + 1 } ) \\| _ j = \\inf _ { p _ j ( z ) = 0 } p _ j ( x - z ) \\leqslant \\inf _ { p _ j ( z ) = 0 } \\left \\{ p _ j ( x - w ) + p _ j ( z - w ) \\right \\} = p _ j ( x - w ) \\end{align*}"}
-{"id": "9348.png", "formula": "\\begin{align*} \\mathcal { V } _ r ( \\gamma ) ( z ) = \\mathcal { V } _ r ( \\sigma ) ( z ) + O _ r ( \\mathcal { V } ^ 2 _ r ( \\sigma ) ( z ) ) , \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{align*} f _ * \\frac { 1 } { 1 + E } \\cap [ E ] = \\frac { 1 } { c ( N _ Z X ) } \\cap [ Z ] = \\prod _ { i = 1 } ^ d \\frac { Z _ i } { 1 + Z _ i } , \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} | c _ k - d _ k | = | c _ 0 - d _ k | = | c _ 0 - d _ 0 - n \\varepsilon | \\geq - | c _ 0 - d _ 0 | + n \\varepsilon \\geq ( n - 1 ) \\varepsilon \\end{align*}"}
-{"id": "8596.png", "formula": "\\begin{align*} u _ 2 ( z ) = \\int _ { D ( 2 | z | ) } + \\int _ { D \\setminus D ( 2 | z | ) } = u _ 2 ^ 1 ( z ) + u _ 2 ^ 2 ( z ) \\end{align*}"}
-{"id": "10013.png", "formula": "\\begin{align*} f ( w ) = g ( w ) \\displaystyle \\prod _ { p = 1 } ^ { q - 1 } ( t _ { w ( p ) } - t _ n ) , \\ \\ \\ f ( v ) = g ( v ) \\displaystyle \\prod _ { p = 1 } ^ { q - 1 } ( t _ { v ( p ) } - t _ n ) \\end{align*}"}
-{"id": "5073.png", "formula": "\\begin{align*} g ( e ^ { i t } ) = \\exp \\left ( \\frac { i } { 2 } \\arg f ( e ^ { 2 i t } ) \\right ) \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} - \\partial _ { y _ i } P ( t ) \\textbf { 1 } _ { ( - \\infty , y ' _ j ] } ( y _ i ) = \\hat { p } _ t ( y _ j ' , y _ i ) \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} M [ G _ \\alpha ] = M [ F _ \\alpha ] [ H _ \\alpha ] \\models | \\mathcal { B } ( T _ \\alpha ) | \\geq | X _ \\alpha | . \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{align*} \\delta = \\frac { \\delta _ 0 } { d _ { l _ 0 } + 1 } \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} \\left \\{ \\ \\begin{aligned} & \\widetilde w _ \\ell ( t ) = \\tilde a _ \\ell \\ , \\Im \\frac { \\zeta ^ \\ell - \\zeta ^ { k \\pi / \\omega } } { \\ell - k \\pi / \\omega } + b _ \\ell \\ , \\Re \\zeta ^ \\ell \\\\ & \\mbox { w i t h } \\tilde a _ \\ell = ( g ^ \\omega _ \\ell - g ^ 0 _ \\ell \\ , \\cos \\ell \\omega ) \\ \\frac { \\ell - k \\pi / \\omega } { \\sin \\ell \\omega } \\quad \\mbox { a n d } b _ \\ell = g ^ 0 _ \\ell \\ , . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3869.png", "formula": "\\begin{gather*} \\left ( \\frac { x - ( - 1 + h ) } { a } \\right ) ^ 2 + \\left ( \\frac { y } { b } \\right ) ^ 2 = 1 , a = h \\frac { \\rho _ 0 + 1 / \\rho _ 0 } { 2 } , b = h \\frac { \\rho _ 0 - 1 / \\rho _ 0 } { 2 } , \\end{gather*}"}
-{"id": "2981.png", "formula": "\\begin{align*} P _ { l } T ^ { \\ , j } P _ { l } \\ , x = \\sum _ { k = b _ { l } } ^ { b _ { l + 1 } - 1 } x _ { k } \\ , P _ l T ^ { \\ , j } \\ , e _ { k } = \\sum _ { k \\ , \\in \\ , [ b _ { l } , b _ { l + 1 } ) \\backslash I _ { j } } x _ { k } \\ , P _ l T ^ { \\ , j } \\ , e _ { k } + \\sum _ { k \\ , \\in \\ , I _ { j } } x _ { k } \\ , P _ l T ^ { \\ , j } \\ , e _ { k } \\end{align*}"}
-{"id": "3385.png", "formula": "\\begin{align*} d X _ t = b ( t , X _ t , \\mu _ t , \\gamma _ t ) d t + \\sigma ( t , X _ t , \\mu _ t , \\gamma _ t ) d W _ t + \\beta ( t , X _ { t - } , \\mu _ { t - } , \\gamma _ { t } ) d \\widetilde N _ t , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "5577.png", "formula": "\\begin{align*} b = \\sum _ { F \\nsubseteq H _ i } w ( F ) \\mbox { a n d } \\sum _ { F \\nsubseteq H _ i \\cup H _ j } w ( F ) = d . \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{align*} { } _ { \\mu , \\sigma \\ast } D _ { v , q ; z } ^ { \\alpha , \\eta , p } ( a ^ { ( z - \\xi ) ^ { r } } ) : = e ^ { \\frac { \\ln a ( - \\xi ) ^ { r } B ( \\eta , \\alpha - 1 ) z ^ { \\eta + \\alpha } } { \\Gamma ( \\alpha ) } F _ { v , q ; p } ^ { ( \\mu , \\sigma ) } ( - r , \\eta ; \\eta + \\alpha - 1 ; \\frac { z } { \\xi } ) . } . \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t \\theta + u \\cdot \\nabla \\theta = \\Delta \\theta \\\\ & \\partial _ t u + u \\cdot \\nabla u + \\nabla p = \\Delta u + \\theta e _ 3 \\\\ & \\nabla \\cdot u = 0 \\\\ & u | _ { t = 0 } = u _ 0 , \\ ; \\ , \\theta | _ { t = 0 } = \\theta _ 0 \\end{aligned} \\right . x \\in \\R ^ 3 , t \\in \\R _ { + } . \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} z ^ { x ' } _ i = - \\frac { ( \\alpha _ i - \\alpha _ 1 ) ( \\alpha _ i - \\alpha _ 2 ) \\cdots ( \\alpha _ i - \\alpha _ { \\check i } ) \\cdots ( \\alpha _ i - \\alpha _ { n } ) } { ( \\alpha _ i - \\lambda _ 1 ' ) ( \\alpha _ i - \\lambda _ 2 ' ) \\cdots ( \\alpha _ i - \\lambda _ { n + 3 } ' ) } \\in \\C . \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} \\rho _ { F } ( \\Omega _ { 2 } ) = \\frac { 1 } { \\Lambda } < \\rho _ { F } ( \\Omega ) = \\rho _ { F } ( \\Omega _ { 1 } ) , \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} \\sharp T _ K ( \\varepsilon , \\eta , d , B ) = \\frac { ( \\varepsilon - \\eta ) K ^ 2 } { 2 \\alpha ^ 2 d ^ 2 } B ^ { 2 - \\frac { 1 } { r } } + O _ { \\varepsilon , \\eta } \\left ( \\frac { K ^ 2 B ^ { 2 - \\frac { 1 } { r } } } { N d ^ 2 } \\right ) + O \\left ( \\varDelta ( \\alpha ) N \\log \\left ( K B \\right ) \\right ) . \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{align*} L ^ q _ \\sigma ( \\Omega ) = \\{ u \\in L ^ q ( \\Omega ) ; \\ , \\mbox { d i v $ u $ } = 0 , \\ , \\nu \\cdot u | _ { \\partial \\Omega } = 0 \\} , \\end{align*}"}
-{"id": "1320.png", "formula": "\\begin{align*} \\frac { f ^ { 3 } } { \\lambda _ { 4 } p \\dot { p } } - 2 \\frac { f p } { \\dot { p } } + \\lambda _ { 4 } \\frac { p ^ { 3 } } { f \\dot { p } } = 0 . \\end{align*}"}
-{"id": "9742.png", "formula": "\\begin{align*} \\partial _ t b = b \\otimes b : D _ x ^ 2 b \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{gather*} ( 1 - 2 \\rho _ { j m } \\cos ( \\beta _ { j , m } ( i _ { j } , i _ { m } ) ) + \\rho _ { j m } ^ { 2 } ) ( 1 - 2 \\rho _ { j m } \\cos ( \\beta _ { j , m } ( i _ { j } , - i _ { m } ) ) + \\rho _ { j m } ^ { 2 } ) \\\\ = w _ { 2 } ( \\cos ( \\alpha _ { j } ) , \\cos ( \\alpha _ { m } ) | \\rho _ { j m } ) . \\end{gather*}"}
-{"id": "124.png", "formula": "\\begin{align*} ( S ^ 2 - 1 ) \\alpha - ( S - 1 ) ^ 2 \\alpha = 2 ( S - 1 ) \\alpha \\in \\ell ^ 4 \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} & c _ 1 ( F ) = e + b - c , \\\\ & c _ 2 ( F ) = c ^ 2 - a ^ 2 - b c - e ( c - a - b ) , \\\\ & c _ 3 ( F ) = l ( Z ) = c _ 2 ( E ( - b ) | _ S ) = ( c - a ) ( c - b ) ( c + a - e ) . \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} a _ { i , j } ( k ) = a \\end{align*}"}
-{"id": "8707.png", "formula": "\\begin{align*} \\int _ { \\R ^ { d } } \\frac { | y | ^ { 2 } } { 1 + | y | ^ { 2 } } \\phi ( y ) M ( x , \\d y ) & = \\int _ { U } \\frac { | c _ { n } ( x , u ) | ^ { 2 } } { 1 + | c _ { n } ( x , u ) | ^ { 2 } } \\phi ( c _ { n } ( x , u ) ) \\nu ( \\d u ) \\\\ & \\le \\| \\phi \\| _ { \\infty } \\int _ { U } \\psi _ { n } ( x ) ^ { 2 } | c ( x , u ) | ^ { 2 } \\nu ( \\d u ) \\le \\kappa \\psi _ { n } ( x ) ^ { 2 } \\| \\phi \\| _ { \\infty } [ | x | ^ { 2 } \\zeta ( | x | ^ { 2 } ) + 1 ] \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} \\| \\cdot \\| _ p : = \\left ( \\int _ \\Omega | \\cdot | ^ p d V _ { 2 n } \\right ) ^ \\frac { 1 } { p } \\mbox { a n d } \\| \\cdot \\| _ \\infty : = \\sup _ \\Omega | \\cdot | , \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} Z _ i = 0 \\end{align*}"}
-{"id": "9874.png", "formula": "\\begin{align*} A \\coloneqq A ( G , s ) = \\max _ { v \\in V } \\min _ { \\gamma \\in \\Gamma ( { s , v } ) } \\sum _ { x y \\in E ( \\Gamma ) } \\min \\{ Y _ { x , y } , Y _ { y , x } \\} . \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{align*} ( n - 1 ) t ^ 2 = ( n - 1 ) \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{align*} \\mathcal { B } ^ * : = \\{ \\mathbf { x _ i ^ * } : i \\leqslant u \\} \\cup \\{ \\Xi ( \\mathbf { w _ j } ) ^ * : j \\leqslant d - u \\} \\end{align*}"}
-{"id": "6222.png", "formula": "\\begin{gather*} ( e _ 0 ^ - ) ^ i ( e _ 1 ^ - ) ^ i = ( q - q ^ { - 1 } ) ^ { - 2 i } ( X _ { 0 1 } - X _ { 3 1 } ) ^ i ( X _ { 2 3 } - X _ { 1 3 } ) ^ i ( i \\in \\mathbb { N } ) . \\end{gather*}"}
-{"id": "4100.png", "formula": "\\begin{align*} d ( l _ 1 \\otimes x _ 2 \\otimes \\dots \\otimes x _ { n - 1 } \\otimes l _ n ) & = l _ 1 x _ 2 \\otimes x _ 3 \\otimes \\dots x _ { n - 1 } \\otimes l _ n \\\\ & + \\sum _ 2 ^ { n - 2 } ( - 1 ) ^ { i + 1 } l _ 1 \\otimes \\dots \\otimes x _ i x _ { i + 1 } \\otimes \\dots \\otimes l _ n \\\\ & + ( - 1 ) ^ n l _ 1 \\otimes x _ 2 \\otimes \\dots \\otimes x _ { n - 1 } l _ n \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} \\alpha ' ( \\mathbf u ) = \\sum ( a \\alpha _ i + b ) ( c \\alpha _ i + d ) v _ i w _ i ' = a c \\alpha ^ 2 ( \\mathbf v ) + ( a d + b c ) \\alpha ( \\mathbf v ) + b d \\mathbf v , \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} \\pi _ { P ( \\theta _ k ) } q _ k \\rightharpoonup \\langle q , e ^ { \\i \\langle \\theta , \\cdot \\rangle _ { \\C ^ d } } \\rangle e ^ { \\i \\langle \\theta _ , \\cdot \\rangle _ { \\C ^ d } } - \\sum _ { z \\in \\Z ^ d \\backslash \\{ 0 \\} } \\frac { ( \\theta + 2 \\pi z ) ( \\theta + 2 \\pi z ) ^ T } { \\vert \\theta + 2 \\pi z \\vert ^ 2 } c ^ { ( z ) } _ { q } e ^ { \\i \\langle \\theta + 2 \\pi z , \\cdot \\rangle _ { \\C ^ d } } = \\pi _ { P ( \\theta ) } q . \\end{align*}"}
-{"id": "9423.png", "formula": "\\begin{align*} \\partial _ t b + ( v \\cdot \\nabla ) b = ( b \\cdot \\nabla ) v - \\tau \\nabla \\times d , \\ ; \\ ; \\partial _ t d + ( v \\cdot \\nabla ) d = ( d \\cdot \\nabla ) v + \\tau \\nabla \\times b , \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} \\int _ { E _ j ^ - } ^ { E _ j ^ + } \\frac { P _ E ( x ) } { \\sqrt { Q _ E ( x ) } } d x = 0 , \\ ; \\ ; j = 1 , 2 , . . . , n . \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } t } ( x + y ) = a + c - b x - d y < a + c - \\frac { b d } { b + d } ( x + y ) \\leq 0 \\mbox { p r o v i d e d } x + y \\geq \\frac { ( b + d ) ( a + c ) } { b d } . \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{align*} a \\ , \\mathcal { H } \\ , b & \\Rightarrow a a ^ { - 1 } = b b ^ { - 1 } a ^ { - 1 } a = b ^ { - 1 } b , \\\\ & \\Rightarrow ( a a ^ { - 1 } ) \\theta = ( b b ^ { - 1 } ) \\theta ( a ^ { - 1 } a ) \\theta = ( b ^ { - 1 } b ) \\theta , \\\\ & \\Rightarrow ( a \\theta ) ( a \\theta ) ^ { - 1 } = ( b \\theta ) ( b \\theta ) ^ { - 1 } ( a \\theta ) ^ { - 1 } ( a \\theta ) = ( b \\theta ) ^ { - 1 } ( b \\theta ) \\\\ & \\Rightarrow a \\theta \\ , \\mathcal { H } \\ , b \\theta , \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} \\langle b , b ' \\rangle _ { \\mu } = \\sum _ { g \\in G } \\langle b ( g ) , b ' ( g ) \\rangle \\mu ( g ) \\end{align*}"}
-{"id": "5886.png", "formula": "\\begin{align*} 0 & < p \\big ( \\phi ( p - 1 ) - \\phi ( p + 1 ) \\big ) + \\phi ( p - 1 ) + \\phi ( p + 1 ) \\\\ & \\leq p \\big ( \\phi ( p - 1 ) - \\phi ( p + 1 ) \\big ) + \\tfrac { 1 } { 2 } ( p - 1 ) + \\tfrac { 1 } { 2 } ( p + 1 ) \\\\ & = p \\big ( \\phi ( p - 1 ) - \\phi ( p + 1 ) + 1 \\big ) \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} \\langle a ( D ) u , \\psi \\rangle : = \\langle u , a ( D ) \\psi \\rangle , \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} ( f \\otimes g ) _ { A _ i \\otimes B _ j } = ( - 1 ) ^ { i | g | } f _ i \\otimes g _ j . \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 3 } { x ^ 2 } + \\frac { A _ 2 } { x } + A _ 1 + A _ 0 x \\right ) Y , \\\\ \\frac { \\partial Y } { \\partial t _ 1 } = \\left ( x B _ { 1 1 } + B _ { 1 0 } \\right ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = \\left ( - \\frac { A _ 3 } { t _ 2 x } + B _ { 2 0 } \\right ) Y , \\end{gather*}"}
-{"id": "9489.png", "formula": "\\begin{align*} \\deg _ { \\Omega _ { k } } ( S ) = \\sum _ { \\ell = 2 k } ^ n \\deg _ { \\Omega _ { k , \\ell } } ( S ) \\le 2 k ^ 2 \\Delta ( \\Omega _ { k - 1 , \\ell - 1 } ) \\le 2 ^ { k } ( k ! ) ^ 2 b _ k , \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} \\alpha _ { i } ( s ) = \\left ( a _ { i } ( s ) \\delta _ { 0 } , 0 \\right ) , 1 \\leq i \\leq | A | , \\ \\beta _ { i } ( s ) = \\left ( b _ { i } ( s ) \\delta _ { k _ { s } } , 0 \\right ) , 1 \\leq i \\leq | B | . \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} n _ { i j } ^ { m } = \\left \\{ \\begin{array} { l r } ( - 1 ) ^ { j - i } { { k + j - i } \\choose { k } } , & { \\rm i f } ~ ~ i \\leq j , \\\\ 0 , & { \\rm i f } ~ ~ i > j , \\end{array} \\right . \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} h ( W , \\nabla _ X Z ) & = X ^ { 0 , 1 } h ( W , Z ) + h ( W , [ X ^ { 0 , 1 } , Z ] ) + h ( [ W , X ^ { 0 , 1 } ] , Z ) , \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{align*} \\Lambda ( t _ k ) = R ^ { - 1 } \\Lambda ( 0 ) R , \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} L _ { 2 n + 1 } - 4 ( - 1 ) ^ { n - 1 } = L _ { 2 n + 1 } - ( - 1 ) ^ { n - 1 } L _ 3 = 5 F _ { n - 1 } F _ { n + 2 } \\ , , \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} L = R ^ 2 p _ { 2 * } \\mathbb { E } _ { \\mathcal { Y } } ( c - e - 4 ) , \\ \\ \\ L ' = R ^ 2 p _ { 2 * } \\mathbb { E } _ { \\mathcal { Y } } ( b - e - 4 ) , \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} f ( p ^ * ( \\lambda y _ 1 + ( 1 - \\lambda ) y _ 2 ) | \\lambda y _ 1 + ( 1 - \\lambda ) y _ 2 ) = \\lambda f ( p ^ * ( y _ 1 ) | y _ 1 ) + ( 1 - \\lambda ) f ( p ^ * ( y _ 2 ) | y _ 2 ) , \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{align*} \\lambda _ { c o s t } = \\Lambda ( 0 ) = B ( 0 ) = \\mu _ 1 ( 0 ) = \\mu _ 2 ( 0 ) = 0 \\end{align*}"}
-{"id": "6771.png", "formula": "\\begin{align*} 2 ^ { k t } & \\| \\tilde \\Pi _ 1 ( ( S ^ { k - 2 } \\varphi ) ^ \\Gamma , 1 _ { \\Lambda , k } ^ \\Gamma ) \\| _ { B ^ s _ { p , \\infty } ( \\real ^ { d _ s } ) } \\le 2 ^ { k t } \\sum _ { n = k - 1 } ^ { k + 1 } \\| ( \\tilde S ^ { n - 2 } ( S ^ { k - 2 } \\varphi ) ^ \\Gamma ) \\cdot \\tilde S _ n ( 1 _ { \\Lambda , k } ^ \\Gamma ) ) \\| _ { B ^ s _ { p , \\infty } } \\ , . \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{gather*} { \\rm { t r } } ( X _ { 0 1 } X _ { 2 3 } ) = 2 + ( q - q ^ { - 1 } ) ^ 2 a ^ { - 1 } , \\\\ { \\rm { t r } } ( X _ { 1 2 } X _ { 3 0 } ) = 2 + ( q - q ^ { - 1 } ) ^ 2 a , \\end{gather*}"}
-{"id": "7896.png", "formula": "\\begin{align*} 0 = F ^ \\ast \\Phi + \\lambda \\Psi . \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} s p _ \\lambda ( \\{ z \\} _ N | \\{ \\overline { \\alpha } \\} ) = \\frac { G _ { \\lambda + \\delta } ( \\{ z \\} _ N | \\{ \\overline { \\alpha } \\} ) } { \\mathrm { d e t } _ N ( z _ j ^ { N - k + 1 } - z _ j ^ { - N + k - 1 } ) } , \\end{align*}"}
-{"id": "9785.png", "formula": "\\begin{align*} v _ { \\epsilon } ^ j = F ^ j _ { y _ j } \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{align*} \\mathbf { Q } _ { H _ { d B } } = \\mathbf { Q } _ { A _ { d B } } + K ^ 2 \\mathbf { Q } _ \\varphi + i K \\left [ \\mathbf { Q } _ { A _ { d B } , \\varphi } ^ T - \\mathbf { Q } _ { A _ { d B } , \\varphi } \\right ] , \\end{align*}"}
-{"id": "9398.png", "formula": "\\begin{align*} T _ 1 ( \\theta v ) = \\pi ( \\theta ) T _ 1 ( v ) , v \\in L ^ 2 ( \\Omega ; E ) , \\ , \\theta \\in L ^ \\infty . \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{gather*} P _ { x , y } ( z ) = z ^ d Q _ V ( z ^ { - 1 } ) . \\end{gather*}"}
-{"id": "2171.png", "formula": "\\begin{align*} \\begin{bmatrix} X _ 1 & X _ 2 \\\\ X _ 3 & X _ 4 \\end{bmatrix} ^ { - 1 } = \\begin{bmatrix} X _ 1 ^ \\dag & X _ 3 ^ \\dag \\\\ X _ 2 ^ \\dag & X _ 4 ^ \\dag \\end{bmatrix} . \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{align*} f _ { p l a n a r } & = \\sqrt { f _ { m a x } ^ 2 - ( m g ) ^ 2 } \\\\ \\mathbf { f } _ { w } & = K _ d | | \\mathbf { v } _ { w } | | ^ 2 ( - \\mathbf { x } _ W ) \\\\ K _ d & = \\frac { 1 } { 2 } \\rho C _ { D } A _ { x _ W } \\end{align*}"}
-{"id": "778.png", "formula": "\\begin{align*} & \\sum _ { ( i , j ) \\in N } \\left ( 1 - \\cos \\left ( 2 \\pi x _ { ( i , j ) } \\right ) \\right ) \\\\ & ( x _ { ( i , j ) } ) _ { ( i , j ) \\in N } \\in \\left [ 0 , \\frac { 1 } { 2 } \\right ] ^ N , \\\\ & \\forall \\ , ( k , \\ell ) \\in S , \\ ; 4 x _ { ( k , \\ell ) } + \\sum _ { \\| ( i , j ) - ( k , \\ell ) \\| _ 1 = 1 } x _ { ( i , j ) } \\geq | v _ { ( k , \\ell ) } | . \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} \\Theta ( x , t , w ) = e ^ { - \\int _ 0 ^ t \\sigma ( x + s w , w ) d s } \\rho ( x + t w ) . \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} \\mathbb { P } ( w _ { n + 1 } & \\leq x , w _ { n - k } < K , w _ { n - j } = K , j = 0 , \\ldots k - 1 ) \\\\ & = \\int _ { 0 - } ^ { K - 0 } [ G ( a _ k ^ x ( w ) ) - G ( b _ k ( w ) ) ] d W _ { n - k } ( w ) , k = 0 , \\ldots , n , \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} \\varphi _ \\nu ( x ) & = \\frac { \\sqrt x } { \\sqrt { 2 \\pi } } \\frac { \\Gamma ( \\alpha + \\beta ) } { \\Gamma ( \\alpha ) \\Gamma ( \\beta ) } \\int _ 0 ^ \\infty e ^ { - u x / 2 } u ^ { \\beta - 1 } d u = \\frac { 2 ^ { \\beta } \\Gamma ( \\alpha + \\beta ) } { \\sqrt { 2 \\pi } \\Gamma ( \\alpha ) } x ^ { 1 / 2 - \\beta } \\\\ & = \\frac { x ^ { \\delta / 2 - 1 } } { 2 ^ { \\delta / 2 } \\Gamma ( \\delta / 2 ) } . \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} [ [ L _ { - 1 } , L _ 1 ] , \\ , [ L _ { r } , \\ , L _ { - 1 } ] ] = [ [ L _ { - 1 } , L _ 1 ] , \\ , S _ s ] ] = S _ s = [ L _ { - 1 } , L _ r ] \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{gather*} z \\phi _ { - 1 } z ^ { - 1 } = y \\phi _ { - 1 } y ^ { - 1 } + O ( t ^ { - 3 / 2 } \\mathcal L ) \\\\ z \\phi _ { < - 1 } z ^ { - 1 } = O ( t ^ { ( - 1 - \\epsilon ) / 2 } \\mathcal L ) \\end{gather*}"}
-{"id": "489.png", "formula": "\\begin{align*} F = \\Z { p ^ { f _ 1 } } \\times \\Z { p ^ { f _ 1 + f _ 2 } } \\times \\cdots \\times \\Z { p ^ { f _ 1 + f _ 2 + \\cdots + f _ r } } \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} \\begin{bmatrix} \\pi & b d ^ { - 1 } \\pi ^ n \\\\ 0 & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\beta \\\\ & 2 \\beta \\end{matrix} \\bigg | \\ , z \\bigg ] = \\bigg ( 1 - \\frac z 2 \\bigg ) ^ { - \\alpha } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 \\alpha & \\frac 1 2 + \\frac 1 2 \\alpha \\\\ & \\frac 1 2 + \\beta \\end{matrix} \\bigg | \\ , \\frac { z ^ 2 } { ( 2 - z ) ^ 2 } \\bigg ] . \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{align*} * _ 0 ( 1 ) = \\frac { 1 } { 6 \\delta _ 0 } ( d \\eta ) ^ 3 , \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} L _ { j + 2 } \\cap X _ { 1 , \\ , j + 2 , \\ , 1 } = 0 . \\end{align*}"}
-{"id": "10018.png", "formula": "\\begin{align*} S _ t f ( x ) = K _ t \\star f ( x ) , \\ \\ t > 0 , \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} \\begin{bmatrix} a _ 1 \\\\ a _ 2 \\\\ \\vdots \\\\ a _ n \\end{bmatrix} & = A _ n ^ { - 1 } \\left ( \\underset { : = B _ { b , m } } { \\underbrace { b _ { m + 1 } - \\sum _ { s = \\pm 1 } \\sum _ { k = 1 } ^ { \\lfloor \\frac { \\sqrt { 2 4 m + 1 } - s } { 6 } \\rfloor } ( - 1 ) ^ { k + 1 } b _ { m + 1 - k ( 3 k + s ) / 2 } } } \\right ) _ { 0 \\leq m < n } . \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\xi ^ { \\mathbf { c } } } L = \\dfrac { d } { d \\varepsilon } | _ { \\varepsilon = 0 } ( e ^ { \\sigma _ { \\varepsilon } } L ) = \\mu L , \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { 2 ^ { n - 1 } - 1 } f _ { s _ 2 \\left ( k \\right ) + 1 } = \\left [ z ^ n \\right ] f \\left ( \\frac { z } { 1 - z } \\right ) = \\sum _ { k = 1 } ^ { n } f _ k \\binom { n - 1 } { n - k } . \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{align*} \\mu ^ { s , N + 1 } _ { H P } \\Lambda _ N ^ { N + 1 } = \\mu ^ { s , N } _ { H P } . \\end{align*}"}
-{"id": "4761.png", "formula": "\\begin{align*} \\begin{bmatrix} 4 m & 4 m & 4 m \\\\ 4 m & 4 m & 4 m \\\\ 4 m & 4 m & 4 m \\end{bmatrix} , \\end{align*}"}
-{"id": "3346.png", "formula": "\\begin{align*} \\partial _ t ^ \\beta f ( t ) = \\frac { 1 } { \\Gamma ( 1 - \\beta ) } \\frac { d } { d t } \\int _ 0 ^ t ( t - s ) ^ { - \\beta } \\left ( f ( s ) - f ( 0 ) \\right ) d s , \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} F ( x , t ) = F _ 1 + F _ 2 + F _ 3 + F _ 4 + F _ 5 \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{align*} \\mu _ g = \\beta A _ \\sigma ^ { 2 \\sigma } . \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} \\tau = \\kappa _ { \\beta } \\cos \\theta , \\ , \\kappa = - \\kappa _ { \\beta } \\sin \\theta , \\ , \\theta ' = \\tau _ { \\beta } . \\end{align*}"}
-{"id": "859.png", "formula": "\\begin{align*} Z _ H ^ k ( t ) : = a _ { k , d } \\int _ { \\mathbb { R } ^ k } ^ { ' } \\int _ 0 ^ t \\prod _ { j = 1 } ^ k ( s - x _ j ) _ + ^ { d - 1 } d s W ( d x _ 1 ) \\ldots W ( d x _ k ) , \\end{align*}"}
-{"id": "5200.png", "formula": "\\begin{align*} ( g \\circ f ) \\cdot x = ( g \\cdot x ) \\circ f = g \\circ ( f \\cdot x ) . \\end{align*}"}
-{"id": "975.png", "formula": "\\begin{align*} \\gamma _ { p , q } = \\gamma _ { a ' , b ' } + 4 . \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} \\varepsilon = \\sup _ { [ 0 , T ^ * ] } \\int _ { [ \\gamma > 0 ] } | A | ^ 4 d \\mu \\le \\varepsilon _ 0 \\end{align*}"}
-{"id": "1.png", "formula": "\\begin{align*} \\hat c _ 0 ( t ) & = - \\mu \\hat h _ { 0 0 } e ^ { - t } - \\frac { \\Gamma } { K _ 1 } t + d _ 0 , \\\\ \\hat h _ 0 ( t ) & = \\hat h _ { 0 0 } e ^ { - t } . \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} U _ { 2 m + 1 } ^ { ( 2 ) } & = U _ { m } U _ { m + 1 } \\\\ & = p U _ { m } U _ { m } - q U _ { m } U _ { m - 1 } \\\\ & = p U _ { 2 m } ^ { ( 2 ) } - q U _ { 2 m - 1 } ^ { ( 2 ) } . \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{align*} J _ T ( x ) = { } & \\begin{pmatrix} 1 & 0 \\\\ i ( x ^ 2 - 1 ) ^ { 1 / 2 } _ - e ^ { - N \\xi _ - ( x ) } & 1 \\end{pmatrix} \\begin{pmatrix} 0 & \\frac { 1 } { \\sqrt { 1 - x ^ 2 } } \\\\ - \\sqrt { 1 - x ^ 2 } & 0 \\end{pmatrix} \\\\ & \\times \\begin{pmatrix} 1 & 0 \\\\ - i ( x ^ 2 - 1 ) ^ { 1 / 2 } _ + e ^ { - N \\xi _ + ( x ) } & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "4677.png", "formula": "\\begin{align*} \\prod _ { n = \\lceil \\gamma R \\rceil + 1 } ^ { \\infty } \\left ( 1 - \\frac { \\gamma ^ 2 R ^ 2 t ^ 2 } { n ^ 2 } \\right ) ^ 2 \\geq e ^ { - \\gamma ^ 2 N V ( t ) } . \\end{align*}"}
-{"id": "8980.png", "formula": "\\begin{align*} \\varphi _ x ^ { - 1 } ( \\sigma ) = ( h ( \\lambda _ 2 , \\lambda _ 3 ; s ) - r , \\lambda _ 2 , \\lambda _ 3 ) . \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} V _ F : = ( \\Omega \\cap V \\cap \\{ F = 0 \\} ) ^ { m - 1 } . \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} \\psi ( x ) = \\frac { ( 1 - a ) _ d ( 1 - a + c + b - x ) _ d } { ( 1 - a + c ) _ d ( 1 - a + b - x ) _ d } \\cdot { } _ 4 F _ 3 \\bigg [ \\begin{matrix} 1 + e & - b + x & - c & - d \\\\ & 1 & 1 - a + e & a - c - d - b + x \\end{matrix} \\bigg | \\ , 1 \\bigg ] \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} Q = \\frac { v _ \\epsilon ''' } { v _ \\epsilon '' } + ( 1 - \\alpha ) ( \\log \\theta ) ' - A t \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} ( K + K ^ { - \\epsilon } ) ^ { - \\frac { \\epsilon } { p } + \\sum _ { i = 1 } ^ m \\frac { \\epsilon } { p _ i } } \\le C \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{align*} [ a ^ + , a ^ - ] = k \\mathbb { I } - 2 N , [ a ^ + , N ] = a ^ + , [ a ^ - , N ] = - a ^ - , \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} F _ 0 & = \\left [ \\overline { \\Phi } ( P _ { j , 0 } ) \\right ] _ { j = 1 } ^ 6 \\\\ & = { \\scriptscriptstyle \\tiny \\sqrt { 2 } \\left [ \\begin{array} { c c c c } 1 & 1 & - 1 & - 1 \\\\ - 1 & 1 & - 1 & 1 \\\\ 0 & 0 & 0 & 0 \\\\ 1 & - 1 & - 1 & 1 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ \\end{array} \\right ] } \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} | v ^ { ( k ) } | \\prod _ { i = m ' + 1 } ^ { \\Delta ^ { ( k ) } - 1 } | w ^ { ( k ) } _ i | & > C \\textrm { f o r e v e r y } \\ 0 \\le m ' \\le \\alpha m . \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{gather*} \\chi _ { 0 , 2 } ^ { n , m } ( x , y | \\rho ) \\allowbreak = \\sum _ { j \\geq 0 } \\rho ^ { j } U _ { j + n } ( x ) U _ { j + m } ( y ) \\allowbreak = \\\\ ( U _ { n } ( x ) U _ { m } ( y ) ( w _ { 2 } ( x , y | \\rho ) - \\rho ^ { 4 } ) \\\\ + \\rho U _ { n + 1 } ( x ) U _ { m + 1 } ( y ) ( 1 - 2 \\rho ^ { 2 } + 4 \\rho ^ { 2 } ( x ^ { 2 } + y ^ { 2 } ) - 4 \\rho x y ) \\\\ + \\rho ^ { 2 } U _ { n + 2 } ( x ) U _ { m + 2 } ( y ) ( 1 - 4 \\rho x y ) + \\rho ^ { 3 } U _ { n + 3 } ( x ) U _ { m + 3 } ( y ) ) / w _ { 2 } ( x , y | \\rho ) \\end{gather*}"}
-{"id": "6818.png", "formula": "\\begin{align*} N ( Q ; F _ 1 , F _ 2 ) & : = \\{ ( i , j ) \\in H ( Q ; F _ 1 ) \\times H ( Q ; F _ 2 ) \\mid i + j = v _ Q ( G ) + 1 \\} , \\\\ \\nu ( Q ; F _ 1 , F _ 2 ) & : = | N ( Q ; F _ 1 , F _ 2 ) | . \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{align*} \\eta _ { X , Y } ^ \\lambda ( \\mathsf { P } ) = \\sum _ { i , j } q ^ { ( \\lambda , \\lambda _ { i } ) } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) \\sum _ { m , n , o , p } \\pi ( S ( X _ { ( 1 ) } ) ) _ { m } ^ { j } c _ { n } ^ { m } \\pi ( X _ { ( 2 ) } S ( Y _ { ( 1 ) } ) ) _ { o } ^ { n } c _ { p } ^ { o } \\pi ( Y _ { ( 2 ) } ) _ { i } ^ { p } . \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} A _ 2 = A _ 2 ( r _ 1 , r _ 2 ) = r _ 1 ( n - r _ 1 ) + r _ 2 ( n - r _ 1 - r _ 2 ) . \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} \\beta ^ u \\alpha _ i ^ { u + t } \\in K , \\ ; t = 0 , 1 , \\dots , p _ i - 2 . \\end{align*}"}
-{"id": "2792.png", "formula": "\\begin{align*} x = \\frac { u ( b v - u a ) } { D _ 1 d _ 2 d _ 3 } , y = \\frac { b v ( u - v ) } { D _ 1 d _ 2 d _ 3 } , s = \\frac { v ( b v - u a ) } { d _ 1 D _ 2 d _ 3 } , t = \\frac { u a ( u - v ) } { d _ 1 D _ 2 d _ 3 } . \\end{align*}"}
-{"id": "8223.png", "formula": "\\begin{align*} \\min \\{ F ( \\nabla u ( x _ { 0 } ) ) - \\overline \\Lambda u ( x _ { 0 } ) , - \\mathcal Q _ { \\infty } u ( x _ { 0 } ) \\} = 0 . \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} g ( t ) \\equiv u _ 0 ( t ) - u _ { f } ( t ) - u _ { \\partial } ( t ) \\quad \\mbox { f o r } | t | = \\rho ' _ 1 \\ , . \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} h ^ { n , 0 } \\ = \\ 1 \\ , . \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} R ( \\alpha , s , h ) : = \\{ z = x + i y \\in \\mathbb { C } | \\ , | x | \\leq h , \\ , y \\in [ \\alpha , \\alpha + s ] \\} ; \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} S _ { 1 } & = & S \\setminus \\{ x , b _ { 1 } \\} = ( S \\setminus \\{ b _ { 1 } \\} ) \\setminus \\{ x \\} \\preceq ( S \\setminus \\{ b _ { 1 } \\} ) \\setminus \\{ x ^ \\prime \\} = ( S \\setminus \\{ x ^ \\prime \\} ) \\setminus \\{ b _ { 1 } \\} \\preceq \\\\ & \\preceq & ( S \\setminus \\{ x ^ \\prime \\} ) \\setminus \\{ x ^ { \\prime \\prime } \\} = S _ { 2 } . \\\\ \\end{align*}"}
-{"id": "654.png", "formula": "\\begin{align*} a _ i ^ { - } | n _ 1 , \\cdots , n _ i , \\cdots , n _ r \\rangle \\ = \\sqrt { n _ i ( k + 1 - ( n _ 1 + n _ 2 + \\cdots + n _ r ) ) } | n _ 1 , \\cdots , n _ i - 1 , \\cdots , n _ r \\rangle , \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{align*} \\frac { q _ i } { q _ j } = \\frac { p _ i + W _ i } { p _ j + W _ j } \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} X ( L ) = 0 , ~ \\ \\ \\forall X \\in \\Gamma ( H A ) . \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} \\big [ [ J _ i ^ + , J _ j ^ - ] , J _ k ^ - \\big ] = - \\delta _ { i k } J _ j ^ - - \\delta _ { i j } J _ k ^ - . \\end{align*}"}
-{"id": "3425.png", "formula": "\\begin{align*} \\xi = h \\exp \\left ( - \\int \\dfrac { \\tau } { \\tau ^ 2 - \\tau - K } \\ , d \\tau \\right ) , \\omega = h \\tau \\exp \\left ( - \\int \\dfrac { \\tau } { \\tau ^ 2 - \\tau - K } \\ , d \\tau \\right ) , \\end{align*}"}
-{"id": "5020.png", "formula": "\\begin{align*} F ( c ) = D ( c ) + \\frac { 1 } { 2 } \\langle ( L _ c - \\partial _ y ^ 2 ) v , v \\rangle _ { L ^ 2 } + N _ c ( ( b e ^ { i y } + \\bar { b } e ^ { - i y } ) \\psi _ * + v ) , \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} ( \\mathbb E \\| M _ { N + 1 } \\| ^ p ) ^ { \\frac 1 p } & \\geq ( \\mathbb E \\| f _ { N } \\| ^ p ) ^ { \\frac 1 p } - \\sum ^ N _ { n = 1 } \\Bigl ( \\mathbb E \\Bigl \\| ( M ^ n _ 1 - \\sigma _ n ) \\phi _ { n } ( \\sigma _ 1 , \\ldots , \\sigma _ { n - 1 } ) \\Bigr \\| ^ p \\Bigr ) ^ { \\frac 1 p } \\\\ & \\geq 2 - \\frac { \\delta } { K L } \\cdot N \\cdot \\max \\{ \\| \\phi _ 1 \\| , \\| \\phi _ 2 \\| _ { \\infty } , \\ldots , \\| \\phi _ N \\| _ { \\infty } \\} > 1 . \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} h _ t + Q \\log | f _ t ' | - b _ t \\overset d = h _ 0 \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} ( \\sum _ { g \\in G } a _ g g ) x = \\sum _ { g \\in G } a _ g g ( x ) , \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{gather*} f _ m ( 1 ) = 1 , f _ m ( 2 ) = m , f _ m ( 3 ) = m ^ 2 + 1 , \\\\ f _ m ( n + 3 ) = m f _ m ( n + 2 ) + f _ m ( n + 1 ) + f _ m ( n ) , ( n > 1 ) . \\end{gather*}"}
-{"id": "7068.png", "formula": "\\begin{align*} q ^ { \\prime } \\left ( t \\right ) = - 4 \\left ( a + \\frac { b ^ { 2 } } { a } \\right ) q ^ { 2 } \\left ( t \\right ) , q \\left ( 0 \\right ) = \\gamma , \\gamma \\geq 0 . \\end{align*}"}
-{"id": "4888.png", "formula": "\\begin{align*} \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\textbf { v } _ { 0 } & \\ ! = \\sum \\limits _ { k \\neq 0 } { { \\mathbf { h } } _ { k } ^ { } } x _ { k } = \\sum \\limits _ { k \\neq 0 } \\sqrt { P _ k \\mathit { l } \\left ( { \\Vert \\bold { z } _ k \\Vert } _ 2 \\right ) } \\mathbf { \\breve { h } } _ { k } ^ { } U _ k , \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} \\left ( e ^ { t \\ , d / d x } \\phi \\right ) ( s ) = \\sum _ { n = 0 } ^ { \\infty } \\dfrac { t ^ n } { n ! } \\phi ^ { ( n ) } ( s ) = \\phi ( s + t ) \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} \\begin{cases} - \\div ( | x _ { n + 1 } | ^ a \\nabla u ) = 0 & B _ 1 \\setminus \\Lambda ( u ) \\cr u = 0 & \\Lambda ( u ) \\end{cases} \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{align*} v ( v + 2 u ) - 8 \\pi \\psi = \\Lambda + 8 \\pi \\rho . \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{align*} \\partial _ \\lambda \\Lambda ( \\gamma , \\lambda ) = - \\frac { c } { 2 \\pi } y ^ 2 \\int _ 0 ^ 1 y \\ln ( 1 - x ) e ^ { ( y ^ 2 + \\lambda y ) \\ln ( 1 - x ) + x y ^ 2 + \\ln x } \\frac { d x } { 1 - x } ; \\end{align*}"}
-{"id": "8421.png", "formula": "\\begin{align*} v a r ( Z _ n ) \\leq 4 \\frac { n ^ 2 C } { n ^ { 2 \\epsilon - 1 } } + 4 n ^ 2 \\eta = 4 \\frac { C } { n ^ { 2 \\epsilon - 3 } } + 4 n ^ 2 \\eta \\leq 4 \\frac { C } { n ^ 3 } + 4 n ^ 2 \\eta \\leq 5 n ^ 2 \\eta . \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } u _ { t } = A u , t \\in \\R \\\\ u ( 0 ) = u _ 0 \\end{array} \\right . . \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{align*} C = \\frac { e ^ { \\int _ 0 ^ { T } \\alpha ( s ) d s } \\cdotp \\int _ 0 ^ { T } \\beta ( s ) e ^ { \\int _ 0 ^ s - \\alpha ( r ) d r } d s } { 1 - e ^ { \\int _ 0 ^ { T } \\alpha ( s ) d s } } . \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} f ( x ) = ( 2 - k ) \\ , x ^ { \\frac { p ^ l + 3 } { 2 } } + 6 \\ , x ^ { \\frac { p ^ l + 1 } { 2 } } + k \\ , x ^ { \\frac { p ^ l - 1 } { 2 } } + 2 ( 3 - k ) \\ , x , \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} & \\Psi ( 0 , z ) = { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & 1 + a \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] = { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - \\frac 1 2 a & \\frac 1 2 + \\frac 1 2 a \\\\ & 1 \\end{matrix} \\bigg | \\ , 4 z ( 1 - z ) \\bigg ] = \\Phi ( 0 , z ) . \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} S = \\left ( s _ { i j } \\right ) C = \\left ( c _ { i j } \\right ) . \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} \\begin{array} { r r c l } N \\otimes _ B M = & ( A f \\otimes h B g \\otimes e A ) & \\oplus & ( A f \\otimes h B g ' \\otimes e ' A ) \\ \\ \\oplus \\\\ & ( A f ' \\otimes h ' B g \\otimes e A ) & \\oplus & ( A f ' \\otimes h ' B g ' \\otimes e ' A ) \\end{array} \\end{align*}"}
-{"id": "9519.png", "formula": "\\begin{align*} \\tau ( b E ( x ) ) = \\tau ( E ( b x ) ) = \\tau ( b x ) , \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} X ^ { \\mathrm { s s } } _ + = X - ( 0 \\oplus \\det V ^ \\vee ) X ^ { \\mathrm { s s } } _ - = X - ( V \\oplus 0 ) , \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{align*} \\langle x ^ * , x ( t ) \\rangle = \\lim _ { k \\to \\infty } \\langle x ^ * , x _ { n ( k ) } ( t ) \\rangle & = \\lim _ { k \\to \\infty } \\int _ 0 ^ t \\langle x ^ * , x _ { n ( k ) } ' ( s ) \\rangle d s + \\langle x ^ * , \\xi _ 0 \\rangle \\\\ & = \\int _ 0 ^ t \\langle x ^ * , u ( s ) \\rangle d s + \\langle x ^ * , \\xi _ 0 \\rangle \\\\ & = \\left \\langle x ^ * , \\int _ 0 ^ t u ( s ) d s + \\xi _ 0 \\right \\rangle \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} K _ i ( m + s ) = \\sum _ { j = 1 } ^ n K _ j ( s ) K _ { i - j + 1 } ( m ) , \\enskip i = 1 , . . . , n . \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{align*} x _ c ( z ) = \\frac { \\rho _ c ( z ) \\ , z + c } { \\rho ( x _ c ( z ) ) } \\end{align*}"}
-{"id": "1780.png", "formula": "\\begin{align*} T _ j ( t ) = e ^ { t A _ j } = \\sum _ { n = 0 } ^ { \\infty } \\dfrac { t ^ n } { n ! } A _ j ^ n , \\end{align*}"}
-{"id": "1502.png", "formula": "\\begin{align*} S _ k ( x ) : = \\prod _ { \\substack { j = 1 \\\\ j \\neq k } } ^ { n - 1 } \\frac { x - t _ j } { t _ k - t _ j } = \\frac { S ( x ) } { S ' ( t _ k ) ( x - t _ k ) } , k = 1 , 2 , \\dots , n - 1 , \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{align*} d \\boldsymbol { Y } _ t = d \\boldsymbol { \\tilde { W } } _ t \\sqrt { \\frac { I + \\boldsymbol { Y } _ t ^ 2 } { 2 } } + \\sqrt { \\frac { I + \\boldsymbol { Y } _ t ^ 2 } { 2 } } d \\boldsymbol { \\tilde { W } } _ t ^ * + \\left [ ( 1 - N - 2 \\Re ( s ) ) \\boldsymbol { Y } _ t + 2 \\Im ( s ) \\boldsymbol { I } + T r \\left ( \\boldsymbol { Y } _ t \\right ) \\boldsymbol { I } \\right ] d t , \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} A ^ 3 \\limsup _ { t \\to + \\infty } \\sqrt t \\| u _ { n + 1 } ( t ) \\| _ { L ^ \\infty ( B _ { A \\sqrt t } ^ c ) } & = A ^ 3 \\limsup _ { t \\to + \\infty } \\sqrt { 4 t } \\| u _ { n + 1 } ( 4 t ) \\| _ { L ^ \\infty ( B _ { A \\sqrt { 4 t } } ^ c ) } \\\\ & \\le C _ 0 \\Bigl ( \\| u _ 0 \\| _ 3 + \\| \\theta _ 0 \\| _ 1 \\Bigr ) + 2 C \\varepsilon ( \\varepsilon + \\kappa _ n ) . \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} \\widetilde { [ \\gamma ] } = \\Vec { \\gamma } + m \\iota _ * ( 1 ) \\in H _ 1 ( U \\Sigma ) ^ { ( 2 ) } , \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{align*} E ^ { - \\frac { 1 } { 2 } } [ G ( k , 0 ) ^ \\dag G ( k , 0 ) + G ' ( k , 0 ) ^ \\dag G ' ( k , 0 ) ] E ^ { - \\frac { 1 } { 2 } } = I _ n . \\end{align*}"}
-{"id": "6972.png", "formula": "\\begin{align*} i \\cdot f _ { i - 1 } = n \\cdot f _ { i - 1 , 0 } \\\\ \\mbox { w i t h } \\\\ 0 \\leq i \\leq d . \\end{align*}"}
-{"id": "7556.png", "formula": "\\begin{align*} v : = \\lim _ { i \\rightarrow \\infty } g \\gamma _ i v _ j / { \\norm { g \\gamma _ i v _ j } } , \\norm { v } = 1 \\lim _ { i \\to \\infty } \\norm { g \\gamma _ i v _ j } = \\infty . \\end{align*}"}
-{"id": "5275.png", "formula": "\\begin{align*} A \\cdot N = ( c \\cdot N ) ( c \\cdot x ) \\textrm { o n } S . \\end{align*}"}
-{"id": "3388.png", "formula": "\\begin{align*} \\bar X _ t = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n X _ t ^ i \\ , , \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} L u _ { 0 } - { \\rm i } \\ , \\omega \\ , B u _ { 0 } - \\omega ^ 2 u _ { 0 } = g \\ , , \\end{align*}"}
-{"id": "7621.png", "formula": "\\begin{align*} 0 = Q ' = \\frac { v _ \\epsilon ^ { ( 4 ) } } { v _ \\epsilon '' } - \\frac { ( v _ \\epsilon ''' ) ^ 2 } { ( v _ \\epsilon '' ) ^ 2 } + ( 1 - \\alpha ) ( \\log \\theta ) '' , \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} s _ k = \\frac { a \\ , P ( t _ k ) } { \\prod \\limits _ { \\substack { j = 1 \\\\ j \\neq k } } ^ { n - 1 } ( t _ k - t _ j ) ^ 2 } , k = 1 , \\dots , n - 1 . \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} \\epsilon _ 0 = \\frac { 1 } { 4 d } \\alpha = \\frac { 1 } { 2 } - \\frac { 1 } { 1 6 d } , \\end{align*}"}
-{"id": "6443.png", "formula": "\\begin{align*} \\varphi ^ * ( u ) = \\varphi ( u ) \\quad \\varphi ^ * ( \\varepsilon ) = \\varphi ( \\varepsilon ) \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = \\frac { \\alpha _ 2 \\gamma ^ 2 _ 3 } { \\alpha ^ 2 _ 5 } y _ 5 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\alpha _ 5 } y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 2 ] = \\frac { \\beta _ 2 \\beta ^ 2 _ 3 } { \\alpha ^ 2 _ 5 } y _ 5 , [ y _ 1 , y _ 3 ] = y _ 4 + \\theta _ 3 y _ 5 , \\\\ [ y _ 3 , y _ 1 ] = \\frac { \\beta _ 6 \\gamma _ 3 } { \\alpha _ 5 } y _ 5 , [ y _ 2 , y _ 3 ] = \\frac { \\beta _ 3 \\gamma _ 2 } { \\alpha _ 5 } y _ 5 , [ y _ 3 , y _ 2 ] = y _ 4 , [ y _ 3 , y _ 3 ] = \\gamma _ 6 y _ 5 . \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} \\P _ { \\psi , \\nu } ( \\xi \\in d s ) = \\frac { \\Psi _ { \\nu } ( b + s ) s ^ { \\nu - 1 } } { \\Gamma ( \\nu ) ( \\Psi ( b + 1 ) - \\Psi ( b ) ) } d s \\ ; . \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } i \\partial _ t u ( t , x ) + \\Delta ^ 2 u ( t , x ) & = - \\mu | u | ^ { \\nu - 1 } u ( t , x ) , ( t , x ) \\in \\R \\times \\R ^ d , \\\\ u ( 0 , x ) & = u _ 0 ( x ) , x \\in \\R ^ d . \\end{array} \\right . \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} F ( t ) \\ ! \\ ! = \\ ! \\ ! \\left \\{ \\begin{array} { l l } Q ( t ) + 2 A _ 3 Q ( t + \\frac { 1 } { 2 } ) , & t \\in [ 0 , 1 - b ] , \\\\ Q ( t ) \\ ! \\ ! + \\ ! \\ ! 2 A _ 3 Q ( t + \\frac { 1 } { 2 } ) \\ ! \\ ! + \\ ! \\ ! 2 A _ 4 Q ( 1 - t ) , & t \\in [ 1 - b , b - \\frac { 1 } { 2 } ] , \\\\ Q ( t ) + 2 A _ 4 Q ( 1 - t ) , & t \\in [ b - \\frac { 1 } { 2 } , \\frac { 1 } { 2 } ] , \\\\ 2 A _ 2 Q ( t ) , & t \\in [ \\frac { 1 } { 2 } , b ] ; \\end{array} \\right . \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} \\Psi ( 0 ) = & { } _ 3 F _ 2 \\bigg [ \\begin{matrix} - a & \\beta & \\gamma \\\\ & 1 & \\delta \\end{matrix} \\bigg | \\ , 1 \\bigg ] \\\\ = & \\frac { \\Gamma ( \\delta ) \\Gamma ( 1 + \\delta + a - \\beta - \\gamma ) } { \\Gamma ( \\delta + a ) \\Gamma ( 1 + \\delta - \\beta - \\gamma ) } \\cdot { } _ 3 F _ 2 \\bigg [ \\begin{matrix} - a & 1 - \\beta & 1 - \\gamma \\\\ & 1 & 1 + \\delta - \\beta - \\gamma \\end{matrix} \\bigg | \\ , 1 \\bigg ] . \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 4 & \\frac 1 4 \\\\ & 1 \\end{matrix} \\bigg | \\ , 4 \\bigg ] _ { p - 1 } \\equiv \\begin{cases} a + b \\sqrt { - 3 } \\pmod { p ^ 2 } , & p \\equiv 1 \\pmod { 4 } , \\\\ a \\sqrt { - 3 } - 3 b \\pmod { p ^ 2 } , & p \\equiv 3 \\pmod { 4 } . \\end{cases} \\end{align*}"}
-{"id": "9963.png", "formula": "\\begin{align*} x \\in \\mathcal { C } ( E _ i , A _ i ) \\mbox { i f a n d o n l y i f } C _ i x = 0 \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ^ p u ( x ) = 0 , & x \\in \\widehat { B } _ p \\\\ u ( x ) = f ( x ) , & \\mbox { } x \\in \\widehat { S } _ p . \\end{cases} \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{align*} x _ { n + 2 } ^ 2 = - \\sum _ { i = 1 } ^ { n + 1 } \\frac { ( \\lambda _ { n + 3 } - \\lambda _ i ) } { ( \\lambda _ { n + 3 } - \\lambda _ { n + 2 } ) } x _ i ^ 2 \\ , \\ , \\textrm { a n d } \\ , \\ , x _ { n + 3 } ^ 2 = - \\sum _ { i = 1 } ^ { n + 1 } \\frac { ( \\lambda _ { n + 2 } - \\lambda _ i ) } { ( \\lambda _ { n + 2 } - \\lambda _ { n + 3 } ) } x _ i ^ 2 . \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} \\left | \\sum _ { j = 1 } ^ N \\ ; \\omega _ j k _ j \\ ; + \\ ; \\sigma \\omega _ l \\right | & \\geq \\frac { \\gamma } { N ^ \\tau } \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{align*} 2 M ^ \\dag J ( k ) M T = & - i k { \\rm d i a g } \\{ I _ { n - n _ D } , 0 _ { n _ D } \\} \\\\ & + i { \\rm d i a g } \\{ ( X _ 1 - I _ { n - n _ D } ) ( X _ 1 + I _ { n - n _ D } ) ^ { - 1 } , - I _ { n _ D } \\} \\\\ & + Q ' { \\rm d i a g } \\{ I _ { n - n _ D } , 0 _ { n _ D } \\} + o ( 1 ) \\\\ = & - i k { \\rm d i a g } \\{ I _ { n - n _ D } , 0 _ { n _ D } \\} + i { \\rm d i a g } \\{ 0 _ { n - n _ D } , - I _ { n _ D } \\} \\\\ & + Q { \\rm d i a g } \\{ I _ { n - n _ D } , 0 _ { n _ D } \\} + o ( 1 ) , \\end{align*}"}
-{"id": "9633.png", "formula": "\\begin{align*} q ^ { ( j ) } _ n = \\frac { H ^ { ( j ) } _ { n - 1 } H ^ { ( j + 1 ) } _ { n } } { H ^ { ( j ) } _ { n } H ^ { ( j + 1 ) } _ { n - 1 } } . \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} \\int _ I f ^ 2 & = 2 ^ m \\left ( ( \\mu _ \\theta ) ^ { ( m ) } ( I ) \\right ) ^ 2 \\le 2 ^ m \\left ( \\Pi _ \\theta ( \\mu ^ { ( m ) } ) ( 5 I ) \\right ) ^ 2 \\lesssim \\int _ { 5 I } g ^ 2 , \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{align*} \\Lambda ^ { \\infty } _ { N + 1 } \\Lambda ^ { N + 1 } _ { N } = \\Lambda _ N ^ { \\infty } . \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{gather*} a _ 0 = p _ 1 q _ 1 ( q _ 1 - 1 ) + \\theta ^ \\infty _ 2 q _ 1 + \\theta ^ 0 , \\\\ a _ 1 = \\frac { 1 } { t _ 2 } \\big ( \\big ( p _ 1 q _ 1 + \\theta ^ \\infty _ 2 \\big ) ( 1 - q _ 1 ) - p _ 2 q _ 2 + \\theta ^ 1 \\big ) , \\\\ ( B _ 1 ) _ { 3 2 } = ( q _ 1 - 1 ) ( p _ 1 q _ 1 - p _ 2 q _ 2 ) + \\big ( \\theta ^ \\infty _ 2 - \\theta ^ \\infty _ 1 \\big ) q _ 1 + \\frac { t _ 2 } { t _ 1 } ( q _ 1 - p _ 2 ) - \\theta ^ 1 - \\theta ^ \\infty _ 2 , \\\\ ( B _ 1 ) _ { 3 3 } = ( p _ 1 + t _ 1 ) ( q _ 1 - 1 ) + ( p _ 1 + q _ 2 ) q _ 1 - p _ 2 q _ 2 . \\end{gather*}"}
-{"id": "5368.png", "formula": "\\begin{align*} c _ 1 ( n , k ) = \\sum _ { i = 0 } ^ { n - k } b ^ { n - k - i } { n - i - 1 \\choose k - 1 } { k + i - 1 \\choose k - 1 } . \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{align*} ( * * ) & = P _ N ( \\phi \\geq 0 \\ , o n \\ , A ^ c | \\phi \\in [ 0 , a ] \\ , o n \\ , A ) = \\int _ { [ 0 , a ] ^ { A } } P _ N ( \\phi \\geq 0 \\ , o n \\ , A ^ c | \\phi = \\psi \\ , o n \\ , A ) g ( \\psi ) d \\psi \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} a \\ , P ( x ) = a \\prod _ { k = 1 } ^ { n - 1 } ( x - t _ k ) ^ 2 + \\sum _ { k = 1 } ^ { n - 1 } s _ k \\prod _ { \\substack { j = 1 \\\\ j \\neq k } } ^ { n - 1 } ( x - t _ j ) ^ 2 , \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{align*} \\tilde { \\eta } _ n ^ { h } = \\eta _ n ^ h - S _ { \\Delta t } ^ { n } h \\quad , \\tilde { \\eta } _ 0 ^ h = 0 . \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} \\partial _ t { v } _ i - d _ i \\Delta _ { x _ 1 , \\vartheta } v _ i = f _ i ( { \\bf v } ) + \\delta _ { 1 , \\varrho } \\ , g _ i ( { \\bf v } ) . \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} b _ j = \\sum _ { i = 1 } ^ n b _ { i j } \\ , g _ i , \\end{align*}"}
-{"id": "1822.png", "formula": "\\begin{align*} \\left | \\dfrac { d ^ { n + m } } { d x ^ { n + m } } \\Big ( e ^ { t \\ , d / d x } \\phi \\Big ) ( x ) \\right | \\leqslant \\sum _ { k = 0 } ^ { \\infty } \\dfrac { t ^ k } { k ! } c ( \\phi , m , j ) M ^ { k + n } = c \\ , M ^ n \\ , e ^ { t \\ , M } , \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{align*} \\widehat { \\mu ^ { } _ 1 } ( k ) \\ , = \\ , \\tfrac { 1 } { 3 } \\bigl ( 1 + 2 \\cos ( \\pi k ) \\bigr ) \\ , = \\ , \\begin{cases} 1 , & , \\\\ - \\frac { 1 } { 3 } , & , \\end{cases} \\end{align*}"}
-{"id": "5327.png", "formula": "\\begin{align*} m - 2 r + i _ 0 + 2 \\sum _ { p = 1 } ^ { m - 1 } i _ p & \\le 2 ( m - r ) < 2 \\left ( \\frac { m } { 2 } \\right ) = m \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} z ^ { ( s + 1 ) ( p - 1 ) } = & \\sum _ { k = 0 } ^ \\infty \\binom { s + 1 } { k } ( z ^ { p - 1 } - 1 ) ^ k = \\sum _ { k = 0 } ^ \\infty \\binom { s } { k } ( z ^ { p - 1 } - 1 ) ^ k + \\sum _ { k = 1 } ^ \\infty \\binom { s } { k - 1 } ( z ^ { p - 1 } - 1 ) ^ k \\\\ = & z ^ { s ( p - 1 ) } + ( z ^ { p - 1 } - 1 ) \\cdot z ^ { s ( p - 1 ) } = z ^ { p - 1 } \\cdot z ^ { s ( p - 1 ) } , \\end{align*}"}
-{"id": "8029.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\| w ( t ) \\| _ 2 ^ 2 + \\int _ { \\bar t } ^ t \\| \\nabla w \\| _ 2 ^ 2 d \\tau \\\\ & = \\frac { 1 } { 2 } \\| w ( \\bar t ) \\| _ 2 ^ 2 + \\int _ { \\bar t } ^ t \\Big [ \\langle ( u _ s + \\widetilde U ) \\otimes w , \\nabla w \\rangle + \\langle f , w \\rangle \\Big ] d \\tau \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} ( u ^ 2 + k ^ 2 , u ^ 3 + k ^ 3 ) = \\left \\{ \\begin{array} { l l } 2 , & 2 \\mid ( k - u ) ; \\\\ 1 , & 2 \\nmid ( k - u ) . \\end{array} \\right . \\end{align*}"}
-{"id": "2187.png", "formula": "\\begin{align*} P _ j N _ { - , k _ j } ^ \\dagger = N _ { - , k _ j } ^ \\dagger , \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} \\begin{cases} ( M ^ { \\otimes m } ) _ { 1 } f = p \\\\ ( M ^ { \\otimes m } ) _ { 2 } f = \\mu _ 2 \\end{cases} \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\binom { 2 k } { k } a _ k x ^ k = \\frac { 1 } { \\sqrt { 1 - 4 x } } \\sum _ { k = 0 } ^ { \\infty } \\binom { 2 k } { k } b _ k \\left ( \\frac { - x } { 1 - 4 x } \\right ) ^ k , \\end{align*}"}
-{"id": "3711.png", "formula": "\\begin{align*} b _ 2 = C - ( b _ 0 + b _ 1 ) . \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} \\log _ { i } ( \\omega ) \\circ ( g \\circ x ) = \\log _ { i } ( g \\circ x ) = \\log _ { i } ( g ) \\circ x = ( \\log _ { i } ( \\omega ) \\circ g ) \\circ x \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} Q ( x ) : = \\prod \\limits _ { k = 1 } ^ n ( x - x _ k ) . \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} \\gamma = \\mathbf { s } ^ { \\vee } \\ \\ \\ a n d \\ \\ \\ Z = ( \\mathbf { s } ) _ 0 . \\end{align*}"}
-{"id": "6699.png", "formula": "\\begin{align*} Q ( 2 \\tau ) : = \\int \\limits _ { | x - e | + | x | = 2 \\tau } \\frac { q ( x ) } { | 2 \\tau x - | x | e | } d S _ { x } . \\end{align*}"}
-{"id": "9584.png", "formula": "\\begin{align*} \\mbox { v a r } \\left ( R _ { n , i } V _ { n , i } - \\rho _ n \\mu _ { n , i } \\right ) & = \\rho _ n E [ V _ { n , i } ^ 2 ] - \\rho _ n ^ 2 \\mu _ { n , i } ^ 2 \\\\ & = \\rho _ n \\left ( \\mbox { v a r } ( V _ { n , i } ) + ( 1 - \\rho _ n ) \\mu _ { n , i } ^ 2 \\right ) . \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} ( z + x + \\frac { k } { 2 } ) ^ { M - 1 } - ( z + x + \\frac { k } { 2 } - 1 ) ^ { M - 1 } = \\sum _ { r = 1 } ^ { M - 1 } ( z + x + \\frac { k } { 2 } - 1 ) ^ { M - r } \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} \\partial _ y L _ y = \\hat { L } ^ * _ y \\partial _ y . \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} 0 < \\varepsilon _ { 0 } < \\Big ( { 2 \\sqrt { 2 \\pi } L \\sum \\limits _ { i = 1 } ^ n \\sqrt { k _ i } } \\Big ) ^ { - 2 } \\ln ^ 2 \\bigg ( \\tfrac { L d } { M _ F } + 1 \\bigg ) . \\end{align*}"}
-{"id": "6904.png", "formula": "\\begin{align*} \\ , ^ { C F } \\ , _ { a } D ^ { \\alpha } _ { t } f ( t ) = \\dfrac { B ( \\alpha ) } { 1 - \\alpha } \\int _ { a } ^ { t } f ' ( s ) \\exp \\left [ \\dfrac { - \\alpha } { 1 - \\alpha } ( t - s ) \\right ] d s , \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} H _ i ( m + s , n ) = \\sum _ { j = 1 } ^ n H _ j ( s , n ) H _ { i - j + 1 } ( m , n ) , \\enskip i = 1 , . . . , n ; \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} ( A \\rho ) _ \\mu & = \\mathcal { F } \\big \\{ \\bar { d } _ \\ell ( s , t ) \\rho ( s , t ) \\big \\} ( p , q ) \\\\ ( A ^ * y ) _ \\nu & = \\sum _ { l = 1 } ^ L \\sum _ { p = 0 } ^ { M - 1 } \\sum _ { q = 0 } ^ { N - 1 } d _ \\ell ( s , t ) e ^ { 2 \\pi i p s / M } e ^ { 2 \\pi i q t / N } y _ \\ell ( p , q ) \\\\ & = M N ~ \\sum _ { l = 1 } ^ L d _ \\ell ( s , t ) ~ \\mathcal { F } ^ { - 1 } \\big \\{ y _ \\ell ( p , q ) \\big \\} ( s , t ) , \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} \\langle \\Phi ^ * ( s , \\omega ) \\tilde x _ n ^ * , \\Phi ^ * ( s , \\omega ) \\tilde x _ m ^ * \\rangle = a _ n \\delta _ { m n } , \\ ; \\ ; \\ ; m , n = 1 , \\ldots , d . \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{gather*} ( B _ { 2 0 } ) _ { 3 2 } = - p _ 1 p _ 2 + \\big ( p _ 2 q _ 2 - \\theta ^ \\infty _ 2 \\big ) ( q _ 1 - t _ 2 ) , ( B _ { 2 0 } ) _ { 3 3 } = p _ 1 - t _ 1 - { t _ 2 } ^ 2 - q _ 2 ( q _ 1 - t _ 2 ) . \\end{gather*}"}
-{"id": "3365.png", "formula": "\\begin{align*} w _ { i , j } + w _ { k , l } = w _ { i , l } + w _ { k , j } . \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{align*} \\rho _ c ( z ) = \\abs { \\rho ( x _ c ) \\ , x _ c - c } \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{align*} M _ p : = \\{ [ D ] _ \\Lambda \\in k _ p \\ ; | \\ ; [ D ] _ \\Lambda ^ 2 \\equiv 0 \\bmod 2 p ^ 2 \\ ; \\} . \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} m _ 0 : = \\int _ { S ^ { n - 1 } } ( u \\cdot v ) _ + ^ { - q } d u > 0 . \\end{align*}"}
-{"id": "4654.png", "formula": "\\begin{align*} \\rho _ R ( t ) & = \\prod _ { | p _ n | > R } \\left ( 1 - \\frac { t } { p _ n } \\right ) ^ 2 \\\\ & = \\lim _ { S \\to \\infty } \\prod _ { R < | p _ n | < S } \\left ( 1 - \\frac { t } { p _ n } \\right ) ^ 2 . \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} \\sum _ { D } \\pi _ D ( x ) = \\pi ( x ) \\bigg ( 1 + O \\bigg ( \\frac { 1 } { \\sqrt { \\log \\log y } } \\bigg ) \\bigg ) , \\end{align*}"}
-{"id": "9022.png", "formula": "\\begin{align*} C & = \\frac { N - 2 B } { N } \\cdot \\frac { 1 - \\frac { T } { N - 2 B } } { 1 - \\left ( \\frac { T } { N - 2 B } \\right ) ^ M } \\\\ & = \\frac { N - 2 B } { N } \\cdot \\left ( 1 + \\frac { T } { N - 2 B } + \\frac { T ^ 2 } { ( N - 2 B ) ^ 2 } + \\cdots + \\frac { T ^ { M - 1 } } { ( N - 2 B ) ^ { M - 1 } } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} f ( s ) = \\sum _ { n = 1 } ^ \\infty a _ n n ^ { - s } \\end{align*}"}
-{"id": "3884.png", "formula": "\\begin{align*} h _ i = \\frac { p _ i ^ 2 } { 2 } + \\frac { \\epsilon _ i } { 2 } q _ i ^ 2 + \\frac { 1 } { 4 } q _ i ^ 4 + \\frac { 1 } { 4 W } \\left ( q _ { i + 1 } - q _ i \\right ) ^ 2 \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} \\beta _ \\chi ( z ^ 2 ) V ^ + ( w ) & = V ^ + ( z ) H ^ { \\beta } ( z ^ 2 ) z ^ { 2 h _ 0 } V ^ - ( z ) V ^ + ( w ) = V ^ + ( z ) H ^ { \\beta } ( z ^ 2 ) z ^ { 2 h _ 0 } i _ { z , w } \\frac { z ^ 2 } { z ^ 2 - w ^ 2 } V ^ + ( w ) V ^ - ( z ) \\\\ & = i _ { z , w } \\frac { z ^ 2 } { z ^ 2 - w ^ 2 } V ^ + ( w ) V ^ + ( z ) H ^ { \\beta } ( z ^ 2 ) z ^ { 2 h _ 0 } V ^ - ( z ) = i _ { z , w } \\frac { z ^ 2 } { z ^ 2 - w ^ 2 } V ^ + ( w ) \\beta _ \\chi ( z ^ 2 ) . \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} w = \\frac { 1 \\pm \\sqrt { 5 } } { 2 } . \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} \\phi ( [ b _ 0 , b _ 1 ] ) \\leq \\phi ( [ a _ { k - 1 } \\wedge b _ 1 , b _ 1 ] ) = \\phi ( [ a _ { k - 1 } , a _ { k - 1 } \\vee b _ 1 ] ) \\leq \\phi ( [ a _ { k - 1 } , a _ k ] ) \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{gather*} P _ { x _ 1 , x _ 3 } ( z ) = z ^ d P _ V ( z ^ { - 1 } ) . \\end{gather*}"}
-{"id": "6359.png", "formula": "\\begin{align*} \\tilde \\varphi ^ \\infty ( \\alpha _ A ( a ) ) \\psi _ t ( \\xi ) = \\psi _ t ( a \\cdot \\xi ) \\quad \\psi _ t ( \\xi ) \\tilde \\varphi ^ \\infty ( \\alpha _ A ( a ) ) = \\psi _ t ( \\xi \\cdot a ) \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} g & = e ^ { 2 u } \\bar g , R [ g ] = e ^ { - 2 u } \\left ( R [ \\bar { g } ] - 2 \\Delta u \\right ) \\\\ H [ g ] & = e ^ { - u } [ H [ \\bar g ] + \\partial _ \\nu u ] \\ , \\ , n = 2 , \\end{align*}"}
-{"id": "4815.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) ^ { p - 1 } = I + \\begin{pmatrix} - \\frac { a } { w } & 0 \\\\ 0 & - \\frac { d } { z } \\end{pmatrix} p \\end{align*}"}
-{"id": "1206.png", "formula": "\\begin{align*} d X _ t ^ \\delta = A X _ t ^ \\delta d t + G _ \\delta ( X _ t ^ \\delta ) d t + \\sigma _ \\delta ( X _ t ^ \\delta ) d W ( t ) , X ^ \\delta ( 0 ) = x . \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} L _ { k } u = 0 y \\in \\left ( 0 , 1 \\right ) , \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} \\sqrt { \\mu _ n } = \\gamma _ n + \\frac { \\omega _ 0 } { n \\pi } + o \\left ( \\frac { 1 } { n } \\right ) , \\gamma _ n : = \\left ( n + \\frac { 1 } { 2 } \\right ) \\pi + ( - 1 ) ^ n \\arcsin a , \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{gather*} A ( z ) = \\frac { 1 } { z ^ { r + 1 } } ( A _ 0 + A _ 1 z + \\cdots ) , r \\in \\mathbb { Z } _ { > 0 } , \\end{gather*}"}
-{"id": "4426.png", "formula": "\\begin{align*} \\pi _ 1 \\big ( C ' \\setminus \\mathcal B , q _ 0 \\big ) = \\big \\langle \\gamma _ 1 , \\ldots , \\gamma _ r , \\alpha _ 1 , \\beta _ 1 , \\ldots , \\alpha _ { g ' } , \\beta _ { g ' } ~ \\big \\vert ~ \\gamma _ 1 \\cdots \\gamma _ r \\cdot \\prod _ { i = 1 } ^ { g ' } [ \\alpha _ i , \\beta _ i ] \\big \\rangle . \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 2 e _ 5 , [ e _ 2 , e _ 1 ] = \\alpha _ 5 e _ 4 + \\alpha _ 6 e _ 5 , [ e _ 2 , e _ 3 ] = \\gamma _ 2 e _ 5 , [ e _ 3 , e _ 2 ] = \\gamma _ 4 e _ 5 . \\end{align*}"}
-{"id": "2154.png", "formula": "\\begin{align*} V ( x ) = - 2 \\frac { d K ( x , x ) } { d x } , \\end{align*}"}
-{"id": "9426.png", "formula": "\\begin{align*} \\partial _ { t } B + \\nabla \\times \\left ( B \\times v + d \\right ) = 0 , \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{align*} \\frac { x } { \\log _ C ( x ) } = \\sum _ { n = 0 } ^ \\infty \\frac { C C _ n } { \\Pi ( n ) } x ^ n \\ , . \\end{align*}"}
-{"id": "7592.png", "formula": "\\begin{align*} \\theta = 1 + \\epsilon ^ 2 e ^ \\psi + \\epsilon ^ 2 e ^ { \\psi - \\rho } , \\end{align*}"}
-{"id": "1233.png", "formula": "\\begin{align*} \\phi _ n \\le C _ 1 \\bigl ( 1 + t _ n ^ { - 1 + \\mu } \\bigr ) + C _ 2 \\Delta t \\sum _ { j = 0 } ^ { n - 1 } t _ { n - j } ^ { - 1 + \\nu } \\phi _ j . \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} U _ i = \\max _ { \\mu \\in [ \\mu _ i , \\mu _ { i + 1 } ] } \\bar { q } ( \\mu ) . \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} \\begin{cases} 1 \\leq x _ { r - k } \\leq x _ { r - k - 1 } & \\quad \\mbox { f o r } 0 \\leq k \\leq k ^ * - 1 , \\\\ 1 \\leq x _ { r - k ^ * } \\leq \\left ( \\frac { \\Xi } { x _ 1 \\cdots x _ { r - k ^ * - 1 } } \\right ) ^ { 1 / ( k ^ * + 1 ) } & \\mbox { f o r } k = k ^ * , \\\\ x _ { r - k } \\geq \\left ( \\frac { \\Xi } { x _ 1 \\cdots x _ { r - k - 1 } } \\right ) ^ { 1 / ( k + 1 ) } & \\quad \\mbox { f o r } k ^ * + 1 \\leq k \\leq r - 2 , \\\\ x _ { 1 } \\geq \\Xi ^ { 1 / r } & \\quad \\mbox { f o r } k = r - 1 . \\end{cases} \\end{align*}"}
-{"id": "9371.png", "formula": "\\begin{align*} B ( \\mathcal { H } , \\mathcal { K } ) = S ^ 1 ( \\mathcal { K } , \\mathcal { H } ) ^ * . \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\overbrace { \\begin{matrix} 0 & 0 \\\\ 0 & 0 \\\\ 1 & \\theta ^ 0 \\end{matrix} } & \\overbrace { \\begin{matrix} 0 & \\sqrt { - t _ 2 } & \\theta ^ \\infty _ 1 / 2 \\\\ 0 & - \\sqrt { - t _ 2 } & \\theta ^ \\infty _ 1 / 2 \\\\ - t _ 1 & 0 & \\theta ^ \\infty _ 2 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "4783.png", "formula": "\\begin{align*} \\Lambda ( Q , f , m _ { \\bullet } , c _ { \\bullet } , b _ { \\bullet } ) = K Q / I ( Q , f , m _ { \\bullet } , c _ { \\bullet } , b _ { \\bullet } ) , \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{align*} b ( z _ 1 \\otimes \\dots \\otimes z _ n ) & = z _ 1 \\otimes \\dots \\otimes z _ n \\\\ & + \\sum _ { 1 } ^ { n - 1 } ( - 1 ) ^ i z _ 1 \\otimes \\dots \\otimes z _ i z _ { i + 1 } \\otimes \\dots \\ \\otimes z _ n \\\\ & + ( - 1 ) ^ n z _ n \\otimes z _ 1 \\otimes \\dots \\otimes z _ { n - 1 } \\end{align*}"}
-{"id": "3589.png", "formula": "\\begin{align*} e _ n ( z ) = \\frac { 1 } { \\sqrt { \\pi } } \\frac { z ^ n } { \\sqrt { n ! } } . \\end{align*}"}
-{"id": "1446.png", "formula": "\\begin{align*} & \\sum _ { \\tiny \\begin{matrix} \\beta + \\gamma = \\alpha , b + c = a \\\\ | \\gamma | + | c | \\leq [ ( \\alpha + | a | ) / 2 ] \\end{matrix} } \\| \\nabla U ^ { ( \\beta , b ) } \\| ^ 2 _ { L ^ 2 } \\| \\langle t \\rangle \\nabla U ^ { ( \\gamma , c ) } \\| ^ 2 _ { L ^ \\infty ( r \\leq \\langle t \\rangle / 2 ) } \\\\ & \\lesssim E _ { | \\alpha | + | a | + 1 } X _ { [ ( | \\alpha | + | a | ) / 2 ] + 3 } . \\end{align*}"}
-{"id": "4614.png", "formula": "\\begin{align*} { \\displaystyle \\left ( \\partial ^ { 2 } \\theta ^ { k } _ { \\mathbf { A } } , \\mathbf { v } \\right ) + D ( \\widetilde { \\theta } ^ { k } _ { \\mathbf { A } } , \\mathbf { v } ) = I _ 1 ^ { k } ( \\mathbf { v } ) + I _ 2 ^ { k } ( \\mathbf { v } ) + I _ 3 ^ { k } ( \\mathbf { v } ) + I _ 4 ^ { k } ( \\mathbf { v } ) . } \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} P _ j = \\sum _ { i = 1 } ^ { j + 1 } \\theta _ { i j } P ' _ i = \\sum _ { i = 1 } ^ { j + 1 } \\theta _ { i j } \\sum _ { k = 0 } ^ { i - 1 } \\eta _ { k i } P _ k . \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { j - 1 } \\beta _ i - j \\beta _ j = ( \\beta - \\alpha ) ( \\frac { j } { 2 j + \\gamma } - \\frac { j ( j + 1 ) } { 2 j + \\gamma + 2 } + \\frac { j ^ 2 } { 2 j + \\gamma } ) = \\frac { 2 j ( j + 1 ) ( \\beta - \\alpha ) } { ( 2 j + \\gamma ) ( 2 j + \\gamma + 2 ) } \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} b _ C = \\tfrac 1 { | G _ C | } \\sum _ { i = 0 } ^ { m - 1 } \\sigma _ m ^ i C . \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{align*} \\left . \\begin{array} { r c l } \\dot x & = & a - b x - x y \\\\ \\dot y & = & c - d y + x y . \\end{array} \\right \\} \\ , \\ , : = \\ , \\ , F ( x , y ) \\ , . \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} \\chi ( R , t ) = R ^ { 2 } \\exp \\left [ - \\frac { R ^ { 2 } } { 2 \\rho ( t ) } \\right ] , \\end{align*}"}
-{"id": "6770.png", "formula": "\\begin{align*} 2 ^ { \\ell t } & \\| \\psi _ \\ell ^ { O p } ( \\Pi _ 1 ( \\varphi , 1 _ \\Lambda ) ) \\| ^ s _ { p , \\Gamma } \\le \\sum _ { k = \\ell - 1 } ^ { \\ell + 3 } 2 ^ { \\ell t } \\| S ^ { k - 2 } \\varphi \\cdot S _ k 1 _ \\Lambda \\| ^ s _ { p , \\Gamma } \\ , . \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\int _ { - \\infty } ^ { \\infty } q _ t ( x , y ) f ( y ) d y & = \\lim _ { t \\to 0 } \\int _ { \\infty } ^ { x } m ( z ) d z \\int _ { - \\infty } ^ { \\infty } p _ t ( z , y ) \\frac { f ' ( y ) } { m ( y ) } d y \\\\ & = \\int _ { - \\infty } ^ { x } m ( z ) \\frac { f ' ( z ) } { m ( z ) } d z = f ( x ) . \\end{align*}"}
-{"id": "4348.png", "formula": "\\begin{align*} F ( x _ 0 ) & = s x _ 0 ^ { 2 / s } + t A ^ { 1 / t } x _ 0 ^ { - 1 / t } = ( s + 2 t ) x _ 0 ^ { 2 / s } = ( s + 2 t ) 2 ^ { - 2 t / ( s + 2 t ) } A ^ { 2 / ( s + 2 t ) } \\\\ & = n 2 ^ { - 2 t / n } A ^ { 2 / n } = n 2 ^ { s / n - 1 } A ^ { 2 / n } . \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} \\mathfrak { K } : = \\left \\{ \\mathbf { x } = ( x ^ { ( 1 ) } , x ^ { ( 2 ) } ) \\in \\mathfrak { L } ^ { 2 } \\ : \\ | x ^ { ( 1 ) } - x ^ { ( 2 ) } | = 1 \\right \\} \\end{align*}"}
-{"id": "9720.png", "formula": "\\begin{align*} D C G = & D C [ A _ 1 | A _ 2 | . . . | A _ T | P ] , \\\\ = & D [ C A _ 1 | C A _ 2 | . . . | C A _ T | C P ] , \\\\ = & [ D A _ 1 C | D A _ 2 C | . . . | D A _ T C | D C P ] , \\\\ \\approx & B _ X ^ X , \\end{align*}"}
-{"id": "1095.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ m a _ i \\lambda ^ { \\prime } _ { i j } = 0 , \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} v _ { \\lambda } = \\det [ v _ { \\lambda _ i - i + j } ] \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} { \\displaystyle \\Psi ( \\mathbf { x } , 0 ) = \\Psi _ 0 ( \\mathbf { x } ) , \\quad \\mathbf { A } ( \\mathbf { x } , 0 ) = \\mathbf { A } _ { 0 } ( \\mathbf { x } ) , \\mathbf { A } _ { t } ( \\mathbf { x } , 0 ) = \\mathbf { A } _ { 1 } ( \\mathbf { x } ) , } \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{gather*} U _ i = V _ 0 + V _ 1 + \\dots + V _ i ( 0 \\le i \\le d ) . \\end{gather*}"}
-{"id": "3510.png", "formula": "\\begin{align*} \\sum _ { n = - \\infty } ^ { \\infty } I _ { n + \\frac { 1 } { 2 } } ( - n + \\frac { 1 } { 2 } ; x ) z ^ { n } = \\sqrt { \\frac { 2 } { \\pi x } } \\left ( \\frac { z e ^ { x z } } { z - 2 } \\right ) . \\end{align*}"}
-{"id": "8105.png", "formula": "\\begin{align*} \\angle L : = \\max _ { i = 1 , \\ldots , k } \\frac { \\norm { a _ i } _ \\infty } { \\norm { a _ 0 } _ \\infty } . \\end{align*}"}
-{"id": "6997.png", "formula": "\\begin{align*} \\left ( \\sum _ { j _ { 1 } = 1 } ^ { \\infty } \\left ( \\ldots \\left ( \\sum _ { j _ { m } = 1 } ^ { \\infty } \\left \\Vert A \\left ( e _ { j _ { 1 } } , \\dots , e _ { j _ { m } } \\right ) \\right \\Vert ^ { s _ { m } } \\right ) ^ { \\frac { s _ { m - 1 } } { s _ { m } } } \\dots \\right ) ^ { \\frac { s _ { 1 } } { s _ { 2 } } } \\right ) ^ { \\frac { 1 } { s _ { 1 } } } \\leq D _ { m , p , \\mathbf { s } } ^ { \\mathbb { K } } \\Vert A \\Vert \\end{align*}"}
-{"id": "9683.png", "formula": "\\begin{align*} \\Theta ( \\pi - \\Theta ( \\theta , \\phi , \\psi ) + \\theta , \\phi , \\psi ) + \\Theta ( \\theta , \\phi , \\psi ) = \\pi . \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} v ^ + = \\limsup _ { t \\to \\infty } v ( t ) , v ^ - = \\liminf _ { t \\to \\infty } v ( t ) . \\end{align*}"}
-{"id": "600.png", "formula": "\\begin{align*} | U _ t f ( x ) - U _ t f ( x _ 0 ) | = | f ( \\phi _ t ( x ) ) - f ( \\phi _ t ( x _ 0 ) ) | \\leq | f | _ L d ( \\phi _ t ( x ) , \\phi _ t ( x _ 0 ) ) , \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} 1 _ { Z ( \\mu , \\nu ) } = 1 _ { Z ( \\mu , s ( \\mu ) ) } 1 _ { Z ( \\nu , s ( \\nu ) ) } ^ * = \\pi ( s _ { \\mu } s _ { \\nu } ^ * ) ; \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{gather*} \\frac { t } { u } \\frac { { \\rm d } u } { { \\rm d } t } = p _ 1 q _ 1 - 2 p _ 2 q _ 2 + q _ 2 , \\frac { t } { v } \\frac { { \\rm d } v } { { \\rm d } t } = 2 p _ 1 q _ 1 - p _ 2 q _ 2 - q _ 1 + 1 . \\end{gather*}"}
-{"id": "6418.png", "formula": "\\begin{align*} U = \\begin{bmatrix} a & \\pi ^ n \\\\ 1 & d \\end{bmatrix} . \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} \\tilde { F } ( x , t ) = t ^ { - 2 } \\tilde { F } ( \\textstyle \\frac { x } { \\sqrt t } , 1 ) , \\qquad \\hbox { a n d } \\tilde { F } ( \\cdot , 1 ) \\in L ^ 1 \\cap L ^ \\infty \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} H _ w H _ s = \\begin{cases} H _ { w s } & \\\\ ( v ^ { - 1 } - v ) H _ w + H _ { w s } & \\end{cases} \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} \\vec { \\gamma _ 3 } ( \\phi , p _ j ) = \\left ( 0 , 0 , \\frac { 1 } { 4 \\pi } \\Im \\int _ { \\gamma } \\beta \\frac { d z } { z } \\right ) = \\frac { 1 } { 2 } ( 0 , 0 , \\beta ) . \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} \\tilde { f } _ { i j k } ^ { \\alpha a } = \\sum _ { \\tiny \\begin{matrix} \\beta + \\gamma = \\alpha , b + c = a \\\\ | \\beta | + | b | , | \\gamma | + | c | < | \\alpha | + | a | \\end{matrix} } C _ { \\alpha } ^ \\beta C _ a ^ b \\nabla _ k ( \\nabla _ i V ^ { ( \\beta , b ) } \\nabla _ j V ^ { ( \\gamma , c ) } - \\nabla _ i H ^ { ( \\beta , b ) } \\cdot \\nabla _ j H ^ { ( \\gamma , c ) } ) , \\end{align*}"}
-{"id": "9566.png", "formula": "\\begin{align*} \\vec { X } _ { l } = \\left ( \\begin{array} { c c c c c } m _ { - \\delta } ^ { l } , & m _ { - \\delta + \\frac { 1 } { 2 } } ^ { l } , & \\dots , & m _ { \\delta - \\frac { 1 } { 2 } } ^ { l } , & m _ { \\delta } ^ { l } \\end{array} \\right ) ^ { T } \\end{align*}"}
-{"id": "6806.png", "formula": "\\begin{align*} S _ { t } + \\left [ S , K \\right ] = - 2 i \\tilde { \\gamma } \\left ( \\nabla \\varphi . \\partial _ { t } \\nabla + \\Delta \\varphi \\partial _ { t } \\right ) + \\end{align*}"}
-{"id": "7053.png", "formula": "\\begin{align*} \\left \\Vert \\frac { \\partial u } { \\partial t } \\right \\Vert _ { X _ { \\mathbf { p } , q } } + \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert = 2 l } \\left \\Vert D ^ { \\alpha } u \\right \\Vert _ { X _ { \\mathbf { p } , q } } + \\left \\Vert A u \\right \\Vert _ { X _ { \\mathbf { p } , q } } \\leq C \\left \\Vert f \\right \\Vert _ { X _ { \\mathbf { p } , q } } . \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} \\Lambda ( Q , f , m _ { \\bullet } , c _ { \\bullet } ) = K Q / I ( Q , f , m _ { \\bullet } , c _ { \\bullet } ) , \\end{align*}"}
-{"id": "3500.png", "formula": "\\begin{align*} I _ { \\upsilon } ( q ; x ) = \\frac { \\left ( \\frac { x } { 2 } \\right ) ^ { \\upsilon } } { 2 ^ { q + \\frac { 1 } { 2 } } } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\Gamma ( 2 \\upsilon + 2 q + 2 n ) } { \\Gamma ( \\upsilon + q + n + \\frac { 1 } { 2 } ) \\ \\Gamma ( 2 \\upsilon + q + n + \\frac { 1 } { 2 } ) \\ n ! } \\left ( \\frac { x } { 2 } \\right ) ^ { n } . \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} H ^ { \\gamma } ( z ^ 2 ) | 0 \\rangle = V ^ + ( z ) \\gamma _ \\chi ( z ^ 2 ) | 0 \\rangle = \\chi _ { - 1 / 2 } | 0 \\rangle + O ( z ^ 2 ) . \\end{align*}"}
-{"id": "4203.png", "formula": "\\begin{align*} W _ { k } = \\int _ { 0 } ^ { 1 } \\frac { \\phi \\left ( \\gamma \\right ) } { \\left ( \\Delta t \\right ) ^ { \\gamma } } \\omega _ { k } \\left ( \\gamma \\right ) \\mathrm { d } \\gamma , \\ ; \\ ; k = 0 , 1 , \\ldots , n . \\end{align*}"}
-{"id": "5083.png", "formula": "\\begin{align*} \\widehat { \\mu ^ { } _ { n } } ( k ) \\ , = \\prod _ { m = 1 } ^ { n } \\tfrac { 1 } { 3 } \\ ! \\left ( 1 + 2 \\cos \\bigl ( \\tfrac { 2 \\pi k } { 2 ^ m } \\bigr ) \\right ) \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} \\mbox { d i v $ w $ } = f \\ ; \\ ; \\mbox { i n $ D $ } , w | _ { \\partial D } = 0 , \\end{align*}"}
-{"id": "9598.png", "formula": "\\begin{align*} W ^ i _ { j m n } = R ^ i _ { j m n } + \\frac 1 { N + 1 } \\delta ^ i _ j R _ { [ m n ] } + \\frac N { N ^ 2 - 1 } \\delta ^ i _ { [ m } R _ { j n ] } + \\frac 1 { N ^ 2 - 1 } \\delta ^ i _ { [ m } R _ { n ] j } . \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} R ( \\bar { \\psi } _ i ) = R ( \\psi _ i ) + R \\big ( \\psi _ i ^ { ( 1 ) } \\big ) + R \\big ( \\psi _ i ^ { ( 2 ) } \\big ) + R \\big ( \\psi _ i ^ { ( 3 ) } \\big ) = 0 . \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} m _ { j } ( E ) : = \\int _ { E _ j ^ - } ^ { E _ j ^ + } \\frac { x ^ n } { \\sqrt { Q _ E ( x ) } } d x \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{align*} ( \\sigma ( a _ { n - 1 } ) , \\sigma ( a _ 0 ) , \\ldots , \\sigma ( a _ { n - 2 } ) ) \\Sigma \\left ( \\begin{array} { c | c } \\begin{smallmatrix} \\sigma ( e _ 1 ) \\\\ \\vdots \\\\ \\sigma ( e _ { \\nu } ) \\end{smallmatrix} & E ^ { \\mu } \\end{array} \\right ) = 0 . \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{align*} S _ j & = y _ j + y _ { m - j + 1 } = 2 y + ( m - 1 ) d , \\\\ \\mathbb { E } [ Y | \\tau _ j ] & = \\frac { S _ j } { 2 } = y + \\frac { ( m - 1 ) } { 2 } d = \\mathbb { E } [ Y ] , \\ \\forall j \\in [ m ] . \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\mathcal { C } \\left ( n \\right ) } \\frac { 1 } { \\vert \\pi \\vert ! } \\frac { \\kappa _ \\pi } { \\pi ! } = \\frac { \\mu _ n } { n ! } . \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} \\dot { y } y ^ { - 1 } = \\left ( \\dot { x } e ^ F + x \\dot { F } e ^ F \\right ) e ^ { - F } x ^ { - 1 } = \\dot { x } x ^ { - 1 } + \\dot { F } \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} \\langle \\vec { a } _ i , J _ { i + 1 } \\rangle + c _ { i + 1 } = p _ i . \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{align*} a ^ * _ i ( e _ k ) = \\delta _ { i k } = \\begin{cases} 1 & i = k \\\\ 0 & i \\neq k , \\end{cases} \\end{align*}"}
-{"id": "8734.png", "formula": "\\begin{align*} P D \\mathbf { x } = \\mathbf { y } \\ ; \\ ; \\ ; \\mbox { \\rm i f a n d o n l y i f } \\ ; \\ ; \\ ; P D \\mathbf { y } = \\mathbf { x } . \\end{align*}"}
-{"id": "3149.png", "formula": "\\begin{align*} C _ { i j k } = - \\frac { n - 2 } { n - 3 } \\nabla _ { l } W _ { i j k l } . \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} t _ 0 = T \\ne \\infty ; \\end{align*}"}
-{"id": "9137.png", "formula": "\\begin{gather*} h _ 3 ( u , v , x _ 2 , x _ 3 ) : = u ^ 4 + 4 u ^ 2 v + 2 v ^ 2 + \\lambda v x _ 2 x _ 3 , h _ 4 ( u , v , x _ 2 , x _ 3 ) : = v \\big ( u ^ 2 + 2 v \\big ) + \\lambda v x _ 2 x _ 3 . \\end{gather*}"}
-{"id": "7436.png", "formula": "\\begin{align*} u _ 1 = \\frac { m _ 1 ( m _ 2 + m _ 4 ) } { M } , u _ 2 = \\frac { m _ 2 m _ 3 - m _ 1 m _ 4 } { M } , u _ 3 = \\frac { ( m _ 1 + m _ 3 ) m _ 4 } { M } \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{align*} D _ { \\xi _ 1 , \\xi _ 2 } ( s ) = \\epsilon _ { \\xi _ 1 , \\xi _ 2 } \\left ( \\frac { N } { N _ 1 N _ 2 } \\right ) ^ { \\frac 1 2 - s } \\overline { D _ { \\xi _ 1 , \\xi _ 2 } ( 1 - \\bar { s } ) } \\end{align*}"}
-{"id": "1391.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ f ( \\underbar { X } ) \\big | g ( \\underbar { X } ) \\right ] = \\mathbb { E } \\left [ f ( \\underbar { X } ) \\right ] , \\ \\forall g ( \\underbar { X } ) . \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } y _ { 1 } ^ { 2 } = ( x - 1 ) ( x - \\rho ^ 3 ) ( x - \\rho ^ 5 ) ( x - \\rho ^ 6 ) \\\\ \\\\ y _ { 2 } ^ { 2 } = ( x - \\rho ^ { 2 } ) ( x - \\rho ^ 4 ) ( x - \\rho ^ 5 ) ( x - \\rho ^ 6 ) \\\\ \\\\ y _ { 3 } ^ { 2 } = ( x - \\rho ) ( x - \\rho ^ 3 ) ( x - \\rho ^ 4 ) ( x - \\rho ^ 5 ) \\end{array} \\right \\} \\subset { \\mathbb C } ^ { 4 } . \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} \\mathbf { U } : = \\mathbb { P } ( \\mathcal { A } ^ { \\vee } ) \\xrightarrow { \\boldsymbol { \\lambda } } \\mathbf { T } \\end{align*}"}
-{"id": "8337.png", "formula": "\\begin{align*} f _ { n } ( Z ) = 2 R e \\enskip i ^ { n + 1 } ( \\zeta _ 0 Z ^ { n + 1 } - \\zeta _ 1 Z ^ { n } \\overline { Z } + \\dots + ( - 1 ) ^ { n / 2 } \\zeta _ { n / 2 } Z ^ { ( n + 2 ) / 2 } \\overline { Z } ^ { n / 2 } ) . \\end{align*}"}
-{"id": "5818.png", "formula": "\\begin{align*} \\nabla f \\neq 0 \\ , \\ , \\partial \\Omega ^ + = \\{ x \\in M ^ n | f ( x ) = 0 \\} , \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{align*} \\Lambda _ { N + 1 } ^ { \\infty } \\Lambda _ N ^ { N + 1 } ( \\omega , d y ) & = \\left ( \\prod _ { k = 1 } ^ { N - 1 } \\frac { 1 } { k ! } \\right ) \\Delta _ { N } ( y ) \\int _ { - \\infty } ^ { y _ 1 } \\cdots \\int _ { y _ N } ^ { \\infty } \\det \\left ( \\phi _ { \\omega } ^ { ( j - 1 ) } ( x _ { N + 2 - i } ) \\right ) ^ { N + 1 } _ { i , j = 1 } d x _ 1 \\cdots d x _ { N + 1 } d y \\\\ & = \\left ( \\prod _ { k = 1 } ^ { N - 1 } \\frac { 1 } { k ! } \\right ) \\Delta _ { N } ( y ) \\det \\left ( \\phi _ { \\omega } ^ { ( j - 1 ) } ( y _ { N + 1 - i } ) \\right ) ^ N _ { i , j = 1 } d y . \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} f \\circ x = \\sum _ { i < \\alpha } r _ { i } e ^ { \\gamma _ { i } \\circ x } \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} T ( f _ 1 , f _ 2 , f _ 3 ) = \\frac { 1 } { N ^ 2 } \\sum \\limits _ { \\substack { n _ 1 , n _ 2 , n _ 3 \\leqslant N \\\\ \\vert n _ 1 - \\sqrt { 2 } n _ 2 + ( - 1 + \\sqrt { 2 } ) n _ 3 \\vert \\leqslant \\frac { 1 } { 2 } } } f _ 1 ( n _ 1 ) f _ 2 ( n _ 2 ) f _ 3 ( n _ 3 ) . \\end{align*}"}
-{"id": "2860.png", "formula": "\\begin{align*} H : = \\left ( \\frac { \\partial ^ 2 } { \\partial x _ 1 ^ 2 } + \\frac { \\partial ^ 2 } { \\partial x _ 2 ^ 2 } \\right ) - \\sum \\limits _ { \\alpha \\in \\Pi } \\dfrac { \\mu _ \\alpha ( \\mu _ \\alpha + 1 ) } { l _ \\alpha ^ 2 ( \\vec { x } - \\vec { \\xi } ) } , \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} \\sup _ { f \\in L ^ 1 ( \\Gamma _ - , d \\xi ) \\atop \\| f \\| _ { L ^ 1 ( \\Gamma _ - , d \\xi ) } = 1 } \\big | \\int _ { \\Gamma _ + } \\phi ( z , v ) j _ + K J f ( z , v ) d \\xi ( z , v ) \\big | \\le C _ 1 e ^ { 2 \\| \\tau \\sigma _ - \\| _ \\infty } \\| k \\| _ { \\infty } \\eta ^ { n - 1 } , \\end{align*}"}
-{"id": "317.png", "formula": "\\begin{align*} \\Xi ^ { \\lambda } ( K _ { a } F _ { a } \\otimes 1 \\otimes E _ { a } K _ { a } ^ { - 1 } \\otimes K _ { a } ) = \\sum _ { j , m , n , o } c _ { j } ^ { o } \\pi ( K _ { a } F _ { a } ) _ { m } ^ { j } c _ { n } ^ { m } \\pi ( E _ { a } K _ { \\lambda } ) _ { o } ^ { n } . \\end{align*}"}
-{"id": "2639.png", "formula": "\\begin{align*} \\widetilde { \\lambda } _ { n } : = n _ { 1 } ^ 2 + n _ { 2 } ^ 2 , \\phi _ { n } ( x ) : = { 2 \\over \\pi } \\sin ( n _ { 1 } x _ { 1 } ) \\sin ( n _ { 2 } x _ { 2 } ) , \\mbox { f o r } \\ , n \\in { \\Bbb N } ^ * \\times { \\Bbb N } ^ * . \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{gather*} A ( x ) = \\frac { A _ 0 } { x ^ 2 } + \\frac { A _ 1 } { x } + \\cdots \\in M _ 2 ( \\mathbb { C } ( \\ ! ( x ) \\ ! ) ) , \\end{gather*}"}
-{"id": "7355.png", "formula": "\\begin{gather*} \\begin{array} { l c l } \\star _ t \\tau _ 3 ( t ) & = & \\frac { \\sqrt { 6 } } { 7 y ( t ) ^ 5 } ( - f ^ { 1 3 6 7 } - f ^ { 1 4 5 7 } - f ^ { 2 3 5 7 } + f ^ { 2 4 6 7 } ) + \\frac { 4 \\sqrt { 6 } } { 2 1 y ( t ) ^ 5 } ( f ^ { 1 2 3 4 } + f ^ { 1 2 5 6 } + f ^ { 3 4 5 6 } ) , \\end{array} \\end{gather*}"}
-{"id": "1615.png", "formula": "\\begin{align*} \\left \\{ \\ \\begin{aligned} & w _ \\ell ( t ) = a _ \\ell \\ , \\Im \\zeta ^ \\ell + b _ \\ell \\ , \\Re \\zeta ^ \\ell \\\\ & \\mbox { w i t h } a _ \\ell = \\frac { g ^ \\omega _ \\ell - g ^ 0 _ \\ell \\ , \\cos \\ell \\omega } { \\sin \\ell \\omega } \\quad \\mbox { a n d } b _ \\ell = g ^ 0 _ \\ell \\ , . \\end{aligned} \\right . \\end{align*}"}
-{"id": "4035.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { p - 1 } \\bigg ( \\frac { ( \\frac 1 2 ) _ k } { k ! } \\bigg ) ^ 2 \\cdot ( - 1 ) ^ k \\equiv p ^ 2 \\sum _ { k = \\frac { p - 1 } 2 } ^ { p - 1 } \\bigg ( \\frac { k ! } { ( \\frac 3 2 ) _ k } \\bigg ) ^ 2 \\cdot ( - 1 ) ^ k \\pmod { p ^ 2 } \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} D = \\left \\langle \\begin{pmatrix} w _ 0 & 0 \\\\ 0 & w _ 0 \\end{pmatrix} , I + \\begin{pmatrix} 1 & 0 \\\\ 0 & d \\end{pmatrix} p \\right \\rangle D ' = \\left \\langle \\begin{pmatrix} w _ 0 & 0 \\\\ 0 & w _ 0 \\end{pmatrix} , I + \\begin{pmatrix} d & 0 \\\\ 0 & 1 \\end{pmatrix} p \\right \\rangle . \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle \\mathcal { E } ( t ) = \\int _ { \\Omega } \\left ( \\frac { 1 } { 2 } \\big \\vert \\left [ i \\nabla + q \\mathbf { A } \\right ] \\Psi \\big \\vert ^ { 2 } + V | \\Psi | ^ { 2 } + \\frac { 1 } { 2 } \\big | \\frac { \\partial \\mathbf { A } } { \\partial t } \\big | ^ { 2 } + \\frac { 1 } { 2 } | \\nabla \\times \\mathbf { A } | ^ { 2 } + \\frac { 1 } { 2 } | \\nabla \\phi | ^ { 2 } \\right ) \\ ; \\mathrm { d } \\mathbf { x } . } \\end{array} \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} W _ { k } = \\int _ { 0 } ^ { 1 } \\frac { 1 } { \\left ( \\Delta t \\right ) ^ { \\gamma } } \\omega _ { k } \\left ( \\gamma \\right ) \\mathrm { d } \\gamma , \\ ; \\ ; k = 0 , 1 , \\ldots , n . \\end{align*}"}
-{"id": "9790.png", "formula": "\\begin{align*} Q _ k ( M , g ) : = \\sup \\{ \\frac { \\| h \\| _ { L ^ { \\infty } } ^ 2 } { \\| h \\| _ { L ^ 2 } ^ 2 } \\mid h \\in \\mathcal { H } ^ k \\} \\cdot \\sup _ { y \\in M } \\int _ M d i s t ( x , y ) ^ { 2 - n } , \\end{align*}"}
-{"id": "158.png", "formula": "\\begin{align*} \\cos ^ 3 \\theta = \\tfrac { 1 } { 8 } ( e ^ { i \\theta } + e ^ { - i \\theta } ) ^ 3 = \\tfrac { 1 } { 4 } \\cos 3 \\theta + \\tfrac { 3 } { 4 } \\cos \\theta \\end{align*}"}
-{"id": "9411.png", "formula": "\\begin{align*} \\psi ( s , t ) = \\langle a ( s ) , b ( t ) \\rangle \\end{align*}"}
-{"id": "9661.png", "formula": "\\begin{align*} \\mathcal { P } ( K ) = \\abs { K } \\ , \\abs { K ^ \\circ } \\geq \\frac { 4 ^ n } { n ! } . \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{align*} f ^ \\# _ k = \\int _ { 0 } ^ { 2 \\pi } e ^ { - i k \\theta } f ( e ^ { i \\theta } ) \\frac { d \\theta } { 2 \\pi } \\end{align*}"}
-{"id": "6233.png", "formula": "\\begin{align*} | G ( w , x - x _ 0 ) | = O ( | ( w , x - x _ 0 ) | ) \\end{align*}"}
-{"id": "709.png", "formula": "\\begin{align*} e _ i ^ 2 e _ { i \\pm 1 } - 2 e _ i e _ { i \\pm 1 } e _ i + e _ { i \\pm 1 } e _ i ^ 2 = 0 , f _ i ^ 2 f _ { i \\pm 1 } - 2 f _ i f _ { i \\pm 1 } f _ i + f _ { i \\pm 1 } f _ i ^ 2 = 0 \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} M f ( s ) = F ( s ) M g ( s ) , \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} R _ { x _ 0 , y _ 0 } ( x , y ) = ( x - x _ 0 ) ^ \\top \\widehat G ( x , y ) ( y - y _ 0 ) , \\end{align*}"}
-{"id": "2015.png", "formula": "\\begin{gather*} A ( x ) \\mapsto A ^ P ( x ) : = P ^ { - 1 } A ( x ) P - P ^ { - 1 } P ' \\end{gather*}"}
-{"id": "3649.png", "formula": "\\begin{align*} [ \\chi _ m , \\chi _ n ] = ( - 1 ) ^ { m - \\frac { 1 } { 2 } } \\delta _ { m , - n } 1 . \\end{align*}"}
-{"id": "9939.png", "formula": "\\begin{align*} | \\xi + \\eta | ^ 2 + | \\xi - \\eta | ^ 2 = 2 | \\xi | ^ 2 + 2 | \\eta | ^ 2 \\end{align*}"}
-{"id": "6267.png", "formula": "\\begin{align*} \\pi _ { P ( \\theta _ k ) } q _ k & = \\langle q _ k , e ^ { \\i \\langle \\theta _ k , \\cdot \\rangle _ { \\C ^ d } } \\rangle e ^ { \\i \\langle \\theta _ k , \\cdot \\rangle _ { \\C ^ d } } - \\sum _ { z \\in \\Z ^ d \\backslash \\{ 0 \\} } \\frac { ( \\theta _ k + 2 \\pi z ) ( \\theta _ k + 2 \\pi z ) ^ T } { \\vert \\theta _ k + 2 \\pi z \\vert ^ 2 } c ^ { ( z ) } _ { q _ k } e ^ { \\i \\langle \\theta _ k + 2 \\pi z , \\cdot \\rangle _ { \\C ^ d } } . \\end{align*}"}
-{"id": "2200.png", "formula": "\\begin{align*} \\left \\| x ^ v \\ ! \\ ! \\int _ x ^ \\infty [ K _ x ( x , s ) - K _ s ( x , s ) ] F ( s + x ) d s \\right \\| & \\le c x ^ v \\ ! \\ ! \\int _ x ^ \\infty \\ ! \\ ! \\sigma ( x ) \\sigma \\left ( \\ ! \\frac { x + s } { 2 } \\ ! \\right ) \\| F ( s + x ) \\| d s \\\\ & \\le c \\int _ x ^ \\infty \\sigma ( y ) \\| F ( 2 y ) \\| d y \\\\ & \\le c \\sigma ( x ) \\in L ^ 1 ( \\mathbb { R } ^ + ) , \\ ; \\ ; v = 0 , 1 . \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} \\sigma = \\frac { 1 } { 4 ( \\gamma - 1 ) } , \\tau = \\frac { 1 } { \\sqrt \\sigma } = 2 \\sqrt { \\gamma - 1 } \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} f \\left ( g _ 1 \\left ( z \\right ) \\right ) = \\sum _ { n = 1 } ^ { \\infty } \\kappa _ n \\frac { z ^ n } { n ! } , \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{align*} \\left | \\frac { \\prod _ { i = 1 } ^ { N } ( x ^ { ( n ) , m } _ { i + 1 } - x ^ { ( n ) , m } _ i ) \\Delta _ N ( \\xi ^ { ( n ) , m } ) } { \\Delta _ { N + 1 } ( x ^ { ( n ) , m } ) } \\right | \\le 1 , \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} \\left \\langle b ( g ) , \\xi \\right \\rangle = \\left \\langle b ( g x _ { \\xi } ) , \\xi \\right \\rangle - \\left \\langle \\pi _ { g } b ( x _ { \\xi } ) , \\xi \\right \\rangle = \\left \\langle b ( g x _ { \\xi } ) , \\xi \\right \\rangle - \\left \\langle b ( x _ { \\xi } ) , \\pi _ { g } ^ { \\ast } \\xi \\right \\rangle . \\end{align*}"}
-{"id": "7387.png", "formula": "\\begin{align*} \\xi _ 0 = [ 0 , T ^ { d _ 1 - d _ 0 } , T ^ { d _ 2 - d _ 1 } , \\dots ] . \\end{align*}"}
-{"id": "6993.png", "formula": "\\begin{align*} \\left ( \\sum _ { j _ { 1 } = 1 } ^ { \\infty } \\left ( \\ldots \\left ( \\sum _ { j _ { m } = 1 } ^ { \\infty } \\left \\Vert T \\left ( e _ { j _ { 1 } } , \\dots , e _ { j _ { m } } \\right ) \\right \\Vert ^ { s _ { m } } \\right ) ^ { \\frac { s _ { m - 1 } } { s _ { m } } } \\dots \\right ) ^ { \\frac { s _ { 1 } } { s _ { 2 } } } \\right ) ^ { \\frac { 1 } { s _ { 1 } } } \\leq \\left ( \\sqrt { 2 } \\right ) ^ { m - 1 } \\Vert T \\Vert \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} a _ { j k } = \\begin{cases} 1 & \\mbox { i f t h e v e r t e x } k - 1 \\in F _ j \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "6389.png", "formula": "\\begin{align*} F ^ 0 _ n = \\{ v _ 1 , \\dots , v _ n \\} \\quad F ^ 1 _ n = \\{ e _ 1 , f _ 1 , \\dots , e _ { n - 1 } , f _ { n - 1 } \\} . \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{align*} t _ { o , n } & = \\mathrm { m a x } ( t _ i , t _ { o , n - 1 } + K _ { n - 1 } \\tau _ { f , n - 1 } ) \\\\ h _ n & = \\left \\{ \\begin{array} { l l } \\frac { a _ { m a x } } { ( 1 - K ) \\tau _ { f , n } } , & t _ i \\geq t _ { o , n - 1 } + \\tau _ { f , n - 1 } \\\\ h _ { n - 1 } , & t _ i < t _ { o , n - 1 } + \\tau _ { f , n - 1 } \\end{array} \\right . \\\\ a _ { m a x } & = 1 / m \\left ( f _ { p l a n a r } - K _ d v _ { w , m a x } ^ 2 \\right ) \\\\ S _ { t r a j } & = ( c _ 1 ( d _ 1 H + d _ 4 ) ( 1 - H ^ 2 ) ) ^ 2 + ( d _ 1 ( 1 - H ^ 2 ) ) ^ 2 \\end{align*}"}
-{"id": "3191.png", "formula": "\\begin{align*} \\mathsf { M } _ { H P } ^ { s , N } ( d X ) = c o n s t \\times \\det \\left ( ( I + i X ) ^ { - s - N } \\right ) \\det \\left ( ( I - i X ) ^ { - \\bar { s } - N } \\right ) \\times d X , \\end{align*}"}
-{"id": "6114.png", "formula": "\\begin{align*} L _ { j + 1 } \\cap X _ { 1 , \\ , j + 1 , \\ , 1 } = 0 . \\end{align*}"}
-{"id": "2282.png", "formula": "\\begin{align*} \\gamma _ i ( z , t ) : = R ^ i \\gamma _ 0 ( z , t ) R ^ { - i } . \\end{align*}"}
-{"id": "1352.png", "formula": "\\begin{align*} f _ { V } : V \\longrightarrow \\R , f _ { V } ( v ) : = \\sigma ( v , v ) \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} \\smash { n = n _ { j _ { m } } + k d _ { j _ { m } } + k ' \\textrm { p e r } ( x _ { l } ) , \\quad \\textrm { w h e r e } \\ 0 \\le k \\le \\dfrac { \\alpha d _ { j _ { m } + 1 } } { d _ { j _ { m } } } - 2 \\ \\textrm { a n d } \\ 0 \\le k ' \\le \\dfrac { \\alpha d _ { j _ { m } } } { \\textrm { p e r } ( x _ { j } ) } } \\cdot \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} \\sup _ { j \\ge 0 } \\ \\| P _ { n } T ^ { \\ , j } \\ , e _ { k } \\| \\le | v _ { l } | \\ \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } | w _ { s } | \\Bigr ) \\ \\sup _ { j \\ge 0 } \\ \\| P _ { n } T ^ { \\ , j } e _ { b _ { \\varphi ( l ) } } \\| \\end{align*}"}
-{"id": "9433.png", "formula": "\\begin{align*} \\mathrm { L } _ { h } ( w ^ * ) = \\partial _ { t } \\big ( { h ^ * } ^ { - 1 } \\big ) - { h ^ * } ^ { - 1 } \\nabla \\cdot v ^ * + v ^ * \\cdot \\nabla \\big ( { h ^ * } ^ { - 1 } \\big ) , \\end{align*}"}
-{"id": "9956.png", "formula": "\\begin{align*} E _ { \\sigma ( t ) } \\dot x = A _ { \\sigma ( t ) } x \\mbox { w h e r e } \\sigma ( t ) \\in \\{ 1 , \\cdots , N \\} \\ , . \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} \\left \\langle f , g \\right \\rangle _ { \\widehat { S } _ p } : = \\frac { 1 } { p } \\int \\limits _ S \\sum _ { j = 0 } ^ { p - 1 } f ( e ^ { \\frac { j \\pi i } { p } } \\zeta ) \\overline { g ( e ^ { \\frac { j \\pi i } { p } } \\zeta ) } \\ , d \\sigma ( \\zeta ) , \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { k = 0 } ^ { \\infty } a _ { k } z ^ { k } , \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{align*} { \\sf M S } ^ { \\omega , * } = \\begin{cases} \\ ( q ^ * _ P - q ^ * _ W \\ ) \\sqrt { 3 } \\sigma , & \\ \\ \\omega \\leq \\mu , \\\\ \\ ( q ^ * _ W - q ^ * _ P \\ ) \\sqrt { 3 } \\sigma , & . \\end{cases} \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} \\left ( \\sum _ { n = 0 } ^ \\infty \\frac { | a _ n | ^ 2 } { n + 1 } \\right ) ^ \\frac { 1 } { 2 } \\leq \\| f \\| _ { H ^ 1 } \\end{align*}"}
-{"id": "3735.png", "formula": "\\begin{align*} P ( z ) = z ^ { m + 1 } - c _ 1 z ^ m - \\cdots - c _ m z \\end{align*}"}
-{"id": "4965.png", "formula": "\\begin{align*} \\widetilde { r } _ { s } - \\widetilde { r } _ { c } = r _ { s } - r _ { c } + \\left ( t _ { 1 } - t _ { 2 } \\right ) \\overline { F } \\mathbf { e } _ { 1 } \\pm \\left ( t _ { 1 } - t _ { 2 } \\right ) \\overline { F } \\mathbf { e } _ { m + 2 } . \\end{align*}"}
-{"id": "9549.png", "formula": "\\begin{align*} \\sigma ( t ^ { - 1 } g ^ { - 1 } , g ) v _ { t ^ { - 1 } } ^ * \\mu ( t ^ { - 1 } g ^ { - 1 } , g ) ^ * = \\alpha _ { t ^ { - 1 } g ^ { - 1 } } ( v _ g ^ * ) v _ { t ^ { - 1 } g ^ { - 1 } } ^ * . \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{align*} P _ { \\infty } ( t ) \\Lambda ^ { \\infty } _ { N } = \\Lambda ^ { \\infty } _ { N } P _ N ( t ) . \\end{align*}"}
-{"id": "9349.png", "formula": "\\begin{align*} \\Big ( \\sum _ { j = 0 } ^ { K - 1 } d ( \\gamma _ { N _ { j + 1 } } , \\gamma _ { N _ j } ) ^ r \\Big ) ^ { \\frac 1 r } = \\Big ( \\sum _ { j = 0 } ^ { K - 1 } \\| \\sigma _ { N _ { j + 1 } } - \\sigma _ { N _ { j } } \\| _ { } ^ r \\Big ) ^ { \\frac 1 r } + O ( \\mathcal { V } ^ 2 _ r ( \\sigma ; [ - R , R ] ) ) . \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} \\| \\sum _ { j = 2 } ^ \\infty \\tilde S ^ { j - 2 } f \\tilde S _ j g \\| _ { B ^ s _ { p , \\infty } ( \\real ^ { d _ s } ) } & \\le C \\sup _ { k \\ge 2 } 2 ^ { k s } \\sum _ { \\ell = - 1 } ^ { + 1 } \\| \\tilde S ^ { k + \\ell - 2 } f \\tilde S _ { k + \\ell } g \\| _ { L _ p ( \\real ^ { d _ s } ) } \\ , . \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} \\partial _ t w _ k ^ 2 - \\Delta w _ k ^ 2 + \\nabla p _ k ^ 2 = - ( J _ k w _ k ) \\cdot \\nabla w _ k , w _ k ^ 2 ( \\cdot , 0 ) = 0 , \\end{align*}"}
-{"id": "6554.png", "formula": "\\begin{align*} a = \\frac { 3 } { 2 } , \\ , \\ , b = \\frac { 1 } { 2 } , \\ , \\ , c = \\frac { 1 } { 2 } - k , \\ , \\ , d = \\frac { 3 } { 2 } - k , \\ , \\ , \\gamma = 1 , \\ , \\ , \\alpha = 1 \\ , \\ , \\mbox { a n d } \\ , \\ , \\beta = 3 . \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} \\langle L _ { c _ * } \\hat { u } _ 0 , \\hat { u } _ 0 \\rangle _ { L ^ 2 ( \\mathbb { R } ) } \\geq A \\| \\hat { u } _ 0 \\| _ { H ^ 1 ( \\mathbb { R } ) } ^ 2 \\mbox { \\rm i f } \\ ; \\ ; \\langle u _ { c _ * } , \\hat { u } _ 0 \\rangle _ { L ^ 2 ( \\mathbb { R } ) } = \\langle \\partial _ { \\xi } ^ { - 1 } \\partial _ c u _ { c } | _ { c = c _ * } , \\hat { u } _ 0 \\rangle _ { L ^ 2 ( \\mathbb { R } ) } = 0 \\end{align*}"}
-{"id": "840.png", "formula": "\\begin{align*} \\begin{cases} { \\varepsilon ^ 2 } \\Delta u - V ( x ) u + \\psi u = 0 , \\ , & x \\in \\R ^ { 3 } , \\\\ \\varepsilon ^ 2 \\Delta \\psi + \\frac { | u | ^ 2 } { 2 } = 0 , \\ , & x \\in \\R ^ { 3 } . \\end{cases} \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{gather*} \\phi _ t + b \\phi _ x + \\phi _ { x x x } + \\phi _ { x y y } = 0 , \\\\ \\phi \\big | _ { t = T } = \\phi _ 0 ( x , y ) , \\phi \\big | _ { x = 0 } = \\phi _ x \\big | _ { x = 0 } = \\phi \\big | _ { x = R } = 0 \\end{gather*}"}
-{"id": "9573.png", "formula": "\\begin{align*} V ^ { \\mathrm { d e s i g n } } ( N _ 1 , N _ 0 , n _ 1 , n _ 0 ) & = E \\left [ \\mbox { v a r } \\big ( \\widehat { \\theta } \\ , | \\ , { \\mathbf { R } } , N _ 1 , N _ 0 \\big ) \\ , | \\ , N _ 1 , N _ 0 \\right ] = \\frac { S ^ 2 _ 1 } { N _ 1 } + \\frac { S ^ 2 _ 0 } { N _ 0 } - \\frac { S ^ 2 _ { \\theta } } { N _ 0 + N _ 1 } . \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} t ^ { \\star } = \\frac { 1 + \\nu ' } { ( \\xi _ 1 + \\xi _ 2 ) } t , \\ , \\ , \\ , T ( c ) = \\frac { \\tau ( c ) } { 1 + \\nu ' } , \\end{align*}"}
-{"id": "8581.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } & \\vec { \\gamma } _ 0 ( p ) = 0 \\qquad & & \\ ; \\ , 1 \\leq \\theta _ 0 ( p ) \\leq 3 \\\\ & r ( p ) \\leq \\theta _ 0 ( p ) - 2 \\qquad & & \\ ; \\ , \\theta _ 0 ( p ) \\geq 2 . \\end{alignedat} \\right . \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\lambda \\dot x ( t ) + \\nabla ^ 2 \\phi ( x ( t ) ) ( \\dot x ( t ) ) + \\nabla \\phi ( x ( t ) ) + \\nabla \\psi ( x ( t ) ) = 0 \\\\ x ( 0 ) = x _ 0 , v ( 0 ) = v _ 0 = \\nabla \\phi ( x _ 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{align*} ( \\Lambda ( x _ 2 , x _ 5 ) , d x _ 2 = 0 , d x _ 5 = x _ 2 ^ 3 ) , ( \\Lambda ( z _ 1 , z _ 2 , z _ 9 ) , d z _ 1 = d z _ 2 = 0 , d z _ 9 = z _ 2 ^ 5 ) . \\end{align*}"}
-{"id": "8725.png", "formula": "\\begin{align*} L ( \\gamma ) : = \\underset { i = 1 , \\ldots , d } { \\max } | | \\Lambda ^ { 2 } \\rho _ { i } ( \\gamma ) | | . | | \\rho _ { i } ( \\gamma ) ^ { - 1 } | | ^ { 2 } \\in [ 1 , \\infty [ . \\end{align*}"}
-{"id": "3230.png", "formula": "\\begin{align*} f ^ { ( N ) } _ { \\mathcal { A } , t } ( x ) = \\int _ { \\mathcal { A } } ^ { } e ^ { - \\lambda _ { N , s } t } \\frac { \\Delta _ N ( y ) } { \\Delta _ N ( x ) } \\det \\left ( p ^ { ( N ) , s } _ t ( x _ i , y _ j ) \\right ) ^ N _ { i , j = 1 } d y > 0 \\ , \\ \\forall x \\in \\mathring { W } ^ N . \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} S _ n = n ^ \\theta l ( \\theta , n ) . \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{align*} \\rho ^ { \\circ } ( X ) = \\pi ( K _ { 2 \\rho } S ^ { - 1 } ( X ) K _ { 2 \\rho } ^ { - 1 } ) . \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} \\begin{aligned} & u _ { 1 i } ( t ) { = } u _ { 1 i } ^ \\varepsilon ( t ) { = } 2 \\sqrt { { \\pi k _ i } { \\varepsilon ^ { - 1 } } } \\cos \\big ( { 2 \\pi k _ i t } { \\varepsilon ^ { - 1 } } \\big ) , \\\\ & u _ { 2 i } ( t ) { = } u _ { 2 i } ^ \\varepsilon ( t ) { = } 2 \\sqrt { { \\pi k _ i } { \\varepsilon ^ { - 1 } } } \\sin \\big ( { 2 \\pi k _ i t } { \\varepsilon ^ { - 1 } } \\big ) , \\end{aligned} \\end{align*}"}
-{"id": "2026.png", "formula": "\\begin{gather*} A ^ P ( z ) = B _ 1 ( z ) \\oplus \\cdots \\oplus B _ n ( z ) , B _ k ( z ) = \\frac { 1 } { z ^ { r + 1 } } \\big ( B ^ k _ 0 + B ^ k _ 1 z + \\cdots \\big ) , \\end{gather*}"}
-{"id": "6448.png", "formula": "\\begin{align*} : a _ k a _ m : = \\begin{cases} a _ k a _ m , ~ m \\leqslant k \\\\ a _ m a _ k , ~ m \\geqslant k \\end{cases} \\end{align*}"}
-{"id": "7658.png", "formula": "\\begin{align*} b _ j = \\sum _ { i = 1 } ^ n b _ { i j } g _ i \\in \\Z [ G ] . \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} \\| f \\| _ { w \\mathcal { M } ^ p _ q } : = \\sup _ { a \\in \\R ^ d , \\ , r , \\gamma > 0 } | B ( a , r ) | ^ { \\frac { 1 } { q } - \\frac { 1 } { p } } \\ , \\gamma \\left | \\{ x \\in B ( a , r ) \\ , : \\ , | f ( x ) | > \\gamma \\} \\right | ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "1273.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mu } \\left ( Z _ { 0 } = \\mathsf { o } \\right ) = 1 . \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{align*} v _ 1 ^ 2 + v _ 2 ^ 2 + \\cdots + v _ { n + 3 } ^ 2 = 0 , \\ , \\mathrm { a n d } \\ , \\lambda _ 1 v _ 1 ^ 2 + \\lambda _ 2 v _ 2 ^ 2 + \\cdots + \\lambda _ { n + 3 } v _ { n + 3 } ^ 2 = 0 . \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{align*} \\sum _ { \\substack { 1 \\leq i , j \\leq M \\\\ d ( v _ i , v _ j ) < \\log m } } \\left | \\hat { \\mu } ( \\xi _ i - \\xi _ j ) \\right | ^ N = O \\left ( \\frac { m ^ 2 } { R ^ 2 } \\right ) + e ^ { 2 c } O \\left ( \\frac { m } { R ^ 4 } \\right ) , \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} \\binom { n + 1 - m } { m - z } \\leq \\frac { n ^ { m - z } } { ( m - z ) ! } \\leq \\frac { m ^ z n ^ { m - z } } { m ! } , \\end{align*}"}
-{"id": "8498.png", "formula": "\\begin{align*} \\aligned M _ r - m _ r = 0 \\mathrm { f o r \\ a l l \\ } r > 0 . \\endaligned \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} B ( x , y ) = \\sum _ { i = 0 } ^ { n - 1 } \\sum _ { j = 0 } ^ { n - 1 } b _ { i j } x ^ i y ^ j = \\frac { Q ( x ) S ( y ) - S ( x ) Q ( y ) } { x - y } . \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} 1 = s f \\end{align*}"}
-{"id": "4460.png", "formula": "\\begin{align*} \\mathbf { F } \\left ( r \\right ) = \\left ( 1 + r \\right ) ^ { - ( d + \\epsilon ) } \\mathbf { F } \\left ( r \\right ) = \\mathrm { e } ^ { - \\varsigma r } ( 1 + r ) ^ { - ( d + \\epsilon ) } \\end{align*}"}
-{"id": "5728.png", "formula": "\\begin{align*} \\mathcal { B } _ \\eta \\circ \\left ( \\mathcal { N } _ A ( s ) \\otimes \\mathcal { N } _ B ( t ) \\right ) = \\mathcal { N } _ C ( \\eta s + | 1 - \\eta | t ) \\circ \\mathcal { B } _ \\eta \\ ; . \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\dim Z _ { [ i ] } ^ { + } \\geq C n ^ { 2 g } . \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{align*} \\eta _ { n + 1 } ^ { h } = S _ { \\Delta t } \\eta _ n ^ h + \\Delta t S _ { \\Delta t } G _ \\delta ' ( X _ n ) . \\eta _ n ^ h + S _ { \\Delta t } e ^ { \\tau A } \\bigl ( \\sigma _ \\delta ' ( X _ n ) . \\eta _ n ^ h \\bigr ) \\Delta W _ n . \\end{align*}"}
-{"id": "7764.png", "formula": "\\begin{align*} \\sigma _ { t } ^ { \\prime } [ l ] = \\sigma _ { t - 1 } ^ { \\prime } [ l ] \\mbox { a n d } O P T _ { t } ^ { \\prime } [ l ] = O P T _ { t - 1 } ^ { \\prime } [ l + 1 ] . \\end{align*}"}
-{"id": "7095.png", "formula": "\\begin{align*} A ^ + = \\{ s ^ { \\frown } i : s \\in \\max ( A ) , \\ i \\in \\{ 0 , 1 \\} , \\mathrm { \\ a n d \\ } s ^ { \\frown } i \\in \\widehat { T } \\} . \\end{align*}"}
-{"id": "6360.png", "formula": "\\begin{align*} \\pi ' _ 1 ( T _ \\xi ) ^ * \\pi _ 1 ' ( T _ \\eta ) = \\pi ' _ 1 ( T ^ * _ \\xi T _ \\eta ) ) = \\pi ' _ 1 ( \\langle \\xi , \\eta \\rangle _ A ) = \\tilde { P } \\tilde \\varphi ^ \\infty ( \\alpha _ A ( \\langle \\xi , \\eta \\rangle _ A ) ) \\tilde { P } . \\end{align*}"}
-{"id": "2287.png", "formula": "\\begin{align*} \\kappa = ( ( 0 , t _ k ) , ( - 1 , t _ k ) , ( - 2 , t _ k ) , \\ldots , ( - 3 k , t _ k ) ) , \\end{align*}"}
-{"id": "7629.png", "formula": "\\begin{align*} \\lim _ { t \\to T ^ - } \\mathrm { d i a m } ( F _ y , \\omega _ \\epsilon ( t ) ) = 0 \\end{align*}"}
-{"id": "3023.png", "formula": "\\begin{align*} \\prod _ { j = 0 } ^ { m - 1 } \\varepsilon _ { n ( j ) } \\ge ( \\Delta b _ { n ( m ) } ) ^ { - m } \\prod _ { j = 0 } ^ { m - 1 } \\Delta b _ { n ( j ) } \\quad \\hbox { f o r i n f i n i t e l y m a n y $ m \\geq 1 $ . } \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} Q ' ( x _ k ) P ' ( x _ k ) = Q '' ( x _ k ) P ( x _ k ) , k = 1 , 2 , \\dots , n . \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} Y _ 1 : = M + L , Y _ 2 : = L . \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} s _ { n , k } ^ { ( - 1 ) } & : = \\sum _ { d | n } p ( d - k ) \\mu ( n / d ) \\end{align*}"}
-{"id": "8309.png", "formula": "\\begin{align*} h & = ( 0 , \\frac { \\Lambda _ 1 ^ \\vee + \\Lambda _ 2 ^ \\vee + \\Lambda _ 3 ^ \\vee + \\Lambda _ 4 ^ \\vee } { 5 } ) , \\\\ 2 [ h ] & = ( 0 , \\frac { - 3 \\Lambda _ 1 ^ \\vee + 2 \\Lambda _ 2 ^ \\vee + 2 \\Lambda _ 3 ^ \\vee - 3 \\Lambda _ 4 ^ \\vee } { 5 } ) . \\end{align*}"}
-{"id": "4459.png", "formula": "\\begin{align*} \\mathbf { D } : = \\underset { x , y \\in \\mathfrak { L } } { \\sup } \\sum _ { z \\in \\mathfrak { L } } \\frac { \\mathbf { F } \\left ( \\left \\vert x - z \\right \\vert \\right ) \\mathbf { F } \\left ( \\left \\vert z - y \\right \\vert \\right ) } { \\mathbf { F } \\left ( \\left \\vert x - y \\right \\vert \\right ) } < \\infty \\ . \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 4 & \\frac 1 4 \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac 1 4 \\bigg ] _ { p - 1 } \\equiv \\begin{cases} \\big ( \\frac { 2 } { p } \\big ) \\cdot ( a + b \\sqrt { - 3 } ) \\pmod { p ^ 2 } , & p \\equiv 1 \\pmod { 4 } , \\\\ \\big ( \\frac { 2 } { p } \\big ) \\cdot 2 ( a \\sqrt { - 3 } - 3 b ) \\pmod { p ^ 2 } , & p \\equiv 3 \\pmod { 4 } , \\end{cases} \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} x _ 1 = X , x _ 2 = x _ 2 ( X , Y ) = Y + \\eta ( X ) a ( Y ) . \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} C = \\operatorname { s p a n } _ { \\R _ { \\geq 0 } } \\{ w _ 1 , w _ 2 , \\cdots , w _ j \\} . \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} \\beta = 1 - \\frac { n } { r } , \\gamma = \\kappa - 1 . \\end{align*}"}
-{"id": "9353.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { J + 1 } \\mathcal { V } _ r ( \\gamma _ j ^ s ) \\lesssim _ r \\mathcal { V } _ r ^ r ( \\sigma ) . \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{align*} b _ \\infty ^ { ( g ) } ( r ) \\ , v _ 0 ( r ) + b _ 0 ^ { ( g ) } ( r ) \\ , v _ \\infty ( r ) \\ ; = \\ ; o ( r ^ { 1 / 2 } ) \\qquad \\textrm { a s } \\ ; r \\downarrow 0 \\ , . \\end{align*}"}
-{"id": "8878.png", "formula": "\\begin{align*} [ f _ j , g _ k ] \\bigg | _ { - 1 } ^ 1 & = \\langle \\ell ^ n [ f _ j ] , g _ k \\rangle - \\langle f _ j , \\ell ^ n [ g _ k ] \\rangle = [ j ^ n ( j + 1 ) ^ n - k ^ n ( k + 1 ) ^ n ] \\langle f _ j , g _ k \\rangle \\\\ & = - [ k ^ n ( k + 1 ) ^ n - j ^ n ( j + 1 ) ^ n ] \\langle f _ j , g _ k \\rangle = \\langle f _ j , \\ell ^ n [ g _ k ] \\rangle - \\langle \\ell ^ n [ f _ j ] , g _ k \\rangle \\\\ & = - [ g _ k , f _ j ] \\bigg | _ { - 1 } ^ 1 . \\end{align*}"}
-{"id": "3115.png", "formula": "\\begin{align*} h ^ { \\tau } _ i = \\begin{cases} h _ i & - N + 1 \\leq i \\leq N , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{align*} \\gamma _ k = 2 ^ { \\ , ( 2 \\ , \\cdot \\ , 2 ^ { C ( k - 1 ) } - 2 ^ { C k } ) } \\ , \\cdot \\ , 1 0 ^ { 1 - \\frac { 1 } { p } } \\cdot 2 ^ { C k ( 1 - \\frac { 1 } { p } ) } \\leq 1 0 \\cdot 2 ^ { - \\frac 1 2 \\ , 2 ^ { C k } } \\cdot 2 ^ { C k } \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{gather*} d = \\deg \\xi : = \\max _ { i = 1 , \\ldots , N } \\deg \\xi _ i , \\\\ \\norm \\xi _ \\infty : = \\max _ { i = 1 , \\ldots , N } \\norm { \\xi _ i } _ \\infty . \\end{gather*}"}
-{"id": "1199.png", "formula": "\\begin{gather*} ( - A ) ^ { - \\alpha } = \\frac { \\sin ( \\pi \\alpha ) } { \\pi } \\int _ { 0 } ^ { \\infty } t ^ { - \\alpha } ( t I - A ) ^ { - 1 } d t , \\\\ ( - A ) ^ { \\alpha } = \\frac { \\sin ( \\pi \\alpha ) } { \\pi } \\int _ { 0 } ^ { \\infty } t ^ { \\alpha - 1 } ( - A ) ( t I - A ) ^ { - 1 } d t , \\end{gather*}"}
-{"id": "8176.png", "formula": "\\begin{align*} X ^ 2 + X + ( \\alpha + \\lambda ^ 4 ) = 0 . \\end{align*}"}
-{"id": "6636.png", "formula": "\\begin{align*} \\mathbb { R } ^ + \\ni r \\mapsto W _ r ( \\psi , \\phi ) \\ ; : = \\ ; \\det \\begin{pmatrix} \\psi ^ { + } ( r ) & \\phi ^ + ( r ) \\\\ \\psi ^ { - } ( r ) & \\phi ^ - ( r ) \\end{pmatrix} , \\quad \\psi , \\phi \\in L ^ 2 ( \\mathbb { R } ^ + , \\mathbb { C } ^ 2 ) \\ , , \\end{align*}"}
-{"id": "7573.png", "formula": "\\begin{align*} 0 < \\alpha = \\frac 2 k - ( 1 + \\frac 2 k ) \\frac a b < 1 \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 1 & 0 \\\\ c a ^ { - 1 } & 1 \\end{bmatrix} \\begin{bmatrix} a & 0 \\\\ 0 & d - c a ^ { - 1 } b \\pi ^ n \\end{bmatrix} \\begin{bmatrix} 1 & a ^ { - 1 } b \\pi ^ n \\\\ 0 & 1 \\end{bmatrix} \\end{align*}"}
-{"id": "5914.png", "formula": "\\begin{align*} s _ { i , j } ^ { ( - 1 ) } & = - \\sum _ { k = 1 } ^ i \\sum _ { k _ 2 = 1 } ^ { k - 1 } \\sum _ { k _ 3 = 1 } ^ { k _ 2 - 1 } s _ { i , k - 1 } \\cdot s _ { k - 1 , k _ 2 - 1 } \\cdot s _ { k _ 2 - 1 , k _ 3 - 1 } \\cdot s _ { k _ 3 - 1 , j } ^ { ( - 1 ) } \\\\ & \\phantom { = \\sum \\sum } + \\sum _ { k = j + 2 } ^ i s _ { i , k - 1 } \\cdot s _ { k - 1 , j } - s _ { i , j } + \\delta _ { i , j } . \\end{align*}"}
-{"id": "4416.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } \\binom { 2 k } { k } \\frac { H _ k ^ { ( 2 ) } } { k } x ^ { k } \\equiv - 2 x ^ p \\sum _ { k = 1 } ^ { p - 1 } \\frac { v _ k ( 2 - 1 / x ) } { k ^ 3 } \\pmod { p } , \\end{align*}"}
-{"id": "9751.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) e ^ { - r t } + \\int ^ { t } _ { 0 } e ^ { - r t ' } \\left [ \\int _ { \\mathbb { R / Z } } \\frac { 1 } { 2 } ( \\widetilde { W } ^ { { \\rm T } } Q _ { r } \\widetilde { W } ) ( t ' , s ) d s - \\mathcal { R } ( t ' ) \\right ] { \\rm d } t ' = \\mathcal { E } ( 0 ) \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} \\lbrack \\Upsilon ] _ { \\mathrm { + } } : = \\frac { 1 } { 2 } \\left ( \\Upsilon + \\Upsilon ^ { \\mathrm { t } } \\right ) [ \\Upsilon ] _ { \\mathrm { - } } : = \\frac { 1 } { 2 } \\left ( \\Upsilon - \\Upsilon ^ { \\mathrm { t } } \\right ) . \\end{align*}"}
-{"id": "683.png", "formula": "\\begin{align*} \\frac { \\partial { D _ { n } ( \\vec { \\theta } ) } } { \\partial \\eta } = \\sqrt { n ( k + 1 - n ) } ~ D _ { n - 1 } ( \\vec { \\theta } ) . \\end{align*}"}
-{"id": "162.png", "formula": "\\begin{align*} \\gamma \\equiv ( S - 1 ) \\beta = ( S ^ 2 - 1 ) \\alpha \\in \\ell ^ 3 \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} L _ { 2 n + 1 } + 4 ( - 1 ) ^ { n - 1 } = L _ { 2 n + 1 } + ( - 1 ) ^ { n - 1 } L _ 3 = L _ { n - 1 } L _ { n + 2 } \\ , , \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} K ( a ) N ^ { - \\frac \\alpha 2 } e ^ { - N ^ \\alpha J ( a ) } N ^ { \\alpha - 1 } J '' \\left ( \\frac 1 \\lambda \\right ) \\left ( a - \\frac 1 \\lambda \\right ) = \\frac { \\lambda } { \\sqrt { 2 \\pi } } N ^ { \\frac \\alpha 2 - 1 } e ^ { - N ^ \\alpha J ( a ) } . \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ s ^ t \\langle ( J _ k w _ k ) \\cdot \\nabla w _ k , \\varphi \\rangle d \\tau = \\int _ s ^ t \\langle w \\cdot \\nabla w , \\varphi \\rangle d \\tau . \\end{align*}"}
-{"id": "9381.png", "formula": "\\begin{align*} m ( \\Delta ) = E ^ { A _ 1 } ( F _ 1 ) X _ 1 E ^ { A _ 2 } ( F _ 2 ) \\cdots E ^ { A _ { n - 1 } } ( F _ { n - 1 } ) X _ { n - 1 } E ^ { A _ n } ( F _ n ) \\end{align*}"}
-{"id": "3625.png", "formula": "\\begin{align*} M f ( s ) = \\int _ 0 ^ \\infty f ( x ) x ^ { s - 1 } d x . \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} u \\in \\textnormal { r } ( \\theta ) \\iff c ^ { ( 0 ) } = 0 \\& \\forall z \\in \\mathbb { Z } ^ 3 \\colon c ^ { ( z ) } \\bot ( \\theta + 2 \\pi z ) . \\end{align*}"}
-{"id": "1678.png", "formula": "\\begin{align*} F _ * ^ e ( \\lambda x _ 1 ^ { k d _ 1 + \\alpha _ 1 } \\dots x _ n ^ { k d _ n + \\alpha _ n } ) = x _ 1 ^ { d _ 1 } \\dots x _ n ^ { d _ n } F _ * ^ e ( \\lambda x _ 1 ^ { s _ 1 } \\dots x _ n ^ { s _ n } ) \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} 0 < \\vert \\sum \\limits _ { j = 1 } ^ { m } \\lambda _ j \\rho _ j - \\varphi _ i \\vert < c _ 1 C _ 3 \\tau _ 2 ^ { C _ 2 } , \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{align*} { \\bf 1 } _ { X _ { \\rm r e g } } \\nabla _ f ( v \\wedge \\omega ) = \\nabla _ f ( v \\wedge \\omega ) . \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} d i m _ q \\mathit { F _ { \\chi } } = \\frac { 1 } { \\prod _ { j \\in \\mathbb { Z } _ { + } } \\big ( 1 - q ^ { \\frac { 2 j - 1 } { 2 } } \\big ) } \\end{align*}"}
-{"id": "6831.png", "formula": "\\begin{align*} R = \\left ( \\gamma ( d , \\eta ) \\right ) ^ { q / 2 } \\int ^ \\prime _ { \\R ^ { d q } } \\int _ W e ^ { i \\langle \\lambda _ 1 + \\ldots + \\lambda _ q , u \\rangle } d u \\frac { \\tilde { B } ( d \\lambda _ 1 ) \\ldots \\tilde { B } ( d \\lambda _ q ) } { \\left ( \\| \\lambda _ 1 \\| \\cdot \\ldots \\cdot \\| \\lambda _ q \\| \\right ) ^ { ( d - \\eta ) / 2 } } , \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{align*} \\eta _ { f , \\phi , \\epsilon , \\delta } ^ T : = \\frac { 1 } { T } \\sum _ { j \\neq k } \\sigma _ j \\sigma _ k \\langle \\Delta _ { \\epsilon , \\delta } ^ f ( x ^ j + \\xi ^ j , x ^ k + \\xi ^ k ; T ) , \\phi \\rangle , \\end{align*}"}
-{"id": "326.png", "formula": "\\begin{align*} \\eta _ a ^ \\lambda ( \\mathsf { P } ) & = \\Xi ^ { \\lambda } ( K _ a F _ a \\otimes 1 \\otimes E _ a K _ a ^ { - 1 } \\otimes K _ a ) - \\Xi ^ { \\lambda } ( K _ { a } F _ { a } \\otimes 1 \\otimes 1 \\otimes E _ { a } ) \\\\ & - \\Xi ^ { \\lambda } ( K _ { a } \\otimes F _ { a } \\otimes E _ { a } K _ { a } ^ { - 1 } \\otimes K _ { a } ) + \\Xi ^ { \\lambda } ( K _ { a } \\otimes F _ { a } \\otimes 1 \\otimes E _ { a } ) . \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} \\{ n \\mid | p _ n | > R \\} = S _ 1 \\cup S _ 2 \\cup ( - S _ 2 ) \\cup S _ 3 \\cup ( - S _ 3 ) , \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} \\vert S _ 1 ( \\lambda _ 1 \\alpha ) \\vert > X ^ { 5 / 6 } ( \\log X ) ^ 5 , \\vert S _ 1 ( \\lambda _ 2 \\alpha ) \\vert > X ^ { 5 / 6 } ( \\log X ) ^ 5 , T \\le \\vert \\alpha \\vert \\le \\eta ^ { - 2 } ( \\log X ) ^ { 3 / 2 } = R , \\end{align*}"}
-{"id": "3134.png", "formula": "\\begin{align*} \\mathfrak { L } _ { g } ^ { * } ( f ) = - ( \\Delta f ) g + H e s s \\ , f - f R i c = g . \\end{align*}"}
-{"id": "5506.png", "formula": "\\begin{align*} f \\circ \\psi _ p ( \\zeta ) = \\psi _ { f ( p ) } ( \\lambda _ p \\cdot \\zeta ) \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 1 e _ 4 + \\alpha _ 2 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 3 e _ 3 + \\alpha _ 4 e _ 4 + \\alpha _ 5 e _ 5 , [ e _ 2 , e _ 1 ] = - \\alpha _ 3 e _ 3 + \\beta _ 1 e _ 4 + \\beta _ 2 e _ 5 , \\\\ [ e _ 2 , e _ 2 ] = \\beta _ 3 e _ 4 + \\beta _ 4 e _ 5 , [ e _ 1 , e _ 3 ] = \\gamma _ 1 e _ 5 , [ e _ 3 , e _ 1 ] = \\gamma _ 2 e _ 5 , [ e _ 2 , e _ 3 ] = \\gamma _ 3 e _ 5 , [ e _ 3 , e _ 2 ] = \\gamma _ 4 e _ 5 , \\\\ [ e _ 3 , e _ 3 ] = \\gamma _ 5 e _ 5 , [ e _ 1 , e _ 4 ] = \\gamma _ 6 e _ 5 , [ e _ 2 , e _ 4 ] = \\gamma _ 7 e _ 5 , [ e _ 3 , e _ 4 ] = \\gamma _ 8 e _ 5 . \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{align*} A _ 2 = \\sum _ { j , k , \\ell , m } V ^ \\ell _ m P ^ m _ j ( 2 P ^ j _ k - \\delta ^ j _ k ) \\otimes P ^ k _ \\ell = \\sum _ { k , \\ell , m } V ^ \\ell _ m P ^ m _ k \\otimes P ^ k _ \\ell . \\end{align*}"}
-{"id": "8577.png", "formula": "\\begin{align*} \\psi _ { \\phi } = \\psi _ { \\phi } ^ { ( 1 , 0 ) } + \\psi _ { \\phi } ^ { ( 0 , 1 ) } = \\psi _ { \\phi } ^ { ( 1 , 0 ) } + \\bar { \\psi _ { \\phi } ^ { ( 1 , 0 ) } } \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} \\widehat { e ^ { - t A } u } = e ^ { t a ( \\xi ) } \\hat { u } , \\mbox { f o r e v e r y } t \\geqslant 0 , \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} r _ { m + 1 } [ A , S ] = \\{ B \\in \\mathcal { A } _ { m + 1 } : r _ m ( B ) = A \\mathrm { \\ a n d \\ } B \\mathrm { \\ i s \\ v a l i d \\ i n \\ } S \\} . \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} \\dim _ F I _ f \\geq \\dim _ F I _ { t f } & = p - \\deg ( \\gcd ( t f , X ^ p - 1 ) ) \\\\ & \\geq p - \\deg ( \\gcd ( \\tilde { f } , X ^ p - 1 ) ) = \\dim _ { E } I _ { \\tilde { f } } . \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} \\partial _ { t } ^ { 2 } [ \\Xi _ { \\mathrm { p } , l } ^ { ( \\omega ) } ( t ) ] _ { + } = - \\int \\nolimits _ { \\mathbb { R } } \\cos \\left ( t \\nu \\right ) \\mathbf { \\mu } _ { l } ^ { ( \\omega ) } ( \\mathrm { d } \\nu ) \\ , t \\in \\mathbb { R } \\ , \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} & \\| f \\| _ { L ^ r ( S _ t , d \\sigma ) } ^ r = \\frac { 1 } { | S _ t | } \\sum _ { \\textbf { x } \\in S _ t } | f ( \\textbf { x } ) | ^ r \\mbox { f o r } ~ ~ 1 \\le r < \\infty , \\\\ & \\| f \\| _ { L ^ \\infty ( S _ t , d \\sigma ) } = \\max _ { \\textbf { x } \\in S _ t } | f ( \\textbf { x } ) | . \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} \\textstyle \\frac 3 4 ( \\hat c _ 0 - 2 \\hat c _ 1 + \\hat c _ 2 ) = \\frac { S ( g , n ) } { ( g - \\frac 1 2 + \\frac n 2 ) ( g - 1 + \\frac n 2 ) ( g - \\frac 3 2 + \\frac n 2 ) ( g - 2 + \\frac n 2 ) } , \\end{align*}"}
-{"id": "649.png", "formula": "\\begin{align*} { \\cal F } = \\bigoplus _ { n = 0 } ^ { \\infty } { \\cal H } ^ n , \\end{align*}"}
-{"id": "9845.png", "formula": "\\begin{align*} W ^ { \\chi } ( \\phi ) = \\int _ { U ^ - } \\phi ( u ) \\psi ( u ) \\ , d u , \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} { \\displaystyle \\| u \\| _ { L ^ p } \\leq C \\| u \\| _ { { H } ^ 1 } , \\| \\mathbf { v } \\| _ { \\mathbf { L } ^ p } \\leq C \\| \\mathbf { v } \\| _ { \\mathbf { H } ^ 1 } , 1 \\leq p \\leq 6 \\ , \\ , ( d = 2 , 3 ) , } \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} \\left \\vert \\mathcal { A } _ { k } \\right \\vert = \\left \\vert \\left \\{ 2 w _ { k } \\leq 0 \\right \\} \\cap B _ { r } \\right \\vert = \\left \\vert \\left \\{ w _ { k } \\leq 0 \\right \\} \\cap B _ { r } \\right \\vert \\geq \\frac { 1 } { 2 } \\left \\vert B _ { r } \\right \\vert , \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} D _ { \\lambda } ^ { - 1 } \\left ( \\varepsilon \\right ) = - d ^ { - 1 } Q _ { \\lambda } ^ { - 1 } \\left ( \\varepsilon \\right ) \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} Y _ { i } = f ( w _ { i } ) + \\varepsilon _ { i } \\quad \\mbox { a v e c } \\varepsilon _ { i } \\mbox { d e l o i } r \\cdot \\lambda , \\end{align*}"}
-{"id": "7754.png", "formula": "\\begin{align*} \\mu ^ { \\ast } ( \\delta ( [ \\nu ^ { - k } \\rho , \\nu ^ { l } \\rho ] ) \\rtimes \\sigma ) & = \\sum _ { i = - k - 1 } ^ { l } \\sum _ { j = i } ^ { l } \\sum _ { \\tau , \\sigma ' } \\delta ( [ \\nu ^ { - i } \\rho , \\nu ^ { k } \\rho ] ) \\times \\\\ & \\times \\delta ( [ \\nu ^ { j + 1 } \\rho , \\nu ^ { l } \\rho ] ) \\times \\tau \\otimes \\delta ( [ \\nu ^ { i + 1 } \\rho , \\nu ^ { j } \\rho ] ) \\rtimes \\sigma ' . \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} \\begin{cases} H _ k [ u ] = f ( x , u ) & , \\\\ [ 5 p t ] u = \\varphi & . \\end{cases} \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} \\mu ^ X ( x ) = \\mu ^ { X ' } ( f ( x ) ) \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{align*} ( { \\rm e } _ { t } ) _ { \\sharp } \\nu ( B _ { 4 \\bar r } ( \\bar x ) ) = 1 , \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{align*} X ( 0 ) \\in \\sl , X ( t _ f ) = R ^ { - 1 } X ( 0 ) R \\in \\sl . \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} \\left [ \\frac { d } { d f } \\left ( \\frac { p } { f } \\right ) \\right ] \\left [ 1 - \\lambda _ { 4 } \\left ( \\frac { p } { f } \\right ) ^ { 2 } \\right ] = 0 . \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} a = u \\overline { u } \\delta \\varepsilon u _ 1 \\cdots u _ n \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} \\tau = \\delta ( e , a , b , c ) + t ( e , a , b ) - m ( e , a , b , c ) . \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} \\ell ^ n [ f ] ( x ) = \\sum _ { k = 1 } ^ n ( - 1 ) ^ k { n \\brace k } _ 2 ( ( 1 - x ^ 2 ) ^ k f ^ { ( k ) } ( x ) ) ^ { ( k ) } , \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{align*} \\eta _ { a } ( C ( \\mathsf { Q } ) ) = \\sum _ { i , j , k } q ^ { ( 2 \\rho , \\lambda _ { i } ) } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) \\varepsilon ( F _ { a } \\triangleright \\mathsf { Q } _ { k } ^ { j } ) \\varepsilon ( E _ { a } \\triangleright \\mathsf { Q } _ { i } ^ { k } ) . \\end{align*}"}
-{"id": "3215.png", "formula": "\\begin{align*} d Y ( t ) = \\sqrt { 2 } d W ( t ) + \\left [ ( \\beta - 1 ) t a n h ( Y ( t ) ) + \\gamma s e c h ( Y ( t ) ) \\right ] d t . \\end{align*}"}
-{"id": "1380.png", "formula": "\\begin{align*} \\tau _ { 1 } \\left ( x , y \\right ) = \\tau _ { 2 } \\left ( x , y \\right ) , \\forall x , y \\in S . \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| w _ k - w \\| _ { L ^ 2 ( 0 , T ; L ^ 2 ( \\Omega _ L ) ) } = 0 , \\forall L \\in [ R , \\infty ) , \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { ( p - 1 ) / 2 } \\binom { 2 k } { k } \\frac { \\bigl ( \\beta ( 1 - \\beta ) \\bigr ) ^ k } { k } \\equiv 2 \\pounds _ 1 \\left ( \\beta \\right ) \\pmod { \\beta ^ p } \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} I ( C : Z | M ) _ { \\hat { \\sigma } _ { C M Z } } & \\le I ( A B : Z | M ) _ { \\hat { \\sigma } _ { A B M Z } } \\\\ & = I ( A : Z | M ) _ { \\hat { \\sigma } _ { A M Z } } + I ( B : Z | M ) _ { \\hat { \\sigma } _ { B M Z } } \\\\ & \\phantom { = } + I ( A : B | M Z ) _ { \\hat { \\sigma } _ { A B M Z } } - I ( A : B | M ) _ { \\hat { \\sigma } _ { A B M } } \\\\ & \\le I ( A : Z | M ) _ { \\hat { \\sigma } _ { A M Z } } + I ( B : Z | M ) _ { \\hat { \\sigma } _ { B M Z } } \\ ; . \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{align*} G ( s ) = - \\frac { \\beta } { 2 } s ^ 2 + \\frac { 1 } { \\alpha + 2 } | s | ^ { \\alpha + 2 } , s \\in \\R \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} [ n ] P = O \\Longleftrightarrow ( \\Psi ^ \\prime _ n ( x , y ) ) ^ 2 = 0 , \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} \\frac { 2 } { x _ 1 - x _ 2 } - \\left ( \\frac { 1 } { x _ 1 - \\zeta } + \\frac { 1 } { x _ 1 - \\overline { \\zeta } } \\right ) = 0 , \\frac { 2 } { x _ 2 - x _ 1 } - \\left ( \\frac { 1 } { x _ 2 - \\zeta } + \\frac { 1 } { x _ 2 - \\overline { \\zeta } } \\right ) = 0 . \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} \\mathfrak { b } : = \\left \\{ \\{ x , x ^ { \\prime } \\} \\subset \\mathfrak { L } : | x - x ^ { \\prime } | = 1 \\right \\} \\end{align*}"}
-{"id": "9576.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } \\theta ^ { \\mathrm { d e s c r } } _ n \\\\ \\gamma ^ { \\mathrm { d e s c r } } _ n \\end{array} \\right ) = \\left ( \\begin{array} { c c } W ^ { X X } _ n & W ^ { X Z } _ n \\\\ W ^ { Z X } _ n & W ^ { Z Z } _ n \\end{array} \\right ) ^ { - 1 } \\left ( \\begin{array} { c } W ^ { X Y } _ n \\\\ W ^ { Z Y } _ n \\end{array} \\right ) , \\end{align*}"}
-{"id": "4754.png", "formula": "\\begin{align*} A _ { \\alpha } & = \\Big ( \\alpha g ( \\alpha ) \\dots g ^ { n _ { \\alpha } - 1 } ( \\alpha ) \\Big ) ^ { m _ { \\alpha } - 1 } \\alpha g ( \\alpha ) \\dots g ^ { n _ { \\alpha } - 2 } ( \\alpha ) , \\mbox { i f } n _ { \\alpha } \\geq 2 , \\\\ A _ { \\alpha } & = \\alpha ^ { m _ { \\alpha } - 1 } , \\mbox { i f } n _ { \\alpha } = 1 , \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} \\nu _ 2 \\left ( { 2 ^ k \\choose 2 i + 1 } \\right ) & = \\nu _ 2 \\left ( \\frac { 2 ^ k } { 2 i + 1 } { 2 ^ k - 1 \\choose 2 i } \\right ) \\\\ & = k - \\nu _ 2 ( 2 i + 1 ) + s _ 2 ( 2 i ) + s _ 2 ( 2 ^ k - 2 i - 1 ) - s _ 2 ( 2 ^ k - 1 ) = k . \\end{align*}"}
-{"id": "6980.png", "formula": "\\begin{align*} L = \\{ 0 , b , 2 b , \\ldots ( o ( b ) - 1 ) b \\} \\subseteq V . \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{align*} C _ j \\ge 0 , \\ ; \\ ; { \\rm r a n k } C _ j = \\dim ( \\ker J ( i k _ j ) ) , \\ ; \\ ; J ( i k _ j ) ^ \\dagger C _ j = 0 _ n , j = \\overline { 1 , N } . \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } \\binom { 2 k } { k } H _ k x ^ k \\equiv - 2 ( \\alpha - \\beta ) ^ { p - 1 } \\pounds _ 1 \\left ( \\frac { \\beta } { \\beta - \\alpha } \\right ) \\pmod { p } , \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} \\{ \\lambda _ 1 ' , \\lambda _ 2 ' , \\ldots , \\lambda _ { n + 3 } ' \\} = \\{ \\lambda _ 1 , \\lambda _ 2 , \\ldots , \\lambda _ { n + 3 } \\} , \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} q ( \\mu ) \\le q ( \\mu _ { i + 1 } ) + \\mu - \\mu _ { i + 1 } + \\eta _ { i + 1 } ( \\mu _ { i + 1 } - \\mu ) \\lambda _ { \\max } ( W ) : = q _ 2 ( \\mu ) . \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{align*} u _ a ( x , t ) = \\frac { 1 } { 2 \\pi } \\int \\limits _ 0 ^ \\infty \\left [ \\cos \\left ( \\xi \\left ( x + \\frac { t } { \\sqrt { 1 + a _ \\alpha \\xi ^ { \\alpha } } } \\right ) \\right ) + \\cos \\left ( \\xi \\left ( x - \\frac { t } { \\sqrt { 1 + a _ \\alpha \\xi ^ { \\alpha } } } \\right ) \\right ) \\right ] e ^ { - \\frac { a ^ 2 \\xi ^ 2 } { 4 } } d \\xi . \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} q ( y ) = ( n - 2 ) ^ 2 \\abs { y } ^ 2 + ( 3 n - 4 ) H ^ 2 + O ( \\abs { y } ^ 3 ) \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{align*} Z _ { A , B } : y ^ 2 = x ^ 3 - 3 A s ^ 4 x + s ^ 5 ( t ^ 2 - 2 B s + 1 ) , \\end{align*}"}
-{"id": "9523.png", "formula": "\\begin{align*} W = \\{ C ^ * X C : C \\in { \\mathrm { C o l } } _ { \\infty , 1 } ( M ) \\} , \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{align*} a _ { v c } v _ c ^ 2 + b _ { v c } v _ c + c _ { v c } = 0 \\end{align*}"}
-{"id": "1402.png", "formula": "\\begin{align*} \\widehat { \\mathcal { G } } _ j : = \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ n g _ i \\hat \\psi _ j ( y _ i , z _ i ) , \\ \\ j = 1 , \\ldots , p , \\end{align*}"}
-{"id": "8430.png", "formula": "\\begin{align*} M _ t = \\int _ 0 ^ t Z _ s d W _ s + \\int _ 0 ^ t \\int _ { E } l _ t ( e ) \\tilde N ( d t , d e ) + h _ t \\ , , \\quad \\forall \\ , t \\in [ 0 , T ] a . s . \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{align*} \\chi _ { i j } ( \\lambda ) = H ( \\lambda , e _ i ) - H ( \\lambda , e _ j ) . \\end{align*}"}
-{"id": "4115.png", "formula": "\\begin{align*} \\tau _ k ( p ) = \\sum _ { i = 1 } ^ \\infty \\frac { ( i + 1 ) ^ k - i ^ k } { p ^ i } . \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\ \\int _ 0 ^ 1 \\cos ( 2 \\pi n x ) f ( x ) \\ , d x = 0 . \\end{align*}"}
-{"id": "7006.png", "formula": "\\begin{align*} Q _ { 1 , \\lambda } \\left ( \\varepsilon \\right ) = \\frac { 1 } { 2 \\varepsilon } \\left [ B + \\left ( B ^ { 2 } + 4 \\varepsilon A _ { \\lambda } \\right ) ^ { \\frac { 1 } { 2 } } \\right ] , Q _ { 2 , \\lambda } \\left ( \\varepsilon \\right ) = \\frac { 1 } { 2 \\varepsilon } \\left [ B - \\left ( B ^ { 2 } + 4 \\varepsilon A _ { \\lambda } \\right ) ^ { \\frac { 1 } { 2 } } \\right ] . \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } \\binom { 2 k } { k } \\frac { x ^ k } { k } \\equiv \\pounds _ 1 ( \\alpha ) + \\pounds _ 1 ( \\beta ) \\pmod { p } , \\end{align*}"}
-{"id": "9604.png", "formula": "\\begin{align*} W { } ^ { ( s ) . i } _ { j m n } = W ^ i _ { j m n } + \\widetilde { \\mathcal D } { } ^ { ( s _ 2 ) . ( s _ 3 ) . i } _ { j m n } - \\widetilde { \\mathcal D } { } ^ { ( s _ 2 ) . ( s _ 3 ) . i } _ { j n m } , \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} \\begin{cases} f _ { n } ( X ) = n X ^ { ( n ^ 2 - 1 ) / 2 } + c X ^ { ( n ^ 2 - 1 ) / 2 - 2 } + \\cdot \\cdot \\cdot , n \\textrm { o d d } ; \\\\ f _ { n } ( X ) = n X ^ { ( n ^ 2 - 4 ) / 2 } + c X ^ { ( n ^ 2 - 4 ) / 2 - 2 } + \\cdot \\cdot \\cdot , n \\textrm { e v e n } . \\end{cases} \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} \\varrho ( p ^ { e _ 1 } , p ^ { e _ 2 } , p ^ { e _ 3 } ) = p ^ { 2 e _ i + e _ j + e _ k } . \\end{align*}"}
-{"id": "3201.png", "formula": "\\begin{align*} e ^ { - \\lambda _ { N , s } t } \\frac { \\Delta _ N ( y ) } { \\Delta _ N ( x ) } \\det \\left ( p ^ { ( N ) , s } _ t ( x _ i , y _ j ) \\right ) ^ N _ { i , j = 1 } d y . \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} X ( s ) = \\begin{pmatrix} ( \\Phi _ 1 ( s ) , \\delta _ 1 ( s ) ) & \\cdots & ( \\Phi _ 1 ( s ) , \\delta _ m ( s ) ) \\\\ & \\ddots & \\\\ ( \\Phi _ m ( s ) , \\delta _ 1 ( s ) ) & \\cdots & ( \\Phi _ m ( s ) , \\delta _ m ( s ) ) \\end{pmatrix} . \\end{align*}"}
-{"id": "8709.png", "formula": "\\begin{align*} \\begin{aligned} \\nu \\big \\{ u \\in U : & x , z \\in \\R ^ { d } x - z \\neq 0 \\\\ & \\qquad | x - z + c ( x , u ) - c ( z , u ) | \\le \\delta | x - z | \\big \\} = 0 . \\end{aligned} \\end{align*}"}
-{"id": "6766.png", "formula": "\\begin{align*} & \\| \\sum _ { j = 2 } ^ \\infty \\tilde S _ j f \\tilde S ^ { j - 2 } g \\| _ { B ^ s _ { p , \\infty } ( \\real ^ { d _ s } ) } \\le C \\sup _ { k \\ge 2 } 2 ^ { k s } \\sum _ { \\ell = - 1 } ^ { + 3 } \\| \\tilde S _ { k + \\ell } f \\tilde S ^ { k + \\ell - 2 } g \\| _ { L _ p ( \\real ^ { d _ s } ) } \\ , . \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} \\Big \\{ \\psi _ 1 ^ { d _ 1 } \\cdots \\psi _ n ^ { d _ n } \\kappa _ { e _ 1 , \\dots , e _ { m } } \\mid m \\geq 0 , d _ i \\geq 0 , e _ j \\geq 1 , \\sum _ { i = 1 } ^ n d _ i + \\sum _ { j = 1 } ^ { m } e _ j = g - 1 \\Big \\} \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{align*} \\theta ( a ) = K _ { 2 \\rho } \\triangleright a \\triangleleft K _ { 2 \\rho } . \\end{align*}"}
-{"id": "1886.png", "formula": "\\begin{align*} C ( M , J , [ g ] ) = \\frac { 2 s ^ H _ S + 8 c + 4 p } { 2 c + p } . \\end{align*}"}
-{"id": "1978.png", "formula": "\\begin{align*} & t ^ N \\langle 1 \\cdots M | \\mathcal { B } ^ \\prime ( z _ 1 ) \\cdots \\mathcal { B } ^ \\prime ( z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle \\\\ = & \\prod _ { j = 1 } ^ N z _ j ^ { j - 1 - N } ( 1 + t ^ { - 1 } z _ j ^ 2 ) \\prod _ { 1 \\le j < k \\le N } ( 1 + t ^ { - 1 } z _ j z _ k ) ( t ^ { - 1 } + z _ j z _ k ^ { - 1 } ) s p _ { \\overline { \\lambda } } ( \\{ z \\} _ N ) . \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\psi \\left ( \\sqrt { \\frac { t ( X _ { i } ) } { u ( X _ { i } ) } } \\right ) \\quad \\mbox { n e s ' \\ ' e c r i t p a s } \\sum _ { i = 1 } ^ { n } \\psi \\left ( \\sqrt { t ( X _ { i } ) } \\right ) - \\sum _ { i = 1 } ^ { n } \\psi \\left ( \\sqrt { u ( X _ { i } ) } \\right ) . \\end{align*}"}
-{"id": "5727.png", "formula": "\\begin{align*} T : = \\bigoplus _ { k = 1 } ^ n \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} \\omega _ M = P ( \\mu ) \\pi ^ * \\omega _ S + \\langle d \\mu \\wedge \\theta ^ 0 \\rangle . \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{gather*} \\frac { 1 } { z ^ { 4 / 3 } } \\begin{pmatrix} - t ^ { 1 / 3 } & 0 & 0 \\\\ 0 & - \\omega t ^ { 1 / 3 } & 0 \\\\ 0 & 0 & - \\omega ^ 2 t ^ { 1 / 3 } \\end{pmatrix} + \\frac { 1 } { z } \\begin{pmatrix} \\theta ^ \\infty _ 1 / 3 & 0 & 0 \\\\ 0 & \\theta ^ \\infty _ 1 / 3 & 0 \\\\ 0 & 0 & \\theta ^ \\infty _ 1 / 3 \\end{pmatrix} . \\end{gather*}"}
-{"id": "5194.png", "formula": "\\begin{align*} \\left ( f \\circ \\left ( g \\circ h \\right ) \\right ) _ { n } = \\sum _ { \\pi \\models n } f _ { \\vert \\pi \\vert } \\left ( g \\circ h \\right ) _ { \\pi } \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} A ' = \\begin{cases} A \\setminus A _ { \\alpha } & r = 1 , \\\\ ( A \\setminus A _ { \\alpha } ) \\cup \\langle a _ { \\alpha , 2 } \\rangle _ I \\oplus \\langle a _ { \\alpha , 3 } \\rangle _ I \\oplus \\cdots \\oplus \\langle a _ { \\alpha , r } \\rangle _ I & r > 1 . \\end{cases} \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i = 1 } ^ N | \\langle \\Theta _ i , \\xi \\rangle | \\geq \\frac { c _ 2 } { \\sqrt { n } } \\textrm { a n d } \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\exp \\left \\{ \\left ( \\frac { \\sqrt { n } | \\langle \\Theta _ i , \\xi \\rangle | + 1 } { 2 } \\right ) ^ { \\alpha } \\right \\} \\leq 3 C _ 3 . \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} u ( x ) = \\begin{cases} P _ p [ f ] ( x ) & \\textrm { f o r } x \\in \\widehat { B } _ p \\\\ f ( x ) & \\textrm { f o r } x \\in \\widehat { S } _ p , \\end{cases} \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} u _ { a , r , R } ( x ) : = U _ { r , R } ( | x - a | ) . \\end{align*}"}
-{"id": "6898.png", "formula": "\\begin{align*} \\frac { z ^ { l } t ^ { q } ( 1 - t ) ^ { l } } { ( 1 - t ) ^ { n + 1 } } = \\frac { t ^ { q } z ^ { l } } { ( 1 - t ) ^ { n - l + 1 } } = \\sum _ { p = 0 } ^ { \\infty } z ^ { l } { n - l + p - q \\choose n - l } t ^ { p } \\end{align*}"}
-{"id": "1504.png", "formula": "\\begin{align*} P ( x ) - S ( x ) ^ 2 = \\sum _ { k = 1 } ^ { n - 1 } P ( t _ k ) g _ k ( x ) + \\sum _ { k = 1 } ^ { n - 1 } P ' ( t _ k ) \\mathfrak { g } _ k ( x ) , \\end{align*}"}
-{"id": "4595.png", "formula": "\\begin{align*} \\big ( \\frac { 1 } { 2 \\tau } ( I _ { h } \\phi ^ { k } - I _ { h } \\phi ^ { k - 2 } ) , \\ , \\nabla \\cdot \\overline { \\partial _ { \\tau } \\theta } _ { \\mathbf { A } } ^ { k } \\big ) = 0 , \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} t \\in [ 0 , T ] \\mapsto \\{ x : | \\tilde w ( x , t ) | \\ge c \\} \\cup ( Y \\cup Z ) ^ c = \\{ x : | \\tilde w ( x , t ) | < c \\} ^ c \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} \\tau ( p ) = \\sup _ { \\psi _ p \\in \\widehat \\Psi _ p } o r d ( \\psi _ p ) \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} \\mathbb { E } \\langle : X _ T \\otimes \\ldots \\otimes X _ T : , \\Phi \\rangle ^ 2 = I _ 1 + I _ 2 + R , \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} I ^ N : = \\{ ( j , k ) \\in \\N ^ * \\times \\{ 1 , \\dots , N \\} \\ : \\ j > k \\} . \\end{align*}"}
-{"id": "30.png", "formula": "\\begin{align*} t _ { \\rm T U R N I N G } = \\ln \\left ( \\frac { \\mu K _ 1 h _ F - \\lambda T _ s } { \\Gamma - \\lambda T _ s - \\lambda \\theta _ F } \\right ) . \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{align*} \\langle ( L _ c - \\partial _ y ^ 2 ) \\tilde { u } , \\tilde { u } \\rangle _ { L ^ 2 ( \\mathbb { R } \\times \\mathbb { T } ) } = \\sum _ { k \\in \\mathbb { Z } } \\langle ( L _ c + k ^ 2 ) \\hat { u } _ k , \\hat { u } _ k \\rangle _ { L ^ 2 ( \\mathbb { R } ) } . \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} T _ { 2 , k } = \\sum _ { v \\leqslant \\frac { ( k - 1 ) K B } { \\alpha d N } } \\mathtt { 1 } _ { \\{ \\alpha v \\} \\in I _ k } = \\frac { ( \\varepsilon - \\eta ) k ( k - 1 ) K ^ 2 } { \\alpha ^ 2 d ^ 2 } \\frac { B ^ { 2 - \\frac { 1 } { r } } } { N ^ 2 } + \\frac { \\eta ( k - 1 ) K ^ 2 } { \\alpha ^ 2 d ^ 2 } \\frac { B ^ { 2 - \\frac { 1 } { r } } } { N ^ 2 } + O \\left ( \\frac { k ^ \\sigma K ^ \\sigma B ^ \\sigma } { d ^ \\sigma N ^ \\sigma } \\right ) . \\end{align*}"}
-{"id": "7683.png", "formula": "\\begin{align*} W ( \\lambda ^ { w _ 1 } x _ 1 , \\lambda ^ { w _ 2 } x _ 2 , \\dots , \\lambda ^ { w _ n } x _ n ) = \\lambda ^ d W ( x _ 1 , x _ 2 , \\dots , x _ n ) . \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} d X ^ { i , n } _ t = b ( t , X ^ { i , n } _ t , \\bar \\mu _ t ^ n , \\gamma _ t ^ { i , n } ) d t + \\sigma ( t , X ^ { i , n } _ t , \\bar \\mu _ t ^ n , \\gamma _ t ^ { i , n } ) d W ^ i _ t + \\beta ( t , X _ t ^ { i , n } , \\bar \\mu ^ n _ { t } , \\gamma ^ { i , n } _ t ) d \\widetilde N ^ i _ t , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "1531.png", "formula": "\\begin{align*} \\frac { c d \\ , P ( x ) } { Q ( x ) ^ 2 f ( x ) ^ 2 } = \\frac { a \\ , P ( x ) } { S ( x ) ^ 2 } , \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a & b \\pi ^ n \\\\ c & d \\end{bmatrix} = \\underbrace { \\begin{bmatrix} e & f \\pi ^ n \\\\ g & h \\end{bmatrix} } _ B \\underbrace { \\begin{bmatrix} p & q \\pi ^ n \\\\ r & s \\end{bmatrix} } _ C = \\begin{bmatrix} e p + f r \\pi ^ n & ( e q + f s ) \\pi ^ n \\\\ g p + h r & g q \\pi ^ n + h s \\end{bmatrix} , \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} g _ { n } = \\sum _ { \\pi \\in \\mathcal { C } \\left ( n \\right ) } \\left ( - 1 \\right ) ^ { \\pi + 1 } x _ { \\pi } = \\sum _ { p = 1 } ^ n \\left ( - 1 \\right ) ^ { p + 1 } \\sum _ { \\underset { k _ { i } \\ge 1 } { k _ { 1 } + \\dots + k _ { p } = n } } x _ { k _ { 1 } } \\dots x _ { k _ { p } } \\end{align*}"}
-{"id": "6264.png", "formula": "\\begin{align*} u ( y ) = \\sum _ { z \\in \\mathbb { Z } ^ d } c ^ { ( z ) } e ^ { \\i \\langle ( \\theta + 2 \\pi z ) , y \\rangle } ( y \\in Y , c ^ { ( z ) } \\in \\mathbb { C } ^ n ) , \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} d _ { n } ( x | q ) \\allowbreak & = \\allowbreak b _ { n } ( x | q ) , \\\\ f _ { n } ( x | q ) & = h _ { n } ( x | q ) . \\end{align*}"}
-{"id": "3978.png", "formula": "\\begin{align*} & 2 \\sum _ { k = 0 } ^ a \\frac { ( - a - p ) _ { p + k } ( \\frac 1 2 ) _ { p + k } } { ( 1 ) _ { p + k } ^ 2 } \\cdot \\bigg ( \\frac { 4 z } { ( 1 + z ) ^ 2 } \\bigg ) ^ k \\\\ \\equiv & - \\frac { \\Phi ( 0 ) } { 4 ^ { p - 1 } ( 1 + z ) ^ { 2 a - 1 } } - \\frac { p } { ( 1 + z ) ^ { 2 a } } \\cdot \\frac { \\Psi _ 1 ' ( 0 ) + z \\cdot \\Psi _ 2 ' ( 0 ) } { 2 } \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\eta _ 2 } ~ f _ { n _ 1 , n _ 2 } ( \\eta _ 1 , \\eta _ 2 ) = \\sqrt { n _ 2 ( k + 1 - ( n _ 1 + n _ 2 ) ) } f _ { n _ 1 , n _ 2 - 1 } ( \\eta _ 1 , \\eta _ 2 ) . \\end{align*}"}
-{"id": "1553.png", "formula": "\\begin{align*} H ( c ) = V D V ^ \\top . \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} u _ 1 ( x ) & = \\frac { - 1 } { m - 1 } \\left ( A c ^ { n - 1 } x ^ { - 1 } + B c ^ { n - 2 } x \\right ) + A \\cdot O ( c ^ { n - 2 } ) + O ( c ^ { n - 3 } ) \\\\ u _ 2 ( x ) & = \\frac { 1 } { m - 1 } \\left ( \\frac { B c ^ { n - 2 } } { 2 - m } x ^ { 2 - m } + \\frac { A c ^ { n - 1 } } { m } x ^ { - m } \\right ) + A \\cdot O ( c ^ { n - 2 } ) + O ( c ^ { n - 3 } ) \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} \\Phi ( a _ 1 , \\ldots , a _ n ) = \\sum _ { i = 1 } ^ n a _ i e _ i . \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} \\begin{bmatrix} \\sigma _ 1 \\int _ 0 ^ T Y _ s ^ { \\frac { 1 } { 2 } } \\ , \\dd W _ s \\\\ - \\sigma _ 1 \\int _ 0 ^ T Y _ s ^ { \\frac { 3 } { 2 } } \\ , \\dd W _ s \\end{bmatrix} = \\begin{bmatrix} T & 0 \\\\ 0 & T ^ 2 \\end{bmatrix} \\begin{bmatrix} \\frac { \\sigma _ 1 } { T } \\int _ 0 ^ T Y _ s ^ { \\frac { 1 } { 2 } } \\ , \\dd W _ s \\\\ - \\frac { \\sigma _ 1 } { T ^ 2 } \\int _ 0 ^ T Y _ s ^ { \\frac { 3 } { 2 } } \\ , \\dd W _ s \\end{bmatrix} . \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{align*} \\mu _ i \\left ( \\mathcal { A } \\right ) = \\int _ { W ^ N } ^ { } \\mu _ i ( d x ) \\int _ { \\mathcal { A } } ^ { } e ^ { - \\lambda _ { N , s } t } \\frac { \\Delta _ N ( y ) } { \\Delta _ N ( x ) } \\det \\left ( p ^ { ( N ) , s } _ t ( x _ i , y _ j ) \\right ) ^ N _ { i , j = 1 } d y = \\int _ { W ^ N } ^ { } \\mu _ i ( d x ) f ^ { ( N ) } _ { \\mathcal { A } , t } ( x ) > 0 , \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} s _ \\lambda ( t _ 1 , t _ 2 , \\dots , t _ { k } ) = \\sum _ { \\mu \\preceq \\lambda ' } K _ { \\lambda ' \\mu } f _ { \\mu } ( t _ 1 , t _ 2 , \\dots , t _ k ) . \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} \\left ( e ^ { t \\frac { d } { d x } } \\phi \\right ) ( s ) = \\phi ( s + t ) , \\textrm { f o r e v e r y } s \\in \\R . \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} \\sum _ { d = 0 } ^ \\infty ( d + 1 ) ^ a \\Pr { \\mathbf { Z } _ n = d + 1 } = \\sum _ { d = 0 } ^ \\infty \\big ( ( d + 1 ) ^ a - d ^ a \\big ) G _ { d + 1 } ( n ) . \\end{align*}"}
-{"id": "6686.png", "formula": "\\begin{align*} \\begin{aligned} ( \\Box - q ( x ) ) V ( x , t ) & = 0 , \\ \\ \\ \\ t > \\lvert x - e \\rvert \\\\ V ( x , \\lvert x - e \\rvert ) & = \\frac { 1 } { 8 \\pi } \\int \\limits _ { 0 } ^ { 1 } q ( s x + ( 1 - s ) e ) d s . \\end{aligned} \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} K ( s , t ) = \\begin{cases} w ( s ) - \\left ( w ( a ) + \\psi ( \\lambda ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ a , t ) , \\\\ w ( s ) - \\left ( w ( a ) + ( 1 + \\psi ( 1 - \\lambda ) ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ t , b ] , \\end{cases} \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} F ( u ) = \\frac { A u ^ 2 + B u + C } { 1 + k u } . \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} K _ { 0 } = \\frac { f g f ^ { \\prime \\prime } g ^ { \\prime \\prime } - \\left ( f ^ { \\prime } g ^ { \\prime } \\right ) ^ { 2 } } { \\left [ 1 - \\left ( f g ^ { \\prime } \\right ) ^ { 2 } \\right ] ^ { 2 } } , \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} \\frac { x _ { k , j } } { B _ { k , j } } = - i \\int _ { 0 } ^ T u ( s ) e ^ { - i ( \\lambda _ j - \\lambda _ k ) s } d s , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\forall j , k \\in \\N ^ * , \\ k \\leq N , \\\\ \\end{align*}"}
-{"id": "4005.png", "formula": "\\begin{align*} \\Psi ( 0 ) = { } _ 3 F _ 2 \\bigg [ \\begin{matrix} - a & 1 + a & \\beta \\\\ & 1 & 2 \\beta \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { \\Gamma ( \\frac 1 2 ) \\Gamma ( \\frac 1 2 + \\beta ) \\Gamma ( \\beta ) } { \\Gamma ( \\frac 1 2 - \\frac 1 2 a ) \\Gamma ( 1 + \\frac 1 2 a ) \\Gamma ( \\beta - \\frac 1 2 a ) \\Gamma ( \\frac 1 2 + \\beta + \\frac 1 2 a ) } . \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} I ( G , x ) = \\sum \\limits _ { i = 0 } ^ { n } f _ { i - 1 } x ^ { i } , \\end{align*}"}
-{"id": "5479.png", "formula": "\\begin{align*} \\Theta _ { r , s } ( z , \\tau ) = \\sum _ { m \\in \\Z } w ^ { s ( m + \\frac { r } { 2 s } ) } q ^ { s ( m + \\frac { r } { 2 s } ) ^ 2 } \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{align*} \\Phi ( x _ 1 , x _ 2 ; \\rho ) ^ { ( 2 l - 1 ) } _ \\alpha : = \\left . \\frac { \\partial ^ { 2 l - 1 } \\Phi } { \\partial \\varphi ^ { 2 l - 1 } } \\right | _ { \\varphi = \\alpha } = 0 \\mbox { \\rm f o r a l l } \\ ; \\alpha \\in \\Pi \\ ; \\mbox { \\rm a n d } \\ ; 1 \\le l \\le \\mu _ \\alpha . \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} C F ( L _ { i } , L _ { i } ) = H ( L _ i ; \\Lambda _ 0 ) \\oplus \\bigoplus \\Lambda _ 0 ( p , q ) \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} { \\frac { { d ^ { 2 } H _ C } } { { d z ^ { 2 } } } } + \\left ( \\alpha + { \\frac { { \\gamma + 1 } } { { z - 1 } } } + { \\frac { { \\beta + 1 } } { { z } } } \\right ) { \\frac { { d H _ C } } { { d z } } } + \\left ( { \\frac { { \\mu } } { { z } } } + { \\frac { { \\nu } } { { z - 1 } } } \\right ) H _ C = 0 , \\end{align*}"}
-{"id": "4734.png", "formula": "\\begin{align*} \\begin{aligned} \\dot x & = \\sqrt { J _ 1 ( x ) } \\sin \\left ( \\ln ( J _ 1 ( x ) ) \\right ) u _ { 1 } ^ \\varepsilon ( t ) \\\\ & + \\sqrt { J _ 1 ( x ) } \\cos \\left ( \\ln ( J _ 1 ( x ) ) \\right ) u _ { 2 } ^ \\varepsilon ( t ) , \\end{aligned} \\end{align*}"}
-{"id": "5576.png", "formula": "\\begin{align*} q _ 1 q _ 2 = a _ 2 q _ 1 \\frac { q _ 2 } { a _ 2 } \\gg \\frac { r q } { \\vert \\alpha \\vert } \\gg r q y ^ { - 1 } . \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} L _ { v _ 0 } ( f ) \\ ; & = \\ ; \\lim _ { r \\downarrow 0 } W _ r ( \\overline { v _ 0 } , f ) \\ ; = \\ ; a _ 0 ^ { ( f ) } \\ , \\lim _ { r \\downarrow 0 } W _ r ( \\overline { v _ 0 } , v _ 0 ) + a _ \\infty ^ { ( f ) } \\ , \\lim _ { r \\downarrow 0 } W _ r ( \\overline { v _ 0 } , v _ \\infty ) \\\\ & \\qquad + \\lim _ { r \\downarrow 0 } b _ \\infty ^ { ( f ) } ( r ) \\ , W _ r ( \\overline { v _ 0 } , v _ 0 ) + \\lim _ { r \\downarrow 0 } b _ 0 ^ { ( f ) } ( r ) \\ , W _ r ( \\overline { v _ 0 } , v _ \\infty ) \\\\ & \\ ; = \\ ; a _ \\infty ^ { ( f ) } \\ , W _ 0 ^ { \\infty } \\ , . \\end{align*}"}
-{"id": "3022.png", "formula": "\\begin{align*} \\lambda x _ { b _ n } & = - \\frac { 1 } { \\prod _ { \\nu = b _ n + 1 } ^ { b _ { n + 1 } - 1 } w _ { \\nu } } x _ { b _ { n + 1 } - 1 } + \\sum _ { l \\in \\varphi ^ { - 1 } ( n ) } v _ l x _ { b _ { l + 1 } - 1 } \\\\ & = - \\frac { 1 } { \\lambda ^ { \\Delta b _ { n } - 1 } } x _ { b _ { n } } + \\sum _ { l \\in \\varphi ^ { - 1 } ( n ) } v _ l \\ , \\frac { \\prod _ { \\nu = b _ l + 1 } ^ { b _ { l + 1 } - 1 } w _ { \\nu } } { \\lambda ^ { \\Delta b _ l - 1 } } \\ , x _ { b _ { l } } . \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{align*} u _ t = - \\frac { 1 } { \\alpha t } u - \\frac { 1 } { 2 t } x \\cdot \\nabla u , \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{align*} I ^ { r , s } _ { \\ell + 1 } = I ^ { r - 1 , s } _ { \\ell } + I ^ { r - 1 , s + 1 } _ { \\ell + 1 } + I ^ { r , s - 1 } _ { \\ell } + o ( 1 ) \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} p p ' m ^ 2 + n ^ 2 & = \\frac { p ' } { \\Delta } x ^ 2 + \\frac { p } { \\Delta } y ^ 2 \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} \\left | \\dfrac { d ^ { n + m } } { d x ^ { n + m } } \\big ( \\phi \\ast f \\big ) ( x ) \\right | \\leqslant \\int _ { B ( 0 , R ) } \\left | \\dfrac { d ^ { n + m } } { d y ^ { n + m } } \\phi ( y ) \\right | | f ( x - y ) | \\ , d y = c ( \\phi , f , m , j ) \\ , M ( \\phi , f , m , j ) ^ n , \\end{align*}"}
-{"id": "6076.png", "formula": "\\begin{align*} L = L _ { - 5 } \\oplus L _ { - 4 } \\oplus \\cdots \\oplus L _ { r } \\end{align*}"}
-{"id": "9516.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = 1 } ^ n \\lambda _ i v _ i ^ * x u v _ i \\right \\| < \\delta . \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{align*} R & < - \\left ( \\log \\frac { 1 - p } { 2 } + \\beta ^ * ( p ) \\right ) \\\\ & = 1 - \\log ( 1 - p ) - \\frac { \\sqrt { p } } { \\sqrt { p } + \\sqrt { 1 - p } } \\log \\frac { p } { 1 - p } - 2 h \\left ( \\frac { \\sqrt { p } } { \\sqrt { p } + \\sqrt { 1 - p } } \\right ) \\\\ & = 1 - h _ { \\frac { 1 } { 2 } } ( p ) , \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} \\left \\| \\prod \\limits _ { i = 1 } ^ m f _ i \\right \\| _ { w \\mathcal { M } ^ { p } _ { \\phi } } \\ge \\frac { C } { ( K + K ^ { - \\epsilon } ) ^ { \\frac { \\epsilon } { p } } \\phi ( K + K ^ { - \\epsilon } ) } . \\end{align*}"}
-{"id": "1388.png", "formula": "\\begin{align*} \\begin{array} { l l } { \\rm { m a x i m i z e } } & \\sum \\limits _ { i = 1 } ^ m S _ i ^ 2 \\\\ { \\rm { s u b j e c t \\ t o } } & \\sum \\limits _ { i = 1 } ^ { m } S _ i = \\ m 2 ^ k \\mathbb { E } [ Y ] \\\\ & \\mathbb { E } [ Y ] 2 ^ k - d \\ \\leq S _ i \\leq \\ \\mathbb { E } [ Y ] 2 ^ k + d , \\ \\forall i \\in [ m ] . \\end{array} \\end{align*}"}
-{"id": "3558.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } y _ { 1 } ^ { 2 } = ( x - \\mu _ { 4 } ) ( x - \\mu _ { 6 } ) ( x - \\mu _ { 7 } ) \\\\ \\\\ y _ { 2 } ^ { 2 } = ( x - 1 ) ( x - \\mu _ { 5 } ) ( x - \\mu _ { 6 } ) ( x - \\mu _ { 7 } ) \\\\ \\\\ y _ { 3 } ^ { 2 } = x ( x - \\mu _ { 4 } ) ( x - \\mu _ { 5 } ) ( x - \\mu _ { 6 } ) \\end{array} \\right \\} \\subset { \\mathbb C } ^ { 4 } . \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{gather*} \\theta ^ 0 _ 1 + \\theta ^ 0 _ 2 + \\theta ^ 1 + \\theta ^ t + \\theta ^ \\infty _ 1 + \\theta ^ \\infty _ 2 + \\theta ^ \\infty _ 3 = 0 . \\end{gather*}"}
-{"id": "733.png", "formula": "\\begin{align*} C ^ 1 ( X ) & = \\left \\{ f \\in C ^ 0 ( X ) : \\sum _ { x \\in X } f ( x ) = 0 \\right \\} , \\\\ C ^ 2 ( X ) & = \\left \\{ f \\in C ^ 0 ( X ) : \\sum _ { x \\in X } f ( x ) = 0 , \\ , \\sum _ { x \\in X } f ( x ) x = 0 \\right \\} \\end{align*}"}
-{"id": "635.png", "formula": "\\begin{gather*} \\lim _ { \\theta \\downarrow 0 ^ + } ( \\sin \\theta ) ^ { 1 - 2 s } y ' ( \\theta ) = 0 \\quad \\lim _ { \\theta \\uparrow \\pi ^ - } ( \\sin \\theta ) ^ { 1 - 2 s } y ' ( \\theta ) = 0 , \\end{gather*}"}
-{"id": "6685.png", "formula": "\\begin{align*} \\begin{aligned} U ( x , t ) = \\frac { \\delta ( t - \\lvert x - e \\rvert ) } { 4 \\pi \\lvert x - e \\rvert } + V ( x , t ) \\end{aligned} \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{align*} U _ { 1 , 2 } ( \\varepsilon ) = \\left ( \\begin{array} { c } \\frac { 0 . 0 4 \\varepsilon } { - 0 . 3 \\varepsilon - 0 . 9 \\mp \\sqrt { 2 5 \\varepsilon ^ 2 + 5 . 4 \\varepsilon + 0 . 8 1 } } \\\\ \\\\ 0 . 0 5 \\end{array} \\right ) . \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} H S ^ n ( G , M ) \\cong H _ \\lambda ^ n ( G , M ) \\oplus H ^ n _ \\delta ( G , M ) \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} s ^ C & = \\Lambda ( \\rho ) = \\Lambda ( r ) = R _ { \\alpha \\enskip \\gamma } ^ { \\enskip \\alpha \\enskip \\gamma } = - \\frac 1 4 R _ { e _ i \\enskip \\ ; \\ , e _ j } ^ { \\enskip J e _ i \\enskip J e _ j } \\end{align*}"}
-{"id": "9810.png", "formula": "\\begin{align*} { \\int _ N { U V ' d x } = \\left . { U V } \\right | _ { - 1 } ^ 1 - \\int _ N { U ' V d x } } \\quad ( S u m m a t i o n \\ ; B y \\ ; P a r t s ) . \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} ( \\lambda ^ { 5 } + \\lambda ^ { 3 } + 1 ) y = b . \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} B + 1 \\leq \\gamma ' = B + 3 - \\frac { 2 } { \\| u \\| _ { L ^ \\infty } } u \\leq B + 5 , \\ \\gamma '' = - \\frac { 2 } { \\| u \\| _ { L ^ \\infty } } u . \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} \\int _ M ( f \\mu ) \\frac { \\omega _ M ^ n } { n ! } = \\int _ 0 ^ 1 f ( x ) \\operatorname { v o l } ( \\mu ^ { - 1 } x ) d x . \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} X ' ( [ b , c ] ) : = \\sum _ { k = 1 } ^ n X ( [ b _ k , c _ k ] ) . \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} g _ j ( E , t ) = \\frac { | P _ E ( t ) | } { \\sqrt { \\left | \\prod _ { k \\neq j } ( t - E _ { k - 1 } ^ + ) ( t - E _ k ^ - ) \\right | } } \\end{align*}"}
-{"id": "5745.png", "formula": "\\begin{align*} \\hat { \\rho } _ { A M } ( s ) : = ( \\mathcal { N } _ A ( s ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) \\ ; . \\end{align*}"}
-{"id": "2536.png", "formula": "\\begin{align*} p _ { 1 } p _ { 2 } e ^ { \\alpha + \\beta } = c \\frac { N } { B _ { 2 } ^ { 2 } } h , \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} \\int _ T | C _ X ( t ) | \\ , d t = \\sum _ { k = 1 } ^ { \\infty } \\frac { \\langle G , H _ k \\rangle _ { \\varphi } ^ 2 } { k ! } \\int _ T C _ Y ^ k ( t ) \\ , d t < + \\infty . \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} \\Lambda ( y , \\lambda ) = 1 - \\frac { y ^ 2 c } { 2 \\pi } J _ 1 ( y , - \\lambda y ) = 0 . \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} \\Omega ^ - : = \\{ \\omega : \\mu - \\sqrt { 3 } \\sigma \\leq \\omega < \\mu ( 1 - \\rho ) - { \\rho } q ^ * \\sqrt { 3 } \\sigma \\} . \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{align*} d \\delta \\left ( X ^ { ( N ) } ; t \\right ) = \\frac { 1 } { N ^ 2 } \\left [ N ^ 2 d t + 2 \\sqrt { N ^ 2 \\delta \\left ( X ^ { ( N ) } ; t \\right ) } d \\tilde { \\beta } ^ N ( t ) \\right ] = d t + \\frac { 1 } { N } 2 \\sqrt { \\delta \\left ( X ^ { ( N ) } ; t \\right ) } d \\tilde { \\beta } ^ N ( t ) , \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{align*} & c _ 1 ( t ) = x _ 1 - \\frac 1 \\lambda ( e ^ { \\lambda t y } \\cos ( \\lambda t \\xi ) - 1 ) \\sin \\theta + \\frac 1 \\lambda e ^ { \\lambda t y } \\sin ( \\lambda t \\xi ) \\ , \\cos \\theta \\\\ & c _ 2 ( t ) = x _ 2 + \\frac 1 \\lambda e ^ { \\lambda t y } \\sin ( \\lambda t \\xi ) \\sin \\theta + \\frac 1 \\lambda ( e ^ { \\lambda t y } \\cos ( \\lambda t \\xi ) - 1 ) \\ , \\cos \\theta \\ , . \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} Y = \\cup _ { t \\in ( - \\delta , \\delta ) } M ( t ) \\cap G . \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} [ e _ 1 , e _ 2 ] = \\alpha _ 2 e _ 4 + \\alpha _ 3 e _ 5 , [ e _ 2 , e _ 1 ] = - \\alpha _ 2 e _ 4 + \\alpha _ 4 e _ 5 , [ e _ 1 , e _ 3 ] = \\beta _ 2 e _ 5 , [ e _ 3 , e _ 1 ] = \\beta _ 3 e _ 5 , [ e _ 3 , e _ 3 ] = \\beta _ 7 e _ 5 . \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} R = \\frac { L } { \\sum _ { n = 1 } ^ N H ( A _ n ^ { [ i ] } ) } \\end{align*}"}
-{"id": "9719.png", "formula": "\\begin{align*} G _ { r \\times K } = [ A _ 1 | A _ 2 | . . . | A _ T | P ] \\end{align*}"}
-{"id": "4949.png", "formula": "\\begin{align*} \\sigma ( M ) = \\sigma ( S ) \\cup \\sigma ( C ) . \\end{align*}"}
-{"id": "9831.png", "formula": "\\begin{align*} T _ i f ( \\mathbf { z } ) = \\frac { f ( \\mathbf { z } ) - f ( s _ i z ) } { \\mathbf { z } ^ { \\alpha _ i ^ { \\vee } } - 1 } - v \\frac { f ( \\mathbf { z } ) - \\mathbf { z } ^ { \\alpha _ i ^ { \\vee } } f ( s _ i \\mathbf { z } ) } { \\mathbf { z } ^ { \\alpha _ i ^ { \\vee } } - 1 } , \\theta _ { \\lambda } f ( \\mathbf { z } ) = \\mathbf { z } ^ { - \\lambda } f ( \\mathbf { z } ) . \\end{align*}"}
-{"id": "4736.png", "formula": "\\begin{align*} \\dot x = f ( x ) + g ( x ) u , \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} f = \\Bigl ( \\int f \\Bigr ) \\delta + \\div V , \\end{align*}"}
-{"id": "2252.png", "formula": "\\begin{align*} \\frac { C C _ { 3 , 2 7 0 } } { \\Pi ( 2 7 0 ) } & = \\left ( ( - L _ 3 ) ^ 2 \\frac { ( - 1 ) ^ 3 \\cdot 2 } { L _ 4 L _ 5 } + ( - L _ 3 ) ^ 5 \\frac { ( - 1 ) ^ 5 } { L _ 4 ^ 5 } \\right ) \\\\ & = \\frac { L _ 3 ^ 5 } { L _ 4 ^ 5 } - \\frac { 2 L _ 3 ^ 2 } { L _ 4 L _ 5 } \\ , . \\end{align*}"}
-{"id": "2888.png", "formula": "\\begin{align*} \\tau _ { \\widehat { \\alpha } } ( \\gamma ) = \\psi _ 1 ^ { \\widehat { \\alpha } _ 1 - 1 } \\cdots \\psi _ { \\hat { \\ell } } ^ { \\widehat { \\alpha } _ { \\hat { \\ell } } - 1 } \\cdot ^ * _ { 1 , \\ldots , \\hat { \\ell } } ( \\gamma \\cdot \\Delta _ { \\mathsf { r e l } } ) \\ . \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{align*} E _ { \\mu } \\left ( w , m , s \\right ) & = \\frac { \\Gamma \\left ( 1 - s + w \\right ) } { \\Gamma \\left ( 1 - s \\right ) } \\left ( m + \\frac { i \\mu } { 2 \\pi N } \\right ) ^ { s - 1 - w } \\left ( \\frac { i \\mu } { 2 \\pi N } \\right ) ^ { w } - \\delta _ { w 0 } m ^ { s - 1 } \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} \\sigma ( P ) = V P V ^ { - 1 } , \\quad \\mathrm { T r } ( V P ) = c \\cdot 1 , \\end{align*}"}
-{"id": "963.png", "formula": "\\begin{align*} \\textrm { V a r } [ Y \\mid Y \\in ( q ( 0 ) , q ( \\alpha _ 1 ) ) ] & = \\textrm { V a r } [ Y \\mid Y \\in [ q ( \\alpha _ 1 ) , q ( \\alpha _ 2 ) ) ] \\\\ & = \\ldots \\\\ & = \\textrm { V a r } [ Y \\mid Y \\in [ q ( \\alpha _ k ) , q ( 1 ) ) ] \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} \\Big ( \\frac { \\gamma } { p ^ + } \\lim _ { r \\downarrow 0 } \\ ; r ^ { - B } g _ { \\mathrm { r e g } } ^ + ( r ) \\Big ) \\ ; = \\ ; ( c _ \\nu \\beta + d _ \\nu ) \\ , \\Big ( \\lim _ { r \\downarrow 0 } \\ ; r ^ { B } g _ { \\mathrm { s i n g } } ^ + ( r ) \\Big ) \\ , , \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} F _ * ^ e ( j u ^ s v ^ s ( f + u v ) ) = \\bigoplus _ { i \\in \\Delta _ e } f _ { ( i , j ) } F _ * ^ e ( i u ^ s v ^ s ) \\oplus F ^ e ( j u ^ { s + 1 } v ^ { s + 1 } ) \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} \\# \\{ \\pi _ { a _ 1 } ( E _ { a _ 1 } [ \\infty ] ) \\cap \\pi _ { a _ 2 } ( E _ { a _ 2 } [ \\infty ] ) \\} = 6 + 4 n \\ge 1 0 , \\end{align*}"}
-{"id": "1239.png", "formula": "\\begin{align*} d \\tilde { X } ( t ) = S _ { \\Delta t } A X _ n d t + S _ { \\Delta t } G ( X _ n ) d t + S _ { \\Delta t } \\sigma ( X _ n ) d W ( t ) . \\end{align*}"}
-{"id": "290.png", "formula": "\\begin{align*} ( Y \\triangleright \\phi ) ( X ) = \\phi ( X Y ) , ( \\phi \\triangleleft Y ) ( X ) = \\phi ( Y X ) . \\end{align*}"}
-{"id": "5543.png", "formula": "\\begin{align*} \\Phi ( u ) = \\frac { \\left \\langle u ^ { ( n ) } \\ ; u ^ { ( n ) } \\right \\rangle } { \\left \\langle u ^ { ( n - p ) } \\ ; u ^ { ( n - p ) } \\right \\rangle } . \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} \\dot { x } = A x = \\left [ \\begin{array} { l l } C & 0 \\\\ 0 & B \\end{array} \\right ] \\left [ \\begin{array} { l } y \\\\ z \\end{array} \\right ] , \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} [ y _ 1 , y _ 2 ] = y _ 4 + \\theta y _ 5 , [ y _ 2 , y _ 1 ] = - y _ 4 , [ y _ 2 , y _ 2 ] = \\frac { \\alpha _ 5 } { \\alpha _ 2 \\gamma _ 1 } y _ 5 , [ y _ 1 , y _ 3 ] = y _ 5 , [ y _ 1 , y _ 4 ] = y _ 5 = - [ y _ 4 , y _ 1 ] . \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{align*} G ( k , 0 ) ^ \\dag G ' ( k , 0 ) - G ' ( k , 0 ) ^ \\dag G ( k , 0 ) = [ G ( k , x ) ^ \\dag ; G ( k , x ) ] = 0 _ n . \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} | v ^ + _ \\infty ( r ) \\ , v ^ + _ 0 ( \\rho ) \\ , \\mathbf { 1 } _ { ( 1 , + \\infty ) } ( r ) \\ , \\mathbf { 1 } _ { ( 1 , + \\infty ) } ( \\rho ) | \\ ; & \\lesssim \\ ; e ^ { - r } \\ , e ^ \\rho \\ , ( \\rho / r ) ^ B \\qquad \\textrm { i f } \\ , 0 < \\rho < r \\\\ | v ^ + _ 0 ( r ) \\ , v ^ + _ \\infty ( \\rho ) \\ , \\mathbf { 1 } _ { ( 1 , + \\infty ) } ( r ) \\ , \\mathbf { 1 } _ { ( 1 , + \\infty ) } ( \\rho ) | \\ ; & \\lesssim \\ ; e ^ r \\ , e ^ { - \\rho } \\ , ( r / \\rho ) ^ B \\qquad \\textrm { i f } \\ , 0 < r < \\rho \\ , , \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} \\lambda _ p ( \\alpha ) : = - \\langle - \\alpha \\rangle _ p + \\frac { \\alpha + \\langle - \\alpha \\rangle _ p } { p } \\cdot ( p - 1 ) . \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } f ^ { 4 } - \\lambda _ { 1 } p ^ { 4 } = \\lambda _ { 2 } f p \\dot { p } \\\\ 2 f ^ { 2 } p ^ { 2 } - \\lambda _ { 3 } p ^ { 4 } = \\lambda _ { 4 } f p \\dot { p } . \\end{array} \\right . \\end{align*}"}
-{"id": "9610.png", "formula": "\\begin{align*} \\aligned \\widetilde { \\mathcal W } { } ^ i _ { j m n } & = R ^ i _ { j m n } + \\frac 1 { N + 1 } \\delta ^ i _ j R _ { [ m n ] } - \\frac 1 2 \\big ( F ^ i _ j \\sigma _ m + F ^ i _ m \\sigma _ j \\big ) _ { | n } + \\frac 1 2 \\big ( F ^ i _ j \\sigma _ n + F ^ i _ n \\sigma _ j \\big ) _ { | m } - \\delta ^ i _ { [ m } \\zeta _ { j n ] } , \\endaligned \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{align*} g ( x ) : = \\left ( x + 1 \\right ) \\ln \\left ( x + 1 \\right ) - x \\ln x \\ ; , \\end{align*}"}
-{"id": "5730.png", "formula": "\\begin{align*} \\mathrm { d } p _ { X } ( t ) ( \\mathbf { x } ) = \\mathrm { e } ^ { - \\frac { | \\mathbf { x } | ^ 2 } { 2 t } } \\frac { \\mathrm { d } ^ { 2 n } x } { ( 2 \\pi \\ , t ) ^ n } \\ ; , \\mathbf { x } \\in \\mathbb { R } ^ { 2 n } \\ ; , \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} f ^ * _ { s o s } & : = \\mbox { m a x i m i z e } \\gamma \\\\ & \\mbox { s u b j e c t t o } f ( x ) - \\gamma \\mbox { i s a S O S i n } \\mathbb R [ X ] . \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} X _ { 2 , \\ , j + 2 , \\ , 7 } \\cap L _ { j + 2 } = 0 . \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} K ( s , t ) = \\begin{cases} s - \\left ( a + \\psi ( \\lambda ) \\frac { b - a } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ a , t ) , \\\\ s - \\left ( a + ( 1 + \\psi ( 1 - \\lambda ) ) \\frac { b - a } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ t , b ] , \\end{cases} \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{align*} \\| K ( x , x ) F ( 2 x ) \\| = \\frac { 1 } { 2 } \\left \\| \\int _ x ^ \\infty V ( t ) F ( 2 x ) d t \\right \\| \\le c \\| F ( 2 x ) \\| \\in L ^ 1 ( \\mathbb { R } ^ + ) , \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} \\frac { \\lambda _ 2 } { s _ 2 } = \\frac { \\lambda _ 2 } { s _ 1 u _ 1 - 1 } < \\frac { \\lambda _ 2 u _ 2 } { u _ 2 u _ 1 s _ 1 - u _ 2 - s _ 1 } = \\frac { \\lambda _ 2 u _ 2 } { s _ 3 } \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} s _ { i - 2 } = \\sum _ { k = 1 } ^ m a _ k \\cdot s _ { k - 2 } . \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{align*} g _ k ( r ) = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } g ( r e ^ { i \\theta } ) \\ , e ^ { - i k \\theta } \\ , d \\theta . \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} \\gamma ^ k _ { \\Omega , o } ( \\xi _ 1 \\cdots \\xi _ k ) ( u ) \\ = \\ \\xi _ 1 \\cdots \\xi _ k ( u ) \\ , , \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle | \\sum \\limits _ { k = 1 } ^ { m } \\mathrm { R e } [ J _ 2 ^ { k , 2 } ] | = | - V _ 0 ( \\| \\theta _ { \\psi } ^ { m } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } - \\| \\theta _ { \\psi } ^ { 0 } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } ) | \\leq C \\| \\theta _ { \\psi } ^ { m } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } + C h ^ { 2 r + 2 } . } \\end{array} \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} \\int _ a ^ b f ( x ) \\ , d x = g ( b ) - g ( a ) . \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} \\Vert \\lim _ { k \\to \\infty } B \\phi _ k \\Vert _ { H _ 1 } \\le \\liminf _ { k \\rightarrow \\infty } \\Vert B \\phi _ k \\Vert _ { H _ 1 } = 0 , \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { \\nu } \\varepsilon _ { n } ^ { \\sigma _ { i } } \\alpha _ { i } \\frac { \\partial ^ { i } u } { \\partial x _ { n } ^ { i } } \\left ( x ^ { \\prime } , 0 , t \\right ) = 0 \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} { } _ 5 F _ 4 \\bigg [ \\begin{matrix} \\alpha & \\alpha & 1 + \\frac 1 2 \\alpha & \\beta & \\gamma \\\\ & 1 & \\frac 1 2 \\alpha & \\alpha - \\beta + 1 & \\alpha - \\gamma + 1 \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { \\Gamma ( 1 + \\alpha - \\beta ) \\Gamma ( 1 + \\alpha - \\gamma ) \\Gamma ( 1 - \\beta - \\gamma ) } { \\Gamma ( 1 + \\alpha ) \\Gamma ( 1 - \\beta ) \\Gamma ( 1 - \\gamma ) \\Gamma ( 1 + \\alpha - \\beta - \\gamma ) } . \\end{align*}"}
-{"id": "226.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { j = 1 } ^ n A _ j z ^ { m _ j } [ 1 + \\epsilon ( z ) ] e ^ { \\omega _ j z } , \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} \\| x ( t ) - x ^ * \\| \\le \\sqrt [ 2 m _ 1 ] { \\tfrac { \\gamma _ 2 } { \\gamma _ 1 } } \\| x ^ 0 - x ^ * \\| \\varphi _ { \\tilde m } ( \\lambda t ) , t = 0 , \\varepsilon , 2 \\varepsilon , \\dots , \\end{align*}"}
-{"id": "1904.png", "formula": "\\begin{align*} A _ c & = 2 m \\lambda ( 1 + \\lambda ) + O ( c ^ { - 1 } ) , & B _ c & = 2 ( m - 2 ) c + O ( c ^ 0 ) , \\\\ C _ c & = 2 ( 1 + 2 \\lambda ) c ^ { n - 1 } + O ( c ^ { n - 2 } ) , & D _ c & = O ( c ^ { n - 2 } ) . \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} \\begin{bmatrix} p + r & 2 r & r \\\\ 2 r & 4 r & 2 r \\\\ r & 2 r & q + r \\end{bmatrix} , \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { r + 1 } m _ i = | ( l _ 1 \\dots l _ { 2 n - 2 } ) ^ { 2 ^ { n - 2 } } | + | c _ { 2 ^ { n - 1 } } \\cdots c _ { r + 1 } | & \\leq 2 ^ { n - 2 } ( 2 n - 2 ) + j ( 2 ^ { n + 1 } + 2 n - 2 ) \\\\ & \\leq ( 2 ^ { n - 2 } + j ) ( 2 n - 2 ) + j 2 ^ { n + 1 } . \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { y } _ 1 & = e ^ { y _ 2 - y _ 1 } - \\frac { 1 } { 2 } , \\dot { y } _ 2 = - e ^ { y _ 2 - y _ 1 } - e ^ { y _ 2 - y _ 3 } + \\frac { 1 } { 2 } , \\\\ \\dot { y } _ 3 & = e ^ { y _ 2 - y _ 3 } + e ^ { y _ 4 - y _ 3 } - \\frac { 1 } { 2 } , \\dot { y } _ 4 = - e ^ { y _ 4 - y _ 3 } + \\frac { 1 } { 2 } . \\end{aligned} \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} [ \\omega _ \\epsilon ( t ) ] = 2 \\pi \\big ( b _ t [ D _ \\infty ] - a _ t [ D _ 0 ] ) \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{align*} \\left | \\sum _ { n = 1 } ^ W e \\left ( \\overline { \\xi } _ { x + n w } \\right ) \\right | ^ 2 \\leq \\frac { W ( W + 1 ) } { 2 } + \\frac { W + 1 } { 2 } \\left | \\sum _ { n = 1 } ^ { W - 1 } e \\left ( \\overline { \\xi } _ { x + ( n + 1 ) w } - \\overline { \\xi } _ { x + n w } \\right ) \\right | . \\end{align*}"}
-{"id": "6932.png", "formula": "\\begin{align*} \\int _ \\Omega | t _ \\varepsilon | ^ q = o ( 1 ) \\int _ \\Omega t _ \\varepsilon ^ 2 = o ( \\varepsilon ^ 3 ) , \\end{align*}"}
-{"id": "7393.png", "formula": "\\begin{align*} S ( h ) : = \\sum _ { \\alpha : i \\to j } \\mathrm { t r } ( h _ i ^ { - 1 } \\phi _ \\alpha ^ * h _ j \\phi _ \\alpha ) \\end{align*}"}
-{"id": "7198.png", "formula": "\\begin{align*} \\left ( \\frac { 1 + \\sqrt { 4 \\ell + 1 } } { 2 q } \\right ) ^ { - 1 } = \\frac { - 1 + \\sqrt { 4 \\ell + 1 } } { 2 q } , \\end{align*}"}
-{"id": "7820.png", "formula": "\\begin{align*} U _ t + b U _ x + U _ { x x x } + U _ { x y y } + \\left ( g _ h ( U ) \\right ) _ x + ( \\psi U ) _ x = F \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} \\mathbb { P } ( { \\cal E } _ 1 = { \\cal T } ) \\geq p _ d ^ { r - 1 } ( 1 - p _ u ) ^ { r ( n - r ) + { r \\choose 2 } - r + 1 } \\end{align*}"}
-{"id": "9339.png", "formula": "\\begin{align*} \\log ( \\gamma _ M ^ { - 1 } \\cdot \\gamma _ N ) + \\frac 1 2 \\Big ( \\sum _ { M < n \\leq N } \\log ( 1 - | F _ n | ^ 2 ) \\Big ) I & = \\log \\Big ( \\prod _ { M < n \\leq N } \\Big ( I + ( 0 , F _ n z ^ n ) \\Big ) \\Big ) \\\\ & = \\log ( I + ( \\sigma _ N - \\sigma _ M ) + H _ M ^ N ) , \\end{align*}"}
-{"id": "8231.png", "formula": "\\begin{align*} \\rho _ { F } ( \\Omega _ { 2 } ) = \\frac { 1 } { \\overline \\Lambda } \\le \\rho _ { 2 , F } ( \\Omega ) \\le \\rho _ { F } ( \\Omega ) = \\rho _ { F } ( \\Omega _ { 1 } ) , \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} G ^ - \\circ \\psi _ q ( \\zeta ) = \\frac { 1 } { d ^ n } G ^ - \\circ \\psi _ p \\left ( \\frac { \\zeta } { \\lambda _ { p , n } } \\right ) . \\end{align*}"}
-{"id": "9596.png", "formula": "\\begin{align*} R ^ i _ { j m n } = L ^ i _ { \\underline { j m } , n } - L ^ i _ { \\underline { j n } , m } + L ^ \\alpha _ { \\underline { j m } } L ^ i _ { \\underline { \\alpha n } } - L ^ \\alpha _ { \\underline { j n } } L ^ i _ { \\underline { \\alpha m } } . \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} ( A u ) _ { d - 1 } g _ 1 \\cdots g _ { d - 1 } = \\theta _ { d - 1 } \\left ( u _ { ( d - 1 ) + } g _ 1 \\cdots g _ { d - 2 } g _ { d - 1 } ' + \\sum _ { i = 1 } ^ { d - 2 } ( A u ) _ i g _ 1 \\cdots g _ { i - 1 } g _ i ' g _ { i + 1 } \\cdots g _ { d - 1 } \\right ) . \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ 3 } e ( u ( \\cdot , t ) , v ( \\cdot , t ) ) d x + \\int _ { 0 } ^ { t } \\int _ { \\mathbb R ^ 3 } ( | \\nabla v | ^ 2 + | \\partial _ t u + ( v \\cdot \\nabla u ) | ^ 2 d x = \\int _ { \\mathbb { R } ^ 3 } e ( u _ 0 , v _ 0 ) d x , \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{align*} \\langle \\Delta ( x + \\xi ^ 1 , y + \\xi ^ 2 ; T ) , \\phi \\rangle = \\int _ 0 ^ T \\int _ 0 ^ T \\phi ( x + \\xi ^ 1 _ r ) | y + \\xi ^ 2 _ s - x - \\xi ^ 1 _ r | ^ { \\frac { \\beta - \\alpha } { 2 } - 1 } d r d s , \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} \\hat { I _ 2 } = \\int _ \\Omega b ( t ) | u _ t | ^ { p + 1 } | u | d x \\quad \\hat { I _ b } = \\int _ \\Omega | u _ t | ^ { p + 2 } d x . \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} \\partial _ { x ^ \\prime } \\eta = O \\left ( \\frac { 1 } { | x ^ \\prime | ^ { 2 + \\varepsilon } } \\right ) , \\textrm { a s } | x ^ \\prime | \\to \\infty . \\end{align*}"}
-{"id": "2319.png", "formula": "\\begin{align*} \\chi _ { 1 3 } & = \\mu _ 1 \\frac { 6 y ^ 3 } { 1 - 3 x ^ 2 - 3 y ^ 2 } ; \\\\ \\chi _ { 2 3 } & = \\mu _ 2 \\frac { 1 2 \\sqrt { 3 } y ^ 3 } { ( 1 - \\sqrt { 3 } x ) ( 1 + \\sqrt { 3 } x ) ^ 2 } . \\end{align*}"}
-{"id": "5363.png", "formula": "\\begin{gather*} c _ 2 ( 2 n , k ) = \\sum _ { i = k } ^ { 2 n } { i - 1 \\choose k - 1 } c _ 1 ( 2 n , i ) = \\sum _ { j = \\lceil \\frac k 2 \\rceil } ^ { n } { 2 j - 1 \\choose k - 1 } c _ 1 ( 2 n , 2 j ) \\\\ = \\sum _ { j = \\lceil \\frac k 2 \\rceil } ^ { n } a ^ { n - j } { 2 j - 1 \\choose k - 1 } { n + j - 1 \\choose n - j } . \\end{gather*}"}
-{"id": "8037.png", "formula": "\\begin{align*} \\| R ( e ^ { i \\theta } , T ) \\| = O ( | \\theta | ^ { - \\alpha } ) , \\theta \\to 0 . \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} U _ { t } ^ { r } \\left ( \\theta _ { 0 } \\right ) : = U _ { t } ^ { - } \\left ( \\theta _ { 0 } \\right ) + Z _ { t } ^ { U } \\left ( U _ { t } \\left ( \\theta _ { 0 } \\right ) - U _ { t } ^ { - } \\left ( \\theta _ { 0 } \\right ) \\right ) , \\end{align*}"}
-{"id": "9565.png", "formula": "\\begin{align*} \\sum _ { \\left | j - 2 \\omega \\right | \\leq 2 \\delta } \\ ! \\ ! e ^ { \\pi i l j } \\overline { \\hat { f } \\left ( \\frac { j } { 2 } \\right ) } \\overline { \\hat { g } \\left ( \\frac { j } { 2 } - \\omega \\right ) } = e ^ { 2 \\pi i l \\omega } \\vec { X } _ { l } ^ { * } \\vec { Y _ { \\omega } } \\end{align*}"}
-{"id": "3677.png", "formula": "\\begin{align*} [ h _ \\chi ( z ) , H ^ { \\beta } ( w ^ 2 ) ] & = [ h _ \\chi ( z ) , V ^ + ( w ) ^ { - 1 } \\beta ( w ^ 2 ) w ^ { - 2 h _ 0 } V ^ - ( w ) ^ { - 1 } ] \\\\ & = \\Big ( \\sum _ { n > 0 } \\frac { z ^ { 2 n - 2 } } { w ^ { 2 n } } - i _ { z , w } \\frac { 1 } { z ^ 2 - w ^ 2 } - i _ { w , z } \\frac { 1 } { w ^ 2 - z ^ 2 } + \\sum _ { n > 0 } \\frac { w ^ { 2 n } } { z ^ { 2 n + 2 } } \\Big ) H ^ { \\beta } ( w ^ 2 ) = - \\frac { 1 } { z ^ 2 } H ^ { \\beta } ( w ^ 2 ) . \\end{align*}"}
-{"id": "9168.png", "formula": "\\begin{align*} y _ t = H ( q ^ { - 1 } ) r _ t + e _ t , \\end{align*}"}
-{"id": "10015.png", "formula": "\\begin{align*} X _ j ^ { n - j + 1 } = - \\displaystyle \\sum _ { i _ 1 + i _ 2 + \\cdots + i _ j = n - j + 1 \\atop i _ j \\leq n - j } X _ 1 ^ { i _ 1 } X _ 2 ^ { i _ 2 } \\cdots X _ j ^ { i _ j } \\ \\ \\ \\ \\ \\ { \\rm i n } \\ M ^ * ( h ) \\end{align*}"}
-{"id": "6914.png", "formula": "\\begin{align*} u _ k ( 0 ) = 0 . \\end{align*}"}
-{"id": "9677.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { F } ( 1 , \\phi , \\psi ) & = - \\hat { F } ( 0 , \\phi , \\psi ) , \\\\ \\hat { G } ( 1 , \\phi , \\psi ) & = - \\hat { H } ( 0 , \\phi , \\psi ) , \\\\ \\hat { H } ( 1 , \\phi , \\psi ) & = \\hat { G } ( 0 , \\phi , \\psi ) \\end{aligned} \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{align*} \\eta _ { X , Y } ^ \\lambda ( \\mathsf { P } ) : = \\sum _ { i , j , k } q ^ { ( \\lambda , \\lambda _ { i } ) } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) \\varepsilon ( X \\triangleright \\mathsf { P } _ { k } ^ { j } ) \\varepsilon ( Y \\triangleright \\mathsf { P } _ { i } ^ { k } ) . \\end{align*}"}
-{"id": "4289.png", "formula": "\\begin{align*} \\deg ( f _ { m - 2 } ( X ) f _ { m + 1 } ^ 2 ( X ) ) = \\frac { 1 } { 2 } ( ( m - 2 ) ^ 2 - 1 ) + ( ( m + 1 ) ^ 2 - 4 ) = \\frac { 3 } { 2 } ( m ^ 2 - 1 ) , \\end{align*}"}
-{"id": "380.png", "formula": "\\begin{align*} \\eta _ { j - 2 k , j } = \\frac { \\alpha _ { j - 2 k - 1 } \\alpha _ { j - 2 k - 2 } \\cdots \\alpha _ { 1 } } { \\alpha _ { j - 1 } \\alpha _ { j - 2 } \\cdots \\alpha _ { 2 k + 1 } } \\eta _ { 0 , 2 k } \\end{align*}"}
-{"id": "3802.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } N \\int _ { t = 0 } ^ 1 \\left ( \\int _ X M A _ { \\theta } ( ( 1 - t ) \\phi _ 0 + t \\phi ) \\right ) t ^ { N - 1 } d t = \\int _ X M A _ { \\theta } ( \\phi ) , \\end{align*}"}
-{"id": "9854.png", "formula": "\\begin{align*} \\mathcal { F } _ i ^ { \\mathbf { z } } ( \\phi _ w ) ( \\pi ^ \\lambda ) = \\left [ T _ w \\cdot \\begin{pmatrix} \\vdots \\\\ \\mathcal { F } _ i ^ { \\mathbf { z } } ( \\phi _ 1 ) ( \\pi ^ \\lambda ) \\\\ \\vdots \\\\ - \\\\ \\vdots \\\\ - \\\\ \\vdots \\\\ \\mathcal { F } _ i ^ { w _ 0 \\mathbf { z } } ( \\phi _ 1 ) ( \\pi ^ \\lambda ) \\\\ \\vdots \\end{pmatrix} \\right ] _ i , \\end{align*}"}
-{"id": "3444.png", "formula": "\\begin{align*} A \\Big \\| \\sum _ { i = 1 } ^ k a _ i e _ { \\gamma _ i } \\Big \\| \\le \\Big \\| \\sum _ { i = 1 } ^ k a _ i v _ i \\Big \\| \\le B \\Big \\| \\sum _ { i = 1 } ^ k a _ i e _ { \\gamma _ i } \\Big \\| \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} \\alpha A _ { \\bar { \\alpha } } & = B _ { \\alpha } = B _ { \\bar { \\alpha } } = A _ { \\bar { \\alpha } } \\alpha , & \\big ( \\alpha - a _ { s ( \\alpha ) } A _ { \\bar { \\alpha } } \\big ) ^ 2 & = 0 \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{align*} \\dot { z } = \\dot { y } \\left ( 1 + \\frac { 1 } { 2 } G ( y ^ { - 1 } k y ) \\right ) + y O ( t ^ { - 2 } \\mathcal L ) \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} D _ { n , k } ( a , x ) & = k \\ , \\Big [ \\displaystyle \\frac { y ^ n ( a - y ) - y ( a - y ) ^ n } { 2 y - a } \\Big ] + D _ n ( a , x ) , \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{align*} I _ n : = I _ { n , n } . \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} \\| t \\nabla \\cdot f _ { \\alpha a } ^ 2 \\| ^ 2 _ { L ^ 2 ( r \\leq \\langle t \\rangle / 2 ) } & \\lesssim \\sum \\limits _ { \\tiny \\begin{matrix} b + c = a \\\\ \\beta + \\gamma = \\alpha \\end{matrix} } \\big \\| \\langle t - r \\rangle | \\nabla ^ 2 U ^ { ( \\beta , b ) } | | \\nabla U ^ { ( \\gamma , c ) } | \\big \\| ^ 2 _ { L ^ 2 ( r \\leq \\langle t \\rangle / 2 ) } \\\\ & \\lesssim E _ { | \\alpha | + | a | + 2 } X _ { [ ( | \\alpha | + | a | ) / 2 ] + 4 } . \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} \\bigcap _ { j = 1 } ^ { n } B _ { \\alpha _ { j } } ( x ; \\varepsilon _ { j } ) \\end{align*}"}
-{"id": "9328.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + A ( \\mu ) x + B ( \\mu ) , \\ \\ \\mbox { w h e r e } \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} s _ { i - 1 } = \\sum _ { k = 1 } ^ m a _ k \\cdot s _ { k - 1 } . \\end{align*}"}
-{"id": "1639.png", "formula": "\\begin{align*} K ^ s _ \\beta ( \\Omega ) = \\big [ K ^ m _ \\beta ( \\Omega ) , K ^ { m + 1 } _ \\beta ( \\Omega ) \\big ] _ \\tau \\ , . \\end{align*}"}
-{"id": "8392.png", "formula": "\\begin{align*} V _ n = \\{ T _ n = \\hat { T } ^ { ( n ) } _ n \\} W _ n = \\{ \\hat { T } ^ { ( ( n + 1 ) ^ 2 ) } _ { n ^ 2 } = \\hat { T } ^ { ( n ^ 2 ) } _ { n ^ 2 } \\} , \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{align*} \\frac { h ( t + \\Delta t ) - h ( t ) } { \\Delta t } = & - \\int _ x ^ \\infty h ( s ) \\frac { F ( t + \\Delta t + s ) - F ( t + s ) } { \\Delta t } d s . \\\\ = & - \\int _ x ^ \\infty h ( s ) F ' ( t + \\theta \\Delta t + s ) d s \\\\ = & - \\int _ { x + t + \\theta \\Delta t } ^ \\infty h ( s - t ) F ' ( s ) d s \\\\ & - \\int _ { x + t + \\theta \\Delta t } ^ \\infty [ h ( s - t - \\theta \\Delta t ) - h ( s - t ) ] F ' ( s ) d s \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{align*} \\frac { 2 ^ { - \\lambda _ p ( \\alpha ) } \\cdot \\Gamma _ p ( \\frac 1 2 ) } { \\Gamma _ p ( 1 - \\frac 1 2 \\alpha ) \\Gamma _ p ( \\frac 1 2 + \\frac 1 2 \\alpha ) } = - \\frac { 2 \\Gamma _ p ( 1 + \\frac 1 2 \\alpha ) } { \\Gamma _ p ( 1 + \\alpha ) \\Gamma _ p ( 1 - \\frac 1 2 \\alpha ) } . \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} \\langle 1 \\cdots M | \\mathcal { B } ^ \\prime ( z ) | \\overline { x } \\rangle = \\frac { ( 1 + t ^ { - 1 } z ^ 2 ) } { t z } \\frac { z ^ { \\overline { \\lambda } + 1 } - z ^ { - \\overline { \\lambda } - 1 } } { z - z ^ { - 1 } } , \\ \\ \\ \\overline { \\lambda } = \\overline { x } - 1 . \\end{align*}"}
-{"id": "6534.png", "formula": "\\begin{align*} Z ( t ) = A ( t ) e ^ { t B } \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{align*} \\mu ^ { 2 ^ { k } + 1 } + \\mu ^ { 2 ^ { k } } + 1 = 0 . \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} \\hat { \\sigma } _ { A B M | Z = \\mathbf { z } } = \\hat { D } _ A \\left ( \\frac { \\lambda \\mathbf { z } } { \\sqrt { \\eta } } \\right ) \\hat { D } _ B \\left ( \\frac { ( 1 - \\lambda ) \\mathbf { z } } { \\sqrt { 1 - \\eta } } \\right ) \\hat { \\rho } _ { A B M } { \\hat { D } _ A \\left ( \\frac { \\lambda \\mathbf { z } } { \\sqrt { \\eta } } \\right ) } ^ \\dag { \\hat { D } _ B \\left ( \\frac { ( 1 - \\lambda ) \\mathbf { z } } { \\sqrt { 1 - \\eta } } \\right ) } ^ \\dag \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} \\Big \\vert I _ u ( 0 , 1 ) - I _ u ( x , 1 ) \\Big \\vert & = \\Big \\vert \\frac { D _ u ( 0 , 1 ) } { H _ u ( x , 1 ) } \\left ( \\frac { H _ u ( x , 1 ) } { H _ u ( 0 , 1 ) } - \\frac { D _ u ( x , 1 ) } { D _ u ( 0 , 1 ) } \\right ) \\Big \\vert \\stackrel { \\eqref { e : H l i m i t a t o } } { \\leq } C . \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ a \\frac { ( - a ) _ { k } ( \\beta ) _ { k } } { ( 1 ) _ { k } ^ 2 } \\cdot z ^ k = & { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & \\beta \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] = ( 1 - z ) ^ a { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & 1 - \\beta \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac { z } { z - 1 } \\bigg ] \\\\ = & ( 1 - z ) ^ a \\sum _ { k = 0 } ^ a \\frac { ( - a ) _ { k } ( 1 - \\beta ) _ { k } } { ( 1 ) _ { k } ^ 2 } \\cdot \\frac { z ^ k } { ( z - 1 ) ^ k } . \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} h ^ p & = \\begin{pmatrix} 1 & p \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a p + \\frac { c ( p - 1 ) p } { 2 } & \\frac { ( a + d + c ( p - 1 ) ) ( p - 1 ) p } { 2 } - c \\sum _ { k = 0 } ^ { p - 1 } k ^ 2 + b p \\\\ c p & d p + \\frac { c ( p - 1 ) p } { 2 } \\end{pmatrix} p \\\\ & = \\begin{pmatrix} 1 & p \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} 0 & - c \\sum _ { k = 0 } ^ { p - 1 } k ^ 2 \\\\ 0 & 0 \\end{pmatrix} p . \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} & \\partial = \\partial ^ N + \\partial ^ \\top , \\bar { \\partial } = \\bar { \\partial } ^ { N } + \\bar { \\partial } ^ \\top \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{align*} & _ { \\mu , \\sigma } D _ { v , q ; z } ^ { \\alpha , \\eta , p } ( f ( z ) ) = \\sum _ { s = 1 } ^ { 3 } ( z _ { s } \\frac { B ( \\eta , \\alpha - 1 ) z ^ { \\eta + \\alpha } } { ( 3 - i ) ) ) ^ { s } \\Gamma ( \\alpha ) } F _ { v , q ; p } ^ { ( \\mu , \\sigma ) } ( s , \\eta ; \\eta + \\alpha - 1 ; \\frac { z } { i - 3 } ) + \\\\ & + \\bar { z } _ { s } \\frac { B ( \\eta , \\alpha - 1 ) z ^ { \\eta + \\alpha } } { ( 3 + i ) ) ) ^ { s } \\Gamma ( \\alpha ) } F _ { v , q ; p } ^ { ( \\mu , \\sigma ) } ( s , \\eta ; \\eta + \\alpha - 1 ; \\frac { z } { - i - 3 } ) ) \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} \\tilde { G } _ n = - \\frac { c } { 2 \\pi } \\tilde { G } _ 1 \\frac { 1 } { \\sqrt { n ! } } \\left ( \\frac { i } { \\gamma } \\right ) ^ { n + 1 } J _ n ( 1 / \\gamma , - \\lambda / \\gamma ) , \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\biggl \\| \\int _ 0 ^ { t _ A } \\tilde { F } ( 4 t - s ) * ( u _ { n } \\theta _ { n } ) ( s ) \\dd s \\biggr \\| _ 1 = 0 . \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} \\int K _ { r } \\left ( x , y \\right ) ~ d x \\approx \\int _ { y _ { 1 } - r } ^ { y _ { 1 } } \\left \\{ \\int _ { y _ { 2 } - h _ { x , y } } ^ { y _ { 2 } + h _ { x , y } } \\frac { 1 } { h _ { x , y } } d x _ { 2 } \\right \\} d x _ { 1 } \\approx \\int _ { x _ { 1 } } ^ { x _ { 1 } + r } d y _ { 1 } = r \\ . \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{gather*} F _ A - ( n + 1 ) \\psi \\prod _ { i = 0 } ^ n x _ i . \\end{gather*}"}
-{"id": "9133.png", "formula": "\\begin{gather*} f _ 1 ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) : = x _ 0 ^ 4 + x _ 1 ^ 4 + \\lambda x _ 0 x _ 1 x _ 2 x _ 3 , \\\\ f _ 2 ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) : = x _ 0 x _ 1 \\big ( x _ 0 ^ 2 + x _ 1 ^ 2 \\big ) + \\lambda x _ 0 x _ 1 x _ 2 x _ 3 , \\\\ g _ 1 ( x _ 2 , x _ 3 ) : = x _ 2 ^ 4 + x _ 3 ^ 4 , \\\\ g _ 2 ( x _ 2 , x _ 3 ) : = x _ 2 x _ 3 \\big ( x _ 2 ^ 2 + x _ 3 ^ 2 \\big ) , \\\\ g _ 3 ( x _ 2 , x _ 3 ) : = x _ 2 ^ 3 x _ 3 + x _ 3 ^ 4 , \\\\ h _ 1 ( u , v , x _ 2 , x _ 3 ) : = u ^ 4 - 4 u ^ 2 v + 2 v ^ 2 + \\lambda v x _ 2 x _ 3 , \\\\ h _ 2 ( u , v , x _ 2 , x _ 3 ) : = v \\big ( u ^ 2 - 2 v \\big ) + \\lambda v x _ 2 x _ 3 . \\end{gather*}"}
-{"id": "6575.png", "formula": "\\begin{align*} [ \\mathcal { A } , \\mathcal { B } ] = \\sum _ { \\tau = i , j , k } \\left ( \\dfrac { 1 } { 3 } [ \\mathcal { A } _ 1 , \\mathcal { B } _ 1 ] _ \\tau + [ \\mathcal { A } _ 1 , \\mathcal { B } _ \\tau ] _ \\tau + [ \\mathcal { A } _ \\tau , \\mathcal { B } _ 1 + \\mathcal { B } _ \\tau ] _ \\tau \\right ) , \\mbox { ~ ~ f o r a l l ~ ~ } \\mathcal { A } , \\mathcal { B } \\in \\mathfrak { h } _ { 2 4 } . \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} 0 \\neq [ L _ { - 4 } , \\ , L _ 2 ] = [ [ L _ { - 2 } , \\ , L _ { - 2 } ] , \\ , L _ 2 ] \\subseteq [ L _ { - 2 } , \\ , [ L _ { - 2 } , \\ , L _ 2 ] ] . \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} H _ { n + 1 } ( x , m ) = \\phi ' _ { \\mu } ( m ) ( x - m ) H _ { n } ( x , m ) + R _ { n + 1 } ( x , m ) , \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} \\textstyle A _ 0 = \\big \\{ \\frac { a + \\alpha ( a ) } { 2 } : a \\in A \\big \\} \\quad A _ 1 = \\big \\{ \\frac { a - \\alpha ( a ) } { 2 } : a \\in A \\big \\} . \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} P _ m t ( \\xi ) = e ( \\xi \\boxtimes \\overline \\rho ) t ( \\xi ) \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} g ( x ) = a x + b - \\frac { s _ 1 } { x - t _ 1 } - \\dots - \\frac { s _ { n - 1 } } { x - t _ { n - 1 } } \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{gather*} f _ m ( n ) = m ^ { n - 1 } + \\sum _ { k = 1 } ^ { n - 1 } \\sum _ { i = k } ^ { n - 1 } \\sum _ { j = 0 } ^ { i - 1 } ( m - 1 ) ^ { i - k } a ^ { i - j } ( a - 1 ) ^ { n - 2 i + j } { i - 1 \\choose k - 1 } { i \\choose j } { n - i - 1 \\choose i - j - 1 } . \\end{gather*}"}
-{"id": "4141.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\lambda } ^ { d } ( x \\in I _ { \\infty } ) = E \\prod _ { i = 0 } ^ { n - 1 } [ \\frac { \\lambda \\rho ( x _ i , \\omega ) \\rho ( x _ { i + 1 } , \\omega ) } { 1 + \\lambda \\rho ( x _ i , \\omega ) \\rho ( x _ { i + 1 } , \\omega ) } ] > 0 . \\end{align*}"}
-{"id": "921.png", "formula": "\\begin{align*} Q ^ { 1 0 } ( \\sigma x _ 2 ) = \\sigma x _ 7 . \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\left . \\partial \\ , ^ k R _ n \\right | _ { \\xi = 1 } = 0 & \\quad & k \\in \\mathcal { N } = \\{ n - p , \\dots , n - 1 \\} \\\\ \\left . \\partial \\ , ^ k R _ m \\right | _ { \\xi = 1 } = 0 & & k \\in \\mathcal { M } = \\{ m - p , \\dots , m - 1 \\} \\end{array} \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} \\Gamma ( N ) : = \\left \\{ \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} : a \\equiv d \\equiv \\pm 1 ( \\ N ) , b \\equiv c \\equiv 0 ( \\ N ) \\right \\} \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi ( t ) & = - ( f ^ { 1 2 7 } + f ^ { 3 4 7 } + f ^ { 5 6 7 } + f ^ { 1 3 5 } - f ^ { 1 4 6 } - f ^ { 2 3 6 } - f ^ { 2 4 5 } ) . \\end{aligned} \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} d _ 2 \\big ( e _ { s ( \\mu ^ { ( j ) } ) } \\otimes e _ { t ( \\mu ^ { ( j ) } ) } \\big ) = \\varrho ( \\mu ^ { ( j ) } ) , \\end{align*}"}
-{"id": "6474.png", "formula": "\\begin{align*} p _ b ^ * ( v _ k ) = e w _ { k e } + \\sum _ { i = 1 } ^ { e - 1 } ( w _ { k e - i } + w _ { k e + i } ) , \\end{align*}"}
-{"id": "9330.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } Y ^ 2 = X ^ 3 + \\left ( \\sum _ { i = 0 } ^ 8 a _ i t ^ i s ^ { 8 - i } \\right ) g _ 2 ^ 2 ( y _ 0 : y _ 1 : y _ 2 : y _ 3 ) X + \\left ( \\sum _ { i = 0 } ^ { 1 2 } b _ i t ^ i s ^ { 1 2 - i } \\right ) g _ 2 ^ 3 ( y _ 0 : y _ 1 : y _ 2 : y _ 3 ) \\\\ g _ 3 ( y _ 0 : y _ 1 : y _ 2 : y _ 3 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} x _ { n } = \\frac { B _ { n } ^ { \\left ( p \\right ) } } { n ! } \\end{align*}"}
-{"id": "5276.png", "formula": "\\begin{align*} \\mathbf { I I } & = \\sum _ i \\int _ { \\partial \\tilde B ^ i _ \\delta } ( c \\cdot x ) ( c \\cdot N ) \\ , d S = O ( \\delta ) . \\end{align*}"}
-{"id": "644.png", "formula": "\\begin{align*} u ( x ) = | x _ { n + 1 } | ^ { 2 s } \\ , q ( x ' ) . \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} \\begin{array} { c c c c } \\mathbf { Q } & = & \\mathbf { L } & \\mathbf { R } \\\\ N _ { \\rm e x t } \\times N _ { \\rm e x t } & & N _ { \\rm e x t } \\times k & k \\times N _ { \\rm e x t } \\end{array} \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} \\dim H ^ 0 ( X , K _ X + t L ) = \\sum _ { j = 0 } ^ n \\binom { t - 1 } { n - j } A _ j ( X , L ) . \\end{align*}"}
-{"id": "7692.png", "formula": "\\begin{align*} Q _ i ( X _ m \\in A ) = \\int _ { X _ m ^ { - 1 } ( A ) } Y _ i \\d Q . \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} \\theta = \\big ( f ^ 2 ( \\alpha ) , f ^ 2 ( \\bar { \\alpha } ) \\big ) \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { q - 1 } k \\binom { 2 k } { k } x ^ k \\equiv x D ( 1 - 4 x ) ^ { ( q - 1 ) / 2 } \\equiv 2 x ( 1 - 4 x ) ^ { ( q - 3 ) / 2 } \\pmod { p } \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} T _ { F , \\widetilde { G } , N } ^ L ( f _ 1 , \\dots , f _ 4 ) \\approx \\frac { 1 } { N ^ 2 } \\sum \\limits _ { \\substack { \\mathbf { n } \\in \\mathbb { Z } ^ 4 \\\\ \\Vert L \\mathbf { n } \\Vert _ \\infty \\leqslant 1 0 - \\delta \\\\ ( \\begin{smallmatrix} 5 & 1 \\end{smallmatrix} ) L \\mathbf { n } = 0 } } \\Big ( \\prod \\limits _ { j = 1 } ^ 4 f _ j ( n _ j ) \\Big ) F ( \\mathbf { n } ) . \\end{align*}"}
-{"id": "7767.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { E [ k \\mbox { - w d } ( W _ n ) ] } { n } = d _ { k } \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} A _ { i j } = \\begin{pmatrix} a _ { i j } & b _ { i j } \\\\ b _ { i j } & a _ { i j } \\end{pmatrix} a _ { i j } b _ { i j } \\in K . \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{align*} S ( X _ { ( 1 ) } ) \\otimes X _ { ( 2 ) } \\otimes S ( Y _ { ( 1 ) } ) \\otimes Y _ { ( 2 ) } & = K _ { a } F _ { a } \\otimes 1 \\otimes E _ { a } K _ { a } ^ { - 1 } \\otimes K _ { a } - K _ { a } F _ { a } \\otimes 1 \\otimes 1 \\otimes E _ { a } \\\\ & - K _ { a } \\otimes F _ { a } \\otimes E _ { a } K _ { a } ^ { - 1 } \\otimes K _ { a } + K _ { a } \\otimes F _ { a } \\otimes 1 \\otimes E _ { a } . \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\big ( \\deg _ { \\widetilde { X } } ( v _ { i } ) - 2 \\big ) & = \\sum _ { i = 1 } ^ { n } \\deg _ { X } ( v _ { i } ) - 2 n \\leq N \\cdot \\deg _ { Y } ( a ) \\cdot \\deg _ { Z } ( b ) - 2 N \\cdot \\deg _ { Y } ( a ) \\\\ & = N \\deg _ { Y } ( a ) \\cdot \\big ( \\deg _ { Z } ( b ) - 2 \\big ) = N \\big ( \\deg _ { \\widetilde { Y } } ( a ) - 2 \\big ) \\big ( \\deg _ { \\widetilde { Z } } ( b ) - 2 \\big ) . \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{align*} \\lim _ { r \\to 0 } r w ( r ) \\leq \\lim _ { r \\to 0 } \\int _ 0 ^ r w ( s ) d s = 0 . \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} ( { \\rm P } _ 1 ) ~ ~ \\max _ { \\mu \\in [ \\underline { \\mu } , \\bar { \\mu } ] } ~ q ( \\mu ) : = \\mu + g ( \\mu ) , \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} R _ { \\alpha \\bar \\beta \\gamma } ^ { \\enskip \\quad \\delta } + R _ { \\bar \\beta \\gamma \\alpha } ^ { \\quad \\enskip \\delta } & = ( \\nabla _ { \\bar \\beta } T ) _ { \\gamma \\alpha } ^ { \\quad \\delta } - T _ { \\bar \\beta , T ( \\gamma , \\alpha ) } ^ { \\quad \\qquad \\delta } . \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} \\bar { d } ( x , y ) : = d ( x , y ) \\vee d ' ( x , y ) . \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { a } \\frac { ( - a - p ) _ { p + k } ^ 2 ( \\beta ) _ { p + k } } { ( 1 ) _ { p + k } ^ 2 ( 1 - \\beta - a - p ) _ { p + k } } \\cdot z ^ { k } = \\frac { ( \\beta ) _ p ( - \\beta - p - a + 1 ) _ a } { ( - \\beta - p - a + 1 ) _ p ( \\beta ) _ a } \\sum _ { k = 0 } ^ { a } \\frac { ( - a - p ) _ k ^ 2 ( \\beta ) _ { k } } { ( 1 ) _ k ^ 2 ( 1 - \\beta - a - p ) _ { k } } \\cdot z ^ { a - k } , \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} \\widetilde { \\chi } ( \\Delta ) = \\sum _ { i = 0 } ^ { d } ( - 1 ) ^ { i - 1 } f _ { i - 1 } \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} \\psi ^ c \\left ( s , \\theta ^ { ( 1 ) } _ 1 , \\ldots , \\theta ^ { ( 1 ) } _ m , \\theta ^ { ( 2 ) } _ 1 , \\ldots , \\theta ^ { ( 2 ) } _ n \\right ) = \\left ( c _ { 1 1 } s \\theta ^ { ( 1 ) } _ 1 \\theta ^ { ( 2 ) } _ 1 , \\ldots , c _ { i j } s \\theta ^ { ( 1 ) } _ i \\theta ^ { ( 2 ) } _ j , \\ldots , c _ { m n } s \\theta ^ { ( 1 ) } _ m \\theta ^ { ( 2 ) } _ n \\right ) \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } d v _ { t } & = & f ( v _ { t } , w _ { t } ) d t + \\sigma d W _ { t } , \\\\ d w _ { t } & = & \\varepsilon g ( v _ { t } , w _ { t } ) d t , \\end{array} \\right . \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} \\frac { d } { d x } \\bigg ( { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a + x & \\beta \\\\ & \\gamma \\end{matrix} \\bigg | \\ , 1 \\bigg ] _ c \\bigg ) \\bigg | _ { x = 0 } \\equiv & \\frac { d } { d x } \\bigg ( { } _ 2 F _ 1 \\bigg [ \\begin{matrix} p - a + x & - b \\\\ & p - c \\end{matrix} \\bigg | \\ , 1 \\bigg ] _ c \\bigg ) \\bigg | _ { x = 0 } \\\\ = & \\frac { \\Gamma ( p - c ) } { \\Gamma ( p - c + b ) } \\cdot \\frac { d } { d x } \\bigg ( \\frac { \\Gamma ( a + b - c - x ) } { \\Gamma ( a - c - x ) } \\bigg ) \\bigg | _ { x = 0 } \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} I ( A : Z | M ) _ { \\hat { \\sigma } _ { A M Z } ( t ) } = S ( A | M ) _ { \\hat { \\sigma } _ { A M } ( t ) } - S ( A | M ) _ { \\hat { \\rho } _ { A M } } \\ ; , \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{gather*} \\frac { { \\rm d } Z } { { \\rm d } z } = A ( z ) Z , \\\\ A ( z ) = \\left ( \\frac { A _ 0 } { z ^ 2 } + \\frac { A _ 1 } { z } + A _ 2 \\right ) \\end{gather*}"}
-{"id": "8352.png", "formula": "\\begin{align*} \\tilde { p } _ { n } ( T , U ) = T ^ { n + 1 } + \\zeta _ 1 T ^ { n } U + \\zeta _ 2 T ^ { n - 1 } U ^ 2 + \\cdot \\cdot \\cdot + \\zeta _ n T U ^ { n } + U ^ { n + 1 } . \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) = \\mathcal { E } ( 0 ) . \\end{align*}"}
-{"id": "9564.png", "formula": "\\begin{align*} \\left | \\mathcal { F } \\left [ f \\cdot S _ { l } g \\right ] \\left ( \\omega \\right ) \\right | ^ { 2 } = \\frac { 1 } { 4 } \\sum _ { ( j , k ) \\in S _ { \\omega } } A _ { k } \\overline { A _ { j } } + e r r o r . \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} \\frac 1 K \\| \\sum _ { i = 1 } ^ n | a _ i | e _ { \\gamma _ i } \\| \\le \\| \\sum _ { i = 1 } ^ n a _ i e _ { \\gamma _ i } \\| \\le K \\| \\sum _ { i = 1 } ^ n | a _ i | e _ { \\gamma _ i } \\| . \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} f _ 0 ( x ) = \\max ( - 2 x + 1 , 2 x - 1 ) . \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} \\int _ { B ( 0 , r ) } \\left \\vert w \\right \\vert d x & = \\int _ { B ( 0 , r ) } \\left \\vert w ( x ) - \\frac { 1 } { | E \\cap B | } \\int _ { E \\cap B } w ( y ) d y \\right \\vert d x \\\\ & \\leq \\frac { 1 } { | E \\cap B | } \\int \\int _ { B \\times E \\cap B } \\left \\vert w ( x ) - w ( y ) \\right \\vert d x d y \\lesssim \\frac { 1 } { | B | } \\int \\int _ { B \\times B } \\left \\vert w ( x ) - w ( y ) \\right \\vert d x d y . \\end{align*}"}
-{"id": "9555.png", "formula": "\\begin{align*} \\vec { b } = \\left ( \\begin{array} { c } b _ { 1 , 1 } , \\dots , b _ { 1 , N } , b _ { 2 , 1 } , \\dots , b _ { K , N } \\end{array} \\right ) ^ T \\in [ 0 , \\infty ) ^ { N K } , \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} \\rho _ d ( t ) : = \\left \\{ \\begin{array} { l l } - \\bar R _ 1 \\dfrac { ( 2 t ) ^ \\frac 3 2 } { 3 \\sqrt { i \\pi } } , & d = 1 , \\\\ & \\\\ \\bar R _ 2 \\left [ \\dfrac { i t } { 2 \\pi } \\log \\left ( \\dfrac { t } { e } \\right ) - \\dfrac { t } { 4 } \\right ] + \\bar R _ 2 ' \\dfrac { i t } { \\pi } , & d = 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} p _ { j } \\left ( \\frac { e ^ { t A } x - x } { t } - A x \\right ) \\leqslant \\frac { 1 } { t } \\sum _ { n = 2 } ^ { \\infty } \\frac { \\big ( t \\ , p ^ { X } _ { j } ( A ) \\big ) ^ { n } } { n ! } p _ { j } ( x ) = \\left ( \\frac { e ^ { t p ^ { X } _ { j } ( A ) } - 1 } { t } - p ^ { X } _ { j } ( A ) \\right ) p _ { j } ( x ) , \\end{align*}"}
-{"id": "9893.png", "formula": "\\begin{align*} h : = d r ^ 2 + \\sigma ^ 2 ( r ) d \\theta ^ 2 + \\sigma ^ 2 ( r ) \\sin ^ 2 ( \\theta ) d \\phi ^ 2 + \\tau ^ 2 ( r ) d z ^ 2 , \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} \\partial _ t f = - v \\partial _ x f + \\partial _ x \\phi [ f ] f ' _ 0 ( v ) + \\partial _ x \\phi [ f ] \\partial _ v f + \\gamma \\partial _ v \\left ( v f + \\partial _ v f \\right ) ~ , ~ \\Delta \\phi = c \\int f d v ~ . \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} { \\varrho _ k } = \\frac { 1 } { p _ { 2 n - 2 } B ( x _ k , x _ k ) } = \\frac { 1 } { p ( x _ k ) } , k = 1 , \\dots , n . \\end{align*}"}
-{"id": "6978.png", "formula": "\\begin{align*} \\Big \\{ t : t \\mbox { o d d a n d } ( t , p ) = 1 \\Big \\} \\cup \\Big \\{ 2 k : k = 1 , \\ldots , \\frac { p ^ k - 1 } { 2 } \\Big \\} \\subseteq S . \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{gather*} \\# \\{ ( i , j ) \\in I \\times ( [ m ] \\setminus I ) : i > j \\} = \\Sigma ( I ) - \\binom { m - k + 1 } { 2 } , \\\\ \\# \\{ ( i ' , j ' ) \\in ( [ n ] ' \\setminus J ' ) \\times J ' : i ' > j ' \\} = k ( n - k ) + \\binom { n - k + 1 } { 2 } - \\Sigma ( J ) . \\end{gather*}"}
-{"id": "7533.png", "formula": "\\begin{gather*} \\left [ ( z \\phi _ 0 z ^ { - 1 } ) ^ * , z \\phi _ { - 1 } z ^ { - 1 } \\right ] = O ( t ^ { - 3 / 2 } \\mathcal L ) \\\\ \\left [ ( z \\phi _ { - 1 } z ^ { - 1 } ) ^ * , z \\phi _ 0 z ^ { - 1 } \\right ] = O ( t ^ { - 3 / 2 } \\mathcal L ) \\end{gather*}"}
-{"id": "6911.png", "formula": "\\begin{align*} u ( x , t ) = \\sum _ { k = 1 } ^ { \\infty } u _ k ( t ) \\sin ( k \\pi x ) , \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} U _ { 2 ( n + l - 1 ) } ^ { ( 2 ) } = q ^ { n + l - 2 } + U _ { n + l } U _ { n + l - 2 } , \\end{align*}"}
-{"id": "9533.png", "formula": "\\begin{align*} g a g ^ { - 1 } = \\alpha _ g ( a ) , \\ \\ \\ a \\in A , \\ g \\in G . \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} [ x _ { i _ { 1 } } ^ { a _ { 1 } } x _ { i _ { 2 } } ^ { a _ { 2 } } \\cdots x _ { i _ { k } } ^ { a _ { k } } ] \\ , f = [ x _ { j _ { 1 } } ^ { a _ { 1 } } x _ { j _ { 2 } } ^ { a _ { 2 } } \\cdots x _ { j _ { k } } ^ { a _ { k } } ] \\ , f . \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{align*} \\rho _ { \\varepsilon } : = \\begin{cases} \\varepsilon ^ { - \\frac { 2 m } { N } } , & { \\rm i n \\ } \\omega _ { \\varepsilon } , \\\\ \\varepsilon ^ { 2 - \\frac { 2 m } { N } } , & { \\rm i n \\ } \\Omega \\setminus \\overline \\omega _ { \\varepsilon } . \\end{cases} \\end{align*}"}
-{"id": "6976.png", "formula": "\\begin{align*} V _ i = \\{ r k + i \\ | \\ k = 0 , \\ldots , q - 1 \\} \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} \\mathbb { E } ( S _ a | { \\cal F } _ { l - 1 } ) = S _ b \\omega . \\end{align*}"}
-{"id": "904.png", "formula": "\\begin{align*} | \\Phi ( x _ 1 , \\ldots , x _ k ) | \\leq p ( \\Phi ) \\frac { 1 } { 1 + | x _ 1 | ^ 2 } \\ldots \\frac { 1 } { 1 + | x _ k | ^ 2 } = p ( \\Phi ) \\theta ( x _ 1 ) \\ldots \\theta ( x _ k ) , \\end{align*}"}
-{"id": "6891.png", "formula": "\\begin{align*} \\frac { P _ { n , j , k } ( y , t ) } { ( 1 - t ) ^ { n + 1 } } = \\sum _ { p = 1 } ^ { \\infty } R _ { n , j , k } ( p , y ) t ^ { p } \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} [ \\Delta ( \\mu ) \\otimes M ] = \\sum _ { \\lambda \\in X ^ + } ( \\sum _ { w \\in W } ( - 1 ) ^ { \\ell ( w ) } \\dim M _ { w \\cdot \\lambda - \\mu } ) [ \\Delta ( \\lambda ) ] . \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{align*} \\eta _ { j , j + 2 } = \\frac { 1 } { \\alpha _ { j + 1 } } \\left [ \\alpha _ { j - 1 } \\eta _ { j - 1 , j + 1 } + \\frac { j + 1 } { \\alpha _ { j } } ( \\beta _ { j } - \\beta _ { j + 1 } ) \\right ] \\end{align*}"}
-{"id": "9997.png", "formula": "\\begin{align*} f ( v ) & = \\sum _ { i = - d } ^ { d } ( \\left \\vert v - i \\right \\vert + 1 ) + \\sum _ { i = - ( k - 1 ) } ^ { k - 1 } ( \\left \\vert v - i \\right \\vert + 2 ) + \\\\ & ( \\left \\vert x ^ { \\prime } - v \\right \\vert + 2 + \\left \\vert - x ^ { \\prime } - v \\right \\vert + 2 ) \\end{align*}"}
-{"id": "6010.png", "formula": "\\begin{align*} A _ { i j } : = ( \\lambda _ { n + 2 } s - t ) \\frac { ( \\lambda _ { n + 3 } - \\lambda _ i ) ( \\lambda _ { n + 3 } - \\lambda _ j ) x _ i x _ j } { ( \\lambda _ { n + 3 } - \\lambda _ { n + 2 } ) ^ 2 x _ { n + 2 } ^ 2 } , \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} Y _ i ( k ) | _ { \\boldsymbol { c } = \\boldsymbol { 0 } } \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} \\dot x = x + \\mu u , \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{align*} V ' ( t ) = - \\Bigl ( \\frac { 4 t } { \\alpha } - \\frac { 8 } { \\alpha } - 2 - t e ^ { - t } \\Bigr ) v ' ( t ) ^ 2 - \\Bigl ( \\frac { 2 ( \\alpha + 1 ) } { \\alpha ^ 2 t ^ 2 } + \\frac { 1 } { 2 \\alpha } e ^ { - t } \\Bigr ) v ( t ) ^ 2 . \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} \\sup _ { f \\in L ^ 1 ( \\Gamma _ - , d \\xi ) \\atop \\| f \\| _ { L ^ 1 ( \\Gamma _ - , d \\xi ) } = 1 } \\Big | \\int _ { \\Gamma _ + } \\phi ( x , v ) j _ + K ^ { n + 2 } ( I - K ) ^ { - 1 } J f ( x , v ) d \\xi ( x , v ) \\Big | \\le C _ { n + 2 } e ^ { ( n + 2 ) \\| \\tau \\sigma _ - \\| _ \\infty } \\| k \\| _ { \\infty } ^ { n + 2 } \\eta ^ { 2 n - 2 } , \\end{align*}"}
-{"id": "6987.png", "formula": "\\begin{align*} \\xi _ { k } = \\gamma _ { k } + \\gamma _ { k + 1 } \\end{align*}"}
-{"id": "72.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial \\tau } = J \\frac { \\partial u } { \\partial t } . \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} \\omega ( p ) = \\sum _ { i = 1 } ^ { \\ell - 1 } \\omega ( u _ i , u _ { i + 1 } ) . \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} \\sum \\limits _ { n \\in S _ 2 ^ * \\cup S _ 3 ^ * } \\left ( \\frac { 1 } { p _ n } + \\frac { 1 } { p _ { - n } } \\right ) ^ 2 & \\leq \\sum _ { n = \\lceil \\gamma R \\rceil + 1 } ^ { \\infty } \\left ( \\frac { \\gamma ^ 2 - 1 } { n } \\right ) ^ 2 \\\\ & \\leq ( \\gamma ^ 2 - 1 ) ^ 2 \\int _ { \\gamma R } ^ { \\infty } \\frac { d x } { x ^ 2 } \\\\ & \\leq \\frac { ( \\gamma ^ 2 - 1 ) ^ 2 } { R } \\\\ & \\leq \\frac { 3 ( \\gamma ^ 2 - 1 ) } { R } . \\end{align*}"}
-{"id": "8625.png", "formula": "\\begin{align*} D _ { n , k } ( 1 , x ) & = k \\ , x \\ , D _ { n - 2 , 1 } ( 1 , x ) + D _ n ( 1 , x ) , \\ , \\ , \\ , \\ , n \\geq 2 \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{align*} \\left \\langle D , \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} \\right \\rangle , \\left \\langle D ' , \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} \\right \\rangle , \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} ( \\omega ) = - Q _ 0 - D + ( q ^ { 2 e + 2 } - q ^ { e + 3 } + 2 q ^ { e + 2 } - q ^ e + q ^ 2 - 1 ) Q _ \\infty . \\end{align*}"}
-{"id": "9913.png", "formula": "\\begin{align*} f _ { k ; 0 } ^ { + \\pm \\pm } = f _ { k ; 0 } ^ { - \\pm \\pm } ( \\stackrel { } { = : } f _ { k ; 0 } ^ { \\circ \\pm \\pm } ) , f _ { k ; - k p } ^ { \\pm + \\pm } = f _ { k ; - k p } ^ { \\pm - \\pm } ( \\stackrel { } { = : } f _ { k ; 0 } ^ { \\pm \\circ \\pm } ) , f _ { k ; k d } ^ { \\pm \\pm + } = f _ { k ; k d } ^ { \\pm \\pm - } ( \\stackrel { } { = : } f _ { k ; 0 } ^ { \\pm \\pm \\circ } ) , \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { \\nu _ 1 = 1 } ^ \\infty \\cdots \\sum _ { \\nu _ r = 1 } ^ \\infty \\frac { \\prod _ { k = 1 } ^ r \\nu _ k ^ { 2 m a } } { \\left [ 1 + \\lambda \\prod _ { k = 1 } ^ r \\nu _ k ^ { 2 m } ( 1 + \\nu _ s ^ 2 ) ^ { - 1 } \\right ] ^ { 2 } } \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\geq \\sum _ { \\nu _ 1 = 1 } ^ \\infty \\cdots \\sum _ { \\nu _ r = 1 } ^ \\infty \\frac { \\prod _ { k = 1 } ^ r \\nu _ k ^ { 2 m a } } { \\left ( 1 + \\lambda \\prod _ { k = 1 } ^ r \\nu _ k ^ { 2 m } \\right ) ^ { 2 } } . \\end{aligned} \\end{align*}"}
-{"id": "36.png", "formula": "\\begin{align*} Z \\left ( T _ { 1 } , \\ldots , T _ { d } , \\mathcal { O } \\right ) : = { \\textstyle \\sum \\nolimits _ { \\underline { n } \\in S } } \\left [ \\mathcal { I } _ { \\underline { n } } \\right ] \\mathbb { L } ^ { - \\left \\Vert \\underline { n } \\right \\Vert } T ^ { \\underline { n } } \\in \\mathcal { M } _ { k } [ \\ ! [ T _ { 1 } , \\ldots , T _ { d } ] \\ ! ] , \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{align*} \\nu ^ { ( i ) } _ 2 ( \\lambda _ 0 , t ) = t ^ d a _ d + O ( t ^ { d + 1 } ) , d = d ^ { ( i ) } ( \\lambda _ 0 ) , a _ d = a _ d ^ { ( i ) } ( \\lambda _ 0 ) \\ne 0 . \\end{align*}"}
-{"id": "1437.png", "formula": "\\begin{align*} \\Lambda = \\{ \\nabla , \\widetilde \\Omega , r \\partial _ r - 1 \\} , \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} W _ { \\widetilde { M } ^ n } ( \\varphi \\circ \\phi ) = W _ { M ^ n } ( \\phi ) . \\end{align*}"}
-{"id": "7865.png", "formula": "\\begin{align*} R _ n ( C ) : = \\sum _ { r = n + 1 } ^ { \\infty } \\frac { T _ r e ^ { - r } } { ( r - 1 ) ! } e ^ { - \\delta r } = \\sum _ { r = n + 1 } ^ { \\infty } \\frac { T _ r C ^ { r - 1 } e ^ { - C r } } { ( r - 1 ) ! } , \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} \\widetilde E = E \\setminus \\{ W _ 0 \\cup F _ 0 \\} , \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} \\left | \\sum _ { x \\in S } e ( \\xi _ x ) \\right | = \\left | \\sum _ { x \\in S } e ( \\overline { \\xi } _ x ) \\right | , \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} \\sum _ { x _ 3 \\in F ^ \\times } \\hat f ^ { ( 3 , 4 ) } ( ( 0 , 0 , x _ 3 , 0 ) , g ) = \\sum _ { x _ 2 \\in F ^ \\times } \\int _ { A ^ { \\oplus 2 } } \\check f ( ( 0 , x _ 2 , y _ 3 , y _ 4 ) , g ) \\ , \\d y _ 3 \\ , \\d y _ 4 , \\end{align*}"}
-{"id": "7969.png", "formula": "\\begin{align*} F ( x , t ) : = \\phi _ 0 g - \\partial _ t E + \\Delta E - h u _ \\infty \\cdot \\nabla E , \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} U H U ^ * \\ ; = \\ ; \\bigoplus _ { j \\in \\frac { 1 } { 2 } \\mathbb { N } } \\ ; \\ ; \\ ; \\bigoplus _ { m _ j = - j } ^ j \\ ; \\bigoplus _ { \\kappa _ j = \\pm ( j + \\frac { 1 } { 2 } ) } \\ ; h _ { m _ j , \\kappa _ j } \\ , , \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{align*} \\frac { { \\partial H ( \\tilde p , \\tilde q ) } } { { \\partial { \\psi _ j } } } = \\frac { { \\partial \\tilde p } } { { \\partial { \\psi _ j } } } \\ln \\frac { { \\tilde p \\left ( { 1 - \\tilde q } \\right ) } } { { \\tilde q \\left ( { 1 - \\tilde p } \\right ) } } - \\frac { { \\tilde p - \\tilde q } } { { \\tilde q \\left ( { 1 - \\tilde q } \\right ) } } \\frac { { \\partial \\tilde q } } { { \\partial { \\psi _ j } } } . \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} c _ { i j } = a _ { i j } - b _ { i j } , \\ 1 \\leq i , j \\leq n \\end{align*}"}
-{"id": "550.png", "formula": "\\begin{align*} Q _ m ( a ) \\coloneqq \\frac { ( - 1 ) ^ m } { 2 ^ m m ! } \\prod _ { k = 1 } ^ { 2 m } \\Big ( a + 1 - \\frac { k } { 2 } \\Big ) \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} \\begin{array} { c c c } ( A u ) _ 1 L ^ k & = & ( u _ { + } k b _ 1 ) \\theta _ { 1 } L ^ { k - 1 } \\\\ ( A u ) _ 2 L ^ k & = & ( u _ { + } k b _ 2 ) \\theta _ { 2 } L ^ { k - 1 } \\\\ & \\vdots \\\\ ( A u ) _ { d - 1 } L ^ k & = & ( u _ { + } k b _ { d - 1 } ) \\theta _ { d - 1 } L ^ { k - 1 } . \\end{array} \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t V - \\mu \\Delta V - \\nabla \\cdot H = \\nabla ^ \\perp \\cdot \\nabla \\cdot \\Delta ^ { - 1 } \\big ( - \\nabla ^ \\perp V \\otimes \\nabla ^ \\perp V + \\nabla ^ \\perp H \\otimes \\nabla ^ \\perp H \\big ) , \\\\ [ - 4 m m ] \\\\ \\partial _ t H - \\nabla V = \\nabla ^ \\perp H \\nabla V , \\\\ \\end{cases} \\end{align*}"}
-{"id": "9783.png", "formula": "\\begin{align*} \\nu = \\lim _ { \\epsilon _ j \\to 0 } \\frac { 1 } { 2 } \\frac { | d ^ * \\zeta _ { \\epsilon _ j } | ^ 2 } { | \\log \\epsilon _ j | } d v _ g . \\end{align*}"}
-{"id": "9649.png", "formula": "\\begin{align*} L ( E , s ) = \\prod _ { p | \\Delta _ E } ( 1 - a _ p p ^ { - s } ) ^ { - 1 } \\cdot \\prod _ { { p \\nmid } \\Delta _ E } ( 1 - a _ p p ^ { - s } + p ^ { 1 - 2 s } ) ^ { - 1 } , \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} U _ { A } ( A _ { n } , B _ { n } ) = \\Pr ( A _ { n } > B _ { n } ) + \\frac { 1 } { 2 } \\Pr ( A _ { n } = B _ { n } ) \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} \\| f \\| _ { A ^ p _ \\alpha } : = \\left ( \\int _ { \\mathbb { D } } | f ( z ) | ^ p \\ , ( \\alpha - 1 ) ( 1 - | z | ^ 2 ) ^ \\alpha \\ , d \\mu ( z ) \\right ) ^ \\frac { 1 } { p } . \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} g \\left ( z \\right ) = \\sum _ { n \\ge 1 } g _ { n } z ^ { n } = \\sum _ { n \\ge 1 } C _ { n - 1 } z ^ { n } = z \\frac { 1 - \\sqrt { 1 - 4 z } } { 2 z } = \\frac { 1 - \\sqrt { 1 - 4 z } } { 2 } , \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} \\sum _ { d = 0 } ^ \\infty \\big ( ( d + 1 ) ^ a - d ^ a \\big ) G _ { d + 1 } ( n ) & = \\sum _ { i = 0 } ^ { a - 1 } \\left ( ( - 1 ) ^ { a - 1 - i } \\binom { a } { i } \\sum _ { d = 0 } ^ \\infty ( d + 1 ) ^ i G _ { d + 1 } ( n ) \\right ) \\\\ & \\rightarrow \\sum _ { i = 0 } ^ { a - 1 } \\left ( ( - 1 ) ^ { a - 1 - i } \\binom { a } { i } \\sum _ { d = 0 } ^ \\infty ( d + 1 ) ^ i G _ { d + 1 } ( \\infty ) \\right ) \\\\ & = \\sum _ { d = 0 } ^ \\infty ( d + 1 ) ^ a \\Pr { \\mathbf { Z } = d + 1 } , \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} g ' ( x ) = \\frac { a P ( x ) } { S ( x ) } , \\tilde { g } ' ( x ) = \\frac { \\tilde { a } P ( x ) } { S ( x ) } . \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{align*} \\chi _ a ( \\mathsf { P } ) : = \\sum _ i c ^ i _ i q ^ { ( \\alpha _ a - 2 \\rho , \\lambda _ i ) } [ d _ a ^ { - 1 } ( \\alpha _ a , \\lambda _ i ) ] _ { q _ a } . \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} \\partial _ t ^ j \\partial _ x ^ n \\partial _ y ^ m J _ 1 ( t , x , y ; \\nu ) = \\sum \\limits _ { l = 1 } ^ { + \\infty } \\frac 1 { 2 \\pi } \\int _ { \\mathbb R } ( i \\theta ) ^ j \\frac { r _ 1 ^ n e ^ { r _ 1 x } - r _ 2 ^ n e ^ { r _ 2 x } } { r _ 1 - r _ 2 } \\widehat \\nu ( \\theta , l ) \\ , d \\theta \\ , \\psi _ l ^ { ( m ) } ( y ) \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} \\pi _ { R ( \\theta ) } A ( \\theta ) = \\pi _ { R ( \\theta ) } A ( \\theta ) \\pi _ { R ( \\theta ) } , & & \\pi _ { N ( \\theta ) } A ( \\theta ) = \\pi _ { N ( \\theta ) } A ( \\theta ) \\pi _ { N ( \\theta ) } . \\end{align*}"}
-{"id": "5273.png", "formula": "\\begin{align*} \\nabla \\cdot A & = u \\cdot \\nabla \\varphi + c \\cdot V - { u - c \\over \\gamma _ n } \\cdot \\nabla \\left ( { m \\cdot x \\over | x | ^ n } \\right ) \\\\ & = | u | ^ 2 - ( u - c ) \\cdot \\left ( V + { 1 \\over \\gamma _ n } \\nabla \\left ( { m \\cdot x \\over | x | ^ n } \\right ) \\right ) . \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{align*} \\abs { z } ^ 2 + 1 = \\frac { 1 } { z _ i } H + z _ i ^ 2 \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{align*} \\int \\vert \\eta , z \\rangle d \\mu ( \\eta , \\bar \\eta , z , \\bar z ) \\langle \\eta , z \\vert = \\sum _ { n = 0 } ^ { 2 j } \\vert n \\rangle \\langle n \\vert , \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} \\binom { n + \\beta - 1 } { n } = \\prod _ { j = 1 } ^ n \\frac { j + \\beta - 1 } { j } & = \\binom { n _ 1 + \\beta - 1 } { n _ 1 } \\prod _ { j = 1 } ^ { n _ 2 } \\frac { n _ 1 + j + \\beta - 1 } { n _ 1 + j } \\\\ & \\leq \\binom { n _ 1 + \\beta - 1 } { n _ 1 } \\binom { n _ 2 + \\beta - 1 } { n _ 2 } , \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{align*} x ' & = \\frac { y ( 1 + x ^ 2 - y ^ 2 ) } { 1 + x ^ 2 + y ^ 2 } \\\\ y ' & = \\frac { - 2 x y ^ 2 } { 1 + x ^ 2 + y ^ 2 } , \\end{align*}"}
-{"id": "9625.png", "formula": "\\begin{align*} A _ l = A ^ { ( j ) } _ n + \\sum ^ n _ { k = 1 } \\bar { \\alpha } _ k g _ k ( l ) , \\ \\ l = j , j + 1 , \\ldots , j + n , \\end{align*}"}
-{"id": "9361.png", "formula": "\\begin{align*} \\varphi ( z ) = h \\psi ( z ) . \\end{align*}"}
-{"id": "3495.png", "formula": "\\begin{align*} B _ { p } ^ { \\left ( \\alpha , \\beta \\right ) } \\left ( x , y \\right ) & = \\int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } { } _ { 1 } F _ { 1 } ( \\alpha ; \\beta ; \\frac { - p } { t ( 1 - t ) } ) d t \\\\ ( R e ( p ) & > 0 , R e ( x ) > 0 , R e ( y ) > 0 , R e ( \\alpha ) > 0 , R e ( \\beta ) > 0 ) . \\end{align*}"}
-{"id": "884.png", "formula": "\\begin{align*} \\eta ^ T _ { \\phi } : = \\frac { 1 } { T } \\sum _ { j \\neq k } \\sigma _ j \\sigma _ k \\langle \\Delta ( x ^ j + \\xi ^ j , x ^ k + \\xi ^ k ; T ) , \\phi \\rangle , \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} f _ \\mathrm { p a r t } ( r ) \\ ; = \\ ; \\Theta _ \\infty ^ { ( g ) } ( r ) \\ , v _ 0 ( r ) + \\Theta _ 0 ^ { ( g ) } ( r ) \\ , v _ \\infty ( r ) \\ , , \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{gather*} t H _ { \\mathrm { K F S } } ^ { \\frac 3 2 + \\frac 3 2 } \\left ( \\theta ^ \\infty _ 1 - \\theta ^ \\infty _ 2 , \\ , 1 - \\theta ^ \\infty _ 1 ; t ; { q _ 1 , p _ 1 \\atop q _ 2 , p _ 2 } \\right ) = t H _ { \\mathrm { I I I } ( D _ 7 ) } \\big ( \\theta ^ \\infty _ 1 - \\theta ^ \\infty _ 2 ; t ; q _ 1 , p _ 1 \\big ) \\\\ \\qquad { } + t H _ { \\mathrm { I I I } ( D _ 7 ) } \\big ( 1 - \\theta ^ \\infty _ 1 ; t ; q _ 2 , p _ 2 \\big ) - p _ 1 q _ 1 p _ 2 q _ 2 - t ( p _ 1 p _ 2 + p _ 1 + p _ 2 ) . \\end{gather*}"}
-{"id": "8881.png", "formula": "\\begin{align*} \\langle Q _ j , Q _ k \\rangle = \\dfrac { [ \\psi ( j + 1 ) - \\psi ( k + 1 ) ] [ 1 + \\cos ( j \\pi ) \\cos ( k \\pi ) ] + \\frac { 1 } { 2 } \\pi \\sin ( ( k - j ) \\pi ) } { ( k - j ) ( j + k + 1 ) } , \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} \\sigma _ { \\mathrm { d } } ^ { ( \\omega ) } \\left ( \\mathbf { x } \\right ) : = \\varrho ^ { ( \\beta , \\omega , \\vartheta , \\lambda ) } \\left ( P _ { \\mathbf { x } } ^ { ( \\omega , \\vartheta ) } \\right ) \\ , \\mathbf { x } \\in \\mathfrak { L } ^ { 2 } \\ . \\end{align*}"}
-{"id": "3377.png", "formula": "\\begin{align*} B _ \\infty & = \\left ( \\frac { 1 } { 2 } - p \\right ) \\sum _ { k = 0 } ^ { \\infty } \\left ( \\frac { 1 - p } { 2 } \\right ) ^ k \\sum _ { j = 0 } ^ k { k \\choose j } ^ 2 \\left ( \\frac { p } { 1 - p } \\right ) ^ { j } \\\\ & = \\left ( \\frac { 1 } { 2 } - p \\right ) \\sum _ { k = 0 } ^ \\infty \\left ( \\frac { 1 } { 2 } - p \\right ) ^ k P _ k \\left ( \\frac { 1 } { 1 - 2 p } \\right ) \\\\ & = \\left ( \\frac { 1 } { 2 } - p \\right ) \\frac { 1 } { \\sqrt { 1 + ( 1 / 2 - p ) ^ 2 - 2 ( 1 / 2 - p ) / ( 1 - 2 p ) } } \\\\ & = 1 . \\end{align*}"}
-{"id": "8675.png", "formula": "\\begin{align*} \\rho \\in R , \\rho \\ge \\lambda _ 2 , \\rho < \\lambda _ 1 , ( \\alpha + 1 ) \\rho > \\sup R , ( \\alpha + 1 ) \\rho \\not = \\lambda _ 1 . \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} \\underline { \\mathbf { E } } _ { \\mathbf { Y } } \\simeq \\mathbb { E } _ { \\mathbf { Y } } , \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} \\begin{aligned} \\Phi [ e ^ { - \\alpha t } w ] & = e ^ { - \\alpha t } ( \\Phi [ w ] - \\alpha w ) \\\\ & \\le e ^ { - \\alpha t } ( C w - \\alpha w ) \\\\ & < 0 . \\end{aligned} \\end{align*}"}
-{"id": "622.png", "formula": "\\begin{gather*} z _ \\alpha : = y _ { p _ { m _ \\alpha } } \\ ; \\textrm { w i t h $ m _ \\alpha = \\min \\Big \\{ h \\ ; : \\ ; y _ { p _ h } \\not \\in \\cup _ { j = 1 } ^ { \\alpha - 1 } B _ { s _ { z _ j } } ( z _ j ) \\Big \\} $ } , \\end{gather*}"}
-{"id": "5871.png", "formula": "\\begin{align*} \\mu = \\frac { N ^ 2 } { V } , \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} S \\left ( \\mathcal { N } ( t ) ( \\hat { \\rho } _ A ) \\right ) \\le S \\left ( \\mathcal { N } ( t ) ( \\hat { \\omega } _ A ) \\right ) = n \\ , g ( E + t ) \\ ; , \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{align*} \\abs { \\nabla ^ 2 f \\otimes \\nabla f + \\nabla f \\otimes \\nabla ^ 2 f } ^ 2 = 2 \\abs { \\nabla ^ 2 f } \\ , \\abs { \\nabla f } + 2 \\abs { \\nabla f [ \\nabla f ] } ^ 2 \\le C \\epsilon \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} Q \\log | f _ t ' | = h _ t - h _ 0 \\circ f _ t \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} p ^ { X } _ { j } ( e ^ { t A } - I ) = p ^ { X } _ { j } \\left ( \\sum _ { n = 1 } ^ { \\infty } \\dfrac { ( t A ) ^ n } { n ! } \\right ) \\leqslant \\sum _ { n = 1 } ^ { \\infty } \\frac { \\big ( t \\ , p ^ { X } _ { j } ( A ) \\big ) ^ { n } } { n ! } = e ^ { t p ^ { X } _ { j } ( A ) } - 1 , \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} \\left \\Vert \\partial _ { t } u \\right \\Vert _ { F } + \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert = 2 l } \\left \\Vert D ^ { \\alpha } u \\right \\Vert _ { F } + \\left \\Vert A u \\right \\Vert _ { F } \\leq C \\left \\Vert f \\right \\Vert _ { F } . \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} f ( s ) = \\frac { \\lambda \\ , g ( s ) } { \\mu + g ( s ) } ( g ' ( s ) \\ge 0 , \\ \\ g ( 0 ) = 0 , \\ \\ \\mu > 0 ) . \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} \\bar { \\varphi } ( u + \\bar { I } ) = \\left \\{ \\begin{array} { c l } c _ { \\alpha } ^ { - 1 } & \\mbox { i f $ u = B _ { \\alpha } $ f o r a n a r r o w $ \\alpha \\in Q _ 1 $ } , \\\\ b _ i c _ { \\alpha } ^ { - 1 } & \\mbox { i f $ u = \\alpha ^ 2 $ f o r s o m e b o r d e r l o o p $ \\alpha \\in Q _ 1 $ } , \\\\ 0 & \\mbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "8383.png", "formula": "\\begin{align*} B _ m : = [ - m , m ] ^ d \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} & \\| g \\| _ { L ^ r ( \\mathbb F _ q ^ d , d \\textbf { m } ) } ^ r = \\sum _ { \\textbf { m } \\in \\mathbb F _ q ^ d } | g ( \\textbf { m } ) | ^ r , \\\\ & \\| f \\| _ { L ^ r ( \\mathbb F _ q ^ d , d \\textbf { x } ) } ^ r = q ^ { - d } \\ , \\sum _ { \\textbf { x } \\in \\mathbb F _ q ^ d } | f ( \\textbf { x } ) | ^ r , \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} z ^ 3 + a z + p = 0 , a \\in \\mathbb R , p = \\varepsilon + i \\theta \\in \\mathbb C . \\end{align*}"}
-{"id": "8252.png", "formula": "\\begin{align*} R _ n = C _ n R _ 1 C _ n ^ { - 1 } . \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} ( c - u ) ^ \\perp \\omega = \\nabla \\left ( \\frac { 1 } { 2 } | u - c | ^ 2 + P + g x _ 2 - \\frac { 1 } { 2 } | c | ^ 2 \\right ) , \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} z = R + P _ { 2 n - 2 p - 1 } , \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} \\frac { d s } { d t } = \\alpha \\beta \\sigma ^ { 2 } \\left ( t \\right ) d t = \\left ( \\alpha \\beta \\right ) ^ { - 1 } \\sigma ^ { - 2 } \\left ( t \\right ) d s . \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{align*} A u = \\frac { 1 } { \\pi i } \\int _ 0 ^ \\infty R _ 0 ^ - ( \\lambda ^ 2 ) v Q D _ 0 Q v \\big [ R _ 0 ^ + ( \\lambda ^ 2 ) - R _ 0 ^ - ( \\lambda ^ 2 ) \\big ] \\lambda \\chi ( \\lambda ) u \\ , d \\lambda \\end{align*}"}
-{"id": "137.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { j = - 1 } ^ { \\infty } | \\alpha _ { j + 1 } - \\alpha _ j | ^ 2 + \\sum _ { j = 0 } ^ { \\infty } \\left [ - \\log ( 1 - | \\alpha _ j | ^ 2 ) - | \\alpha _ j | ^ 2 \\right ] + C \\end{align*}"}
-{"id": "347.png", "formula": "\\begin{align*} \\eta ( C ( \\mathsf { P } _ k ) ) = [ ( \\alpha , \\lambda _ k ) ] _ q = [ k ] _ q . \\end{align*}"}
-{"id": "5788.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\exp \\left ( S ( C | A ' B ' ) _ { \\hat { \\gamma } _ { C A ' B ' } ^ { ( n ) } } - 1 \\right ) = \\eta \\ , a + \\left | 1 - \\eta \\right | b \\ ; , \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} \\varphi = o \\left ( \\frac { 1 } { | x | ^ { n - 2 } } \\right ) , \\textrm { a s } | x | \\to \\infty . \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{gather*} x ^ 3 y - [ 3 ] _ q x ^ 2 y x + [ 3 ] _ q x y x ^ 2 - y x ^ 3 = 0 , \\\\ y ^ 3 x - [ 3 ] _ q y ^ 2 x y + [ 3 ] _ q y x y ^ 2 - x y ^ 3 = 0 . \\end{gather*}"}
-{"id": "7177.png", "formula": "\\begin{align*} u + u ^ { - 1 } = \\tilde { z } + \\tilde { z } ^ { - 1 } \\in k ( P ) , \\end{align*}"}
-{"id": "8606.png", "formula": "\\begin{align*} D _ { p ^ l + 2 , k } ( 1 , x ) & = \\frac { 1 } { 2 } \\ , ( 1 - 4 x ) ^ { \\frac { p ^ l + 1 } { 2 } } + \\frac { k } { 2 } \\ , x \\ , ( 1 - 4 x ) ^ { \\frac { p ^ l - 1 } { 2 } } - \\Big ( 1 - \\frac { k } { 2 } \\Big ) x + \\frac { 1 } { 2 } ; \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} 1 \\le s _ { p q } = \\left \\{ \\begin{array} { l l } \\frac { p q } { p - q } & \\textnormal { i f } 1 \\le q < p < \\infty \\\\ q & \\textnormal { i f } 1 \\le q < p = \\infty \\\\ \\infty & \\textnormal { i f } 1 \\le p \\le q \\le \\infty \\end{array} \\right . . \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} g ( \\chi _ e ) & : = ( q - 1 ) ( q ^ { e + 1 } + q ^ e - q ^ 2 ) / 2 , \\\\ N _ e & : = q ^ { 2 e + 2 } - q ^ { e + 3 } + q ^ { e + 2 } + 1 . \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} 0 & = ( a _ 0 , a _ 1 , \\ldots , a _ { n - 1 } ) \\Sigma E ^ { \\mu + 1 } \\\\ & = ( a _ 0 , a _ 1 , \\ldots , a _ { n - 1 } ) \\left ( \\begin{array} { c | c } 0 & I _ { n - 1 } \\\\ \\hline 1 & 0 \\end{array} \\right ) \\left ( \\begin{array} { c | c } 0 & 1 \\\\ \\hline I _ { n - 1 } & 0 \\end{array} \\right ) \\Sigma E ^ { \\mu + 1 } \\\\ & = ( a _ { n - 1 } , a _ 0 , \\ldots , a _ { n - 2 } ) \\left ( \\begin{array} { c | c } 0 & 1 \\\\ \\hline I _ { n - 1 } & 0 \\end{array} \\right ) \\Sigma E ^ { \\mu + 1 } . \\end{align*}"}
-{"id": "1106.png", "formula": "\\begin{align*} F ( \\mathbf { x } ) = \\int \\limits _ { \\substack { \\boldsymbol { \\xi } \\in \\mathbb { R } ^ m \\\\ \\Vert \\boldsymbol { \\xi } \\Vert _ \\infty \\leqslant X } } c _ X ( \\boldsymbol { \\xi } ) e ( \\frac { \\boldsymbol { \\xi } \\cdot \\mathbf { x } } { N } ) \\ , d \\boldsymbol { \\xi } + O _ { C } \\left ( M N \\frac { \\log X } { X } \\right ) \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{align*} \\nabla ^ \\perp \\cdot H ^ { ( \\alpha , a ) } = f ^ 3 _ { \\alpha a } . \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} 2 H _ { 0 } = \\frac { f _ { 0 } g ^ { \\prime \\prime } } { \\left [ \\left ( f _ { 0 } g ^ { \\prime } \\right ) ^ { 2 } - 1 \\right ] ^ { \\frac { 3 } { 2 } } } . \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} \\| u \\| _ { X _ p } = \\| u \\| _ X + \\sup _ { t > 0 } ( 1 + t ) ^ { \\frac { 1 } { 2 } ( 1 - \\frac { 3 } { p } ) } \\| u ( t ) \\| _ p . \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\widetilde { E } \\big [ \\frac { E \\prod _ { i = 0 } ^ { n - 1 } F ( S _ i , \\widehat { { S } } _ i ; S _ { i + 1 } , \\widehat { S } _ { i + 1 } ) } { ( E \\prod _ { i = 0 } ^ { n - 1 } \\frac { \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } { 1 + \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } ) ^ 2 } \\big ] < + \\infty . \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{align*} \\Lambda _ { N , N + 1 } ( x , d y ) = \\prod _ { i = 1 } ^ { N } \\hat { m } _ s ^ { ( N + 1 ) } ( y _ i ) \\boldmath { 1 } ( y \\in W ^ { N , N + 1 } ( x ) ) d y . \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { a _ n q ^ { 2 n } } { 1 - q ^ n } & = \\frac { 1 } { ( q ; q ) _ { \\infty } } \\sum _ { n \\geq 1 } \\sum _ { k = 1 } ^ { \\lfloor n / 2 \\rfloor } \\left ( s _ o ( n - k , k ) - s _ e ( n - k , k ) \\right ) a _ k \\cdot q ^ n . \\end{align*}"}
-{"id": "1335.png", "formula": "\\begin{align*} g ( z ) = \\frac { a z + b } { c z + d } \\end{align*}"}
-{"id": "1401.png", "formula": "\\begin{align*} \\hat \\sigma _ j ^ 2 = \\hat \\Sigma _ j ^ { - 2 } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\psi _ j ^ 2 ( y _ i , z _ i , \\hat \\beta _ j , \\hat \\eta ^ j ) . \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} L _ { 1 } u = \\sum \\limits _ { i = 0 } ^ { m _ { 1 } } \\varepsilon ^ { \\frac { i } { 2 } } \\alpha _ { i } u ^ { \\left ( i \\right ) } \\left ( 0 , \\varepsilon \\right ) = f _ { 1 } \\left ( \\varepsilon \\right ) , L _ { 2 } u = \\sum \\limits _ { i = 0 } ^ { m _ { 2 } } \\varepsilon ^ { \\frac { i } { 2 } } \\beta _ { i } u ^ { \\left ( i \\right ) } \\left ( T , \\varepsilon \\right ) = f _ { 2 } \\left ( \\varepsilon \\right ) , \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} \\lambda ( U ) : = \\sup \\{ \\| g \\| : g \\in C ( [ 0 , 1 ] ) \\wedge 0 \\leq g \\leq 1 \\wedge ( \\forall x \\in [ 0 , 1 ] \\setminus U ) ( g ( x ) = 0 ) \\} . \\end{align*}"}
-{"id": "3185.png", "formula": "\\begin{align*} ( \\Lambda _ N ^ { \\infty } f ) ( \\omega ) = C o n s t \\times \\int _ { \\mathbb { R } ^ N } ^ { } \\Delta ^ 2 _ N ( x ) F _ { \\omega } ( x _ 1 ) \\cdots F _ { \\omega } ( x _ N ) \\bar { \\hat { f } } ( x ) d x . \\end{align*}"}
-{"id": "1140.png", "formula": "\\begin{align*} \\frac { \\left ( m - s + 1 \\right ) s } { s + 1 } = m + 2 - s - \\frac { m + 2 } { s + 1 } , \\end{align*}"}
-{"id": "7976.png", "formula": "\\begin{align*} & \\| \\widetilde U ( t ) \\| _ r = o ( t ^ { - 1 / 2 + 3 / 2 r } ) , \\quad \\forall r \\in ( 3 , \\infty ] , \\\\ & \\| \\widetilde U ( t ) \\| _ { 3 , \\infty } = o ( 1 ) , \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} B _ K : = \\lim _ { n \\rightarrow \\infty } \\mathbb { P } ( w _ n = K ) = \\int _ { 0 ^ - } ^ { K - 0 } \\sum _ { n = 1 } ^ { \\infty } \\left [ 1 - G ( K - w + n T ) \\right ] d W ( w ) . \\end{align*}"}
-{"id": "3330.png", "formula": "\\begin{align*} N a \\log \\left ( \\frac { N a + N ^ { \\alpha } - 1 } { a \\lambda ( N ^ { \\alpha } + { N } / { \\lambda } ) } \\right ) \\\\ = N a \\log \\left ( 1 + \\frac 1 a ( N ^ { \\alpha - 1 } - N ^ { - 1 } ) \\right ) - N a \\log \\left ( 1 + \\lambda N ^ { \\alpha - 1 } \\right ) . \\end{align*}"}
-{"id": "4206.png", "formula": "\\begin{align*} W _ { k } = \\sum _ { m = 0 } ^ { M - 1 } \\frac { 1 } { \\left ( \\Delta t \\right ) ^ { \\frac { 2 m + 1 } { 2 } \\Delta \\gamma } } \\omega _ { k } \\left ( \\frac { 2 m + 1 } { 2 } \\Delta \\gamma \\right ) \\Delta \\gamma , \\ ; \\ ; k = 0 , 1 , \\ldots , n , \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{align*} 0 = 2 \\left \\langle \\sum _ { x \\in H ^ G } b _ n ( x ) \\theta ^ { ( k ) } ( x ) , v \\right \\rangle = \\sum _ { x \\in H ^ G } \\left \\langle b _ n ( x ) , v - \\pi _ n ( x ) v \\right \\rangle \\theta ^ { ( k ) } ( x ) \\\\ = \\sum _ { x \\in H ^ G } | | b _ n ( x ) | | ^ 2 _ { V _ n } \\theta ^ { ( k ) } ( x ) > 0 , \\end{align*}"}
-{"id": "8300.png", "formula": "\\begin{align*} Z ( k _ 1 , \\dots , k _ n ) = \\det \\Big [ \\mathbb { I } _ { n ! | \\mathcal { E } | ^ n } - E ( k _ n ) & Y _ { n - 1 } ( k _ { n } - k _ { n - 1 } ) \\dots Y _ 1 ( k _ { n } - k _ { { 1 } } ) \\\\ \\left ( \\mathbb { I } _ { n ! } \\otimes S _ v ( k _ { n } ) \\otimes \\mathbb { I } _ { | \\mathcal { E } | ^ { n - 1 } } \\right ) & Y _ 1 ( k _ { 1 } + k _ { n } ) \\dots Y _ { n - 1 } ( k _ { n - 1 } + k _ { n } ) \\Big ] , \\end{align*}"}
-{"id": "9387.png", "formula": "\\begin{align*} \\forall \\ , t \\in \\Omega , { \\phi _ { n + 1 } ( t ) } _ { \\vert E _ n } = \\phi _ { n } ( t ) . \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} D _ E = \\sup _ { I _ 1 , \\cdots , I _ d \\in \\mathcal { J } } \\# \\big \\{ E \\cap ( I _ 1 \\times \\cdots \\times I _ d ) \\big \\} < \\infty , \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} d _ L \\partial _ { \\nu } L = \\alpha _ 0 \\frac { L } { k _ 0 + L } + { \\large { \\copyright _ 1 } } \\ \\ \\omega , \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} w _ \\epsilon ( q , Y ) = - \\epsilon \\frac { q ^ { \\epsilon ^ * } } { q - 1 } \\frac { 1 } { 1 - q ^ { \\epsilon ^ * } Y } . \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} y = ( y \\odot \\lambda _ a ( x ) ) \\oplus x . \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} R ^ e _ n ( s _ { v _ j } ) ( w _ { ( j + 1 ) e } ) = \\sum _ { i = 0 } ^ e w _ { j e + i } \\mbox { a n d } R ^ e _ n ( s _ { v _ j } ) ( w _ { ( j - 1 ) e } ) = \\sum _ { i = 0 } ^ e w _ { j e - i } . \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} M _ J M _ K = \\sum _ L b ^ { L } _ { J , K } M _ L . \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} \\varphi ( T ( a ) ) ( x ) = \\lim _ { j \\in J } \\varphi ( T ( e _ j a ) ) ( x ) = \\lim _ { j \\in J } \\varphi ( T ( e _ j ) a ) ( x ) = \\lim _ { j \\in J } \\varphi ( T ( e _ j ) ) ( \\varphi ( a ) ( x ) ) . \\end{align*}"}
-{"id": "4363.png", "formula": "\\begin{align*} | a _ i | < \\binom { n } { i } ( s + t ) ^ { i / 2 } \\le \\binom { n } { i } n ^ { i / 2 } , i = 1 , 2 , \\ldots , n . \\end{align*}"}
-{"id": "6957.png", "formula": "\\begin{align*} & \\int _ { \\prod _ { k = 1 } ^ r x _ { d - r + k } \\leq ( n b ) ^ { 1 / m } , x _ { d - r + k } \\geq 1 } \\left ( b \\prod _ { k = 1 } ^ r x _ { d - r + k } ^ m - n ^ { - 1 } \\prod _ { k = 1 } ^ r x _ { d - r + k } ^ { 2 m } \\right ) \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\times \\left ( 1 + \\sum _ { j = d - r + 1 } ^ p x _ j ^ 2 \\right ) ^ { - 1 } d x _ { d - r + 1 } \\cdots d x _ { d } \\asymp 1 . \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{align*} \\tilde { f } ^ { \\alpha a } _ 2 = \\sum _ { \\tiny \\begin{matrix} \\beta + \\gamma = \\alpha , b + c = a \\\\ | \\beta | + | b | , | \\gamma | + | c | < | \\alpha | + | a | \\end{matrix} } C _ \\alpha ^ \\beta C _ a ^ b \\nabla ( \\nabla ^ \\perp H ^ { ( \\beta , b ) } \\nabla V ^ { ( \\gamma , c ) } ) . \\end{align*}"}
-{"id": "8961.png", "formula": "\\begin{align*} & \\dot { c } _ 1 ( t ) = ( \\xi \\cos \\theta - y \\sin \\theta ) + \\lambda ( \\xi ( c _ 2 - x _ 2 ) + y ( c _ 1 - x _ 1 ) ) \\\\ & \\dot { c } _ 2 ( t ) = ( \\xi \\sin \\theta + y \\cos \\theta ) + \\lambda ( - \\xi ( c _ 1 - x _ 1 ) + y ( c _ 2 - x _ 2 ) ) \\\\ & \\dot { c } _ 3 ( t ) = z \\ , f ( c ( t ) ) + ( ( \\xi \\sin \\theta + y \\cos \\theta ) + \\lambda ( - \\xi ( c _ 1 - x _ 1 ) + y ( c _ 2 - x _ 2 ) ) ) c _ 1 \\\\ & - ( ( \\xi \\cos \\theta - y \\sin \\theta ) + \\lambda ( \\xi ( c _ 2 - x _ 2 ) + y ( c _ 1 - x _ 1 ) ) ) c _ 2 \\ , . \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} f ( x ) = W ( 0 ) \\lambda \\alpha e ^ { \\lambda \\sigma } e ^ { \\lambda ( \\alpha e ^ { \\lambda \\sigma } - 1 ) x } , 0 < x < K . \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} \\rho c \\frac { \\partial } { \\partial t } T \\left ( x , t \\right ) = - \\frac { \\partial } { \\partial x } q \\left ( x , t \\right ) , \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} g ( \\alpha y , \\alpha ) = \\alpha ^ 2 k ( y , \\alpha ) , \\end{align*}"}
-{"id": "9517.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = 1 } ^ n ( \\sqrt { \\lambda _ i } v _ i ) ^ * x \\tilde { \\alpha } ( \\sqrt { \\lambda _ i } v _ i ) \\right \\| = \\left \\| \\sum _ { i = 1 } ^ n \\lambda _ i v _ i ^ * x u v _ i u ^ * \\right \\| < \\delta , \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{align*} D _ q ( A _ { u _ \\infty } ) = D _ q ( A ) , A _ { u _ \\infty } f = - \\mathbb P [ \\Delta f - u _ \\infty \\cdot \\nabla f ] . \\end{align*}"}
-{"id": "1231.png", "formula": "\\begin{align*} X _ n & = S _ { \\Delta t } ^ { n - \\ell } X _ { \\ell } + \\Delta t S _ { \\Delta t } ^ { n - \\ell } G ( X _ \\ell ) + S _ { \\Delta t } ^ { n - \\ell } e ^ { \\tau A } \\sigma ( X _ \\ell ) \\Delta W _ \\ell \\\\ \\ & ~ + \\Delta t \\sum _ { m = \\ell + 1 } ^ { n - 1 } S _ { \\Delta t } ^ { n - m } G ( X _ m ) + \\sum _ { m = \\ell + 1 } ^ { n - 1 } S _ { \\Delta t } ^ { n - m } e ^ { \\tau A } \\sigma ( X _ m ) \\Delta W _ m , \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} \\rho ^ { \\varepsilon } _ { \\theta } : = ( \\tau ^ { \\varepsilon } _ { \\theta } , H ^ \\varepsilon ) \\delta ^ { \\varepsilon } _ { \\theta } : = ( \\sigma ^ { \\varepsilon } _ { \\theta } , G ^ \\varepsilon ) . \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} I = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & 1 \\\\ \\end{array} \\right ) \\ ; , T = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\\\ \\end{array} \\right ) \\ ; , \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} \\Vert T ^ { n + k } x \\Vert & \\leq \\Vert T ^ { n + k } x _ 0 - T ^ k x _ 0 \\Vert + \\Vert T ^ k x _ 0 - T ^ { n ' + k } z ' \\Vert < \\varepsilon / 2 + \\varepsilon / 2 = \\varepsilon . \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{align*} \\Bigl ( \\exp \\bigl ( - \\rho \\rho _ 0 \\cos ( \\varphi - \\varphi _ 0 ) \\bigr ) \\cdot w _ { ( p _ 1 , p _ 2 ) } ( \\rho , \\varphi ) \\Bigr ) ^ { ( 2 l - 1 ) } _ \\alpha = 0 \\ ; \\mbox { \\rm f o r a l l } \\ ; \\alpha \\in \\Pi \\ ; \\mbox { \\rm a n d } \\ ; 1 \\le l \\le \\mu _ \\alpha , \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} \\chi _ { n } ^ { ( l _ { 1 } , \\ldots l _ { n } ) } ( x _ { 1 } , \\ldots , x _ { n } | \\rho , q ) = \\sum _ { j \\geq 0 } \\frac { \\rho ^ { j } } { ( q ) _ { j } } \\prod _ { k = 1 } ^ { n } h _ { j + l _ { k } } ( x _ { k } | q ) , \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{align*} \\lambda _ { 1 } ( p , \\Omega ) = \\lambda _ 1 ( p , \\Omega _ 1 ) \\le \\lambda _ { 2 } ( p , \\Omega ) , \\lambda _ { 2 } ( p , \\Omega ) = \\lambda _ { 1 } ( p , \\Omega _ { 2 } ) , \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} \\int _ { y \\prec x } ^ { } d y \\prod _ { i = 1 } ^ { N } \\hat { m } ( y _ i ) q _ t ^ { N , N + 1 } \\left ( \\left ( x , y \\right ) , \\left ( x ' , y ' \\right ) \\right ) & = \\det \\left ( { A } _ t ( x , x ' ) _ { i j } \\right ) _ { i , j = 1 } ^ { N + 1 } \\prod _ { i = 1 } ^ { N } \\hat { m } ( y _ i ' ) \\\\ & = \\mathcal { P } ^ { ( N + 1 ) } _ { s } ( t ) ( x , x ' ) \\prod _ { i = 1 } ^ { N } \\hat { m } ( y _ i ' ) . \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} \\alpha = \\liminf _ { k \\to \\infty } H ( x _ k , t _ k ) R ( x _ k , t _ k ) = H R > 0 . \\end{align*}"}
-{"id": "3888.png", "formula": "\\begin{align*} \\sqrt { \\lambda _ n } = n \\pi + \\frac { \\omega + ( - 1 ) ^ n \\omega _ 1 } { n \\pi } + o \\left ( \\frac { 1 } { n } \\right ) , \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} \\Omega ( 0 ) = ( \\alpha + a ) \\cdot \\Upsilon ( 0 ) , \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} \\phi _ 1 ( a _ 1 , a _ 2 , \\ldots , a _ { n - 1 } ) = \\phi _ 2 ( a _ 2 , \\ldots , a _ { n - 1 } ) = \\ldots = \\phi _ { n - 1 } ( a _ { n - 1 } ) . \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} P _ C ( h ) = \\begin{pmatrix} a & b \\\\ & \\\\ c & d \\end{pmatrix} \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{align*} N \\vert n \\rangle = n ~ \\vert n \\rangle . \\end{align*}"}
-{"id": "6341.png", "formula": "\\begin{align*} h ( F ( a _ 1 , \\ldots , a _ n ) ) = f ( h ( a _ 1 ) , \\ldots , h ( a _ n ) ) . \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} c _ \\nu \\ ; & = \\ ; p ^ + { \\textstyle \\Big ( \\frac { \\Gamma ( 2 B ) } { \\Gamma ( B ) } \\ , \\frac { 1 + \\nu + B } { 1 + \\nu } \\Big ) ^ { \\ ! - 1 } } \\\\ d _ \\nu \\ ; & = \\ ; q ^ + { \\textstyle \\Big ( \\frac { \\Gamma ( 2 B ) } { \\Gamma ( B ) } \\ , \\frac { 1 + \\nu + B } { 1 + \\nu } \\Big ) ^ { \\ ! - 1 } } , \\end{align*}"}
-{"id": "9512.png", "formula": "\\begin{align*} w ^ * { } \\lim _ \\beta \\sum _ { j \\in J } x m _ j ^ * x _ 0 \\tilde \\alpha ( m _ j ) = 0 , \\ \\ \\ x \\in M , \\end{align*}"}
-{"id": "2661.png", "formula": "\\begin{align*} \\prod _ { \\ell = 1 } ^ { \\infty } ( 1 + \\epsilon ^ { \\ell } t _ { d \\ell } ) ^ { X _ { \\ell } } = \\sum _ { \\mu } \\binom { X } { \\mu } \\epsilon ^ { 1 \\mu _ 1 + 2 \\mu _ 2 + \\ldots } t _ d ^ { \\mu _ 1 } t _ { 2 d } ^ { \\mu _ 2 } \\ldots = \\sum _ { \\mu } \\binom { X } { \\mu } \\epsilon ^ { \\| \\mu \\| } t _ d ^ { \\mu _ 1 } t _ { 2 d } ^ { \\mu _ 2 } \\ldots . \\end{align*}"}
-{"id": "1444.png", "formula": "\\begin{align*} & r ( \\partial _ r V ^ { ( \\alpha , a ) } + \\partial _ r H ^ { ( \\alpha , a ) } \\cdot \\omega ) \\\\ & = \\omega \\cdot \\Large [ r \\nabla V ^ { ( \\alpha , a ) } + ( r \\partial _ r H ^ { ( \\alpha , a ) } \\cdot \\omega ) \\omega \\Large ] \\\\ & = \\omega \\cdot \\Large [ r \\nabla V ^ { ( \\alpha , a ) } + r \\nabla \\cdot H ^ { ( \\alpha , a ) } \\omega - ( \\Omega H ^ { ( \\alpha , a ) } \\cdot \\omega ^ \\perp ) \\omega \\Large ] , \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} c h g \\big ( | 0 \\rangle \\big ) = 0 ; c h g \\Big ( \\left ( \\chi _ { - j _ k } \\right ) ^ { m _ k } \\dots \\left ( \\chi _ { - j _ 2 } \\right ) ^ { m _ 2 } \\left ( \\chi _ { - j _ 1 } \\right ) ^ { m _ 1 } | 0 \\rangle \\Big ) = \\sum _ { j _ i \\in 2 \\mathbb { Z } + 1 / 2 } m _ i - \\sum _ { j _ i \\in 2 \\mathbb { Z } - 1 / 2 } m _ i . \\end{align*}"}
-{"id": "9531.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = 1 } ^ k a _ i ^ * x \\alpha ( a _ i ) \\right \\| < \\delta . \\end{align*}"}
-{"id": "9977.png", "formula": "\\begin{align*} E _ { \\sigma ( t ) } \\dot x = A _ { \\sigma ( t ) } x \\ , , \\sigma ( t ) \\in \\{ 1 , 2 \\} \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} [ \\widetilde L ] _ s - [ \\widetilde L ] _ u = ( [ L ] \\circ \\tau ) _ s - ( [ L ] \\circ \\tau ) _ u & \\leq ( [ L ] \\circ \\tau ) _ s + \\tau _ s - ( [ L ] \\circ \\tau ) _ u - \\tau _ u \\\\ & = ( [ L ] _ { \\tau _ s } + \\tau _ s ) - ( [ L ] _ { \\tau _ u } + \\tau _ u ) = s - u . \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} S _ 1 = \\frac { \\lambda _ { n + 1 } } { ( \\lambda _ { n + 1 } - \\alpha _ 1 ) } S _ 1 ' - \\frac { 1 } { ( \\lambda _ { n + 1 } - \\alpha _ 1 ) } S _ 1 '' \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{align*} M _ { N + 1 } \\Lambda _ { N } ^ { N + 1 } = M _ N \\ , \\forall N \\ge 1 , \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{align*} 1 ) f ( \\underbar { X } ) & = \\sum \\limits _ { i = 1 } ^ n f _ i ( X _ i ) , \\mbox { o r } \\\\ 2 ) f ( \\underbar { X } ) & = \\prod _ { i = 1 } ^ n f _ i ( X _ i ) \\mbox { a n d } \\ \\prod _ { i = 1 } ^ n \\mathbb { E } [ f _ i ( X _ i ) ] . v a r [ f _ i ( X _ i ) ] \\neq 0 , \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\frac { b _ { k - 1 } ( \\mu ) } { k ! } \\int _ T | C _ Y ( t ) | C _ Y ^ { k - 1 } ( t ) \\ , d t < + \\infty \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} \\frac { 1 } { r ! r ! } | \\psi ^ { r + 1 , r - 1 } | ^ 2 = \\frac { 1 } { ( r - 1 ) ! ( r + 1 ) ! } | \\psi ^ { r , r } | ^ 2 . \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N ^ \\alpha } ( N X _ i ) \\ , N ^ { - \\alpha } \\ , f _ i ( 1 , N ^ { - \\alpha } ) = N ^ { 1 - \\alpha } \\sum _ { i = 1 } ^ { N ^ \\alpha } X _ i \\ , \\omega _ i ( N ^ { \\alpha } ) , \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{align*} A _ N = \\sum _ { \\sum _ i k _ i = N } \\left [ \\frac { y _ { \\# _ 1 } ^ { k _ { \\# _ 1 } } . . . y _ { \\# _ R } ^ { k _ { \\# _ R } } } { R ! } \\binom { R } { \\# _ 1 , . . . \\# _ R } \\sum _ { \\sigma \\in S _ R / \\mathrm { S t a b } ( k _ { r _ 1 } , . . . k _ { r _ l } ) } \\sigma ( k _ 2 ( k _ 2 + k _ 3 ) . . . ( k _ 2 + . . . + k _ R ) ) \\right ] \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} M = \\begin{pmatrix} 4 & 0 & 3 & 0 & 5 \\\\ 0 & 4 & 0 & 3 & 5 \\\\ 1 & 4 & 4 & 0 & 3 \\\\ 4 & 1 & 0 & 4 & 3 \\\\ 3 & 0 & 5 & 0 & 4 \\end{pmatrix} \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { i = 0 } ^ \\infty c _ i \\alpha ^ i \\end{align*}"}
-{"id": "8907.png", "formula": "\\begin{align*} z ( t ) = e _ \\alpha ^ { A t } z ^ 0 + \\int _ { 0 } ^ { t } e _ \\alpha ^ { A ( t - \\tau ) } f ( \\tau ) d \\tau , \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} x _ 1 = - \\frac { n } { 2 y ^ 2 } ~ ~ n \\gg \\lambda y ~ , ~ x _ 1 = - \\frac { \\lambda } { 2 y } ~ ~ n \\ll \\lambda y . \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{align*} 4 s ^ 2 z '' + 4 ( 1 + \\lambda _ 1 + \\lambda _ 2 - 2 \\sigma ) s z ' - z ' + 4 ( \\sigma - \\lambda _ 1 ) ( \\sigma - \\lambda _ 2 ) z + \\frac { \\sigma } { s } z + \\frac { 1 } { s ^ { \\alpha \\sigma } } | z | ^ \\alpha z = 0 . \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} - [ X , \\phi , F , \\alpha _ X ] = [ X , \\phi \\circ \\alpha _ A , - F , - \\alpha _ X ] . \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} d X _ t & = ( r _ t ( X _ t - \\varphi _ t ^ 1 - \\varphi _ t ^ 2 ) + \\varphi _ t ^ 1 \\mu ^ 1 _ t + \\varphi _ t ^ 2 \\mu ^ 2 _ t ) d t + ( \\varphi _ t ^ 1 \\sigma ^ 1 _ t + \\varphi _ t ^ 2 \\sigma ^ 2 _ t ) d W _ t + ( \\varphi _ t ^ 1 \\beta ^ 1 _ t + \\varphi _ t ^ 2 \\beta ^ 2 _ t ) d \\tilde N _ t \\\\ & = ( r _ t X _ t + \\varphi _ t ' ( \\mu _ t - r _ t { \\bf 1 } ) ) d t + \\varphi _ t ' \\sigma _ t d W _ t + \\varphi _ t ' \\beta _ t d \\tilde N _ t , \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} Z ( \\mu , \\nu ) : = \\{ ( x , d ( \\mu ) - d ( \\nu ) , y ) : x \\in Z ( \\mu ) , y \\in Z ( \\nu ) , \\sigma ^ { d ( \\mu ) } ( x ) = \\sigma ^ { d ( \\nu ) } ( y ) \\} . \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} \\inf _ { Q } \\int _ { \\Theta } \\chi ^ 2 ( P _ { \\theta } \\| Q ) d w ( \\theta ) & \\leq \\int _ { \\Theta } \\chi ^ 2 ( P _ { \\theta } \\| P _ { \\theta ^ * } ) d w ( \\theta ) \\\\ & = \\int _ { U ( \\theta ^ * ) } \\chi ^ 2 ( P _ { \\theta } \\| P _ { \\theta ^ * } ) d w ( \\theta ) \\\\ & \\leq \\sup _ { \\theta \\in U ( \\theta ^ * ) } \\chi ^ 2 ( P _ { \\theta } \\| P _ { \\theta ^ * } ) \\leq \\exp ( \\rho ^ 2 t ^ 2 _ { \\theta ^ * } ) - 1 \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} \\frac { \\d F _ { i , \\epsilon } } { \\d F _ \\epsilon } = \\frac { \\d F _ i } { \\d F } \\frac { \\d F _ { i , \\epsilon } / \\d F _ i } { \\d F _ \\epsilon / \\d F } . \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} \\mathfrak { P } _ { t d o } = \\{ \\mathfrak { p } = ( T _ m , \\lambda _ 1 , \\lambda _ 2 , \\dots , \\lambda _ k ) \\ | \\ T _ m - , \\ \\lambda _ 1 > \\lambda _ 2 > \\dots > \\lambda _ k , \\lambda _ i \\in \\frac { 1 } { 2 } + \\mathbb { Z } _ { \\geq 0 } , \\ i = 1 , \\dots , k \\} . \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} { | g ( x ) - g ( y ) | \\over | x - y | ^ \\alpha } \\lesssim { | x | ^ k \\over | x | ^ { \\theta \\alpha } } = 1 . \\end{align*}"}
-{"id": "9621.png", "formula": "\\begin{align*} A _ l = A ^ { ( j ) } _ n + \\sum ^ n _ { k = 1 } \\bar { \\alpha } _ k u _ { k + l - 1 } , \\ \\ l = j , j + 1 , \\ldots , j + n , \\end{align*}"}
-{"id": "9748.png", "formula": "\\begin{align*} \\partial _ t X ( t , s ) = \\partial ^ 2 _ { s s } X ( t , s ) . \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} K _ { 0 } = \\frac { f g f ^ { \\prime \\prime } g ^ { \\prime \\prime } - \\left ( f ^ { \\prime } g ^ { \\prime } \\right ) ^ { 2 } } { \\left [ \\left ( f g ^ { \\prime } \\right ) ^ { 2 } - \\left ( f ^ { \\prime } g \\right ) ^ { 2 } \\right ] ^ { 2 } } , \\end{align*}"}
-{"id": "10053.png", "formula": "\\begin{align*} \\int f \\ , d ( { \\mathcal K } ^ * \\mu ) = \\int { \\mathcal K } f d \\mu \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{align*} V ^ * = \\{ t ^ * _ i : i \\le d \\} \\cup \\{ v ^ * : v \\in V _ 0 \\setminus X \\} . \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} i ( P _ { k , n - 2 } , 1 ) = ( - 1 ) \\left ( \\frac { 1 } { k } - 1 \\right ) + ( - 1 ) \\left ( - \\frac { 1 } { k } ( k + 1 ) ^ { n - 2 } \\right ) = \\frac { ( k + 1 ) ^ { n - 2 } + ( k - 1 ) } { k } . \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{align*} \\eta = Z _ \\eta ^ { - 1 } \\prod _ { j = 1 } ^ { k } [ 1 - \\cos ( \\theta - \\theta _ j ) ] ^ { m _ j } \\frac { d \\theta } { 2 \\pi } , \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} h _ F ( c ) & = \\frac { \\Gamma K } { \\mu K _ 1 } \\frac { 1 } { 1 - b } \\frac { ( 1 + c ) ^ 2 } { ( K + c ) ^ 2 } , \\\\ \\theta _ F ( c ) & = \\frac { \\Gamma } { \\lambda } \\left ( \\frac { c } { K + c } - \\frac { K } { 1 - b } \\frac { ( 1 + c ) ( b + c ) } { ( K + c ) ^ 2 } \\right ) . \\end{align*}"}
-{"id": "9409.png", "formula": "\\begin{align*} \\sum _ k \\| \\Gamma ^ { A , B } ( a _ k ) ( X ) \\| _ 2 ^ 2 & = \\sum _ k \\left \\langle \\Gamma ^ { A , B } ( a _ k ) ( X ) , \\Gamma ^ { A , B } ( a _ k ) ( X ) \\right \\rangle \\\\ & = \\sum _ n \\left \\langle \\Gamma ^ { A , B } ( \\overline { a _ k } ) \\Gamma ^ { A , B } ( a _ k ) ( X ) , X \\right \\rangle \\\\ & = \\left \\langle \\Gamma ^ { A , B } ( \\vert a \\vert ^ 2 ) ( X ) , X \\right \\rangle \\\\ & \\leq \\| \\vert a \\vert ^ 2 \\| _ \\infty \\| X \\| _ 2 ^ 2 = \\| a \\| _ \\infty ^ 2 \\| X \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} \\hat \\gamma = \\theta ( m ( t ) - x ) - y - \\frac { r } { \\lambda } \\ , . \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} I _ \\mathrm { F } = \\sum _ { p = 0 } ^ { M - 1 } \\sum _ { q = 0 } ^ { M - 1 } | \\hat { \\rho } ( p , q ) | ^ 2 = \\sum _ { x = 0 } ^ { M - 1 } \\sum _ { y = 0 } ^ { M - 1 } { \\rho } ^ 2 ( x , y ) . \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{align*} N = [ \\eta _ { i , j } ] _ { i , j \\geq 0 } , \\eta _ { i , j } = \\frac { 1 } { | | P _ i | | ^ 2 } < P _ i , \\frac { d } { d x } P _ j > , \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} \\partial _ z ^ 4 \\zeta = - 1 2 ( \\partial _ z \\zeta ) ( \\partial _ z ^ 2 \\zeta ) . \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} \\dim Z = \\dim ( \\mathbf { s } ) _ 0 = 0 . \\end{align*}"}
-{"id": "511.png", "formula": "\\begin{align*} \\delta _ j = \\sup _ { g \\in L ^ 1 ( \\Gamma _ - ) } W _ { 1 , \\kappa } ( g , g _ j ) \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} { \\bf A } _ 1 = { \\bf G } \\left [ \\begin{array} { c c } { \\bf I } _ r & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf 0 } \\end{array} \\right ] { \\bf G } ' \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} \\int _ { \\R ^ { d } } \\frac { | y | ^ { 2 } } { 1 + | y | ^ { 2 } } M ( x , \\d y ) & = \\int _ { U } \\frac { | c _ { n } ( x , u ) | ^ { 2 } } { 1 + | c _ { n } ( x , u ) | ^ { 2 } } \\nu ( \\d u ) \\le \\int _ { U } \\psi _ { n } ( x ) ^ { 2 } | c ( x , u ) | ^ { 2 } \\nu ( \\d u ) < \\infty . \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} G = I \\oplus K ^ * \\oplus \\mathbb { Q } ^ n \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} ( A u ) _ { d - 1 } f = u _ + \\theta _ { d - 1 } \\left ( g _ { d - 1 } ' + \\sum _ { i = 1 } ^ { d - 2 } \\theta _ i g _ i ' \\right ) . \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{align*} z = \\frac { ( 2 \\ell + 1 ) \\pm \\sqrt { 4 \\ell + 1 } } { 2 \\ell } . \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } t _ l ^ j y _ { j k } = \\delta _ { k l } , k , l = 1 , \\dots , n - 1 , \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} e _ \\chi = \\frac { 1 } { | G | } \\ , \\sum _ { g \\in G } \\chi ( g ^ { - 1 } ) \\ , g . \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } \\binom { 2 k } { k } \\frac { x ^ k } { k ^ 2 } \\equiv 2 \\pounds _ 2 ( \\alpha ) + 2 \\pounds _ 2 ( \\beta ) \\pmod { p } , \\end{align*}"}
-{"id": "5463.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\rho u _ i ^ 2 d x = \\int _ { 2 A _ i \\cap \\Omega } \\rho u _ i ^ 2 d x \\geq \\int _ { A _ i \\cap \\Omega } \\rho u _ i ^ 2 d x = \\int _ { A _ i \\cap \\Omega } \\rho d x \\geq c _ N \\frac { \\int _ { \\Omega } \\rho d x } { 2 j } . \\end{align*}"}
-{"id": "9926.png", "formula": "\\begin{align*} \\hat f ( \\xi ) = \\int d x \\ e ^ { - i \\xi \\cdot x } f ( x ) \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{align*} \\Omega ^ 1 ( D ^ 2 _ { \\ast } , \\C ^ n ) = \\Omega ^ { ( 1 , 0 ) } ( D ^ 2 _ { \\ast } , \\C ^ n ) \\oplus \\Omega ^ { ( 0 , 1 ) } ( D ^ 2 _ { \\ast } , \\C ^ n ) . \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} & \\limsup _ { t \\to + \\infty } \\| \\theta ( t ) \\| _ { L ^ 1 ( B _ { A \\sqrt t } ^ c ) } \\le \\kappa A ^ { - 1 } \\\\ & \\limsup _ { t \\to + \\infty } \\| u ( t ) \\| _ { L ^ 3 ( B _ { A \\sqrt t } ^ c ) } \\le \\kappa A ^ { - 2 } \\qquad \\\\ & \\limsup _ { t \\to + \\infty } \\sqrt t \\| u ( t ) \\| _ { L ^ \\infty ( B _ { A \\sqrt t } ^ c ) } \\le \\kappa A ^ { - 3 } . \\end{align*}"}
-{"id": "9548.png", "formula": "\\begin{align*} \\mu ( t ^ { - 1 } g ^ { - 1 } , g ) = v _ { t ^ { - 1 } g ^ { - 1 } } \\alpha _ { t ^ { - 1 } g ^ { - 1 } } ( v _ g ) \\sigma ( t ^ { - 1 } g ^ { - 1 } , g ) v _ { t ^ { - 1 } } ^ * \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{align*} R ^ e _ n ( s _ { v _ 0 } ) ( p _ b ^ * ( v _ 0 ) ) & = R ^ e _ n ( s _ { v _ 0 } ) \\left ( e w _ 0 + \\sum _ { i = 1 } ^ { e - 1 } ( e - i ) ( w _ { - i } + w _ i ) \\right ) \\left ( \\mbox { b y E q u a t i o n \\ref { b * } } \\right ) \\\\ & = - e \\left ( \\sum _ { i = - ( e - 1 ) } ^ { e - 1 } w _ { i } \\right ) + \\sum _ { i = 1 } ^ { e - 1 } ( e - i ) ( w _ { e - i } + w _ { - ( e - i ) } ) \\left ( \\mbox { b y E q u a t i o n s \\ref { S . 1 } , \\ref { S . 3 } } \\right ) \\\\ & = - \\left ( e w _ 0 + \\sum _ { i = 1 } ^ { e - 1 } ( e - i ) ( w _ { - i } + w _ i ) \\right ) = - p _ b ^ * ( v _ 0 ) . \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} Q _ { r , a } ( ( X _ i ) ) = \\sum _ { \\epsilon \\ , \\in \\{ - 1 , 1 \\} } w _ \\epsilon ( p ^ a , X _ r ) Q _ { r - 1 , a + \\epsilon } \\bigl ( ( p ^ { \\epsilon ^ * ( a + r - i ) } X _ i ) _ i \\bigr ) . \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{gather*} \\sum _ { i = 0 } ^ n \\prod _ { j = 0 } ^ n x _ j ^ { a _ { i , j } } \\end{gather*}"}
-{"id": "7040.png", "formula": "\\begin{align*} \\int \\limits _ { 0 } ^ { 1 } \\left \\Vert \\sum \\limits _ { j = 1 } ^ { m } r _ { j } \\left ( y \\right ) B _ { i } \\left ( \\eta _ { j } , h \\right ) u _ { j } \\right \\Vert _ { E } d y \\leq C \\int \\limits _ { 0 } ^ { 1 } \\left \\Vert \\sum \\limits _ { j = 1 } ^ { m } r _ { j } \\left ( y \\right ) u _ { j } \\right \\Vert _ { E } d y , \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla \\psi | ^ 2 = o ( 1 ) . \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} v _ \\epsilon ( t , 0 ) = 0 \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} \\left | \\sum _ { \\| y \\| _ 1 \\leq R } e ( \\overline { \\xi } _ { x + y } ) \\right | = O \\left ( \\frac { B R ^ 3 } { R _ 2 } \\right ) + \\left | \\sum _ { \\| y \\| _ 1 \\leq R } e ( \\xi ^ i _ { x + y } ) \\right | . \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} T ^ { m + m ' } z = T ^ { m ' } x - \\sum _ { l < 2 ^ { k _ { 0 } } } \\ \\sum _ { j = b _ { l } } ^ { b _ { l + 1 } - 1 } & \\Biggl ( v ^ { ( k ) } \\prod _ { i = j - b _ l + m ' + 1 } ^ { \\Delta ^ { ( k ) } - 1 } w _ { i } ^ { ( k ) } \\Biggr ) ^ { - 1 } \\cdot \\\\ & \\Bigl ( \\ , \\ , \\prod _ { i = 1 } ^ { j - b _ l } w _ { b _ { l } + i } \\Bigr ) ^ { - 1 } \\ , x _ { j } e _ { b _ { 2 ^ { k - 1 } + l } + j - b _ { l } + m ' } . \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} u ( x ) = \\sum _ { k = 0 } ^ { m } p ^ + _ k ( x ^ { \\prime \\prime } ) \\ , \\Psi _ { m - k } ( x \\cdot e , x _ { n + 1 } ) + \\sum _ { k = 0 } ^ { m } p ^ { - } _ k ( x ^ { \\prime \\prime } ) \\ , \\Psi _ { m - k } ( - x \\cdot e , x _ { n + 1 } ) , \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} ( \\partial _ t f ) ( p , t ) = - ( \\Delta ^ 2 f ) ( p , t ) \\ , , \\end{align*}"}
-{"id": "2092.png", "formula": "\\begin{gather*} ( B _ 1 ) _ { 3 2 } = q _ 1 \\left ( - p _ 1 + \\frac { q _ 2 } { t _ 1 } + 1 \\right ) + \\left ( p _ 2 q _ 2 - \\theta ^ \\infty _ 1 - \\frac { t _ 2 } { t _ 1 } \\right ) , \\\\ ( B _ 1 ) _ { 3 3 } = - q _ 2 \\left ( p _ 2 + \\frac { q _ 1 } { t _ 1 } \\right ) - q _ 1 + \\theta ^ \\infty _ 1 - \\theta ^ \\infty _ 2 + \\frac { t _ 2 } { t _ 1 } . \\end{gather*}"}
-{"id": "5318.png", "formula": "\\begin{align*} Y _ \\alpha [ s ] [ x _ \\alpha ^ + \\otimes t ^ { s } , Y _ \\alpha [ s ] ^ { ( r ) } ] & = Y _ \\alpha [ s ] \\partial _ { u _ s } ( Y _ \\alpha [ s ] ^ { ( r - 1 ) } ) \\mod { \\mathcal I } \\\\ & = Y _ \\alpha [ s ] Y _ \\alpha [ s ] ^ { ( r - 2 ) } \\partial _ { u _ s } ( Y _ \\alpha [ s ] ) \\mod { \\mathcal I } \\\\ & = ( r - 1 ) Y _ \\alpha [ s ] ^ { ( r - 1 ) } \\partial _ { u _ s } Y _ \\alpha [ s ] \\mod { \\mathcal { I } } . \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{align*} \\sum _ { n \\leqslant x } \\varphi ( n ) = \\frac { 3 } { \\pi ^ 2 } x ^ 2 + O ( x ( \\log x ) ^ \\frac { 2 } { 3 } ( \\log \\log x ) ^ \\frac { 4 } { 3 } ) , \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} \\begin{cases} s _ { i - j } s _ { n - j } \\leq s _ { i - j + 1 } s _ { n - j - 1 } - s _ { n - 1 } , & \\mbox { i f } j \\mbox { i s e v e n } \\\\ s _ { i - j } s _ { n - j } + s _ { n - 1 } \\leq s _ { i - j + 1 } s _ { n - j - 1 } , & \\mbox { i f } j \\mbox { i s o d d } . \\\\ \\end{cases} \\end{align*}"}
-{"id": "5719.png", "formula": "\\begin{align*} \\hat { \\omega } = \\frac { \\mathrm { e } ^ { - \\beta \\hat { H } } } { \\mathrm { T r } \\ , \\mathrm { e } ^ { - \\beta \\hat { H } } } \\ ; , h = \\beta \\ , I _ { 2 n } \\ ; , \\beta > 0 \\ ; . \\end{align*}"}
-{"id": "4952.png", "formula": "\\begin{align*} S = \\begin{pmatrix} 4 & 3 & 5 \\\\ 5 & 4 & 3 \\\\ 3 & 5 & 4 \\end{pmatrix} C = \\begin{pmatrix} 4 & 3 \\\\ - 3 & 4 \\end{pmatrix} \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} | - t _ k ^ { j , 1 } \\log p _ r - \\theta _ r ^ j - 2 \\pi q | < 2 ^ { - k } ; j = 1 , \\dots , N r = 1 , \\dots , k . \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{align*} x = \\sum _ { i = 1 } ^ n a ^ * _ i ( x ) e _ i , \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} p _ { \\alpha } \\left ( s - \\sum _ { j = 1 } ^ { n } x _ { j } \\right ) \\underset { n \\to \\infty } { \\longrightarrow } 0 . \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} K = T P , \\end{align*}"}
-{"id": "1135.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\sigma , \\alpha , \\gamma } f ( z ) : = \\sup _ { \\substack { I \\subset \\mathbb { R } \\\\ z \\in Q _ I } } \\frac { 1 } { | Q _ I | _ { \\sigma , \\alpha } ^ { 1 - \\frac { \\gamma } { 2 + \\alpha } } } \\int _ { Q _ I } | f ( w ) | \\sigma ( w ) d V _ { \\alpha } ( w ) . \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} \\sum _ { J \\subseteq [ N ] } p _ J ( L ) a _ { J , 2 } - a _ 2 = \\sum _ { J \\subseteq [ N ] } p _ J ( L ) a _ { J , 1 } ^ 2 - a _ 1 ^ 2 , \\end{align*}"}
-{"id": "5422.png", "formula": "\\begin{gather*} \\partial _ { x _ 2 } u _ m = 0 , \\ \\ \\partial _ { x _ 2 } u _ c = 0 \\ \\ x _ 2 = 0 , \\\\ \\partial _ { x _ 2 } u _ m = f _ 1 ( u _ c ) , \\ \\ \\partial _ { x _ 2 } u _ c = 0 \\ \\ x _ 2 = h , \\end{gather*}"}
-{"id": "877.png", "formula": "\\begin{align*} V f ( x ) = \\lim _ { \\delta \\rightarrow 0 _ + } V ^ \\delta f ( x ) ; \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} \\partial _ { t } u _ { m } = i \\left [ \\Delta u _ { m } + \\sum \\limits _ { j = 1 } ^ { N } \\left ( a _ { m j } + b _ { m j } \\left ( x , t \\right ) \\right ) u _ { j } \\right ] , x \\in R ^ { n } , t \\in \\left ( 0 , 1 \\right ) . \\end{align*}"}
-{"id": "3288.png", "formula": "\\begin{align*} h ( x ) = \\sqrt { \\frac { 1 + x ^ 2 } { 2 } } , g ( x ) \\equiv 1 , b ( x ) = ( 1 - N - 2 \\Re ( s ) ) x + 2 \\Im ( s ) , \\alpha = 1 , \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{align*} a ^ 2 - \\frac { 4 \\ell + 3 } { 1 6 a ^ 2 ( \\ell + 1 ) ^ 2 } = \\frac { 2 \\ell + 1 } { 2 ( \\ell + 1 ) } , \\end{align*}"}
-{"id": "1916.png", "formula": "\\begin{align*} J d u = \\sum _ { i , l = 1 } ^ n a _ i P _ { l i } d z _ i + a _ i H _ { i l } d t _ l . \\end{align*}"}
-{"id": "7300.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 5 } { \\alpha _ 1 } y _ 4 + \\theta y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\beta _ 4 } { \\gamma _ 4 } y _ 5 , [ y _ 2 , y _ 3 ] = \\frac { \\gamma _ 2 } { \\gamma _ 4 } y _ 5 , [ y _ 3 , y _ 2 ] = y _ 5 . \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} \\omega _ m - \\omega _ l & = j c \\ ; + \\ ; \\frac { 1 } { 2 } \\left ( \\frac { v _ m } { m } - \\frac { v _ l } { l } \\right ) \\ ; + \\ ; \\frac { c ^ 3 } { 2 \\lambda _ l ^ { 1 / 2 } } - \\frac { c ^ 3 } { 2 \\lambda _ m ^ { 1 / 2 } } + O \\left ( \\frac { 1 } { m ^ 3 } \\right ) + O \\left ( \\frac { 1 } { l ^ 3 } \\right ) , \\end{align*}"}
-{"id": "9558.png", "formula": "\\begin{align*} \\left | \\mathcal { F } \\left [ f \\cdot S _ { l } g \\right ] \\left ( \\omega \\right ) \\right | ^ { 2 } = \\left | \\int _ { - \\infty } ^ { \\infty } \\ ! \\ ! \\ ! f \\left ( t \\right ) g \\left ( t - l \\right ) e ^ { - 2 \\pi i \\omega t } d t \\right | ^ { 2 } = \\left | \\left \\langle f , h \\right \\rangle \\right | ^ { 2 } \\end{align*}"}
-{"id": "10008.png", "formula": "\\begin{align*} \\left \\vert W ^ { i n t } [ \\mathcal { T } _ { n } ] \\right \\vert & \\geq ( W ( G _ { 3 } ( n , d _ { 3 } ^ { \\max } , 1 , s ) ) - W ( G _ { 4 } ( n , d _ { 4 } ^ { \\min } , x _ { 4 } ^ { \\max } , s ) ) ) / 2 = \\\\ & = \\frac { n ^ { 3 } } { 1 2 } - \\frac { \\sqrt { n ^ { 5 } - 3 n ^ { 4 } } } { 2 \\sqrt { 2 } } - n ^ { 2 } + \\frac { 5 } { 3 } \\sqrt { 2 n ^ { 3 } - 6 n ^ { 2 } } + \\frac { 8 3 n } { 1 2 } + \\frac { 1 1 \\sqrt { n - 3 } } { 6 \\sqrt { 2 } } - 1 3 . \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{align*} w _ j = \\begin{cases} 2 & \\quad \\ \\ b _ n < j \\le b _ n + \\delta ^ { ( k ) } \\\\ 1 & \\quad \\ \\ b _ n + \\delta ^ { ( k ) } < j < b _ { n + 1 } - a _ k - l - 2 \\delta ^ { ( k ) } \\\\ 1 / 2 & \\quad \\ \\ b _ { n + 1 } - a _ k - l - 2 \\delta ^ { ( k ) } \\le j < b _ { n + 1 } - a _ k - l - \\delta ^ { ( k ) } \\\\ 2 & \\quad \\ \\ b _ { n + 1 } - a _ k - l - \\delta ^ { ( k ) } \\le j < b _ { n + 1 } - a _ k - l \\\\ 1 & \\quad \\ \\ b _ { n + 1 } - a _ k - l \\le j < b _ { n + 1 } . \\end{cases} \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} E = E _ { - \\lfloor \\frac q 2 \\rfloor } \\oplus \\cdots \\oplus E _ 0 \\oplus \\cdots \\oplus E _ { \\lfloor \\frac r 2 \\rfloor } \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} \\lambda = \\frac { m _ 2 + m _ 4 } { m _ 1 + m _ 2 + m _ 3 + m _ 4 } . \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{align*} U n i f o r m _ { \\mathbb { G T } _ c ( N ) } ^ { x ^ { ( N ) } } ( d x ^ { ( 1 ) } , \\cdots , d x ^ { ( N - 1 ) } ) = \\frac { \\prod _ { j = 1 } ^ { N - 1 } j ! } { \\Delta _ N ( x ^ { ( N ) } ) } \\textbf { 1 } \\left ( x ^ { ( 1 ) } \\prec x ^ { ( 2 ) } \\prec \\cdots \\prec x ^ { ( N - 1 ) } \\prec x ^ { ( N ) } \\right ) d x ^ { ( 1 ) } \\cdots d x ^ { ( N - 1 ) } , \\end{align*}"}
-{"id": "2715.png", "formula": "\\begin{align*} | \\nabla f ( x ) | = \\limsup _ { y \\to x } \\frac { | f ( y ) - f ( x ) | } { | y - x | } . \\end{align*}"}
-{"id": "4345.png", "formula": "\\begin{align*} \\bar B ( z , M ) = \\{ x \\in E : \\phi ( z , x ) \\leq M \\} \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} c ( \\omega ) : = \\left ( { \\beta _ { k } \\lambda _ { k } \\over | \\omega | } + ( 1 + \\beta _ { k } \\lambda _ { k } ) ( \\alpha _ { k - 1 } ( \\omega ) + \\alpha _ { k + 1 } ( \\omega ) ) ^ 2 ( 1 + | \\omega | ) \\right ) c ^ * \\ , , \\end{align*}"}
-{"id": "7072.png", "formula": "\\begin{align*} \\partial _ { s } u = \\left ( a + i b \\right ) \\left [ \\Delta u + A u + Q \\left ( y , s \\right ) \\right ] , y \\in R ^ { n } , s \\in \\left [ 0 , 1 \\right ] \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{align*} & D _ m \\Big \\{ \\big [ v ( t ) , w ( t ) \\big ] ; ( 0 , 0 ) \\Big \\} = \\Bigg [ \\frac { 1 6 \\varepsilon ^ 2 + 3 . 2 \\varepsilon - 0 . 0 9 } { \\varepsilon } w ( t ) ^ 2 \\\\ & + ( 0 . 4 - 1 6 \\varepsilon ) v ( t ) w ( t ) + ( 4 \\varepsilon - 0 . 1 ) v ( t ) ^ 2 \\Bigg ] ^ { 1 / 2 } . \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A u = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "5711.png", "formula": "\\begin{align*} \\hat { D } ( \\mathbf { x } ) : = \\exp \\left ( i \\sum _ { i = 1 } ^ { 2 n } x ^ i \\ , \\Delta ^ { - 1 } _ { i j } \\ , \\hat { R } ^ j \\right ) \\ ; , \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{align*} 0 = [ L _ { - j } , \\ , T _ j ] = [ [ L _ { - j + 1 } , \\ , L _ { - 1 } ] , \\ , T _ j ] = [ [ L _ { - j + 1 } , \\ , T _ j ] , \\ , L _ { - 1 } ] \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} \\varphi ( r ) \\ ; : = \\ ; { \\textstyle \\frac { 1 } { 2 } } ( \\mathbf { A } u ) ( { \\textstyle \\frac { r } { 2 } } ) \\ , e ^ { r / 2 } \\ , , \\mathbf { A } : = \\frac { 1 } { \\sqrt { 2 } } \\begin{pmatrix} 1 & 1 \\\\ 1 & - 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{align*} A _ { i j } = \\sigma _ { i j } - \\nabla ^ 2 _ { i j } f + f \\sigma _ { i j } + O _ { \\sqrt { \\epsilon } } ( \\norm { f } _ { 2 , \\ , p } ) \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} \\begin{aligned} & \\| V ( \\cdot , 0 , \\cdot ) \\| _ { \\widetilde H ^ { ( k + 1 ) / 3 , k + 1 } ( B _ \\delta ) } \\\\ & \\leq c ( T , k , b , L ) \\left ( \\| J ( \\cdot , R , \\cdot ; \\mu ) \\| _ { \\widetilde H ^ { ( k + 1 ) / 3 , k + 1 } ( B _ \\delta ) } + \\| \\partial _ x J ( \\cdot , R , \\cdot ; \\mu ) \\| _ { \\widetilde H ^ { k / 3 , k } ( B _ \\delta ) } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} \\eta ( t _ 0 ) = \\delta \\ , . \\end{align*}"}
-{"id": "9311.png", "formula": "\\begin{align*} | q ( \\hat \\mu ) | < \\left ( 1 - \\frac { L ( 1 - \\hat \\mu ) } { L } \\right ) ^ { k + 1 } = \\hat \\mu ^ { k + 1 } , \\end{align*}"}
-{"id": "3059.png", "formula": "\\begin{align*} ( n - 1 ) a b ^ 2 + a = a + ( n - 1 ) b \\Rightarrow a b = 1 \\end{align*}"}
-{"id": "4284.png", "formula": "\\begin{align*} x ( [ n ] P ) = x - \\frac { \\Psi ^ \\prime _ { n - 1 } ( x , y ) \\Psi ^ \\prime _ { n + 1 } ( x , y ) } { \\left ( \\Psi ^ \\prime _ { n } ( x , y ) \\right ) ^ 2 } ; \\\\ \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} T = \\mathrm { s u p } \\{ t > 0 \\big | [ \\omega _ 0 ] - t ( c _ 1 ( X ) - ( 1 - \\alpha ) [ D ] ) \\} \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} M = \\begin{pmatrix} 4 & 0 & 3 & 0 & 5 \\\\ 0 & 4 & 0 & 3 & 5 \\\\ 1 & 4 & 4 & 0 & 3 \\\\ 4 & 1 & 0 & 4 & 3 \\\\ \\frac { 3 } { 2 } & \\frac { 3 } { 2 } & \\frac { 5 } { 2 } & \\frac { 5 } { 2 } & 4 \\end{pmatrix} \\end{align*}"}
-{"id": "9975.png", "formula": "\\begin{align*} P A _ i ^ { - 1 } E _ i + ( A _ i ^ { - 1 } E _ i ) ^ T P & < 0 , i = 1 , \\dots , N _ 1 \\\\ P A _ j ^ { - 1 } E _ j + ( A _ j ^ { - 1 } E _ j ) ^ T P & \\le 0 , j = N _ 1 + 1 , . . , N \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} \\mathbf { s } \\left ( t \\right ) : = \\underset { ( \\eta , l ^ { - 1 } ) \\rightarrow ( 0 , 0 ) } { \\lim } \\left \\{ \\left ( \\eta ^ { 2 } \\left \\vert \\Lambda _ { l } \\right \\vert \\right ) ^ { - 1 } \\mathbf { S } ^ { ( \\omega , \\eta \\mathbf { A } _ { l } ) } \\left ( t \\right ) \\right \\} \\ . \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} \\xi ( z ) = \\int _ 1 ^ z \\frac { \\rho _ { \\alpha , \\varepsilon } ( s ) } { ( s ^ 2 - 1 ) ^ { 1 / 2 } } d s , z \\in U _ { \\alpha } \\setminus ( - \\infty , 1 ] , \\end{align*}"}
-{"id": "9379.png", "formula": "\\begin{align*} \\Gamma ^ { A , B , C } ( f _ 1 \\otimes f _ 2 \\otimes f _ 3 ) ( X , Y ) = f _ 1 ( A ) X f _ 2 ( B ) Y f _ 3 ( C ) . \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} \\frac { \\partial u _ { m } } { \\partial t } = i \\left [ \\Delta u _ { m } + \\sum \\limits _ { j = 1 } ^ { N } a _ { m j } u _ { j } + \\sum \\limits _ { j = 1 } ^ { N } b _ { m j } u _ { j } \\right ] , x \\in R ^ { n } , t \\in \\left ( 0 , T \\right ) , \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{align*} \\hat { \\phi } ( \\xi ) = \\hat { \\phi } _ { \\omega } ( \\xi ) = F _ { \\omega } ( \\xi ) , \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} ( I - \\tilde K ) u = \\tilde J \\phi . \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} b _ n = \\sum _ { j = 0 } ^ n ( - 1 ) ^ { \\lceil j / 2 \\rceil } b _ { n - G _ j } , \\end{align*}"}
-{"id": "9551.png", "formula": "\\begin{align*} \\beta _ s ( x \\otimes E _ { p , q } ) = \\sigma ( s , p ) ^ * \\alpha _ s ^ { * * } ( x ) \\sigma ( s , q ) \\otimes E _ { s p , s q } . \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} V ( \\gamma , \\epsilon ) : = \\{ ( 1 - q ( C ) - \\gamma ) n \\leq Y _ n ( \\epsilon ) \\leq ( 1 - q ( C ) + \\gamma ) n \\} . \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} f ' ( x ) = \\frac { r _ 1 } { ( x - x _ 1 ) ^ 2 } + \\frac { r _ 2 } { ( x - x _ 2 ) ^ 2 } + \\dots + \\frac { r _ n } { ( x - x _ n ) ^ 2 } = \\frac { c P ( x ) } { Q ( x ) ^ 2 } \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{align*} t _ k = \\frac { | \\ln ( ( 1 + 3 y _ k ^ 2 ) / 4 ) | } { \\sqrt { 3 } y _ k } \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} Q ^ s ( [ 1 ] ) \\circ x = \\sum Q ^ { s + i } ( [ 1 ] \\circ P _ i x ) \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} \\psi ( x ) = & { } _ 7 F _ 6 \\bigg [ \\begin{matrix} - a & - a & 1 - \\frac 1 2 a & - b + x & - c & - d & - e \\\\ & 1 & - \\frac 1 2 a & 1 - a + b - x & 1 - a + c & 1 - a + d & 1 - a + e \\end{matrix} \\bigg | \\ , 1 \\bigg ] \\\\ = & \\frac { ( 1 - a ) _ d ( 1 - a + c + b - x ) _ d } { ( 1 - a + c ) _ d ( 1 - a + b - x ) _ d } \\cdot { } _ 4 F _ 3 \\bigg [ \\begin{matrix} 1 + e & - b + x & - c & - d \\\\ & 1 & 1 - a + e & a - c - d - b + x \\end{matrix} \\bigg | \\ , 1 \\bigg ] \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{align*} M _ 0 J ( k ) T _ 0 T _ 2 = \\begin{bmatrix} k I _ \\mu + o ( k ) & k B _ 0 D _ 0 ^ { - 1 } + o ( k ) \\\\ k C _ 0 A _ 0 ^ { - 1 } + o ( k ) & I _ { n - \\mu } + o ( 1 ) \\end{bmatrix} , \\ ; \\ ; k \\to 0 , \\ ; \\ ; k \\in \\overline { \\mathbb { C } } ^ + . \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{align*} & V ( t ) = \\int _ 0 ^ t e ^ { ( t - \\tau ) \\Delta } ( \\mathbb P _ { \\mathbb R ^ 3 } G ) ( \\tau ) d \\tau , \\\\ & W ( t ) = e ^ { t \\Delta } \\bar v _ 0 + V ( t ) . \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} U _ { t _ k } = S _ { \\Delta t } ^ { k - \\ell - 1 } h + \\Delta t \\sum _ { m = \\ell + 1 } ^ { k - 1 } S _ { \\Delta t } ^ { k - m } B F ' ( X _ m ) . U _ { t _ m } + \\sum _ { m = \\ell + 1 } ^ { k - 1 } S _ { \\Delta t } ^ { k - m } \\bigl ( \\sigma ' ( X _ m ) . U _ { t _ m } \\bigr ) \\Delta W _ m , \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} \\widehat { T } _ { ( \\alpha , k ) } ( h _ 1 + h _ 2 ) = \\widehat { T } _ { ( \\alpha , k ) } ( h _ 1 ) \\cdot \\widehat { T } _ { ( \\alpha , k ) } ( h _ 2 ) . \\end{align*}"}
-{"id": "9457.png", "formula": "\\begin{align*} B ( t , x ) = G \\big ( t , \\Phi ( 0 , t , x ) \\big ) \\exp \\left \\{ - \\int ^ t _ 0 \\nabla \\cdot v \\big ( s , \\Phi ( s , t , x ) \\big ) { \\rm d } s \\right \\} \\end{align*}"}
-{"id": "2434.png", "formula": "\\begin{align*} g ( x ) = G ( 0 ) \\alpha e ^ { \\lambda \\sigma } \\lambda e ^ { - \\lambda x } + \\lambda \\alpha e ^ { \\lambda \\sigma } \\int _ { 0 } ^ x e ^ { - \\lambda ( x - w ) } g ( w ) d w , \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} S = \\left ( s _ { i j } \\right ) C = \\left ( c _ { i j } \\right ) . \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} f _ { T } ( \\mathbf { x } | K _ { n } ) & = \\sum _ { S } ( \\prod _ { 1 \\leq i < j \\leq n } \\left ( \\rho _ { i j } \\right ) ^ { s _ { i j } } ) \\prod _ { m = 1 } ^ { n } T _ { \\sigma _ { m } } ( x _ { m } ) , \\\\ f _ { U } ( \\mathbf { x } | K _ { n } ) & = \\sum _ { S } ( \\prod _ { 1 \\leq i < j \\leq n } \\left ( \\rho _ { i j } \\right ) ^ { s _ { i j } } ) \\prod _ { m = 1 } ^ { n } U _ { \\sigma _ { m } } ( x _ { m } ) , \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} \\psi = \\bigvee \\{ ( \\psi _ m \\wedge \\neg \\psi _ { m + 1 } ) : m \\ge 1 0 ( \\log _ * ( m ) ) \\} \\end{align*}"}
-{"id": "8586.png", "formula": "\\begin{align*} s = f s _ U . \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} ( g ( U ) ) _ x U = \\partial _ x \\Bigl ( \\int _ 0 ^ U g ' ( \\theta ) \\theta \\ , d \\theta \\Bigr ) \\end{align*}"}
-{"id": "6331.png", "formula": "\\begin{align*} A _ R = R ^ \\vee / R . \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} h ^ { ( 2 ) } ( x ) = \\frac { 4 - 2 p x - p ^ { 2 } x ^ { 2 } + 2 p q x ^ { 3 } } { 1 - p x + p q x ^ { 3 } - q ^ { 2 } x ^ { 4 } } . \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{align*} W ( \\ , \\cdot \\ , , \\ , \\cdot \\ , , \\ , \\cdot \\ , , \\nabla f ) = 0 , \\end{align*}"}
-{"id": "9560.png", "formula": "\\begin{align*} \\mathcal { F } \\left [ \\hat { g } \\left ( \\cdot \\right ) e ^ { - 2 \\pi i l \\left ( \\cdot \\right ) } \\right ] \\left ( \\xi \\right ) = \\hat { \\hat { g } } \\left ( \\xi + l \\right ) = g \\left ( - l - \\xi \\right ) , \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{align*} p _ m ( t ) = \\frac { m } { 2 } t ^ 2 - ( m + 1 ) t + \\frac { m } { 2 } . \\end{align*}"}
-{"id": "4485.png", "formula": "\\begin{align*} \\mathfrak { i } _ { \\mathrm { p } } \\left ( t \\right ) : = \\underset { ( \\eta , l ^ { - 1 } ) \\rightarrow ( 0 , 0 ) } { \\lim } \\left \\{ \\left ( \\eta ^ { 2 } \\left \\vert \\Lambda _ { l } \\right \\vert \\right ) ^ { - 1 } \\mathfrak { I } _ { \\mathrm { p } } ^ { ( \\omega , \\eta \\mathbf { A } _ { l } ) } \\left ( t \\right ) \\right \\} \\ . \\end{align*}"}
-{"id": "563.png", "formula": "\\begin{align*} a _ 2 = a _ 3 = \\dots = a _ s = 0 , a _ i \\geq 1 , i = s + 1 , \\dots , n . \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{align*} J = \\left ( \\begin{array} { c c } - b - y & - x \\\\ y & - d + x \\end{array} \\right ) . \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} C = \\begin{pmatrix} 2 c _ { 0 0 } & c _ { 0 1 } & \\cdots & c _ { 0 ( d - 1 ) } \\\\ c _ { 0 1 } & 2 c _ { 1 1 } & \\cdots & c _ { 1 ( d - 1 ) } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ c _ { 0 ( d - 1 ) } & c _ { 1 ( d - 1 ) } & \\cdots & 2 c _ { ( d - 1 ) ( d - 1 ) } \\end{pmatrix} , \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} \\left | \\frac { \\d p _ { k , n } } { \\d v ' _ h } \\right | = \\left | \\frac { k _ h h ^ { - s } } { 2 \\sqrt { 1 + \\lambda _ h / c ^ 2 } } \\right | & \\gtrsim \\ ; \\frac { 1 } { 2 h ^ s \\sqrt { 1 + h ^ { m a x ( s , 2 ) } } } \\geq \\frac { 1 } { 2 N ^ s \\sqrt { 1 + N ^ { m a x ( s , 2 ) } } } \\ ; > \\ ; 0 , \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{align*} 2 ( 1 - \\cos \\theta ) = - ( e ^ { i \\theta / 2 } - e ^ { - i \\theta / 2 } ) ^ 2 \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\log \\left ( \\sqrt { \\frac { t ( X _ { i } ) } { u ( X _ { i } ) } } \\right ) \\quad \\mbox { p a r } \\quad \\sum _ { i = 1 } ^ { n } \\psi \\left ( \\sqrt { \\frac { t ( X _ { i } ) } { u ( X _ { i } ) } } \\right ) , \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} ( k - 1 ) s _ { n - j } s _ { i - j } + \\sum _ { t = 0 } ^ { i - 1 } b _ t \\cdot s _ { t - j } \\leq ( k - 1 ) s _ { n - j - 1 } s _ { i - j + 1 } + \\sum _ { t = 0 } ^ { i - 1 } b _ t \\cdot s _ { t - j + 1 } \\end{align*}"}
-{"id": "3765.png", "formula": "\\begin{align*} c _ j s _ { j - 1 } = c _ { j - 1 } s _ j + \\gcd ( s _ j , s _ { j + 1 } ) \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{align*} \\mathcal { A } _ y = \\mathcal { A } \\otimes _ { O _ \\mathcal { C } } k ( y ) \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} \\ 1 _ V ( i _ * \\tau ) = i _ * ( \\ 1 _ V \\tau ) , \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} \\psi ( x _ 1 , \\dots , x _ n ) = \\sum _ { P \\in \\mathcal { W } _ n } \\mathcal { A } ^ P e ^ { i ( k _ { P ( 1 ) } x _ 1 + \\dots + k _ { P ( n ) } x _ n ) } . \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} \\partial _ t ^ j S ( t , x , y ; u _ 0 ) + \\partial _ t ^ j K ( t , x , y ; f ) = S ( t , x , y ; \\widetilde \\Phi _ j ) + K ( t , x , y ; \\partial _ t ^ j f ) . \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k } \\frac { t ^ 2 } { ( \\epsilon n ) ^ 2 } \\frac { C _ u ^ j } { 1 - C _ u } \\leq \\sum _ { j \\geq 1 } \\frac { t ^ 2 } { ( \\epsilon n ) ^ 2 } \\frac { C _ u ^ j } { 1 - C _ u } = \\frac { t ^ 2 } { \\epsilon ^ 2 n ^ 2 } \\frac { C _ u } { ( 1 - C _ u ) ^ 2 } \\leq D _ 1 \\frac { t ^ 2 } { n ^ 2 } , \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} T ^ i = f ^ i - N ^ i ( L ) . \\end{align*}"}
-{"id": "19.png", "formula": "\\begin{align*} & \\textrm { D e t } = 0 \\implies - R _ { 1 c } - R _ { 1 h } R _ { 3 c } - \\lambda T ' ( c ) = 0 \\\\ & \\implies \\mu \\frac { K _ 1 } { ( 1 + c ) ( 1 + c ^ 2 ) } \\left ( \\frac { 1 - b } { 1 + c } - 2 c ( b + c ) \\right ) + \\lambda T ' ( c ) = \\frac { \\Gamma K } { ( K + c ) ^ 2 } . \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} d ^ c \\psi = J \\psi J ^ { - 1 } \\psi , \\end{align*}"}
-{"id": "3846.png", "formula": "\\begin{align*} \\nu _ 2 ( t _ 5 ( 4 n + j ) ) = 4 \\left \\lceil \\frac { \\nu _ 2 ( n + 1 ) } { 2 } \\right \\rceil - ( \\nu _ 2 ( n + 1 ) \\pmod { 2 } ) , j \\in \\{ 0 , 1 , 2 , 3 \\} , \\\\ \\nu _ 2 ( t _ 9 ( 8 n + j ) ) = 5 \\left \\lceil \\frac { \\nu _ 2 ( n + 1 ) } { 2 } \\right \\rceil - 2 ( \\nu _ 2 ( n + 1 ) \\pmod { 2 } ) , j \\in \\{ 0 , 1 , . . . , 7 \\} . \\end{align*}"}
-{"id": "9502.png", "formula": "\\begin{align*} \\begin{pmatrix} C _ Q & 0 \\\\ [ 6 p t ] 0 & C _ Q \\end{pmatrix} , \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} \\dim \\mathrm { R } = \\dim \\mathrm { P } + \\dim \\mathrm { r } ^ { - 1 } ( S ) = \\binom { c - b + 3 } { 3 } + \\binom { c - a + 3 } { 3 } - \\binom { b - a + 3 } { 3 } - 2 . \\end{align*}"}
-{"id": "1323.png", "formula": "\\begin{align*} 2 \\frac { f p } { \\dot { p } } = \\lambda _ { 3 } \\frac { p ^ { 3 } } { f \\dot { p } } + \\lambda _ { 4 } , \\lambda _ { 4 } \\in \\mathbb { R } . \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} d _ { j , i } = 0 , ~ ~ \\mbox { f o r a l l } ~ j = 1 , \\cdots , k , ~ i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{align*} \\hat { \\sigma } _ { A M | X = \\mathbf { x } } ( t ) = \\hat { D } _ A ( \\mathbf { x } ) \\ , \\hat { \\rho } _ { A M } \\ , { \\hat { D } _ A ( \\mathbf { x } ) } ^ \\dag \\ ; . \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{gather*} \\sum _ { i = 0 } ^ n y _ i ^ d - ( n + 1 ) \\psi \\prod _ { i = 0 } ^ n y _ i ^ { b _ i } . \\end{gather*}"}
-{"id": "10030.png", "formula": "\\begin{align*} g ( \\varphi X , \\varphi Y ) = g ( \\varphi X , Y ) + g ( X , Y ) , \\forall X , Y \\in { \\mathfrak X } ( M ) . \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} \\begin{gathered} S ( t , x , y ; u _ 0 ) \\equiv \\sum _ { l = 1 } ^ { + \\infty } \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb R } \\ , e ^ { i t ( \\xi ^ 3 - b \\xi + \\lambda _ l \\xi ) } e ^ { i \\xi x } \\widehat { u } _ 0 ( \\xi , l ) \\ , d \\xi \\psi _ l ( y ) , \\\\ K ( t , x , y ; f ) \\equiv \\int ^ t _ 0 S ( t - \\tau , x , y ; f ( \\tau , \\cdot , \\cdot ) ) \\ , d \\tau , \\end{gathered} \\end{align*}"}
-{"id": "6082.png", "formula": "\\begin{align*} 0 \\neq [ V _ { - 2 } , \\ , L _ 1 ] = [ V _ { - 2 } , \\ , [ L _ { - 4 } , \\ , L _ 5 ] ] = [ [ L _ { - 4 } , \\ , [ V _ { - 2 } , \\ , L _ 5 ] ] , \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} \\dot { b } = \\mathcal { O } ( ( | \\delta | + | b | ^ 2 ) | b | ) , \\dot { h } = \\mathcal { O } ( | b | ^ 2 ) , \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} - \\frac { \\partial ^ 2 u } { \\partial \\xi ^ 2 } - \\frac { \\partial ^ 2 u } { \\partial y ^ 2 } + 4 c u - 6 u ^ 2 = 0 , ( x , y ) \\in \\mathbb { R } \\times \\mathbb { T } , \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} \\mathbf { e } ( x ) \\coloneqq \\sum _ { n = 0 } ^ { \\infty } \\mathbf { e } _ { n } x ^ { n } \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{align*} [ - i \\partial \\overline { \\partial } \\psi , \\Lambda _ { \\omega } ] \\alpha = \\sum _ { | J | = q } \\left ( \\sum _ { 1 \\leq j \\leq n , j \\not \\in J } \\lambda _ { j } \\right ) \\alpha _ { J } \\overline { \\sigma _ { J } } \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} \\det ( M _ c ) = \\det ( M _ \\infty ) c ^ { n _ 1 + \\cdots + n _ N } + O ( c ^ { n _ 1 + \\cdots + n _ N - 1 } ) . \\end{align*}"}
-{"id": "9557.png", "formula": "\\begin{align*} \\mathcal { F } \\left [ f \\cdot S _ { l } g \\right ] \\left ( \\omega \\right ) = \\int _ { - \\infty } ^ { \\infty } \\ ! \\ ! \\ ! f \\left ( t \\right ) g \\left ( t - l \\right ) e ^ { - 2 \\pi i \\omega t } d t . \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} 1 = m _ 1 < m _ 2 < \\ldots < m _ h < m _ { h + 1 } \\end{align*}"}
-{"id": "9390.png", "formula": "\\begin{align*} \\bigl \\langle [ \\phi ( t ) ] ( x ) , y \\bigr \\rangle = \\bigl \\langle [ \\alpha ( t ) ] ( x ) , [ \\beta ( t ) ] ( y ) \\bigr \\rangle , \\hbox { f o r a . e . } \\ t \\in \\Omega . \\end{align*}"}
-{"id": "9678.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { F } ( s , \\pi , \\psi ) & = \\hat { F } ( T _ \\psi ( s ) , 0 , \\psi ) , \\\\ \\hat { G } ( s , \\pi , \\psi ) & = - \\hat { H } ( T _ \\psi ( s ) , 0 , \\psi ) , \\\\ \\hat { H } ( s , \\pi , \\psi ) & = - \\hat { G } ( T _ \\psi ( s ) , 0 , \\psi ) \\end{aligned} \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} g _ { t , \\ell , s , m } = \\begin{cases} \\sum _ { j = 0 } ^ { \\ell - s } H ( m - 2 \\ell - t j - ( t - 1 ) s ) g _ { t + 1 , \\ell - s , j , m } & m - 2 \\ell \\geq ( t - 1 ) s , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "6229.png", "formula": "\\begin{gather*} x _ { h i } ^ 3 x _ { j k } - [ 3 ] _ q x _ { h i } ^ 2 x _ { j k } x _ { h i } + [ 3 ] _ q x _ { h i } x _ { j k } x _ { h i } ^ 2 - x _ { j k } x _ { h i } ^ 3 = 0 . \\end{gather*}"}
-{"id": "6658.png", "formula": "\\begin{gather*} \\alpha _ 1 \\alpha _ 2 = 0 , \\alpha _ 2 \\alpha _ 3 = 0 , \\alpha _ 3 \\alpha _ 1 = 0 , \\beta _ 1 \\beta _ 3 = 0 , \\beta _ 3 \\beta _ 2 = 0 , \\beta _ 2 \\beta _ 1 = 0 , \\\\ ( \\alpha _ 1 \\beta _ 1 ) ^ { m _ 1 } = ( \\beta _ 3 \\alpha _ 3 ) ^ { m _ 3 } , ( \\beta _ 1 \\alpha _ 1 ) ^ { m _ 1 } = ( \\alpha _ 2 \\beta _ 2 ) ^ { m _ 2 } , ( \\beta _ 2 \\alpha _ 2 ) ^ { m _ 2 } = ( \\alpha _ 3 \\beta _ 3 ) ^ { m _ 3 } . \\end{gather*}"}
-{"id": "4074.png", "formula": "\\begin{align*} { Y _ 2 } \\left ( { \\tilde q , 0 } \\right ) = { Z _ 1 } \\left ( { \\tilde q , 0 } \\right ) - { Z _ 2 } \\left ( { \\tilde q , 0 } \\right ) = 0 . \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} h _ { n } ( x ) = \\left ( \\pi ^ { \\frac { 1 } { 2 } } 2 ^ { n } n ! \\right ) ^ { - \\frac { 1 } { 2 } } \\exp \\left [ - \\frac { x ^ { 2 } } { 2 } \\right ] H _ { n } ( x ) \\end{align*}"}
-{"id": "2872.png", "formula": "\\begin{align*} P = \\sum \\limits _ { \\underline { k } , \\underline { i } \\ , \\ge \\ , \\underline { 0 } } \\alpha _ { \\underline { k } , \\underline { i } } \\ , \\underline { x } ^ { \\underline { i } } \\underline { \\partial } ^ { \\underline { k } } . \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} [ X , \\phi , F , \\alpha _ X ] + [ Y , \\psi , G , \\alpha _ Y ] = [ X \\oplus Y , \\phi \\oplus \\psi , F \\oplus G , \\alpha _ X \\oplus \\alpha _ Y ] , \\end{align*}"}
-{"id": "9996.png", "formula": "\\begin{align*} B _ { 1 } ( n , d _ { 1 } ^ { \\min } , x _ { 1 } ^ { \\max } ) = B _ { 2 } ( n , d _ { 2 } ^ { \\max } , 1 ) . \\end{align*}"}
-{"id": "6345.png", "formula": "\\begin{align*} A _ 0 : = \\{ a \\in A : \\alpha ( a ) = a \\} \\qquad A _ 1 : = \\{ a \\in A : \\alpha ( a ) = - a \\} . \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} \\mathbb { S } ( R ^ { \\bigstar } ) = \\frac { 2 } { d ^ { n + 1 } } \\left [ \\frac { d _ 1 d _ 2 \\dots d _ n } { n + 1 } + \\frac { W ^ { ( n ) } _ 1 } { n } + \\dots + \\frac { W ^ { ( n ) } _ s } { n - s + 1 } + \\dots + \\frac { W ^ { ( n ) } _ { n - 1 } } { 2 } \\right ] \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} \\mathcal { M } ^ { 2 - \\textrm { c o n v } } ( S ^ { n - 1 } \\times S ^ 1 ) = \\textrm { E m b } ^ { 2 - \\textrm { c o n v } } ( S ^ { n - 1 } \\times S ^ 1 , \\mathbb { R } ^ { n + 1 } ) / \\textrm { D i f f } ( S ^ { n - 1 } \\times S ^ 1 ) \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} E : = \\frac { 1 } { n } \\mathrm { T r } _ A \\left [ \\hat { H } _ A \\ , \\hat { \\rho } _ A \\right ] \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} g ( \\mathsf { x } , t , \\mathsf { y } ) = \\sum _ { \\mathsf { n } \\in \\mathbb { N } ^ { d } } \\exp \\left [ - t E _ { \\mathsf { n } } \\right ] \\mathsf { f } _ { \\mathsf { n } } ( \\mathsf { x } ) \\mathsf { f } _ { \\mathsf { n } } ( \\mathsf { y } ) \\end{align*}"}
-{"id": "6073.png", "formula": "\\begin{align*} [ L _ { - 2 } , L _ 1 ] = L _ { - 1 } . \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{align*} k \\widehat { h } ( - k ) [ J _ 1 ( - \\bar { k } ) ^ \\dag ] ^ { - 1 } = k \\widehat { h } ( k ) [ J _ 1 ( \\bar { k } ) ^ \\dag ] ^ { - 1 } , k \\in \\mathbb { R } . \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} { \\displaystyle \\| u \\| _ { L ^ p } \\leq C \\| u \\| _ { { H } ^ 1 } , \\| \\mathbf { v } \\| _ { \\mathbf { L } ^ p } \\leq C \\| \\mathbf { v } \\| _ { \\mathbf { H } ^ 1 } , 1 \\leq p \\leq 6 \\ , \\ , ( d = 2 , 3 ) , } \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} \\psi _ { 1 } ( x ) = { x - 1 \\over x + 1 } \\qquad \\mbox { e t } \\qquad \\psi _ { 2 } ( x ) = { x - 1 \\over \\sqrt { x ^ { 2 } + 1 } } , \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{align*} W _ { C \\bar { C } } & = w _ { 1 , 3 } + w _ { 2 , 4 } \\\\ & = w _ { 1 , 4 } + w _ { 2 , 3 } \\\\ & = W _ { C C } . \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\Big ( ( - 1 ) ^ { n - k } \\big ( \\sum _ { i = 1 } ^ { n + 1 } ( \\lambda _ i - \\lambda _ { n + 2 } ) ( \\lambda _ i - \\lambda _ { n + 3 } ) x _ i ^ 2 \\Gamma _ i ^ k \\big ) s ^ k t ^ { n - k } \\Big ) \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} \\bar { W } _ { n - q } ( K ) = \\left ( \\frac { 1 } { n \\omega _ n } \\int _ { S ^ { n - 1 } } \\rho _ K ^ q ( u ) d u \\right ) ^ { \\frac { 1 } { q } } , \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} { \\mathbf V } ^ { \\sigma , c } | _ { \\sigma ' \\times \\{ 1 , \\ldots , M \\} } & = { \\mathbf V } ^ { \\sigma ' , c ' } { \\mathbf E } ^ { \\sigma ' , \\sigma , c ' , c } . \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} [ L _ { - r } , \\ , S _ { r - 2 } ] = L _ { - 2 } \\end{align*}"}
-{"id": "6456.png", "formula": "\\begin{align*} B _ N = \\sum _ { \\sum _ i l _ i = N } ( - 1 ) ^ { n - 1 } \\frac { \\binom { N } { l _ 1 , . . . , l _ n } v _ { l _ 1 } . . . v _ { l _ n } } { n } \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{align*} \\partial _ t U = A \\Delta U = F ( U ) , \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} \\phi ( x , v ) : = e ^ { \\int _ 0 ^ { \\tau _ - ( x , v ) } ( \\sigma _ 1 - \\sigma _ 2 ) ( x - s v , v ) d s - { \\tau _ - ( x , v ) \\over \\tau ( x , v ) } \\int ^ { \\tau _ + ( x , v ) } _ { - \\tau _ - ( x , v ) } ( \\sigma _ 1 - \\sigma _ 2 ) ( x + s v , v ) d s } , \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} J ( \\phi ) = \\int _ 0 ^ 1 \\int _ X \\dot { \\phi _ t } ( \\omega ^ n - \\omega ^ n _ { \\phi _ t } ) \\frac { 1 } { n ! } \\wedge d t , \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{gather*} \\sigma ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) : = ( x _ 1 , x _ 0 , x _ 2 , x _ 3 ) , \\tau ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) : = ( - x _ 1 , - x _ 0 , x _ 2 , x _ 3 ) . \\end{gather*}"}
-{"id": "7664.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } c _ 1 \\\\ \\vdots \\\\ c _ n \\end{array} \\right ) = A \\left ( \\begin{array} { c } h _ 1 \\\\ \\vdots \\\\ h _ n \\end{array} \\right ) . \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{align*} \\Delta & \\leq \\frac { 1 } { 2 ^ { 2 k } } ( 2 ^ { 2 k } \\mathbb { E } [ Y ] ^ 2 + d ^ 2 ) - \\mathbb { E } [ Y ] ^ 2 = \\frac { d ^ 2 } { 2 ^ { 2 k } } . \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{align*} \\chi _ { 3 2 } = 2 \\sqrt { 3 } \\nu _ 2 y ^ 2 / ( 1 - 3 x ^ 2 ) , \\end{align*}"}
-{"id": "7262.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = \\theta _ 1 y _ 5 , [ y _ 1 , y _ 2 ] = y _ 3 , [ y _ 2 , y _ 1 ] = - y _ 3 + \\theta _ 2 y _ 4 + \\theta _ 3 y _ 5 , [ y _ 2 , y _ 2 ] = \\theta _ 4 y _ 4 + \\theta _ 5 y _ 5 , \\\\ [ y _ 1 , y _ 3 ] = y _ 4 = - [ y _ 3 , y _ 1 ] , [ y _ 2 , y _ 3 ] = y _ 5 = - [ y _ 3 , y _ 2 ] . \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} y _ i = \\left \\{ \\begin{array} { c c } 0 & i \\in U \\\\ r _ i & \\end{array} \\right . \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{align*} x ~ = ~ \\frac { p + p ' y } { q + q ' y } \\end{align*}"}
-{"id": "6755.png", "formula": "\\begin{align*} \\frac { d u } { d t } + O _ { \\varepsilon } u = F u + P f , u \\left ( 0 \\right ) = 0 , t > 0 , F u = - P \\left ( u , \\nabla \\right ) u . \\end{align*}"}
-{"id": "5628.png", "formula": "\\begin{gather*} \\sum _ { s = 1 } ^ { n } l _ { s } ( ( j + 1 ) \\alpha _ { s } + t _ { s } \\alpha _ { s } ) \\allowbreak + \\allowbreak \\sum _ { s = n + 1 } ^ { n + k } l _ { s } ( j \\alpha _ { s } + t _ { s } \\alpha _ { s } ) \\allowbreak \\\\ = \\allowbreak j \\sum _ { s = 1 } ^ { n + k } l _ { s } \\alpha _ { s } + \\sum _ { s = 1 } ^ { n } l _ { s } ( t _ { s } + 1 ) \\alpha _ { s } + \\sum _ { s = n + 1 } ^ { n + k } l _ { s } t _ { s } \\alpha _ { s } . \\end{gather*}"}
-{"id": "1552.png", "formula": "\\begin{align*} c _ k = \\sum _ { j = 1 } ^ n \\varrho _ j x _ j ^ k , k = 0 , \\dots , 2 n - 2 . \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} [ a _ n , a _ m ] = n \\delta _ { m + n , 0 } \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} T ^ { m \\ , \\textrm { p e r } ( x _ l ) } \\ , \\Bigl ( \\ , \\sum _ { i < j } z _ { i } \\Bigr ) - y _ j = T ^ { m \\ , \\textrm { p e r } ( x _ l ) } \\Bigl ( \\sum _ { i < j } z _ i - y _ j \\Bigr ) . \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} \\tilde { \\rho } & = \\rho - n \\cdot d d ^ c f , \\\\ \\tilde { r } & = r - \\Lambda ( d d ^ c f ) \\cdot F , \\\\ \\tilde { \\sigma } & = \\sigma - d d ^ c f . \\end{align*}"}
-{"id": "7947.png", "formula": "\\begin{align*} \\mbox { d i v $ \\mathbb B f $ } = f \\mbox { i f } \\ ; \\ ; \\int _ D f ( x ) \\ , d x = 0 , \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} C : = \\frac { 1 } { 1 2 \\left ( b ^ 2 + ( 2 \\sqrt { 8 4 } ) b ^ { 3 / 2 } + 8 4 b \\right ) + 3 2 } . \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} h = - d x _ 0 ^ 2 + d x _ 1 ^ 2 + d x _ 2 ^ 2 + d x _ 3 ^ 2 + d x _ 4 ^ 2 + d x _ 5 ^ 2 , \\end{align*}"}
-{"id": "7459.png", "formula": "\\begin{align*} Z ( [ x , y ] ) = \\sum _ { \\lambda \\in \\mathrm { s u p p } ( a ) } ( 1 + \\lambda i ) X ( [ x _ \\lambda , y _ \\lambda ] ) \\end{align*}"}
-{"id": "9541.png", "formula": "\\begin{align*} \\alpha _ r ( \\sigma ( s , t ) ) \\sigma ( r , s t ) = \\sigma ( r , s ) \\sigma ( r s , t ) , \\ \\ \\ r , s , t \\in G , \\end{align*}"}
-{"id": "6145.png", "formula": "\\begin{align*} [ [ L _ { - r + 1 } , \\ , L _ { - 2 } ] , \\ , S _ r ] = [ L _ { - r + 1 } , \\ , [ L _ { - 2 } , \\ , S _ r ] ] \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} \\Phi ( \\mathbf { x } ) : = \\sum \\limits _ { i = 1 } ^ { h - m } x _ i \\mathbf { v } ^ { \\mathbf { ( i ) } } , \\end{align*}"}
-{"id": "9439.png", "formula": "\\begin{align*} \\int \\frac { \\big | \\widetilde { U } ( t ) \\big | ^ 2 } { 2 h ( t ) } + \\int ^ { t } _ { 0 } e ^ { r ( t - s ) } \\left [ \\int \\frac { \\widetilde { W } ^ { { \\rm T } } Q _ r ( w ^ * ) \\widetilde { W } } { 2 h } + \\widetilde { W } \\cdot \\mathrm { L } ( w ^ * ) \\right ] { \\rm d } s = e ^ { r t } \\int \\frac { \\big | \\widetilde { U } ( 0 ) \\big | ^ 2 } { 2 h ( 0 ) } . \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} u _ t + b u _ x + u _ { x x x } + u _ { x y y } + u u _ x = f ( t , x , y ) \\end{align*}"}
-{"id": "2918.png", "formula": "\\begin{align*} \\mu ( C ) = \\pmb \\mu \\Bigl ( \\Bigl \\{ ( x _ n ) _ { n \\ge 1 } \\in \\mathbf H ; \\ ; \\sum _ { n = 1 } ^ \\infty x _ n \\in C \\Bigr \\} \\Bigr ) . \\end{align*}"}
-{"id": "993.png", "formula": "\\begin{align*} f _ * E ^ n = ( - 1 ) ^ { n - 1 } [ t ^ n ] \\left ( \\prod _ { i = 1 } ^ d \\frac { t Z _ i } { 1 + t Z _ i } \\right ) = ( - 1 ) ^ { d + 1 } h _ { n - d } ( Z _ 1 , \\cdots , Z _ d ) Z _ 1 \\cdots Z _ d , \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} H _ s ( m , n ) = \\frac { 1 } { n } \\sum _ { j = 1 } ^ n \\omega ^ { ( 1 - s ) j } ( \\omega ^ j + 1 ) ^ m , \\enskip s = 1 , . . . , n ; \\end{align*}"}
-{"id": "8887.png", "formula": "\\begin{align*} { \\bf B } _ 5 = \\begin{pmatrix} 3 2 & 0 & 4 1 4 7 2 & 0 & 1 6 2 0 0 0 0 \\\\ 0 & 3 8 7 2 & 0 & 3 5 5 5 5 2 & 0 \\\\ 3 8 7 2 & 0 & 8 0 3 5 2 & 0 & 2 0 2 4 3 5 2 \\\\ 0 & 8 0 3 5 2 & 0 & 7 3 7 7 9 2 & 0 \\\\ 3 5 5 5 5 2 & 0 & 7 3 7 7 9 2 & 0 & 4 2 2 0 0 0 0 \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "5010.png", "formula": "\\begin{align*} \\tilde { u } ( \\xi , y , t ) = \\left ( b ( t ) e ^ { i y } + \\bar { b } ( t ) e ^ { - i y } \\right ) \\psi _ * ( \\xi ) + v ( \\xi , y , t ) , \\end{align*}"}
-{"id": "9833.png", "formula": "\\begin{align*} R _ { 1 2 } R _ { 1 3 } R _ { 2 3 } = R _ { 2 3 } R _ { 1 3 } R _ { 1 2 } , \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} ( \\nabla \\dot { c } ^ { v } ) _ { ( c ( t ) , \\dot { c } ( t ) ) } = 0 . \\end{align*}"}
-{"id": "3736.png", "formula": "\\begin{align*} Q _ { \\tilde { E } } ( z ) = Q _ { m - n } ^ 2 ( z ) Q _ E ( z ) . \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} Q ( x ) = B c ^ { n - 2 } x ^ 2 - A c ^ { n - 1 } + A \\cdot O ( c ^ { n - 2 } ) + O ( c ^ { n - 3 } ) \\end{align*}"}
-{"id": "5678.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } \\Theta _ { i j } ( t ) = \\mu ( t ) P _ { i j } ( t ) , \\\\ \\Theta _ { i j } ^ { + } ( t ) = \\frac { 1 } { \\mu ( t ) } P _ { i j } ( t ) , \\end{array} \\right . \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\liminf _ n \\hat { F } _ n \\right ) = 1 . \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} \\nu = \\max \\limits _ { t } \\sum _ { \\underset { s = 1 , 2 } { i = 1 } } ^ n | u _ { s i } ^ \\varepsilon ( t ) | = 2 \\sqrt { 2 \\pi } \\varepsilon ^ { - 1 / 2 } \\sum \\limits _ { i = 1 } ^ n \\sqrt { { k _ i } } . \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & j \\\\ 1 & - j \\end{pmatrix} \\begin{pmatrix} 0 & j \\\\ - j & 0 \\end{pmatrix} \\begin{pmatrix} 1 & j \\\\ 1 & - j \\end{pmatrix} ^ { - 1 } = \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} \\mathbf { \\Xi } _ { \\mathrm { p } } \\left ( t \\right ) : = \\underset { l \\rightarrow \\infty } { \\lim } \\mathbb { E } \\left [ \\Xi _ { \\mathrm { p } , l } ^ { ( \\omega ) } \\left ( t \\right ) \\right ] \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} h : = R e s ^ { \\mathcal { G } } _ { \\mathcal { G } ^ \\prime } ( f ) - g . \\end{align*}"}
-{"id": "9732.png", "formula": "\\begin{align*} A _ { t } \\left ( n \\right ) = A _ { t } \\otimes _ { K } K \\left ( < 1 > \\bot n \\times T _ { P } \\right ) , n \\in \\mathbb { N } - \\{ 0 \\} . \\end{align*}"}
-{"id": "7679.png", "formula": "\\begin{align*} b _ E = \\tfrac 1 { | G _ E | } \\sum _ { i = 0 } ^ { m - 1 } \\sigma _ m ^ i E . \\end{align*}"}
-{"id": "6634.png", "formula": "\\begin{align*} g \\ ; = \\ ; a _ 0 ^ { ( g ) } \\ , v _ 0 + a _ \\infty ^ { ( g ) } \\ , v _ \\infty + b _ \\infty ^ { ( g ) } \\ , v _ 0 + b _ 0 ^ { ( g ) } \\ , v _ \\infty \\ , , \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{align*} V _ k ( \\theta ) = \\frac { 1 } { k } \\cos ( k \\theta ) \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{align*} h _ { i j } y ^ { i } = 0 , ~ \\ \\ h ^ { i j } y _ { i } = 0 . \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} & \\left | \\int _ 0 ^ t \\langle f , w \\rangle d \\tau \\right | \\\\ & \\leq \\frac { 1 } { 4 } \\int _ 0 ^ t \\| \\nabla w \\| _ 2 ^ 2 d \\tau + C \\int _ 0 ^ t \\frac { d \\tau } { 1 + \\tau } + C \\int _ 0 ^ { T _ \\varepsilon } \\tau ^ { - 1 / 2 } d \\tau + 2 \\varepsilon \\int _ { T _ \\varepsilon } ^ t \\tau ^ { - 1 / 2 } d \\tau \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} \\begin{aligned} & \\arg \\underset { \\theta } { \\max } & & \\log p ( y | \\theta ) & & & \\theta \\in \\Theta , \\end{aligned} \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{align*} \\bigg ( \\frac { \\partial } { \\partial \\eta _ 1 } \\bigg ) ^ { k + 1 - l } \\bigg ( \\frac { \\partial } { \\partial \\eta _ 2 } \\bigg ) ^ { l } = 0 ; l = 0 , 1 , 2 , \\cdots , k + 1 , \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} F ( v ) = f \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N } \\big | c _ { k } ^ 3 \\big | \\le C \\Delta t ^ { \\frac 1 2 - \\kappa } ( 1 + | x | _ { L ^ { \\max ( p , 2 q ) } } ) ^ { K + 1 } \\int _ { 0 } ^ { T } \\bigl ( 1 + \\frac { 1 } { ( T - t ) ^ { 1 - \\kappa } } \\bigr ) d t . \\end{align*}"}
-{"id": "3836.png", "formula": "\\begin{align*} b _ { 2 ^ { k } - 1 } ( n ) = \\sum _ { j = 0 } ^ { n } t _ { n - j } b _ { 2 ^ { k } } ( j ) , \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} \\psi _ * ( \\xi ) = { \\rm s e c h } ^ 3 ( \\sqrt { c _ * } \\xi ) , \\eta _ * ( \\xi ) = \\int _ { - \\infty } ^ { \\xi } { \\rm s e c h } ^ 3 ( \\sqrt { c _ * } \\xi ' ) d \\xi ' , \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} f : = e ^ s ( 2 c - ( \\alpha n _ 1 + \\beta n _ 2 ) c - u \\cdot \\nabla c ) \\in L ^ p ( s _ 0 , t ; L ^ p ( \\Omega ) ) . \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} \\sigma ( x ) = \\dfrac { 2 } { n } \\ln J ( x ) . \\end{align*}"}
-{"id": "3544.png", "formula": "\\begin{align*} a ( 0 ) = a _ { 0 } \\ ; ; \\ ; \\dot { a } ( 0 ) = \\dot { a } _ { 0 } \\ ; ; \\ ; b ( 0 ) = b _ { 0 } \\ ; ; \\ ; \\dot { b } ( 0 ) = \\dot { b } _ { 0 } \\ ; ; \\ ; \\phi ( 0 ) = \\phi _ { 0 } \\ ; ; \\ ; \\dot { \\phi } ( 0 ) = \\dot { \\phi } _ { 0 } \\ ; ; \\ ; \\rho ( 0 ) = \\rho _ { 0 } . \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\P ( \\eta _ n \\geq k ) = \\P ( \\eta \\geq k ) \\leq ( \\P ( \\eta \\geq 1 ) ) ^ k < 1 . \\end{align*}"}
-{"id": "5998.png", "formula": "\\begin{align*} \\nabla ^ 2 _ { \\vec { x } } w + k ^ 2 _ 0 w = 0 \\mbox { i n } \\ F ^ { ( \\alpha ) } . \\end{align*}"}
-{"id": "9803.png", "formula": "\\begin{align*} { \\ell _ l } \\left ( s \\right ) = \\prod \\limits _ { i = 0 ; i \\ne l } ^ N { \\frac { { s - { s _ i } } } { { { s _ l } - { s _ i } } } } \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{align*} \\bar { A } _ { n , j , k } ( t , - z ) = ( - 1 ) ^ { n } t ^ { n + 1 } A _ { n , j , k } ( 1 / t , - z ) . \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{align*} q _ { i , i } - \\sum _ { j = 1 , j \\neq i } ^ { N } | q _ { i , j } | & = \\sum _ { j = 1 } ^ { N } q _ { i , j } \\\\ & = \\sum _ { j = 1 } ^ { N } \\left ( w _ { i , j } + w _ { i + 1 , j + 1 } - w _ { i , j + 1 } - w _ { i + 1 , j } \\right ) \\\\ & = w _ { i , 1 } - w _ { i + 1 , 1 } + w _ { i + 1 , N } - w _ { i , N } \\\\ & \\geq 0 , \\end{align*}"}
-{"id": "1924.png", "formula": "\\begin{align*} P _ { q } ^ { n } ( e _ { 1 } \\odot \\dots \\odot e _ { n } ) = \\sum _ { \\sigma \\in S _ { n } } q ^ { i ( \\sigma ) } e _ { \\sigma ( 1 ) } \\odot \\dots \\odot e _ { \\sigma ( n ) } , \\end{align*}"}
-{"id": "4597.png", "formula": "\\begin{align*} \\Vert \\nabla { \\theta } ^ { k } _ { \\phi } \\Vert _ { \\mathbf { L } ^ { 2 } } ^ { 2 } \\leq C h ^ { 2 r } + C \\Vert \\nabla { \\theta } ^ { k } _ { \\Psi } \\Vert _ { \\mathbf { L } ^ { 2 } } ^ { 2 } , k = 0 , 1 , \\cdots , m . \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{align*} \\sum _ k \\mathsf { P } _ { k } ^ { i } \\mathsf { P } _ { j } ^ { k } = \\sum _ { m , n , o , p } c _ { n } ^ { m } c _ { p } ^ { o } \\sum _ { k } ( \\mathsf { M } _ { m } ^ { n } ) _ { k } ^ { i } ( \\mathsf { M } _ { o } ^ { p } ) _ { j } ^ { k } = \\sum _ { m , n , p } c _ { n } ^ { m } c _ { p } ^ { n } ( \\mathsf { M } _ { m } ^ { p } ) _ { j } ^ { i } . \\end{align*}"}
-{"id": "71.png", "formula": "\\begin{align*} \\dim R ( \\Sigma _ g ; \\mathcal P _ { \\Sigma _ g } ) = 3 ( g - 1 ) = \\frac { 1 } { 2 } \\dim R ( H ^ i _ g ; \\mathcal P _ { H ^ i _ g } ) \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{align*} U n i f o r m _ { \\mathbb { G T } _ c ( N ) } ^ { x ^ { ( N ) } } ( d x ^ { ( 1 ) } , \\cdots , d x ^ { ( N - 1 ) } ) = \\Lambda ^ N _ { N - 1 } ( x ^ { ( N ) } , d x ^ { ( N - 1 ) } ) \\cdots \\Lambda _ 2 ^ { 3 } ( x ^ { ( 3 ) } , d x ^ { ( 2 ) } ) \\Lambda _ 1 ^ { 2 } ( x ^ { ( 2 ) } , d x ^ { ( 1 ) } ) . \\end{align*}"}
-{"id": "9957.png", "formula": "\\begin{align*} E _ i \\dot x = A _ i x \\ , ; \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} H ^ p _ { - } ( \\R ) = \\{ f - i H ( f ) : f \\in L ^ p ( \\R ) \\} . \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{align*} x _ t ' : = \\exp _ x \\left ( - B ^ N _ t N + B ^ T _ t T + A _ t Z \\right ) . \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{align*} F _ 1 ( \\mathbf u ^ { ( l ) } ) = - \\sum _ { k = 1 } ^ { n } \\Big ( ( \\alpha _ k ) ^ { l } \\sum _ { j = 1 } ^ { n } \\big ( \\frac { ( \\lambda _ { n + 1 } - \\lambda _ j ) ( \\lambda _ { n + 2 } - \\lambda _ j ) ( \\lambda _ { n + 3 } - \\lambda _ j ) } { ( \\lambda _ j - \\alpha _ 1 ) ( \\lambda _ j - \\alpha _ k ) ( \\lambda _ { n + 1 } - \\alpha _ k ) } x _ j ^ 2 \\big ) \\Big ) . \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{align*} \\sigma ( x ) = \\sqrt { 2 ( 1 + x ^ 2 ) } , b ( x ) = 2 \\Im ( s ) - 2 \\Re ( s ) x , H ( x , y ) = 2 ( 1 + x y ) \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{align*} f _ c ( x + \\lambda ) = 2 c ^ { n - 1 } x ( 1 - x ) + O ( c ^ { n - 2 } ) . \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{align*} q _ { i , j } & = w _ { i , j } + w _ { i + 1 , j + 1 } - w _ { i + 1 , j } \\\\ & = ( 1 - p ) p _ { j + 1 } - p p _ i \\\\ & \\leq 0 , \\end{align*}"}
-{"id": "5172.png", "formula": "\\begin{align*} g _ { n } = - \\frac { 1 } { \\left ( n + 1 \\right ) ! } , \\thinspace \\thinspace g \\left ( z \\right ) = 1 - \\frac { e ^ { z } - 1 } { z } , \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} { } _ { 2 } F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\beta \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] = ( 1 - z ) ^ { - \\alpha } \\cdot { } _ { 2 } F _ 1 \\bigg [ \\begin{matrix} \\alpha & 1 - \\beta \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac { z } { z - 1 } \\bigg ] . \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{align*} \\check \\beta _ j : = \\frac { \\hat \\Sigma _ j ^ { - 1 } } { n } \\sum _ { i = 1 } ^ n ( z _ { i j } - z _ { i , - j } ^ T \\hat \\mu ^ j ) ( y _ i - z _ i ^ T \\hat \\beta _ { - j } ) - ( \\hat \\mu ^ j ) ^ T \\hat \\Gamma _ { - j , - j } \\hat \\beta _ { - j } \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{align*} I _ b ( x , t ) = \\int _ \\Omega b ( t ) | u _ t | ^ { p + 2 } d x , \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{align*} \\xi _ t ^ T = \\frac { 1 } { T } \\sum _ { j \\neq k } \\sigma _ j \\sigma _ k \\langle \\Lambda ( x ^ j + \\xi ^ j , x ^ k + \\xi ^ k ; T ) , \\mathbf { 1 } _ { [ 0 , t ] } \\rangle , t \\geq 0 , \\end{align*}"}
-{"id": "4098.png", "formula": "\\begin{align*} \\nu ( 1 _ N \\otimes \\mu ) = 0 \\mbox { a n d } \\mu ( 1 _ M \\otimes \\nu ) = 0 . \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} \\phi ( \\rho ) = \\begin{cases} 1 & \\mathrm { i f } 0 \\leq \\rho \\leq 1 , \\\\ 0 & \\mathrm { i f } \\rho \\geq 2 . \\end{cases} \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} R \\circ \\alpha = \\alpha \\circ R R \\circ \\beta = \\beta \\circ R . \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} L _ { - i } = S \\otimes \\langle x _ 1 ^ { k _ 1 } \\dots x _ n ^ { k _ n } , \\ , k _ 1 + \\dots \\ , + k _ n = i \\rangle , \\end{align*}"}
-{"id": "8838.png", "formula": "\\begin{align*} Z ( \\Gamma ) = \\sum _ { \\delta \\in K } ( n - | \\delta | ) = 2 d n - d ( d + 1 ) = 2 d ( n - \\tfrac { 1 } { 2 } ( d + 1 ) ) . \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} N _ 1 ( b ) = . \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} \\omega _ { k } = \\begin{cases} \\delta & \\textrm { f o r e v e r y } \\ 1 \\le k \\le r \\\\ M & \\textrm { f o r e v e r y } \\ k > r ; \\end{cases} \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{align*} \\eta _ { a } ( C ( \\mathsf { P } ) ) = \\sum _ { i , j } c _ { j } ^ { i } \\pi ( K _ { 2 \\rho } ^ { - 1 } K _ { a } [ E _ { a } , F _ { a } ] ) _ { i } ^ { j } . \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} \\zeta _ k \\in { \\mathbb C } , \\zeta _ k = { \\bar \\zeta } _ { n + 1 - k } , k = 0 , \\cdot \\cdot \\cdot , n + 1 . \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} i \\partial _ t \\phi + \\frac 1 2 \\Delta \\phi - V ( x ) \\phi = 0 , \\ \\ \\phi ( 0 , x ) = \\phi _ 0 ( x ) , \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} y _ i ( m + 1 ) = y _ i ( m ) + y _ { i - 1 } ( m ) . \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} L ^ { 2 n } = L ^ { 2 n } ( \\Lambda ) = ( - 1 ) ^ n d ^ { 2 n } - \\Lambda ( - 1 ) ^ { n - p } d ^ { 2 n - 2 p } , d = \\frac { d } { d t } . \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} \\mathcal { A } ^ { P T _ { i } } = s _ p ( k _ { P ( i ) } - k _ { P { ( i + 1 ) } } ) \\mathcal { A } ^ P , \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\prod _ { j = b _ n + 1 } ^ { b _ { n + 1 } - 1 } | w _ { j } | = \\infty . \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } x _ { n + 1 } ^ { 1 } = x _ { n } ^ { 1 } + h x _ { n } ^ { 2 } , \\\\ x _ { n + 1 } ^ { 2 } = - h x _ { n } ^ { 1 } + \\left ( 1 - h ^ { 2 } \\right ) x _ { n } ^ { 2 } . \\end{array} \\right . \\end{align*}"}
-{"id": "8525.png", "formula": "\\begin{align*} W _ { \\R ^ n } ( \\iota _ a \\circ \\phi ) = W _ { \\R ^ n } ( \\phi ) , \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} \\xi ( z ) & = \\int _ 1 ^ z \\frac { \\rho _ { \\alpha , \\varepsilon } ( s ) - \\rho ( \\cos \\theta ) } { ( s ^ 2 - 1 ) ^ { 1 / 2 } } d s + \\rho ( \\cos \\theta ) \\int _ 1 ^ z \\frac { 1 } { ( s ^ 2 - 1 ) ^ { 1 / 2 } } d s \\\\ & = \\int _ 1 ^ z \\frac { \\rho _ { \\alpha , \\varepsilon } ( s ) - \\rho ( \\cos \\theta ) } { ( s ^ 2 - 1 ) ^ { 1 / 2 } } d s + \\rho ( \\cos \\theta ) \\log \\varphi ( z ) . \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} A ( x ) = \\frac { \\frac { n - 2 } { 4 ( n - 1 ) } R [ \\bar { g } ] u ( x ) \\left ( u ( x ) ^ { \\frac { 4 } { n - 2 } } - 1 \\right ) } { u ( x ) - 1 } . \\end{align*}"}
-{"id": "633.png", "formula": "\\begin{align*} \\begin{cases} \\div ( | x _ 2 | ^ a \\nabla u ) = 0 & B _ 1 \\setminus \\Lambda ( u ) , \\\\ u = 0 & \\Lambda ( u ) , \\end{cases} \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} r G _ 1 ' = \\langle X \\rangle ^ { 2 + \\varepsilon } G _ 1 ' \\in L ^ \\infty . \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} \\widetilde W _ 0 : = \\{ ( i , j ) \\in U \\mid f _ { i j } ( \\tilde p _ { i j } ) < 0 \\} . \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 2 } & \\vec { \\gamma } _ 0 ( p ) = 0 \\qquad & & \\ ; \\ , 1 \\leq \\theta _ 0 ( p ) \\leq 3 \\\\ & r ( p ) \\leq \\theta _ 0 ( p ) - 2 \\qquad & & \\ ; \\ , \\theta _ 0 ( p ) \\geq 2 . \\end{alignedat} \\right . \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n C _ { s , j } ^ { ( 2 ) } \\omega ^ { ( 1 - s ) j } ( \\omega ^ j + 1 ) ^ m , \\enskip s = 1 , . . . , n . \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{align*} N P ^ T D M = ( { P ^ T _ 1 } D _ 1 ) \\bigoplus ( { P ^ T _ 1 } D _ 1 ) \\bigoplus \\cdots \\bigoplus ( { P ^ T _ 1 } D _ 1 ) \\bigoplus \\cdots , \\end{align*}"}
-{"id": "3700.png", "formula": "\\begin{align*} X _ { t } ( y ) = ( X _ { t } ( y ^ c ) , X _ t ( y ^ { s s } ) , X _ { t } ( y ^ u ) ) = ( e ^ { B _ 1 t } y ^ c , e ^ { B _ 2 t } y ^ { s s } , e ^ { C t } y ^ u ) . \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} u _ 0 = u _ { f } + u _ { \\partial } + u _ { \\sf r m } \\ , . \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} \\lim _ n R _ n ( C ) = 0 . \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle | \\Delta t \\sum _ { k = 1 } ^ { m } Q _ { 1 } ^ { k } ( \\partial { \\theta _ { \\psi } ^ { k } } ) | \\leq C \\big \\{ h ^ { 2 r + 2 } + ( \\Delta t ) ^ { 4 } \\big \\} + C \\| \\theta _ { \\psi } ^ { m } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } + C \\Delta t \\sum _ { k = 1 } ^ { m - 1 } { \\| \\theta _ { \\psi } ^ { k } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } } . } \\end{array} \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{align*} \\beta _ { n } ( t ) = | u _ { n + 1 } ( t ) - u _ { n } ( t ) | + | v _ { n + 1 } ( t ) - v _ { n } ( t ) | + | \\rho _ { n + 1 } ( t ) - \\rho _ { n } ( t ) | + | \\psi _ { n + 1 } ( t ) - \\psi _ { n } ( t ) | + | \\phi _ { n + 1 } ( t ) - \\phi _ { n } ( t ) | , \\end{align*}"}
-{"id": "2521.png", "formula": "\\begin{align*} \\left ( f ^ { \\prime } f _ { c _ { 2 } } + B _ { 2 } \\right ) \\left ( z _ { 0 } \\right ) = 0 . \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} 0 \\le \\rho \\le 1 , \\ \\rho ( x ) = 1 \\textrm { f o r } x \\in { \\rm s u p p } _ X k , \\ { \\rm s u p p } \\rho \\subset X . \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} W ( e _ { k ( 0 ) } ) \\dots W ( e _ { k ( d ) } ) = W ( e _ { k ( 0 ) } \\odot \\dots \\odot e _ { k ( d ) } ) + q _ 0 ^ { l - 1 } W ( e _ { k ( 0 ) } ) \\dots \\widehat { W ( e _ { k ( l ) } ) } \\dots W ( e _ { k ( d ) } ) \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} T ^ { n } z & = \\sum _ { k = 1 } ^ { r } a _ { k } T ^ { n } z _ { k } = \\sum _ { k = 1 } ^ { r } a _ { k } T ^ { l _ { 1 } + \\cdots + l _ { k } } z _ { k } \\qquad \\hbox { b y ( i i i ) } , \\intertext { s o t h a t } \\| T ^ { n } z - x \\| & \\le \\sum _ { k = 1 } ^ { r } | a _ { k } | \\ , \\| T ^ { l _ { 1 } + \\cdots + l _ { k } } z _ { k } - x _ { k } \\| < \\varepsilon \\hbox { b y ( i i ) . } \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} H \\cap H ^ { g ^ { - 1 } } = ( H \\cap H ^ g ) ^ { g ^ { - 1 } } = \\langle a \\rangle ^ { g ^ { - 1 } } = \\langle a ^ { g ^ { - 1 } } \\rangle = \\langle a b \\rangle . \\end{align*}"}
-{"id": "8018.png", "formula": "\\begin{align*} T ( t ) = \\left \\{ \\begin{array} { l l } e ^ { - t A _ { u _ \\infty } } & ( \\mbox { s t a r t i n g p r o b l e m } ) , \\\\ e ^ { - t A } & ( \\mbox { l a n d i n g p r o b l e m } ) . \\end{array} \\right . \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} \\| U \\| _ { p l } : = \\inf \\{ \\sum _ { k = 1 } ^ n \\| a _ k \\| \\| u _ k \\| \\| v _ k \\| \\} , \\end{align*}"}
-{"id": "10038.png", "formula": "\\begin{align*} \\nabla _ X J _ { \\varphi } Y = \\frac { 2 } { \\sqrt 5 } \\nabla _ X \\varphi Y - \\frac { 1 } { \\sqrt 5 } \\nabla _ X Y = \\frac { 2 } { \\sqrt 5 } \\varphi ( \\nabla _ X Y ) - \\frac { 1 } { \\sqrt 5 } \\nabla _ X Y = J _ { \\varphi } ( \\nabla _ X Y ) , \\forall X , Y \\in { \\mathfrak X } ( M ) , \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{align*} \\| w ( t ) \\| _ \\infty = O ( t ^ { - 1 / 2 } ) \\mbox { a s $ t \\to \\infty $ } \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{align*} | e ^ { i \\frac { t } { 2 \\pi R } \\Delta } f _ { d k } ( x _ 1 ) | & = \\frac { R ^ { - 1 / 2 } } { ( 2 \\pi ) ^ { 1 / 2 } } \\Big | \\int \\widehat { \\phi } \\big ( R ^ { - 1 / 2 } ( \\xi _ 1 - \\pi R ) \\big ) e ^ { i x _ 1 \\xi _ 1 - i \\frac { t } { 2 \\pi R } \\xi _ 1 ^ { 2 } } d \\xi _ 1 \\Big | \\\\ & = \\frac { 1 } { ( 2 \\pi ) ^ { 1 / 2 } } \\Big | \\int _ { - \\rho } ^ \\rho e ^ { i R ^ { 1 / 2 } ( x _ 1 - t ) y - i \\frac { t } { 2 \\pi } y ^ { 2 } } d y \\Big | \\simeq 1 \\ , , \\end{align*}"}
-{"id": "2781.png", "formula": "\\begin{align*} T _ { \\varepsilon , d , B } = \\frac { \\varepsilon } { \\alpha ^ 2 d ^ 2 } B ^ { \\frac { 2 } { 3 } + \\delta } + O ( \\frac { B ^ { \\varepsilon + \\frac { 1 } { 3 } } } { d ^ \\varepsilon } ) . \\end{align*}"}
-{"id": "2070.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 0 ^ { ( 1 ) } } { x ^ 2 } + \\frac { A _ 0 ^ { ( 0 ) } } { x } + \\frac { A _ { t _ 1 } } { x - t _ 1 } + N \\right ) Y , \\\\ \\frac { \\partial Y } { \\partial t _ 1 } = \\left ( N _ 1 - \\frac { A _ { t _ 1 } } { x - t _ 1 } \\right ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = \\left ( N _ 2 - \\frac { \\frac { 1 } { t _ 2 } A _ 0 ^ { ( 1 ) } } { x } \\right ) Y . \\end{gather*}"}
-{"id": "4185.png", "formula": "\\begin{align*} \\frac { \\partial T } { \\partial t } = \\mathcal { D } \\frac { \\partial ^ { 2 } T } { \\partial x ^ { 2 } } , \\ ; \\ ; \\tau \\frac { \\partial ^ { 2 } T } { \\partial t ^ { 2 } } + \\frac { \\partial T } { \\partial t } = \\mathcal { D } \\frac { \\partial ^ { 2 } T } { \\partial x ^ { 2 } } \\ ; \\ ; \\ ; \\ ; \\tau \\ , { } _ { 0 } ^ { c } \\mathrm { D } _ { t } ^ { \\alpha + 1 } T + \\frac { \\partial T } { \\partial t } = \\mathcal { D } \\frac { \\partial ^ { 2 } T } { \\partial x ^ { 2 } } , \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} \\begin{array} { c } \\ , ^ { A B C } \\ , _ { 0 } D ^ { \\alpha } _ { t } u ( t ) - \\lambda u ( t ) = f ( t ) , t \\geq 0 , \\\\ u ( 0 ) = u _ { 0 } , \\end{array} \\end{align*}"}
-{"id": "5945.png", "formula": "\\begin{align*} y g ( y ) \\underbrace { g ( x ) y } _ { u } \\underbrace { g ( g ( x ) y ) } _ { g ( u ) } \\stackrel { ( c ) } { = } \\underbrace { g ( x ) y } _ { u } \\underbrace { g ( g ( x ) y ) } _ { g ( u ) } y g ( y ) . \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} 0 = \\lim _ { y \\to \\infty } \\frac { { B } ( { t } _ k , y ) } { y ^ { n - 1 } } = \\lim _ { y \\to \\infty } \\frac { { S } ( y ) \\prod _ { l = 1 } ^ n ( { t } _ k - { x } _ l ) - { S } ( { t } _ k ) \\prod _ { l = 1 } ^ n ( y - { x } _ l ) } { ( { t } _ k - y ) y ^ { n - 1 } } = { S } ( { t } _ k ) \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} \\Delta ( f _ { i } ) = \\sum _ { j , k } c _ { j , k } ^ { i } f _ { j } \\otimes f _ { k } . \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} c _ P = K ^ * + 2 q ^ * . \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} \\det ( g ) & = ( \\alpha _ 1 \\delta _ 1 - \\beta _ 1 \\gamma _ 1 ) + ( \\alpha _ 2 \\delta _ 2 - \\beta _ 2 \\gamma _ 2 ) \\epsilon + ( \\alpha _ 1 \\delta _ 2 + \\alpha _ 2 \\delta _ 1 - \\beta _ 1 \\gamma _ 2 - \\beta _ 2 \\gamma _ 1 ) \\sqrt { \\epsilon } \\\\ & = ( \\alpha _ 1 \\delta _ 2 + \\alpha _ 2 \\delta _ 1 - \\beta _ 1 \\gamma _ 2 - \\beta _ 2 \\gamma _ 1 ) \\sqrt { \\epsilon } . \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} A Q '' + 2 B Q ' + C Q = 0 \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} \\widehat { e ^ { - t ( 1 - \\Delta ) } u } = e ^ { t a ( \\xi ) } \\hat { u } , \\mbox { f o r e v e r y } t \\geqslant 0 , \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} \\gamma _ v ( t ) = \\delta v + ( \\alpha _ { \\delta v } + e ^ { 2 t } ) i e _ 1 \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} A _ n = \\begin{pmatrix} \\frac { 1 } { r _ n } & & & & \\\\ & \\frac { 1 } { z _ n } & & & \\\\ & & 1 & & \\\\ & & & \\ddots & \\\\ & & & & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{gather*} \\frac { q x _ i x _ { i + 1 } - q ^ { - 1 } x _ { i + 1 } x _ i } { q - q ^ { - 1 } } = 1 , \\\\ x _ i ^ 3 x _ { i + 2 } - [ 3 ] _ q x _ i ^ 2 x _ { i + 2 } x _ i + [ 3 ] _ q x _ i x _ { i + 2 } x _ i ^ 2 - x _ { i + 2 } x _ i ^ 3 = 0 , \\end{gather*}"}
-{"id": "2760.png", "formula": "\\begin{align*} \\sum _ { d \\leqslant \\varepsilon _ { 1 } ^ \\frac { 1 } { 2 } \\Xi ( \\alpha ) ^ { - \\frac { 1 } { 2 } } B ^ { 1 - \\frac { 1 } { 2 r } } } O _ { \\varepsilon _ i } \\left ( \\frac { \\varDelta ( \\alpha ) \\Xi ( \\alpha ) ^ { - \\frac { 1 } { 2 } } } { d } B ^ { 1 - \\frac { 1 } { 2 r } } \\log ( B ) \\right ) = O _ { \\varepsilon _ i } \\left ( \\varDelta ( \\alpha ) \\Xi ( \\alpha ) ^ { - \\frac { 1 } { 2 } } B ^ { 1 - \\frac { 1 } { 2 r } } \\log ( B ) \\log ( \\Xi ( \\alpha ) ^ { - \\frac { 1 } { 2 } } B ^ { 1 - \\frac { 1 } { 2 r } } ) \\right ) . \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} \\mathbb { E } _ { H _ f } \\left [ F ^ { ( \\tau ) } ( \\textbf { t } ) \\right ] = \\phi \\left [ \\exp \\left ( \\sum _ { k = 1 } ^ \\infty \\sum _ { \\{ i : d _ i | k \\} } \\epsilon _ i ^ { k / d _ i } d _ i \\ , \\frac { t _ k } { k } \\right ) \\right ] . \\end{align*}"}
-{"id": "7882.png", "formula": "\\begin{align*} \\mathbb { P } ( W ( \\epsilon ) ) \\leq \\sum _ { i = 1 } ^ { n } \\mathbb { P } \\left ( \\# { \\cal E } _ i \\geq \\epsilon n \\right ) \\leq D \\frac { ( \\log { n } ) ^ 3 } { n } \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\xi ^ { \\mathbf { c } } } L = 0 ~ C . \\end{align*}"}
-{"id": "2419.png", "formula": "\\begin{align*} D _ i \\ , T ^ i = 0 \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} U ' _ f ( \\alpha ) : = \\frac { \\partial } { \\partial \\alpha } U _ f ( \\alpha ) = & \\int _ { \\mathbb { D } } | f ( z ) | ^ { 2 \\alpha } ( 1 - | z | ^ 2 ) ^ { \\alpha } d \\mu ( z ) \\\\ & - ( \\alpha - 1 ) \\int _ { \\mathbb { D } } | f ( z ) | ^ { 2 \\alpha } ( 1 - | z | ^ 2 ) ^ { \\alpha } \\log \\frac { 1 } { | f ( z ) | ^ { 2 } ( 1 - | z | ^ 2 ) } d \\mu ( z ) . \\end{align*}"}
-{"id": "3887.png", "formula": "\\begin{align*} y \\left ( \\frac { 1 } { 2 } \\ ! + \\ ! 0 \\right ) \\ ! = \\ ! a _ 1 y \\left ( \\frac { 1 } { 2 } \\ ! - \\ ! 0 \\right ) , \\ y ' \\left ( \\frac { 1 } { 2 } \\ ! + \\ ! 0 \\right ) \\ ! = \\ ! a _ 1 ^ { - 1 } y ' \\left ( \\frac { 1 } { 2 } \\ ! - \\ ! 0 \\right ) \\ ! + \\ ! a _ 2 y \\left ( \\frac { 1 } { 2 } \\ ! - \\ ! 0 \\right ) . \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to 0 ^ + } \\int _ { \\Gamma _ - } \\psi _ { \\rho , x _ 0 ' , \\theta _ 0 ' } ( x ' , \\theta ' ) f ( x ' , \\theta ' ) d \\xi ( x ' , \\theta ' ) = f ( x ' _ 0 , \\theta _ 0 ' ) , \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{align*} V ( x , y ) = U \\Bigl ( \\frac { x - y } { 2 } , \\frac { x + y } 2 \\Bigr ) , \\ ; \\ ; \\ ; x , y \\in X . \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} | \\omega _ m - \\omega _ l | & = c \\frac { \\lambda _ m - \\lambda _ l } { \\sqrt { c ^ 2 + \\lambda _ m } + \\sqrt { c ^ 2 + \\lambda _ l } } \\stackrel { m > l } { \\geq } \\frac { c \\lambda _ l ^ { 1 / 2 } } { \\sqrt { c ^ 2 + \\lambda _ m } + \\sqrt { c ^ 2 + \\lambda _ l } } \\\\ & \\gtrsim \\frac { N ^ { \\delta / 3 } \\lambda ^ { 1 / 2 } } { \\sqrt { N ^ { 2 \\delta / 3 } + \\lambda _ m ^ { 1 / 2 } } + \\sqrt { N ^ { 2 \\delta / 3 } + \\lambda _ l ^ { 1 / 2 } } } > 0 , \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} v ( \\xi , y , t ) = \\left ( b ( t ) ^ 2 e ^ { 2 i y } + \\bar { b } ( t ) ^ 2 e ^ { - 2 i y } \\right ) w _ 2 ( \\xi ) + | b ( t ) | ^ 2 w _ 0 ( \\xi ) + w ( \\xi , y , t ) , \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} \\tilde \\Lambda = \\lim _ { p _ { j } \\to \\infty } \\lambda _ { 1 } ( p _ { j } , \\Omega ) ^ { \\frac { 1 } { p _ { j } } } \\le \\frac { 1 } { \\rho _ { F } ( \\Omega ) } . \\end{align*}"}
-{"id": "1530.png", "formula": "\\begin{align*} \\frac { d \\ , f ' ( x ) } { f ( x ) ^ 2 } = g ' ( x ) , \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{align*} F _ 1 ( \\mathbf u ^ { ( l ) } ) = - \\sum _ { k = 1 } ^ { n } \\big ( ( \\alpha _ k ) ^ { l } S _ k \\big ) . \\end{align*}"}
-{"id": "5624.png", "formula": "\\begin{gather*} \\sin ( \\alpha _ { 1 } ) \\cos ( \\alpha _ { 2 } ) = \\frac { - 1 } { 4 } ( \\sin ( - \\alpha _ { 1 } - \\alpha _ { 2 } ) \\allowbreak + \\allowbreak \\sin ( - \\alpha _ { 1 } + \\alpha _ { 2 } ) - \\sin ( \\alpha _ { 1 } - \\alpha _ { 2 } ) - \\sin \\left ( \\alpha _ { 1 } + \\alpha _ { 2 } ) \\right ) \\\\ = \\frac { 1 } { 2 } ( \\sin ( \\alpha _ { 1 } + \\alpha _ { 2 } ) + \\sin ( \\alpha _ { 1 } - \\alpha _ { 2 } ) ) . \\end{gather*}"}
-{"id": "2471.png", "formula": "\\begin{align*} \\delta _ { \\tau ( \\Omega ) } ( r _ n i e _ 1 ; e _ 2 ) = C _ n r _ n ^ { 1 / ( 2 + \\alpha ) + \\epsilon _ n } . \\end{align*}"}
-{"id": "3503.png", "formula": "\\begin{align*} { _ { 1 } F _ { 1 } } ( \\alpha , \\beta , 2 x ) = \\frac { \\Gamma ( \\beta ) } { \\Gamma ( \\alpha ) \\ \\Gamma ( \\beta - \\alpha ) } \\int _ { 0 } ^ { 1 } t ^ { \\alpha - 1 } \\left ( 1 - t \\right ) ^ { \\beta - \\alpha - 1 } \\exp ( 2 x t ) \\ d t . \\end{align*}"}
-{"id": "9417.png", "formula": "\\begin{align*} \\bigl ( T _ \\phi ( X _ 1 , \\ldots , X _ { n - 1 } ) Z \\bigr ) = \\int _ \\Sigma \\bigl ( a _ 1 ( t , A _ 1 ) X _ 1 a _ 2 ( t , A _ 2 ) X _ 2 \\cdots X _ { n - 1 } a _ n ( t , A _ n ) Z \\bigr ) \\ , \\mu ( t ) \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{align*} \\xi R ^ E _ p a _ 0 = \\xi a _ p R ^ F _ p . \\end{align*}"}
-{"id": "5596.png", "formula": "\\begin{gather*} T _ { i } ( 1 ) = 1 , T _ { j } ( - 1 ) = ( - 1 ) ^ { j - 2 \\left \\lfloor j / 2 \\right \\rfloor } , \\\\ U _ { j } ( \\pm 1 ) = \\pm ( j + 1 ) , \\end{gather*}"}
-{"id": "8965.png", "formula": "\\begin{align*} - \\xi = h ( y , z ; s ) - r , \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} [ f , w _ k ] _ n \\bigg | _ a ^ b = 0 , k = 1 , 2 , \\dots , m , \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} \\sum _ { d | k } d N _ d = q ^ k . \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ N \\Lambda _ X \\big ( \\ , \\tau \\omega _ i ( N ) \\big ) = \\\\ & \\qquad \\sum _ { i = 1 } ^ N \\Lambda _ X \\left ( \\tau \\overline F \\left ( \\frac { i - 1 } { N } \\right ) \\right ) + \\sum _ { i = 1 } ^ N \\Lambda _ X ' \\left ( \\tau \\overline F \\left ( \\frac { i - 1 } { N } \\right ) \\right ) \\left [ \\frac { \\tau } { 2 N } \\overline F \\ , ' \\left ( \\frac { i - 1 } { N } \\right ) + O \\left ( \\frac { 1 } { N ^ 2 } \\right ) \\right ] + O \\left ( \\frac { 1 } { N ^ 2 } \\right ) . \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{align*} \\varepsilon ' v _ 1 = \\varepsilon u _ 1 \\delta _ 2 ^ { - 1 } \\quad v _ i = \\delta _ i u _ i \\delta _ { i + 1 } ^ { - 1 } \\quad \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} C ( \\mathbf { d } , \\mathbf { D } , e ) = \\prod _ { p } \\sigma _ p ( \\mathbf { d } , \\mathbf { D } ; e L _ 1 , e L _ 2 , e L _ 3 ) . \\end{align*}"}
-{"id": "629.png", "formula": "\\begin{align*} \\limsup _ { i \\to + \\infty } I _ { w _ { j _ i } } ( y _ i , 0 ^ + ) \\leq \\inf _ { s > 0 } \\limsup _ { i \\to + \\infty } I _ { w _ { j _ i } } ( y _ i , s ) = \\inf _ { s > 0 } I _ { w } ( y , s ) = I _ { w } ( y , 0 ^ + ) . \\end{align*}"}
-{"id": "9806.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 { g d s } \\approx \\sum \\limits _ { i = 0 } ^ N { { g \\left ( s _ { i } \\right ) } { \\omega _ i } } \\equiv \\int _ N { g d s } , \\end{align*}"}
-{"id": "9743.png", "formula": "\\begin{align*} \\partial _ t \\rho + \\nabla \\cdot ( \\rho v ) = 0 . \\end{align*}"}
-{"id": "9275.png", "formula": "\\begin{align*} \\frac { 3 \\alpha } { 3 - 2 \\alpha } = \\frac { 1 2 } { 1 2 + \\beta } \\frac { 3 } { 3 - 2 \\frac { 1 2 } { 1 2 + \\beta } } = \\frac { 3 6 } { 3 6 + 3 \\beta - 2 4 } = \\frac { 1 2 } { 4 + \\beta } \\end{align*}"}
-{"id": "1964.png", "formula": "\\begin{align*} R _ { a b } ( z _ 1 / z _ 2 , t ) L _ { a j } ( z _ 1 , t ) L _ { b j } ( z _ 2 , t ) = L _ { b j } ( z _ 2 , t ) L _ { a j } ( z _ 1 , t ) R _ { a b } ( z _ 1 / z _ 2 , t ) , \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{align*} b ^ { \\frac { 5 } { 2 } } \\sqrt { a b ^ { - 1 } } ( \\sqrt { b a ^ { - 1 } } - 1 ) \\prod _ { i = 1 } ^ { 3 } e _ i ^ 2 f _ i & \\leqslant b ^ { \\frac { 5 } { 2 } } \\frac { \\tau _ 1 - 1 } { \\tau _ 2 } ( b a ( b - a ) ) ^ 2 \\\\ & = b ^ { \\frac { 1 7 } { 2 } } \\frac { \\tau _ 1 - 1 } { \\tau _ 2 } \\left ( \\frac { a } { b } \\left ( 1 - \\frac { a } { b } \\right ) \\right ) ^ 2 \\\\ & \\leqslant b ^ { \\frac { 1 7 } { 2 } } \\frac { \\tau _ 1 - 1 } { 1 6 \\tau _ 2 } \\leqslant B ^ { \\frac { 1 } { 2 } - \\frac { 1 } { r } } , \\end{align*}"}
-{"id": "5063.png", "formula": "\\begin{align*} \\cal { L } _ m : = \\left ( 1 + 2 \\cdot ( K ^ { m - 1 } _ 0 \\cdot 2 \\ , ) ^ { 1 / ( m - 1 ) } \\right ) \\cdot L . \\end{align*}"}
-{"id": "6480.png", "formula": "\\begin{align*} R ^ e _ n ( s _ { v _ 0 } ) ( w _ { e } ) & = s _ { w ( 0 , e - 1 ) } ( w _ e ) = w _ e + ( w _ e , w ( 0 , e - 1 ) ) w ( 0 , e - 1 ) \\\\ & = \\sum _ { i = 0 } ^ e w _ { i } . \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} S ( \\hat { \\sigma } _ A ) - S ( \\hat { \\rho } _ A ) & = I ( A : X ) _ { \\hat { \\sigma } _ { A X } } \\ ; , \\\\ S ( \\Phi ( \\hat { \\sigma } _ A ) ) - S ( \\Phi ( \\hat { \\rho } _ A ) ) & = I ( B : X ) _ { ( \\Phi \\otimes \\mathbb { I } _ X ) ( \\hat { \\sigma } _ { A X } ) } \\ ; , \\end{align*}"}
-{"id": "9112.png", "formula": "\\begin{gather*} B = \\left ( \\begin{matrix} 2 & 0 & 0 & 0 \\\\ 0 & 2 & 0 & 0 \\\\ 0 & 0 & 3 & - 1 \\\\ 0 & 0 & - 1 & 3 \\end{matrix} \\right ) \\mathbf { w } = ( 2 , 2 , 2 , 2 ) . \\end{gather*}"}
-{"id": "7994.png", "formula": "\\begin{align*} \\rho = 2 C _ 0 \\left ( \\| w _ 0 \\| _ 2 + \\kappa _ f \\sqrt { T } \\right ) , T _ * = \\min \\Big \\{ \\left ( 4 C _ 1 k ^ { 3 / 4 } \\rho \\right ) ^ { - 2 } , \\ , ( 1 6 C _ 2 ^ { ( \\varepsilon ) } ) ^ { - 2 } , \\ , 1 \\Big \\} . \\end{align*}"}
-{"id": "5125.png", "formula": "\\begin{align*} x _ { n } = \\frac { B _ { n } ^ { \\left ( a , b \\right ) } } { n ! } \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} \\Pr { \\mathbf { Y } = d + 2 } = \\begin{cases} 0 & d < 0 \\\\ e ^ { - d ^ 2 - d } - e ^ { - ( d + 1 ) ^ 2 - ( d + 1 ) } & d \\geq 0 . \\end{cases} \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} d s ^ { 2 } = \\frac { \\Delta } { r ^ { 2 } } d t ^ { 2 } - \\frac { r ^ { 2 } } { \\Delta } d r ^ { 2 } - r ^ { 2 } \\left ( d \\theta ^ { 2 } + \\sin ^ { 2 } \\theta d \\phi ^ { 2 } \\right ) , \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} \\phi ( t ) : = S ( C | M ) _ { \\hat { \\rho } _ { C M } ( t ) } - \\lambda \\ , S ( A | M ) _ { \\hat { \\rho } _ { A M } ( t ) } - ( 1 - \\lambda ) \\ , S ( B | M ) _ { \\hat { \\rho } _ { B M } ( t ) } \\ ; . \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} t _ { i } + x _ { i + 1 } + y _ { i + 1 } = t _ { i + 1 } p + z _ { i + 1 } \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{align*} 2 \\pi f ( s ) = \\int _ \\Omega g ( t ) \\log | s - t | + \\int _ { \\partial \\Omega } \\Big ( f ( t ) \\frac { \\partial \\log | s - t | } { \\partial n _ t } - \\frac { \\partial f ( t ) } { \\partial n _ t } \\log | s - t | \\Big ) \\end{align*}"}
-{"id": "974.png", "formula": "\\begin{align*} p = \\frac { 8 ( \\nu + 1 ) } { ( \\nu - 1 ) ( d - 2 \\gamma ) } , q = \\frac { d ( \\nu + 1 ) } { d + ( \\nu - 1 ) \\gamma } . \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} P ( \\varepsilon ) : = \\sum _ { i \\in \\N } a _ { i } \\varepsilon ^ { i } \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} [ e _ r , e _ r ] = \\theta _ r e _ n , [ e _ i , e _ j ] = \\sum ^ { n - 1 } _ { m = k + 1 } \\alpha ^ t _ { i j } e _ t + \\beta _ { i j } e _ n , [ e _ j , e _ i ] = - \\sum ^ { n - 1 } _ { m = k + 1 } \\alpha ^ t _ { i j } e _ t + \\gamma _ { j i } e _ n , \\\\ [ e _ p , e _ s ] = \\sum ^ { n - 1 } _ { m = n - t + 1 } \\alpha ^ t _ { i j } e _ t + \\theta _ { i j } e _ n , [ e _ s , e _ p ] = - \\sum ^ { n - 1 } _ { m = n - t + 1 } \\alpha ^ t _ { i j } e _ t + \\delta _ { j i } e _ n , \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{align*} \\ ! \\ ! \\widetilde { J } _ 1 ( k ) ^ { - 1 } \\ ! = \\ ! \\frac { 1 } { k } \\widetilde { T } _ 6 ^ { - 1 } \\widetilde { T } _ 1 \\begin{bmatrix} I _ \\mu + o ( 1 ) & o ( k ) \\\\ o ( k ) & k I _ { n - \\mu } + o ( k ) \\end{bmatrix} \\widetilde { M } _ 1 , \\ ; \\ ; k \\to 0 , \\ ; \\ ; k \\in \\overline { \\mathbb { C } } ^ + . \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{align*} \\widetilde U ( x , t ) = ( 1 - \\phi ( x ) ) U ( x , t ) + \\mathbb B [ U ( \\cdot , t ) \\cdot \\nabla \\phi ] ( x ) = U ( x , t ) + E ( x , t ) , \\end{align*}"}
-{"id": "9403.png", "formula": "\\begin{align*} L ^ \\infty _ \\sigma \\bigl ( \\Omega _ 2 ; B ( L ^ 1 ( \\Omega _ 1 ) , H ) \\bigr ) \\ , & = L ^ \\infty _ \\sigma \\bigl ( \\Omega _ 2 ; ( L ^ 1 ( \\Omega _ 1 ) \\overset { \\wedge } { \\otimes } H ^ * ) ^ * \\bigr ) \\ , \\\\ & = \\bigl ( L ^ 1 ( \\Omega _ 2 ) \\overset { \\wedge } { \\otimes } L ^ 1 ( \\Omega _ 1 ) \\overset { \\wedge } { \\otimes } H ^ * \\bigr ) ^ * \\\\ & = L ^ 1 ( \\Omega _ 1 \\times \\Omega _ 2 ; H ^ * ) ^ * \\\\ & = L ^ \\infty _ \\sigma ( \\Omega _ 1 \\times \\Omega _ 2 ; H ) . \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{align*} \\lambda _ { \\infty } = \\int _ { g \\in \\mathcal { F } ( 1 ) } g \\mu _ { G _ 1 } \\dd \\lambda _ { \\infty } ( 1 ) ( g ) , \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} \\lim _ { t \\to \\bar t + 0 } \\| w ( t ) - w ( \\bar t ) \\| _ 3 = 0 , \\| w ( t ) \\| _ 3 \\leq C \\| w ( \\bar t ) \\| _ 3 ( t \\geq \\bar t ) . \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} \\varphi ( \\omega , y ) & = \\gamma ( 1 _ G , \\widehat { \\phi } ( \\mathfrak { C } ( ( \\bar { \\omega } , y ^ { 1 _ G } ) , 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ) ) \\\\ & = \\gamma ( 1 _ G , \\widehat { \\phi } ( c ^ { 1 _ G } ) ) \\\\ & = \\gamma ( 1 _ G , \\widetilde { \\Psi } ( x ) ) \\\\ & = x . \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} f ( x ) & = \\frac { b _ k } { ( \\lambda ) _ k } + \\frac { l _ k \\Gamma ( \\lambda ) } { ( k - 1 ) ! x ^ { \\lambda } } + \\int _ 0 ^ { \\infty } M _ 1 ( u ) u ^ { k - 1 } e ^ { - x u } u ^ { \\lambda - 1 } \\ , d u \\\\ & = \\frac { b _ k } { ( \\lambda ) _ k } + \\frac { l _ k \\Gamma ( \\lambda ) } { ( k - 1 ) ! x ^ { \\lambda } } + \\int _ 0 ^ { \\infty } N _ 1 ( s ) s ^ { - k - \\lambda } e ^ { - x / s } \\ , d s . \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} A ^ { 1 - \\alpha - \\mu } D ^ { \\alpha } u = F ^ { - 1 } F A ^ { 1 - \\varkappa - \\mu } D ^ { \\alpha } u = F ^ { - 1 } A ^ { 1 - \\varkappa - \\mu } F D ^ { \\alpha } u = \\end{align*}"}
-{"id": "2262.png", "formula": "\\begin{align*} \\exp ( J t ) = \\begin{pmatrix} \\cos t & - \\sin t \\\\ \\sin t & \\cos t \\end{pmatrix} . \\end{align*}"}
-{"id": "5622.png", "formula": "\\begin{align*} w _ { 2 } ( \\cos \\theta , \\cos \\varphi | \\rho ) \\allowbreak = \\allowbreak w _ { 1 } ( \\cos ( \\theta + \\varphi ) | \\rho ) ( w _ { 1 } ( \\theta - \\varphi ) | \\rho ) \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{align*} \\bigl ( \\Xi ( \\vec { \\xi } ) \\bigr ) ( f ) = S \\cdot f ( \\partial _ 1 , \\partial _ 2 ) \\cdot S ^ { - 1 } , \\end{align*}"}
-{"id": "9421.png", "formula": "\\begin{align*} \\partial _ t h + \\nabla \\cdot P = 0 , \\ ; \\ ; \\partial _ t B + \\nabla \\times \\left ( \\frac { B \\times P + D } { h } \\right ) = 0 , \\ ; \\ ; \\nabla \\cdot B = \\nabla \\cdot D = 0 , \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} h _ { 1 , k + 1 , 2 } ( x ) & = \\sum _ { j = 0 } ^ k a _ { j , k } x ^ j ( 1 + x ) ^ { 2 k - 2 j } , k \\in \\N , \\\\ h _ { 2 , k + 1 , 4 } ( x ) & = \\sum _ { j = 0 } ^ k b _ { j , k } x ^ j ( 1 + x ) ^ { 2 k - 2 j } , k \\in \\N _ + , \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { M } K _ 3 ^ { ( k ) } \\leq \\frac { C } { \\Delta t } \\left [ h ^ { 2 r } + ( \\Delta t ) ^ { 4 } \\right ] + C \\sum _ { k = 0 } ^ { M } \\left ( D ( { \\theta } _ { \\mathbf { A } } ^ { k } , { \\theta } _ { \\mathbf { A } } ^ { k } ) + \\| \\nabla \\theta _ { \\Psi } ^ { k } \\| _ { \\mathbf { L } ^ 2 } ^ { 2 } + \\| \\mathbf { v } \\| _ { \\mathbf { L } ^ 2 } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "1116.png", "formula": "\\begin{align*} N \\min _ i \\mu ( B ( x _ i , r ' ) ) \\le \\sum _ { i = 1 } ^ N \\mu ( B ( x _ i , r ' ) ) \\le \\mu ( 2 B ) \\le C _ \\mu ^ 4 \\min _ i \\mu ( B ( x _ i , r ' ) ) , \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} P ( x ) = S ( x ) ^ 2 + \\sum _ { k = 1 } ^ { n - 1 } P ( t _ k ) S _ k ( x ) ^ 2 . \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} b ( g ) = b ( g ) + \\pi _ g b ( h ' ) = b ( g h ' ) = b ( h g ) = b ( h ) + \\pi _ h b ( g ) = \\pi _ h b ( g ) . \\end{align*}"}
-{"id": "9207.png", "formula": "\\begin{align*} [ \\langle M , x ^ * \\rangle ] _ t = [ \\langle M , x ^ * \\rangle ] ^ a _ t + [ \\langle M , x ^ * \\rangle ] ^ q _ t = [ \\langle M ^ a , x ^ * \\rangle ] _ t + [ \\langle M ^ q , x ^ * \\rangle ] _ t , \\end{align*}"}
-{"id": "5519.png", "formula": "\\begin{align*} \\max _ { | \\zeta | \\leq 1 } G ^ + \\circ \\widehat \\psi _ n ( \\zeta ) & = \\max _ { | \\zeta | \\leq 1 } G ^ + \\circ \\phi _ n \\circ h _ n ( \\zeta ) \\\\ & = \\max _ { \\zeta \\in h _ n ( \\Delta ) } G ^ + \\circ \\phi _ n ( \\zeta ) \\\\ & \\leq \\max _ { | \\zeta | \\leq m \\varepsilon } G ^ + \\circ \\phi _ n ( \\zeta ) \\\\ & < 1 , \\end{align*}"}
-{"id": "4396.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } \\binom { 2 k } { k } \\left ( \\frac { H _ k ^ { ( 2 ) } } { k } + \\frac { 1 } { k ^ 3 } \\right ) x ^ k \\equiv 4 \\pounds _ 3 ( \\alpha ) + 4 \\pounds _ 3 ( \\beta ) \\pmod { p } . \\end{align*}"}
-{"id": "3394.png", "formula": "\\begin{align*} \\begin{cases} d Y _ t ^ { i , k } = - \\frac { \\partial H ^ i ( t , X _ t , \\gamma _ t , Y ^ i _ t , Q ^ i _ t , R ^ i _ t ) } { \\partial x ^ k } \\ , d t + \\sum _ { j = 1 } ^ n Q ^ { i , k , j } _ t \\ , d W ^ j _ t + \\sum _ { j = 1 } ^ n R ^ { i , k , j } _ { t } \\ , d \\widetilde N ^ j _ t \\\\ Y ^ { i , k } _ T = \\frac { \\partial g ^ i } { \\partial x ^ k } ( X _ T ) \\ , , \\end{cases} \\end{align*}"}
-{"id": "8983.png", "formula": "\\begin{align*} & M ^ N _ \\tau = \\frac { B ^ N _ { \\tau _ t } - B ^ N _ { \\tau _ t + \\tau ( t - \\tau _ t ) } } { \\sqrt { t - \\tau _ t } } , M ^ T _ \\tau = \\frac { B ^ T _ { \\tau _ t + \\tau ( t - \\tau _ t ) } - B ^ T _ { \\tau _ t } } { \\sqrt { t - \\tau _ t } } . \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} \\Phi = e ^ { - i \\omega t } e ^ { i n \\phi } F ( r ) S ( \\theta ) . \\end{align*}"}
-{"id": "8211.png", "formula": "\\begin{align*} \\lambda _ { 2 } ( p , \\Omega ) = \\lambda _ 1 ( p , \\Omega _ 1 ) = \\lambda _ { 1 } ( p , \\Omega _ { 2 } ) , \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} f _ { n _ 1 , n _ 2 , \\cdots , n _ r } ( \\eta _ 1 , \\eta _ 2 , \\cdots , \\eta _ r ) = \\sqrt { \\frac { ( k - n _ 1 - n _ 2 \\cdots - n _ r ) ! } { k ! n _ 1 ! n _ 2 ! \\cdots n _ r ! } } { { \\eta _ 1 } } ^ { n _ 1 } { { \\eta _ 2 } } ^ { n _ 2 } \\cdots { { \\eta _ r } } ^ { n _ r } . \\end{align*}"}
-{"id": "6761.png", "formula": "\\begin{align*} F = F _ { \\omega ' \\omega } = \\kappa ^ { - 1 } _ \\omega \\circ T ^ { - 1 } \\circ \\kappa _ { \\omega ' } : \\kappa _ { \\omega ' } ^ { - 1 } ( V _ { \\omega ' \\omega } ) \\to U _ \\omega \\ , , \\end{align*}"}
-{"id": "4050.png", "formula": "\\begin{align*} A = ( a _ { i j } ) : ( M ^ i , N ^ i , P ^ i ) \\mapsto ( a _ { 1 1 } & M ^ i + a _ { 1 2 } N ^ { i } + a _ { 1 3 } P ^ { i } , \\\\ & a _ { 2 1 } M ^ i + a _ { 2 2 } N ^ { i } + a _ { 2 3 } P ^ { i } , a _ { 3 1 } M ^ i + a _ { 3 2 } N ^ { i } + a _ { 3 3 } P ^ { i } ) . \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} X _ { 2 , \\ , j + 2 , \\ , 1 } \\cap L _ { j + 2 } = 0 . \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} B _ K = W ( 0 ) \\alpha _ { \\lambda } e ^ { - \\lambda ( K - \\sigma - \\alpha _ { \\lambda } e ^ { \\lambda \\sigma } ) } . \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{gather*} l _ { k , n } ^ { ( t _ { 1 } , . . . , t _ { n + k } ) } ( x _ { 1 } , . . . , x _ { n + k } | \\rho ) = \\\\ \\sum _ { j = 0 } ^ { 2 ^ { n + k } - 1 } \\rho ^ { j } \\sum _ { m = 0 } ^ { j } \\frac { 1 } { m ! } \\left . \\frac { d ^ { m } } { d \\rho ^ { m } } w _ { k + n } ( x _ { 1 } , . . . , x _ { k + n } | \\rho ) \\right \\vert _ { \\rho = 0 } \\\\ \\times \\prod _ { s = 1 } ^ { k } T _ { ( j - m ) + t _ { s } } ( x _ { s } ) \\prod _ { s = 1 + k } ^ { n + k } U _ { ( j - m ) + t _ { s } } ( x _ { s } ) . \\end{gather*}"}
-{"id": "8329.png", "formula": "\\begin{align*} X \\ ; : \\ ; y ^ 2 = a _ 6 x ^ 6 + a _ 4 x ^ 4 + a _ 2 x ^ 2 + a _ 0 , \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} \\left \\langle \\sum _ { k = 0 } ^ { n } \\frac { t ^ { k } a ( D ) ^ { k } } { k ! } u , \\phi \\right \\rangle \\underset { n \\to \\infty } { \\longrightarrow } \\left \\langle e ^ { t a ( D ) } u , \\phi \\right \\rangle . \\end{align*}"}
-{"id": "9538.png", "formula": "\\begin{align*} E _ H ( x ) = \\sum _ { h \\in H } x _ h h . \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , X _ { 2 , \\ , j + 2 , \\ , 2 } ] = 0 . \\end{align*}"}
-{"id": "490.png", "formula": "\\begin{align*} \\sigma _ a ( F ) = \\sum _ { \\substack { H \\leq F , \\\\ } } | H | ^ a . \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} \\frac { C ( a \\mid \\nu + N ^ { - \\frac { \\alpha } { 2 } } \\sigma y ) } { C ( a \\mid \\nu ) } = \\frac { \\log ( \\frac { a } { \\nu } ) \\sqrt { 2 \\pi a } } { \\log ( \\frac { a } { \\nu + N ^ { - \\frac { \\alpha } { 2 } } \\sigma y } ) \\sqrt { 2 \\pi a } } \\to 1 . \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{align*} g _ n = \\sum _ { i _ 1 < i _ 2 < \\cdots < i _ n } \\partial _ { \\theta _ { i _ 1 } } \\partial _ { \\theta _ { i _ 2 } } \\cdots \\partial _ { \\theta _ { i _ n } } , ~ ~ { \\rm f o r } ~ ~ n = 1 , 2 , \\cdots , k ~ ~ { \\rm a n d } ~ ~ g _ 0 = 1 . \\end{align*}"}
-{"id": "7729.png", "formula": "\\begin{align*} \\left \\{ s \\in 2 ^ \\omega \\ | \\ | \\{ j : \\ s _ { | [ n _ j , n _ { j + 1 } ) } \\equiv r _ { | [ n _ j , n _ { j + 1 } ) } \\} | = \\infty \\right \\} \\subseteq R . \\end{align*}"}
-{"id": "3837.png", "formula": "\\begin{align*} | R _ { k } ( 2 ^ { k } m + j ) | = | R _ { 1 } ( 2 m ) | \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{align*} { \\gamma _ k } = \\frac { { P _ k } { \\left | { { h _ k } } \\right | ^ 2 } } { \\mathcal N _ 0 } , \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} \\hat \\theta _ 0 ( t ) = T _ s + A _ 1 e ^ { - t } , \\ , \\ , \\ , \\hat h _ 0 = \\hat h _ { 0 0 } e ^ { - t } , \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} \\sup _ { j \\ge 0 } \\ \\| P _ { n } T ^ { \\ , j } P _ { l } \\ , x \\| \\le \\sum _ { k = b _ { l } } ^ { b _ { l + 1 } - 1 } | x _ { k } | \\ \\sup _ { j \\ge 0 } \\| P _ { n } T ^ { \\ , j } \\ , e _ { k } \\| \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{align*} - c \\tilde { R } S & = - c ( P ' S ' - P S '' ) = P ' ( Q ' f + Q f ' ) - P ( Q '' f + 2 Q ' f ' + Q f '' ) \\\\ & = ( P ' Q ' - P Q '' ) f + ( P ' Q - 2 P Q ' ) f ' - P Q f '' = R Q f = - c R S , \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} ( \\log \\theta ) ''' = \\frac { \\theta ''' } { \\theta } - 3 \\frac { \\theta '' \\theta ' } { \\theta ^ 2 } + 2 \\frac { ( \\theta ' ) ^ 3 } { \\theta ^ 3 } , \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} \\langle 1 \\cdots M | \\mathcal { B } ^ \\prime ( z _ 1 ) \\cdots \\mathcal { B } ^ \\prime ( z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle = \\frac { ( 1 + t ^ { - 1 } z _ 1 ^ 2 ) } { t z _ 1 } \\sum _ { \\overline { y } } \\frac { z _ 1 ^ { \\overline { y } } - z _ 1 ^ { - \\overline { y } } } { z _ 1 - z _ 1 ^ { - 1 } } \\langle \\overline { y } | \\mathcal { B } ^ \\prime ( z _ 2 ) \\cdots \\mathcal { B } ^ \\prime ( z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle , \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{align*} f ^ { \\alpha a } _ { i j } = \\sum \\limits _ { \\tiny \\begin{matrix} b + c = a \\\\ \\beta + \\gamma = \\alpha \\end{matrix} } C _ { \\alpha } ^ \\beta C _ a ^ b ( \\partial _ i V ^ { ( \\beta , b ) } \\partial _ j V ^ { ( \\gamma , c ) } - \\partial _ i H ^ { ( \\beta , b ) } \\cdot \\partial _ j H ^ { ( \\gamma , c ) } ) , \\end{align*}"}
-{"id": "8628.png", "formula": "\\begin{align*} X _ { \\alpha _ { n } } = \\bigcup _ { i = 1 } ^ { n } X _ { \\alpha _ { i } } \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} f \\circ ( y + \\delta ) = \\sum _ { i \\in \\N } a _ { i } \\delta ^ { i } . \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } S ( B | A ' B ' ) _ { \\hat { \\gamma } _ { B A ' B ' } ^ { ( n ) } } = 1 + \\ln b \\ ; . \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} x _ t = L ( q ^ { - 1 } ) v _ t , \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} \\omega ( g , h ) \\ ; : = \\ ; \\langle S ^ * g , h \\rangle - \\langle g , S ^ * h \\rangle \\ , , g , h \\in \\mathcal { D } ( S ^ * ) \\ , . \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} \\alpha = \\frac { a - d + \\sqrt { ( a + d ) ^ 2 - 4 } } { 2 c } , \\beta = \\frac { a - d - \\sqrt { ( a + d ) ^ 2 - 4 } } { 2 c } \\end{align*}"}
-{"id": "5874.png", "formula": "\\begin{align*} i _ 2 = \\frac { \\langle | \\hat { \\rho } | ^ 4 \\rangle } { \\langle | \\hat { \\rho } | ^ 2 \\rangle ^ 2 } , \\end{align*}"}
-{"id": "714.png", "formula": "\\begin{align*} ( j ^ + _ 1 ) ^ { n _ { 1 } } ( j ^ + _ 2 ) ^ { n _ { 2 } } \\cdots ( j ^ + _ r ) ^ { n _ { r } } = 0 , ( j ^ - _ 1 ) ^ { n _ { 1 } } ( j ^ - _ 2 ) ^ { n _ { 2 } } \\cdots ( j ^ - _ r ) ^ { n _ { r } } = 0 , { \\rm f o r } ~ ~ n _ 1 + n _ 2 + \\cdots + n _ { r } = k + 1 . \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} \\tau _ { \\widehat { \\alpha } } ( \\gamma ) = \\sum _ { j _ 1 , \\ldots , j _ { \\hat { \\ell } } } c ^ \\gamma _ { j _ 1 , \\ldots , j _ { \\hat { \\ell } } } \\ , \\tau _ { \\widehat { \\alpha } _ 1 - 1 } ( \\theta _ { j _ 1 } ) \\cdots \\tau _ { \\widehat { \\alpha } _ { \\hat { \\ell } } - 1 } ( \\theta _ { j _ { \\hat { \\ell } } } ) \\ . \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} q _ { p , \\texttt { h o r } } ( 0 , 0 ) = p _ 2 ( 1 _ { H _ 1 } , 1 _ { H _ 2 } ) & & q _ { p , \\texttt { h o r } } ( 1 , 0 ) = p _ 2 ( \\texttt { r i g h t } _ 1 ( p _ 1 ( 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ) , 1 _ { H _ 2 } ) \\\\ q _ { p , \\texttt { v e r } } ( 0 , 0 ) = p _ 2 ( 1 _ { H _ 1 } , 1 _ { H _ 2 } ) & & q _ { p , \\texttt { v e r } } ( 0 , 1 ) = p _ 2 ( 1 _ { H _ 1 } , \\texttt { r i g h t } _ 2 ( p _ 1 ( 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ) ) \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\frac { B _ { n } \\left ( x \\right ) } { n ! } z ^ { n } = \\frac { z e ^ { z x } } { e ^ { z } - 1 } . \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{align*} I : = \\mathbb { E } \\left ( \\int _ { \\mathbb { R } ^ 2 } \\left ( \\int _ 0 ^ T \\int _ 0 ^ T \\phi ( x + \\xi ^ 1 _ r ) | y + \\xi ^ 2 _ s - x - \\xi ^ 1 _ r | ^ { \\gamma - 1 } d s d r \\right ) ^ 2 d x d y \\right ) \\leq C T ^ 2 < \\infty , \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} \\ B u ^ { \\left ( 1 \\right ) } \\left ( t \\right ) + A u \\left ( t \\right ) = f \\left ( t \\right ) , \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{align*} P ^ { s , N } _ { H P } ( t ) ( x _ 1 , \\cdots , x _ N ; y _ 1 , \\cdots , y _ N ) = e ^ { - \\lambda _ { N , s } t } \\frac { \\Delta _ N ( y ) } { \\Delta _ N ( x ) } F _ t ( x _ 1 , \\cdots , x _ N ; y _ 1 , \\cdots , y _ N ) , \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} T ( \\pi ) = T ( \\pi , \\omega ) = \\sum _ { i = 1 } ^ { r } t ( e _ i , \\omega ) . \\end{align*}"}
-{"id": "9776.png", "formula": "\\begin{align*} e _ { \\epsilon } ( \\tilde { u } _ { \\epsilon } ) = e _ { \\epsilon } ( u _ { \\epsilon } ) - \\frac { 1 } { 2 } | j \\phi _ { \\epsilon } | ^ 2 - \\langle d ^ * \\xi _ { \\epsilon } + h _ { \\epsilon } ' , j \\phi _ { \\epsilon } \\rangle - \\frac { 1 } { 2 } ( 1 - | u _ { \\epsilon } | ^ 2 ) | j \\phi _ { \\epsilon } | ^ 2 , \\end{align*}"}
-{"id": "7085.png", "formula": "\\begin{align*} B _ { j } u = \\sum \\limits _ { \\left \\vert \\beta \\right \\vert \\leq m _ { j } } \\ b _ { j \\beta } \\left ( y \\right ) D _ { y } ^ { \\beta } u \\left ( x , y , t \\right ) = 0 x \\in R ^ { n } , y \\in \\partial G , j = 1 , 2 , . . . , m , \\end{align*}"}
-{"id": "5034.png", "formula": "\\begin{align*} ( _ E ( n , k , \\Omega ) ) ^ \\bot = _ E ( n , n - k , \\Omega , v ) \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} \\omega ( y ) ( x \\cdot y ) = ( \\omega ( y ) \\cdot x ) y - ( \\omega ( y ) \\times y ) \\times x . \\end{align*}"}
-{"id": "8713.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { L } f ( x ) & = \\frac { 1 } { 2 } \\sum _ { j , k = 1 } ^ { d } a _ { j k } ( x ) \\frac { \\partial ^ { 2 } } { \\partial x _ { j } \\partial x _ { k } } f ( x ) + \\sum _ { j = 1 } ^ { d } b _ { j } ( x ) \\frac { \\partial } { \\partial x _ { j } } f ( x ) \\\\ & + \\int _ { \\R ^ { d } _ { 0 } } [ f ( x + y ) - f ( x ) - y \\cdot D f ( x ) ] \\nu ( x , \\d y ) , \\end{aligned} \\end{align*}"}
-{"id": "6928.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ \\Omega \\left [ | \\nabla t _ \\varepsilon | ^ 2 + c _ 1 \\frac { t _ \\varepsilon ^ 2 } { \\varepsilon ^ 2 } \\right ] \\leq & C \\varepsilon \\left ( 1 + \\int _ \\Omega \\left [ | \\nabla t _ \\varepsilon | ^ 2 + | \\nabla \\psi _ \\varepsilon | ^ 2 + \\frac { t _ \\varepsilon ^ 2 } { \\varepsilon ^ 2 } \\right ] \\right ) ^ { 1 / 2 } + C \\| t _ \\varepsilon \\| _ \\infty \\left ( 1 + \\int _ \\Omega | \\nabla \\psi _ \\varepsilon | ^ 2 \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{align*} y ( t ) = \\sqrt [ 1 0 ] { 1 - \\frac { 5 } { 3 } t } . \\end{align*}"}
-{"id": "1990.png", "formula": "\\begin{align*} \\langle \\overline { x _ 1 } \\cdots \\overline { x _ { k } } | \\widetilde { A } ^ \\prime ( z ) | \\overline { y _ 1 } \\cdots \\overline { y _ { k } } \\rangle = & \\prod _ { j = 1 } ^ k \\delta _ { \\overline { x _ j } \\overline { y _ j } } , \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} R ^ e _ n ( s _ { v _ k } ) = \\prod _ { w ( i , j ) \\in I ( n , e , k ) } s _ { w ( i , j ) } , \\end{align*}"}
-{"id": "9775.png", "formula": "\\begin{align*} j \\tilde { u } _ { \\epsilon } : = j u _ { \\epsilon } - | u _ { \\epsilon } | ^ 2 j \\phi _ { \\epsilon } , \\end{align*}"}
-{"id": "7161.png", "formula": "\\begin{align*} \\rho ( \\mu _ i ) = \\begin{pmatrix} w & t _ i \\\\ & \\\\ 0 & w ^ { - 1 } \\end{pmatrix} , \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{align*} \\pi ( K _ { k } ) v _ { i } = q ^ { \\delta _ { i , k } - \\delta _ { i , k + 1 } } v _ { i } , \\pi ( E _ { k } ) v _ { i } = \\delta _ { i } ^ { k + 1 } q ^ { - 1 / 2 } v _ { i - 1 } , \\pi ( F _ { k } ) v _ { i } = \\delta _ { i } ^ { k } q ^ { 1 / 2 } v _ { i + 1 } . \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} D _ { n , k } ( a , x ) = \\sum _ { i = 0 } ^ { \\lfloor \\frac n 2 \\rfloor } \\frac { n - k i } { n - i } \\dbinom { n - i } { i } ( - x ) ^ { i } a ^ { n - 2 i } , \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{align*} \\| f \\| _ 2 = \\| f _ { d k } \\| _ 2 \\| f _ \\theta \\| _ 2 \\simeq R ^ { - 1 / 4 } | \\Omega | ^ { 1 / 2 } \\ , , \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} X _ l : = \\mathbb { E } ( { U } _ n | { \\cal G } _ l ) - \\mathbb { E } ( { U } _ n | { \\cal G } _ { l - 1 } ) . \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{align*} a _ { v c } & = \\frac { m } { r _ { m i n } } + K _ d \\\\ b _ { v c } & = 2 K _ d v _ { a i r } \\\\ c _ { v c } & = K _ d v _ { a i r } ^ 2 - f _ { p l a n a r } \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} c _ { \\alpha } ^ - = \\alpha ^ 2 \\psi _ { \\alpha , 0 } ( 0 ) . \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} g _ { n } = \\begin{cases} C _ { n - 1 } & n \\ge 1 \\\\ 1 & n = 0 \\end{cases} \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} & D _ { n , k } ( 1 , x ) \\cr & = \\frac { k } { 8 } \\ , ( 1 - 4 x ) ^ \\frac { p ^ { l _ 1 } + p ^ { l _ 2 } + p ^ { l _ 3 } - 1 } { 2 } + \\frac { ( 2 - k ) } { 8 } \\ , [ ( 1 - 4 x ) ^ \\frac { p ^ { l _ 1 } + p ^ { l _ 2 } } { 2 } + ( 1 - 4 x ) ^ \\frac { p ^ { l _ 1 } + p ^ { l _ 3 } } { 2 } + ( 1 - 4 x ) ^ \\frac { p ^ { l _ 2 } + p ^ { l _ 3 } } { 2 } ] \\cr & + \\frac { k } { 8 } [ ( 1 - 4 x ) ^ \\frac { p ^ { l _ 1 } - 1 } { 2 } + ( 1 - 4 x ) ^ \\frac { p ^ { l _ 2 } - 1 } { 2 } + ( 1 - 4 x ) ^ \\frac { p ^ { l _ 3 } - 1 } { 2 } ] + \\frac { ( 2 - k ) } { 8 } . \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} B _ i = \\frac { 1 } { | C | } | \\{ ( u , v ) \\in C \\times C : | u \\cap v | = k - i \\} | . \\end{align*}"}
-{"id": "5366.png", "formula": "\\begin{align*} f _ 0 ( 1 ) = 1 , f _ 0 ( 2 ) = a , f _ 0 ( n + 2 ) = a f _ 0 ( n + 1 ) - b f _ 0 ( n ) , ( n \\geq 1 ) . \\end{align*}"}
-{"id": "9725.png", "formula": "\\begin{align*} | | | x | | | = \\| x \\| _ \\infty + \\sum _ { n \\in \\N } r _ n \\left | \\langle x , v _ n \\rangle \\right | \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{align*} ( k - 1 ) a ^ 2 + ( n - k - 1 ) b ^ 2 + 1 = a b \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} H ( Y _ 1 , Y _ 3 | M ) = 1 , ~ H ( Y _ 1 , Y _ 3 ) = \\frac { 1 } { 2 } \\log 2 + \\frac { 1 } { 2 } \\log 4 = \\frac { 3 } { 2 } , \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} K _ i ^ * = K _ i , E _ i ^ * = K _ i F _ i , F _ i ^ * = E _ i K _ i ^ { - 1 } . \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} u ( t ) = e ^ { t \\Delta } [ u _ 0 + t \\P \\theta _ 0 e _ 3 ] - B _ 1 ( u , u ) - B _ 2 ( u , \\theta ) , \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} j = k \\leqq 2 . \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} t = \\frac 1 3 d ^ c F . \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{align*} I ( A : B | M ) : = S ( A | M ) + S ( B | M ) - S ( A B | M ) = 0 \\ ; , \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} B _ K = \\int _ { 0 ^ - } ^ { K - 0 } \\sum _ { n = 1 } ^ { \\infty } \\left [ 1 - V ( K - w - \\sigma + n T ) \\right ] d W ( w ) . \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} \\overset { \\cdot } { x } \\ = f \\left ( x \\right ) , \\end{align*}"}
-{"id": "688.png", "formula": "\\begin{align*} \\bigg ( \\frac { \\partial } { \\partial \\eta } \\bigg ) ^ { k + 1 } = 0 . \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { N - 1 } \\big | \\mathcal { E } _ { N , \\ell } ^ { h , k , 2 , 2 } \\big | \\le \\frac { C ( 1 + | x | _ { L ^ p } ) ^ { K } } { \\tau ^ { 2 \\kappa } } \\bigl ( 1 + t _ { N - 1 } ^ { \\frac { 1 } { 2 } - \\frac { 1 } { 2 q } - \\kappa - \\beta - \\gamma } \\bigr ) \\Bigl ( | ( - A ) ^ { - \\beta } h | _ { L ^ { 2 q } } | ( - A ) ^ { - \\gamma } k | _ { L ^ { 2 q } } + \\frac { \\Delta t } { t _ { N } } | h | _ { L ^ { 2 q } } | k | _ { L ^ { 2 q } } \\Bigr ) . \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} \\dot { y } y ^ { - 1 } + \\left ( \\dot { y } y ^ { - 1 } \\right ) ^ * = \\left [ \\left ( y \\phi y ^ { - 1 } \\right ) ^ * , y \\phi y ^ { - 1 } \\right ] - \\rho + f \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{align*} g ( t ) = \\begin{pmatrix} b _ 0 + a _ 0 t & ( \\sqrt { 3 } - ( a _ 0 t + b _ 0 ) s ) / { \\sqrt { 3 } } \\\\ - 1 & s / { \\sqrt { 3 } } \\end{pmatrix} . \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{align*} r ( x ) : = \\abs { D ( x ) - \\delta } ^ 2 \\ , \\abs { D ( x ) + \\delta } ^ 2 \\end{align*}"}
-{"id": "9896.png", "formula": "\\begin{align*} h : = d r ^ 2 + \\sigma ^ 2 ( r ) d \\theta _ { S ^ { 2 n - 1 } } ^ 2 + \\tau ^ 2 ( r ) d z ^ 2 , \\end{align*}"}
-{"id": "2774.png", "formula": "\\begin{align*} | \\sharp S ^ \\prime ( \\varepsilon , K , \\Lambda _ d , B ) - \\sharp T ( \\varepsilon , K , \\Lambda _ d , B ) | & \\ll b ^ 2 \\det ( \\Lambda _ d ) B ^ { 2 - \\frac { 2 } { r } } + 1 . \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} X ^ { [ \\xi ] } \\cap U = X ^ { [ \\eta ] } \\cap U \\quad T _ x ^ 2 \\Phi ^ \\xi = T _ x ^ 2 \\Phi ^ \\eta \\end{align*}"}
-{"id": "7816.png", "formula": "\\begin{align*} u _ t + b u _ x + u _ { x x x } + u _ { x y y } = f - ( g ( v ) ) _ x - ( \\psi v ) _ x \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} \\Phi [ e ^ { - \\alpha t } w ] = e ^ { - \\alpha t } ( \\Phi [ w ] - \\alpha w ) . \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{align*} \\langle \\overline { x _ 1 } \\cdots \\overline { x _ { k - 1 } } | \\widetilde { B } ^ \\prime ( z ) | \\overline { y _ 1 } \\cdots \\overline { y _ { k } } \\rangle = & ( - 1 ) ^ { j - 1 } z ^ { \\overline { y _ j } - 1 } , \\end{align*}"}
-{"id": "5302.png", "formula": "\\begin{align*} { I _ K } = \\sum \\nolimits _ { k = 1 } ^ K { { { \\log } _ 2 } \\left ( { 1 + { \\gamma _ k } } \\right ) } . \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} V : = \\nabla ^ \\perp \\left [ \\left ( \\frac { 1 } { \\gamma _ 2 } \\log { | \\cdot | } \\right ) * \\omega - \\frac { 1 } { \\gamma _ 2 } \\varpi \\log { | \\cdot - \\xi ^ * | } \\right ] \\textrm { i f } n = 2 , \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} [ x , y ] = x _ 1 y _ 4 - \\frac { 1 } { 3 } x _ 2 y _ 3 + \\frac { 1 } { 3 } x _ 3 y _ 2 - x _ 4 y _ 1 . \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} \\overline { R } _ 1 ( v ) \\leq R _ 0 = R _ 0 ( B _ 0 ) . \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{align*} y _ i = x _ i ^ T \\beta _ 0 + \\xi _ i , \\ \\ \\ z _ i = x _ i + w _ i , \\ \\ \\ i = 1 , \\ldots , n \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} \\phi ^ { o } ( f ) ( v ) = f ( \\phi ( v ) ) \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{align*} \\left ( f ^ { \\prime } f _ { c _ { 1 } } + B _ { 1 } \\right ) \\left ( z _ { 0 } \\right ) = 0 \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} J ( A | M ) _ { \\hat { \\rho } _ { A M } } = \\left . \\frac { \\mathrm { d } } { \\mathrm { d } t } S ( A | M ) _ { ( \\mathcal { N } _ A ( t ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) } \\right | _ { t = 0 } \\ ; . \\end{align*}"}
-{"id": "1602.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\big ( \\deg _ { \\widetilde { X } } ( v _ { i } ) - 2 \\big ) \\leq \\theta \\cdot N \\cdot \\big ( \\deg _ { \\widetilde { Y } } ( a ) - 2 \\big ) \\cdot \\big ( \\deg _ { \\widetilde { Z } } ( b ) - 2 \\big ) \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} \\int _ { 0 + } \\frac { \\d r } { \\rho ( r ) } = \\infty , \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow \\partial \\Omega } s _ \\Omega ( z ) = 1 . \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{align*} E _ { T _ * , \\rho } = \\{ w \\in E _ { T _ * } ; \\| w \\| _ { E _ { T _ * } } \\leq \\rho \\} \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } a _ n t ^ n = ( 1 - t ) ^ { - m ^ + } ( 1 + t ) ^ { - m ^ { - } } = \\sum _ { i = 1 } ^ { m ^ + } \\frac { A _ i } { ( 1 - t ) ^ i } + \\sum _ { i = 1 } ^ { m ^ - } \\frac { B _ i } { ( 1 + t ) ^ i } , \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} E [ \\int _ 0 ^ T \\textbf { 1 } _ { D ^ c } d A _ t ] = E [ \\int _ 0 ^ T \\textbf { 1 } _ { D } d A ' _ t ] = 0 , \\end{align*}"}
-{"id": "6070.png", "formula": "\\begin{align*} [ L _ 2 , \\ , [ L _ { - 2 } , \\ , L _ { - 2 } ] ] = 0 \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{align*} \\| \\sum _ { j = 0 } ^ \\infty \\tilde S _ { j + \\ell } f \\tilde S _ j g \\| _ { B ^ s _ { p , \\infty } ( \\real ^ { d _ s } ) } \\le C \\sum _ { j = 0 } ^ { \\infty } \\sup _ { k \\ge 0 } 2 ^ { k s } \\| \\tilde S _ k ( \\tilde S _ { k + j + \\ell } f \\tilde S _ { k + j } g ) \\| _ { L _ p ( \\real ^ { d _ s } ) } \\ , . \\end{align*}"}
-{"id": "9734.png", "formula": "\\begin{align*} \\ \\sum \\limits _ { i = 1 } ^ { n } x _ { i 1 } u _ { i } ^ { \\prime \\prime } = 0 . \\end{align*}"}
-{"id": "8362.png", "formula": "\\begin{align*} t ^ { ( k ) } ( q _ i ) = t ^ { ( k ) } ( q _ i , \\omega ) = \\min ( t ( q _ i , \\omega ) , k ^ { \\alpha } ) , \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} p _ n ( T , U ) = \\mathop \\Pi _ { j = 1 } ^ { n + 1 } ( u ' _ j T - t ' _ j U ) , \\\\ q _ n ( X , Y ) = \\mathop \\Pi _ { j = 1 } ^ { n + 1 } ( y ' _ j X - x ' _ j Y ) . \\end{align*}"}
-{"id": "9362.png", "formula": "\\begin{align*} \\sum _ { l , m } \\sum _ { i = 1 } ^ n | T _ l | | C _ { k , l , m } ^ { ( i ) } | & \\geq \\sum _ { l , m } | T _ l | | C _ { k , l , m } | - M n \\sum _ l | T _ l | \\\\ & \\geq \\bigg | \\bigcup _ l T _ l C _ { k , l } \\bigg | - \\beta | S _ k | \\\\ & \\geq ( 1 - 2 \\beta ) | S _ k | . \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{align*} \\mathcal { M } = \\underbrace { \\mathbb { S } ^ 1 \\times \\cdots \\times \\mathbb { S } ^ 1 } _ { N \\ \\mbox { t i m e s } } , \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} { } _ 3 F _ 2 \\bigg [ \\begin{matrix} \\alpha & \\beta & \\gamma \\\\ & \\alpha - \\beta + 1 & \\alpha - \\gamma + 1 \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { \\Gamma ( \\frac 1 2 \\alpha + 1 ) \\Gamma ( \\alpha - \\beta + 1 ) \\Gamma ( \\alpha - \\gamma + 1 ) \\Gamma ( \\frac 1 2 \\alpha - \\beta - \\gamma + 1 ) } { \\Gamma ( \\alpha + 1 ) \\Gamma ( \\frac 1 2 \\alpha - \\beta + 1 ) \\Gamma ( \\frac 1 2 \\alpha - \\gamma + 1 ) \\Gamma ( \\alpha - \\beta - \\gamma + 1 ) } . \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{align*} h = g - \\dfrac { 1 } { 4 L } p \\otimes p : A ^ { o } \\rightarrow T _ { 2 } ^ { 0 } M \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} | f ( x ) | + | x \\cdot \\nabla f ( x ) | \\le C | x | ^ { - \\frac { 2 } { \\alpha } } x \\not = 0 , \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{align*} \\gamma _ 2 ( t ) = v + \\left ( \\alpha + i ( e ^ { 2 t } + \\norm { v } ^ 2 ) \\right ) e _ 1 \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 \\ , \\left ( \\frac 1 3 \\right ) & x = \\infty \\ , \\left ( \\frac 1 3 \\right ) \\\\ \\overbrace { \\begin{matrix} 1 & 0 \\\\ \\omega & 0 \\\\ \\omega ^ 2 & 0 \\end{matrix} } & \\overbrace { \\begin{matrix} - t ^ { \\frac 1 3 } & 0 \\\\ - \\omega t ^ { \\frac 1 3 } & 0 \\\\ - \\omega ^ 2 t ^ { \\frac 1 3 } & 0 \\end{matrix} } \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "6313.png", "formula": "\\begin{align*} \\lefteqn { ( - 1 ) ^ n \\sum _ { j = 0 } ^ k ( \\lambda - 1 ) _ { k - j } \\binom { k } { j } \\left ( x ^ { j } f ( x ) \\right ) ^ { ( j + n ) } } \\\\ & = ( - 1 ) ^ n \\sum _ { j = 0 } ^ k ( \\lambda - 1 ) _ { k - j } \\binom { k } { j } \\sum _ { m = 0 } ^ { j } \\binom { n + j } { j - m } \\frac { j ! } { m ! } x ^ { m } f ^ { ( n + m ) } ( x ) \\\\ & = ( - 1 ) ^ n \\sum _ { m = 0 } ^ k \\left \\{ \\sum _ { j = m } ^ { k } ( \\lambda - 1 ) _ { k - j } \\binom { k } { j } \\binom { n + j } { j - m } \\frac { j ! } { m ! } \\right \\} x ^ { m } f ^ { ( n + m ) } ( x ) . \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\left \\lfloor ( n - 1 ) / 2 \\right \\rfloor } ( ( n - j - 1 ) - j + 1 ) = \\sum _ { j = 0 } ^ { \\left \\lfloor ( n - 1 ) / 2 \\right \\rfloor } ( n - 2 j ) , \\end{align*}"}
-{"id": "2462.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow \\partial \\Omega } \\max _ { v \\in T _ z \\Omega \\setminus \\{ 0 \\} } \\abs { H _ { b _ \\Omega } ( v ) - \\frac { - 4 } { d + 1 } } = 0 \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} D _ { n , k } ( x _ 1 + x _ 2 , x _ 1 x _ 2 ) & = k \\ , \\Big [ \\displaystyle \\frac { x _ 1 ^ n x _ 2 - x _ 1 x _ 2 ^ n } { x _ 1 - x _ 2 } \\Big ] + D _ n ( x _ 1 + x _ 2 , x _ 1 x _ 2 ) . \\end{align*}"}
-{"id": "5743.png", "formula": "\\begin{align*} \\hat { \\tau } _ { A M X } ( s , t ) = ( \\mathcal { N } _ A ( s ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\sigma } _ { A M X } ( t ) ) \\ ; , \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} \\lambda = - \\frac { X _ 1 + X _ 3 } { X _ 1 + X _ 2 + X _ 3 + X _ 4 } \\in ( - 1 , 0 ) \\end{align*}"}
-{"id": "9561.png", "formula": "\\begin{align*} \\mathcal { F } \\left [ \\frac { \\sin \\pi \\left ( m + 2 x \\right ) } { \\pi \\left ( m + 2 x \\right ) } \\right ] \\left ( \\xi \\right ) = \\frac { e ^ { \\pi i m \\xi } } { 2 } \\chi _ { \\left ( - 1 , 1 \\right ) } \\left ( \\xi \\right ) . \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} m - 2 r + i _ 0 + \\sum _ { p = 1 } ^ { m - 1 } i _ p = m - 2 r + r = m - r \\leq \\frac { m } { 2 } \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{align*} \\frac { 1 } { N ^ 2 } \\sum \\limits _ { \\mathbf { m } \\in \\mathbb { Z } ^ 3 } \\Big ( \\prod \\limits _ { j = 1 } ^ 4 f _ j ( \\Xi ( \\mathbf { m } ) _ j + \\widetilde { \\mathbf { r } } _ j ) \\Big ) F ( \\Xi ( \\mathbf { m } ) + \\widetilde { \\mathbf { r } } ) 1 _ { [ - 1 0 , 1 0 ] ^ 2 } ( L ( \\Xi \\mathbf { m } + \\widetilde { \\mathbf { r } } ) ) \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} M = \\begin{bmatrix} R _ { 1 c } & R _ { 1 \\theta } & R _ { 1 h } \\\\ R _ { 2 c } & R _ { 2 \\theta } & R _ { 2 h } \\\\ R _ { 3 c } & R _ { 3 \\theta } & R _ { 3 h } \\end{bmatrix} = \\begin{bmatrix} R _ { 1 c } & \\lambda & R _ { 1 h } \\\\ T ' ( c ) & - 1 & 0 \\\\ R _ { 3 c } & 0 & - 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} S ( \\hat { \\rho } \\| \\hat { \\sigma } ) = \\mathrm { T r } \\left [ \\hat { \\rho } \\left ( \\ln \\hat { \\rho } - \\ln \\hat { \\sigma } \\right ) \\right ] \\end{align*}"}
-{"id": "6687.png", "formula": "\\begin{align*} \\Bigg \\lvert \\int \\limits _ { \\lvert x - e \\rvert + \\lvert x \\rvert = 2 \\tau } \\frac { q ( x ) } { \\lvert 2 \\tau x - \\lvert x \\rvert e \\rvert } d S _ { x } \\Bigg \\rvert \\leq K \\int \\limits _ { \\lvert x - e \\rvert + \\lvert x \\rvert \\leq 2 \\tau } \\frac { \\lvert q ( x ) \\rvert } { \\lvert x \\rvert \\lvert x - e \\rvert } d x , \\ \\forall \\tau \\in ( 1 / 2 , T / 2 ] . \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{align*} n _ { i j } ^ { 2 } = \\left \\{ \\begin{array} { l r } ( - 1 ) ^ { j - i } { { k + j - i } \\choose { k } } , & { \\rm i f } ~ ~ i \\leq j , \\\\ 0 , & { \\rm i f } ~ ~ i > j , \\end{array} \\right . \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| w _ k ( t ) - w ( t ) \\| _ { 2 , \\Omega _ { R + i } } = 0 , \\forall t \\in ( 0 , T ) \\setminus J _ i . \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{gather*} ( 1 - 2 \\rho \\cos ( \\alpha + \\beta ) + \\rho ^ { 2 } ) ( ( 1 - 2 \\rho \\cos ( \\alpha - \\beta ) + \\rho ^ { 2 } ) \\allowbreak = \\allowbreak \\\\ ( 1 + \\rho ^ { 2 } ) ^ { 2 } - 2 \\rho ( 1 + \\rho ^ { 2 } ) ( \\cos ( \\alpha + \\beta ) + \\cos ( \\alpha - \\beta ) ) + 4 \\rho ^ { 2 } \\cos ( \\alpha + \\beta ) \\cos ( \\alpha - \\beta ) . \\end{gather*}"}
-{"id": "6887.png", "formula": "\\begin{align*} v = \\frac { ( 1 + t ) ^ { 2 } - 2 y t - ( 1 + t ) \\sqrt { ( 1 + t ) ^ { 2 } - 4 y t } } { 2 y t } , \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} x ^ 3 y = y x ^ 3 \\ , . \\end{align*}"}
-{"id": "595.png", "formula": "\\begin{align*} f ( a _ { \\max } ) = a _ { \\max } \\left ( 1 + \\frac { 2 } { Q \\sqrt \\kappa } - \\frac { a _ { \\max } } { Q ^ 2 } \\right ) = \\frac { Q ^ 2 } { 4 } \\left ( 1 + \\frac { 2 } { Q \\sqrt \\kappa } \\right ) ^ 2 = \\frac { 4 } { \\kappa } + \\frac { \\kappa } { 1 6 } + 1 = q \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\sum _ { l = 1 } ^ { N } X _ l \\right ) ^ 2 = \\sum _ { l = 1 } ^ { N } \\mathbb { E } X _ l ^ 2 + 2 \\sum _ { i = 1 } ^ { N - 1 } \\sum _ { j = i + 1 } ^ { N } \\mathbb { E } X _ i X _ j . \\end{align*}"}
-{"id": "7915.png", "formula": "\\begin{align*} t \\mapsto \\{ x : u ( x , t ) = c \\} \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{align*} \\Omega _ 1 ( p ) = & { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 ( 1 - p ) & a + 1 - p \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] \\\\ = & ( 1 - z ) ^ { \\frac { p - 1 } { 2 } } \\cdot { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 ( 1 - p ) & p - a \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac { z } { z - 1 } \\bigg ] = ( 1 - z ) ^ { \\frac { p - 1 } { 2 } - a } \\cdot \\Psi _ 2 ( p , p ) . \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{align*} L ^ q ( \\Omega ) = L ^ q _ \\sigma ( \\Omega ) \\oplus \\{ \\nabla p \\in L ^ q ( \\Omega ) ; \\ , p \\in L ^ q _ { l o c } ( \\overline { \\Omega } ) \\} \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} ( b _ { \\ell + 1 , i } ^ x - a _ { \\ell + 1 , i } ^ x ) \\leq \\overline { q } ( b _ { \\ell , i } ^ x - a _ { \\ell , i } ^ x ) , \\quad i = 1 , \\ldots , d \\ell = 0 , \\ldots , L - 1 . \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} ( I _ C \\otimes \\Delta _ C ) \\Delta _ C ( c ) = ( \\Delta _ C \\otimes I _ C ) \\Delta _ C ( c ) . \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} \\sum _ { d = 0 } ^ { \\infty } ( d + 2 ) ^ u F _ { d + 2 } ( n ) \\rightarrow \\sum _ { d = 0 } ^ { \\infty } ( d + 2 ) ^ u F _ { d + 2 } ( \\infty ) \\end{align*}"}
-{"id": "6137.png", "formula": "\\begin{align*} X _ { 2 , \\ , j + 2 , \\ , 1 4 } \\cap L _ { j + 2 } = 0 . \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} \\frac { 1 } { N ^ { h - m } } \\int \\limits _ { \\mathbf { x } \\in \\mathbb { R } ^ h } F ( \\sum \\limits _ { i = 1 } ^ { h } x _ i \\mathbf { v ^ { ( i ) } } ) G ( L ( \\sum \\limits _ { i = 1 } ^ { h } x _ i \\mathbf { v ^ { ( i ) } } ) ) ( \\prod \\limits _ { j = 1 } ^ { d } g _ j ( \\xi _ j ( \\Phi ( \\mathbf { x _ 1 ^ { h - m } } ) + \\sum \\limits _ { i = h - m + 1 } ^ { h } x _ i \\mathbf { v } ^ { \\mathbf { ( i ) } } ) + \\widetilde { \\mathbf { r } } _ j ) ) \\ , d \\mathbf { x } , \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{align*} W \\cong P \\bigl ( - \\xi _ 1 z _ 1 - \\xi _ 2 z _ 2 \\bigr ) : = \\bigl \\{ f \\in R \\ , \\big | \\ , \\exp ( - \\xi _ 1 z _ 1 - \\xi _ 2 z _ 2 ) f \\ ; \\mbox { \\rm i s } \\ ; ( \\Pi , \\underline { \\mu } ) - \\mbox { \\rm q u a s i - - i n v a r i a n t } \\bigr \\} . \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} L ^ { g _ 0 } f = C ( J , [ g ] ) - s ^ H _ { g _ 0 } \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{align*} \\binom { m - k + 1 } { 2 } + k ( n - k ) + \\binom { n - k + 1 } { 2 } & = \\binom { m } { 2 } + \\binom { n } { 2 } - ( k + 1 ) m + m + n \\\\ & \\equiv \\binom { m } { 2 } + \\binom { n } { 2 } \\bmod 2 . \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{align*} \\nabla _ i ^ \\perp ( \\partial _ t H _ j - \\nabla _ j V ) & = - \\nabla _ l ^ \\perp V \\nabla _ l \\nabla _ i ^ \\perp H _ j + \\nabla _ l \\nabla _ i ^ \\perp V \\nabla _ l ^ \\perp H _ j \\\\ & = \\nabla _ l V \\nabla _ l ^ \\perp \\nabla _ i ^ \\perp H _ j + \\nabla _ l \\nabla _ i ^ \\perp V \\nabla _ l ^ \\perp H _ j \\\\ & = \\nabla _ i ^ \\perp ( \\nabla _ l V \\nabla _ l ^ \\perp H _ j ) , \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{align*} d X ^ { ( n ) } _ i ( t ) = \\sqrt { 2 } | X _ i ^ { ( n ) } ( t ) | d \\beta ^ { ( n ) } _ i ( t ) + \\left [ \\left ( 2 - 2 n - 2 \\Re ( s ) \\right ) X _ i ^ { ( n ) } ( t ) \\right ] d t + \\frac { 1 } { 2 } d K _ i ^ { ( n ) , - } ( t ) - \\frac { 1 } { 2 } d K _ i ^ { ( n ) , + } ( t ) . \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} \\phi _ { X Y } = g ( A ( X ) , Y ) . \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{align*} S ( A ' B ' ) _ { \\hat { \\gamma } ^ { ( n ) } _ { C A ' B ' } } = g \\left ( \\frac { n ^ 2 } { a } - \\frac { 1 } { 2 } \\right ) + g \\left ( \\frac { n ^ 2 } { b } - \\frac { 1 } { 2 } \\right ) = \\ln \\frac { n ^ 4 } { a \\ , b } + 2 + \\mathcal { O } \\left ( \\frac { 1 } { n ^ 4 } \\right ) \\ ; . \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{gather*} X _ { 1 3 } X _ { 3 1 } = X _ { 3 1 } X _ { 1 3 } = 1 , \\\\ \\frac { q X _ { 0 1 } X _ { 1 3 } - q ^ { - 1 } X _ { 1 3 } X _ { 0 1 } } { q - q ^ { - 1 } } = 1 , \\quad \\frac { q X _ { 1 3 } X _ { 3 0 } - q ^ { - 1 } X _ { 3 0 } X _ { 1 3 } } { q - q ^ { - 1 } } = 1 , \\\\ \\frac { q X _ { 2 3 } X _ { 3 1 } - q ^ { - 1 } X _ { 3 1 } X _ { 2 3 } } { q - q ^ { - 1 } } = 1 , \\quad \\frac { q X _ { 3 1 } X _ { 1 2 } - q ^ { - 1 } X _ { 1 2 } X _ { 3 1 } } { q - q ^ { - 1 } } = 1 ; \\end{gather*}"}
-{"id": "4201.png", "formula": "\\begin{align*} \\frac { y _ { j } ^ { n + 1 } - y _ { j } ^ { n - 1 } } { 2 \\Delta t } = f _ { j } ^ { n } , \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} \\begin{aligned} E ( u ) & \\ge \\lambda _ 0 \\| u \\| _ { L ^ 1 ( R \\ , W _ \\phi ) } + \\lambda \\| u - f \\| _ { L ^ 1 ( \\R ^ n ) } \\\\ & \\ge \\lambda \\| f \\| _ { L ^ 1 ( \\R ^ n ) } = E ( 0 ) . \\end{aligned} \\end{align*}"}
-{"id": "5923.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { s _ { n , k } ^ { ( - 1 ) } q ^ n } { 1 - q ^ n } & = \\frac { q ^ k } { ( q ; q ) _ { \\infty } } . \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{align*} d X ( t ) = \\sqrt { 2 ( X ^ 2 ( t ) + 1 ) } d W ( t ) + \\left [ \\left ( 2 n + 2 \\Re ( s ) \\right ) X ( t ) - 2 \\Im ( s ) \\right ] d t \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} S ( a + b + 4 , b + 3 ) = - \\frac { 2 0 1 } 8 - \\frac { 1 7 3 } 8 a - \\frac { 2 1 } 2 b - 6 a ^ 2 - \\frac { 3 9 } 8 a b - \\frac 9 8 b ^ 2 - \\frac 1 2 a ^ 3 - \\frac 3 4 a ^ 2 b \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{align*} { \\mathbb E } _ { \\mathbb Q } \\left [ { \\rm e } ^ { \\vartheta S _ N } \\right ] = { \\displaystyle \\prod _ { i = 1 } ^ N M _ X \\left ( \\omega _ i ( N ) ( { \\rm e } ^ { \\vartheta + \\vartheta ^ \\star } - 1 ) \\right ) } \\left / { \\displaystyle \\prod _ { i = 1 } ^ N M _ X \\left ( \\omega _ i ( N ) ( { \\rm e } ^ { \\vartheta ^ \\star } - 1 ) \\right ) } \\right . . \\end{align*}"}
-{"id": "4766.png", "formula": "\\begin{align*} ( b _ 1 , f _ 1 ) ( b _ 2 , f _ 2 ) = ( b _ 1 b _ 2 , b _ 1 f _ 2 + f _ 1 b _ 2 ) \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{align*} \\ 1 _ V f _ * \\tau = f _ * ( \\ 1 _ { f ^ { - 1 } V } \\tau ) . \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} R _ p = \\left ( \\begin{array} { r r r r r r } 0 & 0 & 0 & 0 & 0 . 5 & 0 . 2 5 \\\\ 0 & 0 & 0 . 5 & 0 . 2 5 & 0 & 0 \\\\ 0 & 0 . 5 & 0 & 0 . 2 5 & 0 & 0 \\\\ 0 . 5 & 0 & 0 & 0 & 0 & 0 . 2 5 \\\\ 0 . 5 & 0 & 0 & 0 . 2 5 & 0 & 0 \\\\ 0 & 0 . 5 & 0 & 0 & 0 & 0 . 2 5 \\\\ 0 & 0 & 0 . 5 & 0 & 0 & 0 . 2 5 \\\\ 0 & 0 & 0 & 0 . 2 5 & 0 . 5 & 0 \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{align*} \\mathbf { k } ( Y ) = \\kappa ( \\mbox { D i a m } ( Y ) ) , \\end{align*}"}
-{"id": "9733.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { n } x _ { i 1 } ^ { 2 } + \\sum \\limits _ { i = 1 } ^ { n } ( u _ { i } ^ { \\prime \\prime } ) ^ { 2 } = 0 \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} \\eta ( 1 ) = 6 , \\eta ( 2 ) = 7 , \\eta ( 3 ) = 1 3 , \\eta ( 4 ) = 1 , \\eta ( 5 ) = 2 , \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} \\begin{pmatrix} M z \\\\ [ . 1 c m ] \\overline { N } z \\end{pmatrix} + \\begin{pmatrix} N \\bar { z } \\\\ [ . 1 c m ] \\overline { M } \\bar { z } \\end{pmatrix} = \\begin{pmatrix} p \\\\ [ . 1 5 c m ] \\overline { p } \\end{pmatrix} \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} x _ { n } = \\sum _ { m = 1 } ^ { n } \\frac { \\left ( - 1 \\right ) ^ { m } } { n ! } \\sum _ { p = 1 } ^ { m } \\left ( - 1 \\right ) ^ { m - p } \\binom { m } { p } S _ { p , n } = \\sum _ { p = 1 } ^ { n } \\frac { \\left ( - 1 \\right ) ^ { p } } { n ! } S _ { p , n } \\sum _ { m = p } ^ { n } \\binom { m } { p } . \\end{align*}"}
-{"id": "9629.png", "formula": "\\begin{align*} \\psi ^ { ( j ) } _ n ( b ) = \\frac { f ^ { ( j ) } _ n ( b ) } { G ^ { ( j ) } _ { n + 1 } } . \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{align*} \\pi _ * 1 = 0 ~ ~ , ~ ~ \\pi _ * H = 0 ~ ~ , ~ ~ \\pi _ * H ^ { i + 2 } = \\left [ - 2 ( - 2 ) ^ i + 3 ( - 3 ) ^ i \\right ] L ^ i \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{align*} R ( x ) = S ' ( x ) Q '' ( x ) - S '' ( x ) Q ' ( x ) . \\end{align*}"}
-{"id": "4635.png", "formula": "\\begin{align*} { \\displaystyle \\Psi ^ \\varepsilon ( \\mathbf { x } , t ) = 0 , \\mathbf { E } ^ { \\varepsilon } ( \\mathbf { x } , t ) \\times \\mathbf { n } = 0 , ( \\mathbf { x } , t ) \\in \\partial \\Omega \\times ( 0 , T ) . } \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} t _ 2 ( 2 ^ k - 4 ) = & t _ 2 ( 2 ^ { k - 1 } - 2 ) + t _ 2 ( 2 ^ { k - 1 } - 3 ) = t _ 2 ( 2 ^ { k - 2 } - 1 ) + t _ 2 ( 2 ^ { k - 2 } - 2 ) - 2 t _ 2 ( 2 ^ { k - 2 } - 2 ) \\\\ = & t _ 2 ( 2 ^ { k - 2 } - 1 ) - t _ 2 ( 2 ^ { k - 2 } - 2 ) = \\frac { 1 } { 3 } \\left ( ( - 2 ) ^ k - 1 ) \\right ) , \\\\ t _ 2 ( 2 ^ k - 5 ) = & - 2 t _ 2 ( 2 ^ { k - 1 } - 3 ) = 4 t _ 2 ( 2 ^ { k - 2 } - 2 ) = \\frac { 1 } { 3 } \\left ( 4 - ( - 2 ) ^ k \\right ) = 1 - t _ 2 ( 2 ^ k - 4 ) , \\\\ t _ 2 ( 2 ^ k - 3 ) = & - 2 t _ 2 ( 2 ^ { k - 1 } - 2 ) = \\frac { 1 } { 3 } \\left ( - 2 - ( - 2 ) ^ k \\right ) = - 1 - t _ 2 ( 2 ^ k - 4 ) . \\end{align*}"}
-{"id": "16.png", "formula": "\\begin{align*} \\omega ^ 2 - \\omega ( R _ { 1 c } - 1 ) - R _ { 1 c } - R _ { 1 h } R _ { 3 c } - \\lambda T ' ( c ) = 0 . \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} ( \\widehat { S } - z I ) ^ { - 1 } = ( S _ D - z I ) ^ { - 1 } + \\eta ( z ) | \\Phi ( z ) \\rangle \\langle \\Phi ( z ) | \\ , . \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{gather*} \\omega _ { \\mathbf { m } } : = \\frac { \\prod \\limits _ { i = 0 } ^ n y _ i ^ { m _ i - 1 } } { F _ { D , \\mu } ^ t } \\Omega \\end{gather*}"}
-{"id": "1937.png", "formula": "\\begin{align*} A ' ( t , x ) & : = \\beta ^ { e _ 1 - 2 } A ( \\beta ^ { e _ 1 } t , \\beta x ) , \\\\ f ' ( t , x ) & : = \\alpha \\beta ^ { e _ 1 } f ( \\beta ^ { e _ 1 } t , \\beta x ) . \\end{align*}"}
-{"id": "8435.png", "formula": "\\begin{align*} X ^ { f } _ t : = E [ A _ T - A _ t + C _ { T - } - C _ { t - } \\ , \\vert \\ , { \\cal F } _ t ] X ^ { ' f } _ t : = E [ A ' _ T - A ' _ t + C ' _ { T - } - C ' _ { t - } \\ , \\vert \\ , { \\cal F } _ t ] , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} B ( t _ k , y ) = \\sum _ { i = 0 } ^ { n - 1 } \\sum _ { j = 0 } ^ { n - 1 } b _ { i j } t _ k ^ i y ^ j = \\frac { Q ( t _ k ) S ( y ) - S ( t _ k ) Q ( y ) } { t _ k - y } = - Q ( t _ k ) S ' ( t _ k ) S _ k ( y ) , \\end{align*}"}
-{"id": "9191.png", "formula": "\\begin{align*} ( \\Phi \\cdot M ) _ t = \\sum _ { k = 1 } ^ K \\sum _ { m = 1 } ^ M \\mathbf 1 _ { B _ { m k } } \\sum _ { n = 1 } ^ N \\langle ( M ( t _ k \\wedge t ) - M ( t _ { k - 1 } \\wedge t ) ) , h _ n \\rangle x _ { k m n } , \\ ; \\ ; t \\geq 0 . \\end{align*}"}
-{"id": "2734.png", "formula": "\\begin{align*} | a u ^ 2 - b v ^ 2 | & = a v ^ 2 \\left | \\left ( \\frac { u } { v } + \\sqrt { \\frac { b } { a } } \\right ) \\left ( \\frac { u } { v } - \\sqrt { \\frac { b } { a } } \\right ) \\right | \\\\ & \\leqslant a v ^ 2 \\left ( 2 \\sqrt { \\frac { b } { a } } + \\frac { \\varepsilon } { B ^ 2 } \\right ) \\frac { \\varepsilon } { B ^ 2 } \\\\ & \\leqslant 2 \\varepsilon \\sqrt { a b } + \\frac { a \\varepsilon ^ 2 } { B ^ 2 } = 2 \\varepsilon \\sqrt { a b } + o ( 1 ) . \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} F _ { \\varepsilon } \\left ( x , t \\right ) = \\frac { i } { \\varepsilon + i } e ^ { \\varepsilon t W } V _ { 2 } \\left ( x , t \\right ) u \\left ( x , t \\right ) \\end{align*}"}
-{"id": "1377.png", "formula": "\\begin{align*} F _ { n } ( x , y , ( n - 1 ) \\pi / \\alpha ) = - \\frac { \\pi } { 2 } + O \\left ( \\frac { 1 } { n } \\right ) , F _ { n } ( x , y , n \\pi / \\alpha ) = \\frac { \\pi } { 2 } + O \\left ( \\frac { 1 } { n } \\right ) . \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} A \\Delta B = \\left ( A \\setminus B \\right ) \\bigcup \\left ( B \\setminus A \\right ) = ( A \\bigcup B ) \\setminus ( A \\bigcap B ) . \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} \\dim V _ { 1 , n } ^ + = ( 1 / 2 ) \\dim V _ { 1 , n } + o ( \\dim V _ { 1 , n } ) \\geq C n ^ { 2 g - 1 } \\end{align*}"}
-{"id": "2498.png", "formula": "\\begin{align*} m ( k , q , \\gamma ) : = k ( d - 1 ) - \\log ( \\gamma \\alpha ^ { - 1 } ) - \\lceil { 1 0 0 \\log ( \\log ( \\log ( \\alpha ^ { - 1 } ) ) ) } \\rceil . \\end{align*}"}
-{"id": "706.png", "formula": "\\begin{align*} \\sigma ( \\eta , \\bar \\eta ) = \\sum _ { n = 0 } ^ { k } \\frac { 1 } { k ! ( k - n ) ! } \\eta ^ { k - n } \\bar \\eta ^ { k - n } . \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} \\theta ' = \\epsilon ^ 2 e ^ \\psi ( 1 + e ^ { - \\rho } ) \\psi ' - \\epsilon ^ 2 e ^ \\psi e ^ { - \\rho } , \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{align*} E _ { \\lambda } ^ { \\omega } \\xi _ t ( O ) = e ^ { - t } \\sum _ { n = 0 } ^ { \\infty } \\sum _ { x : x \\in T } \\frac { t ^ n G _ { \\omega } ^ n ( O , x ) } { n ! } . \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{align*} F _ t ( x _ 1 , \\cdots , x _ N ; y _ 1 , \\cdots , y _ N ) = \\det \\left ( p ^ { ( N ) , s } _ t ( x _ i , y _ j ) \\right ) ^ N _ { i , j = 1 } . \\end{align*}"}
-{"id": "9451.png", "formula": "\\begin{align*} \\partial _ { t } B + \\nabla \\times \\left ( B \\times v + d \\right ) = 0 , \\ ; \\ ; \\ ; \\nabla \\cdot B = 0 \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mu } \\left ( Z _ { t } \\in A _ { R _ { 1 } , R _ { 2 } } \\right ) = \\exp \\left [ - \\frac { R _ { 1 } ^ { 2 } } { 2 \\rho ( t ) } \\right ] - \\exp \\left [ - \\frac { R _ { 2 } ^ { 2 } } { 2 \\rho ( t ) } \\right ] \\end{align*}"}
-{"id": "9215.png", "formula": "\\begin{align*} \\sum _ { n , m = 1 } ^ d V ( e _ n , e _ m ) W ( e _ n ^ * , e _ m ^ * ) \\end{align*}"}
-{"id": "1575.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } t _ k ^ i p _ { 2 n - 2 } \\sigma _ k \\sum _ { j = 0 } ^ { n - 1 } b _ { i j } t _ l ^ j = \\delta _ { k l } , k , l = 1 , \\dots , n - 1 , \\end{align*}"}
-{"id": "1550.png", "formula": "\\begin{align*} H ( c ) = H ( c _ 0 , \\dots c _ { 2 n - 2 } ) : = \\begin{bmatrix} c _ 0 & c _ 1 & \\cdots & c _ { n - 1 } \\\\ c _ 1 & c _ 2 & \\cdots & c _ { n } \\\\ \\vdots & \\vdots & & \\vdots \\\\ c _ { n - 1 } & c _ { n } & \\cdots & c _ { 2 n - 2 } \\end{bmatrix} \\end{align*}"}
-{"id": "9136.png", "formula": "\\begin{gather*} 8 a ^ 4 + \\lambda ^ 2 a ^ 2 + 8 = 0 . \\end{gather*}"}
-{"id": "5879.png", "formula": "\\begin{align*} A _ { \\mu \\nu } = \\bar { d } _ \\ell ( s , t ) e ^ { - 2 \\pi i p s / M } e ^ { - 2 \\pi i q t / N } . \\end{align*}"}
-{"id": "2518.png", "formula": "\\begin{align*} T \\left ( r , f \\right ) = N \\left ( r , \\frac { 1 } { f } \\right ) + O \\left ( r ^ { \\rho - 1 + \\varepsilon } \\right ) + O \\left ( \\log r \\right ) . \\end{align*}"}
-{"id": "4432.png", "formula": "\\begin{align*} H ^ { p , q } ( C _ 1 \\times C _ 2 \\times C _ 3 ) \\simeq \\bigoplus _ { \\substack { s _ 1 + s _ 2 + s _ 3 = p \\\\ t _ 1 + t _ 2 + t _ 3 = q } } H ^ { s _ 1 , t _ 1 } ( C _ 1 ) \\otimes H ^ { s _ 2 , t _ 2 } ( C _ 2 ) \\otimes H ^ { s _ 3 , t _ 3 } ( C _ 3 ) , \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} \\dot { x } = f ( y ) , ~ ~ ~ ~ \\dot { y } = g _ 2 ( x ) y ^ 2 + g _ 1 ( x ) y + g _ 0 ( x ) , \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{align*} \\mathbf { \\dot { p } } _ d = \\left [ \\begin{array} { c } v \\cos \\phi \\\\ v \\sin \\phi \\end{array} \\right ] \\end{align*}"}
-{"id": "1356.png", "formula": "\\begin{align*} \\Delta v + k ^ { 2 } v - q \\left ( x \\right ) v = - \\delta \\left ( x - y \\right ) , x \\in \\mathbb { R } ^ { 3 } \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} \\frac { \\rho _ 1 + 1 / \\rho _ 1 } { 2 r _ x } = \\left ( \\frac { 1 } { r _ x } + h \\left ( 1 - \\frac { 1 } { r _ x } \\right ) \\right ) . \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} Q _ E ( z ) = \\prod _ { j = 0 } ^ n ( z - E _ j ^ + ) ( z - E _ j ^ - ) . \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } W ( \\phi _ k ) = W ( \\phi _ { \\infty } ) + \\sum _ { i = 1 } ^ { p } W ( \\vec { \\Psi } _ i ) + \\sum _ { j = 1 } ^ { q } \\left ( W ( \\vec { \\xi } _ j ) - 4 \\pi \\theta _ j \\right ) \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} ( A \\circ \\tau ) ( t ) = ( A \\circ \\tau ) ( A _ { S _ t } ) = ( A \\circ ( \\tau \\circ A ) ) ( S _ t ) = A _ { S _ t } = t . \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{align*} f _ { c _ { 1 } } \\left ( z \\right ) = \\frac { B _ { 1 } } { B _ { 2 } } f _ { c _ { 2 } } \\left ( z \\right ) + \\frac { h } { B _ { 2 } } f \\left ( z \\right ) . \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{align*} P u = u _ 0 \\otimes \\delta ' + v _ 0 \\otimes \\delta \\end{align*}"}
-{"id": "7239.png", "formula": "\\begin{align*} \\sum _ { J \\subseteq [ N ] } p _ J ( L ) L _ J ^ { - 1 } = ( I + L ) ^ { - 1 } . \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{align*} x _ l ( \\omega ) = \\begin{cases} \\alpha _ l ^ + ( \\omega ) \\textnormal { i f } l > 0 \\\\ - \\alpha _ l ^ - ( \\omega ) \\textnormal { i f } l < 0 \\\\ \\end{cases} . \\end{align*}"}
-{"id": "4676.png", "formula": "\\begin{align*} \\prod _ { n = \\lceil \\gamma R \\rceil + 1 } ^ { \\infty } \\left ( 1 - \\frac { \\gamma ^ 2 R ^ 2 t ^ 2 } { n ^ 2 } \\right ) ^ 2 \\geq e ^ { - 2 \\lceil \\gamma R \\rceil V ( \\frac { \\gamma R } { \\lceil \\gamma R \\rceil } t ) } \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{align*} \\Psi ( \\vec { x } ; \\vec { z } ; \\vec { \\xi } ) : = \\frac { 1 } { 2 ^ \\mu \\cdot \\mu ! } \\Bigl ( H _ { ( x _ 1 , x _ 2 ) } - z _ 1 ^ 2 - z _ 2 ^ 2 \\Bigr ) ^ \\mu \\diamond \\bigl ( \\delta ( x _ 1 - \\xi _ 1 , x _ 2 - \\xi _ 2 ) \\cdot \\exp ( x _ 1 z _ 1 + x _ 2 z _ 2 ) \\bigr ) . \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{align*} \\partial _ t H _ j - \\nabla _ j V = \\nabla _ l V \\nabla _ l ^ \\perp H _ j . \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\liminf _ n V _ n \\right ) = 1 . \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} \\sum _ { i \\in E } m _ i v _ i = - \\sum _ { \\substack { \\alpha : k \\to i \\\\ k \\notin E , i \\in E } } u _ \\alpha \\leq 0 . \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} | v ^ { ( k ) } | \\prod _ { i = \\Delta ^ { ( k ) } - m } ^ { \\Delta ^ { ( k ) } - 1 } | w ^ { ( k ) } _ { i } | & > C \\label { E q u a t i o n 9 } \\end{align*}"}
-{"id": "9578.png", "formula": "\\begin{align*} Y _ { n , i } ^ * ( u ) = u ^ { \\prime } \\theta _ { n , i } + \\xi _ { n , i } , \\end{align*}"}
-{"id": "3055.png", "formula": "\\begin{align*} z _ i ( z _ k - z _ j ) = ( z _ j + z _ k ) ( z _ j - z _ k ) \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} \\mathbb { Q } \\left ( M \\otimes \\mathbb { I } _ { | \\mathcal { E } | ^ { n - 1 } } \\right ) \\mathbb { Q } ^ { - 1 } = \\mathbb { I } _ { | \\mathcal { E } | } \\otimes \\dots \\otimes \\mathbb { I } _ { | \\mathcal { E } | } \\otimes M \\otimes \\mathbb { I } _ { | \\mathcal { E } | } \\otimes \\dots \\otimes \\mathbb { I } _ { | \\mathcal { E } | } \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} \\tau ( \\tilde \\sigma _ 1 - \\sigma _ 2 ) ( x , v ) = \\int _ { - \\tau _ - ( x , v ) } ^ { \\tau _ + ( x , v ) } ( \\sigma _ 1 - \\sigma _ 2 ) ( x + s v , v ) d s , \\end{align*}"}
-{"id": "3401.png", "formula": "\\begin{gather*} \\dot \\phi - a \\phi - \\frac { \\theta + \\left ( 1 - \\frac { 1 } { n } \\right ) \\phi } { 1 + \\frac { 1 } { \\lambda } \\left ( 1 - \\frac { 1 } { n } \\right ) ^ 2 \\phi } \\phi = \\frac { \\lambda \\theta ^ 2 + \\lambda \\theta \\left ( 1 - \\frac { 1 } { n } \\right ) \\phi } { 1 + \\frac { 1 } { \\lambda } \\left ( 1 - \\frac { 1 } { n } \\right ) ^ 2 \\phi } - \\lambda \\varepsilon + a \\phi _ t \\end{gather*}"}
-{"id": "6795.png", "formula": "\\begin{align*} w _ { n } \\left ( x , t \\right ) = e ^ { \\mu } \\upsilon _ { n } \\left ( x , t \\right ) , \\mu = - i 4 \\beta ^ { 2 } \\varphi _ { n } ^ { 4 } \\left ( x _ { 1 } \\right ) t . \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} i ( P _ { k , n } , t ) = ( - 1 ) ^ t \\sum _ { p = 0 } ^ t { { \\frac { 1 } { k } - 1 } \\choose { t - p } } { { - \\frac { 1 } { k } \\ , } \\choose p } ( k p + 1 ) ^ { n } . \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} \\partial _ t J & = [ P ( J ) ^ + - P ( J ) ^ - , J ] . \\end{align*}"}
-{"id": "3781.png", "formula": "\\begin{align*} s _ i = \\ell \\cdot s _ { i - 1 } + \\sum _ { k = 1 } ^ { i - 2 } a _ k \\cdot s _ k \\end{align*}"}
-{"id": "9982.png", "formula": "\\begin{align*} \\tilde { E } _ { \\sigma ( t ) } \\dot { z } = z \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} \\begin{bmatrix} B ( \\lambda ) & 0 \\\\ D ( B ( \\lambda ) ) & B ( \\lambda ) \\end{bmatrix} = \\left \\{ \\begin{bmatrix} b _ 1 & 0 \\\\ f & b _ 2 \\end{bmatrix} \\ \\Big | \\ b _ 1 , b _ 2 \\in B ( \\lambda ) , f \\in D ( B ( \\lambda ) ) \\right \\} \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} \\partial _ { t } u = \\left ( a + i b \\right ) \\left [ \\Delta u + A u + V \\left ( x , t \\right ) u + F \\left ( x , t \\right ) \\right ] , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - n & \\beta \\\\ & \\gamma \\end{matrix} \\bigg | \\ , z \\bigg ] = \\frac { ( \\gamma - \\beta ) _ n } { ( \\gamma ) _ n } \\cdot { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - n & \\beta \\\\ & \\beta - \\gamma - n + 1 \\end{matrix} \\bigg | \\ , 1 - z \\bigg ] , \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{align*} \\mathbf { z } _ { \\Delta \\phi _ l } = \\mathrm { s i g n } \\left ( \\left ( \\mathbf { \\dot { p } } _ { d } \\times ( \\mathbf { p } _ { h } - \\mathbf { p } _ { d } ) \\right ) \\cdot \\mathbf { z } _ I \\right ) \\mathbf { z } _ I \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} 0 = a _ 0 < a _ 1 < \\ldots < a _ n = 1 \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} g \\star x \\star g ^ { - 1 } = \\psi _ g ( x ) x \\in \\mu _ 2 ( B [ \\epsilon ] ) = W _ G ( \\mu _ 2 ) ( B ) \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} c _ m ( n , k ) = \\sum _ { i _ 1 + i _ 2 + \\cdots + i _ k = n } f _ { m - 1 } ( i _ 1 ) \\cdot f _ { m - 1 } ( i _ 2 ) \\cdots f _ { m - 1 } ( i _ k ) , \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} \\phi _ 1 ( y ) : = - I ( a \\ , | \\ , b _ + - y ) + I ( a \\ , | \\ , b _ + ) = a \\log \\left ( 1 - \\frac { y } { b _ + } \\right ) + y , \\ : \\ : \\ : \\ : \\ : \\ : \\phi _ 2 ( y ) : = - I _ X ( b _ + - y ) + I _ X ( b _ + ) . \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{align*} \\Delta \\phi _ { v o } = \\sin ^ { - 1 } \\left ( \\frac { | | \\mathbf { \\dot { p } } _ o | | } { | | \\mathbf { \\dot { p } } _ { d } ' | | } \\right ) - \\sin ^ { - 1 } \\left ( \\frac { v _ { v o } } { | | \\mathbf { \\dot { p } } _ { d } ' | | } \\right ) \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} H _ { d B } ( f ) = A _ { d B } ( f ) + i K \\varphi ( f ) , \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{align*} \\rho _ { \\varepsilon } : = \\varepsilon ^ { 2 m - N - \\delta } + \\varepsilon ^ { - N } \\chi _ { B ( 0 , \\varepsilon ) } , \\end{align*}"}
-{"id": "2775.png", "formula": "\\begin{align*} & \\sum _ { n \\leqslant \\frac { ( \\lambda _ 2 - \\alpha \\mu _ 2 ) ( k - 1 ) K } { N \\det ( \\Lambda _ d ) } B } \\mathtt { 1 } _ { 1 - \\{ n \\theta \\} \\leqslant \\frac { k \\varepsilon K } { N ( \\lambda _ 2 - \\mu _ 2 \\alpha ) } B ^ { 1 - \\frac { 1 } { r } } } \\\\ & = \\frac { k ( k - 1 ) \\varepsilon K ^ 2 } { N ^ 2 \\det ( \\Lambda _ d ) } B ^ { 2 - \\frac { 1 } { r } } + O \\left ( b \\det ( \\Lambda _ d ) \\log \\left ( \\frac { ( k - 1 ) K ( \\lambda _ 2 - \\alpha \\mu _ 2 ) } { N \\det ( \\Lambda _ d ) } B \\right ) \\right ) . \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} \\partial _ { t } u _ { i } \\left ( t , x \\right ) \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert = 2 l } a _ { \\alpha } D ^ { \\alpha } u _ { i } \\left ( t , x \\right ) + \\sum \\limits _ { j = 1 } ^ { \\infty } a _ { i j } u _ { j } \\left ( t , x \\right ) = f _ { i } \\left ( t , x \\right ) x \\in R ^ { n } t \\in \\left ( 0 , \\infty \\right ) , \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} \\vert x _ { n _ k } - x ( [ \\ell ] P _ { n _ k } ) \\vert _ w = \\Bigl \\vert \\frac { \\Psi _ { \\ell - 1 } ^ \\prime ( x _ n , y _ n ) \\Psi _ { \\ell + 1 } ^ \\prime ( x _ n , y _ n ) } { \\left ( \\Psi _ \\ell ^ \\prime ( x _ n , y _ n ) \\right ) ^ 2 } \\Bigr \\vert _ w \\leq C ^ \\prime \\vert x _ { n _ k } - x _ * \\vert _ w , \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} \\left [ \\left ( \\frac { r } { g } \\right ) ^ { 2 } - \\left ( \\frac { p } { f } \\right ) ^ { 2 } \\right ] ^ { \\frac { 1 } { 2 } } \\left [ 2 + \\left ( \\frac { p } { f } \\right ) ^ { 2 } \\left ( \\frac { r } { g } \\right ) ^ { - 2 } \\right ] = \\frac { \\frac { d } { d f } \\left ( \\dfrac { f \\dot { p } } { p } \\right ) \\left ( \\dfrac { p } { f } \\right ) } { 2 H _ { 0 } \\frac { d } { d f } \\left ( \\dfrac { f } { p } \\right ) } . \\end{align*}"}
-{"id": "7718.png", "formula": "\\begin{align*} r \\cdot ( g \\oplus h ) & = ( r \\cdot g ) \\oplus ( r \\cdot h ) \\\\ ( r + s ) \\cdot g & = ( r \\cdot g ) \\oplus ( s \\cdot g ) \\\\ ( r * s ) \\cdot g & = r \\cdot ( s \\cdot g ) \\\\ 1 \\cdot g & = g . \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} \\mathbb { R } ^ + \\ni r \\mapsto W _ r ( v _ 0 , v _ \\infty ) \\ ; : = \\ ; \\det \\begin{pmatrix} v _ 0 ^ + ( r ) & v _ \\infty ^ + ( r ) \\\\ v _ 0 ^ - ( r ) & v _ \\infty ^ - ( r ) \\end{pmatrix} . \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} d f ^ i = 0 , 1 \\leq i \\leq 6 , d f ^ 7 = \\frac { \\sqrt { 6 } } { 6 } \\ , y ^ { - 5 } ( t ) ( f ^ { 1 2 } + f ^ { 3 4 } + f ^ { 5 6 } ) . \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} \\textbf { R } _ { r } & = \\left ( \\boldsymbol { \\gamma } _ { r } \\right ) + \\sigma ^ 2 \\mathbf { I } , & & \\\\ \\textbf { R } _ { d } ^ b & = \\left ( \\boldsymbol { \\gamma } _ { d } ^ b \\right ) + \\sigma ^ 2 \\mathbf { I } , & & \\\\ \\textbf { R } _ { d } ^ m & = \\left ( \\boldsymbol { \\gamma } _ { d } ^ m \\right ) + \\sigma ^ 2 \\mathbf { I } , & & \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} \\langle 1 \\otimes \\alpha _ i ^ \\vee , 1 \\otimes \\alpha _ j ^ \\vee \\rangle = \\alpha _ i ( \\alpha _ j ^ \\vee ) \\epsilon _ i = \\alpha _ j ( \\alpha _ i ^ \\vee ) \\epsilon _ j . \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{align*} \\frac { B C _ { 2 , 1 8 } } { \\Pi ( 1 8 ) } & = \\sum _ { k = 1 } ^ { 1 8 } \\binom { 1 9 } { k + 1 } ( - D _ 2 ) ^ k \\frac { k } { D _ 2 ^ { k - 1 } D _ 3 } \\\\ & = \\frac { D _ 2 } { D _ 3 } \\sum _ { k = 1 } ^ { 1 8 } ( - 1 ) ^ k \\binom { 1 9 } { k + 1 } k = - \\frac { D _ 2 } { D _ 3 } \\ , . \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{align*} w _ { i , j } + w _ { k , l } & = ( 1 - p ) p _ i - p p _ j \\\\ & \\quad + ( 1 - p ) p _ k - p p _ l \\\\ & = ( 1 - p ) p _ i - p p _ l \\\\ & \\quad + ( 1 - p ) p _ k - p p _ j \\\\ & = w _ { i , l } + w _ { k , j } . \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} \\begin{array} { l } \\int _ 0 ^ { \\frac { 1 } { 2 } } Q ( x ) [ \\cos ( 2 k x ) + \\int _ 0 ^ x K ( x , t ) \\cos ( 2 k t ) d t ] d x \\\\ \\qquad + \\int _ { \\frac { 1 } { 2 } } ^ b Q ( x ) [ 2 A _ 2 \\cos ( 2 k x ) + 2 A _ 3 \\cos 2 k ( x - \\frac { 1 } { 2 } ) \\\\ \\qquad \\quad + 2 A _ 4 \\cos 2 k ( x - 1 ) + \\int _ 0 ^ x K ( x , t ) \\cos ( 2 k t ) d t ] d x = 0 . \\end{array} \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} _ { 1 } F _ { 1 } \\left ( \\begin{array} { c } a \\\\ c \\end{array} ; z \\right ) = \\sum _ { n \\ge 0 } \\frac { \\left ( a \\right ) _ { n } } { \\left ( c \\right ) _ { n } } \\frac { z ^ n } { n ! } \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} \\left ( \\frac { m } { r _ { m i n } } + K _ d \\right ) v _ { c , v } ^ 2 + 2 K _ d v _ { a i r } v _ { c , v } + v _ { a i r } ^ 2 - f _ { p l a n a r } = 0 \\end{align*}"}
-{"id": "1691.png", "formula": "\\begin{align*} F _ * ^ e ( j u ^ { q - 1 } v ^ { q - k - 1 } ( f + u v ) ) = \\bigoplus _ { i \\in \\Delta _ e } f _ { ( i , j ) } F _ * ^ e ( i u ^ { q - 1 } v ^ { q - k - 1 } ) \\oplus u F _ * ^ e ( j v ^ { q - k } ) . \\end{align*}"}
-{"id": "2725.png", "formula": "\\begin{align*} \\Phi ( n ) = \\sum _ { d | n } \\frac { d \\Psi _ 1 ( g ( d ) ) } { g ( d ) } \\sum _ { e | d } \\frac { \\mu ( e ) } { e } = \\sum _ { d | n } \\frac { d \\Psi _ 1 ( g ( d ) ) } { g ( d ) } \\phi ( d ) . \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} \\widehat { \\phi } ( \\mathfrak { C } ( ( \\omega ^ 0 , y ^ 0 ) , 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ) ( z _ 1 , z _ 2 ) & = \\widehat { \\Phi } ( \\mathfrak { C } ( ( \\omega ^ 0 , y ^ 0 ) , 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ( z _ 1 , z _ 2 ) ) \\\\ & = \\widehat { \\Phi } ( y ^ 0 ( [ \\omega ] ( ( z _ 1 , z _ 2 ) , ( 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ) ) ) \\\\ & = \\widehat { \\Phi } ( y ( 1 _ G , [ \\omega ] ( ( z _ 1 , z _ 2 ) , ( 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ) ) ) . \\\\ \\end{align*}"}
-{"id": "8745.png", "formula": "\\begin{align*} n _ { i j } ^ { \\infty } = \\left \\{ \\begin{array} { l r } ( - 1 ) ^ { j - i } { { \\lfloor { { i - 1 } \\over 2 } \\rfloor + j - i } \\choose { \\lfloor { { i - 1 } \\over 2 } \\rfloor } } , & { \\rm i f } ~ ~ i \\leq j , \\\\ 0 , & { \\rm i f } ~ ~ i > j \\end{array} \\right . \\end{align*}"}
-{"id": "9779.png", "formula": "\\begin{align*} \\Theta ^ { n - 2 } ( \\nu , x ) = \\lim _ { r \\to 0 } \\frac { \\nu ( B _ r ( x ) ) } { \\omega _ { n - 2 } r ^ { n - 2 } } \\geq \\eta x \\in s p t ( \\nu ) , \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} f '' ( x ) + A _ { \\lambda , \\mu } f ' ( x ) + B _ { \\lambda , \\mu } f ( x ) = 0 , \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} \\| N ^ c _ 0 \\| = 0 & \\leq 0 = \\| M ^ c _ 0 \\| , \\\\ \\| N ^ d _ 0 \\| = \\| N _ 0 \\| & \\leq \\| M _ 0 \\| = \\| M ^ d _ 0 \\| . \\end{align*}"}
-{"id": "9697.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } n ^ j t ^ n \\ = \\ \\frac { A _ j ( t ) } { ( 1 - t ) ^ { j + 1 } } \\ , , \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 5 \\theta } { \\alpha _ 2 } y _ 4 , [ y _ 2 , y _ 1 ] = \\frac { \\beta _ 1 } { \\beta _ 4 \\theta } y _ 3 , [ y _ 2 , y _ 2 ] = y _ 3 , [ y _ 2 , y _ 3 ] = y _ 5 , [ y _ 1 , y _ 4 ] = y _ 5 . \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{align*} \\mathsf { L } _ s ^ { ( N ) } = \\Delta ^ { - 1 } _ N ( x ) \\circ \\left ( \\sum _ { i = 1 } ^ { N } L ^ { ( N ) } _ { s , x _ i } \\right ) \\circ \\Delta _ N ( x ) - \\lambda _ { N , s } . \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} Z ^ { ( r ) } ( \\psi , \\bar \\psi ) & = \\sum _ { n \\in \\N ^ \\N } Z _ { n } \\psi ^ n \\bar \\psi ^ n , \\\\ Z _ { n } \\neq 0 & \\Longrightarrow \\sum _ { l \\geq N + 1 } n _ l \\leq 2 , \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} w = \\Psi w ( t \\geq \\bar t ) \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} a ^ - = \\sum _ { i = 1 } ^ { k } q _ i ^ - a ^ + = \\sum _ { i = 1 } ^ { k } q _ i ^ + \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{align*} h = h _ { i j } d x ^ { i } \\otimes d x ^ { j } , ~ \\ \\ \\ \\ \\ h _ { i j } = g _ { i j } - \\dfrac { y _ { i } y _ { j } } { L } . \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} { { \\cal H } _ 0 } : & \\ ; x _ { k } ^ { ( n ) } \\sim { f _ 0 ^ { ( n ) } } \\left ( x \\right ) , \\\\ { } { { \\cal H } _ 1 } : & \\ ; x _ { k } ^ { ( n ) } \\sim { f _ 1 ^ { ( n ) } } \\left ( x \\right ) . \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} F _ * ^ e ( j u ^ { k + r } v ^ r ( f + u v ) ) = \\bigoplus _ { i \\in \\Delta _ e } f _ { ( i , j ) } F _ * ^ e ( i u ^ { k + r } v ^ { r } ) \\oplus F _ * ^ e ( j u ^ { k + r + 1 } v ^ { r + 1 } ) \\end{align*}"}
-{"id": "10034.png", "formula": "\\begin{align*} T _ p ( M ) = ( D _ { \\phi } ) _ p \\oplus ( D _ { \\bar \\phi } ) _ p = T _ p ^ + ( M ) \\oplus T ^ - _ p ( M ) , \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} & \\int _ M s ^ H \\frac { \\omega _ M ^ n } { n ! } = s ^ H _ S \\frac { ( p + c ) ^ { n - 1 } - c ^ { n - 1 } } { p ( n - 1 ) } - \\left [ ( P \\varphi ) ' ( x ) P ( x ) ^ { n - 2 } \\right ] ^ 1 _ 0 + \\int _ 0 ^ 1 ( P \\varphi ) ' ( x ) P ( x ) ^ { n - 2 } d x \\\\ & = s ^ H _ S \\frac { ( p + c ) ^ { n - 1 } - c ^ { n - 1 } } { p ( n - 1 ) } + 2 c ^ { n - 1 } + 2 ( p + c ) ^ { n - 1 } + p \\int _ 0 ^ 1 \\varphi ' ( x ) P ( x ) ^ { n - 2 } d x + \\int _ 0 ^ 1 \\varphi ( x ) P ( x ) ^ { n - 1 } d x \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} \\frac { e ^ { - \\phi } } { \\int _ { \\mathbb { R } ^ n } e ^ { - \\phi } } = l ( \\nabla \\phi ) \\det ( \\nabla ^ 2 \\phi ) \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} \\textsf { D P P } ( L _ 1 ) = \\textsf { D P P } ( L _ 2 ) \\iff \\exists D \\in \\mathcal D , L _ 2 = D L _ 1 D . \\end{align*}"}
-{"id": "6775.png", "formula": "\\begin{align*} D \\left ( A \\right ) = \\left \\{ u \\in W ^ { 2 , 2 } \\left ( 0 , 1 \\right ) , B _ { j } u = A u \\left ( j \\right ) = 0 j = 0 , 1 \\right \\} , \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} \\mu ^ - _ { | M } ( \\varphi ) = \\int _ M G ^ - d d ^ c \\varphi , \\varphi \\in C ^ \\infty _ 0 ( M ) . \\end{align*}"}
-{"id": "2390.png", "formula": "\\begin{align*} \\partial _ t \\psi = - \\mathrm i \\sqrt { - \\Delta } \\psi \\pm \\psi | \\psi | ^ { p - 1 } . \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} F _ { t - 1 } : = \\{ { \\cal N } _ { t - 1 } = S _ { t - 1 } , \\ldots , { \\cal N } _ 1 = S _ 1 , { \\cal N } _ 0 = S _ 0 \\} \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} \\frac { d } { d x } \\bigg ( \\frac { \\Gamma ( a + b - c - x ) } { \\Gamma ( a - c - x ) } \\bigg ) \\bigg | _ { x = 0 } = & \\frac { d \\big ( ( a - c - x ) _ b \\big ) } { d x } \\bigg | _ { x = 0 } \\\\ = & - \\prod _ { \\substack { 0 \\leq j < b \\\\ j \\neq c - a } } ( a - c + j ) = ( - 1 ) ^ { b - 1 } \\cdot \\frac { \\Gamma _ p ( a + b - c ) } { \\Gamma _ p ( a - c ) } . \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} f ( t ) = t ^ 8 - 3 t ^ 7 + 5 t ^ 6 - 7 t ^ 5 + 9 t ^ 4 - 7 t ^ 3 + 5 t ^ 2 - 3 t + 1 \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} C ^ { \\lambda } _ m ( t ) = \\sum _ { k = 0 } ^ { [ m / 2 ] } ( - 1 ) ^ k \\frac { \\Gamma ( m + \\lambda - k ) } { k ! ( m - 2 k ) ! \\Gamma ( \\lambda ) } ( 2 t ) ^ { m - 2 k } \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{align*} \\vec { \\gamma } _ 2 ( \\phi , p ) = - \\vec { \\gamma } _ 2 ( \\vec { \\Psi } , p ) . \\end{align*}"}
-{"id": "5483.png", "formula": "\\begin{align*} S ^ \\chi _ { ( r , s ) , ( r ' , s ' ) } = - 2 \\sqrt { \\frac { 2 } { u p } } ( - 1 ) ^ { r s ' + s r ' } \\sin \\left ( \\frac { \\pi p } { u } r r ' \\right ) \\sin \\left ( \\frac { \\pi u } { p } s s ' \\right ) . \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} \\nu _ 1 \\equiv \\Gamma \\bigl ( u _ T - P u _ 0 \\big | _ { t = T } - P _ 2 f _ 1 \\big | _ { t = T } \\bigr ) , u \\equiv P u _ 0 + P _ 1 \\nu _ 1 + P _ 2 f _ 1 . \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} m _ 1 ( n ) = \\frac { n } { \\sigma _ 1 } ( 1 + \\epsilon _ { 1 , n } ) , \\ m _ 2 ( n ) = \\frac { \\gamma _ n } { \\sigma _ 2 } ( 1 + \\epsilon _ { 2 , n } ) , \\ \\epsilon _ { i , n } \\to 0 , \\ n \\to \\infty . \\end{align*}"}
-{"id": "5053.png", "formula": "\\begin{align*} o \\ = \\ K / K \\in \\Omega \\ = \\ G / K \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} A ^ 2 \\limsup _ { t \\to + \\infty } \\| u _ { n + 1 } ( t ) \\| _ { L ^ 3 ( B _ { A \\sqrt t } ^ c ) } & = A ^ 2 \\limsup _ { t \\to + \\infty } \\| u _ { n + 1 } ( 4 t ) \\| _ { L ^ 3 ( B _ { A \\sqrt { 4 t } } ^ c ) } \\\\ & \\le C _ 0 \\Bigl ( \\| u _ 0 \\| _ 3 + \\| \\theta _ 0 \\| _ 1 \\Bigr ) + 2 C \\varepsilon ( \\varepsilon + \\kappa _ n ) . \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{align*} \\norm { \\nabla \\rho \\otimes \\nabla \\rho } _ \\infty = \\norm { \\nabla \\rho } ^ 2 _ \\infty \\le C \\epsilon \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{align*} S ( \\hat { \\rho } ) : = - \\mathrm { T r } \\left [ \\hat { \\rho } \\ln \\hat { \\rho } \\right ] \\ ; . \\end{align*}"}
-{"id": "9491.png", "formula": "\\begin{align*} T _ A : = T _ { 1 , A } = T _ { 2 , A } \\mbox { a n d } T _ { A \\cup L } & : = T _ { 1 , A \\cup L } = T _ { 2 , A \\cup L } . \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{align*} \\chi ^ { } _ \\mathrm { m a x } ( k ) & = \\log \\sqrt { 5 } \\approx 0 . 8 0 4 7 \\\\ \\chi ^ { } _ \\mathrm { m i n } ( k ) & = \\log \\sqrt { 5 } - 2 \\log \\left ( 1 . 3 5 6 2 \\right ) \\approx 0 . 1 9 5 3 . \\end{align*}"}
-{"id": "6679.png", "formula": "\\begin{align*} P x = ( \\Phi A \\Phi ^ * ) ( x ) = \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ m a _ { i j } e ^ * _ j ( x ) e _ i , \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n ( - 1 ) ^ { \\lceil k / 2 \\rceil } s _ { n , G _ k } ^ { ( - 1 ) } = 0 . \\end{align*}"}
-{"id": "5408.png", "formula": "\\begin{align*} A x = b , ~ x \\in \\mathbb { R } ^ m . \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} \\partial _ { \\xi } L _ c \\partial _ { \\xi } u _ c = 0 , \\partial _ { \\xi } L _ c \\partial _ c u _ c = - 4 \\partial _ { \\xi } u _ c , \\end{align*}"}
-{"id": "9946.png", "formula": "\\begin{align*} ( ( A ^ { - 1 } E ) ^ { k ^ * + 1 } ) = ( ( A ^ { - 1 } E ) ^ { k ^ * } ) \\end{align*}"}
-{"id": "2442.png", "formula": "\\begin{align*} \\dfrac { \\Phi ( \\theta ) } { G ( 0 ) } = \\sum _ { i = 1 } ^ 2 \\dfrac { C _ i } { \\theta - \\gamma _ i } . \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} x _ i = a _ i \\log t + \\log b _ i \\end{align*}"}
-{"id": "9446.png", "formula": "\\begin{align*} \\int _ { \\mathbb { T } ^ 3 } \\nabla u ( t ) \\cdot B ( t ) = 0 \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} \\phi ( t , \\theta ) = \\frac { 1 } { e ^ { 2 t } + 1 } ( 2 e ^ t \\cos ( \\theta ) , 2 e ^ t \\sin ( \\theta ) , e ^ { 2 t } - 1 ) . \\end{align*}"}
-{"id": "6049.png", "formula": "\\begin{align*} L _ 0 = S + T \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{align*} ( \\varphi u ) _ t = \\Delta ( \\varphi u ) - 2 \\nabla u \\cdot \\nabla \\varphi - u \\Delta \\varphi + \\varphi | u | ^ \\alpha u . \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} \\delta \\eta \\beta & = 0 , & \\varrho \\omega \\gamma & = 0 , & \\sigma \\varepsilon \\eta & = 0 , & \\beta \\varrho \\mu & = 0 , & \\eta \\gamma \\nu & = 0 , & \\omega \\beta \\varepsilon & = 0 , \\end{align*}"}
-{"id": "1706.png", "formula": "\\begin{align*} \\mathbf { z } _ { \\phi , s l o w } & = \\mathrm { s i g n } \\left ( \\left ( \\mathbf { p } _ { m i n } \\times \\mathbf { \\dot { p } } _ d \\right ) \\cdot \\mathbf { z } _ I \\right ) \\mathbf { z } _ I \\\\ \\mathbf { z } _ { \\phi , f a s t } & = \\mathrm { s i g n } \\left ( ( \\mathbf { \\dot { p } } _ o \\times \\mathbf { \\dot { p } } _ d ) \\cdot \\mathbf { z } _ I \\right ) \\mathbf { z } _ I \\end{align*}"}
-{"id": "9826.png", "formula": "\\begin{align*} s _ i \\mathcal { I } ^ \\circ = \\left ( \\frac { 1 - v \\mathbf { z } ^ { \\alpha _ i ^ { \\vee } } } { 1 - v \\mathbf { z } ^ { - \\alpha _ i ^ { \\vee } } } \\right ) \\mathcal { I } ^ \\circ . \\end{align*}"}
-{"id": "729.png", "formula": "\\begin{align*} T _ { ( i , j ) } f ( k , \\ell ) = f ( k - i , \\ell - j ) . \\end{align*}"}
-{"id": "9552.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = 1 } ^ k a _ i ^ * y _ { g _ 1 } \\sigma ( g _ 1 g _ 0 ^ { - 1 } , g _ 0 ) ^ * \\alpha _ { g _ 1 g _ 0 ^ { - 1 } } ( a _ i ) \\right \\| < \\delta , \\end{align*}"}
-{"id": "6641.png", "formula": "\\begin{align*} L _ { v _ 0 } ( g ) \\ ; = \\ ; a _ 0 ^ { ( g ) } \\ , L _ { v _ 0 } ( v _ 0 ) + a _ \\infty ^ { ( g ) } \\ , L _ { v _ 0 } ( v _ \\infty ) + L _ { v _ 0 } ( b _ \\infty ^ { ( g ) } \\ , v _ 0 + b _ 0 ^ { ( g ) } \\ , v _ \\infty ) \\ , ; \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{align*} s _ 1 ( x ) & = \\sqrt { ( x - 1 ) ^ 2 + b ^ 2 - ( b / a ) ^ 2 ( x + 1 - h ) ^ 2 } , & s _ 2 ( x ) & \\strut = \\sqrt { ( x + 1 ) ^ 2 + b ^ 2 - ( b / a ) ^ 2 ( x + 1 - h ) ^ 2 } . \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} ( 1 - p _ d ) ^ { r ( n - r ) } \\leq \\exp \\left ( - \\frac { C _ d r ( n - r ) } { n } \\right ) = e ^ { - C _ d r } \\exp \\left ( \\frac { C _ d r ^ 2 } { n } \\right ) . \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} Z ( \\Gamma ) = Z ( \\Gamma _ 1 ) Z ( \\Gamma _ 2 ) . \\end{align*}"}
-{"id": "7010.png", "formula": "\\begin{align*} u \\left ( x , \\varepsilon \\right ) = D _ { \\lambda } ^ { - 1 } \\left ( \\varepsilon \\right ) \\left [ U _ { 1 , \\lambda } \\left ( x , \\varepsilon \\right ) D _ { 1 , \\lambda } \\left ( \\varepsilon \\right ) + U _ { 2 , \\lambda } \\left ( x , \\varepsilon \\right ) D _ { 2 , \\lambda } \\left ( \\varepsilon \\right ) \\right ] = \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { L } f ( x ) = & \\ \\frac { 1 } { 2 } \\sum _ { j , k = 1 } ^ { d } a _ { j k } ( x ) \\frac { \\partial ^ { 2 } } { \\partial x _ { j } \\partial x _ { k } } f ( x ) + \\sum _ { j = 1 } ^ { d } b _ { j } ( x ) \\frac { \\partial } { \\partial x _ { j } } f ( x ) \\\\ & + \\int _ { U } [ f ( x + c ( x , u ) ) - f ( x ) - c ( x , u ) \\cdot D f ( x ) ] M ( \\d u ) . \\end{aligned} \\end{align*}"}
-{"id": "610.png", "formula": "\\begin{align*} \\div \\ , V ( x ) & = \\phi \\big ( \\textstyle { \\frac { | x | } { t } } \\big ) | \\nabla u ( x ) | ^ 2 \\ , | x _ { n + 1 } | ^ a + \\phi ' \\big ( \\textstyle { \\frac { | x | } { t } } \\big ) u ( x ) \\nabla u ( x ) \\cdot \\frac { x } { t \\ , | x | } \\ , | x _ { n + 1 } | ^ a . \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } S ( A | A ' B ' ) _ { \\hat { \\gamma } _ { A A ' B ' } ^ { ( n ) } } = \\lim _ { n \\to \\infty } S ( A | A ' ) _ { \\hat { \\gamma } _ { A A ' } ^ { ( n ) } } = 1 + \\ln a \\ ; . \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} Y _ { \\delta ^ { \\varepsilon } _ { \\theta } } = Y _ { \\sigma ^ { \\varepsilon } _ \\theta + } \\ , \\ , { \\rm a n d } \\ , \\ , \\zeta ^ { ^ l } _ { \\delta ^ { \\varepsilon } _ { \\theta } } = \\ , \\ , \\check { \\zeta } _ { \\sigma ^ { \\varepsilon } _ \\theta } . \\end{align*}"}
-{"id": "8308.png", "formula": "\\begin{align*} \\lambda ^ { ( 4 ) } & = ( [ 2 ] , [ 0 , 0 , 0 ] , [ 0 , 0 , 0 ] , [ 0 , 0 , 4 ] ) , \\\\ \\lambda ^ { ( 5 ) } & = ( [ 2 ] , [ 0 , 0 , 0 ] , [ 0 , 0 , 4 ] , [ 0 , 0 , 0 ] ) , \\\\ \\lambda ^ { ( 6 ) } & = ( [ 2 ] , [ 0 , 0 , 4 ] , [ 0 , 0 , 0 ] , [ 0 , 0 , 0 ] ) \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{align*} R ^ e _ n ( s _ { v _ 0 } ) ( w _ { 0 } ) & = s _ { w ( - e + 1 , 0 ) } s _ { w ( 0 , e - 1 ) } ( w _ 0 ) \\\\ & = s _ { w ( - e + 1 , 0 ) } \\left ( w _ 0 + ( w ( 0 , e - 1 ) , w _ 0 ) w ( 0 , e - 1 ) \\right ) \\\\ & = s _ { w ( - e + 1 , 0 ) } ( w _ 0 - w ( 0 , e - 1 ) ) = - s _ { w ( - e + 1 , 0 ) } ( w _ 1 + \\cdots w _ { e - 1 } ) \\\\ & = - \\left ( w _ 2 + \\cdots w _ { e - 1 } + w _ 1 + ( w _ 1 , w ( - e + 1 , 0 ) ) w ( - e + 1 , 0 ) \\right ) \\\\ & = - \\sum _ { i = - e + 1 } ^ { e - 1 } w _ i . \\end{align*}"}
-{"id": "7799.png", "formula": "\\begin{gather*} v _ t + b v _ x + v _ { x x x } + v _ { x y y } = 0 , \\\\ v \\big | _ { t = 0 } = 0 , v \\big | _ { x = 0 } = \\widetilde \\mu _ 0 , v \\big | _ { x = R } = v _ x \\big | _ { x = R } = 0 \\end{gather*}"}
-{"id": "8499.png", "formula": "\\begin{align*} \\aligned \\lim _ { i \\rightarrow \\infty } \\left ( \\sup _ { \\mathbf { x } \\in Q } d _ { Q } ( \\mathbf { x } ) | D ^ 2 f _ i ( \\mathbf { x } ) | \\right ) = \\infty . \\endaligned \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{align*} \\mathsf { P } ^ i _ j \\triangleleft K _ \\lambda = \\sum _ { m , n } c ^ m _ n ( \\mathsf { M } ^ n _ m ) ^ i _ j \\triangleleft K _ \\lambda = \\sum _ { m , n } q ^ { - ( \\lambda , \\lambda _ m - \\lambda _ n ) } c ^ m _ n ( \\mathsf { M } ^ n _ m ) ^ i _ j . \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} \\begin{cases} \\Psi _ { - 1 } ( X , Y ) = - 1 ; \\\\ \\Psi _ { 0 } ( X , Y ) = 0 ; \\\\ \\Psi _ { 1 } ( X , Y ) = 1 ; \\\\ \\Psi _ { 2 } ( X , Y ) = 2 Y ; \\\\ \\Psi _ { 3 } ( X , Y ) = 3 X ^ 4 + 6 A X ^ 2 + 1 2 B X - A ^ 2 ; \\\\ \\Psi _ { 4 } ( X , Y ) = 4 Y ( X ^ 6 + 5 A X ^ 4 + 2 0 B X ^ 3 - 5 A ^ 2 X ^ 2 - 4 A B X - 8 B ^ 2 - A ^ 3 ) . \\end{cases} \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{align*} \\langle \\overline { x _ 1 } \\cdots \\overline { x _ { k } } | A ^ \\prime ( z ) | \\overline { y _ 1 } \\cdots \\overline { y _ { k } } \\rangle = & ( - 1 ) ^ k \\prod _ { j = 1 } ^ k \\delta _ { \\overline { x _ j } \\overline { y _ j } } . \\end{align*}"}
-{"id": "7047.png", "formula": "\\begin{align*} \\ L u = \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert = 2 l } a _ { \\alpha } D ^ { \\left [ \\alpha \\right ] } u + A u + \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert < 2 l } A _ { \\alpha } \\left ( x \\right ) D ^ { \\left [ \\alpha \\right ] } u + \\lambda u = f , \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} I ( [ \\lambda , A , B ] \\cdot F ) = ( \\lambda \\det A \\det B ) ^ { p } I ( F ) \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} \\underset { \\epsilon , \\delta \\rightarrow 0 } { l i m } \\quad \\underset { T \\geq 1 } { s u p } \\underset { \\kappa \\in ( 0 , 1 ) } { s u p } \\mathbb { E } \\left | \\eta _ { f , \\psi _ \\kappa , \\epsilon , \\delta } ^ T - \\eta _ { \\psi _ \\kappa } ^ T \\right | ^ 2 = 0 , \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{align*} [ L _ { - r } , \\ , S _ { r - 1 } ] = L _ { - 1 } \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} r \\dot { r } \\left [ \\frac { d } { d f } \\left ( \\frac { f ^ { 3 } } { p \\dot { p } } \\right ) \\right ] - 2 g \\left [ \\frac { d } { d f } \\left ( \\frac { f p } { \\dot { p } } \\right ) \\right ] + \\left [ \\frac { d } { d f } \\left ( \\frac { p ^ { 3 } } { f \\dot { p } } \\right ) \\right ] \\left [ \\left ( \\frac { g ^ { 2 } } { r } \\right ) \\frac { d } { d g } \\left ( \\frac { g ^ { 2 } } { r } \\right ) \\right ] = 0 . \\end{align*}"}
-{"id": "2414.png", "formula": "\\begin{align*} \\frac { 3 } { 2 } d ( d + 1 ) & = \\begin{pmatrix} 2 d + 2 \\\\ 2 \\end{pmatrix} - \\begin{pmatrix} d + 2 \\\\ 2 \\end{pmatrix} \\\\ & = \\begin{pmatrix} 2 d + 2 \\\\ 2 \\end{pmatrix} - \\begin{pmatrix} 2 d - k + 2 \\\\ 2 \\end{pmatrix} + \\alpha ( d - k ) + ( d - k ) ( k + 3 ) - 1 \\\\ & \\geq \\begin{pmatrix} 2 d + 2 \\\\ 2 \\end{pmatrix} - \\begin{pmatrix} 2 d - k + 2 \\\\ 2 \\end{pmatrix} + \\alpha ( d - k ) . \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} \\sum _ { \\overrightarrow { x } \\in V _ n } ( E { \\widetilde { \\rho } } ^ 2 ) ^ { | x _ n | } \\leq ( d + 1 ) ^ n \\sum _ { \\overrightarrow { x } \\in V _ n } \\Big ( \\prod _ { j = 0 } ^ { n - 1 } \\frac { 1 } { \\deg ( x _ j ) } \\Big ) \\Big ( E { \\widetilde { \\rho } } ^ 2 \\Big ) ^ { | x _ n | } \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} \\langle h , \\int _ 0 ^ \\cdot \\int _ E l _ s ( e ) \\tilde N ( d s , d e ) \\rangle _ t = 0 , \\ , \\ , \\ , 0 \\leq t \\leq T { \\rm a . s . } \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{align*} \\Delta & = ( f ( \\underbar { X } ) ) - \\mathbb { E } \\left [ ( f ( \\underbar { X } ) ) \\big | g ( \\underbar { X } ) \\right ] \\\\ & = \\left ( \\mathbb { E } \\left [ f ( \\underbar { X } ) \\big | g ( \\underbar { X } ) \\right ] \\right ) . \\end{align*}"}
-{"id": "4097.png", "formula": "\\begin{align*} \\nu ( 1 _ N \\otimes \\mu ) = \\alpha \\otimes 1 _ X \\mbox { a n d } \\mu ( 1 _ M \\otimes \\nu ) = \\beta \\otimes 1 _ Y . \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} m _ { i , k } = { \\cal O } ( k ^ { { a _ i } k } ) \\ \\mbox { a s } \\ k \\to \\infty , \\ \\mbox { f o r } \\ i = 1 , \\ldots , n , \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} \\frac { \\partial ^ n } { \\partial m ^ n } h _ { \\mu } ( x , m ) = H _ n ( x , m ) h _ { \\mu } ( x , m ) . \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} [ q ^ - , q ^ + ] = \\mathbb { I } _ 2 - 2 N _ q , [ N _ q , q ^ + ] = - q ^ + , [ N _ q , q ^ - ] = + q ^ - , \\end{align*}"}
-{"id": "7710.png", "formula": "\\begin{align*} U _ { 2 m } ^ { ( 2 ) } & = U _ { m } ^ { 2 } \\\\ & = U _ { m } \\left ( p U _ { m - 1 } - q U _ { m - 2 } \\right ) \\\\ & = p U _ { m } U _ { m - 1 } - q \\left ( p U _ { m - 1 } - q U _ { m - 2 } \\right ) U _ { m - 2 } \\\\ & = p U _ { 2 m - 1 } ^ { ( 2 ) } - p q U _ { 2 m - 3 } ^ { ( 2 ) } + q ^ { 2 } U _ { 2 m - 4 } ^ { ( 2 ) } . \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} s _ { i j } = \\left \\{ \\begin{tabular} { c c } $ a _ { i j } + b _ { i j } $ & $ 1 \\leq i , j \\leq n $ \\\\ $ a _ { i j } $ & $ 1 \\leq i \\leq n , \\ j = n + 1 $ \\\\ $ a _ { i j } + b _ { i j } $ & $ i = n + 1 , $ \\ $ 1 \\leq j \\leq n $ \\\\ $ a _ { i j } $ & $ i = n + 1 , $ \\ $ \\ j = n + 1 $ \\end{tabular} \\right . \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} \\mathcal { Q } ( [ x ] \\circ b _ 1 ^ { \\circ n } ) \\equiv \\sum _ { i = 0 } ^ \\infty [ c _ i ] \\circ ( b _ 1 ) ^ { \\circ ( i + 2 n ) } u ^ { 2 ( i + n ) } . \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{align*} \\mu _ t ( X ) = \\frac { d } { d t } X ^ p _ t , \\sigma ^ 2 _ t ( X ) = \\frac { d } { d t } \\langle X ^ m \\rangle _ t . \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{align*} c \\left ( \\overline { \\xi } ( x _ i + y ) \\right ) = c \\left ( \\overline { \\xi } _ i ( x _ i + y ) \\right ) + O \\left ( \\frac { \\left | s ( \\overline { \\xi } _ i ( x _ i + y ) ) \\right | } { R ^ 2 } \\right ) + O \\left ( \\frac { 1 } { R ^ 4 } \\right ) . \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} [ f ] ( x ) = \\int _ { \\R } \\tau d \\nu ^ f _ x . \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} \\lambda _ 1 \\lambda _ 2 x _ 2 ^ 2 + \\lambda _ 0 \\lambda _ 2 x _ 3 ^ 2 + \\lambda _ 0 \\lambda _ 1 x _ 4 ^ 2 + F ( \\lambda _ 0 , \\lambda _ 1 , \\lambda _ 2 ) x _ 5 ^ 2 = 0 , \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} p ^ 2 \\sum _ { k = \\frac { p - 1 } 2 } ^ { p - 1 } \\frac { ( 1 ) _ k ^ 2 } { ( \\frac 3 2 ) _ k ^ 2 } \\cdot ( - 1 ) ^ k \\equiv & \\frac { p ^ 2 ( 1 ) _ { p - 1 } ^ 2 } { ( \\frac 3 2 ) _ { p - 1 } ^ 2 } \\cdot \\big ( \\Psi ( 0 ) + p \\cdot \\Psi ' ( 0 ) \\big ) \\\\ \\equiv & ( 1 - 2 p H _ { \\frac { p - 1 } 2 } ) \\cdot \\Psi ( 0 ) ( 1 + 2 p H _ { \\frac { p - 1 } 2 } ) \\equiv \\Psi ( 0 ) \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} e _ k : = \\sum _ { j = 1 } ^ n \\varrho _ j x _ j ^ k \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{align*} I _ { \\upsilon } ( q ; x ) = \\frac { \\left ( \\frac { x } { 2 } \\right ) ^ { \\upsilon } \\ 2 ^ { 2 \\upsilon + q - \\frac { 3 } { 2 } } \\ \\Gamma ( \\upsilon + q ) } { \\sqrt { \\pi } \\ \\Gamma \\left ( 2 \\upsilon + q + \\frac { 1 } { 2 } \\right ) } { _ { 1 } F _ { 1 } } \\left ( \\upsilon + q , 2 \\upsilon + q + \\frac { 1 } { 2 } , 2 x \\right ) \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} \\varphi ' ( x ) = y ^ 2 \\left ( - x - x ^ 2 \\right ) - \\lambda y + \\frac { n } { x } - \\lambda y x + \\ldots \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} \\{ \\xi ^ 1 , \\ldots , \\xi ^ M \\} = : \\Xi \\subset \\Omega \\end{align*}"}
-{"id": "5910.png", "formula": "\\begin{align*} s _ { i , j } & = s _ e ( i , j ) - s _ o ( i , j ) = [ q ^ i ] \\frac { q ^ j } { 1 - q ^ j } ( q ; q ) _ { \\infty } . \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} \\dim R ( M ; \\mathcal P _ M ) = \\frac { 1 } { 2 } \\dim R ( \\Sigma ; \\mathcal P _ { \\Sigma } ) \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{align*} \\varphi ( k , x ) = A \\cos k x + B \\frac { \\sin k x } { k } + \\int _ 0 ^ x \\frac { \\sin k ( x - t ) \\cos k t } { k } V ( t ) d t A + O \\left ( \\frac { e ^ { | { \\rm I m } k | x } } { k ^ 2 } \\right ) . \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} \\sigma ( T ) = \\sum \\limits _ { \\underline { k } \\ge \\underline { 0 } } \\dfrac { \\underline { x } ^ { \\underline { k } } } { \\underline { k } ! } \\sigma \\bigl ( T _ { ( \\underline { k } ) } \\bigr ) . \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} \\lbrack \\Xi _ { \\mathrm { p } , l } ^ { ( \\omega ) } ( t ) ] _ { + } = \\int _ { \\mathbb { R } } \\left ( \\cos \\left ( t \\nu \\right ) - 1 \\right ) \\mu _ { \\mathrm { p } , l } ^ { ( \\omega ) } ( \\mathrm { d } \\nu ) \\ , t \\in \\mathbb { R } \\ . \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} \\begin{array} { r l } y ( x , \\lambda ) \\tilde { y } ( x , \\lambda ) \\ ! \\ ! \\ ! \\ ! & = A _ 1 ( \\lambda ) + A _ 2 \\cos ( 2 k x ) + A _ 3 \\cos 2 k ( x - \\frac { 1 } { 2 } ) \\\\ & \\quad + A _ 4 \\cos 2 k ( x - 1 ) + \\frac { 1 } { 2 } \\int _ 0 ^ x K ( x , t ) \\cos ( 2 k t ) d t \\end{array} \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{align*} \\mathfrak { L } _ { g } ^ { * } ( f ) = - ( \\Delta f ) g + H e s s \\ , f - f R i c = 0 . \\end{align*}"}
-{"id": "7935.png", "formula": "\\begin{align*} L ^ { q , r } ( D ) = \\big ( L ^ 1 ( D ) , L ^ \\infty ( D ) \\big ) _ { 1 - 1 / q , r } \\end{align*}"}
-{"id": "6236.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac { 1 } { T } \\int _ 0 ^ T | f ( \\sigma + i t ) | ^ 2 d t = \\sum _ { n = 1 } ^ \\infty | a _ n | ^ 2 n ^ { - 2 \\sigma } \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} d \\nu = \\frac { x } { \\pi \\sqrt { 1 - x ^ 2 } } d x . \\end{align*}"}
-{"id": "7216.png", "formula": "\\begin{align*} \\alpha & = \\alpha ( T , R ) \\\\ & = T ^ 2 - 4 \\\\ \\beta & = \\beta ( T , R ) \\\\ & = 2 T ^ 2 + R ^ 2 - R T ^ 2 - 4 \\\\ & = R - 3 \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{align*} D _ { n , k } ( 1 , y ( 1 - y ) ) & = k \\ , y ( 1 - y ) \\ , \\Big [ \\displaystyle \\frac { y ^ { n - 1 } - ( 1 - y ) ^ { n - 1 } } { 2 y - 1 } \\Big ] + D _ n ( 1 , y ( 1 - y ) ) , \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} \\mathcal { R } ( M , N ) : = \\{ p \\in \\mathbb { C } ^ n ~ | ~ \\exists z \\in \\mathbb { C } ^ n : M z + N \\bar { z } = p \\} . \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{align*} \\theta ( s ) = \\begin{cases} s ^ { \\lambda _ 1 - ( \\alpha + 1 ) \\rho } & ( \\alpha + 1 ) \\rho < \\lambda _ 1 \\\\ \\log ( 1 + s ) & ( \\alpha + 1 ) \\rho = \\lambda _ 1 \\\\ 1 & ( \\alpha + 1 ) \\rho > \\lambda _ 1 \\end{cases} \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{align*} d X ^ N _ i ( t ) = d W ^ N _ i ( t ) + \\sum _ { j \\ne i } ^ { } \\frac { 1 } { X ^ N _ i ( t ) - X ^ N _ j ( t ) } d t , \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} \\widehat { \\pi } ^ + _ n ( - s ) : = \\widehat { \\pi } ^ { ( r _ n , \\theta ^ n _ 0 ) } ( \\theta ^ n _ 0 - s ) - l _ n \\widehat { \\pi } ^ - _ n ( - s ) : = \\widehat { \\pi } ^ { ( l _ n , \\theta ^ n _ 0 ) } ( \\theta ^ n _ 0 - s ) - l _ n s \\geq 0 . \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} Q = \\frac { v _ \\epsilon ''' } { v _ \\epsilon '' } + ( \\log \\theta ) ' - A t \\leq C . \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} \\widetilde { W } _ { n - i } ( K ) = \\frac { \\omega _ n } { \\omega _ i } \\int _ { G ( n , i ) } \\mathcal { H } ^ i ( K \\cap \\xi ) d \\xi . \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} \\big ( \\frac { \\partial \\phi } { \\partial t } , \\ , \\Delta \\varphi \\big ) + \\big ( \\mathbf { J } , \\ , \\nabla \\varphi \\big ) = 0 , \\forall \\varphi \\in H _ { 0 } ^ { 1 } ( \\Omega ) \\cap H ^ { 2 } ( \\Omega ) . \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{align*} J _ 1 ( - k ) J _ 1 ( k ) ^ { - 1 } = [ J _ 1 ( - { k } ) ^ \\dag ] ^ { - 1 } J _ 1 ( { k } ) ^ \\dag = [ J _ 1 ( - \\bar { k } ) ^ \\dag ] ^ { - 1 } J _ 1 ( \\bar { k } ) ^ \\dag , k \\in \\mathbb { R } . \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} \\begin{aligned} \\vartheta _ 3 ( z ; \\tau ) & : = \\vartheta _ 4 ( z + \\tfrac 1 2 \\pi ; \\tau ) , \\\\ \\vartheta _ 1 ( z ; \\tau ) & : = - i e ^ { i z + \\tfrac 1 4 \\pi i \\tau } \\vartheta _ 4 ( z + \\tfrac 1 2 \\pi \\tau ; \\tau ) , \\\\ \\vartheta _ 2 ( z ; \\tau ) & : = \\vartheta _ 1 ( z + \\tfrac 1 2 \\pi ; \\tau ) . \\end{aligned} \\end{align*}"}
-{"id": "4759.png", "formula": "\\begin{align*} \\varphi ( u + J ) = \\left \\{ \\begin{array} { c l } c ^ { - 1 } _ { \\alpha } & \\mbox { i f $ u = B _ { \\alpha } $ f o r a n a r r o w $ \\alpha \\in Q _ 1 $ } , \\\\ 0 & \\mbox { o t h e r w i s e } , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{align*} I _ k ( a ) : = \\int _ 0 ^ a \\frac { x ^ k } { \\sqrt { x ( a - x ) } } d x \\end{align*}"}
-{"id": "7502.png", "formula": "\\begin{align*} \\dot { x } x ^ { - 1 } + \\left ( \\dot { x } x ^ { - 1 } \\right ) ^ * = \\left [ \\left ( x \\phi x ^ { - 1 } \\right ) ^ * , x \\phi x ^ { - 1 } \\right ] - \\rho . \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} z _ n = R _ n + \\sum _ { k = 0 } ^ \\nu \\gamma _ { n k } ( x ^ 2 - 1 ) ^ k ; z _ m = R _ m + \\sum _ { k = 0 } ^ \\mu \\gamma _ { m k } ( x ^ 2 - 1 ) ^ k . \\end{align*}"}
-{"id": "5297.png", "formula": "\\begin{align*} \\int _ \\Omega \\omega ( y ) ( x \\cdot y ) \\ , d y & = - \\int _ \\Omega \\omega ( y ) ( x \\cdot y ) \\ , d y - \\int _ \\Omega ( \\omega ( y ) \\times y ) \\times x \\ , d y \\\\ & = - \\frac 1 2 \\left ( \\int _ \\Omega ( \\omega ( y ) \\times y ) \\ , d y \\right ) \\times x . \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} u ( t + s ) = u ( t ) u ( s ) \\mbox { f o r e v e r y } s , t \\in \\R . \\end{align*}"}
-{"id": "3368.png", "formula": "\\begin{align*} \\mathbf { A } = \\begin{bmatrix} 1 & 0 & 0 & 0 & \\dots & 0 & 0 & 0 \\\\ - 1 & 1 & 0 & 0 & \\dots & 0 & 0 & 0 \\\\ 0 & - 1 & 1 & 0 & \\dots & 0 & 0 & 0 \\\\ 0 & 0 & - 1 & 1 & \\dots & 0 & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots \\\\ 0 & 0 & 0 & 0 & \\dots & - 1 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & \\dots & 0 & - 1 & 1 \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{gather*} \\sum _ { k = 0 } ^ { p - 1 } e ^ { \\frac { - 2 k j \\pi i } { p } } | x | ^ { 2 j } = 0 \\quad \\textrm { f o r } j = 1 , \\dots , p - 1 , \\end{gather*}"}
-{"id": "5948.png", "formula": "\\begin{align*} x ' x x & = x \\ , , \\\\ x x ' x ' & = x ' \\ , , \\end{align*}"}
-{"id": "6078.png", "formula": "\\begin{align*} 0 \\neq [ L _ { - 1 } , \\ , L _ { - 3 } ] = [ L _ { - 3 } , \\ , [ L _ { - 5 } , \\ , [ L _ { - 1 } , \\ , L _ 5 ] ] ] = [ L _ { - 5 } , \\ , [ L _ { - 3 } , \\ , [ L _ { - 1 } , \\ , L _ 5 ] ] ] \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} \\eta _ k = - a _ k ( ( A + u _ 0 P _ { \\phi _ k } ^ { \\bot } B - \\lambda _ k ^ { u _ 0 } ) \\big | _ { \\phi _ k ^ { \\bot } } ) ^ { - 1 } u _ 0 P _ { \\phi _ k } ^ { \\bot } B \\phi _ k . \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{align*} O = [ \\theta _ { i j } ] _ { i , j \\geq 0 } , \\theta _ { i j } = \\frac { 1 } { | | P _ i | | ^ 2 } < P _ i , \\int P _ j > , \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{gather*} \\frac { 1 } { \\pi } \\int _ { - 1 } ^ { 1 } \\frac { ( 1 + \\rho ^ { 2 } ) ^ { 3 } + 1 6 \\rho ^ { 3 } x y z - 4 \\rho ^ { 2 } ( 1 + \\rho ^ { 2 } ) ( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } ) } { \\sqrt { 1 - z ^ { 2 } } w _ { 3 } ( x , y , z | \\rho ) } d z \\\\ = \\frac { ( 1 - \\rho ^ { 2 } ) ^ { 3 } + 4 \\rho ^ { 2 } ( 1 - \\rho ^ { 2 } ) ( x ^ { 2 } + y ^ { 2 } ) } { ( 1 - \\rho ^ { 2 } ) ^ { 4 } + 1 6 \\rho ^ { 4 } ( x ^ { 4 } + y ^ { 4 } ) + 8 \\rho ^ { 2 } ( 1 - \\rho ^ { 2 } ) ^ { 2 } ( x ^ { 2 } + y ^ { 2 } ) - 1 6 \\rho ^ { 2 } ( 1 + \\rho ^ { 4 } ) x ^ { 2 } y ^ { 2 } } , \\end{gather*}"}
-{"id": "316.png", "formula": "\\begin{align*} \\Xi ^ { \\lambda } ( K _ { a } F _ { a } \\otimes 1 \\otimes E _ { a } K _ { a } ^ { - 1 } \\otimes K _ { a } ) = \\sum _ { i , j , m , n , o } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) \\pi ( K _ { a } F _ { a } ) _ { m } ^ { j } c _ { n } ^ { m } \\pi ( E _ { a } K _ { \\lambda } ) _ { o } ^ { n } c _ { i } ^ { o } . \\end{align*}"}
-{"id": "7190.png", "formula": "\\begin{align*} \\frac { ( 2 \\ell + 1 ) \\pm i \\sqrt { 4 \\ell + 3 } } { 2 ( \\ell + 1 ) } = ( a + b i \\sqrt { 4 \\ell + 3 } ) ^ 2 \\end{align*}"}
-{"id": "9675.png", "formula": "\\begin{align*} \\Theta ( 0 , \\pi , \\psi ) = \\pi - \\Theta ( 0 , 0 , \\psi ) . \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{align*} \\mu ( R ( z _ { 3 0 } ) ) = Q ^ { 2 0 } Q ^ 6 y _ 4 + x ^ 4 Q ^ { 1 2 } Q ^ 6 y _ 4 \\end{align*}"}
-{"id": "631.png", "formula": "\\begin{align*} \\begin{cases} \\div ( | x _ { n + 1 } | ^ a \\nabla u ) = 0 & B _ 1 \\setminus \\Lambda ( u ) \\cr u = 0 & \\Lambda ( u ) , \\end{cases} \\end{align*}"}
-{"id": "2849.png", "formula": "\\begin{align*} \\delta ^ { ( 2 k + 1 ) } _ { \\alpha } = ( 2 k + 1 ) ( \\delta _ \\circ ) ^ { ( 2 k ) } _ { \\alpha } ( l _ 0 ^ { 2 k } ) ' _ \\alpha + ( \\delta _ \\circ ) ^ { ( 2 k + 1 ) } _ { \\alpha } ( l _ 0 ^ { 2 k } ) _ \\alpha = 0 . \\end{align*}"}
-{"id": "5025.png", "formula": "\\begin{align*} \\dot { h } = 1 2 | b | ^ 2 + \\mathcal { O } ( | b | ^ 4 ) \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{align*} \\sum _ { j \\in T ^ c } g _ j u _ j = \\sum _ { j \\in \\hat S } g _ j u _ j + \\sum _ { j \\notin \\hat S \\cup T } g _ j u _ j \\le \\sum _ { j \\in \\hat S } | g _ j | | u _ j | + g _ s ^ * \\sum _ { j \\notin \\hat S \\cup T } | u _ j | . \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{gather*} \\frac { c _ 0 I _ { m _ i } } { z ^ { r + 1 } } + \\frac { c _ 1 I _ { m _ i } } { z ^ { r } } + \\cdots + \\frac { c _ { r - j } I _ { m _ i } } { z ^ { j + 1 } } + \\frac { d _ i I _ { m _ i } } { z ^ j } + \\frac { * } { z ^ { j - 1 } } + \\cdots , i = 1 \\ldots , l . \\end{gather*}"}
-{"id": "4705.png", "formula": "\\begin{align*} T ( z ) = \\begin{pmatrix} e ^ { - \\ell _ { \\alpha } } & 0 \\\\ 0 & e ^ { \\ell _ { \\alpha } } \\end{pmatrix} Y ( z ) \\begin{pmatrix} e ^ { - N ( g ( z ) - \\ell _ { \\alpha } ) } & 0 \\\\ 0 & e ^ { N ( g ( z ) - \\ell _ { \\alpha } ) } \\end{pmatrix} \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} y _ \\alpha = - ( x _ j - x _ i + \\log c _ \\alpha ) \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} E [ C _ { n , m } ] = 2 \\left [ n + 3 + ( m - 2 ) h _ n - ( m + 2 ) h _ m + ( n - m + 3 ) ( h _ n - h _ { n - m + 1 } ) \\right ] \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ \\infty \\overline { f } _ 1 ( i ) x ^ i \\right ) ^ k = \\sum _ { n = k } ^ \\infty \\overline { c } _ 2 ( n , k ) x ^ n . \\end{align*}"}
-{"id": "2873.png", "formula": "\\begin{align*} P \\cdot Q : = \\sum _ { \\underline { k } \\in \\Sigma } \\sum _ { \\underline { l } \\in \\Sigma } \\sum _ { \\underline { 0 } \\ , \\le \\ , \\underline { i } \\ , \\le \\ , \\underline { k } } \\binom { k _ 1 } { i _ 1 } \\dots \\binom { k _ n } { i _ n } a _ { \\underline { k } } \\frac { \\partial ^ { | \\underline { i } | } \\ , b _ { \\underline { l } } } { \\partial x _ 1 ^ { i _ 1 } \\dots \\partial x _ n ^ { i _ n } } \\partial ^ { \\underline { k } + \\underline { l } - \\underline { i } } . \\end{align*}"}
-{"id": "3095.png", "formula": "\\begin{align*} \\left | \\sum _ { n \\ge 1 } \\frac { ( - 1 ) ^ n q ^ { \\frac { n ( 3 n + 1 ) } { 2 } } } { 1 + q ^ n } \\right | \\le \\frac { 1 } { 1 - | q | } \\sum _ { n \\ge 1 } | q | ^ { \\frac { n ( 3 n + 1 ) } { 2 } } \\ll \\frac { 1 } { y } \\cdot y ^ { - \\frac 1 2 } = y ^ { - \\frac 3 2 } . \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{align*} \\vec { C } _ 1 = \\lambda \\ , \\vec { A } _ 1 . \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} | F ( t ) | & \\leq \\kappa _ 2 \\max \\lbrace | x _ i ( t ) - z _ i | , \\ : i \\in \\lbrace - N , - N + 1 , N , N + 1 \\rbrace \\rbrace , \\\\ \\kappa _ 2 & = 8 \\max \\lbrace | V _ 1 ( x , y ) | , | V _ 2 ( x , y ) | , \\ : | x - y | \\leq 2 \\rbrace \\cdot \\max \\lbrace | V _ { 1 2 } ( x , y ) | , | x - y | \\leq 2 \\rbrace . \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} ( g H g ^ { - 1 } \\cap H ) ( H \\cap g ^ { - 1 } H g ) = \\langle a \\rangle \\langle a b \\rangle = H . \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} \\beta ^ { u + t } \\alpha _ i ^ { u + p _ i - 1 } = \\beta ^ { u + t } \\alpha _ i ^ { u + t + \\lfloor \\frac { p _ i - 1 - t } { s } \\rfloor s } \\alpha _ i ^ { p _ i - 1 - t - \\lfloor \\frac { p _ i - 1 - t } { s } \\rfloor s } \\in S _ i \\alpha _ i ^ { p _ i - 1 - t - \\lfloor \\frac { p _ i - 1 - t } { s } \\rfloor s } \\subseteq K \\end{align*}"}
-{"id": "9635.png", "formula": "\\begin{align*} \\varepsilon ^ { ( j ) } _ { k + 1 } = \\varepsilon ^ { ( j + 1 ) } _ { k - 1 } + \\frac { 1 } { \\varepsilon ^ { ( j + 1 ) } _ { k } - \\varepsilon ^ { ( j ) } _ { k } } , \\ \\ j , k = 0 , 1 , \\ldots , \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{align*} \\eta _ { a } ( C ( \\mathsf { Q } ) ) = \\sum _ { i , j , k } q ^ { ( 2 \\rho - 2 \\alpha _ { a } , \\lambda _ { i } ) } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) \\varepsilon ( F _ { a } \\triangleright \\mathsf { P } _ { k } ^ { j } ) \\varepsilon ( E _ { a } \\triangleright \\mathsf { P } _ { i } ^ { k } ) . \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} & w \\in L ^ 2 _ { l o c } ( [ \\bar t , \\infty ) ; L ^ \\infty ( \\Omega ) ) , \\nabla w \\in L ^ \\infty _ { l o c } ( [ \\bar t , \\infty ) ; L ^ 2 ( \\Omega ) ) , \\\\ & \\partial _ t w , \\ , A w \\in L ^ 2 _ { l o c } ( [ \\bar t , \\infty ) ; L ^ 2 _ \\sigma ( \\Omega ) ) . \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} \\prod _ { j = 1 } l _ j & = \\Big | \\prod _ { j = 1 } ^ n A _ j \\Big | \\\\ & = \\sum _ { i = 1 } ^ n \\sum _ { a \\in \\prod _ { j = 1 , j \\not = i } ^ n A _ j } \\big | \\{ a \\in A _ i : i ( a _ 1 , a _ 2 , \\ldots , a _ { i - 1 } , a , a _ { i + 1 } , \\ldots a _ n ) = i \\} \\big | \\\\ & \\le \\sum _ { i = 1 } ^ n K _ i \\prod _ { j = 1 , j \\not = i } ^ n l _ j \\le \\frac { n } { n + 1 } \\prod _ { j = 1 } ^ n l _ j \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} i ^ * : \\operatorname { Q u a d } ( \\Sigma ) \\to H ^ 1 ( \\partial \\Sigma ) ^ { ( 2 ) } , \\omega \\mapsto i ^ * \\omega = \\omega \\circ i _ * . \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} f _ { a , b } ( X , Y , Z ) = [ a X Z + X ^ 2 , b Y Z + Y ^ 2 , Z ^ 2 ] \\quad \\end{align*}"}
-{"id": "8939.png", "formula": "\\begin{align*} [ T _ { \\phi ( r ) e ^ { - i p \\theta } } , T _ { z ^ l } ] ( z ^ k ) = \\begin{cases} \\big ( G ( k + l ) - G ( k ) \\big ) z ^ { k + l - p } & k \\geq p \\\\ G ( k + l ) \\ , z ^ { k + l - p } & p - l \\leq k \\leq p - 1 . \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{gather*} Z _ m ( e ^ { \\varphi i } \\zeta , e ^ { \\psi i } \\eta ) = Z _ m ( e ^ { - \\psi i } \\eta , e ^ { - \\varphi i } \\zeta ) \\quad \\textrm { a n d } \\overline { Z _ m ( e ^ { \\varphi i } \\zeta , e ^ { \\psi i } \\eta ) } = Z _ m ( e ^ { \\psi i } \\eta , e ^ { \\varphi i } \\zeta ) . \\end{gather*}"}
-{"id": "6981.png", "formula": "\\begin{align*} V _ { 1 } = 1 0 1 0 1 0 1 0 1 0 . 1 0 1 0 1 0 1 0 1 0 . 1 0 1 0 1 0 1 0 1 0 . 1 0 1 0 1 0 1 0 1 0 . 1 0 1 0 1 0 1 0 1 0 \\end{align*}"}
-{"id": "9773.png", "formula": "\\begin{align*} \\delta V ( X ) = \\int _ { A _ m ( M ) } \\langle S , \\nabla X \\rangle d V ( S ) ; \\end{align*}"}
-{"id": "9701.png", "formula": "\\begin{align*} h _ P ^ 2 ( t ) \\ = \\ t \\sum _ { V } x ^ { 2 } + t ^ 2 \\left ( \\sum _ { E } ( y + z ) ^ { 2 } - \\sum _ { V } x ^ { 2 } \\right ) + t ^ 3 \\left ( \\sum _ { E ^ o } ( y + z ) ^ { 2 } - \\sum _ { V ^ o } x ^ { 2 } \\right ) + t ^ 4 \\sum _ { V ^ o } x ^ { 2 } . \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} \\mathcal { F } _ { s } ^ { + } = \\sigma \\left \\{ Z _ { \\tau } ^ { - 1 } \\left ( F \\right ) : \\tau \\leq s , F \\in \\mathcal { B } _ { d } \\right \\} \\end{align*}"}
-{"id": "9597.png", "formula": "\\begin{align*} T ^ i _ { j k } = L ^ i _ { \\underline { j k } } - \\frac 1 { N + 1 } \\big ( \\delta ^ i _ k L ^ \\alpha _ { \\underline { j \\alpha } } + \\delta ^ i _ j L ^ \\alpha _ { \\underline { k \\alpha } } \\big ) . \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t v + v \\cdot \\nabla v + \\nabla p = b \\cdot \\nabla b , \\\\ [ - 4 m m ] \\\\ \\partial _ t b + v \\cdot \\nabla b = b \\cdot \\nabla v , \\\\ [ - 4 m m ] \\\\ \\nabla \\cdot v = 0 \\nabla \\cdot b = 0 . \\end{cases} \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{align*} M _ 1 = \\frac { Z _ 2 Z _ 3 } { ( Z _ 2 - Z _ 1 ) ( Z _ 3 - Z _ 1 ) } , M _ 2 = \\frac { Z _ 1 Z _ 3 } { ( Z _ 1 - Z _ 2 ) ( Z _ 3 - Z _ 2 ) } , M _ 3 = \\frac { Z _ 1 Z _ 2 } { ( Z _ 1 - Z _ 3 ) ( Z _ 2 - Z _ 3 ) } . \\end{align*}"}
-{"id": "9700.png", "formula": "\\begin{align*} h _ { T _ 0 } ^ 1 ( t ) & \\ = \\ t ( v _ 1 + v _ 2 + v _ 3 ) \\\\ h _ { T _ 1 } ^ 1 ( t ) & \\ = \\ t v _ 1 + t ^ 2 ( v _ 2 + v _ 3 ) \\\\ h _ { T _ 2 } ^ 1 ( t ) & \\ = \\ t ^ 2 ( ( v _ 1 + v _ 2 ) + ( v _ 1 + v _ 3 ) - 2 v _ 1 ) + t ^ 3 v _ 1 \\end{align*}"}
-{"id": "9399.png", "formula": "\\begin{align*} \\pi ( \\theta ) ^ * = \\pi ( \\overline { \\theta } ) \\end{align*}"}
-{"id": "1660.png", "formula": "\\begin{align*} F \\cdot \\widetilde { a } = \\sum _ { \\gamma \\in F } \\gamma \\cdot \\widetilde a = \\sum _ { \\gamma \\in F } \\sum _ { \\tau \\in K } \\sum _ { s \\in S } { f _ { s \\cdot \\widetilde { \\tau } } \\cdot \\gamma ^ { - 1 } \\otimes \\gamma \\cdot s \\cdot \\widetilde { \\tau } } . \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} T = X + i Y \\enskip \\mbox { a n d } \\enskip U = X - i Y . \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) \\left ( I + \\begin{pmatrix} \\alpha & 0 \\\\ \\gamma & \\delta \\end{pmatrix} p \\right ) = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a + \\alpha + \\gamma & b + \\delta \\\\ c + \\gamma & d + \\delta \\end{pmatrix} p . \\end{align*}"}
-{"id": "874.png", "formula": "\\begin{align*} \\langle : X _ T \\otimes X _ T : , \\Phi \\rangle : = \\sum _ { j = 1 } ^ m \\Big ( \\langle X _ T , \\phi _ j \\rangle \\langle X _ T , \\psi _ j \\rangle - \\mathbb { E } ( \\langle X _ T , \\phi _ j \\rangle \\langle X _ T , \\psi _ j \\rangle ) \\Big ) . \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{align*} \\langle A ^ { 1 / 2 } f , A ^ { 1 / 2 } g \\rangle = \\langle \\nabla f , \\nabla g \\rangle , \\mbox { f o r $ f , \\ , g \\in D _ 2 ( A ^ { 1 / 2 } ) = H _ { 0 , \\sigma } ^ 1 ( \\Omega ) $ } , \\end{align*}"}
-{"id": "6727.png", "formula": "\\begin{align*} A \\left ( j \\right ) u \\left ( j \\right ) = a \\left ( j \\right ) u _ { y y } \\left ( x , j \\right ) + b \\left ( j \\right ) u _ { y } \\left ( x , j \\right ) = 0 \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ z \\zeta _ j & = \\zeta _ { j + 1 } , j = 0 , 1 , 2 , \\\\ \\partial _ z \\zeta _ 3 & = - 1 2 \\zeta _ 1 \\zeta _ 2 , \\\\ \\partial _ z \\tilde \\vartheta _ 1 & = ( \\zeta - \\eta z ) \\tilde \\vartheta _ 1 \\end{aligned} \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} C _ n ( k _ 1 , k _ 2 , \\dots , k _ { n - 1 } , k _ n ) = ( k _ n , k _ 1 , \\dots , k _ { n - 2 } , k _ { n - 1 } ) , \\end{align*}"}
-{"id": "7033.png", "formula": "\\begin{align*} p _ a ( Z ) = \\frac { Z ( Z + K _ S ) } { 2 } + 1 = \\frac { Z ( 2 Z + R ) } { 2 } + 1 = - n + 4 . \\end{align*}"}
-{"id": "6868.png", "formula": "\\begin{align*} F _ { J } F _ { K } = \\sum _ { L } c _ { J , K } ^ { L } F _ { L } . \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{align*} \\varepsilon ( E _ a \\triangleright ( \\mathsf { N } ^ n _ m ) ^ i _ j ) & = - q ^ { - ( \\alpha _ a , \\lambda _ j ) } \\varepsilon ( E _ a \\triangleright ( \\mathsf { M } ^ n _ m ) ^ i _ j ) , \\\\ \\varepsilon ( F _ a \\triangleright ( \\mathsf { N } ^ n _ m ) ^ i _ j ) & = - q ^ { - ( \\alpha _ a , \\lambda _ i ) } \\varepsilon ( F _ a \\triangleright ( \\mathsf { M } ^ n _ m ) ^ i _ j ) . \\end{align*}"}
-{"id": "8381.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\liminf _ n E ^ c _ n \\right ) = 1 \\end{align*}"}
-{"id": "9821.png", "formula": "\\begin{align*} \\psi _ { w \\mathbf { z } } = D _ i ( w \\mathbf { z } ) ( ( w \\mathbf { z } ) ^ { n \\lambda } - ( w \\mathbf { z } ) ^ { n s _ i \\lambda } ) . \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} \\langle \\xi , \\eta \\rangle _ A = \\langle \\alpha _ X ( \\xi ) , \\eta \\rangle _ A = \\langle \\xi , \\alpha _ X ( \\eta ) \\rangle _ A = \\langle \\xi , - \\eta \\rangle _ A = - \\langle \\xi , \\eta \\rangle _ A . \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} u _ { n + 1 } = u _ 1 - B _ 1 ( u _ n , u _ n ) - B _ 2 ( u _ n , \\theta _ n ) , \\theta _ { n + 1 } = \\theta _ 1 - B _ 3 ( u _ n , \\theta _ n ) . \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{gather*} \\frac { { \\rm d } Y } { { \\rm d } x } = \\left ( \\sum _ { \\nu = 1 } ^ n \\sum _ { k = 0 } ^ { r _ { \\nu } } \\frac { A _ { \\nu } ^ { ( k ) } } { ( x - u _ { \\nu } ) ^ { k + 1 } } + \\sum _ { k = 1 } ^ { r _ { \\infty } } A _ { \\infty } ^ { ( k ) } x ^ { k - 1 } \\right ) Y , A _ * ^ { ( k ) } \\in M ( m , \\mathbb { C } ) \\end{gather*}"}
-{"id": "4085.png", "formula": "\\begin{align*} \\begin{array} { r c l } ( 1 + \\lambda - \\lambda ^ { 1 / 2 + \\alpha } / 2 ) ^ { - 1 } & = & ( 1 + \\lambda ) ^ { - 1 } ( 1 - ( 1 + \\lambda ) ^ { - 1 } \\lambda ^ { 1 / 2 + \\alpha } / 2 ) ^ { - 1 } \\\\ & \\leq & ( 1 + \\lambda ) ^ { - 1 } ( 1 + 2 ( 1 + \\lambda ) ^ { - 1 } \\lambda ^ { 1 / 2 + \\alpha } / 2 ) \\\\ & \\leq & ( 1 + \\lambda ) ^ { - 1 } + ( 1 + \\lambda ) ^ { - 3 / 2 + \\alpha } \\end{array} \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} u _ 2 ^ 2 ( z ) = \\int _ { D \\setminus D ( 0 , 2 | z | ) } \\frac { \\mu ( \\zeta ) f ( \\zeta ) } { \\zeta - z } d \\zeta \\wedge d \\bar { \\zeta } & = \\sum _ { n \\in \\N } ^ { } \\int _ { D \\setminus D ( 0 , 2 | z | ) } \\left ( \\frac { \\mu ( \\zeta ) f ( \\zeta ) } { \\zeta ^ { n + 1 } } d \\zeta \\wedge d \\bar { \\zeta } \\right ) z ^ n \\\\ & = \\sum _ { n \\in \\N } ^ { } d _ n ( z ) \\ , z ^ n \\end{align*}"}
-{"id": "4799.png", "formula": "\\begin{align*} v = u f ( \\bar { \\alpha } ) ^ { - 1 } \\bar { u } f ( \\alpha ) ^ { - 1 } \\end{align*}"}
-{"id": "9501.png", "formula": "\\begin{align*} \\begin{pmatrix} C _ Q & 0 \\\\ [ 6 p t ] 0 & C _ Q \\end{pmatrix} , \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} \\textrm { m i n i m i z e } \\ , \\ , J ( u ) : = \\frac { 1 } { p } \\displaystyle \\int | \\nabla u | ^ p \\ , d x \\textrm { i n } \\ \\mathbb { K } _ p , \\end{align*}"}
-{"id": "9441.png", "formula": "\\begin{align*} \\Lambda ( \\rho , U ) = \\begin{cases} \\displaystyle { \\frac { 1 } { 2 } \\int _ { \\mathbb { T } ^ 3 } | u | ^ 2 \\rho } , & \\rho \\geq 0 , \\ ; U \\ll \\rho , \\ ; U = u \\rho , \\ ; u \\in L ^ 2 _ { \\rho } \\\\ + \\infty , & { \\rm o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} { p _ { o u t , K } } & \\simeq { W _ { \\bf { 0 } } } { \\gamma ^ { - m K } } { g _ { \\bf { 0 } } } \\left ( 2 ^ { \\cal R } \\right ) \\prod \\limits _ { k = 1 } ^ K { \\frac { 1 } { { \\Gamma \\left ( m \\right ) } } { { \\left ( { \\frac { m } { { { \\theta _ k } { \\sigma _ k } ^ 2 \\left ( { 1 - { \\lambda _ k } ^ 2 } \\right ) } } } \\right ) } ^ m } } \\triangleq { p _ { o u t \\_ { a s y } , K } } . \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{gather*} A _ 0 = \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} , A _ k = \\big ( a ^ { ( k ) } _ { i j } \\big ) \\in M _ 2 ( \\mathbb { C } ) . \\end{gather*}"}
-{"id": "8346.png", "formula": "\\begin{align*} D i s ( \\tilde { g } _ { n } ( X , 1 ) ) = c _ 0 ^ { 2 n } \\mathop { \\Pi } _ { 1 \\le j < k \\le n + 1 } ( x _ j - x _ k ) ^ 2 , \\end{align*}"}
-{"id": "6434.png", "formula": "\\begin{align*} q ( x e _ 1 + y e _ 2 + z e _ 3 ) = a x ^ 2 + b y ^ 2 + c z ^ 2 + u y z + v x z + w x y \\quad \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{align*} \\Psi _ 1 ( 0 ) + ( 1 - z ) \\Psi _ 2 ( 0 ) = & { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & \\frac 1 2 \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] + ( 1 - z ) ^ { a + 1 } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & \\frac 1 2 \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac { z } { z - 1 } \\bigg ] \\\\ = & 2 \\bigg ( 1 - \\frac z 2 \\bigg ) ^ { a + 1 } \\cdot { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - \\frac 1 2 a & \\frac 1 2 - \\frac 1 2 a \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac { z ^ 2 } { ( z - 2 ) ^ 2 } \\bigg ] = ( 2 - z ) \\cdot \\Phi ( 0 ) . \\end{align*}"}
-{"id": "1703.png", "formula": "\\begin{align*} \\mathbb { S } ( R ^ { \\bigstar } ) = \\frac { 2 } { d ^ { n + 1 } } \\Big [ \\frac { d _ 1 d _ 2 \\dots d _ n } { n + 1 } + \\frac { W ^ { ( n ) } _ 1 } { n } + \\dots + \\frac { W ^ { ( n ) } _ s } { n - s + 1 } + \\dots + \\frac { W ^ { ( n ) } _ { n - 1 } } { 2 } \\Big ] . \\end{align*}"}
-{"id": "3787.png", "formula": "\\begin{align*} 0 = ( k - 1 ) s _ { n - i } s _ { 0 } \\leq ( k - 1 ) s _ { n - i - 1 } s _ 1 = ( k - 1 ) s _ { n - i - 1 } \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} U _ { n } ^ { ( 2 ) } = p U _ { n - 1 } ^ { ( 2 ) } - p q U _ { n - 3 } ^ { ( 2 ) } + q ^ { 2 } U _ { n - 4 } ^ { ( 2 ) } , \\ n \\geq 4 , \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{align*} & W \\in C ^ 1 ( ( 0 , \\infty ) ; L ^ { 3 , \\infty } ( \\mathbb R ^ 3 ) \\cap L ^ q _ \\sigma ( \\mathbb R ^ 3 ) ) , \\quad \\forall q \\in ( 3 , \\infty ) , \\\\ & \\nabla ^ 2 W \\in C ( ( 0 , \\infty ) ; L ^ { 3 , \\infty } ( \\mathbb R ^ 3 ) \\cap L ^ q ( \\mathbb R ^ 3 ) ) , \\quad \\forall q \\in ( 3 , \\infty ) . \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} \\forall k \\geq 1 , { 1 \\over \\beta _ { k } } : = \\min \\left \\{ \\langle B \\phi , \\phi \\rangle \\ ; ; \\ ; \\phi \\in N ( L - \\lambda _ { k } I ) , \\ ; \\| \\phi \\| = 1 \\right \\} > 0 , \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{align*} G \\circ \\sum \\limits _ { m = 0 } ^ \\infty a _ m x ^ m = \\sum \\limits _ { m = 0 } ^ \\infty \\frac { a _ m } { m + 1 } x ^ { m + 1 } , \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{align*} P ( \\theta , \\mu ) ( d x ) = \\exp ( \\theta x - K _ \\mu ( \\theta ) ) \\mu ( d x ) . \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\liminf _ n { A } _ n ( n ) \\right ) = 1 . \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} N [ J ] \\models \\psi ^ * ( b ' [ J ] ( \\delta ) = i . \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{align*} \\partial _ j \\mathbb B [ \\eta f ] - \\mathbb B [ \\partial _ j ( \\eta f ) ] = { \\cal B } _ j [ \\eta f ] \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & \\pm j x \\\\ \\mp j y & 0 \\end{pmatrix} \\begin{pmatrix} 0 & \\frac { 1 } { y } \\\\ \\frac { 1 } { x } & 0 \\end{pmatrix} = \\begin{pmatrix} \\pm j & 0 \\\\ 0 & \\mp j \\end{pmatrix} , \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) ^ { p ^ 2 } = \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} + \\begin{pmatrix} 0 & b \\\\ c & 0 \\end{pmatrix} p \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{align*} a ^ { 2 ^ { 2 k } - 1 } + a ^ { 2 ^ { 2 k } - 2 ^ { k } } + 1 = 0 , \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} \\mathbf { P } ( y ) = \\{ S \\in \\mathbf { P } | \\ h ^ 0 ( E _ y ( - b ) | _ S ) = 1 \\} . \\end{align*}"}
-{"id": "7151.png", "formula": "\\begin{align*} P _ C ( g ) = \\begin{pmatrix} u & 0 \\\\ & \\\\ 0 & 1 / u \\end{pmatrix} \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{align*} f ' ( t , x ) & : = \\alpha ^ p \\beta ^ p f ( \\alpha ^ { p - 1 } \\beta ^ p t , \\beta x ) , \\\\ A ' ( t , x ) & : = \\alpha ^ { p - 1 } \\beta ^ { p - 2 } A ( \\alpha ^ { p - 1 } \\beta ^ p t , \\beta x ) . \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} \\| \\nabla \\psi _ { 3 , \\varepsilon } \\| _ 4 \\leq C \\| \\Delta \\psi _ { 3 , \\varepsilon } \\| _ { 4 / 3 } \\leq C \\left ( \\int _ \\Omega \\frac { t _ \\varepsilon ^ { 8 / 3 } } { \\varepsilon ^ { 8 / 3 } } \\right ) ^ { 3 / 4 } \\leq \\frac { C } { \\varepsilon ^ 2 } \\cdot o ( \\varepsilon ^ { 9 / 4 } ) = o ( \\varepsilon ^ { 1 / 4 } ) . \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} I _ 1 & : = \\oint _ { \\delta _ 1 } \\omega & I _ 2 & : = \\oint _ { \\delta _ 2 } \\omega \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} T r \\left ( \\left ( 1 + \\lambda ^ 2 \\frac { ( \\mu x + 1 + x y ) ^ 2 } { y ^ 2 } \\right ) \\left ( \\alpha + ( \\mu + y ) ^ 2 \\lambda ^ 2 \\right ) \\right ) = 0 . \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} Q = \\{ ( a , b ) \\colon a , b \\in A , \\ , \\exists c , d , e \\in A , \\ , b = w ( c , d , e ) , \\ , a \\in \\{ c , d , e \\} \\} . \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } | \\alpha _ j | ^ 2 < \\infty \\iff \\int \\log ( w ( \\theta ) ) \\frac { d \\theta } { 2 \\pi } > - \\infty \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} W ( t ) = \\sum _ { i \\in \\N ^ * } \\beta _ i ( t ) f _ i , \\end{align*}"}
-{"id": "8313.png", "formula": "\\begin{align*} T = \\sum _ { n \\in \\N } T \\psi _ n = \\sum _ { n \\in \\N } D ^ { a _ n } f _ n \\end{align*}"}
-{"id": "5964.png", "formula": "\\begin{align*} [ ( \\epsilon + a ) \\lambda ^ { 2 } + ( a + b ) \\lambda + \\epsilon ] z ^ { 2 } + [ \\epsilon ( \\epsilon + a ) \\lambda + \\epsilon ^ { 2 } ] z = 0 . \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} F _ * ^ e ( \\lambda x _ 1 ^ { k d _ 1 + \\alpha _ 1 } \\dots x _ n ^ { k d _ n + \\alpha _ n } ) = x _ 1 ^ { c _ 1 } \\dots x _ n ^ { c _ n } F _ * ^ e ( \\lambda x _ 1 ^ { s _ 1 } \\dots x _ n ^ { s _ n } ) \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} [ 0 , N ] = \\bigcup \\limits _ { i = 0 } ^ { ( C _ 1 C _ 2 \\eta ) ^ { - 1 } - 1 } I _ { x , i } , \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} \\int \\limits _ { 0 } ^ { x _ { k } } \\gamma _ { k } ^ { - 1 } \\left ( y \\right ) d y < \\infty k = 1 , 2 , . . . , n . \\end{align*}"}
-{"id": "7780.png", "formula": "\\begin{align*} u ( T , x , y ) = u _ T ( x , y ) , ( x , y ) \\in \\Omega . \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{align*} h _ s ( x , n ) = \\frac { 1 } { n } \\sum _ { t = 1 } ^ n \\omega ^ { ( 1 - s ) t } \\exp ( \\omega ^ t x ) , \\enskip s = 1 , . . . , n , \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} \\frac { 1 } { K } X _ T ^ 2 & = \\frac { 1 } { K + 1 } - ( 1 - X _ T ) ^ 2 + \\frac { 1 } { K ( K + 1 ) } \\left [ ( K + 1 ) X _ T - K \\right ] ^ 2 \\\\ & \\ge \\frac { 1 } { K + 1 } - ( 1 - X _ T ) ^ 2 . \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{align*} \\| \\nabla \\psi _ { 3 , \\varepsilon _ n } \\| _ 4 \\leq \\frac { C } { \\varepsilon _ n ^ 2 } \\cdot o ( \\varepsilon _ n ) \\cdot \\varepsilon _ n \\| \\nabla \\varphi _ { \\varepsilon _ n } \\| _ 4 = o ( 1 ) \\cdot \\left ( \\| \\nabla \\psi _ { \\varepsilon _ n } \\| _ 4 + 1 \\right ) . \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} \\Pr { \\mathbf { Y } \\geq d + 2 } = e ^ { - d ^ 2 - d } . \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} N _ 1 : = 2 C _ 1 , N _ 2 : = C _ 3 ( b , B , 2 C _ 1 ) . \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} \\frac { 1 } { \\rho _ { F } ( \\Omega ) } = \\Lambda _ 1 ( \\infty , \\Omega ) \\le \\lim _ { j \\to \\infty } \\lambda _ { 2 } ( p _ { j } , \\Omega ) ^ { \\frac 1 p _ { j } } = \\overline { \\Lambda } \\le \\Lambda _ 2 ( \\infty , \\Omega ) = \\frac { 1 } { \\rho _ { 2 , F } ( \\Omega ) } . \\end{align*}"}
-{"id": "9586.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n E \\left [ | V _ { n , i } | ^ { 2 + \\delta } \\right ] & \\leq \\frac { \\| a \\| ^ { 2 + \\delta } } { n } \\sum _ { i = 1 } ^ n E \\left [ \\| X _ { n , i } \\| ^ { 2 + \\delta } \\big ( | Y _ { n , i } | + \\| X _ { n , i } \\| \\| \\theta _ n \\| + \\| Z _ { n , i } \\| \\| \\gamma _ n \\| \\big ) ^ { 2 + \\delta } \\right ] . \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} 0 = \\sum _ { \\substack { d _ 1 + \\cdots + d _ n + e = g - 1 \\\\ d _ i \\geq 0 , e \\geq 1 } } \\prod _ { i = 1 } ^ { n } Q _ { d _ i } ( a _ i ) Q _ { e + 1 } ( a _ { n + 1 } ) \\prod _ { i = 1 } ^ { n } \\psi _ i ^ { d _ i } \\kappa _ e \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} \\lambda = - \\frac { X _ 1 } { X _ 1 + X _ 2 } < \\mu = - \\frac { X _ 3 } { X _ 3 + X _ 4 } < \\lambda + 1 < \\mu + 1 \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\big \\Vert u _ \\varepsilon - u _ \\varepsilon ^ { [ N _ k ] } \\big \\Vert = 0 \\ , . \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{align*} \\partial _ t \\psi = - ( \\alpha + \\mathrm i \\beta ) \\Delta \\psi + F ( \\psi , \\overline { \\psi } ) , \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } m _ { 1 } ^ { n + 1 } \\\\ m _ { 2 } ^ { n + 1 } \\\\ m _ { 3 } ^ { n + 1 } \\end{array} \\right ] = M \\cdot N \\cdot P \\cdot \\left [ \\begin{array} { c } m _ { 1 } ^ { n } \\\\ m _ { 2 } ^ { n } \\\\ m _ { 3 } ^ { n } \\end{array} \\right ] , \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} \\ L u = \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert = 2 l } a _ { \\alpha } D ^ { \\alpha } u + A u + \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert < 2 l } A _ { \\alpha } \\left ( x \\right ) D ^ { \\alpha } u + \\lambda u = f , \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} \\partial _ { \\xi } ( L _ { c _ * } + 1 ) \\psi _ 1 + \\partial _ { \\xi } L _ { c _ * } ' \\psi _ * = \\lambda _ 1 \\psi _ * . \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{align*} \\mu ^ { l } ( B _ t ( w _ 0 ) ) & \\stackrel { \\eqref { e : s t i m a b a n a n a } } { = } \\sum _ { z _ \\alpha \\in Z } \\mu ^ { l } ( W ^ { z _ \\alpha } ) + \\mu ^ { l } _ 1 \\big ( W ^ { ( 2 ) } \\setminus E \\big ) \\stackrel { \\eqref { e : s t i m a c h i a v a } } { \\leq } 2 \\sum _ { z _ \\alpha \\in Z } s _ { z _ \\alpha } ^ { n - 1 } + \\mu ^ { l } _ 1 \\big ( W ^ { ( 2 ) } \\setminus E \\big ) \\\\ & \\leq 2 \\ , \\mu ^ l _ 1 ( B _ { 1 1 t } ( w _ 0 ) ) . \\end{align*}"}
-{"id": "704.png", "formula": "\\begin{align*} \\sigma _ { 2 j } ( \\eta , \\bar \\eta ) = \\sum _ { n = 0 } ^ { 2 j } \\frac { 1 } { ( 2 j ) ! ( 2 j - n ) ! } \\eta ^ { 2 j - n } \\bar \\eta ^ { 2 j - n } . \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} d \\eta ( \\theta ) = \\left ( 1 - \\tfrac { 1 } { 2 } \\cos \\theta - \\cos 2 \\theta + \\tfrac { 1 } { 2 } \\cos 3 \\theta \\right ) \\tfrac { d \\theta } { 2 \\pi } \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} U _ 0 = \\bigcup _ { k \\in \\omega } [ n _ { 2 k } , n _ { 2 k + 1 } ) , \\ \\ U _ 1 = \\bigcup _ { k \\in \\omega } [ n _ { 2 k + 1 } , n _ { 2 k + 2 } ) , \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} \\big ( ( { \\bf A } ^ \\frac { 1 } { 2 } ) ^ { - } \\big ) ' ( { \\bf A } _ 1 ^ \\frac { 1 } { 2 } ) ^ { - } = ( { \\bf A } _ 1 ^ { - } ) ' = { \\bf A } _ 1 ^ { - } . \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} \\mathfrak { D } _ { K _ n / \\mathbb { Q } } = ( \\ell ^ n \\ell ^ { - 1 / ( \\ell - 1 ) } ) . \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} \\lambda _ k ( A ) \\leqslant \\lambda _ k ( M - N \\overline { M } ^ { - 1 } \\overline { N } ) \\leqslant \\lambda _ { k + n } ( A ) ~ , ~ ~ ~ k = 1 , \\hdots , n . \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} \\Sigma _ 1 ~ \\cap ~ \\Sigma _ 1 ' ~ \\cap ~ \\Sigma _ 1 '' = \\lbrace 1 _ { H } \\rbrace . \\end{align*}"}
-{"id": "9828.png", "formula": "\\begin{align*} \\phi _ { w _ 0 } = \\prod _ { \\alpha \\in \\Phi ^ + } ( 1 - \\mathbf { z } ^ { \\alpha _ i ^ { \\vee } } ) ^ { - 1 } . \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} ( u - c ) \\cdot N = 0 , P = \\sigma \\nabla \\cdot N \\textrm { o n } S , \\end{align*}"}
-{"id": "51.png", "formula": "\\begin{align*} \\bar { W } _ n ( K ) = \\exp \\left ( \\frac { 1 } { n \\omega _ n } \\int _ { S ^ { n - 1 } } \\log \\rho _ K ( u ) d u \\right ) . \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} n _ { i j } ^ { m } = \\left \\{ \\begin{array} { l r } ( - 1 ) ^ { j - i } { { k + j - i } \\choose { k } } , & { \\rm i f } ~ ~ i \\leq j , \\\\ 0 , & { \\rm i f } ~ ~ i > j \\end{array} \\right . \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{gather*} A _ { t _ i } = \\begin{pmatrix} - q _ i \\\\ 1 \\end{pmatrix} \\begin{pmatrix} - p _ i & - p _ i q _ i + \\theta ^ { t _ i } \\end{pmatrix} , N = \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} , \\\\ A _ { \\infty \\ , 1 } = \\begin{pmatrix} 0 & - p _ 1 - p _ 2 \\\\ - 1 & 0 \\end{pmatrix} , N _ i = - p _ i N , i = 1 , 2 . \\end{gather*}"}
-{"id": "2743.png", "formula": "\\begin{align*} \\varDelta ( \\alpha ) = \\frac { \\Xi ( \\alpha ) ^ { - 1 } } { \\log ( \\Xi ( \\alpha ) ^ { - 1 } ) } , \\end{align*}"}
-{"id": "3175.png", "formula": "\\begin{align*} \\underset { \\leftarrow } { \\lim } \\mathcal { M } _ p \\left ( W ^ N \\right ) = \\{ ( \\mu _ N ) _ { N \\ge 1 } \\in \\mathcal { W } : \\mu _ { N + 1 } \\Lambda _ N ^ { N + 1 } = \\mu _ N \\ , \\forall N \\} , \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{align*} \\sum _ { i } q ^ { - ( 2 \\rho , \\lambda _ { i } ) } ( \\mathsf { M } _ { m } ^ { n } ) _ { i } ^ { i } = q ^ { - ( 2 \\rho , \\lambda _ { n } ) } \\sum _ { i } q ^ { ( 2 \\rho , \\lambda _ { n } - \\lambda _ { i } ) } u _ { i } ^ { m * } u _ { i } ^ { n } = \\delta _ { m } ^ { n } q ^ { - ( 2 \\rho , \\lambda _ { m } ) } . \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} V _ { n } ^ { ( k ) } = ( \\alpha ^ { m + 1 } + \\beta ^ { m + 1 } ) ^ { r } ( \\alpha ^ { m } + \\beta ^ { m } ) ^ { k - r } \\end{align*}"}
-{"id": "6462.png", "formula": "\\begin{align*} E ( K _ s ) = \\overline { N S ( X ) } / \\overline { T ( X ) } . \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} \\rho _ c ( \\mathbf { x } ) = \\sum _ { \\mathbf { y } \\in S } \\tilde { \\rho } ( \\mathbf { x } - \\mathbf { y } ) , \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{align*} f _ { j i } ( - ( p _ { i j } - m _ { i j } ^ * ) ) = r _ j . \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} 1 + \\sum _ { n \\ge 1 } \\chi ( C ( \\Sigma _ g , n ) ) t ^ n = ( 1 + t ) ^ { 2 - 2 g } . \\end{align*}"}
-{"id": "2197.png", "formula": "\\begin{align*} 2 F ' ( 2 x ) + \\frac { d K ( x , x ) } { d x } - K ( x , x ) F ( 2 x ) & + \\int _ x ^ \\infty K _ x ( x , s ) F ( s + x ) d s \\\\ & + \\int _ x ^ \\infty K ( x , s ) F ' ( s + x ) d s = 0 . \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} Z ( [ a _ { k - 1 } , a _ k ] ) = \\lambda _ k z + \\mu _ k w \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} \\frac { 1 } { \\kappa _ { i , n } } = O \\left ( \\frac { 1 } { n ^ 2 } \\right ) , i = 1 , 2 , n \\to \\infty , \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} \\mathrm { V o l } ( E _ i , \\omega ( t ) ) = E _ i \\cdot [ \\omega ( t ) ] \\to 0 . \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} \\alpha ( y ) = k ^ { \\sum _ { i = 1 } ^ \\ell c _ i \\langle \\alpha _ i , \\alpha \\rangle } \\end{align*}"}
-{"id": "5843.png", "formula": "\\begin{align*} 2 5 \\sum _ { k = 1 } ^ n { ( - 1 ) ^ { k - 1 } F _ { m k } { } ^ 4 } & = \\frac { { ( - 1 ) ^ { n - 1 } L _ { 4 m n + 2 m } + L _ { 2 m } } } { { L _ { 2 m } } } \\\\ & + \\frac { { 4 \\left \\{ { ( - 1 ) ^ { n ( m - 1 ) } L _ { 2 m n + m } - L _ m } \\right \\} } } { { L _ m } } \\\\ & + 3 \\left \\{ { ( - 1 ) ^ { n - 1 } + 1 } \\right \\} \\ , , \\end{align*}"}
-{"id": "5079.png", "formula": "\\begin{align*} s ( 2 n ) \\ , = \\ , s ( n ) s ( 2 n + 1 ) \\ , = \\ , s ( n ) + s ( n + 1 ) \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} \\psi _ i ^ { ( 1 ) } & = ( b _ i c _ { \\alpha } ^ { - 1 } ) ( \\alpha \\otimes \\alpha + e _ i \\otimes \\alpha ^ 2 ) , \\\\ \\psi _ i ^ { ( 2 ) } & = ( b _ i c _ { \\alpha } ^ { - 1 } ) ^ 2 ( \\alpha \\otimes \\alpha ^ 2 + e _ i \\otimes \\alpha ^ 3 ) , \\\\ \\psi _ i ^ { ( 3 ) } & = ( b _ i c _ { \\alpha } ^ { - 1 } ) ^ 3 ( \\alpha \\otimes \\alpha ^ 3 ) . \\end{align*}"}
-{"id": "7585.png", "formula": "\\begin{align*} a _ t = a - ( \\frac 2 k - \\alpha ) t . \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} \\mathsf { c h } _ a \\mathsf { c h } _ b ( \\mathsf { H } ) = \\mathsf { c h } _ a ( \\mathsf { p } ) \\mathsf { c h } _ b ( \\mathsf { H } ) + \\mathsf { c h } _ a ( \\mathsf { L } ) \\mathsf { c h } _ b ( \\mathsf { L } ) + \\mathsf { c h } _ a ( \\mathsf { H } ) \\mathsf { c h } _ b ( \\mathsf { p } ) \\end{align*}"}
-{"id": "10050.png", "formula": "\\begin{align*} g ( \\mathrm { T } ^ { 0 } ( X , Y ) , Z ) - g ( \\mathrm { T } ^ { 0 } ( Z , Y ) , X ) + g ( \\mathrm { T } ^ { 0 } ( J _ { \\varphi } X , Y ) , J _ { \\varphi } Z ) - g ( \\mathrm { T } ^ { 0 } ( J _ { \\varphi } Z , Y ) , J _ { \\varphi } X ) = \\frac { 1 } { 2 } g ( N _ { J _ { \\varphi } } ( X , Z ) , Y ) , \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} & \\| U ( t ) \\| _ { r , \\mathbb R ^ 3 } = o ( t ^ { - 1 / 2 + 3 / 2 r } ) , \\quad \\forall r \\in ( 3 , \\infty ] , \\\\ & \\| U ( t ) \\| _ { 3 , \\infty , \\mathbb R ^ 3 } = o ( 1 ) , \\end{align*}"}
-{"id": "2408.png", "formula": "\\begin{align*} \\frac d { d t } U ( t ) = C U ( t ) ^ { \\frac { p + 1 } 2 } , \\end{align*}"}
-{"id": "6244.png", "formula": "\\begin{align*} \\left | \\frac { 2 ^ k \\| \\lambda ^ { ( k - 1 ) } \\| + 2 ^ { k + 1 } q } { \\lambda [ 0 , T _ { k + 1 } ] + 2 ^ { k + 1 } ( q + 1 ) } - 1 \\right | & = \\frac { 2 ^ { k + 1 } } { \\lambda [ 0 , T _ { k + 1 } ] + 2 ^ { k + 1 } ( q + 1 ) } \\\\ & < \\frac { 2 ^ { k + 1 } } { 2 ^ { \\frac { k ( k + 1 ) } { 2 } } + 2 ^ { k + 1 } ( q + 1 ) } . \\end{align*}"}
-{"id": "4271.png", "formula": "\\begin{align*} \\begin{cases} f ( X ) = 3 X ^ 2 + 4 A ; \\\\ \\phi ( X ) = X ^ 4 - 2 A X ^ 2 - 8 B X + A ^ 2 ; \\\\ g ( X ) = 3 X ^ 3 - 5 A X - 2 7 B ; \\\\ \\psi ( X ) = X ^ 3 + A X + B ; \\\\ \\Delta ^ \\prime = 4 A ^ 3 + 2 7 B ^ 2 . \\end{cases} \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\infty } ( 1 - q ^ { i } ) ( 1 + z q ^ { i - 1 } ) ( 1 + z ^ { - 1 } q ^ { i } ) = \\sum _ { m \\in \\mathbb { Z } } z ^ m q ^ { \\frac { m ( m - 1 ) } { 2 } } . \\end{align*}"}
-{"id": "1956.png", "formula": "\\begin{align*} \\widetilde F = \\{ j \\in F | j \\mbox { i s m a t c h e d i n } X \\} . \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{align*} \\beta ( t ) = \\frac { \\varepsilon ^ 2 ( b \\bar { v } - c \\bar { w } ) ^ 2 } { ( - \\bar { v } ^ 3 + ( a + 1 ) \\bar { v } ^ 2 - a \\bar { v } - \\bar { w } ) ^ 2 + \\varepsilon ^ 2 ( b \\bar { v } - c \\bar { w } ) ^ 2 } . \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} H _ i ( ( S ^ d ) ^ p , L ; \\mathbb { F } _ p ) = \\begin{cases} M , & i = d , \\\\ N ^ { \\oplus \\frac { 1 } { p } { p \\choose i / d } } , & i = 2 d , \\ldots , ( p - 1 ) d , \\\\ \\mathbb { Z } / p , & i = p d , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{align*} A ( r ) = r \\mathbb \\sigma _ 0 \\end{align*}"}
-{"id": "9383.png", "formula": "\\begin{align*} \\langle \\phi , \\varphi \\rangle : = \\int _ { \\Omega } \\langle \\phi ( t ) , \\varphi ( t ) \\rangle \\ , \\mu ( t ) \\ , . \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} \\operatorname * { R e } \\sum _ { i = 1 } ^ d z _ i ^ 2 & = \\operatorname * { R e } \\sum _ { i = 1 } ^ d ( x _ i + \\zeta _ i ) ^ 2 \\ge \\sum _ { i = 1 } ^ d \\left ( x _ i ^ 2 - 2 | x _ i | | \\zeta _ i | - | \\zeta _ i | ^ 2 \\right ) \\ge \\| x \\| ^ 2 - \\delta \\| x \\| ^ 2 - \\delta ^ { - 1 } \\| \\zeta \\| ^ 2 - \\| \\zeta \\| ^ 2 \\\\ & > ( 1 - \\delta - D ^ 2 / \\delta - D ^ 2 ) | x | ^ 2 = ( 1 - 2 D - D ^ 2 ) \\| x \\| ^ 2 \\stackrel { D = \\sqrt { 2 } - 1 } { = } 0 . \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{gather*} \\theta ^ \\infty _ 1 = - \\varepsilon ^ { - 1 } , \\theta ^ \\infty _ 2 = \\tilde { \\theta } ^ \\infty _ 1 + \\varepsilon , t = \\varepsilon \\tilde { t } , H = \\varepsilon ^ { - 1 } \\tilde { H } , \\\\ q _ 1 = \\tilde { q } _ 1 , p _ 1 = \\tilde { p } _ 1 + \\frac { \\tilde { \\theta } ^ \\infty _ 1 + \\varepsilon ^ { - 1 } } { \\tilde { q } _ 1 } , Y = t ^ { - \\varepsilon ^ { - 1 } } \\begin{pmatrix} t & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} \\tilde { Y } . \\end{gather*}"}
-{"id": "431.png", "formula": "\\begin{align*} \\mathbf { T } = \\{ t = ( x , \\xi ) \\ | \\ x = ( S , C ) \\in \\mathrm { R } , \\ \\xi \\in P ( \\mathrm { E x t } ^ 1 ( x ) ) \\} \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} L = L _ { 1 } ^ { \\alpha } L _ { 2 } ^ { 1 - \\alpha } , \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} \\chi ( \\mathcal O _ X ) = - \\frac { ( g ( C ) - 1 ) ^ 3 } { 2 1 6 } = - 1 . \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{align*} \\tau ( x , y ) = \\left \\vert x - y \\right \\vert . \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} \\left \\Vert e ^ { \\gamma \\left \\vert x \\right \\vert ^ { 2 } } \\tilde { u } \\left ( . , t \\right ) \\right \\Vert _ { X } = \\left \\Vert e ^ { \\mu ^ { 2 } \\left ( t \\right ) \\left \\vert x \\right \\vert ^ { 2 } } u \\left ( . , s \\right ) \\right \\Vert _ { X } , \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} \\frac { d \\theta _ i } { d t } = \\omega _ i - \\sum _ { j = 0 } ^ { n } a _ { i , j } \\sin ( \\theta _ { i } - \\theta _ { j } ) i = 0 , \\dots , n \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} f ( 1 ) = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { 4 ^ n } = \\frac 4 3 . \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{align*} x _ t ' = \\tilde { x } _ t + t ^ { 3 / 2 } P _ t \\end{align*}"}
-{"id": "6836.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty ( y ^ 2 - 1 ) \\varphi ( y ) d y = 0 \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{align*} B _ { n } ^ { \\left ( q \\right ) } & = \\binom { n + q } { q - 1 } \\sum _ { p = 1 } ^ { n } \\left ( - 1 \\right ) ^ { p } \\frac { \\left \\{ \\begin{array} { c } n + p \\\\ p \\end{array} \\right \\} } { \\binom { n + p } { p } } \\left ( \\frac { n + 1 } { p + q } \\right ) \\binom { n } { p } \\\\ & = \\sum _ { p = 1 } ^ { n } \\left ( - 1 \\right ) ^ { p } \\frac { \\left \\{ \\begin{array} { c } n + p \\\\ p \\end{array} \\right \\} } { \\binom { n + p } { p } } \\binom { n + q } { n - k } \\binom { q + k - 1 } { k } . \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{gather*} A _ 0 ^ { ( 1 ) } = \\begin{pmatrix} 0 & 0 \\\\ q _ 2 & - t _ 2 \\end{pmatrix} , A _ 0 ^ { ( 0 ) } = \\begin{pmatrix} - p _ 2 q _ 2 & t _ 2 p _ 2 \\\\ 1 - p _ 1 & p _ 2 q _ 2 + \\theta ^ 0 \\end{pmatrix} , N = \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} , \\\\ A _ { t _ 1 } = \\begin{pmatrix} q _ 1 \\\\ 1 \\end{pmatrix} \\begin{pmatrix} p _ 1 & \\theta ^ { t _ 1 } - p _ 1 q _ 1 \\end{pmatrix} , N _ 1 = \\frac { q _ 1 ( p _ 1 q _ 1 - \\theta ^ { t _ 1 } ) } { t _ 1 } N , N _ 2 = - p _ 2 N . \\end{gather*}"}
-{"id": "7958.png", "formula": "\\begin{align*} \\| \\mathbb P _ { \\mathbb R ^ 3 } G ( t ) \\| _ { q , \\mathbb R ^ 3 } & \\leq \\| G ( t ) \\| _ { q , \\mathbb R ^ 3 } + \\| \\nabla Q ( t ) \\| _ { q , \\mathbb R ^ 3 } \\\\ & \\leq C \\| G ( t ) \\| _ { q , \\mathbb R ^ 3 } = C \\| \\bar g ( t ) \\| _ { q , \\mathbb R ^ 3 } \\leq C \\| g ( t ) \\| _ q \\leq C M _ q \\end{align*}"}
-{"id": "1411.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t v + v \\cdot \\nabla v + \\nabla p = \\mu \\Delta v + \\nabla \\cdot ( \\frac { \\partial W ( F ) } { \\partial F } F ^ \\top ) , \\\\ [ - 4 m m ] \\\\ \\partial _ t F + v \\cdot \\nabla F = \\nabla v F , \\\\ [ - 4 m m ] \\\\ \\nabla \\cdot v = 0 , \\nabla \\cdot F ^ \\top = 0 . \\end{cases} \\end{align*}"}
-{"id": "9974.png", "formula": "\\begin{align*} x ( t _ * ^ + ) ^ T P _ 2 x ( t _ * ^ + ) - x ( t _ * ^ - ) ^ T P _ 1 x ( t _ * ^ - ) = - \\frac { m _ 1 m _ 2 } { m } [ x _ 2 ( t _ * ^ - ) - x _ 4 ( t _ * ^ - ) ] ^ 2 \\le 0 \\end{align*}"}
-{"id": "8770.png", "formula": "\\begin{align*} \\frac { | H ^ 0 ( G _ { \\Sigma } , M ) | \\cdot | H ^ 2 ( G _ { \\Sigma } , M ) | } { | H ^ 1 ( G _ { \\Sigma } , M ) | } = \\prod _ { v : \\ \\mathrm { i n f i n i t e } } | H ^ 0 ( G _ v , M ) | \\cdot p ^ { - ( r _ 1 + r _ 2 ) \\cdot v _ p ( | M | ) } . \\end{align*}"}
-{"id": "8654.png", "formula": "\\begin{align*} \\phi _ 1 ( s ) = s ^ { - \\lambda _ 1 } \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{align*} \\sum _ { m = p } ^ { n } \\binom { m } { p } \\binom { m + q - 1 } { q - 1 } = \\frac { n - p + 1 } { p + q } \\binom { n + 1 } { p } \\binom { n + q } { q - 1 } \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{align*} \\gamma _ L ^ { ( i , j ) } ( v _ m \\otimes f ^ n ) : = ( \\mathsf { N } ^ n _ m ) ^ i _ j , \\gamma _ R ^ { ( i , j ) } ( v _ m \\otimes f ^ n ) : = ( \\mathsf { M } ^ n _ m ) ^ i _ j . \\end{align*}"}
-{"id": "9842.png", "formula": "\\begin{align*} [ x , y ] = \\sigma ( t , u ) \\sigma ( u , t ) ^ { - 1 } . \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } f _ { n } = h \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} D ( ( g _ { 1 ^ { z ( \\varphi ^ { - 1 } ) } } ) ^ \\varphi , \\dots , ( g _ { k ^ { z ( \\varphi ^ { - 1 } ) } } ) ^ \\varphi ) = D ( g _ 1 , \\dots , g _ k ) ^ { \\delta ( \\varphi ) } = D ( g _ 1 , \\dots , g _ k ) \\end{align*}"}
-{"id": "9392.png", "formula": "\\begin{align*} \\bigl \\langle T ( x \\otimes w _ 1 ) , y \\otimes w _ 2 \\bigr \\rangle = \\int _ \\Omega \\Bigl ( \\bigl [ \\widehat { T } ( w _ 1 , w _ 2 ) \\bigr ] ( x ) \\Bigr ) ( t ) \\ , y ( t ) \\ , \\mu ( t ) \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} \\frac { m _ i v _ i } { t } = \\sum _ { i \\xrightarrow [ \\alpha ] { } j } c _ \\alpha t ^ { v _ j - v _ i } e ^ { b _ j - b _ i } - \\sum _ { k \\xrightarrow [ \\alpha ] { } i } c _ \\alpha t ^ { v _ i - v _ k } e ^ { b _ i - b _ k } . \\end{align*}"}
-{"id": "8207.png", "formula": "\\begin{align*} \\lambda _ { 1 } ( p , \\Omega ) = \\lambda _ { 1 } ( p , \\mathcal W _ { r _ { 2 } } ) . \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{align*} \\frac { \\textrm { V a r } _ { B } [ Y ] } { \\textrm { V a r } _ { \\hat B } [ Y ] } = \\frac { k ( B ) } { k ( B ' ) } = \\frac { K ( \\psi , F _ Y ( B _ 1 ) ) } { K ( \\psi , F _ Y ( \\hat B _ 1 ) ) } . \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} q ( x e _ 1 + y e _ 2 + z e _ 3 ) = a x ^ 2 + b y ^ 2 + c z ^ 2 \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{align*} \\int _ \\mathbb { R } | y - x | ^ { \\gamma - 1 } | w - y | ^ { \\alpha - 1 } d y = C ' ( \\alpha , \\beta ) | w - x | ^ { \\frac { \\alpha + \\beta } { 2 } - 1 } , \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} L = L _ { - 4 } \\oplus L _ { - 3 } \\oplus \\cdots \\oplus L _ { r } \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{gather*} y _ 1 = - \\frac { 1 } { 2 } \\log \\frac { s } { 2 } + \\frac { 1 } { s } , y _ 2 = - \\frac { 1 } { 2 } \\log ( 2 s ) , \\\\ y _ 3 = \\frac { 1 } { 2 } \\log ( 2 s ) , y _ 4 = \\frac { 1 } { 2 } \\log \\frac { s } { 2 } - \\frac { 1 } { s } \\end{gather*}"}
-{"id": "2025.png", "formula": "\\begin{gather*} A _ 0 = J _ 1 ( \\lambda _ 1 ) \\oplus \\cdots \\oplus J _ n ( \\lambda _ n ) \\end{gather*}"}
-{"id": "7201.png", "formula": "\\begin{align*} \\rho ( a ) & = \\begin{pmatrix} x & 1 \\\\ 0 & 1 / x \\end{pmatrix} \\\\ \\rho ( b ) & = \\begin{pmatrix} y & 0 \\\\ r & 1 / y \\end{pmatrix} \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} { } _ 3 F _ 2 \\bigg [ \\begin{matrix} \\alpha & \\beta & \\gamma \\\\ & 2 \\beta & \\frac 1 2 ( \\alpha + \\gamma + 1 ) \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { \\Gamma ( \\frac 1 2 ) \\Gamma ( \\frac 1 2 + \\beta ) \\Gamma ( \\frac 1 2 + \\frac 1 2 ( \\alpha + \\gamma ) ) \\Gamma ( \\frac 1 2 + \\beta - \\frac 1 2 ( \\alpha + \\gamma ) ) } { \\Gamma ( \\frac 1 2 + \\frac 1 2 \\alpha ) \\Gamma ( \\frac 1 2 + \\frac 1 2 \\gamma ) \\Gamma ( \\frac 1 2 + \\beta - \\frac 1 2 \\alpha ) \\Gamma ( \\frac 1 2 + \\beta - \\frac 1 2 \\gamma ) } . \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 2 } { x ^ 2 } + \\frac { A _ 1 } { x } + A _ 0 \\right ) Y , \\frac { \\partial Y } { \\partial t } = \\left ( \\frac { 1 } { t } A _ 0 x + B _ 0 \\right ) Y , \\end{gather*}"}
-{"id": "2638.png", "formula": "\\begin{align*} \\begin{cases} \\Omega : = ( 0 , \\pi ) \\times ( 0 , \\pi ) , \\\\ \\Omega _ { 1 } : = ( \\ell _ { 1 } , \\ell _ { 1 } + \\delta _ { 1 } ) \\times ( 0 , \\pi ) , \\\\ \\Omega _ { 2 } : = ( \\ell _ { 2 } , \\ell _ { 2 } + \\delta _ { 2 } ) \\times ( 0 , \\pi ) , \\end{cases} \\end{align*}"}
-{"id": "7872.png", "formula": "\\begin{align*} \\mathbb { E } Z ^ 2 _ n ( \\epsilon ) = \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ { n } \\mathbb { E } X _ i X _ j = \\sum _ { i = 1 } ^ { n } \\mathbb { E } X ^ 2 _ i + \\sum _ { i \\neq j } \\mathbb { E } ( X _ i X _ j ) . \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{align*} s _ i & = \\sum _ { j = 0 } ^ { n - 1 } y _ j \\norm { j } { \\sigma ^ i ( \\beta ) } = \\sum _ { j = 1 } ^ \\nu e _ j \\norm { k _ j } { \\sigma ^ i ( \\beta ) } \\\\ & = \\sum _ { j = 1 } ^ \\nu e _ j \\sigma ^ i ( \\alpha ^ { - 1 } ) \\sigma ^ { i + k _ j } ( \\alpha ) = \\sigma ^ i ( \\alpha ^ { - 1 } ) \\sum _ { j = 1 } ^ \\nu e _ j \\sigma ^ { i + k _ j } ( \\alpha ) . \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{align*} f ( k , x ) : = e ^ { i k x } + \\int _ x ^ \\infty K ( x , t ) e ^ { i k t } d t , \\ ; \\ ; x \\ge 0 \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} \\varphi ( t ) = \\frac { 1 } { y ( t ) } \\left ( e ^ { 1 2 7 } + e ^ { 3 4 7 } + e ^ { 5 6 7 } \\right ) + y ( t ) ^ 3 \\left ( e ^ { 1 3 5 } - e ^ { 1 4 6 } - e ^ { 2 3 6 } - e ^ { 2 4 5 } \\right ) , \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} \\partial _ { t } u = \\left ( a + i b \\right ) \\left [ \\Delta u + A u + V _ { 1 } \\left ( x \\right ) u \\right ] , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} F _ * ^ e ( \\lambda x _ 1 ^ { k d _ 1 + \\alpha _ 1 } \\dots x _ n ^ { k d _ n + \\alpha _ n } ) = x _ 1 ^ { d _ 1 } \\dots x _ n ^ { d _ n } F _ * ^ e ( \\lambda x _ 1 ^ { s _ 1 } \\dots x _ n ^ { s _ n } ) \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} [ h _ \\chi ( z ) , H ^ { \\gamma } ( w ^ 2 ) ] & = [ h _ \\chi ( z ) , V ^ + ( w ) \\gamma ( w ^ 2 ) w ^ { 2 h _ 0 } V ^ - ( w ) ] \\\\ & = \\Big ( - \\sum _ { n > 0 } \\frac { z ^ { 2 n - 2 } } { w ^ { 2 n } } + i _ { z , w } \\frac { 1 } { z ^ 2 - w ^ 2 } + i _ { w , z } \\frac { 1 } { w ^ 2 - z ^ 2 } - \\sum _ { n > 0 } \\frac { w ^ { 2 n } } { z ^ { 2 n + 2 } } \\Big ) H ^ { \\gamma } ( w ^ 2 ) = \\frac { 1 } { z ^ 2 } H ^ { \\gamma } ( w ^ 2 ) . \\end{align*}"}
-{"id": "6560.png", "formula": "\\begin{align*} \\frac { 1 } { 1 6 } ( f ^ 1 _ { u u u } + f ^ 1 _ { u v v } + f ^ 2 _ { u u v } + f ^ 2 _ { v v v } ) + \\frac { 1 } { 1 6 \\omega } ( f ^ 1 _ { u v } ( f ^ 1 _ { u u } + f ^ 1 _ { v v } ) - f ^ 2 _ { u v } ( f ^ 2 _ { u u } + f ^ 2 _ { v v } ) - f ^ 1 _ { u u } f ^ 2 _ { u u } + f ^ 1 _ { v v } f ^ 2 _ { v v } ) = - \\frac { 1 } { 8 } < 0 . \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\dim F } b _ { i - k } ( \\Omega M ) \\leq B i ^ m \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} \\sigma ( S ) \\cup \\left ( - \\gamma \\sigma ( C ) \\right ) : = \\left ( \\lambda _ { 1 } , \\lambda _ { 2 } , \\ldots , \\lambda _ { n } , - \\gamma \\mu _ { 1 } , - \\gamma \\mu _ { 2 } , \\ldots , - \\gamma \\mu _ { n } \\right ) . \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} \\Lambda _ { 1 } ( \\infty , \\Omega ) = \\Lambda _ { 1 } ( \\infty , \\Omega _ { 1 } ) \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{align*} d \\big ( e _ { s ( \\alpha ) } \\otimes e _ { t ( \\alpha ) } \\big ) = \\alpha \\otimes e _ { t ( \\alpha ) } - e _ { s ( \\alpha ) } \\otimes \\alpha \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} a _ \\ell \\ , \\big ( \\Im \\zeta ^ \\ell - \\Im \\zeta ^ { k \\pi / \\omega } \\big ) = ( g ^ \\omega _ \\ell - g ^ 0 _ \\ell \\ , \\cos \\ell \\omega ) \\Im \\ , \\frac { \\zeta ^ \\ell - \\zeta ^ { k \\pi / \\omega } } { \\sin \\ell \\omega } \\ ; . \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} c ( \\log ( x ) ) = c \\left ( \\gamma + \\log ( r ) + \\sum _ { i = 1 } ^ { \\infty } ( - 1 ) ^ { n } \\frac { \\varepsilon ^ { n } } { n } \\right ) = c ( \\gamma ) + \\log ( r ) + \\sum _ { i = 1 } ^ { \\infty } ( - 1 ) ^ { n } \\frac { c ( \\varepsilon ) ^ { n } } { n } . \\end{align*}"}
-{"id": "9472.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & \\alpha \\\\ . & - \\alpha \\end{bmatrix} s = \\begin{bmatrix} \\alpha \\lambda \\\\ 0 \\end{bmatrix} \\end{align*}"}
-{"id": "7314.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\gamma _ 1 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 1 \\alpha _ 6 } { \\gamma _ 1 \\gamma _ 6 } y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\beta _ 4 } { \\gamma _ 6 } y _ 5 , [ y _ 3 , y _ 1 ] = \\frac { \\beta _ 6 } { \\gamma _ 6 } y _ 5 , \\\\ [ y _ 2 , y _ 3 ] = y _ 4 + \\theta _ 3 y _ 5 , [ y _ 3 , y _ 2 ] = - y _ 4 , [ y _ 3 , y _ 3 ] = y _ 5 . \\end{align*}"}
-{"id": "1153.png", "formula": "\\begin{align*} A + u ( t ) B = ( A + u _ 0 B ) + u _ 1 ( t ) B , \\ \\ \\ \\ \\ \\ \\ u _ 0 \\in \\R , \\ u _ 1 \\in L ^ 2 ( ( 0 , T ) , \\R ) . \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} \\Phi ( x _ 1 , x _ 2 ; z _ 1 , z _ 2 ) : = \\frac { 1 } { 2 ^ \\mu \\cdot \\mu ! } \\Bigl ( H _ { ( x _ 1 , x _ 2 ) } - z _ 1 ^ 2 - z _ 2 ^ 2 \\Bigr ) ^ \\mu \\diamond \\bigl ( \\delta ( x _ 1 , x _ 2 ) \\cdot \\exp ( x _ 1 z _ 1 + x _ 2 z _ 2 ) \\bigr ) . \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} p _ n = n + O ( | n | ^ { 1 / 2 } \\log ^ 2 | n | ) n \\to \\pm \\infty \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} x _ { n } = \\frac { H _ { n } \\left ( x \\right ) } { n ! } . \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} | v _ { n } ( t ) | ^ { 2 } & = \\exp \\ ! \\left \\{ - f ( t ) n ^ { 2 } - 2 \\mu n w _ { t } \\right \\} \\\\ & = \\exp \\biggl \\{ - f ( t ) \\left ( n + \\frac { \\mu w _ { t } } { f ( t ) } \\right ) ^ { \\ ! 2 } + \\frac { \\mu ^ { 2 } | w _ { t } | ^ { 2 } } { f ( t ) } \\biggr \\} , \\end{align*}"}
-{"id": "5218.png", "formula": "\\begin{align*} c = \\left ( u - \\frac { 1 } { 2 \\pi } \\varpi ^ i \\nabla ^ \\perp \\log { | \\cdot - \\xi ^ i | } \\right ) \\Big | _ { \\xi ^ i } , \\textrm { f o r } i = 1 , \\ldots , M . \\end{align*}"}
-{"id": "8877.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 ( Q _ k ( x ) ) ^ 2 d x = \\dfrac { \\pi ^ 2 - 2 ( 1 + \\cos ^ 2 ( k \\pi ) ) \\psi ' ( k + 1 ) } { 2 ( 2 k + 1 ) } , \\end{align*}"}
-{"id": "1608.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\big \\Vert u _ \\varepsilon - u _ \\varepsilon ^ { [ N ] } \\big \\Vert = 0 \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} h _ 2 ^ 2 = \\begin{pmatrix} \\alpha ^ 2 + \\beta \\gamma & ( \\alpha + \\delta ) \\beta \\\\ ( \\alpha + \\delta ) \\gamma & \\beta \\gamma + \\delta ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "4699.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\widehat { K } _ N \\left ( \\frac { x } { N } , \\frac { y } { N } ; w _ { R , \\alpha } ^ { \\pm } \\right ) = \\frac { \\sin \\pi c _ { \\alpha } ^ { \\pm } ( x - y ) } { \\pi ( x - y ) } \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{align*} S ( k ) = { U _ 0 } + O \\left ( \\frac { 1 } { k } \\right ) , | k | \\to \\infty , \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} W = \\begin{bmatrix} I _ { k - 1 } & \\xi & \\mathbf { 0 } \\\\ \\mathbf { 0 } & - 1 & \\mathbf { 0 } \\\\ \\mathbf { 0 } & \\eta & I _ { n - k } \\end{bmatrix} \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} \\Tilde { \\xi } _ t ^ { f } : = \\xi _ t - E [ \\xi _ T + \\int _ t ^ T f _ s d s | \\cal { F } _ t ] , \\ ; \\Tilde { \\zeta } _ t ^ { f } : = \\zeta _ t - E [ \\zeta _ T + \\int _ t ^ T f _ s d s | \\cal { F } _ t ] , 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} \\lambda _ { 1 } ( p _ { j } , \\tilde \\Omega ) = \\lambda _ { 1 } ( p _ { j } , \\Omega ) , \\quad \\quad \\frac { 1 } { \\rho _ { F } ( \\tilde \\Omega ) } = \\tilde \\Lambda . \\end{align*}"}
-{"id": "6991.png", "formula": "\\begin{align*} \\frac { n } { 2 } \\gamma _ { 1 } + \\frac { n + 2 } { 2 } \\gamma _ { 2 } & = n + 1 \\\\ \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} [ A , S ] = \\{ U \\le S : r _ m ( U ) = A \\} , \\end{align*}"}
-{"id": "7344.png", "formula": "\\begin{align*} \\begin{aligned} f ^ i & = f ^ i ( t ) = y ( t ) \\ , e ^ i , 1 \\leq i \\leq 6 , \\\\ f ^ 7 & = f ^ { 7 } ( t ) = y ( t ) ^ { - 3 } \\ , e ^ 7 , \\end{aligned} \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} \\int _ { \\R ^ { n } } | e ^ { t a ( \\xi ) } \\hat { u } ( \\xi ) | ^ { 2 } \\ , d \\xi = \\int _ { \\R ^ { n } } e ^ { 2 t \\ , \\Re \\ , a ( \\xi ) } | \\hat { u } ( \\xi ) | ^ { 2 } \\ , d \\xi , \\textrm { f o r } u \\in L ^ { 2 } ( \\R ^ { n } ) , \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} ( \\mu _ - \\otimes \\mu _ + ) ( \\Omega ) = 1 . \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} \\| \\nabla ( \\widetilde \\eta - \\widetilde H ) \\| _ { L ^ p } & \\leq A _ p \\left ( \\| ( a - 1 ) \\nabla ( \\widetilde \\eta - \\widetilde H ) \\| _ { L ^ p } + \\| G _ 1 \\| _ { L ^ p } + \\| G _ 2 \\| _ { L ^ p } + \\| a \\nabla \\widetilde H \\| _ { L ^ p } \\right ) \\\\ & \\phantom { = } + B _ p \\left ( \\| F _ 1 \\| _ { L ^ 1 } + \\| F _ 2 \\| _ { L ^ 1 } \\right ) \\leq A _ p c _ 0 \\delta _ 2 ( p ) \\| \\nabla ( \\widetilde \\eta - \\widetilde H ) \\| _ { L ^ p } + C . \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{align*} L ^ { ( n ) } _ s = ( x ^ 2 + 1 ) \\frac { d ^ 2 } { d x ^ 2 } + \\left [ \\left ( 2 - 2 n - 2 \\Re ( s ) \\right ) x + 2 \\Im ( s ) \\right ] \\frac { d } { d x } , \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} Q _ E ( z ) = \\prod _ { j = 0 } ^ n ( z - E _ j ^ + ) ( z - E _ j ^ - ) \\end{align*}"}
-{"id": "3772.png", "formula": "\\begin{align*} c = ( 0 , \\cdots , 0 , a _ { j + 1 } - ( k j + 1 ) , a _ { j + 2 } - ( k ( j + 1 ) + 1 ) , \\cdots , a _ { n - 1 } - ( k ( n - 2 ) + 1 ) , a _ n - ( k ( n - 1 ) + 1 ) ) . \\end{align*}"}
-{"id": "6586.png", "formula": "\\begin{align*} \\mathcal { K } _ { m _ j , \\kappa _ j } : = \\ ; \\mathrm { s p a n } \\{ \\Psi ^ + _ { m _ j , \\kappa _ j } , \\Psi ^ - _ { m _ j , \\kappa _ j } \\} \\ ; \\cong \\ ; \\mathbb { C } ^ 2 \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} { \\mathbf K } _ { i , j } = \\int _ { x \\in \\Gamma } \\int _ { y \\in \\Gamma } k _ { H e l m } ( x , y ) \\varphi _ j ( y ) \\varphi _ i ( x ) \\ , d y \\ , d x . \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} \\dfrac { 1 + \\psi ( 1 - \\lambda ) - \\psi ( \\lambda ) } { 2 } f ( t ) & = \\dfrac { 1 } { \\int _ a ^ b \\nu ( t ) \\Delta t } \\int _ a ^ b K ( s , t ) f ^ { \\Delta } ( s ) \\Delta s + \\dfrac { 1 } { \\int _ a ^ b \\nu ( t ) \\Delta t } \\int _ a ^ b \\nu ( s ) f ( \\sigma ( s ) ) \\Delta s \\\\ & - \\dfrac { \\psi ( \\lambda ) f ( a ) + \\left ( 1 - \\psi ( 1 - \\lambda ) \\right ) f ( b ) } { 2 } , \\end{align*}"}
-{"id": "7689.png", "formula": "\\begin{align*} & P _ 2 ( \\mathbf W _ i \\in [ h _ i 2 ^ { - \\ell } , ( h _ i + 1 ) 2 ^ { - \\ell } ) , ~ 1 \\leq i \\leq m | \\mathbf Y = \\mathbf y ) \\\\ & = P _ 2 ( \\mathbf W _ { \\ell , i } = h _ i 2 ^ { - \\ell } , ~ 1 \\leq i \\leq m | \\mathbf Y = \\mathbf y ) \\\\ & = \\psi _ { \\ell , h } ( \\mathbf y ) = P '' ( \\mathbf X '' _ i \\in [ h _ i 2 ^ { - \\ell } , ( h _ i + 1 ) 2 ^ { - \\ell } ) , ~ 1 \\leq i \\leq m | \\mathbf Y '' = \\mathbf y ) . \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} \\sum _ { i \\in I } \\sum _ { j \\in J } a _ { i , j } = \\sum _ { j \\in J } \\sum _ { i \\in I } a _ { i , j } = \\sum _ { ( i , j ) \\in I \\times J } a _ { i , j } . \\end{align*}"}
-{"id": "6052.png", "formula": "\\begin{align*} L _ i = L _ 1 ^ i = S _ 1 ^ i \\subseteq S _ i \\subseteq L _ i , \\end{align*}"}
-{"id": "2357.png", "formula": "\\begin{align*} e ^ { i \\frac { t } { 2 \\pi R } \\Delta } f ( x ) = e ^ { i \\frac { t } { 2 \\pi R } \\Delta } f _ { d k } ( x _ 1 ) e ^ { i \\frac { t } { 2 \\pi R } \\Delta } f _ { \\theta } ( \\bar { x } ) \\ , , \\end{align*}"}
-{"id": "9754.png", "formula": "\\begin{align*} \\Delta u _ { \\epsilon } = - \\epsilon ^ { - 2 } ( 1 - | u _ { \\epsilon } | ^ 2 ) u _ { \\epsilon } . \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{align*} n ( z ) = \\mathrm { N o r m } _ { F ( z ) / F } ( z ) = 1 \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } w _ { R , \\alpha } ^ - \\left ( \\frac { x } { R } \\right ) = \\lim _ { R \\to \\infty } w _ { R , \\alpha } ^ + \\left ( \\frac { x } { R } \\right ) = 1 \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} \\sigma ( S ) \\cup \\left ( \\pm \\gamma \\sigma ( C ) \\right ) = \\left ( \\lambda _ { 1 } , \\lambda _ { 2 } , \\ldots , \\lambda _ { n } , \\pm \\gamma \\mu _ { 1 } , \\pm \\gamma \\mu _ { 2 } , \\ldots , \\pm \\gamma \\mu _ { n } \\right ) . \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} \\mu _ * ( M A ( \\Phi ) ) = M A _ U ( ( 1 - | x | ) u + | x | v ) N ! d x _ { | \\Sigma ^ { \\circ } _ N } . \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{align*} g _ j ^ * - \\mu _ { s } \\le \\mu _ j - \\mu _ { s } & \\le \\sqrt 2 ( 1 + \\delta _ { s } ) ( \\sqrt { \\log ( 9 e p / j ) } - \\sqrt { \\log ( 9 e p / ( j + s ) ) } ) , \\\\ & = \\sqrt 2 ( 1 + \\delta _ { s } ) \\frac { \\log ( 1 + s / j ) } { \\sqrt { \\log ( 9 e p / j ) } + \\sqrt { \\log ( 9 e p / ( j + s ) ) } } , \\\\ & \\le \\frac { \\sqrt 2 ( 1 + \\delta _ { s } ) } { \\sqrt { \\log ( 9 e p / s ) } } \\log ( 1 + s / j ) . \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} G _ k = \\begin{cases} M M , & \\mbox { w i t h p r o b a b i l i t y } p _ 1 = K ( \\frac { \\theta _ M } { 2 \\pi } ) ^ 2 \\\\ m M , & \\mbox { w i t h p r o b a b i l i t y } p _ 2 = 2 K ( \\frac { 2 \\pi - K \\theta _ M } { 2 \\pi } ) ( \\frac { \\theta _ M } { 2 \\pi } ) + ( K ^ 2 - K ) ( \\frac { \\theta _ M } { 2 \\pi } ) ^ 2 \\\\ m m , & \\mbox { w i t h p r o b a b i l i t y } p _ 3 = ( \\frac { 2 \\pi - K \\theta _ M } { 2 \\pi } ) ^ 2 \\end{cases} \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} \\mathbb { F } : = \\mathbb { F } _ p ( \\alpha _ 1 , \\dots , \\alpha _ n ) . \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} \\| \\Theta \\| _ 1 = & \\| \\Theta _ \\lambda + \\lambda \\bar { \\theta } \\| _ 1 \\leq \\| \\Theta _ \\lambda \\| _ 1 + \\lambda \\| \\bar { \\theta } \\| _ 1 \\\\ \\leq & \\sqrt { C ( b , B , \\theta _ L ) + C ( b , B , N _ 1 ) ( 1 + \\theta _ L ^ 2 ) } + 2 \\theta _ L = : M _ 2 . \\end{align*}"}
-{"id": "9571.png", "formula": "\\begin{align*} V ^ \\mathrm { t o t a l } ( N _ 1 , N _ 0 , n _ 1 , n _ 0 ) & = \\mbox { v a r } \\big ( \\widehat { \\theta } \\ , | \\ , N _ 1 , N _ 0 \\big ) = \\frac { S ^ 2 _ 1 } { N _ 1 } + \\frac { S ^ 2 _ 0 } { N _ 0 } - \\frac { S ^ 2 _ { \\theta } } { n _ 0 + n _ 1 } , \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} { \\bf E } S _ n = n ^ \\theta l ( \\theta , n ) + o ( \\sqrt { { \\bf E } S _ n } ) , \\ \\ \\frac { { \\bf V a r } S _ n } { { \\bf E } S _ n } \\to \\sigma ^ 2 , \\ \\ \\frac { S _ n } { { \\bf E } S _ n } \\stackrel { a . s . } { \\to } 1 , \\ \\ \\frac { S _ n - { \\bf E } S _ n } { \\sqrt { { \\bf V a r } S _ n } } \\Rightarrow { \\bf N } _ { 0 , 1 } , \\end{align*}"}
-{"id": "4916.png", "formula": "\\begin{align*} s _ { i j } = a _ { i j } + b _ { i j } , \\ 1 \\leq i , j \\leq n \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{align*} h _ k ( x ) : = \\left ( 1 - \\frac { Q '' ( x _ k ) } { Q ' ( x _ k ) } ( x - x _ k ) \\right ) Q _ k ( x ) ^ 2 , \\mathfrak { h } _ k ( x ) : = ( x - x _ k ) Q _ k ( x ) ^ 2 \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} z ( t ) = E _ { \\frac { 1 } { 2 } , 1 } ( B \\sqrt { t } ) z _ 0 + \\int _ 0 ^ t E _ { \\frac { 1 } { 2 } , \\frac { 1 } { 2 } } \\left ( B \\sqrt { t - \\tau } \\right ) C \\frac { f ( \\tau ) d \\tau } { \\sqrt { t - \\tau } } , \\end{align*}"}
-{"id": "6816.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta _ { x } u + \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert \\leq 2 m } a _ { \\alpha } \\left ( y \\right ) D _ { y } ^ { \\alpha } u \\left ( x , y , t \\right ) + F \\left ( u , \\bar { u } \\right ) u = 0 , \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} - y '' ( x ) + q ( x ) y ( x ) = \\lambda y ( x ) , \\ 0 < x < 1 , \\end{align*}"}
-{"id": "2683.png", "formula": "\\begin{align*} \\frac { k ^ 2 + u ^ 2 } { k } = k + \\frac { u ^ 2 } { k } , \\end{align*}"}
-{"id": "463.png", "formula": "\\begin{align*} \\vec { Y } _ { i , j } = \\sum _ { i ' = 1 } ^ { a } K _ { i , j ; i ' } \\vec { X } _ { i ' } . \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} ( \\gamma _ i ( 1 - x _ i ) ^ T - x _ i ^ T ) f _ p = 0 . \\end{align*}"}
-{"id": "8144.png", "formula": "\\begin{align*} f _ p ( 0 , 0 ) = q _ 1 q _ 2 ( 1 + \\theta _ { 1 2 } p _ 1 p _ 2 ) . \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} [ V _ 1 , \\ , V _ 1 ] = 0 . \\end{align*}"}
-{"id": "8531.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 1 } \\vec { \\gamma } _ 0 ( \\phi , p ) & = \\vec { \\gamma } _ 3 ( \\vec { \\Psi } , p ) \\\\ \\vec { \\gamma } _ 1 ( \\phi , p ) & = \\vec { \\gamma } _ 1 ( \\vec { \\Psi } , p ) \\\\ \\vec { \\gamma } _ 2 ( \\phi , p ) & = - \\vec { \\gamma } _ 2 ( \\vec { \\Psi } , p ) \\\\ \\vec { \\gamma } _ 3 ( \\phi , p ) & = \\vec { \\gamma } _ 0 ( \\vec { \\Psi } , p ) . \\end{alignedat} \\right . \\end{align*}"}
-{"id": "7308.png", "formula": "\\begin{align*} [ y _ 1 , y _ 2 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 5 } { \\alpha _ 3 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 2 ] = \\frac { \\beta _ 2 } { \\gamma _ 6 } y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\beta _ 4 \\gamma _ 1 } { \\alpha _ 3 \\gamma _ 6 } y _ 5 , [ y _ 3 , y _ 1 ] = \\frac { \\beta _ 6 \\gamma _ 1 } { \\alpha _ 3 \\gamma _ 6 } y _ 5 , \\\\ [ y _ 2 , y _ 3 ] = y _ 4 , [ y _ 3 , y _ 2 ] = \\frac { \\gamma _ 4 } { \\gamma _ 6 } y _ 5 , [ y _ 3 , y _ 3 ] = y _ 5 . \\end{align*}"}
-{"id": "8946.png", "formula": "\\begin{align*} \\beta _ { i _ n , i _ n + 1 } ( S / I _ \\Delta ) \\sim C n \\binom { n } { i _ n } \\sim C n \\frac { 2 ^ { n + 1 } } { \\sqrt { 2 \\pi n } } e ^ { - a ^ 2 / 2 } . \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} \\mathbf { e } ( x ) = \\mathbf { h } ( - x ) ^ { - 1 } , \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} { \\displaystyle - 2 \\mathrm { i } ( \\partial \\theta ^ { k } _ { \\psi } , \\partial { \\theta _ { \\psi } ^ { k } } ) + B \\left ( \\overline { \\mathbf { A } } ^ { k } _ { h } ; \\overline { \\theta } _ { \\psi } ^ { k } , \\partial { \\theta _ { \\psi } ^ { k } } \\right ) = \\sum _ { j = 1 } ^ { 5 } Q _ { j } ^ { k } ( \\partial { \\theta _ { \\psi } ^ { k } } ) . } \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 0 } { x } + \\frac { A _ 1 } { x - 1 } + A _ \\infty \\right ) Y , \\frac { \\partial Y } { \\partial t } = \\left ( \\frac { 1 } { t } A _ \\infty x + B _ 0 \\right ) Y . \\end{gather*}"}
-{"id": "383.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\theta _ { j + 1 , j } = { \\displaystyle \\frac { \\alpha _ j } { j + 1 } } \\\\ \\theta _ { i + 1 , j } = { \\displaystyle - \\frac { \\alpha _ i } { i + 1 } \\sum _ { k = i + 2 } ^ { j + 1 } \\eta _ { i k } \\theta _ { k j } } , i = j - 1 , \\ldots , 1 , 0 \\end{array} \\right . \\end{align*}"}
-{"id": "1658.png", "formula": "\\begin{align*} \\widetilde { a } & = \\sum _ { \\tau \\in K } \\sum _ { s \\in S ( \\tau ) } { f _ { s \\cdot \\widetilde { \\tau } } \\otimes s \\cdot \\widetilde { \\tau } } = \\sum _ { \\tau \\in K } \\sum _ { s \\in S } { f _ { s \\cdot \\widetilde { \\tau } } \\otimes s \\cdot \\widetilde { \\tau } } , \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} L _ j \\subseteq L _ { - 1 } ^ * \\otimes S _ { j - 1 } = X _ { 1 , \\ , j , \\ , 1 } \\oplus X _ { 1 , \\ , j , \\ , 3 } , \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} L u _ { 0 n } - { \\rm i } \\ , \\omega \\ , B u _ { 0 n } - \\omega ^ 2 u _ { 0 n } = g _ { n } . \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{align*} \\int \\log \\left ( \\prod _ { j = 1 } ^ k [ 1 - \\cos ( \\theta - \\theta _ j ) ] ^ { m _ j } \\right ) \\prod _ { j = 1 } ^ k [ 1 - \\cos ( \\theta - \\theta _ j ) ] ^ { m _ j } \\frac { d \\theta } { 2 \\pi } \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} \\sup \\limits _ { k , i } \\sum \\limits _ { j = 1 } ^ { N } \\left \\vert \\frac { \\lambda _ { k } } { D \\left ( \\lambda _ { k } \\right ) } A _ { j i } \\left ( \\lambda _ { k } \\right ) \\right \\vert ^ { q } \\leq C . \\end{align*}"}
-{"id": "1040.png", "formula": "\\begin{align*} E ( I u ( T ) ) & = \\lambda ^ k E ( I u _ \\lambda ( \\lambda ^ k T ) ) = \\lambda ^ k E ( I u _ \\lambda ( C _ 1 N ^ { \\gamma _ 0 ( k ) - } \\delta ) ) \\\\ & \\sim \\lambda ^ k \\leq N ^ { \\frac { k ( k / 2 - \\gamma ) } { \\gamma } } \\sim T ^ { \\frac { k ( k - 2 \\gamma ) } { 2 ( \\gamma _ 0 ( k ) + k ) \\gamma - k ^ 2 } + } . \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} L _ { - 1 } ^ * \\otimes S _ j \\subseteq L _ { - 1 } ^ * \\otimes L _ j \\subseteq L _ { - 1 } ^ * \\otimes ( X _ { 1 , \\ , j , \\ , 1 } \\oplus X _ { 1 , \\ , j , \\ , 3 } ) = L _ { - 1 } ^ * \\otimes X _ { 1 , \\ , j , \\ , 1 } \\oplus L _ { - 1 } ^ * \\otimes X _ { 1 , \\ , j , \\ , 3 } \\end{align*}"}
-{"id": "2321.png", "formula": "\\begin{align*} \\mu _ 1 ( 0 ) = \\mu _ 1 ' ( 0 ) = 0 , \\quad \\mu _ 2 ( 0 ) = \\mu _ 2 ' ( 0 ) = 0 . \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} d _ m \\ , \\partial _ { \\nu } m = \\frac { \\alpha _ 1 \\ , L _ { o x } } { k _ 1 + L _ { o x } } + { \\large { \\copyright _ 2 } } \\ \\ \\Omega \\supset \\omega ; \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( A _ \\infty + \\frac { A _ 0 } { x } + \\frac { A _ 1 } { x - 1 } \\right ) Y , \\frac { \\partial Y } { \\partial t _ 1 } = ( B _ { 1 1 } x + B _ { 1 0 } ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = ( B _ { 2 1 } x + B _ { 2 0 } ) Y . \\end{gather*}"}
-{"id": "8124.png", "formula": "\\begin{align*} \\partial _ t \\phi = \\frac { i } { 2 } \\Delta \\phi - i \\delta V ( x ) \\phi , \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 3 } { x } + A _ 2 + A _ 1 x + A _ 0 x ^ 2 \\right ) Y , \\\\ \\frac { \\partial Y } { \\partial t _ 1 } = ( A _ 0 x + B _ { 1 0 } ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = \\left ( - A _ 0 x ^ 2 + B _ { 2 1 } x + B _ { 2 0 } \\right ) Y . \\end{gather*}"}
-{"id": "7338.png", "formula": "\\begin{align*} * _ t ( 1 ) = \\star _ t \\left ( \\frac { 1 } { \\varepsilon _ t } \\eta \\right ) , \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} { \\bf C } ' { \\bf A } ^ { \\star } { \\bf C } = \\left [ \\begin{array} { c c } { \\bf D } _ r & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf 0 } \\end{array} \\right ] \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} f & = \\sum _ m f _ m , \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} M f ^ { 2 } + N f ^ { \\prime } f - q = B e ^ { b _ { m } z ^ { m } } , \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} H = h _ 1 ^ { - 1 } H h _ 1 = h _ 1 ^ { - 1 } \\pi _ 1 ( G _ 1 ) h _ 1 = \\pi _ 1 ( h ) ^ { - 1 } \\pi _ 1 ( G _ 1 ) \\pi _ 1 ( h ) = \\pi _ 1 ( h ^ { - 1 } G _ 1 h ) , \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} Q _ { 1 , a } ( X _ 1 ) = \\frac { 1 } { ( 1 - X _ 1 ) ( 1 - p ^ a X _ 1 ) } , Q _ { 0 , a } = 1 . \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} \\abs { \\zeta - z } ^ 2 \\le | 1 - r + \\delta | | 1 - r + \\delta | < 8 ( 1 - r ) ( 2 \\delta ) = 1 6 ( 1 - r ) \\delta \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{align*} \\mathcal { B } ( z , \\{ \\overline { \\alpha } \\} ) & = \\widetilde { B } ( z , \\{ \\alpha \\} ) { } _ a \\langle 0 | K ( z , t , \\alpha _ 0 ) | 0 \\rangle _ a A ( z ) + \\widetilde { A } ( z , \\{ \\alpha \\} ) { } _ a \\langle 1 | K ( z , t , \\alpha _ 0 ) | 1 \\rangle _ a B ( z , t , \\{ \\alpha \\} ) \\\\ & = ( t z - \\alpha _ 0 ) \\widetilde { B } ( z , \\{ \\alpha \\} ) A ( z , \\{ \\alpha \\} ) + ( z ^ { - 1 } + \\alpha _ 0 ) \\widetilde { A } ( z , \\{ \\alpha \\} ) B ( z , \\{ \\alpha \\} ) . \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{align*} \\psi \\Big ( \\sum _ { p = 1 } ^ { \\infty } R _ { n , j , k } ( p , y ) t ^ { p } x ^ { n } \\Big ) = R _ { n , j , k } ( p , y ) x ^ { n } \\end{align*}"}
-{"id": "6431.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 1 & 0 \\\\ 0 & \\pi \\end{bmatrix} \\begin{bmatrix} \\pi & \\pi ^ n \\\\ 1 & \\pi + \\pi ^ { n - 1 } \\end{bmatrix} , \\end{align*}"}
-{"id": "9914.png", "formula": "\\begin{align*} f _ { n } ^ { \\pm \\pm \\pm } = f _ { k ; n } ^ { \\pm \\pm \\pm } , f _ { 0 } ^ { \\circ \\pm \\pm } = f _ { k ; 0 } ^ { \\circ \\pm \\pm } , f _ { - k p } ^ { \\pm \\circ \\pm } = f _ { k ; - k p } ^ { \\pm \\circ \\pm } , f _ { k d } ^ { \\pm \\pm \\circ } = f _ { k ; k d } ^ { \\pm \\pm \\circ } . \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} \\frac { \\bar { A } _ { n , j , k } ( t , z ) } { ( 1 - t ) ^ { n + 1 } } = - \\frac { A _ { n , j , k } ( 1 / t , z ) } { ( 1 - ( 1 / t ) ) ^ { n + 1 } } , \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = \\alpha _ 2 y _ 5 , [ y _ 1 , y _ 2 ] = y _ 4 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 5 } { \\alpha _ 3 } y _ 4 + \\theta y _ 5 , [ y _ 2 , y _ 2 ] = \\beta _ 2 y _ 5 , [ y _ 2 , y _ 3 ] = \\gamma _ 2 y _ 5 = - [ y _ 3 , y _ 2 ] . \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} 0 = \\sum _ { \\substack { 0 \\leq d _ i \\leq a _ i \\\\ 1 \\leq e _ j \\leq b _ { j } - 1 } } \\Big ( \\prod _ { i = 1 } ^ n Q _ { d _ i } ( a _ i ) \\psi _ { i } ^ { d _ i } \\Big ) \\Big ( \\prod _ { j = 1 } ^ m Q _ { e _ j + 1 } ( b _ j ) \\Big ) \\kappa _ { e _ 1 , \\dots , e _ m } \\end{align*}"}
-{"id": "9961.png", "formula": "\\begin{align*} Q _ i = - P _ i A _ i ^ { - 1 } E _ i - E _ i ^ T A _ i ^ { - T } P _ i \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{align*} \\bar u = ( \\bar u _ 0 , \\bar u _ 1 , \\bar u _ 2 ) = ( u _ { \\pi 0 } , u _ { \\pi 1 } , u _ { \\pi 2 } ) . \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} \\mathbb { E } ( Z _ 1 | { \\cal F } _ { l - 1 } ) \\leq C S _ b = C \\mathbb { P } ( q _ l \\in \\hat { \\pi } ^ { ( n ) } _ n | { \\cal F } _ { l - 1 } ) . \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} Q _ l ( x ) = \\sum _ { j = 0 } ^ l ( - 1 ) ^ j { x \\choose j } { k - x \\choose l - j } { n - k - x \\choose l - j } , \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} A ' = \\begin{bmatrix} a ' & b ' \\pi ^ n \\\\ c ' & d ' \\end{bmatrix} = \\pi ^ { - m } A \\in R ^ \\bullet . \\end{align*}"}
-{"id": "4026.png", "formula": "\\begin{align*} \\Psi ( 0 ) = \\frac { a } { a - b } \\cdot \\Upsilon ( 0 ) \\Phi ( 0 ) . \\end{align*}"}
-{"id": "2004.png", "formula": "\\begin{align*} & t ^ { N ( M - N ) } \\prod _ { j = 1 } ^ N z _ j ^ { j - 1 - N } ( 1 + t z _ j ^ 2 ) \\prod _ { 1 \\le j < k \\le N } ( 1 + t z _ j z _ k ) ( 1 + t z _ j z _ k ^ { - 1 } ) s p _ { \\overline { \\lambda } } ( \\{ t z \\} _ N ) \\\\ = & \\sum _ { \\{ \\overline { x ^ 1 } \\} , \\dots , \\{ \\overline { x ^ { N - 1 } } \\} } \\prod _ { k = 1 } ^ N ( \\alpha ( z _ k , \\{ \\overline { x ^ { k - 1 } } \\} , \\{ \\overline { x ^ { k } } \\} ) + \\beta ( z _ k , \\{ \\overline { x ^ { k - 1 } } \\} , \\{ \\overline { x ^ { k } } \\} ) ) , \\end{align*}"}
-{"id": "5260.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left ( u - c \\right ) \\times \\omega \\ , d x = 0 . \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{align*} W V _ 1 ^ { m _ 1 } \\cdots V _ n ^ { m _ n } f = ( V _ 1 ^ { i _ 1 + m _ 1 - 1 } \\cdots V _ 1 ^ { i _ n + m _ n - 1 } f ) . \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} \\sum \\limits _ { \\substack { \\beta \\in \\Pi \\\\ \\beta \\ne \\alpha } } \\frac { \\mu _ \\beta \\bigl ( e ( \\beta ) + e ( \\alpha ) \\bigr ) ^ { 2 k - 1 } } { \\bigl ( e ( \\beta ) - e ( \\alpha ) \\bigr ) ^ { 2 k - 1 } } = 0 \\mbox { \\rm a n d } \\sum \\limits _ { \\substack { \\beta \\in \\Pi \\\\ \\beta \\ne \\alpha } } \\frac { \\mu _ \\beta ( \\mu _ \\beta + 1 ) e ( \\beta ) \\bigl ( e ( \\beta ) + e ( \\alpha ) \\bigr ) ^ { 2 k - 1 } } { \\bigl ( e ( \\beta ) - e ( \\alpha ) \\bigr ) ^ { 2 k + 1 } } = 0 ; \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} D _ y ( p ^ 2 - 1 ) = m _ 1 m _ 2 , \\end{align*}"}
-{"id": "8141.png", "formula": "\\begin{align*} \\begin{cases} H f = 0 \\\\ ( M ^ { \\otimes m } ) _ { 2 } f = \\mu _ 2 \\end{cases} \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} \\tau _ 2 ( x ) = \\frac { 3 x - 1 } { ( x - 1 ) ^ 2 } , \\tau _ 3 ( x ) = \\frac { 7 x ^ 2 - 2 x + 1 } { ( x - 1 ) ^ 3 } . \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} B = \\left \\{ u _ 1 + u _ 2 + \\dots + u _ { 2 k + 1 } : \\ k \\in \\omega , \\{ u _ 1 , u _ 2 , \\dots u _ { 2 k + 1 } \\} \\in [ U ] ^ { 2 k + 1 } \\right \\} , \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\frac { \\d r } { r \\zeta ( r ) + 1 } = \\infty , \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\overset { \\cdot } { m } _ { 1 } = a _ { 1 } m _ { 2 } m _ { 3 , } \\\\ \\overset { \\cdot } { m } _ { 2 } = - a _ { 1 } m _ { 1 } m _ { 3 } , \\\\ \\overset { \\cdot } { m } _ { 3 } = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} \\widetilde { \\Delta } _ { k _ 1 , \\cdots , k _ d } ( f ) ( x _ 1 , \\cdots , x _ d ) & = \\widetilde { \\Delta } _ { k _ 1 } \\otimes \\cdots \\otimes \\widetilde { \\Delta } _ { k _ d } ( f ) ( x _ 1 , \\cdots , x _ d ) \\\\ & = \\sum _ { r _ 1 , \\cdots , r _ d \\in \\Z } \\phi _ { k _ 1 } ( r _ 1 ) \\cdots \\phi _ { k _ d } ( r _ d ) \\widehat { f } ( r _ 1 , \\cdots , r _ d ) e ^ { i 2 \\pi ( r _ 1 x _ 1 + \\cdots + r _ d x _ d ) } \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left [ | u | ^ 2 - ( u - c ) \\cdot \\left ( V + { 1 \\over \\gamma _ n } \\nabla \\left ( { m \\cdot x \\over | x | ^ n } \\right ) \\right ) \\right ] \\ , d x = - \\frac { \\gamma _ n } { 2 } \\ ; c \\cdot p , \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} ( B _ { 1 / n } , B _ 0 ) | \\{ \\gamma _ n < - 1 \\} \\stackrel { d } { = } \\Gamma ^ n ( B _ 1 , B _ 2 ) . \\end{align*}"}
-{"id": "9870.png", "formula": "\\begin{align*} G ( \\rho ) = \\dfrac { 1 } { 2 \\pi ^ 2 \\sigma } \\dfrac { ( \\rho ) } { \\rho } + C _ 3 G ( 0 ) = \\dfrac { 1 } { 2 \\pi ^ 2 \\sigma } \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{align*} \\sup _ { j \\ge 0 } \\ \\| P _ { n } T ^ { \\ , j } P _ { l } \\ , x \\| & \\le 2 ^ { \\ , \\delta ^ { ( k - 1 ) } - \\tau ^ { ( k ) } } \\ , \\cdot \\ , \\bigl ( \\Delta ^ { ( k ) } \\bigr ) ^ { 1 - \\frac 1 { p } } \\ , \\cdot \\ , \\Big \\| \\sum _ { i = b _ { l } } ^ { b _ { l + 1 } - 1 } \\Big ( \\prod _ { s = i + 1 } ^ { b _ { l + 1 } - 1 } w _ s \\Big ) \\ , x _ { i } e _ { i } \\Big \\| \\\\ & \\le \\dfrac { \\beta _ { l } } { 4 } \\cdot \\Big \\| \\sum _ { i = b _ { l } } ^ { b _ { l + 1 } - 1 } \\Big ( \\prod _ { s = i + 1 } ^ { b _ { l + 1 } - 1 } w _ s \\Big ) \\ , x _ { i } e _ { i } \\Big \\| \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{align*} \\partial ( b _ { k , \\gamma _ 0 } ' + r _ k ) = \\sum _ { j = 1 } ^ m \\chi _ { X } \\cdot ( a _ j \\cdot \\gamma _ 0 ) \\otimes \\gamma _ 0 ^ { - 1 } \\cdot \\partial \\sigma _ j = \\sum _ { j = 1 } ^ m a _ j \\otimes \\partial \\sigma _ j = \\partial b = c . \\end{align*}"}
-{"id": "1841.png", "formula": "\\begin{align*} R = \\left ( \\frac 1 2 R _ { \\alpha \\beta \\gamma } ^ { \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\delta } \\ , z ^ \\alpha \\wedge z ^ \\beta + R _ { \\alpha \\bar \\beta \\gamma } ^ { \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\delta } \\ , z ^ \\alpha \\wedge \\bar { z } ^ { \\bar { \\beta } } + \\frac 1 2 R _ { \\bar \\alpha \\bar \\beta \\gamma } ^ { \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\delta } \\ , \\bar { z } ^ { \\bar \\alpha } \\wedge \\bar { z } ^ { \\bar \\beta } \\right ) \\otimes z ^ \\gamma \\otimes z _ \\delta . \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} B ( M , \\epsilon ) : = \\bigcup _ { 1 \\leq i \\leq n } \\{ M \\log { n } + 1 \\leq \\# { \\cal E } _ i \\leq \\epsilon n \\} \\end{align*}"}
-{"id": "2964.png", "formula": "\\begin{align*} \\sup _ { j \\ge 0 } \\ \\sum _ { l > n } \\| P _ { n } T ^ { \\ , j } P _ { l } \\ , x \\| = \\infty . \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{gather*} Q ^ { i , k , j } _ t = \\sigma \\left ( \\frac { 1 } { n } - \\delta _ { i , k } \\right ) \\left ( \\frac { 1 } { n } - \\delta _ { i , j } \\right ) \\phi _ t , \\\\ R ^ { i , k , j } _ t = \\left ( \\frac { 1 } { n } - \\delta _ { i , k } \\right ) \\left ( \\frac { 1 } { n } - \\delta _ { i , j } \\right ) \\phi _ { t } \\gamma ^ j _ t , \\end{gather*}"}
-{"id": "5356.png", "formula": "\\begin{gather*} c _ 1 ( n , k ) = \\sum _ { i = 0 } ^ { k - 1 } { k \\choose i } \\sum _ { j _ 1 + j _ 2 + \\cdots + j _ { k - i } = n - i } f ( j _ 1 ) f ( j _ 2 ) \\cdots f ( j _ { k - i } ) \\\\ = \\sum _ { i = 0 } ^ { k - 1 } { k \\choose i } a ^ { k - i } ( a - 1 ) ^ { n - 2 j + i } \\sum _ { j _ 1 + j _ 2 + \\cdots + j _ { k - i } = n - i } 1 \\\\ = \\sum _ { i = 0 } ^ { k - 1 } a ^ { k - i } ( a - 1 ) ^ { n - 2 k + i } { k \\choose i } { n - k - 1 \\choose k - i - 1 } . \\end{gather*}"}
-{"id": "8944.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\rho _ { s + 1 } ( S / I _ \\Delta ) \\geq \\lim _ { n \\to \\infty } \\frac { n - n ^ { 1 - \\epsilon } + 1 } { n } = 1 . \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{align*} \\frac { d t } { d s } = \\frac { 1 } { s } . \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} \\begin{array} { l } { \\displaystyle \\widetilde { \\Psi } ^ 0 = \\Psi ^ 0 , \\ , \\ , \\ , \\widetilde { \\mathbf { E } } ^ 0 = \\mathbf { E } ^ 0 , \\ , \\ , \\ , \\widetilde { \\mathbf { H } } ^ 0 = \\mathbf { H } ^ 0 . } \\end{array} \\end{align*}"}
-{"id": "3899.png", "formula": "\\begin{align*} \\begin{array} { l } \\int _ { \\frac { 1 } { 2 } } ^ b 2 A _ 4 Q ( x ) \\cos 2 k ( x - 1 ) d x = \\int _ { 1 - b } ^ { \\frac { 1 } { 2 } } 2 A _ 4 Q ( 1 - t ) \\cos ( 2 k t ) d t . \\end{array} \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{gather*} \\partial _ t u - \\sum \\nolimits _ { i , j = 1 } ^ n a _ { i j } ( x , t , u ) \\partial ^ 2 _ { x _ i x _ j } u = b ( x , t , u , \\nabla _ x u ) \\mbox { i n } \\ \\Omega \\times ( 0 , T ] , \\\\ \\sum \\nolimits _ { i , j = 1 } ^ n a _ { i j } ( x , t , u ) \\nu _ j \\partial _ { x _ i } u = \\psi _ 1 ( x , t , u ) \\mbox { o n } \\ \\partial \\Omega \\times ( 0 , T ] , \\\\ u | _ { t = 0 } = \\psi _ 0 ( x ) , \\end{gather*}"}
-{"id": "8446.png", "formula": "\\begin{align*} Y _ { \\theta } = \\overline { V } ( \\theta ) = \\underline { V } ( \\theta ) \\mbox { a . s . } \\end{align*}"}
-{"id": "939.png", "formula": "\\begin{align*} \\textrm { V a r } [ X | Y \\in B ] = k ( B ) \\textrm { V a r } [ X ] + \\left ( \\textrm { V a r } [ Y | Y \\in B ] - k ( B ) \\textrm { V a r } [ Y ] \\right ) \\beta \\beta ^ { T } , \\end{align*}"}
-{"id": "7858.png", "formula": "\\begin{align*} \\alpha _ n C ^ { - 1 } ( r - 1 ) + \\alpha ^ 2 _ n C ^ { - 2 } ( r - 1 ) + \\alpha _ n r \\leq \\frac { \\omega } { 3 } ( r - 1 ) + \\frac { \\omega } { 3 } ( r - 1 ) + \\frac { \\omega } { 3 } r = \\omega r - \\frac { 2 \\omega } { 3 } . \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{align*} F _ * ^ e ( L / ( f + u v ) ) = F _ * ^ e ( L ) / F _ * ^ e ( J ) = \\bigoplus _ { k = 0 } ^ { q - 1 } M _ k / M _ k F _ * ^ e ( f + u v ) . \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} R _ { } & = R _ { o D } ( p _ o ) \\mathbb { P } \\{ \\bar { E } _ { } | r _ 1 \\} + R _ { o R } ( p _ o ) \\mathbb { P } \\{ E _ { } | r _ 1 \\} & & \\end{align*}"}
-{"id": "3150.png", "formula": "\\begin{align*} f W _ { i j k l } R _ { i k } R _ { j l } = \\frac { n - 3 } { 2 ( n - 1 ) } f | C _ { i j k } | ^ { 2 } . \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} I ( u ) = \\frac { 1 } { 2 } \\sum _ { i , j } a _ { i j } | u _ i - u _ j | . \\end{align*}"}
-{"id": "9679.png", "formula": "\\begin{align*} \\begin{aligned} \\hat { F } ( s , \\phi , \\pi ) & = \\hat { F } ( s , \\pi - \\phi , 0 ) , \\\\ \\hat { G } ( s , \\phi , \\pi ) & = - \\hat { G } ( s , \\pi - \\phi , 0 ) , \\\\ \\hat { H } ( s , \\phi , \\pi ) & = - \\hat { H } ( s , \\pi - \\phi , 0 ) \\end{aligned} \\end{align*}"}
-{"id": "6743.png", "formula": "\\begin{align*} a \\left ( j \\right ) u _ { y y } \\left ( x , j \\right ) + b \\left ( j \\right ) u _ { y } \\left ( x , j \\right ) = 0 = 0 j = 0 , 1 , x \\in G , \\end{align*}"}
-{"id": "9546.png", "formula": "\\begin{align*} \\mu ( s , t ) = v _ s \\alpha _ s ( v _ t ) \\sigma ( s , t ) v _ { s t } ^ * , \\ \\ \\ s , t \\in G . \\end{align*}"}
-{"id": "495.png", "formula": "\\begin{align*} \\sum _ { \\substack { H \\leq F _ r , \\\\ H \\leq G _ r } } | H | ^ a = \\sum _ { \\substack { H \\leq G _ r } } | H | ^ a = \\sigma _ a ( G _ r ) . \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} h ^ 0 ( E ( m ) ) = 0 , \\ \\ \\ m \\le 0 . \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} \\rho ( g ) ( F _ j ) = F _ { j ^ { \\prime } } , \\ \\rho ( g ) ( e _ j ) = e _ { j ^ { \\prime } } . \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} \\dot x = J ( x ) \\sqrt \\omega \\cos ( \\omega t ) + \\sqrt \\omega \\sin ( \\omega t ) . \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} V = V + j , \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} \\int _ { 0 } ^ 1 \\frac { \\big ( \\beta _ { \\mu , 2 } ^ { ( k ) } ( x , r ) \\big ) ^ 2 } { r } \\ , \\d r & = \\sum _ { q = 0 } ^ { \\infty } \\int _ { \\lambda ^ { q + 1 } } ^ { \\lambda ^ q } \\frac { \\big ( \\beta _ { \\mu , 2 } ^ { ( k ) } ( x , r ) \\big ) ^ 2 } { r } \\ , \\d r \\leq C \\ , \\sum _ { q = 0 } ^ { \\infty } \\big ( \\beta _ { \\mu , 2 } ^ { ( k ) } ( x , \\lambda ^ q ) \\big ) ^ 2 . \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} ( \\nabla \\ , \\partial _ { \\tau } { \\phi } _ { h } ^ { k } , \\ , \\nabla u ) = ( \\partial _ { \\tau } \\vert \\Psi _ { h } ^ { k } \\vert ^ { 2 } , \\ , u ) , \\quad \\forall u \\in X _ { h } ^ { r } . \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} ( 1 + \\omega ) ( \\lambda T ' ( c ) + ( R _ { 1 c } - \\omega ) ( 1 + \\omega ) + R _ { 1 h } R _ { 3 c } ) = 0 , \\end{align*}"}
-{"id": "4933.png", "formula": "\\begin{align*} S = \\begin{pmatrix} 1 & 2 & 3 & 4 \\\\ 2 & 1 & 3 & 4 \\\\ 3 & 2 & 1 & 4 \\\\ 4 & 2 & 3 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "6515.png", "formula": "\\begin{align*} \\sup _ { \\xi , \\left \\Vert \\xi \\right \\Vert = 1 } \\frac { 1 } { n } \\sum _ { x } \\mu ^ { ( n ) } ( x ) \\left | \\left \\langle b ( x ) , \\xi \\right \\rangle \\right | ^ { 2 } \\longrightarrow 0 . \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} \\mathbf { \\tilde { K } } _ { l } : = 2 d n ^ { d } \\left ( 8 + \\left \\Vert \\mathbf { F } \\right \\Vert _ { 1 , \\mathfrak { L } } \\right ) \\underset { z \\in \\mathfrak { L } , | z | = 1 } { \\sum } \\frac { 1 } { \\left \\vert \\Lambda _ { l } \\right \\vert } \\underset { x \\in \\mathfrak { L } } { \\sum } \\mathbf { 1 } \\left [ \\left ( x , x + z \\right ) \\in \\partial ( l b _ { 0 } ) \\right ] \\ . \\end{align*}"}
-{"id": "3595.png", "formula": "\\begin{align*} \\Lambda ( \\gamma , \\lambda ) = 1 - \\frac { c } { 2 \\pi \\gamma ^ 2 } J _ 1 ( 1 / \\gamma , - \\lambda / \\gamma ) = 0 , \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} \\tau ( a ^ * ) = \\overline { \\tau ( a ) } , \\tau ( a b ) = \\tau ( b a ) , \\tau ( a a ^ * ) > 0 a \\neq 0 \\end{align*}"}
-{"id": "638.png", "formula": "\\begin{align*} ( 1 - x ^ 2 ) h '' ( x ) - 2 x h ' ( x ) + \\left ( \\nu ^ 2 + \\nu - \\frac { s ^ 2 } { 1 - x ^ 2 } \\right ) h ( x ) = 0 x \\in ( - 1 , 1 ) , \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{align*} \\underset { n \\to \\infty } { \\lim } \\underset { m \\to \\infty } { \\lim } \\frac { 1 } { \\left ( x _ { N + 1 } ^ { ( n ) , m } - x _ { N - 1 } ^ { ( n ) , m } \\right ) } = 0 \\end{align*}"}
-{"id": "2392.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u = - \\mathrm i | D | u + | u | ^ { p - 1 } u , & t \\in \\lbrack 0 , T ) , x \\in \\mathbb R ^ n , \\\\ u ( 0 , x ) = u _ 0 ( x ) , & x \\in \\mathbb R ^ n , \\end{cases} \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} \\mathcal { H } ^ { n - 1 } ( \\vec { \\alpha } _ { K ' } ^ * ( \\eta ' ) ) = \\mathcal { H } ^ { n - 1 } ( \\vec { \\alpha } _ K ^ * ( \\eta ' ) ) > 0 . \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c c c c c c } 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\\\ 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\\\ 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{align*} k ' ( B ) = k ' ( \\hat B ) \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} e ^ { t \\Re \\ , a ( z ) } | \\hat { u } ( z ) | = | \\widehat { e ^ { t a ( D ) } u } ( z ) | = | V _ { ( t , u ) } ( z ) | \\leq C _ { ( t , u ) } ( 1 + | z | ) ^ { N _ { ( t , u ) } } e ^ { R _ { ( t , u ) } | \\Im \\ ; z | } \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} \\Phi x = \\Phi ' x \\cdot ( u x ) \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} u \\left ( t \\right ) = u _ { 0 } \\left ( t \\right ) + G _ { \\varepsilon } u \\left ( t \\right ) , \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} ( 1 - 2 \\beta ) \\sum _ { k = 1 } ^ { ( p - 1 ) / 2 } \\binom { 2 k } { k } H _ k \\bigl ( \\beta ( 1 - \\beta ) \\bigr ) ^ k \\equiv 2 \\pounds _ 1 ( 2 \\beta ) - 2 \\pounds _ 1 ( \\beta ) - 2 \\pounds _ 1 ( - 1 ) \\beta ^ p \\pmod { p } . \\end{align*}"}
-{"id": "3266.png", "formula": "\\begin{align*} d X ^ N _ i ( t ) = d W ^ N _ i ( t ) + \\left [ - c X ^ N _ i ( t ) + \\sum _ { j \\ne i } ^ { } \\frac { 1 } { X ^ N _ i ( t ) - X ^ N _ j ( t ) } \\right ] d t , \\end{align*}"}
-{"id": "3264.png", "formula": "\\begin{align*} \\delta ( t ) = t + \\delta ( 0 ) , \\end{align*}"}
-{"id": "3307.png", "formula": "\\begin{align*} { q } _ t ^ { N , N + 1 } ( ( x , y ) , ( x ' , y ' ) ) = \\det \\ \\begin{pmatrix} { A } _ t ( x , x ' ) & { B } _ t ( x , y ' ) \\\\ { C } _ t ( y , x ' ) & { D } _ t ( y , y ' ) \\end{pmatrix} \\ \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} { } _ a \\langle 0 | K _ { a } ( z , t ) | 0 \\rangle _ a & = t z , \\\\ { } _ a \\langle 1 | K _ { a } ( z , t ) | 1 \\rangle _ a & = z ^ { - 1 } . \\end{align*}"}
-{"id": "49.png", "formula": "\\begin{align*} \\widetilde { W } _ { n - q } ( K ) = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } \\rho _ K ^ q ( u ) d u . \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} \\omega _ { t _ k ( \\omega ) } \\dots \\omega _ { t _ k ( \\omega ) + | w | - 1 } = w . \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} f ( q ) = \\frac 4 3 + O \\ ( y \\ ) . \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} \\mathbb P ( M _ 1 = 0 ) = \\mathbb P ( N W _ 1 ^ { \\tau _ N } = 0 ) \\leq \\mathbb P ( N W _ 1 = 0 ) = 0 , \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{align*} \\varepsilon ( F _ { a } \\triangleright \\mathsf { Q } _ { k } ^ { j } ) \\varepsilon ( E _ { a } \\triangleright \\mathsf { Q } _ { i } ^ { k } ) = \\sum _ { m , n , o , p } c _ { n } ^ { m } c _ { p } ^ { o } \\varepsilon ( F _ { a } \\triangleright ( \\mathsf { N } _ { m } ^ { n } ) _ { k } ^ { j } ) \\varepsilon ( E _ { a } \\triangleright ( \\mathsf { N } _ { o } ^ { p } ) _ { i } ^ { k } ) . \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} e ^ { ( 1 + o ( 1 ) ) k \\log k } \\le l \\le e ^ { ( 1 + o ( 1 ) ) n \\log n } . \\end{align*}"}
-{"id": "1938.png", "formula": "\\begin{align*} \\norm { f ' } _ m & = \\alpha \\beta ^ { e _ 1 } ( \\beta ^ { e _ 1 } \\beta ^ n ) ^ { - 1 / m } \\norm { f } _ m \\\\ & = \\alpha \\beta ^ { e _ 1 ( 1 - \\frac { 1 } { m } ) - \\frac { n } { m } } \\norm { f } _ m \\\\ & \\leq \\norm { f } _ m . \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} S ( t ) H ( \\cdot , A ) ( \\eta ) = E ^ { A } H ( \\eta , A _ t ) , \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{align*} x _ \\infty = \\big ( 2 m \\lambda ( 1 + \\lambda ) , 2 ( m - 2 ) , 2 ( 1 + 2 \\lambda ) , 0 \\big ) \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} S ( x ) ^ 2 + \\sum _ { k = 1 } ^ { n - 1 } P ( t _ k ) S _ k ( x ) ^ 2 = \\tilde { S } ( x ) ^ 2 + \\sum _ { k = 1 } ^ { n - 1 } P ( \\tilde { t } _ k ) \\tilde { S } _ k ( x ) ^ 2 . \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} 0 \\geq Q '' = \\frac { v _ \\epsilon ^ { ( 5 ) } } { v _ \\epsilon '' } - 3 \\frac { v _ \\epsilon ''' v _ \\epsilon ^ { ( 4 ) } } { ( v _ \\epsilon '' ) ^ 2 } + 2 \\frac { ( v _ \\epsilon ''' ) ^ 2 } { ( v _ \\epsilon '' ) ^ 2 } + ( 1 - \\alpha ) ( \\log \\theta ) ''' . \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 3 } K _ i \\leq C , \\quad k = 3 . \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{align*} \\mathbf { \\dot { p } } _ { d } ' & = \\mathbf { R } _ { \\Delta \\phi _ l } \\mathbf { \\dot { p } } _ { d } \\\\ v _ { v o } & = \\mathbf { \\dot { p } } _ { d } ' \\cdot \\frac { \\mathbf { \\dot { p } } _ { o } } { | | \\mathbf { \\dot { p } } _ { o } | | } \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} \\sigma _ k = \\frac { 1 } { p ( t _ k ) } \\ ( k = 1 , \\dots , n - 1 ) , \\ \\varrho _ k = \\frac { 1 } { p ( x _ k ) } \\ ( k = 1 , \\dots , n ) , \\ \\lambda = 1 / p _ { 2 n - 2 } , \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} \\mathrm { R e } \\left [ B ( \\overline { \\mathbf { A } } _ { h } ^ { k } ; \\Psi _ { h } ^ { k - 1 } , \\Psi _ { h } ^ { k } ) - B ( \\overline { \\mathbf { A } } _ { h } ^ { k } ; \\Psi _ { h } ^ { k } , \\Psi _ { h } ^ { k - 1 } ) \\right ] = 0 . \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } \\left ( \\left ( n j - { j + 1 \\choose 2 } \\right ) - { j + 1 \\choose 2 } + 1 \\right ) = \\sum _ { j = 0 } ^ { n - 1 } \\left ( n j - 2 { j + 1 \\choose 2 } + 1 \\right ) , \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} H z _ i - z _ i ^ 2 = H z _ j - z _ j ^ 2 \\Rightarrow H = z _ i + z _ j \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{align*} d _ i = \\sum _ { \\substack { j \\ \\in \\ 2 \\mathbb { N } _ 0 - 2 \\\\ j \\ < \\ i - 1 \\ \\ \\ } } \\left ( \\frac { \\zeta ( j - i ) } { j - i } - \\frac { 2 ^ j E _ { i - j - 1 } } { 2 ^ { i - j + 1 } - 1 } \\right ) \\frac { 2 ^ { i - j + 3 } - 4 } { 2 ^ { i + 1 } - 1 } \\begin{cases} \\binom { i } { j + 1 } d _ { j + 1 } & j \\neq - 2 \\\\ 1 / ( i + 1 ) & \\end{cases} \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} \\det L _ { J \\setminus \\{ i \\} , J \\setminus \\{ j \\} } ^ * & = \\sum _ { \\sigma \\in \\mathcal M _ { i , j } } \\varepsilon ( \\sigma ) L _ { i , \\sigma ( i ) } ^ * L _ { k _ 1 , \\sigma ( k _ 1 ) } ^ * \\ldots L _ { k _ m , \\sigma ( k _ m ) } ^ * \\ , , \\end{align*}"}
-{"id": "878.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 _ + } V f _ \\epsilon ( x ) = | x | ^ { \\gamma - 1 } , x \\neq 0 ; \\end{align*}"}
-{"id": "6051.png", "formula": "\\begin{align*} \\mathfrak { g } = \\mathfrak { g } _ 1 + \\cdots + \\mathfrak { g } _ k + \\mathfrak { z } ( \\mathfrak { g } ) \\end{align*}"}
-{"id": "4946.png", "formula": "\\begin{align*} \\sigma ( M ) = \\left ( 3 , 0 , 1 , 1 , 1 \\right ) . \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} u ( t , x ) = t ^ { - \\frac { 1 } { \\alpha } } f \\Bigl ( \\frac { x } { \\sqrt t } \\Bigr ) , \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} & \\big | a _ 1 ( T ) { \\mathrm { e } } ^ { \\omega _ 1 / T } + \\cdots + a _ n ( T ) { \\mathrm { e } } ^ { \\omega _ n / T } \\big | _ \\infty \\prod _ { i = 2 } ^ n | a _ i ( T ) | _ \\infty \\ge C ( n ) ^ { - 1 } , \\\\ & | a _ 1 ( T ) | _ \\infty \\prod _ { i = 2 } ^ n \\big | a _ 1 ( T ) { \\mathrm { e } } ^ { \\omega _ i / T } - a _ i ( T ) { \\mathrm { e } } ^ { \\omega _ 1 / T } \\big | _ \\infty \\ge C ( n ) ^ { - ( n - 1 ) } . \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} \\rho _ i ( x _ 0 , x _ { t _ i } ) = \\rho _ i ( x _ { t _ i } , x _ { 2 t _ i } ) = \\ldots = \\rho _ i ( x _ { ( p - 1 ) t _ i } , x _ 0 ) = c _ i . \\end{align*}"}
-{"id": "8331.png", "formula": "\\begin{align*} f _ { n } ( T , U ) = \\zeta _ 0 T ^ { n + 1 } - \\zeta _ 1 T ^ { n } U + \\zeta _ 2 T ^ { n - 1 } U ^ 2 + \\cdot \\cdot \\cdot + ( - 1 ) ^ n \\zeta _ n T U ^ { n } + ( - 1 ) ^ { n + 1 } \\zeta _ { n + 1 } U ^ { n + 1 } . \\end{align*}"}
-{"id": "8530.png", "formula": "\\begin{align*} \\widetilde { \\vec { \\gamma } _ 0 } ( \\phi , p ) = - 4 \\ , \\vec { \\gamma } _ 0 ( \\phi , p ) . \\end{align*}"}
-{"id": "4066.png", "formula": "\\begin{align*} { p _ n } \\buildrel \\Delta \\over = \\int _ { x \\in { { \\cal D } _ n } } { f _ 1 ^ { ( n ) } \\left ( x \\right ) d x } { q _ n } \\buildrel \\Delta \\over = \\int _ { x \\in { { \\cal D } _ n } } { f _ 0 ^ { ( n ) } \\left ( x \\right ) d x } . \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\tau \\geq t \\right ) \\leq \\mathbb { P } \\left ( \\# { \\cal N } _ t \\geq 1 \\right ) \\leq \\mathbb { E } \\# { \\cal N } _ t \\leq C _ u ^ { t } = \\exp \\left ( - 2 \\frac { \\log { C _ u } } { \\log { C _ 1 } } \\log { n } \\right ) \\leq \\frac { 1 } { n ^ 2 } \\end{align*}"}
-{"id": "5639.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } \\frac { b } { c } v & = & - v ^ 3 + ( a + 1 ) v ^ 2 - a v , \\\\ w & = & \\frac { b } { c } v , \\end{array} \\right . \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { k - 1 } \\binom { k - 1 } { t } ( \\frac { - p } { q } ) ^ { t } U _ { m k + t } ^ { ( k ) } & = U _ { m } \\sum _ { t = 0 } ^ { k - 1 } \\binom { k - 1 } { t } ( - q ) ^ { - t } p ^ { t } U _ { m } ^ { k - 1 - t } U _ { m + 1 } ^ { t } \\\\ & = ( - q ) ^ { 1 - k } U _ { m } \\sum _ { t = 0 } ^ { k - 1 } \\binom { k - 1 } { t } ( - q U _ { m } ) ^ { k - 1 - t } ( p U _ { m + 1 } ) ^ { t } . \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} \\mathbf { t } ' ( s ) = \\kappa _ { 1 } ( s ) \\ , \\mathbf { n } _ 1 ( s ) + \\kappa _ { 2 } ( s ) \\ , \\mathbf { n } _ 2 ( s ) , \\ , \\mathbf { n } _ i ' ( s ) = - \\kappa _ { i } ( s ) \\ , \\mathbf { t } ( s ) \\\\ . \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} { \\displaystyle \\mathcal { W } _ h ( \\Omega ) = \\{ \\mathbf { w } _ h \\in \\mathbf { H } ( \\mathbf { c u r l } ; \\Omega ) : \\ , \\ , \\ , \\mathbf { w } _ h | _ e \\in R _ k , \\ , \\ , \\mathbf { w } _ h \\times \\mathbf { n } = 0 \\ , \\ , \\hbox { o n } \\ , \\ , \\partial \\Omega \\} , } \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} \\tau \\left ( I + \\begin{pmatrix} \\alpha & 0 \\\\ 0 & \\delta \\end{pmatrix} p \\right ) ^ { - \\frac { a - d } { \\alpha - \\delta } } = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a - \\alpha \\frac { a - d } { \\alpha - \\delta } & * \\\\ c & d - \\delta \\frac { a - d } { \\alpha - \\delta } \\end{pmatrix} p \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{align*} \\liminf _ { l \\to \\infty } \\ \\inf _ { J \\ge N _ { l } } \\ \\dfrac { 1 } { J + 1 } \\ & \\# \\Bigl \\{ 0 \\le j \\le J \\ , ; \\ , \\Vert P _ { l } T ^ { \\ , j } P _ { l } \\ , x \\Vert \\ge X _ { l } / 2 \\Bigr \\} \\\\ & \\ge \\liminf _ { k \\to \\infty } \\ \\biggl ( 1 - 2 \\delta ^ { ( k ) } \\Bigl ( \\dfrac { 1 } { \\Delta ^ { ( k ) } - \\delta ^ { ( k ) } + 1 } + \\dfrac { 1 } { \\Delta ^ { ( k ) } } \\Bigr ) \\biggr ) = 1 , \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} T ^ { \\ , b _ { N + 1 } - b _ { N } + k - b _ { n } } \\ , e _ { b _ { N } } & = v _ { N } \\ , \\Bigl ( \\prod _ { j = b _ { N } + 1 } ^ { b _ { N + 1 } - 1 } w _ { j } \\Bigr ) \\ , T ^ { \\ , k - b _ { n } } \\ , e _ { b _ { n } } - T ^ { \\ , k - b _ { n } } \\ , e _ { b _ { N } } \\\\ & = v _ { N } \\ , \\Bigl ( \\prod _ { j = b _ { N } + 1 } ^ { b _ { N + 1 } - 1 } w _ { j } \\Bigr ) \\Bigl ( \\prod _ { j = b _ { n } + 1 } ^ { k } w _ { j } \\Bigr ) \\ , e _ { k } - \\Bigl ( \\prod _ { j = b _ { N } + 1 } ^ { b _ { N } + k - b _ { n } } w _ { j } \\Bigr ) \\ , e _ { b _ { N } + k - b _ { n } } . \\end{align*}"}
-{"id": "9346.png", "formula": "\\begin{align*} \\| \\log ( \\gamma _ { L - 1 } ^ { - 1 } \\cdot \\gamma _ L ) - ( \\sigma _ L - \\sigma _ { L - 1 } ) \\| _ { } = | g ( F _ L ) - 1 | | F _ L | , \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} ( z , w ) = \\frac { 1 } { 2 } ( z + w , z + w ) + \\frac { 1 } { 2 } ( z - w , w - z ) = \\frac { 1 } { 2 } ( z + w ) 1 + \\frac { 1 } { 2 } ( z - w ) u . \\end{align*}"}
-{"id": "2622.png", "formula": "\\begin{align*} D ( A ^ * ) = \\left \\{ v \\in { \\Bbb H } \\ ; : \\ ; v _ { 1 } \\in H _ { 1 } , \\ ; - L v _ { 0 } + B v _ { 1 } \\in H _ { 0 } \\right \\} , \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{gather*} \\sum _ { k _ { 1 } \\geq 0 } . . . \\sum _ { k _ { n } \\geq 0 } ( \\prod _ { j = 1 } ^ { n } \\rho _ { j } ^ { k _ { j } } ) \\cos ( \\beta + \\sum _ { j = 1 } ^ { n } i k _ { j } \\alpha _ { j } ) \\\\ = \\frac { \\exp ( i \\beta ) \\prod _ { j = 1 } ^ { n } ( 1 - \\rho _ { j } \\exp ( - i \\alpha _ { j } ) ) + \\exp ( - i \\beta ) \\prod _ { j = 1 } ^ { n } ( 1 - \\rho _ { j } \\exp ( i \\alpha _ { j } ) ) } { 2 \\prod _ { j = 1 } ^ { n } ( 1 + \\rho _ { j } ^ { 2 } - 2 \\rho _ { j } \\cos ( \\alpha _ { j } ) ) } . \\end{gather*}"}
-{"id": "5456.png", "formula": "\\begin{align*} \\begin{cases} U _ { r , R } ( r / 2 ) = 0 , & \\\\ U _ { r , R } ( r ) = 1 , & \\\\ U _ { r , R } ( R ) = 1 , & \\\\ U _ { r , R } ( 2 R ) = 0 , & \\\\ U _ { r , R } ^ { ( l ) } ( r ) = U _ { r , R } ^ { ( l ) } ( r / 2 ) = U _ { r , R } ^ { ( l ) } ( R ) = U _ { r , R } ^ { ( l ) } ( 2 R ) = 0 , & \\forall l = 1 , . . . , m - 1 . \\end{cases} \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} \\Phi ^ { - 1 } ( \\mathcal W ) = & \\bigl \\{ \\bar \\alpha ; \\ ; \\forall n \\ge 1 \\ ; : \\ ; \\alpha _ n \\in \\mathbf Q \\bigr \\} = W , \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} \\mathsf { Z } ^ \\Omega _ { \\mathsf { P } } \\left ( X ; \\frac { 1 } { q } \\right ) _ \\beta = q ^ { - d _ \\beta } \\mathsf { Z } ^ \\Omega _ { \\mathsf { P } } ( X ; q ) _ \\beta \\ , . \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} \\| v ( \\cdot , t ) \\| _ { L ^ { 2 } ( \\mathbf { T } ) } ^ { 2 } & = 2 \\pi \\sum _ { n \\in \\mathbf { Z } } | v _ { n } ( t ) | ^ { 2 } \\\\ & = 2 \\pi \\sum _ { n \\in \\mathbf { Z } } \\exp \\biggl \\{ - f ( t ) \\left ( n + \\frac { \\mu w _ { t } } { f ( t ) } \\right ) ^ { \\ ! 2 } + \\frac { \\mu ^ { 2 } | w _ { t } | ^ { 2 } } { f ( t ) } \\biggr \\} \\\\ & \\ge 2 \\pi \\exp \\biggl \\{ - f ( t ) + \\frac { \\mu ^ { 2 } | w _ { t } | ^ { 2 } } { f ( t ) } \\biggr \\} . \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 2 } \\phi \\left ( \\gamma \\right ) \\ , { } _ { 0 } ^ { c } \\mathrm { D } _ { t } ^ { \\gamma } T \\ , \\mathrm { d } \\gamma = \\mathcal { D } \\frac { \\partial ^ { 2 } T } { \\partial x ^ { 2 } } , \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} V = - { 1 \\over \\gamma _ n } \\nabla \\left ( \\frac { m \\cdot x } { | x | ^ n } \\right ) + O \\left ( { 1 \\over | x | ^ { n + \\varepsilon } } \\right ) , \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } h _ { Q _ i ^ * } ( u _ 0 ) = h _ { K _ 0 } ( u _ 0 ) = 0 . \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} v ( i , j ) v ( i + k , j + k ) & = v _ { i + k - 1 } v _ { i + k } + v ( i + k , j ) ^ 2 + v _ j v _ { j + 1 } \\\\ & = 1 - 2 + 1 = 0 . \\end{align*}"}
-{"id": "3211.png", "formula": "\\begin{align*} \\underset { x _ 1 , \\cdots , x _ N \\to x \\mathbf { 1 } } { \\lim } e ^ { - \\lambda _ N t } \\frac { \\det \\left ( y _ i ^ { j - 1 } \\right ) ^ N _ { i , j = 1 } } { \\det \\left ( x _ i ^ { j - 1 } \\right ) ^ N _ { i , j = 1 } } \\det \\left ( p ^ { ( N ) , s } _ t ( x _ i , y _ j ) \\right ) ^ N _ { i , j = 1 } d y = e ^ { - \\lambda _ N t } \\Delta _ N ( y ) \\det \\left ( \\partial ^ { ( i - 1 ) } _ x p ^ { ( N ) , s } _ t ( x , y _ j ) \\right ) _ { i , j = 1 } ^ N d y . \\end{align*}"}
-{"id": "4413.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { ( p - 1 ) / 2 } C _ k H _ k ^ { ( 2 ) } \\bigl ( \\beta ( 1 - \\beta ) \\bigr ) ^ { k + 1 } \\equiv 2 \\beta \\pounds _ 2 ( \\beta ) + 2 ( 1 - \\beta ) \\pounds _ 2 ( 1 - \\beta ) \\pmod { ( \\beta ^ { p + 1 } , p ) } \\end{align*}"}
-{"id": "468.png", "formula": "\\begin{align*} Y _ 3 : = Y _ 1 + Y _ 2 + L ' = M _ L ' , Y _ 4 : = L ' . \\end{align*}"}
-{"id": "2369.png", "formula": "\\begin{align*} y ~ > ~ \\frac { 1 } { 1 + \\frac { 1 } { a _ { n + 1 } } } ~ \\geq ~ \\frac { 1 } { 1 + \\frac { 1 } { a _ n - 1 } } ~ = ~ 1 - \\frac { 1 } { a _ n } \\end{align*}"}
-{"id": "285.png", "formula": "\\begin{align*} \\mathrm { a d } _ { L } ^ { \\circ } ( X ) ( \\mathsf { N } _ { m } ^ { n } ) = \\pi ( K _ { 2 \\rho } S ^ { - 1 } ( X _ { ( 1 ) } ) K _ { 2 \\rho } ^ { - 1 } X _ { ( 2 ) } ) \\mathsf { N } _ { m } ^ { n } \\pi ( S ( X _ { ( 3 ) } ) K _ { 2 \\rho } S ^ { - 2 } ( X _ { ( 4 ) } ) K _ { 2 \\rho } ^ { - 1 } ) . \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} \\partial _ t { \\bf R } ^ \\varepsilon - \\mathfrak { D } _ \\varepsilon \\Delta _ { x _ 1 , r , \\vartheta } { \\bf R } ^ \\varepsilon - { \\bf F } ( { \\bf R } ^ \\varepsilon ) = { \\bf \\Psi } ^ \\varepsilon \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} \\begin{aligned} Q ( 2 \\tau ) = \\frac { 1 } { 8 } \\int \\limits _ { 0 } ^ { \\pi } \\int \\limits _ { 0 } ^ { 2 \\pi } a ( 2 \\tau ) \\sin \\phi d \\theta d \\phi = \\frac { \\pi } { 2 } a ( 2 \\tau ) . \\end{aligned} \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} \\| f * g \\| _ p \\le \\| f \\| _ r \\| g \\| _ q , 1 + \\frac 1 p = \\frac 1 r + \\frac 1 q , 1 \\le p , q , r \\le \\infty . \\end{align*}"}
-{"id": "4068.png", "formula": "\\begin{align*} { { \\tilde L } _ t } \\buildrel \\Delta \\over = \\sum \\limits _ { s = 1 } ^ t { { { \\tilde l } _ s } } , \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{align*} X \\land g \\circ f \\land L = f \\land K \\circ Y \\land g , \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} 0 . 9 9 9 9 9 9 9 7 \\sum _ { p \\geq x } \\frac { 1 } { ( p - 1 ) ^ 2 } < \\sum _ { p \\geq x } \\frac { 1 } { p ^ 2 } = - \\frac { \\theta ( x ) } { x ^ 2 \\log x } + \\int _ x ^ \\infty \\frac { \\theta ( y ) } { y ^ 3 } \\frac { 1 + 2 \\log y } { ( \\log y ) ^ 2 } d y . \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{align*} \\begin{aligned} \\zeta _ { n + 1 } ^ { h , k } & = S _ { \\Delta t } \\zeta _ { n } ^ { h , k } + \\Delta t S _ { \\Delta t } G _ \\delta ' ( X _ n ) . \\zeta _ { n } ^ { h , k } + S _ { \\Delta t } e ^ { \\tau A } \\bigl ( \\sigma _ \\delta ' ( X _ n ) . \\zeta _ { n } ^ { h , k } \\bigr ) \\Delta W _ n \\\\ & ~ + \\Delta t S _ { \\Delta t } G _ \\delta '' ( X _ n ) . ( \\eta _ n ^ h , \\eta _ n ^ k ) + S _ { \\Delta t } e ^ { \\tau A } \\bigl ( \\sigma _ \\delta '' ( X _ n ) . ( \\eta _ n ^ h , \\eta _ n ^ k ) \\bigr ) \\Delta W _ n . \\end{aligned} \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} \\int _ { \\gamma _ \\lambda } \\frac 1 { 1 - | z | ^ 2 } \\frac { \\partial } { \\partial n } | z | ^ 2 \\ , \\frac { d s } { 4 \\pi } & = \\int _ { \\gamma _ \\lambda } \\frac { | z | ^ 2 } { 1 - | z | ^ 2 } \\frac { \\partial } { \\partial n } \\log | z | ^ 2 \\ , \\frac { d s } { 4 \\pi } \\\\ & = \\int _ { \\gamma _ \\lambda } \\left ( \\frac { | f ( z ) | ^ 2 } { \\lambda } - 1 \\right ) \\frac { \\partial } { \\partial n } \\log | z | ^ 2 \\ , \\frac { d s } { 4 \\pi } . \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} & H ^ 1 ( \\mathrm { G a l } ( \\mathbb { Q } ( i ) / \\mathbb { Q } ) , E ( \\mathbb { Q } ( i ) ) ) \\\\ = \\ , \\ , & \\mathrm { K e r } \\{ H ^ 1 ( \\mathbb { Q } , E ( \\overline { \\mathbb { Q } } ) ) \\to H ^ 1 ( \\mathbb { Q } ( i ) , E ( \\overline { \\mathbb { Q } } ) ) \\} . \\end{align*}"}
-{"id": "9993.png", "formula": "\\begin{align*} X _ 2 = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 1 \\end{bmatrix} ^ T , Y _ 2 = Y _ 1 \\ , . \\end{align*}"}
-{"id": "9488.png", "formula": "\\begin{align*} \\mathbf { v ' } : = ( v _ { \\ell } , v _ { j _ 1 } , \\dots , v _ { j _ { k - 1 } } , v _ { j _ { \\mathbf { v } } } , v _ { j _ { k } } , \\dots , v _ { j _ { 2 k - 2 } } ) \\in U _ { \\ell } \\times U _ { \\ell - 1 } ^ { 2 k - 1 } , \\end{align*}"}
-{"id": "5603.png", "formula": "\\begin{align*} \\chi _ { k , n } ^ { ( t _ { 1 } , . . . , t _ { n + k } ) } ( x _ { 1 } , . . . , x _ { n + k } | \\rho ) = \\allowbreak \\frac { l _ { k , n } ^ { ( t _ { 1 } , . . . , t _ { n + k } ) } ( x _ { 1 } , . . . , x _ { n + k } | \\rho ) } { w _ { n + k } ( x _ { 1 } , . . . , x _ { n + k } | \\rho ) } , \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} G ^ - \\circ \\psi ( \\zeta ) = 0 , \\forall \\zeta \\in U _ 0 . \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} B _ { r } ( x ) = \\big \\{ y \\in \\R ^ { d } : | y - x | < r \\big \\} , Q _ { r } ( x , t ) = B _ { r } ( x ) \\times ( t - r ^ { 2 } , t ] , \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} \\iint \\mu _ \\theta ( x ) \\nu _ \\theta ( x ) \\ , d x \\ , d \\sigma ( \\theta ) = C _ 1 \\int | x - y | ^ { - 1 } \\ , d \\mu ( x ) \\ , d \\nu ( y ) \\ge C _ 2 , \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} \\delta _ { V , T } = \\sup \\ , \\{ \\delta _ { V _ p , T } ; \\ ; V _ p \\subseteq V \\} . \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} { \\displaystyle \\mathbf { A } ^ { - 1 } = \\mathbf { A } ( \\cdot , 0 ) - \\tau \\frac { \\partial \\mathbf { A } } { \\partial t } ( \\cdot , 0 ) = \\mathbf { A } _ { 0 } - \\tau \\mathbf { A } _ { 1 } } , \\end{align*}"}
-{"id": "6893.png", "formula": "\\begin{align*} \\frac { A _ { n , j , k } ( t , z ) } { ( 1 - t ) ^ { n + 1 } } = \\sum _ { p = 1 } ^ { \\infty } Q _ { n , j , k } ( p , z ) t ^ { p } . \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} \\begin{array} { c c | c c } \\geq 2 & = 1 & = 1 & \\geq 1 \\\\ = 1 & \\geq 1 & \\geq 1 & = 0 \\\\ \\hline = 1 & \\geq 1 & \\geq 1 & = 0 \\\\ \\geq 1 & = 0 & = 0 & \\geq 0 \\end{array} \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{align*} \\frac { m ( y ) } { m ( z ) } p _ t ( y , z ) = p _ t ( z , y ) \\end{align*}"}
-{"id": "2088.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( A _ 0 x + A _ 1 + \\frac { A _ 2 } { x } \\right ) Y , \\frac { \\partial Y } { \\partial t _ 1 } = ( B _ { 1 1 } x + B _ { 1 0 } ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = ( B _ { 2 1 } x + B _ { 2 0 } ) Y . \\end{gather*}"}
-{"id": "2042.png", "formula": "\\begin{gather*} \\left \\{ \\frac { { \\zeta _ d } ^ { k b } T _ 0 } { z ^ { \\frac { b } { d } + 1 } } + \\frac { { \\zeta _ d } ^ { k ( b - 1 ) } T _ 1 } { z ^ { \\frac { b - 1 } { d } + 1 } } + \\cdots + \\frac { { \\zeta _ d } ^ { k } T _ { b - 1 } } { z ^ { \\frac { 1 } { d } + 1 } } + \\frac { \\Theta } { z } \\ , \\Bigg | \\ , k = 0 , \\ldots , d - 1 \\right \\} , \\end{gather*}"}
-{"id": "129.png", "formula": "\\begin{align*} d \\eta ( \\theta ) = ( 1 - \\cos \\theta ) \\ , \\frac { d \\theta } { 2 \\pi } \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{align*} \\pi _ { n + k } ( \\Sigma ^ { k } f ) \\circ \\pi _ { n + k } ( \\Sigma ^ { k } g ) = \\pi _ { n + k } ( \\Sigma ^ { k } f \\circ g ) , \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} D _ \\tau j = \\frac { j ^ 2 - 1 9 6 8 j + 2 6 4 5 2 0 8 } { 2 j ^ 2 ( j - 1 7 2 8 ) ^ 2 } . \\end{align*}"}
-{"id": "3956.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & \\beta \\\\ & \\gamma \\end{matrix} \\bigg | \\ , 1 \\bigg ] _ { c } = { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & \\beta \\\\ & \\gamma \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { \\Gamma ( \\gamma ) \\Gamma ( \\gamma + a - \\beta ) } { \\Gamma ( \\gamma + a ) \\Gamma ( \\gamma - \\beta ) } . \\end{align*}"}
-{"id": "8225.png", "formula": "\\begin{align*} - \\mathcal Q _ { \\infty } \\varphi ( x _ { j } ) \\le \\frac { F ^ { 2 } ( \\nabla \\varphi ( x _ { j } ) ) \\mathcal Q _ { 2 } \\varphi ( x _ { j } ) } { p _ { j } - 2 } + \\left ( \\frac { \\lambda _ { 2 } ( p _ { j } , \\Omega ) ^ { \\frac { 1 } { p _ { j } - 4 } } | u _ { j } ( x _ { j } ) | } { F ( \\nabla \\varphi ( x _ { j } ) ) } \\right ) ^ { p _ { j } - 4 } \\frac { u _ { j } ( x _ { j } ) ^ { 3 } } { p _ { j } - 2 } = : \\ell _ { j } . \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} + \\frac { B _ { 2 } ^ { \\prime } q _ { 3 } } { B _ { 2 } } \\left ( f ^ { \\prime } \\right ) ^ { 2 } = - B _ { 2 } ^ { \\prime } . \\end{align*}"}
-{"id": "7454.png", "formula": "\\begin{gather*} \\rho _ 1 ( a ) = \\min \\left \\{ | \\lambda - \\mu | \\mid \\lambda \\neq \\mu , \\lambda , \\mu \\in \\mathrm { s u p p } ( a ) \\right \\} \\\\ \\rho _ 2 ( a ) = \\min \\left \\{ | \\lambda - \\mu | - 1 \\mid | \\lambda - \\mu | > 1 , \\lambda , \\mu \\in \\mathrm { s u p p } ( a ) \\right \\} \\\\ \\rho ( a ) = \\frac { 1 } { 2 } \\min ( \\rho _ 1 ( a ) , \\rho _ 2 ( a ) , 1 ) > 0 . \\end{gather*}"}
-{"id": "1894.png", "formula": "\\begin{align*} x ^ 2 f '' ( x ) - m x f ' ( x ) + m f ( x ) = - A \\cdot P ( x - \\lambda ) ^ { n - 1 } + B \\cdot x ^ 2 \\cdot P ( x - \\lambda ) ^ { n - 2 } \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} f \\circ ( x + \\varepsilon ) = \\hat { f } ( x + \\varepsilon ) = \\sum _ { i = 0 } ^ { \\infty } \\frac { \\hat { f } ^ { ( i ) } ( x ) } { i ! } \\varepsilon ^ { i } . \\end{align*}"}
-{"id": "8028.png", "formula": "\\begin{align*} \\partial _ t w + A w + \\mathbb P [ ( u _ \\infty + u _ s + \\widetilde U ) \\cdot \\nabla w + w \\cdot \\nabla ( u _ s + \\widetilde U ) + w \\cdot \\nabla w ] = \\mathbb P f \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{gather*} t H _ { \\mathrm { K F S } } ^ { \\frac 3 2 + \\frac 4 3 } \\left ( \\theta ^ \\infty _ 1 ; t ; { q _ 1 , p _ 1 \\atop q _ 2 , p _ 2 } \\right ) = t H _ { \\mathrm { I I I } ( D _ 7 ) } \\big ( \\theta ^ \\infty _ 1 ; t ; q _ 1 , p _ 1 \\big ) \\\\ \\qquad { } + t H _ { \\mathrm { I I I } ( D _ 7 ) } \\big ( 1 - \\theta ^ \\infty _ 1 ; t ; q _ 2 , p _ 2 \\big ) - p _ 1 q _ 1 p _ 2 q _ 2 - t \\left ( \\frac { p _ 2 } { q _ 1 } + p _ 1 + p _ 2 \\right ) . \\end{gather*}"}
-{"id": "4911.png", "formula": "\\begin{align*} \\sigma = \\left ( \\lambda _ { 1 } , \\lambda _ { 2 } , \\ldots , \\lambda _ { n } \\right ) \\end{align*}"}
-{"id": "406.png", "formula": "\\begin{align*} \\tau : = \\underset { y \\in \\mathcal { Y } } { \\min } ~ \\tau ( y ) , \\ \\ \\ Y : = \\{ y \\in \\mathcal { Y } | \\ \\tau ( y ) = \\tau \\} . \\end{align*}"}
-{"id": "9442.png", "formula": "\\begin{align*} \\widetilde { \\Lambda } ( \\rho , W , Q ; s , t ) = \\sup \\left \\{ \\int _ s ^ t \\int _ { \\mathbb { T } ^ 3 } a \\rho + A \\cdot W , \\ ; \\ ; \\ ; a + \\frac { 1 } { 2 } | \\sqrt { Q ^ { - 1 } } A | ^ 2 \\leq 0 \\right \\} \\in [ 0 , + \\infty ] , \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} { p _ { o u t , K } } = \\Pr \\left ( { { I _ K } < \\mathcal R } \\right ) = F _ { { I _ K } } ( \\mathcal R ) . \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{align*} f _ m = f _ m ( q ) : = \\frac { q ^ { \\frac { m ( m + 1 ) } { 2 } } } { \\left ( q ^ 2 ; q ^ 2 \\right ) _ m } . \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} F _ * ^ e ( \\lambda x _ 1 ^ { k d _ 1 + \\alpha _ 1 } \\dots x _ n ^ { k d _ n + \\alpha _ n } ) = x _ 1 ^ { c _ 1 } \\dots x _ n ^ { c _ n } F _ * ^ e ( \\lambda x _ 1 ^ { s _ 1 } \\dots x _ n ^ { s _ n } ) \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{align*} \\Pr \\left ( \\frac { q _ i } { q _ j } = c \\right ) = 0 \\end{align*}"}
-{"id": "3152.png", "formula": "\\begin{align*} \\mu _ { N + 1 } \\Lambda _ N ^ { N + 1 } = \\mu _ N \\ , \\forall N \\ge 1 , \\end{align*}"}
-{"id": "5435.png", "formula": "\\begin{align*} d _ m \\partial _ { \\nu } m = \\frac { \\alpha _ 1 L _ { o x } } { k _ 1 + L _ { o x } } + \\frac { \\alpha _ 3 M _ 1 ^ { p _ 1 } } { ( k _ 3 + M _ 2 ) ( k _ 4 ^ { p _ 1 } + M _ 1 ^ { p _ 1 } ) } \\ \\ \\Omega ( p _ 1 > 1 ) . \\end{align*}"}
-{"id": "7484.png", "formula": "\\begin{align*} \\langle a , b \\rangle : = \\tau ( a ^ * b ) \\end{align*}"}
-{"id": "9671.png", "formula": "\\begin{align*} \\begin{aligned} F ( \\pi - \\Theta ( 0 , \\phi , \\psi ) , \\phi , \\psi ) & = - F ( 0 , \\phi , \\psi ) , \\\\ G ( \\pi - \\Theta ( 0 , \\phi , \\psi ) , \\phi , \\psi ) & = - H ( 0 , \\phi , \\psi ) , \\\\ H ( \\pi - \\Theta ( 0 , \\phi , \\psi ) , \\phi , \\psi ) & = G ( 0 , \\phi , \\psi ) . \\end{aligned} \\end{align*}"}
-{"id": "9588.png", "formula": "\\begin{align*} \\widehat \\Lambda _ n - \\Lambda _ n = \\left ( \\frac { 1 } { N } \\sum _ { i = 1 } ^ n R _ { n , i } X _ { n , i } Z _ { n , i } ^ { \\prime } \\right ) \\left ( \\frac { 1 } { N } \\sum _ { i = 1 } ^ n R _ { n , i } Z _ { n , i } Z _ { n , i } ^ { \\prime } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{align*} _ 3 F _ 2 \\left ( \\left . \\begin{array} { c } \\frac { 1 } { 3 } , \\frac { 1 } { 2 } , \\frac { 2 } { 3 } \\\\ 1 , 1 \\\\ \\end{array} \\right | \\frac { 2 7 u ^ { 2 } } { 4 ( 1 - u ) ^ { 3 } } \\right ) = ( 1 - u ) \\sum _ { n = 0 } ^ \\infty \\frac { D _ n } { 4 ^ n } u ^ n . \\end{align*}"}
-{"id": "4646.png", "formula": "\\begin{align*} \\rho _ { I , X } ( t ) = \\prod _ { p \\in X \\setminus I } \\left ( 1 - \\frac { t } { p } \\right ) ^ 2 , t \\in I , \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle \\left ( \\partial ^ { 2 } \\theta ^ { k } _ { \\phi } , \\ , \\eta \\right ) + \\left ( \\nabla \\widetilde { \\theta } ^ { k } _ { \\phi } , \\ , \\nabla \\eta \\right ) = J _ 1 ^ { k } ( \\eta ) + J _ 2 ^ { k } ( \\eta ) + J _ 3 ^ { k } ( \\eta ) . } \\end{array} \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} \\min _ { x \\in \\R ^ n } \\frac { 1 } { m } \\sum _ { i = 1 } ^ m f _ i ( x ) + g ( x ) , \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{align*} \\ddot { \\phi } \\dot { \\phi } + \\left ( \\dfrac { \\dot { a } } { a } + 2 \\dfrac { \\dot { b } } { b } \\right ) \\dot { \\phi } ^ { 2 } = 0 \\end{align*}"}
-{"id": "9850.png", "formula": "\\begin{align*} \\tau \\tilde R ^ { \\mathbf { \\gamma } } ( x ) \\tau \\tilde R ^ { \\mathbf { \\gamma } } ( x ^ { - 1 } ) = \\left ( \\frac { - v + x } { 1 - v x } \\right ) \\left ( \\frac { - v + x ^ { - 1 } } { 1 - v x ^ { - 1 } } \\right ) I = I . \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{align*} f ^ { ( v ) } ( k , x ) = ( i k ) ^ v e ^ { i k x } [ 1 + o ( 1 ) ] , x \\to + \\infty , v = 0 , 1 . \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} \\{ \\left ( \\chi _ { - j _ k } \\right ) ^ { m _ k } \\dots \\left ( \\chi _ { - j _ 2 } \\right ) ^ { m _ 2 } \\left ( \\chi _ { - j _ 1 } \\right ) ^ { m _ 1 } | 0 \\rangle \\ \\arrowvert \\ \\ j _ k > \\dots > j _ 2 > j _ 1 > 0 , \\ j _ i \\in \\mathbb { Z } + \\frac { 1 } { 2 } , \\ m _ i > 0 , m _ i \\in \\mathbb { Z } , \\ i = 1 , 2 , \\dots , k \\} . \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} \\mathbb E | M _ { \\infty } - M ^ n _ { \\infty } | ^ p \\eqsim _ { p } \\mathbb E [ M - M ^ n ] _ { \\infty } ^ { \\frac p 2 } = \\mathbb E \\Bigl ( [ M ^ a - M ^ n ] _ { \\infty } + [ M ^ q ] _ { \\infty } \\Bigr ) ^ { \\frac p 2 } \\geq \\mathbb E [ M ^ q ] _ { \\infty } ^ { \\frac p 2 } , \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{align*} s _ { n , k } ^ { ( - 1 ) } & : = \\sum _ { d | n } p ( d - k ) \\mu ( n / d ) . \\end{align*}"}
-{"id": "9998.png", "formula": "\\begin{align*} W ( G _ { 1 } ( n , d , x , 1 ) ) & = W ( G _ { 1 } ( n , d , x , 0 ) ) + 1 , \\\\ W ( G _ { 1 } ( n , d , x , 2 ) ) & = W ( G _ { 1 } ( n , d , x , 0 ) ) + 6 , \\\\ W ( G _ { 1 } ( n , d , x , - 1 ) ) & = W ( G _ { 1 } ( n , d , x , 0 ) ) + 3 . \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} 0 = \\sum _ { k _ 1 + k _ 2 = k + 1 } \\sum _ { i = 0 } ^ { k _ 1 - 1 } ( - 1 ) ^ * \\frak m _ { k _ 1 } ( x _ 1 , \\dots , x _ i , \\frak m _ { k _ 2 } ( x _ { i + 1 } , \\dots , x _ { k _ 2 } ) , \\dots , x _ k ) , \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} ( g , h ) + ( g ' , h ) & = ( g + g ' , h ) \\\\ ( g , h ) + ( g , h ' ) & = ( g , h + h ' ) \\\\ r \\ , ( g , h ) & = ( r \\ , g , h ) \\\\ r \\ , ( g , h ) & = ( g , r \\ , h ) . \\end{align*}"}
-{"id": "5454.png", "formula": "\\begin{align*} \\mu _ { j } [ \\rho ] = \\inf _ { \\substack { V \\leq H ^ m ( \\Omega ) \\\\ { \\rm d i m } V = j } } \\sup _ { \\substack { 0 \\ne u \\in V } } \\frac { \\int _ { \\Omega } | D ^ m u | ^ 2 d x } { \\int _ { \\Omega } \\rho u ^ 2 d x } . \\end{align*}"}
-{"id": "1145.png", "formula": "\\begin{align*} \\gamma _ 1 = x _ 1 , \\gamma _ k = [ \\gamma _ { k - 1 } , x _ k ] = [ x _ 1 , \\ldots , x _ k ] , \\end{align*}"}
-{"id": "6723.png", "formula": "\\begin{align*} \\eta u \\left ( y \\right ) + \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert \\leq 2 l } b _ { \\alpha } \\left ( y \\right ) D ^ { \\alpha } u \\left ( y \\right ) = f \\left ( y \\right ) \\end{align*}"}
-{"id": "1348.png", "formula": "\\begin{align*} \\bar { \\varphi } _ { x } ( U _ { x } \\times \\Lambda _ { x } \\times D ^ { k } \\times \\{ 0 \\} ) \\bigcap \\varphi ^ { i } _ { \\alpha } ( U ^ { i } _ { \\alpha } \\times \\Lambda ^ { i } _ { \\alpha } \\times D ^ { k } \\times \\{ 0 \\} ) \\ ; = \\ ; \\emptyset , \\end{align*}"}
-{"id": "8539.png", "formula": "\\begin{align*} Q ( \\alpha ) = \\partial \\bar { \\partial } \\alpha \\otimes \\alpha - \\partial \\alpha \\otimes \\bar { \\partial } \\alpha = \\alpha ^ 2 \\otimes \\partial \\bar { \\partial } \\log ( \\alpha ) . \\end{align*}"}
-{"id": "689.png", "formula": "\\begin{align*} \\vert n _ 1 , n _ 2 \\rangle \\longrightarrow f _ { n _ 1 , n _ 2 } ( \\eta _ 1 , \\eta _ 2 ) = c _ { n _ 1 , n _ 2 } { { \\eta _ 1 } } ^ { n _ 1 } { { \\eta _ 2 } } ^ { n _ 2 } a _ i ^ + \\longrightarrow { { \\eta _ i } } . \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} \\begin{cases} i \\dd _ t \\psi _ j ( t ) = A \\psi _ j ( t ) + u ( t ) B \\psi _ j ( t ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ & t \\in ( 0 , T ) , \\ T > 0 , \\\\ \\psi _ j ( 0 ) = \\psi _ j ^ 0 \\in L ^ 2 ( ( 0 , 1 ) , \\C ) , \\ \\ & j \\in \\N ^ * . \\\\ \\end{cases} \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty \\ ! [ \\widehat { h } ( k ) + \\widehat { h } ( - k ) S ( k ) ] \\widehat { h } ( k ) ^ \\dag d k + 2 \\pi \\sum _ { j = 1 } ^ N [ \\widehat { h } ( - i k _ j ) C _ j ] [ \\widehat { h } ( - i k _ j ) C _ j ] ^ \\dag \\ ! = 0 . \\end{align*}"}
-{"id": "7260.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 1 e _ 4 + \\alpha _ 2 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 3 e _ 3 + \\alpha _ 4 e _ 4 + \\alpha _ 5 e _ 5 , [ e _ 2 , e _ 1 ] = - \\alpha _ 3 e _ 3 + \\beta _ 1 e _ 4 + \\beta _ 2 e _ 5 , \\\\ [ e _ 2 , e _ 2 ] = \\beta _ 3 e _ 4 + \\beta _ 4 e _ 5 , [ e _ 1 , e _ 3 ] = \\beta _ 5 e _ 4 + \\beta _ 6 e _ 5 , [ e _ 2 , e _ 3 ] = \\gamma _ 1 e _ 4 + \\gamma _ 2 e _ 5 , \\\\ [ e _ 3 , e _ 1 ] = \\gamma _ 3 e _ 4 + \\gamma _ 4 e _ 5 , [ e _ 3 , e _ 2 ] = \\gamma _ 5 e _ 4 + \\gamma _ 6 e _ 5 , [ e _ 3 , e _ 3 ] = \\gamma _ 7 e _ 4 + \\gamma _ 8 e _ 5 . \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{align*} \\partial _ t \\hat { g } _ k = L _ k \\cdot \\hat { g } _ k + \\widehat { N ( g ) } _ k , \\end{align*}"}
-{"id": "9200.png", "formula": "\\begin{align*} \\mathbb E \\| M ^ c _ t \\| ^ p - \\beta _ { p , X } ^ p \\mathbb E \\| M _ t \\| ^ p \\leq \\mathbb E U ( M _ t , M ^ c _ t ) = \\mathbb E V ( M ^ d + 2 M ^ c , - M ^ d ) \\leq 0 \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} \\iint _ { ( 0 , T ) \\times ( 0 , R ) } v _ l \\bigl ( \\vartheta _ t + ( b - \\lambda _ l ) \\vartheta _ x + \\vartheta _ { x x x } \\bigr ) \\ , d x d t + \\int _ 0 ^ R v _ { 0 l } \\vartheta \\big | _ { t = 0 } \\ , d x = 0 . \\end{align*}"}
-{"id": "9570.png", "formula": "\\begin{align*} E \\big [ \\widehat { \\theta } \\ , | \\ , { \\mathbf { R } } , N _ 1 , N _ 0 \\big ] = \\theta ^ { \\mathrm { c a u s a l } , \\mathrm { s a m p l e } } , \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{align*} T = N + ( d ^ c F ) ^ { 2 , 0 } , t = \\frac 1 3 d ^ c F . \\end{align*}"}
-{"id": "9126.png", "formula": "\\begin{gather*} \\dim H ^ n _ c ( V _ \\lambda ) ^ { G _ { \\max } } = \\dim H ^ n ( V _ \\lambda ) ^ { G _ { \\max } } . \\end{gather*}"}
-{"id": "9950.png", "formula": "\\begin{align*} ( A ^ { - 1 } E ) ^ k = T \\left [ \\begin{array} { c c } J ^ { - k } & 0 \\\\ 0 & N ^ k \\end{array} \\right ] T ^ { - 1 } \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} \\pi _ 2 ( M ) = \\pi _ 2 ( M ^ a ) = ( \\pi _ 2 ( M ) ) ^ a = G _ 1 ^ a = G _ 2 , \\end{align*}"}
-{"id": "10019.png", "formula": "\\begin{align*} t = \\frac { x _ 1 } { 2 R } + \\tau \\end{align*}"}
-{"id": "8093.png", "formula": "\\begin{align*} \\vartheta _ 1 ' ( 0 ) = \\vartheta _ 2 ( 0 ) \\vartheta _ 3 ( 0 ) \\vartheta _ 4 ( 0 ) . \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} ( K + K ^ { - \\epsilon } ) ^ { - \\frac { \\epsilon } { p } + \\sum _ { i = 1 } ^ m \\frac { \\epsilon } { p _ i } } \\le C . \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{align*} \\| x _ j \\| _ 2 ^ 2 = \\Big ( \\Big \\| \\sum _ { \\substack { k = 1 \\\\ k \\ne j } } ^ t P _ { j k } x _ k \\Big \\| _ 2 \\Big ) ^ 2 \\le \\Big ( \\sum _ { \\substack { k = 1 \\\\ k \\ne j } } ^ t \\| P _ { j k } \\| _ 2 \\| x _ k \\| _ 2 \\Big ) ^ 2 \\le ( t - 1 ) \\sum _ { \\substack { k = 1 \\\\ k \\ne j } } ^ t \\| P _ { j k } \\| _ 2 ^ 2 \\| x _ k \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} { \\displaystyle \\Psi ( \\mathbf { x } , t ) = 0 , \\mathbf { A } ( \\mathbf { x } , t ) \\times \\mathbf { n } = 0 , \\nabla \\cdot \\mathbf { A } ( \\mathbf { x } , t ) = 0 , ( \\mathbf { x } , t ) \\in \\partial \\Omega \\times ( 0 , T ) , } \\end{align*}"}
-{"id": "346.png", "formula": "\\begin{align*} \\tilde { \\chi } _ a ( \\mathsf { Q } ) : = \\sum _ i c ^ i _ i q ^ { - ( \\alpha _ a - 2 \\rho , \\lambda _ i ) } [ d _ a ^ { - 1 } ( \\alpha _ a , \\lambda _ i ) ] _ { q _ a } . \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} L _ { c _ * } ' = 4 - 1 2 { \\rm s e c h } ^ 2 ( \\sqrt { c _ * } \\xi ) + 1 2 \\sqrt { c _ * } \\xi \\ ; { \\rm s e c h } ^ 2 ( \\sqrt { c _ * } \\xi ) \\ ; \\tanh ( \\sqrt { c _ * } \\xi ) \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} R _ { k j } ( \\alpha , s , h ) : = \\{ z | \\ , \\Im ( z / e _ k ) + \\mu _ { k j } \\arg z \\in [ \\alpha , \\alpha + s ] , \\ , | \\Re ( z / e _ k ) + \\mu _ { k j } \\log | z | | \\leq h \\} , \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} S ( X / e , \\mathbf { d } , \\mathbf { D } ; e L _ 1 , e L _ 2 , e L _ 3 ) = \\sum _ { \\mathbf { x } \\in \\Lambda _ { e } ( \\mathbf { D } ) \\cap ( X / e ) \\mathcal { R } } \\tau \\left ( \\frac { e L _ 1 ( \\mathbf { x } ) } { d _ 1 } \\right ) \\tau \\left ( \\frac { e L _ 2 ( \\mathbf { x } ) } { d _ 2 } \\right ) \\tau \\left ( \\frac { e L _ 3 ( \\mathbf { x } ) } { d _ 3 } \\right ) , \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{align*} E _ i \\triangleright ( a b ) = ( E _ i \\triangleright a ) ( K _ i \\triangleright b ) + a ( E _ i \\triangleright b ) = ( E _ i \\triangleright a ) b + a ( E _ i \\triangleright b ) , \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} A : = u _ n ( c - u ) + \\left ( \\frac { 1 } { 2 } | u | ^ 2 - c \\cdot u \\right ) e _ n , \\end{align*}"}
-{"id": "9310.png", "formula": "\\begin{align*} \\varphi ( p ) \\ge \\min \\{ \\varphi ( 0 ) , \\varphi ( 1 / 2 ) , \\varphi ( 1 ) \\} = 0 , \\end{align*}"}
-{"id": "9547.png", "formula": "\\begin{align*} V \\pi _ \\alpha ( a ) V ^ * ( \\xi \\otimes \\delta _ t ) & = V \\pi _ \\alpha ( a ) ( v _ { t ^ { - 1 } } ^ * \\xi \\otimes \\delta _ t ) = V ( \\alpha _ { t ^ { - 1 } } ( a ) v _ { t ^ { - 1 } } ^ * \\xi \\otimes \\delta _ t ) \\\\ & = v _ { t ^ { - 1 } } \\alpha _ { t ^ { - 1 } } ( a ) v _ { t ^ { - 1 } } ^ * \\xi \\otimes \\delta _ t = \\beta _ { t ^ { - 1 } } ( a ) \\xi \\otimes \\delta _ t \\\\ & = \\pi _ \\beta ( a ) ( \\xi \\otimes \\delta _ t ) , \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\left \\lfloor ( k + 1 ) / 2 \\right \\rfloor } & w _ { k + 2 - 2 j , k + 1 } - \\sum _ { j = 1 } ^ { \\left \\lfloor k / 2 \\right \\rfloor } w _ { k + 1 - 2 j , k + 1 } \\\\ & = \\sum _ { j = 1 } ^ { \\left \\lfloor ( k + 1 ) / 2 \\right \\rfloor } ( w _ { k + 2 - 2 j , k + 1 } - w _ { k + 1 - 2 j , k + 1 } ) \\\\ & \\geq 0 , \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} \\Psi = \\left ( \\left ( \\psi _ { j _ 1 \\dots j _ n } ^ { ( s , Q ) } \\right ) _ { j _ 1 , \\dots , j _ n = 1 } ^ { | \\mathcal { E } | } \\right ) _ { Q \\in S _ n } \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} d _ H ( C , C ' ) : = \\max \\bigl ( \\delta ( C , C ' ) , \\delta ( C ' , C ) \\bigr ) . \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} v = v ( \\xi ) : = \\Delta \\xi ' . \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} \\dfrac { p _ j ( A x ) } { p _ j ( x ) } = p _ { j } ( A x _ 0 ) \\leqslant \\sup _ { p _ { j } ( z ) = 1 } p _ { j } ( A z ) = p _ { j } ^ { X } ( A ) \\end{align*}"}
-{"id": "9627.png", "formula": "\\begin{align*} G ^ { ( j ) } _ n = | g _ 1 ( j ) \\ , g _ 2 ( j ) \\ , \\cdots \\ , g _ n ( j ) | , \\ \\ G ^ { ( j ) } _ 0 = 1 . \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} \\Delta _ K ( t ) = t ^ 8 - 3 t ^ 7 + 5 t ^ 6 - 7 t ^ 5 + 9 t ^ 4 - 7 t ^ 3 + 5 t ^ 2 - 3 t + 1 \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} H _ \\epsilon '' = \\frac { v _ \\epsilon ^ { ( 4 ) } } { v _ \\epsilon '' } - \\frac { ( v _ \\epsilon ''' ) ^ 2 } { ( v _ \\epsilon '' ) ^ 2 } - \\frac { v _ \\epsilon ''' } { v _ \\epsilon ' - a _ t } + \\frac { ( v _ \\epsilon '' ) ^ 2 } { ( v _ \\epsilon ' - a _ t ) ^ 2 } + \\frac { v _ \\epsilon ''' } { b _ t - v _ \\epsilon ' } + \\frac { ( v _ \\epsilon '' ) ^ 2 } { ( b _ t - v _ \\epsilon ' ) ^ 2 } , \\end{align*}"}
-{"id": "9830.png", "formula": "\\begin{align*} \\omega _ { \\psi } ^ { \\mathbf { z } ^ { - 1 } } ( \\pi _ { \\mathbf { z } ^ { - 1 } } ( t _ { - \\lambda } ) \\Phi _ { w } ^ { \\mathbf { z } ^ { - 1 } } ) = \\mathcal { T } _ w \\ , \\omega _ { \\psi } ^ { \\mathbf { z } ^ { - 1 } } ( \\pi _ { \\mathbf { z } ^ { - 1 } } ( t _ { - \\lambda } ) \\Phi _ { 1 } ^ { \\mathbf { z } ^ { - 1 } } ) \\end{align*}"}
-{"id": "9642.png", "formula": "\\begin{align*} - \\tau ^ n - \\tau ^ { - n } = T \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} \\boldsymbol { c } _ i = \\mathbb { I } _ { | \\mathcal { E } | ^ { i - 1 } } \\otimes \\boldsymbol { c } \\otimes \\mathbb { I } _ { | \\mathcal { E } | ^ { n - i - 1 } } , \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{align*} H ( t ) = e ^ { - t A } w _ 0 + \\int _ 0 ^ t e ^ { - ( t - \\tau ) A } \\mathbb P f ( \\tau ) d \\tau , \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} \\Bigl ( \\mathbb E \\Bigl \\| \\sum ^ N _ { n = 1 } \\varepsilon _ n d f _ n \\Bigr \\| ^ p \\Bigr ) ^ { \\frac 1 p } \\geq \\gamma _ { p , X } ^ { \\delta } ( \\mathbb E \\| f _ N \\| ^ p ) ^ { \\frac 1 p } . \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} b ( x _ 0 \\otimes x _ 1 \\otimes \\dots \\otimes x _ n ) & = x _ 0 x _ 1 \\otimes x _ 2 \\otimes \\dots x _ n \\\\ & + \\sum _ { 0 } ^ { n - 1 } ( - 1 ) ^ i x _ 0 \\otimes \\dots \\otimes x _ i x _ { i + 1 } \\otimes \\dots \\otimes x _ n \\\\ & + ( - 1 ) ^ n x _ n x _ 0 \\otimes x _ 2 \\otimes \\dots \\otimes x _ { n - 1 } . \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{align*} B K _ - B ^ { - 1 } = - 2 \\alpha e ^ { - \\beta } \\cdot K _ 0 + e ^ { - \\beta } \\cdot K _ - + e ^ { - \\beta } \\alpha ^ 2 \\cdot K _ + \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} x ^ \\prime + \\theta y ^ \\prime = ( x + \\theta y ) ( u + \\theta v ) . \\end{align*}"}
-{"id": "7028.png", "formula": "\\begin{align*} K _ S = f ^ * K _ { \\SS } + R = f ^ * E + R = Z + R . \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} I ^ { \\mathfrak { m } } _ L = \\prod _ { v \\mid \\ell } U _ v ^ { ( 1 + e _ { v } \\ell / ( \\ell - 1 ) ) } \\times \\prod _ { v \\mid v _ 0 \\in \\Sigma _ { E , p . m . } } U _ v ^ { ( 1 ) } \\times \\prod _ { v \\not \\in T } U _ v , \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} \\mathbb { T } ^ { ( i ) } \\left ( \\mathcal { A } _ { j _ 1 \\dots j _ n } \\right ) _ { j _ 1 , \\dots , j _ n = 1 } ^ { | \\mathcal { E } | } = \\left ( \\mathcal { A } _ { j _ 1 \\dots j _ n } \\right ) _ { j _ 1 , \\dots , j _ { i - 1 } , j _ { i + 1 } , j _ { i } , j _ { i + 2 } , \\dots , j _ n = 1 } ^ { | \\mathcal { E } | } \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} c ( X _ 0 ) & = ( 1 + H ) ( 1 + H + 3 \\pi ^ * L ) ( 1 + H + 2 \\pi ^ * L ) c ( B ) \\\\ c ( Y _ 0 ) & = ( 3 H + 6 \\pi ^ * L ) \\frac { c ( X _ 0 ) } { 1 + 3 H + 6 \\pi ^ * L } . \\end{align*}"}
-{"id": "9865.png", "formula": "\\begin{align*} C _ 1 = \\frac { 1 } { 2 } L \\Leftrightarrow G ( \\rho ) \\rightarrow - \\frac { 1 } { 2 } ( r - L ) \\rho \\rightarrow \\infty \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} F _ { \\alpha } ( X ) = \\sum _ { \\mathrm { f l a t } ( b ) \\ \\mathrm { r e f i n e s } \\ \\alpha } X ^ { b } . \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} { \\displaystyle \\mathbf { E } ^ { 0 , h _ 0 } = - \\nabla \\phi ^ { 0 , h _ 0 } \\stackrel { \\mathrm { w } } { \\rightharpoonup } - \\nabla \\widetilde { \\phi } ^ 0 = \\widetilde { \\mathbf { E } } ^ 0 \\hbox { w e a k l y \\ , \\ , \\ , i n } ( L ^ 2 ( \\Omega ) ) ^ 3 \\hbox { a s } h _ 0 \\rightarrow 0 . } \\end{align*}"}
-{"id": "10051.png", "formula": "\\begin{align*} \\mathrm { T } ^ { 0 } ( X , Y ) = - \\frac { 1 } { 2 } ( \\nabla ^ { \\mathrm { g } } _ X J _ { \\varphi } ) J _ { \\varphi } Y + \\frac { 1 } { 2 } ( \\nabla ^ { \\mathrm { g } } _ Y J _ { \\varphi } ) J _ { \\varphi } X , \\forall X , Y \\in { \\mathfrak X } ( M ) . \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{align*} f ( q ) = \\frac { 2 } { ( q ) _ \\infty } \\sum _ { n \\in \\Z } \\frac { ( - 1 ) ^ n q ^ { \\frac { n ( 3 n + 1 ) } { 2 } } } { 1 + q ^ n } , \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} [ h _ \\chi ( z ) , H ^ { \\beta } ( w ^ 2 ) ] = - \\frac { 1 } { z ^ 2 } H ^ { \\beta } ( w ^ 2 ) , [ h _ \\chi ( z ) , H ^ { \\gamma } ( w ^ 2 ) ] = \\frac { 1 } { z ^ 2 } H ^ { \\gamma } ( w ^ 2 ) . \\end{align*}"}
-{"id": "9577.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } \\theta ^ { \\mathrm { c a u s a l } } _ n \\\\ \\gamma ^ { \\mathrm { c a u s a l } } _ n \\end{array} \\right ) = \\left ( \\begin{array} { c c } \\Omega ^ { X X } _ n & \\Omega ^ { X Z } _ n \\\\ \\Omega ^ { Z X } _ n & \\Omega ^ { Z Z } _ n \\end{array} \\right ) ^ { - 1 } \\left ( \\begin{array} { c } \\Omega ^ { X Y } _ n \\\\ \\Omega ^ { Z Y } _ n \\end{array} \\right ) . \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} H ( \\psi , \\bar \\psi ) & = H _ 0 ( \\psi , \\bar \\psi ) + N ( \\psi , \\bar \\psi ) , \\end{align*}"}
-{"id": "7923.png", "formula": "\\begin{align*} h ( t ) = 1 \\quad \\mbox { o n $ [ T _ 0 , \\infty ) $ f o r s o m e $ T _ 0 > 0 $ } . \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} d \\big ( e _ { s ( \\alpha ) } \\otimes e _ { t ( \\alpha ) } \\big ) = \\alpha \\otimes e _ { t ( \\alpha ) } - e _ { s ( \\alpha ) } \\otimes \\alpha \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} \\sigma ^ 2 _ { \\mu , X } = + \\infty \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{align*} \\sum _ { j \\ne i } ^ { } \\left ( \\frac { 2 ( 1 + x _ i x _ j ) } { x _ i - x _ j } \\right ) + ( 2 N - 2 ) x _ i = \\sum _ { j \\ne i } ^ { } \\left ( \\frac { 2 ( 1 + x ^ 2 _ i ) } { x _ i - x _ j } \\right ) , \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\int _ { B _ R ( 0 ) \\cap S } \\eta \\ , d x ' = \\int _ { \\mathbb R } \\eta \\ , d x ' = 0 , \\end{align*}"}
-{"id": "9391.png", "formula": "\\begin{align*} \\norm { M _ \\Delta } = \\norm { \\Delta } _ \\infty . \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} F ( \\theta , \\phi ) = \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\cosh \\rho \\cos \\phi , \\frac { 1 } { 2 } \\sinh \\rho \\sin \\theta \\sin \\phi , \\frac { 1 } { 2 } \\sinh \\rho \\cos \\theta \\sin \\phi \\right ) \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} \\rho ( x , y ) = \\frac { 1 } { \\sqrt { M ^ 2 } } \\sum _ { p = 0 } ^ { M - 1 } \\sum _ { q = 0 } ^ { M - 1 } e ^ { i \\frac { 2 \\pi } { M } ( p x + q y ) } \\ ; | \\hat { \\rho } ( p , q ) | e ^ { i \\phi ( p , q ) } . \\end{align*}"}
-{"id": "8068.png", "formula": "\\begin{align*} { \\cal S } _ n ( \\widehat { B } ^ + , \\widehat { B } ^ - ) & : = \\{ \\pi \\in ( { \\cal S } _ n ( \\widehat { B } ^ + , \\widehat { B } ^ - ) ) ^ { ( \\lambda ^ n _ 0 - \\lambda _ 0 ) - } : \\pi ( \\lambda ^ n _ 0 - \\lambda _ 0 ) = y ^ n \\} \\\\ \\widehat { \\cal S } _ n ( \\widehat { B } ^ + , \\widehat { B } ^ - ) & : = \\{ \\widehat { \\pi } \\in \\widehat { \\cal S } _ n ( \\widehat { B } ^ + , \\widehat { B } ^ - ) : \\sigma _ { \\widehat { \\pi } } \\leq ( \\lambda ^ n _ 0 - \\lambda _ 0 ) \\} . \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{align*} I _ { \\alpha } ^ * & : = \\{ g _ { \\alpha } \\in I _ { \\alpha } : o ( g _ { \\alpha } ) = o ( g _ { \\alpha } \\psi _ { \\alpha , \\beta } ) \\beta < \\alpha \\} , \\\\ K _ { \\alpha } ^ * & : = \\{ a _ { \\alpha } \\in G _ { \\alpha } : a _ { \\alpha } \\psi _ { \\alpha , \\beta } = e _ { \\beta } \\beta < \\alpha \\} = \\bigcap _ { \\beta < \\alpha } K _ { \\alpha , \\beta } . \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} \\bigg ( { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\beta \\\\ & \\frac 1 2 + \\frac 1 2 ( \\alpha + \\beta ) \\end{matrix} \\bigg | \\ , z \\bigg ] \\bigg ) ^ 2 = { } _ 3 F _ 2 \\bigg [ \\begin{matrix} \\alpha & \\beta & \\frac 1 2 ( \\alpha + \\beta ) \\\\ & \\alpha + \\beta & \\frac 1 2 + \\frac 1 2 ( \\alpha + \\beta ) \\end{matrix} \\bigg | \\ , 4 z ( 1 - z ) \\bigg ] . \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{align*} ( G _ i ) _ { i , j } = \\left ( \\frac { E _ { 2 ( i - j + 1 ) } } { 4 ^ { 2 - j } } + \\frac { \\zeta ( 2 j - 2 i - 3 ) } { 2 j - 2 i - 3 } ( 1 - 4 ^ { i - j + 2 } ) \\right ) \\frac { 4 } { 1 - 4 ^ i } \\begin{cases} \\binom { 2 i - 1 } { 2 j - 3 } & j \\neq 1 \\\\ 1 / ( 2 i ) & \\end{cases} \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} D ( a ^ { ( 1 ) } \\dots a ^ { ( k ) } ) = \\sum _ { j = 1 } ^ { k } a ^ { ( 1 ) } \\dots a ^ { ( j - 1 ) } ( D a ^ { ( j ) } ) S ^ p a ^ { ( j + 1 ) } \\dots S ^ p a ^ { ( k ) } \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} ( Y _ \\alpha [ s ] ^ { ( r ) } ) _ 0 & = \\left . Y _ \\alpha [ s ] ^ { ( r ) } \\right | _ { u _ s = 0 } \\\\ & = \\left . _ w ( G [ s ] ( w ) ) X _ \\alpha ( w ^ { - 1 } ) ) ^ { ( r ) } \\right | _ { u _ s = 0 } \\\\ & = \\left . _ w \\left ( \\frac { G [ s - 1 ] ( w ) } { 1 + u _ { s } w ^ { s } G [ s - 1 ] ( w ) } X _ \\alpha ( w ^ { - 1 } ) \\right ) ^ { ( r ) } \\right | _ { u _ s = 0 } \\\\ & = _ w ( G [ s - 1 ] ( w ) X _ \\alpha ( w ^ { - 1 } ) ) ^ { ( r ) } \\\\ & = Y _ \\alpha [ s - 1 ] ^ { ( r ) } \\end{align*}"}
-{"id": "9199.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ d U _ { \\tilde x _ i , \\tilde x _ i } ( M _ { s - } ( \\omega ) , M ^ d _ { s - } ( \\omega ) ) \\| \\Phi ^ * ( s , \\omega ) \\tilde x _ i ^ * \\| ^ 2 \\leq 0 . \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} A _ 4 ( r ) = C _ u ^ { r - 1 } e ^ { - C _ d r } e ^ { 2 C _ d \\epsilon r } \\leq C ^ { r - 1 } e ^ { - C r } e ^ { 2 C \\epsilon r } e ^ { \\omega r } = \\frac { e ^ { - r } } { C } e ^ { - \\delta _ 1 r } \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} { \\cal X } ^ { f } _ { \\cdot } : = \\lim _ { n \\rightarrow + \\infty } { \\cal X } _ { \\cdot } ^ { n } \\ , \\ , { \\rm a n d } \\ , \\ , { \\cal X } ^ { ' f } _ { \\cdot } : = \\lim _ { n \\rightarrow + \\infty } { \\cal X } _ { \\cdot } ^ { ' n } \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} \\sum _ { t \\in \\mathbb F _ q } \\nu _ k ^ 2 ( t ) \\le \\frac { 1 } { q } \\left ( \\prod _ { j = 1 } ^ k | E _ j | \\right ) ^ 2 + q ^ { 2 d k - d } \\sum _ { r \\in \\mathbb F _ q } \\left | \\sum _ { { \\bf v } \\in S _ r } \\left ( \\prod _ { j = 1 } ^ k \\widehat { E _ j } ( { \\bf v } ) \\right ) \\right | ^ 2 . \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} 2 e ^ { ( n + 2 ) f } \\frac { \\int _ { \\partial \\Delta } e ^ { n f } d \\mu } { \\int _ \\Delta e ^ { ( n + 2 ) f } d v } = - \\sum _ { i , j = 1 } ^ n \\left ( e ^ { n f } H _ { i j } \\right ) _ { , i j } \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{align*} y ( t ) = \\frac { K } { \\alpha } \\left [ 1 + \\frac { 2 h _ { 1 } ( t _ { 0 } ) } { h _ { 2 } ( t _ { 0 } ) \\exp ( 2 \\alpha K t ) - h _ { 1 } ( t _ { 0 } ) } \\right ] \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} C ( P ) = \\mathrm { T r } \\left ( V ( 2 P - \\mathrm { I d } ) \\otimes P \\otimes P \\right ) . \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} \\begin{pmatrix} \\alpha _ i & \\beta _ i \\\\ \\gamma _ i & \\delta _ i \\end{pmatrix} h _ 1 = h _ 2 \\begin{pmatrix} \\alpha _ i & \\beta _ i \\\\ \\gamma _ i & \\delta _ i \\end{pmatrix} . \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{gather*} B ' : = \\{ b \\in B \\mid [ r , b ] = 0 , [ \\phi _ 0 , b ] = 0 \\} \\\\ M ' : = \\{ m \\in M \\mid [ r , m ] = - m , [ \\phi _ 0 ^ * , m ] = 0 \\} \\end{gather*}"}
-{"id": "7670.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } c _ { 1 , j } \\\\ \\vdots \\\\ c _ { k , j } \\end{array} \\right ) = B \\left ( \\begin{array} { c } b _ { 1 , j } \\\\ \\vdots \\\\ b _ { n , j } \\end{array} \\right ) . \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} { f _ T } \\left ( t \\right ) = \\frac { { { t ^ { m - 1 } } } } { { \\Gamma \\left ( m \\right ) } } { e ^ { - t } } , \\ , t \\ge 0 . \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} \\max \\{ - F ( \\nabla u ( x _ { 0 } ) ) - \\overline \\Lambda u ( x _ { 0 } ) , - \\mathcal Q _ { \\infty } u ( x _ { 0 } ) \\} = 0 . \\end{align*}"}
-{"id": "9660.png", "formula": "\\begin{align*} x ^ 2 + y ^ 2 - z ^ 2 = 0 , z < 0 . \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} 1 - | m _ { y _ 0 } ( x + i y ) | ^ 2 = 1 - \\frac { x ^ 2 + ( y - y _ 0 ) ^ 2 } { x ^ 2 + ( y + y _ 0 ) ^ 2 } = \\frac { 4 y _ 0 y } { x ^ 2 + ( y + y _ 0 ) ^ 2 } . \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{align*} A ( r ) = A ( r ) ^ \\dagger , ~ \\sigma ( \\widehat { n } ) ^ 2 = \\sigma _ 0 , ~ \\sigma ( \\widehat { n } ) ^ \\dagger = \\sigma ( \\widehat { n } ) \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} C _ { \\mu } ( h ^ { \\mu } \\left \\Vert F ^ { - 1 } A \\hat { u } \\right \\Vert _ { X } + \\sum \\limits _ { k = 1 } ^ { n } \\left \\Vert F ^ { - 1 } \\left [ \\left ( i \\xi _ { k } \\right ) ^ { l _ { k } } \\hat { u } \\right ] \\right \\Vert _ { X } + h ^ { - \\left ( 1 - \\mu \\right ) } \\left \\Vert F ^ { - 1 } \\hat { u } \\right \\Vert _ { X } ) . \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} \\left | \\sum _ { j = 1 } ^ N \\frac { k _ j \\lambda _ j } { 1 + \\sqrt { 1 + \\lambda _ j / c ^ 2 } } \\right | & \\leq \\frac { r \\lambda _ N } { 2 } \\leq \\frac { c ^ 2 } { 2 } , \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{align*} \\begin{cases} 1 \\le p < \\frac { N \\alpha } { 2 ( \\alpha + 1 ) } < \\frac { N \\alpha } { 2 } < q & \\alpha > \\frac { 2 } { N - 2 } \\\\ 1 = p < \\frac { N \\alpha } { 2 } < q & \\alpha = \\frac { 2 } { N - 2 } . \\\\ \\end{cases} \\end{align*}"}
-{"id": "23.png", "formula": "\\begin{align*} \\frac { d c _ 0 } { d \\tau } & = \\mu h _ 0 \\frac { b + c _ 0 } { 1 + c _ 0 } - \\frac { \\Gamma } { K _ 1 } \\frac { c _ 0 } { K + c _ 0 } + \\Lambda \\theta _ 0 \\\\ \\frac { d \\theta _ 0 } { d \\tau } & = 0 \\implies \\theta _ 0 = \\theta _ { 0 0 } = { \\rm c o n s t a n t , } \\ , \\ , \\ , \\frac { d h _ 0 } { d \\tau } = 0 \\implies h _ 0 = h _ { 0 0 } = { \\rm c o n s t a n t } . \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} \\Bigl ( \\mathbb E \\Bigl \\| \\sum ^ n _ { j = 1 } \\varepsilon _ j d _ j \\Bigr \\| ^ p \\Bigr ) ^ { \\frac 1 p } \\leq C \\Bigl ( \\mathbb E \\Bigl \\| \\sum ^ n _ { j = 1 } d _ j \\Bigr \\| ^ p \\Bigr ) ^ { \\frac 1 p } . \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} & \\left | \\int _ 0 ^ t \\langle ( h u _ s + \\widetilde U ) \\otimes w , \\nabla w \\rangle d \\tau \\right | \\\\ & \\leq \\frac { 1 } { 4 } \\int _ { T _ 1 } ^ t \\| \\nabla w \\| _ 2 ^ 2 d \\tau + C \\left ( | h | _ \\infty \\| u _ s \\| _ 3 + \\| v _ 0 \\| _ { 3 , \\infty } + M _ 3 \\right ) \\int _ 0 ^ { T _ 1 } \\| \\nabla w \\| _ 2 ^ 2 d \\tau \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} S _ D ^ { - 1 } \\Phi \\ ; = \\ ; a _ 0 ^ { ( S _ D ^ { - 1 } \\Phi ) } \\ , v _ 0 + a _ \\infty ^ { ( S _ D ^ { - 1 } \\Phi ) } \\ , v _ \\infty + b _ \\infty ^ { ( S _ D ^ { - 1 } \\Phi ) } \\ , v _ 0 + b _ 0 ^ { ( S _ D ^ { - 1 } \\Phi ) } \\ , v _ \\infty \\ , . \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} i \\partial _ { t } \\hat { u } \\left ( \\xi , t \\right ) + A _ { \\xi } \\hat { u } \\left ( \\xi , t \\right ) = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} \\delta b _ { n - 1 , n - 1 } = b _ { n , n - 1 } , \\delta b _ { n - 1 , n } = b _ { n n } , \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} C _ X ( a ) = \\frac { 1 } { \\vartheta _ X ^ \\star \\sqrt { 2 \\pi \\Lambda '' _ X ( \\vartheta _ X ^ \\star ) } } , \\end{align*}"}
-{"id": "9007.png", "formula": "\\begin{align*} t ^ { - 1 } \\int _ 0 ^ t \\phi \\phi _ t d t = \\frac { 1 } { 2 } t ^ { - 1 } ( \\phi ^ 2 ( t ) - \\phi ^ 2 ( 0 ) ) < \\frac { M _ 1 ^ 2 } { 2 } t ^ { 2 m - 1 } + C t ^ { - 1 } . \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\mathit { q } _ { 0 } \\right ] & = { \\nu } { \\zeta _ 1 } { } { R _ c ^ { 2 - \\alpha } } , & \\left [ \\mathit { q } _ { 0 } \\right ] & = 2 \\tilde { \\nu } { \\zeta _ 2 } { } { R _ c ^ { 2 ( 1 - \\alpha ) } } . \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} \\phi _ A ( e ^ { 2 \\pi i x } , e ^ { 2 \\pi i y } ) = ( e ^ { 2 \\pi i ( a x + b y ) } , e ^ { 2 \\pi i d y } ) \\ , , \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\Delta _ { A | M } ( ( \\mathcal { N } ( t ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) ) = \\Delta _ { A | M } ( \\hat { \\rho } _ { A M } ) \\ ; . \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} { \\cal P } _ { 2 n - 1 } : = \\left \\{ ( p _ 0 , \\dots , p _ { 2 n - 2 } ) \\in \\mathbb R ^ { 2 n - 1 } : p _ 0 + p _ 1 x + \\dots + p _ { 2 n - 2 } x ^ { 2 n - 2 } \\geq 0 , \\ \\ \\mathbb R \\right \\} , \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{align*} C _ j = P _ { 1 j } C _ j = P _ { 1 j } ( C _ j + I _ n - P _ { 1 j } ) : = P _ { 1 j } D _ j , \\end{align*}"}
-{"id": "825.png", "formula": "\\begin{align*} u ( t ) & = e ^ { t \\Delta } [ u _ 0 + t \\P \\theta _ 0 e _ 3 ] - B _ 1 ( u , u ) - B _ 2 ( u , \\theta ) \\\\ & = \\biggl ( \\int \\theta _ 0 \\Bigr ) t \\ , K ( t ) + t \\ , F ( t ) * V + e ^ { t \\Delta } u _ 0 - [ B _ 1 ( u , u ) + B _ 2 ( u , \\theta ) ] , \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} S _ j = \\{ i \\in W \\mid i \\mbox { a n d } j \\mbox { a r e m a t c h e d } \\} . \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} \\vert n \\rangle = \\sqrt { \\frac { n ! ( k - n ) ! } { k ! } } \\sum _ { \\sigma \\varepsilon S _ { k } } \\vert \\underbrace { - , - , \\cdots , - } _ { k - n } , \\underbrace { + , + , \\cdots , + } _ n \\rangle \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} & \\sup \\{ | f _ 1 ( \\lambda _ 0 ^ n - s ) - f _ 1 ( \\lambda _ 0 - s ) | \\vee | f _ 2 ( \\lambda _ 0 ^ n - s ) - f _ 2 ( \\lambda _ 0 - s ) | : s \\in [ 0 , \\lambda _ 0 - \\lambda _ 1 ] \\} < \\epsilon \\\\ & \\Gamma _ n ( f _ 1 , f _ 2 ) ( - s ) = \\Gamma ( f _ 1 , f _ 2 ) ( - s ) s \\geq \\lambda _ 0 - \\lambda _ 1 . \\end{align*}"}
-{"id": "7699.png", "formula": "\\begin{align*} \\{ U _ { n } ^ { ( 3 ) } \\} _ { n = 0 } ^ { 3 } = \\{ 0 , 0 , 0 , 1 \\} , \\{ V _ { n } ^ { ( 3 ) } \\} _ { n = 0 } ^ { 3 } = \\{ 8 , 4 p , 2 p ^ { 2 } , 8 \\} \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} \\lim _ { x \\to b ^ - } [ a _ k ( x ) f ^ { ( k ) } ( x ) ] ^ { ( k - 1 ) } - \\lim _ { x \\to a ^ + } [ a _ k ( x ) f ^ { ( k ) } ( x ) ] ^ { ( k - 1 ) } = 0 , \\end{align*}"}
-{"id": "1218.png", "formula": "\\begin{align*} \\sum _ k \\lambda _ k ^ { 2 \\beta } \\frac { 1 } { ( 1 + \\lambda _ k \\Delta t ) ^ { 2 n } } { = } \\sum _ k \\lambda _ { k } ^ { - \\frac 1 2 - 2 \\kappa } \\big | ( - A ) ^ { \\frac 1 4 + \\kappa + \\beta } S _ { \\Delta t } ^ n e _ k | _ { L ^ 2 } ^ { 2 } \\le C _ \\kappa t _ { n } ^ { - \\frac 1 2 - 2 \\kappa - 2 \\beta } , \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{gather*} f = a _ 0 + a _ 1 z + \\dots + a _ d z ^ d , a _ 0 = 1 , a _ d \\ne 0 . \\end{gather*}"}
-{"id": "5776.png", "formula": "\\begin{align*} S ( A | M ) _ { ( \\mathcal { N } ( t ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) } \\ge \\lim _ { \\nu \\to \\infty } S ( A | A ' ) _ { ( \\mathcal { N } ( t ) \\otimes \\mathbb { I } _ { A ' } ) ( \\hat { \\omega } _ { A A ' } ( \\nu ) ) } = n \\ln t + n \\ ; . \\end{align*}"}
-{"id": "4202.png", "formula": "\\begin{align*} T _ { j } ^ { 0 } = \\left ( T _ { 0 } \\right ) _ { j } , \\ ; \\ ; q _ { j } ^ { 0 } = 0 , \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{align*} d \\boldsymbol { X } _ t = d \\boldsymbol { W } _ t \\sqrt { \\frac { I + \\boldsymbol { X } _ t ^ 2 } { 2 } } + \\sqrt { \\frac { I + \\boldsymbol { X } _ t ^ 2 } { 2 } } d \\boldsymbol { W } _ t ^ * + \\left [ ( - N - 2 \\Re ( s ) ) \\boldsymbol { X } _ t + 2 \\Im ( s ) \\boldsymbol { I } + T r \\left ( \\boldsymbol { X } _ t \\right ) \\boldsymbol { I } \\right ] d t . \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} h ^ { p , q } ( X ) = \\frac { 1 } { | G | } \\sum _ { g \\in G } \\chi _ { p , q } ( g ) . \\end{align*}"}
-{"id": "7089.png", "formula": "\\begin{align*} A ( n ) = \\{ t \\in A : | t | = n \\} . \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} M _ { a , b } ' ( r ) \\ , = \\ , \\frac { a } { b } \\ , M _ { a + 1 , b + 1 } ( r ) \\ , , U _ { a , b } ' ( r ) \\ , = \\ , - a \\ , U _ { a + 1 , b + 1 } ( r ) \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} v _ \\pm = \\frac { a + 1 } { 3 } \\pm \\sqrt { \\frac { ( a + 1 ) ^ 2 } { 9 } - \\frac { a } { 3 } } , \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} \\lim _ { m \\longrightarrow 0 } \\int \\mathcal { D } _ 0 ^ n f _ { \\nu } ( x , m ) \\nu ( d x ) = \\int \\lim _ { m \\longrightarrow 0 } \\mathcal { D } _ 0 ^ n f _ { \\nu } ( x , m ) \\nu ( d x ) . \\end{align*}"}
-{"id": "9486.png", "formula": "\\begin{align*} \\binom { r - i } { k - i } \\phi ( S ) = \\sum _ { S ' \\in \\binom { V } { k } : S \\subseteq S ' } \\phi ( S ' ) . \\end{align*}"}
-{"id": "718.png", "formula": "\\begin{align*} { \\cal F } = \\bigoplus _ { n = 0 } ^ { \\infty } { \\cal H } ^ n , \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{align*} J ( i k _ j ) U ^ \\dagger [ f ( i k _ j , 0 ) - i f ' ( i k _ j , 0 ) ] P _ j = 0 _ n . \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} w _ R ( t ) & = \\prod _ { | p _ n | \\geq R } \\left ( 1 - \\frac { R t } { p _ n } \\right ) ^ 2 \\\\ & = \\prod _ { n \\in S _ 1 } \\left ( 1 - \\frac { R t } { p _ n } \\right ) ^ 2 \\ , \\left ( \\prod _ { n \\in S _ 2 \\cup S _ 3 } \\left ( 1 - \\frac { R t } { p _ n } \\right ) ^ 2 \\left ( 1 - \\frac { R t } { p _ { - n } } \\right ) ^ 2 \\right ) \\end{align*}"}
-{"id": "4497.png", "formula": "\\begin{align*} \\mathfrak { T } ( a _ { x } ) = a _ { x } x \\in \\mathfrak { L } \\ . \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} \\mathbf { x } _ { \\widetilde { a } } : = \\sum _ { i = 0 } ^ n \\widetilde { a } _ i x _ i , \\end{align*}"}
-{"id": "4953.png", "formula": "\\begin{align*} \\sigma ( S ) = \\left ( 1 2 , i \\sqrt { 3 } , - i \\sqrt { 3 } \\right ) \\sigma ( C ) = \\left ( 4 + 3 i , 4 - 3 i \\right ) . \\end{align*}"}
-{"id": "6034.png", "formula": "\\begin{align*} F _ 1 ( \\mathbf u ^ { ( l ) } ) = - ( \\alpha _ k ) ^ { l } S _ 1 . \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} \\mu ^ { l } ( B _ t ( w _ 0 ) ) = \\sum _ { z \\in Z } \\mu ^ { l } ( W ^ z ) + \\mu ^ { l } \\big ( W ^ { ( 2 ) } \\setminus E \\big ) = \\sum _ { z \\in Z } \\mu ^ { l } ( W ^ z ) + \\mu ^ { l } _ 1 \\big ( W ^ { ( 2 ) } \\setminus E \\big ) . \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} \\left \\Vert u _ { + } \\right \\Vert _ { L ^ { \\infty } \\left ( B _ { \\frac { r } { 2 } } \\right ) } \\leq \\frac { 1 } { 2 \\sqrt { \\delta \\left ( r \\right ) } } \\left ( \\frac { 1 } { \\left \\vert B _ { 3 r } \\right \\vert } \\int _ { B _ { r } } u _ { + } ^ { 2 } \\right ) ^ { \\frac { 1 } { 2 } } \\ , \\ \\ \\ \\ \\ L u = 0 B _ { r } \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} \\log h _ t ( v ) = \\log h _ K ( v ) + t g ( v ) , \\end{align*}"}
-{"id": "4369.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { q - 1 } \\binom { 2 k } { k } x ^ k \\equiv ( 1 - 4 x ) ^ { ( q - 1 ) / 2 } \\pmod { p } \\end{align*}"}
-{"id": "243.png", "formula": "\\begin{align*} \\mathrm { T r } ( M _ 0 \\otimes M _ 1 \\otimes \\cdots \\otimes M _ n ) : = \\sum _ { i _ 0 , \\cdots , i _ n } ( M _ 0 ) ^ { i _ 0 } _ { i _ 1 } \\otimes ( M _ 1 ) ^ { i _ 1 } _ { i _ 2 } \\otimes \\cdots \\otimes ( M _ n ) ^ { i _ n } _ { i _ 0 } . \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} S = \\begin{pmatrix} 2 & 2 & 1 \\\\ 1 & 2 & 2 \\\\ 2 & 1 & 2 \\end{pmatrix} C = \\begin{pmatrix} 0 & 0 & - 1 \\\\ - 1 & 0 & 0 \\\\ 0 & - 1 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} \\left . \\partial \\ , ^ k z _ l \\right | _ { \\xi = 1 } = \\left . \\partial \\ , ^ k R _ l \\right | _ { \\xi = 1 } + \\gamma _ { l k } \\cdot k ! , l = m , n . \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} Y _ { n } ^ { 2 , \\ell } = \\Delta t \\sum _ { m = \\ell } ^ { n - 1 } S _ { \\Delta t } ^ { n - m } B F _ 2 ' ( X _ m ) . Y _ m ^ \\ell + \\Delta t \\sum _ { m = \\ell } ^ { n - 1 } S _ { \\Delta t } ^ { n - m } B F _ { n , 2 } , \\end{align*}"}
-{"id": "470.png", "formula": "\\begin{align*} ( \\varphi ^ { ( 3 ) } ( i , j ) , \\varphi ^ { ( 4 ) } ( i , j ) ) : = \\varphi ( i , j ) . \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} x _ { n } = \\frac { 1 } { n } = \\sum _ { k = 0 } ^ { 2 ^ { n - 1 } - 1 } \\frac { \\left ( - 1 \\right ) ^ { s _ { 2 } \\left ( k \\right ) } } { s _ { 2 } \\left ( k \\right ) + 1 } , \\thinspace \\thinspace n \\ge 1 \\end{align*}"}
-{"id": "6284.png", "formula": "\\begin{align*} \\epsilon ^ { \\textnormal { h o m } } ( \\theta ) E = f . \\end{align*}"}
-{"id": "2796.png", "formula": "\\begin{align*} \\sharp E ( a , b , \\varepsilon , B , r ) = \\frac { 3 } { 2 \\pi ^ 2 } \\frac { \\Phi ( b ) \\Phi ( a ) \\Psi ( b - a ) } { b a ^ { \\frac { 1 } { 2 } } } \\varepsilon B ^ { \\frac { 1 } { 2 } - \\frac { 1 } { r } } + O _ { \\tau _ i , \\varepsilon , \\delta } \\left ( b ^ { \\frac { 2 3 } { 8 } + \\delta } B ^ { \\frac { 3 } { 4 } ( \\frac { 1 } { 2 } - \\frac { 1 } { r } ) } \\log B \\right ) \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} R ( z ) = S ( z ) M ^ { - 1 } ( z ) , z \\in \\mathbb C \\setminus ( \\Gamma _ { \\tau } \\cup [ - 1 , 1 ] ) . \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} \\theta ''' = \\epsilon ^ 2 e ^ { - \\psi } ( 1 + e ^ { - \\rho } ) ( \\psi ''' + 3 \\psi '' \\psi ' + ( \\psi ' ) ^ 3 ) + \\epsilon ^ 2 e ^ \\psi e ^ { - \\rho } ( - \\psi '' - 3 ( \\psi ' ) ^ 2 + 3 \\psi ' - 1 ) , \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{align*} p _ { n } ( T , U ) = \\zeta _ 0 T ^ { n + 1 } + \\zeta _ 1 T ^ { n } U + \\zeta _ 2 T ^ { n - 1 } U ^ 2 + \\cdot \\cdot \\cdot + \\zeta _ n T U ^ { n } + \\zeta _ { n + 1 } U ^ { n + 1 } , \\end{align*}"}
-{"id": "5870.png", "formula": "\\begin{align*} I _ \\mathrm { S } = \\sum _ { ( x , y ) \\in S } \\rho ^ 2 ( x , y ) , \\end{align*}"}
-{"id": "2755.png", "formula": "\\begin{align*} \\sharp S ( \\varepsilon , B ) = & \\sum _ { d = O ( B ^ { 1 - \\frac { 1 } { ( 2 + \\lambda ) r } } ) } \\mu ( d ) \\sharp S ( d , \\varepsilon , B ) \\\\ & = T _ { \\varepsilon , B } + O ( B ^ { 1 - \\frac { 1 } { ( 2 + \\lambda ) r } } ) . \\end{align*}"}
-{"id": "5779.png", "formula": "\\begin{align*} \\hat { \\rho } _ { A M } ( t ) & = \\left ( \\mathcal { N } _ A \\left ( \\frac { \\lambda \\ , t } { \\eta } \\right ) \\otimes \\mathbb { I } _ M \\right ) ( \\hat { \\rho } _ { A M } ) \\ ; , \\\\ \\hat { \\rho } _ { B M } ( t ) & = \\left ( \\mathcal { N } _ B \\left ( \\frac { ( 1 - \\lambda ) \\ , t } { | 1 - \\eta | } \\right ) \\otimes \\mathbb { I } _ M \\right ) ( \\hat { \\rho } _ { B M } ) \\ ; , \\\\ \\hat { \\rho } _ { C M } ( t ) & = \\left ( \\mathcal { N } _ C ( t ) \\otimes \\mathbb { I } _ M \\right ) ( \\hat { \\rho } _ { C M } ) \\ ; , \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{gather*} x _ { i j } x _ { j i } = 1 . \\end{gather*}"}
-{"id": "6017.png", "formula": "\\begin{align*} ( & P ' + Q ' ) S ' - Q ' ( R ' + S ' ) = \\\\ & - ( \\lambda _ { n + 3 } - \\lambda _ { n + 2 } ) x _ { n + 1 } ^ 2 \\sum _ { i = 1 } ^ { n + 1 } \\Big ( ( \\lambda _ { n + 3 } - \\lambda _ i ) ( \\lambda _ { n + 2 } - \\lambda _ i ) \\frac { ( \\lambda _ 1 s - t ) ( \\lambda _ 2 s - t ) ( \\lambda _ { n + 1 } s - t ) } { ( \\lambda _ i s - t ) } x _ i ^ 2 \\Big ) . \\end{align*}"}
-{"id": "4427.png", "formula": "\\begin{align*} h _ 1 \\cdots h _ r \\cdot \\prod _ { i = 1 } ^ { g ' } [ a _ i , b _ i ] = 1 ( \\ast ) . \\end{align*}"}
-{"id": "9666.png", "formula": "\\begin{align*} \\begin{aligned} | O * \\beta ( \\mathcal { A } K ) | = | O * b ( \\mathcal { A } K ) | , & | O * \\gamma ( \\mathcal { A } K ) | = | O * c ( \\mathcal { A } K ) | , \\\\ | ( \\mathcal { A } K ) _ { + + + } | + | ( \\mathcal { A } K ) _ { - + + } | & = | ( \\mathcal { A } K ) _ { - - + } | + | ( \\mathcal { A } K ) _ { + - + } | , \\end{aligned} \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{align*} \\nabla = \\omega \\partial _ r + \\frac { \\omega ^ \\perp } { r } \\partial _ \\theta , \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} & \\hat c _ 0 ( t ) = ( \\mu h _ F - \\Lambda T _ s ) ( 1 - e ^ { - t } ) + ( - \\frac { \\Gamma } { K _ 1 } + \\Lambda T _ s + \\Lambda \\theta _ F ) t , \\\\ & \\hat h _ 0 ( t ) = h _ F e ^ { - t } , \\ , \\ , \\hat \\theta _ 0 ( t ) = T _ s ( 1 - e ^ { - t } ) + \\theta _ F e ^ { - t } . \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{align*} J ( x , y ) : = D F ( x , y ) = \\left ( \\begin{array} { c c } - g _ 1 ' ( x ) + n _ X f _ x ( x , y ) & n _ X f _ y ( x , y ) \\\\ n _ Y f _ x ( x , y ) & - g _ 2 ' ( y ) + n _ Y f _ y ( x , y ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} - \\Delta u = e ^ { 2 \\lambda } K _ g \\in L ^ { \\infty } ( D ^ 2 ) \\end{align*}"}
-{"id": "2769.png", "formula": "\\begin{align*} & 0 < n _ 2 - n _ 1 \\theta \\leqslant \\frac { 3 } { 2 } \\frac { \\det ( \\Lambda _ d ) n _ 1 } { ( \\lambda _ 2 - \\alpha \\mu _ 2 ) ^ 2 } \\varepsilon B ^ { - \\frac { 1 } { r } } \\\\ & \\frac { \\lambda _ 2 - \\alpha \\mu _ 2 } { \\det ( \\Lambda _ d ) } K B - \\frac { \\alpha } { 2 } \\leqslant n _ 1 \\leqslant \\frac { \\lambda _ 2 - \\alpha \\mu _ 2 } { \\det ( \\Lambda _ d ) } K B + \\frac { \\alpha } { 2 } \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} T _ { F , G , N } ^ L ( f _ 1 , \\dots , f _ d ) = \\sum \\limits _ { \\widetilde { \\mathbf { r } } \\in \\widetilde { R } } \\frac { 1 } { N ^ { d - m } } \\sum \\limits _ { \\mathbf { n } \\in \\mathbb { Z } ^ { d - u } } \\Big ( \\prod \\limits _ { j = 1 } ^ d f _ j ( \\xi _ j ( \\mathbf { n } ) + \\widetilde { \\mathbf { r } } _ j ) \\Big ) F ( \\Xi ( \\mathbf { n } ) + \\widetilde { \\mathbf { r } } ) G ( L \\Xi ( \\mathbf { n } ) + L \\widetilde { \\mathbf { r } } ) , \\end{align*}"}
-{"id": "6827.png", "formula": "\\begin{align*} b _ k ( \\mu ) = \\Big ( \\int _ { I m ( G ) } H _ { k } ( G ^ - ( u ) ) \\varphi ( G ^ - ( u ) ) \\ , \\mu ( d u ) \\Big ) ^ 2 \\end{align*}"}
-{"id": "4295.png", "formula": "\\begin{align*} \\vert x \\vert _ w = p ^ { - w ( x ) / e _ w } , \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} \\varphi ( t ) = \\frac { y ( t ) } { z ( t ) ^ 2 } e ^ { 1 2 7 } - { y ( t ) ^ { - 1 } } \\left ( e ^ { 3 4 7 } + e ^ { 5 6 7 } \\right ) + y ( t ) z ( t ) ^ 2 \\left ( e ^ { 1 3 5 } - e ^ { 1 4 6 } + e ^ { 2 3 6 } + e ^ { 2 4 5 } \\right ) , \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{align*} \\Omega _ { i j } ^ { - 1 } = \\left ( \\begin{array} { c c } 4 \\varepsilon - 0 . 1 & - 8 \\varepsilon + 0 . 2 \\\\ - 8 \\varepsilon + 0 . 2 & \\frac { 1 6 \\varepsilon ^ 2 + 3 . 2 \\varepsilon - 0 . 0 9 } { \\varepsilon } \\end{array} \\right ) , \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} p \\cdot \\phi _ x ' ( 0 , 0 ) = & \\sum _ { k = 0 } ^ a \\frac { p \\cdot ( - a ) _ k ( \\delta - \\beta ) _ k ( \\delta - \\gamma ) _ k } { ( \\delta ) _ k ( - f - a ) _ k ( 1 - f - a - \\epsilon ) _ k } \\cdot \\frac { d \\big ( ( - f - x ) _ k \\big ) } { d x } \\bigg | _ { x = 0 } \\\\ & + \\sum _ { k = 0 } ^ a \\frac { ( - f ) _ k ( - a ) _ k ( \\delta - \\beta ) _ k ( \\delta - \\gamma ) _ k } { ( \\delta ) _ k ( - f - a ) _ k ( 1 - f - a - \\epsilon ) _ k } \\sum _ { j = 0 } ^ { k - 1 } \\bigg ( \\frac { p } { j - f - a } + \\frac { p } { j + 1 - f - a - \\epsilon } \\bigg ) \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{align*} \\left ( X ^ { ( N ) } _ 1 ( t ) , \\cdots , X ^ { ( N ) } _ N ( t ) ; t \\ge 0 \\right ) = N \\bigg ( - \\alpha _ 1 ^ { - } \\left ( X ^ { ( N ) } ; t \\right ) , \\cdots , - \\alpha _ { N + 1 - \\mathfrak { n } ( t ) } ^ { - } \\left ( X ^ { ( N ) } ; t \\right ) , \\\\ \\alpha _ { \\mathfrak { n } ( t ) - 1 } ^ { + } \\left ( X ^ { ( N ) } ; t \\right ) , \\cdots , \\alpha _ 1 ^ { + } \\left ( X ^ { ( N ) } ; t \\right ) ; t \\ge 0 \\bigg ) . \\end{align*}"}
-{"id": "2897.png", "formula": "\\begin{align*} \\| ( T - A ) f _ { k } \\| < \\varepsilon \\qquad \\hbox { f o r $ k = - r , \\dots , r $ . } \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} C N ^ + _ { r } ( \\gamma , p ) = \\{ \\gamma ( s ) + u ^ { \\textrm { s t } } \\big ( 1 - \\tfrac { s _ 0 - s } { r } \\big ) \\ ! \\cdot \\ ! \\left ( \\bar { B } _ r ^ { n + 1 } \\cap \\gamma ' ( s ) ^ { \\perp } \\right ) | \\ ; s \\in [ a _ 0 , a _ 1 ] \\} . \\end{align*}"}
-{"id": "8809.png", "formula": "\\begin{align*} g ^ { \\xi } : = g \\circ \\xi : M \\rightarrow T _ { 2 } ^ { 0 } M \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} T ^ { \\ , m - ( j - b _ l ) } e _ { b _ { 2 ^ { k - 1 } + l + 1 } - m + j - b _ { l } } = v ^ { ( k ) } \\ , & \\Bigl ( \\prod _ { i = \\Delta ^ { ( k ) } - m + j - b _ l + 1 } ^ { \\Delta ^ { ( k ) } - 1 } w _ i ^ { ( k ) } \\Bigr ) \\ , e _ { b _ l } \\\\ & - \\Biggl ( \\ ; \\prod _ { i = 1 } ^ { \\Delta ^ { ( k ) } - m + j - b _ l } w _ i ^ { ( k ) } \\Biggr ) ^ { - 1 } e _ { b _ { 2 ^ { k - 1 } + l } } , \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{align*} \\langle X _ T , \\phi \\rangle = \\frac { 1 } { \\sqrt { T } } \\sum _ j \\sigma _ j \\int _ 0 ^ T \\phi ( x _ j + \\xi _ s ^ j ) d s , \\phi \\in \\mathcal { S } ( \\mathbb { R } ) . \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} b _ { m } ( x ) = b _ { n } ( x ) = b ( x ) , \\ \\sigma _ { m } ( x ) = \\sigma _ { n } ( x ) = \\sigma ( x ) , c _ { m } ( x , u ) = c _ { n } ( x , u ) = c ( x , u ) \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{align*} ( \\Lambda _ N ^ { \\infty } f ) ( \\gamma _ 1 ) = \\tilde { C } \\times \\left ( \\phi ^ { \\otimes N } _ { \\gamma _ 2 } * \\phi ^ { \\otimes N } _ { \\alpha ^ + _ 1 } * \\cdots * \\phi ^ { \\otimes N } _ { \\alpha ^ - _ 1 } * \\cdots * g \\right ) ( \\gamma _ 1 , \\cdots , \\gamma _ 1 ) , \\end{align*}"}
-{"id": "9612.png", "formula": "\\begin{align*} g _ { \\underline { i j } } = \\left [ \\begin{array} { c c c } u ^ 2 & 0 & 0 \\\\ 0 & v ^ 2 & 0 \\\\ 0 & 0 & w ^ 2 \\end{array} \\right ] \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} A = \\sum _ i \\frac { p _ i ^ 2 } { 2 } , B = \\sum _ i \\frac { \\epsilon _ i } { 2 } q _ i ^ 2 + \\frac { 1 } { 4 } q _ i ^ 4 + \\frac { 1 } { 2 W } \\left ( q _ { i + 1 } - q _ i \\right ) ^ 2 \\end{align*}"}
-{"id": "4581.png", "formula": "\\begin{align*} \\vec { a } = ( a _ 1 , \\cdots , a _ N ) ^ { T } \\in \\mathbb { R } ^ { N } , \\ , \\vec { b } = ( b _ 1 , \\cdots , b _ N ) ^ { T } \\in \\mathbb { C } ^ { N } , \\ , \\vec { c } = ( c _ 1 , \\cdots , c _ N ) ^ { T } \\in \\mathbb { C } ^ { N } . \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{align*} \\tilde { F } ( x _ 1 , \\cdots , x _ { k - 1 } , x _ k ) = \\prod _ { l = 0 } ^ { k - 2 } \\prod _ { i = l + 1 } ^ { k - 1 } ( \\xi ^ l _ { i + 1 } - \\xi _ i ^ l ) \\partial _ 2 \\partial _ 3 ^ { 2 } \\cdots \\partial ^ { k - 2 } _ { k - 1 } \\partial ^ { k - 1 } _ k \\tilde { F } ( \\xi _ 1 ^ 0 , \\xi _ 2 ^ 1 , \\cdots , \\xi ^ { k - 2 } _ { k - 1 } , \\xi ^ { k - 1 } _ k ) , \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} T ( t ) = \\sum _ { n = 0 } ^ { \\infty } \\dfrac { t ^ n } { n ! } A ^ n = e ^ { t A } , \\mbox { f o r e v e r y } t \\in \\R . \\end{align*}"}
-{"id": "8741.png", "formula": "\\begin{align*} k = \\left \\{ \\begin{array} { l r } \\lfloor { { i - 1 } \\over 2 } \\rfloor , & { \\rm i f } ~ i = 1 , 2 , \\ldots , 2 m ; ~ ~ j = 1 , 2 , \\ldots , \\\\ m - 1 , & { \\rm i f } ~ i = 2 m + 1 , 2 m + 2 , \\ldots ; ~ j = 1 , 2 , \\ldots \\end{array} \\right . \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\frac { 1 } { 2 } } \\ ! Q ( x ) \\cos ( 2 k x ) d x \\ ! + \\ ! \\int _ { \\frac { 1 } { 2 } } ^ b \\ ! 2 A _ 2 Q ( x ) \\ ! \\cos ( 2 k x ) d x \\ ! = \\ ! \\int _ 0 ^ b \\widehat { Q } ( t ) \\cos ( 2 k t ) d t , \\end{align*}"}
-{"id": "2687.png", "formula": "\\begin{align*} A : = \\left [ \\begin{smallmatrix} A _ 1 & & & \\\\ & A _ 2 & & \\\\ & & \\ddots & \\\\ & & & A _ m \\end{smallmatrix} \\right ] \\end{align*}"}
-{"id": "3031.png", "formula": "\\begin{align*} \\psi _ c ( z ) = \\psi ( x _ c ) - c \\end{align*}"}
-{"id": "6850.png", "formula": "\\begin{align*} \\frac { \\alpha - 1 } { \\left ( [ \\alpha ] - 1 \\right ) ^ { 1 - \\{ \\alpha \\} } \\left ( [ \\alpha ] \\right ) ^ { \\{ \\alpha \\} } } & \\leq \\frac { 1 } { \\beta ( k ) ( k - 1 ) ^ { k - 1 / \\beta ( k ) } k ^ { 1 + 1 / \\beta ( k ) - k } } \\\\ & = \\exp \\left ( k \\beta ( k ) - \\log \\left ( k \\beta ( k ) \\right ) - 1 \\right ) . \\end{align*}"}
-{"id": "2446.png", "formula": "\\begin{align*} x ' ( t ) \\in \\Gamma ( t , x ( t ) ) x ( 0 ) = \\xi _ 0 \\in X \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} \\beta : = \\sum _ { k = 0 } ^ \\infty C _ k x ^ { k + 1 } = \\frac { 1 - \\sqrt { 1 - 4 x } } { 2 } . \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} q _ { 3 } \\left ( 4 q _ { 1 } q _ { 3 } - q _ { 2 } ^ { 2 } \\right ) \\frac { B _ { 2 } ^ { \\prime } } { B _ { 2 } } + q _ { 2 } \\left ( 4 q _ { 1 } q _ { 3 } - q _ { 2 } ^ { 2 } \\right ) - q _ { 3 } \\left ( 4 q _ { 1 } q _ { 3 } - q _ { 2 } ^ { 2 } \\right ) ^ { \\prime } + q _ { 3 } ^ { \\prime } \\left ( 4 q _ { 1 } q _ { 3 } - q _ { 2 } ^ { 2 } \\right ) = 0 . \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} a ^ { \\rm h o m } ( 0 ) = a ^ { \\rm h o m } = \\langle a \\rangle \\coloneqq \\int _ Y a ( y ) d y . \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} U _ t + b U _ x + U _ { x x x } + U _ { x y y } + U U _ x + ( \\psi U ) _ x = F \\equiv f - \\widetilde \\psi - \\psi \\psi _ x , \\end{align*}"}
-{"id": "9609.png", "formula": "\\begin{align*} \\rho _ j = \\frac 1 { N + 1 } \\Big ( L ^ \\alpha _ { \\underline { j \\alpha } } + \\frac 1 2 F \\sigma _ j + \\frac 1 2 F ^ \\alpha _ j \\sigma _ \\alpha \\Big ) , \\end{align*}"}
-{"id": "8926.png", "formula": "\\begin{align*} G _ 0 f ( x ) = - \\frac { 1 } { 2 \\pi } \\int _ { \\R ^ 2 } \\log | x - y | f ( y ) \\ , d y , \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{align*} \\frac { e ^ { i a x } } { 1 - i a x } = 1 - \\frac { 3 } { 2 } a ^ 2 x ^ 2 + O \\left ( a ^ 3 \\right ) \\ \\textnormal { a s } a \\to 0 . \\end{align*}"}
-{"id": "3554.png", "formula": "\\begin{align*} p _ { i _ 1 \\cdots i _ r i _ s \\cdots i _ k } = - p _ { i _ 1 \\cdots i _ s i _ r \\cdots i _ k } . \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{align*} d _ q ( A _ { \\alpha } ) = z ^ 2 q + ( 2 z + 1 ) ( \\alpha - z q + 1 ) - 1 , \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} c = \\big | R / ( \\epsilon ^ k ) \\big | = \\big | \\widetilde { R } / ( \\epsilon ^ k ) \\big | . \\end{align*}"}
-{"id": "776.png", "formula": "\\begin{align*} x _ 2 & : = x _ 1 - \\sqrt { \\frac { - 2 } { N } } \\left ( i - \\frac { a } { 2 } \\right ) \\\\ y _ 2 & : = y _ 1 - \\sqrt { \\frac { - 2 } { N } } \\left ( j - \\frac { b } { 2 } \\right ) , \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{align*} \\pi _ B ^ \\omega = d - \\mu , \\pi _ P ^ \\omega = ( \\mu - \\omega ) ^ + / \\rho , \\pi _ W ^ \\omega = \\mu - ( \\mu - \\omega ) ^ + / \\rho . \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} n _ F = \\sum _ { i = 1 } ^ \\infty ( 2 i + 1 ) f _ { 2 i + 1 } - \\sum _ { i = 1 } ^ \\infty ( 2 i - 1 ) f _ { 2 i } . \\end{align*}"}
-{"id": "2967.png", "formula": "\\begin{align*} | v _ n | \\ . \\sup _ { j \\in [ b _ { \\varphi ( n ) } , b _ { \\varphi ( n ) + 1 } ) } \\ \\Bigl ( \\prod _ { s = b _ { \\varphi ( n ) } + 1 } ^ { j } | w _ { s } | \\Bigr ) \\le C _ { n } \\quad \\hbox { f o r e v e r y $ n \\ge 0 $ . } \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} \\left . z ^ { ( i ) } \\right | _ { x = 1 } = 0 j = 0 , 1 , \\dots , l \\Leftrightarrow \\left . \\partial \\ , ^ { j } z \\right | _ { \\xi = 1 } = 0 j = 0 , 1 , \\dots , l . \\end{align*}"}
-{"id": "2121.png", "formula": "\\begin{gather*} \\theta ^ 0 = - \\varepsilon ^ { - 1 } , \\theta ^ t = \\tilde { \\theta } ^ 0 + \\varepsilon ^ { - 1 } , t _ 2 = - \\varepsilon \\tilde { t } _ 2 , \\\\ q _ 2 = \\varepsilon \\tilde { t } _ 2 \\tilde { p } _ 2 , p _ 2 = - \\frac { \\tilde { q } _ 2 } { \\varepsilon \\tilde { t } _ 2 } , H _ { t _ 2 } = - \\varepsilon ^ { - 1 } \\left ( \\tilde { H } _ { 2 } - \\frac { \\tilde { p } _ 2 \\tilde { q } _ 2 } { \\tilde { t } _ 2 } \\right ) . \\end{gather*}"}
-{"id": "8957.png", "formula": "\\begin{align*} [ N , T ] = - 2 Z , \\ [ N , Z ] = 0 , \\ [ T , Z ] = - 2 \\lambda Z \\end{align*}"}
-{"id": "2245.png", "formula": "\\begin{align*} i = \\sum _ { j = 0 } ^ m c _ j r ^ j ( 0 \\le c _ j < r ) \\ , . \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} h _ n ( \\zeta ) = a _ M ( n ) \\zeta ^ M + \\dots + a _ { m _ p } ( n ) \\zeta ^ { m _ p } . \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} w _ 1 = w _ 2 = \\{ 1 , 2 \\} , w _ 3 = w _ 4 = w _ 4 = \\{ 1 , 2 , 5 \\} . \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} 1 - \\mathbb { P } \\{ C _ 1 > R _ { o R } ( p _ o ) | r _ 1 \\} \\mathbb { P } \\{ C _ 2 > R _ { o R } ( p _ o ) | r _ 1 \\} & = p _ o & & \\end{align*}"}
-{"id": "2180.png", "formula": "\\begin{align*} \\ ! \\ ! f ( - k , x ) + f ( k , x ) ( S ( k ) - { U _ 0 } ) = & e ^ { - i k x } I _ n + \\int _ { - \\infty } ^ \\infty K ( x , t ) e ^ { - i k t } d t \\\\ & + e ^ { i k x } \\int _ { - \\infty } ^ \\infty F _ S ( s ) e ^ { - i k s } d s \\\\ & + \\int _ { - \\infty } ^ \\infty K ( x , t ) e ^ { i k t } d t \\int _ { - \\infty } ^ \\infty F _ S ( s ) e ^ { - i k s } d s . \\end{align*}"}
-{"id": "509.png", "formula": "\\begin{align*} W : = \\big \\{ u \\in L ^ 1 ( X \\times V ) | v \\cdot \\nabla _ x u \\in L ^ 1 ( X \\times V ) , \\tau ^ { - 1 } u \\in L ^ 1 ( X \\times V ) \\big \\} , \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} p _ { j } \\left ( \\sum _ { k = m } ^ { n } x _ { k } \\right ) \\leqslant \\sum _ { k = m } ^ { n } p _ { j } ( x _ { k } ) \\underset { m , n \\to \\infty } { \\longrightarrow } 0 . \\end{align*}"}
-{"id": "9574.png", "formula": "\\begin{align*} E \\left [ \\sum _ { i = 1 } ^ n X _ { n , i } Z _ { n , i } ^ { \\prime } \\right ] = \\sum _ { i = 1 } ^ n E [ X _ { n , i } ] Z _ { n , i } ^ { \\prime } = 0 , \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} f ( u , v ) \\cdot l = \\det ( l ) \\ , f ( ( u , v ) l ^ { - 1 } ) ( l \\in L ) \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} ( 1 + z ) ^ { p + 1 } \\Phi ( p ) = & ( 1 + z ) ^ { a + p + 1 } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - \\frac { 1 } 2 ( a + p ) & - \\frac { 1 } 2 ( a + p ) + \\frac 1 2 \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac { 4 z } { ( 1 + z ) ^ 2 } \\bigg ] \\\\ = & { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a - p & - a - p \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] + z ^ { a + p + 1 } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a - p & - a - p \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac 1 z \\bigg ] . \\end{align*}"}
-{"id": "5489.png", "formula": "\\begin{align*} \\sum _ { \\substack { l = 1 \\\\ l \\in 2 \\Z + 1 } } ^ { k + 1 } \\sin ^ 2 \\left ( \\frac { \\pi l } { k + 2 } \\right ) = \\frac { k + 2 } { 4 } \\sum _ { { l = 1 } } ^ { k + 1 } \\sin ^ 2 \\left ( \\frac { \\pi l } { k + 2 } \\right ) = \\frac { k + 2 } { 2 } . \\end{align*}"}
-{"id": "9119.png", "formula": "\\begin{gather*} \\frac { G _ { y _ i } } { F ^ t } \\Omega = \\frac { t G F _ { y _ i } } { F ^ { t + 1 } } \\Omega \\end{gather*}"}
-{"id": "1655.png", "formula": "\\begin{align*} \\limsup _ { \\ell \\to \\infty } | \\sin \\ell \\omega | ^ { - 1 / \\ell } = \\limsup _ { \\ell \\to \\infty } | \\tfrac \\ell { k ( \\ell ) } - \\kappa | ^ { - 1 / \\ell } \\ , . \\end{align*}"}
-{"id": "8496.png", "formula": "\\begin{align*} P ( z , w ) = ( 1 - z ^ 2 \\overline { w } ^ 2 ) H ( z , w ) \\quad \\textrm { a n d } P _ p ( z , w ) = ( 1 - ( z ^ 2 \\overline { w } ^ 2 ) ^ p ) H ( z , w ) . \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} \\operatorname { r c e f } ( S ^ t ) = \\left ( \\begin{array} { c | c } \\begin{array} { c } I _ \\mu \\\\ \\hline a _ 0 \\cdots a _ { \\mu - 1 } \\\\ \\hline H ' \\end{array} & 0 _ { ( t + 1 ) \\times ( t - \\mu ) } \\end{array} \\right ) . \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} i _ 0 + 2 i _ 1 + 2 \\sum _ { p = 2 } ^ { m - 1 } i _ p \\le m . \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\big ( \\deg _ { \\widetilde { X } } ( v _ { i } ) - 2 \\big ) \\leq N \\cdot \\big ( \\deg _ { \\widetilde { Y } } ( a ) - 2 \\big ) \\cdot \\big ( \\deg _ { \\widetilde { Z } } ( b ) - 2 \\big ) \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{align*} \\sum _ { j \\neq i } b _ { i , j } x _ j = \\lambda x _ i - b _ { i , i } x _ i , \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} \\left \\| \\prod \\limits _ { i = 1 } ^ m f _ i \\right \\| _ { \\mathcal { M } ^ { p } _ { \\phi } } & \\ge \\frac { 1 } { \\phi \\left ( K + K ^ { - \\epsilon } \\right ) } \\Biggl ( \\frac { 1 } { | B ( 0 , K + K ^ { - \\epsilon } ) | } \\int \\limits _ { B ( 0 , K + K ^ { - \\epsilon } ) } g _ { \\epsilon , K } ( x ) \\ d x \\Biggr ) ^ { \\frac { 1 } { p } } \\\\ & \\ge \\frac { C } { ( K + K ^ { - \\epsilon } ) ^ { \\frac { \\epsilon } { p } } \\phi ( K + K ^ { - \\epsilon } ) } . \\end{align*}"}
-{"id": "4807.png", "formula": "\\begin{align*} \\left \\langle I + \\begin{pmatrix} 1 & 0 \\\\ 0 & d \\end{pmatrix} p , I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle , \\left \\langle I + \\begin{pmatrix} d & 0 \\\\ 0 & 1 \\end{pmatrix} I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle . \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} L _ { v } ( f ) \\ ; & = \\ ; L _ { v \\chi } ( f ) \\ ; = \\ ; \\lim _ { r \\downarrow 0 } W _ r ( \\overline { v \\chi } , f ) \\ ; = \\ ; - \\omega ( v \\chi , f ) \\\\ & = \\ ; \\langle v \\chi , S ^ * f \\rangle - \\langle S ^ * ( v \\chi ) , f \\rangle \\ ; = \\ ; \\langle v \\chi , \\overline { S } f \\rangle - \\langle v \\chi , \\overline { S } f \\rangle \\ ; = \\ ; 0 \\ , , \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} m _ i v _ i = \\sum _ { \\substack { \\alpha : i \\to j \\\\ v _ i - v _ j = 1 } } u _ \\alpha - \\sum _ { \\substack { \\alpha : k \\to i \\\\ v _ k - v _ i = 1 } } u _ \\alpha \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} M z + N \\bar { z } = p ~ , ~ ~ ~ z \\in \\mathbb { C } ^ n . \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{align*} 0 = B ' ( 0 ) = ( 3 / 2 ) \\lambda _ { c o s t } [ J , X ( 0 ) ] . \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{align*} P Q f '' = \\frac { c P } { Q ^ 2 } ( P ' Q - 2 P Q ' ) = f ' ( P ' Q - 2 P Q ' ) . \\end{align*}"}
-{"id": "7438.png", "formula": "\\begin{align*} a \\leq b \\implies ( x \\wedge b ) \\vee a = ( x \\vee a ) \\wedge b \\qquad x \\in L . \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} g ^ 0 = \\sum _ { \\ell \\in \\N _ * } g ^ 0 _ \\ell \\rho ^ \\ell , g ^ \\omega = \\sum _ { \\ell \\in \\N _ * } g ^ \\omega _ \\ell \\rho ^ \\ell , | g ^ 0 _ \\ell | + | g ^ \\omega _ \\ell | \\le C \\rho _ 1 ^ { - \\ell } . \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} H _ 2 ( X ; \\Q ) = H _ 2 ( X _ 1 ; \\Q ) \\oplus H _ 2 ( X _ 2 ; \\Q ) . \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} ( F ( t ) ) ( \\beta \\cdot \\omega + n ) = ( G ( t _ { \\beta } ) ) ( n ) \\ ( \\beta \\leq \\alpha , \\ n \\in \\omega ) . \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} \\mathrm { b } _ \\sigma C ( P ) & = \\sum _ { i , j , k , \\ell } V ^ i _ j ( 2 P ^ j _ k - \\delta ^ j _ k ) P ^ k _ \\ell \\otimes P ^ \\ell _ i - \\sum _ { i , j , k , \\ell } V ^ i _ j ( 2 P ^ j _ k - \\delta ^ j _ k ) \\otimes P ^ k _ \\ell P ^ \\ell _ i \\\\ & + \\sum _ { i , j , k , \\ell } \\sigma ( P ^ \\ell _ i ) V ^ i _ j ( 2 P ^ j _ k - \\delta ^ j _ k ) \\otimes P ^ k _ \\ell . \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{align*} \\sup _ { 0 \\le j \\le N } \\ \\Vert P _ { n } T ^ { \\ , j } P _ { l } \\ , x \\Vert \\le \\dfrac { 1 } { 4 } \\ , \\beta _ { l } \\ , \\Bigl ( \\ \\sup _ { b _ { l + 1 } - N \\le k < b _ { l + 1 } } \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } | w _ s | \\Bigr ) \\ , \\Vert P _ { l } \\ , x \\Vert \\end{align*}"}
-{"id": "7797.png", "formula": "\\begin{gather*} p \\widetilde u ( p , x , l ) + b \\widetilde u _ x ( p , x , l ) + \\widetilde u _ { x x x } ( p , x , l ) - \\lambda _ l \\widetilde u _ x ( p , x , l ) = 0 , \\\\ \\widetilde u ( p , 0 , l ) = \\widetilde \\nu _ 0 ( p , l ) \\equiv \\int _ { \\mathbb R _ + } \\ ! \\ ! \\int _ 0 ^ L e ^ { - p t } \\psi _ l ( y ) \\nu _ 0 ( t , y ) \\ , d y d t , \\widetilde u _ x ( p , 0 , l ) = \\widetilde \\nu _ 1 ( p , l ) , \\end{gather*}"}
-{"id": "7842.png", "formula": "\\begin{align*} \\delta _ 0 = \\delta - C \\epsilon - \\omega , \\delta _ 1 = \\delta - 2 C \\epsilon - \\omega \\delta _ 2 = \\delta - \\log ( 1 - \\epsilon ) + \\omega . \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} \\begin{aligned} Q ( 2 \\tau ) & = \\frac { 1 } { 8 } \\int \\limits _ { 0 } ^ { \\pi } \\int \\limits _ { 0 } ^ { 2 \\pi } \\frac { q ( \\rho , \\theta , \\phi ) \\sinh \\rho \\sin \\phi \\sqrt { \\cosh ^ { 2 } \\rho - \\cos ^ { 2 } \\phi } } { \\sqrt { ( 4 \\tau ^ { 2 } - \\cos ^ { 2 } \\phi ) ( 4 \\tau ^ { 2 } - 1 ) } } d \\theta d \\phi , \\end{aligned} \\end{align*}"}
-{"id": "9369.png", "formula": "\\begin{align*} L ^ 1 ( \\Omega ; F ) ^ * = B ( L ^ 1 ( \\Omega ) , F ^ * ) . \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} E _ { y } ( \\lambda ) = y + \\sum _ { l = 1 } ^ { r } \\dfrac { 1 } { \\omega _ { l } } \\left \\langle ( \\lambda - A ) y , e _ { l } \\right \\rangle \\ , \\biggl ( e _ { r + l } + \\sum _ { p \\ge 2 } \\dfrac { \\lambda ^ { p - 1 } } { \\omega _ { ( p - 1 ) r + l } \\dots \\omega _ { r + l } } \\ , e _ { p r + l } \\biggr ) \\ ! \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{align*} | e ^ { i \\frac { t } { 2 \\pi R } \\Delta } g ( \\bar { x } ) | \\gtrsim | \\Omega | , \\mbox { f o r a l l } ( \\bar { x } , t ) \\in X _ 0 \\times R ^ { 2 \\sigma - 1 } \\mathbb { Z } \\cap ( 0 , 1 ) \\ , , \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ d \\frac { 1 } { x - a _ i } = \\sum _ { i = 1 } ^ d \\frac { 1 } { x - a _ i } \\prod ^ d _ { \\substack { j = 1 \\\\ j \\neq i } } \\frac { 1 } { a _ i - a _ j } . \\end{align*}"}
-{"id": "9473.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & - q _ m - a _ { m - 1 } & - a _ { m - 2 } & \\ldots & - a _ 1 & - a _ 0 \\\\ & 1 & - q _ { m - 1 } & 0 & \\ldots & 0 \\\\ & & \\ddots & \\ddots & \\ddots & \\vdots \\\\ & & & 1 & - q _ 2 & 0 \\\\ & & & & 1 & - q _ 1 \\\\ & & & & & 1 \\end{bmatrix} s = \\begin{bmatrix} 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix} \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} \\mu _ 2 = A _ { 2 p } \\lambda \\end{align*}"}
-{"id": "10017.png", "formula": "\\begin{align*} S _ t f ( x ) = \\int \\limits _ { \\R ^ n } e ^ { i \\xi \\cdot x } e ^ { i t | \\xi | ^ 2 } \\widehat { f } ( \\xi ) d \\xi , \\ \\ x \\in \\R ^ n , \\ t \\in \\R . \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} \\lim _ { l \\rightarrow \\infty } \\mathbf { \\mu } _ { l } ^ { ( \\omega ) } ( \\mathbb { R } ) = \\lim _ { l \\rightarrow \\infty } \\partial _ { t } ^ { 2 } [ \\Xi _ { \\mathrm { p } , l } ^ { ( \\omega ) } ( 0 ) ] _ { + } = \\partial _ { t } ^ { 2 } [ \\mathbf { \\Xi } _ { \\mathrm { p } } \\left ( 0 \\right ) ] _ { + } = \\mathbf { \\mu } ( \\mathbb { R } ) \\in \\mathcal { B } _ { + } ( \\mathbb { R } ^ { d } ) \\ . \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} \\nabla _ { i j } T = \\nabla _ { j i } T + T * A * A \\ , . \\end{align*}"}
-{"id": "2432.png", "formula": "\\begin{align*} W ( x ) & = 1 - e ^ { - \\lambda ( x - \\sigma + T ) } + \\alpha _ { \\lambda } [ e ^ { - \\lambda ( K - \\sigma ) } - e ^ { - \\lambda ( x - \\sigma + T ) } ] \\\\ & + \\int _ { 0 ^ + } ^ { K - 0 } 1 - e ^ { - \\lambda ( x - w - \\sigma + T ) } d W ( w ) + \\alpha _ { \\lambda } \\int _ { 0 ^ + } ^ { K - 0 } e ^ { - \\lambda ( K - w - \\sigma ) } - e ^ { - \\lambda ( x - w - \\sigma + T ) } d W ( w ) , \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{align*} 2 s - s ^ g & = \\frac 1 2 | d F | ^ 2 - 2 \\delta \\theta - | \\theta | ^ 2 \\\\ s ^ H - 2 s & = | \\theta | ^ 2 + \\delta \\theta \\\\ s ^ H - s ^ g & = \\frac 1 2 | d F | ^ 2 - \\delta \\theta \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\log P _ N ( a ) = - \\sup _ \\vartheta \\left ( \\vartheta a - \\Lambda _ X ( { \\rm e } ^ \\vartheta - 1 ) \\right ) = : I _ Z ( a ) . \\end{align*}"}
-{"id": "5128.png", "formula": "\\begin{align*} x _ { n } = \\sum _ { p = 1 } ^ { n } \\binom { n + 1 } { p + 1 } \\sum _ { \\underset { k _ { i } \\ge 0 } { k _ { 1 } + \\dots + k _ { p } = n } } g _ { k _ { 1 } } \\dots g _ { k _ { p } } . \\end{align*}"}
-{"id": "9443.png", "formula": "\\begin{align*} \\widetilde { \\Lambda } ( \\rho , W , Q ; s , t ) = \\begin{cases} \\displaystyle { \\frac { 1 } { 2 } \\int _ s ^ t \\int _ { \\mathbb { T } ^ 3 } | \\sqrt { Q } w | ^ 2 \\rho } , & { \\rm o n \\ } [ s , t ] , \\ ; \\rho \\geq 0 , \\ ; W \\ll \\rho , \\ ; W = w \\rho , \\ ; w \\in L ^ 2 _ { \\rho } \\\\ + \\infty , & { \\rm o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{align*} 4 s ^ 2 z '' + 4 s z ' - z ' + \\frac { \\lambda _ 1 } { s } z + \\frac { 1 } { s } | z | ^ \\alpha z = 0 . \\end{align*}"}
-{"id": "2362.png", "formula": "\\begin{align*} S _ { N } ( t ) u _ 0 ( x ) : = \\frac { 1 } { ( 2 \\pi ) ^ { n / 2 } } \\int _ { \\mathbb { R } ^ n } \\Psi ( N ^ { - 1 } \\xi ) \\ , \\widehat { u } _ 0 ( \\xi ) \\ , e ^ { i x \\cdot \\xi - i t | \\xi | ^ { 2 } } d \\xi \\ , , \\end{align*}"}
-{"id": "4730.png", "formula": "\\begin{align*} { \\dot { \\bar x } } = - \\sum _ { i = 1 } ^ n \\beta _ { 2 i , 1 i } \\frac { \\partial J ( \\bar x ) } { \\partial \\bar x _ i } F _ { 0 i } ( J ( \\bar x ) ) e _ i . \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{align*} \\frac { ( \\alpha _ 1 - \\lambda _ i ) } { ( \\alpha _ 1 - \\lambda _ 1 ) } \\cdot \\mathbf M _ { i j } = \\frac { ( \\lambda _ { n + 2 } - \\lambda _ i ) ( \\lambda _ { n + 3 } - \\lambda _ i ) } { ( \\lambda _ { n + 2 } - \\lambda _ 1 ) ( \\lambda _ { n + 3 } - \\lambda _ 1 ) } \\cdot \\mathbf M _ { 1 j } . \\end{align*}"}
-{"id": "3809.png", "formula": "\\begin{align*} f _ { n } ( t ) = \\sum _ { i = 0 } ^ { n } a ( i , n ) t ^ { i } & = \\frac { t } { n } \\sum _ { j = 0 } ^ { n - 1 } \\left ( 1 - 2 ^ { \\nu _ { 2 } ( n - j ) + 1 } \\right ) \\sum _ { i = 0 } ^ { k } a ( i , j ) t ^ { i } \\\\ & = \\frac { 1 } { n } \\sum _ { i = 0 } ^ { n - 1 } \\left ( \\sum _ { j = i } ^ { n - 1 } ( 1 - 2 ^ { \\nu _ { 2 } ( n - j ) + 1 } ) a ( i , j ) \\right ) t ^ { i + 1 } \\\\ & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\left ( \\sum _ { j = i - 1 } ^ { n - 1 } ( 1 - 2 ^ { \\nu _ { 2 } ( n - j ) + 1 } ) a ( i - 1 , j ) \\right ) t ^ { i } . \\end{align*}"}
-{"id": "8583.png", "formula": "\\begin{align*} 0 = \\mathrm { d i v } ( s ^ { - 1 } \\otimes s ' ) = \\mathrm { d i v } ( s ^ { - 1 } ) + \\mathrm { d i v } ( s ' ) = - \\mathrm { d i v } ( s ) + \\mathrm { d i v } ( s ' ) . \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} u ( t , x , y ) = w ( t , x , y ) + v ( t , x , y ) , \\end{align*}"}
-{"id": "4214.png", "formula": "\\begin{align*} v _ i = \\frac { \\sum _ { j = 1 } ^ n \\sigma _ { i j } v _ j } { \\sum _ { j = 1 } ^ n \\sigma _ { i j } } . \\end{align*}"}
-{"id": "8469.png", "formula": "\\begin{gather*} u ( x ) = \\sum _ { k = 0 } ^ { p - 1 } e ^ { \\frac { 2 j k \\pi i } { p } } u _ k ( x ) \\quad \\textrm { a n d } v ( x ) = \\sum _ { k = 0 } ^ { p - 1 } e ^ { \\frac { 2 j k \\pi i } { p } } v _ k ( x ) \\end{gather*}"}
-{"id": "1562.png", "formula": "\\begin{align*} N ( c ) ( x ) = H ^ * \\left ( H ( c ) ^ { - 1 } \\right ) ( x ) = \\left \\langle u ( x ) , u ( x ) H ( c ) ^ { - \\top } \\right \\rangle > 0 , \\end{align*}"}
-{"id": "9530.png", "formula": "\\begin{align*} { \\mathrm { R e } } \\ , \\sum _ { i = 1 } ^ r \\langle C ^ * ( X ( 1 - z _ 0 ) ) C \\xi _ i , \\eta _ i \\rangle = { \\mathrm { R e } } \\ , \\sum _ { i = 1 } ^ r \\langle \\tilde { C } ^ * X \\tilde { C } \\xi _ i , \\eta _ i \\rangle \\geq 1 \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{align*} o ( P ) : = \\left \\{ \\begin{array} { c l } \\sup \\bigl \\{ | \\underline { k } | - \\upsilon ( a _ { \\underline { k } } ) \\ ; \\big | \\ ; \\underline { k } \\in \\Sigma \\bigr \\} & P \\ne 0 \\\\ - \\infty & P = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} { \\displaystyle \\mathrm { R e } \\left [ V _ 0 ( \\overline { \\Psi } _ { h } ^ { k } , \\partial \\Psi _ { h } ^ { k } ) \\right ] = \\frac { V _ { 0 } } { 2 } \\partial ( \\Psi _ { h } ^ { k } , \\Psi _ { h } ^ { k } ) . } \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} 2 \\left ( \\frac { r } { g } \\right ) ^ { 4 } - \\left ( \\frac { r } { g } \\right ) ^ { 2 } \\left ( \\frac { p } { f } \\right ) ^ { 2 } + 2 \\left ( \\frac { p } { f } \\right ) ^ { 4 } = 0 . \\end{align*}"}
-{"id": "9681.png", "formula": "\\begin{align*} \\begin{aligned} & \\abs { ( B _ m K _ m ) _ { \\pm \\pm \\pm } } = \\frac { \\abs { B _ m K _ m } } { 8 } , \\\\ & \\overline { \\alpha } ( B _ m K _ m ) = \\overline { a } ( B _ m K _ m ) , \\overline { \\beta } ( B _ m K _ m ) = \\overline { b } ( B _ m K _ m ) , \\overline { \\gamma } ( B _ m K _ m ) = \\overline { c } ( B _ m K _ m ) . \\end{aligned} \\end{align*}"}
-{"id": "8528.png", "formula": "\\begin{align*} \\phi ^ \\ast _ { \\C } T \\R ^ n = \\phi ^ { \\ast } T \\R ^ n \\otimes _ { \\R } \\C \\end{align*}"}
-{"id": "3848.png", "formula": "\\begin{align*} t _ { 2 } ( n ) = \\sum _ { a + b = n } ( - 1 ) ^ { s _ { 2 } ( a ) + s _ { 2 } ( b ) } . \\end{align*}"}
-{"id": "8999.png", "formula": "\\begin{align*} \\int _ { \\mbox { } _ { \\scriptstyle \\mathbb { R } ^ { n } } } \\ ! \\ ! \\ ! \\ : \\ ! u ( x , t ) \\ : d x \\ ; = \\ ; \\ ! \\int _ { \\mbox { } _ { \\scriptstyle \\mathbb { R } ^ { n } } } \\ ! \\ ! \\ ! \\ : \\ ! u _ 0 ( x ) \\ , d x , \\ ; \\ , \\forall \\ ; \\ , 0 \\ : \\ ! < \\ : \\ ! t \\ : \\ ! < \\ : \\ ! \\mbox { \\small $ T $ } _ { \\ ! \\ast } . \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} \\langle \\alpha _ X ( \\xi ) , \\eta \\rangle _ A = \\langle \\alpha _ X ( \\xi ) , \\alpha ^ 2 _ X ( \\eta ) \\rangle _ A = \\alpha _ A ( \\langle \\xi , \\alpha _ X ( \\eta ) \\rangle _ A ) = \\langle \\xi , \\alpha _ X ( \\eta ) \\rangle _ A . \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{align*} H = \\left ( \\frac { \\partial ^ 2 } { \\partial x _ 1 ^ 2 } + \\frac { \\partial ^ 2 } { \\partial x _ 2 ^ 2 } \\right ) - 2 \\left ( \\frac { 1 } { ( x _ 1 - \\xi _ 1 ) ^ 2 } + \\frac { 1 } { ( x _ 2 - \\xi _ 2 ) ^ 2 } \\right ) , \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} f ^ { 4 } - 2 \\lambda _ { 2 } f ^ { 2 } p ^ { 2 } + \\left ( \\lambda _ { 2 } \\lambda _ { 3 } - \\lambda _ { 1 } \\right ) p ^ { 4 } = 0 \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ a \\frac { ( - a - p ) _ { p + k } ( \\frac 1 2 ) _ { p + k } } { ( 1 ) _ { p + k } ^ 2 } \\cdot z ^ k \\equiv ( 1 - z ) ^ a \\sum _ { k = 0 } ^ a \\frac { ( - a - p ) _ { p + k } ( \\frac 1 2 ) _ { p + k } } { ( 1 ) _ { p + k } ^ 2 } \\cdot \\frac { z ^ k } { ( z - 1 ) ^ k } \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "4569.png", "formula": "\\begin{align*} ( - \\Delta \\psi _ { n } , \\ , - \\Delta u _ j ) = ( - \\Delta \\psi _ { n } , \\ , \\lambda _ j u _ j ) = \\lambda _ j ( \\nabla \\psi _ { n } , \\ , \\nabla u _ j ) = \\lambda _ j ( f , \\ , u _ j ) = ( f , \\ , - \\Delta u _ j ) . \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} J ^ \\star _ { m ^ s , m ^ u } : = \\left \\{ p \\in J ^ \\star | \\ , \\tau ^ s ( p ) = m ^ s , \\ , \\tau ^ u ( p ) = m ^ u \\right \\} . \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} \\dot { \\bar x } { = } f _ 0 ( \\bar x ) { + } \\sum \\limits _ { i { < } j } { \\beta _ { j , i } } [ f _ i , f _ j ] ( \\bar x ) , \\quad \\bar x ( 0 ) = x ^ 0 . \\end{align*}"}
-{"id": "7359.png", "formula": "\\begin{align*} d f ^ i = 0 , 1 \\leq i \\leq 6 , d f ^ 7 = \\frac { \\sqrt { 6 } } { 6 } \\ , y ( t ) ^ { - 5 } ( f ^ { 1 2 } + f ^ { 3 4 } + f ^ { 5 6 } ) . \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} ( 1 - \\rho ) \\tilde J \\phi ( x , v ) = ( 1 - \\rho ( x ) ) e ^ { - \\int _ 0 ^ { \\tau _ + ( x , v ) } \\sigma ( x + s v , v ) d s } \\phi ( x + \\tau _ + ( x , v ) v , v ) , \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{align*} C : = \\begin{bmatrix} G ' ( k , 0 ) E ^ { - \\frac { 1 } { 2 } } & G ( k , 0 ) E ^ { - \\frac { 1 } { 2 } } \\\\ G ( k , 0 ) E ^ { - \\frac { 1 } { 2 } } & - G ' ( k , 0 ) E ^ { - \\frac { 1 } { 2 } } \\end{bmatrix} . \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} w _ { b _ { n } + i } = w _ { i } ^ { ( k ) } \\quad \\hbox { f o r e v e r y $ 1 \\le i < \\Delta ^ { ( k ) } $ a n d e v e r y $ n \\in [ 2 ^ { k - 1 } , 2 ^ { k } ) $ . } \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} \\min \\ , \\ , & \\varphi _ 0 ( x ( 0 ) , x ( T ) ) , \\\\ & \\dot { x } = F ( x , u , v ) , { \\rm a . e . } \\ { \\rm o n } \\ [ 0 , T ] , \\\\ & \\eta _ j ( x ( 0 ) , x ( T ) ) = 0 , \\mathrm { f o r } \\ j = 1 \\hdots , d _ { \\eta } , \\\\ & \\varphi _ i ( x ( 0 ) , x ( T ) ) \\leq 0 , \\mathrm { f o r } \\ i = 1 , \\hdots , d _ { \\varphi } , \\\\ & u ( t ) \\in U , \\ , \\ , v ( t ) \\in V , { \\rm a . e . } \\ { \\rm o n } \\ [ 0 , T ] , \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{align*} { \\mathbb E } \\left [ { \\rm e } ^ { \\vartheta S _ N } \\right ] = \\prod _ { i = 1 } ^ N M _ X \\left ( \\omega _ i ( N ) ( \\mathrm { e } ^ \\vartheta - 1 ) \\right ) . \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} \\rho ( 1 ) = ( 1 - \\lambda _ { \\min } ^ + ) ^ 2 . \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{align*} a ( 2 \\tau ) = 0 , \\ \\mbox { f o r } \\ \\tau \\in [ 1 / 2 , T / 2 ] . \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{align*} X ^ { k , \\delta } _ { \\lambda ^ { k - j } t \\theta _ { k } } : = \\lambda ^ { k ( \\sigma - 1 ) } \\mathbb { Z } ^ { n - 1 } + Q ( \\lambda ^ { k - j } t \\theta _ { k } , \\varepsilon _ { 2 } \\lambda ^ { - k ( 1 - 2 \\delta ) } ) \\ , . \\end{align*}"}
-{"id": "2779.png", "formula": "\\begin{align*} d \\leqslant \\left ( \\frac { \\varepsilon } { C ( \\alpha , \\lambda ) } \\right ) ^ \\frac { 1 } { 2 } B ^ { 1 - \\frac { 1 } { 2 r } } = O ( B ^ { \\frac { 1 } { 3 } + \\frac { \\delta } { 2 } } ) . \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} X \\triangleright u ^ i _ j = \\sum _ k \\pi ( X ) ^ k _ j u ^ i _ k , X \\triangleright u ^ { i * } _ j = \\sum _ k \\pi ( S ( X ) ) ^ j _ k u ^ { i * } _ k . \\end{align*}"}
-{"id": "9889.png", "formula": "\\begin{align*} ( { \\mathcal F } D ^ \\gamma _ t w ) ( s ) = ( 2 \\pi i s ) ^ \\gamma \\ , ( { \\mathcal F } w ) ( s ) \\forall s \\in ( - \\infty , \\infty ) . \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{align*} \\sum _ k S ^ 2 ( u ^ k _ b ) S ( u ^ a _ k ) = \\delta ^ a _ b 1 = \\sum _ k S ( u ^ k _ b ) S ^ 2 ( u ^ a _ k ) . \\end{align*}"}
-{"id": "1284.png", "formula": "\\begin{align*} g ( \\mathsf { x } , t , \\mathsf { y } ) = \\sum _ { \\mathsf { n } \\in \\mathbb { N } ^ { d } } \\exp \\left [ - t E _ { \\mathsf { n } } \\right ] \\mathsf { h } _ { \\mathsf { n } } \\left ( \\mathsf { x } \\right ) \\mathsf { h } _ { \\mathsf { n } } \\left ( \\mathsf { y } \\right ) \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { 2 } \\left \\vert \\lambda \\right \\vert ^ { 1 - \\frac { i } { 2 } } \\left \\Vert t ^ { i \\nu } u ^ { \\left ( i \\right ) } \\right \\Vert _ { L _ { p } \\left ( 0 , a ; E \\right ) } + \\left \\Vert A u \\right \\Vert _ { L _ { p } \\left ( 0 , a ; E \\right ) } \\leq C \\left \\Vert f \\right \\Vert _ { L _ { p } \\left ( 0 , a ; E \\right ) } . \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} ( A ^ \\star ) _ { i } = ( A _ { - i } ) ^ \\star \\ \\ \\ \\ \\ \\ \\ \\ d _ { ( A ^ \\star ) _ i } : = ( - 1 ) ^ { i + 1 } ( d _ { A _ { - i - 1 } } ^ \\star ) \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{align*} \\lambda _ 2 ( p , \\Omega ) = \\min \\{ \\lambda : \\lambda \\} . \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{gather*} \\partial _ t L - d _ L \\Delta _ x L = - \\Theta ( x , t ) \\ , L + \\psi ( x , t ) , \\\\ [ 2 m m ] \\partial _ t L _ { o x } - d _ { L _ { o x } } \\Delta _ x L _ { o x } = \\Theta ( x , t ) L - \\underbrace { \\lambda _ { L _ { o x } M _ 1 } \\frac { L _ { o x } } { K _ { M _ 1 } + L _ { o x } } \\ , M _ 1 } _ { i n t a k e \\ , o f \\ , L _ { o x } \\ , b y \\ , M _ 1 } - \\underbrace { \\lambda _ { L _ { o x } M _ 2 } \\frac { L _ { o x } } { K _ { M _ 2 } + L _ { o x } } \\ , M _ 2 } _ { i n t a k e \\ , o f \\ , L _ { o x } \\ , b y \\ , M _ 2 } \\end{gather*}"}
-{"id": "5061.png", "formula": "\\begin{align*} D = q D _ 0 + j \\left ( \\left ( \\sum _ { P \\in B } ( P ) \\right ) - n ( Q ) \\right ) . \\end{align*}"}
-{"id": "5091.png", "formula": "\\begin{align*} \\mathbb { S } ^ \\rho _ { \\mathcal { Q } } ( f _ 1 , f _ 2 , f _ 3 ) : = \\sum _ { Q \\in \\mathcal { Q } } S _ Q ( f _ 1 , f _ 2 , f _ 3 ) . \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{align*} { \\alpha ^ { ( m ) } } = \\tilde q { \\alpha ^ { ( m + 1 ) } } + \\left ( { 1 - \\tilde q } \\right ) { \\alpha ^ { ( m - 1 ) } } \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{align*} \\phi = \\hat \\phi _ 0 \\otimes 1 + \\hat \\phi _ 1 \\otimes w ^ { \\alpha _ 1 } + \\cdots + \\hat \\phi _ { \\nu - 1 } \\otimes w ^ { \\alpha _ { \\nu - 1 } } , \\end{align*}"}
-{"id": "7323.png", "formula": "\\begin{align*} \\varphi = e ^ { 1 2 7 } + e ^ { 3 4 7 } + e ^ { 5 6 7 } + e ^ { 1 3 5 } - e ^ { 1 4 6 } - e ^ { 2 3 6 } - e ^ { 2 4 5 } , \\end{align*}"}
-{"id": "9475.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & - x & . & . & . & . \\\\ . & 1 & - y & - 1 & . & . \\\\ . & . & 1 & x & . & . \\\\ . & . & . & 1 & - y & - 3 \\\\ . & . & . & . & 1 & x \\\\ . & . & . & . & . & 1 \\end{bmatrix} s = \\begin{bmatrix} . \\\\ . \\\\ . \\\\ . \\\\ . \\\\ 1 \\end{bmatrix} , s = \\begin{bmatrix} x ( 1 - y x ) ( 3 - y x ) \\\\ ( 1 - y x ) ( 3 - y x ) \\\\ - x ( 3 - y x ) \\\\ 3 - y x \\\\ - x \\\\ 1 \\end{bmatrix} \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} ( z ) = ( g _ 1 ( z ) ) ~ \\mbox { a n d } ~ ( z ) \\ne ( g _ i ( z ) ) ~ \\mbox { f o r a n y } ~ i \\ne 1 . \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{align*} \\dfrac { \\Phi ( \\theta ) } { G ( 0 ) } = \\dfrac { \\theta \\lambda \\mu ( \\alpha _ { \\lambda } - \\alpha _ { \\mu } ) + \\lambda \\mu ( \\mu \\alpha _ { \\lambda } - \\lambda \\alpha _ { \\mu } ) } { \\theta ^ 2 ( \\mu - \\lambda ) + \\theta \\left [ ( \\mu ^ 2 - \\lambda ^ 2 ) + \\lambda \\mu ( \\alpha _ { \\mu } - \\alpha _ { \\lambda } ) \\right ] + \\lambda \\mu \\left [ \\mu ( 1 - \\alpha _ { \\lambda } ) - \\lambda ( 1 - \\alpha _ { \\mu } ) \\right ] } . \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} { } _ 1 { F _ 1 } \\left ( { \\alpha , \\beta ; \\frac { 1 } { { { \\Omega _ k } } } } \\right ) = 1 + \\frac { \\beta } { \\alpha } \\frac { 1 } { { { \\Omega _ k } } } + o \\left ( { \\frac { 1 } { { { \\Omega _ k } } } } \\right ) . \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} d X _ t & = ( r _ t ( X _ t - \\varphi _ t ^ 1 - \\varphi _ t ^ 2 ) ^ + - R _ t ( X _ t - \\varphi _ t ^ 1 - \\varphi _ t ^ 2 ) ^ - + \\varphi _ t ^ 1 \\mu ^ 1 _ t + \\varphi _ t ^ 2 \\mu ^ 2 _ t ) d t + \\varphi _ t ' \\sigma _ t d W _ t + \\varphi _ t ' \\beta _ t d \\tilde N _ t \\\\ & = ( r _ t X _ t + \\varphi _ t ' ( \\mu _ t - r _ t { \\bf 1 } ) - ( R _ t - r _ t ) ( X _ t - \\varphi _ t ^ 1 - \\varphi _ t ^ 2 ) ^ - ) d t + \\varphi _ t ' \\sigma _ t d W _ t + \\varphi _ t ' \\beta _ t d \\tilde N _ t . \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} F _ * ^ e ( L / ( f + u v ) ) = F _ * ^ e ( L ) / F _ * ^ e ( J ) = \\bigoplus _ { k = 0 } ^ { q - 1 } M _ k / M _ k F _ * ^ e ( f + u v ) . \\end{align*}"}
-{"id": "1510.png", "formula": "\\begin{align*} \\frac { B ( x ) } { A ( x ) } = \\frac { \\varrho _ 0 } { x - a _ 0 } + \\frac { \\varrho _ 1 } { x - a _ 1 } + \\dots + \\frac { \\varrho _ p } { x - a _ p } \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{align*} \\| f g \\| _ { r , s , D } \\leq \\| f \\| _ { 3 , \\infty , D } \\| g \\| _ { q , s , D } , \\frac { 1 } { r } = \\frac { 1 } { 3 } + \\frac { 1 } { q } , q , \\ , r \\in ( 1 , \\infty ) , \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} \\big ( ( \\lambda , g ) ( \\mu , h ) \\big ) ( \\nu , k ) & = ( \\lambda \\rho _ g ( \\mu ) , g + h ) ( \\nu , k ) = ( \\lambda \\rho _ g ( \\mu ) \\rho _ { g + h } ( \\nu ) , g + h + k ) \\\\ & = ( \\lambda , g ) ( \\mu \\rho _ h ( \\nu ) , h + k ) = ( \\lambda , g ) \\big ( ( \\mu , h ) ( \\nu , k ) \\big ) . \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{align*} \\int _ { \\Omega } D ^ m u : D ^ m \\varphi + \\rho u \\varphi d x = \\Lambda \\int _ { \\Omega } \\rho u \\varphi d x \\ , , \\ \\ \\ \\forall \\varphi \\in H ^ m ( \\Omega ) . \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{align*} G ( s ) - G ( t ) & \\le ( 1 + \\alpha ^ 2 ) s r - t r + s ( t - s ) , \\\\ & = r ^ 2 [ ( 1 + \\alpha ^ 2 ) ( 1 - \\alpha ) - ( 1 - \\alpha ^ 2 ) + ( 1 - \\alpha ) ^ 2 \\alpha ] . \\end{align*}"}
-{"id": "2776.png", "formula": "\\begin{align*} & \\sharp T ( \\varepsilon , K , \\Lambda _ d , b ) \\\\ & = \\sum _ { 1 \\leqslant k \\leqslant N } \\left ( \\frac { k \\varepsilon K ^ 2 } { N ^ 2 \\det ( \\Lambda _ d ) } B ^ { 2 - \\frac { 1 } { r } } + O \\left ( \\frac { K ^ 2 } { N ^ 2 \\det ( \\Lambda _ d ) } B ^ { 2 - \\frac { 1 } { r } } \\right ) + O \\left ( b \\det ( \\Lambda _ d ) \\log \\left ( \\frac { k K ( \\lambda _ 2 - \\alpha \\mu _ 2 ) } { N \\det ( \\Lambda _ d ) } B \\right ) \\right ) \\right ) . \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} P ^ * \\tilde K u ( x , w ) = \\int _ { \\R ^ n \\times V } { \\Phi ( x , y - x , \\theta , w , v ' ) _ { | \\theta = { y - x \\over | y - x | } } \\over | x - y | ^ { n - 1 } } \\rho ( u ) ( y , v ' ) d y d v ' , \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\lambda } ^ d \\xi _ t ( O ) \\leq e ^ { - t } \\sum _ { n = 0 } ^ { \\infty } \\frac { t ^ n \\lambda ^ n M ^ { 2 n } ( d + 1 ) ^ n } { n ! E { \\widetilde { \\rho } } ^ 2 } { \\widehat { E } } [ ( E { \\widetilde { \\rho } } ^ 2 ) ^ { | X _ n | } ] \\end{align*}"}
-{"id": "6600.png", "formula": "\\begin{align*} f ( r ) \\ ; = \\ ; o ( r ^ { 1 / 2 } ) \\textrm { a s } r \\downarrow 0 \\ , . \\end{align*}"}
-{"id": "2355.png", "formula": "\\begin{align*} \\Omega : = \\big \\{ \\bar { \\xi } \\in 2 \\pi R ^ { 1 - \\sigma } \\mathbb { Z } ^ { n - 1 } \\ , : \\ , | \\bar { \\xi } | \\le R \\big \\} + B ( 0 , \\rho ) \\ , , \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} ( H ^ * B ) ( x ) = B ( x , x ) = \\lim _ { y \\to x } \\frac { Q ( x ) S ( y ) - S ( x ) Q ( y ) } { x - y } = - Q ( x ) S ' ( x ) + S ( x ) Q ' ( x ) . \\end{align*}"}
-{"id": "7663.png", "formula": "\\begin{align*} \\C G = \\C e _ 1 \\oplus \\cdots \\oplus \\C e _ n . \\end{align*}"}
-{"id": "6439.png", "formula": "\\begin{align*} q ( x e _ 1 + y e _ 2 + z e _ 3 ) = a x ^ 2 + b y ^ 2 + c z ^ 2 + u y z \\end{align*}"}
-{"id": "8802.png", "formula": "\\begin{align*} 2 \\tilde { G } ^ { i } \\left ( x , y \\right ) = 2 G ^ { i } \\left ( x , y \\right ) + P \\left ( x , y \\right ) y ^ { i } , ~ \\ \\forall \\left ( x , y \\right ) \\in A . \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} \\textrm { e r r o r } = \\min _ { \\phi \\in [ 0 , 2 \\pi ) } \\| \\rho - e ^ { i \\phi } \\rho _ 0 \\| _ 2 / \\| \\rho _ 0 \\| _ 2 . \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} \\delta ^ 2 \\vartheta _ 2 ( 0 ) = 2 \\vartheta _ 2 ( 0 ) ( \\psi _ 2 \\psi _ 3 + \\psi _ 2 \\psi _ 4 - \\psi _ 3 \\psi _ 4 ) + ( \\delta \\vartheta _ 2 ) ^ 2 / \\vartheta _ 2 ( 0 ) \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} u = \\Theta v \\equiv P u _ 0 + ( P _ 1 \\circ \\Gamma ) \\bigl ( u _ T - P u _ 0 \\big | _ { t = T } + P _ 2 ( v ^ 2 / 2 ) \\big | _ { t = T } \\bigr ) - P _ 2 ( v ^ 2 / 2 ) , \\end{align*}"}
-{"id": "7099.png", "formula": "\\begin{align*} U ^ * = \\{ u ^ * _ i : i \\le d \\} \\cup \\{ u ^ * : u \\in \\max ( r _ k ( T ) ) \\setminus A _ e \\} . \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} t = \\frac { \\sqrt { 6 } } { 5 A } ( y ( t ) ^ 5 - 1 ) + \\frac { 1 } { 1 0 A ^ 2 } \\log \\left \\vert \\frac { 1 - 2 \\sqrt { 6 } A y ( t ) ^ 5 } { 1 - 2 \\sqrt { 6 } A } \\right \\vert . \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} d \\boldsymbol { X } _ t = g ( \\boldsymbol { X } _ t ) d \\boldsymbol { W } _ t h ( \\boldsymbol { X } _ t ) + h ( \\boldsymbol { X } _ t ) d \\boldsymbol { W } _ t ^ * g ( \\boldsymbol { X } _ t ) + \\left ( b ( \\boldsymbol { X } _ t ) + \\alpha T r \\left ( \\boldsymbol { X } _ t \\right ) \\boldsymbol { I } \\right ) d t , \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} w _ { R , \\alpha } ^ + \\left ( \\frac { x } { R } \\right ) = \\left ( 1 - \\frac { x ^ 2 } { R ^ 2 } \\right ) ^ { - 1 / 2 } \\exp \\left ( - N V \\left ( \\frac { x } { \\alpha R } \\right ) - \\varepsilon _ R \\frac { N x } { R } \\right ) . \\end{align*}"}
-{"id": "7584.png", "formula": "\\begin{align*} b _ t = b - ( 1 + \\frac 2 k ) t \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { \\Gamma \\left ( k + \\alpha \\right ) } { \\Gamma \\left ( k + 1 \\right ) } { _ { 1 } F _ { 1 } } \\left ( \\alpha + k , \\beta , x \\right ) z ^ { k } = \\left ( 1 - z \\right ) ^ { - \\alpha } \\Gamma \\left ( \\alpha \\right ) { _ { 1 } F _ { 1 } } \\left ( \\alpha , \\beta , \\frac { x } { 1 - z } \\right ) . \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} \\int \\varphi _ j ( t ; w ) \\varphi _ k ( t ; w ) w ( t ) d t = \\delta _ { j k } . \\end{align*}"}
-{"id": "9368.png", "formula": "\\begin{align*} L ^ 1 ( \\Omega ; F ) = L ^ 1 ( \\Omega ) \\overset { \\wedge } { \\otimes } F . \\end{align*}"}
-{"id": "5905.png", "formula": "\\begin{align*} 0 & < ( p + \\ell ) \\phi ( p - \\ell ) - ( p - \\ell ) \\phi ( p + \\ell ) \\\\ & = p [ \\phi ( p - \\ell ) - \\phi ( p + \\ell ) ] + \\ell \\phi ( p - \\ell ) + \\ell \\phi ( p + \\ell ) \\\\ & \\leq p [ \\phi ( p - \\ell ) - \\phi ( p + \\ell ) ] + \\ell ( p - \\ell ) + \\ell ( p + \\ell ) \\\\ & = p [ \\phi ( p - \\ell ) - \\phi ( p + \\ell ) + 2 \\ell ] . \\end{align*}"}
-{"id": "9969.png", "formula": "\\begin{align*} \\begin{array} { r c l } x _ 1 ( t _ * ^ + ) = x _ 3 ( t _ * ^ + ) & = & x _ 1 ( t _ * ^ - ) = x _ 3 ( t _ * ^ - ) \\\\ x _ 2 ( t _ * ^ + ) = x _ 4 ( t _ * ^ + ) & = & \\frac { m _ 1 x _ 2 ( t _ * ^ - ) + m _ 2 x _ 4 ( t _ * ^ - ) } { m } \\end{array} \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} { \\displaystyle | \\Delta t \\sum _ { k = 1 } ^ { m } Q _ { 3 } ^ { k } ( \\partial { \\theta _ { \\psi } ^ { k } } ) | \\leq C \\{ h ^ { 2 r } + ( \\Delta t ) ^ { 4 } \\} + C \\Vert \\theta _ { \\psi } ^ { m } \\Vert _ { \\mathcal { L } ^ 2 } ^ { 2 } + \\frac { 1 } { 1 6 } \\Vert \\nabla \\theta _ { \\psi } ^ { m } \\Vert _ { \\mathbf { L } ^ 2 } ^ { 2 } + C \\Delta t \\sum _ { k = 0 } ^ { m } { \\Vert \\nabla \\theta _ { \\psi } ^ { k } \\Vert _ { \\mathbf { L } ^ 2 } ^ { 2 } } . } \\end{align*}"}
-{"id": "1490.png", "formula": "\\begin{align*} Q _ k ( x ) : = \\prod _ { \\substack { j = 1 \\\\ j \\neq k } } ^ n \\frac { x - x _ j } { x _ k - x _ j } = \\frac { Q ( x ) } { Q ' ( x _ k ) ( x - x _ k ) } , k = 1 , 2 , \\dots , n , \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 1 } \\frac { ( 4 x ^ { 2 } - 4 x y - 1 + \\rho ^ { 2 } ) d y } { \\pi \\sqrt { 1 - y ^ { 2 } } ( ( 1 - \\rho ^ { 2 } ) ^ { 2 } - 4 x y \\rho ( 1 + \\rho ^ { 2 } ) + 4 \\rho ^ { 2 } ( x ^ { 2 } + y ^ { 2 } ) ) } = \\frac { 4 x ^ { 2 } - 1 - \\rho ^ { 2 } } { ( 1 + \\rho ^ { 2 } ) ^ { 2 } - 4 x ^ { 2 } \\rho ^ { 2 } } . \\end{align*}"}
-{"id": "6335.png", "formula": "\\begin{align*} w _ i ' . v _ j \\geq 0 \\ ; \\ ; \\ ; \\ ; \\ ; j = 1 , \\hdots , n . \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} [ \\omega _ \\epsilon ( t ) ] = [ \\omega _ 0 ] - 2 \\pi t \\big ( c _ 1 ( X ) - ( 1 - \\alpha ) [ D _ 0 ] \\big ) , \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t v + v \\cdot \\nabla v + \\nabla p = \\nabla \\cdot ( \\frac { \\partial W ( F ) } { \\partial F } F ^ \\top ) , \\\\ [ - 4 m m ] \\\\ \\partial _ t F + v \\cdot \\nabla F = \\nabla v F , \\\\ [ - 4 m m ] \\\\ \\nabla \\cdot v = 0 , \\nabla \\cdot F ^ \\top = 0 , \\end{cases} \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} \\alpha + \\lambda + 1 = \\alpha ' + \\lambda ' , \\lambda , \\lambda ' \\in \\mathrm { s u p p } ( a ) , \\qquad \\alpha , \\alpha ' \\in ( - \\rho , \\rho ) . \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{align*} \\begin{aligned} \\Bigg | \\int \\limits _ { | x - e | + | x | = 2 \\tau } \\frac { q ( x ) } { | 2 \\tau x - | x | e | } d S _ { x } \\Bigg | \\leq K \\int \\limits _ { | x - e | + | x | \\leq 2 \\tau } \\frac { | q ( x ) | } { | x | | x - e | } d x , \\ \\forall \\tau \\in ( 1 / 2 , T / 2 ] . \\end{aligned} \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} E _ { \\mathbf { A } } ( t , x ) : = - \\partial _ { t } \\mathbf { A } ( t , x ) \\ , t \\in \\mathbb { R } , \\ x \\in \\mathbb { R } ^ { d } \\ , \\end{align*}"}
-{"id": "4069.png", "formula": "\\begin{align*} \\tilde { L } _ t = \\frac { \\tilde { \\eta } } { \\eta } L _ t . \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{align*} S ( X | M ) = \\int _ M S ( X | M = m ) \\ , \\mathrm { d } p _ M ( m ) \\ ; . \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{align*} S _ { M , k , q , \\gamma , i } : = \\bigcup _ j A _ { j , i } \\end{align*}"}
-{"id": "1159.png", "formula": "\\begin{align*} \\sup _ { n \\in \\N ^ * } \\| u _ n ^ l & \\| _ { B V ( T _ n ) } < \\infty , \\ \\ \\ \\ \\ \\ \\ \\ \\sup _ { n \\in \\N ^ * } \\| u _ n ^ l \\| _ { L ^ \\infty ( ( 0 , T _ n ) , \\R ) } < \\infty , \\\\ & \\sup _ { n \\in \\N ^ * } T _ n \\| u _ n ^ l \\| _ { L ^ \\infty ( ( 0 , T _ n ) , \\R ) } < \\infty . \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} \\mathbb { E } _ { H _ f } [ F ^ { ( \\tau ) } \\circ m ] = \\mathbb { E } _ { \\prod _ { i = 1 } ^ n S _ { r _ i } } [ F ^ { ( \\tau ) } ] = \\prod _ { i = 1 } ^ n \\mathbb { E } _ { S _ { r _ i } } [ \\prod _ { \\ell = 1 } ^ { \\infty } ( 1 + t _ { d _ i \\ell } ) ^ { X _ { \\ell } } ] . \\end{align*}"}
-{"id": "1634.png", "formula": "\\begin{align*} \\mathfrak { V } _ { \\pm } \\equiv \\bigg \\{ \\phi \\in H ^ { 1 / 2 } ( \\partial \\Omega ) \\colon \\pm \\tfrac { 1 } { 2 } \\phi + \\mathcal { K } _ { \\partial \\Omega } [ \\phi ] = 0 \\bigg \\} \\ , . \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} m _ { k } : = { \\rm c a r d } ( J _ { k } ) , \\qquad \\mbox { w h e r e } \\ ; J _ { k } : = \\left \\{ n \\in { \\Bbb N } ^ * \\times { \\Bbb N } ^ * \\ ; ; \\ ; n _ { 1 } ^ 2 + n _ { 2 } ^ 2 = \\lambda _ { k } \\right \\} . \\end{align*}"}
-{"id": "1185.png", "formula": "\\begin{align*} d ^ { \\rm s u p } ( 1 0 ^ { - 3 } ) \\ , \\leq \\ , \\left \\{ \\begin{array} { l l } 2 & \\mbox { f o r \\ } a = 4 , \\\\ 3 & \\mbox { f o r \\ } a = 3 , \\\\ 6 & \\mbox { f o r \\ } a = 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "3286.png", "formula": "\\begin{align*} G ( x , y ) = g ^ 2 ( x ) h ^ 2 ( y ) + g ^ 2 ( y ) h ^ 2 ( x ) . \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = \\alpha _ 2 y _ 5 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\beta _ 1 } y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = - \\frac { \\alpha _ 3 } { \\beta _ 1 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 2 ] = y _ 4 , \\\\ [ y _ 2 , y _ 3 ] = \\gamma _ 2 y _ 5 , [ y _ 3 , y _ 2 ] = \\gamma _ 4 y _ 5 . \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} j = 1 \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} \\iota _ { N _ 2 ( \\theta ) } ^ * ( \\iota _ \\theta ^ * a \\iota _ \\theta ) ^ { - 1 } \\iota _ { N _ 2 ( \\theta ) } = a ^ { \\textnormal { h o m } } ( \\theta ) ^ { - 1 } ( \\theta \\in \\Theta ) , \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} d ^ b ( { \\bf 1 } ) = 0 . \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{align*} \\left [ P ^ { s , N } _ { H P } ( t ) f \\right ] ( x _ n ) = \\left [ \\mathcal { S } ^ N ( t ) f \\circ \\mathsf { e v a l } _ N \\right ] ( U ^ * x _ n U ) \\to \\left [ \\mathcal { S } ^ N ( t ) f \\circ \\mathsf { e v a l } _ N \\right ] ( U ^ * x U ) = \\left [ P ^ { s , N } _ { H P } ( t ) f \\right ] ( x ) . \\end{align*}"}
-{"id": "7489.png", "formula": "\\begin{gather*} ( m n ^ * ) ^ * = n m ^ * , m m ^ * \\geq 0 , m m ^ * = 0 \\implies m = 0 \\\\ ( m ^ * n ) ^ * = n ^ * m , m ^ * m \\geq 0 , m ^ * m = 0 \\implies m = 0 \\end{gather*}"}
-{"id": "4538.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle I _ 3 ^ { ( k ) } = B ( \\mathbf { A } ^ { k - \\frac { 1 } { 2 } } ; ( I _ { h } \\overline { \\Psi } ^ { k } - \\overline { \\Psi } ^ { k } ) , \\overline { \\theta } _ { \\Psi } ^ { k } ) + B ( \\mathbf { A } ^ { k - \\frac { 1 } { 2 } } ; ( \\overline { \\Psi } ^ { k } - \\Psi ^ { k - \\frac { 1 } { 2 } } ) , \\overline { \\theta } _ { \\Psi } ^ { k } ) . } \\end{array} \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} b _ { n + 1 } = F ( n , b _ n ) \\end{align*}"}
-{"id": "9894.png", "formula": "\\begin{align*} R _ { \\theta r \\theta r } & = - \\sigma { \\sigma '' } & R _ { \\phi r \\phi r } & = - \\sigma \\sigma '' \\sin ^ 2 ( \\theta ) \\\\ R _ { z r z r } & = - \\tau { \\tau '' } & R _ { \\phi \\theta \\phi \\theta } & = \\left ( 1 - ( \\sigma ' ) ^ 2 \\right ) \\sigma ^ 2 \\sin ^ 2 ( \\theta ) \\\\ R _ { \\theta z \\theta z } & = - \\sigma \\sigma ' \\tau \\tau ' & R _ { \\phi z \\phi z } & = - \\sigma \\sigma ' \\tau \\tau ' \\sin ^ 2 ( \\theta ) \\end{align*}"}
-{"id": "889.png", "formula": "\\begin{align*} \\langle : X \\otimes X : , \\Phi \\rangle = \\int _ { \\mathbb { R } ^ 2 } ^ { '' } \\widehat { \\Phi } ( x , y ) Z _ G ( d x ) Z _ G ( d y ) , \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} \\begin{aligned} d \\varphi ( t ) & = \\frac { \\sqrt { 6 } } { 6 } y ( t ) ^ { - 1 } z ( t ) ^ { - 2 } \\Big ( \\big ( z ( t ) ^ { - 2 } - y ( t ) ^ { - 2 } \\big ) ( f ^ { 1 2 3 4 } + f ^ { 1 2 5 6 } ) - 2 z ( t ) ^ { - 2 } f ^ { 3 4 5 6 } \\Big ) \\\\ & = \\tau _ 0 ( t ) \\star _ t \\varphi ( t ) + \\star _ t \\tau _ 3 ( t ) , \\end{aligned} \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{align*} R ( e _ j ) _ { x _ 1 , \\dots , x _ m } = R ^ { ( S \\circ F ) ( x _ { r + 1 } , \\dots , x _ m ) } ( e _ j ) _ { x _ 1 , \\dots , x _ r } , \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{align*} \\Psi _ 1 ( 0 , y ) = { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & \\frac 1 2 ( 1 - y ) \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] = ( 1 - z ) ^ a { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & \\frac 1 2 ( 1 + y ) \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac z { z - 1 } \\bigg ] = \\Psi _ 2 ( 0 , - y ) , \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} C _ \\ell ( f ) ^ i = A ^ { i + 1 } \\oplus B ^ i , d _ { C _ \\ell ( f ) } ^ i = \\begin{bmatrix} - d _ A ^ { i + 1 } & \\\\ f ^ { i + 1 } & d _ B ^ i \\end{bmatrix} . \\end{align*}"}
-{"id": "6113.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , X _ { 1 , \\ , j + 1 , \\ , 2 } ] = 0 . \\end{align*}"}
-{"id": "9994.png", "formula": "\\begin{align*} W ( B _ { 1 } ( n , d , x ) ) & = \\frac { n ^ { 3 } } { 3 } + ( - \\frac { 3 d } { 2 } - \\frac { 5 } { 4 } ) n ^ { 2 } + ( 4 d ^ { 2 } + 1 0 d + \\frac { 1 3 } { 3 } - 2 x ) n + \\\\ & + 2 x ^ { 2 } - \\frac { 8 } { 3 } d ^ { 3 } - 1 2 d ^ { 2 } - \\frac { 4 6 d } { 3 } - 7 . \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\sigma _ { i j } ( v _ i - v _ j ) = 0 \\ \\ \\hbox { f o r a l l } i \\in \\hbox { i n t } ( V ) , \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 f } { \\partial y ^ 2 } = - 1 0 - 5 v + 2 v / y + 7 y - 2 v \\ln 4 + 6 y \\ln 4 + 4 v \\ln v - 2 ( v + 3 y ) \\ln y \\ge 0 . \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{align*} | v ^ { ( k ) } | \\ ; \\ ; \\cdot \\prod _ { i = \\Delta ^ { ( k ) } - 2 \\delta ^ { ( k ) } } ^ { \\Delta ^ { ( k ) } - 1 } | w _ { i } ^ { ( k ) } | = 2 ^ { \\ , \\delta ^ { ( k ) } - \\tau ^ { ( k ) } } > 2 ^ { \\ , 3 \\alpha \\delta ^ { ( k ) } } > C \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} S ( B | M ) _ { ( \\Phi \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) } = S \\left ( ( \\Phi \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) \\right ) - S ( \\hat { \\rho } _ M ) = S \\left ( \\tilde { \\Phi } ( \\hat { \\rho } _ A ) \\right ) - S ( \\hat { \\rho } _ A ) \\ ; , \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} - P \\bigtriangleup _ { \\varepsilon } u + P A u + \\lambda u = f \\left ( x \\right ) , x \\in R _ { + } ^ { n } , \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} 0 = a _ 0 < a _ 1 < \\ldots < a _ m = 1 , 0 = b _ 0 < b _ 1 < \\ldots < b _ n = 1 \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} 0 \\neq [ [ L _ { - 1 } , \\ , L _ 1 ] , \\ , S _ r ] = [ [ L _ { - j } , \\ , S _ j ] , \\ , S _ r ] = [ [ L _ { - j } , \\ , S _ r ] , \\ , S _ j ] \\end{align*}"}
-{"id": "3145.png", "formula": "\\begin{align*} \\nabla _ i \\nabla _ j R _ { k l } - \\nabla _ j \\nabla _ i R _ { k l } = R _ { i j k s } R _ { s l } + R _ { i j l s } R _ { k s } , \\end{align*}"}
-{"id": "2564.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to 0 } \\left | \\bigcup _ { \\stackrel { 0 < | k | \\leq r } { ( k ) \\subseteq \\{ 1 , \\ldots , N } } \\ ; \\bigcup _ { l \\geq N } \\ ; \\{ c ^ 2 \\in [ c ^ 2 _ { l , \\alpha , - } , c ^ 2 _ { l , \\alpha , + } ] \\ ; : | p _ { k , l } ( c ^ 2 ) | \\leq \\gamma \\} \\right | & = 0 . \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{align*} \\Phi = \\sum _ { j = 1 } ^ m \\phi _ j \\otimes \\psi _ j , \\end{align*}"}
-{"id": "4259.png", "formula": "\\begin{align*} - \\Delta \\xi _ m + \\xi _ m = u _ m , \\quad \\xi _ m \\rightarrow 0 , \\quad \\ , x \\rightarrow \\infty . \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} S _ D ^ { - 1 } \\Phi ( r ) \\ ; \\sim \\ ; \\begin{pmatrix} p ^ + \\ ! \\\\ p ^ - \\ ! \\end{pmatrix} r ^ B + o ( r ^ { 1 / 2 } ) \\textrm { a s } \\ ; r \\downarrow 0 \\ , , \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} [ p _ v ] = \\big [ \\sum _ { e \\in v E ^ 1 } s _ e s ^ * _ e \\big ] = \\sum _ { e \\in v E ^ 1 } ( - 1 ) ^ { \\delta ( e ) } [ s _ e ^ * s _ e ] = \\sum _ { w \\in E ^ 0 } ( A ^ \\delta _ E ) ^ t ( v , w ) [ p _ w ] \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} A = ( \\pi _ U + \\pi _ V ) A ( \\pi _ U + \\pi _ V ) = \\pi _ U A \\pi _ U + \\pi _ V A \\pi _ V . \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{align*} J ( k ) ^ { - 1 } = \\frac { N _ { - , \\kappa } } { k - i \\kappa } + N _ { 0 , \\kappa } + O ( k - i \\kappa ) , k \\to i \\kappa \\ ; \\ ; \\overline { \\mathbb { C } } ^ + , \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{align*} \\sigma _ { k \\ast f _ { q _ i } } = g _ { k , i , q _ i , 1 } + g _ { k , i , q _ i , 2 } + g _ { k , i , q _ i , 3 } , \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} | \\Delta | _ \\infty = | a _ 1 | _ \\infty \\cdots | a _ n | _ \\infty . \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} \\begin{aligned} & \\| u \\| _ { Y _ k ( \\Pi _ { t _ 0 } ) } \\\\ & \\leq c ( T , k , b ) \\Bigl ( \\| u _ 0 \\| _ { \\widetilde H ^ k ( \\Sigma ) } + t _ 0 ^ { 1 / 6 } \\| f \\| _ { M _ k ( \\Pi _ { t _ 0 } ) } + \\sum _ { j = 0 } ^ { j _ 0 - 1 } \\| \\partial _ t ^ j f \\big | _ { t = 0 } \\| _ { \\widetilde H ^ { k - 3 ( j + 1 ) } ( \\Sigma ) } \\Bigr ) . \\end{aligned} \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} u : = ( u _ { 0 } , u _ { 1 } ) \\mapsto A u : = \\left ( - u _ { 1 } , L u _ { 0 } + B u _ { 1 } \\right ) \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{align*} N _ { - , k _ j } P _ j = N _ { - , k _ j } . \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{align*} x ~ = ~ \\frac { p + p ' y } { q + q ' y } \\end{align*}"}
-{"id": "8006.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| w _ k ( t ) - w ( t ) \\| _ { 2 , \\Omega _ L } = 0 , \\forall L \\in [ R , \\infty ) , \\ , \\forall t \\in ( 0 , T ) \\setminus J , \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} \\widehat \\tau _ { P _ 1 } ( H _ { P _ 1 } ( m ) - T ) = 1 \\ ; \\ ; \\Leftrightarrow \\ ; \\ ; | b | > e ^ { T _ 1 } , \\widehat \\tau _ { P _ 2 } ( H _ { P _ 2 } ( m ) - T ) = 1 \\ ; \\ ; \\Leftrightarrow \\ ; \\ ; | a b | > e ^ { T _ 2 } . \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} B ' = \\{ b \\in B \\mid [ \\phi , b ] = 0 \\} , M ' = \\{ m \\in M \\mid [ \\phi ^ * , m ] = 0 \\} \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} \\sum _ { \\mu \\models \\pi } x _ { \\mu } = \\sum _ { \\mu _ { 1 } \\models \\pi _ { 1 } } \\dots \\sum _ { \\mu _ { p } \\models \\pi _ { p } } x _ { \\mu _ { 1 } } \\dots x _ { \\mu _ { p } } \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{align*} R ^ e _ n ( s _ { v _ j } ) ( w _ { i } ) = w _ { i } . \\end{align*}"}
-{"id": "1910.png", "formula": "\\begin{align*} g ( \\delta A , X ) = \\delta \\langle A , X \\rangle + g ( A , D ^ g X ) , X \\in \\mathfrak { X } ( M ) \\end{align*}"}
-{"id": "5801.png", "formula": "\\begin{align*} \\exp \\frac { S ( C | M ) _ { ( \\Phi \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) } } { n } & \\ge \\eta \\exp \\frac { S ( A | M ) _ { \\hat { \\rho } _ { A M } } } { n } + \\left | 1 - \\eta \\right | \\exp S _ 0 \\\\ & = \\eta \\exp \\frac { - S ( \\hat { \\rho } _ A ) } { n } + \\left | 1 - \\eta \\right | \\exp S _ 0 \\\\ & \\ge \\eta \\exp ( - g ( E ) ) + \\left | 1 - \\eta \\right | \\exp S _ 0 \\ ; , \\end{align*}"}
-{"id": "1397.png", "formula": "\\begin{align*} f _ i ( X _ i ) & = h \\left ( g ( x _ 1 ) , \\ldots , g ( X _ i ) , \\ldots , g ( x _ m ) \\right ) - \\sum \\limits _ { j \\neq i } f _ j ( x ) . \\end{align*}"}
-{"id": "6624.png", "formula": "\\begin{align*} f \\ ; = \\ ; A _ 0 v _ 0 + A _ \\infty v _ \\infty + f _ \\mathrm { p a r t } \\ , , \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{align*} \\Sigma _ { \\sigma , x + 1 } & = \\sum _ { s = \\pm 1 } \\left ( \\sum _ { 0 \\leq n \\leq x } \\sum _ { k = 1 } ^ { \\left \\lfloor \\frac { \\sqrt { 2 4 n + 2 5 } - s } { 6 } \\right \\rfloor } ( - 1 ) ^ { k + 1 } \\frac { k ( 3 k + s ) } { 2 } \\cdot p ( x - n ) \\right ) . \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} A x : = \\lim _ { t \\to 0 } \\dfrac { T ( t ) x - x } { t } , \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} \\alpha \\xi ^ { m } + \\alpha \\eta ^ { m } + q _ { m } ( \\xi , \\eta ) = ( \\alpha \\xi ^ { m } + \\alpha \\eta ^ { m } ) \\left ( 1 + \\frac { q _ { m } ( \\xi , \\eta ) } { \\alpha \\xi ^ { m } + \\alpha \\eta ^ { m } } \\right ) \\geq \\frac { \\alpha } { 2 } ( \\xi ^ { m } + \\eta ^ { m } ) > c | \\eta | . \\end{align*}"}
-{"id": "2309.png", "formula": "\\begin{align*} \\sigma _ 0 ( t ) = ( \\xi ( t ) , - 1 ) = \\begin{pmatrix} 1 & - \\xi ( t ) \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} c = \\sum _ { i \\in \\Z , \\ , \\Z g \\in \\Z \\backslash G } c _ { g , i } \\gamma _ 1 ^ i g \\ , , \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} f ( \\lambda ) & = f ( 1 + \\lambda ) = 0 & f ' ( \\lambda ) & = 2 c ^ { n - 1 } & f ' ( 1 + \\lambda ) & = - 2 ( p + c ) ^ { n - 1 } . \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} [ y _ 1 , y _ 2 ] = - y _ 3 + \\frac { \\alpha _ 4 } { \\beta _ 3 } y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = y _ 3 , [ y _ 2 , y _ 2 ] = y _ 4 , [ y _ 2 , y _ 3 ] = - \\theta _ 2 y _ 5 , [ y _ 3 , y _ 2 ] = ( \\theta _ 2 - 1 ) y _ 5 , [ y _ 1 , y _ 4 ] = y _ 5 . \\end{align*}"}
-{"id": "1031.png", "formula": "\\begin{align*} ( p , q ) \\in [ 2 , \\infty ] ^ 2 , ( q , p ) \\ne ( 2 , \\infty ) , \\frac { 1 } { p } + \\frac { 1 } { q } = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "3701.png", "formula": "\\begin{align*} P _ r ( \\hat { N } _ { x , r } ) \\subset \\mathcal { C } _ { \\varepsilon } : = \\{ ( u , w ) \\in N _ { z _ 2 } \\mid u \\in L , w \\in L ^ { \\perp } , \\parallel w \\parallel \\leq \\varepsilon \\parallel u \\parallel \\} . \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{align*} M _ L ( f x _ { n + 1 } ^ d , e ) = \\left [ \\begin{array} { c c c } C _ { ( 0 , 0 ) } & \\ldots & C _ { ( 0 , q - 1 ) } \\\\ \\vdots & & \\vdots \\\\ C _ { ( q - 1 , 0 ) } & \\ldots & C _ { ( q - 1 , q - 1 ) } \\\\ \\end{array} \\right ] \\end{align*}"}
-{"id": "9363.png", "formula": "\\begin{align*} | S _ { k , 1 } | \\geq ( 1 - \\gamma ) \\sum _ { l , m } \\sum _ { i = 1 } ^ n | T _ l | | C _ { k , l , m } ^ { ( i ) } | \\geq \\lambda | S _ k | . \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} \\chi ( Z , D ) & = \\chi ( Z ' , D ) + \\chi ( E _ i , D - Z ' ) \\\\ & = \\frac { ( 2 D - K _ Y - Z ' ) \\cdot Z ' } { 2 } + \\frac { ( 2 ( D - Z ' ) - K _ Y - E _ i ) \\cdot E _ i } { 2 } \\\\ & = \\frac { ( 2 D - K _ Y - Z ) \\cdot Z } { 2 } . \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} V = \\Delta _ \\delta \\cup \\bigcup _ { m = 1 } ^ { \\infty } V _ m \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} G _ 1 : = \\langle h , G ^ 0 \\rangle \\leq G . \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} \\Phi ( \\hat { \\rho } _ A ) = \\mathcal { B } _ \\eta ( \\hat { \\rho } _ A \\otimes \\hat { \\sigma } _ B ) \\ ; , \\eta \\ge 0 \\ ; . \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} { \\Bbb F } _ { k } : = \\bigoplus _ { j \\geq k + 1 } N ( L - \\lambda _ { j } I ) \\ , , \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} \\tau '^ { \\frac { 1 } { a ' } } = \\begin{pmatrix} 1 & \\frac { 1 } { a ' } \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} p \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} c ( Y ) = ( 3 f ^ * H + 6 f ^ * \\pi ^ * L - 2 E _ 1 ) ( 1 + E _ 1 ) \\frac { ( 1 + f ^ * Z _ 1 - E _ 1 ) ( 1 + f ^ * Z _ 2 - E _ 1 ) ( 1 + f ^ * Z _ 3 - E _ 1 ) } { ( 1 + f ^ * Z _ 1 ) ( 1 + f ^ * Z _ 2 ) ( 1 + f ^ * Z _ 3 ) } f ^ * c ( X _ 0 ) . \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} B _ i = \\frac { 1 } { | C | } | R _ i \\cap C ^ 2 | , \\ i = 0 , 1 , \\ldots , n . \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} \\left | \\left ( \\prod _ { k = k _ { 0 } + 1 } ^ { k _ { 0 } + n _ j } \\omega _ { k } \\right ) x _ { k _ { 0 } + n _ j } - x _ { k _ { 0 } } \\right | < \\varepsilon \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } d ^ { - n } f ^ { - n } _ * ( [ M ] ) = \\mu ^ - _ { | W ^ s ( p ) } ( M ) \\mu ^ + , \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} \\sigma _ { t } ^ { \\prime } [ l ] = \\sigma _ { t - 1 } ^ { \\prime } [ l + 1 ] \\mbox { a n d } O P T _ { t } ^ { \\prime } [ l ] = O P T _ { t - 1 } ^ { \\prime } [ l + 1 ] . \\end{align*}"}
-{"id": "8061.png", "formula": "\\begin{align*} \\xi _ K : = \\{ \\pi ( 0 ) : \\pi \\in K ^ { ( - 1 ) - } \\pi ( 0 ) \\in [ 0 , 1 ] \\} . \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{gather*} \\dim V _ { x _ 0 } ( \\theta ) = \\dim V _ { x _ 1 } ( \\theta ^ { - 1 } ) = \\dim V _ { x _ 2 } ( \\theta ) = \\dim V _ { x _ 3 } ( \\theta ^ { - 1 } ) . \\end{gather*}"}
-{"id": "1979.png", "formula": "\\begin{align*} & \\prod _ { j = 1 } ^ N z _ j ^ { N + 1 - j } ( 1 + t ^ { - 1 } z _ j ^ 2 ) ^ { - 1 } \\prod _ { 1 \\le j < k \\le N } ( 1 + t ^ { - 1 } z _ j z _ k ) ^ { - 1 } ( t ^ { - 1 } + z _ j z _ k ^ { - 1 } ) ^ { - 1 } \\\\ \\times & t ^ N \\langle 1 \\cdots M | \\mathcal { B } ^ \\prime ( z _ 1 ) \\cdots \\mathcal { B } ^ \\prime ( z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } - \\Delta u + b _ { 1 } ( x ) u _ { t } - { \\rm d i v } \\left ( b _ { 2 } ( x ) \\nabla u _ { t } \\right ) = 0 & \\mbox { i n } ( 0 , \\infty ) \\times \\Omega , \\\\ u ( t , \\sigma ) = 0 & \\mbox { o n } ( 0 , \\infty ) \\times \\partial \\Omega , \\\\ u ( 0 , x ) = u _ { 0 } ( x ) & \\mbox { i n } \\Omega \\\\ u _ { t } ( 0 , x ) = u _ { 1 } ( x ) & \\mbox { i n } \\Omega . \\end{cases} \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} X _ r ^ { ( \\tau ) } = \\sum _ { \\substack { 1 \\leq i \\leq n \\\\ d _ i \\mid r } } X _ { \\frac { r } { d _ i } } \\circ m _ i . \\end{align*}"}
-{"id": "9107.png", "formula": "\\begin{gather*} Z \\left ( \\sum _ { i = 0 } ^ n y _ i ^ d + \\lambda \\prod _ { i = 0 } ^ n y _ i ^ { b _ i } \\right ) \\subset \\mathbf { P } ^ n . \\end{gather*}"}
-{"id": "4703.png", "formula": "\\begin{align*} K _ { N } ( x , y ; W _ N ) = \\frac { \\sqrt { W _ N ( x ) W _ N ( y ) } } { 2 \\pi i ( x - y ) } \\begin{pmatrix} 0 & 1 \\end{pmatrix} Y _ + ( y ) ^ { - 1 } Y _ + ( x ) \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{align*} \\Delta = D _ { m a x } - D _ { a c h } ( k ) \\leq \\frac { d ^ 2 } { 2 ^ { 2 k } } . \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{align*} \\lambda ^ m \\ ; \\big | \\ ; c _ { \\bar { m } - 1 } \\prod \\limits _ { j = 1 } ^ { \\bar { m } - 1 } \\bigl ( ( 1 + \\lambda ) ^ 2 - \\zeta _ j \\bigr ) . \\end{align*}"}
-{"id": "914.png", "formula": "\\begin{align*} \\mathcal { Q } ( b _ 1 ) = b _ 1 \\circ b ( u ^ 2 ) \\circ [ 2 ] _ { [ F ] } b ( u ^ 2 ) \\circ \\dots \\circ [ p - 1 ] _ { [ F ] } b ( u ^ 2 ) . \\end{align*}"}
-{"id": "9795.png", "formula": "\\begin{align*} e _ { \\epsilon } ( u ) = \\frac { 1 } { 2 } ( 1 - | u | ^ 2 ) | d u | ^ 2 + \\frac { 1 } { 2 } | j u | ^ 2 + \\frac { 1 } { 8 } | d | u | ^ 2 | ^ 2 + \\frac { W ( u ) } { \\epsilon ^ 2 } , \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} B _ { 1 } \\left ( x \\right ) = \\frac { 1 } { 2 } \\left [ x ^ { 2 } - \\left ( x - 1 \\right ) ^ { 2 } \\right ] \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { U ^ { d } ( \\mathbb { R } ) } ^ { 2 ^ { d } } = \\int \\limits _ { ( x , \\mathbf { h } ) \\in \\mathbb { R } ^ { d + 1 } } \\prod \\limits _ { \\boldsymbol { \\omega } \\in \\{ 0 , 1 \\} ^ d } \\mathcal { C } ^ { \\vert \\boldsymbol { \\omega } \\vert } f ( x + \\sum \\limits _ { i = 1 } ^ d h _ i \\omega _ i ) \\ , d x \\ , d h _ 1 \\cdots d h _ d \\end{align*}"}
-{"id": "1569.png", "formula": "\\begin{align*} B ( x , x _ k ) = \\sum _ { i = 0 } ^ { n - 1 } \\sum _ { j = 0 } ^ { n - 1 } b _ { i j } x ^ i x _ k ^ j = \\frac { Q ( x ) S ( x _ k ) - S ( x ) Q ( x _ k ) } { x - x _ k } = Q _ k ( x ) Q ' ( x _ k ) S ( x _ k ) , \\end{align*}"}
-{"id": "6226.png", "formula": "\\begin{gather*} \\{ x _ { i j } \\mid i , j \\in \\mathbb { Z } _ 4 , j - i = 1 \\ , o r \\ , j - i = 2 \\} \\end{gather*}"}
-{"id": "9591.png", "formula": "\\begin{align*} \\widetilde T _ n ^ \\alpha & = \\frac { 1 } { N } \\sum _ { i = 1 } ^ n \\widetilde T _ { n , i } ^ \\alpha = \\frac { 1 } { N } \\sum _ { i = 1 } ^ n R _ { n , i } T _ { n , i } ^ \\alpha , \\\\ \\shortintertext { a n d } \\Psi _ n ^ \\alpha & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n E [ W _ { n , i } ^ \\alpha ] . \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{align*} \\prod _ { j = b _ n + \\delta ^ { ( k ) } + 1 } ^ { b _ { n + 1 } - 1 } w _ { j } = 1 . \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{align*} \\partial _ t \\Psi = \\mathrm i \\mathcal { H } \\Psi - ( \\alpha + \\mathrm i \\beta ) \\Delta \\Psi - ( \\alpha _ 1 + \\mathrm i \\beta _ 1 ) \\Psi | \\Psi | ^ 2 , \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} [ L _ { - r + 1 } , \\ , S _ { r - 2 } ] = 0 \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{align*} \\left ( M _ i \\left ( \\begin{array} { c } I _ i \\\\ R _ i \\\\ \\end{array} \\right ) \\right ) _ { i , j } = ( M _ i ) _ { i , j } + ( M _ i ) _ { i , i + 1 } ( R _ i ) _ j = \\ & \\binom { 2 i } { 2 j - 2 } \\frac { 1 } { 4 ^ i i } + \\frac { 1 } { 4 ^ i i } \\binom { 2 i } { 2 j - 2 } \\frac { 1 / ( 4 ^ i - 1 ) } { 2 i - 2 j + 3 } = \\\\ & \\binom { 2 i - 1 } { 2 j - 2 } \\frac { 2 ^ { 1 - 2 i } ( j - i - 1 ) + 2 i - 2 j + 3 } { ( 1 - 4 ^ i ) ( j - i - 1 ) ( 2 i - 2 j + 3 ) } \\end{align*}"}
-{"id": "4881.png", "formula": "\\begin{align*} W _ 1 + \\ldots + W _ r = W - r . \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{align*} \\sigma ( \\widehat { n } ) = \\left ( \\begin{array} { c c } \\cos { \\phi } & \\sin { \\phi } e ^ { i \\psi } \\\\ \\sin { \\phi } e ^ { - i \\psi } & - \\cos { \\phi } \\end{array} \\right ) . \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} \\frac { y _ { j } ^ { n + 1 } - y _ { j } ^ { n } } { \\Delta t } = \\frac { 1 } { 1 2 } \\left ( 2 3 f _ { j } ^ { n } - 1 6 f _ { j } ^ { n - 1 } + 5 f _ { j } ^ { n - 2 } \\right ) , \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} \\sup \\limits _ { k , i } \\sum \\limits _ { j = 1 } ^ { N } \\left \\vert \\frac { \\lambda _ { k } } { D \\left ( \\lambda _ { k } \\right ) } A _ { j i } \\left ( \\lambda _ { k } \\right ) \\right \\vert ^ { q } \\int \\limits _ { \\Omega } \\left \\vert \\sum \\limits _ { k = 1 } ^ { \\mu } r _ { k } \\left ( y \\right ) u _ { k j } \\right \\vert ^ { q } d y . \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} [ n ] \\left ( f ( \\sigma ) - [ \\sigma - 1 ] ( \\widetilde Q + R ) \\right ) = [ \\sigma - 1 ] R ^ \\prime , \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} a \\cdot ( x \\pm \\alpha _ X ( x ) ) \\cdot b = a \\cdot x \\cdot b \\pm a \\cdot \\alpha _ X ( x ) \\cdot b = a \\cdot x \\cdot b \\pm \\alpha _ X ( a \\cdot x \\cdot b ) , \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} m ( L ) \\leq \\sum _ { k = 1 } ^ n m ( [ a _ { k - 1 } , a _ k ] ) . \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} \\begin{gathered} | D u ( t , x ) \\cdot h | \\le C ( T , \\varphi ) t ^ { - \\alpha } | ( - A ) ^ { - \\alpha } h | , \\\\ | D ^ 2 u ( t , x ) \\cdot ( h , k ) | \\le C ( T , \\varphi ) t ^ { - ( \\beta + \\gamma ) } | ( - A ) ^ { - \\beta } h | | ( - A ) ^ { - \\gamma } k | , \\end{gathered} \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{align*} \\alpha _ \\ell = \\frac { [ ( \\ell - 1 ) ! ] ^ { 2 } } { 4 ^ { \\ell - 1 } } \\sum ^ { \\ell - 1 } _ { k = 0 } { \\ell - 1 \\choose k } ^ 2 { 2 ( \\ell - 1 - k ) \\choose \\ell - 1 - k } { 2 k \\choose k } = \\frac { [ ( \\ell - 1 ) ! ] ^ { 2 } } { 4 ^ { \\ell - 1 } } D _ { \\ell - 1 } , \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial F } { \\partial \\theta } & = \\left ( 0 , \\frac { 1 } { 2 } \\sinh \\rho \\cos \\theta \\sin \\phi , - \\frac { 1 } { 2 } \\sinh \\rho \\sin \\theta \\sin \\phi \\right ) \\\\ \\frac { \\partial F } { \\partial \\phi } & = \\left ( - \\frac { 1 } { 2 } \\cosh \\rho \\sin \\phi , \\frac { 1 } { 2 } \\sinh \\rho \\sin \\theta \\cos \\phi , \\frac { 1 } { 2 } \\sinh \\rho \\cos \\theta \\cos \\phi \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} \\lambda _ 0 ^ n ( f _ 1 , f _ 2 ) : = \\inf \\{ & t \\geq 0 : | f _ 1 ( t ) - f _ 2 ( t ) | = 1 / n , ( f _ 1 ( s _ 1 ) - f _ 2 ( s _ 1 ) ) ( f _ 1 ( s _ 2 ) - f _ 2 ( s _ 2 ) ) > 0 \\\\ & s _ 1 , s _ 2 \\in ( t , t + 1 ) \\} \\\\ \\lambda _ 1 ^ n ( f _ 1 , f _ 2 ) : = \\inf \\{ & t > \\lambda _ 0 ^ n : f _ 1 ( t ) = f _ 2 ( t ) \\} . \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{align*} F ( x ) : = \\sum _ { j = 1 } ^ N C _ j ^ 2 e ^ { - k _ j x } + \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } [ S ( k ) - { U _ 0 } ] e ^ { i k x } d k , \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} d ^ p + 1 \\ : = \\ \\dim \\ , F ^ p \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} \\begin{pmatrix} M & N \\\\ [ . 1 c m ] \\overline { N } & \\overline { M } \\end{pmatrix} \\begin{pmatrix} z - w \\\\ [ . 2 c m ] \\overline { z - w } \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ [ . 2 c m ] 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "156.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { k - \\ell } \\binom { k } { j } \\binom { k } { k - \\ell - j } = \\binom { 2 k } { k - \\ell } \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} u _ 2 u _ 1 s _ 1 - u _ 2 - s _ 1 \\leq ( u _ 1 - 2 ) s _ 1 + ( u _ 2 - 1 ) ( u _ 1 s _ 1 - 1 ) = ( u _ 2 u _ 1 s _ 1 - u _ 2 - s _ 1 ) - s _ 1 + 1 , \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{align*} \\sum \\limits _ { \\widetilde { \\mathbf { r } } \\in \\widetilde { R } } \\frac { 1 } { N ^ { d - m } } \\sum \\limits _ { \\mathbf { n } \\in \\mathbb { Z } ^ { d - u } } \\Big ( \\prod \\limits _ { j = 1 } ^ d f _ j ( \\xi _ j ( \\mathbf { n } ) + \\widetilde { \\mathbf { r } } _ j ) \\Big ) F _ { \\widetilde { \\mathbf { r } } } ( \\mathbf { n } ) G _ { \\widetilde { \\mathbf { r } } } ( \\pi _ { m - u } P L \\Xi ( \\mathbf { n } ) ) . \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{align*} \\varphi ( k , 0 ) = A , \\varphi ' ( k , 0 ) = B . \\end{align*}"}
-{"id": "1518.png", "formula": "\\begin{align*} | x _ 1 - \\zeta | ^ 2 + | x _ 2 - \\zeta | ^ 2 = ( x _ 1 - x _ 2 ) ^ 2 . \\end{align*}"}
-{"id": "7026.png", "formula": "\\begin{align*} K \\left [ N \\left ( . , \\varepsilon \\right ) - U _ { 0 , } \\left ( . \\right ) \\right ] \\upsilon = U _ { 0 } \\ast \\left [ N \\left ( . , \\varepsilon \\right ) - \\varepsilon ^ { - 1 } B M \\left ( . , \\varepsilon \\right ) \\right ] A _ { 0 } \\upsilon \\end{align*}"}
-{"id": "9687.png", "formula": "\\begin{align*} \\begin{aligned} \\overline { \\alpha } ( A ( \\varphi ( \\theta _ 0 + \\pi / 2 ) ) X ( \\theta _ 0 + \\pi / 2 ) K ) & = \\overline { \\alpha } ( X ( \\theta _ 0 + \\pi / 2 ) K ) = \\overline { a } ( A ( \\varphi ( \\theta _ 0 ) ) X ( \\theta _ 0 ) K ) , \\\\ \\overline { a } ( A ( \\varphi ( \\theta _ 0 + \\pi / 2 ) ) X ( \\theta _ 0 + \\pi / 2 ) K ) & = \\overline { a } ( X ( \\theta _ 0 + \\pi / 2 ) K ) = \\overline { \\alpha } ( A ( \\varphi ( \\theta _ 0 ) ) X ( \\theta _ 0 ) K ) . \\end{aligned} \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{align*} \\frac { { 1 - p } } { { 1 - \\tilde p } } = \\frac { 1 } { { 1 - { \\psi _ 0 } + { \\psi _ 1 } \\frac { p } { { 1 - p } } } } \\le \\frac { 1 } { { 1 - { \\psi _ 0 } + { \\psi _ 1 } \\frac { q } { { 1 - q } } } } = \\frac { { 1 - q } } { { 1 - \\tilde q } } , \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 1 } \\vec { \\gamma } _ 0 ( \\phi , p ) & = \\vec { \\gamma } _ 3 ( \\vec { \\Psi } , p ) \\\\ \\vec { \\gamma } _ 1 ( \\phi , p ) & = \\vec { \\gamma } _ 1 ( \\vec { \\Psi } , p ) \\\\ \\vec { \\gamma } _ 2 ( \\phi , p ) & = - \\vec { \\gamma } _ 2 ( \\vec { \\Psi } , p ) \\\\ \\vec { \\gamma } _ 3 ( \\phi , p ) & = \\vec { \\gamma } _ 0 ( \\vec { \\Psi } , p ) . \\end{alignedat} \\right . \\end{align*}"}
-{"id": "8935.png", "formula": "\\begin{align*} T _ { \\bar { z } ^ m } ( z ^ k ) & = \\begin{cases} \\frac { k - m + 1 } { k + 1 } z ^ { k - m } & k \\geq m - 1 \\\\ 0 & 0 \\leq k \\leq m - 2 \\end{cases} , \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 1 } \\frac { 2 ( y ( 1 + \\rho ^ { 2 } ) - 2 \\rho x ) \\sqrt { 1 - y ^ { 2 } } d y } { \\pi ( ( 1 - \\rho ^ { 2 } ) ^ { 2 } - 4 x y \\rho ( 1 + \\rho ^ { 2 } ) + 4 \\rho ^ { 2 } ( x ^ { 2 } + y ^ { 2 } ) ) } = - \\rho x . \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} 0 = { } & Q ^ { 2 0 } y _ { 1 0 } + Q ^ { 1 8 } y _ { 1 2 } + Q ^ { 1 7 } y _ { 1 3 } + x ^ 4 ( Q ^ { 1 2 } y _ { 1 0 } ) + y _ 9 ^ 2 ( Q ^ 4 x ) ^ 2 + \\\\ & y _ 7 ^ 2 Q ^ 9 Q ^ 5 x + y _ 8 ^ 2 Q ^ 8 Q ^ 4 x + ( Q ^ 9 y _ 9 ) ( Q ^ 4 x ) ^ 2 + ( Q ^ { 1 0 } y _ 8 ) ( Q ^ 4 x ) ^ 2 + \\\\ & y _ 5 ^ 2 ( Q ^ { 1 1 } Q ^ 7 x + Q ^ { 1 0 } Q ^ 8 x + x ^ 4 Q ^ 6 Q ^ 4 x ) \\end{align*}"}
-{"id": "5371.png", "formula": "\\begin{gather*} f _ 0 ( 1 ) = 1 , f _ 0 ( 2 ) = 0 , f _ 0 ( 3 ) = 1 ; \\\\ f _ 0 ( n + 3 ) = f _ 0 ( n + 1 ) + f _ 0 ( n ) , ( n \\geq 1 ) . \\end{gather*}"}
-{"id": "1315.png", "formula": "\\begin{align*} r = \\frac { g ^ { 2 } } { \\sqrt { \\lambda _ { 4 } g ^ { 2 } + \\lambda _ { 5 } } } , \\lambda _ { 5 } \\in \\mathbb { R } , \\end{align*}"}
-{"id": "6577.png", "formula": "\\begin{align*} e ^ { A } K _ 0 e ^ { - A } = - \\sinh ( 2 r ) ( e ^ { i \\theta \\sigma ( \\hat n ) } \\cdot K _ + + e ^ { - i \\theta \\sigma ( \\hat n ) } \\cdot K _ - ) + \\cosh ( 2 r ) K _ 0 . \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{align*} \\varepsilon ( ( a b ) \\triangleleft X ) = \\varepsilon ( a \\triangleleft X ) \\varepsilon ( b ) + \\varepsilon ( a ) \\varepsilon ( b \\triangleleft X ) . \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{align*} \\partial _ { \\theta _ i } \\theta _ j = \\delta _ { i j } , \\partial _ i 1 = 0 , \\partial _ { \\theta _ i } \\partial _ { \\theta _ j } = \\partial _ { \\theta _ j } \\partial _ { \\theta _ i } . \\end{align*}"}
-{"id": "9553.png", "formula": "\\begin{align*} E _ H ( x ) = \\sum _ { h \\in H } x _ h \\lambda _ \\sigma ( h ) , \\end{align*}"}
-{"id": "9935.png", "formula": "\\begin{align*} I ( \\tau , \\xi ) : = \\sqrt { \\tau + | \\xi | ^ 2 } \\int d \\eta \\ \\delta ( \\tau + | \\xi - \\eta | ^ 2 + | \\xi + \\eta | ^ 2 ) . \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{align*} \\begin{array} { r l } | \\widetilde Z ^ * - \\widetilde Z | & \\leq \\left | \\widetilde Z ^ * - \\bar Z ^ * \\right | + \\left | \\bar Z ^ * - \\widetilde Z \\right | \\\\ & \\leq { \\displaystyle \\max _ { j \\in S } } \\left | \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ n g _ i \\{ \\hat \\psi _ j ( y _ i , z _ i ) - \\bar \\psi _ j ( y _ i , z _ i ) \\} \\right | + \\left | \\bar Z ^ * - \\widetilde Z \\right | \\\\ \\end{array} \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} ( p ^ a - 1 ) Q _ { r , a } ( & X _ 1 , \\cdots , X _ r ) = \\\\ & p ^ a Q _ { r - 1 , a - 1 } ( p ^ { a + r - 1 } X _ 1 , p ^ { a + r - 2 } X _ 2 , \\cdots , p ^ { a + 1 } X _ { r - 1 } ) \\sum _ { f _ r \\ge 0 } ( p ^ a X _ r ) ^ { f _ r } \\\\ & - Q _ { r - 1 , a + 1 } ( X _ 1 , \\cdots , X _ { r - 1 } ) \\sum _ { f _ r \\ge 0 } X _ r ^ { f _ r } . \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} f ( \\lambda , \\mu ) + f ( \\lambda \\mu , \\nu ) = f ( \\mu , \\nu ) + f ( \\lambda , \\mu \\nu ) \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} \\partial h ( \\sigma ) = \\sum _ { j = 0 } ^ { k + 1 } ( - 1 ) ^ j \\cdot h ( \\sigma ) \\circ i _ j ^ { k + 1 } = \\sigma + \\sum _ { j = 1 } ^ { k + 1 } ( - 1 ) ^ j \\cdot h ( \\sigma \\circ i _ { j - 1 } ^ { k } ) = \\sigma + h ( - \\partial \\sigma ) , \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} W ( e _ 1 \\odot \\dots \\odot e _ n ) = \\sum _ { k = 0 } ^ { n } \\sum _ { i _ 1 , \\dots , i _ { n - k } , j _ { n - k + 1 } , \\dots , j _ n } a _ q ^ { \\ast } ( e _ { i _ 1 } ) \\dots a _ q ^ { \\ast } ( e _ { i _ { n - k } } ) a _ q ( I e _ { j _ { n - k + 1 } } ) \\dots a _ q ( I e _ { j _ n } ) q ^ { i ( I _ 1 , I _ 2 ) } , \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{align*} \\left | \\overline { \\xi } ( z ) \\right | ^ 3 \\leq \\left ( \\sum _ { i = 1 } ^ k \\left | \\overline { \\xi } _ i ( z ) \\right | \\right ) ^ 3 \\leq k ^ 2 \\sum _ { i = 1 } ^ k \\left | \\overline { \\xi } _ i ( z ) \\right | ^ 3 , \\end{align*}"}
-{"id": "9148.png", "formula": "\\begin{align*} H _ { i } ( ( S ^ { d } ) ^ p ; \\mathbb { F } _ p ) \\cong \\begin{cases} \\mathbb { F } _ p , & i = 0 , p d , \\\\ N ^ { \\oplus \\frac { 1 } { p } { p \\choose i / d } } , & i = d , 2 d , \\ldots , ( p - 1 ) d , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} T = i ( X + i Y ) \\mbox { a n d } U = i ( X - i Y ) . \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} [ y _ 1 , y _ 2 ] = y _ 3 , [ y _ 2 , y _ 1 ] = - y _ 3 + \\frac { \\alpha _ 4 + \\beta _ 1 } { \\beta _ 3 } y _ 4 + \\frac { ( \\alpha _ 5 + \\beta _ 2 ) \\beta _ 3 - ( \\alpha _ 4 + \\beta _ 1 ) \\beta _ 4 } { \\beta _ 3 } y _ 5 , [ y _ 2 , y _ 2 ] = y _ 4 , \\\\ [ y _ 3 , y _ 1 ] = \\alpha _ 3 \\gamma _ 2 y _ 5 , [ y _ 2 , y _ 3 ] = \\alpha _ 3 \\gamma _ 3 y _ 5 , [ y _ 3 , y _ 2 ] = \\alpha _ 3 \\gamma _ 4 y _ 5 , [ y _ 1 , y _ 4 ] = \\beta _ 3 \\gamma _ 6 y _ 5 . \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N } \\big | a _ { k } ^ 1 \\big | \\le C \\Delta ^ { \\frac 1 2 - 5 \\kappa } ( 1 + | x | _ { L ^ { \\max ( p , 2 q ) } } ) ^ { K + 1 } \\int _ { 0 } ^ { T } \\frac { 1 } { ( T - t ) ^ { 1 - \\kappa } } \\bigl ( 1 + \\frac { 1 } { t ^ { 1 - \\kappa } } \\bigr ) d t . \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{align*} W _ { C \\bar { C } } & = w _ { 1 , 3 } + w _ { 2 , 4 } \\\\ & = ( 1 - p ) p _ 3 - p p _ 1 + ( 1 - p ) p _ 4 - p p _ 2 \\\\ & = ( 1 - p ) p _ 3 - p p _ 2 + ( 1 - p ) p _ 4 - p p _ 1 \\\\ & = w _ { 2 , 3 } + ( 1 - p ) p _ 4 - p p _ 1 \\\\ & \\leq w _ { 2 , 3 } \\\\ & = W _ { C C } . \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{align*} \\int _ { \\partial ( \\Omega \\cap B _ R ( 0 ) ) } y _ i \\omega \\cdot N \\ , d S = \\int _ { \\Omega \\cap B _ R ( 0 ) } \\nabla \\cdot ( y _ i \\omega ) \\ , d y & = \\int _ { \\Omega \\cap B _ R ( 0 ) } \\omega _ i \\ , d y . \\end{align*}"}
-{"id": "5616.png", "formula": "\\begin{gather*} f _ { T } ( \\cos ( \\alpha _ { 1 } ) , . . . , \\cos ( \\alpha _ { n } ) | K _ { n } ) \\allowbreak = \\\\ \\allowbreak \\frac { 1 } { 2 ^ { n } } \\sum _ { i _ { 1 } \\in \\{ - 1 , 1 \\} } . . . \\sum _ { i _ { n } \\in \\{ - 1 , 1 \\} } \\frac { \\sum _ { k = 0 } ^ { n } ( - 1 ) ^ { k } \\sum _ { S _ { k } \\subseteq \\mathcal { K } _ { n } } ^ { \\prime } \\rho _ { S _ { k } } \\cos ( b _ { S _ { k } } ) } { \\prod _ { j = 1 } ^ { n } \\prod _ { m = j + 1 } ^ { n } ( 1 - 2 \\rho _ { j m } \\cos ( \\beta _ { j , m } ( i _ { j } , i _ { m } ) ) + \\rho _ { j m } ^ { 2 } ) } , \\end{gather*}"}
-{"id": "1732.png", "formula": "\\begin{align*} \\mathbf { f } _ { p l a n a r } = \\mathbf { f } _ n + \\mathbf { f } _ { w } \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{align*} g ( t ) = \\exp ( J t ) , g ' = g J , t \\in [ 0 , t _ f ] , t _ f = \\pi / 3 . \\end{align*}"}
-{"id": "9460.png", "formula": "\\begin{align*} \\int h _ n ( t ) v _ n ( t ) \\cdot \\psi { \\rm d } x - \\int P _ 0 \\cdot \\psi { \\rm d } x = \\int _ 0 ^ t \\int \\mathcal { N } _ { \\varepsilon } ( h _ n , B _ n , d _ n , v _ n ) \\cdot \\psi { \\rm d } x { \\rm d } s \\end{align*}"}
-{"id": "5994.png", "formula": "\\begin{align*} ( \\Lambda V _ B , D ) = ( \\Lambda ( z _ 1 , z _ 2 , x _ 2 , x _ 5 , z _ 9 ) , D ) , \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} S _ \\beta ^ { - 1 } \\ ; = \\ ; S _ D ^ { - 1 } + \\frac { 1 } { \\ , \\beta \\| \\Phi \\| ^ { 2 } } \\ : | \\Phi \\rangle \\langle \\Phi | \\ , . \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} ( x ^ k y ^ \\ell z ^ m ) = \\left ( k ( q ^ e + 1 ) + \\ell \\frac { q ^ e + 1 } { q + 1 } + m \\right ) Q _ 0 - \\left ( k ( q ^ e + 1 ) + \\ell q \\frac { q ^ e + 1 } { q + 1 } + m q ^ 3 \\right ) Q _ \\infty + E , \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} \\sigma ( S ) \\cup \\left ( \\pm \\gamma \\sigma ( C ) \\right ) = \\left ( \\lambda _ { 1 } , \\lambda _ { 2 } , \\ldots , \\lambda _ { n } , \\pm \\gamma \\mu _ { 1 } , \\pm \\gamma \\mu _ { 2 } , \\ldots , \\pm \\gamma \\mu _ { n } \\right ) . \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} \\left ( x ^ { j } f ( x ) \\right ) ^ { ( j + n ) } & = \\sum _ { l = 0 } ^ { n + j } \\binom { n + j } { l } \\left ( x ^ j \\right ) ^ { ( l ) } f ^ { ( n + j - l ) } ( x ) \\\\ & = \\sum _ { l = 0 } ^ { j } \\binom { n + j } { l } \\frac { j ! } { ( j - l ) ! } x ^ { j - l } f ^ { ( n + j - l ) } ( x ) \\\\ & = \\sum _ { m = 0 } ^ { j } \\binom { n + j } { j - m } \\frac { j ! } { m ! } x ^ { m } f ^ { ( n + m ) } ( x ) . \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} F _ k = \\bigoplus _ { i = k } ^ n M _ i \\subset F _ 0 \\end{align*}"}
-{"id": "7614.png", "formula": "\\begin{align*} \\Phi ^ * \\rho = k \\log r ^ 2 \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{align*} f ( x ) : = \\left \\{ \\begin{array} { l l } ( c _ s , 0 ) & \\textrm { i f } x = k _ s , s \\ge 2 , \\\\ ( c _ 1 , 1 ) & \\textrm { i f } x = k _ 1 , \\\\ ( e , 0 ) & \\textrm { o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "491.png", "formula": "\\begin{align*} \\sigma _ a ( F ) = \\sum _ { \\substack { H \\leq F , \\\\ } } q ^ { \\log | H | / \\log p } . \\end{align*}"}
-{"id": "7432.png", "formula": "\\begin{align*} x _ i = v _ i \\log t + O ( 1 ) . \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} \\inf _ { \\sigma \\in { \\cal T } _ 0 } \\sup _ { \\tau \\in { \\cal T } _ 0 } E \\ , [ I ( \\tau , \\sigma ) ] = \\sup _ { \\tau \\in { \\cal T } _ 0 } \\inf _ { \\sigma \\in { \\cal T } _ 0 } E \\ , [ I ( \\tau , \\sigma ) ] . \\end{align*}"}
-{"id": "9385.png", "formula": "\\begin{align*} L ^ \\infty _ \\sigma ( \\Omega ; F ^ * ) = B ( L ^ 1 ( \\Omega ) , F ^ * ) , \\end{align*}"}
-{"id": "8960.png", "formula": "\\begin{align*} \\dot { c } ( t ) = - \\xi \\ , N ( c ( t ) ) + y \\ , T ( c ( t ) ) + z \\ , Z ( c ( t ) ) , c ( 0 ) = x . \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { \\infty } \\frac { b _ { 2 k - 1 } ( \\mu ) } { ( 2 k ) ! k } < + \\infty \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} \\mathcal P : = \\{ h \\in B \\mid h ^ * = h , \\mathrm { S p e c } ( h ) \\in ( 0 , \\infty ) \\} \\subset B \\end{align*}"}
-{"id": "9934.png", "formula": "\\begin{align*} \\vect { S } _ \\pm \\Gamma ( t , x , y ) = [ v _ N \\ast \\rho _ \\Gamma ( t , x ) - v _ N \\ast \\rho _ \\Gamma ( t , y ) ] \\Gamma ( t , x , y ) = F \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} \\begin{cases} H _ k [ u ] = \\sigma _ 0 ^ k & , \\\\ [ 5 p t ] u = \\varphi - \\delta & \\end{cases} \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} \\sigma _ { 2 } ^ { 2 } \\left ( Z _ { 1 } \\right ) & = \\frac { 1 7 - \\sqrt { 1 7 ^ { 2 } - 4 \\cdot 3 \\cdot 2 \\cdot 3 } } { 2 } = \\frac { 1 7 - \\sqrt { 2 1 7 } } { 2 } > \\frac { 1 7 - 1 5 } { 2 } = 1 \\\\ \\sigma _ { 2 } ^ { 2 } \\left ( Z _ { 2 } \\right ) & = \\frac { 1 8 - \\sqrt { 1 8 ^ { 2 } - 4 \\cdot 2 \\cdot 3 \\cdot 3 } } { 2 } = \\frac { 1 8 - \\sqrt { 2 5 2 } } { 2 } > \\frac { 1 8 - 1 6 } { 2 } = 1 . \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} D ^ { ( 1 / 2 ) } z = B z + C f , \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} \\tilde { u } _ t = \\partial _ { \\xi } ( L _ { c } - \\partial _ y ^ 2 + 4 ( \\dot { a } - c ) ) \\tilde { u } + 4 ( \\dot { a } - c ) \\partial _ { \\xi } u _ { c } - \\dot { c } \\partial _ c u _ { c } - 6 \\partial _ { \\xi } \\tilde { u } ^ 2 , \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{align*} 2 i k _ j \\varphi ( i k _ j , x ) N _ { - , k _ j } \\ ! = \\ ! - f ( i k _ j , x ) C _ j ^ 2 \\ ! = \\ ! \\left [ - e ^ { - k _ j x } - \\int _ x ^ \\infty K ( x , t ) e ^ { - k _ j t } d t \\right ] C _ j ^ 2 . \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} \\mathcal { L } \\ ; : = \\ ; \\{ v : \\mathbb { R } ^ + \\to \\mathbb { C } ^ 2 \\ , | \\ , \\widetilde { S } v = 0 \\} \\ ; = \\ ; \\mathrm { s p a n } \\{ v _ 0 , v _ \\infty \\} \\ , , \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{align*} \\| u \\| _ { X ^ { \\gamma , b } } = \\| e ^ { - i t \\Lambda ^ k } u \\| _ { H ^ b _ t H ^ \\gamma _ x } . \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{align*} \\frac { d A } { d t } = \\lambda A + c _ 3 ( \\lambda ) | A | ^ 2 A + O ( A ^ 5 ) . \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{align*} [ L _ { - i } , \\ , S _ { k } ] = 0 , \\ , 2 \\leq i \\leq k + 1 , \\ , 1 \\leq k \\leq j - 1 . \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} A _ { k ( E ) } = \\left ( \\frac { T ^ 2 - 4 , R - 3 } { k ( E ) } \\right ) = \\left ( \\frac { z ^ 3 - 4 z ^ 2 + 6 z - 3 , z - 2 } { k ( E ) } \\right ) . \\end{align*}"}
-{"id": "7496.png", "formula": "\\begin{align*} [ \\phi ^ * , b m ] = [ \\phi ^ * , b ] m + b [ \\phi ^ * , m ] = 0 \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} I = \\{ x \\in G \\mid \\exists \\ , x _ i , y _ j \\in I _ 0 , x = x _ 1 + \\cdots + x _ n - y _ 1 - \\cdots - y _ m \\} . \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} [ z ^ n ] F ( z ) = & \\frac { \\partial } { \\partial x } \\bigg ( [ z ^ n ] { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a - x & 1 + a + x \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] \\bigg ) \\bigg | _ { x = 0 } \\\\ = & \\frac { \\partial } { \\partial x } \\bigg ( [ z ^ n ] { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - \\frac 1 2 ( a + x ) & \\frac 1 2 + \\frac 1 2 ( a + x ) \\\\ & 1 \\end{matrix} \\bigg | \\ , 4 z ( 1 - z ) \\bigg ] \\bigg ) \\bigg | _ { x = 0 } = [ z ^ n ] G ( z ) . \\end{align*}"}
-{"id": "9600.png", "formula": "\\begin{align*} P ^ i _ { j k } = \\overline L ^ i _ { j k } - L ^ i _ { j k } = \\overline \\omega { } ^ i _ { j k } - \\omega ^ i _ { j k } + \\overline \\tau { } ^ i _ { j k } - \\tau ^ i _ { j k } , \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} \\frac { x _ { n + 1 } - x _ { n } } { h } = f \\left ( x _ { n } \\right ) + f \\left ( x _ { n + 1 } \\right ) . \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} \\widetilde { \\varphi } ( T ) ( \\varphi ( a ) ( x ) ) = \\varphi ( T ( a ) ) ( x ) . \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial s } \\Big [ \\frac { 1 } { s ^ m } \\Big ] = - m \\Big [ \\frac { 1 } { s ^ { m + 1 } } \\Big ] , s \\Big [ \\frac { 1 } { s ^ { m + 1 } } \\Big ] = \\Big [ \\frac { 1 } { s ^ m } \\Big ] . \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} \\int _ M | A ^ o | ^ 2 d \\mu \\bigg | _ { t = 0 } \\le \\varepsilon _ 2 < 8 \\pi \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} u _ 2 u _ 1 s _ 1 - u _ 2 - s _ 1 \\leq ( u _ 1 - 1 ) s _ 1 + ( u _ 2 - 2 ) ( u _ 1 s _ 1 - 1 ) = ( u _ 2 u _ 1 s _ 1 - u _ 2 - s _ 1 ) - u _ 1 s _ 1 + 2 , \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} L _ f ( p ) : = [ f , p ] = \\left [ \\sum \\nolimits _ i c _ i f _ { \\lambda _ i } , p \\right ] = \\sum _ { i = 1 } ^ k c _ i p ( \\lambda _ i ) , \\end{align*}"}
-{"id": "4044.png", "formula": "\\begin{align*} F ( x _ 1 , x _ 2 ; y _ 1 , y _ 2 ) = \\begin{pmatrix} x _ 1 ^ 2 & x _ 1 x _ 2 & x _ 2 ^ 2 \\end{pmatrix} \\begin{pmatrix} a _ { 1 1 } & a _ { 1 2 } & a _ { 1 3 } \\\\ a _ { 2 1 } & a _ { 2 2 } & a _ { 2 3 } \\\\ a _ { 3 1 } & a _ { 3 2 } & a _ { 3 3 } \\end{pmatrix} \\begin{pmatrix} y _ 1 ^ 2 \\\\ y _ 1 y _ 2 \\\\ y _ 2 ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} \\lambda ^ * _ 1 = \\lambda _ n ^ { - 1 } = \\lambda _ 1 \\cdots \\lambda _ { n - 1 } V \\ge \\lambda _ 1 ^ { n - 1 } V \\ge C ( n ) ^ { - ( n - 1 ) / n } V ^ { 1 / n } . \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} & d x = \\frac { 1 } { 8 } \\sinh \\rho \\sin \\phi ( \\cosh ^ { 2 } \\rho - \\cos ^ { 2 } \\phi ) d \\rho d \\theta d \\phi , \\\\ & \\ \\cosh \\rho \\leq r , \\ \\theta \\in [ 0 , 2 \\pi ] \\ \\mbox { a n d } \\ \\phi \\in [ 0 , \\pi ] . \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{align*} P _ { Y _ 1 , Y _ 3 } ( 0 , 0 ) = \\frac { 1 } { 2 } , P _ { Y _ 1 , Y _ 3 } ( 1 , 0 ) = P _ { Y _ 1 , Y _ 3 } ( 1 , 1 ) = \\frac { 1 } { 4 } , \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} \\frac { \\lambda _ 3 - u _ 2 } { s _ 3 } = \\frac { ( \\alpha - 1 ) \\cdot u _ 2 + \\beta } { s _ 3 } \\leq \\frac { ( \\alpha - 1 ) \\cdot ( u _ 2 u _ 3 - 1 ) + \\beta u _ 3 } { s _ 4 } \\leq \\frac { \\lambda _ 4 - ( u _ 2 u _ 3 - 1 ) } { s _ 4 } . \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} M = \\begin{pmatrix} 1 & 0 & \\frac { 1 } { 2 } & \\frac { 1 } { 2 } & 0 \\\\ 0 & 1 & \\frac { 1 } { 2 } & \\frac { 1 } { 2 } & 0 \\\\ \\frac { 1 } { 2 } & \\frac { 1 } { 2 } & \\frac { 3 } { 2 } & \\frac { 1 } { 2 } & 1 \\\\ \\frac { 1 } { 2 } & \\frac { 1 } { 2 } & \\frac { 1 } { 2 } & \\frac { 3 } { 2 } & 1 \\\\ 0 & 0 & \\frac { 1 } { 2 } & \\frac { 1 } { 2 } & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} \\underset { l \\rightarrow \\infty } { \\lim } \\mathbf { X } _ { l } ^ { ( \\omega ) } \\left ( s _ { 1 } , s _ { 2 } \\right ) = \\mathbf { X } _ { \\infty } \\left ( s _ { 1 } , s _ { 2 } \\right ) \\ . \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} \\gamma = 2 . 8 6 8 1 1 4 0 1 3 ( 4 ) \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} P x = \\Phi A \\Phi ^ * ( x ) = \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ m a _ { i j } e ^ * _ j ( x ) e _ i , x \\in X , \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} \\sigma _ { \\mathrm { e s s } } ( S _ \\beta ) \\ ; = \\ ; \\sigma _ { \\mathrm { e s s } } ( S _ D ) \\ ; = \\ ; ( - \\infty , - 1 ] \\cup [ 1 , + \\infty ) \\ , , \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} \\pi _ { \\textnormal { n } ( \\theta ) } \\epsilon \\tilde E = \\tilde f \\in \\textnormal { n } _ 1 ( \\theta ) \\oplus \\textnormal { r } ( \\theta ) , \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} \\langle \\Phi ( z _ 1 , \\dots , z _ N ) | = \\langle 1 \\cdots M | \\mathcal { B } ( z _ 1 ) \\cdots \\mathcal { B } ( z _ N ) . \\end{align*}"}
-{"id": "6984.png", "formula": "\\begin{align*} \\Phi _ { i , j } ( D _ { n } ) = \\big \\{ [ d _ { 1 } , \\dots , d _ { i - 1 } , d _ { i } - 1 , d _ { i + 1 } , \\dots , d _ { j - 1 } , d _ { j } + 1 , d _ { j + 1 } , \\dots , d _ { n } ] \\big \\} \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} u ( z ) = P ( z ) + | \\mu ( z ) | T ( z ) \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} ( \\pi _ X ) _ * M A _ { \\tilde { \\theta } } ( \\Phi ) = N \\int _ { t = 0 } ^ 1 M A _ { \\theta } ( ( 1 - t ) \\phi _ 0 + t \\phi ) t ^ { N - 1 } d t . \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} \\mathbb { C } ^ { 2 n } = \\mathcal { R } \\begin{pmatrix} N \\\\ [ . 1 c m ] \\overline { M } \\end{pmatrix} \\oplus \\left [ \\mathcal { R } \\begin{pmatrix} N \\\\ [ . 1 c m ] \\overline { M } \\end{pmatrix} \\right ] ^ { \\perp } \\end{align*}"}
-{"id": "4564.png", "formula": "\\begin{align*} \\nabla \\cdot \\mathbf { A } = 0 . \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} \\psi _ 1 ( t , u , v ) & = u + v \\\\ \\psi _ 2 ( t , u , v ) & = v + t \\\\ \\psi _ 3 ( t , u , v ) & = u + t \\\\ \\psi _ 4 ( t , u , v ) & = u + v + t \\end{align*}"}
-{"id": "4433.png", "formula": "\\begin{align*} G _ 1 \\setminus G ^ 0 = \\tau ( G _ 2 \\setminus G ^ 0 ) \\tau ^ { - 1 } = \\tau ^ 2 ( G _ 3 \\setminus G ^ 0 ) \\tau ^ { - 2 } \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} \\int _ { | x | \\le R } | V ( x ) | \\dd x & \\le \\int _ 0 ^ 1 \\ ! \\ ! \\int _ { | x | \\le R / \\lambda } | x | \\ , | f ( x ) | \\dd x \\dd \\lambda \\\\ & = \\int _ { | x | \\le R } \\int _ 0 ^ 1 | x | \\ , | f ( x ) | \\dd \\lambda \\dd x + \\int _ { | x | \\ge R } \\int _ 0 ^ { R / | x | } | x | \\ , | f ( x ) | \\dd \\lambda \\dd x \\\\ & \\le R \\int | f ( x ) | \\dd x . \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{align*} R = H ^ 2 - \\abs { A } ^ 2 \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} \\partial _ { t } u = i \\left [ \\Delta u + A u + V \\left ( x , t \\right ) u \\right ] , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "1358.png", "formula": "\\begin{align*} n ^ { 2 } ( x ) v _ { t t } = \\Delta v . \\end{align*}"}
-{"id": "3484.png", "formula": "\\begin{align*} \\begin{cases} p \\in [ 2 , 6 ) , d = 3 , \\\\ p \\in [ 2 , \\infty ) , d = 2 , \\\\ p \\in [ 2 , \\infty ] , d = 1 . \\end{cases} \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} [ e _ i , e _ j ] = 0 [ f _ i , f _ j ] = 0 \\textrm { f o r } \\vert i - j \\vert > 1 \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\deg _ { X } ( v _ { i } ) \\leq N \\cdot \\deg _ { Y } ( a ) \\cdot \\deg _ { Z } ( b ) . \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} R = \\bigl \\langle h _ 1 , \\dots , h _ m \\rangle _ A . \\end{align*}"}
-{"id": "4585.png", "formula": "\\begin{align*} { \\rm R e } \\langle g ( \\vec { e } ) , \\ , \\vec { e } \\rangle > 0 , { \\rm i f } \\ ; \\Vert \\vec { e } \\Vert _ { H } = \\Vert \\vec { c } \\Vert _ { H } + 1 . \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} H ^ * : \\mathbb { R } ^ { n \\times n } \\mapsto \\mathbb R ^ { 2 n - 1 } , \\ X = ( x _ { i j } ) \\mapsto y = ( y _ k ) , y _ k = \\sum _ { \\substack { i , j = 0 \\\\ i + j = k } } ^ { n - 1 } x _ { i j } , \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{align*} F ( z ) = - K ( x , z - x ) - \\int _ z ^ \\infty K ( x , s + x - z ) F ( s ) d s . \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{align*} f _ * E = 0 , f _ * E ^ 2 = - Z _ 1 Z _ 2 , f _ * E ^ 3 = - ( Z _ 1 + Z _ 2 ) Z _ 1 Z _ 2 , f _ * E ^ 4 = - ( Z _ 1 ^ 2 + Z _ 2 ^ 2 + Z _ 1 Z _ 2 ) Z _ 1 Z _ 2 . \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} u \\left ( 0 , x \\right ) = 0 , x \\in R ^ { n } i = 1 , 2 , . . . , N , N \\in \\mathbb { N } , \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{align*} r \\nabla V ^ { ( \\alpha , a ) } + t ( \\nabla \\cdot H ^ { ( \\alpha , a ) } ) \\omega = ( r \\partial _ r V ^ { ( \\alpha , a ) } + t \\nabla \\cdot H ^ { ( \\alpha , a ) } ) \\omega + \\Omega V \\omega ^ \\perp . \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A u = V \\left ( x , t \\right ) u + F \\left ( x , t \\right ) , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "800.png", "formula": "\\begin{align*} & \\limsup _ { t \\to + \\infty } \\| \\theta ( t ) \\| _ { L ^ 1 ( B ^ c _ { A \\sqrt t } ) } \\le \\kappa A ^ { - 1 } , \\\\ & \\limsup _ { t \\to + \\infty } \\| u ( t ) \\| _ { L ^ 3 ( B ^ c _ { A \\sqrt t } ) } \\le \\kappa A ^ { - 2 } , \\qquad \\\\ & \\limsup _ { t \\to + \\infty } \\sqrt t \\ , \\| u ( t ) \\| _ { L ^ \\infty ( B _ { A \\sqrt t } ^ c ) } \\le \\kappa A ^ { - 3 } . \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n K _ j ( s ) K _ { i - j + 1 } ( m ) + \\sum _ { j = 1 } ^ n K _ { j - 1 } ( s ) K _ { i - j + 1 } ( m ) = \\Sigma _ 1 + \\Sigma _ 2 . \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} \\gamma _ \\chi ( z ^ 2 ) V ^ - ( w ) & = V ^ + ( z ) ^ { - 1 } H ^ { \\gamma } ( z ^ 2 ) z ^ { - 2 h _ 0 } V ^ - ( z ) ^ { - 1 } V ^ - ( w ) = V ^ + ( z ) ^ { - 1 } V ^ - ( w ) H ^ { \\gamma } ( z ^ 2 ) z ^ { - 2 h _ 0 } V ^ - ( z ) ^ { - 1 } \\\\ & = \\left ( 1 - \\frac { z ^ 2 } { w ^ 2 } \\right ) V ^ - ( w ) V ^ + ( z ) ^ { - 1 } H ^ { \\gamma } ( z ^ 2 ) z ^ { - 2 h _ 0 } V ^ - ( z ) ^ { - 1 } = \\frac { w ^ 2 - z ^ 2 } { w ^ 2 } V ^ - ( w ) \\gamma _ \\chi ( z ^ 2 ) . \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\log ( a + b \\cos ( \\pi x ) ) \\ > d x & = \\frac { 1 } { \\pi } \\int _ { 0 } ^ { \\pi } \\log ( a + b \\cos x ) \\ > d x \\\\ & = \\log \\ ( \\frac { a + \\sqrt { a ^ 2 - b ^ 2 } } { 2 } \\ ) \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} f ( x ) = \\sigma ( T x , x ) g ( x ) = \\sigma ( z , x ) - \\sigma ( z , T x ) \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{align*} G ( \\zeta + l ) - G ( \\zeta ) = \\frac { m n } { ( \\zeta + 1 ) ( \\zeta + n + 1 ) } \\ \\zeta = k \\geq \\max \\{ m , p \\} . \\end{align*}"}
-{"id": "6543.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } \\dot x \\\\ \\dot y \\end{array} \\right ) = \\left ( \\begin{array} { c } \\Gamma \\\\ 0 \\end{array} \\right ) w ( x , y ) . \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} \\hat { \\omega } = \\frac { \\mathrm { e } ^ { - \\beta \\hat { H } } } { \\mathrm { T r } \\ , \\mathrm { e } ^ { - \\beta \\hat { H } } } \\ ; . \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} \\tau \\sum _ { k = 1 } ^ { m } U _ 4 ^ { k } ( \\overline { \\partial _ { \\tau } \\theta } _ { \\mathbf { A } } ^ { k } ) \\leq C \\big ( h ^ { 2 r } + \\tau ^ { 4 } \\big ) + C \\tau \\sum _ { k = 0 } ^ { m } \\big ( \\Vert \\nabla \\theta _ { \\Psi } ^ { k } \\Vert _ { \\mathbf { L } ^ { 2 } } ^ { 2 } + \\Vert \\partial _ { \\tau } \\theta _ { \\mathbf { A } } ^ { k } \\Vert _ { \\mathbf { L } ^ { 2 } } ^ { 2 } \\big ) . \\end{align*}"}
-{"id": "9027.png", "formula": "\\begin{align*} d ^ { \\mathcal { K } _ i } - z - 1 & = \\frac { N - T } { T } \\alpha _ i - \\frac { N - 2 B - T } { T } \\cdot \\frac { N } { N - 2 B } \\alpha _ i \\\\ & = \\frac { 2 B } { N - 2 B } \\alpha _ i \\\\ & = 2 B ( N - 2 B - T ) ^ { | \\mathcal { K } _ i | - 1 } T ^ { M - | \\mathcal { K } _ i | } > 0 \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} \\| \\tilde X ( \\phi ^ { ( 1 ) } , \\ldots , \\phi ^ { ( r ) } ) \\| _ s & \\leq K _ s \\sum _ { l = 1 } ^ r \\| \\phi ^ { ( 1 ) } \\| _ 1 \\ldots \\| \\phi ^ { ( l - 1 ) } \\| _ 1 \\| \\phi ^ { ( l ) } \\| _ s \\| \\phi ^ { ( l + 1 ) } \\| _ 1 \\ldots \\| \\phi ^ { ( r ) } \\| _ 1 , \\\\ & \\forall \\ ; \\phi ^ { ( 1 ) } , \\ldots , \\phi ^ { ( r ) } \\in H ^ s \\oplus H ^ s . \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} u ( x ) = \\frac { 1 } { p } \\sum _ { k = 0 } ^ { p - 1 } \\int \\limits _ { S } \\frac { 1 - | x | ^ { 2 p } } { | e ^ \\frac { - k \\pi i } { p } x - \\zeta | ^ n } f ( e ^ \\frac { k \\pi i } { p } \\zeta ) \\ , d \\sigma ( \\zeta ) . \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} \\mathcal { M } _ 1 & = \\lbrace \\hat { \\boldsymbol { x } } ^ { \\boldsymbol { \\omega } } , \\ : \\boldsymbol { \\omega } \\in \\Omega \\rbrace , \\\\ \\hat { \\mathcal { D } } _ { \\hat { \\boldsymbol { x } } ^ { \\boldsymbol { \\omega } } } & = \\cap _ { k \\in \\mathbb { Z } } \\hat { \\mathcal { D } } _ { \\omega _ k , k } , \\\\ \\mathcal { M } _ 2 & = \\cup _ { \\boldsymbol { \\omega } \\in \\Omega } \\hat { \\mathcal { D } } _ { \\hat { \\boldsymbol { x } } ^ { \\boldsymbol { \\omega } } } . \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} T ^ { \\ , m } e _ { b _ { 2 ^ { k - 1 } + l + 1 } - m + j - b _ { l } } = \\Biggl ( v ^ { ( k ) } & \\ ! \\ ! \\ ! \\ ! \\ ! \\ ; \\ ; \\ ; \\ ; \\prod _ { i = \\Delta ^ { ( k ) } - m + j - b _ { l } + 1 } ^ { \\Delta ^ { ( k ) } - 1 } w ^ { ( k ) } _ { i } \\Biggr ) \\ \\Bigl ( \\ , \\ , \\prod _ { i = 1 } ^ { j - b _ l } w _ { b _ l + i } \\Bigr ) \\ , e _ { j } \\\\ & - \\Biggl ( \\ ! \\ ! \\ ! \\ ! \\ ! \\ ; \\ ; \\ ; \\ ; \\prod _ { i = j - b _ l + 1 } ^ { \\Delta ^ { ( k ) } - m + j - b _ { l } } w ^ { ( k ) } _ { i } \\Biggr ) ^ { - 1 } \\ , e _ { b _ { 2 ^ { k - 1 } + l } + j - b _ { l } } \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} T ^ { \\ , m ' } \\ , e _ { b _ { 2 ^ { k - 1 } + l } + j - b _ { l } } = \\Biggl ( \\ ; \\ , \\prod _ { i = j - b _ l + 1 } ^ { j - b _ { l } + m ' } w _ { i } ^ { ( k ) } \\Biggr ) \\ , e _ { b _ { 2 ^ { k - 1 } + l } + j - b _ { l } + m ' } , \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} f _ { 0 } ( t ) & = 1 , \\\\ f _ { 1 } ( t ) & = - t , \\\\ f _ { 2 } ( t ) & = \\frac { 1 } { 2 } ( t - 3 ) t , \\\\ f _ { 3 } ( t ) & = - \\frac { 1 } { 6 } t \\left ( t ^ 2 - 9 t + 2 \\right ) , \\\\ f _ { 4 } ( t ) & = \\frac { 1 } { 2 4 } t \\left ( t ^ 3 - 1 8 t ^ 2 + 3 5 t - 4 2 \\right ) , \\\\ f _ { 5 } ( t ) & = - \\frac { 1 } { 1 2 0 } t \\left ( t ^ 4 - 3 0 t ^ 3 + 1 5 5 t ^ 2 - 2 7 0 t + 2 4 \\right ) . \\end{align*}"}
-{"id": "6461.png", "formula": "\\begin{align*} V _ s '^ 2 = - 2 \\sum _ { i = 0 } ^ { n - 1 } a _ i ^ 2 + 2 \\sum _ { i = 0 } ^ { n - 1 } a _ i a _ { i + 1 } = - \\sum _ { i = 0 } ^ { n - 1 } ( a _ i - a _ { i + 1 } ) ^ 2 . \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} { \\bf u } ^ \\varepsilon \\approx { \\bf R } ^ \\varepsilon : = { \\bf v } ( x _ 1 , \\vartheta , t ) + { \\varepsilon ^ 2 } \\ , { \\bf W } ( x _ 1 , \\tfrac { r - 1 } { { \\varepsilon } } , \\vartheta , t ) , \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} h ^ 1 ( E ( - b ) ) = \\binom { c - b + 3 } { 3 } - 1 , \\ \\ \\ b > 0 . \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} w ( \\xi , y , t ) = 1 2 | b | ^ 2 \\partial _ { c } u _ { c _ * } ( \\xi ) + \\tilde { w } ( \\xi , y , t ) , \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} { } _ { 2 } F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\beta \\\\ & \\gamma \\end{matrix} \\bigg | \\ , z \\bigg ] = ( 1 - z ) ^ { - \\alpha } \\cdot { } _ { 2 } F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\gamma - \\beta \\\\ & \\gamma \\end{matrix} \\bigg | \\ , \\frac { z } { z - 1 } \\bigg ] . \\end{align*}"}
-{"id": "7273.png", "formula": "\\begin{align*} [ y _ 1 , y _ 2 ] = y _ 3 , [ y _ 2 , y _ 1 ] = - y _ 3 + y _ 4 , [ y _ 2 , y _ 2 ] = \\beta _ 4 y _ 5 , [ y _ 3 , y _ 1 ] = \\alpha _ 3 \\gamma _ 2 y _ 5 , [ y _ 2 , y _ 3 ] = \\alpha _ 3 \\gamma _ 3 y _ 5 = - [ y _ 3 , y _ 2 ] , \\\\ [ y _ 1 , y _ 4 ] = ( \\alpha _ 4 + \\beta _ 1 ) \\gamma _ 6 y _ 5 . \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} W _ { k } = \\frac { 1 } { \\left ( \\Delta t \\right ) ^ { \\alpha _ { 0 } } } \\omega _ { k } \\left ( \\alpha _ { 0 } \\right ) + \\sum _ { \\nu = 1 } ^ { N } \\frac { \\tau _ { \\nu } } { \\left ( \\Delta t \\right ) ^ { \\alpha _ { \\nu } } } \\omega _ { k } \\left ( \\alpha _ { \\nu } \\right ) , \\ ; \\ ; k = 0 , 1 , \\ldots , n , \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} \\left [ \\beta \\left ( x \\right ) , \\alpha \\left ( y \\right ) \\right ] = - \\left [ \\beta \\left ( y \\right ) , \\alpha \\left ( x \\right ) \\right ] , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\forall \\ ; x , y \\in L , \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{align*} r _ k ( 1 , \\tilde { \\lambda } ) & = H ( m _ k , \\delta _ k + n _ k ) - H ( u _ 0 , v _ 0 ) \\\\ & = H ( 0 , \\delta _ k ) + A ( m _ k , n _ k ) - H ( u _ 0 , v _ 0 ) \\\\ & = H ( 0 , \\delta _ k ) - H ( u _ 0 , v _ 0 ) + A \\big ( u _ 0 + m _ k - u _ 0 , m _ k \\lambda - \\delta _ k + v _ 0 - u _ 0 \\lambda \\big ) \\\\ & = H ( 0 , \\delta _ k ) - H ( u _ 0 , v _ 0 ) + A ( u _ 0 , v _ 0 ) - A ( 0 , \\delta _ k ) + ( m _ k - u _ 0 ) A ( 1 , \\lambda ) \\\\ & = H ( 0 , \\delta _ k ) - H ( u _ 0 , v _ 0 ) + A ( u _ 0 , v _ 0 ) - A ( 0 , \\delta _ k ) + ( m _ k - u _ 0 ) c ( 1 , \\tilde { \\lambda } ) \\end{align*}"}
-{"id": "2336.png", "formula": "\\begin{align*} \\max _ { j = 1 , . . . , s } \\left [ \\frac { g _ j ^ * } { \\mu _ j } \\right ] \\le 1 , \\sum _ { j = 1 } ^ s ( g _ j ^ * - \\mu _ { s } ) _ + ^ 2 \\le \\mu _ s ^ 2 s \\frac { 3 } { \\log ( 9 e p / s ) } . \\end{align*}"}
-{"id": "417.png", "formula": "\\begin{align*} \\mathbf { E } = \\mathbb { E } _ { \\mathbf { X } } \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} \\Psi ' ( 0 ) \\equiv & \\frac { d } { d x } \\bigg ( { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a + x & - b \\\\ & p - c \\end{matrix} \\bigg | \\ , z \\bigg ] \\bigg ) \\bigg | _ { x = 0 } \\equiv \\frac { d \\big ( \\omega ( x ) \\phi ( x ) \\big ) } { d x } \\bigg | _ { x = 0 } \\\\ = & \\omega ' ( 0 ) \\phi ( 0 ) + \\omega ( 0 ) \\phi ' ( 0 ) \\equiv \\frac { d \\big ( \\Omega ( x ) \\Phi ( x ) \\big ) } { d x } \\bigg | _ { x = 0 } \\pmod { p } . \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} g _ { m - 1 , \\ell , s , m } & = g _ { m - 1 , \\ell , s + 1 , m - 1 } + g _ { m - 1 , \\ell - 1 , s - 1 , m - 1 } = 0 , ( \\ell , s ) \\neq ( 1 , 1 ) \\\\ g _ { m - 1 , 1 , 1 , m } & = g _ { m - 1 , 1 , 2 , m - 1 } + g _ { m - 1 , 0 , 0 , m - 1 } = 1 . \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} \\P ( M \\in d m ) = h ( m ( \\Omega ) ) \\P ( N \\in d m ) \\end{align*}"}
-{"id": "9388.png", "formula": "\\begin{align*} \\bigl [ T _ \\phi ( x ) \\bigr ] ( t ) = \\bigl [ \\phi ( t ) \\bigr ] ( x ) , t \\in \\Omega , \\end{align*}"}
-{"id": "9875.png", "formula": "\\begin{align*} \\frac { t ^ L } { L ! } \\prod _ { i = 1 } ^ L \\bigg ( \\frac { 1 } { \\deg ( \\gamma _ { i - 1 } ) } + \\frac { 1 } { \\deg ( \\gamma _ i ) } \\bigg ) \\leq \\left ( \\frac { 2 e t } { L } \\right ) ^ L Q ( \\gamma ) , \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} L ( f ) = f ( - 1 , 0 ) + f ( 1 , 0 ) , f \\in E . \\end{align*}"}
-{"id": "1996.png", "formula": "\\begin{align*} \\widetilde { A } ( z , \\{ \\alpha \\} ) = { } _ a \\langle 1 | \\widetilde { T } _ { a } ( z , \\{ \\alpha \\} ) | 1 \\rangle _ { a } , \\\\ \\widetilde { B } ( z , \\{ \\alpha \\} ) = { } _ a \\langle 0 | \\widetilde { T } _ { a } ( z , \\{ \\alpha \\} ) | 1 \\rangle _ { a } . \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} v \\Lambda ^ p : = \\{ \\lambda \\in \\Lambda : r ( \\lambda ) = v , d ( \\lambda ) = p \\} \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{align*} J _ k = e ^ { - \\frac { 1 } { k } A } , ( k = 1 , 2 , . . . ) \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} \\delta \\mathcal { D } = \\int _ { \\mathbb { R } ^ n } \\delta \\phi \\Bigl ( \\frac { e ^ { - \\phi } } { \\int _ { \\mathbb { R } ^ n } e ^ { - \\phi } } - l ( \\nabla \\phi ) \\det ( \\nabla ^ 2 \\phi ) \\Bigr ) d \\xi , \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} \\tau _ n = ( n _ 1 , \\dots , n _ t ) \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\eta _ i } ~ f _ { n _ 1 , \\cdots , n _ i , \\cdots , n _ r } ( \\eta _ 1 , \\cdots , \\eta _ i , \\cdots , \\eta _ r ) = \\sqrt { n _ i ( k + 1 - ( n _ 1 + \\cdots + n _ i + \\cdots + n _ r ) ) } f _ { n _ 1 , \\cdots , n _ i - 1 , \\cdots , n _ r } ( \\eta _ 1 , \\cdots , \\eta _ i , \\cdots . \\eta _ r ) \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} P _ { f } = q _ { f } \\cdot ( P _ { W } \\otimes \\lambda ) \\mbox { e t } q _ { f } ( w , y ) = q ( y - f ( w ) ) . \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} \\ \\partial _ { t } u + \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert = 2 l } a _ { \\alpha } D ^ { \\left [ \\alpha \\right ] } u + A u = f \\left ( t , x \\right ) , t \\in \\left ( 0 , \\infty \\right ) x \\in R ^ { n } , \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} N _ { \\Lambda _ i } ( r ) = \\frac { r } { \\pi } [ 1 + o ( 1 ) ] , r \\to \\infty , i = 1 , 2 . \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\binom { 2 k + e } { k } x ^ k = \\frac { 1 } { ( 1 - 2 \\beta ) ( 1 - \\beta ) ^ e } , \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{align*} 2 = 2 \\chi ( Z _ f , 0 ) = - ( K _ Y + Z _ f ) \\cdot Z _ f = - K _ Y \\cdot Z _ f - Z _ f ^ 2 = \\Delta \\cdot Z _ f - Z _ f ^ 2 , \\end{align*}"}
-{"id": "8858.png", "formula": "\\begin{align*} G _ { d + 1 } ( n ) = \\left ( 1 + \\sum _ { i = 1 } ^ { 4 u + 3 } p _ i ( d ) / n ^ i \\right ) e ^ { - 2 d } + O ( q ( d ) e ^ { 2 d } / n ^ { 4 u + 4 } ) \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{align*} u ( t , x ) : = \\phi ( t , \\delta x ) . \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} & W = \\{ ( [ E ] , S , C ) \\ | \\ ( [ E ] , S ) \\in X , \\ C = ( s ) _ 0 \\ i s \\ a \\ s m o o t h \\ c u r v e , \\\\ & w h e r e \\ 0 \\ne s \\in H ^ 0 ( F ( E , S ) ( c - a - b ) ) \\} . \\end{align*}"}
-{"id": "2372.png", "formula": "\\begin{align*} H _ A ( \\phi ) = \\frac 1 { 8 d } \\sum _ { x , y \\in A \\cup \\partial A \\ , : \\ , | x - y | = 1 } ( \\phi _ { x } - \\phi _ { y } ) ^ 2 \\end{align*}"}
-{"id": "5580.png", "formula": "\\begin{align*} x _ 1 = x _ 2 = x _ 3 < x _ 4 = x _ 5 < x _ 6 = x _ 7 = \\ldots = x _ n \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} \\hat { \\tau } _ { A M | X = \\mathbf { x } } ( s , t ) & = \\hat { D } _ A ( \\mathbf { x } ) \\ , ( \\mathcal { N } _ A ( s ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) \\ , { \\hat { D } _ A ( \\mathbf { x } ) } ^ \\dag \\\\ & = ( \\mathcal { N } _ A ( s ) \\otimes \\mathbb { I } _ M ) \\left ( \\hat { D } _ A ( \\mathbf { x } ) \\ , \\hat { \\rho } _ { A M } \\ , { \\hat { D } _ A ( \\mathbf { x } ) } ^ \\dag \\right ) \\\\ & = ( \\mathcal { N } _ A ( s ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\sigma } _ { A M | X = \\mathbf { x } } ( t ) ) \\ ; . \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{align*} \\partial _ t \\psi = - \\mathrm i \\beta \\Delta \\psi - \\alpha _ 1 \\psi | \\psi | ^ { p - 1 } . \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{align*} Q _ 2 ( t ) = \\int _ { \\mathbb R ^ n } | x | ^ 2 | u ( t , x ) | ^ 2 d x , \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} d i m _ { q , z } \\mathit { F _ { \\chi } } : = t r _ { \\mathit { F _ { \\chi } } } q ^ { 2 L _ 0 } z ^ { h _ 0 } . \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} Y ^ { r } _ { h } = \\{ u _ { h } \\in C ( \\overline { \\Omega } ) : \\ ; u _ { h } | _ { K } \\in P _ { r } ( K ) , \\ ; \\forall \\ ; K \\in \\mathcal { T } _ { h } \\} . \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} \\langle \\gamma \\rangle ^ G = \\bigcup _ { i = 1 } ^ r \\langle \\gamma \\rangle ^ { g _ i } \\textrm { , a n d } \\langle \\gamma \\rangle ^ { g _ i } \\cap \\langle \\gamma \\rangle ^ { g _ j } = \\{ e \\} \\textrm { w h e n e v e r } i \\neq j , \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} f _ 1 & = C _ { 1 2 } z ( B _ { 3 1 } B _ { 2 3 } y ^ 2 - A _ { 3 3 } C _ { 2 1 } x z ) \\\\ f _ 2 & = - A _ { 3 3 } z ( B _ { 2 2 } C _ { 2 1 } y ^ 2 - B _ { 3 1 } C _ { 1 2 } x z ) \\\\ f _ 3 & = - A _ { 3 3 } C _ { 1 2 } y z ( B _ { 2 2 } y + B _ { 2 3 } z ) \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} \\Omega _ { 1 } = ( \\ell _ { 1 } , \\ell _ { 1 } + \\delta _ { 1 } ) , \\Omega _ { 2 } : = ( \\ell _ { 2 } , \\ell _ { 2 } + \\delta _ { 2 } ) \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{align*} \\frac { \\ddot { a } } { a } = u + u ^ 2 \\ ; ; \\ ; \\frac { \\ddot { b } } { b } = v + v ^ 2 . \\end{align*}"}
-{"id": "6720.png", "formula": "\\begin{align*} \\frac { d u } { d y } + O _ { t } u \\left ( y \\right ) = f \\left ( y \\right ) , u \\left ( 0 \\right ) = 0 , y \\in R _ { + } . \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{gather*} \\frac { { \\rm d } Y } { { \\rm d } x } = \\big ( Q ( x - T ) ^ { - 1 } P + S \\big ) Y . \\end{gather*}"}
-{"id": "2719.png", "formula": "\\begin{align*} \\mu _ { U _ { H \\leqslant B } } = \\frac { 1 } { \\sharp U _ { H \\leqslant B } } \\sum _ { x \\in U ( k ) , H ( x ) \\leqslant B } \\delta _ x , \\end{align*}"}
-{"id": "1293.png", "formula": "\\begin{align*} \\mathcal { N } ^ { 2 } \\hat { \\alpha } _ { \\mathsf { 0 } } \\exp \\left [ - T E _ { \\mathsf { 0 } } \\right ] \\hat { \\beta } _ { \\mathsf { 0 } } = 1 \\end{align*}"}
-{"id": "2758.png", "formula": "\\begin{align*} \\sharp T ( \\varepsilon _ 1 , \\varepsilon _ 2 , B ) = \\frac { 6 ( \\varepsilon _ 1 - \\varepsilon _ 2 ) } { \\pi ^ 2 ( \\alpha ^ \\prime ) ^ 2 } B ^ { 2 - \\frac { 1 } { r } } + O _ { \\varepsilon _ i , \\sigma } ( B ^ { 1 - \\frac { 1 } { 2 r } + \\sigma ( \\frac { 3 } { 4 } + \\frac { 1 } { 8 r } ) } ) . \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} Q = \\frac { Z ^ 2 } { X ( \\log X ) ^ { 1 0 } } \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} \\alpha f ( \\alpha ) f ^ 2 ( \\alpha ) = \\big ( \\alpha f ( \\alpha ) \\big ) f ^ 2 ( \\alpha ) = c _ { \\bar { \\alpha } } A _ { \\bar { \\alpha } } f ^ 2 ( \\alpha ) = c _ { \\bar { \\alpha } } B _ { \\bar { \\alpha } } = \\bar { \\alpha } f ( \\bar { \\alpha } ) f ^ 2 ( \\bar { \\alpha } ) . \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} \\sigma ( w , z ) = \\sigma ( w , T z ) = \\sigma ( w , S T z ) . \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{align*} c _ { 1 1 } ( Z _ 0 ) = \\pm 1 / \\sqrt { 3 } , c _ { 1 2 } ( Z _ 0 ) = - 2 / 3 , c _ { 2 1 } ( Z _ 0 ) = 0 , m = c _ { 1 2 } / ( 2 c _ { 1 1 } ) = \\pm 1 / \\sqrt { 3 } . \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} L ( u ) = T , \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} \\mu _ t = ( e _ t ) _ \\# \\eta \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} \\alpha ( \\nu , t ) : = \\left ( \\begin{array} { c c | c c } \\nu + t & 0 & \\sqrt { \\nu ^ 2 - \\frac { 1 } { 4 } } & 0 \\\\ 0 & \\nu + t & 0 & - \\sqrt { \\nu ^ 2 - \\frac { 1 } { 4 } } \\\\ \\hline \\sqrt { \\nu ^ 2 - \\frac { 1 } { 4 } } & 0 & \\nu & 0 \\\\ 0 & - \\sqrt { \\nu ^ 2 - \\frac { 1 } { 4 } } & 0 & \\nu \\\\ \\end{array} \\right ) \\ ; . \\end{align*}"}
-{"id": "9608.png", "formula": "\\begin{align*} \\aligned \\psi _ j & = \\frac 1 { N + 1 } \\Big ( \\overline L ^ \\alpha _ { \\underline { j \\alpha } } + \\frac 1 2 \\overline F ^ i _ k \\overline \\sigma _ j + \\frac 1 2 \\overline F ^ i _ j \\overline \\sigma _ k \\Big ) - \\frac 1 { N + 1 } \\Big ( L ^ \\alpha _ { \\underline { j \\alpha } } + \\frac 1 2 F ^ i _ k \\sigma _ j + \\frac 1 2 F ^ i _ j \\sigma _ k \\Big ) . \\endaligned \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} \\frac { 1 } { N ^ { h - m } } \\sum \\limits _ { \\mathbf { n } \\in \\mathbb { Z } ^ h } \\Big ( \\prod \\limits _ { j = 1 } ^ d f _ j ( \\xi _ j ( \\mathbf { n } ) + \\widetilde { \\mathbf { r } } _ j ) \\Big ) \\frac { 1 } { C _ { \\Xi , \\chi } \\eta ^ h } \\int \\limits _ { \\mathbf { y } \\in \\mathbb { R } ^ h } ( F ( \\mathbf { n } ) + O _ C ( \\eta / \\sigma _ F N ) ) G ( L \\mathbf { y } ) \\boldsymbol { \\chi } ( \\Xi ( \\mathbf { y } - \\mathbf { n } ) ) \\ , d \\mathbf { y } . \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{align*} \\nabla ^ 2 _ { \\vec { x } } v _ 0 + k ^ 2 _ 0 v _ 0 = 0 \\mbox { a n d } \\nabla ^ 2 _ { \\vec { x } } v _ n - k ^ 2 _ n v _ n = 0 \\ \\mbox { f o r } \\ n = 1 , 2 , \\dots \\ , . \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} \\begin{array} { l l } \\Psi = 1 & \\{ u _ 0 > f \\} \\\\ \\Psi = - 1 & \\{ u _ 0 < f \\} . \\end{array} \\end{align*}"}
-{"id": "9698.png", "formula": "\\begin{align*} A _ j ( t ) \\ = \\ \\sum _ { n = 0 } ^ j \\sum _ { i = 0 } ^ n ( - 1 ) ^ i \\binom { j + 1 } { i } ( n - i ) ^ j t ^ n \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} M _ { \\pm \\gamma } = \\left ( M _ { { i j } _ { \\pm \\gamma } } \\right ) , M _ { { i j } _ { \\pm \\gamma } } = \\begin{pmatrix} \\frac { s _ { i j } \\pm \\gamma c _ { i j } } { 2 } & \\frac { s _ { i j } \\mp \\gamma c _ { i j } } { 2 } \\\\ \\frac { s _ { i j } \\mp \\gamma c _ { i j } } { 2 } & \\frac { s _ { i j } \\pm \\gamma c _ { i j } } { 2 } \\end{pmatrix} , \\ , 1 \\leq i , j \\leq n \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} \\big ( a ( D ) \\psi \\big ) ( x ) : = \\int _ { \\mathbb { R } ^ n } e ^ { 2 \\pi i x \\cdot \\xi } a ( \\xi ) \\widehat { \\psi } ( \\xi ) \\ , d \\xi , \\textrm { f o r e v e r y } x \\in \\mathbb { R } ^ { n } , \\end{align*}"}
-{"id": "5835.png", "formula": "\\begin{align*} F _ { 2 m n } L _ { 2 m n + 2 m } & = F _ { 4 m n + 2 m } - F _ { 2 m } \\\\ & = F _ { 2 m n + m } L _ { 2 m n + m } - F _ { 2 m } \\end{align*}"}
-{"id": "4524.png", "formula": "\\begin{align*} \\left \\langle e _ { k } , \\mu _ { \\Upsilon } \\left ( \\mathcal { X } \\right ) e _ { q } \\right \\rangle _ { \\mathbb { R } ^ { d } } : = \\frac { 1 } { 4 } \\left ( \\mu _ { e _ { k } + e _ { q } } \\left ( \\mathcal { X } \\right ) - \\mu _ { e _ { k } - e _ { q } } \\left ( \\mathcal { X } \\right ) \\right ) \\ . \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} \\mathbf { p } \\left ( t \\right ) : = \\underset { \\eta \\rightarrow 0 } { \\lim } \\ \\underset { l \\rightarrow \\infty } { \\lim } \\left \\{ \\left ( \\eta ^ { 2 } \\left \\vert \\Lambda _ { l } \\right \\vert \\right ) ^ { - 1 } \\mathbf { P } ^ { ( \\omega , \\eta \\mathbf { A } _ { l } ) } \\left ( t \\right ) \\right \\} \\ . \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{align*} J _ { i j } \\Omega _ { i j } + \\Omega _ { i j } J _ { i j } ^ { \\top } + G _ { i j } = \\textbf { 0 } , \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{align*} - \\Delta _ g f = n f \\mbox { f o r s o m e } f \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} s p _ \\lambda ( \\{ z \\} _ N ) = \\frac { \\mathrm { d e t } _ N ( z _ j ^ { \\lambda _ k + N - k + 1 } - z _ j ^ { - \\lambda _ k - N + k - 1 } ) } { \\mathrm { d e t } _ N ( z _ j ^ { N - k + 1 } - z _ j ^ { - N + k - 1 } ) } , \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{align*} & \\partial _ t U = \\Delta U - \\nabla P - h u _ \\infty \\cdot \\nabla U + \\bar g , \\quad \\mbox { d i v $ U $ } = 0 ( x \\in \\mathbb R ^ 3 , \\ , t > 0 ) , \\\\ & U \\to 0 \\quad \\mbox { a s $ | x | \\to \\infty $ } , \\\\ & U ( x , 0 ) = \\bar v _ 0 ( x ) . \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{align*} f \\left ( z \\right ) = \\left ( \\frac { 1 } { 1 - z } \\right ) ^ { p } - 1 = \\sum _ { k \\ge 1 } \\binom { k + p - 1 } { p - 1 } z ^ { k } . \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\lambda _ { 1 } ( p , \\Omega ) ^ { \\frac 1 p } = \\frac { 1 } { \\rho _ { F } ( \\Omega ) } . \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} c = ( 1 , 3 , 5 , 6 , 7 , \\dots , n ) ( n + 1 , n + 2 , \\dots , 2 n ) . \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{align*} \\sum _ { t \\geq 0 } ( - 1 ) ^ t \\binom { m } { n t + s - 1 } = \\frac { 1 } { n } \\sum _ { j = 1 } ^ n ( \\mu ^ { 2 j - 1 } + 1 ) ^ m \\mu ^ { ( 2 j - 1 ) ( 1 - s ) } , \\enskip s = 1 , . . . , n , \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} \\delta _ { n m } = \\langle \\tilde e _ n , \\tilde e _ m ^ * \\rangle = \\Bigl \\langle \\sum _ { i = 1 } ^ d a _ { n i } e _ i , \\sum _ { j = 1 } ^ d b _ { m j } e _ j ^ * \\Bigr \\rangle = \\sum _ { i = 1 } ^ d a _ { n i } b _ { m i } . \\end{align*}"}
-{"id": "7322.png", "formula": "\\begin{align*} g _ { \\varphi } ( X , Y ) \\ , v o l = \\frac { 1 } { 6 } \\iota _ { X } \\varphi \\wedge \\iota _ { Y } \\varphi \\wedge \\varphi , \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} \\psi _ { n } \\left ( m \\right ) = \\sqrt { m + 2 \\sqrt { m - 2 } } . \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} \\frac { v _ \\epsilon ''' } { v _ \\epsilon '' } = \\frac { v _ { 0 , \\epsilon } ''' e ^ { 2 \\rho } + 3 v _ { 0 , \\epsilon } '' e ^ \\rho + v _ { 0 , \\epsilon } ' } { v _ { 0 , \\epsilon } '' e ^ \\rho + v _ { 0 , \\epsilon } ' } \\to 1 . \\end{align*}"}
-{"id": "327.png", "formula": "\\begin{align*} \\Xi ^ { \\lambda } ( K _ { a } F _ { a } \\otimes 1 \\otimes 1 \\otimes E _ { a } ) & = \\Xi ^ { \\lambda } ( K _ { a } F _ { a } \\otimes 1 \\otimes E _ { a } K _ { a } ^ { - 1 } \\otimes K _ { a } ) + \\Xi ^ { \\lambda } ( K _ { a } \\otimes F _ { a } \\otimes 1 \\otimes E _ { a } ) \\\\ & - \\sum _ { i , j } c _ { j } ^ { i } \\pi ( E _ { a } K _ { \\lambda } K _ { a } F _ { a } ) _ { i } ^ { j } . \\end{align*}"}
-{"id": "960.png", "formula": "\\begin{align*} \\textrm { V a r } _ { \\{ Y \\in B _ 1 \\} } [ X ] = \\ldots = \\textrm { V a r } _ { \\{ Y \\in B _ k \\} } [ X ] . \\end{align*}"}
-{"id": "4559.png", "formula": "\\begin{align*} \\rho = | \\Psi | ^ { 2 } , \\ ; \\ ; \\mathbf { J } = - \\frac { \\mathrm { i } } { 2 } ( \\Psi ^ { * } \\nabla { \\Psi } - \\Psi \\nabla { \\Psi } ^ { * } ) - | \\Psi | ^ { 2 } \\mathbf { A } . \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} \\begin{aligned} & K ( a ) N ^ { - \\frac \\alpha 2 } e ^ { - N ^ \\alpha J ( a ) } \\left ( 1 - e ^ { - N ^ { \\alpha - 1 } J ' ( a ) } \\right ) - K ' ( a ) N ^ { - \\frac \\alpha 2 - 1 } e ^ { - N ^ { \\alpha } J ( a ) - N ^ { \\alpha - 1 } J ' ( a ) } \\\\ & \\qquad \\qquad \\sim K ( a ) N ^ { - \\frac \\alpha 2 } e ^ { - N ^ \\alpha J ( a ) } \\left ( 1 - e ^ { - N ^ { \\alpha - 1 } J ' ( a ) } \\right ) \\ , . \\end{aligned} \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} U = \\begin{bmatrix} a & 0 \\\\ 0 & 1 \\end{bmatrix} \\end{align*}"}
-{"id": "7720.png", "formula": "\\begin{align*} ( u , v ) + ( u ' , v ) & = ( u + u ' , \\ , v ) \\\\ ( u , v ) + ( u , v ' ) & = ( u , \\ , v + v ' ) \\\\ \\alpha \\ , ( u , v ) & = ( \\alpha u , \\ , v ) \\\\ \\alpha \\ , ( u , v ) & = ( u , \\ , \\alpha v ) . \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{align*} \\partial _ t u = \\mathrm i \\Delta u + \\mathrm i | u | ^ { p - 1 } u , \\end{align*}"}
-{"id": "625.png", "formula": "\\begin{gather*} I ( z _ \\alpha , s _ { \\bar w } ) - I ( \\bar w , s _ { \\bar w } ) > L - L + \\tau = \\tau . \\end{gather*}"}
-{"id": "7303.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 , [ y _ 1 , y _ 2 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 5 } { \\alpha _ 3 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 2 ] = y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\alpha _ 3 \\beta _ 4 } { \\alpha _ 1 \\gamma _ 2 } y _ 5 , [ y _ 2 , y _ 3 ] = y _ 5 = - [ y _ 3 , y _ 2 ] . \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} \\mathop { \\lim } \\limits _ { | t | \\to \\infty , t \\in \\mathbb { R } } E ( i t ) = 0 . \\end{align*}"}
-{"id": "8830.png", "formula": "\\begin{align*} \\binom { m - 1 } { m - z - 1 } = \\binom { m - 1 } { z } \\leq m ^ z . \\end{align*}"}
-{"id": "1525.png", "formula": "\\begin{align*} \\tilde { g } ( x ) = \\tilde { a } x + \\tilde { b } - \\sum _ { k = 1 } ^ { n - 1 } \\frac { \\tilde { s } _ k } { x - \\tilde { t } _ k } , \\tilde { a } : = \\bigg ( \\sum _ { k = 1 } ^ { n - 1 } \\tilde { r } _ k \\bigg ) ^ { - 1 } > 0 , \\tilde { s } _ k > 0 , \\tilde { b } \\in \\mathbb R . \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} D _ { n ^ 2 } = \\max _ { n ^ 2 \\leq k < ( n + 1 ) ^ 2 } | S _ k - S _ { n ^ 2 } | \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{align*} T ( v _ j , 0 ) & = U ( v _ j , 0 ) = s _ { w _ { j e } } \\\\ U ( v _ j , i ) & = s _ { w _ { j e - i } } s _ { w _ { j e + i } } \\\\ T ( v _ j , i ) & = U ( v _ j , 0 ) \\cdots U ( v _ j , i ) = T ( v _ j , i - 1 ) U ( v _ j , i ) \\\\ S ( v _ j , i ) & = T ( v _ j , i - 1 ) \\cdots T ( v _ j , 0 ) = T ( v _ j , i - 1 ) S ( v _ j , i - 1 ) . \\end{align*}"}
-{"id": "2406.png", "formula": "\\begin{align*} \\mathrm i \\partial _ t v + D v + \\frac { 1 } { h } [ D , h ] v = \\mathrm i h ^ { p - 1 } | v | ^ { p - 1 } v . \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{align*} \\Xi ^ { \\lambda } ( K _ { a } F _ { a } \\otimes 1 \\otimes E _ { a } K _ { a } ^ { - 1 } \\otimes K _ { a } ) = \\sum _ { i , j , m , n , o , p } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) \\pi ( K _ { a } F _ { a } ) _ { m } ^ { j } c _ { n } ^ { m } \\pi ( E _ { a } K _ { a } ^ { - 1 } ) _ { o } ^ { n } c _ { p } ^ { o } \\pi ( K _ { a } K _ { \\lambda } ) _ { i } ^ { p } . \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} ( \\tau g ) ^ { - 1 } ( x , y , z ) = \\big ( \\psi _ 1 ( g ^ { - 1 } \\tau ^ { - 3 } ) z , \\psi _ 1 ( \\tau g ^ { - 1 } \\tau ^ { - 1 } ) x , \\psi _ 1 ( \\tau ^ 2 g ^ { - 1 } \\tau ^ { - 2 } ) y \\big ) . \\end{align*}"}
-{"id": "4804.png", "formula": "\\begin{align*} \\alpha = I d + \\beta p ^ l \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{align*} \\lambda ( O _ 1 ) = ( 2 n - k \\geq 2 n - k - 1 \\geq 2 n - k - 2 \\geq . . . \\geq 2 n - 2 k + 2 \\geq 0 ) \\ / ; \\\\ \\lambda ( O _ 2 ) = ( 2 n - k + 1 \\geq 2 n - k - 1 \\geq . . . \\geq 2 n - 2 k + 2 \\geq - 1 ) \\ / . \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} \\partial _ z \\zeta ( z ; \\tau ) = - \\wp ( z ; \\tau ) , \\lim _ { z \\to 0 } \\zeta ( z ; \\tau ) - z ^ { - 1 } = 0 . \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} w _ 2 ( \\xi ) = 5 { \\rm s e c h } ^ 2 ( \\sqrt { c _ * } \\xi ) + \\frac { 1 5 } { 4 } { \\rm s e c h } ^ 4 ( \\sqrt { c _ * } \\xi ) - \\tilde { w } _ 2 ( \\xi ) , \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} \\partial _ n u = n \\cdot \\nabla u \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} E ( I u ( t ) ) = E ( I u _ 0 ) + O ( N ^ { - \\gamma _ 0 ( k ) + } ) , \\end{align*}"}
-{"id": "7631.png", "formula": "\\begin{align*} \\big ( [ \\omega _ 0 ] + T ( K _ X + D ) \\big ) \\cdot E _ i = 0 . \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) \\left ( I + \\begin{pmatrix} 0 & \\beta \\\\ 0 & 0 \\end{pmatrix} p \\right ) = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a & b + \\beta \\\\ c & d \\end{pmatrix} p . \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} \\| ( \\theta + 2 \\pi z ) \\times c ^ { ( z ) } \\| = \\| ( \\theta + 2 \\pi z ) \\| \\| c ^ { ( z ) } \\| \\geq \\pi \\| c ^ { ( z ) } \\| \\quad ( z \\in \\mathbb { Z } ^ 3 \\setminus \\{ 0 \\} ) . \\end{align*}"}
-{"id": "3575.png", "formula": "\\begin{align*} \\phi b = \\sum _ i ( - 1 ) ^ { ( \\deg \\xi _ i ) ( \\deg \\phi ) } \\xi _ i \\wedge \\phi h _ i . \\end{align*}"}
-{"id": "2114.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 2 } { x ^ 2 } + \\frac { A _ 1 } { x } + A _ 0 \\right ) Y , \\frac { \\partial Y } { \\partial t } = \\left ( \\frac { 1 } { t } A _ 0 x + B _ 0 \\right ) Y , \\end{gather*}"}
-{"id": "7458.png", "formula": "\\begin{align*} [ b _ + ( \\alpha + \\lambda ) , b _ + ( \\alpha ' + \\lambda ' ) ] = [ x _ + ( \\alpha ) _ \\lambda , x _ + ( \\alpha ' ) _ { \\lambda ' } ] \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{align*} { \\displaystyle \\psi ( \\mathbf { x } , t ) = 0 , \\ , \\ , \\phi ( \\mathbf { x } , t ) = 0 , \\ , \\ , \\mathbf { A } ( \\mathbf { x } , t ) \\times \\mathbf { n } = 0 , \\ , \\ , \\nabla \\cdot \\mathbf { A } ( \\mathbf { x } , t ) = 0 , ( \\mathbf { x } , t ) \\in \\partial \\Omega \\times ( 0 , T ) , } \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { p - 1 } \\binom { 2 k } { k } ^ 2 \\binom { 4 k } { 2 k } \\cdot \\frac { 1 } { 1 4 4 ^ k } \\equiv \\frac { 9 \\Gamma _ p ( \\frac 4 3 ) ^ 2 } { 4 \\Gamma _ p ( \\frac 3 2 ) ^ 2 \\Gamma _ p ( \\frac 5 6 ) ^ 2 } \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} W _ { l } : = \\hat { U } ^ { ( n ) } _ n ( \\omega _ a ) - \\hat { U } ^ { ( n ) } _ n ( \\omega _ b ) = a ( \\gamma _ a ) - b ( \\gamma _ b ) \\end{align*}"}
-{"id": "5955.png", "formula": "\\begin{align*} \\hat { S } _ { 2 T } \\left ( u \\right ) = S _ { 2 T } \\left ( u \\right ) + T ^ { 1 / 2 } \\left ( \\hat { \\theta } _ { T } - \\theta _ { 0 } \\right ) ^ { \\prime } \\frac { 1 } { T } \\sum _ { t = 2 } ^ { T } \\nabla _ { 2 , t } \\left ( u \\right ) + o _ { p } \\left ( 1 \\right ) , \\end{align*}"}
-{"id": "1125.png", "formula": "\\begin{align*} \\eqref { e q - l o c a l - d i m - c o n d } B ' \\subset B \\subset B ( x , r _ 0 ) s = s ( x ) > p , \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} f _ 1 & = - B _ { 3 1 } B _ { 2 2 } B _ { 1 3 } y ^ 3 - C _ { 1 2 } C _ { 2 1 } A _ { 3 3 } x z ^ 2 + ( \\ , \\ , \\cdots ) y ^ 2 z \\\\ f _ 2 & = A _ { 3 3 } z ( B _ { 3 1 } C _ { 1 2 } x z - C _ { 2 1 } y ( B _ { 2 2 } y + B _ { 3 2 } z ) ) \\\\ f _ 3 & = A _ { 3 3 } z ( B _ { 1 3 } C _ { 2 1 } x z - C _ { 1 2 } y ( B _ { 2 2 } y + B _ { 2 3 } z ) ) \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} m _ i h _ i ^ { - 1 } \\frac { d h _ i } { d t } = \\sum _ { \\alpha : i \\to j } h _ i ^ { - 1 } \\phi _ \\alpha ^ * h _ j \\phi _ \\alpha - \\sum _ { \\alpha : j \\to i } \\phi _ \\alpha h _ j ^ { - 1 } \\phi _ \\alpha ^ * h _ i - \\theta _ i . \\end{align*}"}
-{"id": "7659.png", "formula": "\\begin{align*} \\Z [ G ] t = \\sum _ { j = 1 } ^ n \\Z [ G ] b _ j . \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{align*} N ( a ) = 2 m a _ m ^ - < a < a _ m ^ + . \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\lambda } ^ d \\xi _ t ( O ) = e ^ { - t } \\sum _ { n = 0 } ^ { \\infty } \\frac { t ^ n \\lambda ^ n } { n ! } \\Big ( \\sum _ { \\overrightarrow { x } \\in V _ n } E \\prod _ { j = 0 } ^ { n - 1 } \\rho ( x _ j , \\omega ) \\rho ( x _ { j + 1 } , \\omega ) \\Big ) . \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} \\chi _ { \\tau } ( t , r ) : = \\int _ r ^ t \\left [ \\int _ 0 ^ r \\nabla q _ { i \\tau + s - r } ( B _ r - B _ u ) d u \\right ] d s , 0 \\le r \\le t . \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} \\left \\Vert e ^ { k \\left \\vert x \\right \\vert ^ { p } } u \\left ( x , 0 \\right ) \\right \\Vert _ { X } = A _ { k } \\leq a _ { 2 } e ^ { 2 a _ { 1 } k ^ { \\frac { q } { q - p } } } = a _ { 2 } e ^ { 2 a _ { 1 } k ^ { \\frac { 1 } { 2 - p } } } . \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} \\Psi _ { U U ' } ( z , w ) = ( \\frac 1 z , w z ^ k ) : = ( z ' , w ' ) . \\end{align*}"}
-{"id": "5962.png", "formula": "\\begin{align*} ( \\epsilon + a ) y ^ { 2 } + \\epsilon z ^ { 2 } + ( a + b ) y z + \\epsilon ( \\epsilon + a ) y + \\epsilon ^ { 2 } z = 0 . \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} \\langle s e ^ { \\i \\langle \\theta , \\cdot \\rangle _ { \\mathbb { C } ^ d } } \\alpha , e ^ { \\i \\langle \\theta , \\cdot \\rangle _ { \\mathbb { C } ^ d } } \\beta \\rangle = \\sum _ { i , j \\in \\{ 1 , \\ldots , n \\} } \\left ( \\int _ { Y } s _ { i j } \\right ) \\alpha _ i \\beta _ j , ( \\alpha , \\beta \\in \\mathbb { C } ^ n ) . \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} u _ c ( \\xi ) = c \\ ; { \\rm s e c h } ^ 2 ( \\sqrt { c } \\xi ) , \\xi = x - 4 c t - x _ 0 , \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} f ( T , U ) = ( - 1 ) ^ { n + 1 } \\hat { f } ( U , T ) , \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} \\mu ( \\mathsf { x , y } ) = \\varphi ( \\mathsf { x ) } g ( \\mathsf { x } , T , \\mathsf { y } ) \\psi ( \\mathsf { y ) , } \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} \\{ U _ { n } ^ { ( 1 ) } \\} _ { n = 0 } ^ { 3 } = \\{ 0 , 1 , p , p ^ { 2 } - q \\} , \\{ V _ { n } ^ { ( 1 ) } \\} _ { n = 0 } ^ { 3 } = \\{ 2 , p , p ^ { 2 } - 2 q , p ^ { 3 } - 3 p q \\} , \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} ( ( 1 - \\varphi ) u ) _ t = \\Delta ( ( 1 - \\varphi ) u ) + 2 \\nabla u \\cdot \\nabla \\varphi + u \\Delta \\varphi + ( 1 - \\varphi ) | u | ^ \\alpha u . \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} D = \\left [ \\begin{array} { l l } 0 & 0 \\\\ 0 & B \\end{array} \\right ] , \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} \\mathbb { P } ( \\hat { T } ^ { ( k ) } ( \\pi ) \\leq \\beta m ) = \\mathbb { P } \\left ( \\sum _ { i = 1 } ^ { m } t ^ { ( k ) } ( e _ i ) \\leq \\beta m \\right ) \\leq e ^ { s \\beta m } \\prod _ { i = 1 } ^ { m } \\mathbb { E } ( e ^ { - s t ^ { ( k ) } ( e _ i ) } ) . \\end{align*}"}
-{"id": "1987.png", "formula": "\\begin{align*} \\langle 1 \\cdots M | \\mathcal { B } ^ \\prime ( z _ 1 ) \\cdots \\mathcal { B } ^ \\prime ( z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle = ( - 1 ) ^ { N ( N - 1 ) / 2 } \\mathrm { d e t } _ N ( z _ j ^ { \\lambda _ k + N - k + 1 } - z _ j ^ { - \\lambda _ k - N + k - 1 } ) . \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} \\mathfrak G ^ { x } _ \\infty = \\bigcap \\limits _ { \\varepsilon \\in \\Q _ { + } ^ { * } } \\bigcap \\limits _ { R \\in \\Q _ { + } ^ { * } } \\mathfrak H ^ { x } _ { \\varepsilon , R } , \\end{align*}"}
-{"id": "9404.png", "formula": "\\begin{align*} \\int _ { \\Omega _ 1 \\times \\Omega _ 3 } \\langle a ( t _ 1 , t _ 2 ) , b ( t _ 2 , t _ 3 ) \\rangle \\ , & r _ 1 ( t _ 1 ) \\ , r _ 3 ( t _ 3 ) \\ , \\mu _ 1 ( t _ 1 ) \\mu _ 3 ( t _ 3 ) \\\\ & = \\bigl \\langle \\bigl [ \\widetilde { \\phi } ( t _ 2 ) \\bigr ] ( r _ 1 ) , r _ 3 \\bigr \\rangle \\\\ & = \\int _ { \\Omega _ 1 \\times \\Omega _ 3 } \\phi ( t _ 1 , t _ 2 , t _ 3 ) r _ 1 ( t _ 1 ) \\ , r _ 3 ( t _ 3 ) \\ , \\mu _ 1 ( t _ 1 ) \\mu _ 3 ( t _ 3 ) \\end{align*}"}
-{"id": "1365.png", "formula": "\\begin{align*} u _ { s c } ( x , y , k ) = A ( x , y ) e ^ { i k \\tau \\left ( x , y \\right ) } - A _ { 0 } ( x , y ) e ^ { i k \\left \\vert x - y \\right \\vert } + \\widehat { u } ( x , y , k ) , \\end{align*}"}
-{"id": "9435.png", "formula": "\\begin{align*} \\mathrm { L } _ { D } ( w ^ * ) = d ^ * - { h ^ * } ^ { - 1 } \\nabla \\times b ^ * , \\end{align*}"}
-{"id": "4351.png", "formula": "\\begin{align*} | X | = x _ 0 = 2 ^ { - s t / ( s + 2 t ) } . \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{align*} \\left \\{ \\hat { X } , \\ ; \\hat { Y } \\right \\} : = \\hat { X } \\ , \\hat { Y } + \\hat { Y } \\ , \\hat { X } \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} D ( \\alpha ^ 4 ) = ( D \\alpha ) ( S ^ 2 \\alpha ) ^ 3 + \\alpha ( D \\alpha ) ( S ^ 2 \\alpha ) ^ 2 + \\alpha ^ 2 ( D \\alpha ) ( S ^ 2 \\alpha ) + \\alpha ^ 3 ( D \\alpha ) \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{align*} [ f ( - \\bar { k } , x ) ^ \\dagger ; f ( k , x ) ] = 0 _ n , k \\in \\overline { \\mathbb { C } } ^ + . \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} W ( 0 ) & = [ 1 - B _ K ] \\left [ 1 + \\int _ { 0 ^ - } ^ K \\lambda \\alpha e ^ { \\lambda \\sigma } e ^ { \\lambda ( \\alpha e ^ { \\lambda \\sigma } - 1 ) x } d x \\right ] ^ { - 1 } \\\\ & = [ 1 - B _ K ] \\left [ \\dfrac { \\alpha _ { \\lambda } e ^ { \\lambda \\sigma } - 1 } { \\alpha _ { \\lambda } e ^ { \\lambda ( K \\alpha _ { \\lambda } e ^ { \\lambda \\sigma } - K + \\sigma ) } - 1 } \\right ] , \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} B ^ p ( \\Sigma ^ k , M ^ n ) = \\prod _ { j = 1 } ^ { k } ( T ^ \\ast \\Sigma ^ k ) ^ { \\otimes j } \\otimes T M ^ n . \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} \\mathcal { A } ^ P = & - e ^ { - 2 i k _ { P ( n ) } l } \\mathcal { A } ^ { P R _ n } \\\\ = & - e ^ { - 2 i k _ { P ( n ) } l } \\mathcal { A } ^ { P C _ n R _ 1 C _ n ^ { - 1 } } , \\end{align*}"}
-{"id": "3073.png", "formula": "\\begin{align*} \\sum \\limits _ { \\left \\{ i , j \\right \\} \\in E \\left ( G ^ { \\prime } \\right ) } x _ { i } x _ { j } & = \\sum \\limits _ { \\left \\{ i , j \\right \\} \\in E \\left ( G \\right ) } x _ { i } x _ { j } - x _ { k + s - 1 } x _ { k + s } - x _ { k + s + 2 } x _ { k + s + 3 } + x _ { k + s } x _ { k + s + 2 } + x _ { k + s - 1 } x _ { k + s + 3 } \\\\ & = \\sum \\limits _ { \\left \\{ i , j \\right \\} \\in E \\left ( G \\right ) } x _ { i } x _ { j } . \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) \\longmapsto \\chi _ { x } ^ { ( \\Omega ) } \\left ( \\omega _ { 1 } , \\omega _ { 2 } \\right ) : = \\left ( \\chi _ { x } ^ { ( \\mathfrak { L } ) } \\left ( \\omega _ { 1 } \\right ) , \\chi _ { x } ^ { ( \\mathfrak { b } ) } \\left ( \\omega _ { 2 } \\right ) \\right ) \\ , x \\in \\mathbb { Z } ^ { d } \\ , \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = 1 & x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\begin{matrix} 0 \\\\ \\theta ^ 0 _ 1 \\\\ \\theta ^ 0 _ 2 \\end{matrix} & \\begin{matrix} 0 \\\\ 0 \\\\ \\theta ^ 1 \\end{matrix} & \\overbrace { \\begin{matrix} \\sqrt { - t } & \\theta ^ \\infty _ 1 / 2 \\\\ - \\sqrt { - t } & \\theta ^ \\infty _ 1 / 2 \\\\ 0 & \\theta ^ \\infty _ 2 \\\\ \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "4580.png", "formula": "\\begin{align*} \\overline { \\phi } _ { h } ^ { k } = \\sum _ { j = 1 } ^ { N } { a _ { j } u _ j } , \\overline { \\Psi } _ { h } ^ { k } = \\sum _ { j = 1 } ^ { N } { b _ { j } u _ j } , { \\Psi } _ { h } ^ { k - 1 } = \\sum _ { j = 1 } ^ { N } { c _ { j } u _ j } , \\end{align*}"}
-{"id": "647.png", "formula": "\\begin{align*} u ( x ) & = u ( 0 , x _ { n + 1 } ) + | x _ { n + 1 } | ^ { 2 s } \\int _ 0 ^ 1 \\sum _ { i = 1 } ^ { n } \\sum _ { k = 0 } ^ { \\lfloor \\frac { ( m - 1 ) } 2 \\rfloor } x _ { n + 1 } ^ { 2 k } q _ { i , m - 1 - 2 k } ( t x ' ) \\ , x _ i \\ , \\d t \\\\ & = u ( 0 , x _ { n + 1 } ) + | x _ { n + 1 } | ^ { 2 s } \\sum _ { k = 0 } ^ { \\lfloor \\frac { ( m - 1 ) } 2 \\rfloor } x _ { n + 1 } ^ { 2 k } q _ { m - 2 k } ( x ) , \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} \\rho ' ( g ) & = \\begin{pmatrix} i & 0 \\\\ 0 & - i \\end{pmatrix} \\\\ \\rho ' ( h ) & = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{gather*} P _ { A , A ^ * } = e \\sum _ { i = 0 } ^ d \\zeta _ i p _ { i + 1 } p _ { i + 2 } \\dots p _ d , \\end{gather*}"}
-{"id": "4912.png", "formula": "\\begin{align*} \\sigma ( X ) = \\left \\{ { \\displaystyle \\sum \\limits _ { i = 1 } ^ { n } } x _ { i } , x _ { 1 } - x _ { 2 } , x _ { 1 } - x _ { 3 } , \\ldots , x _ { 1 } - x _ { n } \\right \\} . \\end{align*}"}
-{"id": "9101.png", "formula": "\\begin{gather*} \\gcd ( q , ( n + 1 ) \\det ( A ) \\det ( A ' ) ) = 1 . \\end{gather*}"}
-{"id": "2248.png", "formula": "\\begin{align*} \\frac { x ^ { r ^ N } / D _ N } { e _ C ( x ) - \\sum _ { i = 0 } ^ { N - 1 } x ^ { r ^ i } / D _ i } = \\sum _ { n = 0 } ^ \\infty \\frac { B C _ { N , n } } { \\Pi ( n ) } x ^ n \\end{align*}"}
-{"id": "1645.png", "formula": "\\begin{align*} \\nabla E ( x ) = - \\frac { x } { 2 \\pi | x | ^ 2 } \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } x + z + \\frac { y z } { x } = a , \\\\ y + x + \\frac { x z } { y } = b , \\\\ z + y + \\frac { x y } { z } = c . \\end{array} \\right . \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} \\Delta u + k ^ { 2 } n ^ { 2 } ( x ) u = - \\delta ( x - y ) , x \\in \\mathbb { R } ^ { 3 } , \\end{align*}"}
-{"id": "6843.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 - z ) ^ \\alpha } = \\sum _ { n = 0 } ^ \\infty c _ \\alpha ( n ) z ^ n , c _ \\alpha ( n ) : = \\binom { n + \\alpha - 1 } { n } . \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{align*} \\mathcal { N } ( t ) ( \\hat { \\rho } ) : = \\int _ { \\mathbb { R } ^ { 2 n } } \\hat { D } ( \\mathbf { x } ) \\ , \\hat { \\rho } \\ , { \\hat { D } ( \\mathbf { x } ) } ^ \\dag \\ , \\mathrm { e } ^ { - \\frac { | \\mathbf { x } | ^ 2 } { 2 t } } \\frac { \\mathrm { d } ^ { 2 n } x } { ( 2 \\pi \\ , t ) ^ n } \\ ; . \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{align*} \\frak m _ k ^ { ( b _ 0 , \\cdots , b _ k ) } ( x _ 1 , \\cdots , x _ k ) = \\sum _ { \\ell _ 0 , \\cdots , \\ell _ k } \\frak m _ { k + \\ell _ 0 + \\cdots + \\ell _ k } ( b _ 0 ^ { \\ell _ 0 } , x _ 1 , b _ 1 ^ { \\ell _ 1 } , \\cdots , b _ { k - 1 } ^ { \\ell _ { k - 1 } } , x _ k , b _ k ^ { \\ell _ k } ) . \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ N \\ ; c \\sqrt { c ^ 2 + \\lambda _ j } k _ j & \\leq r \\sqrt { c ^ 4 + c ^ 2 \\lambda _ N } \\ ; \\leq \\ ; \\sqrt { 2 } \\ ; r ^ 2 \\lambda _ N , \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} \\varphi ( t ) = - \\frac { 1 } { y ( t ) } \\left ( e ^ { 1 2 7 } + e ^ { 3 4 7 } + e ^ { 5 6 7 } \\right ) - y ( t ) ^ 3 \\left ( e ^ { 1 3 5 } - e ^ { 1 4 6 } - e ^ { 2 3 6 } - e ^ { 2 4 5 } \\right ) , \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{gather*} \\dim V _ C ( \\theta ) = \\dim V _ D ( \\theta ^ { - 1 } ) . \\end{gather*}"}
-{"id": "8378.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } t ( f _ i ) \\leq \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } t ( f _ i ) + \\mu \\leq 2 \\mu . \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{align*} \\lambda _ { \\pm } ( k ) : = \\frac { k \\pm \\sqrt { k ^ 2 + 6 k - 3 } } { 2 } \\ , . \\end{align*}"}
-{"id": "6377.png", "formula": "\\begin{align*} \\alpha _ \\delta ( s _ \\lambda ) = ( - 1 ) ^ { \\delta ( \\lambda ) } s _ \\lambda \\quad \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} J ( A | M ) _ { \\hat { \\rho } _ { A M } } : = \\lim _ { t \\to 0 } \\frac { \\Delta _ { A | M } ( \\hat { \\rho } _ { A M } ) ( t ) } { t } \\ ; . \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} z ( t ) = t ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( A t ^ \\alpha ) z ^ 0 + \\int _ { 0 } ^ { t } ( t - \\tau ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( A ( t - \\tau ) ^ \\alpha ) f ( \\tau ) d \\tau , \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} \\operatorname { A r f } ( \\omega ^ { 0 , 0 } + x \\cdot ) = \\sum ^ g _ { i = 1 } ( x \\cdot [ \\alpha _ i ] ) ( x \\cdot [ \\beta _ i ] ) = \\omega ^ { 0 , 0 } ( x ) . \\end{align*}"}
-{"id": "7983.png", "formula": "\\begin{align*} \\| \\widetilde U \\otimes \\widetilde U + h ( \\widetilde U \\otimes u _ s + u _ s \\otimes \\widetilde U ) \\| _ 2 \\left \\{ \\begin{array} { l l } \\leq C t ^ { - 1 / 4 } & \\mbox { f o r a l l $ t > 0 $ } , \\\\ = o ( t ^ { - 1 / 4 } ) & \\mbox { a s $ t \\to \\infty $ } , \\end{array} \\right . \\end{align*}"}
-{"id": "7336.png", "formula": "\\begin{align*} \\omega ( t ) = \\sum _ { i = 1 } ^ 3 \\lambda _ i ( t ) f ^ i \\wedge J _ 0 f ^ i , \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{align*} \\alpha _ i ^ + ( t ) & = \\alpha _ i ^ + ( 0 ) , \\forall i \\ge 1 , \\\\ \\alpha _ i ^ - ( t ) & = \\alpha _ i ^ - ( 0 ) , \\forall i \\ge 1 , \\\\ \\gamma _ 1 ( t ) & = \\gamma _ 1 ( 0 ) , \\\\ \\gamma _ 2 ( t ) & = t + \\gamma _ 2 ( 0 ) . \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} I _ c & : = \\{ ( \\psi , \\bar \\psi ) \\in H ^ s : ( I _ j ( \\psi , \\bar \\psi ) , \\bar I _ j ( \\psi , \\bar \\psi ) = ( I _ j ( 0 ) , \\bar I _ j ( 0 ) ) , \\ ; \\ ; \\ ; \\ ; j \\geq 1 \\} , \\end{align*}"}
-{"id": "552.png", "formula": "\\begin{align*} \\Delta _ { m + 1 } ( a ) \\coloneqq P _ { m + 1 } \\big ( \\frac { 1 } { 2 } , a \\big ) - Q _ { m + 1 } ( a ) = - \\frac { c } { 2 } a ^ 2 + \\frac { 2 m - 3 } { 4 } a c + d \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { k = 1 } ^ n P ( x _ k ) Q _ k ( x ) ^ 2 + \\sum _ { k = 1 } ^ n \\left ( P ' ( x _ k ) - P ( x _ k ) \\frac { Q '' ( x _ k ) } { Q ' ( x _ k ) } \\right ) ( x - x _ k ) Q _ k ( x ) ^ 2 . \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} L ^ j = a ^ j _ 0 ( t ) \\partial _ t ^ { k _ j } + \\cdots + a ^ j _ { k _ j } ( t ) y , a ^ j _ 0 , \\ldots , a ^ j _ { k _ j } \\in \\C [ t ] , \\norm { a ^ j _ 0 } _ \\infty = 1 . \\end{align*}"}
-{"id": "2273.png", "formula": "\\begin{align*} X ( t + t _ f ) = R ^ { - 1 } X ( t ) R , z ( t + t _ f ) = R ^ { - 1 } . z ( t ) . \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} W ( \\epsilon ) : = \\bigcup _ { 1 \\leq i \\leq n } \\{ \\# { \\cal E } _ i \\geq \\epsilon n \\} \\end{align*}"}
-{"id": "834.png", "formula": "\\begin{align*} \\Phi _ { N } \\left ( t \\right ) \\equiv \\left \\{ \\begin{array} { c c c } t ( \\ln t ) ^ { N } & & t \\geq E = E _ { N } = e ^ { 2 N } \\\\ \\left ( \\ln E \\right ) ^ { N } t & & 0 \\leq t \\leq E = E _ { N } = e ^ { 2 N } \\end{array} \\right . . \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} M _ { i j } = \\begin{pmatrix} a _ { i j } & b _ { i j } \\\\ b _ { i j } & a _ { i j } \\end{pmatrix} \\mathbf { x } _ { i } = \\left ( x _ { i } , x _ { i } \\right ) ^ { T } \\mathbf { y } _ { i } = \\left ( y _ { i } , y _ { i } \\right ) . \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{align*} \\widetilde { H } ' ( s ) = - \\frac { \\alpha \\lambda _ 2 | z | ^ { \\alpha + 2 } } { ( \\alpha + 2 ) s ^ { \\alpha \\lambda _ 2 + 1 } } - \\frac { \\lambda _ 2 | z | ^ 2 } { 2 s ^ 2 } + ( 1 - 2 ( N - 2 ) s ) | z ' | ^ 2 \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} K ( s , t ) = \\begin{cases} w ( s ) - \\left ( w ( a ) + \\psi ( \\lambda ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ a , t ) , \\\\ w ( s ) - \\left ( w ( a ) + ( 1 + \\psi ( 1 - \\lambda ) ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ t , b ] , \\end{cases} \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{align*} L ^ { s , ( N ) } _ { \\theta _ i } = 4 \\cos ^ 2 \\left ( \\frac { \\theta _ i } { 2 } \\right ) \\frac { d ^ 2 } { d \\theta _ i ^ 2 } + \\left [ \\left ( - 4 N - 4 \\Re ( s ) \\right ) \\sin \\left ( \\frac { \\theta _ i } { 2 } \\right ) \\cos \\left ( \\frac { \\theta _ i } { 2 } \\right ) + 4 \\Im ( s ) \\cos ^ 2 \\left ( \\frac { \\theta _ i } { 2 } \\right ) \\right ] \\frac { d } { d \\theta _ i } , \\end{align*}"}
-{"id": "2266.png", "formula": "\\begin{align*} g ( 0 ) = I . \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\rho _ { \\varepsilon , j } u _ i ^ 2 d x \\geq \\int _ { 2 ^ { - 1 } B _ i ^ { \\varepsilon } } \\rho _ { \\varepsilon , j } u _ i ^ 2 d x = \\varepsilon ^ { - N } | 2 ^ { - 1 } B _ i ^ { \\varepsilon } | = 2 ^ { - N } \\omega _ N . \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} u ( z ) = P ( z ) + O ( | z | ^ { a + 1 } \\log ^ { b + 2 } | z | ) . \\end{align*}"}
-{"id": "9613.png", "formula": "\\begin{align*} \\mathcal F ^ i _ { j k } = \\left \\{ \\begin{array} { l l } 0 , & j , k \\in \\{ 1 , 2 \\} \\\\ \\sin u \\ln ( 1 + u ^ 2 + v ^ 2 + w ^ 2 ) , & i = j = 1 , k = 3 \\mbox { o r } i = k = 1 , j = 3 , \\\\ \\cos v \\ln ( 1 + u ^ 2 + v ^ 2 + w ^ 2 ) , & i = j = 2 , k = 3 \\mbox { o r } i = k = 2 , j = 3 , \\\\ 2 w \\ln ( 1 + u ^ 2 + v ^ 2 + w ^ 2 ) , & i = j = k = 3 . \\end{array} \\right . \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} s _ { n , j } ^ { ( - 1 ) } & = - \\sum _ { k = 1 } ^ { n - j } s _ { n , n + 1 - k } ^ { ( - 1 ) } \\cdot s _ { n + 1 - k , j } + \\delta _ { n , j } \\\\ & = - \\sum _ { k = 1 } ^ { n - j } s _ { n , n - k } \\cdot s _ { n - k , j } ^ { ( - 1 ) } + \\delta _ { n , j } \\\\ & = - \\sum _ { k = 1 } ^ { n } s _ { n , k - 1 } \\cdot s _ { k - 1 , j } ^ { ( - 1 ) } + \\delta _ { n , j } . \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} \\mathbf { A } _ { l } ( t , x ) : = \\mathbf { A } ( t , l ^ { - 1 } x ) \\ , t \\in \\mathbb { R } , \\ x \\in \\mathbb { R } ^ { d } \\ . \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} g ( g ( x ) y ) y = y g ( g ( x ) y ) \\ , . \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } q _ t ( x , y ) = - \\partial _ y \\textbf { 1 } ( y \\le x ) = \\delta ( x = y ) . \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} ( I - K ) u = J \\phi , \\end{align*}"}
-{"id": "4140.png", "formula": "\\begin{align*} P _ { \\lambda } ^ { \\omega } ( x \\in I _ { \\infty } ) & = P _ { \\lambda } ^ { \\omega } ( U _ { x _ i , x _ { i + 1 } } < H _ { x _ i } , \\forall i \\in [ 0 , n - 1 ] ) \\\\ & = \\prod _ { i = 0 } ^ { n - 1 } [ \\frac { \\lambda \\rho ( x _ i , \\omega ) \\rho ( x _ { i + 1 } , \\omega ) } { 1 + \\lambda \\rho ( x _ i , \\omega ) \\rho ( x _ { i + 1 } , \\omega ) } ] . \\end{align*}"}
-{"id": "5615.png", "formula": "\\begin{align*} \\frac { f _ { n } ( x _ { 1 } , . . . , x _ { n } | K _ { n } ) } { \\prod _ { m = 1 } ^ { n } f _ { 1 } ( x _ { m } ) } = \\sum _ { S } ( \\prod _ { 1 \\leq i < j \\leq n } \\frac { \\left ( \\rho _ { i j } \\right ) ^ { s _ { i j } } } { s _ { i j } ! } \\prod _ { m = 1 } ^ { n } H _ { \\sigma _ { m } } ( x _ { m } ) ) , \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{align*} J ( k ) = J ( i \\kappa ) + \\dot { J } ( i \\kappa ) ( k - i \\kappa ) + O ( ( k - i \\kappa ) ^ 2 ) , k \\to i \\kappa , \\ ; \\ ; \\overline { \\mathbb { C } } ^ + , \\end{align*}"}
-{"id": "9917.png", "formula": "\\begin{align*} & f ^ { \\pm \\pm \\pm } _ { k ; n } = ( \\pm | n | , \\pm | n + k p | , \\pm | n - k d | ) , \\\\ & f ^ { \\pm \\pm \\pm } _ { k ; - n } = ( \\pm | n | , \\pm | n - k p | , \\pm | n + k d | ) , \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{gather*} \\Lambda _ { \\eta } ( q ) = q ( \\eta ) \\quad \\textrm { f o r } q \\in \\mathcal { H } _ m ^ p ( \\widehat { S } _ p ) . \\end{gather*}"}
-{"id": "9051.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } C _ { s , j } ^ { ( 4 ) } \\mu ^ { ( 1 - s ) ( 2 j - 1 ) } ( \\mu ^ { 2 j - 1 } + 1 ) ^ m , \\enskip s = 1 , . . . , n . \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} \\int _ { x _ i } ^ { x _ { i + 1 } } \\hat { m } ( y _ i ) C _ { t } ( y , x ' ) _ { i j } d y _ i & = - A _ t ( x , x ' ) _ { i + 1 j } + A _ t ( x , x ' ) _ { i j } , \\\\ \\int _ { x _ i } ^ { x _ { i + 1 } } \\hat { m } ( y _ i ) D _ t ( y , y ' ) _ { i j } d y _ i & = - B _ t ( x , y ' ) _ { i + 1 j } + B _ t ( x , y ' ) _ { i j } + \\hat { m } ( y _ j ' ) \\textbf { 1 } \\left ( j = i \\right ) , \\end{align*}"}
-{"id": "5643.png", "formula": "\\begin{align*} - v ^ 3 + v = \\frac { b } { c } v , \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} u [ \\eta , \\psi , \\Psi ] ( \\eta X ) = w ( \\eta X ) + W ( X ) \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} n _ 1 ( \\cdot , t ) & = e ^ { ( t - t _ 0 ) \\Delta } n _ 1 ( t _ 0 ) - \\int _ { t _ 0 } ^ t e ^ { ( t - s ) \\Delta } \\nabla \\cdot \\left ( \\chi n _ 1 ( \\cdot , s ) \\nabla c ( \\cdot , s ) + n _ 1 ( \\cdot , s ) u ( \\cdot , s ) \\right ) d s \\\\ & \\quad \\ , + \\mu _ 1 \\int _ { t _ 0 } ^ t e ^ { ( t - s ) \\Delta } n _ 1 ( 1 - n _ 1 - a _ 1 n _ 2 ) d s \\\\ & = : I _ 1 ( \\cdot , t ) + I _ 2 ( \\cdot , t ) + I _ 3 ( \\cdot , t ) . \\end{align*}"}
-{"id": "1304.png", "formula": "\\begin{align*} f ^ { \\prime } \\left \\vert 1 - \\left ( f g ^ { \\prime } \\right ) ^ { 2 } \\right \\vert - \\frac { 3 f } { 2 } \\left \\vert - 2 f f ^ { \\prime } \\left ( g ^ { \\prime } \\right ) ^ { 2 } \\right \\vert = 0 , \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { u } = i \\left [ \\Delta \\tilde { u } + A \\tilde { u } + \\tilde { V } \\left ( x , t \\right ) \\tilde { u } + \\tilde { F } \\left ( x , t \\right ) \\right ] , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] \\end{align*}"}
-{"id": "9714.png", "formula": "\\begin{align*} G ' = \\left ( \\begin{array} { c | c } G & A \\\\ \\hline A & G \\end{array} \\right ) \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{align*} \\sigma ( w , z ) = \\sigma ( w , S T z ) \\leq \\sigma ( w , T z ) \\leq \\sigma ( w , z ) . \\end{align*}"}
-{"id": "5594.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 1 } T _ { i } ( x ) T _ { j } ( x ) \\frac { 1 } { \\pi \\sqrt { 1 - x ^ { 2 } } } d x & = \\left \\{ \\begin{array} [ c ] { c c c } 0 & i f & i \\neq j \\\\ 1 / 2 & i f & i = j \\neq 0 \\\\ 1 & i f & i = j = 0 \\end{array} \\right . , \\\\ \\int _ { - 1 } ^ { 1 } U _ { i } ( x ) U _ { j } ( x ) \\frac { 2 } { \\pi } \\sqrt { 1 - x ^ { 2 } } d x \\allowbreak & = \\allowbreak \\left \\{ \\begin{array} [ c ] { c c c } 0 & i f & i \\neq j \\\\ 1 & i f & i = j \\end{array} \\right . . \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} \\ - \\varepsilon u ^ { \\left ( 2 \\right ) } \\left ( t , \\varepsilon \\right ) + A u \\left ( t , \\varepsilon \\right ) + B u ^ { \\left ( 1 \\right ) } \\left ( t , \\varepsilon \\right ) + \\lambda u \\left ( t , \\varepsilon \\right ) = f \\left ( t \\right ) , \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle \\tau \\sum _ { k = 1 } ^ { m } | U _ 1 ^ { k } ( \\overline { \\partial _ { \\tau } \\theta } _ { \\mathbf { A } } ^ { k } ) | \\leq C \\left ( h ^ { 2 r } + \\tau ^ { 4 } \\right ) + C \\tau \\sum _ { k = 1 } ^ { m } \\| \\overline { \\partial _ { \\tau } \\theta } _ { \\mathbf { A } } ^ { k } \\| _ { \\mathbf { L } ^ 2 } ^ { 2 } . } \\end{array} \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} d \\bar X _ t = \\frac { 1 } { n } \\sum _ { k = 1 } ^ n d X ^ k _ t = \\frac { \\lambda } { n } \\sum _ { k = 1 } ^ n \\gamma _ t ^ k \\ , d t + \\frac { \\sigma } { n } \\sum _ { k = 1 } ^ n d W ^ k _ t + \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\gamma _ { t } ^ k \\ , d \\widetilde N ^ k _ t . \\end{align*}"}
-{"id": "465.png", "formula": "\\begin{align*} Y _ 3 = L M , Y _ 4 = L M + M , \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} f _ m ( n ) = \\sum _ { k = 1 } ^ n c _ m ( n , k ) . \\end{align*}"}
-{"id": "3249.png", "formula": "\\begin{align*} \\gamma _ 2 \\left ( \\omega \\right ) = \\delta \\left ( \\omega \\right ) - \\sum _ { i = 1 } ^ { \\infty } \\left ( \\alpha _ i ^ + \\left ( \\omega \\right ) \\right ) ^ 2 - \\sum _ { i = 1 } ^ { \\infty } \\left ( \\alpha _ i ^ - \\left ( \\omega \\right ) \\right ) ^ 2 . \\end{align*}"}
-{"id": "9756.png", "formula": "\\begin{align*} j u _ { \\epsilon } : = u _ { \\epsilon } ^ * ( r ^ 2 d \\theta ) = u _ { \\epsilon } ^ 1 d u _ { \\epsilon } ^ 2 - u _ { \\epsilon } ^ 2 d u _ { \\epsilon } ^ 1 \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} g _ i = \\frac { | G _ i | } { 2 k _ i } \\bigg ( 2 g _ i ' - 2 + \\sum _ { j = 1 } ^ { r _ i } \\frac { m _ { i , j } - 1 } { m _ { i , j } } \\bigg ) + 1 , \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{align*} D _ { m a x } = ( f ( \\underbar { X } ) ) . \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} \\textrm { C o v } _ B [ X , Y ] & = \\textrm { V a r } _ B [ X ] a ^ T \\\\ & = k ( B ) \\textrm { V a r } [ X ] a ^ T + ( \\textrm { V a r } _ B [ Y ] - k ( B ) \\textrm { V a r } [ Y ] ) \\beta \\\\ & = k ( B ) ( \\textrm { C o v } [ X , Y ] - \\textrm { V a r } [ Y ] \\beta ) + \\textrm { V a r } _ B [ Y ] \\beta \\\\ & = \\textrm { V a r } _ B [ Y ] \\beta . \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} \\Lambda = \\Lambda _ 1 ( \\infty , \\Omega ^ + ) \\Lambda = \\Lambda _ 1 ( \\infty , \\Omega ^ - ) . \\end{align*}"}
-{"id": "3955.png", "formula": "\\begin{align*} \\frac { \\Gamma ' ( x ) } { \\Gamma ( x ) } = - \\frac 1 x - \\gamma _ 0 + \\sum _ { n = 1 } ^ { \\infty } \\bigg ( \\frac 1 n - \\frac 1 { n + x } \\bigg ) , \\end{align*}"}
-{"id": "5548.png", "formula": "\\begin{align*} c _ n = L ^ { 2 n - 2 } z _ n \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{align*} x = x ' e _ 1 ~ , ~ y = y ' e _ 1 ~ , ~ s = s ' e _ 1 . \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\nu } \\ell _ i ( \\ell _ i - 1 ) ^ 2 ( \\ell _ i + 1 ) \\ell _ i ^ { 4 ( n _ i - 1 ) } . \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} \\widetilde { \\tau _ 0 } = - \\tau _ 0 , \\widetilde { \\tau _ 3 } = \\tau _ 3 . \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{align*} \\xi ^ l _ { i + 1 } - \\xi _ { i } ^ l \\le x _ { i + 1 } - x _ { i - l } = \\xi ^ 0 _ { i + 1 } - \\xi _ { i - l } ^ 0 . \\end{align*}"}
-{"id": "1137.png", "formula": "\\begin{align*} \\sigma _ { 1 } ^ { 2 } \\left ( A \\right ) & = \\frac { p q + r s + \\sqrt { ( p q + r s ) ^ { 2 } - 4 ( q - s ) p r s } } { 2 } , \\\\ \\sigma _ { 2 } ^ { 2 } \\left ( A \\right ) & = \\frac { p q + r s - \\sqrt { ( p q + r s ) ^ { 2 } - 4 ( q - s ) p r s } } { 2 } , \\\\ \\sigma _ { 3 } \\left ( A \\right ) & = \\cdots = \\sigma _ { q } \\left ( A \\right ) = 0 . \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} m _ { i j } ^ { \\infty } = \\left \\{ \\begin{array} { l r } { { \\lfloor { j \\over 2 } \\rfloor } \\choose { j - i } } , & { \\rm i f } ~ ~ i \\leq j , \\\\ 0 , & { \\rm i f } ~ ~ i > j \\end{array} \\right . ( i , j = 1 , 2 , \\ldots ) . \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} p _ { k , l , m } ( c ^ 2 ) & = \\alpha c ^ 2 \\ ; + \\ ; \\sum _ { h = 1 } ^ N \\frac { \\lambda _ h k _ h } { 1 + \\sqrt { 1 + \\frac { \\lambda _ j } { c ^ 2 } } } \\ ; + \\ ; \\omega _ m \\ ; - \\ ; \\omega _ l , \\end{align*}"}
-{"id": "4137.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\lambda } ^ d \\xi _ t ( O ) & \\leq e ^ { - t } \\sum _ { n = 0 } ^ { \\infty } \\frac { t ^ n \\lambda ^ n M ^ { 2 n } ( d + 1 ) ^ n } { n ! E { \\widetilde { \\rho } } ^ 2 } [ \\frac { d E ( { \\widetilde { \\rho } } ^ 2 ) } { d + 1 } + \\frac { 1 } { ( d + 1 ) E ( { \\widetilde { \\rho } } ^ 2 ) } ] ^ n \\\\ & = \\big ( E ( { \\widetilde { \\rho } } ^ 2 ) \\big ) ^ { - 1 } \\exp \\big \\{ t \\big [ \\lambda M ^ 2 \\big ( d E ( { \\widetilde { \\rho } } ^ 2 ) + \\frac { 1 } { E ( { \\widetilde { \\rho } } ^ 2 ) } \\big ) - 1 \\big ] \\big \\} . \\end{align*}"}
-{"id": "9653.png", "formula": "\\begin{align*} F ( x , y , z ) : = x X + y Y + z Z = \\begin{pmatrix} x & y + z \\\\ y - z & - x \\end{pmatrix} . \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} t ^ { - 1 } h h ' = h t ^ { - 1 } . \\end{align*}"}
-{"id": "7105.png", "formula": "\\begin{align*} g _ { \\alpha } \\theta _ { \\alpha } \\psi _ { \\alpha \\pi , \\beta \\pi } & = ( h _ { \\alpha } \\theta _ { \\alpha } ^ H \\psi _ { \\alpha \\pi , \\beta \\pi } ^ H ) ( k _ { \\alpha } \\theta _ { \\alpha } ^ K \\psi _ { \\alpha \\pi , \\beta \\pi } ^ K ) \\\\ & = ( h _ { \\alpha } \\psi _ { \\alpha , \\beta } ^ { H } \\theta _ { \\beta } ^ { H } ) ( k _ { \\alpha } \\psi _ { \\alpha , \\beta } ^ { K } \\theta _ { \\beta } ^ { K } ) \\\\ & = g _ { \\alpha } \\psi _ { \\alpha , \\beta } \\theta _ { \\beta } \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} \\| u _ { y y y } \\| _ { X ( Q _ T ) } + \\| u _ { x y y y } \\big | _ { x = 0 } \\| _ { L _ 2 ( B _ T ) } \\leq c . \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} r ( n ) : = \\mathbb { E } ( \\xi _ n \\xi _ 0 ) = n ^ { \\frac { 2 H - 2 } { k } } L ( n ) , \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} Q _ t = \\sigma \\phi _ t \\ , , R _ t = \\frac { \\theta + \\phi _ t } { 1 + \\frac { 1 } { \\lambda } \\phi _ t } \\phi _ t ( m ( t - ) - X _ { t - } ) , \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{align*} V = O ( 1 / R ^ n ) , \\varphi = O ( 1 / R ^ { n - 1 } ) , \\nabla \\varphi = O ( 1 / R ^ n ) , \\ \\partial B _ R ( 0 ) \\cap \\Omega . \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 2 } { x ^ 2 } + \\frac { A _ 1 } { x } + A _ 0 \\right ) Y , \\frac { \\partial Y } { \\partial t } = \\left ( \\frac { 1 } { t } A _ 0 x + B _ 0 \\right ) Y , \\end{gather*}"}
-{"id": "6127.png", "formula": "\\begin{align*} X _ { 2 , \\ , j + 2 , \\ , 5 } \\cap L _ { j + 2 } = 0 . \\end{align*}"}
-{"id": "6892.png", "formula": "\\begin{align*} A _ { n , j , k } ( t , z ) & \\coloneqq ( 1 - z ) ^ { n } P _ { n , j , k } \\left ( \\frac { 1 + z } { 1 - z } , t \\right ) \\\\ & = ( 1 - z ) ^ { n } t ^ { j + 1 } \\left ( \\frac { 1 + z } { 1 - z } + t \\right ) ^ { k - j } \\left ( 1 + \\frac { 1 + z } { 1 - z } t \\right ) ^ { n - j - k - 1 } \\left ( 1 + \\frac { 1 + z } { 1 - z } \\right ) ^ { 2 j + 1 } \\\\ & = 2 ^ { 2 j + 1 } t ^ { j + 1 } ( 1 + t + z ( 1 - t ) ) ^ { k - j } ( 1 + t - z ( 1 - t ) ) ^ { n - j - k - 1 } , \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} | \\psi | ^ 2 _ { \\Omega ^ p ( M ; E ) } = \\frac { 1 } { r ! s ! } \\langle \\psi _ { \\alpha _ 1 \\ldots \\alpha _ r , \\bar \\beta _ 1 \\ldots \\bar \\beta _ s } , \\psi ^ { \\bar \\alpha _ 1 \\ldots \\bar \\alpha _ r , \\beta _ 1 \\ldots \\beta _ s } \\rangle . \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} L ^ { \\prime } ( f ( x ) , d f _ { x } ( y ) ) = e ^ { \\sigma ( x ) } L ( x , y ) , ~ \\ \\ \\forall ( x , y ) \\in \\tilde { A } . \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} u ( 0 , x , y ) = u _ 0 ( x , y ) , ( x , y ) \\in \\Omega , \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} ( \\hat { U } M + \\hat { V } \\overline { N } ) z = \\hat { U } p + \\hat { V } \\overline { p } . \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} L Q ^ { T \\downarrow } ( 0 | 0 ) = P ^ { T \\downarrow } + L \\left [ \\begin{array} { c } { \\mathbf 0 } ^ T \\\\ P ^ { T \\downarrow } \\end{array} \\right ] . \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} \\| g _ 1 - g _ 2 \\| _ { L _ { 2 } ( P ) } & = \\sqrt { P ( ( f _ 1 ( X ) - Y ) ^ 2 - ( f _ 2 ( X ) - Y ) ^ 2 ) ^ 2 } \\\\ & \\le 2 \\sqrt { P ( ( f _ 1 ( X ) - f _ 2 ( X ) ) ^ 2 } \\\\ & = 2 \\| f _ 1 - f _ 2 \\| _ { L _ { 2 } ( P ) } . \\end{align*}"}
-{"id": "9726.png", "formula": "\\begin{align*} B & = \\{ x \\in X \\colon \\| x \\| _ X + \\| R x \\| _ Y \\leq 1 \\} , \\\\ A & = \\{ x ^ { * * } \\in X ^ { * * } \\colon \\| x ^ { * * } \\| _ { X ^ { * * } } + \\| R ^ { * * } x ^ { * * } \\| _ { Y ^ { * * } } \\leq 1 \\} , \\\\ A _ 0 & = \\{ x ^ { * * } \\in X ^ { * * } \\colon \\| x ^ { * * } \\| _ { X ^ { * * } } + \\| R ^ { * * } x ^ { * * } \\| _ { Y ^ { * * } } < 1 \\} . \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} D i s ( \\tilde { g } _ { n } ) = ( - 1 ) ^ { n ( n + 1 ) / 2 } c _ 0 ^ { 2 n } \\times 2 ^ { n ( n + 1 ) } \\times \\prod _ { j < k } ( { t _ j } - { t _ k } ) ^ 2 / \\prod _ { j = 1 } ^ { n + 1 } ( 1 - { t _ j } ) ^ { 2 n } . \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} \\lim _ { q \\to 0 ^ - } \\| g \\| _ { H ^ q } = \\exp \\left ( \\int _ 0 ^ { 2 \\pi } \\log | g ( e ^ { i \\theta } ) | \\ , \\frac { d \\theta } { 2 \\pi } \\right ) , \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} L _ v { } ^ 2 = L _ { 2 v } + ( - 1 ) ^ v 2 \\ , , v \\in \\Z \\ , , \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{align*} \\mathrm { I m } ( z _ k ( 0 ) ) = \\mathrm { I m } ( Z ( [ x _ { k - 1 } , x _ k ] ) ) \\end{align*}"}
-{"id": "9819.png", "formula": "\\begin{align*} D _ i ( w \\mathbf { z } ) + D _ i ( s _ i w \\mathbf { z } ) = v - 1 . \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} \\mathbb { P } _ c \\left ( | R _ { d i f } - \\mathbb { E } R _ { d i f } | > \\sqrt { n } \\right ) \\leq \\frac { v a r ( R _ { d i f } ) } { n } \\leq \\frac { 2 \\sqrt { n } } { n } = \\frac { 2 } { \\sqrt { n } } \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} \\omega _ b = ( \\omega _ 1 , \\omega _ 2 , \\ldots , \\omega _ { l - 1 } , \\sigma _ { l } , \\sigma _ { l + 2 } , \\ldots , \\sigma _ N ) . \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} w ' ( \\theta ) = \\left . \\frac { V ^ { \\tfrac { 1 } { m _ 1 } - 1 } ( x ) } { m _ 1 } \\left ( \\frac { 1 - m _ 1 } { m _ 1 V ( x ) } \\Big ( \\sum \\limits _ { i = 1 } ^ n \\frac { \\partial V ( x ) } { \\partial x _ i } y _ i \\Big ) ^ 2 + \\sum _ { i , j = 1 } ^ n \\frac { \\partial ^ 2 V ( x ) } { \\partial x _ i \\partial x _ j } y _ i y _ j \\right ) \\right | _ { x = x ^ 0 + \\theta y } . \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\| w ( t ) \\| _ 2 ^ 2 + \\| \\nabla w ( t ) \\| _ 2 ^ 2 = \\langle ( h u _ s + \\widetilde U ) \\otimes w , \\nabla w \\rangle + \\langle f , w \\rangle . \\end{align*}"}
-{"id": "9505.png", "formula": "\\begin{align*} \\mathrm { R e } \\ , \\langle y \\xi , \\eta \\rangle = \\mathrm { R e } \\ , \\phi ( y ) \\geq 1 , \\ \\ \\ y \\in W _ A ( x , \\alpha ) . \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n { ( - 1 ) ^ { k - 1 } L _ { m k } { } ^ 4 } & = \\sum _ { k = 1 } ^ n { ( - 1 ) ^ { k - 1 } L _ { 4 m k } } \\\\ & - 4 \\sum _ { k = 1 } ^ n { ( - 1 ) ^ { k ( m - 1 ) } L _ { 2 m k } } + 3 ( ( - 1 ) ^ { n - 1 } + 1 ) \\ , , \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( \\widehat { \\pi } ^ { ( l _ n , \\theta ^ n _ 0 ) } , \\widehat { \\pi } ^ { ( r _ n , \\theta ^ n _ 0 ) } ) = \\begin{cases} ( \\{ \\widehat { \\pi } _ 0 ( s ) : s \\leq \\theta _ 0 \\} , \\widehat { \\pi } _ 1 ) & \\widehat { \\pi } _ 0 ( \\theta _ 0 - 1 ) < \\widehat { \\pi } _ 1 ( \\theta _ 0 - 1 ) \\\\ ( \\widehat { \\pi } _ 1 , \\{ \\widehat { \\pi } _ 0 ( s ) : s \\leq \\theta _ 0 \\} ) & , \\end{cases} \\end{align*}"}
-{"id": "6454.png", "formula": "\\begin{align*} \\binom { R } { l } \\binom { l } { \\# _ 1 , . . . \\# _ l } \\sum _ { \\sigma \\in S _ l / \\mathrm { S t a b } ( k _ { r _ 1 } , . . . k _ { r _ l } ) } \\sigma ( k _ { r _ 2 } ( k _ { r _ 2 } + k _ { r _ 3 } ) . . . ( k _ { r _ 2 } + . . . + k _ { r _ l } ) ) \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} \\underset { l \\rightarrow \\infty } { \\lim } \\xi _ { l } \\left ( x \\right ) = 1 \\ . \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} { \\mathcal O } ^ - _ { r , c } ( x , y ) = \\bigcap _ { z \\in B _ x ( r ) } { \\mathcal O } _ { c } ( z , y ) \\subset { \\mathcal O } _ { c } ( x , y ) \\subseteq \\bigcup _ { z \\in B _ x ( r ) } { \\mathcal O } _ { c } ( z , y ) = { \\mathcal O } ^ + _ { r , c } ( x , y ) ; \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} \\dim X = \\dim \\overline { N } + \\tau , \\end{align*}"}
-{"id": "5219.png", "formula": "\\begin{align*} [ f ] _ { \\alpha } ( x ) : = \\sup _ { \\substack { \\abs y < 1 \\\\ x + y \\in U } } \\frac { \\abs { f ( x + y ) - f ( x ) } } { \\abs y ^ \\alpha } . \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{align*} - \\lambda _ 1 = h ( \\lambda _ 2 , \\lambda _ 3 ; s ) - r . \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} \\omega ( B _ { i , j } ) = \\begin{cases} \\omega ( B _ { 0 , j } ) + 1 & \\mbox { i f $ j \\leq 1 + r + r ^ 2 + \\cdots + r ^ { n - 1 } $ , } \\\\ \\omega ( B _ { 0 , j } ) & \\mbox { i f $ j > 1 + r + r ^ 2 + \\cdots + r ^ { n - 1 } $ . } \\\\ \\end{cases} \\end{align*}"}
-{"id": "9626.png", "formula": "\\begin{align*} | u _ 1 ( j ) \\ , u _ 2 ( j ) \\ , \\cdots \\ , u _ n ( j ) | = \\left | \\begin{array} { c c c c } u _ 1 ( j ) & u _ 2 ( j ) & \\cdots & u _ n ( j ) \\\\ u _ 1 ( j + 1 ) & u _ 2 ( j + 1 ) & \\cdots & u _ n ( j + 1 ) \\\\ \\vdots & \\vdots & & \\vdots \\\\ u _ 1 ( j + n - 1 ) & u _ 2 ( j + n - 1 ) & \\cdots & u _ n ( j + n - 1 ) \\end{array} \\right | , \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} \\lim _ { x \\to b ^ - } [ a _ k ( x ) f ^ { ( k ) } ] ^ { ( k - j ) } P _ s ( x ) = \\sum _ { h = 0 } ^ s \\left ( \\lim _ { x \\to b ^ - } \\alpha _ h x ^ h [ a _ k ( x ) f ^ { ( k ) } ] ^ { ( k - j ) } \\right ) = 0 , \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} S ( \\hat { \\gamma } ) = S ( \\hat { \\rho } ) + S ( \\hat { \\rho } \\| \\hat { \\gamma } ) \\ ; . \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} 0 \\ge & \\triangle \\beta - \\lambda \\beta + \\frac { \\nabla u _ 1 } { u _ 1 } \\cdot \\nabla \\beta + A ( x ) \\beta \\\\ 0 \\ge & \\triangle \\beta + ( A ( x ) - \\lambda ) \\beta + \\frac { \\nabla u _ 1 } { u _ 1 } \\cdot \\nabla \\beta . \\\\ \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} p ^ { \\pm } \\ ; : = \\ ; q ^ { \\pm } \\ , ( W _ 0 ^ \\infty ) ^ { - 1 } \\| v _ \\infty \\| _ { L ^ 2 ( \\mathbb { R } ^ + , \\mathbb { C } ^ 2 ) } ^ 2 \\end{align*}"}
-{"id": "9670.png", "formula": "\\begin{align*} \\frac { \\partial \\Gamma _ \\psi } { \\partial \\theta } ( \\theta ) = - \\frac { \\partial \\Theta } { \\partial \\theta } ( \\theta , 0 , \\psi ) + 1 > 0 . \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} \\tau = - \\langle \\mathbf { n } , \\mathbf { b } ' \\rangle _ 1 = - \\langle \\frac { \\epsilon _ n } { \\kappa } \\alpha '' , \\frac { \\epsilon _ n } { \\kappa } \\alpha ' \\times _ 1 \\alpha ''' \\rangle _ 1 = \\frac { \\epsilon _ t \\epsilon _ u } { r ^ 2 \\kappa ^ 2 } \\langle \\alpha - J _ 1 \\alpha ' \\times _ 1 \\alpha , \\alpha ' \\times _ 1 \\alpha ''' \\rangle _ 1 . \\end{align*}"}
-{"id": "6817.png", "formula": "\\begin{align*} B _ { j } u = \\sum \\limits _ { \\left \\vert \\beta \\right \\vert \\leq m _ { j } } \\ b _ { j \\beta } \\left ( y \\right ) D _ { y } ^ { \\beta } u \\left ( x , y , t \\right ) = 0 x \\in R ^ { n } , y \\in \\partial G , j = 1 , 2 , . . . , m , \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} ( \\Delta f ) ( i , j ) : = 4 f ( i , j ) - f ( i - 1 , j ) - f ( i + 1 , j ) - f ( i , j - 1 ) - f ( i , j + 1 ) \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{align*} \\phi ( z ) = \\Re \\left ( \\frac { \\alpha _ 1 } { z ^ k } , \\cdots , \\frac { \\alpha _ { n } } { z ^ k } \\right ) + O \\left ( \\frac { 1 } { z ^ { k - 1 } } \\right ) \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{align*} \\mu _ 1 \\left ( A _ 1 \\right ) = \\int _ { W ^ N } ^ { } \\mu _ 1 ( d x ) \\int _ { A _ 1 } ^ { } P _ { H P } ^ { s , N } ( t ) ( x , y ) d y = 0 . \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} \\widetilde { \\chi } ( \\Delta ( G ) ) = - I ( G , - 1 ) \\end{align*}"}
-{"id": "6297.png", "formula": "\\begin{align*} \\| U \\| _ { p l } \\le \\inf \\{ \\sum _ { k = 1 } ^ n \\| x _ k \\| \\| v _ k \\| \\} . \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} \\bigl ( \\ell y ( P ) \\bigr ) ^ 2 \\ell ^ 3 \\bigl [ 4 f ( x ( P ) ) x ( [ 2 ] P ) - g ( x ( P ) ) \\bigr ] = \\Delta ^ \\prime \\ell ^ 5 . \\end{align*}"}
-{"id": "1184.png", "formula": "\\begin{align*} d ^ { \\rm s u p } ( 1 0 ^ { - 2 } ) \\ , \\leq \\ , \\left \\{ \\begin{array} { l l } 2 & \\mbox { f o r \\ } a = 4 , \\\\ 2 & \\mbox { f o r \\ } a = 3 , \\\\ 4 & \\mbox { f o r \\ } a = 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} \\| u ( t ) \\| _ { L ^ \\infty ( \\Omega ) } = O ( t ^ { - 1 / 2 } ) \\end{align*}"}
-{"id": "7158.png", "formula": "\\begin{align*} A _ K = \\mathrm { S p a n } _ K [ 1 , I , J , I J ] , \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} \\left ( Y _ i ( k ) | _ { \\boldsymbol { c } = \\boldsymbol { 0 } } \\right ) _ { [ Q ] [ Q ' ] } = \\mathbb { T } ^ { ( i ) } \\delta _ { [ Q T _ i ] [ Q ' ] } . \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{align*} \\gamma _ i ( z , T _ 1 , T _ 2 , t ) : = \\begin{cases} I , & t \\le T _ 1 ; \\\\ \\gamma _ i ( z , t - T _ 1 ) , & T _ 1 \\le t \\le T _ 2 ; \\\\ \\gamma _ i ( z , T _ 2 - T _ 1 ) , & T _ 2 \\le t . \\end{cases} \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} \\frak { c s } ( a ) = \\frac { 1 } { 4 \\pi ^ 2 } \\int _ M T r \\left ( \\frac { 1 } { 2 } a \\wedge d a + \\frac { 1 } { 3 } a \\wedge a \\wedge a \\right ) . \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} P _ 0 \\equiv 1 , P _ 1 ( x ) = x , P _ 2 ( x ) = \\frac { 1 } { 2 } ( 3 x ^ 2 - 1 ) . \\end{align*}"}
-{"id": "5529.png", "formula": "\\begin{align*} \\Vert d f ^ n _ p \\psi ' _ p ( 0 ) \\Vert & = \\Vert \\lambda _ { p , n } \\psi ' _ n ( 0 ) \\Vert \\\\ & \\geq k \\kappa ^ n . \\end{align*}"}
-{"id": "6747.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } - \\bigtriangleup _ { \\varepsilon } u + \\left ( u . \\nabla \\right ) u + \\nabla \\varphi + a \\frac { \\partial ^ { 2 } u } { \\partial y ^ { 2 } } + b \\frac { \\partial u } { \\partial y } + c u = f \\left ( x , y , t \\right ) , \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} [ a ( D ) u ] \\ \\widehat { } \\ ( \\xi ) = \\lim _ { \\substack { l \\to \\infty \\\\ L ^ { 2 } ( B ( 0 , j ) ) } } \\big ( a ( D ) u _ { l } \\big ) \\ \\widehat { } \\ ( \\xi ) = \\lim _ { \\substack { l \\to \\infty \\\\ L ^ { 2 } ( B ( 0 , j ) ) } } a ( \\xi ) \\ , \\widehat { u _ { l } } ( \\xi ) = a ( \\xi ) \\ , \\widehat { [ u ] } ( \\xi ) , \\end{align*}"}
-{"id": "9781.png", "formula": "\\begin{align*} \\int _ M [ e _ { \\epsilon } ( u _ { \\epsilon } ) - \\frac { 1 } { 2 } | j u _ { \\epsilon } | ^ 2 ] = \\int _ M [ \\frac { 1 } { 2 } ( 1 - | u _ { \\epsilon } | ^ 2 ) | d u _ { \\epsilon } | ^ 2 + \\frac { 1 } { 2 } | d | u _ { \\epsilon } | | ^ 2 + \\frac { W ( u _ { \\epsilon } ) } { \\epsilon ^ 2 } \\leq C ; \\end{align*}"}
-{"id": "519.png", "formula": "\\begin{align*} \\rho ( u ) ( x , v ) : = \\left \\lbrace \\begin{array} { l } \\rho ( x ) u ( x , v ) \\textrm { w h e n } ( x , v ) \\in X \\times V , \\\\ 0 \\textrm { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "1296.png", "formula": "\\begin{align*} \\hat { g } ( \\mathsf { x } , T , \\mathsf { y } ) = \\pi ^ { - \\frac { d } { 2 } } \\exp \\left [ - \\frac { 1 } { 2 } \\left ( \\left \\vert \\mathsf { x } \\right \\vert ^ { 2 } + \\left \\vert \\mathsf { y } \\right \\vert ^ { 2 } + d T \\right ) \\right ] \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} r \\xi '' + ( 1 - 2 B - r ) \\xi ' + B \\ , \\xi \\ ; = \\ ; 0 \\ , . \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} N ( a ) = . \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{align*} G ( k , x ) : = f ( - k , x ) + f ( k , x ) S ( k ) , k \\in \\mathbb { R } . \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{align*} \\frac { z _ 0 ( w _ 0 - 1 ) ^ 2 } { w _ 0 ( z _ 0 - 1 ) ^ 2 } = \\frac { a _ 0 } { b _ 0 } . \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } b _ { i , n - 1 } t _ k ^ i = 0 , k = 1 , \\dots , n - 1 . \\end{align*}"}
-{"id": "2576.png", "formula": "\\begin{align*} \\phi & : = ( \\phi ^ { ( 1 ) } , \\ldots , \\phi ^ { ( r ) } ) = ( \\psi ^ { ( 1 ) } , \\bar \\psi ^ { ( 1 ) } \\ldots , \\psi ^ { ( r ) } , \\bar \\psi ^ { ( r ) } ) \\end{align*}"}
-{"id": "9639.png", "formula": "\\begin{align*} | a _ { d - k - 1 , n } | \\le \\binom { d - 2 } { k } ( | a _ { d - 1 , n } | + d - 2 ) + \\binom { d - 2 } { k - 1 } + \\binom { d - 2 } { k + 1 } . \\end{align*}"}
-{"id": "9335.png", "formula": "\\begin{align*} \\sigma ( F ) ( N ; z ) = \\sum _ { n = - \\infty } ^ N \\left ( \\begin{array} { c c } 0 & F _ n z ^ n \\\\ \\overline { F _ n } z ^ { - n } & 0 \\end{array} \\right ) , \\ ; \\ ; \\gamma [ F ] ( N ; z ) = \\prod _ { n = - \\infty } ^ N T _ n ( z ) . \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} \\mu _ 2 \\rho _ g ( \\lambda _ 2 ) \\rho _ g ( \\lambda _ 1 ) = \\lambda _ 2 \\mu _ 1 \\rho _ g ( \\lambda _ 1 ) = \\lambda _ 2 \\lambda _ 1 \\mu _ 0 , \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} Z ^ { k , n } _ H ( t ) : = \\frac { 1 } { n ^ H } \\sum _ { l = 1 } ^ { [ n t ] } g ( \\xi _ l ) , \\quad , n \\in \\mathbb { N } , t \\geq 0 , \\end{align*}"}
-{"id": "9836.png", "formula": "\\begin{align*} \\tau R ( ( x y ^ { - 1 } ) \\circ \\tau R ( ( ( x y ^ { - 1 } ) ^ { - 1 } ) = \\bigl ( \\sqrt { v } - \\frac { x y ^ { - 1 } } { \\sqrt { v } } \\bigr ) \\bigl ( \\sqrt { v } - \\frac { 1 } { x y ^ { - 1 } \\sqrt { v } } \\bigr ) \\ , . \\end{align*}"}
-{"id": "2805.png", "formula": "\\begin{align*} \\Psi _ 1 ( n ) = \\prod _ { p | n } \\left ( 1 + \\frac { 1 } { p } \\right ) ^ { - 1 } , \\Psi ( n ) = \\sum _ { d | n } \\Psi _ 1 ( d ) \\sum _ { e | d } \\frac { \\mu ( e ) } { e } = \\sum _ { d | n } \\Psi _ 1 ( d ) \\phi ( d ) , \\end{align*}"}
-{"id": "9437.png", "formula": "\\begin{align*} Q _ r ( w ^ * ) = Q ( w ^ * ) + r I _ { 1 0 : 4 } . \\end{align*}"}
-{"id": "3535.png", "formula": "\\begin{align*} \\left ( \\dfrac { \\dot { b } } { b } \\right ) ^ 2 + 2 \\dfrac { \\dot { a } } { a } \\dfrac { \\dot { b } } { b } - \\Lambda = 8 \\pi \\rho + 4 \\pi \\dot { \\phi } ^ 2 , \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{align*} \\left | \\frac { \\left ( x _ { N + 1 } ^ { ( n ) , m } - x _ { N - 1 } ^ { ( n ) , m } \\right ) \\prod _ { i = 1 } ^ { N } ( x ^ { ( n ) , m } _ { i + 1 } - x ^ { ( n ) , m } _ i ) \\Delta _ N ( \\xi ^ { ( n ) , m } ) f \\left ( \\xi ^ { ( n ) } \\right ) } { \\Delta _ { N + 1 } ( x ^ { ( n ) , m } ) } \\right | \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} K _ { \\eta } ( \\alpha ) = \\Bigl ( \\frac { \\sin ( \\pi \\alpha \\eta ) } { \\pi \\alpha } \\Bigr ) ^ 2 \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} \\Delta _ A = \\langle & c _ { 1 0 0 } c _ { 1 1 1 } c _ { 2 0 1 } c _ { 2 1 0 } - c _ { 1 0 1 } c _ { 1 1 0 } c _ { 2 0 0 } c _ { 2 1 1 } , \\\\ & c _ { 0 0 0 } c _ { 0 1 1 } c _ { 1 0 1 } c _ { 1 1 0 } - c _ { 0 0 1 } c _ { 0 1 0 } c _ { 1 0 0 } c _ { 1 1 1 } , \\\\ & c _ { 0 0 0 } c _ { 0 1 1 } c _ { 2 0 1 } c _ { 2 1 0 } - c _ { 0 0 1 } c _ { 0 1 0 } c _ { 2 0 0 } c _ { 2 1 1 } \\rangle . \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} \\tilde { L } ( y ) = L ( d f ( y ) ) = J ( x ) ^ { \\tfrac { 2 } { n } } L ( y ) , ~ \\ \\forall y \\in A , \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{align*} \\int _ M ( 2 s - s ^ g + | \\theta | ^ 2 ) = \\int _ M \\left ( | N | ^ 2 + \\frac 1 2 | ( d ^ c F ) ^ { 2 , 0 } | ^ 2 \\right ) \\frac { F ^ n } { n ! } \\geq 0 \\end{align*}"}
-{"id": "7904.png", "formula": "\\begin{align*} w ( x , t ) = ( 1 - \\phi ( x ) ) d ( x , t ) + \\phi ( x ) f ( x , t ) . \\end{align*}"}
-{"id": "63.png", "formula": "\\begin{align*} Z ( c _ 1 , \\dots , c _ k ; \\mathcal P _ X ) = \\int _ { \\mathcal P _ X } \\mu ( c _ 1 ) \\wedge \\dots \\wedge \\mu ( c _ k ) \\in \\Z \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} { \\displaystyle - 2 \\mathrm { i } ( \\partial _ { \\tau } \\theta ^ { k } _ { \\Psi } , \\partial _ { \\tau } { \\theta _ { \\Psi } ^ { k } } ) + B \\Big ( \\overline { \\mathbf { A } } ^ { k } _ { h } ; \\ , \\overline { \\theta } _ { \\Psi } ^ { k } , \\partial _ { \\tau } { \\theta _ { \\Psi } ^ { k } } \\Big ) = \\sum _ { j = 1 } ^ { 5 } V _ { j } ^ { k } ( \\partial _ { \\tau } { \\theta _ { \\Psi } ^ { k } } ) . } \\end{align*}"}
-{"id": "5307.png", "formula": "\\begin{align*} d = - \\mathop { \\lim } \\limits _ { \\gamma \\to \\infty } \\frac { { \\log \\left ( { { p _ { o u t , K } } } \\right ) } } { { \\log \\left ( \\gamma \\right ) } } . \\end{align*}"}
-{"id": "1056.png", "formula": "\\begin{align*} L = \\left ( \\begin{matrix} I & - \\boldsymbol { \\theta } & - 1 + \\boldsymbol { \\theta } \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "5031.png", "formula": "\\begin{align*} \\dot { b } \\langle \\eta _ * , \\psi _ * \\rangle _ { L ^ 2 } = - 1 2 | b | ^ 2 b \\langle \\psi _ * ^ 2 , \\tilde { w } _ 2 \\rangle _ { L ^ 2 } + \\frac { 1 6 } { 3 \\sqrt { c _ * } } \\left ( \\delta _ 0 + \\frac { 2 0 } { 3 } | b | ^ 2 \\right ) b + \\mathcal { O } ( \\delta _ 0 ^ 2 | b | + | b | ^ 5 ) . \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} p = j \\left ( \\frac { ( j + 2 \\ell ) t } { g } + 1 \\right ) + \\ell , \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} \\begin{array} { l l } ( \\alpha d ) _ { k + q + \\delta } = & ( \\beta c ) _ { k - q } \\quad \\forall k : \\ ; 0 \\leq k \\leq n - p - 1 + q \\\\ ( a d ) _ { q + \\delta - 1 } = & \\sum \\limits _ { k = 0 } ^ { q + \\delta - 1 } \\epsilon ^ { q + \\delta - 1 - k } ( \\alpha d ) _ k = 0 . \\end{array} \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} L _ 0 \\eta = \\int _ { [ - r , 0 ] } d \\mu ( \\theta ) \\eta ( \\theta ) , \\forall \\eta \\in \\C . \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{align*} \\Lambda _ \\chi ( s ) = \\epsilon _ \\chi ( N q ^ 2 ) ^ { \\frac 1 2 - s } \\overline { \\Lambda _ \\chi ( 1 - \\bar { s } ) } , \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] = ( 1 - z ) ^ { - \\alpha } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & 1 - \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac { z } { z - 1 } \\bigg ] , \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} \\theta = \\theta ( \\alpha ) = - \\frac { \\lambda _ 1 - \\alpha \\mu _ 1 } { \\lambda _ 2 - \\alpha \\mu _ 2 } . \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t y ( x , t ) - \\Delta y ( x , t ) = \\chi _ \\omega \\chi _ { ( \\tau , T ) } u ( x , t ) , & \\mbox { i n } \\Omega \\times ( 0 , T ) , \\\\ y ( x , t ) = 0 , & \\mbox { o n } \\partial \\Omega \\times ( 0 , T ) , \\\\ y ( x , 0 ) = y _ 0 ( x ) , & \\mbox { i n } \\Omega . \\end{cases} \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{align*} \\Lambda ( t _ f ) & = R ^ { - 1 } \\Lambda ( 0 ) R , \\\\ F ^ t \\lambda ( t _ f ) & = \\lambda ( 0 ) , \\\\ \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} M ^ a _ s : = \\mathbb E ( \\xi ^ a | \\mathcal F _ s ) , \\ ; \\ ; M ^ q _ s : = \\mathbb E ( \\xi ^ q | \\mathcal F _ s ) , \\ ; \\ ; \\ ; s \\geq 0 . \\end{align*}"}
-{"id": "2689.png", "formula": "\\begin{align*} \\omega _ i - \\sum _ { j = 0 } ^ { n } a _ { i , j } \\sin ( \\theta _ { i } - \\theta _ { j } ) = 0 i = 0 , \\dots , n \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} \\langle \\pi _ { \\textnormal { n } ( \\theta ) } \\epsilon E , \\phi \\rangle = \\langle f , \\phi \\rangle = 0 . \\end{align*}"}
-{"id": "9583.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\mu _ { n , i } = 0 , \\end{align*}"}
-{"id": "484.png", "formula": "\\begin{align*} P _ { Y _ 1 , Y _ 3 | M } ( 0 , 0 | 0 ) = P _ { Y _ 1 , Y _ 3 | M } ( 1 , 0 | 0 ) = \\frac { 1 } { 2 } , \\\\ P _ { Y _ 1 , Y _ 3 | M } ( 0 , 0 | 1 ) = P _ { Y _ 1 , Y _ 3 | M } ( 1 , 1 | 1 ) = \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} g ( \\nabla _ X Y , Z ) = g ( D ^ g _ X Y , Z ) + \\frac 3 2 t _ { X Y Z } - g ( X , T _ { Y Z } ) , \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} ( b _ { \\ell + 1 , i } ^ x - a _ { \\ell + 1 , i } ^ x ) \\leq \\overline q ( b _ { \\ell , i } ^ x - a _ { \\ell , i } ^ x ) , i = 1 , \\ldots , d , \\ell = - L , \\ldots , L - 1 , \\\\ ( b _ { \\ell + 1 , i } ^ y - a _ { \\ell + 1 , i } ^ y ) \\leq \\overline q ( b _ { \\ell , i } ^ y - a _ { \\ell , i } ^ y ) , i = 1 , \\ldots , d , \\ell = - L , \\ldots , L - 1 . \\end{align*}"}
-{"id": "4515.png", "formula": "\\begin{align*} \\mathfrak { F } _ { t , k , q } ^ { ( \\omega ) } \\left ( \\left \\{ x \\right \\} \\right ) = \\sum \\limits _ { y \\in \\mathfrak { L } } \\sigma _ { \\mathrm { p } } ^ { ( \\omega ) } \\left ( y , y - e _ { q } , x , x - e _ { k } \\mathbf { , } t \\right ) \\ . \\end{align*}"}
-{"id": "9592.png", "formula": "\\begin{align*} \\widehat \\Delta ^ Z _ n & = \\widehat \\Delta ^ \\mathrm { e h w } _ n - \\widehat \\Delta _ n ^ \\mathrm { p r o j } , \\ \\ \\ \\mathrm { w h e r e } \\ \\ \\widehat \\Delta _ n ^ \\mathrm { p r o j } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ n R _ { n , i } \\widehat G _ n Z _ { n , i } Z _ { n , i } ^ { \\prime } \\widehat G _ n ^ { \\prime } , \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{align*} 2 a b & = \\pm \\frac { 1 } { 2 ( \\ell + 1 ) } \\\\ a ^ 2 - ( 4 \\ell + 3 ) b ^ 2 & = \\frac { 2 \\ell + 1 } { 2 ( \\ell + 1 ) } \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} G _ { \\delta } = e ^ { \\delta A } G \\bigl ( e ^ { \\delta A } \\cdot \\bigr ) = e ^ { \\delta A } F _ 1 \\bigr ( e ^ { \\delta A } \\cdot \\bigr ) + B e ^ { \\delta A } F _ 2 \\bigl ( e ^ { \\delta A } \\cdot \\bigr ) \\quad , \\sigma _ { \\delta } = e ^ { \\delta A } \\sigma \\bigl ( e ^ { \\delta A } \\cdot \\bigr ) e ^ { \\delta A } . \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{align*} \\Delta f + \\frac { R } { n - 1 } f + \\frac { n \\kappa } { n - 1 } = 0 . \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{gather*} \\ln \\rho \\left ( \\rho \\cos \\varphi - 1 \\right ) + \\rho \\varphi \\sin \\varphi = 0 , \\\\ \\rho \\ln \\rho \\sin \\varphi - \\varphi \\left ( \\rho \\cos \\varphi - 1 \\right ) = 0 . \\end{gather*}"}
-{"id": "9572.png", "formula": "\\begin{align*} V ^ \\mathrm { s a m p l i n g } ( N _ 1 , N _ 0 , n _ 1 , n _ 0 ) & = E \\left [ \\mbox { v a r } \\big ( \\widehat { \\theta } \\ , | \\ , { \\mathbf { X } } , N _ 1 , N _ 0 \\big ) \\ , \\big | \\ , N _ 1 , N _ 0 \\right ] = \\frac { S ^ 2 _ 1 } { N _ 1 } \\left ( 1 - \\frac { N _ 1 } { n _ 1 } \\right ) + \\frac { S ^ 2 _ 0 } { N _ 0 } \\left ( 1 - \\frac { N _ 0 } { n _ 0 } \\right ) , \\end{align*}"}
-{"id": "9832.png", "formula": "\\begin{align*} R = \\sum _ i { \\sqrt { v } } \\ , e _ { i i } \\otimes e _ { i i } + \\sum _ { i \\neq j } e _ { i i } \\otimes e _ { j j } + \\bigl ( { \\sqrt { v } } - \\frac 1 { \\sqrt { v } } \\bigr ) \\sum _ { i > j } e _ { i j } \\otimes e _ { j i } \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{align*} S ( X | M ) : = S ( X M ) - S ( M ) \\end{align*}"}
-{"id": "7999.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\| J _ k w _ k - w \\| _ { L ^ 2 ( 0 , T ; L ^ 2 ( \\Omega _ L ) ) } = 0 , \\forall L \\in [ R , \\infty ) , \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} d _ { H } ( A , B ) = \\max \\left \\{ \\max _ { a \\in A } \\min _ { b \\in B } \\norm { a - b } , \\max _ { b \\in B } \\min _ { a \\in A } \\norm { b - a } \\right \\} . \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} \\theta = T _ { \\alpha \\beta } ^ { \\quad \\beta } z ^ \\alpha + T _ { \\bar \\alpha \\bar \\beta } ^ { \\quad \\bar \\beta } \\bar { z } ^ { \\bar \\alpha } . \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} ( B ^ \\prime _ 1 ( s ) , B ^ \\prime _ 2 ( s ) ) : = \\begin{cases} ( B _ 1 ( \\lambda ^ n _ 0 + s ) - y ^ n _ 0 , B _ 2 ( \\lambda ^ n _ 0 + s ) - y ^ n _ 0 ) & s \\in [ 0 , \\lambda ^ n _ 1 - \\lambda ^ n _ 0 ] \\\\ ( B _ 1 ( \\lambda ^ n _ 0 + s ) - y ^ n _ 0 , B _ 1 ( \\lambda ^ n _ 0 + s ) - y ^ n _ 0 ) & s > \\lambda ^ n _ 1 - \\lambda ^ n _ 0 . \\end{cases} \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{align*} \\ell = \\left \\lfloor \\frac { R _ 2 } { 3 ^ B \\| w \\| _ 1 } \\right \\rfloor , \\end{align*}"}
-{"id": "1659.png", "formula": "\\begin{align*} \\sum _ { s \\in S } { f _ { s \\cdot \\widetilde { \\tau } } \\cdot s } = 0 . \\end{align*}"}
-{"id": "973.png", "formula": "\\begin{align*} \\gamma _ { p , q } = \\frac { d } { 2 } - \\frac { d } { q } - \\frac { 4 } { p } . \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} [ b _ i ^ - , b _ j ^ + ] = \\delta _ { i j } \\mathbb { I } , [ b _ i ^ - , b _ j ^ - ] = [ b _ i ^ + , b _ j ^ + ] = 0 , \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{align*} f ( p ) = \\sup _ { x \\in \\langle H _ o \\cdot p \\rangle } f ( x ) \\geq \\sup _ { x \\in \\langle H _ o \\cdot p \\rangle \\cap C } f ( x ) \\geq \\inf _ { x \\in \\langle H _ o \\cdot p \\rangle \\cap C } f ( x ) \\geq \\inf _ { \\gamma \\in C } f ( \\gamma ) . \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} P _ { \\perp } \\begin{pmatrix} M z \\\\ [ . 1 c m ] \\overline { N } z \\end{pmatrix} + P _ { \\perp } \\begin{pmatrix} N \\bar { z } \\\\ [ . 1 c m ] \\overline { M } \\bar { z } \\end{pmatrix} = P _ { \\perp } \\begin{pmatrix} p \\\\ [ . 1 5 c m ] \\overline { p } \\end{pmatrix} \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} \\widehat \\psi _ n ^ { ( m _ p ) } ( 0 ) & = \\psi _ n ' ( 0 ) h _ n ^ { ( m _ p ) } ( 0 ) \\\\ & = \\psi _ n ' ( 0 ) a _ { m _ p } ( n ) m _ p ! \\end{align*}"}
-{"id": "1637.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ m \\big \\langle \\gamma _ { 1 } u \\ , , \\ , \\chi _ { \\partial \\Omega ^ \\complement _ { ( j ) } } \\big \\rangle = \\int _ { \\partial \\Omega } \\partial _ n u \\ , \\mathrm { d } s = 0 \\ , . \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} g ( k ) : = & \\int _ 0 ^ b [ \\tilde { q } ( x ) - q ( x ) ] y ( x , \\lambda ) \\tilde { y } ( x , \\lambda ) d x + ( \\tilde { h } - h ) \\\\ & + a _ 1 ( \\tilde { a } _ 2 - a _ 2 ) y ( \\textstyle { \\frac { 1 } { 2 } - 0 } , \\lambda ) \\tilde { y } ( \\textstyle { \\frac { 1 } { 2 } - 0 } , \\lambda ) . \\end{align*}"}
-{"id": "5591.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } ( j + 1 ) \\rho ^ { j } U _ { j } ( x ) U _ { j } ( y ) = \\frac { ( 1 + \\rho ^ { 2 } ) ( 1 - \\rho ^ { 2 } ) ^ { 2 } - 4 \\rho ^ { 2 } ( 1 + \\rho ^ { 2 } ) ( x ^ { 2 } + y ^ { 2 } ) + 1 6 \\rho ^ { 3 } x y } { ( ( 1 - \\rho ^ { 2 } ) ^ { 2 } - 4 x y \\rho ( 1 + \\rho ^ { 2 } ) + 4 \\rho ^ { 2 } ( x ^ { 2 } + y ^ { 2 } ) ) ^ { 2 } } , \\end{align*}"}
-{"id": "3702.png", "formula": "\\begin{align*} ( J u ) _ n = a _ n u _ { n - 1 } + b _ n u _ n + a _ { n + 1 } u _ { n + 1 } . \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} \\nabla ^ \\perp \\cdot \\widetilde { S } ^ \\alpha \\Gamma ^ a H = f ^ 3 _ { \\alpha a } , \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} \\xi _ { x + y } ^ i = \\frac { - c ( a y _ 1 + b y _ 2 ) } { y _ 1 ^ 2 + y _ 2 ^ 2 } + O _ { B , R _ 1 } \\left ( r ^ { - 2 } \\right ) , \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} \\lambda = \\frac { \\eta \\ , J ( B | M ) _ { \\hat { \\rho } _ { B M } } } { \\eta \\ , J ( B | M ) _ { \\hat { \\rho } _ { B M } } + | 1 - \\eta | \\ , J ( A | M ) _ { \\hat { \\rho } _ { A M } } } \\ ; , \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} \\chi _ { - \\frac { 1 } { 2 } } ^ n | 0 \\rangle & = H ^ { \\gamma } _ { ( - n ) } \\dots H ^ { \\gamma } _ { ( - 2 ) } H ^ { \\gamma } _ { ( - 1 ) } | 0 \\rangle \\\\ \\chi _ { - \\frac { 3 } { 2 } } ^ n | 0 \\rangle & = H ^ { \\beta } _ { ( - n ) } \\dots H ^ { \\beta } _ { ( - 2 ) } H ^ { \\beta } _ { ( - 1 ) } | 0 \\rangle . \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} p _ i \\equiv 1 \\ ; \\ , s \\ ; \\ ; i = 1 , 2 , \\dots , n . \\end{align*}"}
-{"id": "2799.png", "formula": "\\begin{align*} \\max ( w ^ \\prime , z ^ \\prime ) = z ^ \\prime = z - 1 \\leqslant \\varepsilon _ 1 B ^ { - \\frac { 1 } { r } } . \\end{align*}"}
-{"id": "1037.png", "formula": "\\begin{align*} \\| u \\| _ { X ^ { \\gamma , b } _ \\delta } : = \\inf \\{ \\| w \\| _ { X ^ { \\gamma , b } } \\ | \\ w _ { \\vert [ 0 , \\delta ] \\times \\R ^ k } = u \\} . \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} ( r + 1 ) ^ 2 - 4 r \\frac { \\gamma _ 1 } { \\gamma _ 2 } = \\left ( r + \\left ( 1 - 2 \\frac { \\gamma _ 1 } { \\gamma _ 2 } \\right ) \\right ) ^ 2 + 1 - \\left ( 1 - 2 \\frac { \\gamma _ 1 } { \\gamma _ 2 } \\right ) ^ 2 . \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} d g _ t ( z ) = \\frac { 2 } { g _ t ( z ) - \\xi _ t } d t , \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{align*} F ( \\psi , \\overline { \\psi } ) = - \\sum _ { j = 1 } ^ K ( \\alpha _ j + \\mathrm i \\gamma _ j ) \\psi | \\psi | ^ { p _ j - 1 } , \\end{align*}"}
-{"id": "10028.png", "formula": "\\begin{align*} N _ J ( X , Y ) = ( \\nabla _ X J ) J Y + ( \\nabla _ { J X } J ) Y - ( \\nabla _ Y J ) J X - ( \\nabla _ { J Y } J ) X , \\forall X , Y \\in { \\mathfrak X } ( M ) . \\end{align*}"}
-{"id": "9673.png", "formula": "\\begin{align*} \\begin{aligned} F ( \\theta , \\phi , \\pi ) & = - F ( - \\theta , - \\phi , 0 ) = F ( \\theta , \\pi - \\phi , 0 ) , \\\\ G ( \\theta , \\phi , \\pi ) & = G ( - \\theta , - \\phi , 0 ) = - G ( \\theta , \\pi - \\phi , 0 ) , \\\\ H ( \\theta , \\phi , \\pi ) & = - H ( - \\theta , - \\phi , 0 ) = - H ( \\theta , \\pi - \\phi , 0 ) . \\end{aligned} \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} \\nu _ k ( t ) & = \\left | \\{ ( { \\bf x } ^ 1 , { \\bf x } ^ 2 , \\ldots , { \\bf x } ^ k ) \\in E _ 1 \\times \\ldots \\times E _ k : \\| { \\bf x } ^ 1 + \\cdots + { \\bf x } ^ k \\| = t \\} \\right | \\\\ & = \\sum _ { ( { \\bf x } ^ 1 , { \\bf x } ^ 2 , \\ldots , { \\bf x } ^ k ) \\in E _ 1 \\times E _ 2 \\times \\cdots \\times E _ k } S _ t ( { \\bf x } ^ 1 + { \\bf x } ^ 2 + \\cdots + { \\bf x } ^ k ) . \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} \\xi ^ { \\mathbf { c } } = \\xi ^ { h } + \\nabla ( \\xi ^ { v } ) . \\end{align*}"}
-{"id": "8638.png", "formula": "\\begin{align*} u ( x ) = \\beta ^ { \\frac { 1 } { \\alpha } } | x | ^ { - \\frac { 2 } { \\alpha } } \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{gather*} A _ 0 ^ { ( 1 ) } = \\begin{pmatrix} 0 & 0 \\\\ - q _ 2 & 0 \\end{pmatrix} , A _ 0 ^ { ( 0 ) } = \\begin{pmatrix} - p _ 2 q _ 2 & - t _ 2 / q _ 2 \\\\ 1 - p _ 1 & p _ 2 q _ 2 \\end{pmatrix} , N = \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} , \\\\ A _ { t _ 1 } = \\begin{pmatrix} q _ 1 \\\\ 1 \\end{pmatrix} \\begin{pmatrix} p _ 1 & \\theta ^ { t _ 1 } - p _ 1 q _ 1 \\end{pmatrix} , N _ 1 = \\frac { q _ 1 ( p _ 1 q _ 1 - \\theta ^ { t _ 1 } ) } { t _ 1 } N , N _ 2 = \\frac { 1 } { q _ 2 } N . \\end{gather*}"}
-{"id": "9281.png", "formula": "\\begin{align*} \\int _ { f ^ { - 1 } ( B _ \\frac \\rho 2 ( x ) ) } | A | ^ 4 + | \\nabla A | ^ 2 \\ , d \\mu \\le \\int _ { f ^ { - 1 } ( B _ \\rho ( x ) ) } | A | ^ 4 + | \\nabla A | ^ 2 \\ , d \\mu \\bigg | _ { t = 0 } + c _ 0 t \\rho ^ { - 4 } \\varepsilon _ 1 \\ , , t \\in [ 0 , t _ 0 ) \\ , . \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{align*} \\varepsilon ( \\sigma _ { \\lambda , \\lambda ^ \\prime } ( a ) ) = \\varepsilon ( K _ \\lambda \\triangleright a \\triangleleft K _ { \\lambda ^ \\prime } ) = \\varepsilon ( K _ \\lambda K _ { \\lambda ^ \\prime } \\triangleright a ) . \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} \\sum _ { a + b = 2 n , a < b } ( - 1 ) ^ { s _ { 2 } ( a ) + s _ { 2 } ( b ) } \\equiv | \\{ ( a , b ) \\in \\Z \\times \\Z : \\ ; 0 \\leq a < b , a + b = 2 n \\} | \\equiv n \\pmod { 2 } . \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} ( H _ { ( 1 ) } P ^ T D U _ { ( 1 ) } ) _ { i j } = ( - 1 ) ^ { j - 1 } { { j - 2 } \\choose { i - 1 } } + ( - 1 ) ^ j { { j - 1 } \\choose { i - 1 } } = ( - 1 ) ^ { j - 2 } { { j - 2 } \\choose { i - 2 } } , \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} | u + v \\sqrt { D } | \\leqslant | u - v \\sqrt { D } | + 2 | v | \\sqrt { D } = 2 | v | \\sqrt { D } + O _ { \\varepsilon , \\eta } \\left ( \\frac { 1 } { B } \\right ) \\leqslant 2 \\sqrt { \\frac { A ^ \\prime | m | D } { \\eta ^ \\prime ( 2 \\sqrt { D } - \\frac { \\varepsilon ^ \\prime } { B ^ 2 } ) } } B + O _ { \\varepsilon , \\eta } \\left ( \\frac { 1 } { B } \\right ) . \\end{align*}"}
-{"id": "9410.png", "formula": "\\begin{align*} \\bigl [ \\widetilde { \\phi } ( t _ 2 ) \\bigr ] ( X _ J ) = X _ { \\phi ( \\cdotp , t _ 2 , \\cdotp ) J } \\hbox { f o r a . e . } \\ , t _ 2 . \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{align*} X ( \\Q _ \\ell ) _ 1 = \\left \\{ P \\in X ( \\Q _ \\ell ) : \\int _ { m P - D _ 0 } \\omega _ i = 0 \\right \\} . \\end{align*}"}
-{"id": "7244.png", "formula": "\\begin{align*} \\gamma _ 2 \\sum _ { i , j } c _ { i j } \\gamma _ 1 ^ i \\gamma _ 2 ^ j = \\sum _ { i , j } c _ { i j } \\gamma _ 1 ^ i \\gamma _ 2 ^ j + ( 1 - \\gamma _ 1 ) \\sum _ { i , j } a _ { i j } \\gamma _ 1 ^ i \\gamma _ 2 ^ j \\end{align*}"}
-{"id": "6538.png", "formula": "\\begin{align*} \\dot y = R ^ { - 1 } Q v ( x _ 0 ' + \\Gamma _ 0 ' y ) \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} g ^ 1 _ n ( \\omega , t , y , z ) : = \\inf \\limits _ { u \\in \\R ^ d } \\left [ g ^ 1 ( \\omega , t , y , u ) + ( n + 2 \\lambda ) | u - z | ^ \\alpha \\right ] \\end{align*}"}
-{"id": "5210.png", "formula": "\\begin{align*} \\lambda _ { n m } ^ 2 = z _ { n m } \\tanh \\left ( \\frac { z _ { n m } h } { a } \\right ) . \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} \\begin{array} { l l l } S _ 1 & = & \\{ a \\} , \\\\ S _ 2 & = & S ( v ) \\setminus S _ 1 , \\\\ E _ 2 & = & \\{ c \\in E ( v ) \\vert c a \\in I \\} , \\\\ E _ 1 & = & E ( v ) \\setminus E _ 2 . \\end{array} \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{align*} \\begin{aligned} ( H _ i - q ) ( H _ i + q ^ { - 1 } ) = 0 , & 1 \\leq i \\leq r - 1 , \\\\ H _ i H _ j = H _ j H _ i , & | i - j | \\geq 2 , \\\\ H _ i H _ { i + 1 } H _ i = H _ { i + 1 } H _ i H _ { i + 1 } , & 1 \\leq i \\leq r - 2 . \\end{aligned} \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } t ^ { \\frac 3 2 ( 1 - 1 / p ) } \\| F ( \\cdot , t ) * V \\| _ p = 0 , 1 \\le p \\le \\infty . \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} & \\int _ 0 ^ \\infty G ( y ) ( y ^ 2 - 1 ) \\varphi ( y ) d y \\leq G ( y _ 0 ) \\int _ 0 ^ 1 ( y ^ 2 - 1 ) \\varphi ( y ) d y \\\\ & + G ( 1 ) \\int _ 1 ^ \\infty ( y ^ 2 - 1 ) \\varphi ( y ) d y = ( G ( y _ 0 ) - G ( 1 ) ) \\int _ 0 ^ 1 ( y ^ 2 - 1 ) \\varphi ( y ) d y < 0 \\ ; . \\end{align*}"}
-{"id": "1606.png", "formula": "\\begin{align*} k _ { 2 } ^ { - 1 } k _ { 1 } = g _ { i } g _ { \\mu ( 2 ) } a _ { 4 } g _ { s ( 2 ) } ^ { - 1 } g _ { s ( 1 ) } a _ { 2 } ^ { - 1 } g _ { \\mu ( 1 ) } ^ { - 1 } g _ { i } ^ { - 1 } \\in K \\cap g _ { i } G _ { j } g _ { i } ^ { - 1 } = K \\cap G _ { w _ { i } } = K \\cap g _ { i } A g _ { i } ^ { - 1 } = 1 . \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} a ( n ) & = ( - 2 ) ^ k ( a ( l ) + a ( l + 1 ) ) , \\\\ a ( n - 1 ) & = \\left ( b ( k ) - 2 \\right ) a ( l ) + b ( k ) a ( l + 1 ) , \\\\ a ( n + 1 ) & = \\left ( b ( k ) - 2 \\right ) a ( l + 1 ) + b ( k ) a ( l ) , \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{align*} f ' + C _ 1 f = C _ 2 f ^ q \\mbox { o n $ [ 0 , T ) $ f o r s o m e $ T > 0 $ } , \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} \\lambda _ k & = \\lambda _ { k - 1 } + \\gamma _ k = \\sum _ { \\ell = 1 } ^ { k } \\gamma _ \\ell \\\\ \\| \\lambda _ k \\| & = \\lambda _ k \\left ( [ 0 , T _ k ] \\right ) = ( 2 ^ k + 1 ) \\| \\lambda _ { k - 1 } \\| \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{align*} \\log _ C ( x ) = \\sum _ { i = 0 } ^ \\infty ( - 1 ) ^ i \\frac { x ^ { r ^ i } } { L _ i } \\ , , \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} & \\{ H ^ { \\beta } ( z ^ 2 ) , H ^ { \\gamma } ( w ^ 2 ) \\} = i _ { z , w } \\frac { 1 } { ( z ^ 2 - w ^ 2 ) ^ 2 } - i _ { w , z } \\frac { 1 } { ( w ^ 2 - z ^ 2 ) ^ 2 } ; \\\\ & \\{ H ^ { \\beta } ( z ^ 2 ) , H ^ { \\beta } ( w ^ 2 ) \\} = 0 ; \\{ H ^ { \\gamma } ( z ^ 2 ) , H ^ { \\gamma } ( w ^ 2 ) \\} = 0 . \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( A _ \\infty + \\frac { A _ 0 } { x } + \\frac { A _ 1 } { x - 1 } \\right ) Y , \\frac { \\partial Y } { \\partial t _ 1 } = ( E _ 3 x + B _ 1 ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = \\left ( \\frac { 1 } { t _ 2 } N x + B _ 2 \\right ) Y . \\end{gather*}"}
-{"id": "6171.png", "formula": "\\begin{align*} 0 = [ L _ { - q } , \\ , [ L _ { - 2 } , \\ , S _ 2 ] ] = [ L _ { - q } , \\ , [ L _ { - 1 } , \\ , S _ 1 ] ] = [ L _ { - 1 } , \\ , [ L _ { - q } , \\ , S _ 1 ] ] = [ L _ { - 1 } , \\ , L _ { - q + 1 } ] \\end{align*}"}
-{"id": "4255.png", "formula": "\\begin{align*} ( p - c ) \\phi = \\mathcal { R } _ i \\mathcal { R } _ j ( F ^ { i j } \\phi ) + [ \\phi , \\mathcal { R } _ i \\mathcal { R } _ j ] ( F ^ { i j } ) - c \\phi \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} \\pi _ D ( x ) = \\frac { 2 ^ { \\omega ( D ) + 1 } \\pi ( x ) } { D } \\prod _ { \\substack { r \\leq y \\\\ r \\nmid D } } \\left ( \\frac { r - 3 } { r - 1 } \\right ) + O ( 2 ^ { \\omega ( D ) } x L _ y ^ 2 e ^ { - C { \\sqrt { \\log x } } } ) . \\end{align*}"}
-{"id": "474.png", "formula": "\\begin{align*} & I ( M ; Y _ i Y _ j ) = H ( M ) - H ( M | Y _ i Y _ j ) \\\\ = & H ( M | Y _ i ) - H ( M | Y _ i Y _ j ) = I ( M ; Y _ j | Y _ i ) \\\\ = & H ( Y _ j | Y _ i ) - H ( Y _ j | M Y _ i ) = H ( Y _ j | Y _ i ) . \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} \\hat { F } _ n = E _ { n } ^ c ( n ) \\cap A _ n ( n ) , \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} \\psi _ n ( \\lambda _ { p , - n } \\zeta ) = f ^ { - n } \\circ \\psi _ 0 ( \\zeta ) . \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{align*} \\mathrm { T r } D G ( x , y ) = \\frac { 1 } { y } ( - b - y - d + x ) - \\frac { 1 } { y ^ 2 } ( c - d y + x y ) = \\frac { 1 } { y } \\left ( - b - y - \\frac { c } { y } \\right ) < 0 \\ , . \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} H _ \\varphi f ( x ) = \\int _ 0 ^ \\infty f \\left ( \\frac { x } { t } \\right ) \\frac { \\varphi ( t ) } { t } d t , x \\in \\R . \\end{align*}"}
-{"id": "8324.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ \\Omega u \\varphi _ t - \\nabla v \\cdot \\nabla \\varphi d x d t + \\int _ { \\Omega } u _ 0 ( x ) \\varphi ( 0 , x ) d x = 0 \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} Z ( k _ { P ( 1 ) } , \\dots , k _ { P ( n ) } ) = 0 , \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{align*} \\left [ \\frac { A _ { 1 } h } { B _ { 2 } } - \\left ( \\frac { h } { B _ { 2 } } \\right ) ^ { \\prime } \\right ] f + \\left [ - \\frac { 2 h } { B _ { 2 } } \\right ] f ^ { \\prime } + \\left [ \\frac { A _ { 1 } B _ { 1 } } { B _ { 2 } } - \\left ( \\frac { B _ { 1 } } { B _ { 2 } } \\right ) ^ { \\prime } - \\frac { B _ { 1 } A _ { 2 } } { B _ { 2 } } \\right ] f _ { c _ { 2 } } = 0 . \\end{align*}"}
-{"id": "9448.png", "formula": "\\begin{align*} d ( \\rho , \\rho ' ) = \\sum _ { n \\geq 0 } 2 ^ { - n } \\frac { \\left | \\big \\langle \\rho - \\rho ' , f _ { n } \\big \\rangle \\right | } { 1 + \\left | \\big \\langle \\rho - \\rho ' , f _ { n } \\big \\rangle \\right | } \\end{align*}"}
-{"id": "3734.png", "formula": "\\begin{align*} Q _ E ^ { 1 / 2 } ( z ) & = - z ^ { n + 1 } \\sum _ { j = 0 } ^ \\infty c _ j ( E ) z ^ { - j } . \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} g ( E ) = \\sum _ { \\vec { m } \\in [ 1 , 2 ( n + 1 ) ] ^ n } \\det ( ( D \\omega ( E ) ) _ { \\vec { m } } ) ^ 2 \\end{align*}"}
-{"id": "9921.png", "formula": "\\begin{align*} \\vect { S } _ \\pm \\Gamma = F \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} \\frac { \\bar { A } _ { n , j , k } ( t , z ) } { ( 1 - t ) ^ { n + 1 } } = \\sum _ { p = 0 } ^ { \\infty } Q _ { n , j , k } ( - p , z ) t ^ { p } . \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{align*} g ( 0 ) = I , g ( t _ f ) = R : = \\exp ( J \\pi / 3 ) . \\end{align*}"}
-{"id": "5573.png", "formula": "\\begin{align*} R = \\eta ^ { - 2 } ( \\log X ) ^ { 3 / 2 } \\end{align*}"}
-{"id": "10000.png", "formula": "\\begin{align*} f ( v ) & = \\sum _ { i = - d } ^ { d } ( \\left \\vert v - i \\right \\vert + 1 ) + \\sum _ { i = - ( k - 1 ) } ^ { k - 1 } ( \\left \\vert v - i \\right \\vert + 2 ) + \\\\ & + x ( \\left \\vert v + ( d - 1 ) \\right \\vert + 2 ) + x ( \\left \\vert v - ( d - 1 ) \\right \\vert + 2 ) + \\\\ & + r ( \\left \\vert v + d \\right \\vert + 2 ) + r ( \\left \\vert v - d \\right \\vert + 2 ) . \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} & \\log B _ n ( { \\cal Y } ) = - \\frac { z ^ 2 } { 2 n ^ { d } } \\sum _ { t \\in W _ n } \\chi ( Y _ t ) + o _ p ( 1 ) \\ ; . \\end{align*}"}
-{"id": "7551.png", "formula": "\\begin{align*} ( n ! ) ^ { 2 } \\left ( \\frac { D _ n } { 4 ^ n } - \\frac { D _ { n - 1 } } { 4 ^ { n - 1 } } \\right ) = \\alpha _ { n + 1 } - n ^ { 2 } \\alpha _ { n } , \\end{align*}"}
-{"id": "6916.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\nabla \\left ( \\frac { m t ^ 2 } { 2 } - \\varphi \\right ) \\right | ^ 2 = & m ^ 2 t ^ 2 | \\nabla t | ^ 2 - 2 m t \\nabla \\varphi \\cdot \\nabla t + | \\nabla \\varphi | ^ 2 \\ge m ^ 2 t ^ 2 | \\nabla t | ^ 2 - 2 m | t | | \\nabla \\varphi | | \\nabla t | + | \\nabla \\varphi | ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "4093.png", "formula": "\\begin{align*} \\tau _ { K , N } ^ { ( t ) } ( \\theta ) : = t ^ { 1 / N } \\sigma _ { K , N - 1 } ^ { ( t ) } ( \\theta ) ^ { ( N - 1 ) / N } . \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} ( x _ \\alpha ^ + \\otimes t ^ r ) ^ { ( s ) } v _ \\lambda = \\Lambda _ { \\alpha _ i , s } v _ \\lambda = h - \\lambda ( h ) v _ \\lambda = ( x _ \\alpha ^ - ) ^ { ( k ) } v _ \\lambda = 0 , \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} { Q } ( x ) { S } ( { x } _ k ) - { Q } ( { x } _ k ) { S } ( x ) = { S } ( { x } _ k ) \\prod _ { l = 1 } ^ n ( x - { x } _ l ) , k = 1 , \\dots , n . \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{gather*} P _ V = \\sum _ { i = 0 } ^ { d } \\frac { ( - 1 ) ^ i \\sigma _ i z ^ i } { ( [ i ] _ q ^ { ! } ) ^ 2 } , \\end{gather*}"}
-{"id": "5988.png", "formula": "\\begin{align*} a ^ { 2 } + b ^ { 2 } + c + 1 = 0 . \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{align*} [ \\cdot , \\cdot ] _ \\tau : \\mathfrak { h } _ { 6 } ^ { ( \\tau ) } \\times \\mathfrak { h } _ { 6 } ^ { ( \\tau ) } \\longrightarrow \\mathfrak { h } _ { 6 } ^ { ( \\tau ) } \\quad { } [ \\mathcal { A } , \\mathcal { B } ] _ \\tau = \\mathcal { A } \\mathcal { B } - \\mathcal { B } \\mathcal { A } , \\mbox { ~ ~ f o r a l l ~ ~ } \\mathcal { A } , \\mathcal { B } \\in \\mathfrak { h } _ { 6 } ^ { ( \\tau ) } . \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} \\mu = T . \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle ( n = 2 ) \\qquad \\lim _ { t \\rightarrow T } \\int _ { f ^ { - 1 } ( B _ { \\rho ( t ) } ( x ) ) } | A | ^ 2 d \\mu \\ge \\varepsilon _ 1 \\ , , \\\\ \\displaystyle ( n = 4 ) \\qquad \\lim _ { t \\rightarrow T } \\int _ { f ^ { - 1 } ( B _ { \\rho ( t ) } ( x ) ) } | A | ^ 4 + | \\nabla A | ^ 2 d \\mu \\ge \\varepsilon _ 1 \\ , , \\end{cases} \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} | F _ 2 + F _ 5 + F _ 6 | & = | V _ 2 ( z _ { - N } , x _ { - N + 1 } ) - V _ 2 ( x _ { - N } , x _ { - N + 1 } | \\cdot | V _ 2 ( z _ { - N } , x _ { - N + 1 } ) + V _ 1 ( x _ { - N + 1 } , x _ { - N + 2 } ) | \\\\ & \\leq \\frac { \\kappa _ 2 } { 4 } | x _ { - N } - z _ { - N } | . \\end{align*}"}
-{"id": "9723.png", "formula": "\\begin{align*} C _ 1 = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} Z ( \\mathrm { C a } ( X ) , T ) = { \\textstyle \\prod \\limits _ { P \\in X } } Z _ { \\mathrm { C a } ( X ) } ( T , q , \\mathcal { O } _ { P , X } ) , \\end{align*}"}
-{"id": "6339.png", "formula": "\\begin{align*} N _ 0 : = \\{ x \\in N _ 0 ( M ) \\mid \\exists \\ , x _ 0 \\in M \\ x = x _ 0 \\ x = x _ 0 ^ * \\} . \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{align*} \\Lambda ( B , \\phi ) : = \\{ p \\in \\Lambda ( B ) \\mid p \\phi p = \\phi p \\} \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} \\Delta _ { X } = [ X \\times z _ X ] + B _ X , \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} \\chi _ { x } ( a _ { y } ) = a _ { y + x } \\ , x , y \\in \\mathfrak { L } = \\mathbb { Z } ^ { d } \\ . \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} \\nabla _ { ( k ) } A = \\Big ( \\nabla _ { ( k ) } A ^ o + \\frac 1 n g \\nabla _ { ( k ) } \\vec { H } \\Big ) = \\Big ( \\nabla _ { ( k ) } A ^ o + \\frac 1 { n - 1 } g \\nabla _ { ( k - 1 ) } \\nabla ^ * A ^ o \\Big ) , \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{align*} Z ( \\kappa , \\lambda ) \\cap Z ( \\mu , \\nu ) = \\bigsqcup _ { \\alpha \\in A } Z ( \\kappa \\alpha , \\lambda \\alpha ) \\quad Z ( \\kappa , \\lambda ) \\setminus Z ( \\mu , \\nu ) = \\bigsqcup _ { \\alpha \\in B } Z ( \\kappa \\alpha , \\lambda \\alpha ) . \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} h ' ( g ( \\alpha ) ) g ' ( \\alpha ) = h ' ( \\alpha ) g ' ( \\alpha ) = k h ' ( \\alpha ) \\end{align*}"}
-{"id": "721.png", "formula": "\\begin{align*} a x ^ 2 - ( a + c + 1 ) x + c = 0 . \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} U ( \\theta ^ * ) : = \\Theta \\cap \\{ \\theta \\in \\R ^ n : \\| \\theta - \\theta ^ * \\| _ 2 \\leq \\rho t _ { \\theta ^ * } \\} \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{align*} h \\left ( z \\right ) = \\frac { B _ { 2 } f _ { c _ { 1 } } - B _ { 1 } f _ { c _ { 2 } } } { f \\left ( z \\right ) } . \\end{align*}"}
-{"id": "6134.png", "formula": "\\begin{align*} X _ { 2 , \\ , j + 2 , \\ , 1 2 } \\cap L _ { j + 2 } = 0 . \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} 0 & = Q ^ 6 b _ 1 + b _ 1 ^ 4 \\\\ 0 & = Q ^ { 1 0 } b _ 1 + ( Q ^ 4 b _ 1 ) ^ 2 \\\\ Q ^ 6 b _ 2 & = Q ^ 8 b _ 1 + b _ 1 ^ 2 Q ^ 4 b _ 1 \\\\ 0 & = Q ^ { 1 0 } b _ 2 + b _ 1 ^ 2 Q ^ 6 b _ 2 \\\\ \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} R _ i = T _ { i - 1 } \\dots T _ 1 R _ 1 T _ 1 \\dots T _ { i - 1 } \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} c ^ { \\star } = \\frac { \\delta - K } { 1 - \\alpha \\delta } , \\ , \\ , \\textrm { w h e r e } \\delta = \\frac { \\Gamma } { \\alpha \\lambda } , \\end{align*}"}
-{"id": "9954.png", "formula": "\\begin{align*} \\dot { V } = - x ^ T \\tilde { Q } x \\mbox { w h e r e } \\tilde { Q } = - \\tilde { A } ^ T Q \\tilde { A } . \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} F ( X , n ) = \\{ ( x _ 1 , x _ 2 , \\ldots , x _ n ) \\in X ^ n : x _ i \\neq x _ j i \\neq j \\} . \\end{align*}"}
-{"id": "9307.png", "formula": "\\begin{align*} H _ b ' ( p ) = \\log _ 2 ( 1 - p ) - \\log _ 2 p , \\\\ H _ b '' ( p ) = - \\frac { 1 } { \\ln ( 2 ) p ( 1 - p ) } . \\end{align*}"}
-{"id": "5433.png", "formula": "\\begin{align*} \\partial _ t M _ 2 - d _ { M _ 2 } \\Delta _ x M _ 2 = \\lambda _ { m M _ 2 } \\frac { m } { k _ { M _ 2 } + m } + \\lambda _ { L _ { o x } M _ 2 } \\frac { L _ { o x } } { K _ { M _ 2 } + L _ { o x } } \\ , M _ 2 \\ - \\ \\beta _ { M _ 2 } M _ 2 , \\end{align*}"}
-{"id": "5001.png", "formula": "\\begin{align*} \\beta = 1 2 \\langle \\psi _ * ^ 2 , w _ 0 + w _ 2 \\rangle _ { L ^ 2 } = - 1 2 \\langle \\psi _ * ^ 2 , \\tilde { w } _ 2 \\rangle _ { L ^ 2 } < 0 . \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} t _ 3 ( 4 n + 3 ) & = - 3 t _ 3 ( 2 n + 1 ) - t _ 3 ( 2 n ) = 9 t _ 3 ( n ) + 3 t _ 3 ( n - 1 ) - t _ 3 ( n ) - 3 t _ 3 ( n - 1 ) = 8 t _ 3 ( n ) , \\\\ t _ 3 ( 4 n + 2 ) & = t _ 3 ( 2 n + 1 ) + 3 t _ 3 ( 2 n ) = - 3 t _ 3 ( n ) - t _ 3 ( n - 1 ) + 3 t _ 3 ( n ) + 9 t _ 3 ( n - 1 ) = 8 t _ 3 ( n - 1 ) . \\end{align*}"}
-{"id": "9918.png", "formula": "\\begin{align*} J _ \\nu ( z ) = \\Big ( \\frac z 2 \\Big ) ^ \\nu \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ n \\big ( \\tfrac z 2 \\big ) ^ { 2 n } } { n ! \\ , \\Gamma ( \\nu + n + 1 ) } . \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} \\phi _ k ( u ) = \\mathtt { 1 } _ { u \\in I _ k } ( u ) . \\end{align*}"}
-{"id": "6126.png", "formula": "\\begin{align*} X _ { 2 , \\ , j + 2 , \\ , 4 } \\cap L _ { j + 2 } = 0 . \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} D _ n = \\{ F \\subset \\lambda : | F | = n \\} , \\ ; \\ ; n \\in \\N \\end{align*}"}
-{"id": "3327.png", "formula": "\\begin{align*} P _ N ( a ) & = \\frac { 1 } { \\sqrt { N } } C ( a \\mid \\nu ) \\mathrm { e } ^ { - N I ( a \\mid \\nu ) + \\frac 1 2 N ^ { 2 - \\alpha } ( \\frac { a } { \\nu } - 1 ) ^ 2 \\sigma ^ 2 } . \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} \\frac { C _ 2 } { 2 } & \\le \\iint _ { \\nu _ \\theta ( x ) \\ge \\tau } \\mu _ \\theta ( x ) | \\phi _ K * \\nu _ \\theta ( x ) | \\ , d x \\ , d \\sigma ( \\theta ) \\\\ & \\lesssim K \\int \\mu _ \\theta \\{ x : \\nu _ \\theta ( x ) \\ge \\tau \\} \\ , d \\sigma ( \\theta ) \\\\ & = K \\int \\mu _ \\theta \\{ x : | \\nu _ { \\theta , x } | \\ge \\tau \\} \\ , d \\sigma ( \\theta ) \\\\ & = K \\int \\mu \\{ y : | \\nu _ { \\theta , y } | \\ge \\tau \\} \\ , d \\sigma ( \\theta ) . \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} \\partial _ t ^ j \\partial _ x ^ n \\partial _ y ^ m J _ 0 ( t , x , y ; \\nu ) = \\sum \\limits _ { l = 1 } ^ { + \\infty } \\frac 1 { 2 \\pi } \\int _ { \\mathbb R } ( i \\theta ) ^ j \\frac { r _ 1 r _ 2 ^ n e ^ { r _ 2 x } - r _ 2 r _ 1 ^ n e ^ { r _ 1 x } } { r _ 1 - r _ 2 } \\widehat \\nu ( \\theta , l ) \\ , d \\theta \\ , \\psi _ l ^ { ( m ) } ( y ) \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{align*} \\beta _ { \\pm } \\circ e _ { \\pm } = s _ { \\pm } ~ . \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{align*} u _ \\lambda ( t , x ) = \\lambda ^ { - \\frac { 4 } { \\nu - 1 } } u ( \\lambda ^ { - 4 } t , \\lambda ^ { - 1 } x ) , \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} R _ i ( k _ 1 , \\dots , k _ n ) = ( k _ 1 , \\dots , k _ { i - 1 } , - k _ i , k _ { i + 1 } , \\dots , k _ n ) , \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} f ^ { \\varepsilon } ( x ) = \\mathop { \\rm s u p } _ { y \\in \\mathcal { H } } \\left \\{ f ( y ) - \\frac { | y - x | ^ { 2 } } { 2 \\varepsilon } \\right \\} , f _ { \\varepsilon } ( x ) = \\mathop { \\rm i n f } _ { y \\in \\mathcal { H } } \\left \\{ f ( y ) + \\frac { | y - x | ^ { 2 } } { 2 \\varepsilon } \\right \\} . \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{align*} \\vartheta \\ ( z ; \\tau \\ ) : = \\sum _ { n \\in \\frac 1 2 + \\Z } e ^ { \\pi i n ^ 2 \\tau + 2 \\pi i n \\ ( z + \\frac 1 2 \\ ) } . \\end{align*}"}
-{"id": "2756.png", "formula": "\\begin{align*} \\sharp T _ 1 ( \\varepsilon _ 1 , \\varepsilon _ 2 , d , B ) = \\frac { \\varepsilon _ 1 - \\varepsilon _ 2 } { 2 ( \\alpha ^ \\prime ) ^ 2 d ^ 2 } B ^ { 2 - \\frac { 1 } { r } } + O _ { \\varepsilon _ i } \\left ( \\frac { B ^ { 2 - \\frac { 1 } { r } } } { N d ^ 2 } \\right ) + O _ { \\sigma , \\varepsilon _ i } \\left ( \\frac { B ^ \\sigma N } { d ^ \\sigma } \\right ) , \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} x _ 0 - x _ 1 + y _ 0 - \\varphi ( x ) & = c + y _ 0 - \\varphi ( x ) \\\\ & = y _ 1 - \\varphi ( x ) \\\\ & = \\varphi ( x _ 1 ) - \\varphi ( x ) \\\\ & \\leq x - x _ 1 . \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} \\hat \\gamma ^ i = \\theta ( \\bar x - x ^ i ) - y ^ { i , i } - \\frac { 1 } { \\lambda } r ^ { i , i , i } \\ , . \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} q = q ( C ) : = \\sum _ { r = 1 } ^ { \\infty } \\frac { T _ r e ^ { - r } } { C ( r - 1 ) ! } e ^ { - \\delta r } = \\sum _ { r = 1 } ^ { \\infty } \\frac { T _ r C ^ { r - 1 } e ^ { - C r } } { ( r - 1 ) ! } \\end{align*}"}
-{"id": "4267.png", "formula": "\\begin{align*} \\begin{cases} c _ 4 = b _ 2 ^ 2 - 2 4 b _ 4 ; \\\\ c _ 6 = - b _ 2 ^ 3 + 3 6 b _ 2 b _ 4 - 2 1 6 b _ 6 . \\end{cases} \\end{align*}"}
-{"id": "9169.png", "formula": "\\begin{align*} y _ t = H ( q ^ { - 1 } ) r _ t + e _ t + w _ t . \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{gather*} \\| u _ 0 u _ { 0 x } \\| _ { L _ 2 } \\leq c \\| u _ 0 \\| _ { L _ { \\infty } } \\| u _ { 0 x } \\| _ { L _ 2 } \\leq c _ 1 \\| u _ 0 \\| _ { \\widetilde H ^ 3 } ^ 2 , \\\\ \\| \\psi \\big | _ { t = 0 } u _ { 0 x } + \\psi _ x \\big | _ { t = 0 } u _ 0 \\| _ { L _ 2 } \\leq c \\| \\psi \\big | _ { t = 0 } \\| _ { H ^ 1 } \\| u _ 0 \\| _ { W ^ 1 _ \\infty } \\leq c \\| \\psi \\| _ { X ^ 3 ( Q _ T ) } \\| u _ 0 \\| _ { \\widetilde H ^ 3 } ; \\end{gather*}"}
-{"id": "9356.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { J } \\mathcal { V } _ r ( \\sigma _ j ^ \\ell ) \\lesssim _ r \\| F \\| _ { \\ell ^ r } ^ { r - 1 } \\Big ( \\sum _ { j = 1 } ^ J \\mathcal { V } _ r ^ r ( \\sigma _ j ^ \\ell ) \\Big ) ^ { \\frac 1 r } = { \\| F \\| _ { \\ell ^ r } ^ { r - 1 } } \\Big ( \\sum _ { n \\in \\mathbb { J } } | F _ n | ^ r \\Big ) ^ { \\frac 1 r } \\leq \\| F \\| _ { \\ell ^ r } ^ r \\leq \\mathcal { V } _ r ^ r ( \\gamma ) . \\end{align*}"}
-{"id": "2099.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 0 } { x } + \\frac { A _ 1 } { x - 1 } + A _ \\infty \\right ) Y , \\frac { \\partial Y } { \\partial t } = \\left ( \\frac { 1 } { t } A _ \\infty x + B _ 0 \\right ) Y . \\end{gather*}"}
-{"id": "9484.png", "formula": "\\begin{align*} | T _ { k , S ^ \\ast } ( S ) | & \\equiv \\begin{cases} ( - 1 ) ^ { z ( S ) } | F ' ( \\pi ( S ) ) | \\mod { D e g ( F ^ \\ast ) _ { | S | } } & \\mbox { i f } S \\mbox { i s n o n - d e g e n e r a t e } \\\\ & \\mbox { a n d } | I ( S ) | = k ; \\\\ 0 \\mod { D e g ( F ^ \\ast ) _ { | S | } } & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} \\sum _ { \\substack { d | n \\\\ d \\leq \\left \\lfloor \\frac { n } { m + 1 } \\right \\rfloor } } \\scriptstyle { \\binom { \\frac { n } { d } - 1 - m + k } { k } } a _ d & = \\sum _ { q = 0 } ^ n \\sum _ { i = 1 } ^ { \\left \\lfloor \\frac { n - q } { m + 1 } \\right \\rfloor } \\sum _ { j = 0 } ^ { \\left \\lfloor \\frac { n - q - m } { i } \\right \\rfloor } \\binom { k - 1 + j } { k - 1 } s _ { n - q - m - j i , i } \\cdot a _ i \\cdot p ( q ) . \\end{align*}"}
-{"id": "2391.png", "formula": "\\begin{align*} \\partial _ t \\psi = - \\mathrm i | D | \\psi - \\psi | \\psi | ^ { p - 1 } , | D | = \\sqrt { - \\Delta } , \\end{align*}"}
-{"id": "9655.png", "formula": "\\begin{align*} S = { \\frac 1 2 } ( Y + Z ) = \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} , T = { \\frac 1 2 } ( Y - Z ) = \\begin{pmatrix} 0 & 0 \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "933.png", "formula": "\\begin{align*} \\langle p , f , \\beta \\rangle \\alpha ( z _ { 3 0 } ) = ( \\langle p , f , \\beta \\rangle ( z _ { 1 5 } ) ) ^ 2 \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} I ( A : B | M Z ) _ { \\hat { \\sigma } _ { A B M Z } } & = \\int _ { \\mathbb { R } ^ { 2 n } } I ( A : B | M ) _ { \\hat { \\sigma } _ { A B M | Z = \\mathbf { z } } } \\ , \\mathrm { d } p _ Z ( \\mathbf { z } ) \\\\ & = \\int _ { \\mathbb { R } ^ { 2 n } } I ( A : B | M ) _ { \\hat { \\rho } _ { A B M } } \\ , \\mathrm { d } p _ Z ( \\mathbf { z } ) = 0 \\ ; . \\end{align*}"}
-{"id": "1029.png", "formula": "\\begin{align*} E ( u ( t ) ) : = \\int _ { \\R ^ k } \\frac { 1 } { 2 } | \\Lambda ^ { k / 2 } u ( t , x ) | ^ 2 + \\frac { 1 } { 4 } | u ( t , x ) | ^ 4 d x = E ( u _ 0 ) . \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} \\Lambda = \\Lambda _ { 1 } ( \\infty , \\Omega _ { 2 } ) . \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} \\rho _ j ( f ( x _ 0 ) , f ( x _ { j } ) ) = \\rho _ j ( f ( x _ { j } ) , f ( x _ { 2 j } ) ) = \\cdots = \\rho _ j ( f ( x _ { ( p - 1 ) j } ) , f ( x _ 0 ) ) , \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} C ( a \\mid x ) = \\frac { 1 } { \\tau \\sqrt { 2 \\pi \\Lambda '' _ X ( \\tau ) } } = \\frac { 1 } { \\log ( \\frac { a } { x } ) \\sqrt { 2 \\pi a } } . \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ \\infty F _ { i - 1 } x ^ i \\right ) ^ k = \\sum _ { n = k } ^ \\infty \\overline { c } _ 3 ( n , k ) x ^ n . \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} ( 1 + \\beta ) q & = p + 2 \\frac { p } { n } , n > 1 \\\\ ( 1 + \\beta ) q & = 2 p , n = 1 . \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{align*} \\tau ^ { ( k ) } = 2 ^ { C k ^ 2 } , \\delta ^ { ( k ) } = 2 ^ { C k ^ 2 + 1 } { \\rm a n d } \\Delta ^ { ( k ) } = 2 ^ { 2 C k ^ 2 + 4 } , k \\ge 1 , \\end{align*}"}
-{"id": "1342.png", "formula": "\\begin{align*} \\left | \\alpha + \\frac { d } { c } \\right | = \\left | \\beta + \\frac { d } { c } \\right | = \\frac { 1 } { | c | } . \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} \\mathfrak { i } _ { \\mathrm { p } } \\left ( t \\right ) : = \\underset { ( \\eta , l ^ { - 1 } ) \\rightarrow ( 0 , 0 ) } { \\lim } \\left \\{ \\left ( \\eta ^ { 2 } l ^ { d } \\right ) ^ { - 1 } \\mathfrak { I } _ { \\mathrm { p } } ^ { ( \\omega , \\eta \\mathbf { A } _ { l } ) } \\left ( t \\right ) \\right \\} = \\int \\nolimits _ { t _ { 0 } } ^ { t } \\int \\nolimits _ { t _ { 0 } } ^ { s _ { 1 } } \\mathbf { X } _ { \\infty } ( s _ { 1 } , s _ { 2 } ) \\ \\mathrm { d } s _ { 2 } \\mathrm { d } s _ { 1 } \\ . \\end{align*}"}
-{"id": "9537.png", "formula": "\\begin{align*} \\Psi ( z ) = \\sum _ { i = 1 } ^ k a _ i ^ * z \\alpha _ { g _ 0 } ^ { - 1 } ( a _ i ) , \\ \\ \\ z \\in A \\rtimes _ { \\alpha , r } G , \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{align*} - u ^ { \\left ( 2 \\right ) } \\left ( \\tau \\right ) + A u \\left ( \\tau \\right ) = \\tilde { f } \\left ( \\tau \\right ) , \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} G _ { i + 1 } = M _ { i + 1 } \\left ( \\begin{array} { c } I _ { i + 1 } \\\\ R _ { i + 1 } \\\\ \\end{array} \\right ) \\left ( \\begin{array} { c } 2 e _ 1 ^ { i + 1 } \\\\ M _ i \\\\ \\end{array} \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { k - \\ell } \\binom { k } { j } \\binom { k } { \\ell + j } = \\binom { 2 k } { k + \\ell } \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{align*} d _ q ( A _ { \\alpha - 1 } ) = z ^ 2 q + ( 2 z + 1 ) ( \\alpha - z q ) - 1 , \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{align*} L = [ l _ { i j } ] _ { i , j \\geq 0 } , l _ { i j } = \\frac { 1 } { | | P _ i | | ^ 2 } < P _ i , l P _ j > , \\end{align*}"}
-{"id": "8391.png", "formula": "\\begin{align*} \\mathbb { P } ( \\hat { A } ^ c _ { n _ { \\epsilon } } ( n ) ) \\leq \\frac { \\mathbb { E } \\left ( \\sum _ { i = 1 } ^ { n } X _ i \\right ) ^ 2 } { \\mu ^ 2 n ^ { 2 + 2 \\epsilon } } \\leq \\frac { C _ 1 n } { n ^ { 2 + 2 \\epsilon } } \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} H ^ { \\beta } ( z ^ 2 ) = V ^ + ( z ) ^ { - 1 } \\beta _ \\chi ( z ^ 2 ) z ^ { - 2 h _ 0 } V ^ - ( z ) ^ { - 1 } , H ^ { \\gamma } ( z ^ 2 ) = V ^ + ( z ) \\gamma _ \\chi ( z ^ 2 ) z ^ { 2 h _ 0 } V ^ - ( z ) . \\end{align*}"}
-{"id": "2333.png", "formula": "\\begin{align*} | \\sqrt m - \\sqrt { t _ f } | = \\frac { | m - t _ f | } { \\sqrt m + \\sqrt { t _ f } } \\le \\frac { | m - t _ f | } { \\sqrt { \\max ( t _ f , m ) } } \\le \\sqrt { 6 \\tau \\sigma } . \\end{align*}"}
-{"id": "7007.png", "formula": "\\begin{align*} \\left ( \\varepsilon \\alpha _ { 1 } Q _ { 1 , \\lambda } \\left ( \\varepsilon \\right ) + \\alpha _ { 0 } \\right ) g _ { 1 } + \\left ( \\varepsilon \\alpha _ { 1 } Q _ { 2 , \\lambda } \\left ( \\varepsilon \\right ) + \\alpha _ { 0 } \\right ) g _ { 2 } = f _ { 1 } , \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} & \\int _ { \\Omega _ { 2 } } \\partial _ { 2 } \\phi _ { n } ( x ) \\partial _ { 2 } \\phi _ { m } ( x ) \\ , d x = & \\\\ & { 4 n _ { 2 } m _ { 2 } \\over \\pi ^ 2 } \\int _ { 0 } ^ \\pi \\int _ { \\ell _ { 2 } } ^ { \\ell _ { 2 } + \\delta _ { 2 } } \\sin ( n _ { 1 } x _ { 1 } ) \\sin ( m _ { 1 } x _ { 1 } ) \\ , d x _ { 1 } \\cos ( n _ { 2 } x _ { 2 } ) \\cos ( m _ { 2 } x _ { 2 } ) \\ , d x _ { 2 } = 0 . \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{align*} \\begin{array} { c c c } { \\rm s } _ \\delta ( t ) : = \\begin{cases} \\sin ( \\sqrt { \\delta } t ) / \\sqrt { \\delta } & \\delta > 0 \\\\ t & \\delta = 0 \\\\ \\sinh ( \\sqrt { - \\delta } t ) / \\sqrt { - \\delta } & \\delta < 0 \\end{cases} & , & { \\rm c } _ \\delta ( t ) : = \\begin{cases} \\cos ( \\sqrt { \\delta } t ) & \\delta > 0 \\\\ 1 & \\delta = 0 \\\\ \\cosh ( \\sqrt { - \\delta } t ) & \\delta < 0 \\end{cases} \\end{array} ~ . \\end{align*}"}
-{"id": "5158.png", "formula": "\\begin{align*} \\left ( \\pi + 1 \\right ) ! = \\prod _ { i } \\left ( k _ { i } + 1 \\right ) ! . \\end{align*}"}
-{"id": "3106.png", "formula": "\\begin{align*} \\lambda _ p = \\xi _ 1 ( p ) + \\xi _ 2 ( p ) \\quad p . \\end{align*}"}
-{"id": "6582.png", "formula": "\\begin{align*} \\sigma ( H _ 0 ) \\ ; = \\ ; \\sigma _ { \\mathrm { a c } } ( H _ 0 ) \\ ; = \\ ; ( - \\infty , - m c ^ 2 ] \\cup [ m c ^ 2 , + \\infty ) \\ , . \\end{align*}"}
-{"id": "8943.png", "formula": "\\begin{align*} \\frac { \\overline { \\beta } _ { i _ n , i _ n + 1 } ( S / I _ \\Delta ) } { C \\binom { n } { i _ n } } \\longrightarrow 1 \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} Z _ { j } & : = Z _ { j , j } = f _ { j , j } , \\\\ \\chi _ { j , l } & : = \\frac { f _ { j , l } } { i \\omega \\cdot ( j - l ) } , \\ ; \\ ; \\ ; j \\neq l , \\ ; \\ ; | \\omega \\cdot ( j - l ) | \\geq \\frac { \\gamma } { N ^ \\tau } . \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\int _ { 0 } ^ { t } \\frac { z ( \\tau ) } { ( t - \\tau ) ^ \\alpha } d \\tau \\Bigr | _ { t = 0 } = z ^ 0 , \\end{align*}"}
-{"id": "2312.png", "formula": "\\begin{align*} s ' & = r , \\\\ r ' & = \\frac { 2 ( - 1 + 2 u ) r ^ 2 } { - 1 - s + 2 u s } . \\end{align*}"}
-{"id": "6088.png", "formula": "\\begin{align*} 0 \\neq [ L _ 1 , \\ , L _ { - 5 } ] = [ L _ 1 , \\ , [ L _ { - 2 } , \\ , [ L _ { - 2 } , \\ , [ L _ { - 2 } , \\ , [ L _ { - 2 } , \\ , [ L _ { - 2 } , \\ , L _ 5 ] ] ] \\subseteq [ L _ { - 2 } , \\ , [ L _ { - 2 } , \\ , L _ 0 ] ] , \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} c _ { 0 } ( \\lambda _ { i } ) = e ^ { c _ { 0 } ( \\mu _ { i } ) } = e ^ { c _ { 0 } ( \\mu _ { i } ) ^ { \\uparrow = } } e ^ { c _ { 0 } ( \\mu _ { i } ) ^ { \\downarrow } } . \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} \\partial _ t w _ k ^ 1 - \\Delta w _ k ^ 1 + \\nabla p _ k ^ 1 = - h u _ \\infty \\cdot \\nabla w _ k + f , w _ k ^ 1 ( \\cdot , 0 ) = w _ 0 , \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} \\rho _ { t } ^ { ( \\beta , \\omega , \\vartheta , \\lambda , \\mathbf { A } ) } : = \\left \\{ \\begin{array} { l l l } \\varrho ^ { ( \\beta , \\omega , \\vartheta , \\lambda ) } & , & t \\leq t _ { 0 } \\ , \\\\ \\varrho ^ { ( \\beta , \\omega , \\vartheta , \\lambda ) } \\circ \\tau _ { t , t _ { 0 } } ^ { ( \\omega , \\vartheta , \\lambda , \\mathbf { A } ) } & , & t \\geq t _ { 0 } \\ , \\end{array} \\right . \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} W _ { n + 1 } ( x ) = \\sum _ { k = 0 } ^ n \\int _ { 0 ^ - } ^ { K - 0 } \\left \\lbrace G ( a _ k ^ x ( w ) ) - G ( b _ k ( w ) ) \\right \\rbrace d W _ { n - k } ( w ) . \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} i \\partial _ { t } P _ { \\varepsilon } P _ { + } w _ { n } + \\Delta P _ { \\varepsilon } P _ { + } w _ { n } + A P _ { \\varepsilon } P _ { + } w _ { n } = P _ { \\varepsilon } P _ { + } \\left ( V w _ { n } \\right ) + P _ { \\varepsilon } P _ { + } \\left ( h w _ { n } \\right ) + \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} O ( p ) = \\overline { \\bigcup _ { n \\in \\mathbb N } f ^ n ( p ) } \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{align*} \\omega _ { \\varepsilon } : = \\left \\{ x \\in \\Omega : { \\rm d i s t } ( x , \\partial \\Omega ) < \\varepsilon \\right \\} \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} H F ( M ; \\mathcal P _ M ) = \\mathcal C ( H F _ { ( M _ 1 , \\mathcal P _ { M _ 1 } ) } , H F _ { ( M _ 2 , \\mathcal P _ { M _ 2 } ) } ) . \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{align*} A ( n + 1 ) = { } & \\frac { 1 } { 2 ^ { 4 ( n - 1 ) } } \\sum _ { m = 1 } ^ { n } \\sum _ { \\ell = 1 } ^ { m } \\sum _ { k = 1 } ^ { \\ell } \\frac { [ ( 2 n - 2 m ) ! ] ^ 3 } { [ ( n - m ) ! ] ^ 4 } \\frac { [ ( 2 m - 2 \\ell ) ! ] ^ 3 } { [ ( m - \\ell ) ! ] ^ 4 } \\frac { [ ( 2 \\ell - 2 k ) ! ] ^ 3 } { [ ( \\ell - k ) ! ] ^ 4 } \\frac { [ ( 2 k - 2 ) ! ] ^ 3 } { [ ( k - 1 ) ! ] ^ 4 } . \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{align*} \\mathcal { N } _ { [ \\ ] } ( 2 \\delta B \\gamma ^ \\beta , \\gamma ) = \\sup \\limits _ { g \\in \\mathcal { G } } \\mathcal { N } _ { [ \\ ] } ( \\mathcal G \\cap B _ { P } ( g , 2 \\delta B \\gamma ^ \\beta ) , \\gamma ) . \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} ( R ^ { - 1 } ) ^ { a \\ , b } _ { a \\ , b } = 1 , \\ ; a \\neq b , ( R ^ { - 1 } ) ^ { a \\ , a } _ { a \\ , a } = q _ a ^ { - 1 } , ( R ^ { - 1 } ) ^ { a \\ , b } _ { b \\ , a } = - ( q _ b - q _ b ^ { - 1 } ) , \\ ; a < b . \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} s _ { i - j } = \\sum _ { k = 1 } ^ m a _ k \\cdot s _ { k - j } . \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} \\left ( \\bigcap _ \\alpha I _ \\alpha \\right ) L = \\bigcap _ \\alpha I _ \\alpha L . \\end{align*}"}
-{"id": "8811.png", "formula": "\\begin{align*} \\tilde { g } ^ { \\xi } = f ^ { \\ast } ( g ^ { \\tilde { \\xi } } ) . \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } C ^ { \\lambda } _ m ( t ) w ^ m = ( 1 - 2 t w + w ^ 2 ) ^ { - \\lambda } , \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{align*} \\tilde \\rho _ { \\varepsilon } : = 1 + \\varepsilon ^ { - 2 m + \\delta } \\chi _ { B ( 0 , \\varepsilon ) } . \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{align*} h _ \\alpha ( \\xi ) = \\frac { \\left \\vert \\xi \\right \\vert } { \\sqrt { 1 + a _ \\alpha \\left \\vert \\xi \\right \\vert ^ \\alpha } } \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} d ^ { u _ i } & = \\frac { N } { N - 2 B } \\alpha _ i - \\alpha _ i + 1 \\\\ & = \\frac { 2 B } { N - 2 B } \\alpha _ i + 1 \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{align*} \\frac { d y } { d t } ( t ) ( y ( t ) ) ^ { - 1 } & = \\left ( \\frac { r } { 2 } t ^ { r / 2 - 1 } x ( \\log t ) + t ^ { r / 2 } \\frac { d x } { d t } ( \\log t ) t ^ { - 1 } \\right ) ( x ( \\log t ) ) ^ { - 1 } t ^ { - r / 2 } \\\\ & = t ^ { - 1 } \\left ( \\frac { r } { 2 } + \\frac { d x } { d t } ( \\log t ) ( x ( \\log t ) ) ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "9676.png", "formula": "\\begin{align*} \\Theta ( 0 , \\phi , \\pi ) = \\Theta ( 0 , \\pi - \\phi , 0 ) . \\end{align*}"}
-{"id": "9691.png", "formula": "\\begin{align*} h _ 1 ^ { r } ( P ) + h _ 2 ^ { r } ( P ) + \\ldots + h _ { d + r } ^ { r } ( P ) \\ = \\ ( d + r ) ! \\int _ P x ^ r \\mathrm { d } x \\ , . \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} g ( x ) = \\alpha x ^ { \\lambda } + \\beta + \\int _ 0 ^ { \\infty } \\int _ 0 ^ { x t } e ^ { - u } u ^ { \\lambda - 1 } \\ , d u \\frac { d \\mu ( t ) } { t ^ { \\lambda } } . \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} g _ n ( z ) = d ^ n g _ 0 \\left ( \\frac { z } { \\lambda _ { p , - n } } \\right ) . \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{align*} f \\left ( z \\right ) = \\frac { 2 z + 1 } { \\left ( z ^ { 2 } + 6 z + 1 0 \\right ) ^ { 3 } } . \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} F _ { 1 2 } & = f _ 1 ( x _ 1 , x _ 2 ) y _ 1 ^ 2 + f _ 2 ( x _ 1 , x _ 2 ) y _ 1 y _ 2 + f _ 3 ( x _ 1 , x _ 2 ) y _ 2 ^ 2 \\\\ F _ { 1 3 } & = g _ 1 ( x _ 1 , x _ 2 ) z _ 1 ^ 2 + g _ 2 ( x _ 1 , x _ 2 ) z _ 1 z _ 2 + g _ 3 ( x _ 1 , x _ 2 ) z _ 2 ^ 2 \\end{align*}"}
-{"id": "7844.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\{ { \\cal E } _ 1 = \\{ 1 , 2 , \\ldots , r \\} \\} \\cap \\{ { \\cal T } \\subseteq { \\cal E } _ 1 \\} \\right ) \\leq p _ u ^ { r - 1 } ( 1 - p _ d ) ^ { r ( n - r ) } . \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} \\prod _ { n = R } ^ { \\infty } \\left ( 1 - \\frac { R ^ 2 t ^ 2 } { n ^ 2 } \\right ) ^ 2 \\leq e ^ { - 2 R V ( t ) } \\leq \\prod _ { n = R + 1 } ^ { \\infty } \\left ( 1 - \\frac { R ^ 2 t ^ 2 } { n ^ 2 } \\right ) ^ 2 , t \\in [ - 1 , 1 ] . \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { \\nu _ 2 \\cdots \\nu _ r \\leq \\tau ^ { - 1 / ( 2 m ) } } \\tau ^ { - 1 / ( 2 m ) } / ( \\nu _ 2 \\cdots \\nu _ r ) & = \\tau ^ { - 1 / ( 2 m ) } \\left ( \\sum _ { \\nu \\leq \\tau ^ { - 1 / ( 2 m ) } } 1 / \\nu \\right ) ^ { r - 1 } \\\\ & \\asymp \\tau ^ { - 1 / ( 2 m ) } ( \\log 1 / \\tau ) ^ { r - 1 } . \\end{aligned} \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} \\Gamma ^ 2 = n \\delta ^ 2 = - 4 n , Z ^ 2 = n E ^ 2 = - n , \\Gamma Z = n ( \\delta E ) = 6 n . \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} f _ { F , a } ^ \\lambda ( x ) : = \\bigvee _ { y \\in F } \\bigl [ a ( y ) ( 1 - \\mbox { $ \\frac { 1 } { \\lambda } $ } d ( x , y ) ) ^ + \\bigr ] . \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{align*} \\left | \\sum _ { y \\in B _ { R ' } ( x ) } e ( \\xi ^ i _ y + \\xi ^ e _ y ) \\right | = \\left | \\sum _ { y \\in B _ { R ' } ( x ) } e ( \\xi ^ i _ y ) \\right | + O \\left ( \\frac { 1 } { R _ 1 } \\right ) . \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} \\eta ( 1 ) = 2 , \\eta ( 2 ) = 1 , \\eta ( 3 ) = 7 , \\eta ( 4 ) = 6 , \\eta ( 5 ) = 1 3 , \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} A _ { 2 } f f _ { c _ { 2 } } - f ^ { \\prime } f _ { c _ { 2 } } - f f _ { c _ { 2 } } ^ { \\prime } = B _ { 2 } , \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} \\frac { \\operatorname * { d i a m } _ i B _ { \\sigma ' } } { \\operatorname * { d i a m } _ i B _ { \\sigma } } \\leq \\frac { \\operatorname * { d i a m } _ i B ^ { o c t } _ { \\sigma ' } + C h } { \\operatorname * { d i a m } _ i B ^ { o c t } _ { \\sigma } } = \\frac { 1 } { 2 } + C \\frac { h } { \\operatorname * { d i a m } _ i B ^ { o c t } _ { \\sigma } } . \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} \\| e ^ { t \\Delta } \\bar v _ 0 \\| _ { r , \\mathbb R ^ 3 } \\leq C t ^ { - 1 / 2 + 3 / 2 r } \\| e ^ { \\frac { t } { 2 } \\Delta } \\bar v _ 0 \\| _ { 3 , \\infty , \\mathbb R ^ 3 } = o ( t ^ { - 1 / 2 + 3 / 2 r } ) \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} B ' _ k : = \\frac { 1 } { | C | } \\sum _ { i = 0 } ^ n q _ k ( i ) B _ i \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} x P _ n ( x ) = \\beta ^ 0 _ { n , 1 } P _ { n - 1 } ( x ) + \\displaystyle \\sum _ { n \\leq s \\leq n + 1 } \\beta ^ s _ { n , 1 } P _ { s } ( x ) . \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} { \\displaystyle \\left | \\int _ { \\Omega } g _ j ( \\frac { \\displaystyle \\mathbf { x } } { \\displaystyle \\varepsilon } , \\mathbf { x } , t ) v \\mathnormal { d } \\mathbf { x } \\right | \\leq C \\varepsilon \\| v \\| _ { \\mathbb { H } ^ 1 _ 0 ( \\Omega ) } , \\forall v \\in \\mathbb { H } ^ 1 _ 0 ( \\Omega ) , j = 1 , 2 , } \\end{align*}"}
-{"id": "9731.png", "formula": "\\begin{align*} \\lim _ m \\| y _ m \\| _ \\infty = \\max \\left \\{ a \\| x \\| _ \\infty , \\ \\lim _ m \\| y _ m - a x \\| _ \\infty \\right \\} , \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{gather*} U _ i \\cap U _ j ^ \\prime = 0 ( 0 \\le i , j \\le d , i + j < d ) \\end{gather*}"}
-{"id": "7316.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\alpha _ 1 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 1 ] = \\theta _ 3 y _ 4 + \\theta _ 4 y _ 5 , [ y _ 2 , y _ 2 ] = \\theta _ 5 y _ 4 + \\theta _ 6 y _ 5 , [ y _ 1 , y _ 3 ] = y _ 5 , \\\\ [ y _ 2 , y _ 3 ] = \\frac { \\gamma _ 1 } { \\gamma _ 3 } y _ 4 + \\theta _ 7 y _ 5 , [ y _ 3 , y _ 2 ] = y _ 4 . \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} \\mathbb { S } ^ \\rho ( f _ 1 , f _ 2 , f _ 3 ) & : = \\sum _ { Q \\in \\mathcal { D } } S _ Q ( f _ 1 , f _ 2 , f _ 3 ) \\\\ & : = \\sum _ { Q \\in \\mathcal { D } } \\iiint _ { Q \\times Q \\times Q } K _ Q ( x _ 1 , x _ 2 , x _ 3 ) \\prod _ { j = 1 } ^ 3 f _ j ( x _ j ) \\ , d x _ 1 \\ , d x _ 2 \\ , d x _ 3 , \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} \\widehat { S _ 0 } ( \\textbf { m } ) = q ^ { - 1 } \\delta _ 0 ( \\textbf { m } ) + q ^ { - d - 1 } \\ , G ^ d \\sum _ { \\ell \\in { \\mathbb F } _ q ^ * } \\chi ( \\| \\textbf { m } \\| \\ell ) \\mbox { f o r } ~ ~ \\textbf { m } \\in \\mathbb F _ q ^ d . \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} \\frac { \\bar { q } _ 0 ^ { ( i ) } } { G ^ { ( i ) } } = \\sum \\limits _ { k > 0 : \\bold { z } _ k \\in \\mathcal { N } _ i } { \\mathit { l } \\left ( { \\Vert \\bold { z } _ k \\Vert } _ 2 \\right ) } g _ k P _ k , \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left ( S ( A | M ) _ { ( \\mathcal { N } ( t ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) } - n \\ln t - n \\right ) = 0 \\ ; . \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} t _ { 2 } ( N ) = - t _ { 2 } ( N ' ) & \\Longleftrightarrow t _ { 2 } ( N _ { 1 } ) + t _ { 2 } ( N _ { 1 } - 2 ) = - 2 t _ { 2 } ( N _ { 1 } - 1 ) \\\\ & \\Longleftrightarrow - 2 t _ { 2 } \\left ( \\frac { N _ { 1 } - 1 } { 2 } \\right ) - 2 t _ { 2 } \\left ( \\frac { N _ { 1 } - 3 } { 2 } \\right ) = - 2 t _ { 2 } ( N _ { 1 } - 1 ) \\\\ & \\Longleftrightarrow t _ { 2 } ( N _ { 1 } - 1 ) = t _ { 2 } \\left ( \\frac { N _ { 1 } - 1 } { 2 } \\right ) + t _ { 2 } \\left ( \\frac { N _ { 1 } - 3 } { 2 } \\right ) . \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} V ^ \\infty = \\Psi ( v ^ - ) = \\Psi ( v ^ + ) . \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} p ( \\sqrt { r ^ 2 + h ^ 2 } ) = \\varphi ( h ) \\ , p \\Big ( { r \\over \\psi ( h ) } \\Big ) \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} \\frac { a _ { n - 1 } } { k ( n - 2 ) + 1 } \\leq \\frac { k ( n - 2 ) } { k ( n - 2 ) + 1 } < \\frac { k ( n - 1 ) } { k ( n - 1 ) + 1 } = \\frac { a _ n - 1 } { k ( n - 1 ) + 1 } \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} a \\left ( j \\right ) u _ { y y } \\left ( x , j , t \\right ) + b \\left ( j \\right ) u _ { y } \\left ( x , j , t \\right ) = 0 j = 0 , 1 x \\in R ^ { n } , t \\in \\left ( 0 , 1 \\right ) . \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{align*} G : B \\to B , 1 = P + \\Delta G = P + G \\Delta , P G = G P = 0 \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} \\{ y ^ { ( 2 ) } , \\ , y ^ { ( j ) } \\} = ( j + 1 ) ^ 2 x ^ { ( 2 ) } y ^ { ( j + 1 ) } \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} ( g - \\sigma \\partial ^ 2 _ X ) ^ { - 1 } f _ 2 & = { 1 \\over 2 \\sqrt { \\sigma g } } e ^ { - \\sqrt { g / \\sigma } | \\cdot | } \\ast f _ 2 = : G _ 2 * f _ 2 , \\\\ \\widehat { G } _ 1 ( k ) & = { | c | | k | \\over ( \\sigma k ^ 2 - | c | | k | + g ) ( \\sigma k ^ 2 + g ) } . \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} R e s _ z \\Big ( \\beta _ \\chi ( z ) \\otimes \\gamma _ \\chi ( z ) - \\gamma _ \\chi ( z ) \\otimes \\beta _ \\chi ( z ) \\Big ) = 0 . \\end{align*}"}
-{"id": "9493.png", "formula": "\\begin{align*} C ' ( \\tau _ { r } ^ { - 1 } ( \\ell ) , \\tau _ { f } ^ { - 1 } ( k ) ) : = \\binom { k } { \\ell } \\binom { f - i - k } { r - i - \\ell } . \\end{align*}"}
-{"id": "351.png", "formula": "\\begin{align*} ( Y \\triangleright c _ { f , v } ^ { \\Lambda } ) ( X ) & = c _ { f , v } ^ { \\Lambda } ( X Y ) = f ( X \\triangleright Y \\triangleright v ) = c _ { f , Y \\triangleright v } ^ { \\Lambda } ( X ) , \\\\ ( c _ { f , v } ^ { \\Lambda } \\triangleleft Y ) ( X ) & = c _ { f , v } ^ { \\Lambda } ( Y X ) = f ( Y \\triangleright X \\triangleright v ) = c _ { f \\triangleleft Y , v } ^ { \\Lambda } ( X ) . \\end{align*}"}
-{"id": "8143.png", "formula": "\\begin{align*} p = A _ { 1 \\mu _ 2 } \\lambda \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} \\mathrm { P } : = \\{ S \\in \\mathbf { P } \\ | \\ S \\ \\} . \\end{align*}"}
-{"id": "10041.png", "formula": "\\begin{align*} I d = J _ { \\varphi } ^ + + J _ { \\varphi } ^ - , J _ { \\varphi } = J _ { \\varphi } ^ + - J _ { \\varphi } ^ - , \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} D l _ n ( f ) g & = \\langle D l _ n ( f ) , g \\rangle _ R , D l _ \\infty ( f ) g = \\langle D l _ \\infty ( f ) , g \\rangle _ R , \\\\ D ^ 2 l _ n ( f ) g h & = \\langle D ^ 2 l _ n ( f ) g , h \\rangle _ R , D ^ 2 l _ \\infty ( f ) g h = \\langle D ^ 2 l _ \\infty ( f ) g , h \\rangle _ R . \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} ( \\nabla ^ \\perp \\cdot G ) _ i = G _ { l 2 } \\nabla _ l G _ { i 1 } - G _ { l 1 } \\nabla _ l G _ { i 2 } . \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{align*} { \\mathcal H } _ { \\tau } = { \\mathcal P } _ { \\tau } ^ { \\perp } \\end{align*}"}
-{"id": "3089.png", "formula": "\\begin{align*} f ( q ) = f ( 1 ) + O \\ ( | \\tau | \\ ) , \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 0 } { x } + \\frac { A _ 1 } { x - 1 } + A _ \\infty \\right ) Y , \\frac { \\partial Y } { \\partial t } = \\left ( \\frac { 1 } { t } A _ \\infty x + B _ 0 \\right ) Y . \\end{gather*}"}
-{"id": "7811.png", "formula": "\\begin{align*} \\| u _ { 0 n } \\| _ { L _ 2 } = 1 \\forall \\ n , \\lim \\limits _ { n \\to + \\infty } \\bigl \\| \\partial _ x ( P u _ { 0 n } ) \\big | _ { x = 0 } \\bigr \\| _ { L _ 2 ( B _ T ) } = 0 . \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} & \\langle \\Phi ( z _ 1 , \\dots , z _ N , \\{ \\overline { \\alpha } \\} ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle \\\\ = & t ^ { N ( M - N ) } \\prod _ { j = 1 } ^ N z _ j ^ { j - 1 - N } ( 1 + t z _ j ^ 2 ) \\prod _ { 1 \\le j < k \\le N } ( 1 + t z _ j z _ k ) ( 1 + t z _ j z _ k ^ { - 1 } ) s p _ { \\overline { \\lambda } } ( \\{ t z \\} _ N | \\{ - \\overline { \\alpha } \\} ) . \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} \\| T ^ { n } x - x \\| = \\Bigl | \\Bigl | \\ , \\sum _ { k = 1 } ^ { r } a _ { k } \\bigl ( \\lambda _ { k } ^ { n } - 1 \\bigr ) u _ { k } \\ , \\Bigr | \\Bigr | < \\dfrac { \\delta } { 2 } , \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} \\alpha _ { n + 1 } ( P ) = \\frac { \\max _ w \\alpha _ n ( P _ w ) + \\min _ b \\alpha _ n ( P _ b ) } 2 , \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} \\sum _ { i \\in G _ 0 } m _ i v _ i = 0 . \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\infty } [ c _ i ] \\circ ( b _ 1 ) ^ { i + 2 n } u ^ { 2 ( i + n ) } . \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} \\vert T _ { F , G , N } ^ L ( f _ 1 - 1 , 1 , \\dots , 1 ) \\vert & = \\vert T _ { F , G , N } ^ L ( f _ 1 , 1 , \\dots , 1 ) - T _ { F , G , N } ^ L ( 1 , 1 , \\dots , 1 ) \\vert \\\\ & = \\vert 0 - T _ { F , G , N } ^ L ( 1 , 1 , \\dots , 1 ) \\vert \\gg _ { c , C , \\varepsilon } 1 \\end{align*}"}
-{"id": "9767.png", "formula": "\\begin{align*} j \\tilde { u } = j u - | u | ^ 2 j \\phi \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} b \\colon Y \\ni ( y _ 1 , y _ 2 ) \\mapsto \\begin{pmatrix} \\phi ( y _ 2 ) & 0 \\\\ 0 & \\psi ( y _ 1 ) \\end{pmatrix} . \\end{align*}"}
-{"id": "7080.png", "formula": "\\begin{align*} \\tilde { F } _ { \\varepsilon } \\left ( x , t \\right ) = \\frac { i } { \\varepsilon + i } e ^ { \\varepsilon t \\left ( \\Delta + A \\right ) } \\tilde { V } \\left ( x , t \\right ) \\tilde { u } \\left ( x , t \\right ) , \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{align*} \\lambda _ 1 = \\frac { 1 } { \\alpha } , \\lambda _ 2 = \\frac { 1 } { \\alpha } - \\frac { N - 2 } { 2 } . \\end{align*}"}
-{"id": "4162.png", "formula": "\\begin{align*} P _ { \\lambda } ^ { \\omega } ( C _ t \\neq \\varnothing ) = P _ { \\lambda } ^ { \\omega } ( \\eta _ t ( O ) = 1 ) , \\end{align*}"}
-{"id": "8004.png", "formula": "\\begin{align*} & \\limsup _ { k \\to \\infty } \\left | \\int _ 0 ^ t \\langle ( h u _ s + \\widetilde U ) \\otimes ( w _ k - w ) , \\nabla w _ k \\rangle d \\tau \\right | \\\\ & \\leq ( C Y ( T ) + 1 ) \\ , \\varepsilon + \\lim _ { k \\to \\infty } \\int _ 0 ^ T \\| h u _ s + \\widetilde U _ \\varepsilon \\| _ \\infty \\| \\chi _ { B _ L } ( w _ k - w ) \\| _ 2 \\| \\nabla w _ k \\| _ 2 d \\tau \\\\ & = ( C Y ( T ) + 1 ) \\ , \\varepsilon . \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} \\frac { 2 } { \\pi } \\int _ { - 1 } ^ { 1 } \\frac { ( 1 + \\rho ^ { 2 } ) ^ { 3 } + 1 6 \\rho ^ { 3 } x y z - 4 \\rho ^ { 2 } ( 1 + \\rho ^ { 2 } ) ( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } ) } { w _ { 3 } ( x , y , z | \\rho ) } \\sqrt { 1 - z ^ { 2 } } d z = 1 , \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{align*} \\dot x = \\Gamma v ( x ) + \\epsilon I _ n ( x _ 0 - x ) . \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{align*} & B _ 1 ( u , \\tilde u ) = \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\P \\nabla \\cdot ( u \\otimes \\tilde u ) ( s ) \\dd s , \\\\ & B _ 2 ( u , \\theta ) = \\biggl ( \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } ( t - s ) \\P \\nabla \\cdot ( u \\theta ) ( s ) \\dd s \\biggr ) e _ 3 , \\\\ & B _ 3 ( u , \\theta ) = \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\nabla \\cdot ( u \\theta ) ( s ) \\dd s . \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} \\theta _ { C , D } ( x ) = \\begin{cases} \\frac { | x | ^ 2 } { 2 C } & | x | \\le C D , \\\\ D | x | - \\frac { C D ^ 2 } { 2 } & | x | > C D . \\end{cases} \\end{align*}"}
-{"id": "8549.png", "formula": "\\begin{align*} \\bar { h ( \\zeta , \\kappa ) } = h ( \\zeta , \\bar { \\kappa } ) \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} \\Phi ( X ) = \\phi ( \\phi ^ { - 1 } ( Y ) ) = Y \\cap \\R ^ d _ { \\ge 0 } , \\end{align*}"}
-{"id": "9789.png", "formula": "\\begin{align*} A _ k ( M , c ^ 2 g ) = A _ k ( M , g ) c > 0 . \\end{align*}"}
-{"id": "7606.png", "formula": "\\begin{align*} v _ \\epsilon ( \\rho , t ) & = v _ { 0 , \\epsilon } ( e ^ \\rho , t ) + a _ t \\rho \\\\ & = v _ { \\infty , \\epsilon } ( e ^ { - \\rho } , t ) + b _ t \\rho \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} K _ i ( m + s + 1 ) = K _ i ( ( m + s ) + 1 ) = K _ i ( m + s ) + K _ { i - 1 } ( m + s ) . \\end{align*}"}
-{"id": "8314.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\int _ { \\Omega } \\Psi ( f _ n ( x ) ) \\varphi ( x ) d x = \\int _ { \\Omega } \\left ( \\int _ { \\R } \\Psi d \\nu _ x \\right ) \\varphi ( x ) d x = \\int _ { \\Omega } \\overline { \\Psi } ( x ) \\varphi ( x ) d x \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 1 e _ 3 + \\alpha _ 2 e _ 4 + \\alpha _ 3 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 4 e _ 3 + \\alpha _ 5 e _ 4 + \\alpha _ 6 e _ 5 , [ e _ 2 , e _ 1 ] = \\beta _ 1 e _ 3 + \\beta _ 2 e _ 4 + \\beta _ 3 e _ 5 , \\\\ [ e _ 2 , e _ 2 ] = \\beta _ 4 e _ 3 + \\beta _ 5 e _ 4 + \\beta _ 6 e _ 5 , [ e _ 1 , e _ 3 ] = \\gamma _ 1 e _ 5 , [ e _ 2 , e _ 3 ] = \\gamma _ 2 e _ 5 . \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} \\left \\langle D , I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle \\left \\langle D ' , I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle \\end{align*}"}
-{"id": "9347.png", "formula": "\\begin{align*} \\| \\log ( \\gamma _ M ^ { - 1 } \\cdot \\gamma _ N ) - \\log ( \\gamma _ M ^ { - 1 } \\cdot \\gamma _ { L - 1 } ) - \\log ( \\gamma _ { L - 1 } ^ { - 1 } \\cdot \\gamma _ L ) - \\log ( \\gamma _ L ^ { - 1 } \\cdot \\gamma _ N ) \\| _ { } = O ( | F _ L | ^ 2 ) = O ( A ^ 2 ) . \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} \\langle \\epsilon E , \\phi \\rangle = \\langle f , \\phi \\rangle ( \\phi \\in \\textnormal { n } ( \\theta ) ) , \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} \\psi ( f ) = \\sum _ { p = 0 } ^ { \\infty } f ( p ) t ^ { p } . \\end{align*}"}
-{"id": "6829.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\frac { b _ { 2 k - 1 } ( \\mu ) } { ( 2 k ) ! } \\int _ T C _ Y ^ { 2 k } ( t ) \\ , d t < + \\infty . \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} X \\left ( z \\right ) = \\frac { g \\left ( z \\right ) } { 1 - g \\left ( z \\right ) } = \\frac { 1 - \\sqrt { 1 - 4 z } } { 1 + \\sqrt { 1 - 4 z } } = - 1 + \\frac { 1 - \\sqrt { 1 - 4 z } } { 2 z } = \\sum _ { n \\ge 1 } C _ { n } z ^ { n } , \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} \\tau _ { k , i } = \\tau _ k ( p _ i ) = \\sum _ { j = 1 } ^ \\infty \\frac { \\ell _ k ( p _ i ^ j ) } { p _ i ^ j } = \\sum _ { j = 1 } ^ \\infty \\frac { ( j + 1 ) ^ k - j ^ k } { p _ i ^ k } = \\left [ \\left ( \\frac { 1 } { x } - 1 \\right ) \\left ( x \\frac { \\partial } { \\partial x } \\right ) ^ k \\frac { 1 } { 1 - x } - 1 \\right ] _ { x = \\frac { 1 } { p _ i } } . \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\limsup _ { t \\to T ^ - } \\mathrm { d i a m } ( W _ \\delta , d _ { \\omega ( t ) } ) = 0 , \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} \\Delta \\phi _ g & = \\cos ^ { - 1 } \\left ( \\frac { ( \\mathbf { p } _ g - \\mathbf { p } _ { d } ) \\cdot \\mathbf { \\dot { p } } _ { d } } { | | \\mathbf { p } _ g - \\mathbf { p } _ { d } | | ~ | | \\mathbf { \\dot { p } } _ { d } | | } \\right ) \\\\ \\mathbf { z } _ { \\phi , g } & = \\mathrm { s i g n } \\left ( \\left ( \\mathbf { \\dot { p } } _ { d } \\times ( \\mathbf { p } _ g - \\mathbf { p } _ { d } ) \\right ) \\cdot \\mathbf { z } _ I \\right ) \\mathbf { z } _ I \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} f \\otimes f & = \\sum _ { m , n = 0 } ^ \\infty a _ m t ^ m \\otimes a _ n t ^ n \\\\ & = \\sum _ { m , n = 0 } ^ \\infty ( a _ m \\otimes a _ n ) ( t ^ m * t ^ n ) \\\\ & = \\sum _ { n = 0 } ^ \\infty ( a _ n \\otimes a _ n ) t ^ n \\shortintertext { a n d } \\Delta f & = \\sum _ { n = 0 } ^ \\infty \\Delta a _ n t ^ n . \\end{align*}"}
-{"id": "1018.png", "formula": "\\begin{align*} & t ^ { d } \\sum _ { r = 0 } ^ \\infty h _ r ( a _ 1 , \\ldots , a _ d ) t ^ r = t \\sum _ { i = 1 } ^ d \\sum _ { k = 0 } ^ \\infty a _ i ^ k t ^ k \\prod ^ d _ { \\substack { j = 1 \\\\ j \\neq i } } \\frac { 1 } { a _ i - a _ j } \\\\ & \\sum _ { r = 0 } ^ \\infty h _ r ( a _ 1 , \\ldots , a _ d ) t ^ { r + d - 1 } = \\sum _ { k = 0 } ^ \\infty \\Big ( \\sum _ { i = 1 } ^ d a _ i ^ k \\prod ^ d _ { \\substack { j = 1 \\\\ j \\neq i } } \\frac { 1 } { a _ i - a _ j } \\Big ) t ^ k . \\end{align*}"}
-{"id": "1662.png", "formula": "\\begin{align*} r _ k : = \\sum _ { j = 1 } ^ m \\chi _ { \\gamma _ 0 \\cdot B _ k } \\cdot a _ j \\otimes \\sigma _ j = \\sum _ { j = 1 } ^ m \\chi _ { B _ k } \\cdot ( a _ j \\cdot \\gamma _ 0 ) \\otimes \\gamma _ 0 ^ { - 1 } \\cdot \\sigma _ j \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} A = A \\bigl ( \\Pi , \\underline { \\mu } \\bigr ) : = \\Bigl \\{ f \\in R \\ ; \\big | \\ ; l _ \\alpha ^ { 2 \\mu _ \\alpha + 1 } \\ ; \\mbox { d i v i d e s } \\ ; \\bigl ( f - s _ \\alpha ( f ) \\bigr ) \\ ; \\mbox { f o r a l l } \\ ; \\alpha \\in \\Pi \\Bigr \\} , \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} f _ j : M ^ 2 \\times \\big [ - r _ j ^ { - 4 } t _ j , r _ j ^ { - 4 } ( T - t _ j ) \\big ) \\rightarrow \\R ^ N , f _ j ( p , t ) = \\frac { 1 } { r _ j } \\big ( f ( p , t _ j + r _ j ^ 4 t ) - x _ j \\big ) . \\end{align*}"}
-{"id": "8927.png", "formula": "\\begin{align*} A u = \\frac { 1 } { \\pi i } \\int _ 0 ^ \\infty h ^ - ( \\lambda ) R _ 0 ^ - ( \\lambda ^ 2 ) v S _ 1 D _ 1 S _ 1 v \\big [ R _ 0 ^ + ( \\lambda ^ 2 ) - R _ 0 ^ - ( \\lambda ^ 2 ) \\big ] \\lambda \\chi ( \\lambda ) u \\ , d \\lambda \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{align*} \\tau _ { f , n } = \\left \\{ \\begin{array} { l l } \\frac { 2 c _ 3 } { a _ { m a x } } \\sqrt { S _ { t r a j } } , & t _ i \\geq t _ { o , n - 1 } + \\tau _ { f , n - 1 } \\\\ \\sqrt { \\frac { 2 c _ 3 \\sqrt { S _ { t r a j } } } { h _ { n - 1 } } } , & t _ i < t _ { o , n - 1 } + \\tau _ { f , n - 1 } \\end{array} \\right . \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} ( 1 + k \\phi _ t ) \\dot \\phi _ t = A \\phi ^ 2 _ t + B \\phi _ t + C , \\phi _ T = c \\end{align*}"}
-{"id": "7281.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = e _ 4 , [ e _ 1 , e _ 2 ] = \\alpha _ 1 e _ 3 + \\alpha _ 2 e _ 4 + \\alpha _ 3 e _ 5 , [ e _ 2 , e _ 1 ] = - \\alpha _ 1 e _ 3 + \\beta _ 1 e _ 4 + \\beta _ 2 e _ 5 , [ e _ 2 , e _ 2 ] = e _ 5 . \\end{align*}"}
-{"id": "6857.png", "formula": "\\begin{align*} \\int _ { E _ f ( \\lambda ) } \\Delta \\log \\left ( \\frac 1 { 1 - | z | ^ 2 } \\right ) \\frac { d x d y } { 4 \\pi } & = \\int _ { \\partial E _ f ( \\lambda ) } \\frac { \\partial } { \\partial n } \\log \\left ( \\frac 1 { 1 - | z | ^ 2 } \\right ) \\frac { d s } { 4 \\pi } \\\\ & = \\int _ { \\partial E _ f ( \\lambda ) } \\frac 1 { 1 - | z | ^ 2 } \\frac { \\partial } { \\partial n } | z | ^ 2 \\frac { d s } { 4 \\pi } , \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} ( \\pi ^ + , \\pi ^ - ) \\stackrel { d } { = } ( B ^ + , B ^ - ) , \\end{align*}"}
-{"id": "2102.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 2 } { x ^ 2 } + \\frac { A _ 1 } { x } + A _ 0 \\right ) Y , \\frac { \\partial Y } { \\partial t } = \\left ( \\frac { 1 } { t } A _ 0 x + B _ 0 \\right ) Y , \\end{gather*}"}
-{"id": "1072.png", "formula": "\\begin{align*} T ^ { L , \\Xi , \\widetilde { \\mathbf { r } } } _ { F , G , N } ( f _ 1 , \\dots , f _ { d } ) : = \\frac { 1 } { N ^ { h - m } } \\sum \\limits _ { \\mathbf { n } \\in \\mathbb { Z } ^ h } \\Big ( \\prod \\limits _ { j = 1 } ^ { d } f _ j ( \\xi _ j ( \\mathbf { n } ) + \\widetilde { \\mathbf { r } } _ j ) \\Big ) F ( \\mathbf { n } ) G ( L \\mathbf { n } ) . \\end{align*}"}
-{"id": "1613.png", "formula": "\\begin{align*} \\zeta = t _ 1 + i t _ 2 , \\ \\ \\bar \\zeta = t _ 1 - i t _ 2 \\quad \\mbox { i . e . } \\zeta = \\rho e ^ { i \\vartheta } . \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{align*} C ( \\mathsf { P } ) & : = \\mathrm { T r } \\left ( \\pi ( K _ { 2 \\rho } ^ { - 1 } ) ( 2 \\mathsf { P } - \\mathrm { I d } ) \\otimes \\mathsf { P } \\otimes \\mathsf { P } \\right ) , \\\\ C ( \\mathsf { Q } ) & : = \\mathrm { T r } \\left ( \\pi ( K _ { 2 \\rho } ) ( 2 \\mathsf { Q } - \\mathrm { I d } ) \\otimes \\mathsf { Q } \\otimes \\mathsf { Q } \\right ) . \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\frac { 1 } { y ( [ a ] Q _ { n _ k } ) } = 0 , \\end{align*}"}
-{"id": "9823.png", "formula": "\\begin{align*} T _ i \\mathbf { z } ^ \\rho = - \\mathbf { z } ^ \\rho . \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} G _ i = a _ { i , 1 } P _ { k _ 1 } + a _ { i , 2 } P _ { k _ 2 } + \\dots + a _ { i , m } P _ { k _ m } + a _ { i , m + 1 } f _ 1 + a _ { i , m + 2 } f _ 2 + \\dots + a _ { i , 2 m } f _ m , \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{align*} \\kappa _ { \\alpha } ( s _ 0 ) = \\frac { 1 } { R _ S ( s _ 0 ) } \\sqrt { 1 + J ^ 2 ( s _ 0 ) } \\ , , \\end{align*}"}
-{"id": "9880.png", "formula": "\\begin{align*} Q ( \\sigma ) = \\prod _ { i = 1 } ^ { \\ell ^ - } \\frac { 1 } { \\deg ( v _ i ) } \\prod _ { i = \\ell ^ - } ^ { \\ell - 1 } \\frac { 1 } { \\deg ( v _ i ) } . \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\partial \\Omega } [ \\chi _ { \\partial \\Omega } ] ( x ) = \\left \\{ \\begin{array} { l l } 0 & \\\\ - 1 & \\end{array} \\right . \\mathcal { K } _ { \\partial \\Omega } [ \\chi _ { \\partial \\Omega } ] = - \\tfrac 1 2 \\chi _ { \\partial \\Omega } \\ , . \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} b _ 2 ( 2 n - 1 ) ^ 2 - b _ 2 ( 2 n - 2 ) b _ 2 ( 2 n ) = \\left ( \\sum _ { j = 0 } ^ { n - 1 } b _ 2 ( j ) \\right ) ^ 2 - b _ 2 ( 2 n - 2 ) b _ 2 ( n ) . \\end{align*}"}
-{"id": "9528.png", "formula": "\\begin{align*} \\psi ( w ) = \\sum _ { i = 1 } ^ r \\sum _ { n , j \\leq J } & \\int _ { F _ { n , j } } \\langle C _ { n , j } ( \\omega ) ^ * X ( \\omega ) C _ { n , j } ( \\omega ) \\xi _ i ( \\omega ) , \\eta _ i ( \\omega ) \\rangle \\ , d \\mu ( \\omega ) \\\\ + \\sum _ { i = 1 } ^ r & \\int _ F \\langle ( 1 - z ) \\xi _ i ( \\omega ) , \\eta _ i ( \\omega ) \\rangle \\ , d \\mu ( \\omega ) \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} \\mathrm { c h } \\Big ( \\sum _ { q = 0 } ^ r ( - 1 ) ^ q { \\bigwedge } ^ { q } E ^ \\vee \\Big ) & = \\sum _ { q = 0 } ^ r ( - 1 ) ^ q \\mathrm { c h } \\Big ( { \\bigwedge } ^ q E ^ \\vee \\Big ) = \\prod _ { i = 1 } ^ r ( 1 - e ^ { - a _ { i } } ) \\\\ & = ( a _ 1 \\ldots a _ r ) \\prod _ { i = 1 } ^ r \\frac { ( 1 - e ^ { - a _ { i } } ) } { a _ i } = c _ r ( E ) \\mathrm { t d } ^ { - 1 } ( E ) . \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{align*} N : = \\Sigma / \\operatorname { i m } ( \\Delta ) . \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{gather*} Q _ V = \\sum _ { i = 0 } ^ { d } \\frac { ( - 1 ) ^ i \\mu _ i z ^ i } { ( [ i ] _ q ^ { ! } ) ^ 2 } , \\end{gather*}"}
-{"id": "1274.png", "formula": "\\begin{align*} \\rho ( t ) = \\sinh ( t ) \\exp \\left [ - t \\right ] . \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} | t _ m ( n ) | & = | t _ m ( 2 n ' + 1 ) | = \\left | \\sum _ { j = 0 } ^ { \\lfloor \\frac { m - 1 } { 2 } \\rfloor } { m \\choose 2 j + 1 } t _ m ( n ' - j ) \\right | \\leq \\sum _ { j = 0 } ^ { \\lfloor \\frac { m - 1 } { 2 } \\rfloor } { m \\choose 2 j + 1 } | t _ m ( n ' - j ) | \\\\ & < \\sum _ { j = 0 } ^ { \\lfloor \\frac { m - 1 } { 2 } \\rfloor } { m \\choose 2 j + 1 } m ( n ' ) ^ { \\frac { m } { 2 } } = m ( 2 n ' ) ^ { \\frac { m } { 2 } } < m n ^ { \\frac { m } { 2 } } . \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} { } _ A N _ B = \\bigoplus _ { ( f , h ) \\in E \\times F } { } _ f n _ h \\left ( A f \\otimes h B \\right ) \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } i \\partial _ t \\phi ( t , x ) + \\delta ^ 4 \\Delta ^ 2 \\phi ( t , x ) & = - \\mu | \\phi | ^ { \\nu - 1 } \\phi ( t , x ) , ( t , x ) \\in \\R \\times \\R ^ d , \\\\ \\phi ( 0 , x ) & = \\phi _ 0 ( x ) , x \\in \\R ^ d . \\end{array} \\right . \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{align*} \\begin{array} { r l } \\sqrt { n } \\hat \\Sigma _ j ( \\check \\beta _ j - \\beta _ { 0 j } ) & = \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ n \\psi _ j ( y _ i , z _ i ) + \\sqrt { n } ( \\hat \\Gamma _ { j j } - \\Gamma _ { j j } ) \\beta _ { 0 j } \\\\ & - ( \\mu ^ j _ 0 ) ^ T \\sqrt { n } ( \\hat \\Gamma _ { - j , - j } - \\Gamma _ { - j , - j } ) \\beta _ { 0 , - j } + O _ \\P ( \\delta _ n ) \\end{array} \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} \\int _ { \\R _ { 0 } ^ { d } } | y | ^ { 2 } \\nu ( x , \\d y ) = \\int _ { U } | c ( x , u ) | ^ { 2 } M ( \\d u ) . \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} \\beta ^ { ( k ) } _ { \\mu , 2 } ( x _ 0 , r ) = \\Big ( r ^ { - k - 2 } \\sum _ { l = k + 1 } ^ { n + 1 } \\lambda _ { l } \\Big ) ^ { \\frac 1 2 } . \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} \\big ( \\partial _ { \\tau } \\nabla { \\theta } ^ { k } _ { \\phi } , \\ , \\nabla u \\big ) = \\big ( \\partial _ { \\tau } \\nabla e _ { \\phi } ^ { k } , \\ , \\nabla u \\big ) + \\frac { 1 } { \\tau } \\big ( \\vert \\Psi ^ { k } _ { h } \\vert ^ { 2 } - \\vert \\Psi ^ { k } \\vert ^ { 2 } - \\vert \\Psi ^ { k - 1 } _ { h } \\vert ^ { 2 } + \\vert \\Psi ^ { k - 1 } \\vert ^ { 2 } , \\ , u \\big ) , \\ ; \\forall u \\in X _ { h } ^ { r } . \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} \\int _ 0 ^ b \\ ! Q ( x ) y ( x , \\lambda ) \\tilde { y } ( x , \\lambda ) d x + \\ ! M _ 5 y ( \\textstyle { 1 \\over 2 } - 0 , \\lambda ) \\tilde { y } ( \\textstyle { 1 \\over 2 } - 0 , \\lambda ) + M _ 6 = 0 , \\quad \\forall \\lambda \\ge 0 , \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{align*} ( - 1 ) ^ { s - 1 } a _ n = \\sum _ { k = n } ^ { \\infty } { k \\choose n } ( - 1 ) ^ k a _ k ; \\ ; \\ ; s = 1 \\ ; \\ ; \\mbox { o r } \\ ; \\ ; s = 2 . \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} \\begin{cases} b _ 2 = a _ 1 ^ 2 + 4 a _ 2 ; \\\\ b _ 4 = 2 a _ 4 + a _ 1 a _ 3 ; \\\\ b _ 6 = a _ 3 ^ 2 + 4 a _ 6 ; \\\\ b _ 8 = a _ 1 ^ 2 a _ 6 + 4 a _ 2 a _ 6 - a _ 1 a _ 3 a _ 4 + a _ 2 a _ 3 ^ 2 - a _ 4 ^ 2 . \\end{cases} \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{align*} w _ { n - k } + \\sigma _ { n - k } + v _ { N ( n - k + 1 ) } - ( j + 1 ) T = K , j = 0 , \\ldots , k - 1 . \\end{align*}"}
-{"id": "5240.png", "formula": "\\begin{align*} \\Delta \\Psi _ 1 = b f \\textup { ~ i n ~ } R , \\Psi _ 1 = 0 \\textup { ~ o n ~ } T , \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} u ( t , x , y ) = S ( t , x , y ; u _ 0 ) + K ( t , x , y ; f ) , \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{align*} x \\mapsto \\sum _ { \\chi \\in X ( T ) } a _ \\chi \\cdot \\chi ( x \\mu ^ { - 1 } ) = \\sum _ { \\chi \\in X ( T ) } a _ \\chi \\cdot \\chi ( \\mu ^ { - 1 } ) \\chi ( x ) . \\end{align*}"}
-{"id": "1328.png", "formula": "\\begin{align*} 2 H _ { 0 } \\left [ \\left ( f g ^ { \\prime } \\right ) ^ { 2 } - \\left ( f ^ { \\prime } g \\right ) ^ { 2 } \\right ] ^ { \\frac { 3 } { 2 } } = \\left ( f g ^ { \\prime } \\right ) ^ { 2 } f ^ { \\prime \\prime } g - 2 f g \\left ( f ^ { \\prime } g ^ { \\prime } \\right ) ^ { 2 } + \\left ( f ^ { \\prime } g \\right ) ^ { 2 } f g ^ { \\prime \\prime } . \\end{align*}"}
-{"id": "8437.png", "formula": "\\begin{align*} X ^ { ' f } _ t = - \\int _ { t } ^ T \\pi ' _ s d W _ s - \\int _ { t } ^ T \\int _ E l ' _ s ( e ) \\tilde N ( d s , d e ) - ( h ^ 2 _ T - h ^ 2 _ t ) + A ' _ T - A ' _ t + C ' _ { T - } - C ' _ { t - } . \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{align*} i e _ 1 = ( B _ n A _ n \\tau ) ( \\xi + r _ n \\vect { n } ( \\xi ) ) . \\end{align*}"}
-{"id": "749.png", "formula": "\\begin{align*} \\sqrt { a ^ 2 + b ^ 2 } - \\sqrt { a ^ 2 } = \\int _ { a ^ 2 } ^ { a ^ 2 + b ^ 2 } \\frac { d t } { 2 \\sqrt { t } } \\leq \\frac { b ^ 2 } { 2 a } . \\end{align*}"}
-{"id": "1845.png", "formula": "\\begin{align*} \\begin{aligned} R _ { X Y } Z + R _ { Z X } Y + R _ { Y Z } X = ( & \\nabla _ X T ) _ { Y Z } + ( \\nabla _ Z T ) _ { X Y } + ( \\nabla _ Y T ) _ { Z X } \\\\ - & T ( X , T _ { Y Z } ) - T ( Z , T _ { X Y } ) - T ( Y , T _ { Z X } ) . \\end{aligned} \\end{align*}"}
-{"id": "6617.png", "formula": "\\begin{align*} r ( \\varphi ^ - \\ ! ) '' + ( 1 - r ) ( \\varphi ^ - \\ ! ) ' - \\textstyle { \\frac { \\nu ^ 2 - 1 } { r } } \\ , \\varphi ^ - \\ ! \\ ; = \\ ; 0 \\ , , \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\log Q _ N ( a ) = - \\sup _ \\vartheta \\left ( \\vartheta a - \\int _ 0 ^ 1 \\Lambda _ X \\left ( ( { \\rm e } ^ \\vartheta - 1 ) \\overline F ( x ) \\right ) { \\rm d } x \\right ) . \\end{align*}"}
-{"id": "4417.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } \\binom { 2 k } { k } \\frac { x ^ { k } } { k ^ 3 } \\equiv 4 \\pounds _ 3 ( \\alpha ) + 4 \\pounds _ 3 ( \\beta ) + 2 x ^ p ( \\pounds _ 3 ( 1 - 1 / \\alpha ) + \\pounds _ 3 ( 1 - 1 / \\beta ) ) \\pmod { p } . \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} { S } ( x ) = \\prod _ { k = 1 } ^ { n - 1 } ( x - { t } _ k ) . \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} \\tilde { f } _ { n } ( 1 , 1 ) = c _ 0 ( - i ) ^ { n + 1 } . \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow \\partial \\Omega } s _ \\Omega ( z ) = 1 , \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} X _ { l } = \\Bigl \\| \\sum _ { k = b _ { l } } ^ { b _ { l + 1 } - 1 } \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Bigr ) \\ , x _ { k } e _ { k } \\Bigr \\| . \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} \\hat { T } ^ { ( k _ 1 ) } _ n \\leq T _ n \\leq \\sum _ { i = 1 } ^ { n } t ( f _ i ) \\end{align*}"}
-{"id": "6982.png", "formula": "\\begin{align*} D _ { n } & = \\big \\{ [ d _ { 1 } , d _ { 2 } , \\dots , d _ { n } ] \\big \\} \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( f ( Z _ { r } ) \\left \\vert \\mathcal { F } _ { s } ^ { + } \\vee \\mathcal { F } _ { t } ^ { - } \\right . \\right ) = \\mathbb { E } \\left ( f ( Z _ { r } ) \\left \\vert Z _ { s } , Z _ { t } \\right . \\right ) \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{align*} \\left [ \\Lambda _ N ^ { N + 1 } f \\right ] ( x _ 1 , \\cdots , x _ { N + 1 } ) = \\mathbb { E } _ { \\mathbb { U } ( N + 1 ) } \\left [ f \\left [ \\mathsf { e v a l } _ N \\left ( \\pi _ N ^ { N + 1 } \\left [ U ^ * \\textnormal { d i a g } \\left ( x _ 1 , \\cdots , x _ { N + 1 } \\right ) U \\right ] \\right ) \\right ] \\right ] . \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} w _ f ( t ) & = \\sup \\big \\{ d _ 2 \\big ( f ( x ) , f ( y ) \\big ) : x , y \\in M _ 1 , \\ , d _ 1 ( x , y ) \\le t \\big \\} \\\\ \\rho _ f ( t ) & = \\inf \\big \\{ d _ 2 \\big ( f ( x ) , f ( y ) \\big ) : x , y \\in M _ 1 , \\ , d _ 1 ( x , y ) \\ge t \\big \\} . \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{align*} s _ { \\lambda ^ * } = ( s _ \\lambda ^ * s _ \\lambda ) s _ { \\lambda ^ * } = s _ \\lambda ^ * c _ \\Lambda ( \\lambda , \\lambda ^ * ) s _ { \\lambda \\lambda ^ * } \\ = s _ \\lambda ^ * s _ { r ( \\lambda ) } = s _ \\lambda ^ * ( s _ \\lambda s _ \\lambda ^ * ) = s _ \\lambda ^ * \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} l ^ * = \\begin{cases} 0 , & \\lambda _ { 1 } \\leq \\gamma _ 2 c ^ 2 , \\\\ \\sqrt { 1 / \\sqrt { \\gamma \\lambda _ 1 } - c / \\lambda _ 1 } v _ { 1 } , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "789.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\theta ( t ) = e ^ { t \\Delta } \\theta _ 0 - \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\nabla \\cdot ( \\theta u ) ( s ) \\dd s \\\\ & u ( t ) = e ^ { t \\Delta } [ u _ 0 + t \\P \\theta _ 0 e _ 3 ] - \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } ( t - s ) \\P \\nabla \\cdot ( \\theta u ) ( s ) \\dd s \\ , \\ , e _ 3 - \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\P \\nabla \\cdot ( u \\otimes u ) ( s ) \\dd s \\\\ & \\nabla \\cdot u _ 0 = 0 \\end{aligned} \\right . \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} \\sum _ { l = - r } ^ { r } \\dfrac { 1 } { \\omega _ { l } } \\ , \\biggl ( u _ { ( 2 r + 1 ) + l } & + \\sum _ { p \\ge 2 } \\dfrac { \\overline { \\lambda } ^ { p - 1 } } { \\omega _ { ( p - 1 ) ( 2 r + 1 ) + l } \\dots \\omega _ { ( 2 r + 1 ) + l } } u _ { p ( 2 r + 1 ) + l } \\\\ & + \\sum _ { p \\ge 1 } \\dfrac { \\omega _ { l - p ( 2 r + 1 ) } \\dots \\omega _ { l - ( 2 r + 1 ) } } { \\overline { \\lambda } ^ { p } } u _ { - p ( 2 r + 1 ) + l } \\biggr ) { ( \\lambda - A ) } ^ { * } { e _ { l } } = 0 \\end{align*}"}
-{"id": "8306.png", "formula": "\\begin{align*} f _ { 1 / 4 } ( \\tau ) & : = t _ 4 ( \\tau ) = \\frac { \\eta ( \\tau ) ^ 8 } { \\eta ( 4 \\tau ) ^ 8 } , \\\\ f _ { 1 / 2 } ( \\tau ) & : = \\frac { 1 } { t _ 4 ( \\tau ) + 1 6 } = \\frac { \\eta ( \\tau ) ^ 8 \\eta ( 4 \\tau ) ^ { 1 6 } } { \\eta ( 2 \\tau ) ^ { 2 4 } } , \\\\ f _ { 1 / 1 } ( \\tau ) & : = \\frac { 1 } { t _ 4 ( \\tau ) } = \\frac { \\eta ( 4 \\tau ) ^ 8 } { \\eta ( \\tau ) ^ 8 } \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} \\rho _ { F } ( \\Omega ^ { + } ) = \\rho _ { F } ( \\Omega ^ { - } ) = \\frac { 1 } { \\Lambda } \\le \\rho _ { 2 , F } ( \\Omega ) , \\end{align*}"}
-{"id": "391.png", "formula": "\\begin{align*} O _ a ^ x = [ \\vartheta _ { i j } ] _ { i , j \\geq 0 } , \\left \\{ \\begin{array} { l } \\vartheta _ { 0 j } = { \\displaystyle - \\sum _ { i = 1 } ^ { j + 1 } \\theta _ { i j } P _ i ( a ) } \\\\ \\vartheta _ { i j } = \\theta _ { i j } , \\ i > 0 \\end{array} \\right . \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} \\prod _ { k = 0 } ^ { m - 1 } \\Gamma _ p \\bigg ( x + \\frac k m \\bigg ) = m ^ { - \\lambda _ p ( m x ) } \\Gamma _ p ( m x ) \\prod _ { k = 0 } ^ { m - 1 } \\Gamma _ p \\bigg ( \\frac k m \\bigg ) , \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} \\hat { \\rho } _ { A B M } ( t ) & : = \\left ( \\mathcal { N } _ A \\left ( \\frac { \\lambda \\ , t } { \\eta } \\right ) \\otimes \\mathcal { N } _ B \\left ( \\frac { ( 1 - \\lambda ) \\ , t } { | 1 - \\eta | } \\right ) \\otimes \\mathbb { I } _ M \\right ) ( \\hat { \\rho } _ { A B M } ) \\ ; , \\\\ \\hat { \\rho } _ { C M } ( t ) & : = ( \\mathcal { B } _ \\eta \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A B M } ( t ) ) \\ ; . \\end{align*}"}
-{"id": "4696.png", "formula": "\\begin{align*} K _ N ( x , y , w _ { R , \\alpha } ^ - ) = \\alpha ^ 2 K _ N \\left ( \\alpha ^ 2 x , \\alpha ^ 2 y ; ( 1 - t ^ 2 ) ^ { 1 / 2 } e ^ { - N V _ { \\alpha , \\widetilde { \\varepsilon } _ R } ( t ) } \\right ) . \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} \\lefteqn { ( - 1 ) ^ n \\left ( \\sum _ { j = 0 } ^ k ( \\lambda - 1 ) _ { k - j } \\binom { k } { j } \\left ( x ^ { j } f ( x ) \\right ) ^ { ( j ) } \\right ) ^ { ( n ) } } \\\\ & = ( - 1 ) ^ n \\sum _ { m = 0 } ^ k \\binom { k } { m } \\frac { \\Gamma ( n + k + \\lambda ) } { \\Gamma ( n + m + \\lambda ) } x ^ { m } f ^ { ( n + m ) } ( x ) = T ^ { \\lambda } _ { n , k } ( f ) ( x ) . \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} \\omega \\cdot ( x _ 0 , x _ 1 , \\ldots , x _ { p - 1 } ) = ( x _ { 1 } , x _ { 2 } , \\ldots , x _ { p - 1 } , x _ { 0 } ) . \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{align*} F _ * ^ e ( j u ^ { q - 1 } v ^ { q - k - 1 } ( f + u v ) ) = \\bigoplus _ { i \\in \\Delta _ e } f _ { ( i , j ) } F _ * ^ e ( i u ^ { q - 1 } v ^ { q - k - 1 } ) \\oplus u F _ * ^ e ( j v ^ { q - k } ) . \\end{align*}"}
-{"id": "2482.png", "formula": "\\begin{align*} { E _ q ^ { \\gamma _ N } } : = \\bigcup _ { Q , \\ , l ( Q ) \\le 2 ^ { 1 0 } l ( q ) : \\ , \\int _ Q | f _ q | \\ge \\gamma _ N \\cdot l ( Q ) } Q , \\end{align*}"}
-{"id": "5222.png", "formula": "\\begin{align*} u = \\nabla \\left ( \\frac { p \\cdot x } { | x | ^ n } \\right ) + O \\left ( \\frac { 1 } { | x | ^ { n + \\varepsilon } } \\right ) , \\textrm { a s } | x | \\to \\infty . \\end{align*}"}
-{"id": "2179.png", "formula": "\\begin{align*} S ( k ) - { U _ 0 } = \\int _ { - \\infty } ^ \\infty F _ S ( x ) e ^ { - i k x } d x , F _ S ( x ) = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ \\infty ( S ( k ) - { U _ 0 } ) e ^ { i k x } d k . \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} [ L _ { - r + 1 } , \\ , S _ r ] = S _ 1 \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} \\begin{array} { c } \\alpha \\notin C , \\\\ M [ G ] \\models | \\mathcal { B } ( T ) | ] = \\omega _ \\alpha . \\end{array} \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} X _ i = \\beta _ i Y + \\varepsilon _ i , \\ ; \\ ; \\ ; \\ ; \\textrm { C o v } ( Y , \\varepsilon _ i ) = 0 , \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} n = n _ { 1 } + \\dots + n _ { m } . \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} \\sigma ( S ) \\cup \\gamma \\sigma ( C ) : = \\left ( \\lambda _ { 1 } , \\lambda _ { 2 } , \\ldots , \\lambda _ { n } , \\gamma \\mu _ { 1 } , \\gamma \\mu _ { 2 } , \\ldots , \\gamma \\mu _ { n } \\right ) \\end{align*}"}
-{"id": "9365.png", "formula": "\\begin{align*} \\phi ( t _ 1 , t _ 2 , t _ 3 ) = \\left \\langle a ( t _ 1 , t _ 2 ) , b ( t _ 2 , t _ 3 ) \\right \\rangle , a . e . \\hbox { - } ( t _ 1 , t _ 2 , t _ 3 ) . \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{align*} \\Lambda \\cdot \\left [ \\begin{array} { c } x _ 1 ^ 2 \\\\ x _ 2 ^ 2 \\\\ \\vdots \\\\ x _ { n + 1 } ^ 2 \\\\ \\end{array} \\right ] = \\left [ \\begin{array} { c } a _ 0 \\\\ a _ { 1 } \\\\ \\vdots \\\\ a _ n \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{align*} \\{ f ( x , y ; e _ k ) \\mid k = 1 , 2 , 3 \\} . \\end{align*}"}
-{"id": "1349.png", "formula": "\\begin{align*} W _ { m , \\alpha } : = \\zeta ^ { - 1 } ( V _ { m , \\alpha } ) \\ ; \\subset \\ ; X . \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{align*} [ m _ { t } ] ( P _ i ) _ t = ( Q _ i ) _ t \\end{align*}"}
-{"id": "1786.png", "formula": "\\begin{align*} \\sup _ { p _ j ( x ) \\leqslant 1 } p _ j ( A x ) & = \\sup _ { p _ j ( x ) \\leqslant 1 } \\left \\{ \\inf _ { p _ j ( z ) = 0 } p _ j ( A x ) - p _ j ( z ) \\right \\} \\\\ & \\leqslant \\sup _ { p _ j ( x ) \\leqslant 1 } \\left \\{ \\inf _ { p _ j ( z ) = 0 } p _ j ( A x - z ) \\right \\} = \\sup _ { p _ j ( x ) \\leqslant 1 } \\| [ A x ] _ j \\| _ j \\\\ & \\leqslant \\sup _ { \\| [ x ] _ j \\| _ j \\leqslant 1 } \\| A _ j [ x ] _ j \\| _ j < \\infty \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} \\widehat { f } - \\tilde { f } ^ * = G _ \\lambda ^ { - 1 } \\left [ \\frac { 1 } { 2 } D ^ 2 l _ \\infty ( \\bar { f } ) ( \\widehat { f } - \\bar { f } ) - \\frac { 1 } { 2 } D ^ 2 l _ { n } ( \\bar { f } ) ( \\widehat { f } - \\bar { f } ) \\right ] . \\end{align*}"}
-{"id": "6259.png", "formula": "\\begin{align*} \\begin{aligned} a ^ { \\textnormal { h o m } } _ { i j r s } ( \\theta ) : = \\sum _ { k = 1 } ^ n \\sum _ { l = 1 } ^ d \\int _ Y a _ { i j k l } \\big ( \\partial _ { l } N ^ { ( r s ) } _ { \\theta k } ( y ) + e ^ { \\i \\theta \\cdot y } \\delta _ { k r } \\delta _ { l s } \\big ) & e ^ { - \\i \\theta \\cdot y } \\ , { \\textnormal { d } } y \\\\ & ( { i , r \\in \\{ 1 , \\ldots , n \\} , j , s \\in \\{ 1 , \\ldots , d \\} } ) , \\end{aligned} \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} \\sup \\limits _ { k , i } \\sum \\limits _ { j = 1 } ^ { N } \\left \\vert \\frac { \\lambda _ { k } } { D \\left ( \\lambda _ { k } \\right ) } A _ { j i } \\left ( \\lambda _ { k } \\right ) \\right \\vert ^ { q } \\leq C . \\end{align*}"}
-{"id": "2132.png", "formula": "\\begin{gather*} t H _ { \\mathrm { F S } } ^ { A _ 3 } \\left ( { \\alpha , \\beta \\atop \\gamma , \\delta } ; t ; { q _ 1 , p _ 1 \\atop q _ 2 , p _ 2 } \\right ) = t H _ { \\mathrm { I I I } ( D _ 6 ) } \\big ( { \\theta ^ \\infty _ 3 , \\theta ^ \\infty _ 3 - \\theta ^ \\infty _ 1 + 1 } ; t ; q _ 1 , p _ 1 \\big ) \\\\ \\qquad { } + t H _ { \\mathrm { I I I } ( D _ 6 ) } \\big ( { - } \\theta ^ 0 _ 2 - \\theta ^ \\infty _ 2 , \\theta ^ \\infty _ 1 - \\theta ^ \\infty _ 2 ; t ; q _ 2 , p _ 2 \\big ) - p _ 1 p _ 2 ( q _ 1 q _ 2 + t ) . \\end{gather*}"}
-{"id": "1309.png", "formula": "\\begin{align*} K _ { 0 } \\left [ 2 \\frac { f ^ { 3 } } { p \\dot { p } } r \\dot { r } - 4 \\frac { f p } { \\dot { p } } g + 2 \\frac { p ^ { 3 } } { f \\dot { p } } \\left ( \\frac { g ^ { 2 } } { r } \\right ) \\left \\{ \\frac { d } { d g } \\left ( \\frac { g ^ { 2 } } { r } \\right ) \\right \\} \\right ] = \\frac { d } { d g } \\left ( \\frac { g \\dot { r } } { r } \\right ) . \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} \\gamma _ { 1 } + \\gamma _ { 2 } = \\gamma _ { 2 } + \\gamma _ { 3 } = \\gamma _ { 3 } + \\gamma _ { 4 } = \\cdots = \\gamma _ { n - 1 } + \\gamma _ { n } \\end{align*}"}
-{"id": "4354.png", "formula": "\\begin{align*} N _ j = t / k + O ( \\sqrt { n } ) . \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} \\hat c _ i : = c _ i \\cdot \\left ( ( Q _ { g + 1 - n } ( a _ 1 ) - Q _ { g + 1 - n } ( \\textstyle { a _ 1 - \\frac 1 2 } ) ) \\prod _ { i = 2 } ^ n Q _ 1 ( a _ i ) Q _ 2 ( \\frac 3 2 ) \\right ) ^ { - 1 } , i = 0 , 1 , 2 , \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{align*} A _ { k ( C ) } = \\left ( \\frac { I _ g ^ 2 - 4 \\ , , \\ , I _ { [ g , h ] } - 2 } { k ( C ) } \\right ) . \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} D ' = \\pi _ * D = \\sum _ i ^ \\ell \\beta _ i \\pi _ * F _ i . \\end{align*}"}
-{"id": "2348.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } v ( t ) = \\nabla \\phi ( x ( t ) ) \\\\ \\lambda \\dot x ( t ) + \\dot v ( t ) + \\nabla \\phi ( x ( t ) ) + \\nabla \\psi ( x ( t ) ) = 0 \\\\ x ( 0 ) = x _ 0 , v ( 0 ) = v _ 0 = \\nabla \\phi ( x _ 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} & D _ { n , k } ( 1 , x ) \\cr & = \\frac { k } { 8 } \\ , ( 1 - 4 x ) ^ \\frac { p ^ { l _ 1 } + p ^ { l _ 2 } + p ^ { l _ 3 } - 1 } { 2 } + \\frac { ( 2 - k ) } { 8 } \\ , [ ( 1 - 4 x ) ^ \\frac { p ^ { l _ 1 } + p ^ { l _ 2 } } { 2 } + ( 1 - 4 x ) ^ \\frac { p ^ { l _ 1 } + p ^ { l _ 3 } } { 2 } + ( 1 - 4 x ) ^ \\frac { p ^ { l _ 2 } + p ^ { l _ 3 } } { 2 } ] \\cr & + \\frac { k } { 8 } [ ( 1 - 4 x ) ^ \\frac { p ^ { l _ 1 } - 1 } { 2 } + ( 1 - 4 x ) ^ \\frac { p ^ { l _ 2 } - 1 } { 2 } + ( 1 - 4 x ) ^ \\frac { p ^ { l _ 3 } - 1 } { 2 } ] + \\frac { ( 2 - k ) } { 8 } . \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} \\ell [ y ] ( x ) = k ( k + 1 ) y ( x ) , \\end{align*}"}
-{"id": "6645.png", "formula": "\\begin{align*} \\omega ( f , g ) \\ ; = \\ ; a _ 0 ^ { ( g ) } \\omega ( f , v _ 0 ) + a _ \\infty ^ { ( g ) } \\omega ( f , v _ \\infty ) + \\omega ( f , b _ \\infty ^ { ( g ) } \\ , v _ 0 ) + \\omega ( f , b _ 0 ^ { ( g ) } \\ , v _ \\infty ) \\ , . \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{align*} X \\triangleright a ^ { * } = ( S ( X ) ^ { * } \\triangleright a ) ^ { * } , a ^ { * } \\triangleleft X = ( a \\triangleleft S ( X ) ^ { * } ) ^ { * } . \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} H _ 1 = \\oplus _ { \\nu = 1 } ^ { n _ 1 } C _ \\nu , \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} C _ { } = \\frac { 1 - \\frac { T } { N - U } } { 1 - \\left ( \\frac { T } { N - U } \\right ) ^ M } \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} \\mu ^ { 2 ^ { k + 1 } } + \\mu ^ { 2 ^ { k } - 1 } + 1 = 0 . \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{align*} e _ n ( \\vec { \\theta } ) = \\sum _ { i _ 1 < i _ 2 < \\cdots < i _ n } \\theta _ { i _ 1 } \\theta _ { i _ 2 } \\cdots \\theta _ { i _ n } , ~ ~ { \\rm f o r } ~ ~ n = 1 , 2 , \\cdots , k ~ ~ { \\rm a n d } ~ ~ e _ 0 ( \\vec { \\theta } ) = 1 . \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{align*} \\mathcal { P } ^ { ( N + 1 ) } _ { s } ( t ) \\Lambda _ { N , N + 1 } = \\Lambda _ { N , N + 1 } \\hat { \\mathcal { P } } ^ { ( N ) } _ { s } ( t ) , \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} \\sum _ { d = 0 } ^ { \\kappa } ( d + 2 ) ^ u F _ { d + 2 } ( n ) = \\left ( \\sum _ { d = 0 } ^ \\kappa ( d + 2 ) ^ u e ^ { - d ^ 2 - d } \\right ) + O \\left ( \\sum _ { d = 0 } ^ \\kappa ( d + 2 ) ^ u \\frac { ( \\log n ) ^ 6 } { n } e ^ { d ^ 2 + d } \\right ) , \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} \\left ( 2 ^ { u _ { i _ 1 } } + \\cdots + 2 ^ { u _ { i _ M } } \\right ) ^ n \\left ( 2 ^ { u _ { j _ 1 } } + \\cdots + 2 ^ { u _ { j _ N } } \\right ) ^ n = \\left ( \\sum _ { ( p , q ) \\in \\{ 1 , \\ldots , M \\} \\times \\{ 1 , \\ldots , N \\} } 2 ^ { u _ { i _ p } + u _ { j _ q } } \\right ) ^ n \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 , [ y _ 1 , y _ 2 ] = y _ 3 , [ y _ 2 , y _ 1 ] = - y _ 3 + ( \\alpha _ 5 + \\beta _ 2 ) y _ 5 , [ y _ 2 , y _ 2 ] = \\beta _ 4 y _ 5 , \\\\ [ y _ 3 , y _ 1 ] = \\alpha _ 3 \\gamma _ 2 y _ 5 , [ y _ 2 , y _ 3 ] = \\alpha _ 3 \\gamma _ 3 y _ 5 = - [ y _ 3 , y _ 2 ] , [ y _ 1 , y _ 4 ] = \\alpha _ 1 \\gamma _ 6 y _ 5 . \\end{align*}"}
-{"id": "2707.png", "formula": "\\begin{align*} \\sum \\limits _ { l = 1 } ^ n | \\langle \\phi _ j , \\phi _ l \\rangle | ^ 2 = \\frac { n } { m } , j \\in \\{ 1 , 2 , . . . , n \\} . \\end{align*}"}
-{"id": "6323.png", "formula": "\\begin{align*} \\begin{array} { l l l } g ( \\mu ) = & \\max _ { x \\in \\Bbb R ^ n } & x ^ T D x \\\\ & { \\rm s . t . } & \\| x \\| = 1 \\\\ & \\ & x ^ T ( B - \\mu W ) x \\ge 0 \\end{array} \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} \\tilde { \\psi } ( Y _ 1 , Y _ 1 ' , Y _ 3 ) = M , \\tilde { \\psi } ( Y _ 1 , Y _ 1 ' , Y _ 4 ) = M , \\\\ \\tilde { \\psi } ( Y _ 2 , Y _ 2 ' , Y _ 3 ) = M , \\tilde { \\psi } ( Y _ 2 , Y _ 2 ' , Y _ 4 ) = M , \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} \\limsup _ { i \\to + \\infty } I _ u ( x + r _ { j _ i } y _ j , 0 ^ + ) & \\leq \\inf _ { s > 0 } \\limsup _ { i \\to + \\infty } I _ u ( x + r _ { j _ i } y _ j , r _ { j _ i } \\ , s ) \\\\ & = \\inf _ { s > 0 } \\limsup _ { i \\to + \\infty } I _ { u _ { x , r _ { j _ i } } } ( y _ j , s ) = \\inf _ { s > 0 } I _ { w } ( y , s ) = I _ w ( y , 0 ^ + ) . \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{align*} 2 s - s ^ g = | T | ^ 2 - \\frac 9 2 | t | ^ 2 - 2 \\delta \\theta - | \\theta | ^ 2 \\end{align*}"}
-{"id": "947.png", "formula": "\\begin{align*} Z : = \\hat a ^ { 1 / 2 } R O U , \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} \\frac { d } { d t } E _ { \\lambda } ^ { \\omega } \\xi _ t ( x ) = - E _ { \\lambda } ^ { \\omega } \\xi _ t ( x ) + \\sum _ { y : y \\thicksim x } \\lambda \\rho ( x , \\omega ) \\rho ( y , \\omega ) E _ { \\lambda } ^ { \\omega } \\xi _ t ( y ) . \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} F = \\frac { 1 } { 2 } \\int f \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{align*} I _ { 1 / 2 } ( q ; x ) = ( - x ) ^ { - q - \\frac { 1 } { 2 } } \\sqrt { \\frac { x } { 2 \\pi } } \\ \\left ( \\Gamma ( q + \\frac { 1 } { 2 } ) - \\Gamma ( q + \\frac { 1 } { 2 } , - 2 x ) \\right ) , \\ \\ \\ R e ( q ) > - \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{align*} ( \\mathsf { N } _ { m } ^ { n } ) _ { j } ^ { i } \\triangleleft X & = ( u _ { m } ^ { i } \\triangleleft X _ { ( 1 ) } ) ( u _ { n } ^ { j * } \\triangleleft X _ { ( 2 ) } ) = \\sum _ { k , \\ell } \\pi ( X _ { ( 1 ) } ) _ { k } ^ { i } u _ { m } ^ { k } \\pi ( S ( X _ { ( 2 ) } ) ) _ { j } ^ { \\ell } u _ { n } ^ { \\ell * } \\\\ & = \\sum _ { k , \\ell } \\pi ( X _ { ( 1 ) } ) _ { k } ^ { i } ( \\mathsf { N } _ { m } ^ { n } ) _ { \\ell } ^ { k } \\pi ( S ( X _ { ( 2 ) } ) ) _ { j } ^ { \\ell } . \\end{align*}"}
-{"id": "3349.png", "formula": "\\begin{align*} \\phi ( \\lambda ) = \\kappa \\lambda + \\phi _ 0 ( \\lambda ) \\hbox { a n d } S _ t = \\kappa t + \\bar S _ t . \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} 0 = [ [ L _ { - q + i } , \\ , L _ { 1 } ] , \\ , L _ { q - i } ] = [ [ L _ { - q + i } , \\ , L _ { q - i } ] , \\ , L _ { 1 } ] , \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} { \\bf G L } _ 2 \\left ( \\mathbb { Z } / m \\mathbb { Z } \\right ) \\xrightarrow { \\approx } \\prod _ { i = 1 } ^ { \\nu } { \\bf G L } _ 2 \\left ( \\mathbb { Z } / \\ell _ i ^ { n _ i } \\mathbb { Z } \\right ) . \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} z _ b ( s ) = & z _ b ( \\sigma ) - \\sigma ^ \\gamma e ^ { \\frac { 1 } { 4 \\sigma } } \\Bigl ( \\int _ s ^ \\sigma t ^ { - \\gamma } e ^ { - \\frac { 1 } { 4 t } } \\ , d t \\Bigr ) z _ b ' ( \\sigma ) \\\\ & - \\frac { 1 } { 4 } \\int _ s ^ \\sigma t ^ { - \\gamma } e ^ { - \\frac { 1 } { 4 t } } \\int _ t ^ \\sigma r ^ { \\gamma - 2 } e ^ { \\frac { 1 } { 4 r } } g ( z _ { b } ( r ) ) \\ , d r d t \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{align*} \\liminf _ { l \\to \\infty } \\ , \\inf _ { J \\ge \\delta ^ { ( k + 1 ) } } \\ \\dfrac { 1 } { J + 1 } \\ \\# \\ , \\Bigl \\{ 0 \\le j \\le J \\ , ; \\ , & \\Vert P _ { l } T ^ { \\ , j } T _ { l } \\ , x \\Vert \\ge X _ { l } / 2 \\Bigr \\} \\\\ & \\ge \\liminf _ { k \\to \\infty } \\ , \\Bigl ( 1 - \\dfrac { 8 \\delta ^ { ( k ) } } { \\delta ^ { ( k + 1 ) } } - \\dfrac { 8 \\delta ^ { ( k ) } } { \\Delta ^ { ( k ) } } \\Bigr ) = 1 , \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} \\left ( Q _ { t } + \\lambda \\right ) ^ { - 1 } = \\left [ I + L _ { t } \\left ( O _ { t } + \\lambda \\right ) ^ { - 1 } \\right ] \\left ( O _ { t } + \\lambda \\right ) , \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{align*} \\int _ { \\overline { C } _ r } [ f ( k , x ) { U _ 0 } + e ^ { - i k x } I _ n ] e ^ { i k y } d k = 0 . \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} \\alpha _ \\delta ( 1 _ v \\cdot 1 _ e ) = \\delta _ { v , r ( e ) } \\alpha _ \\delta ( 1 _ e ) = \\delta _ { v , r ( e ) } ( - 1 ) ^ { \\delta ( e ) } 1 _ e = 1 _ v \\cdot \\alpha _ \\delta ( 1 _ e ) \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} ( w + w ^ { - 1 } ) ^ 2 - 4 = ( w - w ^ { - 1 } ) ^ 2 , \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} \\textrm { I } _ n = \\widetilde { E } \\big { [ } \\frac { \\textrm { I } _ { n , 1 } } { \\textrm { I } _ { n , 2 } } I _ { \\{ \\tau < n \\} } \\big { ] } . \\end{align*}"}
-{"id": "1860.png", "formula": "\\begin{align*} 2 s - s ^ g & = | N | ^ 2 , & s ^ H - 2 s = | N | ^ 2 . \\end{align*}"}
-{"id": "9306.png", "formula": "\\begin{align*} \\delta ( n ) = \\begin{cases} \\frac { M ^ 2 } { \\lambda n ^ 2 } & \\frac { M } { \\lambda n } \\le 1 \\\\ \\frac { 2 M } { n } - \\lambda & \\end{cases} . \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{align*} A _ 1 \\ ! : = \\ ! \\sum _ { 1 \\leq k < j } S _ { N } \\Big ( \\frac { t _ { j } ( x ) } { 2 \\pi \\lambda ^ { j } } \\Big ) f _ k ( x ) A _ 2 \\ ! : = \\ ! \\sum _ { k > j } S _ { N } \\Big ( \\frac { t _ { j } ( x ) } { 2 \\pi \\lambda ^ { j } } \\Big ) f _ k ( x ) \\ , . \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} \\vert n \\rangle \\longrightarrow f _ { n } ( \\eta ) = c _ { n } { { \\eta } } ^ { n } , ~ a ^ + \\longrightarrow { { \\eta } } . \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} \\Vert \\Psi _ n ( t ) \\Vert ^ { 2 } _ { \\mathcal { L } ^ { 2 } } = \\Vert \\Psi _ n ( 0 ) \\Vert ^ { 2 } _ { \\mathcal { L } ^ { 2 } } , \\mathcal { E } _ { n } ( t ) = \\mathcal { E } _ { n } ( 0 ) . \\end{align*}"}
-{"id": "7075.png", "formula": "\\begin{align*} u _ { \\varepsilon } \\left ( x , t \\right ) = e ^ { \\varepsilon t W } u \\left ( x , t \\right ) t \\in \\left [ 0 , 1 \\right ] . \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{align*} \\varphi _ 1 = e ^ { 1 2 7 } + e ^ { 3 4 7 } + e ^ { 5 6 7 } + e ^ { 1 3 5 } - e ^ { 1 4 6 } - e ^ { 2 3 6 } - e ^ { 2 4 5 } . \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} S _ x ( X ) = \\bigl ( F _ 1 ^ { \\sigma _ 1 } \\bigr ) ^ \\natural \\cong \\bigl ( F _ 2 ^ { \\sigma _ 2 } \\bigr ) ^ \\natural = F _ 2 ^ { \\sigma _ 2 } \\big / \\bigl ( F _ 2 ^ { \\sigma _ 2 } \\cap F _ 2 \\bigr ) . \\end{align*}"}
-{"id": "5610.png", "formula": "\\begin{align*} \\chi _ { 1 , 0 } ^ { m , 0 } ( x | \\rho ) & = \\sum _ { i = 0 } ^ { \\infty } \\rho ^ { i } T _ { i + m } ( x ) \\allowbreak = \\allowbreak \\frac { T _ { m } ( x ) - \\rho T _ { m - 1 } ( x ) } { w _ { 1 } ( x | \\rho ) } , \\\\ \\chi _ { 0 , 1 } ^ { 0 , m } ( x | \\rho ) & = \\sum _ { i = 0 } ^ { \\infty } \\rho ^ { i } U _ { i + m } ( x ) = \\frac { U _ { m } ( x ) - \\rho U _ { m - 1 } ( x ) } { w _ { 1 } ( x | \\rho ) } , \\end{align*}"}
-{"id": "8459.png", "formula": "\\begin{align*} u _ 0 : = \\inf \\{ x \\in \\R , \\ , \\ , \\exists ( \\sigma , \\varphi ) \\in { \\cal S } ( x ) \\} . \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{align*} T f = \\sum _ { k = 1 } ^ n f ( t _ { k } ) p _ k , \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} \\sqrt { a c } = y _ 2 ^ { - 1 } y _ 0 + y _ 2 ^ { - 1 } y _ 1 \\sqrt { b c } . \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} \\partial _ { u _ { s } } G [ s ] ( w ) = - w ^ s G [ s ] ( w ) ^ 2 . \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} \\widehat { \\tilde { T } } \\left ( \\xi , s \\right ) = \\hat { T } _ { 0 } \\left ( \\xi \\right ) \\frac { \\Phi \\left ( s \\right ) } { \\xi ^ { 2 } + s \\Phi \\left ( s \\right ) } \\ ; \\ ; \\ ; \\ ; \\widehat { \\tilde { q } } \\left ( \\xi , s \\right ) = - \\hat { T } _ { 0 } \\left ( \\xi \\right ) \\frac { \\mathrm { i } \\xi } { \\xi ^ { 2 } + s \\Phi \\left ( s \\right ) } . \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{align*} G ^ \\pm _ y ( y _ 1 ) = e ^ { \\mp i \\lambda | y | } R _ 0 ^ \\pm ( \\lambda ^ 2 ) ( y _ 1 , y ) . \\end{align*}"}
-{"id": "1318.png", "formula": "\\begin{align*} K _ { 0 } \\left [ \\frac { f ^ { 3 } } { \\lambda _ { 4 } p \\dot { p } } - 2 \\frac { f p } { \\dot { p } } + \\lambda _ { 4 } \\frac { p ^ { 3 } } { f \\dot { p } } \\right ] g ^ { 2 } + \\frac { p } { f \\dot { p } } - 1 = 0 . \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} f _ 1 & = 0 ; & & \\\\ \\sum _ { i = 1 } ^ { n + m } f _ i & = d - g + m ; & & \\\\ a _ i + f _ i & = d _ i , & & 2 \\leq i \\leq n ; \\\\ a _ { n + j } + f _ { n + j } & = e _ j + 1 , & & 1 \\leq j \\leq m ; \\\\ a _ 1 & = 2 g - 1 - \\sum _ { j = 2 } ^ m a _ j . & & \\end{align*}"}
-{"id": "2766.png", "formula": "\\begin{align*} \\sup _ { | x - \\alpha | < 1 } | f ^ \\prime ( x ) | = \\sup _ { | x - \\alpha | < 1 } 2 | a x | \\leqslant 4 \\sqrt { a b } , \\end{align*}"}
-{"id": "2241.png", "formula": "\\begin{align*} \\frac { 1 } { { } _ 2 F _ 1 ( 1 , N ; N + 1 ; - x ) } = \\frac { ( - 1 ) ^ { N - 1 } x ^ N / N } { \\log ( 1 + t ) - \\sum _ { n = 1 } ^ { N - 1 } ( - 1 ) ^ { N - 1 } x ^ n / n } = \\sum _ { n = 0 } ^ \\infty c _ { N , n } \\frac { x ^ n } { n ! } \\ , , \\end{align*}"}
-{"id": "9755.png", "formula": "\\begin{align*} \\mu _ { \\epsilon } : = \\frac { e _ { \\epsilon } ( u _ { \\epsilon } ) } { | \\log \\epsilon | } d v _ g \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{align*} R _ { k + 1 } = \\alpha _ { k + 1 } R _ k \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} \\sum Q ^ j b _ k = \\left ( \\sum _ { n = k } ^ \\infty \\sum _ { u = 0 } ^ k \\binom { n - k + u - 1 } { u } b _ { n + u } b _ { k - u } \\right ) \\left ( \\sum _ { n = 0 } ^ \\infty b _ n \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} t _ { 2 } ( N ) = - t _ { 2 } ( N ' ) & \\Longleftrightarrow t _ { 2 } ( N _ { 1 } - 1 ) + t _ { 2 } ( N _ { 1 } + 1 ) = - 2 t _ { 2 } ( N _ { 1 } ) \\\\ & \\Longleftrightarrow - 2 t _ { 2 } \\left ( \\frac { N _ { 1 } } { 2 } - 1 \\right ) - 2 t _ { 2 } \\left ( \\frac { N _ { 1 } } { 2 } \\right ) = - 2 t _ { 2 } ( N _ { 1 } ) \\\\ & \\Longleftrightarrow t _ { 2 } ( N _ { 1 } ) = t _ { 2 } \\left ( \\frac { N _ { 1 } } { 2 } \\right ) + t _ { 2 } \\left ( \\frac { N _ { 1 } } { 2 } - 1 \\right ) . \\end{align*}"}
-{"id": "477.png", "formula": "\\begin{align*} & H ( M | Y _ i Y _ 3 ) \\le H ( Y _ 3 Y _ 4 | Y _ i Y _ 3 ) = H ( Y _ 4 | Y _ i Y _ 3 ) \\le H ( Y _ 4 | Y _ i ) . \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{align*} s ^ 2 z '' ( s ) = - \\Bigl ( t ^ { \\frac { 1 } { 2 } } v ' ( t ) + \\frac { 1 } { 2 } t ^ { - \\frac { 1 } { 2 } } v ( t ) \\Bigr ) + \\Bigl ( t ^ { \\frac { 1 } { 2 } } v '' ( t ) + t ^ { - \\frac { 1 } { 2 } } v ' ( t ) - \\frac { 1 } { 4 } t ^ { - \\frac { 3 } { 2 } } v ( t ) \\Bigr ) . \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} \\varsigma _ { f } \\left ( T \\right ) = \\mathcal { Z } \\left ( T _ { 1 } , \\ldots , T _ { d } , U , S \\right ) \\mid _ { \\begin{array} [ c ] { l } T _ { 1 } = \\ldots = T _ { d } = T \\\\ U = 1 \\end{array} . } \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\overbrace { \\begin{matrix} 0 & 0 \\\\ t _ 2 & \\theta ^ 0 \\end{matrix} } & \\overbrace { \\begin{matrix} 1 & 0 & - t _ 1 / 2 & \\theta ^ \\infty _ 1 / 2 \\\\ - 1 & 0 & t _ 1 / 2 & \\theta ^ \\infty _ 1 / 2 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "9706.png", "formula": "\\begin{align*} h _ 2 ^ 2 ( P ) & = ( v _ 1 + v _ 2 ) ^ { 2 } + ( v _ 2 + v _ 3 ) ^ { 2 } + ( v _ 3 + v _ 4 ) ^ { 2 } + ( v _ 1 + v _ 4 ) ^ { 2 } \\\\ & + \\tfrac { 1 } { 2 } ( v _ 1 + v _ 3 ) ^ { 2 } + \\tfrac { 1 } { 2 } ( v _ 2 + v _ 4 ) ^ { 2 } - v _ 1 ^ { 2 } - v _ 2 ^ { 2 } - v _ 3 ^ { 2 } - v _ 4 ^ { 2 } \\\\ & = \\tfrac { 1 } { 2 } ( v _ 1 + v _ 2 + v _ 3 + v _ 4 ) ^ { 2 } + \\tfrac { 1 } { 2 } ( v _ 1 + v _ 2 ) ^ { 2 } + \\tfrac { 1 } { 2 } ( v _ 2 + v _ 3 ) ^ { 2 } + \\tfrac { 1 } { 2 } ( v _ 3 + v _ 4 ) ^ { 2 } + \\tfrac { 1 } { 2 } ( v _ 1 + v _ 4 ) ^ { 2 } . \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} \\frac { d u } { d t } ( t , x ) = \\frac 1 2 { \\rm T r } \\left ( \\sigma ( x ) \\sigma ^ * ( x ) D ^ 2 u ( t , x ) \\right ) + \\langle A x + G ( x ) , D u ( t , x ) \\rangle , u ( 0 , x ) = \\varphi ( x ) . \\end{align*}"}
-{"id": "6612.png", "formula": "\\begin{align*} c \\ ; & = \\ ; ( \\textstyle { \\frac { \\Gamma ( B ) } { \\Gamma ( 2 B ) } \\ , \\frac { 1 + \\nu } { 1 + \\nu + B } } ) \\cdot \\displaystyle { \\lim _ { r \\downarrow 0 } } \\ ; r ^ B g ^ + \\ ! ( r ) \\\\ f \\ ; & = \\ ; g - c ( \\beta S _ D ^ { - 1 } \\Phi + \\Phi ) \\ , . \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} \\operatorname { s i n c } ( x ) = \\frac { \\sin { ( \\pi x ) } } { \\pi x } . \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\left ( S ( A | M ) _ { ( \\mathcal { N } ( t ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) } - n \\ln t - n \\right ) & \\le \\limsup _ { t \\to \\infty } n \\left ( g ( E + t ) - \\ln t - 1 \\right ) \\\\ & = 0 \\ ; . \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} F = \\mathbf { F } | _ { \\mathbb { P } ^ 3 \\times \\{ \\mathbf { x } \\} } , \\ \\ \\ \\mathbf { x } \\in \\mathbf { X } , \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{align*} \\mathrm { T r } ( V P ) = c \\cdot 1 , \\sigma ( P ) = V P V ^ { - 1 } . \\end{align*}"}
-{"id": "9877.png", "formula": "\\begin{align*} ( \\deg ( \\gamma _ 0 ) , \\dots , \\deg ( \\gamma _ 7 ) ) = ( \\boldsymbol 5 , 5 , 7 , \\boldsymbol 3 , 4 , 4 , \\boldsymbol 2 , 5 ) . \\end{align*}"}
-{"id": "7226.png", "formula": "\\begin{align*} \\langle \\rho _ i \\mid i \\in I \\rangle \\cap \\langle \\rho _ i \\mid i \\in J \\rangle = \\langle \\rho _ i \\mid i \\in I \\cap J \\rangle . \\end{align*}"}
-{"id": "2.png", "formula": "\\begin{align*} \\hat c _ 0 ( t ) = \\mu h _ M ( 1 - e ^ { - t } ) - \\frac { \\Gamma } { K _ 1 } t . \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n ( - 1 ) ^ { n - k } \\binom { n } { k } \\lambda ^ k \\mp 1 = ( \\lambda - 1 ) ^ n \\mp 1 = 0 \\end{align*}"}
-{"id": "2261.png", "formula": "\\begin{align*} J = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} \\in \\sl . \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} m - 2 r + i _ 0 & = m + i _ 0 - 2 \\sum _ { p = 0 } ^ { m - 1 } i _ p = m - i _ 0 - 2 \\sum _ { p = 1 } ^ { m - 1 } i _ p \\ge 0 , \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} c _ { \\alpha _ 1 + \\alpha _ 2 + \\cdots + \\alpha _ k } ( n ) = \\sum _ { n _ 1 + n _ 2 + \\cdots + n _ k = n } c _ { \\alpha _ 1 } ( n _ 1 ) c _ { \\alpha _ 2 } ( n _ 2 ) \\cdots c _ { \\alpha _ k } ( n _ k ) . \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{align*} \\det ( X ) = \\frac { a _ 0 r } { \\sqrt { 3 } } > 0 , c _ { 2 1 } ( X ) = a _ 0 > 0 . \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} \\frac { d q _ i } { d t } = p _ i , \\frac { d p _ i } { d t } = - \\epsilon _ i q _ i - q _ i ^ 3 + \\frac { 1 } { W } \\left ( q _ { i + 1 } + q _ { i - 1 } - 2 q _ i \\right ) \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} h _ 0 ' = h _ t + Q \\log | f _ t ' | - b _ t \\end{align*}"}
-{"id": "55.png", "formula": "\\begin{align*} \\Phi ( c h ) = \\Phi ( h ) , \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} T _ { 0 } \\left ( x \\right ) = \\frac { T _ { 0 } } { 2 \\sqrt { \\pi \\varepsilon } } \\mathrm { e } ^ { - \\frac { x ^ { 2 } } { 4 \\varepsilon } } , \\ ; \\ ; x \\in \\mathbb { R } , \\end{align*}"}
-{"id": "3204.png", "formula": "\\begin{align*} H ( x , y ) = 2 ( 1 + x y ) . \\end{align*}"}
-{"id": "7364.png", "formula": "\\begin{align*} \\frac { \\sqrt { 6 } } { 5 A } \\left ( 1 - y ^ 5 ( t ) \\right ) + \\frac { 1 } { 1 0 A ^ 2 } \\mathrm { l o g } \\left ( \\frac { 2 A \\sqrt { 6 } \\ , y ^ 5 ( t ) + 1 } { 2 A \\sqrt { 6 } + 1 } \\right ) = t . \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 G ( y ) ( y ^ 2 - 1 ) \\varphi ( y ) d y = G ( y _ 0 ) \\int _ 0 ^ 1 ( y ^ 2 - 1 ) \\varphi ( y ) d y \\ ; . \\end{align*}"}
-{"id": "8846.png", "formula": "\\begin{align*} \\sum _ { d = 0 } ^ \\infty ( d + 2 ) ^ a \\Pr { \\mathbf { Y } _ n = d + 2 } = 1 + \\sum _ { d = 0 } ^ \\infty \\big ( ( d + 2 ) ^ a - ( d + 1 ) ^ a \\big ) F _ { d + 2 } ( n ) . \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{align*} g \\left ( z \\right ) = \\frac { x \\left ( z \\right ) } { 1 + x \\left ( z \\right ) } . \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{align*} \\widetilde { Q } ( H ) = \\sum _ { i = 0 } ^ { \\infty } \\pi ^ * \\alpha _ i H ^ i = \\pi ^ * \\alpha _ 0 + ( \\pi ^ * \\alpha _ 1 ) H + H ^ 2 \\sum _ { k = 0 } ^ { \\infty } ( \\pi ^ * \\alpha _ k ) H ^ k , \\end{align*}"}
-{"id": "6318.png", "formula": "\\begin{align*} x ^ { \\lambda - 1 + k } f ( x ) & = \\frac { l _ k } { ( k - 1 ) ! } x ^ { k - 1 } + \\frac { b _ k } { ( \\lambda ) _ k } x ^ { \\lambda + k - 1 } \\\\ & { } + x ^ { \\lambda + k - 1 } \\int _ 0 ^ { \\infty } M _ 1 ( u ) u ^ { \\lambda + k - 2 } e ^ { - x u } \\ , d u , \\end{align*}"}
-{"id": "785.png", "formula": "\\begin{align*} R ( e _ j ) _ { x _ 1 , \\dots , x _ m } = R ^ { ( c _ 1 + f _ 1 ( x _ { r + 1 } , \\dots , x _ m ) , \\dots , c _ s + f _ s ( x _ { r + 1 } , \\dots , x _ m ) ) } ( e _ j ) _ { x _ 1 , \\dots , x _ r } , \\ \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} U = \\{ ( i , j ) \\in \\widetilde E _ W \\mid i \\notin S _ j \\} . \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} _ { x _ 0 } M _ i \\subseteq \\big \\{ u _ { x _ 0 } = 0 \\big \\} . \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle \\sum _ { k = 1 } ^ { M } K _ 4 ^ { ( k ) } \\leq \\frac { C h ^ { 2 r } } { \\Delta t } - \\sum _ { k = 1 } ^ { M } \\left ( f ( \\theta _ { \\Psi } ^ { k - 1 } , \\theta _ { \\Psi } ^ { k - 1 } ) , \\mathbf { v } \\right ) + C \\sum _ { k = 1 } ^ { M } \\left ( \\| \\nabla \\theta _ { \\Psi } ^ { k - 1 } \\| _ { \\mathbf { L } ^ 2 } ^ { 2 } + \\| \\mathbf { v } \\| _ { \\mathbf { L } ^ 2 } ^ { 2 } \\right ) . } \\end{array} \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{align*} ( n - 2 ) b = 0 \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a & b \\pi ^ n \\\\ c & d \\end{bmatrix} \\in R ^ \\bullet . \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} \\Psi ( e ^ { i z } ) = \\exp ( i \\chi _ e ( z ) ) , \\Psi ( 0 ) = 0 \\end{align*}"}
-{"id": "6656.png", "formula": "\\begin{gather*} \\alpha \\beta = 0 , \\ \\beta \\gamma = 0 , \\ \\gamma \\alpha = 0 , \\ \\alpha ^ r = ( \\beta \\eta \\gamma ) ^ s , \\ ( \\gamma \\beta \\eta ) ^ s = ( \\eta \\gamma \\beta ) ^ s , \\ \\eta ^ 2 = b ( \\eta \\gamma \\beta ) ^ s . \\end{gather*}"}
-{"id": "7917.png", "formula": "\\begin{align*} M ( T ) = \\lim _ { \\tau \\uparrow T } \\partial F _ \\tau ( U ) . \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{align*} \\varepsilon ( F _ { a } \\triangleright ( \\mathsf { N } _ { m } ^ { n } ) _ { j } ^ { i } ) & = \\pi ( F _ { a } ) _ { m } ^ { i } \\pi ( 1 ) _ { j } ^ { n } - \\pi ( K _ { a } ^ { - 1 } ) _ { m } ^ { i } \\pi ( K _ { a } F _ { a } ) _ { j } ^ { n } \\\\ & = \\pi ( F _ { a } ) _ { m } ^ { i } \\pi ( 1 ) _ { j } ^ { n } - \\pi ( 1 ) _ { m } ^ { i } \\pi ( F _ { a } ) _ { j } ^ { n } , \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} f _ { n } ( t ) = \\frac { t } { n } \\sum _ { k = 0 } ^ { n - 1 } ( 1 - 2 ^ { \\nu _ { 2 } ( n - k ) + 1 } ) f _ { k } ( t ) . \\end{align*}"}
-{"id": "5755.png", "formula": "\\begin{align*} & \\hat { \\sigma } _ { A B M | Z = \\mathbf { z } } \\\\ & = \\hat { D } _ A \\left ( \\frac { \\lambda \\mathbf { z } } { \\sqrt { \\eta } } \\right ) \\hat { D } _ B \\left ( \\frac { ( 1 - \\lambda ) T \\mathbf { z } } { \\sqrt { \\eta - 1 } } \\right ) \\hat { \\rho } _ { A B M } { \\hat { D } _ A \\left ( \\frac { \\lambda \\mathbf { z } } { \\sqrt { \\eta } } \\right ) } ^ \\dag { \\hat { D } _ B \\left ( \\frac { ( 1 - \\lambda ) T \\mathbf { z } } { \\sqrt { \\eta - 1 } } \\right ) } ^ \\dag \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } \\binom { 2 k } { k } \\frac { H _ k } { k } x ^ k \\equiv 2 ( \\alpha - \\beta ) ^ { p } \\left ( \\pounds _ 2 \\left ( \\frac { \\alpha } { \\alpha - \\beta } \\right ) - \\pounds _ 2 \\left ( \\frac { \\beta } { \\beta - \\alpha } \\right ) \\right ) \\pmod { p } , \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} L _ y = e ^ { y + o ( y ) } < e ^ { 2 y } = ( \\log x ) ^ 2 \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{align*} \\left \\Vert e ^ { \\gamma \\left \\vert x \\right \\vert ^ { p } } \\tilde { u } _ { k } \\left ( x , 0 \\right ) \\right \\Vert _ { X } = \\left \\Vert e ^ { \\gamma \\left ( \\frac { \\alpha } { \\beta } \\right ) ^ { p / 2 } \\left \\vert x \\right \\vert ^ { p } } u _ { k } \\left ( x , 0 \\right ) \\right \\Vert _ { X } = B _ { 0 } , \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} g _ { \\mathrm { r e g } } \\ ; & : = \\ ; f + a \\ , S _ D ^ { - 1 } \\Phi \\in \\mathcal { D } ( S _ D ) \\\\ g _ { \\mathrm { s i n g } } \\ ; & : = \\ ; \\frac { b } { \\gamma } \\ , \\Phi \\in \\ker S ^ * \\ , , \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} E ( u ) = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } \\times \\mathbb { T } } \\left [ u _ x ^ 2 + u _ y ^ 2 - 4 u ^ 3 \\right ] d x d y \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} [ X _ { c ( t ) } ^ * ] ^ { \\perp } = \\left \\{ \\tilde { u } \\in L ^ 2 _ { \\mu } ( \\mathbb { R } \\times \\mathbb { T } ) : \\langle u _ { c ( t ) } , \\tilde { u } \\rangle _ { L ^ 2 ( \\mathbb { R } \\times \\mathbb { T } ) } = \\langle \\partial _ { \\xi } ^ { - 1 } \\partial _ c u _ { c ( t ) } , \\tilde { u } \\rangle _ { L ^ 2 ( \\mathbb { R } \\times \\mathbb { T } ) } = 0 \\right \\} . \\end{align*}"}
-{"id": "5062.png", "formula": "\\begin{align*} [ \\Z / m \\Z ] _ { G , \\alpha } \\cong \\bigoplus _ { i = 1 } ^ t [ \\Z / p _ i ^ { e _ i } \\Z ] _ { G , \\alpha _ { e _ i } } \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} ( x , y ) = \\mbox { t h e c o e f f i c i e n t o f $ \\omega _ i $ i n $ x y $ } , \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{gather*} \\left ( \\prod _ { i = 0 } ^ n y _ i \\right ) \\left ( \\sum _ { i = 0 } ^ n ( - 1 ) ^ { i } \\frac { d y _ 0 } { y _ 0 } \\wedge \\dots \\wedge \\widehat { \\frac { d y _ i } { y _ i } } \\wedge \\dots \\wedge \\frac { d y _ n } { y _ n } \\right ) . \\end{gather*}"}
-{"id": "4727.png", "formula": "\\begin{align*} \\dot { \\bar x } = - \\beta _ { 2 , 1 } \\nabla J ( \\bar x ) F _ 0 ( J ( \\bar x ) ) . \\end{align*}"}
-{"id": "839.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\varepsilon ^ 2 } { 2 m } \\Delta u - V ( x ) u + \\psi u = 0 , \\ , & x \\in \\R ^ { 3 } , \\\\ \\Delta \\psi + 4 \\pi \\tau | u | ^ 2 = 0 , \\ , & x \\in \\R ^ { 3 } , \\end{cases} \\end{align*}"}
-{"id": "5682.png", "formula": "\\begin{align*} & q ( t ) = \\left ( \\begin{array} { c } - \\varepsilon ( b \\bar { v } - c \\bar { w } ) \\\\ - \\bar { v } ^ 3 + ( a + 1 ) \\bar { v } ^ 2 - a \\bar { v } - \\bar { w } \\end{array} \\right ) \\\\ & \\times \\frac { 1 } { \\sqrt { ( - \\bar { v } ^ 3 + ( a + 1 ) \\bar { v } ^ 2 - a \\bar { v } - \\bar { w } ) ^ 2 + \\varepsilon ^ 2 ( b \\bar { v } - c \\bar { w } ) ^ 2 } } . \\end{align*}"}
-{"id": "8203.png", "formula": "\\begin{align*} \\int _ { A _ k } F ^ p ( \\nabla u ) \\ d x = \\lambda \\int _ { A _ k } | u | ^ { p - 2 } u ( u - k ) \\ d x \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} v ( \\mathsf { x , } t ) = \\mathcal { N } \\exp \\left [ - \\frac { \\left \\vert \\mathsf { x } \\right \\vert ^ { 2 } + d ( T - t ) } { 2 } \\right ] \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} \\mathcal { V } _ \\theta ^ { - 1 } \\big ( c ^ { ( z ) } \\big ) _ { z \\in \\mathbb { Z } ^ 3 } = \\sum _ { z \\in \\mathbb { Z } ^ 3 } c ^ { ( z ) } e ^ { \\i \\langle \\theta + 2 \\pi z , \\cdot \\rangle _ { \\C ^ 3 } } , \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} \\| \\widetilde { E } \\| _ { L ^ { \\frac { 1 2 d - 8 } { 3 d + 4 } , \\infty } ( S _ t , d \\sigma ) } \\lesssim \\| E \\| _ { L ^ { \\frac { 4 } { 3 } } ( \\mathbb F _ q ^ d , d \\textbf { m } ) } = | E | ^ { \\frac { 3 } { 4 } } \\quad \\mbox { f o r a l l } ~ ~ E \\subset \\mathbb F _ q ^ d , ~ t \\ne 0 . \\end{align*}"}
-{"id": "2579.png", "formula": "\\begin{align*} \\tilde X : = \\sum _ { l \\in \\Z \\setminus \\{ 0 \\} } \\tilde { X _ l } ( \\psi , \\bar \\psi ) e ^ { i l \\cdot } , \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} h \\tau h ^ { - 1 } \\tau ^ { - \\frac { w } { z } } & = \\left ( \\begin{pmatrix} 1 & - 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a & - b \\\\ - c & d \\end{pmatrix} p \\right ) \\left ( \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a & * \\\\ c & d \\end{pmatrix} p \\right ) \\\\ & = I + \\begin{pmatrix} 2 a - c & * \\\\ 0 & 2 d - c \\end{pmatrix} p . \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} R _ 1 ( k _ 1 , \\dots , k _ n ) = ( - k _ 1 , k _ 2 , \\dots , k _ n ) . \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{align*} \\hat \\gamma ^ i _ t = \\frac { \\theta + \\left ( 1 - \\frac { 1 } { n } \\right ) \\phi _ t } { 1 + \\frac { 1 } { \\lambda } \\left ( 1 - \\frac { 1 } { n } \\right ) ^ 2 \\phi _ t } ( \\bar X _ { t - } - X ^ i _ { t - } ) , t \\in [ 0 , T ] \\ , , \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{align*} - 2 i k \\varphi ( k , x ) J ( k ) ^ { - 1 } = f ( - k , x ) + f ( k , x ) S ( k ) , k \\in \\mathbb { R } . \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} \\iint _ { \\Sigma _ - } \\bigl ( v _ { k t } \\varphi _ i ( x ) \\psi _ m ( y ) - v _ k ( b \\varphi _ i ' \\psi _ m + \\varphi ''' _ i \\psi _ m + \\varphi ' _ i \\psi '' _ m ) \\bigr ) \\ , d x d y - \\iint _ { \\Sigma _ - } f \\varphi _ i \\psi _ m \\ , d x d y = 0 , \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{align*} \\lambda = \\lim _ { n \\to \\infty } \\lambda _ n . \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} \\mathbf { h } ( x ) \\coloneqq \\sum _ { n = 0 } ^ { \\infty } \\mathbf { h } _ { n } x ^ { n } \\end{align*}"}
-{"id": "9895.png", "formula": "\\begin{align*} g : = d r ^ 2 + \\sigma ^ 2 ( r ) d \\theta ^ 2 + \\tau ^ 2 ( r ) d z ^ 2 . \\end{align*}"}
-{"id": "908.png", "formula": "\\begin{align*} e ^ { \\frac { 1 } { 2 ( n + 1 ) } } \\frac { 1 } { \\sqrt 2 } \\left ( \\sqrt { p _ n } + \\sqrt { 1 - p _ n } \\right ) = e ^ { \\frac { 1 } { 2 ( n + 1 ) } } \\sqrt { \\frac { 1 } { 2 } + \\sqrt { p _ n ( 1 - p _ n ) } } = e ^ { \\frac { 1 } { 2 ( n + 1 ) } } \\sqrt { e ^ { - \\frac { 1 } { n + 1 } } } = 1 . \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} \\Psi _ { \\pi ( p ) } = \\left [ \\begin{array} { l l } C & 0 \\\\ 0 & B \\end{array} \\right ] \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{align*} \\Gamma _ \\kappa ( z , y ) = z \\int _ 1 ^ \\infty \\kappa ( y + \\tau z ) \\tau ^ 2 d \\tau , \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ f ( \\underbar { X } ) \\big | g ( \\underbar { X } ) \\right ] & = \\mathbb { E } \\left [ f ( \\underbar { X } ) \\right ] , \\forall g ( \\underbar { X } ) . \\end{align*}"}
-{"id": "10029.png", "formula": "\\begin{align*} g ( \\varphi X , Y ) = g ( X , \\varphi Y ) , \\forall X , Y \\in { \\mathfrak X } ( M ) . \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} \\| f _ \\varepsilon \\| _ { \\mathcal H ^ p _ a ( \\mathbb C _ + ) } = \\left ( \\int _ { - \\infty } ^ { \\infty } \\frac { 1 } { { \\sqrt { x ^ 2 + \\varepsilon ^ 2 } } ^ { 1 + p \\varepsilon } } d x \\right ) ^ { 1 / p } < \\infty \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} K _ S ^ 2 = ( Z + R ) ^ 2 = Z ^ 2 + 2 R Z + R ^ 2 = - n + 1 2 - 2 = 1 0 - n . \\end{align*}"}
-{"id": "6344.png", "formula": "\\begin{align*} X = \\left \\{ \\kappa ^ n : \\kappa \\in Y \\right \\} . \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t t } u - \\partial _ { x } ( a \\partial _ { x } u ) + b _ { 1 } \\partial _ { t } u - \\partial _ { x } ( b _ { 2 } \\partial _ { x } \\partial _ { t } u ) = 0 & \\mbox { i n } ( 0 , \\infty ) \\times ( 0 , \\pi ) , \\\\ u ( t , 0 ) = u ( t , \\pi ) = 0 & \\mbox { o n } ( 0 , \\infty ) , \\\\ u ( 0 , x ) = u _ { 0 } ( x ) & \\mbox { i n } ( 0 , \\pi ) \\\\ u _ { t } ( 0 , x ) = u _ { 1 } ( x ) & \\mbox { i n } ( 0 , \\pi ) , \\end{cases} \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} \\int K _ { r } \\left ( x , y \\right ) ~ d x \\approx \\int _ { y _ { 1 } - r } ^ { y _ { 1 } } \\left \\{ \\int _ { y _ { 2 } - h _ { x , y } } ^ { y _ { 2 } + h _ { x , y } } \\frac { 1 } { h _ { x , y } } d x _ { 2 } \\right \\} d x _ { 1 } \\approx \\int _ { x _ { 1 } } ^ { x _ { 1 } + r } d y _ { 1 } = r \\ . \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\bar { v } _ 0 = \\bar { \\textbf { w } } _ { 0 } ^ { \\ast } \\textbf { v } _ { 0 } & \\ ! = \\sum \\limits _ { k \\neq 0 } \\bar { \\textbf { w } } _ { 0 } ^ { \\ast } { \\mathbf { { H } } _ { k } ^ { } } \\textbf { w } _ { k } x _ { k } = \\sum \\limits _ { k \\neq 0 } \\sqrt { G _ k P _ k \\mathit { l } \\left ( { \\Vert \\bold { z } _ k \\Vert } _ 2 \\right ) } a _ k U _ k , \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{align*} & Y ( t ) : = \\left ( \\| w _ 0 \\| _ 2 ^ 2 + C \\| f \\| _ { L ^ 2 ( 0 , t ; H ^ { - 1 } ( \\Omega ) ) } ^ 2 \\right ) e ^ { C N t } , \\\\ & N : = 1 + | h | _ \\infty ^ 2 \\| u _ s \\| _ \\infty ^ 2 + \\| \\widetilde U _ { \\varepsilon _ 0 } \\| _ { L ^ \\infty ( 0 , \\infty ; L ^ \\infty ( \\Omega ) ) } ^ 2 , \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\widetilde { S } ^ \\alpha \\Gamma ^ a V - \\mu \\Delta \\sum \\limits _ { l = 0 } ^ \\alpha C _ \\alpha ^ l ( - 1 ) ^ { \\alpha - l } \\widetilde { S } ^ l \\Gamma ^ a V - \\nabla \\cdot \\widetilde { S } ^ \\alpha \\Gamma ^ a H = f ^ 1 _ { \\alpha a } , \\\\ \\partial _ t \\widetilde { S } ^ \\alpha \\Gamma ^ a H - \\nabla \\widetilde { S } ^ \\alpha \\Gamma ^ a V = f ^ 2 _ { \\alpha a } , \\end{cases} \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} B _ 1 ( u _ n , u _ n ) ( 4 t ) = \\biggl ( \\int _ 0 ^ { t _ A } + \\int _ { t _ A } ^ { 4 t } \\biggr ) F ( 4 t - s ) * ( u _ n \\otimes u _ n ) ( s ) \\dd s \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} \\eta _ n ^ h = \\Pi _ { n - 1 : 0 } h , \\end{align*}"}
-{"id": "9932.png", "formula": "\\begin{align*} \\frac { 2 } { k } + \\frac { 1 } { \\ell } = \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{align*} \\frac { \\sup _ { \\| X _ i \\| _ { \\ell _ \\mathbb { R } ^ 2 } = 1 } \\sum _ { i , j = 1 } ^ l a _ { i j } \\langle X _ i , X _ j \\rangle } { \\| P _ { A _ k } \\| _ { M _ p ( \\mathbb { Z } ^ l ) } } \\geq \\frac { 2 k ( k - 1 ) ^ 2 } { ( 2 k ( k - 1 ) ( 2 k - 3 ) ) ^ { 1 - \\frac { 2 } { p ^ \\prime } } \\big ( \\frac { 2 k ( k - 1 ) ( 2 k - 1 ) } { 3 } \\big ) ^ { \\frac { 2 } { p ^ \\prime } } } . \\end{align*}"}
-{"id": "876.png", "formula": "\\begin{align*} \\widehat { V ^ \\delta \\phi } ( x ) = \\widehat { h _ \\delta } ( x ) \\widehat { \\phi } ( x ) , x \\in \\mathbb { R } , \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a & b \\pi ^ n \\\\ c & d \\end{bmatrix} \\in R ^ \\bullet \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} H _ g ( v ) : = \\frac { R ( v , J v , J v , v ) } { \\norm { v } _ g ^ 4 } . \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} K _ i ( m + s , n ) = \\sum _ { j = 1 } ^ n K _ j ( s , n ) K _ { i - j + 1 } ( m , n ) , \\enskip i = 1 , . . . , n . \\end{align*}"}
-{"id": "9279.png", "formula": "\\begin{align*} t _ 0 = \\sup \\{ 0 \\le t \\le \\min ( T , \\lambda ) : \\eta ( \\tau ) \\le \\delta \\ \\ 0 \\le \\tau \\le t \\} , \\end{align*}"}
-{"id": "5131.png", "formula": "\\begin{align*} \\tilde { S } _ { m , n } = \\sum _ { \\underset { k _ { i } \\ge 1 } { k _ { 1 } + \\dots + k _ { m } = n } } \\binom { n } { k _ { 1 } , \\dots , k _ { m } } w _ { k _ { 1 } } \\dots w _ { k _ { m } } \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{align*} \\eta _ { X , Y } ( C ( \\mathsf { P } ) ) = \\sum _ { i , j , k } q ^ { - ( 2 \\rho , \\lambda _ i ) } ( 2 c ^ i _ j - \\delta ^ i _ j ) \\varepsilon ( X \\triangleright \\mathsf { P } ^ j _ k ) \\varepsilon ( Y \\triangleright \\mathsf { P } ^ k _ i ) . \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} { \\displaystyle - \\nabla \\cdot \\mathbf { A } ( \\mathbf { x } , t ) + \\nabla \\cdot \\mathbf { A } ( \\mathbf { x } , 0 ) - \\int _ { 0 } ^ { t } \\rho ( \\mathbf { x } , \\tau ) d \\tau = 0 \\ , , } \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{align*} H \\cdot \\Omega = \\{ h B \\ ; | \\ ; h \\in H , \\ ; B \\in \\Omega \\ ; \\} , \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{align*} \\nabla _ i ^ \\perp ( \\nabla ^ \\perp \\cdot H ) & = \\nabla _ l ^ \\perp H _ 2 \\nabla _ l \\nabla _ i ^ \\perp H _ 1 - \\nabla _ l ^ \\perp H _ 1 \\nabla _ l \\nabla _ i ^ \\perp H _ 2 \\\\ & = \\nabla _ l ^ \\perp H _ 2 \\nabla _ l \\nabla _ i ^ \\perp H _ 1 + \\nabla _ l H _ 1 \\nabla _ l ^ \\perp \\nabla _ i ^ \\perp H _ 2 \\\\ & = \\nabla _ i ^ \\perp ( \\nabla _ l ^ \\perp H _ 2 \\nabla _ l H _ 1 ) , \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{align*} c o n s t \\times \\prod _ { i = 1 } ^ { N } \\cos ^ { 2 N - 2 } \\left ( \\frac { \\theta _ i } { 2 } \\right ) \\prod _ { 1 \\le i < j \\le N } ^ { } \\left ( \\tan \\left ( \\frac { \\theta _ j } { 2 } \\right ) - \\tan \\left ( \\frac { \\theta _ i } { 2 } \\right ) \\right ) ^ 2 d \\theta _ 1 \\cdots d \\theta _ N , \\end{align*}"}
-{"id": "3861.png", "formula": "\\begin{align*} C _ 1 : = \\frac { 2 d } { \\rho - 1 } ( 1 + \\Lambda _ m ) ^ d \\exp \\left ( \\gamma \\left ( \\frac { \\rho + 1 / \\rho } { 2 } + 1 \\right ) \\right ) \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} \\Delta ( E ) : = - b _ 2 ^ 2 b _ 8 - 8 b _ 4 ^ 3 - 2 7 b _ 6 ^ 2 + 9 b _ 2 b _ 4 b _ 6 , \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} Y ^ { r } _ { h } = \\{ u _ { h } \\in C ( \\Omega ) : \\ ; u _ { h } | _ { K } \\in P _ { r } ( K ) , \\ ; \\forall \\ ; K \\in \\mathcal { T } _ { h } \\} . \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} Y = \\bigcap _ { w , i } Y ( w , U _ i ) , \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} M = \\begin{pmatrix} 1 & 1 & 1 & 1 & 0 & 1 \\\\ 1 & 1 & 1 & 1 & 1 & 0 \\\\ 0 & 1 & 1 & 1 & 1 & 1 \\\\ 1 & 0 & 1 & 1 & 1 & 1 \\\\ 1 & 1 & 0 & 1 & 1 & 1 \\\\ 1 & 1 & 1 & 0 & 1 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} \\Lambda _ { 1 } ( \\infty , \\Omega ) = \\Lambda _ { 1 } ( \\infty , \\Omega _ { 1 } ) \\end{align*}"}
-{"id": "4893.png", "formula": "\\begin{align*} \\tilde { \\textbf { y } } _ { 0 } ^ { } & = \\mathbf { \\tilde { h } } _ { 0 } ^ { } \\sqrt { P } U _ { 0 } + \\tilde { \\textbf { z } } _ { 0 } , { \\mathbf { \\tilde { h } } _ { 0 } ^ { } } = { \\left ( \\textbf { R } _ { 0 } \\right ) ^ { - 1 / 2 } \\textbf { H } _ { 0 } ^ { } \\textbf { w } _ { 0 } } . & & \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} d _ { } ( X _ 1 , X _ 2 ) : = 1 / 2 \\inf _ { { \\cal R } \\in C ( X _ 1 , X _ 2 ) } ( { \\cal R } ) \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} \\Gamma R _ 0 = f ^ * \\delta \\cdot R _ 0 = \\delta \\cdot f _ * R _ 0 = \\delta \\cdot ( n - 2 ) \\delta = 8 - 4 n . \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} w & 0 \\\\ 0 & w \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) ^ p = \\begin{pmatrix} w & 0 \\\\ 0 & w \\end{pmatrix} . \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} \\rho _ s ^ { - 1 } = \\limsup _ { \\ell \\to \\infty } | \\sin \\ell \\omega | ^ { - 1 / \\ell } , \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} { S } ^ { } _ 0 \\ , = \\ , \\left ( \\begin{matrix} 1 & 0 \\\\ 1 & 1 \\end{matrix} \\right ) , { S } ^ { } _ 1 \\ , = \\ , \\left ( \\begin{matrix} 1 & 1 \\\\ 0 & 1 \\end{matrix} \\right ) , { v } \\ , = \\ , \\left ( \\begin{matrix} 1 \\\\ 0 \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "9351.png", "formula": "\\begin{align*} \\mathcal { V } _ r ( \\gamma ) \\leq \\sum _ { j = 1 } ^ { J } \\mathcal { V } _ r ( \\gamma _ j ^ \\ell ) + \\sum _ { j = 1 } ^ { J + 1 } \\mathcal { V } _ r ( \\gamma _ j ^ s ) . \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} c _ { \\alpha } ^ + = \\psi _ { \\alpha , \\varepsilon } ( 0 ) = \\psi _ { \\alpha , 0 } ( 0 ) , \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} H _ 2 = H _ 2 ( M , \\gamma , \\epsilon ) : = V ( \\gamma , \\epsilon ) \\bigcap \\bigcap _ { 1 \\leq j \\leq n } \\left ( \\{ \\# { \\cal E } _ j \\leq M \\log { n } \\} \\bigcup \\{ \\# { \\cal E } _ j > \\epsilon n \\} \\right ) \\end{align*}"}
-{"id": "2095.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\overbrace { \\begin{matrix} 1 & 0 & 0 & 0 & t _ 1 & 0 & \\theta ^ \\infty _ 1 \\\\ 0 & 0 & 0 & 0 & 0 & \\sqrt { t _ 2 } & \\theta ^ \\infty _ 2 / 2 \\\\ 0 & 0 & 0 & 0 & 0 & - \\sqrt { t _ 2 } & \\theta ^ \\infty _ 2 / 2 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "5012.png", "formula": "\\begin{align*} a ( t ) = \\int _ 0 ^ t c ( t ' ) d t ' + h ( t ) , c ( t ) = c _ * + \\delta ( t ) , \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{align*} \\pi _ { n + k } ( \\Sigma ^ { k } f ) = \\Phi _ { k } ^ { k ' } \\circ \\pi _ { n + k ' } ( \\Sigma ^ { k ' } f ) \\circ ( \\Phi _ { k } ^ { k ' } ) ^ { - 1 } . \\end{align*}"}
-{"id": "6396.png", "formula": "\\begin{align*} \\Delta ( H ) = \\bigcup _ { a \\in H } \\Delta ( a ) \\end{align*}"}
-{"id": "9929.png", "formula": "\\begin{align*} q = \\frac { 1 - 2 \\varepsilon } { \\frac { 1 } { 2 } - \\varepsilon ' - \\varepsilon } \\geq 2 \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} 0 > ( E _ i + E _ j ) ^ 2 = 2 E _ i \\cdot E _ j + E _ i ^ 2 + E _ j ^ 2 \\end{align*}"}
-{"id": "13.png", "formula": "\\begin{align*} \\mu K _ 1 \\frac { 1 } { 1 + c ^ 2 } \\frac { b + c } { 1 + c } - \\frac { \\Gamma c } { K + c } + \\lambda T ( c ) = 0 . \\end{align*}"}
-{"id": "7082.png", "formula": "\\begin{align*} \\partial _ { t } u - i \\left [ \\Delta u + A u + V \\left ( x , t \\right ) u \\right ] = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} ( 1 - x ) ^ { m } H _ { m } ( x ) & = \\left ( \\sum _ { j = 0 } ^ { m } { m \\choose j } ( - 1 ) ^ { j } x ^ { j } \\right ) \\left ( \\sum _ { n = 0 } ^ { \\infty } b _ { m } ( n ) x ^ { n } \\right ) \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\left ( \\sum _ { j = 0 } ^ { m } { m \\choose j } ( - 1 ) ^ { j } b _ { m } ( n - j ) \\right ) x ^ { n } = \\sum _ { n = 0 } ^ { \\infty } b _ { m } ( n ) x ^ { 2 n } . \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} & \\dim \\mathbf { T } = \\binom { c - b + 3 } { 3 } + \\binom { c - a + 3 } { 3 } - \\binom { b - a + 3 } { 3 } + \\binom { c + a - e + 3 } { 3 } \\\\ & - \\binom { a + b - e + 3 } { 3 } - \\binom { 2 a - e + 3 } { 3 } + \\delta ( e , a , b , c ) - 3 . \\end{align*}"}
-{"id": "9490.png", "formula": "\\begin{align*} \\phi _ { \\ell - 1 } ( S ) = \\phi _ { \\ell } ( S ) + \\tau _ { \\Omega ' _ { k , \\ell } } ( S ) \\equiv 0 + 0 \\equiv 0 \\mod { b _ k } . \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} u ( \\mathsf { x , } t ) = \\sum _ { \\mathsf { n } \\in \\mathbb { N } ^ { d } } \\alpha _ { \\mathsf { n } } \\exp \\left [ - t E _ { \\mathsf { n } } \\right ] \\mathsf { h } _ { \\mathsf { n } } \\left ( \\mathsf { x } \\right ) \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} \\tilde h _ n ( \\zeta ) = a _ { m + 1 } ( n ) e ^ { i ( m + 1 ) \\theta } \\zeta ^ { m + 1 } + \\dots + a _ 1 ( n ) e ^ { i \\theta } \\zeta . \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} \\Re \\xi ( z ) = \\Re \\left ( \\int _ { \\cos \\theta } ^ z \\frac { \\rho _ { \\alpha , \\varepsilon } ( s ) - \\rho _ { \\alpha , \\varepsilon } ( \\cos \\theta ) } { ( s ^ 2 - 1 ) ^ { 1 / 2 } } d s \\right ) + \\rho _ { \\alpha , \\varepsilon } ( \\cos \\theta ) \\log \\tau . \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{gather*} c _ 1 ( k , k ) = 1 , \\\\ c _ 1 ( n , k ) = \\sum _ { i = 1 } ^ k \\sum _ { j = i } ^ { \\left \\lfloor \\frac { n - k } { 2 } \\right \\rfloor } { k \\choose i } { j - 1 \\choose i - 1 } { n - k - j - 1 \\choose j - 1 } , ( n > k ) . \\end{gather*}"}
-{"id": "6919.png", "formula": "\\begin{align*} - \\Delta t + ( 2 \\alpha + \\alpha _ t t ) \\frac { t } { \\varepsilon ^ 2 } = c \\left ( | \\nabla \\psi | ^ 2 + 2 \\nabla \\psi \\cdot \\nabla \\zeta + | \\nabla \\zeta | ^ 2 \\right ) . \\end{align*}"}
-{"id": "5161.png", "formula": "\\begin{align*} g \\left ( z \\right ) = e ^ { z } - 1 = \\sum _ { n \\ge 1 } \\frac { z ^ { n } } { n ! } \\end{align*}"}
-{"id": "9611.png", "formula": "\\begin{align*} \\widetilde W ^ i _ { j m n } = W ^ i _ { j m n } - \\frac 1 2 \\big ( F ^ i _ j \\sigma _ m + F ^ i _ m \\sigma _ j \\big ) _ { | n } + \\frac 1 2 \\big ( F ^ i _ j \\sigma _ n + F ^ i _ n \\sigma _ j \\big ) _ { | m } , \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} \\lambda _ { k } = \\int _ { \\Omega } F ^ { p } ( \\nabla u _ { k } ) \\ , d x \\quad \\int _ { \\Omega } | u _ k | ^ { p } \\ , d x = 1 \\end{align*}"}
-{"id": "6959.png", "formula": "\\begin{align*} \\| \\widehat { f } - f _ 0 \\| _ 0 ^ 2 & = O \\left \\{ \\lambda J ( f _ 0 ) \\right \\} + O _ \\P \\left \\{ n ^ { - 1 } \\lambda ^ { - 1 / ( 2 m - 2 ) } \\right \\} + o _ \\P \\left \\{ n ^ { - 1 } \\lambda ^ { - 1 / ( 2 m - 2 ) } \\right \\} \\\\ & = O _ \\P \\left \\{ n ^ { - 2 ( m - 1 ) / ( 2 m - 1 ) } \\right \\} . \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} A = A \\bigl ( \\Pi , \\underline { \\mu } \\bigr ) = \\bigl \\{ f \\in R \\ , \\big | \\ , f ' _ \\alpha = f ^ { ''' } _ \\alpha = \\dots = f ^ { ( 2 \\mu _ \\alpha - 1 ) } _ { \\alpha } = 0 \\ ; \\ ; \\mbox { \\rm f o r a l l } \\ ; \\ ; \\alpha \\in \\Pi \\bigr \\} . \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { r } ( - 1 ) ^ m \\frac { ( 2 d ) ^ m } { m ! } \\overline { p } _ i = p _ i e ^ { - 2 d } + o ( n ^ { - t } e ^ { 2 d } ) , \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{align*} | \\partial _ x ^ { ( i ) } p _ t ( x , y ) | \\lesssim \\left ( \\frac { 1 } { \\sqrt { 1 + x ^ 2 } } \\right ) ^ { i } \\left ( \\frac { 1 } { \\sqrt { 1 + y ^ 2 } } \\right ) \\sum _ { j = 0 } ^ { i } c ( j , i , t ) \\exp \\left ( - \\left ( C _ j t - \\frac { ( a r s i n h ( x ) - a r s i n h ( y ) ) ^ 2 } { C _ j t } \\right ) ^ - \\right ) + \\mathsf { l } . \\mathsf { o } . \\mathsf { t } , \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { n } \\sum \\limits _ { i = 0 } ^ { 2 } \\left \\vert \\lambda \\right \\vert ^ { 1 - \\frac { i } { 2 } } \\left \\Vert x _ { k } ^ { i \\alpha } \\frac { \\partial ^ { i } u } { \\partial x _ { k } ^ { i } } \\right \\Vert _ { L _ { p } \\left ( G ; E \\right ) } + \\left \\Vert A u \\right \\Vert _ { L _ { p } \\left ( G ; E \\right ) } \\leq M \\left \\Vert f \\right \\Vert _ { L _ { p } \\left ( G ; E \\right ) } . \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{align*} \\eta _ { a } ( C ( \\mathsf { Q } ) ) = \\sum _ { i , j } c _ { j } ^ { i } \\pi ( E _ { a } K _ { 2 \\rho } K _ { a } ^ { - 1 } F _ { a } ) _ { i } ^ { j } - \\sum _ { i , j } c _ { j } ^ { i } \\pi ( K _ { a } F _ { a } E _ { a } K _ { 2 \\rho } K _ { a } ^ { - 2 } ) _ { i } ^ { j } . \\end{align*}"}
-{"id": "1628.png", "formula": "\\begin{align*} K ^ m _ \\beta ( \\Omega ) = \\{ u \\in L ^ 2 _ { \\rm l o c } ( \\Omega ) , \\rho ^ { \\beta + | \\alpha | } \\partial ^ \\alpha _ t u \\in L ^ 2 ( \\Omega ) , \\ \\ \\forall \\alpha \\in \\N ^ 2 , \\ | \\alpha | \\le m \\} . \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A u + \\left \\Vert u \\right \\Vert ^ { \\frac { 4 } { n } } u = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { k = 1 } ^ n P ( x _ k ) Q _ k ( x ) ^ 2 . \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} \\begin{cases} \\gamma _ { k , j } ^ { u _ 0 } = \\widehat \\gamma _ { k , j } = - i \\int _ { 0 } ^ T u _ 1 ( s ) e ^ { - i ( \\lambda _ j ^ { u _ 0 } - \\lambda _ k ^ { u _ 0 } ) s } d s B ^ { u _ 0 } _ { k , j } , \\ \\ \\ \\ \\ & \\forall j , k \\in \\N ^ * , \\ k \\neq j , \\\\ \\gamma _ { k , k } ^ { u _ 0 } = \\Re ( \\widehat \\gamma _ { k , k } ) = 0 , & \\forall k \\in \\N ^ * . \\\\ \\end{cases} \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{gather*} \\alpha \\circ \\beta = \\beta \\circ \\alpha , \\\\ \\alpha ( x y ) = \\alpha ( x ) \\alpha ( y ) \\beta ( x y ) = \\beta ( x ) \\beta ( y ) , \\\\ \\alpha ( x ) ( y z ) = ( x y ) \\beta ( z ) . \\end{gather*}"}
-{"id": "7604.png", "formula": "\\begin{align*} v _ \\epsilon ' v _ \\epsilon '' = \\frac 1 2 ( ( v _ \\epsilon ' ) ^ 2 ) ' \\leq C e ^ { \\alpha \\rho } , \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} \\dot x = J _ 1 ( x ) u _ { 1 } ^ \\varepsilon ( t ) + u _ { 2 } ^ \\varepsilon ( t ) . \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} \\chi ^ { \\prime } _ + ( t ) & = \\max _ { \\alpha \\in \\partial \\rho ( v _ 0 + t v _ 1 ) } \\alpha ^ T v _ 1 \\\\ \\chi ^ { \\prime } _ - ( t ) & = \\min _ { \\alpha \\in \\partial \\rho ( v _ 0 + t v _ 1 ) } \\alpha ^ T v _ 1 \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{align*} x _ { n } = \\frac { B _ { n } \\left ( x \\right ) } { n ! } = \\sum _ { p = 1 } ^ { n } \\binom { n + 1 } { p + 1 } \\sum _ { k _ { 1 } + \\dots + k _ { p } = n } g _ { k _ { 1 } } \\dots g _ { k _ { p } } \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} v _ \\epsilon '' ( t , \\rho _ t ) = \\sup _ { \\rho \\in \\mathbb R } v _ \\epsilon '' ( t , \\rho ) . \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} - \\log ( 1 - x ) + \\log ( 1 - y ) = \\log \\left ( \\frac { 1 - y } { 1 - x } \\right ) \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} ( f ' \\otimes g ' ) \\circ ( f \\otimes g ) = ( - 1 ) ^ { | g ' | | f | } ( f ' \\circ f ) \\otimes ( g ' \\circ g ) . \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} \\iiint _ { Q _ T } P \\widetilde u _ 0 ( \\phi _ t + b \\phi _ x + \\phi _ { x x x } + \\phi _ { x y y } ) \\ , d x d y d t + \\iint \\widetilde u _ 0 \\phi \\big | _ { t = 0 } \\ , d x d y = 0 . \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{align*} \\mathfrak { n } ( t ) = \\underset { 1 \\le i \\le N } { a r g m i n } \\{ \\alpha _ i ^ { + } \\left ( X ^ { ( N ) } ; t \\right ) = 0 \\} , \\end{align*}"}
-{"id": "1347.png", "formula": "\\begin{align*} \\varphi ^ { i } _ { \\alpha } ( U ^ { i } _ { \\alpha } \\times \\Lambda ^ { i } _ { \\alpha } \\times D ^ { k } \\times \\{ 0 \\} ) \\bigcap \\varphi ^ { j } _ { \\beta } ( U ^ { j } _ { \\beta } \\times \\Lambda ^ { j } _ { \\beta } \\times D ^ { k } \\times \\{ 0 \\} ) = \\emptyset , \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} \\| \\chi _ { B ( 0 , R ) } \\| _ { w \\mathcal { M } ^ { p } _ { \\phi } } \\le m \\prod \\limits _ { i = 1 } ^ m \\| \\chi _ { B ( 0 , R ) } \\| _ { w \\mathcal { M } ^ { p _ i } _ { \\phi _ i } } . \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} S ^ * ( X , \\mathbf { d } , \\mathbf { d } ) & = C ^ * ( \\mathbf { d } ) \\operatorname { v o l } ( \\mathcal { R } ) X ^ 2 ( \\log X ) ^ 3 + O _ \\varepsilon ( ( d _ 1 d _ 2 d _ 3 ) ^ { \\varepsilon - 1 } X ^ 2 ( \\log X ) ^ 2 ) + O _ \\varepsilon ( ( d _ 1 d _ 2 d _ 3 ) ^ \\varepsilon X ^ { \\frac { 2 3 } { 1 2 } + \\varepsilon } ) . \\end{align*}"}
-{"id": "10011.png", "formula": "\\begin{align*} d _ h ( w ) = & \\ | \\{ ( j , i ) \\mid 2 \\leq j < i \\leq n - 1 , \\ w ( j ) > w ( i ) \\} | \\\\ & \\qquad + | \\{ ( 1 , i ) \\mid 1 < i < n , \\ a > w ( i ) \\} | \\\\ & \\qquad + | \\{ ( j , n ) \\mid 1 < j < n , \\ w ( j ) > b \\} | \\\\ = & \\ | \\{ ( j , i ) \\mid 2 \\leq j < i \\leq n - 1 , \\ w ( j ) > w ( i ) \\} | + ( a - 1 ) + ( n - b ) . \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} \\alpha _ { j } ( \\omega ) : = { \\lambda _ { j } \\over | \\omega ^ 2 - \\lambda _ { j } | } . \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} q ( C ) = 1 . \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} \\langle f - \\Bigl ( \\int f \\Bigr ) \\delta , \\varphi \\rangle & = \\int f ( x ) [ \\varphi ( x ) - \\varphi ( 0 ) ] \\dd x = \\int f ( x ) \\int _ 0 ^ 1 x \\cdot \\nabla \\varphi ( \\lambda x ) \\dd \\lambda \\dd x \\\\ & = \\int \\ ! \\ ! \\ ! \\int _ 0 ^ 1 f ( x / \\lambda ) \\frac { \\dd \\lambda } { \\lambda ^ { n + 1 } } x \\cdot \\nabla \\varphi ( x ) \\dd x = \\langle \\div V , \\varphi \\rangle . \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} \\{ \\chi _ m , H _ 0 \\} + f ^ { ( m ) } _ 0 \\ ; & = \\ ; Z _ { m + 1 } , \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{align*} x _ j = \\sqrt { - 1 } \\frac { t _ j + 1 } { t _ j - 1 } . \\end{align*}"}
-{"id": "5900.png", "formula": "\\begin{align*} 1 + \\frac { 2 } { ( r - 3 ) / ( r - 1 ) } \\frac { 1 } { ( r - 1 ) ( 1 - 1 / r ) } & = 1 + \\frac { 2 r } { ( r - 1 ) ( r - 3 ) } \\\\ & = \\frac { r ^ 2 - 2 r + 3 } { ( r - 1 ) ( r - 3 ) } \\\\ & = \\frac { ( r - 1 ) } { ( r - 3 ) } \\left ( 1 + \\frac { 2 } { ( r - 1 ) ^ 2 } \\right ) . \\end{align*}"}
-{"id": "7321.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\psi ( t ) = \\Delta _ { t } \\psi ( t ) + 2 d \\Big ( \\big ( A - { \\rm { T r } _ t } ( \\tau ( t ) ) \\big ) \\varphi ( t ) \\Big ) , d \\psi ( t ) \\ , = \\ , 0 , \\psi ( 0 ) = \\psi , \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} \\mathbb { S } ( R ^ { \\bigstar } ) = \\frac { 2 } { d ^ { n + 1 } } \\left [ \\frac { d _ 1 d _ 2 \\dots d _ n } { n + 1 } + \\frac { W ^ { ( n ) } _ 1 } { n } + \\dots + \\frac { W ^ { ( n ) } _ s } { n - s + 1 } + \\dots + \\frac { W ^ { ( n ) } _ { n - 1 } } { 2 } \\right ] \\end{align*}"}
-{"id": "5508.png", "formula": "\\begin{align*} \\widehat \\psi _ { p _ 1 } ( \\mathbb C ) \\cap \\widehat \\psi _ { p _ 2 } ( \\mathbb C ) \\neq \\emptyset \\Rightarrow \\widehat \\psi _ { p _ 1 } ( \\mathbb C ) = \\widehat \\psi _ { p _ 2 } ( \\mathbb C ) . \\end{align*}"}
-{"id": "3600.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to 0 } \\partial _ \\lambda \\Lambda ( \\gamma , \\lambda ) = \\partial _ \\lambda \\Lambda _ 0 ( \\lambda ) . \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{align*} \\Phi ( u ) ( t ) = e ^ { i t \\Delta ^ 2 } u _ 0 + i \\mu \\int _ 0 ^ t e ^ { i ( t - s ) \\Delta ^ 2 } | u ( s ) | ^ { \\nu - 1 } u ( s ) d s = : u _ { } ( t ) + u _ { } ( t ) \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} u ( t ) - \\int _ { 0 } ^ { t } u ( s ) K ( t , s ) d s = \\widehat { f } ( t ) , \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} \\Omega _ 2 ( x ) = \\frac { \\Gamma ( 1 - a - a x - \\gamma ) \\Gamma ( 1 - a - a x - \\delta ) \\Gamma ( 1 - a - a x + b + b x - \\gamma - \\delta ) } { \\Gamma ( 1 - a - a x + b + b x - \\gamma ) \\Gamma ( 1 - a - a x + b + b x - \\delta ) \\Gamma ( 1 - a - a x - \\gamma - \\delta ) } , \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} d _ k = c _ k = \\frac { 1 } { k + 1 } \\mbox { f o r a l l } k \\geq 2 . \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} \\left \\Vert \\tilde { u } _ { \\varepsilon } \\left ( . , t _ { i } \\right ) \\right \\Vert _ { X } \\geq \\frac { 1 } { 4 N _ { 1 } } \\left \\Vert u \\left ( . , 0 \\right ) \\right \\Vert _ { X } , 0 < \\varepsilon \\leq \\varepsilon _ { 0 } , i = 1 , 2 , . . . m \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} P _ z ( \\zeta ) = \\frac { 1 } { 2 \\pi } \\frac { 1 - r ^ 2 } { \\abs { \\zeta - z } ^ 2 } \\ge \\frac { 1 } { 2 \\pi } \\frac { 1 - r } { 1 5 ( 1 - r ) \\delta } = \\frac { 1 } { 3 0 \\pi \\delta } \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ r ( - 1 ) ^ m \\frac { \\lambda ^ m } { m ! } = e ^ { - \\lambda } + O \\left ( ( \\log n ) ^ 6 e ^ \\lambda / n \\right ) , \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} e ^ { 2 f } \\tilde { s } ^ H & = s ^ H + m \\Delta ^ g ( f ) + m g ( \\theta , d f ) \\\\ \\tilde { s } ^ H & = u ^ 2 s ^ H - m u \\left ( \\Delta ^ g ( u ) + g ( \\theta , d u ) \\right ) - m | d u | ^ 2 _ g \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{align*} A _ n = \\begin{pmatrix} \\frac { 1 } { r _ n } & & & & \\\\ & \\frac { 1 } { z _ n } & & & \\\\ & & 1 & & \\\\ & & & \\ddots & \\\\ & & & & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} { } _ 3 F _ 2 \\bigg [ \\begin{matrix} \\alpha & \\alpha & \\beta \\\\ & 1 & \\alpha - \\beta + 1 \\end{matrix} \\bigg | \\ , z \\bigg ] = ( 1 - z ) ^ { - \\alpha } { } _ 3 F _ 2 \\bigg [ \\begin{matrix} 1 - \\beta & \\frac 1 2 \\alpha & \\frac 1 2 \\alpha + \\frac 1 2 \\\\ & 1 & \\alpha - \\beta + 1 \\end{matrix} \\bigg | \\ , - \\frac { 4 z } { ( 1 - z ) ^ 2 } \\bigg ] . \\end{align*}"}
-{"id": "4994.png", "formula": "\\begin{align*} ( L _ { c _ * } + 4 ) w _ 2 = 6 \\psi _ * ^ 2 , \\end{align*}"}
-{"id": "337.png", "formula": "\\begin{align*} \\varepsilon ( F _ { a } \\triangleright ( \\mathsf { M } _ { m } ^ { n } ) _ { j } ^ { i } ) = - \\pi ( K _ { a } F _ { a } ) _ { m } ^ { i } \\pi ( 1 ) _ { j } ^ { n } + \\pi ( K _ { a } ) _ { m } ^ { i } \\pi ( F _ { a } ) _ { j } ^ { n } . \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{align*} \\frac { d } { d z } f ( z ) = - \\frac { 1 + w ^ 2 } { ( z - w ) ^ 2 } , \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{align*} r ^ i ( \\hat { \\rho } ) : = \\mathrm { T r } \\left [ \\hat { R } ^ i \\ , \\hat { \\rho } \\right ] \\ ; , i = 1 , \\ , \\ldots , \\ , 2 n \\ ; . \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} D ( N ^ { - 1 } ) ^ T D \\ , ( P D ) \\ , D N ^ T D = ( P _ 1 D _ 1 ) \\bigoplus ( P _ 1 D _ 1 ) \\bigoplus \\cdots \\bigoplus ( P _ 1 D _ 1 ) \\bigoplus \\cdots , \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} u _ \\varepsilon ( x ) = O \\big ( e ^ { - \\theta _ 2 / \\varepsilon } \\big ) , ~ \\mbox { f o r } ~ x \\in \\R ^ 3 \\backslash \\bigcup _ { j = 1 } ^ k B _ { d } ( x _ { j , \\varepsilon } ) . \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} \\dot x = P ( x , y ) , \\dot y = Q ( x , y ) , \\end{align*}"}
-{"id": "4649.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\left ( \\frac { 1 } { p _ n } + \\frac { 1 } { p _ { - n } } \\right ) . \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{align*} \\sum _ { k } q ^ { ( 2 \\rho , \\lambda _ { k } - \\lambda _ { b } ) } u _ { b } ^ { k } S ( u _ { k } ^ { a } ) = \\delta _ { b } ^ { a } 1 = \\sum _ { k } q ^ { ( 2 \\rho , \\lambda _ { a } - \\lambda _ { k } ) } S ( u _ { b } ^ { k } ) u _ { k } ^ { a } . \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} E \\ : : \\ : y ^ 2 = x ^ 3 + a x ^ 2 + b x , \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} a x ^ 2 - b y ^ 2 = a \\end{align*}"}
-{"id": "5092.png", "formula": "\\begin{align*} | \\mathbb { S } _ { \\mathcal { G } ' } ^ \\rho ( b _ 1 , h _ 2 , h _ 3 ) | & \\leq \\sum _ { R \\in \\mathcal { G } ' } \\mathop { \\sum _ { Q \\in \\mathcal { E } } } _ { R ( Q ) = R } | S _ R ( b _ { 1 , Q } , h _ 2 , h _ 3 ) | \\\\ & \\le \\sum _ { R \\in \\mathcal { G } ' } \\mathop { \\sum _ { Q \\in \\mathcal { E } } } _ { R ( Q ) = R } \\frac { \\| b _ { 1 , Q } \\| _ { L ^ 1 } \\| h _ 2 1 _ R \\| _ { L ^ 1 } \\| h _ 3 1 _ R \\| _ { L ^ 1 } } { | R | ^ 2 } , \\end{align*}"}
-{"id": "995.png", "formula": "\\begin{align*} f _ * E = 0 , f _ * E ^ 2 = 0 , f _ * E ^ 3 = Z _ 1 Z _ 2 Z _ 3 , f _ * E ^ 4 = ( Z _ 1 + Z _ 2 ) Z _ 1 Z _ 2 Z _ 3 . \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{align*} f \\ ( d \\ ) = 2 \\ ( \\pi \\lambda \\ ) ^ 2 d ^ 3 \\exp \\ ( - \\pi \\lambda d ^ 2 \\ ) . \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} f ( s ) = \\frac { \\lambda \\ , s ^ p } { \\mu + s ^ p } f ( s ) = \\frac { \\alpha + \\lambda \\ , s ^ p } { \\mu + s ^ p } ( p > 0 , \\ \\ \\mu > \\frac { \\alpha } { \\lambda } , \\ \\ \\lambda > 0 ) ; \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{align*} ( n - 1 ) t ^ 3 = ( n - 1 ) t \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 - \\eqref { 1 1 1 . 1 a } \\times \\bigg ( \\ln \\frac { u _ { \\alpha \\beta } ^ 2 } { \\bar { n } } \\bigg ) _ + ^ k d x , k = 1 , 2 , 3 , \\cdots , \\quad \\ ( \\cdot ) _ + : = \\max \\{ 0 , \\cdot \\} . \\end{align*}"}
-{"id": "2107.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 \\ , \\left ( \\frac 1 2 \\right ) & x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\overbrace { \\begin{matrix} 0 & 0 \\\\ 1 & \\theta ^ 0 / 2 \\\\ - 1 & \\theta ^ 0 / 2 \\end{matrix} } & \\overbrace { \\begin{matrix} \\sqrt { t } & \\theta ^ \\infty _ 1 / 2 \\\\ - \\sqrt { t } & \\theta ^ \\infty _ 1 / 2 \\\\ 0 & \\theta ^ \\infty _ 2 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "2420.png", "formula": "\\begin{align*} d s ^ { 2 } = \\epsilon d t ^ { 2 } + d x ^ 2 + d y ^ 2 , \\epsilon = \\pm 1 . \\end{align*}"}
-{"id": "5289.png", "formula": "\\begin{align*} V ( x ) = { 1 \\over \\gamma _ 3 } \\int _ \\Omega \\omega ( y ) \\times \\left [ f ( x , 0 ) + K _ 0 ( x , y ) \\right ] \\ , d y . \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} \\varphi ( t + \\varepsilon ) & = \\tilde G \\circ V ( t + \\varepsilon ) \\\\ & \\leq \\tilde G \\circ G ( \\varphi ( t ) + \\varepsilon ) \\\\ & \\leq \\varphi ( t ) + \\varepsilon , \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} A _ { \\rm b a c k } = \\tilde j ( I - \\tilde K ) ^ { - 1 } \\rho \\tilde J + \\tilde j ( 1 - \\rho ) \\tilde J . \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{align*} 2 \\Phi ( p ) = & 2 ( 1 - z ) ^ { \\frac a 2 } \\cdot { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 + \\frac 1 2 ( p - 1 - a ) & - \\frac 1 2 ( p - 1 - a ) \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac { z ^ 2 } { 4 z - 4 } \\bigg ] _ { p - 1 } \\\\ = & ( 1 - z ) ^ { a - \\frac { p - 1 } { 2 } } \\cdot \\big ( \\Omega _ 1 ( 0 ) + \\Omega _ 2 ( 0 ) \\big ) , \\end{align*}"}
-{"id": "4196.png", "formula": "\\begin{align*} \\oint _ { \\Gamma } \\tilde { P } \\left ( x , s \\right ) \\mathrm { e } ^ { s t } \\mathrm { d } s = 0 , \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} h _ { m _ j } \\ ; : = \\ ; h _ { m _ j , \\kappa _ j = j + \\frac { 1 } { 2 } } \\oplus h _ { m _ j , \\kappa _ j = - ( j + \\frac { 1 } { 2 } ) } \\end{align*}"}
-{"id": "8812.png", "formula": "\\begin{align*} f ^ { \\ast } ( g ^ { \\tilde { \\xi } } ) = e ^ { \\sigma } g ^ { \\xi } . \\end{align*}"}
-{"id": "6714.png", "formula": "\\begin{align*} \\left ( L _ { t } + \\lambda \\right ) u = P _ { t } \\left ( D \\right ) u + A u + \\lambda u = f \\left ( x \\right ) , x \\in R ^ { n } , \\end{align*}"}
-{"id": "7019.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 0 } ^ { 2 } \\varepsilon ^ { \\frac { j } { 2 } } \\left \\vert \\lambda \\right \\vert ^ { 1 - \\frac { j } { 2 } } \\left \\Vert u _ { 2 } ^ { \\left ( j \\right ) } \\right \\Vert _ { X } + \\left \\Vert A u _ { 2 } \\right \\Vert _ { X } \\leq C \\sum \\limits _ { k = 1 } ^ { 2 } \\left [ \\left \\Vert f _ { k } \\right \\Vert _ { E _ { k } } + \\left \\vert \\lambda \\right \\vert ^ { 1 - \\theta _ { k } } \\left \\Vert f _ { k } \\right \\Vert _ { E } + \\right . \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} p _ s ( x ^ 2 , y ^ 2 ) & = \\frac { 2 s y } { \\pi ( s ^ 2 + ( y - x ) ^ 2 ) ( s ^ 2 + ( y + x ) ^ 2 ) } \\\\ & = \\frac { 1 } { 2 \\pi x } \\left ( \\frac { s } { s ^ 2 + ( y - x ) ^ 2 } - \\frac { s } { s ^ 2 + ( y + x ) ^ 2 } \\right ) . \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} i \\omega \\cdot ( j - l ) \\chi _ { j , l } + Z _ { j , l } & = f _ { j , l } . \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} \\phi \\left ( \\gamma \\right ) = \\tau ^ { \\gamma } , \\ ; \\ ; \\tau > 0 . \\end{align*}"}
-{"id": "9535.png", "formula": "\\begin{align*} Y _ S : = \\{ y \\in A \\rtimes _ { \\alpha , r } G : { \\mathrm { s u p p } } ( y ) \\subseteq S \\} . \\end{align*}"}
-{"id": "9373.png", "formula": "\\begin{align*} L ^ 2 ( \\Omega _ 1 \\times \\Omega _ 2 ) = S ^ 2 ( L ^ 2 ( \\Omega _ 1 ) , L ^ 2 ( \\Omega _ 2 ) ) . \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{align*} \\mathbb { Z } [ \\Gamma / \\Gamma ' ] = \\mathbb { Z } [ t , t ^ { - 1 } ] . \\end{align*}"}
-{"id": "2329.png", "formula": "\\begin{align*} \\lambda = \\sigma ( 1 + 2 \\gamma ) \\sqrt { 2 \\log ( p / s ) } . \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} { } _ 3 F _ 2 \\bigg [ \\begin{matrix} - n & \\alpha & \\beta \\\\ & \\gamma & \\delta \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { ( \\gamma - \\alpha ) _ n ( \\gamma - \\beta ) _ n } { ( \\gamma ) _ n ( \\gamma - \\alpha - \\beta ) _ n } , \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} \\omega _ 2 ( 0 ) = \\frac { ( 1 - a + c + d ) _ b } { ( 1 - a + c ) _ b ( 1 - a + d ) _ b } \\equiv \\frac { ( 1 - a - \\gamma - \\delta ) _ b } { ( 1 - a - \\gamma ) _ b ( 1 - a - \\delta ) _ b } = \\Omega _ 2 ( 0 ) \\pmod { p } . \\end{align*}"}
-{"id": "8620.png", "formula": "\\begin{align*} D _ { n , k } ( 1 , y ( 1 - y ) ) & = k \\ , \\Big [ \\displaystyle \\frac { y ^ n ( 1 - y ) - y ( 1 - y ) ^ n } { 2 y - 1 } \\Big ] + D _ n ( 1 , y ( 1 - y ) ) , \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} \\beta _ { \\mathsf { n } } = \\mathcal { N } \\int _ { \\mathbb { R } ^ { d } } \\mathsf { d x } \\psi _ { T } ( \\mathsf { x } ) \\mathsf { h } _ { \\mathsf { n } } \\left ( \\mathsf { x } \\right ) , \\end{align*}"}
-{"id": "9955.png", "formula": "\\begin{align*} x \\in \\mathcal { C } ( E , A ) \\mbox { i f a n d o n l y i f } C x = 0 \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} \\begin{aligned} \\nu u , x \\mapsto x + c ( x , u ) . \\end{aligned} \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{gather*} \\chi _ { 2 , 0 } ^ { n , m } ( x , y | \\rho ) \\allowbreak = \\allowbreak \\sum _ { k \\geq 0 } \\rho ^ { k } T _ { k + n } ( x ) T _ { k + m } ( y ) \\allowbreak = \\allowbreak \\\\ ( T _ { n } ( x ) T _ { m } ( y ) ( w _ { 2 } ( x , y | \\rho ) - \\rho ^ { 4 } ) \\\\ + \\rho T _ { n + 1 } ( x ) T _ { m + 1 } ( y ) ( 1 - 2 \\rho ^ { 2 } + 4 \\rho ^ { 2 } ( x ^ { 2 } + y ^ { 2 } ) - 4 \\rho x y ) \\\\ + \\rho ^ { 2 } T _ { n + 2 } ( x ) T _ { m + 2 } ( y ) ( 1 - 4 \\rho x y ) + \\rho ^ { 3 } T _ { n + 3 } ( x ) T _ { m + 3 } ( y ) ) / w _ { 2 } ( x , y | \\rho ) , \\end{gather*}"}
-{"id": "9395.png", "formula": "\\begin{align*} \\bigl \\langle T ( x \\otimes w _ 1 ) , y \\otimes w _ 2 \\bigr \\rangle & = \\int _ { \\Omega } \\bigl \\langle [ \\Delta ( t ) ] ( w _ 1 ) , w _ 2 \\bigr \\rangle \\ , x ( t ) y ( t ) \\ , \\mu ( t ) \\\\ & = \\bigl \\langle M _ \\Delta ( x \\otimes w _ 1 ) , y \\otimes w _ 2 \\bigr \\rangle \\end{align*}"}
-{"id": "352.png", "formula": "\\begin{align*} ( S ( c _ { f , v } ^ { \\Lambda } ) \\triangleleft Y ) ( X ) & = S ( c _ { f , v } ^ { \\Lambda } ) ( Y X ) = c _ { f , v } ^ { \\Lambda } ( S ( Y X ) ) = f ( S ( X ) \\triangleright S ( Y ) \\triangleright v ) \\\\ & = f ( S ( X ) \\triangleright ( v \\triangleleft Y ) ) = c _ { f , v \\triangleleft Y } ^ \\Lambda ( S ( X ) ) = S ( c _ { f , v \\triangleleft Y } ^ \\Lambda ) ( X ) . \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{align*} u _ { x x x } = f - u _ t - b u _ x - u _ { x y y } . \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 2 e _ 4 , [ e _ 1 , e _ 2 ] = \\alpha _ 5 e _ 4 , [ e _ 2 , e _ 1 ] = \\beta _ 1 e _ 3 , [ e _ 2 , e _ 3 ] = \\gamma _ 2 e _ 5 , [ e _ 1 , e _ 4 ] = \\gamma _ 3 e _ 5 . \\end{align*}"}
-{"id": "4464.png", "formula": "\\begin{align*} \\tau ^ { ( \\omega , \\vartheta , \\lambda ) } : = \\{ \\tau _ { t } ^ { ( \\omega , \\vartheta , \\lambda ) } \\} _ { t \\in { \\mathbb { R } } } \\ . \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{align*} \\sum _ { l > n } C _ { l } \\ , ( b _ { l + 1 } - b _ { l } ) ^ { 1 - \\frac { 1 } { p } } \\ , \\Bigl ( \\ \\sup _ { b _ { l + 1 } - 2 ( b _ { n + 1 } - b _ n ) \\le i < b _ { l + 1 } } \\prod _ { s = i + 1 } ^ { b _ { l + 1 } - 1 } | w _ s | \\Bigr ) \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} w _ 1 = w _ 2 = \\{ 1 , 2 \\} , w _ 3 = w _ 4 = \\{ 1 , 2 , 3 , 4 \\} , w _ 5 = \\{ 1 , 2 , 3 , 4 , 5 \\} . \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} U _ f ( \\alpha ) = \\alpha ( \\alpha - 1 ) \\int _ 0 ^ { 1 } \\lambda ^ { \\alpha - 1 } \\mu ( E _ f ( \\lambda ) ) \\ , d \\lambda . \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { \\psi _ V ( 0 ) N } K _ N \\left ( \\frac { x } { \\psi _ V ( 0 ) N } , \\frac { y } { \\psi _ V ( 0 ) N } ; e ^ { - N V } \\right ) = \\frac { \\sin \\pi ( x - y ) } { \\pi ( x - y ) } . \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = 1 & x = \\infty \\ , \\left ( \\frac 1 3 \\right ) \\\\ \\begin{matrix} 0 \\\\ \\theta ^ 0 _ 1 \\\\ \\theta ^ 0 _ 2 \\end{matrix} & \\begin{matrix} 0 \\\\ 0 \\\\ \\theta ^ 1 \\end{matrix} & \\overbrace { \\begin{matrix} t ^ { \\frac 1 3 } & \\theta ^ \\infty _ 1 / 3 \\\\ \\omega t ^ { \\frac 1 3 } & \\theta ^ \\infty _ 1 / 3 \\\\ \\omega ^ 2 t ^ { \\frac 1 3 } & \\theta ^ \\infty _ 1 / 3 \\\\ \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "4032.png", "formula": "\\begin{align*} \\Psi ( 0 ) = \\Omega ( 0 ) \\Phi ( 0 ) . \\end{align*}"}
-{"id": "9482.png", "formula": "\\begin{align*} \\det ( A ) = \\frac { \\prod _ { 1 \\le i < j \\le n } ( x _ j - x _ i ) ( y _ i - y _ j ) } { \\prod _ { 1 \\le i , j \\le n } ( x _ i - y _ j ) } . \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{align*} L _ { m } \\left \\{ f \\left ( t \\right ) \\right \\} & = \\exp \\left ( \\int _ { 0 } ^ { \\infty } e ^ { - s t } \\ln \\left ( f \\left ( t \\right ) \\right ) d t \\right ) \\\\ ( f \\left ( t \\right ) & \\in \\left [ 0 , \\infty \\right ) ) . \\end{align*}"}
-{"id": "966.png", "formula": "\\begin{align*} \\frac { \\textrm { V a r } _ { B } [ X ] } { \\textrm { V a r } _ { B } [ Y ] } = k ' ( B ) \\frac { \\textrm { V a r } [ X ] } { \\textrm { V a r } [ Y ] } + \\left ( 1 - k ' ( B ) \\right ) \\beta \\beta ^ { T } , \\end{align*}"}
-{"id": "1289.png", "formula": "\\begin{align*} \\sum _ { \\mathsf { n } \\in \\mathbb { N } ^ { d } } \\alpha _ { \\mathsf { n } } \\exp \\left [ - T E _ { \\mathsf { n } } \\right ] \\beta _ { \\mathsf { n } } = 1 . \\end{align*}"}
-{"id": "9757.png", "formula": "\\begin{align*} j u _ { \\epsilon } = d ^ * \\xi _ { \\epsilon } + h _ { \\epsilon } . \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} \\varsigma ( s ) = \\tau ( s ) \\rho ( s ) + \\frac { { \\rm d } } { { \\rm d } s } \\left ( \\frac { \\rho ' ( s ) } { \\tau ( s ) } \\right ) \\ , \\mbox { a n d } \\rho = \\frac { 1 } { \\kappa } , \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} \\alpha ( 2 d ) : = \\frac { 3 } { 2 } d ( d - 1 ) + 1 . \\end{align*}"}
-{"id": "2338.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ s \\log ( 1 + s / j ) ^ 2 \\le s \\int _ 0 ^ 1 \\log ( 1 + 1 / x ) ^ 2 d x \\le 3 s . \\end{align*}"}
-{"id": "9900.png", "formula": "\\begin{align*} \\rho _ i ^ * = \\frac { 1 } { K } e ^ { \\frac { F _ i ( \\rho ^ * ) } { \\beta } } \\ , ~ \\textrm { f o r a n y $ i \\in S $ \\ , w h e r e } K = \\sum _ { j = 1 } ^ n e ^ { \\frac { F _ j ( \\rho ^ * ) } { \\beta } } \\ . \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} \\gamma _ { k } & = | \\big \\{ b _ { i } | b _ { i } = k \\big \\} | \\end{align*}"}
-{"id": "4869.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a _ i & 0 \\\\ c _ i & a _ i \\end{pmatrix} p \\right ) ^ { \\frac { 1 } { \\sqrt { c _ i } } } = \\begin{pmatrix} 1 & \\frac { 1 } { \\sqrt { c _ i } } \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a ' _ i & * \\\\ \\sqrt { c _ i } & a ' _ i \\end{pmatrix} p \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} 0 < { \\bf u } ' { \\bf u } + { \\bf v } ' { \\bf v } = { \\bf x } ' { \\bf A } _ 1 ^ { - } { \\bf x } + { \\bf y } ' { \\bf A } _ 2 ^ { - } { \\bf y } + ( { \\bf x } + { \\bf y } ) ' ( { \\bf A } _ 1 + { \\bf A } _ 2 ) ^ { - } ( { \\bf x } + { \\bf y } ) , \\end{align*}"}
-{"id": "3752.png", "formula": "\\begin{align*} \\left ( \\sum _ { i \\in \\N } \\log _ { i } ( \\omega ) \\right ) \\circ \\left ( \\sum _ { i \\in \\N } \\log _ { i } ( \\omega ) \\right ) = \\sum _ { i \\in \\N } \\log _ { i } \\left ( \\sum _ { j \\in \\N } \\log _ { j } ( \\omega ) \\right ) . \\end{align*}"}
-{"id": "161.png", "formula": "\\begin{align*} \\left [ ( S - 1 ) ^ 2 ( S + 1 ) \\alpha \\right ] _ j = \\alpha _ { j + 3 } - \\alpha _ { j + 2 } - \\alpha _ { j + 1 } + \\alpha _ j \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} { \\widetilde { \\mu } ^ X ( x ) } = { \\widetilde { \\mu } ^ { X ' } ( f ( x ) ) } \\end{align*}"}
-{"id": "3119.png", "formula": "\\begin{align*} \\begin{aligned} x _ { - j } ( t ) & \\leq \\tilde { z } _ { - j + 1 } , & j & \\geq 0 , \\\\ x _ j ( t ) & \\geq \\tilde { z } _ j , & j & \\geq 1 , \\end{aligned} \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } z _ { 1 } ^ { 2 } = ( u - A ( \\rho ^ { 3 } ) ) ( u - A ( \\rho ^ { 5 } ) ) ( u - A ( \\rho ^ { 6 } ) ) \\\\ \\\\ z _ { 2 } ^ { 2 } = ( u - 1 ) ( u - A ( \\rho ^ { 4 } ) ) ( u - A ( \\rho ^ { 5 } ) ) ( u - A ( \\rho ^ { 6 } ) ) \\\\ \\\\ z _ { 3 } ^ { 2 } = u ( u - A ( \\rho ^ { 3 } ) ) ( u - A ( \\rho ^ { 4 } ) ) ( u - A ( \\rho ^ { 5 } ) ) \\end{array} \\right \\} \\subset { \\mathbb C } ^ { 4 } . \\end{align*}"}
-{"id": "9902.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathcal { H } ( \\rho ( t ) | \\rho ^ \\infty ) = - \\beta \\mathcal { I } ( \\rho ( t ) | \\rho ^ \\infty ) \\ . \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} A : = \\{ & ( f _ 1 , f _ 2 ) : f _ i \\in C [ 0 , \\infty ) f _ i ( 0 ) = 0 i = 1 , 2 \\lambda _ 0 ( f _ 1 , f _ 2 ) , \\lambda _ 1 ( f _ 1 , f _ 2 ) < \\infty \\} \\\\ A ^ n : = \\{ & ( f _ 1 , f _ 2 ) : f _ i \\in C [ 0 , \\infty ) f _ i ( 0 ) = 0 i = 1 , 2 , \\lambda ^ n _ 0 ( f _ 1 , f _ 2 ) , \\lambda ^ n _ 1 ( f _ 1 , f _ 2 ) < \\infty \\} . \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} L _ 0 \\Big ( \\left ( \\chi _ { - j _ k } \\right ) ^ { m _ k } \\dots \\left ( \\chi _ { - j _ 2 } \\right ) ^ { m _ 2 } \\left ( \\chi _ { - j _ 1 } \\right ) ^ { m _ 1 } | 0 \\rangle \\Big ) = \\frac { 1 } { 2 } \\left ( m _ k \\cdot j _ k + \\dots m _ 2 \\cdot j _ 2 + m _ 1 \\cdot j _ 1 \\right ) \\Big ( \\left ( \\chi _ { - j _ k } \\right ) ^ { m _ k } \\dots \\left ( \\chi _ { - j _ 2 } \\right ) ^ { m _ 2 } \\left ( \\chi _ { - j _ 1 } \\right ) ^ { m _ 1 } | 0 \\rangle \\Big ) . \\end{align*}"}
-{"id": "2943.png", "formula": "\\begin{align*} \\| T ^ { \\ , n } z - x _ { l } \\| < \\sum _ { u \\ge 0 } 2 ^ { - ( j _ { m } + u ) } = 2 ^ { - ( j _ { m } - 1 ) } \\end{align*}"}
-{"id": "682.png", "formula": "\\begin{align*} \\frac { \\partial e _ { n } ( \\vec { \\theta } ) } { \\partial \\eta } = ( k - n ) ~ e _ { n - 1 } ( \\vec { \\theta } ) ~ ~ { \\rm f o r } ~ ~ n = 1 , 2 , \\cdots , k ~ ~ { \\rm a n d } ~ ~ \\frac { \\partial e _ { 0 } ( \\vec { \\theta } ) } { \\partial \\eta } = 0 , \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{align*} c _ k = \\int _ { - \\infty } ^ \\infty t ^ k \\ , d \\sigma ( k = 0 , \\dots , 2 n - 3 ) , c _ { 2 n - 2 } = \\int _ { - \\infty } ^ \\infty t ^ { 2 n - 2 } \\ , d \\sigma + \\lambda , \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{align*} I _ 1 = \\int _ \\Omega g \\cdot \\nabla u u _ t d x , I _ 2 = \\int _ \\Omega b ( t ) | u _ t | ^ { p } u _ t u d x \\quad I _ 3 = \\int _ \\Omega | u _ t | ^ 2 d x . \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} \\theta = 1 + \\epsilon ^ 2 e ^ \\psi + \\epsilon ^ 2 e ^ { \\psi - \\rho } \\geq \\epsilon ^ 2 C ^ { - 1 } ( 1 + e ^ { - \\rho } ) . \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} \\abs { \\zeta - z } ^ 2 \\le | 1 - r + 2 \\delta | | 1 - r + 2 \\delta | < 5 ( 1 - r ) ( 3 \\delta ) = 1 5 ( 1 - r ) \\delta \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} C _ { n } = \\frac { 1 } { 2 n + 1 } \\binom { 2 n } { n } . \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} v _ A = ( 0 , \\cdots , 0 , a _ 1 , a _ 2 , \\cdots , a _ l , a _ k + 1 ) . \\end{align*}"}
-{"id": "1966.png", "formula": "\\begin{align*} T _ { a } ( z ) = L _ { a M } ( z , t ) \\cdots L _ { a 1 } ( z , t ) , \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{align*} \\left [ \\sum _ { \\ell = 0 } ^ n \\frac { ( \\ell ! ) ^ 2 D _ \\ell } { 4 ^ { \\ell } } x ^ \\ell \\right ] ^ { m - 1 } \\sum _ { k = 0 } ^ n \\frac { [ ( 2 k ) ! ] ^ 3 } { 2 ^ { 4 k } ( k ! ) ^ 4 } x ^ { k } \\in \\frac { 1 } { 2 ^ { 2 ( n - 1 ) } } \\mathbb Z [ x ] , \\end{align*}"}
-{"id": "8355.png", "formula": "\\begin{align*} m _ A ( I _ A \\otimes \\lambda _ A ) ( a \\otimes r ) = r a = m _ A ( \\lambda _ A \\otimes I _ A ) ( r \\otimes a ) . \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} A ^ o _ { i j } = A _ { i j } - \\frac 1 n \\vec { H } g _ { i j } \\ , . \\end{align*}"}
-{"id": "6945.png", "formula": "\\begin{align*} W _ G ( H ) = W ' \\ltimes H \\ , . \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} ( - 1 ) ^ k + \\sum _ { j = 1 } ^ k ( - 1 ) ^ { k - j } \\left ( t ^ { n _ j } + t ^ { - n _ j } \\right ) , \\end{align*}"}
-{"id": "9532.png", "formula": "\\begin{align*} \\lambda ( g ) \\pi _ \\alpha ( a ) \\lambda ( g ) ^ * = \\pi _ \\alpha ( g ( a ) ) , \\ \\ \\ a \\in A , \\ g \\in G , \\end{align*}"}
-{"id": "169.png", "formula": "\\begin{align*} \\widetilde { g } ( \\textbf { x } ) = \\sum _ { \\textbf { m } \\in \\mathbb F _ q ^ d } g ( \\textbf { m } ) \\ , \\chi ( - \\textbf { x } \\cdot \\textbf { m } ) \\mbox { f o r } ~ ~ \\textbf { x } \\in \\mathbb F _ q ^ d . \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} P _ 0 ( r , a ) & = 1 ; \\\\ P _ m ( r , a ) - P _ { m } ( r , a - 1 ) & = ( ( m - \\textstyle { \\frac 1 2 } ) r - a ) P _ { m - 1 } ( r , a - 1 ) ; \\\\ P _ m ( r , 0 ) & = P _ m ( r , r - 2 ) . \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} ( M - N \\overline { M } ^ { - 1 } \\overline { N } ) z = p - N \\overline { M } ^ { - 1 } \\overline { p } . \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} M ^ \\omega \\cap M ' = M ^ \\omega \\cap M , \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} \\begin{cases} u _ { t t } + L u + B u _ { t } = 0 & \\mbox { i n } ( 0 , \\infty ) \\\\ u ( t ) \\in D ( L ) & \\\\ u ( 0 ) = u _ { 0 } \\in D ( L ) & \\\\ u _ { t } ( 0 ) = u _ { 1 } \\in H _ { 1 } . & \\end{cases} \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{align*} [ T _ { \\phi ( r ) e ^ { - i p \\theta } } , T _ { z ^ l } ] = [ T _ { \\bar { z } ^ m } , T _ { z ^ n } ] . \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} { 1 \\over \\gamma _ 3 } \\int _ A \\omega ( y ) \\times K _ 0 ( x , y ) \\ , d y = { 1 \\over \\gamma _ 3 } \\int _ \\Omega \\omega ( y ) \\times \\left [ - { y \\over | x | ^ 3 } + 3 { x \\cdot y \\over | x | ^ 5 } x \\right ] \\ , d y + O \\left ( { 1 \\over | x | ^ 4 } \\right ) . \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} u \\left ( x , 0 \\right ) = a \\left ( x \\right ) , x \\in R _ { + } ^ { n } , t \\in \\left ( 0 , T \\right ) , \\end{align*}"}
-{"id": "6000.png", "formula": "\\begin{align*} ( R \\otimes j ) ( x ) = x . \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} \\zeta ( z ; \\tau ) = \\eta z + \\partial _ z ( \\log \\vartheta _ 1 ( \\frac { \\pi z } 2 ; \\tau ) ) , \\eta : = - \\frac { \\pi ^ 2 } { 1 2 } \\frac { \\vartheta _ 1 ''' ( 0 ) } { \\vartheta _ 1 ' ( 0 ) } . \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{align*} \\mathrm { a d } ^ \\circ _ L ( X ) ( M ) = \\rho ^ \\circ ( X _ { ( 1 ) } ) ( M \\triangleleft X _ { ( 2 ) } ) \\rho ^ \\circ ( S ^ { - 1 } ( X _ { ( 3 ) } ) ) , X \\in \\mathcal { U } , \\ M \\in \\mathrm { M a t } _ { N \\times N } ( \\mathcal { B } ) . \\end{align*}"}
-{"id": "9416.png", "formula": "\\begin{align*} \\bigl \\langle \\Gamma ( \\phi ) ( X _ 1 , \\ldots , X _ { n - 1 } ) , X _ n \\bigr \\rangle & = \\langle \\phi , \\varphi \\rangle \\\\ & = \\int _ { \\Sigma } \\langle \\widetilde { \\phi } ( t ) , \\varphi \\rangle \\ , \\mu ( t ) \\\\ & = \\int _ { \\Sigma } \\bigl \\langle \\Gamma ( \\widetilde { \\phi } ( t ) ) ( X _ 1 , \\ldots , X _ { n - 1 } ) , X _ n \\bigr \\rangle \\ , \\mu ( t ) \\ , . \\end{align*}"}
-{"id": "9458.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } G ( t , x ) = \\nabla v \\big ( t , \\Phi ( t , 0 , x ) \\big ) \\cdot G ( t , x ) & 0 \\leq t \\leq T \\\\ \\qquad \\qquad - ( \\nabla \\times d ) \\big ( t , \\Phi ( t , 0 , x ) \\big ) \\exp \\left \\{ \\int ^ t _ 0 \\nabla \\cdot v \\big ( s , \\Phi ( s , 0 , x ) \\big ) { \\rm d } s \\right \\} & \\\\ G ( 0 , x ) = B _ 0 ( x ) & x \\in \\mathbb { T } ^ 3 \\end{cases} \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} \\sum _ { \\substack { k = 1 \\\\ k \\neq k _ 0 } } ^ { n - 1 } \\gamma _ k S _ k ( x ) ^ 2 = \\sum _ { \\substack { j = 1 \\\\ j \\neq j _ 0 } } ^ { n - 1 } \\delta _ j \\tilde { S } _ j ( x ) ^ 2 . \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} f ( y & ) + t \\alpha ( | x - y | / t ) \\\\ & = f ( z ) + t \\alpha ( | x - z | / t ) + ( f ( y ) - f ( z ) ) + t \\alpha ( | x - y | / t ) - t \\alpha ( | x - z | / t ) \\\\ & \\ge f ( z ) + t \\alpha ( | x - z | / t ) - L | z - y | + t \\alpha ( | x - y | / t ) - t \\alpha ( | x - z | / t ) \\\\ & = f ( z ) + t \\alpha ( | x - z | / t ) , \\end{align*}"}
-{"id": "8291.png", "formula": "\\begin{align*} ( \\mathbb { I } _ { | \\mathcal { E } | ^ n } - \\boldsymbol { c } _ i ) \\mathbb { Q } ^ { - 1 } \\mathcal { A } ^ { ( P Q ^ { - 1 } , Q ) } = ( \\mathbb { I } _ { | \\mathcal { E } | ^ n } - \\boldsymbol { c } _ i ) \\mathbb { Q } ^ { - 1 } \\mathcal { A } ^ { ( P Q ^ { - 1 } , Q T _ i ) } \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} & \\frac 1 2 \\frac { d } { d t } \\int _ { \\mathbb { R } ^ 2 } ( | V ^ { ( \\alpha , a ) } | ^ 2 + | H ^ { ( \\alpha , a ) } | ^ 2 ) d x \\\\ & + \\frac 1 2 \\mu \\int _ { \\mathbb { R } ^ 2 } | \\nabla V ^ { ( \\alpha , a ) } | ^ 2 d x - \\sum \\limits _ { l = 0 } ^ { \\alpha - 1 } \\mu C \\int _ { \\mathbb { R } ^ 2 } | \\nabla V ^ { ( l , a ) } | ^ 2 d x \\lesssim \\langle t \\rangle ^ { - \\frac 3 2 } E _ { \\kappa - 3 } E _ { \\kappa - 1 } ^ { \\frac 1 2 } . \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} \\begin{array} { c c c } 1 & = & L _ { f _ \\Gamma } \\\\ ( A u ) _ { 1 } & = & u _ { + } L _ 1 ' \\\\ ( A u ) _ { 2 } & = & u _ { + } L _ 2 ' \\\\ & \\vdots \\\\ ( A u ) _ { d - 1 } & = & u _ { + } L _ { d - 1 } ' . \\end{array} \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{align*} \\dot x = F ( x ) \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ { \\rho } = \\int _ \\Omega D ^ m u : D ^ m v + \\rho u v d x , \\ \\ \\ \\mathcal 8 u , v \\in H ^ m ( \\Omega ) . \\end{align*}"}
-{"id": "9860.png", "formula": "\\begin{align*} \\widehat { G } ( k ) = \\frac { \\widehat { \\zeta } ( k ) } { k ^ 2 } \\Leftrightarrow \\boldsymbol { \\nabla } ^ 2 G ( r ) = - \\zeta ( r ) \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{align*} z ' ( s ) = \\frac { 1 } { s } \\Bigl ( t ^ { \\frac { 1 } { 2 } } v ' ( t ) + \\frac { 1 } { 2 } t ^ { - \\frac { 1 } { 2 } } v ( t ) \\Bigr ) \\end{align*}"}
-{"id": "3627.png", "formula": "\\begin{align*} F ( s ) = \\sum _ n \\frac { c _ n } { \\mu _ n ^ s } . \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} f _ T ( z ) & = \\hat r _ T ( z + \\xi _ T ) \\\\ & = r _ T ( z + \\xi _ T ) - \\xi _ { T - T } \\\\ & = r _ T ( z + \\xi _ T ) \\\\ & = g ^ { - 1 } _ T ( z + \\xi _ T ) . \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} & \\left ( \\frac { c ( x ) + c ( y ) } { 2 } \\right ) ^ N e \\left ( x \\left ( i - \\frac { a } { 2 } \\right ) + y \\left ( j - \\frac { b } { 2 } \\right ) \\right ) \\\\ & = \\exp \\left ( - \\frac { x _ 2 ^ 2 + y _ 2 ^ 2 } { 2 } - \\frac { \\left ( i - \\frac { a } { 2 } \\right ) ^ 2 + \\left ( j - \\frac { b } { 2 } \\right ) ^ 2 } { N } \\right ) \\\\ & \\times \\left ( 1 + O \\left ( \\frac { x _ 2 ^ 4 + y _ 2 ^ 4 } { N } + \\frac { i ^ 4 + j ^ 4 + 1 } { N ^ 3 } \\right ) \\right ) . \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 \\rho ^ + ( \\hat { E } ^ { 0 i } ) \\rho ^ + ( \\hat { E } ^ { 0 i } ) = - \\xi _ { \\rho ^ + } I _ { \\rho ^ + } , \\\\ \\sum _ { i = 1 } ^ 3 \\rho ^ - ( \\hat { E } ^ { \\tau ( i ) } ) \\rho ^ - ( \\hat { E } ^ { \\tau ( i ) } ) = - \\xi _ { \\rho ^ - } I _ { \\rho ^ - } , \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} \\Bigl ( \\mathbb E \\Bigl \\| \\sum ^ N _ { n = 1 } \\varepsilon _ n \\ , \\sigma _ n \\phi _ { n } ( \\sigma _ 1 , \\ldots , \\sigma _ { n - 1 } ) \\Bigr \\| ^ p \\Bigr ) ^ { \\frac 1 p } \\geq \\gamma _ { p , X } ^ { \\delta } \\Bigl ( \\mathbb E \\Bigl \\| \\sum ^ N _ { n = 1 } \\sigma _ n \\phi _ { n } ( \\sigma _ 1 , \\ldots , \\sigma _ { n - 1 } ) \\Bigr \\| ^ p \\Bigr ) ^ { \\frac 1 p } . \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} \\lim _ { n \\to \\pm \\infty } \\frac { p _ n } { n } = 1 . \\end{align*}"}
-{"id": "2991.png", "formula": "\\begin{align*} | v _ { n } | \\ , \\cdot \\sup _ { j \\ , \\in \\ , [ b _ { \\varphi ( n ) } , b _ { \\varphi ( n ) + 1 } ) } \\ \\prod _ { s = b _ { \\varphi ( n ) + 1 } } ^ { j } \\ ! \\ ! \\ ! | w _ { s } | \\le 2 ^ { \\ , \\delta ^ { ( k - 1 ) } - \\tau ^ { ( k ) } } . \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} a = - \\frac { b } { c } , \\end{align*}"}
-{"id": "7902.png", "formula": "\\begin{align*} f ( x , t ) = \\begin{cases} u ( x ) - t & , \\\\ t - u ( x ) & . \\end{cases} \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{align*} S _ { p , n } = \\sum _ { \\underset { k _ { i } \\ge 0 } { k _ { 1 } + \\dots + k _ { p } = n } } \\binom { n } { k _ { 1 } , \\dots , k _ { p } } w _ { k _ { 1 } } \\dots w _ { k _ { p } } = \\left ( - 1 \\right ) ^ { p } n ! \\sum _ { \\underset { k _ { i } \\ge 0 } { k _ { 1 } + \\dots + k _ { p } = n } } g _ { k _ { 1 } } \\dots g _ { k _ { p } } \\end{align*}"}
-{"id": "1351.png", "formula": "\\begin{align*} T ( \\mathcal { R } _ { \\sigma _ { 2 } } ( \\sigma _ { 1 } ) ) \\circ T ( \\sigma _ { 2 } ) \\ ; = \\ ; T ( \\sigma _ { 1 } , \\sigma _ { 2 } ) \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} \\| \\phi \\| _ { s , 1 } & : = \\frac { 1 } { r } \\sum _ { l = 1 } ^ r \\| \\phi ^ { ( 1 ) } \\| _ 1 \\ldots \\| \\phi ^ { ( l - 1 ) } \\| _ 1 \\| \\phi ^ { ( l ) } \\| _ s \\| \\phi ^ { ( l + 1 ) } \\| _ 1 \\ldots \\| \\phi ^ { ( r ) } \\| _ 1 . \\end{align*}"}
-{"id": "680.png", "formula": "\\begin{align*} \\eta ^ n = \\sqrt { \\frac { n ! k ! } { ( k - n ) ! } } D _ { n } ( \\vec { \\theta } ) = n ! e _ { n } ( \\vec { \\theta } ) \\end{align*}"}
-{"id": "9760.png", "formula": "\\begin{align*} | h | _ b : = \\max _ { 1 \\leq i \\leq k } | \\int _ { \\gamma _ i } h | , \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} B _ { m , \\ell } & = g _ { 1 , \\ell , 0 , m } + \\sum _ { s = 1 } ^ \\ell g _ { 1 , \\ell , s , m } \\\\ & = g _ { 1 , \\ell , 1 , m - 1 } + g _ { 1 , \\ell - 1 , - 1 , m - 1 } + \\sum _ { s = 1 } ^ \\ell g _ { 1 , \\ell , s + 1 , m } + \\sum _ { s = 1 } ^ \\ell g _ { 1 , \\ell - 1 , s - 1 , m } \\\\ & = g _ { 1 , \\ell , 1 , m - 1 } + g _ { 1 , \\ell , 0 , m - 1 } + \\sum _ { s = 2 } ^ { \\ell + 1 } g _ { 1 , \\ell , s , m } + \\sum _ { s = 0 } ^ { \\ell - 1 } g _ { 1 , \\ell - 1 , s , m } \\\\ & = B _ { m - 1 , \\ell } + B _ { m - 1 , \\ell - 1 } . \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} \\frac { \\| F ( \\nabla v ) \\| _ { L ^ { \\infty } ( \\Omega ) } } { \\| v \\| _ { L ^ { \\infty } ( \\Omega ) } } = \\min _ { \\varphi \\in W _ { 0 } ^ { 1 , \\infty } ( \\Omega ) \\setminus \\{ 0 \\} } \\frac { \\| F ( \\nabla \\varphi ) \\| _ { L ^ { \\infty } ( \\Omega ) } } { \\| \\varphi \\| _ { L ^ { \\infty } ( \\Omega ) } } = \\Lambda _ 1 ( \\infty , \\Omega ) = \\frac { 1 } { \\rho _ { F } ( \\Omega ) } . \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} 0 = [ P _ { k _ 1 } , G _ 1 ] | _ a ^ b = 0 + \\dots + 0 + & a _ { i , m + 1 } [ P _ { k _ 1 } , f _ 1 ] | _ a ^ b + \\dots + a _ { i , 2 m } [ P _ { k _ 1 } , f _ m ] | _ a ^ b , \\\\ 0 = [ P _ { k _ 2 } , G _ 1 ] | _ a ^ b = 0 + \\dots + 0 + & a _ { i , m + 1 } [ P _ { k _ 2 } , f _ 1 ] | _ a ^ b + \\dots + a _ { i , 2 m } [ P _ { k _ 2 } , f _ m ] | _ a ^ b , \\\\ & \\vdots \\\\ 0 = [ P _ { k _ m } , G _ 1 ] | _ a ^ b = 0 + \\dots + 0 + & a _ { i , m + 1 } [ P _ { k _ m } , f _ 1 ] | _ a ^ b + \\dots + a _ { i , 2 m } [ P _ { k _ m } , f _ m ] | _ a ^ b . \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{align*} \\Xi ^ { \\lambda } ( X \\otimes X ^ { \\prime } \\otimes Y \\otimes Y ^ { \\prime } ) : = \\sum _ { i , j , m , n , o , p } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) \\pi ( X ) _ { m } ^ { j } c _ { n } ^ { m } \\pi ( X ^ { \\prime } Y ) _ { o } ^ { n } c _ { p } ^ { o } \\pi ( Y ^ { \\prime } K _ { \\lambda } ) _ { i } ^ { p } . \\end{align*}"}
-{"id": "8287.png", "formula": "\\begin{align*} \\mathbb { Q } ^ { - 1 } \\mathcal { A } ^ { ( P R _ { Q ( 1 ) } , Q ) } = \\left ( S _ v ( - k _ { P Q ( 1 ) } ) \\otimes \\mathbb { I } _ { | \\mathcal { E } | ^ { n - 1 } } \\right ) \\mathbb { Q } ^ { - 1 } \\mathcal { A } ^ { ( P , Q ) } . \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} d i m _ { q , z } \\mathit { F _ { \\chi } } = \\frac { 1 } { \\prod _ { j \\in \\mathbb { Z } _ { + } } \\big ( 1 - z q ^ { 2 j - \\frac { 3 } { 2 } } \\big ) \\big ( 1 - z ^ { - 1 } q ^ { 2 j - \\frac { 1 } { 2 } } \\big ) } . \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} B _ { \\alpha } = \\Big ( \\alpha g ( \\alpha ) \\dots g ^ { n _ { \\alpha } - 1 } ( \\alpha ) \\Big ) ^ { m _ { \\alpha } } \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{align*} \\hat { m } _ s ^ { ( N + 1 ) } ( x ) \\circ \\widehat { L ^ { ( N + 1 ) } _ s } \\circ \\left ( \\hat { m } _ s ^ { ( N + 1 ) } \\right ) ^ { - 1 } ( x ) - c _ { N , s } = L _ s ^ { ( N ) } \\end{align*}"}
-{"id": "3147.png", "formula": "\\begin{align*} f \\nabla _ { i } R _ { j k } = - ( \\nabla _ { i } f ) R _ { j k } + \\nabla _ { i } \\nabla _ { j } \\nabla _ { k } f + \\frac { R } { n - 1 } \\nabla _ { i } f g _ { j k } . \\end{align*}"}
-{"id": "9960.png", "formula": "\\begin{align*} \\dot { v } = - x ^ T \\tilde { Q } _ i x \\mbox { w h e r e } \\tilde { Q } _ i = \\tilde { A } _ i ^ T Q _ i \\tilde { A } _ i \\end{align*}"}
-{"id": "3525.png", "formula": "\\begin{align*} { } _ { \\ast } D ^ { \\alpha } ( f ( z ) ) = e ^ { D ^ { \\alpha } ( l n ( f ( z ) ) ) } = e ^ { \\frac { 1 } { \\Gamma ( n - \\alpha ) } \\frac { d ^ { n } } { d x ^ { n } } \\int _ { 0 } ^ { z } ( \\ln f ( t ) ) ( z - t ) ^ { n - \\alpha - 1 } d t } , \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} X ^ 2 + X + ( \\alpha + \\lambda _ 1 ^ 2 \\lambda _ 2 ^ 2 ) = 0 . \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{align*} \\widehat { \\psi _ \\kappa } ( x ) = \\widehat { \\psi } ( x ) \\widehat { g _ \\kappa } ( x ) = \\widehat { \\psi } ( x ) \\widehat { g } ( \\kappa x ) , \\end{align*}"}
-{"id": "9015.png", "formula": "\\begin{align*} A ( B , \\gamma ) : = \\left [ \\frac { \\gamma ^ p \\left | \\left \\{ x \\in B \\ , : \\ , \\prod \\limits _ { i = 1 } ^ m \\bigl | \\frac { f _ i ( x ) } { \\| f _ i \\| _ { w \\mathcal { M } ^ { p _ i } _ { \\phi _ i } } } \\bigr | > \\gamma \\right \\} \\right | } { \\phi ^ { p } ( R ) | B | } \\right ] ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "9792.png", "formula": "\\begin{align*} \\Delta u = - \\epsilon ^ { - 2 } ( 1 - | u | ^ 2 ) u , \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} \\partial _ t J & = [ P ( J ) ^ + - P ( J ) ^ - , J ] . \\end{align*}"}
-{"id": "9179.png", "formula": "\\begin{align*} \\hat h = ( R ^ \\top R + \\eta K ^ { - 1 } ) ^ { - 1 } R ^ \\top y = C y \\ , , \\end{align*}"}
-{"id": "7449.png", "formula": "\\begin{align*} m ( L ) \\leq | z | + | w | = m ( [ 0 , x ] ) + m ( [ x , 1 ] ) \\end{align*}"}
-{"id": "5961.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\lambda ^ { 2 ^ { 2 k + 1 } - 1 } + \\lambda + 1 = 0 , \\\\ \\lambda ^ { 3 } + \\lambda + 1 = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "1570.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } x _ l ^ i p _ { 2 n - 2 } \\varrho _ k \\sum _ { j = 0 } ^ { n - 1 } b _ { i j } x _ k ^ j = Q _ k ( x _ l ) = \\delta _ { k l } , k , l = 1 , \\dots , n , \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{align*} \\mathbb B [ \\eta f ] ( x ) = \\int _ D \\Gamma _ \\kappa ( x - y , y ) ( \\eta f ) ( y ) d y \\end{align*}"}
-{"id": "1483.png", "formula": "\\begin{align*} f ( x ) = - \\frac { r _ 1 } { x - x _ 1 } - \\frac { r _ 2 } { x - x _ 2 } - \\dots - \\frac { r _ n } { x - x _ n } \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{align*} \\mathfrak { L } _ { g } ^ { * } ( f ) = - ( \\Delta f ) g + H e s s \\ , f - f R i c = \\kappa g , \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{align*} A _ \\kappa : = \\int _ 0 ^ \\infty f ( i \\kappa , x ) ^ \\dagger f ( i \\kappa , x ) d x = \\frac { \\dot { f } ' ( i \\kappa , 0 ) ^ \\dagger f ( i \\kappa , 0 ) - \\dot { f } ( i \\kappa , 0 ) ^ \\dagger f ' ( i \\kappa , 0 ) } { - 2 i \\kappa } ; \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{align*} \\alpha _ { n + 1 } \\alpha _ { n - 1 } = ( \\alpha _ n + \\eta _ n ) ( \\alpha _ n - \\eta _ { n - 1 } ) \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} \\tilde { \\textbf { y } } _ 0 = \\textbf { R } _ 0 ^ { - 1 / 2 } \\textbf { y } _ { 0 } = \\textbf { R } _ 0 ^ { - 1 / 2 } \\textbf { H } _ { 0 } \\textbf { w } _ { 0 } \\sqrt { P } U _ 0 + \\tilde { \\textbf { z } } _ { 0 } \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{gather*} A ^ { S _ 1 G _ 1 S _ 2 G _ 2 P } ( z ) = \\frac { 1 } { z ^ { 4 / 3 } } \\begin{pmatrix} - t ^ { 1 / 3 } & 0 & 0 \\\\ 0 & - \\omega t ^ { 1 / 3 } & 0 \\\\ 0 & 0 & - \\omega ^ 2 t ^ { 1 / 3 } \\end{pmatrix} \\\\ \\hphantom { A ^ { S _ 1 G _ 1 S _ 2 G _ 2 P } ( z ) = } { } + \\frac { 1 } { z } \\begin{pmatrix} \\theta ^ \\infty _ 1 / 3 - 2 / 3 & 0 & 0 \\\\ 0 & \\theta ^ \\infty _ 1 / 3 - 2 / 3 & 0 \\\\ 0 & 0 & \\theta ^ \\infty _ 1 / 3 - 2 / 3 \\end{pmatrix} + \\cdots . \\end{gather*}"}
-{"id": "6324.png", "formula": "\\begin{align*} \\underline { \\mu } = \\min _ { \\| x \\| = 1 } \\dfrac { x ^ T B x } { x ^ T W x } , \\ \\ \\bar { \\mu } = \\max _ { \\| x \\| = 1 } \\dfrac { x ^ T B x } { x ^ T W x } \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} \\beta ( t ) = r _ 0 \\frac { \\alpha ( t ) - a _ 0 } { \\Vert \\alpha ( t ) - a _ 0 \\Vert } \\ , , \\end{align*}"}
-{"id": "5507.png", "formula": "\\begin{align*} \\widehat \\Psi _ p : = \\{ \\widehat \\psi \\in \\widehat \\Psi _ \\mathcal S | \\ , \\widehat \\psi ( 0 ) = p \\} . \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} \\beta _ \\chi ( z ^ 2 ) = \\sum _ { m \\in \\mathbb { Z } } \\chi _ { - 2 m + \\frac { 1 } { 2 } } ( z ^ 2 ) ^ { m - 1 } ; \\gamma _ \\chi ( z ^ 2 ) = \\sum _ { m \\in \\mathbb { Z } } \\chi _ { - 2 m - \\frac { 1 } { 2 } } ( z ^ 2 ) ^ { m } \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} \\eta = \\varepsilon ^ { \\pi / \\omega } , \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} p \\left ( \\mathsf { x } , t ; \\mathsf { z } , r ; \\mathsf { y } , s \\right ) = g ^ { - 1 } ( \\mathsf { x } , t - s , \\mathsf { y } ) g ( \\mathsf { x } , t - r , \\mathsf { z } ) g ( \\mathsf { z } , r - s , \\mathsf { y } ) \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{align*} \\left . \\frac { \\mathrm { d } } { \\mathrm { d } k } ( \\mathrm { R e } \\ , \\lambda _ { \\pm } ( k ) ) \\right | _ { k = 0 } = \\frac { 1 } { 2 } > 0 \\ , , \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{align*} Q _ { - 1 } ( t ) = \\int _ { \\mathbb R ^ n } \\langle x \\rangle ^ { - 1 } | u ( t , x ) | ^ 2 d x . \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} P S '' - P ' S ' + \\tilde { R } S = 0 , \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} F ( \\lambda _ 0 , \\lambda _ 1 , \\lambda _ 2 ) : = \\lambda _ 0 ^ 2 + \\lambda _ 1 ^ 2 + \\lambda _ 2 ^ 2 - 2 ( \\lambda _ 0 \\lambda _ 1 + \\lambda _ 0 \\lambda _ 2 + \\lambda _ 1 \\lambda _ 2 ) \\end{align*}"}
-{"id": "9662.png", "formula": "\\begin{align*} \\abs { K _ { + + } } = \\abs { K _ { - + } } = \\frac { \\abs { K } } { 4 } . \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{align*} ( \\mathsf { N } _ { m } ^ { n } ) ^ { * } = \\mathsf { N } _ { n } ^ { m } , \\quad \\mathsf { N } _ { m } ^ { n } \\mathsf { N } _ { o } ^ { p } = \\delta _ { o } ^ { n } \\mathsf { N } _ { m } ^ { p } , \\quad \\mathrm { T r } ( \\pi ( K _ { 2 \\rho } ) \\mathsf { N } _ { m } ^ { n } ) = \\delta _ { m } ^ { n } q ^ { ( 2 \\rho , \\lambda _ { m } ) } . \\end{align*}"}
-{"id": "3485.png", "formula": "\\begin{align*} E _ g ( \\beta , \\sigma ) \\approx E _ g ^ { T F } = \\frac { A _ 0 ^ { 2 \\sigma } } { \\sigma + 1 } \\beta , \\mu _ g ( \\beta , \\sigma ) \\approx \\mu _ g ^ { T F } = A _ 0 ^ { 2 \\sigma } \\beta , \\beta \\gg 1 . \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} \\begin{array} { l l l } S _ 1 & = & \\{ a _ i \\} , \\\\ S _ 2 & = & S ( v ) \\setminus S _ 1 , \\\\ E _ 2 & = & \\{ c \\in E ( v ) \\vert c a _ i \\in I \\} , \\\\ E _ 1 & = & E ( v ) \\setminus E _ 2 . \\end{array} \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty | t | ^ k \\ , d \\sigma < \\infty , k = 0 , \\dots , 2 n - 2 . \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{align*} a ^ { ( 2 ) } _ \\pm ( \\lambda _ 1 , \\lambda _ 2 ) = b _ \\pm ( \\lambda _ 2 ) _ { \\lambda _ 1 } \\in [ a _ - ( \\lambda _ 1 ) , a _ + ( \\lambda _ 1 ) ] \\end{align*}"}
-{"id": "6387.png", "formula": "\\begin{align*} A ^ \\delta _ E ( v , w ) = \\sum _ { e \\in v E ^ 1 w } ( - 1 ) ^ { \\delta ( e ) } \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} \\mathbb { P } ( w _ n = K ) & = \\sum _ { k = 1 } ^ { n } \\mathbb { P } ( w _ n = K , w _ { n - k } < K , w _ { n - j } = K , j = 1 , \\ldots , k - 1 ) \\\\ & = \\sum _ { k = 1 } ^ { n } \\int _ { 0 ^ - } ^ { K - 0 } \\left [ 1 - G ( b _ k ( w ) ) \\right ] d W _ { n - k } . \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} x ' _ j = \\sqrt { - 1 } \\frac { t ' _ j + 1 } { t ' _ j - 1 } . \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} ( 1 - 4 x ) ^ { 1 / 2 } = ( 1 - 4 x ) ^ { - 1 / 2 } ( 1 - 4 x ) \\equiv ( 1 - 4 x ) ^ { ( q + 1 ) / 2 } + 2 x ^ q \\pmod { ( x ^ { q + 1 } , p ) } . \\end{align*}"}
-{"id": "4408.png", "formula": "\\begin{align*} ( 1 - 2 \\beta ) ^ { p + 1 } \\cdot \\frac { \\mathrm { d } } { \\mathrm { d } \\beta } \\pounds _ 1 \\left ( \\frac { \\beta } { 2 \\beta - 1 } \\right ) = \\frac { ( 1 - 2 \\beta ) ^ p } { \\beta } \\cdot \\pounds _ 0 \\left ( \\frac { \\beta } { 2 \\beta - 1 } \\right ) \\equiv - \\pounds _ 0 ( \\beta ) \\pmod { p } . \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} H ( x ) : = \\frac { 1 } { 2 \\pi } \\varpi ^ i \\log { | x | } + \\Psi ( x ) \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } z ^ n \\sum _ { \\pi \\in \\mathcal { C } _ n } \\sum _ { k _ i \\in \\pi } g _ { k _ i } . = \\left ( \\frac { 1 - z } { 1 - 2 z } \\right ) ^ 2 g \\left ( z \\right ) . \\end{align*}"}
-{"id": "4671.png", "formula": "\\begin{align*} \\begin{aligned} S _ 1 & = \\{ n \\in \\mathbb Z \\mid | p _ n | > R | n | < \\lfloor \\alpha R \\rfloor \\} , \\\\ S _ 2 & = \\{ n \\in \\mathbb N \\mid n \\geq \\lfloor \\alpha R \\rfloor p _ n \\geq - p _ { - n } \\} , \\\\ S _ 3 & = \\{ n \\in \\mathbb N \\mid n \\geq \\lfloor \\alpha R \\rfloor p _ n < - p _ { - n } \\} . \\end{aligned} \\end{align*}"}
-{"id": "2448.png", "formula": "\\begin{align*} Q \\left ( v , m \\right ) & = \\sum _ { w = 0 } ^ { v - 1 } \\frac { m ^ { w } } { w ! } e ^ { - m } \\end{align*}"}
-{"id": "783.png", "formula": "\\begin{align*} R ( e _ j ) _ { x _ 1 , \\dots , x _ m } = R ^ { S ( x _ { r + 1 } , \\dots , x _ m ) } ( e _ j ) _ { x _ 1 , \\dots , x _ r } , \\end{align*}"}
-{"id": "2379.png", "formula": "\\begin{align*} \\eqref { p r o p 3 } & \\leq - \\frac 1 { N ^ d } \\log \\sum _ { A \\subset \\Lambda _ N \\ , : \\ , A \\ , i s \\ , \\Delta - s p a r s e } \\exp ( J ' { | A | } ) P _ { A ^ c } ( \\Omega _ { A ^ c } ^ + ) \\\\ & \\leq - \\frac 1 { N ^ { d } } \\left ( \\left ( \\frac N \\Delta \\right ) ^ d [ ( d \\log \\Delta + c _ 0 ) + J ' - d \\log \\Delta + c _ 1 \\log \\log \\Delta ] \\right ) \\\\ & = - \\frac { J ' + c _ 0 + c _ 1 \\log \\log \\Delta } { \\Delta ^ d } < 0 \\mbox { f o r $ \\Delta = \\Delta ( J ' ) $ l a r g e e n o u g h . } \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} \\varphi ( z ) = ( z _ 0 + j z _ 1 , z _ 2 + j z _ 3 ) , \\end{align*}"}
-{"id": "985.png", "formula": "\\begin{align*} \\pi _ * \\left ( \\frac { 1 } { 1 - H } \\right ) = \\frac { 1 } { ( 1 + 2 L ) ( 1 + 3 L ) } = \\frac { - 2 } { 1 + 2 L } + \\frac { 3 } { 1 + 3 L } , \\end{align*}"}
-{"id": "6412.png", "formula": "\\begin{align*} J ( R ) = \\begin{bmatrix} \\pi D & \\pi ^ n D \\\\ D & \\pi D \\end{bmatrix} . \\end{align*}"}
-{"id": "8447.png", "formula": "\\begin{align*} \\tau ^ * _ { \\theta } : = \\inf \\{ t \\geq \\theta , \\ , \\ , Y _ t = \\xi _ t \\} ; \\ , \\sigma ^ * _ { \\theta } : = \\inf \\{ t \\geq \\theta , \\ , \\ , Y _ t = \\zeta _ t \\} , \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} \\ell [ f ] ( x ) : = - \\dfrac { 1 } { w ( x ) } \\left ( \\dfrac { d } { d x } \\left [ p ( x ) \\dfrac { d f } { d x } ( x ) \\right ] + q ( x ) f ( x ) \\right ) . \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} T ^ { \\ , b _ { N + 1 } - b _ { N } + k - b _ { n } } \\ , z = e _ k - \\Bigl ( \\prod _ { j = b _ { N } + 1 } ^ { b _ { N } + k - b _ { n } } w _ { j } \\Bigr ) \\ , v _ { N } ^ { - 1 } \\ , \\Bigl ( \\ , \\prod _ { j = b _ { N } + 1 } ^ { b _ { N + 1 } - 1 } w _ { j } \\Bigr ) ^ { - 1 } \\Bigl ( \\prod _ { j = b _ { N } + 1 } ^ { k } w _ { j } \\Bigr ) ^ { - 1 } e _ { b _ { N } + k - b _ { n } } ; \\end{align*}"}
-{"id": "9313.png", "formula": "\\begin{align*} | q _ { m } | = \\left | \\frac { q ( 0 ) } { \\mu ^ { m } } \\right | = \\left | \\lim _ { \\mu \\to 0 ^ + } \\frac { q ( \\mu ) } { \\mu ^ { m } } \\right | & \\le \\lim _ { t \\to 0 ^ + } \\mu ^ { k + 1 - m } = 0 . \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } w _ { 1 } ^ { 2 } = ( z - \\mu _ { 4 } ) ( z - \\mu _ { 5 } ) ( z - \\mu _ { 6 } ) \\\\ \\\\ w _ { 2 } ^ { 2 } = ( z - \\mu _ { 4 } ) ( z - \\mu _ { 5 } ) ( z - \\mu _ { 6 } ) \\\\ \\\\ w _ { 3 } ^ { 2 } = z ( z - 1 ) ( z - \\mu _ { 4 } ) ( z - \\mu _ { 6 } ) \\end{array} \\right \\} \\subset { \\mathbb C } ^ { 4 } . \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} 0 < q ( C ) < q ( 1 ) = 1 \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{gather*} \\frac { y _ 0 ^ { m _ 0 } G ( y _ 1 , \\dots , y _ n ) } { F _ 0 ^ { t + 1 } } \\Omega = \\frac { ( m _ 0 - d + 1 ) y _ 0 ^ { m _ 0 - d } G ( y _ 1 , \\dots , y _ n ) } { t F _ 0 ^ t } \\Omega \\end{gather*}"}
-{"id": "6372.png", "formula": "\\begin{align*} \\gamma ^ \\Lambda _ \\chi ( s _ \\lambda ) = \\chi ( d ( \\lambda ) ) s _ \\lambda . \\end{align*}"}
-{"id": "7455.png", "formula": "\\begin{align*} x _ \\pm ( \\alpha ) _ \\lambda = b _ \\pm ( \\alpha + \\lambda ) \\quad \\in [ a _ - ( \\lambda ) , a _ + ( \\lambda ) ] \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} L _ 2 ^ { \\beta \\gamma ; \\ ( \\lambda , \\mu ) } ( z ) = \\lambda : \\left ( \\partial _ z \\beta ( z ) \\right ) \\gamma ( z ) : + ( \\lambda + 1 ) : \\beta ( z ) \\left ( \\partial _ z \\gamma ( z ) \\right ) : + \\frac { \\mu } { z } : \\beta ( z ) \\gamma ( z ) : + \\frac { ( 2 \\lambda + 1 ) \\mu - \\mu ^ 2 } { 2 z ^ 2 } , \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} \\alpha : = - 4 \\| \\psi _ * \\| ^ 2 _ { L ^ 2 } + 1 2 \\langle \\psi _ * ^ 2 , \\partial _ c u _ { c _ * } \\rangle _ { L ^ 2 } \\end{align*}"}
-{"id": "2722.png", "formula": "\\begin{align*} \\phi ( n ) = \\prod _ { p | n } \\left ( 1 - \\frac { 1 } { p } \\right ) , \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} k ' = t k t ^ { - 1 } k ^ { - 1 } & = \\left ( I + \\begin{pmatrix} a + c & - a + b - c + d \\\\ c & - c + d \\end{pmatrix} p \\right ) \\left ( I - \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) \\\\ & = I + \\begin{pmatrix} c & - a - c + d \\\\ 0 & - c \\end{pmatrix} p . \\end{align*}"}
-{"id": "9813.png", "formula": "\\begin{align*} \\int _ { E , N } { \\nabla V d \\xi d \\eta d \\zeta } = \\int _ { \\partial E , N } { V \\hat n d S } \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} V ( | x + y | ) - V ( | x | ) - D V ( | x | ) \\cdot y = \\frac { 1 } { | x + y | ^ { 2 } } - \\frac { 1 } { | x | ^ { 2 } } + \\frac { 2 x \\cdot y } { | x | ^ { 4 } } \\le K \\frac { | y | ^ { 2 } \\vee | x \\cdot y | } { | x | ^ { 4 } } , \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , X _ { 2 , \\ , j + 2 , \\ , 8 } ] = 0 . \\end{align*}"}
-{"id": "3750.png", "formula": "\\begin{align*} c \\left ( \\sum _ { i < \\beta } r _ { i } e ^ { \\gamma _ { i } } \\right ) = \\sum _ { i < \\beta } r _ { i } e ^ { c ( \\gamma _ { i } ) } . \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} \\beta _ { \\epsilon } ( \\varphi _ k - u _ { k , \\epsilon } ) = \\frac { ( \\omega + \\sqrt { - 1 } \\partial \\bar \\partial u _ { k , \\epsilon } ) ^ n } { \\omega ^ n } \\leq \\frac { ( \\omega + \\partial \\bar \\partial \\varphi _ { k } ) ^ n } { \\omega ^ n } \\leq C \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{gather*} F _ A : = \\sum _ { i = 0 } ^ n \\prod _ { j = 0 } ^ n x _ i ^ { a _ { i , j } } . \\end{gather*}"}
-{"id": "440.png", "formula": "\\begin{align*} \\delta ( e , a , b , c ) = \\left \\{ \\begin{array} { c c } \\binom { 2 a + b - c - e + 3 } { 3 } , & \\mathrm { i f } \\ c \\le 2 a + b - e , \\\\ & \\\\ 0 , & \\mathrm { i f } \\ c > 2 a + b - e . \\end{array} \\right . \\end{align*}"}
-{"id": "6096.png", "formula": "\\begin{align*} [ L _ { - i } , \\ , S _ { j - 1 } ] = 0 , \\ , 2 \\leq i \\leq j . \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} v = F ( x , y ) \\frac { \\partial } { \\partial { x } } + G ( x , y ) \\frac { \\partial } { \\partial { y } } \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{align*} N _ { 1 - , k _ j } = N _ { 1 - , k _ j } [ D _ j ^ { - 1 } ] ^ \\dag C _ j , \\widetilde { N } _ { 1 - , k _ j } = \\widetilde { N } _ { 1 - , k _ j } [ \\widetilde { D } _ j ^ { - 1 } ] ^ \\dag C _ j . \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} X ( x , y ) : = \\big \\langle u ( x ) , u ( y ) X ^ \\top \\big \\rangle = \\sum _ { i = 0 } ^ { n - 1 } \\sum _ { j = 0 } ^ { n - 1 } x _ { i j } x ^ i y ^ j , \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} \\chi _ { c _ { \\kappa ^ \\sigma } } ( e _ i , e _ j ) = \\begin{cases} 1 & i = j \\\\ - 1 & \\end{cases} \\end{align*}"}
-{"id": "1679.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { n } ( d _ j r + \\frac { q ( d - d _ j ) } { d } + \\frac { d _ j t } { d } ) = \\sum _ { j = 0 } ^ { n } \\frac { q ^ j } { d ^ j } W ^ { ( n ) } _ j r ^ { n - j } + \\sum _ { c = 0 } ^ { n - 1 } g ^ { ( n ) } _ c ( q ) r ^ { c } \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathbb { P } ( \\frac { 1 } { n } \\kappa ( Y _ { n } ) \\in O _ { 2 } ) \\geq \\mathbb { P } ( \\kappa ( Y _ { k _ { n } n _ { 0 } + ( n - k _ { n } n _ { 0 } ) } ) \\in k _ { n } n _ { 0 } O _ { 1 } + M _ { n _ { 0 } } ) \\geq \\\\ & \\mathbb { P } ( \\kappa ( Y _ { k _ { n } n _ { 0 } } ) \\in k _ { n } n _ { 0 } O _ { 1 } ) . \\mathbb { P } ( Y _ { n - k _ { n } n _ { 0 } } \\in L _ { n _ { 0 } } ) \\geq e ^ { - n _ { 0 } k _ { n } \\alpha } \\frac { 1 } { 2 } \\end{aligned} \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} H _ \\epsilon = \\log \\frac { v _ \\epsilon '' } { ( v ' _ \\epsilon - a _ t ) ( b _ t - v ' _ \\epsilon ) } . \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} U _ { r , R } ( t ) : = \\begin{cases} 0 & { \\rm i f \\ } t \\in [ 0 , r / 2 [ \\cup [ 2 R , + \\infty [ , \\\\ \\sum _ { i = 0 } ^ { 2 m - 1 } \\frac { a _ i } { r ^ i } t ^ i & { \\rm i f \\ } t \\in [ r / 2 , r [ , \\\\ 1 & { \\rm i f \\ } t \\in [ r , R [ , \\\\ \\sum _ { i = 0 } ^ { 2 m - 1 } \\frac { b _ i } { R ^ i } t ^ i & { \\rm i f \\ } t \\in [ R , 2 R [ . \\end{cases} \\end{align*}"}
-{"id": "3137.png", "formula": "\\begin{align*} B _ { i j } = \\nabla ^ k C _ { k i j } , \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} \\| H _ \\varepsilon \\| _ { L ^ \\infty ( \\Omega ) } = \\| \\eta _ \\varepsilon \\| _ { L ^ \\infty ( \\partial \\Omega ) } \\leq C ( g ) . \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} f ^ { - 1 } ( f ( 1 ) ) & = f ^ { - 1 } ( f ( \\infty ) ) = \\{ 1 , \\infty \\} , \\\\ f ^ { - 1 } ( f ( 0 ) ) & = f ^ { - 1 } ( f ( - 1 ) ) = \\{ 0 , - 1 \\} , \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} d _ { M _ 2 } \\partial _ { \\nu } M _ 2 = - \\frac { \\alpha _ 2 \\ , M _ 2 ^ { p _ 2 } } { k ^ { p _ 2 } _ 2 + M _ 2 ^ { p _ 2 } } \\ \\ \\Omega \\supset \\omega ( p _ 2 > 1 ) ; \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} m ( L ) \\leq \\sum _ { k = 1 } ^ n | Z ( [ a _ { k - 1 } , a _ k ] ) | = : M \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{align*} x ' & = f _ 1 ( x , y ; u ) : = \\frac { y ( b + 2 a x - c x ^ 2 + c y ^ 2 ) } { b + 2 a x - c x ^ 2 - c y ^ 2 } \\\\ y ' & = f _ 2 ( x , y ; u ) : = \\frac { 2 ( a - c x ) y ^ 2 } { b + 2 a x - c x ^ 2 - c y ^ 2 } . \\end{align*}"}
-{"id": "9825.png", "formula": "\\begin{align*} w \\widetilde { \\mathcal { I } } ^ \\circ = \\widetilde { \\mathcal { I } } ^ \\circ \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} \\frac { 1 } { x } \\int _ 0 ^ { \\infty } e ^ { - x s } \\ , d \\mu _ { k + 1 } ( s ) & = ( \\lambda + k ) \\frac { 1 } { x } \\int _ 0 ^ { \\infty } e ^ { - x s } \\ , d \\mu _ k ( s ) \\\\ & { } - \\int _ 0 ^ { \\infty } s e ^ { - x s } \\ , d \\mu _ k ( s ) . \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} a = \\pi _ * ( \\omega / f ) , \\end{align*}"}
-{"id": "9645.png", "formula": "\\begin{align*} F _ 1 ( s _ 1 ) = F _ 3 ( s _ 1 ) \\Rightarrow s _ 1 = \\pm \\sqrt { 3 0 } / 3 , \\end{align*}"}
-{"id": "8616.png", "formula": "\\begin{align*} f ( x ) = & \\displaystyle \\sum _ { i = 0 } ^ { \\frac { s } { 2 } } \\left [ \\binom { s } { s - 2 i } \\ , k + \\ , \\binom { s } { s - ( 2 i + 1 ) } \\ , ( 2 - k ) \\right ] \\ , x ^ \\frac { p ^ l + s - 2 i - 1 } { 2 } \\cr & + \\displaystyle \\sum _ { i = 0 } ^ { \\frac { s } { 2 } } \\left [ \\binom { s } { s - 2 i + 1 } \\ , k + \\ , \\binom { s } { s - 2 i } \\ , ( 2 - k ) \\right ] \\ , x ^ \\frac { s - 2 i } { 2 } \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} \\eta ~ f _ { n } ( \\eta ) = \\sqrt { ( n + 1 ) ( k - n ) } f _ { n - 1 } ( \\eta ) , \\frac { \\partial } { \\partial \\eta } ~ f _ { n } ( \\eta ) = \\sqrt { n ( k + 1 - n ) } f _ { n - 1 } ( \\eta ) . \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} \\int _ { \\Omega _ { 2 } } \\ ! \\ ! | \\nabla \\phi ( x ) | ^ 2 d x & = \\sum _ { n \\in J _ { k } } | c _ { n } | ^ 2 \\int _ { \\Omega _ { 2 } } \\ ! \\ ! | \\nabla \\phi _ { n } ( x ) | ^ 2 d x \\\\ & + \\sum _ { n , m \\in J _ { k } \\atop n \\neq m } c _ { n } \\overline { c _ { m } } \\int _ { \\Omega _ { 2 } } \\ ! \\ ! \\nabla \\phi _ { n } ( x ) \\cdot \\nabla \\phi _ { m } ( x ) \\ , d x . \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{align*} \\chi ^ { } _ \\mathrm { m i n } ( k ) & = \\log \\sqrt { L } - \\underset { n \\rightarrow \\infty } { \\lim } \\dfrac { 1 } { n } \\log \\left \\Vert \\big ( B ^ { \\left ( n \\right ) } ( k ) \\big ) ^ { - 1 } \\right \\Vert \\\\ \\chi ^ { } _ \\mathrm { m a x } ( k ) & = \\log \\sqrt { L } + \\underset { n \\rightarrow \\infty } { \\lim } \\dfrac { 1 } { n } \\log \\left \\Vert B ^ { \\left ( n \\right ) } ( k ) \\right \\Vert . \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} T ^ { \\ , b _ { n + 1 } - b _ n } \\ , e _ { b _ n } = \\smash [ t ] { v _ n \\ , \\Big ( \\prod _ { j = b _ n + 1 } ^ { b _ { n + 1 } - 1 } w _ j \\Big ) \\ , e _ { b _ { \\varphi ( n ) } } - e _ { b _ n } \\quad \\hbox { f o r e v e r y $ n \\geq 1 $ } . } \\end{align*}"}
-{"id": "6584.png", "formula": "\\begin{align*} E _ { n , \\kappa } \\ ; = \\ ; m c ^ 2 \\Big ( 1 + \\frac { ( Z \\alpha _ \\mathrm { f } ) ^ 2 } { \\big ( n + \\sqrt { \\kappa ^ 2 - ( Z \\alpha _ \\mathrm { f } ) ^ 2 } \\ , \\big ) ^ { \\ ! 2 } } \\Big ) ^ { \\ ! - \\frac { 1 } { 2 } } , n \\in \\mathbb { N } _ 0 , \\ , \\kappa \\in \\mathbb { Z } \\ ! \\setminus \\ ! \\{ 0 \\} \\ , . \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} \\langle ( L _ { c _ * } + 1 ) \\hat { u } _ { \\pm 1 } , \\hat { u } _ { \\pm 1 } \\rangle _ { L ^ 2 ( \\mathbb { R } ) } \\geq A \\| \\hat { u } _ { \\pm 1 } \\| _ { H ^ 1 ( \\mathbb { R } ) } ^ 2 \\mbox { \\rm i f } \\ ; \\ ; \\langle \\eta _ * , \\hat { u } _ { \\pm 1 } \\rangle _ { L ^ 2 ( \\mathbb { R } ) } = 0 . \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} 1 + \\frac { i c } { 2 \\pi \\gamma } \\psi _ 1 ^ 0 ( \\sqrt { 2 } / \\gamma , - \\lambda / \\gamma ) = 0 . \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} T _ n \\leq T ( \\pi _ n ) \\leq \\sum _ { i = 1 } ^ { n } t ( f _ i ) \\hat { T } ^ { ( n ) } _ n \\leq \\hat { T } ^ { ( n ) } ( \\hat { \\pi } ^ { ( n ) } _ n ) \\leq \\sum _ { i = 1 } ^ { n } t ^ { ( n ) } ( f _ i ) . \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} P _ { Y _ 2 , Y _ 3 | M } ( 0 , 0 | 0 ) = P _ { Y _ 2 , Y _ 3 | M } ( 1 , 0 | 0 ) = \\frac { 1 } { 2 } , \\\\ P _ { Y _ 2 , Y _ 3 | M } ( 0 , 1 | 1 ) = P _ { Y _ 2 , Y _ 3 | M } ( 1 , 0 | 1 ) = \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} [ [ L _ { - i } , \\ , S _ { q - 1 } ] , \\ , M _ { - q + i } ] = 0 , \\ , 2 \\leqq i < q - 1 \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{align*} - c \\ , S ' = Q ' f + Q f ' , - c \\ , S '' = Q '' f + 2 Q ' f ' + Q f '' . \\end{align*}"}
-{"id": "5933.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { a _ n q ^ { ( m + 1 ) n } } { ( 1 - q ^ n ) ^ { k + 1 } } & = \\frac { 1 } { ( q ; q ) _ { \\infty } } \\sum _ { n \\geq 1 } \\sum _ { i = 1 } ^ { \\left \\lfloor \\frac { n } { m + 1 } \\right \\rfloor } \\sum _ { j = 0 } ^ { \\left \\lfloor \\frac { n - m } { i } \\right \\rfloor } \\binom { k - 1 + j } { k - 1 } s _ { n - m - j i , i } \\cdot a _ i \\cdot q ^ n , \\end{align*}"}
-{"id": "9506.png", "formula": "\\begin{align*} \\delta ( \\pi ( m ) ) = \\pi ( m ) b - b \\pi ( m ) , \\ \\ \\ m \\in M . \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} S ( p ) & = \\frac { \\phi ( m _ 1 ) } { m _ 1 } \\Bigg ( 1 + O \\bigg ( \\frac { 1 } { y { \\sqrt { \\log y } } } \\bigg ) \\Bigg ) - \\frac { \\phi ( m _ 2 ) } { m _ 2 } \\Bigg ( 1 + O \\bigg ( \\frac { 1 } { y { \\sqrt { \\log y } } } \\bigg ) \\Bigg ) \\\\ & = \\frac { \\phi ( m _ 1 ) } { m _ 1 } - \\frac { \\phi ( m _ 2 ) } { m _ 2 } + O \\left ( \\frac { 1 } { y { \\sqrt { \\log y } } } \\right ) , \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\displaystyle \\int _ \\Omega x ^ \\perp \\cdot \\rho u \\ , d x & n = 2 , \\\\ \\\\ \\displaystyle \\int _ \\Omega x \\times \\rho u \\ , d x & n = 3 \\end{array} \\right . \\end{align*}"}
-{"id": "2886.png", "formula": "\\begin{align*} \\overline { \\tau _ { \\alpha _ 1 - 1 } ( \\gamma _ 1 ) \\cdots \\tau _ { \\alpha _ { \\ell } - 1 } ( \\gamma _ { \\ell } ) } = \\sum _ { P \\{ 1 , \\ldots , \\ell \\} } \\ \\prod _ { S \\in P } \\ \\sum _ { \\widehat { \\alpha } } \\tau _ { \\widehat { \\alpha } } ( \\widetilde { \\mathsf { K } } _ { \\alpha _ S , \\widehat { \\alpha } } \\cdot \\gamma _ S ) \\ . \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} \\partial _ X ^ j G _ Z , \\ [ \\partial _ X ^ j G _ Z ] _ \\alpha \\le \\frac { C _ k } { \\abs { W - X } ^ 2 + 1 } , \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{align*} B _ { n } \\left ( x \\right ) = \\sum _ { p = 1 } ^ { n } \\binom { n + 1 } { p + 1 } \\left ( - 1 \\right ) ^ { p } B _ { n } ^ { \\left ( - p \\right ) } \\left ( - p x \\right ) \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{gather*} \\frac { { \\rm d } Y } { { \\rm d } x } = \\big ( { - } P ( x - S ) ^ { - 1 } Q - T \\big ) Y \\end{gather*}"}
-{"id": "4814.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} w & 0 \\\\ 0 & w \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) ^ { p ^ 2 } = \\begin{pmatrix} w & 0 \\\\ 0 & w \\end{pmatrix} . \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} c _ 0 \\int _ { B _ r ( x _ 0 ) } | \\nabla ( u _ \\varepsilon - h ) ( x ) | ^ p \\ , d x \\leq \\frac { 1 } { \\varepsilon } | \\{ u _ \\varepsilon = 0 \\} \\cap B _ r ( x _ 0 ) | , \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} ( a x ^ { 3 } + x ^ { 2 } + a x + 1 ) y ^ { 2 } + ( b x ^ { 2 } + c x ^ { 2 } + x ^ { 2 } + b + 1 ) y + b = 0 . \\end{align*}"}
-{"id": "7959.png", "formula": "\\begin{align*} M _ q = | h ^ \\prime | _ \\infty \\| u _ s \\| _ q + ( | h | _ \\infty + | h | _ \\infty ^ 2 ) ( \\| u _ s \\| _ \\infty + | u _ \\infty | ) \\| \\nabla u _ s \\| _ q , \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} \\frak h = \\left ( 0 , 0 , 0 , 0 , 0 , 0 , \\frac { \\sqrt { 6 } } { 6 } ( e ^ { 1 2 } + e ^ { 3 4 } + e ^ { 5 6 } ) \\right ) . \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} R _ { Q ( 1 ) } = Q R _ 1 Q ^ { - 1 } , \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} K _ N ( x , y ; w ) = \\sqrt { w ( x ) w ( y ) } \\sum _ { j = 0 } ^ { N - 1 } \\varphi _ { j } ( x ; w ) \\varphi _ j ( y ; w ) , N \\geq 1 . \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} ( a b ) ^ g = ( a b ) ^ { a c } = ( a b ) ^ c = a \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{align*} J & = - 2 \\int _ { \\mathbb { R } ^ 2 } ( r \\partial _ r V ^ { ( \\alpha , a ) } + t \\nabla \\cdot H ^ { ( \\alpha , a ) } ) \\cdot \\mu t \\Delta \\sum \\limits _ { l = 0 } ^ { \\alpha } C _ \\alpha ^ l ( - 1 ) ^ { \\alpha - l } V ^ { ( l , a ) } d x \\\\ & \\quad + \\| S V ^ { ( \\alpha , a ) } - t f ^ 1 _ { \\alpha a } \\| ^ 2 _ { L ^ 2 } . \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} U ^ { \\nu } ( x ) = x , x \\in [ - 1 , 1 ] . \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} \\dfrac { d } { d x } \\left [ p ( x ) \\dfrac { d y } { d x } \\right ] + q ( x ) y = - \\lambda w ( x ) y , \\end{align*}"}
-{"id": "2183.png", "formula": "\\begin{align*} K ( x , y ) + F _ S ( x + y ) \\ ! + \\ ! \\int _ x ^ \\infty \\ ! \\ ! K ( x , t ) F _ S \\ ! ( t + y ) d t \\ ! = \\ ! 2 i \\sum _ { j = 1 } ^ N k _ j \\varphi ( i k _ j , x ) N _ { - , k _ j } e ^ { - k _ j y } . \\end{align*}"}
-{"id": "9053.png", "formula": "\\begin{align*} K _ s ( m , n ) = \\frac { 1 } { n } \\sum _ { j = 1 } ^ { n } \\mu ^ { ( 1 - s ) ( 2 j - 1 ) } ( \\mu ^ { 2 j - 1 } + 1 ) ^ m , \\enskip s = 1 , . . . , n . \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} \\begin{cases} - \\varepsilon ^ 2 \\Delta \\omega + \\omega = 0 & B _ R ( 0 ) \\\\ \\omega = 1 & \\partial B _ R ( 0 ) \\end{cases} \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{align*} M _ 1 J ( k ) T _ 1 = \\begin{bmatrix} k I _ \\mu + o ( k ) & o ( k ) \\\\ o ( k ) & I _ { n - \\mu } + o ( 1 ) \\end{bmatrix} , k \\to 0 , \\ ; \\ ; k \\in \\overline { \\mathbb { C } } ^ + , \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} \\overline { \\tau _ { \\alpha _ 1 - 1 } ( \\gamma _ 1 ) \\cdots \\tau _ { \\alpha _ { \\ell } - 1 } ( \\gamma _ { \\ell } ) } = \\sum _ { P \\{ 1 , \\ldots , l \\} } \\ \\prod _ { S \\in P } \\ \\sum _ { \\widehat { \\alpha } } \\tau _ { \\widehat { \\alpha } } ( \\widetilde { \\mathsf { K } } _ { \\alpha _ S , \\widehat { \\alpha } } \\cdot \\gamma _ S ) \\ \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} \\hat { u } \\left ( \\xi , 0 \\right ) = \\hat { f } \\left ( \\xi \\right ) , \\xi \\in R ^ { n } , \\end{align*}"}
-{"id": "9925.png", "formula": "\\begin{align*} \\widetilde { f } ( \\tau , \\xi ) = \\int d t d x \\ e ^ { - i ( \\tau t + \\xi \\cdot x ) } f ( t , x ) . \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} X ^ { \\mathrm { s s } } _ + = X - ( 0 \\oplus V ^ \\vee ) X ^ { \\mathrm { s s } } _ - = X - ( V \\oplus 0 ) , \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{align*} + \\left [ \\frac { A _ { 1 } B _ { 1 } } { B _ { 2 } } - \\left ( \\frac { B _ { 1 } } { B _ { 2 } } \\right ) ^ { \\prime } \\right ] f f _ { c _ { 2 } } - \\frac { B _ { 1 } } { B _ { 2 } } f ^ { \\prime } f _ { c _ { 2 } } - \\frac { B _ { 1 } } { B _ { 2 } } f f _ { c _ { 2 } } ^ { \\prime } = B _ { 1 } . \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} \\mathcal { A } f ( \\xi ) = \\sum _ { x : x \\in T } [ f ( \\xi _ { x \\delta } ) - f ( \\xi ) ] + \\sum _ { x : x \\in T } \\sum _ { y : y \\thicksim x } \\lambda \\rho ( x ) \\rho ( y ) [ f ( \\xi _ { x y } ) - f ( \\xi ) ] \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} F \\colon X = \\Sigma \\times I \\to X , \\ ; ( \\omega , x ) { \\mapsto } ( \\sigma \\omega , f _ { \\omega _ 0 } ( x ) ) , \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} G ( u ( t ) ) & \\le G ( v ( t ) ) = G ( | \\Delta _ { 0 } | ) + \\int _ { 0 } ^ { t } G ' ( v ( s ) ) v ' ( s ) \\d s \\\\ & = G ( | \\Delta _ { 0 } | ) + \\frac { 5 \\kappa _ { R } } { 2 } \\int _ { 0 } ^ { t } \\frac { \\rho ( u ( s ) ) } { \\rho ( v ( s ) ) } \\d s \\le G ( | \\Delta _ { 0 } | ) + \\frac { 5 \\kappa _ { R } } { 2 } t , \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{align*} \\mathrm { a d } ^ \\circ _ R ( X ) ( M ) = \\rho ^ \\circ ( S ^ { - 1 } ( X _ { ( 1 ) } ) ) ( S ^ { - 2 } ( X _ { ( 2 ) } ) \\triangleright M ) \\rho ^ \\circ ( X _ { ( 3 ) } ) , X \\in \\mathcal { U } , \\ M \\in \\mathrm { M a t } _ { N \\times N } ( \\mathcal { B } ) . \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frak m _ k ( b , \\dots , b ) = 0 . \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{align*} P _ i ( x ) = F ( 2 ^ { - i } ( 1 + x ) ) - ( - 1 ) ^ i F ( 2 ^ { - i } ( 1 - x ) ) \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} \\mu _ { \\alpha } : = \\alpha f ( \\alpha ) - c _ { \\bar { \\alpha } } A _ { \\bar { \\alpha } } . \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} V ^ \\delta \\phi ( x ) : = \\int _ \\mathbb { R } h _ \\delta ( x - y ) \\phi ( y ) d y , \\phi \\in \\mathcal { S } , x \\in \\mathbb { R } . \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} g _ 1 \\left ( z \\right ) = \\sum _ { n = 1 } ^ { \\infty } \\mu _ n \\frac { z ^ n } { n ! } , \\thinspace \\thinspace f \\left ( z \\right ) = \\log \\left ( 1 + z \\right ) \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 0 } ^ { 2 } \\varepsilon ^ { \\frac { j } { 2 } } \\left \\vert \\lambda \\right \\vert ^ { 1 - \\frac { j } { 2 } } \\left \\Vert u _ { 1 } ^ { \\left ( j \\right ) } \\right \\Vert _ { X } + \\left \\Vert A u _ { 1 } \\right \\Vert _ { X } \\leq C \\left \\Vert f \\right \\Vert _ { X } . \\end{align*}"}
-{"id": "8691.png", "formula": "\\begin{align*} \\| e ^ { ( t - s ) \\Delta } F _ 1 ( s ) \\| _ { L ^ 1 } & \\le \\| F _ 1 ( s ) \\| _ { L ^ 1 } = \\int _ { \\{ | x | < \\sqrt s \\} } s ^ { - \\frac { \\alpha + 1 } { \\alpha } } \\Bigl | f \\Bigl ( \\frac { x } { \\sqrt s } \\Bigr ) \\Bigr | ^ { \\alpha + 1 } d x \\\\ & = \\int _ { \\{ | x | < 1 \\} } | f ( x ) | ^ { \\alpha + 1 } d x \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} & \\big ( \\nabla { \\theta } ^ { k } _ { \\phi } , \\ , \\nabla u \\big ) = \\big ( \\nabla ( \\phi ^ { k } - I _ { h } \\phi ^ { k } ) , \\nabla u \\big ) + \\big ( \\vert \\Psi ^ { k } _ { h } \\vert ^ { 2 } - \\vert \\Psi ^ { k } \\vert ^ { 2 } , \\ , u \\big ) , \\forall u \\in X _ { h } ^ { r } . \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{align*} ( x , y , \\omega ) \\mapsto \\Delta ( x , y , \\omega ) = \\int _ 0 ^ T \\int _ 0 ^ T \\phi ( x + \\xi ^ 1 _ r ( \\omega ) ) | y + \\xi ^ 2 _ s ( \\omega ) - x - \\xi ^ 2 _ r ( \\omega ) | ^ { \\frac { \\beta - \\alpha } { 2 } - 1 } d r d s . \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} g \\left ( \\nu - \\frac { 1 } { 2 } \\right ) = \\ln \\nu + 1 + \\mathcal { O } \\left ( \\frac { 1 } { \\nu ^ 2 } \\right ) \\ ; . \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , S _ 1 ] = 0 \\end{align*}"}
-{"id": "9762.png", "formula": "\\begin{align*} j u = d \\psi + d ^ * \\xi + h _ u . \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} [ a , b ] ^ { \\operatorname { g r } } = a b - \\alpha _ A ( b ) a . \\end{align*}"}
-{"id": "9651.png", "formula": "\\begin{align*} E ^ { ( D ) } \\ : \\ \\ D y ^ 2 = f ( x ) . \\end{align*}"}
-{"id": "9170.png", "formula": "\\begin{align*} \\hat h = ( R ^ \\top R ) ^ { - 1 } R ^ \\top y . \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } b ^ { s - 1 } _ { 1 } F _ { 1 } \\left ( \\alpha , \\beta , - b \\right ) d b = \\frac { \\Gamma \\left ( \\alpha - s \\right ) \\Gamma \\left ( \\beta \\right ) \\Gamma \\left ( s \\right ) } { \\Gamma \\left ( \\alpha \\right ) \\Gamma \\left ( \\beta - s \\right ) } . \\end{align*}"}
-{"id": "2138.png", "formula": "\\begin{gather*} \\theta ^ 0 _ 1 = \\tilde { \\theta } ^ 0 - \\varepsilon ^ { - 1 } , \\theta ^ 0 _ 2 = \\varepsilon ^ { - 1 } , t = \\varepsilon \\tilde { t } , H = \\varepsilon ^ { - 1 } \\tilde { H } , \\\\ q _ 1 = \\tilde { q } _ 1 , p _ 1 = \\tilde { p } _ 1 + \\frac { - \\tilde { \\theta } ^ 0 + \\varepsilon ^ { - 1 } } { \\tilde { q } _ 1 } , x = \\varepsilon ^ { - 1 } \\tilde { x } , Y = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & \\varepsilon & 0 \\\\ 0 & 0 & \\varepsilon q _ 1 \\end{pmatrix} \\tilde { Y } . \\end{gather*}"}
-{"id": "6494.png", "formula": "\\begin{align*} h _ { \\mu } ( x , m ) = \\exp ( \\phi _ \\mu ( m ) x - k _ \\mu ( \\phi _ \\mu ( m ) ) ) . \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} D + E = \\sum _ { j = 0 } ^ { n + 1 } \\frac { q ^ j } { d ^ j } W ^ { ( n + 1 ) } _ j r ^ { n + 1 - j } . \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} & { \\rm D i s c r } = 0 \\implies ( R _ { 1 c } + 1 ) ^ 2 + 4 R _ { 1 h } R _ { 3 c } + 4 \\lambda T ' ( c ) = 0 , \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} P _ r ( t ) = t ^ { 3 + r } - t ^ { 2 + r } + t ^ r - t ^ { r - 1 } + t ^ { r - 2 } - \\ldots - t ^ 4 + t ^ 3 - t + 1 . \\end{align*}"}
-{"id": "5867.png", "formula": "\\begin{align*} \\rho ( \\mathbf { x } ) = g \\left ( \\rho ( \\mathbf { x } ) \\right ) , g \\in G . \\end{align*}"}
-{"id": "4891.png", "formula": "\\begin{align*} \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! C _ { I U } = { \\log _ 2 } { \\left ( 1 + \\frac { | \\lambda _ { m a x } ( \\textbf { H } _ { 0 } ) | ^ 2 } { \\sigma ^ 2 + \\mathbf { v } _ 1 ^ \\ast \\boldsymbol { \\Sigma } _ { 0 } \\mathbf { v } _ 1 } P _ { } \\right ) } , \\end{align*}"}
-{"id": "9476.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & - x & 3 & - 3 x \\\\ . & 1 & - y & 1 \\\\ . & . & 1 & - x \\\\ . & . & . & 1 \\end{bmatrix} s = \\begin{bmatrix} . \\\\ . \\\\ . \\\\ 1 \\end{bmatrix} , s = \\begin{bmatrix} x y x - x \\\\ y x - 1 \\\\ x \\\\ 1 \\end{bmatrix} \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} u ( z ) = G _ { \\nu } ( f ( z ) ) . \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} p : = \\frac { N ^ { 1 - \\alpha } } { \\lambda + N ^ { 1 - \\alpha } } , \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{align*} g ' = C _ 2 e ^ { - C _ 1 ( q - 1 ) t } g ^ q . \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} \\phi _ 2 = \\phi ( [ x , 1 ] ) \\geq \\phi ( [ x , x \\vee a _ 1 ] ) = \\phi ( [ x \\wedge a _ 1 , a _ 1 ] ) \\geq \\phi ( [ 0 , a _ 1 ] ) \\end{align*}"}
-{"id": "5179.png", "formula": "\\begin{align*} x _ { n } = F _ { n } \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{align*} F ( x + y ) + K ( x , y ) + \\int _ x ^ \\infty K ( x , t ) F ( t + y ) d t = 0 _ n , y > x \\ge 0 , \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} b = ( 0 , a _ 2 , \\cdots , a _ j , j , j + 1 , \\cdots , n - 1 ) \\end{align*}"}
-{"id": "9268.png", "formula": "\\begin{align*} \\nabla _ j \\vec { H } = 2 \\nabla _ i ( A ^ o ) ^ i _ j = : \\frac { n } { n - 1 } ( \\nabla ^ * A ^ o ) _ j \\ , . \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} D _ 1 f ( i , j ) : = f ( i + 1 , j ) - f ( i , j ) , D _ 2 f ( i , j ) : = f ( i , j + 1 ) - f ( i , j ) . \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{align*} b _ q = f _ q - \\Pi _ q [ f _ q ] . \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} 0 = [ L _ { - 4 } , \\ , L _ 2 ] = [ [ L _ { - 3 } , \\ , L _ { - 1 } ] , \\ , L _ 2 ] = [ L _ { - 3 } , \\ , [ L _ { - 1 } , \\ , L _ 2 ] ] = [ L _ { - 3 } , \\ , L _ 1 ] \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} \\mathbb { E } ( f ( X - \\mu ) | Y \\in B _ 1 ) = \\mathbb { E } ( f ( R A U ) | Z _ 1 + a \\mu ) . \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} U _ X : = \\Pi _ X / \\Pi _ X ^ { ( 3 ) } \\end{align*}"}
-{"id": "6392.png", "formula": "\\begin{align*} \\langle \\alpha _ \\delta ( 1 _ e ) , \\alpha _ \\delta ( 1 _ f ) \\rangle _ { C _ 0 ( E ^ 0 ) } = ( - 1 ) ^ { \\delta ( e ) + \\delta ( f ) } \\langle 1 _ e , 1 _ f \\rangle _ { C _ 0 ( E ^ 0 ) } = \\begin{cases} 1 _ { s ( e ) } & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{align*} & U ( x , t ) = W \\left ( x - u _ \\infty \\int _ 0 ^ t h ( \\tau ) d \\tau , t \\right ) , \\\\ & P ( x , t ) = Q \\left ( x - u _ \\infty \\int _ 0 ^ t h ( \\tau ) d \\tau , t \\right ) , \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} & \\int _ { \\Omega _ { 2 } } \\partial _ { 1 } \\phi _ { n } ( x ) \\partial _ { 1 } \\phi _ { m } ( x ) \\ , d x = & \\\\ & { 4 n _ { 1 } m _ { 1 } \\over \\pi ^ 2 } \\int _ { 0 } ^ \\pi \\int _ { \\ell _ { 2 } } ^ { \\ell _ { 2 } + \\delta _ { 2 } } \\cos ( n _ { 1 } x _ { 1 } ) \\cos ( m _ { 1 } x _ { 1 } ) \\ , d x _ { 1 } \\sin ( n _ { 2 } x _ { 2 } ) \\sin ( m _ { 2 } x _ { 2 } ) \\ , d x _ { 2 } = 0 . \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} v a r ( J _ y ) = \\sum _ { z \\in U _ { t - 1 } } v a r ( X _ { y , z } ) \\leq \\sum _ { z \\in U _ { t - 1 } } \\mathbb { E } X ^ 2 _ { y , z } = \\sum _ { z \\in U _ { t - 1 } } \\mathbb { E } X _ { y , z } \\leq C _ u . \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{align*} \\varepsilon ( E _ { a } \\triangleright ( \\mathsf { N } _ { m } ^ { n } ) _ { j } ^ { i } ) = \\pi ( E _ { a } ) _ { m } ^ { i } \\pi ( K _ { a } ^ { - 1 } ) _ { j } ^ { n } - \\pi ( 1 ) _ { m } ^ { i } \\pi ( E _ { a } K _ { a } ^ { - 1 } ) _ { j } ^ { n } . \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} \\theta _ 0 ( \\phi , p ) = \\lim _ { r \\rightarrow 0 } \\frac { \\mathrm { A r e a } ( \\phi ( \\Sigma ) \\cap B _ r ( p ) ) } { \\pi r ^ 2 } \\in \\N . \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} \\Delta ( f g ) = ( \\Delta f ) ( \\Delta g ) . \\end{align*}"}
-{"id": "4513.png", "formula": "\\begin{align*} E _ { \\mathbf { A } } ( t , x ) : = - \\partial _ { t } \\mathbf { A } ( t , x ) \\ , t \\in \\mathbb { R } , \\ x \\in \\mathbb { R } ^ { d } \\ . \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} T _ { F , G , N } ^ L ( f _ 1 , \\dots , f _ 4 ) = \\frac { 1 } { N ^ 2 } \\sum \\limits _ { \\substack { \\mathbf { n } \\in \\mathbb { Z } ^ 4 \\\\ \\Vert L \\mathbf { n } \\Vert _ \\infty \\leqslant 1 0 \\\\ ( \\begin{smallmatrix} 5 & 1 \\end{smallmatrix} ) L \\mathbf { n } = - 1 , 0 , 1 } } \\Big ( \\prod \\limits _ { j = 1 } ^ 4 f _ j ( n _ j ) \\Big ) F ( \\mathbf { n } ) . \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 \\ , \\left ( \\frac 1 2 \\right ) & x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\overbrace { \\begin{matrix} \\sqrt { t _ 2 } & 0 \\\\ - \\sqrt { t _ 2 } & 0 \\end{matrix} } & \\overbrace { \\begin{matrix} 1 & 0 & - t _ 1 / 2 & 0 \\\\ - 1 & 0 & t _ 1 / 2 & 0 \\end{matrix} } \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "2679.png", "formula": "\\begin{align*} ( 1 + k , 1 + k ^ 2 ) = \\left \\{ \\begin{array} { l l } 2 , & 2 \\nmid k ; \\\\ 1 , & 2 \\mid k . \\end{array} \\right . \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{align*} ( w t _ \\lambda ) \\cdot _ p \\mu = w ( \\mu + p \\lambda + \\rho ) - \\rho \\end{align*}"}
-{"id": "8842.png", "formula": "\\begin{align*} Z ^ * ( \\Gamma ) = \\sum _ { f = 0 } ^ { | E | } ( - 1 ) ^ f \\sum _ { \\substack { F \\subseteq E \\\\ | F | = f } } Z ( \\Gamma _ F ) . \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} F _ * ^ e ( j u ^ { r } v ^ { q - k - r } ( f + u v ) ) = \\bigoplus _ { i \\in \\Delta _ e } f _ { ( i , j ) } F _ * ^ e ( i u ^ { r } v ^ { q - k - r } ) \\oplus F _ * ^ e ( j u ^ { r + 1 } v ^ { q - k - r + 1 } ) \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} E _ \\delta : x ^ 2 - \\delta y ^ 2 = 1 . \\end{align*}"}
-{"id": "8134.png", "formula": "\\begin{align*} \\chi _ { \\tau } ( t , r ) = - \\int _ 0 ^ r \\frac { ( B _ r - B _ u ) } { \\pi | B _ r - B _ u | ^ 2 } \\Big ( e ^ { - \\frac { | B _ r - B _ u | ^ 2 } { 2 ( i \\tau + t - r ) } } - e ^ { - \\frac { | B _ r - B _ u | ^ 2 } { 2 i \\tau } } \\Big ) d u , \\end{align*}"}
-{"id": "1619.png", "formula": "\\begin{align*} \\frac { \\zeta ^ \\ell - \\zeta ^ { k \\pi / \\omega } } { \\sin \\ell \\omega } = \\ \\frac { \\zeta ^ \\ell - \\zeta ^ { k \\pi / \\omega } } { \\ell - k \\pi / \\omega } \\ \\frac { \\ell - k \\pi / \\omega } { \\sin \\ell \\omega - \\sin k \\pi } \\ ; . \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} V ^ { \\tfrac { 1 } { m _ 1 } } ( x ( \\varepsilon ) ) \\le V ^ { \\tfrac { 1 } { m _ 1 } } ( x ^ 0 ) { + } \\frac { 1 } { m _ 1 } V ^ { \\tfrac { 1 } { m _ 1 } { - } 1 } ( x ^ 0 ) \\sum _ { i = 1 } ^ n \\frac { \\partial V ( x ^ 0 ) } { \\partial x _ i } y _ i + \\frac { \\bar L } { 2 } . \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} \\left ( \\frac { f } { p } \\right ) ^ { 2 } + \\left ( \\lambda _ { 2 } \\lambda _ { 3 } - \\lambda _ { 1 } \\right ) \\left ( \\frac { f } { p } \\right ) ^ { - 2 } = 2 \\lambda _ { 2 } . \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{align*} A ( z ) = { } _ a \\langle 0 | T _ { a } ( z ) | 0 \\rangle _ { a } , \\\\ B ( z ) = { } _ a \\langle 0 | T _ { a } ( z ) | 1 \\rangle _ { a } , \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} u _ \\varepsilon ( t ) = u _ 0 ( t ) + \\sum _ { \\beta \\in \\frac { \\pi } { \\omega } \\N _ * } \\varepsilon ^ \\beta \\ , u ^ \\beta ( t ) + \\sum _ { \\beta \\in \\frac { \\pi } { \\omega } \\N _ * } \\varepsilon ^ \\beta \\ , U ^ \\beta ( \\tfrac { t } { \\varepsilon } ) . \\end{align*}"}
-{"id": "2621.png", "formula": "\\begin{align*} D ( A ) : = \\left \\{ u \\in { \\Bbb H } \\ ; ; \\ ; u _ { 1 } \\in H _ { 1 } , \\ ; L u _ { 0 } + B u _ { 1 } \\in H _ { 0 } \\right \\} . \\end{align*}"}
-{"id": "1301.png", "formula": "\\begin{align*} \\frac { K _ { 0 } \\left [ 1 - \\left ( f g ^ { \\prime } \\right ) ^ { 2 } \\right ] ^ { 2 } } { f f ^ { \\prime \\prime } \\left ( g ^ { \\prime } \\right ) ^ { 2 } } = \\frac { g g ^ { \\prime \\prime } } { \\left ( g ^ { \\prime } \\right ) ^ { 2 } } - \\frac { \\left ( f ^ { \\prime } \\right ) ^ { 2 } } { f f ^ { \\prime \\prime } } . \\end{align*}"}
-{"id": "7997.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\sup _ { 0 \\leq t \\leq T } | \\langle w _ k ( t ) - w ( t ) , \\phi \\rangle | = 0 , \\forall \\phi \\in L ^ 2 _ \\sigma ( \\Omega ) , \\end{align*}"}
-{"id": "9438.png", "formula": "\\begin{align*} \\left ( \\frac { { \\rm d } } { { \\rm d } t } - r \\right ) \\int \\frac { \\big | \\widetilde { U } \\big | ^ 2 } { 2 h } + \\int \\frac { \\widetilde { W } ^ { \\rm T } Q _ r ( w ^ * ) \\widetilde { W } } { 2 h } + \\int \\widetilde { W } \\cdot \\mathrm { L } ( w ^ * ) = 0 . \\end{align*}"}
-{"id": "3305.png", "formula": "\\begin{align*} \\partial _ t p _ t ( y , z ) = L _ { y } p _ t ( y , z ) , \\ \\ t > 0 , y , z \\in \\mathbb { R } , \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} X _ F & : = X \\cap \\Delta \\cap \\{ F = 0 \\} , & Y _ F & : = \\pi _ z ( X _ F ) . \\end{align*}"}
-{"id": "367.png", "formula": "\\begin{align*} { R _ 1 ^ * } ^ { m _ 1 } \\cdots { R _ n ^ * } ^ { m _ n } W f = ( V _ 1 ^ { i _ 1 + m _ 1 - 1 } \\cdots V _ 1 ^ { i _ n + m _ n - 1 } f ) \\end{align*}"}
-{"id": "4500.png", "formula": "\\begin{align*} 0 < \\mathfrak { m } _ { \\mathrm { A C } } \\left ( \\mathbb { R } \\backslash \\left \\{ 0 \\right \\} \\right ) < \\infty D _ { \\left \\{ 0 \\right \\} } = 0 \\ , \\end{align*}"}
-{"id": "7061.png", "formula": "\\begin{align*} u \\left ( x , 0 \\right ) = f \\left ( x \\right ) . \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } G _ n ^ x ( w ) = \\sum _ { n \\geq 0 } \\mathbb { P } ( b _ n ( w ) \\leq \\sigma _ 0 + v _ 1 \\leq a _ n ^ x ( w ) ) , \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} a ( n ) ^ 2 & - a ( n + 1 ) a ( n - 1 ) \\\\ & = 2 ^ { 2 k } - ( b ( k ) - 2 ) b ( k ) \\geq 2 ^ { 2 k } - 1 / 9 ( 2 ^ k + 1 ) ( 2 ^ k + 3 ) = 1 / 9 ( 2 ^ { 2 k + 3 } - 2 ^ { k + 2 } - 3 ) > 0 \\end{align*}"}
-{"id": "3540.png", "formula": "\\begin{align*} u = \\frac { \\dot { a } } { a } \\ ; ; \\ ; v = \\frac { \\dot { b } } { b } \\ ; ; \\ ; \\psi = \\frac { 1 } { 2 } \\dot { \\phi } ^ { 2 } . \\end{align*}"}
-{"id": "304.png", "formula": "\\begin{align*} \\sum _ { k } \\varepsilon ( X \\triangleright \\mathsf { P } _ { k } ^ { j } ) \\varepsilon ( Y \\triangleright \\mathsf { P } _ { i } ^ { k } ) = \\sum _ { m , n , o , p } \\pi ( S ( X _ { ( 1 ) } ) ) _ { m } ^ { j } c _ { n } ^ { m } \\pi ( X _ { ( 2 ) } S ( Y _ { ( 1 ) } ) ) _ { o } ^ { n } c _ { p } ^ { o } \\pi ( Y _ { ( 2 ) } ) _ { i } ^ { p } . \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{align*} \\eta = \\sum _ { i = 1 } ^ { k } \\theta _ i \\bar { \\eta } = \\sum _ { i = 1 } ^ { k } \\bar { \\theta } _ i \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} \\ell [ y ] ( x ) = - ( ( 1 - x ^ 2 ) y ' ( x ) ) ' \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{align*} I _ l ^ Z = \\{ i < \\tilde { i } : | z _ i | > l \\mathrm { \\ a n d \\ } z _ i ( l ) = 1 \\} , \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} w _ k ^ i = 2 M _ i ^ k ( X _ i \\cos ( \\theta _ i k ) + Y _ i \\sin ( \\theta _ i k ) ) , k \\geq 0 , \\end{align*}"}
-{"id": "9991.png", "formula": "\\begin{align*} & E _ 1 = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} , A _ 1 = \\begin{bmatrix} - 1 & 0 & 4 \\pi \\\\ 0 & - 1 & 0 \\\\ - 4 \\pi & 0 & - 4 \\end{bmatrix} & \\\\ & E _ 2 = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 1 \\end{bmatrix} , A _ 2 = \\begin{bmatrix} - k _ 2 k _ 3 & 0 & 0 \\\\ 0 & - 1 & 0 \\\\ - 4 k _ 1 k _ 2 k _ 3 & - 1 & - 4 k _ 3 \\end{bmatrix} & \\end{align*}"}
-{"id": "5411.png", "formula": "\\begin{align*} \\lambda _ k ( A ) \\leqslant \\lambda _ k ( M ) \\leqslant \\lambda _ { k + n } ( A ) ~ , ~ ~ ~ k = 1 , \\hdots , n . \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ \\tau \\tilde \\vartheta _ 1 & = - \\frac { i } \\pi \\partial _ z ^ 2 \\tilde \\vartheta _ 1 , \\\\ \\partial _ \\tau \\zeta & = \\partial _ \\tau ( \\eta z ) + \\partial _ \\tau \\frac { \\partial _ z \\tilde \\vartheta _ 1 } { \\tilde \\vartheta _ 1 } \\\\ \\partial _ \\tau \\zeta _ j & = \\partial _ z ^ j ( \\partial _ \\tau \\zeta ) , j = 1 , 2 , 3 , \\end{aligned} \\end{align*}"}
-{"id": "5647.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } \\frac { 1 } { \\varepsilon } ( - 3 v _ \\star ^ 2 + 2 ( a + 1 ) v _ \\star - a ) + c - 2 \\sqrt { \\frac { b } { \\varepsilon } } < 0 , \\\\ \\frac { 1 } { \\varepsilon } ( - 3 v _ \\star ^ 2 + 2 ( a + 1 ) v _ \\star - a ) + c + 2 \\sqrt { \\frac { b } { \\varepsilon } } > 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{align*} p _ - ( \\lambda ) h p _ - ( \\lambda ) = o \\left ( t ^ { \\lambda _ 1 } ( \\log t ) ^ { \\lambda _ 2 } \\cdots \\left ( \\log ^ { ( n - 1 ) } t \\right ) ^ { \\lambda _ n } \\right ) . \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} K _ \\varepsilon = \\| w _ 0 \\| _ 2 ^ 2 + C \\left ( | h | _ \\infty \\| u _ s \\| _ 3 + \\| v _ 0 \\| _ { 3 , \\infty } + M _ 3 \\right ) \\int _ 0 ^ { T _ 1 } \\| \\nabla w \\| _ 2 ^ 2 d \\tau + C \\sqrt { T _ \\varepsilon } . \\end{align*}"}
-{"id": "2335.png", "formula": "\\begin{align*} \\mu _ j \\coloneqq \\sigma ( 1 + \\delta _ j ) ( 1 + \\sqrt { 2 \\log ( 9 e p / j ) } ) , j = 1 , . . . , s . \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} \\eta = e ^ { - \\psi } \\eta _ 0 \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} \\mathbf { E } = - \\nabla \\phi - \\frac { \\partial \\mathbf { A } } { \\partial t } , \\mathbf { B } = \\nabla \\times \\mathbf { A } , \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{align*} a _ 0 \\ne 0 \\enskip \\mbox { a n d } \\enskip a _ k = \\mu \\bar a _ { n - k } \\enskip \\mbox { f o r e v e r y } \\enskip 0 \\le k \\le n . \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{align*} \\log ( | \\Omega | ) & \\ge ( d / 2 ) \\log \\left ( \\frac { p } { 5 d } \\right ) , \\sum _ { j = 1 } ^ p \\mathbf 1 _ { w _ j \\ne w _ j ' } = \\| w - w ' \\| ^ 2 > d , \\end{align*}"}
-{"id": "1092.png", "formula": "\\begin{align*} \\frac { 1 } { N ^ { s + 1 } } \\int \\limits _ { \\mathbf { w } \\in \\mathbb { R } ^ { s + 1 } } ^ * g _ 1 ( \\psi _ 1 ( \\mathbf { w } ) ) \\prod \\limits _ { k = 1 } ^ { s + 1 } b _ k ^ \\prime ( \\mathbf { w } ) b _ k ( \\mathbf { w } ) \\ , d \\mathbf { w } . \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{gather*} \\nabla ^ 2 \\phi = 0 \\mbox { i n } \\ W , \\\\ \\phi _ y - \\nu \\phi = 0 \\mbox { o n } \\ F , \\\\ \\partial \\phi / \\partial n = 0 \\mbox { o n } \\ \\partial W \\setminus \\bar F . \\end{gather*}"}
-{"id": "1797.png", "formula": "\\begin{align*} \\alpha \\xi ^ { m } + \\alpha \\eta ^ { m } + q _ { m } ( \\xi , \\eta ) = ( \\alpha \\xi ^ { m } + \\alpha \\eta ^ { m } ) \\left ( 1 + \\frac { q _ { m } ( \\xi , \\eta ) } { \\alpha \\xi ^ { m } + \\alpha \\eta ^ { m } } \\right ) \\leq \\frac { \\alpha } { 2 } ( \\xi ^ { m } + \\eta ^ { m } ) < 0 < | \\eta | . \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{align*} - \\Psi '' ( x ) + V ( x ) \\Psi ( x ) = k ^ 2 \\Psi ( x ) , x \\in \\mathbb { R } ^ + : = ( 0 , + \\infty ) , \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{align*} \\sum _ { t \\geq 0 } \\binom { m } { n t + s - 1 } = \\frac { 1 } { n } \\sum _ { j = 1 } ^ n ( \\omega ^ j + 1 ) ^ m \\omega ^ { j ( 1 - s ) } , \\enskip s = 1 , . . . , n ; \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} u _ { k + 1 } = u _ { 0 } + G _ { \\varepsilon } u _ { k } u _ { k } = u _ { k \\varepsilon } \\left ( t \\right ) k = 0 1 , 2 , . . . . , \\end{align*}"}
-{"id": "4254.png", "formula": "\\begin{align*} - \\int _ { \\mathbb R ^ 3 } \\partial _ t \\nabla _ \\beta u ^ i \\nabla _ \\beta \\Delta u ^ i \\phi ^ 6 d x = \\frac { 1 } { 2 } \\frac { d } { d t } \\int _ { \\mathbb R ^ 3 } | \\nabla ^ 2 u | ^ 2 \\phi ^ 6 d x + 6 \\int _ { \\mathbb R ^ 3 } \\partial _ t \\nabla _ \\beta u ^ i \\nabla _ { \\beta \\alpha } ^ 2 u ^ i \\phi ^ 5 \\nabla _ \\alpha \\phi d x . \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} { \\rm P r } \\{ f _ R ( x ) = y \\} \\le \\frac { | { \\cal Y } | } { | { \\cal X } | } \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} \\epsilon _ g ( f ) = \\begin{cases} 1 & g | f \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "1808.png", "formula": "\\begin{align*} \\Big \\| \\big [ [ u ] \\big ] _ { j } \\Big \\| _ { j } = \\Big \\| \\widehat { [ u ] } \\Big \\| _ { L ^ { 2 } ( B [ 0 , j ] ) } . \\end{align*}"}
-{"id": "3260.png", "formula": "\\begin{align*} \\alpha _ i ^ - ( t ) = \\alpha _ i ^ - ( 0 ) \\ , \\forall t \\ge 0 . \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} \\overline s _ { T } ( x ) = { \\theta e ^ { - \\theta x } \\over 1 - e ^ { - \\theta T } } \\ 1 _ { [ 0 , T ] } . \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{align*} [ L _ { - k } , \\ , S _ k ] = 0 , \\ , 2 \\leqq k \\leqq j - 1 , \\end{align*}"}
-{"id": "4773.png", "formula": "\\begin{align*} \\pi _ 1 ( \\varphi ) & = \\alpha f ( \\alpha ) - c _ { \\bar { \\alpha } } \\bar { \\alpha } A ' _ { \\bar { \\alpha } } = \\alpha f ( \\alpha ) - c _ { \\bar { \\alpha } } A _ { \\bar { \\alpha } } = 0 , \\\\ \\pi _ 1 ( \\psi ) & = - c _ { { \\alpha } } { \\alpha } A ' _ { { \\alpha } } + \\bar { \\alpha } f ( \\bar { \\alpha } ) = - c _ { { \\alpha } } A _ { { \\alpha } } + \\bar { \\alpha } f ( \\bar { \\alpha } ) = 0 , \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} ( - \\gamma H _ { \\rm { O H } } - 2 \\lambda ) U = - i \\sum _ n G _ n a ^ \\dagger e _ n . \\end{align*}"}
-{"id": "9797.png", "formula": "\\begin{align*} \\alpha : = \\phi ( | u | ^ 2 ) j u , \\end{align*}"}
-{"id": "915.png", "formula": "\\begin{align*} P ( y ) = \\sum _ { i = 0 } ^ \\infty c _ i \\alpha ^ i \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} \\breve { g } ( t ) = \\frac { \\operatorname { S i } \\Big ( 2 \\pi B \\big ( t + \\frac { T _ c } { 2 } \\big ) \\Big ) - \\operatorname { S i } \\Big ( 2 \\pi B \\big ( t - \\frac { T _ c } { 2 } \\big ) \\Big ) } { \\pi \\sqrt { T _ c } } , \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a & b \\pi ^ n \\\\ c & d \\end{bmatrix} \\in R ^ \\bullet \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} \\mu ^ + : = d d ^ c G ^ + , \\mu ^ - : = d d ^ c G ^ - . \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} \\varphi _ 2 ' ( \\sum z _ i \\mathbf w _ i ) = \\sum ( c \\alpha _ i + d ) z _ i ^ 2 . \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} \\langle \\epsilon E , \\phi \\rangle = 0 \\quad ( \\phi \\in \\textnormal { n } _ 2 ( \\theta ) ) . \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} \\zeta _ \\ell \\eta _ m + \\eta _ \\ell \\zeta _ m = { } & \\eta _ { \\ell + m } , \\\\ \\zeta _ \\ell \\zeta _ m - \\eta _ { \\ell } \\eta _ { m } = { } & \\zeta _ { \\ell + m } . \\end{align*}"}
-{"id": "9805.png", "formula": "\\begin{align*} { \\left . { \\frac { { \\partial U } } { { \\partial \\xi } } } \\right | _ { n m l } } = \\sum \\limits _ { i , j , k = 0 } ^ N { { { { U } } _ { i j k } } { \\ell ' _ i } ( { \\xi _ n } ) { \\ell _ j } ( { \\eta _ m } ) { \\ell _ k } ( { \\zeta _ l } ) } = \\sum \\limits _ { i = 0 } ^ N { { U _ { i m l } } { { \\ell ' } _ i } ( { \\xi _ n } ) } \\equiv \\sum \\limits _ { i = 0 } ^ N { { U _ { i m l } } { \\mathcal { D } _ { n i } } } , \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{align*} { \\cal R } _ n ( g ) = \\sum _ { x \\in \\xi } g \\Bigl ( \\phi _ { l ^ x _ n } \\bigl ( { \\cal M } ( { \\cal S } ^ \\prime ( \\widehat { \\pi } ^ { ( r ^ x _ n , 0 ) } , \\widehat { \\pi } ^ { ( l ^ x _ n , 0 ) } ) ) \\bigr ) , \\phi ^ \\prime _ { l ^ x _ n } \\bigl ( { \\cal M } ( \\widehat { \\cal S } ( \\widehat { \\pi } ^ { ( r ^ x _ n , 0 ) } , \\widehat { \\pi } ^ { ( l ^ x _ n , 0 ) } ) ) \\bigr ) \\Bigr ) . \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{align*} z _ i \\ , z _ j ( z _ i + z _ j ) = H \\end{align*}"}
-{"id": "9223.png", "formula": "\\begin{align*} x _ { \\alpha , \\alpha ' } = \\sup _ { \\beta \\geq \\alpha , \\beta ' \\geq \\alpha ' } \\lvert x _ { \\beta } - x _ { \\beta ' } \\rvert \\wedge y . \\end{align*}"}
-{"id": "9606.png", "formula": "\\begin{align*} L ^ i _ { \\underline { j k } } = \\overline L ^ i _ { \\underline { j k } } - \\delta ^ i _ k \\psi _ j - \\delta ^ i _ j \\psi _ k - F ^ i _ k \\sigma _ j - F ^ i _ j \\sigma _ k . \\end{align*}"}
-{"id": "9449.png", "formula": "\\begin{align*} \\begin{array} { l l } \\displaystyle { h ^ { * } ( \\theta , x ) = h ' ( \\sqrt { 2 \\theta } , x ) } & \\displaystyle { b ^ { * } ( \\theta , x ) = b ' ( \\sqrt { 2 \\theta } , x ) } \\\\ \\displaystyle { d ^ { * } ( \\theta , x ) = \\frac { d ' ( \\sqrt { 2 \\theta } , x ) } { \\sqrt { 2 \\theta } } } & \\displaystyle { v ^ { * } ( \\theta , x ) = \\frac { v ' ( \\sqrt { 2 \\theta } , x ) } { \\sqrt { 2 \\theta } } } \\end{array} \\end{align*}"}
-{"id": "10045.png", "formula": "\\begin{align*} g ( \\mathrm { T } ^ { \\mathrm { w } } ( X , Y ) , Z ) - g ( \\mathrm { T } ^ { \\mathrm { w } } ( Z , Y ) , X ) = g ( \\mathrm { T } ^ { \\mathrm { w } } ( J _ { \\varphi } Z , Y ) , J _ { \\varphi } X ) - g ( \\mathrm { T } ^ { \\mathrm { w } } ( J _ { \\varphi } X , Y ) , J _ { \\varphi } Z ) , \\forall X , Y , Z \\in { \\mathfrak X } ( M ) , \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} \\Pi _ N \\psi ( x ) : = \\int _ { | k | \\leq N } \\hat \\psi ( k ) e ^ { i k \\cdot x } d k , \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} \\| \\widetilde { E } \\| _ { L ^ { \\infty , \\infty } ( S _ t , d \\sigma ) } \\lesssim \\| E \\| _ { L ^ 1 ( \\mathbb F _ q ^ d , d { \\bf m } ) } = | E | \\quad \\mbox { f o r a l l } ~ ~ E \\subset \\mathbb F _ q ^ d , ~ t \\ne 0 . \\end{align*}"}
-{"id": "2507.png", "formula": "\\begin{align*} v _ 1 & = ( 0 , 1 2 8 6 4 , - 1 7 1 5 2 , - 1 4 5 8 0 , 2 9 1 6 3 , - 1 4 5 8 4 , 4 2 8 8 , 0 , 1 ) ^ T , \\\\ v _ 2 & = ( 0 , 1 9 2 , - 2 5 6 , - 2 2 0 , 4 4 1 , - 2 2 2 , 6 4 , 1 , 0 ) ^ T , \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} \\omega _ a = ( \\omega _ 1 , \\omega _ 2 , \\ldots , \\omega _ { l } , \\sigma _ { l + 1 } , \\sigma _ { l + 2 } , \\ldots , \\sigma _ N ) \\end{align*}"}
-{"id": "847.png", "formula": "\\begin{align*} | x ^ { ( 1 ) } _ { j , \\varepsilon } - x ^ { ( 2 ) } _ { j , \\varepsilon } | = o \\big ( \\varepsilon ^ { 3 } \\big ) . \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{align*} I ( \\mu ) = \\sup _ L ( I _ L ( \\mu ) ) I _ L ( \\mu ) = \\inf _ { \\{ \\nu \\ | \\ , \\pi _ L ( \\nu ) = \\pi _ L ( \\mu ) \\} } I ( \\nu ) \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} \\Bigl \\vert a _ 2 q _ 1 \\frac { \\lambda _ 1 } { \\lambda _ 2 } - a _ 1 q _ 2 \\Bigr \\vert & = \\Bigl \\vert \\frac { a _ 2 } { \\lambda _ 2 \\alpha } ( q _ 1 \\lambda _ 1 \\alpha - a _ 1 ) - \\frac { a _ 1 } { \\lambda _ 2 \\alpha } ( q _ 2 \\lambda _ 2 \\alpha - a _ 2 ) \\Bigr \\vert \\\\ & \\ll q _ 2 \\vert q _ 1 \\lambda _ 1 \\alpha - a _ 1 \\vert + q _ 1 \\vert q _ 2 \\lambda _ 2 \\alpha - a _ 2 \\vert \\\\ & \\ll Q _ 2 \\frac { X ( \\log X ) ^ { 1 0 } } { Z _ 1 ^ 2 } + Q _ 1 \\frac { X ( \\log X ) ^ { 1 0 } } { Z _ 2 ^ 2 } . \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{gather*} t H _ { \\mathrm { S u z } } ^ { 2 + \\frac { 3 } { 2 } } \\left ( { - \\theta ^ 0 _ 1 , \\ , \\theta ^ 0 _ 2 - \\theta ^ 0 _ 1 \\atop \\theta ^ \\infty _ 2 } ; t ; { q _ 1 , p _ 1 \\atop q _ 2 , p _ 2 } \\right ) = t H _ { \\mathrm { I I I } ( D _ 7 ) } \\big ( { - } \\theta ^ 0 _ 1 ; t ; q _ 1 , p _ 1 \\big ) \\\\ \\qquad { } + t H _ { \\mathrm { I I I } ( D _ 7 ) } \\big ( \\theta ^ 0 _ 2 - \\theta ^ 0 _ 1 ; t ; q _ 2 , p _ 2 \\big ) + p _ 2 q _ 1 \\big ( p _ 1 ( q _ 1 + q _ 2 ) + \\theta ^ \\infty _ 2 \\big ) - q _ 1 . \\end{gather*}"}
-{"id": "7452.png", "formula": "\\begin{align*} a \\wedge b = 0 , a \\vee b = 1 . \\end{align*}"}
-{"id": "1283.png", "formula": "\\begin{align*} H _ { n } ( x ) = \\left ( - 1 \\right ) ^ { n } \\exp \\left [ x ^ { 2 } \\right ] \\frac { \\mathsf { d } ^ { n } } { \\mathsf { d } x ^ { n } } \\exp \\left [ - x ^ { 2 } \\right ] . \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} S ( e _ i \\otimes e _ i ) = \\psi _ i \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} \\widehat { \\mu } ( - k ) \\ , = \\ , \\overline { \\widehat { \\mu } ( k ) } \\ , = \\ , \\widehat { \\mu } ( k ) \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} \\int _ T | C _ X ( t ) | \\ , d t = + \\infty , \\end{align*}"}
-{"id": "6955.png", "formula": "\\begin{align*} \\psi _ 1 ( t ) = 1 , \\psi _ { 2 \\nu } ( t ) = \\sqrt { 2 } \\cos 2 \\pi \\nu t , \\psi _ { 2 \\nu + 1 } ( t ) = \\sqrt { 2 } \\sin 2 \\pi \\nu t , \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} & \\Psi ( 0 ) = \\frac { ( \\delta - \\varrho _ a ) _ a ( \\epsilon - \\varrho _ a ) _ a } { ( \\delta ) _ a ( \\epsilon ) _ a } \\cdot \\Phi ( 0 ) . \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} f : \\mathbb { R } ^ { n } \\rightarrow \\mathbb { R } ^ { n } , ~ \\ \\ x = \\left ( x ^ { 0 } , x ^ { 1 } , . . . , x ^ { n - 1 } \\right ) \\mapsto ( f ^ { 0 } ( x ^ { 0 } ) , f ^ { 1 } ( x ^ { 1 } ) , . . . , f ^ { n - 1 } ( x ^ { n - 1 } ) ) , \\end{align*}"}
-{"id": "3743.png", "formula": "\\begin{align*} \\left ( \\sum _ { i \\in I } x _ { i } \\right ) ^ { n } = \\sum _ { \\tau : n \\to I } \\prod _ { m < n } x _ { \\tau ( m ) } . \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} \\| [ x ] _ j \\| _ j : = \\inf _ { p _ j ( z ) = 0 } p _ j ( x - z ) , \\end{align*}"}
-{"id": "8167.png", "formula": "\\begin{align*} { \\bf C } = \\left [ \\begin{array} { c c } { \\bf C } _ { 1 1 } & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf C } _ { 2 2 } \\end{array} \\right ] _ , \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { a _ n q ^ { ( m + 1 ) n } } { ( 1 - q ^ n ) ^ { k + 1 } } & = \\frac { 1 } { ( q ; q ) _ { \\infty } } \\sum _ { n \\geq 1 } \\sum _ { i = 1 } ^ { \\lfloor n / ( m + 1 ) \\rfloor } s _ { n - m , i } \\frac { a _ i } { ( 1 - q ^ i ) ^ k } \\cdot q ^ n , \\end{align*}"}
-{"id": "4519.png", "formula": "\\begin{align*} \\mathbf { \\Xi } _ { \\mathrm { p } } \\left ( t \\right ) = \\ \\underset { l \\rightarrow \\infty } { \\lim } \\Xi _ { \\mathrm { p } , l } ^ { ( \\omega ) } \\left ( t \\right ) \\in \\mathcal { B } ( \\mathbb { R } ^ { d } ) \\ . \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{gather*} _ { a _ 1 , a _ 2 , \\dots , a _ s } Y [ s ] ^ { ( k - \\ell ( a ) ) } v _ m = 0 0 \\le \\ell ( a ) \\le k , \\ k > m \\end{gather*}"}
-{"id": "7532.png", "formula": "\\begin{align*} [ ( z \\phi _ 0 z ^ { - 1 } ) ^ * & , z \\phi _ 0 z ^ { - 1 } ] = \\\\ & = [ \\phi _ 0 ^ * , \\phi _ 0 ] + \\frac { 1 } { 2 } [ \\phi _ 0 ^ * , [ G ( k ) , \\phi _ 0 ] ] + \\frac { 1 } { 2 } [ [ G ( k ) , \\phi _ 0 ] ^ * , \\phi _ 0 ] + O ( t ^ { - 2 } \\mathcal L ) \\\\ & = \\theta - \\Delta G ( k ) + O ( t ^ { - 2 } \\mathcal L ) \\\\ & = \\theta + ( P - 1 ) ( k ) + O ( t ^ { - 2 } \\mathcal L ) \\end{align*}"}
-{"id": "4350.png", "formula": "\\begin{align*} | \\tau _ j ( \\alpha ) | = | Y | ^ { 1 / t } = | X | ^ { - 1 / ( 2 t ) } \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } y ( t ) = \\frac { 2 A \\sqrt { 6 } \\ , y ( t ) ^ 5 \\ , - \\ , 1 } { 1 2 \\ , y ( t ) ^ 9 } , \\\\ y ( 0 ) = 1 . \\end{cases} \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} \\left | \\prod _ { i = 1 } ^ m z _ i - \\prod _ { i = 1 } ^ m w _ i \\right | \\leq \\sum _ { i = 1 } ^ m | z _ i - w _ i | \\ ; . \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} \\frac { x _ { k , j } } { B _ { k , j } } & = - i \\int _ { 0 } ^ T u ( s ) e ^ { - i ( \\lambda _ j - \\lambda _ k ) s } d s = - i \\int _ { 0 } ^ T u ( s ) e ^ { - i ( \\lambda _ n - \\lambda _ m ) s } d s = \\frac { x _ { n , m } } { B _ { n , m } } . \\\\ \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ { \\delta } r ^ s = \\frac { q ^ { s + 1 } } { ( s + 1 ) d ^ { s + 1 } } + V _ s ( q ) \\end{align*}"}
-{"id": "713.png", "formula": "\\begin{align*} j ^ - _ i \\vert n _ 1 , n _ 2 , \\dots , n _ i , \\dots , n _ { r + 1 } \\rangle = \\sqrt { n _ i ( k + 1 - ( n _ 1 + n _ 2 + \\cdots + n _ { r } ) ) } \\vert n _ 1 , n _ 2 , \\dots , n _ i - 1 , \\dots , n _ { r + 1 } \\rangle . \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} V ( t ) = t ^ 2 + \\sum _ { k = 2 } ^ { \\infty } \\frac { t ^ { 2 k } } { k ( 2 k - 1 ) } \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{align*} s + t = \\frac { 1 } { 2 } ( 3 n + 1 ) . \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} X _ { n + 1 } = S _ { \\Delta t } X _ n + \\Delta t S _ { \\Delta t } G ( X _ n ) + S _ { \\Delta t } \\sigma ( X _ n ) \\Delta W _ n , \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} | \\{ v \\leq 0 \\} \\cap Q _ 2 | & \\geq \\frac { | Q _ 2 | } { 2 } , \\\\ | \\{ v \\geq 1 \\} \\cap \\overline { Q } _ 2 | & \\geq \\delta _ 0 , \\textrm { a n d } \\\\ | \\{ 0 < v < 1 \\} \\cap Q _ 2 | & = 0 . \\end{align*}"}
-{"id": "9115.png", "formula": "\\begin{gather*} \\tilde { \\omega } _ { \\mathbf { m } } : = \\frac { \\prod \\limits _ { i = 0 } ^ n x _ i ^ { m _ i - 1 } } { ( F _ { A , \\mu } ( x _ 0 ^ { w _ 0 } , \\dots , x _ n ^ { w _ n } ) ) ^ t } \\Omega \\end{gather*}"}
-{"id": "6058.png", "formula": "\\begin{align*} [ L _ { - q + 1 } , \\ , [ V _ { - 2 } ^ j , \\ , [ V _ { - 2 } , \\ , L _ { q } ] ] ] = [ V _ { - 2 } ^ j , [ V _ { - 2 } , \\ , [ L _ { - q + 1 } , \\ , L _ { q } ] ] = [ V _ { - 2 } ^ j , L _ { - 1 } ] \\neq 0 , \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} \\omega = \\sum _ { i = 1 } ^ M \\varpi ^ i \\delta _ { \\xi ^ i } + \\omega _ { \\mathrm { a c } } , \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} \\partial _ { t } u = i \\left [ \\Delta u + L u + V \\left ( x , t \\right ) u \\right ] , x \\in R ^ { n } , y \\in \\Omega , t \\in \\left [ 0 , T \\right ] . \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\frac { B _ { n } ^ { \\left ( p \\right ) } \\left ( x \\right ) } { n ! } z ^ { n } = e ^ { z x } \\left ( \\frac { z } { e ^ { z } - 1 } \\right ) ^ { p } . \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N } \\big | b _ { k } ^ 2 \\big | \\le C \\Delta ^ { \\frac 1 2 - \\kappa } ( 1 + | x | _ { L ^ { \\max ( p , 8 q ) } } ) ^ { K + 2 } \\int _ { 0 } ^ { T } \\bigl ( 1 + \\frac 1 { t ^ { 3 \\kappa } } \\bigr ) \\bigl ( 1 + \\frac { 1 } { ( T - t ) ^ { 1 - \\kappa } } \\bigr ) d t . \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{align*} \\eta _ { a } ( C ( \\mathsf { Q } ) ) = \\sum _ { i , j } c _ { j } ^ { i } \\pi ( K _ { 2 \\rho } K _ { a } ^ { - 1 } [ E _ { a } , F _ { a } ] ) _ { i } ^ { j } . \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} i , j = 1 , 2 , \\dots , d ; \\alpha , \\beta = 1 , 2 , \\dots , N ; k = 1 , 2 , \\dots , \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} \\widehat v '' ( \\rho ) = ( b - a ) \\frac { e ^ \\rho } { ( 1 + e ^ \\rho ) ^ 2 } . \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , X _ { 2 , \\ , j + 2 , \\ , 6 } ] = 0 . \\end{align*}"}
-{"id": "3886.png", "formula": "\\begin{align*} y ' ( 0 ) - h y ( 0 ) = 0 , y ' ( 1 ) + H y ( 1 ) = 0 , \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} \\partial _ c u _ { c _ * } ( \\xi ) = { \\rm s e c h } ^ 2 ( \\sqrt { c _ * } \\xi ) - \\sqrt { c _ * } \\xi \\tanh ( \\sqrt { c _ * } \\xi ) { \\rm s e c h } ^ 2 ( \\sqrt { c _ * } \\xi ) , \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} q ^ { 2 d k - d } \\left | \\sum _ { { \\bf m } \\in S _ 0 } \\left ( \\prod _ { j = 1 } ^ k \\widehat { E _ j } ( { \\bf m } ) \\right ) \\right | ^ 2 - \\nu ^ 2 _ k ( 0 ) \\le \\frac { 4 } { q } \\left ( \\prod _ { j = 1 } ^ k | E _ j | \\right ) ^ 2 . \\end{align*}"}
-{"id": "4667.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\frac { 1 } { N } K _ N \\left ( \\frac { x } { N } , \\frac { y } { N } ; w _ { R , \\alpha } ^ { \\pm } \\right ) = \\frac { \\sin \\pi c _ { \\alpha } ^ { \\pm } ( x - y ) } { \\pi ( x - y ) } \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} P = \\sum \\limits _ { \\underline { i } \\ge \\underline { 0 } } \\frac { \\underline { x } ^ { \\underline { i } } } { \\underline { i } ! } \\ , P _ { ( \\underline { i } ) } , \\mbox { \\rm w h e r e } P _ { ( \\underline { i } ) } = \\underline { i } ! \\sum _ { \\substack { \\underline { k } \\ge \\underline { 0 } \\\\ | \\underline { k } | - | \\underline { i } | \\le d } } \\alpha _ { \\underline { k } , \\underline { i } } \\ , \\underline { \\partial } ^ { \\underline { k } } . \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { \\delta } r ^ s = \\frac { 1 } { s + 1 } \\sum _ { j = 0 } ^ s ( - 1 ) ^ j \\binom { s + 1 } { j } B _ j \\delta ^ { s + 1 - j } \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} B _ r ( p , q , X _ 1 , \\cdots , X _ r ) = \\prod _ { t = 1 } ^ r \\prod _ { j = 0 } ^ { r - t + 1 } \\bigl ( 1 - q ^ j p ^ { ( r - t + 1 ) j - j ^ 2 } X _ t \\bigr ) \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , [ L _ { - 2 } , \\ , L _ i ] ] = 0 , \\ , 0 \\leqq i \\leqq 2 j - 1 \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{align*} \\eta _ { X , Y } ^ \\lambda ( \\mathsf { P } ) = \\sum _ { i , j , k , \\ell , m , n } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) \\pi ( S ( X _ { ( 1 ) } ) ) _ { k } ^ { j } c _ { \\ell } ^ { k } \\pi ( X _ { ( 2 ) } S ( Y _ { ( 1 ) } ) ) _ { m } ^ { \\ell } c _ { n } ^ { m } \\pi ( Y _ { ( 2 ) } K _ { \\lambda } ) _ { i } ^ { n } . \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} u ( z ) = P ( z ) + | \\mu ( z ) | T ( z ) \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} y _ { n - 1 , k } = p _ { 2 n - 2 } \\sigma _ k \\sum _ { i = 0 } ^ { n - 1 } b _ { i , n - 1 } t _ k ^ i = 0 , k = 1 , \\dots , n - 1 . \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} T _ i ( k _ 1 , \\dots , k _ n ) \\equiv ( k _ { T _ i ( 1 ) } , \\dots , k _ { T _ i ( n ) } ) = ( k _ 1 , \\dots , k _ { i - 1 } , k _ { i + 1 } , k _ i , k _ { i + 2 } , \\dots , k _ n ) . \\end{align*}"}
-{"id": "9394.png", "formula": "\\begin{align*} B _ 2 ( W _ 1 \\times W _ 2 , L ^ \\infty ( \\Omega ) ) & = B ( W _ 1 \\overset { \\wedge } { \\otimes } W _ 2 , L ^ \\infty ( \\Omega ) ) \\\\ & = B ( L ^ 1 ( \\Omega ) , ( W _ 1 \\overset { \\wedge } { \\otimes } W _ 2 ) ^ * ) \\\\ & = B ( L ^ 1 ( \\Omega ) , B ( W _ 1 , W _ 2 ^ * ) ) \\\\ & = L ^ \\infty _ \\sigma \\bigl ( \\Omega ; B ( W _ 1 , W _ 2 ^ * ) \\bigr ) . \\end{align*}"}
-{"id": "5969.png", "formula": "\\begin{align*} [ ( \\epsilon + a ) \\lambda ^ { 2 } + ( a + b ) \\lambda + \\epsilon ] ( \\epsilon \\lambda ^ { 2 } + b \\lambda + \\epsilon ) + [ \\epsilon ( \\epsilon + a ) \\lambda + \\epsilon ^ { 2 } ] ( \\lambda ^ { 3 } + \\lambda + 1 ) = 0 . \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} S _ v ( k ) = - ( A + i k B ) ^ { - 1 } ( A - i k B ) , \\end{align*}"}
-{"id": "4840.png", "formula": "\\begin{align*} h ( h k h ^ { - 1 } k ^ { - 1 } ) h ^ { - 1 } = I + \\begin{pmatrix} 0 & b \\left ( \\frac { w } { z } - 1 \\right ) \\frac { w } { z } \\\\ c \\left ( \\frac { z } { w } - 1 \\right ) \\frac { w } { z } & 0 \\end{pmatrix} p \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{align*} H ^ { ( 2 ) } = - ( i / \\gamma ) R ( - 2 \\sqrt { 2 } / \\gamma , - 2 \\lambda / \\gamma ) a ^ \\dagger G + \\frac { i c } { 4 \\pi \\gamma } H ^ { ( 2 ) } _ 0 R ( - 2 \\sqrt { 2 } / \\gamma , - 2 \\lambda / \\gamma ) e _ 1 . \\end{align*}"}
-{"id": "9603.png", "formula": "\\begin{align*} \\aligned W { } ^ { ( s ) . i } _ { j m n } & = R ^ i _ { j m n } + \\frac 1 { N + 1 } \\delta ^ i _ j R _ { [ m n ] } + \\frac N { N ^ 2 - 1 } \\delta ^ i _ { [ m } R _ { j n ] } + \\frac 1 { N ^ 2 - 1 } \\delta ^ i _ { [ m } R _ { n ] j } \\\\ & + \\widetilde { \\mathcal D } { } ^ { ( s _ 2 ) . ( s _ 3 ) . i } _ { j m n } - \\widetilde { \\mathcal D } { } ^ { ( s _ 2 ) . ( s _ 3 ) . i } _ { j n m } . \\endaligned \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} & Q _ { g + 2 - n } ( a _ 1 ) - Q _ { g + 2 - n } ( a _ 1 - \\textstyle { \\frac 1 2 } ) = \\frac { ( a _ 1 ) ( a _ 1 - \\frac 1 2 ) \\cdots ( a _ 1 - g - 1 + n ) } { ( g + 1 - n ) ! } \\\\ & Q _ { g + 2 - n } ( a _ 1 + 1 ) - Q _ { g + 2 - n } ( a _ 1 + \\textstyle { \\frac 3 2 } ) = \\frac { - ( a _ 1 + \\frac 3 2 ) ( a _ 1 + 1 ) \\cdots ( a _ 1 - g + \\frac 1 2 + n ) } { ( g + 1 - n ) ! } \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} { \\displaystyle \\mathbf { J } _ q ( \\mathbf { x } , t ) = \\frac { - e \\hbar } { 2 i m } \\big ( \\overline { \\Psi } \\nabla \\Psi - \\Psi \\nabla \\overline { \\Psi } \\big ) } , \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} m \\mathbf { \\ddot { p } } & = \\mathbf { f } + m \\mathbf { g } + \\mathbf { d } _ { p } \\\\ \\mathbf { J } \\mathbf { \\dot { \\omega } } & = \\mathbf { \\omega } \\times \\mathbf { J } \\mathbf { \\omega } + \\mathbf { u } + \\mathbf { R } _ { I B } \\mathbf { d } _ { \\omega } \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{align*} | h | _ \\infty = \\sup _ { t \\geq 0 } | h ( t ) | , | h ^ \\prime | _ \\infty = \\sup _ { t \\geq 0 } | h ^ \\prime ( t ) | . \\end{align*}"}
-{"id": "9507.png", "formula": "\\begin{align*} \\| \\ , | z | + j v ^ * + a u v ^ * \\| < \\| z \\| = \\| \\ , | z | \\ , \\| , \\end{align*}"}
-{"id": "4042.png", "formula": "\\begin{align*} I & = 1 2 a e - 3 b d + c ^ 2 \\\\ J & = 7 2 a c e - 2 7 a d ^ 2 - 2 7 b ^ 2 e + 9 b c d - 2 c ^ 3 \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{align*} q ( x e _ 1 + y e _ 2 + z e _ 3 ) = a x ^ 2 + b y ^ 2 + c z ^ 2 \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{align*} h = - d x _ 0 ^ 2 + \\sum _ { j = 1 } ^ { n + 1 } d x _ j ^ 2 \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} y ( f ) = a f ^ b + c , \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{gather*} \\chi _ { n , 0 } ( x _ { 1 } , . . . , x _ { n } | \\rho ) \\geq 0 , \\\\ \\underset { j } { \\int _ { - 1 } ^ { 1 } . . . \\int _ { - 1 } ^ { 1 } ( } \\prod _ { s = 1 } ^ { n } \\frac { 1 } { \\pi \\sqrt { 1 - x _ { s } ^ { 2 } } } ) \\chi _ { n , 0 } ( x _ { 1 } , . . . , x _ { n } | \\rho ) d x _ { 1 } . . . d x _ { j } \\allowbreak = \\allowbreak \\prod _ { s = j + 1 } ^ { n } \\frac { 1 } { \\pi \\sqrt { 1 - x _ { s } ^ { 2 } } } , \\end{gather*}"}
-{"id": "44.png", "formula": "\\begin{align*} W _ { n - i } ( K ) = \\frac { \\omega _ n } { \\omega _ i } \\int _ { G ( n , i ) } \\mathcal { H } ^ i ( K | \\xi ) d \\xi , \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle \\sum _ { j = 1 } ^ { 4 } | I _ 3 ^ { k , j } ( \\overline { \\partial \\theta } _ { \\mathbf { A } } ^ { k } ) | \\leq C \\left \\{ ( \\Delta t ) ^ { 4 } + h ^ { 2 r + 2 } + D ( \\widetilde { \\theta } _ { \\mathbf { A } } ^ { k } , \\widetilde { \\theta } _ { \\mathbf { A } } ^ { k } ) + \\| \\overline { \\partial \\theta } _ { \\mathbf { A } } ^ { k } \\| _ { \\mathbf { L } ^ 2 } ^ { 2 } \\right \\} . } \\end{array} \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{align*} \\langle [ \\phi , b ] , m \\rangle & = \\tau ( [ b ^ * , \\phi ^ * ] m ) \\\\ & = \\tau ( b ^ * \\phi ^ * m - \\phi ^ * b ^ * m ) \\\\ & = \\tau ( b ^ * [ \\phi ^ * , m ] ) \\\\ & = \\langle b , [ \\phi ^ * , m ] \\rangle \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{align*} A _ \\sigma ^ 2 = \\left ( \\frac { \\pi ^ 2 } { 8 \\beta ( x _ c ^ \\sigma ) ^ 2 } \\right ) ^ { 1 / \\sigma } . \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} j & = j ( \\tau ) & j _ 1 & = j ' ( \\tau ) & j _ 2 & = j '' ( \\tau ) \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { m _ { k } } \\varepsilon ^ { \\frac { i } { 2 } } \\alpha _ { i } u ^ { \\left ( i \\right ) } \\left ( 0 , \\varepsilon \\right ) = f _ { 1 } \\left ( \\varepsilon \\right ) , \\sum \\limits _ { i = 0 } ^ { m _ { k } } \\varepsilon ^ { \\frac { i } { 2 } } \\beta _ { i } u ^ { \\left ( i \\right ) } \\left ( T , \\varepsilon \\right ) = f _ { 2 } \\left ( \\varepsilon \\right ) , \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } f ' _ \\alpha + t _ { ( \\alpha , 1 ) } f _ \\alpha & = & 0 \\\\ \\dfrac { f ''' _ \\alpha } { 3 ! } + \\dfrac { f '' _ \\alpha } { 2 ! } \\cdot \\dfrac { t _ { ( \\alpha , 1 ) } } { 1 ! } + \\dfrac { f ' _ \\alpha } { 1 ! } \\cdot \\dfrac { t _ { ( \\alpha , 2 ) } } { 2 ! } + f _ \\alpha \\cdot \\dfrac { t _ { ( \\alpha , 3 ) } } { 3 ! } & = & 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "451.png", "formula": "\\begin{align*} \\boldsymbol { \\pi } \\circ \\mathbf { h } = \\mathbf { g } , \\ \\ \\ \\ \\pi \\circ h = g \\ \\ \\ \\textrm { a n d } \\ \\ \\ \\tilde { \\Phi } \\circ \\mathbf { h } = h \\circ \\Phi , \\end{align*}"}
-{"id": "6153.png", "formula": "\\begin{align*} [ L _ { - r } , \\ , S _ { r - 2 } ] = 0 \\end{align*}"}
-{"id": "9324.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + A x + B \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} ( \\omega + \\sqrt { - 1 } \\partial \\bar \\partial \\varphi ) ^ n = e ^ { F ( x , \\varphi ( x ) ) } \\omega ^ n , \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{align*} z = \\frac { ( 2 \\ell + 1 ) \\pm \\sqrt { 4 \\ell + 1 } } { 2 \\ell } = \\left ( a + b \\sqrt { 4 \\ell + 1 } \\right ) ^ 2 \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} X _ { 2 , \\ , j + 1 , \\ , 6 } \\cap L _ { j + 1 } = 0 . \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{align*} \\omega _ { N } : = \\frac { \\pi ^ { \\frac { N } { 2 } } } { \\Gamma \\left ( \\frac { N } { 2 } + 1 \\right ) } , \\end{align*}"}
-{"id": "9948.png", "formula": "\\begin{align*} \\dot { x } = \\tilde { A } x \\end{align*}"}
-{"id": "9616.png", "formula": "\\begin{align*} \\widetilde { \\mathcal T } { } ^ i _ { j k } = T ^ i _ { j k } - \\frac 1 2 \\mathcal F ^ i _ { j k } + \\frac 1 { 8 } \\delta ^ i _ j \\mathcal F ^ \\alpha _ { k \\alpha } + \\frac 1 8 \\delta ^ i _ k \\mathcal F ^ \\alpha _ { j \\alpha } , \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} E _ t = \\{ x \\in 2 B _ 0 : M ^ * _ { 2 B _ 0 , 2 B _ 0 } g ^ p ( x ) > t ^ p \\} . \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{align*} \\Xi ^ \\lambda ( K _ a F _ a \\otimes 1 \\otimes 1 \\otimes E _ a ) & = 2 \\Xi ^ \\lambda ( K _ a \\otimes F _ a \\otimes 1 \\otimes E _ a ) - \\sum _ { i , j } c _ { j } ^ { i } \\pi ( E _ { a } K _ { \\lambda } K _ { a } F _ { a } ) _ { i } ^ { j } , \\\\ \\Xi ^ \\lambda ( K _ a \\otimes F _ a \\otimes E _ a K _ a ^ { - 1 } \\otimes K _ a ) & = \\sum _ { i , j } c _ { j } ^ { i } \\pi ( K _ { a } F _ { a } E _ { a } K _ { \\lambda } ) _ { i } ^ { j } . \\end{align*}"}
-{"id": "7381.png", "formula": "\\begin{align*} E = { \\lambda } ^ { 2 } \\left ( u + 2 \\ , M \\right ) u . \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} T _ x ( N _ x \\cdot x ) = \\{ 0 \\} \\times T _ v ( K \\cdot v ) \\subset T _ y Y \\times V . \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} n _ 1 | \\mu _ 1 + \\theta \\mu _ 2 | = | n _ 1 \\mu _ 1 + n _ 2 \\mu _ 2 - ( n _ 2 - n _ 1 \\theta ) \\mu _ 2 | \\leqslant K B + | \\mu _ 2 | ( n _ 2 - n _ 1 \\theta ) , \\end{align*}"}
-{"id": "9575.png", "formula": "\\begin{align*} ( \\widehat \\theta _ n , \\widehat \\gamma _ n ) = { \\arg \\min } _ { ( \\theta , \\gamma ) } \\ , \\sum _ { i = 1 } ^ n R _ { n , i } \\big ( Y _ { n , i } - X _ { n , i } ^ { \\prime } \\theta - Z _ { n , i } ^ { \\prime } \\gamma \\big ) ^ 2 . \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} \\hat \\Phi ( L ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\log p _ { Z _ i } ( L ) = \\sum _ { J \\subseteq [ N ] } \\hat p _ J \\log \\det ( L _ J ) - \\log \\det ( I + L ) \\ , , \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} \\Vert \\Delta \\psi _ { n } \\Vert ^ { 2 } _ { { L } ^ { 2 } } = ( - \\Delta \\psi _ { n } , \\ , - \\Delta \\psi _ { n } ) = ( f , \\ , - \\Delta \\psi _ { n } ) \\leq \\Vert f \\Vert _ { L ^ { 2 } } \\Vert \\Delta \\psi _ { n } \\Vert _ { { L } ^ { 2 } } , \\end{align*}"}
-{"id": "9769.png", "formula": "\\begin{align*} \\Delta u _ { \\epsilon } = - \\epsilon ^ { - 2 } ( 1 - | u _ { \\epsilon } | ^ 2 ) u _ { \\epsilon } \\end{align*}"}
-{"id": "3104.png", "formula": "\\begin{align*} \\sum _ { \\substack { a \\pmod * { q } \\\\ ( a , q ) = 1 } } \\bigl ( f | \\gamma - \\overline { \\xi ( q ) } f \\bigr ) \\left | \\begin{pmatrix} 1 & a / q \\\\ & 1 \\end{pmatrix} \\right . = 0 . \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} J _ { \\alpha B } ( x _ { k + 1 } ) = \\alpha ( b ^ k + D u ^ { k + 1 } - d ^ { k + 1 } ) , \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 5 } { \\alpha _ 1 } y _ 4 + \\theta y _ 5 , [ y _ 2 , y _ 2 ] = y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\beta _ 4 } { \\gamma _ 2 } y _ 5 , [ y _ 2 , y _ 3 ] = y _ 5 = - [ y _ 3 , y _ 2 ] . \\end{align*}"}
-{"id": "1629.png", "formula": "\\begin{align*} K ^ m _ \\beta ( \\Omega ) = \\{ u \\in L ^ 2 _ { \\rm l o c } ( \\Omega ) , \\rho ^ { \\beta } ( \\rho \\partial _ \\rho ) ^ { \\alpha _ 1 } \\partial ^ { \\alpha _ 2 } _ \\vartheta u \\in L ^ 2 ( \\Omega ) , \\ \\ \\forall \\alpha \\in \\N ^ 2 , \\ | \\alpha | \\le m \\} . \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} \\dot { b } \\langle \\eta _ * , \\psi _ * \\rangle _ { L ^ 2 } = - 1 2 | b | ^ 2 b \\langle \\psi _ * ^ 2 , \\tilde { w } _ 2 \\rangle _ { L ^ 2 } + \\frac { 6 4 } { \\sqrt { c _ * } } | b | ^ 2 b + \\frac { 1 6 } { 3 \\sqrt { c _ * } } b \\delta + \\mathcal { O } ( \\delta ^ 2 | b | + | b | ^ 5 ) , \\end{align*}"}
-{"id": "4902.png", "formula": "\\begin{align*} { { \\mu _ { } } } & = \\mu _ Y , & { { b _ { } } } & = \\frac { \\mu _ Y } { \\Gamma ( 1 - 1 / { { c _ { } } } ) } . \\end{align*}"}
-{"id": "2410.png", "formula": "\\begin{align*} \\dot { c } _ 1 & = - 2 & \\dot { c } _ 2 & = 2 \\\\ c _ 1 \\cdot \\dot { x } _ 1 ^ { ( 1 ) } & = x _ 1 ^ { ( 1 ) } - x _ 2 ^ { ( 1 ) } & c _ 2 \\cdot \\dot { x } _ 2 ^ { ( 1 ) } & = x _ 1 ^ { ( 1 ) } - x _ 2 ^ { ( 1 ) } \\\\ c _ 1 \\cdot \\dot { x } _ 1 ^ { ( 2 ) } & = x _ 1 ^ { ( 2 ) } - x _ 2 ^ { ( 2 ) } & c _ 2 \\cdot \\dot { x } _ 2 ^ { ( 2 ) } & = x _ 1 ^ { ( 2 ) } - x _ 2 ^ { ( 2 ) } \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{align*} f _ * E ^ n = \\sum _ { \\ell = 1 } ^ d Z _ \\ell ^ { n } M _ \\ell , M _ \\ell = \\prod _ { \\substack { m = 1 \\\\ m \\neq \\ell } } \\frac { Z _ m } { Z _ m - Z _ \\ell } . \\end{align*}"}
-{"id": "8801.png", "formula": "\\begin{align*} f : \\mathbb { R } ^ { n } \\rightarrow \\mathbb { R } ^ { n } , ~ f : = ( f _ { 1 } , i d _ { \\mathbb { R } ^ { n - k } } ) \\end{align*}"}
-{"id": "8325.png", "formula": "\\begin{align*} \\mu ( t ) = \\left ( \\int _ { \\Omega } u _ 0 ( x ) d x - \\int _ { \\Omega } u _ r ( t , x ) d x \\right ) \\tilde { \\mu } ( t ) , \\end{align*}"}
-{"id": "7524.png", "formula": "\\begin{gather*} k : = \\left [ \\left ( y \\phi _ { - 1 } y ^ { - 1 } \\right ) ^ * , y \\phi _ { - 1 } y ^ { - 1 } \\right ] \\\\ z : = y \\left ( 1 + \\frac { 1 } { 2 } G ( y ^ { - 1 } k y ) \\right ) . \\end{gather*}"}
-{"id": "1303.png", "formula": "\\begin{align*} H _ { 0 } = \\frac { f g ^ { \\prime \\prime } } { 2 \\left \\vert 1 - \\left ( f g ^ { \\prime } \\right ) ^ { 2 } \\right \\vert ^ { \\frac { 3 } { 2 } } } . \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{align*} P ' ( x ) = Q '' ( x ) S ( x ) - Q ( x ) S '' ( x ) , \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} X : = \\bigcup _ { i , j } B _ i ^ j \\cup \\bigcup _ { i , j , k } B _ i ^ { j k } \\ , . \\end{align*}"}
-{"id": "5132.png", "formula": "\\begin{align*} S _ { m , n } = \\sum _ { \\underset { k _ { i } \\ge 0 } { k _ { 1 } + \\dots + k _ { m } = n } } \\binom { n } { k _ { 1 } , \\dots , k _ { m } } w _ { k _ { 1 } } \\dots w _ { k _ { m } } \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} F ^ n _ { \\nu , \\mu , \\alpha } ( x ) = \\begin{pmatrix} \\nu \\cr n \\end{pmatrix} \\frac { \\Gamma ( \\mu + n ) } { \\Gamma ( \\mu - \\alpha + n + 1 ) } x ^ { \\mu - \\alpha + n } , \\end{align*}"}
-{"id": "5338.png", "formula": "\\begin{align*} g _ { t , \\ell , s , m } = \\begin{cases} g _ { t , \\ell , s + 1 , m - 1 } + g _ { t , \\ell - 1 , s - 1 , m - 1 } , m - 2 \\ell \\geq ( t - 1 ) s \\\\ 0 , . \\end{cases} \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} ( 1 - 2 \\beta ) \\sum _ { k = 1 } ^ { ( p - 1 ) / 2 } \\binom { 2 k } { k } H _ k ^ { ( 2 ) } \\bigl ( \\beta ( 1 - \\beta ) \\bigr ) ^ k = 2 \\pounds _ 2 ( \\beta ) - 2 \\pounds _ 2 ( 1 - \\beta ) \\pmod { ( \\beta ^ p , p ) } \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} ( J _ 0 u ) _ n = a _ n u _ { n - 1 } + a _ { n + 1 } u _ { n + 1 } . \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} v _ { \\star \\star } = \\frac { a + 1 } { 3 } \\pm \\sqrt { \\frac { ( a + 1 ) ^ 2 } { 9 } - \\frac { a + \\varepsilon c } { 3 } } , \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} u ( t ) : = 1 + t + \\dfrac { t ^ 2 } { 2 ! } + \\dfrac { t ^ 3 } { 3 ! } + \\cdots + \\dfrac { t ^ n } { n ! } + \\cdots , \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{align*} \\Delta \\phi _ { m i n } & = \\Delta \\phi _ r , ~ ~ \\Delta \\phi _ { m a x } = \\Delta \\phi _ { \\theta } , ~ ~ \\mathrm { f o r } ~ \\mathbf { z } _ { \\phi , r } = \\mathbf { z } _ I \\\\ \\Delta \\phi _ { m i n } & = \\Delta \\phi _ { \\theta } , ~ ~ \\Delta \\phi _ { m a x } = \\Delta \\phi _ r , ~ ~ \\mathrm { f o r } ~ \\mathbf { z } _ { \\phi , r } = - \\mathbf { z } _ I \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{align*} k = \\lfloor n ^ { 1 / 4 } \\rfloor \\end{align*}"}
-{"id": "7185.png", "formula": "\\begin{align*} z = \\frac { ( 2 \\ell + 1 ) \\pm i \\sqrt { 4 \\ell + 3 } } { 2 ( \\ell + 1 ) } . \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\frac { \\alpha ^ 2 } { N } K _ { N } \\left ( \\frac { x \\alpha ^ 2 } { N } , \\frac { y \\alpha ^ 2 } { N } ; ( 1 - t ^ 2 ) ^ { 1 / 2 } e ^ { - N V _ { \\alpha , \\widetilde { \\varepsilon } _ R } ( t ) } \\right ) = \\frac { \\sin \\pi c _ { \\alpha } ^ - ( x - y ) } { \\pi ( x - y ) } \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{align*} \\mathbf { R } _ { A _ { d B } } ^ { i , j } = \\frac { \\mathbf { R } _ { A _ { d B } } ^ { i , i } \\mathbf { R } _ { A _ { d B } } ^ { j , j } / N _ f + \\mathbf { R } _ { A _ { d B } } ^ { i , i } \\circ \\mathbf { R } _ { A _ { d B } } ^ { j , j } } { 2 } , \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} [ L _ { - 4 } , \\ , L _ 5 ] = L _ 1 . \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} \\| \\mathfrak { a } \\| _ { L ^ \\infty ( \\mathcal I ) } : = \\sup _ { t \\in \\mathcal I } | a _ t | \\le | a _ { t _ 0 } | + \\| \\mathfrak { a } \\| _ { v _ \\rho ( \\mathcal I ) } \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{align*} g _ { k } = \\begin{cases} 1 & k = 1 , 2 \\\\ 0 & \\end{cases} . \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} d ^ { x } & = N ( N - 2 B ) ^ { M - 1 } - ( N - 2 B ) ^ M + 1 \\\\ & = 2 B ( N - 2 B ) ^ { M - 1 } + 1 \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{align*} x _ 1 = \\sqrt { 2 \\pi ^ 2 N } x , y _ 1 = \\sqrt { 2 \\pi ^ 2 N } y . \\end{align*}"}
-{"id": "9397.png", "formula": "\\begin{align*} \\forall \\ , v \\in V , T ( v ) = T _ 2 \\bigl ( T _ 1 ( v ) \\bigr ) . \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} & \\left ( 2 g - \\textstyle \\frac 3 2 - \\sum _ { i = 2 } ^ n d _ i , d _ 2 , \\dots , d _ { M - 1 } , d _ M - 1 , d _ { M + 1 } , \\dots , d _ n , \\ , d _ { n + 1 } = \\frac 3 2 , \\ , d _ { n + 2 } = - \\frac 1 2 \\right ) . \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { m - 1 } \\prod _ { \\nu = b _ { n ( j ) } + 1 } ^ { b _ { n ( j ) + 1 } - 1 } \\vert w _ { \\nu } \\vert \\leq A ^ { \\sum \\limits _ { j = 1 } ^ { m - 1 } \\Delta ^ { ( k _ j ) } } \\leq A ^ { \\sum \\limits _ { i = 0 } ^ { k _ m - 1 } \\Delta ^ { ( i ) } } \\quad \\textrm { f o r e v e r y } m \\ge 1 . \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { r } ( - 1 ) ^ m S _ m & = \\sum _ { m = 0 } ^ r ( - 1 ) ^ m \\frac { \\lambda ^ m } { m ! } + O \\left ( \\sum _ { m = 0 } ^ r d ^ 2 m ^ 2 / n \\frac { \\lambda ^ m } { m ! } \\right ) \\\\ & = \\sum _ { m = 0 } ^ r ( - 1 ) ^ m \\frac { \\lambda ^ m } { m ! } + O \\left ( d ^ 2 r ^ 2 e ^ \\lambda / n \\right ) . \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} \\forall r \\in R _ 0 ^ * \\cap R _ 1 \\bmod Q _ 1 , \\mu _ 1 ( r ) = \\frac { 1 } { | R _ 1 \\bmod Q _ 1 | } . \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} h ^ 1 ( F ( c - a - b ) ) = 0 . \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} d X _ t ^ i = a ( \\bar X _ t - X _ t ^ i ) d t + \\sigma \\ , d W _ t ^ i + \\gamma _ { t } ^ i d N ^ i _ t \\ , , i = 1 , \\dots , n , \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} \\mathbf { P } \\bigl ( [ E ] \\bigr ) = \\{ S \\in \\mathbf { P } | \\ h ^ 0 ( E ( - b ) | _ S ) = 1 \\} \\ \\ \\ \\ \\ \\ \\underset { [ E ] \\in N _ s } { \\min } \\tau _ E = \\tau , \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} - u ^ { \\left ( 2 \\right ) } \\left ( t \\right ) + A u \\left ( t \\right ) = f \\left ( t \\right ) , \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} C _ G ( y ) ^ 0 = \\langle T , U _ { \\alpha } : \\alpha \\in \\Psi \\rangle \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{align*} z _ i z _ k = ( n - 1 ) \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} V ( \\theta ) = - 2 c _ k \\sum _ { j = k + 1 } ^ { 2 k } \\frac { 1 } { j - k } \\binom { 2 k } { j } ( - 1 ) ^ { j - k } \\cos ( ( j - k ) \\theta ) \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} P _ j ^ i ( T ) = \\sum _ { k _ 1 + \\ldots + k _ j = i } c _ { i j } \\nabla _ { ( k _ 1 ) } T * \\cdots * \\nabla _ { ( k _ j ) } T , \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{align*} d _ q ( A _ { \\alpha - 2 } ) = \\left \\{ \\begin{array} { l l } z ^ 2 q + ( 2 z + 1 ) ( \\alpha - 1 - z q ) - 1 & \\textrm { i f } \\ \\delta > 0 \\\\ ( z - 1 ) ^ 2 q + ( 2 z - 1 ) ( \\alpha - 1 - ( z - 1 ) q ) - 1 & \\textrm { i f } \\ \\delta = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} \\frac { q _ { 1 } } { q _ { 3 } } - \\frac { M ^ { \\prime } - A _ { 2 } M } { N } = \\frac { H } { 3 B _ { 2 } N } . \\end{align*}"}
-{"id": "5180.png", "formula": "\\begin{align*} F _ { 0 } = 1 , \\thinspace \\thinspace F _ { 1 } = 1 , \\thinspace \\thinspace F _ { 2 } = 2 , \\thinspace \\thinspace F _ { 3 } = 3 , \\thinspace \\thinspace F _ { 4 } = 5 , \\thinspace \\thinspace F _ { 5 } = 8 \\dots \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{align*} \\partial _ t ^ 2 \\hat { u } + \\frac { \\xi ^ 2 } { 1 + a _ \\alpha | \\xi | ^ \\alpha } \\ , \\hat { u } = \\hat { v _ 0 } \\otimes \\delta + \\hat { u _ 0 } \\otimes \\delta ' . \\end{align*}"}
-{"id": "7126.png", "formula": "\\begin{align*} T _ h ^ n = O ( h ) + G _ h ^ { n } - Q ^ n _ h \\ , . \\end{align*}"}
-{"id": "655.png", "formula": "\\begin{align*} a _ i ^ { + } | n _ 1 , \\cdots , n _ i , \\cdots , n _ r \\rangle \\ = \\sqrt { ( n _ i + 1 ) ( k + 1 - ( n _ 1 + n _ 2 + \\cdots + n _ r + 1 ) ) } | n _ 1 , \\cdots , n _ i + 1 , \\cdots , n _ r \\rangle . \\end{align*}"}
-{"id": "3472.png", "formula": "\\begin{align*} \\partial _ t ^ 2 \\hat { u } + \\frac { \\xi ^ 2 } { 1 + a _ \\alpha | \\xi | ^ \\alpha } \\ , \\hat { u } = 0 . \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} \\alpha + \\frac { d } { c } = \\frac { a + d + \\sqrt { ( a + d ) ^ 2 - 4 } } { 2 c } , \\beta + \\frac { d } { c } = \\frac { a + d - \\sqrt { ( a + d ) ^ 2 - 4 } } { 2 c } \\end{align*}"}
-{"id": "9809.png", "formula": "\\begin{align*} { \\left ( { f , g } \\right ) _ { E , N } } = { \\left ( { { \\mathbb { I } ^ N } \\left ( f \\right ) , { \\mathbb { I } ^ N } \\left ( g \\right ) } \\right ) _ { E , N } } . \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} 0 \\neq [ L _ { - 2 } , \\ , S _ 3 ] = [ L _ { - 2 } , \\ , [ L _ 1 , \\ , M _ 2 ] ] = [ L _ 1 , \\ , [ L _ { - 2 } , \\ , M _ 2 ] ] \\end{align*}"}
-{"id": "175.png", "formula": "\\begin{align*} \\left ( \\prod \\limits _ { j = 1 } ^ k | E _ j | - \\nu _ k ( 0 ) \\right ) ^ 2 \\ge \\frac { 1 } { 9 } \\left ( \\prod \\limits _ { j = 1 } ^ k | E _ j | \\right ) ^ 2 . \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} \\| \\nabla ^ j U ( t ) \\| _ { r , \\mathbb R ^ 3 } = \\| \\nabla ^ j W ( t ) \\| _ { r , \\mathbb R ^ 3 } , \\| \\nabla ^ j U ( t ) \\| _ { 3 , \\infty , \\mathbb R ^ 3 } = \\| \\nabla ^ j W ( t ) \\| _ { 3 , \\infty , \\mathbb R ^ 3 } , \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{align*} L \\left ( s , \\chi \\right ) & = G \\left ( \\chi \\right ) q ^ { - s } \\left ( 2 \\pi \\right ) ^ { s - 1 } \\Gamma \\left ( 1 - s \\right ) \\sum _ { \\mu = \\pm 1 } L \\left ( 1 - s , \\overline { \\chi } \\right ) \\chi \\left ( - \\mu \\right ) e ^ { i \\mu \\frac { \\pi } { 2 } \\left ( 1 - s \\right ) } \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{align*} 1 - \\log ( 2 ) + \\left ( - \\frac { 1 } { 2 } \\right ) & - \\sum _ { n = 0 } ^ { \\infty } \\frac { 1 } { ( n + 2 ) ( n + 3 ) } \\\\ & - \\log \\left ( \\prod _ { n = 0 } ^ { \\infty } \\left [ 1 - \\left ( \\frac { 1 } { n + 2 } \\right ) ^ 2 \\right ] \\right ) \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} [ b _ k ^ { - } , b _ l ^ { + } ] = \\delta _ { k l } , \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} v _ { N } ( \\mathsf { x , } t ) = \\sum _ { \\mathsf { n } : 0 \\leq n _ { j } \\leq N - 1 } \\beta _ { \\mathsf { n } } \\exp \\left [ - ( T - t ) E _ { \\mathsf { n } } \\right ] \\mathsf { h } _ { \\mathsf { n } } \\left ( \\mathsf { x } \\right ) \\end{align*}"}
-{"id": "9274.png", "formula": "\\begin{align*} t = \\frac { \\mu ( f _ 0 ) ^ \\frac 4 n } { C _ n } \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{align*} \\lambda _ { 2 } ( p , \\Omega ) = \\tilde \\lambda _ { 2 } ( p , \\Omega ) = \\inf _ { \\gamma \\in \\Gamma _ \\Omega ( u _ 1 , - u _ 1 ) } \\max _ { u \\in \\gamma ( [ 0 , 1 ] ) } \\int _ \\Omega F ^ p ( \\nabla u ( x ) ) \\ d x \\end{align*}"}
-{"id": "7017.png", "formula": "\\begin{align*} \\sup \\limits _ { \\lambda , \\varepsilon } R \\left \\{ \\xi ^ { i } \\frac { d } { d \\xi ^ { i } } \\sigma \\left ( \\lambda , \\varepsilon , \\xi \\right ) : \\xi \\in R \\backslash \\left \\{ 0 \\right \\} \\right \\} \\leq M _ { 2 } i = 0 1 . \\end{align*}"}
-{"id": "7534.png", "formula": "\\begin{align*} \\left [ ( z \\phi _ { - 1 } z ^ { - 1 } ) ^ * , z \\phi _ { - 1 } z ^ { - 1 } \\right ] = \\left [ ( y \\phi _ { - 1 } y ^ { - 1 } ) ^ * , y \\phi _ { - 1 } y ^ { - 1 } \\right ] + O ( t ^ { - 2 } \\mathcal L ) . \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} \\bigg ( { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & 1 - \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] \\bigg ) ^ 2 = { } _ 3 F _ 2 \\bigg [ \\begin{matrix} \\alpha & 1 - \\alpha & \\frac 1 2 \\\\ & 1 & 1 \\end{matrix} \\bigg | \\ , 4 z ( 1 - z ) \\bigg ] \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{align*} \\bar { \\varphi } = \\big ( f ( \\alpha ) , - c _ { \\bar { \\alpha } } A ' _ { \\bar { \\alpha } } - b _ i B ' _ { \\bar { \\alpha } } \\big ) \\mbox { a n d } \\bar { \\psi } = \\big ( - c _ { { \\alpha } } A ' _ { { \\alpha } } - b _ i A _ { \\alpha } , f ( \\bar { \\alpha } ) \\big ) , \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} 1 & = \\omega ( m _ 1 ) + \\omega ( m _ 2 ) - 1 \\\\ & = \\omega ( m _ 1 m _ 2 ) = \\omega ( D _ y ( p ^ 2 - 1 ) ) \\\\ & = \\omega _ y ( p ^ 2 - 1 ) \\in \\big [ 1 . 5 \\log \\log y , 2 . 5 \\log \\log y \\big ] \\end{align*}"}
-{"id": "7129.png", "formula": "\\begin{align*} E ^ n & = O ( 1 ) \\int _ { I _ P } { \\omega _ { 1 - \\alpha } } ( s - x _ { n - 1 } ) d s = O ( h ) \\omega _ { 1 - \\alpha } ( \\xi - x _ { n - 1 } ) , \\quad { \\rm f o r ~ s o m e } ~ ~ \\xi \\in I _ P . \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} \\eta & = G _ 1 * f _ 2 + G _ 2 * f _ 2 , \\eta _ X = G _ 1 ' * f _ 2 + G _ 2 ' * f _ 2 . \\end{align*}"}
-{"id": "9938.png", "formula": "\\begin{align*} | \\hat K ( t - s , \\xi ) | = & \\ \\left | \\int ^ \\infty _ { - \\infty } e ^ { i \\tau ( t - s ) } \\frac { e ^ { i | \\xi | ^ 2 ( t - s ) } } { | \\tau | ^ { 1 / 2 } } \\ d \\tau \\right | \\lesssim \\frac { 1 } { | t - s | ^ { 1 / 2 } } \\end{align*}"}
-{"id": "6830.png", "formula": "\\begin{align*} \\frac { b _ { 2 k - 1 } ( \\mu ) } { ( 2 k ) ! } = O \\left ( \\frac { 1 } { \\sqrt { k } } \\right ) , k \\to + \\infty . \\end{align*}"}
-{"id": "9412.png", "formula": "\\begin{align*} \\Gamma ^ { A , C , B } ( \\widetilde { \\psi } ) ( X , Y ) = \\Gamma ^ { A , B } ( \\psi ) ( X Y ) . \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{align*} \\left | \\int _ X \\alpha \\cdot ( \\beta \\cdot g ) d \\mu - \\left ( \\int _ X \\alpha d \\mu \\right ) \\left ( \\int _ X \\beta d \\mu \\right ) \\right | = O ( \\| \\alpha \\| _ l \\| \\beta \\| _ l \\| g \\| ^ { - \\delta _ 0 ' } ) \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} \\tilde p _ { i j } : = \\left \\{ \\begin{array} { l l } \\max \\{ a _ { i j } , p _ { i j } - m _ { i j } \\} & \\mbox { i f } ( i , j ) \\in U \\\\ p _ { i j } & \\mbox { o t h e r w i s e } \\end{array} \\right . \\begin{array} { l } ( i , j ) \\in E . \\end{array} \\end{align*}"}
-{"id": "2208.png", "formula": "\\begin{align*} E : = G ( k , 0 ) ^ \\dag G ( k , 0 ) + G ' ( k , 0 ) ^ \\dag G ' ( k , 0 ) \\end{align*}"}
-{"id": "3424.png", "formula": "\\begin{align*} K = - \\frac { ( C k ^ 2 - B k + A ) A } { ( 2 A - k B ) ^ 2 } . \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{align*} \\varphi _ x ( \\xi , y , z ) = \\exp _ x ( - \\xi N + y T + z Z ) , \\end{align*}"}
-{"id": "9903.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mathcal { H } ( \\rho | \\rho ^ \\infty ) = & - \\beta \\frac { d } { d t } \\mathcal { \\bar F } ( \\rho ( t ) ) = \\beta ( \\nabla \\bar F , \\nabla \\bar F ) _ \\rho \\\\ = & \\beta \\sum _ { ( i , j ) \\in E } [ ( \\bar F _ j ( \\rho ) - \\bar F _ i ( \\rho ) ) _ + ] ^ 2 \\rho _ i \\\\ = & \\beta \\sum _ { ( i , j ) \\in E } [ ( \\log \\frac { \\rho _ i } { e ^ { F _ i ( \\rho ) / \\beta } } - \\log \\frac { \\rho _ j } { e ^ { F _ j ( \\rho ) / \\beta } } ) _ + ] ^ 2 \\rho _ i \\ . \\\\ \\end{align*}"}
-{"id": "824.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } t ^ { \\frac 1 2 ( 1 - 3 / p ) } \\| e ^ { t \\Delta } u _ 0 \\| _ p = 0 . \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} D _ { p ^ l , k } ( 1 , x ) & = \\frac { k } { 2 } \\ , ( 1 - 4 x ) ^ { \\frac { p ^ l - 1 } { 2 } } + 1 - \\frac { k } { 2 } , \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{align*} & \\partial _ t \\widetilde U = \\Delta \\widetilde U - \\nabla P - h u _ \\infty \\cdot \\nabla \\widetilde U + g - F , \\mbox { d i v $ \\widetilde U $ } = 0 ( x \\in \\Omega , \\ , t > 0 ) , \\\\ & \\widetilde U | _ { \\partial \\Omega } = 0 , \\\\ & \\widetilde U \\to 0 \\quad \\mbox { a s $ | x | \\to \\infty $ } , \\\\ & \\widetilde U ( \\cdot , 0 ) = ( 1 - \\phi ) v _ 0 + \\mathbb B [ v _ 0 \\cdot \\nabla \\phi ] , \\end{align*}"}
-{"id": "1215.png", "formula": "\\begin{align*} d Z _ { t , s } = A Z _ { t , s } d t + \\sigma ' ( X ( t , x ) ) . Z _ { t , s } d W ( t ) \\quad , Z _ { s , s } = z , \\end{align*}"}
-{"id": "9605.png", "formula": "\\begin{align*} \\overline L ^ i _ { \\underline { j k } } = L ^ i _ { \\underline { j k } } + \\delta ^ i _ k \\psi _ j + \\delta ^ i _ j \\psi _ k + F ^ i _ k \\sigma _ j + F ^ i _ j \\sigma _ k , \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} \\{ f _ i ^ - , f _ j ^ + \\} = \\delta _ { i j } \\mathbb { I } , \\{ f _ i ^ + , f _ j ^ + \\} = \\{ f _ i ^ - , f _ j ^ - \\} = 0 . \\end{align*}"}
-{"id": "5048.png", "formula": "\\begin{align*} A _ 2 h _ { 2 , i } = A _ 1 i \\in [ k ] \\setminus \\{ j \\} . \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} X ^ { f } _ t = - \\int _ { t } ^ T \\pi _ s d W _ s - \\int _ { t } ^ T \\int _ E l _ s ( e ) \\tilde N ( d s , d e ) - ( h ^ 1 _ T - h ^ 1 _ t ) + A _ T - A _ t + C _ { T - } - C _ { t - } ; \\end{align*}"}
-{"id": "1355.png", "formula": "\\begin{align*} - \\Lambda \\tilde { y } _ { \\varepsilon } '' = - c _ { \\varepsilon } n _ { 0 } \\hat { m } ' _ { \\varepsilon } , \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{align*} S _ { b } ( x , x , y ) = S _ { b } ( y , y , x ) \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} H _ { ( p - 1 ) / 2 } = 2 \\sum _ { k = 1 } ^ { p - 1 } \\frac { 1 } { 2 k } = \\pounds _ 1 ( 1 ) + \\pounds _ 1 ( - 1 ) \\equiv \\pounds _ 1 ( - 1 ) \\pmod { p } . \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} d _ A ( ( y _ 1 , s _ 1 ) , ( y _ 2 , s _ 2 ) ) : = 2 \\gamma ( \\pi ^ { ( y _ 1 , s _ 1 ) } , \\pi ^ { ( y _ 2 , s _ 2 ) } ) - ( s _ 1 + s _ 2 ) . \\end{align*}"}
-{"id": "2815.png", "formula": "\\begin{align*} S ( X / e , \\mathbf { d } , \\mathbf { D } ; e L _ 1 , e L _ 2 , e L _ 3 ) \\ll e ^ \\varepsilon \\left ( \\frac { X } { e } \\right ) ^ { 2 + \\varepsilon } = \\frac { X ^ { 2 + \\varepsilon } } { e ^ 2 } . \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} g ( \\mathsf { x } , T , \\mathsf { y } ) = \\hat { g } ( \\mathsf { x } , T , \\mathsf { y } ) + \\sum _ { \\mathsf { n } \\in \\mathbb { N } ^ { d } , \\mathsf { n \\neq 0 } } \\exp \\left [ - T E _ { \\mathsf { n } } \\right ] \\mathsf { h } _ { \\mathsf { n } } \\left ( \\mathsf { x } \\right ) \\mathsf { h } _ { \\mathsf { n } } \\left ( \\mathsf { y } \\right ) \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} \\left | \\sum _ { j = 1 } ^ N \\ ; \\omega _ j k _ j \\ ; + \\ ; \\sigma _ 1 \\omega _ l \\ ; + \\ ; \\sigma _ 2 \\omega _ m \\right | & \\geq \\frac { \\gamma } { N ^ \\tau } \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} z y ^ 2 = x ^ 3 - \\tfrac { 1 } { 4 8 } c _ 4 x z ^ 2 - \\tfrac { 1 } { 8 6 4 } c _ 6 z ^ 3 . \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} t ( l _ n ) = 1 \\ \\ \\longleftrightarrow \\ \\ n \\in F . \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} \\frac { 2 x ^ { 3 } } { 2 y ^ { 3 } } = \\frac { x ^ { 4 } - 1 } { y ^ { 4 } - 1 } = \\frac { - 2 x } { - 2 y } . \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} \\big | M \\big | = \\big | \\mathbb { F } \\big | ^ { \\ell _ A ( M ) } , \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} V _ { \\alpha , \\varepsilon } ( x ) = V \\left ( \\frac { x } { \\alpha } \\right ) + \\varepsilon x , \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} \\left [ \\begin{array} { l } \\dot { x } ^ c \\\\ \\dot { x } ^ { s s } \\\\ \\dot { x } ^ u \\end{array} \\right ] = A \\left [ \\begin{array} { l } x ^ c \\\\ x ^ { s s } \\\\ x ^ u \\end{array} \\right ] = \\left [ \\begin{array} { l l l } B _ 1 & 0 & 0 \\\\ 0 & B _ 2 & 0 \\\\ 0 & 0 & C \\end{array} \\right ] \\left [ \\begin{array} { l } x ^ c \\\\ x ^ { s s } \\\\ x ^ u \\end{array} \\right ] , \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} e ^ 1 _ \\tau & = \\int _ { 0 } ^ { t } e ^ { ( t - s ) A } \\bigl ( F _ 1 ( X _ { s } ^ { \\tau } ) - F _ 1 ( X _ { s } ^ { 0 } ) \\bigr ) d s + ( \\int _ { 0 } ^ { t } e ^ { ( t - s + \\tau ) A } \\bigl ( \\sigma ( X _ { s } ^ { \\tau } ) - \\sigma ( X _ { s } ^ { 0 } ) \\bigr ) d W ( s ) \\\\ & + \\int _ { 0 } ^ { t } \\bigl ( e ^ { \\tau A } - I \\bigr ) e ^ { ( t - s ) A } \\sigma ( X _ { s } ^ { 0 } ) d W ( s ) , \\end{align*}"}
-{"id": "1820.png", "formula": "\\begin{align*} p _ { ( m , j ) } ( f \\ast \\psi _ t - f ) = \\sup _ { | x | \\leqslant j } \\big | f ^ { ( m ) } \\ast \\psi _ t ( x ) - f ^ { ( m ) } ( x ) \\big | \\ { \\underset { \\R } { \\overset { t \\to 0 } { \\longrightarrow } } } \\ 0 , \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} \\mu ( [ u ] ) = c \\int _ { D ^ 2 } u ^ * \\omega \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} p ^ 2 \\cdot \\frac { ( 1 ) _ { p - 1 } ^ 2 } { ( \\frac 3 2 ) _ { p - 1 } ^ 2 } = & \\frac { \\Gamma _ p ( 1 + p ) ^ 2 \\Gamma _ p ( \\frac 1 2 ) ^ 2 } { \\Gamma _ p ( 1 ) ^ 2 \\Gamma _ p ( \\frac 1 2 + p ) ^ 2 } \\\\ \\equiv & 1 + p \\cdot \\frac { \\Gamma _ p ( \\frac 1 2 ) ^ 2 } { \\Gamma _ p ( 1 ) ^ 2 } \\cdot \\frac { d } { d x } \\bigg ( \\frac { \\Gamma ( 1 + x ) ^ 2 } { \\Gamma ( \\frac { p + 1 } { 2 } + x ) ^ 2 } \\bigg ) \\bigg | _ { x = 0 } = 1 - 2 p H _ { p - 1 } \\pmod { p ^ 2 } , \\end{align*}"}
-{"id": "1058.png", "formula": "\\begin{align*} T _ { F , G , N } ^ L ( f _ 1 , \\dots , f _ d ) : = \\frac { 1 } { N ^ { d - m } } \\sum \\limits _ { \\mathbf { n } \\in \\mathbb { Z } ^ d } \\Big ( \\prod \\limits _ { j = 1 } ^ d f _ j ( n _ j ) \\Big ) F ( \\mathbf { n } ) G ( L \\mathbf { n } ) . \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} g \\left ( x , t \\right ) = \\theta _ { R } \\left ( x \\right ) \\theta \\left ( \\psi \\left ( x , t \\right ) \\right ) \\tilde { u } \\left ( x , t \\right ) , \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} g ^ i ( x ) = \\frac { c } { 2 } ( \\bar x - x ^ i ) ^ 2 . \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} \\log ( \\mu + \\varepsilon ) & = \\log ( \\mu ) + \\log \\left ( 1 + \\frac { \\varepsilon } { \\mu } \\right ) = \\log ( \\mu ) + P \\left ( \\frac { \\varepsilon } { \\mu } \\right ) \\\\ & = \\mu _ { 1 } + \\varepsilon _ { 1 } \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} \\mathcal { M } f ( x ) = \\sup _ { k \\in \\mathbb { Z } } | \\mathcal { A } _ k f ( x ) | . \\end{align*}"}
-{"id": "9951.png", "formula": "\\begin{align*} \\Pi = T \\left [ \\begin{array} { c c } I & 0 \\\\ 0 & 0 \\end{array} \\right ] T ^ { - 1 } \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} & \\partial _ t u + u \\cdot \\nabla u = \\Delta u - \\nabla p _ u - h u _ \\infty \\cdot \\nabla u , \\\\ & \\mbox { d i v $ u $ } = 0 , \\\\ & u | _ { \\partial \\Omega } = - h u _ \\infty , \\\\ & u \\to 0 \\quad \\mbox { a s $ | x | \\to \\infty $ } , \\\\ \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\liminf _ n { E } ^ c _ n ( n ) \\right ) = 1 . \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{align*} \\frac { d \\Theta _ { i j } } { d t } & = J _ { i j } ( t ) \\Theta _ { i j } ( t ) + \\Theta _ { i j } ( t ) J _ { i j } ( t ) ^ { \\top } + P _ { i j } ( t ) G _ { i j } P _ { i j } ( t ) , \\end{align*}"}
-{"id": "3250.png", "formula": "\\begin{align*} \\left ( \\bar { X } ^ { ( N ) } ( t ) ; t \\ge 0 \\right ) \\overset { d e f } { = } \\left ( \\mathfrak { r } _ N \\left ( X ^ { ( N ) } \\right ) ( t ) ; t \\ge 0 \\right ) = \\left ( \\alpha _ i ^ { \\pm } \\left ( X ^ { ( N ) } ; t \\right ) , \\gamma _ 1 \\left ( X ^ { ( N ) } ; t \\right ) , \\delta \\left ( X ^ { ( N ) } ; t \\right ) , i \\in \\mathbb { N } ; t \\ge 0 \\right ) , \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 5 , [ y _ 1 , y _ 2 ] = y _ 4 + \\theta y _ 5 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 3 \\alpha _ 6 } { \\alpha _ 2 \\gamma _ 1 } y _ 5 , [ y _ 2 , y _ 2 ] = \\frac { \\beta _ 2 \\alpha ^ 2 _ 3 } { \\alpha _ 2 \\gamma ^ 2 _ 1 } y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\alpha _ 3 \\beta _ 4 } { \\alpha _ 2 \\gamma _ 1 } y _ 5 = - [ y _ 3 , y _ 1 ] , \\\\ [ y _ 2 , y _ 3 ] = y _ 4 , [ y _ 3 , y _ 2 ] = \\frac { \\alpha ^ 2 _ 3 \\gamma _ 4 } { \\alpha _ 2 \\gamma ^ 2 _ 1 } y _ 5 . \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} L _ { 1 \\varepsilon } u = \\sum \\limits _ { i = 0 } ^ { \\nu } \\varepsilon _ { n } ^ { \\sigma _ { i } } \\alpha _ { i } \\frac { \\partial ^ { i } u } { \\partial x _ { n } ^ { i } } \\left ( x ^ { \\prime } , 0 \\right ) = 0 \\nu \\in \\left \\{ 0 , 1 \\right \\} , \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} R - \\left | \\sum _ { j = 1 } ^ R e ( \\xi _ { x + ( j , 0 ) } ^ i - \\xi _ x ^ i ) \\right | \\to \\infty R \\to \\infty , \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} \\mathcal { B } ^ \\prime ( z ) & = \\widetilde { B } ^ \\prime ( z ) { } _ a \\langle 0 | K ^ \\prime ( z , t ) | 0 \\rangle _ a A ^ \\prime ( z ) + \\widetilde { A } ^ \\prime ( z ) { } _ a \\langle 1 | K ^ \\prime ( z , t ) | 1 \\rangle _ a B ^ \\prime ( z ) \\\\ & = t ^ { - 1 } z \\widetilde { B } ^ \\prime ( z ) A ^ \\prime ( z ) + z ^ { - 1 } \\widetilde { A } ^ \\prime ( z ) B ^ \\prime ( z ) . \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} \\rho ( y ) = \\int _ 0 ^ { + \\infty } \\exp \\left \\{ - ( \\lambda y ) ^ 2 / 2 \\right \\} \\mu ( d \\lambda ) , y \\in \\R , \\end{align*}"}
-{"id": "4979.png", "formula": "\\begin{align*} L _ c = L _ { c _ * } + L _ { c _ * } ' ( c - c _ * ) + \\tilde { L } _ c ( c - c _ * ) ^ 2 , \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } y ( t ) = - \\frac { 1 } { 1 2 } \\frac { y ( t ) ^ 4 + z ( t ) ^ 4 } { y ( t ) ^ 5 z ( t ) ^ 8 } , \\ , & \\ , \\ , \\frac { d } { d t } z ( t ) = \\frac { 1 } { 1 2 } \\frac { z ( t ) ^ 2 - y ( t ) ^ 2 } { y ( t ) ^ 4 z ( t ) ^ 7 } , \\\\ y ( 0 ) = 1 , \\ , & \\ , \\ , z ( 0 ) = 1 . \\end{cases} \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{align*} - \\left ( \\dfrac { 1 } { f f ^ { \\prime \\prime } } \\right ) ^ { \\prime } + \\left ( \\frac { f ^ { 3 } } { f ^ { \\prime \\prime } } \\right ) ^ { \\prime } \\left ( g ^ { \\prime } \\right ) ^ { 4 } = 0 . \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{align*} [ L _ { - 4 } , \\ , [ V _ { - 2 } , \\ , L _ 5 ] ] = L _ { - 1 } , \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} X \\left ( \\begin{array} { c | c } \\operatorname { r c e f } ( \\Sigma E ^ \\mu ) & \\begin{array} { c } 0 \\\\ \\hline I _ { n - ( \\mu + 1 ) } \\end{array} \\end{array} \\right ) = 0 \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} f \\left ( z \\right ) f \\left ( z + c _ { 2 } \\right ) - q \\left ( z \\right ) = p _ { 2 } \\left ( z \\right ) e ^ { \\beta \\left ( z \\right ) } . \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{align*} E = \\begin{pmatrix} e _ 1 & \\sigma ^ { - 1 } ( e _ 1 ) & \\cdots & \\sigma ^ { - \\nu + 1 } ( e _ 1 ) \\\\ e _ 2 & \\sigma ^ { - 1 } ( e _ 2 ) & \\ldots & \\sigma ^ { - \\nu + 1 } ( e _ 2 ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ e _ \\nu & \\sigma ^ { - 1 } ( e _ \\nu ) & \\cdots & \\sigma ^ { - \\nu + 1 } ( e _ \\nu ) \\end{pmatrix} . \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{align*} x _ { n } = \\sum _ { \\pi \\in \\mathcal { C } \\left ( n \\right ) } g _ { \\pi } = \\sum _ { p = 1 } ^ n \\sum _ { \\underset { k _ { i } \\ge 1 } { k _ { 1 } + \\dots + k _ { p } = n } } g _ { k _ { 1 } } \\dots g _ { k _ { p } } . \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} - \\log ( 1 - x ) = \\log \\left ( 1 - \\frac { x } { x - 1 } \\right ) \\end{align*}"}
-{"id": "7137.png", "formula": "\\begin{align*} _ x { \\rm D } _ 1 ^ \\alpha u _ { \\rm e x } ( x ) = \\sum _ { n = 0 } ^ \\infty ( - 1 ) ^ n \\left ( \\mu F ^ n _ { \\mu - 1 , \\nu + 1 , \\alpha } ( 1 - x ) - \\nu F ^ n _ { \\mu , \\nu , \\alpha } ( 1 - x ) \\right ) . \\end{align*}"}
-{"id": "1870.png", "formula": "\\begin{align*} - \\Lambda ( d d ^ c f ) = \\Delta ^ g ( f ) + g ( \\theta , d f ) . \\end{align*}"}
-{"id": "1528.png", "formula": "\\begin{align*} t _ k = T ( \\eta _ k ) = \\mathrm { i } \\ , \\frac { 1 + \\mathrm { e } ^ { 2 \\pi \\mathrm { i } \\ , k / n } } { 1 - \\mathrm { e } ^ { 2 \\pi \\mathrm { i } \\ , k / n } } \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} \\left [ C _ { x y } \\right ] & = \\mathbb { E } \\left [ C _ { x y } ^ 2 \\right ] = \\kappa \\frac { ( \\alpha - 2 ) ^ 2 \\mathbb { E } \\left [ \\mathit { L } _ { s } ^ 2 \\right ] } { ( \\alpha - 1 ) \\left ( \\mathbb { E } \\left [ \\mathit { L } _ { s } \\right ] \\right ) ^ 2 } = \\kappa \\frac { ( \\alpha - 2 ) ^ 2 } { ( \\alpha - 1 ) } e ^ { \\frac { \\sigma _ { S F } ^ 2 } { \\zeta ^ 2 } } , & \\kappa & = \\frac { \\zeta _ 2 } { 8 \\pi \\lambda _ 1 \\zeta _ 1 ^ 2 R _ c ^ 2 } . \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} F _ { \\kappa + 2 } ( n ) \\leq e ^ { - \\kappa ^ 2 - \\kappa } + O \\left ( \\frac { ( \\log n ) ^ 6 } { n } e ^ { \\kappa ^ 2 + \\kappa } \\right ) = o ( n ^ { - 1 / 8 } ) . \\end{align*}"}
-{"id": "8813.png", "formula": "\\begin{align*} \\varphi _ { \\varepsilon } ^ { \\ast } ( g ^ { \\xi } ) = e ^ { \\sigma } g ^ { \\xi } , \\end{align*}"}
-{"id": "9300.png", "formula": "\\begin{align*} p _ { i i , i } = 1 \\ \\ \\mbox { a n d } \\ \\ p _ { i j , i } + p _ { i j , j } = 1 , \\ \\ \\textrm { f o r a l l } \\ i , j \\in I , \\ ( i \\neq j ) . \\end{align*}"}
-{"id": "9798.png", "formula": "\\begin{align*} \\phi ( t ) = \\frac { 1 } { t } t \\geq \\frac { 3 } { 4 } \\phi ( t ) = 1 t \\leq \\frac { 1 } { 4 } , | \\phi ' | \\leq 2 . \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} I ( \\rho , \\delta ) : = \\xi ^ { ^ u } _ { \\rho } { \\bf 1 } _ { \\tau \\leq \\sigma } + \\zeta ^ { ^ l } _ { \\delta } { \\bf 1 } _ { \\sigma < \\tau } . \\end{align*}"}
-{"id": "3160.png", "formula": "\\begin{align*} \\hat { M } ( A ) = F _ { M } ( A ) = \\int _ { H } e ^ { i T r ( A X ) } M ( d X ) \\ \\textnormal { f o r } \\ A \\in H ( \\infty ) . \\end{align*}"}
-{"id": "1400.png", "formula": "\\begin{align*} \\begin{array} { r l } \\psi _ j ( y _ i , z _ i , \\theta , \\hat \\eta ^ j ) & = z _ { i j } ( y _ i - z _ { i j } \\theta - z _ { i } ^ T \\hat \\beta _ { - j } ) + \\Gamma _ { j j } \\theta \\\\ & - ( \\hat \\mu ^ j ) ^ T \\{ z _ { i , - j } ( y _ i - z _ { i j } \\theta - z _ { i , - j } ^ T \\hat \\beta _ { - j } ) + \\Gamma _ { - j , - j } \\hat \\beta _ { - j } \\} \\\\ & = ( e ^ j - \\hat \\mu ^ j ) ^ T \\{ z _ { i } ( y _ i - z _ { i j } \\theta - z _ { i , - j } ^ T \\hat \\beta _ { - j } ) + \\Gamma ( \\theta e ^ j + \\hat \\beta _ { - j } ) \\} \\end{array} \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{align*} U = [ G ( k , 0 ) - i G ' ( k , 0 ) ] [ G ( k , 0 ) + i G ' ( k , 0 ) ] ^ { - 1 } , k \\in \\mathbb { R } \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} z _ { n _ l } \\sim \\frac { { n _ l } \\pi } { \\omega _ { l } - \\omega _ { l - 1 } } + O ( 1 ) , \\ , l = 1 , \\ldots , 2 J + 2 ; \\ , n _ { l } \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{align*} \\phi ( \\lambda ) = \\kappa \\lambda + \\int _ 0 ^ \\infty ( 1 - e ^ { - \\lambda x } ) \\mu ( d x ) . \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} w _ 0 ( \\xi ) = - 1 5 { \\rm s e c h } ^ 2 ( \\sqrt { c _ * } \\xi ) + \\frac { 1 5 } { 2 } { \\rm s e c h } ^ 4 ( \\sqrt { c _ * } \\xi ) , \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} \\| ( T - A ) e _ { k } \\| < \\varepsilon \\qquad \\hbox { f o r $ k = 1 , \\dots , r $ . } \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{align*} \\Xi ^ \\lambda ( K _ a \\otimes F _ a \\otimes 1 \\otimes E _ a ) & = \\Xi ^ \\lambda ( K _ a F _ a \\otimes 1 \\otimes E _ a K _ a ^ { - 1 } \\otimes K _ a ) \\\\ & = \\sum _ { i , j , k , \\ell } c ^ i _ j \\pi ( K _ a F _ a ) ^ j _ k c ^ k _ \\ell \\pi ( E _ a K _ \\lambda ) ^ \\ell _ i . \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} \\Phi ( t , \\omega ) = \\sum _ { k = 1 } ^ K \\sum _ { m = 1 } ^ M \\mathbf 1 _ { ( t _ { k - 1 } , t _ k ] \\times B _ { m k } } ( t , \\omega ) \\sum _ { n = 1 } ^ N h _ n \\otimes x _ { k m n } , \\ ; \\ ; \\ ; t \\geq 0 , \\omega \\in \\Omega , \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{align*} Z _ H ^ k ( t ) = c _ { k , d } \\int _ { \\mathbb { R } ^ k } ^ { '' } \\frac { e ^ { i ( u _ 1 + \\ldots + u _ k ) t } - 1 } { i ( u _ 1 + \\ldots + u _ k ) } | u _ 1 | ^ { - d } \\ldots | u _ k | ^ { - d } \\widehat { W } ( d u _ 1 ) \\ldots \\widehat { W } ( d u _ k ) ; \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} | \\psi | ^ 2 _ { \\Omega ^ p ( M ; E ) } = \\frac { 1 } { p ! } \\langle \\psi _ { i _ 1 \\ldots i _ p } , \\psi ^ { i _ 1 \\ldots i _ p } \\rangle = \\frac { 1 } { p ! } g ^ { i _ 1 j _ 1 } \\cdots g ^ { i _ p j _ p } \\langle \\psi _ { i _ 1 \\ldots i _ p } , \\psi _ { j _ 1 \\ldots j _ p } \\rangle . \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} \\omega ( \\alpha E ) = \\omega ( E ) , \\ ; \\ ; \\alpha > 0 . \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{align*} \\mu _ { H P } ^ { s , N } P _ { H P } ^ { s , N } ( t ) = \\mu _ { H P } ^ { s , N } \\ \\ t \\ge 0 , \\forall N \\ge 1 . \\end{align*}"}
-{"id": "7000.png", "formula": "\\begin{align*} \\ \\varepsilon u ^ { \\left ( 2 \\right ) } \\left ( x , \\varepsilon \\right ) + A u \\left ( x , \\varepsilon \\right ) = f \\left ( x , \\varepsilon \\right ) , \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} \\hat { \\sigma } _ { C M Z } ( t ) = ( \\mathcal { B } _ \\eta \\otimes \\mathbb { I } _ { M Z } ) ( \\hat { \\sigma } _ { A B M Z } ( t ) ) \\ ; . \\end{align*}"}
-{"id": "8535.png", "formula": "\\begin{align*} & \\int _ { \\Sigma } K _ { g _ { \\phi _ { \\infty } } } d \\mathrm { v o l } _ { g _ { \\phi _ { \\infty } } } = 2 \\pi \\chi ( \\Sigma ) + 2 \\pi ( 3 - 1 ) = 2 \\pi \\chi ( \\Sigma ) + 4 \\pi \\\\ & \\int _ { S ^ 2 } K _ { g _ { \\vec { \\chi } } } d \\mathrm { v o l } _ { g _ { \\vec { \\chi } } } = - 8 \\pi \\\\ & \\int _ { S ^ 2 } K _ { g _ { \\vec { \\Psi } } } d \\mathrm { v o l } _ { g _ { \\vec { \\Psi } } } = 4 \\pi , \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} \\lim _ { \\nu \\to \\infty } S ( A | A ' ) _ { ( \\mathcal { N } ( t ) \\otimes \\mathbb { I } _ { A ' } ) ( \\hat { \\omega } _ { A A ' } ( \\nu ) ) } = n \\ln t + n \\ ; . \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{align*} \\alpha ^ 2 + | \\alpha _ 2 | ^ 2 \\geq \\vartheta ^ { - 2 } + \\vartheta = 1 . 8 9 4 5 5 8 \\ldots , \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} h _ { n - 1 } = \\frac { 1 / m ( f _ { p l a n a r } - K _ d ( v _ c + v _ { a i r } ) ^ 2 ) } { 1 / 2 \\tau _ { f , 1 } } \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{align*} [ L _ { - j - 1 } , \\ , S _ j ] = L _ { - 1 } . \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} F _ i ( n _ 1 , \\cdots , n _ i + 1 , \\cdots , n _ r ) - F _ i ( n _ 1 , \\cdots , n _ i + 1 , \\cdots , n _ r ) = k - ( n _ 1 + n _ 2 + \\cdots + n _ r ) - n _ i , \\end{align*}"}
-{"id": "5373.png", "formula": "\\begin{gather*} \\overline { f } _ 1 ( 1 ) = 0 , \\overline { f } _ 1 ( 2 ) = 1 , \\overline { f } _ 1 ( 3 ) = 1 , \\\\ \\overline { f } _ 1 ( n + 3 ) = \\overline { f } _ 0 ( n + 1 ) + \\overline { f } _ 0 ( n ) . \\end{gather*}"}
-{"id": "5892.png", "formula": "\\begin{align*} p - 1 = m _ 1 n _ 1 , p + 1 = m _ 2 n _ 2 , \\gcd ( m _ 1 , m _ 2 ) = 2 , \\end{align*}"}
-{"id": "1409.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t F + v \\cdot \\nabla F = \\nabla v F , \\\\ [ - 4 m m ] \\\\ F _ { m j } \\nabla _ m F _ { i k } = F _ { l k } \\nabla _ l F _ { i j } , i , j , m , k , l \\in \\{ 1 , 2 , \\cdots , n \\} . \\end{cases} \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} \\lim _ { \\nu \\rightarrow \\infty } \\left \\{ \\nu ^ { 2 } \\mu _ { \\mathrm { A C } } \\left ( \\left [ \\nu , \\infty \\right ) \\right ) \\right \\} = 0 \\ , \\end{align*}"}
-{"id": "605.png", "formula": "\\begin{align*} | h _ { \\lambda , K ' } - h _ { \\lambda , F } | & = \\bigl | \\bigl [ 1 - \\mbox { $ \\frac { 1 } { \\lambda } $ } d ( x , K ' ) \\bigr ] ^ + - \\bigl [ 1 - \\mbox { $ \\frac { 1 } { \\lambda } $ } d ( x , F ) \\bigr ] ^ + | \\\\ & \\leq \\bigl | \\bigl [ 1 - \\mbox { $ \\frac { 1 } { \\lambda } $ } d ( x , K ' ) \\bigr ] - \\bigl [ 1 - \\mbox { $ \\frac { 1 } { \\lambda } $ } d ( x , F ) \\bigr ] \\bigr | \\\\ & = \\mbox { $ \\frac { 1 } { \\lambda } $ } | d ( x , K ' ) - d ( x , F ) | \\leq \\mbox { $ \\frac { 1 } { \\lambda } $ } \\cdot d _ H ( K ' , F ) . \\end{align*}"}
-{"id": "3717.png", "formula": "\\begin{align*} ( A ) _ { j k } & = \\int _ { E _ j ^ - } ^ { E _ j ^ + } \\frac { x ^ { n - k } } { \\sqrt { Q _ E ( x ) } } d x \\\\ ( \\vec { b } ) _ j & = \\int _ { E _ j ^ - } ^ { E _ j ^ + } \\frac { x ^ n } { \\sqrt { Q _ E ( x ) } } d x . \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} \\frac { \\| v \\| _ 2 ^ 2 } { m ^ 2 } = \\frac { \\| \\Delta \\xi ^ * \\| _ 2 ^ 2 } { m ^ 2 } \\ll \\frac { \\| \\xi ^ * \\| _ 2 ^ 2 } { m ^ 2 } \\ll 1 - | \\hat { \\mu } ( \\xi ) | \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} B : = \\frac { 3 } { 2 } \\sqrt { n _ l } \\ , e ^ { 2 | D | _ 0 / \\theta _ { L } } , b : = \\frac { 1 } { 2 } \\sqrt { n _ l } \\ , e ^ { - ( B ^ 2 + 2 | D | _ 0 ) / \\theta _ { L } } . \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} t _ { 2 } ( 2 n ) = 2 \\sum _ { a + b = 2 n , a < b } ( - 1 ) ^ { s _ { 2 } ( a ) + s _ { 2 } ( b ) } + 1 \\end{align*}"}
-{"id": "5030.png", "formula": "\\begin{align*} \\delta = \\delta _ 0 - \\frac { 1 6 } { 3 } | b | ^ 2 + \\mathcal { O } ( \\delta _ 0 ^ 2 + | b | ^ 4 ) , \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{align*} \\begin{cases} \\left ( 1 + \\frac { 1 } { \\lambda } \\phi _ t \\right ) \\dot \\phi _ t = \\left ( \\lambda + \\frac { 2 a } { \\lambda } \\right ) \\phi ^ 2 _ t + ( 2 ( a + \\lambda \\theta ) - \\varepsilon ) \\phi _ t - \\lambda ( \\varepsilon - \\theta ^ 2 ) \\\\ \\phi _ T = c \\end{cases} \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} Q ( 2 \\tau ) = 0 , \\ \\ \\ \\tau \\in [ 1 / 2 , T / 2 ] . \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} \\overline { C } _ 0 = ( s ) _ 0 , \\ \\ \\ 0 \\ne s \\in H ^ 0 ( E ( c - a - b ) ) . \\end{align*}"}
-{"id": "144.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } \\alpha _ j ^ 2 ( \\alpha _ { j + 1 } ^ 2 + \\alpha _ { j - 1 } ^ 2 ) < \\infty \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{align*} H ^ { ( n ) } \\left ( \\frac { 1 } { f } \\right ) & = \\sum _ { k = 1 } ^ n \\frac { ( - 1 ) ^ k } { f ^ { k + 1 } } \\sum _ { i _ 1 , \\dots , i _ k \\ge 1 \\atop i _ 1 + \\cdots + i _ k = n } H ^ { ( i _ 1 ) } ( f ) \\cdots H ^ { ( i _ k ) } ( f ) \\\\ & = \\sum _ { k = 1 } ^ n \\binom { n + 1 } { k + 1 } \\frac { ( - 1 ) ^ k } { f ^ { k + 1 } } \\sum _ { i _ 1 , \\dots , i _ k \\ge 0 \\atop i _ 1 + \\cdots + i _ k = n } H ^ { ( i _ 1 ) } ( f ) \\cdots H ^ { ( i _ k ) } ( f ) \\ , . \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} \\partial [ a ] = \\sum _ { [ b ] ; \\mu ( b ) = \\mu ( a ) - 1 } \\langle \\partial ( [ a ] ) , [ b ] \\rangle [ b ] . \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} \\tilde { g } = T _ { 2 } ^ { 0 } f \\circ g ^ { \\prime } \\circ d f , \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} | \\omega _ { j , r } - z _ { k , r } ^ { j , m } | < 2 \\cdot 2 ^ { - k } = 2 ^ { - k + 1 } \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} u = O \\left ( \\frac { 1 } { | x | ^ { n - 1 + \\varepsilon } } \\right ) \\textrm { a s } | x | \\to \\infty , \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{align*} X = X ( x , y ) = \\begin{pmatrix} x / y & - x ^ 2 / y - y \\\\ 1 / y & - x / y \\end{pmatrix} = \\hat z J \\hat z ^ { - 1 } , \\hat z = \\hat z ( x , y ) = \\begin{pmatrix} y & x \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} \\Delta _ K ( t ) = t ^ 2 - 3 t + 1 . \\end{align*}"}
-{"id": "3411.png", "formula": "\\begin{align*} \\tilde \\gamma ^ i _ t = \\tilde \\gamma ^ i ( t , X _ { t - } ) = \\frac { \\theta + \\left ( 1 - \\frac { 1 } { n } \\right ) \\eta _ { t } } { 1 + \\frac { 1 } { \\lambda } \\left ( 1 - \\frac { 1 } { n } \\right ) ^ 2 \\eta _ { t } } ( \\bar X _ { t - } - X ^ i _ { t - } ) , i = 1 , \\ldots , n , \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & 1 - \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] = { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 \\alpha & \\frac 1 2 - \\frac 1 2 \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , 4 z ( 1 - z ) \\bigg ] . \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { N - 1 } \\big | \\mathcal { D } _ { N , \\ell } ^ { h , 2 , 1 } \\big | \\le \\frac { C ( 1 + | x | _ { L ^ p } ) ^ K } { \\tau ^ { 2 \\kappa } } \\bigl ( t _ { N - 1 } ^ { - \\frac 1 2 + \\kappa } \\Delta t | h | _ { L ^ q } + t _ { N - 1 } ^ { \\frac 1 2 + \\kappa - \\beta } \\big | ( - A ) ^ { - \\beta } h \\big | _ { L ^ q } \\bigr ) . \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{align*} g _ k ^ { i , \\gamma } = \\sum _ { \\delta - \\omega _ l } \\sigma _ { k , \\omega _ l } ^ { i , \\gamma } \\ast { ( f _ q ^ { \\gamma } \\chi _ { \\omega _ l } ) } . \\end{align*}"}
-{"id": "200.png", "formula": "\\begin{align*} \\norm { i } { \\sigma ^ k ( \\gamma ) } & = \\sigma ^ k ( \\norm { i } { \\gamma } ) , \\\\ \\norm { i } { \\sigma ^ k ( \\beta ) } & = \\sigma ^ k ( \\alpha ) ^ { - 1 } \\sigma ^ { k + i } ( \\alpha ) . \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} M z + N \\bar { z } = p ~ , ~ ~ z \\in \\mathbb { C } ^ n , \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} 0 = d ( e ^ { 2 f } F ) = e ^ { 2 f } \\left ( ( d F ) _ 0 + \\left ( 2 d f + \\frac { \\theta } { n - 1 } \\right ) \\wedge F \\right ) \\end{align*}"}
-{"id": "41.png", "formula": "\\begin{align*} Z _ { \\mathrm { C a } ( X ) } \\big ( q ^ { - 1 } T , q , \\mathcal { O } _ { P , X } \\big ) \\mid _ { q \\rightarrow 1 } = \\varsigma _ { f } ( T ) . \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{align*} \\beta = ( \\beta _ 1 , \\ldots , \\beta _ n ) ^ { T } , \\beta _ i = \\frac { \\emph { C o v } [ Y , X _ i ] } { \\emph { V a r } [ Y ] } . \\end{align*}"}
-{"id": "8185.png", "formula": "\\begin{align*} [ n ] = \\hat h [ n ] \\hat h ^ { - 1 } \\end{align*}"}
-{"id": "9384.png", "formula": "\\begin{align*} L ^ p ( \\Omega ; E ) ^ * = L ^ q _ { \\sigma } ( \\Omega ; E ^ * ) . \\end{align*}"}
-{"id": "4153.png", "formula": "\\begin{align*} f _ n ( \\tilde { \\omega } ) & = \\big { ( } E \\prod _ { i = 0 } ^ { n - 1 } \\frac { \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } { 1 + \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } \\big { ) } ^ { - 1 } \\\\ & \\leq \\big ( E \\prod _ { i = 0 } ^ { n - 1 } \\frac { \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } { 1 + \\lambda M ^ 2 } \\big ) ^ { - 1 } \\\\ & = [ \\frac { ( 1 + \\lambda M ^ 2 ) } { \\lambda E \\rho ^ 2 } ] ^ n \\frac { E \\rho ^ 2 } { ( E \\rho ) ^ 2 } . \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} u \\left ( t , \\varepsilon \\right ) = M \\left ( t , \\varepsilon \\right ) f _ { 1 } \\left ( \\varepsilon \\right ) + N \\left ( t , \\varepsilon \\right ) f _ { 2 } \\left ( \\varepsilon \\right ) + r _ { \\left [ 0 , T \\right ] } F ^ { - 1 } \\Phi \\left ( \\xi , \\varepsilon \\right ) F \\bar { f } \\left ( \\xi \\right ) , \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} x = M s + v , \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} \\dim C _ H ( h ) = \\dim C _ { H / C _ 1 } ( h C _ 1 ) = \\dim C _ { H / C _ 2 } ( h C _ 2 ) = \\dim C _ { H / C } ( h C ) . \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 1 } T _ { j } ( x ) \\frac { 2 \\sqrt { 1 - x ^ { 2 } } } { \\pi } d x = \\left \\{ \\begin{array} [ c ] { c c c } 1 & i f & j = 0 \\\\ - 1 / 2 & i f & j = 2 \\\\ 0 & i f & j \\notin \\{ 0 , 2 \\} \\end{array} \\right . , \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{gather*} v _ t + ( b - \\lambda _ l ) v _ x + v _ { x x x } = 0 , \\\\ v \\big | _ { t = 0 } = v _ { 0 l } , v \\big | _ { x = 0 } = v _ x \\big | _ { x = 0 } = v \\big | _ { x = R } = v _ x \\big | _ { x = R } = 0 . \\end{gather*}"}
-{"id": "5696.png", "formula": "\\begin{align*} \\chi ^ { } _ { \\mathrm { m a x } } \\left ( k \\right ) & = \\log \\sqrt { L } \\\\ \\chi ^ { } _ { \\mathrm { m i n } } \\left ( k \\right ) & = \\log \\sqrt { L } - m \\left ( Q - R \\right ) . \\end{align*}"}
-{"id": "2464.png", "formula": "\\begin{align*} K _ { \\Omega } ( z _ 1 , z _ 2 ) = K _ { V \\cap \\Omega } ( z _ 1 , z _ 2 ) \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} \\Delta _ { z ' } \\times \\partial ( \\Delta _ { w ' } \\times \\Delta _ w ) = \\big [ \\Delta _ { z ' } \\times \\partial \\Delta _ { w ' } \\times \\Delta _ w \\big ] \\cup \\big [ \\Delta ' _ z \\times \\partial \\Delta _ w \\big ] . \\end{align*}"}
-{"id": "6692.png", "formula": "\\begin{align*} \\begin{aligned} x _ { 1 } & = \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\cosh \\rho \\cos \\phi \\\\ x _ { 2 } & = \\frac { 1 } { 2 } \\sinh \\rho \\sin \\theta \\sin \\phi \\\\ x _ { 3 } & = \\frac { 1 } { 2 } \\sinh \\rho \\cos \\theta \\sin \\phi \\\\ \\end{aligned} \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{align*} \\int _ X \\phi _ { \\varepsilon } \\ , \\ , d \\mu _ X = \\textrm { V o l } Y + O ( \\varepsilon ^ { p ' } ) \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} 4 P ' ( c _ * ) \\dot { h } = 1 2 | b | ^ 2 \\langle \\partial _ c u _ { c _ * } , \\psi _ * ^ 2 \\rangle _ { L ^ 2 } + \\mathcal { O } ( | b | ^ 4 ) = \\frac { 4 8 } { 5 \\sqrt { c _ * } } | b | ^ 2 + \\mathcal { O } ( | b | ^ 4 ) , \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} F ( \\varepsilon ) = \\begin{cases} 1 & \\mathcal { P } ( \\varepsilon ) = \\varnothing ; \\\\ \\min _ { \\lambda , \\beta \\in \\Phi _ D ( \\mathcal { P } ( \\varepsilon ) ) } | \\lambda - \\beta | & \\end{cases} \\end{align*}"}
-{"id": "7374.png", "formula": "\\begin{align*} H ^ { 2 } = \\frac { r ^ { 2 } - 2 M r } { r ^ { 2 } f \\left ( r \\right ) } . \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} ( \\epsilon _ C \\otimes I _ C ) \\Delta _ C = 1 \\otimes c , \\quad ( I _ C \\otimes \\epsilon _ C ) \\Delta _ C ( c ) = c \\otimes 1 . \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} F _ x = F _ { u ( x ) q ( x ) } : 2 ^ { \\leq \\omega _ 1 } \\to 2 ^ { \\leq \\omega _ 1 } , \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} \\sharp S ( \\varepsilon _ 1 , \\varepsilon _ 2 , B ) = \\sharp T ( \\varepsilon _ 1 , \\varepsilon _ 2 , B ) + O _ { \\varepsilon _ i } ( 1 ) . \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} & \\sum _ { ( i , j ) \\in N } \\left ( 1 - \\cos \\left ( 2 \\pi x _ { ( i , j ) } \\right ) \\right ) \\\\ & ( x _ { ( i , j ) } ) _ { ( i , j ) \\in N } \\in \\left [ - \\frac { 1 } { 2 } , \\frac { 1 } { 2 } \\right ] ^ N , \\\\ & \\forall \\ , ( k , \\ell ) \\in S , \\ ; 4 x _ { ( k , \\ell ) } - \\sum _ { \\| ( i , j ) - ( k , \\ell ) \\| _ 1 = 1 } x _ { ( i , j ) } = v _ { ( k , \\ell ) } , \\end{align*}"}
-{"id": "534.png", "formula": "\\begin{align*} \\sup _ { f \\in L ^ 1 ( \\Gamma _ - , d \\xi ) \\atop \\| f \\| _ { L ^ 1 ( \\Gamma _ - , d \\xi ) } = 1 } \\Big | \\int _ { \\Gamma _ + } \\phi ( x , v ) j _ + K ^ { n + 2 } ( I - K ) ^ { - 1 } J f ( x , v ) d \\xi ( x , v ) \\Big | \\le C _ { n + 2 } e ^ { ( n + 2 ) \\| \\tau \\sigma _ - \\| _ \\infty } \\| k \\| _ { \\infty } ^ { n + 2 } \\eta ^ { n - 2 } , \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} S ^ * ( X , \\mathbf { d } , \\mathbf { D } ) = C ^ * ( \\mathbf { d } , \\mathbf { D } ) \\operatorname { v o l } ( \\mathcal { R } ) X ^ 2 ( \\log X ) ^ 3 + O _ { \\varepsilon } \\left ( \\left ( \\frac { D ^ \\varepsilon } { \\delta ( \\mathbf { D } ) } + 1 \\right ) X ^ { \\frac { 2 3 } { 1 2 } + \\varepsilon } + \\frac { D ^ \\varepsilon } { \\operatorname { d e t } ( \\Lambda ( \\mathbf { D } ) ) } X ^ 2 ( \\log X ) ^ 2 \\right ) , \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} \\mathbb { S } ( R ) = \\lim _ { e \\rightarrow \\infty } \\frac { \\sharp ( F _ * ^ e ( R ) , R ) } { p ^ { e ( d + \\alpha ( R ) ) } } \\end{align*}"}
-{"id": "3391.png", "formula": "\\begin{align*} J ^ i ( \\gamma ) = J ^ i ( \\gamma ^ 1 , \\ldots , \\gamma ^ n ) = \\mathbb { E } \\bigg [ \\int _ 0 ^ T f ^ i ( X _ t , \\gamma _ t ^ i ) \\ , d N ^ i _ t + g ^ i ( X _ T ) \\bigg ] = \\mathbb { E } \\bigg [ \\int _ 0 ^ T \\lambda f ^ i ( X _ t , \\gamma _ t ^ i ) \\ , d t + g ^ i ( X _ T ) \\bigg ] \\ , , \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ 0 ^ t \\langle ( h u _ s + \\widetilde U ) \\otimes w _ k , \\nabla w _ k \\rangle d \\tau = \\int _ 0 ^ t \\langle ( h u _ s + \\widetilde U ) \\otimes w , \\nabla w \\rangle d \\tau . \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} \\frac { y _ { j } ^ { n + 1 } - y _ { j } ^ { n } } { \\Delta t } = f _ { j } ^ { n } , \\end{align*}"}
-{"id": "3158.png", "formula": "\\begin{align*} P _ { H P } ^ { s , \\infty } ( t ) \\Lambda _ N ^ { \\infty } = \\Lambda _ N ^ { \\infty } P _ { H P } ^ { s , N } ( t ) \\ , \\ t \\ge 0 , \\forall N \\ge 1 , \\end{align*}"}
-{"id": "6585.png", "formula": "\\begin{align*} \\sigma _ \\mathrm { e s s } ( \\widetilde { H } ) \\ ; & = \\ ; \\sigma ( \\overline { H _ 0 } ) \\ ; = \\ ; ( - \\infty , - 1 ] \\cup [ 1 , + \\infty ) \\\\ \\sigma _ \\mathrm { d i s c } ( \\widetilde { H } ) \\ ; & \\subset \\ ; ( - 1 , 1 ) \\ , . \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} \\int O ( c ^ n ) d x & = O ( c ^ n ) , & \\frac { d } { d x } O ( c ^ n ) & = O ( c ^ n ) , & x ^ k O ( c ^ n ) & = O ( c ^ n ) . \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} u ( t , 0 , y ) = \\nu _ 0 ( t , y ) , u _ x ( t , 0 , y ) = \\nu _ 1 ( t , y ) , ( t , y ) \\in B _ T . \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} C ^ k ( X ) = \\{ \\delta _ 1 * f + \\delta _ 2 * g : f , g \\in C ^ { k - 1 } ( X ) \\} . \\end{align*}"}
-{"id": "677.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\eta } = \\sum _ { i = 1 } ^ { k } \\frac { \\partial } { \\partial \\theta _ i } , \\frac { \\partial } { \\partial \\bar { \\eta } } = \\sum _ { i = 1 } ^ { k } \\frac { \\partial } { \\partial \\bar { \\theta } _ i } . \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} \\langle \\tilde { G } , G \\rangle & = G _ 0 \\tilde { G } _ 1 ^ \\ast \\frac { i c y } { 2 \\pi } \\left ( J _ 0 ( y , - \\lambda y ) \\left ( 1 + \\lambda y \\dfrac { c } { 2 \\pi } J _ 0 ( y , - \\lambda y ) \\right ) - \\lambda y \\frac { c } { 2 \\pi } \\sum _ { n \\geq 1 } \\dfrac { J _ n ( y , - \\lambda y ) } { n } \\right ) \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} \\frac { d c } { d t } & = \\mu h K _ 1 \\frac { b + c } { 1 + c } - \\frac { \\Gamma c } { K + c } + \\lambda \\theta = R _ 1 ( c , \\theta , h ; \\mu , \\lambda ) , \\\\ \\frac { d \\theta } { d t } & = - \\theta + T ( c ) = R _ 2 ( c , \\theta ) , \\\\ \\frac { d h } { d t } & = \\frac { 1 } { 1 + c ^ 2 } - h = R _ 3 ( c , h ) . \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} \\begin{bmatrix} \\pi & \\lambda \\pi ^ n \\\\ 0 & 1 \\end{bmatrix} \\begin{bmatrix} a & b \\pi ^ n \\\\ c & d \\end{bmatrix} = \\begin{bmatrix} a \\pi + c \\lambda \\pi ^ n & ( b \\pi + d \\lambda ) \\pi ^ n \\\\ c & d \\end{bmatrix} \\qquad ( ) , \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} G _ { \\kappa } ( t ) = \\sum _ { | \\alpha | + | a | \\leq \\kappa - 1 } \\sum _ { 1 \\leq i \\leq 2 } \\int _ { \\mathbb { R } ^ 2 } \\frac { | \\nabla _ i V ^ { ( \\alpha , a ) } + \\nabla _ i H ^ { ( \\alpha , a ) } \\cdot \\omega | ^ 2 + | \\nabla _ i H ^ { ( \\alpha , a ) } \\cdot \\omega ^ \\perp | ^ 2 } { \\langle t - r \\rangle ^ 2 } e ^ q d x . \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} \\left [ \\lambda , \\begin{pmatrix} r & s \\\\ t & u \\end{pmatrix} \\right ] : G ( x _ 1 , x _ 2 ) \\mapsto \\lambda ^ 2 G ( r x _ 1 + t x _ 2 , s x _ 1 + u x _ 2 ) . \\end{align*}"}
-{"id": "1290.png", "formula": "\\begin{align*} g _ { N } ( \\mathsf { x } , t , \\mathsf { y } ) = \\sum _ { \\mathsf { n } : 0 \\leq n _ { j } \\leq N - 1 } \\exp \\left [ - t E _ { \\mathsf { n } } \\right ] \\mathsf { h } _ { \\mathsf { n } } \\left ( \\mathsf { x } \\right ) \\mathsf { h } _ { \\mathsf { n } } \\left ( \\mathsf { y } \\right ) \\end{align*}"}
-{"id": "9815.png", "formula": "\\begin{align*} \\mathfrak { M } ^ J = \\bigoplus _ { w \\in W } \\mathcal { M } ( w \\mathbf { z } ) , \\mathcal { M } ( w \\mathbf { z } ) = M ( w \\mathbf { z } ) ^ J \\end{align*}"}
-{"id": "9366.png", "formula": "\\begin{align*} B _ 2 ( E \\times F , G ) = B ( E \\overset { \\wedge } { \\otimes } F , G ) . \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} \\alpha _ n ^ 2 ( \\alpha _ n ^ 2 - \\alpha _ { n + 1 } \\alpha _ { n - 1 } ) = \\alpha _ n ^ 2 \\eta _ n \\eta _ { n - 1 } - \\alpha _ n ^ 3 ( \\eta _ n - \\eta _ { n - 1 } ) \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} \\mu _ 3 & : z = w = 0 , x ^ 2 + y ^ 3 = 0 \\\\ \\mu _ 2 & : x = w = 0 , y ^ 3 + z ^ 9 = 0 . \\end{align*}"}
-{"id": "5672.png", "formula": "\\begin{align*} & P _ g \\Big \\{ \\big [ v ( t ) , w ( t ) \\big ] ; [ \\bar { v } ( t ) , \\bar { w } ( t ) \\big ] \\Big \\} \\\\ & = \\frac { 1 } { Z } \\exp \\Bigg [ - \\frac { 1 } { 2 \\sigma ^ 2 } \\left ( \\begin{array} { c } v ( t ) - \\bar { v } ( t ) \\\\ w ( t ) - \\bar { w } ( t ) \\end{array} \\right ) ^ { \\top } \\Theta _ { i j } ^ { - 1 } ( t ) \\\\ & \\times \\left ( \\begin{array} { c } v ( t ) - \\bar { v } ( t ) \\\\ w ( t ) - \\bar { w } ( t ) \\end{array} \\right ) \\Bigg ] . \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 1 + \\lambda ^ 2 & 0 \\\\ 0 & 1 + \\mu ^ 2 \\end{bmatrix} \\quad B = \\begin{bmatrix} 0 & - \\mu \\\\ \\lambda & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} h _ 1 ( x ) : = \\begin{cases} - \\frac { 1 } { x } \\log ( 1 - x ) & \\\\ 1 & \\end{cases} \\end{align*}"}
-{"id": "3344.png", "formula": "\\begin{align*} q _ N ( a ) = \\left ( { N } \\varrho ( 1 ) \\right ) ^ { N a } { \\rm e } ^ { - N \\varrho ( 1 ) } \\frac { 1 } { ( N a ) ! } \\sim \\left ( \\frac { \\varrho ( 1 ) } { a } \\right ) ^ { N a } { \\rm e } ^ { N \\left ( a - \\varrho ( 1 ) \\right ) } \\frac { 1 } { \\sqrt { 2 \\pi N a } } = \\check q _ N ( a ) . \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{align*} f \\left ( z \\right ) = \\frac { \\log \\left ( 1 + z \\right ) } { z } = \\sum _ { n \\ge 0 } \\frac { \\left ( - 1 \\right ) ^ { n } } { n + 1 } z ^ { n } \\end{align*}"}
-{"id": "5449.png", "formula": "\\begin{align*} ( - \\Delta ) ^ m u = \\mu \\rho u \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} ( u - c ) \\cdot \\nabla u + \\nabla P + g e _ n = 0 , \\nabla \\cdot u = 0 \\textrm { i n } \\Omega , \\end{align*}"}
-{"id": "510.png", "formula": "\\begin{align*} \\begin{array} { r c l c l c l } A g & = & J g \\big | _ { \\Gamma _ + } & + & K J g \\big | _ { \\Gamma _ + } & + & K ^ 2 ( I - K ) ^ { - 1 } J g \\big | _ { \\Gamma _ + } \\\\ & : = & B g & + & S g & + & M g . \\end{array} \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{align*} \\sum _ { k _ { 1 } + \\dots + k _ { p } = n } \\frac { \\left ( - 1 \\right ) ^ { p } } { \\left ( k _ { 1 } + 1 \\right ) ! \\dots \\left ( k _ { p } + 1 \\right ) ! } = \\left ( - 1 \\right ) ^ { p } \\frac { p ! } { \\left ( n + p \\right ) ! } \\left \\{ \\begin{array} { c } n + p \\\\ p \\end{array} \\right \\} \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} ( e _ { i } \\ast e _ { j } ) \\ast e _ { m } = \\sum _ { k = 1 } ^ { n } a _ { i , j } ( k ) ( e _ { k } \\ast e _ { m } ) = \\sum _ { k = 1 } ^ { n } a _ { i , j } ( k ) a _ { k , m } \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} S ^ { \\sigma } _ { 0 T } ( \\varphi ) = \\begin{cases} \\int _ 0 ^ T L ( \\dot { \\varphi } _ s ) d s , \\varphi \\\\ \\infty . \\end{cases} \\end{align*}"}
-{"id": "4423.png", "formula": "\\begin{align*} G _ 1 : = ( \\alpha \\circ \\nu ) ^ { - 1 } \\big ( \\langle ( 2 , 3 ) \\rangle \\big ) \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} \\begin{cases} \\alpha _ 5 \\gamma _ 3 = \\alpha _ 1 \\gamma _ 4 \\alpha _ 7 \\gamma _ 3 = \\alpha _ 3 \\gamma _ 4 \\\\ \\beta _ 5 \\gamma _ 3 = \\beta _ 1 \\gamma _ 4 \\beta _ 3 \\gamma _ 3 = \\alpha _ 1 \\gamma _ 5 \\\\ \\beta _ 7 \\gamma _ 3 = \\alpha _ 3 \\gamma _ 5 \\gamma _ 1 \\gamma _ 3 = \\beta _ 1 \\gamma _ 5 \\\\ \\beta _ 3 \\gamma _ 4 = \\alpha _ 5 \\gamma _ 5 \\beta _ 7 \\gamma _ 4 = \\alpha _ 7 \\gamma _ 5 \\\\ \\gamma _ 1 \\gamma _ 4 = \\beta _ 5 \\gamma _ 5 \\end{cases} \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} | | \\nabla E ( \\boldsymbol { h } ) | | ^ 2 _ { l ^ 2 ( \\mathbb { Z } ) } = & ( V _ 2 ( z _ { - N - 1 } , z _ { - N } ) + V _ 1 ( z _ { - N } , x _ { - N + 1 } ) ) ^ 2 + ( V _ 2 ( z _ { - N } , x _ { - N + 1 } ) + V _ 1 ( x _ { - N + 1 } , x _ { - N + 2 } ) ) ^ 2 \\\\ & + \\sum _ { j = - N + 2 } ^ { N - 1 } ( V _ 2 ( x _ { j - 1 } , x _ j ) + V _ 1 ( x _ j , x _ { j + 1 } ) ) ^ 2 \\\\ & + ( V _ 2 ( x _ { N - 1 } , x _ N ) + V _ 1 ( x _ N , z _ { N + 1 } ) ) ^ 2 + ( V _ 2 ( ( x _ N , z _ { N + 1 } ) ) + V _ 1 ( z _ { N + 1 } , z _ { N + 2 } ) ) ^ 2 . \\end{align*}"}
-{"id": "4960.png", "formula": "\\begin{align*} \\sigma _ { c , t _ { 2 } } = \\left ( \\beta _ { 1 } + t _ { 2 } , \\beta _ { 2 } , \\beta _ { 3 } , \\ldots , \\beta _ { m + 1 } , \\beta _ { m + 2 } \\pm t _ { 2 } , \\overline { \\beta } _ { m + 1 } , \\ldots , \\overline { \\beta } _ { 3 } , \\overline { \\beta } _ { 2 } \\right ) . \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} R ^ { ( k + 1 ) } & = \\frac { N - \\tilde { B } - 2 ( B - \\tilde { B } ) } { N - \\tilde { B } } \\cdot \\frac { 1 - \\frac { T } { N - \\tilde { B } - 2 ( B - \\tilde { B } ) } } { 1 - ( \\frac { T } { N - \\tilde { B } - 2 ( B - \\tilde { B } ) } ) ^ M } \\\\ & = \\frac { N + \\tilde { B } - 2 B } { N - \\tilde { B } } \\cdot \\frac { 1 - \\frac { T } { N + \\tilde { B } - 2 B } } { 1 - ( \\frac { T } { N + \\tilde { B } - 2 B } ) ^ M } \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{align*} \\frac { ( \\beta ) _ { p + k } } { ( 1 ) _ { p + k } } = & \\frac { \\Gamma _ p ( \\beta + p + k ) \\Gamma _ p ( 1 ) } { \\Gamma _ p ( \\beta ) \\Gamma _ p ( p + k + 1 ) } \\cdot \\frac { t p } { p } \\cdot \\frac { ( t + 1 ) p } { p } \\equiv ( t + 1 ) \\cdot \\frac { ( \\beta ) _ k } { ( 1 ) _ k } \\\\ \\equiv & t \\cdot \\frac { ( \\beta ) _ k } { ( 1 ) _ k } + t p \\cdot \\frac { ( \\beta ) _ k } { ( 1 ) _ k } \\cdot \\bigg ( H _ { p - 1 - b } + \\sum _ { j = 0 } ^ { k - 1 } \\frac { 1 } { j + \\beta } - H _ k \\bigg ) \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "1370.png", "formula": "\\begin{align*} f ^ { \\ast } ( x , y ) : = \\lim _ { k ^ { \\prime } \\rightarrow \\infty } \\sup _ { k \\in ( k ^ { \\prime } , \\infty ) } f \\left ( x , y , k \\right ) = \\left ( A ( x , y ) + A _ { 0 } ( x , y ) \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} H ^ { ( 0 ) } & : = H _ 0 + N ^ { ( 1 ) } . \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} \\lambda _ { 2 } ( p , \\Omega ) = \\lambda _ { 1 } ( p , \\Omega ) = \\lambda _ { 1 } \\left ( p , \\mathcal W _ { r _ { i } } \\right ) . \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} \\wp ' & = \\wp _ 1 & \\wp _ 1 ' & = \\wp _ 2 & \\wp _ 2 ' & = 1 2 \\wp \\wp _ 1 . \\end{align*}"}
-{"id": "1455.png", "formula": "\\begin{align*} \\| \\tilde { f } _ { i j k } ^ { \\alpha a } \\| _ { L ^ 2 ( r \\leq \\langle t \\rangle / 2 ) } \\leq \\sum _ { \\tiny \\begin{matrix} \\beta + \\gamma = \\alpha , b + c = a \\\\ | \\beta | + | b | , | \\gamma | + | c | < | \\alpha | + | a | \\end{matrix} } \\big \\| | \\nabla ^ 2 U ^ { ( \\beta , b ) } | | \\nabla U ^ { ( \\gamma , c ) } | \\big \\| _ { L ^ 2 ( r \\leq \\langle t \\rangle / 2 ) } . \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} \\langle P _ j , Q _ k \\rangle = \\dfrac { 2 \\sin ( j \\pi ) \\cos ( k \\pi ) [ \\psi ( j + 1 ) - \\psi ( k + 1 ) ] + \\pi \\cos ( ( k - j ) \\pi ) - \\pi } { \\pi ( k - j ) ( j + k + 1 ) } , \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{align*} \\mathbf { \\Omega } _ J ( h ^ * _ J , \\dot { J } ) = d \\mu ^ h ( \\dot { J } ) . \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\overbrace { \\begin{matrix} 1 & 0 & 0 & 0 & \\frac 3 2 t _ 1 & 0 & - \\frac { t _ 2 } { 2 } & 0 \\\\ - 1 & 0 & 0 & 0 & - \\frac 3 2 t _ 1 & 0 & \\frac { t _ 2 } { 2 } & 0 \\end{matrix} } \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "3102.png", "formula": "\\begin{align*} L _ f ( s ) = \\sum _ { ( \\xi _ 1 , \\xi _ 2 ) \\in X } D _ { \\xi _ 1 , \\xi _ 2 } ( s ) L _ { \\xi _ 1 , \\xi _ 2 } ( s ) + \\sum _ { g \\in C } D _ g ( s ) L _ g ( s ) . \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} \\textrm { I I } _ n \\leq [ \\frac { ( 1 + \\lambda M ^ 2 ) } { \\lambda E \\rho ^ 2 } ] ^ n \\frac { E \\rho ^ 2 } { ( E \\rho ) ^ 2 } P ( \\tau \\geq n ) = [ \\frac { ( 1 + \\lambda M ^ 2 ) } { d \\lambda E \\rho ^ 2 } ] ^ n \\frac { E \\rho ^ 2 } { ( E \\rho ) ^ 2 } . \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} u ( x ) = P _ p [ f ] ( x ) = \\frac { 1 } { p } \\sum _ { k = 0 } ^ { p - 1 } \\int _ S \\frac { 1 - | x | ^ { 2 p } } { | e ^ { \\frac { - k \\pi i } { p } } x - \\zeta | ^ n } f ( e ^ { \\frac { k \\pi i } { p } } ) d \\sigma ( \\zeta ) . \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} \\sum _ { s \\in S } { f _ { s \\cdot \\widetilde { \\tau } } \\cdot s \\cdot \\gamma ^ { - 1 } \\otimes \\gamma \\cdot \\widetilde { \\tau } } & = \\biggl ( \\sum _ { s \\in S } f _ { s \\cdot \\widetilde { \\tau } } \\cdot s \\biggr ) \\cdot \\gamma ^ { - 1 } \\otimes \\gamma \\cdot \\widetilde { \\tau } \\stackrel { ( \\ref { e q : l i f t o f z e r o } ) } { = } 0 . \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} & \\ p x ' + p y ' + \\delta z ' + \\log { \\lambda _ { G \\times G ; \\varphi } ( x ' , y ' , z ' ) } \\\\ = & \\ 2 p \\bar { x } + \\delta z ' + \\log { \\lambda _ { G \\times G ; \\varphi } ( x ' , y ' , z ' ) } \\\\ \\ge & \\ 2 p \\bar { x } + \\delta z ' + \\log { \\lambda _ { G \\times G ; \\varphi } ( \\bar { x } , \\bar { x } , z ' ) } , \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} \\nabla _ { A } = B \\left ( x \\right ) \\nabla \\ . \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} \\langle 1 \\cdots M | \\mathcal { B } ^ \\prime ( z _ 1 ) \\cdots \\mathcal { B } ^ \\prime ( z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle = \\sum _ { \\overline { y } } \\langle 1 \\cdots M | \\mathcal { B } ^ \\prime ( z _ 1 ) | \\overline { y } \\rangle \\langle \\overline { y } | \\mathcal { B } ^ \\prime ( z _ 2 ) \\cdots \\mathcal { B } ^ \\prime ( z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle , \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} \\| ( T - A ) e _ { k } \\| < \\varepsilon \\quad { \\rm a n d } \\| ( T - A ) ^ { * } e _ { k } \\| < \\varepsilon \\qquad \\hbox { f o r $ k = 1 , \\dots , r . $ } \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{align*} w ( s ) = s ^ { - \\frac { 1 } { \\alpha } } f \\Bigl ( \\frac { 1 } { \\sqrt { s } } \\Bigr ) , \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} \\| U \\| _ { p o p } : = \\inf \\{ \\sum _ { k = 1 } ^ n \\| a _ k \\| \\| u _ k \\| \\| v _ k \\| \\| b _ k \\| \\} , \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} { \\displaystyle { \\| \\Psi _ { h } ^ { k } \\| } _ { \\mathcal { L } ^ 2 } ^ { 2 } = { \\| \\Psi _ { h } ^ { 0 } \\| } _ { \\mathcal { L } ^ 2 } ^ { 2 } , \\mathcal { G } _ { h } ^ { k } \\leq C , } \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} K _ b ( f , \\eta , \\varpi ) = K _ { b , 1 } f + K _ { b , 2 } ( \\eta , \\varpi ) , \\end{align*}"}
-{"id": "6776.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + a \\frac { \\partial ^ { 2 } u } { \\partial y ^ { 2 } } + b \\frac { \\partial u } { \\partial y } = F \\left ( u , \\bar { u } \\right ) , \\end{align*}"}
-{"id": "5286.png", "formula": "\\begin{align*} K _ { \\xi ^ * } ( x , y ) = - \\frac { y - \\xi ^ * } { | x | ^ 2 } + 2 \\frac { ( y - \\xi ^ * ) \\cdot x } { | x | ^ 4 } x + O \\left ( \\frac { | y - \\xi ^ * | ^ 2 } { | x | ^ 3 } \\right ) , | x | \\to \\infty . \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{align*} \\mu _ g = \\frac { \\pi ^ 2 } { 8 ( x _ c ^ \\sigma ) ^ 2 } . \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\lambda } ^ d | L _ n | ^ 2 = \\sum _ { x : | x | = n } \\sum _ { y : | y | = n } \\mathbb { P } _ { \\lambda } ^ d ( x \\in L _ n , y \\in L _ n ) . \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} \\tilde { G } = - \\frac { i c } { 2 \\pi \\gamma } \\tilde { G } _ 1 [ B ( i \\sqrt { 2 } / \\gamma ) + \\lambda / \\gamma ] ^ { - 1 } \\cdot e _ 0 ; \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\beta \\\\ & \\alpha - \\beta + 1 \\end{matrix} \\bigg | \\ , z ^ 2 \\bigg ] = ( 1 + z ) ^ { - 2 \\alpha } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\alpha - \\beta + \\frac 1 2 \\\\ & 2 \\alpha - 2 \\beta + 1 \\end{matrix} \\bigg | \\ , \\frac { 4 z } { ( 1 + z ) ^ 2 } \\bigg ] . \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} & \\partial _ t v + v \\cdot \\nabla v + h ( u _ s \\cdot \\nabla v + v \\cdot \\nabla u _ s ) = \\Delta v - \\nabla p _ v - h u _ \\infty \\cdot \\nabla v + g , \\\\ & \\mbox { d i v $ v $ } = 0 , \\\\ & v | _ { \\partial \\Omega } = 0 , \\\\ & v \\to 0 \\quad \\mbox { a s $ | x | \\to \\infty $ } , \\\\ & v ( \\cdot , 0 ) = v _ 0 : = u _ 0 - h ( 0 ) u _ s \\\\ \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} \\| ( C _ { N } - A ) e _ { k } \\| < \\varepsilon \\quad { \\rm a n d } \\| ( C _ { N } - A ) ^ { * } e _ { k } \\| < \\varepsilon \\qquad \\hbox { f o r e v e r y $ k = 1 , \\dots , r $ } \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} f ( x + \\varepsilon ) = \\sum _ { i = 0 } ^ { \\infty } \\frac { f ^ { ( i ) } ( x ) } { i ! } \\varepsilon ^ { i } . \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} \\lambda _ 1 \\langle \\eta _ * , \\psi _ * \\rangle _ { L ^ 2 } = \\langle \\eta _ * , \\partial _ { \\xi } L _ { c _ * } ' \\psi _ * \\rangle _ { L ^ 2 } = - \\langle \\psi _ * , L _ { c _ * } ' \\psi _ * \\rangle _ { L ^ 2 } . \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} 1 & \\asymp \\int _ { \\prod _ { k = 1 } ^ r x _ { k } \\leq ( n b ) ^ { 1 / m } , x _ { k } \\geq 1 } \\left ( b \\prod _ { k = 1 } ^ r x _ { k } ^ m - n ^ { - 1 } \\prod _ { k = 1 } ^ r x _ { k } ^ { 2 m } \\right ) \\left ( 1 + \\sum _ { k = 1 } ^ r x _ k ^ 2 \\right ) ^ { - 1 } d x _ { 1 } \\cdots d x _ { r } \\\\ & \\asymp \\int _ { \\prod _ { k = 1 } ^ r x _ { k } \\leq ( n b ) ^ { 1 / m } , x _ { k } \\geq 1 } b \\prod _ { k = 1 } ^ r x _ { k } ^ m \\left ( 1 + \\sum _ { k = 1 } ^ r x _ k ^ 2 \\right ) ^ { - 1 } d x _ { 1 } \\cdots d x _ { r } \\end{align*}"}
-{"id": "5496.png", "formula": "\\begin{align*} f _ { i , \\Delta _ i } ( { x } ^ * ) \\le 0 \\textrm { f o r a l l } i \\in I _ 1 \\textrm { a n d } f _ { i , \\Delta _ i } ( { x } ^ * ) = 0 \\textrm { f o r a l l } i \\in I _ 2 . \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} \\mathbf u = \\sum ( c \\alpha _ i + d ) ^ { 2 } v _ i w _ i ' = c ^ 2 \\alpha ^ 2 ( \\mathbf v ) + 2 c d \\alpha ( \\mathbf v ) + d ^ 2 \\mathbf v , \\end{align*}"}
-{"id": "9962.png", "formula": "\\begin{align*} \\alpha _ i = \\frac { 1 } { 2 } \\min \\{ x ^ T \\tilde { Q } _ i x : x \\in \\mathcal { C } _ i \\mbox { a n d } x ^ T P _ i x = 1 \\} . \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} S ( p ) : = \\frac { \\phi ( p - 1 ) } { p - 1 } - \\frac { \\phi ( p + 1 ) } { p + 1 } , \\end{align*}"}
-{"id": "9658.png", "formula": "\\begin{align*} x ^ 2 + y ^ 2 - z ^ 2 = 0 , ( x , y , z ) \\neq ( 0 , 0 , 0 ) \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} \\tilde { s } ^ H - \\tilde { s } ^ { \\tilde { g } } = 2 | N | _ { \\tilde { g } } ^ 2 = 2 e ^ { - 2 f } | N | _ { g } ^ 2 \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} \\frac 1 D \\max _ j | a _ i | \\le \\| \\sum _ { i = 1 } ^ k a _ i e _ { \\gamma _ i } \\| \\le D \\max _ j | a _ j | . \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} \\ell _ \\Omega ( \\sigma ) = \\int _ a ^ b k _ \\Omega ( \\sigma ( t ) ; \\sigma ^ \\prime ( t ) ) d t . \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} c _ { s , t } c _ { t , s } = 1 . \\end{align*}"}
-{"id": "4048.png", "formula": "\\begin{align*} \\begin{matrix} \\geq 4 & \\geq 3 & \\ge 2 \\\\ \\geq 2 & \\geq 1 & = 0 \\\\ = 0 & \\geq 0 & \\geq 0 \\end{matrix} \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} \\big ( [ \\omega _ 0 ] + T ( K _ X + D ) \\big ) \\cdot E = 0 , \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} { p _ { o u t , K } } = \\Pr \\left ( { { { \\log } _ 2 } { { \\left ( { 1 + \\frac { { P } { { \\left | { { h } } \\right | } ^ 2 } } { \\mathcal N _ 0 } } \\right ) } ^ K } < \\mathcal R } \\right ) = \\frac { 1 } { { \\Gamma \\left ( m \\right ) } } \\Upsilon \\left ( { m , \\frac { { m \\mathcal N _ 0 \\left ( { { 2 ^ { \\mathcal R / K } } - 1 } \\right ) } } { { { P } { \\sigma } ^ 2 } } } \\right ) . \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ t | \\beta _ j | ^ { q } = t + \\frac { q } { 2 } \\log 2 + O ( n ^ { - 1 / 4 } ) , \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} \\int _ { y ' \\prec x ' } ^ { } d x ' { q } _ t ^ { N , N + 1 } ( ( x , y ) , ( x ' , y ' ) ) = \\det \\left ( { D } _ t ( y , y ' ) _ { i j } \\right ) _ { i , j = 1 } ^ { N } = \\hat { \\mathcal { P } } ^ { ( N ) } _ { s } ( t ) ( y , y ' ) . \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} \\beta = \\frac { 2 } { \\alpha } \\Bigl ( N - 2 - \\frac { 2 } { \\alpha } \\Bigr ) . \\end{align*}"}
-{"id": "7335.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left ( \\frac { 1 } { 2 } \\omega ( t ) ^ 2 \\right ) = - \\varepsilon ^ { - 2 } _ t \\left [ ( d \\eta , \\omega ( t ) ) _ t \\ , \\omega ( t ) \\wedge d \\eta - d \\eta \\wedge d \\eta \\right ] , \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{align*} s ^ g ( p ) - 2 s ( p ) & = T _ { i k i } T _ { j k j } + \\frac 9 4 t _ { i j k } ^ 2 + 3 t _ { i j } ^ { \\ ; \\ ; k } T _ { i \\ , \\ , j } ^ { \\ , \\ , k } - T _ { j k i } T _ { i k j } + 2 \\delta \\theta \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} \\delta \\in ( 0 , 3 ) \\mbox { a n d } \\delta ' = 3 - \\delta , \\end{align*}"}
-{"id": "9882.png", "formula": "\\begin{align*} \\sum _ { | \\gamma | = L } Q ( \\gamma ) \\leq \\sum _ { s = 1 } ^ { L / 2 + 1 } 2 ^ L n ^ s / s ! \\leq \\left ( \\frac { 8 e n } { L } \\right ) ^ { L / 2 + 1 } . \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} \\widetilde { r } _ { s } + \\widetilde { r } _ { c } & = \\frac { 1 } { n } \\overline { F } \\left ( \\sigma _ { s , t _ { 1 } } ^ { T } + \\sigma _ { c , t _ { 2 } } ^ { T } \\right ) \\\\ & = r _ { s } + r _ { c } + \\left ( t _ { 1 } + t _ { 2 } \\right ) \\overline { F } \\mathbf { e } _ { 1 } \\pm \\left ( t _ { 1 } + t _ { 2 } \\right ) \\overline { F } \\mathbf { e } _ { m + 2 } \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} m _ 1 & = - 2 - i + ( 1 + i ) z \\\\ m _ 2 & = i \\\\ m _ 3 & = 1 - i - z \\\\ r & = m _ 1 + m _ 2 I + m _ 3 J \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{align*} d X ^ { ( n ) } _ i ( t ) = \\sqrt { 2 ( ( X _ i ^ { ( n ) } ) ^ 2 ( t ) + 1 ) } d \\beta ^ { ( n ) } _ i ( t ) + \\left [ \\left ( 2 - 2 n - 2 \\Re ( s ) \\right ) X _ i ^ { ( n ) } ( t ) + 2 \\Im ( s ) \\right ] d t + \\frac { 1 } { 2 } d K _ i ^ { ( n ) , - } ( t ) - \\frac { 1 } { 2 } d K _ i ^ { ( n ) , + } ( t ) , \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} \\operatorname { r c e f } ( S ^ t ) = \\left ( \\begin{array} { c | c } \\operatorname { r c e f } ( S ^ \\mu ) & 0 _ { ( t + 1 ) \\times ( t - \\mu ) } \\end{array} \\right ) . \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} \\tau _ A ( m n ^ * ) = \\tau _ B ( n ^ * m ) . \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} u _ { \\rm e x } ( x ) = x ^ { 4 - \\theta ( 1 - \\alpha ) } ( 1 - x ) ^ { 4 - ( 1 - \\theta ) ( 1 - \\alpha ) } . \\end{align*}"}
-{"id": "1994.png", "formula": "\\begin{align*} \\widetilde { T } _ { a } ( z , \\{ \\alpha \\} ) = \\widetilde { L } _ { a 1 } ( z , t , \\alpha _ 1 ) \\cdots \\widetilde { L } _ { a M } ( z , t , \\alpha _ M ) , \\end{align*}"}
-{"id": "8659.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } u ( t , x ) = \\lim _ { t \\to 0 } t ^ { - \\frac { 1 } { \\alpha } } f \\Bigl ( \\frac { x } { \\sqrt t } \\Bigr ) = | x | ^ { - \\frac { 2 } { \\alpha } } \\lim _ { r \\to \\infty } r ^ { \\frac { 2 } { \\alpha } } f \\Bigl ( \\frac { x } { | x | } r \\Bigr ) = \\omega ( x ) | x | ^ { - \\frac { 2 } { \\alpha } } , \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} S _ X ( \\sigma ) = \\{ x \\in H ^ 2 ( X , \\mathbb Z ) : \\sigma ^ * x = x \\} \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} { \\cal P } _ { 2 n - 1 } ^ * = \\overline { \\cal M } _ { 2 n - 1 } , { \\cal M } _ { 2 n - 1 } ^ * = { \\cal P } _ { 2 n - 1 } , ( \\mathcal { S } _ + ^ n ) ^ * = \\mathcal { S } _ + ^ n . \\end{align*}"}
-{"id": "10006.png", "formula": "\\begin{align*} W ( G _ { 4 } ( n , d , x , s ) ) & = W ( B _ { 2 } ( n - 1 , d , x ) ) + \\sum _ { i = - d } ^ { d } ( \\left \\vert s - i \\right \\vert + 1 ) + \\\\ & + \\sum _ { i = - ( k - 1 ) } ^ { k - 1 } ( \\left \\vert s - i \\right \\vert + 2 ) + ( s - x ^ { \\prime } + 2 ) + \\\\ & + 2 x ( d + 1 ) + 2 r ( d + 2 ) . \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} & H ( M | Y _ i Y _ 3 ) + H ( M | Y _ i Y _ 4 ) \\le H ( Y _ 4 | Y _ i ) + H ( Y _ 3 | Y _ i Y _ 4 ) \\\\ = & H ( Y _ 3 Y _ 4 | Y _ i ) \\stackrel { ( a ) } { \\le } H ( Y _ i Y _ { i ' } | Y _ i ) \\\\ = & H ( Y _ { i ' } | Y _ i ) \\le \\log d , \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} \\tilde L _ s ( y ( \\gamma ( s ) , t ) ) = s ^ { - \\nu } L _ s ( y ( \\gamma ( s ) , t ) ) = 0 \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} [ p , f _ \\lambda ] = \\sum _ \\alpha \\gamma _ \\alpha a _ \\alpha \\lambda ^ \\alpha = p ( \\lambda ) . \\end{align*}"}
-{"id": "9987.png", "formula": "\\begin{align*} Y _ j ^ T x ( t _ * ^ { + } ) = Y _ i ^ T x ( t _ * ^ - ) . \\end{align*}"}
-{"id": "7186.png", "formula": "\\begin{align*} z + z ^ { - 1 } = \\frac { 2 \\ell + 1 } { \\ell + 1 } \\in \\Q . \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} P _ { l } T ^ { \\ , j } \\ , e _ { k } = \\zeta _ { j , k } \\ , \\cdot \\ , \\Big ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Big ) \\ ; e _ { b _ { l } + n } . \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{gather*} J _ \\infty : = \\left \\langle _ { a _ 1 , a _ 2 , \\dots , a _ s } Y _ \\infty [ s ] ^ { ( k - \\ell ( a ) ) } \\ \\mid \\ 0 \\leq \\ell ( a ) \\leq k , \\ m + 1 \\leq k \\right \\rangle \\\\ J _ m : = \\left \\langle _ { a _ 1 , a _ 2 , \\dots , a _ s } \\mathsf { Y } [ s ] ^ { ( k - \\ell ( a ) ) } \\ \\mid \\ 0 \\leq \\ell ( a ) \\leq k , \\ m + 1 \\leq k \\right \\rangle . \\end{gather*}"}
-{"id": "335.png", "formula": "\\begin{align*} \\varepsilon ( E _ { a } \\triangleright ( \\mathsf { M } _ { m } ^ { n } ) _ { j } ^ { i } ) & = - \\pi ( E _ { a } K _ { a } ^ { - 1 } ) _ { m } ^ { i } \\pi ( K _ { a } ) _ { j } ^ { n } + \\pi ( 1 ) _ { m } ^ { i } \\pi ( E _ { a } ) _ { j } ^ { n } \\\\ & = - \\pi ( E _ { a } ) _ { m } ^ { i } \\pi ( 1 ) _ { j } ^ { n } + \\pi ( 1 ) _ { m } ^ { i } \\pi ( E _ { a } ) _ { j } ^ { n } , \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} \\begin{vmatrix} \\alpha _ { k _ 1 } & \\sigma ( \\alpha _ { k _ 1 } ) & \\dots & \\sigma ^ { t - 1 } ( \\alpha _ { k _ 1 } ) \\\\ \\alpha _ { k _ 2 } & \\sigma ( \\alpha _ { k _ 2 } ) & \\dots & \\sigma ^ { t - 1 } ( \\alpha _ { k _ 2 } ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\alpha _ { k _ { t } } & \\sigma ( \\alpha _ { k _ { t } } ) & \\dots & \\sigma ^ { t - 1 } ( \\alpha _ { k _ { t } } ) \\end{vmatrix} \\neq 0 . \\end{align*}"}
-{"id": "9357.png", "formula": "\\begin{align*} \\int _ { I _ k ^ n } | g ( x ) | ^ p \\d x = 2 ^ { - n } . \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} \\angle L : = \\max _ { i = 1 , \\ldots , k } \\frac { \\norm { a _ i } _ \\infty } { \\norm { a _ 0 } _ \\infty } \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{align*} \\ ! \\ ! \\ ! f ( k , x ) \\ ! = e ^ { i k x } \\left [ \\ ! I _ n \\ ! + \\ ! \\frac { Q ( x ) } { i k } \\ ! + \\ ! o \\left ( \\ ! \\frac { 1 } { k } \\ ! \\right ) \\ ! \\right ] \\ ! , \\ ; Q ( x ) \\ ! : = \\frac { \\int _ x ^ \\infty V ( t ) d t } { 2 } , \\ ; | k | \\to \\infty \\ ; \\ ; \\overline { \\mathbb { C } } ^ + . \\end{align*}"}
-{"id": "9816.png", "formula": "\\begin{align*} \\mathfrak { M } ^ J = \\bigoplus _ { w \\in W } \\mathcal { M } ' ( w \\mathbf { z } ) \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} \\bar \\partial _ X Z & = [ X ^ { 0 , 1 } , Z ] ^ { 1 , 0 } , X \\in T M , Z \\in C ^ \\infty ( M , T ^ { 1 , 0 } ) . \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} { _ a I _ x ^ { 1 - \\alpha } } u ' ( x _ { n + 1 } ) & = \\sum _ { j = 1 } ^ { n + 1 } \\int _ { I _ j } \\omega _ { 1 - \\alpha } ( x _ { n + 1 } - s ) u ' ( s ) \\ , d s \\\\ & = \\int _ { I _ 1 } \\omega _ { 1 - \\alpha } ( x _ { n + 1 } - s ) u ' ( s ) \\ , d s + \\sum _ { j = 1 } ^ { n } \\int _ { I _ j } \\omega _ { 1 - \\alpha } ( x _ { n } - q ) u ' ( q + h ) \\ , d q . \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} I _ { \\mathbf { x } } ^ { ( \\omega , \\vartheta ) } : = - 2 \\mathrm { I m } \\left ( \\langle \\mathfrak { e } _ { x ^ { ( 1 ) } } , \\Delta _ { \\omega , \\vartheta } \\mathfrak { e } _ { x ^ { ( 2 ) } } \\rangle a _ { x ^ { ( 1 ) } } ^ { \\ast } a _ { x ^ { ( 2 ) } } \\right ) \\ , \\mathbf { x } : = ( x ^ { ( 1 ) } , x ^ { ( 2 ) } ) \\in \\mathfrak { L } ^ { 2 } \\ . \\end{align*}"}
-{"id": "6994.png", "formula": "\\begin{align*} \\left ( \\sum _ { j _ { 3 } = 1 } ^ { \\infty } \\left ( \\sum _ { j _ { 1 } , j _ { 2 } = 1 } ^ { \\infty } \\left \\Vert T \\left ( x _ { \\mathbf { j } } \\right ) \\right \\Vert ^ { r } \\right ) ^ { \\frac { 1 } { r } \\cdot r } \\right ) ^ { \\frac { 1 } { r } } = \\left \\Vert \\left ( T ( x _ { \\mathbf { j } } ) \\right ) _ { \\mathbf { j } \\in \\mathbb { N } ^ { 3 } } \\right \\Vert _ { r } \\leq C \\prod _ { k = 1 } ^ { 3 } \\Vert x ^ { k } \\Vert _ { w , p _ { k } } , \\end{align*}"}
-{"id": "7148.png", "formula": "\\begin{align*} P ( x ) = x _ 2 ^ 2 x _ 3 ^ 2 + 1 8 x _ 1 x _ 2 x _ 3 x _ 4 - 4 x _ 2 ^ 3 x _ 4 - 4 x _ 1 x _ 3 ^ 3 - 2 7 x _ 1 ^ 2 x _ 4 ^ 2 . \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{align*} \\varphi _ 2 | _ { W _ x } ( \\sum z _ i \\mathbf e _ i ' ) = \\sum _ { i = 1 } ^ n \\Big ( \\frac { ( \\lambda _ { n + 3 } - \\lambda _ i ) ^ 2 x _ i ^ 2 } { ( \\lambda _ { n + 3 } - \\lambda _ { n + 2 } ) ^ 2 x _ { n + 2 } ^ 2 } + \\frac { ( \\lambda _ { n + 2 } - \\lambda _ i ) ^ 2 x _ i ^ 2 } { ( \\lambda _ { n + 3 } - \\lambda _ { n + 2 } ) ^ 2 x _ { n + 3 } ^ 2 } + 1 \\Big ) z _ i ^ 2 \\qquad \\qquad \\end{align*}"}
-{"id": "6590.png", "formula": "\\begin{align*} \\frac { g _ 1 ^ + } { g _ 0 ^ + } \\ ; = \\ ; c _ \\nu \\ , \\beta + d _ \\nu \\end{align*}"}
-{"id": "9504.png", "formula": "\\begin{align*} W _ A ( x , \\alpha ) = \\left \\{ \\sum _ { i = 1 } ^ n a _ i ^ * x \\alpha ( a _ i ) : n \\geq 1 , \\ a _ i \\in A , \\ \\sum _ { i = 1 } ^ n a _ i ^ * a _ i = 1 \\right \\} \\subseteq A . \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} \\delta \\vartheta _ j = - \\tfrac 1 4 \\partial _ z ^ 2 \\vartheta _ j \\end{align*}"}
-{"id": "4644.png", "formula": "\\begin{align*} \\chi ( \\Sigma _ 0 ( G , d _ t , \\sigma _ 0 ) ) = \\chi ( G ) , \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\rho ^ { n } \\sin ( n \\alpha + \\beta ) & = ( \\sin ( \\beta ) - \\rho \\sin ( \\beta - \\alpha ) ) / ( 1 - 2 \\rho \\cos ( \\alpha ) + \\rho ^ { 2 } ) , \\\\ \\sum _ { n \\geq 0 } \\rho ^ { n } \\cos ( n \\alpha + \\beta ) & = ( \\cos ( \\beta ) - \\rho \\cos ( \\beta - \\alpha ) / ( 1 - 2 \\rho \\cos ( \\alpha ) + \\rho ^ { 2 } ) . \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{align*} \\left [ \\frac { 1 } { \\left ( \\Lambda _ { N , N + 1 } h _ { N , s } \\right ) ( x ) } \\Lambda _ { N , N + 1 } \\circ h _ { N , s } ( y ) \\right ] ( x , d y ) = \\Lambda _ { N } ^ { N + 1 } ( x , d y ) . \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} \\partial _ t a _ n & = a _ n ( b _ { n + 1 } - b _ n ) , \\\\ \\partial _ t b _ n & = 2 ( a _ n ^ 2 - a _ { n - 1 } ^ 2 ) . \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{align*} \\phi _ { t } ^ { \\ast } \\left \\{ f , g \\right \\} = \\left \\{ \\phi _ { t } ^ { \\ast } f , \\phi _ { t } ^ { \\ast } g \\right \\} , \\ \\ \\left ( \\forall \\right ) \\ f , g \\in C ^ { \\infty } \\left ( P , \\mathbb { R } \\right ) . \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} \\dot x = \\sum _ { i = 1 } ^ { \\ell } f _ i ( x ) u _ i ( t ) , x \\in D \\subseteq \\mathbb R ^ n , \\ ; f _ i : D \\to \\mathbb R ^ n . \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} u _ 2 ( z ) & = \\sum _ { n = 0 } ^ a \\left ( \\int _ { D ^ 2 } \\frac { \\mu ( \\zeta ) f ( \\zeta ) } { \\zeta ^ { n + 1 } } d \\zeta \\wedge d \\bar { \\zeta } \\right ) z ^ n - \\sum _ { n = 0 } ^ { a } \\left ( \\int _ { D ( 0 , 2 | z | ) } \\frac { \\mu ( \\zeta ) f ( \\zeta ) } { \\zeta ^ { n + 1 } } d \\zeta \\wedge d \\bar { \\zeta } \\right ) z ^ n \\\\ & + \\sum _ { n = a + 1 } ^ { \\infty } \\left ( \\int _ { D \\setminus D ( 0 , 2 | z | ) } \\frac { \\mu ( \\zeta ) f ( \\zeta ) } { \\zeta ^ { n + 1 } } d \\zeta \\wedge d \\bar { \\zeta } \\right ) z ^ n . \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} F _ 2 : = \\int _ \\Omega n _ 1 ^ 2 + \\int _ \\Omega ( n _ 2 - 1 ) ^ 2 \\end{align*}"}
-{"id": "1311.png", "formula": "\\begin{align*} K _ { 0 } \\left [ \\lambda _ { 1 } r ^ { 2 } - 2 \\lambda _ { 3 } g ^ { 2 } + \\lambda _ { 2 } \\left ( \\frac { g ^ { 2 } } { r } \\right ) ^ { 2 } \\right ] - \\frac { g \\dot { r } } { r } = \\frac { - \\lambda _ { 2 } } { p ^ { 2 } } . \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} C _ { \\nu , r } \\boxtimes C _ { \\lambda , r ' } \\cong \\bigoplus _ { r '' = 1 } ^ { 2 k + 2 } { N _ { r , r ' } } ^ { r '' } C _ { \\lambda + \\nu , r '' } . \\end{align*}"}
-{"id": "4918.png", "formula": "\\begin{align*} \\sigma \\left ( A \\right ) = \\sigma \\left ( S \\right ) \\cup \\sigma \\left ( C \\right ) \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{align*} \\pi ( X K _ { \\lambda } ) _ { j } ^ { i } c _ { k } ^ { j } \\pi ( K _ { \\lambda ^ { \\prime } } Y ) _ { \\ell } ^ { k } = \\pi ( X ) _ { j } ^ { i } \\pi ( K _ { \\lambda } K _ { \\lambda ^ { \\prime } } ) _ { j } ^ { j } c _ { k } ^ { j } \\pi ( Y ) _ { \\ell } ^ { k } = \\pi ( X K _ { \\lambda } K _ { \\lambda ^ { \\prime } } ) _ { j } ^ { i } c _ { k } ^ { j } \\pi ( Y ) _ { \\ell } ^ { k } . \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} B _ 1 ( x _ 1 , x _ 2 , x _ 3 ; y _ 1 , y _ 2 , y _ 3 ) & = x _ 2 y _ 1 - x _ 1 y _ 2 , \\\\ B _ 2 ( x _ 1 , x _ 2 , x _ 3 ; y _ 1 , y _ 2 , y _ 3 ) & = x _ 3 y _ 1 + a _ 1 x _ 2 y _ 1 + a _ 3 x _ 1 y _ 1 - x _ 2 y _ 3 , \\\\ B _ 3 ( x _ 1 , x _ 2 , x _ 3 ; y _ 1 , y _ 2 , y _ 3 ) & = x _ 2 y _ 2 + a _ 2 x _ 2 y _ 1 + a _ 4 x _ 1 y _ 1 - x _ 3 y _ 3 . \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} S ( A | X ) _ { \\hat { \\sigma } _ { A X } } & = \\int _ { \\mathbb { R } ^ { 2 m } } S \\left ( \\hat { \\sigma } _ { A | X = \\mathbf { x } } \\right ) \\mathrm { d } p _ X ( \\mathbf { x } ) = S ( \\hat { \\rho } _ A ) \\ ; , \\\\ S ( B | X ) _ { ( \\Phi \\otimes \\mathbb { I } _ X ) ( \\hat { \\sigma } _ { A X } ) } & = \\int _ { \\mathbb { R } ^ { 2 m } } S \\left ( \\Phi ( \\hat { \\sigma } _ { A | X = \\mathbf { x } } ) \\right ) \\mathrm { d } p _ X ( \\mathbf { x } ) = S ( \\Phi ( \\hat { \\rho } _ A ) ) \\ ; , \\end{align*}"}
-{"id": "9766.png", "formula": "\\begin{align*} \\tilde { u } = \\phi ^ { - 1 } \\cdot u \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty G ( y ) ( y ^ 2 & - 1 ) \\varphi ( y ) d y \\\\ & \\geq G ( y _ 0 ) \\int _ 0 ^ 1 ( y ^ 2 - 1 ) \\varphi ( y ) d y + G ( 1 ) \\int _ 1 ^ \\infty ( y ^ 2 - 1 ) \\varphi ( y ) d y \\\\ & = - G ( y _ 0 ) \\int _ 1 ^ \\infty ( y ^ 2 - 1 ) \\varphi ( y ) d y + G ( 1 ) \\int _ 1 ^ \\infty ( y ^ 2 - 1 ) \\varphi ( y ) d y \\\\ & = ( G ( 1 ) - G ( y _ 0 ) ) \\int _ 1 ^ \\infty ( y ^ 2 - 1 ) \\varphi ( y ) d y > 0 \\ ; . \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{align*} \\alpha ^ { \\pm } _ i ( t ) = \\alpha _ i ^ { \\pm } ( 0 ) e ^ { - c t } \\ , \\gamma _ 1 ( t ) = \\gamma _ 1 ( 0 ) e ^ { - c t } \\ , \\ \\delta ( t ) = \\frac { 1 } { 2 c } \\left ( 1 - e ^ { - 2 c t } \\right ) + \\delta ( 0 ) e ^ { - 2 c t } , \\end{align*}"}
-{"id": "2824.png", "formula": "\\begin{align*} q = f _ { \\omega _ { - n } } ^ { - 1 } \\circ \\dots \\circ f _ { \\omega _ { - 1 } } ^ { - 1 } ( p ) . \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{align*} \\eta _ { a } ( C ( \\mathsf { P } ) ) = \\sum _ { i , j } c _ { j } ^ { i } \\pi \\left ( K _ { 2 \\rho } ^ { - 1 } K _ { a } \\frac { K _ { a } - K _ { a } ^ { - 1 } } { q _ { a } - q _ { a } ^ { - 1 } } \\right ) _ { i } ^ { j } . \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{align*} \\Xi ^ { \\lambda } ( K _ { a } \\otimes F _ { a } \\otimes 1 \\otimes E _ { a } ) = \\sum _ { i , j , n , o , p } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) c _ { n } ^ { j } \\pi ( K _ { a } F _ { a } ) _ { o } ^ { n } c _ { p } ^ { o } \\pi ( E _ { a } K _ { \\lambda } ) _ { i } ^ { p } . \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} D ^ { ( \\alpha ) } g ( t ) = \\frac { 1 } { \\Gamma ( m - \\alpha ) } \\int _ 0 ^ t \\frac { g ^ { ( m ) } ( \\tau ) } { ( t - \\tau ) ^ { \\alpha - m + 1 } } d \\tau , \\ m - 1 < \\alpha \\leq m . \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 , [ y _ 1 , y _ 2 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 5 } { \\alpha _ 3 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\alpha _ 3 \\beta _ 4 } { \\alpha _ 1 \\gamma _ 4 } y _ 5 , [ y _ 2 , y _ 3 ] = \\frac { \\gamma _ 2 } { \\gamma _ 4 } y _ 5 , [ y _ 3 , y _ 2 ] = y _ 5 . \\end{align*}"}
-{"id": "9301.png", "formula": "\\begin{align*} ( \\mathbf { x } \\circ \\mathbf { y } ) _ k = \\sum ^ m _ { i , j = 1 } p _ { i j , k } x _ i y _ j \\end{align*}"}
-{"id": "5664.png", "formula": "\\begin{align*} \\Omega _ { i j } = \\Omega _ { j i } = \\left ( \\begin{array} { c c } \\frac { 4 \\varepsilon + 0 . 9 } { 3 . 6 \\varepsilon - 0 . 0 9 } & \\frac { \\varepsilon } { 1 . 8 \\varepsilon - 0 . 0 4 5 } \\\\ \\frac { \\varepsilon } { 1 . 8 \\varepsilon - 0 . 0 4 5 } & \\frac { \\varepsilon } { 3 . 6 \\varepsilon - 0 . 0 9 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n L _ k { } ^ 4 = F _ { 2 n + 1 } ( L _ { 2 n + 1 } - 4 ( - 1 ) ^ { n - 1 } ) + 6 n - 5 \\ , . \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} I _ 0 = \\ < P \\ > , I _ { k + 1 } = \\ < I _ k , \\xi ^ k P \\ > . \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{align*} f ( q ) = ( \\cos \\theta + \\lambda ( q _ 2 - x _ 2 ) ) ^ 2 + ( \\sin \\theta - \\lambda ( q _ 1 - x _ 1 ) ) ^ 2 . \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} T ^ { \\ , n } \\Bigl ( \\sum _ { s = 1 } ^ { j _ { m } + j } z _ { s } \\Bigr ) - x _ { l } & = T ^ { \\ , k d _ { j _ { m } + j } + i } \\ , \\Bigl ( \\sum _ { s = 1 } ^ { j _ { m } + j } z _ { s } \\Bigr ) - x _ { l } = T ^ { \\ , i } \\ , \\Bigl ( \\sum _ { s = 1 } ^ { j _ { m } + j } z _ { s } \\Bigr ) - x _ { l } \\qquad \\hbox { b y ( i ) } \\\\ & \\smash [ b ] { = T ^ { \\ , i } \\ , \\Bigl ( \\sum _ { s = 1 } ^ { j _ { m } + j - 1 } z _ { s } \\Bigr ) - x _ { l } + T ^ { \\ , i } z _ { j _ { m } + j } . } \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{align*} \\kappa ^ { s } \\Omega ( s p ) - \\kappa \\Omega ( p ) = & ( \\kappa ^ { s - 1 } - 1 ) \\cdot \\kappa \\Omega ( s p ) + \\kappa \\cdot \\big ( \\Omega ( s p ) - \\Omega ( p ) \\big ) \\\\ \\equiv & \\frac { s - 1 } { 2 } \\cdot \\big ( ( \\kappa ^ { 2 } - 1 ) \\cdot \\kappa \\Omega ( 3 p ) + \\kappa \\cdot ( \\Omega ( 3 p ) - \\Omega ( p ) ) \\big ) \\\\ = & \\frac { s - 1 } { 2 } \\cdot \\big ( \\kappa ^ { 3 } \\Omega ( 3 p ) - \\kappa \\Omega ( p ) \\big ) \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , L _ { - 2 } ] = 0 \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} = \\mathcal { Z } \\big ( T _ { 1 } , \\ldots , T _ { d } , U , S \\big ) \\mid _ { \\begin{array} [ c ] { l } { \\small T _ { 1 } = \\ldots = T _ { d } = T } \\\\ { \\small U = q } \\end{array} . } \\end{align*}"}
-{"id": "7346.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi ( t ) & = f ^ { 1 2 7 } + f ^ { 3 4 7 } + f ^ { 5 6 7 } + f ^ { 1 3 5 } - f ^ { 1 4 6 } - f ^ { 2 3 6 } - f ^ { 2 4 5 } . \\end{aligned} \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} \\begin{aligned} x _ 1 & = \\frac { 1 } { 2 } \\log t + y _ 1 ( \\log t ) , x _ 2 = - \\frac { 1 } { 2 } \\log t + y _ 2 ( \\log t ) , \\\\ x _ 3 & = \\frac { 1 } { 2 } \\log t + y _ 3 ( \\log t ) , x _ 4 = - \\frac { 1 } { 2 } \\log t + y _ 4 ( \\log t ) \\end{aligned} \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi ( t ) & = f ^ { 1 2 7 } + f ^ { 3 4 7 } + f ^ { 5 6 7 } + f ^ { 1 3 5 } - f ^ { 1 4 6 } - f ^ { 2 3 6 } - f ^ { 2 4 5 } . \\end{aligned} \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} \\bar R _ 2 ' = 2 \\pi \\int _ 0 ^ { + \\infty } \\frac { \\log \\lambda } { \\lambda ^ d } \\mu ( d \\lambda ) . \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\eta _ 1 } ~ f _ { n _ 1 , n _ 2 } ( \\eta _ 1 , \\eta _ 2 ) = \\sqrt { n _ 1 ( k + 1 - ( n _ 1 + n _ 2 ) ) } f _ { n _ 1 - 1 , n _ 2 } ( \\eta _ 1 , \\eta _ 2 ) , \\end{align*}"}
-{"id": "4420.png", "formula": "\\begin{align*} X = ( C _ 1 \\times C _ 2 \\times C _ 3 ) / G . \\end{align*}"}
-{"id": "9500.png", "formula": "\\begin{align*} T _ q X _ z = \\left \\{ p ( \\lambda ) \\vert ~ \\deg p ( \\lambda ) = d - 1 p ( z ) = 0 \\right \\} . \\end{align*}"}
-{"id": "223.png", "formula": "\\begin{align*} p ( x ) = \\sum _ { m = 0 } ^ M p _ m ( x ) + r _ M ( x ) \\end{align*}"}
-{"id": "5111.png", "formula": "\\begin{align*} g \\left ( z \\right ) = \\sum _ { k \\ge 1 } \\frac { - 1 } { \\left ( k + 1 \\right ) ! } z ^ { k } = - \\frac { 1 } { z } \\left ( e ^ { z } - 1 - z \\right ) = 1 - \\frac { e ^ { z } - 1 } { z } \\end{align*}"}
-{"id": "7928.png", "formula": "\\begin{align*} g ( x , t ) : = - h ^ \\prime u _ s + ( h - h ^ 2 ) ( u _ s + u _ \\infty ) \\cdot \\nabla u _ s , \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} \\rho ( p _ 1 . . . p _ j ) = \\sum _ { i = 0 } ^ j X ( i ) e _ i ( \\pi _ { p _ 1 } , . . . , \\pi _ { p _ j } ) . \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{align*} M _ { 1 1 } = - \\frac { \\Gamma K } { ( K + c ^ { \\star } ) ^ 2 } , \\ , \\ , M _ { 1 2 } = \\lambda , \\ , \\ , M _ { 2 1 } = T ' ( c ^ { \\star } ) , \\ , \\ , M _ { 2 2 } = - 1 . \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{align*} \\gamma _ 1 ( t ) = \\gamma _ 1 ( 0 ) \\ , \\forall t \\ge 0 . \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} ( w _ \\nu ( \\omega _ k ) - w _ \\lambda ( \\omega _ k ) ) c _ { \\lambda , \\mu } ^ { \\nu , d } & = \\sum _ \\eta 2 ^ { A ( \\lambda , \\eta ) } c _ { \\eta , \\mu } ^ { \\nu , d } - \\sum _ \\xi 2 ^ { A ( \\xi , \\nu ) } c _ { \\lambda , \\mu } ^ { \\xi , d } \\\\ & + ( c _ { \\lambda ^ * , \\mu } ^ { \\nu , d - 1 } - c _ { \\lambda , \\mu } ^ { \\nu ^ + , d - 1 } ) + ( c _ { \\lambda ^ { * * } , \\mu } ^ { \\nu , d - 1 } - c _ { \\lambda , \\mu } ^ { \\nu ^ { + + } , d - 1 } ) \\ / , \\end{align*}"}
-{"id": "6527.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sum _ { x \\in \\mathcal { S } , g \\in \\rm { D A } } \\left ( h ( x . g ) - h ( x ) \\right ) ^ 2 \\mu ( g ) = \\sum _ { n = 0 } ^ \\infty \\left ( \\frac { a } { q ^ n } \\right ) ^ 2 \\frac { q ^ n } { 4 } + \\left ( \\frac { a } { 2 q ^ { n + 1 } } \\right ) ^ 2 \\frac { q ^ n } { 2 q } \\frac { q ( q - 1 ) } { 2 } < \\infty . \\end{align*}"}
-{"id": "9292.png", "formula": "\\begin{align*} \\delta _ { g ^ { - 1 } } \\rho _ g \\circ \\rho _ { g ^ { - 1 } } \\delta _ g = \\Phi \\phi _ { g ^ { - 1 } , g } ^ { - 1 } \\circ \\delta _ 1 \\circ \\phi _ { g ^ { - 1 } , g } \\Phi . \\end{align*}"}
-{"id": "3870.png", "formula": "\\begin{align*} \\sup _ { x \\in [ - 1 + h - a , - 1 + h + a ] } \\sqrt { ( x - 1 ) ^ 2 + y ( x ) ^ 2 } + \\sqrt { ( x + 1 ) ^ 2 + y ( x ) ^ 2 } = : \\widetilde M \\stackrel { ! } { \\leq } \\rho _ 1 + 1 / \\rho _ 1 , \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{align*} z _ i \\ , z _ k ( z _ i + z _ k ) = H \\end{align*}"}
-{"id": "3634.png", "formula": "\\begin{align*} p ( x ) = ( 1 / 6 ) \\left [ | x | ^ { - 1 / 2 } \\ 1 _ { ( 0 , 1 ] } ( | x | ) + x ^ { - 2 } \\ 1 _ { ] 1 ; + \\infty [ } ( | x | ) \\right ] , \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} D ^ \\alpha g ( t ) = \\frac { 1 } { \\Gamma ( m - \\alpha ) } \\frac { d ^ m } { d t ^ m } \\int _ 0 ^ t \\frac { g ( \\tau ) } { ( t - \\tau ) ^ { \\alpha - m + 1 } } d \\tau . \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} y _ n \\rightarrow y ^ * & \\mbox { w e a k l y i n } L ^ 2 ( 0 , T ; H _ 0 ^ 1 ( \\Omega ) ) \\cap H ^ 1 ( 0 , T ; L ^ 2 ( \\Omega ) ) , \\\\ & \\mbox { s t r o n g l y i n } C ( [ \\delta , T ] ; L ^ 2 ( \\Omega ) ) \\mbox { a s } n \\rightarrow \\infty , \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} \\rho ( \\mathbf { x } ) = \\rho ( \\mathbf { x } + \\mathbf { y } ) , \\mathbf { y } \\in \\Lambda , \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} \\sum _ \\eta ( - 1 ) ^ { \\ell ( \\eta ) } \\frac { \\ell ( \\eta ) ! } { \\prod _ { i = 1 } ^ { s } m _ i ( \\eta ) ! } x _ { | \\eta | } \\prod _ { i = 1 } ^ { s } u _ i ^ { m _ i ( \\eta ) } \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{align*} s _ { t , v } = t + \\frac { 1 } { 2 } \\log \\left ( 1 - \\frac { \\alpha _ { \\delta v } } { e ^ { 2 t } } \\right ) . \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} D _ { \\Gamma } ^ { ( s , * ) } = \\bigsqcup _ { j _ 1 , \\dots , j _ n = 1 } ^ { | \\mathcal { E } | } \\left ( \\bigsqcup _ { Q \\in S _ n } { D } _ { j _ 1 \\dots j _ n } ^ { ( s , Q ) } \\right ) , \\end{align*}"}
-{"id": "9132.png", "formula": "\\begin{gather*} y ^ 2 = x ^ 3 - 2 7 s ^ 4 \\big ( \\lambda ^ 4 + 1 4 4 \\big ) x - 5 4 s \\big ( { - } s ^ 5 \\lambda ^ 6 + 8 6 4 s ^ { 1 0 } + 6 4 8 s ^ 5 \\lambda ^ 2 + 8 6 4 \\big ) . \\end{gather*}"}
-{"id": "7465.png", "formula": "\\begin{align*} y _ \\beta = a _ - ( \\beta + \\delta ) \\wedge b _ + ( \\beta ) \\end{align*}"}
-{"id": "1509.png", "formula": "\\begin{align*} A ( x ) = ( x - a _ 0 ) ( x - a _ 1 ) \\cdots ( x - a _ p ) \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} E = & \\{ w \\in C ( ( \\bar t , \\infty ) ; L ^ 6 _ \\sigma ( \\Omega ) \\cap L ^ \\infty ( \\Omega ) ) ; \\ , \\nabla w \\in C ( ( \\bar t , \\infty ) ; L ^ 3 ( \\Omega ) ) , \\\\ & \\qquad \\qquad \\| w \\| _ E : = \\sup _ { t \\in ( \\bar t , \\infty ) } \\phi _ w ( t ) < \\infty , \\ , \\lim _ { t \\to \\bar t + 0 } \\phi _ w ( t ) = 0 \\} \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} \\kappa _ { \\beta } = \\sqrt { \\kappa ^ 2 + \\tau ^ 2 } \\mbox { a n d } \\tau _ { \\beta } = \\frac { \\kappa ^ 2 } { \\kappa ^ 2 + \\tau ^ 2 } \\left ( \\frac { \\tau } { \\kappa } \\right ) ' . \\end{align*}"}
-{"id": "9180.png", "formula": "\\begin{align*} r = r _ a : = \\{ a , - a , a ^ { - 1 } , - a ^ { - 1 } \\} \\subset \\P ^ 1 ( k ) \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{align*} \\beta _ k ^ k ( i ) = \\sum _ { m | Q _ i } \\ell _ k ( m ) \\max _ { b \\bmod m } \\mu _ i ( ( b \\bmod m ) ) . \\end{align*}"}
-{"id": "4817.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) ^ { p ^ 2 - 1 } = I + \\begin{pmatrix} - \\frac { a } { w } & 0 \\\\ 0 & - \\frac { d } { z } \\end{pmatrix} p \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} t ^ { \\ast } = \\tau _ { 0 } ^ { \\frac { 1 } { \\alpha _ { 0 } } } , \\ ; \\ ; \\ ; \\ ; t ^ { \\ast } = \\tau , \\end{align*}"}
-{"id": "10023.png", "formula": "\\begin{align*} \\sum \\limits _ { i = k + 1 } ^ \\infty v _ i \\leq 2 v _ { k + 1 } \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} K ( x , t ) = t ^ { - 3 / 2 } K ( \\textstyle \\frac { x } { \\sqrt t } , 1 ) , \\qquad \\hbox { a n d } K ( \\cdot , 1 ) \\in \\bigcap _ { 1 < p \\le \\infty } L ^ p ( \\R ^ 3 ) . \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} T \\left ( x , t \\right ) = T _ { 0 } \\left ( x \\right ) \\ast _ { x } P \\left ( x , t \\right ) \\ ; \\ ; \\ ; \\ ; q \\left ( x , t \\right ) = T _ { 0 } \\left ( x \\right ) \\ast _ { x } Q \\left ( x , t \\right ) , \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} z : = v _ { N } ^ { - 1 } \\ \\Bigl ( \\prod _ { j = b _ N + 1 } ^ { b _ { N + 1 } - 1 } w _ j \\Bigr ) ^ { - 1 } \\ , \\Bigl ( \\prod _ { j = b _ n + 1 } ^ { k } w _ j \\Bigr ) ^ { - 1 } \\ e _ { b _ N } \\end{align*}"}
-{"id": "3963.png", "formula": "\\begin{align*} [ z ^ n ] G ( z ) = & ( - 1 ) ^ n \\sum _ { k = 0 } ^ { p - 1 } \\binom { k } { n - k } ( - 4 ) ^ k \\cdot \\frac { d } { d x } \\bigg ( \\frac { ( - \\frac 1 2 ( a + x ) ) _ k } { ( 1 ) _ k } \\cdot \\frac { ( \\frac 1 2 + \\frac 1 2 ( a + x ) ) _ k } { k ! } \\bigg ) \\bigg | _ { x = 0 } \\\\ = & ( - 1 ) ^ n \\sum _ { \\frac 1 2 n \\leq k \\leq p - 1 } \\binom { k } { n - k } ( - 4 ) ^ k \\cdot \\frac { ( \\frac 1 2 + \\frac 1 2 a ) _ k } { k ! } \\cdot \\frac { d } { d x } \\bigg ( \\frac { ( - \\frac 1 2 ( a + x ) ) _ k } { ( 1 ) _ k } \\bigg ) \\bigg | _ { x = 0 } . \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} \\mathcal { Q } _ { r , T } ( x ) : = B _ { r } ( x ) \\times ( 0 , T ] , \\quad \\mathcal { Q } _ { r , T } = \\mathcal { Q } _ { r , T } ( 0 ) \\quad \\quad \\mathcal { Q } _ { T } : = \\R ^ { d } \\times ( 0 , T ] . \\end{align*}"}
-{"id": "1157.png", "formula": "\\begin{align*} { x _ { k , j } ^ { u _ 0 } } / { B ^ { u _ 0 } _ { k , j } } = - i \\int _ { 0 } ^ T u ( s ) e ^ { - i ( \\lambda _ j ^ { u _ 0 } - \\lambda _ k ^ { u _ 0 } ) s } d s , \\ \\ \\ \\ \\ \\ \\forall j , k \\in \\N ^ * , \\ k \\leq N \\\\ \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} \\left | \\frac { \\ell - k \\pi / \\omega } { \\sin \\ell \\omega } \\right | = \\frac 1 \\omega \\ , \\left | \\frac { \\ell \\omega - k \\pi } { \\sin ( \\ell \\omega - k \\pi ) } \\right | \\le \\frac \\pi { 2 \\omega } \\ , . \\end{align*}"}
-{"id": "191.png", "formula": "\\begin{align*} \\| ( f d \\sigma ) ^ \\vee \\| _ { L ^ r ( \\mathbb F _ q ^ d , d { \\bf m } ) } & \\le \\sum _ { j = 1 } ^ N a _ j \\ , \\| ( F _ j d \\sigma ) ^ \\vee \\| _ { L ^ r ( \\mathbb F _ q ^ d , d { \\bf m } ) } \\\\ & \\lesssim \\sum _ { j = 1 } ^ N a _ j \\ , \\| F _ j \\| _ { L ^ p ( S _ t , d \\sigma ) } \\sim \\int _ { 0 } ^ \\infty s ^ { \\frac { 1 } { p } - 1 } f ^ * ( s ) ~ d s = \\| f \\| _ { L ^ { p , 1 } ( S _ t , d \\sigma ) } . \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} ( f , g ) _ k = ( - 1 ) ^ k \\left ( f _ k g _ 0 + f _ { k - 1 } g _ 1 + \\ldots + f _ 0 g _ k \\right ) ; \\quad ( f , g ) _ k = 0 , \\ ; k < 0 . \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} p _ { k , l } ( c , ( v _ j ) _ { j \\geq 1 } ) \\ ; & = \\alpha c ^ 2 \\ ; + \\ ; \\sum _ { j = 1 } ^ N \\frac { k _ j \\lambda _ j } { 1 + \\sqrt { 1 + \\lambda _ j / c ^ 2 } } \\ ; - \\frac { \\lambda _ l } { 1 + \\sqrt { 1 + \\lambda _ l / c ^ 2 } } , \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} c _ { \\delta , \\delta ' } = M ^ { \\delta , \\delta ' } ( X _ 1 ^ \\delta ) = \\frac { 2 ^ { \\delta / 2 } \\Gamma ( ( \\delta + \\delta ' ) / 2 ) } { \\Gamma ( \\delta ' / 2 ) } . \\end{align*}"}
-{"id": "6925.png", "formula": "\\begin{align*} \\| t \\| _ 2 = o ( \\varepsilon ) . \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} D _ y ( n ) : = \\max \\big \\{ d : \\big \\} . \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} K _ { \\lambda , \\mu } ( t ) = \\sum _ { \\mathrm { s o r t } ( \\mathrm { f l a t } ( a ) ) = \\lambda ^ { \\prime } } K _ { a , b } ( t ) . \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} G [ s ] ( w ) = \\frac { G [ s - 1 ] ( w ) } { 1 + u _ { s } w ^ { s } G [ s - 1 ] ( w ) } . \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} \\begin{pmatrix} z \\\\ [ . 1 c m ] \\bar { z } \\end{pmatrix} \\in \\mathcal { R } \\begin{pmatrix} N \\\\ [ . 1 c m ] \\overline { M } \\end{pmatrix} . \\end{align*}"}
-{"id": "7852.png", "formula": "\\begin{align*} \\mathbb { P } ( { \\cal E } _ 1 = \\{ 1 , 2 , \\ldots , r \\} ) \\geq T _ r p _ d ^ { r - 1 } ( 1 - p _ u ) ^ { r ( n - r ) + { r \\choose 2 } - r + 1 } \\end{align*}"}
-{"id": "8886.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 6 & 3 4 5 6 \\\\ 6 4 0 & 6 4 8 0 \\end{pmatrix} \\quad \\begin{pmatrix} 1 6 & 6 4 0 \\\\ 3 4 5 6 & 6 4 8 0 \\end{pmatrix} \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{gather*} u ( x ) = \\left \\langle u , P _ p ( \\cdot , x ) \\right \\rangle _ { \\widehat { S } _ p } = \\frac { 1 } { p } \\sum _ { j = 0 } ^ { p - 1 } \\int _ S u ( e ^ { \\frac { j \\pi i } { p } } \\zeta ) \\overline { P _ p ( e ^ { \\frac { j \\pi i } { p } } \\zeta , x ) } \\ , d \\sigma ( \\zeta ) . \\end{gather*}"}
-{"id": "4800.png", "formula": "\\begin{align*} g ^ { ( m ) } ( x ) = f ^ { ( r m ) } ( r x ) . \\end{align*}"}
-{"id": "1450.png", "formula": "\\begin{align*} I _ 1 = I _ { 1 1 } + I _ { 1 2 } + I _ { 1 3 } + I _ { 1 4 } + I _ { 1 5 } , \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} \\alpha ( \\left [ x , y \\right ] ) = \\left [ \\alpha \\left ( x \\right ) , \\alpha \\left ( y \\right ) \\right ] \\ ; \\ ; \\ ; \\ ; \\beta ( \\left [ x , y \\right ] ) = \\left [ \\beta \\left ( x \\right ) , \\beta \\left ( y \\right ) \\right ] , \\ ; \\ ; \\ ; \\forall \\ ; x , y \\in L , \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{align*} h \\in Q _ s \\subset ( 1 + J ^ s ) / ( 1 + J ^ { s + 1 } ) = J ^ s / J ^ { s + 1 } . \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} \\beta _ \\chi ( z ^ 2 ) = V ^ + ( z ) H ^ { \\beta } ( z ^ 2 ) V ^ - ( z ) z ^ { 2 h _ 0 } , \\gamma _ \\chi ( z ^ 2 ) = V ^ + ( z ) ^ { - 1 } H ^ { \\gamma } ( z ^ 2 ) V ^ - ( z ) ^ { - 1 } z ^ { - 2 h _ 0 } . \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} \\widetilde { E } s ^ { \\tau } = \\sum _ { k = 0 } ^ { \\infty } \\frac { s ^ k } { d ^ k } ( 1 - \\frac { 1 } { d } ) < \\infty . \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{align*} ( \\alpha ' ) ^ 2 ( \\mathbf u ) = \\sum ( a \\alpha _ i + b ) ^ 2 v _ i w _ i ' = a ^ 2 \\alpha ^ 2 ( \\mathbf v ) + 2 a b \\alpha ( \\mathbf v ) + b ^ 2 \\mathbf v . \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} \\varphi _ * f _ * \\mu _ X ( ( - \\infty , t ] ) & = \\tilde G _ * F _ * f _ * \\mu _ X ( ( - \\infty , t ] ) \\\\ & = F _ * f _ * \\mu _ X ( \\tilde G ^ { - 1 } ( ( - \\infty , t ] ) ) \\\\ & = F _ * f _ * \\mu _ X ( \\tilde G ^ { - 1 } ( ( - \\infty , t ] ) \\setminus \\{ 0 \\} ) \\\\ & = F _ * f _ * \\mu _ X ( ( 0 , G ( t ) ] ) \\\\ & = F _ * f _ * \\mu _ X ( ( - \\infty , G ( t ) ] ) \\\\ & = G ( t ) \\\\ & = g _ * \\mu _ X ( ( - \\infty , t ] ) . \\end{align*}"}
-{"id": "4905.png", "formula": "\\begin{align*} { \\mathbf { \\hat { h } } _ { s d } ^ { ( b ) } } & = { \\left ( \\textbf { R } _ { d } ^ b \\right ) ^ { - 1 / 2 } \\textbf { H } _ { s d } ^ { } \\textbf { w } _ { 0 s } } , & & \\quad \\ ; \\\\ { \\mathbf { \\hat { h } } _ { s d } ^ { ( m ) } } & = { \\left ( \\textbf { R } _ { d } ^ m \\right ) ^ { - 1 / 2 } \\textbf { H } _ { s d } ^ { } \\textbf { w } _ { 0 s } } . & & \\end{align*}"}
-{"id": "3013.png", "formula": "\\begin{align*} \\beta _ { l } : = 4 \\ , \\gamma _ k \\quad \\textrm { f o r e v e r y } \\ l \\in J _ k \\textrm { a n d e v e r y } k \\ge 1 . \\end{align*}"}
-{"id": "3037.png", "formula": "\\begin{align*} R = H ^ 2 - \\abs { A } ^ 2 \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{gather*} t H _ { \\mathrm { S u z } } ^ { 2 + \\frac { 3 } { 2 } } \\left ( { \\theta ^ 0 _ 2 - \\theta ^ 0 _ 1 + \\theta ^ 1 + 1 , \\ , \\theta ^ 0 _ 2 + \\theta ^ 1 + 1 \\atop \\theta ^ 0 _ 2 - \\theta ^ 0 _ 1 } ; t ; { q _ 1 , p _ 1 \\atop q _ 2 , p _ 2 } \\right ) = t H _ { \\mathrm { I I I } ( D _ 7 ) } \\big ( \\theta ^ 0 _ 2 - \\theta ^ 0 _ 1 + \\theta ^ 1 + 1 ; t ; q _ 1 , p _ 1 \\big ) \\\\ \\qquad { } + t H _ { \\mathrm { I I I } ( D _ 7 ) } \\big ( \\theta ^ 0 _ 2 + \\theta ^ 1 + 1 ; t ; q _ 2 , p _ 2 \\big ) + p _ 2 q _ 1 ( p _ 1 ( q _ 1 + q _ 2 ) + \\theta ^ 0 _ 2 - \\theta ^ 0 _ 1 ) - q _ 1 . \\end{gather*}"}
-{"id": "2345.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } v ( t ) \\in \\partial \\phi ( x ( t ) ) \\\\ \\lambda \\dot x ( t ) + \\dot v ( t ) + v ( t ) + \\nabla \\psi ( x ( t ) ) = 0 \\\\ x ( 0 ) = x _ 0 , v ( 0 ) = v _ 0 \\in \\partial \\phi ( x _ 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "2874.png", "formula": "\\begin{align*} P \\cdot Q : = \\sum \\limits _ { p = - ( d + e ) } ^ \\infty \\left ( \\sum \\limits _ { \\substack { m + l = p \\\\ m \\ge - d \\\\ l \\ge - e } } P _ m \\cdot Q _ l \\right ) . \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} E \\prod _ { i = \\tau + 1 } ^ { n - 1 } [ \\frac { \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } { 1 + \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } ] = E \\prod _ { i = \\tau + 1 } ^ { n - 1 } [ \\frac { \\lambda \\rho ( \\widehat { S } _ i ) \\rho ( \\widehat { S } _ { i + 1 } ) } { 1 + \\lambda \\rho ( \\widehat { S } _ i ) \\rho ( \\widehat { S } _ { i + 1 } ) } ] , \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} \\hat { \\rho } _ { C M } : = ( \\mathcal { B } _ \\eta \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A B M } ) \\ ; . \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty \\ ! \\widehat { h } ( k ) \\widehat { h } ( k ) ^ \\dag d k \\ ! + \\ ! \\int _ { - \\infty } ^ \\infty \\ ! \\widehat { h } ( - k ) & [ S ( k ) \\ ! - U _ 0 ] \\widehat { h } ( k ) ^ \\dag d k \\\\ & + 2 \\pi \\sum _ { j = 1 } ^ N [ \\widehat { h } ( - i k _ j ) C _ j ] [ \\widehat { h } ( - i k _ j ) C _ j ] ^ \\dag \\ ! = 0 . \\end{align*}"}
-{"id": "4355.png", "formula": "\\begin{align*} \\delta \\leq 2 \\cdot 3 ^ { 1 / n } \\sin ( \\pi / ( 2 k ) ) + 3 ^ { 1 / n } - 1 < \\pi 3 ^ { 1 / n } / k + 3 ^ { 1 / n } - 1 = O ( 1 / k ) . \\end{align*}"}
-{"id": "4126.png", "formula": "\\begin{align*} 0 < R ' ( t ) ( \\beta - \\alpha ) = R ( \\beta ) - R ( \\alpha ) = \\frac { \\alpha ^ { 2 k + 1 } } { n - \\alpha ^ 2 } < 2 n ^ { ( 2 k + 1 ) ( 1 - k ) - 1 } . \\end{align*}"}
-{"id": "3911.png", "formula": "\\begin{align*} | \\Phi _ 0 ( i t ) | \\ge C _ 1 \\sqrt { | t | } e ^ { ( \\sigma _ 1 + \\sigma _ 2 ) | { \\rm I m } \\sqrt { i t } | } = C _ 1 \\sqrt { | t | } e ^ { 2 b | { \\rm I m } \\sqrt { i t } | } . \\end{align*}"}
-{"id": "4316.png", "formula": "\\begin{align*} f ( \\sigma _ 1 \\sigma _ 2 ) = f ( \\sigma _ 1 ) + \\ , ^ { \\sigma _ 1 } f ( \\sigma _ 2 ) = f ( \\sigma _ 1 ) + \\ , ^ { \\sigma _ 1 } g ( \\sigma _ 2 ) = f ( \\sigma _ 1 ) . \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} I ( A : Z | M ) _ { \\hat { \\sigma } _ { A M Z } ( t ) } = \\Delta _ { A | M } ( \\hat { \\rho } _ { A M } ) \\left ( \\frac { \\lambda ^ 2 \\ , t } { \\eta } \\right ) \\ ; . \\end{align*}"}
-{"id": "7871.png", "formula": "\\begin{align*} 0 \\leq R _ { 2 , n } ( C , \\epsilon ) = \\sum _ { r = \\epsilon n + 1 } ^ { \\infty } \\frac { T _ r C ^ { r - 1 } e ^ { - C r } } { ( r - 1 ) ! } e ^ { - \\delta _ 2 r } \\leq \\sum _ { r = \\epsilon n + 1 } ^ { \\infty } \\frac { T _ r C ^ { r - 1 } e ^ { - C r } } { ( r - 1 ) ! } = R _ { \\epsilon n } ( C ) . \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{gather*} H ^ { \\frac 5 2 + \\frac 3 2 } _ { \\mathrm { G a r } , t _ 1 } \\left ( { t _ 1 \\atop t _ 2 } ; { q _ 1 , p _ 1 \\atop q _ 2 , p _ 2 } \\right ) = H _ { \\mathrm { I I } } \\left ( { 0 } ; t _ 1 ; q _ 1 , p _ 1 \\right ) - 2 p _ 2 q _ 2 q _ 1 - q _ 2 - \\frac { t _ 2 } { q _ 2 } , \\\\ t _ 2 H ^ { \\frac 5 2 + \\frac 3 2 } _ { \\mathrm { G a r } , t _ 2 } \\left ( { t _ 1 \\atop t _ 2 } ; { q _ 1 , p _ 1 \\atop q _ 2 , p _ 2 } \\right ) = { p _ 2 } ^ 2 { q _ 2 } ^ 2 - p _ 1 q _ 2 + \\frac { t _ 2 } { q _ 2 } \\big ( p _ 1 - { q _ 1 } ^ 2 - t _ 1 \\big ) . \\end{gather*}"}
-{"id": "4308.png", "formula": "\\begin{align*} K _ 3 : = K _ 2 ( \\ell ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } \\frac { ( - \\phi ^ 3 _ + ) ^ k } { k ^ 3 } \\binom { 2 k } { k } + \\sum _ { k = 1 } ^ { p - 1 } \\frac { ( - \\phi ^ 3 _ - ) ^ k } { k ^ 3 } \\binom { 2 k } { k } - \\sum _ { k = 1 } ^ { p - 1 } \\frac { ( - 1 ) ^ k } { k ^ 3 } \\binom { 2 k } { k } . \\end{align*}"}
-{"id": "9466.png", "formula": "\\begin{align*} \\frac { { \\rm d } } { { \\rm d } t } \\int \\frac { \\big | \\widetilde { U } \\big | ^ 2 } { 2 h } + \\int \\frac { W ^ { \\rm T } Q ( w ^ * ) W } { 2 h } + \\int \\widetilde { W } \\cdot \\mathrm { L } ( w ^ * ) = \\int \\Big [ \\phi \\cdot \\big ( b - b ^ * \\big ) + \\psi \\cdot \\big ( d - d ^ * \\big ) + \\varphi \\cdot \\big ( v - v ^ * \\big ) \\Big ] \\end{align*}"}
-{"id": "1724.png", "formula": "\\begin{align*} a _ { m a x } & = \\frac { 1 } { m } \\left ( f _ { p l a n a r } - K _ d v _ { w , m a x } ^ 2 \\right ) \\\\ S _ { t r a j } & = ( c _ 1 ( d _ 1 H + d _ 4 ) ( 1 - H ^ 2 ) ) ^ 2 + ( d _ 1 ( 1 - H ^ 2 ) ) ^ 2 \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{gather*} \\frac { d u _ 1 } { d t } = g ( u _ n ) - k _ 1 u _ 1 , \\\\ \\frac { d u _ i } { d t } = u _ { i - 1 } - k _ i u _ i , i = 2 , \\ldots , n ( k _ i > 0 ) , \\end{gather*}"}
-{"id": "5593.png", "formula": "\\begin{align*} U _ { n } ( \\cos ( \\alpha ) ) \\allowbreak = \\allowbreak \\sin ( ( n + 1 ) \\alpha ) / \\sin ( \\alpha ) T _ { n } ( \\cos ( \\alpha ) ) \\allowbreak = \\allowbreak \\cos ( n \\alpha ) \\end{align*}"}
-{"id": "9256.png", "formula": "\\begin{align*} \\Big ( \\sum _ { ( k _ 1 , \\cdots , k _ d ) \\in \\Z ^ d } | m ( k _ 1 , \\cdots , k _ d ) \\widehat { f } ( k _ 1 , \\cdots , k _ d ) | ^ 2 \\Big ) ^ { 1 / 2 } & \\geq ( \\sum _ { 1 \\leq k _ 1 , \\cdots , k _ d \\leq 2 ^ N } \\frac { 1 } { k _ 1 \\cdots k _ d } \\Big ) ^ { 1 / 2 } \\\\ & = \\prod _ { i = 1 } ^ d \\Big ( \\sum _ { 1 \\leq k _ i \\leq 2 ^ N } \\frac { 1 } { k _ i } \\Big ) ^ { 1 / 2 } \\\\ & \\sim N ^ { d / 2 } , \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} Q _ 1 & = \\{ ( 0 , 0 ) , ( 0 , 2 ) , ( 2 , 0 ) , ( 2 , 2 ) \\} \\\\ Q _ 2 & = \\{ ( 0 , 0 ) , ( 1 , 2 ) , ( 2 , 1 ) \\} \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} \\max \\{ f ( - 1 + h - a ) , f ( - 1 + h + a ) \\} = f ( - 1 + h - a ) . \\end{align*}"}
-{"id": "7789.png", "formula": "\\begin{align*} \\frac d { d t } \\iint _ { \\Sigma _ - } ( \\partial _ t ^ j \\partial _ y ^ n v _ k ) ^ 2 \\ , d x d y + \\int _ 0 ^ L ( \\partial _ t ^ j \\partial _ y ^ n v _ { k x } ) ^ 2 \\big | _ { x = 0 } \\ , d y = 2 \\iint _ { \\Sigma _ - } \\partial _ t ^ j \\partial _ y ^ n f \\partial _ t ^ j \\partial _ y ^ n v _ k \\ , d x d y , \\end{align*}"}
-{"id": "9563.png", "formula": "\\begin{align*} S _ { \\omega } = \\left \\{ \\left ( j , k \\right ) \\in \\mathbb { Z \\times \\mathbb { Z } } | \\left | k - 2 \\omega \\right | \\leq 2 \\delta \\left | j - 2 \\omega \\right | \\leq 2 \\delta \\right \\} , \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{align*} b = \\frac { 1 } { 4 \\gamma + 8 } . \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} \\left \\{ \\mathbb { J } _ { \\mathrm { d } } ^ { ( \\omega , \\eta \\mathbf { \\bar { A } } _ { l } ) } \\left ( t \\right ) \\right \\} _ { k } : = \\left \\vert \\Lambda _ { l } \\right \\vert ^ { - 1 } \\underset { x \\in \\Lambda _ { l } } { \\sum } \\rho _ { t } ^ { ( \\beta , \\omega , \\vartheta , \\lambda , \\eta \\mathbf { \\bar { A } } _ { l } ) } \\left ( \\mathrm { I } _ { ( x + e _ { k } , x ) } ^ { ( \\omega , \\vartheta , \\eta \\mathbf { \\bar { A } } _ { l } ) } \\right ) \\ . \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} \\partial _ t g = L \\cdot g + N ( g ) , \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{align*} X \\triangleright ( \\mathsf { N } ^ n _ m ) ^ i _ j = ( X _ { ( 1 ) } \\triangleright u ^ i _ m ) ( X _ { ( 2 ) } \\triangleright u _ n ^ { j * } ) = ( X _ { ( 1 ) } \\triangleright u ^ i _ m ) ( X _ { ( 2 ) } \\triangleright S ( u ^ n _ j ) ) , \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} [ w _ j , w _ k ] \\bigg | _ { - 1 } ^ 1 = \\int _ { - 1 } ^ 1 \\ell ^ 3 [ w _ j ] w _ k d x - \\int _ { - 1 } ^ 1 w _ j \\ell ^ 3 [ w _ k ] d x \\end{align*}"}
-{"id": "3945.png", "formula": "\\begin{align*} { } _ 3 F _ 2 \\bigg [ \\begin{matrix} \\alpha & \\alpha & \\beta \\\\ & 1 & \\alpha - \\beta + 1 \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { \\Gamma ( 1 + \\frac 1 2 \\alpha ) \\Gamma ( 1 + \\alpha - \\beta ) \\Gamma ( 1 - \\frac 1 2 \\alpha - \\beta ) } { \\Gamma ( \\alpha + 1 ) \\Gamma ( 1 - \\frac 1 2 \\alpha ) \\Gamma ( 1 - \\beta ) \\Gamma ( 1 + \\frac 1 2 \\alpha - \\beta ) } . \\end{align*}"}
-{"id": "7735.png", "formula": "\\begin{align*} \\alpha _ k ^ { - 1 } = \\int _ { - 1 } ^ 1 \\left ( 1 - s ^ 2 \\right ) ^ { k / 2 } d s \\geq \\int _ { - 1 / \\sqrt { k } } ^ { 1 / \\sqrt { k } } \\left ( 1 - s ^ 2 \\right ) ^ { k / 2 } d s \\geq \\frac { c } { \\sqrt { k } } . \\end{align*}"}
-{"id": "7733.png", "formula": "\\begin{align*} \\int _ { S ^ { n - 1 } } \\rho _ D ( x ) g ( x ) d x = \\int _ { S ^ { n - 1 } } R g ( x ) d \\nu _ D ( x ) , \\forall g \\in C ( S ^ { n - 1 } ) . \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} u ( \\mathsf { x } , t ) = \\int _ { \\mathbb { R } ^ { d } } \\mathsf { d y } g ( \\mathsf { x } , t , \\mathsf { y } ) \\varphi ( \\mathsf { y ) } \\end{align*}"}
-{"id": "5262.png", "formula": "\\begin{align*} \\int _ { \\Omega \\cap B _ R ( 0 ) } ( u - c ) \\times \\omega \\ , d x & = \\int _ { \\partial B _ R ( 0 ) \\cap \\Omega } B N \\ , d S . \\end{align*}"}
-{"id": "9892.png", "formula": "\\begin{align*} \\mathrm { S K } \\colon \\mathcal { B } = \\bigoplus _ { h \\in \\mathbb { N } } \\mathcal { B } ^ h \\rightarrow \\bigcup _ { \\lambda } \\big [ \\mathcal { S U T } ( \\lambda ) \\times \\mathcal { S T } ( \\lambda ) \\big ] . \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} \\Pr [ & \\abs { S } \\le 1 ] \\\\ & = \\frac { 1 } { 2 ^ { K - 1 } } \\Pr [ \\abs { S } \\le 1 \\mid T = K ] + \\Bigl ( 1 - \\frac { 1 } { 2 ^ { K - 1 } } \\Bigr ) \\Pr [ \\abs { S } \\le 1 \\mid T \\ge K + 2 ] \\\\ & \\ge \\frac { 1 } { 2 ^ { K - 1 } } F \\left ( \\frac { ( K + 1 ) ^ 2 - K } { ( 2 K + 1 ) ^ 2 } \\right ) + \\Bigl ( 1 - \\frac { 1 } { 2 ^ { K - 1 } } \\Bigr ) F \\left ( \\frac { ( K + 1 ) ^ 2 - ( K + 2 ) } { ( 2 K + 1 ) ^ 2 } \\right ) . \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} B _ { j } u = \\sum \\limits _ { \\left \\vert \\beta \\right \\vert \\leq l _ { j } } \\ b _ { j \\beta } \\left ( y \\right ) D _ { y } ^ { \\beta } u \\left ( x , y \\right ) = 0 x \\in R ^ { n } \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{gather*} \\frac { q x _ { h i } x _ { i j } - q ^ { - 1 } x _ { i j } x _ { h i } } { q - q ^ { - 1 } } = 1 . \\end{gather*}"}
-{"id": "5668.png", "formula": "\\begin{align*} & P _ g \\Big \\{ \\big [ v ( t ) , w ( t ) \\big ] ; ( v _ 0 , w _ 0 ) \\Big \\} \\\\ & = \\frac { 1 } { Z } \\exp \\left [ - \\frac { \\bigg ( D _ m \\Big \\{ \\big [ v ( t ) , w ( t ) \\big ] ; ( v _ 0 , w _ 0 ) \\Big \\} \\bigg ) ^ 2 } { 2 \\sigma ^ 2 } \\right ] . \\end{align*}"}
-{"id": "9508.png", "formula": "\\begin{align*} \\Phi | _ { ( J + A u ) } = 0 . \\end{align*}"}
-{"id": "9580.png", "formula": "\\begin{align*} \\theta _ n ^ { \\mathrm { c a u s a l } } & = \\left ( \\sum _ { i = 1 } ^ n E \\left [ W _ { n , i } ^ { X X } \\right ] \\right ) ^ { - 1 } \\sum _ { i = 1 } ^ n E \\left [ W _ { n , i } ^ { X X } \\varphi _ { n , i } \\right ] , \\\\ \\shortintertext { a n d , w i t h p r o b a b i l i t y a p p r o a c h i n g o n e , } \\theta _ n ^ { \\mathrm { c a u s a l } , \\mathrm { s a m p l e } } & = \\left ( \\sum _ { i = 1 } ^ n R _ { n , i } E \\left [ W _ { n , i } ^ { X X } \\right ] \\right ) ^ { - 1 } \\sum _ { i = 1 } ^ n R _ { n , i } E \\left [ W _ { n , i } ^ { X X } \\varphi _ { n , i } \\right ] , \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{align*} X _ k = S _ { \\Delta t } ^ k x + \\Delta t \\sum _ { \\ell = 0 } ^ { k - 1 } S _ { \\Delta t } ^ { k - \\ell } G ( X _ \\ell ) + \\int _ { 0 } ^ { t _ k } S _ { \\Delta t } ^ { k - \\ell _ s } \\sigma ( X _ { \\ell _ s } ) d W ( s ) . \\end{align*}"}
-{"id": "9695.png", "formula": "\\begin{align*} \\Pi _ { S ^ { \\ast } } \\ = \\ \\left \\{ \\sum _ { i = 1 } ^ { d + 1 } \\lambda _ i \\bar { v } _ i : 0 < \\lambda _ i \\leq 1 i \\in I , 0 \\leq \\lambda _ i < 1 i \\not \\in I \\right \\} \\ , . \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} \\texttt { l e f t } _ i ( \\mathsf { s } ) = s _ 1 , & & \\texttt { r i g h t } _ i ( \\mathsf { s } ) = s _ 2 . \\end{align*}"}
-{"id": "5834.png", "formula": "\\begin{align*} 2 5 \\sum _ { k = 1 } ^ n { F _ { m k } { } ^ 4 } = \\frac { { ( F _ { 2 m n } L _ { 2 m n + 2 m } + 4 ( - 1 ) ^ { m n - 1 } L _ m F _ { m n } L _ { m n + m } ) } } { { F _ { 2 m } } } + 6 n \\ , , \\end{align*}"}
-{"id": "9014.png", "formula": "\\begin{align*} ( K + K ^ { - \\epsilon } ) ^ { - \\frac { \\epsilon } { p } + \\sum _ { i = 1 } ^ m \\frac { \\epsilon } { p _ i } } \\le C , \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} u = F _ 1 ( V ( x ) ) \\sqrt \\omega u _ 1 ( \\omega t ) + 2 \\alpha F _ 2 ( V ( x ) ) \\sqrt \\omega u _ 2 ( \\omega t ) , \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{align*} ( x , y ) = \\mbox { \\ , t h e c o e f f i c i e n t o f $ \\omega _ i $ i n $ x y $ } , \\end{align*}"}
-{"id": "3427.png", "formula": "\\begin{align*} \\xi = h \\exp \\left ( - \\dfrac { \\tan ^ { - 1 } \\left ( \\frac { - 1 + 2 \\tau } { \\sqrt { - 1 - 4 K } } \\right ) } { \\sqrt { - 1 - 4 K } } \\frac { 1 } { \\sqrt { \\tau ^ 2 - \\tau - K } } \\right ) . \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} \\gamma ( \\alpha ) = \\lim _ { t \\uparrow \\alpha } \\gamma ( t ) = \\sup \\{ \\gamma ( t ) : t \\in [ 0 , \\alpha ) \\} \\leq g ( x ) . \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} g ^ { ( 2 ) } ( x ) = \\frac { x ^ { 2 } } { 1 - p x + p q x ^ { 3 } - q ^ { 2 } x ^ { 4 } } . \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} \\bigg \\| \\bigg ( \\ln \\frac { u _ { \\alpha \\beta } ^ 2 } { \\underbar { n } } \\bigg ) _ - \\bigg \\| _ { L ^ { 2 k } ( \\Omega ) } \\leq \\frac { 2 \\big ( B ^ 2 + 2 | D | _ 0 \\big ) } { \\theta _ L } , k = 1 , 2 , 3 , \\cdots . \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} h _ s ( x , n ) = \\sum _ { t = 0 } ^ { \\infty } \\frac { x ^ { n t + s - 1 } } { ( n t + s - 1 ) ! } , \\enskip s = 1 , . . . , n . \\end{align*}"}
-{"id": "1036.png", "formula": "\\begin{align*} \\widehat { J ^ \\alpha _ N f } ( \\xi ) : = ( \\eta ( N ^ { - 1 } \\xi ) ) ^ \\alpha \\hat { f } ( \\xi ) . \\end{align*}"}
-{"id": "5495.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n q _ j x _ j \\frac { \\partial f _ \\Delta } { \\partial x _ j } ( x ) = d f _ \\Delta ( x ) . \\end{align*}"}
-{"id": "9763.png", "formula": "\\begin{align*} F _ i | _ { S ^ 1 \\times 0 } = \\gamma _ i \\end{align*}"}
-{"id": "7703.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { k - 1 } \\binom { k - 1 } { t } ( \\frac { - p } { q } ) ^ { t } U _ { m k + t } ^ { ( k ) } & = ( - q ) ^ { 1 - k } U _ { m } ( p U _ { m + 1 } - q U _ { m } ) ^ { k - 1 } \\\\ & = ( - q ) ^ { 1 - k } U _ { m } U _ { m + 2 } ^ { k - 1 } \\\\ & = ( - q ) ^ { 1 - k } U _ { m } U _ { ( m + 2 ) ( k - 1 ) } ^ { ( k - 1 ) } , \\end{align*}"}
-{"id": "2618.png", "formula": "\\begin{align*} { \\Bbb E } _ { k } : = \\left ( N ( L - \\lambda _ { k } I ) \\oplus { \\Bbb F } _ { k } \\right ) ^ \\perp = \\bigoplus _ { 1 \\leq j \\leq k - 1 } N ( L - \\lambda _ { j } I ) \\ , , \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{align*} V ( z ) = \\sum _ { \\ell = - k } ^ { k } c _ \\ell z ^ \\ell \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} g ' ( x ) = a + \\frac { s _ 1 } { ( x - t _ 1 ) ^ 2 } + \\dots + \\frac { s _ { n - 1 } } { ( x - t _ { n - 1 } ) ^ 2 } = \\frac { a \\ , P ( x ) } { S ( x ) ^ 2 } . \\end{align*}"}
-{"id": "8266.png", "formula": "\\begin{align*} \\mathcal { A } ^ { P R } = - \\mathcal { A } ^ P \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{align*} ( \\alpha , g ) * ( \\beta , h ) = \\begin{cases} ( \\alpha , g h ) & \\alpha = \\beta , \\\\ ( \\beta , h ) & \\alpha > \\beta , \\\\ ( \\alpha , g ) & \\alpha < \\beta , \\\\ ( \\alpha \\beta , 1 ) & \\alpha \\bot \\beta . \\end{cases} \\end{align*}"}
-{"id": "4075.png", "formula": "\\begin{align*} { Z _ d } \\left ( { \\tilde q , \\delta } \\right ) \\buildrel \\Delta \\over = - 4 \\left ( { 1 - 2 \\tilde q - \\delta } \\right ) H \\left ( { \\tilde q , { \\rm { } } 1 - \\tilde q - \\delta } \\right ) + { ( 1 - 2 \\tilde q ) ^ 2 } \\left ( { 2 \\ln \\frac { { 1 - \\tilde q - \\delta } } { { \\tilde q + \\delta } } + \\ln \\frac { { 1 - \\tilde q } } { { \\tilde q } } } \\right ) . \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} \\widetilde M = 2 r _ x \\left ( \\frac { 1 } { r _ x } + h \\left ( 1 - \\frac { 1 } { r _ x } \\right ) \\right ) , r _ x : = \\frac { 1 } { 2 } \\left ( \\rho _ 0 + \\frac { 1 } { \\rho _ 0 } \\right ) . \\end{align*}"}
-{"id": "160.png", "formula": "\\begin{align*} V ( \\theta ) = \\tfrac { 1 } { 2 } \\cos \\theta + \\tfrac { 1 } { 2 } \\cos 2 \\theta - \\tfrac { 1 } { 6 } \\cos 3 \\theta \\end{align*}"}
-{"id": "9988.png", "formula": "\\begin{align*} E _ { \\sigma ( t ) } \\dot x = A _ { \\sigma ( t ) } x \\ , , \\sigma ( t ) \\in \\{ 1 , 2 \\} , \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} \\operatorname { r c e f } ( S ^ \\mu _ 0 ) = \\left ( \\begin{array} { c } I _ \\mu \\\\ \\hline a _ 0 \\cdots a _ { \\mu - 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} ( \\lambda ^ { 3 } + \\lambda + 1 ) z + ( \\epsilon \\lambda ^ { 2 } + b \\lambda + \\epsilon ) = 0 . \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} F _ * ^ e ( j u ^ { k - 1 } v ^ { q - 1 } ( f + u v ) ) = \\bigoplus _ { i \\in \\Delta _ e } f _ { ( i , j ) } F _ * ^ e ( i u ^ { k - 1 } v ^ { q - 1 } ) \\oplus v F _ * ^ e ( j u ^ { k } ) . \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} x _ { n } = \\sum _ { p = 1 } ^ { n } \\left ( - 1 \\right ) ^ { p + 1 } \\binom { n + 1 } { p + 1 } \\sum _ { \\underset { k _ { i } \\ge 0 } { k _ { 1 } + \\dots + k _ { p } = n } } x _ { k _ { 1 } } \\dots x _ { k _ { p } } . \\end{align*}"}
-{"id": "9471.png", "formula": "\\begin{align*} A _ { 1 , 1 } U + A _ { 1 , 2 } + T = 0 . \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} \\hat { \\varphi } ( \\omega , r ) { ( g , h _ 1 , h _ 2 ) } & = \\upsilon ^ { - 1 } \\left ( \\varphi ( \\sigma ^ { ( g , h _ 1 , h _ 2 ) ^ { - 1 } } ( \\omega , r ) ) | _ { 0 , \\dots , \\kappa - 1 } \\right ) \\\\ & = \\upsilon ^ { - 1 } \\left ( T ^ { g ^ { - 1 } } ( \\rho ( y ) ) | _ { 0 , \\dots , \\kappa - 1 } \\right ) \\\\ & = \\upsilon ^ { - 1 } \\left ( ( \\rho ( \\sigma ^ { g ^ { - 1 } } ( y ) ) ) | _ { 0 , \\dots , \\kappa - 1 } \\right ) \\\\ & = \\upsilon ^ { - 1 } \\left ( \\upsilon ( y ( g ) ) \\right ) \\\\ & = y ( g ) \\end{align*}"}
-{"id": "9296.png", "formula": "\\begin{align*} M _ \\phi ( f ) ( x ) = \\begin{cases} e ^ { - x ^ 2 } , & \\mbox { $ x \\in [ - \\frac { \\sqrt { 2 } } 2 , \\frac { \\sqrt { 2 } } 2 ] $ , } \\\\ \\frac 1 { \\sqrt { 2 e } } \\frac 1 x , & \\mbox { $ x \\in ( - \\infty , - \\frac { \\sqrt { 2 } } 2 ) \\bigcup ( \\frac { \\sqrt { 2 } } 2 , + \\infty ) $ . } \\end{cases} \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{align*} \\Omega ^ { j } = \\big \\{ \\bar { \\xi } \\in 2 \\pi \\lambda ^ { j ( 1 - \\sigma ) } \\mathbb { Z } ^ { n - 1 } \\ , : \\ , \\lambda ^ { j } \\leq | \\xi _ { m } | < \\lambda ^ { j + 1 } , \\ , m = 2 , \\ldots , n \\big \\} + Q \\Big ( 0 , \\frac { \\varepsilon _ { 1 } } { \\sqrt { n \\ ! - \\ ! 1 } } \\Big ) \\ , , \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{align*} | p _ i - q _ i | & \\leq \\left | \\frac { p _ i \\sum _ { j = 1 } ^ N W _ j - W _ i } { 1 + \\sum _ { j = 1 } ^ N W _ j } \\right | \\\\ & \\leq \\delta ( N + 1 ) . \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{gather*} Z ^ p _ m ( \\zeta , \\eta ) = \\sum _ { k = 0 } ^ { p - 1 } e ^ { \\frac { - 2 j k \\pi i } { p } } Z _ { m - 2 k } ( \\zeta , \\eta ) \\quad \\textrm { f o r } \\quad \\zeta \\in S . \\end{gather*}"}
-{"id": "6046.png", "formula": "\\begin{align*} u ^ { y } _ 1 = u ^ { x } _ 1 , u ^ { f ( y ) } _ 1 = u ^ { f ( x ) } _ 1 , \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} \\log \\left ( \\abs { f } ( \\theta _ T ( \\lambda ) ) \\right ) = \\max _ { \\chi \\in X ( T ) } \\left ( \\log \\abs { a _ \\chi } + \\log \\abs { \\varpi } \\cdot \\langle \\chi , \\lambda \\rangle \\right ) \\end{align*}"}
-{"id": "9012.png", "formula": "\\begin{align*} \\frac { 1 } { \\phi ( R ) } \\left ( \\frac { 1 } { | B | } \\int \\limits _ { B } \\prod \\limits _ { i = 1 } ^ m | f _ i ( x ) | ^ { p } d x \\right ) ^ { \\frac { 1 } { p } } & \\le \\frac { 1 } { \\phi ( R ) } \\left ( \\frac { 1 } { | B | } \\int \\limits _ { B } \\prod \\limits _ { i = 1 } ^ m | f _ i ( x ) | ^ { p ^ * } d x \\right ) ^ { \\frac { 1 } { p ^ * } } \\\\ & \\le \\prod \\limits _ { i = 1 } ^ m \\frac { 1 } { \\phi _ i ( R ) } \\left ( \\frac { 1 } { | B | } \\int \\limits _ { B } | f _ i ( x ) | ^ { p _ i } d x \\right ) ^ { \\frac { 1 } { p _ i } } . \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } x P _ j = \\alpha _ j P _ { j + 1 } + \\beta _ j P _ { j } + \\gamma _ j P _ { j - 1 } , \\ j \\geq 0 \\\\ P _ { - 1 } = 0 , \\ P _ 0 = 1 \\end{array} \\right . , \\end{align*}"}
-{"id": "9334.png", "formula": "\\begin{align*} d ( X , Y ) = \\log ( 1 + \\| X ^ { - 1 } \\cdot Y - I \\| _ { } ) , \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} \\frac 1 p + \\frac 1 { p ^ * } \\ , = \\ , 1 . \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{align*} \\sigma ( x , t ) - \\ell ^ 2 \\partial _ x ^ 2 \\sigma ( x , t ) = E \\ , \\varepsilon ( x , t ) , \\end{align*}"}
-{"id": "9688.png", "formula": "\\begin{align*} h ^ { r } _ P ( t ) \\ = \\ \\sum _ { i = 0 } ^ { d + r } h ^ { r } _ i ( P ) t ^ i . \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} f ( \\xi _ p ( \\mathbb C ) ) = \\xi _ { f ( p ) } ( \\mathbb C ) , \\end{align*}"}
-{"id": "7522.png", "formula": "\\begin{align*} \\Delta : B \\to B , \\Delta ( b ) = \\left [ \\phi _ 0 ^ * , [ \\phi _ 0 , b ] \\right ] \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} | { \\cal C } | \\leq \\sum _ { i = 0 } ^ { d n } { n \\choose i } \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} R ^ e _ n ( s _ { v _ 0 } ) & \\left ( e w _ e + \\sum _ { i = 1 } ^ { e - 1 } ( e - i ) ( w _ { e - i } + w _ { e + i } ) \\right ) \\\\ & = e R ^ e _ n ( s _ { v _ 0 } ) ( w _ e ) + R ^ e _ n ( s _ { v _ 0 } ) \\left ( \\sum _ { i = 1 } ^ { e - 1 } ( e - i ) w _ { e - i } \\right ) + R ^ e _ n ( s _ { v _ 0 } ) \\left ( \\sum _ { i = 1 } ^ { e - 1 } ( e - i ) w _ { e + i } ) \\right ) \\\\ & = e \\sum _ { i = 0 } ^ e w _ { i } + \\sum _ { i = 1 } ^ { e - 1 } ( e - i ) w _ { i } + \\sum _ { i = 1 } ^ { e - 1 } ( e - i ) w _ { e + i } \\\\ & = p _ b ^ * ( v _ 0 ) + p _ b ^ * ( v _ 1 ) . \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{gather*} V _ { c } ( \\frac { \\displaystyle \\mathbf { x } } { \\displaystyle \\varepsilon } ) = \\left \\{ \\begin{array} { l } 0 , \\ , \\hbox { i n e a c h c u b e } \\\\ 1 , \\ , \\hbox { o t h e r s } \\end{array} \\right . \\end{gather*}"}
-{"id": "7500.png", "formula": "\\begin{align*} b : = h \\left [ h ^ { - 1 } \\phi ^ * h , \\phi \\right ] - g \\left [ g ^ { - 1 } \\phi ^ * g , \\phi \\right ] - ( h - g ) \\rho \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} f \\dot { p } = p \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( u - c ) \\omega \\ , d x = 0 , \\end{align*}"}
-{"id": "8060.png", "formula": "\\begin{align*} ( \\pi ^ + , \\pi ^ - ) \\stackrel { d } { = } ( B ^ + , B ^ - ) . \\end{align*}"}
-{"id": "3283.png", "formula": "\\begin{align*} h _ N ( \\theta ) = \\prod _ { 1 \\le i < j \\le N } ^ { } \\left ( \\tan \\left ( \\frac { \\theta _ j } { 2 } \\right ) - \\tan \\left ( \\frac { \\theta _ i } { 2 } \\right ) \\right ) . \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} \\omega = y ( 1 + \\lambda x ^ { k } ) d x + x ^ { k + 1 } d y , \\end{align*}"}
-{"id": "267.png", "formula": "\\begin{align*} K _ \\lambda \\triangleright ( \\mathsf { M } ^ n _ m ) ^ i _ j \\triangleleft K _ { \\lambda ^ { \\prime } } = q ^ { - ( \\lambda , \\lambda _ i - \\lambda _ j ) } q ^ { - ( \\lambda ^ \\prime , \\lambda _ m - \\lambda _ n ) } ( \\mathsf { M } ^ n _ m ) ^ i _ j . \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{align*} 0 \\ ; = \\ ; \\omega ( f , g ) \\ ; = \\ ; \\langle S ^ * f , g \\rangle - \\langle f , S ^ * g \\rangle \\qquad \\forall g \\in \\mathcal { D } ( S ^ * ) \\end{align*}"}
-{"id": "5420.png", "formula": "\\begin{align*} f _ 1 ( u _ c ) = \\frac { \\alpha _ 1 + \\beta _ 1 \\ , u _ c } { 1 + u _ c / \\tau _ 1 } , f _ 2 ( u _ m ) = \\frac { \\alpha _ 2 \\ , u _ m } { 1 + u _ m / \\tau _ 2 } . \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} E _ 2 ^ { s , t } = H ^ s ( \\mathbb { Z } / p ; H ^ t ( X _ { d , p } ; \\mathbb { F } _ p ) ) . \\end{align*}"}
-{"id": "2353.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } | m _ i | = \\infty . \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\psi _ N ( a \\ , | \\ , x ) { \\rm e } ^ { N \\ , I ( a \\ , | \\ , x ) } \\sqrt { N } = C ( a \\ , | \\ , x ) , \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} { } _ 4 F _ 3 \\bigg [ \\begin{matrix} - n & \\alpha & \\beta & \\gamma \\\\ & 1 & \\delta & \\epsilon \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { ( \\delta - \\alpha ) _ n ( \\epsilon - \\alpha ) _ n } { ( \\delta ) _ n ( \\epsilon ) _ n } \\cdot { } _ 4 F _ 3 \\bigg [ \\begin{matrix} - n & \\alpha & 1 - \\beta & 1 - \\gamma \\\\ & 1 & 1 + \\alpha - n - \\delta & 1 + \\alpha - n - \\epsilon \\end{matrix} \\bigg | \\ , 1 \\bigg ] , \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} p _ n = n + O ( | n | ^ { 1 / 2 } \\log ^ { 1 + \\varepsilon } | n | ) n \\to \\pm \\infty . \\end{align*}"}
-{"id": "2236.png", "formula": "\\begin{align*} J _ 1 ( k ) = J ( k ) T _ 6 = k M _ 1 ^ { - 1 } \\begin{bmatrix} I _ \\mu + o ( 1 ) & o ( 1 ) \\\\ o ( 1 ) & \\frac { 1 } { k } ( I _ { n - \\mu } + o ( 1 ) ) \\end{bmatrix} T _ 1 ^ { - 1 } T _ 6 , k \\to 0 , \\ ; \\ ; k \\in \\overline { \\mathbb { C } } ^ + . \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} B _ { n } = - \\sum _ { k = 1 } ^ { n } \\binom { n } { k } \\frac { B _ { n - k } } { k + 1 } . \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{align*} \\begin{cases} w ( s ) \\sim \\phi _ 1 ( s ) = s ^ { - \\frac { 1 } { \\alpha } } \\\\ w ( s ) \\sim \\phi _ 2 ( s ) = \\begin{cases} s ^ { - \\frac { 1 } { \\alpha } + \\frac { N - 2 } { 2 } } & N \\ge 3 \\\\ s ^ { - \\frac { 1 } { \\alpha } } \\log s & N = 2 . \\end{cases} \\end{cases} \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} \\tilde { W } ( x ) = \\int _ { 0 - } ^ { K - 0 } G ( x - w + T ) d \\tilde { W } ( w ) + \\int _ { K - 0 } ^ { x + T } d \\tilde { W } ( w ) , x \\geq 0 . \\end{align*}"}
-{"id": "139.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } | \\alpha _ j | ^ { 2 M + 2 } < \\infty \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 2 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 3 e _ 4 + \\alpha _ 4 e _ 5 , [ e _ 2 , e _ 1 ] = \\alpha _ 5 e _ 4 + \\alpha _ 6 e _ 5 , [ e _ 2 , e _ 2 ] = \\beta _ 2 e _ 5 , [ e _ 3 , e _ 3 ] = \\gamma _ 6 e _ 5 . \\end{align*}"}
-{"id": "9868.png", "formula": "\\begin{align*} C _ 2 = - \\frac { 1 } { 2 \\pi } \\left ( \\left ( \\frac { 2 \\sigma } { L } \\right ) - \\gamma \\right ) \\Leftrightarrow G ( \\rho ) \\rightarrow - \\frac { 1 } { 2 \\pi } \\left ( \\frac { r } { L } \\right ) \\rho \\rightarrow \\infty \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{align*} A u = \\frac { 1 } { \\pi i } \\int _ 0 ^ \\infty R _ 0 ^ - ( \\lambda ^ 2 ) v S _ 3 D _ 3 S _ 3 v \\big [ R _ 0 ^ + ( \\lambda ^ 2 ) - R _ 0 ^ - ( \\lambda ^ 2 ) \\big ] \\lambda ^ { - 1 } \\chi ( \\lambda ) u \\ , d \\lambda \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{align*} \\Delta & = D _ { m a x } - D = 0 . \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} ( \\ell _ { \\Psi ^ * F } ) ' ( \\alpha ^ 2 y , \\alpha ) = ( \\ell _ F ) ' ( \\alpha k ^ { - 1 } ( y , \\alpha ) ) ( k ^ { - 1 } ) ' ( y , \\alpha ) + \\tilde h ( y , \\alpha ) \\cdot \\langle 2 \\rangle _ F ( \\alpha ) \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} X _ { h } ^ { r } = Y ^ { r } _ { h } \\cap H _ { 0 } ^ { 1 } ( \\Omega ) , \\mathcal { X } _ { h } ^ { r } = X _ { h } ^ { r } \\oplus { \\rm i } X _ { h } ^ { r } , \\mathbf { X } _ { h } = \\big ( Y ^ { 2 } _ { h } \\big ) ^ { 3 } \\cap \\mathbf { H } ^ { 1 } _ { \\rm t } ( \\Omega ) . \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{align*} \\rho _ c ( z ) \\ , z = \\rho ( x _ c ) \\ , x _ c - c \\end{align*}"}
-{"id": "9440.png", "formula": "\\begin{align*} \\Lambda ( \\rho , U ) = \\sup \\left \\{ \\int _ { \\mathbb { T } ^ 3 } a \\rho + A \\cdot U , \\ ; \\ ; \\ ; a + \\frac { 1 } { 2 } | A | ^ 2 \\le 0 \\right \\} \\in [ 0 , + \\infty ] , \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} { s } ^ H _ { g _ { f , J } } = - u ^ { \\frac { 2 } { n } } \\sum _ { i , j = 1 } ^ n H _ { i j , i j } + 2 u ^ { \\frac { 2 } { n } - 1 } \\sum _ { i , j = 1 } ^ n a _ i H _ { i j , j } - 2 u ^ { \\frac { 2 } { n } - 2 } \\sum _ { i , j = 1 } ^ n a _ i a _ j H _ { i j } , \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} P _ i ' ( x ) = & 2 ^ { - i } F ' ( 2 ^ { - i } ( 1 + x ) ) + ( - 1 ) ^ i 2 ^ { - i } F ' ( 2 ^ { - i } ( 1 - x ) ) = \\\\ & 2 ^ { 1 - i } ( F ( 2 ^ { 1 - i } ( 1 + x ) ) - ( - 1 ) ^ { i - 1 } F ( 2 ^ { 1 - i } ( 1 - x ) ) = 2 ^ { 1 - i } P _ { i - 1 } ( x ) \\end{align*}"}
-{"id": "752.png", "formula": "\\begin{align*} 1 - \\left | \\hat { \\mu } ( \\xi ( v ) ) \\right | = O \\left ( \\frac { \\log ( 1 + R ) } { R ^ 2 m ^ 2 } \\right ) + \\sum _ { i = 1 } ^ k \\Big ( 1 - \\left | \\hat { \\mu } ( \\xi ( v _ i ) ) \\right | \\Big ) . \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} \\psi = \\frac { 1 } { r ! s ! } \\psi _ { \\alpha _ 1 \\ldots \\alpha _ r , \\bar \\beta _ 1 \\ldots \\bar \\beta _ s } z ^ { \\alpha _ 1 } \\cdots z ^ { \\alpha _ r } \\wedge \\bar { z } ^ { \\bar { \\beta } _ 1 } \\cdots \\bar { z } ^ { \\bar { \\beta } _ s } , \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} U = \\sum _ { \\mu \\in F v _ n } ( - 1 ) ^ { \\delta ( \\mu ) } s _ \\mu s ^ * _ \\mu . \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} \\begin{aligned} \\chi ( X ) & = \\int _ { X \\times X } \\Delta \\cdot { \\Delta } = \\int c ( N _ \\Delta { X \\times X } ) \\cap [ \\Delta ] = \\int c ( T X ) \\cap [ X ] . \\end{aligned} \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{align*} & P ^ { s , N } _ { H P } ( t ) f \\in C _ 0 ( W ^ N ) , \\ \\ \\forall t > 0 \\ , \\\\ & \\lim _ { t \\to 0 } P ^ { s , N } _ { H P } ( t ) f = f . \\end{align*}"}
-{"id": "2069.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 \\ , \\left ( \\frac 1 2 \\right ) & x = t _ 1 & x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\overbrace { \\begin{matrix} \\sqrt { t _ 2 } & 0 \\\\ - \\sqrt { t _ 2 } & 0 \\end{matrix} } & \\begin{matrix} 0 \\\\ \\theta ^ { t _ 1 } \\end{matrix} & \\overbrace { \\begin{matrix} 1 & \\theta ^ \\infty _ 1 / 2 \\\\ - 1 & \\theta ^ \\infty _ 1 / 2 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "2279.png", "formula": "\\begin{align*} g ( s ) = I + \\frac { s - 1 } { \\alpha ^ 2 c _ 0 s } \\begin{pmatrix} c _ 0 - m & * \\\\ 1 & m - c _ 0 s \\end{pmatrix} , \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{align*} z w z ^ { - 1 } = z w w z w ^ { - 1 } = z ^ 2 w , \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} \\alpha & = \\chi _ \\rho ( a ) ^ 2 - 4 \\\\ \\beta & = \\chi _ \\rho ( [ a , b ] ) - 2 \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{align*} h ( w _ 2 , w _ 3 ; s ) & = h ( \\lambda _ 2 , \\lambda _ 3 ; s ) + h _ y ( \\lambda _ 2 , \\lambda _ 3 ; s ) ( w _ 2 - \\lambda _ 2 ) + R ( s , w ' , \\lambda ' ) , \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{align*} M = [ \\mu _ { i , j } ] _ { i , j \\geq 0 } , \\mu _ { i , j } = \\frac { 1 } { | | P _ i | | ^ 2 } < P _ i , x P _ j > , \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} ( 1 - \\varrho _ a ) _ a = ( 1 + \\beta + \\gamma - \\delta - \\epsilon - a ) _ a \\equiv 0 \\pmod { p } . \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} W ( x ) = \\int _ { 0 ^ - } ^ { K - 0 } \\sum _ { n \\geq 0 } G _ n ^ x ( w ) d W ( w ) , \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} \\partial _ { t } u = i \\left [ \\Delta u + L u + V \\left ( x , t \\right ) u \\right ] , x \\in R ^ { n } , y \\in G , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{align*} f ( g _ 0 , \\cdots , g _ n ) & = g _ 0 ( \\psi ^ n ) ^ { - 1 } ( \\phi ) ( g _ 0 ^ { - 1 } g _ 1 , \\cdots , g _ { n - 1 } ^ { - 1 } g _ n ) \\\\ & = - g _ 0 \\cdot g _ 0 ^ { - 1 } g _ 1 \\cdot ( \\psi ^ n ) ^ { - 1 } ( \\phi ) ( ( g _ 0 ^ { - 1 } g _ 1 ) ^ { - 1 } , g _ 0 ^ { - 1 } g _ 1 g _ 1 ^ { - 1 } g _ 2 , \\cdots , g _ { n - 1 } ^ { - 1 } g _ n ) \\\\ & = - g _ 1 \\cdot ( \\psi ^ n ) ^ { - 1 } ( \\phi ) ( g _ 1 ) ^ { - 1 } g _ 0 , g _ 0 ^ { - 1 } g _ 2 , \\cdots , g _ { n - 1 } ^ { - 1 } g _ n ) \\\\ & = - f ( g _ 1 , g _ 0 , \\cdots , g _ n ) . \\end{align*}"}
-{"id": "6926.png", "formula": "\\begin{align*} \\| \\nabla \\psi _ 3 \\| _ p \\leq \\frac { C } { \\varepsilon ^ 2 } \\| t ^ 2 \\| _ q = \\frac { C } { \\varepsilon ^ 2 } \\| t \\| _ { 2 q } ^ 2 \\leq \\frac { C } { \\varepsilon ^ 2 } \\| t \\| _ 2 \\| t \\| _ p . \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\langle F ^ { p - 1 } ( \\nabla u ) \\nabla _ { \\xi } F ( \\nabla u ) , \\nabla \\varphi \\rangle \\ d x = \\lambda \\int _ \\Omega | u | ^ { p - 2 } u \\varphi \\ d x \\end{align*}"}
-{"id": "5524.png", "formula": "\\begin{align*} f \\circ \\widehat \\psi _ p ( \\zeta ) = \\widehat \\psi _ { f ( p ) } ( \\lambda _ p \\cdot \\zeta ) . \\end{align*}"}
-{"id": "4962.png", "formula": "\\begin{align*} F = \\left ( \\omega ^ { \\left ( k - 1 \\right ) \\left ( j - 1 \\right ) } \\right ) _ { 1 \\leq k , j \\leq n } \\omega = \\exp \\left ( \\tfrac { 2 \\pi i } { n } \\right ) \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} \\alpha _ n = \\frac { n + 1 } { 2 ( n + \\lambda ) } , \\ \\beta _ n = 0 , \\ \\gamma _ n = \\frac { n + 2 \\lambda - 1 } { 2 ( n + \\lambda ) } = 1 - \\alpha _ n \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} \\mathcal { A } ^ { P R _ 1 } = \\mathbb { I } _ { n ! } \\otimes S _ v ( - k _ { P ( 1 ) } ) \\otimes \\mathbb { I } _ { | \\mathcal { E } | ^ { n - 1 } } \\mathcal { A } ^ P . \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} \\sum \\limits _ { t = 1 } ^ { \\infty } C _ Y ^ { 2 k } ( t ) \\le c _ 0 \\int \\limits _ { 1 } ^ { \\infty } \\frac { L ^ { 2 k } ( t ) } { t ^ { 2 k \\eta } } \\ , d t \\le \\int \\limits _ { 1 } ^ { \\infty } \\frac { c _ 1 \\ , d t } { t ^ { 2 k ( \\eta - \\delta ) } } , k \\in \\N , \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{align*} d \\varphi ( t ) = - \\frac { \\sqrt { 6 } } { 3 \\ , y ( t ) ^ 5 } \\ , ( f ^ { 1 2 3 4 } + f ^ { 1 2 5 6 } + f ^ { 3 4 5 6 } ) = \\tau _ 0 ( t ) \\star _ t \\varphi ( t ) + \\star _ t \\tau _ 3 ( t ) , \\end{align*}"}
-{"id": "4592.png", "formula": "\\begin{align*} { \\displaystyle | \\tau \\sum _ { k = 1 } ^ { m } V _ { 3 } ^ { k } ( \\partial _ { \\tau } { \\theta _ { \\Psi } ^ { k } } ) | \\leq C \\big ( h ^ { 2 r } + \\tau ^ { 4 } \\big ) + C \\Vert \\theta _ { \\Psi } ^ { m } \\Vert _ { \\mathcal { L } ^ 2 } ^ { 2 } + \\frac { 1 } { 3 2 } \\Vert \\nabla \\theta _ { \\Psi } ^ { m } \\Vert _ { \\mathbf { L } ^ 2 } ^ { 2 } + C \\tau \\sum _ { k = 0 } ^ { m } { \\Vert \\nabla \\theta _ { \\Psi } ^ { k } \\Vert _ { \\mathbf { L } ^ 2 } ^ { 2 } } } \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} r ( t ) = \\int _ { - \\infty } ^ { \\infty } { \\eta } ( \\alpha ) { \\eta } ( \\alpha - t ) { \\rm d } \\alpha , \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{gather*} | r _ j ( \\theta , a ) | \\leq c _ 1 ( | \\theta | ^ { 1 / 3 } + | a | ^ { 1 / 2 } ) , j = 1 , 2 , \\\\ | r _ 1 ( \\theta , a ) - r _ 2 ( \\theta , a ) | \\geq c _ 0 ( | \\theta | ^ { 1 / 3 } + | a | ^ { 1 / 2 } ) \\end{gather*}"}
-{"id": "7930.png", "formula": "\\begin{align*} \\mbox { d i v $ \\widetilde U $ } = 0 , \\widetilde U | _ { \\partial \\Omega } = 0 , \\widetilde U \\to 0 \\ ; ( | x | \\to \\infty ) . \\end{align*}"}
-{"id": "3808.png", "formula": "\\begin{align*} a ( n , n ) & = \\frac { ( - 1 ) ^ { n } } { n ! } , \\\\ a ( n - 1 , n ) & = \\frac { ( - 1 ) ^ { n + 1 } } { 2 ! ( n - 2 ) ! } 3 , \\ ; n \\geq 2 , \\\\ a ( n - 2 , n ) & = \\frac { ( - 1 ) ^ { n } } { 4 ! ( n - 3 ) ! } ( 2 7 n - 7 3 ) , \\ ; n \\geq 3 , \\\\ a ( 1 , n ) & = \\frac { 1 - 2 ^ { \\nu _ { 2 } ( n ) + 1 } } { n } , \\ ; n \\geq 1 . \\\\ \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{align*} G ( \\varphi ) : = \\frac { \\exp ( \\mathfrak { g } ^ \\vee ) } { _ { } ( \\varphi ) } \\times U ( 1 ) ^ { \\mathrm { r k } \\ ; ( \\varphi ) } , \\end{align*}"}
-{"id": "2936.png", "formula": "\\begin{align*} A _ { m , 0 } : = \\Bigl \\{ n _ { j _ { m } } + k d _ { j _ { m } } + k ' \\textrm { p e r } ( x _ { l } ) \\ , ; \\ , 0 \\le k ' \\le \\dfrac { \\alpha d _ { j _ { m } } } { \\textrm { p e r } ( x _ { l } ) } , \\ 0 \\le k \\le \\dfrac { \\alpha d _ { j _ { m } + 1 } } { d _ { j _ { m } } } - 2 \\Bigr \\} , \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{align*} \\rho _ { \\varepsilon , j } : = \\varepsilon + \\sum _ { i = 1 } ^ j \\varepsilon ^ { - N } \\chi _ { B _ i ^ { \\varepsilon } } . \\end{align*}"}
-{"id": "2677.png", "formula": "\\begin{align*} u ^ 2 + k ^ 2 = x ^ 3 , ( u , k , x ) = 1 , u , k , x \\in \\mathbb { Z } \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} K _ 1 \\cap \\tau K _ 1 \\tau ^ { - 1 } = \\lbrace 1 _ G \\rbrace . \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} ( u - c ) \\times \\omega = \\nabla \\left ( \\frac 1 2 | u - c | ^ 2 + P + g x _ 3 - \\frac 1 2 | c | ^ 2 \\right ) , \\end{align*}"}
-{"id": "8401.png", "formula": "\\begin{align*} U _ n = U _ n ( \\omega ) = \\min _ { \\pi \\subset B _ { 8 \\mu \\beta ^ { - 1 } _ 1 n ^ { 1 + \\epsilon } } } T ( \\pi , \\omega ) \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} \\mathcal { B } : = \\{ \\mathbf { x _ i } : i \\leqslant u \\} \\cup \\{ \\Xi ( \\mathbf { w _ j } ) : j \\leqslant d - u \\} \\end{align*}"}
-{"id": "2326.png", "formula": "\\begin{align*} h ( \\cdot ) = \\sqrt n \\lambda \\| \\cdot \\| _ 1 , \\end{align*}"}
-{"id": "9835.png", "formula": "\\begin{align*} R _ { 1 2 } ( x ) R _ { 1 3 } ( x y ) R _ { 2 3 } ( y ) = R _ { 2 3 } ( y ) R _ { 1 3 } ( x y ) R _ { 1 2 } ( x ) . \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} \\lambda ( c ) = \\lambda _ 1 ( c - c _ * ) + \\mathcal { O } ( ( c - c _ * ) ^ 2 ) , \\psi ( c ) = \\psi _ * + \\psi _ 1 ( c - c _ * ) + \\mathcal { O } _ { H ^ 3 _ { \\mu } } ( ( c - c _ * ) ^ 2 ) , \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\exp \\left ( S ( A | A ' B ' ) _ { \\hat { \\gamma } _ { A A ' B ' } ^ { ( n ) } } - 1 \\right ) & = a \\ ; , \\\\ \\lim _ { n \\to \\infty } \\exp \\left ( S ( B | A ' B ' ) _ { \\hat { \\gamma } _ { B A ' B ' } ^ { ( n ) } } - 1 \\right ) & = b \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} B = \\{ ( x _ { 1 } , x _ { 2 } ) \\in \\textbf { R } ^ 2 \\mbox { } | \\mbox { } l _ { i } ^ { B } \\leq x _ { i } \\leq r _ { i } ^ { B } , i = 1 , 2 \\} \\end{align*}"}
-{"id": "7633.png", "formula": "\\begin{align*} 2 g ( E ) - 2 = ( K _ X + E ) \\cdot E . \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} R ' \\hookrightarrow R ^ \\vee = \\bigoplus _ i R _ i ^ \\vee . \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} \\frac { B _ { n } } { n ! } = \\sum _ { \\pi \\in \\mathcal { C } \\left ( n \\right ) } \\frac { \\left ( - 1 \\right ) ^ { \\vert \\pi \\vert } } { \\left ( \\pi + 1 \\right ) ! } = \\sum _ { p = 1 } ^ { n } \\binom { n + 1 } { p + 1 } \\left ( - 1 \\right ) ^ { p } \\sum _ { k _ { 1 } + \\dots + k _ { p } = n } \\frac { 1 } { \\left ( k _ { 1 } + 1 \\right ) ! \\dots \\left ( k _ { p } + 1 \\right ) ! } \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} G ^ + ( f ( z ) ) = d G ^ + ( z ) , G ^ - ( f ^ { - 1 } ( z ) ) = d G ^ - ( z ) \\end{align*}"}
-{"id": "3933.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] = ( 1 - z ) ^ { - \\alpha } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 \\alpha & \\frac 1 2 - \\frac 1 2 \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , - \\frac { 4 z } { ( 1 - z ) ^ 2 } \\bigg ] . \\end{align*}"}
-{"id": "9753.png", "formula": "\\begin{align*} E _ { \\epsilon } ( u ) = \\int _ M e _ { \\epsilon } ( u ) : = \\int _ M \\frac { | d u | ^ 2 } { 2 } + \\frac { ( 1 - | u | ^ 2 ) ^ 2 } { 4 \\epsilon ^ 2 } \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{gather*} ( \\phi \\psi ) ( X ) = ( \\phi \\otimes \\psi ) \\Delta ( X ) , 1 ( X ) = \\varepsilon ( X ) , \\\\ \\Delta ( \\phi ) ( X \\otimes Y ) = \\phi ( X Y ) , \\varepsilon ( \\phi ) = \\phi ( 1 ) , \\\\ S ( \\phi ) ( X ) = \\phi ( S ( X ) ) , \\phi ^ * ( X ) = \\overline { \\phi ( S ( X ) ^ * ) } . \\end{gather*}"}
-{"id": "3795.png", "formula": "\\begin{align*} \\frac { s _ 1 ^ 2 - 3 s _ 1 - 2 } { 2 } + 2 s _ 1 + 5 = \\frac { ( s _ 1 + 1 ) ( s _ 1 - 2 ) } { 2 } + 5 . \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\alpha ^ k P _ k ( x ) = \\frac { 1 } { \\sqrt { 1 + \\alpha ^ 2 - 2 \\alpha x } } . \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle g _ b '' + \\frac { r } { 2 } g _ b ' + \\frac { 1 } { \\alpha } g _ b + | g _ b | ^ \\alpha g _ b = 0 r \\ge 0 \\\\ g _ b ( 0 ) = 0 , g ' _ b ( 0 ) = b . \\end{cases} \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} G _ j = \\frac { 1 } { 2 } \\left \\lceil \\frac { j } { 2 } \\right \\rceil \\left \\lceil \\frac { 3 j + 1 } { 2 } \\right \\rceil , \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} \\begin{aligned} u _ k = \\mathcal { V } _ { \\theta _ k } ^ { - 1 } \\big ( c ^ { ( z ) } _ k \\big ) _ { z \\in \\Z ^ 3 } = \\sum _ { z \\in \\Z ^ 3 } c ^ { ( z ) } _ k e ^ { \\i \\langle \\theta _ k + 2 \\pi z , \\cdot \\rangle _ { \\C ^ 3 } } , & & u = \\mathcal { V } _ \\theta ^ { - 1 } \\big ( c ^ { ( z ) } \\big ) _ { z \\in \\Z ^ 3 } = \\sum _ { z \\in \\Z ^ 3 } c ^ { ( z ) } e ^ { \\i \\langle \\theta + 2 \\pi z , \\cdot \\rangle _ { \\C ^ 3 } } , \\end{aligned} \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} V ( t ) = 2 t v ' ( t ) ^ 2 + \\Bigl ( \\frac { 2 ( \\alpha + 1 ) } { \\alpha ^ 2 t } + \\frac { 1 } { 2 \\alpha } e ^ { - t } \\Bigr ) v ( t ) ^ 2 + \\Psi ( v ( t ) ) \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} d \\tilde { \\eta } ^ { h , x } ( t ) = \\Bigl ( A \\tilde { \\eta } ^ { h , x } ( t ) d t + \\sigma ' ( X ( t , x ) ) . \\tilde { \\eta } ^ { h , x } d W ( t ) \\Bigr ) + \\sigma ' ( X ( t , x ) ) . e ^ { t A } h d W ( t ) . \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{align*} \\left | \\hat { \\mu } ( \\xi _ i ) \\right | ^ N = \\frac { e ^ c } { m } \\left [ 1 + O \\left ( \\frac { \\log m } { m ^ 2 } \\right ) \\right ] . \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } x _ n z ^ n = \\frac { \\sum _ { j \\in J } z ^ j } { 1 - \\sum _ { j \\in J } z ^ j } , \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} ( \\sigma ( a _ { n - 1 } ) , \\sigma ( a _ 0 ) , \\ldots , \\sigma ( a _ { n - 2 } ) ) \\left ( \\begin{array} { c | c } 0 & 1 \\\\ \\hline I _ { n - 1 } & 0 \\end{array} \\right ) \\sigma ( \\Sigma ) \\sigma ( E ^ { \\mu + 1 } ) = 0 . \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{align*} A = \\begin{bmatrix} \\pi & \\pi ^ n \\\\ \\pi & \\pi ^ 2 + \\pi ^ n \\end{bmatrix} \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} z _ \\omega ( \\Gamma ) = n ^ c + \\sum _ { x = 1 } ^ c ( - 1 ) ^ x y _ x n ^ { c - x } \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} \\gamma _ { 0 } \\mathcal { D } _ { \\partial \\Omega } [ \\phi ] = - \\tfrac { 1 } { 2 } \\phi + \\mathcal { K } _ { \\partial \\Omega } [ \\phi ] \\ , , \\gamma _ { 0 } ^ \\complement \\mathcal { D } _ { \\partial \\Omega } [ \\phi ] = \\tfrac { 1 } { 2 } \\phi + \\mathcal { K } _ { \\partial \\Omega } [ \\phi ] \\ , , \\forall \\phi \\in H ^ { 1 / 2 + \\tau } ( \\partial \\Omega ) \\ , . \\end{align*}"}
-{"id": "1496.png", "formula": "\\begin{align*} Q ' P ' - Q '' P = Q R . \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} G _ n = \\left \\{ \\hat { \\pi } ^ { ( n ) } _ n \\subseteq B _ { 8 \\mu \\beta _ 1 ^ { - 1 } n ^ { 1 + \\epsilon _ 0 } } \\right \\} \\end{align*}"}
-{"id": "9745.png", "formula": "\\begin{align*} \\partial _ t B + \\nabla \\cdot \\left ( \\frac { B \\otimes P - P \\otimes B } { | B | } \\right ) = 0 , \\ ; \\ ; \\ ; \\nabla \\cdot B = 0 , \\ ; \\ ; \\ ; P = \\nabla \\cdot \\left ( \\frac { B \\otimes B } { | B | } \\right ) , \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{align*} \\sum _ { b _ { n ( 1 ) } \\leq j < b _ { n ( 1 ) + 1 } } c _ { 1 , j } \\lambda ^ { b _ { n ( 1 ) + 1 } - 1 - j } y _ j = 0 \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{align*} & \\Phi w \\in C ^ { \\mu } _ { l o c } ( ( 0 , T _ * ] ; L ^ q _ \\sigma ( \\Omega ) ) , \\forall q \\in [ 2 , \\infty ) , \\ , \\forall \\mu \\in ( 0 , \\mu _ 0 ) , \\\\ & \\nabla \\Phi w \\in C ^ { \\mu } _ { l o c } ( ( 0 , T _ * ] ; L ^ q ( \\Omega ) ) , \\forall q \\in [ 2 , 6 ) , \\ , \\forall \\mu \\in ( 0 , \\mu _ 0 - 1 / 2 ) , \\end{align*}"}
-{"id": "4480.png", "formula": "\\begin{align*} \\underset { l \\rightarrow \\infty } { \\lim } J _ { \\mathrm { d } , l } ^ { ( \\omega , \\mathcal { A } ) } ( t ) = \\underset { l \\rightarrow \\infty } { \\lim } \\left ( \\left . \\partial _ { \\eta } \\mathbb { J } _ { \\mathrm { d } } ^ { ( \\omega , \\eta \\mathbf { \\bar { A } } _ { l } ) } \\left ( t \\right ) \\right \\vert _ { \\eta = 0 } \\right ) = \\left ( \\mathbf { \\Xi } _ { \\mathrm { d } } \\vec { w } \\right ) \\int _ { t _ { 0 } } ^ { t } \\mathcal { E } _ { s } \\mathrm { d } s \\ . \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} \\mathbf { \\Xi } _ { \\mathrm { d } } : = \\underset { l \\rightarrow \\infty } { \\lim } \\mathbb { E } \\left [ \\Xi _ { \\mathrm { d } , l } ^ { ( \\omega ) } \\right ] \\ . \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} \\widetilde { \\phi } _ A ( r e ^ { 2 \\pi i x } , e ^ { 2 \\pi i y } ) = ( r e ^ { 2 \\pi i ( a x + b y ) } , e ^ { 2 \\pi i d y } ) \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} ( H ^ * X ) ( x ) = X ( x , x ) . \\end{align*}"}
-{"id": "6441.png", "formula": "\\begin{align*} q ( x e _ 1 + y e _ 2 + z e _ 3 ) = a x ^ 2 + b y ^ 2 + c z ^ 2 + u y z \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} T ^ { 0 , 2 } & = N , & T ^ { 1 , 1 } & = 0 , & T ^ { 2 , 0 } & = ( d ^ c F ) ^ { 2 , 0 } . \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} \\| \\psi _ \\varepsilon \\| _ \\infty \\leq 2 M : = 2 \\| \\varphi _ 0 \\| _ \\infty . \\end{align*}"}
-{"id": "366.png", "formula": "\\begin{align*} \\| W f \\| ^ p _ { \\ell ^ p ( v , L ^ p ( \\Omega , \\mathbb { F } , \\mu ) ) } = M \\| f \\| ^ p _ { L ^ p ( \\Omega , \\mathbb { F } , \\mu ) } \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} f ^ * ( x ) = & k \\ , x ^ { \\frac { r p ^ l - 1 } { 2 } } + ( 2 - k ) \\ , \\binom { r } { r - 1 } \\ , x ^ { \\frac { ( r - 1 ) p ^ l } { 2 } } + k \\ , \\binom { r } { e - 2 } \\ , x ^ { \\frac { ( r - 2 ) p ^ l - 1 } { 2 } } \\cr & + ( 2 - k ) \\ , \\binom { r } { r - 3 } \\ , x ^ { \\frac { ( r - 3 ) p ^ l } { 2 } } + \\cdots + ( 2 - k ) \\ , \\binom { r } { 2 } \\ , x ^ { \\frac { 2 p ^ l } { 2 } } + k \\ , \\binom { r } { 1 } \\ , x ^ { \\frac { p ^ l - 1 } { 2 } } \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} s ^ C - s = R _ { \\alpha \\enskip \\gamma } ^ { \\enskip \\alpha \\enskip \\gamma } + R ^ { \\alpha \\quad \\gamma } _ { \\enskip \\gamma \\alpha } & = ( \\nabla _ { \\bar \\alpha } T ) _ \\gamma ^ { \\enskip \\bar \\alpha \\gamma } - T ^ { \\alpha \\qquad \\ ; \\gamma } _ { \\enskip T ( \\gamma , \\alpha ) } = ( \\nabla _ \\alpha T ) _ { \\bar \\gamma } ^ { \\enskip \\alpha \\bar \\gamma } - T ^ { \\bar \\alpha \\qquad \\ ; \\bar \\gamma } _ { \\enskip T ( \\bar \\gamma , \\bar \\alpha ) } . \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} f _ { c _ { 1 } } ^ { \\prime } \\left ( z \\right ) = \\left ( \\frac { h } { B _ { 2 } } \\right ) ^ { \\prime } f \\left ( z \\right ) + \\frac { h } { B _ { 2 } } f ^ { \\prime } \\left ( z \\right ) + \\left ( \\frac { B _ { 1 } } { B _ { 2 } } \\right ) ^ { \\prime } f _ { c _ { 2 } } \\left ( z \\right ) + \\frac { B _ { 1 } } { B _ { 2 } } f _ { c _ { 2 } } ^ { \\prime } \\left ( z \\right ) . \\end{align*}"}
-{"id": "9544.png", "formula": "\\begin{align*} \\lambda _ \\sigma ( s ) \\lambda _ \\sigma ( t ) = \\pi _ \\alpha ( \\sigma ( s , t ) ) \\lambda _ \\sigma ( s t ) , \\ \\ \\ s , t \\in G . \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} \\begin{cases} \\alpha _ 3 ( \\gamma _ 1 + \\gamma _ 2 ) + \\alpha _ 1 \\gamma _ 7 - \\beta _ 1 \\gamma _ 6 = 0 \\gamma _ 5 = 0 \\\\ \\alpha _ 3 ( \\gamma _ 3 + \\gamma _ 4 ) + \\alpha _ 4 \\gamma _ 7 - \\beta _ 3 \\gamma _ 6 = 0 \\gamma _ 6 = 0 \\end{cases} \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{align*} \\Delta _ { A | M } ( \\hat { \\rho } _ { A M } ) ( t ) = S ( A | M ) _ { ( \\mathcal { N } _ A ( t ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) } - S ( A | M ) _ { \\hat { \\rho } _ { A M } } \\ ; . \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} S _ m = \\frac { ( 2 d ) ^ m } { m ! } \\left ( \\sum _ { i = 0 } ^ { t - 1 } \\overline { p } _ i / n ^ i + O ( \\overline { q } / n ^ t ) \\right ) . \\end{align*}"}
-{"id": "6927.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\int _ \\Omega | \\nabla \\zeta _ \\varepsilon | ^ 2 = \\int _ { \\O } | \\nabla \\zeta | ^ 2 . \\end{align*}"}
-{"id": "439.png", "formula": "\\begin{align*} \\dim \\mathcal { W } = \\dim X + m ( e , a , b , c ) . \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} \\Omega \\cong \\bigl \\{ f \\in C \\ , \\big | \\ , f ^ { ( 0 ) } _ \\alpha = f ^ { ( 2 ) } _ \\alpha = \\dots = f ^ { ( 2 ( \\nu _ \\alpha - 1 ) ) } _ { \\alpha } = 0 \\ ; \\mbox { \\rm f o r a l l } \\ ; \\alpha \\in \\Lambda _ n \\bigr \\} . \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} < \\tilde { G } , G > = G _ 0 \\tilde { G } _ 1 ^ \\ast \\frac { i c } { 2 \\pi } \\partial _ \\lambda \\Lambda ( \\gamma , \\lambda ) . \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} \\begin{cases} \\beta _ 1 \\gamma _ 1 + \\beta _ 2 \\gamma _ 3 = \\alpha _ 1 \\gamma _ 2 + \\alpha _ 2 \\gamma _ 4 \\\\ \\beta _ 4 \\gamma _ 1 + \\beta _ 5 \\gamma _ 3 = \\alpha _ 4 \\gamma _ 2 + \\alpha _ 5 \\gamma _ 4 \\end{cases} \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} ( q ^ { \\varepsilon } ) _ { \\varepsilon } = q , \\end{align*}"}
-{"id": "268.png", "formula": "\\begin{align*} \\theta ( \\mathsf { P } ) = \\pi ( K _ { 2 \\rho } ^ { - 1 } ) \\mathsf { P } \\pi ( K _ { 2 \\rho } ) , \\quad \\theta ( \\mathsf { Q } ) = \\pi ( K _ { 2 \\rho } ) \\mathsf { Q } \\pi ( K _ { 2 \\rho } ^ { - 1 } ) , \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} \\lambda _ { 1 } ( p , \\Omega ) = \\lambda _ { 1 } ( p , \\mathcal W _ { r _ { i } } ) , i = 1 , 2 . \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { \\nu } \\varepsilon _ { n } ^ { \\sigma _ { i } } \\alpha _ { i } \\frac { \\partial ^ { i } u } { \\partial x _ { n } ^ { i } } \\left ( x ^ { \\prime } , 0 , y , t \\right ) = 0 \\nu \\in \\left \\{ 0 , 1 \\right \\} , x ^ { \\prime } \\in R ^ { n - 1 } y \\in \\left ( 0 , 1 \\right ) \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{align*} \\Lambda ^ { N + 1 } _ { N } ( x , d y ) = \\frac { N ! \\Delta _ N ( y ) } { \\Delta _ { N + 1 } ( x ) } \\boldmath { 1 } ( y \\prec x ) d y , \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} \\psi = \\frac { | \\Phi [ w ] | } { | w | } \\qquad \\end{align*}"}
-{"id": "2818.png", "formula": "\\begin{align*} C _ 1 = \\prod _ { p } \\left ( 1 - \\frac { 1 } { p } \\right ) ^ 3 \\left ( 1 + \\frac { 3 } { p } - \\frac { 1 } { p ^ 2 } - \\frac { 1 8 } { p ( p + 2 ) } \\right ) . \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} \\widehat \\varepsilon _ { m , L } \\leq C e \\widetilde q ^ m L = C e \\exp ( m \\ln ( \\widetilde q ) + \\ln L ) \\leq C e \\exp ( m \\ln ( \\widetilde q ) + m / K ) \\leq C ' \\exp ( - b m ) , \\end{align*}"}
-{"id": "6851.png", "formula": "\\begin{align*} U _ f ( \\alpha ) : = \\| f \\| _ { A ^ { 2 \\alpha } _ \\alpha } ^ { 2 \\alpha } = \\int _ { \\mathbb { D } } | f ( z ) | ^ { 2 \\alpha } \\ , ( \\alpha - 1 ) ( 1 - | z | ^ 2 ) ^ \\alpha \\ , d \\mu ( z ) . \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} f _ { \\vec { w } } \\left ( t \\right ) = \\int \\nolimits _ { \\mathbb { R } } \\mathrm { e } ^ { i t \\nu } \\mu _ { \\vec { w } } \\left ( \\mathrm { d } \\nu \\right ) \\ , t \\in \\mathbb { R } \\ . \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} T ^ { \\ , 2 ( b _ { n + 1 } - b _ n ) } e _ k = e _ k \\textrm { i f } \\ k \\in [ b _ n , b _ { n + 1 } ) , \\ n \\ge 0 . \\end{align*}"}
-{"id": "4968.png", "formula": "\\begin{align*} | b _ * | = \\sqrt { \\frac { \\lambda ' ( c _ * ) ( c _ + - c _ * ) } { | \\gamma | } } , c _ + \\geq c _ * . \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} \\sup _ { \\xi , \\left \\Vert \\xi \\right \\Vert = 1 } \\sum _ { x } \\mu ^ { ( n ) } ( x ) \\left | \\left \\langle b ( x ) , \\xi \\right \\rangle \\right | ^ { 2 } \\le \\epsilon ^ { 2 } \\delta n . \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} \\widetilde { P } ( \\tau = k ) = \\frac { 1 } { d ^ k } ( 1 - \\frac { 1 } { d } ) , \\ \\forall k \\geq 0 . \\end{align*}"}
-{"id": "7373.png", "formula": "\\begin{align*} \\begin{aligned} - 1 4 4 y ^ 7 z ^ { 1 6 } y '' & = 4 8 A ^ 2 ( z ^ { 1 2 } y ^ 2 - z ^ { 1 0 } y ^ 4 ) + 2 A \\sqrt { 6 } \\ , ( 1 0 z ^ 8 y - 1 1 z ^ 6 y ^ 3 - 8 z ^ 4 y ^ 5 ) \\\\ & + 1 2 z ^ 4 - 4 z ^ 2 y ^ 2 - 7 y ^ 4 . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\frac { x y } { z } + \\frac { y } { x } + x = a , \\\\ \\frac { y z } { x ^ { 2 } } + \\frac { z } { y } + y = b , \\\\ \\frac { x ^ { 2 } z } { y ^ { 2 } } + \\frac { x ^ { 2 } } { z } + z = c . \\end{array} \\right . \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} \\max _ { | \\zeta | \\leq \\lambda _ { p _ n } } G ^ + \\circ \\psi _ { f ( p _ n ) } ( \\zeta ) & = \\max _ { | \\zeta | \\leq 1 } G ^ + \\circ \\psi _ { f ( p _ n ) } ( \\lambda _ { p _ n } \\zeta ) \\\\ & = \\max _ { | \\zeta | \\leq 1 } G ^ + \\circ f \\circ \\psi _ { p _ n } ( \\zeta ) \\\\ & = d \\max _ { | \\zeta | \\leq 1 } G ^ + \\circ \\psi _ { p _ n } ( \\zeta ) \\\\ & = d . \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{align*} c = 1 - 6 \\frac { ( u - p ) ^ 2 } { u p } \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 2 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 3 e _ 4 + \\alpha _ 4 e _ 5 , [ e _ 2 , e _ 1 ] = \\alpha _ 6 e _ 5 , [ e _ 2 , e _ 2 ] = \\beta _ 2 e _ 5 , [ e _ 3 , e _ 3 ] = \\gamma _ 6 e _ 5 . \\end{align*}"}
-{"id": "9656.png", "formula": "\\begin{align*} [ X , S ] = 2 S \\ , , \\qquad [ X , T ] = - 2 T \\ , , \\qquad [ S , T ] = X \\ , \\end{align*}"}
-{"id": "7472.png", "formula": "\\begin{align*} L ' = \\{ x \\in L \\mid x = 0 \\phi ( [ 0 , x ] ) = \\phi ( L ) \\} \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} f ( \\xi ^ * ) = 2 . 8 6 8 1 1 4 0 1 3 ( 4 ) . \\end{align*}"}
-{"id": "9422.png", "formula": "\\begin{align*} \\partial _ t D + \\nabla \\times \\left ( \\frac { D \\times P - B } { h } \\right ) = 0 , \\ ; \\partial _ t P + \\nabla \\cdot \\left ( \\frac { P \\otimes P - B \\otimes B - D \\otimes D - I _ 3 } { h } \\right ) = 0 , \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{align*} \\alpha = 1 - \\frac { 4 } { \\chi _ { \\rho } ( \\gamma ) ^ 2 } . \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} \\varphi f ^ { 2 } + \\psi f ^ { \\prime } f - q = A e ^ { \\frac { 1 } { 2 } b _ { m } z ^ { m } } \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} \\mathbb { P } ( \\# { \\cal E } _ i \\geq M \\log { n } ) \\leq \\sum _ { r = M \\log { n } } ^ { n } \\frac { T _ r e ^ { - r } } { C ( r - 1 ) ! } e ^ { - \\delta _ 0 r } \\leq e ^ { - \\delta _ 0 M \\log { n } } \\sum _ { r = M \\log { n } } ^ { n } \\frac { T _ r e ^ { - r } } { C ( r - 1 ) ! } . \\end{align*}"}
-{"id": "1794.png", "formula": "\\begin{align*} p _ { j } ^ { * } \\big ( a ( D ) [ u ] \\big ) & = p _ { j } ^ { * } \\Big ( \\big [ a ( D ) u \\big ] \\Big ) = \\left ( \\int _ { | \\xi | \\leqslant j } | a ( \\xi ) | ^ { 2 } \\ , \\big | \\widehat { [ u ] } ( \\xi ) \\big | ^ { 2 } \\ , d \\xi \\right ) ^ { 1 / 2 } \\\\ & \\leqslant \\| a \\| _ { L ^ { \\infty } ( B ( 0 , j ) ) } \\ , \\big \\| \\widehat { [ u ] } \\big \\| _ { L ^ { 2 } ( B ( 0 , j ) ) } \\\\ & = \\| a \\| _ { L ^ { \\infty } ( B ( 0 , j ) ) } \\ , p _ { j } ^ { * } \\big ( [ u ] \\big ) , \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{align*} \\int \\prod _ { j = 1 } ^ { k } ( 1 - \\cos ( \\theta - \\theta _ j ) ) ^ { m _ j } \\log ( w ( \\theta ) ) \\ , d \\theta > - \\infty \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} \\varphi \\ = \\ \\tau ( o ) \\ , . \\end{align*}"}
-{"id": "719.png", "formula": "\\begin{align*} J _ i ^ { \\pm } | m _ 1 , \\cdots , m _ i , \\cdots , m _ r \\rangle \\ = \\sqrt { F _ i ( m _ 1 , \\cdots , m _ i \\pm 1 , \\cdots , m _ r ) } | m _ 1 , \\cdots , m _ i \\pm 1 , \\cdots , m _ r \\rangle \\ \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{align*} \\begin{bmatrix} \\ ! X _ 1 & X _ 2 \\\\ X _ 3 & X _ 4 \\end{bmatrix} ^ { - 1 } \\ ! \\ ! \\ ! = \\begin{bmatrix} X ^ { - 1 } & - X ^ { - 1 } X _ 2 X _ 4 ^ { - 1 } \\\\ - X _ 4 ^ { - 1 } X _ 3 X ^ { - 1 } & X _ 4 ^ { - 1 } X _ 3 X ^ { - 1 } X _ 2 X _ 4 ^ { - 1 } + X _ 4 ^ { - 1 } \\ ! \\end{bmatrix} . \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} u _ \\varepsilon ( x ) = O \\big ( e ^ { - \\theta _ 1 R } \\big ) , ~ \\mbox { f o r } ~ x \\in \\R ^ 3 \\backslash \\bigcup _ { j = 1 } ^ k B _ { R \\varepsilon } ( x _ { j , \\varepsilon } ) . \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} [ Y _ { c ( t ) } ^ * ] ^ { \\perp } = \\left \\{ v \\in [ X _ { c ( t ) } ^ * ] ^ { \\perp } : \\langle \\eta _ * e ^ { i y } , v \\rangle _ { L ^ 2 ( \\mathbb { R } \\times \\mathbb { T } ) } = \\langle \\eta _ * e ^ { - i y } , v \\rangle _ { L ^ 2 ( \\mathbb { R } \\times \\mathbb { T } ) } = 0 \\right \\} . \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} A _ 1 = A _ 1 ( r _ 1 , r _ 2 ) = p _ u ^ { r _ 1 - 1 } ( 1 - p _ d ) ^ { r _ 1 ( n - r _ 1 ) } p _ u ^ { r _ 2 - 1 } ( 1 - p _ d ) ^ { r _ 2 ( n - r _ 1 - r _ 2 ) } . \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} \\begin{aligned} \\int \\limits _ { | x - e | + | x | \\leq 2 \\tau } \\frac { q ( x ) } { | x | | x - e | } d x \\leq C \\int \\limits _ { 1 } ^ { 2 \\tau } Q ( r ) d r . \\end{aligned} \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} S ( A A ' ) _ { \\hat { \\gamma } _ { A A ' } ^ { ( n ) } } & = 2 \\ , g \\left ( n - \\frac { 1 } { 2 } \\right ) = \\ln n ^ 2 + 2 + \\mathcal { O } \\left ( \\frac { 1 } { n ^ 2 } \\right ) \\ ; , \\\\ S ( A ' ) _ { \\hat { \\gamma } _ { A A ' } ^ { ( n ) } } & = g \\left ( \\frac { n ^ 2 } { a } - \\frac { 1 } { 2 } \\right ) = \\ln \\frac { n ^ 2 } { a } + 1 + \\mathcal { O } \\left ( \\frac { 1 } { n ^ 4 } \\right ) \\ ; , \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{align*} \\beta ^ * _ T = - t _ 0 u _ T , \\beta ^ * _ { T ^ c } = 0 , \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} { } _ 3 F _ 2 \\bigg [ \\begin{matrix} - n & \\alpha & \\beta & \\\\ & 1 & \\alpha + \\beta - n \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { ( 1 - \\alpha ) _ n ( 1 - \\beta ) _ n } { n ! \\cdot ( 1 - \\alpha - \\beta ) _ n } . \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{align*} \\mathcal { F } ^ { - 1 } \\left [ \\frac { 1 } { \\xi ^ { 2 } + \\lambda } \\right ] \\left ( x \\right ) = \\frac { 1 } { 2 \\sqrt { \\lambda } } \\mathrm { e } ^ { - \\left \\vert x \\right \\vert \\sqrt { \\lambda } } , \\ ; \\ ; x \\in \\mathbb { R } , \\ ; \\lambda \\in \\mathbb { C } \\backslash \\left ( - \\infty , 0 \\right ] , \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} \\nabla X _ { ( x , y ) } : = ( S ( X ^ { j } ) + G _ { ~ j } ^ { i } X ^ { j } ) _ { ( x , y ) } \\dot { \\partial } _ { i } , ~ \\ \\ \\forall ( x , y ) \\in A , \\end{align*}"}
-{"id": "4435.png", "formula": "\\begin{align*} \\big ( ( \\tau g ) ^ { - 1 } \\big ) ^ { \\ast } \\omega = \\varphi _ 1 ( \\tau g \\tau ^ { - 1 } ) \\omega _ 2 \\otimes \\varphi _ 1 ( \\tau ^ 2 g \\tau ^ { - 2 } ) ^ { \\ast } \\omega _ 3 \\otimes \\varphi _ 1 ( \\tau ^ { 3 } g ) ^ { \\ast } \\omega _ 1 . \\end{align*}"}
-{"id": "9641.png", "formula": "\\begin{align*} P _ n ( x ) = ( x - \\tau ^ n ) ( x - \\tau ^ { - n } ) ( x ^ { d - 2 } + 1 + \\sum _ { j = 1 } ^ { d / 2 - 2 } \\epsilon _ { j } ( x ^ { d - 2 - j } + x ^ { j } ) + \\epsilon _ { d / 2 - 1 } x ^ { d / 2 - 1 } ) , \\end{align*}"}
-{"id": "10007.png", "formula": "\\begin{align*} W ( G _ { 4 } ( n , d , x - 1 , s ) ) - W ( G _ { 4 } ( n , d , x , s ) ) & \\leq W ( G _ { 4 } ( n , d , x _ { 4 } ^ { \\max } - 1 , s ) ) - W ( G _ { 4 } ( n , d , x _ { 4 } ^ { \\max } , s ) ) = \\\\ & = 2 ( n - 4 ) + 2 . \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{align*} & D _ { z } ^ { \\mu , \\eta , p } { f ( z ) } : = \\frac { 1 } { \\Gamma ( - \\mu ) } \\int _ { 0 } ^ { z } f ( t ) ( z - t ) ^ { - \\mu - 1 } e x p \\big ( \\frac { - p z ^ { 2 } } { t ( z - t ) } ) d t , \\\\ & ( R e ( \\mu ) < 0 , R e ( p ) > 0 ) . \\end{align*}"}
-{"id": "9370.png", "formula": "\\begin{align*} ( E \\hat { \\otimes } _ { \\gamma _ 2 ^ * } F ) ^ * = \\Gamma _ { 2 } ( E , F ^ * ) . \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} \\partial _ t q _ t ( x , y ) & = \\hat { L } _ y ^ * q _ t ( x , y ) , \\ \\ t > 0 , x , y \\in \\mathbb { R } , \\\\ \\lim _ { t \\to 0 } q _ t ( x , y ) & = \\delta ( x = y ) . \\end{align*}"}
-{"id": "10032.png", "formula": "\\begin{align*} J _ { \\varphi } = \\frac { 1 } { \\sqrt { 5 } } ( 2 \\varphi - I d ) \\end{align*}"}
-{"id": "1809.png", "formula": "\\begin{align*} A : = 1 - \\Delta \\colon H ^ { 2 } ( \\R ^ { n } ) \\subset L ^ { 2 } ( \\R ^ { n } , \\C ) \\to L ^ { 2 } ( \\R ^ { n } , \\C ) , \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} e _ { i } \\ast ( e _ { j } \\ast e _ { m } ) = \\sum _ { p = 1 } ^ { n } a _ { j , m } ( p ) ( e _ { i } \\ast e _ { p } ) = \\sum _ { p = 1 } ^ { n } a _ { j , m } ( p ) a _ { i , p } \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} \\tilde \\phi _ { \\tilde t } + \\frac { p _ \\textrm { f l u i d } } { \\rho } + \\frac { 1 } { 2 } | \\nabla \\tilde \\phi | ^ 2 + g \\tilde z = H ( \\tilde t ) , \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} \\ 1 _ V \\ 1 _ W \\mu = \\ 1 _ { V \\cap W } \\mu \\end{align*}"}
-{"id": "9342.png", "formula": "\\begin{align*} \\| \\log ( \\gamma _ M ^ { - 1 } \\cdot \\gamma _ L ) \\| _ { } & \\leq \\| \\sigma _ L - \\sigma _ M \\| _ { } + K ( \\alpha ^ { \\frac 1 r } A ) ^ 2 \\leq \\mathcal { V } _ r ( \\sigma ; [ M , N ] ) + K A ^ 2 \\\\ & = A ( 1 + K A ) \\leq A ( 1 + 2 K \\delta ) = O ( A ) , \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} L _ { - 2 } = [ L _ { - 1 } , \\ , L _ { - 1 } ] = [ L _ { - 1 } , \\ , [ V _ { - 2 } , \\ , [ V _ { - 2 } , \\ , [ V _ { - 2 } , \\ , L _ 5 ] ] ] ] \\subseteq [ V _ { - 2 } , \\ , L _ 0 ] \\subseteq V _ { - 2 } , \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{gather*} Q _ 1 = \\tilde { q } _ 1 + \\frac { 1 } { 8 } \\left ( \\frac { \\tilde { p } _ 1 } { \\varepsilon ^ 6 } - \\frac { 3 \\tilde { p } _ 1 { } ^ 2 } { \\varepsilon ^ 2 } - \\frac { 3 \\tilde { p } _ 2 } { \\varepsilon ^ 2 } - \\frac { 2 \\tilde { t } _ 1 } { \\varepsilon ^ 2 } - \\frac { 3 } { \\varepsilon ^ { 1 0 } } \\right ) , Q _ 2 = \\tilde { q } _ 2 - \\frac { 3 \\tilde { p } _ 1 } { 8 \\varepsilon ^ 2 } + \\frac { 1 } { 8 \\varepsilon ^ 6 } . \\end{gather*}"}
-{"id": "4725.png", "formula": "\\begin{align*} \\dot x = F _ { 1 } ( J ( x ) ) { \\sqrt \\omega } u _ { 1 } ( \\omega t ) + F _ { 2 } ( J ( x ) ) { \\sqrt \\omega } u _ { 2 } ( \\omega t ) . \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{align*} \\mathbb { C } _ q [ G / T ] : = \\{ a \\in \\mathbb { C } _ q [ G ] : K _ \\lambda \\triangleright a = a \\} , \\mathbb { C } _ q [ T \\backslash G ] : = \\{ a \\in \\mathbb { C } _ q [ G ] : a \\triangleleft K _ \\lambda = a \\} . \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} X _ O : w _ 0 ^ 2 - q w _ 1 ^ 2 = p _ 1 p _ 2 w _ 2 ^ 2 . \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} \\left ( A u \\right ) _ { i } = u _ { + } s \\theta _ { i } \\frac { \\partial f } { \\partial \\theta _ { i } } i = 1 , \\ldots , d - 1 . \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} \\begin{cases} \\ , ( \\varphi ^ + \\ ! ) ' \\ , = \\ , \\varphi ^ + \\ ! - \\frac { 1 - \\nu } { r } \\ , \\varphi ^ - \\\\ \\ , ( \\varphi ^ - \\ ! ) ' \\ , = \\ , - \\frac { 1 + \\nu } { r } \\ , \\varphi ^ + \\ , . \\end{cases} \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} s _ t = t + \\frac { 1 } { 2 } \\log \\left ( 1 - \\frac { \\norm { v } ^ 2 } { e ^ { 2 t } } \\right ) . \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} W _ 1 ( \\lambda ) \\coloneqq \\begin{bmatrix} \\pi & \\lambda \\pi ^ n \\\\ 0 & 1 \\end{bmatrix} \\quad W _ 2 ( \\lambda ) \\coloneqq \\begin{bmatrix} 1 & 0 \\\\ \\lambda & \\pi \\end{bmatrix} . \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\underset { \\kappa \\in ( 0 , 1 ) } { s u p } \\mathbb { E } \\left | \\langle : X _ T \\otimes \\ldots \\otimes X _ T : , \\Phi _ { \\epsilon , \\psi _ \\kappa } ^ f \\rangle - \\rho _ { f , \\epsilon , \\psi _ \\kappa } ^ T \\right | ^ 2 = 0 , \\epsilon > 0 , \\end{align*}"}
-{"id": "4482.png", "formula": "\\begin{align*} \\underset { l \\rightarrow \\infty } { \\lim \\sup } \\ \\mathbb { E } \\left [ \\left ( \\mathbb { F } ^ { ( l ) } ( I _ { ( e _ { k } , 0 ) } ^ { ( \\omega , \\vartheta ) } ) , \\mathbb { F } ^ { ( l ) } ( I _ { ( e _ { q } , 0 ) } ^ { ( \\omega , \\vartheta ) } ) \\right ) _ { \\sim } ^ { ( \\omega ) } \\right ] = \\infty \\end{align*}"}
-{"id": "9607.png", "formula": "\\begin{align*} \\overline L ^ i _ { \\underline { j k } } = L ^ i _ { \\underline { j k } } + \\delta ^ i _ k \\psi _ j + \\delta ^ i _ j \\psi _ k - \\frac 1 2 \\overline F ^ i _ k \\overline \\sigma _ j - \\frac 1 2 \\overline F ^ i _ j \\overline \\sigma _ k + \\frac 1 2 F ^ i _ k \\sigma _ j + \\frac 1 2 F ^ i _ j \\sigma _ k . \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} a \\left ( j \\right ) u _ { y y } \\left ( t , j , \\varepsilon \\right ) + b \\left ( j \\right ) u _ { y } \\left ( t , j , \\varepsilon \\right ) = 0 j = 0 , 1 t \\in \\left ( 0 , T \\right ) , \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{gather*} K _ i K _ i ^ { - 1 } = K _ i ^ { - 1 } K _ i = 1 , \\ \\ K _ i K _ j = K _ j K _ i , \\\\ K _ i E _ j K _ i ^ { - 1 } = q _ i ^ { a _ { i j } } E _ j , \\ \\ K _ i F _ j K _ i ^ { - 1 } = q _ i ^ { - a _ { i j } } F _ j , \\\\ E _ i F _ j - F _ j E _ i = \\delta _ { i j } \\frac { K _ i - K _ i ^ { - 1 } } { q _ i - q _ i ^ { - 1 } } , \\end{gather*}"}
-{"id": "3153.png", "formula": "\\begin{align*} \\mu _ { H P } ^ { s , N + 1 } \\Lambda _ N ^ { N + 1 } = \\mu _ { H P } ^ { s , N } \\ , \\forall N \\ge 1 , \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} \\log ( 1 + \\varepsilon ) = \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n + 1 } \\frac { \\varepsilon ^ { n } } { n } . \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} R ^ L & \\asymp n ^ { - 1 } \\int _ { \\prod _ { k = 1 } ^ r x _ { p + k } \\leq ( n b ) ^ { 1 / m } , x _ { p + k } \\geq 1 } \\left ( 1 - \\frac { 1 } { n b } \\prod _ { k = 1 } ^ r x _ { p + k } ^ m \\right ) \\\\ & \\asymp [ \\log ( n b ) ] ^ { r - 1 } n ^ { - 1 + 1 / m } b ^ { 1 / m } \\asymp [ n ( \\log n ) ^ { 1 - r } ] ^ { - 2 m / ( 2 m + 1 ) } , \\end{align*}"}
-{"id": "9857.png", "formula": "\\begin{align*} \\boldsymbol { \\nabla } ^ 2 A ( \\boldsymbol { x } ) = - B ( \\boldsymbol { x } ) A ( \\boldsymbol { x } ) \\rightarrow 0 | \\boldsymbol { x } | \\rightarrow \\infty \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{gather*} t H _ { \\mathrm { F S } } ^ { A _ 3 } \\left ( { \\alpha _ , \\beta \\atop \\gamma , \\delta } ; t ; { q _ 1 , p _ 1 \\atop q _ 2 , p _ 2 } \\right ) = t H _ { \\mathrm { I I I } ( D _ 6 ) } ( \\alpha , \\beta ; t ; q _ 1 , p _ 1 ) + t H _ { \\mathrm { I I I } ( D _ 6 ) } ( \\gamma , \\delta ; t ; q _ 2 , p _ 2 ) - p _ 1 p _ 2 ( q _ 1 q _ 2 + t ) . \\end{gather*}"}
-{"id": "6358.png", "formula": "\\begin{align*} \\pi _ 1 ( a ) \\rho = \\begin{cases} \\pi _ 0 ( a ) \\rho & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8282.png", "formula": "\\begin{align*} \\mathbb { T } ^ { ( i ) } ( & M _ 1 \\otimes \\dots \\otimes M _ { i - 1 } \\otimes M _ i \\otimes M _ { i + 1 } \\otimes M _ { i + 2 } \\otimes \\dots \\otimes M _ n ) \\mathbb { T } ^ { ( i ) } \\\\ = & M _ 1 \\otimes \\dots \\otimes M _ { i - 1 } \\otimes M _ { i + 1 } \\otimes M _ { i } \\otimes M _ { i + 2 } \\otimes \\dots \\otimes M _ n \\end{align*}"}
-{"id": "9847.png", "formula": "\\begin{align*} \\tau ^ 1 _ { \\mu , \\mu } = ( 1 - q ^ { - 1 } ) \\frac { \\mathbf { z } ^ { \\left ( n _ \\alpha \\lceil \\frac { B ( \\alpha ^ \\vee , \\mu ) } { n _ \\alpha Q ( \\alpha ^ \\vee ) } \\rceil - \\frac { B ( \\alpha ^ \\vee , \\mu ) } { Q ( \\alpha ^ \\vee ) } \\right ) \\alpha ^ \\vee } } { 1 - q ^ { - 1 } \\mathbf { z } ^ { n _ \\alpha \\alpha ^ \\vee } } \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\frac { 1 } { 2 } } & Q ( x ) \\ ! \\int _ 0 ^ x K ( x , t ) \\cos ( 2 k t ) d t d x \\ ! + \\ ! \\int _ { \\frac { 1 } { 2 } } ^ b Q ( x ) \\ ! \\int _ 0 ^ x \\ ! K ( x , t ) \\cos ( 2 k t ) d t d x \\\\ & = \\int _ 0 ^ b \\int _ t ^ b K ( x , t ) Q ( x ) d x \\cos ( 2 k t ) d t . \\end{align*}"}
-{"id": "3867.png", "formula": "\\begin{align*} \\operatorname * { R e } z & \\in \\operatorname * { R e } \\Omega _ x , \\\\ \\left | \\operatorname * { I m } z _ i \\right | & \\leq \\frac { \\rho - 1 / \\rho } { 2 } D + \\varepsilon \\delta _ B \\leq \\left ( \\eta \\frac { \\rho - 1 / \\rho } { 2 } + \\varepsilon \\right ) \\delta _ B \\qquad i = 1 , \\ldots , d , \\end{align*}"}
-{"id": "8367.png", "formula": "\\begin{align*} | \\hat { T } ^ { ( k ) } _ { n } - \\hat { T } ^ { ( k ) } _ { n _ 1 } | \\leq \\hat { T } ^ { ( k ) } _ { n , n _ 1 } \\leq \\sum _ { i = n + 1 } ^ { n _ 1 } t ^ { ( k ) } ( f _ i ) \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} \\vert n _ 1 , n _ 2 , \\cdots , n _ r \\rangle = \\sqrt { \\frac { ( k - n _ 1 - n _ 2 \\cdots - n _ r ) ! } { k ! n _ 1 ! n _ 2 ! \\cdots n _ r ! } } { { ( a ^ + _ 1 ) } } ^ { n _ 1 } { { ( a ^ + _ 2 ) } } ^ { n _ 2 } \\cdots { { ( a ^ + _ r ) } } ^ { n _ r } \\vert 0 , 0 , \\cdots , 0 \\rangle n _ 1 + n _ 2 + \\cdots + n _ r \\leq k , \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{align*} & \\alpha \\in Y , \\psi _ { \\alpha , \\alpha } = 1 _ { G _ { \\alpha } } , \\textit { t h e i d e n t i t y a u t o m o r p h i s m o f } G _ { \\alpha } , \\\\ & \\alpha , \\beta , \\gamma \\in Y \\alpha \\geq \\beta \\geq \\gamma , \\end{align*}"}
-{"id": "7056.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A u + V \\left ( x , t \\right ) u = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , T \\right ] , \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} y \\mapsto { Q } ( x ) { S } ( y ) - { Q } ( y ) { S } ( x ) - { S } ( y ) \\prod _ { l = 1 } ^ n ( x - { x } _ l ) , k = 1 , \\dots , n , \\end{align*}"}
-{"id": "151.png", "formula": "\\begin{align*} c _ k \\sum _ { j = 0 } ^ { 2 k } \\binom { 2 k } { j } ( - 1 ) ^ { j - k } e ^ { i ( j - k ) \\theta } \\ , \\frac { d \\theta } { 2 \\pi } \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} Y ( \\chi _ { - 1 / 2 } | 0 \\rangle , z ) = H ^ { \\gamma } ( z ) , Y ( \\chi _ { - 3 / 2 } | 0 \\rangle , z ) = H ^ { \\beta } ( z ) . \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} & \\left ( ( 1 + \\sqrt { x } ) ^ { 4 k + 4 } + ( 1 - \\sqrt { x } ) ^ { 4 k + 4 } \\right ) = 2 \\sum _ { j = 0 } ^ { 2 k + 2 } { 4 k + 4 \\choose 2 j } x ^ j \\\\ = & 2 ( 1 + x ) ^ { 2 k + 2 } + 2 \\sum _ { j = 1 } ^ { 2 k + 1 } \\left ( { 4 k + 4 \\choose 2 j } - { 2 k + 2 \\choose j } \\right ) x ^ j = 2 ( 1 + x ) ^ { 2 k + 2 } + 8 x \\sum _ { j = 0 } ^ { 2 k } \\alpha _ j x ^ j \\\\ = & 2 ( 1 + x ) ^ { 2 k + 2 } + 8 x \\sum _ { j = 0 } ^ { k } a ' _ j x ^ j ( 1 + x ) ^ { 2 k - j } , \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} E _ 2 ^ { p , q } = H ^ p ( G ; H ^ q ( X ; R ) ) . \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} \\hat { \\sigma } _ A : = \\int _ { \\mathbb { R } ^ { 2 m } } \\hat { D } _ A ( \\mathbf { x } ) \\ , \\hat { \\rho } _ A \\ , { \\hat { D } _ A ( \\mathbf { x } ) } ^ \\dag \\ , \\mathrm { d } p _ X ( \\mathbf { x } ) \\ ; . \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} \\{ u , k , x \\} = \\{ s ( s ^ 2 - 3 t ^ 2 ) , t ( 3 s ^ 2 - t ^ 2 ) , t ^ 2 + s ^ 2 \\} \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{align*} A J ( k ) ^ { - 1 } = - \\frac { 1 } { 2 i k } [ I _ n + { U _ 0 } ] + O \\left ( \\frac { 1 } { k ^ 2 } \\right ) , \\ ; B J ( k ) ^ { - 1 } = \\frac { 1 } { 2 } [ I _ n - { U _ 0 } ] + O \\left ( \\frac { 1 } { k } \\right ) . \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} s _ \\lambda ( t _ 1 , t _ 2 , \\dots , t _ r ) = \\sum _ { \\mu \\preceq \\lambda } K _ { \\lambda \\mu } M _ { \\mu } ( t _ 1 , t _ 2 , \\dots , t _ r ) , \\end{align*}"}
-{"id": "10052.png", "formula": "\\begin{align*} ( I ( X _ { \\gamma _ { t } } ) , t \\geq 0 ) \\stackrel { ( d ) } { = } ( X ^ h _ t , t \\geq 0 ) , \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{align*} d \\mu ^ \\xi ( X ) = \\omega ( X , \\xi ^ * ) \\forall \\xi \\in \\mathfrak { g } , X \\in T S . \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} [ \\omega ] = 2 \\pi ( b [ D _ \\infty ] - a [ D _ 0 ] ) . \\end{align*}"}
-{"id": "4553.png", "formula": "\\begin{align*} \\rho = q | \\Psi | ^ { 2 } , \\mathbf { J } = - \\frac { \\mathrm { i } q \\hbar } { 2 m } \\big ( \\Psi ^ { \\ast } \\nabla { \\Psi } - \\Psi \\nabla { \\Psi } ^ { \\ast } \\big ) - \\frac { \\vert q \\vert ^ { 2 } } { m } \\vert \\Psi \\vert ^ { 2 } \\mathbf { A } , \\end{align*}"}
-{"id": "8351.png", "formula": "\\begin{align*} D i s ( \\tilde { q } _ { n } ( X , 1 ) ) = d _ 0 ^ { 2 n } \\mathop { \\Pi } _ { 1 \\le j < k \\le n + 1 } ( x ' _ j - x ' _ k ) ^ 2 , \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} S ( p ) : = \\frac { \\phi ( p - \\ell ) } { p - \\ell } - \\frac { \\phi ( p + \\ell ) } { p + \\ell } \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} d ^ { \\rm s u p } ( 1 0 ^ { - 1 } ) \\ , \\leq \\ , \\left \\{ \\begin{array} { l l } 1 & \\mbox { f o r \\ } a = 4 , \\\\ 1 & \\mbox { f o r \\ } a = 3 , \\\\ 3 & \\mbox { f o r \\ } a = 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} \\mu \\left ( G \\right ) = 3 / 2 + \\sqrt { n - 7 / 4 } . \\end{align*}"}
-{"id": "6057.png", "formula": "\\begin{align*} 0 \\neq [ V _ { - j } , \\ , L _ { - 1 } ] = [ V _ { - j } , \\ , [ L _ { - q + 1 } , \\ , L _ { q - 2 } ] ] = [ L _ { - q + 1 } , \\ , [ V _ { - j } , \\ , L _ { q - 2 } ] ] . \\end{align*}"}
-{"id": "3576.png", "formula": "\\begin{align*} \\phi \\wedge \\alpha = \\sum ( - 1 ) ^ { ( \\deg a _ i ) ( \\deg \\phi ) } a _ i \\wedge ( \\phi \\wedge h _ i ) . \\end{align*}"}
-{"id": "9924.png", "formula": "\\begin{align*} \\rho _ { \\Gamma } ( t , x ) : = \\Gamma ( t , x , x ) \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{align*} g ( s , \\cdot ) : = ( u ( s , \\cdot ) - u ( 0 , \\cdot ) ) w ( s ) + \\int _ { ( 0 , s ] } ( u ( s , \\cdot ) - u ( s - r , \\cdot ) ) \\mu ( d r ) , s > 0 , \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } \\frac { t ^ { j } } { \\left ( q \\right ) _ { j } } h _ { j } \\left ( x | q \\right ) = \\frac { 1 } { \\prod _ { k = 0 } ^ { \\infty } v \\left ( x | t q ^ { k } \\right ) } , \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} \\mathbb { P } ( { \\cal E } _ 1 = \\{ 1 , 2 , \\ldots , r \\} ) \\geq \\sum _ { { \\cal T } \\in { \\cal T } _ r } \\mathbb { P } ( { \\cal E } _ 1 = { \\cal T } ) . \\end{align*}"}
-{"id": "2139.png", "formula": "\\begin{align*} M ( V ) = \\begin{pmatrix} V & A ^ T \\\\ A & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} \\sum _ { \\substack { H \\leq F _ r , \\\\ H \\leq G _ r } } | F _ r / H | ^ a = p ^ a \\sum _ { \\substack { H \\leq G _ r } } | G _ r / H | ^ a = p ^ a \\sigma _ a ( G _ r ) \\end{align*}"}
-{"id": "6773.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A u = F \\left ( u , \\bar { u } \\right ) , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d t } \\star _ t \\varphi ( t ) & = \\frac { A \\sqrt { 6 } \\big ( y ( t ) ^ 3 z ( t ) ^ 2 - y ( t ) z ( t ) ^ 4 \\big ) - 1 } { 3 \\ , y ( t ) ^ 2 z ( t ) ^ 4 } ( e ^ { 1 2 3 4 } + e ^ { 1 2 5 6 } ) \\\\ & - \\frac { 2 A \\sqrt { 6 } \\ , y ( t ) z ( t ) ^ 4 + 1 } { 3 \\ , y ( t ) ^ 2 z ( t ) ^ 4 } e ^ { 3 4 5 6 } , \\end{aligned} \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{align*} - B ^ \\dag \\Psi ( 0 ) + A ^ \\dag \\Psi ' ( 0 ) = 0 , \\end{align*}"}
-{"id": "3623.png", "formula": "\\begin{align*} \\varphi ^ { \\prime \\prime } ( x ^ \\ast ) = - 2 y ^ 2 + o \\left ( y ^ 2 \\right ) . \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{align*} \\chi _ R ( x ) = \\chi ( \\tfrac x R ) \\ , ; \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} \\| u \\| _ { X ^ 2 ( Q _ T ) } + \\| u _ x \\big | _ { x = 0 } \\| _ { L _ 2 ( B _ T ) } + \\| u _ t \\| _ { X ( Q _ T ) } + \\| u _ { t x } \\big | _ { x = 0 } \\| _ { L _ 2 ( B _ T ) } \\leq c . \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} K _ B = \\left ( \\begin{array} { c c c c } 1 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 1 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} \\epsilon _ g \\epsilon _ { g ' } = \\epsilon _ { g g ' } . \\end{align*}"}
-{"id": "6545.png", "formula": "\\begin{align*} \\dot z = Q v ( x _ 0 + \\Gamma _ 0 z ) \\ , . \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{align*} \\aligned 0 = & \\sum _ { \\ell = 0 } ^ { k } \\frak n _ { k - \\ell } ( \\frak n _ { \\ell } ( y ; x _ 1 , \\dots , x _ { \\ell } ) , x _ { \\ell + 1 } , \\dots , x _ k ) ) \\\\ & + \\sum _ { i = - 1 } ^ { k } \\sum _ { j = i } ^ { k } ( - 1 ) ^ * \\frak n _ { k - j + i + 1 } ( y ; x _ 1 , \\dots , \\frak m ( x _ { i + 1 } , \\dots , x _ j ) , x _ { j + 1 } , \\dots , x _ k ) , \\endaligned \\end{align*}"}
-{"id": "9492.png", "formula": "\\begin{align*} \\phi ( R _ { 1 , A } ) = R _ { 2 , A } \\mbox { a n d } \\phi ( R _ { 1 , A \\cup L } ) = R _ { 2 , A \\cup L } . \\end{align*}"}
-{"id": "4192.png", "formula": "\\begin{gather*} T \\left ( x , 0 \\right ) = T _ { 0 } \\left ( x \\right ) , \\ ; \\ ; q \\left ( x , 0 \\right ) = 0 , \\ ; \\ ; x \\in \\mathbb { R } , \\\\ \\lim _ { x \\rightarrow \\pm \\infty } T \\left ( x , t \\right ) = 0 , \\ ; \\ ; \\lim _ { x \\rightarrow \\pm \\infty } q \\left ( x , t \\right ) = 0 , \\ ; \\ ; t > 0 . \\end{gather*}"}
-{"id": "10033.png", "formula": "\\begin{align*} N _ { J _ { \\varphi } } ( X , Y ) = \\frac { 4 } { 5 } N _ { \\varphi } ( X , Y ) , \\forall X , Y \\in { \\mathfrak X } ( M ) . \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} 1 + \\sum _ { j = 1 } ^ { \\left \\lfloor n / 2 \\right \\rfloor } ( ( n - j ) - j + 1 ) = 1 + \\sum _ { j = 1 } ^ { \\left \\lfloor n / 2 \\right \\rfloor } ( n - 2 j + 1 ) , \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} [ n ] P = O \\Longleftrightarrow f _ n ( x ) = 0 , \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} 3 c _ i + 6 b _ i = 6 k _ i + 9 x _ i \\wedge 2 d _ i + 6 b _ i = 6 k _ i + 4 y _ i \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} \\delta _ 0 = x _ 1 , \\delta _ k = [ \\delta _ { k - 1 } ( x _ 1 , \\ldots , x _ { 2 ^ { k - 1 } } ) , \\delta _ { k - 1 } ( x _ { 2 ^ { k - 1 } + 1 } , \\ldots , x _ { 2 ^ k } ) ] , \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} \\left . P _ n ( x ) = \\displaystyle \\lim _ { m \\longrightarrow 0 } \\mathcal { D } _ 0 ^ n f _ { \\nu } ( x , m ) = \\frac { 1 } { n ! } \\frac { \\partial ^ n } { \\partial m ^ n } f _ { \\nu } ( x , m ) \\right | _ { m = 0 } . \\end{align*}"}
-{"id": "3229.png", "formula": "\\begin{align*} e ^ { - \\lambda _ { N , s } t } \\frac { \\Delta _ N ( y ) } { \\Delta _ N ( x ) } \\det \\left ( p ^ { ( N ) , s } _ t ( x _ i , y _ j ) \\right ) ^ N _ { i , j = 1 } > 0 \\ , \\ \\forall \\left ( t , x , y \\right ) \\in ( 0 , \\infty ) \\times \\mathring { W } ^ N \\times \\mathring { W } ^ N , \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} y ^ 2 + a _ 1 x y + a _ 3 y = x ^ 3 + a _ 2 x ^ 2 + a _ 4 x \\end{align*}"}
-{"id": "9744.png", "formula": "\\begin{align*} ( B , \\rho , P ) ( t , x ) = \\int _ { a \\in \\mathcal { A } } \\left ( \\int _ { \\mathbb { R / Z } } \\delta ( x - X ( t , s , a ) ) ( \\partial _ s X , 1 , \\partial _ t X ) ( t , s , a ) d s \\right ) d \\mu ( a ) \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{align*} d \\lambda _ i ( t ) = 2 h ( \\lambda _ i ( t ) ) g ( \\lambda _ i ( t ) ) d \\beta _ i ( t ) + \\left ( b ( \\lambda _ i ( t ) ) + \\alpha \\sum _ { k = 1 } ^ { N } \\lambda _ k ( t ) + 2 \\sum _ { k \\ne i } ^ { } \\frac { G \\left ( \\lambda _ i ( t ) , \\lambda _ k ( t ) \\right ) } { \\lambda _ i ( t ) - \\lambda _ k ( t ) } \\right ) d t , \\end{align*}"}
-{"id": "9386.png", "formula": "\\begin{align*} L ^ p ( \\Omega ; E _ n ) ^ * = L ^ q ( \\Omega ; E _ n ^ * ) \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} 0 = P x = \\sum _ { i = 1 } ^ { n } \\sum _ { j = 1 } ^ n a _ { i j } e ^ * _ j ( x ) e _ i . \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{align*} P ^ T D \\mathbf { x } = \\mathbf { y } \\ ; \\ ; \\ ; \\mbox { \\rm i f a n d o n l y i f } \\ ; \\ ; \\ ; P ^ T D \\mathbf { y } = \\mathbf { x } , \\end{align*}"}
-{"id": "4380.png", "formula": "\\begin{align*} ( 1 - \\beta ) ^ e \\sum _ { 0 \\le k < q } \\binom { 2 k + e } { k } x ^ k \\equiv ( 1 - 2 \\beta ) ^ { q - 1 } \\pmod { ( \\beta ^ q , p ) } . \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} S ( C | M ) = \\int _ M S ( C | M = m ) \\ , \\mathrm { d } p _ M ( m ) \\ ; , \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = e _ 4 , [ e _ 1 , e _ 2 ] = \\alpha _ 2 e _ 4 + \\alpha _ 3 e _ 5 , [ e _ 2 , e _ 1 ] = \\beta _ 1 e _ 3 + \\beta _ 2 e _ 4 + \\beta _ 3 e _ 5 , [ e _ 2 , e _ 2 ] = e _ 5 . \\end{align*}"}
-{"id": "9481.png", "formula": "\\begin{align*} | H '' ( S ) | = \\frac { | H ' ( S ) | } { s _ { r - 1 } } = \\frac { r - | S | } { s _ { r - 1 } } | H ( S ) | . \\end{align*}"}
-{"id": "5586.png", "formula": "\\begin{align*} \\sum _ { j \\geq 0 } \\rho ^ { 2 j } U _ { 2 j } ( x ) = \\chi _ { 0 , 2 } ^ { 0 , 0 } ( x , 0 , i \\rho ) = \\frac { 1 + \\rho ^ { 2 } } { ( 1 + \\rho ^ { 2 } ) ^ { 2 } - 4 \\rho ^ { 2 } x ^ { 2 } } \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} F ( \\tau , x ) & : = \\frac { 1 } { 4 \\pi } \\Big ( \\lvert x - e \\rvert V ( x , 2 \\tau - \\lvert x \\rvert ) + \\lvert x \\rvert v _ { 2 } ( x , 2 \\tau - \\lvert x - e \\rvert ) \\\\ & \\ \\ \\ \\ \\ \\ + 4 \\pi \\lvert x \\rvert \\lvert x - e \\rvert \\int \\limits _ { \\lvert x \\rvert } ^ { 2 \\tau - \\lvert x - e \\rvert } V ( x , 2 \\tau - t ) v _ { 2 } ( x , t ) d t \\Big ) \\end{align*}"}
-{"id": "9177.png", "formula": "\\begin{align*} f = H l , \\end{align*}"}
-{"id": "2846.png", "formula": "\\begin{align*} \\Pi = \\Lambda _ n : = \\left \\{ 0 , \\frac { 1 } { n } \\pi , \\dots , \\frac { n - 1 } { n } \\pi \\right \\} \\end{align*}"}
-{"id": "5358.png", "formula": "\\begin{align*} f _ 0 ( 1 ) = 1 , f _ 0 ( 2 ) = 0 , f _ 0 ( n + 2 ) = a f _ 0 ( n ) , ( n \\geq 1 ) . \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} y ( x , \\lambda ) \\tilde { y } ( x , \\lambda ) = \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\cos ( 2 k x ) + \\frac { 1 } { 2 } \\int _ 0 ^ x K ( x , t ) \\cos ( 2 k t ) d t \\end{align*}"}
-{"id": "9477.png", "formula": "\\begin{align*} c \\le 0 . 9 ( p / 2 ) ^ { h } / ( q ^ r 4 ^ q ) , \\mbox { w h e r e } q : = 2 f \\cdot f ! \\mbox { a n d } h : = 2 ^ r \\binom { q + r } { r } , \\end{align*}"}
-{"id": "1955.png", "formula": "\\begin{align*} \\widehat { E } _ W = \\{ ( i , j ) \\in \\widetilde { E } _ W \\mid f _ { j i } ( - p _ { i j } ) \\geq r _ j \\} . \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} \\tilde j ( 1 - \\rho ) \\tilde J \\phi ( x , v ) = e ^ { - \\int _ 0 ^ { \\tau _ + ( x , v ) } \\sigma ( x + s v , v ) d s } \\phi ( x + \\tau _ + ( x , v ) v , v ) , \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} \\rho = \\overset { n } { \\sum \\limits _ { j = 1 } } \\mu _ { j } \\varrho ^ { ( \\beta _ { j } ) } \\end{align*}"}
-{"id": "2653.png", "formula": "\\begin{align*} \\epsilon _ g ( f ) = \\begin{cases} 1 g | f \\\\ 0 . \\end{cases} \\end{align*}"}
-{"id": "8866.png", "formula": "\\begin{align*} { \\bf M } { \\bf \\alpha } = { \\bf 0 } \\quad { \\bf M } = \\left ( \\begin{matrix} [ w _ 1 , w _ 1 ] \\big | _ a ^ b & \\dots & [ w _ 1 , w _ { r } ] \\big | _ a ^ b \\\\ \\vdots & \\ddots & \\vdots \\\\ [ w _ { r } , w _ 1 ] \\big | _ a ^ b & \\dots & [ w _ { r } , w _ { r } ] \\big | _ a ^ b \\end{matrix} \\right ) , { \\bf \\alpha } = \\left ( \\begin{matrix} \\alpha _ 1 \\\\ \\vdots \\\\ \\alpha _ r \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "3110.png", "formula": "\\begin{align*} \\Delta _ 2 = | | D ^ 2 E ( \\boldsymbol { 0 } ) ^ { - 1 } | | _ { l ^ 2 ( \\mathbb { Z } ) } ^ { - 1 } \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{align*} v _ c = \\frac { - b _ { v c } \\pm \\sqrt { b _ { v c } ^ 2 - 4 a _ { v c } c _ { v c } } } { 2 a _ { v c } } \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} \\left \\Vert e ^ { \\gamma \\left \\vert x \\right \\vert ^ { p } } \\tilde { u } _ { k } \\left ( x , 0 \\right ) \\right \\Vert _ { X } = \\left \\Vert e ^ { \\gamma \\left ( \\frac { \\alpha } { \\beta } \\right ) ^ { p / 2 } \\left \\vert x \\right \\vert ^ { p } } \\upsilon \\left ( x , 0 \\right ) \\right \\Vert _ { X } = a _ { 0 } , \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} W _ \\beta = \\left \\{ f \\in R \\left | \\begin{array} { l } \\dfrac { \\partial f } { \\partial z _ 1 } ( 0 , \\rho ) = \\dfrac { \\xi _ 1 ^ 2 \\rho } { \\xi _ 1 + \\beta \\rho } f ( 0 , \\rho ) \\\\ \\dfrac { \\partial f } { \\partial z _ 2 } ( \\rho , 0 ) = \\dfrac { \\xi _ 2 ^ 2 \\rho } { \\xi _ 2 + \\beta \\rho } f ( \\rho , 0 ) \\end{array} \\right . \\right \\} . \\end{align*}"}
-{"id": "4455.png", "formula": "\\begin{align*} t _ { 1 } : = \\min \\left \\{ t \\geq t _ { 0 } : \\int \\nolimits _ { t _ { 0 } } ^ { t ^ { \\prime } } E _ { \\mathbf { A } } ( s , x ) \\mathrm { d } s = 0 x \\in \\mathbb { R } ^ { d } t ^ { \\prime } \\geq t \\right \\} \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} [ e _ 1 , e _ 2 ] = \\alpha _ 3 e _ 4 + \\alpha _ 4 e _ 5 , [ e _ 2 , e _ 1 ] = \\alpha _ 5 e _ 4 + \\alpha _ 6 e _ 5 , [ e _ 2 , e _ 2 ] = \\beta _ 2 e _ 5 , [ e _ 3 , e _ 3 ] = \\gamma _ 6 e _ 5 . \\end{align*}"}
-{"id": "4976.png", "formula": "\\begin{align*} \\langle \\eta _ * , \\psi _ * \\rangle _ { L ^ 2 } = \\frac { 1 } { 2 } \\left ( \\int _ { \\mathbb { R } } { \\rm s e c h } ^ 3 ( \\sqrt { c _ * } \\xi ) d \\xi \\right ) ^ 2 = \\frac { \\pi ^ 2 } { 8 c _ * } . \\end{align*}"}
-{"id": "6230.png", "formula": "\\begin{align*} b = b _ a ( n - 1 ) a _ { n - 1 } + b _ a ( n - 2 ) a _ { n - 2 } + \\cdots + b _ a ( 1 ) a _ { 1 } + b _ a ( 0 ) a _ { 0 } . \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} \\dot \\psi & = i \\nabla _ { \\bar \\psi } \\chi ( \\psi , \\bar \\psi ) = : X _ \\chi ( \\psi , \\bar \\psi ) \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{align*} L ( a _ m ^ - ) = L ( a _ m ^ + ) = 0 L ( a ) > 0 a _ m ^ - < a < a _ m ^ + . \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { t ^ { n } a ( D ) ^ { n } } { n ! } u \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} U _ { n } = \\frac { \\alpha ^ { n } - \\beta ^ { n } } { \\alpha - \\beta } , \\ V _ { n } = \\alpha ^ { n } + \\beta ^ { n } , \\end{align*}"}
-{"id": "7376.png", "formula": "\\begin{align*} \\delta = \\mu + \\nu - \\alpha \\bigg ( \\frac { { \\beta + \\gamma + 2 } } { { 2 } } \\bigg ) , \\end{align*}"}
-{"id": "6074.png", "formula": "\\begin{align*} 0 \\neq [ L _ { - 4 } , L _ { 2 } ] = [ [ L _ { - 2 } , L _ { - 2 } ] , L _ { 2 } ] \\subseteq [ L _ { - 2 } , \\ , [ L _ { - 2 } , L _ { 2 } ] ] \\end{align*}"}
-{"id": "9970.png", "formula": "\\begin{align*} P _ 1 = P _ 2 = P : = \\left [ \\begin{array} { c c c c } k _ 1 & \\epsilon m _ 1 & 0 & 0 \\\\ \\epsilon m _ 1 & m _ 1 & 0 & 0 \\\\ 0 & 0 & k _ 2 & \\epsilon m _ 2 \\\\ 0 & 0 & \\epsilon m _ 2 & m _ 2 \\end{array} \\right ] \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} W ( p , q ) ^ { n } = \\left [ \\begin{array} { c c } p & - q \\\\ 1 & 0 \\end{array} \\right ] ^ { n } = \\left [ \\begin{array} { c c } U _ { n + 1 } & - q U _ { n } \\\\ U _ { n } & - q U _ { n - 1 } \\end{array} \\right ] , \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} x _ { n } = \\sum _ { m = 1 } ^ { n } \\frac { \\left ( - 1 \\right ) ^ { m } } { n ! } \\tilde { S } _ { m , n } \\end{align*}"}
-{"id": "9111.png", "formula": "\\begin{gather*} A = \\left ( \\begin{matrix} 4 & 0 & 0 & 0 \\\\ 0 & 4 & 0 & 0 \\\\ 0 & 0 & 3 & 1 \\\\ 0 & 0 & 1 & 3 \\end{matrix} \\right ) \\mathbf { a } = ( 1 , 1 , 1 , 1 ) . \\end{gather*}"}
-{"id": "7955.png", "formula": "\\begin{align*} \\mbox { d i v $ \\bar g $ } = ( 1 - \\phi _ 0 ) ( h - h ^ 2 ) \\sum _ j ( \\partial _ j u _ s ) \\cdot \\nabla u _ { s j } - g \\cdot \\nabla \\phi _ 0 , \\end{align*}"}
-{"id": "9431.png", "formula": "\\begin{align*} \\frac { { \\rm d } } { { \\rm d } t } \\int _ { x \\in \\mathbb { T } ^ 3 } \\frac { B ^ 2 + 1 } { 2 h } + \\int _ { x \\in \\mathbb { T } ^ 3 } \\frac { D ^ 2 + P ^ 2 } { h } = 0 \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{align*} w _ 1 & = [ E _ 1 ] _ \\Lambda \\\\ w _ 2 & = [ E _ 2 ] _ \\Lambda , \\end{align*}"}
-{"id": "180.png", "formula": "\\begin{align*} \\sum _ { t \\in \\mathbb F _ q } \\nu _ k ^ 2 ( t ) & = \\sum _ { t \\in \\mathbb F _ q } \\nu _ k ( t ) \\ , \\overline { \\nu _ k } ( t ) \\\\ & = q ^ { 2 d k } \\sum _ { { \\bf m } , { \\bf v } \\in \\mathbb F _ q ^ d } \\left ( \\prod _ { j = 1 } ^ k \\overline { \\widehat { E _ j } } ( { \\bf m } ) \\right ) \\left ( \\prod _ { j = 1 } ^ k \\widehat { E _ j } { ( { \\bf v } ) } \\right ) \\sum _ { t \\in \\mathbb F _ q } \\widehat { S _ t } ( { \\bf m } ) \\ , \\overline { \\widehat { S _ t } } ( { \\bf v } ) . \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} V ^ \\perp = \\begin{cases} \\mathbf 0 & \\\\ V & \\end{cases} \\end{align*}"}
-{"id": "7305.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = \\alpha _ 2 y _ 5 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\beta _ 1 } y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = - \\frac { \\alpha _ 3 } { \\beta _ 1 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 2 ] = y _ 4 , [ y _ 1 , y _ 3 ] = \\beta _ 4 y _ 5 , \\\\ [ y _ 3 , y _ 2 ] = \\gamma _ 4 y _ 5 . \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} 0 = [ L _ { - q + 2 } , \\ , L _ { - 2 } ] \\supseteq [ [ L _ { - q } , \\ , S _ 2 ] , \\ , L _ { - 2 } ] = [ L _ { - q } , \\ , [ L _ { - 2 } , \\ , S _ 2 ] ] \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} D _ { n , k } ( 1 , y ( 1 - y ) ) & = k \\ , \\Big [ \\displaystyle \\frac { y ^ n ( 1 - y ) - y ( 1 - y ) ^ n } { 2 y - 1 } \\Big ] + D _ n ( 1 , y ( 1 - y ) ) , \\end{align*}"}
-{"id": "3587.png", "formula": "\\begin{align*} H _ { \\rm { O H } } = \\frac 1 2 \\left ( - \\partial _ p ^ 2 + p ^ 2 - 1 \\right ) ~ , \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} V _ m : = U _ { n _ { k _ m } } \\cap \\Delta _ m \\end{align*}"}
-{"id": "6416.png", "formula": "\\begin{align*} \\begin{bmatrix} \\pi & b d ^ { - 1 } \\pi ^ n \\\\ 0 & 1 \\end{bmatrix} \\begin{bmatrix} 1 & x \\pi ^ n \\\\ 0 & 1 \\end{bmatrix} = \\begin{bmatrix} \\pi & \\lambda \\pi ^ n \\\\ 0 & 1 \\end{bmatrix} , \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} J ( u ) = \\int _ 0 ^ 1 \\left ( \\int _ 0 ^ x f ( t ) d t \\right ) ^ 2 + ( f ( x ) ^ 2 - 1 ) ^ 2 d x \\end{align*}"}
-{"id": "6332.png", "formula": "\\begin{align*} ( a _ 3 ^ \\vee ) ^ 2 = - \\frac { 3 ( n - 2 ) } { n + 1 } \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} Y _ T ^ 2 = Y _ 0 ^ 2 + ( 2 a + \\sigma _ 1 ^ 2 ) \\int _ 0 ^ T Y _ s \\ , \\dd s + 2 \\sigma _ 1 \\int _ 0 ^ T Y _ s ^ { \\frac { 3 } { 2 } } \\ , \\dd W _ s . \\end{align*}"}
-{"id": "1513.png", "formula": "\\begin{align*} \\sum _ { \\substack { j = 1 \\\\ j \\neq k } } ^ n \\frac { 2 } { x _ k - x _ j } - \\sum _ { j = 1 } ^ { n - 1 } \\left ( \\frac { 1 } { x _ k - \\zeta _ j } + \\frac { 1 } { x _ k - \\overline { \\zeta } _ j } \\right ) = 0 , k = 1 , 2 , \\dots , n . \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} X ^ { f } = \\ , \\Ref [ ( X ^ { ' f } + \\Tilde \\xi ^ { f } ) \\mathbb { I } _ { [ 0 , T ) } ] ; X ^ { ' f } = \\ , \\Ref [ ( X ^ { f } - \\Tilde \\zeta ^ { f } ) \\mathbb { I } _ { [ 0 , T ) } ] , \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} { \\rm T r } \\left ( \\sigma ( x ) \\sigma ^ * ( x ) D ^ 2 u ( t , x ) \\right ) = \\sum _ { n } D ^ 2 u ( t , x ) . ( \\sigma ^ 2 ( x ) e _ n , e _ n ) , \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} E [ 1 \\mbox { - w d ( R ) } ] = \\frac { 2 } { \\sqrt { \\pi } } \\sqrt { n } ( 1 + o ( 1 ) ) . \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} d X _ t & = - \\frac { 2 X _ t } { X _ t ^ 2 + Y _ t ^ 2 } d t - d \\xi _ t , \\end{align*}"}
-{"id": "2440.png", "formula": "\\begin{align*} g '' ( x ) + A _ { \\lambda , \\mu } g ' ( x ) + B _ { \\lambda , \\mu } g ( x ) = 0 . \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{align*} B _ { i j } = \\frac { 1 } { n - 3 } \\nabla ^ { k } \\nabla ^ { l } W _ { i k j l } + \\frac { 1 } { n - 2 } R _ { k l } W _ { i } \\ , ^ { k } \\ , _ { j } \\ , ^ { l } , \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{align*} S ^ r = \\begin{pmatrix} s _ 0 \\alpha & \\sigma ^ { - 1 } ( s _ 1 ) \\alpha & \\ldots & \\sigma ^ { - r + 1 } ( s _ { r - 1 } ) \\alpha \\\\ s _ 1 \\sigma ( \\alpha ) & \\sigma ^ { - 1 } ( s _ 2 ) \\sigma ( \\alpha ) & \\ldots & \\sigma ^ { - r + 1 } ( s _ r ) \\sigma ( \\alpha ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ s _ t \\sigma ^ t ( \\alpha ) & \\sigma ^ { - 1 } ( s _ { t + 1 } ) \\sigma ^ t ( \\alpha ) & \\ldots & \\sigma ^ { - r + 1 } ( s _ { t + r - 1 } ) \\sigma ^ t ( \\alpha ) \\\\ \\end{pmatrix} _ { ( t + 1 ) \\times r } . \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{align*} \\begin{gathered} \\sum _ { k } u _ { a } ^ { k * } u _ { b } ^ { k } = \\delta _ { b } ^ { a } 1 = \\sum _ { k } u _ { k } ^ { a } u _ { k } ^ { b * } , \\\\ \\sum _ { k } q ^ { ( 2 \\rho , \\lambda _ { k } - \\lambda _ { b } ) } u _ { b } ^ { k } u _ { a } ^ { k * } = \\delta _ { b } ^ { a } 1 = \\sum _ { k } q ^ { ( 2 \\rho , \\lambda _ { a } - \\lambda _ { k } ) } u _ { k } ^ { b * } u _ { k } ^ { a } . \\end{gathered} \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} U _ { 2 m } ^ { ( 2 ) } & = U _ { m + 1 } U _ { m - 1 } + q ^ { m - 1 } \\\\ & = U _ { m - 1 } \\left ( p U _ { m } - q U _ { m - 1 } \\right ) + q ^ { m - 1 } \\\\ & = p U _ { m - 1 } U _ { m } - q U _ { m - 1 } ^ { 2 } + q ^ { m - 1 } \\\\ & = p U _ { 2 m - 1 } ^ { ( 2 ) } - q U _ { 2 m - 2 } ^ { ( 2 ) } + q ^ { m - 1 } . \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} \\operatorname { L i } _ { 2 } ( z ) : = \\sum _ { n = 1 } ^ \\infty \\frac { z ^ n } { n ^ 2 } . \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} u = \\nabla ^ \\perp \\psi + V ( \\varpi ) \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{align*} \\big \\langle \\gamma _ { 1 } u \\ , , \\ , \\chi _ { \\partial \\Omega ^ \\complement _ { ( j ) } } \\big \\rangle = 0 \\forall j \\in \\{ 1 , \\dots , m \\} \\ , . \\end{align*}"}
-{"id": "537.png", "formula": "\\begin{align*} G ( x , v ) : = \\left \\lbrace \\begin{array} { l } ( 1 - ( v _ 0 \\cdot v ) ^ 2 ) ( \\tilde E _ 1 k _ 1 - \\tilde E _ 2 k _ 2 ) ( x - { ( x - x _ 0 ) \\cdot ( v - ( v \\cdot v _ 0 ) v _ 0 ) \\over 1 - ( v \\cdot v _ 0 ) ^ 2 } v , v _ 0 , v ) \\textrm { w h e n } v \\not = \\pm v _ 0 , \\\\ 0 \\textrm { o t h e r w i s e } , \\end{array} \\right . \\end{align*}"}
-{"id": "7478.png", "formula": "\\begin{align*} \\langle v , w \\rangle = \\sum _ { i \\in Q _ 0 } m _ i \\mathrm { t r } \\left ( h _ i ^ { - 1 } v h _ i ^ { - 1 } w \\right ) , v , w \\in T _ h ( G / K ) \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{align*} J _ 1 ( k ) = J _ 1 ^ 0 ( k ) - Q \\frac { J _ 1 ^ 0 ( k ) } { i k } + \\frac { Q B _ 1 } { i k } + o ( 1 ) , | k | \\to \\infty , \\ ; \\ ; k \\in \\overline { \\mathbb { C } } ^ { + } . \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{align*} \\kappa = ( ( 1 , t _ k ) , ( 2 , t _ k ) , ( 3 , t _ k ) , \\ldots , ( 3 k - 1 , t _ k ) ) . \\end{align*}"}
-{"id": "3505.png", "formula": "\\begin{align*} I _ { 1 / 2 } ( 1 / 2 ; x ) = e ^ { x } I _ { 1 / 2 } ( x ) = \\sqrt { \\frac { 2 } { \\pi x } } e ^ { x } \\sinh x , \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\dfrac { T ( t ) x - x } { t } = A x , \\mbox { f o r e v e r y } x \\in X . \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} J _ T = \\begin{pmatrix} 1 & 0 \\\\ e ^ { - N \\xi _ - } & 1 \\end{pmatrix} \\begin{pmatrix} 0 & 1 \\\\ - 1 & 0 \\end{pmatrix} \\begin{pmatrix} 1 & 0 \\\\ e ^ { - N \\xi _ + } & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } S ( A | M ) _ { ( \\mathcal { N } ( t ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) } = S ( A | M ) _ { \\hat { \\rho } _ { A M } } \\ ; . \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} C _ k = \\frac { 1 } { k + 1 } \\binom { 2 k } { k } = \\binom { 2 k } { k } - \\binom { 2 k } { k + 1 } \\end{align*}"}
-{"id": "8804.png", "formula": "\\begin{align*} \\sigma _ { , k } y ^ { k } y ^ { i } - \\dfrac { 1 } { 2 } g ^ { i k } \\sigma _ { , k } L = P y ^ { i } . \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{align*} \\left ( \\dfrac { \\dot { b } } { b } \\right ) ^ 2 + 2 \\dfrac { \\ddot { b } } { b } - \\Lambda = - \\dfrac { 8 \\pi \\rho } { 3 } - 4 \\pi \\dot { \\phi } ^ 2 , \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} \\epsilon ^ { \\textnormal { h o m } } ( \\theta ) \\tilde { E } = \\tilde { f } . \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} [ P ( J ) ^ + - P ( J ) ^ - , J ] = 0 , \\end{align*}"}
-{"id": "4872.png", "formula": "\\begin{align*} ( h _ 1 h _ 2 ) ^ 3 & = \\begin{pmatrix} \\alpha w & \\beta w \\\\ 0 & - \\alpha z \\end{pmatrix} ^ 3 \\\\ & = \\begin{pmatrix} \\alpha ^ 3 w ^ 3 & * \\\\ 0 & - \\alpha ^ 3 z ^ 3 \\end{pmatrix} . \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{align*} | t _ f - m | = | \\sqrt { t _ f } - \\sqrt { m } | ( \\sqrt { t _ f } + \\sqrt { m } ) \\le C ( 2 \\sqrt { t _ f } + C ) . \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} Z ^ p _ m ( x , \\zeta ) = \\left [ C ^ { n / 2 } _ m ( \\frac { x \\cdot \\overline { \\zeta } } { | x | | \\overline { \\zeta } | } ) - C ^ { n / 2 } _ { m - 2 p } ( \\frac { x \\cdot \\overline { \\zeta } } { | x | | \\overline { \\zeta } | } ) \\right ] | x | ^ m | \\overline { \\zeta } | ^ m , \\end{align*}"}
-{"id": "1224.png", "formula": "\\begin{align*} \\Pi _ { n } z = S _ { \\Delta t } z + \\Delta t S _ { \\Delta t } B e ^ { \\tau A } G ' ( X _ n ) . z + S _ { \\Delta t } e ^ { \\tau A } \\bigl ( \\sigma ' ( X _ n ) . z \\bigr ) \\Delta W _ n . \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { N - 1 } \\big | \\mathcal { E } _ { N , \\ell } ^ { h , k , 2 , 1 } \\big | \\le \\frac { C ( 1 + | x | _ { L ^ p } ) ^ K } { \\tau ^ { 2 \\kappa } } \\bigl ( t _ { N - 1 } ^ { - \\frac 1 2 + \\kappa } \\Delta t | h | _ { L ^ { 2 q } } | k | _ { L ^ { 2 q } } + t _ { N - 1 } ^ { \\frac 1 2 + \\kappa - \\beta - \\gamma } | ( - A ) ^ { - \\beta } h | _ { L ^ { 2 q } } | ( - A ) ^ { - \\gamma } k | _ { L ^ { 2 q } } \\bigr ) . \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{align*} \\eta _ a ( C ( \\mathsf { P } ) ) = \\sum _ { i } c _ { i } ^ { i } q ^ { ( \\alpha _ { a } - 2 \\rho , \\lambda _ { i } ) } \\frac { q ^ { ( \\alpha _ { a } , \\lambda _ { i } ) } - q ^ { - ( \\alpha _ { a } , \\lambda _ { i } ) } } { q _ { a } - q _ { a } ^ { - 1 } } . \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} \\frac { ( a - 1 ) ^ 2 } { 4 } = \\frac { b } { c } , \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} x ^ 4 = x \\implies x ^ 2 = x \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} \\mathbf { Q } ( p , q ) = E \\left [ \\left ( \\widetilde { H } _ { i , j } ( n ) - \\eta _ { i , j } ( n ) \\right ) \\left ( \\widetilde { H } _ { \\ell , r } ( m ) - \\eta _ { \\ell , r } ( m ) \\right ) ^ * \\right ] , \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } \\binom { 2 k } { k } \\frac { x ^ k } { k } \\equiv \\pounds _ 1 ( \\beta ) + \\pounds _ 1 ( 1 - \\beta ) \\pmod { p } , \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} \\tilde { p } _ { k , l } ( c ^ 2 ) & = 0 ; \\\\ c ^ 2 = c ^ 2 _ { l , \\alpha } & : = \\frac { \\lambda _ l } { \\alpha ( \\alpha + 2 ) } , \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{align*} \\mathsf { P } ^ * = \\mathsf { P } , \\mathsf { P } ^ 2 = \\mathsf { P } , \\mathrm { T r } ( K _ { 2 \\rho } ^ { - 1 } \\mathsf { P } ) = q ^ { r - N } [ r ] _ q . \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} | t | ^ 2 _ { \\Omega ^ 2 } = | t ^ { 2 , 0 } | ^ 2 + | t ^ { 1 , 1 } | ^ 2 = 3 | t ^ { 2 , 0 } | ^ 2 = \\frac 1 3 | ( d ^ c F ) ^ { 2 , 0 } | ^ 2 . \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to \\infty } \\frac { \\alpha - 1 } { \\left ( [ \\alpha ] - 1 \\right ) ^ { 1 - \\{ \\alpha \\} } \\left ( [ \\alpha ] \\right ) ^ { \\{ \\alpha \\} } } = 1 . \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} \\alpha _ { \\mathsf { n } } = \\mathcal { N } \\int _ { \\mathbb { R } ^ { d } } \\mathsf { d x } \\varphi _ { 0 } ( \\mathsf { x } ) \\mathsf { h } _ { \\mathsf { n } } \\left ( \\mathsf { x } \\right ) , \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{align*} g ( z ) = g ( r e ^ { i \\theta } ) = \\sum _ { k = - \\infty } ^ { \\infty } g _ k ( r ) e ^ { i k \\theta } \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{align*} ( 1 - t ^ { 2 } \\mathbf { h } ( x ) \\mathbf { e } ( x y ) ) ^ { - 1 } ( 1 + t \\mathbf { h } ( x ) ) = \\frac { 1 } { 1 - t } + \\sum _ { n = 1 } ^ { \\infty } \\sum _ { L \\vDash n } \\frac { N _ L } { ( 1 - t ) ( 1 - t ^ { 2 } ) ^ { n } } x ^ { n } \\mathbf { r } _ { L } \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} M [ G _ { \\alpha } ] = M [ F _ { \\alpha } ] [ H _ { \\alpha } ] \\end{align*}"}
-{"id": "93.png", "formula": "\\begin{align*} \\frak { F u k } ( - \\Sigma ) = \\frak { F u k } ( \\Sigma ) ^ { \\rm o p } . \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{align*} | D ^ m u ( x ) | ^ 2 \\leq C _ { N , m } \\sum _ { k = 1 } ^ m \\frac { ( U ^ { ( k ) } ( x - a ) ) ^ 2 } { | x - a | ^ { 2 ( m - k ) } } . \\end{align*}"}
-{"id": "7977.png", "formula": "\\begin{align*} & \\| \\widetilde U ( t ) \\| _ r \\leq C ( t - \\bar t ) ^ { - 1 / 2 + 3 / 2 r } \\| U ( \\bar t ) \\| _ { 3 , \\infty , \\mathbb R ^ 3 } , \\forall r \\in ( 3 , \\infty ] , \\\\ & \\| \\nabla \\widetilde U ( t ) \\| _ { 3 , \\infty } \\leq C ( t - \\bar t ) ^ { - 1 / 2 } \\| U ( \\bar t ) \\| _ { 3 , \\infty , \\mathbb R ^ 3 } , \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} S _ T \\ ; & = \\ ; S ^ * \\upharpoonright \\mathcal { D } ( S _ T ) \\\\ \\mathcal { D } ( S _ T ) \\ ; & = \\ ; \\left \\{ f + S _ D ^ { - 1 } ( T v + w ) + v \\left | \\ ! \\ ! \\begin{array} { c } f \\in \\mathcal { D } ( \\overline { S } ) \\ , , \\ ; v \\in \\mathcal { D } ( T ) \\\\ w \\in \\ker S ^ * \\cap \\mathcal { D } ( T ) ^ \\perp \\end{array} \\ ! \\ ! \\right . \\right \\} . \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{align*} & D _ m \\Big \\{ \\big [ v ( t ) , w ( t ) \\big ] ; \\big [ \\bar { v } ( t ) , \\bar { w } ( t ) \\big ] \\Big \\} \\\\ & = \\sqrt { \\left ( \\begin{array} { c } v ( t ) - \\bar { v } ( t ) \\\\ w ( t ) - \\bar { w } ( t ) \\end{array} \\right ) ^ { \\top } \\Theta _ { i j } ^ { + } ( t ) \\left ( \\begin{array} { c } v ( t ) - \\bar { v } ( t ) \\\\ w ( t ) - \\bar { w } ( t ) \\end{array} \\right ) } . \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} \\widehat { P _ { \\leq M } f } ( \\xi ) & : = \\varphi ( M ^ { - 1 } \\xi ) \\hat { f } ( \\xi ) , \\\\ \\widehat { P _ { > M } f } ( \\xi ) & : = ( 1 - \\varphi ( M ^ { - 1 } \\xi ) ) \\hat { f } ( \\xi ) , \\\\ \\widehat { P _ M f } ( \\xi ) & : = ( \\varphi ( M ^ { - 1 } \\xi ) - \\varphi ( 2 M ^ { - 1 } \\xi ) ) \\hat { f } ( \\xi ) , \\end{align*}"}
-{"id": "4922.png", "formula": "\\begin{align*} A _ { i j } y _ { j } = \\begin{pmatrix} a _ { i j } & b _ { i j } \\\\ b _ { i j } & a _ { i j } \\end{pmatrix} x _ { j } \\mathbf { f = } \\begin{pmatrix} a _ { i j } - b _ { i j } \\\\ b _ { i j } - a _ { i j } \\end{pmatrix} x _ { j } = c _ { i j } x _ { j } \\mathbf { f } \\end{align*}"}
-{"id": "6086.png", "formula": "\\begin{align*} [ V _ { - 2 } , \\ , [ V _ { - 2 } , \\ , [ V _ { - 2 } , \\ , L _ 5 ] ] ] = L _ { - 1 } , \\end{align*}"}
-{"id": "9154.png", "formula": "\\begin{align*} E _ 2 ^ { s , t } = H ^ s ( \\mathbb { Z } / p ; N ^ { \\oplus \\frac { 1 } { p } { p \\choose t / p } } ) = \\begin{cases} ( \\mathbb { Z } / p ) ^ { \\frac { 1 } { p } { p \\choose t / p } } , & s = 0 , \\\\ 0 , & s > 0 . \\end{cases} \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { B C _ { 2 , n } } { \\Pi ( n ) } x ^ n & = \\dfrac { \\dfrac { x ^ 9 } { D _ 2 } } { \\sum _ { i = 2 } ^ \\infty \\dfrac { x ^ { 3 ^ i } } { D _ i } } \\\\ & = 1 - \\frac { D _ 2 } { D _ 3 } x ^ { 1 8 } + \\frac { D _ 2 ^ 2 } { D _ 3 ^ 2 } x ^ { 3 6 } - \\frac { D _ 2 ^ 3 } { D _ 3 ^ 3 } x ^ { 5 4 } \\\\ & \\quad + \\left ( \\frac { D _ 2 ^ 4 } { D _ 3 ^ 4 } - \\frac { D _ 2 } { D _ 4 } \\right ) x ^ { 7 2 } - \\left ( \\frac { D _ 2 ^ 5 } { D _ 3 ^ 5 } - \\frac { 2 D _ 2 ^ 2 } { D _ 3 D _ 4 } \\right ) x ^ { 7 2 } + \\cdots \\ , . \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{align*} E _ k \\triangleright w = \\sum _ { m = 1 } ^ r E _ k \\triangleright ( v _ m \\otimes f ^ m ) = q ^ { 1 / 2 } v _ { k } \\otimes f ^ { k + 1 } - q ^ { 1 / 2 } v _ k \\otimes f ^ { k + 1 } = 0 . \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{gather*} - d _ i \\partial _ r \\big ( v _ i + \\varepsilon ^ 2 W _ i \\big ) | _ { r = 1 } - \\varepsilon g _ i ( { \\bf R } ^ \\varepsilon ) = \\varepsilon \\big ( g _ i ( { \\bf v } ) - g _ i ( { \\bf R } ^ \\varepsilon ) \\big ) , i \\in \\{ 1 , 3 , 4 \\} , \\\\ - \\partial _ r v _ i | _ { r = 1 } = 0 , i \\in \\{ 2 , 5 , 6 \\} , \\end{gather*}"}
-{"id": "5234.png", "formula": "\\begin{align*} \\Delta \\Psi = 0 \\textup { ~ i n ~ } R , \\psi = c _ 1 ( \\varpi ) \\eta - \\psi _ V ( \\varpi ) \\textup { ~ o n ~ } T . \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} \\frac { d } { d z } W _ z ( k _ 1 , . . . k _ R ) = \\sum _ { 1 \\leqslant r _ 1 < r _ 2 < . . . < r _ l \\leqslant R } k _ { r _ 2 } ( k _ { r _ 2 } + k _ { r _ 3 } ) . . . ( k _ { r _ 2 } + . . . + k _ { r _ l } ) W _ z ( k _ 1 , . . . \\widehat { k _ { r _ 1 } } , . . . \\widehat { k _ { r _ l } } , . . . k _ R ) \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} \\widehat { N ( g ) } _ 1 = | A | ^ 2 A \\left ( - i a ^ \\dagger ( U + U ^ \\ast ) + i a ^ \\dagger H ^ { ( 2 ) } + i c \\sqrt { 2 } \\pi ^ { 1 / 4 } \\frac 1 2 \\langle e _ 0 , H ^ { ( 2 ) } \\rangle a ^ \\dagger G ^ \\ast \\right ) \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{align*} g ( x , y , k ) = \\cos ( k \\alpha ) - p ( x , y , k ) , \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} V _ 1 = \\lbrace 2 k \\ | \\ k = 0 , \\ldots , q - 1 \\rbrace \\mbox { a n d } V _ 2 = \\lbrace 2 k + 1 \\ | \\ k = 0 , \\ldots , q - 1 \\rbrace . \\end{align*}"}
-{"id": "9931.png", "formula": "\\begin{align*} q = \\frac { 2 } { ( 2 - \\varepsilon ) ( \\frac { 1 } { 2 } - \\varepsilon ) } \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} b _ 2 = | \\max \\{ \\omega ( p ) \\} | = \\max \\{ \\omega ( p ) \\} . \\end{align*}"}
-{"id": "10009.png", "formula": "\\begin{align*} h ' & : = ( h ( 1 ) , \\ldots , h ( r - 1 ) , h ( r ) + 1 , h ( r + 1 ) , \\ldots , h ( n ) ) , \\\\ h '' & : = ( h ( 1 ) , \\ldots , h ( r - 1 ) , h ( r ) + 2 , h ( r + 1 ) , \\ldots , h ( n ) ) \\end{align*}"}
-{"id": "1871.png", "formula": "\\begin{align*} \\Lambda ( d d ^ c f ) = & \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ m \\left ( d d ^ c f \\right ) ( e _ i , J e _ i ) \\\\ = & \\sum _ { i = 1 } ^ m e _ i \\left ( d ^ c f ( J e _ i ) \\right ) - d ^ c f ( J D ^ g _ { e _ i } e _ i ) - \\left ( d ^ c f \\right ) \\left ( ( D ^ g _ { e _ i } J ) e _ i \\right ) \\\\ = & - \\Delta ^ g ( f ) - g ( \\theta , d f ) . \\end{align*}"}
-{"id": "9534.png", "formula": "\\begin{align*} X _ S : = \\overline { { \\mathrm { s p a n } } } ^ { \\| \\cdot \\| } \\{ A g : g \\in S \\} \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{gather*} \\frac { { \\rm d } Y } { { \\rm d } x } = \\left \\{ \\begin{pmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} + \\frac { A _ 0 } { x } + \\frac { A _ 1 } { x - 1 } \\right \\} Y , A _ * \\sim \\operatorname { d i a g } ( 0 , \\theta ^ * _ 1 , \\theta ^ * _ 2 ) , * = 0 , 1 . \\end{gather*}"}
-{"id": "2354.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } u ( x , t ) = u _ 0 ( x ) , x \\in \\R ^ n , \\forall \\ u _ 0 \\in H ^ s ( \\R ^ { n } ) \\ , , \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} \\tilde { V } \\left ( x , t \\right ) = \\alpha \\beta \\sigma ^ { 2 } \\left ( t \\right ) V \\left ( \\sqrt { \\alpha \\beta } x \\sigma \\left ( t \\right ) , \\beta t \\sigma \\left ( t \\right ) \\right ) , \\end{align*}"}
-{"id": "3710.png", "formula": "\\begin{align*} b _ { n + 1 } + b _ n + b _ { n - 1 } & = C \\\\ b _ { n + 1 } & = b _ { n - 1 } \\end{align*}"}
-{"id": "4372.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { 2 q - 1 } \\binom { 2 k } { k } x ^ k \\equiv ( 1 - 4 x ) ^ { ( q - 1 ) / 2 } ( 1 + 2 x ^ q ) \\pmod { p } . \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} d \\leqslant \\left ( \\frac { \\varepsilon _ 1 } { C ( \\alpha , \\lambda ) } \\right ) ^ { \\frac { 1 } { 2 + \\lambda } } B ^ { 1 - \\frac { 1 } { ( 2 + \\lambda ) r } } . \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} A ( x ) = \\big ( - \\varepsilon ^ 2 | \\nabla u _ { \\varepsilon } ( x ) | ^ 2 + u _ \\varepsilon ^ 2 ( x ) ( V ( x ) - \\frac { 1 } { 8 \\pi \\varepsilon ^ 2 } \\int _ { \\R ^ 3 } \\frac { u ^ 2 _ { \\varepsilon } ( \\xi ) } { | x - \\xi | } d \\xi ) \\big ) \\nu _ i ( x ) . \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{gather*} \\frac { q C D - q ^ { - 1 } D C } { q - q ^ { - 1 } } = I . \\end{gather*}"}
-{"id": "5708.png", "formula": "\\begin{align*} \\left [ \\hat { R } ^ i , \\ ; \\hat { R } ^ { j } \\right ] = i \\ , \\Delta ^ { i j } \\ , \\hat { \\mathbb { I } } \\ ; , i , \\ , j = 1 , \\ , \\ldots , \\ , 2 n \\ ; , \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\bigg [ \\mathrm { i } \\big ( \\Psi , \\ , \\frac { \\partial \\widetilde { \\Psi } } { \\partial t } \\big ) + \\frac { 1 } { 2 } \\big ( \\left ( \\mathrm { i } \\nabla + \\mathbf { A } \\right ) \\Psi , \\ , \\left ( \\mathrm { i } \\nabla + \\mathbf { A } \\right ) \\widetilde \\Psi \\big ) + \\big ( V \\Psi , \\widetilde \\Psi \\big ) + \\big ( \\phi \\Psi , \\widetilde \\Psi \\big ) \\bigg ] \\mathrm { d } t = 0 , \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} S _ { A , \\pmb { \\omega } } f _ { k } = \\begin{cases} A f _ { k } + \\omega _ { k - ( 2 r + 1 ) } f _ { k - ( 2 r + 1 ) } & \\textrm { f o r e v e r y } \\ | k | \\le r \\\\ \\omega _ { k - ( 2 r + 1 ) } f _ { k - ( 2 r + 1 ) } & \\textrm { f o r e v e r y } \\ | k | > r . \\end{cases} \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} T = \\frac { r _ 0 ^ 4 } { 4 n ^ 2 } \\ , . \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} \\Bigl ( \\mathbb E \\Bigl \\| \\sum ^ n _ { j = 1 } \\varepsilon _ j d _ j \\Bigr \\| ^ p \\Bigr ) ^ { \\frac 1 p } \\leq \\beta \\Bigl ( \\mathbb E \\Bigl \\| \\sum ^ n _ { j = 1 } d _ j \\Bigr \\| ^ p \\Bigr ) ^ { \\frac 1 p } . \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{align*} x ^ { \\boldsymbol { \\omega } , - } _ i & = \\tilde { z } _ j + k & i & = 2 i _ k N + j , \\ : k \\in \\mathbb { Z } , \\ : j = 1 , . . . , 2 ( i _ { k + 1 } - i _ k ) N , \\\\ x ^ { \\boldsymbol { \\omega } , + } _ i & = \\tilde { z } _ { - j + 1 } + k & i & = 2 i _ k N - j , \\ : k \\in \\mathbb { Z } , \\ : j = 0 , . . . , 2 ( i _ k - i _ { k - 1 } ) N - 1 . \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} f ^ { \\prime \\prime } = \\frac { H } { 3 B _ { 2 } N } f + \\frac { \\left [ - 2 B _ { 2 } N ^ { \\prime } + B _ { 2 } A _ { 2 } N - 2 B _ { 2 } M + B _ { 2 } ^ { \\prime } N \\right ] } { 3 B _ { 2 } N } f ^ { \\prime } . \\end{align*}"}
-{"id": "2903.png", "formula": "\\begin{align*} B _ { A , \\pmb { \\omega } } e _ { k } = A e _ k \\qquad { \\rm a n d } B _ { A , \\pmb { \\omega } } ^ * e _ k = A ^ { * } e _ { k } + \\delta e _ { k + r } \\ , , \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} U _ { 2 m + 1 } ^ { ( 2 ) } & = U _ { m } U _ { m + 1 } \\\\ & = U _ { m } \\left ( p U _ { m } - q U _ { m - 1 } \\right ) \\\\ & = p U _ { m } U _ { m } - q \\left ( p U _ { m - 1 } - q U _ { m - 2 } \\right ) U _ { m - 1 } \\\\ & = p U _ { 2 m } ^ { ( 2 ) } - p q U _ { 2 m - 2 } ^ { ( 2 ) } + q ^ { 2 } U _ { 2 m - 3 } ^ { ( 2 ) } . \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\frac { B _ { n } ^ { \\left ( a , b \\right ) } } { n ! } z ^ { n } = \\frac { 1 } { _ { 1 } F _ { 1 } \\left ( \\begin{array} { c } a \\\\ a + b \\end{array} ; z \\right ) } \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} \\log ( 1 - x ) = - x - R ( x ) \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} \\partial _ i \\circ h _ i ( g _ 0 \\wedge \\cdots \\wedge g _ i ) & = g _ 0 \\wedge \\cdots \\wedge g _ i - \\sum ^ i _ { j = 0 } ( - 1 ) ^ j 1 \\wedge g _ 0 \\wedge \\cdots \\wedge \\hat { g _ j } \\wedge \\cdots \\wedge g _ i \\\\ & = g _ 0 \\wedge \\cdots \\wedge g _ i - h _ { i - 1 } \\circ \\partial _ { i - 1 } ( g _ 0 \\wedge \\cdots \\wedge g _ i ) . \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow + \\infty } \\lambda \\dot x ( t ) + \\dot v ( t ) = 0 . \\end{align*}"}
-{"id": "9536.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = 1 } ^ k a _ i ^ * y _ { g _ 1 } \\alpha _ { g _ 1 g _ 0 ^ { - 1 } } ( a _ i ) \\right \\| < \\delta . \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{align*} ( x ^ { - 2 } x ^ 2 ) \\theta _ n = y _ n ^ { - 2 } y _ n ^ 2 = x ^ { - n } x ^ n , \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{align*} ( x ) _ { n , k } : = x ( x + k ) ( x + 2 k ) \\cdots ( x + ( n - 1 ) k ) = \\Gamma _ k ( x + n k ) / \\Gamma _ k ( x ) . \\end{align*}"}
-{"id": "9807.png", "formula": "\\begin{align*} \\int _ { E , N } { g d \\xi d \\eta d \\zeta } \\equiv \\sum \\limits _ { i , j , k = 0 } ^ N { { g _ { i j k } } { \\omega _ { i j k } } } , \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{align*} g \\left ( \\rho ( \\mathbf { x } ) \\right ) = \\rho ( R _ g \\cdot \\mathbf { x } + T _ g ) . \\end{align*}"}
-{"id": "545.png", "formula": "\\begin{align*} \\beta _ 2 ( x , v , x ' , v ' ) = \\int _ 0 ^ { \\tau _ - ( x , v ) } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! E ( x , x - t v , x ' ) { k ( x - t v , v _ 1 , v ) k ( x ' , v ' , v _ 1 ) _ { | v _ 1 = \\widehat { x - t v - x ' } } \\over | x - t v - x ' | ^ { n - 1 } } d t , \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} d ( x , y ) : = \\sum _ { j = 1 } ^ { \\infty } \\frac { 1 } { 2 ^ { j } } \\frac { p _ { j } ( x - y ) } { 1 + p _ { j } ( x - y ) } . \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} y ( P ) = \\Bigl ( \\frac { a } { b } \\Bigr ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } u _ { t } + \\Delta u = u , t > 0 \\\\ u ( 0 ) = u _ 0 \\in L ^ { 2 } \\end{array} \\right . , \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} [ a _ - , a _ + ] = k ~ \\mathbb { I } - 2 N [ N , a _ { \\pm } ] = \\pm a _ { \\pm } . \\end{align*}"}
-{"id": "1214.png", "formula": "\\begin{align*} \\begin{gathered} \\tilde { \\eta } ^ { h , x } ( t ) = \\int _ { 0 } ^ { t } \\Pi ( t , s ) \\sigma ' ( X ( s , x ) ) . e ^ { s A } h d W ( s ) , \\\\ \\zeta ^ { h , x } ( t ) = \\int _ 0 ^ t \\Pi ( t , s ) \\sigma '' ( X ( s , x ) ) \\cdot ( \\eta ^ { h , x } ( s ) , \\eta ^ { h , x } ( s ) ) d W ( s ) , \\end{gathered} \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} [ \\theta { \\bf W } _ \\alpha + ( 1 - \\theta ) { \\bf W } _ \\alpha ^ T ] \\Phi = \\psi { \\bf K } , \\end{align*}"}
-{"id": "8347.png", "formula": "\\begin{align*} \\tilde { f } _ { n } ( T , U ) = T ^ { n + 1 } - \\zeta _ 1 T ^ { n } U + \\zeta _ 2 T ^ { n - 1 } U ^ 2 + \\cdot \\cdot \\cdot + ( - 1 ) ^ n \\zeta _ n T U ^ { n } + ( - 1 ) ^ { n + 1 } U ^ { n + 1 } . \\end{align*}"}
-{"id": "10025.png", "formula": "\\begin{align*} \\sum \\limits _ { i = K } ^ { k - 1 } | S _ { t _ k ( x ) } h _ { v _ i } ( x ) | \\lesssim v _ k ^ 2 \\sum \\limits _ { K } ^ { k - 1 } \\frac { 1 } { v _ i ^ \\beta } \\lesssim \\frac { v _ k ^ 2 } { v _ { k - 1 } ^ \\beta } . \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} \\int _ { M \\times \\R _ { \\tau } } \\Vert F _ { A } \\Vert ^ 2 \\ , \\ , { \\rm V o l } _ M d \\tau = E < \\infty . \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} \\min _ { \\{ u \\in H , d \\in \\mathcal { H } ( E ) \\} } F ( d ) \\ \\ \\ \\ \\hbox { s u b j e c t t o } \\ \\ \\ \\ D u = d , \\end{align*}"}
-{"id": "1028.png", "formula": "\\begin{align*} \\| u ( t ) \\| _ { L ^ 2 ( \\R ^ k ) } = \\| u _ 0 \\| _ { L ^ 2 ( \\R ^ k ) } , \\end{align*}"}
-{"id": "5451.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta ) ^ m u = \\mu \\rho u , & { \\rm i n \\ } \\Omega , \\\\ \\mathcal N _ 0 u = \\cdots = \\mathcal N _ { m - 1 } u = 0 , & { \\rm o n \\ } \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} ( \\pi ^ + ( s ) , \\pi ^ - ( s ) ) : = \\begin{cases} ( \\pi _ 0 ( \\theta _ 0 + s ) - \\pi _ 0 ( \\theta _ 0 ) , \\pi _ 1 ( \\theta _ 0 + s ) - \\pi _ 0 ( \\theta _ 0 ) ) & \\pi _ 0 ( \\theta _ 0 + 1 ) > \\pi _ 1 ( \\theta _ 0 + 1 ) \\\\ ( \\pi _ 1 ( \\theta _ 0 + s ) - \\pi _ 0 ( \\theta _ 0 ) , \\pi _ 0 ( \\theta _ 0 + s ) - \\pi _ 0 ( \\theta _ 0 ) ) & \\pi _ 0 ( \\theta _ 0 + 1 ) < \\pi _ 1 ( \\theta _ 0 + 1 ) . \\end{cases} \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{gather*} \\chi _ { 0 , 3 } ( x , y , z | \\rho ) = \\sum _ { i \\geq 0 } \\rho ^ { i } U _ { i } ( x ) U _ { i } ( y ) U _ { i } ( z ) = \\\\ ( ( 1 + \\rho ^ { 2 } ) ^ { 3 } + 1 6 \\rho ^ { 3 } x y z - 4 \\rho ^ { 2 } ( 1 + \\rho ^ { 2 } ) ( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } ) ) / w _ { 3 } ( x , y , z | \\rho ) , \\end{gather*}"}
-{"id": "9879.png", "formula": "\\begin{align*} \\sum _ { | \\sigma _ 1 | + \\dots + | \\sigma _ s | = L } \\ \\prod _ { i = 1 } ^ { s } Q ( \\sigma _ i ) & = \\sum _ { j = 1 } ^ m \\ \\sum _ { ( \\pi ( \\sigma _ 1 ) , \\dots , \\pi ( \\sigma _ s ) ) = T _ j } \\ \\prod _ { i = 1 } ^ { s } Q ( \\sigma _ i ) \\\\ & \\leq \\sum _ { j = 1 } ^ m \\ \\prod _ { k = 1 } ^ { s } \\left ( \\sum _ { \\pi ( \\sigma _ k ) = T _ { j , k } } Q ( \\sigma _ k ) \\right ) . \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} \\widetilde H ^ \\prime = \\oplus _ { i = 0 } ^ { d _ 0 - 1 } \\widetilde H ^ \\prime _ { ( \\chi ^ i ) } , \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} \\Psi _ { \\mathrm { b v } } & = \\left ( \\psi _ 1 ( 0 ) , \\dots , \\psi _ { | \\mathcal { E } | } ( 0 ) , \\psi _ 1 ( l _ 1 ) , \\dots , \\psi _ { | \\mathcal { E } | } ( l _ { | \\mathcal { E } | } ) \\right ) , \\\\ \\Psi _ { \\mathrm { b v } } ' & = \\left ( \\psi _ 1 ' ( 0 ) , \\dots , \\psi _ { | \\mathcal { E } | } ' ( 0 ) , - \\psi _ 1 ' ( l _ 1 ) , \\dots , - \\psi _ { | \\mathcal { E } | } ' ( l _ { | \\mathcal { E } | } ) \\right ) , \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{align*} \\sum _ { k } ( \\mathsf { M } _ { m } ^ { n } ) _ { k } ^ { i } ( \\mathsf { M } _ { o } ^ { p } ) _ { j } ^ { k } = u _ { i } ^ { m * } \\left ( \\sum _ { k } u _ { k } ^ { n } u _ { k } ^ { o * } \\right ) u _ { j } ^ { p } = \\delta _ { o } ^ { n } u _ { i } ^ { m * } u _ { j } ^ { p } = \\delta _ { o } ^ { n } ( \\mathsf { M } _ { m } ^ { p } ) _ { j } ^ { i } . \\end{align*}"}
-{"id": "5579.png", "formula": "\\begin{align*} x _ i = x _ j . \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} & ( \\nu _ 2 ( t _ { 2 ^ k } ( 2 ^ l n + j ) ) ) _ { n \\in \\N } = ( \\nu _ 2 ( t _ { 2 ^ k } ( 2 ^ { l - 1 } ( 2 n + s ) + j ' ) ) ) _ { n \\in \\N } \\\\ & = ( \\alpha + \\beta \\nu _ 2 ( 2 n + s + 1 ) ) _ { n \\in \\N } = ( \\alpha + \\beta s + \\beta s \\nu _ 2 ( n + 1 ) ) _ { n \\in \\N } , \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} & t ^ N \\langle 1 \\cdots M | \\mathcal { B } ^ \\prime ( z _ 1 ) \\cdots \\mathcal { B } ^ \\prime ( z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle / D ^ \\prime \\\\ = & t ^ N \\langle 1 \\cdots M | \\mathcal { B } ^ \\prime ( z _ 1 ) \\cdots \\mathcal { B } ^ \\prime ( z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle / D ^ \\prime \\Big | _ { z _ 1 \\longleftrightarrow z _ 1 ^ { - 1 } } , \\end{align*}"}
-{"id": "6912.png", "formula": "\\begin{align*} f ( x , t ) = \\sum _ { k = 1 } ^ { \\infty } f _ k ( t ) \\sin ( k \\pi x ) , \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{align*} x = y , s = t , x s = y t , x t = y s . \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\lambda _ k X ( E _ k / E _ { k - 1 } ) = 0 \\end{align*}"}
-{"id": "9867.png", "formula": "\\begin{align*} G ( \\rho ) = - \\dfrac { 1 } { 2 \\pi } ( \\rho ) + C _ 2 \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{align*} ( b _ 1 , \\ldots , b _ k ) = ( q ( \\alpha _ 1 ) , \\ldots , q ( \\alpha _ k ) ) , \\end{align*}"}
-{"id": "5460.png", "formula": "\\begin{align*} { \\partial ^ { \\alpha } u ( x ) } = \\sum _ { k = 1 } ^ m \\frac { c _ { N , k , \\alpha } } { | x - a | ^ { m - k } } U ^ { ( k ) } ( | x - a | ) , \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} x _ 1 = \\frac { m _ 2 } { m _ 1 + m _ 2 } \\log t + \\log C _ 1 , x _ 2 = - \\frac { m _ 1 } { m _ 1 + m _ 2 } \\log t + \\log C _ 2 \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} & \\partial _ t w + w \\cdot \\nabla w + \\widetilde U \\cdot \\nabla w + w \\cdot \\nabla \\widetilde U + h ( u _ s \\cdot \\nabla w + w \\cdot \\nabla u _ s ) \\\\ & \\qquad = \\Delta w - \\nabla p _ w - h u _ \\infty \\cdot \\nabla w + f , \\\\ & \\mbox { d i v $ w $ } = 0 , \\\\ & w | _ { \\partial \\Omega } = 0 , \\\\ & w \\to 0 \\quad \\mbox { a s $ | x | \\to \\infty $ } , \\\\ & w ( \\cdot , 0 ) = w _ 0 : = v _ 0 - \\widetilde U ( \\cdot , 0 ) , \\end{align*}"}
-{"id": "4170.png", "formula": "\\begin{align*} \\operatorname { S i } ( x ) = \\int _ { 0 } ^ x \\frac { \\sin ( u ) } { u } { \\rm d } u . \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} \\widetilde g ( \\sigma ) : = [ \\sigma - 1 ] \\widetilde Q , \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{align*} \\Re ( r ( n , x + i 0 , J ) ) = 0 \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{align*} \\tilde { g } _ { n } ( X , Y ) = c _ 0 X ^ { n + 1 } + c _ 1 X ^ { n } Y + \\cdot \\cdot \\cdot + c _ n X Y ^ { n } + c _ { n + 1 } Y ^ { n + 1 } , \\\\ c _ j \\in { \\mathbb R } , j = 0 , 1 , \\dots , n + 1 . \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{align*} w = \\pm \\frac { 1 } { 2 q } ( 1 \\pm \\sqrt { 4 \\ell + 1 } ) \\in \\Q ( z ) \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} E \\prod _ { i = 0 } ^ { n - 1 } F ( S _ i , \\widehat { { S } } _ i ; S _ { i + 1 } , \\widehat { S } _ { i + 1 } ) = E \\prod _ { i = 0 } ^ { n - 1 } \\frac { \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } { 1 + \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } . \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} \\frac { { \\rm d } } { { \\rm d } s } \\left ( \\frac { \\alpha ( s ) - P _ S ( s ) } { R _ S ( s ) } \\right ) = \\frac { 1 } { R _ S } \\left [ \\mathbf { t } + \\frac { \\rho \\rho ' } { \\tau R _ S ^ 2 } \\varsigma \\ , \\mathbf { n } + \\left ( \\frac { \\rho ' \\ , ^ 2 } { \\tau ^ 2 R _ S ^ 2 } - 1 \\right ) \\varsigma \\ , \\mathbf { b } \\right ] . \\end{align*}"}
-{"id": "8483.png", "formula": "\\begin{gather*} f ( x ) = \\sum _ { k = 0 } ^ { m } g _ k ( x ) \\quad \\textrm { f o r } x \\in \\widehat { S } _ p . \\end{gather*}"}
-{"id": "2373.png", "formula": "\\begin{align*} P _ A ( d \\phi ) = \\frac 1 { Z _ A } e ^ { - H _ A ( \\phi ) } d \\phi _ A \\delta _ 0 ( d \\phi _ { A ^ c } ) \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{gather*} V _ n = U _ n \\cap U _ { d - n } ^ \\prime ( 0 \\le n \\le d ) . \\end{gather*}"}
-{"id": "4291.png", "formula": "\\begin{align*} ( \\Psi ^ \\prime _ { n } ( X , Y ) ) ^ 2 = n ^ 2 X ^ { n ^ 2 - 1 } + c X ^ { n ^ 2 - 3 } + \\cdot \\cdot \\cdot . \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{align*} x ^ { \\boldsymbol { \\omega } } _ { 2 m N + j } = \\begin{cases} y _ 0 + k & \\omega _ m = 0 , \\\\ z _ j + k & \\omega _ m = 1 . \\end{cases} \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{align*} H ( t ) = H ( ( 1 - \\alpha ^ 2 ) r ) \\ge \\underline C \\lambda \\sqrt s = r \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} \\widehat { K } _ N ( x , y ; w ) = \\sum _ { j = 0 } ^ { N - 1 } \\varphi _ { j } ( x ; w ) \\varphi _ j ( y ; w ) , \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} \\dot \\phi & = X _ { \\Phi ^ \\ast g } ( \\phi , \\bar \\phi ) . \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} \\vec { D } _ 0 = 0 . \\end{align*}"}
-{"id": "6328.png", "formula": "\\begin{align*} \\underline { \\mu } = \\mu _ 1 < \\cdots < \\mu _ { k + 1 } = \\bar { \\mu } . \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} \\sup _ { \\eta \\in X } | c ( x , \\eta ) - c ( x , \\eta _ u ) | = \\sup _ { \\eta \\in X } 0 = 0 . \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} ( a ^ + ) ^ n \\vert 0 \\rangle = \\sqrt { F ( n ) ! } ~ \\vert n \\rangle \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} T ( f _ 1 , f _ 2 , f _ 3 ) : = \\frac { 1 } { N ^ 2 } \\sum \\limits _ { x , d \\in \\mathbb { Z } } f _ 3 ( x ) f _ 2 ( x + d ) f _ 1 ( [ x + \\sqrt { 2 } d ] ) . \\end{align*}"}
-{"id": "6465.png", "formula": "\\begin{align*} v ( i , j ) ^ 2 & = \\sum _ { k = i } ^ j v _ k ^ 2 + v _ i v _ { i + 1 } + v _ { j } v _ { j - 1 } + \\sum _ { k = i + 1 } ^ { j - 1 } ( v _ k v _ { k - 1 } + v _ k v _ { k + 1 } ) \\\\ & = - 2 ( j - i + 1 ) + 2 + 2 ( j - i - 1 ) = - 2 . \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} W ( X ) = \\sum _ { k = 1 } ^ \\infty w _ k \\Big ( \\frac R { R _ 0 } \\Big ) ^ { - k } \\sin k \\theta \\ , . \\end{align*}"}
-{"id": "1831.png", "formula": "\\begin{align*} \\Omega ^ { ( r , s + 1 ) + ( s + 1 , r ) } ( M ) = \\left [ \\Omega ^ { r , s } ( M ; T M ) \\oplus \\Omega ^ { s + 1 , r - 1 } ( M ; T M ) \\right ] \\cap \\Omega ^ { p + 1 } ( M ) , \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{align*} h _ 1 = t ^ { 1 / 2 } , h _ 2 = t ^ { - 1 / 2 } \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{align*} H _ s ( u , v ) = H _ s ( 0 , 0 ) + A ( u , v ) + \\kappa _ s ( u , v ) ( 1 , \\widetilde { \\lambda } ) \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{align*} T = \\{ ( y , v ) \\in [ 2 \\sqrt { 2 } , 4 ] ^ 2 \\mid y \\le v \\} . \\end{align*}"}
-{"id": "4525.png", "formula": "\\begin{align*} \\left \\langle \\vec { w } , \\mu _ { \\Upsilon } \\left ( \\mathcal { X } \\right ) \\vec { w } \\right \\rangle _ { \\mathbb { R } ^ { d } } = \\mu _ { \\vec { w } } \\left ( \\mathcal { X } \\right ) \\ . \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} K _ N ( x , y ; \\rho _ R ) = \\frac { 1 } { R } K _ N \\left ( \\frac { x } { R } , \\frac { y } { R } ; w _ R \\right ) . \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 1 } \\frac { 2 ( 4 x ^ { 2 } - 4 x y - 1 + \\rho ^ { 2 } ) \\sqrt { 1 - y ^ { 2 } } d y } { \\pi ( ( 1 - \\rho ^ { 2 } ) ^ { 2 } - 4 x y \\rho ( 1 + \\rho ^ { 2 } ) + 4 \\rho ^ { 2 } ( x ^ { 2 } + y ^ { 2 } ) ) } = 4 x ^ { 2 } - 1 , \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} \\beta ( [ 0 , t ] _ < ^ 2 ) = \\int _ 0 ^ t \\int _ 0 ^ s \\delta ( B _ s - B _ u ) d u d s . \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} \\tilde { W } _ { n + 1 } ( x ) = \\int _ { 0 - } ^ { K - 0 } G ( x - w + T ) d \\tilde { W } _ n ( w ) + \\int _ { K - 0 } ^ { T + x } d \\tilde { W } _ n ( w ) . \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} \\frac { \\ ( | q | ^ 2 ; | q | ^ 2 \\ ) _ \\infty } { ( | q | ) ^ 2 _ \\infty } = \\sqrt { \\frac { y } { 2 } } e ^ { \\frac { \\pi } { 8 y } } \\ ( 1 + O \\ ( e ^ { - \\frac { \\pi } { y } } \\ ) \\ ) . \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{gather*} \\begin{array} { l c l } \\star _ t \\tau _ 3 ( t ) & = & \\frac { \\sqrt { 6 } } { 7 y ( t ) ^ 5 } ( - f ^ { 1 3 6 7 } - f ^ { 1 4 5 7 } - f ^ { 2 3 5 7 } + f ^ { 2 4 6 7 } ) + \\frac { 4 \\sqrt { 6 } } { 2 1 y ( t ) ^ 5 } ( f ^ { 1 2 3 4 } + f ^ { 1 2 5 6 } + f ^ { 3 4 5 6 } ) , \\end{array} \\end{gather*}"}
-{"id": "6232.png", "formula": "\\begin{align*} f _ { ( r , n ) } = ( 1 + z + z ^ 2 + \\cdots + z ^ { r - 1 } ) f _ { ( r , n - 1 ) } . \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ k ( - 1 ) ^ n p _ { i _ 1 i _ 2 \\cdots i _ { k - 1 } j _ r } p _ { j _ 0 \\cdots \\widehat { j _ r } \\cdots j _ k } = 0 , \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} w _ { R , \\alpha } ^ - ( \\alpha ^ { - 2 } t ) & = ( 1 - t ^ 2 ) ^ { 1 / 2 } e ^ { - N ( V ( \\frac { t } { \\alpha } ) + \\widetilde { \\varepsilon } _ R t ) } , t \\in [ - 1 , 1 ] , \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} \\partial _ \\theta ( V ^ { ( \\alpha , a ) } + H ^ { ( \\alpha , a ) } \\cdot \\omega ) & = \\widetilde { \\Omega } V ^ { ( \\alpha , a ) } + \\widetilde { \\Omega } H ^ { ( \\alpha , a ) } \\cdot \\omega \\\\ & = \\widetilde { S } ^ \\alpha \\widetilde { \\Omega } \\Gamma ^ a V + \\widetilde { S } ^ \\alpha \\widetilde { \\Omega } \\Gamma ^ a H \\cdot \\omega \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} \\pi _ 1 ( E _ 1 [ 4 ] ) = \\pi _ 2 ( E _ 2 [ 4 ] ) = \\{ 0 , 1 , - 1 , i , - i , \\infty \\} \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} { } _ { 2 } F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\beta \\\\ & 1 \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { \\Gamma ( 1 - \\alpha - \\beta ) } { \\Gamma ( 1 - \\alpha ) \\Gamma ( 1 - \\beta ) } . \\end{align*}"}
-{"id": "7535.png", "formula": "\\begin{align*} \\left [ ( z \\phi z ^ { - 1 } ) ^ * , z \\phi z ^ { - 1 } \\right ] = \\theta + P ( k ) + O ( t ^ { - 1 - \\epsilon } \\mathcal L ) \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{align*} \\Psi _ 1 ' ( 0 ) = & 2 \\cdot \\frac { d } { d x } \\bigg ( { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & - a - x \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] _ { p - 1 } \\bigg ) \\bigg | _ { x = 0 } \\\\ = & 2 \\cdot \\frac { d } { d x } \\bigg ( ( 1 - z ) ^ a { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & 1 + a + x \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac z { z - 1 } \\bigg ] _ { p - 1 } \\bigg ) \\bigg | _ { x = 0 } \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} P _ 1 ( y ) & = A A ^ \\dag \\ , y , \\\\ P _ 2 ( y ) & = | y _ 0 | e ^ { i \\arg { y } } , \\end{align*}"}
-{"id": "988.png", "formula": "\\begin{align*} \\pi _ * \\widetilde { Q } ( H ) & = - 2 \\sum _ { k = 0 } ^ { \\infty } \\alpha _ k ( - 2 L ) ^ k + 3 \\sum _ { k = 0 } ^ { \\infty } \\alpha _ k ( - 3 L ) ^ k \\\\ & = - 2 \\frac { Q ( H ) - \\alpha _ 1 H - \\alpha _ 0 } { H ^ 2 } \\Big | _ { H = - 2 L } + 3 \\frac { Q ( H ) - \\alpha _ 1 H - \\alpha _ 0 } { H ^ 2 } \\Big | _ { H = - 3 L } \\\\ & = - 2 \\frac { Q ( H ) } { H ^ 2 } \\Big | _ { H = - 2 L } + 3 \\frac { Q ( H ) } { H ^ 2 } \\Big | _ { H = - 3 L } + \\frac { Q ( 0 ) } { 6 L ^ 2 } . \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{align*} { \\sf v a r } [ \\Pi ^ \\omega _ i \\ ( q _ i ^ * , K _ i ^ * , \\Delta _ i ^ * \\ ) ] - { \\sf v a r } [ \\pi ^ \\omega _ i ] = - 3 K ^ * _ i \\sigma ^ 2 / 2 < 0 . \\end{align*}"}
-{"id": "9652.png", "formula": "\\begin{align*} L ' ( E , 1 ) \\ , L ( E ^ { ( D ) } , 1 ) = c _ E \\ , u ^ { - 2 } \\ , | D | ^ { - { \\frac 1 2 } } \\ , { \\hat h } ( P _ { D , r } ) , \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} u _ { n , j } = u _ { \\alpha } = \\frac { t ^ { j } } { ( 1 - t ) ^ { n + 1 } } x ^ { n } \\end{align*}"}
-{"id": "8753.png", "formula": "\\begin{align*} \\Delta ^ 1 ( A ) = 0 . \\end{align*}"}
-{"id": "9752.png", "formula": "\\begin{align*} \\mathcal { E } ( t , a ) e ^ { - r t } + \\int ^ { t } _ { 0 } e ^ { - r t ' } \\left [ \\int _ { \\mathbb { R / Z } } \\frac { 1 } { 2 } ( \\widetilde { W } ^ { { \\rm T } } Q _ { r } \\widetilde { W } ) ( t ' , s , a ) d s - \\mathcal { R } ( t ' , a ) \\right ] { \\rm d } t ' = \\mathcal { E } ( 0 , a ) . \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\dim H ^ 2 ( G _ { L , T } , Z _ i ( U _ X ) ) \\leq B n ^ { 2 g - 1 } , \\end{align*}"}
-{"id": "2810.png", "formula": "\\begin{align*} L _ \\infty ( M _ i ) = \\max _ { 1 \\leqslant i \\leqslant 3 } ( \\| M _ i \\| ) \\leqslant D _ 1 D _ 2 D _ 3 L _ \\infty ( L _ i ) = D L _ \\infty . \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N } \\big | a _ { k } ^ 2 \\big | \\le C \\Delta ^ { \\frac 1 2 - 2 \\kappa } ( 1 + | x | _ { L ^ { \\max ( p , 2 q ) } } ) ^ { K + 1 } \\int _ { 0 } ^ { T } \\frac { 1 } { ( T - t ) ^ { 1 - \\kappa } } \\bigl ( 1 + \\frac { 1 } { t ^ { 1 - \\kappa } } \\bigr ) d t . \\end{align*}"}
-{"id": "1363.png", "formula": "\\begin{align*} u _ { s c } \\left ( x , y , k \\right ) = u \\left ( x , y , k \\right ) - u _ { 0 } \\left ( x , y , k \\right ) . \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} \\| u \\| _ { ( N , \\alpha ) } : = \\sup _ { x \\in \\R ^ { n } } ( 1 + | x | ) ^ { N } \\big | \\partial ^ { \\alpha } u ( x ) \\big | \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } a _ k T ^ { \\lambda _ k } \\end{align*}"}
-{"id": "2184.png", "formula": "\\begin{align*} \\varphi ( i k _ j , x ) U ^ \\dagger [ f ( i k _ j , 0 ) - i f ' ( i k _ j , 0 ) ] P _ j = f ( i k _ j , x ) P _ j , \\end{align*}"}
-{"id": "7156.png", "formula": "\\begin{align*} \\rho ( g ) & = \\begin{pmatrix} u & 1 \\\\ 0 & u ^ { - 1 } \\end{pmatrix} \\\\ \\rho ( h ) & = \\begin{pmatrix} u & 0 \\\\ z & u ^ { - 1 } \\end{pmatrix} \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} \\frac { p _ { 1 } } { p _ { 2 } } e ^ { \\alpha - \\beta } = k \\frac { B _ { 1 } } { B _ { 2 } } = k \\frac { f _ { c _ { 1 } } } { f _ { c _ { 2 } } } , \\end{align*}"}
-{"id": "8501.png", "formula": "\\begin{align*} p _ i = c i ^ { - 1 / \\theta } ( 1 + o ( i ^ { - 1 / 2 } ) ) , \\end{align*}"}
-{"id": "6950.png", "formula": "\\begin{align*} K ( t , t ' ) = \\sum _ { \\nu = 1 } ^ \\infty \\lambda _ \\nu \\psi _ \\nu ( t ) \\psi _ \\nu ( t ' ) , \\end{align*}"}
-{"id": "1054.png", "formula": "\\begin{align*} L : = \\left ( \\begin{matrix} 1 & 0 & - \\sqrt { 2 } & - 1 + \\sqrt { 2 } \\\\ 0 & 1 & - \\sqrt { 3 } & - 1 + \\sqrt { 3 } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "9849.png", "formula": "\\begin{align*} + \\frac { 1 - v } { 1 - v x } \\sum _ { i > j } e _ { i j } \\otimes e _ { j i } + x \\frac { 1 - v } { 1 - v x } \\sum _ { i < j } e _ { i j } \\otimes e _ { j i } , \\ , \\ , \\ , x = z ^ n _ i z _ j ^ { - n } . \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{align*} \\begin{cases} \\gamma _ 3 = \\gamma _ 5 = \\gamma _ 7 = \\gamma _ 8 = 0 \\\\ \\gamma _ 4 = - \\gamma _ 1 \\\\ \\gamma _ 6 = - \\gamma _ 2 \\end{cases} \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} y ^ 2 z + a _ 1 x y z + a _ 3 y z ^ 2 - ( x ^ 3 + a _ 2 x ^ 2 z + a _ 4 x z ^ 2 + a _ 6 z ^ 3 ) = 0 , \\end{align*}"}
-{"id": "9323.png", "formula": "\\begin{align*} w ^ 2 = f _ 6 ( x _ 0 : x _ 1 : x _ 2 ) \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{align*} \\sum \\limits _ { \\left \\{ i , j \\right \\} \\in E \\left ( G ^ { \\prime } \\right ) } x _ { i } x _ { j } & = \\sum \\limits _ { \\left \\{ i , j \\right \\} \\in E \\left ( G \\right ) } x _ { i } x _ { j } - x _ { k + s - 1 } x _ { k + s } - x _ { k + s + 2 } x _ { k + s + 3 } + x _ { k + s } x _ { k + s + 2 } + x _ { k + s - 1 } x _ { k + s + 3 } \\\\ & = \\sum \\limits _ { \\left \\{ i , j \\right \\} \\in E \\left ( G \\right ) } x _ { i } x _ { j } + \\left ( x _ { k + s - 1 } - x _ { k + s } \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} \\dim C _ H ( h ) = \\dim C _ { H / C _ 1 } ( h C _ 1 ) = \\dim C _ { H / C _ 2 } ( h C _ 2 ) = \\dim C _ { H / C } ( h C ) . \\end{align*}"}
-{"id": "10043.png", "formula": "\\begin{align*} \\nabla ^ { 0 } _ X Y = \\nabla ^ { \\mathrm { g } } _ X Y - \\frac { 1 } { 2 } ( \\nabla ^ { \\mathrm { g } } _ X J _ { \\varphi } ) J _ { \\varphi } Y , \\forall X , Y \\in { \\mathfrak X } ( M ) . \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} & { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\beta \\\\ & \\gamma \\end{matrix} \\bigg | \\ , 1 \\bigg ] _ { c } - { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - 2 & \\beta \\\\ & \\gamma \\end{matrix} \\bigg | \\ , 1 \\bigg ] _ { c } = \\Psi ( s p ) - \\Psi ( 0 ) \\equiv \\Psi ' ( 0 ) \\cdot s p \\equiv \\psi ' ( 0 ) \\cdot s p \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} I ( C : M ) _ { ( \\Phi \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) } & = S ( \\Phi ( \\hat { \\rho } _ A ) ) - S ( C | M ) _ { ( \\Phi \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) } \\\\ & \\le n \\ ; g \\left ( \\eta \\ , E + | 1 - \\eta | \\ , E _ 0 + \\frac { \\eta + | 1 - \\eta | - 1 } { 2 } \\right ) \\\\ & \\phantom { \\le } - n \\ln \\left ( \\eta \\ , \\mathrm { e } ^ { - g ( E ) } + \\left | 1 - \\eta \\right | \\mathrm { e } ^ { S _ 0 } \\right ) \\ ; , \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} \\left \\langle \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} p , H ' \\cap \\ker \\varphi \\right \\rangle \\end{align*}"}
-{"id": "9886.png", "formula": "\\begin{align*} \\| ( B _ k ^ T ) ^ { - 1 } \\| \\ , \\| B _ k ^ T \\| \\leq C , k = 1 , \\ldots , N _ K \\ , . \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} \\tilde q _ i = \\max \\{ f _ { i j } ( p _ { i j } ) \\mid ( i , j ) \\in \\widetilde E \\} ( \\forall i \\in W ) . \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial y } + P _ { t } \\left ( D \\right ) u + A u = f \\left ( y , x \\right ) , u ( 0 , x ) = 0 , \\end{align*}"}
-{"id": "5106.png", "formula": "\\begin{align*} g _ { n } = \\frac { - 1 } { \\left ( n + 1 \\right ) ! } . \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} 1 + \\frac 1 p = \\frac 1 r + \\frac 1 q = \\frac 1 { \\tilde r } + \\frac 1 { \\tilde q } . \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{gather*} - 4 x y U _ { s } ( y ) U _ { m } ( x ) + 2 ( 2 x ^ { 2 } + 2 y ^ { 2 } - 1 ) U _ { s + 1 } ( y ) U _ { m + 1 } ( x ) \\\\ - 4 x y U _ { s + 2 } ( y ) U _ { m + 3 } ( x ) + U _ { s + 3 } ( y ) U _ { m + 3 } ( x ) = 0 , \\end{gather*}"}
-{"id": "929.png", "formula": "\\begin{align*} 0 & = Q ^ 6 \\xi _ 1 ^ 2 + \\xi _ 1 ^ 8 \\\\ 0 & = Q ^ 8 \\xi _ 1 ^ 2 + \\xi _ 1 ^ 4 Q ^ 4 \\xi _ 1 ^ 2 \\\\ 0 & = Q ^ { 1 0 } \\xi _ 1 ^ 2 + ( Q ^ 4 \\xi _ 1 ^ 2 ) ^ 2 \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} \\Big | \\sum _ { m = 2 } ^ { n + 1 } \\int _ { \\Gamma _ + } ( \\alpha _ { m , 1 } - \\alpha _ { m , 2 } ) ( x , v , x '' , v '' ) \\phi ( x , v ) d \\xi ( x , v ) \\Big | \\le C \\| \\phi \\| _ \\infty \\eta ^ { n - 1 } , \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{align*} \\displaystyle \\sum _ i P ( a _ i ^ { ( 1 ) } , b _ j | A _ 1 , B ) = \\displaystyle \\sum _ i P ( a _ i ^ { ( 2 ) } , b _ j | A _ 2 , B ) . \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} \\Big | \\sum _ { m = 2 } ^ { n + 1 } \\int _ { \\Gamma _ + } ( \\alpha _ { m , 1 } - \\alpha _ { m , 2 } ) ( x , v , x '' , v '' ) \\phi ( x , v ) d \\xi ( x , v ) \\Big | \\le \\left \\lbrace \\begin{array} { l } C \\| \\phi \\| _ \\infty \\eta ( 1 + | \\ln ( \\eta ) | ) , \\textrm { } n = 3 , \\\\ C \\| \\phi \\| _ \\infty \\eta \\textrm { w h e n } n \\ge 4 , \\end{array} \\right . \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } t _ 1 \\\\ \\vdots \\\\ t _ n \\end{array} \\right ) = A ^ { - 1 } \\left ( \\begin{array} { c } d _ 1 \\\\ \\vdots \\\\ d _ n \\end{array} \\right ) . \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} \\frac { 1 } { \\rho _ { F } ( \\Omega ) } = \\frac { 1 } { \\| d _ { F } \\| _ { L ^ { \\infty } ( \\Omega ) } } = \\min _ { \\varphi \\in W _ { 0 } ^ { 1 , \\infty } ( \\Omega ) \\setminus \\{ 0 \\} } \\frac { \\| F ( \\nabla \\varphi ) \\| _ { L ^ { \\infty } ( \\Omega ) } } { \\| \\varphi \\| _ { L ^ { \\infty } ( \\Omega ) } } . \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} P & = \\frac { Q } { ( Q ^ 2 - 1 ) } \\\\ R & = \\frac { ( 1 - 2 Q ^ 2 ) } { Q ^ 2 ( Q ^ 2 - 1 ) } . \\end{align*}"}
-{"id": "9568.png", "formula": "\\begin{align*} \\mathbf { F } _ { i , j } = \\begin{cases} \\overline { \\hat { f } \\left ( \\frac { i - 2 n - 1 } { 2 } \\right ) } \\hat { f } \\left ( \\frac { j - 2 n - 1 } { 2 } \\right ) , & \\left | i - j \\right | \\leq 2 \\delta , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} \\int \\nolimits _ { \\mathbb { R } } \\varphi ^ { \\prime } \\left ( s \\right ) \\mathrm { d } s = 0 \\in \\mathbb { R } ^ { d } \\ . \\end{align*}"}
-{"id": "5119.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\frac { B _ { n } ^ { \\left ( p \\right ) } } { n ! } z ^ { n } = \\left ( \\frac { z } { e ^ { z } - 1 } \\right ) ^ { p } . \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{align*} B _ { n } \\left ( x \\right ) = \\sum _ { p = 1 } ^ { n } \\binom { n + 1 } { p + 1 } \\left ( - 1 \\right ) ^ { p } B _ { n } ^ { \\left ( - p \\right ) } \\left ( - p x \\right ) , \\thinspace \\thinspace n \\ge 1 . \\end{align*}"}
-{"id": "5809.png", "formula": "\\begin{align*} \\hat { \\sigma } _ { A | X = \\mathbf { x } } = \\hat { D } _ A ( \\mathbf { x } ) \\ , \\hat { \\rho } _ A \\ , { \\hat { D } _ A ( \\mathbf { x } ) } ^ \\dag \\ ; . \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{align*} & G _ t ( x ) = \\frac { e ^ { - | x | ^ 2 / ( 4 t ) } } { ( 4 \\pi t ) ^ { 3 / 2 } } . \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} a ( t ) = \\sum a _ k ( - t ) ^ k , \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} \\Delta _ H ^ { \\perp } = \\begin{cases} \\left \\{ I + \\begin{pmatrix} 0 & b \\\\ c & 0 \\end{pmatrix} p \\in H \\cap \\ker \\varphi \\right \\} & H N ( C _ s ) \\\\ \\left \\{ I + \\begin{pmatrix} 0 & b \\\\ c & 0 \\end{pmatrix} p \\in H \\cap K \\right \\} & H N ( C _ { n s } ) . \\end{cases} \\end{align*}"}
-{"id": "1194.png", "formula": "\\begin{align*} \\| ( f _ t - h _ 0 ) e ^ { \\lambda _ 1 t } - h _ 1 \\| _ k = & \\| h _ 2 e ^ { ( \\lambda _ 1 - \\lambda _ 2 ) t } + \\cdots \\| _ k \\\\ = & e ^ { ( \\lambda _ 1 - \\lambda _ 2 ) t } \\left \\| \\sum _ { j \\geq 2 } e ^ { ( \\lambda _ 2 - \\lambda _ j ) t } h _ j \\right \\| _ k \\\\ \\leq & e ^ { ( \\lambda _ 1 - \\lambda _ 2 ) t } \\sum _ { j \\geq 2 } e ^ { ( \\lambda _ 2 - \\lambda _ j ) t } \\| h _ j \\| _ k \\\\ \\leq & e ^ { ( \\lambda _ 1 - \\lambda _ 2 ) t } C \\sum _ { j \\geq 2 } ( 1 + j ^ N ) e ^ { ( \\lambda _ 2 - \\lambda _ j ) t } . \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{align*} \\mathrm { o r d } _ P ( y ^ 2 - \\alpha ) > \\mathrm { o r d } _ P ( ( 2 y ) ^ 2 ) = 2 \\ , \\mathrm { o r d } _ P ( 2 y ) \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} \\Upsilon ( x ) = & \\frac { \\Gamma _ p ( 1 - a - a x + b + b x ) \\Gamma _ p ( 1 - a - a x - \\gamma ) } { \\Gamma _ p ( 1 - a - a x ) \\Gamma _ p ( 1 - a - a x + b + b x - \\gamma ) } \\\\ & \\cdot \\frac { \\Gamma _ p ( 1 - a - a x - \\delta ) \\Gamma _ p ( 1 - a - a x + b + b x - \\gamma - \\delta ) } { \\Gamma _ p ( 1 - a - a x + b + b x - \\delta ) \\Gamma _ p ( 1 - a - a x - \\gamma - \\delta ) } . \\end{align*}"}
-{"id": "5920.png", "formula": "\\begin{align*} p ( n - k ) & : = \\sum _ { d | n } s _ { d , k } ^ { ( - 1 ) } . \\end{align*}"}
-{"id": "2669.png", "formula": "\\begin{align*} b ^ 2 = x ^ 3 \\wedge b ^ 3 = y ^ 2 . \\end{align*}"}
-{"id": "9589.png", "formula": "\\begin{align*} \\widehat \\Gamma _ n - \\widetilde W ^ { X X } _ n & = ( \\widehat \\Lambda _ n - \\Lambda _ n ) \\widetilde W ^ { Z Z } _ n ( \\widehat \\Lambda _ n - \\Lambda _ n ) ^ { \\prime } - \\widetilde W ^ { X Z } _ n ( \\widehat \\Lambda _ n - \\Lambda _ n ) ^ { \\prime } - ( \\widehat \\Lambda _ n - \\Lambda _ n ) \\widetilde W ^ { X Z } _ n \\overset { p } { \\rightarrow } 0 . \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} u ( x , y , t ) = u _ { c _ * + \\delta ( t ) } ( \\xi ) + \\left ( b ( t ) e ^ { i y } + \\bar { b } ( t ) e ^ { - i y } \\right ) \\psi _ * ( \\xi ) + v ( \\xi , y , t ) . \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} h _ u ( f _ v ) : = h ( f _ v ) = \\langle v , u \\rangle , v \\in \\R ^ m . \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{align*} d _ 2 = u _ j + 2 v _ j & \\le 2 ( s + t ) / ( 3 s _ 1 ) + v _ j \\\\ & = 2 d _ 2 ( s + t ) / ( 3 ( s + 2 t ) ) + v _ j \\\\ & \\le d _ 2 - d _ 2 / 3 + v _ j , \\end{align*}"}
-{"id": "4875.png", "formula": "\\begin{align*} h _ 2 \\left ( I + \\begin{pmatrix} a & 0 \\\\ 0 & d \\end{pmatrix} p \\right ) h _ 2 ^ { - 1 } = I + \\frac { \\begin{pmatrix} a \\alpha ^ 2 + d \\beta \\gamma & a \\alpha \\beta - d \\alpha \\beta \\\\ a \\alpha \\gamma - d \\alpha \\gamma & a \\beta \\gamma - d \\alpha ^ 2 \\end{pmatrix} p } { - \\alpha ^ 2 - \\beta \\gamma } . \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{align*} \\bar { \\psi } _ i & = \\psi _ i & & \\mbox { i f $ i \\notin \\partial ( Q , f ) $ } , \\\\ \\bar { \\psi } _ i & = \\psi _ i + \\psi _ i ^ { ( 1 ) } + \\psi _ i ^ { ( 2 ) } + \\psi _ i ^ { ( 3 ) } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! & & \\mbox { i f $ i \\in \\partial ( Q , f ) $ } . \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} \\dim _ { F _ i } S _ i = p _ i , S _ i + S _ i \\alpha _ i + \\dots + S _ i \\alpha _ i ^ { s - 1 } = \\mathbb { K } , \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} \\Delta _ { A | M } ( ( \\mathcal { N } _ A ( s ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) ) ( t ) = I ( A : X | M ) _ { ( \\mathcal { N } _ A ( s ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\sigma } _ { A M X } ( t ) ) } \\ ; , \\end{align*}"}
-{"id": "8126.png", "formula": "\\begin{align*} F _ 2 ( z ) = \\sum _ { n = 0 } ^ \\infty F _ { 2 , n } ( z ) , z \\in \\bar { \\mathbb D } _ 0 , \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{gather*} \\theta ^ 0 _ 1 = \\tilde { \\theta } ^ 0 - \\varepsilon ^ { - 1 } , \\theta ^ 0 _ 2 = \\varepsilon ^ { - 1 } , t = - \\varepsilon \\tilde { t } , H = - \\varepsilon ^ { - 1 } \\tilde { H } , \\\\ q _ 1 = 1 - \\tilde { p } _ 1 - \\tilde { p } _ 2 , p _ 1 = \\tilde { q } _ 2 , q _ 2 = 1 - \\tilde { p } _ 1 , p _ 2 = \\tilde { q } _ 1 - \\tilde { q } _ 2 . \\end{gather*}"}
-{"id": "4212.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { | \\partial V | } g _ i = 0 , \\end{align*}"}
-{"id": "2342.png", "formula": "\\begin{align*} g ( r _ 2 ) \\le f ( r _ 1 ) + ( r _ 1 - r _ 2 ) ^ 2 / 2 + ( r _ 2 - t _ f ) ^ 2 / 2 = g ( r _ 1 ) + ( t _ f - r _ 1 ) ( r _ 2 - r _ 1 ) . \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} \\Phi ( L ) = \\sum _ { J \\subseteq [ N ] } p _ J ^ * \\log | \\det ( K - I _ { \\bar J } ) | . \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} N : = \\begin{pmatrix} - i & 5 i & - 3 + i & 3 - 3 i & 7 + i \\\\ 0 & - 3 + 2 i & 0 & - 2 & - 1 + 3 i \\\\ 0 & - 1 & 0 & - 1 + 2 i & 5 \\\\ 0 & 1 & 0 & i & - i \\\\ 0 & 5 & 0 & 7 + 4 i & 1 + 2 i \\end{pmatrix} , \\end{align*}"}
-{"id": "901.png", "formula": "\\begin{align*} \\rho _ { \\Phi } ^ T : = \\frac { 1 } { T ^ { k / 2 } } \\sum _ { j _ 1 \\neq \\ldots \\neq j _ k } \\sigma _ { j _ 1 } \\ldots \\sigma _ { j _ k } \\int _ { [ 0 , T ] ^ k } \\Phi ( x ^ { j _ 1 } + \\xi ^ { j _ 1 } _ { s _ 1 } , \\ldots , x ^ { j _ k } + \\xi ^ { j _ k } _ { s _ k } ) d s _ 1 \\ldots d s _ k , \\end{align*}"}
-{"id": "2680.png", "formula": "\\begin{align*} \\frac { k ^ 2 + 1 } { k + 1 } = k - 1 + \\frac { 2 } { k + 1 } \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} \\mu _ 2 ^ { ( \\alpha ) } = A _ { 2 p } ^ { ( \\alpha ) } \\lambda . \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Xi _ 1 ^ { ( k ) } ( t ) + c \\Pi _ 1 ^ { ( k ) } ( t ) \\leq C \\Gamma _ 1 ^ { ( k ) } ( t ) , k = 0 , 1 , \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{align*} u ( t ) = u _ { 0 } \\ast _ { x } E _ 0 ( t ) + v _ 0 \\ast _ x E _ 1 ( t ) , \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = e _ 4 , [ e _ 1 , e _ 2 ] = \\alpha _ 1 e _ 3 + \\alpha _ 2 e _ 4 + \\alpha _ 3 e _ 5 , [ e _ 2 , e _ 1 ] = \\beta _ 1 e _ 3 + \\beta _ 2 e _ 4 + \\beta _ 3 e _ 5 , [ e _ 2 , e _ 2 ] = e _ 5 . \\end{align*}"}
-{"id": "8827.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { r } ( - 1 ) ^ m \\frac { ( 2 d ) ^ m } { m ! } \\overline { p } _ i = b _ i ( - 2 d ) e ^ { - 2 d } + O \\left ( \\frac { M ( - 2 d ) ^ { r + 1 } } { ( r + 1 ) ! } \\right ) , \\end{align*}"}
-{"id": "9595.png", "formula": "\\begin{align*} a ^ i _ { j | k } = a ^ i _ { j , k } + L ^ i _ { \\underline { \\alpha k } } a ^ \\alpha _ j - L ^ \\alpha _ { \\underline { j k } } a ^ i _ \\alpha , \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{gather*} \\frac { { \\rm d } Y } { { \\rm d } x } = \\left ( \\frac { A _ 0 } { x } + \\frac { A _ 1 } { x - 1 } + \\frac { A _ t } { x - t } \\right ) Y \\end{gather*}"}
-{"id": "9721.png", "formula": "\\begin{align*} G ^ { { \\cal I } _ i } = \\begin{cases} I , & ~ { \\cal I } _ i \\in ~ T y p e _ A \\\\ C , & ~ { \\cal I } _ i \\in ~ T y p e _ B \\\\ I + C - C _ 1 , & ~ { \\cal I } _ i \\in ~ T y p e _ C , \\end{cases} \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} H ( z , w ) = \\frac { 1 } { ( \\overline { w } ^ 2 z ^ 2 - 2 \\overline { w } \\cdot z + 1 ) ^ { n / 2 } } . \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} g ( t ) = \\sum _ { k \\ge 1 } g _ k \\sin \\frac { k \\pi } { \\omega } \\vartheta , t \\in \\rho ' _ 1 \\Pi , \\end{align*}"}
-{"id": "9303.png", "formula": "\\begin{align*} p _ { i j , i } p _ { i k , k } + p _ { i j , j } p _ { j k , k } = p _ { j k , k } p _ { i k , k } . \\end{align*}"}
-{"id": "5317.png", "formula": "\\begin{align*} [ x _ \\alpha ^ + \\otimes t ^ { s } , ( r + 1 ) Y _ \\alpha [ s ] ^ { ( r + 1 ) } ] & = [ x _ \\alpha ^ + \\otimes t ^ { s } , Y _ \\alpha [ s ] Y _ \\alpha [ s ] ^ { ( r ) } ] \\\\ & = [ x _ \\alpha ^ + \\otimes t ^ { s } , Y _ \\alpha [ s ] ] Y _ \\alpha [ s ] ^ { ( r ) } + Y _ \\alpha [ s ] [ x _ \\alpha ^ + \\otimes t ^ { s } , Y _ \\alpha [ s ] ^ { ( r ) } ] \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} \\frac { x _ { n + 1 } - x _ { n } } { h } = f \\left ( x _ { n } \\right ) . \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} c _ { j } = \\prod _ { i = k _ { 0 } + 1 } ^ { j } ( \\lambda _ { k _ 0 } - \\lambda _ { i } ) ^ { - 1 } \\quad \\textrm { f o r e v e r y } j > k _ { 0 } . \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{align*} T _ 4 T _ 3 M _ 0 J ( k ) T _ 0 T _ 5 = \\begin{bmatrix} k I _ \\mu + o ( k ) & o ( k ) \\\\ o ( k ) & I _ { n - \\mu } + o ( 1 ) \\end{bmatrix} , \\ ; \\ ; k \\to 0 , \\ ; \\ ; k \\in \\overline { \\mathbb { C } } ^ + , \\end{align*}"}
-{"id": "1626.png", "formula": "\\begin{align*} \\zeta = t _ 1 + i t _ 2 \\quad \\mbox { a n d } z = x _ 1 + i x _ 2 \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{align*} H = 0 \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} ( H _ { ( 1 ) } P ^ T D ) _ { i k } = ( - 1 ) ^ k \\biggl ( { { k } \\choose { i } } - { { k } \\choose { i + 1 } } + \\cdots + ( - 1 ) ^ { k } { { k } \\choose { k } } \\biggr ) = ( - 1 ) ^ k { { k - 1 } \\choose { i - 1 } } , \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{gather*} \\left ( Z ( X _ { \\psi } , T ) \\prod _ { i = 0 } ^ { n - 1 } \\big ( 1 - q ^ i T \\big ) \\right ) ^ { ( - 1 ) ^ { n } } \\left ( Z ( X ' _ { \\psi } , T ) \\prod _ { i = 0 } ^ { n - 1 } \\big ( 1 - q ^ i T \\big ) \\right ) ^ { ( - 1 ) ^ { n } } \\end{gather*}"}
-{"id": "3814.png", "formula": "\\begin{align*} \\nu _ 2 \\left ( { 2 ^ k \\choose 2 i } \\right ) & = \\nu _ 2 \\left ( \\frac { 2 ^ k } { 2 i } { 2 ^ k - 1 \\choose 2 i - 1 } \\right ) \\\\ & = k - 1 - \\nu _ 2 ( i ) + s _ 2 ( 2 i - 1 ) + s _ 2 ( 2 ^ k - 2 i ) - s _ 2 ( 2 ^ k - 1 ) = k - 1 - \\nu _ 2 ( i ) , i > 0 \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} \\| \\widetilde { E _ j } \\| _ { L ^ { k } ( S _ r , d \\sigma ) } \\lesssim q ^ \\alpha \\| E _ j \\| _ { L ^ { \\ell } ( \\mathbb F _ q ^ d , d \\textbf { m } ) } \\quad \\mbox { f o r a l l } ~ ~ r \\in \\mathbb F _ q ^ * , ~ ~ j = 1 , 2 , \\ldots , k . \\end{align*}"}
-{"id": "3019.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } 2 ^ { - \\tau ^ { ( k ) } } \\ , M ^ { \\Delta ^ { ( k - 1 ) } } = 0 \\quad \\hbox { f o r e v e r y $ M > 0 $ } , \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{align*} S _ 2 = \\frac { \\lambda _ { n + 1 } } { ( \\lambda _ { n + 1 } - \\alpha _ 2 ) } S _ 2 ' - \\frac { 1 } { ( \\lambda _ { n + 1 } - \\alpha _ 2 ) } S _ 2 '' \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} B ' _ l = \\frac { 1 } { | C | } \\sum _ { j = 0 } ^ n \\frac { \\mu _ l } { v _ j } Q _ j ( l ) B _ j l \\in [ 0 . . n ] , \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} L _ { 2 m k + m } - L _ { 2 m k - m } = L _ m L _ { 2 m k } \\ , , \\quad \\mbox { $ m $ o d d } \\ , , \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} \\nabla _ { A } = \\overline { \\left \\{ c \\in \\left ( \\mathbb { C } ^ { * } \\right ) ^ { n } \\mid \\exists \\theta \\in \\left ( \\mathbb { C } ^ { * } \\right ) ^ { d - 1 } f \\left ( \\theta \\right ) = \\frac { \\partial f } { \\partial \\theta _ { i } } \\left ( \\theta \\right ) = 0 i \\right \\} } . \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} g _ 1 : = \\prod _ { \\epsilon \\in I } g _ { 1 \\epsilon } \\ \\ \\ \\ g _ 2 : = \\prod _ { \\epsilon \\in I } g _ { 2 \\epsilon } . \\end{align*}"}
-{"id": "8275.png", "formula": "\\begin{align*} D _ { j _ 1 \\dots j _ n } ^ { ( s , Q ) } = \\{ ( x _ 1 , \\dots , x _ n ) \\in D _ { j _ 1 \\dots j _ n } ^ { ( s ) } ; \\ x _ { Q ( 1 ) } < \\dots < x _ { Q ( n ) } \\} , \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a - p & p - a \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] \\equiv \\sum _ { k = 0 } ^ a \\frac { ( - a ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } \\cdot z ^ k - z ^ p \\sum _ { k = 0 } ^ a \\frac { ( - a ) _ k ^ 2 } { ( 1 ) _ k ^ 2 } \\cdot \\big ( 1 + 2 p ( H _ { a - k } - H _ k ) \\big ) \\cdot z ^ k \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{align*} \\frac { d | | \\mathbf { \\ddot { p } } _ d | | } { d t } & = 0 \\\\ \\frac { d } { d t } \\sqrt { v ^ 2 \\dot { \\phi } ^ 2 + \\dot { v } ^ 2 } & = 0 \\\\ v ^ 2 \\dot { \\phi } \\ddot { \\phi } + \\dot { \\phi } ^ 2 v \\dot { v } + \\dot { v } \\ddot { v } & = 0 \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} X _ { 2 , \\ , j + 2 , \\ , 3 } \\cap L _ { j + 2 } = 0 . \\end{align*}"}
-{"id": "9947.png", "formula": "\\begin{align*} \\mathcal { C } = \\mathcal { C } ( E , A ) : = ( ( A ^ { - 1 } E ) ^ { k ^ { * } } ) \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} S _ n ( d ) = S _ n ( [ 0 , d ] ) : = \\{ L _ G ( e _ 1 ) L _ G ( e _ 2 ) \\cdots L _ G ( e _ n ) : e _ 1 e _ 2 \\cdots e _ n \\in \\Gamma _ n ( G ; f , [ 0 , d ] ) \\} . \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\lambda } ^ d | L _ n | = \\mathbb { E } _ { \\lambda } ^ d \\sum _ { x : | x | = n } I _ { \\{ x \\in I _ { \\infty } \\} } = \\sum _ { x : | x | = n } \\mathbb { P } _ { \\lambda } ^ d ( x \\in I _ { \\infty } ) . \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{align*} ( u , v ) \\longmapsto \\left ( \\frac { b ( \\frac { u } { v } - 1 ) } { \\frac { u } { v } ( b - \\frac { u } { v } a ) } , \\frac { \\frac { u } { v } a ( \\frac { u } { v } - 1 ) } { b - \\frac { u } { v } a } \\right ) = \\left ( \\frac { b v ( u - v ) } { u ( b v - u a ) } , \\frac { u a ( u - v ) } { v ( b v - u a ) } \\right ) . \\end{align*}"}
-{"id": "9933.png", "formula": "\\begin{align*} \\vect { S } _ \\pm \\Gamma = F \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} - \\gamma [ B ( i \\sqrt { 2 } / \\gamma ) + \\lambda / \\gamma ] \\tilde { G } = \\frac { i c } { 2 \\pi } \\tilde { G } _ 1 e _ 0 , \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} \\begin{aligned} \\Xi _ { N } & : = \\{ ( a _ 1 , a _ 2 , \\dots , a _ N ) \\mid 1 \\leq a _ 1 < a _ 2 < \\dots < a _ N \\leq K \\} , \\\\ \\Xi _ { N } ^ { > } & : = \\{ ( c _ 1 , c _ 2 , \\dots , c _ N ) \\mid K \\geq c _ 1 > c _ 2 > \\dots > c _ N \\geq 1 \\} . \\end{aligned} \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} C _ { n - 1 } = \\sum _ { m = 1 } ^ { n } \\left ( - 1 \\right ) ^ { m + 1 } \\sum _ { \\underset { k _ { i } \\ge 1 } { k _ { 1 } + \\dots + k _ { m } = n } } C _ { k _ { 1 } } \\dots C _ { k _ { m } } . \\end{align*}"}
-{"id": "3507.png", "formula": "\\begin{align*} I _ { \\upsilon } ( 1 / 2 ; x ) = e ^ { x } J _ { \\upsilon } ( x ) , \\ \\ \\ R e ( \\upsilon ) > - \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{gather*} f _ 2 ( u ) > 0 \\ \\ \\ \\ u > 0 , f _ 2 ( 0 ) = 0 , f _ 2 ( u ) \\to f _ 2 ^ + \\ \\ \\ \\ u \\to + \\infty ; \\\\ f _ 1 ( u ) > 0 \\ \\ \\ \\ u > c _ 0 , f _ 1 ( c _ 0 ) = 0 , f _ 1 ( u ) \\to f _ 1 ^ + \\ \\ \\ \\ u \\to + \\infty , \\end{gather*}"}
-{"id": "9144.png", "formula": "\\begin{align*} f _ i ( x _ 0 , \\ldots , x _ { p - 1 } ) = ( \\rho _ i ( x _ 0 , x _ { t _ i } ) , \\rho _ i ( x _ { t _ i } , x _ { 2 t _ i } ) , \\ldots , \\rho _ i ( x _ { ( p - 1 ) t _ i } , x _ 0 ) ) . \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} f ( z ) = y . \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} \\left \\langle I + \\begin{pmatrix} 1 & 0 \\\\ \\gamma & 1 \\end{pmatrix} p , I + \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} p , I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} g _ { n } ( X , Y ) = a _ 0 X ^ { n + 1 } + a _ 1 X ^ { n } Y + \\cdot \\cdot \\cdot + a _ n X Y ^ { n } + a _ { n + 1 } Y ^ { n + 1 } , \\\\ \\mbox { w i t h } a _ 0 , a _ 1 , \\cdot \\cdot \\cdot , a _ { n + 1 } \\in { \\mathbb R } . \\end{align*}"}
-{"id": "6886.png", "formula": "\\begin{align*} u = \\frac { 1 + t ^ { 2 } - 2 y t - ( 1 - t ) \\sqrt { ( 1 + t ) ^ { 2 } - 4 y t } } { 2 ( 1 - y ) t } \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} g ( z ) : = \\int _ { - 1 } ^ 1 \\log ( z - t ) \\psi _ { \\alpha , \\varepsilon } ( t ) d t , z \\in \\mathbb { C } \\setminus [ - \\infty , 1 ] , \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{align*} \\tau ^ { ( k ) } = 2 ^ { C k } , \\quad \\delta ^ { ( k ) } = 2 \\ , \\cdot \\ , 2 ^ { C k } \\quad \\textrm { a n d } \\Delta ^ { ( k ) } = 1 0 \\ , \\cdot \\ , 2 ^ { C k } , k \\ge 1 . \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} \\widetilde { T } ^ { L , \\Xi , \\widetilde { \\mathbf { r } } } _ { F , G , N } ( g _ 1 , \\dots , g _ { d } ) : = \\frac { 1 } { N ^ { h - m } } \\int \\limits _ { \\mathbf { x } \\in \\mathbb { R } ^ h } \\Big ( \\prod \\limits _ { j = 1 } ^ { d } g _ j ( \\xi _ j ( \\mathbf { x } ) + \\widetilde { \\mathbf { r } _ j } ) \\Big ) F ( \\mathbf { x } ) G ( L \\mathbf { x } ) \\ , d \\mathbf { x } . \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} \\big ( \\frac { \\partial \\phi _ { n } } { \\partial t } , \\ , f \\big ) = \\big ( \\nabla \\frac { \\partial \\phi _ { n } } { \\partial t } , \\ , \\nabla \\psi _ { n } \\big ) = \\big ( \\frac { \\partial } { \\partial t } | \\Psi _ { n } | ^ { 2 } , \\ , \\psi _ { n } \\big ) . \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} b _ k = b _ k ^ 1 + b _ k ^ 2 = b _ k ^ 1 + \\bigl ( b _ { k } ^ { 2 , 1 } + b _ { k } ^ { 2 , 2 } + b _ { k } ^ { 2 , 3 } + b _ k ^ { 2 , 4 } \\bigr ) , \\end{align*}"}
-{"id": "9971.png", "formula": "\\begin{align*} Q _ 1 = \\left [ \\begin{array} { c c c c } 2 c _ 1 \\ ! - \\ ! 2 \\epsilon m _ 1 & \\epsilon \\frac { m _ 1 c _ 1 } { k _ 1 } & 0 & 0 \\\\ * & 2 \\epsilon \\frac { m _ 1 ^ 2 } { k _ 1 } & 0 & 0 \\\\ 0 & 0 & 2 c _ 2 \\ ! - \\ ! 2 \\epsilon m _ 2 & \\epsilon \\frac { m _ 2 c _ 2 } { k _ 2 } \\\\ 0 & 0 & * & 2 \\epsilon \\frac { m _ 2 ^ 2 } { k _ 2 } \\end{array} \\right ] \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} r _ { k + 1 } = \\left \\{ \\begin{array} { l l l } r ^ { \\ast } \\left ( x _ { 1 } , r _ { k } \\right ) & & r _ { k } \\geq \\frac { 1 } { \\left \\vert F ^ { \\prime } \\left ( x _ { 1 } \\right ) \\right \\vert } \\\\ \\frac { 1 } { 2 } r _ { k } & & r _ { k } < \\frac { 1 } { \\left \\vert F ^ { \\prime } \\left ( x _ { 1 } \\right ) \\right \\vert } \\end{array} \\right . \\end{align*}"}
-{"id": "1969.png", "formula": "\\begin{align*} \\widetilde { A } ( z ) = { } _ a \\langle 1 | \\widetilde { T } _ { a } ( z ) | 1 \\rangle _ { a } , \\\\ \\widetilde { B } ( z ) = { } _ a \\langle 0 | \\widetilde { T } _ { a } ( z ) | 1 \\rangle _ { a } . \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} \\psi _ { \\alpha , \\varepsilon } ( x ) = \\frac { 2 \\sqrt { \\alpha ^ 2 - 1 } - \\alpha \\varepsilon x } { 2 \\alpha \\pi \\sqrt { 1 - x ^ 2 } } + \\frac { 1 } { \\alpha \\pi } \\arctan \\left ( \\frac { \\sqrt { 1 - x ^ 2 } } { \\sqrt { \\alpha ^ 2 - 1 } } \\right ) , x \\in ( - 1 , 1 ) . \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 2 } { x ^ 2 } + \\frac { A _ 1 } { x } + A _ 0 \\right ) Y , \\frac { \\partial Y } { \\partial t } = \\left ( \\frac { 1 } { t } A _ 0 x + B _ 0 \\right ) Y , \\end{gather*}"}
-{"id": "3619.png", "formula": "\\begin{align*} \\varphi ( x ) = y ^ 2 \\left ( - \\frac { x ^ 2 } { 2 } - \\frac { x ^ 3 } { 3 } \\right ) - \\lambda x y + n \\ln x - \\lambda y \\frac { x ^ 2 } { 2 } + \\ldots \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} 2 \\lambda H ^ { ( 0 ) } = L _ 0 H ^ { ( 0 ) } + ( i a ^ \\dagger G + c c ) ; \\end{align*}"}
-{"id": "1161.png", "formula": "\\begin{align*} x _ k = \\int _ 0 ^ T u ( s ) e ^ { - i \\lambda _ k s } d s , \\ \\ \\ \\ \\ \\ \\ \\ \\forall k \\in \\N ^ * . \\\\ \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} ( U M + V \\overline { N } ) z + ( U N + V \\overline { M } ) \\bar { z } = U p + V \\overline { p } \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{align*} y ( t ) : = t ^ { r / 2 } x ( \\log t ) \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} E ^ r = \\begin{pmatrix} e _ 1 & \\sigma ^ { - 1 } ( e _ 1 ) & \\cdots & \\sigma ^ { - r + 1 } ( e _ 1 ) \\\\ e _ 2 & \\sigma ^ { - 1 } ( e _ 2 ) & \\ldots & \\sigma ^ { - r + 1 } ( e _ 2 ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ e _ \\nu & \\sigma ^ { - 1 } ( e _ \\nu ) & \\cdots & \\sigma ^ { - r + 1 } ( e _ \\nu ) \\end{pmatrix} _ { \\nu \\times r } \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} \\omega & = \\omega ^ 0 + \\varepsilon \\omega ^ 1 + \\o ( \\varepsilon ) , \\\\ \\Phi & = \\Phi ^ 0 + \\varepsilon \\Phi ^ 1 + \\o ( \\varepsilon ) , \\\\ \\xi & = \\xi ^ 0 + \\varepsilon \\xi ^ 1 + \\o ( \\varepsilon ) . \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} v ( \\theta ) = & v ( 0 ) + \\int _ 0 ^ \\theta w ( s ) d s \\le v ( 0 ) + \\int _ 0 ^ \\theta ( w ( 0 ) + \\bar L s ) d s \\\\ = & v ( 0 ) + v ' ( 0 ) \\theta + \\frac { \\bar L } { 2 } \\theta ^ 2 . \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} 0 \\neq [ L _ { - 4 } , \\ , L _ { - 1 } ] = [ L _ { - 4 } , \\ , [ L _ { - 3 } , \\ , [ L _ { - 3 } , \\ , L _ 5 ] ] ] = [ L _ { - 3 } , \\ , [ L _ { - 3 } , \\ , [ L _ { - 4 } , \\ , L _ 5 ] ] ] , \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{align*} s ^ C - s = \\frac 1 2 ( | \\theta | ^ 2 + \\delta \\theta ) - \\frac 1 2 \\left ( T ^ { \\alpha \\qquad \\ ; \\gamma } _ { \\enskip T ( \\gamma , \\alpha ) } + T ^ { \\bar \\alpha \\qquad \\ ; \\bar \\gamma } _ { \\enskip T ( \\bar \\gamma , \\bar \\alpha ) } \\right ) . \\end{align*}"}
-{"id": "9667.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\Theta ( K ) } d \\eta \\int _ 0 ^ { \\pi } \\rho _ { K } ^ 3 ( P ( \\xi , \\eta ) ) \\sin \\xi \\ , d \\xi = \\int _ { \\Theta ( K ) } ^ \\pi d \\eta \\int _ 0 ^ { \\pi } \\rho _ { K } ^ 3 ( P ( \\xi , \\eta ) ) \\sin \\xi \\ , d \\xi . \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} \\widetilde { r } _ { s } = \\frac { 1 } { n } \\overline { F } \\sigma _ { s , t _ { 1 } } ^ { T } \\widetilde { r } _ { c } = \\frac { 1 } { n } \\overline { F } \\sigma _ { c , t _ { 2 } } ^ { T } . \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} \\omega _ i - \\sum _ { j \\in \\{ i ^ + , i ^ - \\} } a _ { i , j } \\sin ( \\theta _ { i } - \\theta _ { j } ) = 0 i = 0 , \\dots , n \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{align*} ( 1 - \\varphi ) u ( t ) = e ^ { ( t - \\varepsilon ) \\Delta } ( 1 - \\varphi ) u ( \\varepsilon ) + \\int _ \\varepsilon ^ t e ^ { t - s ) \\Delta } [ 2 \\nabla u \\cdot \\nabla \\varphi + u \\Delta \\varphi + ( 1 - \\varphi ) | u | ^ \\alpha u ] \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} \\tilde f _ h ^ n : = \\frac { h ^ 2 } { \\omega _ { 2 - \\alpha } ( h ) } f ^ n . \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{gather*} \\dot { h _ 1 } / h _ 1 = \\frac { 1 } { 2 } h _ 2 / h _ 1 , \\dot { h _ 2 } / h _ 2 = - \\frac { 1 } { 2 } h _ 2 / h _ 1 - h _ 2 / h _ 3 \\\\ \\dot { h _ 3 } / h _ 3 = \\frac { 1 } { 2 } h _ 4 / h _ 3 + h _ 2 / h _ 3 , \\dot { h _ 4 } / h _ 4 = - \\frac { 1 } { 2 } h _ 4 / h _ 3 \\end{gather*}"}
-{"id": "8678.png", "formula": "\\begin{align*} \\Psi ( v ) = \\frac { 1 } { \\alpha + 2 } | v | ^ { \\alpha + 2 } - \\frac { 2 } { \\alpha ^ 2 } | v | ^ 2 \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{align*} \\sigma _ k \\ast { f _ q ^ { \\gamma } } = \\sigma _ k \\ast { f _ q ^ { \\gamma } } - g _ k ^ { m ( k , q , \\gamma ) , \\gamma } + \\sum _ { i \\ge m ( k , q , \\gamma ) } ( g _ k ^ { i , \\gamma } - g _ k ^ { i + 1 , \\gamma } ) . \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{align*} \\Gamma ( Q , f , m _ { \\bullet } , c _ { \\bullet } ) = K Q / L ( Q , f , m _ { \\bullet } , c _ { \\bullet } ) , \\end{align*}"}
-{"id": "9846.png", "formula": "\\begin{align*} c _ { s } ^ { ( n ) } ( \\chi _ \\mathbf { z } ) = c _ { s } ^ { ( n ) } ( \\mathbf { z } ) = \\frac { 1 - q ^ { - 1 } \\mathbf { z } ^ { n _ \\alpha \\alpha ^ \\vee } } { 1 - \\mathbf { z } ^ { n _ \\alpha \\alpha ^ \\vee } } = \\frac { 1 - q ^ { - 1 } ( \\mathbf { z } ' ) ^ { \\alpha ' } } { 1 - ( \\mathbf { z } ' ) ^ { \\alpha ' } } . \\end{align*}"}
-{"id": "8518.png", "formula": "\\begin{align*} X = \\cup \\{ X _ \\delta : \\ \\delta \\in S \\} , \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! C _ { I A } = { \\log _ 2 } { \\left ( 1 + \\norm { \\tilde { \\textbf { s } } _ d ^ \\star } ^ 2 P _ { } \\right ) } . \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{align*} \\mathrm { b } _ \\sigma C ( P ) = A _ 1 + A _ 2 = 1 \\otimes \\sum _ { i , j } V ^ i _ j P ^ j _ i . \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} \\big ( \\frac { \\partial \\mathbf { A } } { \\partial t } , \\ , \\nabla \\varphi ) = 0 , \\ ; \\forall \\varphi \\in H _ { 0 } ^ { 1 } ( \\Omega ) \\cap H ^ { 2 } ( \\Omega ) . \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} g ( u ) = \\left ( \\prod _ { i = 1 } ^ n u _ i \\right ) \\left ( a _ 1 \\otimes \\ldots \\otimes a _ m \\right ) \\left ( \\begin{array} { c } \\theta _ 0 \\\\ \\theta _ m \\\\ \\theta _ { m - 1 } \\\\ \\ldots \\\\ \\theta _ { 1 2 \\ldots m } \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "2607.png", "formula": "\\begin{align*} g _ { i \\bar j } = \\delta _ { i \\bar j } , \\ g _ { i \\bar j l } = 0 \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} \\hat { S } _ { 1 T } \\left ( u \\right ) = S _ { 1 T } \\left ( u \\right ) + T ^ { 1 / 2 } \\left ( \\hat { \\theta } _ { T } - \\theta _ { 0 } \\right ) ^ { \\prime } \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\nabla \\left ( F _ { t , \\theta _ 0 } \\left ( \\cdot \\mid \\Omega _ t \\right ) , u \\right ) + o _ { p } \\left ( 1 \\right ) , \\end{align*}"}
-{"id": "4172.png", "formula": "\\begin{align*} { y } ( t ) & = \\breve { y } ( t ) \\ast h ( t ) \\\\ & = \\gamma { x } ( t - \\tau ) + { \\eta } ( t ) . \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} \\begin{aligned} G ' _ 1 & = [ \\lambda \\det B , A ] \\cdot G _ 1 \\\\ G ' _ 2 & = [ \\lambda \\det A , B ] \\cdot G _ 2 \\end{aligned} \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } i u _ { t } + \\Delta u = \\pm | u | ^ 4 u , \\ , \\ , \\ , \\ , x \\in { \\mathbb { R } } , \\ , \\ , \\ , \\ , t \\in \\mathbb { R } , \\\\ u ( x , 0 ) = u _ 0 ( x ) \\in H ^ { s } ( \\mathbb { R } ) . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} 0 = a _ 0 < a _ 1 < \\ldots < a _ n = 1 \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{align*} \\Theta ( \\alpha ) = \\left ( \\begin{array} { c c } \\bar { \\alpha } & \\rho \\\\ \\rho & - \\alpha \\\\ \\end{array} \\right ) \\rho = \\sqrt { 1 - | \\alpha | ^ 2 } \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} \\left | | F ( w ) | ^ 2 - | f ( i t ) | ^ 2 \\right | = \\left | | F ( w ) | ^ 2 - | F ( z ) | ^ 2 \\right | < \\epsilon . \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} t \\in [ 0 , T ] \\mapsto \\{ x \\in Y : w ( x , t ) = c \\} \\end{align*}"}
-{"id": "3217.png", "formula": "\\begin{align*} p _ t ( x , y ) = q _ t ( f ( x ) , f ( y ) ) \\partial _ y f ( y ) \\ \\ \\textnormal { w h e r e } \\ \\ f ( x ) = a r s i n h ( x ) . \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} K _ 2 = K _ 1 ( p _ 1 ^ { 1 / n } , . . . , p _ \\nu ^ { 1 / n } ) , \\end{align*}"}
-{"id": "3146.png", "formula": "\\begin{align*} ( \\nabla _ { i } f ) R _ { j k } + f \\nabla _ { i } R _ { j k } = \\nabla _ { i } \\nabla _ { j } \\nabla _ { k } f - ( \\nabla _ { i } \\Delta f ) g _ { j k } . \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } h _ g ( \\theta ) d \\theta \\leq \\frac { 2 b } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } | \\sin \\theta | d \\theta = \\frac { 4 b } { \\pi } . \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} f _ { c _ { 2 } } \\left ( z \\right ) = M \\left ( z \\right ) f \\left ( z \\right ) + N \\left ( z \\right ) f ^ { \\prime } \\left ( z \\right ) \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{align*} \\frac { u } { v } - \\sqrt { \\frac { b } { a } } \\leqslant \\frac { b a ^ { - 1 } - u v ^ { - 1 } } { u v ^ { - 1 } + \\sqrt { b a ^ { - 1 } } } \\varepsilon B ^ { - \\frac { 1 } { r } } \\leqslant \\frac { b a ^ { - 1 } - \\sqrt { b a ^ { - 1 } } } { 2 \\sqrt { b a ^ { - 1 } } } \\varepsilon B ^ { - \\frac { 1 } { r } } = \\frac { 1 } { 2 } \\left ( \\sqrt { \\frac { b } { a } } - 1 \\right ) \\varepsilon B ^ { - \\frac { 1 } { r } } . \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} \\psi = \\psi ^ { ( 4 ) } + \\psi ^ { ( 3 ) } \\wedge \\eta , \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{align*} \\widetilde { \\mathbf { H } } = [ H _ { 1 , 1 } ( \\mathbf { f } ) , \\dots , H _ { 1 , N _ R } ( \\mathbf { f } ) , H _ { 2 , 1 } ( \\mathbf { f } ) , \\dots , H _ { N _ T , N _ R } ( \\mathbf { f } ) ] ^ T , \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} f = \\sum _ { j = 1 } ^ N a _ j 1 _ { F _ j } \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} \\langle [ D ] _ T , \\gamma _ 2 , \\ldots , \\gamma _ r \\rangle _ d = ( [ D ] \\cap d ) \\langle \\gamma _ 2 , \\ldots , \\gamma _ r \\rangle _ d \\ / . \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} \\Omega _ { R , \\delta } : = \\left ( \\Omega \\cap B _ R ( 0 ) \\right ) \\setminus \\bigcup _ { i = 1 } ^ M \\tilde B _ \\delta ^ i , \\end{align*}"}
-{"id": "2821.png", "formula": "\\begin{align*} A _ k = \\{ \\omega : F ^ { t _ k } ( \\ ; ( \\omega , p ) \\ ; ) \\in U _ w \\} , \\ ; k \\in \\mathbb N . \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla \\psi _ \\varepsilon | ^ 4 = o ( 1 ) , \\end{align*}"}
-{"id": "7827.png", "formula": "\\begin{align*} \\nu _ 1 \\equiv \\Gamma ( u _ T - P u _ 0 \\big | _ { t = T } ) , u \\equiv P u _ 0 + P _ 1 \\nu _ 1 \\end{align*}"}
-{"id": "6598.png", "formula": "\\begin{align*} \\mathcal { D } ( S ^ * ) \\ ; & = \\ ; \\mathcal { D } ( S _ D ) \\dotplus \\ker S ^ * \\ , , \\\\ & = \\ ; \\mathcal { D } ( \\overline { S } ) \\dotplus S _ D ^ { - 1 } \\ker S ^ * \\dotplus \\ker S ^ * \\ , . \\end{align*}"}
-{"id": "8593.png", "formula": "\\begin{align*} \\frac { 1 } { \\zeta - z } = \\sum _ { n = 0 } ^ { \\infty } z ^ n \\zeta ^ { - ( n + 1 ) } \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{align*} & P _ g \\Big \\{ \\big [ v ( t ) , w ( t ) \\big ] ; ( v _ 0 , w _ 0 ) \\Big \\} \\\\ & = \\frac { 1 } { Z } \\exp \\Bigg [ - \\frac { 1 } { 2 \\sigma ^ 2 } \\left ( \\begin{array} { c } v ( t ) - v _ 0 \\\\ w ( t ) - w _ 0 \\end{array} \\right ) ^ { \\top } \\Omega _ { i j } ^ { - 1 } \\left ( \\begin{array} { c } v ( t ) - v _ 0 \\\\ w ( t ) - w _ 0 \\end{array} \\right ) \\Bigg ] , \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} \\mathbf { t } = ( A ' - \\kappa _ 1 B ) \\mathbf { t } + ( B ' + \\kappa _ 1 A ) \\mathbf { n } _ 1 + \\kappa _ 2 A \\ , \\mathbf { n } _ 2 \\Rightarrow \\left \\{ \\begin{array} { c } A ' - \\kappa _ 1 B = 1 \\\\ B ' + \\kappa _ 1 A = 0 \\\\ \\kappa _ 2 A = 0 \\\\ \\end{array} \\right . . \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} d ( u , v ) : = \\sum _ { j = 1 } ^ { \\infty } \\frac { 1 } { 2 ^ { j } } \\frac { p _ { j } ^ { * } ( u - v ) } { 1 + p _ { j } ^ { * } ( u - v ) } \\end{align*}"}
-{"id": "8750.png", "formula": "\\begin{align*} ( N P ^ T D M + I ) \\mathbf y = \\biggl ( \\bigoplus _ { i = 1 } ^ { \\infty } \\left [ \\begin{array} { c c } 2 & - 1 \\\\ 0 & 0 \\end{array} \\right ] \\biggr ) \\mathbf y = \\mathbf 0 , \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} x = y & \\Rightarrow \\phi ( x , y ) = 0 \\\\ \\intertext { a n d } ( \\norm { x } - \\norm { y } ) ^ 2 & \\leq \\phi ( x , y ) \\leq \\left ( \\norm { x } + \\norm { y } \\right ) ^ 2 \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{align*} \\phi ^ { ( a , 0 ) } ( t , x ) = a \\phi _ 0 ( x ) \\exp ( - i \\mu a ^ { \\nu - 1 } t | \\phi _ 0 ( x ) | ^ { \\nu - 1 } ) \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} [ M ] = \\sum _ { \\lambda \\in X ^ + } [ M : L ( \\lambda ) ] [ L ( \\lambda ) ] \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{align*} & \\{ ( ( k - 2 ) e + 1 , ( k - 1 ) e + 1 ) \\cdots ( ( k - 1 ) e , k e ) \\} \\\\ & \\{ ( - e + 1 , ( k - 1 ) e + 1 ) \\cdots ( 0 , ( k - 1 ) e ) \\} \\\\ & \\{ ( ( k - 2 ) e + 1 , ( k - 1 ) e + 1 ) \\cdots ( ( k - 1 ) e , k e ) \\} \\\\ = & ( - e + 1 , k e + 1 ) \\cdots ( 0 , k e ) , \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{align*} R = g _ k ( R ^ { - 1 } g _ k R ^ 1 ) ( R ^ { - 2 } g _ k R ^ 2 ) \\cdots ( R ^ { - 3 k } g _ k R ^ { 3 k } ) , \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{align*} \\dot { x } x ^ { - 1 } + \\left ( \\dot { x } x ^ { - 1 } \\right ) ^ * = P \\left ( \\left [ \\left ( x \\phi _ { - 1 } x ^ { - 1 } \\right ) ^ * , x \\phi _ { - 1 } x ^ { - 1 } \\right ] \\right ) - r \\end{align*}"}
-{"id": "9709.png", "formula": "\\begin{align*} r ( f ) = | C _ 0 | + | P _ 1 | + | C _ 1 | + | P _ 0 | = | C _ 0 ^ * | + | P _ 1 ^ * | + | C _ 1 ^ * | + | P _ 0 ^ * | . \\end{align*}"}
-{"id": "6069.png", "formula": "\\begin{align*} G = E / { \\mathcal T } = G _ { - { \\lfloor \\frac q 2 \\rfloor } } \\oplus \\cdots \\oplus G _ { - 1 } \\oplus G _ 0 \\oplus G _ 1 \\oplus \\cdots \\oplus G _ { \\lfloor \\frac r 2 \\rfloor } \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} \\| Q - \\theta _ L \\| _ 1 \\leq & \\| Q - \\bar { \\theta } \\| _ 1 + \\| \\bar { \\theta } - \\theta _ L \\| _ 1 \\\\ \\leq & C ( b , B , \\theta _ L ) \\delta + C ( b , B , N _ 1 ) \\delta \\big ( \\delta + \\| \\bar { \\theta } - \\theta _ L \\| \\big ) + \\| \\bar { \\theta } - \\theta _ L \\| _ 1 \\\\ \\leq & C ( b , B , \\theta _ L ) \\delta + C ( b , B , N _ 1 ) \\delta ^ 2 \\\\ = & C _ 1 \\delta + C _ 2 ( b , B , N _ 1 ) \\delta ^ 2 , \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} \\mu ^ - _ { | W ^ s ( p ) } ( M _ { r _ n } ) > 0 , \\ ; \\ ; \\mathrm { a n d } \\ ; \\ ; \\mu ^ - _ { | W ^ s ( p ) } ( \\partial M _ { r _ n } ) = 0 . \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} \\Delta \\Omega = o ( \\gamma ) . \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} Q \\left ( x \\right ) = b _ { 1 } ^ { - \\frac { n } { 2 } } e ^ { - b _ { 2 } \\left \\vert x \\right \\vert ^ { p } } b _ { 1 } b _ { 2 } > 0 p > 1 \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} \\vec { a } ^ { ( j ) } : = \\left ( 2 g - 1 - A , a _ 2 , \\dots , a _ { j - 1 } , a _ j - \\textstyle \\frac 1 2 , a _ { j + 1 } , \\dots , a _ n , \\frac 3 2 , - \\frac 1 2 \\right ) , \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} 2 H _ { 0 } = \\frac { \\lambda _ { 1 } ^ { 2 } \\left ( \\dfrac { g ^ { \\prime } } { g } \\right ) ^ { \\prime } } { \\left [ \\left ( \\dfrac { g ^ { \\prime } } { g } \\right ) ^ { 2 } - \\lambda _ { 1 } ^ { 2 } \\right ] ^ { \\frac { 3 } { 2 } } } . \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{align*} u _ { 0 , a } ( x ) : = \\frac { 1 } { a \\sqrt { \\pi } } e ^ { - \\frac { x ^ 2 } { a ^ 2 } } . \\end{align*}"}
-{"id": "9707.png", "formula": "\\begin{align*} a ( v ) & = \\tfrac { 1 } { 2 } ( v + v _ 1 ) ^ { 2 } + \\tfrac { 1 } { 2 } ( v + v _ 2 ) ^ { 2 } - v ^ { 2 } + \\sum _ { u \\in N ( v ) \\setminus \\{ v _ 1 , v _ 2 \\} } \\tfrac { 1 } { 2 } ( v + u ) ^ { 2 } \\\\ & = \\tfrac { 1 } { 2 } ( v _ 1 + v _ 2 ) ^ { 2 } + \\tfrac { 1 } { 2 } v _ 1 ^ { 2 } + \\tfrac { 1 } { 2 } v _ 2 ^ { 2 } + \\sum _ { u \\in N ( v ) \\setminus \\{ v _ 1 , v _ 2 \\} } \\tfrac { 1 } { 2 } ( v + u ) ^ { 2 } \\ , . \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} \\frac { \\phi ( p - 1 ) } { p - 1 } = \\prod _ { q | ( p - 1 ) } \\left ( 1 - \\frac { 1 } { q } \\right ) \\qquad \\frac { \\phi ( p + 1 ) } { p + 1 } = \\prod _ { q | ( p + 1 ) } \\left ( 1 - \\frac { 1 } { q } \\right ) , \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} [ h _ n , H ^ { \\gamma } ( w ^ 2 ) ] = 0 , n \\neq 0 ; [ h _ 0 , H ^ { \\gamma } ( w ^ 2 ) ] = H ^ { \\gamma } ( w ^ 2 ) . \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{align*} \\| \\alpha \\| _ { \\psi , \\omega } ^ { 2 } : = \\int _ { \\Omega } | \\alpha | _ { \\omega } ^ { 2 } e ^ { \\psi } d V _ { \\omega } . \\end{align*}"}
-{"id": "5294.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( y _ j \\omega _ i + y _ i \\omega _ j ) \\ , d y = 0 . \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} \\Psi _ 1 ( n ) = \\prod _ { p | n } \\left ( 1 + \\frac { 1 } { p } \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{align*} \\begin{cases} z ( t ) \\sim z _ 1 ( t ) = e ^ { - \\frac { t } { \\alpha } } \\\\ z ( t ) \\sim z _ 2 ( t ) = \\begin{cases} e ^ { - t \\big ( \\frac { 1 } { \\alpha } - \\frac { N - 2 } { 2 } \\big ) } & N \\ge 3 \\\\ t e ^ { - \\frac { t } { \\alpha } } & N = 2 . \\end{cases} \\end{cases} \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{gather*} \\{ k \\in \\mathbb { Z } _ { > 0 } \\ , | \\ , k l _ j \\in \\mathbb { Z } \\ ( j = 0 , \\ldots , s - 1 ) \\} . \\end{gather*}"}
-{"id": "3727.png", "formula": "\\begin{align*} h _ \\alpha ^ { - 1 } ( E , \\beta ) = \\frac { \\beta } { \\alpha } E . \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} { } _ 3 F _ 2 \\bigg [ \\begin{matrix} \\alpha & 1 - \\alpha & \\beta \\\\ & 1 & 2 \\beta \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { \\Gamma ( \\frac 1 2 ) \\Gamma ( \\frac 1 2 + \\beta ) \\Gamma ( \\beta ) } { \\Gamma ( \\frac 1 2 + \\frac 1 2 \\alpha ) \\Gamma ( 1 - \\frac 1 2 \\alpha ) \\Gamma ( \\frac 1 2 + \\beta - \\frac 1 2 \\alpha ) \\Gamma ( \\beta + \\frac 1 2 \\alpha ) } . \\end{align*}"}
-{"id": "8717.png", "formula": "\\begin{align*} \\psi _ { n } ( r ) : = \\int _ { 0 } ^ { | r | } \\int _ { 0 } ^ { y } \\rho _ { n } ( u ) \\d u \\d y , r \\in \\R , n \\ge 1 . \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} \\tilde { X } ( t ) = X _ n + ( t - t _ n ) S _ { \\Delta t } A X _ n + ( t - t _ n ) S _ { \\Delta t } G ( X _ n ) + S _ { \\Delta t } \\sigma ( X _ n ) \\bigl ( W ( t ) - W ( t _ n ) \\bigr ) . \\end{align*}"}
-{"id": "2759.png", "formula": "\\begin{align*} \\sum _ { d \\leqslant \\varepsilon _ { 1 } ^ \\frac { 1 } { 2 } \\Xi ( \\alpha ) ^ { - \\frac { 1 } { 2 } } B ^ { 1 - \\frac { 1 } { 2 r } } } \\mu ( d ) \\frac { \\varepsilon _ 1 - \\varepsilon _ 2 } { 2 ( \\alpha ^ \\prime ) ^ 2 d ^ 2 } B ^ { 2 - \\frac { 1 } { r } } & = \\frac { 6 ( \\varepsilon _ 1 - \\varepsilon _ 2 ) } { \\pi ^ 2 ( \\alpha ^ \\prime ) ^ 2 } B ^ { 2 - \\frac { 1 } { r } } + O _ { \\varepsilon _ i } ( \\Xi ( \\alpha ) ^ \\frac { 1 } { 2 } B ^ { 1 - \\frac { 1 } { 2 r } } ) , \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{align*} \\eta _ { a } ( C ( \\mathsf { Q } ) ) = \\sum _ { i } c _ { i } ^ { i } q ^ { - ( \\alpha _ { a } - 2 \\rho , \\lambda _ { i } ) } [ d _ { a } ^ { - 1 } ( \\alpha _ { a } , \\lambda _ { i } ) ] _ { q _ { a } } . \\end{align*}"}
-{"id": "9771.png", "formula": "\\begin{align*} h _ { \\epsilon } = j \\phi _ { \\epsilon } + h _ { \\epsilon } ' , \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} \\nu _ \\pm ( \\nu , t ) = \\frac { 1 } { 2 } \\sqrt { 4 \\nu \\ , t \\pm 2 t \\sqrt { 4 \\nu \\ , t + t ^ 2 + 1 } + 2 t ^ 2 + 1 } = \\sqrt { \\nu \\ , t } + \\mathcal { O } ( 1 ) \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} b _ n = \\max _ { r \\in R _ { i } ^ * } \\max _ { b \\bmod n } \\frac { n | R _ { i + 1 } \\cap ( r \\bmod Q _ i ) \\cap ( b \\bmod n ) \\bmod Q _ { i + 1 } | } { | R _ { i + 1 } \\cap ( r \\bmod Q _ i ) \\bmod Q _ { i + 1 } | } . \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} { V } ( x ) : = \\frac { 1 } { \\gamma _ 3 } \\int _ { \\Omega } \\omega ( y ) \\times \\frac { x - y } { | x - y | ^ 3 } \\ , d y \\textrm { i f } n = 3 . \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{align*} \\mathrm { a d } _ { R } ^ { \\circ } ( X ) ( \\mathsf { M } _ { m } ^ { n } ) = \\pi ( K _ { 2 \\rho } ^ { - 1 } X _ { ( 1 ) } K _ { 2 \\rho } S ^ { - 1 } ( X _ { ( 2 ) } ) ) \\mathsf { M } _ { m } ^ { n } \\pi ( S ^ { - 2 } ( X _ { ( 3 ) } ) K _ { 2 \\rho } ^ { - 1 } S ( X _ { ( 4 ) } ) K _ { 2 \\rho } ) . \\end{align*}"}
-{"id": "1762.png", "formula": "\\begin{align*} p _ { m , j } ( \\phi ) : = \\sum _ { | \\alpha | \\leqslant m } \\sup _ { K _ j } \\big | \\partial ^ { \\alpha } \\phi \\big | , \\textrm { f o r } \\phi \\in C ^ { \\infty } ( \\Omega ) , \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} \\xi ( f _ 2 ) - \\xi ( f _ 1 \\varphi _ 0 ) = 2 ( \\sum ^ g _ { i = 1 } \\lambda _ i [ \\alpha _ i ] + \\sum ^ g _ { i = 1 } \\mu _ i [ \\beta _ i ] ) \\cdot \\in H ^ 1 ( \\Sigma ) \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} w ( t ) = \\widetilde w ( t ) \\qquad \\mbox { o n $ [ \\bar t , \\infty ) $ } , \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{align*} s ^ g ( p ) = e _ i g ( D ^ g _ { e _ j } e _ j , e _ i ) - e _ j g ( D ^ g _ { e _ i } e _ j , e _ i ) \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} \\Delta _ p u = g ' _ \\sigma ( u - \\varphi ) + f ' _ \\varepsilon \\bigg ( \\int _ { \\Omega ^ c } h _ \\delta ( u ) \\bigg ) h ' _ \\delta ( u ) \\chi _ { \\Omega ^ c } , \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} \\Xi ( \\alpha ) = ( 4 \\sqrt { a b } ) ^ { - 1 } , \\xi ( \\theta ) = ( 1 6 2 b \\det ( \\Gamma ) ) ^ { - 1 } . \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{align*} x ' = y ; y ' = \\frac { y ^ 2 } { m + x } . \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} s _ { i , j } ^ { ( - 1 ) } & = \\delta _ { i , j } - s _ { i , j } + \\Sigma _ 1 ( i , j ) - \\Sigma _ 2 ( i , j ) + \\cdots + ( - 1 ) ^ { i + j + 1 } \\Sigma _ { i - j } ( i , j ) . \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{align*} \\xi _ 1 = x , \\xi _ 2 = \\frac { 3 y ^ 2 } { 1 + \\sqrt { 3 } x } + \\sqrt { 3 } x = \\frac { 3 ( x ^ 2 + y ^ 2 ) + \\sqrt { 3 } x } { 1 + \\sqrt { 3 } x } . \\end{align*}"}
-{"id": "2257.png", "formula": "\\begin{align*} X ' = X ( \\delta ( u , X ) Z _ 0 ( u ) - X ) , \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} \\eta = \\frac { \\pi ^ 2 } { 3 } \\frac { \\delta ( \\vartheta _ 2 ( 0 ) \\vartheta _ 3 ( 0 ) \\vartheta _ 4 ( 0 ) ) } { \\vartheta _ 2 ( 0 ) \\vartheta _ 3 ( 0 ) \\vartheta _ 4 ( 0 ) } \\end{align*}"}
-{"id": "9619.png", "formula": "\\begin{align*} F ( x ) = \\int ^ x _ a f ( t ) \\ , d t . \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} X _ { 2 , \\ , j + 2 , \\ , 1 5 } \\cap L _ { j + 2 } = 0 . \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{align*} K _ i ( m + ( s + 1 ) ) = \\sum _ { j = 1 } ^ n K _ j ( s + 1 ) K _ { i - j + 1 } ( m ) , \\enskip i = 1 , . . . , n . \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{align*} \\dfrac { 1 } { J + 1 } \\ \\# \\bigl \\{ 0 \\le j \\le J \\ , ; \\ , \\| P _ { l } T ^ { \\ , j } P _ { l } \\ , x \\| \\ge X _ { l } / 2 \\bigr \\} = 1 \\textrm { f o r e v e r y } \\ J \\ge 0 . \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} s = \\Lambda ( \\sigma ) = R _ { \\alpha \\enskip \\beta } ^ { \\enskip \\beta \\enskip \\alpha } = \\frac 1 2 R _ { i \\ ; \\ , j } ^ { \\ ; \\ , j \\ ; \\ , i } = - \\frac 1 2 R _ { i j } ^ { \\ ; \\ , \\ ; \\ , i j } \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{align*} \\begin{array} { r l } \\displaystyle \\sqrt { n } \\sigma _ j ^ { - 1 } ( \\check \\beta _ j - \\beta _ { 0 j } ) = \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ n \\{ \\bar \\psi _ j ( y _ i , z _ i ) - \\sigma _ j ^ { - 1 } \\Sigma _ j ^ { - 1 } ( e ^ j - \\mu ^ j _ 0 ) ^ T \\varphi ( z _ i ) \\beta _ 0 \\} \\\\ \\\\ \\displaystyle + O _ \\P ( \\sigma _ j ^ { - 1 } \\Sigma _ j ^ { - 1 } \\delta _ n ) \\end{array} \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} ( x ^ { 2 } + 1 ) y = a ^ { - 1 } ( b x + c x + x + a ) . \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{align*} c _ 3 & = d _ 3 = \\tanh ^ { - 1 } - \\varepsilon _ 1 = 3 . 8 \\\\ c _ 2 & = d _ 2 = 2 c _ 3 / \\tau _ { f _ n } = 7 . 6 / \\tau _ { f _ n } \\\\ c _ 1 & = c _ 4 = 0 . 5 \\Delta \\phi _ n \\\\ d _ 1 & = d _ 4 = 0 . 5 \\Delta v _ n \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} h _ { r } ( a _ 1 , \\ldots , a _ d ) = \\sum _ { i = 1 } ^ d a _ i ^ { r + d - 1 } \\prod ^ d _ { \\substack { j = 1 \\\\ j \\neq i } } \\frac { 1 } { a _ i - a _ j } . \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} D _ \\tau f _ j = \\frac { P _ j ( f _ 1 , \\ldots , f _ k ) } { Q _ j ( f _ 1 , \\ldots , f _ k ) } \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = \\infty \\\\ \\overbrace { \\begin{matrix} 1 & \\theta ^ 0 _ 1 \\\\ 0 & \\theta ^ 0 _ 2 \\\\ 0 & 0 \\end{matrix} } & \\overbrace { \\begin{matrix} t & \\theta ^ \\infty _ 1 \\\\ 0 & \\theta ^ \\infty _ 2 \\\\ 0 & \\theta ^ \\infty _ 3 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "879.png", "formula": "\\begin{align*} V f _ \\epsilon ( x ) = \\int _ { - \\epsilon } ^ \\epsilon \\frac { 1 } { | x - y | ^ { 1 - \\gamma } } \\frac { 1 } { \\epsilon } f \\left ( \\frac { y } { \\epsilon } \\right ) d y \\leq \\norm { f } _ \\infty \\frac { 1 } { \\epsilon } \\int _ { - \\epsilon } ^ \\epsilon \\frac { 1 } { | y | ^ { 1 - \\gamma } } d y = \\norm { f } _ \\infty \\frac { 2 ^ { 2 - \\gamma } } { \\gamma } | x | ^ { \\gamma - 1 } , \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} \\left \\{ \\Xi _ { \\mathrm { d } , l } ^ { ( \\omega ) } \\right \\} _ { k , q } : = \\frac { \\delta _ { k , q } } { \\left \\vert \\Lambda _ { l } \\right \\vert } \\underset { x \\in \\Lambda _ { l } } { \\sum } \\sigma _ { \\mathrm { d } } ^ { ( \\omega ) } \\left ( x + e _ { k } , x \\right ) \\in \\left [ - 2 \\left ( \\vartheta + 1 \\right ) , 2 \\left ( \\vartheta + 1 \\right ) \\right ] \\ . \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} L _ { v _ \\infty } ( f ) \\ ; = \\ ; - a _ 0 ^ { ( f ) } \\ , W _ 0 ^ { \\infty } \\ , . \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{align*} J ( k ) : = [ f ( - \\bar { k } , x ) ^ \\dagger ; \\varphi ( k , x ) ] = f ( - \\bar { k } , 0 ) ^ \\dagger B - f ' ( - \\bar { k } , 0 ) ^ \\dagger A . \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{align*} \\textrm { V a r } [ Y \\mid Y \\in ( - \\infty , b _ 1 ) ] = \\textrm { V a r } [ Y \\mid Y \\in [ b _ 1 , b _ 2 ) ] = \\ldots = \\textrm { V a r } [ Y \\mid Y \\in [ b _ k , \\infty ) ] . \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} t _ 1 + \\ldots + t _ r = k _ 2 - 1 . \\end{align*}"}
-{"id": "5000.png", "formula": "\\begin{align*} ( L _ { c _ * } + 4 ) \\tilde { w } _ 2 ( \\xi ) = 2 0 { \\rm s e c h } ^ 2 ( \\sqrt { c _ * } \\xi ) . \\end{align*}"}
-{"id": "507.png", "formula": "\\begin{align*} \\Gamma _ \\pm = \\{ ( x , v ) \\in \\partial X \\times V , \\mbox { s . t . } \\pm v \\cdot \\nu ( x ) > 0 \\} , \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{align*} { \\hat { D } ( \\mathbf { x } ) } ^ \\dag \\ , \\hat { R } ^ i \\ , \\hat { D } ( \\mathbf { x } ) = \\hat { R } ^ i + x ^ i \\ , \\hat { \\mathbb { I } } \\ ; . \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} t _ 2 ( 2 ^ m n + 4 ) & = t _ 2 ( 2 ^ { m - 1 } n + 2 ) + t _ 2 ( 2 ^ { m - 1 } n + 1 ) \\\\ & = t _ 2 ( 2 ^ { m - 2 } n + 1 ) + t _ 2 ( 2 ^ { m - 2 } n ) - 2 t _ 2 ( 2 ^ { m - 2 } n ) \\\\ & = - 2 t _ 2 ( 2 ^ { m - 3 } n ) - t _ 2 ( 2 ^ { m - 3 } n ) - t _ 2 ( 2 ^ { m - 3 } n - 1 ) \\\\ & = - 3 t _ 2 ( 2 ^ { m - 3 } n ) - t _ 2 ( 2 ^ { m - 3 } n - 1 ) \\\\ & = - 3 ( t _ 2 ( 2 ^ { m - 4 } n + t _ 2 ( 2 ^ { m - 4 } n - 1 ) ) + 2 t _ 2 ( 2 ^ { m - 4 } n - 1 ) \\\\ & = - 3 t _ 2 ( 2 ^ { m - 4 } n ) - t _ 2 ( 2 ^ { m - 4 } n - 1 ) \\\\ & = \\ldots \\\\ & = - 3 t _ 2 ( n ) - t _ 2 ( n - 1 ) . \\\\ \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} \\mathcal { B } ( z ) & = \\widetilde { B } ( z ) { } _ a \\langle 0 | K ( z , t ) | 0 \\rangle _ a A ( z ) + \\widetilde { A } ( z ) { } _ a \\langle 1 | K ( z , t ) | 1 \\rangle _ a B ( z ) \\\\ & = t z \\widetilde { B } ( z ) A ( z ) + z ^ { - 1 } \\widetilde { A } ( z ) B ( z ) . \\end{align*}"}
-{"id": "6711.png", "formula": "\\begin{align*} \\left \\lvert \\ \\int \\limits _ { | x - e | + | x | \\leq 2 \\tau } \\frac { q ( x ) } { | x | | x - e | } d x \\right \\rvert & = \\left \\lvert \\frac { 1 } { 2 } \\int \\limits _ { 1 } ^ { 2 \\tau } \\int \\limits _ { 0 } ^ { \\pi } \\int \\limits _ { 0 } ^ { 2 \\pi } a ( r ) \\sin \\phi d \\theta d \\phi d r \\right \\rvert \\\\ & \\leq C \\int \\limits _ { 1 } ^ { 2 \\tau } | a ( r ) | d r . \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} f _ 1 & = - B _ { 2 3 } B _ { 3 2 } C _ { 1 1 } y ^ 2 z + z ^ 2 ( \\ , \\ , \\cdots ) \\\\ f _ 2 & = - A _ { 3 3 } B _ { 3 2 } z ^ 2 ( C _ { 1 1 } x + C _ { 2 1 } y + C _ { 3 1 } z ) \\\\ f _ 3 & = - A _ { 3 3 } B _ { 2 3 } z ^ 2 ( C _ { 1 1 } x + C _ { 1 2 } y + C _ { 1 3 } z ) \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} g _ { i } \\cdot g _ { j } = a _ { i , j } ( k ) = g _ { k } \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} F ^ { - 1 } \\xi ^ { \\alpha } A ^ { 1 - \\varkappa - \\mu } \\hat { u } = F ^ { - 1 } \\xi ^ { \\alpha } A ^ { 1 - \\varkappa - \\mu } G ^ { - 1 } \\left ( \\xi \\right ) \\left [ h ^ { \\mu } \\left ( A + \\sum \\limits _ { k = 1 } ^ { n } \\left \\vert \\xi _ { k } \\right \\vert ^ { l _ { k } } \\right ) + h ^ { - \\left ( 1 - \\mu \\right ) } \\right ] \\hat { u } . \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} \\tilde { L } : = L ^ { \\prime } \\circ d f , \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{align*} \\chi ^ { } _ \\mathrm { m i n } ( k ) & = \\log \\sqrt { L } - \\underset { n \\rightarrow \\infty } { \\lim } \\dfrac { 1 } { n } \\log \\left \\Vert B ( k ) \\cdot \\ldots \\cdot B ( L ^ { n - 1 } k ) \\right \\Vert \\\\ \\chi ^ { } _ \\mathrm { m a x } ( k ) & = \\log L - \\chi ^ { } _ \\mathrm { m i n } ( k ) - m \\left ( Q - R \\right ) , \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} X _ l = X _ l ( \\omega ) = \\int \\nu _ l ( d \\sigma ) W _ { l } \\ ; \\ ; \\ ; \\omega \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{align*} \\int _ \\Omega \\left [ | \\nabla t _ \\varepsilon | ^ 2 + \\frac { t _ \\varepsilon ^ 2 } { \\varepsilon ^ 2 } \\right ] = o ( 1 ) . \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{align*} & \\| U ( t ) \\| _ { \\infty , \\mathbb R ^ 3 } \\leq C ( \\| v _ 0 \\| _ { 3 , \\infty } + M _ 3 ) \\ , t ^ { - 1 / 2 } , \\\\ & \\| \\nabla ^ j U ( t ) \\| _ { r , \\mathbb R ^ 3 } \\leq C ( \\| v _ 0 \\| _ { 3 , \\infty } + M _ 3 ) \\ , t ^ { - 1 / 2 + 3 / 2 r - j / 2 } , \\forall r \\in ( 3 , \\infty ) , \\\\ & \\| \\nabla ^ j U ( t ) \\| _ { 3 , \\infty , \\mathbb R ^ 3 } \\leq C ( \\| v _ 0 \\| _ { 3 , \\infty } + M _ 3 ) \\ , t ^ { - j / 2 } \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} j ^ + _ i \\vert n _ 1 , n _ 2 , \\dots , n _ i , \\dots , n _ { r + 1 } \\rangle = \\sqrt { ( n _ i + 1 ) ( k - ( n _ 1 + n _ 2 + \\cdots + n _ { r } ) ) } \\vert n _ 1 , n _ 2 , \\dots , n _ i + 1 , \\dots , n _ { r + 1 } \\rangle \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{align*} C : = \\sqrt { \\eta } \\ , A + \\sqrt { | 1 - \\eta | } \\ , B \\ ; , \\eta \\ge 0 \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} E = [ E _ 0 ^ + , E _ 0 ^ - ] \\setminus \\bigcup _ { j = 1 } ^ n ( E _ j ^ - , E _ j ^ + ) . \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} \\| S ^ n x \\| = O \\bigg ( \\frac { ( \\log n ) ^ { 1 / \\alpha } } { n ^ { 1 / \\alpha } } \\bigg ) , n \\to \\infty , \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\beta \\\\ & \\alpha - \\beta + 1 \\end{matrix} \\bigg | \\ , z \\bigg ] = ( 1 + z ) ^ { - \\alpha } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 \\alpha & \\frac 1 2 + \\frac 1 2 \\alpha \\\\ & \\alpha - \\beta + 1 \\end{matrix} \\bigg | \\ , \\frac { 4 z } { ( 1 + z ) ^ 2 } \\bigg ] , \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} n - 1 & = ( z _ i + z _ j ) z _ i - z _ i ^ 2 \\\\ n - 1 & = ( z _ i + z _ j ) z _ j - z _ j ^ 2 \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} w . x = g ( w ) . g ( x ) = v ^ \\vee . g ( x ) \\in \\{ - 1 , 0 , 1 \\} \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} \\sum _ { \\alpha } [ \\phi _ \\alpha ^ * , \\phi _ \\alpha ] = 0 . \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{align*} \\alpha _ i = 1 - \\alpha _ { k + 1 - i } , \\quad \\textrm { f o r } i = 1 , 2 , \\ldots , k . \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} \\left [ \\frac { 1 } { \\pi } \\begin{pmatrix} 1 & 0 \\\\ 0 & \\pi \\end{pmatrix} , \\begin{pmatrix} 1 & 0 \\\\ 0 & \\pi \\end{pmatrix} , \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} \\right ] \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} v _ 0 ( e ^ \\rho ) = v ( \\rho ) - a \\rho v _ \\infty ( e ^ { - \\rho } ) = v ( \\rho ) - b \\rho \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a ' & * \\\\ 1 & a ' \\end{pmatrix} p , \\end{align*}"}
-{"id": "4010.png", "formula": "\\begin{align*} \\Psi ( s p ) = & \\Psi ( 0 ) + \\frac { s } { t } \\cdot \\big ( \\Psi ( t p ) - \\Psi ( 0 ) \\big ) \\\\ = & \\Omega ( 0 ) \\Phi ( 0 ) + \\frac { s } { t } \\cdot \\big ( \\Omega ( t p ) \\Phi ( t p ) - \\Omega ( 0 ) \\Phi ( 0 ) \\big ) \\equiv \\Omega ( s p ) \\Phi ( s p ) \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} \\Bigl | \\Bigl | T ^ { \\ , n } \\Bigl ( \\sum _ { s = 1 } ^ { j _ { m } + j } z _ { s } \\Bigr ) - x _ { l } \\Bigr | \\Bigr | < \\sum _ { u = 0 } ^ { j } 2 ^ { - ( j _ { m } + u ) } , \\end{align*}"}
-{"id": "7299.png", "formula": "\\begin{align*} [ y _ 1 , y _ 2 ] = y _ 4 + \\theta y _ 5 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 5 } { \\alpha _ 3 } y _ 4 , [ y _ 2 , y _ 2 ] = y _ 5 , [ y _ 1 , y _ 3 ] = y _ 5 , [ y _ 2 , y _ 3 ] = y _ 5 = - [ y _ 3 , y _ 2 ] . \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} \\left ( Y _ i ( k ) \\right ) _ { [ Q ] [ Q ' ] } = \\frac { - i \\alpha } { k + i \\alpha } \\boldsymbol { c } _ i \\delta _ { [ Q ] [ Q ' ] } + \\left ( \\frac { k } { k + i \\alpha } \\boldsymbol { c } _ i + \\mathbb { T } ^ { ( i ) } \\left ( \\mathbb { I } _ { | \\mathcal { E } | ^ n } - \\boldsymbol { c } _ i \\right ) \\right ) \\delta _ { [ Q T _ i ] [ Q ' ] } . \\end{align*}"}
-{"id": "1975.png", "formula": "\\begin{align*} & \\langle \\Phi ( t ^ { - 1 } z _ 1 , \\dots , t ^ { - 1 } z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle \\\\ = & t ^ { M N } \\prod _ { j = 1 } ^ N z _ j ^ { j - 1 - N } ( 1 + t ^ { - 1 } z _ j ^ 2 ) \\prod _ { 1 \\le j < k \\le N } ( 1 + t ^ { - 1 } z _ j z _ k ) ( t ^ { - 1 } + z _ j z _ k ^ { - 1 } ) s p _ { \\overline { \\lambda } } ( \\{ z \\} _ N ) . \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{align*} ( P ^ { s , N + 1 } _ { H P } ( t ) \\Lambda ^ { N + 1 } _ N ) ( x , d y ) = ( \\Lambda ^ { N + 1 } _ N P ^ { s , N } _ { H P } ( t ) ) ( x , d y ) . \\end{align*}"}
-{"id": "3496.png", "formula": "\\begin{align*} B _ { p } ^ { ( \\alpha , \\beta ; m ) } ( x , y ) & = \\int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } { } _ { 1 } F _ { 1 } ( \\alpha ; \\beta ; \\frac { - p } { t ^ { m } ( 1 - t ) ^ { m } } ) d t . \\\\ ( R e ( p ) & > 0 , R e ( x ) > 0 , R e ( y ) > 0 , R e ( \\alpha ) > 0 , R e ( \\beta ) > 0 ) \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} | \\sigma | _ { \\eta _ 0 } ^ 2 = \\frac { e ^ \\rho } { 1 + e ^ \\rho } \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} ( \\alpha \\gamma - \\epsilon \\beta \\delta ) + ( - \\alpha \\delta + \\beta \\gamma ) \\sqrt { \\epsilon } & = ( \\gamma - \\delta \\sqrt { \\epsilon } ) ( \\alpha + \\beta \\sqrt { \\epsilon } ) \\\\ & = ( \\gamma + \\delta \\sqrt { \\epsilon } ) ( \\alpha - \\beta \\sqrt { \\epsilon } ) \\\\ & = ( \\alpha \\gamma - \\epsilon \\beta \\delta ) + ( \\alpha \\delta - \\beta \\gamma ) \\sqrt { \\epsilon } . \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} s _ { 2 } = \\sum _ { k = 1 } ^ m a _ k \\cdot s _ { k - i + 2 } = a _ { i - 1 } \\cdot s _ 1 = a _ { i - 1 } \\end{align*}"}
-{"id": "6175.png", "formula": "\\begin{align*} W _ { [ n ] } : = W / W ^ { [ n + 1 ] } . \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} p _ { j } ( T x ) \\leqslant C \\sum _ { r = 1 } ^ { k } p _ { l _ { r } } ( x ) , \\textrm { f o r a l l } x \\in X . \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} \\psi _ { \\alpha , 0 } ( x ) & = \\frac { 1 } { 2 \\alpha } + \\frac { 1 } { 2 \\alpha \\pi \\sqrt { 1 - x ^ 2 } } \\left ( \\int _ { - \\alpha } ^ { - 1 } + \\int _ 1 ^ { \\alpha } \\right ) \\frac { \\sqrt { t ^ 2 - 1 } } { | t - x | } d t , \\\\ & = \\frac { 1 } { 2 \\alpha } + \\frac { 1 } { \\alpha \\pi \\sqrt { 1 - x ^ 2 } } \\int _ 1 ^ \\alpha \\frac { t \\sqrt { t ^ 2 - 1 } } { t ^ 2 - x ^ 2 } \\ , d t , x \\in [ - 1 , 1 ] . \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{align*} d X _ t ^ { \\delta , \\tau } = A X _ t ^ { \\delta , \\tau } d t + G _ \\delta ( X _ t ^ { \\delta , \\tau } ) d t + e ^ { \\tau A } \\sigma _ \\delta ( X _ t ^ { \\delta , \\tau } ) d W ( t ) , X ^ { \\delta , \\tau } ( 0 ) = x . \\end{align*}"}
-{"id": "1705.png", "formula": "\\begin{align*} \\mathbf { O } _ { a u g } = \\left [ \\mathbf { O } , ~ \\left [ \\begin{array} { c c } \\theta _ { + , 1 } & \\theta _ { - , 1 } \\\\ \\vdots & \\vdots \\\\ \\theta _ { + , n _ 1 + \\cdots + n _ k } & \\theta _ { - , n _ 1 + \\cdots + n _ k } \\end{array} \\right ] \\right ] \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} \\sup _ { n \\in \\N } p _ { ( m , j ) } \\left ( M ^ { - n } \\dfrac { d ^ n } { d x ^ n } \\phi \\right ) = \\sup _ { n \\in \\N } \\sup _ { | x | \\leqslant j } \\left | M ^ { - n } \\dfrac { d ^ { n + m } } { d x ^ { n + m } } \\phi ( x ) \\right | < \\infty . \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} \\lim _ { r \\downarrow 0 } \\ , r ^ B g ( r ) \\ ; & = \\ ; g _ 0 \\\\ \\lim _ { r \\downarrow 0 } \\ , r ^ { - B } ( g ( r ) - g _ 0 r ^ { - B } ) \\ ; & = \\ ; g _ 1 \\end{align*}"}
-{"id": "6794.png", "formula": "\\begin{align*} i \\partial _ { t } \\upsilon _ { n } + \\Delta \\upsilon _ { n } + A \\upsilon _ { n } = V _ { n } \\left ( x , t \\right ) \\upsilon _ { n } + \\phi _ { n } \\left ( x _ { 1 } \\right ) F \\left ( x , t \\right ) , \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} L _ 1 ( \\mathbf { x } ) = x _ 1 , L _ 2 ( \\mathbf { x } ) = x _ 2 , L _ 3 ( \\mathbf { x } ) = x _ 2 - x _ 1 , \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} \\{ U _ { n } ^ { ( 2 ) } \\} _ { n = 0 } ^ { 3 } = \\{ 0 , 0 , 1 , p \\} , \\{ V _ { n } ^ { ( 2 ) } \\} _ { n = 0 } ^ { 3 } = \\{ 4 , 2 p , p ^ { 2 } , p ^ { 3 } - 2 p q \\} , \\end{align*}"}
-{"id": "1055.png", "formula": "\\begin{align*} \\frac { 1 } { N ^ 2 } \\sum \\limits _ { x , d \\in \\mathbb { Z } } 1 _ { A _ N } ( x ) 1 _ { A _ N } ( x + d ) \\Big ( \\prod \\limits _ { i = 1 } ^ s 1 _ { A _ N } ( [ x + \\theta _ i d ] ) \\Big ) \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} W ( \\phi _ { \\infty } ) = \\lim \\limits _ { k \\rightarrow \\infty } W ( \\phi _ k ) - 4 \\pi m . \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{align*} \\hat { \\sigma } _ { A M } = ( \\mathcal { N } ( t ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) \\ ; . \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} & \\frac 1 2 \\frac { d } { d t } \\int _ { \\mathbb { R } ^ 2 } ( | V ^ { ( \\alpha , a ) } | ^ 2 + | H ^ { ( \\alpha , a ) } | ^ 2 ) d x \\\\ & - \\int _ { \\mathbb { R } ^ 2 } \\mu \\Delta \\sum \\limits _ { l = 0 } ^ { \\alpha } C _ \\alpha ^ l ( - 1 ) ^ { \\alpha - l } V ^ { ( l , a ) } \\cdot V ^ { ( \\alpha , a ) } d x \\lesssim \\langle t \\rangle ^ { - \\frac 3 2 } E _ { \\kappa - 3 } E _ { \\kappa - 1 } ^ { \\frac 1 2 } . \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{align*} f _ { i } = f _ { 0 } \\circ f _ { i - 1 } = \\max ( - 2 f _ { i - 1 } + 1 , 2 f _ { i - 1 } - 1 ) . \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{gather*} \\omega ( t ) = \\lambda _ 1 ( t ) f ^ { 1 2 } + \\lambda _ 2 ( t ) f ^ { 3 4 } + \\lambda _ 3 ( t ) f ^ { 5 6 } , \\\\ [ 3 p t ] \\widehat \\rho ( t ) = \\sqrt { \\lambda _ 1 ( t ) \\lambda _ 2 ( t ) \\lambda _ 3 ( t ) } \\left ( - f ^ { 2 4 6 } + f ^ { 1 3 6 } + f ^ { 1 4 5 } + f ^ { 2 3 5 } \\right ) , \\\\ \\end{gather*}"}
-{"id": "9869.png", "formula": "\\begin{align*} \\boldsymbol { K } ( \\rho ) = - K ( \\rho ) \\ , \\boldsymbol { e } _ r K ( \\rho ) = - \\frac { 1 } { \\sigma } \\frac { d G } { d \\rho } = \\frac { 1 } { 2 \\pi \\sigma } \\frac { 1 - J _ 0 ( \\rho ) } { \\rho } \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{align*} \\mu _ i \\rho _ g ( \\lambda _ i ) = \\lambda _ i \\mu _ { i - 1 } i = 1 , 2 . \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } \\Omega _ 1 ( x ) = \\lim _ { x \\to 0 } \\frac { \\Gamma ( 1 - a x + b x ) } { \\Gamma ( 1 - a x ) } \\cdot \\frac { ( 1 - a - a x ) ( 2 - a - a x ) \\cdots ( - a x ) } { ( 1 - a - a x + b + b x ) \\cdots ( - a x + b x ) } = \\frac { a \\cdot ( 1 - a ) _ b } { a - b } . \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{align*} ( * ) & = P _ N ( \\phi \\in [ 0 , a ] \\ , o n \\ , A ) \\\\ & = \\prod _ { i = 1 } ^ { | A | } P _ N ( \\phi _ { x _ i } \\in [ 0 , a ] | \\phi _ { x _ { i + 1 } } , \\ldots , \\phi _ { x _ { | A | } } \\in [ 0 , a ] ) \\\\ & = \\prod _ { i = 1 } ^ { | A | } \\int _ { [ 0 , a ] ^ { A _ i } } P _ N ( \\phi _ { x _ i } \\in [ 0 , a ] | \\phi = \\psi \\ , o n \\ , A _ i ) g _ i ( \\psi ) d \\psi \\end{align*}"}
-{"id": "3202.png", "formula": "\\begin{align*} d X _ i ( t ) = \\sqrt { 2 ( X _ i ^ 2 ( t ) + 1 ) } d W _ i ( t ) + \\left [ \\left ( 2 - 2 N - 2 \\Re ( s ) \\right ) X _ i ( t ) + 2 \\Im ( s ) + \\sum _ { j \\ne i } ^ { } \\frac { 2 ( X ^ 2 _ i ( t ) + 1 ) } { X _ i ( t ) - X _ j ( t ) } \\right ] d t , \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{align*} A ( z , \\{ \\alpha \\} ) = { } _ a \\langle 0 | T _ { a } ( z , \\{ \\alpha \\} ) | 0 \\rangle _ { a } , \\\\ B ( z , \\{ \\alpha \\} ) = { } _ a \\langle 0 | T _ { a } ( z , \\{ \\alpha \\} ) | 1 \\rangle _ { a } , \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} | y _ \\ell ( p , q ) | = \\bigg | \\sum _ { s = 0 } ^ { M - 1 } \\sum _ { t = 0 } ^ { N - 1 } \\bar { d } _ \\ell ( s , t ) \\rho ( s , t ) e ^ { - 2 \\pi i p s / M } e ^ { - 2 \\pi i q t / N } \\bigg | , \\ell = 1 , \\dots , L , \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{align*} f _ q = \\sum _ { j = 0 } ^ N f _ q ^ { \\gamma _ j } . \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} \\{ ( a , b ) \\colon a , b \\in A , \\ , \\exists c _ 1 , c _ 2 , \\dots , c _ k \\in A , \\ , b = w ( c _ 1 , \\dots , c _ k ) , \\ , a \\in \\{ c _ 1 , \\dots , c _ k \\} \\} . \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} G = \\begin{cases} ( \\bigoplus _ { p \\in P _ 1 } \\mathbb { Z } _ { p ^ { m _ p } } ^ { n _ p } ) \\oplus ( \\bigoplus _ { p \\in P _ 2 } \\mathbb { Z } [ p ^ { \\infty } ] ^ { n _ p } ) & \\mbox { i f } G \\\\ ( \\bigoplus _ { p \\in \\mathbb { P } } \\mathbb { Z } [ p ^ { \\infty } ] ^ { n _ p } ) \\oplus ( \\mathbb { Q } ^ { n } ) & \\mbox { o t h e r w i s e , } \\end{cases} \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{align*} k ( P ) ( u ) = k ( P ) ( u + u ^ { - 1 } ) = k ( P ) , \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} \\gamma : = \\frac { 9 6 } { \\pi ^ 2 \\sqrt { c _ * } } \\left ( - \\frac { 1 } { 5 } \\sqrt { c _ * } \\langle \\psi _ * ^ 2 , \\tilde { w } _ 2 \\rangle _ { L ^ 2 } + \\frac { 1 6 } { 2 7 } \\right ) . \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} \\chi _ j \\left ( \\sum _ { i = 1 } ^ n a _ i \\ , g _ i \\right ) = \\sum _ { i = 1 } ^ n a _ i \\ , \\chi _ j ( g _ i ) \\in \\Z [ \\zeta _ { q _ j } ] . \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} A ( \\nabla \\psi _ 0 ) = \\frac { e ^ { - \\psi _ 0 } } { \\int _ { \\mathbb { R } ^ n } e ^ { - \\psi _ 0 } } \\det ( \\nabla ^ 2 \\psi _ 0 ) ^ { - 1 } . \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { p - 1 } \\binom { 2 k } { k } \\binom { 4 k } { 2 k } \\cdot \\frac { 1 } { 4 8 ^ k } \\equiv 2 a - \\frac { p } { 2 a } \\pmod { p ^ 2 } , \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} f ( x , y , z ) : = & \\ x y ( x ^ 3 y + 4 x ^ 2 y ^ 2 + x y ^ 3 + 2 x ^ 3 z + 1 0 x ^ 2 y z + 1 0 x y ^ 2 z \\\\ & + 2 y ^ 3 z + 4 x ^ 2 z ^ 2 + 7 x y z ^ 2 + 4 y ^ 2 z ^ 2 ) . \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} \\left [ K _ { 0 } \\left ( g ^ { \\prime } \\right ) ^ { 4 } \\right ] f ^ { 4 } - \\left [ 2 K _ { 0 } \\left ( f _ { 0 } g g ^ { \\prime } \\right ) ^ { 2 } \\right ] f ^ { 2 } + \\left ( f _ { 0 } g \\right ) ^ { 4 } + \\left ( f _ { 0 } g ^ { \\prime } \\right ) ^ { 2 } = 0 . \\end{align*}"}
-{"id": "3775.png", "formula": "\\begin{align*} s _ i = \\sum _ { k = 1 } ^ m a _ k \\cdot s _ k \\end{align*}"}
-{"id": "9106.png", "formula": "\\begin{gather*} X _ \\lambda : = Z \\left ( \\sum _ { i = 0 } ^ n \\prod _ { j = 0 } ^ n x _ j ^ { a _ { i , j } } + \\lambda \\prod _ { i = 0 } ^ n x _ i ^ { a _ i } \\right ) \\end{gather*}"}
-{"id": "9208.png", "formula": "\\begin{align*} [ \\langle N , x ^ * \\rangle ] _ t = [ \\langle N , x ^ * \\rangle ] ^ a _ t + [ \\langle N , x ^ * \\rangle ] ^ q _ t = [ \\langle M ^ a , x ^ * \\rangle ] _ t . \\end{align*}"}
-{"id": "9360.png", "formula": "\\begin{align*} h = \\sum _ { k , l , m } \\ , \\sum _ { i = 1 } ^ n \\sum _ { c \\in C _ { k , l , m } ^ { ( i ) } } h _ { k , l , c , i , i } . \\end{align*}"}
-{"id": "3302.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } q _ t ( x , y ) f ( y ) d y & = \\int _ { - \\infty } ^ { x } d z \\int _ { - \\infty } ^ { \\infty } - \\partial _ y p _ t ( y , z ) f ( y ) d y = \\int _ { - \\infty } ^ { x } d z \\int _ { - \\infty } ^ { \\infty } p _ t ( y , z ) f ' ( y ) d y \\\\ & = \\int _ { \\infty } ^ { x } d z \\int _ { - \\infty } ^ { \\infty } \\frac { m ( z ) } { m ( y ) } \\frac { m ( y ) } { m ( z ) } p _ t ( y , z ) f ' ( y ) d y \\\\ & = \\int _ { \\infty } ^ { x } m ( z ) d z \\int _ { - \\infty } ^ { \\infty } p _ t ( z , y ) \\frac { f ' ( y ) } { m ( y ) } d y . \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{align*} \\mu ^ { \\mathfrak { M } } _ { N + 1 } \\Lambda _ N ^ { N + 1 } = \\mu ^ { \\mathfrak { M } } _ N . \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} K _ 0 ( x , y ) = - { y \\over | x | ^ 3 } + 3 { x \\cdot y \\over | x | ^ 5 } x + O \\left ( { | y | ^ 2 \\over | x | ^ 4 } \\right ) , \\textrm { a s $ | x | \\to \\infty $ . } \\end{align*}"}
-{"id": "4587.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle V _ 3 ^ { k } ( \\overline { \\theta } _ { \\Psi } ^ { k } ) = B ( \\mathbf { A } ^ { k - \\frac { 1 } { 2 } } ; ( \\overline { \\Psi } ^ { k } - \\mathcal { I } _ { h } \\overline { \\Psi } ^ { k } ) , \\overline { \\theta } _ { \\Psi } ^ { k } ) + B ( \\mathbf { A } ^ { k - \\frac { 1 } { 2 } } ; ( \\Psi ^ { k - \\frac { 1 } { 2 } } - \\overline { \\Psi } ^ { k } ) , \\overline { \\theta } _ { \\Psi } ^ { k } ) , } \\end{array} \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{align*} ( * * ) & = \\int _ { [ 0 , a ] ^ { A } } P _ N ( \\phi + \\tilde { \\psi } \\geq 0 \\ , o n \\ , A ^ c | \\phi = 0 \\ , o n \\ , A ) g ( \\psi ) d \\psi \\\\ & = \\int _ { [ 0 , a ] ^ { A } } P _ { A ^ c } ( \\phi + \\tilde { \\psi } \\geq 0 \\ , o n \\ , A ^ c ) g ( \\psi ) d \\psi \\\\ & \\geq P _ { A ^ c } ( \\Omega _ { A ^ c } ^ + ) \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{align*} \\hat { E } ^ { 0 1 } = ( \\breve { e } _ 1 , 0 ) , \\ \\ \\hat { E } ^ { 0 2 } = ( \\breve { e } _ 2 , 0 ) , \\ \\ \\hat { E } ^ { 0 3 } = ( \\breve { e } _ 3 , 0 ) , \\\\ \\hat { E } ^ { 2 3 } = ( 0 , \\breve { e } _ 1 ) , \\ \\ \\hat { E } ^ { 3 1 } = ( 0 , \\breve { e } _ 2 ) , \\ \\ \\hat { E } ^ { 1 2 } = ( 0 , \\breve { e } _ 3 ) , \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} \\underset { l \\rightarrow \\infty } { \\lim } \\left \\{ \\frac { n ^ { d } } { \\left \\vert \\Lambda _ { l } \\right \\vert } \\mathfrak { F } _ { t , k , q } ^ { ( \\omega ) } \\left ( l b _ { j } \\right ) \\right \\} = \\mathbb { E } \\left [ \\mathfrak { F } _ { t , k , q } ^ { ( \\omega ) } \\left ( \\left \\{ 0 \\right \\} \\right ) \\right ] \\ . \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{align*} \\frac { d v } { d \\tau } = \\frac { b v - c ( - v ^ 3 + ( a + 1 ) v ^ 2 - a v ) } { - 3 v ^ 2 + 2 ( a + 1 ) v - a } \\ , \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{align*} T ( s ) = \\frac { - n _ 2 X _ 1 + n _ 1 X _ 2 } { | ( n _ 1 , n _ 2 ) | } . \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{align*} z ( z _ k , t _ f ) = R ^ { - ( 3 k + 1 ) } . z _ k = R ^ { - 1 } . z _ k . \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} D _ { n , k } ( 1 , x ) & = k \\ , x \\ , D _ { n - 1 , 2 } ( 1 , x ) + D _ n ( 1 , x ) , \\ , \\ , \\ , \\ , n \\geq 1 . \\end{align*}"}
-{"id": "9543.png", "formula": "\\begin{align*} \\pi _ \\alpha ( \\alpha _ s ( a ) ) = \\lambda _ \\sigma ( s ) \\pi _ \\alpha ( a ) \\lambda _ \\sigma ( s ) ^ * , \\ \\ \\ a \\in A , \\ s \\in G , \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{align*} \\int _ { \\Omega } D ^ m u : D ^ m \\varphi d x = \\sigma \\int _ { \\partial \\Omega } u \\varphi d x \\ , , \\ \\ \\ \\forall \\varphi \\in H ^ m ( \\Omega ) \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} \\left \\langle z _ m h _ n ^ k \\right \\rangle - ( - 1 ) ^ \\Delta \\left \\langle z _ n h _ m ^ { k + \\Delta } \\right \\rangle = \\left ( \\Lambda _ m - \\Lambda _ n \\right ) \\cdot ( - 1 ) ^ k \\left \\langle \\ldots \\right \\rangle \\end{align*}"}
-{"id": "7832.png", "formula": "\\begin{align*} H _ 1 = H _ 1 ( M ) : = \\bigcap _ { 1 \\leq j \\leq n } \\{ \\# { \\cal E } _ j \\leq M \\log { n } \\} \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} D ^ \\alpha z = A z + f , \\ 0 < \\alpha \\le 1 . \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} g = \\begin{pmatrix} \\alpha _ 1 + \\alpha _ 2 \\sqrt { \\epsilon } & \\beta _ 1 + \\beta _ 2 \\sqrt { \\epsilon } \\\\ \\gamma _ 1 + \\gamma _ 2 \\sqrt { \\epsilon } & \\delta _ 1 + \\delta _ 2 \\sqrt { \\epsilon } \\end{pmatrix} , \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{align*} ( \\mathrm { b } _ { \\sigma _ { \\lambda , \\lambda ^ \\prime } } \\eta _ { X , Y } ) ( a _ { 0 } \\otimes a _ { 1 } \\otimes a _ { 2 } \\otimes a _ { 3 } ) & = \\varepsilon ( a _ { 0 } ) \\varepsilon ( X \\triangleright a _ { 1 } ) \\varepsilon ( Y \\triangleright a _ { 2 } ) \\varepsilon ( a _ { 3 } ) \\\\ & - \\varepsilon ( \\sigma _ { \\lambda , \\lambda ^ { \\prime } } ( a _ { 3 } ) ) \\varepsilon ( a _ { 0 } ) \\varepsilon ( X \\triangleright a _ { 1 } ) \\varepsilon ( Y \\triangleright a _ { 2 } ) . \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{align*} \\partial _ t \\psi = - \\mathrm i \\beta \\Delta \\psi - \\sum _ { j = 1 } ^ N \\mathrm i \\gamma _ j \\psi | \\psi | ^ { p _ j - 1 } . \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} \\alpha _ 1 | _ { A _ { M _ 2 } } = 1 \\alpha _ 2 | _ { A _ { M _ 1 } } = 1 . \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{align*} \\mathbb { E } \\langle X , \\phi \\rangle \\langle X , \\psi \\rangle = \\frac { 1 } { \\pi } \\int _ \\mathbb { R } \\widehat { \\phi } ( x ) \\overline { \\widehat { \\psi } ( x ) } | x | ^ { - \\alpha } d x , \\phi , \\psi \\in \\mathcal { S } ( \\mathbb { R } ) , \\end{align*}"}
-{"id": "9258.png", "formula": "\\begin{align*} ( \\Delta ^ 2 f ) ( p ) = ( \\Delta \\vec { H } ) ( p ) \\ , . \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} \\Psi \\longrightarrow \\Psi ^ { \\prime } = e ^ { \\mathrm { i } \\chi } \\Psi , \\mathbf { A } \\longrightarrow \\mathbf { A } ^ { \\prime } = \\mathbf { A } + \\nabla \\chi , \\phi \\longrightarrow \\phi ^ { \\prime } = \\phi - \\frac { \\partial \\chi } { \\partial t } , \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} h _ { \\alpha \\bar \\beta } = h ( z _ \\alpha , z _ \\beta ) . \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} \\langle \\overline { x ^ N _ N } \\cdots \\overline { x ^ N _ 1 } | \\mathcal { B } ( z ) | \\overline { x ^ { N + 1 } _ 1 } \\cdots \\overline { x ^ { N + 1 } _ { N + 1 } } \\rangle = \\alpha ( z , \\{ \\overline { x ^ N } \\} , \\{ \\overline { x ^ { N + 1 } } \\} ) + \\beta ( z , \\{ \\overline { x ^ N } \\} , \\{ \\overline { x ^ { N + 1 } } \\} ) , \\end{align*}"}
-{"id": "7641.png", "formula": "\\begin{align*} \\left [ \\dfrac { 1 + \\psi ( 1 - \\lambda ) - \\psi ( \\lambda ) } { 2 } f ( t ) + \\dfrac { \\psi ( \\lambda ) f ( a ) + \\left ( 1 - \\psi ( 1 - \\lambda ) \\right ) f ( b ) } { 2 } \\right ] \\int _ a ^ b \\nu ( t ) \\Delta t \\\\ = \\int _ a ^ b K ( s , t ) f ^ { \\Delta } ( s ) \\Delta s + \\int _ a ^ b \\nu ( s ) f ( \\sigma ( s ) ) \\Delta s , \\end{align*}"}
-{"id": "841.png", "formula": "\\begin{align*} | x ^ { ( 1 ) } _ { j , \\varepsilon } - x ^ { ( 2 ) } _ { j , \\varepsilon } | = o ( \\varepsilon ) , \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} r ( t ) & = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ { \\infty } \\frac { N _ { 0 } } { 2 } \\left | H ( \\omega ) \\right | ^ 2 e ^ { - { \\rm j } \\omega t } { \\rm d } \\omega \\\\ & = B N _ 0 \\operatorname { s i n c } { ( 2 B t ) } , \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} F _ * ^ e ( j u ^ { q - 1 } v ^ { q - 1 } ( f + u v ) ) = \\bigoplus _ { i \\in \\Delta _ e } f _ { ( i , j ) } F _ * ^ e ( i u ^ { q - 1 } v ^ { q - 1 } ) \\oplus u v F _ * ^ e ( j ) , \\end{align*}"}
-{"id": "3826.png", "formula": "\\begin{align*} a ( n ) ^ 2 & - a ( n - 1 ) a ( n + 1 ) \\\\ & = ( 2 ^ { 2 k } - b ( k ) ( b ( k ) - 2 ) ) ( a ( l ) ^ 2 + a ( l + 1 ) ^ 2 ) + ( 2 ^ { 2 k + 1 } - b ( k ) ^ 2 - ( b ( k ) - 2 ) ^ 2 ) a ( l ) a ( l + 1 ) \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} 0 \\neq [ L _ { - 1 } , \\ , L _ { - 1 } ] = [ ( B ( V _ 1 ) ) _ 1 , \\ , ( B ( V _ 1 ) ) _ 1 ] \\end{align*}"}
-{"id": "6178.png", "formula": "\\begin{align*} \\dim H ^ 1 ( G _ T , Z _ { [ n ] } ) = \\dim H ^ 2 ( G _ T , Z _ { [ n ] } ) + \\dim Z _ { [ n ] } - \\dim Z _ { [ n ] } ^ { + } , \\end{align*}"}
-{"id": "731.png", "formula": "\\begin{align*} \\delta _ 1 ( i , j ) & = \\left \\{ \\begin{array} { l l l } - 1 & & ( i , j ) = ( 0 , 0 ) \\\\ 1 & & ( i , j ) = ( - 1 , 0 ) \\\\ 0 & & \\end{array} \\right . \\\\ \\delta _ 2 ( i , j ) & = \\left \\{ \\begin{array} { l l l } - 1 & & ( i , j ) = ( 0 , 0 ) \\\\ 1 & & ( i , j ) = ( 0 , - 1 ) \\\\ 0 & & \\end{array} \\right . \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} \\dim _ K ( \\varphi \\Lambda + \\psi \\Lambda ) & = \\dim _ K \\varphi \\Lambda + \\dim _ K \\psi \\Lambda - \\dim _ K ( \\varphi \\Lambda \\cap \\psi \\Lambda ) \\\\ & = m _ { f ( \\alpha ) } n _ { f ( \\alpha ) } + m _ { f ( \\bar { \\alpha } ) } n _ { f ( \\bar { \\alpha } ) } + 1 . \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} i \\partial _ t s _ t + \\frac 1 2 \\Delta s _ t = 0 , \\ \\ s _ 0 ( x ) = \\delta ( x ) , \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{align*} X _ t : = \\varphi ^ { - 1 } _ x ( x _ t ' ) = \\bigg ( - B ^ N _ t , \\ , B ^ T _ t , \\ , A _ t \\bigg ) . \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} 0 < \\tilde \\lambda _ { 1 } ( p , \\Omega ) = \\lambda _ { 1 } ( p , \\Omega ) \\le \\tilde \\lambda _ { 2 } ( p , \\Omega ) \\le \\ldots \\le \\tilde \\lambda _ { k } ( p , \\Omega ) \\le \\tilde \\lambda _ { k + 1 } ( p , \\Omega ) \\le \\ldots , \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} \\binom { u ( \\varepsilon ) } { U ( \\varepsilon ) } = \\mathcal { V } [ \\varepsilon , \\varepsilon ^ { \\pi / \\omega } ] \\forall \\ , \\varepsilon \\in ( 0 , \\varepsilon _ 1 ) \\ , . \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} \\mathbf { V } : = \\mathbf { Y } / / \\mathbf { k } ^ * \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} \\vert \\widetilde { T } _ { F , G , N } ^ { L , \\Xi , \\widetilde { \\mathbf { r } } } ( g _ 1 , \\dots , g _ d ) \\vert \\ll _ { c , C , \\varepsilon } \\Big \\vert \\frac { 1 } { N ^ { h - m } } \\int \\limits _ { \\mathbf { x } \\in \\mathbb { R } ^ { h - m } } F _ 1 ( \\mathbf { x } ) \\prod \\limits _ { j = 1 } ^ d g _ j ( \\psi _ j ( \\mathbf { x } ) + a _ j ) \\ , d \\mathbf { x } \\Big \\vert , \\end{align*}"}
-{"id": "1451.png", "formula": "\\begin{align*} \\nabla ^ \\perp _ j V ^ { ( \\alpha , a ) } \\nabla _ k \\nabla _ j V + \\nabla _ k \\nabla ^ \\perp _ j V \\nabla _ j V ^ { ( \\alpha , a ) } = 0 . \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} \\tau _ { u _ i } \\leq \\left \\lfloor \\frac { d ^ { u _ i } - 1 } { 2 } \\right \\rfloor = \\frac { B } { N - 2 B } \\alpha _ i \\end{align*}"}
-{"id": "7444.png", "formula": "\\begin{align*} \\phi : = \\max \\{ \\phi ( [ 0 , a ] ) \\mid a > 0 \\} \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{gather*} \\exp ( - i \\beta ) \\prod _ { j = 1 } ^ { n } ( 1 - \\rho _ { j } \\exp ( i \\alpha _ { j } ) ) \\allowbreak = \\\\ \\allowbreak \\sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j } \\sum _ { M _ { j , n } \\subseteq S _ { n } } \\prod _ { k \\in M _ { j , n } } \\rho _ { k } \\exp ( - i \\beta + i \\sum _ { k \\in M _ { j , n } } \\alpha _ { k } ) . \\end{gather*}"}
-{"id": "1532.png", "formula": "\\begin{align*} - c \\ , S ( x ) = Q ( x ) f ( x ) . \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\lambda _ { 1 } ( p , \\Omega ) ^ { \\frac 1 p } = \\frac { 1 } { \\rho _ { F } ( \\Omega ) } , \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\lambda ^ { - d } | \\log \\lambda | \\mu ( d \\lambda ) = + \\infty , \\end{align*}"}
-{"id": "7377.png", "formula": "\\begin{align*} \\eta = \\frac { { \\alpha ( \\beta + 1 ) } } { { 2 } } - \\mu - \\frac { { \\beta + \\gamma + \\beta \\gamma } } { { 2 } } . \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{align*} L = \\{ ( i , j ) \\in U \\mid p _ { i j } - m _ { i j } < a _ { i j } \\} . \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} f ^ * & = \\mbox { m a x i m i z e } \\gamma \\\\ & \\mbox { s u b j e c t t o } f ( x ) - \\gamma \\geq 0 \\mbox { f o r a l l } x \\in \\mathbb R ^ n . \\end{align*}"}
-{"id": "3187.png", "formula": "\\begin{align*} P _ { N + 1 } ( t ) \\Lambda _ N ^ { N + 1 } = \\Lambda _ N ^ { N + 1 } P _ N ( t ) \\ , \\ \\forall { t } \\ge 0 , \\ \\forall N \\ge 1 . \\end{align*}"}
-{"id": "4190.png", "formula": "\\begin{align*} \\phi \\left ( \\gamma \\right ) = \\delta \\left ( \\gamma - \\alpha _ { 0 } \\right ) + \\sum _ { \\nu = 1 } ^ { N } \\tau _ { \\nu } \\ , \\delta \\left ( \\gamma - \\alpha _ { \\nu } \\right ) \\ ; \\ ; \\ ; \\ ; \\phi \\left ( \\gamma \\right ) = 1 , \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} \\psi = \\frac { | \\nabla \\Phi [ w ] | } { | \\nabla w | } \\qquad \\end{align*}"}
-{"id": "467.png", "formula": "\\begin{align*} \\tilde { \\psi } ( Y _ 1 , Y _ 3 ) = M , \\tilde { \\psi } ( Y _ 1 , Y _ 4 ) = M , \\\\ \\tilde { \\psi } ( Y _ 2 , Y _ 3 ) = M , \\tilde { \\psi } ( Y _ 2 , Y _ 4 ) = M . \\end{align*}"}
-{"id": "2988.png", "formula": "\\begin{align*} v ^ { ( k ) } = 2 ^ { - \\tau ^ { ( k ) } } \\quad \\textrm { a n d } w _ { i } ^ { ( k ) } = \\begin{cases} 2 & \\textrm { i f } \\ \\ 1 \\le i \\le \\delta ^ { ( k ) } \\\\ 1 & \\textrm { i f } \\ \\ \\delta ^ { ( k ) } < i < \\Delta ^ { ( k ) } \\end{cases} \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} \\langle - ( 1 + \\varrho _ a - a - \\epsilon ) \\rangle _ p = p + d - b - c - 1 \\geq d - c = \\langle - ( \\delta - \\gamma ) \\rangle _ p . \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} Q '' ( x _ k ) = 2 \\ , Q ' ( x _ k ) \\sum _ { \\substack { j = 1 \\\\ j \\neq k } } ^ n \\frac { 1 } { x _ k - x _ j } , k = 1 , 2 , \\dots , n , \\end{align*}"}
-{"id": "9964.png", "formula": "\\begin{align*} \\begin{array} { r c l } m _ 1 \\ddot { q } _ 1 + c _ 1 \\dot { q } _ 1 + k _ 1 q _ 1 & = & 0 \\\\ m _ 2 \\ddot { q } _ 2 + c _ 2 \\dot { q } _ 2 + k _ 2 q _ 2 & = & 0 \\end{array} \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} \\| u \\| _ W : = \\| \\tau ^ { - 1 } u \\| _ { L ^ 1 } + \\| v \\cdot \\nabla _ x u \\| _ { L ^ 1 } . \\end{align*}"}
-{"id": "9599.png", "formula": "\\begin{align*} W ^ i _ { j m n } = R ^ i _ { j m n } + \\frac 1 { N - 1 } \\big ( \\delta ^ i _ { m } R _ { j n } - \\delta ^ i _ { n } R _ { j m } \\big ) . \\end{align*}"}
-{"id": "9044.png", "formula": "\\begin{align*} H _ { - ( s - 1 ) } ( m , n ) = H _ { n - s + 1 } ( m , n ) , \\enskip K _ { - ( s - 1 ) } ( m , n ) = - K _ { n - s + 1 } ( m , n ) . \\end{align*}"}
-{"id": "8920.png", "formula": "\\begin{align*} \\left ( \\frac { d } { d t } \\right ) ^ m E _ { \\alpha , \\beta } ( t ) = m ! E _ { \\alpha , \\beta + \\alpha m } ^ { m + 1 } ( t ) , m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "6746.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { \\nu } \\varepsilon _ { n } ^ { \\sigma _ { i } } \\alpha _ { i } \\frac { \\partial ^ { i } u } { \\partial x _ { n } ^ { i } } \\left ( x ^ { \\prime } , 0 , t \\right ) = 0 \\nu \\in \\left \\{ 0 , 1 \\right \\} , u \\left ( x , 0 \\right ) = a \\left ( x \\right ) \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} F ' ( A ) = \\bigcup _ { a \\in A } F ' ( P _ a ) . \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{gather*} p _ i = ( t ^ i - t ^ { - i } ) ^ 2 ( b b ^ * t ^ { 2 i - 2 d } + c c ^ * t ^ { 2 d - 2 i } - z ) . \\end{gather*}"}
-{"id": "9983.png", "formula": "\\begin{align*} \\tilde { E } _ i = Y _ i ^ T A _ i ^ { - 1 } X _ i \\ , . \\end{align*}"}
-{"id": "81.png", "formula": "\\begin{align*} \\frak m _ { k + \\ell + 1 } ( x _ 1 , \\cdots , x _ { \\ell } , _ c , y _ 1 , \\cdots , y _ { k } ) = 0 . \\end{align*}"}
-{"id": "8843.png", "formula": "\\begin{align*} Z ( \\Gamma _ F ) = Z ( \\Gamma ' _ { F ' } ) \\ , ( n - \\tfrac { 1 } { 2 } ( d + 1 ) ) ^ { m ' - s } . \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{align*} R = \\frac { N - 2 B - U } { N - U } \\cdot \\frac { 1 - \\frac { T } { N - 2 B - U } } { 1 - ( \\frac { T } { N - 2 B - U } ) ^ M } \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} \\mathrm { d } p _ { Z } ( t ) ( \\mathbf { z } ) = \\mathrm { e } ^ { - \\frac { | \\mathbf { z } | ^ 2 } { 2 t } } \\frac { \\mathrm { d } ^ { 2 n } z } { ( 2 \\pi \\ , t ) ^ n } \\ ; , \\mathbf { z } \\in \\mathbb { R } ^ { 2 n } \\ ; , \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{align*} \\pi ( X K _ { \\lambda } ) _ { j } ^ { i } c _ { k } ^ { j } \\pi ( K _ { \\lambda ^ { \\prime } } Y ) _ { \\ell } ^ { k } = \\pi ( X ) _ { j } ^ { i } \\pi ( K _ { \\lambda } ) _ { j } ^ { j } c _ { k } ^ { j } \\pi ( K _ { \\lambda ^ { \\prime } } ) _ { k } ^ { k } \\pi ( Y ) _ { \\ell } ^ { k } . \\end{align*}"}
-{"id": "1906.png", "formula": "\\begin{align*} \\frac { f ' _ c ( x + \\lambda ) } { 2 c ^ { n - 1 } } = 1 - 2 x + O ( c ^ { - 1 } ) \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} \\gamma ( [ 0 , t ] _ < ^ 2 ) = \\int _ 0 ^ t \\chi _ { 0 } ( t , r ) d B _ r . \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{align*} f _ q = \\sum _ { j = 0 } ^ N f _ q ^ { \\gamma _ j } . \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} S ( x ) : = \\prod _ { k = 1 } ^ { n - 1 } ( x - t _ k ) . \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{align*} & \\quad \\ , \\sum _ { K \\geq 1 } \\sum _ { V \\geq A K } \\exp \\left ( K \\log ( m ^ 2 ) - V ( ( c / 2 ) \\log m ) \\right ) \\\\ & = \\sum _ { K \\geq 1 } \\exp \\left ( 2 K \\log m - A K ( c / 2 ) \\log m \\right ) ( 1 + o ( 1 ) ) \\\\ & = m ^ { 2 - A c / 2 } ( 1 + o ( 1 ) ) , \\end{align*}"}
-{"id": "9294.png", "formula": "\\begin{align*} \\| \\mathfrak { a } \\| _ { v _ \\rho ( \\mathcal I ) } = \\sup \\big ( \\sum _ { k \\geq 1 } | a _ { t _ k } - a _ { t _ { k - 1 } } | ^ \\rho \\big ) ^ { \\frac { 1 } { \\rho } } , \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} \\tilde { u } \\left ( x , t \\right ) = \\left ( \\sqrt { \\alpha \\beta } \\sigma \\left ( t \\right ) \\right ) ^ { \\frac { n } { 2 } } u \\left ( \\sqrt { \\alpha \\beta } x \\sigma \\left ( t \\right ) , \\beta t \\sigma \\left ( t \\right ) \\right ) e ^ { \\eta } . \\end{align*}"}
-{"id": "9843.png", "formula": "\\begin{align*} [ x , y ] = ( a , b ) ^ { 2 k } . \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{align*} J _ 1 ( k ) = - i k A _ 1 + B _ 1 + Q A _ 1 + o ( 1 ) , | k | \\to \\infty , \\ ; \\ ; k \\in \\overline { \\mathbb { C } } ^ { + } , \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} W ^ p = 1 + \\lambda ^ p T ^ { - N } . \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} A = \\left ( u + 2 \\ , M \\right ) ^ { 2 } { u } ^ { 2 } , \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{align*} f \\mathring { R i c } = \\mathring { H e s s f } , \\end{align*}"}
-{"id": "2744.png", "formula": "\\begin{align*} \\sharp T _ K ( \\varepsilon , \\eta , d , B ) = \\frac { ( \\varepsilon - \\eta ) K ^ 2 } { 2 \\alpha ^ 2 d ^ 2 } B ^ { 2 - \\frac { 1 } { r } } + O _ { \\varepsilon , \\eta } \\left ( \\frac { K ^ 2 B ^ { 2 - \\frac { 1 } { r } } } { N d ^ 2 } \\right ) + O _ { \\sigma } \\left ( \\frac { K ^ \\sigma B ^ \\sigma N } { d ^ \\sigma } \\right ) . \\end{align*}"}
-{"id": "219.png", "formula": "\\begin{align*} X \\left ( \\begin{array} { c | c } \\operatorname { r c e f } ( S ^ \\mu _ 0 ) & 0 \\\\ \\hline 0 & I _ { n - ( \\mu + 1 ) } \\end{array} \\right ) = 0 . \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} c _ 1 = l _ 1 l _ 2 , \\ \\ c _ 2 = l _ 3 l _ 4 \\ \\ \\ \\ c _ { n } = l _ 1 l _ 2 l _ 3 l _ 4 . \\end{align*}"}
-{"id": "1366.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial k } \\emph { \\ } \\widehat { u } ( x , y , k ) = O \\left ( \\frac { 1 } { k } \\right ) , \\ > k \\rightarrow \\infty . \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{align*} Z = X + X ^ { - 1 } X ' . \\end{align*}"}
-{"id": "9450.png", "formula": "\\begin{align*} \\partial _ { t } h + \\nabla \\cdot ( h v ) = 0 , \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , L _ 2 ] \\subseteq L _ 0 \\cap M ( B ( L _ { - 2 } ) ) = 0 , \\end{align*}"}
-{"id": "9545.png", "formula": "\\begin{align*} \\sigma ( s , t ) : = \\phi ( s ) \\phi ( t ) \\phi ( s t ) ^ { - 1 } \\in N \\subseteq \\mathcal { U } ( A \\rtimes _ { \\alpha , r } N ) , \\ \\ \\ \\ \\ s , t \\in G / N , \\end{align*}"}
-{"id": "9401.png", "formula": "\\begin{align*} \\phi ( t _ 1 , t _ 2 , t _ 3 ) = \\bigl \\langle a ( t _ 1 , t _ 2 ) , b ( t _ 2 , t _ 3 ) \\bigr \\rangle \\qquad \\hbox { f o r a . e . } \\ ( t _ 1 , t _ 2 , t _ 3 ) \\in \\Omega _ 1 \\times \\Omega _ 2 \\times \\Omega _ 3 . \\end{align*}"}
-{"id": "7849.png", "formula": "\\begin{align*} C _ u ^ { r - 1 } e ^ { - C _ d r } e ^ { C _ d \\epsilon r } \\leq C ^ { r - 1 } e ^ { - C r } e ^ { C \\epsilon r } e ^ { \\omega r } = \\frac { 1 } { C } e ^ { - \\delta _ 0 r } e ^ { - r } \\end{align*}"}
-{"id": "7736.png", "formula": "\\begin{align*} N _ 1 = n ^ 3 , N _ 2 = \\lceil n \\log ^ 3 n \\rceil , R _ 1 = n / \\sqrt { \\log n } R _ 2 = n / \\sqrt { \\log \\log n } . \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} t _ { m } ( n ) = \\sum _ { i _ { 1 } + i _ { 2 } + \\ldots + i _ { m } = n } ( - 1 ) ^ { \\sum _ { k = 1 } ^ { m } s _ { 2 } ( i _ { k } ) } . \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} I _ 2 = \\frac { C _ u ^ 2 } { n ^ 2 } \\left ( n r - \\frac { r ^ 2 } { 2 } - \\frac { 3 r } { 2 } + 1 \\right ) \\leq \\frac { C _ u ^ 2 } { n ^ 2 } \\left ( n r + 1 \\right ) = \\frac { C _ u ^ 2 r } { n ^ 2 } + \\frac { C _ u ^ 2 } { n ^ 2 } \\leq \\frac { C _ u ^ 2 \\epsilon } { n } + \\frac { C _ u ^ 2 } { n ^ 2 } . \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} u ^ i _ j \\triangleleft X = \\sum _ k \\pi ( X ) ^ i _ k u ^ k _ j , u ^ { i * } _ j \\triangleleft X = \\sum _ k \\pi ( S ( X ) ) ^ k _ i u ^ { k * } _ j . \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} q = e ^ { - C ( 1 - q ) } . \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} \\left \\vert \\mathbf { K } _ { l , \\delta } ^ { \\leq } \\right \\vert = \\mathcal { O } \\left ( \\delta ^ { d + 1 } l ^ { d } \\right ) \\ , \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} \\hat { U } ^ { ( n ) } _ n = \\hat { U } ^ { ( n ) } _ n ( \\omega ) = \\min _ { \\pi \\subset B _ { 8 \\mu \\beta ^ { - 1 } _ 1 n ^ { 1 + \\epsilon } } } \\hat { T } ^ { ( n ) } ( \\pi , \\omega ) \\end{align*}"}
-{"id": "2294.png", "formula": "\\begin{align*} \\chi _ { k , e _ 1 } ( t _ k - t ) = \\chi _ { k , e _ 2 } ( t ) , t \\in [ 0 , t _ k ] . \\end{align*}"}
-{"id": "9494.png", "formula": "\\begin{align*} H _ \\ell ' & : = H _ { \\ell - 1 } ^ \\ast \\cup H _ \\ell , \\\\ H _ \\ell ^ \\ast & : = H _ { \\ell - 1 } ^ \\ast \\cup H _ \\ell \\cup J _ \\ell = H _ \\ell ' \\cup J _ \\ell , \\\\ H ^ \\ast & : = H _ { r - i - 1 } ^ \\ast . \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} [ L _ { - q + 1 } , \\ , A _ 1 ] = 0 . \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} \\psi _ N ( a \\ , | \\ , x ) : = { \\mathbb P } \\left ( { \\rm P o i s } \\left ( N x \\right ) \\geqslant N a \\right ) , \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} W _ 0 = \\{ ( i , j ) \\in E \\mid f _ { i j } ( p _ { i j } ) < 0 \\} . \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{align*} A _ { j , i } : = \\bigg ( \\sigma _ { k , j , i } \\ast \\chi _ { \\bigcup _ { R \\in \\mathcal { R } _ { j , i } } } 1 0 0 R \\bigg ) . \\end{align*}"}
-{"id": "6119.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , X _ { 1 , \\ , j + 2 , \\ , 2 } ] = 0 , \\end{align*}"}
-{"id": "2633.png", "formula": "\\begin{align*} \\langle B \\phi _ k , \\phi _ k \\rangle \\geq \\epsilon _ { 2 } \\int _ { \\ell _ { 2 } } ^ { \\ell _ { 2 } + \\delta _ { 2 } } | \\phi _ { k } ' ( x ) | ^ 2 d x = { 2 \\epsilon _ { 2 } k ^ 2 \\over \\pi } \\int _ { \\ell _ { 2 } } ^ { \\ell _ { 2 } + \\delta _ { 2 } } \\ ! \\ ! \\ ! \\cos ^ 2 ( k x ) \\ , d x \\geq c \\ , k ^ 2 . \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{gather*} \\frac { 1 } { f } \\frac { \\partial f } { \\partial \\tilde { t } _ 1 } = - \\tilde { q } _ 1 , \\frac { 1 } { f } \\frac { \\partial f } { \\partial \\tilde { t } _ 2 } = \\frac { \\tilde { p } _ 2 \\tilde { q } _ 2 + 1 } { \\tilde { t } _ 2 } . \\end{gather*}"}
-{"id": "2153.png", "formula": "\\begin{align*} \\ ! \\ ! | | K ( x , t ) | | \\le c \\sigma \\left ( \\frac { x + t } { 2 } \\right ) , \\ ; \\ ; \\sigma ( x ) : = \\int _ x ^ \\infty | | V ( t ) | | d t \\in L ^ p ( \\mathbb { R } ^ + ) , \\ ; \\ ; p = 1 , 2 , \\end{align*}"}
-{"id": "118.png", "formula": "\\begin{align*} d \\mu ( \\theta ) = w ( \\theta ) \\frac { d \\theta } { 2 \\pi } + d \\mu _ s \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} p _ { ( m , j ) } \\Big ( e ^ { t \\ , d / d x } \\phi \\Big ) = \\sup _ { | x | \\leqslant j } \\left | \\dfrac { d ^ m } { d x ^ m } \\phi ( t + x ) \\right | \\leqslant \\sup _ { | x | \\leqslant j + \\lceil | t | \\rceil } \\left | \\dfrac { d ^ m } { d x ^ m } \\phi ( t + x ) \\right | = p _ { ( m , j + \\lceil | t | \\rceil ) } ( \\phi ) , \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} g = ( & 1 , n + 2 , 2 , n + 3 , 5 , n + 6 , 7 , n + 8 , \\dots , 2 i - 1 , n + 2 i , \\dots , n - 2 , 2 n - 1 , n , \\\\ & n + 1 , 3 , n + 4 , 4 , n + 5 , 6 , n + 7 , \\dots , 2 j , n + 2 j + 1 , \\dots , n - 1 , 2 n ) . \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{align*} a ^ - \\vert 0 \\rangle = 0 a ^ + \\vert k \\rangle = 0 . \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} 0 \\le I ( A : B | M ) _ { \\hat { \\rho } _ { A B M } ( t ) } \\le I ( A : B | M ) _ { \\hat { \\rho } _ { A B M } } = 0 \\ ; . \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{align*} D _ { n , k } ( 1 , y ( 1 - y ) ) = k \\ , \\Big [ \\displaystyle \\frac { y ^ n ( 1 - y ) - y ( 1 - y ) ^ n } { 2 y - 1 } \\Big ] + D _ n ( 1 , y ( 1 - y ) ) , \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} m _ { i j } = \\max \\left \\{ 1 , \\lceil m _ { i j } ^ { * } \\rceil \\right \\} \\forall ( i , j ) \\in U . \\end{align*}"}
-{"id": "6551.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\mathrm { d e t } \\ , J ( x , y ) & = & g _ 1 ' ( x ) g _ 2 ' ( y ) - g _ 1 ' ( x ) n _ Y f _ y ( x , y ) - g _ 2 ' ( y ) n _ X f _ x ( x , y ) \\ , , \\\\ \\mathrm { T r } \\ , J ( x , y ) & = & - g _ 1 ' ( x ) - g _ 2 ' ( y ) + n _ X f _ x ( x , y ) + n _ Y f _ y ( x , y ) \\ , . \\end{array} \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} H = \\left \\langle I + \\begin{pmatrix} a & 0 \\\\ 0 & d \\end{pmatrix} p , I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle , a , d \\in \\mathbb { Z } / p \\mathbb { Z } \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} v \\in H ^ { h } _ 0 ( \\Omega ) \\sum _ { 0 \\leq | \\alpha | , | \\beta | \\leq h } \\int a _ { \\alpha , \\beta } D ^ \\beta v D ^ \\alpha w = f ( w ) w \\in H ^ h _ 0 ( \\Omega ) . \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t V ^ { ( \\alpha , a ) } - \\mu \\Delta \\sum \\limits _ { l = 0 } ^ \\alpha C _ \\alpha ^ l ( - 1 ) ^ { \\alpha - l } V ^ { ( l , a ) } - \\nabla \\cdot H ^ { ( \\alpha , a ) } = f ^ 1 _ { \\alpha a } , \\\\ \\partial _ t H ^ { ( \\alpha , a ) } - \\nabla V ^ { ( \\alpha , a ) } = f ^ 2 _ { \\alpha a } \\end{cases} \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} | h ^ { - 1 } ( p ) - g ^ { n _ k } \\circ h ^ { - 1 } ( p _ 0 ) | & = | h ^ { - 1 } ( p ) - h ^ { - 1 } ( p _ { n _ k } ) | \\\\ [ 0 . 2 e m ] & \\leq \\left ( \\left | \\frac { \\alpha - \\bar { \\alpha } } { ( p - 1 ) ^ 2 } \\right | + \\delta \\right ) | p - p _ { n _ k } | \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} \\tilde { S } _ { m , n } = \\sum _ { p = 1 } ^ { m } \\left ( - 1 \\right ) ^ { m - p } \\binom { m } { p } S _ { p , n } , \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} \\frac { ( \\Psi ^ \\prime _ { \\ell ^ n } ( X , Y ) ) ^ 2 } { ( \\Psi ^ \\prime _ { \\ell ^ { n - 1 } } ( X , Y ) ) ^ 2 } = \\ell ^ 2 X ^ { \\ell ^ { 2 n - 2 } ( \\ell ^ 2 - 1 ) } + c X ^ { \\ell ^ { 2 n - 2 } ( \\ell ^ 2 - 1 ) - 2 } + \\cdot \\cdot \\cdot , \\end{align*}"}
-{"id": "1779.png", "formula": "\\begin{align*} T _ j ( t ) \\big ( T _ j ( s ) [ x ] _ j \\big ) = [ T ( t ) \\circ T ( s ) x ] _ j = T _ j ( t + s ) [ x ] _ j \\end{align*}"}
-{"id": "9503.png", "formula": "\\begin{align*} q ( z ) & = z ^ 2 - Q _ 1 z - Q _ 0 \\\\ [ 6 p t ] w ( z ) & = W _ 1 z + W _ 0 . \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{align*} H ( \\textbf { x } ) = \\sum _ { j = 0 } ^ { \\infty } G ( x _ { j + 1 } , \\dots , x _ { j + k } ) \\end{align*}"}
-{"id": "2065.png", "formula": "\\begin{gather*} ( \\hat { A } _ 2 ) _ { 2 1 } = - { p _ 2 } ^ 2 q _ 1 ( p _ 2 q _ 1 - t _ 2 ) - p _ 2 \\big ( 2 p _ 1 q _ 1 + p _ 2 q _ 2 + 2 \\theta ^ \\infty _ 1 \\big ) + 1 . \\end{gather*}"}
-{"id": "1725.png", "formula": "\\begin{align*} h _ n = \\left \\{ \\begin{array} { l l } \\frac { a _ { m a x } } { ( 1 - K ) \\tau _ { f , n } } , & t _ i \\geq t _ { o , n - 1 } + \\tau _ { f , n - 1 } \\\\ h _ { n - 1 } , & t _ i < t _ { o , n - 1 } + \\tau _ { f , n - 1 } \\end{array} \\right . \\end{align*}"}
-{"id": "9614.png", "formula": "\\begin{align*} \\mathcal F ^ \\alpha _ { j \\alpha } = \\big ( \\sin u + \\cos v + w ) \\sigma _ j + F ^ { ( j ) } _ j \\sigma _ { ( j ) } \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N ^ \\gamma } \\log Q _ N ( a ) = - r _ Q . \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { 1 - \\varepsilon } 2 \\left ( \\frac r { \\sqrt n } \\right ) ^ 2 \\right ) ^ { n - 1 } & = \\left ( 1 - \\frac { 1 - \\varepsilon } 2 \\cdot \\frac { r ^ 2 } n \\right ) ^ { \\left ( - \\frac { 2 n } { ( 1 - \\varepsilon ) r ^ 2 } \\right ) \\cdot \\left ( - \\frac { ( 1 - \\varepsilon ) r ^ 2 } 2 \\right ) - 1 } \\\\ & \\to e ^ { - \\frac { r ^ 2 } 2 \\cdot ( 1 - \\varepsilon ) } \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} \\mathcal { H } _ n ^ { ( s ) } = \\bigotimes _ { i = 1 } ^ { n } \\left ( \\bigoplus _ { j = 1 } ^ { | \\mathcal { E } | } L ^ 2 ( 0 , \\infty ) \\right ) . \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} \\partial _ { t } \\upsilon = i \\left ( \\Delta \\upsilon + A \\upsilon + F \\left ( x , t \\right ) \\right ) , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } \\dot u \\\\ \\dot v \\end{array} \\right ) = \\left ( \\begin{array} { c c } 0 & - \\omega \\\\ \\omega & 0 \\end{array} \\right ) \\left ( \\begin{array} { c } u \\\\ v \\end{array} \\right ) + \\left ( \\begin{array} { c } f ^ 1 ( u , v ) \\\\ f ^ 2 ( u , v ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} \\Phi _ { U ' } : \\mathbb C ^ 2 \\setminus \\{ x _ 1 = 0 \\} \\to U ' \\end{align*}"}
-{"id": "9995.png", "formula": "\\begin{align*} W ( B _ { 1 } ( n , d , x ) ) & = \\sum _ { i = - d } ^ { d } \\sum _ { j = i + 1 } ^ { d } ( j - i ) + \\sum _ { i = - ( k - 1 ) } ^ { k - 1 } \\sum _ { j = i + 1 } ^ { k - 1 } ( j - i + 2 ) + \\\\ & + ( 2 d + 2 - 2 ( x - 1 ) ) + \\sum _ { i = - d } ^ { d } \\sum _ { j = - ( k - 1 ) } ^ { k - 1 } ( \\left \\vert i - j \\right \\vert + 1 ) + \\\\ & + \\sum _ { i = - d } ^ { d } 2 ( \\left \\vert i - x ^ { \\prime } \\right \\vert + 1 ) + \\sum _ { i = - ( k - 1 ) } ^ { k - 1 } ( 2 \\left \\vert i - x ^ { \\prime } \\right \\vert + 2 ) . \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} B _ 3 ( u _ n , \\theta _ n ) ( 4 t ) = \\biggl ( \\int _ 0 ^ { t _ A } + \\int _ { t _ A } ^ { 4 t } \\biggr ) \\tilde { F } ( 4 t - s ) * ( u _ n \\theta _ n ) ( s ) \\dd s , \\end{align*}"}
-{"id": "5170.png", "formula": "\\begin{align*} \\sum _ { k _ { 1 } + \\dots + k _ { p } = n } g _ { k _ { 1 } } \\dots g _ { k _ { p } } = \\frac { \\left ( - 1 \\right ) ^ { p } } { n ! } B _ { n } ^ { \\left ( - p \\right ) } \\left ( - p x \\right ) \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} \\Gamma ^ { 1 } ( \\mathcal { H } , ( F , \\nabla F ) ) = \\Gamma ^ { 1 } ( S , ( \\alpha , v ) ) , \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} S _ \\lambda : = t _ { \\lambda _ 1 ^ 1 } \\cdots t _ { \\lambda _ 1 ^ { m _ 1 } } t _ { \\lambda _ 2 ^ 1 } \\cdots t _ { \\lambda _ 2 ^ { m _ 2 } } \\cdots t _ { \\lambda _ { k + l } ^ 1 } \\cdots t _ { \\lambda _ { k + l } ^ { m _ { k + l } } } . \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} \\frac { \\phi ( n ) } { n } = \\prod _ { q | n } \\bigg ( \\frac { q - 1 } { q } \\bigg ) , \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} \\varepsilon _ { \\alpha } : = 2 \\sqrt { 1 - \\alpha ^ { - 2 } } . \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} \\chi _ { c _ { \\kappa ^ \\sigma } } ( m , n ) = c _ { \\kappa ^ \\sigma } ( m , n ) c _ { \\kappa ^ \\sigma } ( n , m ) ^ { - 1 } = c _ { \\kappa ^ \\sigma } ( m , n ) c _ { \\kappa ^ \\sigma } ( n , m ) \\end{align*}"}
-{"id": "6199.png", "formula": "\\begin{gather*} V _ L ( \\theta ) = \\{ v \\in V \\mid L v = \\theta v \\} . \\end{gather*}"}
-{"id": "8642.png", "formula": "\\begin{align*} \\gamma = \\frac { 2 } { \\alpha } - \\frac { N - 4 } { 2 } . \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} [ . . . \\{ x _ 1 , \\dots , x _ k , x _ { k + 1 } \\} ] \\ , = \\ , \\{ x _ 1 , \\dots , x _ k , x _ k + 1 \\} , \\dots , \\{ x _ 1 , \\dots , x _ k , x _ { k + 1 } \\} . \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} \\exp ( \\varepsilon ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\varepsilon ^ { n } } { n ! } \\end{align*}"}
-{"id": "8354.png", "formula": "\\begin{align*} m _ A ( I _ A \\otimes m _ A ) ( a \\otimes b \\otimes c ) = m _ A ( m _ A \\otimes I _ A ) ( a \\otimes b \\otimes c ) . \\end{align*}"}
-{"id": "9037.png", "formula": "\\begin{align*} k _ s ( x , n ) = \\sum _ { t = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ t x ^ { n t + s - 1 } } { ( n t + s - 1 ) ! } , \\enskip s = 1 , . . . , n , \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = \\frac { \\alpha _ 2 \\beta ^ 2 _ 1 } { \\alpha ^ 2 _ 3 \\gamma _ 6 } y _ 5 , [ y _ 1 , y _ 2 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = - y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 2 ] = y _ 4 , [ y _ 3 , y _ 3 ] = y _ 5 . \\end{align*}"}
-{"id": "9297.png", "formula": "\\begin{align*} V : x _ k ^ { \\prime } = \\sum _ { i , j = 1 } ^ m p _ { i j , k } x _ i x _ j , \\ \\ k = 1 , 2 , \\ldots , m \\end{align*}"}
-{"id": "8578.png", "formula": "\\begin{align*} g = e ^ { 2 \\lambda } | d z | ^ 2 = \\lambda | z | ^ { 2 \\theta _ 0 - 2 } \\left ( 1 + O ( | z | ) \\right ) | d z | ^ 2 \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} E _ 2 : = \\int _ \\Omega n _ 1 + k _ 2 \\int _ \\Omega \\left ( n _ 2 - \\log n _ 2 \\right ) + \\frac { \\ell _ 2 } { 2 } \\int _ \\Omega c ^ 2 \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} x _ n = \\mathcal { C } _ { n } ^ { \\left \\{ J \\right \\} } , \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{align*} \\sum _ { k } S ( u _ { k } ^ { a } ) u _ { b } ^ { k } = \\delta _ { b } ^ { a } 1 = \\sum _ { k } u _ { k } ^ { a } S ( u _ { b } ^ { k } ) . \\end{align*}"}
-{"id": "9090.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Xi _ 4 ( t ) + \\frac { 2 } { 3 } \\Pi _ 4 ( t ) = \\mathcal { I } _ 4 ( t ) , \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} & \\langle \\Phi ( z _ 1 , \\dots , z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle \\\\ = & t ^ { N ( M - N ) } \\prod _ { j = 1 } ^ N z _ j ^ { j - 1 - N } ( 1 + t z _ j ^ 2 ) \\prod _ { 1 \\le j < k \\le N } ( 1 + t z _ j z _ k ) ( 1 + t z _ j z _ k ^ { - 1 } ) s p _ { \\overline { \\lambda } } ( \\{ t z \\} _ N ) . \\end{align*}"}
-{"id": "3569.png", "formula": "\\begin{align*} a \\wedge \\psi = \\lim _ { \\epsilon \\to 0 ^ + } \\chi ( | h | ^ 2 / \\epsilon ) a \\wedge \\psi \\end{align*}"}
-{"id": "9885.png", "formula": "\\begin{align*} 2 s _ 1 ( s _ 1 - s _ 2 ) = s _ 1 ^ 2 - s _ 2 ^ 2 + ( s _ 1 - s _ 2 ) ^ 2 \\forall s _ 1 , s _ 2 \\in \\R . \\end{align*}"}
-{"id": "5021.png", "formula": "\\begin{align*} N _ c ( ( b e ^ { i y } + \\bar { b } e ^ { - i y } ) \\psi _ * ) = 0 , \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} { \\mathcal H } _ { c _ 1 } ^ { a \\star } \\cap { \\mathcal A } _ { T ^ 2 , k _ 1 ^ { \\star } } = \\varnothing \\mbox { a n d } { \\mathcal H } _ { c _ 1 - \\epsilon } ^ { a \\star } \\cap { \\mathcal A } _ { T ^ 2 , k _ 1 ^ { \\star } } \\neq \\varnothing \\end{align*}"}
-{"id": "1127.png", "formula": "\\begin{align*} \\phi _ j : = \\psi _ j \\biggl ( 1 - \\sum _ { k = 1 } ^ { j - 1 } \\phi _ k \\biggr ) , j \\ge 2 . \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } S ( C | A ' B ' ) _ { \\hat { \\gamma } ^ { ( n ) } _ { C A ' B ' } } = \\ln \\left ( \\eta \\ , a + \\left ( 1 - \\eta \\right ) b \\right ) + 1 \\ ; . \\end{align*}"}
-{"id": "6736.png", "formula": "\\begin{align*} \\ - t ^ { 2 \\nu } u ^ { \\left ( 2 \\right ) } \\left ( t \\right ) + \\left ( A + \\lambda \\right ) u \\left ( t \\right ) = f , t \\in \\left ( 0 , a \\right ) , \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} U = \\begin{bmatrix} a \\pi ^ { - 1 } + \\pi ^ { n - 1 } & \\pi ^ n \\\\ 1 & \\pi \\end{bmatrix} \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{align*} \\partial _ t u = - \\mathrm i | D | u + | u | ^ { p - 1 } u \\end{align*}"}
-{"id": "9648.png", "formula": "\\begin{align*} L ( f , s ) = \\sum _ { n = 1 } ^ { \\infty } c _ n ( f ) \\ , n ^ { - s } . \\end{align*}"}
-{"id": "4516.png", "formula": "\\begin{align*} \\mathfrak { F } _ { t , k , q } ^ { ( \\omega ) } \\left ( \\Lambda \\right ) = \\sum \\limits _ { x \\in \\Lambda } \\mathfrak { F } _ { t , k , q } ^ { ( \\omega ) } \\left ( \\left \\{ x \\right \\} \\right ) \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{align*} a _ { m a x , n } = h _ { n - 1 } \\tau _ { f , n } \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{align*} \\det \\begin{bmatrix} & \\int _ { y _ N } ^ { \\infty } \\phi _ { \\omega } ( x _ { N + 1 } ) d x _ { N + 1 } & \\cdots & - \\phi ^ { ( N - 1 ) } _ { \\omega } ( y _ N ) \\\\ & \\vdots & \\ddots & \\vdots \\\\ & \\int _ { y _ 1 } ^ { y _ 2 } \\phi _ { \\omega } ( x _ 2 ) d x _ 2 & \\cdots & \\phi ^ { ( N - 1 ) } _ { \\omega } ( y _ 2 ) - \\phi ^ { ( N - 1 ) } _ { \\omega } ( y _ 1 ) \\\\ & \\int _ { - \\infty } ^ { y _ 1 } \\phi _ { \\omega } ( x _ 1 ) d x _ 1 & \\cdots & \\phi ^ { ( N - 1 ) } _ { \\omega } ( y _ 1 ) \\end{bmatrix} _ { ( N + 1 ) \\times ( N + 1 ) } . \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} & H ( Y _ 3 | Y _ 2 ) = H ( Y _ 3 | Y _ 2 = y _ 2 ) = H ( Y _ 4 | Y _ 1 ) = H ( Y _ 4 | Y _ 1 = y _ 1 ) \\\\ = & \\frac { d + 2 } { 2 } \\cdot \\frac { 1 } { d } \\log \\frac { 2 d } { d + 2 } + \\frac { d - 2 } { 2 } \\cdot \\frac { 1 } { d } \\log d \\\\ & H ( Y _ 3 | Y _ 1 ) = H ( Y _ 3 | Y _ 1 = y _ 1 ) = H ( Y _ 4 | Y _ 2 ) = H ( Y _ 4 | Y _ 2 = y _ 2 ) \\\\ = & \\frac { 1 } { 2 } \\log 2 + \\frac { 1 } { 2 } \\log d \\end{align*}"}
-{"id": "8548.png", "formula": "\\begin{align*} \\frac { \\partial L _ 0 } { \\partial z _ 2 } = \\bar { \\frac { \\partial L _ 0 } { \\partial z _ 1 } } , \\frac { \\partial L _ 0 } { \\partial w _ 2 } = \\bar { \\frac { \\partial L _ 0 } { \\partial w _ 1 } } \\end{align*}"}
-{"id": "7011.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { 2 } \\varepsilon ^ { \\frac { i } { 2 } } \\left \\vert \\lambda \\right \\vert ^ { 1 - \\frac { i } { 2 } } \\left \\Vert u ^ { \\left ( i \\right ) } \\right \\Vert _ { X } + \\left \\Vert A u \\right \\Vert _ { X } \\leq C \\sum \\limits _ { i = 0 } ^ { 2 } \\varepsilon ^ { \\frac { i } { 2 } } \\left \\vert \\lambda \\right \\vert ^ { 1 - \\frac { i } { 2 } } \\end{align*}"}
-{"id": "6910.png", "formula": "\\begin{align*} X _ { k } = \\sin ( k \\pi x ) k = 1 , 2 , 3 , \\cdots . \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} \\pi _ * ( w ^ m \\mu ) = ( 2 \\pi i ) ^ p \\mu _ m . \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} \\lim _ { x \\to b ^ - } [ a _ k ( x ) f ^ { ( k ) } ( x ) ] ^ { ( k - 2 ) } = 0 . \\end{align*}"}
-{"id": "1278.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mu } \\left ( Z _ { 0 } = \\mathsf { o } \\right ) = \\mathbb { P } _ { \\mu } \\left ( Z _ { T } = \\mathsf { o } \\right ) = 1 \\end{align*}"}
-{"id": "9772.png", "formula": "\\begin{align*} \\| V \\| ( M ) = \\lim _ { \\epsilon _ j \\to 0 } \\frac { 1 } { | \\log \\epsilon _ j | } ( E _ { \\epsilon _ j } ( u _ { \\epsilon _ j } ) - \\frac { 1 } { 2 } \\| h _ { \\epsilon _ j } \\| _ { L ^ 2 } ^ 2 ) . \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} V _ { n } ^ { ( 2 ) } = p V _ { n - 1 } ^ { ( 2 ) } - p q V _ { n - 3 } ^ { ( 2 ) } + q ^ { 2 } V _ { n - 4 } ^ { ( 2 ) } , \\ n \\geq 4 , \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} \\frac { d c } { d \\tau } & = \\mu h \\frac { b + c } { 1 + c } - \\frac { \\Gamma } { K _ 1 } \\frac { c } { K + c } + \\Lambda \\theta = F _ 1 , \\\\ \\frac { d \\theta } { d \\tau } & = \\epsilon ( - \\theta + T ( c ) ) = \\epsilon R _ 2 , \\\\ \\frac { d h } { d \\tau } & = \\epsilon \\left ( \\frac { 1 } { 1 + c ^ 2 } - h \\right ) = \\epsilon R _ 3 . \\end{align*}"}
-{"id": "9699.png", "formula": "\\begin{align*} h _ P ^ 1 ( t ) \\ = \\ t \\sum _ { V } x + t ^ 2 \\left ( \\sum _ { E ^ o } ( y + z ) - 2 \\sum _ { V ^ o } x \\right ) + t ^ 3 \\sum _ { V ^ o } x . \\end{align*}"}
-{"id": "6303.png", "formula": "\\begin{align*} [ M ] = \\sum _ { \\lambda \\in X ^ + } ( \\sum _ { w \\in W } ( - 1 ) ^ { \\ell ( w ) } \\dim M _ { w \\cdot \\lambda } ) [ \\Delta ( \\lambda ) ] = \\sum _ { \\lambda \\in X ^ + } ( \\sum _ { w \\in W } ( - 1 ) ^ { \\ell ( w ) } \\dim M _ { \\lambda - w \\cdot 0 } ) [ \\Delta ( \\lambda ) ] \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} [ a _ i ^ { + } , a _ j ^ - ] ~ | n _ 1 , \\cdots , n _ i , \\cdots , n _ j , \\cdots , n _ r \\rangle \\ = \\sqrt { n _ j ( n _ i + 1 ) ) } ~ | n _ 1 , \\cdots , n _ i + 1 , \\cdots , n _ j - 1 , \\cdots , n _ r \\rangle , \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{align*} f _ i ( X _ i ) & = \\frac { h \\left ( g ( x _ 1 ) , \\ldots , g ( x _ { i - 1 } ) , g ( X _ i ) , g ( x _ { i + 1 } ) , \\ldots , g ( x _ n ) \\right ) } { \\prod \\limits _ { j \\neq i } f _ j ( x _ j ) } , \\end{align*}"}
-{"id": "1069.png", "formula": "\\begin{align*} \\psi _ 1 ( u _ 1 , u _ 2 , u _ 3 ) & = ( 1 + \\iota ( N ) ^ { - 1 } ) u _ 1 + u _ 2 \\\\ \\psi _ 2 ( u _ 1 , u _ 2 , u _ 3 ) & = u _ 1 + u _ 2 \\\\ \\psi _ 3 ( u _ 1 , u _ 2 , u _ 3 ) & = u _ 3 . \\end{align*}"}
-{"id": "1857.png", "formula": "\\begin{align*} & e _ i g ( \\nabla _ { e _ j } e _ j , e _ i ) + e _ i T _ { j i j } - e _ j g ( \\nabla _ { e _ i } e _ j , e _ i ) - e _ j T _ { j i i } \\\\ = & 2 s ( p ) + g ( \\nabla _ { e _ i } e _ i , \\nabla _ { e _ j } e _ j ) - g ( \\nabla _ { e _ i } e _ j , \\nabla _ { e _ j } e _ i ) - 2 e _ j T _ { j i i } . \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} \\delta \\left ( R \\right ) \\geq C _ { \\mu } \\left \\Vert u \\left ( . , 0 \\right ) \\right \\Vert _ { X } e ^ { - \\left ( M + 1 0 \\varkappa \\right ) } = C _ { \\mu } \\left \\Vert u \\left ( . , 0 \\right ) \\right \\Vert _ { X } e ^ { - \\left ( M + 2 0 \\right ) a _ { 1 } k ^ { \\frac { 1 } { 2 - p } } } \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{align*} G ( \\alpha _ 0 , \\dots , \\alpha _ d ) = G _ R ( \\alpha _ 0 , \\dots , \\alpha _ d ) + \\Gamma ( \\alpha _ 0 , \\dots , \\alpha _ { d - 1 } ) - \\Gamma ( \\alpha _ 1 , \\dots , \\alpha _ d ) \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\alpha _ 1 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 6 \\gamma _ 3 } { \\alpha _ 1 \\beta _ 6 } y _ 5 , [ y _ 2 , y _ 2 ] = \\frac { \\beta _ 2 \\gamma _ 3 } { \\alpha _ 1 \\beta _ 6 } y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\beta _ 4 } { \\beta _ 6 } y _ 5 , \\\\ [ y _ 3 , y _ 1 ] = y _ 5 , [ y _ 2 , y _ 3 ] = \\frac { \\gamma _ 1 } { \\gamma _ 3 } y _ 4 + \\theta _ 3 y _ 5 , [ y _ 3 , y _ 2 ] = y _ 4 . \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} N w - M \\overline { w } = 0 . \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{align*} A _ D v = b , \\end{align*}"}
-{"id": "6252.png", "formula": "\\begin{align*} \\pi _ { N ( \\theta ) } M ( \\theta ) \\pi _ { N ( \\theta ) } = \\pi _ { N ( \\theta ) } \\widetilde { M } ( \\theta ) \\pi _ { N ( \\theta ) } ( \\theta \\in \\Theta ) . \\end{align*}"}
-{"id": "1788.png", "formula": "\\begin{align*} \\left [ A x - \\dfrac { T ( t ) x - x } { t } \\right ] _ j = [ A x ] _ j - \\dfrac { [ T ( t ) x ] _ j - [ x ] _ j } { t } = \\left ( A _ j [ x ] _ j - \\dfrac { T _ j ( t ) [ x ] _ j - [ x ] _ j } { t } \\right ) { \\underset { t \\to 0 } { \\overset { X _ j } { \\longrightarrow } } } [ 0 ] _ j , \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} g ( r ) \\ ; & = \\ ; g _ 0 r ^ { - \\sqrt { 1 - \\nu ^ 2 } } + g _ 1 r ^ { \\sqrt { 1 - \\nu ^ 2 } } + o ( r ^ { 1 / 2 } ) \\quad \\textrm { a s } \\ ; r \\downarrow 0 \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} f = ( \\widetilde { \\mu } ^ { X ' } | _ { M ' } ) ^ { - 1 } \\circ \\widetilde { \\mu } ^ X | _ { M } : M \\to M ' . \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} \\left ( L _ { \\varepsilon } + \\lambda \\right ) u = \\bar { f } \\left ( x \\right ) , x \\in \\mathbb { R } = \\left ( - \\infty , \\infty \\right ) \\end{align*}"}
-{"id": "1654.png", "formula": "\\begin{align*} \\limsup _ { \\ell \\to \\infty } ( c \\ , \\ell ^ d ) ^ { 1 / \\ell } = 1 \\ ; \\mbox { f o r a n y $ c > 0 $ , $ d \\in \\R $ . } \\end{align*}"}
-{"id": "231.png", "formula": "\\begin{align*} | N _ Y ( R _ l ( \\alpha , s , h ) ) - s ( \\omega _ { l } - \\omega _ { l - 1 } ) / \\pi | < \\tilde { n } _ l - 1 + \\epsilon , \\ , l = 1 , \\ldots , 2 J + 2 , \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} d _ L \\partial _ { \\nu } L = \\alpha _ 0 \\ , \\frac { L } { k _ 0 + L } + \\frac { \\alpha _ 5 F ^ { p _ 3 } } { k _ 5 ^ { p _ 3 } + F ^ { p _ 3 } } \\ \\ \\omega ( p _ 3 > 1 ) . \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} U = \\begin{bmatrix} a & b \\pi ^ n \\\\ c & d \\end{bmatrix} \\end{align*}"}
-{"id": "5058.png", "formula": "\\begin{align*} \\eta ^ a _ 0 \\ = \\ 0 \\ , , \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } v ( t ) \\in T ( x ( t ) ) \\\\ \\lambda ( t ) \\dot x ( t ) + \\dot v ( t ) + v ( t ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} d ^ { \\rm s u p } ( 1 0 ^ { - 1 } ) \\ , \\le \\ , \\left \\{ \\begin{array} { l l } 1 & \\mbox { f o r \\ } a = 4 , \\\\ 1 & \\mbox { f o r \\ } a = 3 , \\\\ 2 & \\mbox { f o r \\ } a = 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{gather*} u _ t + b u _ x + u _ { x x x } + u _ { x y y } = f _ { 1 x } , f _ 1 \\in L _ 2 ( Q _ T ) , \\\\ u \\big | _ { t = 0 } = u _ 0 \\in L _ 2 , u \\big | _ { t = T } = u _ T \\in L _ 2 , u \\big | _ { x = 0 } = u \\big | _ { x = R } = 0 , u _ x \\big | _ { x = R } = \\nu _ 1 \\end{gather*}"}
-{"id": "1003.png", "formula": "\\begin{align*} h ^ { 1 , 1 } ( Y ) = h ^ { 1 , 1 } ( B ) + f + 1 , h ^ { 2 , 1 } ( Y ) = h ^ { 1 , 1 } ( Y ) - \\frac { 1 } { 2 } \\chi ( Y ) , \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{gather*} Z _ m ( e ^ { \\varphi i } \\zeta , e ^ { \\psi i } \\eta ) : = e ^ { m ( \\varphi - \\psi ) i } Z _ m ( \\zeta , \\eta ) \\qquad \\textrm { f o r } \\quad \\zeta , \\eta \\in S . \\end{gather*}"}
-{"id": "5842.png", "formula": "\\begin{align*} 2 5 \\sum _ { k = 1 } ^ n { ( - 1 ) ^ { k - 1 } F _ { m k } { } ^ 4 } & = \\sum _ { k = 1 } ^ n { ( - 1 ) ^ { k - 1 } L _ { 4 m k } } \\\\ & + 4 \\sum _ { k = 1 } ^ n { ( - 1 ) ^ { k ( m - 1 ) } L _ { 2 m k } } + 3 ( ( - 1 ) ^ { n - 1 } + 1 ) \\ , . \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} B _ { A , \\pmb { \\omega } } x = A P _ { r } x + \\sum _ { k = 1 } ^ { r } x _ { k + r } \\ , \\delta e _ { k } + \\sum _ { k > r } x _ { k + r } \\ , M e _ { k } \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} ( \\rho _ u - \\rho _ b ) \\ u _ p = \\rho _ u u _ u - \\rho _ b u _ b , \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\log \\left ( \\frac { t ( X _ { i } ) } { u ( X _ { i } ) } \\right ) = 2 \\sum _ { i = 1 } ^ { n } \\log \\left ( \\sqrt { \\frac { t ( X _ { i } ) } { u ( X _ { i } ) } } \\right ) \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} \\Psi ( 0 ) = \\frac { \\Gamma _ p ( \\beta + \\gamma - \\delta ) } { \\Gamma _ p ( \\delta - \\varrho _ a ) } \\cdot \\frac { \\Gamma _ p ( \\beta + \\gamma - \\epsilon ) } { \\Gamma _ p ( \\epsilon - \\varrho _ a ) } \\cdot \\frac { \\Gamma _ p ( \\delta ) } { \\Gamma _ p ( \\delta + a ) } \\cdot \\frac { \\Gamma _ p ( \\epsilon ) } { \\Gamma _ p ( \\epsilon + a ) } \\cdot \\Phi ( 0 ) = \\Omega ( 0 ) \\Phi ( 0 ) , \\end{align*}"}
-{"id": "1665.png", "formula": "\\begin{align*} | c _ M | _ 1 & \\leq \\binom { n } { n - 1 } \\cdot | c _ N | _ 1 \\cdot | c | _ 1 \\leq n \\cdot | c _ N | _ 1 \\cdot \\epsilon . \\end{align*}"}
-{"id": "2619.png", "formula": "\\begin{align*} { \\Bbb H } : = H _ { 1 } \\times H _ { 0 } , \\end{align*}"}
-{"id": "4895.png", "formula": "\\begin{align*} \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\sigma _ 0 ^ 2 = \\bar { q } _ 0 = \\sum \\limits _ { k > 0 : \\bold { z } _ k \\in \\Phi _ 2 } { G _ k \\mathit { l } \\left ( { \\Vert \\bold { z } _ k \\Vert } _ 2 \\right ) } g _ k P _ k = \\bar { q } _ 0 ^ { ( 1 ) } + \\bar { q } _ 0 ^ { ( 2 ) } + \\bar { q } _ 0 ^ { ( 3 ) } , \\end{align*}"}
-{"id": "970.png", "formula": "\\begin{align*} \\textrm { \\emph { C o r } } _ { B _ 1 } [ X ] = \\ldots = \\textrm { \\emph { C o r } } _ { B _ k } [ X ] , \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{gather*} \\frac { { \\rm d } Y } { { \\rm d } x } = \\left ( \\frac { A _ 0 } { x } + \\frac { A _ 1 } { x - 1 } + \\sum _ { j = 1 } ^ N \\frac { A _ { t _ j } } { x - t _ j } \\right ) Y . \\end{gather*}"}
-{"id": "1722.png", "formula": "\\begin{align*} H & = \\tanh ( c _ 2 \\tau - c _ 3 ) = \\tanh ( d _ 2 \\tau - d _ 3 ) \\\\ & = \\tanh \\left ( \\frac { 2 c _ 3 } { \\tau _ f } K \\tau _ f - c _ 3 \\right ) \\\\ & = \\tanh \\left ( c _ 3 ( 2 K - 1 ) \\right ) \\\\ K & = \\frac { 1 } { 2 } \\left ( \\frac { \\tanh ^ { - 1 } H } { c _ 3 } + 1 \\right ) \\end{align*}"}
-{"id": "8076.png", "formula": "\\begin{align*} \\Omega ^ 1 _ S [ z _ 1 , \\ldots , z _ k ] : = \\Omega ^ 1 _ S \\otimes \\C [ z _ 1 , \\ldots , z _ k ] \\end{align*}"}
-{"id": "2301.png", "formula": "\\begin{align*} \\bar \\eta _ j : = \\eta _ j | _ N = \\bar \\eta _ 1 a _ j ^ 1 + \\bar \\eta _ 1 ^ 2 a _ j ^ 2 + O ( \\bar \\eta _ 1 ^ 3 ) . \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} c _ { \\kappa ^ \\sigma } ( m , n ) = \\prod _ { j < i } ( - 1 ) ^ { m _ { \\sigma ( i ) } n _ { \\sigma ( j ) } } . \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} A _ n ^ { - 1 } = \\sum _ { i = 1 } ^ { n - 1 } \\binom { n - 1 } { i } ( - 1 ) ^ { i + 1 } A _ n ^ { i - 1 } , \\ n \\geq 2 . \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} f _ k : = \\sum _ { j = 1 } ^ { n - 1 } \\sigma _ j t _ j ^ k + \\lambda \\delta _ { k , 2 n - 2 } , k = 0 , \\dots , 2 n - 2 , \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { u } = i \\left [ \\Delta \\tilde { u } + A \\tilde { u } + \\tilde { V } \\left ( x , t \\right ) \\tilde { u } + \\tilde { F } \\left ( x , t \\right ) \\right ] , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } \\varepsilon \\frac { d \\varphi } { d \\tau } & = & - 3 v _ \\star ^ 2 \\varphi + 2 ( a + 1 ) v _ \\star \\varphi - a \\varphi - \\eta , \\\\ \\\\ \\frac { d \\eta } { d \\tau } & = & b \\varphi - c \\eta . \\end{array} \\right . \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{align*} c _ { Y _ 4 } B ( \\log B ) ^ { \\operatorname { r g } \\operatorname { P i c } ( Y _ 4 ) - 1 } \\times O ( B ^ { - \\frac { \\dim Y _ 4 } { r } } ) = O ( B ^ { 1 - \\frac { 2 } { r } } ( \\log B ) ^ 5 ) . \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\mathbb { E } Z _ n ( \\epsilon _ 0 ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { P } ( 1 \\leq \\# { \\cal E } _ i \\leq \\epsilon _ 0 n ) \\leq \\sum _ { r = 1 } ^ { \\epsilon _ 0 n } \\frac { 1 } { C } \\frac { T _ r e ^ { - r } } { ( r - 1 ) ! } e ^ { - \\delta _ 0 r } \\leq q _ 0 ( C , \\epsilon _ 0 , \\omega _ 0 ) . \\end{align*}"}
-{"id": "5069.png", "formula": "\\begin{align*} \\chi ( t + \\pi ) = \\chi ( t ) + \\pi \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\lambda } ^ d | L _ n | ^ 2 & = \\sum _ { \\overrightarrow { x } \\in \\Lambda _ n } \\sum _ { \\overrightarrow { y } \\in \\Lambda _ n } E [ \\prod _ { i = 0 } ^ { n - 1 } F ( x _ i , y _ i ; x _ { i + 1 } , y _ { i + 1 } ) ] \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ a \\frac { ( - a ) _ { k } ( \\beta ) _ { k } } { ( 1 ) _ { k } ^ 2 } \\cdot z ^ k \\cdot \\bigg ( \\sum _ { j = 0 } ^ { k - 1 } \\frac { 1 } { j + \\beta } - 2 H _ k \\bigg ) \\\\ = & ( 1 - z ) ^ a \\sum _ { k = 0 } ^ a \\frac { ( - a ) _ { k } ( 1 - \\beta ) _ { k } } { ( 1 ) _ { k } ^ 2 } \\cdot \\frac { z ^ k } { ( z - 1 ) ^ k } \\cdot \\bigg ( \\sum _ { j = 0 } ^ { k - 1 } \\frac { 1 } { j + 1 - \\beta } - 2 H _ k \\bigg ) . \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} \\int _ { \\gamma _ \\lambda } \\left ( \\frac { | f ( z ) | ^ 2 } { \\lambda } - 1 \\right ) \\frac { \\partial } { \\partial n } \\log | z | ^ 2 \\ , \\frac { d s } { 4 \\pi } = & - \\frac 1 { \\lambda } \\int _ { \\gamma _ \\lambda } \\log | z | ^ 2 \\frac { \\partial } { \\partial n } | f ( z ) | ^ 2 \\ , \\frac { d s } { 4 \\pi } \\\\ & + \\int _ { E _ f ( \\lambda ) } \\frac 1 { \\lambda } \\log | z | ^ 2 \\Delta | f ( z ) | ^ 2 \\frac { d x d y } { 4 \\pi } + \\frac { | f ( 0 ) | ^ 2 } { \\lambda } - 1 . \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} P ( L \\mathbf { x _ i } ) & : = \\mathbf { b _ i } , 1 \\leqslant i \\leqslant u \\\\ P ( \\mathbf { y _ j } ) & : = \\mathbf { b _ { j + u } } , 1 \\leqslant j \\leqslant m - u , \\end{align*}"}
-{"id": "8868.png", "formula": "\\begin{align*} [ P _ i , P _ j ] \\bigg | _ a ^ b & = \\int _ a ^ b \\ell [ P _ i ] P _ j w d x - \\int _ a ^ b P _ i \\ell [ P _ j ] w d x = ( \\lambda _ i - \\lambda _ j ) \\int _ a ^ b P _ i P _ j w d x = 0 . \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} \\{ h , g \\} & = \\sum _ { j , k } \\{ h _ j , g _ k \\} , \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} ( U M + V \\overline { N } ) z = U p + V \\overline { p } . \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 } ^ k c _ i ( x \\cdot \\lambda _ i ) ^ { 2 d } \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} 0 = \\int _ \\Omega \\Big ( y ( \\omega ( y ) \\cdot x ) + ( y \\cdot x ) \\omega ( y ) \\Big ) \\ , d y . \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{align*} \\Delta _ g y = 0 \\end{align*}"}
-{"id": "8694.png", "formula": "\\begin{align*} L _ 1 ( b ) = \\lim _ { r \\to \\infty } r ^ { \\frac { 2 } { \\alpha } } g _ b ( r ) \\in \\R \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} D _ { n , k } ( 1 , y ( 1 - y ) ) & = k \\ , \\Big [ \\displaystyle \\frac { y ^ n ( 1 - y ) - y ( 1 - y ) ^ n } { 2 y - 1 } \\Big ] + D _ n ( 1 , y ( 1 - y ) ) , \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} e _ j ^ 2 = e _ j , \\ \\sum _ { j \\in J } e _ j = 1 \\in \\mathcal { A } , \\ e _ j e _ { j ^ { \\prime } } = 0 \\ \\forall j \\ne j ^ { \\prime } . \\end{align*}"}
-{"id": "3408.png", "formula": "\\begin{align*} d Y _ t = \\left ( \\lambda ( \\varepsilon - \\theta ^ 2 ) - ( a + \\lambda \\theta ) \\phi _ t + \\theta \\frac { \\theta + \\phi _ t } { 1 + \\frac { 1 } { \\lambda } \\phi _ t } \\phi _ t \\right ) ( m ( t ) - X _ t ) d t + Q _ t \\ , d W _ t + R _ { t } \\ , d \\widetilde N _ t \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } S ( C | A ' B ' ) _ { \\hat { \\gamma } ^ { ( n ) } _ { C A ' B ' } } = \\ln \\left ( \\eta \\ , a + \\left ( \\eta - 1 \\right ) b \\right ) + 1 \\ ; . \\end{align*}"}
-{"id": "2305.png", "formula": "\\begin{align*} 3 \\int _ 0 ^ { \\pi / 3 } d t = \\pi , \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} & ( \\widetilde { E } \\times E ) [ \\prod _ { i = 0 } ^ { n - 1 } F ( S _ i , \\widehat { { S } } _ i ; S _ { i + 1 } , \\widehat { S } _ { i + 1 } ) ] \\\\ = & \\sum _ { \\overrightarrow { x } \\in \\Lambda _ n } \\sum _ { \\overrightarrow { y } \\in \\Lambda _ n } \\frac { 1 } { d ^ { 2 n } } E [ \\prod _ { i = 0 } ^ { n - 1 } F ( x _ i , y _ i ; x _ { i + 1 } , y _ { i + 1 } ) ] . \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{align*} L ^ { q , r } ( D ) = \\big ( L ^ { q _ 0 , r _ 0 } ( D ) , L ^ { q _ 1 , r _ 1 } ( D ) \\big ) _ { \\theta , r } \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} N = \\{ [ E ] \\in N _ s \\ | \\ \\tau _ E = \\tau \\} . \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{align*} \\vec { \\gamma } _ 3 ( \\phi , p ) = \\vec { \\gamma } _ 0 ( \\vec { \\Psi } , p ) . \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} \\Delta u + \\frac { 1 } { 2 t } x \\cdot \\nabla u + \\frac { 1 } { \\alpha t } u + | u | ^ \\alpha u = 0 \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} \\mathbf { F } ^ { \\vee } = \\mathbf { F } ( c - e - b ) . \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{align*} C _ j : = P _ j B _ j ^ { - \\frac { 1 } { 2 } } , j = \\overline { 1 , N } , \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} | t _ 2 ( 2 n ) | = \\frac { | t _ 2 ( 2 n - 1 ) + t _ 2 ( 2 n + 1 ) | } { 2 } \\Longleftrightarrow | t _ 2 ( n ) + t _ 2 ( n - 1 ) | = \\frac { | - 2 t _ 2 ( n - 1 ) - 2 t _ 2 ( n ) | } { 2 } \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} \\left \\Vert \\partial _ { x _ { 1 } } ^ { j } h \\left ( x _ { 1 } , t \\right ) \\right \\Vert _ { L ^ { \\infty } \\left ( \\mathbb { R } \\times \\left [ 0 , 1 \\right ] \\right ) } \\leq C _ { j } n ^ { - \\left ( j + 1 \\right ) } , j = 1 , 2 , . . . , \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{gather*} q _ 1 : = - 8 x _ 2 ^ 4 + \\lambda ^ 2 x _ 2 ^ 2 x _ 3 ^ 2 - 8 \\lambda x _ 2 x _ 3 u ^ 2 - 8 x _ 3 ^ 4 + 8 u ^ 4 , \\\\ q _ 2 : = - 8 x _ 2 ^ 3 x _ 3 + \\lambda ^ 2 x _ 2 ^ 2 x _ 3 ^ 2 - 8 x _ 2 x _ 3 ^ 3 - 8 \\lambda x _ 2 x _ 3 u ^ 2 + 8 u ^ 4 , \\\\ q _ 3 : = 8 x _ 2 ^ 3 x _ 3 + \\lambda ^ 2 x _ 2 ^ 2 x _ 3 ^ 2 t + 8 x _ 2 x _ 3 ^ 3 + 2 \\lambda x _ 2 x _ 3 u ^ 2 + u ^ 4 , \\\\ q _ 6 : = - 8 x _ 2 ^ 4 + \\lambda ^ 2 x _ 2 ^ 2 x _ 3 ^ 2 - 8 x _ 2 ^ 3 x _ 3 - 8 \\lambda x _ 2 x _ 3 u ^ 2 + 8 u ^ 4 , \\\\ q _ 7 : = 8 x _ 2 ^ 4 + \\lambda ^ 2 x _ 2 ^ 2 x _ 3 ^ 2 + 8 x _ 2 ^ 3 x _ 3 + 2 \\lambda x _ 2 x _ 3 u ^ 2 + u ^ 4 . \\end{gather*}"}
-{"id": "3010.png", "formula": "\\begin{align*} \\bigcup _ { k \\ge k _ 0 } \\bigcup _ { m \\in \\varphi ^ { - 1 } ( n ) \\cap J _ k } \\Big \\{ s \\in [ 1 , b _ { m + 1 } - b _ m ) ; \\ ; \\prod _ { i = b _ { m + 1 } - s } ^ { b _ { m + 1 } - 1 } | w _ { i } | = 2 ^ { \\delta ^ { ( k ) } } \\Big \\} \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} h = \\sum _ { i = 0 } ^ \\infty X _ i \\bar f _ i , \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{align*} X \\mapsto U = \\frac { i - X } { i + X } . \\end{align*}"}
-{"id": "6611.png", "formula": "\\begin{align*} r ^ { - B } & ( g ^ + \\ ! ( r ) - b r ^ { - B } ) \\ ; = \\ ; r ^ { - B } \\big ( f ^ + \\ ! ( r ) + a ( S _ D ^ { - 1 } \\Phi ) ^ + \\ ! ( r ) + b \\gamma ^ { - 1 } \\Phi ^ + \\ ! ( r ) - b r ^ { - B } \\big ) \\\\ & = \\ ; a \\ , p ^ + + b \\ , q ^ + \\gamma ^ { - 1 } + o ( r ^ { 1 / 2 - B } ) \\ , , \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{align*} f ( A ) = Z f ( J ) Z ^ { - 1 } = Z \\ ; \\mathrm { d i a g } \\{ f ( J _ 1 ) , f ( J _ 2 ) , \\ldots , f ( J _ s ) \\} Z ^ { - 1 } , \\end{align*}"}
-{"id": "9062.png", "formula": "\\begin{align*} d _ u ( G ) = \\dim C _ { C _ G ( y ) ^ 0 } ( x ) . \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} \\tilde { X } _ { 2 , \\ , j + 1 , \\ , 2 } \\cap L _ { j + 1 } = 0 . \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} g & = u ^ \\frac 4 { n - 2 } \\bar g , R [ g ] = u ^ { - \\frac { n + 2 } { n - 2 } } \\left ( R [ \\bar { g } ] u - \\frac { 4 ( n - 1 ) } { n - 2 } \\Delta u \\right ) \\\\ H [ g ] & = u ^ { - \\frac n { n - 2 } } \\Big [ H [ \\bar g ] + \\frac { 2 } { n - 2 } \\partial _ \\nu u \\Big ] \\ , n \\ge 3 . \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{align*} w z w ^ { - 1 } = a - b d \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = \\infty \\ , \\left ( \\frac 1 3 \\right ) \\\\ \\overbrace { \\begin{matrix} 1 & \\theta ^ 0 _ 2 \\\\ 0 & \\theta ^ 0 _ 1 \\\\ 0 & 0 \\end{matrix} } & \\overbrace { \\begin{matrix} - t ^ { \\frac 1 3 } & \\theta ^ \\infty _ 1 / 3 \\\\ - \\omega t ^ { \\frac 1 3 } & \\theta ^ \\infty _ 1 / 3 \\\\ - \\omega ^ 2 t ^ { \\frac 1 3 } & \\theta ^ \\infty _ 1 / 3 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "1993.png", "formula": "\\begin{align*} T _ { a } ( z , \\{ \\alpha \\} ) = L _ { a M } ( z , t , \\alpha _ M ) \\cdots L _ { a 1 } ( z , t , \\alpha _ 1 ) , \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} h _ 2 \\left ( I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right ) h _ 2 ^ { - 1 } = I + \\frac { \\begin{pmatrix} \\alpha \\gamma & - \\alpha ^ 2 \\\\ \\gamma ^ 2 & - \\alpha \\gamma \\end{pmatrix} p } { - \\alpha ^ 2 - \\beta \\gamma } \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\phi \\left ( \\gamma \\right ) \\ , { } _ { 0 } ^ { c } \\mathrm { D } _ { t } ^ { \\gamma } q \\left ( x , t \\right ) \\mathrm { d } \\gamma = - \\lambda \\frac { \\partial } { \\partial x } T \\left ( x , t \\right ) , \\end{align*}"}
-{"id": "9298.png", "formula": "\\begin{align*} p _ { i j , k } \\geq 0 , \\ \\ p _ { i j , k } = p _ { j i , k } , \\ \\ \\sum _ { k = 1 } ^ { m } p _ { i j , k } = 1 , \\ \\ i , j , k \\in \\{ 1 , 2 , \\ldots , m \\} \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{align*} \\dot z = R ^ { - 1 } [ Q D v ( \\theta ( t ) ) \\Gamma _ 0 ] R z . \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} \\tilde S _ k : = ( \\psi _ k ^ { ( d _ s ) } ) ^ { O p } f \\ , , \\tilde S ^ k : = \\sum _ { j = 0 } ^ k ( \\psi _ j ^ { ( d _ s ) } ) ^ { O p } f = \\sum _ { j = 0 } ^ k \\tilde S _ j f \\ , . \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} \\int _ A \\omega ( y ) K _ { \\xi ^ * } ( x , y ) ^ \\perp \\ , d y & = \\left ( - { m \\over | x | ^ 2 } + 2 { ( x \\cdot m ) \\over | x | ^ { 4 } } x \\right ) + O \\left ( { 1 \\over | x | ^ { 3 } } \\right ) , \\textrm { a s } | x | \\to \\infty , \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{align*} f _ a ( t ) = t ^ 4 - a t ^ 3 + ( 2 a - 1 ) t ^ 2 - a t + 1 \\end{align*}"}
-{"id": "362.png", "formula": "\\begin{align*} \\chi _ r ( \\mathsf { P } ) = \\sum _ { i = 1 } ^ r q ^ { ( \\alpha _ r - 2 \\rho , \\lambda _ i ) } [ ( \\alpha _ r , \\lambda _ i ) ] _ q = q ^ { - ( 2 \\rho , \\lambda _ r ) + 1 } . \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{align*} D _ m \\Big \\{ & \\big [ v ( t ) , w ( t ) \\big ] ; \\big [ \\bar { v } ( t ) , \\bar { w } ( t ) \\big ] \\Big \\} \\\\ & = \\sqrt { \\frac { ( v ( t ) - \\bar { v } ( t ) ) ^ 2 + ( w ( t ) - \\bar { w } ( t ) ) ^ 2 } { \\mu ( t ) } } . \\end{align*}"}
-{"id": "9009.png", "formula": "\\begin{align*} \\sup \\limits _ { a \\in \\R ^ d , r > 0 } | B ( a , r ) | ^ { \\frac { 1 } { q _ i } - \\frac { 1 } { p _ i } } \\Bigl ( { \\displaystyle \\int _ { B ( a , r ) } | f _ i ( x ) | ^ { p _ i } \\ , d x } \\Bigr ) ^ { \\frac { 1 } { p _ i } } = | B ( 0 , L ) | ^ { \\frac { 1 } { q _ i } - \\frac { 1 } { p _ i } } \\Bigl ( { \\displaystyle \\int _ { B ( 0 , L ) } | f _ i ( x ) | ^ { p _ i } \\ , d x } \\Bigr ) ^ { \\frac { 1 } { p _ i } } \\end{align*}"}
-{"id": "5515.png", "formula": "\\begin{align*} \\widehat \\psi _ p ( \\zeta ) = \\xi _ p \\circ h _ p ( \\zeta ) . \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} p _ { j } ^ { X } ( S _ N - S _ M ) = p _ { j } ^ { X } \\left ( \\sum _ { n = M + 1 } ^ { N } \\dfrac { ( t A ) ^ n } { n ! } \\right ) \\leqslant \\sum _ { n = M + 1 } ^ { N } \\dfrac { \\left ( t \\ , p _ { j } ^ { X } ( A ) \\right ) ^ n } { n ! } < \\varepsilon , \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{align*} \\psi _ { \\alpha , \\varepsilon } ( 0 ) = \\psi _ { \\alpha , 0 } ( 0 ) = \\frac { 1 } { 2 \\alpha } + \\frac { \\sqrt { \\alpha ^ 2 - 1 } - \\arctan ( \\sqrt { \\alpha ^ 2 - 1 } ) } { \\alpha \\pi } , \\end{align*}"}
-{"id": "355.png", "formula": "\\begin{align*} ( \\mathsf { M } ^ n _ m ) ^ i _ j \\triangleleft X = ( u ^ { m * } _ i \\triangleleft X _ { ( 1 ) } ) ( u ^ n _ j \\triangleleft X _ { ( 2 ) } ) = ( S ( u ^ i _ m ) \\triangleleft X _ { ( 1 ) } ) ( u ^ n _ j \\triangleleft X _ { ( 2 ) } ) . \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} C F ( ( M ; \\mathcal P _ M ) ; L ) = C _ * ( R ( M ; \\mathcal P _ M ) \\times _ { ( \\Sigma ; \\mathcal P _ { \\Sigma } ) } L ) \\ , \\ , \\hat \\otimes \\ , \\ , \\Lambda _ 0 . \\end{align*}"}
-{"id": "6907.png", "formula": "\\begin{align*} \\ , ^ { A B C } \\ , _ { 0 } D ^ { \\alpha } _ { t } u ( t ) - \\lambda u ( t ) = f ( t ) , 0 \\leq t \\leq T , \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} \\dfrac { \\ddot { a } } { a } + \\dfrac { \\dot { a } \\dot { b } } { a b } + \\dfrac { \\ddot { b } } { b } - \\Lambda = - \\dfrac { 8 \\pi \\rho } { 3 } - 4 \\pi \\dot { \\phi } ^ 2 , \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} F _ i ( n _ 1 , \\cdots , 0 , \\cdots , n _ r ) = 0 \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{align*} K _ i ( m + 1 ) = K _ i ( m ) + K _ { i - 1 } ( m ) = \\sum _ { j = 1 } ^ n K _ j ( 1 ) K _ { i - j + 1 } ( m ) . \\end{align*}"}
-{"id": "620.png", "formula": "\\begin{align*} \\mu ^ l : = \\sum _ { x \\in A ^ { ( l ) } } s _ { x } ^ { n - 1 } \\delta _ { x } . \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{align*} x _ \\pm ( \\lambda \\pm \\kappa ) = x _ { N , \\pm } ( \\lambda \\pm \\kappa ) , \\qquad \\lambda \\in Y \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{align*} \\dfrac { d ^ { 2 } x ^ { i } } { d t ^ { 2 } } + 2 G ^ { i } ( x , \\dot { x } ) = 0 , \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} \\mu _ { k j } : = \\frac { m _ { k j } - m _ { k j + 1 } } { ( \\omega _ { k j } - \\omega _ { k j + 1 } ) e _ k } \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} ( u ^ 2 + k ^ 2 ) b ^ 2 = k ^ 2 x ^ 3 \\wedge ( u ^ 3 + k ^ 3 ) b ^ 3 = k ^ 3 y ^ 2 . \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} & \\prod _ { i = 1 } ^ d \\frac { t } { 1 - a _ i t } = \\sum _ { i = 1 } ^ d \\frac { t } { 1 - a _ i t } \\prod ^ d _ { \\substack { j = 1 \\\\ j \\neq i } } \\frac { 1 } { a _ i - a _ j } . \\end{align*}"}
-{"id": "288.png", "formula": "\\begin{align*} \\mathrm { T r } ( \\pi ( K _ { 2 \\rho } ^ { - 1 } ) \\mathsf { P } ) = q ^ { - ( 2 \\rho , \\lambda _ { m } ) } , \\quad \\mathrm { T r } ( \\pi ( K _ { 2 \\rho } ) \\mathsf { Q } ) = q ^ { ( 2 \\rho , \\lambda _ { m } ) } . \\end{align*}"}
-{"id": "9872.png", "formula": "\\begin{align*} \\boldsymbol { K } ( \\rho ) = - K ( \\rho ) \\ , \\boldsymbol { e } _ r K ( \\rho ) = - \\frac { 1 } { \\sigma } \\frac { d G } { d \\rho } = \\dfrac { 1 } { 2 \\pi ^ 2 \\sigma ^ 2 } \\dfrac { ( \\rho ) - \\sin ( \\rho ) } { \\rho ^ 2 } \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} 2 g ( E ) - 2 & = ( K _ X + E ) \\cdot E \\\\ & < - D ' \\cdot E + ( 1 - \\beta ) E ^ 2 \\leq ( 1 - \\beta ) E ^ 2 < 0 . \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} s ^ H = - \\sum _ { i , j = 1 } ^ n H _ { i j , i j } . \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} \\left ( \\frac { g } { r } \\right ) \\frac { d } { d g } \\left ( \\frac { g ^ { 2 } } { r } \\right ) = \\lambda _ { 4 } , \\lambda _ { 4 } \\in \\mathbb { R } . \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} 2 a ^ 2 = x ^ 3 \\wedge 2 a ^ 3 = y ^ 2 . \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} \\d X = A \\cdot X \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} v = a u + \\log \\left ( 2 a \\right ) . \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} g ^ 1 _ n ( \\omega , t , y , z ) : = \\sup \\limits _ { u \\in \\R ^ d } \\left [ g ^ 1 ( \\omega , t , y , u ) - ( n + 2 \\lambda ) | u - z | ^ \\alpha \\right ] \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{gather*} \\partial _ t \\tilde { u } _ i - d _ i \\Delta \\tilde { u } _ i \\ge f _ i \\big ( t , x , \\tilde { u } _ i , [ { \\bf \\tilde { u } } ] _ { a _ i } , [ { \\bf 0 } ] _ { b _ i } \\big ) , \\\\ d _ i \\partial _ { \\nu } \\tilde { u } _ i \\ge g _ i \\big ( x , \\tilde { u } _ i , [ { \\bf \\tilde { u } } ] _ { c _ i } , [ { \\bf 0 } ] _ { d _ i } \\big ) , i = 1 , \\ldots , 6 . \\end{gather*}"}
-{"id": "2649.png", "formula": "\\begin{align*} \\Pr \\{ X = u , Y = v \\} = \\begin{cases} P ( ( u , v ) ) & ( u , v ) \\in E _ G \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1373.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } k _ { n } = \\infty . \\end{align*}"}
-{"id": "5533.png", "formula": "\\begin{align*} g _ p ( \\zeta ) & = d ^ n g _ n ( \\lambda _ { p , n } \\cdot \\zeta ) \\\\ & \\leq d ^ { 1 - \\frac { \\log ( C ' _ p ) } { \\log | \\delta | } } \\ , d ^ { - \\frac { \\log | \\zeta | } { \\log | \\delta | } } \\\\ & \\leq M | \\zeta | ^ { - \\frac { \\log | d | } { \\log | \\delta | } } . \\end{align*}"}
-{"id": "4848.png", "formula": "\\begin{align*} \\left \\langle D , I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle \\left \\langle D ' , I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle \\end{align*}"}
-{"id": "3597.png", "formula": "\\begin{align*} \\psi _ n ^ 1 ( \\xi , \\lambda ) = \\frac { 1 } { n ! } \\left ( \\frac { i \\xi } { \\sqrt { 2 } } \\right ) ^ { n - 1 } ( - \\lambda ) J _ n ( | \\xi | / \\sqrt { 2 } , \\lambda ) . \\end{align*}"}
-{"id": "7884.png", "formula": "\\begin{align*} \\frac { 4 } { q ^ 2 \\gamma ^ { 2 } n ^ 2 } \\left ( n q \\left ( 1 + \\frac { \\gamma ^ 3 } { 6 4 } \\right ) + n ^ 2 q ^ 2 \\frac { \\gamma ^ 3 } { 1 6 } \\right ) = \\frac { 4 } { q \\gamma ^ { 2 } n } \\left ( 1 + \\frac { \\gamma ^ 3 } { 6 4 } \\right ) + \\frac { \\gamma } { 4 } \\leq \\frac { \\gamma } { 2 } \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 \\ , \\left ( \\frac 1 2 \\right ) & x = \\infty \\ , \\left ( \\frac 1 3 \\right ) \\\\ \\overbrace { \\begin{matrix} 0 & 0 \\\\ 1 & \\theta ^ 0 / 2 \\\\ - 1 & \\theta ^ 0 / 2 \\end{matrix} } & \\overbrace { \\begin{matrix} - t ^ { \\frac 1 3 } & \\theta ^ \\infty _ 1 / 3 \\\\ - \\omega t ^ { \\frac 1 3 } & \\theta ^ \\infty _ 1 / 3 \\\\ - \\omega ^ 2 t ^ { \\frac 1 3 } & \\theta ^ \\infty _ 1 / 3 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "3828.png", "formula": "\\begin{align*} b _ { m } ( 2 n ) = & \\sum _ { j = 0 } ^ { m - 1 } { m \\choose j + 1 } ( - 1 ) ^ { j } b _ { m } ( 2 n - j - 1 ) + b _ { m } ( n ) , \\\\ b _ { m } ( 2 n + 1 ) = & \\sum _ { j = 0 } ^ { m - 1 } { m \\choose j + 1 } ( - 1 ) ^ { j } b _ { m } ( 2 n - j ) . \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{align*} h & = ( 0 , \\frac { \\Lambda _ 4 ^ \\vee } { 6 } + \\frac { 2 \\Lambda ^ \\vee _ 5 } { 3 } , \\frac { \\Lambda ^ \\vee _ 1 } { 3 } ) , \\\\ 2 [ h ] & = ( 0 , \\frac { \\Lambda _ 4 ^ \\vee } { 3 } - \\frac { 2 \\Lambda ^ \\vee _ 5 } { 3 } , \\frac { 2 \\Lambda ^ \\vee _ 1 } { 3 } - \\Lambda _ 2 ^ \\vee ) , \\\\ 3 [ h ] & = ( 0 , - \\Lambda _ 1 ^ \\vee + \\Lambda _ 2 ^ \\vee - \\Lambda _ 3 ^ \\vee + \\frac { \\Lambda _ 4 ^ \\vee } { 2 } , 0 ) . \\end{align*}"}
-{"id": "3441.png", "formula": "\\begin{align*} \\| \\sum _ { i = 1 } ^ n a _ i e _ { \\gamma _ i } \\| = \\| \\sum _ { i = 1 } ^ n a _ i e _ { \\beta _ i } \\| \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} \\left | \\sum _ { j = 1 } ^ N k _ j f ( x _ j ) \\right | & \\leq \\frac { r } { 2 } f ( x _ N ) \\leq \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} \\mathcal { B } _ \\eta ( \\hat { \\rho } _ { A B } ) : = \\mathrm { T r } _ D \\left [ \\hat { U } _ \\eta \\ , \\hat { \\rho } _ { A B } \\ , \\hat { U } _ \\eta ^ \\dag \\right ] \\ ; . \\end{align*}"}
-{"id": "400.png", "formula": "\\begin{align*} M : = \\{ ( f _ 1 , f _ 2 , f _ 3 , f _ 4 ) \\in \\Pi _ { i = 1 } ^ 4 V _ i \\ | \\ \\eqref { l a m b d a , m u } \\ \\ \\cap _ { i = 1 } ^ 4 \\{ f _ i ( x ) = 0 \\} = \\emptyset \\} . \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} L ^ * \\varphi & = \\sum \\limits _ { i = 1 } ^ u \\lambda _ i \\mathbf { x _ i ^ * } + \\sum \\limits _ { j = 1 } ^ { d - u } \\mu _ j \\Xi ( \\mathbf { w _ j } ) ^ * \\\\ \\omega & = \\sum \\limits _ { i = 1 } ^ u \\lambda _ i ^ \\prime \\mathbf { x _ i ^ * } + \\sum \\limits _ { j = 1 } ^ { d - u } \\mu _ j ^ \\prime \\Xi ( \\mathbf { w _ j } ) ^ * . \\end{align*}"}
-{"id": "2489.png", "formula": "\\begin{align*} \\lambda ( f _ q ^ { \\gamma } ) : = \\sum _ { Q \\in \\mathcal { Q } } l ( Q ) . \\end{align*}"}
-{"id": "810.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\biggl \\| \\int _ 0 ^ { t _ A } F ( 4 t - s ) * ( u _ n \\otimes u _ n ) ( s ) \\dd s \\biggr \\| _ 3 = \\lim _ { t \\to + \\infty } \\sqrt { 4 t } \\biggl \\| \\int _ 0 ^ { t _ A } F ( 4 t - s ) * ( u _ n \\otimes u _ n ) ( s ) \\dd s \\biggr \\| _ \\infty = 0 . \\end{align*}"}
-{"id": "5203.png", "formula": "\\begin{align*} \\hat { g } _ { s , t } \\circ \\hat { g } _ { t , s } \\circ \\hat { g } _ { s , t } = \\hat { g } _ { s , t } . \\end{align*}"}
-{"id": "9278.png", "formula": "\\begin{align*} \\eta ( t ) = \\sup _ { x \\in \\R ^ N } \\int _ { f ^ { - 1 } ( B _ \\rho ( x ) ) } | A | ^ 4 + | \\nabla A | ^ 2 \\ , d \\mu \\ , . \\end{align*}"}
-{"id": "9801.png", "formula": "\\begin{align*} \\left ( { { \\mathbf { u } } , { \\mathbf { v } } } \\right ) = \\int _ \\Omega { { { \\mathbf { u } } ^ T } { \\mathbf { v } } d x d y d z } , \\left \\| { \\mathbf { u } } \\right \\| = \\sqrt { \\left ( { { \\mathbf { u } } , { \\mathbf { u } } } \\right ) } . \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} \\abs { f ( W ( z ) ) - f ( W ( \\zeta ) ) } \\ge 2 \\ell \\cos \\left ( \\frac { \\pi } { 2 } - \\frac { r } { 4 L } \\right ) \\abs { z - \\zeta } = 2 \\ell \\sin \\left ( \\frac { r } { 4 L } \\right ) \\abs { z - \\zeta } . \\end{align*}"}
-{"id": "3296.png", "formula": "\\begin{align*} \\boldsymbol { U } ( t ) = \\mathfrak { C } ( \\boldsymbol { X } ) ( t ) = \\frac { i - \\boldsymbol { X } ( t ) } { i + \\boldsymbol { X } ( t ) } \\in \\mathbb { U } ( N ) , \\end{align*}"}
-{"id": "8556.png", "formula": "\\begin{align*} \\vec { \\gamma } _ 0 ( \\phi , p ) = \\vec { \\gamma } _ 3 ( \\vec { \\Psi } , p ) \\end{align*}"}
-{"id": "9620.png", "formula": "\\begin{align*} F ( x + i h ) = G _ n ( x ; h ) + \\sum ^ n _ { k = 1 } \\bar { \\alpha } _ k f ( x + ( i + k - 1 ) h ) , \\ \\ i = 0 , 1 , \\ldots , n , \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } \\Theta _ { i j } ( 0 ) = \\Theta _ { i j } ( T ) , \\\\ \\\\ \\Theta _ { i j } ( t ) \\left ( \\begin{array} { c } f \\big [ \\bar { v } ( t ) , \\bar { w } ( t ) \\big ] \\\\ \\\\ g \\big [ \\bar { v } ( t ) , \\bar { w } ( t ) \\big ] \\end{array} \\right ) \\equiv 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{align*} \\mathcal { N } ( s ) \\circ \\mathcal { N } ( t ) = \\mathcal { N } ( s + t ) \\ ; . \\end{align*}"}
-{"id": "9375.png", "formula": "\\begin{align*} \\Gamma ( f _ 1 \\otimes f _ 2 \\otimes f _ 3 ) ( X , Y ) = f _ 1 ( A ) X f _ 2 ( B ) Y f _ 3 ( C ) \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} \\langle x , y \\rangle : = \\sum _ { k = 1 } ^ p x _ k y _ k , \\langle X , Y \\rangle : = \\sum _ { k = 1 } ^ p \\sum _ { j = 1 } ^ q x _ { k j } y _ { k j } . \\end{align*}"}
-{"id": "4389.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k } \\binom { k } { j } ( - 1 ) ^ j H _ j ^ { ( 2 ) } = - \\frac { H _ k } { k } , \\end{align*}"}
-{"id": "9026.png", "formula": "\\begin{align*} \\tau _ a \\leq \\left \\lfloor \\frac { d ^ a - 1 } { 2 } \\right \\rfloor = 3 \\end{align*}"}
-{"id": "9852.png", "formula": "\\begin{align*} \\mathcal { F } _ i ^ { \\mathbf { z } } ( \\phi ^ \\circ ) ( \\pi ^ \\lambda ) = \\sum _ { w \\in W } \\mathcal { F } _ i ^ { \\mathbf { z } } ( \\phi _ w ) ( \\pi ^ \\lambda ) . \\end{align*}"}
-{"id": "8142.png", "formula": "\\begin{align*} ( \\gamma _ { i j } ( 1 - x _ { i j } ) ^ T - x _ { i j } ^ T ) f _ { \\mu _ 2 } = 0 . \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} A ^ * v = \\left ( v _ { 1 } , - L v _ { 0 } + B v _ { 1 } \\right ) . \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} \\left ( P _ 1 ( q _ i ) , \\dots , \\widehat { P _ { \\ell _ { i } } ( q _ i ) } , \\dots , P _ n ( q _ i ) \\right ) = \\left ( P _ 1 ( q _ { i - 1 } ) , \\dots , \\widehat { P _ { k _ { i - 1 } } ( q _ { i - 1 } ) } , \\dots , P _ n ( q _ { i - 1 } ) \\right ) \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} y _ 2 - y _ 1 = y _ 4 - y _ 3 = - \\log 2 . \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} \\Lambda _ 2 ( \\infty , \\Omega ) = \\frac { 1 } { \\rho _ { 2 , F } ( \\Omega ) } . \\end{align*}"}
-{"id": "9521.png", "formula": "\\begin{align*} \\left | \\tau \\left ( \\sum _ { i = 1 } ^ s c _ i ^ * x c _ i \\right ) \\right | & = \\left | \\tau E \\left ( \\sum _ { i = 1 } ^ s c _ i ^ * x c _ i \\right ) \\right | = \\left | \\tau \\left ( \\sum _ { i = 1 } ^ s c _ i ^ * E ( x ) c _ i \\right ) \\right | \\\\ & = | \\tau ( b E ( x ) ) | < \\delta . \\end{align*}"}
-{"id": "6500.png", "formula": "\\begin{align*} V _ { \\nu } ( m ) = \\int ( x - m ) ^ 2 Q _ m ( d x ) \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} [ h _ n , H ^ { \\beta } ( w ^ 2 ) ] = 0 , n \\neq 0 ; [ h _ 0 , H ^ { \\beta } ( w ^ 2 ) ] = - H ^ { \\beta } ( w ^ 2 ) . \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} ( u v ^ { - 1 } ) \\hat { \\phi } = ( s t ^ { - 1 } ) \\hat { \\phi } & \\Leftrightarrow ( u \\phi ) ( v \\phi ) ^ { - 1 } = ( s \\phi ) ( t \\phi ) ^ { - 1 } \\\\ & \\Leftrightarrow u \\phi t \\phi = s \\phi v \\phi \\\\ & \\Leftrightarrow u t = s v \\\\ & \\Leftrightarrow u v ^ { - 1 } = s t ^ { - 1 } . \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} \\operatorname { A r f } ( \\omega ) : = \\sum ^ g _ { i = 1 } \\omega ( [ \\alpha _ i ] ) \\omega ( [ \\beta _ i ] ) \\in \\Z / 2 \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} \\eta _ 0 = \\frac h { 1 + e ^ \\rho } . \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} \\Lambda ( c ) = \\frac 1 2 c _ { e _ i , J e _ i } = \\frac 1 2 \\Lambda ( R _ { e _ i , J e _ i } ) = \\frac 1 4 R _ { e _ i , J e _ i , e _ j , J e _ j } \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\mathrm { T r } _ C \\left [ \\hat { H } _ C \\ , \\Phi ( \\hat { \\rho } _ A ) \\right ] & = \\frac { \\eta } { n } \\mathrm { T r } _ A \\left [ \\hat { H } _ A \\ , \\hat { \\rho } _ A \\right ] + | 1 - \\eta | \\ , E _ 0 + \\frac { \\eta + | 1 - \\eta | - 1 } { 2 } \\\\ & \\le \\eta \\ , E + | 1 - \\eta | \\ , E _ 0 + \\frac { \\eta + | 1 - \\eta | - 1 } { 2 } \\ ; . \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{gather*} W ^ { ( 1 ) } : = \\big \\{ w \\in W \\ ; : \\ ; I _ u ( w , s _ w ) < L - \\tau \\big \\} \\quad W ^ { ( 2 ) } : = W \\setminus W ^ { ( 1 ) } , \\end{gather*}"}
-{"id": "6239.png", "formula": "\\begin{align*} \\gamma _ 1 & = \\lambda _ 1 \\\\ \\gamma _ k & = \\sum _ { m = 1 } ^ { M _ k } \\sum _ { j = 1 } ^ { N } c _ j \\delta _ { t _ k ^ { j , m } } , \\| \\gamma _ k \\| = 2 ^ k \\| \\lambda _ { k - 1 } \\| \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{align*} T _ { e ^ { - i p \\theta } \\phi ( r ) } ( z ^ k ) & = \\begin{cases} ( 2 n - 2 p + 2 ) \\widehat { \\phi } ( 2 k - p + 2 ) z ^ { k - p } & k \\geq \\max \\{ 0 , p \\} \\\\ 0 & 0 \\leq k \\leq p - 1 \\end{cases} , \\end{align*}"}
-{"id": "7136.png", "formula": "\\begin{align*} _ 0 { \\rm D } _ x ^ \\alpha u _ { \\rm e x } ( x ) = \\sum _ { n = 0 } ^ \\infty ( - 1 ) ^ n \\left ( \\mu F ^ n _ { \\nu , \\mu , \\alpha } ( x ) - \\nu F ^ n _ { \\nu - 1 , \\mu + 1 , \\alpha } ( x ) \\right ) , \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} l ( Z ) = c _ 2 ( E ( - b ) | _ S ) = ( c - a ) ( c - b ) ( c + a - e ) > 0 , \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{align*} x _ k = \\frac { \\prod _ { \\nu = b _ n + 1 } ^ { k } w _ { \\nu } } { \\lambda ^ { k - b _ n } } x _ { b _ n } \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} e ^ { - 2 i k _ { P ( n ) } l } = \\prod _ { i = 1 } ^ { n - 1 } s _ p ( k _ { P ( n ) } + k _ { P ( i ) } ) s _ p ( k _ { P ( n ) } - k _ { P ( i ) } ) \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} \\gamma ( t ) = e ^ { 2 t } i e _ 1 \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} \\frac { B _ { 2 } ^ { \\prime } } { B _ { 2 } } \\frac { q _ { 1 } } { q _ { 2 } } - \\frac { q _ { 1 } ^ { \\prime } } { q _ { 2 } } = \\frac { H } { 3 B _ { 2 } N } . \\end{align*}"}
-{"id": "3572.png", "formula": "\\begin{align*} \\omega = b \\vartheta . \\end{align*}"}
-{"id": "3609.png", "formula": "\\begin{align*} 2 \\lambda H ^ { ( 2 ) } = L _ 2 H ^ { ( 2 ) } - i a ^ \\dagger G . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} d _ { i + 1 } [ 1 ] \\circ d _ i = \\pi _ { i + 2 } [ 2 ] \\circ \\Delta _ { i + 1 } [ 1 ] \\circ \\pi _ { i + 1 } [ 1 ] \\circ \\Delta _ i \\end{align*}"}
-{"id": "7207.png", "formula": "\\begin{align*} R T ^ 2 - 2 T ^ 2 - R ^ 2 + R + 1 = 0 \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} \\left \\langle u , v \\right \\rangle _ { \\widehat { S } _ p } = \\frac { 1 } { p } \\int \\limits _ S \\sum _ { j = 0 } ^ { p - 1 } e ^ { \\frac { m j \\pi i } { p } } u ( \\zeta ) e ^ { \\frac { - m j \\pi i } { p } } \\overline { v ( \\zeta ) } \\ , d \\sigma ( \\zeta ) = \\left \\langle u , v \\right \\rangle _ S . \\end{align*}"}
-{"id": "9272.png", "formula": "\\begin{align*} \\Delta A = \\nabla _ { ( 2 ) } H + A * A * A . \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{align*} d X ^ { ( n + 1 ) } _ i ( t ) = \\sqrt { 2 ( ( X _ i ^ { ( n + 1 ) } ) ^ 2 ( t ) + 1 ) } d W ^ { ( n + 1 ) } _ i ( t ) + \\bigg [ \\left ( 2 - 2 ( n + 1 ) - 2 \\Re ( s ) \\right ) X _ i ^ { ( n + 1 ) } ( t ) + 2 \\Im ( s ) + \\\\ \\sum _ { j \\ne i } ^ { } \\frac { 2 ( ( X _ i ^ { ( n + 1 ) } ( t ) ) ^ 2 + 1 ) } { X _ i ^ { ( n + 1 ) } ( t ) - X _ j ^ { ( n + 1 ) } ( t ) } \\bigg ] d t , \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} \\varphi ( x ) = \\frac { q \\cdot x } { | x | ^ n } + O \\left ( \\frac { 1 } { | x | ^ { n - 1 + \\varepsilon } } \\right ) , \\nabla \\varphi ( x ) = \\nabla \\left ( \\frac { q \\cdot x } { | x | ^ n } \\right ) + O \\left ( \\frac { 1 } { | x | ^ { n + \\varepsilon } } \\right ) , \\end{align*}"}
-{"id": "9429.png", "formula": "\\begin{align*} \\partial _ t \\tau + v \\cdot \\nabla \\tau = \\tau \\nabla \\cdot v , \\ ; \\ ; \\ ; \\partial _ t b + ( v \\cdot \\nabla ) b = ( b \\cdot \\nabla ) v - \\tau \\nabla \\times d , \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle Q _ 3 ^ { k } ( \\overline { \\theta } _ { \\psi } ^ { k } ) = B ( \\mathbf { A } ^ { k - \\frac { 1 } { 2 } } ; ( \\overline { \\psi } ^ { k } - R _ h \\overline { \\psi } ^ { k } ) , \\overline { \\theta } _ { \\psi } ^ { k } ) + B ( \\mathbf { A } ^ { k - \\frac { 1 } { 2 } } ; ( \\psi ^ { k - \\frac { 1 } { 2 } } - \\overline { \\psi } ^ { k } ) , \\overline { \\theta } _ { \\psi } ^ { k } ) . } \\end{array} \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} D _ { n , k } ( 1 , y ( 1 - y ) ) & = \\frac { k } { 2 ^ { i + 1 } } \\ , \\Big [ \\displaystyle \\frac { ( u + 1 ) ^ { p ^ { l _ 1 } + p ^ { l _ 2 } + \\cdots + p ^ { l _ i } } ( 1 - u ) - ( u + 1 ) ( 1 - u ) ^ { p ^ { l _ 1 } + p ^ { l _ 2 } + \\cdots + p ^ { l _ i } } } { u } \\Big ] \\cr & + \\frac { 1 } { 2 ^ i } \\ , \\Big [ ( u + 1 ) ^ { p ^ { l _ 1 } + p ^ { l _ 2 } + \\cdots + p ^ { l _ i } } + ( 1 - u ) ^ { p ^ { l _ 1 } + p ^ { l _ 2 } + \\cdots + p ^ { l _ i } } \\Big ] \\cr \\end{align*}"}
-{"id": "1232.png", "formula": "\\begin{align*} Y _ { n } ^ { \\ell } = \\Pi _ { n - 1 } Y _ { n - 1 } ^ \\ell + \\Delta t S _ { \\Delta t } G _ { n - 1 } + S _ { \\Delta t } e ^ { \\tau A } \\bigl ( \\sigma '' ( X _ { n - 1 } ) . ( Z _ { n - 1 } ^ { 1 } , Z _ { n - 1 } ^ 2 ) \\bigr ) \\Delta W _ { n - 1 } , \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} d _ { } ( ( X , d ) , ( X ^ \\prime , d ^ \\prime ) ) : = \\inf \\{ d _ { \\cal H } ( \\phi ( X ) , \\phi ^ \\prime ( X ^ \\prime ) ) \\} \\end{align*}"}
-{"id": "9758.png", "formula": "\\begin{align*} E _ { \\epsilon } ( u _ { \\epsilon } ) - \\frac { 1 } { 2 } \\| h _ { \\epsilon } \\| _ { L ^ 2 } ^ 2 = o ( | \\log \\epsilon | ) \\epsilon \\to 0 . \\end{align*}"}
-{"id": "3499.png", "formula": "\\begin{align*} I _ { \\upsilon } ( q ; x ) = \\frac { \\left ( \\frac { x } { 2 } \\right ) ^ { \\upsilon } } { 2 ^ { q + \\frac { 1 } { 2 } } } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\Gamma ( 2 \\upsilon + 2 q + 2 n ) } { \\Gamma ( \\upsilon + q + n + \\frac { 1 } { 2 } ) \\ \\Gamma ( 2 \\upsilon + q + n + \\frac { 1 } { 2 } ) \\ n ! } \\left ( \\frac { x } { 2 } \\right ) ^ { n } . \\end{align*}"}
-{"id": "7564.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ 3 \\int _ { I ^ 2 } & \\epsilon ^ { i j k } \\frac { \\kappa ^ 2 } { 1 6 \\pi } e ^ { - \\kappa ^ 2 | \\hat { y } _ { k , \\bar { s } } ^ { \\bar { u } } - \\hat { y } _ { k , s } ^ { u } | ^ 2 / 1 6 } [ - \\zeta _ { \\kappa } ^ k ( \\hat { s } ) ] \\eta _ \\kappa ( \\hat { s } ) \\ y _ { i , s } ^ { u , \\prime } y _ { j , \\bar { s } } ^ { \\bar { u } , \\prime } \\ d \\hat { s } . \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} \\exp ( t A ) : = \\sum _ { n = 0 } ^ { \\infty } \\dfrac { t ^ n } { n ! } A ^ n , \\end{align*}"}
-{"id": "8728.png", "formula": "\\begin{align*} - \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log \\mu _ { n } ( O _ { 2 } ^ { 1 2 } ) \\geq \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { 2 } - \\frac { 1 } { m } \\log \\mu _ { m } ( O ^ { i } _ { 1 } ) + 2 \\xi \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{align*} \\ell ' = \\left \\lfloor \\frac { R _ 3 } { 3 ^ B \\| w \\| _ 1 } \\right \\rfloor \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} \\beta _ k ^ k ( 1 ) = \\sum _ { m | Q _ 1 } \\ell _ k ( m ) \\max _ { b \\bmod m } \\frac { | R _ 1 \\cap ( b \\bmod m ) \\bmod Q _ 1 | } { | R _ 1 \\bmod Q _ 1 | } . \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{gather*} w _ { m + 1 } ( x _ { 1 } , . . . , x _ { m - 1 } , \\cos ( \\alpha ) , \\cos ( \\beta ) | \\rho ) = \\\\ w _ { m } ( x _ { 1 } , . . . , x _ { m - 1 } , \\cos ( \\alpha + \\beta ) | \\rho ) w _ { m } ( x _ { 1 } , . . . , x _ { m - 1 } , \\cos ( \\alpha - \\beta ) | \\rho ) , \\end{gather*}"}
-{"id": "6905.png", "formula": "\\begin{align*} \\ , ^ { A B C } \\ , _ { a } D ^ { \\alpha } _ { t } f ( t ) = \\dfrac { B ( \\alpha ) } { 1 - \\alpha } \\int _ { a } ^ { t } f ' ( s ) E _ { \\alpha } \\left [ \\dfrac { - \\alpha } { 1 - \\alpha } ( t - s ) ^ { \\alpha } \\right ] d s , \\end{align*}"}
-{"id": "9952.png", "formula": "\\begin{align*} \\tilde { A } = T \\left [ \\begin{array} { c c } J & 0 \\\\ 0 & 0 \\end{array} \\right ] T ^ { - 1 } \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} X ^ 3 + A X + B = y ^ 2 , \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} [ w _ j , w _ k ] _ n \\bigg | _ a ^ b = 0 , j , k = 1 , 2 , \\dots , m \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = \\frac { \\alpha _ 2 } { \\gamma _ 6 } y _ 5 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\beta _ 1 } y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = - \\frac { \\alpha _ 3 } { \\beta _ 1 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 2 ] = y _ 4 , [ y _ 2 , y _ 3 ] = \\frac { \\gamma _ 2 } { \\gamma _ 6 } y _ 5 , [ y _ 3 , y _ 3 ] = y _ 5 . \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} \\mathcal { D } ( \\widetilde { S } ) \\ ; & = \\ ; \\overline { C ^ \\infty _ 0 ( \\mathbb { R } ^ + , \\mathbb { C } ^ 2 ) } ^ { \\| \\cdot \\| _ S } \\\\ \\mathcal { D } ( S ^ * ) \\ ; & = \\ ; \\{ \\psi \\in L ^ 2 ( \\mathbb { R } ^ + , \\mathbb { C } ^ 2 ) \\ , | \\ , \\widetilde { S } \\psi \\in L ^ 2 ( \\mathbb { R } ^ + , \\mathbb { C } ^ 2 ) \\} \\ , , \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} h _ 2 \\left ( I + \\begin{pmatrix} 0 & b \\\\ c & 0 \\end{pmatrix} p \\right ) h _ 2 ^ { - 1 } = I + \\frac { \\begin{pmatrix} c \\alpha \\beta + b \\alpha \\gamma & c \\beta ^ 2 - b \\alpha ^ 2 \\\\ - c \\alpha ^ 2 + b \\gamma ^ 2 & - c \\alpha \\beta - b \\alpha \\gamma \\end{pmatrix} p } { - \\alpha ^ 2 - \\beta \\gamma } . \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} \\sigma _ { s , t _ { 1 } } = \\left ( \\lambda _ { 1 } + t _ { 1 } , \\lambda _ { 2 } , \\lambda _ { 3 } , \\ldots , \\lambda _ { m + 1 } , \\lambda _ { m + 2 } \\pm t _ { 1 } , \\overline { \\lambda } _ { m + 1 } , \\ldots , \\overline { \\lambda } _ { 3 } , \\overline { \\lambda } _ { 2 } \\right ) \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} z = \\frac { s \\ , y _ F + 1 - y _ O } { 1 + s } , \\mbox { w i t h } s = \\frac { \\nu _ O W _ O } { \\nu _ F W _ F } . \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{align*} b ( a + b ) & = ( k - 1 ) a ^ 2 + ( n - k ) b ^ 2 + 1 \\\\ a ( a + b ) & = k a ^ 2 + ( n - k - 1 ) b ^ 2 + 1 \\end{align*}"}
-{"id": "10024.png", "formula": "\\begin{align*} \\sum \\limits _ { i = K } ^ { k - 1 } \\frac { 1 } { v _ i ^ \\beta } \\lesssim \\frac { 1 } { v _ { k - 1 } ^ \\beta } \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{align*} & T _ s = \\{ | x | = \\alpha _ s , | y | = b _ 1 \\} ; \\\\ & \\widetilde { T } _ s = \\{ | \\tilde { x } | = \\tilde { \\alpha } _ s , | \\tilde { y } | = \\tilde { b } _ 1 \\} . \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} \\begin{array} { l c l } \\star _ t \\tau _ 3 ( t ) & = & - \\frac { \\sqrt { 6 } \\ , \\big ( 5 y ( t ) ^ 2 + z ( t ) ^ 2 \\big ) } { 2 1 \\ , y ( t ) ^ 3 z ( t ) ^ 4 } f ^ { 3 4 5 6 } + \\frac { \\sqrt { 6 } \\ , \\big ( 3 y ( t ) ^ 2 - 5 z ( t ) ^ 2 \\big ) } { 4 2 \\ , y ( t ) ^ 3 z ( t ) ^ 4 } ( f ^ { 1 2 3 4 } + f ^ { 1 2 5 6 } ) \\\\ & & + \\frac { \\sqrt { 6 } \\ , \\big ( 2 y ( t ) ^ 2 - z ( t ) ^ 2 \\big ) } { 2 1 \\ , y ( t ) ^ 3 z ( t ) ^ 4 } ( - f ^ { 1 3 6 7 } - f ^ { 1 4 5 7 } + f ^ { 2 3 5 7 } - f ^ { 2 4 6 7 } ) , \\end{array} \\end{align*}"}
-{"id": "7889.png", "formula": "\\begin{align*} E ( u ) = \\int _ { \\R ^ n } \\phi ( \\nabla u ) + \\lambda \\| u - f \\| _ { L ^ 1 ( \\R ^ n ) } , \\end{align*}"}
-{"id": "6865.png", "formula": "\\begin{align*} q r ^ \\ast = 4 . \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{align*} c ( T ) = \\overline { \\textrm { d e n s } } \\ , \\bigl ( x , B ( 0 , \\varepsilon ) \\bigr ) \\quad \\textrm { f o r e v e r y } \\varepsilon > 0 . \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{align*} \\begin{aligned} ( \\omega + i \\partial \\bar \\partial w ) ^ { n + 1 } & = 0 \\qquad \\bar D \\times X \\\\ \\omega + i \\partial \\bar \\partial w | \\{ s \\} \\times X & > 0 s \\in \\bar D \\\\ w | \\partial D \\times X & = \\varphi \\end{aligned} \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} \\tilde \\omega _ 0 = \\omega _ { - n } , \\ ; \\dots , \\ ; \\tilde \\omega _ { n - 1 } = \\omega _ { - 1 } , \\ ; \\dots \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} | t - \\zeta _ j | \\geq C ( 1 + | t | ) , j = 1 , \\dots , n - 1 , t \\in \\mathbb R . \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} I ( A : B | M ) _ { \\hat { \\rho } _ { A B M } } = 0 \\ ; , \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} \\varrho ( \\alpha _ 1 \\alpha _ 2 \\dots \\alpha _ m ) = \\sum _ { k = 1 } ^ m \\alpha _ 1 \\alpha _ 2 \\dots \\alpha _ { k - 1 } \\otimes \\alpha _ { k + 1 } \\dots \\alpha _ m \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{gather*} \\begin{array} { l c l } \\tau _ 3 ( t ) & = & \\frac { \\sqrt { 6 } } { 7 \\ , y ( t ) ^ 5 } ( - f ^ { 1 3 5 } + f ^ { 1 4 6 } + f ^ { 2 3 6 } + f ^ { 2 4 5 } ) + \\frac { 4 \\sqrt { 6 } } { 2 1 y ( t ) ^ 5 } ( f ^ { 1 2 7 } + f ^ { 3 4 7 } + f ^ { 5 6 7 } ) , \\\\ \\end{array} \\end{gather*}"}
-{"id": "1091.png", "formula": "\\begin{align*} \\Big \\vert \\frac { 1 } { N ^ { s + 1 } } \\int \\limits _ { \\mathbf { w } \\in \\mathbb { R } ^ { s + 1 } } \\prod \\limits _ { j = 1 } ^ d g _ j ( \\psi _ j ( \\mathbf { w } ) + a _ j ) F ( \\mathbf { w } ) \\ , d \\mathbf { w } \\Big \\vert \\ll _ C \\rho ^ { - \\Omega ( 1 ) } \\sigma _ F ^ { - 1 } . \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} \\prod _ { s = b _ { l } + n + 1 } ^ { b _ { l + 1 } - 1 } | w _ { s } | = \\prod _ { s = b _ l + k _ { 0 } + 1 } ^ { b _ { l + 1 } - 1 } | w _ { s } | = 1 , \\end{align*}"}
-{"id": "8369.png", "formula": "\\begin{align*} \\hat { T } ^ { ( n ) } _ n = \\hat { T } ^ { ( n ) } ( \\hat { \\pi } ^ { ( n ) } _ n ) . \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{align*} \\overline { \\cal M } _ { 2 n - 1 } = \\left \\{ c \\in \\mathbb R ^ { 2 n - 1 } : H ( c ) \\succeq 0 \\right \\} , \\end{align*}"}
-{"id": "3725.png", "formula": "\\begin{align*} \\omega _ j = \\rho _ E ( ( - \\infty , E _ j ^ - ] ) , \\ ; \\ ; j = 1 , 2 , \\cdots , n . \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} \\begin{aligned} & \\| B _ 1 ( u , \\tilde u ) \\| _ X \\le C \\| u \\| _ X \\| \\tilde u \\| _ { X } , \\\\ & \\| B _ 2 ( u , \\theta ) \\| _ X \\le C \\| u \\| _ X \\| \\theta \\| _ Y , \\\\ & \\| B _ 3 ( u , \\tilde \\theta ) \\| _ Y \\le C \\| u \\| _ X \\| \\tilde \\theta \\| _ Y . \\end{aligned} \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} \\ L _ { 0 } u = \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert = 2 l } a _ { \\alpha } D ^ { \\alpha } u + A u + \\lambda u = f , \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{align*} X _ { \\tau _ u } = \\varphi _ x ^ { - 1 } ( x ' _ { \\tau _ u } ) = ( - \\lambda _ 1 - \\beta ^ N _ { \\tau _ u } , \\lambda _ 2 + \\beta ^ T _ { \\tau _ u } , \\lambda _ 3 + \\mathcal { A } _ { \\tau _ u } ) , \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} U : = \\sum _ { \\substack { \\delta X < n _ 1 ^ k , n _ 2 ^ k , n _ 3 ^ k , n _ 4 ^ k \\le X \\\\ \\vert n _ 1 ^ k + n _ 2 ^ k - n _ 3 ^ k - n _ 4 ^ k \\vert \\le 1 / \\tau } } 1 , \\textrm { a n d } V : = \\sum _ { \\substack { \\delta X < n _ 1 ^ k , n _ 2 ^ k , n _ 3 ^ k , n _ 4 ^ k \\le X \\\\ \\vert n _ 1 ^ k + n _ 2 ^ k - n _ 3 ^ k - n _ 4 ^ k \\vert > 1 / \\tau } } \\frac { 1 } { \\vert n _ 1 ^ k + n _ 2 ^ k - n _ 3 ^ k - n _ 4 ^ k \\vert } , \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} S _ D ^ { - 1 } \\Phi \\ ; = \\ ; \\Theta _ \\infty ^ { ( v _ \\infty ) } ( r ) \\ , v _ 0 ( r ) + \\Theta _ 0 ^ { ( v _ \\infty ) } ( r ) \\ , v _ \\infty ( r ) \\ , . \\end{align*}"}
-{"id": "3413.png", "formula": "\\begin{align*} \\psi _ t = \\frac { \\theta + \\left ( 1 - \\frac { 1 } { n } \\right ) \\phi _ t } { 1 + \\frac { 1 } { \\lambda } \\left ( 1 - \\frac { 1 } { n } \\right ) ^ 2 \\phi _ t } , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} a _ + ( \\alpha ) \\wedge b _ + ( \\beta ) = a _ + ( \\alpha ) > a _ + ( \\alpha ) \\wedge b _ - ( \\beta ) = a _ + ( \\alpha ) \\wedge b _ + ( \\beta ) \\wedge ( a _ - ( \\alpha ) \\vee b _ - ( \\beta ) ) \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k { k \\choose j } ^ 2 \\alpha ^ j = ( 1 - \\alpha ) ^ k P _ k \\left ( \\frac { 1 + \\alpha } { 1 - \\alpha } \\right ) \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{gather*} \\phi _ t + b \\phi _ x + \\phi _ { x x x } + \\phi _ { x y y } = f \\in C _ 0 ^ \\infty ( Q _ T ) , \\\\ \\phi \\big | _ { t = T } = 0 , \\phi \\big | _ { x = 0 } = \\phi _ x \\big | _ { x = 0 } = \\phi \\big | _ { x = R } = 0 \\end{gather*}"}
-{"id": "5223.png", "formula": "\\begin{align*} u _ \\pm = \\nabla \\left ( \\frac { p _ \\pm \\cdot x } { | x | ^ n } \\right ) + O \\left ( \\frac { 1 } { | x | ^ { n + \\varepsilon } } \\right ) , \\textrm { a s } | x | \\to \\infty . \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} \\sup _ { v '' \\in V } \\int _ { v \\in V , \\ \\min ( | v - v '' | , | v + v '' | ) \\ge \\eta ' } { d v \\over 1 - ( v \\cdot v '' ) ^ 2 } \\le \\left \\lbrace \\begin{array} { l } C ( 1 + | \\ln ( \\eta ' ) | ) \\textrm { w h e n } n = 3 , \\\\ C \\textrm { w h e n } n \\ge 4 . \\end{array} \\right . \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{align*} \\tilde { x } _ t = \\exp _ x \\left ( - B ^ N _ t N + B ^ T _ t T + t R _ t Z \\right ) \\end{align*}"}
-{"id": "6218.png", "formula": "\\begin{gather*} \\theta _ i = a + b t ^ { 2 i - d } + c t ^ { d - 2 i } , \\\\ \\theta _ i ^ * = a ^ * + b ^ * t ^ { 2 i - d } + c ^ * t ^ { d - 2 i } \\end{gather*}"}
-{"id": "8955.png", "formula": "\\begin{align*} & N ( q ) = - \\bigg ( \\cos \\theta + \\lambda ( q _ 2 - x _ 2 ) \\bigg ) X _ 1 ( q ) - \\bigg ( \\sin \\theta - \\lambda ( q _ 1 - x _ 1 ) \\bigg ) X _ 2 ( q ) , \\\\ & T ( q ) = - \\bigg ( \\sin \\theta - \\lambda ( q _ 1 - x _ 1 ) \\bigg ) X _ 1 ( q ) + \\bigg ( \\cos \\theta + \\lambda ( q _ 2 - x _ 2 ) \\bigg ) X _ 2 ( q ) , \\\\ & Z ( q ) = f ( q ) X _ 3 ( q ) , \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} \\tau _ { \\widehat { \\alpha } } ( \\mathsf { p } ) = \\tau _ { \\widehat { \\alpha } _ 1 - 1 } ( \\mathsf { p } ) \\cdots \\tau _ { \\widehat { \\alpha } _ { \\hat { \\ell } } - 1 } ( \\mathsf { p } ) . \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{align*} \\left [ \\Lambda _ N ^ { N + 1 } f \\right ] ( z _ 1 , \\cdots , z _ { N + 1 } ) = \\frac { N ! \\prod _ { i = 1 } ^ { N } ( z _ { i + 1 } - z _ i ) \\Delta _ N ( \\xi ) f ( \\xi ) } { \\Delta _ { N + 1 } ( z ) } \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} s _ { \\lambda , k } ( x _ 0 , x _ 1 , \\dots , x _ { m - 1 } ) = \\sum _ { \\mu \\preceq \\lambda ' } K _ { \\lambda ' \\mu } f _ { \\mu , k } ( x _ 0 , x _ 1 , \\dots , x _ { m - 1 } ) . \\end{align*}"}
-{"id": "469.png", "formula": "\\begin{align*} Y _ 1 = \\phi ^ { ( 1 ) } ( M , L ) : = M + L , Y _ 2 = \\phi ^ { ( 2 ) } ( M , L ) : = L . \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\int \\limits _ { 0 } ^ N \\sum \\limits _ { n \\in [ N ] } u ( n ) 1 _ { E _ { x , \\delta } } ( n ) \\ , d x & = \\frac { 1 } { N } \\sum \\limits _ { n \\in [ N ] } u ( n ) \\int \\limits _ { 0 } ^ N 1 _ { E _ { x , \\delta } } ( n ) \\ , d x \\\\ & = \\sum \\limits _ { n \\in [ N ] } u ( n ) 2 \\delta \\\\ & = 2 \\delta N ^ { d - m } T _ { F , G , N } ^ L ( 1 , \\dots , 1 ) \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} V _ { 1 } = \\{ 0 , 2 , 4 , 6 , \\ldots , 2 p ^ k - 2 \\} \\ \\ \\mbox { a n d } \\ V _ { 2 } = \\{ 1 , 3 , 5 , 7 , \\ldots , 2 p ^ k - 1 \\} . \\end{align*}"}
-{"id": "6969.png", "formula": "\\begin{align*} I _ \\Delta = ( \\{ x _ { j _ 1 } x _ { j _ 2 } \\cdots x _ { j _ r } : \\{ x _ { j _ 1 } , x _ { j _ 2 } , \\ldots , x _ { j _ r } \\} \\notin \\Delta \\} ) . \\end{align*}"}
-{"id": "7263.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 1 e _ 4 + \\alpha _ 2 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 3 e _ 3 + \\alpha _ 4 e _ 4 + \\alpha _ 5 e _ 5 , [ e _ 2 , e _ 1 ] = - \\alpha _ 3 e _ 3 + \\beta _ 1 e _ 4 + \\beta _ 2 e _ 5 , \\\\ [ e _ 2 , e _ 2 ] = \\beta _ 3 e _ 4 + \\beta _ 4 e _ 5 , [ e _ 1 , e _ 4 ] = \\beta _ 5 e _ 5 , [ e _ 2 , e _ 4 ] = \\beta _ 6 e _ 5 . \\end{align*}"}
-{"id": "8104.png", "formula": "\\begin{align*} L = a _ 0 ( t ) \\partial _ t ^ k + \\cdots + a _ k ( t ) y , a _ 0 , \\ldots , a _ k \\in \\C [ t ] , a _ 0 \\not \\equiv 0 . \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} \\sum _ { \\substack { H \\leq F _ r , \\\\ H \\not \\leq G _ r } } | H | ^ a = \\sum _ { \\substack { K \\leq F _ { r - 1 } } } p ^ { a \\ell } | F _ { r - 1 } / K | | K | ^ a = p ^ { a \\ell } | F _ { r - 1 } | \\sigma _ { a - 1 } ( F _ { r - 1 } ) . \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} g _ p ( \\zeta ) = d ^ n g _ n ( \\lambda _ { p , n } \\cdot \\zeta ) . \\end{align*}"}
-{"id": "3546.png", "formula": "\\begin{align*} \\dot { \\psi } _ { n + 1 } = - 2 ( u _ { n } + 2 v _ { n } ) \\rho _ { n } \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} 2 H _ { 0 } = \\frac { f _ { 0 } g ^ { \\prime \\prime } } { \\left [ 1 - \\left ( f _ { 0 } g ^ { \\prime } \\right ) ^ { 2 } \\right ] ^ { \\frac { 3 } { 2 } } } . \\end{align*}"}
-{"id": "1683.png", "formula": "\\begin{align*} \\mathbb { S } ( R ^ { \\bigstar } ) = \\frac { 2 } { d ^ { n + 1 } } \\Big [ \\frac { d _ 1 d _ 2 \\dots d _ n } { n + 1 } + \\frac { W ^ { ( n ) } _ 1 } { n } + \\dots + \\frac { W ^ { ( n ) } _ s } { n - s + 1 } + \\dots + \\frac { W ^ { ( n ) } _ { n - 1 } } { 2 } \\Big ] . \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\mathcal { C } _ n } f _ { \\vert \\pi \\vert } g _ { \\pi } = \\sum _ { p = 1 } ^ { n } f _ p \\sum _ { \\underset { k _ { i } \\ge 1 } { k _ { 1 } + \\dots + k _ { p } = n } } g _ { k _ { 1 } } \\dots g _ { k _ { p } } = \\sum _ { p = 1 } ^ { n } \\left ( \\sum _ { m = p } ^ { n } f _ m \\binom { m } { p } \\right ) \\sum _ { k _ { 1 } + \\dots + k _ { p } = n } g _ { k _ { 1 } } \\dots g _ { k _ { p } } . \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} u \\left ( 0 \\right ) = u _ { 0 } , \\end{align*}"}
-{"id": "6378.png", "formula": "\\begin{align*} \\frac { s _ \\mu s ^ * _ \\nu + ( - 1 ) ^ j \\alpha ( s _ \\mu s ^ * _ \\nu ) } { 2 } = \\begin{cases} s _ \\mu s ^ * _ \\nu & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} \\tfrac { 1 } { 3 } \\bigl ( 1 + 2 \\cos ( x ) \\bigr ) \\ , = \\ , 1 - \\tfrac { 1 } { 3 } x ^ 2 + O ( x ^ 4 ) \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} F = F ( x ) = F ( E , S ) . \\end{align*}"}
-{"id": "10022.png", "formula": "\\begin{align*} | S _ t h _ v ( x ) | \\lesssim \\frac { | t | } { v ^ { 4 + ( n - 1 ) ^ 2 / 2 ( n + 1 ) } } \\lesssim \\frac { | t | } { v ^ { 4 + n / 2 } } = \\frac { | t | } { v ^ \\beta } \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{gather*} \\theta ^ 0 = 2 \\varepsilon ^ { - 1 } , \\theta ^ \\infty _ 1 = - 2 \\varepsilon ^ { - 1 } , t _ 2 = \\varepsilon \\tilde { t } _ 2 , H _ { t _ 2 } = \\varepsilon ^ { - 1 } \\tilde { H } _ 2 , \\\\ q _ 2 = \\tilde { q } _ 2 , p _ 2 = \\tilde { p } _ 2 + \\frac { 1 } { \\varepsilon \\tilde { q } _ 2 } , Y = \\tilde { x } ^ { \\varepsilon ^ { - 1 } } \\tilde { Y } . \\end{gather*}"}
-{"id": "3049.png", "formula": "\\begin{align*} \\abs { z } ^ 2 - z _ j ^ 2 + 1 = 0 \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{align*} g \\left ( x , y , k \\right ) = 0 \\Leftrightarrow \\cos \\left ( k \\alpha ( x , y ) \\right ) = p \\left ( x , y , k \\right ) . \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} N ( P ^ T D ) N ^ { - 1 } = ( P _ 1 ^ T D _ 1 ) \\bigoplus ( P _ 1 ^ T D _ 1 ) \\bigoplus \\cdots \\bigoplus ( P _ 1 ^ T D _ 1 ) \\bigoplus \\cdots \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} k = d ^ { N ^ { O ( N ^ 2 ) } } . \\end{align*}"}
-{"id": "6625.png", "formula": "\\begin{align*} y ' + \\mathbf { V } ( r ) y \\ , = \\ , g \\ , , y \\ , : = \\ , \\mathbf { E } f \\ , , \\end{align*}"}
-{"id": "9782.png", "formula": "\\begin{align*} d \\alpha _ { \\epsilon } = d [ j ( u _ { \\epsilon } / | u _ { \\epsilon } | ) ] = 0 \\{ | u _ { \\epsilon } | \\geq \\frac { 1 } { 2 } \\} . \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} a _ k = a _ k ^ 1 + a _ k ^ 2 = \\bigl ( a _ { k } ^ { 1 , 1 } + a _ { k } ^ { 1 , 2 } + a _ { k } ^ { 1 , 3 } \\bigr ) + \\bigl ( a _ { k } ^ { 2 , 1 } + a _ { k } ^ { 2 , 2 } + a _ { k } ^ { 2 , 3 } \\bigr ) , \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} E ( t ) = \\frac { 1 } { 2 } \\int _ { \\Omega } | u _ t ( t , x ) | ^ 2 d x + { 1 \\over 2 } \\int _ { \\Omega } | \\nabla u ( t , x ) | ^ 2 \\ , d x . \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} \\xi ( r ) \\ , : = \\ , r ^ B \\varphi ^ - \\ ! ( r ) \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} \\partial \\tilde { B } _ \\delta ^ i = \\{ ( \\tilde { r } _ \\delta ^ i ( \\theta ) \\cos ( \\theta ) , \\ , \\tilde r _ \\delta ^ i ( \\theta ) \\sin ( \\theta ) ) : \\theta \\in [ 0 , 2 \\pi ) \\} , \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{align*} \\varphi _ 1 ' = l _ 1 z _ 1 ^ 2 + l _ 2 z _ 2 ^ 2 + \\cdots + l _ { n + 3 } z _ { n + 3 } ^ 2 = 0 \\ , \\ , { \\rm a n d } \\ , \\ , \\varphi _ 2 ' = z _ 1 ^ 2 + z _ 2 ^ 2 + \\cdots + z _ { n + 3 } ^ 2 = 0 , \\end{align*}"}
-{"id": "2519.png", "formula": "\\begin{align*} T \\left ( r , f \\right ) = N _ { 1 ) } \\left ( r , \\frac { 1 } { f } \\right ) + O \\left ( r ^ { \\rho - 1 + \\varepsilon } \\right ) + O \\left ( \\log r \\right ) , \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} \\displaystyle \\frac { \\displaystyle \\left ( k \\cdot s _ i + \\sum _ { t = 0 } ^ { i - 1 } b _ t \\cdot s _ t \\right ) - s _ i } { s _ { n - 1 } } \\leq \\frac { \\displaystyle \\left ( k \\cdot s _ { i + 1 } + \\sum _ { t = 0 } ^ { i - 1 } b _ t \\cdot s _ { t + 1 } \\right ) - s _ { i + 1 } } { s _ n } \\leq \\frac { \\lambda _ n - s _ { i + 1 } } { s _ n } . \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{align*} ( I _ n - U _ 0 ) ( U + I _ n ) = 0 . \\end{align*}"}
-{"id": "9690.png", "formula": "\\begin{align*} h ^ { r } _ P ( t ) \\ = \\ \\sum _ { i = 0 } ^ { d + r } h ^ { r } _ i t ^ i \\end{align*}"}
-{"id": "1423.png", "formula": "\\begin{align*} \\partial _ t \\nabla _ i ^ \\perp H _ j - \\nabla _ j \\nabla _ i ^ \\perp V = - \\nabla _ l ^ \\perp V \\nabla _ l \\nabla _ i ^ \\perp H _ j + \\nabla _ l \\nabla _ i ^ \\perp V \\nabla _ l ^ \\perp H _ j . \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} \\mu \\left ( G \\right ) = \\int _ { G } \\mathsf { d x d y } \\varphi ( \\mathsf { x ) } g ( \\mathsf { x } , T , \\mathsf { y } ) \\psi ( \\mathsf { y ) } \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} [ e _ 1 , e _ 2 ] = \\alpha _ 2 e _ 4 + \\alpha _ 3 e _ 5 , [ e _ 2 , e _ 1 ] = - \\alpha _ 2 e _ 4 + \\alpha _ 4 e _ 5 , [ e _ 3 , e _ 1 ] = \\beta _ 3 e _ 5 , [ e _ 3 , e _ 2 ] = \\beta _ 6 e _ 5 , [ e _ 3 , e _ 3 ] = \\beta _ 7 e _ 5 . \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} \\left \\lvert \\nabla _ { x } ( 2 \\tau - | x | - | x - e | ) \\right \\rvert = \\left \\lvert \\frac { x } { | x | } + \\frac { x - e } { | x - e | } \\right \\rvert = \\left \\lvert \\frac { | x - e | x + ( x - e ) | x | } { | x | | x - e | } \\right \\rvert . \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} X ^ \\sigma = C \\cup R _ 1 \\cup \\ldots \\cup R _ k \\cup \\{ p _ 1 , \\ldots , p _ n \\} . \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} \\| \\widetilde \\Phi _ j \\| _ { \\widetilde H ^ { k - 3 j } ( \\Sigma ) } \\leq c ( k , b ) \\Bigl ( \\| u _ 0 \\| _ { \\widetilde H ^ k ( \\Sigma ) } + \\sum _ { m = 0 } ^ { j - 1 } \\| \\partial _ t ^ m f \\big | _ { t = 0 } \\| _ { \\widetilde H ^ { k - 3 ( m + 1 ) } ( \\Sigma ) } \\Bigr ) . \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} 0 & = \\Delta _ { m + 1 } \\big ( - \\frac { 1 } { 2 } \\big ) - \\Delta _ { m + 1 } ( 0 ) = - \\frac { c } { 8 } - \\frac { 2 m - 3 } { 8 } c = \\frac { 1 - m } { 4 } c \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{gather*} \\frac { 1 } { u } \\frac { \\partial u } { \\partial t _ 1 } = - \\frac { 2 } { t _ 1 } \\big ( p _ 1 q _ 1 + p _ 2 q _ 2 + \\theta ^ \\infty _ 1 \\big ) , \\frac { 1 } { u } \\frac { \\partial u } { \\partial t _ 2 } = - 2 p _ 2 q _ 1 . \\end{gather*}"}
-{"id": "3746.png", "formula": "\\begin{align*} c _ { k } = \\sum _ { n = 1 } ^ { k } a _ { n } \\sum \\limits _ { m _ { 1 } + \\ldots + m _ { n } = k } b _ { m _ { 1 } } \\cdots b _ { m _ { n } } . \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{align*} p _ + ( \\lambda ) h p _ + ( \\lambda ) = O \\left ( t ^ { \\lambda _ 1 } ( \\log t ) ^ { \\lambda _ 2 } \\cdots \\left ( \\log ^ { ( n - 1 ) } t \\right ) ^ { \\lambda _ n } \\right ) \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} w _ { k + 1 } ^ i = \\gamma \\left ( 1 - \\omega \\lambda _ i \\right ) w _ k ^ i + ( 1 - \\gamma ) ( 1 - \\omega \\lambda _ i ) w _ { k - 1 } ^ i , i = 1 , 2 , \\dots , n . \\end{align*}"}
-{"id": "4101.png", "formula": "\\begin{align*} d s ( l \\otimes x \\otimes l ' ) = & l _ I \\otimes x \\otimes l ' - 1 \\otimes l _ I x \\otimes l ' + 1 \\otimes l _ I \\otimes x l ' , \\\\ s d ( l \\otimes x \\otimes l ' ) = & 1 \\otimes \\left ( l x \\right ) _ I \\otimes l ' - 1 \\otimes l _ I \\otimes x l ' . \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{align*} X = \\begin{pmatrix} c _ { 1 1 } ( X ) & c _ { 1 2 } ( X ) \\\\ c _ { 2 1 } ( X ) & - c _ { 1 1 } ( X ) \\end{pmatrix} . \\end{align*}"}
-{"id": "6661.png", "formula": "\\begin{align*} H ( m _ k , \\lambda m _ k ) & = H ( m _ k , \\delta _ k + n _ k ) = H ( 0 , \\delta _ k ) + A ( m _ k , n _ k ) \\\\ & = H ( 0 , \\delta _ k ) + A ( m _ k , m _ k \\lambda - \\delta _ k ) \\\\ & = H ( 0 , \\delta _ k ) + m _ k A ( 1 , \\lambda ) - A ( 0 , \\delta _ k ) . \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{align*} f ( y , v ) = 3 \\sqrt { 3 } y _ 0 ^ 5 v ^ 2 \\nu ( k , t ) _ 2 . \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} \\int \\nolimits _ { t _ { 0 } } ^ { t } E _ { \\mathbf { A } } ( s , x ) \\mathrm { d } s = 0 \\ , x \\in \\mathbb { R } ^ { d } \\ , \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} q ^ - = \\vert - \\rangle \\langle + \\vert , q ^ + = \\vert + \\rangle \\langle - \\vert , N _ q = \\vert - \\rangle \\langle + \\vert . \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{align*} g ( r _ 1 ) \\le f ( r _ 2 ) + ( r _ 1 - r _ 2 ) ^ 2 / 2 + ( r _ 1 - t _ f ) ^ 2 / 2 . = g ( r _ 2 ) + ( t _ f - r _ 1 ) ( r _ 2 - r _ 1 ) . \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} \\lfloor X \\rceil ( \\psi , \\bar \\psi ) & : = \\sum _ { l \\in \\Z \\setminus \\{ 0 \\} } \\lfloor X _ l \\rceil ( \\psi , \\bar \\psi ) e ^ { i l \\cdot } . \\end{align*}"}
-{"id": "9367.png", "formula": "\\begin{align*} ( E \\overset { \\wedge } { \\otimes } F ) ^ * = B ( E , F ^ * ) . \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} \\frac { n + 1 } { 2 } \\gamma _ { 1 } + \\frac { n - 1 } { 2 } \\gamma _ { 2 } & = n \\\\ \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} W : = \\{ ( s , v ) \\in ( 0 , \\tau _ + ( x _ 0 , y _ 0 ) ) \\times V \\ | \\ | G ( x _ 0 + s v _ 0 + \\tau _ + ( x _ 0 + s v _ 0 , v ) v , v ) | \\le ( { \\rm L i p } ( G ) + 1 ) \\eta \\} , \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} & \\left ( \\int _ 0 ^ \\infty \\left \\{ t ^ { 1 / q } f ^ * ( t ) \\right \\} ^ r \\frac { d t } { t } \\right ) ^ { 1 / r } < \\infty ( 1 \\leq r < \\infty ) , \\\\ & \\sup _ { t > 0 } \\ , t ^ { 1 / q } \\ , f ^ * ( t ) < \\infty ( r = \\infty ) . \\end{align*}"}
-{"id": "5088.png", "formula": "\\begin{align*} { u } ^ { } _ { b _ k } \\ , : = \\ , \\dfrac { b ^ { } _ k } { 2 } \\left ( \\begin{matrix} 1 & 0 \\\\ 1 & 1 \\end{matrix} \\right ) \\left ( \\begin{matrix} 1 \\\\ 1 \\end{matrix} \\right ) \\ , = \\ , \\dfrac { b ^ { } _ k } { 2 } \\left ( \\begin{matrix} 1 \\\\ 2 \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "1600.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\big ( \\deg _ { \\widetilde { X } } ( v _ { i } ) - 2 \\big ) & \\leq N \\big ( \\deg _ { \\widetilde { Y } } ( a ) - 2 \\big ) \\cdot \\big ( \\deg _ { \\widetilde { Z } } ( b ) - 2 \\big ) \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} V ( t , \\eta , b ) = \\inf _ { \\substack { \\int _ 0 ^ t f ( t - s ) \\sigma _ 0 u _ s d s \\ , \\ , = \\ , \\ , b - T ( t ) \\eta ( 0 ) \\\\ | u _ s | \\leq 1 } } \\int _ 0 ^ t \\frac { g } { 2 } \\left ( 1 - \\sqrt { 1 - u ^ 2 _ s } \\right ) d s . \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} \\d X ( t ) = \\psi ( X ( t - ) ) \\d L ( t ) , X ( 0 ) = x \\in \\R ^ { d } , \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} \\aligned & \\partial { \\mathcal M } ( L _ 1 , L _ 2 ; a , c ; 2 , E ) \\\\ & = \\bigcup _ { E _ 1 + E _ 2 = E } { \\mathcal M } ( L _ 1 , L _ 2 ; a , c ; 1 , E _ 1 ) \\times { \\mathcal M } ( L _ 1 , L _ 2 ; c , b ; 1 , E _ 2 ) , \\endaligned \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{align*} \\left | D _ 1 ^ a D _ 2 ^ b \\xi _ x ^ { e } \\right | \\ll \\frac { 1 } { R } , . \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} g _ { 1 } = D _ { 1 , \\lambda } \\left ( \\varepsilon \\right ) D _ { \\lambda } ^ { - 1 } \\left ( \\varepsilon \\right ) , g _ { 2 } = D _ { 2 , \\lambda } \\left ( \\varepsilon \\right ) D _ { \\lambda } ^ { - 1 } \\left ( \\varepsilon \\right ) , \\end{align*}"}
-{"id": "7488.png", "formula": "\\begin{align*} b m ^ * a : = ( a ^ * m b ^ * ) ^ * \\end{align*}"}
-{"id": "9871.png", "formula": "\\begin{align*} C _ 3 = 0 \\Leftrightarrow G ( \\rho ) \\rightarrow \\frac { 1 } { 4 \\pi r } \\rho \\rightarrow \\infty \\end{align*}"}
-{"id": "9428.png", "formula": "\\begin{align*} \\nabla \\cdot B = 0 . \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} \\left ( Q + \\lambda \\right ) ^ { - 1 } \\left ( X \\right ) = \\left ( Q + \\lambda \\right ) ^ { - 1 } \\left ( X , Y \\right ) \\times I \\left ( Y , X \\right ) . \\end{align*}"}
-{"id": "9304.png", "formula": "\\begin{align*} \\delta ( n ) & = 2 ( F _ 1 ( \\mathbf { 0 } ) - F _ { 1 } ^ * ) . \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{align*} q _ t ( x , y ) = - \\partial _ y \\int _ { - \\infty } ^ { x } p _ t ( y , z ) d z . \\end{align*}"}
-{"id": "9308.png", "formula": "\\begin{align*} \\varphi '' ( p ) = \\frac { - 1 } { \\ln ( 2 ) x ( 1 - x ) } + 8 , \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} f \\left ( g \\left ( z \\right ) \\right ) = - \\log \\left ( 1 - z \\right ) \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} B \\ ; : = \\ ; \\sqrt { 1 - \\nu ^ 2 } \\ , . \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} \\Psi ( 0 ) = \\frac { ( 1 - \\varrho _ a ) _ a ( \\epsilon - \\varrho _ a ) _ a } { a ! \\cdot ( \\epsilon ) _ a } \\cdot { } _ 4 F _ 3 \\bigg [ \\begin{matrix} \\varrho _ a & - a & \\delta - \\beta & \\delta - \\gamma \\\\ & \\delta & \\varrho _ a - a & 1 + \\varrho _ a - a - \\epsilon \\end{matrix} \\bigg | \\ , 1 \\bigg ] . \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{align*} \\eta _ a \\left ( \\sum _ { i = 1 } ^ r b _ i C ( \\mathsf { P } _ i ) \\right ) = \\sum _ { i = 1 } ^ r b _ i \\chi _ a ( \\mathsf { P } _ i ) = b _ a q ^ { ( \\alpha _ a - 2 \\rho , \\omega _ a ) } . \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N - 1 } | b _ k ^ 1 | \\le C \\Delta t ^ { \\frac 1 2 - 2 \\kappa } ( 1 + | x | _ { L ^ { \\max ( p , 2 q ) } } ) ^ { K + 1 } . \\end{align*}"}
-{"id": "3530.png", "formula": "\\begin{align*} D _ { z } ^ { \\mu , p } { f ( z ) } : = \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ { 0 } ^ { z } f ( t ) ( z - t ) ^ { \\alpha - 1 } e x p \\big ( \\frac { - p z ^ { 2 } } { t ( z - t ) } \\big ) d t \\end{align*}"}
-{"id": "8609.png", "formula": "\\begin{align*} D _ { n , k } ( 1 , y ( 1 - y ) ) & = k \\ , \\Big [ \\displaystyle \\frac { y ^ n ( 1 - y ) - y ( 1 - y ) ^ n } { 2 y - 1 } \\Big ] + D _ n ( 1 , y ( 1 - y ) ) , \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} \\tilde { u } \\left ( x , t \\right ) = \\left ( \\sqrt { \\alpha \\beta } \\sigma \\left ( t \\right ) \\right ) ^ { \\frac { n } { 2 } } u \\left ( \\sqrt { \\alpha \\beta } x \\sigma \\left ( t \\right ) , \\beta t \\sigma \\left ( t \\right ) \\right ) e ^ { \\eta } . \\end{align*}"}
-{"id": "3142.png", "formula": "\\begin{align*} \\displaystyle { C _ { i j k } = \\nabla _ { i } R _ { j k } - \\nabla _ { j } R _ { i k } - \\frac { 1 } { 2 ( n - 1 ) } \\big ( \\nabla _ { i } R g _ { j k } - \\nabla _ { j } R g _ { i k } ) . } \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\lambda _ { 2 } ( p , \\Omega ) ^ { \\frac 1 p } = \\Lambda _ 2 ( \\infty , \\Omega ) = \\frac { 1 } { \\rho _ { 2 , F } ( \\Omega ) } . \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} \\underset { \\epsilon , \\delta , \\kappa \\rightarrow \\infty } { l i m } \\mathbb { E } \\left | \\langle : X \\otimes X : , \\Psi _ { \\epsilon , \\delta , \\psi _ \\kappa } ^ f \\rangle - \\int _ { \\mathbb { R } ^ 2 } ^ { '' } \\widehat { \\psi } ( x + y ) | y | ^ { - \\gamma } Z _ G ( d x ) Z _ G ( d y ) \\right | ^ 2 = 0 , \\end{align*}"}
-{"id": "2049.png", "formula": "\\begin{gather*} \\frac { 1 } { z ^ { 3 / 2 } } \\begin{pmatrix} \\sqrt { t } & 0 \\\\ 0 & - \\sqrt { t } \\end{pmatrix} + \\frac { 1 } { z } \\begin{pmatrix} \\theta ^ \\infty _ 1 / 2 & 0 \\\\ 0 & \\theta ^ \\infty _ 1 / 2 \\end{pmatrix} \\end{gather*}"}
-{"id": "4181.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\pm \\infty } T \\left ( x , t \\right ) = 0 , \\ ; \\ ; \\lim _ { x \\rightarrow \\pm \\infty } q \\left ( x , t \\right ) = 0 , \\ ; \\ ; t > 0 . \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { n } ( d _ j r + \\frac { q ( d - d _ j ) } { d } + \\frac { d _ j t } { d } ) = \\sum _ { j = 0 } ^ { n } \\frac { q ^ j } { d ^ j } W ^ { ( n ) } _ j r ^ { n - j } + \\sum _ { c = 0 } ^ { n - 1 } g ^ { ( n ) } _ c ( q ) r ^ { c } \\end{align*}"}
-{"id": "3574.png", "formula": "\\begin{align*} \\phi = \\widehat { \\phi } _ 0 \\otimes 1 + \\widehat { \\phi } _ 1 \\otimes w ^ { \\alpha _ 1 } + \\dots + \\widehat { \\phi } _ { \\nu - 1 } \\otimes w ^ { \\alpha _ { \\nu - 1 } } , \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} I _ { n , d } : = | \\{ \\tau \\in S _ n \\mid \\tau ^ 2 = 1 , \\tau \\zeta ^ d = \\zeta ^ d \\tau \\} | \\end{align*}"}
-{"id": "6301.png", "formula": "\\begin{align*} [ M ] = \\sum _ { \\lambda \\in X ^ + } ( M : T ( \\lambda ) ) [ T ( \\lambda ) ] . \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\psi ( t ) = - \\Delta _ { t } \\psi ( t ) , d \\psi ( t ) \\ , = \\ , 0 , \\psi ( 0 ) = \\psi , \\end{align*}"}
-{"id": "1487.png", "formula": "\\begin{align*} P ( x ) = \\prod _ { k = 1 } ^ { n - 1 } ( x - \\zeta _ k ) ( x - \\overline { \\zeta } _ k ) = \\prod _ { k = 1 } ^ { n - 1 } \\big ( ( x - a _ k ) ^ 2 + b _ k ^ 2 \\big ) \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\lambda } ^ d | L _ n | = d ^ n E \\prod _ { i = 0 } ^ { n - 1 } [ \\frac { \\lambda \\rho ( S _ i , \\omega ) \\rho ( S _ { i + 1 } , \\omega ) } { 1 + \\lambda \\rho ( S _ i , \\omega ) \\rho ( S _ { i + 1 } , \\omega ) } ] . \\end{align*}"}
-{"id": "2989.png", "formula": "\\begin{align*} | v ^ { ( k ) } | \\ , \\cdot \\prod _ { i = 1 } ^ { \\Delta ^ { ( k ) } - 1 } | w ^ { ( k ) } _ i | = 2 ^ { \\ , \\delta ^ { ( k ) } - \\tau ^ { ( k ) } } > 2 ^ { \\ , 2 \\alpha \\Delta ^ { ( k ) } } \\ge C , \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} H _ s ( m , n ) = \\sum _ { t \\geq 0 } \\binom { m } { n t + s - 1 } , \\enskip s = 1 , . . . , n ; \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} { \\frac { \\kappa ^ 2 } { ( \\kappa ^ 2 + \\tau ^ 2 ) ^ { \\frac { 3 } { 2 } } } \\left ( \\frac { \\tau } { \\kappa } \\right ) ' = \\frac { \\kappa _ { \\gamma } } { \\kappa _ { \\gamma } } = \\mbox { c o n s t a n t } . } \\end{align*}"}
-{"id": "2570.png", "formula": "\\begin{align*} p _ { k , l , m } & = \\sum _ { h = 1 } ^ N \\omega _ h k _ h \\ ; \\pm 2 j l \\pm j ^ 2 \\pm a _ { l m } . \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} \\mathfrak { F } _ { t , k , q } ^ { ( \\chi _ { x } ^ { ( \\Omega ) } ( \\omega ) ) } \\left ( \\Lambda \\right ) = \\mathfrak { F } _ { t , k , q } ^ { ( \\omega ) } \\left ( \\Lambda + x \\right ) \\ , \\Lambda \\in \\mathcal { P } _ { f } ( \\mathfrak { L } ) , \\ x \\in \\mathbb { Z } ^ { d } \\ . \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{align*} \\frac { \\partial V ( x + t u , y + t v ) } { \\partial t } = \\langle \\partial _ x V ( x , y ) , u \\rangle + \\langle \\partial _ y V ( x , y ) , v \\rangle , \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{align*} [ T , [ T , \\cdots [ T , N ] ] ] = ( - 1 ) ^ { k - 1 } 2 ^ k \\lambda ^ { k - 1 } Z , [ T , [ T , \\cdots [ T , Z ] ] ] = ( - 2 \\lambda ) ^ { k } Z . \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} { \\cal X } ^ { n + 1 } = \\Ref [ { \\cal X } ^ { ' n } + \\Tilde { \\xi } ^ { f } ] . \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} \\frac { k ^ 3 + u ^ 3 } { k ^ 2 + u ^ 2 } = k + \\frac { u ^ 2 ( k - u ) } { k ^ 2 + u ^ 2 } , \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} \\sharp T _ K ( \\varepsilon , \\eta , d , B ) = \\frac { ( \\varepsilon - \\eta ) K ^ 2 } { 2 \\alpha ^ 2 d ^ 2 } B ^ { 2 - \\frac { 1 } { r } } + O _ { \\varepsilon , \\eta } \\left ( \\frac { K ^ 2 B ^ { 2 - \\frac { 1 } { r } } } { N d ^ 2 } \\right ) + O _ { \\sigma } \\left ( \\frac { K ^ \\sigma B ^ \\sigma N } { d ^ \\sigma } \\right ) . \\end{align*}"}
-{"id": "9834.png", "formula": "\\begin{align*} R ( x ) = R - x R _ { 2 1 } ^ { - 1 } , \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} \\beta ^ { u ' } \\alpha _ i ^ { u + p _ i - 1 } \\in K , \\ ; u ' = 0 , 1 , \\dots , u - 1 . \\end{align*}"}
-{"id": "3310.png", "formula": "\\begin{align*} \\int _ { y ' \\prec x ' } ^ { } d x ' \\int _ { y \\prec x } ^ { } d y \\prod _ { i = 1 } ^ { N } \\hat { m } ( y _ i ) q _ t ^ { N , N + 1 } \\left ( \\left ( x , y \\right ) , \\left ( x ' , y ' \\right ) \\right ) = \\int _ { y \\prec x } ^ { } \\prod _ { i = 1 } ^ { N } \\hat { m } ( y _ i ) d y \\int _ { y ' \\prec x ' } ^ { } d x ' q _ t ^ { N , N + 1 } \\left ( \\left ( x , y \\right ) , \\left ( x ' , y ' \\right ) \\right ) . \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{gather*} \\mathfrak { X } _ m ^ l ( [ 0 , T ] ) : = \\bigcap _ { k = 0 } ^ m C ^ k ( [ 0 , T ] ; H ^ { l + m - k } ( \\Omega ) ) , \\\\ \\mathfrak { Y } _ m ^ l ( [ 0 , T ] ) : = \\bigcap _ { k = 0 } ^ { [ m / 2 ] } C ^ k ( [ 0 , T ] ; H ^ { l + m - 2 k } ( \\Omega ) ) , \\quad \\ m , l = 0 , 1 , 2 , \\cdots , \\\\ \\mathfrak { Z } ( [ 0 , T ] ) : = C ^ 2 ( [ 0 , T ] ; H ^ 2 ( \\Omega ) ) . \\end{gather*}"}
-{"id": "4076.png", "formula": "\\begin{align*} X = X _ 0 + X ^ m + X ^ p \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} B = \\mathrm { E n d } ( V _ 1 ) \\times \\ldots \\times \\mathrm { E n d } ( V _ n ) \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{align*} A ( z ) = \\frac 1 z \\left ( A _ 0 + A _ 1 z + \\cdots \\right ) , \\end{align*}"}
-{"id": "924.png", "formula": "\\begin{align*} Q ^ 2 b _ 1 & = b _ 1 ^ 2 \\\\ Q ^ 4 b _ 1 & = b _ 3 + b _ 1 b _ 2 + b _ 1 ^ 3 \\\\ Q ^ 6 b _ 1 & = b _ 1 ^ 4 \\\\ Q ^ 8 b _ 1 & = b _ 5 + b _ 1 b _ 4 + b _ 2 b _ 3 + b _ 1 ^ 2 b _ 3 + b _ 1 b _ 2 ^ 2 + b _ 1 ^ 3 b _ 2 + b _ 1 ^ 5 \\\\ Q ^ { 1 0 } b _ 1 & = b _ 3 ^ 2 + b _ 1 ^ 2 b _ 2 ^ 2 + b _ 1 ^ 6 \\\\ Q ^ 6 b _ 2 & = b _ 5 + b _ 1 b _ 4 + b _ 2 b _ 3 + b _ 1 b _ 2 ^ 2 \\\\ Q ^ { 1 0 } b _ 2 & = b _ 1 ^ 2 b _ 5 + b _ 1 ^ 3 b _ 4 + b _ 1 ^ 2 b _ 2 b _ 3 + b _ 1 ^ 3 b _ 2 ^ 2 \\end{align*}"}
-{"id": "3309.png", "formula": "\\begin{align*} \\det \\left ( { A } _ t ( x , x ' ) _ { i j } \\right ) _ { i , j = 1 } ^ { N + 1 } & = \\mathcal { P } ^ { ( N + 1 ) } _ { s } ( t ) ( x , x ' ) , \\\\ \\det \\left ( { D } _ t ( y , y ' ) _ { i j } \\right ) _ { i , j = 1 } ^ { N } & = \\hat { \\mathcal { P } } ^ { ( N ) } _ { s } ( t ) ( y , y ' ) . \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{align*} a _ \\alpha = - \\cos ( \\frac { \\alpha \\pi } { 2 } ) . \\end{align*}"}
-{"id": "2445.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\int _ \\Omega \\hat { \\varphi } _ i ( t ) d t & = \\int _ \\Omega \\varphi ( t ) d t \\\\ & \\ge \\int _ \\Omega L ( t , x ( t ) , x ' ( t ) ) F \\left ( t , \\int _ 0 ^ t \\psi ( s ) d s \\right ) \\\\ & \\ge \\int _ \\Omega L ( t , x ( t ) , x ' ( t ) ) F \\left ( t , \\int _ 0 ^ t f ( s , x ( s ) , x ' ( s ) ) d s \\right ) . \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } v ( t ) \\in \\partial \\phi ( x ( t ) ) \\\\ \\lambda ( t ) \\dot x ( t ) + \\dot v ( t ) + v ( t ) + \\nabla \\psi ( x ( t ) ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} G _ { \\alpha } = F _ { \\alpha } \\ast H _ { \\alpha } , \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{align*} \\left ( \\mathsf { X } ^ { D B M } _ { \\infty } ( t ) ; t \\ge 0 \\right ) = \\left ( \\alpha ^ { \\pm } _ i ( t ) , \\gamma _ 1 ( t ) , \\gamma _ 2 ( t ) ; t \\ge 0 \\right ) \\end{align*}"}
-{"id": "1813.png", "formula": "\\begin{align*} \\underset { | \\xi | \\to \\infty } { \\lim } e ^ { - 2 t ( 1 + 4 \\pi ^ { 2 } | \\xi | ^ { 2 } ) } ( 1 + | \\xi | ) ^ { 2 M } = \\infty , t < 0 M \\in \\N . \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} I n d _ { \\mathbb { Z } / p , \\mathbb { F } _ p } S ( \\Delta ^ { r } ) ^ { \\perp } = \\langle b ^ { r ( p - 1 ) / 2 } \\rangle \\subseteq \\mathbb { F } _ p [ a , b ] / \\langle a ^ 2 \\rangle = H ^ { \\ast } ( B \\mathbb { Z } / p ; \\mathbb { F } _ p ) . \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} 1 < q < \\frac { s } { s - 1 } 3 q = \\frac { s q } { s - s q + q } . \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{align*} \\begin{pmatrix} a & b \\\\ c & d \\\\ \\end{pmatrix} & = \\pm \\textrm { i d } . \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} u _ N = \\mathcal { B } _ { \\epsilon , N } ( \\theta ) ^ { - 1 } \\big ( \\iota ^ * _ { N ( \\theta ) } - \\iota ^ * _ { N ( \\theta ) } M ( \\theta ) \\pi _ { R ( \\theta ) } \\mathcal { B } _ { \\epsilon } ( \\theta ) ^ { - 1 } \\big ) f , \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} \\sup _ { \\eta \\in X } | c ( x , \\eta ) - c ( x , \\eta _ u ) | = \\lambda \\rho ( x ) \\rho ( u ) \\leq \\lambda M ^ 2 , \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} - \\Delta _ 1 \\Psi = \\left ( - \\psi _ j '' \\right ) ) _ { j = 1 } ^ { | \\mathcal { E } | } . \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} \\Delta _ { x _ 1 , \\xi _ 2 , \\vartheta } u = \\frac { \\partial ^ 2 u } { \\partial x _ 1 ^ 2 } + \\frac { 1 } { \\varepsilon ^ 2 } \\frac { \\partial ^ 2 u } { \\partial \\xi _ 2 ^ 2 } + \\frac { 1 } { \\varepsilon \\xi _ 2 + 1 } \\frac { 1 } { \\varepsilon } \\frac { \\partial u } { \\partial \\xi _ 2 } + \\frac { 1 } { ( \\varepsilon \\xi _ 2 + 1 ) ^ 2 } \\frac { \\partial ^ 2 u } { \\partial \\vartheta ^ 2 } , \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} c P ( x ) = \\sum _ { k = 1 } ^ n r _ k \\prod _ { \\substack { j = 1 \\\\ j \\neq k } } ^ n ( x - x _ j ) ^ 2 . \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 2 \\pi } \\max \\left \\{ | f ( r e ^ { i \\theta } ) | ^ 2 \\ , : \\ , | f ( r e ^ { i \\theta } ) | ^ 2 ( 1 - r ^ 2 ) = \\lambda \\right \\} \\ , \\frac { d \\theta } { 2 \\pi } \\le \\| f \\| _ { H ^ 2 } ^ 2 \\end{align*}"}
-{"id": "7155.png", "formula": "\\begin{align*} \\rho ( a ) = \\begin{pmatrix} 1 & x \\\\ & \\\\ 0 & 1 \\end{pmatrix} ~ \\hbox { a n d } ~ \\rho ( b ) = \\begin{pmatrix} 1 & 0 \\\\ & \\\\ y & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} \\begin{aligned} & \\arg \\underset { { { { \\theta } } } } { \\max } & & Q ( { { { \\theta } } } | { { { \\theta } } } ^ { ( m ) } ) & & & \\sum _ { j = 1 } ^ 2 w _ j = 1 , w _ j \\geq 0 \\quad , j \\in \\{ 1 , 2 \\} . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "9299.png", "formula": "\\begin{align*} p _ { i j , k } = 0 \\ \\ \\mbox { i f } \\ \\ k \\not \\in \\{ i , j \\} , \\ \\ \\textrm { f o r a l l } \\ i , j , k \\in I . \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} \\left \\langle f , g \\right \\rangle _ { \\widehat { S } _ p } : = \\frac { 1 } { p } \\int \\limits _ S \\sum _ { j = 0 } ^ { p - 1 } f ( e ^ { \\frac { j \\pi i } { p } } \\zeta ) \\overline { g ( e ^ { \\frac { j \\pi i } { p } } \\zeta ) } \\ , d \\sigma ( \\zeta ) , \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{align*} \\binom { n } { s + 1 } p ^ { \\binom { s + 1 } { 2 } } \\leq n ^ { s + 1 } \\left ( p ^ { s / 2 } \\right ) ^ { s + 1 } \\ll n ^ { s + 1 } ( n ^ { - 1 } ) ^ { s + 1 } = 1 . \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} \\langle D \\sigma _ { e _ n , e _ m } ^ 2 ( x ) , h \\rangle = \\int _ { ( 0 , 1 ) } ( \\sigma ^ 2 ) ' \\bigl ( x ( \\xi ) \\bigr ) e _ n ( \\xi ) e _ m ( \\xi ) h ( \\xi ) d \\xi = \\langle D \\sigma _ { e _ n , h } ^ 2 ( x ) , e _ m \\rangle . \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{align*} E _ { \\alpha , \\beta } ^ \\rho ( z ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\rho ) _ k } { \\Gamma ( \\alpha k + \\beta ) } \\frac { z ^ k } { k ! } , \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{align*} \\dot x = \\Gamma v ( x ) , v \\in \\mathcal { K } \\ , . \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} \\left \\Vert \\mathbf { F } \\right \\Vert _ { 1 , \\mathfrak { L } } : = \\underset { y \\in \\mathfrak { L } } { \\sup } \\sum _ { x \\in \\mathfrak { L } } \\mathbf { F } \\left ( \\left \\vert x - y \\right \\vert \\right ) = \\sum _ { x \\in \\mathfrak { L } } \\mathbf { F } \\left ( \\left \\vert x \\right \\vert \\right ) < \\infty \\ . \\end{align*}"}
-{"id": "4760.png", "formula": "\\begin{align*} \\varphi ( u + I ) = \\left \\{ \\begin{array} { c l } c ^ { - 1 } _ { \\alpha } & \\mbox { i f $ u = B _ { \\alpha } $ f o r a n a r r o w $ \\alpha \\in Q _ 1 $ } , \\\\ 0 & \\mbox { o t h e r w i s e } , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2850.png", "formula": "\\begin{align*} T _ { ( \\alpha , k ) } \\bigl ( \\exp ( h ) \\bigr ) = \\widehat { T } _ { ( \\alpha , k ) } ( h ) \\cdot \\exp \\bigl ( h _ \\alpha ^ { ( 0 ) } \\bigr ) . \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} f ^ * _ { N , g r a d } : = \\mbox { m a x i m i z e } \\gamma \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} 1 + \\xi _ 1 + Q ^ 1 \\xi _ 1 + Q ^ 2 \\xi _ 1 + Q ^ 3 \\xi _ 1 + \\dots = ( 1 + \\xi _ 1 + \\xi _ 2 + \\dots ) ^ { - 1 } \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} H = \\{ r e ^ { i \\phi } \\mid r > 0 , \\phi \\in I \\} \\end{align*}"}
-{"id": "10039.png", "formula": "\\begin{align*} \\nabla _ X \\varphi Y = \\frac { 1 } { 2 } \\nabla _ X Y + \\frac { \\sqrt { 5 } } { 2 } \\nabla _ X J _ { \\varphi } Y = \\frac { 1 } { 2 } \\nabla _ X Y + \\frac { \\sqrt { 5 } } { 2 } J _ { \\varphi } \\nabla _ X Y = \\varphi ( \\nabla _ X Y ) , \\forall X , Y \\in { \\mathfrak X } ( M ) , \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} \\partial _ t v _ 5 = f _ 5 ( { \\bf v } ) \\partial _ t v _ 6 = f _ 6 ( { \\bf v } ) . \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} Z _ t ^ T : = \\eta _ { T , \\mathbf { 1 } _ { [ 0 , t ] } } , t \\geq 0 , \\end{align*}"}
-{"id": "5980.png", "formula": "\\begin{align*} u ^ { 3 } + ( \\alpha ^ { - 2 } \\beta ^ { 2 } + \\alpha ^ { - 1 } \\gamma ) u + ( \\alpha ^ { - 2 } \\beta \\gamma + \\alpha ^ { - 1 } \\theta ) = 0 . \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} \\frak m _ 1 ^ { b , b ' } ( x ) = \\sum _ { k , \\ell = 0 } ^ { \\infty } \\frak m _ { k + \\ell + 1 } ( \\underbrace { b , \\dots , b } _ { k } , x , \\underbrace { b ' , \\dots , b ' } _ { \\ell } ) . \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{align*} D _ m \\Big \\{ \\big [ v ( t ) , w ( t ) \\big ] ; \\big [ \\bar { v } ( t ) , \\bar { w } ( t ) \\big ] \\Big \\} = \\frac { \\left \\lVert \\left ( \\begin{array} { c } v ( t ) - \\bar { v } ( t ) \\\\ w ( t ) - \\bar { w } ( t ) \\end{array} \\right ) \\right \\rVert } { \\sqrt { \\mu ( t ) } } . \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} 2 \\pi \\chi ( \\Sigma ) = \\int _ { \\Sigma } K _ { g _ { \\phi _ { \\infty } } } d \\mathrm { v o l } _ { g _ { \\phi _ { \\infty } } } + \\int _ { S ^ 2 } K _ { g _ { \\vec { \\chi } } } d \\mathrm { v o l } _ { g _ { \\vec { \\chi } } } + \\int _ { S ^ 2 } K _ { g _ { \\vec { \\Psi } } } d \\mathrm { v o l } _ { g _ { \\vec { \\Psi } } } . \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{align*} d \\gamma _ 1 \\left ( X ^ { ( N ) } ; t \\right ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } d W ^ N _ i ( t ) = \\frac { 1 } { \\sqrt { N } } d \\beta ^ N ( t ) , \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{gather*} G _ 1 = \\begin{pmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ - t & 0 & 0 \\end{pmatrix} . \\end{gather*}"}
-{"id": "8226.png", "formula": "\\begin{align*} \\overline \\Lambda = \\Lambda _ 1 ( \\infty , \\Omega ^ + ) \\overline \\Lambda = \\Lambda _ 1 ( \\infty , \\Omega ^ - ) . \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} B _ s : = R \\otimes _ { R ^ s } R \\{ 1 \\} \\end{align*}"}
-{"id": "8909.png", "formula": "\\begin{align*} D ^ { ( \\alpha ) } z = A z + f , \\ 0 < \\alpha \\le 1 , \\end{align*}"}
-{"id": "8947.png", "formula": "\\begin{align*} \\frac { \\sqrt { 2 \\pi n } } { C n 2 ^ { n + 1 } } \\beta _ { i _ n , i _ n + 1 } ( S / I _ \\Delta ) = \\frac { \\sqrt { 2 \\pi } } { ( 1 - c ) 2 ^ n \\sqrt { n } } \\beta _ { i _ n , i _ n + 1 } ( S / I _ \\Delta ) \\sim e ^ { - a ^ 2 / 2 } . \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} \\alpha \\xi ^ { m } - \\beta \\eta ^ { m } + q _ { m } ( \\xi , \\eta ) = \\eta ^ { m } \\left ( \\frac { \\alpha \\xi ^ { m } } { \\eta ^ { m } } - \\beta + \\frac { q _ { m } ( \\xi , \\eta ) } { \\eta ^ { m } } \\right ) \\geq \\frac { - \\beta } { 2 } \\eta ^ { m } > c | \\eta | . \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Vert \\nabla { \\theta } _ { \\phi } ^ { m } \\Vert _ { \\mathbf { L } ^ { 2 } } ^ { 2 } \\leq C h ^ { 2 r } + C \\tau \\sum _ { k = 0 } ^ { m } \\Vert \\nabla { \\theta } _ { \\phi } ^ { k } \\Vert _ { \\mathbf { L } ^ { 2 } } ^ { 2 } + \\sum _ { k = 1 } ^ { m } J _ 2 ^ { k } . \\end{align*}"}
-{"id": "3531.png", "formula": "\\begin{align*} { } _ { \\mu , \\sigma \\ast } D _ { v , q ; z } ^ { \\alpha , \\eta , p } ( e ^ { z ^ { \\lambda } } ) : = \\exp ( \\frac { z ^ { \\eta + \\lambda + \\alpha } } { \\Gamma ( \\alpha ) } B _ { v , q } ^ { \\left ( \\mu , \\sigma \\right ) } ( \\eta + \\lambda , \\alpha - 1 ; p ) ) . \\end{align*}"}
-{"id": "9524.png", "formula": "\\begin{align*} \\Phi ( x ) ( \\omega ) = \\phi ( \\omega ) ( x ( \\omega ) ) \\end{align*}"}
-{"id": "3429.png", "formula": "\\begin{align*} \\min \\{ p ^ { d - 1 } , 4 p ^ { d - 2 } \\} & \\le L L ( Z B ) = p ^ { a _ 1 } + \\ldots + p ^ { a _ s } - s + 1 \\\\ & \\le p ^ { a _ 1 } + p ^ { a _ 2 } + p ^ { a _ 3 + \\ldots + a _ s } - 2 \\le p ^ { d - 2 } + 2 ( p - 1 ) . \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{align*} \\langle A ^ { - 1 } _ { \\varepsilon , \\delta } \\alpha , \\alpha \\rangle _ { \\omega _ { \\varepsilon } } \\leq \\sum _ { j = 1 } ^ { k } \\frac { | \\alpha _ { j } | ^ { 2 } } { \\varepsilon + \\lambda _ { j } } \\frac { \\varepsilon } { n - k } + \\sum _ { l = k + 1 } ^ { n } \\frac { | \\alpha _ { l } | ^ { 2 } } { \\varepsilon } \\frac { \\varepsilon } { n - k - 1 } \\leq ( n - k - 1 ) ^ { - 1 } | \\alpha | _ { i \\partial \\overline { \\partial } \\psi } ^ { 2 } . \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{align*} 4 z '' ( t ) + 4 ( \\gamma - 1 ) z ' ( t ) - e ^ { - t } z ' ( t ) - \\beta z ( t ) + | z ( t ) | ^ \\alpha z ( t ) = 0 . \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} { \\bf A } _ 1 = { \\bf G } \\left [ \\begin{array} { c c } { \\bf I } _ r & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf 0 } \\end{array} \\right ] { \\bf G } ' \\mbox { a n d } { \\bf A } _ 2 = { \\bf G } \\left [ \\begin{array} { c c } { \\bf D } _ r & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf 0 } \\end{array} \\right ] { \\bf G } ' \\end{align*}"}
-{"id": "8024.png", "formula": "\\begin{align*} H _ { f _ 0 , 2 } ( t ) = \\int _ { ( \\bar t + t ) / 2 } ^ { t - 1 } + \\int _ { t - 1 } ^ t = : H _ { f _ 0 , 2 1 } ( t ) + H _ { f _ 0 , 2 2 } ( t ) . \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} & ( - b ) _ k \\prod _ { \\substack { 1 \\leq j \\leq a - k \\\\ j \\neq a - b } } ( j - a + b ) = \\lim _ { z \\to 0 } \\frac { ( - b - z ) _ k ( 1 - a + b + z ) _ { a - k } } { z } \\\\ = & ( - 1 ) ^ a \\lim _ { z \\to 0 } \\frac { ( - b - z ) _ { a - k } ( 1 - a + b + z ) _ { k } } { z } = ( - 1 ) ^ a ( - b ) _ { a - k } \\prod _ { \\substack { 1 \\leq j \\leq k \\\\ j \\neq a - b } } ( j - a + b ) . \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} \\tilde { g } : = \\left ( d f \\right ) ^ { \\ast } g ^ { \\prime } . \\end{align*}"}
-{"id": "1612.png", "formula": "\\begin{align*} u _ 0 = u _ { f } + u _ { \\partial } + u _ { \\sf r m } \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} \\dfrac { d } { d t } ( g ( \\dot { c } , \\xi ) ) = \\left ( \\nabla g ^ { v } \\right ) ( \\dot { c } ^ { v } , \\xi ^ { v } ) + g ^ { v } ( ( \\nabla \\dot { c } ^ { v } , \\xi ^ { v } ) + g ^ { v } ( \\dot { c } ^ { v } , \\nabla \\xi ^ { v } ) . \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} \\mu ( E _ f ( \\lambda ) ) \\le \\frac { 1 } { \\lambda } \\int _ { 0 } ^ { 2 \\pi } \\max \\left \\{ | f ( r e ^ { i \\theta } ) | ^ 2 \\ , : \\ , | f ( r e ^ { i \\theta } ) | ^ 2 ( 1 - r ^ 2 ) = \\lambda \\right \\} \\ , \\frac { d \\theta } { 2 \\pi } - 1 . \\end{align*}"}
-{"id": "9182.png", "formula": "\\begin{align*} x ^ 4 + 2 a x ^ 3 - ( 2 / a ) x - 1 = 0 \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} ( 1 - p _ d ) ^ { A _ 2 } \\leq \\exp \\left ( - \\frac { C _ d A _ 2 } { n } \\right ) = e ^ { - C _ d ( r _ 1 + r _ 2 ) } \\exp \\left ( \\frac { C _ d } { n } ( r _ 1 ^ 2 + r _ 1 r _ 2 + r _ 2 ^ 2 ) \\right ) . \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } b _ { i } a _ { i j } + b _ { j } a _ { j i } - b _ { i } b _ { j } = 0 , \\\\ 1 \\leq i , \\ j \\leq s . \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\end{array} \\right . \\end{align*}"}
-{"id": "4790.png", "formula": "\\begin{align*} \\bar { \\mu } _ { \\alpha } & = \\alpha f ( \\alpha ) - c _ { \\bar { \\alpha } } A _ { \\bar { \\alpha } } & & \\mbox { i f $ \\alpha $ i s n o t a b o r d e r l o o p } , \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\\\ \\bar { \\mu } _ { \\alpha } & = \\alpha ^ 2 - c _ { \\bar { \\alpha } } A _ { \\bar { \\alpha } } - b _ i B _ { \\bar { \\alpha } } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! & & \\mbox { i f $ \\alpha $ i s a b o r d e r l o o p } . \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} ( q ; q ) _ \\infty = \\frac { 1 } { \\sqrt { - i \\tau } } e ^ { - \\frac { \\pi i \\tau } { 1 2 } - \\frac { \\pi i } { 1 2 \\tau } } \\ ( 1 + O \\ ( e ^ { - \\frac { 2 \\pi i } { \\tau } } \\ ) \\ ) . \\end{align*}"}
-{"id": "8990.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\int _ { \\Omega } \\langle A _ { \\varepsilon , \\delta + \\delta ' } ^ { - 1 } ( \\alpha - \\alpha _ { j } ) , \\alpha - \\alpha _ { j } \\rangle _ { \\omega _ { \\varepsilon } } e ^ { \\psi - ( \\delta - \\delta ' ) \\phi } d V _ { \\omega _ { \\varepsilon } } = 0 . \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} \\hat { \\Delta } ( M ) = \\bigcup _ { t \\ge 0 } \\bigl ( t \\Delta ( X ) \\times \\{ t \\} \\bigr ) . \\end{align*}"}
-{"id": "3290.png", "formula": "\\begin{align*} \\sum _ { k \\ne i } ^ { } \\frac { \\lambda _ k ^ 2 ( t ) } { \\lambda _ i ( t ) - \\lambda _ k ( t ) } = \\sum _ { k \\ne i } ^ { } \\frac { \\lambda _ i ^ 2 ( t ) } { \\lambda _ i ( t ) - \\lambda _ k ( t ) } - ( N - 2 ) \\lambda _ i ( t ) - \\sum _ { k = 1 } ^ { N } \\lambda _ k ( t ) , \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{align*} K \\phi = a _ 0 ( z ) \\otimes 1 + \\cdots + a _ m ( z ) \\otimes w ^ m \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} a u ^ 2 - b v ^ 2 = m \\lambda B \\leqslant u \\leqslant B . \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} \\partial ^ i ( \\phi ) ( g _ 0 , g _ 1 , \\cdots , g _ i ) & = g _ 0 \\cdot \\phi ( g _ 1 , \\cdots , g _ i ) \\\\ & + \\sum _ { j = 1 } ^ i ( - 1 ) ^ j \\phi ( g _ 0 , \\cdots , g _ { j - 2 } , g _ { j - 1 } g _ j , g _ { j + 1 } , \\cdots , g _ i ) \\\\ & + ( - 1 ) ^ { i + 1 } \\phi ( g _ 0 , \\cdots , g _ { i - 1 } ) . \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} \\omega = x ( \\lambda _ 1 + a ( x , y ) ) d y - y ( \\lambda _ 2 + b ( x , y ) ) d x , \\end{align*}"}
-{"id": "2398.png", "formula": "\\begin{align*} f ( t ) = e ^ { - C _ 1 t } \\bigg ( f ( 0 ) ^ { - ( q - 1 ) } + C _ 1 ^ { - 1 } C _ 2 e ^ { - C _ 1 ( q - 1 ) t } - C _ 1 ^ { - 1 } C _ 2 \\bigg ) ^ { - \\frac { 1 } { q - 1 } } . \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} \\Gamma ( Q _ \\infty ) = \\biggl \\{ k ( q ^ e + 1 ) + \\ell & q \\frac { q ^ e + 1 } { q + 1 } + m q ^ 3 \\mid \\\\ & 0 \\le \\ell \\le q , 0 \\le m < \\frac { q ^ e + 1 } { q + 1 } , k < 0 , k ( q ^ e + 1 ) + \\ell q \\frac { q ^ e + 1 } { q + 1 } + m q ^ 3 \\ge 0 \\biggr \\} , \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\left \\| ( \\mathcal { N } ( t ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) - \\hat { \\rho } _ { A M } \\right \\| _ 1 = 0 \\ ; . \\end{align*}"}
-{"id": "9515.png", "formula": "\\begin{align*} C _ M ( x ) = V _ { M , J } ( x ) \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{align*} Z _ 1 ^ { ( 1 ) } = [ x ] = H + 2 \\pi ^ * L , Z _ 2 ^ { ( 1 ) } = [ y ] = H + 3 \\pi ^ * L , Z _ 3 ^ { ( 1 ) } = [ s ] = \\pi ^ * S . \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} Q ( u ) = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } \\times \\mathbb { T } } u ^ 2 d x d y \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} B _ 2 ( u _ n , \\theta _ n ) ( 4 t ) = \\biggl ( \\int _ 0 ^ { t _ A } + \\int _ { t _ A } ^ { 4 t } \\biggr ) ( 4 t - s ) F ( 4 t - s ) * ( u _ n \\theta _ n ) ( s ) \\dd s \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} & d S _ { x } = \\frac { 1 } { 4 } \\sinh \\rho \\sin \\phi \\sqrt { \\cosh ^ { 2 } \\rho - \\cos ^ { 2 } \\phi } d \\theta d \\phi , \\\\ & \\ \\ \\cosh \\rho = r , \\ \\theta \\in ( 0 , 2 \\pi ) \\ a n d \\ \\phi \\in ( 0 , \\pi ) . \\end{align*}"}
-{"id": "9637.png", "formula": "\\begin{align*} | a _ { d - 1 , n } | > \\frac { 1 } { 2 } \\left ( \\frac { d } { d - 2 } \\right ) \\left ( 2 + \\sum _ { k = 2 } ^ { d - 2 } | a _ { k , n } | \\right ) . \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} D S _ k ( C , X ) & = ( \\partial _ { c _ 1 } S _ { k } , \\partial _ { x _ 1 ^ { ( 1 ) } } S _ { k } , . . . , \\partial _ { x _ 1 ^ { ( n ) } } S _ { k } , \\partial _ { c _ 2 } S _ { k } , . . . , \\partial _ { x _ k ^ { ( n ) } } S _ { k } ) \\\\ & = ( s ( x _ 1 ) , c _ 1 \\partial _ 1 s | _ { x = x _ 1 } , . . . , c _ 1 \\partial _ d s | _ { x = x _ 1 } , s ( x _ 2 ) , . . . , c _ 1 \\partial _ n s | _ { x = x _ k } ) . \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{align*} { \\cal M } _ { 2 n - 1 } : = \\left \\{ c = ( c _ 0 , \\dots , c _ { 2 n - 2 } ) \\in \\mathbb R ^ { 2 n - 1 } : c _ k = \\int _ { - \\infty } ^ \\infty t ^ k \\ , d \\sigma , \\ \\sigma \\in M _ { 2 n - 1 } \\right \\} , \\end{align*}"}
-{"id": "2778.png", "formula": "\\begin{align*} \\sharp S ( \\varepsilon , B ) = T _ { \\varepsilon , B } + O _ { \\varepsilon } ( B ^ { 1 - \\frac { 1 } { r } } ) = T _ { \\varepsilon , B } + O ( 1 ) . \\end{align*}"}
-{"id": "6400.png", "formula": "\\begin{align*} A = \\begin{bmatrix} \\pi & 0 \\\\ 0 & 1 \\end{bmatrix} \\begin{bmatrix} \\pi ^ { - 1 } a & b \\pi ^ { - 1 } \\pi ^ n \\\\ c & d \\end{bmatrix} \\end{align*}"}
-{"id": "1001.png", "formula": "\\begin{align*} Z = \\begin{pmatrix} Z _ 1 ^ { ( 1 ) } & Z _ 2 ^ { ( 1 ) } & Z _ 3 ^ { ( 1 ) } \\\\ Z _ 1 ^ { ( 2 ) } & Z _ 2 ^ { ( 2 ) } & \\end{pmatrix} = \\begin{pmatrix} H + 2 \\pi ^ * L & H + 3 \\pi ^ * L & \\pi ^ * S \\\\ f _ 1 ^ * ( H + 3 \\pi ^ * L ) - E _ 1 & E _ 1 & \\end{pmatrix} . \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} & E [ \\prod _ { i = \\tau + 1 } ^ { n - 1 } F ( S _ i , \\widehat { { S } } _ i ; S _ { i + 1 } , \\widehat { S } _ { i + 1 } ) ] \\\\ = & E \\prod _ { i = \\tau + 1 } ^ { n - 1 } [ \\frac { \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } { 1 + \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } ] E \\prod _ { i = \\tau + 1 } ^ { n - 1 } [ \\frac { \\lambda \\rho ( \\widehat { S } _ i ) \\rho ( \\widehat { S } _ { i + 1 } ) } { 1 + \\lambda \\rho ( \\widehat { S } _ i ) \\rho ( \\widehat { S } _ { i + 1 } ) } ] . \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} \\frac { | H | \\cdot | C _ \\mu | } { N ! } \\Phi \\left [ \\prod _ { k = 1 } ^ { N } \\left ( \\sum _ { \\{ i : d _ i | k \\} } \\epsilon _ i ^ { \\frac { k } { d _ i } } d _ i \\right ) ^ { \\mu _ k } \\right ] \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} \\sigma ( F ) = ( V _ { y _ 1 } , \\dots , V _ { y _ { 2 ^ n } } ) , \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} \\iint u _ { x x y } ^ 2 \\ , d x d y \\leq \\iint ( u _ { x x x } ^ 2 + u _ { x y y } ^ 2 ) \\ , d x d y + \\int _ 0 ^ L u ^ 2 _ { x y y } \\big | _ { x = 0 } \\ , d y + c \\iint u _ { x x } ^ 2 \\ , d x d y . \\end{align*}"}
-{"id": "9468.png", "formula": "\\begin{align*} s & = ( s _ 1 , s _ 2 , \\ldots , s _ n ) = ( s _ 1 ^ p q , \\ldots , s _ { k - 1 } ^ p q , q , s _ 2 ^ q , \\ldots , s _ { n _ q } ^ q ) , \\\\ t & = ( t _ 1 , t _ 2 , \\ldots , t _ n ) = ( t _ 1 ^ p , \\ldots , t _ { k - 1 } ^ p , p , p t _ 2 ^ q , \\ldots , p t _ { n _ q } ^ q ) \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} { g } ' ( x ) = \\frac { { Q } ' _ 1 ( x ) { S } ( x ) - { Q } _ 1 ( x ) { S } ' ( x ) } { { S } ( x ) ^ 2 } = \\frac { P ( x ) } { { S } ( x ) ^ 2 } , \\end{align*}"}
-{"id": "9858.png", "formula": "\\begin{align*} \\boldsymbol { v } ( \\boldsymbol { x } ) = \\boldsymbol { \\nabla } \\times \\boldsymbol { \\psi } ( \\boldsymbol { x } ) - \\boldsymbol { \\nabla } \\phi ( \\boldsymbol { x } ) \\mbox { w h e r e } \\boldsymbol { \\nabla } \\cdot \\boldsymbol { \\psi } ( \\boldsymbol { x } ) = 0 \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{align*} { \\mathcal J } _ { H , K , N } ( t ) : = \\begin{cases} \\chi _ { \\{ t = 0 \\} } & N = 1 , K > 0 \\\\ \\chi _ { \\{ H t \\geq 0 } \\} & N = 1 , K \\leq 0 \\\\ \\left ( { \\rm c } _ \\delta ( t ) + \\frac { H } { N - 1 } { \\rm s } _ \\delta ( t ) \\right ) _ + ^ { N - 1 } & N \\in ( 1 , \\infty ) \\\\ \\end{cases} ~ . \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} \\varphi ( t ) = \\frac { 1 } { y ( t ) } \\ , ( e ^ { 1 2 7 } + e ^ { 3 4 7 } + e ^ { 5 6 7 } ) + y ( t ) ^ 3 \\ , ( e ^ { 1 3 5 } - e ^ { 1 4 6 } - e ^ { 2 3 6 } - e ^ { 2 4 5 } ) , t \\in \\left ( - \\infty , \\ , \\frac { 3 } { 5 } \\right ) , \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} ( \\log \\theta ) ' = \\frac { \\theta ' } { \\theta } , \\end{align*}"}
-{"id": "9617.png", "formula": "\\begin{align*} & \\aligned \\widetilde { \\mathcal W } { } ^ i _ { j m n } = R ^ i _ { j m n } - \\delta ^ i _ { [ m } \\zeta _ { j n ] } - \\frac 1 2 \\mathcal F ^ i _ { j m | n } + \\frac 1 2 \\mathcal F ^ i _ { j n | m } , \\endaligned \\\\ & \\aligned \\widetilde { W } { } ^ i _ { j m n } & = R ^ i _ { j m n } + \\frac 1 { N - 1 } \\big ( \\delta ^ i _ m R _ { j n } - \\delta ^ i _ n R _ { j m } \\big ) - \\frac 1 2 \\mathcal F ^ i _ { j m | n } + \\frac 1 2 \\mathcal F ^ i _ { j n | m } . \\endaligned \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} \\frac { k ^ 3 + u ^ 3 } { k } = k ^ 2 + \\frac { u ^ 3 } { k } . \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} S _ \\beta \\ ; & : = \\ ; S ^ * \\upharpoonright \\mathcal { D } ( S _ \\beta ) \\\\ \\mathcal { D } ( S _ \\beta ) \\ ; & : = \\ ; \\left \\{ g = f + c ( \\beta S _ D ^ { - 1 } \\Phi + \\Phi ) \\left | \\ ! \\begin{array} { c } f \\in \\mathcal { D } ( \\overline { S } ) \\\\ c \\in \\mathbb { C } \\end{array} \\right . \\ ! \\ ! \\right \\} . \\end{align*}"}
-{"id": "6266.png", "formula": "\\begin{align*} \\pi _ { P ( \\theta ) } \\phi = \\langle \\phi , e ^ { \\i \\langle \\theta , \\cdot \\rangle _ { \\C ^ d } } \\rangle e ^ { \\i \\langle \\theta , \\cdot \\rangle _ { \\C ^ d } } - \\sum _ { z \\in \\Z ^ d \\backslash \\{ 0 \\} } \\frac { ( \\theta + 2 \\pi z ) ( \\theta + 2 \\pi z ) ^ T } { \\vert \\theta + 2 \\pi z \\vert ^ 2 } c ^ { ( z ) } _ \\phi e ^ { \\i \\langle \\theta + 2 \\pi z , \\cdot \\rangle _ { \\C ^ d } } , \\end{align*}"}
-{"id": "8672.png", "formula": "\\begin{align*} L w = 4 s ^ 2 w '' + 4 \\gamma s w ' - \\beta w \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{gather*} \\| E _ { y _ { 0 } } \\pi - \\widehat { \\mathfrak I } ^ { B ^ x _ 1 } _ { y _ { - 1 } } ( E _ { y _ { 0 } } \\pi ) \\| _ { L ^ \\infty ( B _ 1 ^ x ) } \\leq C _ T \\widehat q ^ m \\| \\pi \\| _ { L ^ \\infty ( B _ 0 ^ x ) } \\pi \\in { \\mathcal Q } _ m , \\\\ C _ T : = \\frac { 2 d } { \\rho - 1 } ( 1 + \\Lambda _ m ) ^ d \\exp \\left ( \\gamma \\left ( \\frac { \\rho + 1 / \\rho } { 2 } + 1 \\right ) \\right ) . \\end{gather*}"}
-{"id": "4109.png", "formula": "\\begin{align*} \\rho ( T ) = 1 - \\sum _ { \\emptyset \\neq S _ 1 \\subset T } \\pi _ { S _ 1 } + \\sum _ { \\substack { \\emptyset \\neq S _ 1 , S _ 2 \\subset T \\\\ S _ 1 < S _ 2 } } \\pi _ { S _ 1 } \\pi _ { S _ 2 } - \\sum _ { \\substack { \\emptyset \\neq S _ 1 , S _ 2 , S _ 3 \\subset T \\\\ S _ 1 < S _ 2 < S _ 3 } } \\pi _ { S _ 1 } \\pi _ { S _ 2 } \\pi _ { S _ 3 } + . . . . \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} \\| ( T - A ) f _ { k } \\| < \\varepsilon \\quad { \\rm a n d } \\| ( T - A ) ^ { * } f _ { k } \\| < \\varepsilon \\qquad \\hbox { f o r $ k = - r , \\dots , r $ } , \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} \\| w ( t ) \\| _ \\infty = O ( t ^ { - 1 / 2 } ) \\qquad \\mbox { a s $ t \\to \\infty $ } . \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} \\left ( \\sum _ { j _ { 1 } = 1 } ^ { \\infty } \\left ( \\sum _ { j _ { 2 } = 1 } ^ { \\infty } \\left ( \\sum _ { j _ { 3 } = 1 } ^ { \\infty } \\left \\| T \\left ( x _ { \\mathbf { j } } \\right ) \\right \\| ^ { s _ { 3 } } \\right ) ^ { \\frac { s _ { 2 } } { s _ { 3 } } } \\right ) ^ { \\frac { s _ { 1 } } { s _ { 2 } } } \\right ) ^ { \\frac { 1 } { s _ { 1 } } } \\leq C \\prod _ { k = 1 } ^ { 3 } \\left \\| x ^ { k } \\right \\| _ { w , q _ { k } } , \\end{align*}"}
-{"id": "2856.png", "formula": "\\begin{align*} H _ { ( x _ 1 , x _ 2 ) } \\diamond \\Phi ( x _ 1 , x _ 2 ; z _ 1 , z _ 2 ) = ( z _ 1 ^ 2 + z _ 2 ^ 2 ) \\cdot \\Phi ( x _ 1 , x _ 2 ; z _ 1 , z _ 2 ) . \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} 4 R R _ 0 + ( R _ 0 ) ^ 2 = - 4 R ^ 2 + \\Gamma ^ 2 = 8 - 4 n . \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} T _ { \\C } S ^ n = T S ^ n \\otimes _ { \\R } \\C \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{align*} \\theta ( \\mathsf { N } _ { m } ^ { n } ) _ { j } ^ { i } & = q ^ { ( 2 \\rho , \\lambda _ { m } - \\lambda _ { n } ) } q ^ { ( 2 \\rho , \\lambda _ { i } - \\lambda _ { j } ) } ( \\mathsf { N } _ { m } ^ { n } ) _ { j } ^ { i } \\\\ & = q ^ { ( 2 \\rho , \\lambda _ { m } - \\lambda _ { n } ) } \\pi ( K _ { 2 \\rho } ) _ { i } ^ { i } ( \\mathsf { N } _ { m } ^ { n } ) _ { j } ^ { i } \\pi ( K _ { 2 \\rho } ^ { - 1 } ) _ { j } ^ { j } . \\end{align*}"}
-{"id": "9266.png", "formula": "\\begin{align*} K = \\frac 1 2 ( | \\vec { H } | ^ 2 - | A | ^ 2 ) \\ , . \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} \\left \\Vert F \\left ( u , \\upsilon \\right ) \\right \\Vert _ { L ^ { q } } \\leq C \\left \\Vert u \\right \\Vert _ { L ^ { r } } \\left \\Vert \\nabla \\upsilon \\right \\Vert _ { L ^ { s } } \\frac { 1 } { q } = \\frac { 1 } { r } + \\frac { 1 } { s } . \\end{align*}"}
-{"id": "8397.png", "formula": "\\begin{align*} \\hat { T } ^ { ( n ) } ( \\hat { \\pi } ^ { ( n ) } _ n ) = \\hat { T } ^ { ( n ) } ( Q _ 1 ) + { t } ^ { ( n ) } ( e ) + \\hat { T } ^ { ( n ) } ( Q _ 2 ) \\end{align*}"}
-{"id": "8559.png", "formula": "\\begin{align*} \\vec { \\gamma } _ 1 ( \\phi , p ) = \\vec { \\gamma } _ 1 ( \\vec { \\Psi } , p ) \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} \\mathcal { N } = \\prod _ { v \\in T } U _ v ^ \\ell \\times \\prod _ { v \\in S } U _ v \\cdot ( L _ v ^ * ) ^ \\ell \\times \\prod _ { v \\not \\in S \\cup T } U _ v . \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{align*} z _ i ^ { x ' } - z _ i ^ { x } \\approx \\Big ( v ^ { x } _ i \\sum _ { j = 1 } ^ { n + 3 } \\frac { 1 } { \\alpha _ { i } - \\lambda _ j } \\Big ) \\epsilon _ 0 , \\end{align*}"}
-{"id": "2868.png", "formula": "\\begin{align*} t _ { n + 1 } ( u , v ) : = - v \\frac { \\partial t _ n } { \\partial u } ( u , v ) + u \\frac { \\partial t _ n } { \\partial v } ( u , v ) + v t _ n ( u , v ) \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} & H ( Y _ j | Y _ i ) = H ( Y _ j | Y _ i = y _ i ) \\\\ = & \\frac { d + 1 } { 2 } \\cdot \\frac { 1 } { d } \\log \\frac { 2 d } { d + 1 } + \\frac { d - 1 } { 2 } \\cdot \\frac { 1 } { d } \\log d \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} E ^ n _ 1 & = O ( 1 ) \\int _ { I _ 1 } { \\omega _ { 1 - \\alpha } } ( x _ { n + 1 } - s ) d s = O ( h ) ( x _ { n + 1 } - \\xi ) ^ { - \\alpha } , { \\rm f o r ~ s o m e } ~ ~ \\xi \\in I _ 1 . \\end{align*}"}
-{"id": "9320.png", "formula": "\\begin{align*} ( n , d ) = ( 5 8 , 1 0 ) , ( 7 4 , 1 4 ) ( 9 8 , 1 8 ) . \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 3 } { x ^ 2 } + \\frac { A _ 2 } { x } + A _ 1 + A _ 0 x \\right ) Y , \\\\ \\frac { \\partial Y } { \\partial t _ 1 } = \\left ( x B _ { 1 1 } + B _ { 1 0 } \\right ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = \\left ( - \\frac { A _ 3 } { t _ 2 x } + B _ { 2 0 } \\right ) Y , \\end{gather*}"}
-{"id": "9729.png", "formula": "\\begin{align*} \\lim _ m \\| x + y _ m \\| _ \\infty = \\| x \\| _ \\infty + \\lim _ m \\| y _ m \\| _ \\infty \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , [ L _ { - q + 1 } , \\ , A _ 1 ] ] = [ L _ { - q + 1 } , \\ , [ L _ { - 2 } , \\ , A _ 1 ] ] = 0 , \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} \\lim _ { x \\to b ^ - } [ a _ k ( x ) f ^ { ( k ) } ] ^ { ( k - j ) } = 0 \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} D _ { j } = \\prod \\limits _ { k = 1 } ^ { n } \\left [ 2 ^ { j _ { k } } , \\left . 2 ^ { j _ { k } + 1 } \\right ) \\right . \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{align*} \\Lambda _ N ^ { \\infty } ( \\omega , d x ) = \\left ( \\prod _ { k = 1 } ^ { N - 1 } \\frac { 1 } { k ! } \\right ) \\det \\left ( \\phi _ { \\omega } ^ { ( j - 1 ) } ( x _ { N + 1 - i } ) \\right ) ^ N _ { i , j = 1 } \\Delta _ N ( x ) d x , \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} \\pi _ 1 ( g ^ 2 ) = ( 1 , 2 , 5 , 7 , \\dots , 2 i - 1 , \\dots , n , 3 , 4 , 6 , \\dots , 2 j , \\dots , n - 1 ) \\end{align*}"}
-{"id": "9824.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\circ } ( \\mathbf { z } ^ { \\lambda } ) = \\prod _ { \\alpha ^ \\vee \\in \\Phi ^ { \\vee } _ + } ( 1 - v \\mathbf { z } ^ { - \\alpha ^ \\vee } ) \\chi _ { \\lambda } ( \\mathbf { z } ) . \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} ( 1 - p _ u ) ^ { n - 1 } \\leq \\mathbb { P } ( \\# { \\cal E } _ i = 1 ) \\leq ( 1 - p _ d ) ^ { n - 1 } . \\end{align*}"}
-{"id": "9923.png", "formula": "\\begin{align*} \\mathcal { X } ^ s = \\{ ( \\varphi , \\Gamma , \\Lambda ) \\in H ^ s \\times H ^ s _ \\times H ^ s _ { } \\} \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} [ 2 ] _ F ( \\alpha ) = \\alpha \\cdot \\langle 2 \\rangle _ F ( \\alpha ) , \\end{align*}"}
-{"id": "9899.png", "formula": "\\begin{align*} \\bar F _ i ( \\rho ) : = F _ { i } ( \\rho ) - \\beta \\log \\rho _ { i } \\ . \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{align*} u _ t \\ , + \\ : \\mbox { d i v } \\ ; \\ ! \\mbox { \\boldmath $ f $ } ( x , t , u ) \\ , + \\ : \\mbox { d i v } \\ , \\mbox { \\boldmath $ g $ } ( t , u ) \\ : = \\ ; \\mu ( t ) \\ ; \\ ! \\ ; \\ ! \\mbox { d i v } \\ , ( \\ ; \\ ! | \\ ; \\ ! \\nabla u \\ , | ^ { \\ : \\ ! p - 2 } \\ , \\nabla u \\ ; \\ ! ) , \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} A _ { n , j , k } ( t , z ) & = 2 ^ { 2 j + 1 } t ^ { j + 1 } ( t + 1 - z ( t - 1 ) ) ^ { k - j } ( t + 1 + z ( t - 1 ) ) ^ { n - j - k + 1 } \\\\ & = 2 ^ { 2 j + 1 } t ^ { n + 1 } \\left ( \\frac { 1 } { t } \\right ) ^ { j + 1 } \\left ( 1 + \\frac { 1 } { t } - z \\left ( 1 - \\frac { 1 } { t } \\right ) \\right ) ^ { k - j } \\left ( 1 + \\frac { 1 } { t } + z \\left ( 1 - \\frac { 1 } { t } \\right ) \\right ) ^ { n - j - k - 1 } \\\\ & = t ^ { n + 1 } A _ { n , j , k } ( 1 / t , - z ) , \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} c _ i = c _ i ( T _ X ) , \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} c _ 2 ( S ) = \\chi _ { \\rm t o p } ( S ) & = \\chi _ { \\rm t o p } ( S - R - R _ 0 ) + \\chi _ { \\rm t o p } ( R ) + \\chi _ { \\rm t o p } ( R _ 0 ) \\\\ & = n \\cdot \\chi _ { \\rm t o p } ( \\SS - \\delta ) + ( - 2 ) + ( - 2 ) ( n - 2 ) \\\\ & = 3 n - 2 - 2 ( n - 2 ) = n + 2 . \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} \\Vert \\Xi ^ * ( \\mathbf { e _ i } ^ * - \\lambda \\mathbf { e _ j } ^ * - L ^ * ( P ^ * \\pi _ { m - u } ^ * ( \\varphi ) ) ) \\Vert _ \\infty = O ( \\eta ) . \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{align*} f \\left ( z \\right ) f \\left ( z + c _ { 1 } \\right ) - q \\left ( z \\right ) = p _ { 1 } \\left ( z \\right ) e ^ { \\alpha \\left ( z \\right ) } \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{align*} f ( k , x ) = e ^ { i k x } + \\int _ x ^ \\infty K ( x , t ) e ^ { i k t } d t , x \\ge 0 , k \\in \\overline { \\mathbb { C } } ^ + , \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} v ^ 2 & = v ( k , l ) ^ 2 + 2 v ( k , l ) v ' + v '^ { 2 } \\\\ & = - 2 - 2 \\sum _ { k \\leq i \\leq l } b _ i + b _ k + b _ l + 2 \\sum _ { k < i < l } b _ i + v '^ { 2 } \\\\ & = - 2 - b _ k - b _ l + v '^ 2 . \\end{align*}"}
-{"id": "9862.png", "formula": "\\begin{align*} \\sigma \\frac { \\pi } { h } = 1 \\Leftrightarrow \\sigma = \\frac { h } { \\pi } \\Rightarrow s = \\frac { h k } { \\pi } \\rho = \\frac { \\pi r } { h } \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{gather*} \\frac { q X _ { i , i + 1 } X _ { i + 1 , i + 2 } - q ^ { - 1 } X _ { i + 1 , i + 2 } X _ { i , i + 1 } } { q - q ^ { - 1 } } = 1 ( i \\in \\mathbb { Z } _ 4 ) , \\\\ X _ { i , i + 1 } ^ 3 X _ { i + 2 , i + 3 } - [ 3 ] _ q X _ { i , i + 1 } ^ 2 X _ { i + 2 , i + 3 } X _ { i , i + 1 } + [ 3 ] _ q X _ { i , i + 1 } X _ { i + 2 , i + 3 } X _ { i , i + 1 } ^ 2 \\\\ - X _ { i + 2 , i + 3 } X _ { i , i + 1 } ^ 3 = 0 ( i \\in \\mathbb { Z } _ 4 ) . \\end{gather*}"}
-{"id": "9838.png", "formula": "\\begin{align*} \\mathfrak { T } _ i \\Delta ^ \\ast ( \\phi ) = \\Delta ^ \\ast ( T _ i ^ \\ast \\phi ) . \\end{align*}"}
-{"id": "5159.png", "formula": "\\begin{align*} \\sum _ { k _ { 1 } + \\dots + k _ { p } = n } \\binom { n } { k _ { 1 } , \\dots , k _ { p } } \\frac { 1 } { \\left ( k _ { 1 } + 1 \\right ) \\dots \\left ( k _ { p } + 1 \\right ) } = \\frac { n ! p ! } { \\left ( n + p \\right ) ! } \\left \\{ \\begin{array} { c } n + p \\\\ p \\end{array} \\right \\} . \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} ( K _ g ) \\ , g \\otimes \\partial \\phi = O ( | z | ^ { \\theta _ 0 - 1 } ) \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} \\tau _ { - 2 } ( \\mathsf { H } ^ j ) = - \\delta _ { j , 3 } \\ , , \\tau _ { - 1 } ( \\gamma ) = 0 \\ , . \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} R e s ^ { \\mathcal { G } } _ { \\mathcal { G } ^ \\prime } ( f ) = g + I n f ^ { \\mathcal { G } ^ \\prime } _ { G ^ \\prime } ( \\widetilde h ) , \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} \\iota ( x ) : = { } & \\pi I _ 0 ( x ) ; & \\kappa ( x ) : = { } & K _ 0 ( | x | ) ; \\\\ \\iota _ + ( x ) : = { } & \\iota ( x ) e ^ { - x } \\mathbb 1 _ { ( 0 , \\infty ) } ( x ) ; & \\kappa _ + ( x ) : = { } & \\kappa ( x ) e ^ { - x } ; \\\\ \\iota _ - ( x ) : = { } & \\iota ( x ) e ^ x \\mathbb 1 _ { ( - \\infty , 0 ) } ( x ) ; & \\kappa _ - ( x ) : = { } & \\kappa ( x ) e ^ { x } , \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{align*} g ( \\bar \\eta , t _ f ( \\bar \\eta _ 1 ) ) & = R ; \\\\ z ( \\bar \\eta , t _ f ( \\bar \\eta _ 1 ) ) & = R ^ { - 1 } . ( i y _ 1 + ( \\bar \\eta _ 5 + i \\bar \\eta _ 6 ) ) . \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} u _ 1 = \\frac { m _ 1 m _ 2 } { m _ 1 + m _ 2 } , u _ 2 = 0 , u _ 3 = \\frac { m _ 3 m _ 4 } { m _ 3 + m _ 4 } \\end{align*}"}
-{"id": "9632.png", "formula": "\\begin{align*} D ^ { ( j ) } _ n = \\frac { H ^ { ( j ) } _ { n + 1 } H ^ { ( j + 1 ) } _ { n - 1 } } { H ^ { ( j ) } _ { n } H ^ { ( j + 1 ) } _ { n } } = e ^ { ( j ) } _ n . \\end{align*}"}
-{"id": "35.png", "formula": "\\begin{align*} \\mathcal { I } _ { m } : = { \\textstyle \\bigcup \\nolimits _ { \\substack { \\underline { n } \\in S \\\\ \\left \\Vert \\underline { n } \\right \\Vert = m } } } \\mathcal { I } _ { \\underline { n } } , \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} w & 0 \\\\ 0 & w \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) ^ { p - 1 } = I - \\frac { 1 } { w } \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} ( \\textstyle { \\frac 1 2 } + \\alpha + \\beta + \\gamma ) ( \\alpha + \\beta + \\gamma ) + ( \\textstyle { \\frac 1 2 } + \\alpha ) ( \\alpha ) + ( \\textstyle { \\frac 1 2 } + \\beta ) ( \\beta ) + ( \\textstyle { \\frac 1 2 } + \\gamma ) ( \\gamma ) & \\\\ - ( \\textstyle { \\frac 1 2 } + \\alpha + \\beta ) ( \\alpha + \\beta ) - ( \\textstyle { \\frac 1 2 } + \\alpha + \\gamma ) ( \\alpha + \\gamma ) - ( \\textstyle { \\frac 1 2 } + \\beta + \\gamma ) ( \\beta + \\gamma ) & = 0 . \\end{align*}"}
-{"id": "5095.png", "formula": "\\begin{align*} B ' _ k = \\frac { 1 } { | C | } \\sum _ { i = 0 } ^ n P _ k ( i ) B _ i k \\in [ 0 . . n ] . \\end{align*}"}
-{"id": "2533.png", "formula": "\\begin{align*} + \\left ( 2 N ^ { \\prime } - A _ { 2 } N + 2 M \\right ) \\left ( \\left ( f ^ { \\prime } \\right ) ^ { 2 } + f f ^ { \\prime \\prime } \\right ) + N \\left ( 3 f ^ { \\prime } f ^ { \\prime \\prime } + f f ^ { \\prime \\prime \\prime } \\right ) = - B _ { 2 } ^ { \\prime } . \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{gather*} A _ \\infty : = - ( A _ 0 + A _ 1 + A _ t ) = \\operatorname { d i a g } \\big ( \\theta ^ \\infty _ 1 , \\theta ^ \\infty _ 2 , \\theta ^ \\infty _ 3 \\big ) . \\end{gather*}"}
-{"id": "7071.png", "formula": "\\begin{align*} \\tilde { u } \\left ( x , t \\right ) = \\left ( \\sqrt { \\alpha \\beta } \\mu \\left ( t \\right ) \\right ) ^ { \\frac { n } { 2 } } u \\left ( \\sqrt { \\alpha \\beta } x \\mu \\left ( t \\right ) , \\beta t \\mu \\left ( t \\right ) \\right ) e ^ { \\eta } . \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} f _ { \\beta } f _ { \\gamma } = \\sum _ { \\alpha } c _ { \\beta , \\gamma } ^ { \\alpha } f _ { \\alpha } . \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} f ^ * : = \\inf f _ 0 ( x ) \\textrm { s u c h t h a t } x \\in S . \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} \\ge \\int _ { S ^ { n - 1 } } \\left ( \\int _ 0 ^ { \\rho _ K ( \\theta ) } r ^ { n - 1 } f ( r \\theta ) d r \\right ) d \\theta = \\int _ K f = \\mu ( K ) , \\end{align*}"}
-{"id": "9483.png", "formula": "\\begin{align*} c \\le p ^ { h } / ( q ^ r 4 ^ q ) , \\mbox { w h e r e } h : = 2 ^ r \\binom { q + r } { r } \\mbox { a n d } q : = 2 f \\cdot f ! . \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{align*} u ( t ) = e ^ { ( t - \\varepsilon ) \\Delta } u ( \\varepsilon ) + \\int _ \\varepsilon ^ t e ^ { ( t - s ) \\Delta } | u ( s ) | ^ \\alpha u ( s ) \\ , d s \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} W _ 1 ^ 2 = \\frac { 1 } { ( P \\log P ) ^ 2 } , \\forall k \\geq 2 , W _ k ^ 2 = \\frac { k } { ( P \\log P ) ^ k } , \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} Z ^ * ( \\Gamma ) = \\sum _ \\omega z ^ * _ \\omega ( \\Gamma ) , \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} \\begin{cases} \\beta _ 1 \\gamma _ 1 = \\alpha _ 1 \\gamma _ 3 & \\beta _ 1 \\gamma _ 2 = \\alpha _ 1 \\gamma _ 4 \\\\ \\beta _ 4 \\gamma _ 1 = \\alpha _ 4 \\gamma _ 3 & \\beta _ 4 \\gamma _ 2 = \\alpha _ 4 \\gamma _ 4 \\end{cases} \\end{align*}"}
-{"id": "8731.png", "formula": "\\begin{align*} ( - 1 ) ^ { s - 1 } \\ , a _ n = \\sum _ { k = 0 } ^ { n } { n \\choose k } ( - 1 ) ^ k a _ k ; \\ ; \\ ; s = 1 \\ ; \\ ; \\mbox { o r } \\ ; \\ ; s = 2 . \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} \\langle : X \\otimes X : , \\Psi _ { \\epsilon , \\delta , \\psi _ \\kappa } ^ f \\rangle = \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { R } ^ 2 } ^ { '' } \\widehat { \\psi _ \\kappa } ( x + y ) \\widehat { f _ \\epsilon } ( y ) \\widehat { h _ \\delta } ( y ) Z _ G ( d x ) Z _ G ( d y ) . \\end{align*}"}
-{"id": "3705.png", "formula": "\\begin{align*} E = [ E _ 0 ^ + , E _ 0 ^ - ] \\setminus \\bigcup _ { j = 1 } ^ n ( E _ j ^ - , E _ j ^ + ) . \\end{align*}"}
-{"id": "4428.png", "formula": "\\begin{align*} \\Sigma = \\bigcup _ { h \\in H } ~ \\bigcup _ { i \\in \\mathbb Z } ~ \\bigcup _ { j = 1 } ^ r \\left \\{ h h _ j ^ i h ^ { - 1 } \\right \\} . \\end{align*}"}
-{"id": "2939.png", "formula": "\\begin{align*} T ^ { \\ , n } \\Bigl ( \\sum _ { s = 1 } ^ { j _ { m } } z _ { s } \\Bigr ) - x _ { l } & = T ^ { \\ , n _ { j _ { m } } + k ' \\textrm { p e r } ( x _ { l } ) } \\ , \\Bigl ( \\sum _ { s = 1 } ^ { j _ { m } } z _ { s } \\Bigr ) - x _ { l } \\qquad \\hbox { b y ( i ) } \\\\ & = T ^ { \\ , n _ { j _ { m } } + k ' \\textrm { p e r } ( x _ { l } ) } \\ , z _ { j _ { m } } - T ^ { \\ , k ' \\textrm { p e r } ( x _ { l } ) } \\ , \\Bigl ( x _ { l } - \\sum _ { s = 1 } ^ { j _ { m } - 1 } z _ { s } \\Bigr ) \\qquad \\hbox { b y ( i v ) . } \\end{align*}"}
-{"id": "4341.png", "formula": "\\begin{align*} \\bar { B } ( z , M ) = \\{ x \\in X : \\sigma ( z , x ) \\leq M \\} \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} { } _ { 0 } ^ { c } \\mathrm { D } _ { t } ^ { \\alpha } T = \\mathcal { D } \\frac { \\partial ^ { 2 } T } { \\partial x ^ { 2 } } , \\end{align*}"}
-{"id": "9350.png", "formula": "\\begin{align*} \\gamma & = \\gamma _ 1 ^ s \\oplus \\gamma _ 1 ^ \\ell \\oplus \\gamma _ 2 ^ s \\oplus \\gamma _ 2 ^ \\ell \\oplus \\ldots \\oplus \\gamma _ { J } ^ s \\oplus \\gamma _ { J } ^ \\ell \\oplus \\gamma _ { J + 1 } ^ s , \\\\ \\sigma & = \\sigma _ 1 ^ s \\oplus \\sigma _ 1 ^ \\ell \\oplus \\sigma _ 2 ^ s \\oplus \\sigma _ 2 ^ \\ell \\oplus \\ldots \\oplus \\sigma _ { J } ^ s \\oplus \\sigma _ { J } ^ \\ell \\oplus \\sigma _ { J + 1 } ^ s . \\end{align*}"}
-{"id": "2625.png", "formula": "\\begin{align*} L _ { \\omega } u : = L u - { \\rm i } \\ , \\omega B u - \\omega ^ 2 u . \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{align*} - \\Delta u = e ^ { 2 \\lambda } K _ g , \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} 2 5 F _ { m k } { } ^ 4 = L _ { 4 m k } + ( - 1 ) ^ { m k - 1 } 4 L _ { 2 m k } + 6 \\ , . \\end{align*}"}
-{"id": "643.png", "formula": "\\begin{align*} u ( x ) = q ( x '' ) \\ , \\left ( x _ n + \\sqrt { x _ n ^ 2 + x _ { n + 1 } ^ 2 } \\right ) ^ s + c \\ , \\Psi _ 1 ( x _ n , x _ { n + 1 } ) ; \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{align*} I = \\{ \\alpha ^ { x n } \\rightarrow \\beta _ 1 ^ { x n } \\mid n \\in N \\} = \\{ \\alpha ^ { n x } \\rightarrow \\beta _ 1 ^ { n x } \\mid n \\in N \\} \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} \\tilde { F } \\left ( x , t \\right ) = \\left ( \\sqrt { \\alpha \\beta } \\sigma \\left ( t \\right ) \\right ) ^ { \\frac { n } { 2 } + 2 } \\left ( \\sqrt { \\alpha \\beta } x \\sigma \\left ( t \\right ) , \\beta t \\sigma \\left ( t \\right ) \\right ) . \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi ( t ) & = f ^ { 1 2 7 } - f ^ { 3 4 7 } - f ^ { 5 6 7 } + f ^ { 1 3 5 } - f ^ { 1 4 6 } + f ^ { 2 3 6 } + f ^ { 2 4 5 } . \\end{aligned} \\end{align*}"}
-{"id": "9414.png", "formula": "\\begin{align*} \\phi ( t _ 1 , t _ 2 , \\ldots , t _ n ) = \\int _ { \\Sigma } a _ 1 ( t , t _ 1 ) a _ 2 ( t , t _ 2 ) \\cdots a _ n ( t , t _ n ) \\ , \\mu ( t ) \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} \\partial _ t ^ 2 u - \\Delta u = Q ( \\partial u , \\partial ^ 2 u ) . \\end{align*}"}
-{"id": "6445.png", "formula": "\\begin{align*} \\begin{cases} U _ r v _ k = ( z + \\frac { k } { r } + \\frac 1 2 ) v _ { k + r } \\\\ L _ r v _ k = ( z + w + 2 \\frac { k } { r } ) v _ { k } \\\\ D _ r v _ k = ( w + \\frac { k } { r } - \\frac 1 2 ) v _ { k - r } \\\\ \\end{cases} \\end{align*}"}
-{"id": "9227.png", "formula": "\\begin{align*} M ^ { \\delta , \\delta ' } = \\frac { c _ { \\delta , \\delta ' } } { X _ 1 ^ \\delta } P _ 0 ^ { { \\rm B E S } ( \\delta + \\delta ' ) } , \\delta , \\delta ' > 0 , \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{align*} \\Q ( \\sqrt { \\lambda } ) = \\Q ( \\sqrt { u ^ { 2 n } } ) = \\Q ( u ^ n ) = \\Q ( \\lambda ) . \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} | x _ { \\alpha } | \\wedge y = | x _ { \\alpha } | \\wedge ( y - a + a ) \\leq | x _ { \\alpha } | \\wedge \\lvert y - a \\rvert + | x _ { \\alpha } | \\wedge a \\in U + U \\subseteq V , \\end{align*}"}
-{"id": "9108.png", "formula": "\\begin{gather*} ( g _ 1 , \\dots , g _ n ) ( x _ 0 : x _ 1 : \\dots : x _ n ) : = \\big ( x _ 0 : \\zeta ^ { g _ 1 } x _ 1 : \\zeta ^ { g _ 2 } x _ 2 : \\dots : \\zeta ^ { g _ n } x _ n \\big ) , \\end{gather*}"}
-{"id": "6334.png", "formula": "\\begin{align*} - 2 < ( a _ 2 ^ \\vee ) ^ 2 = - \\frac { 2 ( n - 1 ) } { n + 1 } < - \\frac 3 2 . \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{align*} \\chi ( Z , D ) = \\frac { ( 2 D - K _ Y - Z ) \\cdot Z } { 2 } = \\chi ( Z , 0 ) + D \\cdot Z . \\end{align*}"}
-{"id": "9622.png", "formula": "\\begin{align*} r ^ { ( j ) } _ n = \\frac { H ^ { ( j ) } _ n } { K ^ { ( j ) } _ n } , \\ \\ s ^ { ( j ) } _ n = \\frac { K ^ { ( j ) } _ { n + 1 } } { H ^ { ( j ) } _ n } , \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} ( j _ { Q ( 1 ) } ' , \\dots , j _ { Q ( n ) } ' ) = ( j _ { Q ( 1 ) } , \\dots , j _ { Q ( n - 1 ) } , j _ { Q ( n ) } + | \\mathcal { E } | ) . \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{align*} \\mu P _ { \\infty } ( t ) \\Lambda _ N ^ { \\infty } = \\mu \\Lambda _ N ^ { \\infty } P _ N ( t ) = \\mu _ N P _ N ( t ) . \\end{align*}"}
-{"id": "9672.png", "formula": "\\begin{align*} \\begin{aligned} F ( \\theta , \\pi , \\psi ) & = F ( \\tilde { \\theta } , 0 , \\psi ) , \\\\ G ( \\theta , \\pi , \\psi ) & = - H ( \\tilde { \\theta } , 0 , \\psi ) , \\\\ H ( \\theta , \\pi , \\psi ) & = - G ( \\tilde { \\theta } , 0 , \\psi ) , \\end{aligned} \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{align*} K _ { \\lambda } \\triangleright u _ { b } ^ { a * } \\triangleleft K _ { \\lambda ^ { \\prime } } = ( K _ { \\lambda } ^ { - 1 } \\triangleright u _ { b } ^ { a } \\triangleleft K _ { \\lambda ^ { \\prime } } ^ { - 1 } ) ^ { * } = q ^ { - ( \\lambda , \\lambda _ { b } ) } q ^ { - ( \\lambda ^ { \\prime } , \\lambda _ { a } ) } u _ { b } ^ { a * } . \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{align*} \\partial _ t x = \\xi ( t , x ) \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{align*} \\Psi _ 1 ( 0 ) - z \\cdot \\Psi _ 2 ( 0 ) = ( 1 - z ) \\cdot \\Phi ( 0 ) . \\end{align*}"}
-{"id": "887.png", "formula": "\\begin{align*} \\underset { T \\rightarrow \\infty } { l i m } \\underset { \\kappa \\in ( 0 , 1 ) } { s u p } \\mathbb { E } \\left | \\langle : X _ T \\otimes X _ T : , \\Psi _ { \\epsilon , \\delta , \\phi } ^ f \\rangle - \\eta _ { f , \\phi , \\epsilon , \\delta } ^ T \\right | ^ 2 = 0 , \\epsilon , \\delta > 0 , \\phi \\in \\mathcal { S } , \\end{align*}"}
-{"id": "9579.png", "formula": "\\begin{align*} \\theta _ n ^ { \\mathrm { c a u s a l } } & = \\left ( \\sum _ { i = 1 } ^ n E \\left [ W _ { n , i } ^ { X X } \\right ] \\right ) ^ { - 1 } \\sum _ { i = 1 } ^ n E \\left [ W _ { n , i } ^ { X X } \\right ] \\theta _ { n , i } , \\\\ \\shortintertext { a n d , w i t h p r o b a b i l i t y a p p r o a c h i n g o n e , } \\theta _ n ^ { \\mathrm { c a u s a l } , \\mathrm { s a m p l e } } & = \\left ( \\sum _ { i = 1 } ^ n R _ { n , i } E \\left [ W _ { n , i } ^ { X X } \\right ] \\right ) ^ { - 1 } \\sum _ { i = 1 } ^ n R _ { n , i } E \\left [ W _ { n , i } ^ { X X } \\right ] \\theta _ { n , i } , \\end{align*}"}
-{"id": "8718.png", "formula": "\\begin{align*} D \\psi _ { n } ( | x | ) = \\psi ' _ { n } ( | x | ) \\frac { x } { | x | } , \\ \\ D ^ { 2 } \\psi _ { n } ( | x | ) = \\psi '' _ { n } ( | x | ) \\frac { x x ^ { T } } { | x | ^ { 2 } } + \\psi ' _ { n } ( | x | ) \\biggl [ \\frac { I } { | x | } - \\frac { x x ^ { T } } { | x | ^ { 3 } } \\biggr ] . \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{align*} c ( T ) = \\inf _ { n \\ge 1 } c ( T _ { 1 } \\oplus \\cdots \\oplus T _ { n } ) \\le \\inf _ { n \\ge 1 } c ( T _ { n } ) . \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{align*} \\theta ( \\mathsf { M } ^ n _ m ) ^ i _ j & = q ^ { - ( 2 \\rho , \\lambda _ i - \\lambda _ j ) } q ^ { - ( 2 \\rho , \\lambda _ m - \\lambda _ n ) } ( \\mathsf { M } ^ n _ m ) ^ i _ j \\\\ & = q ^ { - ( 2 \\rho , \\lambda _ m - \\lambda _ n ) } \\pi ( K _ { 2 \\rho } ^ { - 1 } ) _ { i } ^ { i } ( \\mathsf { M } ^ n _ m ) ^ i _ j \\pi ( K _ { 2 \\rho } ) _ { j } ^ { j } . \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} L ( y ) = \\varepsilon \\left \\vert y ^ { 0 } y ^ { 1 } . . . . y ^ { n - 1 } \\right \\vert ^ { \\tfrac { 2 } { n } } , \\end{align*}"}
-{"id": "7648.png", "formula": "\\begin{align*} K ( s , t ) = \\begin{cases} s - \\left ( a + \\lambda \\frac { b - a } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ a , t ) , \\\\ s - \\left ( a + ( 2 - \\lambda ) \\frac { b - a } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ t , b ] , \\end{cases} \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} \\epsilon \\frac { d c } { d t } & = \\mu h \\frac { b + c } { 1 + c } - \\frac { \\Gamma } { K _ 1 } \\frac { c } { K + c } + \\frac { \\lambda } { K _ 1 } \\theta = R _ 1 / K _ 1 = F _ 1 , \\\\ \\frac { d \\theta } { d t } & = - \\theta + T ( c ) = R _ 2 , \\\\ \\frac { d h } { d t } & = \\frac { 1 } { 1 + c ^ 2 } - h = R _ 3 . \\end{align*}"}
-{"id": "9478.png", "formula": "\\begin{align*} c \\le 0 . 9 p ^ { h } / ( q ^ r 4 ^ q ) , \\mbox { w h e r e } q : = 2 f \\cdot f ! \\mbox { a n d } h : = 2 ^ r \\binom { q + r } { r } , \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{align*} C _ 2 = C _ 2 ( r ) = \\frac { 6 } { \\pi ^ 2 } \\left ( \\eta \\left ( 1 - \\frac { 2 } { r } \\right ) \\right ) ^ 3 \\prod _ { p } \\left ( 1 - \\frac { 1 } { p } \\right ) ^ 3 \\left ( 1 + \\frac { 3 } { p } - \\frac { 1 } { p ^ 2 } - \\frac { 1 8 } { p ( p + 2 ) } \\right ) . \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} u _ \\lambda ( t , x ) = \\lambda ^ { - k / 2 } u ( \\lambda ^ { - k } t , \\lambda ^ { - 1 } x ) , \\end{align*}"}
-{"id": "5204.png", "formula": "\\begin{align*} ( \\mathcal { S } _ u ^ \\vee : \\Delta _ v ^ \\vee \\langle n \\rangle ) = 0 \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} h ^ 2 \\bigl ( E _ y ( c - e - 4 ) | _ S \\bigr ) = h ^ 0 \\bigl ( E _ y ( - b ) | _ S \\bigr ) . \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} & \\int _ { x _ 1 \\cdots x _ r \\geq \\Xi , x _ k \\geq 1 } \\prod _ { k = 1 } ^ r x _ k ^ \\beta ( x _ 1 ^ \\alpha + x _ 2 ^ \\alpha + \\cdots + x _ r ^ \\alpha ) ^ { - 1 } d x _ 1 \\cdots d x _ r \\asymp \\Xi ^ { \\beta + 1 - \\alpha / r } . \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} a ( i + 1 , n ) = \\frac { 1 } { n } \\sum _ { j = i } ^ { n - 1 } ( 1 - 2 ^ { \\nu _ { 2 } ( n - j ) + 1 } ) a ( i , j ) . \\end{align*}"}
-{"id": "7778.png", "formula": "\\begin{align*} \\mbox { w h e t h e r } & a ) \\mbox { } u ( t , x , 0 ) = u ( t , x , L ) = 0 , \\\\ \\mbox { o r } & b ) \\mbox { } u _ y ( t , x , 0 ) = u _ y ( t , x , L ) = 0 , \\\\ \\mbox { o r } & c ) \\mbox { } u ( t , x , 0 ) = u _ y ( t , x , L ) = 0 , \\\\ \\mbox { o r } & d ) \\mbox { } u \\mbox { i s a n $ L $ - p e r i o d i c f u n c t i o n w i t h r e s p e c t t o $ y $ . } \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\downarrow 0 } q _ i ( C , \\epsilon , \\epsilon ) = q ( C ) . \\end{align*}"}
-{"id": "7225.png", "formula": "\\begin{align*} ( \\rho _ i \\rho _ j ) ^ 2 = 1 , \\textrm { f o r $ i , j \\in \\{ 0 , \\ldots , n - 1 \\} $ w i t h $ | i - j | \\geq 2 $ } . \\end{align*}"}
-{"id": "9654.png", "formula": "\\begin{align*} x ^ 2 + y ^ 2 - z ^ 2 = \\alpha ^ 2 . \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{align*} e ^ { A } K _ - e ^ { - A } = \\sinh ^ 2 ( r ) e ^ { i 2 \\theta \\sigma ( \\hat n ) } \\cdot K _ + + \\cosh ^ 2 ( r ) K _ - - \\sinh ( 2 r ) e ^ { i \\theta \\sigma ( \\hat n ) } \\cdot K _ 0 . \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} { } _ 3 F _ 2 \\bigg [ \\begin{matrix} \\alpha & \\beta & \\gamma \\\\ & 1 & \\delta \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { \\Gamma ( \\delta ) \\Gamma ( 1 + \\delta - \\alpha - \\beta - \\gamma ) } { \\Gamma ( \\delta - \\alpha ) \\Gamma ( 1 + \\delta - \\beta - \\gamma ) } \\cdot { } _ 3 F _ 2 \\bigg [ \\begin{matrix} \\alpha & 1 - \\beta & 1 - \\gamma \\\\ & 1 & 1 + \\delta - \\beta - \\gamma \\end{matrix} \\bigg | \\ , 1 \\bigg ] . \\end{align*}"}
-{"id": "950.png", "formula": "\\begin{align*} Y - a \\mu = a ( X - \\mu ) \\stackrel { d } { = } R ( a A ) U = \\hat a ^ { 1 / 2 } R e _ 1 O U = e _ 1 Z = Z _ 1 . \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{align*} A = \\frac { d ^ 2 } { d x ^ 2 } + \\left [ ( \\beta - 1 ) t a n h ( x ) + \\gamma s e c h ( x ) \\right ] \\frac { d } { d x } . \\end{align*}"}
-{"id": "8507.png", "formula": "\\begin{align*} \\mathbb { S } = \\left \\{ \\begin{array} { l l } \\mathbb { L } \\times \\mathbb { R } , & 2 \\notin C , \\\\ \\mathbb { R } & 2 \\in C . \\end{array} \\right . \\end{align*}"}
-{"id": "2636.png", "formula": "\\begin{align*} L u : = - \\rho ( x ) ^ { - 1 } ( a ( x ) u ' ) ' + \\rho ( x ) ^ { - 1 } q ( x ) u , \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u ^ { 2 } + v ^ { 2 } + \\epsilon v = a ^ { 2 } , \\\\ u ^ { 2 } + \\epsilon u + \\epsilon v = b ^ { 2 } . \\end{array} \\right . \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( A _ 0 x + A _ 1 + \\frac { A _ 2 } { x } \\right ) Y , \\frac { \\partial Y } { \\partial t _ 1 } = ( B _ { 1 1 } x + B _ { 1 0 } ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = ( B _ { 2 1 } x + B _ { 2 0 } ) Y , \\end{gather*}"}
-{"id": "5419.png", "formula": "\\begin{align*} \\lambda ( M - N \\overline { M } ^ { - 1 } \\overline { N } ) = \\left ( \\begin{array} { r } 0 . 0 4 6 3 \\\\ 0 . 2 4 8 8 \\\\ 2 . 0 0 7 3 \\end{array} \\right ) \\end{align*}"}
-{"id": "9624.png", "formula": "\\begin{align*} K ^ { ( j ) } _ n = \\left | \\begin{array} { c c c c } 1 & 1 & \\cdots & 1 \\\\ u _ { j } & u _ { j + 1 } & \\cdots & u _ { j + n - 1 } \\\\ u _ { j + 1 } & u _ { j + 2 } & \\cdots & u _ { j + n } \\\\ \\vdots & \\vdots & & \\vdots \\\\ u _ { j + n - 2 } & u _ { j + n - 1 } & \\cdots & u _ { j + 2 n - 3 } \\end{array} \\right | , \\ \\ K ^ { ( j ) } _ 0 = 1 . \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{align*} ( \\omega + i \\partial \\bar \\partial w ) ^ { n + 1 } = 0 . \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{gather*} \\sum _ { n = 0 } ^ { \\infty } V _ C ( q ^ { - 2 n } \\theta ) = \\sum _ { n = 0 } ^ { \\infty } V _ D ( q ^ { 2 n } \\theta ^ { - 1 } ) . \\end{gather*}"}
-{"id": "3564.png", "formula": "\\begin{align*} ( \\ 1 _ V \\tau ) \\otimes \\ 1 _ { V ' } \\tau ' = \\ 1 _ { V \\times V ' } ( \\tau \\otimes \\tau ' ) . \\end{align*}"}
-{"id": "2405.png", "formula": "\\begin{align*} \\| \\frac 1 { h _ R } [ D , h _ R ] f \\| _ 2 = R ^ { - 1 + n / 2 } \\| \\frac { 1 } { h } [ D , h ] f _ { R ^ { - 1 } } \\| _ 2 \\leq R ^ { - 1 } \\| \\frac { 1 } { h } [ D , h ] \\| \\| f \\| _ 2 . \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} \\begin{bmatrix} 4 m & 4 m & 4 m \\\\ 4 m & 4 m & 4 m \\\\ 4 m & 4 m & 4 m \\end{bmatrix} , \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} \\varepsilon ^ 3 & = \\varepsilon ( \\alpha \\eta \\beta \\mu \\gamma + b _ 1 \\alpha \\eta \\beta \\mu \\gamma \\varepsilon ) = \\varepsilon \\alpha \\eta \\beta \\mu \\gamma , \\\\ \\varepsilon ^ 3 & = ( \\alpha \\eta \\beta \\mu \\gamma + b _ 1 \\alpha \\eta \\beta \\mu \\gamma \\varepsilon ) \\varepsilon = \\alpha \\eta \\beta \\mu \\gamma \\varepsilon . \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} w _ t = L ( q ^ { - 1 } ) v _ t , \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} \\frac 1 { 2 C } v _ \\epsilon '' ( t , \\rho _ t ) = \\int _ { | \\rho - \\rho _ t | \\leq 1 / 2 C } { \\frac { v _ \\epsilon '' ( t , \\rho _ t ) } 2 d \\rho } < \\int _ { - \\infty } ^ { \\infty } { v _ \\epsilon '' ( t , \\rho ) d \\rho } = b _ t - a _ t = ( 1 + \\alpha ) ( T - t ) \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 \\ , \\left ( \\frac 1 2 \\right ) & x = \\infty \\\\ \\overbrace { \\begin{matrix} \\sqrt { t _ 1 } & 0 \\\\ - \\sqrt { t _ 1 } & 0 \\end{matrix} } & \\overbrace { \\begin{matrix} 0 & 0 & \\theta ^ \\infty _ 1 \\\\ 1 & - t _ 2 & \\theta ^ \\infty _ 2 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "95.png", "formula": "\\begin{align*} P = \\Big \\{ \\ , g \\mathrel { \\Big | } \\underset { t \\to 0 } { \\operatorname { l i m } } \\ , ( \\lambda _ t \\cdot g \\cdot \\lambda _ t ^ { - 1 } ) L \\ , \\Big \\} \\subseteq G . \\end{align*}"}
-{"id": "9898.png", "formula": "\\begin{align*} \\frac { d \\rho _ i } { d t } & = \\sum _ { j \\in N ( i ) } \\rho _ j [ F _ i ( \\rho ) - F _ j ( \\rho ) + \\beta ( \\log \\rho _ j - \\log \\rho _ i ) ] _ + \\\\ & - \\sum _ { j \\in N ( i ) } \\rho _ { i } [ F _ j ( \\rho ) - F _ i ( \\rho ) + \\beta ( \\log \\rho _ i - \\log \\rho _ j ) ] _ + \\ , \\\\ \\end{align*}"}
-{"id": "9329.png", "formula": "\\begin{align*} Y ^ 2 = X ^ 3 + A ( t : s ) f _ 6 ^ 2 ( x _ 0 : x _ 1 : x _ 2 ) X + B ( t : s ) f _ 6 ^ 3 ( x _ 0 : x _ 1 : x _ 2 ) . \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{gather*} \\left ( { Z ( X _ { i , \\lambda } , T ) } { \\prod _ { j = 0 } ^ { n - 1 } \\big ( 1 - q ^ j T \\big ) } \\right ) ^ { ( - 1 ) ^ n } . \\end{gather*}"}
-{"id": "3661.png", "formula": "\\begin{align*} L _ 0 = \\frac { 1 } { 2 } \\Big ( \\frac { 1 } { 2 } : \\chi _ { - \\frac { 1 } { 2 } } \\chi _ { \\frac { 1 } { 2 } } : - \\frac { 3 } { 2 } : \\chi _ { - \\frac { 3 } { 2 } } \\chi _ { \\frac { 3 } { 2 } } : + \\frac { 5 } { 2 } : \\chi _ { - \\frac { 5 } { 2 } } \\chi _ { \\frac { 5 } { 2 } } : - \\dots \\Big ) . \\end{align*}"}
-{"id": "1343.png", "formula": "\\begin{align*} f ( z ) - \\alpha & = f ( z ) - f ( \\alpha ) \\\\ & = \\frac { a z + b } { c z + d } - \\frac { a \\alpha + b } { c \\alpha + d } = \\frac { ( a d - b c ) ( z - \\alpha ) } { ( c z + d ) ( c \\alpha + d ) } = \\frac { z - \\alpha } { ( c z + d ) ( c \\alpha + d ) } \\end{align*}"}
-{"id": "3487.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi _ \\sigma ( x ) \\to \\varphi _ { \\infty } ( x ) = \\begin{cases} \\sin ( \\sqrt { 2 } x ) , & 0 \\le x < \\frac { \\sqrt { 2 } \\pi } { 4 } , \\\\ 1 , & x \\ge \\frac { \\sqrt { 2 } \\pi } { 4 } . \\\\ \\end{cases} \\end{aligned} \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} \\bigg ( \\frac { c _ 2 h ( x ) } { r } \\bigg ) ^ p | \\{ u _ \\varepsilon = 0 \\} \\cap B _ r ( x _ 0 ) | \\leq c _ 3 \\int _ { B _ { r } ( x ) } | \\nabla ( h - u _ \\varepsilon ) ( x ) | ^ p \\ , d x . \\end{align*}"}
-{"id": "764.png", "formula": "\\begin{align*} \\sum _ { n = U } ^ { U + V } e \\left ( \\overline { \\xi } _ { x + n w } \\right ) = O \\left ( \\frac { V } { R _ 1 } + \\frac { V ^ 2 } { U } + \\frac { 1 } { d } \\right ) = o _ R ( V ) . \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{align*} d X ^ { ( n ) } _ i ( t ) = \\sqrt { 2 ( ( X _ i ^ { ( n ) } ) ^ 2 ( t ) + 1 ) } d W ^ { ( n ) } _ i ( t ) + \\left [ \\left ( 2 - 2 n - 2 \\Re ( s ) \\right ) X _ i ^ { ( n ) } ( t ) + 2 \\Im ( s ) + \\sum _ { j \\ne i } ^ { } \\frac { 2 ( ( X _ i ^ { ( n ) } ( t ) ) ^ 2 + 1 ) } { X _ i ^ { ( n ) } ( t ) - X _ j ^ { ( n ) } ( t ) } \\right ] d t , \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{align*} \\tau _ { \\alpha } ( s _ 0 ) = \\frac { \\langle \\alpha ( s _ 0 ) - P _ S ( s _ 0 ) , \\alpha ' ( s _ 0 ) \\times \\alpha ''' ( s _ 0 ) \\rangle } { 1 + J ^ 2 ( s _ 0 ) } = \\frac { J ' ( s _ 0 ) } { 1 + J ^ 2 ( s _ 0 ) } + \\frac { \\kappa ( s _ 0 ) \\varsigma ( s _ 0 ) } { 1 + J ^ 2 ( s _ 0 ) } \\ , , \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{align*} K _ 1 = \\mathbb { Q } ( \\mu _ { 4 } ) = \\mathbb { Q } ( \\sqrt { - 1 } ) ; \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} & \\overline { V _ 1 ( k , l ) } = N ( 0 , a , b , c ) \\ \\ \\ \\ \\ { f o r } \\ \\ \\ a = 0 , \\ \\ b = k + 1 - l , \\ \\ c = k + 1 , \\\\ & \\overline { V _ 2 ( k , l ) } = N ( 0 , a , b , c ) \\ \\ \\ \\ \\ { f o r } \\ \\ \\ a = b = l , \\ \\ c = k + 1 , \\\\ & \\overline { V _ 3 ( k , l ) } = N ( - 1 , a , b , c ) \\ \\ \\ { f o r } \\ \\ \\ a = b = l - 1 , \\ \\ c = k + 1 , \\\\ & \\overline { V _ 4 ( k ) } = N ( 0 , a , b , c ) \\ \\ \\ \\ \\ \\ \\ { f o r } \\ \\ \\ a = b = 0 , \\ \\ \\ c = k + 1 . \\\\ \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} Z _ { \\mathrm { C a } ( X ) } \\big ( q ^ { - 1 } T , q , \\mathcal { O } _ { P , X } \\big ) = \\# \\left ( Z \\big ( T _ { 1 } , \\ldots , T _ { d } , \\mathcal { O } _ { P , X } \\big ) \\right ) \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} \\frac { 1 } { \\varepsilon } ( - 3 v _ \\star ^ 2 + 2 ( a + 1 ) v _ \\star - a ) - c = 0 . \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\frac { 1 } { N } \\widehat { K } _ N \\left ( \\frac { x } { N } , \\frac { x } { N } ; w _ R \\right ) = \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "141.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } | \\alpha _ { j + 1 } - \\alpha _ j | ^ 2 + | \\alpha _ j | ^ 4 < \\infty \\end{align*}"}
-{"id": "6509.png", "formula": "\\begin{align*} \\int \\left ( \\sum _ { n \\in \\mathbb { N } } T _ n ( x ) T _ 1 ( x ) z ^ n \\right ) \\nu ( d x ) = \\left ( \\int T _ 1 ( x ) ^ 2 \\nu ( d x ) \\right ) z . \\end{align*}"}
-{"id": "9497.png", "formula": "\\begin{align*} \\begin{pmatrix} \\beta & 1 & 0 & 0 \\\\ [ 6 p t ] 0 & \\beta & 0 & 0 \\\\ [ 6 p t ] 0 & 0 & \\beta & 1 \\\\ [ 6 p t ] 0 & 0 & 0 & \\beta \\end{pmatrix} \\end{align*}"}
-{"id": "3877.png", "formula": "\\begin{align*} f ^ \\prime ( - 1 + h - a ) & = s _ 1 ^ \\prime ( - 1 + h - a ) + s _ 2 ^ \\prime ( - 1 + h - a ) = \\left ( - 1 + \\frac { b ^ 2 / a } { 2 + a - h } \\right ) + \\sqrt { 1 - b ^ 2 / a ^ 2 } . \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{align*} u ( x ) - \\sum _ { | \\alpha | \\leq k } \\frac { \\partial ^ { \\alpha } u ( 0 ) } { \\alpha ! } x ^ { \\alpha } = \\sum _ { | \\beta | = k + 1 } \\frac { | \\beta | } { \\beta ! } \\int _ 0 ^ 1 ( 1 - t ) ^ k \\partial ^ { \\beta } u ( t x ) d t x ^ { \\beta } . \\end{align*}"}
-{"id": "5325.png", "formula": "\\begin{align*} j i _ { k } + ( j + 1 ) i _ { k + 1 } + 2 \\sum _ { p = k + 2 } ^ { m - 1 } i _ p \\le m - k + j + 1 \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} \\ell _ k ( p ^ j ) = ( j + 1 ) ^ k - j ^ k . \\end{align*}"}
-{"id": "4898.png", "formula": "\\begin{align*} \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\mathbb { E } \\left [ \\tilde { \\mathit { q } } _ { 0 } \\right ] & = 0 , \\mathbb { E } \\left [ ( \\tilde { \\mathit { q } } _ { 0 } ) ^ 2 \\right ] = \\frac { 1 } { 2 } \\left [ \\mathit { q } _ { 0 } \\right ] . \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\binom { y } { k } a _ k x ^ k = ( 1 + x ) ^ y \\sum _ { k = 0 } ^ { \\infty } \\binom { y } { k } b _ k \\left ( \\frac { x } { 1 + x } \\right ) ^ k , \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{align*} & u _ s \\in L ^ q ( \\Omega ) \\cap C ^ \\infty ( \\Omega ) , \\| u _ s \\| _ q \\leq C | u _ \\infty | ^ { 1 / 2 } , \\forall q \\in ( 2 , \\infty ] , \\\\ & \\nabla u _ s \\in L ^ r ( \\Omega ) , \\| \\nabla u _ s \\| _ r \\leq C ^ \\prime | u _ \\infty | ^ { 1 / 2 } , \\forall r \\in ( 4 / 3 , \\infty ] , \\\\ & \\mbox { p r o v i d e d $ 0 < | u _ \\infty | \\leq \\delta _ 0 $ } . \\end{align*}"}
-{"id": "7124.png", "formula": "\\begin{align*} T _ h ^ n = - \\frac { h } { 2 } f ' ( \\zeta _ n ) + E _ 1 ^ n + \\sum _ { j = 1 } ^ { n } \\int _ { I _ j } \\omega _ { 1 - \\alpha } ( x _ { n } - q ) E _ 2 ^ { n , j } ( q ) \\ , d q , { \\rm f o r } ~ ~ n \\ge 1 , \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{align*} & c _ 3 ( t ) = x _ 3 + \\frac 1 { 2 \\lambda ^ 2 y } ( e ^ { 2 \\lambda t y } - 1 ) ( - \\xi + \\lambda z ) + \\frac 1 { \\lambda ^ 2 } e ^ { \\lambda t y } \\sin ( \\lambda t \\xi ) \\\\ & + \\frac 1 { \\lambda } \\left ( x _ 1 \\ , ( e ^ { \\lambda t y } \\cos ( \\theta - \\lambda t \\xi ) - \\cos \\theta ) + x _ 2 \\ , ( e ^ { \\lambda t y } \\sin ( \\theta - \\lambda t \\xi ) - \\sin \\theta ) \\right ) . \\end{align*}"}
-{"id": "1497.png", "formula": "\\begin{align*} P Q '' - P ' Q ' + R Q = 0 , \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{align*} W f = ( V _ 1 ^ { i _ 1 - 1 } \\cdots V _ n ^ { i _ n - 1 } f ) , \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} E _ \\rho = \\{ w \\in E ; \\| w \\| _ E \\leq \\rho \\} \\end{align*}"}
-{"id": "5804.png", "formula": "\\begin{align*} S ( C | M = m ) \\ge \\frac { k } { 2 } \\ln \\left ( \\eta \\exp \\frac { 2 S ( A | M = m ) } { k } + \\left | 1 - \\eta \\right | \\exp \\frac { 2 S ( B | M = m ) } { k } \\right ) \\ ; . \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{gather*} \\mathbf { v } = \\mathbf { v } _ { 0 } + \\nabla \\varphi , \\mathbf { v } _ { 0 } \\in \\mathbf { H } ^ { 1 } _ { t , 0 } ( \\Omega ) , \\varphi \\in H _ { 0 } ^ { 1 } ( \\Omega ) \\cap H ^ { 2 } ( \\Omega ) , \\\\ \\Vert \\mathbf { v } _ { 0 } \\Vert _ { \\mathbf { H } ^ { 1 } } \\leq C \\Vert \\mathbf { v } \\Vert _ { \\mathbf { H } ^ { 1 } } , \\ ; \\ ; \\Vert \\varphi \\Vert _ { H ^ { 2 } } \\leq C \\Vert \\mathbf { v } \\Vert _ { \\mathbf { H } ^ { 1 } } . \\end{gather*}"}
-{"id": "9664.png", "formula": "\\begin{align*} \\frac { 1 } { 6 } \\int _ 0 ^ 1 \\left ( \\begin{vmatrix} P \\\\ \\alpha ^ \\circ ( t ) \\\\ { \\alpha ^ \\circ } ' ( t ) \\end{vmatrix} + \\begin{vmatrix} P \\\\ \\beta ^ \\circ ( t ) \\\\ { \\beta ^ \\circ } ' ( t ) \\end{vmatrix} + \\begin{vmatrix} P \\\\ \\gamma ^ \\circ ( t ) \\\\ { \\gamma ^ \\circ } ' ( t ) \\end{vmatrix} \\right ) \\ , d t , \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} f ' \\left ( \\frac { z } { 1 - z } \\right ) g ( z ) = \\sum _ { n \\geq 1 } z ^ n \\sum _ { \\pi \\in \\mathcal { C } _ n } f _ { \\vert \\pi \\vert } \\sum _ { k _ i \\in \\pi } g _ { k _ i } . \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} ( a d H ' + b F ) ^ { \\dim X } = & ( a d H ' ) ^ { \\dim X } + ( \\dim X ) ( a d ) ^ { \\dim X - 1 } b ( H ' ) ^ { \\dim X - 1 } \\cdot F \\\\ = & ( a d ) ^ { \\dim X - 1 } ( a d \\cdot H '^ { \\dim X } + ( \\dim X ) b ( H '^ { \\dim X - 1 } \\cdot F ) ) , \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{align*} \\widetilde { \\mathbf { A } } _ { d B } = \\mathbf { Q } _ { A _ { d B } } ^ { 1 / 2 } \\mathbf { N } + \\boldsymbol { \\mu } _ { A _ { d B } } , \\end{align*}"}
-{"id": "1359.png", "formula": "\\begin{align*} d \\tau = n ( x ) \\left \\vert d x \\right \\vert , | d x | = \\sqrt { ( d x _ { 1 } ) ^ { 2 } + ( d x _ { 2 } ) ^ { 2 } + ( d x _ { 3 } ) ^ { 2 } } . \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} \\beta _ 0 = & \\sum _ { i = 1 } ^ { p } W ( \\phi _ i ) + \\sum _ { j = 1 } ^ { q } \\left ( W ( \\vec { \\Psi } _ j ) - 4 \\pi \\theta _ j \\right ) \\in 4 \\pi \\N , \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} \\dot x = f _ 0 ( x ) + \\sum _ { j = 1 } ^ \\ell f _ j ( x ) { \\sqrt \\omega } u _ j ( { \\omega } t ) , \\end{align*}"}
-{"id": "5681.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } \\alpha ( t ) = q ( t ) ^ \\top \\big [ J ( t ) ^ \\top + J ( t ) \\big ] q ( t ) , \\\\ \\beta ( t ) = q ( t ) ^ \\top G _ { i j } q ( t ) . \\end{array} \\right . \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} \\varphi u ( t ) = e ^ { ( t - \\varepsilon ) \\Delta } \\varphi u ( \\varepsilon ) + \\int _ \\varepsilon ^ t e ^ { t - s ) \\Delta } [ - 2 \\nabla u \\cdot \\nabla \\varphi - u \\Delta \\varphi + \\varphi | u | ^ \\alpha u ] \\end{align*}"}
-{"id": "9102.png", "formula": "\\begin{gather*} \\left ( Z ( X _ { A , \\psi } , T ) \\prod _ { i = 0 } ^ { n - 1 } \\big ( 1 - q ^ i T \\big ) \\right ) ^ { ( - 1 ) ^ { n } } \\mbox { a n d } \\left ( Z ( X _ { A ' , \\psi } , T ) \\prod _ { i = 0 } ^ { n - 1 } \\big ( 1 - q ^ i T \\big ) \\right ) ^ { ( - 1 ) ^ { n } } \\end{gather*}"}
-{"id": "7803.png", "formula": "\\begin{align*} \\| u \\| _ { X ( Q _ T ) } + \\| u _ { x } \\big | _ { x = 0 } \\| _ { L _ 2 ( B _ T ) } \\leq c . \\end{align*}"}
-{"id": "2914.png", "formula": "\\begin{align*} G _ { T } = \\bigcap _ { \\varepsilon \\in \\Q ^ { + * } } \\bigcap _ { N \\ge 1 } G _ { T , \\varepsilon , N } \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{gather*} \\sum _ { k _ { 1 } \\geq 0 } . . . \\sum _ { k _ { n } \\geq 0 } ( \\prod _ { j = 1 } ^ { n } \\rho _ { j } ^ { k _ { j } } ) \\sin ( \\beta + \\sum _ { j = 1 } ^ { n } k _ { j } \\alpha _ { j } ) \\allowbreak = \\allowbreak \\\\ \\frac { 1 } { 2 i } \\frac { \\exp ( i \\beta ) \\prod _ { j = 1 } ^ { n } ( 1 - \\rho _ { j } \\exp ( - i \\alpha _ { j } ) ) - \\exp ( - i \\beta ) \\prod _ { j = 1 } ^ { n } ( 1 - \\rho _ { j } \\exp ( i \\alpha _ { j } ) ) } { \\prod _ { j = 1 } ^ { n } ( 1 + \\rho _ { j } ^ { 2 } - 2 \\rho _ { j } \\cos ( \\alpha _ { j } ) ) } \\allowbreak . \\end{gather*}"}
-{"id": "9514.png", "formula": "\\begin{align*} V _ { M , J } ( x ) : = \\{ \\Phi ( x ) : \\Phi \\in S _ { \\Z ( M ) } ( M ) , \\ \\Phi | _ J = 0 \\} . \\end{align*}"}
-{"id": "5584.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 1 } \\frac { 2 ( 1 - \\rho ^ { 2 } ) \\sqrt { 1 - y ^ { 2 } } d y } { \\pi ( ( 1 - \\rho ^ { 2 } ) ^ { 2 } - 4 x y \\rho ( 1 + \\rho ^ { 2 } ) + 4 \\rho ^ { 2 } ( x ^ { 2 } + y ^ { 2 } ) ) } = 1 , \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } f ( v , w ) & = & v ( a - v ) ( v - 1 ) - w , \\\\ g ( v , w ) & = & b v - c w , \\end{array} \\right . \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} 0 \\neq [ V _ { - 2 } , \\ , L _ { - 1 } ] = [ V _ { - 2 } , \\ , [ L _ { - 4 } , \\ , [ V _ { - 2 } , \\ , L _ 5 ] ] ] = [ L _ { - 4 } , \\ , [ V _ { - 2 } , \\ , [ V _ { - 2 } , \\ , L _ 5 ] ] ] , \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} \\Psi ( x , v , v ' , r ) : = e ^ { - \\int _ 0 ^ r \\sigma ( x + s v , v ) d s } k ( x + r v , v , v ' ) , \\ \\ ( x , v , v ' , r ) \\in \\R ^ n \\times V ^ 2 \\times \\R . \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{align*} B ^ { \\prime } ( k ) = \\left ( \\begin{array} { c | c } \\begin{matrix} 1 + e ^ { 2 \\pi i k } & 0 \\\\ - e ^ { 2 \\pi i k } & 0 \\end{matrix} & \\boldsymbol { O } \\\\ \\hline \\boldsymbol { O } & \\begin{matrix} 1 & 1 \\\\ e ^ { 2 \\pi i k } & - e ^ { 2 \\pi i k } \\end{matrix} \\end{array} \\right ) , \\end{align*}"}
-{"id": "9663.png", "formula": "\\begin{align*} \\abs { K ^ \\circ _ { + + } } = \\frac { 2 - b - c } { \\abs { K } } , \\abs { K ^ \\circ _ { - + } } = \\frac { 2 + b + c } { \\abs { K } } . \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{align*} Z _ 1 ^ { ( 2 ) } = [ y ] = f _ 1 ^ * ( H + 3 \\pi ^ * L ) - E _ 1 , Z _ 2 ^ { ( 2 ) } = [ e _ 1 ] = E _ 1 . \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} \\int _ { \\Omega _ { 2 } } | \\nabla \\phi _ { n } ( x ) | ^ 2 d x \\geq c _ { * } \\ , \\delta _ { 2 } \\ , ( n _ { 1 } ^ 2 + n _ { 2 } ^ 2 ) = c _ { * } \\ , \\delta _ { 2 } \\ , \\lambda _ { k } . \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} \\displaystyle \\int _ { 0 } ^ { T } \\left [ \\big ( \\nabla \\phi , \\ , \\nabla \\widetilde { \\phi } \\big ) - \\big ( | \\Psi | ^ { 2 } , \\ , \\widetilde { \\phi } \\big ) \\right ] \\mathrm { d } t = 0 , \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} \\phi ( x , y ) = 0 \\Rightarrow x = y \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} h ( x ) = & \\displaystyle \\sum _ { i = 0 } ^ { \\frac { s + 1 } { 2 } } \\left [ \\binom { s } { s - 2 i + 1 } \\ , k + \\ , \\binom { s } { s - 2 i } \\ , ( 2 - k ) \\right ] \\ , x ^ \\frac { p ^ l + s - 2 i } { 2 } \\cr & + \\displaystyle \\sum _ { i = 0 } ^ { \\frac { s + 1 } { 2 } } \\left [ \\binom { s } { s - 2 i } \\ , k + \\ , \\binom { s } { s - ( 2 i + 1 ) } \\ , ( 2 - k ) \\right ] \\ , x ^ \\frac { s - 2 i - 1 } { 2 } \\end{align*}"}
-{"id": "8545.png", "formula": "\\begin{align*} | \\alpha | _ { W P } ^ 2 = g ^ { - 2 } \\otimes \\alpha \\otimes \\bar { \\alpha } = e ^ { - 4 \\lambda } | f ( z ) | ^ 2 , \\end{align*}"}
-{"id": "1833.png", "formula": "\\begin{align*} \\Lambda ( \\phi ) = \\frac 1 2 h ( \\phi , F ) = \\frac { i } { 2 } \\left ( \\phi ^ \\alpha _ { \\enskip \\alpha } - \\phi ^ { \\enskip \\alpha } _ \\alpha \\right ) \\end{align*}"}
-{"id": "3567.png", "formula": "\\begin{align*} \\hat \\phi = \\hat \\phi _ 0 + \\hat \\phi _ 1 \\otimes w ^ { \\alpha _ 1 } + \\cdots + \\hat \\phi _ { \\nu - 1 } \\otimes w ^ { \\alpha _ { \\nu - 1 } } , \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} \\xi ^ * _ x = \\xi ' _ x - C ( \\xi ) , \\end{align*}"}
-{"id": "10044.png", "formula": "\\begin{align*} \\{ \\nabla ^ { 0 } + Q \\colon Q \\in \\mathcal L , g ( Q ( X , Y ) , Z ) + g ( Q ( X , Z ) , Y ) = 0 , \\forall X , Y , Z \\in { \\mathfrak X } ( M ) \\} . \\end{align*}"}
-{"id": "6464.png", "formula": "\\begin{align*} s _ { v ( i , j ) } = s _ { v _ j } s _ { v ( i , j - 1 ) } s _ { v _ j } . \\end{align*}"}
-{"id": "2030.png", "formula": "\\begin{gather*} S = \\operatorname { d i a g } \\big ( 1 , z ^ s , \\ldots , z ^ { ( m - 1 ) s } \\big ) , \\end{gather*}"}
-{"id": "5522.png", "formula": "\\begin{align*} V = \\Delta _ \\delta \\cup \\bigcup _ { m \\geq M } V _ m \\end{align*}"}
-{"id": "6257.png", "formula": "\\begin{align*} u _ R = \\mathcal { B } _ { \\epsilon , R } ( \\theta ) ^ { - 1 } \\big ( \\iota ^ * _ { R ( \\theta ) } - \\iota ^ * _ { R ( \\theta ) } M ( \\theta ) \\pi _ { N ( \\theta ) } \\mathcal { B } _ { \\epsilon } ( \\theta ) ^ { - 1 } \\big ) f . \\end{align*}"}
-{"id": "4217.png", "formula": "\\begin{align*} ( D u ) _ { i j } = u _ i - u _ j \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} ( \\epsilon _ i , \\epsilon _ j ) = ( - 1 ) ^ { \\theta } \\delta _ { i j } , ( \\delta _ \\mu , \\delta _ \\nu ) = - ( - 1 ) ^ { \\theta } \\delta _ { \\mu \\nu } , ( \\epsilon _ i , \\delta _ \\mu ) = ( \\delta _ \\mu , \\epsilon _ i ) = 0 , \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} \\psi ( \\rho ) = \\begin{cases} 1 & \\mathrm { i f } 0 \\leq \\rho \\leq 1 , \\\\ 0 & \\mathrm { i f } \\rho \\geq 2 . \\end{cases} \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{align*} \\R ( X , \\ , Y ) Z : = \\nabla ^ 2 _ { Y , \\ , X } Z - \\nabla ^ 2 _ { X , \\ , Y } Z \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle | \\tau \\sum _ { k = 1 } ^ { m } V _ { 1 } ^ { k } ( \\partial _ { \\tau } { \\theta _ { \\Psi } ^ { k } } ) | \\leq C \\big ( h ^ { 2 r } + \\tau ^ { 4 } \\big ) + C \\| \\theta _ { \\Psi } ^ { m } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } + C \\tau \\sum _ { k = 1 } ^ { m - 1 } { \\| \\theta _ { \\Psi } ^ { k } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } } . } \\end{array} \\end{align*}"}
-{"id": "9888.png", "formula": "\\begin{align*} \\| \\pi _ h [ g ( q ) ] - g ( q ) \\| _ { L ^ \\infty ( K _ k ) } \\leq \\max _ { i , \\ , j = 0 , \\ldots , P _ d ^ k } | g ( q ( P _ i ^ k ) ) - g ( q ( P _ j ^ k ) ) | , \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} \\left ( \\sum _ { n = 0 } ^ \\infty \\frac { | a _ n | ^ 2 } { c _ { 2 / p } ( n ) } \\right ) ^ \\frac { 1 } { 2 } \\leq \\| f \\| _ { H ^ p } . \\end{align*}"}
-{"id": "1967.png", "formula": "\\begin{align*} \\widetilde { T } _ { a } ( z ) = \\widetilde { L } _ { a 1 } ( z , t ) \\cdots \\widetilde { L } _ { a M } ( z , t ) , \\end{align*}"}
-{"id": "6514.png", "formula": "\\begin{align*} \\mathcal { S } _ { n } ( \\delta ) : = \\left \\{ g \\in G : \\ \\frac { 1 } { 2 } \\left \\Vert \\mu ^ { ( n ) } - g \\mu ^ { ( n ) } \\right \\Vert _ { 1 } < \\delta \\right \\} . \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{align*} m _ { i i } = 1 , ~ m _ { i ( i + 1 ) } = 3 \\mbox { a n d } m _ { i j } = 2 \\mbox { f o r } | i - j | \\geq 2 , i , j \\in \\Z / n \\Z . \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} X _ { 2 , \\ , j + 1 , \\ , 2 } \\cap L _ { j + 1 } = 0 . \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{gather*} u _ k ( e ^ { \\frac { j \\pi i } { p } } \\zeta ) = e ^ { \\frac { ( m - 2 k ) j \\pi i } { p } } u _ k ( \\zeta ) , v _ k ( e ^ { \\frac { j \\pi i } { p } } \\zeta ) = e ^ { \\frac { ( l - 2 k ) j \\pi i } { p } } v _ k ( \\zeta ) \\end{gather*}"}
-{"id": "6798.png", "formula": "\\begin{align*} a ^ { 2 } \\left ( x _ { 1 } \\right ) \\geq 0 , \\left \\Vert \\partial _ { x _ { 1 } } ^ { j } a \\left ( x _ { 1 } \\right ) \\right \\Vert _ { L ^ { \\infty } \\left ( \\mathbb { R } \\right ) } \\leq C _ { j } n ^ { - j } , j = 1 , 2 , . . . , \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} T _ { \\varepsilon , \\eta , d , k , B } = \\frac { ( \\varepsilon - \\eta ) k K ^ 2 } { \\alpha ^ 2 d ^ 2 } \\frac { B ^ { 2 - \\frac { 1 } { r } } } { N ^ 2 } + \\frac { \\eta K ^ 2 } { \\alpha ^ 2 d ^ 2 } \\frac { B ^ { 2 - \\frac { 1 } { r } } } { N ^ 2 } + O \\left ( \\frac { k ^ \\sigma K ^ \\sigma B ^ \\sigma } { d ^ \\sigma N ^ \\sigma } \\right ) . \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} ( x _ 1 , x _ 2 , x _ 3 ) & = \\sigma ^ { ( g _ 1 , g _ 2 , g _ 3 ) } ( x _ 1 , x _ 2 , x _ 3 ) \\\\ & = ( \\sigma ^ { ( g _ 1 , 1 _ { G _ 2 } , 1 _ { G _ 3 } ) } ( x _ 1 ) , \\sigma ^ { ( 1 _ { G _ 1 } , g _ 2 , 1 _ { G _ 3 } ) } ( x _ 2 ) \\sigma ^ { ( 1 _ { G _ 1 } , 1 _ { G _ 2 } , g _ 3 ) } ( x _ 3 ) ) . \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} \\left \\langle \\vec { w } , \\Upsilon \\left ( t \\right ) \\vec { w } \\right \\rangle _ { \\mathbb { R } ^ { d } } = \\int \\nolimits _ { \\mathbb { R } } \\cos \\left ( t \\nu \\right ) \\left \\langle \\vec { w } , \\mu _ { \\Upsilon } \\left ( \\mathrm { d } \\nu \\right ) \\vec { w } \\right \\rangle _ { \\mathbb { R } ^ { d } } \\ , t \\in \\mathbb { R } \\ , \\ \\vec { w } \\in \\mathbb { R } ^ { d } \\ . \\end{align*}"}
-{"id": "10001.png", "formula": "\\begin{align*} W ( G _ { 2 } ( n , d , x , 1 ) ) & = W ( G _ { 2 } ( n , d , x , 0 ) ) + 1 , \\\\ W ( G _ { 2 } ( n , d , x , 2 ) ) & = W ( G _ { 2 } ( n , d , x , 0 ) ) + 6 , \\\\ W ( G _ { 2 } ( n , d , x , - 1 ) ) & = W ( G _ { 2 } ( n , d , x , 0 ) ) + 3 . \\end{align*}"}
-{"id": "1354.png", "formula": "\\begin{align*} - \\tilde { u } _ { \\varepsilon } '' = \\frac { 1 } { \\Lambda } ( \\tilde { y } _ { \\varepsilon } - \\tilde { u } _ { \\varepsilon } ) \\exp ( \\hat { u } _ { \\varepsilon } ) , \\label { e q : a p p r o x i m a t e s y s t e m u } \\end{align*}"}
-{"id": "9901.png", "formula": "\\begin{align*} & \\textrm { P r } ( X _ \\beta ( t + h ) = j \\mid X _ \\beta ( t ) = i ) \\\\ = & \\begin{cases} ( \\bar F _ j ( \\rho ) - \\bar F _ i ( \\rho ) ) _ + h + o ( h ) \\ , \\quad & \\textrm { i f } ~ j \\in N ( i ) \\ ; \\\\ 1 - \\sum _ { j \\in N ( i ) } ( \\bar F _ j ( \\rho ) - \\bar F _ i ( \\rho ) ) _ + h + o ( h ) \\ , & \\textrm { i f } ~ j = i \\ ; \\\\ 0 \\ , & \\textrm { o t h e r w i s e } \\ , \\end{cases} \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} X _ k = S _ { \\Delta t } ^ { k } x + \\Delta t \\sum _ { \\ell = 0 } ^ { k - 1 } S _ { \\Delta t } ^ { k - \\ell } G ( X _ { \\ell } ) + \\sum _ { \\ell = 0 } ^ { k - 1 } S _ { \\Delta t } ^ { k - \\ell } \\sigma ( X _ \\ell ) \\Delta W _ \\ell . \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} \\inf _ { ( u ^ 1 , u ^ 2 , . . . , u ^ k ) \\in \\mathcal { A } ^ k } \\sum _ { l = 1 } ^ { k } I ^ { l } ( v ^ l ) + \\Phi ^ k ( u ^ 1 , u ^ 2 , . . . , u ^ k ) , \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} F _ { 2 u } = F _ u L _ u \\end{align*}"}
-{"id": "17.png", "formula": "\\begin{align*} \\textrm { T r } = 0 \\implies R _ { 1 c } - 1 = 0 \\implies \\frac { \\mu K _ 1 ( 1 - b ) } { ( 1 + c ^ 2 ) ( 1 + c ) ^ 2 } - \\frac { \\Gamma K } { ( K + c ) ^ 2 } - 1 = 0 . \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{align*} \\dot { z } z ^ { - 1 } = \\dot { y } y ^ { - 1 } + O ( t ^ { - 2 } \\mathcal L ) . \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{align*} d \\lambda _ i ( t ) = \\sqrt { 2 ( 1 + \\lambda ^ 2 _ i ( t ) ) } d \\beta _ i ( t ) + \\left ( 2 \\Im ( s ) + \\left ( 2 - 2 N - 2 \\Re ( s ) \\right ) \\lambda _ i ( t ) + \\sum _ { j \\ne i } ^ { } \\frac { 2 \\left ( 1 + \\lambda ^ 2 _ i ( t ) \\right ) } { \\lambda _ i ( t ) - \\lambda _ j ( t ) } \\right ) d t . \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} \\Phi _ 0 ( \\lambda ) : & = \\sigma _ 1 ^ { - 1 } \\sqrt { \\lambda } \\sin ( \\sigma _ 1 \\sqrt { \\lambda } ) [ \\cos ( \\sigma _ 2 \\sqrt { \\lambda } ) + a ] \\\\ & = ( 1 + a ) \\lambda \\prod _ { n = 1 } ^ \\infty \\left ( 1 - \\frac { \\lambda } { \\alpha _ n } \\right ) \\prod _ { n = 0 } ^ \\infty \\left ( 1 - \\frac { \\lambda } { \\beta _ n } \\right ) . \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} A & : = \\left ( \\lambda + \\frac { 2 a } { \\lambda } \\left ( 1 - \\frac { 1 } { n } \\right ) \\right ) \\left ( 1 - \\frac { 1 } { n } \\right ) > 0 , \\\\ B & : = \\lambda \\theta \\left ( 2 - \\frac { 1 } { n } \\right ) - \\varepsilon \\left ( 1 - \\frac { 1 } { n } \\right ) ^ 2 + 2 a , \\\\ C & : = \\lambda ( \\theta ^ 2 - \\varepsilon ) < 0 . \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} \\phi _ w ( t ) : = ( t - \\bar t ) ^ { 1 / 2 } \\big ( \\| w ( t ) \\| _ \\infty + \\| \\nabla w ( t ) \\| _ 3 \\big ) + ( t - \\bar t ) ^ { 1 / 4 } \\| w ( t ) \\| _ 6 . \\end{align*}"}
-{"id": "9657.png", "formula": "\\begin{align*} \\theta ( X ) = - X \\ , , \\qquad \\theta ( S ) = - T , \\qquad \\theta ( T ) = - S . \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} F _ 2 ( z ) = { - } F _ 1 ( z ) \\int { \\frac { F _ 0 ( z ) } { F _ 1 ( z ) ^ 2 } } d z , \\end{align*}"}
-{"id": "8187.png", "formula": "\\begin{align*} ( g \\circ h ) ^ * \\omega = ( h ' \\circ g ) ^ * \\omega \\implies h ^ * ( g ^ * \\omega ) = g ^ * \\omega \\ , . \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} \\mathbf A : = \\Bigl \\{ ( x _ n ) _ { n \\ge 1 } \\in \\mathbf H ; \\ ; x _ 1 + \\cdots + x _ N \\in B ( a , \\varepsilon ) \\quad { \\rm a n d } \\sum _ { n > N } \\Vert x _ n \\Vert ^ 2 < \\varepsilon ^ 2 \\Bigr \\} . \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} \\mathcal { A } ^ P = \\left ( \\mathbb { Q } ^ { - 1 } \\mathcal { A } ^ { ( P Q ^ { - 1 } , Q ) } \\right ) _ { Q \\in S _ n } . \\end{align*}"}
-{"id": "2540.png", "formula": "\\begin{align*} \\left [ q _ { 1 } ^ { \\prime } + \\frac { q _ { 2 } R } { 2 B _ { 2 } q _ { 3 } } - \\frac { B _ { 2 } ^ { \\prime } } { B _ { 2 } } q _ { 1 } \\right ] f + \\left [ 2 q _ { 1 } + q _ { 2 } ^ { \\prime } - \\frac { 1 } { 2 } \\frac { B _ { 2 } ^ { \\prime } } { B _ { 2 } } q _ { 2 } - \\frac { 1 } { 2 } \\left ( q _ { 2 } + q _ { 3 } ^ { \\prime } \\right ) \\frac { q _ { 2 } } { q _ { 3 } } + \\frac { R } { B _ { 2 } } \\right ] f ^ { \\prime } = 0 . \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} \\Lambda ^ { \\pm } = v \\pm ( b - e ) . \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} \\gamma \\alpha ^ 2 & = 0 , & \\alpha ^ 2 \\sigma & = 0 , & \\sigma \\beta ^ 2 & = 0 , & \\beta ^ 2 \\gamma & = 0 , \\\\ \\alpha \\sigma \\beta & = 0 , & \\beta \\gamma \\alpha & = 0 , & \\sigma \\gamma \\sigma & = 0 , & \\gamma \\sigma \\gamma & = 0 , \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{align*} f \\left ( g \\left ( z \\right ) \\right ) = \\left ( \\frac { z } { e ^ { z } - 1 } \\right ) ^ { q } - 1 = \\left ( \\frac { 1 } { 1 - g \\left ( z \\right ) } \\right ) ^ { q } - 1 \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} \\left \\langle I + \\begin{pmatrix} 1 & 0 \\\\ 0 & d \\end{pmatrix} p , I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle \\left \\langle I + \\begin{pmatrix} d & 0 \\\\ 0 & 1 \\end{pmatrix} p , I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle . \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} m _ { R } ( T ^ { - 1 } A ) & = \\int _ { K _ { u } } m _ { 0 } \\bigl ( R ( T ^ { - 1 } A - x ) \\bigr ) \\ , d \\nu _ { u , T } ( x ) = \\int _ { K _ { u } } m _ { 0 } \\bigl ( R ( T ^ { - 1 } ( A - T x ) ) \\bigr ) \\ , d \\nu _ { u , T } ( x ) \\\\ & = \\int _ { T ( K _ { u } ) } m _ { 0 } \\bigl ( R ( A - x ) \\bigr ) \\ , d \\nu _ { u , T } ( x ) = m _ { R } ( A ) . \\end{align*}"}
-{"id": "845.png", "formula": "\\begin{align*} \\eta _ { 1 , \\varepsilon } ( x ) \\rightarrow \\sum _ { i = 1 } ^ 3 b _ { 1 , i } \\frac { \\partial U _ { a _ 1 } ( x ) } { \\partial x ^ i } \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} \\omega \\cdot N = 0 \\textrm { o n } S . \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} \\hat { 1 } _ D ( e _ j ) = \\sum _ { y \\in D } ( - 1 ) ^ { y _ j } = \\sum _ { y \\in D } ( 1 - 2 y _ j ) = | D | - 2 \\sum _ { y \\in D } y _ j , \\end{align*}"}
-{"id": "6690.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 1 6 \\pi ^ { 2 } } \\int \\limits _ { | x - e | + | x | = 2 \\tau } \\frac { q ( x ) } { | 2 \\tau x - | x | e | } d S _ { x } = - \\int \\limits _ { | x | + | x - e | \\leq 2 \\tau } \\frac { q ( x ) } { | x | | x - e | } F ( \\tau , x ) d x . \\end{aligned} \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} z = \\left ( \\begin{array} { c } - 3 3 . 1 8 0 7 + 2 . 4 5 8 9 i \\\\ - 5 6 . 9 5 7 4 - 3 . 3 3 2 3 i \\\\ - 4 2 . 5 6 8 7 + 5 6 . 7 6 6 9 i \\end{array} \\right ) \\end{align*}"}
-{"id": "6583.png", "formula": "\\begin{align*} \\alpha _ \\mathrm { f } \\ ; = \\ ; \\frac { e ^ 2 } { \\hbar c } \\ ; \\approx \\ ; \\frac { 1 } { 1 3 7 } \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} \\sigma ^ { 2 p } - \\overline { \\mathstrut \\Lambda _ n } = \\left ( \\sigma ^ 2 - \\lambda _ 0 ^ 2 \\right ) \\cdot \\prod \\limits _ { i = 1 } ^ { p - 1 } \\left ( \\sigma - \\lambda _ i \\right ) \\cdot \\left ( \\sigma - \\overline { \\mathstrut \\lambda _ i } \\right ) , \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} \\dot { A } = \\lambda A + \\langle \\tilde { G } , \\widehat { N ( g ) } _ 1 \\rangle = \\lambda A + ( c _ 3 ^ { ( 1 ) } + c _ 3 ^ { ( 2 ) } + c _ 3 ^ { ( 3 ) } ) | A | ^ 2 A . \\end{align*}"}
-{"id": "8916.png", "formula": "\\begin{align*} f ( \\lambda _ k ) , f ' ( \\lambda _ k ) , \\ldots , f ^ { ( m _ k - 1 ) } ( \\lambda _ k ) ( k = 1 , \\ldots , s ) \\end{align*}"}
-{"id": "9302.png", "formula": "\\begin{align*} p _ { j k , j } p _ { i j , i } + p _ { j k , k } p _ { i k , i } = p _ { i j , i } p _ { i k , i } \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{align*} v ( \\lambda ^ { n } k ) = \\lambda ^ { n / 2 } B ^ { - 1 } ( \\lambda ^ { n - 1 } k ) \\cdot \\ldots \\cdot B ^ { - 1 } ( k ) v ( k ) , \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} e ^ { \\alpha } _ y ( z ) & = \\exp ( \\sum _ { n \\ge 1 } y _ n z ^ n ) \\exp ( - \\sum _ { n \\ge 1 } \\frac { \\partial } { n \\partial y _ n } z ^ { - n } ) e ^ { \\alpha } z ^ { \\partial _ { \\alpha } } , \\\\ e ^ { - \\alpha } _ y ( z ) & = \\exp ( - \\sum _ { n \\ge 1 } y _ n z ^ n ) \\exp ( \\sum _ { n \\ge 1 } \\frac { \\partial } { n \\partial y _ n } z ^ { - n } ) e ^ { - \\alpha } z ^ { - \\partial _ { \\alpha } } , \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{align*} \\left [ P ^ { s , N } _ { H P } ( t ) f \\right ] ( x ) = \\left [ \\mathcal { S } ^ N ( t ) f \\circ \\mathsf { e v a l } _ N \\right ] ( H ) = \\left [ \\mathcal { S } ^ N ( t ) f \\circ \\mathsf { e v a l } _ N \\right ] ( U ^ * x U ) \\ , \\forall U \\in \\mathbb { U } ( N ) , \\end{align*}"}
-{"id": "7568.png", "formula": "\\begin{align*} \\Pi ^ \\omega _ W ( q , K , \\Delta ) & : = \\pi _ W ^ \\omega - q \\Delta + \\left ( p ^ { \\omega , * } - K \\right ) ^ + \\Delta , \\\\ \\Pi ^ \\omega _ P ( q , K , \\Delta ) & : = \\pi _ P ^ \\omega + q \\Delta - \\left ( p ^ { \\omega , * } - K \\right ) ^ + \\Delta , \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} B _ { \\alpha } ( x ; \\varepsilon ) : = \\{ y \\in X : p _ { \\alpha } ( y - x ) < \\varepsilon \\} \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} P _ { n } x = \\sum _ { k = b _ { n } } ^ { b _ { n + 1 } - 1 } x _ { k } e _ { k } . \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\beta \\\\ & \\alpha - \\beta + 1 \\end{matrix} \\bigg | \\ , - 1 \\bigg ] = \\frac { \\Gamma ( \\alpha - \\beta + 1 ) \\Gamma ( \\frac 1 2 \\alpha + 1 ) } { \\Gamma ( \\alpha + 1 ) \\Gamma ( \\frac { 1 } { 2 } \\alpha - \\beta + 1 ) } . \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{align*} & \\partial _ t w + A w + \\mathbb P [ S w + ( J _ k w ) \\cdot \\nabla w ] = \\mathbb P f , \\\\ & w ( 0 ) = w _ 0 , \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} a ^ { \\textnormal { h o m } } _ { i j r s } : = \\sum _ { k = 1 } ^ n \\sum _ { l = 1 } ^ d \\int _ Y a _ { i j k l } \\big ( \\partial _ { l } N ^ { ( r s ) } _ { k } ( y ) + \\delta _ { k r } \\delta _ { l s } \\big ) \\ , { \\textnormal { d } } y ( { i , r \\in \\{ 1 , \\ldots , n \\} , j , s \\in \\{ 1 , \\ldots , d \\} } ) , \\end{align*}"}
-{"id": "9829.png", "formula": "\\begin{align*} \\omega _ { \\psi } ^ { \\mathbf { z } ^ { - 1 } } ( \\pi _ { \\mathbf { z } ^ { - 1 } } ( t _ { - \\lambda } ) \\Phi _ { s _ i w } ^ { \\mathbf { z } ^ { - 1 } } ) = \\mathcal { T } _ i \\ , \\omega _ { \\psi } ^ { \\mathbf { z } ^ { - 1 } } ( \\pi _ { \\mathbf { z } ^ { - 1 } } ( t _ { - \\lambda } ) \\Phi _ { w } ^ { \\mathbf { z } ^ { - 1 } } ) . \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} X _ { l } : = \\mathbb { E } ( \\hat { U } ^ { ( n ) } _ n | { \\cal F } _ l ) - \\mathbb { E } ( \\hat { U } ^ { ( n ) } _ n | { \\cal F } _ { l - 1 } ) . \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} L _ { t } ^ { A } \\left ( \\rho \\right ) : = \\int _ { t _ { 0 } } ^ { t } \\rho \\circ \\tau _ { t , t _ { 0 } } \\left ( \\partial _ { t } A _ { t } \\right ) \\mathrm { d } t \\ . \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} \\partial _ \\lambda \\Lambda ( \\gamma = 0 , \\lambda = 0 ) = \\frac { c } { 2 \\sqrt { 2 \\pi } } . \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{align*} Q ( 2 \\tau ) : = \\int \\limits _ { | x | + | x - e | = 2 \\tau } \\frac { q ( x ) } { | 2 \\tau x - | x | e | } d S _ { x } . \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} x ^ n _ i : = \\min \\{ x \\in ( i / n , ( i + 1 ) / n ) : \\pi \\in { \\cal S } ( \\widehat { \\pi } ^ { ( ( i + 1 ) / n , 0 ) } , \\widehat { \\pi } ^ { ( i / n , 0 ) } ) \\sigma _ \\pi \\leq - 1 \\pi ( 0 ) = x \\} . \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{align*} m ( \\Lambda ( a ) ) \\geq | Z ( \\Lambda ( a ) ) | \\geq \\mathrm { R e } Z ( \\Lambda ( a ) ) = X ( L ) \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} J _ { \\alpha A } ( 2 p _ k - x _ k ) = \\alpha ( b ^ k + D u ^ { k + 1 } - d ^ k ) \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{align*} \\psi _ i = \\big ( e _ i \\otimes e _ { t ( f ( \\alpha ) ) } \\big ) f ^ 2 ( \\alpha ) + \\big ( e _ i \\otimes e _ { t ( f ( \\bar { \\alpha } ) ) } \\big ) f ^ 2 ( \\bar { \\alpha } ) - \\alpha \\big ( e _ { t ( \\alpha ) } \\otimes e _ i \\big ) - \\bar { \\alpha } \\big ( e _ { t ( \\bar { \\alpha } ) } \\otimes e _ i \\big ) . \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} \\frac { ( - a - p ) _ { p + k } ( p - a ) _ { p + k } } { ( 1 ) _ { p + k } ^ 2 } = & \\frac { \\Gamma _ p ( - a + k ) } { \\Gamma _ p ( - a - p ) } \\cdot \\frac { \\Gamma _ p ( 2 p - a + k ) } { \\Gamma _ p ( p - a ) } \\cdot \\frac { \\Gamma _ p ( 1 ) ^ 2 } { \\Gamma _ p ( p + k + 1 ) ^ 2 } \\cdot \\frac { ( - p ) \\cdot p } { p ^ 2 } \\\\ \\equiv & - \\frac { ( - a ) _ k ^ 2 } { ( 1 ) _ { k } ^ 2 } \\cdot \\big ( 1 + 2 p ( H _ { p - 1 - a + k } - H _ k ) \\big ) \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} \\sum _ { \\alpha : i \\to j } [ \\phi _ \\alpha ^ * , \\phi _ \\alpha ] = 0 . \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{align*} _ { \\mu , \\sigma } D _ { v , q ; z } ^ { \\alpha , \\eta , p } ( z ^ { \\lambda } ) = \\frac { z ^ { \\eta + \\lambda + \\alpha } } { \\Gamma ( \\alpha ) } B _ { v , q } ^ { \\left ( \\mu , \\sigma \\right ) } ( \\eta + \\lambda , \\alpha - 1 ; p ) \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } \\beta } \\sum _ { k = 1 } ^ { p - 1 } C _ k H _ k x ^ { k + 1 } = ( 1 - 2 \\beta ) \\sum _ { k = 1 } ^ { p - 1 } \\binom { 2 k } { k } H _ k x ^ k . \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} W ^ T = x ^ 2 + y ^ 3 w + z ^ 9 + w ^ { 1 2 } \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { q ^ n } { 1 \\pm q ^ n } & = \\frac { 1 } { ( \\mp q ; q ) _ { \\infty } } \\sum _ { n \\geq 1 } \\left ( s _ o ( n ) \\pm s _ e ( n ) \\right ) q ^ n , \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} \\begin{aligned} & w ( x , t ) < e ^ { \\alpha t } c < e ^ { \\alpha b } c = \\hat { c } \\qquad , \\\\ & w ( y , b ) = e ^ { \\alpha b } c = \\hat { c } \\qquad \\end{aligned} \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\rightarrow 0 } \\left \\Vert M \\left ( t , \\varepsilon \\right ) \\right \\Vert _ { B \\left ( E \\right ) } = 0 , \\lim _ { \\varepsilon \\rightarrow 0 } \\left \\Vert N \\left ( t , \\varepsilon \\right ) \\right \\Vert _ { B \\left ( E \\right ) } \\leq C _ { 0 } . \\end{align*}"}
-{"id": "2281.png", "formula": "\\begin{align*} s \\mapsto H ( \\lambda , u ( s ) ) = \\frac { a s + b } { c s + d } , \\end{align*}"}
-{"id": "9453.png", "formula": "\\begin{align*} \\varepsilon \\Big [ \\partial _ { t } ( h v ) + \\nabla \\cdot ( h v \\otimes v ) - ( h d \\cdot \\nabla ) d + ( - \\triangle ) ^ l v \\Big ] + h v = \\nabla \\cdot \\left ( \\frac { B \\otimes B } { h } \\right ) + \\nabla \\left ( h ^ { - 1 } \\right ) , \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} s ( x ) = \\lim _ { r \\to 0 } \\inf \\{ s > 0 : \\eqref { e q - l o c a l - d i m - c o n d } B ' \\subset B \\subset B ( x , r ) \\} , \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} \\ ! \\ ! \\ ! \\sum _ { \\kappa _ { 1 , n } \\in S _ 1 , \\ n \\ge 0 } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\frac { ( \\kappa _ { 1 , n } - n ^ 2 \\pi ^ 2 / \\sigma _ 1 ^ 2 ) _ + } { n ^ 2 } + \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\sum _ { \\kappa _ { 2 , n } \\in S _ 2 , \\ n \\ge 0 } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\frac { ( \\kappa _ { 2 , n } - \\gamma _ n ^ 2 / \\sigma _ 2 ^ 2 ) _ + } { n ^ 2 } < \\infty , \\sigma _ 1 + \\sigma _ 2 = 2 b , \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} A _ 2 ( { \\tilde X ' } , \\tilde L ' ) = 0 . \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{align*} F _ i ( \\mathbf u ^ { ( l ) } ) = \\frac { ( \\alpha _ i - \\alpha _ 1 ) ( \\alpha _ i - \\alpha _ 2 ) \\cdots ( \\alpha _ i - \\alpha _ { \\check i } ) \\cdots ( \\alpha _ i - \\alpha _ { n } ) } { ( \\alpha _ i - \\lambda _ 1 ) ( \\alpha _ i - \\lambda _ 2 ) \\cdots ( \\alpha _ i - \\lambda _ { n + 1 } ) } \\cdot ( \\alpha _ i ) ^ { l } . \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} D _ { \\Gamma } ^ { ( s ) } = \\bigsqcup _ { j _ 1 , \\dots , j _ n = 1 } ^ { | \\mathcal { E } | } D _ { j _ 1 \\dots j _ n } ^ { ( s ) } \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{align*} m _ s ^ { ( n ) } ( x ) = ( 1 + x ^ 2 ) ^ { - \\Re ( s ) - n } e ^ { 2 \\Im ( s ) A r g ( 1 + i x ) } , \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{align*} f \\left ( f _ { c _ { 1 } } - \\gamma f _ { c _ { 2 } } \\right ) = \\left ( 1 - \\gamma \\right ) q . \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} ( \\Phi \\otimes \\mathbb { I } _ M ) \\left ( \\hat { D } _ A ( \\mathbf { x } ) \\ , \\hat { \\rho } _ { A M } \\ , { \\hat { D } _ A ( \\mathbf { x } ) } ^ \\dag \\right ) = \\hat { D } _ B ( K \\mathbf { x } ) \\ , ( \\Phi \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) \\ , { \\hat { D } _ B ( K \\mathbf { x } ) } ^ \\dag \\ ; . \\end{align*}"}
-{"id": "9746.png", "formula": "\\begin{align*} B ( x ) = \\int _ { \\mathbb { R / Z } } \\delta ( x - X ( s ) ) X ' ( s ) d s , \\ ; \\ ; \\ ; x \\in \\mathbb { T } ^ d , \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} v \\cdot \\nabla _ x \\ln ( \\phi ) ( x , v ) = ( \\sigma _ 1 - \\sigma _ 2 ) ( x , v ) - { \\int _ { - \\tau _ - ( x , v ) } ^ { \\tau _ + ( x , v ) } ( \\sigma _ 1 - \\sigma _ 2 ) ( x + s v , v ) d s \\over \\tau ( x , v ) } , \\ ( x , v ) \\in X \\times V , \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} H ' \\cap \\ker \\varphi = \\left \\langle I + \\begin{pmatrix} \\alpha & 0 \\\\ \\gamma & \\alpha \\end{pmatrix} p , I + \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} , I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{align*} C '' _ { i 1 } : = C ' _ { i 1 } - \\Big ( \\frac { ( \\lambda _ 1 - \\lambda _ i ) x _ i } { ( \\lambda _ 1 - \\lambda _ 2 ) x _ 2 } \\Big ) C ' _ { 2 1 } = \\frac { ( \\lambda _ 2 - \\lambda _ i ) x _ i } { ( \\lambda _ 1 - \\lambda _ 2 ) x _ 1 } ( \\lambda _ 1 s - t ) , \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} I + \\begin{pmatrix} \\alpha & 0 \\\\ 0 & \\delta \\end{pmatrix} p \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} 0 = [ f , 1 ] _ n | _ a ^ b & = \\lim _ { x \\to b ^ - } [ - a _ 1 f ' + ( a _ 2 f '' ) ' - \\dots + ( - 1 ) ^ n ( a _ n f ^ { ( n ) } ) ^ { ( n - 1 ) } ] \\\\ & \\quad \\ , - \\lim _ { x \\to a ^ + } [ - a _ 1 f ' + ( a _ 2 f '' ) ' - \\dots + ( - 1 ) ^ n ( a _ n f ^ { ( n ) } ) ^ { ( n - 1 ) } ] \\\\ & = \\lim _ { x \\to b ^ - } ( a _ n f ^ { ( n ) } ) ^ { ( n - 1 ) } - \\lim _ { x \\to a ^ + } ( a _ n f ^ { ( n ) } ) ^ { ( n - 1 ) } . \\end{align*}"}
-{"id": "3723.png", "formula": "\\begin{align*} m _ { j } ( E ) & = \\int _ { E _ j ^ - } ^ { E _ j ^ + } \\frac { f _ j ( E , x ) } { \\sqrt { ( E _ j ^ + - x ) ( x - E _ j ^ - ) } } d x \\\\ & = \\sum _ { k = 1 } ^ \\infty a _ k ^ j ( E ) \\int _ { E _ j ^ - } ^ { E _ j ^ + } \\frac { x ^ k } { \\sqrt { ( E _ j ^ + - x ) ( E _ j ^ - - x ) } } d x \\\\ & = \\sum _ { \\vec { m } \\in [ 1 , 2 ( n + 1 ) ] ^ n } c _ { \\vec { m } } E ^ { \\vec { m } } , \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} P _ { t } \\left ( D \\right ) u _ { i } + \\sum \\limits _ { j = 1 } ^ { N } \\left ( a _ { i j } + \\lambda \\right ) u _ { j } \\left ( x \\right ) = f _ { i } \\left ( x \\right ) x \\in R ^ { n } , \\end{align*}"}
-{"id": "7892.png", "formula": "\\begin{align*} W _ \\phi & = \\{ x \\in \\R ^ n \\mid | x | _ q \\le 1 \\} \\\\ B _ \\phi & = \\{ x \\in \\R ^ n \\mid | x | _ p \\le 1 \\} . \\end{align*}"}
-{"id": "5109.png", "formula": "\\begin{align*} x \\left ( z \\right ) = \\frac { g \\left ( z \\right ) } { 1 - g \\left ( z \\right ) } , \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} 2 g ( E ) - 2 = ( K _ X + E ) \\cdot E < - D \\cdot E + E ^ 2 \\leq - 1 . \\end{align*}"}
-{"id": "270.png", "formula": "\\begin{align*} \\theta ( \\mathsf { P } _ { j } ^ { i } ) = \\sum _ { m , n } c ^ m _ n q ^ { - ( 2 \\rho , \\lambda _ m - \\lambda _ n ) } \\pi ( K _ { 2 \\rho } ^ { - 1 } ) _ { i } ^ { i } ( \\mathsf { M } _ { m } ^ { n } ) _ { j } ^ { i } \\pi ( K _ { 2 \\rho } ) _ { j } ^ { j } . \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} { \\rm T r } = - \\frac { \\Gamma K } { ( K + c ^ { \\star } ) ^ 2 } - 1 < 0 , \\ , \\ , { \\rm D e t } = \\frac { \\Gamma K } { ( K + c ^ { \\star } ) ^ 2 } - \\lambda T ' ( c ^ { \\star } ) , \\ , \\ , { \\rm D i s c r } = \\left ( \\frac { \\Gamma K } { ( K + c ^ { \\star } ) ^ 2 } - 1 \\right ) ^ 2 + 4 \\lambda T ' ( c ^ { \\star } ) > 0 . \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\biggl \\| \\int _ 0 ^ { t _ A } ( 4 t - s ) F ( 4 t - s ) * ( u _ n \\theta _ n ) ( s ) \\dd s \\biggr \\| _ 3 = \\lim _ { t \\to + \\infty } \\sqrt { 4 t } \\biggl \\| \\int _ 0 ^ { t _ A } ( 4 t - s ) F ( 4 t - s ) * ( u _ n \\theta _ n ) ( s ) \\dd s \\biggr \\| _ \\infty = 0 . \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} P ^ * \\tilde K u ( x , w ) = \\int _ V k ( x , w , v ) \\int _ 0 ^ { \\tau _ + ( x , v ) } e ^ { - \\int _ 0 ^ r \\sigma ( x + s v , v ) d s } \\int _ V k ( x + r v , v , v ' ) \\rho ( u ) ( x + r v , v ' ) d v ' d r d v , \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{align*} t ^ { \\frac { 1 } { 2 } } v ( t ) = z ( s ) , t = \\log s . \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} \\int _ { \\Omega _ { R , \\delta } } ( u - c ) ^ \\perp \\omega _ { \\mathrm { a c } } \\ , d x & = \\int _ { \\Omega \\cap \\partial B _ R ( 0 ) } B N \\ , d S . \\end{align*}"}
-{"id": "3949.png", "formula": "\\begin{align*} & { } _ 3 F _ 2 \\bigg [ \\begin{matrix} \\alpha & \\beta & \\gamma \\\\ & \\alpha - \\beta + 1 & \\alpha - \\gamma + 1 \\end{matrix} \\bigg | \\ , z \\bigg ] \\\\ = & ( 1 - z ) ^ { - \\alpha } { } _ 3 F _ 2 \\bigg [ \\begin{matrix} \\alpha - \\beta - \\gamma + 1 & \\frac 1 2 \\alpha & \\frac 1 2 \\alpha + \\frac 1 2 \\\\ & \\alpha - \\beta + 1 & \\alpha - \\gamma + 1 \\end{matrix} \\bigg | \\ , - \\frac { 4 z } { ( 1 - z ) ^ 2 } \\bigg ] . \\end{align*}"}
-{"id": "5278.png", "formula": "\\begin{align*} \\mathbf { I I I } & = - \\int _ { \\partial B _ R ( 0 ) \\cap \\Omega } \\left [ \\left ( \\varphi - { m \\cdot x \\over \\gamma _ n | x | ^ n } \\right ) ( c \\cdot N ) - ( c \\cdot x ) N \\cdot ( \\nabla \\varphi + V ) \\right ] \\ , d S + O \\left ( \\frac { 1 } { R ^ n } \\right ) . \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} \\begin{cases} \\beta _ 1 \\beta _ 5 = \\alpha _ 1 \\beta _ 6 \\\\ \\alpha _ 4 \\beta _ 6 = \\beta _ 3 \\beta _ 5 \\end{cases} \\end{align*}"}
-{"id": "9345.png", "formula": "\\begin{align*} \\theta ^ { \\frac 1 r } - ( 1 - \\theta ) ^ { \\frac 1 r } < \\frac { \\mathcal { V } _ r ( \\sigma ; [ L - 1 , L ] ) } { \\mathcal { V } _ r ( \\sigma ; [ M , N ] ) } = \\frac { | F _ L | } { A } \\leq 1 , \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{align*} \\frac { f _ m } { f _ { m - 1 } } = \\frac { q ^ m } { 1 - q ^ { 2 m } } \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} \\partial _ { t } f = S f + K f + \\left ( a + i b \\right ) \\left ( V f + e ^ { \\gamma \\varphi } F \\right ) R ^ { n } \\times \\left [ 0 , 1 \\right ] \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{align*} & & \\alpha _ i \\beta _ i & = 0 , & \\beta _ i \\alpha _ i & = 0 , & ( \\alpha _ i \\alpha _ { i + 1 } \\dots \\alpha _ { i - 1 } ) ^ p & = ( \\beta _ { i - 1 } \\beta _ { i - 2 } \\dots \\beta _ { i } ) ^ q , & & \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} \\tilde { u } \\left ( x , t \\right ) = \\left ( \\sqrt { \\alpha \\beta } \\mu \\left ( t \\right ) \\right ) ^ { \\frac { n } { 2 } } u \\left ( \\sqrt { \\alpha \\beta } x \\mu \\left ( t \\right ) , \\beta t \\mu \\left ( t \\right ) \\right ) e ^ { \\eta } . \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} \\phi \\in N ( L - \\lambda _ { k } I ) , \\ ; \\| \\phi \\| = 1 \\iff \\phi = \\sum _ { n \\in J _ { k } } c _ { n } \\phi _ { n } \\mbox { w i t h } \\sum _ { n \\in J _ { k } } | c _ { n } | ^ 2 = 1 . \\end{align*}"}
-{"id": "9161.png", "formula": "\\begin{align*} \\lambda _ { \\max } = \\frac { \\mu ^ { 2 } } { 2 } ( a ^ { 2 } + b ^ { 2 } + 2 c ^ { 2 } + 2 ) + \\frac { \\mu ^ { 2 } } { 2 } \\sqrt { ( a ^ { 2 } - b ^ { 2 } - 4 c ) ^ { 2 } + 4 ( a c + b c + a - b ) ^ { 2 } } . \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} \\partial _ t m - d _ m \\Delta _ x m = - \\underbrace { \\lambda _ { m M _ 1 } \\frac { m } { k _ { M _ 1 } + m } } _ { t r a n s f o r m a t i o n \\ , i n t o \\ , M _ 1 } - \\underbrace { \\lambda _ { m M _ 2 } \\frac { m } { k _ { M _ 2 } + m } } _ { t r a n s f o r m a t i o n \\ , i n t o \\ , M _ 2 } \\ \\ \\underbrace { - \\ \\ \\beta _ m \\ , m } _ { d e a t h \\ , o f \\ , m } , \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} D ( \\alpha ^ 4 ) \\stackrel { . } { = } 4 ( D \\alpha ) \\alpha ^ 3 = 4 \\gamma \\alpha ^ 3 \\end{align*}"}
-{"id": "1266.png", "formula": "\\begin{align*} v ( \\mathsf { x } , t ) = \\int _ { \\mathbb { R } ^ { d } } \\mathsf { d y } g ( \\mathsf { x } , T - t , \\mathsf { y } ) \\psi ( \\mathsf { y ) , } \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} | J _ { i j } | = a _ { i j } \\hbox { f o r e v e r y } i , j \\hbox { w i t h } u _ i \\neq u _ j \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{align*} R _ j ^ * ( F ) = ( f _ { i _ 1 } , \\dots , f _ { i _ j + 1 } , \\dots , f _ { i _ n } ) , \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{align*} & \\mathcal { B } _ \\eta \\left ( \\hat { D } _ A ( \\mathbf { x } ) \\ , \\hat { D } _ B ( \\mathbf { y } ) \\ , \\hat { \\rho } _ { A B } \\ , { \\hat { D } _ A ( \\mathbf { x } ) } ^ \\dag \\ , { \\hat { D } _ B ( \\mathbf { y } ) } ^ \\dag \\right ) \\\\ & = \\hat { D } _ C \\left ( \\sqrt { \\eta } \\ , \\mathbf { x } + \\sqrt { \\eta - 1 } \\ , T \\mathbf { y } \\right ) \\ , \\mathcal { B } _ \\eta ( \\hat { \\rho } _ { A B } ) \\ , { \\hat { D } _ C \\left ( \\sqrt { \\eta } \\ , \\mathbf { x } + \\sqrt { \\eta - 1 } \\ , T \\mathbf { y } \\right ) } ^ \\dag \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{align*} P _ N ( a ) \\sim \\frac { 1 } { 1 - { \\rm e } ^ { \\vartheta ^ \\star _ Z } } \\frac { 1 } { \\sqrt { 2 \\pi N \\Lambda _ Z '' ( \\vartheta _ Z ^ \\star ) } } { \\rm e } ^ { - N I _ Z ( a ) } = \\frac { 1 } { 1 - a \\ , \\frac { 1 + \\lambda } { 1 + a } } \\frac { 1 } { \\sqrt { 2 \\pi N a ( a + 1 ) } } { \\rm e } ^ { - N I _ Z ( a ) } \\ , . \\end{align*}"}
-{"id": "6597.png", "formula": "\\begin{align*} \\mathcal { D } ( S _ D ) \\ ; = \\ ; \\mathcal { D } ( \\overline { S } ) \\dotplus S _ D ^ { - 1 } \\ker S ^ * \\ , . \\end{align*}"}
-{"id": "7888.png", "formula": "\\begin{align*} E ( u ) = \\int _ { \\R ^ n } \\phi ( \\nabla u ) + \\lambda \\| u - f \\| _ { L ^ 1 ( \\R ^ n ) } . \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 1 e _ 4 + \\alpha _ 2 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 3 e _ 4 + \\alpha _ 4 e _ 5 , [ e _ 2 , e _ 1 ] = \\alpha _ 5 e _ 4 + \\alpha _ 6 e _ 5 , [ e _ 2 , e _ 2 ] = \\beta _ 1 e _ 4 + \\beta _ 2 e _ 5 , \\\\ [ e _ 1 , e _ 3 ] = \\beta _ 3 e _ 4 + \\beta _ 4 e _ 5 , [ e _ 3 , e _ 1 ] = \\beta _ 5 e _ 4 + \\beta _ 6 e _ 5 , [ e _ 2 , e _ 3 ] = \\gamma _ 1 e _ 4 + \\gamma _ 2 e _ 5 , [ e _ 3 , e _ 2 ] = \\gamma _ 3 e _ 4 + \\gamma _ 4 e _ 5 , \\\\ [ e _ 3 , e _ 3 ] = \\gamma _ 5 e _ 4 + \\gamma _ 6 e _ 5 . \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{align*} \\left [ \\Lambda _ N ^ { N + 1 } f \\right ] ( z _ 1 , \\cdots , z _ { N + 1 } ) = \\frac { N ! \\int _ { z _ 1 } ^ { z _ 2 } \\cdots \\int _ { z _ { N } } ^ { z _ { N + 1 } } \\Delta _ N ( y ) f ( y ) d y _ 1 \\cdots d y _ N } { \\Delta _ { N + 1 } ( z ) } . \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} z _ \\omega ( \\Gamma ) = \\prod _ { i = 1 } ^ c z _ { \\omega _ i } ( \\Gamma _ i ) . \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} S _ T ^ { - 1 } = S _ D ^ { - 1 } + P _ T T ^ { - 1 } P _ T \\ , . \\end{align*}"}
-{"id": "8985.png", "formula": "\\begin{align*} \\int _ 0 ^ \\tau M _ s ^ N d M _ s ^ T = \\beta _ { \\int _ 0 ^ \\tau ( M ^ N _ s ) ^ 2 d s } , \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{gather*} f ( x ) = u _ j ( x ) \\quad \\textrm { f o r } x \\in e ^ \\frac { j \\pi i } { p } S , j = 0 , 1 , \\dots , p - 1 , \\end{gather*}"}
-{"id": "9615.png", "formula": "\\begin{align*} T ^ i _ { j k } = \\left \\{ \\begin{array} { l l } \\Gamma ^ i _ { \\underline { j k } } , & i \\not \\in \\{ j , k \\} , \\\\ - \\frac 1 { N + 1 } \\Gamma ^ { ( k ) } _ { \\underline { k ( k ) } } , & i = j \\neq k , \\\\ \\frac { N - 1 } { N + 1 } \\big ( \\frac 1 u \\delta ^ i _ 1 + \\frac 1 v \\delta ^ i _ 2 + \\frac 1 w \\delta ^ i _ 3 \\big ) , & i = j = k . \\end{array} \\right . \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} v \\theta _ I w \\textrm { i f a n d o n l y i f } i ^ v = i ^ w \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} \\widetilde Q = Q + Q '' . \\end{align*}"}
-{"id": "9704.png", "formula": "\\begin{align*} h _ 2 ^ 2 ( P ) = ( v _ 1 + v _ 2 ) ^ 2 + ( v _ 1 + v _ 3 ) ^ 2 + ( v _ 2 + v _ 3 ) ^ 2 - v _ 1 ^ 2 - v _ 2 ^ 2 - v _ 3 ^ 2 = ( v _ 1 + v _ 2 + v _ 3 ) ^ 2 . \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{align*} X \\stackrel { d } { = } \\mu + R A U , \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} u ( x , y , t ) = u _ { c ( t ) } ( \\xi ) + \\tilde { u } ( \\xi , y , t ) , \\xi = x - 4 a ( t ) . \\end{align*}"}
-{"id": "9601.png", "formula": "\\begin{align*} & \\aligned { \\widetilde { \\mathcal T } } { } ^ i _ { j k } = L ^ i _ { \\underline { j k } } - \\omega ^ i _ { j k } , \\endaligned \\\\ \\displaybreak [ 0 ] & \\aligned { \\widetilde { \\mathcal W } } { } ^ i _ { j m n } = R ^ i _ { j m n } - \\omega { } ^ i _ { j m | n } + \\omega { } ^ i _ { j n | m } + \\omega { } ^ \\alpha _ { j m } \\omega { } ^ i _ { \\alpha n } - \\omega { } ^ \\alpha _ { j n } \\omega { } ^ i _ { \\alpha m } , \\endaligned \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{align*} \\mathcal { C } _ { n } ^ { \\left \\{ J \\right \\} } = \\# \\left \\{ \\left ( k _ { 1 } , \\dots , k _ { m } \\right ) \\vert k _ { 1 } + \\dots + k _ { m } = n , \\thinspace \\thinspace k _ { i } \\in J \\right \\} \\end{align*}"}
-{"id": "1172.png", "formula": "\\begin{align*} H _ \\varphi ( H ( f ) ) = H ( H _ \\varphi ( f ) ) \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} \\# \\big \\{ p \\leq x : \\phi ( p - 1 ) = \\phi ( p + 1 ) \\big \\} < \\frac { x } { \\exp ( ( \\log x ) ^ { 1 / 3 } ) } + 2 = o ( \\pi ( x ) ) . \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} \\sum _ { r = 1 } ^ { \\delta } r ^ s = \\frac { 1 } { s + 1 } \\sum _ { j = 0 } ^ s ( - 1 ) ^ j \\binom { s + 1 } { j } B _ j \\delta ^ { s + 1 - j } \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{align*} \\Xi ^ { \\lambda } ( K _ { a } \\otimes F _ { a } \\otimes 1 \\otimes E _ { a } ) = \\sum _ { i , n , o , p } c _ { n } ^ { i } \\pi ( K _ { a } F _ { a } ) _ { o } ^ { n } c _ { p } ^ { o } \\pi ( E _ { a } K _ { \\lambda } ) _ { i } ^ { p } . \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{align*} V ( \\theta ) = - \\frac { 1 } { 6 } \\cos ( 2 \\theta ) + \\frac { 4 } { 3 } \\cos ( \\theta ) \\end{align*}"}
-{"id": "6948.png", "formula": "\\begin{align*} \\psi _ g ( 1 + \\epsilon ) = 1 + g + \\epsilon \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( { \\cal R } _ n ) = 0 . \\end{align*}"}
-{"id": "4546.png", "formula": "\\begin{align*} J _ 5 ^ { ( k ) } = \\Big ( f ( \\theta _ { \\Psi } ^ { k - 1 } , \\theta _ { \\Psi } ^ { k - 1 } ) , \\frac { 1 } { 2 } ( \\partial \\mathbf { A } _ { h } ^ { k } + \\partial \\mathbf { A } _ { h } ^ { k - 1 } ) \\Big ) , \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} X _ n = S _ { \\Delta t } ^ { n } x + \\Delta t \\sum _ { k = 0 } ^ { n - 1 } S _ { \\Delta t } ^ { n - k } B G ( X _ k ) + \\sum _ { k = 0 } ^ { n - 1 } S _ { \\Delta t } ^ { n - k } e ^ { \\tau A } \\sigma ( X _ k ) \\Delta W _ k . \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} L ^ \\prime : = L K ^ \\prime . \\end{align*}"}
-{"id": "4470.png", "formula": "\\begin{align*} \\sigma _ { \\mathrm { p } } ^ { ( \\omega ) } \\left ( \\mathbf { x } , \\mathbf { y } , t \\right ) : = \\int \\nolimits _ { 0 } ^ { t } \\varrho ^ { ( \\beta , \\omega , \\vartheta , \\lambda ) } \\left ( i [ I _ { \\mathbf { y } } ^ { ( \\omega , \\vartheta ) } , \\tau _ { s } ^ { ( \\omega , \\vartheta , \\lambda ) } ( I _ { \\mathbf { x } } ^ { ( \\omega , \\vartheta ) } ) ] \\right ) \\mathrm { d } s \\ . \\end{align*}"}
-{"id": "8885.png", "formula": "\\begin{align*} { \\bf B } _ 4 = \\begin{pmatrix} 0 & 1 6 & 0 & 3 4 5 6 \\\\ 1 6 & 0 & 6 4 0 & 0 \\\\ 0 & 6 4 0 & 0 & 6 4 8 0 \\\\ 3 4 5 6 & 0 & 6 4 8 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} P ^ \\phi _ t \\mu ( E ) : = \\mu \\bigl ( \\phi _ t ^ { - 1 } ( E ) \\bigr ) \\end{align*}"}
-{"id": "5391.png", "formula": "\\begin{align*} P _ { \\perp } \\begin{pmatrix} z \\\\ \\bar { z } \\end{pmatrix} = \\begin{pmatrix} P _ { 1 1 } & P _ { 1 2 } \\\\ P _ { 2 1 } & P _ { 2 2 } \\end{pmatrix} \\begin{pmatrix} x + i y \\\\ x - i y \\end{pmatrix} = \\begin{pmatrix} P _ { 1 1 } + P _ { 1 2 } & i ( P _ { 1 1 } - P _ { 1 2 } ) \\\\ P _ { 2 1 } + P _ { 2 2 } & i ( P _ { 2 1 } - P _ { 2 2 } ) \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\end{align*}"}
-{"id": "577.png", "formula": "\\begin{align*} h _ t : = h _ 0 \\circ f _ t + Q \\log | f _ t ' | , \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} \\sigma _ j ( A x ) = \\sigma _ j \\left ( \\lim _ { k \\to \\infty } z _ k \\right ) = \\lim _ { { k \\to \\infty } \\atop { k \\geqslant j } } \\sigma _ j ( z _ k - z _ j ) + \\sigma _ j ( z _ j ) = \\sigma _ j ( z _ j ) = \\left ( A _ j \\circ \\sigma _ j \\right ) ( x ) , \\end{align*}"}
-{"id": "10004.png", "formula": "\\begin{align*} W ( G _ { 3 } ( n , d , x , s ) ) & = W ( B _ { 1 } ( n - 1 , d , x ) ) + \\sum _ { i = - d } ^ { d } ( \\left \\vert s - i \\right \\vert + 1 ) + \\\\ & + \\sum _ { i = - ( k - 1 ) } ^ { k - 1 } ( \\left \\vert s - i \\right \\vert + 2 ) + ( s - x ^ { \\prime } + 2 ) + \\\\ & + ( - x ^ { \\prime } - s + 2 ) . \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} G ( \\lambda ) : = g ( k ) , \\ \\Phi ( \\lambda ) : = \\prod _ { n = 0 } ^ \\infty \\left ( 1 - \\frac { \\lambda } { \\kappa _ { 1 , n } } \\right ) \\left ( 1 - \\frac { \\lambda } { \\kappa _ { 2 , n } } \\right ) , \\ E ( \\lambda ) : = \\frac { G ( \\lambda ) } { \\Phi ( \\lambda ) } . \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} \\xi _ { p _ 1 } ( \\mathbb C ) \\cap \\xi _ { p _ 2 } ( \\mathbb C ) \\neq \\emptyset \\Rightarrow \\xi _ { p _ 1 } ( \\mathbb C ) = \\xi _ { p _ 2 } ( \\mathbb C ) . \\end{align*}"}
-{"id": "9513.png", "formula": "\\begin{align*} w ^ * { } \\lim _ \\beta \\sum _ { j \\in J } m _ j ^ * x _ 0 \\tilde \\alpha ( m _ j ) = 0 . \\end{align*}"}
-{"id": "5492.png", "formula": "\\begin{align*} e ^ { \\pi i \\left ( \\langle \\lambda , \\lambda \\rangle - \\frac { 1 } { 1 2 } \\right ) } T _ { C _ { \\lambda , r } } & = T _ { M _ r } , \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{align*} E ( t ) + \\int _ 0 ^ t I _ b ( x , s ) d s = E ( 0 ) + d ( t ) - d ( 0 ) . \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} \\dd ( Y _ t ^ 2 ) = 2 Y _ t \\ , \\dd Y _ t + \\sigma _ 1 ^ 2 Y _ t \\ , \\dd t = 2 Y _ t ( a \\ , \\dd t + \\sigma _ 1 Y _ t ^ { \\frac { 1 } { 2 } } \\ , \\dd W _ t ) + \\sigma _ 1 ^ 2 Y _ t \\ , \\dd t = ( 2 a + \\sigma _ 1 ^ 2 ) Y _ t \\ , \\dd t + 2 \\sigma _ 1 Y _ t ^ { \\frac { 3 } { 2 } } \\ , \\dd W _ t , \\end{align*}"}
-{"id": "2597.png", "formula": "\\begin{align*} f ^ { ( m ) } & = f ^ { ( m ) } _ 0 + f ^ { ( m ) } _ N , \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} \\left ( \\sigma ^ 2 - \\lambda ^ 2 \\right ) L ^ { ( 2 n - 2 k - 2 ) } = \\overline { A _ { n - k } } A _ { n - k } , \\\\ \\ ; A _ { n - k } = \\sigma ^ { n - p - k - 1 } \\left ( \\sigma ^ 2 - \\lambda ^ 2 \\right ) \\cdot \\prod \\limits _ { i = 1 } ^ { p - 1 } \\left ( \\sigma - \\lambda _ i \\right ) . \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] = ( 1 + z ) ^ { - \\alpha } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 \\alpha & \\frac 1 2 + \\frac 1 2 \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac { 4 z } { ( 1 + z ) ^ 2 } \\bigg ] . \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} h ^ 1 ( E ( - c ) ) = 1 , \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{align*} \\sup _ { 0 \\le j \\le N } \\ \\| P _ { n } T ^ { \\ , j } P _ { l } \\ , x \\| \\le \\dfrac { \\beta _ { l } } { 4 } \\cdot \\Bigl ( \\prod _ { i = \\Delta ^ { ( k ) } - N + 1 } ^ { \\Delta ^ { ( k ) } - 1 } \\ ! \\ ! \\ ! | w _ { i } ^ { ( k ) } | \\ \\Bigr ) \\ , \\cdot \\ , \\| P _ { l } \\ , x \\| . \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{align*} { \\bf W } _ \\alpha = \\begin{bmatrix} b _ 0 & 0 & 0 & 0 & 0 & \\cdots & 0 \\\\ b _ 1 & b _ 0 & 0 & 0 & 0 & \\cdots & 0 \\\\ b _ 2 & b _ 1 & b _ 0 & 0 & 0 & \\cdots & 0 \\\\ b _ 3 & b _ 2 & b _ 1 & b _ 0 & 0 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\cdots & \\vdots \\\\ b _ { P - 1 } & b _ { P - 2 } & w _ { P - 3 } & b _ { P - 4 } & \\cdots & b _ 1 & b _ 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "7311.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\alpha _ 1 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 1 ] = \\alpha _ 6 y _ 5 , [ y _ 2 , y _ 2 ] = \\beta _ 2 y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\alpha _ 1 \\beta _ 4 } { \\gamma _ 1 } y _ 5 , \\\\ [ y _ 2 , y _ 3 ] = y _ 4 = - [ y _ 3 , y _ 2 ] . \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} \\left ( 1 - c \\right ) f f _ { c _ { 1 } } f _ { c _ { 2 } } = q \\left ( f _ { c _ { 2 } } - k f _ { c _ { 1 } } \\right ) . \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} P ' ( t _ k ) S ' ( t _ k ) = P ( t _ k ) S '' ( t _ k ) , k = 1 , \\dots , n - 1 , \\end{align*}"}
-{"id": "636.png", "formula": "\\begin{gather*} \\lim _ { \\theta \\downarrow 0 ^ + } ( \\sin \\theta ) ^ { 1 - 2 s } y ' ( \\theta ) = 0 \\quad y ( \\pi ) = 0 , \\end{gather*}"}
-{"id": "3935.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] - z ^ { 1 - \\alpha } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac 1 z \\bigg ] = ( 1 - z ) ^ { 1 - \\alpha } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 \\alpha & \\frac 1 2 - \\frac 1 2 \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , - \\frac { 4 z } { ( 1 - z ) ^ 2 } \\bigg ] . \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{align*} \\prod _ { j = 0 } ^ { k - 1 } \\left [ 1 - \\cos \\left ( \\theta - \\frac { 2 \\pi j } { k } \\right ) \\right ] = K _ k [ 1 - \\cos ( k \\theta ) ] \\end{align*}"}
-{"id": "3673.png", "formula": "\\begin{align*} V ^ - ( z ) ^ { - 1 } = \\exp \\Big ( \\sum _ { n > 0 } \\frac { 1 } { n } h _ n z ^ { - 2 n } \\Big ) ; V ^ + ( z ) ^ { - 1 } = \\exp \\Big ( - \\sum _ { n > 0 } \\frac { 1 } { n } h _ { - n } z ^ { 2 n } \\Big ) \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} z _ b ( s ) = s ^ { - \\frac { 1 } { \\alpha } } g _ b \\Bigl ( \\frac { 1 } { \\sqrt { s } } \\Bigr ) , 0 < s < \\infty \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} \\tau ( n ) = \\sum _ { d | n } 1 , n \\geqslant 1 , \\end{align*}"}
-{"id": "6855.png", "formula": "\\begin{align*} \\mu ( E _ f ( \\lambda ) ) \\le \\frac { 1 } { \\lambda } \\int _ 0 ^ { 2 \\pi } f ^ * ( \\theta ) ^ 2 \\ , \\frac { d \\theta } { 2 \\pi } - 1 \\le \\frac { C } { \\lambda } \\| f \\| _ { H ^ 2 } ^ 2 - 1 = \\frac { C } { \\lambda } - 1 , \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} \\begin{array} { l l l } S _ 1 & = & \\{ a _ 1 , \\ldots , a _ m \\} , \\\\ S _ 2 & = & S ( v ) \\setminus S _ 1 , \\\\ E _ 2 & = & \\{ c \\in E ( v ) \\vert c a _ i \\in I , \\mbox { f o r a l l } i = 1 , \\ldots m \\} , \\\\ E _ 1 & = & E ( v ) \\setminus E _ 2 . \\end{array} \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{align*} 1 + \\sum _ { m = 0 } ^ j \\frac { B _ { 2 m + 2 } ( 4 ^ { m + 1 } - 1 6 ^ { m + 1 } ) ( 2 j + 1 ) ! } { ( 2 j - 2 m ) ! ( 2 m + 2 ) ! } = 1 - 1 = 0 \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} \\alpha \\xi ^ { m } + \\beta \\eta ^ { m } + q _ { m } ( \\xi , \\eta ) = \\eta ^ { m } \\left ( \\frac { \\alpha \\xi ^ { m } } { \\eta ^ { m } } + \\beta + \\frac { q _ { m } ( \\xi , \\eta ) } { \\eta ^ { m } } \\right ) \\geq \\frac { \\beta } { 2 } \\eta ^ { m } > c | \\eta | . \\end{align*}"}
-{"id": "5104.png", "formula": "\\begin{align*} \\frac { 1 } { 2 ! } & = \\frac { 1 } { 2 } , \\thinspace \\thinspace \\frac { 1 } { 2 ! 2 ! } - \\frac { 1 } { 3 ! } = \\frac { 1 } { 4 } - \\frac { 1 } { 6 } = \\frac { 1 } { 1 2 } , \\thinspace \\thinspace \\frac { 1 } { 4 ! } - \\frac { 1 } { 2 ! 3 ! } - \\frac { 1 } { 3 ! 2 ! } + \\frac { 1 } { 2 ! 2 ! 2 ! } = 0 , \\\\ \\frac { 1 } { 2 ! 2 ! 2 ! 2 ! } & - \\frac { 3 } { 2 ! 2 ! 3 ! } + \\frac { 2 } { 2 ! 4 ! } + \\frac { 1 } { 3 ! 3 ! } - \\frac { 1 } { 5 ! } = - \\frac { 1 } { 7 2 0 } , \\dots \\end{align*}"}
-{"id": "9705.png", "formula": "\\begin{align*} h _ 2 ^ 2 ( P ) & = ( v _ 1 + v _ 2 ) ^ { 2 } + ( v _ 1 + v _ 3 ) ^ { 2 } + ( v _ 2 + v _ 3 ) ^ { 2 } + ( v _ 1 + v _ 4 ) ^ { 2 } + ( v _ 2 + v _ 4 ) ^ { 2 } + ( v _ 3 + v _ 4 ) ^ { 2 } \\\\ & - v _ 1 ^ { 2 } - v _ 2 ^ { 2 } - v _ 3 ^ { 2 } - v _ 4 ^ { 2 } \\\\ & = ( v _ 1 + v _ 2 ) ^ { 2 } + ( v _ 1 + v _ 3 ) ^ { 2 } + ( v _ 2 + v _ 3 ) ^ { 2 } + 2 v _ 4 ^ 2 + 2 v _ 4 ( v _ 1 + v _ 2 + v _ 3 ) \\\\ & = ( v _ 1 + v _ 2 ) ^ { 2 } + ( v _ 1 + v _ 3 ) ^ { 2 } + ( v _ 2 + v _ 3 ) ^ { 2 } + \\tfrac { 8 } { 9 } ( v _ 1 + v _ 2 + v _ 3 ) ^ { 2 } . \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{align*} L _ \\mu ( \\theta ) = \\displaystyle \\int _ { \\mathbb { R } } \\exp ( \\theta x ) \\mu ( d x ) , \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{align*} P _ { H P } ^ { s , N + 1 } ( t ) \\Lambda _ N ^ { N + 1 } = \\Lambda _ N ^ { N + 1 } P _ { H P } ^ { s , N } ( t ) \\ , \\ t \\ge 0 , \\forall N \\ge 1 . \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{align*} \\det ( X ( t ) ) = 1 , t \\in [ 0 , t _ f ] . \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{align*} e ^ { 2 \\lambda } = | z | ^ { 2 \\theta _ 0 - 2 } \\left ( 1 + 2 \\ , \\Re \\left ( \\alpha _ 0 z \\right ) + O ( | z | ^ { 2 - \\epsilon } ) \\right ) . \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{align*} B K _ 0 B ^ { - 1 } = ( 1 - 2 \\alpha \\gamma e ^ { - \\beta } ) \\cdot K _ 0 + \\gamma e ^ { - \\beta } \\cdot K _ - - \\alpha ( 1 - e ^ { - \\beta } \\alpha \\gamma ) \\cdot K _ + . \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{align*} H & = a ( ( k - 1 ) a ^ 2 + ( n - k ) b ^ 2 + 1 ) \\\\ H & = b ( k a ^ 2 + ( n - k - 1 ) b ^ 2 + 1 ) \\\\ H & = a b ( a + b ) \\end{align*}"}
-{"id": "4265.png", "formula": "\\begin{align*} y ^ 2 + a _ 1 x y + a _ 3 y = x ^ 3 + a _ 2 x ^ 2 + a _ 4 x + a _ 6 , \\end{align*}"}
-{"id": "2945.png", "formula": "\\begin{align*} \\# A _ { m , 0 } \\ge \\dfrac { \\alpha d _ { j _ { m } } } { \\textrm { p e r } ( x _ { l } ) } \\Bigl ( \\dfrac { \\alpha d _ { j _ { m } + 1 } } { d _ { j _ { m } } } - 1 \\Bigr ) \\ge \\dfrac { \\alpha d _ { j _ { m } } } { \\textrm { p e r } ( x _ { l } ) } \\ , \\cdot \\ , \\dfrac { \\alpha d _ { j _ { m } + 1 } } { 2 d _ { j _ { m } } } = \\dfrac { \\alpha ^ { 2 } d _ { j _ { m } + 1 } } { 2 \\textrm { p e r } ( x _ { l } ) } \\cdot \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{align*} T = \\begin{pmatrix} \\norm { 0 } { \\sigma ^ { k _ 1 } ( \\beta ) } & \\norm { 0 } { \\sigma ^ { k _ 2 } ( \\beta ) } & \\cdots & \\norm { 0 } { \\sigma ^ { k _ \\nu } ( \\beta ) } \\\\ \\norm { 1 } { \\sigma ^ { k _ 1 } ( \\beta ) } & \\norm { 1 } { \\sigma ^ { k _ 2 } ( \\beta ) } & \\cdots & \\norm { 1 } { \\sigma ^ { k _ \\nu } ( \\beta ) } \\\\ \\vdots & \\vdots & & \\vdots \\\\ \\norm { n - 1 } { \\sigma ^ { k _ 1 } ( \\beta ) } & \\norm { n - 1 } { \\sigma ^ { k _ 2 } ( \\beta ) } & \\cdots & \\norm { n - 1 } { \\sigma ^ { k _ \\nu } ( \\beta ) } \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{align*} ( 2 \\rho , \\alpha _ a ) = ( \\alpha _ a , \\alpha _ a ) \\sum _ i ( \\omega _ i , \\alpha _ a ^ \\vee ) = ( \\alpha _ a , \\alpha _ a ) \\sum _ i \\delta _ { i a } = ( \\alpha _ a , \\alpha _ a ) , \\end{align*}"}
-{"id": "949.png", "formula": "\\begin{align*} X - \\mu \\stackrel { d } { = } R A U = \\hat a ^ { - 1 / 2 } A O ^ T Z \\end{align*}"}
-{"id": "5147.png", "formula": "\\begin{align*} f ( g ( z ) ) = f _ 0 + \\sum _ { n \\geq 1 } z ^ n \\sum _ { \\pi \\in \\mathcal { C } _ n } f _ { \\vert \\pi \\vert } g _ { \\pi } . \\end{align*}"}
-{"id": "2404.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ n } \\phi ( | \\xi | ) | \\xi | e ^ { \\mathrm i | x | \\xi _ 1 } d \\xi & = ( - \\mathrm i | x | ) ^ { - k } \\ ! \\int _ { \\mathbb R ^ n } \\partial _ 1 ^ k ( \\phi ( | \\xi | ) | \\xi | ) e ^ { \\mathrm i | x | \\xi _ 1 } d \\xi \\\\ & = ( - \\mathrm i | x | ) ^ { - k } \\ ! \\int _ { \\mathbb R ^ n } \\partial _ 1 ^ k ( | \\xi | ) \\phi ( | \\xi | ) e ^ { \\mathrm i | x | \\xi _ 1 } + R _ k ( \\xi ) d \\xi , \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} L ^ { \\prime } \\circ d f _ { | A } = e ^ { \\sigma } L . \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} \\Gamma _ p ( \\alpha ) \\Gamma _ p ( 1 - \\alpha ) = ( - 1 ) ^ { \\langle - \\alpha \\rangle _ p - 1 } . \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} q ( \\eta ) = \\left \\langle q , Z ^ p _ m ( \\cdot , \\eta ) \\right \\rangle _ { \\widehat { S } _ p } = \\left \\langle q , Z ^ p _ m ( \\cdot , \\eta ) \\right \\rangle _ S . \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} \\left \\vert \\sum \\limits _ { j = 0 } ^ { 2 } \\varepsilon ^ { \\frac { j } { 2 } } \\left \\vert \\lambda \\right \\vert ^ { 1 - \\frac { j } { 2 } } \\xi ^ { j } \\right \\vert \\leq C \\left ( 1 + \\varepsilon \\xi ^ { 2 } \\left \\vert \\lambda \\right \\vert ^ { - 1 } \\right ) . \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{align*} \\tilde { F } ( x _ 1 , \\cdots , x _ { k - 1 } , x _ k ) = ( x _ k - x _ { k - 1 } ) \\partial _ k \\tilde { F } \\left ( x _ 1 , \\cdots , x _ { k - 1 } , \\xi _ k ^ 1 \\right ) . \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} T _ h ^ n = O ( h ) ( 1 + { ( x _ n - a ) } ^ { - \\alpha } ) , \\mbox { f o r } 1 \\le n \\le P - 1 . \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} ( y _ F ) _ 0 = ( y _ O ) _ 0 = 0 . 4 , ( y _ N ) _ 0 = 0 . 2 , ( y _ P ) _ 0 = 0 , \\theta _ 0 = 3 0 0 ^ \\circ K . \\end{align*}"}
-{"id": "2492.png", "formula": "\\begin{align*} \\sigma _ k \\ast { f _ q } = g _ { 1 , k , q } + g _ { 2 , k , q } \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} \\mu ( c ) & = \\frac { ( 1 + c ^ 2 ) ( 1 + c ) ^ 2 } { K _ 1 ( 1 - b ) } \\left ( 1 + \\frac { \\Gamma K } { ( K + c ) ^ 2 } \\right ) , \\\\ \\lambda ( c ) & = \\frac { 1 } { T ( c ) } \\left ( \\frac { \\Gamma c } { K + c } - \\frac { ( b + c ) ( 1 + c ) } { 1 - b } \\left ( 1 + \\frac { \\Gamma K } { ( K + c ) ^ 2 } \\right ) \\right ) . \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} \\dfrac { 1 + \\psi ( 1 - \\lambda ) - \\psi ( \\lambda ) } { 2 } p ( t ) & = \\dfrac { 1 } { \\int _ a ^ b \\nu ( t ) \\Delta t } \\int _ a ^ b K ( s , t ) p ^ { \\Delta } ( s ) \\Delta s + \\dfrac { 1 } { \\int _ a ^ b \\nu ( t ) \\Delta t } \\int _ a ^ b \\nu ( s ) p ( \\sigma ( s ) ) \\Delta s \\\\ & - \\dfrac { \\psi ( \\lambda ) p ( a ) + \\left ( 1 - \\psi ( 1 - \\lambda ) \\right ) p ( b ) } { 2 } , \\end{align*}"}
-{"id": "9499.png", "formula": "\\begin{align*} p ' ( u ) p ( - u ) + p ( u ) p ' ( - u ) = 0 q ( u ^ 2 ) . \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} R \\left ( \\{ x , y \\} \\right ) = [ R ( x ) , R ( y ) ] , \\ ; \\ ; \\ ; \\forall \\ ; x , y \\in L . \\end{align*}"}
-{"id": "4944.png", "formula": "\\begin{align*} \\sigma ( S ) \\cup \\pm \\gamma \\sigma ( C ) = \\left ( \\lambda _ { 1 } , \\lambda _ { 2 } , \\ldots , \\lambda _ { n + 1 } , \\pm \\gamma \\mu _ { 1 } , \\pm \\gamma \\mu _ { 2 } , \\ldots , \\pm \\gamma \\mu _ { n } \\right ) . \\end{align*}"}
-{"id": "3644.png", "formula": "\\begin{align*} ( u , t ) \\mapsto \\sum _ { i = 1 } ^ { n } \\psi \\left ( \\sqrt { \\frac { t ( X _ { i } ) } { u ( X _ { i } ) } } \\right ) \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} \\| f \\| _ { A ^ 2 _ \\alpha } & = \\left ( \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { c _ \\alpha ( n ) } \\left | \\sum _ { n _ 1 + n _ 2 = n } a _ { n _ 1 } b _ { n _ 2 } \\right | ^ 2 \\right ) ^ \\frac { 1 } { 2 } \\\\ & \\leq \\left ( \\sum _ { n = 0 } ^ \\infty \\sum _ { n _ 1 + n _ 2 = n } \\frac { | a _ { n _ 1 } | ^ 2 } { c _ { \\alpha _ 1 } ( n _ 1 ) } \\frac { | b _ { n _ 2 } | ^ 2 } { c _ { \\alpha _ 2 } ( n _ 2 ) } \\right ) ^ \\frac { 1 } { 2 } = \\| g \\| _ { A ^ 2 _ { \\alpha _ 1 } } \\| h \\| _ { A ^ 2 _ { \\alpha _ 2 } } , \\end{align*}"}
-{"id": "120.png", "formula": "\\begin{align*} ( S \\alpha ) _ n = \\alpha _ { n + 1 } \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} \\int _ { \\R ^ { n } } f ^ { 2 } ( \\xi ) \\ , d \\xi = \\int _ { \\R ^ { n } } \\left ( \\sum _ { N = 1 } ^ { \\infty } f _ { N } ^ { 2 } ( \\xi ) \\right ) \\ , d \\xi = \\sum _ { N = 1 } ^ { \\infty } \\int _ { \\R ^ { n } } f _ { N } ^ { 2 } ( \\xi ) \\ , d \\xi = \\sum _ { N = 1 } ^ { \\infty } \\frac { 1 } { 2 ^ { N } } < \\infty . \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} & \\nabla _ k ( \\nabla _ i V ^ { ( \\beta , b ) } \\nabla _ j V ^ { ( \\gamma , c ) } - \\nabla _ i H ^ { ( \\beta , b ) } \\cdot \\nabla _ j H ^ { ( \\gamma , c ) } ) \\\\ & = \\nabla _ k \\nabla _ i V ^ { ( \\beta , b ) } \\nabla _ j V ^ { ( \\gamma , c ) } - \\nabla _ k \\nabla _ i H ^ { ( \\beta , b ) } \\cdot \\nabla _ j H ^ { ( \\gamma , c ) } \\\\ & \\quad + \\nabla _ i V ^ { ( \\beta , b ) } \\nabla _ k \\nabla _ j V ^ { ( \\gamma , c ) } - \\nabla _ i H ^ { ( \\beta , b ) } \\cdot \\nabla _ k \\nabla _ j H ^ { ( \\gamma , c ) } . \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} { n \\choose i } \\sum _ { w \\in { [ 1 . . n ] \\choose i } } \\hat { 1 } _ D ( w ) ^ 2 = \\left ( | D | P _ i ( k ) \\right ) ^ 2 \\end{align*}"}
-{"id": "9800.png", "formula": "\\begin{align*} \\eta : = \\frac { \\int _ { B _ 1 ( p ) } e _ { \\epsilon } ( u ) } { | \\log \\epsilon | } . \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} A = I + \\frac { B } { 2 } - 1 \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} s _ 2 \\leq n ( n + 1 ) ^ u G _ { \\kappa + 1 } ( n ) \\leq n ^ { u + 1 } ( 1 + o ( 1 ) ) e ^ { - 2 \\kappa } + O ( n ^ { u + 1 } q ( \\kappa ) e ^ { 2 \\kappa } / n ^ { 4 u + 4 } ) = o ( 1 ) . \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{align*} \\sum _ { k } ( \\mathsf { N } _ { m } ^ { n } ) _ { k } ^ { i } ( \\mathsf { N } _ { o } ^ { p } ) _ { j } ^ { k } = u _ { m } ^ { i } \\left ( \\sum _ { k } u _ { n } ^ { k * } u _ { o } ^ { k } \\right ) u _ { p } ^ { j * } = \\delta _ { o } ^ { n } u _ { m } ^ { i } u _ { p } ^ { j * } = \\delta _ { o } ^ { n } ( \\mathsf { N } _ { m } ^ { p } ) _ { j } ^ { i } . \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} \\gamma _ i \\sum _ { g \\in G } c _ g g & = \\Bigg ( \\prod _ { j = 1 } ^ { i - 1 } \\gamma _ j ^ { a _ j } \\Bigg ) \\sum _ { g \\in G } c _ g g \\intertext { i n $ \\ell _ 2 G $ , u s i n g t h e f a c t t h a t t h e $ \\gamma _ j $ ' s p a i r w i s e c o m m u t e . R e i n d e x i n g , } \\sum _ { g \\in G } c _ g g & = \\sum _ { g \\in G } c _ g \\Bigg ( \\prod _ { j = 1 } ^ { i - 1 } \\gamma _ j ^ { a _ j } \\gamma _ i ^ { - 1 } \\Bigg ) g . \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} \\| f \\| _ { H ^ q } \\leq \\left ( \\sum _ { n = 0 } ^ \\infty | a _ n | ^ 2 c _ { q / 2 } ( n ) \\right ) ^ \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "7464.png", "formula": "\\begin{align*} a _ + ( \\alpha ) \\vee a _ - ( \\alpha + 1 ) \\vee b _ + ( \\alpha - \\delta + 1 ) = x _ { \\alpha + 1 } \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} m _ i \\dot { x } _ i = \\sum _ { i \\xrightarrow [ \\alpha ] { } j } c _ \\alpha e ^ { x _ j - x _ i } - \\sum _ { k \\xrightarrow [ \\alpha ] { } i } c _ \\alpha e ^ { x _ i - x _ k } \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{align*} C _ { 2 n - 1 } : = \\left \\{ u ( t ) = ( 1 , t , \\dots , t ^ { 2 n - 2 } ) \\in \\mathbb { R } ^ { 2 n - 1 } : t \\in \\mathbb R \\right \\} , \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{align*} \\bigl ( \\Xi ( \\vec { \\xi } ) ( f ) \\bigr ) _ { ( x _ 1 , x _ 2 ) } \\diamond \\Psi ( \\vec { x } ; \\vec { z } ; \\vec { \\xi } ) = f ( z _ 1 , z _ 2 ) \\cdot \\Psi ( \\vec { x } ; \\vec { z } ; \\vec { \\xi } ) . \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} \\begin{aligned} Q ( 2 \\tau ) = \\int \\limits _ { 0 } ^ { \\pi } \\int \\limits _ { 0 } ^ { 2 \\pi } q ( \\ln ( 2 \\tau + \\sqrt { 4 \\tau ^ { 2 } - 1 } ) , \\theta , \\phi ) \\sin \\phi d \\theta d \\phi . \\end{aligned} \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{align*} \\sum _ { l > j } f _ q ^ { \\gamma _ l } = f _ q \\chi _ { \\bigcup _ { l > j } E _ q ^ { \\gamma _ l } } , \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} \\partial _ t F - d _ { F } \\Delta _ x F = \\lambda _ { M _ 1 F } \\frac { M _ 1 } { K _ F + M _ 1 } \\ - \\ \\beta _ F F \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} \\phi _ j ( x ) = \\sqrt { 2 } \\sin ( j \\pi x ) , \\ \\ \\ \\ \\ \\ \\ \\ \\lambda _ j = \\pi ^ 2 j ^ 2 , \\ \\ \\ \\ \\ \\ \\ \\forall j \\in \\N ^ * . \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{align*} I _ \\ell ( x ) = \\frac { e ^ x } { \\sqrt { 2 \\pi x } } + O \\ ( \\frac { e ^ x } { x ^ { \\frac 3 2 } } \\ ) . \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{align*} \\eta _ { 0 , 2 k } = \\frac { 1 } { \\alpha _ { 2 k - 1 } } ( \\gamma _ { 1 } \\eta _ { 1 , 2 k - 1 } - \\gamma _ { 2 k - 1 } \\eta _ { 0 , 2 k - 2 } ) = 0 \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{align*} \\mathcal { A } = \\mathcal { A } _ 1 + \\mathcal { A } _ \\tau ; \\end{align*}"}
-{"id": "8479.png", "formula": "\\begin{gather*} f ( \\eta ) = \\sum _ { m = 0 } ^ { \\infty } \\left \\langle f , Z ^ p _ m ( \\cdot , \\eta ) \\right \\rangle _ { \\widehat { S } _ p } \\quad \\textrm { i n } L ^ 2 ( \\widehat { S } _ p ) . \\end{gather*}"}
-{"id": "5918.png", "formula": "\\begin{align*} a _ { n , k } ^ { \\prime } & : = \\sum _ { d | n } s _ { d , k } ^ { ( - 1 ) } \\\\ a _ { n , k } ^ { \\prime \\prime } & : = \\sum _ { d | n } p ( d - k ) \\mu ( n / d ) . \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} & M _ s = \\{ ( f _ 1 , \\ldots , f _ 4 ) \\in M \\ | \\ \\ S = \\{ f _ 3 ( x ) = 0 \\} \\ \\ \\eqref { S , S ' } \\\\ & \\ C _ 0 = \\{ f _ 1 ( x ) = f _ 3 ( x ) = 0 \\} \\ \\ \\eqref { C = S c a p S ' } \\ \\} . \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{align*} | v ^ { ( k ) } | \\ , \\cdot \\prod _ { i = m ' + 1 } ^ { \\Delta ^ { ( k ) } - 1 } | w ^ { ( k ) } _ i | \\ge 2 ^ { \\ , \\delta ^ { ( k ) } - m ' - \\tau ^ { ( k ) } } \\ge 2 ^ { \\ , \\delta ^ { ( k ) } - \\alpha \\Delta ^ { ( k ) } - \\tau ^ { ( k ) } } > C . \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} \\sigma ( x ) ^ \\star = \\sigma ( x ) \\quad , \\bigl ( \\sigma ' ( x ) . h \\bigr ) ^ \\star = \\sigma ' ( x ) . h \\quad , \\bigl ( \\sigma '' ( x ) . ( k _ 1 , k _ 2 ) \\bigr ) ^ { \\star } = \\sigma '' ( x ) . ( k _ 1 , k _ 2 ) . \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{align*} \\vec { F } = \\vec { F } _ 0 + O ( | z | ^ { a - \\epsilon } ) \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{align*} U _ t & = U _ { t _ k } + \\int _ { t _ k } ^ { t } \\bigl ( A S _ { \\Delta t } U _ { t _ k } + S _ { \\Delta t } G ' ( X _ k ) . U _ { t _ k } \\bigr ) d r + \\int _ { t _ k } ^ { t } S _ { \\Delta t } \\bigl ( \\sigma ' ( X _ k ) . U _ { t _ k } \\bigr ) d W ( r ) , \\\\ U _ { t _ { k + 1 } } & = S _ { \\Delta t } U _ { t _ k } + \\Delta t S _ { \\Delta t } G ' ( X _ k ) . U _ { t _ k } + S _ { \\Delta t } \\bigl ( \\sigma ' ( X _ k ) . U _ { t _ k } \\bigr ) \\Delta W _ k , \\\\ U _ { t _ { \\ell + 1 } } & = h . \\end{align*}"}
-{"id": "7134.png", "formula": "\\begin{align*} \\int _ 0 ^ x \\frac { ( x - z ) ^ { - \\alpha } } { \\Gamma ( 1 - \\alpha ) } z ^ { \\mu - 1 } ( 1 - z ) ^ \\nu d z & = \\sum _ { n = 0 } ^ \\infty ( - 1 ) ^ n \\begin{pmatrix} \\nu \\cr n \\end{pmatrix} \\frac { \\Gamma ( \\mu + n ) } { \\Gamma ( \\mu - \\alpha + n + 1 ) } x ^ { \\mu - \\alpha + n } . \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{align*} 0 = v _ { s \\bar s } - ( v _ { s \\bar x _ k } ) ( v _ { x _ j \\bar x _ k } ) ^ { - 1 } ( v _ { x _ j \\bar s } ) ^ T = w _ { s \\bar s } - \\sum _ { j = 1 } ^ n | w _ { x _ j \\bar s } | ^ 2 . \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} H ^ { s } ( \\mathbb { Z } / p ; N ) = \\begin{cases} \\mathbb { Z } / p , & s = 0 , \\\\ 0 , & s > 0 . \\end{cases} \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} \\phi \\left ( \\gamma \\right ) = \\delta \\left ( \\gamma \\right ) , \\ ; \\ ; \\phi \\left ( \\gamma \\right ) = \\tau \\delta \\left ( \\gamma - 1 \\right ) + \\delta \\left ( \\gamma \\right ) , \\ ; \\ ; \\phi \\left ( \\gamma \\right ) = \\tau \\delta \\left ( \\gamma - \\alpha \\right ) + \\delta \\left ( \\gamma \\right ) , \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} \\vert n _ 1 , n _ 2 \\rangle = \\sqrt { \\frac { ( k - n _ 1 - n _ 2 ) ! } { k ! n _ 1 ! n _ 2 ! } } { ( { a ^ + _ 1 } ) } ^ { n _ 1 } { ( { a ^ + _ 2 } ) } ^ { n _ 2 } \\vert 0 , 0 \\rangle n _ 1 + n _ 2 \\leq k , \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{align*} x _ { \\mathbf n , \\lambda } : = e _ 0 + \\sum _ { m = 1 } ^ \\infty \\frac { \\prod _ { l = 1 } ^ { m } ( 1 + \\lambda ^ { \\Delta b _ { n ( l - 1 ) } } ) } { \\Big ( \\prod _ { l = 1 } ^ m v _ { n ( l ) } \\Big ) \\Big ( \\prod _ { l = 1 } ^ { m - 1 } \\prod _ { \\nu = b _ { n ( l ) } + 1 } ^ { b _ { n ( l ) + 1 } - 1 } w _ { \\nu } \\Big ) } \\sum _ { j = b _ { n ( m ) } } ^ { b _ { n ( m ) + 1 } - 1 } \\frac { \\lambda ^ { b _ { n ( k ) + 1 } - 1 - j } } { \\prod _ { \\nu = j + 1 } ^ { b _ { n ( k ) + 1 } - 1 } w _ { \\nu } } e _ j , \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} M _ { \\nu } ( \\theta ) = \\int \\frac { 1 } { 1 - \\theta x } \\nu ( d x ) \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} F _ m \\sum _ { k = 1 } ^ n { L _ { 2 m k } } & = F _ { m + 2 m n } - F _ m \\\\ & = F _ { m + m n + m n } - F _ { m + m n - m n } \\\\ & = F _ { m n } L _ { m n + m } \\ , , \\quad \\mbox { $ m $ e v e n } \\ , , \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{align*} ( - 1 ) ^ n \\frac { \\Gamma ( b - a - n ) \\Gamma ( a + n ) } { n ! \\Gamma ( c - a - n ) } \\ , ( - z ) ^ { - a - n } = \\frac { \\Gamma ( a ) \\Gamma ( b - a ) } { \\Gamma ( c - a ) } \\ , ( - z ) ^ { - a } \\ \\frac { ( 1 + a - c ) _ n ( a ) _ n } { ( 1 + a - b ) _ n n ! } \\ , ( - z ) ^ { - n } \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{align*} \\Delta = D _ { m a x } - D _ { a c h } ( k ) \\leq \\frac { D _ { m a x } } { 2 ^ k } . \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} \\smash [ b ] { | v ^ { ( k ) } | \\prod _ { j = \\Delta ^ { ( k ) } - m } ^ { \\Delta ^ { ( k ) } - 1 } | w ^ { ( k ) } _ j | } & \\ge C \\intertext { a n d } \\smash [ t ] { | v ^ { ( k ) } | \\prod _ { j = m ' + 1 } ^ { \\Delta ^ { ( k ) } - 1 } | w ^ { ( k ) } _ j | } & > C \\quad \\textrm { f o r e v e r y } \\ \\ 0 \\le m ' \\le \\alpha \\Delta ^ { ( k ) } . \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} x _ 0 + y _ 0 = t _ 0 p + z _ 0 , \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} M z + N \\bar { z } = p \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} \\bar { \\varrho } ^ { ( \\beta , \\vartheta , \\lambda ) } \\left ( B \\right ) : = \\mathbb { E } \\left [ \\varrho ^ { ( \\beta , \\omega , \\vartheta , \\lambda ) } \\left ( B \\right ) \\right ] \\ , B \\in \\mathcal { U } \\ , \\end{align*}"}
-{"id": "1287.png", "formula": "\\begin{align*} v ( \\mathsf { x , } t ) = \\sum _ { \\mathsf { n } \\in \\mathbb { N } ^ { d } } \\beta _ { \\mathsf { n } } \\exp \\left [ - ( T - t ) E _ { \\mathsf { n } } \\right ] \\mathsf { h } _ { \\mathsf { n } } \\left ( \\mathsf { x } \\right ) \\end{align*}"}
-{"id": "3254.png", "formula": "\\begin{align*} \\mu _ { N + 1 } P _ { N + 1 } ( t ) \\Lambda _ N ^ { N + 1 } = \\mu _ { N + 1 } \\Lambda ^ { N + 1 } _ N P _ N ( t ) = \\mu _ N P _ N ( t ) , \\end{align*}"}
-{"id": "3008.png", "formula": "\\begin{align*} S _ { \\eta , n } = \\bigcup _ { m \\in \\mathcal { N } _ { \\eta , n } } \\Big \\{ s \\in [ 1 , b _ { m + 1 } - b _ m ) ; \\ ; | v _ { m } | \\prod _ { i = b _ { m + 1 } - s } ^ { b _ { m + 1 } - 1 } | w _ { i } | > \\frac { 1 } { \\eta } \\Big \\} , \\end{align*}"}
-{"id": "9990.png", "formula": "\\begin{align*} & E _ { 1 } = \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} , \\ , A _ { 1 } = \\begin{bmatrix} - 1 & 4 \\pi \\\\ - 4 \\pi & - 4 \\end{bmatrix} & \\\\ & E _ 2 = \\begin{bmatrix} 1 & 0 \\\\ 1 & 0 \\end{bmatrix} , A _ 2 = \\begin{bmatrix} - 1 & 4 \\pi \\\\ - \\pi & - 1 \\end{bmatrix} & \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 1 e _ 4 + \\alpha _ 2 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 3 e _ 3 + \\alpha _ 4 e _ 4 + \\alpha _ 5 e _ 5 , [ e _ 2 , e _ 1 ] = - \\alpha _ 3 e _ 3 + \\beta _ 1 e _ 4 + \\beta _ 2 e _ 5 , \\\\ [ e _ 2 , e _ 2 ] = \\beta _ 3 e _ 4 + \\beta _ 4 e _ 5 , [ e _ 1 , e _ 3 ] = \\gamma _ 1 e _ 5 , [ e _ 2 , e _ 3 ] = \\gamma _ 2 e _ 5 , [ e _ 3 , e _ 1 ] = \\gamma _ 3 e _ 4 + \\gamma _ 4 e _ 5 , \\\\ [ e _ 3 , e _ 2 ] = \\gamma _ 5 e _ 4 + \\gamma _ 6 e _ 5 , [ e _ 3 , e _ 3 ] = \\gamma _ 7 e _ 4 + \\gamma _ 8 e _ 5 . \\end{align*}"}
-{"id": "4499.png", "formula": "\\begin{align*} \\digamma _ { t } = \\sqrt { D _ { \\left \\{ 0 \\right \\} } } B _ { t } + \\int \\nolimits _ { 0 } ^ { t } \\int \\nolimits _ { \\left \\vert \\nu \\right \\vert \\geq 1 } \\nu N \\left ( \\mathrm { d } s \\mathrm { d } \\nu \\right ) + \\int \\nolimits _ { 0 } ^ { t } \\int \\nolimits _ { \\left \\vert \\nu \\right \\vert < 1 } \\nu M \\left ( \\mathrm { d } s \\mathrm { d } \\nu \\right ) \\ , t \\in \\mathbb { R } _ { 0 } ^ { + } \\ . \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{gather*} t H _ { \\mathrm { K F S } } ^ { \\frac 3 2 + \\frac 3 2 } \\left ( { \\theta ^ 0 _ 1 - \\theta ^ 0 _ 2 , \\ , \\theta ^ 0 _ 2 + 1 } ; t ; { q _ 1 , p _ 1 \\atop q _ 2 , p _ 2 } \\right ) = t H _ { \\mathrm { I I I } ( D _ 7 ) } \\big ( \\theta ^ 0 _ 1 - \\theta ^ 0 _ 2 ; t ; q _ 1 , p _ 1 \\big ) \\\\ \\qquad { } + t H _ { \\mathrm { I I I } ( D _ 7 ) } \\big ( \\theta ^ 0 _ 2 + 1 ; t ; q _ 2 , p _ 2 \\big ) - p _ 1 q _ 1 p _ 2 q _ 2 - t ( p _ 1 p _ 2 + p _ 1 + p _ 2 ) . \\end{gather*}"}
-{"id": "1102.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { U ^ d [ N ] } ^ { 2 ^ d } = \\frac { 1 } { \\vert R \\vert } \\sum \\limits _ { x , h _ 1 , \\dots , h _ d } \\prod \\limits _ { \\boldsymbol { \\omega } \\in \\{ 0 , 1 \\} ^ d } \\mathcal { C } ^ { \\vert \\boldsymbol { \\omega } \\vert } f 1 _ { [ N ] } ( x + \\mathbf { h } \\cdot \\boldsymbol { \\omega } ) , \\end{align*}"}
-{"id": "9786.png", "formula": "\\begin{align*} E _ { \\epsilon } ( F ^ j _ { y _ j } ) = \\max _ { y \\in D ^ 2 } E _ { \\epsilon } ( F ^ j _ y ) \\to c _ { \\epsilon } ( M ) : = \\inf _ { F \\in \\Gamma ( M ) } \\max _ { y \\in D ^ 2 } E _ { \\epsilon } ( F _ y ) j \\to \\infty . \\end{align*}"}
-{"id": "9602.png", "formula": "\\begin{align*} \\omega ^ i _ { j k } = s _ { ( 1 ) } \\big ( \\delta ^ i _ j \\rho _ k + \\delta ^ i _ k \\rho _ j \\big ) + s _ { ( 2 ) } \\big ( F ^ i _ j \\sigma _ k + F ^ i _ k \\sigma _ j \\big ) + s _ { ( 3 ) } \\sigma _ { j k } \\varphi ^ i , \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} ( \\log \\theta ) '' = \\frac { \\theta '' } { \\theta } - \\frac { ( \\theta ' ) ^ 2 } { \\theta ^ 2 } , \\end{align*}"}
-{"id": "9337.png", "formula": "\\begin{align*} \\log ( 1 + \\| T _ n ( z ) - I \\| _ { } ) = \\frac { 1 } { 2 } \\log \\Big ( \\frac { 1 + | F _ n | } { 1 - | F _ n | } \\Big ) . \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} g _ { k } = \\begin{cases} 1 & k \\in J \\\\ 0 & \\end{cases} . \\end{align*}"}
-{"id": "4270.png", "formula": "\\begin{align*} f ( x ( P ) ) \\phi ( x ( P ) ) - g ( x ( P ) ) \\psi ( x ( P ) ) = \\Delta ^ \\prime , \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{align*} w + w ^ { - 1 } = \\pm \\frac { 1 } { q } \\sqrt { 4 \\ell + 1 } \\notin \\Q . \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} \\mathcal { F } ( u ^ 1 , u ^ 2 ) = I ^ 1 ( u ^ 1 ) + I ^ 2 ( u ^ 2 ) + \\Phi ( u ^ 1 , u ^ 2 ) \\end{align*}"}
-{"id": "2994.png", "formula": "\\begin{align*} X _ { l } : = \\Bigl \\| \\sum _ { i = b _ { l } } ^ { b _ { l + 1 } - 1 } \\Big ( \\prod _ { s = i + 1 } ^ { b _ { l + 1 } - 1 } w _ s \\Big ) \\ , x _ { i } e _ { i } \\Bigr \\| . \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} h _ 1 ( x , 1 ) = e ^ x , h _ 1 ( x , 2 ) = \\cosh x , h _ 2 ( x , 2 ) = \\sinh x . \\end{align*}"}
-{"id": "1321.png", "formula": "\\begin{align*} \\frac { F _ { 1 } \\left ( f \\right ) } { F _ { 2 } \\left ( f \\right ) } + \\frac { G _ { 2 } \\left ( g \\right ) } { G _ { 1 } \\left ( g \\right ) } = 0 , \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} \\Phi _ { \\epsilon , \\phi } ^ f ( x _ 1 , \\ldots , x _ k ) : = \\phi ( x _ 1 ) f _ \\epsilon ( x _ 2 - x _ 1 ) \\ldots f _ \\epsilon ( x _ k - x _ 1 ) . \\end{align*}"}
-{"id": "2858.png", "formula": "\\begin{align*} \\Phi ( x _ 1 , x _ 2 ; z _ 1 , z _ 2 ) = \\bigl ( \\delta ( z _ 1 , z _ 2 ) + \\sum \\limits _ { i _ 1 + i _ 2 < \\mu } c _ { i _ 1 , i _ 2 } ( x _ 1 , x _ 2 ) z _ 1 ^ { i _ 1 } z _ 2 ^ { i _ 2 } \\bigr ) \\cdot \\exp ( x _ 1 z _ 1 + x _ 2 z _ 2 ) , \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} \\left \\| \\xi ^ T _ { t } \\right \\| _ { \\Pi , T } = \\sup _ { \\varepsilon \\in ( 0 , 1 ) \\cap \\mathop { \\mathbb Q } \\nolimits } \\frac { 1 } { \\varepsilon } \\max _ { 1 \\le k \\le N _ { \\Pi } ( \\varepsilon ) } \\sqrt { \\sum _ { t = 1 } ^ { T } \\mathop { \\rm E } \\nolimits \\left [ \\sup _ { u , v \\in \\Psi ( \\varepsilon ; k ) } \\left | \\xi ^ T _ { t } ( u ) - \\xi ^ T _ { t } ( v ) \\right | ^ 2 \\mid \\mathop { \\mathcal F } \\nolimits ^ T _ { t } \\right ] } . \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} P _ { i + 1 } ( x ) = 2 ^ { - i } \\int _ 0 ^ x P _ i ( u ) d u + P _ { i + 1 } ( 0 ) = 2 ^ { - i } \\int _ 0 ^ x P _ i ( u ) d u + \\begin{cases} 0 & i \\\\ 2 F ( 2 ^ { - i - 1 } ) & \\end{cases} \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} \\biggl | \\ , \\prod _ { j = 1 } ^ { n - 1 } \\dfrac { \\lambda - \\lambda _ { j } } { \\omega _ { j } } \\biggr | ^ { 2 } \\ge \\dfrac { \\delta ^ { \\ , 2 ( n - 1 ) } } { ( \\omega _ { 1 } \\dots \\omega _ { n - 1 } ) ^ { 2 } } \\textrm { f o r e v e r y } \\ n \\ge 2 . \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n v _ j \\alpha _ j ^ t h ( \\alpha _ j ) c _ j = 0 t = 0 , \\dots , s - 1 . \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} 1 < s < 1 + \\frac { 1 } { p } , p > 2 1 < s \\leq \\frac { 3 } { 2 } , p = 2 \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} s _ 1 & = \\sum _ { d = 0 } ^ \\kappa ( d + 1 ) ^ u G _ { d + 1 } ( n ) \\\\ s _ 2 & = \\sum _ { d = \\kappa + 1 } ^ { n - 1 } ( d + 1 ) ^ u G _ { d + 1 } ( n ) . \\end{align*}"}
-{"id": "4839.png", "formula": "\\begin{align*} h k h ^ { - 1 } k ^ { - 1 } = I + \\begin{pmatrix} 0 & b \\left ( \\frac { w } { z } - 1 \\right ) \\\\ c \\left ( \\frac { z } { w } - 1 \\right ) & 0 \\end{pmatrix} p . \\end{align*}"}
-{"id": "1999.png", "formula": "\\begin{align*} \\langle \\Phi ( z _ 1 , \\dots , z _ N , \\{ \\overline { \\alpha } \\} ) | = \\langle 1 \\cdots M | \\mathcal { B } ( z _ 1 , \\{ \\overline { \\alpha } \\} ) \\cdots \\mathcal { B } ( z _ N , \\{ \\overline { \\alpha } \\} ) . \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{gather*} H ^ { n - 1 } ( X _ \\psi ) = H ^ { n - 1 } ( Y _ \\psi ) ^ G \\oplus C . \\end{gather*}"}
-{"id": "3319.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\xi _ N ( N a ) \\ , { \\rm e } ^ { N I _ X ( a ) } \\sqrt { N } = C _ X ( a ) I _ X ' ( a ) . \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} ( \\Sigma _ r A ) ^ i = A ^ { i + 1 } , d _ { \\Sigma _ r A } ^ i = d _ A ^ { i + 1 } . \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} t _ j ^ u = \\frac 1 { \\tau _ j ^ u } = \\left \\{ \\begin{array} { l l } 1 & \\textrm { i f } \\ l _ j \\ \\textrm { i s e v e n , o r } \\ u \\ \\textrm { a n d } \\ l _ j \\ \\textrm { a r e b o t h o d d } \\\\ - 1 & \\textrm { i f } \\ u \\ \\textrm { i s e v e n a n d } \\ l _ j \\ \\textrm { i s o d d } . \\end{array} \\right . \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} g ( k ) = ( \\tilde { y } ' y - \\tilde { y } y ' ) ( 1 , \\lambda ) . \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} \\begin{cases} D _ t ^ 2 x - \\Delta _ y x + F ^ { - T } \\nabla _ y p = 0 , \\\\ [ - 4 m m ] \\\\ \\det ( \\nabla _ y x ) = 1 , \\end{cases} \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} \\sum _ { p \\leq x } \\omega _ y ( p \\pm \\ell ) & = \\pi ( x ) \\log \\log y + O ( \\pi ( x ) ) , \\\\ \\sum _ { p \\leq x } \\omega _ y ( p \\pm \\ell ) ^ 2 & = \\pi ( x ) ( \\log \\log y ) ^ 2 + O ( \\pi ( x ) \\log \\log y ) , \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} \\vec { v } _ k : = \\Big ( a _ 1 = 2 g - \\frac { n + 2 + k } { 2 } , \\underbrace { 1 , \\dots , 1 } _ { k } , \\underbrace { \\frac 1 2 , \\dots \\frac 1 2 } _ { n - 1 - k } , \\ , a _ { n + 1 } = \\frac 3 2 , \\ , a _ { n + 2 } = - \\frac 1 2 \\Big ) . \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} F ( \\Sigma ) / \\mathcal { M } ( \\Sigma ) = F ( \\Sigma ) \\overset \\rho \\cong \\{ ( \\nu _ i ) ^ n _ { i = 0 } \\in \\Z ^ { n + 1 } ; \\ , \\ , { \\sum } ^ n _ { i = 0 } \\nu _ i = 2 \\} . \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} \\dot x = \\tfrac { 1 } { 2 } e ^ { J ( x ) } \\sqrt \\omega \\cos ( \\omega t ) + e ^ { - J ( x ) } \\sqrt \\omega \\sin ( \\omega t ) . \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} \\zeta _ 0 : = \\log ( \\lambda a ) + 1 - \\lambda a \\ , , \\zeta _ k : = ( - 1 ) ^ { k } \\left ( \\lambda ^ { k } \\left ( \\frac { 1 } { k } - \\frac { a \\lambda } { k + 1 } \\right ) - a ^ { - k } \\left ( \\frac { 1 } { k } - \\frac { 1 } { k + 1 } \\right ) \\right ) . \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} \\hat \\gamma ^ i _ t = \\theta ( \\bar X _ { t - } - X _ { t - } ^ i ) - \\bigg ( \\frac { 1 } { n } - 1 \\bigg ) \\phi _ { t } ( \\bar X _ { t - } - X ^ i _ { t - } ) - \\frac { 1 } { \\lambda } \\left ( \\frac { 1 } { n } - 1 \\right ) ^ 2 \\phi _ { t } \\hat \\gamma ^ i _ t \\ , , \\end{align*}"}
-{"id": "7879.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\{ \\# { \\cal E } _ 1 \\geq \\epsilon n \\} \\cap \\{ \\tau = k \\} \\right ) \\leq \\mathbb { P } \\left ( \\bigcup _ { j = 1 } ^ { k } \\{ \\# { \\cal N } _ j \\geq \\epsilon n t ^ { - 1 } \\} \\right ) \\leq \\sum _ { j = 1 } ^ { k } \\mathbb { P } \\left ( \\# { \\cal N } _ j \\geq \\epsilon n t ^ { - 1 } \\right ) . \\end{align*}"}
-{"id": "157.png", "formula": "\\begin{align*} \\cos ^ 2 \\theta = \\tfrac { 1 } { 4 } ( e ^ { i \\theta } + e ^ { - i \\theta } ) ^ 2 = \\tfrac { 1 } { 2 } \\cos 2 \\theta + \\tfrac { 1 } { 2 } \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{align*} A = \\frac { 1 } { 2 } \\left ( U + I _ n \\right ) \\quad B = \\frac { i } { 2 } \\left ( U - I _ n \\right ) \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} A _ N v = b , \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} \\int _ { f ^ { - 1 } \\overline { ( B _ { r _ j } ( x _ j ) ) } } | A | ^ 2 d \\mu \\bigg | _ { t = t _ j } \\ge \\varepsilon _ 3 x _ j \\in \\R ^ N . \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} I ( \\tau , \\sigma ) : = \\xi _ { \\tau } { \\bf 1 } _ { \\tau \\leq \\sigma } + \\zeta _ { \\sigma } { \\bf 1 } _ { \\sigma < \\tau } . \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} T ( k ; \\mathbf { l } ) = \\begin{pmatrix} 0 & e ^ { i k \\mathbf { l } } \\\\ e ^ { i k \\mathbf { l } } & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} \\bar { \\theta } = \\big ( f ^ 2 ( \\alpha ) , f ^ 2 ( \\bar { \\alpha } ) \\big ) \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} \\nu ^ 2 _ k ( 0 ) & = \\nu _ k ( 0 ) \\ , \\overline { \\nu _ k ( 0 ) } = ( \\mbox { A } + \\mbox { B } ) ( \\overline { \\mbox { A } } + \\overline { \\mbox { B } } ) = | \\mbox { A } | ^ 2 + | \\mbox { B } | ^ 2 + \\mbox { A } \\overline { \\mbox { B } } + \\overline { \\mbox { A } } \\mbox { B } \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} \\frac { d } { d \\tau } a ( \\tau ) = { \\rm g r a d } _ { a ( \\tau ) } \\frak { c s } . \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{gather*} m = \\min \\{ n \\mid 0 \\le n \\le d , W \\cap V _ n \\ne 0 \\} , \\\\ m ^ \\prime = \\min \\{ n \\mid 0 \\le n \\le d , W \\cap V _ n ^ \\prime \\ne 0 \\} . \\end{gather*}"}
-{"id": "4935.png", "formula": "\\begin{align*} A _ { i j } = \\left \\{ \\begin{tabular} { c c } $ \\begin{pmatrix} a _ { i j } & b _ { i j } \\\\ b _ { i j } & a _ { i j } \\end{pmatrix} $ & $ 1 \\leq i , j \\leq n $ \\\\ $ \\begin{pmatrix} a _ { i j } \\\\ a _ { i j } \\end{pmatrix} $ & $ 1 \\leq i \\leq n , \\ j = n + 1 $ \\\\ $ \\begin{pmatrix} a _ { i j } & b _ { i j } \\end{pmatrix} $ & $ i = n + 1 , $ \\ $ 1 \\leq j \\leq n $ \\\\ $ a _ { i j } $ & $ i = n + 1 , $ \\ $ \\ j = n + 1 . $ \\end{tabular} \\right . \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{align*} \\xi = h \\exp \\left ( - \\int \\dfrac { \\tau } { \\tau ^ 2 - \\tau - K } \\ , d \\tau \\right ) , \\phi = \\left ( h \\tau \\exp \\left ( - \\int \\dfrac { \\tau } { \\tau ^ 2 - \\tau - K } \\ , d \\tau \\right ) \\right ) e ^ \\frac { A } { k } t - \\frac { 1 } { k } \\ , . \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{align*} \\frac { ( - a - p ) _ { p + k } } { ( 1 ) _ { p + k } } = & \\frac { \\Gamma _ p ( - a + k ) \\Gamma _ p ( 1 ) } { \\Gamma _ p ( - a - p ) \\Gamma _ p ( p + k + 1 ) } \\cdot \\frac { - p } { p } \\\\ \\equiv & - \\frac { \\Gamma _ p ( - a + k ) \\Gamma _ p ( 1 ) } { \\Gamma _ p ( - a ) \\Gamma _ p ( k + 1 ) } \\bigg ( 1 + p \\cdot \\frac { \\Gamma _ p ' ( - a ) } { \\Gamma _ p ( - a ) } - p \\cdot \\frac { \\Gamma _ p ' ( k + 1 ) } { \\Gamma _ p ( k + 1 ) } \\bigg ) \\\\ \\equiv & - \\frac { ( - a ) _ k } { ( 1 ) _ k } - p \\cdot \\frac { ( - a ) _ k } { ( 1 ) _ k } \\cdot ( H _ { p - a - 1 } - H _ k ) \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{align*} \\eta ^ T _ t = \\frac { 1 } { T } \\sum _ { j \\neq k } \\sigma _ j \\sigma _ k \\int _ 0 ^ T \\int _ 0 ^ T \\mathbf { 1 } _ { [ 0 , t ] } ( x ^ j + \\xi ^ j _ r ) \\frac { 1 } { | x ^ k + \\xi ^ k _ s - x ^ j - \\xi ^ j _ r | ^ { 1 - \\frac { \\beta - \\alpha } { 2 } } } d r d s \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} { \\displaystyle \\mathbf { A } ( \\cdot , - \\Delta t ) = \\mathbf { A } ( \\cdot , 0 ) - \\Delta t \\frac { \\partial \\mathbf { A } } { \\partial t } ( \\cdot , 0 ) = \\mathbf { A } _ { 0 } - \\Delta t \\mathbf { A } _ { 1 } } , \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { k - 1 } p ^ { - t } U _ { m k + t } ^ { ( k ) } & = U _ { m } ^ { k } ( p U _ { m } ) ^ { 1 - k } \\left ( \\frac { U _ { m + 1 } ^ { k } - p ^ { k } U _ { m } ^ { k } } { U _ { m + 1 } - p U _ { m } } \\right ) \\\\ & = p ^ { 1 - k } U _ { m } \\left ( \\frac { U _ { m + 1 } ^ { k } - p ^ { k } U _ { m } ^ { k } } { - q U _ { m - 1 } } \\right ) \\\\ & = \\left ( - \\frac { p } { q } \\right ) p ^ { - k } \\left ( \\frac { U _ { m } } { U _ { m - 1 } } \\right ) \\left ( U _ { ( m + 1 ) k } ^ { ( k ) } - p ^ { k } U _ { m k } ^ { ( k ) } \\right ) . \\end{align*}"}
-{"id": "10049.png", "formula": "\\begin{align*} \\mathrm { T } ^ { 0 } ( J _ { \\varphi } X , J _ { \\varphi } Y ) = - \\mathrm { T } ^ { 0 } ( X , Y ) , \\forall X , Y \\in { \\mathfrak X } ( M ) . \\end{align*}"}
-{"id": "7945.png", "formula": "\\begin{align*} \\| \\nabla f \\| _ 2 = \\| A ^ { 1 / 2 } f \\| _ 2 \\leq C \\| ( 1 + A _ { u _ \\infty } ) ^ { 1 / 2 } f \\| _ 2 \\end{align*}"}
-{"id": "2201.png", "formula": "\\begin{align*} F _ S ( x ) ^ \\dagger & = \\frac { 1 } { 2 \\pi } \\left [ \\int _ { - \\infty } ^ \\infty ( S ( k ) - { U _ 0 } ) e ^ { i k x } d k \\right ] ^ \\dag \\\\ & = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ \\infty [ S ( k ) - { U _ 0 } ] ^ \\dag e ^ { - i k x } d k \\\\ & = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ \\infty [ S ( - k ) - { U _ 0 } ] e ^ { - i k x } d k \\\\ & = \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ \\infty [ S ( k ) - { U _ 0 } ] e ^ { i k x } d k = F _ S ( x ) . \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\beta \\\\ & \\frac 1 2 + \\frac 1 2 ( \\alpha + \\beta ) \\end{matrix} \\bigg | \\ , z \\bigg ] = { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 \\alpha & \\frac 1 2 \\beta \\\\ & \\frac 1 2 + \\frac 1 2 ( \\alpha + \\beta ) \\end{matrix} \\bigg | \\ , 4 z ( 1 - z ) \\bigg ] , \\end{align*}"}
-{"id": "10012.png", "formula": "\\begin{align*} \\displaystyle \\prod _ { p = 1 } ^ { q - 1 } ( t _ { w ( p ) } - t _ n ) \\equiv \\displaystyle \\prod _ { p = 1 } ^ { q - 1 } ( t _ { v ( p ) } - t _ n ) \\not \\equiv 0 \\ \\ \\ ( { \\rm m o d } \\ t _ { w ( i ) } - t _ { w ( j ) } ) . \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} ( \\nabla \\psi _ { n } , \\ , \\nabla u ) = ( f , \\ , u ) \\forall u \\in Z _ { n } . \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} a ' _ 1 & = a _ 1 + 2 s , \\\\ a ' _ 2 & = a _ 2 - s a _ 1 + 3 r - s ^ 2 , \\\\ a ' _ 3 & = a _ 3 + r a _ 1 + 2 t . \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{align*} R _ 0 ( e _ i ) = \\sum _ { j = 1 } ^ m \\xi _ { i j } ( x _ 1 , \\dots , x _ m ) \\partial _ { x _ j } . \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 2 } { N } K _ N \\left ( \\frac { 2 x } { N } , \\frac { 2 y } { N } ; e ^ { - N V } \\right ) = \\frac { \\sin \\pi ( x - y ) } { \\pi ( x - y ) } . \\end{align*}"}
-{"id": "4057.png", "formula": "\\begin{align*} F _ { 1 2 } = ( \\sum _ { 1 \\leq i , j \\leq 2 } H _ { i j 1 1 } x _ { i } y _ { j } ) ( \\sum _ { 1 \\leq i , j \\leq 2 } H _ { i j 2 2 } x _ { i } y _ { j } ) - ( \\sum _ { 1 \\leq i , j \\leq 2 } H _ { i j 1 2 } x _ { i } y _ { j } ) ( \\sum _ { 1 \\leq i , j \\leq 2 } H _ { i j 2 1 } x _ { i } y _ { j } ) . \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{align*} & \\mathbb { E } \\left [ f ( \\underbar { X } ) \\big | g ( \\underbar { X } ) \\right ] = \\sum \\limits _ { i = 1 } ^ n \\mathbb { E } \\left [ f _ i ( X _ i ) \\big | g ( \\underbar { X } ) \\right ] \\\\ & > \\sum \\limits _ { i = 1 } ^ n \\mathbb { E } \\left [ f _ i ( X _ i ) \\right ] = \\mathbb { E } \\left [ f ( \\underbar { X } ) \\right ] . \\end{align*}"}
-{"id": "9851.png", "formula": "\\begin{align*} A _ { s _ i } ^ { \\mathbf { z } ' } = \\frac { 1 - v z _ i ^ n z _ { i + 1 } ^ { - n } } { 1 - z _ i ^ n z _ { i + 1 } ^ { - n } } I _ { V _ { z ^ n _ 1 } } \\otimes \\cdots \\otimes \\tau \\tilde R ^ { \\gamma } ( z _ i ^ n z _ { i + 1 } ^ { - n } ) \\otimes \\cdots \\otimes I _ { V _ { z ^ n _ r } } . \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} P _ { n } T ^ { \\ , j _ 0 } \\ , e _ { k } & = \\smash [ b ] { P _ { n } T ^ { \\ , j _ 0 - ( b _ { l + 1 } - k ) } \\ \\Bigl ( v _ { l } \\ , \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Bigr ) \\ , e _ { b _ { \\varphi ( l ) } } \\Bigr ) , } \\intertext { s o t h a t } \\| P _ { n } T ^ { \\ , j _ 0 } \\ , e _ { k } \\| & \\smash [ t ] { \\le | v _ { l } | \\ , \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } | w _ { s } | \\ , \\Bigr ) \\ , \\sup _ { j \\ge 0 } \\ \\| P _ { n } T ^ { \\ , j } \\ , e _ { b _ { \\varphi ( l ) } } \\| . } \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} \\lim \\limits _ { R \\rightarrow \\infty } \\left \\Vert V \\right \\Vert _ { L _ { t } ^ { 1 } L _ { x } ^ { \\infty } \\left ( L \\left ( H \\right ) \\right ) } = 0 , \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} h _ 1 k = \\begin{pmatrix} w & 0 \\\\ 0 & - w \\end{pmatrix} + \\begin{pmatrix} \\alpha w & \\beta w + b _ 1 \\\\ - \\gamma w + c _ 1 & - \\delta w \\end{pmatrix} p \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} H _ { d } ( ( S ^ { d } ) ^ p , L ; \\mathbb { F } _ p ) & \\cong H _ { d } ( ( S ^ { d } ) ^ p ; \\mathbb { F } _ p ) / i m \\ \\iota _ { \\ast } \\\\ & = \\langle x _ 1 , \\ldots , x _ p \\rangle / \\langle x _ 1 + \\cdots + x _ p \\rangle \\\\ & = : M . \\\\ \\end{align*}"}
-{"id": "6990.png", "formula": "\\begin{align*} \\gamma _ { 1 } + \\gamma _ { 2 } & = 2 \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{align*} W _ \\pm ^ * W _ \\pm = I , W _ \\pm W _ \\pm ^ * = P _ { a c } ( H ) , \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} \\Delta _ { l ^ { 2 } } \\ , u ( t _ { i } ) = \\Vert u _ { a s } ( \\cdot , t _ { i } ) - u _ { n s } ( \\cdot , t _ { i } ) \\Vert _ { l ^ { 2 } } , \\ ; \\ ; \\delta _ { l ^ { 2 } } \\ , u ( t _ { i } ) = \\frac { \\Delta _ { l ^ { 2 } } \\ , u ( t _ { i } ) } { \\Vert u _ { a s } ( \\cdot , t _ { i } ) \\Vert _ { l ^ { 2 } } } , \\ ; \\ ; \\Delta _ { l ^ { \\infty } } \\ , u = \\Vert u _ { a s } - u _ { n s } \\Vert _ { l ^ { \\infty } } , \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{align*} \\widehat { h } ( - k ) J _ 1 ( - k ) J _ 1 ( k ) ^ { - 1 } = \\widehat { h } ( k ) , k \\in \\mathbb { R } . \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} L ^ \\lambda ( z ^ 2 ) = \\sum _ { n \\in \\mathbb { Z } } L _ n ( z ^ 2 ) ^ { - n - 2 } = - \\sum _ { n \\in \\mathbb { Z } } \\Big ( \\sum _ { k + l = n } \\left ( \\lambda ( k + 1 ) + ( \\lambda + 1 ) l \\right ) : \\chi _ { 2 k + \\frac { 1 } { 2 } } \\chi _ { 2 l - \\frac { 1 } { 2 } } : \\Big ) ( z ^ 2 ) ^ { - n - 2 } , \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} D = \\sum _ { i = 1 } ^ \\ell \\beta _ i F _ i \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} \\dot { y } y ^ { - 1 } + \\left ( \\dot { y } y ^ { - 1 } \\right ) ^ * = \\left [ \\left ( y \\phi y ^ { - 1 } \\right ) ^ * , y \\phi y ^ { - 1 } \\right ] - \\rho + f . \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{align*} K ' ( \\psi , B ^ \\ast _ 1 ) = K ' ( \\psi , B ^ \\ast _ 2 ) = \\ldots = K ' ( \\psi , B ^ \\ast _ k ) . \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle | \\sum _ { k = 1 } ^ { M } J _ 2 ^ { ( k ) , 1 } | \\leq C \\big ( h ^ { 2 r + 2 } + ( \\Delta t ) ^ { 4 } \\big ) + C \\| \\theta _ { \\Psi } ^ { M } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } + C \\Delta t \\sum _ { k = 1 } ^ { M - 1 } { \\| \\theta _ { \\Psi } ^ { k } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } } . } \\end{array} \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} \\tilde { S } = \\{ ( i j ) \\} _ { i - j > 1 } , \\{ i + 1 , i \\} _ { i \\in S } . \\end{align*}"}
-{"id": "9628.png", "formula": "\\begin{align*} f ^ { ( j ) } _ n ( b ) = | g _ 1 ( j ) \\ , g _ 2 ( j ) \\ , \\cdots \\ , g _ n ( j ) \\ , b ( j ) | , \\ \\ f ^ { ( j ) } _ 0 ( b ) = b ( j ) . \\end{align*}"}
-{"id": "7110.png", "formula": "\\begin{align*} & T _ { \\alpha } = \\{ g _ { \\alpha } \\in G _ { \\alpha } : \\exists \\beta < \\alpha g _ { \\alpha } \\in K _ { \\alpha , \\beta } \\} = \\bigcup _ { \\beta < \\alpha } K _ { \\alpha , \\beta } . \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} F _ * ^ e ( j u ^ { k - 1 } v ^ { q - 1 } ( f + u v ) ) = \\bigoplus _ { i \\in \\Delta _ e } f _ { ( i , j ) } F _ * ^ e ( i u ^ { k - 1 } v ^ { q - 1 } ) \\oplus v F _ * ^ e ( j u ^ { k } ) . \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} \\Phi ^ \\ast g ( \\phi , \\bar \\phi ) & : = g \\circ \\Phi ( \\psi , \\bar \\psi ) . \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} T _ { F , G , N } ^ L ( f _ 1 , \\dots , f _ d ) = \\sum \\limits _ { \\mathbf { r } \\in R } \\sum \\limits _ { \\substack { \\mathbf { n } \\in [ N ] ^ d \\\\ S \\mathbf { n } = \\mathbf { r } } } \\prod \\limits _ { j = 1 } ^ d f _ j ( n _ j ) \\ll _ { c , C , \\varepsilon } \\rho ^ { \\Omega ( 1 ) } + o _ { \\rho } ( 1 ) \\end{align*}"}
-{"id": "6940.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta w = v & \\Omega \\\\ w = 0 & \\partial \\Omega \\end{cases} \\end{align*}"}
-{"id": "5452.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u = \\mu \\rho u , & { \\rm i n \\ } \\Omega , \\\\ \\frac { \\partial u } { \\partial \\nu } = 0 , & { \\rm o n \\ } \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} \\mathfrak { i } _ { \\mathrm { d } } \\left ( t \\right ) : = \\underset { \\eta \\rightarrow 0 } { \\lim } \\ \\underset { l \\rightarrow \\infty } { \\lim } \\left \\{ \\left ( \\eta ^ { 2 } \\left \\vert \\Lambda _ { l } \\right \\vert \\right ) ^ { - 1 } \\mathfrak { I } _ { \\mathrm { d } } ^ { ( \\omega , \\eta \\mathbf { A } _ { l } ) } \\left ( t \\right ) \\right \\} \\ . \\end{align*}"}
-{"id": "8551.png", "formula": "\\begin{align*} \\phi ( z ) = \\Re \\left ( \\int _ { \\ast } ^ z \\frac { \\alpha } { w ^ 2 } d w , \\int _ { \\ast } ^ z \\frac { i \\alpha } { w ^ 2 } d w , \\int _ { \\ast } ^ { z } \\frac { \\beta } { w ^ { } } d w \\right ) + O ( 1 ) \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} { \\Vert \\Psi _ { h } ^ { k } \\Vert } _ { \\mathcal { L } ^ 2 } ^ { 2 } = { \\Vert \\Psi _ { h } ^ { 0 } \\Vert } _ { \\mathcal { L } ^ 2 } ^ { 2 } , \\mathcal { E } ^ { k } _ { h } = \\mathcal { E } ^ { 0 } _ { h } . \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 1 & \\pi ^ n \\\\ 1 & \\pi ^ 2 + \\pi ^ n \\end{bmatrix} \\begin{bmatrix} \\pi & 0 \\\\ 0 & 1 \\end{bmatrix} , \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{align*} T _ 0 & = 0 ; \\\\ T _ { i + 1 } & = T _ i + t _ { i + 1 } ; \\\\ z _ { i } & = R ^ { k _ i - k _ { i + 1 } } . z ( z _ { i - 1 } , t _ i ) ; \\\\ \\gamma ( \\kappa , z _ 0 , t ) & = \\gamma _ { k _ 1 } ( z _ 0 , T _ 0 , T _ 1 , t ) \\gamma _ { k _ 2 } ( z _ 1 , T _ 1 , T _ 2 , t ) \\cdots \\gamma _ { k _ n } ( z _ { n - 1 } , T _ { n - 1 } , T _ n , t ) . \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{align*} \\Lambda \\equiv \\Lambda _ 0 = \\frac { 3 } { 2 } J \\lambda _ { c o s t } , \\nu \\equiv 0 . \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{align*} \\phi _ { t } \\left ( x \\right ) = \\left [ \\exp \\left ( t X _ { H _ { 2 } } \\right ) \\circ \\exp \\left ( t X _ { H _ { 1 } } \\right ) \\right ] \\left ( x \\right ) . \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } \\left [ - \\log ( 1 - | \\alpha _ j | ^ 2 ) - \\sum _ { m = 1 } ^ { M } \\frac { | \\alpha _ j | ^ { 2 m } } { m } \\right ] < \\infty \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} J ^ \\star : = s u p p ( \\mu ) . \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} F ( t ) \\ ! \\ ! = \\ ! \\ ! \\left \\{ \\begin{array} { l l } Q ( t ) + 2 A _ 3 Q ( t + \\frac { 1 } { 2 } ) , & t \\in [ 0 , b - \\frac { 1 } { 2 } ] \\\\ Q ( t ) , & t \\in [ b - \\frac { 1 } { 2 } , 1 - b ] \\\\ Q ( t ) + 2 A _ 4 Q ( 1 - t ) , & t \\in [ 1 - b , \\frac { 1 } { 2 } ] \\\\ 2 A _ 2 Q ( t ) , & t \\in [ \\frac { 1 } { 2 } , b ] . \\end{array} \\right . \\end{align*}"}
-{"id": "4446.png", "formula": "\\begin{align*} \\SS ^ d _ + = \\{ y = ( y _ 1 , \\ldots , y _ { d + 1 } ) \\in \\SS ^ d : y _ { d + 1 } > 0 \\} \\ , . \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{align*} \\langle \\overline { x _ 1 } \\cdots \\overline { x _ { k - 1 } } | \\mathcal { B } ^ \\prime ( z ) | \\overline { y _ 1 } \\cdots \\overline { y _ { k } } \\rangle = & ( - 1 ) ^ { k + 1 } ( - 1 ) ^ { j - 1 } ( z ^ { \\overline { y _ j } } - z ^ { - \\overline { y _ j } } ) , \\end{align*}"}
-{"id": "613.png", "formula": "\\begin{align*} D _ u ( x _ 0 , r ) = \\frac { 2 } { r } \\int _ { \\frac { r } { 2 } } ^ r d t \\int _ { B _ t ( x _ 0 ) } | \\nabla u ( x ) | ^ 2 \\ , | x _ { n + 1 } | ^ a \\ , \\d x . \\end{align*}"}
-{"id": "9796.png", "formula": "\\begin{align*} \\zeta ( t ) = 1 t \\leq \\frac { 1 } { 8 } \\zeta ( t ) = 0 t \\geq \\frac { 1 } { 4 } . \\end{align*}"}
-{"id": "547.png", "formula": "\\begin{align*} \\tilde e _ { a _ 1 } \\otimes \\cdots \\otimes \\tilde e _ { a _ v } \\mapsto \\phi ^ { ( h - 1 ) ( r - 2 ) / ( r - 1 ) } ( r - 1 ) ^ h \\delta _ { ( r - 1 ) | h - 1 - \\sum _ { i = 1 } ^ v a _ i } . \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} \\Delta y _ s ( m ) = y _ { s - 1 } ( m ) , \\enskip s = 2 , 3 , . . . , n , \\enskip \\Delta y _ 1 ( m ) = y _ n ( m ) \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} V = \\sum _ { \\mathbf { q } \\in \\Lambda ^ * } e ^ { - b | \\mathbf { q } | ^ 2 } , \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} \\lim _ { x \\to b ^ - } \\dfrac { r _ 2 } { ( 1 - x ^ 2 ) ^ 2 } < \\lim _ { x \\to b ^ - } \\left ( \\sum _ { i = 0 } ^ { k - 2 } \\binom { k - 2 } { i } \\widetilde { a } _ k ^ { ( k - 2 - i ) } ( x ) f ^ { ( k + i ) } ( x ) \\right ) = : S _ 2 , \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} L _ s = s ^ \\nu \\tilde L _ s , \\tilde L _ 0 \\neq 0 . \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} \\hat { H } _ n = \\left \\{ \\hat { \\pi } ^ { ( n ) } _ n \\subseteq B _ { 8 \\mu \\beta _ 1 ^ { - 1 } n ^ { 1 + \\epsilon } } \\right \\} \\end{align*}"}
-{"id": "10014.png", "formula": "\\begin{align*} u _ 0 ( X ) + \\displaystyle \\sum _ { k = 1 } ^ { n - 1 } u _ k ( X ) Y _ k \\end{align*}"}
-{"id": "3269.png", "formula": "\\begin{align*} \\gamma _ 2 ( t ) = \\frac { 1 } { 2 c } \\left ( 1 - e ^ { - 2 c t } \\right ) + \\gamma _ 2 ( 0 ) e ^ { - 2 c t } . \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} i d _ g \\otimes i d _ h = i d _ { g h } . \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} \\Xi \\left ( z _ v : { v \\in V } \\right ) = 1 + \\sum _ { n = 1 } ^ \\infty \\frac { 1 } { n ! } \\sum _ { \\substack { ( v _ 1 , . . . , v _ n ) \\in V ^ n \\\\ \\forall i \\neq j , \\ ; v _ i \\sim v _ j } } z _ { v _ 1 } . . . z _ { v _ n } . \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ N \\int _ { \\partial B _ { \\widetilde r } ( a _ j ) \\cap \\O } | \\nabla \\eta | \\ , d \\sigma \\leq C _ 1 ( r ) . \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} n ! \\big | ( - A ) R ( \\lambda , A ) ^ { n + 1 } x \\big | _ { L ^ p } & = \\big | \\int _ { 0 } ^ { \\infty } t ^ n e ^ { - \\lambda t } ( - A ) e ^ { t A } x d t \\big | _ { L ^ p } \\\\ & \\le C ( p , \\beta ) \\int _ { 0 } ^ { \\infty } e ^ { - \\lambda t } t ^ { n - 1 } d t | x | _ { L ^ p } \\\\ & \\le C ( p , \\beta ) ( n - 1 ) ! \\lambda ^ { - n } | x | _ { L ^ p } . \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} \\int _ U { | \\omega ( y ) | \\over | x - y | ^ s } \\ , d y & = \\int _ { U \\setminus B _ r ( x ) } { | \\omega ( y ) | \\over | x - y | ^ s } \\ , d y + \\int _ { U \\cap B _ r ( x ) } { | \\omega ( y ) | \\over | x - y | ^ s } \\ , d y \\\\ & \\leq { \\| \\omega \\| _ { L ^ 1 ( U ) } \\over r ^ s } + \\left ( { \\gamma _ n \\over n - q s } \\right ) ^ { 1 / q } r ^ { ( n - q s ) / q } \\| \\omega \\| _ { L ^ p ( U ) } . \\end{align*}"}
-{"id": "2456.png", "formula": "\\begin{align*} C _ r ( f ) ^ i = A ^ { i + 1 } \\oplus B ^ i , d _ { C _ r ( f ) } ^ i = \\begin{bmatrix} d _ A ^ { i + 1 } & \\\\ ( - 1 ) ^ i f ^ { i + 1 } & d _ B ^ i \\end{bmatrix} , \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} S = a \\left ( \\Delta + A + \\gamma ^ { 2 } \\left \\vert \\nabla \\varphi \\right \\vert ^ { 2 } \\right ) - i b \\gamma \\left ( 2 \\nabla \\varphi . \\nabla + \\Delta \\varphi \\right ) + \\gamma \\partial _ { t } \\varphi , \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} \\chi ( t + \\epsilon ^ 2 ) = \\frac 1 \\alpha \\int _ 0 ^ t \\frac { ( r + \\epsilon ^ 2 ) ^ \\alpha - \\epsilon ^ { 2 \\alpha } } r d r , \\end{align*}"}
-{"id": "9944.png", "formula": "\\begin{align*} \\mathcal { N } : = N \\int d x \\ \\rho _ \\Gamma ( t , x ) \\end{align*}"}
-{"id": "9559.png", "formula": "\\begin{align*} \\overline { \\mathcal { F } \\left [ \\overline { g \\left ( \\cdot \\right ) } \\right ] \\left ( \\eta \\right ) } = \\hat { g } \\left ( - \\eta \\right ) . \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} \\dot \\omega = \\frac { u _ f } { \\delta } \\ \\eta ( y _ F , y _ O ) \\ ( G - 0 . 5 ) ^ - , \\eta ( y _ F , y _ O ) = \\min ( \\frac { y _ F } { \\nu _ F W _ F } , \\frac { y _ O } { \\nu _ O W _ O } ) , \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} & P ( R _ { n } ( f ^ * _ { \\eta } ) - R _ { n } ( f ^ * ) \\ge R ( f ^ * _ { \\eta } ) - R ( f ^ * ) + \\eta ) \\\\ & = P \\left ( R _ { n } ( f ^ * _ { \\eta } ) - R _ { n } ( f ^ * ) - ( R ( f ^ * _ { \\eta } ) - R ( f ^ * ) ) \\ge \\eta \\right ) \\\\ & \\le \\exp \\left ( - \\frac { n } { 4 } \\left ( \\frac { \\eta ^ { 2 - \\beta } } { B } \\land 3 \\eta \\right ) \\right ) \\\\ & \\le \\exp \\left ( - \\frac { n \\eta ^ { 2 - \\beta } } { 4 B } \\right ) . \\end{align*}"}
-{"id": "8250.png", "formula": "\\begin{align*} C _ n = T _ { n - 1 } T _ { n - 2 } \\dots T _ 1 \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} ( F ^ \\sigma ) ^ \\natural = F ^ \\sigma / ( F ^ \\sigma \\cap F + \\ker ( \\sigma ) ) \\cong ( F + F ^ \\sigma ) / ( F + \\ker ( \\sigma ) ) . \\end{align*}"}
-{"id": "7400.png", "formula": "\\begin{align*} \\log | h ( t ) | = \\lambda _ 1 \\log t + \\lambda _ 2 \\log \\log t + \\cdots + \\lambda _ n \\log ^ { ( n ) } t + O ( 1 ) \\end{align*}"}
-{"id": "5187.png", "formula": "\\begin{align*} \\left \\{ \\vert \\pi \\vert \\right \\} _ { \\pi \\in \\mathcal { C } \\left ( 3 \\right ) } = \\left \\{ 1 , 2 , 2 , 3 \\right \\} \\end{align*}"}
-{"id": "2122.png", "formula": "\\begin{gather*} f = \\tilde { q } _ 1 - \\frac { \\tilde { t } _ 2 } { \\tilde { p } _ 2 } + \\frac { \\tilde { p } _ 1 } { \\tilde { p } _ 2 { } ^ 2 } . \\end{gather*}"}
-{"id": "1946.png", "formula": "\\begin{align*} \\mu = \\alpha \\wedge \\tilde \\mu , \\end{align*}"}
-{"id": "9689.png", "formula": "\\begin{align*} T \\ = \\ \\sum _ { 1 \\leq i _ j \\leq d } T _ { i _ 1 \\dots i _ r } e _ { i _ 1 } \\otimes \\dots \\otimes e _ { i _ r } . \\end{align*}"}
-{"id": "3875.png", "formula": "\\begin{align*} \\frac { f ( - 1 + h + a ) } { 2 r _ x } = \\begin{cases} - \\frac { 1 } { r _ x } + h \\left ( 1 + \\frac { 1 } { r _ x } \\right ) & \\mbox { i f $ a - 2 + h \\ge 0 $ } , \\\\ \\frac { 1 } { r _ x } & \\mbox { i f $ a - 2 + h \\leq 0 $ } \\end{cases} \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} [ e _ r , e _ r ] = \\theta _ r e _ n , [ e _ i , e _ j ] = \\sum ^ { n - 1 } _ { t = 1 } \\alpha ^ t _ { i j } e _ t + \\beta _ { i j } e _ n , [ e _ j , e _ i ] = - \\sum ^ { n - 1 } _ { t = 1 } \\alpha ^ t _ { i j } e _ t + \\gamma _ { j i } e _ n . \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} M _ \\infty = \\begin{pmatrix} \\frac { - 1 } { m } & \\frac { - \\lambda ^ 2 } { m - 2 } & \\lambda & \\lambda ^ m \\\\ \\frac { - 1 } { m } & \\frac { - ( 1 + \\lambda ) ^ 2 } { m - 2 } & 1 + \\lambda & ( 1 + \\lambda ) ^ m \\\\ 0 & \\frac { - \\lambda } { n - 1 } & 1 & m \\lambda ^ { m - 1 } \\\\ 0 & \\frac { - ( 1 + \\lambda ) } { n - 1 } & 1 & m ( 1 + \\lambda ) ^ { m - 1 } \\end{pmatrix} \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{align*} n _ { i j } ^ { \\infty } = \\left \\{ \\begin{array} { l r } ( - 1 ) ^ { j - i } { { \\lfloor { { i - 1 } \\over 2 } \\rfloor + j - i } \\choose { \\lfloor { { i - 1 } \\over 2 } \\rfloor } } , & { \\rm i f } ~ ~ 1 \\leq i \\leq j , \\\\ 0 , & { \\rm i f } ~ ~ i > j \\geq 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = \\frac { \\alpha _ 2 } { \\gamma _ 2 } y _ 5 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\beta _ 1 } y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = - \\frac { \\alpha _ 3 } { \\beta _ 1 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 2 ] = y _ 4 , [ y _ 2 , y _ 3 ] = y _ 5 . \\end{align*}"}
-{"id": "453.png", "formula": "\\begin{align*} \\mathbf { f } \\circ \\mathbf { h } = \\mathrm { i d } _ { \\mathbf { V } } . \\end{align*}"}
-{"id": "1264.png", "formula": "\\begin{align*} \\mathcal { F } _ { t } ^ { - } = \\sigma \\left \\{ Z _ { \\tau } ^ { - 1 } \\left ( F \\right ) : \\tau \\geq t , F \\in \\mathcal { B } _ { d } \\right \\} , \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{align*} M ^ \\omega \\cap M ' = M ^ \\omega \\cap M , \\end{align*}"}
-{"id": "2255.png", "formula": "\\begin{align*} g ' = g X , X : [ 0 , t _ f ] \\to \\sl , \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{align*} f _ * \\widetilde { Q } ( E ) & = f _ * \\sum _ a ( f ^ * Q _ a ) E ^ a = \\sum _ a Q _ a f _ * E ^ a = \\sum _ a Q _ a \\sum _ { \\ell = 1 } ^ d Z _ \\ell ^ { a } M _ \\ell = \\sum _ { a } \\sum _ { \\ell = 1 } ^ d Q _ a Z _ \\ell ^ { a } M _ \\ell = \\sum _ { \\ell = 1 } ^ d { Q } ( Z _ \\ell ) M _ \\ell . \\end{align*}"}
-{"id": "7700.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { k - 1 } \\binom { k - 1 } { t } ( - p ) ^ { - t } U _ { m k + t } ^ { ( k ) } & = ( - p ) ^ { 1 - k } U _ { m } \\sum _ { t = 0 } ^ { k - 1 } \\binom { k - 1 } { t } ( - p ) ^ { k - 1 - t } U _ { m } ^ { k - 1 - t } U _ { m + 1 } ^ { t } \\\\ & = ( - p ) ^ { 1 - k } U _ { m } \\sum _ { t = 0 } ^ { k - 1 } \\binom { k - 1 } { t } ( - p U _ { m } ) ^ { k - 1 - t } U _ { m + 1 } ^ { t } . \\end{align*}"}
-{"id": "10037.png", "formula": "\\begin{align*} \\mathcal G _ { ( \\varphi , g ) } = \\bigcup _ { p \\in M } \\left \\{ ( X _ 1 , \\ldots , X _ r , Y _ 1 , \\ldots , Y _ s ) \\in \\mathcal F _ p ( M ) \\colon \\begin{array} { c } X _ 1 , \\ldots , X _ r \\in ( D _ { \\phi } ) _ p , Y _ 1 , \\ldots , Y _ s \\in ( D _ { \\bar \\phi } ) _ p , \\\\ g ( X _ i , X _ j ) = \\delta _ { i j } , i , j = 1 , \\ldots , r , \\\\ g ( Y _ i , Y _ j ) = \\delta _ { i j } , i , j = 1 , \\ldots , s , \\\\ g ( X _ i , Y _ j ) = 0 , i , = 1 , \\ldots r , j = 1 , \\ldots , s \\\\ \\end{array} \\right \\} , \\end{align*}"}
-{"id": "9593.png", "formula": "\\begin{align*} \\Delta _ n ^ \\mu - \\Delta _ n ^ \\mathrm { p r o j } & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n E [ X _ { n , i } \\varepsilon _ { n , i } ] E [ \\varepsilon _ { n , i } X _ { n , i } ^ { \\prime } ] \\\\ & - \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n E [ X _ { n , i } \\varepsilon _ { n , i } ] Z _ { n , i } ^ { \\prime } \\right ) \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n Z _ { n , i } Z _ { n , i } ^ { \\prime } \\right ) ^ { - 1 } \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n Z _ { n , i } E [ \\varepsilon _ { n , i } X _ { n , i } ^ { \\prime } ] \\right ) . \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} { D } _ { j _ 1 \\dots j _ n } ^ { ( s ) } = ( 0 , \\infty ) ^ n . \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{align*} \\dot { H } = 2 \\Lambda - H ^ 2 + v ( v + 2 u ) - 8 \\pi \\psi . \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{align*} d X _ i ( t ) = \\sqrt { 2 ( X _ i ^ 2 ( t ) + 1 ) } d W _ i ( t ) + \\left [ \\left ( 2 - 2 N - 2 \\Re ( s ) \\right ) X _ i ( t ) + 2 \\Im ( s ) + \\sum _ { j \\ne i } ^ { } \\frac { 2 ( X ^ 2 _ i ( t ) + 1 ) } { X _ i ( t ) - X _ j ( t ) } \\right ] d t . \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = \\frac { \\alpha _ 2 } { \\beta _ 5 } y _ 4 , [ y _ 1 , y _ 2 ] = y _ 3 + \\frac { \\alpha _ 5 } { \\beta _ 5 } y _ 4 , [ y _ 2 , y _ 1 ] = \\frac { \\beta _ 1 } { \\alpha _ 4 } y _ 3 , [ y _ 2 , y _ 2 ] = \\frac { \\beta _ 4 } { \\alpha _ 4 } y _ 3 + y _ 4 , [ y _ 2 , y _ 3 ] = y _ 5 , [ y _ 1 , y _ 4 ] = y _ 5 . \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} \\Upsilon _ { k | l } : = \\Upsilon _ { m - k } \\overline { \\Upsilon } _ { m + l } = \\langle \\ , K _ a \\mid a \\in \\{ 1 , \\dots m - k \\} \\cup \\{ m + l + 1 , \\dots , m + n \\} \\ , \\rangle , \\end{align*}"}
-{"id": "8361.png", "formula": "\\begin{align*} T _ n = T _ n ( \\omega ) = \\inf _ { \\pi \\in { \\cal P } _ n } T ( \\pi , \\omega ) \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} \\mathbb { E } ( \\hat { U } ^ { ( n ) } _ n - \\mathbb { E } \\hat { U } ^ { ( n ) } _ { n } ) ^ 2 = \\sum _ { l = 1 } ^ { N } \\mathbb { E } X _ l ^ 2 . \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{align*} \\eta _ { j , j + 2 } = \\frac { 1 } { \\alpha _ { j + 1 } } \\left [ \\alpha _ { j - 1 } \\eta _ { j - 1 , j + 1 } + ( \\beta _ { j } - \\beta _ { j + 1 } ) \\eta _ { j , j + 1 } + \\gamma _ { j + 1 } \\eta _ { j + 1 , j + 1 } + \\gamma _ { j + 1 } \\eta _ { j , j } \\right ] \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} \\langle \\tilde { G } , G \\rangle & = G _ 0 \\tilde { G } _ 1 ^ \\ast \\frac { i c ^ 2 y } { ( 2 \\pi ) ^ 2 } \\left ( J _ 0 ( y , - \\lambda y ) - \\lambda y \\sum _ { n \\geq 1 } \\dfrac { J _ n ( y , - \\lambda y ) } { n } \\right ) \\\\ & = G _ 0 \\tilde { G } _ 1 ^ \\ast \\frac { i c } { 2 \\pi } \\partial _ \\lambda \\Lambda ( \\lambda ) . \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} { g } ( x ) : = \\frac { { Q _ 1 } ( x ) } { { S } ( x ) } , \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} \\nabla g ^ { v } = 0 , \\end{align*}"}
-{"id": "5676.png", "formula": "\\begin{align*} J _ { i j } ( t ) = \\left ( \\begin{array} { c c } - 3 \\bar { v } ^ 2 + 2 ( a + 1 ) \\bar { v } - a & \\ : \\ : \\ : \\ : \\ : - 1 \\\\ \\\\ \\varepsilon b & \\ : \\ : \\ : \\ : \\ : - \\varepsilon c \\end{array} \\right ) , \\end{align*}"}
-{"id": "5333.png", "formula": "\\begin{align*} m _ i ( \\lambda ) & = \\sum _ { j = 1 } ^ { \\ell ( \\mu ) } m _ i ( { \\eta ' } ^ j ) . \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{align*} B _ { n } = \\sum _ { p = 1 } ^ { n } \\left ( - 1 \\right ) ^ { p } \\frac { \\left \\{ \\begin{array} { c } n + p \\\\ p \\end{array} \\right \\} } { \\binom { n + p } { p } } \\binom { n + 1 } { p + 1 } , \\ , \\ , n \\ge 1 . \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} I ^ n \\subseteq J _ 2 ^ n + \\sum _ { j = 0 } ^ { n - 1 } I ^ { j } ( I J _ 2 ^ { n - j - 1 } \\cap J _ 1 ) , \\hbox { f o r a l l } \\ n \\geq 1 . \\end{align*}"}
-{"id": "9702.png", "formula": "\\begin{align*} h _ { T _ 0 } ^ 2 ( t ) & \\ = \\ t ( v _ 1 ^ 2 + v _ 2 ^ 2 + v _ 3 ^ 2 ) + t ^ 2 ( ( v _ 1 + v _ 2 ) ^ 2 + ( v _ 2 + v _ 3 ) ^ 2 + ( v _ 3 + v _ 1 ) ^ 2 - v _ 1 ^ 2 - v _ 2 ^ 2 - v _ 3 ^ 2 ) \\\\ h _ { T _ 1 } ^ 2 ( t ) & \\ = \\ t v _ 1 ^ 2 + t ^ 2 ( ( v _ 1 + v _ 2 ) ^ 2 + ( v _ 1 + v _ 3 ) ^ 2 - v _ 1 ^ 2 ) + t ^ 3 ( v _ 2 + v _ 3 ) ^ 2 \\\\ h _ { T _ 2 } ^ 2 ( t ) & \\ = \\ t ^ 2 ( v _ 2 + v _ 3 ) ^ 2 + t ^ 3 ( ( v _ 1 + v _ 2 ) ^ 2 + ( v _ 1 + v _ 3 ) ^ 2 - v _ 1 ^ 2 ) + t ^ 4 v _ 1 ^ 2 \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} C ( M , J , [ g ] ) = \\frac { 1 } { \\operatorname { v o l } ( M ) } \\int _ M s ^ H \\frac { \\omega _ M ^ n } { n ! } \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} f ( z ) - \\beta = \\frac { z - \\beta } { ( c z + d ) ( c \\beta + d ) } . \\end{align*}"}
-{"id": "9999.png", "formula": "\\begin{align*} W ( G _ { 1 } ( n , d , x - 1 , s ) ) - W ( G _ { 1 } ( n , d , x , s ) ) & \\leq W ( G _ { 1 } ( n , d , 2 , s ) ) - W ( G _ { 1 } ( n , d , 1 , s ) ) = \\\\ & = 2 ( n - 5 ) + 2 . \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c | c } A _ 0 & A _ 1 \\end{array} \\right ) B = \\left ( \\begin{array} { c } B _ 0 \\\\ \\hline B _ 1 \\end{array} \\right ) , \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{align*} x _ 0 x _ 1 = x _ 2 x _ 3 = x _ 4 ^ 2 , \\end{align*}"}
-{"id": "6790.png", "formula": "\\begin{align*} \\gamma = \\left ( k a _ { 0 } \\right ) ^ { \\frac { 1 } { 2 } } \\beta = k ^ { \\frac { 1 } { p } } , \\alpha = a _ { 0 } ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "6169.png", "formula": "\\begin{align*} [ [ L _ { - q + 2 } , \\ , L _ { - 2 } ] , \\ , S _ { q - 1 } ] = 0 \\end{align*}"}
-{"id": "5862.png", "formula": "\\begin{align*} | \\hat { \\rho } _ c ( \\mathbf { q } ) | ^ 2 = | \\hat { s } ( \\mathbf { q } ) | ^ 2 ~ | \\hat { \\rho } ( \\mathbf { q } ) | ^ 2 \\end{align*}"}
-{"id": "955.png", "formula": "\\begin{align*} \\textrm { C o v } _ B [ \\gamma X , Y ] & = \\gamma \\textrm { C o v } _ B [ X , Y ] = \\textrm { V a r } _ B [ Y ] \\gamma \\beta ( X | Y ) = \\frac { \\textrm { V a r } _ B [ Y ] } { \\textrm { V a r } [ Y ] } \\gamma \\textrm { C o v } [ X , Y ] \\\\ & = \\frac { \\textrm { V a r } _ B [ Y ] } { \\textrm { V a r } [ Y ] } \\textrm { C o v } [ \\gamma X , Y ] = 0 . \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} \\widehat v ' ( \\rho ) = a + ( b - a ) \\frac { e ^ \\rho } { 1 + e ^ \\rho } , \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{align*} 0 = [ T _ { g } , T _ { f } ] = [ T _ { g } , T _ { f _ 1 } ] + [ T _ { g } , T _ { \\bar { z } ^ m \\bar { f _ 2 } } ] = [ T _ { \\psi _ 2 } , T _ { \\varphi _ 1 } ] + [ T _ { g } , T _ { \\bar { z } ^ m \\bar { f _ 2 } } ] . \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} \\varphi \\circ \\tilde { \\mathbf { q } } = q \\circ v , \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{align*} \\psi = \\sum _ { j = 1 } ^ m a _ j \\mathbf { 1 } _ { I _ j } , \\end{align*}"}
-{"id": "2023.png", "formula": "\\begin{align*} A ^ P ( z ) = \\frac { \\tilde { A } _ 0 } { z } . \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { \\nu } \\varepsilon _ { n } ^ { \\sigma _ { i } } \\alpha _ { i } \\frac { \\partial ^ { i } u } { \\partial x _ { n } ^ { i } } \\left ( x ^ { \\prime } , 0 , t \\right ) = 0 \\nu \\in \\left \\{ 0 , 1 \\right \\} , \\end{align*}"}
-{"id": "9976.png", "formula": "\\begin{align*} r a n k ( A _ { i _ j } ^ { - 1 } E _ { i _ j } - A _ j ^ { - 1 } E _ j ) = r a n k ( A _ { i _ j } ^ { - 1 } E _ { i _ j } ) - r a n k ( A _ j ^ { - 1 } E _ j ) \\ , . \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} \\lambda = \\lambda _ 1 ^ 1 \\cdots \\lambda _ 1 ^ { m _ 1 } \\lambda _ 2 ^ 1 \\cdots \\lambda _ 2 ^ { m _ 2 } \\cdots \\lambda _ { k + l } ^ 1 \\cdots \\lambda _ { k + l } ^ { m _ { k + l } } \\end{align*}"}
-{"id": "3486.png", "formula": "\\begin{align*} \\begin{cases} \\varphi _ { \\sigma } ( x ) = - \\frac { 1 } { 2 } \\varphi _ { \\sigma } ^ { \\prime \\prime } ( x ) + \\varphi _ { \\sigma } ^ { 2 \\sigma + 1 } ( x ) , 0 < x < + \\infty , \\\\ \\varphi _ { \\sigma } ( 0 ) = 0 , \\lim \\limits _ { x \\to + \\infty } \\varphi _ { \\sigma } ( x ) = 1 . \\end{cases} \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ p \\psi _ { X _ 1 \\ldots J X _ k \\ldots X _ p } = ( r - s ) J ^ E \\left ( \\psi _ { X _ 1 \\ldots X _ p } \\right ) \\forall X _ i \\in T M . \\end{align*}"}
-{"id": "8773.png", "formula": "\\begin{align*} g _ { ( x , y ) } = g _ { i j } ( x , y ) d x ^ { i } \\otimes d x ^ { j } \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} f _ n ( \\tilde { \\omega } ) = \\frac { E [ \\prod _ { i = 0 } ^ { \\tau } F ( S _ i , \\widehat { { S } } _ i ; S _ { i + 1 } , \\widehat { S } _ { i + 1 } ) ] E [ \\prod _ { i = \\tau + 1 } ^ { n - 1 } F ( S _ i , \\widehat { { S } } _ i ; S _ { i + 1 } , \\widehat { S } _ { i + 1 } ) ] } { \\{ E \\prod _ { i = 0 } ^ { \\tau } [ \\frac { \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } { 1 + \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } ] \\} ^ 2 \\{ E \\prod _ { i = \\tau + 1 } ^ { n - 1 } [ \\frac { \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } { 1 + \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } ] \\} ^ 2 } . \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{gather*} A _ 0 = \\begin{pmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} , A _ 1 = \\begin{pmatrix} p _ 2 q _ 2 & - p _ 2 & p _ 1 p _ 2 \\\\ 0 & p _ 1 q _ 1 - p _ 2 q _ 2 - \\theta ^ 0 _ 2 & 1 \\\\ - t & q _ 1 & - p _ 1 q _ 1 - \\theta ^ 0 _ 1 \\end{pmatrix} , \\\\ A _ 2 = \\begin{pmatrix} 0 & 0 & 0 \\\\ q _ 2 & - 1 & p _ 1 \\\\ 0 & 0 & 0 \\end{pmatrix} . \\end{gather*}"}
-{"id": "3107.png", "formula": "\\begin{align*} \\Delta _ 0 = \\sum _ { k = - \\infty } ^ { \\infty } ( V ( \\tilde { z } _ k , \\tilde { z } _ { k + 1 } ) - V ( z _ k , z _ { k + 1 } ) ) . \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} T ( \\hat { \\pi } ^ { ( n ) } _ n ) = \\hat { T } ^ { ( n ) } ( \\hat { \\pi } ^ { ( n ) } _ n ) = \\hat { T } ^ { ( n ) } _ n \\leq \\hat { T } ^ { ( n ) } ( \\pi ) = \\sum _ { i = 1 } ^ { r } t ^ { ( n ) } ( g _ i ) \\leq \\sum _ { i = 1 } ^ { r } t ( g _ i ) = T ( \\pi ) . \\end{align*}"}
-{"id": "2671.png", "formula": "\\begin{align*} a ^ 2 = 4 \\alpha ^ 3 \\wedge a ^ 3 = 2 \\beta ^ 2 . \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} \\left ( a \\right ) _ { n } = \\frac { \\Gamma \\left ( a + n \\right ) } { \\Gamma \\left ( a \\right ) } . \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{align*} T _ { \\varepsilon , B } = \\frac { 6 \\varepsilon } { \\pi ^ 2 \\alpha ^ 2 } B ^ { \\frac { 2 } { 3 } + \\delta } + O ( B ^ { \\frac { 2 } { 3 } + \\frac { 2 } { 3 } \\varepsilon + ( 1 - \\varepsilon ) \\frac { \\delta } { 2 } } ) = \\frac { 6 \\varepsilon } { \\pi ^ 2 \\alpha ^ 2 } ( B ^ { \\frac { 2 } { 3 } + \\delta } + o ( 1 ) ) . \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{align*} & \\mathcal { B } _ \\eta \\left ( \\hat { D } _ A ( \\mathbf { x } ) \\ , \\hat { D } _ B ( \\mathbf { y } ) \\ , \\hat { \\rho } _ { A B } \\ , { \\hat { D } _ A ( \\mathbf { x } ) } ^ \\dag \\ , { \\hat { D } _ B ( \\mathbf { y } ) } ^ \\dag \\right ) \\\\ & = \\hat { D } _ C \\left ( \\sqrt { \\eta } \\ , \\mathbf { x } + \\sqrt { 1 - \\eta } \\ , \\mathbf { y } \\right ) \\ , \\mathcal { B } _ \\eta ( \\hat { \\rho } _ { A B } ) \\ , { \\hat { D } _ C \\left ( \\sqrt { \\eta } \\ , \\mathbf { x } + \\sqrt { 1 - \\eta } \\ , \\mathbf { y } \\right ) } ^ \\dag \\end{align*}"}
-{"id": "1652.png", "formula": "\\begin{align*} g ^ 0 ( \\rho ) = \\sum _ { \\ell \\in \\N _ * } g ^ 0 _ \\ell \\rho ^ \\ell , g ^ \\omega ( \\rho ) = \\sum _ { \\ell \\in \\N _ * } g ^ \\omega _ \\ell \\rho ^ \\ell \\end{align*}"}
-{"id": "9636.png", "formula": "\\begin{align*} | a _ { d - l } | > \\frac { 1 } { 2 } \\left ( \\frac { d } { d - 2 l } \\right ) \\sum _ { k = 0 , k \\ne l , d - l } ^ { d } | a _ { k } | , \\ ; \\ ; l < d / 2 \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{align*} h ^ { 2 } ( P , Q ) = \\frac { 1 } { 2 } \\int \\left ( \\sqrt { \\frac { d P } { d \\nu } } - \\sqrt { \\frac { d P } { d \\nu } } \\right ) ^ { 2 } d \\nu \\quad \\mbox { e t } \\quad \\rho ( P , Q ) = \\int \\sqrt { \\frac { d P } { d \\nu } \\frac { d Q } { d \\nu } } d \\nu . \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} \\ell \\ = \\ U ^ { n , 0 } \\ , . \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} k _ 1 ( x , 1 ) = e ^ { - x } , k _ 1 ( x , 2 ) = \\cos x , k _ 2 ( x , 2 ) = \\sin x . \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} H _ { f _ 0 } ( t ) = \\left ( \\int _ { \\bar t } ^ { ( \\bar t + t ) / 2 } + \\int _ { ( \\bar t + t ) / 2 } ^ t \\right ) T ( t - \\tau ) \\mathbb P f _ 0 ( \\tau ) d \\tau = : H _ { f _ 0 , 1 } ( t ) + H _ { f _ 0 , 2 } ( t ) . \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} \\sup _ { | x - \\theta | < 1 } | g ^ \\prime ( x ) | & = \\sup _ { | x - \\theta | < 1 } | 2 | ( a \\lambda _ 2 ^ 2 - b \\mu _ 2 ^ 2 ) x + ( a \\lambda _ 1 \\lambda _ 2 - b \\mu _ 1 \\mu _ 2 ) | \\\\ & \\leqslant 2 \\sqrt { a b } | \\lambda _ 2 \\mu _ 1 - \\lambda _ 1 \\mu _ 2 | + 2 | a \\lambda _ 2 ^ 2 - b \\mu _ 2 ^ 2 | + 4 | a \\lambda _ 1 \\lambda _ 2 - b \\mu _ 1 \\mu _ 2 | . \\end{align*}"}
-{"id": "9909.png", "formula": "\\begin{align*} \\dot { x } _ { 1 } & = x _ { 1 } ^ { 3 } + x _ { 3 } , \\\\ \\dot { x } _ { 2 } & = x _ { 1 } + x _ { 3 } , \\\\ \\dot { x } _ { 3 } & = 0 . 1 x _ { 1 } + x _ { 2 } ^ { 2 } + u . \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { \\infty } \\frac { b _ { 2 k - 1 } ( \\mu ) } { ( 2 k ) ! } \\sum \\limits _ { t = 1 } ^ { \\infty } C _ Y ^ { 2 k } ( t ) < + \\infty . \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{align*} X _ 1 = \\frac { \\partial } { \\partial x _ 1 } - x _ 2 \\frac { \\partial } { \\partial x _ 3 } , X _ 2 = \\frac { \\partial } { \\partial x _ 2 } + x _ 1 \\frac { \\partial } { \\partial x _ 3 } , X _ 3 = \\frac { \\partial } { \\partial x _ 3 } \\end{align*}"}
-{"id": "1701.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ { \\delta } r ^ s = \\frac { q ^ { s + 1 } } { ( s + 1 ) d ^ { s + 1 } } + V _ s ( q ) \\end{align*}"}
-{"id": "556.png", "formula": "\\begin{align*} \\sum _ { \\substack { d _ 1 + \\dots + d _ { n + 1 } = g \\\\ 0 \\leq d _ i \\leq a _ i } } \\prod _ { i = 1 } ^ { n + 1 } Q _ { d _ i } ( a _ i ) \\prod _ { i = 1 } ^ { n + 1 } \\psi _ i ^ { d _ i } \\end{align*}"}
-{"id": "9708.png", "formula": "\\begin{align*} P = \\{ x \\in \\R ^ d \\colon A x \\leq 1 \\} \\end{align*}"}
-{"id": "4501.png", "formula": "\\begin{align*} D _ { \\vartheta _ { 0 } } : = \\sup \\left \\{ \\left \\Vert \\Psi ^ { ( \\omega , \\vartheta ) } \\right \\Vert _ { \\mathcal { W } } : \\omega \\in \\Omega , \\ \\vartheta \\in \\lbrack 0 , \\vartheta _ { 0 } ] \\right \\} < \\infty \\ . \\end{align*}"}
-{"id": "5227.png", "formula": "\\begin{align*} u = \\nabla \\varphi + V \\textrm { i n } \\Omega , \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{align*} M _ \\infty : = N \\ , { \\mathbb E } X \\ , \\int _ 0 ^ \\infty \\overline F ( x ) { \\rm d } x = N \\ , \\lambda \\ , E , \\end{align*}"}
-{"id": "9930.png", "formula": "\\begin{align*} \\delta = \\frac { 1 - 2 \\varepsilon } { 2 ( 5 - 2 \\varepsilon ) } \\ \\ \\implies \\ \\ \\frac { \\frac { 1 } { 2 } + \\delta } { \\frac { 1 } { 2 } - \\delta } = \\frac { 3 } { 2 } - \\varepsilon . \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 0 ^ { ( 1 ) } } { x ^ 2 } + \\frac { A _ 0 ^ { ( 0 ) } } { x } + \\frac { A _ { t _ 1 } } { x - t _ 1 } + N \\right ) Y , \\\\ \\frac { \\partial Y } { \\partial t _ 1 } = \\left ( N _ 1 - \\frac { A _ { t _ 1 } } { x - t _ 1 } \\right ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = \\left ( N _ 2 - \\frac { \\frac { 1 } { t _ 2 } A _ 0 ^ { ( 1 ) } } { x } \\right ) Y . \\end{gather*}"}
-{"id": "7255.png", "formula": "\\begin{align*} \\begin{cases} \\beta _ 1 \\gamma _ 1 = \\alpha _ 1 \\gamma _ 3 & \\beta _ 1 \\gamma _ 2 + \\beta _ 2 \\gamma _ 5 = \\alpha _ 1 \\gamma _ 4 + \\alpha _ 2 \\gamma _ 6 \\\\ \\beta _ 4 \\gamma _ 1 = \\alpha _ 4 \\gamma _ 3 & \\beta _ 4 \\gamma _ 2 + \\beta _ 5 \\gamma _ 5 = \\alpha _ 4 \\gamma _ 4 + \\alpha _ 5 \\gamma _ 6 \\\\ \\gamma _ 3 \\gamma _ 5 = \\gamma _ 1 \\gamma _ 6 \\end{cases} \\end{align*}"}
-{"id": "2578.png", "formula": "\\begin{align*} X ( \\psi , \\bar \\psi ) & = \\sum _ { l \\in \\Z \\setminus \\{ 0 \\} } X _ l ( \\psi , \\bar \\psi ) e ^ { i l \\cdot } . \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} y \\Lambda _ n = \\{ x \\in \\Delta _ p \\ | \\ x _ j = y _ j \\ ( j < n ) \\} \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} x \\cdot \\frac { \\mathrm { d } } { \\mathrm { d } \\beta } \\pounds _ 1 ( x ) = \\pounds _ { 0 } ( x ) = \\frac { x - x ^ p } { 1 - x } \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } C _ k H _ k x ^ { k + 1 } \\equiv \\pounds _ 1 ( \\beta ) + ( \\alpha - \\beta ) ^ { p + 1 } \\pounds _ 1 \\left ( \\frac { \\beta } { \\beta - \\alpha } \\right ) \\pmod { p } . \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} L _ m F _ { m n } L _ { m n + m } & = L _ m ( F _ { 2 m n + m } - ( - 1 ) ^ { m n } F _ m ) \\\\ & = L _ m F _ { 2 m n + m } + ( - 1 ) ^ { m n - 1 } F _ { 2 m } \\ , . \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} R = \\begin{bmatrix} D & \\pi ^ n D \\\\ D & D \\end{bmatrix} = \\left \\{ \\ , \\begin{bmatrix} a & b \\pi ^ n \\\\ c & d \\end{bmatrix} : a , b , c , d \\in D \\ , \\right \\} , \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} p ( z ) = \\frac { \\sqrt { 3 } z - 2 } { 2 z - \\sqrt { 3 } } , q ( z ) = \\frac { - 1 } { z - \\sqrt { 3 } } , r ( z ) = \\frac { - z - 1 } { z } . \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} i \\partial _ { t } w _ { n } + \\Delta w _ { n } + A w _ { n } = \\tilde { V } _ { n } \\left ( x , t \\right ) w _ { n } + \\tilde { F } _ { n } \\left ( x , t \\right ) , \\end{align*}"}
-{"id": "2189.png", "formula": "\\begin{align*} f ( i k _ j , x ) P _ j = - 2 i k _ j \\varphi ( i k _ j , x ) N _ { - , k _ j } A _ { k _ j } P _ j = - 2 i k _ j \\varphi ( i k _ j , x ) N _ { - , k _ j } B _ j , \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} I n d _ { \\mathbb Z / p , \\mathbb { F } _ p } F ( \\mathbb R ^ d , p ) = \\langle a b ^ { \\frac { ( d - 1 ) ( p - 1 ) } { 2 } } , b ^ { \\frac { ( d - 1 ) ( p - 1 ) } { 2 } + 1 } \\rangle \\subseteq \\mathbb { F } _ p [ a , b ] / \\langle a ^ 2 \\rangle . \\end{align*}"}
-{"id": "9650.png", "formula": "\\begin{align*} P _ { D , r } = \\sum _ { i = 1 } ^ { h _ K } \\phi _ E ( z _ i ) . \\end{align*}"}
-{"id": "9910.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 0 & 0 & 1 \\\\ 1 & 0 & 1 \\\\ 0 . 1 & 0 & 0 \\end{bmatrix} , \\ ; B = \\begin{bmatrix} 0 \\\\ 0 \\\\ 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{align*} \\limsup _ { m \\to \\infty } | v _ { n ( m ) } | \\ , \\cdot \\ , 2 ^ { n ( m ) } ( \\Delta b _ { n ( m ) } ) ^ m \\left ( \\prod _ { j = 1 } ^ { m - 1 } \\frac { | v _ { n ( j ) } | } { \\Delta b _ { n ( j ) } } \\ , \\prod _ { \\nu = b _ { n ( j ) } + 1 } ^ { b _ { n ( j ) + 1 } - 1 } | w _ { \\nu } | \\right ) < \\infty . \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} h = \\sum _ { i = 1 } ^ n h _ i g _ i = \\sum _ { i = 1 } ^ n c _ i e _ i , \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} \\psi | _ { x _ 1 = 0 } = 0 . \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} \\beta _ 0 = \\sum _ { i = 1 } ^ { p } W ( \\phi _ i ) + \\sum _ { j = 1 } ^ q \\left ( W ( \\vec { \\Psi } _ j ) - 4 \\pi \\theta _ j \\right ) \\in 4 \\pi \\N , \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} \\gamma = \\frac { 1 } { \\sqrt { B N _ 0 } } = \\sqrt { } \\end{align*}"}
-{"id": "9582.png", "formula": "\\begin{align*} E [ X _ { n , i } Y _ { n , i } ] & = E [ X _ { n , i } U _ { n , i } ^ { \\prime } ] \\theta _ { n , i } + E [ X _ { n , i } ] \\xi _ { n , i } \\\\ & = E [ X _ { n , i } X _ { n , i } ^ { \\prime } ] \\theta _ { n , i } . \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} ( \\pi _ U ) _ * M A ( \\Phi _ j ) = N \\int _ { t = 0 } ^ 1 M A ( ( 1 - t ) u + t v _ j ) t ^ { N - 1 } d t . \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , S _ r ] = [ L _ { - 2 } , \\ , S _ 4 ] = [ L _ { - 2 } , \\ , [ S _ 1 , \\ , S _ 3 ] ] = [ S _ 1 , \\ , S _ 1 ] = 0 \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{align*} & \\left | \\frac { d g ( z ) } { d \\bar { z } } \\right | ^ 2 - 2 \\pi \\mathrm { R e } \\left ( \\bar { z } g ( z ) \\frac { d g ( z ) } { d \\bar { z } } \\right ) \\\\ & = \\frac { 1 } { 4 m ^ 3 n ^ 3 } e ^ { - \\left [ \\frac { 2 } { x ( 1 - x ) } + \\frac { 2 } { y ( 1 - y ) } \\right ] } \\Bigg { [ } \\frac { ( 2 x - 1 ) ^ 2 } { x ^ 4 ( 1 - x ) ^ 4 } + \\frac { ( 2 y - 1 ) ^ 2 } { y ^ 4 ( 1 - y ) ^ 4 } + \\frac { 4 \\pi ( 2 x - 1 ) } { x ( 1 - x ) ^ 2 } + \\frac { 4 \\pi ( 2 y - 1 ) } { y ( 1 - y ) ^ 2 } \\Bigg { ] } \\geq 0 . \\end{align*}"}
-{"id": "6067.png", "formula": "\\begin{align*} T _ i = \\{ x \\in E _ i \\mid [ x , E _ { - 1 } ] \\subseteq T _ { i - 1 } \\} . \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{align*} \\beta = - 4 \\lambda _ 1 \\lambda _ 2 , \\gamma = 1 + \\lambda _ 1 + \\lambda _ 2 \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} K _ { i - 1 } ( m + s ) = \\sum _ { j = 1 } ^ n K _ j ( s ) K _ { i - j } ( m ) . \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} F ( x ) = \\prod _ { i = 1 } ^ d \\frac { 1 } { x - a _ i } , \\end{align*}"}
-{"id": "1605.png", "formula": "\\begin{align*} h _ { 2 } ^ { - 1 } h _ { 1 } = g _ { i } g _ { \\lambda ( 2 ) } a _ { 3 } g _ { s ( 2 ) } ^ { - 1 } g _ { s ( 1 ) } a _ { 1 } ^ { - 1 } g _ { \\lambda ( 1 ) } ^ { - 1 } g _ { i } ^ { - 1 } \\in H \\cap g _ { i } G _ { j } g _ { i } ^ { - 1 } = H \\cap G _ { w _ { i } } = H \\cap g _ { i } A g _ { i } ^ { - 1 } = 1 , \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{align*} \\textrm { V a r } [ Y \\mid Y \\in B _ 1 ] = \\ldots = \\textrm { V a r } [ Y \\mid Y \\in B _ k ] , \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} \\vert \\widetilde { T } _ { F , G , N } ^ L ( \\widetilde { f _ 1 } , \\dots , \\widetilde { f _ d } ) \\vert \\ll _ G \\frac { 1 } { N ^ { d - m } } \\int \\limits _ { \\substack { \\mathbf { x } \\in \\mathbb { R } ^ d \\\\ L \\mathbf \\mathbf { x } = \\mathbf { 0 } } } \\Big ( \\prod \\limits _ { j = 1 } ^ d \\widetilde { f _ j } ( x _ j ) \\Big ) F ( \\mathbf { x } ) \\ , d \\mu ( \\mathbf { x } ) , \\end{align*}"}
-{"id": "8925.png", "formula": "\\begin{align*} f ( H ) P _ { a c } ( H ) = W _ { \\pm } f ( - \\Delta ) W _ { \\pm } ^ * , \\end{align*}"}
-{"id": "3212.png", "formula": "\\begin{align*} e ^ { - \\lambda _ N t } \\int _ { W ^ N \\cap [ - R , R ] ^ N } ^ { } \\Delta _ N ( y ) \\det \\left ( \\partial ^ { ( i - 1 ) } _ x p ^ { ( N ) , s } _ t ( x , y _ j ) \\right ) _ { i , j = 1 } ^ N d y \\lesssim C ( N , t , R ) \\prod _ { i = 1 } ^ { N } \\int _ { - R } ^ { R } | \\partial ^ { ( i - 1 ) } _ x p ^ { ( N ) , s } _ t ( x , z ) | d z . \\end{align*}"}
-{"id": "9942.png", "formula": "\\begin{align*} [ \\Gamma ( t , x + z , x ) ] \\ \\widetilde { } = \\frac { 1 } { 4 \\pi | \\xi | } \\widehat { \\Gamma _ 0 } \\left ( \\frac { \\xi ^ 2 - \\tau } { 2 \\xi } , \\frac { \\xi ^ 2 + \\tau } { 2 \\xi } \\right ) e ^ { i z ( \\frac { \\xi ^ 2 - \\tau } { 2 \\xi } ) } \\end{align*}"}
-{"id": "9844.png", "formula": "\\begin{align*} ( \\mathbf { z } ' ) ^ { \\alpha ' } = \\mathbf { z } ^ { n _ \\alpha \\alpha ^ \\vee } . \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} [ \\langle \\widetilde N , x ^ * \\rangle ] _ s = [ \\langle N , x ^ * \\rangle ] _ { \\tau _ s } \\leq [ \\langle M , x ^ * \\rangle ] _ { \\tau _ s } = [ \\langle \\widetilde M , x ^ * \\rangle ] _ s \\end{align*}"}
-{"id": "7728.png", "formula": "\\begin{align*} H _ k = \\left \\{ x \\in \\Delta _ p \\ | \\ \\{ j \\in \\omega \\ | \\ k < ( j + 1 ) ^ 2 - j ^ 2 \\ \\wedge \\ ( x _ { | I ^ k _ j } \\equiv 0 \\vee x _ { | I ^ k _ j } \\equiv p - 1 ) \\} \\in \\mathcal { U } \\right \\} \\end{align*}"}
-{"id": "9454.png", "formula": "\\begin{align*} \\frac { { \\rm d } } { { \\rm d } t } \\int _ { \\mathbb { T } ^ 3 } \\left [ \\frac { 1 + B ^ 2 } { 2 h } + \\varepsilon \\frac { h ( v ^ 2 + d ^ 2 ) } { 2 } \\right ] + \\int _ { \\mathbb { T } ^ 3 } h ( v ^ 2 + d ^ 2 ) + \\varepsilon \\int _ { \\mathbb { T } ^ 3 } \\big | \\nabla ^ l v \\big | ^ 2 + \\big | \\nabla ^ l d \\big | ^ 2 = 0 \\end{align*}"}
-{"id": "7176.png", "formula": "\\begin{align*} \\overline { r } : V = \\Gamma ' / [ \\Gamma ' , \\Gamma ' ] \\to k ( P ) ( \\tilde { z } ) . \\end{align*}"}
-{"id": "10010.png", "formula": "\\begin{align*} w & = w ( 1 ) \\cdots w ( r ) \\cdots w ( h ( r ) ) \\ w ( h ' ( r ) ) \\ w ( h '' ( r ) ) \\cdots w ( h ( r + 1 ) ) \\cdots w ( n ) , \\\\ w ' & = w ( 1 ) \\cdots w ( r ) \\cdots w ( h ( r ) ) \\ w ( h '' ( r ) ) \\ w ( h ' ( r ) ) \\cdots w ( h ( r + 1 ) ) \\cdots w ( n ) . \\end{align*}"}
-{"id": "3098.png", "formula": "\\begin{align*} \\left | 1 - q \\right | ^ 2 = 1 - 2 e ^ { - 2 \\pi y } \\cos ( 2 \\pi x ) + e ^ { - 4 \\pi y } > 1 - 2 e ^ { - 2 \\pi y } \\cos ( 2 \\pi M y ) + e ^ { - 4 \\pi y } . \\end{align*}"}
-{"id": "9165.png", "formula": "\\begin{align*} & A + A ' - B ' B - ( p - 2 ) ( T _ { B } + \\Lambda ) ' ( T _ { B } + \\Lambda ) \\\\ & = { \\rm d i a g } \\Bigl \\{ 2 + \\lambda ^ { 2 } - ( p - 2 ) \\Big ( c + \\frac { \\lambda + \\mu } { 2 } \\Big ) ^ { \\ ! 2 } , \\ 2 + \\mu ^ { 2 } - ( p - 2 ) \\Big ( c - \\frac { \\lambda + \\mu } { 2 } \\Big ) ^ { \\ ! 2 } \\Bigr \\} \\\\ & = : { \\rm d i a g } \\{ g ( c ) , \\ , h ( c ) \\} . \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} \\lambda = \\frac { \\eta \\ , \\exp \\frac { S ( A | M ) _ { \\hat { \\rho } _ { A M } ( t ) } } { n } } { \\eta \\ , \\exp \\frac { S ( A | M ) _ { \\hat { \\rho } _ { A M } ( t ) } } { n } + \\left | 1 - \\eta \\right | \\exp \\frac { S ( B | M ) _ { \\hat { \\rho } _ { B M } ( t ) } } { n } } \\ ; , \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} ( 0 \\neq ) [ L _ { - j } , \\ , S _ j ] = [ [ L _ { - 1 } , L _ { - j + 1 } ] , \\ , S _ j ] = [ L _ { - 1 } , \\ , [ L _ { - j + 1 } , \\ , S _ j ] ] = [ L _ { - 1 } , \\ , L _ 1 ] \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} \\aligned R _ { l _ i } = \\sup _ { \\mathbf { x } \\in Q _ { R _ { l _ i } } } d _ { Q _ { R _ { l _ i } } } ( \\mathbf { x } ) | D ^ 2 \\widehat { f _ i } ( \\mathbf { x } ) | . \\endaligned \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{align*} q _ i = \\left \\{ \\begin{array} { c r } \\frac { \\alpha } { 1 - \\alpha } \\ , q _ { i - 1 } & i \\neq k + 1 \\\\ \\frac { \\beta } { 1 - \\beta } \\ , q _ { i - 1 } & i = k + 1 \\end{array} \\right . . \\end{align*}"}
-{"id": "9841.png", "formula": "\\begin{align*} \\sigma ( g , h ) = \\mathbf { s } ( g ) \\ , \\mathbf { s } ( h ) \\ , \\mathbf { s } ( g h ) ^ { - 1 } . \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} \\hat \\ell & : = L _ \\ell ^ - ( p - \\ell ^ ) = 2 L _ \\ell ^ - p = 2 ( \\ell + \\lceil ( p - \\ell ) / 2 \\rceil ) - p \\\\ & = 2 \\ell + 2 \\lceil ( p - \\ell ) / 2 \\rceil - p = \\begin{cases} \\ell & p - \\ell , \\\\ \\ell + 1 & \\end{cases} \\end{align*}"}
-{"id": "7149.png", "formula": "\\begin{align*} \\check f ^ { ( 3 , 4 ) } ( ( 0 , 0 , 0 , 0 ) , g ) = \\hat f ^ { ( 3 , 4 ) } ( ( 0 , 0 , 0 , 0 ) , g ) . \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} \\sup _ { j \\ge 0 } \\ \\| P _ { n } T ^ { \\ , j } P _ { l } \\ , x \\| & \\le C _ { l } \\ , \\cdot \\ , ( b _ { l + 1 } - b _ { l } ) ^ { 1 - \\frac 1 { p } } \\ , \\cdot \\ , \\Bigl \\| \\sum _ { k = b _ { l } } ^ { b _ { l + 1 } - 1 } \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Bigr ) \\ , x _ { k } e _ { k } \\Bigr \\| , \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{align*} \\Bigl \\| \\sum _ { k \\ , \\in \\ , I _ { j _ { 0 } } } x _ { k } \\ , \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Bigr ) \\ , e _ { k } \\ , \\Bigr \\| \\ge \\dfrac { X _ { l } } { 2 } \\cdot \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{align*} \\Xi ^ { \\lambda } ( K _ a F _ a \\otimes 1 \\otimes 1 \\otimes E _ a ) = \\sum _ { i , j , m , p } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) \\pi ( K _ { a } F _ { a } ) _ { m } ^ { j } c _ { p } ^ { m } \\pi ( E _ { a } K _ { \\lambda } ) _ { i } ^ { p } . \\end{align*}"}
-{"id": "6553.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\dot x & = & a - b x - \\gamma x ^ \\alpha y ^ \\beta \\\\ \\dot y & = & c - d y + \\gamma x ^ \\alpha y ^ \\beta . \\end{array} \\end{align*}"}
-{"id": "4006.png", "formula": "\\begin{align*} \\Psi ( 0 ) = - \\frac { \\Gamma _ p ( \\frac 1 2 ) \\Gamma _ p ( \\frac 1 2 + \\beta ) \\Gamma _ p ( \\beta ) } { \\Gamma _ p ( \\frac 1 2 - \\frac 1 2 a ) \\Gamma _ p ( 1 + \\frac 1 2 a ) \\Gamma _ p ( \\beta - \\frac 1 2 a ) \\Gamma _ p ( \\frac 1 2 + \\beta + \\frac 1 2 a ) } = - \\Phi ( 0 ) . \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{align*} R ^ e _ n ( s _ { v _ 0 } ) ( w _ { i } ) & = s _ { w ( i - e , i - 1 ) } s _ { w ( i - e + 1 , i ) } ( w _ i ) = s _ { w ( i - e , i - 1 ) } ( w _ i - w ( i - e + 1 , i ) ) \\\\ & = w _ i + ( w ( i - e , i - 1 ) , w _ i ) w ( i - e , i - 1 ) \\\\ & ~ ~ ~ ~ - w ( i - e + 1 , i ) - ( w ( i - e + 1 , i ) , w ( i - e , i - 1 ) ) w ( i - e , i + 1 ) \\\\ & = w _ i + w ( i - e , i - 1 ) - w ( i - e + 1 , i ) \\\\ & = w _ { i - e } . \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} \\omega = \\varphi ^ * \\left ( \\frac { d z } { z ^ { k + 1 } } + \\lambda \\frac { d z } { z } \\right ) \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{align*} \\alpha = \\lim _ { n \\rightarrow \\infty } \\frac { \\pi } { k _ { n + 1 } - k _ { n } } . \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} F _ 1 = 0 \\Rightarrow & \\mu K _ 1 h \\frac { b + c } { 1 + c } - \\frac { \\Gamma c } { K + c } + \\lambda \\theta = 0 . \\\\ \\Rightarrow & ( \\mu h K _ 1 - \\Gamma + \\lambda \\theta ) c ^ 2 + ( \\mu h K _ 1 ( b + K ) - \\Gamma + \\lambda \\theta ( K + 1 ) ) c + ( \\mu h K _ 1 b K + \\lambda \\theta ) = 0 \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} \\lambda = \\frac { m _ 4 } { m _ 3 + m _ 4 } , \\mu = \\frac { m _ 2 } { m _ 1 + m _ 2 } . \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} P ( x ) = \\sum _ { k = 1 } ^ n P ( x _ k ) h _ k ( x ) + \\sum _ { k = 1 } ^ n P ' ( x _ k ) \\mathfrak { h } _ k ( x ) , \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mu } \\left ( Z _ { 0 } \\in A _ { R _ { 1 } , R _ { 2 } } \\right ) = 0 \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} \\big [ [ J _ i ^ + , J _ j ^ - ] , J _ k ^ + \\big ] = + \\delta _ { j k } J _ i ^ + + \\delta _ { i j } J _ k ^ + \\end{align*}"}
-{"id": "1429.png", "formula": "\\begin{align*} \\nabla ^ \\perp \\cdot \\widetilde { \\Omega } H = \\nabla ^ \\perp \\widetilde { \\Omega } H _ 2 \\cdot \\nabla H _ 1 + \\nabla ^ \\perp H _ 2 \\cdot \\nabla \\widetilde { \\Omega } H _ 1 , \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { j } \\frac { 1 } { m ! } \\left . \\frac { d ^ { m } } { d \\rho ^ { m } } w _ { k + n } ( x _ { 1 } , . . . , x _ { k + n } | \\rho ) \\right \\vert _ { \\rho = 0 } \\times \\prod _ { s = 1 } ^ { k } T _ { ( j - m ) + t _ { s } } ( x _ { s } ) \\prod _ { s = 1 + k } ^ { n + k } U _ { ( j - m ) + t _ { s } } ( x _ { s } ) = 0 . \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} \\mathbb { P } ( \\# { \\cal E } _ i \\geq M \\log { n } ) \\leq \\frac { 1 } { C } e ^ { - \\delta _ 0 M \\log { n } } = \\frac { 1 } { C } \\frac { 1 } { n ^ { M \\delta _ 0 } } \\leq \\frac { 1 } { n ^ { \\theta + 1 } } \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} ( \\mathcal { L } _ { \\xi ^ { \\mathbf { c } } } L ) \\left ( x , y \\right ) = \\mu L ( x , y ) . \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} H ^ p _ { + } ( \\R ) = \\{ f + i H ( f ) : f \\in L ^ p ( \\R ) \\} \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{align*} \\zeta _ { \\mbox { } _ { R , \\ ; \\ ! { \\scriptstyle \\epsilon } } } ( x ) \\ ; = \\ ; \\ ! \\ ; \\ ! \\Bigl \\{ \\ ; \\ ! e ^ { \\ ! - \\ , \\epsilon \\ ; \\ ! \\ , \\sqrt { \\ : \\ ! 1 \\ , + \\ , | \\ ; \\ ! x \\ ; \\ ! | ^ { 2 } \\ ; \\ ! } } - \\ ; \\ ! \\ ; \\ ! e ^ { \\ ! - \\ , \\epsilon \\ ; \\ ! \\sqrt { \\ ; \\ ! 1 \\ ; \\ ! + \\ ; \\ ! R ^ { \\ : \\ ! 2 } \\ : \\ ! } } \\ ; \\ ! \\Bigr \\} ^ { \\ ! \\ : \\ ! p } \\ , \\mbox { i f } \\ ; | \\ ; \\ ! x \\ ; \\ ! | \\ ; \\ ! < \\ : \\ ! \\mbox { \\footnotesize $ R $ } \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{align*} I _ j : = \\Bigl \\{ k \\in [ b _ { l } , b _ { l + 1 } ) \\ , ; \\ , j + k - b _ { l } \\mod ( b _ { l + 1 } - b _ l ) \\ \\textrm { d o e s n o t b e l o n g t o } \\ [ k _ { 0 } , k _ { 1 } ) \\cup [ k _ 2 , b _ { l + 1 } - b _ l ) \\Bigr \\} . \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} { 4 n \\choose 2 j } - { 2 n \\choose j } & = { 2 n \\choose j } \\left ( \\frac { ( 4 n - 1 ) ( 4 n - 3 ) \\cdots ( 4 n - 2 j + 1 ) } { 1 \\cdot 3 \\cdots ( 2 j - 1 ) } - 1 \\right ) \\\\ & = { 2 n \\choose j } \\frac { ( 4 n - 1 ) ( 4 n - 3 ) \\cdots ( 4 n - 2 j + 1 ) - 1 \\cdot 3 \\cdots ( 2 j - 1 ) } { 1 \\cdot 3 \\cdots ( 2 j - 1 ) } . \\end{align*}"}
-{"id": "1948.png", "formula": "\\begin{align*} p _ { i j } = \\left \\{ \\begin{array} { l l } { b } _ { i j } & \\mbox { i f } f _ { j i } ( - { b } _ { i j } ) \\geq 0 \\mbox { a n d } \\\\ \\max \\left \\{ a _ { i j } , \\left \\lfloor { - f ^ { - 1 } _ { j i } ( 0 ) } \\right \\rfloor \\right \\} & \\mbox { e l s e w h e r e } . \\end{array} \\right . \\begin{array} { l } ( \\forall ( i , j ) \\in E ) . \\end{array} \\\\ \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} E _ t ( i , n ) & : = \\{ \\widehat { \\pi } _ 0 ( - t ) \\in [ i / n , ( i + 1 ) / n ) \\} ; \\\\ E _ t ( i , n , s ) & : = \\{ \\widehat { \\pi } ^ { ( \\frac { i + 1 } { n } , s ) } ( s - 1 ) > \\widehat { \\pi } ^ { ( \\frac { i } { n } , s ) } ( s - 1 ) \\} , \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} d X _ t ^ i = \\frac { a } { n } \\sum _ { j = 1 } ^ n ( X _ t ^ j - X _ t ^ i ) \\ , d t + \\sigma \\ , d W _ t ^ i + \\gamma _ { t } ^ i d N ^ i _ t \\ , , i = 1 , \\dots , n , \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} \\sum _ { n \\in \\Z } \\frac { ( - 1 ) ^ n q ^ { \\frac { n ( 3 n + 1 ) } { 2 } } } { 1 + q ^ n } = \\frac 1 2 + 2 \\sum _ { n \\ge 1 } \\frac { ( - 1 ) ^ n q ^ { \\frac { n ( 3 n + 1 ) } { 2 } } } { 1 + q ^ n } \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{align*} \\left \\vert x - \\frac { p ' } { q ' } \\right \\vert ~ = ~ \\frac { x } { q ' | p + p ' y | } \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} h _ 0 \\Big ( \\left ( \\chi _ { - j _ k } \\right ) ^ { m _ k } \\dots \\left ( \\chi _ { - j _ 2 } \\right ) ^ { m _ 2 } \\left ( \\chi _ { - j _ 1 } \\right ) ^ { m _ 1 } | 0 \\rangle \\Big ) = \\Big ( \\sum _ { j _ i \\in 2 \\mathbb { Z } + 1 / 2 } m _ i - \\sum _ { j _ i \\in 2 \\mathbb { Z } - 1 / 2 } m _ i \\Big ) \\Big ( \\left ( \\chi _ { - j _ k } \\right ) ^ { m _ k } \\dots \\left ( \\chi _ { - j _ 2 } \\right ) ^ { m _ 2 } \\left ( \\chi _ { - j _ 1 } \\right ) ^ { m _ 1 } | 0 \\rangle \\Big ) . \\end{align*}"}
-{"id": "5815.png", "formula": "\\begin{align*} R [ \\bar { g } ] e ^ { 2 u } \\le \\left ( R [ \\bar { g } ] - 2 \\Delta u \\right ) \\ , \\ , n = 2 \\ , \\\\ R [ \\bar { g } ] u ^ { \\frac { n + 2 } { n - 2 } } \\le \\left ( R [ \\bar { g } ] u - \\frac { 4 ( n - 1 ) } { n - 2 } \\Delta u \\right ) \\ , \\ , n \\ge 3 . \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} X _ { 2 , \\ , j + 1 , \\ , 4 } \\cap L _ { j + 1 } = 0 . \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{align*} \\begin{cases} r ^ { \\frac { 2 } { \\alpha } } h _ m ( r ) \\to ( - 1 ) ^ m \\beta ^ { \\frac { 1 } { \\alpha } } & \\frac { 2 } { N - 2 } < \\alpha < \\frac { 4 } { N - 2 } \\\\ | \\log r | ^ { \\frac { 1 } { \\alpha } } r ^ { \\frac { 2 } { \\alpha } } h _ m ( r ) \\to ( - 1 ) ^ m ( { \\frac { 2 } { \\alpha } } ) ^ { \\frac { 2 } { \\alpha } } & \\alpha = \\frac { 2 } { N - 2 } \\end{cases} \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{align*} w _ { ( p _ 1 , p _ 2 ) } ( z _ 1 , z _ 2 ) = z _ 1 ^ { p _ 1 } z _ 2 ^ { p _ 2 } \\cdot \\delta ( z _ 1 , z _ 2 ) + \\ ; \\mbox { \\rm l o w e r o r d e r t e r m s } . \\ ; \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} q ( \\eta ) = \\left \\langle q , Z ^ p _ m ( \\cdot , \\eta ) \\right \\rangle _ { \\widehat { S } _ p } \\quad \\textrm { f o r e v e r y } q \\in \\mathcal { H } ^ p _ m ( \\widehat { S } _ p ) . \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{align*} W _ { \\pm } f = \\lim _ { t \\to \\pm \\infty } e ^ { i t H } e ^ { - i t H _ 0 } f . \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} \\lim _ { y \\to \\infty } y ^ 2 J _ 1 ( y , - \\lambda y ) = \\int _ 0 ^ \\infty e ^ { - s ^ 2 / 2 - \\lambda s } s d s . \\end{align*}"}
-{"id": "8201.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\langle F ^ { p - 1 } ( \\nabla u _ { k } ) \\nabla _ { \\xi } F ( \\nabla u _ { k } ) , \\nabla \\varphi \\rangle d x = \\lambda _ { k } \\int _ \\Omega | u _ { k } | ^ { p - 2 } u _ { k } \\varphi \\ , d x \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} & \\partial _ t w _ k + ( J _ k w _ k ) \\cdot \\nabla w _ k + S w _ k = \\Delta w _ k - \\nabla p _ k + f , \\\\ & \\mbox { d i v $ w _ k $ } = 0 , \\\\ & w _ k | _ { \\partial \\Omega } = 0 , \\\\ & w _ k \\to 0 \\quad \\mbox { a s $ | x | \\to \\infty $ } , \\\\ & w _ k ( \\cdot , 0 ) = w _ 0 . \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} \\min \\{ I ( u ) : \\ \\ u \\in \\R ^ n \\ \\ \\hbox { a n d } \\ \\ u | _ { \\partial V } = f \\} . \\end{align*}"}
-{"id": "8764.png", "formula": "\\begin{align*} \\Sigma f \\circ \\Phi ( Y ) ( q ) = \\Phi ( X ) \\circ S ^ { 1 } \\land f ( q ) . \\end{align*}"}
-{"id": "6628.png", "formula": "\\begin{align*} W _ r ( v _ 0 , v _ \\infty ) \\ ; & = \\ ; \\lim _ { r \\downarrow 0 } W _ r ( v _ 0 , v _ \\infty ) \\ ; = \\ ; \\textstyle { \\frac { 4 ^ B B } { ( 1 + \\nu ) \\cos ( B \\pi ) } } \\ ; = : \\ ; W _ 0 ^ \\infty \\ , . \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} b _ k = \\sum _ { j = 0 } ^ { k } ( - 1 ) ^ j \\binom { k } { j } a _ j = [ z ^ 0 ] \\biggl ( ( - 1 ) ^ k ( 1 - z ) ^ k \\sum _ { j = 0 } ^ { N } a _ j z ^ { - j } \\biggr ) \\end{align*}"}
-{"id": "3550.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\dot { W } = C ^ { 2 } _ { 0 } - W ^ 2 & \\hbox { ; } \\\\ W ( 0 ) = H ( 0 ) & \\hbox { . } \\end{array} \\right . \\end{align*}"}
-{"id": "9916.png", "formula": "\\begin{align*} m = \\pm ( n + k p ) , \\ell = \\pm ( n - k d ) . \\end{align*}"}
-{"id": "3124.png", "formula": "\\begin{align*} | F _ 1 | & = ( V _ 1 ( z _ { - N } , x _ { - N + 1 } ) - V _ 1 ( z _ { - N } , z _ { - N + 1 } ) ) ^ 2 \\\\ & \\leq | V _ { 1 2 } ( z _ { - N } , c _ 1 ) | | x _ { - N + 1 } - z _ { - N + 1 } | \\cdot ( | V _ 1 ( z _ { - N } , x _ { - N + 1 } ) | + | V _ 1 ( z _ { - N } , z _ { N + 1 } ) | ) \\\\ & \\leq \\frac { \\kappa _ 2 } { 4 } | x _ { - N + 1 } - z _ { - N + 1 } | , \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } W _ { n + 1 } ( x ) = \\lim _ { n \\rightarrow \\infty } \\lim _ { m \\rightarrow \\infty } \\sum _ { j = 1 } ^ { l _ m } \\sum _ { k = 0 } ^ n G ^ x _ k ( w _ { m , j - 1 } ) [ W _ { n - k } ( w _ { m , j } ) - W _ { n - k } ( w _ { m , j - 1 } ) ] . \\end{align*}"}
-{"id": "4710.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 \\\\ - i ( z ^ 2 - 1 ) ^ { 1 / 2 } e ^ { - N \\xi ( z ) } & 1 \\end{pmatrix} ; \\end{align*}"}
-{"id": "8114.png", "formula": "\\begin{align*} L = \\partial _ t ^ { k + 1 } + \\sum _ { j = 1 } ^ k a _ j ( t ) \\partial _ t ^ { n - j } , a _ 0 , \\ldots , a _ k : \\bar U \\to \\C \\end{align*}"}
-{"id": "9509.png", "formula": "\\begin{align*} \\Psi ( z + j + a u ) = z , \\ \\ \\ z \\in \\Z ( M ) , \\ \\ j \\in J , \\ \\ a \\in A . \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} y ( P ) ^ 2 \\bigl [ 4 f ( x ( P ) ) x ( [ 2 ] P ) - g ( x ( P ) ) \\bigr ] = \\Delta ^ \\prime , \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{align*} \\Delta \\phi ' _ l = \\left ( \\mathbf { z } _ I \\cdot \\mathbf { z } _ { \\Delta \\phi _ l } \\right ) \\Delta \\phi _ l + \\left ( \\mathbf { z } _ I \\cdot \\mathbf { z } _ { \\Delta \\phi _ { v o , l } } \\right ) \\Delta { \\phi _ { v o , l } } \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} a \\cdot ( x \\otimes y ) = \\tilde \\psi ( \\phi ( a ) ) ( x \\otimes y ) = ( a \\cdot x ) \\otimes y . \\end{align*}"}
-{"id": "8789.png", "formula": "\\begin{align*} \\tilde { L } ( x , y ) = e ^ { \\sigma } L ( x , y ) , ~ ~ \\forall ( x , y ) \\in A ; \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{align*} h & = ( 0 , \\frac { \\Lambda _ 2 ^ \\vee } { 4 } + \\frac { \\Lambda _ { 1 0 } ^ \\vee } { 2 } ) , \\\\ 2 [ h ] & = ( 0 , \\frac { \\Lambda _ 2 ^ \\vee } { 2 } - \\Lambda _ 3 ^ \\vee + \\Lambda _ 4 ^ \\vee - \\Lambda _ 5 ^ \\vee + \\Lambda _ 6 ^ \\vee - \\Lambda _ 7 ^ \\vee + \\Lambda _ 8 ^ \\vee - \\Lambda _ 9 ^ \\vee + \\Lambda _ { 1 0 } ^ \\vee ) . \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} f ( x ) = A \\left ( \\frac { - c ^ { n - 1 } } { m } + O ( c ^ { n - 2 } ) \\right ) + B \\left ( \\frac { - c ^ { n - 2 } } { m - 2 } + O ( c ^ { n - 3 } ) \\right ) x ^ 2 + C x + D x ^ m . \\end{align*}"}
-{"id": "1294.png", "formula": "\\begin{align*} \\mathcal { N } _ { 0 , T } = c \\exp \\left [ \\frac { T E _ { \\mathsf { 0 } } } { 2 } \\right ] \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} \\Psi _ 1 ( 0 ) + z \\cdot \\Psi _ 2 ( 0 ) = & { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & - a \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] + z ^ { a + 1 } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & - a \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac 1 z \\bigg ] \\\\ = & ( 1 + z ) ^ { a + 1 } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - \\frac { a } 2 & - \\frac { a } 2 + \\frac 1 2 \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac { 4 z } { ( 1 + z ) ^ 2 } \\bigg ] = ( 1 + z ) \\Phi ( 0 ) . \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{align*} u ( t ) = e ^ { ( t - \\tau ) \\Delta } u ( \\tau ) + \\int _ \\tau ^ t e ^ { ( t - s ) \\Delta } | u ( s ) | ^ \\alpha u ( s ) \\ , d s \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} { A } _ n ( k ) = \\left \\{ \\sum _ { i = 1 } ^ { n } t ^ { ( k ) } ( f _ i ) \\leq 2 \\mu n \\right \\} , \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} w = w ( X , Y ) : = \\frac { X ^ 2 + ( 1 - Y ) ^ 2 } { 1 - Y } . \\end{align*}"}
-{"id": "9827.png", "formula": "\\begin{align*} s _ i ( 1 + \\mathcal { T } _ i ) = \\left ( \\frac { 1 - v \\mathbf { z } ^ { \\alpha _ i ^ { \\vee } } } { 1 - v \\mathbf { z } ^ { - \\alpha _ i ^ { \\vee } } } \\right ) ( 1 + \\mathcal { T } _ i ) . \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} \\Phi _ { U ' } ( x _ 1 , x _ 2 ) = ( \\frac { x _ 2 } { x _ 1 } , x _ 1 ^ k ) : = ( z ' , w ' ) \\end{align*}"}
-{"id": "7761.png", "formula": "\\begin{align*} r _ { i } ^ { B _ { 1 } } \\leq l _ { i } ^ { B _ { 2 } } \\mbox { f o r } i = 1 , 2 . \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} \\left \\{ \\Xi _ { \\mathrm { p } , l } ^ { ( \\omega ) } \\left ( t \\right ) \\right \\} _ { k , q } : = \\frac { 1 } { \\left \\vert \\Lambda _ { l } \\right \\vert } \\underset { x , y \\in \\Lambda _ { l } } { \\sum } \\sigma _ { \\mathrm { p } } ^ { ( \\omega ) } \\left ( x + e _ { q } , x , y + e _ { k } , y , t \\right ) \\end{align*}"}
-{"id": "2866.png", "formula": "\\begin{align*} S = \\delta ( \\partial _ 1 , \\partial _ 2 ) + \\sum \\limits _ { i _ 1 + i _ 2 < \\mu } b _ { i _ 1 , i _ 2 } ( x _ 1 , x _ 2 ) \\partial _ 1 ^ { i _ 1 } \\partial _ 2 ^ { i _ 2 } . \\end{align*}"}
-{"id": "1094.png", "formula": "\\begin{align*} \\vert \\widetilde { T } _ { F , G , N } ^ { L , \\Xi , \\widetilde { \\mathbf { r } } } ( g _ 1 , \\dots , g _ d ) \\vert \\ll \\Big \\vert \\frac { 1 } { N ^ { h - m } } \\int \\limits _ { \\mathbf { x } \\in \\mathbb { R } ^ { h - m } } F _ 1 ( \\mathbf { x } + \\sum \\limits _ { k = 1 } ^ { s + 1 } w _ k \\mathbf { f _ k } ) \\prod \\limits _ { j = 1 } ^ d g _ j ( \\psi _ j ^ \\prime ( \\mathbf { x } , \\mathbf { w } ) + a _ j ) \\ , d \\mathbf { x } \\Big \\vert , \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} s _ { ( N ) } = s _ N = \\sum _ { \\sum i k _ i = N } \\frac { x _ 1 ^ { k _ 1 } } { k _ 1 ! } . . . \\frac { x _ N ^ { k _ N } } { k _ N ! } \\end{align*}"}
-{"id": "1158.png", "formula": "\\begin{align*} \\begin{cases} \\dot { x } ( t ) = x ( t ) v ( t ) , \\ \\ \\ \\ \\ \\ t \\in ( 0 , \\tau ) , \\\\ x ( 0 ) = I d _ { S U ( { N _ 1 } ) } . \\\\ \\end{cases} \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} B ( z ) = c \\prod _ { k = 1 } ^ n \\frac { z - a _ k } { 1 - \\overline { a } _ k z } \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} \\Psi = \\left ( \\psi _ { j _ 1 \\dots j _ n } ^ { ( s ) } \\right ) _ { j _ 1 , \\dots , j _ n = 1 } ^ { | \\mathcal { E } | } \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{align*} \\left | \\frac { \\det ( \\Lambda _ d ) n _ 1 } { ( \\lambda _ 2 - \\alpha \\mu _ 2 ) ^ 2 } \\varepsilon B ^ { - \\frac { 1 } { r } } - \\frac { n _ 1 \\mu _ 1 + n _ 2 \\mu _ 2 } { \\lambda _ 2 - \\alpha \\mu _ 2 } \\varepsilon B ^ { - \\frac { 1 } { r } } \\right | = \\left | \\frac { \\mu _ 2 ( n _ 2 - \\theta n _ 1 ) } { \\lambda _ 2 - \\alpha \\mu _ 2 } \\varepsilon B ^ { - \\frac { 1 } { r } } \\right | \\leqslant 2 ^ { 1 0 } K \\varepsilon ^ 2 b ^ 2 \\det ( \\Lambda _ d ) ^ { \\frac { 3 } { 2 } } B ^ { 1 - \\frac { 2 } { r } } , \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} \\mu = \\left ( \\frac { N } { 6 4 . 1 7 } \\right ) ^ 2 , \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} 0 = [ f , P _ 0 ] _ 1 | _ a ^ b = [ f , 1 ] _ 1 | _ a ^ b & = \\langle \\ell [ f ] , 1 \\rangle _ { L ^ 2 [ ( a , b ) , w ] } - \\langle f , \\ell [ 1 ] \\rangle _ { L ^ 2 [ ( a , b ) , w ] } \\\\ & = \\int _ a ^ b \\left ( \\dfrac { 1 } { w ( x ) } [ p ( x ) f ' ( x ) ] ' \\right ) w ( x ) d x - 0 \\\\ & = \\lim _ { x \\to b ^ - } ( p ( x ) f ' ( x ) ) - \\lim _ { x \\to a ^ + } ( p ( x ) f ' ( x ) ) . \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} f _ m ( 1 ) = 1 , f _ m ( 2 ) = m + a ; f _ m ( n + 2 ) = ( a + m ) f _ m ( n + 1 ) - b f _ m ( n ) , ( n \\geq 1 ) . \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} b _ 1 : = \\frac { g _ 1 } { x _ 0 ^ { N } } \\ \\ \\ \\ b _ 2 : = \\frac { g _ 2 } { x _ 0 ^ { N } } . \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{align*} x _ { u } \\leq \\frac { 2 \\left ( t - 1 \\right ) \\sqrt { n } } { 2 \\mu ^ { 2 } - n + t - 1 } < \\frac { 2 \\left ( t - 1 \\right ) \\sqrt { n } } { n } = \\frac { 2 \\left ( t - 1 \\right ) } { \\sqrt { n } } . \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { n \\in \\mathbb { N } } T _ n ( x ) z ^ n = \\frac { 1 } { u ( z ) [ f ( z ) - x ] } \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{align*} g ( x , y ) = x ^ { - 1 + \\alpha / 2 } y ^ { - 1 + \\beta / 2 } \\mathbf { 1 } _ { \\{ x > 0 , y > 0 \\} } , \\end{align*}"}
-{"id": "9943.png", "formula": "\\begin{align*} ( v _ N \\Gamma \\circ \\Gamma ) ( x , y ) = \\int d w \\ v _ N ( x - w ) \\Gamma ( x , w ) \\Gamma ( w , y ) = \\int d z \\ v _ N ( z ) \\Gamma ( x , x - z ) \\Gamma ( x - z , y ) . \\end{align*}"}
-{"id": "7982.png", "formula": "\\begin{align*} \\kappa _ f : = \\sup _ { t > 0 } t ^ { 3 / 4 } ( 1 + t ) ^ { - 1 / 4 } \\| f ( t ) \\| _ 2 < \\infty . \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} \\tilde { q } _ { n } ( X , Y ) = d _ 0 X ^ { n + 1 } + d _ 1 X ^ { n } Y + \\cdot \\cdot \\cdot + d _ n X Y ^ { n } + d _ { n + 1 } Y ^ { n + 1 } , \\\\ d _ j \\in { \\mathbb R } , j = 0 , 1 , \\dots , n + 1 . \\end{align*}"}
-{"id": "9710.png", "formula": "\\begin{align*} r ( f ) & = | C _ 0 ^ * | + | P _ 1 ^ * | + | C _ 1 ^ * | + | P _ 0 ^ * | = | C _ 0 ^ * | + | P _ 1 ^ * | + | C _ 1 ^ * | + | N _ 0 ^ * | + | P _ 0 ^ * \\setminus N _ 0 ^ * | \\\\ & = r ( f _ 0 ) + | C _ 1 ^ * | + | P _ 0 ^ * \\setminus N _ 0 ^ * | . \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} J _ 1 ^ { k } = \\Big ( f ( \\theta _ { \\Psi } ^ { k - 1 } , \\theta _ { \\Psi } ^ { k - 1 } ) , \\ ; \\overline { \\partial _ { \\tau } \\theta } _ { \\mathbf { A } } ^ { k } \\Big ) , \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} R ^ e _ n ( s _ { v _ 0 } ) ( p _ b ^ * ( v _ 0 ) ) = p _ b ^ * ( s _ { v _ 0 } ( v _ 0 ) ) = - p _ b ^ * ( v _ 0 ) . \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{align*} \\tilde { X } _ t = \\varphi _ x ^ { - 1 } ( \\tilde { x } _ t ) = ( - B ^ N _ t , B ^ T _ t , t R _ t ) , \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{align*} \\{ x ' _ { \\tau _ t } \\not \\in \\Omega \\} = \\{ h ( B ^ T _ { \\tau _ t } , A _ { \\tau _ t } ; s ) > r - B ^ N _ { \\tau _ t } \\} , \\end{align*}"}
-{"id": "6054.png", "formula": "\\begin{align*} L _ { - q } = [ L _ { - q + i } , \\ , L _ { - i } ] = [ [ L _ { - q } , \\ , L _ { i } ] , \\ , L _ { - i } ] = [ L _ { - q } , [ L _ { - i } , \\ , L _ { i } ] ] , \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} \\sigma _ { t } = \\left ( \\lambda _ { 1 } + t , \\lambda _ { 2 } , \\lambda _ { 3 } , \\ldots , \\lambda _ { m + 1 } , \\lambda _ { m + 2 } \\pm t , \\overline { \\lambda } _ { m + 1 } , \\ldots , \\overline { \\lambda } _ { 3 } , \\overline { \\lambda } _ { 2 } \\right ) \\end{align*}"}
-{"id": "1695.png", "formula": "\\begin{align*} D + E = \\sum _ { j = 0 } ^ { n + 1 } \\frac { q ^ j } { d ^ j } W ^ { ( n + 1 ) } _ j r ^ { n + 1 - j } . \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} [ M ^ c \\circ \\tau ] _ t - [ M ^ c \\circ \\tau ] _ s = [ M ^ c ] _ { \\tau _ t } - [ M ^ c ] _ { \\tau _ s } & \\leq [ M ^ c ] _ { \\tau _ t } - [ M ^ c ] _ { \\tau _ s } + ( \\tau _ t - \\tau _ s ) \\\\ & = ( [ M ^ c ] _ { \\tau _ t } + \\tau _ t ) - ( [ M ^ c ] _ { \\tau _ s } + \\tau _ s ) \\\\ & = A _ { \\tau _ t } - A _ { \\tau _ s } = t - s . \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} \\psi _ 1 ( t , u , v , w ) & = u + v + 2 w \\\\ \\psi _ 2 ( t , u , v , w ) & = v + t - w \\\\ \\psi _ 3 ( t , u , v , w ) & = u + t - w \\\\ \\psi _ 4 ( t , u , v , w ) & = u + v + t , \\end{align*}"}
-{"id": "9659.png", "formula": "\\begin{align*} x ^ 2 + y ^ 2 - z ^ 2 = 0 , z > 0 . \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} & \\gamma _ 1 = \\{ e ^ { i t } \\colon \\theta - 2 \\delta \\le t \\le \\theta - \\delta \\} \\\\ & \\gamma _ 2 = \\{ e ^ { i t } \\colon \\theta - \\delta \\le t \\le \\theta - \\delta / 2 \\} \\\\ & \\gamma _ 3 = \\{ e ^ { i t } \\colon \\theta + \\delta / 2 \\le t \\le \\theta + \\delta \\} \\\\ & \\gamma _ 4 = \\{ e ^ { i t } \\colon \\theta + \\delta \\le t \\le \\theta + 2 \\delta \\} \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} F _ * ^ e ( j u ^ { q - 1 } v ^ { q - 1 } ( f + u v ) ) = \\bigoplus _ { i \\in \\Delta _ e } f _ { ( i , j ) } F _ * ^ e ( i u ^ { q - 1 } v ^ { q - 1 } ) \\oplus u v F _ * ^ e ( j ) , \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} ( x _ { Q ( 1 ) } ' , \\dots , x _ { Q ( n ) } ' ) = ( x _ { Q ( 1 ) } , \\dots , x _ { Q ( n - 1 ) } , l _ { j _ { Q ( n ) } } - x _ { Q ( n ) } ) \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} \\begin{pmatrix} M & N \\\\ [ . 1 c m ] \\overline { N } & \\overline { M } \\end{pmatrix} \\begin{pmatrix} z \\\\ [ . 2 c m ] \\overline { w } \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ [ . 2 c m ] 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} \\tilde { g } ^ { \\xi } = e ^ { \\sigma } g ^ { \\xi } , ~ \\ \\forall \\xi \\in \\Gamma ( A ) . \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} \\deg ( f _ { m + 2 } ( X ) f _ { m - 1 } ^ 2 ( X ) ) = \\frac { 1 } { 2 } ( ( m + 2 ) ^ 2 - 1 ) + ( ( m - 1 ) ^ 2 - 4 ) = \\frac { 3 } { 2 } ( m ^ 2 - 1 ) , \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} \\left [ M ^ { \\prime } - A _ { 2 } M \\right ] f ^ { 2 } + \\left [ N ^ { \\prime } - A _ { 2 } N + 2 M \\right ] f ^ { \\prime } f + N \\left ( \\left ( f ^ { \\prime } \\right ) ^ { 2 } + f f ^ { \\prime \\prime } \\right ) = - B _ { 2 } . \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{align*} \\left ( \\left ( f \\circ g \\right ) \\circ h \\right ) _ { n } & = \\sum _ { \\substack { \\pi \\models n \\\\ \\mu \\models \\vert \\pi \\vert } } f _ { \\vert \\mu \\vert } g _ { \\mu } h _ { \\pi } . \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} \\begin{pmatrix} M & N \\\\ [ . 1 c m ] \\overline { N } & \\overline { M } \\end{pmatrix} \\begin{pmatrix} w \\\\ [ . 2 c m ] \\overline { z } \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ [ . 2 c m ] 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "9341.png", "formula": "\\begin{align*} \\mathcal { V } _ r ( \\sigma _ 1 ) \\vee \\mathcal { V } _ r ( \\sigma _ 2 ) = \\alpha ^ { \\frac 1 r } A , \\end{align*}"}
-{"id": "3506.png", "formula": "\\begin{align*} I _ { 0 } ( 1 / 2 ; x ) = e ^ { x } J _ { 0 } ( x ) , \\ \\ \\ I _ { 1 } ( 1 / 2 ; x ) = e ^ { x } J _ { 1 } ( x ) , \\ \\ \\ I _ { 0 } ( 3 / 2 ; x ) = e ^ { x } ( J _ { 0 } ( x ) + J _ { 1 } ( x ) ) , \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} \\varphi _ { 0 } ( \\mathsf { x } ) = \\delta _ { 0 } ( \\mathsf { x } ) \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} \\begin{pmatrix} M & N \\\\ [ . 1 c m ] \\overline { N } & \\overline { M } \\end{pmatrix} \\begin{pmatrix} w \\\\ [ . 2 c m ] \\overline { z } \\end{pmatrix} = \\begin{pmatrix} p \\\\ [ . 2 c m ] \\overline { p } \\end{pmatrix} \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} \\operatorname { r c e f } ( S ^ \\mu ) = \\left ( \\begin{array} { c } \\operatorname { r c e f } ( S ^ \\mu _ 0 ) \\\\ \\hline H ' \\end{array} \\right ) . \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} I _ j ( t ) & : = \\widehat { I ( t ) } ( j ) , \\ ; \\ ; j \\geq 1 , \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} f : = g ( \\dot { c } , \\xi ) = g ^ { v } ( \\dot { c } ^ { v } , \\xi ^ { v } ) , \\end{align*}"}
-{"id": "1280.png", "formula": "\\begin{align*} \\rho ( t ) = \\frac { \\sinh ( t ) \\sinh ( T - t ) } { \\sinh ( T ) } \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} \\frac { d w } { d \\tau } = - 3 v ^ 2 + 2 ( a + 1 ) v - a , \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{align*} \\tau = - \\int \\limits _ { t } ^ { a } \\gamma ^ { - 1 } \\left ( z \\right ) d z , t = t \\left ( \\tau \\right ) = \\left [ a ^ { 1 - \\nu } - \\left ( \\nu - 1 \\right ) \\tau \\right ] ^ { \\frac { 1 } { 1 - \\nu } } , \\end{align*}"}
-{"id": "1236.png", "formula": "\\begin{align*} Y _ { n } ^ { \\ell } = \\Pi _ { n - 1 } Y _ { n - 1 } ^ { \\ell } = S _ { \\Delta t } Y _ { n - 1 } ^ { \\ell } + \\Delta t S _ { \\Delta t } G ' ( X _ { n - 1 } ) . Y _ { n - 1 } ^ { \\ell } + S _ { \\Delta t } e ^ { \\tau A } \\bigl ( \\sigma ' ( X _ { n - 1 } ) . Y _ { n - 1 } ^ { \\ell } \\bigr ) \\Delta W _ { n - 1 } . \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} m _ { i i } = 1 , ~ m _ { i ( i + 1 ) } = 3 \\mbox { a n d } m _ { i j } = 2 \\mbox { f o r } | i - j | \\geq 2 , i , j \\in \\{ 1 , 2 , \\cdots , n \\} . \\end{align*}"}
-{"id": "3265.png", "formula": "\\begin{align*} \\gamma _ 2 ( t ) = t + \\gamma _ 2 ( 0 ) . \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\frac { d } { d t } \\int _ { \\Omega } | \\nabla \\phi _ { n } | ^ { 2 } \\ , \\mathrm { d } \\mathbf { x } - \\big ( \\frac { \\partial } { \\partial t } | \\Psi _ { n } | ^ { 2 } , \\ , \\phi _ { n } \\big ) = 0 . \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} \\sum \\limits _ { \\substack { \\mathbf { n } \\in [ N ] ^ 4 \\\\ n _ 1 - n _ 2 = n _ 2 - n _ 3 \\\\ \\vert ( n _ 2 - n _ 3 ) - \\sqrt { 2 } ( n _ 3 - n _ 4 ) \\vert \\leqslant \\frac { 1 } { 2 } } } \\mu ( n _ 1 ) \\mu ( n _ 2 ) \\mu ( n _ 3 ) \\mu ( n _ 4 ) = o ( N ^ 2 ) \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} \\langle \\Delta _ { \\epsilon , \\delta } ^ f ( \\eta , \\xi ; T ) , \\phi \\rangle : = \\int _ 0 ^ T \\int _ 0 ^ T \\phi ( \\eta _ u ) ( V ^ \\delta f _ \\epsilon ) ( \\xi _ v - \\eta _ u ) d u d v . \\end{align*}"}
-{"id": "8180.png", "formula": "\\begin{align*} T r \\left ( \\left ( 1 + \\lambda ^ 2 \\frac { ( \\mu x + 1 ) ^ 2 } { y ^ 2 } \\right ) \\left ( \\alpha + \\mu ^ 2 \\lambda ^ 2 \\right ) \\right ) = 0 . \\end{align*}"}
-{"id": "1515.png", "formula": "\\begin{align*} \\frac { 1 } { t - \\zeta } + \\frac { 1 } { t - \\overline { \\zeta } } = 0 . \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} \\dim V = \\dim N + \\delta ( e , a , b , c ) + t ( e , a , b ) , \\end{align*}"}
-{"id": "7474.png", "formula": "\\begin{align*} \\bigoplus _ { \\alpha : i \\to j } \\mathrm { H o m } ( E _ i , E _ j ) { \\bigg { / } } \\prod _ { i \\in Q _ 0 } G L ( E _ i ) = : V / G \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{align*} X \\triangleright ( \\mathsf { M } ^ n _ m ) ^ i _ j = ( X _ { ( 1 ) } \\triangleright u ^ { m * } _ i ) ( X _ { ( 2 ) } \\triangleright u ^ n _ j ) = \\sum _ { k , \\ell } \\pi ( S ( X _ { ( 1 ) } ) ) ^ i _ k \\pi ( X _ { ( 2 ) } ) ^ \\ell _ j u ^ { m * } _ k u ^ n _ \\ell . \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} { \\hat p } = ( \\ , & 0 . 1 5 2 9 3 3 4 2 , \\ , 0 . 0 8 9 7 6 0 6 7 9 , \\ , 0 . 0 2 1 2 6 6 9 7 7 , \\ , 0 . 0 1 5 7 7 8 1 9 1 , \\\\ & 0 . 1 2 9 7 6 9 8 6 , \\ , 0 . 0 7 6 1 6 5 3 7 2 , \\ , 0 . 0 2 0 8 5 3 1 9 9 , \\ , 0 . 0 1 5 4 7 1 2 0 5 , \\\\ & 0 . 1 3 5 3 3 7 9 3 , \\ , 0 . 1 1 7 8 9 4 0 9 , \\ , 0 . 0 1 8 8 2 0 1 4 2 , \\ , 0 . 0 2 0 7 2 3 5 , \\\\ & 0 . 0 8 3 8 5 9 9 1 7 , \\ , 0 . 0 7 3 0 5 1 1 2 5 , \\ , 0 . 0 1 3 4 7 5 7 6 , \\ , 0 . 0 1 4 8 3 8 6 1 9 \\ , ) . \\end{align*}"}
-{"id": "8453.png", "formula": "\\begin{align*} Y _ { \\rho ^ { \\varepsilon } _ { \\theta } } = Y _ { \\tau ^ { \\varepsilon } _ { \\theta } + } \\xi ^ { ^ u } _ { \\rho ^ { \\varepsilon } _ { \\theta } } = \\hat \\xi _ { \\tau ^ { \\varepsilon } _ { \\theta } } . \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{gather*} u ( x ) = \\sum _ { k = 0 } ^ { p - 1 } e ^ { \\frac { 2 j k \\pi i } { p } } u _ k ( x ) \\quad \\textrm { f o r } x \\in e ^ { \\frac { j \\pi i } { p } } S , j = 0 , 1 , \\dots , p - 1 . \\end{gather*}"}
-{"id": "1313.png", "formula": "\\begin{align*} \\left [ \\frac { d } { d f } \\left ( \\frac { f } { p } \\right ) \\right ] \\left [ \\lambda _ { 1 } \\left ( \\frac { f } { p } \\right ) ^ { 2 } - 1 \\right ] = 0 . \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} D _ J ^ { ( j ) } ( L _ J ^ * ) ^ { - 1 } D _ J ^ { ( j ) } & = \\left ( D _ J ^ { ( j ) } L _ J ^ * D _ J ^ { ( j ) } \\right ) ^ { - 1 } = ( L _ J ^ * ) ^ { - 1 } \\end{align*}"}
-{"id": "7552.png", "formula": "\\begin{align*} \\left ( \\sum _ { \\ell = 1 } ^ { n + 1 } \\alpha _ \\ell x ^ { \\ell - 1 } \\right ) ^ 2 = \\left [ \\sum _ { \\ell = 0 } ^ n \\frac { ( \\ell ! ) ^ 2 D _ \\ell } { 4 ^ { \\ell } } x ^ \\ell \\right ] ^ 2 \\in \\mathbb Z [ x ] , \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} & ( 1 - { q ' } ^ { m _ 3 - m _ 0 + m _ 1 - m _ 3 } ) ( 1 - { q ' } ^ { m _ 3 - m _ 0 + m _ 1 - m _ 3 + 1 } ) \\cdots ( 1 - { q ' } ^ { m _ 3 - m _ 0 + m _ 1 - m _ 3 + m _ 0 - m _ 1 - 1 } ) \\\\ = & ( 1 - { q ' } ^ { m _ 1 - m _ 0 } ) ( 1 - { q ' } ^ { m _ 1 - m _ 0 + 1 } ) \\cdots ( 1 - { q ' } ^ { - 1 } ) = 1 - O ( 1 / q ' ) . \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} A u : = ( - u _ { 1 } , L u _ { 0 } + B u _ { 1 } ) , \\end{align*}"}
-{"id": "3611.png", "formula": "\\begin{align*} [ B ( - 2 i \\sqrt { 2 } / \\gamma ) + 2 \\lambda / \\gamma ] H ^ { ( 2 ) } = - ( i / \\gamma ) a ^ \\dagger G + \\frac { i c } { 4 \\pi \\gamma } \\mathbb { P } H ^ { ( 2 ) } . \\end{align*}"}
-{"id": "7922.png", "formula": "\\begin{align*} \\mbox { d i v $ u _ 0 $ } = 0 , \\nu \\cdot ( u _ 0 + h ( 0 ) u _ \\infty ) | _ { \\partial \\Omega } = 0 , \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} x _ i = v _ i \\log t + b _ i \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{align*} \\langle \\overline { x _ 1 } \\cdots \\overline { x _ { k - 1 } } | B ^ \\prime ( z ) | \\overline { y _ 1 } \\cdots \\overline { y _ { k } } \\rangle = & ( - 1 ) ^ k ( - 1 ) ^ { j - 1 } z ^ { 1 - \\overline { y _ j } } , \\end{align*}"}
-{"id": "1258.png", "formula": "\\begin{align*} g ( \\mathsf { x } , t , \\mathsf { y } ) = g ( \\mathsf { y } , t , \\mathsf { x } ) > 0 \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{align*} I _ j : = \\Bigl \\{ k \\in [ b _ { l } , b _ { l + 1 } ) \\ , ; \\ , j + k - b _ { l } \\mod ( b _ { l + 1 } - b _ l ) \\ \\textrm { d o e s n o t b e l o n g t o } \\ [ k _ { 0 } , k _ { 1 } ) \\Bigr \\} , \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{align*} \\vec { \\beta } = \\vec { F } ( z ) d z . \\end{align*}"}
-{"id": "7609.png", "formula": "\\begin{align*} W _ \\delta = \\{ | \\sigma | _ { \\eta _ 0 } ^ 2 = \\frac { e ^ \\rho } { 1 + e ^ \\rho } < \\delta \\} \\end{align*}"}
-{"id": "6576.png", "formula": "\\begin{align*} \\langle \\Delta U \\rangle \\langle \\Delta V \\rangle = \\frac { 1 } { 4 } \\mathbb { I } _ 2 , \\end{align*}"}
-{"id": "4335.png", "formula": "\\begin{align*} P = \\{ 2 , 3 , 5 , 7 , 1 1 , 1 3 , 1 7 , 1 9 , 2 3 , 2 9 , 3 1 , 3 7 , p ^ \\prime \\} . \\end{align*}"}
-{"id": "7808.png", "formula": "\\begin{align*} \\bigl \\| \\partial _ x ( P u _ 0 ) \\big | _ { x = 0 } \\bigr \\| _ { L _ 2 ( B _ T ) } \\leq \\| u _ 0 \\| _ { L _ 2 } . \\end{align*}"}
-{"id": "9550.png", "formula": "\\begin{align*} \\pi _ \\beta ( v _ g ^ * ) \\lambda _ \\mu ( g ) ( \\xi \\otimes \\delta _ t ) & = \\pi _ \\beta ( v _ g ^ * ) ( \\mu ( t ^ { - 1 } g ^ { - 1 } , g ) \\xi \\otimes \\delta _ { g t } ) \\\\ & = \\beta _ { t ^ { - 1 } g ^ { - 1 } } ( v _ g ^ * ) \\mu ( t ^ { - 1 } g ^ { - 1 } , g ) \\xi \\otimes \\delta _ { g t } , \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} c _ 1 ( n , n ) = 1 , c _ 1 ( n , k ) = \\sum _ { i = 0 } ^ { k - 1 } { k \\choose i } { n - k - 1 \\choose k - i - 1 } a ^ { k - i } ( a - 1 ) ^ { n - 2 k + i } , ( k < n ) . \\end{align*}"}
-{"id": "4545.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle \\Delta t \\mathrm { R e } \\left [ B \\left ( \\overline { \\mathbf { A } } _ { h } ^ { k } ; \\overline { \\theta } _ { \\Psi } ^ { k } , \\partial \\theta _ { \\Psi } ^ { k } \\right ) \\right ] = \\mathrm { R e } \\big ( J _ 1 ^ { ( k ) } \\big ) + \\mathrm { R e } \\big ( J _ 2 ^ { ( k ) } \\big ) + \\mathrm { R e } \\big ( J _ 3 ^ { ( k ) } \\big ) + \\mathrm { R e } \\big ( J _ 4 ^ { ( k ) } \\big ) . } \\end{array} \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} g ( x ) = \\sum _ { j = 0 } ^ \\infty c _ j H _ j ( x ) , \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{align*} \\begin{cases} d \\tilde { x } _ t = - N ( \\tilde { x } _ t ) d B _ t ^ N + T ( \\tilde { x } _ t ) d B _ t ^ T \\\\ \\tilde { x } _ 0 = x \\end{cases} \\end{align*}"}
-{"id": "6503.png", "formula": "\\begin{align*} f _ { \\nu } ( x , m ) = \\sum _ { n \\geq 0 } m ^ n P _ n ( x ) . \\end{align*}"}
-{"id": "5458.png", "formula": "\\begin{align*} u _ { i _ k } ( x ) : = u _ { a _ { i _ k } , r _ { i _ k } , R _ { i _ k } } ( x ) . \\end{align*}"}
-{"id": "7333.png", "formula": "\\begin{align*} \\frac { 1 } { 6 } \\omega ( t ) ^ 3 = \\frac { 1 } { 4 } \\rho ( t ) \\wedge \\widehat \\rho ( t ) = \\varepsilon ^ 2 _ t * _ 0 ( 1 ) , \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{align*} L ( f ( x , y ; u ) ) = \\frac { \\ell _ 1 ( u ) } { \\ell _ 2 ( u ) } , \\end{align*}"}
-{"id": "4768.png", "formula": "\\begin{align*} \\eta \\gamma & = \\xi \\alpha , & \\xi \\sigma & = \\lambda \\eta \\beta , & \\mu \\alpha & = \\omega \\gamma , & \\omega \\beta & = \\mu \\sigma . \\end{align*}"}
-{"id": "2181.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { - \\infty } ^ \\infty [ f ( - k , x ) & + f ( k , x ) ( S ( k ) - { U _ 0 } ) - I _ n e ^ { - i k x } ] e ^ { i k y } d k \\\\ = & K ( x , y ) + F _ S ( x + y ) + \\int _ x ^ \\infty K ( x , t ) F _ S ( t + y ) d t . \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} \\max _ { | \\zeta | \\leq 1 } G ^ + \\circ \\widehat \\psi _ n = 1 , \\end{align*}"}
-{"id": "4861.png", "formula": "\\begin{align*} H = \\left \\langle I + \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} p , I + \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} p , I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle , \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} \\chi _ { x } ^ { ( \\mathfrak { L } ) } \\left ( \\omega _ { 1 } \\right ) \\left ( y \\right ) : = \\omega _ { 1 } \\left ( y + x \\right ) \\ , \\ \\chi _ { x } ^ { ( \\mathfrak { b } ) } \\left ( \\omega _ { 2 } \\right ) \\left ( \\{ y , y ^ { \\prime } \\} \\right ) : = \\omega _ { 2 } \\left ( \\{ y + x , y ^ { \\prime } + x \\} \\right ) \\ . \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{align*} ( a ) _ { n } : = \\begin{cases} 1 , & n = 0 \\\\ a ( a + 1 ) \\cdots ( a + n - 1 ) , & n \\in \\mathbb Z _ { > 0 } \\\\ \\end{cases} ; ( a ) _ { n } : = \\frac { \\Gamma ( a + n ) } { \\Gamma ( a ) } , a \\notin \\mathbb Z _ { \\leq 0 } , \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} T r ( \\lambda ) = 0 \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{align*} m ( k , q , \\gamma ) : = k ( d - 1 ) - \\log ( \\gamma \\alpha ^ { - 1 } ) - \\lceil { 1 0 0 \\log ( \\log ( \\log ( \\alpha ^ { - 1 } ) ) ) } \\rceil , \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{gather*} ( x ^ \\pm \\otimes t ^ k ) ^ { ( r ) } \\cdot ( v _ 1 \\otimes v _ 2 ) = \\sum _ { i = 0 } ^ { r } ( x ^ \\pm \\otimes t ^ k ) ^ { ( i ) } \\cdot v _ 1 \\otimes ( x ^ \\pm \\otimes t ^ k ) ^ { ( r - i ) } \\cdot v _ 2 , \\\\ \\Lambda _ r \\cdot ( v _ 1 \\otimes v _ 2 ) = \\sum _ { i = 0 } ^ { r } \\Lambda _ i \\cdot v _ 1 \\otimes \\Lambda _ { r - i } \\cdot v _ 2 , \\\\ \\binom { h } { r } \\cdot ( v _ 1 \\otimes v _ 2 ) = \\sum _ { i = 0 } ^ { r } \\binom { h } { i } \\cdot v _ 1 \\otimes \\binom { h } { r - i } \\cdot v _ 2 . \\end{gather*}"}
-{"id": "10003.png", "formula": "\\begin{align*} \\left \\vert W ^ { i n t } [ \\mathcal { T } _ { n } ] \\right \\vert & \\geq W ( G _ { 1 } ( n , d _ { 1 } ^ { \\max } , 1 , 1 ) ) - W ( G _ { 2 } ( n , d _ { 2 } ^ { \\min } , x _ { 2 } ^ { \\max } , - 1 ) ) = \\\\ & = \\frac { n ^ { 3 } } { 6 } - \\frac { \\sqrt { n ^ { 5 } - n ^ { 4 } } } { \\sqrt { 2 } } - 3 n ^ { 2 } + \\frac { 1 0 } { 3 } \\sqrt { 2 n ^ { 3 } - 8 n ^ { 2 } } + \\frac { 1 4 3 n } { 6 } + \\\\ & + 2 5 \\sqrt { 2 n - 8 } - 2 5 . \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} h \\circ g \\circ h ^ { - 1 } ( x ) = \\frac { x } { ( c \\alpha + d ) ^ 2 } . \\end{align*}"}
-{"id": "3709.png", "formula": "\\begin{align*} ( b _ { n + 1 } - b _ { n - 1 } ) ( b _ { n + 1 } + b _ n + b _ { n - 1 } ) = - c _ 1 ( b _ { n + 1 } - b _ { n - 1 } ) . \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} u _ { 0 } \\left ( x , y , k \\right ) = A _ { 0 } ( x , y ) e ^ { i k \\left \\vert x - y \\right \\vert } , A _ { 0 } ( x , y ) = \\frac { 1 } { 4 \\pi \\left \\vert x - y \\right \\vert } , \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{gather*} f ^ { \\vee } ( z ) = \\frac { z ^ d f ( z ^ { - 1 } ) } { a _ d } . \\end{gather*}"}
-{"id": "3769.png", "formula": "\\begin{align*} c = ( 0 , 0 , \\cdots , 0 , a _ { j + 1 } - j , a _ { j + 2 } - ( j + 1 ) , \\cdots , a _ n - ( n - 1 ) ) . \\end{align*}"}
-{"id": "1656.png", "formula": "\\begin{align*} | \\cdot | _ 1 \\colon C _ n ( M ; A ) & \\longrightarrow \\R _ { \\geq 0 } \\\\ \\sum _ { j = 1 } ^ m f _ j \\otimes \\sigma _ j & \\longmapsto \\sum _ { j = 1 } ^ m | f _ j | \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{align*} ( \\mathsf { M } ^ n _ m ) ^ i _ j : = u ^ { m * } _ i u ^ n _ j , ( \\mathsf { N } ^ n _ m ) ^ i _ j : = u ^ i _ m u ^ { j * } _ n . \\end{align*}"}
-{"id": "9630.png", "formula": "\\begin{align*} A ^ { ( j ) } _ n = \\frac { \\psi ^ { ( j ) } _ n ( a ) } { \\psi ^ { ( j ) } _ n ( I ) } , \\end{align*}"}
-{"id": "848.png", "formula": "\\begin{align*} E _ 1 ( x ) = \\frac { 1 } { 8 \\pi \\varepsilon ^ 2 } \\big ( \\int _ { \\R ^ 3 } U _ { a _ j } ^ 2 ( \\frac { \\xi - x ^ { ( 1 ) } _ { j , \\varepsilon } } { \\varepsilon } ) { | x - \\xi | } ^ { - 1 } d \\xi \\big ) + o ( 1 ) , ~ \\mbox { i n } ~ B _ { d } ( x ^ { ( 1 ) } _ { j , \\varepsilon } ) , \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{align*} \\frac { p } { { \\tilde p } } = \\frac { p } { { \\left ( { 1 - { \\psi _ 0 } - { \\psi _ 1 } } \\right ) p + { \\psi _ 0 } } } \\ge \\frac { q } { { \\left ( { 1 - { \\psi _ 0 } - { \\psi _ 1 } } \\right ) q + { \\psi _ 0 } } } = \\frac { q } { { \\tilde q } } , \\end{align*}"}
-{"id": "3578.png", "formula": "\\begin{align*} i _ * p _ * = \\pi _ * ( i \\times i ) _ * . \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{align*} L = \\bigoplus _ { i = - q } ^ 1 L _ i \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} \\frac { ( - a ) _ k } { ( 1 ) _ k } = ( - 1 ) ^ k \\cdot \\binom { a } { k } = ( - 1 ) ^ k \\cdot \\binom { a } { a - k } = ( - 1 ) ^ a \\cdot \\frac { ( - a ) _ { a - k } } { ( 1 ) _ { a - k } } \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} h _ 0 = \\sum _ { k \\in \\mathbb { Z } _ { \\geq 0 } + 1 / 2 } : \\chi _ { - k } \\chi _ { k } : = \\chi _ { - \\frac { 1 } { 2 } } \\chi _ { \\frac { 1 } { 2 } } + \\chi _ { - \\frac { 3 } { 2 } } \\chi _ { \\frac { 3 } { 2 } } + \\dots \\end{align*}"}
-{"id": "9941.png", "formula": "\\begin{align*} \\vect { N } _ T ( X ) : = \\vect { N } _ T ( \\varphi ) ^ 2 + \\vect { N } _ T ( \\Gamma ) + \\vect { N } _ T ( \\Lambda ) \\lesssim \\ 1 . \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} L _ { 1 \\varepsilon } u = \\sum \\limits _ { i = 0 } ^ { \\nu } \\varepsilon _ { n } ^ { \\sigma _ { i } } \\alpha _ { i } \\frac { \\partial ^ { i } u } { \\partial x _ { n } ^ { i } } \\left ( x ^ { \\prime } , 0 , t \\right ) = 0 \\nu \\in \\left \\{ 0 , 1 \\right \\} \\end{align*}"}
-{"id": "2703.png", "formula": "\\begin{align*} E _ 1 : = \\int _ \\Omega \\left ( n _ 1 - N _ 1 \\log \\frac { n _ 1 } { N _ 1 } \\right ) + k _ 1 \\int _ \\Omega \\left ( n _ 2 - N _ 2 \\log \\frac { n _ 2 } { N _ 2 } \\right ) + \\frac { \\ell _ 1 } { 2 } \\int _ \\Omega c ^ 2 \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} z _ { n _ l } \\sim \\frac { { n _ l } \\pi } { \\omega _ { l } - \\omega _ { l - 1 } } + O ( 1 ) , \\ , l = 1 , \\ldots , 2 J + 2 ; \\ , n _ { l } \\in \\mathbb { N } . \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} [ f _ j , g _ k ] \\bigg | _ { - 1 } ^ 1 = 0 \\qquad j = k . \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{align*} p _ \\lambda : = p _ + ( \\lambda ) - p _ - ( \\lambda ) \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} - 2 \\beta \\binom { p - 1 } { ( p - 1 ) / 2 } H _ { ( p - 1 ) / 2 } ( - \\beta ^ 2 ) ^ { ( p - 1 ) / 2 } \\equiv - 2 H _ { ( p - 1 ) / 2 } \\beta ^ p \\equiv - 2 \\pounds _ 1 ( - 1 ) \\beta ^ p \\pmod { p } , \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} T _ { j } ( 0 ) \\allowbreak = \\allowbreak U _ { j } ( 0 ) = \\left \\{ \\begin{array} [ c ] { c c c } 0 & i f & j \\\\ ( - 1 ) ^ { j / 2 } & i f & j \\end{array} \\right . , \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} G _ n e _ 1 & = \\frac { D ( f ^ n ( p ) ) } { D ( p ) } \\Vert d f ^ n _ p v _ 0 ^ s \\Vert e _ 1 = c _ { 1 , n } ( p ) e _ 1 , \\ ; \\ ; \\mathrm { a n d } \\\\ G _ n e _ 2 & = \\frac { D ( f ^ n ( p ) ) } { D ( p ) } \\Vert d f ^ n _ p v _ 0 ^ u \\Vert e _ 2 = c _ { 2 , n } ( p ) e _ 2 . \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} \\int _ { X ^ n } { d z _ 2 \\ldots d z _ { n + 1 } \\over \\Pi _ { i = 1 } ^ { n + 1 } | z _ i - z _ { i + 1 } | ^ { n - 1 } } \\le C \\int _ { X ^ { n - l + 2 } } { d z _ l \\ldots d z _ { n + 1 } \\over | z _ 1 - z _ l | ^ { n - l + 1 } \\Pi _ { i = l } ^ { n + 1 } | z _ i - z _ { i + 1 } | ^ { n - 1 } } \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} H ^ s _ { \\mu } ( \\mathbb { R } ) = \\left \\{ u \\in H ^ s _ { \\rm l o c } ( \\mathbb { R } ) : e ^ { \\mu \\xi } u \\in H ^ s ( \\mathbb { R } ) \\right \\} , s \\geq 0 , \\mu > 0 . \\end{align*}"}
-{"id": "8000.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\| w ( t ) - w _ 0 \\| _ 2 = 0 . \\end{align*}"}
-{"id": "9364.png", "formula": "\\begin{align*} | S _ { k , 0 } | = | S _ k | - | S _ { k , 1 } | \\leq ( 1 - \\lambda ) | S _ k | \\leq | S _ k ^ \\sharp | , \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{align*} \\vec { Q } & = O ( | z | ^ { 3 - \\theta _ 0 - \\epsilon } ) . \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\begin{matrix} 0 \\\\ \\theta ^ 0 \\end{matrix} & \\overbrace { \\begin{matrix} 1 & 0 & - \\frac { 3 t _ 2 } { 2 } & 0 & \\frac { t _ 1 } { 2 } + \\frac { 3 { t _ 2 } ^ 2 } { 8 } & \\theta ^ \\infty _ 1 / 2 \\\\ - 1 & 0 & \\frac { 3 t _ 2 } { 2 } & 0 & - \\frac { t _ 1 } { 2 } - \\frac { 3 { t _ 2 } ^ 2 } { 8 } & \\theta ^ \\infty _ 1 / 2 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "4029.png", "formula": "\\begin{align*} & \\frac { d } { d x } \\bigg ( \\frac { ( 1 - a - a x + b + b x ) _ p } { ( 1 - a - a x ) _ p } \\bigg ) \\bigg | _ { x = 0 } \\\\ = & \\frac { a - b } { a } \\cdot \\prod _ { \\substack { 1 \\leq j \\leq p \\\\ j \\neq a - b } } ( j - a + b ) \\cdot \\prod _ { \\substack { 1 \\leq j \\leq p \\\\ j \\neq a } } \\frac 1 { j - a } \\cdot \\bigg ( \\sum _ { \\substack { 1 \\leq j \\leq p \\\\ j \\neq a } } \\frac { a } { j - a } - \\sum _ { \\substack { 1 \\leq j \\leq p \\\\ j \\neq a - b } } \\frac { a - b } { j - a + b } \\bigg ) \\equiv 0 \\pmod { p } . \\end{align*}"}
-{"id": "720.png", "formula": "\\begin{align*} F _ i ( m _ 1 , \\cdots , m _ i , \\cdots , m _ r ) = m _ i ( k - ( m _ 1 + m _ 2 + \\cdots + m _ r ) ) , \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} u ^ 3 = X _ { \\alpha _ 2 } , \\ ; \\ ; 3 u ^ 2 v = X _ { \\alpha _ 1 + \\alpha _ 2 } , \\ ; \\ ; 3 u v ^ 2 = X _ { 2 \\alpha _ 1 + \\alpha _ 2 } , \\ ; \\ ; v ^ 3 = X _ { 3 \\alpha _ 1 + \\alpha _ 2 } , \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} H ( { \\bf q } , { \\bf p } ) = \\sum _ i \\frac { p _ i ^ 2 } { 2 } + \\frac { \\epsilon _ i } { 2 } q _ i ^ 2 + \\frac { 1 } { 4 } q _ i ^ 4 + \\frac { 1 } { 2 W } \\left ( q _ { i + 1 } - q _ i \\right ) ^ 2 , \\end{align*}"}
-{"id": "619.png", "formula": "\\begin{gather*} r _ x : = \\begin{cases} \\inf \\big \\{ t \\in ( 0 , 1 ] \\ ; : \\ ; \\Theta _ u ( x , t ) > L \\big \\} & I _ u ( x , 1 ) > L , \\\\ 1 & I _ u ( x , 1 ) \\leq L . \\end{cases} \\end{gather*}"}
-{"id": "1128.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\phi _ k = 1 B _ j . \\end{align*}"}
-{"id": "9682.png", "formula": "\\begin{align*} \\abs { L ^ \\circ } = \\frac { 3 2 } { 3 \\abs { L } } \\leq \\frac { 8 } { 3 } . \\end{align*}"}
-{"id": "6352.png", "formula": "\\begin{gather*} ( F - F ^ * ) \\phi ( a ) = 0 , ( F ^ 2 - 1 ) \\phi ( a ) = 0 , \\quad \\\\ [ F , \\phi ( a ) ] ^ { \\operatorname { g r } } = 0 \\end{gather*}"}
-{"id": "2589.png", "formula": "\\begin{align*} \\Phi ^ \\ast g & = \\sum _ { l = 0 } ^ \\infty g _ l , \\\\ g _ 0 & : = g , \\\\ g _ { l } & : = \\frac { 1 } { l } \\{ \\chi , g _ { l - 1 } \\} , \\ ; \\ ; l \\geq 1 . \\end{align*}"}
-{"id": "225.png", "formula": "\\begin{align*} N _ g ( R ( \\alpha , s , h ) ) : = \\{ \\mbox { t h e n u m b e r o f z e r o s o f } g ( z ) \\mbox { i n } R ( \\alpha , s , h ) \\} , \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} Q ( u _ 0 ) = 2 \\pi \\left [ P ( c _ * ) + \\sqrt { c _ * } \\delta + \\frac { 1 6 } { 1 5 \\sqrt { c _ * } } | b | ^ 2 + \\mathcal { O } ( \\delta ^ 2 + | b | ^ 4 ) \\right ] , \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{align*} u ( 0 ) : = u _ { 0 } = \\dfrac { \\dot { a } _ { 0 } } { a _ { 0 } } ; \\ ; v ( 0 ) : = v _ { 0 } = \\dfrac { \\dot { b } _ { 0 } } { b _ { 0 } } ; \\ , \\rho ( 0 ) = \\rho _ { 0 } \\ ; ; \\psi ( 0 ) : = \\psi _ { 0 } = \\dot { \\phi } _ { 0 } ; \\ ; \\phi ( 0 ) = \\phi _ { 0 } . \\end{align*}"}
-{"id": "7617.png", "formula": "\\begin{align*} | v _ \\epsilon ( t , \\rho ) - a _ t \\rho | & = | \\int _ 0 ^ \\rho { ( v _ \\epsilon ' - a _ t ) d \\rho } | \\\\ & \\leq ( 1 + \\alpha ) ( T - t ) | \\rho | \\to 0 \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} \\binom { u ( \\varepsilon ) } { U ( \\varepsilon ) } = \\mathcal { V } _ 0 [ \\varepsilon ^ { 1 / q } ] + \\mathcal { V } _ 1 [ \\varepsilon ^ { p } ] \\log \\varepsilon \\forall \\ , \\varepsilon \\in ( 0 , \\varepsilon _ 1 ) \\ , . \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} F ( 0 ) = 0 \\quad , F ' ( x ) = 2 F ( 2 x ) \\quad , \\forall x \\in [ 0 , 1 ] \\ : \\ F ( 1 + x ) = 1 - F ( 1 - x ) \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } X _ i = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } t ( f _ i ) - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\mathbb { E } t ( f _ i ) \\longrightarrow 0 \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} \\mathbb { S } ( R ^ { \\bigstar } ) = \\frac { 2 } { d ^ { n + 1 } } \\Big [ \\frac { d _ 1 d _ 2 \\dots d _ n } { n + 1 } + \\frac { W ^ { ( n ) } _ 1 } { n } + \\dots + \\frac { W ^ { ( n ) } _ s } { n - s + 1 } + \\dots + \\frac { W ^ { ( n ) } _ { n - 1 } } { 2 } \\Big ] . \\end{align*}"}
-{"id": "653.png", "formula": "\\begin{align*} F _ i ( n _ 1 , \\cdots , n _ i , \\cdots , n _ r ) = n _ i \\big ( k + 1 - ( n _ 1 + \\cdots + n _ i + \\cdots + n _ r ) \\big ) . \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} \\begin{aligned} S _ 1 ^ * & = \\{ n \\in \\mathbb Z \\mid | p _ n | > R | n | \\leq \\lceil \\gamma R \\rceil \\} , \\\\ S _ 2 ^ * & = \\{ n \\in \\mathbb N \\mid n \\geq \\lceil \\gamma R \\rceil + 1 p _ n \\geq - p _ { - n } \\} , \\\\ S _ 3 ^ * & = \\{ n \\in \\mathbb N \\mid n \\geq \\lceil \\gamma R \\rceil + 1 p _ n < - p _ { - n } \\} . \\end{aligned} \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{align*} D _ { a c h } ( \\underbar { k } ) = \\min _ { \\hat { f } } \\mathbb { E } _ { \\underbar { X } , \\underbar { k } } [ ( f ( \\underbar { X } ) - \\hat { f } ( g ( \\underbar { X } , \\underbar { k } ) ) ) ^ 2 ] , \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} p _ { k , l , m } ( c ^ 2 ) & : = \\sum _ { j = 1 } ^ N \\ ; \\omega _ j ( c ^ 2 ) k _ j \\ ; - \\omega _ l ( c ^ 2 ) \\ ; + \\omega _ m ( c ^ 2 ) . \\end{align*}"}
-{"id": "4713.png", "formula": "\\begin{align*} \\Re \\xi ( z ) = \\Re \\left ( \\int _ { e ^ { i \\theta } } ^ { \\tau e ^ { i \\theta } } \\frac { \\rho _ { \\alpha , \\varepsilon } ( \\frac { 1 } { 2 } ( w + \\frac { 1 } { w } ) ) - \\rho _ { \\alpha , \\varepsilon } ( \\cos \\theta ) } { w } d w \\right ) + \\rho _ { \\alpha , \\varepsilon } ( \\cos \\theta ) \\log \\tau . \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} { \\sigma _ k } = \\frac { 1 } { p _ { 2 n - 2 } B ( t _ k , t _ k ) } = \\frac { 1 } { p ( t _ k ) } , k = 1 , \\dots , n - 1 , \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} \\omega = G ( x , y ) d x - F ( x , y ) d y , \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} ^ { - 1 } = \\begin{pmatrix} a & b \\frac { w } { z } \\\\ c \\frac { z } { w } & d \\end{pmatrix} . \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} \\varkappa = - b + \\left \\{ \\begin{aligned} \\pi ^ 2 ( 1 - \\delta ) \\bigl ( \\frac 3 { R ^ 2 } + \\frac 1 { L ^ 2 } \\bigr ) \\qquad & \\mbox { i n t h e c a s e a ) } , \\\\ \\pi ^ 2 ( 1 - \\delta ) \\bigl ( \\frac 3 { R ^ 2 } + \\frac 1 { 4 L ^ 2 } \\bigr ) \\qquad & \\mbox { i n t h e c a s e c ) } , \\\\ \\pi ^ 2 ( 1 - \\delta ) \\frac 3 { R ^ 2 } \\qquad & \\mbox { i n t h e c a s e s b ) a n d d ) } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{align*} T _ 1 = \\{ \\pi _ 2 ^ { j \\tau } \\pi _ 1 ^ j \\mid 1 \\leq j \\leq p - 1 \\} \\cup \\{ 1 \\} . \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} T ^ { n _ { j } + k } z - x _ { l } = { T ^ { k } \\ , \\Bigl ( \\ , \\sum _ { i < j } z _ { i } \\Bigr ) + T ^ { n _ { j } + k } z _ { j } - y _ { j } + \\sum _ { i > j } T ^ { n _ { j } + k } z _ { i } . } \\end{align*}"}
-{"id": "6680.png", "formula": "\\begin{align*} n = \\alpha ( ( T - \\lambda I ) ^ p ) = \\beta ( ( T - \\lambda I ) ^ p ) = m , \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} u [ \\eta , \\psi , \\Psi ] ( x ) = w ( x ) + W ( x / \\eta ) \\end{align*}"}
-{"id": "5151.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\mathcal { C } \\left ( n \\right ) } \\frac { \\left ( - 1 \\right ) ^ { \\vert \\pi \\vert } } { \\vert \\pi \\vert } \\frac { \\mu _ \\pi } { \\pi ! } = \\frac { \\kappa _ n } { n ! } \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} h _ { 1 , k + 2 , 2 } ( x ) & = \\frac { 1 } { 2 } \\left ( ( 1 + \\sqrt { x } ) ^ { 2 k + 4 } h _ { 1 , k + 1 , 2 } ( \\sqrt { x } ) + ( 1 - \\sqrt { x } ) ^ { 2 k + 4 } h _ { 1 , k + 1 , 2 } ( - \\sqrt { x } ) \\right ) \\\\ & = \\sum _ { j = 0 } ^ k \\frac { 1 } { 2 } a _ { j , k } \\left ( \\sqrt { x } ^ j ( 1 + \\sqrt { x } ) ^ { 4 k - 2 j + 4 } + ( - \\sqrt { x } ) ^ j ( 1 - \\sqrt { x } ) ^ { 4 k - 2 j + 4 } \\right ) . \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} \\begin{cases} \\{ H _ k [ u ] \\} ^ { \\frac { 1 } { k } } = - \\beta _ \\epsilon ( \\varphi - u ) & , \\\\ [ 5 p t ] u = \\varphi - \\delta & \\end{cases} \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{align*} e ^ { 2 u } = | z | ^ { 2 - 2 \\theta _ 0 } e ^ { 2 \\lambda } \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{align*} - \\partial _ y \\int _ { - \\infty } ^ { x } p _ t ( y , z ) d z = \\hat { p } _ t ( x , y ) . \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{align*} \\eta _ { a } ( a _ 0 \\otimes a _ 1 \\otimes a _ 2 ) : = \\eta _ { F _ a , E _ a } ( a _ 0 \\otimes a _ 1 \\otimes a _ 2 ) = \\varepsilon ( a _ { 0 } ) \\varepsilon ( F _ { a } \\triangleright a _ { 1 } ) \\varepsilon ( E _ { a } \\triangleright a _ { 2 } ) . \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} \\dim H ^ 1 ( G _ T , W _ { [ n ] } ) \\leq \\sum _ { i = 1 } ^ { n } \\dim H ^ 1 ( G _ T , Z _ { [ i ] } ) . \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} \\langle \\vec { a } _ i , J _ i \\rangle + c _ i = p _ i . \\end{align*}"}
-{"id": "5199.png", "formula": "\\begin{align*} \\left ( f ^ { \\left ( 1 \\right ) } \\circ \\left ( f ^ { \\left ( 2 \\right ) } \\circ f ^ { \\left ( 3 \\right ) } \\right ) \\right ) \\circ f ^ { \\left ( 4 \\right ) } = \\sum _ { \\substack { \\pi _ 1 \\models n \\\\ \\pi _ 2 \\models \\vert \\pi _ 1 \\vert \\\\ \\pi _ 3 \\models \\pi _ 2 } } f _ { \\vert \\pi _ 2 \\vert } ^ { \\left ( 1 \\right ) } f _ { \\vert \\pi _ 3 \\vert } ^ { \\left ( 2 \\right ) } f _ { \\pi _ 3 } ^ { \\left ( 3 \\right ) } f _ { \\pi _ 1 } ^ { \\left ( 4 \\right ) } . \\end{align*}"}
-{"id": "5393.png", "formula": "\\begin{align*} M : = \\begin{pmatrix} 0 & - i & 0 & 2 - i & 5 i \\\\ 0 & 3 i & 0 & 3 & 9 i \\\\ - 1 & 5 & 1 - 3 i & - 3 + 3 i & 1 + 7 i \\\\ 0 & - 2 i & 0 & - i & - i \\\\ 0 & 1 - i & 0 & i & - 2 - 3 i \\end{pmatrix} , \\end{align*}"}
-{"id": "8613.png", "formula": "\\begin{align*} D _ { n , k } \\Big ( 1 , \\frac { 1 } { 4 } \\Big ) \\ , = \\ , \\frac { k ( n - 1 ) + 2 } { 2 ^ n } . \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} \\left \\langle z _ n \\overline { \\mathstrut A } ( \\sigma ) A ( \\sigma ) z _ n \\right \\rangle = - \\lambda ^ 2 c _ n \\left \\langle z _ n \\right \\rangle . \\end{align*}"}
-{"id": "4514.png", "formula": "\\begin{align*} \\mathbf { Z } _ { l , j , k , q } ^ { ( \\omega ) } ( t ) : = \\frac { n ^ { d } } { \\left \\vert \\Lambda _ { l } \\right \\vert } \\sum \\limits _ { x , y \\in \\mathfrak { L } \\cap ( l b _ { j } ) } \\sigma _ { \\mathrm { p } } ^ { ( \\omega ) } \\left ( y , y - e _ { q } , x , x - e _ { k } \\mathbf { , } t \\right ) \\ . \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} \\begin{aligned} p _ N ( a ) & = P _ N ( a ) - P _ N \\left ( a + \\frac 1 N \\right ) \\\\ & \\sim K ( a ) N ^ { - \\frac \\alpha 2 } e ^ { - N ^ \\alpha J ( a ) } - K \\left ( a + \\frac 1 N \\right ) N ^ { - \\frac \\alpha 2 } e ^ { - N ^ \\alpha J \\left ( a + \\frac 1 N \\right ) } \\ , . \\end{aligned} \\end{align*}"}
-{"id": "5751.png", "formula": "\\begin{align*} \\Delta _ { C | M } ( \\hat { \\rho } _ { C M } ) ( t ) = I ( C : Z | M ) _ { \\hat { \\sigma } _ { C M Z } ( t ) } \\ ; , \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} \\widetilde { \\varepsilon } _ R = \\alpha ^ { - 2 } \\varepsilon _ R . \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} & c ( x ) = 1 + \\lambda \\sum _ { u : u \\sim x } \\rho ( x ) \\rho ( u ) , \\\\ & p ( x , \\varnothing ) = \\frac { 1 } { 1 + \\lambda \\sum _ { u : u \\sim x } \\rho ( x ) \\rho ( u ) } , \\\\ & p ( x , \\{ x , y \\} ) = \\frac { \\lambda \\rho ( x ) \\rho ( y ) } { 1 + \\lambda \\sum _ { u : u \\sim x } \\rho ( x ) \\rho ( u ) } , \\ { \\rm w h e n } \\ y \\sim x , \\\\ & p ( x , A ) = 0 , \\ \\rm o t h e r w i s e , \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{align*} N _ { - , k _ j } J ( i k _ j ) = 0 _ n , N _ { 0 , k _ j } J ( i k _ j ) + N _ { - , k _ j } \\dot { J } ( i k _ j ) = I _ n . \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{align*} u ^ { ( a , \\delta , \\lambda ) } ( t , x ) : = \\lambda ^ { - \\frac { 4 } { \\nu - 1 } } \\phi ^ { ( a , \\delta ) } ( \\lambda ^ { - 4 } t , \\lambda ^ { - 1 } \\delta x ) . \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { U ^ { d } ( N ) } ^ { 2 ^ d } = \\frac { 1 } { N ^ { d + 1 } } \\sum \\limits _ { x , h _ 1 , \\dots , h _ d } \\prod \\limits _ { \\boldsymbol { \\omega } \\in \\{ 0 , 1 \\} ^ d } \\mathcal { C } ^ { \\vert \\boldsymbol { \\omega } \\vert } f ( x + \\mathbf { h } \\cdot \\boldsymbol { \\omega } ) , \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} [ L _ { - 1 } , \\ , L _ 1 ] = [ [ L _ { - 1 } , \\ , A _ 0 ] , \\ , L _ 1 ] = [ [ L _ { - 1 } , \\ , L _ 1 ] , \\ , A _ 0 ] \\subset A _ 0 , \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } x ^ { \\prime } = a _ { 1 } + x , \\\\ y ^ { \\prime } = a _ { 2 } + a _ { 3 } x + \\left ( \\cosh \\theta \\right ) y + \\left ( \\sinh \\theta \\right ) z , \\\\ z ^ { \\prime } = a _ { 4 } + a _ { 5 } x + \\left ( \\sinh \\theta \\right ) y + \\left ( \\cosh \\theta \\right ) z , \\end{array} \\right . \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} u _ { \\beta } u _ { \\gamma } = \\sum _ { \\alpha } c _ { \\beta , \\gamma } ^ { \\alpha } u _ { \\alpha } \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} \\sum Q ^ j b _ 1 = \\left ( \\sum _ { n = 1 } ^ \\infty ( b _ n b _ 1 + ( n - 1 ) b _ { n + 1 } ) \\right ) \\left ( \\sum _ { n = 0 } ^ \\infty b _ n \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle | \\sum _ { k = 1 } ^ { m } J _ 2 ^ { k , 1 } | \\leq C \\big \\{ h ^ { 2 r + 2 } + ( \\Delta t ) ^ { 4 } \\big \\} + C \\| \\theta _ { \\psi } ^ { m } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } + C \\Delta t \\sum _ { k = 1 } ^ { m - 1 } { \\| \\theta _ { \\psi } ^ { k } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } } . } \\end{array} \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} u _ 1 ( x ) & = \\frac { - 1 } { m - 1 } \\int Q ( x ) x ^ { - 2 } d x & u _ 2 ( x ) & = \\frac { 1 } { m - 1 } \\int Q ( x ) x ^ { - m - 1 } d x \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} H ^ s _ { \\rm e v e n } = \\left \\{ u \\in H ^ s ( \\mathbb { R } \\times \\mathbb { T } ) : u ( - \\xi , y ) = u ( \\xi , y ) = u ( \\xi , - y ) \\right \\} , s \\geq 0 . \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 1 e _ 4 + \\alpha _ 2 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 4 e _ 5 , [ e _ 2 , e _ 3 ] = \\beta _ 6 e _ 5 , [ e _ 3 , e _ 2 ] = \\beta _ 8 e _ 5 , [ e _ 1 , e _ 4 ] = \\gamma _ 3 e _ 5 . \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} I \\left ( i ; Q _ \\mathcal { T } ^ { [ i ] } \\right ) = 0 , \\ : \\mathcal { T } \\subset \\{ 1 , \\cdots , N \\} , \\ : | \\mathcal { T } | = T \\end{align*}"}
-{"id": "3433.png", "formula": "\\begin{align*} | \\arg z | \\le \\begin{cases} \\alpha \\mu \\pi / 2 - \\epsilon , & \\ 0 < \\alpha \\mu < 2 \\\\ ( 2 - \\alpha \\mu / 2 ) \\pi & \\ 2 \\le \\alpha \\mu < 4 \\\\ 0 & \\ \\alpha \\mu \\ge 4 , \\end{cases} \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} d ^ Q = \\{ ( x _ 1 , \\dots , x _ n ) \\in \\mathbb { R } _ + ^ n ; \\ x _ { Q ( 1 ) } < \\dots < x _ { Q ( n ) } \\} , \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} F _ { 2 m k + m } - F _ { 2 m k - m } = F _ m L _ { 2 m k } \\ , , \\quad \\mbox { $ m $ e v e n } \\ , , \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} \\Delta \\Psi = f \\textup { ~ i n ~ } R , \\Psi = 0 \\textup { ~ o n ~ } T \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} c _ \\kappa ( m , n ) = ( - 1 ) ^ { \\kappa ( m , n ) } ( m , n ) \\in A \\times A . \\end{align*}"}
-{"id": "110.png", "formula": "\\begin{align*} D ( t _ { k ^ x } t _ { 1 ^ x } ^ { - 1 } , \\dots , t _ { k ^ x } t _ { ( k - 1 ) ^ x } ^ { - 1 } , 1 ) = D ( ( t _ 1 ^ { - 1 } ) ^ \\sigma , \\dots , ( t _ { k - 1 } ^ { - 1 } ) ^ \\sigma , 1 ) . \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} \\Phi _ + = \\{ \\alpha _ 1 , \\ ; \\ ; \\alpha _ 2 , \\ ; \\ ; \\alpha _ 1 + \\alpha _ 2 , \\ ; \\ ; 2 \\alpha _ 1 + \\alpha _ 2 , \\ ; \\ ; 3 \\alpha _ 1 + \\alpha _ 2 , \\ ; \\ ; 3 \\alpha _ 1 + 2 \\alpha _ 2 \\} . \\end{align*}"}
-{"id": "2328.png", "formula": "\\begin{align*} \\underline C & \\coloneqq \\frac { \\sigma } { 1 + \\delta _ s } , c _ 0 = \\frac { 1 + \\gamma + \\sqrt 3 } { \\gamma } , \\\\ \\bar C & \\coloneqq \\frac { \\sigma } { \\kappa ( c _ 0 , s ) } \\bigg ( 1 + \\frac { \\sigma \\kappa ( c _ 0 , s ) ( \\sqrt s + 2 \\sqrt { \\log 3 } ) } { \\lambda \\sqrt s } + \\frac { \\sqrt 3 } { \\sqrt { \\log ( 9 e p / s ) } } \\bigg ) . \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} L = \\left \\{ \\ , \\begin{bmatrix} 1 & 0 \\\\ x & 1 \\end{bmatrix} : x \\in D \\ , \\right \\} , \\ ; U = \\left \\{ \\ , \\begin{bmatrix} 1 & x \\pi ^ n \\\\ 0 & 1 \\end{bmatrix} : x \\in D \\ , \\right \\} , \\ ; \\Delta = \\left \\{ \\ , \\begin{bmatrix} \\varepsilon & 0 \\\\ 0 & \\eta \\end{bmatrix} : \\varepsilon , \\eta \\in D ^ \\times \\ , \\right \\} . \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} L _ { - 1 } ^ * \\otimes X _ { 2 , \\ , j + 1 , \\ , 2 } = L _ { - 1 } ^ * \\otimes \\tilde { X } _ { 2 , \\ , j + 1 , \\ , 2 } + X _ { 2 , \\ , j + 2 , \\ , 6 } = X _ { 2 , \\ , j + 2 , \\ , 4 } + X _ { 2 , \\ , j + 2 , \\ , 5 } + X _ { 2 , \\ , j + 2 , \\ , 6 } \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ 2 } f ^ 1 _ { \\alpha a } V ^ { ( \\alpha , a ) } d x \\lesssim \\sum _ { \\tiny \\begin{matrix} \\beta + \\gamma = \\alpha \\\\ b + c = a \\end{matrix} } \\sum _ { 1 \\leq i , j \\leq 2 } \\| \\nabla _ i V ^ { ( \\beta , b ) } \\nabla _ j V ^ { ( \\gamma , c ) } - \\nabla _ i H ^ { ( \\beta , b ) } \\cdot \\nabla _ j H ^ { ( \\gamma , c ) } \\| _ { L ^ 2 } \\| V ^ { ( \\alpha , a ) } \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} a _ i ^ { \\pm } | n _ 1 , \\cdots , n _ i , \\cdots , n _ r \\rangle \\ = \\sqrt { F _ i ( n _ 1 , \\cdots , n _ i \\pm 1 , \\cdots , n _ r ) } | n _ 1 , \\cdots , n _ i \\pm 1 , \\cdots , n _ r \\rangle \\ \\end{align*}"}
-{"id": "6079.png", "formula": "\\begin{align*} 0 \\neq [ L _ { - 3 } , \\ , L _ 1 ] = [ L _ { - 3 } , \\ , [ L _ { - 3 } , \\ , [ L _ { - 1 } , \\ , L _ 5 ] ] ] , \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} u _ t + 1 2 u u _ x + u _ { x x x } + u _ { x y y } = 0 , \\end{align*}"}
-{"id": "6526.png", "formula": "\\begin{align*} \\forall i , j \\in \\{ 1 , \\dots , q - 1 \\} , \\forall v , \\left \\{ \\begin{array} { l } h ( i v ) - h ( j v ) = 0 , \\\\ h ( i v ) - h ( 0 v ) = \\frac { a } { 2 q ^ { | v | } } , \\\\ h ( 0 v ) - h ( v ) = \\frac { a } { q ^ { | v | } } . \\end{array} \\right . \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} \\sum _ { i \\neq j } ^ n t _ i = \\frac { ( n - 1 ) l } { r } . \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } \\mu ( r ) = \\infty , \\end{align*}"}
-{"id": "10042.png", "formula": "\\begin{align*} \\mathcal { L } = \\{ Q \\in \\mathcal T ^ 1 _ 2 ( M ) \\colon Q ( X , J _ { \\varphi } Y ) = J _ { \\varphi } Q ( X , Y ) , \\forall X , Y \\in { \\mathfrak X } ( M ) \\} . \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} \\widetilde E _ W = \\{ ( i , j ) \\in \\widetilde E \\mid f _ { i j } ( p _ { i j } ) = \\tilde q _ i \\} . \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{align*} A ( 1 , \\lambda ) = \\frac { 1 } { m _ k } H ( 0 , 0 ) - \\frac { 1 } { m _ k } H ( 0 , \\delta _ k ) + \\frac { 1 } { m _ k } A ( 0 , \\delta _ k ) + \\frac { r _ k } { m _ k } ( 1 , \\tilde { \\lambda } ) . \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} { \\bf u } & = ( { \\bf A } _ 1 ^ \\frac { 1 } { 2 } ) ^ { - } { \\bf x } - ( { \\bf A } _ 1 ^ \\frac { 1 } { 2 } ) ' ( { \\bf A } _ 1 + { \\bf A } _ 2 ) ^ { - } ( { \\bf x } + { \\bf y } ) \\\\ { \\bf v } & = ( { \\bf A } _ 2 ^ \\frac { 1 } { 2 } ) ^ { - } { \\bf y } - ( { \\bf A } _ 2 ^ \\frac { 1 } { 2 } ) ' ( { \\bf A } _ 1 + { \\bf A } _ 2 ) ^ { - } ( { \\bf x } + { \\bf y } ) . \\end{align*}"}
-{"id": "2685.png", "formula": "\\begin{align*} u ^ 2 + p ^ 2 = x ^ 3 \\end{align*}"}
-{"id": "3038.png", "formula": "\\begin{align*} - \\frac { 1 } { 2 } \\Delta _ { g } g _ { i j } + Q _ { i j } ( g , \\ , D g ) = c g _ { i j } \\mbox { f o r e v e r y } i , \\ , j \\end{align*}"}
-{"id": "3277.png", "formula": "\\begin{align*} \\mathbb { G T } ( N ) = \\big \\{ \\left ( x ^ 1 , \\cdots , x ^ N \\right ) : x ^ i \\prec x ^ { i + 1 } , \\textnormal { f o r } 1 \\le i \\le N - 1 \\big \\} . \\end{align*}"}
-{"id": "10046.png", "formula": "\\begin{align*} g ( \\mathrm { T } ^ { \\mathrm { w } } ( J _ { \\varphi } ^ + X , Y ) , J _ { \\varphi } ^ + Z ) - g ( \\mathrm { T } ^ { \\mathrm { w } } ( J _ { \\varphi } ^ + Z , Y ) , J _ { \\varphi } ^ + X ) = 0 , g ( \\mathrm { T } ^ { \\mathrm { w } } ( J _ { \\varphi } ^ - X , Y ) , J _ { \\varphi } ^ - Z ) - g ( \\mathrm { T } ^ { \\mathrm { w } } ( J _ { \\varphi } ^ - Z , Y ) , J _ { \\varphi } ^ - X ) = 0 , \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} \\mu = \\sum _ { r = 1 } ^ n c _ r \\mu _ r , \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} m _ i h _ i ^ { - 1 } \\frac { d h _ i } { d t } = \\sum _ { \\alpha : i \\to j } h _ i ^ { - 1 } \\phi _ \\alpha ^ * h _ j \\phi _ \\alpha - \\sum _ { \\alpha : j \\to i } \\phi _ \\alpha h _ j ^ { - 1 } \\phi _ \\alpha ^ * h _ i \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} \\mathbb { P } ( \\# { \\cal E } _ i = r ) \\geq { n - 1 \\choose r - 1 } T _ r p _ d ^ { r - 1 } ( 1 - p _ u ) ^ { r ( n - r ) + { r \\choose 2 } - r + 1 } . \\end{align*}"}
-{"id": "2793.png", "formula": "\\begin{align*} | a u ^ 2 - b v ^ 2 | \\leqslant \\varepsilon a v ^ 2 \\left ( \\frac { b } { a } - \\frac { u } { v } \\right ) B ^ { - \\frac { 1 } { r } } \\leqslant \\varepsilon a \\left ( \\frac { b } { a } - \\sqrt { \\frac { b } { a } } \\right ) \\frac { D _ 1 D _ 2 d _ 3 } { b \\sqrt { a } \\left ( \\sqrt { \\frac { b } { a } } - 1 \\right ) } B ^ { \\frac { 1 } { 2 } - \\frac { 1 } { r } } = \\frac { \\varepsilon D _ 1 D _ 2 d _ 3 } { \\sqrt { b } } B ^ { \\frac { 1 } { 2 } - \\frac { 1 } { r } } , \\end{align*}"}
-{"id": "8532.png", "formula": "\\begin{align*} & \\vec { \\gamma } _ 0 ( \\vec { \\Psi } , p _ j ) = \\vec { \\gamma } _ 1 ( \\vec { \\Psi } , p _ j ) = \\vec { \\gamma } _ 2 ( \\vec { \\Psi } , p _ j ) = 0 , \\end{align*}"}
-{"id": "8954.png", "formula": "\\begin{align*} \\gamma _ { x , v } ^ \\lambda ( t ) = x * \\left ( \\cos \\theta \\frac { \\sin ( \\lambda t ) } { \\lambda } + \\sin \\theta \\frac { 1 - \\cos ( \\lambda t ) } { \\lambda } , - \\cos \\theta \\frac { 1 - \\cos ( \\lambda t ) } { \\lambda } + \\sin \\theta \\frac { \\sin ( \\lambda t ) } { \\lambda } , - \\frac { \\lambda t - \\sin ( \\lambda t ) } { \\lambda ^ 2 } \\right ) . \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{align*} \\frac { 1 } { z _ i } H + z _ i ^ 2 = \\frac { 1 } { z _ j } H + z _ j ^ 2 \\end{align*}"}
-{"id": "3267.png", "formula": "\\begin{align*} \\frac { d } { d t } \\alpha ^ { \\pm } _ i ( t ) = - c \\alpha _ i ^ { \\pm } ( t ) \\ , \\ \\frac { d } { d t } \\gamma _ 1 ( t ) = - c \\gamma _ 1 ( t ) \\ , \\ \\frac { d } { d t } \\delta ( t ) = ( 1 - 2 c \\delta ( t ) ) . \\end{align*}"}
-{"id": "6513.png", "formula": "\\begin{align*} \\forall \\gamma \\in F , b ( \\gamma ) = 0 . \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} \\mathcal { W } _ n : = ( \\mathbb { Z } / 2 \\mathbb { Z } ) ^ n \\rtimes S _ n . \\end{align*}"}
-{"id": "9644.png", "formula": "\\begin{align*} s _ 1 : = 2 \\cos ( t _ 1 ) , s _ 2 : = 2 \\cos ( t _ 2 ) . \\end{align*}"}
-{"id": "9716.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\alpha _ { i , j } g _ i = a _ j , ~ ~ \\sum _ { i = 1 } ^ r \\alpha _ { i , j } a _ i = g _ j . \\end{align*}"}
-{"id": "7243.png", "formula": "\\begin{align*} \\gamma _ 1 \\sum _ { i , j } c _ { i j } \\gamma _ 1 ^ i \\gamma _ 2 ^ j = \\sum _ { i , j } c _ { i j } \\gamma _ 1 ^ i \\gamma _ 2 ^ j , \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { N } X _ l = \\mathbb { E } ( \\hat { U } ^ { ( n ) } _ n | { \\cal F } _ N ) - \\mathbb { E } ( \\hat { U } ^ { ( n ) } _ n | { \\cal F } _ 0 ) = \\hat { U } ^ { ( n ) } _ n - \\mathbb { E } \\hat { U } ^ { ( n ) } _ n \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} E ( x _ 1 , \\ldots , x _ m ) = E ( x _ 1 , \\ldots , x _ { m - 1 } ) E ( x _ { m - 1 } , x _ m ) \\ \\end{align*}"}
-{"id": "9747.png", "formula": "\\begin{align*} B ( x ) = \\int _ { a \\in \\mathcal { A } } \\left ( \\int _ { \\mathbb { R / Z } } \\delta ( x - X ( s , a ) ) \\partial _ s X ( s , a ) d s \\right ) d \\mu ( a ) \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{align*} \\left | \\hat { \\mu } ( \\xi _ i - \\xi _ j ) \\right | ^ N = e ^ { 2 c } m ^ { - 2 + O ( \\log ( 1 + \\rho ) / \\rho ^ 2 ) } . \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} D = \\sum _ { p \\in \\Sigma ^ 2 } ^ { } n _ p ( D ) \\ , p \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{align*} A _ { N , \\varepsilon } \\left ( r \\right ) = C _ { 1 } \\exp \\left ( C _ { 2 } \\left ( \\frac { \\varphi \\left ( r \\right ) } { r } \\right ) ^ { \\frac { 1 } { N - 1 - \\varepsilon } } \\right ) . \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} \\rho ( \\omega ) = \\begin{cases} ( 1 - \\omega \\lambda _ { \\max } ) ^ 2 & \\omega \\leq 0 \\\\ ( 1 - \\omega \\lambda _ { \\min } ^ + ) ^ 2 & 0 \\leq \\omega \\leq \\omega ^ * \\\\ ( 1 - \\omega \\lambda _ { \\max } ) ^ 2 & \\omega \\geq \\omega ^ * \\end{cases} , \\end{align*}"}
-{"id": "2221.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty \\ ! [ \\widehat { h } ( k ) + \\widehat { h } ( - k ) S ( k ) ] \\widehat { h } ( k ) ^ \\dag d k = 0 . \\end{align*}"}
-{"id": "5049.png", "formula": "\\begin{align*} A _ 2 \\gamma = A _ 2 \\gamma \\in \\mathbb { F } _ q ( \\{ \\alpha _ i : i \\in \\{ 1 , 2 , \\dots , k - 1 \\} \\setminus \\{ j \\} \\} ) . \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} P _ { \\perp } \\begin{pmatrix} z \\\\ \\bar { z } \\end{pmatrix} = 0 . \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} z _ \\alpha = \\frac 1 2 ( e _ { 2 \\alpha - 1 } - i e _ { 2 \\alpha } ) \\end{align*}"}
-{"id": "7781.png", "formula": "\\begin{align*} u _ t + b u _ x + u _ { x x x } + u _ { x y y } = f ( t , x , y ) . \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{gather*} P _ { 2 n } = \\sum _ { k = 1 } ^ { 2 n } \\sum _ { j = \\lceil \\frac k 2 \\rceil } ^ { n } { 2 j - 1 \\choose k - 1 } { n + j - 1 \\choose n - j } , \\\\ P _ { 2 n - 1 } = \\sum _ { k = 1 } ^ { 2 n - 1 } \\sum _ { j = \\lceil \\frac { k + 1 } { 2 } \\rceil } ^ { n } { 2 j - 2 \\choose k - 1 } { n + j - 2 \\choose n - j } . \\end{gather*}"}
-{"id": "4626.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\int _ 0 ^ \\infty f ( t ) \\ , \\varphi _ T ( \\dd t ) = f ( \\infty ) . \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{gather*} \\frac { q x _ i x _ { i + 1 } - q ^ { - 1 } x _ { i + 1 } x _ i } { q - q ^ { - 1 } } = 1 , \\\\ x _ i ^ 3 x _ { i + 2 } - [ 3 ] _ q x _ i ^ 2 x _ { i + 2 } x _ i + [ 3 ] _ q x _ i x _ { i + 2 } x _ i ^ 2 - x _ { i + 2 } x _ i ^ 3 = 0 , \\end{gather*}"}
-{"id": "2490.png", "formula": "\\begin{align*} f _ q = \\sum _ { j = 1 } ^ N f _ q ^ { \\gamma _ j } , \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\bigg [ \\big ( \\mathbf { A } , \\frac { \\partial ^ { 2 } \\widetilde { \\mathbf { A } } } { \\partial t ^ { 2 } } ) + \\big ( \\nabla \\times \\mathbf { A } , \\nabla \\times \\widetilde { \\mathbf { A } } \\big ) + \\big ( \\frac { \\mathrm { i } } { 2 } ( \\Psi ^ { * } \\nabla { \\Psi } - \\Psi \\nabla { \\Psi } ^ { * } ) , \\widetilde { \\mathbf { A } } \\big ) + \\big ( | \\Psi | ^ { 2 } \\mathbf { A } , \\widetilde { \\mathbf { A } } \\big ) \\bigg ] \\mathrm { d } t = 0 , \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} P _ p ( x , \\zeta ) = ( 1 - | x | ^ { 2 p } ) \\sum _ { m = 0 } ^ { \\infty } C ^ { n / 2 } _ m ( \\frac { x \\cdot \\overline { \\zeta } } { | x | | \\overline { \\zeta } | } ) | x | ^ m | \\overline { \\zeta } | ^ m , \\end{align*}"}
-{"id": "1123.png", "formula": "\\begin{align*} q ( x ) < \\begin{cases} \\displaystyle \\frac { s ( x ) p } { s ( x ) - p } , & s ( x ) > p , \\\\ \\infty , & s ( x ) \\le p , \\end{cases} \\end{align*}"}
-{"id": "3565.png", "formula": "\\begin{align*} \\tau _ \\gamma \\wedge d z = ( 2 \\pi i ) ^ { - p } \\pi _ * ( w ^ \\gamma \\tau ) . \\end{align*}"}
-{"id": "8591.png", "formula": "\\begin{align*} T ( z ) = O ( | z | ^ { 1 - \\frac { 2 } { p } } \\log ^ { \\frac { 2 } { p ' } } | z | ) . \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} p _ { ( m , j ) } ( S _ M - S _ N ) \\leqslant \\sum _ { n = N + 1 } ^ { M } \\dfrac { t ^ n } { n ! } p _ { ( m , j ) } \\big ( \\phi ^ { ( m ) } \\big ) \\leqslant \\sum _ { n = N + 1 } ^ { M } \\dfrac { ( t M ) ^ n } { n ! } \\ , \\sup _ n M ^ { - n } \\ , p _ { ( m , j ) } \\big ( \\phi ^ { ( m ) } \\big ) \\ { \\underset { \\R } { \\overset { N , M \\to \\infty } { \\longrightarrow } } } \\ 0 . \\end{align*}"}
-{"id": "9352.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { J } \\mathcal { V } _ r ( \\gamma _ j ^ \\ell ) \\leq J ^ { \\frac 1 { r ' } } \\Big ( \\sum _ { j = 1 } ^ { J } \\mathcal { V } ^ r _ r ( \\gamma _ j ^ \\ell ) \\Big ) ^ { \\frac 1 r } \\lesssim _ r \\| F \\| _ { \\ell ^ r } ^ { r - 1 } \\Big ( \\sum _ { j = 1 } ^ J \\mathcal { V } _ r ^ r ( \\gamma _ j ^ \\ell ) \\Big ) ^ { \\frac 1 r } = \\frac { \\| F \\| _ { \\ell ^ r } ^ { r - 1 } } 2 \\Big ( \\sum _ { n \\in \\mathbb { J } } \\log \\Big ( \\frac { 1 + | F _ n | } { 1 - | F _ n | } \\Big ) ^ r \\Big ) ^ { \\frac 1 r } . \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{align*} \\omega = \\left ( \\{ \\alpha ^ + _ i ( \\omega ) \\} , \\{ \\alpha ^ + _ i ( \\omega ) \\} , \\gamma _ 1 ( \\omega ) , \\gamma _ 2 ( \\omega ) \\right ) \\mapsto \\left ( - \\alpha _ 1 ^ - ( \\omega ) , - \\alpha _ 2 ^ - ( \\omega ) , \\cdots , \\alpha _ 2 ^ + ( \\omega ) , \\alpha _ 1 ^ + ( \\omega ) \\right ) \\in C o n f ( \\mathbb { R } ^ * ) . \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{align*} \\mathcal { F } ( \\mathrm { D } ^ { \\alpha } w ) = | \\xi | ^ { \\alpha } \\cos ( \\frac { \\alpha \\pi } { 2 } ) \\hat { w } , \\alpha \\in \\left ( 1 , 3 \\right ) . \\end{align*}"}
-{"id": "8673.png", "formula": "\\begin{align*} \\widetilde { H } ( s ) = 2 s ^ 2 | z ' | ^ 2 + \\frac { \\lambda _ 2 } { 2 s } z ^ 2 + \\frac { 1 } { ( \\alpha + 2 ) s ^ { \\alpha \\lambda _ 2 } } | z | ^ { \\alpha + 2 } \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{align*} [ f ( \\pm k , x ) ^ \\dagger ; f ( \\pm k , x ) ] = \\pm 2 i k I _ n , k \\in \\mathbb { R } , \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} 0 = [ [ L _ { - r } , \\ , S _ { r - 2 } ] , \\ , S _ { r - 1 } ] = [ [ L _ { - r } , \\ , S _ { r - 1 } ] , S _ { r - 2 } ] \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} \\omega ( h , g ) \\ ; = \\ ; - \\overline { \\omega ( h , g ) } \\ , , \\end{align*}"}
-{"id": "437.png", "formula": "\\begin{align*} \\boldsymbol { \\mathcal { W } } : = \\mathbb { P } ( ( p _ * \\mathbf { F } ( c - a - b ) ) ^ { \\vee } ) \\xrightarrow { \\boldsymbol { \\pi } } \\mathbf { X } . \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} L _ m \\sum _ { k = 1 } ^ n { ( - 1 ) ^ { k - 1 } L _ { 2 m k } } = ( - 1 ) ^ { n - 1 } L _ { m + 2 m n } + L _ m \\quad \\mbox { $ m $ e v e n } \\ , . \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} \\| \\alpha \\| ^ 2 = \\sum _ { 1 \\leq h < h \\leq 6 } ^ 6 a _ { h k } ^ 2 . \\end{align*}"}
-{"id": "5121.png", "formula": "\\begin{align*} g _ { n } = - \\frac { p ! } { \\left ( p + n \\right ) ! } \\left \\{ \\begin{array} { c } n + p \\\\ p \\end{array} \\right \\} \\end{align*}"}
-{"id": "8879.png", "formula": "\\begin{align*} \\det { ( { \\bf M } _ n ) } = \\det { [ ( { \\bf 0 } ^ \\top ) ( { \\bf C } _ n ) - ( - { \\bf B } ^ \\top _ n ) ( { \\bf B } _ n ) ] } = \\det { ( { \\bf B } ^ \\top _ n { \\bf B } _ n ) } = [ \\det ( { \\bf B } _ n ) ] ^ 2 . \\end{align*}"}
-{"id": "9911.png", "formula": "\\begin{align*} \\ ; \\big [ A \\mid B \\big ] = 3 . \\end{align*}"}
-{"id": "1568.png", "formula": "\\begin{align*} \\varrho _ k = \\frac { 1 } { p ( x _ k ) } = \\frac { 1 } { p _ { 2 n - 2 } Q ' ( x _ k ) S ( x _ k ) } , k = 1 , \\dots , n . \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} \\lambda = \\frac { m _ 2 + 2 m _ 3 + \\ldots + ( n - 1 ) m _ n } { m _ 1 + m _ 2 + \\ldots + m _ n } \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} \\frac { 1 } { p } \\frac { d } { d t } \\int _ { \\Omega } n _ 1 ^ p & = - \\frac { 1 } { p } \\int _ { \\Omega } u \\cdot \\nabla n _ 1 ^ p + \\int _ { \\Omega } n _ 1 ^ { p - 1 } \\Delta n _ 1 + \\chi _ 1 \\frac { p - 1 } { p } \\int _ { \\Omega } \\nabla n _ 1 ^ p \\cdot \\nabla c \\\\ & \\quad \\ , + \\mu _ 1 \\int _ { \\Omega } n _ 1 ^ p - \\mu _ 1 \\int _ { \\Omega } n _ 1 ^ { p + 1 } - a _ 1 \\mu _ 1 \\int _ { \\Omega } n _ 1 ^ p n _ 2 . \\end{align*}"}
-{"id": "8856.png", "formula": "\\begin{align*} \\sum _ { d = 0 } ^ { \\infty } ( d + 1 ) ^ u G _ { d + 1 } ( n ) = \\sum _ { d = 0 } ^ { n - 1 } ( d + 1 ) ^ u G _ { d + 1 } ( n ) = s _ 1 + s _ 2 , \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} w _ i = v _ i + \\log ( 1 + a ) , i = 1 , 2 , \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} | \\Delta ( \\alpha ) | & \\leq ( 4 c d ) ^ { ( d ^ 2 - d ) / 2 - ( s _ 0 ^ 2 - s _ 0 ) / 2 } D ^ { ( d ^ 2 - d ) / 2 } \\\\ & \\qquad \\qquad \\qquad \\qquad \\ ( \\frac { d } { s _ 0 ^ 2 - s _ 0 } \\ ) ^ { ( s _ 0 ^ 2 - s _ 0 ) / 2 } s _ 0 ^ { ( s _ 0 ^ 2 + s _ 0 ) / 2 + 1 / 1 2 } e ^ { - s _ 0 ^ 2 / 4 + O ( 1 ) } \\\\ & = ( 4 c d ) ^ { ( d ^ 2 - d ) / 2 - ( s _ 0 ^ 2 - s _ 0 ) / 2 } D ^ { ( d ^ 2 - d ) / 2 } \\\\ & \\qquad \\qquad \\qquad \\qquad \\ ( \\frac { d } { s _ 0 - 1 } \\ ) ^ { ( s _ 0 ^ 2 - s _ 0 ) / 2 } s _ 0 ^ { s _ 0 + 1 / 1 2 } e ^ { - s _ 0 ^ 2 / 4 + O ( 1 ) } . \\end{align*}"}
-{"id": "5807.png", "formula": "\\begin{align*} S ( \\hat { \\omega } ) = S ( \\hat { \\rho } ) + S ( \\hat { \\rho } \\| \\hat { \\omega } ) + \\beta \\ , \\mathrm { T r } \\left [ \\hat { H } \\left ( \\hat { \\omega } - \\hat { \\rho } \\right ) \\right ] \\ge S ( \\hat { \\rho } ) \\ ; . \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} \\mathcal { M } ^ { 2 - \\textrm { c o n v } } ( S ^ { n } ) = \\textrm { E m b } ^ { 2 - \\textrm { c o n v } } ( S ^ n , \\mathbb { R } ^ { n + 1 } ) / \\textrm { D i f f } ( S ^ n ) \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} \\Lambda _ c ( u _ c + \\tilde { u } ) - \\Lambda _ c ( u _ c ) = \\frac { 1 } { 2 } \\langle ( L _ c - \\partial _ y ^ 2 ) \\tilde { u } , \\tilde { u } \\rangle _ { L ^ 2 } + N _ c ( \\tilde { u } ) , \\end{align*}"}
-{"id": "9520.png", "formula": "\\begin{align*} \\tau ( b y ) I _ n = \\sum _ { i = 1 } ^ s c _ i ^ * y c _ i , \\ \\ \\ y \\in \\mathbb { M } _ n , \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} g _ { n } = \\frac { \\left ( - 1 \\right ) ^ { n + 1 } } { \\left ( n + 1 \\right ) ! } \\left [ x ^ { n + 1 } - \\left ( x - 1 \\right ) ^ { n + 1 } \\right ] \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{align*} \\frac { B _ { n } ^ { \\left ( q \\right ) } } { n ! } = \\sum _ { p = 1 } ^ { n } \\left ( \\sum _ { m = p } ^ { n } \\binom { m } { p } \\binom { m + q - 1 } { q - 1 } \\right ) \\sum _ { k _ { 1 } + \\dots + k _ { p } = n } \\frac { \\left ( - 1 \\right ) ^ { p } } { \\left ( k _ { 1 } + 1 \\right ) ! \\dots \\left ( k _ { p } + 1 \\right ) ! } . \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} c _ i ( 0 ) = \\begin{cases} 0 & i \\\\ 2 F ( 2 ^ { - i - 1 } ) & \\end{cases} \\quad , c _ i ( j ) = \\begin{cases} 2 ^ { 1 - i } c ( i - 1 , j - 1 ) / j & j < i \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} ( - q ) _ \\infty = \\frac { ( q ^ 2 ; q ^ 2 ) _ \\infty } { ( q ) _ \\infty } = \\frac { 1 } { \\sqrt { 2 } } e ^ { \\frac { \\pi i } { 2 4 \\tau } } + O \\ ( y e ^ { \\frac { \\pi } { 2 4 } \\ ( \\frac { - 1 } { \\tau } \\ ) } \\ ) . \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} \\dot { y } _ \\alpha = \\frac { 1 } { m _ i } \\left ( \\sum _ { \\xleftarrow [ \\alpha ] { } i \\xrightarrow [ \\beta ] { } } e ^ { - y _ \\beta } - \\sum _ { \\xleftarrow [ \\alpha ] { } i \\xleftarrow [ \\beta ] { } } e ^ { - y _ \\beta } \\right ) - \\frac { 1 } { m _ j } \\left ( \\sum _ { \\xrightarrow [ \\alpha ] { } j \\xrightarrow [ \\beta ] { } } e ^ { - y _ \\beta } - \\sum _ { \\xrightarrow [ \\alpha ] { } j \\xleftarrow [ \\beta ] { } } e ^ { - y _ \\beta } \\right ) . \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} \\frac { b } { c } = - a . \\end{align*}"}
-{"id": "2553.png", "formula": "\\begin{align*} x ( P ) = \\frac { \\max _ w x ( P _ w ) + \\min _ b x ( P _ b ) } 2 , \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} [ \\Delta ( \\lambda ) : L ( \\mu ) ] = [ \\Delta ( p \\cdot \\lambda ) : L ( p \\cdot \\mu ) ] , ( T ( \\lambda ) : \\Delta ( \\mu ) ) = ( T ( p \\cdot \\lambda ) : \\Delta ( p \\cdot \\mu ) ) \\end{align*}"}
-{"id": "7237.png", "formula": "\\begin{align*} \\sqrt n ( \\tilde L - L ^ * ) = - V ( L ^ * ) \\frac { 1 } { \\sqrt n } \\sum _ { i = 1 } ^ n \\left ( ( L _ { Z _ i } ^ * ) ^ { - 1 } - ( I + L ^ * ) ^ { - 1 } \\right ) + \\rho _ n , \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} ( f , x ) ( g , y ) = ( f \\tau _ x g , x y ) , \\textrm { w h e r e } \\tau _ x g ( z ) = g ( z x ^ { - 1 } ) . \\end{align*}"}
-{"id": "9407.png", "formula": "\\begin{align*} \\underset { j } { \\lim } \\ ( \\Gamma ^ { B , C } ( v _ j ) ( Y ) Z \\Gamma ^ { A , B } ( u ) ( X ) ) & = ( \\Gamma ^ { B , C } ( v ) ( Y ) Z \\Gamma ^ { A , B } ( u ) ( X ) ) \\\\ & = ( \\Gamma ^ { A , B } ( u ) ( X ) \\Gamma ^ { B , C } ( v ) ( Y ) Z ) . \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} i \\colon \\Omega ^ { p + 1 } ( M ) \\hookrightarrow \\Omega ^ { p } ( M ; T M ) , i ( \\phi ) _ { X _ 1 \\ldots X _ p } = { \\phi _ { - , X _ 1 \\ldots X _ p } } ^ { \\sharp _ g } . \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} M _ { 0 } = M _ { 0 } ^ { ^ { \\prime } } = C \\left \\Vert a \\right \\Vert _ { L ^ { n } \\left ( R _ { + } ^ { n } ; E \\right ) } , \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{align*} d x = \\Big | \\frac { \\partial ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) } { \\partial ( \\rho , \\theta , \\phi ) } \\Big | d \\theta d \\phi d \\rho ; \\ \\ \\cosh \\rho \\leq r , \\ \\theta \\in [ 0 , 2 \\pi ] \\ \\ \\phi \\in [ 0 , \\pi ] \\end{align*}"}
-{"id": "4870.png", "formula": "\\begin{align*} \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} \\tau \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} ^ { - 1 } \\tau ^ { - \\frac { w } { z } } = I + \\begin{pmatrix} * & * \\\\ c _ i \\left ( \\frac { z } { w } - \\frac { w } { z } \\right ) & * \\end{pmatrix} p . \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle | \\sum _ { k = 1 } ^ { M } J _ 1 ^ { ( k ) } | \\leq C \\big ( h ^ { 2 r + 2 } + ( \\Delta t ) ^ { 4 } \\big ) + C \\| \\theta _ { \\Psi } ^ { M } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } + C \\Delta t \\sum _ { k = 1 } ^ { M - 1 } { \\| \\theta _ { \\Psi } ^ { k } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } } . } \\end{array} \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} \\left \\Vert \\mathbf { E } ^ { \\mathbf { A } } \\right \\Vert _ { \\infty } : = \\sup \\left \\{ \\left \\vert E _ { \\mathbf { A } } ( t , x ) \\right \\vert \\ : \\ ( t , x ) \\in \\mathrm { s u p p } ( A ) \\right \\} < \\infty \\ . \\end{align*}"}
-{"id": "7337.png", "formula": "\\begin{align*} & \\sum _ { i < j } \\left \\{ { \\lambda ' _ i } ( t ) \\lambda _ j ( t ) + \\lambda _ i ( t ) { \\lambda ' _ j } ( t ) \\right \\} f ^ i \\wedge J _ 0 f ^ i \\wedge f ^ j \\wedge J _ 0 f ^ j = \\\\ & - \\frac { 1 } { \\varepsilon ^ 2 _ t } \\sum _ { i < j } \\left [ \\left ( \\frac { n _ 1 } { \\lambda _ 1 ( t ) } + \\frac { n _ 2 } { \\lambda _ 2 ( t ) } + \\frac { n _ 3 } { \\lambda _ 3 ( t ) } \\right ) \\left ( n _ i \\lambda _ j ( t ) + n _ j \\lambda _ i ( t ) ) \\right ) - 2 n _ i n _ j \\right ] f ^ i \\wedge J _ 0 f ^ i \\wedge f ^ j \\wedge J _ 0 f ^ j . \\\\ \\end{align*}"}
-{"id": "3483.png", "formula": "\\begin{align*} \\bar { Y } u ^ + = \\left ( D _ { t } E _ 0 ^ + + A _ { x } ^ { \\alpha } E _ 0 ^ + \\right ) \\ast _ x u _ { 0 } = A _ { x } ^ { \\alpha } \\tilde { E } \\ast _ x u _ { 0 } = A _ { x } ^ { \\alpha } \\tilde { u } . \\end{align*}"}
-{"id": "6749.png", "formula": "\\begin{align*} - \\bigtriangleup _ { \\varepsilon } u + \\left ( A + \\lambda \\right ) u = f \\left ( x \\right ) , x \\in R _ { + } ^ { n } , \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} \\frac { d } { d z } s _ { N + 1 } ( X _ 1 , . . . ) = X _ { N + 1 } ' + \\sum _ { k = 0 } ^ { N } s _ { N - k } ( X _ 1 , . . . ) X _ { k + 1 } ' \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} | x | _ { W ^ { 2 \\alpha , p } } = | x | _ { L ^ p } + \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\frac { | x ( \\xi ) - x ( \\eta ) | ^ p } { | \\xi - \\eta | ^ { 1 + 2 \\alpha p } } d \\xi d \\eta . \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} \\mathbb { S } ( R ^ { \\bigstar } ) = \\frac { 2 } { d ^ { n + 1 } } \\Big [ \\frac { d _ 1 d _ 2 \\dots d _ n } { n + 1 } + \\frac { W ^ { ( n ) } _ 1 } { n } + \\dots + \\frac { W ^ { ( n ) } _ s } { n - s + 1 } + \\dots + \\frac { W ^ { ( n ) } _ { n - 1 } } { 2 } \\Big ] . \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{align*} E _ ( u ) = \\int _ { \\Omega } \\phi ( \\nabla u ) + \\lambda \\| u - f \\| _ { L ^ 2 ( \\Omega ) } ^ 2 \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} H _ { 1 } : = D ( L ^ { 1 / 2 } ) \\qquad \\mbox { a n d } H _ { - 1 } : = ( H _ { 1 } ) ' , \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{align*} \\mathrm { a d } _ { R } ^ { \\circ } ( X ) ( \\mathsf { M } _ { m } ^ { n } ) = \\pi ( K _ { 2 \\rho } ^ { - 1 } X _ { ( 1 ) } K _ { 2 \\rho } ) ( S ^ { - 2 } ( X _ { ( 2 ) } ) \\triangleright \\mathsf { M } _ { m } ^ { n } ) \\pi ( K _ { 2 \\rho } ^ { - 1 } S ( X _ { ( 3 ) } ) K _ { 2 \\rho } ) . \\end{align*}"}
-{"id": "3624.png", "formula": "\\begin{align*} f ( t ) = \\sum _ { n \\geq 0 } c _ n g ( \\mu _ n t ) , \\end{align*}"}
-{"id": "2857.png", "formula": "\\begin{align*} \\bigl ( \\Xi ( f ) \\bigr ) _ { ( x _ 1 , x _ 2 ) } \\diamond \\Phi ( x _ 1 , x _ 2 ; z _ 1 , z _ 2 ) = f ( z _ 1 , z _ 2 ) \\cdot \\Phi ( x _ 1 , x _ 2 ; z _ 1 , z _ 2 ) . \\end{align*}"}
-{"id": "9876.png", "formula": "\\begin{align*} Q ( \\gamma ) \\coloneqq \\prod _ { i = 1 } ^ { | \\gamma | } \\frac { 1 } { \\min ( \\deg ( \\gamma _ { i - 1 } ) , \\deg ( \\gamma _ i ) ) } . \\end{align*}"}
-{"id": "1555.png", "formula": "\\begin{align*} E _ n ( x ) : = \\det \\begin{bmatrix} c _ 0 & c _ 1 & \\cdots & c _ { n - 2 } & 1 & 1 \\\\ c _ 1 & c _ 2 & \\cdots & c _ { n - 1 } & x _ { k _ 0 } & x \\\\ \\vdots & \\vdots & & \\vdots & \\vdots & \\vdots \\\\ c _ { n - 1 } & c _ n & \\cdots & c _ { 2 n - 3 } & x _ { k _ 0 } ^ { n - 1 } & x ^ { n - 1 } \\\\ c _ n & c _ { n + 1 } & \\cdots & c _ { 2 n - 2 } & x _ { k _ 0 } ^ n & x ^ n \\end{bmatrix} \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} U ( t ) & : = F _ t ( U ) , \\\\ U ' ( t ) & : = F _ t ( U ' ) , \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} \\widehat { \\mu } ( 2 ^ N k ) \\ , = \\ , \\widehat { \\mu } ( k ) \\ , \\prod _ { m = 1 } ^ { N } \\tfrac { 1 } { 3 } \\bigl ( 1 + 2 \\cos ( 2 ^ m \\pi k ) \\bigr ) . \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\left ( \\frac { \\left ( - 1 \\right ) ^ { n + p } } { n ! } \\int _ { 0 } ^ { 1 } \\dots \\int _ { 0 } ^ { 1 } \\left ( p x - p + u _ { 1 } + \\dots + u _ { p } \\right ) ^ { n } d u _ { 1 } \\dots d u _ { p } \\right ) z ^ { n } & = \\left ( - 1 \\right ) ^ { p } e ^ { - z \\left ( p x - p \\right ) } \\left ( \\frac { 1 - e ^ { - z } } { z } \\right ) ^ { p } \\\\ & = \\left ( - 1 \\right ) ^ { p } e ^ { - z p x } \\left ( \\frac { e ^ { z } - 1 } { z } \\right ) ^ { p } ; \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\left ( \\frac { d } { d t } \\omega ( t ) ^ 2 \\right ) = - \\varepsilon ^ { - 2 } _ t * _ t ( \\omega ( t ) \\wedge d \\eta ) \\wedge d \\eta . \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} h = \\sum _ { i = 1 } ^ \\infty X _ i f _ i , \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{align*} \\langle \\mathring { \\varphi } , \\psi \\rangle _ { e ^ { ( 2 + n ) f } } = \\langle \\varphi , \\mathring { \\psi } \\rangle _ { e ^ { ( 2 + n ) f } } . \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} F _ n = E _ { n } ^ c \\cap A _ n , \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{align*} \\sin ( \\pi / 2 + k \\alpha ) = - p ( x , y , k ) . \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} \\Phi ( h ) = - \\frac { 1 } { | \\mu | } \\int _ { S ^ { n - 1 } } \\log h ( v ) d \\mu ( v ) + \\log \\bar { V } _ q ( [ h ] ) , \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} \\tau _ B U ( \\alpha ) & = U \\big ( f ^ 2 ( \\alpha ) \\big ) , & \\tau _ B U \\big ( f ^ 2 ( \\alpha ) \\big ) & = U \\big ( f ( \\alpha ) \\big ) , & \\tau _ B U \\big ( f ( \\alpha ) \\big ) & = U ( \\alpha ) . \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} \\mu ^ { 2 ^ { k + 1 } - 1 } + 1 = 0 . \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{align*} \\frac { x } { e _ C ( x ) } = \\sum _ { n = 0 } ^ \\infty \\frac { B C _ n } { \\Pi ( n ) } x ^ n \\ , . \\end{align*}"}
-{"id": "2123.png", "formula": "\\begin{gather*} \\frac { 1 } { \\tilde { u } } \\frac { \\partial \\tilde { u } } { \\partial \\tilde { t } _ 1 } = - \\frac { 2 } { \\tilde { t } _ 1 } \\big ( \\tilde { p } _ 1 \\tilde { q } _ 1 + \\tilde { p } _ 2 \\tilde { q } _ 2 + \\tilde { \\theta } ^ \\infty _ 1 \\big ) , \\frac { 1 } { \\tilde { u } } \\frac { \\partial \\tilde { u } } { \\partial \\tilde { t } _ 2 } = - 2 \\tilde { p } _ 2 \\tilde { q } _ 1 . \\end{gather*}"}
-{"id": "7845.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\{ { \\cal E } _ 1 = \\{ 1 , 2 , \\ldots , r \\} \\} \\right ) \\leq T _ r p _ u ^ { r - 1 } ( 1 - p _ d ) ^ { r ( n - r ) } \\end{align*}"}
-{"id": "5685.png", "formula": "\\begin{align*} & e ^ { \\int _ 0 ^ t \\alpha ( s ) d s } \\Bigg [ \\int _ 0 ^ t \\beta ( s ) e ^ { \\int _ 0 ^ s - \\alpha ( r ) d r } d s + C \\Bigg ] \\\\ & = e ^ { \\int _ 0 ^ { t + T } \\alpha ( s ) d s } \\Bigg [ \\int _ 0 ^ { t + T } \\beta ( s ) e ^ { \\int _ 0 ^ s - \\alpha ( r ) d r } d s + C \\Bigg ] \\\\ & = e ^ { \\int _ 0 ^ { t } \\alpha ( s ) d s } e ^ { \\int _ t ^ { t + T } \\alpha ( s ) d s } \\Bigg [ \\int _ 0 ^ { t } \\beta ( s ) e ^ { \\int _ 0 ^ s - \\alpha ( r ) d r } d s \\\\ & + \\int _ t ^ { t + T } \\beta ( s ) e ^ { \\int _ 0 ^ s - \\alpha ( r ) d r } d s + C \\Bigg ] , \\end{align*}"}
-{"id": "5694.png", "formula": "\\begin{align*} \\log L - \\dfrac { 1 } { n } \\underset { m = 0 } { \\overset { n - 1 } { \\sum } } \\log \\left | \\det B \\left ( L ^ { m } k \\right ) \\right | \\underset { a . e . k } { \\xrightarrow { n \\rightarrow \\infty } } \\log L - \\underbrace { \\int _ { 0 } ^ { 1 } \\log \\left | p ^ { } _ { L } \\right | d k } _ { = 0 } - \\underbrace { \\int _ { 0 } ^ { 1 } \\log \\left | Q - R \\right | d k } _ { = m \\left ( Q - R \\right ) } , \\end{align*}"}
-{"id": "9434.png", "formula": "\\begin{align*} \\mathrm { L } _ { B } ( w ^ * ) = \\partial _ { t } b ^ * + ( v ^ * \\cdot \\nabla ) b ^ * - ( b ^ * \\cdot \\nabla ) v ^ * + { h ^ * } ^ { - 1 } \\nabla \\times d ^ * , \\end{align*}"}
-{"id": "5671.png", "formula": "\\begin{align*} D _ m = \\min _ { ( v , w ) \\in \\big [ v ( t ) , w ( t ) \\big ] } D _ m \\Big \\{ \\big [ v ( t ) , w ( t ) \\big ] ; ( 0 , 0 ) \\Big \\} . \\end{align*}"}
-{"id": "9396.png", "formula": "\\begin{align*} \\norm { T _ 1 ( v ) } ^ 2 _ L & = \\ell \\bigl ( \\zeta \\mapsto \\norm { \\zeta \\cdotp v } ^ 2 \\bigr ) \\\\ & \\leq p \\bigl ( \\zeta \\mapsto \\norm { \\zeta \\cdotp v } ^ 2 _ 2 \\bigr ) \\\\ & \\leq C ^ 2 \\norm { v } ^ 2 _ 2 \\end{align*}"}
-{"id": "6769.png", "formula": "\\begin{align*} s _ 1 = s + \\frac { 1 } { p _ 1 } - \\frac { 1 } { p } > 0 \\ , . \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\int _ { \\Omega } \\langle A ^ { - 1 } _ { \\varepsilon , \\delta + \\delta ' } \\beta _ { j } , \\beta _ { j } \\rangle _ { \\omega _ { \\varepsilon } } e ^ { \\psi - ( \\delta - \\delta ' ) \\phi } d V _ { \\omega _ { \\varepsilon } } \\leq \\lim _ { j \\to \\infty } \\frac { 1 } { \\delta ( n - 1 ) } \\int _ { \\Omega } | \\beta _ { j } | ^ { 2 } _ { \\omega _ { \\varepsilon } } e ^ { \\psi - ( \\delta - \\delta ' ) \\phi } d V _ { \\omega _ { \\varepsilon } } = 0 . \\end{align*}"}
-{"id": "10027.png", "formula": "\\begin{align*} g ( J X , Y ) = g ( X , J Y ) , \\forall X , Y \\in { \\mathfrak X } ( M ) . \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} \\sigma _ k = \\frac { 1 } { p ( t _ k ) } = - \\frac { 1 } { p _ { 2 n - 2 } Q ( t _ k ) S ' ( t _ k ) } , k = 1 , \\dots , n - 1 . \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} d ^ { \\rm s u p } ( 1 0 ^ { - 2 } ) \\ , \\le \\ , \\left \\{ \\begin{array} { l l } 2 & \\mbox { f o r \\ } a = 4 , \\\\ 2 & \\mbox { f o r \\ } a = 3 , \\\\ 3 & \\mbox { f o r \\ } a = 2 \\end{array} \\right . \\end{align*}"}
-{"id": "9774.png", "formula": "\\begin{align*} \\langle T _ { \\epsilon } ( \\tilde { u } ) , f \\rangle : = \\int _ M e _ { \\epsilon } ( \\tilde { u } ) f ( S ( p ) ) d v _ g \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} m _ { i , 2 k } = { \\cal O } ( ( 2 k ) ^ { { 2 a _ i } k } ) \\ \\mbox { a s } \\ k \\to \\infty , \\ \\mbox { f o r } \\ i = 1 , \\ldots , n , \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} D ( p ) : = \\frac { | \\det ( v ^ s | v ^ u ) | } { \\Vert v ^ s \\Vert \\ , \\Vert v ^ u \\Vert } . \\end{align*}"}
-{"id": "4717.png", "formula": "\\begin{align*} R ( z ) = I + O ( e ^ { - c N } ) N \\to \\infty , \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} a = \\partial _ z ~ , ~ a ^ \\dagger = z ~ , ~ H _ { \\rm { O H } } = z \\partial _ z . \\end{align*}"}
-{"id": "8173.png", "formula": "\\begin{align*} ( { \\bf A } _ 1 ^ \\frac { 1 } { 2 } ) ^ { - } = \\left [ \\begin{array} { c c } { \\bf I } _ r & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf 0 } \\end{array} \\right ] { \\bf G } ^ { - 1 } \\mbox { a n d } ( { \\bf A } _ 2 ^ \\frac { 1 } { 2 } ) ^ { - } = \\left [ \\begin{array} { c c } { \\bf D } _ r ^ \\frac { - 1 } { 2 } & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf 0 } \\end{array} \\right ] { \\bf G } ^ { - 1 } ~ _ . \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} \\widehat { \\phi } ( \\mathfrak { C } ( ( \\omega ^ 1 , y ^ 1 ) , 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ) ( z _ 1 , z _ 2 ) & = \\widehat { \\Phi } ( \\mathfrak { C } ( ( \\omega ^ 1 , y ^ 1 ) , 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ( z _ 1 , z _ 2 ) ) \\\\ & = \\widehat { \\Phi } ( y ^ 1 ( [ \\omega ] ( ( z _ 1 , z _ 2 ) , ( 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ) ) ) \\\\ & = \\widehat { \\Phi } ( y ( s ^ { - 1 } , [ \\omega ] ( ( z _ 1 , z _ 2 ) , ( 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ) ) ) . \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} ( k - 1 ) s _ i s _ n + s _ n \\cdot \\sum _ { t = 0 } ^ { i - 1 } b _ t \\cdot s _ t \\leq ( k - 1 ) s _ { i + 1 } s _ { n - 1 } + s _ { n - 1 } \\cdot \\sum _ { t = 0 } ^ { i - 1 } b _ t \\cdot s _ { t + 1 } . \\end{align*}"}
-{"id": "9089.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Xi _ 2 ^ { ( k ) } ( t ) + c \\Pi _ 2 ^ { ( k ) } ( t ) \\leq C \\Gamma _ 2 ^ { ( k ) } ( t ) , k = 0 , 1 , \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} X _ t ^ { Y _ 0 , \\varphi ^ * } ( \\omega ) = Y _ 0 - \\int _ 0 ^ t f ( s , X _ s ^ { Y _ 0 , \\varphi ^ * } ( \\omega ) , Z _ s ( \\omega ) , k _ s ( \\omega ) ) d s + M _ t ( \\omega ) , \\ , \\ , 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} \\langle H ( x ) , X \\rangle = \\langle x , H ^ * ( X ) \\rangle , x \\in \\mathbb R ^ { 2 n - 1 } , \\ X \\in \\mathbb R ^ { n \\times n } . \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} \\langle \\chi , \\lambda ( \\mu ) \\rangle = \\log _ { \\abs { \\varpi } } ( \\chi ( \\mu ) ) \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} d ^ I = \\{ ( x _ 1 , \\dots , x _ n ) \\in \\mathbb { R } _ + ^ n ; \\ x _ 1 < \\dots < x _ n \\} . \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{align*} ( 1 + z ) ^ { 2 p } \\cdot \\Phi ( p ) \\equiv & \\Psi _ 1 ( p ) + z ^ { 2 p } \\cdot \\Psi _ 2 ( p ) + 2 z ^ { p } \\cdot \\Phi ( 0 ) - \\frac { z \\cdot \\Psi _ 1 ( 0 ) + z ^ { 2 p } \\cdot \\Psi _ 2 ( 0 ) } { 1 + z } \\\\ = & \\Psi _ 1 ( p ) + z ^ { 2 p } \\cdot \\Psi _ 2 ( p ) - \\frac { z ( z ^ { p - 1 } - 1 ) ^ 2 \\cdot \\Psi _ 1 ( 0 ) } { 1 + z } \\\\ \\equiv & \\Psi _ 1 ( p ) + z ^ { 2 p } \\cdot \\Psi _ 2 ( p ) \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} \\sum _ { D } \\frac { 2 ^ { \\omega ( D ) + 1 } } { D } \\prod _ { \\substack { r \\leq y \\\\ r \\nmid D } } \\left ( \\frac { r - 3 } { r - 1 } \\right ) = 1 + o ( 1 ) , \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} E ( \\overline { \\mathbb { Q } } ) [ 2 ] = \\left \\{ p _ \\infty \\ , , \\ , ( 0 , 1 ) \\ , , \\ , \\left ( 0 , \\frac { - 1 + \\sqrt { 5 } } { 2 } \\right ) \\ , , \\ , \\left ( 0 , \\frac { - 1 - \\sqrt { 5 } } { 2 } \\right ) \\right \\} . \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} \\widetilde { S } ( x ) = \\sum _ { \\substack { \\alpha : i \\to j \\\\ v _ i - v _ j = 1 } } \\left ( c _ \\alpha e ^ { x _ j - x _ i } - u _ \\alpha ( x _ j - x _ i ) \\right ) \\end{align*}"}
-{"id": "6564.png", "formula": "\\begin{align*} \\begin{array} { r c l } x _ 0 & = & \\displaystyle { \\frac { a + c + b d - \\theta } { 2 b } } \\\\ y _ 0 & = & \\displaystyle { \\frac { a + c - b d + \\theta } { 2 d } } \\end{array} \\mbox { w h e r e } \\theta : = \\sqrt { ( a + c + b d ) ^ 2 - 4 a b d } = \\sqrt { ( a + c - b d ) ^ 2 + 4 b c d } \\ , . \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} \\tau \\ , { } _ { 0 } ^ { c } \\mathrm { D } _ { t } ^ { \\alpha } T + { } _ { 0 } ^ { c } \\mathrm { D } _ { t } ^ { \\beta } T = \\mathcal { D } \\frac { \\partial ^ { 2 } T } { \\partial x ^ { 2 } } , \\ ; \\ ; \\ ; \\ ; \\tau \\ , { } _ { 0 } ^ { c } \\mathrm { D } _ { t } ^ { 2 \\alpha } T + { } _ { 0 } ^ { c } \\mathrm { D } _ { t } ^ { \\alpha } T = \\mathcal { D } \\frac { \\partial ^ { 2 } T } { \\partial x ^ { 2 } } , \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} y _ { j k } : = p _ { 2 n - 2 } \\sigma _ k \\sum _ { i = 0 } ^ { n - 1 } b _ { i j } t _ k ^ i , j = 0 , \\dots , n - 1 , k = 1 , \\dots , n - 1 . \\end{align*}"}
-{"id": "2295.png", "formula": "\\begin{align*} y = 1 + 3 y _ k ^ 2 , v = y e ^ { \\sqrt { 3 } y _ k t } , \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} L ^ 2 ( \\Gamma ) = \\bigoplus _ { j = 1 } ^ { | \\mathcal { E } | } L ^ 2 ( 0 , l _ j ) , \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{align*} \\Delta : = \\{ ( x , t ) : t \\in ( 0 , \\tau _ 2 - \\tau _ 1 ) , B ^ - ( t ) < x < B ^ + ( t ) \\} . \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} R ( \\xi , \\lambda ) e _ n = \\sum _ m \\frac { \\sqrt { m ! } } { \\sqrt { n ! } } \\psi _ m ^ n ( \\xi , \\lambda ) e _ m ~ \\mbox { a n d } ~ a ^ \\dagger G = \\sum _ { n \\geq 1 } \\sqrt { n } G _ { n - 1 } e _ n \\end{align*}"}
-{"id": "5313.png", "formula": "\\begin{align*} \\Xi \\left ( { a , 1 , 0 , 1 } \\right ) = \\Gamma \\left ( { a + s } \\right ) = \\mathop { \\lim } \\limits _ { u \\to 0 } { u ^ { a + s } } \\Psi \\left ( { 1 , 1 + a + s ; u } \\right ) , \\ , a + s > 0 . \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} b _ 1 ( 2 n - 1 ) ^ 2 - b _ 1 ( 2 n - 2 ) b _ 1 ( 2 n ) & = b _ 1 ( 2 n - 2 ) ^ 2 - b _ 1 ( 2 n - 2 ) b _ 1 ( 2 n ) \\\\ & = b _ 1 ( 2 n - 2 ) \\left ( b _ 1 ( 2 n - 2 ) - b _ 1 ( 2 n ) \\right ) = - b _ 1 ( 2 n - 2 ) b _ 1 ( n ) . \\end{align*}"}
-{"id": "8026.png", "formula": "\\begin{align*} \\rho = 2 \\Big \\{ c _ 1 \\| w ( \\bar t ) \\| _ 3 + 2 \\gamma ( 1 + \\| \\nabla u _ s \\| _ q + \\| \\nabla u _ s \\| _ r ) \\| U ( \\bar t ) \\| _ { 3 , \\infty , \\mathbb R ^ 3 } \\Big \\} \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} D _ r ^ j F = \\sum _ { k = 1 } ^ n \\partial _ { x _ k } f ( B ( h _ 1 ) , \\ldots , B ( h _ n ) ) h ^ j _ k ( r ) , j = 1 , \\ldots , d \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{align*} F ( x ) = \\mathop { \\rm l i m } _ { \\varepsilon \\nearrow \\frac { 1 } { M } } ( f ^ { \\varepsilon } ) _ { \\varepsilon } ( x ) = \\mathop { \\rm l i m } _ { \\varepsilon \\nearrow \\frac { 1 } { M } } \\mathop { \\rm i n f } _ { z \\in \\mathcal { H } } \\mathop { \\rm s u p } _ { y \\in \\mathcal { H } } \\{ f ( y ) - \\frac { | y - z | ^ { 2 } } { 2 \\varepsilon } + \\frac { | z - x | ^ { 2 } } { 2 \\varepsilon } \\} \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} B _ R ( x ) = \\left \\{ y : \\| y - x \\| _ 1 \\leq R \\right \\} . \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} d e t ( \\Delta ( \\lambda ) ) = 0 , \\lambda \\in \\mathbb { C } \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} \\varrho ( f ) ( \\sigma ) = \\varrho ^ \\prime ( g ) ( \\sigma ) \\mod X _ { ( \\chi ) } , \\end{align*}"}
-{"id": "7297.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = \\alpha _ 2 y _ 5 , [ y _ 1 , y _ 2 ] = y _ 4 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 5 } { \\alpha _ 3 } y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 3 ] = \\gamma _ 2 y _ 5 , [ y _ 3 , y _ 2 ] = \\gamma _ 4 y _ 5 . \\end{align*}"}
-{"id": "6506.png", "formula": "\\begin{align*} x ^ q P _ n ( x ) = \\beta ^ 0 _ { n , q } P _ { n - q } ( x ) + \\displaystyle \\sum _ { n - q + 1 \\leq s \\leq n + q } \\beta ^ s _ { n , q } P _ { s } ( x ) . \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} \\frac { 1 } { T _ o } \\int _ { 0 } ^ { T _ o } | { x } ( t ) | ^ 2 { \\rm d } t = 1 \\end{align*}"}
-{"id": "6731.png", "formula": "\\begin{align*} \\ - u ^ { \\left [ 2 \\right ] } \\left ( t \\right ) + \\left ( A + \\lambda \\right ) u \\left ( t \\right ) = f , t \\in \\left ( 0 , a \\right ) , \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} F = y ^ 2 z + a _ 1 x y z + a _ 3 y z ^ 2 - ( x ^ 3 + a _ 2 x ^ 2 z + a _ 4 x z ^ 2 + a _ 6 z ^ 3 ) , \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} M : = \\bigoplus _ { \\alpha : i \\to j } \\mathrm { H o m } ( E _ i , E _ j ) \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} \\Phi \\ : = \\ : \\begin{pmatrix} \\Phi ^ + \\ ! \\\\ \\Phi ^ - \\ ! \\end{pmatrix} \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{align*} B _ { i j } ' : = B _ { i j } - \\Big ( \\frac { ( \\lambda _ { n + 3 } - \\lambda _ i ) x _ i } { ( \\lambda _ { n + 3 } - \\lambda _ 1 ) x _ 1 } \\Big ) B _ { 1 j } = ( \\lambda _ { n + 3 } s - t ) \\frac { ( \\lambda _ 1 - \\lambda _ i ) ( \\lambda _ { n + 2 } - \\lambda _ j ) x _ i x _ j } { ( \\lambda _ { n + 3 } - \\lambda _ { n + 2 } ) ( \\lambda _ { n + 3 } - \\lambda _ 1 ) x _ { n + 3 } ^ 2 } \\end{align*}"}
-{"id": "9336.png", "formula": "\\begin{align*} \\mathcal { V } _ r ( \\sigma ) ( z ) & = \\sup _ K \\sup _ { N _ 0 < \\ldots < N _ K } \\Big ( \\sum _ { j = 0 } ^ { K - 1 } \\| \\sigma _ { N _ { j + 1 } } ( z ) - \\sigma _ { N _ j } ( z ) \\| _ { } ^ r \\Big ) ^ { \\frac 1 r } , \\\\ \\mathcal { V } _ r ( \\gamma ) ( z ) & = \\sup _ K \\sup _ { N _ 0 < \\ldots < N _ K } \\Big ( \\sum _ { j = 0 } ^ { K - 1 } d ( \\gamma _ { N _ j } ( z ) , \\gamma _ { N _ { j + 1 } } ( z ) ) ^ r \\Big ) ^ { \\frac 1 r } , \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} N _ a ( \\lambda ) \\lesssim \\lambda ^ { - 1 / b } \\left [ \\log ( 1 / \\lambda ) \\right ] ^ { d - p - 1 } = \\lambda ^ { - a - 1 / 2 m } \\left [ \\log ( 1 / { \\lambda } ) \\right ] ^ { d - p - 1 } . \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} B ' \\subset \\bigcup _ { i = 1 } ^ N B ( x _ i , 2 r ' ) \\subset \\bigcup _ { i = 1 } ^ N B \\bigl ( x _ i , \\tfrac { 1 } { 2 } r \\bigr ) . \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} f ( g ( z ) ) = f _ 0 + \\sum _ { n \\geq 1 } z ^ n \\sum _ { \\pi \\in \\mathcal { C } _ n } f _ { \\vert \\pi \\vert } g _ { \\pi } . \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} \\alpha \\xi ^ { m } - \\alpha \\eta ^ { m } + q _ { m } ( \\xi , \\eta ) = - \\alpha \\eta ^ { m } \\left ( \\frac { \\xi ^ { m } } { - \\eta ^ { m } } + 1 + \\frac { q _ { m } ( \\xi , \\eta ) } { - \\alpha \\eta ^ { m } } \\right ) \\geq \\frac { - \\alpha } { 2 } \\eta ^ { m } > c | \\eta | . \\end{align*}"}
-{"id": "434.png", "formula": "\\begin{align*} h ( S , C , \\xi , \\mathbf { k } \\psi ) : = ( [ E ] , S , C ) , \\ \\ \\ w h e r e \\ \\ \\ E = \\ker ( \\psi ) ( c - b ) . \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{align*} P ' S ' - Q ' R ' = ( P ' + Q ' ) S ' - Q ' ( R ' + S ' ) = { ( \\lambda _ { n + 3 } - \\lambda _ { n + 2 } ) ^ 3 x _ { n + 2 } ^ 2 x _ { n + 3 } ^ 2 } ( P S - Q R ) , \\end{align*}"}
-{"id": "8171.png", "formula": "\\begin{align*} { \\bf A } _ 2 = { \\bf G } \\left [ \\begin{array} { c c } { \\bf D } _ r & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf 0 } \\end{array} \\right ] { \\bf G } ' . \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} \\lim _ { t \\to T ^ - } \\mathrm { d i a m } ( F _ y , \\omega _ \\epsilon ( t ) ) = 0 . \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} \\theta = \\Lambda ( d F ) = J \\delta F . \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} f \\ ; = \\ ; a _ 0 ^ { ( f ) } \\ , v _ 0 + a _ \\infty ^ { ( f ) } \\ , v _ \\infty + b _ \\infty ^ { ( f ) } \\ , v _ 0 + b _ 0 ^ { ( f ) } \\ , v _ \\infty \\ , , \\end{align*}"}
-{"id": "5844.png", "formula": "\\begin{align*} L _ { u + v } - ( - 1 ) ^ v L _ { u - v } = 5 F _ v F _ u \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} X _ { 2 , \\ , j + 2 , \\ , 1 1 } \\cap L _ { j + 2 } = 0 . \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{align*} t = ( 1 - \\alpha ^ 2 ) r = ( 2 \\underline C - \\bar C ) \\lambda \\sqrt s . \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { F } _ q } ( A _ u h _ { u , j } ) = l / r , u = 1 , 2 , \\dots , r . \\end{align*}"}
-{"id": "2808.png", "formula": "\\begin{align*} S ( X , \\mathbf { d } , \\mathbf { D } ; L _ 1 , L _ 2 , L _ 3 ) = \\sum _ { \\mathbf { x } \\in \\Lambda ( \\mathbf { D } ) \\cap X \\mathcal { R } } \\tau \\left ( \\frac { L _ 1 ( \\mathbf { x } ) } { d _ 1 } \\right ) \\tau \\left ( \\frac { L _ 2 ( \\mathbf { x } ) } { d _ 2 } \\right ) \\tau \\left ( \\frac { L _ 3 ( \\mathbf { x } ) } { d _ 3 } \\right ) . \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} [ y _ 1 , y _ 2 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 5 } { \\alpha _ 3 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 2 ] = \\beta _ 2 y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\beta _ 4 \\gamma _ 1 } { \\alpha _ 3 } y _ 5 , [ y _ 2 , y _ 3 ] = y _ 4 , [ y _ 3 , y _ 2 ] = \\gamma _ 4 y _ 5 . \\end{align*}"}
-{"id": "3190.png", "formula": "\\begin{align*} c o n s t \\times \\prod _ { 1 \\le j < k \\le N } ^ { } | u _ j - u _ k | ^ 2 \\prod _ { j = 1 } ^ { N } ( 1 + u _ j ) ^ { \\bar { s } } ( 1 + \\bar { u _ j } ) ^ s \\times d \\theta _ j . \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{align*} K _ \\mu ( \\theta ) = \\log ( L _ \\mu ( \\theta ) ) , \\ \\ \\ \\ \\ \\theta \\in \\Theta ( \\mu ) , \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a & b \\pi ^ n \\\\ c & d \\end{bmatrix} \\in R ^ \\bullet \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{align*} V = \\rho ^ { - 1 } ( R ) \\subset U = Y _ 4 \\setminus \\cup _ { i = 1 } ^ 4 Z _ i . \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} p ( x ) = N ( { c } ) ( x ) = H ^ * ( H ( { c } ) ^ { - 1 } ) ( x ) = \\tilde { p } _ { 2 n - 2 } ( H ^ * { B } ) ( x ) = \\tilde { p } _ { 2 n - 2 } { B } ( x , x ) . \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } y ( t ) = - \\frac { 2 A \\sqrt { 6 } y ( t ) ^ 5 + 1 } { 1 2 y ( t ) ^ 9 } , \\\\ y ( 0 ) = 1 . \\end{cases} \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} \\langle \\partial a , b \\rangle = \\sum _ { E } \\# { \\mathcal M } ( L _ 1 , L _ 2 ; a , b ; 1 , E ) [ b ] . \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} [ L _ { - r } , \\ , [ L _ { - 2 } , \\ , S _ r ] ] = L _ { - 2 } \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} \\varphi : X \\rightarrow \\lbrack 0 , \\infty ) , \\varphi ( x ) = d ( x , x _ { 0 } ) , \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} y _ { i k } : = p _ { 2 n - 2 } \\varrho _ k \\sum _ { j = 0 } ^ { n - 1 } b _ { i j } x _ k ^ j , i = 0 , \\dots , n - 1 , \\ ; k = 1 , \\dots , n , \\end{align*}"}
-{"id": "7684.png", "formula": "\\begin{align*} W = x _ 1 ^ m + f ( x _ 2 , x _ 3 , x _ 4 ) . \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} C _ { } = \\frac { 1 - \\frac { T } { N } } { 1 - \\left ( \\frac { T } { N } \\right ) ^ M } \\end{align*}"}
-{"id": "8255.png", "formula": "\\begin{align*} \\left ( - \\Delta _ n + 2 \\alpha \\sum _ { i \\neq j } \\delta ( x _ i - x _ j ) \\right ) \\psi ( x _ 1 , \\dots , x _ n ) = E \\psi ( x _ 1 , \\dots , x _ n ) \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{align*} \\Bigl ( v ' + \\Bigl ( \\frac { 1 } { \\alpha } - \\frac { 2 } { \\alpha t } - \\frac { e ^ { - t } } { 4 } \\Bigr ) v \\Bigr ) ' = \\frac { 1 } { 4 t } \\Bigr ( \\underline { v } ^ \\alpha - | v | ^ \\alpha + \\frac { 4 ( \\alpha - 1 ) } { \\alpha ^ 2 t } - \\frac { 1 } { \\alpha } e ^ { - t } + t e ^ { - t } \\Bigl ) v . \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} i \\partial _ t \\phi _ { \\mathrm { s p d e } } + \\frac 1 2 \\Delta \\phi _ { \\mathrm { s p d e } } - \\dot { W } ( x ) \\cdot \\phi _ { \\mathrm { s p d e } } = 0 . \\end{align*}"}
-{"id": "6626.png", "formula": "\\begin{align*} \\mathbf { V } ( r ) \\ , : = \\frac { 1 } { r } \\begin{pmatrix} - 1 & \\nu \\\\ - \\nu & 1 \\end{pmatrix} + \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , \\mathbf { E } \\ , : = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} \\| \\omega \\wedge \\alpha \\| ^ 2 = \\| \\alpha \\| ^ 2 + \\left ( a _ { 1 2 } + a _ { 3 4 } + a _ { 5 6 } \\right ) ^ 2 = \\| \\alpha \\| ^ 2 + \\left \\| \\frac 1 2 \\ , \\omega ^ 2 \\wedge \\alpha \\right \\| ^ 2 . \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} y & \\coloneqq \\psi _ 1 ^ { g - 2 - A } \\prod _ { i = 2 } ^ n \\psi _ i ^ { a _ i } \\kappa _ 1 , \\\\ x _ \\ell & \\coloneqq \\psi _ 1 ^ { g - 2 - A } \\prod _ { i = 2 } ^ n \\psi _ i ^ { a _ i + \\delta _ { i \\ell } } , \\ell = 2 , \\dots , n . \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{align*} x _ 1 v _ 1 + x _ 2 v _ 2 + \\cdots + x _ { n + 3 } v _ { n + 3 } = 0 , \\ , \\lambda _ 1 x _ 1 v _ 1 + \\lambda _ 2 x _ 2 v _ 2 + \\cdots + \\lambda _ { n + 3 } x _ { n + 3 } v _ { n + 3 } = 0 , \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{align*} ( v _ e , w _ e ) : = \\Big \\{ ( v , w ) \\in \\mathbb { R } ^ 2 : f ( v , w ) = g ( v , w ) = 0 \\Big \\} . \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{align*} X \\land \\kappa _ { i } \\circ f \\land K _ { i + 1 } = f \\land \\Sigma K _ { i } \\circ Y \\land \\kappa _ { i } , \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} t _ { X Y Z } = \\frac 1 3 \\left ( g ( X , T _ { Y Z } ) + g ( Z , T _ { X Y } ) + g ( Y , T _ { Z X } ) \\right ) \\end{align*}"}
-{"id": "3447.png", "formula": "\\begin{align*} \\sum _ { j = 1 , j \\ne i } ^ k | \\psi ^ i ( v _ j ) | \\le 2 \\frac { 2 r } m \\le \\frac 1 4 \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} \\mathcal { W } _ n = \\left \\{ C _ n ^ d ( R _ n ) ^ j X ; \\ X \\in \\mathcal { W } _ { n - 1 } , \\ d = 0 , \\dots , n - 1 , \\ j = 0 , 1 \\right \\} . \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{align*} B ^ \\dag A = A ^ \\dag B , A ^ \\dag A + B ^ \\dag B > 0 . \\end{align*}"}
-{"id": "9859.png", "formula": "\\begin{align*} \\vartheta ( \\boldsymbol { x } ) = - \\boldsymbol { \\nabla } ^ 2 \\phi ( \\boldsymbol { x } ) \\mbox { a n d } \\boldsymbol { \\omega } ( \\boldsymbol { x } ) = - \\boldsymbol { \\nabla } ^ 2 \\boldsymbol { \\psi } ( \\boldsymbol { x } ) \\end{align*}"}
-{"id": "8948.png", "formula": "\\begin{align*} P ( \\Omega ) = \\lim _ { t \\to 0 } \\sqrt { \\frac { 2 \\pi } { t } } \\iint _ { \\Omega \\times \\Omega ^ c } p _ t ( x , y ) \\ , d y \\ , d x \\ , . \\end{align*}"}
-{"id": "8941.png", "formula": "\\begin{align*} G ( \\zeta ) = c - \\sum _ { j = 0 } ^ { \\infty } \\frac { m n } { ( j l + \\zeta + 1 ) ( j l + \\zeta + n + 1 ) } . \\end{align*}"}
-{"id": "4673.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { R t } { p _ n } \\right ) \\left ( 1 - \\frac { R t } { p _ { - n } } \\right ) & = \\left ( 1 - \\frac { R t } { p _ n } \\right ) \\left ( 1 + \\frac { R t } { p _ n } - R t \\left ( \\frac { 1 } { p _ n } + \\frac { 1 } { p _ { - n } } \\right ) \\right ) \\\\ & = \\left ( 1 - \\frac { R ^ 2 t ^ 2 } { p _ n ^ 2 } \\right ) \\left ( 1 + \\frac { R t } { 1 + \\frac { R t } { p _ n } } \\left ( - \\frac { 1 } { p _ n } - \\frac { 1 } { p _ { - n } } \\right ) \\right ) . \\end{align*}"}
-{"id": "4084.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\int _ x ^ { \\infty } t ^ a e ^ { - c t ^ { \\beta } } d t & \\le & ( c - ( 1 + a - \\beta ) x ^ { - \\beta } ) ^ { - 1 } \\int _ x ^ { \\infty } ( c - ( 1 + a - \\beta ) t ^ { - \\beta } ) t ^ a e ^ { - c t ^ { \\beta } } d t \\\\ & = & ( c - ( 1 + a - \\beta ) x ^ { - \\beta } ) ^ { - 1 } x ^ { 1 + a - \\beta } e ^ { - c x ^ { \\beta } } . \\end{array} \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} O ( 2 ^ { \\omega ( D ) } x L _ y ^ 2 e ^ { - C { \\sqrt { \\log x } } } ) & = O ( 2 ^ { 2 . 5 \\log \\log y } x ( \\log x ) ^ 4 e ^ { - C { \\sqrt { \\log x } } } ) \\\\ & = O \\big ( ( \\log x ) ^ 4 ( \\log y ) ^ 2 x e ^ { - C { \\sqrt { \\log x } } } \\big ) \\\\ & = o \\left ( \\frac { \\pi ( x ) } { D ( \\log y ) ^ 3 } \\right ) = o \\left ( \\frac { \\pi _ D ( x ) } { \\log y } \\right ) . \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{align*} \\Pi ( i ) = \\prod _ { j = 0 } ^ m D _ j ^ { c _ j } \\end{align*}"}
-{"id": "2094.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 2 } { x ^ 2 } + \\frac { A _ 1 } { x } + A _ 0 \\right ) Y , \\frac { \\partial Y } { \\partial t _ 1 } = \\left ( \\frac { B _ { 1 1 } } { x } + B _ { 1 0 } \\right ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = \\left ( B _ { 2 1 } x + B _ { 2 0 } \\right ) Y , \\end{gather*}"}
-{"id": "5281.png", "formula": "\\begin{align*} \\int _ { \\partial B _ R ( 0 ) \\cap \\Omega } x \\times u \\ , d S = \\int _ { \\partial B _ R ( 0 ) \\cap \\{ x _ n < 0 \\} } x \\times \\nabla \\left ( \\frac { p \\cdot x } { | x | ^ n } \\right ) \\ , d S + O \\left ( \\frac { 1 } { R ^ { 1 + \\varepsilon } } \\right ) . \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} h ^ y ( z ) = \\sum _ { n \\ge 1 } \\frac { \\partial } { \\partial y _ n } z ^ { - n - 1 } + h ^ y _ 0 z ^ { - 1 } + \\sum _ { n \\ge 1 } n y _ n z ^ { n - 1 } , \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\sum _ { l = 1 } ^ { N } X _ l \\right ) ^ { 2 } = \\sum _ { l = 1 } ^ { N } \\mathbb { E } X _ l ^ 2 . \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} \\lambda _ 0 ( f _ 1 , f _ 2 ) : = \\inf \\{ & t \\geq 0 : ( f _ 1 ( s _ 1 ) - f _ 2 ( s _ 1 ) ) ( f _ 1 ( s _ 2 ) - f _ 2 ( s _ 2 ) ) > 0 s _ 1 , s _ 2 \\in ( t , t + 1 ) \\} \\\\ \\lambda _ 1 ( f _ 1 , f _ 2 ) : = \\inf \\{ & t > \\lambda _ 0 : f _ 1 ( t ) = f _ 2 ( t ) \\} . \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} \\tau _ b \\leq \\left \\lfloor \\frac { d _ p ^ b - 1 } { 2 } \\right \\rfloor = 2 \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} U \\begin{bmatrix} e & f \\pi ^ n \\\\ g & h \\end{bmatrix} = \\begin{bmatrix} a e + g \\pi ^ n & ( a f + h ) \\pi ^ n \\\\ e + d g & f \\pi ^ n + d h \\end{bmatrix} . \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 2 } { x ^ 2 } + \\frac { A _ 1 } { x } + A _ 0 \\right ) Y , \\frac { \\partial Y } { \\partial t } = \\left ( \\frac { 1 } { t } A _ 0 x + B _ 0 \\right ) Y , \\end{gather*}"}
-{"id": "3839.png", "formula": "\\begin{align*} h _ { 1 , 2 , 4 } ( x ) & = 4 ( 3 x + 1 ) ( 3 x ^ 3 + 2 7 x ^ 2 + 3 3 x + 1 ) \\equiv 1 \\pmod { 3 } , \\\\ h _ { 0 , 2 , 2 } ( x ) & = 5 x ^ 2 + 1 0 x + 1 \\equiv 1 \\pmod { 5 } , \\\\ h _ { 2 , 2 , 2 } ( x ) & = x ^ 2 + 1 0 x + 5 \\equiv x ^ 2 \\pmod { 5 } \\end{align*}"}
-{"id": "9420.png", "formula": "\\begin{align*} \\partial _ t h + \\nabla \\cdot P = 0 , \\ ; \\ ; \\ ; \\partial _ t P + \\left ( \\frac { P \\otimes P - B \\otimes B - D \\otimes D } { h } \\right ) = \\nabla \\left ( \\frac { 1 } { h } \\right ) , \\end{align*}"}
-{"id": "4224.png", "formula": "\\begin{align*} G ^ * ( D ^ * b ) = \\left \\{ \\begin{array} { l l } 0 & \\hbox { i f } \\ \\ b \\in \\mathcal { B } \\\\ \\infty & \\hbox { o t h e r w i s e } , \\end{array} \\right . \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{align*} \\delta : = | n _ l - n _ r | + | \\theta _ l - \\theta _ { L } | + | \\theta _ r - \\theta _ { L } | + | \\phi _ r | , \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{align*} Q ( u ) = 2 \\pi \\left [ P ( c _ * + \\delta ) + | b | ^ 2 \\| \\psi _ * \\| ^ 2 _ { L ^ 2 } + \\mathcal { O } ( | \\delta | | b | ^ 2 + | b | ^ 4 ) \\right ] , \\end{align*}"}
-{"id": "5634.png", "formula": "\\begin{align*} T ( k , C ) = \\left \\{ g \\in G ( k ) \\middle | g k g ^ { - 1 } \\subseteq C \\right \\} \\end{align*}"}
-{"id": "4940.png", "formula": "\\begin{align*} w _ { j } = \\left \\{ \\begin{tabular} { c c } $ v _ { j } \\mathbf { e } $ & $ 1 \\leq j \\leq n $ \\\\ $ v _ { n + 1 } $ & $ j = n + 1 . $ \\end{tabular} \\right . \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\dim H ^ 2 ( G _ T , Z _ { [ i ] } ) \\leq d B n ^ { 2 g - 1 } , \\end{align*}"}
-{"id": "1005.png", "formula": "\\begin{align*} Z _ 1 = 2 \\pi ^ * L + H , Z _ 2 = 3 \\pi ^ * L + H , Z _ 3 = \\pi ^ * S . \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} \\gamma = 2 . 8 6 8 1 1 4 0 1 3 ( 4 ) . \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{align*} 0 = \\mathfrak { L } _ K J = D _ { K } ^ g J - [ D ^ g K , J ] . \\end{align*}"}
-{"id": "9668.png", "formula": "\\begin{align*} \\rho _ { L } ( P ( \\xi , \\eta ) ) = \\rho _ { K } ( P ( \\xi , \\pi - \\eta ) ) . \\end{align*}"}
-{"id": "1756.png", "formula": "\\begin{align*} p _ { ( m , j ) } ( \\phi ) : = \\sup _ { | x | \\leqslant j } \\left | \\dfrac { d ^ m \\phi } { d x ^ m } ( x ) \\right | , \\phi \\in C ^ { \\infty } , \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} ( \\Sigma _ \\ell A ) ^ i = A ^ { i + 1 } , d _ { \\Sigma _ \\ell A } ^ i = - d _ A ^ { i + 1 } . \\end{align*}"}
-{"id": "8012.png", "formula": "\\begin{align*} \\partial _ t w _ k ^ 3 - \\Delta w _ k ^ 3 + \\nabla p _ k ^ 3 = - ( h u _ s + \\widetilde U ) \\cdot \\nabla w _ k - w _ k \\cdot \\nabla ( h u _ s + \\widetilde U ) , w _ k ^ 3 ( \\cdot , 0 ) = 0 , \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} \\Vert A \\Vert _ { \\ast } & = \\sqrt { \\sigma _ { 1 } ^ { 2 } \\left ( A \\right ) + \\sigma _ { 2 } ^ { 2 } \\left ( A \\right ) + 2 \\sigma _ { 1 } \\left ( A \\right ) \\sigma _ { 2 } \\left ( A \\right ) } \\\\ & = \\sqrt { p q + r s + 2 \\sqrt { p r s ( q - s ) } } , \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} | \\sigma | ^ 2 _ \\eta = \\frac { e ^ { - \\psi + \\rho } } { 1 + e ^ \\rho } , \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{align*} \\left . \\begin{array} { r c l } \\dot x & = & a - g _ 1 ( x ) + n _ X f ( x , y ) \\\\ \\dot y & = & c - g _ 2 ( y ) + n _ Y f ( x , y ) \\end{array} \\right \\} \\ , \\ , : = \\ , \\ , F ( x , y ) \\end{align*}"}
-{"id": "5450.png", "formula": "\\begin{align*} \\int _ { \\Omega } D ^ m u : D ^ m \\varphi d x = \\mu \\int _ { \\Omega } \\rho u \\varphi d x \\ , , \\ \\ \\ \\forall \\varphi \\in H ^ m ( \\Omega ) \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{align*} \\tilde { F } ( x _ 1 , \\cdots , x _ { k - 1 } , x _ k ) = \\prod _ { i = 1 } ^ { k - 1 } \\left ( \\xi ^ { 0 } _ { i + 1 } - \\xi _ i ^ 0 \\right ) \\partial _ 2 \\cdots \\partial _ k \\tilde { F } ( \\xi _ 1 ^ 0 , \\xi _ 2 ^ 1 , \\cdots , \\xi ^ 1 _ { k - 1 } , \\xi ^ { 1 } _ k ) . \\end{align*}"}
-{"id": "9123.png", "formula": "\\begin{gather*} Y _ \\lambda \\colon \\ \\sum _ { i = 0 } ^ n y _ i ^ { d } + \\lambda \\prod _ { i = 0 } ^ n y _ i ^ { b _ i } . \\end{gather*}"}
-{"id": "9238.png", "formula": "\\begin{align*} \\alpha ( s ) - p = A ( s ) \\ , \\mathbf { t } ( s ) + B ( s ) \\ , \\mathbf { n } _ 1 ( s ) , \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{align*} \\left ( B _ { 2 } f _ { c _ { 1 } } - B _ { 1 } f _ { c _ { 2 } } \\right ) \\left ( z _ { 0 } \\right ) = 0 \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} S _ z ( f ) & : = \\frac { f ''' } { f ' } - \\frac 3 2 \\left ( \\frac { f '' } { f ' } \\right ) ^ 2 & D _ z ( f ) & : = S _ z ( f ) / ( f ' ) ^ 2 . \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} a ^ { * * } = a , ( a + b ) ^ * = a ^ * + b ^ * , ( \\lambda a ) ^ * = \\overline { \\lambda } a ^ * , ( a b ) ^ * = b ^ * a ^ * \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{align*} f _ i = \\frac { 1 } { | | P _ i | | ^ 2 } < P _ i , f > \\end{align*}"}
-{"id": "9518.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = 1 } ^ k m _ i ^ * x m _ i \\right \\| < \\delta . \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} \\int _ \\Omega \\omega \\ , d x = 0 . \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{align*} \\left [ \\phi _ 0 ^ * , \\phi _ \\lambda \\right ] = 0 , \\lambda \\leq - 1 . \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{gather*} \\sum _ { i = 0 } ^ { c - 1 } x _ i ^ d + h ( x _ c , \\dots , x _ n ) \\end{gather*}"}
-{"id": "7140.png", "formula": "\\begin{align*} \\varpi _ 1 = 2 \\alpha _ 1 + \\alpha _ 2 , \\varpi _ 2 = 3 \\alpha _ 1 + 2 \\alpha _ 2 . \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} \\lambda : = N ^ { \\frac { k / 2 - \\gamma } { \\gamma } } \\Big ( \\frac { 1 } { 2 C _ 0 } \\Big ) ^ { - \\frac { 1 } { 2 \\gamma } } ( 1 + \\| u _ 0 \\| _ { H ^ \\gamma _ x } ) ^ { \\frac { 2 } { \\gamma } } \\end{align*}"}
-{"id": "7445.png", "formula": "\\begin{align*} m ( L ) = \\sum _ { k = 1 } ^ n | Z ( [ a _ { k - 1 } , a _ k ] ) | . \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{align*} \\qquad w = e ^ { 2 \\pi i z } , q = e ^ { 2 \\pi i \\tau } , b _ \\pm = \\pm p ' r - p s \\qquad a = p p ' . \\qquad \\qquad \\qquad \\qquad \\qquad { } \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{gather*} f ^ \\vee = \\frac { a _ d + a _ { d - 1 } z + \\dots + a _ 0 z ^ d } { a _ d } . \\end{gather*}"}
-{"id": "6001.png", "formula": "\\begin{align*} | \\langle U ( \\eta \\otimes \\xi _ n ) , \\eta \\otimes \\xi _ n \\rangle - 1 | = n \\int _ { [ 0 , \\frac 1 n ] } \\langle ( \\delta ^ { i t } - 1 ) \\eta , \\eta \\rangle \\ , d t \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} \\frac { d \\hat c _ 0 } { d t } & = \\mu \\hat h _ 0 - \\frac { \\Gamma } { K _ 1 } + \\Lambda \\hat \\theta _ 0 , \\\\ \\frac { d \\hat \\theta _ 0 } { d t } & = - \\hat \\theta _ 0 + T _ s \\\\ \\frac { d \\hat h _ 0 } { d t } & = - \\hat h _ 0 , \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} \\epsilon _ g \\cdot \\epsilon _ h = \\epsilon _ { g h } . \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} \\tilde { E } : - y ^ 2 = z ^ 3 - 2 z + 1 . \\end{align*}"}
-{"id": "2508.png", "formula": "\\begin{align*} T _ { ( C , X ) } S _ { k } ^ { - 1 } ( s ) = \\ker D S _ { k } ( C , X ) . \\end{align*}"}
-{"id": "6949.png", "formula": "\\begin{align*} E ( a , 1 , 6 ) : & & w _ { 2 , 6 } - w _ { 1 , 5 } + w _ { 1 , 3 } w _ { 4 , 6 } \\ & = \\ 0 \\\\ E ( b , 1 , 6 ) : & & w _ { 1 , 3 } w _ { 2 , 6 } + w _ { 1 , 5 } w _ { 4 , 6 } \\ & = \\ 0 \\end{align*}"}
-{"id": "4422.png", "formula": "\\begin{align*} K _ 1 \\cap \\tau K _ 1 \\tau ^ { - 1 } \\cap \\tau ^ 2 K _ 1 \\tau ^ { - 2 } = \\lbrace 1 _ G \\rbrace \\end{align*}"}
-{"id": "9912.png", "formula": "\\begin{align*} r = r ( n , m , \\ell ) \\in \\{ 1 , 2 , 3 \\} \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\| ( \\Phi w ) ( t ) - w _ 0 \\| _ 2 = 0 . \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} { { \\mathbf { { h } } } _ { k } ^ { } } & = { \\mathbf { { H } } _ { k } ^ { } } \\textbf { w } _ { k } = \\sqrt { \\mathit { l } \\left ( { \\Vert \\bold { z } _ k \\Vert } _ 2 \\right ) } \\mathbf { \\tilde { H } } _ { k } \\textbf { w } _ { k } = \\sqrt { \\mathit { l } \\left ( { \\Vert \\bold { z } _ k \\Vert } _ 2 \\right ) } { { \\mathbf { \\breve { h } } } _ { k } ^ { } } , & & \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} t _ { k ^ x } t _ { i ^ x } ^ { - 1 } = ( t _ i ^ { - 1 } ) ^ \\sigma \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} \\hat { \\rho } _ c ( \\mathbf { q } ) & = \\int _ { \\mathbb { R } ^ D } d \\mathbf { x } ~ \\rho _ c ( \\mathbf { x } ) e ^ { - i 2 \\pi \\mathbf { q } \\cdot \\mathbf { x } } \\\\ & = \\left ( \\sum _ { \\mathbf { y } \\in S } e ^ { - i 2 \\pi \\mathbf { q } \\cdot \\mathbf { y } } \\right ) \\left ( \\int _ { \\mathbb { R } ^ D } d \\mathbf { x } ' ~ \\tilde { \\rho } ( \\mathbf { x } ' ) e ^ { - i 2 \\pi \\mathbf { q } \\cdot \\mathbf { x } ' } \\right ) \\\\ & = \\hat { s } ( \\mathbf { q } ) ~ \\hat { \\rho } ( \\mathbf { q } ) . \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\dot x & = & a - b x + \\gamma x \\\\ \\dot y & = & c - d y \\end{array} \\end{align*}"}
-{"id": "6536.png", "formula": "\\begin{align*} \\dot y = Q v ( x _ 0 + \\Gamma _ 0 y ) \\ , . \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} \\tau ( \\phi \\psi ^ * ) = \\tau ( \\phi ^ * \\psi ) . \\end{align*}"}
-{"id": "5299.png", "formula": "\\begin{align*} { \\bf y } _ k = h _ k { \\bf x } _ k + { \\bf n } _ k , \\end{align*}"}
-{"id": "6015.png", "formula": "\\begin{align*} \\det ( \\mathbf M ) = \\det ( \\mathbf M ''' ) = ( P S - Q R ) \\prod _ { j = 3 } ^ n ( \\lambda _ j s - t ) , \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} \\partial _ t q _ t ( x , y ) = - \\partial _ y \\int _ { - \\infty } ^ { x } \\partial _ t p _ t ( y , z ) d z & = - \\partial _ y \\int _ { - \\infty } ^ { x } L _ y p _ t ( y , z ) d z \\\\ & = - \\hat { L } _ y ^ * \\partial _ y \\int _ { - \\infty } ^ { x } p _ t ( y , z ) d z \\\\ & = \\hat { L } _ y ^ * q _ t ( x , y ) . \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} \\Phi \\phi _ 1 \\circ \\delta _ 1 = \\phi _ 1 \\Phi \\colon \\rho _ 1 \\Phi \\to \\Phi . \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} f = a _ 0 + a _ 1 X + \\cdots + a _ { p - 1 } X ^ { p - 1 } \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} L ^ { 2 n } - \\lambda ^ 2 L ^ { 2 n - 2 } = \\overline { \\mathstrut A } ( \\sigma ) A ( \\sigma ) , \\ ; A = \\sigma ^ { n - p - 1 } \\left ( \\sigma ^ 2 - \\lambda ^ 2 \\right ) \\cdot \\prod \\limits _ { i = 1 } ^ { p - 1 } \\left ( \\sigma - \\lambda _ i \\right ) , \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{align*} D _ { n } ( \\vec { \\theta } ) = \\sqrt { \\frac { n ! ( k - n ) ! } { k ! } } e _ { n } ( \\vec { \\theta } ) . \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} { \\displaystyle \\nabla \\cdot \\mathbf { A } _ { 0 } + \\phi _ { 1 } = 0 , \\nabla \\cdot \\mathbf { A } _ { 1 } + \\Delta \\phi _ { 0 } + \\vert \\psi _ { 0 } \\vert ^ { 2 } = 0 . } \\end{align*}"}
-{"id": "902.png", "formula": "\\begin{align*} \\mathbb { E } ( \\langle : X _ T \\otimes \\ldots \\otimes X _ T : , \\Phi \\rangle - \\rho _ { \\Phi } ^ T ) ^ 2 = \\mathbb { E } \\langle : X _ T \\otimes \\ldots \\otimes X _ T : , \\Phi \\rangle ^ 2 - \\mathbb { E } ( \\rho _ { \\Phi } ^ T ) ^ 2 . \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{gather*} { G _ 1 } ^ { - 1 } \\begin{pmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 0 \\\\ - t & 0 & 0 \\end{pmatrix} G _ 1 = \\begin{pmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0 \\end{pmatrix} . \\end{gather*}"}
-{"id": "4229.png", "formula": "\\begin{align*} I _ 1 ( u ) = \\sum _ { i , j } | b ^ k _ { i j } + ( D u ) _ { i j } - d ^ k _ { i j } | ^ 2 \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( A _ 0 x + A _ 1 + \\frac { A _ 2 } { x } \\right ) Y , \\frac { \\partial Y } { \\partial t _ 1 } = ( B _ { 1 1 } x + B _ { 1 0 } ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = ( B _ { 2 1 } x + B _ { 2 0 } ) Y , \\end{gather*}"}
-{"id": "9257.png", "formula": "\\begin{align*} ( \\Delta f ) ( p ) = \\vec { H } ( p ) \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{align*} X _ { l } : = \\Bigl \\| \\sum _ { k = b _ { l } } ^ { b _ { l + 1 } - 1 } \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Bigr ) \\ , x _ { k } e _ { k } \\Bigr \\| . \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{align*} \\textrm { \\emph { V a r } } _ { \\{ Y < q ( \\alpha _ 1 ) \\} } [ X ] & = \\textrm { \\emph { V a r } } _ { \\{ q ( \\alpha _ 1 ) < Y < q ( \\alpha _ 2 ) \\} } [ X ] \\\\ & = \\ldots \\\\ & = \\textrm { \\emph { V a r } } _ { \\{ q ( \\alpha _ { k - 1 } ) < Y < q ( \\alpha _ k ) \\} } [ X ] = \\textrm { \\emph { V a r } } _ { \\{ q ( \\alpha _ k ) < Y \\} } [ X ] , \\end{align*}"}
-{"id": "438.png", "formula": "\\begin{align*} m = m ( e , a , b , c ) : = h ^ 0 ( F ( c - a - b ) ) - 1 , \\end{align*}"}
-{"id": "8517.png", "formula": "\\begin{align*} ( \\nu \\geq \\lambda ) \\rightarrow \\left ( M [ J ] \\models 2 ^ \\nu \\leq ( \\nu ^ { \\nu \\cdot \\nu } ) ^ M = ( \\nu ^ + ) ^ M \\right ) \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} \\int _ \\Omega \\left [ | \\nabla t | ^ 2 + ( 2 \\alpha + \\alpha _ t t ) \\frac { t ^ 2 } { \\varepsilon ^ 2 } \\right ] = \\int _ \\Omega c t \\left ( | \\nabla \\psi | ^ 2 + 2 \\nabla \\psi \\cdot \\nabla \\zeta + | \\nabla \\zeta | ^ 2 \\right ) . \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} L ^ { h w v } ( z ^ 2 ) : = : H ^ { \\gamma } ( z ^ 2 ) H ^ { \\beta } ( z ^ 2 ) : = \\sum _ { n \\in \\mathbb { Z } } L ^ { h w v } _ n z ^ { - 2 n - 4 } \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} \\phi ^ \\ast T _ { \\C } S ^ n = T _ { \\C } \\Sigma ^ 2 \\oplus T _ { \\C } ^ N \\Sigma ^ 2 , \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} a ^ { 2 } + a + 1 = 0 . \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{align*} Y ( t ) = a r s i n h ( X ( t ) ) = \\log \\left ( X ( t ) + \\sqrt { 1 + X ^ 2 ( t ) } \\right ) . \\end{align*}"}
-{"id": "1505.png", "formula": "\\begin{align*} P ( x ) - S ( x ) ^ 2 = \\sum _ { k = 1 } ^ { n - 1 } P ( t _ k ) S _ k ( x ) ^ 2 + \\sum _ { k = 1 } ^ { n - 1 } \\left ( P ' ( t _ k ) - P ( t _ k ) \\frac { S '' ( t _ k ) } { S ' ( t _ k ) } \\right ) ( x - t _ k ) \\ , S _ k ( x ) ^ 2 , \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} \\vec { H } = _ g \\ A = g ^ { i j } A _ { i j } \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} f ( x ) & = \\frac { l _ k } { ( k - 1 ) ! } x ^ { - \\lambda } + \\frac { b _ k } { ( \\lambda ) _ k } + \\int _ 0 ^ { \\infty } M _ 1 ( u ) u ^ { \\lambda + k - 2 } e ^ { - x u } \\ , d u \\\\ & = \\frac { b _ k } { ( \\lambda ) _ k } + \\int _ 0 ^ { \\infty } \\left ( M _ 1 ( u ) u ^ { k - 1 } + \\frac { l _ k } { ( k - 1 ) ! } \\right ) e ^ { - x u } u ^ { \\lambda - 1 } \\ , d u . \\end{align*}"}
-{"id": "1512.png", "formula": "\\begin{align*} \\frac { P ' ( x ) } { P ( x ) } = \\sum _ { j = 1 } ^ { n - 1 } \\left ( \\frac { 1 } { x - \\zeta _ j } + \\frac { 1 } { x - \\overline { \\zeta } _ j } \\right ) . \\end{align*}"}
-{"id": "614.png", "formula": "\\begin{align*} \\int _ { B _ t } | \\nabla u _ k ( x ) | ^ 2 \\ , | x _ { n + 1 } | ^ a \\ , d x & \\leq \\frac { D _ { u _ k } ( 1 ) } { 2 ( 1 - t ) } = \\frac { I _ { u _ k } ( 1 ) \\ , H _ { u _ k } ( 1 ) } { 2 ( 1 - t ) } \\leq \\frac { M } { 2 ( 1 - t ) } , \\end{align*}"}
-{"id": "9730.png", "formula": "\\begin{align*} \\left | \\langle x + \\bar y , v _ n \\rangle \\right | = \\left | \\langle x , v _ n \\rangle \\right | + \\left | \\langle \\bar y , v _ n \\rangle \\right | \\textrm { f o r a l l \\ , } n \\in \\N . \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} f ^ { \\prime \\prime } = \\left ( \\frac { B _ { 2 } ^ { \\prime } } { B _ { 2 } } \\frac { q _ { 1 } } { q _ { 2 } } - \\frac { q _ { 1 } ^ { \\prime } } { q _ { 2 } } \\right ) f + \\frac { 1 } { 2 } \\left ( \\frac { B _ { 2 } ^ { \\prime } } { B _ { 2 } } - \\frac { q _ { 2 } } { q _ { 3 } } - \\frac { N ^ { \\prime } } { N } \\right ) f ^ { \\prime } . \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} K _ 1 \\cap \\tau K _ 1 \\tau ^ { - 1 } = \\lbrace 1 _ G \\rbrace . \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{align*} \\dot \\sigma ( t ) = - \\| \\dot x ( t ) \\| - \\| \\dot v ( t ) \\| , \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} U \\begin{bmatrix} 1 & 0 \\\\ 0 & d ^ { - 1 } \\end{bmatrix} \\begin{bmatrix} 1 & 0 \\\\ - c & 1 \\end{bmatrix} = \\begin{bmatrix} a - b c d ^ { - 1 } \\pi ^ n & b d ^ { - 1 } \\pi ^ n \\\\ 0 & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{gather*} A ( z ) = \\frac { 1 } { z ^ { r + 1 } } \\left ( A _ 0 + A _ 1 z + \\cdots \\right ) , r \\in \\mathbb { Z } _ { > 0 } . \\end{gather*}"}
-{"id": "5332.png", "formula": "\\begin{align*} \\psi ( s _ { \\lambda , r } ( x _ 0 , x _ 1 , \\dots , x _ { m - 1 } ) ) & = \\left ( v _ - ^ { \\otimes r } \\otimes v _ + ^ { \\otimes m - r } \\otimes \\sum _ { \\mu \\preceq \\lambda } K _ { \\lambda \\mu } M _ { \\mu } ( t _ 1 , t _ 2 , \\dots , t _ r ) \\right ) \\\\ & = \\left ( v _ - ^ { \\otimes r } \\otimes v _ + ^ { \\otimes m - r } \\otimes s _ \\lambda ( t _ 1 , t _ 2 , \\dots , t _ r ) \\right ) . \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} \\lbrack \\mathbf { \\Xi } _ { \\mathrm { p } } ( t ) ] _ { + } = - \\frac { t ^ { 2 } } { 2 } \\mathbf { \\mu } \\left ( \\left \\{ 0 \\right \\} \\right ) - \\int \\nolimits _ { 0 } ^ { t } \\mathrm { d } s \\int \\nolimits _ { 0 } ^ { s } \\mathrm { d } \\alpha \\int \\nolimits _ { \\mathbb { R } \\backslash \\left \\{ 0 \\right \\} } \\mathbf { \\mu } \\left ( \\mathrm { d } \\nu \\right ) \\cos \\left ( \\alpha \\nu \\right ) \\ . \\end{align*}"}
-{"id": "9332.png", "formula": "\\begin{align*} W ^ 2 = f _ 6 ( x _ 0 : x _ 1 : x _ 2 ) z ( x ^ 3 + A ( t : s ) x z ^ 2 + B ( t : s ) z ^ 3 ) . \\end{align*}"}
-{"id": "5895.png", "formula": "\\begin{align*} 1 + \\frac { 1 } { r } & < 1 + \\frac { 2 } { r - 1 } \\leq \\bigg ( 1 + \\frac { 1 } { r - 1 } \\bigg ) ^ 2 \\\\ & = \\frac { 1 } { ( 1 - 1 / r ) ^ 2 } = \\bigg ( \\frac { n _ 1 ' } { \\phi ( n _ 1 ' ) } \\cdot \\frac { \\phi ( n _ 1 ) } { n _ 1 } \\bigg ) ^ 2 \\\\ & \\leq 1 + \\frac { \\frac { 3 } { 2 } ( 1 6 + 1 6 ) } { y ( \\log y ) ^ { 1 / 6 } } = 1 + \\frac { 4 8 } { y ( \\log y ) ^ { 1 / 1 6 } } . \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} t k ' t ^ { - 1 } k '^ { - 1 } = I + \\begin{pmatrix} 0 & - 2 c \\\\ 0 & 0 \\end{pmatrix} p \\end{align*}"}
-{"id": "6558.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } x \\\\ y \\end{array} \\right ) = \\left ( \\begin{array} { r c } 1 & 0 \\\\ - \\frac { 1 } { 2 } & \\frac { \\sqrt { 3 } } { 6 } \\end{array} \\right ) \\left ( \\begin{array} { c } u \\\\ v \\end{array} \\right ) + \\left ( \\begin{array} { c } 1 \\\\ 1 \\end{array} \\right ) , \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} v ^ * b v & = v ^ * \\left ( \\phi ^ * \\phi - \\phi \\phi ^ * - \\phi ^ * g \\phi + g \\phi g ^ { - 1 } \\phi ^ * g \\right ) v \\\\ & = ( \\phi v ) ^ * ( 1 - g ) \\phi v + ( \\phi ^ * v ) ^ * \\left ( g ^ { - 1 } - 1 \\right ) \\phi ^ * v \\geq 0 \\end{align*}"}
-{"id": "6398.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a & b \\pi ^ n \\\\ c & d \\end{bmatrix} \\in R . \\end{align*}"}
-{"id": "7211.png", "formula": "\\begin{align*} z = \\frac { 3 \\pm \\sqrt { 5 } } { 2 } , \\end{align*}"}
-{"id": "5.png", "formula": "\\begin{align*} \\nabla . \\mathbf { \\sigma } = 0 \\Rightarrow \\nabla . ( \\underbrace { \\xi _ 1 \\mathbf { e } _ t + \\xi _ 2 \\theta _ t \\mathbf { I } } _ + \\underbrace { E ' ( \\mathbf { e } + \\nu ' \\theta \\mathbf { I } ) } _ - \\underbrace { \\tau ( c ) \\mathbf { I } ) } _ = 0 , \\end{align*}"}
-{"id": "146.png", "formula": "\\begin{align*} d \\eta ( \\theta ) = \\left [ 1 - \\frac { 4 } { 3 } \\cos ( \\theta ) + \\frac { 1 } { 3 } \\cos ( 2 \\theta ) \\right ] \\ , d \\theta \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} \\beta _ 0 = \\min _ { \\phi \\in \\Omega } \\max _ { t \\in [ 0 , 1 ] } W ( \\phi _ t ) , \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{gather*} \\chi _ { 1 , 1 } ^ { n , m } ( x , y | \\rho ) \\allowbreak = \\allowbreak \\sum _ { j \\geq 0 } \\rho ^ { j } U _ { m + j } ( x ) T _ { n + j } ( y ) \\allowbreak = \\\\ ( T _ { n } ( y ) U _ { m } ( x ) ( w _ { 2 } ( x , y | \\rho ) - \\rho ^ { 4 } ) \\\\ + \\rho T _ { n + 1 } ( y ) U _ { m + 1 } ( x ) ( 1 - 2 \\rho ^ { 2 } + 4 \\rho ^ { 2 } ( x ^ { 2 } + y ^ { 2 } ) - 4 \\rho x y ) \\\\ + \\rho ^ { 2 } T _ { n + 2 } ( y ) U _ { m + 2 } ( y ) ( 1 - 4 \\rho x y ) + \\rho ^ { 3 } T _ { n + 3 } ( y ) U _ { m + 3 } ( y ) ) / w _ { 2 } ( x , y | \\rho ) . \\end{gather*}"}
-{"id": "8227.png", "formula": "\\begin{align*} \\rho _ { F } ( \\Omega ^ { + } ) = \\rho _ { F } ( \\Omega ^ { - } ) = \\frac { 1 } { \\overline \\Lambda } \\le \\rho _ { 2 , F } ( \\Omega ) , \\end{align*}"}
-{"id": "4294.png", "formula": "\\begin{align*} \\prod _ { v } \\vert x \\vert _ v = 1 , \\end{align*}"}
-{"id": "4302.png", "formula": "\\begin{align*} \\begin{cases} A _ 1 = - \\frac { 1 } { ( e _ 1 - e _ 3 ) ( e _ 2 - e _ 1 ) } ; \\\\ A _ 2 = - \\frac { 1 } { ( e _ 3 - e _ 2 ) ( e _ 3 - e _ 1 ) } ; \\\\ A _ 3 = - \\frac { 1 } { ( e _ 3 - e _ 2 ) ( e _ 1 - e _ 3 ) } . \\\\ \\end{cases} \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} L _ 1 = x _ 1 , L _ 2 = x _ 2 , L _ 3 = x _ 2 - x _ 1 , \\end{align*}"}
-{"id": "9818.png", "formula": "\\begin{align*} \\mathfrak { T } _ i \\mathfrak { T } _ j \\cdots = \\mathfrak { T } _ j \\mathfrak { T } _ i \\cdots \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} \\widetilde { C } _ q ( K , \\eta ) = \\frac { 1 } { n } \\int _ { \\vec { \\alpha } _ K ^ * ( \\eta ) } \\rho _ K ^ q ( u ) d u . \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} \\max ( A ) = \\{ t \\in A : t = l _ A \\} , \\end{align*}"}
-{"id": "2194.png", "formula": "\\begin{align*} C _ j : = P _ j \\left ( P _ j \\int _ 0 ^ \\infty f ( i k _ j , x ) ^ \\dagger f ( i k _ j , x ) d x P _ j + I _ n - P _ j \\right ) ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} L _ n L _ { n + 1 } = L _ { 2 n + 1 } - ( - 1 ) ^ { n - 1 } \\ , , L _ 2 L _ 3 = L _ 5 + 1 \\ , . \\end{align*}"}
-{"id": "5570.png", "formula": "\\begin{align*} \\mathcal { P } ( X ) = \\{ ( p _ 1 , p _ 2 , p _ 3 ) : \\delta X < p _ 1 , p _ { 2 } \\le X , \\ \\delta X < p _ 3 ^ k \\le X \\} \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} u ( 0 ) = u _ 0 , \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} d _ f ( x , y ) : = d _ 0 ( x , y ) \\vee | f ( x ) - f ( y ) | \\end{align*}"}
-{"id": "2127.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = 1 & x = \\infty \\\\ \\begin{matrix} 0 \\\\ \\theta ^ 0 _ 1 \\\\ \\theta ^ 0 _ 2 \\end{matrix} & \\begin{matrix} 0 \\\\ 0 \\\\ \\theta ^ 1 \\end{matrix} & \\overbrace { \\begin{matrix} t & \\theta ^ \\infty _ 1 \\\\ 0 & \\theta ^ \\infty _ 2 \\\\ 0 & \\theta ^ \\infty _ 3 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "2447.png", "formula": "\\begin{align*} ( i , s , j ) \\eta ^ \\natural ( m , t , n ) \\hbox { i f a n d o n l y i f } i = m \\hbox { a n d } j = n , \\hbox { a n d } \\textsf { 0 } \\eta ^ \\natural \\textsf { 0 } \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 4 & \\frac 3 4 \\\\ & 1 \\end{matrix} \\bigg | \\ , - \\frac 1 3 \\bigg ] _ { p - 1 } \\equiv \\begin{cases} \\big ( \\frac { \\sqrt { - 3 } } { p } \\big ) \\cdot ( a + b \\sqrt { - 3 } ) \\pmod { p ^ 2 } , & p \\equiv 1 \\pmod { 4 } , \\\\ - \\big ( \\frac { \\sqrt { - 3 } } { p } \\big ) \\cdot 3 ( a + b \\sqrt { - 3 } ) \\pmod { p ^ 2 } , & p \\equiv 3 \\pmod { 4 } , \\end{cases} \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} d \\hat r _ t ( z ) = - \\frac { 2 } { \\hat r _ t ( z ) } - d \\xi _ { T - t } \\end{align*}"}
-{"id": "9922.png", "formula": "\\begin{align*} \\left ( \\vect { S } + \\frac { 1 } { N } v _ N ( x - y ) \\right ) \\Lambda = F \\end{align*}"}
-{"id": "4824.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) ^ n = \\begin{pmatrix} 1 & n \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a n + \\frac { c ( n - 1 ) n } { 2 } & \\frac { ( a + d + c ( n - 1 ) ) ( n - 1 ) n } { 2 } - c \\sum _ { k = 0 } ^ { n - 1 } k ^ 2 + b n \\\\ c n & d n + \\frac { c ( n - 1 ) n } { 2 } \\end{pmatrix} p \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} ( \\epsilon - \\varrho _ a ) _ a = ( \\beta + \\gamma - \\delta - a ) _ a \\equiv 0 \\pmod { p } . \\end{align*}"}
-{"id": "1596.png", "formula": "\\begin{align*} c _ k = \\sum _ { j = 1 } ^ { n - 1 } \\sigma _ j t _ j ^ k + \\delta _ { 2 n - 2 , k } \\lambda = \\sum _ { j = 1 } ^ n \\varrho _ j x _ j ^ k , k = 0 , \\dots , 2 n - 2 . \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} T ^ { \\ , 2 ( b _ { n + 1 } - b _ n ) } \\ , e _ { b _ n } = \\smash { v _ n \\ , \\Big ( \\prod _ { j = b _ n + 1 } ^ { b _ { n + 1 } - 1 } w _ j \\Big ) e _ { b _ { \\varphi ( n ) } } - T ^ { \\ , b _ { n + 1 } - b _ { n } } \\ , e _ { b _ { n } } = e _ { b _ { n } } . } \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} \\alpha _ \\infty & = \\frac { 1 } { y ^ 2 } \\alpha ^ \\prime = \\frac { z ^ 3 - 4 z ^ 2 + 6 z - 3 } { z ^ 3 - 2 z + 1 } \\\\ \\beta _ \\infty & = \\frac { z ^ 2 } { y ^ 2 } \\beta = \\frac { z ^ 3 - 2 z ^ 2 } { z ^ 3 - 2 z + 1 } \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} \\gamma _ 4 : = \\min \\bigg \\{ \\frac { \\gamma _ 1 } { 4 \\gamma _ 3 } , \\frac { \\gamma _ 2 } { 4 \\gamma _ 3 } , \\frac { 1 } { 4 } \\bigg \\} > 0 . \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} a = A ^ \\prime ( a ^ \\prime ) ^ 2 , b = B ^ \\prime ( b ^ \\prime ) ^ 2 , \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} Y _ * ( M A ( \\Phi _ { | \\{ p \\} \\times \\mathbb { C } ^ N } ) ) = N ! M A _ { \\mathbb { R } ^ N } ( f _ p ) . \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} { \\bf S } = ( { \\bf D } ^ { ' } ) ^ { - 1 } \\left [ \\begin{array} { c c c c } { \\bf I } & { \\bf 0 } & { \\bf 0 } \\\\ { \\bf 0 } & { \\gamma } { \\bf I } & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf 0 } & { \\bf I } \\\\ \\end{array} \\right ] { \\bf D } ^ { - 1 } . \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} K _ 1 \\cap K _ 2 \\cap \\delta K _ 2 \\delta ^ { - 1 } = \\lbrace 1 _ G \\rbrace \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = \\infty \\ , \\left ( \\frac 1 3 \\right ) \\\\ \\overbrace { \\begin{matrix} 1 & \\theta ^ 0 _ 2 \\\\ 0 & \\theta ^ 0 _ 1 \\\\ 0 & 0 \\end{matrix} } & \\overbrace { \\begin{matrix} - t ^ { \\frac 1 3 } & \\theta ^ \\infty _ 1 / 3 \\\\ - \\omega t ^ { \\frac 1 3 } & \\theta ^ \\infty _ 1 / 3 \\\\ - \\omega ^ 2 t ^ { \\frac 1 3 } & \\theta ^ \\infty _ 1 / 3 \\end{matrix} } \\end{matrix} \\right ) . \\end{gather*}"}
-{"id": "8889.png", "formula": "\\begin{align*} [ f , g ] _ n ( x ) = \\sum _ { k = 1 } ^ n \\sum _ { j = 1 } ^ k ( - 1 ) ^ { k + j } \\Big \\{ ( a _ k ( x ) & \\overline { g } ^ { ( k ) } ( x ) ) ^ { ( k - j ) } f ^ { ( j - 1 ) } ( x ) \\\\ & - ( a _ k ( x ) f ^ { ( k ) } ( x ) ^ { ( k - j ) } ) \\overline { g } ^ { ( j - 1 ) } ( x ) \\Big \\} , \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} E ( \\boldsymbol { h } ) & = \\sum _ { k = - \\infty } ^ { \\infty } ( V ( z _ k + h _ k , z _ { k + 1 } + h _ { k + 1 } ) - V ( z _ k , z _ { k + 1 } ) ) , \\\\ \\nabla E ( \\boldsymbol { h } ) _ k & = V _ 2 ( z _ { k - 1 } + h _ { k - 1 } , z _ k + h _ k ) + V _ 1 ( z _ k + h _ k , z _ { k + 1 } + h _ { k + 1 } ) , \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{align*} S ( k ) = - J ( - k ) J ( k ) ^ { - 1 } , k \\in \\mathbb { R } . \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} u _ t - F u = f \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} & | \\alpha | + | \\beta | = 1 , \\\\ & \\alpha = ( \\alpha _ 0 , \\alpha _ 1 , \\hdots , \\alpha _ { d _ { \\varphi } } ) \\geq 0 , \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} ( 1 + u ^ 2 / k ^ 2 ) b ^ 2 = x ^ 3 \\wedge ( 1 + u ^ 3 / k ^ 3 ) b ^ 3 = y ^ 2 ; \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{align*} \\tilde { u } _ b ( \\xi , y ) = 2 b ^ 2 \\cos ( 2 y ) w _ 2 ( \\xi ) + b ^ 2 w _ 0 ( \\xi ) + \\delta \\partial _ c u _ { c _ * } ( \\xi ) + \\tilde { w } ( \\xi , y ) , \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} | H ^ 1 ( G _ { \\Sigma } , \\widetilde { T } ^ * ( 1 ) ) / \\widetilde { R } \\mathbf { z } _ 1 | = \\frac { r } { | \\widetilde { R } / R | } \\cdot | H ^ 1 ( G _ { \\Sigma } , T ^ * ( 1 ) ) / R \\mathbf { z } _ 1 | , \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{align*} p _ { \\beta , j } ( \\sqrt { r ^ 2 + h ^ 2 } ) & = \\sum _ { k = 0 } ^ j \\varphi _ { k } ( h ) p _ { \\beta , k } \\Big ( { r \\over \\psi ( h ) } \\Big ) \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} < p _ b ^ * x , p _ b ^ * y > = { \\rm d e g } ( b ) < x , y > x , y \\in N S ( X ) , \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} 2 a _ j = a _ { j - 1 } + a _ { j + 1 } , j \\neq k _ i \\mbox { f o r } i \\in \\Z / n \\Z . \\end{align*}"}
-{"id": "7793.png", "formula": "\\begin{align*} r ^ 3 + a r + i \\theta = 0 \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{align*} \\partial _ t ^ w f ( t ) = \\frac { d } { d t } \\int _ 0 ^ t w ( t - s ) \\left ( f ( s ) - f ( 0 ) \\right ) d s , \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{gather*} F _ 0 ( t ) = \\left ( \\frac { C } { k } - \\frac { B } { k ^ 2 } + \\frac { A } { k ^ 3 } \\right ) e ^ { - 2 \\frac { A } { k } t } , F _ 1 ( t ) = \\left ( \\frac { B } { k } - 2 \\frac { A } { k ^ 2 } \\right ) e ^ { - \\frac { A } { k } t } . \\end{gather*}"}
-{"id": "2538.png", "formula": "\\begin{align*} f ^ { \\prime \\prime } = \\frac { R } { 2 B _ { 2 } q _ { 3 } } f + \\frac { B _ { 2 } ^ { \\prime } q _ { 3 } - B _ { 2 } \\left ( q _ { 2 } + q _ { 3 } ^ { \\prime } \\right ) } { 2 B _ { 2 } q _ { 3 } } f ^ { \\prime } . \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + a _ 2 x ^ 2 + a _ 4 x + a _ 6 . \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{align*} x _ { n } = C _ { n } , \\thinspace \\thinspace n \\ge 0 . \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} \\mu ( [ u ] ) = c \\int _ { D ^ 2 } u ^ * \\omega \\end{align*}"}
-{"id": "9318.png", "formula": "\\begin{align*} & C _ { 6 0 , i } \\ ( i = 1 , 2 , \\ldots , 1 3 ) , D _ { 6 0 , i } \\ ( i = 1 , 2 , \\ldots , 7 ) , E _ { 6 0 , i } \\ ( i = 1 , 2 , 3 , 4 ) , F _ { 6 0 } , \\\\ & H _ { 6 0 , i } \\ ( i = 1 , 3 , 4 ) , J _ { 6 0 , i } \\ ( i = 1 , 2 , 3 , 4 , 5 ) , K _ { 6 0 , i } \\ ( i = 1 , 2 ) , L _ { 6 0 , i } \\ ( i = 1 , 2 ) . \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} f ( \\sigma ) = [ \\sigma - 1 ] ( \\widetilde Q + R ) , \\end{align*}"}
-{"id": "1521.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n - 1 } \\alpha _ k S _ k ( x ) ^ 2 = \\sum _ { k = 1 } ^ { n - 1 } \\beta _ k \\tilde { S } _ k ( x ) ^ 2 . \\end{align*}"}
-{"id": "8828.png", "formula": "\\begin{align*} \\binom { m - z } { u _ 1 , u _ 2 , \\ldots , u _ z } \\binom { n - m } { m - z } = \\frac { 1 } { m ! } \\frac { \\prod _ { j = 0 } ^ { z + \\rho - 1 } ( m - j ) \\prod _ { j = 0 } ^ { m - z - 1 } ( n - m - j ) } { \\prod _ { j = 1 } ^ z u _ j ! } . \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{align*} T _ { F , G , N } ^ { \\Xi , L , \\widetilde { \\mathbf { r } } } ( f _ 1 , \\dots , f _ d ) = \\frac { 1 } { C _ { \\Xi , \\chi } \\eta ^ { h } } \\widetilde { T } _ { F , G , N } ^ { \\Xi , L , \\widetilde { \\mathbf { r } } } ( f _ 1 \\ast \\chi , \\dots , f _ d \\ast \\chi ) + O _ { C , c , \\varepsilon } ( \\eta / \\sigma _ G ) + O _ { C , c , \\varepsilon } ( \\eta / \\sigma _ F N ) . \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} n ^ { 2 \\theta } \\mathbb { P } ( H _ n ^ c \\cap \\{ t ( f _ i ) < n ^ { \\theta } \\} ) \\leq n ^ { 2 \\theta } \\mathbb { P } ( H _ n ^ c ) \\leq n ^ { 2 \\theta } \\frac { C } { n ^ { 1 + 2 \\epsilon } } = \\frac { C } { n ^ { 2 \\epsilon - 2 \\theta + 1 } } \\end{align*}"}
-{"id": "108.png", "formula": "\\begin{align*} m ^ k = \\frac { | G | } { | H | } = \\frac { 2 ( n ! / 2 ) ^ 2 } { 4 } = \\frac { ( n ! ) ^ 2 } { 8 } \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{align*} \\langle f \\mid g \\rangle = \\int _ X \\overline { f ( x ) } { g ( x ) } d \\mu ( x ) , \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} \\alpha ( c - c _ * ) b + \\beta b ^ 3 = 0 \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} \\dim ( A + B ) + \\dim ( A \\cap B ) = \\dim A + \\dim B . \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} \\breve { y } ( t ) = \\gamma \\breve { x } ( t - \\tau ) + \\breve { \\eta } ( t ) , \\end{align*}"}
-{"id": "740.png", "formula": "\\begin{align*} C ( \\xi ) = \\frac { 1 } { 2 \\pi } \\arg \\left ( \\hat { \\mu } ( \\xi ) \\right ) \\in \\textstyle { \\left [ - \\frac { 1 } { 2 } , \\frac { 1 } { 2 } \\right ) } \\end{align*}"}
-{"id": "1442.png", "formula": "\\begin{align*} r \\partial _ r H ^ { ( \\alpha , a ) } + t \\nabla V ^ { ( \\alpha , a ) } = S H ^ { ( \\alpha , a ) } - t f ^ 2 _ { \\alpha a } . \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} A : = \\{ ( g , h ) \\in \\mathcal P \\times \\mathcal P \\mid g \\leq h \\} \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{align*} I ( \\mu ) = H ( \\eta \\ , | \\ , \\mu ) \\end{align*}"}
-{"id": "7141.png", "formula": "\\begin{align*} \\alpha _ 1 ^ \\vee = H _ 2 - H _ 1 , \\alpha _ 2 ^ \\vee = H _ 1 . \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} \\Theta _ u ( x , \\rho ) : = \\sup _ { y \\in \\bar { B } _ { \\rho } ( x ) \\cap \\Gamma ( u ) } I _ u ( y , \\rho ) . \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} \\mathbf { F } : = \\ker \\boldsymbol { \\varepsilon } , \\end{align*}"}
-{"id": "6815.png", "formula": "\\begin{align*} \\frac { \\partial u _ { m } } { \\partial t } = i \\left [ \\Delta u _ { m } + \\sum \\limits _ { j = 1 } ^ { N } a _ { m j } u _ { j } + \\sum \\limits _ { j = 1 } ^ { N } b _ { m j } u _ { j } \\right ] , x \\in R ^ { n } , t \\in \\left ( 0 , T \\right ) , \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} Q _ N ( a ) : = { \\mathbb P } \\left ( { \\rm P o i s } \\left ( N ^ { 1 - \\alpha } \\sum _ { i = 1 } ^ { N ^ \\alpha } X _ i \\ , \\omega _ i ( N ^ { \\alpha } ) \\right ) \\geqslant N a \\right ) ; \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} d X _ t = A X _ t d t + G ( X _ t ) d t + \\sigma ( X _ t ) d W _ t , \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{align*} Y \\tilde { u } = \\left ( - D _ { t } + A _ { x } ^ { \\alpha } \\right ) \\tilde { u } = \\left ( - D _ { t } \\tilde { E } + A _ { x } ^ { \\alpha } \\tilde { E } \\right ) \\ast _ x u _ 0 = 0 , \\tilde { u } ( 0 ) = u _ { 0 } . \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{align*} d _ t ( t ) = \\pi \\rho ^ 2 \\left ( a _ h \\phi _ t \\alpha - \\frac { g _ 3 } { 2 } \\phi _ t ^ 2 \\right ) , \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{align*} 2 M ^ \\dag J ( k ) M = & - { i k } M ^ \\dag ( U + I _ n ) M + i M ^ \\dag ( U - I _ n ) M \\\\ & + M ^ \\dag Q ( 0 ) M M ^ \\dag ( U + I _ n ) M + o ( 1 ) \\\\ = & - i k { \\rm d i a g } \\{ I _ { n - n _ D } , 0 _ { n _ D } \\} T ^ { - 1 } + i { \\rm d i a g } \\{ ( X _ 1 - I _ { n - n _ D } ) , - 2 I _ { n _ D } \\} \\\\ & + Q ' { \\rm d i a g } \\{ I _ { n - n _ D } , 0 _ { n _ D } \\} T ^ { - 1 } + o ( 1 ) , \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} \\mu _ n = \\int _ 0 ^ 1 F ( x ) x ^ n d x = \\frac { 1 } { n + 1 } - d _ n \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} \\sup \\limits _ { \\lambda , \\varepsilon } R \\left \\{ \\xi ^ { i } \\frac { d } { d \\xi ^ { i } } \\Psi \\left ( \\lambda , \\varepsilon , \\xi \\right ) \\xi \\in R \\backslash \\left \\{ 0 \\right \\} \\right \\} \\leq M _ { 1 } i = 0 1 . \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} \\overline { \\Lambda } = \\Lambda _ { 2 } ( \\infty , \\Omega ) . \\end{align*}"}
-{"id": "9293.png", "formula": "\\begin{align*} \\rho _ g \\Phi ( c ) = \\rho _ g ( c _ 1 ) \\overset { \\delta _ { g , 1 } } { \\to } c _ g = \\Phi \\rho _ g ( c ) \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} A _ 1 = - \\sum _ { i , j , \\ell } V ^ i _ j P ^ j _ \\ell \\otimes P ^ \\ell _ i + 1 \\otimes \\sum _ { i , j } V ^ i _ j P ^ j _ i . \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{align*} L _ { k } u = \\sum \\limits _ { i = 0 } ^ { m _ { k } } \\left [ \\delta _ { k i } u _ { x _ { k } } ^ { \\left [ i \\right ] } \\left ( a _ { k } , x \\left ( k \\right ) \\right ) + \\sum \\limits _ { j = 0 } ^ { N _ { k } } \\nu _ { k i j } u _ { x _ { k } } ^ { \\left [ i \\right ] } \\left ( x _ { k i j } , x \\left ( k \\right ) \\right ) \\right ] = 0 \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} D ^ g _ \\alpha ( z _ { \\bar \\alpha } ) = D ^ g _ { \\bar \\alpha } ( z _ \\alpha ) & = - \\frac 1 4 \\theta ^ \\beta z _ \\beta - \\frac 1 4 \\theta ^ { \\bar \\beta } z _ { \\bar \\beta } . \\end{align*}"}
-{"id": "184.png", "formula": "\\begin{align*} & q ^ { 2 d k - d } \\left | \\sum _ { { \\bf m } \\in S _ 0 } \\left ( \\prod _ { j = 1 } ^ k \\widehat { E _ j } ( { \\bf m } ) \\right ) \\right | ^ 2 - \\nu ^ 2 _ k ( 0 ) \\\\ & \\le 2 \\left ( q ^ { \\frac { d } { 2 } - 1 } \\left ( \\prod _ { j = 1 } ^ k | E _ j | \\right ) ^ { \\frac { 2 k - 1 } { k } } + q ^ { d - 1 } \\left ( \\prod _ { j = 1 } ^ k | E _ j | \\right ) ^ { \\frac { 2 k - 2 } { k } } \\right ) . \\end{align*}"}
-{"id": "6937.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta w = \\div { \\mathbf g } & \\Omega \\\\ w = 0 & \\partial \\Omega \\end{cases} \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} V _ i = V _ i ( \\gamma ) : = \\{ ( 1 - q ( C ) - \\gamma ) n \\leq \\# { \\cal E } _ i \\leq ( 1 - q ( C ) + \\gamma ) n \\} \\end{align*}"}
-{"id": "4632.png", "formula": "\\begin{align*} { \\displaystyle \\mathbf { D } = \\eta \\mathbf { E } , \\mathbf { B } = \\mu \\mathbf { H } , } \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} 2 p _ a ( E ) - 2 = ( K _ X + E ) \\cdot E \\end{align*}"}
-{"id": "9897.png", "formula": "\\begin{align*} h _ { \\mu _ L } + \\int q \\varphi ^ u d \\mu _ L = h _ { \\mu _ L } - q \\lambda ^ + ( \\mu _ L ) > h _ { \\mu _ L } - \\lambda ^ + ( \\mu _ L ) = 0 . \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} \\begin{aligned} \\dot x { = } \\sqrt { \\tfrac { 1 { - } e ^ { { - } J _ 1 ( x ) } } { 1 { + } e ^ { J _ 1 ( x ) } } } & \\big ( \\sin ( e ^ { J _ 1 ( x ) } { + } 2 \\ln ( e ^ { J _ 1 ( x ) } { - } 1 ) ) u _ { 1 } ^ \\varepsilon ( t ) \\\\ { + } & \\cos ( e ^ { J _ 1 ( x ) } { + } 2 \\ln ( e ^ { J _ 1 ( x ) } { - } 1 ) ) u _ { 2 } ^ \\varepsilon ( t ) \\big ) , \\end{aligned} \\end{align*}"}
-{"id": "4478.png", "formula": "\\begin{align*} \\left \\{ \\mathbb { J } _ { \\mathrm { t h } } ^ { ( \\omega , l ) } \\right \\} _ { k } : = \\left \\vert \\Lambda _ { l } \\right \\vert ^ { - 1 } \\underset { x \\in \\Lambda _ { l } } { \\sum } \\varrho ^ { ( \\beta , \\omega , \\vartheta , \\lambda ) } \\left ( I _ { \\left ( x + e _ { k } , x \\right ) } ^ { ( \\omega , \\vartheta ) } \\right ) \\ . \\end{align*}"}
-{"id": "9953.png", "formula": "\\begin{align*} \\dot { V } = 2 x ^ T P \\dot { x } & = 2 ( A ^ { - 1 } E \\dot { x } ) ^ T P \\dot { x } = \\dot { x } ^ T ( P A ^ { - 1 } E + E ^ T A ^ { - T } P ) \\dot { x } \\\\ & = - \\dot { x } ^ T Q \\dot { x } \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} \\rho _ i ^ 2 = 1 , \\textrm { f o r $ 0 \\leq i \\leq n - 1 $ } , \\end{align*}"}
-{"id": "7189.png", "formula": "\\begin{align*} \\Q ( w ) = \\Q ( w + w ^ { - 1 } ) = \\Q ( z ) . \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} \\Sigma ^ t = \\begin{pmatrix} \\sigma ^ { k _ 1 } ( \\alpha ) & \\sigma ^ { k _ 2 } ( \\alpha ) & \\cdots & \\sigma ^ { k _ \\nu } ( \\alpha ) \\\\ \\sigma ^ { k _ 1 + 1 } ( \\alpha ) & \\sigma ^ { k _ 2 + 1 } ( \\alpha ) & \\cdots & \\sigma ^ { k _ \\nu + 1 } ( \\alpha ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\sigma ^ { k _ 1 + t } ( \\alpha ) & \\sigma ^ { k _ 2 + t } ( \\alpha ) & \\cdots & \\sigma ^ { k _ \\nu + t } ( \\alpha ) \\\\ \\end{pmatrix} _ { ( t + 1 ) \\times \\nu } . \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{align*} \\widehat { h } ( - k ) S ( k ) = - \\widehat { h } ( k ) , k \\in \\mathbb { R } , \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} \\psi ( x _ 1 , \\dots , x _ { n - 1 } , l ) = 0 \\end{align*}"}
-{"id": "8710.png", "formula": "\\begin{align*} D V ( x ) = - 2 | x | ^ { - 4 } x , D ^ 2 V ( x ) = - 2 | x | ^ { - 4 } I + 4 | x | ^ { - 6 } x x ^ { T } . \\end{align*}"}
-{"id": "9981.png", "formula": "\\begin{align*} x ( t ) = T _ { \\sigma ( t ) } z ( t ) \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{align*} \\| ( T - A ) e _ { k } \\| < \\varepsilon \\quad { \\rm a n d } \\| ( T - A ) ^ { * } e _ { k } \\| < \\varepsilon \\qquad \\hbox { f o r $ k = 1 , \\dots , r $ } , \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} { \\rm f o r } ~ n = 0 : y ^ 2 J _ { 1 } + \\lambda y J _ 0 = 1 . \\end{align*}"}
-{"id": "6571.png", "formula": "\\begin{align*} \\mathcal { O } = \\{ \\varphi _ { k } \\ , \\mid \\ , k \\in N \\} , \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} \\bigl ( ( \\pi _ J ) _ * \\mathcal { S } _ w ^ \\vee : \\Delta _ { v , J } ^ \\vee \\langle n \\rangle \\bigr ) = 0 \\end{align*}"}
-{"id": "8363.png", "formula": "\\begin{align*} \\hat { T } ^ { ( k ) } ( \\pi ) = \\hat { T } ^ { ( k ) } ( \\pi , \\omega ) = \\sum _ { i = 1 } ^ { r } t ^ { ( k ) } ( e _ i , \\omega ) . \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} & \\nu _ { 2 } ( b _ { 2 } ( 4 n + 3 ) ) = 3 , \\\\ & \\nu _ { 2 } ( b _ { 2 } ( 8 n + 5 ) ) = 3 , \\\\ & \\nu _ { 2 } ( b _ { 2 } ( 1 6 n + i ) ) = 3 \\ ; f o r \\ ; i \\in \\{ 6 , 9 , 1 2 \\} , \\\\ & \\nu _ { 2 } ( b _ { 2 } ( 3 2 n + i ) ) = 3 \\ ; f o r \\ ; i \\in \\{ 8 , 1 7 , 2 6 \\} , \\\\ & \\nu _ { 2 } ( b _ { 2 } ( 6 4 n + i ) ) = 3 \\ ; f o r \\ ; i \\in \\{ 1 6 , 3 3 , 5 0 \\} , \\\\ & \\nu _ { 2 } ( b _ { 2 } ( 1 2 8 n + i ) ) = 3 \\ ; f o r \\ ; i \\in \\{ 3 2 , 6 5 , 9 8 \\} , \\\\ & \\nu _ { 2 } ( b _ { 2 } ( 2 5 6 n + i ) ) = 3 \\ ; f o r \\ ; i \\in \\{ 6 4 , 1 2 9 , 1 9 4 \\} , \\\\ & \\\\ \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} S : = \\{ f _ 3 ( x ) = 0 \\} \\ \\ \\ \\mathrm { a n d } \\ \\ \\ S ' : = \\{ f _ 1 ( x ) = 0 \\} \\end{align*}"}
-{"id": "9452.png", "formula": "\\begin{align*} \\varepsilon \\Big [ \\partial _ { t } ( h d ) + \\nabla \\cdot [ h ( d \\otimes v - v \\otimes d ) ] + ( - \\triangle ) ^ l d \\Big ] + h d = \\nabla \\times b , \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} \\lambda [ T _ { k + 1 } , T ] & = \\sum _ { \\ell = 1 } ^ { q } \\gamma _ { k + 1 } ^ { ( \\ell ) } [ T _ { k + 1 } ^ { \\ell - 1 } , T _ { k + 1 } ^ { \\ell } ] + \\gamma _ { k + 1 } ^ { ( q + 1 ) } [ T _ { k + 1 } ^ { q } , T ] \\\\ & \\leq 2 ^ { k + 1 } q + 2 ^ { k + 1 } \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{gather*} f _ m ( 1 ) = 1 , f _ m ( 2 ) = m + a , \\\\ f _ m ( n + 2 ) = ( m + a - 1 ) f _ m ( n + 1 ) + m f _ m ( n ) , ( n \\geq 1 ) . \\end{gather*}"}
-{"id": "8072.png", "formula": "\\begin{align*} A ( W ) : = \\{ w \\in W : W _ w \\subset A \\} \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} \\alpha _ n : = \\frac { n ^ 2 \\pi ^ 2 } { \\sigma _ 1 ^ 2 } , \\quad \\beta _ n : = \\frac { \\gamma _ n ^ 2 } { \\sigma _ 2 ^ 2 } , n \\in \\mathbb { N } _ 0 , \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} x _ i = v _ i \\log t + b _ i \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} F _ { 2 m k + m } + F _ { 2 m k - m } = F _ m L _ { 2 m k } \\ , , \\quad \\mbox { $ m $ o d d } \\ , . \\end{align*}"}
-{"id": "3760.png", "formula": "\\begin{align*} \\partial ( \\lambda ) = \\prod _ { n \\in \\N } \\log _ { n } ( \\lambda ) . \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} \\Big ( \\frac { d } { d T } \\Big ) ^ k \\big ( a _ 1 ( T ) { \\mathrm { e } } ^ { \\omega _ 1 T } + \\cdots + a _ n ( T ) { \\mathrm { e } } ^ { \\omega _ n T } \\big ) = a _ { k , 1 } ( T ) { \\mathrm { e } } ^ { \\omega _ 1 T } + \\cdots + a _ { k , n } ( T ) { \\mathrm { e } } ^ { \\omega _ n T } \\end{align*}"}
-{"id": "9093.png", "formula": "\\begin{align*} \\alpha _ { X } : = \\max _ X \\mathrm { p r } ( 1 - e ^ { f _ { \\omega } } ) . \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ { \\infty } ( - 1 ) ^ { n } Q _ { n , j , k } ( p , z ) t ^ { p } & = \\frac { ( - 1 ) ^ { n } A ( t , z ) } { ( 1 - t ) ^ { n + 1 } } \\\\ & = \\frac { ( - 1 ) ^ { n } t ^ { n + 1 } A ( 1 / t , - z ) } { ( 1 - t ) ^ { n + 1 } } \\\\ & = \\frac { \\bar { A } ( t , - z ) } { ( 1 - t ) ^ { n + 1 } } \\\\ & = \\sum _ { p = 1 } ^ { \\infty } Q _ { n , j , k } ( - p , - z ) t ^ { p } . \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} \\begin{cases} \\alpha + i r \\beta = r z \\\\ \\alpha - i r \\beta = r w \\end{cases} \\end{align*}"}
-{"id": "9319.png", "formula": "\\begin{align*} & C _ { 5 8 , 1 } \\cong C _ { 5 8 , i } \\ ( i = 2 , 4 , 5 , 7 , 8 , 1 1 , 1 2 , 1 4 , 1 5 , 1 7 , 1 8 ) , \\\\ & C _ { 5 8 , 3 } \\cong C _ { 5 8 , i } \\ ( i = 6 , 9 , 1 0 , 1 3 , 1 6 ) , \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{align*} f _ 3 ( a _ i , x ) : = a _ i x ^ 4 + 2 a _ i ^ 2 x ^ 3 - 2 x - a _ i \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{align*} & \\phi _ g ^ { \\beta , \\sigma } ( x ) \\to \\phi _ g ^ { \\infty } ( x ) = \\begin{cases} \\sin ( \\frac { \\pi { x } } { 2 ( L - 1 ) } ) , & 0 \\le { x } < L - 1 , \\\\ 1 , & L - 1 \\le { x } \\le 1 , \\\\ \\sin ( \\frac { \\pi ( L + x - 2 ) } { 2 ( L - 1 ) } ) , & 1 < x \\le { L } , \\end{cases} \\\\ & \\mu _ g ( \\beta , \\sigma ) \\to \\frac { \\pi ^ 2 } { 8 ( L - 1 ) ^ 2 } , { E } _ g ( \\beta , \\sigma ) \\to \\frac { \\pi ^ 2 } { 8 ( L - 1 ) } . \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { s - 1 } \\beta ^ t \\alpha _ i ^ { u + p _ i - 1 } = \\Big ( \\sum _ { t = 0 } ^ { s - 1 } \\beta ^ t \\alpha _ i ^ { p _ i - 1 } \\Big ) \\alpha _ i ^ u \\in S _ i ^ { ( 2 ) } \\alpha _ i ^ u \\subseteq K . \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} L _ { m k } { } ^ 4 = L _ { 4 m k } - ( - 1 ) ^ { m k - 1 } 4 L _ { 2 m k } + 6 \\ , , \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} P _ { \\perp } \\begin{pmatrix} N \\bar { z } \\\\ [ . 1 c m ] \\overline { M } \\bar { z } \\end{pmatrix} = 0 . \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} M _ t = \\begin{cases} 0 , & \\ ; \\ ; 0 \\leq t \\leq 1 ; \\\\ M _ { n } + M ^ n _ { t - n } \\phi _ { n } ( \\sigma _ 1 , \\ldots , \\sigma _ { n - 1 } ) , & \\ ; \\ ; t \\in ( n , n + 1 ] , n \\in \\{ 1 \\ldots , N \\} , \\\\ M _ { N + 1 } , & \\ ; \\ ; t > N + 1 , \\end{cases} \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} E _ ( u ) = \\int _ { \\Omega } | \\nabla u | _ 2 + \\lambda \\| u - f \\| _ { L ^ 2 ( \\Omega ) } ^ 2 , \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} \\prod _ { i = 2 } ^ n \\psi _ i ^ { d _ i } & \\left [ D _ { 1 , n + 1 , n + 2 } \\pi ^ * \\psi _ 1 ^ { d _ 1 } ( \\psi _ 0 + \\psi _ 1 + \\psi _ { n + 1 } + \\psi _ { n + 2 } ) \\right . \\\\ & \\left . + D _ { 1 , n + 1 , n + 2 } ( D _ { 1 , n + 1 } + D _ { 1 , n + 2 } + D _ { n + 1 , n + 2 } ) \\pi ^ * \\psi _ 1 ^ { d _ 1 } \\right ] . \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{align*} \\frac { \\partial W } { \\partial r } ( z ) \\geq 0 | z | = r > R _ 0 , \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} & [ Q _ j , Q _ k ] \\bigg | _ { - 1 } ^ 1 \\\\ & = \\left \\{ \\begin{array} { l l } \\dfrac { 2 [ \\psi ( j + 1 ) - \\psi ( k + 1 ) ] [ j ^ 3 ( j + 1 ) ^ 3 - k ^ 3 ( k + 1 ) ^ 3 ] } { ( k - j ) ( j + k + 1 ) } = : \\Phi _ { j k } , & j + k j \\neq k , \\\\ 0 , & j + k j = k . \\end{array} \\right . \\end{align*}"}
-{"id": "7232.png", "formula": "\\begin{align*} \\left | \\left ( ( L ^ * _ J ) ^ { - 1 } \\right ) _ { i , j } \\right | = \\left | \\frac { \\det L ^ * _ { J \\setminus \\{ i \\} , J \\setminus \\{ j \\} } } { \\det L ^ * _ J } \\right | , \\end{align*}"}
-{"id": "10026.png", "formula": "\\begin{align*} N _ { J } ( X , Y ) = J ^ 2 [ X , Y ] + [ J X , J Y ] - J [ J X , Y ] - J [ X , J Y ] , \\forall X , Y \\in { \\mathfrak X } ( M ) . \\end{align*}"}
-{"id": "4544.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle | \\sum _ { k = 1 } ^ { M } J _ 3 ^ { ( k ) , 2 } | \\leq C \\big ( h ^ { 2 r } + ( \\Delta t ) ^ { 4 } \\big ) + C \\| \\theta _ { \\Psi } ^ { M } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } + C \\Delta t \\sum _ { k = 1 } ^ { M - 1 } { \\| \\nabla \\theta _ { \\Psi } ^ { k } \\| _ { \\mathbf { L } ^ 2 } ^ { 2 } } . } \\end{array} \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} f ' ( r ) & = - \\frac r 2 + ( n - 1 ) \\cdot \\frac 1 { \\sqrt n } \\tan \\frac r { \\sqrt n } \\\\ & = - \\frac { \\sqrt n } 2 \\cdot \\frac r { \\sqrt n } + \\left ( \\sqrt n - \\frac 1 { \\sqrt n } \\right ) \\tan \\frac r { \\sqrt n } \\\\ & \\geq \\frac { \\sqrt n } 2 \\left ( \\tan \\frac r { \\sqrt n } - \\frac r { \\sqrt n } \\right ) \\geq 0 , \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} b _ 1 ( 2 n ) ^ 2 - b _ 1 ( 2 n - 1 ) b _ 1 ( 2 n + 1 ) & = b _ 1 ( 2 n ) b _ 1 ( n ) , \\\\ b _ 1 ( 2 n - 1 ) ^ 2 - b _ 1 ( 2 n - 2 ) b _ 1 ( 2 n ) & = - b _ 1 ( 2 n - 2 ) b _ 1 ( n ) , \\\\ b _ 2 ( 2 n ) ^ 2 - b _ 2 ( 2 n - 1 ) b _ 2 ( 2 n + 1 ) & = \\left ( \\sum _ { j = 0 } ^ n b _ 2 ( j ) \\right ) ^ 2 , \\\\ b _ 2 ( 2 n - 1 ) ^ 2 - b _ 2 ( 2 n - 2 ) b _ 2 ( 2 n ) & = \\left ( \\sum _ { j = 0 } ^ n b _ 2 ( j ) \\right ) ^ 2 - b _ 2 ( 2 n - 2 ) b _ 2 ( n ) . \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} Y = Z \\cup ( \\cup \\{ X _ \\delta : \\ \\delta \\in S , \\delta < \\alpha \\} ) . \\end{align*}"}
-{"id": "7345.png", "formula": "\\begin{align*} d f ^ i = 0 , 1 \\leq i \\leq 6 , d f ^ 7 = \\frac { \\sqrt { 6 } } { 6 } \\ , y ( t ) ^ { - 5 } ( f ^ { 1 2 } + f ^ { 3 4 } + f ^ { 5 6 } ) . \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} 2 U ^ { \\mu _ { \\alpha , 0 } - \\frac { \\varepsilon } { 2 } \\nu } ( x ) & = 2 U ^ { \\mu _ { \\alpha , 0 } } ( x ) - \\varepsilon U ^ { \\nu } ( x ) \\\\ & = - V \\left ( \\frac { x } { \\alpha } \\right ) + \\ell - \\varepsilon x , x \\in [ - 1 , 1 ] . \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} \\Phi ^ * \\omega _ T \\leq C \\Phi ^ * \\widetilde \\omega _ T = \\frac C { r ^ { 2 - \\alpha k } } \\omega _ { \\mathrm { e u c l } } , \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} K ( s , t ) = \\begin{cases} w ( s ) - \\left ( w ( a ) + \\psi ( \\lambda ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ a , t ) , \\\\ w ( s ) - \\left ( w ( a ) + ( 1 + \\psi ( 1 - \\lambda ) ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ t , b ] , \\end{cases} \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} \\sum _ { j \\in J } \\sum _ { i \\in I _ { j } } a _ { i } = \\sum _ { i \\in I } a _ { i } . \\end{align*}"}
-{"id": "763.png", "formula": "\\begin{align*} & \\quad \\ , \\sum _ { n = U } ^ { U + V } e \\left ( \\overline { \\xi } _ { x + n w } \\right ) = \\sum _ { n = U } ^ { U + V } e \\left ( \\xi _ { x + n w } ^ e + \\xi _ { x + n w } ^ i \\right ) \\\\ & = \\sum _ { n = U } ^ { U + V } e \\left ( \\xi _ { x } ^ e + \\xi _ { x + U w } ^ i + n \\left ( \\xi _ y ^ e - \\xi _ x ^ e \\right ) + O \\left ( \\frac { 1 } { R _ 1 } \\right ) + O \\left ( \\frac { V } { U } \\right ) \\right ) . \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} \\hat { T } ^ { ( k ) } _ n = \\hat { T } ^ { ( k ) } _ n ( \\omega ) = \\inf _ { \\pi \\in { \\cal P } _ n } \\hat { T } ^ { ( k ) } ( \\pi , \\omega ) \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} g _ { t , \\ell , - 1 , m } & = \\sum _ { j = 0 } ^ { \\ell + 1 } H ( m - 2 \\ell - t j + ( t - 1 ) ) g _ { t + 1 , \\ell + 1 , j , m } \\\\ & = \\sum _ { j = 0 } ^ { \\ell + 1 } H ( m - 2 ( \\ell + 1 ) - t j ) g _ { t + 1 , \\ell + 1 , j , m } \\\\ & = g _ { t , \\ell + 1 , 0 , m } . \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} h ^ { - 1 } \\frac { d h } { d t } = [ h ^ { - 1 } \\phi ^ * h , \\phi ] - \\rho \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } y ( t ) = \\frac { 2 A \\sqrt { 6 } \\ , y ( t ) z ( t ) ^ 6 + 2 z ( t ) ^ 2 + y ( t ) ^ 2 } { 1 2 \\ , y ( t ) ^ 3 z ( t ) ^ 8 } , & \\frac { d } { d t } z ( t ) = - \\frac { 2 A \\sqrt { 6 } \\ , y ( t ) z ( t ) ^ 4 + 1 } { 1 2 \\ , y ( t ) ^ 2 z ( t ) ^ 7 } , \\\\ y ( 0 ) = 1 , & z ( 0 ) = 1 . \\end{cases} \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{align*} 1 - | \\hat { \\mu } ( \\overline { \\xi } _ i ) | = O \\left ( \\frac { 1 } { R ^ 2 m ^ 2 } \\right ) + \\frac { 1 } { m ^ 2 } \\sum _ { \\| y \\| _ 1 \\leq R ' } \\Big ( 1 - c \\left ( \\overline { \\xi } _ i ( x _ i + y ) \\right ) \\Big ) . \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} \\left \\{ \\ \\begin{aligned} & w _ \\ell ( t ) = a _ \\ell \\ , \\Im ( \\zeta ^ \\ell \\log \\zeta ) + b _ \\ell \\ , \\Re \\zeta ^ \\ell \\\\ & \\mbox { w i t h } a _ \\ell = \\frac { g ^ \\omega _ \\ell - g ^ 0 _ \\ell \\ , \\cos \\ell \\omega } { \\omega \\cos \\ell \\omega } \\quad \\mbox { a n d } b _ \\ell = g ^ 0 _ \\ell \\ , . \\end{aligned} \\right . \\end{align*}"}
-{"id": "601.png", "formula": "\\begin{align*} h _ { \\lambda , C } ( x ) : = \\bigl [ 1 - \\mbox { $ \\frac { 1 } { \\lambda } $ } d ( x , C ) \\bigr ] ^ + . \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{align*} \\eta _ { X , Y } ( a _ 0 \\otimes a _ 1 \\otimes a _ 2 ) : = \\varepsilon ( a _ 0 ) \\varepsilon ( X \\triangleright a _ 1 ) \\varepsilon ( Y \\triangleright a _ 2 ) . \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{align*} E _ { \\alpha , \\beta } ( A ) = Z \\ ; \\mathrm { d i a g } \\{ E _ { \\alpha , \\beta } ( J _ 1 ) , E _ { \\alpha , \\beta } ( J _ 2 ) , \\ldots , E _ { \\alpha , \\beta } ( J _ s ) \\} Z ^ { - 1 } , \\end{align*}"}
-{"id": "9631.png", "formula": "\\begin{align*} \\psi ^ { ( j ) } _ n ( b ) = \\frac { \\psi ^ { ( j + 1 ) } _ { n - 1 } ( b ) - \\psi ^ { ( j ) } _ { n - 1 } ( b ) } { D ^ { ( j ) } _ n } , \\ \\ D ^ { ( j ) } _ n = \\frac { G ^ { ( j ) } _ { n + 1 } G ^ { ( j + 1 ) } _ { n - 1 } } { G ^ { ( j ) } _ { n } G ^ { ( j + 1 ) } _ { n } } . \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\big ( A _ 0 x ^ 3 + A _ 1 x ^ 2 + A _ 2 x + A _ 3 \\big ) Y , \\\\ \\frac { \\partial Y } { \\partial t _ 1 } = \\big ( A _ 0 x ^ 2 + A _ 1 x + B _ { 1 0 } \\big ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = ( - A _ 0 x + B _ { 2 0 } ) Y , \\end{gather*}"}
-{"id": "8561.png", "formula": "\\begin{align*} \\phi ( z ) = \\Re \\left ( \\sum _ { j = 1 } ^ { \\theta _ 0 } \\frac { \\alpha _ j ^ 1 } { z ^ j } + \\beta _ 1 \\log ( z ) + O ( 1 ) , \\cdots , \\sum _ { j = 1 } ^ { \\theta _ 0 - 1 } \\frac { \\alpha _ j ^ n } { z ^ j } + \\beta _ n \\log ( z ) + O ( 1 ) \\right ) \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} R = \\begin{bmatrix} D & \\pi ^ n D \\\\ D & D \\end{bmatrix} \\subset M _ 2 ( D ) \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{align*} 4 t v '' + \\Bigl ( \\frac { 4 t } { \\alpha } - \\frac { 8 } { \\alpha } - t e ^ { - t } \\Bigr ) v ' - \\Bigr ( \\frac { 4 } { \\alpha ^ 2 } - \\frac { 4 ( \\alpha + 1 ) } { \\alpha ^ 2 t } - \\frac { e ^ { - t } } { \\alpha } \\Bigl ) v + | v | ^ \\alpha v = 0 . \\end{align*}"}
-{"id": "9804.png", "formula": "\\begin{align*} { \\mathbb { I } ^ N } \\left ( u \\right ) = \\sum \\limits _ { i , j , k = 0 } ^ N { { u _ { i j k } } { \\ell _ i } ( \\xi ) { \\ell _ j } ( \\eta ) { \\ell _ k } ( \\zeta ) } . \\end{align*}"}
-{"id": "9799.png", "formula": "\\begin{align*} \\xi : = \\Delta _ H ^ { - 1 } ( \\omega - H ( \\omega ) ) . \\end{align*}"}
-{"id": "9986.png", "formula": "\\begin{align*} r a n k ( \\tilde { E } _ { i _ j } - \\tilde { E } _ j ) = r a n k ( \\tilde { E } _ { i _ j } ) - r a n k ( \\tilde { E } _ j ) \\ , . \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{align*} f ( q ) : = \\sum _ { n = 0 } ^ \\infty \\frac { q ^ { n ^ 2 } } { ( - q ) ^ 2 _ n } \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{align*} M _ 0 J ( k ) T _ 0 = \\begin{bmatrix} k A _ 0 + o ( k ) & k B _ 0 + o ( k ) \\\\ k C _ 0 + o ( k ) & D _ 0 + o ( 1 ) \\end{bmatrix} , \\ ; \\ ; k \\to 0 , \\ ; \\ ; k \\in \\overline { \\mathbb { C } } ^ + , \\end{align*}"}
-{"id": "3112.png", "formula": "\\begin{align*} \\frac { d x _ j ( t ) } { d t } = - V _ 2 ( x _ { j - 1 } ( t ) , x _ j ( t ) ) - V _ 1 ( x _ j ( t ) , x _ { j + 1 } ( t ) ) . \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} f _ { n _ 1 , n _ 2 } ( \\eta _ 1 , \\eta _ 2 ) = \\sqrt { \\frac { ( k - n _ 1 - n _ 2 ) ! } { k ! n _ 1 ! n _ 2 ! } } { { \\eta _ 1 } } ^ { n _ 1 } { { \\eta _ 2 } } ^ { n _ 2 } . \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{align*} B _ { \\varphi _ 1 } ( \\mathbf w _ i ' , \\mathbf w _ j ' ) = B _ { \\varphi _ 2 } ( \\mathbf w _ i ' , \\mathbf w _ j ' ) = 0 . \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\frac { x _ { n + 1 } - x _ { n } } { h } = \\sum \\limits _ { i = 1 } ^ { s } b _ { i } \\ f \\left ( X _ { i } \\right ) , \\\\ X _ { i } = x _ { n } + \\sum \\limits _ { j = 1 } ^ { s } a _ { i j } \\ f \\left ( X _ { j } \\right ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1079.png", "formula": "\\begin{align*} L ^ * P ^ * \\pi _ { m - u } ^ * ( \\varphi ) - \\sum \\limits _ { j = 1 } ^ { d - u } \\lambda _ j \\Xi ( \\mathbf { w _ j } ) ^ * = \\omega _ V + \\omega _ W , \\end{align*}"}
-{"id": "412.png", "formula": "\\begin{align*} \\mathbb { B } \\otimes \\mathbf { k } ( \\mathbf { x } ) = H ^ 0 ( E _ y ( - b ) | _ S ) . \\end{align*}"}
-{"id": "5282.png", "formula": "\\begin{align*} \\int _ { \\partial B _ R ( 0 ) \\cap \\Omega } x \\times u \\ , d S & \\to - p \\times \\int _ { \\partial B _ 1 ( 0 ) \\cap \\{ x _ n < 0 \\} } x \\ , d S = \\frac { \\pi ^ { \\frac { n - 1 } { 2 } } } { \\Gamma ( \\frac { n + 1 } { 2 } ) } p \\times e _ n , \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} h = f ( x , y ) = f ( u , v ) . \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} \\sum _ { n \\leqslant x } \\tau ( n ) = O ( x \\log x ) . \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } \\frac { n - 1 } { m - n + 1 + i } \\geq \\sum _ { i = 0 } ^ { n - 1 } \\log \\left ( 1 + \\frac { n - 1 } { m - n + 1 + i } \\right ) \\geq \\log 2 . \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{align*} | \\gamma | ^ q & = | 2 \\gamma - 1 | ^ { q / ( n + 1 ) } = \\ ( 5 - 4 \\cos ( \\pi j / k ) + O ( 1 / k ) \\ ) ^ { q / ( 2 n + 2 ) } \\\\ & = 1 + \\frac { q \\log ( 5 - 4 \\cos ( \\pi j / k ) ) } { 2 n + 2 } + O \\ ( \\frac { 1 } { k n } \\ ) \\\\ & = 1 + \\frac { q \\log ( 5 - 4 \\cos ( \\pi j / k ) ) } { 4 t } + O \\ ( \\frac { 1 } { k n } \\ ) . \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\liminf _ n A _ n \\right ) = 1 \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} \\mathbb { E } [ \\epsilon _ g ] = \\begin{cases} \\frac { 1 } { q ^ { \\deg ( g ) } } \\deg ( g ) \\leq N \\\\ 0 . \\end{cases} \\end{align*}"}
-{"id": "10035.png", "formula": "\\begin{align*} ( D _ { \\phi } ) _ p = T _ p ^ + ( M ) , ( D _ { \\bar \\phi } ) _ p = T _ p ^ - ( M ) , \\forall p \\in M , \\end{align*}"}
-{"id": "9820.png", "formula": "\\begin{align*} \\theta _ { \\lambda } ( \\phi _ { w \\mathbf { z } } ) = ( w \\mathbf { z } ) ^ { \\lambda } \\phi _ { w \\mathbf { z } } . \\end{align*}"}
-{"id": "2383.png", "formula": "\\begin{align*} U ( t , x ) = \\left ( \\begin{array} { c } u _ 1 ( t , x ) \\\\ u _ 2 ( t , x ) \\\\ \\end{array} \\right ) , A = \\left ( \\begin{array} { c c } - \\alpha & \\beta \\\\ - \\beta & - \\alpha \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "9159.png", "formula": "\\begin{align*} \\tilde { a } _ { \\alpha \\beta } ^ { i j } ( t ) = a _ { \\alpha \\beta } ^ { i j } ( 0 , t ) , \\quad \\tilde { \\sigma } _ { \\alpha \\beta } ^ { i k } ( t ) = \\sigma _ { \\alpha \\beta } ^ { i k } ( 0 , t ) , \\end{align*}"}
-{"id": "9525.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ r \\langle C ( \\omega _ 0 ) ^ * X ( \\omega _ 0 ) C ( \\omega _ 0 ) \\xi _ i ( \\omega _ 0 ) , \\eta _ i ( \\omega _ 0 ) \\rangle \\right | < \\delta r K ^ 2 . \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{align*} \\widehat { g } ( k ) = \\widehat { h } ( - k ) [ S ( k ) - U _ 0 ] . \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} Z _ { - 1 } = 0 \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} { \\displaystyle \\mathbf { A } ( \\cdot , - \\Delta t ) = \\mathbf { A } ( \\cdot , 0 ) - \\Delta t \\frac { \\partial \\mathbf { A } } { \\partial t } ( \\cdot , 0 ) = \\mathbf { A } _ { 0 } - \\Delta t \\mathbf { A } _ { 1 } } , \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} \\underline { \\vphantom { p } \\textrm { d e n s } } \\ \\mathcal { N } _ { T } ( x , B ( 0 , \\varepsilon ) ^ c ) = 1 \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} \\lambda ' ( c _ * ) = \\frac { 1 2 8 } { 3 \\pi ^ 2 } \\sqrt { c _ * } . \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\mathcal Q _ { p } u = \\lambda | u | ^ { p - 2 } u & \\ \\Omega \\\\ u = 0 & \\ \\partial \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "911.png", "formula": "\\begin{align*} b ( s + t ) = \\sum [ a _ { i , j } ] \\circ b ( s ) ^ { \\circ i } \\circ b ( t ) ^ { \\circ j } . \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} Z ( k _ { i _ 1 } , \\dots , k _ { i _ n } ) = 0 , \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} D _ { n , k } \\Big ( 1 , \\frac { 1 } { 4 } \\Big ) \\ , = \\ , \\frac { k ( n - 1 ) + 2 } { 2 ^ n } ; \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} P _ S = \\alpha + \\frac { 1 } { \\kappa } \\mathbf { n } + \\frac { 1 } { \\tau } \\frac { { \\rm d } } { { \\rm d } s } \\left ( \\frac { 1 } { \\kappa } \\right ) \\mathbf { b } \\ , \\mbox { a n d } \\ , R _ S = \\sqrt { \\frac { 1 } { \\kappa ^ 2 } + \\frac { 1 } { \\tau ^ 2 } \\left [ \\frac { { \\rm d } } { { \\rm d } s } \\left ( \\frac { 1 } { \\kappa } \\right ) \\right ] ^ 2 } . \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} \\mathbb { V = } \\tilde { V } _ { \\chi \\left ( \\left \\vert x > R \\right \\vert \\right ) } \\left ( x , t \\right ) , \\mathbb { F } = \\tilde { V } _ { \\chi \\left ( \\left \\vert x < R \\right \\vert \\right ) } \\left ( x , t \\right ) \\tilde { u } \\left ( x , t \\right ) . \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} Q _ j \\cong X _ { 1 , \\ , j , \\ , 2 } \\cong H ( 2 : \\mathbf { n } , \\omega ) _ j = \\langle y ^ { ( j + 1 ) } , \\ , x y ^ { ( j + 1 ) } , \\ , x ^ { ( 2 ) } y ^ { ( j + 1 ) } \\rangle . \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} L _ { c _ * } w _ 0 = 1 2 \\psi _ * ^ 2 \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{align*} \\kappa = ( ( 0 , t _ 1 + \\eta _ 1 ) , ( - 1 , t _ 1 + \\eta _ 2 ) , ( - 2 , t _ 1 + \\eta _ 3 ) , ( - 3 , t _ 1 + \\eta _ 4 ) ) , \\eta = ( \\eta _ 1 , \\eta _ 2 , \\eta _ 3 , \\eta _ 4 , \\eta _ 5 , \\eta _ 6 ) \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} \\dot { x } x ^ { - 1 } + \\left ( \\dot { x } x ^ { - 1 } \\right ) ^ * = \\left [ \\left ( x \\phi x ^ { - 1 } \\right ) ^ * , x \\phi x ^ { - 1 } \\right ] - \\rho + s \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{align*} X \\triangleright \\mathsf { Q } ^ i _ j = X \\triangleright \\gamma _ { L } ^ { ( i , j ) } ( w ) = \\gamma _ { L } ^ { ( i , j ) } ( X \\triangleright w ) . \\end{align*}"}
-{"id": "1620.png", "formula": "\\begin{align*} \\lim _ { \\ell \\omega \\to k \\pi } \\Im \\frac { \\zeta ^ \\ell - \\zeta ^ { k \\pi / \\omega } } { \\sin \\ell \\omega } = \\Im ( \\zeta ^ \\ell \\log \\zeta ) \\ \\frac { 1 } { \\omega \\cos \\ell \\omega } \\ ; . \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { a _ n q ^ n } { 1 - q ^ n } & = \\sum _ { m \\geq 1 } b _ m q ^ m , \\ | q | < 1 , \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} ( \\lambda , g ) = ( \\mu , h ' ) ( \\nu , k ' ) = ( \\mu \\rho _ { h ' } ( \\nu ) , h ' + k ' ) \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} \\frac { ( z ) _ { a - k } } { ( 1 - a - z ) _ { a - k } } = \\frac { \\Gamma ( z + a - k ) \\Gamma ( 1 - a - z ) } { \\Gamma ( z ) \\Gamma ( 1 - z - k ) } = ( - 1 ) ^ a \\cdot \\frac { \\Gamma ( 1 - a - z ) \\Gamma ( z + k ) } { \\Gamma ( 1 - a - z + k ) \\Gamma ( z ) } = \\frac { ( - 1 ) ^ a ( z ) _ { k } } { ( 1 - a - z ) _ { k } } , \\end{align*}"}
-{"id": "9263.png", "formula": "\\begin{align*} A _ { i j } = ( \\partial ^ 2 _ { i j } f ) ^ \\perp \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} y _ 2 = - \\frac { 1 } { 2 } \\log ( 2 s ) , y _ 3 = \\frac { 1 } { 2 } \\log ( 2 s ) . \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} \\delta _ \\Omega ( z ) = \\inf \\{ \\norm { z - w } : w \\in \\partial \\Omega \\} \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} f _ t : = g _ t ^ { - 1 } \\hat f _ t ( z ) : = f _ t ( z + \\xi _ t ) = g _ t ^ { - 1 } ( z + \\xi _ t ) . \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{gather*} \\sum _ { \\mathbf { m } = ( m _ 0 , \\dots , m _ n ) , m _ i \\geq 1 } a _ { \\mathbf { m } } \\omega _ { \\mathbf { m } } \\end{gather*}"}
-{"id": "8116.png", "formula": "\\begin{align*} V _ F \\cap \\Delta = \\tilde V \\cap \\{ F = 0 \\} . \\end{align*}"}
-{"id": "2699.png", "formula": "\\begin{align*} E : = \\int _ \\Omega \\left ( n _ 1 - N _ 1 - N _ 1 \\log \\frac { n _ 1 } { N _ 1 } \\right ) + b _ 1 \\int _ \\Omega \\left ( n _ 2 - N _ 2 - N _ 2 \\log \\frac { n _ 2 } { N _ 2 } \\right ) + \\frac { b _ 2 } { 2 } \\int _ \\Omega c ^ 2 \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{align*} E ( t ) = \\frac { 1 } { 2 } \\left ( \\| u _ t \\| ^ 2 + k \\| \\Delta u \\| ^ 2 + \\int _ \\Omega a ( x ) | \\nabla u | ^ 2 d x \\right ) , \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} S _ m = \\big ( 1 + O ( d ^ 2 m ^ 2 / n ) \\big ) \\frac { \\lambda ^ m } { m ! } , \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} \\mathbf { \\sigma } _ { \\mathrm { p } } \\left ( \\alpha \\right ) = - \\frac { \\alpha ^ { 2 } } { 2 } D _ { \\left \\{ 0 \\right \\} } + \\int \\nolimits _ { \\mathbb { R } \\backslash \\left \\{ 0 \\right \\} } \\left ( \\mathrm { e } ^ { i \\alpha \\nu } - 1 - i \\alpha \\nu \\mathbf { 1 } \\left [ \\left \\vert \\nu \\right \\vert < 1 \\right ] \\right ) \\mathfrak { m } _ { \\mathrm { A C } } \\left ( \\mathrm { d } \\nu \\right ) \\ , \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} \\Psi ( 0 ) = { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & \\beta \\\\ & \\gamma \\end{matrix} \\bigg | \\ , z \\bigg ] = \\frac { \\Gamma _ p ( \\gamma ) \\Gamma _ p ( \\gamma + a - \\beta ) } { \\Gamma _ p ( \\gamma + a ) \\Gamma _ p ( \\gamma - \\beta ) } \\cdot { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & \\beta \\\\ & 1 - a + \\beta - \\gamma \\end{matrix} \\bigg | \\ , 1 - z \\bigg ] = \\Omega ( 0 ) \\Phi ( 0 ) . \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\begin{matrix} 0 \\\\ 0 \\\\ \\theta ^ 0 \\end{matrix} & \\overbrace { \\begin{matrix} 1 & 0 & t _ 2 & 0 & \\theta ^ \\infty _ 1 \\\\ 0 & 0 & 0 & \\sqrt { t _ 1 } & \\theta ^ \\infty _ 2 / 2 \\\\ 0 & 0 & 0 & - \\sqrt { t _ 1 } & \\theta ^ \\infty _ 2 / 2 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "1066.png", "formula": "\\begin{align*} T _ { F , G , N } ^ L ( 1 , \\dots , 1 ) \\ll _ { c , C , \\varepsilon } \\sigma _ G + \\frac { \\log X } { \\sigma _ G X } + X ^ { O ( 1 ) } \\max \\limits _ { \\substack { \\mathbf { k } \\in \\mathbb { Z } ^ m \\\\ 0 < \\Vert \\mathbf { k } \\Vert _ \\infty \\leqslant X } } \\Big ( \\prod \\limits _ { j = m + 1 } ^ d \\min ( 1 , N ^ { - 1 } \\Vert \\mathbf { k } \\cdot \\mathbf { v _ j } \\Vert ^ { - 1 } _ { \\mathbb { R } / \\mathbb { Z } } ) \\Big ) , \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} & V = \\{ ( S , C , \\xi , \\mathbf { k } \\psi ) \\ | \\ ( S , C , \\xi , \\mathbf { k } \\psi ) \\ s a t i s f i e s \\ t h e \\ a b o v e \\ c o n d i t i o n s \\ ( i ) - ( i i ) \\\\ & a n d \\ t h e \\ o p e n \\ c o n d i t i o n \\ [ \\ker ( \\psi ) ( c - b ) ] \\in N \\} \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac { 1 } { b ( T ) } \\int _ 0 ^ T a ( t ) f ( t ) \\ , \\dd t = f ( \\infty ) . \\end{align*}"}
-{"id": "3668.png", "formula": "\\begin{align*} d e g ( h _ { - m } v ) = 2 m + d e g ( v ) , \\forall \\ m \\in \\mathbb { Z } _ + . \\end{align*}"}
-{"id": "1208.png", "formula": "\\begin{align*} X _ { n + 1 } - X _ n = \\Delta t \\bigl ( A X _ { n + 1 } + G ( X _ n ) \\bigr ) + \\sigma ( X _ n ) \\bigl ( W \\bigl ( ( n + 1 ) \\Delta t \\bigr ) - W ( n \\Delta t ) \\bigr ) , \\ ; X _ 0 = x . \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} \\rho ' ( \\mu _ i ) = \\begin{pmatrix} w & t _ i \\\\ & \\\\ 0 & w ^ { - 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{gather*} \\chi _ { n , 0 } ( \\cos ( \\alpha _ { 1 } ) , . . . , \\cos ( \\alpha _ { n } ) | \\rho ) \\allowbreak = \\allowbreak \\sum _ { k \\geq 0 } \\rho ^ { k } \\prod _ { j = 1 } ^ { n } T _ { k } ( \\cos ( \\alpha _ { j } ) ) \\allowbreak = \\\\ \\allowbreak \\frac { 1 } { 2 ^ { n } } \\sum _ { i _ { 1 } \\in \\{ - 1 , 1 \\} } . . . \\sum _ { i _ { n } \\in \\{ - 1 , 1 \\} } \\frac { ( 1 - \\rho \\cos ( \\sum _ { k = 1 } ^ { n } i _ { k } \\alpha _ { k } ) ) } { ( 1 - 2 \\rho \\cos ( \\sum _ { k = 1 } ^ { n } i _ { k } \\alpha _ { k } ) + \\rho ^ { 2 } ) } , \\end{gather*}"}
-{"id": "3473.png", "formula": "\\begin{align*} \\partial _ { t } ^ { 2 } u ( x , t ) - L ^ \\alpha _ x \\partial _ x ^ 2 u ( x , t ) = 0 , \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} \\alpha ^ 2 & = \\sigma \\gamma , & \\gamma \\sigma & = \\lambda \\beta ^ 2 , & \\gamma \\alpha & = \\beta \\gamma , & \\alpha \\sigma & = \\sigma \\beta . \\end{align*}"}
-{"id": "3354.png", "formula": "\\begin{align*} N ( V ) = C \\cdot V ^ { 2 0 } + O ( V ^ { 2 0 - \\delta } ) \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{align*} \\dot z = D F ( \\theta ( t ) ) z . \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{align*} \\gamma _ 1 ( t ) = i e ^ { 2 t } e _ 1 . \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} h ( x ) = \\left \\{ \\begin{array} { l l } x & \\forall x \\leq 0 , \\\\ x - \\frac { 2 } { 3 } & \\forall x \\geq 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} u ^ 2 - D v ^ 2 = A ^ \\prime m , | m | \\leqslant 2 \\varepsilon \\sqrt { a b } + 1 . \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{align*} \\triangle _ 0 = \\frac { 1 } { f } ( N ^ 2 + T ^ 2 ) , \\end{align*}"}
-{"id": "43.png", "formula": "\\begin{align*} \\bigcup _ { i \\in [ n ] } A ^ H _ { i \\mapsto f ( i ) } = [ n ] . \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} W ( x ) = \\int _ { 0 ^ - } ^ { K - 0 } \\sum _ { n \\geq 0 } V _ n ^ x ( w ) d W ( w ) , \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} B = \\left ( M + u \\right ) \\left ( u + 2 \\ , M \\right ) u , \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\liminf _ n F _ n \\right ) = 1 . \\end{align*}"}
-{"id": "4819.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) ^ { p - 1 } & = I + \\begin{pmatrix} - a w ^ { p - 2 } & 0 \\\\ 0 & - d z ^ { p - 2 } \\end{pmatrix} p \\\\ & = I + \\begin{pmatrix} - \\frac { a } { w } & 0 \\\\ 0 & - \\frac { d } { z } \\end{pmatrix} p . \\end{align*}"}
-{"id": "9643.png", "formula": "\\begin{align*} t _ i = \\left \\{ \\begin{array} { l l } \\theta _ i & \\mbox { i f $ \\theta _ i \\in ( 0 , \\pi ) $ } ; \\\\ 2 \\pi - \\theta _ i & \\mbox { i f $ \\theta _ i \\in ( \\pi , 2 \\pi ) $ } . \\end{array} \\right . , \\ ; i = 1 , 2 . \\end{align*}"}
-{"id": "9764.png", "formula": "\\begin{align*} G _ i : = \\{ y \\in B ^ { n - 1 } \\mid \\int _ { S ^ 1 \\times y } e _ { \\epsilon } ( u \\circ F _ i ) \\leq \\frac { 2 C E _ { \\epsilon } ( u ) } { | B _ 1 ^ { n - 1 } | } \\} , \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} \\delta _ 1 = \\frac { 1 } { 8 c _ 2 } , \\delta _ 2 = \\frac { 1 } { 1 6 c _ 1 c _ 3 } , \\delta _ 3 = \\min \\Big \\{ \\delta _ 1 , \\ ; \\frac { 1 } { 3 2 \\gamma ( 1 + \\delta _ 1 ) c _ 3 } \\Big \\} . \\end{align*}"}
-{"id": "3518.png", "formula": "\\begin{align*} & D _ { z } ^ { \\mu } { f ( z ) } : = \\frac { 1 } { \\Gamma ( - \\mu ) } \\int _ { 0 } ^ { z } f ( t ) ( z - t ) ^ { - \\mu - 1 } d t , \\\\ & ( R e ( \\mu ) < 0 ) . \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{align*} X _ t = e ^ { t A } x + \\int _ { 0 } ^ { t } e ^ { ( t - s ) A } G ( X _ s ) d s + \\int _ { 0 } ^ { t } e ^ { ( t - s ) A } \\sigma ( X _ s ) d W ( s ) , \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} \\partial = \\partial ^ \\top + \\partial ^ N , \\bar { \\partial } = \\bar { \\partial } ^ \\top + \\bar { \\partial } ^ N . \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{align*} \\det \\left ( \\phi ( x _ { i } - y _ j ) \\right ) ^ n _ { i , j = 1 } \\ge 0 . \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & n \\\\ 0 & 1 \\end{pmatrix} \\left ( \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a _ 1 & 0 \\\\ 1 & a _ 1 \\end{pmatrix} p \\right ) \\begin{pmatrix} 1 & n \\\\ 0 & 1 \\end{pmatrix} ^ { - 1 } = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a _ 1 + n & - n ^ 2 \\\\ 1 & a _ 1 - n \\end{pmatrix} p . \\end{align*}"}
-{"id": "2695.png", "formula": "\\begin{align*} \\begin{aligned} T _ 1 & = \\{ ( 2 , 0 , 0 ) , ( 1 , 1 , 0 ) , ( 0 , 2 , 0 ) , ( 1 , 0 , 1 ) , ( 0 , 1 , 1 ) \\} \\\\ T _ 2 & = \\{ ( 2 , 0 , 0 ) , ( 1 , 1 , 0 ) , ( 0 , 2 , 0 ) , ( 1 , 0 , 1 ) , ( 0 , 1 , 1 ) \\} \\\\ T _ 3 & = \\{ ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) , ( 0 , 0 , 0 ) \\} \\end{aligned} \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mu } \\left ( Z _ { t } \\in F \\right ) = \\int _ { F } \\mathsf { d x } u ( \\mathsf { x } , t ) v ( \\mathsf { x } , t ) \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} \\norm { i } { \\gamma } = \\gamma \\sigma ( \\gamma ) \\dots \\sigma ^ { i - 1 } ( \\gamma ) . \\end{align*}"}
-{"id": "6835.png", "formula": "\\begin{align*} f _ { U , V } ( x , y ) = \\sum _ { k = 0 } ^ { \\infty } \\dfrac { \\Phi ^ { ( k + 1 ) } ( x ) \\Phi ^ { ( k + 1 ) } ( y ) } { k ! } ( s i g n ( C _ Y ( t ) ) r ) ^ k , \\ ; \\ ; x , y \\in \\mathbb { R } . \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} R ^ 2 = R ^ 2 U ^ T U = R ^ 2 U ^ T A ^ T \\Sigma ^ { - 1 } A U \\stackrel { d } { = } ( X - \\mu ) ^ T \\Sigma ^ { - 1 } ( X - \\mu ) , \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} r _ 2 < \\lim _ { x \\to b ^ - } [ a _ k ( x ) f ^ { ( k ) } ( x ) ] ^ { ( k - 2 ) } = \\lim _ { x \\to b ^ - } \\left ( \\sum _ { i = 0 } ^ { k - 2 } \\binom { k - 2 } { i } a _ k ^ { ( k - 2 - i ) } ( x ) f ^ { ( k + i ) } ( x ) \\right ) . \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} H ( t ) = \\int _ \\Omega u u _ t d x + \\sigma E ( t ) . \\end{align*}"}
-{"id": "524.png", "formula": "\\begin{align*} \\tilde K ^ 2 u ( x , w ) = \\int _ { \\R } \\Theta ( x , t , w ) \\rho P ^ * \\tilde K u ( x + t w , w ) d t , \\end{align*}"}
-{"id": "5767.png", "formula": "\\begin{align*} \\alpha ( \\nu ) : = \\left ( \\begin{array} { c c | c c } \\nu & 0 & \\sqrt { \\nu ^ 2 - \\frac { 1 } { 4 } } & 0 \\\\ 0 & \\nu & 0 & - \\sqrt { \\nu ^ 2 - \\frac { 1 } { 4 } } \\\\ \\hline \\sqrt { \\nu ^ 2 - \\frac { 1 } { 4 } } & 0 & \\nu & 0 \\\\ 0 & - \\sqrt { \\nu ^ 2 - \\frac { 1 } { 4 } } & 0 & \\nu \\\\ \\end{array} \\right ) \\ ; , \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} Y _ t ^ { i , k } = \\bigg ( \\frac { 1 } { n } - \\delta _ { i , k } \\bigg ) ( \\bar X _ t - X ^ i _ t ) \\phi _ t , \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} 0 < \\varrho ( r ) \\le ( 1 + r ) ^ { 2 } \\varrho ( r / ( 1 + r ) ) r > 0 , \\int _ { 0 ^ { + } } \\frac { \\d r } { \\varrho ( r ) } = \\infty , \\end{align*}"}
-{"id": "7665.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } h _ 1 \\\\ \\vdots \\\\ h _ n \\end{array} \\right ) = A ^ { - 1 } \\left ( \\begin{array} { c } c _ 1 \\\\ \\vdots \\\\ c _ n \\end{array} \\right ) . \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} \\nabla _ i ^ \\perp ( \\nabla ^ \\perp \\cdot H ) = \\nabla _ l ^ \\perp H _ 2 \\nabla _ l \\nabla _ i ^ \\perp H _ 1 - \\nabla _ l ^ \\perp H _ 1 \\nabla _ l \\nabla _ i ^ \\perp H _ 2 . \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} r = \\rho ^ \\kappa \\quad \\mbox { a n d } \\theta = \\kappa \\vartheta . \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} \\sum _ { j = i + 1 } ^ N | \\phi ^ i ( u ^ j ) | \\le \\frac { 2 r } m \\le \\frac 1 8 \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} u ' ( t ) = u ( t ) , \\mbox { f o r e v e r y } t \\in ( a , b ) , \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} H ^ { \\ast } ( X _ { d , p } ; \\mathbb { F } _ p ) = H ^ { \\ast } ( ( S ^ { d } ) ^ p \\setminus L ; \\mathbb { F } _ p ) = H _ { p d - \\ast } ( ( S ^ { d } ) ^ p , L ; \\mathbb { F } _ p ) . \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{align*} \\mu ( t ) = e ^ { \\int _ 0 ^ t \\alpha ( s ) d s } \\Bigg [ \\int _ 0 ^ t \\beta ( s ) e ^ { \\int _ 0 ^ s - \\alpha ( r ) d r } d s + C \\Bigg ] . \\end{align*}"}
-{"id": "462.png", "formula": "\\begin{align*} H _ { 1 + s } ( X | Z ) : = \\frac { - 1 } { s } \\log \\sum _ { z \\in { \\cal Z } } P _ Z ( z ) \\sum _ { x \\in { \\cal X } } P _ { X | Z } ( x | z ) ^ { 1 + s } , \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} r _ { l } : = \\limsup _ { p \\to \\infty } \\bigl ( \\omega _ { l - p ( 2 r + 1 ) } \\dots \\omega _ { l - ( 2 r + 1 ) } \\bigr ) ^ { 1 / p } < 1 . \\end{align*}"}
-{"id": "8769.png", "formula": "\\begin{align*} \\frac { | H ^ 0 ( G _ { K _ v } , M ) | \\cdot | H ^ 2 ( G _ { K _ v } , M ) | } { | H ^ 1 ( G _ { K _ v } , M ) | } = p ^ { - [ K _ v : \\mathbb { Q } _ p ] \\cdot v _ p ( | M | ) } , \\end{align*}"}
-{"id": "9424.png", "formula": "\\begin{align*} \\partial _ t \\tau + ( v \\cdot \\nabla ) \\tau = \\tau \\nabla \\cdot v , \\ ; \\ ; \\partial _ t v + ( v \\cdot \\nabla ) v = ( b \\cdot \\nabla ) b + ( d \\cdot \\nabla ) d + \\tau \\nabla \\tau , \\end{align*}"}
-{"id": "2916.png", "formula": "\\begin{align*} { \\rm s p a n } \\ , [ f _ \\xi ; \\ ; \\xi \\in \\Omega \\cap \\Lambda ] = { \\rm s p a n } \\ , \\Bigl [ \\bigcup _ { \\xi \\in \\Omega \\cap \\Lambda } \\ker ( T _ \\Lambda - \\xi ) \\Bigr ] = { \\rm P e r } ( T _ \\Lambda ) . \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} h ( x , t ) : = \\begin{cases} ( x + k _ 0 , t + 1 ) & B _ { ( x - k _ 0 , t + 1 ) } \\notin V \\\\ ( x - k _ 0 , t + 1 ) & B _ { ( x + k _ 0 , t + 1 ) } \\notin V \\\\ ( x + U _ { ( x , t ) } k _ 0 , t + 1 ) & \\end{cases} \\end{align*}"}
-{"id": "1588.png", "formula": "\\begin{align*} { Q } ( x ) { S } ( y ) - { Q } ( y ) { S } ( x ) = { S } ( y ) \\prod _ { l = 1 } ^ n ( x - { x } _ l ) - { S } ( x ) \\prod _ { l = 1 } ^ n ( y - { x } _ l ) . \\end{align*}"}
-{"id": "2014.png", "formula": "\\begin{gather*} \\frac { { \\rm d } Z } { { \\rm d } x } = \\big ( P ^ { - 1 } A ( x ) P - P ^ { - 1 } P ' \\big ) Z . \\end{gather*}"}
-{"id": "8078.png", "formula": "\\begin{align*} S _ z ( g \\circ f ) & = S _ z ( f ) & D _ z ( f \\circ g ) = D _ z ( f ) \\circ g \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} d X _ t = [ a ( m ( t ) - X _ t ) + \\lambda \\gamma _ t ] \\ , d t + \\sigma \\ , d W _ t + \\gamma _ { t } \\ , d \\widetilde N _ t , X _ 0 = x _ 0 , \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{gather*} \\begin{cases} d \\varphi = \\tau _ 0 \\ , \\psi + 3 \\ , \\tau _ 1 \\wedge \\varphi + \\star _ { \\varphi } \\tau _ 3 , \\\\ d \\psi = 4 \\tau _ 1 \\wedge \\psi - \\star _ { \\varphi } \\tau _ 2 , \\end{cases} \\end{gather*}"}
-{"id": "8442.png", "formula": "\\begin{align*} { \\cal X } ^ { n + 1 } : = \\Ref [ ( { \\cal X } ^ { ' n } + \\Tilde { \\xi } ^ { f } ) { \\bf 1 } _ { [ 0 , T ) } ] ; { \\cal X } ^ { ' n + 1 } : = \\Ref [ ( { \\cal X } ^ { n } - \\Tilde { \\zeta } ^ { f } ) { \\bf 1 } _ { [ 0 , T ) } ] \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} B ( \\vec { \\gamma } ) = \\bigl \\{ f \\in R \\ , | \\ , f ' _ { \\alpha } = \\gamma _ \\alpha f _ \\alpha \\mbox { \\rm f o r a l l } \\alpha \\in \\Pi \\bigr \\} , \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{align*} { { \\cal H } _ 0 } : & \\ ; x _ { k } ^ { ( n ) } = w _ { k } ^ { ( n ) } , \\\\ { } { { \\cal H } _ 1 } : & \\ ; x _ { k } ^ { ( n ) } = \\theta + w _ { k } ^ { ( n ) } , \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} \\partial _ { t } \\upsilon - i \\left [ \\Delta \\upsilon + A \\upsilon + \\tilde { V } \\upsilon \\right ] = \\eta _ { R } ^ { \\prime } \\left ( t \\right ) \\theta _ { M } \\left ( x \\right ) \\tilde { u } \\left ( x , t \\right ) - \\left ( 2 \\nabla \\theta _ { M } . \\nabla \\tilde { u } + \\tilde { u } \\Delta \\theta _ { M } \\right ) \\eta _ { R } . \\end{align*}"}
-{"id": "3804.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } ( n + 1 ) f _ { n + 1 } ( t ) x ^ { n } & = t \\left ( \\sum _ { n = 0 } ^ { \\infty } ( 1 - 2 ^ { \\nu _ { 2 } ( n + 1 ) + 1 } ) x ^ { n } \\right ) \\left ( \\sum _ { n = 0 } ^ { \\infty } f _ { n } ( t ) x ^ { n } \\right ) \\\\ & = t \\sum _ { n = 0 } ^ { \\infty } \\left ( \\sum _ { k = 0 } ^ { n } ( 1 - 2 ^ { \\nu _ { 2 } ( n - k + 1 ) + 1 } ) f _ { k } ( t ) \\right ) x ^ { n } . \\end{align*}"}
-{"id": "5826.png", "formula": "\\begin{align*} L _ { u + v } + ( - 1 ) ^ v L _ { u - v } = L _ v L _ u \\ , , \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} & \\mathfrak { h } \\colon \\mathfrak { U } \\rightarrow \\widetilde { \\mathfrak { U } } , \\\\ & \\mathfrak { h } = \\tilde { \\pi } \\circ h \\circ \\pi ^ { - 1 } \\textrm { o n } U \\backslash \\{ 0 \\} , \\\\ & \\mathfrak { h } ( 0 ) = 0 \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{align*} \\mathbf M _ { i j } - \\frac { ( \\lambda _ { n + 2 } - \\lambda _ i ) ( \\lambda _ { n + 3 } - \\lambda _ i ) ( \\alpha _ 1 - \\lambda _ 1 ) } { ( \\lambda _ { n + 2 } - \\lambda _ 1 ) ( \\lambda _ { n + 3 } - \\lambda _ 1 ) ( \\alpha _ 1 - \\lambda _ i ) } \\mathbf M _ { 1 j } = \\frac { ( \\lambda _ 1 - \\lambda _ i ) } { ( \\alpha _ 1 - \\lambda _ i ) } N _ { i j } . \\end{align*}"}
-{"id": "6075.png", "formula": "\\begin{align*} B ( L _ { - 2 } ) _ { - 2 } = L _ { - 4 } = [ L _ { - 2 } , \\ , L _ { - 2 } ] \\not \\subseteq M ( B ( L _ { - 2 } ) ) , \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} \\vert n _ 1 , n _ 2 , \\cdots , n _ r \\rangle \\longrightarrow f _ { n _ 1 , n _ 2 , \\cdots , n _ r } ( \\eta _ 1 , \\eta _ 2 , \\cdots , \\eta _ r ) = c _ { n _ 1 , n _ 2 , \\cdots , n _ r } { { \\eta _ 1 } } ^ { n _ 1 } { { \\eta _ 2 } } ^ { n _ 2 } \\cdots { { \\eta _ r } } ^ { n _ r } a _ i ^ + \\longrightarrow { { \\eta _ i } } . \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{align*} 4 s ^ 2 w '' ( s ) + 4 \\gamma s w ' ( s ) - w ' ( s ) - \\beta w ( s ) + | w ( s ) | ^ \\alpha w ( s ) = 0 , \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} g _ { n } = - \\frac { 1 } { n ! } \\frac { \\left ( a \\right ) _ { n } } { \\left ( a + b \\right ) _ { n } } . \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} \\left \\Vert \\alpha \\beta \\sigma ^ { 2 } \\left ( t \\right ) V \\left ( \\sqrt { \\alpha \\beta } x \\sigma \\left ( t \\right ) , \\beta t \\sigma \\left ( t \\right ) \\right ) \\right \\Vert _ { L \\left ( H \\right ) } \\leq \\alpha ^ { - 1 } \\beta \\left \\Vert V \\right \\Vert _ { B } = \\left ( k a _ { 0 } ^ { - 1 } \\right ) ^ { \\frac { 1 } { p } } \\left \\Vert V \\right \\Vert _ { B } \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} \\sum _ { r \\geq y } \\frac { 1 } { r ^ 2 } = \\int _ y ^ { \\infty } \\frac { d \\pi ( t ) } { t ^ 2 } = \\frac { \\pi ( t ) } { t ^ 2 } \\Bigg | _ y ^ { \\infty } + 2 \\int _ y ^ { \\infty } \\frac { \\pi ( t ) \\ , d t } { t ^ 3 } \\ll \\frac { \\pi ( y ) } { y ^ 2 } \\ll \\frac { 1 } { y \\log y } , \\end{align*}"}
-{"id": "1666.png", "formula": "\\begin{align*} \\mathbb { S } ( R ) = \\lim _ { e \\rightarrow \\infty } \\frac { \\sharp ( F _ * ^ e ( R ) , R ) } { p ^ { e ( d + \\alpha ( R ) ) } } \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 { W } _ i ( \\xi _ 2 , \\cdot ) \\ , d \\xi _ 2 = 0 , i = 1 , \\ldots , 4 , \\end{align*}"}
-{"id": "3601.png", "formula": "\\begin{align*} L ^ \\dagger \\cdot h = \\gamma \\left [ - H _ { \\rm { O H } } + i \\frac { \\sqrt { 2 } } { \\gamma } \\right ] h - \\frac { i c } { 2 \\pi } \\langle e _ 1 , h \\rangle e _ 0 . \\end{align*}"}
-{"id": "9778.png", "formula": "\\begin{align*} V _ { \\epsilon } : = \\frac { 1 } { | \\log \\epsilon | } T _ { \\epsilon } ( \\tilde { u } _ { \\epsilon } ) , \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} b _ 1 = | \\min \\{ \\omega ( p ) \\} | = - \\min \\{ \\omega ( p ) \\} , \\end{align*}"}
-{"id": "2801.png", "formula": "\\begin{align*} M ( M + k ) N ( N + l ) = L ^ 2 , \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{align*} \\varphi ( Q _ 1 ) = \\psi ( Q _ 2 ) . \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{align*} & \\phi _ g ^ { \\beta , \\sigma } ( x ) \\to \\phi _ g ^ { T F } ( x ) = \\frac { 1 } { \\sqrt { L } } , 0 < x < L , \\\\ & \\mu _ g ( \\beta , \\sigma ) \\approx \\frac { \\beta } { L ^ { \\sigma + 1 } } \\to \\infty , E _ g ( \\beta , \\sigma ) \\approx \\frac { \\beta } { ( \\sigma + 1 ) L ^ { \\sigma + 1 } } \\to \\infty . \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{align*} \\varphi _ 1 | _ { W _ x } ( \\sum z _ i \\mathbf e _ i ' ) = \\sum _ { i = 1 } ^ n \\Big ( \\lambda _ { n + 2 } \\frac { ( \\lambda _ { n + 3 } - \\lambda _ i ) ^ 2 x _ i ^ 2 } { ( \\lambda _ { n + 3 } - \\lambda _ { n + 2 } ) ^ 2 x _ { n + 2 } ^ 2 } + \\lambda _ { n + 3 } \\frac { ( \\lambda _ { n + 2 } - \\lambda _ i ) ^ 2 x _ i ^ 2 } { ( \\lambda _ { n + 3 } - \\lambda _ { n + 2 } ) ^ 2 x _ { n + 3 } ^ 2 } + \\lambda _ i \\Big ) z _ i ^ 2 \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} p ( x ) = \\tilde { p } _ { 2 n - 2 } { B } ( x , x ) = \\tilde { p } _ { 2 n - 2 } ( { S } ( x ) { Q } ' _ 1 ( x ) - { Q } _ 1 ( x ) { S } ' ( x ) ) . \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} \\big ( ( b _ 1 , f _ 1 ) , ( b _ 2 , f _ 2 ) \\big ) = f _ 1 ( b _ 2 ) + f _ 2 ( b _ 1 ) \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} \\mathbf { p } _ { h , i , j } & = \\mathbf { p } _ { d } + p _ { h , i , j } \\mathbf { R } _ { \\phi _ { h , i , j } } \\mathbf { r } _ { i , j } \\\\ \\phi _ { h , i , j } & = \\sin ^ { - 1 } \\frac { r _ c } { r _ { i , j } } \\\\ p _ { h , i , j } & = \\sqrt { ( r _ { i , j } ) ^ 2 - r _ c ^ 2 } \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{gather*} y ( 0 ) = 0 \\quad y ( \\pi ) = 0 . \\end{gather*}"}
-{"id": "673.png", "formula": "\\begin{align*} \\theta _ i ^ 2 = 0 , [ \\theta _ i , \\theta _ j ] = 0 . \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} \\frac { \\Gamma _ p ( x + 1 ) } { \\Gamma _ p ( x ) } = \\begin{cases} - x , & | x | _ p = 1 , \\\\ - 1 , & | x | _ p > 1 . \\end{cases} \\end{align*}"}
-{"id": "2124.png", "formula": "\\begin{gather*} \\theta ^ 0 = - 2 \\varepsilon ^ { - 1 } , \\theta ^ \\infty _ 1 = \\tilde { \\theta } ^ \\infty _ 1 + 2 \\varepsilon ^ { - 1 } , t _ 2 = \\varepsilon \\tilde { t } _ 2 , H _ { t _ 2 } = \\varepsilon ^ { - 1 } \\tilde { H } _ { 2 } , \\\\ q _ 2 = - \\tilde { q } _ 2 , p _ 2 = - \\tilde { p } _ 2 - \\frac { 1 } { \\varepsilon \\tilde { q } _ 2 } , Y = x ^ { - \\varepsilon ^ { - 1 } } \\tilde { Y } . \\end{gather*}"}
-{"id": "808.png", "formula": "\\begin{align*} \\biggl \\| \\int _ 0 ^ { t _ A } F ( 4 t - s ) * ( u _ n \\otimes u _ n ) ( s ) \\biggr \\| _ 3 & \\le C \\int _ 0 ^ { t _ A } \\| F ( 4 t - s ) \\| _ { 3 / 2 } \\| u _ n ( s ) \\| _ 3 ^ 2 \\dd s \\\\ & \\le C ( 4 t - t _ A ) ^ { - 1 } t _ A \\| u _ n \\| _ X ^ 2 \\le C \\varepsilon ^ 2 ( 4 t - t _ A ) ^ { - 1 } t _ A . \\end{align*}"}
-{"id": "4377.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } T _ k x ^ k = \\bigl ( ( 1 - 3 x ) ( 1 + x ) \\bigr ) ^ { - 1 / 2 } = ( 1 - 2 x - 3 x ^ 2 ) ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{align*} z + z ^ { - 1 } = \\frac { 2 \\ell + 1 } { \\ell } \\in \\Q . \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} \\hat { T } ^ { ( n ) } _ n = \\hat { T } ^ { ( n ) } ( \\hat { \\pi } ^ { ( n ) } _ n ) . \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} S _ { \\Delta t } = \\bigl ( I - \\Delta t A \\bigr ) ^ { - 1 } . \\end{align*}"}
-{"id": "8081.png", "formula": "\\begin{align*} D _ \\tau f _ j = R _ j \\circ f \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} \\sum _ { n \\ge 2 } \\ \\biggl | \\ , \\prod _ { j = 1 } ^ { n - 1 } \\dfrac { \\lambda - \\lambda _ { j } } { \\omega _ { j } } \\biggr | ^ { 2 } = \\infty , \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} L \\circ d \\varphi _ { \\varepsilon } = e ^ { \\sigma _ { \\varepsilon } } L , \\end{align*}"}
-{"id": "8449.png", "formula": "\\begin{align*} Y _ { \\sigma ^ * _ { \\theta } } \\ , = \\ , \\zeta _ { \\sigma ^ * _ { \\theta } } { \\rm a n d } Y _ { \\overline \\sigma _ { \\theta } } \\ , = \\ , \\zeta _ { \\overline \\sigma _ { \\theta } } \\mbox { a . s . } \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} \\sum _ { \\| y \\| _ 1 \\leq R } ( 1 - c ( \\xi ^ * _ { x + y } ) ) & = O _ { B , R _ 1 } ( 1 ) , \\\\ \\sum _ { \\| y \\| _ 1 > R } ( 1 - c ( \\xi ^ * _ { x + y } ) ) & = O _ { B , R _ 1 } \\left ( R ^ { - 2 } \\right ) , \\\\ \\left | \\sum _ { \\| y \\| _ 1 \\leq R } s ( \\xi ^ * _ { x + y } ) \\right | \\leq \\sum _ { \\| y \\| _ 1 \\leq R } \\left | s ( \\xi ^ * _ { x + y } ) \\right | & = O _ { B , R _ 1 } ( \\log R ) . \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} P _ { M | Y _ 1 , Y _ 3 } ( 0 | 0 , 0 ) & = P _ { M | Y _ 1 , Y _ 3 } ( 1 | 0 , 0 ) = \\frac { 1 } { 2 } , \\\\ P _ { M | Y _ 1 , Y _ 3 } ( 0 | 1 , 0 ) & = P _ { M | Y _ 1 , Y _ 3 } ( 1 | 1 , 1 ) = 1 , \\end{align*}"}
-{"id": "9189.png", "formula": "\\begin{align*} [ M ] _ t : = \\mathbb P - \\lim _ { { \\rm m e s h } \\to 0 } \\sum _ { n = 1 } ^ N \\| M ( t _ n ) - M ( t _ { n - 1 } ) \\| ^ 2 , \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} a _ i \\bmod m _ i , i = 1 , 2 , . . . , k \\end{align*}"}
-{"id": "1408.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t v + v \\cdot \\nabla v + \\nabla p = \\nabla \\cdot ( \\frac { \\partial W ( F ) } { \\partial F } F ^ \\top ) , \\\\ [ - 4 m m ] \\\\ \\nabla \\cdot v = 0 . \\end{cases} \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{align*} g ^ { i j } = ( 1 - 2 f + O _ { \\sqrt { \\epsilon } } ( \\norm { f } _ { 1 , \\ , p } ) ) \\sigma ^ { i j } \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} ( 1 - \\beta ) ^ e \\sum _ { q - e \\le k < q - e / 2 } \\binom { 2 k + e } { k } x ^ { k } \\equiv \\beta ^ { q - e } \\sum _ { q - e \\le k < q - e / 2 } \\binom { 2 k + e } { k } x ^ { k - q + e } \\pmod { ( \\beta ^ q , p ) } , \\end{align*}"}
-{"id": "6350.png", "formula": "\\begin{align*} \\tilde \\psi ( T ) ( x \\otimes y ) = ( T x ) \\otimes y \\quad \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} \\textrm { \\emph { C o v } } _ { B } [ X , Y ] = \\textrm { \\emph { V a r } } _ { B } [ Y ] \\beta ( X | Y ) . \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} \\begin{cases} b _ 2 & = a _ 1 ^ 2 + 4 a _ 2 \\\\ b _ 4 & = a _ 1 a _ 3 + 2 a _ 4 \\\\ b _ 6 & = a _ 3 ^ 2 + 4 a _ 6 \\\\ b _ 8 & = a _ 1 ^ 2 a _ 6 - a _ 1 a _ 3 a _ 4 + 4 a _ 2 a _ 6 + a _ 2 a _ 3 ^ 2 - a _ 4 ^ 2 \\\\ c _ 4 & = b _ 2 ^ 2 - 2 4 b _ 4 \\\\ c _ 6 & = - b _ 2 ^ 3 + 3 6 b _ 2 b _ 4 - 2 1 6 b _ 6 \\\\ \\Delta & = - b _ 2 ^ 2 b _ 8 - 8 b _ 4 ^ 3 - 2 7 b _ 6 ^ 2 + 9 b _ 2 b _ 4 b _ 6 \\\\ j & = { c _ 4 ^ 3 } / { \\Delta } \\end{cases} \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} \\mathcal { A } ^ P = & E ( - k _ { P ( n ) } ) \\mathcal { A } ^ { P R _ n } \\\\ = & E ( - k _ { P ( n ) } ) \\mathcal { A } ^ { P C _ n R _ 1 C _ n ^ { - 1 } } , \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{align*} & f = F - \\widetilde U \\cdot \\nabla \\widetilde U - h ( u _ s \\cdot \\nabla \\widetilde U + \\widetilde U \\cdot \\nabla u _ s ) , \\\\ & w _ 0 = \\phi v _ 0 - \\mathbb B [ v _ 0 \\cdot \\nabla \\phi ] \\in L ^ 2 _ \\sigma ( \\Omega ) , \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} \\left . \\frac { d } { d t } \\log \\bar { W } _ { n - q } ( [ K , f ] _ t ) \\right | _ { t = 0 } = \\frac { 1 } { \\widetilde { W } _ { n - q } ( K ) } \\int _ { S ^ { n - 1 } } f ( v ) d \\widetilde { C } _ q ( K , v ) , \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} \\left ( L _ { \\varepsilon } + \\lambda \\right ) u = \\ - \\varepsilon u ^ { \\left ( 2 \\right ) } \\left ( x , \\varepsilon \\right ) + A u \\left ( x , \\varepsilon \\right ) + B u ^ { \\left ( 1 \\right ) } \\left ( x , \\varepsilon \\right ) + \\lambda u \\left ( x , \\varepsilon \\right ) = f \\left ( x \\right ) , x \\in \\left ( 0 , T \\right ) , \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} x = \\frac { t } { ( 1 + \\varepsilon ) } = t \\cdot \\sum _ { n = 0 } ^ { \\infty } ( - 1 ) ^ { n } \\varepsilon ^ { n } , \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} \\vert n \\rangle = \\sqrt { \\frac { ( k - n ) ! } { k ! n ! } } { ( { a ^ + } ) } ^ { n } \\vert 0 \\rangle n \\leq k , \\end{align*}"}
-{"id": "2554.png", "formula": "\\begin{align*} V ( x ) & = \\sum _ { j \\geq 1 } \\ ; v _ j \\ ; c o s ( j x ) . \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} [ \\ell ] ( Q _ 0 - \\xi ( Q ) - Q ^ \\prime ) = P _ 0 - \\xi ( P ) - [ \\ell ] Q ^ \\prime = O . \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} \\Phi ( u ^ 1 , u ^ 2 ) = \\sum _ { \\mathcal { C } ^ 2 } { \\left \\lvert \\frac { u ^ 1 _ i - u ^ 1 _ j } { | J _ { i j } ^ 1 | } - \\frac { u ^ 2 _ i - u ^ 2 _ j } { | J _ { i j } ^ 2 | } \\right \\rvert } ^ 2 , \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{align*} z = 1 , w = 1 , z = w , z w = 1 . \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} \\sup _ { j _ { r } \\le i \\le k _ { r } } \\| T ^ { \\ , i } x - T ^ { \\ , i } y _ { r } \\| < \\delta \\quad \\textrm { w h i l e } T ^ { \\ , N _ { \\delta , m } + k _ { s } } x = x . \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{gather*} G _ 2 = \\begin{pmatrix} 1 & 1 & 1 \\\\ - t ^ { 1 / 3 } & - \\omega t ^ { 1 / 3 } & - \\omega ^ 2 t ^ { 1 / 3 } \\\\ t ^ { 2 / 3 } & \\omega ^ 2 t ^ { 2 / 3 } & \\omega t ^ { 2 / 3 } \\end{pmatrix} \\end{gather*}"}
-{"id": "1047.png", "formula": "\\begin{align*} a _ { i , j } ( k ) = a \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle a = \\lim _ { r \\to 0 } f _ a ( r ) = \\lim _ { s \\to \\infty } s ^ { \\frac { 1 } { \\alpha } } w ( s ) , \\\\ \\displaystyle L ( a ) = \\lim _ { r \\to \\infty } r ^ { \\frac { 2 } { \\alpha } } f _ a ( r ) = \\lim _ { s \\to 0 } w ( s ) = w ( 0 ) . \\end{cases} \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} z ( 0 ) = z ^ 0 , \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} k h & = \\left ( I + \\begin{pmatrix} 0 & \\beta \\\\ \\gamma & 0 \\end{pmatrix} \\right ) \\left ( \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} + \\begin{pmatrix} a & 0 \\\\ 0 & d \\end{pmatrix} p \\right ) \\\\ & = \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} + \\begin{pmatrix} a & \\beta z \\\\ \\gamma w & d \\end{pmatrix} p \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} i \\partial _ t \\phi _ { \\hom } + \\frac 1 2 \\Delta \\phi _ { \\hom } + \\frac { \\bar R _ 2 } { \\pi } \\phi _ { \\hom } = 0 , \\ \\ \\phi _ { \\hom } ( 0 , x ) = \\phi _ 0 ( x ) , \\ \\ x \\in \\R ^ 2 , \\end{align*}"}
-{"id": "6108.png", "formula": "\\begin{align*} 0 = [ L _ { - 2 } , \\ , [ L _ { 2 } , \\ , L _ { j } ] ] = [ L _ { 2 } , \\ , [ L _ { - 2 } , \\ , L _ { j } ] ] \\end{align*}"}
-{"id": "10020.png", "formula": "\\begin{align*} | t _ k ( x ) | \\sim \\frac { 1 } { R _ k } = v _ k ^ 2 \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} [ P ( J ) ^ + - P ( J ) ^ - , J ] = 0 . \\end{align*}"}
-{"id": "3410.png", "formula": "\\begin{align*} \\hat \\gamma _ t = \\frac { \\theta + \\phi _ { t } } { 1 + \\frac { 1 } { \\lambda } \\phi _ { t } } ( \\mathbb E [ X _ { t - } ] - X _ { t - } ) , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} \\begin{cases} \\alpha + i \\beta = z \\\\ \\alpha - i \\beta = w \\end{cases} \\end{align*}"}
-{"id": "2696.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n P _ j = a I _ m , \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , X _ { 2 , \\ , j + 2 , \\ , 1 3 } ] = 0 . \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} & q ^ { \\varepsilon } ( x ) = \\alpha ( s ) + \\frac { 1 } { 1 - \\varepsilon M } \\left ( \\frac { M } { 2 } | x - s | ^ { 2 } + \\langle v ( s ) , x - s \\rangle + \\frac { \\varepsilon } { 2 } | v ( s ) | ^ { 2 } \\right ) , \\\\ & q _ { \\varepsilon } ( x ) = \\alpha ( s ) + \\frac { 1 } { 1 + \\varepsilon M } \\left ( \\frac { M } { 2 } | x - s | ^ { 2 } + \\langle v ( s ) , x - s \\rangle - \\frac { \\varepsilon } { 2 } | v ( s ) | ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} & \\Delta \\phi _ 1 = \\cos ^ { - 1 } \\left ( \\frac { r _ s - \\Delta T _ s ( v _ { o , m a x } - v _ { c , s } ) } { r _ s } \\right ) + \\sin ^ { - 1 } \\left ( \\frac { r _ c } { r _ s } \\right ) \\\\ & \\Delta \\phi _ 2 = \\pi / 2 + \\sin ^ { - 1 } \\left ( v _ { o , m a x } / v _ { c , s } \\right ) - \\Delta \\phi _ 1 \\\\ & \\tau _ { f , 1 } = \\frac { c _ 3 m \\Delta \\phi _ 1 v _ { c , s } } { f _ { p l a n a r } - K _ d ( v _ { c , s } + v _ { a i r } ) ^ 2 } \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} Y : = a X = \\sum _ { i = 1 } ^ { n } a _ i X _ i , \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} F ( x , t ) = t ^ { - 2 } F ( \\textstyle \\frac { x } { \\sqrt t } , 1 ) , \\qquad \\hbox { a n d } F ( \\cdot , 1 ) \\in L ^ 1 \\cap L ^ \\infty \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{align*} \\widehat { S } _ { \\mu } = \\left \\{ \\begin{array} { r c l r c l } x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } & = & 0 , & \\mu _ { 4 } \\ ; x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 4 } ^ { 2 } & = & 0 , \\\\ \\mu _ { 5 } \\ ; x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 5 } ^ { 2 } & = & 0 , & \\mu _ { 6 } \\ ; x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 6 } ^ { 2 } & = & 0 , \\\\ \\mu _ { 7 } \\ ; x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 7 } ^ { 2 } & = & 0 \\end{array} \\right \\} \\subset { \\mathbb P } _ { \\mathbb C } ^ { 6 } . \\end{align*}"}
-{"id": "9693.png", "formula": "\\begin{align*} H _ q ( P ) \\ = \\ H _ q ( P _ 1 ) \\sqcup H _ q ( P _ 2 ) \\sqcup \\cdots \\sqcup H _ q ( P _ m ) \\end{align*}"}
-{"id": "9322.png", "formula": "\\begin{align*} g = \\frac { 2 2 - r - a } { 2 } , \\ \\ k = \\frac { r - a } { 2 } + 1 , \\ \\ r = 1 0 + k - g , \\ \\ a = 1 2 - k - g . \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{align*} | f ^ 2 _ { \\alpha a } | \\lesssim & \\sum \\limits _ { \\tiny \\begin{matrix} b + c = a \\\\ \\beta + \\gamma = \\alpha \\end{matrix} } | \\nabla ^ \\perp _ j H ^ { ( \\beta , b ) } \\cdot \\omega ^ \\perp \\nabla _ j V ^ { ( \\gamma , c ) } | \\\\ & + \\sum \\limits _ { \\tiny \\begin{matrix} b + c = a \\\\ \\beta + \\gamma = \\alpha \\end{matrix} } | ( \\nabla ^ \\perp _ j H ^ { ( \\beta , b ) } \\cdot \\omega + \\nabla ^ \\perp _ j V ^ { ( \\beta , b ) } ) \\nabla _ j V ^ { ( \\gamma , c ) } \\omega | , \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} \\mathbb { P } ( \\# { \\cal E } _ 1 = r ) \\geq { n - 1 \\choose r - 1 } T _ r p _ d ^ { r - 1 } ( 1 - p _ u ) ^ { r ( n - r ) + { r \\choose 2 } - r + 1 } . \\end{align*}"}
-{"id": "1480.png", "formula": "\\begin{align*} u ( \\cdot , t + \\tau ) - u ( \\cdot , t ) = u ^ { n _ 1 ( t ) } - u ^ { n _ 0 ( t ) } , \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t v - \\mu \\Delta v - \\nabla \\cdot G = - \\nabla p - v \\cdot \\nabla v + \\nabla \\cdot ( G G ^ \\top ) , \\\\ [ - 4 m m ] \\\\ \\partial _ t G - \\nabla v = - v \\cdot \\nabla G + \\nabla v G , \\\\ [ - 4 m m ] \\\\ \\nabla \\cdot v = 0 , \\nabla \\cdot G ^ \\top = 0 . \\end{cases} \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} \\mathbb { F } ^ { ( l ) } \\left ( B \\right ) : = \\frac { 1 } { \\left \\vert \\Lambda _ { l } \\right \\vert ^ { 1 / 2 } } \\underset { x \\in \\Lambda _ { l } } { \\sum } \\left \\{ \\chi _ { x } \\left ( B \\right ) - \\varrho ^ { ( \\beta , \\omega , \\vartheta , \\lambda ) } \\left ( \\chi _ { x } \\left ( B \\right ) \\right ) \\mathbf { 1 } _ { \\mathcal { U } } \\right \\} \\ . \\end{align*}"}
-{"id": "7327.png", "formula": "\\begin{align*} \\mathrm { T r } ( \\tau ) = \\frac { 1 } { 4 } \\tau _ 0 \\ , \\mathrm { T r } ( g _ { \\varphi } ) = \\frac { 7 } { 4 } \\tau _ 0 . \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} c _ 1 ( t ) & = c _ { 1 , 0 } - 2 t & x _ 1 ^ { ( i ) } ( t ) & = \\gamma _ { 1 , i } + \\frac { \\gamma _ { 2 , i } } { c _ { 1 , 0 } + c _ { 2 , 0 } } \\cdot \\sqrt { \\frac { c _ { 2 , 0 } + 2 t } { c _ { 1 , 0 } - 2 t } } \\\\ c _ 2 ( t ) & = c _ { 2 , 0 } + 2 t & x _ 2 ^ { ( i ) } ( t ) & = \\gamma _ { 2 , i } - \\frac { \\gamma _ { 2 , i } } { c _ { 1 , 0 } + c _ { 2 , 0 } } \\cdot \\sqrt { \\frac { c _ { 1 , 0 } - 2 t } { c _ { 2 , 0 } + 2 t } } \\end{align*}"}
-{"id": "3670.png", "formula": "\\begin{align*} \\sum _ { \\mathfrak { p } \\in \\mathfrak { P } _ { t d o } } q ^ { d e g ( v _ { \\mathfrak { p } ) } } = \\Big ( \\sum _ { m \\in \\mathbb { Z } _ { \\geq 0 } } q ^ { T _ m } \\Big ) \\cdot \\prod _ { i \\in \\mathbb { Z } _ { + } } ( 1 + q ^ { \\frac { 2 i - 1 } { 2 } } ) . \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ m \\int _ { B _ i ^ { \\varepsilon } } \\frac { ( U _ 2 ^ { ( k ) } ( | x - a _ i | ) ) ^ 2 } { | x - a _ i | ^ { 2 ( m - k ) } } d x \\leq C _ { m , \\Omega , j } \\sum _ { i = 0 } ^ { m - 2 } \\varepsilon ^ { - 2 m - 2 i } \\int _ 0 ^ { \\varepsilon } t ^ { 2 i + N - 1 } \\leq C _ { m , \\Omega , j } . \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{align*} R \\big ( e _ { s ( \\alpha ) } \\otimes e _ { t ( f ( \\alpha ) ) } \\big ) = \\varrho ( \\bar { \\mu } _ { \\alpha } ) \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} \\Sigma _ m ( i , j ) & : = \\underset { } { \\underbrace { \\sum _ { k _ 1 = j + 2 } ^ i \\sum _ { k _ 2 = j + 2 } ^ { k _ 1 - 1 } \\cdots \\sum _ { k _ m = j + 2 } ^ { k _ { m - 1 } - 1 } } } s _ { i , k _ 1 - 1 } \\cdot s _ { k _ 1 - 1 , k _ 2 - 1 } \\times \\cdots \\times s _ { k _ m - 1 , j } . \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} ( 1 - x ) ^ p \\pounds _ 1 \\left ( \\frac { 1 - y } { 1 - x } \\right ) = \\sum _ { k = 1 } ^ { p - 1 } \\frac { ( 1 - x ) ^ { p - k } ( 1 - y ) ^ k } { k } \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} a _ + ( \\alpha ) \\wedge a _ + ( \\beta ) \\wedge ( a _ - ( \\alpha ) \\vee a _ - ( \\beta ) ) = a _ + ( \\alpha ) = a _ + ( \\alpha ) \\wedge a _ + ( \\beta ) . \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = 1 & x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\begin{matrix} 0 \\\\ 0 \\\\ \\theta ^ 0 \\end{matrix} & \\begin{matrix} 0 \\\\ 0 \\\\ \\theta ^ 1 \\end{matrix} & \\overbrace { \\begin{matrix} 0 & \\sqrt { t _ 2 } & \\theta ^ \\infty _ 1 / 2 \\\\ 0 & - \\sqrt { t _ 2 } & \\theta ^ \\infty _ 1 / 2 \\\\ - t _ 1 & 0 & \\theta ^ \\infty _ 2 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "2751.png", "formula": "\\begin{align*} \\operatorname { E r } = O _ { \\varepsilon , \\eta } \\left ( \\frac { K ^ 2 B ^ { 2 - \\frac { 1 } { r } } } { N d ^ 2 } \\right ) + O _ { \\sigma } \\left ( \\frac { K ^ \\sigma B ^ \\sigma N } { d ^ \\sigma } \\right ) . \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} \\begin{aligned} d f ^ i & = 0 , 1 \\leq i \\leq 6 , \\\\ d f ^ 7 & = \\frac { \\sqrt { 6 } } { 6 } \\ , y ( t ) ^ { - 1 } z ( t ) ^ { - 2 } \\Big ( y ( t ) ^ { - 2 } f ^ { 1 2 } + z ( t ) ^ { - 2 } f ^ { 3 4 } + z ( t ) ^ { - 2 } f ^ { 5 6 } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "9861.png", "formula": "\\begin{align*} \\widehat { \\zeta } ( s ) = ( 2 \\pi \\sigma ^ 2 ) ^ { \\frac { d } { 2 } } \\int _ 0 ^ \\infty \\zeta ( \\rho ) \\ \\dfrac { J _ { \\frac { d } { 2 } - 1 } ( s \\rho ) } { ( s \\rho ) ^ { \\frac { d } { 2 } - 1 } } \\rho ^ { d - 1 } \\ d \\rho \\end{align*}"}
-{"id": "9408.png", "formula": "\\begin{align*} \\underset { j } { \\lim } \\ , \\underset { i } { \\lim } \\ ( \\Gamma ^ { A , B , C } ( u _ i v _ j ) ( X , Y ) Z ) & = \\underset { j } { \\lim } \\ ( \\Gamma ^ { A , B , C } ( u v _ j ) ( X , Y ) Z ) \\\\ & = ( \\Gamma ^ { A , B , C } ( u v ) ( X , Y ) Z ) . \\end{align*}"}
-{"id": "7688.png", "formula": "\\begin{align*} Q ( X \\in A , \\mathbf Y \\in B ) & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n Q _ i ( X \\in A , \\mathbf Y \\in B ) \\\\ & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n Q _ i ( X \\in A ) Q _ i ( \\mathbf Y \\in B ) \\\\ & = Q ( X \\in A ) \\frac { 1 } { n } \\sum _ { i = 1 } ^ n Q _ i ( \\mathbf Y \\in B ) = Q ( X \\in A ) Q ( \\mathbf Y \\in B ) , \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} \\alpha _ i ^ t \\in K , \\ ; t = 0 , 1 , \\dots , p _ i - 2 . \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} B _ 1 & = x _ 2 y _ 1 + x _ 1 y _ 2 + a _ { 2 2 } x _ 2 y _ 2 + a _ { 3 2 } x _ 3 y _ 2 + a _ { 2 3 } x _ 2 y _ 3 + a _ { 3 3 } x _ 3 y _ 3 \\\\ B _ 2 & = - x _ 3 y _ 1 + a _ { 1 2 } x _ 2 y _ 2 + a _ { 1 3 } x _ 2 y _ 3 , \\\\ B _ 3 & = - x _ 1 y _ 3 + a _ { 2 1 } x _ 2 y _ 2 + a _ { 3 1 } x _ 3 y _ 2 . \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} C _ 0 : = S \\cap S ' . \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} u ( x , t ) = \\frac { \\delta ( t - \\lvert x \\rvert ) } { 4 \\pi \\lvert x \\rvert } + v ( x , t ) \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{align*} \\begin{aligned} & t \\in [ 0 , \\infty ) \\mapsto M _ { \\rm o u t e r } ( t ) : = \\partial F _ t ( U ) , \\\\ & t \\in [ 0 , \\infty ) \\mapsto M _ { \\rm i n n e r } ( t ) : = \\partial F _ t ( \\overline { U ^ c } ) , \\ , \\\\ & t \\in [ 0 , \\infty ) \\mapsto F _ t ( M ) \\end{aligned} \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{align*} \\Delta _ { A | M } ( \\hat { \\rho } _ { A M } ) ( t ) & : = I ( A : X | M ) _ { \\hat { \\sigma } _ { A M X } ( t ) } \\ge 0 \\ ; , t > 0 \\ ; , \\\\ \\Delta _ { A | M } ( \\hat { \\rho } _ { A M } ) ( 0 ) & : = 0 \\ ; , \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} \\sum _ { i , j = 0 } ^ { r + 1 } a _ { i j } z _ i z _ j = 0 , \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} \\mathbf { K } _ { l } = \\mathbf { K } _ { l , \\delta } ^ { \\leq } + \\mathbf { K } _ { l , \\delta } ^ { > } \\ . \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} H _ w H _ s = \\begin{cases} H _ { w s } & \\mbox { i f } w s > w , \\\\ ( v ^ { - 1 } - v ) H _ w + H _ { w s } & \\mbox { i f } w s < w . \\end{cases} \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{align*} 1 + \\sum _ { m = 0 } ^ j \\frac { B _ { 2 m + 2 } ( 4 ^ { m + 1 } - 1 6 ^ { m + 1 } ) ( 2 j + 1 ) ! } { ( 2 j - 2 m ) ! ( 2 m + 2 ) ! } \\end{align*}"}
-{"id": "3913.png", "formula": "\\begin{align*} { } _ { m + 1 } F _ m \\bigg [ \\begin{matrix} \\alpha _ 0 & \\alpha _ 1 & \\ldots & \\alpha _ m \\\\ & \\beta _ 1 & \\ldots & \\beta _ m \\end{matrix} \\bigg | \\ , z \\bigg ] : = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\alpha _ 0 ) _ k ( \\alpha _ 1 ) _ k \\cdots ( \\alpha _ m ) _ k } { ( \\beta _ 1 ) _ k \\cdots ( \\beta _ m ) _ k } \\cdot \\frac { z ^ k } { k ! } , \\end{align*}"}
-{"id": "56.png", "formula": "\\begin{align*} \\left . \\frac { d } { d t } \\log \\bar { V } _ q ( [ K , g , t ] ) \\right | _ { t = 0 } = \\frac { 1 } { \\widetilde { V } _ q ( K ) } \\int _ { S ^ { n - 1 } } g ( v ) d \\widetilde { C } _ q ( K , v ) , \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} \\overline { \\omega ( f , v _ \\infty ) } \\ ; = \\ ; - \\omega ( v _ \\infty , f ) \\ ; = \\ ; \\lim _ { r \\downarrow 0 } W _ r ( \\overline { v _ \\infty } , f ) \\ ; = \\ ; L _ { v _ \\infty } ( f ) \\ ; = \\ ; 0 \\ , . \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} V ^ \\omega _ W ( q , K , \\Delta ) : = \\Pi ^ \\omega _ W ( q , K , \\Delta ) - \\pi ^ \\omega _ W . \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} f _ { n } ( \\eta ) = \\sqrt { \\frac { ( k - n ) ! } { k ! n ! } } { { \\eta } } ^ { n } . \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} \\mathcal { P } _ m ( t ( \\xi ) ) = t ( D _ m \\xi ) , \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} \\rho = \\log | w | ^ 2 _ h \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{gather*} F _ n = \\sum _ { j = 0 } ^ { \\left \\lfloor \\frac { n - 1 } { 2 } \\right \\rfloor } { n - j - 1 \\choose j } , P _ n = \\sum _ { j = 0 } ^ { \\left \\lfloor \\frac { n - 1 } { 2 } \\right \\rfloor } 2 ^ { n - 2 j - 1 } { n - j - 1 \\choose j } , \\\\ J _ n = \\sum _ { j = 0 } ^ { \\left \\lfloor \\frac { n - 1 } { 2 } \\right \\rfloor } 2 ^ { j } { n - j - 1 \\choose j } . \\end{gather*}"}
-{"id": "1364.png", "formula": "\\begin{align*} f \\left ( x , y , k \\right ) = \\left \\vert u _ { s c } \\left ( x , y , k \\right ) \\right \\vert ^ { 2 } , \\ > \\forall x , y \\in S , x \\neq y , \\forall k \\in \\left ( a , b \\right ) . \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A u + V \\left ( x , t \\right ) u = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "8911.png", "formula": "\\begin{align*} z ( t ) = E _ { \\alpha , 1 } ( A t ^ \\alpha ) z ^ 0 + \\int _ { 0 } ^ { t } ( t - \\tau ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( A ( t - \\tau ) ^ \\alpha ) f ( \\tau ) d \\tau . \\end{align*}"}
-{"id": "178.png", "formula": "\\begin{align*} \\prod \\limits _ { j = 1 } ^ k | E _ j | - \\nu _ k ( 0 ) & \\ge \\prod \\limits _ { j = 1 } ^ k | E _ j | - q ^ { - 1 } \\prod _ { j = 1 } ^ k | E _ j | - q ^ { \\frac { d } { 2 } } \\left ( \\prod _ { j = 1 } ^ k | E _ j | \\right ) ^ { \\frac { k - 1 } { k } } \\\\ & \\ge \\frac { 1 } { 3 } \\prod \\limits _ { j = 1 } ^ k | E _ j | + \\left ( \\frac { 1 } { 3 } \\prod \\limits _ { j = 1 } ^ k | E _ j | - q ^ { \\frac { d } { 2 } } \\left ( \\prod _ { j = 1 } ^ k | E _ j | \\right ) ^ { \\frac { k - 1 } { k } } \\right ) . \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} \\cos \\theta _ i = \\frac { \\alpha _ i } { M _ i } , \\sin \\theta _ i = \\frac { \\beta _ i } { M _ i } . \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{align*} \\partial ^ { \\alpha , \\theta } _ x \\phi : = \\theta { _ a { \\rm D } _ x ^ { \\alpha } } \\phi + ( 1 - \\theta ) { _ x { \\rm D } _ b ^ { \\alpha } } \\phi . \\end{align*} % \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} P ( z ) = z ^ { m + 1 } - c _ 1 z ^ m - \\cdots - c _ m z , \\end{align*}"}
-{"id": "6250.png", "formula": "\\begin{align*} \\begin{aligned} \\forall \\ , \\theta \\in \\Theta : & & \\Re M ( \\theta ) : = \\tfrac { 1 } { 2 } ( M ( \\theta ) + M ( \\theta ) ^ * ) \\geq c , & & A ( \\theta ) = - A ( \\theta ) ^ * , \\end{aligned} \\end{align*}"}
-{"id": "8327.png", "formula": "\\begin{align*} \\langle \\Delta _ { \\Lambda } \\phi ( u ( t ) ) , \\varphi _ \\Lambda ( t ) \\rangle = \\langle \\phi ( u ( t ) ) , \\Delta _ \\Lambda \\varphi _ \\Lambda ( t ) \\rangle . \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} C _ { e a } ( \\Phi ) = \\sup \\left \\{ I ( C : M ) _ { ( \\Phi \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) } : \\hat { \\rho } _ { A M } , \\ ; \\mathrm { T r } _ A \\left [ \\hat { H } _ A \\ , \\hat { \\rho } _ A \\right ] \\le n \\ , E \\right \\} \\ ; . \\end{align*}"}
-{"id": "1044.png", "formula": "\\begin{align*} e _ { i } \\ast e _ { j } = \\sum _ { k } a _ { i , j } ( k ) e _ { k } ( i , j , k = 1 , 2 , \\ldots , n ) \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{align*} P ' _ 2 = \\eta _ { 1 , 2 } P _ 1 + \\eta _ { 0 , 2 } \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} \\left | \\int _ { \\partial \\Omega _ r } t _ \\varepsilon \\frac { \\partial t _ \\varepsilon } { \\partial n } \\right | \\leq C \\varepsilon ^ { 1 / 2 } \\cdot \\varepsilon ^ { 3 / 2 } = C \\varepsilon ^ 2 . \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} x ^ 2 - D y ^ 2 = m . \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} B _ 2 ( p , q , X _ 1 , X _ 2 ) \\ , Q _ { 2 , a } ( X _ 1 , X _ 2 ) = 1 + q X _ 1 - q ( q + 1 ) X _ 1 X _ 2 . \\end{align*}"}
-{"id": "9459.png", "formula": "\\begin{align*} \\int h _ n ( t ) d _ n ( t ) \\cdot \\psi { \\rm d } x - \\int D _ 0 \\cdot \\psi { \\rm d } x = \\int _ 0 ^ t \\int \\mathcal { S } _ { \\varepsilon } ( h _ n , B _ n , d _ n , v _ n ) \\cdot \\psi { \\rm d } x { \\rm d } s \\end{align*}"}
-{"id": "9436.png", "formula": "\\begin{align*} \\mathrm { L } _ { P } ( w ^ * ) = v ^ * - ( b ^ * \\cdot \\nabla ) b ^ * - { h ^ * } ^ { - 1 } \\nabla \\big ( { h ^ * } ^ { - 1 } \\big ) . \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} S ( g , n ) = - g + \\frac { 1 1 } 8 n - \\frac 9 4 g ^ 2 + \\frac 9 8 g n - \\frac 1 2 g ^ 3 + \\frac 3 4 g ^ 2 n - \\frac 1 4 n ^ 3 \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} \\Phi = \\sum _ { j = 1 } ^ m \\phi ^ { ( 1 , j ) } \\otimes \\ldots \\otimes \\phi ^ { ( k , j ) } , \\end{align*}"}
-{"id": "9711.png", "formula": "\\begin{align*} F _ { e x t , X } = \\left ( \\begin{array} { r c r c r } F _ X & F _ X ^ X & \\hdots & \\hdots & F _ X ^ X \\\\ F _ X ^ X & F _ X & F _ X ^ X & \\hdots & \\vdots \\\\ \\vdots & F _ X ^ X & \\ddots & \\hdots & \\vdots \\\\ \\vdots & \\vdots & \\vdots & F _ X & F _ X ^ X \\\\ F _ X ^ X & \\hdots & \\hdots & F _ X ^ X & F _ X \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} V ^ - ( z ) V ^ + ( w ) & = \\exp \\left ( - [ \\sum _ { m > 0 } \\frac { 1 } { m } h _ m z ^ { - 2 m } , \\sum _ { n > 0 } \\frac { 1 } { n } h _ { - n } w ^ { 2 n } ] \\right ) \\cdot V ^ + ( w ) V ^ - ( z ) \\\\ & = \\exp \\left ( \\sum _ { m > 0 } \\frac { 1 } { m } \\frac { w ^ { 2 m } } { z ^ { 2 m } } \\right ) \\cdot V ^ + ( w ) V ^ - ( z ) = i _ { z , w } \\exp \\left ( - \\ln \\left ( 1 - \\frac { w ^ 2 } { z ^ 2 } \\right ) \\right ) \\cdot V ^ + ( w ) V ^ - ( z ) . \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} Y _ n ( \\epsilon ) : = \\sum _ { i = 1 } ^ { n } Y _ i \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} \\theta = ( 2 - m ) d f . \\end{align*}"}
-{"id": "7610.png", "formula": "\\begin{align*} \\Phi _ U : \\mathbb C ^ 2 \\setminus \\{ x _ 2 = 0 \\} \\to U \\end{align*}"}
-{"id": "9510.png", "formula": "\\begin{align*} \\phi ( x ) = \\tilde \\alpha ^ { - 1 } ( x ) \\tilde \\alpha ^ { - 1 } ( x _ 0 ) , \\ \\ \\ x \\in M . \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} & g _ \\varepsilon = g \\partial \\Omega \\cap B _ R ( x _ 0 ) \\ , \\\\ & \\| g _ \\varepsilon \\| _ { H ^ 1 ( \\partial B _ R ( x _ 0 ) \\cap \\Omega ) } \\leq C , \\\\ & \\int _ { \\partial B _ R ( x _ 0 ) \\cap \\Omega } W ( g _ \\varepsilon ) \\leq C \\varepsilon ^ 2 , \\\\ & g _ \\varepsilon \\to g H ^ s ( \\partial B _ R ( x _ 0 ) \\cap \\Omega ) , \\ 0 < s < 1 . \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} n ( x ) = 1 \\ > x \\in \\mathbb { R } ^ { 3 } \\setminus \\Omega . \\end{align*}"}
-{"id": "3879.png", "formula": "\\begin{align*} \\sup _ { x \\in [ - 1 + h - a , - 1 + h + a ] } f ( x ) & = \\max \\{ f ( - 1 + h - a ) , f ( - 1 + h + a ) \\} \\stackrel { ( \\ref { e q : f - e n d p o i n t s - m a x - l e f t } ) } { = } f ( - 1 + h - a ) = 2 r _ x \\left ( \\frac { 1 } { r _ x } + h \\left ( 1 - \\frac { 1 } { r _ x } \\right ) \\right ) . \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} ( a ^ { 2 } c + a ^ { 2 } + b ^ { 2 } + c ^ { 2 } + 1 ) x + ( a ^ { 3 } + a b ^ { 2 } + a b c + a c + a ) = 0 . \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{align*} \\phi _ { m , n } ( z ) = \\sqrt { 2 } \\int _ \\mathbb { R } e ^ { 4 \\pi i ( x \\xi + \\frac { 1 } { 2 } x y ) } \\tilde { h } _ m ( \\xi + y ) \\tilde { h } _ n ( \\xi ) \\ , d \\xi , \\end{align*}"}
-{"id": "6498.png", "formula": "\\begin{align*} \\mathcal { K } _ + ( \\nu ) = \\{ P _ \\theta ( d x ) ; \\theta \\in ( 0 , \\theta _ + ) \\} = \\{ Q _ m ( d x ) , m \\in ( m _ 0 , m _ + ) \\} \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} y ^ 2 = z ^ 3 - 2 z + 1 . \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} ( \\pi _ U ) _ * M A ( \\Phi + h ) = N \\int _ { t = 0 } ^ 1 M A ( ( 1 - t ) u + t v ) t ^ { N - 1 } d t \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} \\upsilon \\left ( x , t \\right ) = U \\left ( x , 1 - t \\right ) u \\left ( \\frac { x } { 1 - t } , \\frac { t } { 1 - t } \\right ) \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} \\mathrm { P } ( [ E ] ) = \\mathrm { P } \\cap \\mathbf { P } \\bigl ( [ E ] \\bigr ) , \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} F = F ( \\mu ) = \\{ P ( \\theta , \\mu ) , \\ \\theta \\in \\Theta ( \\mu ) \\} \\end{align*}"}
-{"id": "1784.png", "formula": "\\begin{align*} A x : = \\lim _ { j \\to \\infty } z _ j , \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} u \\big ( r \\cos \\theta , r \\sin \\theta \\big ) & = A _ 1 \\ , r ^ { m + s } \\ , ( \\sin \\theta ) ^ s \\ , P ^ s _ \\nu ( \\cos \\theta ) \\\\ & \\stackrel { \\eqref { e : P s } } { = } A _ 1 \\ , r ^ { m + s } \\ , \\sum _ { k = 0 } ^ { m } \\frac { ( m + 1 ) _ k ( - m ) _ k } { 2 ^ k k ! \\ , ( 1 - s ) _ k } \\big ( 1 - \\cos \\theta \\big ) ^ k \\big ( 1 + \\cos \\theta \\big ) ^ s \\ , . \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} J _ 2 ^ { k } = \\big ( | \\Psi _ { h } ^ { k } | ^ { 2 } - | \\mathcal { I } _ { h } \\Psi ^ { k } | ^ { 2 } , \\ ; \\overline { \\theta } _ { \\phi } ^ { k } \\big ) - \\big ( | \\Psi _ { h } ^ { k - 1 } | ^ { 2 } - | \\mathcal { I } _ { h } \\Psi ^ { k - 1 } | ^ { 2 } , \\ ; \\overline { \\theta } _ { \\phi } ^ { k } \\big ) , \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { - g } } \\partial _ { \\mu } \\left ( \\sqrt { - g } g ^ { \\mu \\nu } \\partial _ { \\nu } \\Phi \\right ) = 0 , \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} 5 F _ u { } ^ 2 = L _ { 2 u } - ( - 1 ) ^ u 2 \\ , , u \\in \\Z \\ , , \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{align*} h ( t ) + \\int _ x ^ \\infty h ( s ) F ( s + t ) d s = 0 , \\end{align*}"}
-{"id": "5118.png", "formula": "\\begin{align*} \\frac { B _ { 2 } \\left ( x \\right ) } { 2 ! } = \\frac { 1 } { 2 } \\left [ x ^ { 2 } - \\left ( x - 1 \\right ) ^ { 2 } \\right ] - \\frac { 1 } { 6 } \\left [ x ^ { 3 } - \\left ( x - 1 \\right ) ^ { 3 } \\right ] . \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} A \\cdot G = G _ { l e f t } ^ { - 1 } \\cdot G = I _ n . \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} \\Bigl | \\Bigl | \\ , T ^ { \\ , n } \\Bigl ( \\sum _ { s = 1 } ^ { j _ { m } + j } z _ { s } \\Bigr ) - x _ { l } \\ , \\Bigr | \\Bigr | < \\sum _ { u = 0 } ^ { j } 2 ^ { - ( j _ { m } + u ) } . \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { \\Gamma \\left ( k + 1 \\right ) } I _ { v - k } ( q + 2 k ; x ) t ^ { k } = \\sum _ { n = 0 } ^ { \\infty } \\frac { 2 ^ { v + q + n - \\frac { 1 } { 2 } } x ^ { n + v } } { \\ \\sqrt { \\pi } \\Gamma ( 2 \\upsilon + q + n + \\frac { 1 } { 2 } ) \\ n ! } \\cdot \\sum _ { k = 0 } ^ { \\infty } \\frac { \\Gamma ( \\upsilon + q + n + k ) 2 ^ { k } t ^ { k } } { \\ \\ k ! x ^ { k } } . \\end{align*}"}
-{"id": "9919.png", "formula": "\\begin{align*} J _ { \\nu - 1 } ( z ) - J _ { \\nu + 1 } ( z ) & = 2 J ' _ \\nu ( z ) , \\\\ J _ { \\nu - 1 } ( z ) + J _ { \\nu + 1 } ( z ) & = \\frac { 2 \\nu } { z } J _ \\nu ( z ) . \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} m _ \\omega ( S _ 1 S _ 2 ) = m _ \\omega ( S _ 1 ) S _ 2 + m _ \\omega ( S _ 2 ) \\end{align*}"}
-{"id": "328.png", "formula": "\\begin{align*} \\eta _ a ( C ( \\mathsf { P } ) ) = \\sum _ i c ^ i _ i q ^ { ( \\alpha _ a - 2 \\rho , \\lambda _ i ) } [ d _ a ^ { - 1 } ( \\alpha _ a , \\lambda _ i ) ] _ { q _ a } . \\end{align*}"}
-{"id": "9928.png", "formula": "\\begin{align*} \\alpha = \\left ( \\frac { \\frac { 1 } { 2 } + \\varepsilon } { \\frac { 1 } { 2 } - \\varepsilon } \\right ) \\varepsilon ' . \\end{align*}"}
-{"id": "422.png", "formula": "\\begin{align*} h ^ 0 ( E ( - b ) | _ S ) = 1 \\end{align*}"}
-{"id": "5424.png", "formula": "\\begin{gather*} \\frac { d s } { d t } = - k _ 1 e \\ , s + k _ { - 1 } c , \\frac { d e } { d t } = - k _ 1 e \\ , s + ( k _ { - 1 } + k _ 2 ) c , \\\\ \\frac { d c } { d t } = k _ 1 e \\ , s - ( k _ { - 1 } + k _ 2 ) c , \\frac { d p } { d t } = k _ 2 c , \\end{gather*}"}
-{"id": "6975.png", "formula": "\\begin{align*} V ' = \\{ v _ { 1 } , v _ { 2 } , \\ldots , v _ { \\frac { q - 1 } { 2 } + 1 } \\} , \\end{align*}"}
-{"id": "7822.png", "formula": "\\begin{align*} \\iint ( g ( U ) ) _ x U \\ , d x d y = 0 . \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} G _ \\nu ( z ) = \\int \\frac { 1 } { z - x } \\nu ( d x ) . \\end{align*}"}
-{"id": "2881.png", "formula": "\\begin{align*} \\gamma _ { 1 n } = \\left [ \\begin{array} { c | c } - g _ { n + 1 } I _ { n } & B _ { n } \\\\ \\hline d _ { 1 n } & ( - 1 ) ^ { n + 1 } \\Delta _ { n + 1 } \\end{array} \\right ] , \\mbox { w i t h } B _ { n } = \\begin{pmatrix} - \\Delta _ { 1 } \\\\ \\vdots \\\\ ( - 1 ) ^ { n } \\Delta _ { n } \\end{pmatrix} . \\end{align*}"}
-{"id": "5783.png", "formula": "\\begin{align*} \\phi ( t ) - \\phi ( 0 ) = \\int _ 0 ^ t \\phi ' ( s ) \\ , \\mathrm { d } s \\le 0 \\ ; . \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\left ( D \\phi _ { t } \\left ( x \\right ) \\right ) \\cdot \\Pi \\cdot \\left ( D \\phi _ { t } \\left ( x \\right ) \\right ) ^ { t } = \\Pi \\left ( \\phi _ { t } \\left ( x \\right ) \\right ) , \\\\ \\left ( \\forall \\right ) \\ t \\in \\mathbb { R } , \\ \\left ( \\forall \\right ) \\ x \\in \\mathbb { R } ^ { n } , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\end{array} \\right . \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} \\Psi ( p ) = & \\lim _ { x \\to 0 } { } _ 3 F _ 2 \\bigg [ \\begin{matrix} - a + p + x & 1 + a - p - x & \\beta \\\\ & 1 & 2 \\beta \\end{matrix} \\bigg | \\ , 1 \\bigg ] \\\\ = & \\lim _ { x \\to 0 } \\frac { \\Gamma ( \\frac 1 2 ) \\Gamma ( \\frac 1 2 + \\beta ) \\Gamma ( \\beta ) } { \\Gamma ( \\frac 1 2 - \\frac { p - 1 - a + x } 2 ) \\Gamma ( 1 + \\frac { { p - 1 - a + x } } 2 ) \\Gamma ( \\beta - \\frac { p - 1 - a + x } 2 ) \\Gamma ( \\frac 1 2 + \\beta + \\frac { p - 1 - a + x } 2 ) } = 0 . \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} \\| \\ , B _ { A , \\pmb { \\omega } } x \\ , | | ^ { 2 } = \\biggl | \\biggl | A P _ { r } x + \\delta \\sum _ { k = 1 } ^ { r } x _ { k + r } e _ { k } \\biggr | \\biggr | ^ { 2 } + M ^ { 2 } \\sum _ { k > r } | x _ { k + r } | ^ { 2 } . \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} ( - a - p ) _ k ( p - a ) _ k = \\frac { \\Gamma _ p ( - a + k - p ) } { \\Gamma _ p ( - a - p ) } \\cdot \\frac { \\Gamma _ p ( - a + k + p ) } { \\Gamma _ p ( - a + p ) } \\equiv \\frac { \\Gamma _ p ( - a + k ) ^ 2 } { \\Gamma _ p ( - a ) ^ 2 } = ( - a ) _ k ^ 2 \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{align*} h ^ x _ 0 e ^ { m \\alpha } P ( x _ 1 , x _ 2 , \\dots , x _ n , \\dots ; y _ 1 , y _ 2 , \\dots , y _ n \\dots ) = m e ^ { m \\alpha } P ( x _ 1 , x _ 2 , \\dots , x _ n , \\dots ; y _ 1 , y _ 2 , \\dots , y _ n \\dots ) . \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} g ' \\kappa g '^ { - 1 } & = g ' ( I + A p ) g '^ { - 1 } \\\\ & = ( g + B p ) ( I + A p ) ( g ^ { - 1 } - g ^ { - 1 } B g ^ { - 1 } p ) \\\\ & = I + ( B g ^ { - 1 } + g A g ^ { - 1 } - B g ^ { - 1 } ) p \\\\ & = I + g A g ^ { - 1 } \\\\ & = g ( I + A p ) g ^ { - 1 } \\\\ & = g \\kappa g ^ { - 1 } . \\end{align*}"}
-{"id": "9542.png", "formula": "\\begin{align*} \\sigma ( e , s ) = \\sigma ( s , e ) = 1 , \\ \\ \\ s \\in G . \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} F ( q ^ { - 1 } ) : = H ( q ^ { - 1 } ) L ( q ^ { - 1 } ) , \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} \\bigl \\lvert \\lvert \\varphi \\rvert ( x ) \\bigl \\rvert \\leq \\lvert \\varphi \\rvert \\bigl ( \\lvert x \\rvert \\bigl ) = \\Bigl \\{ \\bigl \\lvert \\varphi ( y ) \\bigl \\rvert \\ : \\lvert y \\rvert \\leq \\lvert x \\rvert \\Bigl \\} \\leq \\varepsilon . \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{align*} & A = [ Z ' ; H _ { \\gamma } ' \\otimes K _ { \\gamma } ' ; \\psi _ { \\gamma , \\delta } ^ { H ' } \\otimes \\psi _ { \\gamma , \\delta } ^ { K ' } ] \\\\ & B = [ Z '' ; H _ { \\tau } '' \\otimes K _ { \\tau } '' ; \\psi _ { \\tau , \\sigma } ^ { H '' } \\otimes \\psi _ { \\tau , \\sigma } ^ { K '' } ] \\end{align*}"}
-{"id": "8803.png", "formula": "\\begin{align*} 2 \\tilde { G } ^ { i } = 2 G ^ { i } + \\sigma _ { , k } y ^ { k } y ^ { i } - \\dfrac { 1 } { 2 } g ^ { i h } \\sigma _ { , h } L . \\end{align*}"}
-{"id": "5663.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } - 2 a \\Omega _ { 1 1 } - \\Omega _ { 1 2 } - \\Omega _ { 2 1 } + 1 = 0 , \\\\ b \\varepsilon \\Omega _ { 1 1 } + ( - a - c \\varepsilon ) \\Omega _ { 1 2 } - \\Omega _ { 2 2 } = 0 , \\\\ b \\varepsilon \\Omega _ { 1 1 } + ( - c \\varepsilon - a ) \\Omega _ { 2 1 } - \\Omega _ { 2 2 } = 0 , \\\\ b \\varepsilon \\Omega _ { 1 2 } + b \\varepsilon \\Omega _ { 2 1 } - 2 c \\varepsilon \\Omega _ { 2 2 } = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "9873.png", "formula": "\\begin{align*} 0 & = ( ( c _ 1 ( E _ 1 + E _ 2 ) + ( c _ 2 - c _ 1 ) E _ 2 + D ) | _ { E _ 1 } \\cdot A _ 1 ) \\\\ & = - 2 c _ 1 + c _ 2 + b _ 1 + C _ 1 \\cdot A _ 1 \\\\ & \\ge - 3 c _ 1 + ( k - a _ 1 ) \\end{align*}"}
-{"id": "1025.png", "formula": "\\begin{align*} z y ^ 2 = x ^ 3 + \\tfrac { 1 } { 4 } b _ 2 x ^ 2 z + \\tfrac { 1 } { 2 } b _ 4 x z ^ 2 + \\tfrac { 1 } { 4 } b _ 6 z ^ 3 . \\end{align*}"}
-{"id": "9980.png", "formula": "\\begin{align*} x ( t _ * ^ + ) = M _ { j i } x ( t _ * ^ - ) \\end{align*}"}
-{"id": "9788.png", "formula": "\\begin{align*} A _ k ( M , g ) : = \\sup \\{ \\| \\xi \\| _ { L ^ { \\infty } ( M ) } / \\max _ { y \\in M } \\int _ M \\frac { | \\Delta _ H \\xi | ( x ) } { d i s t ( x , y ) ^ { n - 2 } } \\mid \\xi \\perp \\mathcal { H } ^ k \\} < \\infty , \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{align*} \\dot z = Q v ( x _ 0 + \\Gamma _ 0 z ) \\ , . \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{gather*} \\Delta ( K _ i ) = K _ i \\otimes K _ i , \\Delta ( E _ i ) = E _ i \\otimes K _ i + 1 \\otimes E _ i , \\Delta ( F _ i ) = F _ i \\otimes 1 + K _ i ^ { - 1 } \\otimes F _ i , \\\\ S ( K _ i ) = K _ i ^ { - 1 } , S ( E _ i ) = - E _ i K _ i ^ { - 1 } , S ( F _ i ) = - K _ i F _ i , \\varepsilon ( K _ i ) = 1 , \\varepsilon ( E _ i ) = \\varepsilon ( F _ i ) = 0 . \\end{gather*}"}
-{"id": "8554.png", "formula": "\\begin{align*} \\mathrm { ( I I I ) } ^ N = 0 . \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} u _ t = \\Delta u + | u | ^ \\alpha u \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{align*} \\int _ { X \\in H } ^ { } e ^ { i T r ( \\textnormal { d i a g } ( r _ 1 , \\cdots , r _ n , 0 , 0 , \\cdots ) X ) } M _ { \\omega } ( d X ) = \\prod _ { j = 1 } ^ { n } F _ { \\omega } ( r _ j ) . \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} & M _ a ( \\lambda ) \\asymp \\sum _ { \\nu _ 1 = 1 } ^ \\infty \\cdots \\sum _ { \\nu _ r = 1 } ^ \\infty \\frac { \\prod _ { k = 1 } ^ r \\nu _ k ^ { 2 m a } } { \\left [ 1 + \\lambda \\prod _ { k = 1 } ^ r \\nu _ k ^ { 2 m } ( 1 + \\nu _ r ^ 2 ) ^ { - 1 } \\right ] ^ { 2 } } \\\\ & \\asymp \\int _ 1 ^ \\infty \\int _ 1 ^ \\infty \\cdots \\int _ 1 ^ \\infty \\left [ 1 + \\lambda x _ 1 ^ { b } \\cdots x _ { r - 1 } ^ { b } x _ r ^ { b ( m - 1 ) / m } \\right ] ^ { - 2 } d x _ 1 \\cdots d x _ { r - 1 } d x _ r , \\end{align*}"}
-{"id": "9413.png", "formula": "\\begin{align*} \\langle u ( x ) , y \\rangle = \\int _ { \\Sigma } \\bigl [ \\alpha ( x ) ] ( \\omega ) [ \\beta ( y ) ] ( \\omega ) \\ , \\nu ( \\omega ) , x \\in E , y \\in F . \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} \\widehat v ( \\rho ) = a \\rho + ( b - a ) \\log ( e ^ \\rho + 1 ) . \\end{align*}"}
-{"id": "4853.png", "formula": "\\begin{align*} \\begin{pmatrix} w & 0 \\\\ 0 & - w \\end{pmatrix} \\begin{pmatrix} 0 & x \\\\ y & 0 \\end{pmatrix} \\begin{pmatrix} w & 0 \\\\ 0 & - w \\end{pmatrix} ^ { - 1 } \\begin{pmatrix} 0 & x \\\\ y & 0 \\end{pmatrix} ^ { - 1 } = - I \\end{align*}"}
-{"id": "3403.png", "formula": "\\begin{align*} H ( t , x , y , q , r , \\gamma ) = \\lambda \\left ( \\frac { \\gamma ^ 2 } { 2 } - \\theta \\gamma ( m ( t ) - x ) + \\frac { \\varepsilon } { 2 } ( m ( t ) - x ) ^ 2 \\right ) + [ a ( m ( t ) - x ) + \\lambda \\gamma ] y + \\sigma q + \\gamma r \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} \\mathbb { E } _ { S _ N } \\left [ \\binom { X } { \\mu } \\right ] \\Phi \\left [ \\prod _ { k = 1 } ^ { \\infty } \\left ( \\sum _ { \\{ i : d _ i | k \\} } \\epsilon _ i ^ { \\frac { k } { d _ i } } d _ i \\right ) ^ { \\mu _ k } \\right ] \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} & G _ { 0 , 1 } ( x , T ) = \\sum _ { m = 0 } ^ { \\infty } h _ { 0 , 1 , m } ( x ) T ^ { m } = \\sum _ { m = 0 } ^ { \\infty } \\frac { 1 } { 2 } ( ( 1 + \\sqrt { x } ) ^ { m } + ( 1 - \\sqrt { x } ) ^ { m } ) T ^ { m } = \\frac { T - 1 } { ( x - 1 ) T ^ 2 + 2 T - 1 } , \\\\ & G _ { 1 , 1 } ( x , T ) = \\sum _ { m = 0 } ^ { \\infty } h _ { 1 , 1 , m } ( x ) T ^ { m } = \\sum _ { m = 0 } ^ { \\infty } \\frac { 1 } { 2 \\sqrt { x } } ( ( 1 + \\sqrt { x } ) ^ { m } - ( 1 - \\sqrt { x } ) ^ { m } ) T ^ { m } = - \\frac { T } { ( x - 1 ) T ^ 2 + 2 T - 1 } . \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} \\overline \\tau _ { \\theta } : = \\inf \\{ t \\geq \\theta , \\ , \\ , A _ t > A _ { \\theta } \\ , \\ , { \\rm o r } \\ , \\ , C _ { t ^ - } > C _ { \\theta ^ - } \\} ; \\overline \\sigma _ { \\theta } : = \\inf \\{ t \\geq \\theta , \\ , \\ , A ' _ t > A ' _ { \\theta } \\ , \\ , { \\rm o r } \\ , \\ , C ' _ { t ^ - } > C ' _ { \\theta ^ - } \\} . \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} \\phi ( a + b ) & = \\phi ( a ) + \\phi ( b ) \\\\ \\phi ( a b ) & = \\phi ( a ) \\phi ( b ) \\\\ \\phi ( 1 _ R ) & = 1 _ S \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k c _ i f ( x _ i ) = 0 { \\rm f o r } ~ ~ ~ f \\in E . \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} \\mu = - \\frac { X _ 3 + X _ 4 } { X _ 1 + X _ 2 + X _ 3 + X _ 4 } . \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} ^ { - 1 } = \\begin{pmatrix} a + c & - a + b - c + d \\\\ c & - c + d \\end{pmatrix} . \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} L _ { j + 1 } \\cap X _ { 1 , \\ , j + 1 , \\ , 3 } = 0 . \\end{align*}"}
-{"id": "8117.png", "formula": "\\begin{align*} W _ F : = ( W \\cap \\{ F = 0 \\} ) ^ m . \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} \\mathfrak { m } = \\prod _ { v \\mid \\ell } \\mathfrak { p } _ v ^ { 1 + e _ { v } \\ell / ( \\ell - 1 ) } \\prod _ { v \\mid v _ 0 \\in \\Sigma _ { E , p . m . } } \\mathfrak { p } _ v , \\end{align*}"}
-{"id": "7765.png", "formula": "\\begin{align*} \\sigma _ { t } ^ { \\prime } [ l ] = \\sigma _ { t - 1 } ^ { \\prime } [ l ] \\leq O P T _ { t - 1 } ^ { \\prime } [ l ] \\leq O P T _ { t - 1 } ^ { \\prime } [ l + 1 ] = O P T _ { t } ^ { \\prime } [ l ] . \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{align*} \\tilde { F } : = - 4 \\delta u _ { c _ * } - 8 \\delta b \\cos ( y ) \\psi _ * + 6 ( 2 b \\cos ( y ) \\psi _ * + \\tilde { u } _ b ) ^ 2 \\end{align*}"}
-{"id": "3979.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ a \\frac { ( - a - p ) _ { p + k } ( \\frac 1 2 ) _ { p + k } } { ( 1 ) _ { p + k } ^ 2 } \\cdot \\bigg ( \\frac { 4 z } { ( 1 + z ) ^ 2 } \\bigg ) ^ k \\equiv - \\frac { \\phi ( 0 ) } { 2 } \\cdot ( 4 ^ { 1 - p } + p \\cdot H _ { a } ) - \\frac { p } { 2 } \\cdot \\phi ' ( 0 ) \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{align*} I _ { \\alpha , \\beta } & : = \\psi _ { \\alpha , \\beta } = \\{ a _ { \\beta } \\in G _ { \\beta } : \\exists a _ { \\alpha } \\in G _ { \\alpha } , a _ { \\alpha } \\psi _ { \\alpha , \\beta } = a _ { \\beta } \\} , \\\\ K _ { \\alpha , \\beta } & : = \\psi _ { \\alpha , \\beta } = \\{ a _ { \\alpha } \\in G _ { \\alpha } : a _ { \\alpha } \\psi _ { \\alpha , \\beta } = e _ { \\beta } \\} , \\end{align*}"}
-{"id": "3648.png", "formula": "\\begin{align*} \\chi ( z ) = \\sum _ { n \\in \\mathbb { Z } + 1 / 2 } \\chi _ n z ^ { - n - 1 / 2 } \\end{align*}"}
-{"id": "7719.png", "formula": "\\begin{align*} f ' ( x _ 1 , \\dots , x _ m ) = \\sum _ { i = 1 } ^ m \\sum _ { j = 1 } ^ { n _ i } a _ { i , j } \\ , x _ i \\ , b _ { i , j } \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} \\det \\left ( \\mathbb { I } - S _ v ( k ) T ( k ; \\mathbf { l } ) \\right ) = 0 , \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} q ^ b | d , q = P ( q ^ b ) \\leq y , y < q ^ b . \\end{align*}"}
-{"id": "9061.png", "formula": "\\begin{align*} \\Sigma _ 2 = \\sum _ { j = 1 } ^ n K _ { j - 1 } ( s ) K _ { i - j + 1 } ( m ) \\enskip ( j - 1 : = j ) = \\sum _ { j = 0 } ^ { n - 1 } K _ { j } ( s ) K _ { i - j } ( m ) . \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{align*} \\rho ' : = r \\in \\mathrm { c e n t e r } ( B ' ) , \\phi ' : = \\phi _ { - 1 } \\in M ' \\end{align*}"}
-{"id": "8720.png", "formula": "\\begin{align*} D H ( | x | ) & = \\frac { 2 x } { ( 1 + | x | ^ { 2 } ) ^ { 2 } } , \\ \\ D ^ { 2 } H ( | x | ) = \\frac { 2 I } { ( 1 + | x | ^ { 2 } ) ^ { 2 } } - \\frac { 8 x x ^ { T } } { ( 1 + | x | ^ { 2 } ) ^ { 3 } } . \\end{align*}"}
-{"id": "7932.png", "formula": "\\begin{align*} h ( t ) = 0 \\quad \\mbox { o n $ [ T _ 0 , \\infty ) $ f o r s o m e $ T _ 0 > 0 $ } ; h ( 0 ) = 1 . \\end{align*}"}
-{"id": "4240.png", "formula": "\\begin{align*} \\begin{array} { c c c } ( A u ) _ 1 L _ f ( p ) & = & u _ { + } L _ 1 ( p ) \\\\ ( A u ) _ 2 L _ f ( p ) & = & u _ { + } L _ 2 ( p ) \\\\ & \\vdots \\\\ ( A u ) _ { d - 1 } L _ f ( p ) & = & u _ { + } L _ { d - 1 } ( p ) \\end{array} \\end{align*}"}
-{"id": "4921.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ { n } A _ { i j } w _ { j } = \\left ( \\sum \\limits _ { j = 1 } ^ { n } s _ { i j } v _ { j } \\right ) \\mathbf { e = } \\left ( \\lambda v _ { i } \\right ) \\mathbf { e = } \\lambda \\left ( v _ { i } \\mathbf { e } \\right ) = \\lambda w _ { i } \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} M _ { \\pm \\gamma } = \\left ( M _ { { i j } _ { \\pm \\gamma } } \\right ) M _ { { i j } _ { \\pm \\gamma } } = \\begin{pmatrix} \\frac { s _ { i j } \\pm \\gamma c _ { i j } } { 2 } & \\frac { s _ { i j } \\mp \\gamma c _ { i j } } { 2 } \\\\ \\frac { s _ { i j } \\mp \\gamma c _ { i j } } { 2 } & \\frac { s _ { i j } \\pm \\gamma c _ { i j } } { 2 } \\end{pmatrix} , \\ \\ 1 \\leq i , j \\leq n \\end{align*}"}
-{"id": "6596.png", "formula": "\\begin{align*} \\Phi ^ { \\pm } ( r ) \\ ; : = \\ ; e ^ { - r } r ^ { - B } \\big ( \\textstyle { \\frac { \\pm ( 1 + \\nu ) + B } { 1 + \\nu } } \\ , U _ { - B , 1 - 2 B } ( 2 r ) - \\textstyle { \\frac { 2 r B } { 1 + \\nu } } \\ , U _ { 1 - B , 2 - 2 B } ( 2 r ) \\big ) \\ , , \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} \\widehat { I _ N f } ( \\xi ) : = m _ N ( \\xi ) \\hat { f } ( \\xi ) , \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} T ( c ) = \\frac { \\alpha c } { 1 + \\alpha c } , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\alpha > 0 . \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{align*} \\Psi _ 2 ' ( 0 ) = & 2 z ^ a \\cdot \\frac { d } { d x } \\bigg ( { } _ 2 F _ 1 \\bigg [ \\begin{matrix} - a & - a - x \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac 1 z \\bigg ] \\bigg ) \\bigg | _ { x = 0 } \\\\ = & 2 z ^ a \\cdot \\frac { d } { d x } \\bigg ( \\sum _ { k = 0 } ^ a \\frac { ( - a ) _ { a - k } ( - a - x ) _ { a - k } } { ( 1 ) _ { a - k } ^ 2 } \\cdot \\frac { 1 } { z ^ { a - k } } \\bigg ) \\bigg | _ { x = 0 } . \\end{align*}"}
-{"id": "3813.png", "formula": "\\begin{align*} t _ { 2 ^ k } ( 2 ^ { k + 1 } n + 2 j ) = \\sum _ { i = 0 } ^ { 2 ^ { k - 1 } } { 2 ^ k \\choose 2 i } t _ { 2 ^ k } ( 2 ^ k n + j - i ) \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} 2 5 \\sum _ { k = 1 } ^ n F _ k { } ^ 4 = F _ { 2 n + 1 } ( L _ { 2 n + 1 } + 4 ( - 1 ) ^ { n - 1 } ) + 6 n + 3 \\ , . \\end{align*}"}
-{"id": "393.png", "formula": "\\begin{align*} T \\equiv g _ 2 ( M ) N ^ 2 + g _ 1 ( M ) N + D = 0 \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{gather*} \\sum _ { j \\geq 0 } t ^ { j } T _ { 2 j + 1 } ( x ) T _ { 2 j + 1 } ( y ) = \\\\ \\frac { ( 1 - t ) x y ( 1 + 6 t + t ^ { 2 } - 4 t ( x ^ { 2 } + y ^ { 2 } ) ) } { ( 1 - t ) ^ { 4 } + 8 t ( 1 - t ) ^ { 2 } ( x ^ { 2 } + y ^ { 2 } ) - 1 6 t ( 1 + t ^ { 2 } ) x ^ { 2 } y ^ { 2 } + 1 6 t ^ { 2 } ( x ^ { 4 } + y ^ { 4 } ) } . \\end{gather*}"}
-{"id": "5971.png", "formula": "\\begin{align*} \\zeta _ { 1 } \\lambda + \\zeta _ { 2 } = 0 , \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} \\div \\ , W ( x ) & = \\phi ' \\big ( \\textstyle { \\frac { | x | } { t } } \\big ) \\cdot \\ , \\frac { x } { t \\ , | x | } \\Big ( \\frac { | \\nabla u | ^ 2 } { 2 } x - ( \\nabla u \\cdot x ) \\nabla u \\Big ) | x _ { n + 1 } | ^ a + \\phi \\big ( \\textstyle { \\frac { | x | } { t } } \\big ) \\ , \\frac { n + a - 1 } { 2 } \\ , | \\nabla u ( x ) | ^ 2 | x _ { n + 1 } | ^ a . \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{align*} - \\log ( 1 - | \\alpha | ^ 2 ) = \\sum _ { m = 1 } ^ { \\infty } \\frac { | \\alpha | ^ { 2 m } } { m } \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} \\tilde { p } _ { 2 n - 2 } \\sum _ { j = 0 } ^ { n - 1 } { b } _ { k j } { c } _ { j + l } = \\delta _ { k l } , k , l = 0 , \\dots , n - 1 , \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} \\Delta _ { K , 2 } ( t ) = t ^ 2 + t + 1 \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} { { d } \\over { d t } } E ( t ) = - \\int _ { \\Omega } b _ { 1 } ( x ) | u _ { t } ( t , x ) | ^ 2 \\ , d x - \\int _ { \\Omega } b _ { 2 } ( x ) | \\nabla u _ t ( t , x ) | ^ 2 d x . \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{align*} E _ k \\triangleright ( v _ m \\otimes f ^ m ) & = E _ k \\triangleright v _ m \\otimes K _ k \\triangleright f ^ m + v _ m \\otimes E _ k \\triangleright f ^ m \\\\ & = q ^ { - \\delta _ { m , k } + \\delta _ { m , k + 1 } } \\delta _ { m } ^ { k + 1 } q ^ { - 1 / 2 } v _ { m - 1 } \\otimes f ^ { m } - \\delta _ { m } ^ { k } q ^ { 1 / 2 } v _ { m } \\otimes f ^ { m + 1 } \\\\ & = \\delta ^ { k + 1 } _ m q ^ { 1 / 2 } v _ { m - 1 } \\otimes f ^ m - \\delta ^ k _ m q ^ { 1 / 2 } v _ m \\otimes f ^ { m + 1 } . \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} \\mathcal { G } _ { r } ( D , \\gamma ) \\cap B _ { \\delta ( r ) } ( p ) = C N _ { r } ( \\gamma _ p , p ) \\cap B _ { \\delta ( r ) } ( p ) , \\end{align*}"}
-{"id": "8906.png", "formula": "\\begin{align*} E _ { \\alpha , \\beta } ( A ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { A ^ k } { \\Gamma ( \\alpha k + \\beta ) } , \\ \\alpha > 0 , \\ \\beta \\in \\mathbb { C } , \\end{align*}"}
-{"id": "2709.png", "formula": "\\begin{align*} \\rho = \\varrho \\bigl ( \\theta , ( y _ i ) _ { 1 \\leq 1 \\leq N _ s } \\bigr ) = \\frac 1 { \\displaystyle \\frac { R \\theta } { P _ { t h } } \\sum _ { i = O , P , N } \\frac { y _ i } { W _ i } + \\frac { y _ F } { \\rho _ F } } , \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ { 1 } U _ { j } ( x ) \\frac { 1 } { \\pi \\sqrt { 1 - x ^ { 2 } } } d x = \\left \\{ \\begin{array} [ c ] { c c c } 0 & i f & j \\\\ 1 & i f & j \\end{array} \\right . . \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} \\ \\partial _ { t } u + \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert = 2 l } a _ { \\alpha } D ^ { \\alpha } u + A u = f \\left ( t , x \\right ) , t \\in \\left ( 0 , \\infty \\right ) x \\in R ^ { n } , \\end{align*}"}
-{"id": "4370.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\binom { 2 k } { k } x ^ k = \\frac { 1 } { \\sqrt { 1 - 4 x } } , \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} I ( G ) = ( x _ i x _ j : \\{ i , j \\} \\in E ( G ) ) . \\end{align*}"}
-{"id": "3143.png", "formula": "\\begin{align*} \\displaystyle { C _ { i j k } = - \\frac { ( n - 2 ) } { ( n - 3 ) } \\nabla _ { l } W _ { i j k l } , } \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} f = x _ 2 y _ 2 ( \\alpha x _ 1 y _ 2 + \\beta x _ 2 y _ 1 + \\gamma x _ 2 y _ 2 ) \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 1 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 2 e _ 4 + \\alpha _ 3 e _ 5 , [ e _ 2 , e _ 1 ] = - \\alpha _ 2 e _ 4 + \\alpha _ 4 e _ 5 , [ e _ 2 , e _ 2 ] = \\alpha _ 5 e _ 5 , [ e _ 1 , e _ 3 ] = \\beta _ 1 e _ 4 + \\beta _ 2 e _ 5 , \\\\ [ e _ 3 , e _ 1 ] = - \\beta _ 1 e _ 4 + \\beta _ 3 e _ 5 , [ e _ 2 , e _ 3 ] = \\beta _ 4 e _ 4 + \\beta _ 5 e _ 5 , [ e _ 3 , e _ 2 ] = - \\beta _ 4 e _ 4 + \\beta _ 6 e _ 5 , [ e _ 3 , e _ 3 ] = \\beta _ 7 e _ 5 . \\end{align*}"}
-{"id": "2809.png", "formula": "\\begin{align*} N _ i = \\max ( v _ p ( D _ i ) , \\nu _ i + v _ p ( d _ i ) ) . \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{align*} H : = \\sum _ { i = 1 } ^ n x _ i \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} ( B ^ + , B ^ - ) ( t ) : = \\begin{cases} ( B _ 1 ( \\tau _ 1 + t ) - B _ 1 ( \\tau _ 1 ) , B _ 2 ( \\tau _ 1 + t ) - B _ 1 ( \\tau _ 1 ) ) & B _ 1 ( \\tau _ 1 + 1 ) > B _ 2 ( \\tau _ 1 + t ) \\\\ ( B _ 2 ( \\tau _ 1 + t ) - B _ 1 ( \\tau _ 1 ) , B _ 1 ( \\tau _ 1 + t ) - B _ 1 ( \\tau _ 1 ) ) & B _ 2 ( \\tau _ 1 + 1 ) > B _ 1 ( \\tau _ 1 + t ) . \\end{cases} \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{align*} L ^ \\prime : = \\pi _ { m - u } P L \\Xi . \\end{align*}"}
-{"id": "9855.png", "formula": "\\begin{align*} \\mathcal { F } _ i ^ { \\mathbf { z } } ( \\phi ^ \\circ ) ( \\pi ^ \\lambda ) = \\sum _ { w \\in W } \\mathcal { F } _ i ^ { \\mathbf { z } } ( \\phi _ w ) ( \\pi ^ \\lambda ) = \\sum _ { w \\in W } \\left [ T _ w \\cdot \\begin{pmatrix} f _ 1 ^ \\lambda ( \\mathbf { z } ) \\\\ \\vdots \\\\ f _ k ^ \\lambda ( \\mathbf { z } ) \\\\ - \\\\ \\vdots \\\\ - \\\\ f _ 1 ^ \\lambda ( w _ 0 \\mathbf { z } ) \\\\ \\vdots \\\\ f _ k ^ \\lambda ( w _ 0 \\mathbf { z } ) \\end{pmatrix} \\right ] _ i . \\end{align*}"}
-{"id": "6623.png", "formula": "\\begin{align*} v _ 0 \\ ; & : = u _ \\infty - { \\textstyle \\frac { \\Gamma ( 2 B ) } { \\Gamma ( B ) } } \\ , u _ 0 \\\\ v _ \\infty \\ ; & : = \\ ; u _ \\infty \\ , . \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} \\Delta _ { \\widetilde { Y } } = [ \\widetilde { Y } \\times \\tilde z ] + B + C , \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} \\mathcal { A } _ k f ( x ) = f \\ast \\sigma _ k , \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} f ^ { ( 2 k + 1 ) } _ { \\alpha } = ( 2 k + 1 ) ( \\delta _ \\circ ) ^ { ( 2 k ) } _ { \\alpha } g ' _ \\alpha + ( \\delta _ \\circ ) ^ { ( 2 k + 1 ) } _ { \\alpha } g _ \\alpha = 0 . \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} E _ A = \\prod _ { i = 1 } ^ { \\min ( m , n ) } \\prod _ { { 1 \\leq a _ 1 < \\cdots < a _ i \\leq m } \\atop { 1 \\leq b _ 1 < \\cdots < b _ i \\leq n } } \\mathrm { d e t } [ a _ 1 , \\ldots , a _ i ; b _ 1 , \\ldots , b _ i ] \\end{align*}"}
-{"id": "9740.png", "formula": "\\begin{align*} { | | v | | _ \\rho } = \\sqrt { \\int _ { \\mathbb { T } ^ d } v ^ 2 \\rho } , \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{align*} I = T S + B A . \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} \\langle : X \\otimes \\ldots \\otimes X : , \\Phi \\rangle = \\int _ { \\mathbb { R } ^ k } ^ { '' } \\widehat { \\Phi } ( x _ 1 , \\ldots , x _ k ) Z _ G ( d x _ 1 ) \\ldots Z _ G ( d x _ k ) , \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} | x | ^ n N \\cdot V | _ S = - { m \\cdot e _ n \\over \\gamma _ n } + O \\left ( { 1 \\over | x ' | ^ { \\varepsilon } } \\right ) , \\end{align*}"}
-{"id": "9467.png", "formula": "\\begin{align*} U + \\begin{bmatrix} a _ { m - 1 , m } & \\ldots & a _ { m - 1 , n } \\end{bmatrix} - T \\begin{bmatrix} 1 & a _ { m , m + 1 } & \\ldots & a _ { m , n } \\\\ . & \\ddots & \\ddots & \\vdots \\\\ . & . & 1 & a _ { n - 1 , n } \\\\ . & . & . & 1 \\end{bmatrix} = \\begin{bmatrix} 0 & \\ldots & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} X : = \\theta _ s ^ { - 1 } ( N ) \\cap P \\times N , \\ \\ \\ \\ \\ \\ \\theta : = \\theta _ s | _ X : X \\to N . \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{align*} \\frac { 1 } { Q } \\left ( M _ i \\left ( \\begin{array} { c } i _ i \\\\ R _ i \\\\ \\end{array} \\right ) \\right ) _ { i , j } = \\frac { 2 ^ { 1 - 2 i } ( j - i - 1 ) + 2 i - 2 j + 3 } { 4 ( j - i - 1 ) ( 2 i - 2 j + 3 ) } \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} ( \\epsilon ^ { 2 } + a b + a \\epsilon ) \\lambda ^ { 3 } + ( \\epsilon ^ { 2 } + a b + b ^ { 2 } ) \\lambda ^ { 2 } = 0 , \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{align*} M _ f = \\begin{pmatrix} f _ 0 & f _ 1 & \\ldots & f _ m & 0 & \\ldots & 0 \\\\ 0 & \\sigma ( f _ 0 ) & \\ldots & \\sigma ( f _ { m - 1 } ) & \\sigma ( f _ { m } ) & \\ldots & 0 \\\\ & & \\ddots & & & \\ddots & \\\\ 0 & \\ldots & 0 & \\sigma ^ { n - m - 1 } ( f _ 0 ) & \\ldots & \\ldots & \\sigma ^ { n - m - 1 } ( f _ m ) \\end{pmatrix} _ { ( n - m ) \\times n } . \\end{align*}"}
-{"id": "3148.png", "formula": "\\begin{align*} f C _ { i j k } = T _ { i j k } + W _ { i j k l } \\nabla _ { l } f , \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} u ^ 2 + k ^ 2 = x ^ 3 \\wedge u ^ 3 + k ^ 3 = y ^ 2 , ( u , k ) = 1 \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} | v _ N | \\prod _ { j = b _ N + 1 } ^ { b _ { N + 1 } - 1 } | w _ j | > \\dfrac { 1 } { \\varepsilon } \\Bigl ( \\prod _ { j = b _ n + 1 } ^ { k } | w _ j | \\Bigr ) ^ { - 1 } \\ , \\max \\{ 1 , \\| w \\| _ { \\infty } \\} ^ { k - b _ n } . \\end{align*}"}
-{"id": "8859.png", "formula": "\\begin{align*} s _ 1 = \\left ( \\sum _ { d = 0 } ^ \\kappa ( d + 1 ) ^ u e ^ { - 2 d } \\right ) + o ( 1 ) \\rightarrow \\sum _ { d = 0 } ^ \\infty ( d + 1 ) ^ u e ^ { - 2 d } \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = t _ 1 & x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\overbrace { \\begin{matrix} 0 & 0 \\\\ - t _ 2 & \\theta ^ 0 \\end{matrix} } & \\begin{matrix} 0 \\\\ \\theta ^ { t _ 1 } \\end{matrix} & \\overbrace { \\begin{matrix} 1 & \\theta ^ \\infty _ 1 / 2 \\\\ - 1 & \\theta ^ \\infty _ 1 / 2 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "3232.png", "formula": "\\begin{align*} \\widehat { L ^ { ( n ) } _ s } & = ( x ^ 2 + 1 ) \\frac { d ^ 2 } { d x ^ 2 } + \\left [ 2 x - \\left ( 2 - 2 n - 2 \\Re ( s ) \\right ) x - 2 \\Im ( s ) \\right ] \\frac { d } { d x } \\\\ & = ( x ^ 2 + 1 ) \\frac { d ^ 2 } { d x ^ 2 } + \\left [ \\left ( 2 n + 2 \\Re ( s ) \\right ) x - 2 \\Im ( s ) \\right ] \\frac { d } { d x } , \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } k \\gamma _ { k } & = \\frac { n ( n + 1 ) } { 2 } \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{align*} 1 - | \\hat { \\mu } ( \\overline { \\xi } _ i ) | = O \\left ( \\frac { \\log ( 1 + R ) } { R ^ 2 m ^ 2 } \\right ) + \\frac { 1 } { m ^ 2 } \\sum _ { \\| y \\| _ 1 \\leq R ' } \\Big ( 1 - c \\left ( \\overline { \\xi } ( x _ i + y ) \\right ) \\Big ) . \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} y _ t & = H ( q ^ { - 1 } ) ( r _ t + x _ t ) + e _ t \\\\ & = H ( q ^ { - 1 } ) ( r _ t + L ( q ^ { - 1 } ) v _ t ) + e _ t . \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{align*} T ( v _ 0 , k ) & = T ( v _ 0 , k - 1 ) U ( v _ 0 , k ) \\\\ & = ( 0 , - 1 , \\cdots , - k + 1 , 1 , \\cdots , k ) ( k , k + 1 ) ( - k , - k + 1 ) \\\\ & = ( 0 , - 1 , \\cdots , - k , 1 , \\cdots , k + 1 ) . \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{align*} L L ( Z B ) & \\le \\sum _ { i = 0 } ^ { \\alpha } \\sum _ { j = 0 } ^ { e - 1 } { i ( p ^ e - 1 ) + p ^ j } = e ( p ^ e - 1 ) \\Bigl ( \\sum _ { i = 0 } ^ { \\alpha } i \\Bigr ) + ( \\alpha + 1 ) \\frac { p ^ e - 1 } { p - 1 } \\\\ & = e ( p ^ e - 1 ) \\frac { \\alpha ( \\alpha + 1 ) } { 2 } + ( \\alpha + 1 ) \\frac { p ^ e - 1 } { p - 1 } \\\\ & \\le \\Bigl ( \\frac { d } { e } + 1 \\Bigr ) \\Bigl ( \\frac { d } { 2 } + \\frac { 1 } { p - 1 } \\Bigr ) ( p ^ e - 1 ) . \\end{align*}"}
-{"id": "4015.png", "formula": "\\begin{align*} \\langle - ( \\varrho _ a - a ) \\rangle _ p = d + e - b - c \\geq d - b = \\langle - ( \\delta - \\beta ) \\rangle _ p \\end{align*}"}
-{"id": "9447.png", "formula": "\\begin{align*} R ( t ) = \\int ^ { t } _ { 0 } \\int _ { \\mathbb { T } ^ 3 } e ^ { - r s } \\widetilde { W } \\cdot \\mathrm { L } ( w ^ * ) . \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{align*} \\hat { m } _ s ^ { ( n ) } ( x ) = ( 1 + x ^ 2 ) ^ { \\Re ( s ) + n - 1 } e ^ { - 2 \\Im ( s ) A r g ( 1 + i x ) } . \\end{align*}"}
-{"id": "5193.png", "formula": "\\begin{align*} \\left ( f \\circ g \\right ) _ { n } = \\sum _ { \\pi \\models n } f _ { \\vert \\pi \\vert } g _ { \\pi } . \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} \\lim _ { k \\to + \\infty } \\lambda _ { k } = \\lambda _ { 1 } \\end{align*}"}
-{"id": "256.png", "formula": "\\begin{align*} ( \\mathsf { M } _ { m } ^ { n } ) ^ { * } = \\mathsf { M } _ { n } ^ { m } , \\quad \\mathsf { M } _ { m } ^ { n } \\mathsf { M } _ { o } ^ { p } = \\delta _ { o } ^ { n } \\mathsf { M } _ { m } ^ { p } , \\quad \\mathrm { T r } ( \\pi ( K _ { 2 \\rho } ^ { - 1 } ) \\mathsf { M } _ { m } ^ { n } ) = \\delta _ { m } ^ { n } q ^ { - ( 2 \\rho , \\lambda _ { m } ) } . \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} \\vec { b } = \\vec { c } - { \\rm i } \\frac { \\tau } { 4 } W ^ { - 1 } S \\vec { b } - { \\rm i } \\frac { \\tau } { 2 } W ^ { - 1 } Q ( \\vec { b } ) \\vec { b } . \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} C _ n = \\sum _ { k = 0 } ^ { n - 1 } C _ { k } C _ { n - 1 - k } . \\end{align*}"}
-{"id": "322.png", "formula": "\\begin{align*} \\Xi ^ \\lambda ( K _ a F _ a \\otimes 1 \\otimes 1 \\otimes E _ a ) = 2 \\Xi ^ \\lambda ( K _ a \\otimes F _ a \\otimes 1 \\otimes E _ a ) - \\sum _ { m , p } c _ { p } ^ { m } \\pi ( E _ a K _ \\lambda K _ a F _ a ) _ { m } ^ { p } . \\end{align*}"}
-{"id": "9638.png", "formula": "\\begin{align*} p _ 6 = \\frac { 4 } { \\pi ^ 2 } \\left [ \\int _ { \\arccos \\frac { \\sqrt { 3 0 } } { 6 } } ^ { \\arccos \\frac { \\sqrt { 1 9 } - 1 } { 6 } } \\left ( \\arccos \\frac { - 5 - 6 \\cos t } { 6 + 6 \\cos t } - ( \\pi - t ) \\right ) \\mathrm { d } t + \\right . \\\\ \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } x y z + x z + x = a , \\\\ x ^ { 2 } y z + x ^ { 2 } y + y = b , \\\\ x ^ { 2 } y ^ { 2 } z + y ^ { 2 } z + z = c . \\end{array} \\right . \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{align*} \\hat { U } _ \\eta \\left ( \\hat { H } _ A + \\hat { H } _ B \\right ) \\hat { U } _ \\eta ^ \\dag = \\hat { H } _ C + \\hat { H } _ D \\ ; . \\end{align*}"}
-{"id": "9321.png", "formula": "\\begin{align*} 1 6 5 - 2 \\gamma = \\gamma . \\end{align*}"}
-{"id": "5230.png", "formula": "\\begin{align*} & c ( \\varpi ) = \\left ( - \\frac { \\varpi } { 2 \\gamma _ 2 } + o ( \\varpi ^ 2 ) \\right ) e _ 1 , \\| u ( \\varpi ) - V ( \\varpi ) \\| _ { { H } ^ { k - 1 } ( \\Omega ( \\varpi ) ) } = O ( | \\varpi | ^ 3 ) , \\\\ & \\left \\| \\eta ( \\varpi ) - \\frac { \\varpi ^ 2 } { 4 \\pi ^ 2 } ( g - \\sigma \\partial _ { x _ 1 } ^ 2 ) ^ { - 1 } \\left ( \\frac { x _ 1 ^ 2 - 1 } { ( 1 + x _ 1 ^ 2 ) ^ 2 } \\right ) \\right \\| _ { H ^ k ( \\R ) } = O ( | \\varpi | ^ 3 ) . \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{align*} h & = ( 0 , \\frac { \\Lambda _ 1 ^ \\vee + 3 \\Lambda _ 3 ^ \\vee } { 4 } , 0 ) , \\\\ 2 [ h ] & = ( 0 , \\frac { - \\Lambda _ 1 ^ \\vee - \\Lambda _ 3 ^ \\vee } { 2 } + \\Lambda _ 4 ^ \\vee - \\Lambda _ 5 ^ \\vee + \\Lambda _ 6 ^ \\vee - \\Lambda _ 7 ^ \\vee + \\Lambda _ 8 ^ \\vee - \\Lambda _ 9 ^ \\vee , 0 ) . \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} p p ' m ^ 2 + n ^ 2 & = \\frac { \\left ( p p ' d ^ 2 + c ^ 2 \\right ) x ^ 2 + p \\left ( p ' + p \\right ) y ^ 2 - p \\left ( c + p ' d \\right ) x y } { \\Delta ^ { 2 } } \\end{align*}"}
-{"id": "8893.png", "formula": "\\begin{align*} L H S & = \\sum _ { k = 1 } ^ n \\sum _ { j = 1 } ^ k ( - 1 ) ^ { k + j } [ a _ k ( x ) P _ s ^ { ( k ) } ( x ) ] ^ { ( k - j ) } f ^ { ( j - 1 ) } ( x ) , \\\\ R H S & = \\sum _ { k = 1 } ^ n \\sum _ { j = 1 } ^ k ( - 1 ) ^ { k + j + 1 } [ a _ k ( x ) f ^ { ( k ) } ( x ) ] ^ { ( k - j ) } P _ s ^ { ( j - 1 ) } ( x ) . \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{align*} [ G ( k , 0 ) + i G ' ( k , 0 ) ] J ( k ) = - 2 i k I _ n , [ G ( k , 0 ) - i G ' ( k , 0 ) ] J ( k ) = - 2 i k U . \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} & \\vec a \\coloneqq \\left ( a _ 1 = 2 g - \\textstyle \\frac 3 2 - A , a _ 2 , \\dots , a _ n , a _ { n + 1 } = \\frac 3 2 , a _ { n + 2 } = - \\frac 1 2 \\right ) , \\end{align*}"}
-{"id": "1816.png", "formula": "\\begin{align*} \\left ( e ^ { t \\ , d / d x } \\phi \\right ) ( s _ 0 ) = \\sum _ { n = 0 } ^ { \\infty } \\dfrac { t ^ n } { n ! } \\phi ^ { ( n ) } ( s _ 0 ) , \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{align*} \\tau _ 1 & : = \\inf \\{ t \\geq 0 : s _ 1 , s _ 2 \\in [ t , t + 1 ] , ( B _ 1 ( s _ 1 ) - B _ 2 ( s _ 1 ) ) ( B _ 1 ( s _ 2 ) - B _ 2 ( s _ 2 ) ) \\geq 0 \\} \\\\ \\tau _ 2 & : = \\inf \\{ t > \\tau _ 1 : B _ 1 ( t ) = B _ 2 ( t ) \\} . \\end{align*}"}
-{"id": "9812.png", "formula": "\\begin{align*} { \\left ( { { \\nabla ^ 2 } \\Phi , V } \\right ) _ N } - { \\left ( { { \\nabla ^ 2 } V , \\Phi } \\right ) _ N } = \\int _ { \\partial E , N } { \\left ( { \\nabla \\Phi \\cdot \\hat n V - \\nabla V \\cdot \\hat n \\Phi } \\right ) d S } \\quad ( D i s c r e t e \\ ; G r e e n ' s \\ ; S e c o n d \\ ; I d e n t i t y ) \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} ( x ^ { \\lambda - 1 + k } f ( x ) ) ^ { ( k - 2 ) } & = l _ k x + \\frac { b _ k } { \\lambda ( \\lambda + 1 ) } x ^ { \\lambda + 1 } \\\\ & { } + x ^ { \\lambda + 1 } \\int _ 0 ^ { \\infty } M _ { k - 1 } ( u ) u ^ { \\lambda } e ^ { - x u } \\ , d u , \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\big ( A _ 0 x ^ 2 + A _ 1 x + A _ 2 \\big ) Y , \\frac { \\partial Y } { \\partial t _ 1 } = ( B _ { 1 1 } x + B _ { 1 0 } ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = ( B _ { 2 1 } x + B _ { 2 0 } ) Y , \\end{gather*}"}
-{"id": "6510.png", "formula": "\\begin{align*} T _ 1 ( x ) = \\alpha x + \\beta . \\end{align*}"}
-{"id": "9287.png", "formula": "\\begin{align*} - \\int _ M ( \\nabla _ { i j } & A _ { k l } \\nabla ^ j A ^ { k l } ) ( \\nabla ^ i | \\nabla A | ^ 2 ) \\ , \\gamma ^ s \\ , d \\mu \\\\ & = - 2 \\int _ M ( \\nabla _ { i j } A _ { k l } \\nabla ^ j A ^ { k l } ) ( \\nabla ^ i \\nabla _ p A _ { q r } \\nabla ^ p A ^ { q r } ) \\ , \\gamma ^ s \\ , d \\mu = - \\frac 1 2 \\int _ M \\big | \\nabla | \\nabla A | ^ 2 \\big | ^ 2 \\ , \\gamma ^ s \\ , d \\mu \\ , . \\end{align*}"}
-{"id": "9469.png", "formula": "\\begin{align*} ( P , Q ) = \\left ( \\begin{bmatrix} I _ { k - 1 } & . & . \\\\ . & 1 & T \\\\ . & . & I _ { n - k } \\end{bmatrix} , \\begin{bmatrix} I _ { k - 1 } & . & . \\\\ . & 1 & U \\\\ . & . & I _ { n - k } \\end{bmatrix} \\right ) \\end{align*}"}
-{"id": "3247.png", "formula": "\\begin{align*} P ^ { s , \\infty } _ { H P } ( t ) \\Lambda ^ { \\infty } _ N f = \\Lambda ^ { \\infty } _ N P ^ { s , N } _ { H P } ( t ) f \\ , \\ \\forall t \\ge 0 , \\ \\forall N \\ge 1 . \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{align*} \\phi _ 2 ( s ) = \\begin{cases} 1 & N \\ge 3 \\alpha > \\frac { 2 } { N - 2 } \\\\ ( \\log s ) ^ { - \\frac { 1 } { \\alpha } } & N \\ge 3 \\alpha = \\frac { 2 } { N - 2 } \\\\ s ^ { - \\lambda _ 2 } & N \\ge 3 \\alpha < \\frac { 2 } { N - 2 } \\\\ s ^ { - \\lambda _ 1 } \\log s & N = 2 . \\end{cases} \\end{align*}"}
-{"id": "9465.png", "formula": "\\begin{align*} \\Lambda ( h ^ { \\varepsilon } _ { 0 } , U ^ { \\varepsilon } _ { 0 } ) = \\int \\frac { | B ^ { \\varepsilon } _ { 0 } | ^ { 2 } + 1 } { 2 h ^ { \\varepsilon } _ { 0 } } \\leq \\Lambda ( h _ { 0 } , U _ { 0 } ) \\end{align*}"}
-{"id": "3380.png", "formula": "\\begin{align*} \\Pr ( \\| P _ X - Q _ X \\| _ 1 \\geq \\varepsilon ) = 0 . \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} R H S & = \\sum _ { k = 1 } ^ n \\sum _ { j = 1 } ^ k ( - 1 ) ^ { k + j + 1 } [ a _ k ( x ) f ^ { ( k ) } ( x ) ] ^ { ( k - j ) } P _ s ^ { ( j - 1 ) } ( x ) = 0 \\end{align*}"}
-{"id": "8317.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } L ( v ) = f \\Omega \\\\ D ^ { \\alpha } u = 0 | \\alpha | \\leq h - 1 \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "9495.png", "formula": "\\begin{align*} H ^ 0 ( Z , T S \\vert _ Z ) & \\longrightarrow H ^ 0 ( Z , T S \\vert _ Z ) \\\\ [ 6 p t ] \\sigma ( z ) & \\mapsto f ( z ) \\sigma ( z ) , \\end{align*}"}
-{"id": "1203.png", "formula": "\\begin{align*} d X _ t = A X _ t d t + G ( X _ t ) d t + \\sigma ( X _ t ) d W ( t ) , X ( 0 ) = x , \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} h _ k f ( y ) = \\bigvee _ { x \\in X } f ( x ) \\cdot k ( x , y ) , \\textrm { f o r a l l } y \\in Y . \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{align*} k + \\frac 3 2 = \\frac { p } { 2 p ' } \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} \\sum _ { i \\in G _ 0 } m _ i v _ i = 0 . \\end{align*}"}
-{"id": "6018.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n \\Big ( ( - 1 ) ^ { n - k } \\big ( \\sum _ { i = 1 } ^ { n + 1 } ( \\lambda _ i - \\lambda _ { n + 2 } ) ( \\lambda _ i - \\lambda _ { n + 3 } ) x _ i ^ 2 \\Gamma _ i ^ k \\big ) s ^ k t ^ { n - k } \\Big ) \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{align*} W _ \\lambda ( g ) = \\int _ { \\mathbb { C } } g ( z ) \\pi _ \\lambda ( z ) d z , \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} u ( x ) - c & = : \\frac { 1 } { 2 \\pi } \\varpi ^ i \\nabla ^ \\perp \\log { | x | } + G ( x ) , \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{align*} \\Xi ^ \\lambda ( K _ a F _ a \\otimes 1 \\otimes 1 \\otimes E _ a ) = \\sum _ { i , j , m , n , o , p } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) \\pi ( K _ { a } F _ { a } ) _ { m } ^ { j } c _ { n } ^ { m } \\pi ( 1 ) _ { o } ^ { n } c _ { p } ^ { o } \\pi ( E _ { a } K _ { \\lambda } ) _ { i } ^ { p } . \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{align*} \\int _ { \\Omega _ { R , \\delta } } \\left ( u \\cdot \\nabla \\varphi + c \\cdot V \\right ) \\ , d x & = \\int _ { B _ R ( 0 ) \\cap S } A \\cdot N \\ , d S + \\sum _ i \\int _ { \\partial \\tilde B ^ i _ \\delta } A \\cdot N \\ , d S \\\\ & + \\int _ { \\partial B _ R ( 0 ) \\cap \\Omega } A \\cdot N \\ , d S = : \\mathbf { I } + \\mathbf { I I } + \\mathbf { I I I } . \\end{align*}"}
-{"id": "9343.png", "formula": "\\begin{align*} \\| \\exp ( X ) - I - X \\| _ { } = O ( \\| X \\| _ { } ^ 2 ) \\| \\log ( I + X ) - X \\| _ { } = O ( \\| X \\| _ { } ^ 2 ) , \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} \\psi _ { 1 , 0 } ( x ) = \\frac { 1 } { 2 } , x \\in [ - 1 , 1 ] . \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} \\sigma _ a ( F _ r ) = p ^ a \\sigma _ a ( G _ r ) + \\sigma _ { a + 1 } ( F _ { r - 1 } ) \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} A , A ' , B , B ' \\geq 0 , A + A ' = B + B ' : = M > 0 . \\end{align*}"}
-{"id": "3154.png", "formula": "\\begin{align*} e ^ { - \\lambda _ { N , s } t } \\frac { \\Delta _ N ( y ) } { \\Delta _ N ( x ) } \\det \\left ( p ^ { ( N ) , s } _ t ( x _ i , y _ j ) \\right ) ^ N _ { i , j = 1 } d y , \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} F ( \\mathbf { x } ) = \\sum \\limits _ { \\substack { \\mathbf { k } \\in \\mathbb { Z } ^ m \\\\ \\Vert \\mathbf { k } \\Vert _ \\infty \\leqslant X } } c _ X ( \\mathbf { k } ) e ( \\mathbf { k } \\cdot \\mathbf { x } ) + O \\left ( M \\frac { \\log X } { X } \\right ) \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} \\int _ { \\Omega _ { R , \\delta } } \\left ( c _ 1 - u _ 1 \\right ) \\omega \\ , d x & = g \\int _ { B _ R ( 0 ) \\cap S } \\eta \\ , d x ' + \\sigma \\int _ { \\partial B _ R ( 0 ) \\cap S } \\nu \\cdot N \\ , d s \\\\ & + \\sum _ i \\int _ { \\partial \\tilde B ^ i _ \\delta } A \\cdot N \\ , d S + \\int _ { \\partial B _ R ( 0 ) \\cap \\Omega } A \\cdot N \\ , d S , \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{align*} \\Delta \\mathbb B [ \\eta f ] - \\mathbb B [ \\Delta ( \\eta f ) ] = \\sum _ j { \\cal B } _ j [ \\partial _ j ( \\eta f ) ] + \\sum _ j \\partial _ j { \\cal B } _ j [ \\eta f ] . \\end{align*}"}
-{"id": "3226.png", "formula": "\\begin{align*} \\frac { \\prod _ { l = 0 } ^ { k - 2 } \\prod _ { i = l + 1 } ^ { k - 1 } ( \\xi ^ l _ { i + 1 } - \\xi _ i ^ l ) } { \\prod _ { 1 \\le i < j \\le k } ^ { } ( x _ { j } - x _ { i } ) } \\le 1 . \\end{align*}"}
-{"id": "2704.png", "formula": "\\begin{align*} F _ 1 : = \\int _ \\Omega ( n _ 1 - N _ 1 ) ^ 2 + \\int _ \\Omega ( n _ 2 - N _ 2 ) ^ 2 \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} \\left ( I + \\begin{pmatrix} a & 0 \\\\ 0 & d \\end{pmatrix} p \\right ) = \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} + \\begin{pmatrix} a w & 0 \\\\ 0 & d z \\end{pmatrix} p \\end{align*}"}
-{"id": "3075.png", "formula": "\\begin{align*} S ( t _ { a b } ) = ( - 1 ) ^ { [ a ] ( [ a ] + [ b ] ) } \\bar { t } _ { b a } , S ( \\bar { t } _ { a b } ) = ( - 1 ) ^ { [ b ] ( [ a ] + [ b ] ) } q ^ { ( 2 \\rho , \\epsilon _ a - \\epsilon _ b ) } t _ { b a } . \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} \\Delta \\Psi _ 2 = 0 \\textup { ~ i n ~ } R , \\Psi _ 2 = c _ 1 ( \\varpi ) \\eta - \\psi _ V ( \\eta , \\varpi ) \\textup { ~ o n ~ } T . \\end{align*}"}
-{"id": "3329.png", "formula": "\\begin{align*} \\xi _ 0 : = - a \\log ( \\lambda a ) + a - \\frac { 1 } { \\lambda } , \\ : \\ : \\ : \\ : \\xi _ k : = ( - 1 ) ^ k \\left ( \\lambda ^ { - k } \\left ( \\frac { a } { k } - \\frac { 1 / \\lambda } { k + 1 } \\right ) - a ^ { k + 1 } \\left ( \\frac { 1 } { k } - \\frac { 1 } { k + 1 } \\right ) \\right ) , \\end{align*}"}
-{"id": "1884.png", "formula": "\\begin{align*} s ^ H = \\frac { B } { P } - \\frac { ( P ^ { n - 1 } \\mathbf { H } ) '' } { P ^ { n - 1 } } . \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} E ( I u ( t ) ) : = \\frac { 1 } { 2 } \\| I u ( t ) \\| ^ 2 _ { \\dot { H } ^ { k / 2 } _ x } + \\frac { 1 } { 4 } \\| I u ( t ) \\| ^ 4 _ { L ^ 4 _ x } . \\end{align*}"}
-{"id": "9402.png", "formula": "\\begin{align*} \\bigl \\langle [ \\widetilde { \\phi } ( t _ 2 ) ] ( r _ 1 ) , r _ 3 \\bigr \\rangle \\ , = \\ , \\bigl \\langle [ \\alpha ( t _ 2 ) ] ( r _ 1 ) , [ \\beta ( t _ 2 ) ] ( r _ 3 ) \\bigr \\rangle \\hbox { f o r a . e . } \\ t _ 2 \\in \\Omega _ 2 . \\end{align*}"}
-{"id": "3209.png", "formula": "\\begin{align*} e ^ { - \\lambda _ { N , s } t } \\frac { \\Delta _ N ( y ) } { \\Delta _ N ( x ) } \\det \\left ( p ^ { ( N ) , s } _ t ( x _ i , y _ j ) \\right ) ^ N _ { i , j = 1 } , \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} \\frak { F u k } ( \\Sigma \\sqcup \\Sigma ' ) = \\frak { F u k } ( \\Sigma ) \\otimes \\frak { F u k } ( \\Sigma ' ) . \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} \\begin{pmatrix} M & N \\\\ [ . 1 c m ] \\overline { N } & \\overline { M } \\end{pmatrix} \\begin{pmatrix} z + w \\\\ [ . 2 c m ] \\overline { z + w } \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ [ . 2 c m ] 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} \\gamma = \\sup \\{ \\gamma _ i : \\ i \\in \\omega \\} \\end{align*}"}
-{"id": "767.png", "formula": "\\begin{align*} \\sum _ { n = U ' } ^ { U ' + V ' } e \\left ( \\overline { \\xi } _ { x + ( n + 1 ) w } - \\overline { \\xi } _ { x + n w } \\right ) = O \\left ( \\frac { V ' } { R _ 1 } + \\frac { ( V ' ) ^ 2 } { U ' } + \\frac { 1 } { \\delta } \\right ) = o _ R ( V ' ) . \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} \\lambda _ { 1 } ( p , \\Omega ) = \\min _ { \\varphi \\in W _ { 0 } ^ { 1 , p } ( \\Omega ) \\setminus \\{ 0 \\} } \\frac { \\displaystyle \\int _ { \\Omega } F ^ { p } ( \\nabla \\varphi ) \\ , d x } { \\displaystyle \\int _ { \\Omega } | \\varphi | ^ { p } \\ , d x } . \\end{align*}"}
-{"id": "5738.png", "formula": "\\begin{align*} S ( A | M ) _ { ( \\mathcal { N } ( t ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) } = S ( ( \\mathcal { N } ( t ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) ) - S ( \\hat { \\rho } _ M ) \\ ; , \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{align*} \\frac { c m + d n } { a m + b n } = \\frac { n } { m } \\textrm { o r } & \\frac { c m + d n } { a m + b n } = \\frac { m } { n } , \\\\ c m ^ 2 - b n ^ 2 + ( d - a ) m n = 0 \\textrm { o r } & d n ^ 2 - a m ^ 2 + ( c - b ) m n = 0 . \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} d _ m \\ , \\partial _ { x _ 2 } u _ m = \\varepsilon \\ , f _ 1 ( u _ c ) \\ \\ x _ 2 = \\varepsilon , \\ \\ x _ 1 \\in ( 0 , L ) , \\end{align*}"}
-{"id": "2708.png", "formula": "\\begin{align*} \\dot \\omega _ F = - \\nu _ F W _ F \\ , \\dot \\omega , \\dot \\omega _ O = - \\nu _ O W _ 0 \\ , \\dot \\omega , \\dot \\omega _ P = \\nu _ P W _ P \\ , \\dot \\omega \\mbox { a n d } \\dot \\omega _ N = 0 , \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{align*} W _ { 1 } ( x | \\rho , q ) = \\sum _ { n \\geq 0 } \\frac { \\rho ^ { n } } { ( q ) _ { n } } d _ { n } ( x | q ) . \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} d g _ t ( z ) = \\frac { 2 } { g _ t ( z ) - \\xi _ t } d t \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} \\dfrac { d T ( t ) } { d t } = A \\circ T ( t ) = T ( t ) \\circ A , \\mbox { f o r e v e r y } t . \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} \\| u \\| _ { W ^ \\infty } = \\| u \\| _ { L ^ \\infty } + \\| \\tau v \\cdot \\nabla _ x u \\| _ { L ^ \\infty } . \\end{align*}"}
-{"id": "7603.png", "formula": "\\begin{align*} \\frac { \\omega _ \\epsilon ( t ) ^ 2 } { \\widehat \\omega ^ 2 } = \\frac { v _ \\epsilon ' v _ \\epsilon '' } { b ( b - a ) } e ^ { - \\rho } \\leq \\frac C { | \\sigma | _ \\eta ^ { 2 ( 1 - \\alpha ) } } \\leq C e ^ { - ( 1 - \\alpha ) \\rho } \\end{align*}"}
-{"id": "9737.png", "formula": "\\begin{align*} \\partial _ t B + \\nabla \\cdot \\left ( \\frac { B \\otimes P - P \\otimes B } { \\rho } \\right ) = 0 , \\ ; \\ ; \\ ; \\nabla \\cdot B = 0 , \\end{align*}"}
-{"id": "7701.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ { k - 1 } \\binom { k - 1 } { t } ( - p ) ^ { - t } U _ { m k + t } ^ { ( k ) } & = ( - p ) ^ { 1 - k } U _ { m } ( - p U _ { m } + U _ { m + 1 } ) ^ { k - 1 } \\\\ & = ( - p ) ^ { 1 - k } U _ { m } ( - q U _ { m - 1 } ) ^ { k - 1 } \\\\ & = \\left ( \\frac { q } { p } \\right ) ^ { k - 1 } U _ { m } U _ { ( m - 1 ) ( k - 1 ) } ^ { ( k - 1 ) } , \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{align*} \\sum _ { k _ { 1 } + \\dots + k _ { p } = n } g _ { k _ { 1 } } \\dots g _ { k _ { p } } & = \\frac { \\left ( - 1 \\right ) ^ { n + p } } { n ! } \\sum _ { k _ { 1 } + \\dots + k _ { p } = n } \\binom { n } { k _ { 1 } , \\dots , k _ { p } } \\int _ { 0 } ^ { 1 } \\dots \\int _ { 0 } ^ { 1 } \\prod _ { i = 1 } ^ { p } \\left ( x + u _ { i } - 1 \\right ) ^ { k _ { i } } d u _ { 1 } \\dots d u _ { p } \\\\ & = \\frac { \\left ( - 1 \\right ) ^ { n + p } } { n ! } \\int _ { 0 } ^ { 1 } \\dots \\int _ { 0 } ^ { 1 } \\left ( p x - p + u _ { 1 } + \\dots + u _ { p } \\right ) ^ { n } d u _ { 1 } \\dots d u _ { p } . \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} \\tau _ { x } \\leq \\left \\lfloor \\frac { d ^ { x } - 1 } { 2 } \\right \\rfloor = B ( N - 2 B ) ^ { M - 1 } \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} H ^ { \\beta } ( z ^ 2 ) | 0 \\rangle = V ^ + ( z ) ^ { - 1 } \\beta _ \\chi ( z ^ 2 ) | 0 \\rangle = \\chi _ { - 3 / 2 } | 0 \\rangle + O ( z ^ 2 ) , \\end{align*}"}
-{"id": "7611.png", "formula": "\\begin{align*} \\Phi _ U ( x _ 1 , x _ 2 ) = ( \\frac { x _ 1 } { x _ 2 } , x _ 2 ^ k ) : = ( z , w ) , \\end{align*}"}
-{"id": "8208.png", "formula": "\\begin{align*} \\lambda _ { 2 } ( p , \\Omega ) = \\min \\{ \\lambda _ { 1 } ( p , \\mathcal W _ { r _ { 1 } } ) , \\lambda _ { 2 } ( p , \\mathcal W _ { r _ 2 } ) \\} . \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} a ^ { 2 ^ { 2 k } + 2 ^ { k } } + a ^ { 2 ^ { 2 k } + 1 } + a ^ { 2 ^ { k } + 1 } = 0 . \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 1 \\\\ - 1 & 1 \\end{pmatrix} \\begin{pmatrix} 0 & 1 \\\\ - 1 & 0 \\end{pmatrix} \\begin{pmatrix} 1 & 1 \\\\ - 1 & 1 \\end{pmatrix} ^ { - 1 } = \\begin{pmatrix} 0 & 1 \\\\ - 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} | f | _ s & : = \\sup \\frac { \\left \\| \\tilde X _ { \\lfloor f \\rceil } ( \\phi ) \\right \\| _ s } { \\| \\phi \\| _ { s , 1 } } , \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} & t ^ { \\frac { 3 } { 2 } ( 1 - 1 / p ) } \\| K ( t ) \\| _ { L ^ p ( B _ { A \\sqrt t } ^ c ) } \\ge c A ^ { - 3 + 3 / p } . \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} \\| T _ j ( t ) [ x ] _ j \\| _ j & = \\inf _ { p _ j ( z ) = 0 } p _ j \\Big ( T ( t ) x - T ( t ) z - \\big ( z - T ( t ) z \\big ) \\Big ) \\\\ & \\leqslant \\inf _ { p _ j ( z ) = 0 } \\left \\{ p _ j ^ { X } \\big ( T ( t ) \\big ) p _ j ( x - z ) + p _ j ( z ) + p _ j \\big ( T ( t ) z \\big ) \\right \\} \\\\ & = p _ j ^ { X } \\big ( T ( t ) \\big ) \\| [ x ] _ j \\| _ j , \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} w _ { R , \\alpha } ^ + ( t ) & = ( 1 - t ^ 2 ) ^ { - 1 / 2 } e ^ { - N \\left ( V \\left ( \\frac { t } { \\alpha } \\right ) + \\varepsilon _ R t \\right ) } , \\\\ w _ { R , \\alpha } ^ - ( t ) & = ( 1 - \\beta ^ { - 2 } t ^ 2 ) ^ { 1 / 2 } e ^ { - N ( V ( \\alpha t ) + \\varepsilon _ R t ) } \\chi _ { [ - \\beta , \\beta ] } ( t ) , \\beta = \\alpha ^ { - 2 } , \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} \\alpha ( P ) = \\frac { \\max _ w \\alpha ( P _ w ) + \\min _ b \\alpha ( P _ b ) } 2 . \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{align*} z ( z _ k , t _ k ) = R ^ { - 1 } . z _ k , z _ k = 0 + i y _ k . \\end{align*}"}
-{"id": "3026.png", "formula": "\\begin{align*} y _ 0 + \\sum _ { m = 1 } ^ \\infty \\ , \\prod _ { l = 1 } ^ m \\bigl ( 1 + \\lambda ^ { \\Delta ^ { ( k _ { l - 1 } ) } } \\bigr ) \\sum _ { b _ { n ( m ) } \\leq j < b _ { n ( m ) + 1 } } c _ { m , j } \\lambda ^ { b _ { n ( m ) + 1 } - 1 - j } y _ j = 0 \\end{align*}"}
-{"id": "9936.png", "formula": "\\begin{align*} ( T F ) ( t , x ) = \\left ( \\frac { \\widetilde F } { | \\tau + | \\xi | ^ 2 | ^ { 1 / 4 } } \\right ) ^ \\vee : = \\frac { 1 } { 2 \\pi } \\int d \\xi \\ e ^ { i ( \\xi \\cdot x + \\tau t ) } \\frac { \\widetilde { F } ( \\tau , \\xi ) } { | \\tau + | \\xi | ^ 2 | ^ { 1 / 4 } } \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} ( \\varepsilon , y ) ( \\varepsilon ' , y ' ) = \\begin{cases} ( \\varepsilon , u _ 1 \\cdots ( u _ k \\varepsilon ' ) v _ 1 \\cdots v _ l ) & \\\\ ( \\varepsilon \\varepsilon ' , v _ 1 \\cdots v _ l ) & . \\end{cases} \\end{align*}"}
-{"id": "8726.png", "formula": "\\begin{align*} I ( x ) = \\underset { \\underset { x \\in O } { O \\ , } } { \\sup } - \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log \\mu _ { n } ( O ) = \\underset { \\underset { x \\in O } { O \\ , } } { \\sup } - \\liminf _ { n \\to \\infty } \\frac { 1 } { n } \\log \\mu _ { n } ( O ) \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} T _ { F , G , N } ^ L ( 1 , \\dots , 1 ) \\ll _ { c , C , \\varepsilon } \\frac { 1 } { N ^ { d - m } } \\sum \\limits _ { \\substack { n _ { m + 1 } , \\dots , n _ d \\in \\mathbb { Z } \\\\ \\vert n _ { m + 1 } \\vert , \\dots , \\vert n _ { d } \\vert \\leqslant N } } \\widetilde { G } ( \\sum _ { j = m + 1 } ^ d \\mathbf { v _ j } n _ j ) , \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} ( \\partial _ t ^ 2 - \\Delta ) u = N _ { \\alpha \\beta \\mu \\nu } \\partial _ { \\alpha } \\partial _ { \\beta } u \\partial _ { \\mu } \\partial _ { \\nu } u , \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{align*} T _ { k , \\varepsilon , d , B } = \\frac { 2 \\varepsilon k } { \\alpha ^ 2 d ^ 2 } B ^ \\delta + O \\left ( \\left ( \\frac { k } { d } B ^ { \\frac { 2 } { 3 } } \\right ) ^ \\varepsilon \\right ) . \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} ( \\nabla f _ p ) _ * ( M A _ { \\mathbb { R } ^ N } ( f _ p ) ) = d x _ { | \\Sigma _ N } . \\end{align*}"}
-{"id": "9427.png", "formula": "\\begin{align*} D = h d = \\nabla \\times \\left ( \\frac { B } { h } \\right ) , \\ ; \\ ; P = h v = \\nabla \\cdot \\left ( \\frac { B \\otimes B } { h } \\right ) + \\nabla \\left ( h ^ { - 1 } \\right ) , \\end{align*}"}
-{"id": "9968.png", "formula": "\\begin{align*} \\dot { q } _ 2 ( t ^ + ) = \\dot { q } _ 1 ( t ^ + ) = \\frac { m _ 1 \\dot { q } _ 1 ( t ^ - ) + m _ 2 \\dot { q } _ 2 ( t ^ - ) } { m } \\end{align*}"}
-{"id": "4543.png", "formula": "\\begin{align*} { \\displaystyle | \\sum _ { k = 1 } ^ { M } { J _ 3 ^ { ( k ) } } | \\leq C ( h ^ { 2 r } + ( \\Delta t ) ^ { 4 } ) + C \\Vert \\theta _ { \\Psi } ^ { M } \\Vert _ { \\mathcal { L } ^ 2 } ^ { 2 } + \\frac { 1 } { 1 6 } \\Vert \\nabla \\theta _ { \\Psi } ^ { M } \\Vert _ { \\mathbf { L } ^ 2 } ^ { 2 } + C \\Delta t \\sum _ { k = 1 } ^ { M - 1 } { \\Vert \\nabla \\theta _ { \\Psi } ^ { k } \\Vert _ { \\mathbf { L } ^ 2 } ^ { 2 } } . } \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} & L u : = - ( a ( \\cdot ) u ' ) ' \\\\ & D ( L ) : = \\left \\{ u \\in H ^ 1 _ { 0 } ( 0 , \\pi ) \\ ; ; \\ ; L u \\in L ^ 2 ( 0 , \\pi ) \\right \\} , \\end{align*}"}
-{"id": "9405.png", "formula": "\\begin{align*} \\Gamma ^ { A , B , C } ( u v ) ( X , Y ) & = u _ 1 ( A ) X ( u _ 2 v _ 1 ) ( B ) Y v _ 2 ( C ) \\\\ & = u _ 1 ( A ) X u _ 2 ( B ) v _ 1 ( B ) Y v _ 2 ( C ) \\\\ & = \\Gamma ^ { A , B } ( u ) ( X ) \\Gamma ^ { B , C } ( v ) ( Y ) . \\end{align*}"}
-{"id": "9325.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + A ( t : s ) x + B ( t : s ) , \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} \\omega ( g , h ) \\ ; = \\ ; - \\lim _ { r \\downarrow 0 } W _ r ( \\overline { g } , h ) \\ , . \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} a ^ 2 + b ^ 2 = x ^ 3 \\wedge a ^ 3 + b ^ 3 = y ^ 2 \\\\ \\end{align*}"}
-{"id": "4180.png", "formula": "\\begin{align*} T \\left ( x , 0 \\right ) = T _ { 0 } \\left ( x \\right ) , \\ ; \\ ; q \\left ( x , 0 \\right ) = 0 , \\ ; \\ ; x \\in \\mathbb { R } , \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} X ^ 3 + A X + B = y ^ 2 , \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{align*} g _ \\epsilon ( t ) = \\exp \\left ( \\epsilon \\begin{pmatrix} c _ { 1 1 } ( t ) & c _ { 1 2 } ( t ) \\\\ c _ { 1 2 } ( t ) & - c _ { 1 1 } ( t ) \\end{pmatrix} \\right ) e ^ { J t } \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{align*} a = \\left ( 1 - \\frac { \\mu } { \\lambda } \\right ) 0 + \\frac { \\mu } { \\lambda } b \\in B . \\end{align*}"}
-{"id": "864.png", "formula": "\\begin{align*} \\rho ^ T _ t : = \\frac { 1 } { T ^ { k / 2 } } \\sum _ { j _ 1 \\neq \\ldots \\neq j _ k } \\sigma _ { j _ 1 } \\ldots \\sigma _ { j _ k } \\langle \\Lambda ^ { ( k ) } ( x ^ { j _ 1 } + \\xi ^ { j _ 1 } , \\ldots , x ^ { j _ k } + \\xi ^ { j _ k } ; T ) , \\mathbf { 1 } _ { [ 0 , t ] } \\rangle , t \\geq 0 . \\end{align*}"}
-{"id": "9540.png", "formula": "\\begin{align*} \\theta ( a _ 1 x a _ 2 ) = \\theta ( a _ 1 ) \\theta ( x ) \\theta ( a _ 2 ) , \\ \\ \\ x \\in X , \\ a _ 1 , a _ 2 \\in A . \\end{align*}"}
-{"id": "4822.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & x \\\\ y & 0 \\end{pmatrix} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\begin{pmatrix} 0 & x \\\\ y & 0 \\end{pmatrix} ^ { - 1 } = \\begin{pmatrix} d & c \\frac { x } { y } \\\\ b \\frac { y } { x } & a \\end{pmatrix} . \\end{align*}"}
-{"id": "8550.png", "formula": "\\begin{align*} D _ { \\omega } L _ 0 & = 0 . \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} \\partial _ { r } u - i k u = o \\left ( 1 / r \\right ) , \\ > r \\rightarrow \\infty . \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} \\mathbb { E } ( t ) : = \\Xi ( t ) + \\beta \\Big [ \\alpha \\Xi _ 1 ^ { ( 0 ) } ( t ) + \\Xi _ 2 ^ { ( 0 ) } ( t ) \\Big ] + \\beta \\Big [ \\Xi _ 3 ( t ) + \\beta \\Xi _ 4 ( t ) \\Big ] + \\beta ^ 3 \\Big [ \\alpha \\Xi _ 1 ^ { ( 1 ) } ( t ) + \\Xi _ 2 ^ { ( 1 ) } ( t ) \\Big ] , \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\big ( \\deg _ { \\widetilde { X } } ( v _ { i } ) - 2 \\big ) \\leq N \\cdot \\deg _ { Y } ( a ) \\cdot \\big ( M \\deg _ { Z } ( b ) - 2 \\big ) \\ , . \\end{align*}"}
-{"id": "9972.png", "formula": "\\begin{align*} Q _ 2 = \\frac { 1 } { k } \\left [ \\begin{array} { c c c c } 2 k _ 1 c \\ ! - \\ ! 2 \\epsilon m _ 1 k & m k _ 1 \\ ! - \\ ! m _ 1 k \\ ! + \\ ! \\epsilon m _ 1 c & k _ 2 c & \\epsilon m _ 2 c \\\\ * & 2 \\epsilon m m _ 1 & m k _ 2 & \\epsilon m m _ 2 \\\\ * & * & - 2 \\epsilon _ 2 k & - m _ 2 k \\\\ * & * & * & 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} { \\mathcal T } = T _ 0 \\oplus T _ 1 \\oplus \\ldots \\oplus T _ { \\lfloor \\frac r 2 \\rfloor } \\end{align*}"}
-{"id": "9966.png", "formula": "\\begin{align*} m : = m _ 1 + m _ 2 \\ , , c : = c _ 1 + c _ 2 \\ , , k : = k _ 1 + k _ 2 \\ , . \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} A _ n = \\left \\{ \\sum _ { i = 1 } ^ { n } t ( f _ i ) \\leq 2 \\mu n \\right \\} , \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} - d \\tilde Y _ t = - Z _ t d W _ t - \\int _ E k _ t ( e ) \\tilde N ( d t , d e ) - d h _ t + d A _ t - d A ' _ t + d C _ { t - } - d C ' _ { t - } , \\end{align*}"}
-{"id": "9498.png", "formula": "\\begin{align*} p ' ( u ) & = p ' _ 0 + p ' _ 1 u + \\dots + p ' _ { 2 d - 1 } u ^ { 2 d - 1 } \\\\ q ' ( u ^ 2 ) & = q ' _ 0 + q ' _ 1 u ^ 2 + \\dots + q ' _ { d - 1 } u ^ { 2 d - 1 } \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{align*} \\mathcal { S } _ { + + } ^ { n } : = \\{ M \\in \\mathbb R ^ { n \\times n } : M = M ^ \\top M \\succ 0 \\} . \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} Z _ { \\mathbf { x } } = \\mathcal { Z } \\cap S _ { \\mathbf { x } } \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} L _ c : = - \\partial ^ 2 _ { \\xi } + 4 c - 1 2 c \\ ; { \\rm s e c h } ^ 2 ( \\sqrt { c } \\xi ) . \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} V ( x + a z , y + b z ) & \\leq V ( x , y ) + \\frac { \\partial V ( x + a t z , y + b t z ) } { \\partial t } \\\\ & = V ( x , y ) + a \\langle \\partial _ x V ( x , y ) , z \\rangle + b \\langle \\partial _ y V ( x , y ) , z \\rangle . \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{align*} S = \\{ ( 0 , 0 ) , ( - 1 , 0 ) , ( 1 , 0 ) , ( 0 , - 1 ) , ( 0 , 1 ) \\} \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{align*} w _ { 1 } ( x | \\rho ) & = 1 - 2 \\rho x + \\rho ^ { 2 } , \\\\ w _ { 2 } ( x , y | \\rho ) & = ( 1 - \\rho ^ { 2 } ) ^ { 2 } - 4 x y \\rho ( 1 + \\rho ^ { 2 } ) + 4 \\rho ^ { 2 } ( x ^ { 2 } + y ^ { 2 } ) . \\end{align*}"}
-{"id": "8088.png", "formula": "\\begin{align*} \\begin{aligned} \\delta \\psi _ 2 & = 2 ( \\psi _ 2 \\psi _ 3 + \\psi _ 2 \\psi _ 4 - \\psi _ 3 \\psi _ 4 ) , \\\\ \\delta \\psi _ 3 & = 2 ( \\psi _ 2 \\psi _ 3 + \\psi _ 3 \\psi _ 4 - \\psi _ 2 \\psi _ 4 ) , \\\\ \\delta \\psi _ 4 & = 2 ( \\psi _ 2 \\psi _ 4 + \\psi _ 3 \\psi _ 4 - \\psi _ 2 \\psi _ 3 ) . \\end{aligned} \\end{align*}"}
-{"id": "6542.png", "formula": "\\begin{align*} Z _ \\epsilon ( t ) = \\left ( \\begin{array} { c c } \\hat { Z } _ \\epsilon ( t ) & A ( t ) \\\\ 0 & e ^ { - \\epsilon t } I _ { n - r } \\end{array} \\right ) , \\mbox { a n d h e n c e , } Z _ \\epsilon ( T _ { \\epsilon } ) = \\left ( \\begin{array} { c c } \\hat { Z } _ \\epsilon ( T _ \\epsilon ) & A ( T _ \\epsilon ) \\\\ 0 & e ^ { - \\epsilon T _ \\epsilon } I _ { n - r } \\end{array} \\right ) . \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{align*} \\int _ { Y a _ t } w \\ , \\ , d \\mu _ { Y a _ t } = \\frac { \\textrm { V o l } Y } { \\textrm { V o l } X } \\int _ X w \\ , \\ , d \\mu _ X + O ( \\| w \\| _ l \\ , \\ , e ^ { - \\delta _ 0 t } ) \\end{align*}"}
-{"id": "8743.png", "formula": "\\begin{align*} k = \\left \\{ \\begin{array} { l r } \\lfloor { { i - 1 } \\over 2 } \\rfloor , & { \\rm i f } ~ i = 1 , 2 , 3 , 4 ; ~ ~ j = 1 , 2 , \\ldots , \\\\ 1 , & { \\rm i f } ~ i = 5 , 6 , \\ldots ; ~ j = 1 , 2 , \\ldots , \\end{array} \\right . \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{align*} \\vert T _ { F , G , N } ^ L ( f _ 1 , \\dots , f _ 4 ) - T _ { F , \\widetilde { G } , N } ^ L ( f _ 1 , \\dots , f _ 4 ) \\vert = \\kappa ( \\delta ) . \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} A _ { i j } y _ { j } = \\begin{pmatrix} a _ { i j } & b _ { i j } \\\\ b _ { i j } & a _ { i j } \\end{pmatrix} x _ { j } \\mathbf { f = } \\begin{pmatrix} a _ { i j } - b _ { i j } \\\\ b _ { i j } - a _ { i j } \\end{pmatrix} x _ { j } = c _ { i j } x _ { j } \\mathbf { f } \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} w _ k = w _ k ^ 1 + w _ k ^ 2 + w _ k ^ 3 , p _ k = p _ k ^ 1 + p _ k ^ 2 + p _ k ^ 3 , \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} h _ n ^ k & = ( - 1 ) ^ k L ^ { 2 n - 2 k - 2 } z _ n ; \\\\ h _ n ^ k & = c _ { 0 n } \\frac { z ^ { 2 k } } { ( 2 k ) ! } + c _ { 1 n } \\frac { z ^ { 2 k - 2 } } { ( 2 k - 2 ) ! } + \\ldots + c _ { k n } ; \\\\ h _ n ^ k & = 0 , k < 0 . \\end{align*}"}
-{"id": "1009.png", "formula": "\\begin{align*} \\chi ( Y ) = \\int _ { Y } c ( T Y ) = \\int _ { X _ 0 } f _ * c ( T Y ) = \\int _ B \\pi _ * f _ * c ( T Y ) = \\int _ B 6 \\frac { 2 L + 3 L S - S ^ 2 } { ( 1 + S ) ( 1 + 6 L - 2 S ) } c ( T B ) . \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{gather*} \\frac { 1 } { f } \\frac { \\partial f } { \\partial \\tilde { t } _ 1 } = - \\tilde { p } _ 1 \\tilde { q } _ 1 - 1 , \\frac { 1 } { f } \\frac { \\partial f } { \\partial \\tilde { t } _ 2 } = - \\tilde { q } _ 2 - \\tilde { t } _ 2 . \\end{gather*}"}
-{"id": "183.png", "formula": "\\begin{align*} q ^ { 2 d k - d } \\left | \\sum _ { { \\bf m } \\in S _ 0 } \\left ( \\prod _ { j = 1 } ^ k \\widehat { E _ j } ( { \\bf m } ) \\right ) \\right | ^ 2 - \\nu ^ 2 _ k ( 0 ) \\le - \\mbox { A } \\overline { \\mbox { B } } - \\overline { \\mbox { A } } \\mbox { B } \\le 2 | \\mbox { A } | | \\mbox { B } | . \\end{align*}"}
-{"id": "2806.png", "formula": "\\begin{align*} C _ 1 = \\prod _ { p } \\left ( 1 - \\frac { 1 } { p } \\right ) ^ 3 \\left ( 1 + \\frac { 3 } { p } - \\frac { 1 } { p ^ 2 } - \\frac { 1 8 } { p ( p + 2 ) } \\right ) . \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} 0 = [ [ L _ { - q + i } , \\ , L _ { q - i } ] , \\ , L _ { - q + i + 1 } ] = [ L _ { - q + i } , \\ , [ L _ { q - i } , \\ , L _ { - q + i + 1 } ] ] . \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} { \\bf C } _ n = \\begin{pmatrix} 0 & 0 & \\Phi _ { 0 2 } & 0 & \\dots & 0 \\\\ 0 & 0 & 0 & \\Phi _ { 1 3 } & \\dots & \\Phi _ { 1 ( n - 1 ) } \\\\ \\Phi _ { 2 0 } & 0 & 0 & 0 & \\dots & 0 \\\\ 0 & \\Phi _ { 3 1 } & 0 & 0 & \\dots & \\Phi _ { 3 ( n - 1 ) } \\\\ 0 & 0 & \\Phi _ { 4 2 } & 0 & \\dots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & \\Phi _ { ( n - 1 ) 1 } & 0 & \\Phi _ { ( n - 1 ) 3 } & \\dots & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "9358.png", "formula": "\\begin{align*} M n \\bigg ( \\sum _ { l = 1 } ^ L | T _ l | \\bigg ) \\leq \\beta | S _ k | . \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} B ( \\vec \\gamma ) : = \\Theta ^ { - 1 } ( \\vec \\gamma ) \\cong \\bigl \\{ f \\in R \\ ; \\big | \\ ; T _ { ( \\alpha , 2 \\mu _ \\alpha ) } ^ - ( f ) = \\gamma _ \\alpha \\cdot T _ { ( \\alpha , 2 \\mu _ \\alpha ) } ^ + ( f ) \\ ; \\ ; \\mbox { \\rm f o r a l l } \\ ; \\ ; \\alpha \\in \\Pi \\bigr \\} . \\end{align*}"}
-{"id": "6475.png", "formula": "\\begin{align*} R ^ e _ n ( s _ { v _ j } ) ( w _ { j e } ) = - \\sum _ { i = ( j - 1 ) e + 1 } ^ { ( j + 1 ) e - 1 } w _ i . \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{align*} u ^ { ( \\delta , \\lambda ) } ( t , x ) : = \\lambda ^ { - \\frac { 4 } { \\nu - 1 } } \\phi ^ { ( \\delta ) } ( \\lambda ^ { - 4 } t , \\lambda ^ { - 1 } \\delta x ) . \\end{align*}"}
-{"id": "4688.png", "formula": "\\begin{align*} \\psi _ { \\alpha , 0 } ( x ) = \\frac { 1 } { 2 \\alpha } + \\frac { 1 } { \\alpha \\pi \\sqrt { 1 - x ^ 2 } } \\left [ \\sqrt { \\alpha ^ 2 - 1 } - \\sqrt { 1 - x ^ 2 } \\arctan \\left ( \\frac { \\sqrt { \\alpha ^ 2 - 1 } } { \\sqrt { 1 - x ^ 2 } } \\right ) \\right ] , \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} \\| F ( t - s ) \\| _ { L ^ r ( B ^ c _ { A \\sqrt t } ) } & = ( t - s ) ^ { - 2 + 3 / ( 2 r ) } \\Bigl ( \\int _ { | x | \\ge A \\sqrt t / \\sqrt { t - s } } | F ( x , 1 ) | ^ r \\dd x \\Bigr ) ^ { 1 / r } \\\\ & \\le C A ^ { - 4 + 3 / r } t ^ { - 2 + 3 / ( 2 r ) } . \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} V ^ { r } _ { h } = Y ^ { r } _ { h } \\cap H _ { 0 } ^ { 1 } ( \\Omega ) , \\mathcal { V } ^ { r } _ { h } = V ^ { r } _ { h } \\oplus { \\rm i } V ^ { r } _ { h } , \\mathbf { V } ^ { r } _ { h } = \\big ( Y ^ { r } _ { h } \\big ) ^ { 3 } \\cap \\mathbf { H } ^ { 1 } _ { \\rm t } ( \\Omega ) . \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} c \\cdot V | _ S = - \\frac { 1 } { \\gamma _ n } c ^ \\prime \\cdot \\nabla \\left ( \\frac { m ^ \\prime \\cdot x ^ \\prime } { | x ^ \\prime | ^ n } \\right ) + O \\left ( \\frac { 1 } { | x ^ \\prime | ^ { n + \\varepsilon } } \\right ) \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} P _ { l } T ^ { \\ , j } \\ , e _ { k } & = \\pm \\ , \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Bigr ) \\ \\Bigl ( \\prod _ { s = b _ l + 1 } ^ { b _ { l } + n } w _ s \\Bigr ) \\ \\Bigl ( \\prod _ { s = b _ l + 1 } ^ { b _ { l + 1 } - 1 } w _ s \\Bigr ) ^ { - 1 } \\ e _ { b _ { l } + n } \\\\ & = \\pm \\ , \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Bigr ) \\ \\Bigl ( \\prod _ { s = b _ { l } + n + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Bigr ) ^ { - 1 } e _ { b _ { l } + n } \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{align*} K _ { \\mathsf { S i n e _ 2 } } ( x , y ) = \\frac { \\sin \\left ( \\pi \\left ( y - x \\right ) \\right ) } { \\pi \\left ( y - x \\right ) } , \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} e ^ { - M } \\left \\Vert u \\left ( . , 0 \\right ) \\right \\Vert _ { X } \\leq \\left \\Vert u \\left ( . , t \\right ) \\right \\Vert _ { X } = \\left \\Vert \\tilde { u } \\left ( . , s \\right ) \\right \\Vert _ { X } \\leq e ^ { M } \\left \\Vert u \\left ( . , 0 \\right ) \\right \\Vert _ { X } t , s \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} ( z ) _ { k } ( 1 - a - z ) _ { a - k } = ( - 1 ) ^ a ( z ) _ { a - k } ( 1 - a - z ) _ { k } . \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} u ( \\varepsilon ) = v [ \\varepsilon ^ { \\pi / \\omega } ] \\ , , U ( \\varepsilon ) = V [ \\varepsilon ^ { \\pi / \\omega } ] \\forall \\ , \\varepsilon \\in ( 0 , \\eta _ 1 ^ { \\omega / \\pi } ) \\ , . \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\big ( \\nabla \\cdot \\mathbf { A } , \\ , \\varphi ) \\frac { { \\rm d } ^ { 2 } \\eta } { { \\rm d } t ^ { 2 } } \\ , \\mathrm { d } t = 0 , \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} R ^ { ( k + 1 ) } = \\frac { 1 - \\frac { T } { N - B } } { 1 - ( \\frac { T } { N - B } ) ^ M } \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} \\| \\pi _ j ( [ x ] _ { j + 1 } ) \\| _ j \\leqslant \\inf _ { p _ { j + 1 } ( w ) = 0 } p _ { j + 1 } ( x - w ) = \\| [ x ] _ { j + 1 } \\| _ { j + 1 } . \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} c = \\frac { \\lambda - \\mu } { 2 ( p - 2 ) } . \\end{align*}"}
-{"id": "8256.png", "formula": "\\begin{align*} - \\Delta _ n \\psi = - \\sum _ { j = 1 } ^ n \\frac { \\partial ^ 2 \\psi } { \\partial { x _ j } ^ 2 } . \\end{align*}"}
-{"id": "9562.png", "formula": "\\begin{align*} \\left | \\mathcal { F } \\left [ f \\cdot S _ { l } g \\right ] \\left ( \\omega \\right ) \\right | ^ { 2 } = \\frac { 1 } { 4 } \\sum _ { k \\in \\mathbb { Z } } \\sum _ { j \\in \\mathbb { Z } } A _ { k } \\overline { A _ { j } } \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} \\mathbf { f } & = ( k _ s + 1 ) \\mathbf { e } _ 2 - ( k _ s + 1 ) \\mathbf { e } _ 2 ( 0 ) + \\nu \\\\ \\mathbf { \\dot { \\nu } } & = ( k _ s + 1 ) \\alpha _ 2 \\mathbf { e } _ 2 + \\beta \\mathrm { s i g n } ( \\mathbf { e } _ 2 ) \\\\ \\mathbf { e } _ 2 & = \\mathbf { \\dot { e } } _ 1 + \\alpha _ 1 ( \\mathbf { p } _ d - \\mathbf { p } ) \\\\ \\mathbf { u } & = k _ p \\mathbf { q } _ d + k _ i \\int \\mathbf { q } _ d d t + k _ d \\mathbf { \\dot { q } _ d } \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} \\underline { \\mathbf { E } } _ { \\mathbf { V } } \\simeq \\mathbb { E } _ { \\mathbf { V } } . \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} \\xi ^ { k + 1 } P = \\sum _ { j = 0 } ^ k c _ j \\cdot \\xi ^ { k - j } P , c _ j \\in R \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} 2 - ( K + 1 ) ( 1 - X _ T ) ^ 2 & = \\frac { K + 1 } { 2 K + 1 } ( 1 + X _ T ) ^ 2 - \\frac { 2 } { 2 K + 1 } \\left [ ( K + 1 ) X _ T - K \\right ] ^ 2 \\\\ & \\le \\frac { K + 1 } { 2 K + 1 } ( 1 + X _ T ) ^ 2 . \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{gather*} A ^ { S _ 1 G _ 1 S _ 2 G _ 2 } ( z ) = \\frac { 1 } { z ^ { 4 / 3 } } \\begin{pmatrix} - t ^ { 1 / 3 } & 0 & 0 \\\\ 0 & - \\omega t ^ { 1 / 3 } & 0 \\\\ 0 & 0 & - \\omega ^ 2 t ^ { 1 / 3 } \\end{pmatrix} \\\\ \\hphantom { A ^ { S _ 1 G _ 1 S _ 2 G _ 2 } ( z ) = } { } + \\frac { 1 } { z } \\begin{pmatrix} \\theta ^ \\infty _ 1 / 3 - 2 / 3 & * & * \\\\ * & \\theta ^ \\infty _ 1 / 3 - 2 / 3 & * \\\\ * & * & \\theta ^ \\infty _ 1 / 3 - 2 / 3 \\end{pmatrix} + \\cdots . \\end{gather*}"}
-{"id": "7987.png", "formula": "\\begin{align*} S w = \\{ h ( u _ \\infty + u _ s ) + \\widetilde U \\} \\cdot \\nabla w + w \\cdot \\nabla ( h u _ s + \\widetilde U ) . \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} E : = \\{ \\# { \\cal E } _ 1 = r _ 1 \\} \\cap \\{ \\# { \\cal E } _ 2 = r _ 2 \\} \\cap \\{ { \\cal E } _ 1 \\neq { \\cal E } _ 2 \\} \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{align*} \\mu _ Y ( Y \\setminus Y _ { \\varepsilon } ) = O ( \\varepsilon ) \\end{align*}"}
-{"id": "5255.png", "formula": "\\begin{align*} \\varphi _ \\pm ( x ) = \\frac { p _ \\pm \\cdot x } { | x | ^ n } + O \\left ( \\frac { 1 } { | x | ^ { n - 1 + \\varepsilon } } \\right ) , \\nabla \\varphi _ \\pm ( x ) = \\nabla \\left ( \\frac { p _ \\pm \\cdot x } { | x | ^ n } \\right ) + O \\left ( \\frac { 1 } { | x | ^ { n + \\varepsilon } } \\right ) , \\end{align*}"}
-{"id": "6100.png", "formula": "\\begin{align*} [ [ S _ { j } , \\ , S _ { j + 1 } ] , \\ , L _ { - j - 1 } ] = [ [ L _ { - j - 1 } , \\ , S _ { j } ] , \\ , S _ { j + 1 } ] = [ L _ { - 1 } , \\ , S _ { j + 1 } ] = S _ j \\neq 0 . \\end{align*}"}
-{"id": "7542.png", "formula": "\\begin{gather*} h _ 1 = t ^ { 1 / 2 } ( \\log t ) ^ { - 1 / 2 } \\left ( 1 + ( \\log t ) ^ { - 1 } \\right ) , h _ 2 = t ^ { - 1 / 2 } ( \\log t ) ^ { - 1 / 2 } \\\\ h _ 3 = t ^ { 1 / 2 } ( \\log t ) ^ { 1 / 2 } , h _ 4 = t ^ { - 1 / 2 } ( \\log t ) ^ { 1 / 2 } \\left ( 1 + ( \\log t ) ^ { - 1 } \\right ) \\end{gather*}"}
-{"id": "6292.png", "formula": "\\begin{align*} \\varphi ( \\sigma ^ s ( \\omega , y ) ) & = \\gamma ( 1 _ G , \\widehat { \\phi } ( \\mathfrak { C } ( ( \\omega ^ 1 , y ^ 1 ) , 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ) ) \\\\ & = \\gamma ( s , \\widehat { \\phi } ( \\mathfrak { C } ( ( \\omega ^ 0 , y ^ 0 ) , 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ) ) \\\\ & = T ^ s ( \\gamma ( 1 _ G , \\widehat { \\phi } ( \\mathfrak { C } ( ( \\omega ^ 0 , y ^ 0 ) , 1 _ { H _ 1 } , 1 _ { H _ 2 } ) ) ) ) \\\\ & = T ^ s ( \\varphi ( \\omega , y ) ) . \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{align*} \\lambda _ k = \\int _ \\Omega F ^ { p } ( \\nabla u _ k ) \\ , d x , \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} \\| f \\| _ { H ^ q } \\leq \\left ( \\sum _ { n = 0 } ^ \\infty | a _ n | ^ 2 c _ \\beta ( n ) \\right ) ^ \\frac { 1 } { 2 } = : \\| f \\| _ { D _ \\beta } . \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} \\begin{aligned} f ^ { + } _ { V } ( v ) & = \\psi ^ { - 1 } _ { x } ( t ) , \\\\ f ^ { - } _ { V } ( v ) & = ( - \\psi _ { x } ) ^ { - 1 } ( - t ) , \\end{aligned} \\end{align*}"}
-{"id": "6254.png", "formula": "\\begin{align*} \\pi _ V A \\pi _ U \\subseteq A \\pi _ V \\pi _ U = 0 \\pi _ U A \\pi _ V \\subseteq A \\pi _ U \\pi _ V = 0 . \\end{align*}"}
-{"id": "9496.png", "formula": "\\begin{align*} A _ { ( q ( z ) , p ( z ) ) } \\coloneqq \\begin{pmatrix} q _ 1 & 1 & 0 & 0 \\\\ [ 6 p t ] q _ 0 & 0 & 0 & 0 \\\\ [ 6 p t ] 0 & 0 & q _ 1 & 1 \\\\ [ 6 p t ] 0 & 0 & q _ 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} & \\{ \\theta _ 0 = - t \\} \\cap \\{ \\theta ^ n _ 0 = - t ^ n _ 0 \\} \\cap E _ t ( i , n ) \\cap O _ n = \\\\ & ( F _ t ) ^ c \\cap \\{ \\widehat { \\pi } _ 0 ( s ) , \\widehat { \\pi } _ 1 ( s ) \\in [ i / n , ( i + 1 ) / n ) s \\in [ - t ^ n _ 0 , - t ] \\} \\cap E _ t ( i , n , - t ^ n _ 0 ) \\cap O _ n . \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} \\det ( I + L ) = \\sum _ { J \\subseteq [ N ] } \\det ( L _ J ) . \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} w _ 1 = P _ 0 , w _ 2 = P _ 1 , w _ 3 = P _ 2 , w _ 4 = Q _ 1 , w _ 5 = Q _ 2 , w _ 6 = Q _ 3 . \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} t ( l _ n ) = 1 \\ \\ \\longleftrightarrow \\ \\ n \\in F . \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} \\mathcal { U } [ N _ 1 , N _ 2 ] : = \\Big \\{ q \\in C ^ 2 ( \\overline { \\Omega } ) \\ , \\Big | \\ , \\| q - \\theta _ L \\| _ 1 \\leq N _ 1 \\delta , \\quad \\| q _ { x x } \\| \\leq N _ 2 \\delta , q ( 0 ) = \\theta _ l , \\ q ( 1 ) = \\theta _ r \\Big \\} . \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{align*} I _ { 0 } ( q ; x ) = \\frac { 2 ^ { q - \\frac { 1 } { 2 } } } { \\sqrt { \\pi } } \\ { _ { 1 } F _ { 1 } } ( q , q + \\frac { 1 } { 2 } , 2 x ) , \\ \\ \\ R e ( q ) > 0 . \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} \\int _ { \\R ^ n } \\phi ^ \\circ ( f ) g & \\le \\int _ { \\R ^ n } \\phi ^ \\circ ( f ) | g | \\\\ & = \\int _ { \\R ^ n } f \\cdot \\eta _ { \\phi ^ \\circ } ( f ) | g | \\le \\int _ { \\R ^ n } \\phi \\left ( \\eta _ { \\phi ^ \\circ } ( f ) | g | \\right ) = \\int _ { \\R ^ n } | g | . \\end{align*}"}
-{"id": "3121.png", "formula": "\\begin{align*} h _ { \\hat { \\mu } _ { \\Omega } } ( \\hat { S } ^ { 2 N } ) = \\log 2 . \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} \\begin{cases} i \\dd _ t \\psi _ j ( t ) = ( A + u _ 0 B ) \\psi _ j ( t ) + u _ 1 ( t ) B \\psi _ j ( t ) , \\ \\ \\ \\ \\ & t \\in ( 0 , T ) , \\\\ \\psi _ j ^ 0 = \\psi _ j ( 0 ) , \\ & j \\in \\N ^ * . \\\\ \\end{cases} \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{align*} \\dot { y } y ^ { - 1 } + \\left ( \\dot { y } y ^ { - 1 } \\right ) ^ * = P \\left ( \\left [ \\left ( y \\phi _ { - 1 } y ^ { - 1 } \\right ) ^ * , y \\phi _ { - 1 } y ^ { - 1 } \\right ] \\right ) . \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} \\rho _ { T , \\Phi } : = \\frac { 1 } { T ^ { k / 2 } } \\sum _ { j _ 1 \\neq \\ldots \\neq j _ k } \\sigma _ { j _ 1 } \\ldots \\sigma _ { j _ k } \\int _ { [ 0 , T ] ^ k } \\Phi ( x ^ { j _ 1 } + \\xi ^ { j _ 1 } _ { s _ 1 } , \\ldots , x ^ { j _ k } + \\xi ^ { j _ k } _ { s _ k } ) d s _ 1 \\ldots d s _ k , \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} A \\limsup _ { t \\to + \\infty } \\| \\theta _ { n + 1 } ( t ) \\| _ { L ^ 1 ( B _ { A \\sqrt t } ^ c ) } & = A \\limsup _ { t \\to + \\infty } \\| \\theta _ { n + 1 } ( 4 t ) \\| _ { L ^ 1 ( B _ { A \\sqrt { 4 t } } ^ c ) } \\\\ & \\le C _ 0 \\Bigl ( \\| u _ 0 \\| _ 3 + \\| \\theta _ 0 \\| _ 1 \\Bigr ) + 2 C \\varepsilon ( \\varepsilon + \\kappa _ n ) . \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} \\langle v , w \\rangle : = \\sum _ { i \\in Q _ 0 } m _ i \\mathrm { t r } \\left ( h _ i ^ { - 1 } v h _ i ^ { - 1 } w \\right ) \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} b \\leq a = \\left ( m - c \\right ) / b \\leq \\left ( 4 n + 1 \\right ) / 5 < n . \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} \\widetilde u _ { \\varepsilon _ n } ( x ) : = u _ { \\varepsilon _ n } ( x _ n + \\varepsilon _ n x ) . \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} s _ { 2 } \\left ( 0 \\right ) = 0 , \\thinspace \\thinspace s _ { 2 } \\left ( 1 \\right ) = 1 , \\thinspace \\thinspace s _ { 2 } \\left ( 2 \\right ) = 1 , \\thinspace \\thinspace s _ { 2 } \\left ( 3 \\right ) = 2 . \\end{align*}"}
-{"id": "9444.png", "formula": "\\begin{align*} e ^ { - r t } \\Lambda ( h ( t ) , \\widetilde { U } ( t ) ) + \\widetilde { \\Lambda } ( h , \\widetilde { W } , e ^ { - r s } Q _ r ( w ^ * ) ; 0 , t ) + R ( t ) = \\Lambda ( h ( 0 ) , \\widetilde { U } ( 0 ) ) . \\end{align*}"}
-{"id": "7898.png", "formula": "\\begin{align*} U = B _ s \\coloneqq \\left \\{ x + s W _ \\phi \\mid x \\in B x + s W _ \\phi \\subset B \\right \\} . \\end{align*}"}
-{"id": "7043.png", "formula": "\\begin{align*} \\left [ K \\left ( \\xi \\right ) + A + \\lambda \\right ] \\hat { u } \\left ( \\xi \\right ) = f ^ { \\symbol { 9 4 } } \\left ( \\xi \\right ) , \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} H ^ i ( X _ { d , p } ; \\mathbb { F } _ p ) = \\begin{cases} \\mathbb { Z } / p , & i = 0 , \\\\ N ^ { \\oplus \\frac { 1 } { p } { p \\choose i / d } } , & i = d , \\ldots , ( p - 2 ) d , \\\\ M , & t = ( p - 1 ) d , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} x ^ 3 = y ^ 3 \\implies x = y \\end{align*}"}
-{"id": "6063.png", "formula": "\\begin{align*} 0 = [ [ L _ { - 1 } , \\ , L _ 1 ] , \\ , L _ r ] = [ L _ 1 , [ L _ { - 1 } \\ , L _ r ] ] \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} M _ c \\cdot \\vec { x } _ c = ( 0 , 0 , 2 c ^ { n - 1 } , - 2 ( p + c ) ^ { n - 1 } ) ^ \\mathrm { T } . \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} d \\zeta ^ { h , k , x } ( t ) = A \\zeta ^ { h , k , x } ( t ) d t + \\sigma ' ( X ( t , x ) ) . \\zeta ^ { h , k , x } ( t ) d W ( t ) + \\sigma '' ( X ( t , x ) ) . \\bigl ( \\eta ^ { h , x } ( t ) , \\eta ^ { k , x } ( t ) \\bigr ) d W ( t ) , \\end{align*}"}
-{"id": "9927.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { ( 2 \\pi ) ^ d } \\int d \\xi \\ e ^ { i \\xi \\cdot x } \\hat f ( \\xi ) . \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} 0 & = d _ 1 ( d _ 2 + 1 ) ^ 2 - d _ 2 ( d _ 1 + 1 ) ^ 2 \\\\ & = ( d _ 1 d _ 2 - 1 ) ( d _ 2 - d _ 1 ) , \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{gather*} u ( x ) = \\int _ S \\frac { 1 - | x | ^ { 2 } } { | x - \\zeta | ^ n } d \\sigma ( \\zeta ) = 1 \\end{gather*}"}
-{"id": "7101.png", "formula": "\\begin{align*} \\{ k < \\tilde { k } : e _ k ( | e _ j | ) = 1 \\} = \\{ k < \\tilde { k } : g ( e _ k ) ( | g ( e _ j ) | ) = 1 \\} = \\{ k < \\tilde { k } : f _ k ( | f _ j | ) = 1 \\} . \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} - \\frac { \\partial H ^ i ( t , x , \\gamma , y ^ i , q ^ i , r ^ i ) } { \\partial x ^ k } = \\lambda \\theta \\bigg ( \\frac { 1 } { n } - \\delta _ { i , k } \\bigg ) \\gamma ^ i - \\lambda \\varepsilon \\bigg ( \\frac { 1 } { n } - \\delta _ { i , k } \\bigg ) ( \\bar x - x ^ i ) - \\frac { a } { n } \\sum _ { j = 1 } ^ n ( y ^ { i , j } - y ^ { i , k } ) \\ , , \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} ( f * g ) ( i , j ) = \\sum _ { ( k , \\ell ) \\in X } f ( i - k , j - \\ell ) g ( k , \\ell ) \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} \\Phi _ { L ^ * } ( L ) = \\sum _ { J \\subseteq [ N ] } p _ J ( L ^ * ) \\log \\det ( L _ J ) - \\log \\det ( I + L ) . \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{gather*} d _ { n } ^ { ( 2 ) } ( x , y | q ) = \\\\ ( - 1 ) ^ { n } \\sum _ { j = 0 } ^ { \\left \\lfloor n / 2 \\right \\rfloor } ( - 1 ) ^ { j } q ^ { - \\binom { n - 2 j } { 2 } - j + \\binom { j } { 2 } } \\frac { ( q ) _ { n } } { ( q ) _ { j } ( q ) _ { n - 2 j } } b _ { n - 2 j } ( x | q ) b _ { n - 2 j } ( y | q ) , \\\\ f _ { n } ^ { ( 2 ) } ( x , y | q ) \\allowbreak = \\allowbreak \\sum _ { j = 0 } ^ { \\left \\lfloor n / 2 \\right \\rfloor } \\frac { ( q ) _ { n } } { ( q ) _ { j } ( q ) _ { n - 2 j } } h _ { n - 2 j } ( x | q ) h _ { n - 2 j } ( y | q ) . \\end{gather*}"}
-{"id": "9712.png", "formula": "\\begin{align*} { G } _ { e x t } = ( \\underbrace { G | G | . . . | G } _ { m ~ } ) . \\end{align*}"}
-{"id": "7315.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\alpha _ 1 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 1 ] = \\alpha _ 6 y _ 5 , [ y _ 2 , y _ 2 ] = \\beta _ 2 y _ 5 , [ y _ 2 , y _ 3 ] = \\frac { \\gamma _ 1 } { \\gamma _ 3 } y _ 4 + \\theta _ 3 y _ 5 , \\\\ [ y _ 3 , y _ 2 ] = y _ 4 . \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{align*} \\psi \\sum _ { j = 1 } ^ P k _ j \\sum _ { i = 1 } ^ P e _ { i , j } = 0 , { \\rm w h e r e } ~ ~ k _ j > 0 . \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} ( A ^ * A ) _ { \\nu \\nu } = M N \\sum _ { l = 1 } ^ L | d _ \\ell ( s , t ) | ^ 2 . \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} \\mathbf { h } _ { L } = \\sum _ { i _ { 1 } , \\dots , i _ { n } } X _ { i _ { 1 } } X _ { i _ { 2 } } \\cdots X _ { i _ { n } } \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} z ^ { - \\alpha } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac 1 z \\bigg ] = ( z - 1 ) ^ { - \\alpha } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 \\alpha & \\frac 1 2 - \\frac 1 2 \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , - \\frac { 4 z } { ( 1 - z ) ^ 2 } \\bigg ] . \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} y & \\coloneqq \\psi _ 1 ^ { g - n - 1 } \\prod _ { i = 2 } ^ n \\psi _ i ^ { 1 } \\kappa _ 1 \\\\ \\tilde { x } _ \\ell & \\coloneqq - \\frac { Q _ { 3 } ( 5 / 2 ) } { Q _ { 1 } ( 1 ) Q _ 2 ( 3 / 2 ) } \\psi _ 2 ^ { 1 } \\dotsb \\psi _ \\ell ^ { 2 } \\dotsb \\psi _ n ^ { 1 } \\psi _ 1 ^ { g - n - 1 } , \\ell = 2 , \\dots , n . \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} K ( \\psi , A ) & = \\textrm { V a r } [ V _ 2 | \\ , F _ { 1 } ( V _ 1 ) \\in A ] , \\\\ K ' ( \\psi , A ) & = \\frac { \\textrm { V a r } [ V _ 2 | \\ , F _ { 1 } ( V _ 1 ) \\in A ] } { \\textrm { V a r } [ V _ 1 | \\ , F _ { 1 } ( V _ 1 ) \\in A ] } , \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} & \\sigma ^ + | 1 \\rangle = | 0 \\rangle , \\ \\sigma ^ + | 0 \\rangle = 0 , \\ \\langle 0 | \\sigma ^ + = \\langle 1 | , \\ \\langle 1 | \\sigma ^ + = 0 , \\\\ & \\sigma ^ - | 0 \\rangle = | 1 \\rangle , \\ \\sigma ^ - | 1 \\rangle = 0 , \\ \\langle 1 | \\sigma ^ - = \\langle 0 | , \\ \\langle 0 | \\sigma ^ - = 0 . \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} s _ { r , r ' } : = ( - 1 ) ^ { r + r ' } \\sqrt { \\frac { 1 } { 2 k + 3 } } \\sin \\left ( \\frac { \\pi r r ' ( k + 2 ) } { ( 2 k + 3 ) } \\right ) . \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\lambda } ^ d | L _ n | = d ^ n E \\prod _ { i = 0 } ^ { n - 1 } [ \\frac { \\lambda \\rho ( x _ i , \\omega ) \\rho ( x _ { i + 1 } , \\omega ) } { 1 + \\lambda \\rho ( x _ i , \\omega ) \\rho ( x _ { i + 1 } , \\omega ) } ] . \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} G * f = \\frac { 1 } { 4 } \\sum _ { n = 0 } ^ \\infty ( \\nu ^ { * n } * f ) , f \\in C ^ 1 ( X ) . \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} \\dfrac { g ^ { \\prime } } { g } = \\frac { \\lambda _ { 1 } \\left ( 2 H _ { 0 } z + \\lambda _ { 2 } \\right ) } { \\sqrt { \\left ( 2 H _ { 0 } z + \\lambda _ { 2 } \\right ) ^ { 2 } - 1 } } . \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} \\langle \\mathfrak { e } _ { x } , \\Delta _ { \\omega , \\vartheta } ^ { ( \\mathbf { A } ) } \\mathfrak { e } _ { y } \\rangle = \\exp \\left ( i \\int \\nolimits _ { 0 } ^ { 1 } \\left [ \\mathbf { A } ( t , \\alpha y + ( 1 - \\alpha ) x ) \\right ] ( y - x ) \\mathrm { d } \\alpha \\right ) \\langle \\mathfrak { e } _ { x } , \\Delta _ { \\omega , \\vartheta } \\mathfrak { e } _ { y } \\rangle \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} \\varepsilon : = | F | \\inf _ { z \\in F } \\mu ^ { ( R ) } ( z ) > 0 . \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} \\int _ { M ^ 2 } \\big ( | \\Delta ^ \\perp \\vec H | ^ 2 + | \\nabla A ^ o | ^ 2 | \\vec H | ^ 2 + | \\vec H | ^ 4 | A ^ o | ^ 2 \\big ) d \\mu = 0 \\end{align*}"}
-{"id": "9837.png", "formula": "\\begin{align*} \\psi _ { w } = \\left \\{ \\begin{array} { l l } ( v - 1 ) \\phi _ { w } + \\sqrt { v } ( \\tau R ) ^ { - 1 } _ { i , i + 1 } \\phi _ { s _ i w } & \\\\ \\sqrt { v } ( \\tau R ) _ { i , i + 1 } \\phi _ { s _ i w } & . \\end{array} \\right . \\end{align*}"}
-{"id": "9890.png", "formula": "\\begin{align*} F _ w ^ B ( \\mathbf { x } ) = 2 ^ { - o ( w ) } F _ w ^ C ( \\mathbf { x } ) , \\end{align*}"}
-{"id": "5074.png", "formula": "\\begin{align*} \\abs { g ( z ) - g ( \\zeta ) } = \\abs { 2 g ( \\zeta ) + g ( - z ) - g ( \\zeta ) } \\ge \\abs { 2 g ( \\zeta ) } - \\abs { g ( - z ) - g ( \\zeta ) } \\ge r \\ge \\frac { r } { 2 } \\abs { z - \\zeta } . \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{align*} u ( \\cdot , 0 ) \\ , = \\ , u _ { 0 } \\in L ^ { 1 } ( \\mathbb { R } ^ { n } ) \\cap L ^ { \\infty } ( \\mathbb { R } ^ { n } ) . \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{align*} \\partial _ x ^ { ( i ) } p _ t ( x , y ) = \\sum _ { } ^ { } \\frac { i ! } { k _ 1 ! \\cdots k _ i ! } \\partial ^ { ( k ) } _ { f ( x ) } q _ t ( f ( x ) , f ( y ) ) \\prod _ { j = 1 } ^ { i } \\left ( \\frac { \\partial _ x ^ { ( j ) } f ( x ) } { j ! } \\right ) ^ { k _ j } \\partial _ y f ( y ) , \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} A \\Psi _ { \\mathrm { b v } } + B \\Psi _ { \\mathrm { b v } } ' = 0 \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} \\tilde N \\cdot \\nabla \\tilde \\varphi + \\tilde a \\tilde \\varphi = \\tilde b \\textrm { o n } S ^ \\sim \\setminus \\{ 0 \\} , \\end{align*}"}
-{"id": "3631.png", "formula": "\\begin{align*} \\partial _ \\lambda \\Lambda ( y , \\lambda ) & = \\frac { y c } { 2 \\pi } \\sum _ { n \\geq 1 } \\dfrac { y ^ 2 J _ { n + 1 } ( y , - \\lambda y ) } { n } = - \\frac { y c } { 2 \\pi } \\sum _ { n \\geq 1 } \\left ( J _ n - J _ { n - 1 } + \\dfrac { \\lambda y } { n } J _ n \\right ) \\\\ & = \\frac { y c } { 2 \\pi } \\left ( J _ 0 ( y , - \\lambda y ) - \\lambda y \\sum _ { n \\geq 1 } \\dfrac { J _ { n } ( y , - \\lambda y ) } { n } \\right ) , \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle | \\mathrm { R e } \\big ( \\sum _ { k = 1 } ^ { M } J _ 2 ^ { ( k ) , 2 } \\big ) | = | - V _ 0 ( \\| \\theta _ { \\Psi } ^ { M } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } - \\| \\theta _ { \\Psi } ^ { 0 } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } ) | \\leq C \\| \\theta _ { \\Psi } ^ { M } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } + C h ^ { 2 r + 2 } . } \\end{array} \\end{align*}"}
-{"id": "1163.png", "formula": "\\begin{align*} \\lambda _ j - \\lambda _ k - \\lambda _ n + \\lambda _ m = 0 , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\forall j , n , k , m \\in \\N ^ * , \\ k , m \\leq N \\end{align*}"}
-{"id": "2513.png", "formula": "\\begin{align*} f \\left ( z \\right ) f \\left ( z + c _ { 1 } \\right ) : = F = q \\left ( z \\right ) + p _ { 1 } \\left ( z \\right ) e ^ { \\alpha \\left ( z \\right ) } \\in S \\left ( f \\right ) . \\end{align*}"}
-{"id": "7307.png", "formula": "\\begin{align*} [ y _ 1 , y _ 2 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 5 } { \\alpha _ 3 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 2 ] = \\frac { \\alpha _ 3 \\beta _ 2 } { \\beta _ 6 \\gamma _ 1 } y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\beta _ 4 } { \\beta _ 6 } y _ 5 , [ y _ 3 , y _ 1 ] = y _ 5 , [ y _ 2 , y _ 3 ] = y _ 4 . \\end{align*}"}
-{"id": "7188.png", "formula": "\\begin{align*} p ( x ) = x ^ 2 - ( w + w ^ { - 1 } ) x + 1 \\in \\Q ( w + w ^ { - 1 } ) [ x ] , \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} H _ \\epsilon ' = \\frac { v _ \\epsilon ''' } { v _ \\epsilon '' } - \\frac { v _ \\epsilon '' } { v _ \\epsilon ' - a _ t } + \\frac { v _ \\epsilon '' } { b _ t - v _ \\epsilon ' } , \\end{align*}"}
-{"id": "6719.png", "formula": "\\begin{align*} \\left \\Vert u \\right \\Vert _ { W _ { p _ { 1 } } ^ { 1 } \\left ( R _ { + } ; D \\left ( O _ { t } \\right ) , X \\right ) } = \\left \\Vert O _ { t } u \\right \\Vert _ { L _ { p } \\left ( R _ { + } ; X \\right ) } + \\left \\Vert u ^ { \\prime } \\right \\Vert _ { L _ { p } \\left ( R _ { + } ; X \\right ) } = \\end{align*}"}
-{"id": "9594.png", "formula": "\\begin{align*} L ^ i _ { \\underline { j k } } = \\frac 1 2 \\big ( L ^ i _ { j k } + L ^ i _ { k j } \\big ) , \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} \\| \\widehat { f } - f _ 0 \\| _ { L _ 2 ( c ) } ^ 2 = O \\{ \\lambda ^ { 1 - c } J ( f _ 0 ) \\} + O _ \\P \\left \\{ n ^ { - 1 } \\lambda ^ { - ( c + 1 / 2 m ) } [ \\log ( 1 / \\lambda ) ] ^ { r \\wedge ( d - p ) - 1 } \\right \\} , \\end{align*}"}
-{"id": "4218.png", "formula": "\\begin{align*} | J _ { i j } | = a _ { i j } \\hbox { f o r e v e r y } i , j \\hbox { w i t h } v _ i \\neq v _ j \\end{align*}"}
-{"id": "850.png", "formula": "\\begin{align*} K _ Y + \\pi ^ { - 1 } _ * \\Delta = \\pi ^ * ( K _ X + \\Delta ) + \\sum _ i a _ i E _ i , \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} \\begin{aligned} m _ 1 \\dot { x } _ 1 & = e ^ { x _ 2 - x _ 1 } , m _ 2 \\dot { x } _ 2 = - e ^ { x _ 2 - x _ 1 } - e ^ { x _ 2 - x _ 3 } , \\\\ m _ 3 \\dot { x } _ 3 & = e ^ { x _ 2 - x _ 3 } + e ^ { x _ 4 - x _ 3 } , m _ 4 \\dot { x } _ 4 = - e ^ { x _ 4 - x _ 3 } . \\end{aligned} \\end{align*}"}
-{"id": "8440.png", "formula": "\\begin{align*} X = \\ , \\Ref [ ( X ^ { ' } + \\Tilde \\xi ^ { f } ) \\mathbb { I } _ { [ 0 , T ) } ] ; X ^ { ' } = \\ , \\Ref [ ( X - \\Tilde \\zeta ^ { f } ) \\mathbb { I } _ { [ 0 , T ) } ] . \\end{align*}"}
-{"id": "7208.png", "formula": "\\begin{align*} T & = \\chi _ \\rho ( a ) \\\\ R & = \\chi _ \\rho ( a b ) \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} \\pi _ 0 \\left ( \\mathcal { M } ^ { 2 - \\textrm { c o n v } } ( S ^ { n - 1 } \\times S ^ 1 ) \\right ) \\cong \\begin{cases} \\mathcal { K } \\left ( \\mathcal { M } ^ { 2 - \\textrm { c o n v } } ( S ^ { 1 } \\times S ^ 1 ) \\right ) , & n = 2 , \\\\ 0 , & n \\geq 3 , \\end{cases} \\end{align*}"}
-{"id": "3144.png", "formula": "\\begin{align*} ( { \\rm d i v } R m ) _ { j k l } = \\nabla _ k R _ { j l } - \\nabla _ l R _ { j k } . \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} { \\rm F o r } ~ n > 0 : ~ n ( J _ n - J _ { n - 1 } ) + y ^ 2 J _ { n + 1 } + \\lambda y J _ n = 0 ~ , \\end{align*}"}
-{"id": "1537.png", "formula": "\\begin{align*} Y ( x ) = \\lambda Q ( x ) + \\mu S ( x ) , \\lambda , \\mu \\in \\mathbb { R } . \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{gather*} t H _ { \\mathrm { K F S } } ^ { \\frac 4 3 + \\frac 4 3 } \\left ( t ; { q _ 1 , p _ 1 \\atop q _ 2 , p _ 2 } \\right ) = t H _ { \\mathrm { I I I } ( D _ 8 ) } ( t ; q _ 1 , p _ 1 ) \\\\ \\qquad { } + t H _ { \\mathrm { I I I } ( D _ 8 ) } ( t ; q _ 2 , p _ 2 ) - p _ 1 q _ 1 p _ 2 q _ 2 + \\left ( \\frac { q _ 1 q _ 2 } { t } + q _ 1 + q _ 2 \\right ) . \\end{gather*}"}
-{"id": "9587.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\mbox { v a r } ( V _ { n , i } ) = a ^ { \\prime } \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\mbox { v a r } \\left ( X _ { n , i } \\varepsilon _ { n , i } \\right ) \\right ) a \\rightarrow a ^ { \\prime c o n d } a > 0 . \\end{align*}"}
-{"id": "9866.png", "formula": "\\begin{align*} K ( \\rho ) = - \\frac { ( \\rho ) } { \\pi } \\end{align*}"}
-{"id": "2555.png", "formula": "\\begin{align*} \\omega _ j : = \\omega _ j ( c ) & = c \\sqrt { c ^ 2 + \\lambda _ j } \\ ; = \\ ; c ^ 2 \\ ; + \\frac { \\lambda _ j } { 1 + \\sqrt { 1 + \\lambda _ j / c ^ 2 } } \\\\ & = \\ ; c ^ 2 \\ ; + \\frac { \\lambda _ j } { 2 } - \\frac { \\lambda _ j ^ 2 } { 2 c ^ 2 } \\frac { 1 } { ( 1 + \\sqrt { 1 + \\lambda _ j / c ^ 2 } ) ^ 2 } , \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} 0 = A \\cdot [ 1 , \\ , f ] + 2 A \\cdot [ d , \\ , x ] + B \\cdot [ 1 , \\ , f ] + B \\cdot [ d , \\ , x ] = 2 A x ^ { ( 2 ) } y ^ { ( j ) } \\end{align*}"}
-{"id": "2984.png", "formula": "\\begin{align*} \\| P _ { l } T ^ { \\ , j } P _ { l } \\ , x \\| \\ge \\Bigl \\| \\sum _ { k \\ , \\in \\ , [ b _ { l } , b _ { l + 1 } ) \\backslash I _ j } \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Bigr ) x _ { k } e _ { k } \\ , \\Bigr \\| \\ge \\Bigl \\| \\sum _ { k \\ , \\in \\ , I _ { j _ { 0 } } } \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Bigr ) x _ { k } e _ { k } \\ , \\Bigr \\| \\ge \\dfrac { X _ { l } } { 2 } \\cdot \\end{align*}"}
-{"id": "9703.png", "formula": "\\begin{align*} ( T _ { i j } ) \\ = \\ \\sum _ { k = 1 } ^ d \\lambda _ k u _ k u _ k ^ t \\ , \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{align*} X = - \\frac i 2 ( T + U ) \\mbox { a n d } Y = - \\frac 1 2 ( T - U ) . \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} W \\vec { b } = W \\vec { c } - { \\rm i } \\frac { \\tau } { 4 } S \\vec { b } - { \\rm i } \\frac { \\tau } { 2 } Q \\vec { b } , W \\vec { a } = \\vec { r } , Q = Q ( \\vec { a } ) , \\vec { r } = \\vec { r } ( \\vec { b } ) , \\end{align*}"}
-{"id": "5578.png", "formula": "\\begin{align*} F F = F \\mbox { a n d } F G F = F G \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} a = \\frac { 1 } { \\sqrt { 2 } } \\left ( \\partial _ p + p \\right ) ~ , ~ a ^ \\dagger = \\frac { 1 } { \\sqrt { 2 } } \\left ( - \\partial _ p + p \\right ) . \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} A _ { 0 } = \\frac { b _ { 2 } } { \\left ( 2 - \\bar { t } \\right ) ^ { p } } A _ { 1 } = \\frac { b _ { 2 } } { \\left ( 1 - \\bar { t } \\right ) ^ { p } } . \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{align*} P ^ { s , N + 1 } _ { H P } ( t ) \\Lambda ^ { N + 1 } _ N f = \\Lambda ^ { N + 1 } _ N P ^ { s , N } _ { H P } ( t ) f . \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} Z _ { \\ell } = 0 , \\ , i < \\ell < 0 . \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} \\begin{aligned} S _ 1 & = \\{ ( 2 , 0 , 0 ) , ( 1 , 1 , 0 ) , ( 0 , 2 , 0 ) , ( 1 , 0 , 1 ) \\} \\\\ S _ 2 & = \\{ ( 2 , 0 , 0 ) , ( 1 , 1 , 0 ) , ( 0 , 2 , 0 ) , ( 0 , 1 , 1 ) \\} \\\\ S _ 3 & = \\{ ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( 0 , 0 , 1 ) , ( 0 , 0 , 0 ) \\} \\end{aligned} \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 3 , [ y _ 2 , y _ 1 ] = y _ 4 + \\theta y _ 5 , [ y _ 2 , y _ 2 ] = \\frac { \\alpha _ 1 \\beta _ 6 \\gamma _ 1 } { \\beta ^ 2 _ 2 \\gamma _ 5 } y _ 5 , [ y _ 1 , y _ 3 ] = y _ 4 , [ y _ 2 , y _ 3 ] = y _ 5 , [ y _ 1 , y _ 4 ] = y _ 5 . \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} P _ { \\perp } = \\begin{pmatrix} P _ { 1 1 } & P _ { 1 2 } \\\\ P _ { 2 1 } & P _ { 2 2 } \\end{pmatrix} \\end{align*}"}
-{"id": "8632.png", "formula": "\\begin{align*} \\Delta f + \\frac { 1 } { 2 } x \\cdot \\nabla f + \\frac { 1 } { \\alpha } f + | f | ^ \\alpha f = 0 . \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} { } _ { 2 } F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\beta \\\\ & \\gamma \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { \\Gamma ( \\gamma ) \\Gamma ( \\gamma - \\alpha - \\beta ) } { \\Gamma ( \\gamma - \\alpha ) \\Gamma ( \\gamma - \\beta ) } \\end{align*}"}
-{"id": "893.png", "formula": "\\begin{align*} \\widehat { \\Phi _ { \\epsilon , \\phi } ^ f } ( x _ 1 , \\ldots , x _ k ) = \\widehat { \\phi } ( x _ 1 + \\ldots + x _ k ) \\widehat { f _ \\epsilon } ( x _ 2 ) \\ldots \\widehat { f _ \\epsilon } ( x _ k ) . \\end{align*}"}
-{"id": "9529.png", "formula": "\\begin{align*} { \\mathrm { R e } } \\ , \\psi ( w ) \\leq | \\psi ( w ) | \\leq \\delta r K ^ 2 \\sum _ { n , j \\leq J } \\mu ( F _ { n , j } ) + \\mu ( F ) r K ^ 2 < 2 \\delta r K ^ 2 = 2 / 3 . \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{align*} P f & = \\left ( a _ { 1 1 } \\delta _ 0 ( f ) + a _ { 1 2 } \\delta _ 1 ( f ) \\right ) \\cdot 1 + \\left ( a _ { 2 1 } \\delta _ 1 ( f ) + a _ { 2 2 } \\delta _ 0 ( f ) \\right ) \\cdot x \\\\ & = \\delta _ 0 ( f ) \\cdot 1 + \\delta _ 1 ( f ) - \\delta _ 0 ( f ) \\cdot x = f ( 0 ) + \\left ( f ( 1 ) - f ( 0 ) \\right ) x . \\end{align*}"}
-{"id": "502.png", "formula": "\\begin{align*} Q _ { r , a } ( X _ 1 , & \\cdots , X _ r ) = \\\\ & \\frac { p ^ a } { ( p ^ a - 1 ) ( 1 - p ^ a X _ r ) } Q _ { r - 1 , a - 1 } ( p ^ { a + r - 1 } X _ 1 , p ^ { a + r - 2 } X _ 2 , \\cdots , p ^ { a + 1 } X _ { r - 1 } ) \\\\ & - \\frac { 1 } { ( p ^ a - 1 ) ( 1 - X _ r ) } Q _ { r - 1 , a + 1 } ( X _ 1 , \\cdots , X _ { r - 1 } ) . \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} \\Delta _ { A | M } ( \\hat { \\rho } _ { A M } ) ( s + t ) & = \\Delta _ { A | M } ( \\hat { \\rho } _ { A M } ) ( s ) + \\Delta _ { A | M } ( ( \\mathcal { N } _ A ( s ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) ) ( t ) \\\\ & \\ge \\Delta _ { A | M } ( \\hat { \\rho } _ { A M } ) ( s ) \\ ; . \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} [ f , g ] _ F = T _ { t _ 0 } S _ { - t _ 0 } [ f , g ] _ F \\subseteq T _ { t _ 0 } [ S _ { - t _ 0 } f , S _ { - t _ 0 } g ] _ E , \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} \\beta \\ll _ { c , C , \\varepsilon } \\Big \\vert \\frac { 1 } { N ^ { h - m } } \\int \\limits _ { \\mathbf { x } \\in \\mathbb { R } ^ { h - m } } F _ 1 ( \\mathbf { x } ) \\prod \\limits _ { j = 1 } ^ d g _ j ( \\xi _ j ( \\Phi ( \\mathbf { x } ) ) + a _ j ) \\ , d \\mathbf { x } \\Big \\vert , \\end{align*}"}
-{"id": "3374.png", "formula": "\\begin{align*} R _ { \\textrm { c u t o f f } } ( p ) = 1 - h _ { \\frac { 1 } { 2 } } ( p ) , \\end{align*}"}
-{"id": "3414.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , T ] } | \\tau _ n ( t ) - t | \\to 0 \\textrm { a n d } \\lim _ { n \\to \\infty } \\sup _ { t \\in [ 0 , T ] } \\rho ( x ^ n ( \\tau _ n ( t ) ) , x ( t ) ) = 0 . \\end{align*}"}
-{"id": "3528.png", "formula": "\\begin{align*} _ { \\mu , \\sigma } D _ { v , q ; z } ^ { \\alpha , \\eta , p } ( f ( z ) ) = _ { \\mu , \\sigma \\ast } D _ { v , q ; z } ^ { \\alpha , \\eta , p } ( e ^ { f \\left ( z \\right ) } ) . \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 1 e _ 5 + \\alpha _ 2 e _ 4 + \\alpha _ 3 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 4 e _ 3 + \\alpha _ 5 e _ 4 + \\alpha _ 6 e _ 5 , [ e _ 2 , e _ 1 ] = \\beta _ 1 e _ 3 + \\beta _ 2 e _ 4 + \\beta _ 3 e _ 5 , \\\\ [ e _ 2 , e _ 2 ] = \\beta _ 4 e _ 3 + \\beta _ 5 e _ 4 + \\beta _ 6 e _ 5 , [ e _ 1 , e _ 3 ] = \\gamma _ 1 e _ 4 + \\gamma _ 2 e _ 5 , [ e _ 2 , e _ 3 ] = \\gamma _ 3 e _ 4 + \\gamma _ 4 e _ 5 , [ e _ 1 , e _ 4 ] = \\gamma _ 5 e _ 5 , [ e _ 2 , e _ 4 ] = \\gamma _ 6 e _ 5 . \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} - 1 \\leq \\rho _ { i j } \\leq 1 , \\ ; i , j = 1 , 2 , 3 , i < j . \\end{align*}"}
-{"id": "9692.png", "formula": "\\begin{align*} H _ q ( P ) \\ = \\ P \\setminus \\bigcup _ { i \\in I _ q ( P ) } F _ i \\end{align*}"}
-{"id": "9474.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & - q _ 1 & 0 & \\ldots & 0 & - a _ 0 \\\\ & 1 & - q _ 2 & \\ddots & \\vdots & - a _ 1 \\\\ & & \\ddots & \\ddots & 0 & \\vdots \\\\ & & & 1 & - q _ { m - 1 } & - a _ { m - 2 } \\\\ & & & & 1 & - q _ m - a _ { m - 1 } \\\\ & & & & & 1 \\end{bmatrix} s = \\begin{bmatrix} 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\\\ 0 \\\\ 1 \\end{bmatrix} \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{align*} \\sum _ { i , j , m , p } \\delta _ { j } ^ { i } \\pi ( K _ a F _ a ) _ { m } ^ { j } c _ { p } ^ { m } \\pi ( E _ a K _ \\lambda ) _ { i } ^ { p } = \\sum _ { m , p } c _ { p } ^ { m } \\pi ( E _ a K _ \\lambda K _ a F _ a ) _ { m } ^ { p } . \\end{align*}"}
-{"id": "4333.png", "formula": "\\begin{align*} j ( E ) = c _ 4 ^ 3 / \\Delta = \\frac { ( 1 6 ) ^ 2 ( a ^ 2 - 3 b ) ^ 3 } { b ^ 2 ( a ^ 2 - 4 b ) } . \\end{align*}"}
-{"id": "5714.png", "formula": "\\begin{align*} \\sigma ^ { i j } ( \\hat { \\rho } ) : = \\frac { 1 } { 2 } \\mathrm { T r } \\left [ \\left \\{ \\hat { R } ^ i - r ^ i ( \\hat { \\rho } ) , \\ ; \\hat { R } ^ j - r ^ j ( \\hat { \\rho } ) \\right \\} \\hat { \\rho } \\right ] \\ ; , i , \\ , j = 1 , \\ , \\ldots , \\ , 2 n \\ ; , \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} H _ \\varphi ( H ( f ) ) = H ( H _ \\varphi ( f ) ) \\end{align*}"}
-{"id": "6026.png", "formula": "\\begin{align*} \\widetilde { q } ( \\{ \\Phi \\} \\times S _ { \\Phi } ) = \\widetilde { q } ( \\{ \\Phi ' \\} \\times S _ { \\Phi ' } ) \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} \\alpha k ' ( y , \\alpha ) ( \\ell _ { \\Psi ^ * F } ) ' ( \\alpha ^ 2 k ( y , \\alpha ) , \\alpha ) = \\alpha \\cdot ( \\ell _ F ) ' ( \\alpha y ) + h ( \\alpha y , \\alpha ) \\cdot \\langle 2 \\rangle _ F ( \\alpha ) \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{align*} d \\rho _ E ( x ) = \\frac { 1 } { \\pi } \\sum _ { j = 0 } ^ n \\frac { | P _ E ( x ) | } { \\sqrt { | Q _ E ( x ) | } } \\chi _ { [ E _ j ^ + , E _ { j + 1 } ^ - ] } ( x ) d x \\end{align*}"}
-{"id": "1945.png", "formula": "\\begin{align*} g ^ k _ \\epsilon = f _ { k + 1 } ( z ) H ^ { k + 1 } U ^ k _ \\epsilon + H ^ { k } U ^ { k - 1 } _ \\epsilon f _ { k } + H ^ k R ^ k _ \\epsilon . \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n \\left ( e ^ { n f } H _ { i j } \\right ) _ { , i j } = \\sum _ { i , j = 1 } ^ n e ^ { n f } H _ { i j , i j } - 2 e ^ { 2 n f } a _ i H _ { i j , j } + 2 e ^ { 3 n f } a _ i a _ j H _ { i j } . \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{align*} f ( x ) : = \\left \\{ \\begin{array} { l l } c _ s & \\textrm { i f } x = k _ s , \\\\ e & \\textrm { o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "6924.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & - \\Delta \\psi _ 1 = b | \\nabla \\psi | ^ 2 , \\ - \\Delta \\psi _ 2 = R , \\ - \\Delta \\psi _ 3 = S \\Omega , \\\\ & \\psi _ 1 = \\psi _ 2 = \\psi _ 3 = 0 \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{align*} & t ^ N \\langle 1 \\cdots M | \\Bigg ( \\frac { \\mathcal { B } ( t ^ { - 1 } z _ 1 ) } { t ^ { M + 1 } } \\Bigg ) \\cdots \\Bigg ( \\frac { \\mathcal { B } ( t ^ { - 1 } z _ N ) } { t ^ { M + 1 } } \\Bigg ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle \\\\ = & \\prod _ { j = 1 } ^ N z _ j ^ { j - 1 - N } ( 1 + t ^ { - 1 } z _ j ^ 2 ) \\prod _ { 1 \\le j < k \\le N } ( 1 + t ^ { - 1 } z _ j z _ k ) ( t ^ { - 1 } + z _ j z _ k ^ { - 1 } ) s p _ { \\overline { \\lambda } } ( \\{ z \\} _ N ) . \\end{align*}"}
-{"id": "5909.png", "formula": "\\begin{align*} A _ n & : = \\left ( s _ e ( i , j ) - s _ o ( i , j ) \\right ) _ { 1 \\leq i , j \\leq n } , \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} q ^ \\pm \\ ; : = \\ ; \\textstyle \\frac { 4 ^ B ( - B \\pm ( 1 + \\nu ) ) \\Gamma ( - 2 B ) } { ( 1 + \\nu ) \\Gamma ( - B ) } \\ , . \\end{align*}"}
-{"id": "9883.png", "formula": "\\begin{align*} \\sum _ { | \\gamma | = L } \\left ( \\frac { 2 e t } { L } \\right ) ^ L Q ( \\gamma ) \\leq \\left ( \\frac { 2 e t } { L } \\right ) ^ L \\left ( \\frac { 8 e n } { L } \\right ) ^ { L / 2 + 1 } . \\leq { C _ 1 n } ( C _ 2 n t ^ 2 / L ^ 3 ) ^ { L / 2 } . \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} \\mathbb { E } ( U _ n - \\mathbb { E } U _ n ) ^ 2 = \\sum _ { l = 1 } ^ { N } \\mathbb { E } X _ l ^ 2 . \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} p ( s x + t y ) = \\sum _ { j = 0 } ^ { \\deg p } p ^ { ( j ) } ( x , y ) s ^ { \\deg p - j } t ^ j , \\end{align*}"}
-{"id": "6706.png", "formula": "\\begin{align*} \\begin{aligned} Q ( 2 \\tau ) \\leq C K \\int \\limits _ { 1 } ^ { 2 \\tau } Q ( r ) d r . \\end{aligned} \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{align*} & D _ { p ^ l + 3 , k } ( 1 , x ) = D _ { p ^ l + 2 , k } ( 1 , x ) - x D _ { p ^ l + 1 , k } ( 1 , x ) \\cr & = \\frac { 1 } { 2 } \\ , ( 1 - 4 x ) ^ { \\frac { p ^ l + 1 } { 2 } } + \\frac { k } { 2 } \\ , x \\ , ( 1 - 4 x ) ^ { \\frac { p ^ l - 1 } { 2 } } - \\Big ( 1 - \\frac { k } { 2 } \\Big ) x + \\frac { 1 } { 2 } \\cr & - \\Big ( \\frac { 1 } { 2 } - \\frac { k } { 4 } \\Big ) \\ , x \\ , ( 1 - 4 x ) ^ { \\frac { p ^ l + 1 } { 2 } } - \\frac { k } { 4 } \\ , x \\ , ( 1 - 4 x ) ^ { \\frac { p ^ l - 1 } { 2 } } - \\frac { 1 } { 2 } \\ , x \\end{align*}"}
-{"id": "9377.png", "formula": "\\begin{align*} \\| \\phi \\| _ { \\infty } = \\sup _ { i , j , k } | m _ { i j k } | . \\end{align*}"}
-{"id": "5087.png", "formula": "\\begin{align*} F ( x ) \\ , : = \\ , \\mu \\bigl ( [ 0 , x ] \\bigr ) \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} - d X _ t = f ( t , X _ t , { \\varphi _ t } ' \\sigma _ t , { \\varphi _ t } ' \\beta _ t ) d t - { \\varphi _ t } ' \\sigma _ t d W _ t - { \\varphi _ t } ' \\beta _ t d \\tilde N _ t , \\ ; ; \\ ; X _ 0 = x , \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{align*} A _ 1 ( t ) & \\le C \\sqrt { t } \\int _ 0 ^ \\infty d \\xi \\int _ { - \\infty } ^ { \\infty } d y \\int _ { - \\infty } ^ { \\infty } d z \\int _ 0 ^ t d u \\ , \\chi _ t ( \\xi , y , z , u ) \\\\ & \\cdot \\int _ 0 ^ \\infty d x _ 0 ( a + b ) ( a + 2 b x _ 0 ) ( 1 + \\log ^ + ( 1 / x _ 0 ) ) e ^ { - C ' x _ 0 ^ 2 } \\\\ & = C \\sqrt { t } \\int _ 0 ^ \\infty d \\xi \\int _ { - \\infty } ^ { \\infty } d y \\int _ { - \\infty } ^ { \\infty } d z \\int _ 0 ^ t d u \\ , \\chi _ t ( \\xi , y , z , u ) ( C _ 1 a ( a + b ) + 2 C _ 2 b ( a + b ) ) \\end{align*}"}
-{"id": "9724.png", "formula": "\\begin{align*} | | | x | | | : = \\| x \\| _ \\infty + \\sum _ { n \\in \\N } 2 ^ { - a _ n ^ 2 } \\left | \\langle x , u _ n - e _ { a _ n } \\rangle \\right | \\bigl ( x \\in c _ 0 \\bigr ) . \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{align*} \\alpha s \\frac { d g } { d s } = g X , g ( 1 ) = I , \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) \\left ( I + \\begin{pmatrix} \\alpha & 0 \\\\ 0 & \\delta \\end{pmatrix} p \\right ) = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a + \\alpha & b + \\delta \\\\ c & d + \\delta \\end{pmatrix} p . \\end{align*}"}
-{"id": "1322.png", "formula": "\\begin{align*} \\frac { f ^ { 3 } } { p \\dot { p } } = \\lambda _ { 1 } \\frac { p ^ { 3 } } { f \\dot { p } } + \\lambda _ { 2 } , \\left ( \\frac { g } { r } \\right ) \\frac { d } { d g } \\left ( \\frac { g ^ { 2 } } { r } \\right ) = - \\lambda _ { 1 } \\frac { r \\dot { r } } { g } + \\lambda _ { 3 } , \\end{align*}"}
-{"id": "9485.png", "formula": "\\begin{align*} \\phi ( S ) : = \\sum _ { S ' \\in \\binom { V } { r } : S \\subseteq S ' } \\phi ( S ' ) . \\end{align*}"}
-{"id": "1297.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mu } \\left ( Z _ { T } \\in F _ { T } \\right ) = \\int _ { F _ { T } } \\mathsf { d x } u _ { N = 1 } ( \\mathsf { x , } T ) \\psi ( \\mathsf { x } ) + O \\left ( \\exp \\left [ - T \\right ] \\right ) \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{align*} F _ { u + v } - ( - 1 ) ^ v F _ { u - v } = F _ v L _ u \\ , , \\end{align*}"}
-{"id": "9811.png", "formula": "\\begin{align*} { \\left ( { { \\nabla ^ 2 } \\Phi , V } \\right ) _ N } + { \\left ( { \\nabla \\Phi , \\nabla V } \\right ) _ N } = \\int _ { \\partial E , N } { \\nabla \\Phi \\cdot \\hat n V d S } \\quad ( D i s c r e t e \\ ; G r e e n ' s \\ ; F i r s t \\ ; I d e n t i t y ) . \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 5 \\int _ { \\mathcal { M } _ { k } } ( u _ { k } ^ i ) ^ 2 d v _ { g _ { k } } \\geq 1 - \\frac { D } { a _ { k } } , \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} U \\big | _ { t = 0 } = U _ 0 \\equiv u _ 0 - \\psi \\big | _ { t = 0 } , U \\big | _ { x = 0 } = U \\big | _ { x = R } = 0 , U _ x \\big | _ { x = R } = V _ 1 \\equiv \\nu _ 1 - \\psi _ x \\big | _ { x = R } \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} 0 < n _ 2 - n _ 1 \\theta \\leqslant \\frac { \\det ( \\Lambda _ d ) \\varepsilon B ^ { - \\frac { 1 } { r } } } { ( \\lambda _ 2 - \\alpha \\mu _ 2 ) ^ 2 } \\times \\frac { ( \\lambda _ 2 - \\alpha \\mu _ 2 ) K B } { \\det ( \\Lambda _ d ) } = \\frac { K \\varepsilon } { \\lambda _ 2 - \\alpha \\mu _ 2 } B ^ { 1 - \\frac { 1 } { r } } \\leqslant \\frac { 1 } { 6 4 } . \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} & \\| B _ 1 ( u , \\tilde u ) \\| _ { X _ p } \\le C _ p \\| u \\| _ { X _ p } \\| \\tilde u \\| _ { X } , \\\\ & \\| B _ 2 ( u , \\tilde \\theta ) \\| _ { X _ p } \\le C _ p \\| u \\| _ { X _ p } \\| \\tilde \\theta \\| _ Y , \\\\ 1 < p \\le 3 . \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} m _ { R } ( B ( u , \\varepsilon ) ) & \\ge \\int _ { K _ { u } \\ , \\cap \\ , B ( 0 , ( 1 - \\gamma ) \\varepsilon ) } m _ { 0 } ( B ( 0 , R \\gamma \\varepsilon ) ) \\ , d \\nu _ { u , T } ( x ) \\\\ & = m _ { 0 } ( B ( 0 , R \\gamma \\varepsilon ) ) \\ , \\cdot \\ , \\nu _ { u , T } ( B ( u , ( 1 - \\gamma ) \\varepsilon ) ) . \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} \\int \\vert \\eta , z _ 1 , z _ 2 , \\cdots , z _ { r } \\rangle d \\mu _ r ( \\eta , \\bar \\eta , \\{ z _ i \\} , \\{ \\bar z _ i \\} ) \\langle \\eta , z _ 1 , z _ 2 , \\cdots , z _ { r } \\vert = { \\rm I d e n t i t y } . \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} \\phi \\left ( \\gamma \\right ) = \\tau _ { 0 } \\ , \\delta \\left ( \\gamma - \\alpha _ { 0 } \\right ) + \\sum _ { \\nu = 1 } ^ { N } \\tau _ { \\nu } \\ , \\delta \\left ( \\gamma - \\alpha _ { \\nu } \\right ) , \\ ; \\ ; \\begin{array} { c } 0 \\leq \\alpha _ { 0 } < \\ldots < \\alpha _ { N } < 1 , \\\\ \\tau _ { 0 } , \\tau _ { 1 } , \\ldots , \\tau _ { N } > 0 , \\end{array} \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} \\left \\langle v L ^ { 2 n } y \\right \\rangle = 0 v \\in \\Omega ^ n , \\end{align*}"}
-{"id": "3111.png", "formula": "\\begin{align*} \\tilde { \\nu } ( \\mathcal { \\mathcal { D } } ) \\geq \\tilde { \\nu } ( \\theta ^ { - 1 } ( \\mathcal { M } _ 1 \\cap \\mathcal { D } ) ) = \\mu ( \\mathcal { M } _ 1 \\cap \\mathcal { D } ) = \\mu ( \\mathcal { D } ) , \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} \\begin{cases} x ( [ a ] Q _ { n _ k } ) = \\frac { x ( \\Psi _ { a } ^ \\prime ) ^ 2 - \\Psi _ { a + 1 } ^ \\prime \\Psi _ { a - 1 } ^ \\prime } { ( \\Psi _ { a } ^ \\prime ) ^ 2 } ; \\\\ y ( [ a ] Q _ { n _ k } ) = \\frac { \\Psi _ { a + 2 } ^ \\prime ( \\Psi _ { a - 1 } ^ \\prime ) ^ 2 - \\Psi _ { a - 2 } ^ \\prime ( \\Psi _ { a + 1 } ^ \\prime ) ^ 2 } { 4 y ( \\Psi _ { a } ^ \\prime ) ^ 3 } , \\end{cases} \\end{align*}"}
-{"id": "3820.png", "formula": "\\begin{align*} | t _ m ( n ) | & = | t _ m ( 2 n ' ) | = \\left | \\sum _ { j = 0 } ^ { \\lfloor \\frac { m } { 2 } \\rfloor } { m \\choose 2 j } t _ m ( n ' - j ) \\right | \\leq \\sum _ { j = 0 } ^ { \\lfloor \\frac { m } { 2 } \\rfloor } { m \\choose 2 j } | t _ m ( n ' - j ) | \\\\ & < \\sum _ { j = 0 } ^ { \\lfloor \\frac { m } { 2 } \\rfloor } { m \\choose 2 j } m ( n ' ) ^ { \\frac { m } { 2 } } = m ( 2 n ' ) ^ { \\frac { m } { 2 } } = m n ^ { \\frac { m } { 2 } } . \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} N \\cdot \\left ( c - V \\right ) = N \\cdot \\nabla \\varphi = - ( n - 2 ) \\frac { x \\cdot N } { | x | ^ n } \\tilde \\varphi + \\frac { 1 } { | x | ^ n } \\tilde { N } \\cdot \\nabla \\tilde \\varphi , \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\frac 1 2 + \\alpha \\\\ & \\frac 3 2 - 2 \\alpha \\end{matrix} \\bigg | \\ , - \\frac 1 3 \\bigg ] = \\frac { 8 ^ { - 2 \\alpha } } { 9 ^ { - 2 \\alpha } } \\cdot \\frac { \\Gamma ( \\frac 4 3 ) \\Gamma ( \\frac 3 2 - 2 \\alpha ) } { \\Gamma ( \\frac 3 2 ) \\Gamma ( \\frac 4 3 - 2 \\alpha ) } , \\end{align*}"}
-{"id": "8265.png", "formula": "\\begin{align*} s _ p ( k ) = \\frac { k - i \\alpha } { k + i \\alpha } \\end{align*}"}
-{"id": "2797.png", "formula": "\\begin{align*} \\sharp F ( a , b , \\varepsilon , B , r ) & = \\sum _ { e _ 1 f _ 1 | b , e _ 2 f _ 2 | a , e _ 3 f _ 3 | b - a } \\left ( \\prod _ { i = 1 } ^ 3 \\mu ( e _ i ) \\right ) \\sharp F ( e _ 1 , e _ 2 , e _ 3 , f _ 1 , f _ 2 , f _ 3 , a , b , \\varepsilon , B , r ) , \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{align*} \\lambda _ { \\mathrm { B } } ^ \\psi & = \\ ( \\delta - 0 . 5 \\delta ^ 2 \\ ) { \\lambda _ { \\mathrm { B } } } , \\\\ \\lambda _ { \\mathrm { B } } ^ \\varphi & = 0 . 5 \\left ( { 1 - \\delta ^ 2 } \\right ) { \\lambda _ { \\mathrm { B } } } . \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{align*} ( x _ 1 , x _ 2 , x _ 3 ) * ( y _ 1 , y _ 2 , y _ 3 ) = ( x _ 1 + y _ 1 , x _ 2 + y _ 2 , x _ 3 + y _ 3 + x _ 1 y _ 2 - x _ 2 y _ 1 ) \\ , . \\end{align*}"}
-{"id": "2637.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t t } u - \\Delta u + b _ { 1 } \\partial _ { t } u - { \\rm d i v } ( b _ { 2 } \\nabla \\partial _ { t } u ) = 0 & \\mbox { i n } ( 0 , \\infty ) \\times \\Omega , \\\\ u ( t , \\sigma ) = 0 & \\mbox { o n } ( 0 , \\infty ) \\times \\partial \\Omega , \\\\ u ( 0 , x ) = u _ { 0 } ( x ) & \\mbox { i n } \\Omega \\\\ u _ { t } ( 0 , x ) = u _ { 1 } ( x ) & \\mbox { i n } \\Omega , \\end{cases} \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} \\| r ^ q f ^ { q - 2 } _ 2 \\| ^ { 1 / q } _ { L ^ \\infty } = \\left \\| r ^ { q / ( q - 2 ) } f _ 2 \\right \\| ^ { ( q - 2 ) / q } _ { L ^ \\infty } \\le \\| f _ 2 \\| ^ { ( q - 2 ) / q } _ { C ^ 0 _ { w ^ 2 } } , { q \\over q - 2 } \\le { 4 \\over 3 + \\varepsilon } . \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} f ( x ) = ( u _ 1 ( x ) + C ) x + ( u _ 2 ( x ) + D ) x ^ m \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{align*} \\prod _ { i = \\delta ^ { ( k ) } + 1 } ^ { \\Delta ^ { ( k ) } - 1 } w _ { i } ^ { ( k ) } = 1 . \\end{align*}"}
-{"id": "4527.png", "formula": "\\begin{align*} - \\frac { \\partial } { \\partial t } \\nabla \\cdot \\mathbf { A } ( \\mathbf { x } , t ) = | \\Psi ( \\mathbf { x } , t ) | ^ { 2 } , \\ , \\ , ( \\mathbf { x } , t ) \\in \\Omega \\times ( 0 , T ) , \\end{align*}"}
-{"id": "9979.png", "formula": "\\begin{align*} E = X Y ^ T \\ , . \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} ( k - 1 ) s _ { n - 2 } s _ i + \\sum _ { t = 0 } ^ { i - 1 } b _ t \\cdot s _ t \\geq ( k - 1 ) s _ { n - 1 } s _ { i - 1 } + \\sum _ { t = 0 } ^ { i - 1 } b _ t \\cdot s _ { t - 1 } . \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} u \\left ( x \\right ) = F ^ { - 1 } \\left [ A + \\lambda + P _ { t } \\left ( \\xi \\right ) \\right ] ^ { - 1 } \\hat { f } . \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} [ f _ 3 ( k + k ' ) ] ^ { - 1 } \\cdot [ h ( f _ 1 + f _ 1 ' ) ] & = [ ( f _ 3 k ) ^ { - 1 } + ( f _ 3 k ' ) ^ { - 1 } ] \\cdot [ h f _ 1 + h f _ 1 ' ] \\\\ & = [ ( f _ 3 k ) ^ { - 1 } \\cdot h f _ 1 ] + [ ( f _ 3 k ' ) ^ { - 1 } \\cdot h f _ 1 ' ] \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{align*} C ' _ { 1 1 } : = ( \\lambda _ 1 s - t ) - \\Big ( \\frac { ( \\lambda _ { n + 2 } - \\lambda _ 1 ) ( \\lambda _ { n + 3 } - \\lambda _ 1 ) x _ 1 } { ( \\lambda _ { n + 3 } - \\lambda _ { n + 2 } ) ( \\lambda _ 1 - \\lambda _ 2 ) x _ 2 } \\Big ) C ' _ { 2 1 } \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{align*} F ' ( 2 x ) = \\frac { 1 } { 4 } V ( x ) + K ( x , x ) F ( 2 x ) - \\frac { 1 } { 2 } \\int _ x ^ \\infty [ K _ x ( x , s ) - K _ s ( x , s ) ] F ( s + x ) d s . \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} B : = \\prod _ { i \\in Q _ 0 } \\mathrm { E n d } ( E _ i ) \\end{align*}"}
-{"id": "5770.png", "formula": "\\begin{align*} & \\lim _ { \\nu \\to \\infty } S ( A | A ' ) _ { ( \\mathcal { N } ( t ) \\otimes \\mathbb { I } _ { A ' } ) ( \\hat { \\omega } _ { A A ' } ( \\nu ) ) } \\\\ & = \\lim _ { \\nu \\to \\infty } n \\left ( g \\left ( \\nu _ + ( \\nu , t ) - \\frac { 1 } { 2 } \\right ) + g \\left ( \\nu _ - ( \\nu , t ) - \\frac { 1 } { 2 } \\right ) - g \\left ( \\nu - \\frac { 1 } { 2 } \\right ) \\right ) \\\\ & = n \\ln t + n \\ ; , \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} u ^ 2 - v ^ 2 D = A ^ \\prime m . \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} [ L _ { - r + 1 } , \\ , S _ { r - 1 } ] = [ L _ { - r } , \\ , S _ r ] \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{align*} \\alpha _ i ^ + ( \\omega ) \\ne 0 , \\alpha _ j ^ - ( \\omega ) \\ne 0 , \\gamma _ 2 ( \\omega ) = 0 \\textnormal { a n d } \\gamma _ 1 ( \\omega ) = \\lim _ { n \\to \\infty } \\left ( \\sum _ { l \\in \\mathbb { Z } ^ * } ^ { } x _ { l } ( \\omega ) \\textbf { 1 } _ { | x _ l ( \\omega ) | > \\frac { 1 } { n ^ 2 } } \\right ) \\end{align*}"}
-{"id": "8484.png", "formula": "\\begin{gather*} P _ p [ f ] ( x ) = \\sum _ { k = 0 } ^ { m } g _ k ( x ) \\quad \\textrm { f o r } x \\in \\widehat { B } _ p . \\end{gather*}"}
-{"id": "4138.png", "formula": "\\begin{align*} \\limsup _ { t \\rightarrow \\infty } \\frac { 1 } { t } \\log \\mathbb { P } _ { \\lambda } ^ d ( C _ t ^ O \\neq \\varnothing ) & \\leq \\lambda M ^ 2 \\big ( d E ( { \\widetilde { \\rho } } ^ 2 ) + \\frac { 1 } { E ( { \\widetilde { \\rho } } ^ 2 ) } \\big ) - 1 \\\\ & = \\lambda ( d E { \\rho } ^ 2 + \\frac { M ^ 4 } { E { \\rho } ^ 2 } ) - 1 < 0 \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} X = \\{ ( S , [ E ] ) \\in \\mathrm { P } \\times N \\ | \\ h ^ 0 ( E ( - b ) | _ S ) = 1 \\} . \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{gather*} x : \\ , \\begin{bmatrix} q ^ { - 1 } & ( q - q ^ { - 1 } ) a ^ { - 1 } \\\\ 0 & q \\end{bmatrix} , y : \\ , \\begin{bmatrix} q & 0 \\\\ q - q ^ { - 1 } & q ^ { - 1 } \\end{bmatrix} . \\end{gather*}"}
-{"id": "444.png", "formula": "\\begin{align*} & \\dim \\mathbf { U } = \\binom { c - b + 3 } { 3 } + \\binom { c - a + 3 } { 3 } - \\binom { b - a + 3 } { 3 } + \\binom { c + a - e + 3 } { 3 } \\\\ & - \\binom { a + b - e + 3 } { 3 } - \\binom { 2 a - e + 3 } { 3 } + \\binom { b + c - e + 3 } { 3 } - \\binom { 2 b - e + 3 } { 3 } + \\delta ( e , a , b , c ) - 3 . \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{align*} \\hat { U } _ \\eta ^ \\dag \\ , \\hat { R } _ C ^ i \\ , \\hat { U } _ \\eta & = \\sqrt { \\eta } \\ , \\hat { R } _ A ^ i + \\sqrt { 1 - \\eta } \\ , \\hat { R } _ B ^ i \\ ; , \\\\ \\hat { U } _ \\eta ^ \\dag \\ , \\hat { R } _ D ^ i \\ , \\hat { U } _ \\eta & = - \\sqrt { 1 - \\eta } \\ , \\hat { R } _ A ^ i + \\sqrt { \\eta } \\ , \\hat { R } _ B ^ i \\ ; , i = 1 , \\ , \\ldots , \\ , 2 n \\ ; . \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} \\sum _ { \\substack { H \\leq F _ r , \\\\ H \\not \\leq G _ r } } | F _ r / H | ^ a = \\sum _ { \\substack { K \\leq F _ { r - 1 } } } | F _ { r - 1 } / K | | F _ { r - 1 } / K | ^ a = \\sigma _ { a + 1 } ( F _ { r - 1 } ) \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} \\sum _ { k = i + 1 } ^ { j + 1 } \\eta _ { i k } \\theta _ { k j } = 0 \\end{align*}"}
-{"id": "9554.png", "formula": "\\begin{align*} b _ { k , j } : = \\left | \\int _ { - \\infty } ^ { \\infty } \\ ! \\ ! \\ ! f \\left ( t \\right ) g \\left ( t - l _ { k } \\right ) e ^ { - 2 \\pi i \\omega _ { j } t } d t \\right | ^ { 2 } \\end{align*}"}
-{"id": "1688.png", "formula": "\\begin{align*} F _ * ^ e ( j u ^ s v ^ s ( f + u v ) ) = \\bigoplus _ { i \\in \\Delta _ e } f _ { ( i , j ) } F _ * ^ e ( i u ^ s v ^ s ) \\oplus F ^ e ( j u ^ { s + 1 } v ^ { s + 1 } ) \\end{align*}"}
-{"id": "1932.png", "formula": "\\begin{align*} ( 1 + \\beta ) q = 2 + ( 1 + \\beta ) p - \\frac { 2 p } { p ' } . \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} W _ { j , \\varepsilon } ( x ) \\big ( U _ { a _ j } ( \\frac { x - x _ { j , \\varepsilon } } { \\varepsilon } ) + w _ \\varepsilon ( x ) \\big ) = O ( e ^ { - \\eta / \\varepsilon } ) , ~ \\mbox { f o r } ~ x \\in \\R ^ 3 . \\end{align*}"}
-{"id": "9340.png", "formula": "\\begin{align*} \\| \\log ( I + ( \\sigma _ N - \\sigma _ M ) + H _ M ^ N ) - ( \\sigma _ N - \\sigma _ M ) \\| _ { } = O ( \\| \\sigma _ N - \\sigma _ M \\| _ { } ^ 2 ) , \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} f ^ { - n } \\circ \\widehat \\psi _ p ( \\zeta ) = \\widehat \\psi _ n ( \\lambda _ { p , - n } \\cdot \\zeta ) . \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} \\varepsilon _ R = \\frac { 2 R } { N } \\sum _ { | p _ n | > R } \\frac { 1 } { p _ n } . \\end{align*}"}
-{"id": "9122.png", "formula": "\\begin{gather*} \\sum _ { \\mathbf { m } = ( m _ 0 , \\dots , m _ n ) , m _ i \\geq - N } a _ { \\mathbf { m } } \\omega _ { \\mathbf { m } } \\end{gather*}"}
-{"id": "7187.png", "formula": "\\begin{align*} ( w + w ^ { - 1 } ) ^ 2 - 2 = w ^ 2 + w ^ { - 2 } = z + z ^ { - 1 } , \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{align*} a ^ \\flat ( e ^ { i \\theta } ) = \\sum _ { k = - \\infty } ^ { \\infty } a _ k e ^ { i k \\theta } \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{align*} \\widetilde { J } _ 1 ( - k ) ^ { - 1 } J _ 1 ( - k ) = \\widetilde { J } _ 1 ( k ) ^ { - 1 } J _ 1 ( k ) , k \\in \\mathbb { R } \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "7486.png", "formula": "\\begin{gather*} ( \\phi , \\psi ) \\mapsto \\phi \\psi ^ * = \\sum _ { \\alpha : i \\to j } \\frac { 1 } { m _ j } \\phi _ \\alpha \\psi _ \\alpha ^ * \\\\ ( \\phi , \\psi ) \\mapsto \\phi ^ * \\psi = \\sum _ { \\alpha : i \\to j } \\frac { 1 } { m _ i } \\phi _ \\alpha ^ * \\psi _ \\alpha \\end{gather*}"}
-{"id": "3618.png", "formula": "\\begin{align*} ( B ( i \\xi ) - \\lambda ) ^ { - 1 } z ^ \\beta = \\sum _ { \\alpha \\in \\mathbb { N } } \\psi _ \\alpha ^ \\beta ( \\xi , \\lambda ) z ^ \\alpha . \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{align*} & \\big \\| t | f ^ 2 _ { \\alpha a } | + ( t + r ) | f ^ 3 _ { \\alpha a } | \\big \\| ^ 2 _ { L ^ 2 ( r \\leq \\langle t \\rangle / 2 ) } \\\\ & \\lesssim \\sum _ { \\tiny \\begin{matrix} \\beta + \\gamma = \\alpha \\\\ b + c = a \\end{matrix} } \\big \\| \\langle t \\rangle | \\nabla U ^ { ( \\beta , b ) } | | \\nabla U ^ { ( \\gamma , c ) } | \\big \\| ^ 2 _ { L ^ 2 ( r \\leq \\langle t \\rangle / 2 ) } . \\end{align*}"}
-{"id": "5288.png", "formula": "\\begin{align*} \\left | \\int _ B \\omega ( y ) ( y - \\xi ^ * ) \\ , d y \\right | & \\leq \\int _ B | y | ^ k | \\omega ( y ) | { | y - \\xi ^ * | \\over | y | ^ k } \\ , d y = O \\left ( { 1 \\over | x | ^ { ( k - 1 ) ( 1 - \\varepsilon ) } } \\right ) , | x | \\to \\infty , \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} \\textrm { b d y } + \\tfrac { 1 } { 2 } \\sum _ { n = 0 } ^ { \\infty } \\left [ | \\alpha _ n | ^ 2 + | \\alpha _ { n + k } | ^ 2 - \\alpha _ n \\bar { \\alpha } _ { n + k } - \\bar { \\alpha } _ n \\alpha _ { n + k } \\right ] \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} [ v ^ * v ] = ( - 1 ) ^ { \\partial v } [ v v ^ * ] \\quad \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} \\psi ( + \\infty ) = \\lim _ { x \\rightarrow + \\infty } \\psi ( x ) = \\lim _ { y \\rightarrow 0 } \\psi ( 1 / y ) = - \\lim _ { y \\rightarrow 0 } \\psi ( y ) = - \\psi ( 0 ) . \\end{align*}"}
-{"id": "3406.png", "formula": "\\begin{align*} \\begin{cases} d Y _ t = \\left [ ( a + \\lambda \\theta ) Y _ t + \\theta R _ t + \\lambda ( \\varepsilon - \\theta ^ 2 ) ( m ( t ) - X _ t ) \\right ] \\ , d t + Q _ t \\ , d W _ t + R _ { t } \\ , d \\widetilde N _ t \\\\ Y _ T = c ( X _ T - m ( T ) ) . \\end{cases} \\end{align*}"}
-{"id": "2785.png", "formula": "\\begin{align*} \\rho : [ x : y ] \\times [ s : t ] \\longmapsto \\left ( \\frac { y } { x } - 1 , \\frac { t } { s } - 1 \\right ) = ( w - 1 , z - 1 ) . \\end{align*}"}
-{"id": "111.png", "formula": "\\begin{align*} D ( t _ { k ^ x } ^ { - 1 } t _ { 1 ^ x } , \\dots , t _ { k ^ x } ^ { - 1 } t _ { ( k - 1 ) ^ x } , 1 ) = D ( t _ 1 ^ \\sigma , \\dots , t _ { k - 1 } ^ \\sigma , 1 ) . \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } u ^ { 2 } + v w + u v = a ^ { 2 } , \\\\ v ^ { 2 } + v w + u w = b ^ { 2 } , \\\\ w ^ { 2 } + u w + u v = c ^ { 2 } . \\end{array} \\right . \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{align*} b & = \\pm \\frac { 1 } { 4 a \\ell } \\\\ a ^ 2 & \\in \\left \\{ \\frac { 1 } { 4 \\ell } , \\frac { 4 \\ell + 1 } { 4 \\ell } \\right \\} \\end{align*}"}
-{"id": "3511.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { \\Gamma \\left ( k + 1 \\right ) } I _ { v - k } ( q + 2 k ; x ) t ^ { k } = \\left ( 1 - \\frac { 2 t } { x } \\right ) ^ { - v - q } I _ { v } ( q ; \\frac { x ^ { 2 } } { x - 2 t } ) . \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{align*} 4 v '' + \\Bigl ( \\frac { 4 } { t } - e ^ { - t } \\Bigr ) v ' - \\Bigl ( \\frac { 1 } { t ^ 2 } + e ^ { - t } \\Bigl ( \\frac { 1 } { 2 t } - \\frac { 1 } { \\alpha } - ( t ^ { \\frac { 1 } { 2 } } | v | ) ^ \\alpha \\Bigr ) \\Bigr ) v = 0 . \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{align*} | E ( i t ) | & \\leq \\frac { C _ 3 } { \\sqrt { | t | } } \\prod _ { \\kappa _ { 1 , n } \\ge \\alpha _ n } \\frac { \\kappa _ { 1 , n } } { \\alpha _ n } \\prod _ { \\kappa _ { 2 , n } \\ge \\beta _ n } \\frac { \\kappa _ { 2 , n } } { \\beta _ n } \\\\ & \\le \\frac { C _ 3 } { \\sqrt { | t | } } \\prod _ { n = 0 } ^ \\infty \\left [ 1 + \\frac { ( \\kappa _ { 1 , n } - \\alpha _ n ) _ + } { \\alpha _ n } \\right ] \\prod _ { n = 0 } ^ \\infty \\left [ 1 + \\frac { ( \\kappa _ { 2 , n } - \\beta _ n ) _ + } { \\beta _ n } \\right ] . \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} A : = \\{ \\alpha \\in s ( \\lambda ) \\Lambda ^ p : Z ( \\kappa \\alpha , \\lambda \\alpha ) = Z ( \\mu \\beta , \\nu \\beta ) \\beta \\in s ( \\nu ) \\Lambda ^ q \\} \\end{align*}"}
-{"id": "5645.png", "formula": "\\begin{align*} J _ { i j } = \\left ( \\begin{array} { c c } \\frac { 1 } { \\varepsilon } ( - 3 v _ \\star ^ 2 + 2 ( a + 1 ) v _ \\star - a ) & \\ : \\ : \\ : \\ : \\ : - \\frac { 1 } { \\varepsilon } \\\\ \\\\ b & \\ : \\ : \\ : \\ : \\ : - c \\end{array} \\right ) . \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} x _ { n } = \\sum _ { j = 1 } ^ { n } g _ { j } x _ { n - j } = g _ { n } + g _ { n - 1 } x _ { 1 } + \\dots + g _ { 1 } x _ { n - 1 } . \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a & b \\pi ^ n \\\\ c & d \\end{bmatrix} \\in R ^ \\times . \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} p _ { i j k } = \\mathrm { p r o b } ( X = i , Y = j , Z = k ) = a _ { i j } b _ { i k } c _ { j k } . \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} d i m _ q \\mathit { F _ { \\chi } } : = t r _ { \\mathit { F _ { \\chi } } } q ^ { 2 L _ 0 } = \\sum _ { k \\in \\frac { 1 } { 2 } \\mathbb { Z } } d i m \\Big ( s p a n \\{ v \\in \\mathit { F _ { \\chi } } \\ | \\ d e g ( v ) = k \\} \\Big ) q ^ k . \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{align*} \\begin{aligned} ( \\omega + i \\partial \\bar \\partial w ) ^ { n + 1 } & = 0 \\qquad \\bar D \\times X \\\\ \\omega + i \\partial \\bar \\partial w | \\{ s \\} \\times X & > 0 s \\in \\bar D , \\end{aligned} \\end{align*}"}
-{"id": "9522.png", "formula": "\\begin{align*} \\left \\| \\sum _ { j = 1 } ^ m \\lambda _ j u _ j ^ * \\left ( \\sum _ { i = 1 } ^ s c _ i ^ * x c _ i \\right ) u _ j \\right \\| < \\delta , \\end{align*}"}
-{"id": "10016.png", "formula": "\\begin{align*} X _ 1 \\displaystyle \\prod _ { \\ell = 2 } ^ { h ( 1 ) } ( X _ 1 - X _ { \\ell } ) = 0 \\ \\ \\ { \\rm i n } \\ M ^ * ( h ) . \\end{align*}"}
-{"id": "9787.png", "formula": "\\begin{align*} \\| h _ { v _ { \\epsilon } ^ j } \\| _ { L ^ 2 } \\leq C ( M ) \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} T r ( \\alpha \\lambda ^ 2 x ^ 2 + \\lambda ^ 2 y ^ 2 + \\lambda ^ 4 x ^ 2 y ^ 2 ) = 1 . \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} \\partial _ t v - \\mu \\Delta v - \\nabla \\cdot G = - \\nabla p - \\nabla \\cdot ( v \\otimes v ) + \\nabla \\cdot ( G G ^ \\top ) . \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} H = H \\bigl ( \\bigl ( \\Lambda _ n , m \\bigr ) ; \\vec \\xi \\bigr ) : = \\left ( \\frac { \\partial ^ 2 } { \\partial x _ 1 ^ 2 } + \\frac { \\partial ^ 2 } { \\partial x _ 2 ^ 2 } \\right ) - \\sum \\limits _ { \\alpha \\in \\Pi } \\dfrac { m ( m + 1 ) } { l _ \\alpha ^ 2 ( \\vec { x } - \\vec \\xi ) } \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} \\tilde { \\alpha } _ X ( T ) = \\alpha _ X \\circ T \\circ \\alpha _ X \\quad \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ d \\frac { 1 } { x - a _ i } = \\sum _ { i = 1 } ^ d \\frac { 1 } { x - a _ i } \\prod ^ d _ { \\substack { j = 1 \\\\ j \\neq i } } \\frac { 1 } { a _ i - a _ j } . \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{align*} 2 G ^ { i } ( x , y ) = \\dfrac { 1 } { 2 } g ^ { i h } ( L _ { \\cdot h , j } y ^ { j } - L _ { , h } ) \\end{align*}"}
-{"id": "3524.png", "formula": "\\begin{align*} \\int _ { a } ^ { b } f ( x ) ^ { d x } = e ^ { \\int _ { a } ^ { b } ( \\ln \\circ f ) ( x ) \\ , d x } . \\end{align*}"}
-{"id": "5640.png", "formula": "\\begin{align*} v _ 0 = 0 v _ { 1 , 2 } = \\frac { a + 1 } { 2 } \\pm \\sqrt { \\frac { ( a - 1 ) ^ 2 } { 4 } - \\frac { b } { c } } , \\end{align*}"}
-{"id": "4967.png", "formula": "\\begin{align*} \\frac { d b } { d t } = \\lambda ' ( c _ * ) ( c _ + - c _ * ) b + \\gamma | b | ^ 2 b , t > 0 , \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{align*} J Z B ^ { 2 p ^ { d - 2 } + p - 1 + \\lambda } & \\subseteq 1 _ B J Z F G ^ { 2 p ^ { d - 2 } + p - 1 + \\lambda } \\subseteq I _ { \\le D } ( G ) \\cdot J Z F G ^ { 2 p ^ { d - 2 } + p - 1 + \\lambda } \\\\ & \\subseteq I _ { < D } ( G ) \\cdot J Z F G ^ { p ^ { d - 2 } + p - 1 + \\lambda } = \\sum _ { Q < D } { I _ { \\le Q } ( G ) \\cdot J Z F G ^ { p ^ { d - 2 } + p - 1 + \\lambda } } \\\\ & \\subseteq \\sum _ { Q < D } { I _ { < Q } ( G ) \\cdot J Z F G ^ { \\lambda } } = 0 . \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} c ( r e ^ { \\gamma } ) = r e ^ { c ( \\gamma ) } . \\end{align*}"}
-{"id": "8254.png", "formula": "\\begin{align*} P ( k _ 1 , \\dots , k _ n ) = ( k _ { P ( 1 ) } , \\dots , k _ { P ( n ) } ) . \\end{align*}"}
-{"id": "9338.png", "formula": "\\begin{align*} \\| \\log ( \\gamma _ { N - 1 } ^ { - 1 } \\cdot \\gamma _ { N } ) - ( \\sigma _ { N } - \\sigma _ { N - 1 } ) \\| _ { } = | g ( F _ { N } ) - 1 | \\| ( 0 , F _ { N } z ^ { N } ) \\| _ { } = | g ( F _ { N } ) - 1 | | F _ { N } | . \\end{align*}"}
-{"id": "6676.png", "formula": "\\begin{align*} \\Phi ^ * ( x ) = \\begin{pmatrix} a ^ * _ 1 ( x ) \\\\ \\vdots \\\\ a ^ * _ n ( x ) \\end{pmatrix} , x \\in X . \\end{align*}"}
-{"id": "2019.png", "formula": "\\begin{gather*} \\tilde { D } _ j = g ^ { - 1 } D _ j g , \\exp \\big ( 2 \\pi i k \\tilde { \\Theta } \\big ) = g ^ { - 1 } \\exp ( 2 \\pi i k \\Theta ) g \\end{gather*}"}
-{"id": "306.png", "formula": "\\begin{align*} \\pi ( X K _ \\lambda ) _ { j } ^ { i } c _ { k } ^ { j } \\pi ( K _ { \\lambda ^ \\prime } Y ) _ { \\ell } ^ { k } = \\pi ( X K _ { \\lambda } K _ { \\lambda ^ { \\prime } } ) _ { j } ^ { i } c _ { k } ^ { j } \\pi ( Y ) _ { \\ell } ^ { k } = \\pi ( X ) _ { j } ^ { i } c _ { k } ^ { j } \\pi ( K _ { \\lambda } K _ { \\lambda ^ { \\prime } } Y ) _ { \\ell } ^ { k } . \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} a A = \\hat a ^ { 1 / 2 } e _ 1 O , \\end{align*}"}
-{"id": "2803.png", "formula": "\\begin{align*} \\sum _ { n \\leqslant x } \\phi ( n ) = \\frac { 6 } { \\pi ^ 2 } x + O ( ( \\log x ) ^ \\frac { 2 } { 3 } ( \\log \\log x ) ^ \\frac { 4 } { 3 } ) , \\end{align*}"}
-{"id": "4728.png", "formula": "\\begin{align*} F _ 2 ( z ) \\frac { d F _ 1 ( z ) } { d z } - F _ 1 ( z ) \\frac { d F _ 2 ( z ) } { d z } = F _ 0 ( z ) , z \\in \\mathbb R , \\end{align*}"}
-{"id": "4053.png", "formula": "\\begin{align*} \\tau _ 1 & = ( 1 , 1 ; 0 , 0 ; 0 , 0 ) , & \\tau _ 2 & = ( 0 , 1 ; 0 , 1 ; 0 , 0 ) , & \\tau _ 3 & = ( 1 , 2 ; 0 , 1 ; 0 , 1 ) , \\\\ \\tau _ 4 & = ( 1 , 1 ; 1 , 1 ; 0 , 1 ) , & \\tau _ 5 & = ( 1 , 2 ; 1 , 2 ; 1 , 1 ) , & \\tau _ 6 & = ( 2 , 3 ; 1 , 2 ; 1 , 2 ) . \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{align*} \\mathsf { X } _ { \\infty } ( t ) \\overset { d } { = } \\underset { N \\to \\infty } { w \\textendash \\lim } \\ \\left ( \\mathfrak { r } _ N \\right ) _ * \\left ( \\mu _ N P _ N ( t ) \\right ) , \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} c _ + \\leq c _ * : \\lim _ { t \\to \\infty } \\| u ( t ) - u _ { c _ { \\infty } } \\| _ { H ^ 1 _ { \\mu } ( \\mathbb { R } \\times \\mathbb { T } ) } = 0 , \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} \\int _ \\Omega \\left [ | \\nabla t | ^ 2 + \\frac { t ^ 2 } { \\varepsilon ^ 2 } \\right ] = o ( 1 ) . \\end{align*}"}
-{"id": "1643.png", "formula": "\\begin{align*} \\lim _ { X \\to \\infty } \\Xi _ j ( X ) = 0 \\ , . \\end{align*}"}
-{"id": "5947.png", "formula": "\\begin{align*} ( x ' ) ^ { k - 1 } x ^ k = x \\ , . \\end{align*}"}
-{"id": "4445.png", "formula": "\\begin{align*} \\varphi _ { k } ( h ) = { j \\choose k } h ^ { 2 ( k - j ) } ( 1 + h ^ 2 ) ^ { - \\beta } ( k = 0 , \\ldots , j ) \\qquad \\qquad \\psi ( h ) = \\sqrt { 1 + h ^ 2 } . \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} c _ k \\equiv \\frac { 1 } { \\binom { 2 k } { k } } = \\frac { ( k ! ) ^ 2 } { ( 2 k ) ! } \\end{align*}"}
-{"id": "9316.png", "formula": "\\begin{align*} \\beta = 0 & \\gamma \\in \\{ 2 m \\mid m = 2 , 3 , 4 , 7 , 1 2 \\} , \\\\ \\beta = 1 & \\gamma \\in \\{ 2 m \\mid m = 8 , 9 , 1 0 , 1 1 , 1 2 , 1 5 \\} , \\\\ \\beta = 2 & \\gamma \\in \\{ 2 m \\mid m = 4 , 6 , 7 , 8 , 9 , 1 0 , 1 1 , 1 2 , 1 3 , 1 4 , 1 5 , 5 1 , 5 2 , 5 3 , 5 4 \\} . \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{align*} & \\limsup _ { c \\to 0 } c ^ 2 \\kappa ( \\| M \\| _ \\infty , C _ R , c ) = 2 C _ R \\Vert M \\Vert _ \\infty < \\infty , \\\\ & \\limsup _ { C _ R \\to \\infty } \\tfrac { \\kappa ( \\| M \\| _ \\infty , C _ R , c ) } { C _ R } = 2 \\big ( 1 + \\tfrac { \\Vert M \\Vert _ \\infty } { c } \\big ) ^ 2 + 1 < \\infty , \\\\ & \\limsup _ { \\| M \\| _ \\infty \\to \\infty } \\tfrac { \\kappa ( \\| M \\| _ \\infty , C _ R , c ) } { \\| M \\| _ \\infty ^ 2 } = \\tfrac { 2 C _ R } { c ^ 2 } < \\infty . \\end{align*}"}
-{"id": "9741.png", "formula": "\\begin{align*} \\partial _ t b + ( v \\cdot \\nabla ) b = ( b \\cdot \\nabla ) v , \\ ; \\ ; \\ ; v = ( b \\cdot \\nabla ) b , \\end{align*}"}
-{"id": "1182.png", "formula": "\\begin{align*} d ^ { \\rm s u p } ( 1 0 ^ { - 3 } ) \\ , \\le \\ , \\left \\{ \\begin{array} { l l } 2 & \\mbox { f o r \\ } a = 4 , \\\\ 3 & \\mbox { f o r \\ } a = 3 , \\\\ 4 & \\mbox { f o r \\ } a = 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "2224.png", "formula": "\\begin{align*} A _ 1 = M { \\rm d i a g } \\{ I _ { n - n _ D } , 0 _ { n _ D } \\} M ^ \\dag , \\ ; B _ 1 = M { \\rm d i a g } \\{ 0 _ { n - n _ D } , - i I _ { n _ D } \\} M ^ \\dag ; \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} I _ u ( x _ 0 , r _ 1 ) - I _ u ( x _ 0 , r _ 0 ) = \\int _ { r _ 0 } ^ { r _ 1 } \\frac { 2 \\ , t } { H _ u ^ 2 ( x _ 0 , t ) } \\Big ( H _ u ( x _ 0 , t ) \\ ; E _ u ( x _ 0 , t ) - D _ u ^ 2 ( x _ 0 , t ) \\Big ) \\ , \\d t \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} \\Big \\{ \\psi _ 1 ^ { d _ 1 } \\cdots \\psi _ n ^ { d _ n } \\mid \\sum _ { i = 1 } ^ n d _ i = g - 1 \\Big \\} . \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{align*} { x } _ t = \\tilde { x } _ { \\mathfrak { t } ^ { - 1 } ( t ) } . \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\lambda _ n n ^ { - s } = \\prod _ p \\frac 1 { 1 - \\lambda _ p p ^ { - s } + \\xi ( p ) p ^ { - 2 s } } , \\qquad \\overline { \\lambda _ p } = \\overline { \\xi ( p ) } \\lambda _ p p \\nmid N . \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{align*} E _ t = 1 + ( E _ - \\cdot ( M - V ) ) _ t + \\frac { 1 } { 2 } ( E _ - \\cdot \\langle M ^ c \\rangle ) _ t + \\sum _ { s \\le t } E _ { s ^ - } ( e ^ { \\Delta M _ s } - 1 - \\Delta M _ s ) . \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} r _ k = \\frac { c P ( x _ k ) } { \\prod \\limits _ { \\substack { j = 1 \\\\ j \\neq k } } ^ n ( x _ k - x _ j ) ^ 2 } , k = 1 , 2 , \\dots , n . \\end{align*}"}
-{"id": "9965.png", "formula": "\\begin{align*} \\begin{array} { r c l } m \\ddot { q } _ 1 + c \\dot { q } _ 1 + k q _ 1 & = & 0 \\\\ q _ 1 - q _ 2 & = & 0 \\end{array} \\end{align*}"}
-{"id": "4442.png", "formula": "\\begin{align*} L ( s ) = L \\left ( \\frac { s } { s ^ \\ast } s ^ \\ast \\right ) > \\frac { s } { s ^ \\ast } L ( s ^ \\ast ) , \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { C C _ { 3 , n } } { \\Pi ( n ) } x ^ n & = \\dfrac { - \\dfrac { x ^ { 2 7 } } { L _ 3 } } { \\sum _ { i = 3 } ^ \\infty ( - 1 ) ^ i \\dfrac { x ^ { 3 ^ i } } { L _ i } } \\\\ & = 1 + \\frac { L _ 3 } { L _ 4 } x ^ { 5 4 } + \\frac { L _ 3 ^ 2 } { L _ 4 ^ 2 } x ^ { 1 0 8 } + \\frac { L _ 3 ^ 3 } { L _ 4 ^ 3 } x ^ { 1 6 2 } \\\\ & \\quad + \\left ( \\frac { L _ 3 ^ 5 } { L _ 4 ^ 5 } - \\frac { 2 L _ 3 ^ 2 } { L _ 4 L _ 5 } \\right ) x ^ { 2 7 0 } + \\left ( \\frac { L _ 3 ^ 6 } { L _ 4 ^ 6 } - \\frac { 3 L _ 3 ^ 3 } { L _ 4 ^ 2 L _ 5 } \\right ) x ^ { 3 2 4 } + \\cdots \\ , . \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{align*} \\alpha _ \\ell : = \\frac { 1 } { 2 ^ { 4 ( \\ell - 1 ) } } \\sum _ { k = 1 } ^ { \\ell } \\frac { [ ( 2 \\ell - 2 k ) ! ] ^ 3 } { [ ( \\ell - k ) ! ] ^ 4 } \\frac { [ ( 2 k - 2 ) ! ] ^ 3 } { [ ( k - 1 ) ! ] ^ 4 } \\in \\mathbb Z , \\forall \\ell \\in \\mathbb Z _ { > 0 } . \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{align*} \\nu _ 2 ( t ) = \\frac { - 1 + e ^ { - 2 \\sqrt { 3 } t } + 2 \\sqrt { 3 } t e ^ { - \\sqrt { 3 } t } } { \\sqrt { 3 } } \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{align*} f _ q = \\sum _ { j = 1 } ^ N f _ q ^ { \\gamma _ j } . \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{gather*} f _ m ( 1 ) = 1 , f _ m ( 2 ) = m , \\\\ f _ m ( n + 2 ) = m f _ m ( n + 1 ) + a f _ m ( n ) , ( n \\geq 1 ) . \\end{gather*}"}
-{"id": "9166.png", "formula": "\\begin{align*} \\max _ { c \\in \\R } \\big \\{ g ( c ) \\wedge h ( c ) \\big \\} = 2 + { \\lambda ^ 2 + \\mu ^ 2 \\over 2 } - \\frac { ( p - 2 ) ( \\lambda + \\mu ) ^ { 2 } } { 4 } - \\frac { ( \\lambda - \\mu ) ^ { 2 } } { 4 ( p - 2 ) } , \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} J _ R ( z ) & = M ( z ) \\begin{pmatrix} 1 & 0 \\\\ - i ( z ^ 2 - 1 ) ^ { 1 / 2 } e ^ { - N \\xi ( z ) } & 1 \\end{pmatrix} M ^ { - 1 } ( z ) \\\\ & = I + O \\left ( e ^ { - c N } \\right ) , N \\to \\infty , \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{align*} E = - \\phi U + \\mathbb B [ U \\cdot \\nabla \\phi ] . \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} \\Delta y _ s ( m ) = y _ { s - 1 } ( m ) , \\enskip s = 2 , 3 , . . . , n , \\enskip \\Delta y _ 1 ( m ) = - y _ n ( m ) \\end{align*}"}
-{"id": "9759.png", "formula": "\\begin{align*} \\| V \\| ( M ) = \\lim _ { \\epsilon _ j \\to 0 } \\frac { E _ { \\epsilon _ j } ( u _ { \\epsilon _ j } ) - \\frac { 1 } { 2 } \\| h _ { \\epsilon _ j } \\| _ { L ^ 2 } ^ 2 } { | \\log \\epsilon _ j | } . \\end{align*}"}
-{"id": "6591.png", "formula": "\\begin{align*} ( - E _ 0 ( \\beta ) , E _ 0 ( \\beta ) ) \\ , , E _ 0 ( \\beta ) \\ , : = \\ , \\frac { | \\beta | } { | \\beta | \\| S _ D ^ { - 1 } \\| + 1 } \\ , . \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{align*} G ( k , x ) = e ^ { - i k x } I _ n + e ^ { i k x } S ( k ) + o ( 1 ) , x \\to \\infty . \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} \\log ( x + \\varepsilon ) = \\log \\left ( x \\left ( 1 + \\frac { \\varepsilon } { x } \\right ) \\right ) = \\log ( x ) + \\log ( 1 + \\delta ) = \\log ( x ) + \\sum _ { i = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { i + 1 } } { i } \\delta ^ { i } . \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} - q U _ { n + l } U _ { n + l - 2 } + q U _ { n + l - 1 } ^ { 2 } = q \\det ( W ( p , q ) ^ { n + l - 2 } ) . \\end{align*}"}
-{"id": "2403.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ n } \\phi ( | \\xi | ) | \\xi | e ^ { \\mathrm i x \\cdot \\xi } d \\xi = \\int _ { \\mathbb R ^ n } \\phi ( | \\xi | ) | \\xi | e ^ { \\mathrm i | x | \\xi _ 1 } d \\xi . \\end{align*}"}
-{"id": "6540.png", "formula": "\\begin{align*} \\dot x = \\Gamma v ( x ) . \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} L _ { 1 } u = 0 , L _ { 2 } u = 0 , \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} \\sum _ { \\substack { j = 1 \\\\ j \\neq k } } ^ { n - 1 } \\frac { 2 } { t _ k - t _ j } - \\sum _ { j = 1 } ^ { n - 1 } \\left ( \\frac { 1 } { t _ k - \\zeta _ j } + \\frac { 1 } { t _ k - \\overline { \\zeta } _ j } \\right ) = 0 , k = 1 , 2 , \\dots , n - 1 . \\end{align*}"}
-{"id": "1096.png", "formula": "\\begin{align*} \\left \\vert \\sum \\limits _ { i = 1 } ^ m a _ i \\lambda _ { i j } \\right \\vert & = \\left \\vert \\sum \\limits _ { i = 1 } ^ m a _ i ( \\lambda _ { i j } - \\lambda ^ \\prime _ { i j } ) \\right \\vert \\\\ & \\ll \\eta . \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} \\alpha _ 1 ( a , b ) = b a ^ { - 1 } , \\alpha _ 2 ( a , b ) = a ^ 2 b ^ { - 1 } ( ( a , b ) \\in M _ 0 ) , \\end{align*}"}
-{"id": "7810.png", "formula": "\\begin{align*} \\iint u _ 0 ^ 2 \\ , d x d y \\leq \\frac 1 T \\iiint _ { Q _ T } ( P u _ 0 ) ^ 2 \\ , d x d y d t + \\iint _ { B _ T } \\bigl ( \\partial _ x ( P u _ 0 ) \\big | _ { x = 0 } \\bigr ) ^ 2 \\ , d y d t . \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } \\lambda _ { j } = \\infty , \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{align*} & w _ k \\to w \\quad \\mbox { w e a k l y - s t a r i n $ L ^ \\infty ( 0 , T ; L ^ 2 _ \\sigma ( \\Omega ) ) $ } , \\\\ & w _ k \\to w \\quad \\mbox { w e a k l y i n $ L ^ 2 ( 0 , T ; H ^ 1 _ { 0 , \\sigma } ( \\Omega ) ) $ } , \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{align*} \\mathbf { f } _ { p l a n a r } = m \\mathbf { \\ddot { p } } + \\mathbf { f } _ { w } \\end{align*}"}
-{"id": "4849.png", "formula": "\\begin{align*} \\left \\langle D _ { H _ i } , I + \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} p \\right \\rangle \\left \\langle D _ { H _ i } , I + \\begin{pmatrix} 0 & 0 \\\\ 1 & 0 \\end{pmatrix} p \\right \\rangle \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} x _ i ^ { \\boldsymbol { \\omega } , - } & \\geq \\tilde { z } _ { - 2 i _ { - 1 } N + i } - 1 \\geq \\tilde { z } _ { 2 N + i } - 1 \\geq \\tilde { z } _ { - i } - 1 \\geq \\tilde { z } _ { | i | - 1 } - 1 , \\\\ x _ i ^ { \\boldsymbol { \\omega } , + } & \\leq \\tilde { z } _ { i + 1 } = \\tilde { z } _ { - ( | i | - 1 ) } , \\end{align*}"}
-{"id": "4171.png", "formula": "\\begin{align*} H ( \\omega ) = \\begin{cases} 1 & \\mbox { i f } | \\omega | \\leq 2 \\pi B \\\\ 0 & \\mbox { e l s e } \\end{cases} \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} B ( i \\xi ) = H _ { \\rm { O H } } - \\frac { i \\xi } { \\sqrt { 2 } } \\left ( a + a ^ \\dagger \\right ) . \\end{align*}"}
-{"id": "4603.png", "formula": "\\begin{align*} \\omega ^ { 0 , h } ( x ) : = \\sum ^ g _ { i = 1 } ( x \\cdot [ \\alpha _ i ] ) ( x \\cdot [ \\beta _ i ] ) + h ( x ) \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 3 } { x ^ 2 } + \\frac { A _ 2 } { x } + A _ 1 + A _ 0 x \\right ) Y , \\frac { \\partial Y } { \\partial t _ 1 } = - \\frac { A _ 3 } { t _ 1 x } Y , \\frac { \\partial Y } { \\partial t _ 2 } = ( E _ 2 x + B _ 1 ) Y . \\end{gather*}"}
-{"id": "1543.png", "formula": "\\begin{align*} \\mathcal { S } _ + ^ { n } : = \\{ M \\in \\mathbb R ^ { n \\times n } : M = M ^ \\top M \\succeq 0 \\} . \\end{align*}"}
-{"id": "9590.png", "formula": "\\begin{align*} \\breve \\Delta ^ \\mathrm { e h w } _ n & = \\frac { 1 } { N } \\sum _ { i = 1 } ^ n R _ { n , i } X _ { n , i } \\widehat \\varepsilon _ { n , i } ^ 2 X _ { n , i } ^ { \\prime } , \\widetilde \\Delta ^ \\mathrm { e h w } _ n = \\frac { 1 } { N } \\sum _ { i = 1 } ^ n R _ { n , i } X _ { n , i } \\varepsilon _ { n , i } ^ 2 X _ { n , i } ^ { \\prime } , \\\\ \\shortintertext { a n d } \\Delta ^ \\mathrm { e h w } _ n & = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n E [ X _ { n , i } \\varepsilon _ { n , i } ^ 2 X _ { n , i } ^ { \\prime } ] . \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} C ( x , t ) & : = \\{ ( y , s ) \\in \\Z ^ 2 : h ^ l ( y , s ) = ( x , t ) l \\geq 0 \\} . \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} Q _ m ( a ) - Q _ m ( a - 1 ) = \\frac { 1 } { 2 } \\big ( m - \\frac { 1 } { 2 } - 2 a \\big ) Q _ { m - 1 } ( a - 1 ) , \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} m _ { R } ( A ) = \\int _ { K _ { u } } m _ { 0 } \\bigl ( R ( A - x ) \\bigr ) \\ , d \\nu _ { u , T } ( x ) \\textrm { f o r e v e r y B o r e l s u b s e t } \\ A \\ \\textrm { o f } \\ X , \\end{align*}"}
-{"id": "1368.png", "formula": "\\begin{align*} f \\left ( x , y , k \\right ) = A ^ { 2 } ( x , y ) + A _ { 0 } ^ { 2 } ( x , y ) - 2 A ( x , y ) A _ { 0 } ( x , y ) \\cos \\left ( k \\alpha \\right ) + \\hat { f } \\left ( x , y , k \\right ) , \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{align*} U _ \\lambda : = \\{ u \\in U \\mid H ( \\lambda , u ) = \\max _ { u \\in U } H ( \\lambda , u ) = H ^ + ( \\lambda ) \\} \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} \\div & \\big ( x _ 2 ^ a \\nabla u ( x ) \\big ) = \\\\ & \\sum _ { k = 0 } ^ { m - 1 } \\Big ( ( m - 2 k ) ( m - 2 k - 1 ) \\ , \\alpha _ k + 2 ( k + 1 ) ( 2 k + 1 + a ) \\ , \\alpha _ { k + 1 } \\Big ) \\ , x _ 1 ^ { m - 2 k - 2 } x _ 2 ^ { 2 k + a } = 0 \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} \\partial _ { x _ i } f ( x , z ) & = - \\partial _ { z _ i } f ( x , z ) = \\frac { 1 } { | x - z | ^ n } e _ i - n \\frac { x _ i - z _ i } { | x - z | ^ { n + 2 } } ( x - z ) . \\end{align*}"}
-{"id": "753.png", "formula": "\\begin{align*} | \\overline { \\xi } _ i ( x _ i + y ) | = O ( 1 / r ^ 2 ) . \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} & u _ s \\cdot \\nabla u _ s = \\Delta u _ s - \\nabla p _ { u _ s } - u _ \\infty \\cdot \\nabla u _ s , \\\\ & \\mbox { d i v $ u _ s $ } = 0 , \\\\ & u _ s | _ { \\partial \\Omega } = - u _ \\infty , \\\\ & u _ s \\to 0 \\quad \\mbox { a s $ | x | \\to \\infty $ } . \\\\ \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} f ( r ) \\ ; & = \\ ; o ( r ^ { 1 / 2 } ) \\ , , \\\\ ( S _ D ^ { - 1 } \\Phi ) ( r ) \\ ; & = \\ ; \\begin{pmatrix} p ^ + \\ ! \\\\ p ^ - \\ ! \\end{pmatrix} r ^ B + o ( r ^ { 1 / 2 } ) \\ , , \\\\ r ^ B \\Phi ( r ) \\ ; & = \\ ; \\begin{pmatrix} 1 \\\\ - \\frac { 1 + \\nu - B } { 1 + \\nu + B } \\end{pmatrix} \\gamma + \\begin{pmatrix} q ^ + \\ ! \\\\ q ^ - \\ ! \\end{pmatrix} r ^ { 2 B } + o ( r ^ { 1 / 2 + B } ) \\end{align*}"}
-{"id": "3529.png", "formula": "\\begin{align*} { } _ { \\mu , \\sigma \\ast } D _ { v , q ; z } ^ { \\alpha , \\eta , p } \\left ( { f ( z ) } \\right ) ) = \\exp ( \\frac { \\sqrt { \\frac { 2 } { \\pi } } } { \\Gamma ( \\alpha ) } \\int _ { 0 } ^ { z } \\left ( \\ln f ( t ) \\right ) ( z - t ) ^ { \\alpha - 1 } e x p \\big ( \\frac { - p z ^ { 2 } } { t ( z - t ) } \\big ) d t ) \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} { \\bf A } ^ { \\star } = \\left ( \\begin{array} { c c } { \\bf A } _ { 1 1 } ^ { \\star } & { \\bf A } _ { 1 2 } ^ { \\star } \\\\ { \\bf A } _ { 2 1 } ^ { \\star } & { \\bf A } _ { 2 2 } ^ { \\star } \\end{array} \\right ) , \\end{align*}"}
-{"id": "8074.png", "formula": "\\begin{align*} L = a _ 0 ( t ) \\partial _ t ^ k + \\cdots + a _ k ( t ) y , a _ 0 , \\ldots , a _ k \\in \\C [ t ] , a _ 0 \\not \\equiv 0 \\end{align*}"}
-{"id": "3951.png", "formula": "\\begin{align*} { } _ 4 F _ 3 \\bigg [ \\begin{matrix} - n & \\alpha & \\beta & \\gamma \\\\ & \\delta & \\epsilon & \\rho \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { ( \\delta - \\alpha ) _ n ( \\epsilon - \\alpha ) _ n } { ( \\delta ) _ n ( \\epsilon ) _ n } \\cdot { } _ 4 F _ 3 \\bigg [ \\begin{matrix} - n & \\alpha & \\rho - \\beta & \\rho - \\gamma \\\\ & \\rho & 1 + \\alpha - n - \\delta & 1 + \\alpha - n - \\epsilon \\end{matrix} \\bigg | \\ , 1 \\bigg ] , \\end{align*}"}
-{"id": "9355.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { J + 1 } \\mathcal { V } _ r ( \\sigma _ j ^ s ) \\lesssim _ r \\mathcal { V } _ r ^ r ( \\gamma ) . \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{gather*} F _ { 2 n } = \\sum _ { k = 1 } ^ n { n + k - 1 \\choose n - k } , F _ { 2 n - 1 } = \\sum _ { k = 1 } ^ n { n + k - 2 \\choose n - k } , \\\\ J _ { 2 n } = \\sum _ { k = 1 } ^ n 2 ^ { n - k } { n + k - 1 \\choose n - k } , J _ { 2 n - 1 } = \\sum _ { k = 1 } ^ n 2 ^ { n - k } { n + k - 2 \\choose n - k } . \\end{gather*}"}
-{"id": "1607.png", "formula": "\\begin{align*} u _ \\varepsilon ^ { [ N ] } ( t ) = \\sum _ { n = 0 } ^ N \\varepsilon ^ { \\beta _ n } \\ , u ^ { \\beta _ n } ( t ) + \\sum _ { n = 0 } ^ N \\varepsilon ^ { \\beta _ n } \\ , U ^ { \\beta _ n } ( \\tfrac { t } { \\varepsilon } ) \\ , . \\end{align*}"}
-{"id": "2516.png", "formula": "\\begin{align*} A _ { 1 } f f _ { c _ { 1 } } - f ^ { \\prime } f _ { c _ { 1 } } - f f _ { c _ { 1 } } ^ { \\prime } = B _ { 1 } , \\end{align*}"}
-{"id": "1739.png", "formula": "\\begin{align*} \\tau _ { f , 1 } = \\frac { c _ 3 m \\Delta \\phi _ 1 v _ c } { f _ { p l a n a r } - K _ d ( v _ c + v _ { a i r } ) ^ 2 } \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 3 } K _ i \\leq C , \\quad k = 4 . \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} y : = x \\exp \\left ( \\frac { 1 } { 2 } \\int f \\right ) \\end{align*}"}
-{"id": "6897.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ { \\infty } Q _ { n , j , k } ( p , z ) t ^ { p } & = \\frac { 2 ^ { 2 j + 1 } t ^ { j + 1 } ( 1 + t + z ( 1 - t ) ) ^ { k - j } ( 1 + t - z ( 1 - t ) ) ^ { n - j - k - 1 } } { ( 1 - t ) ^ { n + 1 } } \\end{align*}"}
-{"id": "4051.png", "formula": "\\begin{align*} F ' _ i ( x , y , z ) = \\det ( A _ j A _ k ) F _ i ( ( x , y , z ) A _ i ) \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} R e s ^ { \\mathcal { G } } _ { \\mathcal { G } ^ \\prime } ( f ) - g = I n f ^ { \\mathcal { G } ^ \\prime } _ { G ^ \\prime } ( \\widetilde h ) , \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } \\binom { 2 k } { k } H _ k ^ { ( 2 ) } x ^ { k } \\equiv \\frac { 2 \\pounds _ 2 ( \\alpha ) - 2 \\pounds _ 2 ( \\beta ) } { \\beta - \\alpha } \\pmod { p } , \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\theta ( t ) = e ^ { t \\Delta } \\theta _ 0 - \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\nabla \\cdot ( \\theta u ) ( s ) \\dd s \\\\ & u ( t ) = e ^ { t \\Delta } u _ 0 - \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\P \\nabla \\cdot ( u \\otimes u ) ( s ) \\dd s + \\int _ 0 ^ t e ^ { ( t - s ) \\Delta } \\P \\theta ( s ) e _ 3 \\dd s . \\\\ & \\nabla \\cdot u _ 0 = 0 \\end{aligned} \\right . \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{align*} \\begin{cases} \\mu _ { 0 , 0 } = \\beta _ 0 , \\ \\mu _ { 1 , 0 } = \\alpha _ 0 \\\\ \\mu _ { j - 1 , j } = \\gamma _ j , \\ \\mu _ { j , j } = \\beta _ j , \\ \\mu _ { j + 1 , j } = \\alpha _ j \\\\ \\mu _ { i , j } = 0 , \\ | i - j | > 1 \\end{cases} , \\ j = 1 , 2 , \\ldots \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} \\| f _ \\varepsilon \\| _ { \\mathcal H ^ p _ a ( \\mathbb C _ + ) } = \\left ( \\int _ { - \\infty } ^ { \\infty } \\frac { 1 } { \\sqrt { x ^ 2 + 1 } ^ { 1 + p \\varepsilon } } d x \\right ) ^ { 1 / p } < \\infty . \\end{align*}"}
-{"id": "9378.png", "formula": "\\begin{align*} B _ 2 ( S ^ 2 ( \\mathcal { H } ) \\times S ^ 2 ( \\mathcal { H } ) , S ^ 2 ( \\mathcal { H } ) ) & = B ( S ^ 2 ( \\mathcal { H } ) \\overset { \\wedge } { \\otimes } S ^ 2 ( \\mathcal { H } ) , S ^ 2 ( \\mathcal { H } ) ) \\\\ & = \\bigl ( S ^ 2 ( \\mathcal { H } ) \\overset { \\wedge } { \\otimes } S ^ 2 ( \\mathcal { H } ) \\overset { \\wedge } { \\otimes } S ^ 2 ( \\mathcal { H } ) \\bigr ) ^ * . \\end{align*}"}
-{"id": "8326.png", "formula": "\\begin{align*} \\langle F ( u ) , \\varphi \\rangle _ { C ^ 1 ( \\Omega ) } = - \\int _ { \\Omega } \\nabla \\phi ( u ) \\cdot \\nabla \\varphi d x \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{align*} \\| e ^ { ( t - s ) \\Delta } F _ 3 ( s ) \\| _ { L ^ r } & \\le \\| F _ 2 ( s ) \\| _ { L ^ r } \\\\ & \\le \\Bigl ( \\int _ { \\{ | x | > \\sqrt T \\} } | x | ^ { - r N } d x \\Bigr ) ^ { \\frac { 1 } { p } } \\le C < \\infty . \\end{align*}"}
-{"id": "9665.png", "formula": "\\begin{align*} | K _ { + + + } | = | K _ { - + + } | = | K _ { - - + } | = | K _ { + - + } | , \\overline { \\alpha } = \\overline { a } , \\overline { \\beta } = \\overline { b } , \\overline { \\gamma } = \\overline { c } . \\end{align*}"}
-{"id": "2048.png", "formula": "\\begin{gather*} G = \\begin{pmatrix} 0 & 1 & 0 \\\\ - 1 / q _ 2 & 0 & - 1 / q _ 2 \\\\ \\varepsilon q _ 1 & 0 & 0 \\end{pmatrix} . \\end{gather*}"}
-{"id": "4415.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } \\binom { 2 k } { k } \\frac { H _ k ^ { ( 2 ) } } { k } x ^ { k } \\equiv - 2 x ^ p ( \\pounds _ 3 ( 1 - 1 / \\alpha ) + \\pounds _ 3 ( 1 - 1 / \\beta ) ) \\pmod { p } , \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} m _ j = ( 0 , \\dots , 0 , 1 , 0 \\dots , 0 ) \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{align*} \\eta D _ { n } ( \\vec { \\theta } ) = \\sqrt { ( n + 1 ) ( k - n ) } ~ D _ { n + 1 } ( \\vec { \\theta } ) . \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{gather*} \\begin{array} { c } x = u _ i \\ \\left ( \\frac { 1 } { d } \\right ) \\\\ \\overbrace { \\begin{array} { c c c c c } t ^ 0 _ 1 & t ^ 1 _ 1 & \\ldots & t ^ { b - 1 } _ 1 & \\theta _ 1 \\\\ \\vdots & \\vdots & & \\vdots & \\vdots \\\\ t ^ 0 _ m & t ^ 1 _ m & \\ldots & t ^ { b - 1 } _ m & \\theta _ m \\end{array} } \\end{array} \\end{gather*}"}
-{"id": "897.png", "formula": "\\begin{align*} \\underset { \\kappa \\rightarrow 0 } { l i m } \\quad \\underset { T \\geq 1 } { s u p } \\mathbb { E } \\left | \\rho _ { \\psi } ^ T - \\rho _ { \\psi _ \\kappa } ^ T \\right | ^ 2 = 0 , \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} p _ i = i ^ { - 1 / \\theta } L _ 0 ( i , \\theta ) , \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ 0 ^ t d s \\int _ 0 ^ { s / \\tau } \\frac { u d u } { 1 + u ^ 2 } = \\frac { 1 } { 4 \\pi } \\int _ 0 ^ t \\log \\frac { \\tau ^ 2 + s ^ 2 } { \\tau ^ 2 } d s . \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} g _ { \\pi } = g _ { n _ { 1 } } \\dots g _ { n _ { m } } . \\end{align*}"}
-{"id": "9940.png", "formula": "\\begin{align*} \\vect { S } \\Lambda = F \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{align*} u _ 0 ( x ) : = \\sum _ { j \\in \\mathbb { N } } e ^ { i \\pi \\lambda ^ j ( 1 , \\theta _ { j } ) \\cdot x } \\phi ( \\lambda ^ { j / 2 } x _ 1 ) g _ { j } ( \\bar { x } ) . \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} C = \\begin{pmatrix} 0 & 2 & 2 & 3 \\\\ 2 & 0 & 2 & 3 \\\\ 2 & 2 & 0 & 3 \\\\ 3 & 2 & 2 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} \\mathbf { h } \\circ \\mathbf { f } = \\mathrm { i d } _ { \\mathbf { W } } . \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} ( 1 + \\beta ) q = p + 2 \\frac { p } { n } , \\end{align*}"}
-{"id": "9569.png", "formula": "\\begin{align*} E \\big [ \\widehat { \\theta } \\ , | \\ , { \\mathbf { X } } , N _ 1 , N _ 0 \\big ] = \\theta ^ { \\mathrm { d e s c r } } , \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{align*} s + t = \\frac { 1 } { 2 } ( n + 1 ) s _ 1 + n t _ 1 . \\end{align*}"}
-{"id": "395.png", "formula": "\\begin{align*} t ( 0 , a , b ) = \\left \\{ \\begin{array} { c l } 4 , & \\ a = b = 0 , \\\\ 1 , & \\ 0 = a < b \\ \\ a = b > 0 , \\\\ 0 , & . \\end{array} \\right . t ( - 1 , a , b ) = \\left \\{ \\begin{array} { c l } 1 , & \\ a = b , \\\\ 0 , & . \\end{array} \\right . \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} a _ { i j } = \\left \\{ \\begin{array} { l r } ( - 1 ) ^ { j - i } , & { \\rm i f } ~ i \\leq j , \\\\ 0 , & { \\rm i f } ~ i > j , \\end{array} \\right . \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} \\tilde { g } ( x , y ) = e ^ { \\sigma } g ( x , y ) , ~ ~ \\forall ( x , y ) \\in A . \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} \\theta _ { 1 2 } = \\frac { f _ p ( 0 , 0 ) - q _ 1 q _ 2 } { q _ 1 q _ 2 p _ 1 p _ 2 } . \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} \\lfloor f \\rceil ( \\psi , \\bar \\psi ) & : = \\sum _ { | j | = r } | f _ j | \\ , z ^ j , \\end{align*}"}
-{"id": "480.png", "formula": "\\begin{align*} A _ 3 \\left [ A _ 1 ~ A _ 2 \\right ] \\left [ \\begin{array} { c c c c } x _ 1 & x _ 2 & \\cdots & x _ b \\\\ y _ 1 & y _ 2 & \\cdots & y _ b \\end{array} \\right ] = \\left [ \\begin{array} { c c c c } x _ 1 & x _ 2 & \\cdots & x _ b \\end{array} \\right ] . \\end{align*}"}
-{"id": "8304.png", "formula": "\\begin{align*} \\frac { h _ i ^ \\vee } { k _ i } = \\frac { \\dim ( V _ 1 ) - 2 4 } { 2 4 } \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} \\frac 1 { | \\sigma | _ \\eta ^ { 2 ( 1 - \\alpha ) } } \\widehat v '' ( \\rho ) = \\frac { e ^ { \\alpha \\rho } } { ( 1 + e ^ \\rho ) ^ { 1 + \\alpha } } \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} & \\left | \\int _ s ^ t \\langle p _ k ^ 2 + c _ k ^ 2 , w _ k \\cdot \\nabla \\psi _ L \\rangle + \\langle p _ k ^ 3 + c _ k ^ 3 , w _ k \\cdot \\nabla \\psi _ L \\rangle d \\tau \\right | \\\\ & \\leq C _ T \\| w _ k \\cdot \\nabla \\psi _ L \\| _ { L ^ 5 ( 0 , T ; L ^ { 1 5 / 8 } ( \\Omega ) ) } + C _ T \\| w _ k \\cdot \\nabla \\psi _ L \\| _ { L ^ { q ^ \\prime } ( 0 , T ; L ^ { ( r _ * ) ^ \\prime } ( \\Omega ) ) } \\\\ & \\leq C _ T \\big ( \\| \\nabla \\psi _ L \\| _ { 3 0 } + \\| \\nabla \\psi _ L \\| _ \\sigma \\big ) , \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} \\langle t t ' , x \\rangle = \\sum _ { ( x ) } \\langle t \\otimes t ' , x _ { ( 1 ) } \\otimes x _ { ( 2 ) } \\rangle = \\sum _ { ( x ) } ( - 1 ) ^ { [ t ' ] [ x _ { ( 1 ) } ] } \\langle t , x _ { ( 1 ) } \\rangle \\langle t ' , x _ { ( 2 ) } \\rangle \\end{align*}"}
-{"id": "9978.png", "formula": "\\begin{align*} A _ 2 ^ { - 1 } E _ 2 = A _ 1 ^ { - 1 } E _ 1 - g h ^ T , \\end{align*}"}
-{"id": "7427.png", "formula": "\\begin{align*} m _ i v _ i = \\sum _ { \\substack { \\alpha : i \\to j \\\\ v _ i - v _ j = 1 } } c _ \\alpha e ^ { b _ j - b _ i } - \\sum _ { \\substack { \\alpha : k \\to i \\\\ v _ k - v _ i = 1 } } c _ \\alpha e ^ { b _ i - b _ k } . \\end{align*}"}
-{"id": "6718.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial y } + P _ { t } \\left ( D \\right ) u + A u = f \\left ( y , x \\right ) , u ( 0 , x ) = 0 , \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } \\Phi _ \\mu ( Q _ i ) = \\sup \\{ \\Phi _ \\mu ( K ) : \\widetilde { V } _ q ( K ) = | \\mu | K \\in \\mathcal { K } _ o ^ n \\} . \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} \\Delta ( G ) = \\{ A \\subset V ( G ) : A \\mbox { i s a n i n d e p e n d e n t s e t o f } G \\} . \\end{align*}"}
-{"id": "5660.png", "formula": "\\begin{align*} \\varphi = - \\lim _ { \\sigma \\rightarrow 0 } \\sigma ^ 2 \\log P \\Big \\{ \\big [ v ( t ) , w ( t ) \\big ] , \\sigma \\Big \\} . \\end{align*}"}
-{"id": "617.png", "formula": "\\begin{align*} \\lim _ { k \\to + \\infty } \\left \\vert I _ u ( x , r _ k ) - I _ u ( x , 0 ^ + ) \\right \\vert & \\leq \\lim _ { k \\to + \\infty } \\left \\vert I _ u ( x , r _ k ) - I _ u ( 0 , r _ k ) \\right \\vert + \\left \\vert I _ u ( x , 0 ^ + ) - I _ u ( 0 , 0 ^ + ) \\right \\vert \\\\ & \\stackrel { \\textup { ( i i i ) } } { = } \\lim _ { k \\to + \\infty } \\left \\vert I _ u ( x , r _ k ) - I _ u ( 0 , r _ k ) \\right \\vert = 0 . \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{align*} J _ 1 & = - 2 \\int _ { \\mathbb { R } ^ 2 } t \\nabla \\cdot H ^ { ( \\alpha , a ) } \\cdot \\mu ( t \\partial _ t \\nabla \\cdot H ^ { ( \\alpha , a ) } - t \\nabla \\cdot f ^ 2 _ { \\alpha a } ) d x \\\\ & = - 2 \\int _ { \\mathbb { R } ^ 2 } t \\nabla \\cdot H ^ { ( \\alpha , a ) } \\cdot \\mu ( - r \\partial _ r \\nabla \\cdot H ^ { ( \\alpha , a ) } + S \\nabla \\cdot H ^ { ( \\alpha , a ) } - t \\nabla \\cdot f ^ 2 _ { \\alpha a } ) d x . \\end{align*}"}
-{"id": "204.png", "formula": "\\begin{align*} \\operatorname { r r e f } ( M _ f N ) = \\left ( \\begin{array} { c } \\varepsilon _ { i _ 1 } \\\\ \\hline \\vdots \\\\ \\hline \\varepsilon _ { i _ { n - m } } \\end{array} \\right ) \\end{align*}"}
-{"id": "8450.png", "formula": "\\begin{align*} Y _ { \\tau ^ * _ { \\theta } } \\ , = \\ , \\xi _ { \\tau ^ * _ { \\theta } } { \\rm a n d } Y _ { \\overline \\tau _ { \\theta } } \\ , = \\ , \\xi _ { \\overline \\tau _ { \\theta } } \\mbox { a . s . } \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} a ^ { 2 } b ^ { 2 } + a ^ { 4 } + b ^ { 2 } + c ^ { 2 } + 1 = 0 . \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} d \\theta = { p } \\cdot \\pi ^ * \\omega _ S \\end{align*}"}
-{"id": "9315.png", "formula": "\\begin{align*} \\beta = 0 & \\gamma \\in \\{ 2 m \\mid m = 0 , 1 , 5 , 6 , 8 , 9 , 1 0 , 1 1 , 1 3 , \\ldots , 6 5 , 6 8 , 7 1 , 7 9 \\} , \\\\ \\beta = 1 & \\gamma \\in \\{ 2 m \\mid m = 1 3 , 1 4 , 1 6 , \\ldots , 5 8 , 6 3 \\} , \\\\ \\beta = 2 & \\gamma \\in \\{ 2 m \\mid m = 0 , 1 6 , \\ldots , 5 0 , 5 5 \\} \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} \\nu : = \\frac { 1 } { 4 } \\left ( \\delta _ { ( - 1 , 0 ) } + \\delta _ { ( 1 , 0 ) } + \\delta _ { ( 0 , - 1 ) } + \\delta _ { ( 0 , 1 ) } \\right ) \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{align*} A ( \\sigma ) z _ n = \\overline { \\mathstrut A } ( \\sigma ) z _ n = 0 . \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{align*} \\int _ \\mathbb { R } | w - x | ^ { \\frac { \\alpha + \\beta } { 2 } - 1 } | w - z | ^ { \\gamma - 1 } d w = C '' ( \\alpha , \\beta ) | z - x | ^ { \\beta - 1 } . \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} \\phi _ A ( e ^ { 2 \\pi i x } , e ^ { 2 \\pi i y } ) = ( e ^ { 2 \\pi i ( a x + b y ) } , e ^ { 2 \\pi i ( c x + d y ) } ) \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} h _ i : = l _ { 2 i - 1 } l _ { 2 i } \\ \\ \\ \\ g _ { j 0 } : = l _ { 2 n - 3 + 2 j } l _ { 2 n - 2 + 2 j } . \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} \\varphi ^ { ( 3 ) } _ { i , j } : = \\varphi ^ { ( 3 ) } ( i , j ) , \\varphi ^ { ( 4 ) } _ { i , j } : = \\varphi ^ { ( 4 ) } ( i , j ) . \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} t = \\sum _ { i = 1 } ^ n t _ i g _ i , \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} \\rho ( \\mathbf { x } ) = \\frac { 1 } { \\mathrm { v o l } ( \\Lambda ) } \\sum _ { \\mathbf { q } \\in \\Lambda ^ * } \\hat { \\rho } ( \\mathbf { q } ) ~ e ^ { i 2 \\pi \\mathbf { q } \\cdot \\mathbf { x } } , \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{align*} w _ { i , j } = \\big { | } ( 1 - p ) \\min \\{ p _ i , p _ j \\} - p \\max \\{ p _ i , p _ j \\} \\big { | } _ + , \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} \\int \\sigma _ { 2 j } ( \\eta , \\bar \\eta ) d \\eta d \\bar \\eta \\eta ^ { n } \\bar \\eta ^ { n } = \\frac { ( 2 j ) ! } { ( 2 j - n ) ! } . \\end{align*}"}
-{"id": "8928.png", "formula": "\\begin{align*} J _ 0 ( \\lambda | y - w | ) - J _ 0 ( \\lambda | y | ) = \\lambda \\int ^ { | y - w | } _ { | y | } J _ 0 ' ( \\lambda s ) \\ , d s . \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} T r \\left ( \\left ( 1 + \\lambda ^ 2 x ^ 2 + \\frac { \\lambda ^ 2 \\mu ^ 2 } { x ^ 2 } \\right ) \\left ( \\alpha + \\frac { \\lambda ^ 2 } { x ^ 2 } \\right ) \\right ) = 0 . \\end{align*}"}
-{"id": "2302.png", "formula": "\\begin{align*} a _ 1 ^ 1 = 1 , a _ 2 ^ 1 = - 1 , a _ 3 ^ 1 = 1 , a _ 4 ^ 1 = - 1 . \\end{align*}"}
-{"id": "8113.png", "formula": "\\begin{align*} \\partial _ t ^ { k + 1 } P ( t , x ( t ) ) = \\sum _ { j = 0 } ^ k c _ j ( t , x ( t ) ) \\cdot \\partial _ t ^ { k - j } P ( t , x ( t ) ) . \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} f ( a ) = a \\left ( 1 + \\frac { 2 } { Q \\sqrt \\kappa } - \\frac { a } { Q ^ 2 } \\right ) . \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{align*} x \\left ( z \\right ) = \\frac { 1 - \\frac { e ^ { z } - 1 } { z } } { \\frac { e ^ { z } - 1 } { z } } = \\frac { z } { e ^ { z } - 1 } - 1 = \\sum _ { n \\ge 1 } \\frac { B _ { n } } { n ! } z ^ { n } . \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} \\Omega : = \\{ x \\in \\R ^ n : x _ n < \\eta ( x ^ \\prime ) \\} , \\end{align*}"}
-{"id": "3815.png", "formula": "\\begin{align*} t _ { 2 ^ k } ( 2 ^ { k + 1 } n + 2 j + 1 ) = \\sum _ { i = 0 } ^ { 2 ^ { k - 1 } - 1 } { 2 ^ k \\choose 2 i + 1 } t _ { 2 ^ k } ( 2 ^ k n + j - i ) \\end{align*}"}
-{"id": "1155.png", "formula": "\\begin{align*} & \\begin{cases} \\alpha _ { k , j } ^ { u _ 0 } ( u _ 1 ) = \\frac { \\overline { \\widehat \\alpha _ { j , j } ( u _ 1 ) } } { | \\widehat \\alpha _ { j , j } ( u _ 1 ) | } \\widehat \\alpha _ { k , j } ( u _ 1 ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ & \\forall j , k \\in \\N ^ * , \\ j , k \\leq N , \\\\ \\alpha ^ { u _ 0 } _ { k , j } ( u _ 1 ) = \\widehat \\alpha _ { k , j } ( u _ 1 ) , \\ & \\forall j , k \\in \\N ^ * , \\ j > N , \\ k \\leq N . \\\\ \\end{cases} \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{align*} \\left ( \\Lambda _ { N , N + 1 } h _ { N , s } \\right ) ( x ) = \\frac { 1 } { N ! } \\Delta _ { N + 1 } ( x ) \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} t \\in [ 0 , T ] \\mapsto \\{ w ( \\cdot , t ) = c \\} \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{align*} \\begin{aligned} D ^ g J d ( h u ^ { - 1 } ) ^ \\sharp = & 2 u ^ { - 3 } h d u \\otimes J d u ^ \\sharp - u ^ { - 2 } d u \\otimes J d h ^ \\sharp - u ^ { - 2 } d h \\otimes J d u ^ \\sharp \\\\ & + u ^ { - 1 } D ^ g ( J d h ^ \\sharp ) - u ^ { - 2 } h D ^ g J d u ^ \\sharp \\\\ d u \\otimes J d ( h u ^ { - 2 } ) ^ \\sharp = & u ^ { - 2 } d u \\otimes J d h ^ \\sharp - 2 u ^ { - 3 } h d u \\otimes J d u ^ \\sharp \\end{aligned} \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{gather*} ( - A ) ^ { - \\alpha } x = \\sum _ { i \\in \\N ^ { \\star } } \\lambda _ i ^ { - \\alpha } \\langle x , e _ i \\rangle e _ i , x \\in H , \\\\ ( - A ) ^ { \\alpha } x = \\sum _ { i \\in \\N ^ { \\star } } \\lambda _ i ^ \\alpha \\langle x , e _ i \\rangle e _ i , x \\in D _ 2 \\bigl ( ( - A ) ^ { \\alpha } \\bigr ) = \\left \\{ x \\in H ; \\sum _ { i = 1 } ^ { \\infty } \\lambda _ { i } ^ { 2 \\alpha } \\langle x , e _ i \\rangle ^ 2 < \\infty \\right \\} . \\end{gather*}"}
-{"id": "4301.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\frac { x ( [ a ] Q _ { n _ k } ) } { y ( [ a ] Q _ { n _ k } ) } = 0 . \\end{align*}"}
-{"id": "8460.png", "formula": "\\begin{align*} Y _ t ( \\omega ) = Y _ 0 - \\int _ 0 ^ t f ( s , \\omega , Y _ s ( \\omega ) , Z _ s ( \\omega ) , k _ s ( \\omega ) ) d s + M _ t ( \\omega ) - A _ t ( \\omega ) - C _ { t - } ( \\omega ) , \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { F } _ q } ( B h _ i ) = t _ i = l / r . \\end{align*}"}
-{"id": "9839.png", "formula": "\\begin{align*} ( \\Delta \\otimes ) F = F _ { 1 3 } F _ { 2 3 } , \\ , \\ , \\ , ( \\otimes \\Delta ) F = F _ { 1 3 } F _ { 1 2 } , \\ , \\ , \\ , F _ { 1 2 } F _ { 1 3 } F _ { 2 3 } = F _ { 2 3 } F _ { 1 3 } F _ { 1 2 } , \\ , \\ , \\ , F F _ { 2 1 } = 1 \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} \\| U \\| _ { o p } = \\inf \\{ \\| a \\| \\| u \\| \\| v \\| \\| b \\| \\} . \\end{align*}"}
-{"id": "4835.png", "formula": "\\begin{align*} K = \\left \\langle I + \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} p , I + \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} p , I + \\begin{pmatrix} \\sqrt { \\epsilon } & 0 \\\\ 0 & - \\sqrt { \\epsilon } \\end{pmatrix} p , I + \\begin{pmatrix} 0 & \\sqrt { \\epsilon } \\\\ - \\sqrt { \\epsilon } & 0 \\end{pmatrix} p \\right \\rangle . \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ { n + 1 } A _ { i j } y _ { j } = \\left ( \\sum \\limits _ { j = 1 } ^ { n } c _ { i j } x _ { j } \\mathbf { f + } A _ { i , n + 1 } y _ { n + 1 } \\right ) \\mathbf { = } \\left ( \\lambda x _ { i } \\right ) \\mathbf { f = } \\lambda \\left ( x _ { i } \\mathbf { f } \\right ) = \\lambda y _ { i } \\end{align*}"}
-{"id": "5786.png", "formula": "\\begin{align*} \\hat { \\gamma } _ { A B A ' B ' } ^ { ( n ) } = \\hat { \\gamma } _ { A A ' } ^ { ( n ) } \\otimes \\hat { \\gamma } _ { B B ' } ^ { ( n ) } \\ ; , n \\in \\mathbb { N } \\ ; , \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{align*} I \\leq \\prod _ { j = 1 } ^ k \\left ( \\int _ { \\mathbb { R } ^ { 2 k } } g ( \\mathbf { z } , \\mathbf { w } ) ^ k d \\mathbf { z } d \\mathbf { w } \\right ) ^ \\frac { 1 } { k } \\end{align*}"}
-{"id": "7666.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } c _ 1 \\\\ \\vdots \\\\ c _ k \\end{array} \\right ) = B \\left ( \\begin{array} { c } h _ 1 \\\\ \\vdots \\\\ h _ n \\end{array} \\right ) . \\end{align*}"}
-{"id": "6930.png", "formula": "\\begin{align*} \\int _ \\Omega | \\nabla \\psi _ \\varepsilon | ^ 2 = o ( 1 ) , \\end{align*}"}
-{"id": "9134.png", "formula": "\\begin{gather*} h _ j ( x _ 0 + x _ 1 , x _ 0 x _ 1 , x _ 2 , x _ 3 ) = f _ j ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) . \\end{gather*}"}
-{"id": "1522.png", "formula": "\\begin{align*} S ( x ) ^ 2 + \\sum _ { k = 1 } ^ { n - 1 } \\big ( P ( t _ k ) + t \\alpha _ k \\big ) S _ k ( x ) ^ 2 = \\tilde { S } ( x ) ^ 2 + \\sum _ { k = 1 } ^ { n - 1 } \\big ( P ( \\tilde { t } _ k ) + t \\beta _ k \\big ) \\tilde { S } _ k ( x ) ^ 2 \\end{align*}"}
-{"id": "4613.png", "formula": "\\begin{align*} { \\displaystyle \\| \\psi _ { h } ^ { k } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } = { \\| \\psi _ { h } ^ { 0 } \\| } _ { \\mathcal { L } ^ 2 } ^ { 2 } , k = 1 , 2 \\cdots , M . } \\end{align*}"}
-{"id": "2975.png", "formula": "\\begin{align*} \\| P _ { n } T ^ { \\ , j } P _ { l } \\ , x \\| & \\le C _ { l } \\ , \\cdot \\ , ( b _ { l + 1 } - b _ { l } ) ^ { 1 - \\frac 1 { p } } \\ , \\cdot \\ , \\Bigl \\| \\sum _ { k = b _ { l + 1 } - N } ^ { b _ { l + 1 } - 1 } \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } w _ { s } \\Bigr ) \\ , x _ { k } e _ { k } \\Bigr \\| \\\\ & \\le C _ { l } \\ , \\cdot \\ , ( b _ { l + 1 } - b _ { l } ) ^ { 1 - \\frac 1 { p } } \\ , \\cdot \\ , \\sup _ { b _ { l + 1 } - N \\le k < b _ { l + 1 } } \\Bigl ( \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } | w _ { s } \\ , | \\Bigr ) \\ , \\cdot \\ , \\| P _ { l } \\ , x \\| . \\end{align*}"}
-{"id": "5320.png", "formula": "\\begin{align*} G [ s ] ( w ) & = \\frac { 1 } { 1 + u _ 1 w + u _ 2 w ^ 2 \\cdots + u _ s w ^ s } \\\\ & = \\sum _ { n = 0 } ^ \\infty ( - 1 ) ^ { n } ( u _ 1 w + u _ 2 w ^ 2 \\cdots + u _ s w ^ s ) ^ n \\\\ & = \\sum _ { n = 0 } ^ \\infty ( - 1 ) ^ { n } \\sum _ { \\nu _ 1 + \\cdots \\nu _ s = n } \\frac { n ! } { \\prod _ { i = 1 } ^ { s } \\nu _ { i } ! } \\prod _ { j = 1 } ^ s ( u _ j w ^ j ) ^ { \\nu _ j } \\\\ & = \\sum _ { \\eta } ( - 1 ) ^ { \\ell ( \\eta ) } \\frac { \\ell ( \\eta ) ! } { \\prod _ { i = 1 } ^ s m _ i ( \\eta ) ! } w ^ { | \\eta | } \\prod _ { i = 1 } ^ s u _ i ^ { m _ i ( \\eta ) } \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{align*} C _ 1 ( t ) = O ( t ^ { 3 / 2 - \\eta } ) , C _ 2 ( t ) = O ( t ^ { 3 / 2 - \\eta } ) . \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} b _ n = \\sum _ { k = 1 } ^ n \\sum _ { j = 0 } ^ { k - 1 } p ( n - k ) ( - 1 ) ^ { \\lceil j / 2 \\rceil } b ( k - G _ j ) . \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} B ( x , y ) = \\sum _ { i = 0 } ^ { n - 1 } \\sum _ { j = 0 } ^ { n - 1 } b _ { i j } x ^ i y ^ j = \\frac { v ( x ) w ( y ) - w ( x ) v ( y ) } { x - y } , \\end{align*}"}
-{"id": "2277.png", "formula": "\\begin{align*} x ( t ) = - m + c _ 0 e ^ { \\alpha t } , y ( t ) = c _ 0 \\alpha e ^ { \\alpha t } . \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} d ' ( q ) = 4 a b c + u v w - a u ^ 2 - b v ^ 2 - c w ^ 2 . \\end{align*}"}
-{"id": "9048.png", "formula": "\\begin{align*} K _ s ( m , n ) = \\sum _ { t \\geq 0 } ( - 1 ) ^ t \\binom { m } { n t + s - 1 } , \\enskip s = 1 , . . . , n . \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { ( p - 1 ) / 2 } C _ k H _ k ^ { ( 2 ) } \\bigl ( \\beta ( 1 - \\beta ) \\bigr ) ^ { k + 1 } \\equiv 2 \\beta \\pounds _ 2 ( \\beta ) - ( 1 - \\beta ) \\pounds _ 1 ( \\beta ) ^ 2 \\pmod { \\beta ^ { p + 1 } } . \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{gather*} \\langle \\nabla _ { \\xi } F ( \\xi ) , \\xi \\rangle = F ( \\xi ) , \\langle \\nabla _ { \\xi } F ^ { o } ( \\xi ) , \\xi \\rangle = F ^ { o } ( \\xi ) , \\\\ F ( \\nabla _ { \\xi } F ^ o ( \\xi ) ) = F ^ o ( \\nabla _ { \\xi } F ( \\xi ) ) = 1 , \\forall \\xi \\in \\R ^ n \\setminus \\{ 0 \\} , \\\\ F ^ o ( \\xi ) \\nabla _ { \\xi } F ( \\nabla _ { \\xi } F ^ o ( \\xi ) ) = F ( \\xi ) \\nabla _ { \\xi } F ^ o ( \\nabla _ { \\xi } F ( \\xi ) ) = \\xi \\forall \\xi \\in \\R ^ n \\setminus \\{ 0 \\} . \\end{gather*}"}
-{"id": "698.png", "formula": "\\begin{align*} ( a ^ + _ 1 ) ^ { l _ 1 } ( a ^ + _ 2 ) ^ { l _ 2 } \\cdots ( a ^ + _ r ) ^ { l _ r } = 0 l _ 1 + l _ 2 + \\cdots l _ r = k + 1 . \\end{align*}"}
-{"id": "4537.png", "formula": "\\begin{align*} { \\displaystyle 2 \\mathrm { i } ( \\partial \\theta ^ { k } _ { \\Psi } , \\overline { \\theta } _ { \\Psi } ^ { k } ) - B \\left ( \\overline { \\mathbf { A } } ^ { k } _ { h } ; \\overline { \\theta } _ { \\Psi } ^ { k } , \\overline { \\theta } _ { \\Psi } ^ { k } \\right ) = I _ 1 ^ { ( k ) } + I _ 2 ^ { ( k ) } + I _ 3 ^ { ( k ) } + I _ 4 ^ { ( k ) } , } \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} M - N \\overline { M } ^ { - 1 } \\overline { N } = \\left ( \\begin{array} { r r r } 0 . 5 7 5 6 - 0 . 0 0 0 0 i & 0 . 3 9 0 6 + 0 . 2 0 6 0 i & - 0 . 1 5 7 6 - 0 . 5 1 9 6 i \\\\ 0 . 3 9 0 6 - 0 . 2 0 6 0 i & 0 . 8 1 3 1 - 0 . 0 0 0 0 i & - 0 . 5 2 2 0 - 0 . 6 0 8 9 i \\\\ - 0 . 1 5 7 6 + 0 . 5 1 9 6 i & - 0 . 5 2 2 0 + 0 . 6 0 8 9 i & 0 . 9 1 3 7 - 0 . 0 0 0 0 i \\end{array} \\right ) \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{align*} u _ { N } ( \\mathsf { x , } t ) = \\sum _ { \\mathsf { n } : 0 \\leq n _ { j } \\leq N - 1 } \\alpha _ { \\mathsf { n } } \\exp \\left [ - t E _ { \\mathsf { n } } \\right ] \\mathsf { h } _ { \\mathsf { n } } \\left ( \\mathsf { x } \\right ) \\end{align*}"}
-{"id": "8700.png", "formula": "\\begin{align*} N _ 1 ( c ) = 2 m c _ m ^ - < c < c _ m ^ + . \\end{align*}"}
-{"id": "6992.png", "formula": "\\begin{align*} \\left ( \\sum _ { j _ { 1 } , \\cdots , j _ { m } = 1 } ^ { \\infty } \\left \\vert T ( e _ { j _ { 1 } } , \\cdots , e _ { j _ { m } } ) \\right \\vert ^ { \\frac { p } { p - m } } \\right ) ^ { \\frac { p - m } { p } } \\leq \\left ( \\sqrt { 2 } \\right ) ^ { m - 1 } \\left \\Vert T \\right \\Vert \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} X _ k ( f ) = \\sum _ { d | k } \\frac { d } { k } \\# \\{ \\} . \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} \\boldsymbol { \\mathcal { X } } = \\mathbb { P } ( \\mathbb { U } ) . \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} ( 1 - 2 \\beta ) \\sum _ { k = 1 } ^ { ( p - 1 ) / 2 } \\binom { 2 k } { k } H _ k \\bigl ( \\beta ( 1 - \\beta ) \\bigr ) ^ k \\equiv 2 \\pounds _ 1 ( 2 \\beta ) - 2 \\pounds _ 1 ( \\beta ) \\pmod { \\beta ^ p } , \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{align*} u _ \\partial ( t ) = \\sum _ { \\ell \\in \\N _ * } w _ \\ell ( t ) = \\sum _ { \\ell \\in \\N _ * } \\big ( \\frac { g ^ \\omega _ \\ell - g ^ 0 _ \\ell \\ , \\cos \\ell \\omega } { \\sin \\ell \\omega } \\ , \\Im \\zeta ^ \\ell + g ^ 0 _ \\ell \\ , \\Re \\zeta ^ \\ell \\big ) \\ , , \\end{align*}"}
-{"id": "6077.png", "formula": "\\begin{align*} 0 = [ L _ { - 1 } , \\ , [ L _ { - 3 } , \\ , [ L _ { - 3 } , \\ , L _ 5 ] ] ] = [ L _ { - 3 } , \\ , [ L _ { - 3 } , \\ , [ L _ { - 1 } , \\ , L _ 5 ] ] ] . \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} | E ^ \\nu | = \\left | \\begin{matrix} e _ 1 & \\sigma ^ { - 1 } ( e _ 1 ) & \\cdots & \\sigma ^ { - \\nu + 1 } ( e _ 1 ) \\\\ e _ 2 & \\sigma ^ { - 1 } ( e _ 2 ) & \\ldots & \\sigma ^ { - \\nu + 1 } ( e _ 2 ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ e _ \\nu & \\sigma ^ { - 1 } ( e _ \\nu ) & \\cdots & \\sigma ^ { - \\nu + 1 } ( e _ \\nu ) \\end{matrix} \\right | = 0 \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} Y _ t : = X _ t - X ^ { ' } _ t + E [ \\xi _ T + \\int _ t ^ T f _ s d s | \\cal { F } _ t ] , \\ ; 0 \\leq t \\leq T , \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{align*} q _ m ( t ) = \\frac { m + 1 } { 2 } t ^ 2 - m t + \\frac { m + 1 } { 2 } . \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} \\varepsilon = \\sup _ { [ 0 , T ^ * ] } \\int _ { [ \\gamma > 0 ] } | A | ^ n d \\mu \\le \\varepsilon _ 0 \\end{align*}"}
-{"id": "5665.png", "formula": "\\begin{align*} \\lambda _ { 1 , 2 } ( \\varepsilon ) = \\frac { 5 \\varepsilon + 0 . 9 \\mp \\sqrt { 2 5 \\varepsilon ^ 2 + 5 . 4 \\varepsilon + 0 . 8 1 } } { 7 . 2 \\varepsilon - 0 . 1 8 } , \\end{align*}"}
-{"id": "7164.png", "formula": "\\begin{align*} ( w + w ^ { - 1 } ) ^ 2 - 4 = w ^ 2 + w ^ { - 2 } - 2 = ( w - w ^ { - 1 } ) ^ 2 , \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} d \\eta ^ { h , x } ( t ) = A \\eta ^ { h , x } ( t ) d t + \\sigma ' ( X ( t , x ) ) . \\eta ^ { h , x } ( t ) d W ( t ) , \\eta ^ { h , x } ( 0 ) = h . \\end{align*}"}
-{"id": "1010.png", "formula": "\\begin{align*} \\begin{aligned} \\chi ( Y ) & = 6 \\frac { 2 L t + 3 L S t ^ 2 - S ^ 2 t ^ 2 } { ( 1 + S t ) ( 1 + 6 L t - 2 S t ) } c _ t ( T B ) \\\\ & = 1 2 L t + 6 t ^ 2 ( 2 c _ 1 L - 1 2 L ^ 2 + 5 L S - S ^ 2 ) + \\\\ & \\ \\ + 6 t ^ 3 ( - 1 2 c _ 1 L ^ 2 + 5 c _ 1 L S - c _ 1 S ^ 2 + 2 c _ 2 L + 7 2 L ^ 3 - 5 4 L ^ 2 S + 1 5 L S ^ 2 - S ^ 3 ) + \\cdots \\end{aligned} \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} \\vert \\vert n _ 0 , n _ 1 , \\dots , n _ { r } \\rangle \\rangle = \\frac { ( b _ 0 ^ + ) ^ { n _ 0 } } { \\sqrt { n _ 0 ! } } \\frac { ( b _ 1 ^ + ) ^ { n _ 1 } } { \\sqrt { n _ 1 ! } } \\cdots \\frac { ( b _ { r } ^ + ) ^ { n _ { r } } } { \\sqrt { n _ { r } ! } } \\vert \\vert 0 , 0 , \\cdots , 0 \\rangle \\rangle . \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\overbrace { \\begin{matrix} 1 & \\theta ^ 0 _ 1 \\\\ 0 & \\theta ^ 0 _ 2 \\\\ 0 & 0 \\end{matrix} } & \\overbrace { \\begin{matrix} \\sqrt { t } & \\theta ^ \\infty _ 1 / 2 \\\\ - \\sqrt { t } & \\theta ^ \\infty _ 1 / 2 \\\\ 0 & \\theta ^ \\infty _ 2 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "7086.png", "formula": "\\begin{align*} E _ \\delta : x ^ 2 + \\delta y ^ 2 = 1 . \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} q _ { 1 } f ^ { 2 } + q _ { 2 } f ^ { \\prime } f + q _ { 3 } \\left ( f ^ { \\prime } \\right ) ^ { 2 } = - B _ { 2 } , \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} \\int _ { \\R } ( H ^ m - H ^ n ) H ' d x = \\left [ \\frac { H ^ { m + 1 } } { m + 1 } \\right ] ^ { + \\infty } _ { - \\infty } - \\left [ \\frac { H ^ { n + 1 } } { n + 1 } \\right ] ^ { + \\infty } _ { - \\infty } = \\frac { 1 } { m + 1 } - \\frac { 1 } { n + 1 } . \\end{align*}"}
-{"id": "9372.png", "formula": "\\begin{align*} S ^ 2 ( \\mathcal { H } , \\mathcal { K } ) = S ^ 2 ( \\mathcal { K } , \\mathcal { H } ) ^ * . \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} \\begin{aligned} & x _ 0 ( t _ 0 ) \\in E _ 0 , & t \\in [ t _ 0 , t _ 0 + \\lambda ] , \\ : \\ : & x _ 0 ( t ) \\leq \\tilde { z } _ 1 , \\\\ & x _ 1 ( t _ 0 ) \\in E _ 1 , & t \\in [ t _ 0 , t _ 0 + \\lambda ] , \\ : \\ : & x _ 1 ( t ) \\geq \\tilde { z } _ 1 . \\end{aligned} \\end{align*}"}
-{"id": "9400.png", "formula": "\\begin{align*} \\bigr [ \\widetilde { \\phi } ( t _ 2 ) \\bigr ] ( r ) = \\int _ { \\Omega _ 1 } \\phi ( t _ 1 , t _ 2 , \\cdotp ) \\ , r ( t _ 1 ) \\ , \\mu _ 1 ( t _ 1 ) , t _ 2 \\in \\Omega _ 2 . \\end{align*}"}
-{"id": "9581.png", "formula": "\\begin{align*} \\varepsilon _ { n , i } = Y _ { n , i } - X _ { n , i } ^ { \\prime } \\theta _ n ^ \\mathrm { c a u s a l } - Z _ { n , i } ^ { \\prime } \\gamma _ n ^ \\mathrm { c a u s a l } . \\end{align*}"}
-{"id": "5707.png", "formula": "\\begin{align*} \\hat { R } ^ { 2 k - 1 } : = \\hat { Q } ^ k \\ ; , \\qquad \\hat { R } ^ { 2 k } : = \\hat { P } ^ k \\ ; , k = 1 , \\ , \\ldots , \\ , n \\ ; , \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} f = f _ 0 - \\widetilde U \\cdot \\nabla \\widetilde U , f _ 0 = F - ( u _ s \\cdot \\nabla \\widetilde U + \\widetilde U \\cdot \\nabla u _ s ) ( t \\geq \\bar t ) . \\end{align*}"}
-{"id": "4410.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } C _ k H _ k ^ { ( 2 ) } x ^ { k + 1 } \\equiv 2 \\alpha \\pounds _ 2 ( \\alpha ) + 2 \\beta \\pounds _ 2 ( \\beta ) \\pmod { p } . \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{align*} E _ q ^ { \\gamma _ 0 } : = q \\setminus \\bigg ( \\bigcup _ { 1 \\le j \\le N } E _ q ^ { \\gamma _ j } \\bigg ) . \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} ( \\Psi ^ \\prime _ { n } ( X , Y ) ) ^ 2 = f _ n ^ 2 ( X ) = n ^ 2 X ^ { n ^ 2 - 1 } + c X ^ { n ^ 2 - 3 } + \\cdot \\cdot \\cdot , \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} v ( x , t ) : = u ( x , t ) - h ( t ) u _ s ( x ) , p _ v ( x , t ) : = p _ u ( x , t ) - h ( t ) p _ { u _ s } ( x ) \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } W ( \\phi _ k ) = W ( \\phi _ { \\infty } ) + \\sum _ { i = 1 } ^ { p } W ( \\vec { \\Psi } _ i ) + \\sum _ { j = 1 } ^ { q } \\left ( W ( \\vec { \\xi } _ j ) - 4 \\pi \\theta _ j \\right ) \\end{align*}"}
-{"id": "9331.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } Y ^ 2 = X ^ 3 + A ( t : s ) { q '' _ 2 } ^ 2 ( y _ 0 : y _ 1 : y _ 2 : y _ 3 : y _ 4 ) X + B ( t : s ) { q '' _ 2 } ^ 3 ( y _ 0 : y _ 1 : y _ 2 : y _ 3 : y _ 4 ) \\\\ q _ 2 ' ( y _ 0 : y _ 1 : y _ 2 : y _ 3 : y _ 4 ) = 0 \\\\ q _ 2 ( y _ 0 : y _ 1 : y _ 2 : y _ 3 : y _ 4 ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} Y = \\varphi ^ { - 1 } \\bigl ( N \\bigr ) = \\mathcal { Y } \\times _ { N _ s } N \\ \\ \\ \\ \\ \\ \\varphi : Y \\to N \\ \\ G . \\end{align*}"}
-{"id": "5997.png", "formula": "\\begin{align*} \\phi _ y - \\nu \\phi = 0 \\mbox { o n } \\ F \\mbox { a n d } \\phi _ y = 0 \\mbox { o n } \\ \\partial W \\cap \\{ y = - 1 \\} . \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} z _ i z _ j = ( n - 1 ) \\end{align*}"}
-{"id": "9461.png", "formula": "\\begin{align*} C _ 0 = \\max \\big \\{ ( \\inf h _ 0 ) ^ { - 1 } , \\sup h _ 0 , \\| h _ 0 \\| _ { H ^ 4 } , \\| B _ 0 \\| _ { H ^ 4 } , \\| P _ 0 \\| _ { L ^ 2 } , \\| D _ 0 \\| _ { L ^ 2 } \\big \\} \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} \\delta _ \\Lambda ( \\lambda ) = \\pi ( d ( \\lambda ) ) \\quad \\end{align*}"}
-{"id": "47.png", "formula": "\\begin{align*} [ h _ K ] = K , \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} \\operatorname { r c e f } ( S ^ t ) = \\left ( \\begin{array} { c | c } \\begin{array} { c } I _ \\mu \\\\ \\hline a _ 0 \\cdots a _ { \\mu - 1 } \\\\ \\hline H ' \\end{array} & 0 _ { ( t + 1 ) \\times ( t - \\mu ) } \\end{array} \\right ) , \\end{align*}"}
-{"id": "3878.png", "formula": "\\begin{align*} \\operatorname * { s i g n } f ^ \\prime ( - 1 + h - a ) & = \\operatorname * { s i g n } \\left [ - \\left ( 1 - \\frac { b ^ 2 / a } { 2 + a - h } \\right ) ^ 2 + ( 1 - b ^ 2 / a ^ 2 ) \\right ] \\\\ & = \\operatorname * { s i g n } \\left [ \\frac { b ^ 2 / a ^ 2 } { ( 2 + a - h ) ^ 2 } \\left ( - ( 2 + a - h ) ^ 2 + 2 a ( 2 - h + a ) - b ^ 2 \\right ) \\right ] \\stackrel { ( \\ref { e q : l e m m a : i n c l u s i o n - 1 0 1 } ) } { \\leq } 0 . \\end{align*}"}
-{"id": "9684.png", "formula": "\\begin{align*} \\pi - \\Theta ( \\Theta _ 1 T _ 0 ( s ) , 0 , 0 ) + \\Theta _ 1 T _ 0 ( s ) & = \\pi - \\Theta _ 0 s , \\\\ \\pi - \\Theta ( \\Theta _ 0 T _ \\pi ( t ) , 0 , \\pi ) + \\Theta _ 0 T _ \\pi ( t ) & = \\pi - \\Theta _ 1 t , \\end{align*}"}
-{"id": "8522.png", "formula": "\\begin{align*} J ' = \\varphi [ J ] = \\{ f : \\ f = \\varphi ( g ) g \\in J \\} \\in N [ J ] \\end{align*}"}
-{"id": "7396.png", "formula": "\\begin{align*} 0 = E _ 0 \\subset E _ 1 \\subset \\ldots \\subset E _ n = E \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} Q _ 0 ( x ) & = \\dfrac { 1 } { 2 } \\ln \\left ( \\dfrac { 1 + x } { 1 - x } \\right ) , \\\\ Q _ 1 ( x ) & = \\dfrac { x } { 2 } \\ln \\left ( \\dfrac { 1 + x } { 1 - x } \\right ) - 1 , \\\\ Q _ 2 ( x ) & = \\dfrac { 3 x ^ 2 - 1 } { 4 } \\ln \\left ( \\dfrac { 1 + x } { 1 - x } \\right ) - \\dfrac { 3 x } { 2 } , \\\\ Q _ 3 ( x ) & = \\dfrac { 5 x ^ 3 - 3 x } { 4 } \\ln \\left ( \\dfrac { 1 + x } { 1 - x } \\right ) - \\dfrac { 5 x ^ 2 } { 2 } + \\dfrac { 2 } { 3 } . \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} X = ( C _ 1 \\times C _ 2 \\times C _ 3 ) / G \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} Y _ { n } ^ { 1 , \\ell } & = S _ { \\Delta t } ^ { n - \\ell } z _ \\ell + \\Delta t \\sum _ { m = \\ell } ^ { n - 1 } S _ { \\Delta t } ^ { n - m } F _ 1 ' ( X _ m ) . Y _ { m } ^ { \\ell } + \\sum _ { m = \\ell } ^ { n - 1 } S _ { \\Delta t } ^ { n - m } e ^ { \\tau A } \\bigl ( \\sigma ' ( X _ m ) . Y _ { m } ^ { \\ell } \\bigr ) \\Delta W _ m \\\\ & + \\Delta t \\sum _ { m = \\ell } ^ { n - 1 } F _ { n , 1 } + \\sum _ { m = \\ell } ^ { n - 1 } S _ { \\Delta t } ^ { n - m } e ^ { \\tau A } \\sigma '' ( X _ m ) . ( Z _ m ^ 1 , Z _ m ^ 2 ) \\Delta W _ m \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } u ' = A u , t \\in \\R \\\\ u ( 0 ) = u _ 0 \\in X \\end{array} \\right . , \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{align*} E \\prod _ { j = 0 } ^ { n - 1 } \\rho ( x _ j , \\omega ) \\rho ( x _ { j + 1 } , \\omega ) & = M ^ { 2 n } E \\prod _ { j = 0 } ^ { n - 1 } \\widetilde { \\rho } ( x _ j , \\omega ) \\widetilde { \\rho } ( x _ { j + 1 } , \\omega ) \\\\ & \\leq M ^ { 2 n } E \\prod _ { j = 0 } ^ { | x _ n | - 1 } \\widetilde { \\rho } ( y _ j , \\omega ) \\widetilde { \\rho } ( y _ { j + 1 } , \\omega ) \\\\ & \\leq M ^ { 2 n } [ E { \\widetilde { \\rho } } ^ 2 ] ^ { | x _ n | - 1 } . \\end{align*}"}
-{"id": "6805.png", "formula": "\\begin{align*} S = - i \\tilde { \\gamma } \\left ( 2 \\nabla \\varphi . \\nabla + \\Delta \\varphi \\right ) K = i \\left ( \\Delta + A + \\tilde { \\gamma } ^ { 2 } \\left \\vert \\nabla \\varphi \\right \\vert ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} f ( [ E ] , S , C ) : = ( S , C , \\xi , \\mathbf { k } \\psi ) . \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} h _ t = h \\circ f _ t + \\frac { 2 } { \\sqrt \\kappa } \\log | f _ t | . \\end{align*}"}
-{"id": "3405.png", "formula": "\\begin{align*} \\begin{cases} d X _ t = [ ( a + \\lambda \\theta ) ( m ( t ) - X _ t ) - \\lambda Y _ t - R _ t ] \\ , d t + \\sigma \\ , d W _ t + \\left [ \\theta ( m ( t - ) - X _ { t - } ) - \\left ( Y _ { t - } + \\frac { R _ { t } } { \\lambda } \\right ) \\right ] \\ , d \\widetilde N _ t \\\\ X _ 0 = x _ 0 \\end{cases} \\end{align*}"}
-{"id": "7128.png", "formula": "\\begin{align*} \\begin{aligned} _ x I _ b ^ { 1 - \\alpha } & u ' ( x _ { n - 1 } ) = \\sum _ { j = n } ^ { P } \\int _ { I _ j } \\omega _ { 1 - \\alpha } ( s - x _ { n - 1 } ) u ' ( s ) \\ , d s \\\\ & = \\int _ { I _ P } \\omega _ { 1 - \\alpha } ( s - x _ { n - 1 } ) u ' ( s ) \\ , d s + \\sum _ { j = n + 1 } ^ { P } \\int _ { I _ j } \\omega _ { 1 - \\alpha } ( q - x _ { n } ) u ' ( q - h ) \\ , d q , \\end{aligned} \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} \\begin{cases} i \\dd _ t \\psi ( t ) = A \\psi ( t ) + u ( t ) \\mu \\psi ( t ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ t \\in ( 0 , T ) , \\\\ \\psi ( 0 ) = \\psi ^ 0 \\in L ^ 2 ( ( 0 , 1 ) , \\C ) . \\\\ \\end{cases} \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} \\mathbb E \\| N _ { \\infty } - N ^ n _ { \\infty } \\| ^ p & \\eqsim _ p \\mathbb E [ N - N ^ n ] _ { \\infty } ^ { \\frac p 2 } = \\mathbb E \\bigl ( [ N - N ^ n ] ^ c _ { \\infty } + [ N - N ^ n ] ^ q _ { \\infty } + [ N - N ^ n ] ^ a _ { \\infty } \\bigr ) ^ { \\frac p 2 } \\\\ & = \\mathbb E \\bigl ( [ N ] ^ c _ { \\infty } + [ N - N ^ n ] ^ q _ { \\infty } + [ N ] ^ a _ { \\infty } \\bigr ) ^ { \\frac p 2 } \\geq \\mathbb E \\bigl ( [ N ] ^ c _ { \\infty } + [ N ] ^ a _ { \\infty } \\bigr ) ^ { \\frac p 2 } , \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} \\mathbb { E } T _ { n + m } \\leq \\mathbb { E } T _ n + \\mathbb { E } T _ { n , m + n } = \\mathbb { E } T _ n + \\mathbb { E } T _ m , \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} [ L _ { - 3 } , \\ , T _ j ] = 0 , \\ , [ L _ { - 4 } , \\ , T _ j ] = 0 , \\ , \\cdots , [ L _ { - j } , \\ , T _ j ] = 0 , \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} \\nabla _ { i j } T = \\nabla _ { j i } T + R ^ \\perp _ { i j } T \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} - u ^ { \\left ( 2 \\right ) } \\left ( y \\right ) + A u \\left ( y \\right ) = \\bar { f } \\left ( \\tau \\right ) , \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} \\pi _ 1 ( g ^ { - ( n + 1 ) } b g ^ { n + 1 } b ) = \\pi _ 1 ( g ^ { - ( n + 1 ) } b g ^ { n + 1 } ) \\pi _ 1 ( b ) = ( 3 , 4 ) ( 2 , 5 ) ( 1 , 2 ) ( 3 , 4 ) = ( 1 , 2 , 5 ) , \\end{align*}"}
-{"id": "1867.png", "formula": "\\begin{align*} \\tilde { R } ^ { \\tilde { \\nabla } } ( Z ) = R ^ \\nabla ( Z ) + i d d ^ c f \\cdot Z . \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} X _ \\theta ( 0 ) = 1 , \\forall i \\geq 1 , \\ ; X _ { \\theta } ( i ) = - \\theta \\sum _ { j = 0 } ^ { i - 1 } \\binom { i - 1 } { j } X _ \\theta ( j ) . \\end{align*}"}
-{"id": "7021.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 0 } ^ { 2 } \\varepsilon ^ { \\frac { j } { 2 } } \\left \\vert \\lambda \\right \\vert ^ { 1 - \\frac { j } { 2 } } \\left \\Vert u _ { 2 } ^ { \\left ( j \\right ) } \\right \\Vert _ { X } + \\left \\Vert A u _ { 2 } \\right \\Vert _ { X } \\leq \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} u ( t , 0 , y ) = \\mu ( t , y ) , ( t , y ) \\in B _ T . \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} [ a , b ] : = \\{ x \\in L \\mid a \\leq x \\leq b \\} . \\end{align*}"}
-{"id": "3898.png", "formula": "\\begin{align*} \\begin{array} { l } \\int _ { \\frac { 1 } { 2 } } ^ b 2 A _ 3 Q ( x ) \\cos 2 k ( x - \\frac { 1 } { 2 } ) d x = \\int _ 0 ^ { b - \\frac { 1 } { 2 } } 2 A _ 3 Q ( t + \\frac { 1 } { 2 } ) \\cos ( 2 k t ) d t , \\end{array} \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{gather*} c _ 1 ( n , k ) = \\sum _ { 2 j _ 1 + 1 + 2 j _ 2 + 1 + \\cdots + 2 j _ k + 1 = n } a ^ { j _ 1 } \\cdot a ^ { j _ 2 } \\cdots a ^ { j _ k } \\\\ = a ^ { \\frac { n - k } { 2 } } \\sum _ { j _ 1 + j _ 2 + \\cdots + j _ k = \\frac { n + k } { 2 } } 1 = a ^ { \\frac { n - k } { 2 } } { \\frac { n + k } { 2 } - 1 \\choose k - 1 } . \\end{gather*}"}
-{"id": "9669.png", "formula": "\\begin{align*} \\rho _ { L } ( P ( \\xi , \\eta ) ) = \\rho _ { K } ( P ( \\pi - \\xi , \\pi - \\eta ) ) = \\rho _ { K } ( P ( \\xi , - \\eta ) ) . \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{align*} - ( \\Delta f ) g _ { i j } + \\nabla _ { i } \\nabla _ { j } f - f R _ { i j } = \\kappa g _ { i j } . \\end{align*}"}
-{"id": "8822.png", "formula": "\\begin{align*} S _ m \\leq \\binom { n - 1 } { m } \\frac { 1 } { n ! } \\lambda ^ m \\left ( \\prod _ { j = 0 } ^ { m - 1 } ( n - 2 j ) \\right ) ( n - 2 m ) ! = \\big ( 1 + O ( d ^ 2 m ^ 2 / n ) \\big ) \\frac { \\lambda ^ m } { m ! } . \\end{align*}"}
-{"id": "2631.png", "formula": "\\begin{align*} B \\phi : = b _ { 1 } \\phi - ( b _ { 2 } \\phi ' ) ' . \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} 0 = a _ 0 < a _ 1 < \\ldots < a _ m = 1 , 0 = b _ 0 < b _ 1 < \\ldots < b _ n = 1 \\end{align*}"}
-{"id": "6044.png", "formula": "\\begin{align*} v ^ { x } _ i = - \\frac { ( \\alpha _ { i } - \\alpha _ 1 ) ( \\alpha _ { i } - \\alpha _ 2 ) \\cdots ( \\alpha _ { i } - \\alpha _ { \\check i } ) \\cdots ( \\alpha _ { i } - \\alpha _ n ) } { ( \\alpha _ { i } - \\lambda _ 1 ) ( \\alpha _ { i } - \\lambda _ 2 ) \\cdots ( \\alpha _ { i } - \\lambda _ { n + 3 } ) } \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} \\sum _ { n , m = 1 } ^ d V ( e _ n , e _ m ) W ( e _ n ^ * , e _ m ^ * ) \\end{align*}"}
-{"id": "8485.png", "formula": "\\begin{align*} P _ p ( x , \\zeta ) = \\sum _ { m = 0 } ^ { \\infty } Z ^ p _ m ( x , \\zeta ) = \\frac { 1 - | x | ^ { 2 p } } { ( x ^ 2 \\overline { \\zeta } ^ 2 - 2 x \\cdot \\overline { \\zeta } + 1 ) ^ { n / 2 } } \\quad \\textrm { f o r } x \\in \\widehat { B } _ p , \\ , \\ , \\zeta \\in \\widehat { S } _ p . \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{align*} \\rho _ { f , \\epsilon , \\phi } ^ T : = \\frac { 1 } { T ^ { k / 2 } } \\sum _ { j _ 1 \\neq \\ldots \\neq j _ k } \\sigma _ { j _ 1 } \\ldots \\sigma _ { j _ k } \\langle \\Lambda _ \\epsilon ^ f ( x ^ { j _ 1 } + \\xi ^ { j _ 1 } , \\ldots , x ^ { j _ k } + \\xi ^ { j _ k } ; T ) , \\phi \\rangle , \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} c _ \\Lambda ( \\lambda \\mu , \\nu ) = c _ \\Lambda ( \\lambda , \\nu ) c _ \\Lambda ( \\mu , \\nu ) \\qquad c _ \\Lambda ( \\lambda , \\mu \\nu ) = c _ \\Lambda ( \\lambda , \\mu ) c _ \\Lambda ( \\lambda , \\nu ) . \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} [ A _ 1 , \\ , [ A _ 1 , \\ , L _ { - q } ] ] \\subseteq [ L _ { - q + 1 } , \\ , A _ 1 ] = 0 \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} 1 + \\sum _ { d = 0 } ^ \\infty \\big ( ( d + 2 ) ^ a - ( d + 1 ) ^ a \\big ) F _ { d + 2 } ( n ) & = 1 + \\sum _ { i = 0 } ^ { a - 1 } \\left ( v _ i \\sum _ { d = 0 } ^ \\infty ( d + 2 ) ^ i F _ { d + 2 } ( n ) \\right ) \\\\ & \\rightarrow 1 + \\sum _ { i = 0 } ^ { a - 1 } \\left ( v _ i \\sum _ { d = 0 } ^ \\infty ( d + 2 ) ^ i F _ { d + 2 } ( \\infty ) \\right ) \\\\ & = \\sum _ { d = 0 } ^ \\infty ( d + 2 ) ^ a \\Pr { \\mathbf { Y } = d + 2 } , \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\widetilde \\psi \\equiv \\psi _ t + b \\psi _ x + \\psi _ { x x x } + \\psi _ { x y y } \\in C ^ \\infty ( \\overline { Q } _ T ) , \\\\ & \\psi \\in Y _ 0 ( Q _ T ) \\cap L _ 2 ( 0 , T ; W ^ 1 _ \\infty ) , \\\\ & \\psi \\big | _ { x = 0 } = \\mu _ 0 , \\psi \\big | _ { x = R } = \\nu _ 0 , \\| \\psi _ x \\big | _ { x = R } \\| _ { L _ 2 ( B _ T ) } \\leq c \\| \\nu _ 0 \\| _ { H ^ { 1 / 3 , 1 } ( B _ T ) } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{align*} \\mathbf { I I } & = \\int _ { \\partial B _ R ( 0 ) \\cap \\Omega _ + } A \\cdot N \\ , d S + \\int _ { \\partial B _ R ( 0 ) \\cap \\Omega _ - } A \\cdot N \\ , d S \\to \\frac { \\gamma _ n } { 2 } \\rho _ + p _ + \\cdot c - \\frac { \\gamma _ n } { 2 } \\rho _ - p _ - \\cdot c , \\end{align*}"}
-{"id": "9793.png", "formula": "\\begin{align*} \\int _ { B _ r ( p ) } \\langle e _ { \\epsilon } ( u ) I d - d u ^ * d u , \\nabla X \\rangle = \\int _ { \\partial B _ r } [ e _ { \\epsilon } ( u ) I d - d u ^ * d u ] ( X , \\nu ) , \\end{align*}"}
-{"id": "7081.png", "formula": "\\begin{align*} \\tilde { u } _ { \\varepsilon } \\left ( x , t \\right ) = e ^ { \\varepsilon t \\left ( \\Delta + A \\right ) } \\tilde { u } \\left ( x , t \\right ) t \\in \\left [ 0 , 1 \\right ] . \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} A ( n , \\delta n , w n ) \\ge \\frac { { n \\choose w n } } { \\sum _ { i = 0 } ^ { \\delta n / 2 - 1 } { w n \\choose i } { n - w n \\choose i } } . \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{gather*} ( K _ 0 - \\gamma q ^ { 2 i - d } I ) U _ i = 0 , ( K _ 1 - \\gamma q ^ { d - 2 i } I ) U _ i = 0 \\end{gather*}"}
-{"id": "262.png", "formula": "\\begin{align*} ( \\mathsf { P } ) _ { j } ^ { i * } = \\sum _ { m , n } \\overline { c _ { n } ^ { m } } ( \\mathsf { M } _ { m } ^ { n } ) _ { j } ^ { i * } = \\sum _ { m , n } \\overline { c _ { n } ^ { m } } ( \\mathsf { M } _ { n } ^ { m } ) _ { i } ^ { j } . \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} & S \\left ( \\tilde { \\Phi } ( \\hat { \\rho } _ A ) \\right ) - S ( \\hat { \\rho } _ A ) \\\\ & = S \\left ( \\tilde { \\Phi } ( \\hat { \\gamma } _ A ) \\right ) - S ( \\hat { \\gamma } _ A ) + S ( \\hat { \\rho } _ A \\| \\hat { \\gamma } _ A ) - S \\left ( \\left . \\tilde { \\Phi } ( \\hat { \\rho } _ A ) \\right \\| \\tilde { \\Phi } ( \\hat { \\gamma } _ A ) \\right ) \\\\ & \\ge S \\left ( \\tilde { \\Phi } ( \\hat { \\gamma } _ A ) \\right ) - S ( \\hat { \\gamma } _ A ) \\ ; , \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} \\bigg ( { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 \\alpha & \\frac 1 2 \\beta \\\\ & \\frac 1 2 + \\frac 1 2 ( \\alpha + \\beta ) \\end{matrix} \\bigg | \\ , z \\bigg ] \\bigg ) ^ 2 = { } _ 3 F _ 2 \\bigg [ \\begin{matrix} \\alpha & \\beta & \\frac 1 2 ( \\alpha + \\beta ) \\\\ & \\alpha + \\beta & \\frac 1 2 + \\frac 1 2 ( \\alpha + \\beta ) \\end{matrix} \\bigg | \\ , z \\bigg ] . \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} \\mathrm { d e g } ( \\mathrm { d i v } ( s ) ) = \\sum \\mathrm { z e r o e s } ( s ) - \\sum \\mathrm { p o l e s } ( s ) < 0 \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} \\alpha ( P ) = \\frac { \\alpha ( P _ w ) + \\alpha ( P _ b ) } 2 . \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} \\sum _ { d = 0 } ^ { \\infty } ( d + 1 ) ^ u G _ { d + 1 } ( n ) \\rightarrow \\sum _ { d = 0 } ^ { \\infty } ( d + 1 ) ^ u G _ { d + 1 } ( \\infty ) \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} T ( \\hat { \\pi } ^ { ( n ) } _ n ) = \\sum _ { i = 1 } ^ { L } t ( h _ i ) = \\sum _ { i = 1 } ^ { L } t ^ { ( n ) } ( h _ i ) = \\hat { T } ^ { ( n ) } ( \\hat { \\pi } ^ { ( n ) } _ n ) = \\hat { T } ^ { ( n ) } _ n . \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta ( \\widetilde \\eta - \\widetilde H ) = F _ 1 + F _ 2 + \\div G _ 1 + \\div G _ 2 + \\div ( a \\nabla \\widetilde H ) + \\div ( ( a - 1 ) \\nabla ( \\widetilde \\eta - \\widetilde H ) ) & \\Omega \\\\ \\widetilde \\eta - \\widetilde H = 0 & \\partial \\Omega \\end{cases} . \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} h ^ 0 ( F ( c - a - b ) ) = \\Biggl \\{ \\begin{array} { c c c } 1 , & \\mathrm { i f } \\ ( e , a ) \\ne ( 0 , 0 ) , \\\\ 2 , & \\ \\ \\ \\ \\mathrm { i f } \\ e = a = 0 , \\ b > 0 , \\\\ 3 , & \\ \\mathrm { i f } \\ e = a = b = 0 , \\end{array} \\end{align*}"}
-{"id": "9738.png", "formula": "\\begin{align*} \\partial _ t \\rho + \\nabla \\cdot P = 0 , \\ ; \\ ; \\ ; P = \\nabla \\cdot \\left ( \\frac { B \\otimes B } { \\rho } \\right ) \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} { c } _ k = \\sum _ { j = 1 } ^ { n - 1 } { \\sigma } _ j { t } _ j ^ k + \\delta _ { 2 n - 2 , k } { \\lambda } = \\sum _ { j = 1 } ^ n { \\varrho } _ j { x } _ j ^ k , k = 0 , \\dots , 2 n - 2 , \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{align*} \\frac { 1 } { N ^ 2 } \\sum \\limits _ { \\substack { x , d \\in \\mathbb { Z } \\\\ 1 \\leqslant x \\leqslant N } } \\mu ( x ) f _ 1 ( x + d ) \\Big ( \\prod \\limits _ { i = 2 } ^ { s + 1 } f _ i ( [ x + \\theta _ { i - 1 } d ] ) \\Big ) = o ( 1 ) \\end{align*}"}
-{"id": "2749.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N } \\frac { k ^ \\sigma K ^ \\sigma B ^ \\sigma } { d ^ \\sigma N ^ \\sigma } = O \\left ( \\frac { K ^ \\sigma B ^ \\sigma N } { d ^ \\sigma } \\right ) . \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} L _ 1 ( c _ m ^ - ) = L _ 1 ( c _ m ^ + ) = 0 L _ 1 ( c ) > 0 c _ m ^ - < c < c _ m ^ + . \\end{align*}"}
-{"id": "3176.png", "formula": "\\begin{align*} W ^ { \\infty } = \\underset { \\leftarrow } { \\lim } W ^ N = \\textnormal { E x } \\left ( \\underset { \\leftarrow } { \\lim } \\mathcal { M } _ p \\left ( W ^ N \\right ) \\right ) . \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} N ( \\psi , \\bar \\psi ) & = \\sum _ { l = 1 } ^ { r } \\hat N _ { l } ( x , \\psi , \\bar \\psi ) \\\\ & + N ( \\psi , \\bar \\psi ) - \\sum _ { l = 1 } ^ { r } \\hat N _ { l } ( x , \\psi , \\bar \\psi ) \\\\ & = : N ^ { ( 1 ) } ( \\psi , \\bar \\psi ) + N ^ { ( 1 , r ) } ( \\psi , \\bar \\psi ) , \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} g ' ( x , \\alpha ) ( \\ell _ { \\Psi ^ * F } ) ' ( g ( x , \\alpha ) , \\alpha ) = \\alpha \\cdot ( \\ell _ F ) ' ( x , \\alpha ) + h ( x , \\alpha ) \\cdot \\langle 2 \\rangle _ F ( \\alpha ) \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} \\lim _ n \\frac { \\mathbb { E } T _ n } { n } = \\inf _ { n \\geq 1 } \\frac { \\mathbb { E } T _ n } { n } = : \\mu _ F . \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\frac { B _ { n } } { n ! } z ^ { n } = \\frac { z } { e ^ { z } - 1 } . \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{align*} \\Xi ^ { \\lambda } ( K _ { a } \\otimes F _ { a } \\otimes E _ { a } K _ { a } ^ { - 1 } \\otimes K _ { a } ) = \\sum _ { i , j , m , n , o , p } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) \\pi ( K _ { a } ) _ { m } ^ { j } c _ { n } ^ { m } \\pi ( F _ { a } E _ { a } K _ { a } ^ { - 1 } ) _ { o } ^ { n } c _ { p } ^ { o } \\pi ( K _ { a } K _ { \\lambda } ) _ { i } ^ { p } . \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} \\langle \\Delta ( \\eta , \\xi ; T ) , \\phi \\rangle = \\int _ 0 ^ T \\int _ 0 ^ T \\phi ( \\eta _ u ) | \\xi _ v - \\eta _ u | ^ { \\gamma - 1 } d u d v , \\phi \\in \\mathcal { S } . \\end{align*}"}
-{"id": "3534.png", "formula": "\\begin{align*} g = - d t ^ { 2 } + a ^ { 2 } ( t ) ( d x ^ { 1 } ) ^ { 2 } + b ^ { 2 } ( t ) \\left [ ( d x ^ { 2 } ) ^ { 2 } + ( d x ^ { 3 } ) ^ { 2 } \\right ] \\end{align*}"}
-{"id": "4951.png", "formula": "\\begin{align*} M = \\begin{pmatrix} 0 & 0 & 1 & 0 & 1 \\\\ 0 & 0 & 0 & 1 & 1 \\\\ 1 & 0 & 0 & 1 & 0 \\\\ 0 & 1 & 1 & 0 & 0 \\\\ 1 & 1 & 0 & 0 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{align*} \\nabla ^ \\perp \\cdot \\widetilde { S } H = \\nabla ^ \\perp \\widetilde { S } H _ 2 \\cdot \\nabla H _ 1 + \\nabla ^ \\perp H _ 2 \\cdot \\nabla \\widetilde { S } H _ 1 . \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{align*} W _ 1 ^ { ( \\alpha ) } = \\{ ( \\vec { x } , y ) : \\vec { x } \\in F ^ { ( \\alpha ) } , \\ , y \\in ( - 1 , 0 ) \\} \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{align*} \\left [ \\prod _ { j = 1 , j \\ne q } ^ { k } ( S - e ^ { i \\theta _ j } ) ^ { m _ j } \\right ] ( S - e ^ { i \\theta _ q } ) ^ { m _ q } \\beta ^ { ( q ) } \\in \\ell ^ 2 \\end{align*}"}
-{"id": "634.png", "formula": "\\begin{align*} { } _ 2 F _ 1 ( \\alpha , \\beta ; \\gamma ; z ) : = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( \\alpha ) _ k ( \\beta ) _ k } { ( \\gamma ) _ k } \\ , \\frac { z ^ k } { k ! } , \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{align*} \\mathrm { o r d } _ { [ x ] } \\mathrm { D i f f } _ { C } ( B ) = \\frac { m - 1 } { m } + \\sum _ j \\frac { b _ j } { m } \\mathrm { o r d } _ { [ x ] } ( m B _ j ) | _ C \\in D ( I ) , \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} \\frac { 1 } { N ^ { h - m } } \\sum \\limits _ { \\mathbf { n } \\in \\mathbb { Z } ^ h } \\Big ( \\prod \\limits _ { j = 1 } ^ d f _ j ( \\xi _ j ( \\mathbf { n } ) + \\widetilde { \\mathbf { r } } _ j ) \\Big ) F ( \\mathbf { n } ) H ( L \\mathbf { n } ) + E \\end{align*}"}
-{"id": "1002.png", "formula": "\\begin{align*} h ^ { 1 , 1 } ( B ) = 1 0 - K ^ 2 . \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{align*} 4 z '' ( t ) + 4 ( \\gamma - 1 ) z ' ( t ) - \\beta z ( t ) + | z ( t ) | ^ \\alpha z ( t ) = 0 . \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} X = \\big ( C _ 1 \\times C _ 2 \\times C _ 3 \\big ) / \\mathbb Z _ 5 ^ 2 \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} P ( h ) : = \\bigl \\{ f \\in R \\ , \\big | \\ , \\exp ( h ) f \\ ; \\mbox { \\rm i s } \\ ; ( \\Pi , \\underline { \\mu } ) - \\mbox { \\rm q u a s i - - i n v a r i a n t } \\bigr \\} = B \\bigl ( \\Upsilon ( h ) \\bigr ) . \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} \\begin{matrix} \\geq 2 & \\geq 2 & \\geq 2 \\\\ \\geq 1 & \\geq 1 & \\geq 1 \\\\ \\geq 1 & \\geq 1 & = 0 \\end{matrix} \\begin{matrix} \\geq 2 & \\geq 1 & \\geq 1 \\\\ \\geq 2 & \\geq 1 & \\geq 1 \\\\ \\geq 2 & \\geq 1 & = 0 \\end{matrix} \\begin{matrix} \\geq 3 & \\geq 2 & \\geq 1 \\\\ \\geq 2 & \\geq 1 & \\geq 1 \\\\ \\geq 1 & \\geq 1 & = 0 \\end{matrix} \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{align*} \\sum _ { l > n } C _ { l } \\ , ( b _ { l + 1 } - b _ { l } ) ^ { 1 - \\frac 1 p } \\Bigl ( \\ \\sup _ { b _ { l + 1 } - 2 ( b _ { n + 1 } - b _ n ) \\le i < b _ { l + 1 } } \\prod _ { s = i + 1 } ^ { b _ { l + 1 } - 1 } | w _ s | \\Bigr ) \\leq K . \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} \\Omega = \\mathcal W _ { r _ 1 } \\cup \\mathcal W _ { r _ 2 } , r _ 1 , r _ 2 > 0 \\mathcal W _ { r _ { 1 } } \\cap \\mathcal W _ { r _ { 2 } } = \\emptyset . \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} \\tau = \\inf \\{ { t \\geq 1 } : { \\cal N } _ t = \\emptyset \\} \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} \\sigma : = \\min ( s , s '' ) . \\end{align*}"}
-{"id": "4280.png", "formula": "\\begin{align*} \\begin{cases} f _ { n } ( X ) = \\Psi ^ \\prime _ n ( X , Y ) , n \\textrm { o d d } ; \\\\ f _ { n } ( X ) = \\Psi ^ \\prime _ n ( X , Y ) / Y , n \\textrm { e v e n } , \\end{cases} \\end{align*}"}
-{"id": "9309.png", "formula": "\\begin{align*} \\varphi '' ( p ) \\begin{cases} > 0 & | p - 1 / 2 | < c , \\\\ = 0 & | p - 1 / 2 | = c , \\\\ < 0 & | p - 1 / 2 | > c , \\end{cases} \\end{align*}"}
-{"id": "3454.png", "formula": "\\begin{align*} c _ i ( j ) = \\begin{cases} \\frac { 2 ^ { ( 1 - i ) j } F ( 2 ^ { j - i } ) } { 2 ^ { ( 1 - j ) j / 2 - 1 } j ! } & 0 \\leq j < i j - i \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{align*} R _ t ( w ) = \\ 1 ( N _ t ( w ) = 0 ) + \\sum _ { v \\in \\C _ t ( w ) } 1 / N _ t ( v ) \\end{align*}"}
-{"id": "5353.png", "formula": "\\begin{align*} c _ m ( n , k ) = \\sum _ { i = k } ^ n ( m - 1 ) ^ { i - k } { i - 1 \\choose k - 1 } c _ 1 ( n , i ) , \\ ; ( 1 \\leq k \\leq n ) . \\end{align*}"}
-{"id": "978.png", "formula": "\\begin{align*} \\frac { 1 } { m } - \\frac { 1 } { p } = 1 - \\frac { ( \\nu - 1 ) ( d - 2 \\gamma ) } { 8 } = : \\theta > 0 . \\end{align*}"}
-{"id": "8555.png", "formula": "\\begin{align*} \\mathrm { ( I V ) } ^ N = 0 , \\end{align*}"}
-{"id": "7276.png", "formula": "\\begin{align*} [ e _ 1 , e _ 1 ] = \\alpha _ 1 e _ 3 + \\alpha _ 2 e _ 4 + \\alpha _ 3 e _ 5 , [ e _ 1 , e _ 2 ] = \\alpha _ 4 e _ 3 + \\alpha _ 5 e _ 4 + \\alpha _ 6 e _ 5 , [ e _ 2 , e _ 1 ] = \\beta _ 1 e _ 3 + \\beta _ 2 e _ 4 + \\beta _ 3 e _ 5 , \\\\ [ e _ 2 , e _ 2 ] = \\beta _ 4 e _ 3 + \\beta _ 5 e _ 4 + \\beta _ 6 e _ 5 , [ e _ 1 , e _ 3 ] = \\gamma _ 1 e _ 5 , [ e _ 2 , e _ 3 ] = \\gamma _ 2 e _ 5 , [ e _ 1 , e _ 4 ] = \\gamma _ 3 e _ 5 , [ e _ 2 , e _ 4 ] = \\gamma _ 4 e _ 5 . \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} V _ { n } ( \\mathbf { x | } K _ { n } ) = \\prod _ { j = 1 } ^ { n - 1 } \\prod _ { k = j + 1 } ^ { n } w _ { 2 } ( x _ { k } , x _ { j } | \\rho _ { k j } ) , \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} \\textrm { \\emph { V a r } } _ { B } [ X ] = k ( B ) \\textrm { \\emph { V a r } } [ X ] + \\left ( \\textrm { \\emph { V a r } } _ { B } [ Y ] - k ( B ) \\textrm { \\emph { V a r } } [ Y ] \\right ) \\beta \\beta ^ { T } , \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} \\nu ( x , \\Gamma ) = \\int _ U I _ { \\Gamma } ( c ( x , u ) ) M ( \\d u ) , \\end{align*}"}
-{"id": "3047.png", "formula": "\\begin{align*} \\abs { x } ^ 2 x _ i - x _ i ^ 3 + x _ i = H \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} \\ 1 _ { V \\times V ' } T \\otimes T ' = \\ 1 _ V T \\otimes \\ 1 _ { V ' } T ' . \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} \\varphi _ \\nu ( x ) = \\frac { \\sqrt { x } e ^ { x / 2 } } { \\sqrt { 2 \\pi } } \\int _ { ( 0 , 1 ] } e ^ { - \\tfrac { x } { 2 v } } v ^ { - 3 / 2 } \\nu ( d v ) . \\end{align*}"}
-{"id": "9863.png", "formula": "\\begin{align*} \\zeta ( \\rho ) = \\frac { J _ { \\frac { d } { 2 } } ( \\rho ) } { ( 2 \\pi \\sigma ^ 2 \\rho ) ^ { \\frac { d } { 2 } } } \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} S _ x ( M ) = F _ 2 ^ { \\sigma _ 2 } \\big / \\bigl ( F _ 2 ^ { \\sigma _ 2 } \\cap F _ 2 \\bigr ) \\oplus F _ 0 ^ { \\sigma _ 0 } \\big / \\bigl ( F _ 0 ^ { \\sigma _ 0 } \\cap F _ 0 \\bigr ) . \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{align*} [ G ( k , 0 ) & + i G ' ( k , 0 ) ] E ^ { - 1 } [ G ( k , 0 ) ^ \\dag - i G ' ( k , 0 ) ^ \\dag ] \\\\ = & G ( k , 0 ) E ^ { - 1 } G ( k , 0 ) ^ \\dag + G ' ( k , 0 ) E ^ { - 1 } G ' ( k , 0 ) ^ \\dag \\\\ & - i [ G ( k , 0 ) E ^ { - 1 } G ' ( k , 0 ) ^ \\dag - G ' ( k , 0 ) E ^ { - 1 } G ( k , 0 ) ^ \\dag ] = I _ n \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} X ^ 3 + A X + B = y ^ 2 , \\end{align*}"}
-{"id": "3771.png", "formula": "\\begin{align*} b = ( a _ 1 , a _ 2 , \\cdots , a _ j , k j + 1 , k ( j + 1 ) + 1 , \\cdots , k ( n - 2 ) + 1 , k ( n - 1 ) + 1 ) \\end{align*}"}
-{"id": "6152.png", "formula": "\\begin{align*} [ [ L _ { - r + 1 } , \\ , S _ { r - 1 } ] , \\ , [ L _ { - 2 } , \\ , S _ r ] ] = 0 \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} ( \\psi _ l , \\bar \\psi _ l ) & : = ( \\Pi _ N \\psi , \\Pi _ N \\bar \\psi ) , \\\\ ( \\psi _ h , \\bar \\psi _ h ) & : = ( ( i d - \\Pi _ N ) \\psi , ( i d - \\Pi _ N ) \\bar \\psi ) . \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{align*} w ^ { ( l ) } _ i = v ^ { x } _ i \\sum _ { j = 1 } ^ { n + 3 } \\lambda _ j ^ l ( \\alpha _ i - \\lambda _ j ) ^ { - 1 } . \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} | c \\alpha + d | ^ 2 & = ( c \\alpha + d ) ( c \\bar { \\alpha } + d ) \\\\ & = c ^ 2 \\alpha \\bar { \\alpha } + c d ( \\alpha + \\bar { \\alpha } ) + d ^ 2 \\\\ & = c ^ 2 \\left ( - \\frac { b } { c } \\right ) + c d \\ , \\frac { a - d } { c } + d ^ 2 \\\\ & = - b c + a d - d ^ 2 + d ^ 2 \\\\ & = a d - b c \\\\ & = 1 . \\end{align*}"}
-{"id": "6177.png", "formula": "\\begin{align*} 0 = H ^ 0 ( G _ T , W _ { [ n - 1 ] } ) \\to H ^ 1 ( G _ T , Z _ { [ n ] } ) \\to H ^ 1 ( G _ T , W _ { [ n ] } ) \\to H ^ 1 ( G _ T , W _ { [ n - 1 ] } ) \\overset { \\delta } { \\to } H ^ 2 ( G _ T , Z _ { [ n ] } ) \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{align*} | v _ n | \\ , \\cdot \\sup _ { j \\in [ b _ { \\varphi ( n ) } , b _ { \\varphi ( n ) + 1 } ) } \\ \\Bigl ( \\prod _ { s = b _ { \\varphi ( n ) } + 1 } ^ { j } | w _ { s } | \\Bigr ) \\le C _ { n } \\quad \\hbox { f o r e v e r y $ n \\ge 0 $ , } \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} u ( x , y , t ) = u _ { c ( t ) } ( \\xi ) + \\tilde { u } ( \\xi , y , t ) , \\xi = x - 4 a ( t ) \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{align*} \\Vert d f _ p ^ n v ^ s _ 0 \\Vert & = c _ { 1 , n } ( p ) \\frac { D ( p ) } { D ( f ^ n ( p ) ) } \\\\ & \\leq C _ p | \\delta | ^ n , \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{align*} \\pi ( s _ v ) = p _ v v \\in \\Lambda ^ 0 \\pi ( s _ \\lambda ) = t _ \\lambda d ( \\lambda ) \\in E _ P . \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{align*} \\sum _ { \\alpha : i \\to j } [ h _ i ^ { - 1 } \\phi _ \\alpha ^ * h _ j , \\phi _ \\alpha ] = \\sum _ { i \\in Q _ 0 } \\theta _ i \\mathrm { p r } _ { E _ i } . \\end{align*}"}
-{"id": "3756.png", "formula": "\\begin{align*} f \\circ ( g \\circ x ) = \\sum _ { i < \\alpha } r _ { i } e ^ { \\gamma _ { i } \\circ ( g \\circ x ) } = \\sum _ { i < \\alpha } r _ { i } e ^ { ( \\gamma _ { i } \\circ g ) \\circ x } = \\left ( \\sum _ { i < \\alpha } r _ { i } e ^ { \\gamma _ { i } \\circ g } \\right ) \\circ x = ( f \\circ g ) \\circ x . \\end{align*}"}
-{"id": "5070.png", "formula": "\\begin{align*} \\chi ( \\theta + \\delta ) - \\chi ( \\theta - \\delta ) \\le \\pi \\left ( \\frac { 2 \\delta } { \\pi } + \\frac { 8 \\delta } { \\pi } \\right ) = 1 0 \\delta . \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{align*} \\frac { 1 } { p ' } = \\frac { 1 } { m } + \\frac { \\nu - 1 } { p } , \\frac { 1 } { q ' } = \\frac { 1 } { q } + \\frac { \\nu - 1 } { n } , \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} | \\sigma _ i ( \\alpha ) | = | X | ^ { 1 / s } , \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} \\mathbb W ^ u ( p ) = \\{ z \\in \\mathbb C ^ 2 | \\ , \\Vert f ^ { - n } ( z ) - f ^ { - n } ( p ) \\Vert \\to 0 \\} \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { a _ n q ^ n } { 1 \\pm q ^ n } & = \\frac { 1 } { ( \\mp q ; q ) _ { \\infty } } \\sum _ { n \\geq 1 } \\left ( \\sum _ { k = 1 } ^ n \\left ( s _ o ( n , k ) \\pm s _ e ( n , k ) \\right ) a _ k \\right ) q ^ n , \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{align*} p ( \\xi , \\tau ) = - \\tau ^ 2 + \\frac { \\xi ^ 2 } { 1 + a _ \\alpha | \\xi | ^ \\alpha } = - \\tau ^ 2 + h _ \\alpha ^ 2 ( \\xi ) , \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} \\kappa _ { ( K , \\widehat { K } ) } ( f ) : = \\begin{cases} 0 & \\xi _ { K } = \\emptyset \\\\ \\sum _ { x \\in \\xi _ { K } \\cap [ 0 , 1 ] } f ( \\phi ( { \\cal M } ^ { ( x , 0 ) } ) , \\phi ^ \\prime ( \\widehat { \\cal M } ^ { ( x , 0 ) } ) ) & 0 < \\# \\xi _ { K } < \\infty \\\\ \\infty & . \\end{cases} \\end{align*}"}
-{"id": "8279.png", "formula": "\\begin{align*} \\mathbb { T } _ { | \\mathcal { E } | ^ 2 } = \\begin{pmatrix} \\mathbb { I } _ { | \\mathcal { E } | } \\otimes m _ 1 \\\\ \\vdots \\\\ \\mathbb { I } _ { | \\mathcal { E } | } \\otimes m _ { | \\mathcal { E } | } \\end{pmatrix} , \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 _ { n - n _ D } & 0 _ { ( n - n _ D ) \\times n _ D } \\\\ 0 _ { n _ D \\times ( n - n _ D ) } & I _ { n _ D } \\end{bmatrix} \\begin{bmatrix} I _ { n - n _ D } + X _ 1 & X _ 2 \\\\ X _ 3 & I _ { n _ D } + X _ 4 \\end{bmatrix} = 0 , \\end{align*}"}
-{"id": "239.png", "formula": "\\begin{align*} u ^ i _ j = c ^ { \\Lambda } _ { f ^ i , v _ j } ( X ) = f ^ i ( X \\triangleright v _ j ) . \\end{align*}"}
-{"id": "8457.png", "formula": "\\begin{align*} - d X _ t = f ( t , X _ t , Z _ t , k _ t ) d t - Z _ t d W _ t - k _ t d \\tilde N _ t ; \\ ; X _ 0 = x . \\end{align*}"}
-{"id": "5485.png", "formula": "\\begin{align*} \\vartheta _ 2 ( z , \\tau ) & = \\sum _ { n \\in \\Z } w ^ { \\left ( n + \\frac 1 2 \\right ) } q ^ { \\frac 1 2 \\left ( n + \\frac 1 2 \\right ) ^ 2 } , \\vartheta _ 1 ( z , \\tau ) = - \\vartheta _ 2 \\left ( z + \\frac { 1 } { 2 } , \\tau \\right ) \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} [ L _ { - 1 } , \\ , M _ 1 ] = [ [ L _ { - 1 } , \\ , X ] , \\ , M _ 1 ] = [ [ L _ { - 1 } , \\ , M _ 1 ] , X ] \\subseteq X . \\end{align*}"}
-{"id": "4390.png", "formula": "\\begin{align*} \\pounds _ d ( x ) = \\sum _ { k = 1 } ^ { p - 1 } \\frac { x ^ k } { k ^ d } . \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{gather*} \\frac { t } { u } \\frac { { \\rm d } u } { { \\rm d } t } = - p _ 1 q _ 2 - t q _ 2 + \\theta ^ \\infty _ 1 - \\theta ^ \\infty _ 2 , \\frac { t } { v } \\frac { { \\rm d } v } { { \\rm d } t } = - t q _ 1 + \\theta ^ \\infty _ 1 - \\theta ^ \\infty _ 3 . \\end{gather*}"}
-{"id": "8984.png", "formula": "\\begin{align*} X _ v = X _ { \\tau _ t } + \\bigg ( \\sqrt { t - \\tau _ t } M ^ N _ \\tau , \\sqrt { t - \\tau _ t } M ^ T _ \\tau , A _ { \\tau _ t + \\tau ( t - \\tau _ t ) } - A _ { \\tau _ t } \\bigg ) , \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} \\nabla _ i A ^ i _ j = \\nabla _ i \\vec { H } = \\nabla _ i \\Big ( ( A ^ o ) ^ i _ j + \\frac 1 n g _ j ^ i \\vec { H } \\Big ) , \\end{align*}"}
-{"id": "9380.png", "formula": "\\begin{align*} \\Gamma ^ { A _ 1 , A _ 2 , \\ldots , A _ n } ( f _ 1 \\otimes \\cdots \\otimes f _ n ) & ( X _ 1 , \\ldots , X _ { n - 1 } ) = \\\\ & f _ 1 ( A _ 1 ) X _ 1 f _ 2 ( A _ 2 ) \\cdots f _ { n - 1 } ( A _ { n - 1 } ) X _ { n - 1 } f _ n ( A _ n ) . \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} \\omega ( \\xi ) \\dot \\omega ( \\xi ) = \\omega ( \\xi ) + K \\xi , \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{gather*} g _ 1 ( { \\bf u } ) = \\varphi _ 1 ( x ) \\left ( \\eta _ 1 \\dfrac { u _ 1 } { 1 + u _ 1 } + \\ , \\eta _ 2 \\dfrac { u _ 6 ^ { p _ 3 } } { 1 + u _ 6 ^ { p _ 3 } } \\right ) , \\\\ g _ 3 ( { \\bf u } ) = \\varphi _ 2 ( x ) \\left ( \\eta _ 3 \\dfrac { u _ 2 } { 1 + u _ 2 } + \\ , \\eta _ 4 \\dfrac { u _ 5 ^ { p _ 1 } } { ( \\lambda _ { 9 } + u _ 4 ) ( 1 + u _ 5 ^ { p _ 1 } ) } \\right ) , \\\\ g _ 4 ( { \\bf u } ) = - \\varphi _ 2 ( x ) \\ , \\eta _ 5 \\dfrac { u _ 4 ^ { p _ 2 } } { 1 + u _ 4 ^ { p _ 2 } } . \\end{gather*}"}
-{"id": "2819.png", "formula": "\\begin{align*} T ( \\tau _ 1 , \\tau _ 2 ) = 4 C _ 1 \\operatorname { v o l } ( \\mathcal { R } ) X ^ { \\frac { 1 } { 2 } } ( \\log X ) ^ 3 + O ( X ^ { \\frac { 1 } { 2 } } ( \\log X ) ^ 2 ) . \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} q ^ { d k } \\ , \\left ( \\max \\limits _ { r \\in \\mathbb F _ q ^ * } \\prod \\limits _ { j = 1 } ^ k \\left ( \\sum \\limits _ { { \\bf v } \\in S _ r } | \\widehat { E _ j } ( { \\bf v } ) | ^ k \\right ) ^ { \\frac { 1 } { k } } \\right ) \\lesssim q ^ { k \\alpha + d - 1 } \\left ( \\prod \\limits _ { j = 1 } ^ k | E _ j | \\right ) ^ { \\frac { 1 } { \\ell } } . \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} \\theta \\mapsto \\sum _ { i = 1 } ^ { n } \\ 1 _ { [ \\theta , \\theta + 1 ] } ( X _ { i } ) . \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} f ^ { \\varepsilon } ( x ) = \\mathop { \\rm s u p } _ { s \\in S } \\{ \\ell _ s ^ { \\varepsilon } ( x ) \\} , \\end{align*}"}
-{"id": "9881.png", "formula": "\\begin{align*} \\sum _ { | \\sigma _ 1 | + \\dots + | \\sigma _ s | = L } \\ \\prod _ { i = 1 } ^ { s } Q ( \\sigma _ i ) \\leq m \\times 1 \\leq 2 ^ L n ^ s . \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} X _ { a c } X _ { b c } = & q X _ { b c } X _ { a c } , & \\ & a > b , \\\\ X _ { a b } X _ { a c } = & q ^ { - 1 } X _ { a c } X _ { a b } , & \\ & b > c , \\\\ X _ { a c } X _ { b d } = & X _ { b d } X _ { a c } , & \\ & a > b , c > d , \\\\ X _ { a c } X _ { b d } = & X _ { b d } X _ { a c } + ( q - q ^ { - 1 } ) X _ { b c } X _ { a d } , & \\ & a > b , c < d . \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} x = \\sum _ { i < \\alpha } r _ { i } e ^ { \\gamma _ { i } } \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{align*} J = \\begin{pmatrix} \\begin{smallmatrix} J _ { n _ 1 } ( \\lambda _ { n _ 1 } , \\overline { \\lambda } _ { n _ 1 } ) & & & & & \\\\ & \\ddots & & & \\\\ & & J _ { n _ p } ( \\lambda _ { n _ p } , \\overline { \\lambda } _ { n _ p } ) & & & & \\\\ & & & J _ { n _ q } ( \\lambda _ { n _ q } ) & & \\\\ & & & & \\ddots & \\\\ & & & & & J _ { n _ r } ( \\lambda _ { n _ r } ) \\end{smallmatrix} \\end{pmatrix} , \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} \\dfrac { 1 + \\psi ( 1 - \\lambda ) - \\psi ( \\lambda ) } { 2 } f ( \\sigma ( t ) ) & = \\dfrac { 1 } { \\int _ a ^ b \\nu ( t ) \\Delta t } \\int _ a ^ b K ( s , t ) f ^ { \\Delta } ( \\sigma ( s ) ) \\Delta s + \\dfrac { 1 } { \\int _ a ^ b \\nu ( t ) \\Delta t } \\int _ a ^ b \\nu ( s ) f ( \\sigma ^ 2 ( s ) ) \\Delta s \\\\ & - \\dfrac { \\psi ( \\lambda ) f ( \\sigma ( a ) ) + \\left ( 1 - \\psi ( 1 - \\lambda ) \\right ) f ( \\sigma ( b ) ) } { 2 } . \\end{align*}"}
-{"id": "9794.png", "formula": "\\begin{align*} F _ { \\epsilon } ( u , p , r ) : = e ^ { \\Lambda r ^ 2 } r ^ { 2 - n } \\int _ { B _ r ( p ) } e _ { \\epsilon } ( u ) \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{align*} \\sum _ { \\substack { 1 \\leq i , j \\leq M \\\\ d ( v _ i , v _ j ) \\geq \\log m } } \\left | \\hat { \\mu } ( \\xi _ i - \\xi _ j ) \\right | ^ N & = \\sum _ { \\substack { 1 \\leq i , j \\leq M \\\\ d ( v _ i , v _ j ) \\geq \\log m } } e ^ { 2 c } m ^ { - 2 } \\left ( 1 + O \\left ( \\frac { \\log \\log m } { \\log m } \\right ) \\right ) \\\\ & \\leq M ^ 2 | \\hat { \\mu } ( \\xi ) | ^ { 2 N } \\left ( 1 + O \\left ( \\frac { \\log \\log m } { \\log m } \\right ) \\right ) . \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{align*} \\eta _ a ( C ( \\mathsf { P } _ i ) ) = \\chi _ a ( \\mathsf { P } _ i ) = q ^ { ( \\alpha _ a - 2 \\rho , \\omega _ i ) } [ d _ a ^ { - 1 } ( \\alpha _ a , \\omega _ i ) ] _ { q _ a } = \\delta _ { i a } q ^ { ( \\alpha _ a - 2 \\rho , \\omega _ a ) } . \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} U _ { t - 1 } = \\{ 1 , 2 , \\ldots , n \\} \\setminus \\left ( \\cup _ { i = 0 } ^ { t - 1 } S _ i \\right ) . \\end{align*}"}
-{"id": "1731.png", "formula": "\\begin{align*} f _ { w , m a x } & = \\frac { 1 } { 2 } \\rho C _ { D } A _ { x _ W } v _ { w - m a x } ^ 2 = K _ d v _ { w - m a x } ^ 2 \\\\ f _ { n , m a x } & = m \\frac { v _ c ^ 2 } { r _ { m i n } } \\\\ f _ { t , m a x } & = 0 \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} ( \\varphi ( f ) ) ( x ) = \\begin{cases} F _ x ( f ( x ) ) \\ x \\in d \\\\ f ( x ) \\ \\ \\ \\end{cases} \\end{align*}"}
-{"id": "2909.png", "formula": "\\begin{align*} \\sum _ { l = - r } ^ { r } \\dfrac { 1 } { \\omega _ { l } } \\ , \\biggl ( u _ { ( 2 r + 1 ) + l } & + \\sum _ { p \\ge 2 } \\dfrac { \\overline { \\lambda } ^ { p - 1 } } { \\omega _ { ( p - 1 ) ( 2 r + 1 ) + l } \\dots \\omega _ { ( 2 r + 1 ) + l } } u _ { p ( 2 r + 1 ) + l } \\\\ & + \\sum _ { p \\ge 1 } \\dfrac { \\omega _ { l - p ( 2 r + 1 ) } \\dots \\omega _ { l - ( 2 r + 1 ) } } { \\overline { \\lambda } ^ { p } } u _ { - p ( 2 r + 1 ) + l } \\biggr ) = 0 \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} f _ n ( T , U ) = \\mathop \\Pi _ { j = 1 } ^ { n + 1 } ( u _ j T - t _ j U ) , \\\\ g _ n ( X , Y ) = \\mathop \\Pi _ { j = 1 } ^ { n + 1 } ( y _ j X - x _ j Y ) . \\end{align*}"}
-{"id": "557.png", "formula": "\\begin{align*} \\sum _ { \\substack { d _ 1 + \\dots + d _ { n } = g - 1 \\\\ 0 \\leq d _ i \\leq a _ i + \\delta _ { i \\ell } ( a _ { n + 1 } - 1 ) } } \\prod _ { i = 1 } ^ { n } Q _ { d _ i + \\delta _ { i \\ell } } ( a _ i + \\delta _ { i \\ell } a _ { n + 1 } ) \\prod _ { i \\not = \\ell , n + 1 } \\psi _ i ^ { d _ i } D _ { i , \\ell } \\pi ^ * ( \\psi _ \\ell ^ { d _ \\ell } ) \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} \\tau ^ { \\varepsilon } _ { \\theta } : = \\inf \\{ t \\geq \\theta , \\ , \\ , Y _ t \\leq \\xi _ t + \\varepsilon \\} ; \\sigma ^ { \\varepsilon } _ { \\theta } : = \\inf \\{ t \\geq \\theta , \\ , \\ , Y _ t \\geq \\zeta _ t - \\varepsilon \\} . \\end{align*}"}
-{"id": "4914.png", "formula": "\\begin{align*} A = \\sum \\limits _ { i = 1 } ^ { k } \\gamma _ { i } A _ { i } . \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} \\sharp ( F _ * ^ e ( R ^ { \\sharp } ) , R ^ { \\sharp } ) = b ^ e [ ( \\frac { q - 1 } { 2 } ) ^ n + ( \\frac { q + 1 } { 2 } ) ^ n ] \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{align*} p ( v ) = \\{ K : v \\in K \\} . \\end{align*}"}
-{"id": "6300.png", "formula": "\\begin{align*} [ M ] = \\sum _ { \\lambda \\in X ^ + } ( M : \\Delta ( \\lambda ) ) [ \\Delta ( \\lambda ) ] , \\end{align*}"}
-{"id": "2018.png", "formula": "\\begin{gather*} A ( z ) = \\frac { A _ 0 } { z ^ { r + 1 } } + \\frac { A _ 1 } { z ^ { r } } + \\cdots \\in M _ m ( \\mathbb { C } ( \\ ! ( z ) \\ ! ) ) , \\end{gather*}"}
-{"id": "7870.png", "formula": "\\begin{align*} R _ { 1 , N } ( C , \\epsilon _ m ) : = \\sum _ { r = N + 1 } ^ { \\infty } b _ r e ^ { \\beta _ m r } = \\sum _ { r = N + 1 } ^ { \\infty } \\frac { T _ r e ^ { - r } } { ( r - 1 ) ! } e ^ { - ( \\delta - \\beta _ m ) r } \\leq \\sum _ { r = N + 1 } ^ { \\infty } \\frac { T _ r e ^ { - r } } { ( r - 1 ) ! } \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{align*} F ( y , \\tau ) = \\bigg ( | h _ y ( \\lambda _ 2 , \\lambda _ 3 ; s ) | | y | + K | \\lambda _ 1 | | y | + K | y | ^ 2 \\bigg ) ^ 2 . \\end{align*}"}
-{"id": "2777.png", "formula": "\\begin{align*} \\sum _ { 1 \\leqslant d \\leqslant K ^ \\frac { 1 } { 2 } b ^ { - \\frac { 1 } { 4 } } \\det ( \\Lambda ) ^ { - \\frac { 1 } { 2 } } B ^ { \\frac { 1 } { 4 } ( 2 - \\frac { 1 } { r } ) } } K b ^ \\frac { 1 } { 2 } B ^ { 1 - \\frac { 1 } { 2 r } } \\log ( B ) = O \\left ( \\frac { K ^ \\frac { 3 } { 2 } b ^ \\frac { 1 } { 4 } } { \\det ( \\Lambda ) ^ \\frac { 1 } { 2 } } B ^ { \\frac { 3 } { 4 } ( 2 - \\frac { 1 } { r } ) } \\log B \\right ) . \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} i e _ 1 = ( B _ n A _ n \\tau _ n ) ( \\xi _ n + r _ n \\vect { n } ( \\xi _ n ) ) . \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} v _ G ( w _ G ( a _ G ) ) = v ( x ) . \\end{align*}"}
-{"id": "3596.png", "formula": "\\begin{align*} G _ n = - \\frac { c } { 2 \\pi } G _ 0 \\frac { 1 } { \\sqrt { n ! } } \\left ( \\frac { - i } { \\gamma } \\right ) ^ { n } ( \\lambda / \\gamma ) J _ n ( 1 / \\gamma , - \\lambda / \\gamma ) , \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} , \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} , \\begin{pmatrix} 0 & 0 \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} \\frac { t ^ N \\langle 1 \\cdots M | \\mathcal { B } ^ \\prime ( z _ 1 ) \\cdots \\mathcal { B } ^ \\prime ( z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle } { t ^ N \\langle 1 \\cdots M | \\mathcal { B } ^ \\prime ( z _ 1 ) \\cdots \\mathcal { B } ^ \\prime ( z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle | _ { z _ 1 \\longleftrightarrow z _ 1 ^ { - 1 } } } = \\frac { ( t ^ \\prime z _ 1 + z _ 1 ^ { - 1 } ) } { ( z _ 1 + t ^ \\prime z _ 1 ^ { - 1 } ) } . \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} F ( x ) = \\sum _ { i = 1 } ^ d \\frac { A _ i } { x - a _ i } . \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} \\tilde { G } _ n = - \\frac { i c } { 2 \\pi \\gamma } \\sqrt { n ! } \\tilde { G } _ 1 \\psi _ n ^ 0 ( \\sqrt { 2 } / \\gamma , - \\lambda / \\gamma ) ~ , ~ \\mbox { w i t h } ~ \\psi _ n ^ 0 ( \\xi , \\lambda ) = \\frac { 1 } { n ! } \\left ( \\frac { i \\xi } { \\sqrt { 2 } } \\right ) ^ { n } J _ n ( | \\xi | / \\sqrt { 2 } , \\lambda ) . \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} c _ { ( 1 ) , \\lambda } ^ \\mu = \\begin{cases} \\tilde { c } _ { ( 1 ) , \\lambda } ^ \\mu & \\textrm { i f } \\lambda \\neq \\mu \\ / ; \\\\ \\tilde { c } _ { ( 1 ) , \\lambda } ^ \\lambda - C ( t ) = w _ 0 ( \\omega _ k ) - w _ \\lambda ( \\omega _ k ) & \\textrm { i f } \\lambda = \\mu \\ / , \\end{cases} \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 1 \\alpha _ 4 } { \\gamma _ 1 } y _ 5 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 1 \\alpha _ 6 } { \\gamma _ 1 } y _ 5 , [ y _ 2 , y _ 2 ] = \\frac { \\alpha ^ 2 _ 1 \\beta _ 2 } { \\gamma ^ 2 _ 1 } y _ 5 , [ y _ 1 , y _ 3 ] = \\beta _ 4 y _ 5 , \\\\ [ y _ 3 , y _ 1 ] = \\beta _ 6 y _ 5 , [ y _ 2 , y _ 3 ] = y _ 4 , [ y _ 3 , y _ 2 ] = \\frac { \\alpha _ 1 \\gamma _ 4 } { \\gamma _ 1 } y _ 5 , [ y _ 3 , y _ 3 ] = \\gamma _ 6 y _ 5 . \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{align*} \\sigma ^ i ( \\alpha ) s _ i = \\sum _ { j = 1 } ^ \\nu e _ j \\sigma ^ { i + k _ j } ( \\alpha ) , \\end{align*}"}
-{"id": "2693.png", "formula": "\\begin{align*} P _ { G , Y } ( v _ 1 , \\dots , v _ n , u _ 1 , \\dots , u _ n ) = \\begin{cases} \\sum _ { k = 0 } ^ { n } \\bar { Y } _ { 1 k } \\ , v _ 1 u _ k - S _ 1 = 0 \\\\ \\vdots \\\\ \\sum _ { k = 0 } ^ { n } \\bar { Y } _ { n k } v _ n u _ k - S _ n = 0 \\\\ [ 1 . 5 e x ] \\sum _ { k = 0 } ^ { n } Y _ { 1 k } \\ , \\ , u _ 1 v _ k - \\bar { S } _ 1 = 0 \\\\ \\vdots \\\\ \\sum _ { k = 0 } ^ { n } Y _ { n k } \\ , u _ n v _ k - \\bar { S } _ n = 0 \\end{cases} \\end{align*}"}
-{"id": "5477.png", "formula": "\\begin{align*} C _ { \\nu , r } \\boxtimes C _ { \\lambda , r ' } \\cong \\bigoplus _ { r '' = 1 } ^ { 2 k + 2 } { N _ { r , r ' } } ^ { r '' } C _ { \\lambda + \\nu , r '' } , \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} 0 = [ 0 , \\ , S _ r ] = [ [ L _ { - r - 1 } , \\ , S _ { r - 1 } ] , \\ , S _ r ] = [ [ L _ { - r - 1 } , \\ , S _ { r } ] , \\ , S _ { r - 1 } ] = [ L _ { - 1 } , \\ , S _ { r - 1 } ] \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} b _ { j } : = j n ^ { - 1 } + n ^ { - 1 } [ - 1 / 2 , 1 / 2 ] ^ { d } [ - 1 / 2 , 1 / 2 ] ^ { d } = \\underset { j \\in \\mathcal { D } _ { n } } { \\bigcup } b _ { j } \\ . \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} p - 2 + I _ { p - 1 } + \\frac { 1 } { p - 1 } \\sum _ { d = 1 } ^ { p - 1 } I _ { p - 1 , d } . \\end{align*}"}
-{"id": "8888.png", "formula": "\\begin{align*} { \\bf B } _ 4 = \\begin{pmatrix} 8 2 1 9 8 8 4 3 2 & 6 6 0 2 1 0 8 2 8 9 2 8 & 0 & 6 5 3 1 9 0 9 7 8 2 8 4 8 0 \\\\ 0 & 0 & 2 1 1 8 1 8 7 2 0 3 3 2 8 & 0 \\\\ 3 8 8 1 1 2 5 0 0 0 0 & 9 6 8 6 2 4 4 0 5 6 3 2 & 0 & 7 0 0 7 8 1 1 1 2 6 7 4 5 6 \\\\ 8 1 2 3 4 1 5 7 5 0 0 0 0 & 1 3 2 8 0 2 5 7 1 4 3 2 3 2 & 0 & 1 2 0 2 9 1 6 7 4 5 7 7 8 5 6 \\end{pmatrix} . \\end{align*}"}
-{"id": "4737.png", "formula": "\\begin{align*} u = V ( x ) \\sqrt \\omega \\cos ( \\omega t ) + 2 \\alpha \\sqrt \\omega \\sin ( \\omega t ) , \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{align*} \\mu ^ { s , N } _ { H P } ( d x ) & = c o n s t \\times \\Delta ^ 2 _ N ( x ) \\prod _ { j = 1 } ^ { N } ( 1 + i x _ j ) ^ { - s - N } ( 1 - i x _ j ) ^ { - \\bar { s } - N } d x _ j \\\\ & = c o n s t \\times \\Delta ^ 2 _ N ( x ) \\prod _ { j = 1 } ^ { N } ( 1 + x ^ 2 _ j ) ^ { - \\Re ( s ) - N } e ^ { 2 \\Im ( s ) A r g ( 1 + i x _ j ) } d x _ j . \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} X = \\big ( C _ 1 \\times \\ldots \\times C _ n \\big ) / G . \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} \\bar { A } _ { n , j , k } ( t , z ) & = ( - 1 ) ^ { n } t ^ { n + 1 } A _ { n , j , k } ( 1 / t , z ) \\\\ & = ( - 1 ) ^ { n } 2 ^ { 2 j + 1 } t ^ { n + 1 } \\left ( \\frac { 1 } { t } \\right ) ^ { j + 1 } \\left ( 1 + \\frac { 1 } { t } + z \\left ( 1 - \\frac { 1 } { t } \\right ) \\right ) ^ { k - j } \\left ( 1 + \\frac { 1 } { t } - z \\left ( 1 - \\frac { 1 } { t } \\right ) \\right ) ^ { n - j - k - 1 } \\\\ & = ( - 1 ) ^ { n } 2 ^ { 2 j + 1 } t ^ { j + 1 } ( 1 + t - z ( 1 - t ) ) ^ { k - j } ( 1 + t + z ( 1 - t ) ) ^ { n - j - k + 1 } . \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} S ( \\boldsymbol { x } ) = \\inf \\sum _ { k = - \\infty } ^ { \\infty } ( V ( x _ k , x _ { k + 1 } ) - V ( y _ 0 , y _ 0 ) ) , \\end{align*}"}
-{"id": "6879.png", "formula": "\\begin{align*} \\Delta \\mathbf { h } _ { n } = \\sum _ { i = 0 } ^ { n } \\mathbf { h } _ { i } \\otimes \\mathbf { h } _ { n - i } \\end{align*}"}
-{"id": "10047.png", "formula": "\\begin{align*} \\mathrm { T } ^ { 0 } ( J _ { \\varphi } X , J _ { \\varphi } Y ) + \\mathrm { T } ^ { 0 } ( X , Y ) = - \\frac { 1 } { 2 } N _ { J _ { \\varphi } } ( X , Y ) , \\forall X , Y \\in { \\mathfrak X } ( M ) , \\end{align*}"}
-{"id": "7652.png", "formula": "\\begin{align*} \\dfrac { 1 + \\psi ( 1 - \\lambda ) - \\psi ( \\lambda ) } { 2 } q ( t ) & = \\dfrac { 1 } { \\int _ a ^ b \\nu ( t ) \\Delta t } \\int _ a ^ b K ( s , t ) q ^ { \\Delta } ( s ) \\Delta s + \\dfrac { 1 } { \\int _ a ^ b \\nu ( t ) \\Delta t } \\int _ a ^ b \\nu ( s ) q ( \\sigma ( s ) ) \\Delta s \\\\ & - \\dfrac { \\psi ( \\lambda ) q ( a ) + \\left ( 1 - \\psi ( 1 - \\lambda ) \\right ) q ( b ) } { 2 } . \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} \\mathbf { f } _ { d } = m \\mathbf { g } + \\mathbf { R } _ { W I } \\mathbf { f } _ { w } + \\overbrace { \\mathbf { f } _ { n } + \\mathbf { f } _ { t } } ^ { m \\mathbf { \\ddot { p } } } \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} u ( z ) = \\int _ 0 ^ 1 t ^ { \\xi ^ 2 / 2 - \\lambda - 1 } \\frac { 1 } { \\sqrt { m ! } } \\left [ t \\left ( z - \\frac { i \\xi } { \\sqrt { 2 } } + i \\frac { \\xi } { \\sqrt { 2 } } \\right ) e ^ { ( 1 - t ) \\left ( \\frac { \\xi ^ 2 } { 2 } + \\frac { i \\xi } { \\sqrt { 2 } } z \\right ) } \\right ] ^ m d t . \\end{align*}"}
-{"id": "9487.png", "formula": "\\begin{align*} \\sum _ { T '' \\subsetneq T ' } f ( T '' ) = \\begin{cases} 1 & \\mbox { i f } T ' = T ; \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} { \\displaystyle \\left ( \\partial ^ { 2 } \\theta ^ { k } _ { \\mathbf { A } } , \\mathbf { v } \\right ) + D ( \\widetilde { \\theta ^ { k } _ { \\mathbf { A } } } , \\mathbf { v } ) = K _ 1 ^ { ( k ) } + K _ 2 ^ { ( k ) } + K _ 3 ^ { ( k ) } + K _ 4 ^ { ( k ) } . } \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} \\dot { \\bar x } = [ J ( \\bar x ) , 1 ] = - \\tfrac { 1 } { 2 } \\nabla J ( \\bar x ) . \\end{align*}"}
-{"id": "9761.png", "formula": "\\begin{align*} j u : = u ^ 1 d u ^ 2 - u ^ 2 d u ^ 1 = u ^ * ( r ^ 2 d \\theta ) , \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{align*} \\big ( A \\phi \\big ) ( x ) : = \\phi ' ( x ) \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} A _ { n + 1 } , _ { n + 1 } w _ { n + 1 } = s _ { n + 1 , n + 1 } v _ { n + 1 } . \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} \\alpha _ g = \\frac { I _ g ^ 2 - 4 } { I _ g ^ 2 } = 1 - \\frac { 4 } { I _ g ^ 2 } , \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{align*} \\Delta \\phi _ 2 = \\pi / 2 + \\sin ^ { - 1 } \\left ( \\frac { v _ o } { v _ c } \\right ) - \\Delta \\phi _ 1 \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} ( K + K ^ { - \\epsilon } ) ^ { - \\frac { \\epsilon } { p } + \\sum _ { i = 1 } ^ m \\frac { \\epsilon } { p _ i } } \\le C . \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{align*} { \\bf a } = \\binom { e ^ { t \\Delta } [ u _ 0 + t \\P \\theta _ 0 e _ 3 ] } { e ^ { t \\Delta } \\theta _ 0 } \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} c ( x ) = r _ { 0 } e ^ { c _ { 0 } ( \\lambda _ { 0 } ) } + \\sum _ { 1 \\leq i < \\beta } r _ { i } e ^ { c _ { 0 } ( \\lambda _ { i } ) } + s , \\end{align*}"}
-{"id": "6298.png", "formula": "\\begin{align*} \\| U \\| _ l = \\inf \\left \\{ \\| a \\| \\left ( \\sum _ { k = 1 } ^ n \\| u \\| ^ 2 \\| v \\| ^ 2 \\right ) ^ { \\frac { 1 } { 2 } } \\right \\} , \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{align*} \\tau _ \\pm ( x , v ) = \\inf \\{ s > 0 | x \\pm s v \\not \\in X \\} \\end{align*}"}
-{"id": "7657.png", "formula": "\\begin{align*} \\beta _ j = \\sum _ { i = 1 } ^ n b _ { i j } \\alpha ' _ i , \\end{align*}"}
-{"id": "7539.png", "formula": "\\begin{align*} \\dot { h _ 1 } / h _ 1 = \\frac { 1 } { 2 } h _ 2 / h _ 1 , \\dot { h _ 2 } / h _ 2 = - \\frac { 1 } { 2 } h _ 2 / h _ 1 \\end{align*}"}
-{"id": "4374.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty C _ k x ^ { k + 1 } = \\frac { 1 - \\sqrt { 1 - 4 x } } { 2 } . \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} \\Pi _ 1 ( \\varphi , \\upsilon ) & = \\sum _ { k = 2 } ^ \\infty S ^ { k - 2 } \\varphi \\cdot S _ k \\upsilon \\ , , \\Pi _ 2 ( \\varphi , \\upsilon ) = \\sum _ { k = 0 } ^ \\infty ( S _ { k - 1 } \\varphi + S _ k \\varphi + S _ { k + 1 } \\varphi ) \\cdot S _ k \\upsilon \\ , , \\\\ & \\mbox { a n d } \\Pi _ 3 ( \\varphi , \\upsilon ) = \\sum _ { j = 2 } ^ \\infty S _ j \\varphi \\cdot S ^ { j - 2 } \\upsilon = \\Pi _ 1 ( \\upsilon , \\varphi ) \\ , . \\end{align*}"}
-{"id": "9856.png", "formula": "\\begin{align*} \\mathcal { T } _ i ( f ) : = D _ i ^ { ( n ) } ( \\mathbf { z } ) f - \\mathbf { z } ^ { n _ { \\alpha _ i } \\alpha _ i ^ \\vee } c ^ { ( n ) } _ { s _ i } ( \\mathbf { z } ) s _ i \\cdot f . \\end{align*}"}
-{"id": "2761.png", "formula": "\\begin{align*} K ^ { 2 } b \\leqslant U ( \\alpha , \\varepsilon ) B ^ { \\frac { 4 } { 5 } ( \\frac { 1 } { r } - 1 ) - \\frac { 3 } { 5 } ( 2 - \\frac { 1 } { r } ) } , U ( \\alpha , \\varepsilon ) = ( 2 ^ { 2 1 } \\times 1 6 2 \\alpha ^ 2 \\varepsilon ^ 2 ) ^ { - \\frac { 2 } { 5 } } \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ^ p u ( x ) = 0 , & x \\in B , \\\\ \\Delta ^ j u ( x ) = f _ j ( x ) , & \\mbox { } x \\in S . \\end{cases} \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} a _ { 0 } + \\sum _ { n = 1 } ^ { \\infty } a _ { n } \\sum _ { k = 1 } ^ { \\infty } d _ { n , k } \\varepsilon ^ { k } = a _ { 0 } + \\sum _ { k = 1 } ^ { \\infty } \\sum _ { n = 1 } ^ { \\infty } a _ { n } d _ { n , k } \\varepsilon ^ { k } = ( P \\circ Q ) ( \\varepsilon ) . \\end{align*}"}
-{"id": "1189.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } i u _ { t } + ( - \\Delta ) ^ { \\alpha } u = \\pm | u | ^ 2 u , \\ , \\ , \\ , \\ , x \\in { \\mathbb { K } } , \\ , \\ , \\ , \\ , t \\in \\mathbb { R } , \\\\ u ( x , 0 ) = u _ 0 ( x ) \\in H ^ { s } ( \\mathbb { K } ) , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} M = \\{ u \\in W _ { 0 } ^ { 1 , p } ( \\Omega ) \\colon \\int _ { \\Omega } | u | ^ { p } d x = 1 \\} . \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{align*} \\frac { B _ { n } } { n ! } = \\sum _ { \\pi \\in \\mathcal { C } \\left ( n \\right ) } \\frac { \\left ( - 1 \\right ) ^ { \\vert \\pi \\vert } } { \\left ( \\pi + 1 \\right ) ! } = \\sum _ { \\pi \\in C \\left ( n \\right ) } \\frac { \\left ( - 1 \\right ) ^ { \\vert \\pi \\vert } } { \\vert \\pi \\vert + 1 } \\frac { 1 } { \\pi ! } . \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} \\sum _ { F \\in { \\cal F } _ s } ( - 1 ) ^ { | F | } Z ( \\Gamma _ F ) = \\binom { m - c _ 0 } { s } ( n - \\tfrac { 1 } { 2 } ( d + 1 ) ) ^ { m ' - s } \\left ( \\sum _ { F ' \\in { \\cal F } ' } ( - 1 ) ^ { | F ' | } Z ( \\Gamma ' _ { F ' } ) \\right ) . \\end{align*}"}
-{"id": "5718.png", "formula": "\\begin{align*} \\hat { \\gamma } = \\frac { \\exp \\left ( - \\frac { 1 } { 2 } \\sum _ { i , \\ , j = 1 } ^ { 2 n } \\left ( \\hat { R } ^ i - r ^ i \\right ) h _ { i j } \\left ( \\hat { R } ^ j - r ^ j \\right ) \\right ) } { \\mathrm { T r } \\exp \\left ( - \\frac { 1 } { 2 } \\sum _ { i , \\ , j = 1 } ^ { 2 n } \\left ( \\hat { R } ^ i - r ^ i \\right ) h _ { i j } \\left ( \\hat { R } ^ j - r ^ j \\right ) \\right ) } \\ ; , \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{align*} e ^ { ( n + 2 ) f } { s } ^ H _ { g _ { f , J } } = - \\sum _ { i , j = 1 } ^ n \\left ( e ^ { n f } H _ { i j } \\right ) _ { , i j } . \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} \\dot { \\bar x } = f ( \\bar x ) - \\alpha g ( \\bar x ) L _ g V ( \\bar x ) , \\end{align*}"}
-{"id": "3873.png", "formula": "\\begin{align*} f ( - 1 + h - a ) & = 2 + a - h + \\sqrt { 1 - b ^ 2 / a ^ 2 } ( h - a ) + \\sqrt { b ^ 2 - h ^ 2 b ^ 2 / a ^ 2 } , \\\\ f ( - 1 + h + a ) & = \\sqrt { ( a - 2 + h ) ^ 2 } + \\sqrt { 1 - b ^ 2 / a ^ 2 } ( h + a ) + \\sqrt { b ^ 2 - h ^ 2 b ^ 2 / a ^ 2 } . \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{align*} \\mathrm { c l } ^ { - 1 } ( O _ \\rho ) = \\bigcup _ { \\mathrm { c l } ( a ) = x } U _ \\rho ( a ) \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} | f | _ L : = \\sup \\left \\{ \\frac { | f ( x ) - f ( y ) | } { d ( x , y ) } \\ , : \\ , x , y \\in S , \\ x \\neq y \\right \\} . \\end{align*}"}
-{"id": "5846.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n ( - 1 ) ^ { k - 1 } F _ k { } ^ 4 = \\frac { F _ n F _ { n + 1 } ( ( - 1 ) ^ { n - 1 } L _ n L _ { n + 1 } + L _ 2 L _ 3 ) } { 1 5 } \\ , . \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{align*} G ( k , 0 ) J ( k ) = - i k ( U + I _ n ) , G ' ( k , 0 ) J ( k ) = k ( U - I _ n ) , \\end{align*}"}
-{"id": "171.png", "formula": "\\begin{align*} S _ t = \\{ \\textbf { x } \\in \\mathbb F _ q ^ d : x _ 1 ^ 2 + \\cdots + x _ d ^ 2 = t \\} . \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} a _ n = \\sum _ { k = n } ^ { \\infty } { k \\choose n } ( - 1 ) ^ k b _ k \\ ; \\ ; \\ ; \\mbox { \\rm i f a n d o n l y i f } \\ ; \\ ; \\ ; b _ n = \\sum _ { k = n } ^ { \\infty } { k \\choose n } ( - 1 ) ^ k a _ k . \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} b _ { m } ( 2 n ) = & \\sum _ { j = 0 } ^ { n } { 2 ( n - j ) + m - 1 \\choose m - 1 } b _ { m } ( j ) , \\\\ b _ { m } ( 2 n + 1 ) = & \\sum _ { j = 0 } ^ { n } { 2 ( n - j ) + m \\choose m - 1 } b _ { m } ( j ) . \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} { \\tilde Y } _ t = X ^ { f } _ t - X ^ { ' f } _ t t \\in [ 0 , T ] \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{align*} \\textrm { \\emph { V a r } } _ { B } [ X ] = \\textrm { \\emph { V a r } } [ X ] + \\left ( \\textrm { \\emph { V a r } } _ { B } [ Y ] - \\textrm { \\emph { V a r } } [ Y ] \\right ) \\beta \\beta ^ { T } , \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} K _ 2 : = K _ 1 ( p _ 1 ^ { 1 / 2 } , . . . , p _ \\nu ^ { 1 / 2 } ) , \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} p ' ( \\alpha \\cdot x ) = \\alpha \\cdot p ( x ) . \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} F _ { m } ( x ) & \\equiv \\prod _ { n = 0 } ^ { \\infty } \\left ( 1 + x ^ { 2 ^ { n } } \\right ) ^ { m } \\equiv \\left ( \\prod _ { n = 0 } ^ { \\infty } \\left ( 1 + x ^ { 2 ^ { n } } \\right ) \\right ) ^ { m } \\\\ & \\equiv \\frac { 1 } { ( 1 - x ) ^ { m } } \\equiv \\sum _ { n = 0 } ^ { \\infty } { n + m - 1 \\choose m - 1 } x ^ { n } \\pmod { 2 } . \\end{align*}"}
-{"id": "7619.png", "formula": "\\begin{align*} v _ \\epsilon ( \\rho , t ) & = v _ { 0 , \\epsilon } ( e ^ \\rho , t ) + a _ t \\rho \\\\ & = v _ { \\infty , \\epsilon } ( e ^ { - \\rho } , t ) + b _ t \\rho . \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} ( 2 \\beta - 1 ) ^ p \\pounds _ 0 \\left ( \\frac { \\beta } { 2 \\beta - 1 } \\right ) = \\frac { \\beta ( 2 \\beta - 1 ) ^ p - \\beta ^ p ( 2 \\beta - 1 ) } { ( \\beta - 1 ) } \\equiv \\frac { \\beta - \\beta ^ p } { ( 1 - \\beta ) } = \\pounds _ 0 ( \\beta ) \\pmod { p } \\end{align*}"}
-{"id": "8494.png", "formula": "\\begin{gather*} t : = \\frac { x \\cdot \\overline { \\zeta } } { | x | | \\overline { \\zeta } | } \\qquad \\textrm { a n d } w : = | x | | \\overline { \\zeta } | \\end{gather*}"}
-{"id": "7167.png", "formula": "\\begin{align*} 1 = \\mathrm { N o r m } _ { k ( \\tilde { z } ) / k } ( \\tilde { z } ) = \\tilde { z } ^ 2 \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{align*} T _ 5 ^ { - 1 } T _ 0 ^ { - 1 } J ( k ) ^ { - 1 } M _ 0 ^ { - 1 } T _ 3 ^ { - 1 } T _ 4 ^ { - 1 } = \\begin{bmatrix} \\frac { 1 } { k } ( I _ \\mu + o ( 1 ) ) & o ( 1 ) \\\\ o ( 1 ) & I _ { n - \\mu } + o ( 1 ) \\end{bmatrix} , \\ ; \\ ; k \\to 0 , \\ ; \\ ; k \\in \\overline { \\mathbb { C } } ^ + . \\end{align*}"}
-{"id": "9907.png", "formula": "\\begin{align*} \\dot { x } _ { 1 } & = x _ { 1 } ^ { 3 } + x _ { 2 } , \\\\ \\dot { x } _ { 2 } & = u . \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} ( q ^ + ) ^ 2 = ( q ^ - ) ^ 2 = 0 \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} \\lambda _ { k } = \\lambda _ { 1 } ( p , \\Omega ^ { + } _ { k } ) \\ge \\frac { C _ { n , F } } { | \\Omega ^ { + } _ { k } | ^ { \\frac p n } } , \\lambda _ { k } = \\lambda _ { 1 } ( p , \\Omega ^ { - } _ { k } ) \\ge \\frac { C _ { n , F } } { | \\Omega ^ { - } _ { k } | ^ { \\frac p n } } . \\end{align*}"}
-{"id": "9462.png", "formula": "\\begin{align*} \\rho _ { \\varepsilon } ( x ) = \\sum _ { \\vec { k } \\in \\mathbb { Z } ^ { 3 } } \\widetilde { \\rho } _ { \\varepsilon } ( x + \\vec { k } ) = \\sum _ { \\vec { k } \\in \\mathbb { Z } ^ { 3 } } \\frac { 1 } { \\varepsilon ^ { 3 } } \\widetilde { \\rho } \\left ( \\frac { x + \\vec { k } } { \\varepsilon } \\right ) . \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a & b \\pi ^ n \\\\ c & d \\end{bmatrix} \\in R ^ \\bullet \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} - \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log \\mu _ { n } ( O _ { 2 } ^ { 1 2 } ) \\geq \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { 2 } - \\liminf _ { n \\to \\infty } \\frac { 1 } { n } \\log \\mu _ { n } ( O ^ { i } _ { 1 } ) + 3 \\xi \\end{align*}"}
-{"id": "9647.png", "formula": "\\begin{align*} y ^ 2 = f ( x ) , \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{align*} \\mathrm { T r } _ A \\left [ \\hat { H } _ A \\ , \\hat { \\rho } _ A \\right ] = E _ 0 < \\infty \\ ; , S ( \\hat { \\rho } _ M ) < \\infty \\end{align*}"}
-{"id": "3170.png", "formula": "\\begin{align*} F _ { N + 1 } ( z _ 1 , \\cdots , z _ { N + 1 } ) = \\int _ { z _ 1 } ^ { z _ 2 } \\int _ { z _ 2 } ^ { z _ 3 } \\cdots \\int _ { z _ N } ^ { z _ { N + 1 } } \\Delta _ N ( y ) f ( y ) d y _ 1 \\cdots d y _ N \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} S ^ { \\chi , C } _ { C _ { \\lambda , r _ 1 } , C _ { \\mu , r _ 2 } } \\cdot e ^ { 2 \\pi i \\langle \\lambda , \\mu \\rangle } \\cdot \\sqrt { k / 2 } & = \\begin{cases} s _ { r , r ' } & \\lambda , \\mu \\in 2 L ' / L , \\\\ ( - 1 ) ^ r s _ { r , r ' } & \\lambda \\in 2 L ' / L , \\mu \\not \\in 2 L ' / L \\mu \\in 2 L ' / L , \\lambda \\not \\in 2 L ' / L , \\\\ ( - 1 ) ^ { r + r ' } s _ { r , r ' } & \\lambda , \\mu \\not \\in 2 L ' / L . \\end{cases} \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} \\forall u \\in D : \\ \\ \\langle p _ \\alpha ^ { ( u ) } : \\ \\alpha \\in S \\rangle = p . \\end{align*}"}
-{"id": "7831.png", "formula": "\\begin{align*} \\delta = \\delta ( C ) : = C - 1 - \\log { C } . \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} L : = \\begin{bmatrix} l _ { n _ l - 1 } & \\ldots & l _ 0 & 0 & 0 & \\ldots & 0 \\\\ 0 & l _ { n _ l - 1 } & \\ldots & l _ 0 & 0 & \\ldots & 0 \\\\ \\vdots & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots & \\vdots \\\\ 0 & \\ldots & 0 & l _ { n _ l - 1 } & \\ldots & l _ 0 & 0 \\\\ 0 & 0 & \\ldots & 0 & l _ { n _ l - 1 } & \\ldots & l _ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "6730.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { n } \\sum \\limits _ { i = 0 } ^ { 2 } \\left \\vert \\lambda \\right \\vert ^ { 1 - \\frac { i } { 2 } } \\left \\Vert x _ { k } ^ { i \\alpha _ { k } } \\frac { \\partial ^ { i } u } { \\partial x _ { k } ^ { i } } \\right \\Vert _ { L _ { p } \\left ( G ; E \\right ) } + \\left \\Vert A u \\right \\Vert _ { L _ { p } \\left ( G ; E \\right ) } \\leq M \\left \\Vert f \\right \\Vert _ { L _ { p } \\left ( G ; E \\right ) } . \\end{align*}"}
-{"id": "9959.png", "formula": "\\begin{align*} E _ j ( I - \\Pi _ j ) \\Pi _ i = 0 \\end{align*}"}
-{"id": "6045.png", "formula": "\\begin{align*} v ^ { f ( x ) } _ i = - \\frac { ( \\alpha _ { i } - \\alpha _ 1 ) ( \\alpha _ { i } - \\alpha _ 2 ) \\cdots ( \\alpha _ { i } - \\alpha _ { \\check i } ) \\cdots ( \\alpha _ { i } - \\alpha _ n ) } { ( \\alpha _ { i } - \\lambda _ 1 ' ) ( \\alpha _ { i } - \\lambda _ 2 ' ) \\cdots ( \\alpha _ { i } - \\lambda _ { n + 3 } ' ) } . \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} \\begin{pmatrix} \\Re P _ { 1 1 } + \\Re P _ { 1 2 } & \\Im P _ { 1 2 } - \\Im P _ { 1 1 } \\\\ \\Re P _ { 2 1 } + \\Re P _ { 2 2 } & \\Im P _ { 2 2 } - \\Im P _ { 2 1 } \\\\ \\Im P _ { 1 1 } + \\Im P _ { 1 2 } & \\Re P _ { 1 1 } - \\Re P _ { 1 2 } \\\\ \\Im P _ { 2 1 } + \\Im P _ { 2 2 } & \\Re P _ { 2 1 } - \\Re P _ { 2 2 } \\end{pmatrix} = \\{ 0 \\} \\end{align*}"}
-{"id": "9432.png", "formula": "\\begin{align*} \\frac { { \\rm d } } { { \\rm d } t } \\int _ { x \\in \\mathbb { T } ^ 3 } \\frac { \\big | \\widetilde { U } \\big | ^ 2 } { 2 h } + \\int _ { x \\in \\mathbb { T } ^ 3 } \\frac { \\widetilde { W } ^ { \\rm T } Q ( w ^ * ) \\widetilde { W } } { 2 h } + \\int _ { x \\in \\mathbb { T } ^ 3 } \\widetilde { W } \\cdot \\mathrm { L } ( w ^ * ) = 0 \\end{align*}"}
-{"id": "4659.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\frac { 2 } { N } K _ N \\left ( \\frac { 2 x } { N } , \\frac { 2 y } { N } ; w _ R \\right ) = \\frac { \\sin \\pi ( x - y ) } { \\pi ( x - y ) } \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} \\Delta ^ { - 1 } = \\frac { 1 } { 4 } \\left ( \\delta _ { ( 0 , 0 ) } - \\nu \\right ) ^ { - 1 } = \\frac { 1 } { 4 } \\sum _ { n = 0 } ^ \\infty \\nu ^ { * n } = G \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{align*} \\tau _ f = \\frac { 2 c _ 3 m c _ 1 v _ c } { f _ { p l a n a r } - K _ d ( v _ c + v _ { a i r } ) ^ 2 } \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} \\underline { g } ( \\cdot , y , z ) : = g ^ 1 ( \\cdot , y , 0 ) - ( f _ \\cdot + \\tilde f _ \\cdot ) - ( \\mu + \\tilde \\mu ) | y | - \\lambda | z | ^ \\alpha - \\tilde \\lambda | z | ^ { \\tilde \\alpha } \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} ( \\Lambda V _ X , d ) = ( \\Lambda ( x _ 5 ) , d = 0 ) , ( \\Lambda V _ F , d ) = ( \\Lambda ( z _ 2 , z _ 9 ) , d z _ 2 = 0 , d z _ 9 = z _ 2 ^ 5 ) . \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{align*} e ^ { t \\Delta } = ( 4 \\pi t ) ^ { - 3 / 2 } e ^ { - | \\cdot | ^ 2 / 4 t } * ( \\cdot ) , \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} a _ { i + 1 } = a _ i + k \\cdot \\frac { a _ i } { k ( i - 1 ) + 1 } \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} \\prod _ { \\mathfrak { P } \\mid p } \\left ( \\mathfrak { D } _ n \\right ) _ { \\mathfrak { P } } = ( 1 ) , \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{align*} \\Gamma ' = [ \\Gamma , \\Gamma ] \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{align*} P _ { 1 j } N _ { - , k _ j } ^ \\dag = N _ { - , k _ j } ^ \\dag , \\end{align*}"}
-{"id": "2763.png", "formula": "\\begin{align*} \\nu _ 1 ^ 2 \\leqslant \\nu _ 1 \\nu _ 2 \\leqslant 4 \\det ( \\Gamma ) = 4 | \\lambda _ 2 \\mu _ 1 - \\lambda _ 1 \\mu _ 2 | . \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{align*} \\theta ( \\mathsf { P } _ { j } ^ { i } ) = q ^ { - ( 2 \\rho , \\lambda _ { i } - \\lambda _ { j } ) } ( q ^ { ( 2 \\rho , \\lambda _ { n } - \\lambda _ { m } ) } ( \\mathsf { M } _ { m } ^ { n } ) _ { j } ^ { i } + q ^ { - ( 2 \\rho , \\lambda _ { n } - \\lambda _ { m } ) } ( \\mathsf { M } _ { n } ^ { m } ) _ { j } ^ { i } ) . \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} \\partial _ t M _ 1 - d _ { M _ 1 } \\Delta _ x M _ 1 = \\lambda _ { m M _ 1 } \\frac { m } { k _ { M _ 1 } + m } + \\lambda _ { L _ { o x } M _ 1 } \\frac { L _ { o x } } { K _ { M _ 1 } + L _ { o x } } \\ , M _ 1 - \\underbrace { \\lambda _ { M _ 1 F } \\frac { M _ 1 } { K _ F + M _ 1 } } _ { t r a n s f o r m a t i o n \\ , i n t o \\ , F } \\ - \\ \\beta _ { M _ 1 } M _ 1 , \\end{align*}"}
-{"id": "2891.png", "formula": "\\begin{align*} \\mathsf { c h } _ i ( \\gamma ) = \\tau _ { i - 2 } ( \\gamma ) \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} { \\cal I } ( t ) : = \\int _ 0 ^ t d r \\int _ { [ r , t ] ^ 2 } \\int _ { [ 0 , r ] ^ 2 } d u d s \\int _ { \\R ^ { 2 d } } F ( \\xi , u , s , r ) d \\xi < + \\infty . \\end{align*}"}
-{"id": "6967.png", "formula": "\\begin{align*} h _ k = \\sum _ { i = 0 } ^ { k } ( - 1 ) ^ { k - i } \\binom { d - i } { k - i } f _ { i - 1 } . \\end{align*}"}
-{"id": "2486.png", "formula": "\\begin{align*} f _ q ^ { \\gamma _ j } = ( f _ q - \\sum _ { l > j } f _ q ^ { \\gamma _ l } ) \\chi _ { E _ q ^ { \\gamma _ j } } . \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} \\big ( T ( t ) \\phi \\big ) ( x ) : = \\phi ( t + x ) , \\mbox { f o r e v e r y } \\phi \\in C _ B , \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{align*} \\iiint _ { \\Pi _ T ^ - } \\Bigl [ v ( \\phi _ t + b \\phi _ x + \\phi _ { x x x } + \\phi _ { x y y } ) + f \\phi \\Bigr ] \\ , d x d y d t = 0 . \\end{align*}"}
-{"id": "9063.png", "formula": "\\begin{align*} C _ G ( g ) ^ 0 / ( C \\cap C _ G ( g ) ^ 0 ) \\cong ( C _ G ( g ) / C ) ^ 0 = C _ { G / C } ( g C ) ^ 0 . \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} \\kappa ^ 2 ( s ) = \\frac { \\epsilon _ u \\epsilon _ n } { r ^ 2 } \\Big ( 1 - \\epsilon _ t \\ , J _ 1 ^ 2 ( s ) \\Big ) . \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} f ( t ) = t ( \\log t - 1 ) \\end{align*}"}
-{"id": "4296.png", "formula": "\\begin{align*} x ( [ \\ell ] P _ * ) = x _ * - \\frac { \\Psi _ { \\ell - 1 } ^ \\prime ( x _ * , 0 ) \\Psi _ { \\ell + 1 } ^ \\prime ( x _ * , 0 ) } { \\left ( \\Psi _ \\ell ^ \\prime ( x _ * , 0 ) \\right ) ^ 2 } , \\end{align*}"}
-{"id": "8481.png", "formula": "\\begin{align*} P _ p [ f ] ( x ) : = \\left \\langle f , P _ p ( \\cdot , x ) \\right \\rangle _ { \\widehat { S } _ p } = \\frac { 1 } { p } \\sum _ { j = 0 } ^ { p - 1 } \\int _ S f ( e ^ { \\frac { j \\pi i } { p } } \\zeta ) \\overline { P _ p ( e ^ { \\frac { j \\pi i } { p } } \\zeta , x ) } \\ , d \\sigma ( \\zeta ) \\end{align*}"}
-{"id": "229.png", "formula": "\\begin{align*} V _ { k j } ( h ) : = \\{ z | \\ , \\Im ( z / e _ k ) \\geq 0 , \\ , | \\Re ( z / e _ k ) + \\mu _ { k j } \\log | z | | \\leq h \\} . \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} y _ { n - 1 , n } = p _ { 2 n - 2 } b _ { n - 1 , n - 1 } = p _ { 2 n - 2 } = \\frac { 1 } { \\lambda } . \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) ^ n & = \\begin{pmatrix} 1 & n \\\\ 0 & 1 \\end{pmatrix} + \\begin{pmatrix} a n + \\frac { c ( n - 1 ) n } { 2 } & \\frac { ( a + d + c ( n - 1 ) ) ( n - 1 ) n } { 2 } - c \\sum _ { k = 0 } ^ { n - 1 } k ^ 2 + b n \\\\ c n & d n + \\frac { c ( n - 1 ) n } { 2 } \\end{pmatrix} p \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} c ( T X _ 1 ) & = ( 1 + E _ 1 ) \\frac { ( 1 + f ^ * Z _ 1 - E _ 1 ) ( 1 + f ^ * Z _ 2 - E _ 1 ) ( 1 + f ^ * Z _ 3 - E _ 1 ) } { ( 1 + f ^ * Z _ 1 ) ( 1 + f ^ * Z _ 2 ) ( 1 + f ^ * Z _ 3 ) } f ^ * c ( T X _ 0 ) \\\\ c ( T Y ) & = \\frac { ( 1 + E _ 1 ) ( 1 + f ^ * Z _ 1 - E _ 1 ) ( 1 + f ^ * Z _ 2 - E _ 1 ) ( 1 + f ^ * Z _ 3 - E _ 1 ) } { ( 1 + 3 H + 6 L - 2 E _ 1 ) ( 1 + f ^ * Z _ 1 ) ( 1 + f ^ * Z _ 2 ) ( 1 + f ^ * Z _ 3 ) } f ^ * c ( T X _ 0 ) \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} \\dot \\omega _ K = 1 0 ^ 4 \\ , y _ F \\ , y _ O \\ e ^ { - 9 0 0 / \\theta } . \\end{align*}"}
-{"id": "9680.png", "formula": "\\begin{align*} \\tilde { G } ( 0 , \\phi , \\pi ) = - \\tilde { G } ( 0 , \\pi - \\phi , 0 ) , \\tilde { H } ( 0 , \\phi , \\pi ) = - \\tilde { H } ( 0 , \\pi - \\phi , 0 ) , \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\frac { N } { R } = 2 . \\end{align*}"}
-{"id": "5310.png", "formula": "\\begin{align*} \\left \\{ { \\mathcal M { f _ { \\left . G _ K \\right | T } } } \\right \\} \\left ( s \\right ) = { \\rm E } \\left \\{ { \\left . { { G _ K ^ { s - 1 } } } \\right | T = t } \\right \\} \\triangleq \\phi \\left ( { \\left . s \\right | t } \\right ) . \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{align*} P L _ b ( s _ { v _ 0 } ) ( ( P ) ) & = R _ { y _ 1 } ( s _ { v _ 0 } ) \\cdots R _ { y _ r } ( s _ { v _ 0 } ) ( ( P ) ) \\\\ & = p _ b ^ * ( P ) + \\sum _ { i = 1 } ^ r \\sum _ { j = 1 } ^ { e _ i } \\left ( e _ i w _ 0 ^ { y _ i } + \\sum _ { j = 1 } ^ { e _ i - 1 } ( k - j ) ( w _ { - j } ^ { y _ i } + w _ j ^ { y _ i } ) \\right ) \\\\ & = p _ b ^ * ( ( P ) + v _ 0 ) = p _ b ^ * ( s _ { v _ 0 } ( ( P ) ) . \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} I ( \\tau , \\sigma ) : = \\xi _ { \\tau } \\textbf { 1 } _ { \\{ \\tau \\leq \\sigma \\} } + \\zeta _ { \\sigma } \\textbf { 1 } _ { \\{ \\sigma < \\tau \\} } . \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} { \\cal M } ^ { ( x , 0 ) } = { \\cal M } ^ { ( x , 0 ) } _ { ( K , \\widehat { K } ) } & : = { \\cal M } ( { \\cal S } ( \\widehat { \\pi } ^ { ( x , 0 ) } _ r , \\widehat { \\pi } ^ { ( x , 0 ) } _ l ) ) \\\\ \\widehat { { \\cal M } } ^ { ( x , 0 ) } = \\widehat { { \\cal M } } ^ { ( x , 0 ) } _ { ( K , \\widehat { K } ) } & : = { \\cal M } ( \\widehat { { \\cal S } } ( \\widehat { \\pi } ^ { ( x , 0 ) } _ r , \\widehat { \\pi } ^ { ( x , 0 ) } _ l ) ) . \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} 0 \\neq [ V _ { - 2 } , \\ , L _ 1 ] = [ V _ { - 2 } , \\ , [ V _ { - 2 } , \\ , [ V _ { - 2 } , \\ , L _ 5 ] ] ] , \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} A _ { i j } w _ { j } = \\begin{pmatrix} a _ { i j } & b _ { i j } \\\\ b _ { i j } & a _ { i j } \\end{pmatrix} v _ { j } \\mathbf { e = } \\begin{pmatrix} a _ { i j } + b _ { i j } \\\\ b _ { i j } + a _ { i j } \\end{pmatrix} v _ { j } = s _ { i j } v _ { j } \\mathbf { e } \\end{align*}"}
-{"id": "2910.png", "formula": "\\begin{align*} \\| ( T - A ) f _ { k } \\| < \\varepsilon \\quad { \\rm a n d } \\| ( T - A ) ^ { * } f _ { k } \\| < \\varepsilon \\qquad \\hbox { f o r e v e r y $ k = - r , \\dots , r $ . } \\end{align*}"}
-{"id": "4805.png", "formula": "\\begin{align*} & x _ 1 \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} + y _ 1 \\begin{pmatrix} 0 & 1 \\\\ 0 & 0 \\end{pmatrix} , \\\\ & x _ 2 \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} + y _ 2 \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , \\\\ & x _ 3 \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} + y _ 3 \\begin{pmatrix} 0 & \\epsilon \\\\ 1 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "2011.png", "formula": "\\begin{align*} \\mathbf { Q } _ { A _ { d B } } = \\mathbf { R } _ { A _ { d B } } \\circ ( \\boldsymbol { \\sigma } _ { A _ { d B } } \\boldsymbol { \\sigma } ^ T _ { A _ { d B } } ) , \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} \\beta = \\sum _ { i = 1 } ^ \\ell \\beta _ i . \\end{align*}"}
-{"id": "7979.png", "formula": "\\begin{align*} F _ 1 = 0 ( t \\geq T _ 0 ) \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} \\begin{pmatrix} g ^ + \\\\ g ^ - \\end{pmatrix} \\ ; = \\ ; \\begin{pmatrix} f ^ + \\\\ f ^ - \\end{pmatrix} + a \\ , S _ D ^ { - 1 } \\begin{pmatrix} \\Phi ^ + \\\\ \\Phi ^ - \\end{pmatrix} + \\frac { b } { \\gamma } \\begin{pmatrix} \\Phi ^ + \\\\ \\Phi ^ - \\end{pmatrix} \\gamma \\ , : = \\ , \\textstyle { \\frac { \\Gamma ( 2 B ) } { \\Gamma ( B ) } \\ , \\frac { 1 + \\nu + B } { 1 + \\nu } } \\end{align*}"}
-{"id": "9634.png", "formula": "\\begin{align*} e ^ { ( j ) } _ n = q ^ { ( j + 1 ) } _ n - q ^ { ( j ) } _ n + e ^ { ( j + 1 ) } _ { n - 1 } , \\ \\ q ^ { ( j ) } _ { n + 1 } = \\frac { e ^ { ( j + 1 ) } _ { n } } { e ^ { ( j ) } _ n } q ^ { ( j + 1 ) } _ { n } , \\ \\ j \\geq 0 , \\ n \\geq 1 , \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{align*} f _ * E ^ n = ( - 1 ) ^ { d + 1 } h _ { n - d } ( Z _ 1 , \\cdots , Z _ d ) Z _ 1 \\cdots Z _ d , \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{align*} n _ 1 \\geqslant \\frac { B ^ { \\frac { 1 } { r } - 1 } \\xi ( \\theta ) } { 1 6 K \\varepsilon b \\sqrt { \\det ( \\Lambda _ d ) } } = \\frac { B ^ { \\frac { 1 } { r } - 1 } } { 1 6 \\times 1 6 2 K \\varepsilon b ^ 2 \\det ( \\Lambda _ d ) ^ { \\frac { 3 } { 2 } } } , \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{align*} \\Phi \\left ( \\mathfrak { M } \\right ) = \\big \\{ \\left ( \\mathsf { e v a l } _ N \\circ \\pi _ N ^ { \\infty } \\right ) _ * \\mathfrak { M } \\ \\big \\} _ { N \\ge 1 } , \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} \\Delta ^ g ( \\mu ) = - \\frac { ( P ^ { n - 1 } \\mathbf { H } ) ' ( \\mu ) } { P ( \\mu ) ^ { n - 1 } } . \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} \\frac { 1 } { Z _ { I , X } } \\prod _ { j < k } ( t _ k - t _ j ) ^ 2 \\prod _ { j = 1 } ^ N \\rho _ { I , X } ( t _ j ) , N = \\# ( X \\cap I ) , \\end{align*}"}
-{"id": "3165.png", "formula": "\\begin{align*} \\det \\left ( \\phi ^ { ( i - 1 ) } ( x _ { n + 1 - j } ) \\right ) ^ n _ { i , j = 1 } \\ge 0 \\ , \\ n = 1 , 2 , \\cdots \\textnormal { a n d } x _ 1 < \\cdots < x _ n . \\end{align*}"}
-{"id": "9464.png", "formula": "\\begin{align*} \\nabla \\cdot B ^ { \\varepsilon } _ { 0 } = 0 . \\end{align*}"}
-{"id": "4387.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { q - 1 } \\binom { 2 k } { k } a _ k x ^ k \\equiv \\frac { 1 } { \\sqrt { 1 - 4 x } } \\sum _ { k = 0 } ^ { q - 1 } \\binom { 2 k } { k } b _ k \\left ( \\frac { - x } { 1 - 4 x } \\right ) ^ k \\pmod { ( x ^ q , p ) } . \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} \\frac { d } { d f } \\left ( \\frac { f } { p } \\right ) \\left [ 1 - \\left ( \\lambda _ { 2 } \\lambda _ { 3 } - \\lambda _ { 1 } \\right ) \\left ( \\frac { f } { p } \\right ) ^ { - 4 } \\right ] = 0 . \\end{align*}"}
-{"id": "3242.png", "formula": "\\begin{align*} e ^ { - c _ { N , s } t } \\hat { p } _ t ^ { ( N + 1 ) , s } ( x , y ) \\frac { \\left ( \\hat { m } _ s ^ { ( N + 1 ) } \\right ) ^ { - 1 } ( y ) } { \\left ( \\hat { m } _ s ^ { ( N + 1 ) } \\right ) ^ { - 1 } ( x ) } = p _ t ^ { ( N ) , s } ( x , y ) . \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} F _ * ^ e ( j u ^ { k + r } v ^ r ( f + u v ) ) = \\bigoplus _ { i \\in \\Delta _ e } f _ { ( i , j ) } F _ * ^ e ( i u ^ { k + r } v ^ { r } ) \\oplus F _ * ^ e ( j u ^ { k + r + 1 } v ^ { r + 1 } ) \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} I _ 2 ( d ) = \\frac { 1 } { 2 } \\sum _ { i , j } a _ { i j } | d _ { i j } + ( D u _ f ) _ { i j } | + \\frac { \\alpha } { 2 } \\sum _ { i , j } | b ^ k _ { i j } + ( D u ^ { k + 1 } ) _ { i j } - d _ { i j } | ^ 2 \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} c \\sqrt { c ^ 2 + \\lambda _ l } = c \\lambda _ l ^ { 1 / 2 } \\sqrt { 1 + \\frac { c ^ 2 } { \\lambda _ l } } & = c \\lambda _ l ^ { 1 / 2 } \\left ( 1 + \\frac { c ^ 2 } { 2 \\lambda _ l } + O \\left ( \\frac { 1 } { \\lambda _ l ^ 2 } \\right ) \\right ) , \\end{align*}"}
-{"id": "9289.png", "formula": "\\begin{align*} g \\otimes h = g h \\end{align*}"}
-{"id": "3273.png", "formula": "\\begin{align*} \\mathfrak { T } _ { W ^ { n + 1 } } = \\inf \\{ t > 0 : \\exists \\ 1 \\le i < j \\le n + 1 \\textnormal { s . t } X _ i ^ { ( n + 1 ) } ( t ) = X _ j ^ { ( n + 1 ) } ( t ) \\} = \\infty \\ \\ a . s . , \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} d ( x , y ) = \\begin{cases} | x _ 2 - y _ 2 | , & j = k j = 0 k = 0 , \\\\ 1 , & j \\ne k , \\ j , k \\ge 1 . \\end{cases} \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} N a \\sum _ { k = 1 } ^ { k _ - + 1 } \\frac { ( - 1 ) ^ { k + 1 } ( a ^ { - k } - { \\lambda } ^ { k } ) } { k } N ^ { ( \\alpha - 1 ) k } - 1 = N a \\sum _ { k = 0 } ^ { k _ - } \\frac { ( - 1 ) ^ { k } ( a ^ { - ( k + 1 ) } - { \\lambda } ^ { k + 1 } ) } { k + 1 } N ^ { ( \\alpha - 1 ) ( k + 1 ) } - 1 \\ , . \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} f ^ * = \\inf _ { x \\in \\mathbb R ^ n } f ( x ) , \\end{align*}"}
-{"id": "10002.png", "formula": "\\begin{align*} & W ( G _ { 2 } ( n , d , x - 1 , s ) ) - W ( G _ { 2 } ( n , d , x , s ) ) \\overset { } { \\leq } \\\\ & \\overset { } { \\leq } W ( G _ { 2 } ( n , d , x _ { 2 } ^ { \\max } - 1 , s ) ) - W ( G _ { 2 } ( n , d , x _ { 2 } ^ { \\max } , s ) ) \\overset { } { = } \\\\ & \\overset { } { = } 2 ( n - 5 ) + 2 . \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{align*} V ( x ) = - 2 \\frac { d } { d x } K ( x , x ) , x \\ge 0 , \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{align*} \\rho _ { \\psi _ \\kappa } ^ T : = \\frac { 1 } { T ^ { k / 2 } } \\sum _ { j _ 1 \\neq \\ldots \\neq j _ k } \\sigma _ { j _ 1 } \\ldots \\sigma _ { j _ k } \\langle \\Lambda ^ { ( k ) } ( x ^ { j _ 1 } + \\xi ^ { j _ 1 } , \\ldots , x ^ { j _ k } + \\xi ^ { j _ k } ; T ) , \\psi _ \\kappa \\rangle , \\end{align*}"}
-{"id": "1792.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } T ' ( t ) = A T ( t ) , \\ ; t \\in \\R \\\\ T ( 0 ) = I \\end{array} \\right . \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} C ( a \\ , | \\ , x ) : = \\frac { 1 } { 1 - \\exp \\left ( \\vartheta ^ \\star \\right ) } \\ , \\frac { 1 } { \\sqrt { 2 \\pi \\Lambda '' ( \\vartheta ^ \\star ) } } = \\frac { 1 } { 1 - a / x } \\ , \\frac { 1 } { \\sqrt { 2 \\pi a } } . \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} \\Psi ( X , Y ) & = \\int _ R f ( W , Z ) G ( X , Y , W , Z ) \\ , d W \\ , d Z , \\end{align*}"}
-{"id": "5761.png", "formula": "\\begin{align*} I ( B : Z | M ) _ { \\hat { \\sigma } _ { B M Z } ( t ) } = \\Delta _ { B | M } ( \\hat { \\rho } _ { B M } ) \\left ( \\frac { ( 1 - \\lambda ) ^ 2 \\ , t } { | 1 - \\eta | } \\right ) \\ ; , \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{align*} d X ( t ) = \\sqrt { 2 ( X ^ 2 ( t ) + 1 ) } d W ( t ) + ( \\beta X ( t ) + \\gamma ) d t , \\end{align*}"}
-{"id": "9253.png", "formula": "\\begin{align*} \\psi \\circ \\phi ( z ) & = a z + a b , \\\\ \\phi \\circ \\psi ( z ) & = a z + b , \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} ( 1 - t ) \\sum _ { k = 0 } ^ a \\frac { ( - a - p ) _ { p + k } ( \\beta ) _ { p + k } } { ( 1 ) _ { p + k } ^ 2 } \\cdot z ^ k \\equiv t ( 1 - z ) ^ a \\sum _ { k = 0 } ^ a \\frac { ( - a - p ) _ { p + k } ( 1 - \\beta ) _ { p + k } } { ( 1 ) _ { p + k } ^ 2 } \\cdot \\frac { z ^ k } { ( z - 1 ) ^ k } \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "3660.png", "formula": "\\begin{align*} L ^ \\lambda _ 0 = - \\sum _ { k \\in \\mathbb { Z } } \\left ( \\lambda + k \\right ) : \\chi _ { - 2 k + \\frac { 1 } { 2 } } \\chi _ { 2 k - \\frac { 1 } { 2 } } : . \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} u _ 1 ( z ) = \\frac { 1 } { 2 \\pi i } \\int _ { S ^ 1 } \\frac { u ( \\zeta ) } { \\zeta - z } d \\zeta = \\sum _ { n = 0 } ^ { \\infty } \\left ( \\frac { 1 } { 2 \\pi i } \\int _ { S ^ 1 } u ( \\zeta ) \\zeta ^ { - ( n + 1 ) } d \\zeta \\right ) z ^ n = \\sum _ { n \\in \\N } ^ { } c _ n z ^ n \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} \\mathcal W = \\{ \\xi \\in \\R ^ n \\colon F ^ o ( \\xi ) < 1 \\} \\end{align*}"}
-{"id": "7057.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A u = F \\left ( u , \\bar { u } \\right ) , x \\in R ^ { n } , t \\in \\left [ 0 , T \\right ] , \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{gather*} ( y _ 0 , y _ 1 , \\dots , y _ n ) \\mapsto \\left ( \\prod _ { i = 0 } ^ n y _ i ^ { b _ { 0 , i } } , \\prod _ { i = 0 } ^ n y _ i ^ { b _ { 1 , i } } , \\dots , \\prod _ { i = 0 } ^ n y _ i ^ { b _ { n , i } } \\right ) \\end{gather*}"}
-{"id": "5658.png", "formula": "\\begin{align*} ( - 3 v ^ 2 + 2 ( a + 1 ) v - a ) \\frac { d v } { d \\tau } = \\frac { d w } { d \\tau } = b v - c w , \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} ( 1 - { q ' } ^ { - m _ 0 } ) ( 1 - { q ' } ^ { - m _ 0 + 1 } ) \\cdots ( 1 - { q ' } ^ { - m _ 0 + m _ 7 - 1 } ) = 1 - O ( { q ' } ^ { - m _ 0 + m _ 7 - 1 } ) , \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{align*} \\dot x = \\Gamma v ( x ) . \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} \\biggl ( \\sum _ { n = 0 } ^ { \\infty } a _ { n } t ^ { n } \\biggr ) * \\biggl ( \\sum _ { n = 0 } ^ { \\infty } b _ { n } t ^ { n } \\biggr ) \\coloneqq \\sum _ { n = 0 } ^ { \\infty } a _ { n } b _ { n } t ^ { n } . \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{align*} \\eta _ { X , Y } ( C ( \\mathsf { P } ) ) = \\sum _ { i , j , k , \\ell } \\pi ( K _ { 2 \\rho } ^ { - 1 } ) ^ i _ j ( 2 \\varepsilon ( \\mathsf { P } ^ j _ k ) - \\delta ^ j _ k ) \\varepsilon ( X \\triangleright \\mathsf { P } ^ k _ \\ell ) \\varepsilon ( Y \\triangleright \\mathsf { P } ^ { \\ell } _ i ) . \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{align*} P Y '' - P ' Y ' + R Y = 0 \\end{align*}"}
-{"id": "1378.png", "formula": "\\begin{align*} \\alpha \\left ( k _ { n + 1 } - k _ { n } \\right ) = \\pi + O \\left ( \\frac { 1 } { n } \\right ) , n \\rightarrow \\infty . \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{align*} a & = c _ { 1 1 } ( Z _ 0 ) = \\frac { u _ 1 - u _ 2 } { \\sqrt { 3 } } , \\\\ b & = c _ { 1 2 } ( Z _ 0 ) = \\frac { u _ 0 } { 3 } - \\frac { 2 u _ 1 } { 3 } - \\frac { 2 u _ 2 } { 3 } , \\\\ c & = c _ { 2 1 } ( Z _ 0 ) = u _ 0 . \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} a _ 1 & = - a _ { 2 2 } , \\\\ a _ 2 & = - ( a _ { 1 1 } a _ { 3 3 } + a _ { 1 2 } a _ { 3 2 } + a _ { 1 3 } a _ { 3 1 } + a _ { 2 1 } a _ { 2 3 } ) , \\\\ a _ 3 & = a _ { 1 2 } a _ { 2 3 } a _ { 3 1 } + a _ { 1 3 } a _ { 2 1 } a _ { 3 2 } - a _ { 1 1 } a _ { 2 3 } a _ { 3 2 } - a _ { 1 2 } a _ { 2 1 } a _ { 3 3 } . \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} a \\in ( - \\infty , 0 ) , ~ b \\in ( a , a + 1 ] , ~ s = { 2 \\over b - a } & n = 2 , \\\\ a \\in ( - \\infty , 1 / 2 ) , ~ b \\in [ a , a + 1 ] , ~ s = { 6 + 2 ( b - a ) } & n = 3 . \\end{align*}"}
-{"id": "4167.png", "formula": "\\begin{align*} \\lambda ^ { ( b + \\xi ) } _ { \\nu } ( s ) = \\frac { \\alpha } { \\Gamma ( 1 - \\alpha ) } s ^ { \\nu - \\alpha - 1 } e ^ { - b s } \\qquad s \\in ( 0 , \\infty ) \\ ; . \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} S _ { \\beta } \\ ; & : = \\ ; S ^ * \\upharpoonright \\mathcal { D } ( S _ { \\beta } ) \\\\ \\mathcal { D } ( S _ { \\beta } ) \\ ; & : = \\ ; \\Big \\{ g \\in \\mathcal { D } ( S ^ * ) \\ , \\Big | \\ , \\frac { g _ 1 ^ + } { g _ 0 ^ + } = c _ \\nu \\beta + d _ \\nu \\Big \\} \\ , , \\end{align*}"}
-{"id": "5271.png", "formula": "\\begin{align*} \\int _ { B _ R ( 0 ) \\cap \\Omega } e _ n \\cdot ( \\omega \\times ( c - u ) ) \\ , d x & = g \\int _ { B _ R ( 0 ) \\cap S } \\eta \\ , d x ' + \\sigma \\int _ { \\partial B _ R \\cap S } N \\cdot \\nu \\ , d s \\\\ & + \\int _ { \\partial B _ R ( 0 ) \\cap \\Omega } A \\cdot N \\ , d S . \\end{align*}"}
-{"id": "10036.png", "formula": "\\begin{align*} \\mathcal G _ { \\varphi } = \\bigcup _ { p \\in M } \\left \\{ ( X _ 1 , \\ldots , X _ r , Y _ 1 , \\ldots , Y _ s ) \\in \\mathcal F _ p ( M ) \\colon X _ 1 , \\ldots , X _ r \\in ( D _ { \\phi } ) _ p , Y _ 1 , \\ldots , Y _ s \\in ( D _ { \\bar \\phi } ) _ p \\right \\} , \\end{align*}"}
-{"id": "5089.png", "formula": "\\begin{align*} C _ { T , K } : = C ( \\| K \\| _ { \\operatorname { C Z } _ { \\alpha } } + \\sup _ { \\varepsilon > 0 } \\| T _ { \\varepsilon } ( 1 , 1 ) \\| _ { \\operatorname { B M O } } & + \\sup _ { \\varepsilon > 0 } \\| T ^ { 1 * } _ { \\varepsilon } ( 1 , 1 ) \\| _ { \\operatorname { B M O } } \\\\ & + \\sup _ { \\varepsilon > 0 } \\| T ^ { 2 * } _ { \\varepsilon } ( 1 , 1 ) \\| _ { \\operatorname { B M O } } + \\sup _ { \\varepsilon > 0 } \\| T _ { \\varepsilon } \\| _ { \\operatorname { W B P } } ) \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\| w ( t ) \\| ^ 2 _ { L ^ 2 ( \\Omega ) } + \\int _ s ^ t \\| \\nabla w \\| ^ 2 _ { L ^ 2 ( \\Omega ) } d \\tau \\\\ & \\leq \\frac { 1 } { 2 } \\| w ( s ) \\| ^ 2 _ { L ^ 2 ( \\Omega ) } + \\int _ s ^ t \\langle ( h u _ s + \\widetilde U ) \\otimes w , \\nabla w \\rangle d \\tau + \\int _ s ^ t \\langle f , w \\rangle d \\tau \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} \\int \\vert \\eta , z _ 1 , z _ 2 \\rangle d \\mu ( \\eta , \\bar \\eta , z _ 1 , \\bar z _ 1 , z _ 2 , \\bar z _ 2 ) \\langle \\eta , z _ 1 , z _ 2 \\vert = \\sum _ { l = 0 } ^ { k } \\sum _ { n = 0 } ^ { k - l } \\vert n , l \\rangle \\langle n , l \\vert \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} A _ { i } A _ { j } = \\sum _ { k = 1 } ^ { n } a _ { i , j } ( k ) A _ { k } \\\\ \\end{align*}"}
-{"id": "9949.png", "formula": "\\begin{align*} S E T = \\left [ \\begin{array} { c c } I & 0 \\\\ 0 & N \\end{array} \\right ] S A T = \\left [ \\begin{array} { c c } J & 0 \\\\ 0 & I \\end{array} \\right ] \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{align*} { n _ R ( k ) \\choose l } = { { k + 2 \\choose 2 } \\left ( { k + 2 \\choose 2 } - 1 \\right ) - 2 k \\choose l } \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} \\partial _ { t } ^ { 2 } [ \\mathbf { \\Xi } _ { \\mathrm { p } } ( t ) ] _ { + } = - \\int \\nolimits _ { \\mathbb { R } } \\cos \\left ( t \\nu \\right ) \\mathbf { \\mu } \\left ( \\mathrm { d } \\nu \\right ) \\ . \\end{align*}"}
-{"id": "2537.png", "formula": "\\begin{align*} q _ { 1 } ^ { \\prime } f ^ { 2 } + \\left ( 2 q _ { 1 } + q _ { 2 } ^ { \\prime } \\right ) f ^ { \\prime } f + \\left ( q _ { 2 } + q _ { 3 } ^ { \\prime } \\right ) \\left ( f ^ { \\prime } \\right ) ^ { 2 } + q _ { 2 } f ^ { \\prime \\prime } f + 2 q _ { 3 } f ^ { \\prime } f ^ { \\prime \\prime } = - B _ { 2 } ^ { \\prime } . \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} 0 = [ [ L _ { - r + 1 } , \\ , [ L _ { - 2 } , \\ , S _ r ] ] , \\ , S _ { r - 2 } ] = [ [ L _ { - r + 1 } , \\ , S _ { r - 2 } ] , \\ , [ L _ { - 2 } , \\ , S _ r ] ] \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} \\left ( L _ { \\varepsilon } + \\lambda \\right ) u = 0 L _ { k } u = f _ { k } - L _ { k } u _ { 1 } . \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} \\left [ P _ { t } \\left ( \\xi \\right ) + A + \\lambda \\right ] \\hat { u } \\left ( \\xi \\right ) = \\hat { f } \\left ( \\xi \\right ) . \\end{align*}"}
-{"id": "4702.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } K _ N \\left ( \\frac { x } { N } , \\frac { y } { N } ; w _ R \\right ) = \\frac { \\sin \\frac { \\pi } { 2 } ( x - y ) } { \\pi ( x - y ) } , \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{align*} \\sharp S _ { d , \\varepsilon , B } = \\frac { 6 \\varepsilon } { \\pi ^ 2 \\alpha ^ 2 } B ^ { \\frac { 2 } { 3 } + \\delta } + O ( B ^ { \\frac { 2 } { 3 } + \\frac { 2 } { 3 } \\varepsilon + ( 1 - \\varepsilon ) \\frac { \\delta } { 2 } } ) = \\frac { 6 } { \\pi ^ 2 \\alpha ^ 2 } B ^ { \\frac { 2 } { 3 } + \\delta _ 0 } ( \\int \\chi ( \\varepsilon ) ( x ) \\operatorname { d } x + o ( 1 ) ) . \\end{align*}"}
-{"id": "3431.png", "formula": "\\begin{align*} ( \\lambda ) _ n = \\frac { \\Gamma ( \\lambda + n ) } { \\Gamma ( \\lambda ) } \\ ( \\Re ( \\lambda ) > 0 ) , \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} \\operatorname { v o l } ( M ) & = \\int _ M \\frac { \\omega _ M ^ n } { n ! } = \\operatorname { v o l } ( B ) \\int _ 0 ^ 1 P ( x ) ^ { n - 1 } d x = \\frac { ( p + c ) ^ n - c ^ n } { p n } \\\\ \\int _ M s ^ H \\frac { \\omega _ M ^ n } { n ! } & \\overset { \\eqref { S c a l C a l a b i } } { = } \\int _ 0 ^ 1 \\frac { s ^ H _ S - ( P \\varphi ) '' ( x ) } { P ( x ) } \\operatorname { v o l } ( \\mu ^ { - 1 } x ) d x \\\\ & = \\operatorname { v o l } ( B ) \\int _ 0 ^ 1 \\left [ s ^ H _ S - ( P \\varphi ) '' ( x ) \\right ] P ( x ) ^ { n - 2 } d x \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} \\chi _ { \\tau } ( t , r ) = \\frac { 1 } { ( 2 \\pi ) ^ \\frac { d } { 2 } } \\int _ 0 ^ r d u \\left \\{ \\int _ 0 ^ { t - r } \\frac { B _ u - B _ r } { ( i \\tau + s ) ^ { \\frac { d } { 2 } + 1 } } e ^ { - \\frac { | B _ r - B _ u | ^ 2 } { 2 ( i \\tau + s ) } } d s \\right \\} . \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} \\phi ( z ) = \\Re \\left ( \\vec { A } _ 0 z ^ { \\theta _ 0 } \\right ) + O \\left ( | z | ^ { \\theta _ 0 + 1 } \\log | z | \\right ) \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ 0 } { x } + \\frac { A _ { t _ 1 } } { x - t _ 1 } + \\frac { A _ { t _ 2 } } { x - t _ 2 } + N \\right ) Y , \\\\ \\frac { \\partial Y } { \\partial t _ 1 } = \\left ( N _ 1 - \\frac { A _ { t _ 1 } } { x - t _ 1 } \\right ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = \\left ( N _ 2 - \\frac { A _ { t _ 2 } } { x - t _ 2 } \\right ) Y . \\end{gather*}"}
-{"id": "2310.png", "formula": "\\begin{align*} \\sigma _ 2 ( t ) = \\begin{pmatrix} 1 & - \\xi ( t ) \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} - \\sqrt { 3 } / 2 \\\\ ( 1 + s ( t ) ) / 2 \\end{pmatrix} \\end{align*}"}
-{"id": "6091.png", "formula": "\\begin{align*} [ [ L _ { - 3 } , \\ , [ L _ { - 4 } , \\ , L _ 2 ] ] = [ [ L _ { - 4 } , \\ , [ L _ { - 3 } , \\ , L _ 2 ] ] = [ L _ { - 4 } , \\ , L _ { - 1 } ] \\neq 0 , \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{align*} & \\partial _ t W = \\Delta W - \\nabla Q + G , \\quad \\mbox { d i v $ W $ } = 0 ( y \\in \\mathbb R ^ 3 , \\ , t > 0 ) , \\\\ & W \\to 0 \\quad \\mbox { a s $ | y | \\to \\infty $ } , \\\\ & W ( y , 0 ) = \\bar v _ 0 ( y ) . \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} x ^ 3 = 1 \\implies x = 1 \\end{align*}"}
-{"id": "9727.png", "formula": "\\begin{align*} \\lim _ m \\| u + u _ m \\| _ \\infty & = \\lim _ m \\| u + ( u _ m - P u _ m ) \\| _ \\infty = \\lim _ m \\max \\bigl \\{ \\| u \\| _ \\infty , \\ , \\| u _ m - P u _ m \\| _ \\infty \\bigr \\} \\\\ & = \\max \\bigl \\{ \\| u \\| _ \\infty , \\ , \\lim _ m \\| u _ m \\| _ \\infty \\bigr \\} , \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} \\mu ( C \\sqrt { n } L _ { \\mu } B ^ n ) = 1 , \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} f ( x ) = W ( 0 ) \\alpha e ^ { \\lambda \\sigma } \\lambda e ^ { - \\lambda x } + \\lambda \\alpha e ^ { \\lambda \\sigma } \\int _ { 0 } ^ x e ^ { - \\lambda ( x - w ) } f ( w ) d w , 0 < x < K . \\end{align*}"}
-{"id": "1104.png", "formula": "\\begin{align*} \\Vert g \\Vert _ { U ^ { d } ( \\mathbb { R } , N ) } ^ { 2 ^ { d } } \\asymp \\frac { 1 } { N ^ { d + 1 } } \\int \\limits _ { ( x , \\mathbf { h } ) \\in \\mathbb { R } ^ { d + 1 } } \\prod \\limits _ { \\boldsymbol { \\omega } \\in \\{ 0 , 1 \\} ^ d } \\mathcal { C } ^ { \\vert \\boldsymbol { \\omega } \\vert } g ( x + \\sum \\limits _ { i = 1 } ^ d h _ i \\omega _ i ) \\ , d x \\ , d h _ 1 \\cdots d h _ d . \\end{align*}"}
-{"id": "9685.png", "formula": "\\begin{align*} ( \\check { G } _ 0 ^ 2 + \\check { H } _ 0 ^ 2 ) ( 0 , 0 , 0 ) = ( G ( 0 , 0 , 0 ) + \\alpha _ 0 ) ^ 2 + ( H ( 0 , 0 , 0 ) ) ^ 2 \\not = 0 . \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} \\theta \\sim \\vartheta _ \\gamma : = ( 1 + \\varepsilon \\gamma \\ , \\mid \\ , \\varepsilon ^ { 2 ( m - j ) + 1 } ) . \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} f _ { i j } ( p _ { i j } ) = f _ { i j } ( c ) > q _ i . \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{gather*} \\mathcal { B } : = \\left \\{ \\omega _ { \\mathbf { m } } \\colon 0 < m _ i < d \\mbox { f o r } i = 0 , \\dots , n \\ \\ \\sum _ { i = 0 } ^ n m _ i \\equiv 0 \\bmod d \\right \\} \\end{gather*}"}
-{"id": "3874.png", "formula": "\\begin{align*} \\frac { f ( - 1 + h - a ) } { 2 r _ x } = \\frac { 1 } { r _ x } + h \\left ( 1 - \\frac { 1 } { r _ x } \\right ) . \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } P _ { 1 1 } = ( - \\bar { v } ^ 3 + ( a + 1 ) \\bar { v } ^ 2 - a \\bar { v } - \\bar { w } ) ^ 2 , \\\\ P _ { 1 2 } = \\varepsilon ( - \\bar { v } ^ 3 + ( a + 1 ) \\bar { v } ^ 2 - a \\bar { v } - \\bar { w } ) ( b \\bar { v } - c \\bar { w } ) , \\\\ P _ { 2 2 } = \\varepsilon ^ 2 ( b \\bar { v } - c \\bar { w } ) ^ 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} \\mu _ { j } [ \\rho ] \\leq \\max _ { i = 1 , . . . , j } \\frac { \\int _ { \\Omega } | D ^ m u _ i | ^ 2 d x } { \\int _ { \\Omega } \\rho u _ i ^ 2 d x } . \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} \\psi _ p ( 0 ) = p , \\max _ { | \\zeta | \\leq 1 } G ^ + \\circ \\psi _ p ( \\zeta ) = 1 . \\end{align*}"}
-{"id": "5221.png", "formula": "\\begin{align*} \\eta = \\frac { 1 } { g } p ^ \\prime \\cdot \\nabla \\left ( \\frac { c ^ \\prime \\cdot x ^ \\prime } { | x ^ \\prime | ^ n } \\right ) + O \\left ( \\frac { 1 } { | x ^ \\prime | ^ { n + \\varepsilon } } \\right ) , \\textrm { a s } | x ^ \\prime | \\to \\infty , \\end{align*}"}
-{"id": "5654.png", "formula": "\\begin{align*} 3 \\frac { b } { c } - 2 = \\varepsilon c . \\end{align*}"}
-{"id": "6315.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } e ^ { - x s } \\ , d \\mu _ { k + 1 } ( s ) + b _ { k + 1 } & = ( \\lambda + k ) \\int _ 0 ^ { \\infty } e ^ { - x s } \\ , d \\mu _ k ( s ) + ( \\lambda + k ) b _ k \\\\ & { } - x \\int _ 0 ^ { \\infty } s e ^ { - x s } \\ , d \\mu _ k ( s ) . \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} \\tilde K u ( x , v ) = \\int _ { V \\times \\R } \\Psi ( x , v , v ' , r ) \\rho ( u ) ( x + r v , v ' ) d r d v \\end{align*}"}
-{"id": "7142.png", "formula": "\\begin{align*} \\alpha _ 1 ( t _ 1 H _ 1 + t _ 2 H _ 2 ) = t _ 2 - t _ 1 , \\alpha _ 2 ( t _ 1 H _ 1 + t _ 2 H _ 2 ) = 2 t _ 1 - t _ 2 . \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} f _ j ( E , x ) & = \\sum _ { k = 0 } ^ \\infty a _ k ^ j ( E ) x ^ k . \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\Delta , f } ( \\xi ) = 2 \\int _ { \\partial \\Delta } e ^ { n f } \\xi d \\mu - 2 \\frac { \\int _ { \\partial \\Delta } e ^ { n f } d \\mu } { \\int _ \\Delta e ^ { ( n + 2 ) f } d v } \\int _ \\Delta \\xi e ^ { ( n + 2 ) f } d v . \\end{align*}"}
-{"id": "1952.png", "formula": "\\begin{align*} \\widetilde E = \\{ ( i , j ) \\in E \\mid f _ { i j } ( p _ { i j } ) \\geq 0 \\mbox { a n d } f _ { j i } ( - p _ { j i } ) \\geq 0 \\} . \\end{align*}"}
-{"id": "8791.png", "formula": "\\begin{align*} \\tilde { g } _ { i j } ( x , y ) = \\dfrac { \\partial \\tilde { x } ^ { k } } { \\partial x ^ { i } } \\dfrac { \\partial \\tilde { x } ^ { l } } { \\partial x ^ { j } } g _ { k l } ^ { \\prime } ( \\tilde { x } , \\tilde { y } ) , ~ \\ \\ \\forall ( x , y ) \\in A . \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ^ 2 u = \\mu \\rho u , & { \\rm i n \\ } \\Omega , \\\\ \\frac { \\partial ^ 2 u } { \\partial \\nu ^ 2 } = 0 , & { \\rm o n \\ } \\partial \\Omega , \\\\ { \\rm d i v } _ { \\partial \\Omega } ( D ^ 2 u \\cdot \\nu ) + \\frac { \\partial \\Delta u } { \\partial \\nu } = 0 , & { \\rm o n \\ } \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "3635.png", "formula": "\\begin{align*} \\rho ( P _ { s } , P _ { t } ) = \\int \\sqrt { s ( x ) t ( x ) } \\ , d \\mu ( x ) < 1 . \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} \\frac { x _ { n + 1 } - x _ { n } } { h } = f \\left ( \\frac { x _ { n + 1 } + x _ { n } } { 2 } \\right ) . \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} \\phi _ U ^ { - 1 } ( \\nu ) = p ^ { - 1 } ( \\phi _ Y ^ { - 1 } ( \\nu ) ) , \\phi _ Y ^ { - 1 } ( \\nu ) = \\psi ^ { - 1 } ( \\nu ) , \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} v _ 1 = - \\frac { 1 } { \\sqrt { 3 } } , \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} I ( A : B | M ) _ { \\hat { \\rho } _ { A B M } } = 0 \\ ; . \\end{align*}"}
-{"id": "5196.png", "formula": "\\begin{align*} \\left ( f \\circ \\left ( g \\circ h \\right ) \\right ) _ { n } = \\sum _ { \\mu \\models \\pi \\models n } f _ { \\vert \\pi \\vert } g _ { \\vert \\mu \\vert } h _ { \\mu } \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { r } \\frac { ( 2 d ) ^ m } { m ! } \\left | \\overline { q } \\right | \\leq \\sum _ { m = 0 } ^ { \\infty } \\frac { ( 2 d ) ^ m \\left ( d ^ { a _ 1 } m ^ { a _ 1 } + a _ 2 \\right ) } { m ! } = ( d ^ { a _ 1 } b ( 2 d ) + a _ 2 ) e ^ { 2 d } . \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} \\underbrace { g ( x ) y } _ { u } \\underbrace { g ( g ( x ) y ) } _ { g ( u ) } & \\stackrel { ( a ) } { = } \\underbrace { g ( g ( x ) y ) } _ { g ( u ) } \\underbrace { g ( x ) y } _ u \\\\ & = \\underbrace { g ( g ( x ) y ) } _ { g ( u ) } \\underbrace { g ( x ) } _ { g ( x ) } y \\\\ & \\stackrel { ( b ) } { = } \\underbrace { g ( x ) } _ { g ( x ) } \\underbrace { g ( g ( x ) y ) } _ { g ( u ) } y , \\end{align*}"}
-{"id": "9777.png", "formula": "\\begin{align*} E _ { \\epsilon } ( \\tilde { u } _ { \\epsilon } ) = E _ { \\epsilon } ( u _ { \\epsilon } ) - \\frac { 1 } { 2 } \\| h _ { \\epsilon } \\| _ { L ^ 2 } ^ 2 + O ( | \\log \\epsilon | ^ { 1 / 2 } ) \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{align*} \\partial \\bar { \\partial } H + \\frac { e ^ { 2 \\lambda } } { 2 } | H _ 0 | ^ 2 H = 0 \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} ( a ^ - ) ^ { k + 1 } = 0 ( a ^ + ) ^ { k + 1 } = 0 \\end{align*}"}
-{"id": "2320.png", "formula": "\\begin{align*} B ' = \\frac { 3 } { 2 } \\lambda _ { c o s t } [ J , X ] + [ B , X ] . \\end{align*}"}
-{"id": "7219.png", "formula": "\\begin{align*} m _ 1 ^ 2 + \\alpha ^ \\prime - m _ 3 ^ 2 \\beta = 0 , \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{align*} s _ 1 + \\ldots + s _ r = s + r . \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} \\| ( \\operatorname * { I } - { \\mathfrak I } _ \\ell ) ( E _ { y _ { - \\ell + 1 } } \\pi ) \\| _ { L ^ \\infty ( B _ { \\ell } ^ x ) } \\leq C _ 1 \\widehat q ^ m \\| \\pi \\| _ { L ^ \\infty ( B _ { \\ell - 1 } ^ x ) } = C _ 1 \\widehat q ^ m \\| E _ { y _ { - \\ell + 1 } } \\pi \\| _ { L ^ \\infty ( B _ { \\ell - 1 } ^ x ) } , \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} V ( t , a , b ) = \\inf _ { \\varphi _ 0 = a , \\varphi _ t = b } S _ { 0 t } ( \\varphi ) . \\end{align*}"}
-{"id": "389.png", "formula": "\\begin{align*} \\begin{array} { r r } \\begin{cases} \\eta _ { i , j } = ( 1 - ( - 1 ) ^ { j - i } ) ( \\lambda + i ) , \\ i < j \\\\ \\eta _ { i , j } = 0 , \\ i \\geq j \\end{cases} & \\begin{cases} \\theta _ { j \\pm 1 , j } = { \\displaystyle \\frac { \\pm 1 } { 2 ( \\lambda + j ) } } \\\\ \\theta _ { i , j } = 0 , \\ i \\neq j \\pm 1 \\end{cases} \\end{array} , j \\geq 0 \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} S _ { 2 T } \\left ( u \\right ) : = \\frac { 1 } { ( T - 1 ) ^ { 1 / 2 } } \\sum _ { t = 2 } ^ { T } \\left \\{ I _ { t , \\theta _ { 0 } } \\left ( u _ { 1 } \\right ) I _ { t - 1 , \\theta _ { 0 } } \\left ( u _ { 2 } \\right ) - u _ { 1 } u _ { 2 } \\right \\} , \\end{align*}"}
-{"id": "582.png", "formula": "\\begin{align*} d \\hat r _ t ( z ) = - \\frac { 2 } { \\hat r _ t ( z ) } - \\sqrt \\kappa d B _ { t } . \\end{align*}"}
-{"id": "1876.png", "formula": "\\begin{align*} e ^ { 2 f } \\left ( \\tilde { s } ^ H - \\tilde { s } ^ { \\tilde { g } } \\right ) = ( s ^ H - s ^ g ) + ( 2 - m ) \\Delta f - ( m - 2 ) | d f | ^ 2 . \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{align*} \\dot { J } ( i \\kappa ) = \\dot { f } ' ( i \\kappa , 0 ) ^ \\dagger A - \\dot { f } ( i \\kappa , 0 ) ^ \\dagger B , \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} F _ 5 = F _ { 5 1 } + F _ { 5 2 } , \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} f _ { N } ( \\xi ) : = \\frac { 2 ^ { - N / 2 } } { \\big ( m ( B _ { N } ) \\big ) ^ { 1 / 2 } } \\chi _ { B _ { N } } ( \\xi ) , \\xi \\in \\mathbb { R } ^ { n } , \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{align*} S ( X ) : = - \\int _ { \\mathbb { R } ^ k } \\ln p _ X ( \\mathbf { x } ) \\ ; \\mathrm { d } p _ X ( \\mathbf { x } ) \\ ; , \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} R \\big ( e _ { s ( \\alpha ) } \\otimes e _ { t ( f ( \\alpha ) ) } \\big ) = \\varrho ( \\mu _ { \\alpha } ) \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{align*} \\hat { H } = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { 2 n } \\left ( \\hat { R } ^ i \\right ) ^ 2 - \\frac { n } { 2 } \\ , \\hat { \\mathbb { I } } \\ ; . \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{align*} \\textstyle x - \\sigma ^ j ( \\beta ) \\mid _ r f \\iff \\sum _ { i = 0 } ^ { n - 1 } f _ i \\norm { i } { \\sigma ^ j ( \\beta ) } = 0 \\iff \\sum _ { i = 0 } ^ { n - 1 } f _ i \\sigma ^ { i + j } ( \\alpha ) = 0 . \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} \\bigg ( \\frac { \\partial } { \\partial \\eta _ 1 } \\bigg ) ^ { l _ 1 } \\bigg ( \\frac { \\partial } { \\partial \\eta _ 2 } \\bigg ) ^ { l _ 2 } \\cdots \\bigg ( \\frac { \\partial } { \\partial \\eta _ r } \\bigg ) ^ { l _ r } = 0 , l _ 1 + l _ 2 + \\cdots l _ r = k + 1 , \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{align*} p _ t ( x , y ) = \\partial _ y \\int _ { x } ^ { r } \\hat { p } _ t ( y , d z ) . \\end{align*}"}
-{"id": "9694.png", "formula": "\\begin{align*} C _ { S ^ { \\ast } } \\ = \\ \\left \\{ \\sum _ { i = 1 } ^ { d + 1 } \\lambda _ i \\bar { v } _ i : \\lambda _ i \\geq 0 i \\in [ d + 1 ] , \\lambda _ i \\neq 0 i \\in I \\right \\} \\subseteq \\mathbb { R } ^ { d + 1 } \\end{align*}"}
-{"id": "669.png", "formula": "\\begin{align*} a ^ + \\vert n \\rangle = \\sqrt { F ( n + 1 ) } ~ \\vert n + 1 \\rangle , a ^ - \\vert n \\rangle = \\sqrt { F ( n ) } ~ \\vert n - 1 \\rangle \\end{align*}"}
-{"id": "8774.png", "formula": "\\begin{align*} L _ { \\cdot i } = 2 y _ { i } , ~ ~ \\ y _ { i \\cdot j } = g _ { i j } , \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{align*} f _ { c _ { 1 } } \\left ( z \\right ) = \\varphi \\left ( z \\right ) f \\left ( z \\right ) + \\psi \\left ( z \\right ) f ^ { \\prime } \\left ( z \\right ) , \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{align*} \\partial _ { t } ^ { 2 } u ( x , t ) - L ^ \\alpha _ x \\partial _ x ^ 2 u ( x , t ) = u _ 0 ( x ) \\otimes \\delta ' ( t ) + v _ 0 ( x ) \\otimes \\delta ( t ) , \\ ; \\ ; ( x , t ) \\in \\mathbb { R } ^ 2 , \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} \\sharp S ( \\varepsilon , K , \\Lambda , B ) = \\frac { \\Theta ( \\Lambda ) \\varepsilon K ^ 2 } { 2 } B ^ { 2 - \\frac { 1 } { r } } + O ( K ^ \\frac { 3 } { 2 } b ^ { \\frac { 1 } { 4 } } \\det ( \\Lambda ) ^ \\frac { 1 } { 2 } B ^ { \\frac { 3 } { 4 } ( 2 - \\frac { 1 } { r } ) } \\log B + K b ^ { \\frac { 3 } { 2 } } B ^ { 1 - \\frac { 1 } { 2 r } } \\log B ) . \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} \\lambda ( c ) = \\lambda ' ( c _ * ) ( c - c _ * ) + \\mathcal { O } ( ( c - c _ * ) ^ 2 ) \\mbox { \\rm a s } c \\to c _ * , \\end{align*}"}
-{"id": "350.png", "formula": "\\begin{align*} v \\triangleleft X : = S ( X ) \\triangleright v , ( f \\triangleleft X ) ( v ) : = f ( X \\triangleright v ) . \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} E ( \\beta ) \\ ; : = \\ ; \\frac { | \\beta | } { \\ , | \\beta | \\| S _ D ^ { - 1 } \\| + 1 \\ , } \\ , . \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & 0 \\\\ - c a ^ { - 1 } & 1 \\end{bmatrix} A = \\begin{bmatrix} a & b \\pi ^ n \\\\ 0 & d - c a ^ { - 1 } b \\pi ^ n \\end{bmatrix} . \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} \\norm { D F ( \\zeta ) } = \\frac { \\abs { \\Phi ' ( r ( z ) ) } \\norm { D \\Psi ( z ) ^ { - 1 } } } { \\abs { \\zeta } ^ 2 \\abs { z } ^ 2 } \\norm { D F ( \\zeta ) ^ { - 1 } } = \\abs { \\zeta } ^ 2 \\abs { z } ^ 2 \\frac { \\norm { D \\Psi ( z ) } } { \\abs { \\Phi ' ( r ( z ) ) } } . \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} \\widehat { K } _ \\eta ( \\alpha ) : = \\max \\{ 0 , \\eta - \\vert \\alpha \\vert \\} , \\quad \\textrm { w h e r e } \\ \\eta > 0 , \\end{align*}"}
-{"id": "5476.png", "formula": "\\begin{align*} K _ { \\alpha , \\beta } = \\frac { 1 } { N + | \\alpha | + | \\beta | } \\int _ { \\partial B } H _ 1 ^ { \\alpha _ 1 + \\beta _ 1 } \\cdots H _ N ^ { \\alpha _ N + \\beta _ N } d \\sigma ( \\theta ) . \\end{align*}"}
-{"id": "2509.png", "formula": "\\begin{align*} U _ { \\tau _ - , x , r } : = \\{ \\tau \\in C ( \\tau _ - ) | d ( x , P ( \\tau _ - , \\tau ) ) < r \\} . \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } u ( t , x ) = \\lim _ { t \\to 0 } t ^ { - \\frac { 1 } { \\alpha } } f _ a \\Bigl ( \\frac { | x | } { \\sqrt t } \\Bigr ) = | x | ^ { - \\frac { 2 } { \\alpha } } \\lim _ { r \\to \\infty } r ^ { \\frac { 2 } { \\alpha } } f _ a ( r ) = L ( a ) | x | ^ { - \\frac { 2 } { \\alpha } } . \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { j = 1 } ^ n \\Delta _ { i j } x _ 1 x _ j ^ 2 - c x _ 1 + 1 & = 0 \\\\ & \\vdotswithin { = } \\\\ \\sum _ { j = 1 } ^ n \\Delta _ { i j } x _ n x _ j ^ 2 - c x _ n + 1 & = 0 \\end{aligned} \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{align*} G _ { ~ j } ^ { i } = G _ { ~ \\cdot j } ^ { i } . \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mu } \\left ( Z _ { t } \\in A _ { R _ { 1 } , R _ { 2 } } \\right ) = \\exp \\left [ - \\frac { R _ { 1 } ^ { 2 } } { 2 \\rho ( t ) } \\right ] - \\exp \\left [ - \\frac { R _ { 2 } ^ { 2 } } { 2 \\rho ( t ) } \\right ] \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} x ( [ 2 ] P ) = \\frac { \\phi ( x ( P ) ) } { 4 \\psi ( x ( P ) ) } . \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} A z = b , ~ ~ z \\in \\mathbb { C } ^ n \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} L _ 1 = \\langle \\partial _ j , \\ , j = 1 , \\dots n \\rangle \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} \\mu = ( ( \\mu _ 1 , \\delta _ 1 ) , \\ldots , ( \\mu _ \\ell , \\delta _ \\ell ) ) \\ , . \\end{align*}"}
-{"id": "4970.png", "formula": "\\begin{align*} \\Lambda _ c ( u ) : = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } } \\left [ \\left ( \\partial _ { \\xi } u \\right ) ^ 2 - 4 u ^ 3 + 4 c u ^ 2 \\right ] d \\xi , \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} h ( x ) = e ^ { - 4 \\pi N G _ 0 ( x ) } , \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{align*} ( \\Phi w ) ( t ) = H ( t ) - \\int _ 0 ^ t e ^ { - ( t - \\tau ) A } \\mathbb P [ S w + ( J _ k w ) \\cdot \\nabla w ] ( \\tau ) d \\tau , \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{align*} \\kappa _ { 2 i } = \\left ( \\frac { d t } { d s _ { 2 i } } \\right ) ^ 3 \\delta ( u , X ) u _ i , \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} b c ( 2 - ( u + 1 / u ) ) = 0 . \\end{align*}"}
-{"id": "5137.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } C _ { n } z ^ { n } = \\frac { 1 - \\sqrt { 1 - 4 z } } { 2 z } . \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} V ( t ) = ( 1 + t ) \\log ( 1 + t ) + ( 1 - t ) \\log ( 1 - t ) , t \\in [ - 1 , 1 ] . \\end{align*}"}
-{"id": "1927.png", "formula": "\\begin{align*} \\| u _ m ( \\xi ) \\| & \\le 2 ( 1 - q _ 0 ) ^ { \\frac { - 1 } { 2 } } m ^ { \\frac { - 1 } { 2 } } \\max \\Big \\{ \\Big \\| \\sum _ { k = 1 } ^ m W ( \\xi \\odot e _ k ) ^ * W ( \\xi \\odot e _ k ) \\Big \\| ^ { \\frac 1 2 } , \\Big \\| \\sum _ { k = 1 } ^ m W ( \\xi \\odot e _ k ) W ( \\xi \\odot e _ k ) ^ * \\Big \\| ^ { \\frac 1 2 } \\Big \\} \\\\ & \\le 2 ( 1 - q _ 0 ) ^ { \\frac { - 1 } { 2 } } \\| W ( \\xi \\odot e _ 1 ) \\| . \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} \\dot x = \\sin ( J _ 1 ( x ) ) u _ { 1 } ^ \\varepsilon ( t ) + \\cos ( J _ 1 ( x ) ) u _ { 2 } ^ \\varepsilon ( t ) . \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{align*} \\| \\phi _ { \\varepsilon } \\| _ l = O ( \\varepsilon ^ { - 2 l - 4 \\textrm { d i m } ( Y ) - \\frac { \\textrm { d i m } ( X ) } { 2 } } ) , \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = t _ 1 & x = t _ 2 & x = \\infty \\ \\left ( \\frac 1 2 \\right ) \\\\ \\begin{matrix} 0 \\\\ \\theta ^ { t _ 1 } \\end{matrix} & \\begin{matrix} 0 \\\\ \\theta ^ { t _ 2 } \\end{matrix} & \\overbrace { \\begin{matrix} \\sqrt { - 1 } & 0 & 0 & \\theta ^ \\infty _ 1 / 2 \\\\ - \\sqrt { - 1 } & 0 & 0 & \\theta ^ \\infty _ 1 / 2 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "1783.png", "formula": "\\begin{align*} \\sigma _ j ( z _ j ) = A _ j \\circ \\sigma _ j ( x ) = \\pi _ j \\big ( A _ { j + 1 } \\circ \\sigma _ { j + 1 } ( x ) \\big ) = \\pi _ j \\big ( \\sigma _ { j + 1 } ( z _ { j + 1 } ) \\big ) = \\sigma _ j ( z _ { j + 1 } ) \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} z ^ { a + s ( p - 1 ) } : = z ^ a \\cdot ( z ^ { p - 1 } ) ^ { a } = z ^ { a } \\sum _ { k = 0 } ^ \\infty \\binom { s } { k } ( z ^ { p - 1 } - 1 ) ^ k . \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{align*} \\begin{smallmatrix} \\Xi ^ \\dag \\eta \\Xi = S = \\begin{pmatrix} \\begin{smallmatrix} S _ { 2 n _ 1 } & & & & & \\\\ & \\ddots & & & \\\\ & & S _ { 2 n _ p } & & & & \\\\ & & & \\epsilon _ { n _ q } S _ { n _ q } & & \\\\ & & & & \\ddots & \\\\ & & & & & \\epsilon _ { n _ r } S _ { n _ r } \\end{smallmatrix} \\end{pmatrix} , \\end{smallmatrix} \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { W - 1 } e \\left ( \\overline { \\xi } _ { x + ( n + 1 ) w } - \\overline { \\xi } _ { x + n w } \\right ) , \\end{align*}"}
-{"id": "6946.png", "formula": "\\begin{align*} K _ 1 = K [ x ] / ( x ^ 2 - t ) K _ 2 = K [ x , y ] / ( x ^ 2 - t , y ^ 2 - x ) \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} \\begin{bmatrix} a ' & \\pi ^ n \\\\ 1 & d ' \\end{bmatrix} \\in R ^ \\bullet \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} A _ { \\rm b a c k } \\phi = \\tilde j ( I - \\tilde K ) ^ { - 1 } \\rho \\tilde J \\phi + \\tilde j ( I - \\tilde K ) ^ { - 1 } ( 1 - \\rho ) \\tilde J \\phi . \\end{align*}"}
-{"id": "6459.png", "formula": "\\begin{align*} \\left < s _ 0 , \\cdots , s _ { n - 1 } \\mid ( s _ i s _ j ) ^ { m _ { i j } } = 1 \\right > , \\end{align*}"}
-{"id": "6107.png", "formula": "\\begin{align*} [ L _ { - 2 } , \\ , [ L _ { - 2 } , \\ , L _ { j + 2 } ] ] = 0 \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} M ^ { d , n } & = ( M ^ { d , n } _ s ) _ { s \\geq 0 } = ( \\mathbb E ( \\xi _ n ^ d | \\mathcal F _ { s } ) ) _ { s \\geq 0 } , \\\\ M ^ { c , n } & = ( M ^ { c , n } _ s ) _ { s \\geq 0 } = ( \\mathbb E ( \\xi _ n ^ c | \\mathcal F _ { s } ) ) _ { s \\geq 0 } \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} m : = \\left \\{ \\begin{array} { l l } - \\displaystyle \\int _ \\Omega \\omega ( x - \\xi ^ * ) ^ { \\perp } \\ , d x & n = 2 , \\\\ \\\\ - \\displaystyle \\frac 1 2 \\int _ \\Omega \\omega \\times x \\ , d x & n = 3 . \\end{array} \\right . \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{align*} \\sum _ { l > n } C _ { l } \\ , ( b _ { l + 1 } - b _ { l } ) ^ { 1 - \\frac 1 { p } } \\Bigl ( \\ \\sup _ { b _ { l + 1 } - 2 ( b _ { n + 1 } - b _ n ) \\le k < b _ { l + 1 } } \\prod _ { s = k + 1 } ^ { b _ { l + 1 } - 1 } | w _ s | \\Bigr ) & = \\sum _ { k > k _ 0 } 2 ^ { k - 1 } 2 ^ { \\delta ^ { ( k - 1 ) } - \\tau ^ { ( k ) } } ( \\Delta ^ { ( k ) } ) ^ { 1 - \\frac 1 { p } } \\\\ & = \\sum _ { k > k _ 0 } 2 ^ { k - 1 } \\gamma _ k , \\end{align*}"}
-{"id": "9840.png", "formula": "\\begin{align*} [ x , y ] = ( a , b ) ^ { B ( \\mu , \\nu ) } \\ ; . \\end{align*}"}
-{"id": "9887.png", "formula": "\\begin{align*} \\Delta t _ n \\leq C \\ , \\Delta t _ { n - 1 } , n = 2 , \\ldots , N , \\mbox { a s } \\Delta t \\rightarrow 0 _ + . \\end{align*}"}
-{"id": "6342.png", "formula": "\\begin{align*} \\left | D _ { i _ 1 } + \\cdots + D _ { i _ M } \\right | = \\left ( 2 ^ { 2 ^ { i _ 1 } } + \\cdots + 2 ^ { 2 ^ { i _ M } } \\right ) ^ n , \\end{align*}"}
-{"id": "168.png", "formula": "\\begin{align*} \\int _ { E _ 0 } \\| u ^ * ( t ) \\| d t \\geq \\int _ { E _ 0 } \\Big ( M ( t ) + \\epsilon _ 0 \\Big ) d t = \\int _ { E _ 0 } M ( t ) d t + \\epsilon _ 0 | E _ 0 | . \\end{align*}"}
-{"id": "2056.png", "formula": "\\begin{gather*} \\frac { \\partial Y } { \\partial x } = \\left ( \\frac { A _ { t _ 1 } } { x - t _ 1 } + \\frac { A _ { t _ 2 } } { x - t _ 2 } + A _ { \\infty \\ , 1 } + N x \\right ) Y , \\\\ \\frac { \\partial Y } { \\partial t _ 1 } = \\left ( N _ 1 - \\frac { A _ { t _ 1 } } { x - t _ 1 } \\right ) Y , \\frac { \\partial Y } { \\partial t _ 2 } = \\left ( N _ 2 - \\frac { A _ { t _ 2 } } { x - t _ 2 } \\right ) Y . \\end{gather*}"}
-{"id": "3704.png", "formula": "\\begin{align*} \\omega _ j = \\omega _ E ( ( - \\infty , E _ j ^ - ] ) . \\end{align*}"}
-{"id": "9915.png", "formula": "\\begin{align*} ( - n + \\sigma m ) d = ( n - \\tau \\ell ) p . \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} \\dot { b } = \\lambda ' ( c _ * ) ( c _ + - c _ * ) b + \\gamma | b | ^ 2 b , t \\in \\mathbb { R } _ + , \\end{align*}"}
-{"id": "10048.png", "formula": "\\begin{align*} \\mathrm { T } ^ { 0 } ( X , Y ) = 0 , \\forall X , Y \\in D _ { \\phi } , \\mathrm { T } ^ { 0 } ( X , Y ) = 0 , \\forall X , Y \\in D _ { \\bar \\phi } . \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} M : = ( m + 1 ) ^ d = \\operatorname * { d i m } { \\mathcal Q } _ m . \\end{align*}"}
-{"id": "8863.png", "formula": "\\begin{align*} \\ell ^ n [ f ] ( x ) = \\sum _ { j = 1 } ^ n ( - 1 ) ^ j ( a _ j ( x ) f ^ { ( j ) } ( x ) ) ^ { ( j ) } , x \\in ( a , b ) . \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} \\max \\limits _ { r \\in \\mathbb F _ q ^ * } \\prod \\limits _ { j = 1 } ^ k \\left ( \\sum \\limits _ { { \\bf v } \\in S _ r } | \\widehat { E _ j } ( { \\bf v } ) | ^ k \\right ) ^ { \\frac { 1 } { k } } . \\end{align*}"}
-{"id": "9848.png", "formula": "\\begin{align*} \\tau ^ 2 _ { s ( \\mu ) + \\alpha ^ \\vee , \\mu } = q ^ { - 1 } g ( B ( \\alpha ^ \\vee , \\mu ) - Q ( \\alpha ^ \\vee ) ) \\mathbf { z } ^ { - \\alpha ^ \\vee } \\frac { 1 - \\mathbf { z } ^ { n _ \\alpha \\alpha ^ \\vee } } { 1 - q ^ { - 1 } \\mathbf { z } ^ { n _ \\alpha \\alpha ^ \\vee } } , \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} w _ { 1 } ( \\cos ( \\alpha + \\beta ) | \\rho ) w _ { 1 } ( \\cos ( \\alpha - \\beta ) | \\rho ) \\allowbreak = w _ { 2 } ( \\cos ( \\alpha ) , \\cos ( \\beta ) | \\rho ) . \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} \\left \\langle z _ n L ^ { 2 n } z _ n \\right \\rangle - \\lambda ^ 2 \\left \\langle z _ n L ^ { 2 n - 2 } z _ n \\right \\rangle = - \\lambda ^ 2 c _ n \\left \\langle z _ n \\right \\rangle , \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{align*} \\sigma _ { \\lambda , \\lambda ^ \\prime } ( \\mathsf { M } ^ n _ m ) ^ i _ j & = q ^ { - ( \\lambda , \\lambda _ i - \\lambda _ j ) } q ^ { - ( \\lambda ^ \\prime , \\lambda _ m - \\lambda _ n ) } ( \\mathsf { M } ^ n _ m ) ^ i _ j , \\\\ \\sigma _ { \\lambda , \\lambda ^ \\prime } ( \\mathsf { N } ^ n _ m ) ^ i _ j & = q ^ { ( \\lambda , \\lambda _ m - \\lambda _ n ) } q ^ { ( \\lambda ^ \\prime , \\lambda _ i - \\lambda _ j ) } ( \\mathsf { N } ^ n _ m ) ^ i _ j . \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{align*} I _ { \\upsilon } ( q ; x ) = \\frac { \\left ( \\frac { x } { 2 } \\right ) ^ { \\upsilon } 2 ^ { 2 \\upsilon + q - \\frac { 1 } { 2 } } } { \\sqrt { \\pi } \\ \\Gamma \\left ( \\upsilon + \\frac { 1 } { 2 } \\right ) } \\int _ { 0 } ^ { 1 } t ^ { \\upsilon + q - 1 } \\left ( 1 - t \\right ) ^ { \\upsilon - \\frac { 1 } { 2 } } \\exp ( 2 x t ) \\ d t . \\end{align*}"}
-{"id": "9206.png", "formula": "\\begin{align*} E \\langle M _ { \\infty } , N _ { \\infty } \\rangle = E \\Bigl \\langle M _ { \\infty } , \\sum _ { k = 1 } ^ d N ^ k _ { \\infty } y _ k \\Bigr \\rangle = \\sum _ { k = 1 } ^ d \\mathbb E \\langle M _ { \\infty } , y _ k \\rangle N ^ k _ { \\infty } \\stackrel { ( * ) } = 0 , \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{gather*} \\tau _ 1 ^ * ( \\tilde { \\omega } _ { a b c d } ) = ( - 1 ) ^ { b + 1 } ( I ) ^ { a + b } \\tilde { \\omega } _ { b a c d } . \\end{gather*}"}
-{"id": "2531.png", "formula": "\\begin{align*} f _ { c _ { 2 } } ^ { \\prime } = M ^ { \\prime } f + \\left ( M + N ^ { \\prime } \\right ) f ^ { \\prime } + N f ^ { \\prime \\prime } . \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} L \\cdot g = - \\gamma B ( - i \\sqrt { 2 } / \\gamma ) \\cdot g + \\frac { i c } { 2 \\pi } \\langle e _ 0 , g \\rangle e _ 1 \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} T _ \\lambda T _ \\mu & = s _ \\lambda s _ \\mu = c _ { \\Lambda } ( \\lambda , \\mu ) s _ { \\lambda \\mu } = ( - 1 ) ^ 1 s _ { \\lambda \\mu } = - s _ { \\lambda \\mu } \\\\ T _ { \\mu ' } T _ { \\lambda ' } & = s _ { \\mu ' } s _ { \\lambda ' } = c _ { \\Lambda } ( \\mu ' , \\lambda ' ) s _ { \\mu ' \\lambda ' } = ( - 1 ) ^ 0 s _ { \\lambda \\mu } = s _ { \\lambda \\mu } . \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} r _ { s } = \\frac { 1 } { n } \\overline { F } \\sigma _ { 1 } ^ { T } r _ { c } = \\frac { 1 } { n } \\overline { F } \\sigma _ { 2 } ^ { T } \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} v ^ * ( y ( 0 ) ) ^ * f ( 0 ) y ( 0 ) v - v ^ * s ( 0 ) v = v ^ * ( f ( 0 ) - s ( 0 ) ) v \\geq 0 , \\end{align*}"}
-{"id": "8639.png", "formula": "\\begin{align*} a _ m = \\inf \\{ a > 0 ; N ( a ) \\ge m + 1 \\} \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} r \\dot { r } = \\frac { \\lambda _ { 4 } g ^ { 5 } + 2 \\lambda _ { 5 } g ^ { 3 } } { \\left ( \\lambda _ { 4 } g ^ { 2 } + \\lambda _ { 5 } \\right ) ^ { 2 } } . \\end{align*}"}
-{"id": "2893.png", "formula": "\\begin{align*} [ - \\mathsf { T } ^ { v i r } ] = \\pi _ { P * } \\Big ( ( \\mathbb { I } ^ \\bullet ) \\cdot ( ( \\mathbb { I } ^ \\bullet ) ^ \\vee ) \\cdot ( X ) \\Big ) - \\pi _ { P * } \\Big ( \\Big ) \\ , . \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{align*} \\dot z = Q D v ( \\theta ( t ) ) \\Gamma _ 0 z \\end{align*}"}
-{"id": "8221.png", "formula": "\\begin{align*} \\lambda _ { 1 } ( p , \\Omega ) ^ { \\frac 1 p } \\le \\lambda _ { 2 } ( p , \\Omega ) ^ { \\frac 1 p } \\le \\lambda _ { 2 } ( p , \\mathcal W _ { 1 } \\cup \\mathcal W _ { 2 } ) ^ { \\frac 1 p } = \\lambda _ { 1 } ( p , \\mathcal W _ { 1 } ) ^ { \\frac 1 p } , \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} g _ 2 ( x ) P _ j '' + g _ 1 ( x ) P _ j ' + a _ j P _ j = 0 , \\ j = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{gather*} \\frac { q r s - q ^ { - 1 } s r } { q - q ^ { - 1 } } = 1 . \\end{gather*}"}
-{"id": "3312.png", "formula": "\\begin{align*} \\int _ { y ' \\prec x ' } ^ { } d x ' \\int _ { y \\prec x } ^ { } d y \\prod _ { i = 1 } ^ { N } \\hat { m } ( y _ i ) q _ t ^ { N , N + 1 } \\left ( \\left ( x , y \\right ) , \\left ( x ' , y ' \\right ) \\right ) . \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} x _ \\alpha = a _ - ( \\alpha ) \\vee b _ + ( \\alpha - \\delta ) \\end{align*}"}
-{"id": "9967.png", "formula": "\\begin{align*} m \\dot { q } _ 2 ( t ^ + ) = m \\dot { q } _ 1 ( t ^ + ) = m _ 1 \\dot { q } _ 1 ( t ^ - ) + m _ 2 \\dot { q } _ 2 ( t ^ - ) \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} ( \\epsilon + a ) \\lambda ^ { 2 } z ^ { 2 } + \\epsilon z ^ { 2 } + ( a + b ) \\lambda z ^ { 2 } + \\epsilon ( \\epsilon + a ) \\lambda z + \\epsilon ^ { 2 } z = 0 , \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} \\sum \\limits _ { j ' = 1 } ^ n | \\langle f _ j , f _ { j ' } \\rangle | ^ 2 = \\frac { n } { m } j \\in \\{ 1 , . . . , n \\} . \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} F ^ n ( y _ n ) = ( \\sigma ^ n \\tilde \\omega , f _ { \\omega _ { - 1 } } \\circ \\dots \\circ f _ { \\omega _ { - n } } ( q ) ) = ( \\sigma ^ n \\tilde \\omega , p ) . \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ 3 } \\nabla _ i p \\Delta v ^ i \\phi ^ 6 d x = - 6 \\int _ { \\mathbb R ^ 3 } ( p - c ) \\Delta v ^ i \\phi ^ 5 \\nabla _ i \\phi d x , \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} S _ { 0 T } ( \\varphi ) = \\begin{cases} \\int _ 0 ^ T L ( \\varphi _ s , \\dot { \\varphi } _ s ) d s , \\varphi \\\\ \\infty . \\end{cases} \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} \\Gamma \\equiv \\Lambda \\circ A ^ { - 1 } = P _ { 1 T } ^ { - 1 } \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} \\mu ^ n = P \\circ ( X ^ n ) ^ { - 1 } \\mu = P \\circ X ^ { - 1 } \\ , , \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} \\rho = 0 . 0 2 9 5 ~ ~ ~ ~ ~ ~ ~ \\beta = 0 . 4 2 ~ ~ ~ ~ ~ ~ ~ \\eta = 1 0 ^ { - 2 0 } ~ ~ ~ ~ ~ ~ ~ b = 5 1 . 5 3 . \\end{align*}"}
-{"id": "9735.png", "formula": "\\begin{align*} A _ { t } ^ { \\prime } \\left ( n \\right ) = A _ { t } \\otimes _ { K } K \\left ( \\left ( 2 ^ { k } + 1 \\right ) \\times < 1 > \\bot \\left ( n - 1 \\right ) \\times T _ { P } \\right ) , n \\in \\mathbb { N } - \\{ 0 \\} . \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } d ^ { - k } f ^ { - k } _ * ( [ M _ { r _ n } ] ) = \\mu ^ - _ { | W ^ s } ( M _ { r _ n } ) \\mu ^ + . \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ \\infty 4 ^ { m - \\sum \\limits _ { k = 1 } ^ { m } ( \\delta ^ { ( k - 1 ) } - \\tau ^ { ( k ) } ) } \\ , \\Delta ^ { ( m ) } \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\partial \\Omega _ \\eta } [ \\tilde { \\mu } ] = u [ \\eta , \\psi , \\Psi ] - \\sum _ { j = 1 } ^ { m ^ { \\# } } c _ j \\Xi _ j ( \\cdot / \\eta ) \\ , . \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ b \\binom { b } { k } \\sum _ { j = 1 } ^ k \\frac { ( - 1 ) ^ j } { j } \\cdot \\binom { a } { k - j } = - \\binom { a + b } { b } \\sum _ { j = a + 1 } ^ { a + b } \\frac 1 j \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} m = a b + c \\end{align*}"}
-{"id": "9585.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\mbox { v a r } \\left ( \\frac { R _ { n , i } V _ { n , i } - \\rho _ n \\mu _ { n , i } } { s _ n \\sqrt { n \\rho _ n } } \\right ) = 1 . \\end{align*}"}
-{"id": "3520.png", "formula": "\\begin{align*} \\frac { P ( z ) } { Q ( z ) } = \\displaystyle \\sum _ { i = 1 } ^ { p } \\displaystyle \\sum _ { r = 1 } ^ { k _ { _ { i } } } \\frac { a _ { i r } } { ( z - z _ { _ { i } } ) ^ { r } } + \\displaystyle \\sum _ { j = 1 } ^ { q } \\displaystyle \\sum _ { s = 1 } ^ { l _ { _ { j } } } \\frac { \\beta _ { j s } z + \\gamma _ { j s } } { \\Big ( z ^ { 2 } - 2 R e ( z _ { _ { j } } ) z + | z _ { _ { j } } | ^ { 2 } \\big ) ^ { s } } \\end{align*}"}
-{"id": "9382.png", "formula": "\\begin{align*} \\Lambda ( \\phi ) ( X , Y ) = \\rho _ A ^ { - 1 } \\circ \\Gamma ^ { A , B , C } ( \\phi ) ( \\widetilde { X } , \\widetilde { Y } ) \\circ \\rho _ C . \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} \\begin{aligned} \\int \\limits _ { | x - e | + | x | \\leq 2 \\tau } \\frac { q ( x ) } { | x | | x - e | } d x & = \\frac { 1 } { 2 } \\int \\limits _ { \\cosh \\rho \\leq 2 \\tau } \\int \\limits _ { 0 } ^ { \\pi } \\int \\limits _ { 0 } ^ { 2 \\pi } q ( \\rho , \\theta , \\phi ) \\sinh \\rho \\sin \\phi d \\theta d \\phi d \\rho . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } v ^ { 2 ^ { 2 k + 1 } } + v ^ 2 + v = a ^ { 2 } / \\epsilon ^ 2 , \\\\ v ^ { 4 } + v ^ { 2 } + v + \\frac { 1 } { \\epsilon ^ { 4 } } ( a ^ { 4 } + b ^ { 4 } + a ^ { 2 } \\epsilon ^ { 2 } ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "7424.png", "formula": "\\begin{align*} \\frac { C _ 2 } { C _ 1 } = \\frac { m _ 1 m _ 2 } { c ( m _ 1 + m _ 2 ) } . \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} W _ { 3 } ( n ) & = 4 5 ( 9 n ^ 2 - 7 3 n + 1 7 6 ) , \\\\ W _ { 4 } ( n ) & = 7 ( 1 2 1 5 n ^ 3 - 1 9 7 1 0 n ^ 2 + 1 2 1 6 8 5 n - 2 6 6 3 9 8 ) , \\\\ W _ { 5 } ( n ) & = 9 4 5 \\left ( 2 4 3 n ^ 4 - 6 5 7 0 n ^ 3 + 7 4 1 6 5 n ^ 2 - 3 9 4 8 7 8 n + 8 0 5 4 4 0 \\right ) , \\\\ W _ { 6 } ( n ) & = 1 6 5 \\left ( 4 5 9 2 7 n ^ 5 - 1 8 6 2 5 9 5 n ^ 4 + 3 3 0 7 0 2 7 5 n ^ 3 - 3 1 0 3 5 9 5 8 1 n ^ 2 + 1 4 9 7 3 9 1 0 1 4 n - 2 9 1 6 6 1 1 7 2 8 \\right ) . \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} \\boldsymbol { \\mathcal { X } } : = \\{ \\mathbf { x } = ( S , y ) \\in \\mathbf { P } \\times \\mathcal { Y } \\ | \\ S \\in \\mathbf { P } ( y ) \\} , \\ \\ \\ \\boldsymbol { \\theta } : \\boldsymbol { \\mathcal { X } } \\to \\mathcal { Y } , \\ ( S , y ) \\mapsto y . \\end{align*}"}
-{"id": "6425.png", "formula": "\\begin{align*} W _ 1 ( \\lambda ) R = \\left \\{ \\ , \\begin{bmatrix} a \\pi & b \\pi ^ n \\\\ c & d \\end{bmatrix} : a , b , c , d \\in D b \\equiv d \\lambda \\mod \\pi D \\ , \\right \\} . \\end{align*}"}
-{"id": "7925.png", "formula": "\\begin{align*} u ( x , t ) = h ( t ) u _ s + \\widetilde U ( x , t ) + w ( x , t ) , \\end{align*}"}
-{"id": "7655.png", "formula": "\\begin{align*} K ( s , t ) = \\begin{cases} s - \\left ( a + \\lambda \\frac { b - a } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ a , t ) , \\\\ s - \\left ( a + ( 2 - \\lambda ) \\frac { b - a } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ t , b ] , \\end{cases} \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} \\| H _ \\varphi \\| _ { H ^ p _ + ( \\R ) \\to H ^ p _ + ( \\R ) } \\leq \\| H _ \\varphi \\| _ { L ^ p ( \\R ) \\to L ^ p ( \\R ) } = \\int _ 0 ^ \\infty t ^ { 1 / p - 1 } \\varphi ( t ) d t . \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} \\frac 1 p + \\frac 1 { p ' } = 1 \\ , , \\mbox { i . e . , } \\ , p ' = \\frac { p } { p - 1 } \\ , . \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\alpha _ 1 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 2 ] = \\frac { \\beta _ 2 \\gamma _ 1 } { \\alpha _ 1 \\beta _ 6 } y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\beta _ 4 } { \\beta _ 6 } y _ 5 , [ y _ 3 , y _ 1 ] = y _ 5 , \\\\ [ y _ 2 , y _ 3 ] = y _ 4 + \\theta _ 3 y _ 5 , [ y _ 3 , y _ 2 ] = - y _ 4 . \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} W _ { n } = p W _ { n - 1 } - q W _ { n - 2 } , \\ n \\geq 2 , \\end{align*}"}
-{"id": "8506.png", "formula": "\\begin{align*} \\widehat { x } = \\{ \\widehat { y } \\times \\{ \\mathbb { 1 } _ \\mathbb { P } \\} : \\ y \\in x \\} \\end{align*}"}
-{"id": "9618.png", "formula": "\\begin{align*} G _ n ( x ; h ) = \\frac { \\left | \\begin{array} { c c c c } F ( x ) & F ( x + h ) & \\cdots & F ( x + n h ) \\\\ f ( x ) & f ( x + h ) & \\cdots & f ( x + n h ) \\\\ \\vdots & \\vdots & & \\vdots \\\\ f ( x + ( n - 1 ) h ) & f ( x + n h ) & \\cdots & f ( x + ( 2 n - 1 ) h ) \\end{array} \\right | } { \\left | \\begin{array} { c c c c } 1 & 1 & \\cdots & 1 \\\\ f ( x ) & f ( x + h ) & \\cdots & f ( x + n h ) \\\\ \\vdots & \\vdots & & \\vdots \\\\ f ( x + ( n - 1 ) h ) & f ( x + n h ) & \\cdots & f ( x + ( 2 n - 1 ) h ) \\end{array} \\right | } , \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{align*} p _ j ( x _ { \\lambda } - x _ { \\eta } ) & = \\inf _ { p _ j ( z ) = 0 } \\left \\{ p _ j ( x _ { \\lambda } - x _ { \\eta } ) - p _ j ( z ) \\right \\} \\\\ & \\leqslant \\inf _ { p _ j ( z ) = 0 } p _ j ( x _ { \\lambda } - x _ { \\eta } - z ) \\\\ & = \\| [ x _ { \\lambda } ] _ j - [ x _ { \\eta } ] _ j \\| _ j < \\varepsilon , \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} \\dot x = \\sum _ { i = 1 } ^ n \\Big ( F _ { 1 i } ( J ( x ) ) u _ { 1 i } ( t ) + F _ { 2 i } ( J ( x ) ) u _ { 2 i } ( t ) \\Big ) e _ i , \\ , x \\in \\mathbb R ^ n , \\end{align*}"}
-{"id": "1479.png", "formula": "\\begin{align*} \\hat u ( \\cdot , t + \\tau ) - \\hat u ( \\cdot , t ) = \\chi ^ 0 _ \\tau ( t ) \\ u ^ 0 + \\sum _ { n = 1 } ^ { N - 1 } \\chi ^ n _ \\tau ( t ) \\left [ u ^ n - u ^ { n - 1 } \\right ] - \\chi ^ N _ \\tau ( t ) \\ u ^ { N - 1 } . \\end{align*}"}
-{"id": "9406.png", "formula": "\\begin{align*} \\underset { i } { \\lim } \\ ( \\Gamma ^ { A , B } ( u _ i ) ( X ) \\Gamma ^ { B , C } ( v _ j ) ( Y ) Z ) & = ( \\Gamma ^ { A , B } ( u ) ( X ) \\Gamma ^ { B , C } ( v _ j ) ( Y ) Z ) \\\\ & = ( \\Gamma ^ { B , C } ( v _ j ) ( Y ) Z \\Gamma ^ { A , B } ( u ) ( X ) ) \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{align*} U _ 0 = M { \\rm d i a g } \\{ I _ { n - n _ D } , - I _ { n _ D } \\} M ^ \\dag \\end{align*}"}
-{"id": "5753.png", "formula": "\\begin{align*} \\hat { \\sigma } _ { C M | Z = \\mathbf { z } } ( t ) = \\hat { D } _ C ( \\mathbf { z } ) \\ , \\hat { \\rho } _ { C M } \\ , { \\hat { D } _ C ( \\mathbf { z } ) } ^ \\dag \\ ; . \\end{align*}"}
-{"id": "7757.png", "formula": "\\begin{align*} k \\mbox { - w d ( Q ) } = n - \\mbox { v c ( $ G _ { Q , k } $ ) } . \\end{align*}"}
-{"id": "5462.png", "formula": "\\begin{align*} \\int _ { \\Omega } | D ^ m u | ^ { 2 p } d x \\leq C _ { N , m } \\int _ { \\Omega } \\sum _ { k = 1 } ^ m \\frac { ( U ^ { ( k ) } ( | x - a | ) ) ^ { 2 p } } { | x - a | ^ { 2 p ( m - k ) } } d x , \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\alpha _ 1 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 6 \\gamma _ 1 } { \\alpha _ 1 ( \\gamma _ 2 + \\gamma _ 4 ) } y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\beta _ 4 } { \\gamma _ 2 + \\gamma _ 4 } y _ 5 , \\\\ [ y _ 2 , y _ 3 ] = y _ 4 + y _ 5 , [ y _ 3 , y _ 2 ] = - y _ 4 . \\end{align*}"}
-{"id": "7098.png", "formula": "\\begin{align*} U _ { m _ j } = U _ { m _ j - 1 } \\cup V ^ * . \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{align*} A _ 2 = \\sum _ { i , j , k , \\ell , m , n } V ^ \\ell _ m ( V ^ { - 1 } ) ^ m _ n \\sigma ( P ^ n _ i ) V ^ i _ j ( 2 P ^ j _ k - \\delta ^ j _ k ) \\otimes P ^ k _ \\ell . \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} W = x ^ 2 + y ^ 3 + z ^ 9 + y w ^ { 1 2 } = 0 \\end{align*}"}
-{"id": "580.png", "formula": "\\begin{align*} d f _ t ( z ) = - \\frac { 2 } { f _ t ( z ) } d t - d \\xi _ t \\end{align*}"}
-{"id": "6095.png", "formula": "\\begin{align*} [ L _ { - k - 1 } , \\ , S _ k ] = 0 , \\ , 1 \\leq k \\leq j - 1 , \\hbox { a n d } \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{align*} \\Delta & = D _ { m a x } - D _ { a c h } ( k ) = \\frac { 1 } { 2 ^ { k } m } \\sum _ { j = 1 } ^ { r } \\frac { S _ j ^ 2 } { N _ j } - \\mathbb { E } [ Y ] ^ 2 . \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{align*} h ( \\alpha ) = \\left \\{ \\begin{array} { c l } \\alpha & \\mbox { f o r a n y a r r o w $ \\alpha \\in Q _ 1 $ w h i c h i s n o t a b o r d e r l o o p } , \\\\ \\alpha - a _ { s ( \\alpha ) } \\alpha ^ 2 & \\mbox { f o r a n y b o r d e r l o o p $ \\alpha \\in Q _ 1 $ } . \\end{array} \\right . \\end{align*}"}
-{"id": "8232.png", "formula": "\\begin{align*} \\overline { \\Lambda } = \\Lambda _ { 2 } ( \\infty , \\Omega ) . \\end{align*}"}
-{"id": "6807.png", "formula": "\\begin{align*} C _ { 0 } k ^ { C _ { p } } A _ { k } ^ { 2 } = C _ { 0 } k ^ { C _ { p } } e ^ { a \\left ( k , p \\right ) } \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } { \\bf C } ' { \\bf F } ^ { - 1 } { \\bf A } _ 1 ( { \\bf F } ^ { - 1 } ) ' { \\bf C } & = & \\left [ \\begin{array} { c c } { \\bf D } _ r & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf 0 } \\end{array} \\right ] \\\\ { \\bf C } ' { \\bf F } ^ { - 1 } { \\bf A } _ 2 ( { \\bf F } ^ { - 1 } ) ' { \\bf C } & = & \\left [ \\begin{array} { c c } { \\bf I } _ r & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf 0 } \\end{array} \\right ] \\end{array} \\right . _ . \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} { n - 2 \\choose r _ 1 - 1 , r _ 2 - 1 } : = \\frac { ( n - 2 ) ! } { ( r _ 1 - 1 ) ! ( r _ 2 - 1 ) ! ( n - r _ 1 - r _ 2 ) ! } . \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{align*} \\Omega _ 1 ( 0 ) - \\Omega _ 1 ( p ) \\equiv - p \\cdot \\Omega _ 1 ' ( 0 ) = p \\cdot \\Omega _ 2 ' ( 0 ) \\equiv \\Omega _ 2 ( p ) - \\Omega _ 2 ( 0 ) \\pmod { p ^ 2 } , \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} \\Omega = \\{ x : M _ { H L } ( f ) ( x ) > \\alpha \\} . \\end{align*}"}
-{"id": "2093.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = \\infty \\ , \\left ( \\frac 1 3 \\right ) \\\\ \\overbrace { \\begin{matrix} 0 & 0 \\\\ 0 & 0 \\\\ t _ 1 & \\theta ^ 0 \\end{matrix} } & \\overbrace { \\begin{matrix} - 1 & \\frac { t _ 2 } { 3 } & \\theta ^ \\infty _ 1 / 3 \\\\ - \\omega & \\frac { \\omega ^ 2 t _ 2 } { 3 } & \\theta ^ \\infty _ 1 / 3 \\\\ - \\omega ^ 2 & \\frac { \\omega t _ 2 } { 3 } & \\theta ^ \\infty _ 1 / 3 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "7598.png", "formula": "\\begin{align*} \\theta '' = \\epsilon ^ 2 e ^ \\psi ( 1 + e ^ { - \\rho } ) ( \\psi '' + ( \\psi ' ) ^ 2 ) + \\epsilon ^ 2 e ^ \\psi e ^ { - \\rho } ( 1 - 2 \\psi ' ) , \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} \\mathbf { K } _ { l } : = \\frac { 1 } { \\left \\vert \\Lambda _ { l } \\right \\vert } \\underset { j , k \\in \\mathcal { D } _ { n } , j \\neq k } { \\sum } \\ \\underset { x \\in \\mathfrak { L } \\cap ( l b _ { j } ) } { \\sum } \\ \\underset { z _ { 1 , 2 } \\in \\mathfrak { L } , | z _ { 1 , 2 } | = 1 } { \\sum } \\ \\underset { y \\in \\mathfrak { L } \\cap ( l b _ { k } ) } { \\sum } \\mathbf { F } \\left ( \\left \\vert x + z _ { 1 } + z _ { 2 } - y \\right \\vert \\right ) \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{align*} \\langle a [ \\nabla N ^ { ( r s ) } _ \\theta + e ^ { \\i \\langle \\theta , \\cdot \\rangle _ { \\mathbb { C } ^ d } } e _ r \\otimes e _ s ] , \\nabla \\phi \\rangle = 0 , ( \\phi \\in [ H ^ 1 _ \\theta ( Y ) \\perp e ^ { \\i \\langle \\theta , \\cdot \\rangle _ { \\mathbb { C } ^ d } } ] ^ n ) \\end{align*}"}
-{"id": "807.png", "formula": "\\begin{align*} \\| \\tilde { F } ( t - s ) \\| _ { L ^ r ( B ^ c _ { A \\sqrt t } ) } & = ( t - s ) ^ { - 2 + 3 / ( 2 r ) } \\Bigl ( \\int _ { | x | \\ge A \\sqrt t / \\sqrt { t - s } } | \\tilde { F } ( x , 1 ) | ^ r \\dd x \\Bigr ) ^ { 1 / r } \\\\ & \\le C A ^ { - 4 + 3 / r } t ^ { - 2 + 3 / ( 2 r ) } . \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} a _ { j } a _ { k } = \\sum _ { i } c _ { j , k } ^ { i } a _ { i } , \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} \\tau = \\inf \\{ t : N ^ + _ t \\geq 1 \\} \\wedge \\inf \\{ t : N ^ - _ t \\geq 1 \\} \\wedge 1 . \\end{align*}"}
-{"id": "9127.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ n g _ i b _ i \\equiv 0 \\bmod d . \\end{gather*}"}
-{"id": "8840.png", "formula": "\\begin{align*} h _ v & \\in \\{ 1 , 2 , \\ldots , n \\} v \\in V \\\\ h _ u & = h _ v + \\omega ( u , v ) \\\\ h _ u & \\not = h _ v . \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{align*} N ( s ) = \\frac { - n _ 1 X _ 1 - n _ 2 X _ 2 } { | ( n _ 1 , n _ 2 ) | } . \\end{align*}"}
-{"id": "9864.png", "formula": "\\begin{align*} G ( \\rho ) = - \\dfrac { \\sigma } { \\pi } \\big ( ( \\rho ) \\ , \\rho + \\cos ( \\rho ) \\big ) + C _ 1 \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} ( a ^ + ) ^ n \\vert 0 \\rangle = \\sqrt { \\frac { n ! k ! } { ( k - n ) ! } } ~ \\vert n \\rangle \\end{align*}"}
-{"id": "4772.png", "formula": "\\begin{align*} \\varphi = \\big ( f ( \\alpha ) , - c _ { \\bar { \\alpha } } A ' _ { \\bar { \\alpha } } \\big ) \\mbox { a n d } \\psi = \\big ( - c _ { { \\alpha } } A ' _ { { \\alpha } } , f ( \\bar { \\alpha } ) \\big ) . \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{align*} \\kappa = ( ( k _ 1 , t _ 1 ) , ( k _ 2 , t _ 2 ) , \\ldots , ( k _ { n } , t _ { n } ) ) \\end{align*}"}
-{"id": "9768.png", "formula": "\\begin{align*} E _ { \\epsilon } ( \\tilde { u } ) = E _ { \\epsilon } ( u ) + \\int _ M \\frac { 1 } { 2 } | u | ^ 2 | j \\phi | ^ 2 - \\langle j u , j \\phi \\rangle . \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} [ e _ 1 , e _ 2 ] = \\alpha _ 2 e _ 4 + \\alpha _ 3 e _ 5 , [ e _ 2 , e _ 1 ] = - \\alpha _ 2 e _ 4 + \\alpha _ 4 e _ 5 , [ e _ 3 , e _ 1 ] = \\beta _ 3 e _ 5 , [ e _ 3 , e _ 3 ] = \\beta _ 7 e _ 5 . \\end{align*}"}
-{"id": "5547.png", "formula": "\\begin{align*} z _ { n s } = R + P _ { 2 n - 2 p - 2 } . \\end{align*}"}
-{"id": "5817.png", "formula": "\\begin{align*} - \\Delta f - \\frac { R [ \\bar g ] } { n - 1 } f = 0 f > 0 \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} \\mathbb { E } J _ n ^ 2 = \\frac { 1 } { n ^ 2 } \\mathbb { E } ( T _ n - \\hat { T } ^ { ( n ) } _ n ) ^ 2 \\leq \\frac { 1 } { n ^ 2 } \\left ( \\mathbb { E } T _ n ^ 2 + \\mathbb { E } \\left ( \\hat { T } ^ { ( n ) } _ n \\right ) ^ 2 \\right ) \\leq 2 C \\end{align*}"}
-{"id": "5680.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } d \\mu = \\alpha ( t ) \\mu ( t ) d t + \\beta ( t ) d t , \\\\ \\mu ( 0 ) = \\mu ( T ) , \\end{array} \\right . \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} X _ { n + 1 } ^ { \\delta , \\tau , \\Delta t } = S _ { \\Delta t } X _ { n } ^ { \\delta , \\tau , \\Delta t } + \\Delta t S _ { \\Delta t } G _ \\delta \\bigl ( X _ { n } ^ { \\delta , \\tau , \\Delta t } \\bigr ) + e ^ { \\tau A } S _ { \\Delta t } \\sigma _ \\delta \\bigl ( X _ { n } ^ { \\delta , \\tau , \\Delta t } \\bigr ) \\Delta W _ n , \\end{align*}"}
-{"id": "7530.png", "formula": "\\begin{align*} z \\phi _ 0 z & = y \\left ( 1 + \\frac { 1 } { 2 } G ( y ^ { - 1 } k y ) \\right ) \\phi _ 0 \\left ( 1 - \\frac { 1 } { 2 } G ( y ^ { - 1 } k y ) \\right ) y ^ { - 1 } + O ( t ^ { - 2 } \\mathcal L ) \\\\ & = \\phi _ 0 + \\frac { 1 } { 2 } \\left [ G ( k ) , \\phi _ 0 \\right ] + O ( t ^ { - 2 } \\mathcal L ) \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} d ^ b ( y ) = \\sum _ { k = 0 } ^ { \\infty } \\frak n _ k ( y ; b , \\dots , b ) . \\end{align*}"}
-{"id": "1350.png", "formula": "\\begin{align*} W ' \\circ W \\ ; = \\ ; \\Phi ( W ' \\cup W ) . \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} L _ { j + 2 } \\cap X _ { 1 , \\ , j + 2 , \\ , 3 } = 0 . \\end{align*}"}
-{"id": "7738.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i = 1 } ^ N | \\langle \\Theta _ i , \\xi \\rangle | \\geq \\frac { c _ 2 } { \\sqrt { n } } - \\delta \\textrm { a n d } \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\exp \\left \\{ \\left ( \\frac { \\sqrt { n } | \\langle \\Theta _ i , \\xi \\rangle | + 1 } { 2 } \\right ) ^ { \\alpha } \\right \\} \\leq 3 C _ 3 + N ^ { 1 / 4 } \\delta . \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} 1 1 9 9 0 8 7 & = 4 \\ast 2 9 9 7 7 2 - 1 , \\\\ 1 7 9 8 6 3 1 & = 6 \\ast 2 9 9 7 7 2 - 1 , \\\\ 3 5 9 7 2 6 3 & = 1 2 \\ast 2 9 9 7 7 2 - 1 \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{align*} g _ c ( y ) = c ( z - y ) ( z - x ) ( y - x ) + ( 1 + x z ) ( z - x ) - ( 1 + x y ) ( y - x ) - ( 1 + y z ) ( z - y ) , \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} \\lim _ { t \\downarrow 0 } \\frac 1 t P _ { 1 \\to 2 } ( t ) = g / 2 = \\lim _ { t \\downarrow 0 } \\frac 1 t P _ { 2 \\to 1 } ( t ) \\end{align*}"}
-{"id": "6660.png", "formula": "\\begin{align*} H ( m _ k , \\lambda m _ k ) - H ( 0 , 0 ) = r _ k ( 1 , \\tilde { \\lambda } ) . \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} \\nu _ k ( t ) = q ^ { d k } \\sum _ { { \\bf m } \\in \\mathbb F _ q ^ d } \\widehat { S _ t } ( { \\bf m } ) \\left ( \\prod _ { j = 1 } ^ k \\overline { \\widehat { E _ j } } ( { \\bf m } ) \\right ) . \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{align*} \\partial _ t \\psi _ k = - \\mathrm i H _ k \\psi _ k + F _ k ( \\Psi , \\overline { \\Psi } ) , \\ \\ k = 1 , \\cdots , N . \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{align*} \\sum _ { \\| y \\| _ 1 \\leq R ' } \\left | s \\left ( \\overline { \\xi } _ i ( x _ i + y ) \\right ) \\right | = O ( \\log ( 1 + R ) ) . \\end{align*}"}
-{"id": "7206.png", "formula": "\\begin{align*} p ( s , t ) = ( - 2 - 3 s + s ^ 2 ) t ^ 4 + ( 4 + 4 s - s ^ 2 - s ^ 3 ) t ^ 2 - 1 , \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} G ^ { + / - } ( z ) : = \\lim _ { n \\to \\infty } \\frac { \\log ^ + \\Vert f ^ { + n / - n } ( z ) \\Vert } { d ^ n } , \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{align*} A & : = \\left ( \\varphi - { m \\cdot x \\over \\gamma _ n | x | ^ n } \\right ) ( u - c ) + ( c \\cdot x ) u , \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} | \\lambda _ { p , n } | & = \\frac { \\Vert d f ^ n _ p \\psi _ p ' ( 0 ) \\Vert } { \\Vert \\psi _ n ' ( 0 ) \\Vert } \\\\ & \\leq C ' _ p | \\delta | ^ n . \\end{align*}"}
-{"id": "9802.png", "formula": "\\begin{align*} { { U } } = \\sum \\limits _ { i , j , k = 0 } ^ N { { { { U } } _ { i j k } } { \\ell _ i } ( \\xi ) { \\ell _ j } ( \\eta ) { \\ell _ k } ( \\zeta ) } , \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} { f _ { \\left . G _ K \\right | T } } \\left ( { \\left . x \\right | t } \\right ) = \\left \\{ { { \\mathcal M ^ { - 1 } } \\phi } \\right \\} \\left ( \\left . x \\right | t \\right ) = \\frac { 1 } { { 2 \\pi \\rm i } } \\int \\nolimits _ { c - { \\rm i } \\infty } ^ { c + { \\rm i } \\infty } { { x ^ { - s } } } \\phi \\left ( { \\left . s \\right | t } \\right ) d s , \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} 2 ( x _ 1 - \\zeta ) ( x _ 1 - \\overline { \\zeta } ) = ( ( x _ 1 - \\overline { \\zeta } ) + ( x _ 1 - \\zeta ) ) ( x _ 1 - x _ 2 ) \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} \\left ( \\sum _ { j _ { 1 } , \\dots , j _ { m } = 1 } ^ { \\infty } \\left \\vert A ( e _ { j _ { 1 } } , \\dots , e _ { j _ { m } } ) \\right \\vert ^ { \\frac { 1 } { 1 - \\left \\vert 1 / \\mathbf { p } \\right \\vert } } \\right ) ^ { 1 - \\left \\vert 1 / \\mathbf { p } \\right \\vert } \\leq D _ { m , \\mathbf { p } } ^ { \\mathbb { K } } \\Vert A \\Vert \\end{align*}"}
-{"id": "4376.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty C _ k x ^ { k + 1 } = \\frac { 1 - \\sqrt { 1 - 4 x } } { 2 } \\equiv \\frac { 1 - ( 1 - 4 x ) ^ { ( q + 1 ) / 2 } } { 2 } - x ^ q \\pmod { ( x ^ { q + 1 } , p ) } , \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{align*} \\textup { O s c } ^ \\lambda _ { \\mu } ( x , r ) & = \\frac { 1 } { ( 4 0 ) ^ { 2 ( n - 1 ) } ( 1 1 t ) ^ { n - 1 } } \\ , \\textup { O s c } ^ \\lambda _ { \\mu _ 1 ^ l } ( w _ 0 + 1 1 t x , 1 1 t r ) \\\\ & \\leq \\frac { 2 ^ { 2 n + 1 } \\ , } { ( 4 0 ) ^ { 2 ( n - 1 ) } ( 1 1 t ) ^ { n - 1 } } \\ , \\bar C ( \\lambda ) ^ 2 \\ , \\tau \\ , ( 1 1 t r ) ^ { n - 1 } \\leq \\delta _ { \\ref { t : R e i f d i s c r e t o } } ^ 2 ( \\lambda ) \\ , r ^ { n - 1 } . \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} \\| \\widetilde { \\varphi } ( T ) \\| & = \\sup \\big \\{ \\| \\widetilde { \\varphi } ( T ) ( x ) \\| : x \\in X , \\| x \\| < 1 \\big \\} \\\\ & = \\sup \\big \\{ \\| \\widetilde { \\varphi } ( T ) ( \\varphi ( a ) ( x ) ) \\| : a \\in A , x \\in X , \\| a \\| \\leq 1 , \\| x \\| < 1 \\big \\} \\\\ & = \\sup \\big \\{ \\| \\varphi ( T ( a ) ) ( x ) \\| : a \\in A , x \\in X , \\| a \\| \\leq 1 , \\| x \\| < 1 \\big \\} \\\\ & \\leq \\ \\| T \\| , \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} \\Delta ^ g u = - \\sum _ { i , j = 1 } ^ n a _ i H _ { i j , j } , | d u | _ g ^ 2 = \\sum _ { i , j = 1 } ^ n a _ i a _ j H _ { i j } . \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} S ( e _ i \\otimes e _ i ) = \\bar { \\psi } _ i \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} I _ 1 \\ , = \\ , [ 1 0 ^ { - 1 } , \\infty ) , \\quad \\mbox { a n d } I _ j \\ , = \\ , [ 1 0 ^ { - j } , 1 0 ^ { - j + 1 } ) \\quad \\mbox { f o r \\ } j = 2 , 3 , \\dots . \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} Q ^ s ( x ) = Q ^ s [ 1 ] \\circ x + \\sum Q ^ { s + i } [ 1 ] \\circ z _ i . \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{gather*} c _ + ( \\alpha , \\beta ) : = a _ + ( \\alpha ) \\wedge b _ + ( \\beta ) \\\\ c _ - ( \\alpha , \\beta ) : = a _ + ( \\alpha ) \\wedge b _ + ( \\beta ) \\wedge ( a _ - ( \\alpha ) \\vee b _ - ( \\beta ) ) . \\end{gather*}"}
-{"id": "2971.png", "formula": "\\begin{align*} \\sup _ { j \\ge 0 \\ } \\ \\| P _ { n } T ^ { \\ , j } \\ , e _ { b _ { n } } \\| & \\le \\smash [ t ] { \\sup _ { j \\in [ b _ { n } , b _ { n + 1 } ) } \\ \\prod _ { s = b _ { n } + 1 } ^ { j } | w _ { s } | . } \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{align*} g ( \\alpha ^ \\sharp , A ( X ) ) & = g ( A ^ * , \\alpha \\otimes X ) , \\\\ \\left . \\frac { d } { d t } \\right | _ 0 g _ t ( \\alpha , \\beta ) & = - g ( \\alpha , \\dot { J } J \\beta ) , \\end{align*}"}
-{"id": "4742.png", "formula": "\\begin{align*} f _ i ( x ( v ) ) & = f _ i ( x ^ 0 ) { + } \\int _ 0 ^ v \\frac { d f _ i ( x ( s ) ) } { d s } d s \\\\ & = f _ i ( x ^ 0 ) { + } \\int _ 0 ^ v \\sum _ { j = 1 } ^ m L _ { f _ j } f _ i ( x ( s ) ) u _ j ( s ) d s . \\end{align*}"}
-{"id": "7495.png", "formula": "\\begin{align*} \\langle [ \\phi , b ] , [ \\phi , b ] \\rangle & = \\langle b , [ \\phi ^ * , [ \\phi , b ] ] \\rangle \\\\ & = \\langle b , [ \\phi , [ \\phi ^ * , b ] ] \\rangle \\\\ & = \\langle [ \\phi ^ * , b ] , [ \\phi ^ * , b ] \\rangle \\end{align*}"}
-{"id": "1038.png", "formula": "\\begin{align*} E ( I u _ \\lambda ( 0 ) ) = \\frac { 1 } { 2 } \\| I u _ \\lambda ( 0 ) \\| ^ 2 _ { \\dot { H } ^ { k / 2 } _ x } + \\frac { 1 } { 4 } \\| I u _ \\lambda ( 0 ) \\| ^ 4 _ { L ^ 4 _ x } . \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{align*} \\mathrm { I m } Z ( [ 0 , x ] ) & = \\sum _ \\alpha \\alpha X ( [ a _ - ( \\alpha ) , x _ \\alpha ] ) \\\\ & > \\sum _ \\beta \\beta X ( [ a _ - ( \\beta + \\delta ) , x _ { \\beta + \\delta } ] ) \\\\ & = \\sum _ \\beta \\beta X ( [ a _ - ( \\beta + \\delta ) \\wedge b _ + ( \\beta ) , b _ + ( \\beta ) ] ) \\\\ & = \\sum _ \\beta \\beta ( X ( [ b _ - ( \\beta ) , b _ + ( \\beta ) ] ) - X ( [ b _ - ( \\beta ) , y _ \\beta ] ) ) \\\\ & = 0 - \\mathrm { I m } Z ( [ 0 , y ] ) \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } b _ { m } ( 2 n ) x ^ { n } & = \\left ( \\sum _ { n = 0 } ^ { \\infty } \\binom { m + 2 n - 1 } { n } x ^ { 2 n } \\right ) \\left ( \\sum _ { n = 0 } ^ { \\infty } b _ { m } ( n ) x ^ { n } \\right ) \\\\ & = \\sum _ { n = 0 } ^ { \\infty } \\left ( \\sum _ { j = 0 } ^ { n } \\binom { m + 2 ( n - j ) - 1 } { 2 ( n - j ) } b _ { m } ( j ) \\right ) x ^ { n } . \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} \\| u ( t ) - u _ s \\| _ { L ^ \\infty ( \\Omega ) } = O ( t ^ { - 1 / 2 } ) \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} k _ { \\nu } ( \\theta ) = \\frac { M _ { \\nu } ( \\theta ) - 1 } { \\theta M _ { \\nu } ( \\theta ) } . \\end{align*}"}
-{"id": "5145.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\frac { H _ { n } \\left ( x \\right ) } { n ! } z ^ { n } = e ^ { 2 x z - z ^ { 2 } } . \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} H _ n = E _ { n _ { \\epsilon } } ^ c ( n ) \\cap \\hat { A } _ { n _ { \\epsilon } } ( n ) , \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} \\begin{pmatrix} - 2 7 . 4 3 1 0 + 5 0 . 9 4 8 3 i \\\\ - 4 . 0 6 4 7 + 5 . 7 5 4 3 i \\\\ \\hphantom { - . } 0 . 0 0 0 0 + 0 . 0 0 0 0 i \\\\ \\hphantom { - . } 2 . 7 6 9 4 - 1 . 2 2 2 0 i \\\\ \\hphantom { - . } 0 . 6 8 7 5 + 0 . 9 2 0 3 i \\end{pmatrix} + \\mathbb { C } \\begin{pmatrix} 1 - 3 i \\\\ 0 \\\\ 1 \\\\ 0 \\\\ 0 \\end{pmatrix} \\end{align*}"}
-{"id": "4261.png", "formula": "\\begin{align*} \\sum _ { v } \\textup { i n v } _ v ( ( ( \\pi ^ a ) ^ * ( \\alpha _ i ) ) ( R _ v ) ) & = \\sum _ v \\textup { i n v } _ v ( \\alpha _ i ( \\pi ^ a ( R _ v ) ) ) = \\sum _ { v \\in S } \\textup { i n v } _ v ( \\alpha _ i ( Q _ v ) ) = \\sum _ { v \\in S } \\textup { i n v } _ v ( \\alpha _ i ( P _ v ) ) \\\\ & = \\sum _ { v } \\textup { i n v } _ v ( \\alpha _ i ( P _ v ) ) = 0 , \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } f \\left ( z \\right ) f \\left ( z + c _ { 1 } \\right ) - q \\left ( z \\right ) = p _ { 1 } \\left ( z \\right ) e ^ { \\alpha \\left ( z \\right ) } , \\\\ f \\left ( z \\right ) f \\left ( z + c _ { 2 } \\right ) - q \\left ( z \\right ) = p _ { 2 } \\left ( z \\right ) e ^ { \\beta \\left ( z \\right ) } , \\end{array} \\right . \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} a \\xi _ i = \\xi _ j a . \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} \\dot { y } _ 2 = - e ^ { y _ 2 - y _ 3 } , \\dot { y } _ 3 = e ^ { y _ 2 - y _ 3 } \\end{align*}"}
-{"id": "8629.png", "formula": "\\begin{align*} ( g \\cdot \\omega ) ( g ' ) = \\omega ( g ^ { - 1 } g ' ) \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} ( y _ 0 + c ) - \\varphi ( x ) & = y _ 1 - \\varphi ( x ) \\\\ & = \\varphi ( x _ 1 ) - \\varphi ( x ) \\\\ & \\leq x _ 1 - x \\\\ & = x _ 0 + c - x . \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{align*} \\| e ^ { ( t - s ) \\Delta } | u ( s ) | ^ \\alpha u ( s ) \\| _ { L ^ r } & \\le \\| | u ( s ) | ^ \\alpha u ( s ) \\| _ { L ^ r } = \\| u ( s ) \\| _ { L ^ { ( \\alpha + 1 ) r } } ^ { \\alpha + 1 } \\\\ & = s ^ { - \\frac { \\alpha + 1 } { \\alpha } + \\frac { N } { 2 r } } \\| f \\| _ { L ^ { ( \\alpha + 1 ) r } } ^ { \\alpha + 1 } . \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} R ' = ( R \\otimes \\Q ) \\cap L \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} ( x ' ) ^ k x ^ k = x ' x \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{align*} d _ l : = l + \\frac { \\textrm { d i m } ( G ) - \\textrm { d i m } ( K ) } { 2 } . \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} E _ 0 : = \\frac { 1 } { n } \\mathrm { T r } _ B \\left [ \\hat { H } _ B \\ , \\hat { \\sigma } _ B \\right ] \\ ; , S _ 0 : = \\frac { S ( \\hat { \\sigma } _ B ) } { n } \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} g _ 1 k = \\begin{pmatrix} 0 & x \\\\ y & 0 \\end{pmatrix} + \\begin{pmatrix} a _ i + \\gamma x & b _ i + \\delta x \\\\ c _ i + \\alpha y & d _ i + \\beta y \\end{pmatrix} p . \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} S = \\sum _ { k = 1 } ^ n g \\left ( \\nu _ k - \\frac { 1 } { 2 } \\right ) \\ ; , \\end{align*}"}
-{"id": "8584.png", "formula": "\\begin{align*} \\mathrm { d e g } ( K _ { \\Sigma ^ 2 } ) = 2 g - 2 \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} T _ h ^ n = O ( h ) ( 1 + { ( b - x _ { n - 1 } ) } ^ { - \\alpha } ) , \\mbox { f o r } 1 \\le n \\le P - 1 . \\end{align*}"}
-{"id": "7745.png", "formula": "\\begin{align*} C = 4 \\sum _ { i : 0 < \\lambda _ i < 1 } ( X _ i ^ 2 + Y _ i ^ 2 ) , \\end{align*}"}
-{"id": "1269.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mu } \\left ( Z _ { 0 } \\in F _ { 0 } , Z _ { T } \\in F _ { T } \\right ) = \\mu ( F _ { 0 } \\times F _ { T } ) \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{align*} F _ { n } = F _ { n - 1 } + F _ { n - 2 } , \\ , \\ , n \\ge 2 , \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} m _ i v _ i = \\sum _ { i \\xrightarrow [ \\alpha ] { } j } u _ \\alpha - \\sum _ { k \\xrightarrow [ \\alpha ] { } i } u _ \\alpha \\end{align*}"}
-{"id": "4281.png", "formula": "\\begin{align*} \\begin{cases} f _ { n } ( X ) = c X ^ { ( n ^ 2 - 1 ) / 2 } + \\cdot \\cdot \\cdot , n \\textrm { o d d } ; \\\\ f _ { n } ( X ) = c X ^ { ( n ^ 2 - 4 ) / 2 } + \\cdot \\cdot \\cdot , n \\textrm { e v e n } , \\end{cases} \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\lambda } ^ d \\xi _ t ( O ) & \\leq e ^ { - t } \\sum _ { n = 0 } ^ { \\infty } \\frac { t ^ n \\lambda ^ n } { n ! } \\big [ \\sum _ { \\overrightarrow { x } \\in V _ n } ( E { \\widetilde { \\rho } } ^ 2 ) ^ { | x _ n | - 1 } \\big ] M ^ { 2 n } \\\\ & = e ^ { - t } \\sum _ { n = 0 } ^ { \\infty } \\frac { t ^ n \\lambda ^ n M ^ { 2 n } } { n ! E { \\widetilde { \\rho } } ^ 2 } \\big [ \\sum _ { \\overrightarrow { x } \\in V _ n } ( E { \\widetilde { \\rho } } ^ 2 ) ^ { | x _ n | } \\big ] \\end{align*}"}
-{"id": "4424.png", "formula": "\\begin{align*} K _ 1 \\cap \\tau K _ 1 \\tau ^ { - 1 } \\cap \\tau ^ 2 K _ 1 \\tau ^ { - 2 } = \\lbrace 1 _ G \\rbrace \\end{align*}"}
-{"id": "9354.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { J + 1 } \\mathcal { V } _ r ( \\gamma _ j ^ s ) \\leq \\mathcal { V } _ r ( \\sigma ) + C _ r ( \\mathcal { V } ^ 2 _ r ( \\sigma ) \\wedge \\mathcal { V } ^ r _ r ( \\sigma ) ) . \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} \\mbox { d i v $ w _ k ^ j $ } = 0 , w _ k ^ j | _ { \\partial \\Omega } = 0 , w _ k ^ j \\to 0 \\quad \\mbox { a s $ | x | \\to \\infty $ } \\end{align*}"}
-{"id": "6253.png", "formula": "\\begin{align*} \\overline { \\pi _ V A \\pi _ U } = \\overline { \\pi _ U A \\pi _ V } = 0 . \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} u ( x ) = q ( x '' ) \\ , x _ n + c \\ , \\Phi _ 2 ( x _ n , x _ { n + 1 } ) ; \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{align*} [ T _ { \\bar { z } ^ m } , T _ { z ^ n } ] ( z ^ k ) = \\begin{cases} \\frac { m n } { ( k + 1 ) ( k + n + 1 ) } \\ , z ^ { k + n - m } & k \\geq m \\\\ \\frac { k + n - m + 1 } { k + n + 1 } \\ , z ^ { k + n - m } & m - n \\leq k \\leq m - 1 . \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} B ( h ) = \\int _ 0 ^ \\infty h ( r ) d B _ r = \\sum _ { j = 1 } ^ d \\int _ 0 ^ \\infty h ^ j ( r ) d B _ r ^ j , h = ( h ^ 1 , \\ldots , h ^ d ) \\in H , \\end{align*}"}
-{"id": "9937.png", "formula": "\\begin{align*} T T ^ \\ast F = \\left ( \\frac { \\widetilde F } { | \\tau + | \\xi | ^ 2 | ^ { 1 / 2 } } \\right ) ^ \\vee = F \\ast \\left ( \\frac { 1 } { | \\tau + | \\xi | ^ 2 | ^ { 1 / 2 } } \\right ) ^ \\vee = : F \\ast K . \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\frac { E [ k \\mbox { - w d } ( R _ n ) ] } { n } = c _ { k } \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} { \\displaystyle - 2 \\mathrm { i } ( \\partial \\theta ^ { k } _ { \\psi } , \\varphi ) + B \\left ( \\overline { \\mathbf { A } } ^ { k } _ { h } ; \\overline { \\theta } _ { \\psi } ^ { k } , \\varphi \\right ) = \\sum \\limits _ { j = 1 } ^ 5 Q _ j ^ { k } ( \\varphi ) , } \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} c _ + > c _ * : \\lim _ { t \\to \\infty } \\| u ( t ) - u _ { b _ { \\infty } } \\| _ { H ^ 1 _ { \\mu } ( \\mathbb { R } \\times \\mathbb { T } ) } = 0 , \\end{align*}"}
-{"id": "2240.png", "formula": "\\begin{align*} \\frac { 1 } { { } _ 1 F _ 1 ( 1 ; N + 1 ; x ) } = \\frac { x ^ N / N ! } { e ^ t - \\sum _ { n = 0 } ^ { N - 1 } x ^ n / n ! } = \\sum _ { n = 0 } ^ \\infty B _ { N , n } \\frac { x ^ n } { n ! } \\ , , \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} { \\mathcal M } ( L _ 1 , L _ 2 ; a , b ) = \\bigcup _ { k , E } { \\mathcal M } ( L _ 1 , L _ 2 ; a , b ; k , E ) \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{align*} \\dot { \\rho } + \\dfrac { 4 } { 3 } \\left ( \\dfrac { \\dot { a } } { a } + \\dfrac { \\dot { b } } { b } \\right ) \\rho = 0 . \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} L ( a ) = \\lim _ { r \\to \\infty } r ^ { \\frac { 2 } { \\alpha } } f _ a ( r ) \\in \\R \\end{align*}"}
-{"id": "5954.png", "formula": "\\begin{align*} T ^ { 1 / 2 } \\left ( \\hat { \\theta } _ { T } - \\theta _ { 0 } \\right ) = \\delta \\xi _ { 0 } + \\frac { 1 } { { T ^ { 1 / 2 } } } \\sum _ { t = 1 } ^ { T } \\ell _ { t } \\left ( Y _ { t } , \\Omega _ { t } \\right ) + o _ { p } \\left ( 1 \\right ) , \\end{align*}"}
-{"id": "4929.png", "formula": "\\begin{align*} M _ { \\pm \\gamma } = \\left ( M _ { { i j } _ { \\pm \\gamma } } \\right ) M _ { { i j } _ { \\pm \\gamma } } = \\begin{pmatrix} \\frac { s _ { i j } \\pm \\gamma c _ { i j } } { 2 } & \\frac { s _ { i j } \\mp \\gamma c _ { i j } } { 2 } \\\\ \\frac { s _ { i j } \\mp \\gamma c _ { i j } } { 2 } & \\frac { s _ { i j } \\pm \\gamma c _ { i j } } { 2 } \\end{pmatrix} , \\ 1 \\leq i , j \\leq n \\end{align*}"}
-{"id": "3617.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } n ^ { 1 / 2 } e ^ { - t \\sqrt { n } } \\sim \\frac { 4 } { t ^ 3 } ~ { \\rm a n d } ~ \\sum _ { n \\geq 1 } ( - 1 ) ^ n n ^ { 1 / 2 } e ^ { - t \\sqrt { n } } = O ( 1 ) . \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} 0 \\neq [ [ L _ { - 1 } , \\ , L _ 1 ] , \\ , L _ r ] = [ [ L _ { - 1 } , \\ , L _ r ] , \\ , L _ 1 ] \\subseteq [ S _ { r - 1 } , \\ , L _ 1 ] \\subseteq S , \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} & \\sum _ { d = O ( B ^ { 1 - \\frac { 1 } { ( 2 + \\lambda ) r } } ) } O _ { \\varepsilon _ i , \\sigma } \\left ( \\frac { B ^ { 1 - \\frac { 1 } { 2 r } + \\sigma ( \\frac { 3 } { 4 } + \\frac { 1 } { 8 r } ) } } { d ^ { 1 + \\frac { \\sigma } { 2 } } } \\right ) = O _ { \\varepsilon _ i , \\sigma } ( B ^ { 1 - \\frac { 1 } { 2 r } + \\sigma ( \\frac { 3 } { 4 } + \\frac { 1 } { 8 r } ) } ) . \\end{align*}"}
-{"id": "4904.png", "formula": "\\begin{align*} \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! C \\ ! \\ ! = \\ ! \\ ! \\alpha _ 1 \\log { \\left ( 1 + \\norm { \\mathbf { \\hat { h } } _ { s d } ^ { ( b ) } } ^ 2 P _ { s } \\right ) } \\ ! \\ ! + \\ ! \\ ! \\alpha _ 2 \\log { \\left ( 1 + \\norm { \\mathbf { \\hat { h } } _ { s d } ^ { ( m ) } } ^ 2 P _ { s } \\right ) } . \\end{align*}"}
-{"id": "9985.png", "formula": "\\begin{align*} P \\tilde { E } _ i + \\tilde { E } _ i ^ T P & < 0 \\ , , i = 1 , \\dots , N _ 1 \\\\ P \\tilde { E } _ j + \\tilde { E } _ j ^ T P & \\le 0 \\ , , j = N _ 1 + 1 , \\dots , N \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} \\partial _ { \\xi } ( L _ c + k ^ 2 ) U = \\lambda U , k \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{align*} d X ( t ) = \\sqrt { 2 ( X ^ 2 ( t ) + 1 ) } d W ( t ) + \\left [ \\left ( 2 - 2 n - 2 \\Re ( s ) \\right ) X ( t ) + 2 \\Im ( s ) \\right ] d t . \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} \\alpha : = - \\liminf _ { n \\to \\infty } \\frac { 1 } { n } \\log \\mathbb { P } ( \\frac { 1 } { n } \\kappa ( Y _ { n } ) \\in O ) \\geq - \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log \\mathbb { P } ( \\frac { 1 } { n } \\lambda ( Y _ { n } ) \\in O ' ) \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{gather*} \\frac { { \\rm d } q _ i } { { \\rm d } t } = \\frac { \\partial H _ { \\mathrm { F S } } ^ { A _ 5 } } { \\partial p _ i } , \\frac { { \\rm d } p _ i } { { \\rm d } t } = - \\frac { \\partial H _ { \\mathrm { F S } } ^ { A _ 5 } } { \\partial q _ i } , i = 1 , 2 , \\end{gather*}"}
-{"id": "5901.png", "formula": "\\begin{align*} S & = \\sum _ { \\substack { r \\in [ 5 , y ] \\\\ b \\geq 1 } } f ( r ^ b ) = \\sum _ { \\substack { b \\geq 1 \\\\ 5 \\leq r \\leq y } } \\left ( \\frac { 2 } { r ^ b } + \\frac { 2 } { r ^ b } \\left ( \\frac { r - 1 } { r - 3 } - 1 \\right ) \\right ) \\\\ & = \\sum _ { \\substack { r \\in [ 2 , y ] \\\\ b \\geq 1 } } \\frac { 2 } { r ^ b } + O \\bigg ( \\sum _ { \\substack { r \\geq 5 \\\\ b \\geq 1 } } \\frac { 1 } { r ^ { b + 1 } } \\bigg ) \\\\ & = 2 \\log \\log y + O ( 1 ) , \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} a P ( b ) - a b = a [ \\phi ^ * , m ] = [ \\phi ^ * , a m ] . \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} \\left \\langle z _ n \\frac { x ^ { 2 k } } { ( 2 k ) ! } \\right \\rangle = a _ k , \\left \\langle z _ m \\frac { x ^ { 2 k } } { ( 2 k ) ! } \\right \\rangle = b _ k , c _ { k n } = c _ { k } , c _ { k m } = d _ k , \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} \\Lambda _ { l } : = \\{ ( x _ { 1 } , \\ldots , x _ { d } ) \\in \\mathfrak { L } \\ , : \\ , | x _ { 1 } | , \\ldots , | x _ { d } | \\leq l \\} \\ , l \\in \\mathbb { R } _ { 0 } ^ { + } \\ . \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} F _ t ( M ) \\subseteq \\{ w ( \\cdot , t ) = 0 \\} = M ( t ) \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} K _ { \\xi ^ * } ( x , y ) : = f ( x , y ) - f ( x , \\xi ^ * ) , f ( x , z ) : = \\frac { x - z } { | x - z | ^ n } . \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} T r \\left ( \\left ( 1 + \\frac { \\lambda ^ 2 \\mu ^ 2 } { x ^ 2 } \\right ) \\left ( \\alpha + \\frac { \\lambda ^ 2 } { x ^ 2 } \\right ) \\right ) = 0 . \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{align*} ( R ^ { - 1 } g _ k ) ^ { 3 k + 1 } = R ^ { - 3 k } = ( - I ) ^ k . \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} & \\lambda _ { k } = \\ell ^ 2 k ^ 2 + C _ { 1 } + c _ { k } , \\\\ & \\phi _ { k } ( x ) = C _ { 2 } \\ , a ( x ) ^ { - 1 / 4 } \\sin ( k \\xi ( x ) ) + O ( k ^ { - 1 } ) , \\\\ & \\phi _ { k } ' ( x ) = C _ { 2 } \\ , a ( x ) ^ { - 3 / 4 } \\ , k \\ , \\cos ( k \\xi ( x ) ) + O ( 1 ) , \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} m = \\prod _ { p } p ^ { m _ p } = \\prod _ { \\substack { p \\in \\operatorname { D e } \\\\ p | m } } p ^ { m _ p } \\cdot \\prod _ { \\substack { p \\notin \\operatorname { D e } \\\\ p | m } } p ^ { m _ p } = m _ 1 \\cdot m _ 2 . \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} Q = W _ G ( H ) \\ltimes G \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} M _ { \\alpha _ 1 } ^ { ( \\varphi ' _ { 1 } , \\varphi ' _ { 2 } ) } : = p _ 1 \\big ( ( p _ 2 | _ { C _ { \\alpha _ 1 } } ) ^ { - 1 } ( ( \\varphi _ 1 ' , \\varphi _ 2 ' ) ) \\big ) = \\{ x \\in M _ { \\alpha _ 1 } \\mid \\theta _ { ( \\varphi ' _ { 1 } , \\varphi ' _ { 2 } ) } ( f ( x ) ) = \\theta _ { ( \\varphi _ { 1 } , \\varphi _ { 2 } ) } ( x ) \\} \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} \\frac { 4 \\beta ( 2 ) } { \\pi } = 1 . 1 6 6 2 \\ldots . \\end{align*}"}
-{"id": "9814.png", "formula": "\\begin{align*} \\theta _ { \\lambda } T _ i - T _ i \\theta _ { s _ i \\lambda } = ( v - 1 ) \\frac { \\theta _ { \\lambda } - \\theta _ { s _ i \\lambda } } { 1 - \\theta _ { - \\alpha _ i ^ { \\vee } } } . \\end{align*}"}
-{"id": "9283.png", "formula": "\\begin{align*} \\int _ { f ^ { - 1 } ( B _ { r _ j } ( x ) ) } | A | ^ 2 d \\mu \\bigg | _ { t = t _ j } \\le \\varepsilon _ 3 x \\in \\R ^ N , \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\lambda } ^ d | L _ n | ^ 2 = d ^ { 2 n } ( \\widetilde { E } \\times E ) [ \\prod _ { i = 0 } ^ { n - 1 } F ( S _ i , \\widehat { { S } } _ i ; S _ { i + 1 } , \\widehat { S } _ { i + 1 } ) ] \\end{align*}"}
-{"id": "5057.png", "formula": "\\begin{align*} E _ { \\ge 0 } \\ : = \\ \\bigoplus _ { \\ell \\ge 0 } E _ \\ell \\ , . \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} E _ { \\xi } = \\left \\{ x \\in G : \\ \\left | \\left \\langle b ( x ) , \\xi \\right \\rangle \\right | \\le \\epsilon n ^ { 1 / 2 } \\right \\} . \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} E [ \\Delta Y _ { \\tau } / { \\cal F } _ { \\tau ^ - } ] = - \\Delta A _ { \\tau } + \\Delta A ' _ { \\tau } = - \\Delta A _ { \\tau } { \\bf 1 } _ { \\{ Y _ { \\tau ^ - } = \\overline { \\xi } _ \\tau \\} \\cap D } + \\Delta A ' _ { \\tau } { \\bf 1 } _ { \\{ Y _ { \\tau ^ - } = \\underline { \\zeta } _ \\tau \\} \\cap D ' } { \\rm a . s . } \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} R _ n = \\sum _ { j = 0 } ^ { 2 p - 1 } r _ { n j } e ^ { i \\lambda _ j x } , R _ m = \\sum _ { j = 0 } ^ { 2 p - 1 } r _ { m j } e ^ { i \\lambda _ j x } ; \\lambda _ j ^ { 2 p } = \\Lambda . \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} d ( \\bar \\lambda ) & = ( e _ i , 0 ) , 1 \\le i \\le k d ( \\bar \\lambda ) = ( 0 , e _ i ) , 1 \\le i \\le l ; \\\\ d ( \\bar \\mu ) & = ( e _ j , 0 ) , 1 \\le j \\le k d ( \\bar \\mu ) = ( 0 , e _ j ) , 1 \\le j \\le l . \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\lambda _ k X ( F _ k / E _ { k - 1 } ) \\leq 0 \\end{align*}"}
-{"id": "7469.png", "formula": "\\begin{align*} \\kappa = \\frac { 1 } { 2 } \\min \\{ | \\lambda - \\mu | \\mid \\lambda , \\mu \\in Y \\} \\cup \\{ 1 - | \\lambda - \\mu | \\mid \\lambda , \\mu \\in Y , | \\lambda - \\mu | < 1 \\} > 0 \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{align*} P ( x ) = S ( x ) Q ' ( x ) - S ' ( x ) Q ( x ) \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} \\psi ( x _ 1 , \\dots , x _ n ) = \\psi ( x _ { Q ( 1 ) } , \\dots , x _ { Q ( n ) } ) , \\end{align*}"}
-{"id": "1440.png", "formula": "\\begin{align*} 2 \\int _ { \\mathbb { R } ^ 2 } t \\nabla \\cdot H ^ { ( \\alpha , a ) } \\cdot \\mu r \\partial _ r \\nabla \\cdot H ^ { ( \\alpha , a ) } d x = - 2 \\int _ { \\mathbb { R } ^ 2 } \\mu t | \\nabla \\cdot H ^ { ( \\alpha , a ) } | ^ 2 d x , \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} \\begin{array} { l c l } \\tau _ 3 ( t ) & = & - \\frac { \\sqrt { 6 } \\ , \\big ( 5 y ( t ) ^ 2 + z ( t ) ^ 2 \\big ) } { 2 1 \\ , y ( t ) ^ 3 z ( t ) ^ 4 } f ^ { 1 2 7 } + \\frac { \\sqrt { 6 } \\ , \\big ( 3 y ( t ) ^ 2 - 5 z ( t ) ^ 2 \\big ) } { 4 2 \\ , y ( t ) ^ 3 z ( t ) ^ 4 } ( f ^ { 3 4 7 } + f ^ { 5 6 7 } ) \\\\ & & + \\frac { \\sqrt { 6 } \\ , \\big ( 2 y ( t ) ^ 2 - z ( t ) ^ 2 \\big ) } { 2 1 \\ , y ( t ) ^ 3 z ( t ) ^ 4 } ( f ^ { 1 3 5 } - f ^ { 1 4 6 } + f ^ { 2 3 6 } + f ^ { 2 4 5 } ) , \\end{array} \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} \\| ( S _ { A , \\pmb { \\omega } } - A ) f _ { k } \\| < \\varepsilon \\quad { \\rm a n d } \\| ( S _ { A , \\pmb { \\omega } } - A ) ^ { * } f _ { k } \\| < \\varepsilon \\qquad \\hbox { f o r e v e r y $ k = - r , \\dots , r $ . } \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{align*} - \\Phi '' + \\Phi V ( x ) = k ^ 2 \\Phi , x > 0 , \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} \\left \\Vert \\Phi \\right \\Vert _ { \\mathcal { W } } : = \\underset { x , y \\in \\mathfrak { L } } { \\sup } \\sum \\limits _ { \\Lambda \\in \\mathcal { P } _ { f } ( \\mathfrak { L } ) , \\ ; \\Lambda \\supset \\{ x , y \\} } \\frac { \\Vert \\Phi _ { \\Lambda } \\Vert _ { \\mathcal { U } } } { \\mathbf { F } \\left ( \\left \\vert x - y \\right \\vert \\right ) } \\ . \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} \\omega = \\frac { x } { r } , r = | x | , \\omega ^ \\perp = ( \\omega ^ \\perp _ 1 , \\omega ^ \\perp _ 2 ) = ( - \\omega _ 2 , \\omega _ 1 ) , \\nabla ^ \\perp = ( - \\partial _ 2 , \\partial _ 1 ) . \\end{align*}"}
-{"id": "1769.png", "formula": "\\begin{align*} \\sup _ { p _ { j } ( x ) = 1 } p _ { j } ( A x ) , \\sup _ { p _ { j } ( x ) < 1 } p _ { j } ( A x ) \\mbox { a n d } \\sup _ { p _ { j } ( x ) \\leqslant 1 } p _ { j } ( A x ) \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} [ L _ { - r - 1 } , \\ , S _ { r - 1 } ] = L _ { - 2 } \\end{align*}"}
-{"id": "8870.png", "formula": "\\begin{align*} G _ i = a _ { i , 1 } P _ { k _ 1 } + a _ { i , 2 } P _ { k _ 2 } + \\dots + a _ { i , m } P _ { k _ m } i = 1 , \\dots , m . \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} \\partial _ \\nu u \\ge 0 . \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} \\widetilde { W } _ { n - q } ( K ) = \\widetilde { C } _ q ( K , S ^ { n - 1 } ) . \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) ^ { p } = \\begin{pmatrix} w & 0 \\\\ 0 & z \\end{pmatrix} + \\begin{pmatrix} 0 & b \\\\ c & 0 \\end{pmatrix} p \\end{align*}"}
-{"id": "5667.png", "formula": "\\begin{align*} & D _ m \\Big \\{ \\big [ v ( t ) , w ( t ) \\big ] ; ( v _ 0 , w _ 0 ) \\Big \\} \\\\ & = \\sqrt { \\left ( \\begin{array} { c } v ( t ) - v _ 0 \\\\ w ( t ) - w _ 0 \\end{array} \\right ) ^ { \\top } \\Omega _ { i j } ^ { - 1 } \\left ( \\begin{array} { c } v ( t ) - v _ 0 \\\\ w ( t ) - w _ 0 \\end{array} \\right ) } , \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} M _ { 0 } + \\sum _ { k = 1 } ^ \\infty 2 ^ { k - 1 } M _ { 0 } ^ { - \\frac 1 2 \\ , C ^ k } \\leq M \\quad \\textrm { a n d } \\sum _ { k = 1 } ^ \\infty 2 ^ k M _ { 0 } ^ { - \\frac 1 { 1 2 } C ^ k } C ^ k \\leq 1 . \\end{align*}"}
-{"id": "1024.png", "formula": "\\begin{align*} 1 7 2 8 \\Delta = c _ 4 ^ 3 - c _ 6 ^ 2 , 4 b _ 8 = b _ 2 b _ 6 - b _ 4 ^ 2 . \\end{align*}"}
-{"id": "2147.png", "formula": "\\begin{align*} f ( k , x ) = e ^ { i k x } + \\int _ x ^ \\infty \\frac { \\sin k ( t - x ) } { k } V ( t ) f ( k , t ) d t , k \\in \\overline { { \\mathbb { C } } } ^ + . \\end{align*}"}
-{"id": "9333.png", "formula": "\\begin{align*} \\| ( a , b ) \\| _ { } = | a | + | b | , \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} \\hat { \\omega } _ A \\left ( \\| \\alpha \\| _ \\infty \\right ) = \\int _ { \\mathbb { R } ^ { 2 m } } \\hat { D } _ A ( \\mathbf { x } ) \\ , \\hat { \\gamma } _ A \\ , { \\hat { D } _ A ( \\mathbf { x } ) } ^ \\dag \\ , \\mathrm { d } p _ X ( \\mathbf { x } ) \\ ; , \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{align*} \\sum _ { m = 2 ^ n } ^ { 2 ^ { n + 1 } - 1 } \\ ! \\ ! s ( m ) \\ , = \\ , 3 ^ n , \\end{align*}"}
-{"id": "8307.png", "formula": "\\begin{align*} \\lambda ^ { ( 1 ) } & = ( [ 0 ] , [ 0 , 0 , 0 ] , [ 0 , 0 , 0 ] , [ 0 , 4 , 0 ] ) , \\\\ \\lambda ^ { ( 2 ) } & = ( [ 0 ] , [ 0 , 0 , 0 ] , [ 0 , 4 , 0 ] , [ 0 , 0 , 0 ] ) , \\\\ \\lambda ^ { ( 3 ) } & = ( [ 0 ] , [ 0 , 4 , 0 ] , [ 0 , 0 , 0 ] , [ 0 , 0 , 0 ] ) \\end{align*}"}
-{"id": "9326.png", "formula": "\\begin{align*} u ^ 2 = z ( x ^ 3 + A ( t : s ) x z ^ 2 + B ( t : s ) z ^ 3 ) \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} R = \\begin{bmatrix} D & \\pi ^ n D \\\\ D & D \\end{bmatrix} \\subset M _ 2 ( D ) \\end{align*}"}
-{"id": "9770.png", "formula": "\\begin{align*} j u _ { \\epsilon } = d ^ * \\xi _ { \\epsilon } + h _ { \\epsilon } , \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} \\Delta _ { Y } = [ Y \\times z ] + B _ Y , \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{align*} V ( x ) ^ \\dag = V ( x ) . \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} k + 2 = k + \\frac { 3 } { 2 } + \\frac { 1 } { 2 } = \\frac { p } { 2 p ' } + \\frac { 1 } { 2 } = \\frac { p + p ' } { 2 p ' } = \\frac { u } { p ' } . \\end{align*}"}
-{"id": "9305.png", "formula": "\\begin{align*} H ( \\sigma \\ , | \\ , X _ 1 , \\dots , X _ k ) & \\stackrel { ( d ) } { = } H ( X _ 1 , \\dots , X _ k \\ , | \\ , \\sigma ) + H ( \\sigma ) - H ( X _ 1 , \\dots , X _ k ) \\\\ & \\stackrel { ( e ) } { = } k H ( X _ 1 \\ , | \\ , \\sigma ) + 1 - H ( X _ 1 , \\dots , X _ k ) \\\\ & \\stackrel { ( f ) } { \\ge } k H ( X _ 1 \\ , | \\ , \\sigma ) + 1 - k H ( X _ 1 ) , \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} \\dim H ^ 1 ( G _ T , W _ { [ n ] } ) & \\leq \\sum _ { i = 1 } ^ { n } \\left ( \\dim Z _ { [ i ] } + \\dim H ^ 2 ( G _ T , Z _ { [ i ] } ) - \\dim Z _ { [ i ] } ^ { + } \\right ) \\\\ & = \\dim W _ { [ n ] } + \\sum _ { i = 1 } ^ { n } \\dim H ^ 2 ( G _ T , Z _ { [ i ] } ) - \\sum _ { i = 1 } ^ { n } \\dim Z _ { [ i ] } ^ { + } . \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{align*} | R _ 3 ( t ) | = O ( t ^ { 3 / 2 } ) ; \\end{align*}"}
-{"id": "10005.png", "formula": "\\begin{align*} G _ { 3 } ( n , d , x - 1 , s ) - G _ { 3 } ( n , d , x , s ) & \\leq G _ { 3 } ( n , d , 2 , s ) - G _ { 3 } ( n , d , 1 , s ) = \\\\ & = 2 ( n - 4 ) + 2 . \\end{align*}"}
-{"id": "9374.png", "formula": "\\begin{align*} L ^ { \\infty } ( \\Omega _ 1 \\times \\Omega _ 2 ) = B ( L ^ 1 ( \\Omega _ 1 ) , L ^ { \\infty } ( \\Omega _ 2 ) ) . \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} \\pi _ 1 ( g ^ { n + 1 } ) = ( 1 , 3 , 2 , 4 , 5 , \\dots , n ) . \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} { \\rm I n d e x } D _ u \\overline { \\partial } = \\mu ( [ u ] ) + \\dim L . \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} f ^ { - 1 } \\circ \\widehat \\psi _ p ( \\zeta ) = \\widehat \\psi _ { f ^ { - 1 } ( p ) } ( \\lambda _ { p , - 1 } \\cdot \\zeta ) . \\end{align*}"}
-{"id": "329.png", "formula": "\\begin{align*} E _ { a } K _ { 2 \\rho } ^ { - 1 } K _ { a } = K _ { 2 \\rho } ^ { - 1 } K _ { a } E _ { a } . \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} \\delta = \\sum _ { i = 0 } ^ { \\infty } ( | G _ i | - 1 ) . \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} a ( \\mathbf { y } ) = a ( - \\mathbf { y } ) = \\int d \\mathbf { x } ~ \\rho ( \\mathbf { x } ) \\rho ( \\mathbf { x } + \\mathbf { y } ) , \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} \\left | \\sum _ { h = 1 } ^ N \\frac { \\lambda _ h k _ h } { 1 + \\sqrt { 1 + \\frac { \\lambda _ j } { c ^ 2 } } } \\right | \\leq \\frac { r \\lambda _ N } { 1 + \\sqrt { 1 + \\lambda _ N / c ^ 2 } } & \\leq \\frac { r \\lambda _ N } { 1 + \\sqrt { 1 + \\lambda _ N / \\lambda _ l } } \\leq \\frac { r \\lambda _ N } { 2 } , \\end{align*}"}
-{"id": "5811.png", "formula": "\\begin{align*} \\Phi \\left ( \\hat { D } _ A ( \\mathbf { x } ) \\ , \\hat { \\rho } _ A \\ , { \\hat { D } _ A ( \\mathbf { x } ) } ^ \\dag \\right ) = \\hat { D } _ B ( K \\mathbf { x } ) \\ , \\Phi ( \\hat { \\rho } _ A ) \\ , { \\hat { D } _ B ( K \\mathbf { x } ) } ^ \\dag \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} \\Psi ( 0 ) = \\lim _ { x \\to 0 } \\Psi ( x ) = \\lim _ { x \\to 0 } \\Omega _ 1 ( x ) \\Omega _ 2 ( x ) \\Phi ( x ) = \\Omega _ 1 ( 0 ) \\Omega _ 2 ( 0 ) \\Phi ( 0 ) . \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{align*} \\mathrm { a d } _ { L } ^ { \\circ } ( X ) ( \\mathsf { N } _ { m } ^ { n } ) = \\pi ( K _ { 2 \\rho } S ^ { - 1 } ( X _ { ( 1 ) } ) K _ { 2 \\rho } ^ { - 1 } ) ( \\mathsf { N } _ { m } ^ { n } \\triangleleft X _ { ( 2 ) } ) \\pi ( K _ { 2 \\rho } S ^ { - 2 } ( X _ { ( 3 ) } ) K _ { 2 \\rho } ^ { - 1 } ) . \\end{align*}"}
-{"id": "9822.png", "formula": "\\begin{align*} ( T _ i f ) ( \\mathbf { z } ) = \\frac { ( 1 - v ) \\mathbf { z } ^ { \\alpha ^ \\vee _ i } } { 1 - \\mathbf { z } ^ { \\alpha ^ \\vee _ i } } f ( \\mathbf { z } ) + \\frac { 1 - v \\mathbf { z } ^ { \\alpha ^ \\vee _ i } } { 1 - \\mathbf { z } ^ { - \\alpha ^ \\vee _ i } } f ( s _ i \\mathbf { z } ) , \\theta _ \\lambda f ( \\mathbf { z } ) = \\mathbf { z } ^ \\lambda \\ , f ( \\mathbf { z } ) . \\end{align*}"}
-{"id": "7723.png", "formula": "\\begin{align*} e & = \\sum _ { j = 0 } ^ { m + 1 } x _ j \\ ; \\ ; \\ ; \\ ; e _ i = \\sum _ { \\substack { j = 0 \\\\ j \\ne i } } ^ { m + 1 } x _ j \\end{align*}"}
-{"id": "5311.png", "formula": "\\begin{align*} \\phi \\left ( { \\left . s \\right | t } \\right ) = \\prod \\nolimits _ { k = 1 } ^ K { { \\rm E } \\left \\{ { \\left . { { { { { \\left ( 1 + \\gamma _ k \\right ) } } } ^ { s - 1 } } } \\right | t } \\right \\} } . \\end{align*}"}
-{"id": "9808.png", "formula": "\\begin{align*} { \\left ( { f , g } \\right ) _ { E , N } } = \\int _ { E , N } { f g d \\xi d \\eta d \\zeta } \\equiv \\sum \\limits _ { i , j , k = 0 } ^ N { { f _ { i j k } } { g _ { i j k } } { \\omega _ { i j k } } } , { \\left \\| f \\right \\| _ { E , N } } = \\sqrt { { { \\left ( { f , f } \\right ) } _ { E , N } } } . \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} \\alpha ( s ) = \\begin{cases} \\frac { s ^ 2 } { 4 C } & | s | \\le 2 C L , \\\\ L | s | - L ^ 2 C & | s | > 2 C L . \\end{cases} \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} { \\bf L } = { \\bf K } ^ { - 1 / 2 } \\left ( { \\bf H } + \\left [ \\begin{array} { c c } { \\bf 0 } & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf G } ^ { - 1 } \\end{array} \\right ] \\right ) ^ { - 1 } \\left [ \\begin{array} { c c } { \\bf 0 } & { \\bf 0 } \\\\ { \\bf 0 } & { \\bf G } ^ { - 1 } \\end{array} \\right ] { \\bf K } ^ { 1 / 2 } . \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N } T _ { \\varepsilon , \\eta , d , k , B } = \\frac { ( \\varepsilon - \\eta ) K ^ 2 } { 2 \\alpha ^ 2 d ^ 2 } B ^ { 2 - \\frac { 1 } { r } } + O _ { \\varepsilon , \\eta } \\left ( \\frac { K ^ 2 B ^ { 2 - \\frac { 1 } { r } } } { N d ^ 2 } \\right ) + O _ { \\sigma } \\left ( \\frac { K ^ \\sigma B ^ \\sigma N } { d ^ \\sigma } \\right ) \\end{align*}"}
-{"id": "5656.png", "formula": "\\begin{align*} \\mathcal { M } _ 0 : = \\Big \\{ ( v , w ) \\in \\mathbb { R } ^ 2 : f ( v , w ) = 0 \\Big \\} . \\end{align*}"}
-{"id": "1865.png", "formula": "\\begin{align*} \\left ( D ^ g _ X J \\right ) Y + \\left ( D ^ g _ Y J \\right ) X = 0 \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} Q = \\left \\{ ( [ \\lambda _ 0 : \\lambda _ 1 : \\lambda _ 2 ] , \\Pi ) | \\quad \\{ \\lambda _ 0 Q _ 0 + \\lambda _ 1 Q _ 1 + \\lambda _ 2 Q _ 2 = 0 \\} \\supset \\Pi \\right \\} . \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{align*} D _ q ( A ) = W ^ { 2 , q } ( \\Omega ) \\cap W ^ { 1 , q } _ 0 ( \\Omega ) \\cap L ^ q _ \\sigma ( \\Omega ) , A f = - \\mathbb P \\Delta f . \\end{align*}"}
-{"id": "6621.png", "formula": "\\begin{align*} M _ { a , b } ( r ) \\ ; & = \\ ; \\frac { \\ , e ^ r \\ , r ^ { a - b } \\ , } { \\Gamma ( a ) } \\ , ( 1 + O ( r ^ { - 1 } ) ) \\qquad \\textrm { a s } \\ , r \\to + \\infty \\\\ M _ { a , b } ( r ) \\ ; & = \\ ; 1 + O ( r ) \\qquad \\qquad \\ ; \\ ; \\textrm { a s } \\ , r \\downarrow 0 \\quad \\textrm { a n d } - \\ ! b \\notin \\mathbb { N } \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{align*} \\frac { ( - 1 ) ^ N x ^ { r ^ N } / L _ N } { \\log _ C ( x ) - \\sum _ { i = 0 } ^ { N - 1 } ( - 1 ) ^ i x ^ { r ^ i } / L _ i } = \\sum _ { n = 0 } ^ \\infty \\frac { C C _ { N , n } } { \\Pi ( n ) } x ^ n \\ , , \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} x \\odot \\lambda _ a ( y ) = x \\odot \\lambda _ a ( x \\wedge y ) \\mbox { a n d } x = ( x \\wedge y ) \\oplus ( x \\odot \\lambda _ a ( y ) ) . \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} \\phi ' ( t ) = J ( C | M ) _ { \\hat { \\rho } _ { C M } ( t ) } - \\frac { \\lambda ^ 2 } { \\eta } J ( A | M ) _ { \\hat { \\rho } _ { A M } ( t ) } - \\frac { ( 1 - \\lambda ) ^ 2 } { | 1 - \\eta | } J ( B | M ) _ { \\hat { \\rho } _ { B M } ( t ) } \\le 0 \\ ; . \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} v \\in C ^ { 2 h } _ b ( \\Omega ) \\cap C ^ { 2 h - 1 } _ b ( \\overline { \\Omega } ) L ( v ) = f . \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{gather*} \\frac { { \\rm d } Y } { { \\rm d } z } = \\left ( \\frac { A _ 0 } { z ^ { r + 1 } } + \\frac { A _ 1 } { z ^ { r } } + \\cdots + A _ { r + 1 } + A _ { r + 2 } z + \\cdots \\right ) Y . \\end{gather*}"}
-{"id": "7950.png", "formula": "\\begin{align*} { \\cal B } _ j [ \\eta f ] ( x ) = \\int _ D \\Gamma _ { \\partial _ j \\kappa } ( x - y , y ) ( \\eta f ) ( y ) d y ( j = 1 , 2 , 3 ) . \\end{align*}"}
-{"id": "5511.png", "formula": "\\begin{align*} f ^ { - 1 } \\circ \\widehat \\psi _ p ( \\zeta ) = \\widehat \\psi _ { f ^ { - 1 } ( p ) } ( \\lambda _ { p , - 1 } \\cdot \\zeta ) . \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{gather*} { \\it \\Upsilon } _ 1 = \\big \\{ ( x _ 1 , \\vartheta ) : \\ \\ x _ 1 \\in ( 0 , \\ell ) , r = 1 , \\vartheta \\in ( 0 , 2 \\pi ) \\big \\} , \\ \\ { \\it \\Upsilon } _ { 1 + \\varepsilon } . \\end{gather*}"}
-{"id": "8270.png", "formula": "\\begin{align*} e ^ { - 2 i k _ j l } = \\prod _ { i \\neq j } s _ p ( k _ { j } + k _ { i } ) s _ p ( k _ { j } - k _ { i } ) , \\end{align*}"}
-{"id": "4297.png", "formula": "\\begin{align*} x ( [ \\ell ] P _ n ) = x _ n - \\frac { \\Psi _ { \\ell - 1 } ^ \\prime ( x _ n , y _ n ) \\Psi _ { \\ell + 1 } ^ \\prime ( x _ n , y _ n ) } { \\left ( \\Psi _ \\ell ^ \\prime ( x _ n , y _ n ) \\right ) ^ 2 } , \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} P ( x , \\zeta ) = \\sum _ { m = 0 } ^ { \\infty } Z _ m ( x , \\zeta ) = \\frac { 1 - | x | ^ 2 | \\overline { \\zeta } | ^ 2 } { ( x ^ 2 \\overline { \\zeta } ^ 2 - 2 x \\cdot \\overline { \\zeta } + 1 ) ^ { n / 2 } } . \\end{align*}"}
-{"id": "4624.png", "formula": "\\begin{align*} \\begin{cases} \\dd Y _ t = ( a - b Y _ t ) \\ , \\dd t + \\sigma _ 1 \\sqrt { Y _ t } \\ , \\dd W _ t , \\\\ \\dd X _ t = ( \\alpha - \\beta Y _ t - \\gamma X _ t ) \\ , \\dd t + \\sigma _ 2 \\sqrt { Y _ t } \\ , ( \\varrho \\ , \\dd W _ t + \\sqrt { 1 - \\varrho ^ 2 } \\ , \\dd B _ t ) + \\sigma _ 3 \\ , \\dd L _ t , \\end{cases} t \\in [ 0 , \\infty ) , \\end{align*}"}
-{"id": "7937.png", "formula": "\\begin{align*} 1 < q _ 0 < q < q _ 1 < \\infty , \\frac { 1 } { q } = \\frac { 1 - \\theta } { q _ 0 } + \\frac { \\theta } { q _ 1 } , 1 \\leq r _ 0 , r _ 1 , r \\leq \\infty . \\end{align*}"}
-{"id": "4279.png", "formula": "\\begin{align*} \\begin{cases} \\Psi _ { 2 n } ( X , Y ) = \\Psi _ n ( X , Y ) \\left ( \\Psi _ { n + 2 } ( X , Y ) \\Psi _ { n - 1 } ^ 2 ( X , Y ) - \\Psi _ { n - 2 } ( X , Y ) \\Psi _ { n + 1 } ^ 2 ( X , Y ) \\right ) / 2 Y ; \\\\ \\Psi _ { 2 n + 1 } ( X , Y ) = \\Psi _ { n + 2 } ( X , Y ) \\Psi _ n ^ 3 ( X , Y ) - \\Psi _ { n + 1 } ^ 3 ( X , Y ) \\Psi _ { n - 1 } ( X , Y ) . \\end{cases} \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} \\dot { p } _ \\alpha = p _ \\alpha \\left ( \\sum _ \\beta a _ { \\alpha \\beta } p _ \\beta + b _ \\alpha \\right ) . \\end{align*}"}
-{"id": "7903.png", "formula": "\\begin{align*} t \\mapsto \\{ f ( \\cdot , t ) = c \\} \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} A = \\begin{bmatrix} \\pi ^ { - 1 } a & b \\pi ^ n \\\\ \\pi ^ { - 1 } c & d \\end{bmatrix} \\begin{bmatrix} \\pi & 0 \\\\ 0 & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "7293.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\alpha _ 5 } y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 2 ] = \\frac { \\beta _ 2 \\gamma _ 6 } { \\gamma ^ 2 _ 2 } y _ 5 , [ y _ 2 , y _ 3 ] = y _ 5 , [ y _ 3 , y _ 3 ] = y _ 5 . \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} M ( z ) = \\begin{pmatrix} 1 & - \\frac { 1 } { i ( z ^ 2 - 1 ) ^ { 1 / 2 } } \\\\ \\frac { z - ( z ^ 2 - 1 ) ^ { 1 / 2 } } { 2 i } & \\frac { z + ( z ^ 2 - 1 ) ^ { 1 / 2 } } { 2 ( z ^ 2 - 1 ) ^ { 1 / 2 } } \\end{pmatrix} , z \\in \\mathbb C \\setminus [ - 1 , 1 ] . \\end{align*}"}
-{"id": "6657.png", "formula": "\\begin{gather*} \\alpha \\beta = 0 , \\beta \\gamma = 0 , \\gamma \\alpha = 0 , \\delta \\xi = 0 , \\xi \\eta = 0 , \\eta \\delta = 0 , \\\\ \\alpha ^ r = ( \\beta \\delta \\eta \\gamma ) ^ s , \\ \\ \\quad ( \\gamma \\beta \\delta \\eta ) ^ s = ( \\delta \\eta \\gamma \\beta ) ^ s , \\ \\ \\quad \\xi ^ t = ( \\eta \\gamma \\beta \\delta ) ^ s . \\end{gather*}"}
-{"id": "7422.png", "formula": "\\begin{align*} m _ 1 \\dot { x } _ 1 = c e ^ { x _ 2 - x _ 1 } , m _ 2 \\dot { x } _ 2 = - c e ^ { x _ 2 - x _ 1 } \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\mu } \\left ( Z _ { 0 } \\in A _ { 0 , R _ { 2 } } \\right ) = \\mathbb { P } _ { \\mu } \\left ( Z _ { T } \\in A _ { 0 , R _ { 2 } } \\right ) = 1 . \\end{align*}"}
-{"id": "9526.png", "formula": "\\begin{align*} \\left | \\sum _ { i = 1 } ^ r \\langle C _ { n , j } ( \\omega _ 0 ) ^ * X ( \\omega _ 0 ) C _ { n , j } ( \\omega _ 0 ) \\xi _ i ( \\omega _ 0 ) , \\eta _ i ( \\omega _ 0 ) \\rangle \\right | < \\delta r K ^ 2 \\end{align*}"}
-{"id": "8160.png", "formula": "\\begin{align*} { \\bf g } = \\left [ \\begin{array} { c c } { \\bf g } _ { 1 1 } & { \\bf g } _ { 1 2 } \\\\ { \\bf 0 } & { \\bf g } _ { 2 2 } \\end{array} \\right ] \\end{align*}"}
-{"id": "8294.png", "formula": "\\begin{align*} \\psi _ { j _ 1 \\dots j _ n } ^ { Q } ( x _ 1 , \\dots , x _ n ) = \\psi _ { j _ 1 ' \\dots j _ n ' } ^ { Q } ( x _ 1 ' , \\dots , x _ n ' ) \\end{align*}"}
-{"id": "9344.png", "formula": "\\begin{align*} \\| \\log ( \\gamma _ M ^ { - 1 } \\cdot \\gamma _ N ) - \\log ( \\gamma _ M ^ { - 1 } \\cdot \\gamma _ L ) - \\log ( \\gamma _ L ^ { - 1 } \\cdot \\gamma _ N ) \\| _ { } = O ( A ^ 2 ) , \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{align*} L ^ \\alpha _ x w ( x , t ) = \\mathcal { F } _ { \\xi \\rightarrow x } ^ { - 1 } ( \\frac { 1 } { \\sqrt { 1 + a _ \\alpha \\left \\vert \\xi \\right \\vert ^ \\alpha } } ) \\ast _ x w ( x , t ) , \\alpha \\in \\ , ] 1 , 3 [ , \\end{align*}"}
-{"id": "992.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty h _ n ( x _ 1 , \\cdots , x _ d ) t ^ n = \\prod _ { \\ell = 1 } ^ { d } \\frac { 1 } { 1 - x _ \\ell t } . \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} \\psi _ { n } ( m ) & \\leq \\left \\Vert B \\right \\Vert _ { \\ast } = \\sqrt { m + 2 \\sqrt { p s \\left ( k - s \\right ) } } \\leq \\sqrt { m + \\sqrt { p k ^ { 2 } } } \\\\ & < \\sqrt { m + \\sqrt { m \\left \\lceil m / n \\right \\rceil } } < \\sqrt { m } + \\sqrt { \\left \\lceil m / n \\right \\rceil } / 2 , \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} p p ' m ^ 2 + n ^ 2 + m b _ \\pm + \\frac { b _ \\pm ^ 2 + p p ' } { 4 p p ' } + n & = \\frac { p ' } { \\Delta } \\left ( x + \\frac { b _ \\pm + p ' } { 2 p ' } \\right ) ^ 2 + \\frac { p } { \\Delta } \\left ( y + \\frac { b _ \\pm - p } { 2 p } \\right ) ^ 2 . \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { ( p - 1 ) / 2 } \\binom { 2 k } { k } \\frac { \\bigl ( \\beta ( 1 - \\beta ) \\bigr ) ^ k } { k ^ 2 } \\equiv 2 \\pounds _ 2 \\left ( \\beta \\right ) + 2 \\pounds _ 2 \\left ( 1 - \\beta \\right ) \\pmod { ( \\beta ^ p , p ) } \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} \\langle \\phi _ k , e ^ { \\i \\langle \\theta , \\cdot \\rangle _ { \\mathbb { C } ^ d } } \\rangle _ { L ^ 2 ( Y ) } = 0 ( k \\in \\N ) , \\end{align*}"}
-{"id": "7825.png", "formula": "\\begin{align*} \\iint P _ { 1 T } \\nu _ 1 \\cdot \\phi _ 0 \\ , d x d y = \\iint _ { B _ T } \\nu _ 1 \\cdot \\Lambda \\phi _ 0 \\ , d y d t . \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} G _ { d , r } ( \\textbf { t } , \\epsilon ) : = \\mathbb { E } _ { S _ { r } } [ \\prod _ { \\ell = 1 } ^ { \\infty } ( 1 + \\epsilon ^ { \\ell } t _ { d \\ell } ) ^ { X _ { \\ell } } ] = \\exp \\left ( \\sum _ { \\ell } \\epsilon ^ { \\ell } \\frac { t _ { d \\ell } } { \\ell } \\right ) . \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{align*} & \\big \\lvert \\mathbb { E } \\left [ f ( \\underbar { X } ) \\big | g ( \\underbar { X } ) \\right ] \\big \\rvert = \\prod \\limits _ { i = 1 } ^ n \\big \\lvert \\mathbb { E } \\left [ f _ i ( X _ i ) \\big | g ( \\underbar { X } ) \\right ] \\big \\rvert \\\\ & > \\prod \\limits _ { i \\in [ n ] } \\big \\lvert \\mathbb { E } \\left [ f _ i ( X _ i ) \\right ] \\big \\rvert = \\big \\lvert \\mathbb { E } \\left [ f ( \\underbar { X } ) \\right ] \\big \\rvert . \\end{align*}"}
-{"id": "5516.png", "formula": "\\begin{align*} \\mathcal H = \\bigcup _ { p \\in J ^ \\star } \\mathcal H _ p \\subset P o l ( m + 1 ) . \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} \\varphi _ 2 = e ^ { 1 2 7 } - e ^ { 3 4 7 } - e ^ { 5 6 7 } + e ^ { 1 3 5 } - e ^ { 1 4 6 } + e ^ { 2 3 6 } + e ^ { 2 4 5 } , \\end{align*}"}
-{"id": "6780.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + A u + F \\left ( u , \\bar { u } \\right ) u = 0 , x \\in R ^ { n } , t \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "5364.png", "formula": "\\begin{gather*} c _ 2 ( 2 n - 1 , k ) = \\sum _ { i = k } ^ { 2 n - 1 } { i - 1 \\choose k - 1 } c _ 1 ( 2 n , i ) = \\sum _ { j = \\lceil \\frac { k + 1 } { 2 } \\rceil } ^ { n } { 2 j - 2 \\choose k - 1 } c _ 1 ( 2 n - 1 , 2 j - 1 ) \\\\ = \\sum _ { j = \\lceil \\frac { k + 1 } { 2 } \\rceil } ^ { n } a ^ { n - j } { 2 j - 2 \\choose k - 1 } { n + j - 2 \\choose n - j } . \\end{gather*}"}
-{"id": "3129.png", "formula": "\\begin{align*} ( \\partial \\bar \\partial v ) ^ { n + 1 } = 0 \\qquad \\det \\big ( v _ { x _ j \\bar x _ k } \\big ) _ { 0 \\le j , k \\le n } = 0 . \\end{align*}"}
-{"id": "134.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sum _ { j = 0 } ^ { N } \\left [ G ( \\alpha _ j , \\dots , \\alpha _ { j + d } ) - \\log ( 1 - | \\alpha _ j | ^ 2 ) \\right ] \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} L H S & = \\sum _ { k = 1 } ^ n \\sum _ { j = 1 } ^ k ( - 1 ) ^ { k + j } [ a _ k ( x ) P _ s ^ { ( k ) } ( x ) ] ^ { ( k - j ) } f ^ { ( j - 1 ) } ( x ) = 0 . \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} ( T _ { B } + \\Lambda ) ' ( T _ { B } + \\Lambda ) = \\mu ^ { 2 } \\begin{bmatrix} a ^ { 2 } + ( c - 1 ) ^ { 2 } & a c + b c + a - b \\\\ a c + b c + a - b & b ^ { 2 } + ( c + 1 ) ^ { 2 } \\end{bmatrix} \\end{align*}"}
-{"id": "6311.png", "formula": "\\begin{align*} T ^ { \\lambda } _ { n , k } ( f ) ( x ) = ( - 1 ) ^ n \\left ( \\sum _ { j = 0 } ^ k ( \\lambda - 1 ) _ { k - j } \\binom { k } { j } \\left ( x ^ { j } f ( x ) \\right ) ^ { ( j ) } \\right ) ^ { ( n ) } , \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} \\gamma ( t ) = \\inf \\{ g ( x ) : x \\in D , \\ , f ( x ) \\geq t \\} \\end{align*}"}
-{"id": "9749.png", "formula": "\\begin{align*} \\partial _ t P + \\nabla \\cdot \\left ( \\frac { P \\otimes P } { \\rho } \\right ) = 0 , \\ ; \\ ; \\ ; \\partial _ t \\rho + \\nabla \\cdot P = 0 . \\end{align*}"}
-{"id": "3009.png", "formula": "\\begin{align*} \\bigcup _ { k \\ge k _ 0 } \\bigcup _ { m \\in \\varphi ^ { - 1 } ( n ) \\cap J _ k } \\Big \\{ s \\in [ 1 , b _ { m + 1 } - b _ m ) ; \\ ; | v _ { m } | \\prod _ { i = b _ { m + 1 } - s } ^ { b _ { m + 1 } - 1 } | w _ { i } | \\ge 2 ^ { \\delta ^ { ( k ) } - \\tau ^ { ( k ) } } \\Big \\} , \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} \\mathfrak { I } _ { \\mathrm { p } } ^ { ( \\omega , \\eta \\mathbf { A } _ { l } ) } \\left ( t \\right ) = \\eta ^ { 2 } l ^ { d } \\int \\nolimits _ { t _ { 0 } } ^ { t } \\int \\nolimits _ { t _ { 0 } } ^ { s _ { 1 } } \\mathbf { X } _ { l } ^ { ( \\omega ) } ( s _ { 1 } , s _ { 2 } ) \\ \\mathrm { d } s _ { 2 } \\mathrm { d } s _ { 1 } + \\mathcal { O } ( \\eta ^ { 3 } l ^ { d } ) \\ , \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} \\tau \\left ( \\mathbf { a } \\right ) = \\begin{pmatrix} \\tau _ { 1 } \\left ( \\mathbf { a } \\right ) \\\\ \\tau _ { 2 } \\left ( \\mathbf { a } \\right ) \\\\ \\vdots \\\\ \\tau _ { n - 1 } \\left ( \\mathbf { a } \\right ) \\\\ \\tau _ { n } \\left ( \\mathbf { a } \\right ) \\end{pmatrix} . \\end{align*}"}
-{"id": "9780.png", "formula": "\\begin{align*} 2 ^ { n - 2 } \\omega _ { n - 2 } \\Theta ^ { n - 2 } ( \\nu , x ) = \\lim _ { r \\to 0 } ( r / 2 ) ^ { 2 - n } \\nu ( B _ r ( x ) ) < \\eta _ 1 . \\end{align*}"}
-{"id": "8953.png", "formula": "\\begin{align*} \\mathrm { r e a c h } ( E ) : = \\inf \\{ \\mathrm { r e a c h } ( E , s ) \\ , | \\ , s \\in E \\} . \\end{align*}"}
-{"id": "7385.png", "formula": "\\begin{align*} \\abs { f / g } = \\exp ( \\deg ( f ) - \\deg ( g ) ) \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} \\sqrt { \\alpha \\beta ^ { - 1 } } \\left \\vert x \\right \\vert \\leq \\left \\vert y \\right \\vert \\sqrt { \\alpha ^ { - 1 } \\beta } \\left \\vert x \\right \\vert = \\left ( k a _ { 0 } ^ { - 1 } \\right ) ^ { \\frac { 1 } { 2 p } } \\left \\vert x \\right \\vert \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} \\sigma _ { t } ^ { \\prime } [ l ] = \\sigma _ { t - 1 } ^ { \\prime } [ l ] \\mbox { a n d } O P T _ { t } ^ { \\prime } [ l ] = O P T _ { t - 1 } ^ { \\prime } [ l ] . \\end{align*}"}
-{"id": "626.png", "formula": "\\begin{align*} \\Delta ^ { 2 s _ { \\bar w } } _ { s _ { \\bar w } } ( y _ { w } ) \\leq I ( y _ { w } , 1 ) - I ( y _ { w } , s _ { w } ) \\leq L + \\eta - L = \\eta . \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} \\begin{array} { c } \\forall i \\in I , ~ \\forall j \\in \\{ 1 , \\hdots , n \\} , ~ U _ { i j } : = \\hat { U } _ { i j } , ~ V _ { i j } : = \\hat { V } _ { i j } . \\end{array} \\end{align*}"}
-{"id": "7646.png", "formula": "\\begin{align*} K ( s , t ) = \\begin{cases} w ( s ) - \\left ( w ( a ) + \\psi ( \\lambda ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ a , t ) , \\\\ w ( s ) - \\left ( w ( a ) + ( 1 + \\psi ( 1 - \\lambda ) ) \\frac { w ( b ) - w ( a ) } { 2 } \\right ) , ~ ~ ~ ~ s \\in [ t , b ] , \\end{cases} \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{align*} \\left . \\begin{array} { r c l } \\dot x & = & \\frac { 3 } { 2 } - \\frac { x } { 2 } - x y ^ 3 \\\\ \\dot y & = & \\left ( \\frac { 1 } { 2 } - k \\right ) - \\left ( \\frac { 3 } { 2 } - k \\right ) y + x y ^ 3 \\end{array} \\right \\} = : F ( x , y ; k ) . \\end{align*}"}
-{"id": "6804.png", "formula": "\\begin{align*} \\partial _ { t } f = S f + K f + i \\left [ A + e ^ { \\tilde { \\gamma } \\varphi } F \\right ] \\left ( x , t \\right ) \\in R ^ { n } \\times \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} K ( \\psi , A ) = \\frac { \\mathbb { P } [ W \\in B ] } { \\mathbb { P } [ V _ 1 \\in B ] } , \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} p - N \\overline { M } ^ { - 1 } \\overline { p } = \\left ( \\begin{array} { r } 0 . 1 7 5 9 - 1 . 9 8 3 0 i \\\\ 2 . 1 3 0 2 + 1 . 2 5 9 7 i \\\\ 1 . 4 6 5 8 - 1 . 7 8 3 5 i \\end{array} \\right ) . \\end{align*}"}
-{"id": "1986.png", "formula": "\\begin{align*} \\mathrm { d e t } _ N ( z _ j ^ { N - k + 1 } - z _ j ^ { - N + k - 1 } ) = ( - 1 ) ^ N \\prod _ { j = 1 } ^ N z _ j ^ { j - 1 - N } ( 1 - z _ j ^ 2 ) \\prod _ { 1 \\le j < k \\le N } ( 1 - z _ j z _ k ) ( 1 - z _ j z _ k ^ { - 1 } ) , \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { R t } { p _ n } \\right ) \\left ( 1 - \\frac { R t } { p _ { - n } } \\right ) = \\left ( 1 - \\frac { R ^ 2 t ^ 2 } { p _ { - n } ^ 2 } \\right ) \\left ( 1 + \\frac { R t } { 1 + \\frac { R t } { p _ { - n } } } \\left ( - \\frac { 1 } { p _ n } - \\frac { 1 } { p _ { - n } } \\right ) \\right ) . \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} \\alpha x ^ { 3 } + \\beta x ^ { 2 } + \\gamma x + \\theta = 0 , \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} | T | _ { \\Omega ^ 2 } ^ 2 - \\frac 9 2 | t | _ { \\Omega ^ 3 } ^ 2 = | N | ^ 2 + \\frac 1 2 | ( d ^ c F ) ^ { 2 , 0 } | ^ 2 . \\end{align*}"}
-{"id": "3965.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac { 1 } { 4 } & \\frac { 1 } { 4 } \\\\ & 1 \\end{matrix} \\bigg | \\ , 4 z ( 1 - z ) \\bigg ] _ { p - 1 } \\equiv \\begin{cases} P _ { \\frac { p - 1 } { 2 } } ( 1 - 2 z ) \\pmod { p ^ 2 } , & p \\equiv 1 \\pmod { 4 } , \\\\ P _ { \\frac { 3 p - 1 } { 2 } } ( 1 - 2 z ) \\pmod { p ^ 2 } , & p \\equiv 3 \\pmod { 4 } . \\end{cases} \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} T _ n ( f \\circ g ; a ) = \\sum _ { \\pi \\in C _ n } T _ { \\vert \\pi \\vert } ( f ; g ( a ) ) \\prod _ { i = 1 } ^ n [ T _ i ( g ; a ) ] ^ { \\pi _ i } . \\end{align*}"}
-{"id": "1121.png", "formula": "\\begin{align*} E _ k & = \\bigcup _ { j = 2 } ^ \\infty \\bigl ( [ k - 2 k ^ { - j } , k - k ^ { - j } ] \\cup [ k + k ^ { - j } , k + 2 k ^ { - j } ] \\bigr ) \\times [ 0 , k ^ { 1 - j } ] , \\\\ \\ X & = \\bigl ( \\R \\times ( - \\infty , 0 ] \\bigr ) \\cup \\bigcup _ { k = 3 } ^ \\infty E _ k , \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{align*} \\alpha _ i ^ + ( t ) = \\alpha _ i ^ + ( 0 ) \\ , \\forall t \\ge 0 . \\end{align*}"}
-{"id": "3044.png", "formula": "\\begin{align*} e ^ { n f } \\sqrt { 1 + \\abs { \\nabla f } ^ 2 } = 1 + n f + O _ { \\sqrt { \\epsilon } } ( \\norm { f } _ { 1 , \\ , p } ) \\end{align*}"}
-{"id": "3516.png", "formula": "\\begin{align*} _ { \\mu , \\sigma } D _ { v , q ; z } ^ { \\alpha , \\eta , p } ( ( z - \\xi ) ^ { r } ) : = \\frac { ( - \\xi ) ^ { r } B ( \\eta , \\alpha - 1 ) z ^ { \\eta + \\alpha } } { \\Gamma ( \\alpha ) } F _ { v , q ; p } ^ { ( \\mu , \\sigma ) } ( - r , \\eta ; \\eta + \\alpha - 1 ; \\frac { z } { \\xi } ) \\end{align*}"}
-{"id": "1529.png", "formula": "\\begin{align*} - \\frac { d } { f ( x ) } + e = g ( x ) \\end{align*}"}
-{"id": "9713.png", "formula": "\\begin{align*} D _ { e x t } G _ { e x t } = \\begin{pmatrix} D G & D G & \\hdots & D G \\\\ D G & D G & \\hdots & D G \\\\ \\vdots & \\vdots & \\hdots & \\vdots \\\\ D G & D G & \\hdots & D G \\end{pmatrix} \\approx F _ { e x t , X } , \\end{align*}"}
-{"id": "8014.png", "formula": "\\begin{align*} & p _ k ^ 1 + c _ k ^ 1 \\to p ^ 1 \\quad \\mbox { w e a k l y i n $ L ^ 2 ( s , T ; L ^ 6 ( \\Omega ) ) $ } , \\\\ & \\nabla p _ k ^ 1 \\to \\nabla p ^ 1 \\quad \\mbox { w e a k l y i n $ L ^ 2 ( s , T ; L ^ 2 ( \\Omega ) ) $ } , \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{align*} - 4 \\delta b \\| \\psi _ * \\| ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } + \\frac { 3 } { \\pi } \\langle \\cos ( y ) \\psi _ * , ( 2 b \\cos ( y ) \\psi _ * + \\tilde { u } _ b ) ^ 2 \\rangle _ { L ^ 2 ( \\mathbb { R } \\times \\mathbb { T } ) } = 0 . \\end{align*}"}
-{"id": "4476.png", "formula": "\\begin{align*} \\mathbf { \\Xi } _ { \\mathrm { d } } = \\underset { l \\rightarrow \\infty } { \\lim } \\Xi _ { \\mathrm { d } , l } ^ { ( \\omega ) } \\ . \\end{align*}"}
-{"id": "7590.png", "formula": "\\begin{align*} c _ t = - \\log v _ \\epsilon ' ( t , 0 ) - \\log v _ \\epsilon '' ( t , 0 ) - ( 1 - \\alpha ) \\log ( 1 + 2 \\epsilon ^ 2 ) \\end{align*}"}
-{"id": "4103.png", "formula": "\\begin{align*} b ( z _ 0 \\otimes \\dots \\otimes z _ n ) & = z _ 0 z _ 1 \\otimes \\dots \\otimes z _ n \\\\ & + \\sum _ { 0 } ^ { n - 1 } ( - 1 ) ^ i z _ 0 \\otimes \\dots \\otimes z _ i z _ { i + 1 } \\otimes \\dots \\ \\otimes z _ n \\\\ & + ( - 1 ) ^ n z _ n z _ 0 \\otimes \\dots \\otimes z _ { n - 1 } \\end{align*}"}
-{"id": "1524.png", "formula": "\\begin{align*} f ( x ) : = - \\frac { 1 } { g ( x ) - g ( x _ { k _ 0 } ) } \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} J _ { \\mathrm { R L S } } ( h ) = \\| y - R h \\| _ 2 ^ 2 + \\eta \\| h \\| _ { K ^ { - 1 } } ^ 2 \\ , , \\end{align*}"}
-{"id": "5254.png", "formula": "\\begin{align*} \\tilde a ( \\tilde x ) : = - ( n - 2 ) \\left ( N \\cdot x \\right ) , \\tilde b ( \\tilde x ) : = { | x | ^ n } N \\cdot ( c - V ) . \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{align*} \\varphi _ 1 ' ( \\sum z _ i \\mathbf w _ i ) = \\sum ( a \\alpha _ i + b ) z _ i ^ 2 \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} \\alpha = \\alpha \\left ( x , y \\right ) = \\tau ( x , y ) - \\left \\vert x - y \\right \\vert . \\end{align*}"}
-{"id": "9425.png", "formula": "\\begin{align*} \\partial _ { t } h + \\nabla \\cdot \\left ( h v \\right ) = 0 , \\end{align*}"}
-{"id": "9623.png", "formula": "\\begin{align*} H ^ { ( j ) } _ n = \\left | \\begin{array} { c c c c } u _ { j } & u _ { j + 1 } & \\cdots & u _ { j + n - 1 } \\\\ u _ { j + 1 } & u _ { j + 2 } & \\cdots & u _ { j + n } \\\\ \\vdots & \\vdots & & \\vdots \\\\ u _ { j + n - 1 } & u _ { j + n } & \\cdots & u _ { j + 2 n - 2 } \\end{array} \\right | , \\ \\ H ^ { ( j ) } _ 0 = 1 , \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} P _ j = \\sum _ { i = 0 } ^ { j } \\left [ \\sum _ { k = i + 1 } ^ { j + 1 } \\eta _ { i k } \\theta _ { k j } \\right ] P _ i . \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} [ L _ 1 , \\ , L _ 1 ] = 0 \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} ( \\pi ^ + , \\pi ^ - ) ( s ) : = \\begin{cases} ( \\pi ^ { ( x , t ) } _ + ( t + s ) - x , \\pi ^ { ( x , t ) } _ - ( t + s ) - x ) & 0 \\leq s \\leq \\gamma ( \\pi ^ { ( x , t ) } _ + , \\pi ^ { ( x , t ) } _ - ) - t \\\\ ( \\pi ^ { ( x , t ) } _ + ( t + s ) - x , \\pi ^ { ( x , t ) } _ + ( t + s ) - x ) & s \\geq \\gamma ( \\pi ^ { ( x , t ) } _ + , \\pi ^ { ( x , t ) } _ - ) - t . \\end{cases} \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{align*} 1 - \\left | \\hat { \\mu } ( \\xi _ i - \\xi _ j ) \\right | = 2 ( 1 - | \\hat { \\mu } ( \\xi ) | ) + O \\left ( \\frac { \\log ( 1 + \\rho ) } { \\rho ^ 2 m ^ 2 } \\right ) \\end{align*}"}
-{"id": "7022.png", "formula": "\\begin{align*} B u ^ { \\prime } \\left ( t \\right ) + A u \\left ( t \\right ) = f _ { 0 } \\left ( t \\right ) , t \\in \\left ( 0 , T \\right ) , \\end{align*}"}
-{"id": "8538.png", "formula": "\\begin{align*} h = - d x _ 0 ^ 2 + d x _ 1 ^ 2 + d x _ 2 ^ 2 + d x _ 3 ^ 2 + d x _ 4 ^ 2 \\end{align*}"}
-{"id": "6336.png", "formula": "\\begin{align*} y \\odot \\lambda _ a ( x ) = y \\odot \\lambda _ b ( x ) \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} m L ^ g ( f ) + S = \\lambda e ^ { 2 f } \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{align*} y = a ^ { 1 - \\nu } - \\left ( \\nu - 1 \\right ) \\tau , \\tau = \\tau \\left ( y \\right ) = \\frac { 1 } { \\nu - 1 } \\left ( a ^ { 1 - \\nu } - y \\right ) \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} \\tau ( s ) = \\frac { \\epsilon _ t \\epsilon _ u } { \\kappa ^ 2 ( s ) r ^ 2 } J _ 1 ' ( s ) = \\frac { J _ 1 ' ( s ) } { 1 - \\epsilon _ t \\ , J _ 1 ^ 2 ( s ) } . \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} ( | ( v M ) | _ 2 ) ^ 2 \\ = \\ & \\left ( \\sum _ { s \\in S _ 1 } \\left ( \\frac { 1 } { 3 } v ( p _ s ) \\right ) ^ 2 \\right ) + \\left ( \\sum _ { s \\in S _ 2 } \\left ( \\frac { 1 } { 3 } v ( q _ s ) \\right ) ^ 2 \\right ) \\\\ & + \\left ( \\sum _ { s \\in S _ 3 } \\left ( \\frac { 1 } { 3 } v ( r _ s a ) + \\frac { 1 } { 3 } v ( r _ s b ) \\right ) ^ 2 \\right ) + \\left ( \\sum _ { s \\in S _ 4 } \\left ( \\frac { 1 } { 3 } v ( t _ s c ) \\right ) ^ 2 \\right ) . \\end{align*}"}
-{"id": "8379.png", "formula": "\\begin{align*} E _ { m } : = \\bigcup _ { r \\geq \\frac { 8 \\mu } { \\beta _ 1 } m } \\ ; \\ ; \\bigcup _ { \\pi \\in { \\cal Q } _ r } \\left \\{ T ( \\pi ) < \\beta _ 1 r \\right \\} \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { s = 1 } ^ { 3 } \\left ( \\frac { z _ { _ { s } } } { \\left ( x - ( - 3 + i ) \\right ) ^ { s } } + \\frac { \\bar { z } _ { s } } { \\left ( x - ( - 3 - i ) \\right ) ^ { s } } \\right ) \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} F _ * ^ e ( j u ^ { r } v ^ { q - k - r } ( f + u v ) ) = \\bigoplus _ { i \\in \\Delta _ e } f _ { ( i , j ) } F _ * ^ e ( i u ^ { r } v ^ { q - k - r } ) \\oplus F _ * ^ e ( j u ^ { r + 1 } v ^ { q - k - r + 1 } ) \\end{align*}"}
-{"id": "7984.png", "formula": "\\begin{align*} & \\varphi \\in C ( [ 0 , \\infty ) ; L ^ 2 _ \\sigma ( \\Omega ) ) \\cap L ^ \\infty _ { l o c } ( [ 0 , \\infty ) ; L ^ { 3 , \\infty } ( \\Omega ) ) , \\\\ & \\nabla \\varphi \\in L ^ 2 _ { l o c } ( [ 0 , \\infty ) ; L ^ 2 ( \\Omega ) ) , \\partial _ t \\varphi \\in L ^ 2 _ { l o c } ( [ 0 , \\infty ) ; L ^ 2 _ \\sigma ( \\Omega ) ) . \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{align*} f ( x ) = e ^ { i \\pi R ( 1 , \\theta ) \\cdot x } \\phi ( R ^ { 1 / 2 } x _ 1 ) g ( \\bar { x } ) , \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{align*} g ( s ) = - \\beta s + | s | ^ \\alpha s , s \\in \\R \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} a d = b c \\cdot t ^ \\Delta + Q , \\ ; \\deg Q \\leq q + \\delta - 2 \\ , \\forall \\deg ( b c ) \\leq n - p - 1 . \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n { ( - 1 ) ^ { k - 1 } L _ { m k } { } ^ 4 } = - \\frac { { ( L _ { 4 m n + 2 m } - L _ { 2 m } ) } } { { L _ { 2 m } } } - \\frac { { 4 ( L _ { 2 m n + m } - L _ m ) } } { { L _ m } } \\ , , \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} \\langle B \\phi _ k , \\phi _ k \\rangle \\geq \\epsilon _ { 1 } \\int _ { \\ell _ { 1 } } ^ { \\ell _ { 1 } + \\delta _ { 1 } } | \\phi _ { k } ( x ) | ^ 2 d x = { 2 \\epsilon _ { 1 } \\over \\pi } \\int _ { \\ell _ { 1 } } ^ { \\ell _ { 1 } + \\delta _ { 1 } } \\ ! \\ ! \\ ! \\sin ^ 2 ( k x ) \\ , d x \\geq c . \\end{align*}"}
-{"id": "8998.png", "formula": "\\begin{align*} = \\ ; \\| \\ , u ( \\cdot , t _ 0 ) \\ , \\| _ { \\mbox { } _ { \\scriptstyle L ^ { q } ( \\mathbb { R } ^ { n } ) } } ^ { \\ : \\ ! q } \\ ! \\ ; \\ ! + \\ : q \\ , ( q - 1 ) \\ ! \\int _ { \\mbox { } _ { \\scriptstyle t _ { \\mbox { } _ { 0 } } } } ^ { \\ : \\ ! t } \\ ! \\ ; \\ ! \\int _ { \\mbox { } _ { \\scriptstyle \\mathbb { R } ^ { n } } } \\ ! \\ ! | \\ , u \\ , | ^ { \\ : \\ ! q - 2 } \\ , \\langle \\ , \\mbox { \\boldmath $ f $ } ( x , \\tau , u ) , \\ ; \\ ! \\nabla u \\ , \\rangle \\ ; d x \\ , d \\tau , \\end{align*}"}
-{"id": "7318.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 + \\theta _ 1 y _ 5 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\gamma _ 3 } y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 1 ] = \\frac { \\alpha _ 1 \\alpha _ 6 } { \\gamma _ 3 \\gamma _ 6 } y _ 5 , [ y _ 2 , y _ 2 ] = \\frac { \\alpha ^ 2 _ 1 \\beta _ 2 } { \\gamma ^ 2 _ 3 \\gamma _ 6 } y _ 5 , [ y _ 1 , y _ 3 ] = \\frac { \\beta _ 4 } { \\gamma _ 6 } y _ 5 , \\\\ [ y _ 2 , y _ 3 ] = \\frac { \\gamma _ 1 } { \\gamma _ 3 } y _ 4 + \\theta _ 3 y _ 5 , [ y _ 3 , y _ 2 ] = y _ 4 , [ y _ 3 , y _ 3 ] = y _ 5 . \\end{align*}"}
-{"id": "2340.png", "formula": "\\begin{align*} H ( t ) \\le \\sigma \\sqrt s + \\sigma 2 \\sqrt { \\log 3 } + \\mu _ s \\sqrt s ( 1 + \\gamma + \\eta ) / \\kappa ( c _ 0 , s ) = \\bar C \\lambda \\sqrt s , \\end{align*}"}
-{"id": "4636.png", "formula": "\\begin{align*} { \\displaystyle \\Psi ^ 0 ( \\mathbf { x } , t ) = 0 , \\mathbf { E } ^ 0 ( \\mathbf { x } , t ) \\times \\mathbf { n } = 0 , ( \\mathbf { x } , t ) \\in \\partial \\Omega \\times ( 0 , T ) . } \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\frac { 1 } { N } \\widehat { K } _ N \\left ( \\frac { x } { N } , \\frac { y } { N } ; w _ R \\right ) = \\frac { \\sin \\frac { \\pi } { 2 } ( x - y ) } { \\pi ( x - y ) } . \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{align*} ( T _ { z + \\bar { z } } ) ^ 2 & = T _ { g _ 1 } ( T _ { z + \\bar { z } } ) ^ 3 = T _ { g _ 2 } , \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} \\Delta _ { A | M } ( ( \\mathcal { N } _ A ( s ) \\otimes \\mathbb { I } _ M ) ( \\hat { \\rho } _ { A M } ) ) ( t ) = I ( A : X | M ) _ { \\hat { \\tau } _ { A M X } ( s , t ) } \\ ; , \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{align*} P ( B ) = \\frac { 1 } { 2 ( n - 1 ) - 1 } = \\frac { 1 } { 2 n - 3 } . \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} \\tilde { A } _ n ^ { [ i ] } \\neq A _ n ^ { [ i ] } , n \\in \\mathcal { B } , \\ : | \\mathcal { B } | = B \\end{align*}"}
-{"id": "2361.png", "formula": "\\begin{align*} X _ 0 = \\big \\{ \\bar { x } \\in R ^ { \\sigma - 1 } \\mathbb { Z } ^ { n - 1 } \\ , : \\ , | \\bar { x } | \\le 2 \\big \\} + B ( 0 , \\varepsilon R ^ { - 1 } ) \\ , . \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{align*} P _ { 1 } & = \\sum _ { z \\in C _ { a } } u ^ { z } & P _ { 2 } & = \\sum _ { z \\in C _ { b } } u ^ { z } \\\\ Q & = \\sum _ { z \\in P _ { a } } u ^ { z } & R & = \\sum _ { z \\in P _ { b } } u ^ { z } \\end{align*}"}
-{"id": "9480.png", "formula": "\\begin{align*} | H ' ( S ) | = ( r - | S | ) | H ( S ) | . \\end{align*}"}
-{"id": "2925.png", "formula": "\\begin{align*} T ^ { n _ { j } } z - x _ { j } & = \\sum _ { i < j } z _ { i } + T ^ { n _ { j } } z _ { j } + \\sum _ { i > j } T ^ { n _ { j } } z _ { i } - x _ { j } . \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{align*} P u : = \\partial _ t ^ 2 u - L _ x ^ \\alpha \\partial _ x ^ 2 u = \\partial _ t ^ 2 u - \\mathcal { F } _ { \\xi \\rightarrow x } ^ { - 1 } ( \\frac { 1 } { \\sqrt { 1 + a _ \\alpha \\left \\vert \\xi \\right \\vert ^ \\alpha } } ) \\ast _ x \\partial _ x ^ 2 u . \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{align*} \\partial ^ w _ t f ( t ) : = \\frac { d } { d t } \\int _ 0 ^ t w ( t - s ) ( f ( s ) - f ( 0 ) ) d s , \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n { ( - 1 ) ^ { k - 1 } L _ { m k } { } ^ 4 } & = - \\frac { { ( L _ { 4 m n + 2 m } - L _ { 2 m } ) } } { { L _ { 2 m } } } \\\\ & - \\frac { { 4 ( - 1 ) ^ { m - 1 } ( L _ { 2 m n + m } - ( - 1 ) ^ m L _ m ) } } { { L _ m } } \\ , . \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{align*} \\mu ^ { s , N + 1 } _ { H P } \\Lambda _ N ^ { N + 1 } = \\mu ^ { s , N } _ { H P } . \\end{align*}"}
-{"id": "10040.png", "formula": "\\begin{align*} \\nabla ^ { 0 } _ X Y = \\nabla _ X Y - \\frac { 1 } { 2 } ( \\nabla _ X J _ { \\varphi } ) J _ { \\varphi } Y , \\forall X , Y \\in { \\mathfrak X } ( M ) , \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} \\dot \\omega \\omega = F _ 1 ( t ) \\omega + F _ 0 ( t ) , \\end{align*}"}
-{"id": "7347.png", "formula": "\\begin{align*} \\varphi ( t ) = \\frac { y ( t ) } { z ( t ) ^ 2 } \\ , e ^ { 1 2 7 } - \\frac { 1 } { y ( t ) } \\ , e ^ { 3 4 7 } - \\frac { 1 } { y ( t ) } \\ , e ^ { 5 6 7 } + y ( t ) z ( t ) ^ 2 \\left ( e ^ { 1 3 5 } - e ^ { 1 4 6 } + e ^ { 2 3 6 } + e ^ { 2 4 5 } \\right ) , \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{align*} s ^ H _ { \\tilde { g } _ t } = e ^ { - 2 f } \\left ( s ^ H _ { g _ t } + m \\Delta ^ { g _ t } ( f ) \\right ) = u ^ { 2 / n } s ^ H _ { g _ t } - 2 u ^ { 2 / n - 1 } \\Delta ^ { g _ t } ( u ) - 2 u ^ { 2 / n - 2 } | d u | ^ 2 _ { g _ t } . \\end{align*}"}
-{"id": "9445.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\int _ { \\mathbb { T } ^ 3 } \\partial _ t u \\ ; h + \\nabla u \\cdot P = \\int _ { \\mathbb { T } ^ 3 } u ( t ) h ( t ) - \\int _ { \\mathbb { T } ^ 3 } u ( 0 ) h ( 0 ) \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} { } _ 4 F _ 3 \\bigg [ \\begin{matrix} - n & \\alpha & \\beta & \\gamma \\\\ & 1 & \\delta & \\epsilon \\end{matrix} \\bigg | \\ , 1 \\bigg ] = \\frac { ( 1 - \\alpha ) _ n ( \\epsilon - \\alpha ) _ n } { n ! \\cdot ( \\epsilon ) _ n } \\cdot { } _ 4 F _ 3 \\bigg [ \\begin{matrix} - n & \\alpha & \\delta - \\beta & \\delta - \\gamma \\\\ & \\delta & \\alpha - n & 1 + \\alpha - n - \\epsilon \\end{matrix} \\bigg | \\ , 1 \\bigg ] \\end{align*}"}
-{"id": "3045.png", "formula": "\\begin{align*} \\int ( \\Delta _ \\sigma f ) ^ 2 \\ , d V _ \\sigma - \\int \\abs { \\nabla ^ 2 f } ^ 2 \\ , d V _ \\sigma = ( n - 1 ) \\int \\abs { \\nabla f } ^ 2 d V _ \\sigma = O _ { \\sqrt { \\epsilon } } ( \\norm { \\nabla f } _ p ) \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} \\bigcup _ { m \\ge m _ { 0 } } \\ , \\bigcup _ { j = 0 } ^ { j _ { m + 1 } - j _ { m } - 1 } A _ { m , j } . \\end{align*}"}
-{"id": "8365.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\frac { \\hat { T } ^ { ( k ) } _ n } { n } \\right ) ^ 2 \\leq \\mathbb { E } \\left ( \\frac { T _ n } { n } \\right ) ^ 2 \\leq \\frac { 1 } { n ^ 2 } \\mathbb { E } \\left ( \\sum _ { i = 1 } ^ { n } t ( f _ i ) \\right ) ^ 2 \\leq C \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } u ' ( t ) = u ( t ) , t \\in \\R \\\\ u ( 0 ) = 1 \\end{array} \\right . \\end{align*}"}
-{"id": "9389.png", "formula": "\\begin{align*} B ( E , L ^ { \\infty } _ { \\sigma } ( \\Omega ; F ^ * ) ) & = \\bigl ( E \\overset { \\wedge } { \\otimes } L ^ 1 ( \\Omega ; F ) \\bigr ) ^ * \\\\ & = \\bigl ( E \\overset { \\wedge } { \\otimes } L ^ 1 ( \\Omega ) \\overset { \\wedge } { \\otimes } F \\bigr ) ^ * \\\\ & = L ^ 1 ( \\Omega ; E \\overset { \\wedge } { \\otimes } F ) ^ * \\\\ & = L ^ { \\infty } _ { \\sigma } ( \\Omega ; B ( E , F ^ * ) ) . \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} ( u ^ 2 + k ^ 2 ) \\frac { b ^ 2 } { k ^ 2 } = x ^ 3 \\wedge ( u ^ 3 + k ^ 3 ) \\frac { b ^ 3 } { k ^ 3 } = y ^ 2 \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} \\psi ^ \\prime _ 1 ( u _ 1 , u _ 2 , u _ 3 , w _ 1 , w _ 2 ) & = ( 1 + \\iota ( N ) ^ { - 1 } ) u _ 1 + u _ 2 + w _ 1 \\\\ \\psi ^ \\prime _ 2 ( u _ 1 , u _ 2 , u _ 3 , w _ 1 , w _ 2 ) & = u _ 1 + u _ 2 + w _ 2 \\\\ \\psi ^ \\prime _ 3 ( u _ 1 , u _ 2 , u _ 3 , w _ 1 , w _ 2 ) & = u _ 3 . \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} I ( u ) = \\frac { 1 } { 2 } \\sum _ { i , j } a _ { i j } | u _ i - u _ j | , \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} \\mathrm { R e } \\left [ \\big ( V \\overline { \\Psi } _ { h } ^ { k } , \\partial _ { \\tau } \\Psi _ { h } ^ { k } \\big ) \\right ] = \\frac { 1 } { 2 } \\partial _ { \\tau } \\big ( V \\Psi _ { h } ^ { k } , \\Psi _ { h } ^ { k } \\big ) , \\mathrm { R e } \\left [ \\big ( \\overline { \\phi } _ { h } ^ { k } \\overline { \\Psi } _ { h } ^ { k } , \\partial _ { \\tau } \\Psi _ { h } ^ { k } \\big ) \\right ] = \\frac { 1 } { 2 } \\big ( \\overline { \\phi } _ { h } ^ { k } , \\ , \\partial _ { \\tau } | \\Psi _ { h } ^ { k } | ^ { 2 } \\big ) . \\end{align*}"}
-{"id": "8371.png", "formula": "\\begin{align*} \\mu : = \\sup _ { i } \\mathbb { E } t ( q _ i ) . \\end{align*}"}
-{"id": "9479.png", "formula": "\\begin{align*} | F ( S ) | = \\begin{cases} s _ { r - 1 } & \\mbox { i f } S \\in F ^ { s h } ; \\\\ 0 & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{align*} S _ { M , k , q , \\gamma } : = \\bigcup _ { i = 1 } ^ N S _ { M , k , q , \\gamma , i } . \\end{align*}"}
-{"id": "3336.png", "formula": "\\begin{align*} h \\left ( \\frac 1 \\lambda \\right ) + h ' \\left ( \\frac 1 \\lambda \\right ) \\left ( a - \\frac 1 \\lambda \\right ) & = 1 - N ^ { \\alpha - 1 } J '' \\left ( \\frac 1 \\lambda \\right ) \\left ( a - \\frac 1 \\lambda \\right ) = 1 - N ^ { \\alpha - 1 } \\lambda ^ 2 \\left ( a - \\frac 1 \\lambda \\right ) \\ , , \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} P _ { \\perp } \\begin{pmatrix} M \\\\ [ . 1 c m ] \\overline { N } \\end{pmatrix} . \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} \\tilde \\sigma _ 1 : = \\sigma _ 1 - v \\cdot \\nabla _ x \\ln ( \\phi ) , \\ \\tilde k _ 1 ( x , v ' , v ) = { \\phi ( x , v ) \\over \\phi ( x , v ' ) } k _ 1 ( x , v ' , v ) , \\end{align*}"}
-{"id": "198.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ q \\alpha _ { t _ i } ^ { [ \\sigma ] } c _ i = \\sum _ { i = q + 1 } ^ m \\alpha _ { t _ i } ^ { [ \\sigma ] } b _ i \\end{align*}"}
-{"id": "9750.png", "formula": "\\begin{align*} \\frac { d \\mathcal { E } } { d t } + \\int _ { \\mathbb { R / Z } } \\frac { \\widetilde { W } ^ { { \\rm T } } Q ( b ^ { * } , v ^ { * } ) \\widetilde { W } } { 2 } ( t , s ) d s - \\mathcal { R } ( t ) = 0 , \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} [ L _ { - 1 } , \\ , S _ 1 ] = [ [ L _ { - r - 1 } , \\ , S _ { r } ] , \\ , L _ 1 ] \\subseteq [ L _ { - r } , \\ , S _ r ] \\subseteq [ L _ { - 1 } , \\ , S _ 1 ] \\end{align*}"}
-{"id": "4287.png", "formula": "\\begin{align*} f _ 4 ( X ) = 4 ( X ^ 6 + 5 A X ^ 4 + 2 0 B X ^ 3 - 5 A ^ 2 X ^ 2 - 4 A B X - 8 B ^ 2 - A ^ 3 ) , \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} \\xi ( f \\varphi ' _ i ) - \\xi ( f ) = 2 [ \\alpha _ i ] \\cdot \\quad \\xi ( f \\varphi '' _ i ) - \\xi ( f ) = 2 [ \\beta _ i ] \\cdot \\end{align*}"}
-{"id": "3966.png", "formula": "\\begin{align*} z ^ { 1 + s ( p - 1 ) } f ( s p ) - z f ( 0 ) = & z ^ { 1 + s ( p - 1 ) } \\cdot \\big ( f ( s p ) - f ( 0 ) \\big ) + \\big ( z ^ { 1 + s ( p - 1 ) } - z \\big ) \\cdot f ( 0 ) \\\\ \\equiv & z ^ { 1 + s ( p - 1 ) } \\cdot s p f ' ( 0 ) + s ( z ^ { p } - z ) \\cdot f ( 0 ) \\\\ \\equiv & z ^ p \\cdot s ( f ( p ) - f ( 0 ) ) + s ( z ^ { p } - z ) \\cdot f ( 0 ) = s \\big ( z ^ p f ( p ) - z f ( 0 ) \\big ) \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} \\begin{cases} \\beta _ 1 \\gamma _ 1 = \\alpha _ 1 \\gamma _ 2 \\\\ \\alpha _ 4 \\gamma _ 2 = \\beta _ 4 \\gamma _ 1 \\end{cases} \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{align*} & \\frac { 1 } { N ^ { h - m } } \\sum \\limits _ { \\mathbf { n } \\in \\mathbb { Z } ^ d } \\Big ( \\prod \\limits _ { j = 1 } ^ d f _ j ( n _ j ) \\Big ) \\frac { 1 } { C _ { \\Xi , \\chi } \\eta ^ h } \\int \\limits _ { \\mathbf { y } \\in \\mathbb { R } ^ h } F ( \\mathbf { y } ) G ( L \\mathbf { y } ) \\boldsymbol { \\chi } ( \\Xi ( \\mathbf { y } ) + \\widetilde { \\mathbf { r } } - \\mathbf { n } ) \\ , d \\mathbf { y } . \\end{align*}"}
-{"id": "7695.png", "formula": "\\begin{align*} U _ { n } ^ { ( k ) } = \\frac { 1 } { ( \\alpha - \\beta ) ^ { k } } ( \\alpha ^ { m + 1 } - \\beta ^ { m + 1 } ) ^ { r } ( \\alpha ^ { m } - \\beta ^ { m } ) ^ { k - r } \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} \\mathcal { K } _ { \\partial \\Omega } [ \\phi ] \\equiv \\frac { 1 } { 2 } \\bigg ( \\gamma _ { 0 } \\mathcal { D } _ { \\partial \\Omega } [ \\phi ] + \\gamma _ { 0 } ^ \\complement \\mathcal { D } _ { \\partial \\Omega } [ \\phi ] \\bigg ) \\ , , \\mathcal { W } _ { \\partial \\Omega } [ \\phi ] \\equiv - \\gamma _ { 1 } \\mathcal { D } _ { \\partial \\Omega } [ \\phi ] = - \\gamma _ { 1 } ^ \\complement \\mathcal { D } _ { \\partial \\Omega } [ \\phi ] \\end{align*}"}
-{"id": "4222.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { | \\partial V | } g _ i = 0 . \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} \\Psi ( n ) = \\sum _ { d | n } \\Psi _ 1 ( d ) \\sum _ { e | d } \\frac { \\mu ( e ) } { e } = \\sum _ { d | n } \\Psi _ 1 ( d ) \\phi ( d ) , \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} T _ n = T ( \\pi _ n ) . \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} & \\int _ 0 ^ { \\infty } e ^ { - x s } s ^ { \\lambda - 1 } \\ , d \\mu _ k ( s ) < \\infty , k = 0 , \\ldots , N . \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} | u _ A - u _ E | & = \\frac { \\| u _ A - u _ E \\| _ { L ^ q ( A \\cap E ) } } { \\mu ( A \\cap E ) ^ { 1 / q } } \\\\ & \\le \\frac { \\| u - u _ A \\| _ { L ^ q ( A ) } + \\| u - u _ E \\| _ { L ^ q ( E ) } } { \\mu ( A \\cap E ) ^ { 1 / q } } \\le \\frac { 2 Q } { \\mu ( A \\cap E ) ^ { 1 / q } } . \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{align*} E _ q ^ { \\gamma _ j } : = \\bigcup _ { Q , \\ , l ( Q ) \\le 2 ^ { 1 0 } l ( q ) : \\ , \\int _ Q | f _ q - f _ q \\chi _ { \\bigcup _ { l > j } E _ q ^ { \\gamma _ l } } | \\ge \\gamma _ j \\cdot l ( Q ) } Q . \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{align*} v _ c = \\mathrm { m i n } \\left ( v _ { c , v } , ~ v _ { c , s } \\right ) \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } d v _ { \\tau } & = & \\frac { 1 } { \\varepsilon } f ( v _ { \\tau } , w _ { \\tau } ) d \\tau + \\frac { \\sigma } { \\sqrt { \\varepsilon } } d W _ { \\tau } , \\\\ d w _ { \\tau } & = & g ( v _ { \\tau } , w _ { \\tau } ) d \\tau , \\end{array} \\right . \\end{align*}"}
-{"id": "3819.png", "formula": "\\begin{align*} \\nu _ 2 ( t _ 3 ( 4 n ) ) = \\nu _ 2 ( t _ 3 ( 4 n + 1 ) ) & = 0 , & \\\\ \\nu _ 2 ( t _ 3 ( 4 n + 3 ) ) = \\nu _ 2 ( t _ 3 ( 4 n + 6 ) ) & = 3 + \\nu _ 2 ( t _ 3 ( n ) ) , \\end{align*}"}
-{"id": "1807.png", "formula": "\\begin{align*} \\int _ { \\R ^ { n } } e ^ { 2 t \\ , \\Re \\ , \\ , a ( \\xi ) } | f ( \\xi ) | ^ { 2 } \\ , d \\xi = \\sum _ { N = 1 } ^ { \\infty } \\int _ { B _ { N } } e ^ { 2 t \\ , \\Re \\ , \\ , a ( \\xi ) } f _ { N } ^ { 2 } ( \\xi ) \\ , d \\xi \\geqslant \\sum _ { N = 1 } ^ { \\infty } \\frac { 1 } { 2 N } = \\infty . \\end{align*}"}
-{"id": "9905.png", "formula": "\\begin{align*} \\ ; \\big [ A - \\lambda I \\mid B \\big ] = n \\ ; \\mbox { f o r a l l } \\ ; \\lambda \\in \\Lambda _ + ( A ) . \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & t _ 1 & \\cdots & t _ 1 ^ { n - 2 } & t _ 1 ^ { n - 1 } \\\\ \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 1 & t _ { n - 1 } & \\cdots & t _ { n - 1 } ^ { n - 2 } & t _ { n - 1 } ^ { n - 1 } \\\\ 0 & 0 & \\cdots & 0 & \\lambda \\end{bmatrix} . \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} \\frac { d E ( \\boldsymbol { h } ( t ) ) } { d t } = F ( t ) - | | \\nabla E ( \\boldsymbol { h } ( t ) ) | | ^ 2 _ { l ^ 2 ( \\mathbb { Z } ) } , \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} \\int _ { \\partial \\tilde B ^ i _ \\delta } { N _ n \\over | x | ^ 2 } \\ , d S = \\int _ { \\partial \\tilde B ^ i _ \\delta } N _ n \\left ( { 1 \\over \\delta ^ 2 } + O ( 1 ) \\right ) \\ , d S = O ( \\delta ) . \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} [ y _ 1 , y _ 1 ] = y _ 4 , [ y _ 1 , y _ 2 ] = \\frac { \\alpha _ 3 } { \\alpha _ 5 } y _ 4 + \\theta _ 1 y _ 5 , [ y _ 2 , y _ 1 ] = y _ 4 + \\theta _ 2 y _ 5 , [ y _ 2 , y _ 3 ] = \\alpha _ 1 \\gamma _ 2 y _ 5 . \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{align*} \\tilde P _ { N , a , b } ( \\tilde \\xi _ N < \\epsilon N ^ d | \\Omega _ N ^ + ) = \\frac { \\tilde P _ { N , a , b } ( \\{ \\tilde \\xi _ N < \\epsilon N ^ d \\} \\cap \\Omega _ N ^ + ) } { \\tilde P _ { N , a , b } ( \\Omega _ N ^ + ) } \\end{align*}"}
-{"id": "6646.png", "formula": "\\begin{align*} \\overline { \\omega ( f , v _ 0 ) } \\ ; = \\ ; - \\omega ( v _ 0 , f ) \\ ; = \\ ; \\lim _ { r \\downarrow 0 } W _ r ( \\overline { v _ 0 } , f ) \\ ; = \\ ; L _ { v _ 0 } ( f ) \\ ; = \\ ; 0 \\ , , \\end{align*}"}
-{"id": "291.png", "formula": "\\begin{align*} \\varepsilon ( Y \\triangleright \\phi ) = ( Y \\triangleright \\phi ) ( 1 ) = \\phi ( Y ) = ( \\phi \\triangleleft Y ) ( 1 ) = \\varepsilon ( \\phi \\triangleleft Y ) . \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} x y ^ 2 x = y x ^ 2 x \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} \\mathbb { P } ( \\# { \\cal E } _ i = r ) \\leq \\frac { T _ r e ^ { - r } } { C ( r - 1 ) ! } e ^ { - \\delta _ 0 r } \\leq \\frac { T _ r e ^ { - r } } { C ( r - 1 ) ! } \\frac { 1 } { n ^ { M \\delta _ 0 } } . \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} \\dim X - \\dim V = \\tau - \\delta ( e , a , b , c ) - t ( e , a , b ) . \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{align*} \\nabla ^ \\perp \\cdot H = \\nabla ^ \\perp H _ 2 \\cdot \\nabla H _ 1 . \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} \\Psi _ { \\Lambda + x } ^ { \\mathrm { I P } } = \\chi _ { x } \\left ( \\Psi _ { \\Lambda } ^ { \\mathrm { I P } } \\right ) \\ , \\Lambda \\in \\mathcal { P } _ { f } ( \\mathfrak { L } ) \\ . \\end{align*}"}
-{"id": "1917.png", "formula": "\\begin{align*} d J d u = \\sum _ { i , l = 1 } ^ n a _ i P _ { l i , j } d z _ j \\wedge d z _ i + a _ i H _ { i l , j } d z _ j \\wedge d t _ l . \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} \\frac { d } { d t } E _ { \\lambda } ^ { \\omega } \\xi _ t = ( G _ { \\omega } - I ) E _ { \\lambda } ^ { \\omega } \\xi _ t . \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} \\mathbb { E } ( J _ y ) = \\sum _ { z \\in U _ { t - 1 } } \\mathbb { E } X _ { y , z } \\leq \\# U _ { t - 1 } p _ u \\leq n p _ u = C _ u . \\end{align*}"}
-{"id": "242.png", "formula": "\\begin{align*} \\mathrm { b } ( m \\otimes a _ 1 \\otimes \\cdots \\otimes a _ n ) & : = m a _ 1 \\otimes \\cdots \\otimes a _ n \\\\ & + \\sum _ { i = 1 } ^ { n - 1 } ( - 1 ) ^ i m \\otimes a _ 1 \\otimes \\cdots \\otimes a _ i a _ { i + 1 } \\otimes \\cdots \\otimes a _ n \\\\ & + ( - 1 ) ^ n a _ n m \\otimes a _ 1 \\otimes \\cdots \\otimes a _ { n - 1 } . \\end{align*}"}
-{"id": "9556.png", "formula": "\\begin{align*} \\left | \\mathcal { F } \\left [ f \\cdot S _ { l } g \\right ] \\left ( \\omega \\right ) \\right | = \\frac { 1 } { 2 } \\left | \\sum _ { m \\in \\mathbb { Z } } \\ ! \\ ! e ^ { - \\pi i l m } \\hat { f } \\left ( \\frac { m } { 2 } \\right ) \\hat { g } \\left ( \\frac { m } { 2 } - \\omega \\right ) \\right | \\end{align*}"}
-{"id": "1856.png", "formula": "\\begin{align*} \\nabla _ { e _ i } e _ j = D ^ g _ { e _ i } ( e _ j ) + \\left ( \\frac 3 2 t _ { i j } ^ { \\enskip \\ ; k } - T _ { j \\enskip i } ^ { \\ , \\ , k } \\right ) e _ k \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} J _ T ( x ) = \\begin{pmatrix} e ^ { - 2 \\pi i N \\int _ x ^ 1 \\psi _ { \\alpha , \\varepsilon } ( s ) d s } & \\frac { 1 } { \\sqrt { 1 - x ^ 2 } } \\\\ 0 & e ^ { 2 \\pi i N \\int _ x ^ 1 \\psi _ { \\alpha , \\varepsilon } ( s ) d s } \\end{pmatrix} ; \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} \\delta _ { U , Q , B , r } = \\sum _ { x \\in U ( k ) ; H ( x ) \\leqslant B } \\delta _ { B ^ { \\frac { 1 } { r } } \\rho ( x ) } . \\end{align*}"}
-{"id": "9419.png", "formula": "\\begin{align*} \\partial _ t D + \\nabla \\times \\left ( \\frac { - \\lambda ^ 2 B + D \\times ( D \\times B ) } { \\sqrt { \\lambda ^ 4 + \\lambda ^ 2 B ^ 2 + \\lambda ^ 2 D ^ 2 + | D \\times B | ^ 2 } } \\right ) , \\quad \\nabla \\cdot D = 0 . \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} D _ * & \\cong L ^ \\infty ( \\alpha ; \\Z ) \\otimes _ { \\Z \\Gamma } C _ * \\bigl ( \\pi ^ { - 1 } ( \\partial M ) ; \\Z \\bigr ) \\\\ & \\cong L ^ \\infty ( \\alpha ; \\Z ) \\otimes _ { \\Z \\Gamma } \\Z \\Gamma \\otimes _ { \\Z \\Gamma _ 0 } C _ * ( U ; \\Z ) \\\\ & \\cong L ^ \\infty ( \\alpha _ 0 ; \\Z ) \\otimes _ { \\Z \\Gamma _ 0 } C _ * ( U ; \\Z ) \\\\ & = C _ * ( \\partial M ; \\alpha _ 0 ) . \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} ( p ^ a - 1 ) \\sigma _ a ( [ p ; f _ 1 , f _ 2 ] ) = p ^ { a ( f _ 1 + f _ 2 + 1 ) + f _ 1 } \\frac { p ^ { ( a - 1 ) ( f _ 1 + 1 ) } - 1 } { p ^ { a - 1 } - 1 } - \\frac { p ^ { ( a + 1 ) ( f _ 1 + 1 ) } - 1 } { p ^ { a + 1 } - 1 } . \\end{align*}"}
-{"id": "826.png", "formula": "\\begin{align*} T ( z ) = \\frac { a z + b } { c z + d } , \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} \\sharp ( F _ * ^ e ( R ^ { \\sharp } ) , R ^ { \\sharp } ) = b ^ e [ ( \\frac { q - 1 } { 2 } ) ^ n + ( \\frac { q + 1 } { 2 } ) ^ n ] \\end{align*}"}
-{"id": "9197.png", "formula": "\\begin{align*} & \\Delta U ( M _ s , M ^ d _ s ) - \\langle \\partial _ x U ( M _ { s - } , M ^ d _ { s - } ) , \\Delta M _ s \\rangle - \\langle \\partial _ y U ( M _ { s - } , M ^ d _ { s - } ) , \\Delta M ^ d _ s \\rangle \\\\ & = V ( M _ { s - } + M ^ d _ { s - } + 2 \\Delta M _ s , M ^ d _ { s - } - M _ { s - } ) - V ( M _ { s - } + M ^ d _ { s - } , M ^ d _ { s - } - M _ { s - } ) \\\\ & \\quad - \\langle \\partial _ x V ( M _ { s - } + M ^ d _ { s - } , M ^ d _ { s - } - M _ { s - } ) , 2 \\Delta M _ s \\rangle \\leq 0 , \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} \\psi ^ Q _ { j _ 1 \\dots j _ n } = \\sum _ { P \\in \\mathcal { W } _ n } \\mathcal { A } _ { j _ 1 \\dots j _ n } ^ { ( P , Q ) } e ^ { i ( k _ { P ( 1 ) } x _ 1 + \\dots + k _ { P ( n ) } x _ n ) } . \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{align*} \\psi ^ { ( D ) } _ n ( \\xi ) = \\chi ( 2 ^ { - n } \\| \\xi \\| ) - \\chi ( 2 ^ { - n + 1 } \\| \\xi \\| ) \\ , , n \\ge 1 \\ , . \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} \\sqrt { 4 t } \\biggl \\| \\int _ 0 ^ { t _ A } F ( 4 t - s ) * ( u _ n \\otimes u _ n ) ( s ) \\biggr \\| _ \\infty & \\le C \\sqrt t \\int _ 0 ^ { t _ A } \\| F ( 4 t - s ) \\| _ { 3 } \\| u _ n ( s ) \\| _ 3 ^ 2 \\dd s \\\\ & \\le C \\sqrt t ( 4 t - t _ A ) ^ { - 3 / 2 } t _ A \\| u _ n \\| _ X ^ 2 \\le C \\varepsilon ^ 2 \\sqrt t ( t - t _ A ) ^ { - 3 / 2 } t _ A . \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{align*} g _ { n } = \\frac { \\left ( - 1 \\right ) ^ { n + 1 } } { \\left ( n + 1 \\right ) ! } \\left ( x ^ { n + 1 } - \\left ( x - 1 \\right ) ^ { n + 1 } \\right ) = \\frac { \\left ( - 1 \\right ) ^ { n + 1 } } { n ! } \\int _ { 0 } ^ { 1 } \\left ( x + u - 1 \\right ) ^ { n } d u , \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { r } \\frac { ( 2 d ) ^ m } { m ! } \\left | \\overline { q } \\right | = O ( q _ 0 e ^ { 2 d } ) \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{gather*} \\left ( \\begin{matrix} x = 0 & x = \\infty \\ , \\left ( \\frac 1 2 \\right ) \\\\ \\begin{matrix} 0 \\\\ 0 \\\\ \\theta ^ 0 \\end{matrix} & \\overbrace { \\begin{matrix} 1 & t _ 2 & - t _ 1 / 2 & \\theta ^ \\infty _ 1 / 2 \\\\ - 1 & t _ 2 & t _ 1 / 2 & \\theta ^ \\infty _ 1 / 2 \\\\ 0 & 0 & 0 & \\theta ^ \\infty _ 2 \\end{matrix} } \\end{matrix} \\right ) , \\end{gather*}"}
-{"id": "4663.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } - \\frac { 1 } { N } \\log w _ R ( t ) = N V ( t ) , t \\in ( - 1 , 1 ) . \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} \\lambda _ k g ( x ) = \\bigwedge _ { y \\in Y } g ( y ) / k ( x , y ) , \\textrm { f o r a l l } x \\in X . \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{align*} \\frac { 1 } { A ( N ) } \\sum _ { n = 1 } ^ { [ N t ] } X ( n ) \\Rightarrow Y ( t ) \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{align*} \\mu _ { \\infty } ( \\pi ( N ( L , W ) ) ) > 0 \\mu _ { \\infty } ( \\pi ( S ( L , W ) ) ) = 0 . \\end{align*}"}
-{"id": "82.png", "formula": "\\begin{align*} \\frak m _ 1 ( \\frak m _ 1 ( x ) ) + \\frak m _ 2 ( \\frak m _ 0 ( 1 ) , x ) + ( - 1 ) ^ { \\deg x + 1 } \\frak m _ 2 ( x , \\frak m _ 0 ( 1 ) ) = 0 . \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} \\Psi ' ( 0 ) = 2 \\Psi ( 0 ) \\cdot H _ { \\frac { p - 1 } { 2 } } \\pmod { p } . \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{align*} m \\left ( f \\right ) = \\int _ { 0 } ^ { 1 } \\log \\left | f \\left ( e ^ { 2 \\pi i k } \\right ) \\right | d k = \\log \\left | a _ { 0 } \\right | + \\sum _ { i = 1 } ^ { n } \\log \\left ( \\max \\left \\{ 1 , \\left | \\xi _ { i } \\right | \\right \\} \\right ) , \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} \\left ( \\begin{pmatrix} w & 0 \\\\ 0 & w \\end{pmatrix} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\right ) ^ { p ^ 2 - 1 } = I - \\frac { 1 } { w } \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} p \\end{align*}"}
-{"id": "1033.png", "formula": "\\begin{align*} m _ N ( \\xi ) : = \\left \\{ \\begin{array} { c l } 1 & | \\xi | \\leq N , \\\\ ( N ^ { - 1 } | \\xi | ) ^ { \\gamma - 2 } & | \\xi | \\geq 2 N . \\end{array} \\right . \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} D ( \\alpha , K ) = \\left ( \\frac { \\varepsilon K ^ 2 } { \\Xi ( \\alpha ) } \\right ) ^ \\frac { 1 } { 2 } \\leqslant 2 \\sqrt { \\varepsilon } K b ^ \\frac { 1 } { 2 } \\ll K b ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "4082.png", "formula": "\\begin{align*} \\begin{array} { r c l } R _ t ( y , z , v ) & = & ( N _ t ( z ) ) ^ { - 1 } \\ 1 ( z \\in \\C _ t ( y ) , \\ z \\ne v ) + \\ 1 ( y = z \\ne v , \\ , N _ t ( y ) = 0 ) ) / ( n - 1 ) \\ \\hbox { a n d } \\\\ R _ t ( y , v ) & = & ( \\ 1 ( N _ t ( y ) = 0 , v \\ne y ) + \\sum _ { z \\in \\C _ t ( y ) \\setminus \\{ v \\} } N _ t ( z ) ^ { - 1 } ) / ( n - 1 ) . \\\\ \\end{array} \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} W _ 2 ( \\lambda ) R = \\left \\{ \\ , \\begin{bmatrix} a & b \\pi ^ n \\\\ c & d \\pi \\end{bmatrix} : a , b , c , d \\in D c \\equiv a \\lambda \\mod \\pi D \\ , \\right \\} . \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} H ^ { \\beta } ( z ^ 2 ) = \\sum _ { n \\in \\mathbb { Z } } H ^ { \\beta } _ { ( n ) } z ^ { - 2 n - 2 } ; H ^ { \\gamma } ( z ^ 2 ) = \\sum _ { n \\in \\mathbb { Z } } H ^ { \\gamma } _ { ( n ) } z ^ { - 2 n - 2 } . \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} \\big ( a ( D ) u \\big ) ( x ) = \\sum _ { | \\alpha | \\leqslant m } a _ { \\alpha } \\int _ { \\mathbb { R } ^ n } e ^ { 2 \\pi i x \\cdot \\xi } \\xi ^ { \\alpha } \\hat { u } ( \\xi ) \\ , d \\xi = \\int _ { \\mathbb { R } ^ n } e ^ { 2 \\pi i x \\cdot \\xi } a ( \\xi ) \\hat { u } ( \\xi ) \\ , d \\xi , \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 - \\eqref { 1 1 1 . 1 a } \\times \\frac { 1 } { u _ { \\alpha \\beta } } \\bigg ( \\ln \\frac { u _ { \\alpha \\beta } ^ 2 } { \\underbar { n } } \\bigg ) _ - ^ { 2 k - 1 } d x , k = 1 , 2 , 3 , \\cdots , \\quad \\ ( \\cdot ) _ - : = \\min \\{ 0 , \\cdot \\} . \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} \\left | \\sum _ { n = 1 } ^ N y _ n \\right | ^ 2 \\leq \\frac { N + H } { H + 1 } \\sum _ { n = 1 } ^ N | y _ n | ^ 2 + \\frac { 2 ( N + H ) } { H + 1 } \\sum _ { h = 1 } ^ H \\left ( 1 - \\frac { h } { H + 1 } \\right ) \\left | \\sum _ { n = 1 } ^ { N - h } y _ { n + h } \\overline { y _ n } \\right | . \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{align*} \\max _ j | a _ i | \\le \\frac 4 3 \\max _ j | \\psi ^ j ( \\sum _ { i = 1 } ^ k a _ i v _ i ) | \\le \\frac 4 3 \\| \\sum _ { i = 1 } ^ k a _ i v _ i \\| \\le \\frac 8 3 r \\max _ j | a _ j | \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{gather*} \\lambda _ 1 + \\cdots + \\lambda _ p = \\mu _ 1 + \\cdots + \\mu _ q = m . \\end{gather*}"}
-{"id": "6581.png", "formula": "\\begin{align*} H ^ 1 ( \\mathbb { R } ^ 3 ) \\otimes \\mathbb { C } ^ 4 \\ ; \\cong \\ ; H ^ 1 ( \\mathbb { R } ^ 3 , \\mathbb { C } ^ 4 ) \\ , , \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} \\mathcal A _ { \\Lambda } ( u , \\nabla u , \\nabla ^ { 2 } u ) = 0 \\quad \\Omega \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} u \\left ( x \\right ) = F ^ { - 1 } \\left [ A + K \\left ( \\xi \\right ) + \\lambda \\right ] ^ { - 1 } f ^ { \\symbol { 9 4 } } . \\end{align*}"}
-{"id": "4510.png", "formula": "\\begin{align*} \\underset { l \\rightarrow \\infty } { \\lim } \\mathbf { \\tilde { K } } _ { l } = 0 \\ . \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{gather*} \\frac { \\partial } { \\partial t } T \\left ( x , t \\right ) = - \\frac { \\partial } { \\partial x } q \\left ( x , t \\right ) , \\ ; \\ ; x \\in \\mathbb { R } , \\ ; t > 0 , \\\\ \\int _ { 0 } ^ { 1 } \\phi \\left ( \\gamma \\right ) \\ , { } _ { 0 } ^ { c } \\mathrm { D } _ { t } ^ { \\gamma } q \\left ( x , t \\right ) \\mathrm { d } \\gamma = - \\frac { \\partial } { \\partial x } T \\left ( x , t \\right ) , \\ ; \\ ; x \\in \\mathbb { R } , \\ ; t > 0 , \\end{gather*}"}
-{"id": "2164.png", "formula": "\\begin{align*} S ( - k ) = S ( k ) ^ \\dagger = S ( k ) ^ { - 1 } , k \\in \\mathbb { R } , \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} h _ v & \\in [ n ] v \\in V \\\\ h _ u & = h _ v + \\omega ( u , v ) \\end{align*}"}
-{"id": "9908.png", "formula": "\\begin{align*} A = \\begin{bmatrix} 0 & 1 \\\\ 0 & 0 \\end{bmatrix} , \\ ; B = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} V ( \\mathsf { x } ) = \\frac { \\left \\vert \\mathsf { x } \\right \\vert ^ { 2 } } { 2 } , \\end{align*}"}
-{"id": "9646.png", "formula": "\\begin{align*} t _ 1 = \\arccos ( s _ 1 / 2 ) , \\ ; t _ 2 = \\arccos ( s _ 2 / 2 ) , \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} Z ^ { ( i ) } | Z _ 1 \\in B _ 1 - a \\mu \\stackrel { d } { = } Z | Z _ 1 \\in B _ 1 - a \\mu , \\ ; \\ ; i = 2 , \\dots , n . \\end{align*}"}
-{"id": "4997.png", "formula": "\\begin{align*} \\beta : = 1 2 \\langle \\psi _ * ^ 2 , w _ 0 + w _ 2 \\rangle _ { L ^ 2 } . \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} \\tilde { s } ^ H = \\varphi ^ 2 \\frac { B } { P } - \\varphi ^ 2 \\frac { ( P ^ { n - 1 } \\mathbf { H } ) '' } { P ^ { n - 1 } } + m \\varphi \\varphi ' \\frac { ( P ^ { n - 1 } \\mathbf { H } ) ' } { P ^ { n - 1 } } + m \\left ( \\varphi \\varphi '' - ( \\varphi ' ) ^ 2 \\right ) \\mathbf { H } \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} \\pm \\sqrt { \\left \\vert K _ { 0 } \\right \\vert } = \\frac { g _ { 0 } f ^ { \\prime } } { 1 - \\left ( g _ { 0 } f \\right ) ^ { 2 } } . \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{align*} ( L _ { c _ * } - \\partial _ y ^ 2 + 4 \\delta ) \\tilde { u } _ b = \\Pi \\tilde { F } , \\end{align*}"}
-{"id": "657.png", "formula": "\\begin{align*} [ a _ j ^ { + } , a _ i ^ - ] ~ | n _ 1 , \\cdots , n _ i , \\cdots , n _ j , \\cdots , n _ r \\rangle \\ = \\sqrt { n _ i ( n _ j + 1 ) ) } ~ | n _ 1 , \\cdots , n _ i - 1 , \\cdots , n _ j + 1 , \\cdots , n _ r \\rangle . \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{align*} _ 3 F _ 2 \\left ( \\left . \\begin{array} { c } \\frac { 1 } { 3 } , \\frac { 1 } { 2 } , \\frac { 2 } { 3 } \\\\ 1 , 1 \\\\ \\end{array} \\right | \\frac { 2 7 u ^ { 2 } } { 4 ( 1 - u ) ^ { 3 } } \\right ) = 1 + 3 \\sum _ { n = 1 } ^ \\infty [ ( 2 n - 1 ) ! ! ] ^ 2 { 3 n - 1 \\choose 2 n } \\frac { 1 } { ( n ! 2 ^ n ) ^ 2 } \\frac { u ^ { 2 n } } { ( 1 - u ) ^ { 3 n } } , \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{align*} \\mathbb P _ { \\mathbb R ^ 3 } G ( t ) = G ( t ) - \\nabla Q ( t ) \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , z ^ 2 \\bigg ] = ( 1 + z ) ^ { - 2 \\alpha } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\frac 1 2 \\\\ & 1 \\end{matrix} \\bigg | \\ , \\frac { 4 z } { ( 1 + z ) ^ 2 } \\bigg ] . \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{align*} v ( t _ k ) + \\nabla \\psi ( x ( t _ k ) ) \\in \\partial \\phi ( x ( t _ k ) ) + \\nabla \\psi ( x ( t _ k ) ) = \\partial _ L ( \\phi + \\psi ) ( x ( t _ k ) ) . \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} \\psi \\left ( s \\right ) = s \\Phi \\left ( s \\right ) + \\xi ^ { 2 } , \\ ; \\ ; s \\in \\mathbb { C } , \\end{align*}"}
-{"id": "1893.png", "formula": "\\begin{align*} x ^ j _ c = x ^ j _ \\infty c ^ { m - n _ j } + O ( c ^ { m - n _ j - 1 } ) . \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} U _ r ( a ) = \\left \\{ b \\in \\mathcal B ( L ) \\mid \\mathrm { d i s t } ( \\lambda , \\mathrm { s u p p } ( a ) ) \\geq r \\implies a _ \\pm ( \\lambda ) = b _ \\pm ( \\lambda ) \\right \\} \\end{align*}"}
-{"id": "9415.png", "formula": "\\begin{align*} \\Gamma ^ { A _ 1 , \\ldots , A _ n } ( \\phi ) ( X _ 1 , \\ldots , X _ { n - 1 } ) = \\int _ { \\Sigma } a _ 1 ( t , A _ 1 ) X _ 1 a _ 2 ( t , A _ 2 ) X _ 2 \\cdots X _ { n - 1 } a _ n ( t , A _ n ) \\ , \\mu ( t ) \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} \\partial _ { t } ^ { 2 } [ \\Xi _ { \\mathrm { p } , l } ^ { ( \\omega ) } ( t ) ] _ { + } = \\underset { l \\rightarrow \\infty } { \\lim } \\partial _ { t } ^ { 2 } [ \\Xi _ { \\mathrm { p } , l } ^ { ( \\omega ) } ( t ) ] _ { + } \\ , t \\in \\mathbb { R } \\ . \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} \\dot \\psi & = X _ g ( \\psi , \\bar \\psi ) , \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} \\widetilde w _ \\ell ( t ) = a _ \\ell \\ , \\big ( \\Im \\zeta ^ \\ell - \\Im \\zeta ^ { k \\pi / \\omega } \\big ) + b _ \\ell \\ , \\Re \\zeta ^ \\ell \\end{align*}"}
-{"id": "9765.png", "formula": "\\begin{align*} \\| d j v \\| _ { W ^ { - 1 , p _ n } } = \\| d j v \\| _ { ( W ^ { 1 , 2 n } ) ^ * } \\leq C \\left ( \\frac { E _ { \\epsilon } ( v ) } { | \\log \\epsilon | } + \\epsilon ^ { \\gamma } \\right ) \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} W ^ T = W = x ^ 2 + y ^ 4 + y z ^ 4 + w ^ { 1 6 } , \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{align*} \\varphi ( k , x ) = \\frac { 1 } { 2 i k } f ( k , x ) J ( - k ) - \\frac { 1 } { 2 i k } f ( - k , x ) J ( k ) , k \\in \\mathbb { R } \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} S = \\begin{pmatrix} 0 & 1 & 1 \\\\ 1 & 1 & 0 \\\\ 2 & 0 & 0 \\end{pmatrix} C = \\begin{pmatrix} 0 & 1 \\\\ 1 & - 1 \\end{pmatrix} \\end{align*}"}
-{"id": "136.png", "formula": "\\begin{align*} d \\eta ( \\theta ) = ( 1 - \\cos \\theta ) \\frac { d \\theta } { 2 \\pi } \\end{align*}"}
-{"id": "3003.png", "formula": "\\begin{align*} \\sup _ { 0 \\le j \\le N } \\ \\Vert P _ { n } T ^ { \\ , j } P _ { l } \\ , x \\Vert & \\le \\dfrac { 1 } { 4 } \\ , \\beta _ { l } \\ , \\cdot \\Bigl ( \\prod _ { i = \\Delta ^ { ( k ) } - N + 1 } ^ { \\Delta ^ { ( k ) } - 1 } \\ ! \\ ! \\ ! | w _ { i } ^ { ( k ) } | \\ \\Bigr ) \\ , \\cdot \\ , \\Vert P _ { l } \\ , x \\Vert . \\end{align*}"}
-{"id": "9992.png", "formula": "\\begin{align*} X _ 1 = Y _ 1 = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} ^ T \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} I _ k ( a ) & = \\frac { \\Gamma ( k + 1 / 2 ) } { \\Gamma ( k + 1 ) } \\sqrt { \\pi } a ^ k \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} \\begin{bmatrix} \\int _ 0 ^ T Y _ u \\ , \\dd u & T \\\\ \\int _ 0 ^ { T / 2 } Y _ u \\ , \\dd u & T / 2 \\end{bmatrix} \\end{align*}"}
-{"id": "9984.png", "formula": "\\begin{align*} z ( t ) = Y _ { \\sigma ( t ) } ^ T x ( t ) \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} \\phi ( 0 ) & \\ge \\lim _ { t \\to \\infty } \\phi ( t ) \\\\ & = \\lim _ { t \\to \\infty } \\left ( S ( C | M ) _ { \\hat { \\rho } _ { C M } ( t ) } - \\lambda \\ , S ( A | M ) _ { \\hat { \\rho } _ { A M } ( t ) } - ( 1 - \\lambda ) \\ , S ( B | M ) _ { \\hat { \\rho } _ { B M } ( t ) } \\right ) \\\\ & = n \\left ( \\lambda \\ln \\frac { \\eta } { \\lambda } + \\left ( 1 - \\lambda \\right ) \\ln \\frac { | 1 - \\eta | } { 1 - \\lambda } \\right ) \\ ; , \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{align*} \\widehat { h } ( - i k _ j ) C _ j = 0 , j = \\overline { 1 , N } , \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} F ( 2 ^ { - i } ) = \\sum _ { j = 0 } ^ i c _ { i + 1 } ( j ) = \\sum _ { j = 0 } ^ { i / 2 } c _ { i + 1 } ( 2 j ) = \\sum _ { j = 0 } ^ { i / 2 } \\frac { 2 ^ { ( 1 - ( i + 1 ) ) 2 j } F ( 2 ^ { 2 j - ( i + 1 ) } ) } { 2 ^ { ( 1 - 2 j ) ( 2 j ) / 2 - 1 } ( 2 j ) ! } = \\sum _ { j = 0 } ^ { i / 2 } \\frac { 4 ^ { j ( j - i ) } F \\left ( 2 ^ { 2 j - i - 1 } \\right ) } { 2 ^ { j - 1 } ( 2 j ) ! } \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} & \\frac { \\Gamma _ p ( \\alpha _ 1 + s _ 1 p ) \\cdots \\Gamma _ p ( \\alpha _ m + s _ m p ) } { \\Gamma _ p ( \\beta _ 1 + t _ 1 p ) \\cdots \\Gamma _ p ( \\beta _ n + t _ n p ) } - \\frac { \\Gamma _ p ( \\alpha _ 1 ) \\cdots \\Gamma _ p ( \\alpha _ m ) } { \\Gamma _ p ( \\beta _ 1 ) \\cdots \\Gamma _ p ( \\beta _ n ) } \\\\ \\equiv & ( - 1 ) ^ { \\delta } p \\cdot \\frac { d } { d x } \\bigg ( \\frac { \\Gamma ( a _ 1 + s _ 1 x ) \\cdots \\Gamma ( a _ m + s _ m x ) } { \\Gamma ( b _ 1 + t _ 1 x ) \\cdots \\Gamma ( b _ n + t _ n x ) } \\bigg ) \\bigg | _ { x = 0 } \\pmod { p ^ 2 } , \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} \\mathcal { A } ^ { P T _ i } = Y _ i ( k _ { P ( i ) } - k _ { P ( i + 1 ) } ) \\mathcal { A } ^ P , \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} & [ \\mathit { q } _ { 0 } ] - \\left ( \\frac { \\Gamma ( 1 - 2 / { { c _ { } } } ) } { \\Gamma ^ 2 ( 1 - 1 / { { c _ { } } } ) } - 1 \\right ) \\left ( \\mathbb { E } [ \\mathit { q } _ { 0 } ] \\right ) ^ 2 = 0 , { { b _ { } } } = \\frac { \\mathbb { E } [ \\mathit { q } _ { 0 } ] } { \\Gamma ( 1 - 1 / { { c _ { } } } ) } . \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} \\bigg \\| \\bigg ( \\ln \\frac { u _ { \\alpha \\beta } ^ 2 } { \\bar { n } } \\bigg ) _ + \\bigg \\| _ { L ^ { k + 1 } ( \\Omega ) } \\leq \\bigg ( \\frac { \\alpha } { \\sqrt { \\bar { n } } } \\bigg ) ^ { \\frac { 1 } { k + 1 } } \\frac { 4 | D | _ 0 } { \\theta _ L } , k = 1 , 2 , 3 , \\cdots . \\end{align*}"}
-{"id": "3350.png", "formula": "\\begin{align*} \\int _ { 0 } ^ a w ( x ) d x = \\int _ { 0 } ^ a \\left ( \\int _ { ( x , \\infty ) } \\mu ( d \\xi ) \\right ) d x = \\int _ { 0 } ^ \\infty \\left ( \\int _ 0 ^ { \\xi \\wedge a } d x \\right ) \\mu ( d \\xi ) = \\int _ 0 ^ \\infty ( \\xi \\wedge a ) \\mu ( d \\xi ) < \\infty . \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{align*} \\left ( ^ { \\rho } I _ { a + } ^ { \\alpha } f \\right ) \\left ( x \\right ) = \\frac { \\rho ^ { 1 - \\alpha } } { \\Gamma \\left ( \\alpha \\right ) } \\int _ { a } ^ { x } \\frac { \\tau ^ { \\rho - 1 } f \\left ( \\tau \\right ) } { \\left ( x ^ { \\rho } - \\tau ^ { \\rho } \\right ) ^ { 1 - \\alpha } } d \\tau , \\end{align*}"}
-{"id": "7062.png", "formula": "\\begin{align*} u \\left ( x , t \\right ) = F ^ { - 1 } \\left [ e ^ { i A _ { \\xi } t } \\hat { f } \\left ( \\xi \\right ) \\right ] , A _ { \\xi } = A + \\left \\vert \\xi \\right \\vert ^ { 2 } . \\end{align*}"}
-{"id": "6156.png", "formula": "\\begin{align*} [ L _ { - r - 1 } , \\ , S _ { r - 1 } ] = [ [ L _ { - r } , \\ , L _ { - 1 } ] , \\ , S _ { r - 1 } ] \\subseteq [ L _ { - r } , \\ , S _ { r - 2 } ] \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{align*} \\bar { \\mathfrak { h } } & = \\mathfrak { h } _ 1 , \\textrm { o n } V _ 1 , \\\\ \\bar { \\mathfrak { h } } & = \\mathfrak { h } _ 2 , \\textrm { o n } V _ 2 , \\textrm { a n d } \\\\ \\bar { \\mathfrak { h } } & = \\mathfrak { h } _ 0 , \\textrm { o n } D \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} \\sum _ { q - e \\le k < q - e / 2 } \\binom { 2 k + e } { k } x ^ { k - q + e } \\equiv \\frac { \\alpha ^ e - \\beta ^ e } { \\alpha - \\beta } \\pmod { p } . \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{align*} I _ { \\upsilon } ( q ; x ) = \\frac { \\left ( \\frac { x } { 2 } \\right ) ^ { \\upsilon } \\ 2 ^ { 2 \\upsilon + q - \\frac { 1 } { 2 } } \\ \\Gamma ( \\upsilon + q ) } { \\sqrt { \\pi } \\ \\Gamma ( 2 \\upsilon + q + \\frac { 1 } { 2 } ) } { _ { 1 } F _ { 1 } } \\left ( \\upsilon + q , 2 \\upsilon + q + \\frac { 1 } { 2 } , 2 x \\right ) . \\end{align*}"}
-{"id": "8358.png", "formula": "\\begin{align*} m _ H ( I _ H \\otimes \\sigma _ H ) \\Delta _ H ( h ) = \\epsilon _ H ( h ) 1 _ H = m _ H ( \\sigma _ H \\otimes I _ H ) \\Delta _ H ( h ) \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} \\bar { \\varrho } ^ { ( \\beta , \\vartheta , \\lambda ) } = \\bar { \\varrho } ^ { ( \\beta , \\vartheta , \\lambda ) } \\circ \\chi _ { x } \\ , x \\in \\mathfrak { L } = \\mathbb { Z } ^ { d } \\ . \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} M ( r ) : = \\sup _ { \\psi \\in \\Psi _ \\mathcal { S } } \\max _ { | \\zeta | \\leq r } G ^ + \\circ \\psi ( \\zeta ) . \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{align*} - \\Delta \\lambda = e ^ { 2 \\lambda } K _ g + 2 \\pi ( \\theta _ 0 - 1 ) \\delta _ 0 , \\end{align*}"}
-{"id": "9640.png", "formula": "\\begin{align*} | n \\omega _ { j } - \\frac { 1 / 2 + j - 1 } { d - 2 } | < \\varepsilon \\ ; \\ ; ( \\mod 1 ) \\ ; \\ ; ( j = 1 , 2 , \\ldots , d / 2 - 1 ) . \\end{align*}"}
-{"id": "6725.png", "formula": "\\begin{align*} \\sup \\limits _ { k , i } \\sum \\limits _ { j = 1 } ^ { N } \\left \\vert \\frac { \\lambda _ { k } } { D \\left ( \\lambda _ { k } \\right ) } A _ { j i } \\left ( \\lambda _ { k } \\right ) \\right \\vert ^ { q } \\int \\limits _ { \\Omega } \\left \\vert \\sum \\limits _ { k = 1 } ^ { \\mu } r _ { k } \\left ( y \\right ) u _ { k j } \\right \\vert ^ { q } d y . \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} \\begin{bmatrix} \\pi & \\lambda \\pi ^ n \\\\ 0 & 1 \\end{bmatrix} \\begin{bmatrix} e & f \\pi ^ n \\\\ g & h \\end{bmatrix} = \\begin{bmatrix} e \\pi + g \\lambda \\pi ^ n & ( f \\pi + h \\lambda ) \\pi ^ n \\\\ g & h \\end{bmatrix} . \\end{align*}"}
-{"id": "9248.png", "formula": "\\begin{align*} f ( x ) = \\mathop { \\rm s u p } _ { s \\in S } \\ ; \\{ \\alpha ( s ) + \\langle v ( s ) , x - s \\rangle \\} \\end{align*}"}
-{"id": "4937.png", "formula": "\\begin{align*} c _ { i j } = a _ { i j } - b _ { i j } , \\ 1 \\leq i , j \\leq n . \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} | u + v \\sqrt { D } | \\geqslant 2 | v | \\sqrt { D } - | u - v \\sqrt { D } | = 2 | v | \\sqrt { D } + O _ { \\varepsilon , \\eta } \\left ( \\frac { 1 } { B } \\right ) \\geqslant 2 \\sqrt { \\frac { A ^ \\prime | m | D } { \\varepsilon ^ \\prime ( 2 \\sqrt { D } + \\frac { \\varepsilon ^ \\prime } { B ^ 2 } ) } } B + O _ { \\varepsilon , \\eta } \\left ( \\frac { 1 } { B } \\right ) . \\end{align*}"}
-{"id": "9686.png", "formula": "\\begin{align*} \\overline { \\alpha } = \\overline { a } , | K _ { + + + } | = | K _ { + - + } | . \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{gather*} \\begin{array} { l c l } \\tau _ 3 ( t ) & = & \\frac { \\sqrt { 6 } } { 7 \\ , y ( t ) ^ 5 } ( - f ^ { 1 3 5 } + f ^ { 1 4 6 } + f ^ { 2 3 6 } + f ^ { 2 4 5 } ) + \\frac { 4 \\sqrt { 6 } } { 2 1 y ( t ) ^ 5 } ( f ^ { 1 2 7 } + f ^ { 3 4 7 } + f ^ { 5 6 7 } ) , \\\\ \\end{array} \\end{gather*}"}
-{"id": "9456.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } \\Phi ( t , s , x ) = v \\big ( t , \\Phi ( t , s , x ) \\big ) & 0 \\leq t \\leq T \\\\ \\Phi ( s , s , x ) = x & 0 \\leq s \\leq T , \\ ; x \\in \\mathbb { T } ^ 3 \\end{cases} \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} \\begin{matrix} = 2 & \\geq 2 & = 1 \\\\ \\geq 2 & \\geq 1 & \\geq 1 \\\\ = 1 & \\geq 1 & = 0 \\end{matrix} \\end{align*}"}
-{"id": "5968.png", "formula": "\\begin{align*} \\eta _ { 1 } z + \\eta _ { 2 } = 0 . \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{align*} \\dot { z } z ^ { - 1 } + \\left ( \\dot { z } z ^ { - 1 } \\right ) ^ * & = \\dot { y } y ^ { - 1 } + \\left ( \\dot { y } y ^ { - 1 } \\right ) ^ * + O ( t ^ { - 2 } \\mathcal L ) \\\\ & = P ( k ) + O ( t ^ { - 2 } \\mathcal L ) \\\\ & = \\left [ ( z \\phi z ^ { - 1 } ) ^ * , z \\phi z ^ { - 1 } \\right ] - \\theta + O ( t ^ { - 1 - \\epsilon } \\mathcal L ) \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} j ( E ) : = c _ 4 ^ 3 / \\Delta . \\end{align*}"}
-{"id": "3399.png", "formula": "\\begin{align*} \\hat \\gamma ^ i _ t = \\frac { \\theta + \\left ( 1 - \\frac { 1 } { n } \\right ) \\phi _ { t } } { 1 + \\frac { 1 } { \\lambda } \\left ( 1 - \\frac { 1 } { n } \\right ) ^ 2 \\phi _ { t } } ( \\bar X _ { t - } - X ^ i _ { t - } ) \\ , . \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} \\mathcal { D } _ { n } : = \\{ - \\left ( n - 1 \\right ) / 2 , - \\left ( n - 3 \\right ) / 2 , \\cdots , \\left ( n - 3 \\right ) / 2 , \\left ( n - 1 \\right ) / 2 \\} ^ { d } \\ . \\end{align*}"}
-{"id": "6864.png", "formula": "\\begin{align*} \\left | \\langle P f , g \\rangle _ { L ^ 2 ( \\mathbb { T } ) } \\right | = \\left | \\langle f , P g \\rangle _ { L ^ 2 ( \\mathbb { T } ) } \\right | \\leq \\| f \\| _ { L ^ { 4 / 3 } } \\| P g \\| _ { H ^ 4 } \\leq \\| f \\| _ { L ^ { 4 / 3 } } \\| g \\| _ { L ^ \\infty } . \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 0 } ^ { 2 } \\varepsilon ^ { \\frac { j } { 2 } } \\left \\vert \\lambda \\right \\vert ^ { 1 - \\frac { j } { 2 } } \\left \\Vert u ^ { \\left ( j \\right ) } \\right \\Vert _ { L _ { p } \\left ( R ; E \\right ) } + \\left \\Vert A u \\right \\Vert _ { L _ { p } \\left ( R ; E \\right ) } = \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{align*} ( * ) & = \\prod _ { i = 1 } ^ { | A | } \\int _ { [ 0 , a ] ^ { A _ i } } P _ N ( \\phi _ { x _ i } + \\tilde { \\psi } _ { x _ i } \\in [ 0 , a ] | \\phi = 0 \\ , o n \\ , A _ i ) g _ i ( \\psi ) d \\psi \\\\ & = \\prod _ { i = 1 } ^ { | A | } \\int _ { [ 0 , a ] ^ { A _ i } } P _ { A _ i ^ c } ( \\phi _ { x _ i } + \\tilde { \\psi } _ { x _ i } \\in [ 0 , a ] ) g _ i ( \\psi ) d \\psi \\\\ & \\geq \\prod _ { i = 1 } ^ { | A | } P _ { A _ i ^ c } ( \\phi _ { x _ i } \\in [ 0 , a ] ) \\\\ & \\geq [ c ( 1 / 2 \\wedge a ) ] ^ { | A | } \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{align*} \\rho ^ { \\circ } ( X ) = \\pi ( K _ { 2 \\rho } ^ { - 1 } S ( X ) K _ { 2 \\rho } ) . \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} I n d _ { ( \\mathbb Z / p ) ^ n , \\mathbb { F } _ p } S ( V ) = \\langle \\alpha _ 1 b _ 1 + \\cdots + \\alpha _ n b _ n \\rangle \\subseteq \\mathbb { F } _ p [ b _ 1 , \\ldots , b _ n ] \\subseteq H ^ { \\ast } ( B ( \\mathbb Z / p ) ^ n ; \\mathbb { F } _ p ) . \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j \\leq p } \\left | \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\bar \\psi _ j ( y _ i , z _ i ) \\right | \\leq C \\sqrt { \\frac { \\log ( p n ) } { n } } \\end{align*}"}
-{"id": "1119.png", "formula": "\\begin{align*} C _ 0 = \\bigl \\{ ( x , y ) \\in \\R ^ 2 : 0 < \\sqrt { 1 - x ^ 2 } - e ^ { - 1 / | x | } \\le y < \\sqrt { 1 - x ^ 2 } \\bigr \\} , \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} X _ { n + 1 } = S _ { \\Delta t } X _ n + \\Delta t S _ { \\Delta t } G ( X _ n ) + S _ { \\Delta t } \\sigma ( X _ n ) \\bigl ( W \\bigl ( ( n + 1 ) \\Delta t \\bigr ) - W ( n \\Delta t ) \\bigr ) , \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} \\pi _ { \\textnormal { r } ( \\theta ) } f = \\pi _ { \\textnormal { r } ( \\theta ) } \\pi _ { \\textnormal { n } ( \\theta ) } \\epsilon E = 0 . \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} X _ { 2 , \\ , j + 2 , \\ , 1 0 } \\cap L _ { j + 2 } = 0 . \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{align*} T ( v _ j , k ) & = ( j e , j e - 1 , \\cdots , j e - k , 1 , \\cdots , j e + k + 1 ) \\\\ S ( v _ j , k ) & = ( j e - k + 1 , j e + 1 ) ( j e - k + 2 , j e + 2 ) \\cdots ( j e , j e + k ) . \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} \\Delta \\Psi = 0 \\textup { ~ i n ~ } R , \\Psi = h \\textup { ~ o n ~ } T \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{align*} \\mathbb { P } \\{ C < R _ { o D } ( p _ o ) | r _ 1 \\} & = p _ o & & \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} \\begin{bmatrix} \\pi & 0 \\\\ 0 & 1 \\end{bmatrix} , \\begin{bmatrix} 1 & 0 \\\\ 0 & \\pi \\end{bmatrix} , \\end{align*}"}
-{"id": "9817.png", "formula": "\\begin{align*} \\mathfrak { T } _ i ^ 2 = ( v - 1 ) \\mathfrak { T } _ i + v \\end{align*}"}
-{"id": "5921.png", "formula": "\\begin{align*} \\sum _ { d | n } s _ { d , k } ^ { ( - 1 ) } & = \\sum _ { m = 0 } ^ n \\sum _ { j = 1 } ^ { n - m } \\left ( s _ o ( n - m , j ) - s _ e ( n - m , j ) \\right ) s _ { j , k } ^ { ( - 1 ) } \\cdot p ( m ) \\\\ & = \\sum _ { m = 0 } ^ n \\delta _ { n - k , m } \\cdot p ( m ) \\\\ & = p ( n - k ) , \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{gather*} P _ { x _ 0 , x _ 2 } ( z ) = z ^ d Q _ V ( z ^ { - 1 } ) . \\end{gather*}"}
-{"id": "3464.png", "formula": "\\begin{align*} \\frac { 1 } { 4 j } + \\sum _ { m = 1 } ^ i \\frac { E _ { 2 m } } { ( 4 m ) } \\binom { 2 j } { 2 m } \\frac { m } { j } = 1 + \\sum _ { m = 1 } ^ i E _ { 2 m } \\binom { 2 j } { 2 m } = \\sum _ { m = 0 } ^ j E _ { 2 m } \\binom { 2 j } { 2 m } \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & 1 - 3 \\alpha \\\\ & \\frac 3 2 - 2 \\alpha \\end{matrix} \\bigg | \\ , \\frac 1 4 \\bigg ] = \\frac { 1 6 ^ { - \\alpha } } { 2 7 ^ { - \\alpha } } \\cdot \\frac { \\Gamma ( \\frac 4 3 ) \\Gamma ( \\frac 3 2 - 2 \\alpha ) } { \\Gamma ( \\frac 3 2 ) \\Gamma ( \\frac 4 3 - 2 \\alpha ) } . \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} \\min _ { \\omega \\in \\R } \\left \\{ \\rho ( \\omega ) = \\max _ { i : \\lambda _ i > 0 } ( 1 - \\omega \\lambda _ i ) ^ 2 \\right \\} . \\end{align*}"}
-{"id": "7133.png", "formula": "\\begin{align*} { _ 0 { \\rm D } _ x ^ { \\alpha } } u _ { \\rm e x } = \\int _ 0 ^ x \\frac { ( x - z ) ^ { - \\alpha } } { \\Gamma ( 1 - \\alpha ) } u ' _ { \\rm e x } ( z ) d z = \\int _ 0 ^ x \\frac { ( x - z ) ^ { - \\alpha } } { \\Gamma ( 1 - \\alpha ) } \\left [ \\mu z ^ { \\mu - 1 } ( 1 - z ) ^ \\nu - \\nu z ^ \\mu ( 1 - z ) ^ { \\nu - 1 } \\right ] d z . \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} \\frac { ( - q ) _ \\infty } { ( q ) _ \\infty } = \\frac { \\ ( q ^ 2 ; q ^ 2 \\ ) _ \\infty } { ( q ) ^ 2 _ \\infty } . \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{align*} K ^ \\prime : = \\prod _ { y ^ { 2 } \\in \\mathcal { U } } K _ 3 ^ { ( y ) } . \\end{align*}"}
-{"id": "3087.png", "formula": "\\begin{align*} \\lambda = \\frac { 1 } { \\sqrt { 2 \\sqrt { 5 } } } , \\qquad \\alpha = 0 , A = \\frac { \\pi ^ 2 } { 1 0 } . \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{align*} \\langle A ^ { - 1 } _ { \\varepsilon , \\delta } \\overline { \\sigma _ { j } } , \\overline { \\sigma _ { s } } \\rangle _ { \\omega _ { \\varepsilon } } = \\langle \\biggl ( \\sum _ { \\substack { 1 \\leq m \\leq k \\\\ m \\neq j } } \\frac { 1 + \\delta \\lambda _ { m } } { \\varepsilon + \\lambda _ { m } } + \\frac { n - k } { \\varepsilon } \\biggl ) ^ { - 1 } \\overline { \\sigma _ { j } } , \\overline { \\sigma _ { s } } \\rangle _ { \\omega _ { \\varepsilon } } \\leq \\frac { \\varepsilon } { n - k } \\delta _ { j s } \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} \\mu ( \\lambda ) = ( 2 d ) ^ m \\left ( n ^ { t + m } \\left ( \\sum _ { i = 0 } ^ { t - 1 } p _ i ^ \\lambda / n ^ i + O ( q ^ \\lambda / n ^ t ) \\right ) \\right ) \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\# { \\cal E } _ 1 = r \\right ) \\leq { n - 1 \\choose r - 1 } T _ r p _ u ^ { r - 1 } ( 1 - p _ d ) ^ { r ( n - r ) } . \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} & h _ { 0 , 1 } ( x ) = \\frac { 1 } { 2 } \\left ( ( 1 + \\sqrt { x } ) ^ m + ( 1 - \\sqrt { x } ) ^ m \\right ) \\equiv \\frac { 1 } { 2 } \\left ( 1 + x ^ { \\frac { m } { 2 } } + 1 - x ^ { \\frac { m } { 2 } } \\right ) = 1 \\pmod { p } , \\\\ & h _ { 1 , 1 } ( x ) = \\frac { 1 } { 2 \\sqrt { x } } \\left ( ( 1 + \\sqrt { x } ) ^ m - ( 1 - \\sqrt { x } ) ^ m \\right ) \\equiv \\frac { 1 } { 2 \\sqrt { x } } \\left ( 1 + x ^ { \\frac { m } { 2 } } - 1 + x ^ { \\frac { m } { 2 } } \\right ) = x ^ { \\frac { m - 1 } { 2 } } \\pmod { p } , \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{gather*} \\rho : = ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) \\mapsto \\big ( \\zeta x _ 0 , \\zeta ^ 3 x _ 1 , \\zeta ^ 4 x _ 2 , \\zeta ^ 2 x _ 1 \\big ) . \\end{gather*}"}
-{"id": "2033.png", "formula": "\\begin{gather*} A ^ { S _ 1 } ( z ) = \\frac { 1 } { z ^ { 5 / 3 } } \\begin{pmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 0 \\\\ - t & 0 & 0 \\end{pmatrix} + \\frac { 1 } { z ^ { 4 / 3 } } \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & q _ 1 & 0 \\end{pmatrix} \\\\ \\hphantom { A ^ { S _ 1 } ( z ) = } { } + \\frac { 1 } { z } \\begin{pmatrix} p _ 2 q _ 2 & 0 & 0 \\\\ 0 & p _ 1 q _ 1 - p _ 2 q _ 2 - \\theta ^ 0 _ 2 - 1 / 3 \\\\ 0 & 0 & - p _ 1 q _ 1 - \\theta ^ 0 _ 1 - 2 / 3 \\end{pmatrix} + \\cdots . \\end{gather*}"}
-{"id": "1950.png", "formula": "\\begin{align*} F _ 0 = \\{ ( i , j ) \\in E \\mid f _ { j i } ( - p _ { i j } ) < 0 \\} . \\end{align*}"}
-{"id": "6273.png", "formula": "\\begin{align*} \\langle a e ^ { \\i \\langle \\theta , \\cdot \\rangle _ { \\C ^ d } } X , \\i \\theta \\phi \\rangle = 0 , ( \\phi \\in [ H ^ 1 _ { \\theta } ( Y ) \\perp e ^ { { \\i \\langle \\theta , \\cdot \\rangle _ { \\C ^ d } } } ] ^ n ) . \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{align*} \\eta _ a ^ { \\lambda } ( \\mathsf { P } ) = \\sum _ { i , j } c _ { j } ^ { i } \\pi ( E _ { a } K _ { \\lambda } K _ { a } F _ { a } ) _ { i } ^ { j } - \\sum _ { i , j } c _ { j } ^ { i } \\pi ( K _ { a } F _ { a } E _ { a } K _ { \\lambda } ) _ { i } ^ { j } . \\end{align*}"}
-{"id": "8754.png", "formula": "\\begin{align*} f ( g _ 0 , \\cdots , g _ i , g _ { i + 1 } , \\cdots , g _ n ) = - f ( g _ 0 , \\cdots , g _ { i + 1 } , g _ i , \\cdots , g _ n ) \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} \\omega _ * = \\beta _ * + 2 \\alpha _ * { \\mathcal D } ' ( \\Omega \\setminus \\{ a _ 1 , \\ldots , a _ N \\} ) . \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} \\sum \\limits _ { k = k _ { 0 } + 1 } ^ { k _ { 0 } + N Q } c _ { k } v _ { k } \\in U . \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\lambda } f ( G ( z , \\lambda ) ) \\Bigr | _ { \\lambda = 1 } = \\frac { \\partial } { \\partial \\lambda } \\sum _ { n \\geq 1 } z ^ n \\sum _ { \\pi \\in \\mathcal { C } _ n } f _ { \\vert \\pi \\vert } \\lambda ^ { \\sum _ { k _ i \\in \\pi } g _ { k _ i } } \\Bigr | _ { \\lambda = 1 } . \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{align*} y ( t ) \\phi _ { - 1 } ( y ( t ) ) ^ { - 1 } = t ^ { - 1 / 2 } x ( \\log t ) \\phi _ { - 1 } ( x ( \\log t ) ) ^ { - 1 } \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} T x = \\left \\{ \\begin{array} { c c c } x & & x \\in C _ { x _ { 0 } , r } \\cup C _ { x _ { 1 } , \\rho } \\\\ \\alpha & & \\end{array} \\right . \\end{align*}"}
-{"id": "7480.png", "formula": "\\begin{align*} \\log ( | h ( t ) | ) = \\lambda _ 1 \\log t + \\lambda _ 2 \\log \\log t + \\ldots + \\lambda _ n \\log ^ { ( n ) } t + O ( 1 ) . \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} { \\mathop { { \\rm { m a x } } } \\limits _ { \\mathcal R \\in \\left \\{ \\mathcal R \\in \\mathbb R _ + : { p _ { o u t , K } } \\le \\epsilon \\right \\} } } & { \\bar { \\mathcal T } = \\frac { { \\mathcal R \\left ( { 1 - { p _ { o u t , K } } } \\right ) } } { { \\sum \\nolimits _ { k = 0 } ^ { K - 1 } { { p _ { o u t , k } } } } } } , \\end{align*}"}
-{"id": "4159.png", "formula": "\\begin{align*} \\textrm { I } _ { n , 2 } & = \\{ E \\prod _ { i = 0 } ^ { \\tau } [ \\frac { \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } { 1 + \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } ] \\} ^ 2 \\\\ & \\geq \\{ E \\prod _ { i = 0 } ^ { \\tau } [ \\frac { \\lambda \\rho ( S _ i ) \\rho ( S _ { i + 1 } ) } { 1 + \\lambda M ^ 2 } ] \\} ^ 2 \\\\ & = ( E \\rho ) ^ 4 ( \\frac { \\lambda } { 1 + \\lambda M ^ 2 } ) ^ { 2 \\tau + 2 } ( E \\rho ^ 2 ) ^ { 2 \\tau } . \\end{align*}"}
-{"id": "1527.png", "formula": "\\begin{align*} - \\frac { 1 } { f ( x ) } + g ( x _ { k _ 0 } ) = g ( x ) = d \\tilde { g } ( x ) + e = - \\frac { d } { \\tilde { f } ( x ) } + e \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} u ( t , 0 , y ) = \\mu _ 0 ( t , y ) , u ( t , R , y ) = \\nu _ 0 ( t , y ) , u _ x ( t , R , y ) = \\nu _ 1 ( t , y ) \\end{align*}"}
-{"id": "3930.png", "formula": "\\begin{align*} \\lambda _ p ^ { ( 2 ) } ( \\alpha ) : = \\frac { \\langle - \\alpha \\rangle _ p - \\langle - \\alpha \\rangle _ { p ^ 2 } } { p } \\cdot ( p - 1 ) - \\langle - \\alpha \\rangle _ p . \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{align*} \\begin{cases} \\widetilde \\Omega V = \\Omega V , \\\\ \\widetilde \\Omega H = \\Omega H - H ^ \\perp . \\end{cases} \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} I = \\sum _ { j = 1 } ^ n \\Z [ G ] b _ j . \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} \\sum _ { { \\bf m } \\in D } \\left ( \\prod _ { i = 1 } ^ k | f _ i ( { \\bf m } ) | \\right ) \\le \\prod _ { i = 1 } ^ k \\left ( \\sum _ { { \\bf m } \\in D } | f _ i ( { \\bf m } ) | ^ { p _ i } \\right ) ^ { \\frac { 1 } { p _ i } } . \\end{align*}"}
-{"id": "2047.png", "formula": "\\begin{gather*} \\begin{pmatrix} - t & & \\\\ & 0 & \\\\ & & 0 \\end{pmatrix} \\to \\begin{pmatrix} 0 & 1 & \\\\ & 0 & \\\\ & & 0 \\end{pmatrix} , t \\to 0 . \\end{gather*}"}
-{"id": "6011.png", "formula": "\\begin{align*} B _ { i j } : = ( \\lambda _ { n + 3 } s - t ) \\frac { ( \\lambda _ { n + 2 } - \\lambda _ i ) ( \\lambda _ { n + 2 } - \\lambda _ j ) x _ i x _ j } { ( \\lambda _ { n + 3 } - \\lambda _ { n + 2 } ) ^ 2 x _ { n + 3 } ^ 2 } , \\end{align*}"}
-{"id": "5076.png", "formula": "\\begin{align*} \\omega = \\left \\{ \\begin{array} { l l } ( a _ 2 , a _ 1 , a _ 3 , a _ 5 , a _ 4 ) & \\textrm { i f } a _ 3 + a _ 5 + a _ 4 \\neq 0 , \\\\ ( a _ 5 , a _ 3 , a _ 2 , a _ 4 , a _ 1 ) & \\textrm { i f } a _ 3 + a _ 5 + a _ 4 = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "4135.png", "formula": "\\begin{align*} \\sum _ { \\overrightarrow { x } \\in V _ n } \\Big ( \\prod _ { j = 0 } ^ { n - 1 } \\frac { 1 } { \\deg ( x _ j ) } \\Big ) \\Big ( E { \\widetilde { \\rho } } ^ 2 \\Big ) ^ { | x _ n | } = \\widehat { { E } } [ ( E { \\widetilde { \\rho } } ^ 2 ) ^ { | X _ n | } ] \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} D _ { n , k } \\Big ( 1 , \\frac { 1 } { 4 } \\Big ) \\ , = \\ , \\frac { k ( p ^ { l _ 1 } + p ^ { l _ 2 } + \\cdots + p ^ { l _ i } - 1 ) + 2 } { 2 ^ { p ^ { l _ 1 } + p ^ { l _ 2 } + \\cdots + p ^ { l _ i } } } = \\frac { 2 - k } { 2 ^ i } . \\end{align*}"}
-{"id": "645.png", "formula": "\\begin{align*} u ( x ) = \\sum _ { k = 0 } ^ { m } p _ k ( x ^ { \\prime \\prime } ) \\ , \\Phi _ { m - k } ( x \\cdot e , x _ { n + 1 } ) , \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{align*} x = \\tan \\left ( \\frac { \\theta } { 2 } \\right ) \\textnormal { o r } \\theta = 2 \\tan ^ { - 1 } ( x ) . \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} \\sharp ( F _ * ^ e ( R ^ { \\bigstar } ) , R ^ { \\bigstar } ) = r _ e + 2 \\sum _ { k = 1 } ^ { q - 1 } \\sharp ( C o k _ S ( A ^ k ) , R ) \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{align*} - \\tfrac 1 2 \\mu + \\mathcal { K } _ { \\partial \\Omega } [ \\mu ] = \\gamma _ { 0 } u \\qquad \\partial \\Omega \\ , . \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} \\theta _ { n } ^ { \\prime } ( \\tau ) = \\beta \\varphi _ { n } ^ { 2 } \\left ( \\tau \\right ) \\leq \\beta , \\theta _ { n } ^ { \\left ( j \\right ) } ( \\tau ) = \\beta C _ { j } n ^ { 1 - j } , j = 1 , 2 , . . . . \\end{align*}"}
-{"id": "1630.png", "formula": "\\begin{align*} \\gamma _ { 0 } ^ \\complement \\mathcal { D } _ { \\partial \\Omega } [ \\phi ] - \\gamma _ { 0 } \\mathcal { D } _ { \\partial \\Omega } [ \\phi ] = \\phi \\ , \\gamma _ { 1 } \\mathcal { D } _ { \\partial \\Omega } [ \\phi ] = \\gamma _ { 1 } ^ \\complement \\mathcal { D } _ { \\partial \\Omega } [ \\phi ] \\ , , \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} \\alpha \\delta b + \\beta b ^ 3 + \\mathcal { O } ( \\delta b ^ 2 , b ^ 4 ) = 0 , \\end{align*}"}
-{"id": "7667.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } h _ 1 \\\\ \\vdots \\\\ h _ n \\end{array} \\right ) = A ^ { - 1 } \\left ( \\begin{array} { c } c _ 1 \\\\ \\vdots \\\\ c _ n \\end{array} \\right ) . \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{align*} \\Psi ( z _ 1 , \\dots , z _ N , \\{ \\overline { \\alpha } \\} ) \\rangle = \\mathcal { B } ( z _ 1 , \\{ \\overline { \\alpha } \\} ) \\cdots \\mathcal { B } ( z _ N , \\{ \\overline { \\alpha } \\} ) | \\Omega \\rangle . \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} T = \\epsilon ( X + i Y ) \\enskip \\mbox { a n d } \\enskip U = \\epsilon ( X - i Y ) . \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} T _ n = T ( \\pi _ n ) \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} a = - \\varepsilon c . \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{align*} a _ 1 = 1 + \\lambda _ 1 + \\lambda _ 2 - 2 \\sigma , a _ 2 = - 2 ( \\sigma - \\lambda _ 1 ) ( \\sigma - \\lambda _ 2 ) , a _ 3 = 4 ( \\lambda _ 1 + \\lambda _ 2 - 2 \\sigma ) \\end{align*}"}
-{"id": "2899.png", "formula": "\\begin{align*} B _ { A , \\pmb { \\omega } } e _ { k } = \\begin{cases} A e _ { k } & \\textrm { f o r e v e r y } \\ 1 \\le k \\le r \\\\ \\omega _ { k - r } e _ { k - r } & \\textrm { f o r e v e r y } \\ k > r . \\end{cases} \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} \\mathbf { p } _ { o , i } ' & = \\mathbf { \\dot { p } } _ { o , i } \\Delta T _ s + \\mathbf { p } _ { o , i } \\\\ r _ { i , j } & = | | \\mathbf { r } _ { i , j } | | = | | \\mathbf { p } _ { o , i } - \\mathbf { p } _ { d } | | \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} \\omega ( f , g ) \\ ; = \\ ; \\langle S ^ * f , g \\rangle - \\langle f , S ^ * g \\rangle \\ ; = \\ ; \\langle \\overline { S } f , g \\rangle - \\langle \\overline { S } f , g \\rangle \\ ; = \\ ; 0 \\ , . \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{align*} | \\vec { A } _ 0 | ^ 2 = \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} A _ 1 \\left ( \\begin{smallmatrix} b _ { q + 1 } \\\\ \\vdots \\\\ b _ m \\end{smallmatrix} \\right ) = \\alpha _ { t _ { q + 1 } } ^ { [ \\sigma ] } b _ { q + 1 } + \\dots + \\alpha _ { t _ { m } } ^ { [ \\sigma ] } b _ { m } \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} { \\bf v } = { \\bf a } + \\bf { B } ( { \\bf v } , { \\bf v } ) , \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{align*} v ^ { 2 } + 2 u v - \\Lambda = 8 \\pi \\rho + 8 \\pi \\psi , \\end{align*}"}
-{"id": "1241.png", "formula": "\\begin{align*} | ( - A ) ^ { - \\beta } ( S _ { \\Delta t } - I ) | _ { \\mathcal { L } ( L ^ q ) } = \\Delta t | ( - A ) ^ { 1 - \\beta } S _ { \\Delta t } | _ { \\mathcal { L } ( L ^ q ) } \\le C _ { \\beta , q } \\Delta t ^ \\beta , \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} \\Big \\langle \\tau _ { k _ 1 } ( \\gamma _ 1 ) \\ldots \\tau _ { k _ r } ( \\gamma _ r ) \\Big \\rangle _ { \\ ! n , \\beta } ^ X = \\int _ { [ P _ { n } ( X , \\beta ) ] ^ { v i r } } \\prod _ { i = 1 } ^ r \\tau _ { k _ i } ( \\gamma _ i ) \\ . \\end{align*}"}
-{"id": "312.png", "formula": "\\begin{align*} \\Xi ^ { \\lambda } ( K _ { a } \\otimes F _ { a } \\otimes 1 \\otimes E _ { a } ) = \\sum _ { i , j , m , n , o , p } ( 2 c _ { j } ^ { i } - \\delta _ { j } ^ { i } ) \\pi ( K _ { a } ) _ { m } ^ { j } c _ { n } ^ { m } \\pi ( F _ { a } ) _ { o } ^ { n } c _ { p } ^ { o } \\pi ( E _ { a } K _ { \\lambda } ) _ { i } ^ { p } . \\end{align*}"}
-{"id": "1476.png", "formula": "\\begin{align*} \\Delta _ p ( \\tilde { u } + \\varphi ) = g ' _ \\sigma ( \\tilde { u } ) + f ' _ \\varepsilon \\bigg ( \\int _ { \\Omega ^ c } h _ \\delta ( \\tilde { u } ) \\bigg ) h ' _ \\delta ( \\tilde { u } ) \\chi _ { \\Omega ^ c } , \\end{align*}"}
-{"id": "6822.png", "formula": "\\begin{align*} ( z f ) = Q _ 0 + D - q ^ 3 ( q ^ { 2 e - 1 } - q ^ e + q ^ { e - 1 } ) Q _ \\infty . \\end{align*}"}
-{"id": "7428.png", "formula": "\\begin{align*} \\widetilde { S } ( x ) = \\sum _ { \\substack { \\alpha : i \\to j \\\\ v _ i - v _ j = 1 } } c _ \\alpha e ^ { x _ j - x _ i } + \\sum _ { i \\in G _ 0 } m _ i v _ i x _ i \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{align*} \\frac { [ ( 2 n ) ! ] ^ 3 } { 2 ^ { 4 n } ( n ! ) ^ 4 } = \\frac { 1 } { 2 ^ { 2 n } } [ ( 2 n - 1 ) ! ! ] ^ { 2 } { 2 n \\choose n } \\in \\frac { 1 } { 2 ^ { 2 n } } \\mathbb Z , \\forall n \\in \\mathbb Z _ { \\geq 0 } , \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{align*} g _ { i j } = ( 1 + 2 f + O _ { \\sqrt { \\epsilon } } ( \\norm { f } _ { 1 , \\ , p } ) ) \\sigma _ { i j } \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{align*} \\Delta : = \\bigoplus _ { k = 1 } ^ n \\left ( \\begin{array} { c c } 0 & 1 \\\\ - 1 & 0 \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{gather*} \\frac { { \\rm d } Y } { { \\rm d } x } = A ( x ) Y . \\end{gather*}"}
-{"id": "9739.png", "formula": "\\begin{align*} \\mathcal { F } [ \\rho , B ] = \\int _ { x \\in \\mathbb { T } ^ d } F ( \\rho ( x ) , B ( x ) ) , \\ ; \\ ; \\ ; F ( \\rho , B ) = \\frac { | B | ^ 2 } { 2 \\rho } , \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} \\delta ( C , C ' ) : = \\sup _ { x \\in C } d ( x , C ' ) . \\end{align*}"}
-{"id": "1795.png", "formula": "\\begin{align*} \\C \\ni z \\mapsto V _ { ( t , u ) } ( z ) = e ^ { t a ( z ) } \\hat { u } ( z ) . \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} a _ { n } ^ { \\prime \\prime \\prime } & : = \\sum _ { k = 1 } ^ n ( - 1 ) ^ { k + 1 } \\left ( s _ { n , k ( 3 k - 1 ) / 2 } ^ { ( - 1 ) } + s _ { n , k ( 3 k + 1 ) / 2 } ^ { ( - 1 ) } \\right ) \\\\ & \\quad \\longmapsto \\{ 1 , 1 , 2 , 3 , 6 , 7 , 1 4 , 1 7 , 2 7 , 3 4 , 5 5 , 6 3 , \\ldots \\} . \\end{align*}"}
-{"id": "3991.png", "formula": "\\begin{align*} \\frac { ( - a ) _ k ^ 2 } { ( 1 ) _ { k } ^ 2 } \\cdot ( H _ { a - k } - H _ k ) = \\frac { d } { d x } \\bigg ( \\frac { ( - a ) _ { a - k } ( - a - x ) _ { a - k } } { ( 1 ) _ { a - k } ^ 2 } - \\frac { ( - a ) _ k ( - a - x ) _ { k } } { ( 1 ) _ k ^ 2 } \\bigg ) \\bigg | _ { x = 0 } \\pmod { p } . \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} \\Pr \\{ x U _ 1 = x ' U _ 1 \\} \\le \\Big ( \\frac { n } { q ' } \\Big ) ^ m \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} D _ J ^ { ( j ) } H _ J ^ { ( j ) } D _ J ^ { ( j ) } = - H _ J ^ { ( j ) } \\ , . \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} \\langle \\sigma ^ n , \\xi \\rangle = \\langle \\sigma - \\Delta u ^ n , \\xi \\rangle = \\langle \\sigma , \\xi \\rangle - \\langle u ^ n , \\Delta \\xi \\rangle . \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{align*} c _ j = \\max ( \\gamma ^ { 2 ^ { - j } / ( d - 1 ) } l ( q ) ^ { 1 - 2 ^ { - j } } , ( \\gamma ^ { 1 / ( d - 1 ) } \\alpha ^ { - 1 } ) ^ { ( 1 - 2 ^ { - j } / ( d - 1 ) ) } ) . \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} \\psi ( k ) = \\begin{cases} ( 3 - 2 k ) / ( 6 k ) & \\\\ 1 / 1 2 & \\\\ ( 3 - k ) / ( 6 k ) & \\\\ 1 / 2 4 & \\end{cases} \\end{align*}"}
-{"id": "6167.png", "formula": "\\begin{align*} [ L _ { - i } , \\ , M _ { - q + i } ] = 0 , \\hbox { f o r a l l } i \\geqq 2 \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} \\underset { \\epsilon \\rightarrow 0 } { l i m } \\quad \\underset { T \\geq 1 } { s u p } \\underset { \\kappa \\in ( 0 , 1 ) } { s u p } \\mathbb { E } \\left | \\rho _ { \\psi _ \\kappa } ^ T - \\rho _ { f , \\epsilon , \\psi _ \\kappa } ^ T \\right | ^ 2 = 0 , \\end{align*}"}
-{"id": "9393.png", "formula": "\\begin{align*} \\bigl [ \\widehat { T } ( w _ 1 , w _ 2 ) \\bigr ] ( \\theta x ) & = \\bigl \\langle \\bigl [ T ( \\theta ( x \\otimes w _ 1 ) ) \\bigr ] ( \\cdotp ) , w _ 2 \\bigr \\rangle \\\\ & = \\bigl \\langle \\theta ( \\cdotp ) \\ , \\bigl [ T ( x \\otimes w _ 1 ) \\bigr ] ( \\cdotp ) , w _ 2 \\bigr \\rangle \\\\ & = \\theta \\bigl [ \\widehat { T } ( w _ 1 , w _ 2 ) \\bigr ] ( x ) , \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{align*} \\dot z = Q v ( x _ 0 + \\Gamma _ 0 z ) = : H ( z ) , y = 1 \\ , , \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} \\mathrm { M \\ddot { o } } ( 2 , 2 k - 1 ) & : = \\left ( \\begin{matrix} k + 1 \\\\ 2 \\end{matrix} \\right ) + \\left \\lfloor \\frac { k } { 2 } \\right \\rfloor . \\end{align*}"}
-{"id": "1601.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\big ( \\deg _ { \\widetilde { X } } ( v _ { i } ) - 2 \\big ) & \\leq 3 N \\big ( \\deg _ { \\widetilde { Y } } ( a ) - 2 \\big ) \\cdot \\big ( \\deg _ { \\widetilde { Z } } ( b ) - 2 \\big ) . \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} P _ E ( z ) = z ^ n - c _ 1 z ^ { n - 1 } - . . . - c _ { n - 1 } z - c _ n , \\end{align*}"}
-{"id": "5003.png", "formula": "\\begin{align*} \\tilde { u } _ t = \\partial _ { \\xi } ( L _ { c } - \\partial _ y ^ 2 + 4 ( \\dot { a } - c ) ) \\tilde { u } + 4 ( \\dot { a } - c ) \\partial _ { \\xi } u _ { c } - \\dot { c } \\partial _ c u _ { c } - 6 \\partial _ { \\xi } \\tilde { u } ^ 2 , \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{gather*} w _ { k + n } ( \\cos ( \\alpha _ { 1 } ) , . . . , \\cos ( \\alpha _ { k + n } ) | \\rho ) = \\\\ \\prod _ { i _ { 1 } \\in \\{ - 1 , 1 \\} } . . . \\prod _ { i _ { k + n } \\in \\{ - 1 , 1 \\} } ( 1 - 2 \\rho \\cos ( \\sum _ { s = 1 } ^ { n + k } i _ { s } \\alpha _ { s } ) + \\rho ^ { 2 } ) . \\end{gather*}"}
-{"id": "3068.png", "formula": "\\begin{align*} ( k - 1 ) a + ( n - k - 1 ) b = 0 \\end{align*}"}
-{"id": "9715.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\alpha _ { i , j } ( g _ i , a _ i ) = ( a _ j , g _ j ) , \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle i \\hbar \\frac { \\partial \\Psi ( \\mathbf { x } , t ) } { \\partial t } = \\hat { H } \\Psi ( \\mathbf { x } , t ) , \\hat { H } = \\hat { H } _ 0 + \\hat { V } ( \\mathbf { x } , t ) } , \\end{array} \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} \\phi _ { 1 , 1 } = \\rho _ { 1 } \\phi _ { 1 } . \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{align*} \\Delta & = D _ { m a x } - D _ { a c h } ( k ) \\\\ & = \\sum _ { j = 1 } ^ { m } \\ ! \\frac { 1 } { p ( \\tau _ j ) } \\left ( \\sum _ { i = 1 } ^ { m } \\ ! y _ i p ( \\tau _ j | y _ i ) p ( y _ i ) \\right ) ^ 2 \\ ! \\ ! - \\ ! \\mathbb { E } [ Y ] ^ 2 . \\end{align*}"}
-{"id": "3789.png", "formula": "\\begin{align*} u _ 2 s _ 1 - u _ 2 - s _ 1 \\leq \\lambda _ 3 ( s _ 1 - 1 ) \\leq ( u _ 2 - 2 ) ( s _ 1 - 1 ) = u _ 2 s _ 1 - u _ 2 - 2 s _ 1 + 2 , \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{align*} \\theta ( \\mathsf { P } ) = \\pi ( K _ { 2 \\rho } ^ { - 1 } ) \\mathsf { P } \\pi ( K _ { 2 \\rho } ) , \\theta ( \\mathsf { Q } ) = \\pi ( K _ { 2 \\rho } ) \\mathsf { Q } \\pi ( K _ { 2 \\rho } ^ { - 1 } ) . \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{align*} & ( 1 - z ) ^ p \\cdot \\Phi ( p ) \\equiv ( 1 - z ) ^ p \\cdot p \\Phi ' ( 0 ) + ( 1 - z ) ^ p \\cdot \\Phi ( 0 ) \\\\ \\equiv & p \\big ( \\Psi _ 1 ' ( 0 ) - z ^ p \\cdot \\Psi _ 2 ' ( 0 ) \\big ) + \\big ( 1 - z ^ p - ( 1 - z ) ^ p \\big ) \\cdot \\Psi _ 1 ( 0 ) + ( 1 - z ) ^ p \\cdot \\Phi ( 0 ) \\\\ \\equiv & \\big ( \\Psi _ 1 ( p ) - \\Psi _ 1 ( 0 ) \\big ) - z ^ p \\cdot \\big ( \\Psi _ 2 ( p ) - \\Psi _ 2 ( 0 ) \\big ) + ( 1 - z ^ p ) \\cdot \\Psi _ 1 ( 0 ) \\\\ = & \\Psi _ 1 ( p ) - z ^ p \\cdot \\Psi _ 2 ( p ) \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} \\{ H _ 0 , \\chi \\} + Z & = f , \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} d Y _ t & = \\frac { 2 Y _ t } { X _ t ^ 2 + Y _ t ^ 2 } d t , \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} u _ k = \\left ( \\begin{matrix} 0 & 2 ^ { - k } \\\\ 2 ^ { - k } & 0 \\end{matrix} \\right ) v _ k = 2 ^ { - k } \\left ( \\begin{matrix} 2 & 1 \\\\ 1 & 2 \\end{matrix} \\right ) \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{align*} \\left ( a I _ { \\ast } ^ { \\alpha } f \\right ) \\left ( x \\right ) = e ^ { \\frac { 1 } { \\Gamma \\left ( \\alpha \\right ) } \\int _ { a } ^ { x } \\left ( x - t \\right ) ^ { \\alpha - 1 } ( \\ln \\circ f ) ( x ) \\ , d x } , x > a . \\end{align*}"}
-{"id": "7809.png", "formula": "\\begin{align*} \\| u _ 0 \\| _ { L _ 2 } \\leq c \\bigl \\| \\partial _ x ( P u _ 0 ) \\big | _ { x = 0 } \\bigr \\| _ { L _ 2 ( B _ T ) } . \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} g \\omega = k d f + f \\eta \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} | \\Delta _ y ( z _ 1 ) - \\Delta _ y ( z _ 2 ) | & = | \\Pi _ { \\theta ( y , z _ 3 ) } ( z _ 1 - z _ 2 ) | \\\\ & \\le | \\Pi _ { \\theta ( y , x _ Q ) } ( z _ 1 - z _ 2 ) | + \\| \\Pi _ { \\theta ( y , z _ 3 ) } - \\Pi _ { \\theta ( y , x _ Q ) } \\| | z _ 1 - z _ 2 | \\\\ & \\lesssim | \\Pi _ { \\theta ( y , x _ Q ) } ( z _ 1 - z _ 2 ) | + 2 ^ { - m _ j } 2 ^ { - m _ j } \\\\ & \\le | \\Pi _ { \\theta ( y , x _ Q ) } ( z _ 1 ) - \\Pi _ { \\theta ( y , x _ Q ) } ( z _ 2 ) | + 2 \\cdot 2 ^ { - m _ { j + 1 } } . \\end{align*}"}
-{"id": "6478.png", "formula": "\\begin{align*} R ^ e _ n ( s _ { v _ j } ) ( w _ { j e - i } ) = w _ { ( j + 1 ) e - i } \\mbox { a n d } R ^ e _ n ( s _ { v _ j } ) ( w _ { j e + i } ) = w _ { ( j - 1 ) e + i } . \\end{align*}"}
-{"id": "4841.png", "formula": "\\begin{align*} \\Delta _ H = \\begin{cases} \\left \\{ I + \\begin{pmatrix} a & 0 \\\\ 0 & d \\end{pmatrix} p \\in H \\cap \\ker \\varphi \\right \\} & H N ( C _ s ) \\\\ \\left \\{ I + \\begin{pmatrix} a & 0 \\\\ 0 & d \\end{pmatrix} p \\in H \\cap K \\right \\} & H N ( C _ { n s } ) . \\end{cases} \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} ( \\psi \\circ \\kappa ) ^ \\sharp = \\kappa ^ \\sharp \\circ \\psi ^ \\sharp . \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} h _ Y ( x ) : = \\frac { 1 - g _ Y ( 1 - x ) } { x } , \\end{align*}"}
-{"id": "8848.png", "formula": "\\begin{align*} \\sum _ { d = 0 } ^ { \\infty } ( d + 2 ) ^ u F _ { d + 2 } ( n ) = \\sum _ { d = 0 } ^ { \\lceil \\sqrt { 2 n } \\rceil } ( d + 2 ) ^ u F _ { d + 2 } ( n ) = s _ 1 + s _ 2 + s _ 3 , \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} f _ j ( E , x ) : = \\frac { x ^ n } { \\sqrt { \\left | \\prod _ { k \\neq j } ( E _ k ^ + - x ) ( E _ k ^ - - x ) \\right | } } . \\end{align*}"}
-{"id": "6103.png", "formula": "\\begin{align*} j = k \\end{align*}"}
-{"id": "5116.png", "formula": "\\begin{align*} x _ { n } = \\frac { B _ { n } \\left ( x \\right ) } { n ! } . \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{align*} R _ n ^ e ( s _ { v _ 0 } ) ( p _ b ^ * ( v _ 1 ) ) = p _ b ^ * ( s _ { v _ 0 } ( v _ 1 ) ) = p _ b ^ * ( v _ 0 + v _ 1 ) . \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} A _ { 2 p } = \\left ( \\begin{array} { r r r r r r } 0 . 5 & 0 & 0 & 0 . 2 5 & 0 . 5 & 0 . 2 5 \\\\ 0 & 0 . 5 & 0 & 0 . 2 5 & 0 . 5 & 0 . 2 5 \\\\ 0 & 0 & 0 . 5 & 0 . 2 5 & 0 . 5 & 0 . 2 5 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "3493.png", "formula": "\\begin{align*} \\begin{cases} \\varphi _ { \\infty } ( x ) = - \\frac { 1 } { 2 } \\varphi _ { \\infty } ^ { \\prime \\prime } ( x ) , x > 0 , \\\\ \\varphi _ { \\infty } ( 0 ) = 0 , \\varphi _ { \\infty } ^ \\prime ( 0 ) = \\sqrt { 2 } . \\end{cases} \\end{align*}"}
-{"id": "479.png", "formula": "\\begin{align*} & I ( M ; Y _ i Y _ 3 ) + I ( M ; Y _ i Y _ 4 ) \\\\ = & 2 H ( M ) - ( H ( M | Y _ i Y _ 3 ) + H ( M | Y _ i Y _ 4 ) ) \\\\ \\ge & 2 H ( M ) - \\log d . \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} \\frac { \\varepsilon } { G ( \\varepsilon ) } = \\frac { \\frac { m \\theta ^ 2 } { 2 \\sqrt { a b } } + \\delta _ 1 } { G ( \\frac { m \\theta ^ 2 } { 2 \\sqrt { a b } } ) } - \\frac { \\delta _ 1 } { 2 G ( \\frac { m \\theta ^ 2 } { 2 \\sqrt { a b } } ) } \\leqslant \\frac { m \\theta ^ 2 } { 2 \\sqrt { a b } } - \\frac { \\delta _ 1 } { 2 G ( \\frac { m \\theta ^ 2 } { 2 \\sqrt { a b } } ) } \\leqslant \\frac { m \\theta ^ 2 } { 2 \\sqrt { a b } } - \\frac { \\delta _ 1 } { 2 | \\varepsilon _ D ^ * | ^ 2 } < \\eta , \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} x ( \\varepsilon ) & = x ^ 0 + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n { [ F _ { 1 i } e _ i , F _ { 2 i } e _ i ] ( J ( x ^ 0 ) ) } \\int \\limits _ 0 ^ \\varepsilon \\int \\limits _ 0 ^ \\tau \\big ( u _ { 2 i } ( \\tau ) u _ { 1 i } ( \\theta ) - u _ { 1 i } ( \\tau ) u _ { 2 i } ( \\theta ) \\big ) d \\theta d \\tau + R ( \\varepsilon ) \\\\ & = x ^ 0 { - } \\varepsilon \\sum _ { i = 1 } ^ n \\frac { \\partial J ( x ^ 0 ) } { \\partial x _ i } F _ { 0 i } ( J ( x ^ 0 ) ) e _ i + R ( \\varepsilon ) . \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{align*} G ( t ) = G ( 0 ) \\exp \\left ( \\sqrt { 2 } \\beta ( t ) + \\left ( 1 - 2 n - \\left ( u + u ' + v + v ' \\right ) \\right ) t \\right ) . \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{gather*} ( e _ 1 ^ + ) ^ i ( e _ 0 ^ + ) ^ i = ( q - q ^ { - 1 } ) ^ { - 2 i } ( X _ { 1 2 } - X _ { 1 3 } ) ^ i ( X _ { 3 0 } - X _ { 3 1 } ) ^ i ( i \\in \\mathbb { N } ) . \\end{gather*}"}
-{"id": "6523.png", "formula": "\\begin{align*} \\mathcal { T } _ { s } = \\left ( \\tau ( s ) , \\alpha _ { 1 } ( s ) , \\ldots , \\alpha _ { | A | } ( s ) , \\beta _ { 1 } ( s ) , \\ldots , \\beta _ { | B | } ( s ) \\right ) \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} \\mathbb R ^ 3 \\setminus \\Omega \\subset B _ R : = \\{ x \\in \\mathbb R ^ 3 ; | x | < R \\} , \\end{align*}"}
-{"id": "9327.png", "formula": "\\begin{align*} B ( t : s ) : = b _ { 1 2 } t ^ { 1 2 } + \\sum _ { i = 5 } ^ { 1 1 } b _ i t ^ i s ^ { 1 2 - i } + ( - a _ 4 + \\frac { a _ 1 ^ 4 } { 1 7 2 8 } + \\frac { a _ 3 a _ 1 } { 6 } + \\frac { a _ 2 ^ 2 } { 1 2 } + \\frac { a _ 2 a _ 1 ^ 2 } { 7 2 } ) t ^ 4 s ^ 8 + \\\\ + ( - a _ 3 + \\frac { a _ 2 a _ 1 } { 6 } + \\frac { a _ 1 ^ 3 } { 2 1 6 } ) t ^ 3 s ^ 9 + ( - a _ 2 + \\frac { a _ 1 ^ 2 } { 1 2 } ) t ^ 2 s ^ { 1 0 } - a _ 1 t ^ 1 s ^ { 1 1 } + 2 s ^ { 1 2 } . \\end{align*}"}
-{"id": "4878.png", "formula": "\\begin{align*} p ( \\vec { x } ) = \\max ( m _ 1 ( \\vec { x } ) , \\ldots , m _ { k } ( \\vec { x } ) ) , \\end{align*}"}
-{"id": "2691.png", "formula": "\\begin{align*} \\omega _ { i } - \\sum _ { j \\in \\{ i ^ + , i ^ - \\} } a _ { i , j } ' ( x _ i x _ j ^ { - 1 } - x _ j x _ i ^ { - 1 } ) = 0 i = 1 , \\dots , n \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 1 + } c _ { \\alpha } ^ { - } = \\lim _ { \\alpha \\to 1 + } c _ { \\alpha } ^ { + } = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "7227.png", "formula": "\\begin{align*} [ p _ 1 , \\ldots , p _ { n - 1 } ] : = \\langle \\rho _ 0 , \\ldots , \\rho _ { n - 1 } \\mid & \\rho _ i ^ 2 = 1 \\textrm { f o r $ 0 \\leq i \\leq n - 1 $ } , \\\\ & ( \\rho _ { i - 1 } \\rho _ i ) ^ { p _ i } = 1 \\textrm { f o r $ 1 \\leq i \\leq n - 1 $ } , \\\\ & ( \\rho _ i \\rho _ j ) ^ 2 = 1 \\textrm { f o r $ i , j \\in \\{ 0 , \\ldots , n - 1 \\} $ w i t h $ | i - j | \\geq 2 $ } \\rangle . \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} ( \\xi _ 1 + \\xi _ 2 ) \\theta _ t + ( 1 + \\nu ' ) \\theta - \\tau ( c ) ) = A , \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{align*} e _ C ( x ) = \\sum _ { i = 0 } ^ \\infty \\frac { x ^ { r ^ i } } { D _ i } \\ , , \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} \\mathcal { N } ( 2 \\delta B \\gamma ^ \\beta , \\gamma ) = \\sup \\limits _ { g \\in \\mathcal { G } } \\mathcal { N } ( \\mathcal G \\cap B _ { P } ( g , 2 \\delta B \\gamma ^ \\beta ) , \\gamma ) \\end{align*}"}
-{"id": "4643.png", "formula": "\\begin{align*} \\zeta ( G ) = \\min \\{ \\xi ( G , T ) | \\textit { $ T $ i s a s p a n n i n g t r e e o f $ G $ } \\} . \\end{align*}"}
-{"id": "6444.png", "formula": "\\begin{align*} \\varphi ( \\varepsilon ' v _ 1 ) = \\varphi ( \\varepsilon u _ 1 ) \\delta _ 2 ^ { - 1 } \\quad \\varphi ( v _ i ) = \\delta _ i \\varphi ( u _ i ) \\delta _ { i + 1 } ^ { - 1 } \\quad \\end{align*}"}
-{"id": "8016.png", "formula": "\\begin{align*} | I I | = \\left | \\int _ s ^ t \\langle ( h u _ s + \\widetilde U ) \\otimes w , ( \\nabla w _ k - \\nabla w ) \\psi _ L \\rangle \\right | \\to 0 ( k \\to \\infty ) \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} f ( x , \\gamma ^ i ) = \\frac { 1 } { 2 } ( \\gamma ^ i ) ^ 2 - \\theta \\gamma ^ i ( \\bar x - x ^ i ) + \\frac { \\varepsilon } { 2 } ( \\bar x - x ^ i ) ^ 2 , \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{align*} Y = \\Big \\{ \\ , x \\mathrel { \\Big | } \\underset { t \\to 0 } { \\operatorname { l i m } } \\ , ( \\lambda _ t \\cdot x ) Z \\ , \\Big \\} \\subset X . \\end{align*}"}
-{"id": "4252.png", "formula": "\\begin{align*} - \\int _ { \\mathbb { R } ^ 3 } | \\Delta v | ^ 2 \\phi ^ 6 d x = - \\int _ { \\mathbb R ^ 3 } | \\nabla ^ 2 v | \\phi ^ 6 d x - 6 \\int _ { \\mathbb R ^ 3 } \\nabla _ i v ( \\nabla _ { i j } ^ 2 v \\nabla _ j \\phi - \\Delta v \\nabla _ i \\phi ) \\phi ^ 5 d x \\end{align*}"}
-{"id": "9674.png", "formula": "\\begin{align*} \\begin{aligned} F ( \\pi - \\Theta ( \\theta , \\phi , \\psi ) + \\theta , \\phi , \\psi ) & = - F ( \\theta , \\phi , \\psi ) , \\\\ G ( \\pi - \\Theta ( \\theta , \\phi , \\psi ) + \\theta , \\phi , \\psi ) & = - H ( \\theta , \\phi , \\psi ) , \\\\ H ( \\pi - \\Theta ( \\theta , \\phi , \\psi ) + \\theta , \\phi , \\psi ) & = G ( \\theta , \\phi , \\psi ) . \\end{aligned} \\end{align*}"}
-{"id": "3443.png", "formula": "\\begin{align*} \\frac 1 2 \\max _ j | a _ i | \\le \\Big \\| \\sum _ { i = 1 } ^ k a _ i v _ i \\Big \\| \\le C \\max _ j | a _ j | . \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} \\lim _ { p \\to \\infty } \\lambda _ { 2 } ( p , \\Omega ) ^ { \\frac 1 p } = \\frac { 1 } { \\rho _ { 2 , F } ( \\Omega ) } , \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{align*} R Q = P ' Q ' - P Q '' = ( Q '' S - Q S '' ) Q ' - ( S Q ' - S ' Q ) Q '' = ( S ' Q '' - S '' Q ' ) Q , \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} m _ { i i } = 1 , ~ m _ { i ( i + 1 ) } = 3 \\mbox { a n d } m _ { i j } = 2 \\mbox { f o r } | i - j | \\geq 2 . \\end{align*}"}
-{"id": "9418.png", "formula": "\\begin{align*} \\partial _ t B + \\nabla \\times \\left ( \\frac { \\lambda ^ 2 D + B \\times ( D \\times B ) } { \\sqrt { \\lambda ^ 4 + \\lambda ^ 2 B ^ 2 + \\lambda ^ 2 D ^ 2 + | D \\times B | ^ 2 } } \\right ) = 0 , \\quad \\nabla \\cdot B = 0 , \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} & \\frac { 1 } { N ^ { h - m } } \\sum \\limits _ { \\mathbf { n } \\in \\mathbb { Z } ^ h } \\Big ( \\prod \\limits _ { j = 1 } ^ d f _ j ( \\xi _ j ( \\mathbf { n } ) + \\widetilde { \\mathbf { r } } _ j ) \\Big ) \\frac { 1 } { C _ { \\Xi , \\chi } \\eta ^ h } \\int \\limits _ { \\mathbf { y } \\in \\mathbb { R } ^ h } F ( \\mathbf { y } ) G ( L \\mathbf { y } ) \\boldsymbol { \\chi } ( \\Xi ( \\mathbf { y } - \\mathbf { n } ) ) \\ , d \\mathbf { y } , \\end{align*}"}
-{"id": "1723.png", "formula": "\\begin{align*} v _ { w , m a x } = \\mathrm { m a x } \\left ( v _ i , ~ v _ i + \\Delta v \\right ) + v _ { a i r } \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{align*} - x ^ { 2 \\alpha _ { 2 } } \\frac { d ^ { 2 } u } { d x _ { 2 } ^ { 2 } } + \\left ( S + \\lambda \\right ) u \\left ( x _ { 2 } \\right ) = f \\left ( x _ { 2 } \\right ) L _ { 2 } u = 0 . \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} { { f _ { G _ K } } \\left ( x \\right ) = { \\rm E } _ { T } \\left \\{ { f _ { \\left . G _ K \\right | T } } \\left ( x | t \\right ) \\right \\} = \\int \\nolimits _ 0 ^ \\infty { { f _ { \\left . G _ K \\right | T } } \\left ( { \\left . x \\right | t } \\right ) { f _ T } \\left ( t \\right ) d t } } , \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} \\underline K ^ n _ \\cdot = n \\int _ 0 ^ \\cdot ( \\underline Y ^ n _ s - L _ s ) ^ - { \\rm d } s \\leq n \\int _ 0 ^ \\cdot ( \\dot Y ^ n _ s - L _ s ) ^ - { \\rm d } s = \\dot K ^ n _ \\cdot \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} ( T - T P ) ^ n & = ( T - T P ) ^ n = ( T ( I - P ) ) ^ n = T ^ n ( I - P ) ^ n \\\\ & = T ^ n ( I - P ) = T ^ n - T ^ n P = T ^ n - \\lambda ^ n P . \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{align*} ( \\varphi ( t ) , \\psi ( t ) , x ' ( t ) ) & \\in \\bigcap _ { \\delta > 0 } \\overline { \\mathrm { c o } } \\ , \\tilde { \\Gamma } \\left ( t , B ^ \\delta _ \\mathit { w } \\left ( \\int _ 0 ^ t \\psi ( s ) d s , x ( t ) \\right ) \\right ) \\\\ & = \\tilde { \\Gamma } \\left ( t , \\int _ 0 ^ t \\psi ( s ) d s , x ( t ) \\right ) \\quad . \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{align*} H _ 3 = H _ 3 ( M , \\gamma ) : = \\bigcup _ { i = 1 } ^ { n } V _ i \\bigcap \\bigcap _ { 1 \\leq j \\leq n , j \\neq i } \\{ \\# { \\cal E } _ j \\leq M \\log { n } \\} \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} d _ 1 ( e _ { s ( \\alpha ) } \\otimes e _ { t ( \\alpha ) } ) = \\alpha \\otimes e _ { t ( \\alpha ) } - e _ { s ( \\alpha ) } \\otimes \\alpha \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{align*} \\bar { A } _ { n , j , k } ( t , z ) = ( - 1 ) ^ { n } t ^ { n + 1 } A _ { n , j , k } ( 1 / t , z ) , \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} [ x _ + ( \\alpha ) _ \\lambda , x _ + ( \\alpha ) _ { \\lambda + 1 } ] = [ b _ + ( \\alpha + \\lambda ) , b _ + ( \\alpha + \\lambda + 1 ) ] \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - q } \\bigg ( g ^ { 1 - q } ( t ) - g ^ { 1 - q } ( 0 ) \\bigg ) = \\frac { C _ 2 } { C _ 1 ( 1 - q ) } ( e ^ { - C _ 1 ( q - 1 ) t } - 1 ) . \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{align*} \\ker ( T ^ * ) = T ( X ) ^ \\bot , \\ker ( T ) = T ^ * ( Y ^ * ) _ \\bot , \\end{align*}"}
-{"id": "9784.png", "formula": "\\begin{align*} \\nu ( B _ { r / 2 } ( x ) ) = \\lim _ { \\epsilon _ j \\to 0 } \\frac { 1 } { 2 } \\frac { \\| d ^ * \\zeta _ { \\epsilon _ j } \\| _ { L ^ 2 ( B _ { r / 2 } ( x ) ) } ^ 2 } { | \\log \\epsilon _ j | } \\leq \\lim _ { \\epsilon _ j \\to 0 } \\frac { C } { | \\log \\epsilon _ j | } = 0 . \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} \\mathcal { A } _ { \\alpha \\beta } ^ { i j } ( p , \\lambda ) : = 2 a _ { \\alpha \\beta } ^ { i j } - \\sigma _ { \\gamma \\alpha } ^ { i k } \\sigma _ { \\gamma \\beta } ^ { j k } - ( p - 2 ) ( \\sigma _ { \\gamma \\alpha } ^ { i k } - \\lambda _ { \\gamma \\alpha } ^ { i k } ) ( \\sigma _ { \\gamma \\beta } ^ { j k } - \\lambda _ { \\gamma \\beta } ^ { j k } ) \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{align*} \\psi ( x ) = \\Omega _ 1 ( x ) \\omega _ 2 ( x ) \\phi ( x ) . \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} | \\Delta _ k ( E _ 1 , \\ldots , E _ k ) | \\gtrsim \\frac { \\left ( \\prod \\limits _ { j = 1 } ^ k | E _ j | \\right ) ^ 2 } { \\sum \\limits _ { t \\in \\mathbb F _ q ^ * } \\nu _ k ^ 2 ( t ) } . \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{align*} Y _ 0 = 1 \\quad \\hbox { a n d } Y _ t \\rightarrow Y _ t + 1 \\quad \\hbox { a t r a t e } Q _ t = b + \\sum _ { i = 2 } ^ { Y _ t } 1 / ( 1 + N _ { t _ i } ^ i ) . \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbf { H } ( 0 ) & = \\mathbf { H } ( 1 ) = 0 , \\\\ \\mathbf { H } ' ( 0 ) & = 2 = - \\mathbf { H } ' ( 1 ) , \\end{aligned} \\\\ \\mathbf { H } ( x ) > 0 \\quad ( 0 < x < 1 ) . \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{align*} N = \\begin{pmatrix} \\norm { 0 } { \\beta } & \\norm { 0 } { \\sigma ( \\beta ) } & \\cdots & \\norm { 0 } { \\sigma ^ { n - 1 } ( \\beta ) } \\\\ \\norm { 1 } { \\beta } & \\norm { 1 } { \\sigma ( \\beta ) } & \\cdots & \\norm { 1 } { \\sigma ^ { n - 1 } ( \\beta ) } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\norm { n - 1 } { \\beta } & \\norm { n - 1 } { \\sigma ( \\beta ) } & \\cdots & \\norm { n - 1 } { \\sigma ^ { n - 1 } ( \\beta ) } \\end{pmatrix} , \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} q ( \\rho , \\theta , \\phi ) = q \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\cosh \\rho \\cos \\phi , \\frac { 1 } { 2 } \\sinh \\rho \\sin \\theta \\sin \\phi , \\frac { 1 } { 2 } \\sinh \\rho \\cos \\theta \\sin \\phi \\right ) . \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\alpha & \\frac 1 2 \\\\ & 1 \\end{matrix} \\bigg | \\ , z \\bigg ] = ( 1 - z ) ^ { - \\frac 1 2 \\alpha } { } _ 2 F _ 1 \\bigg [ \\begin{matrix} \\frac 1 2 \\alpha & \\frac 1 2 - \\frac 1 2 \\alpha \\\\ & 1 \\end{matrix} \\bigg | \\ , - \\frac { z ^ 2 } { 4 z - 4 } \\bigg ] . \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} \\left . \\frac { \\partial } { \\partial x } y \\left ( x , t \\right ) \\right \\vert _ { x = j \\Delta x , \\ ; t = n \\Delta t } \\approx \\frac { y _ { j + 1 } ^ { n } - y _ { j - 1 } ^ { n } } { 2 \\Delta x } , \\end{align*}"}
-{"id": "9312.png", "formula": "\\begin{align*} | q _ 0 | = | q ( 0 ) | = \\left | \\lim _ { \\mu \\to 0 ^ + } q ( \\mu ) \\right | & \\le \\lim _ { \\mu \\to 0 ^ + } \\mu ^ { k + 1 } = 0 . \\end{align*}"}
-{"id": "2605.png", "formula": "\\begin{align*} ( \\omega + \\sqrt { - 1 } \\partial \\bar \\partial u ) ^ n = \\beta _ { \\epsilon } ( \\varphi _ k - u ) \\omega ^ n , \\end{align*}"}
-{"id": "5279.png", "formula": "\\begin{align*} \\int _ { B _ R ( 0 ) \\setminus S } \\rho | \\nabla \\varphi | ^ 2 \\ , d x & = \\int _ { B _ R ( 0 ) \\cap S } \\left ( A _ { + } \\cdot N _ + + A _ { - } \\cdot N _ - \\right ) \\ , d S + \\int _ { \\partial B _ R ( 0 ) \\setminus S } A \\cdot N \\ , d S \\\\ & = : \\mathbf { I } + \\mathbf { I I } . \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} f ( x ) = \\int _ 0 ^ { \\infty } e ^ { - x t } t ^ { \\lambda - 1 } \\varphi ( t ) \\ , d t + c , x > 0 \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} P ( \\lambda ) & = ( 2 , \\sqrt { 2 ( 2 - \\lambda ) } ) & Q ( \\lambda ) & = ( 3 , \\sqrt { 6 ( 3 - \\lambda ) } ) \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{align*} \\mathrm { o r d } _ P \\left ( \\frac { 4 } { \\chi _ { \\rho } ( \\gamma ) ^ 2 } \\right ) = 2 \\ , \\mathrm { o r d } _ P ( 2 ) - 2 \\ , \\mathrm { o r d } _ P ( \\chi _ { \\rho } ( \\gamma ) ) > 2 \\ , \\mathrm { o r d } _ P ( 2 ) . \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{align*} 2 q ^ * + K ^ * = \\frac { 1 } { \\rho } c _ B , \\ \\ \\ \\ \\Delta ^ * = \\sqrt { 3 } \\sigma . \\end{align*}"}
-{"id": "8192.png", "formula": "\\begin{align*} F ( t \\xi ) = | t | F ( \\xi ) , t \\in \\R , \\ , \\xi \\in \\R ^ { n } , \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} e _ k = e ^ { i \\phi _ k } . \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} | a ( \\gamma ) - b ( \\gamma ) | = | Y _ a - Y _ b | , \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} ( M _ i ) _ { k , j } = \\binom { 2 k } { 2 j - 2 } \\frac { 1 } { 4 ^ k k } \\quad , ( R _ i ) _ j = \\binom { 2 i } { 2 j - 2 } \\frac { 1 / ( 4 ^ i - 1 ) } { 2 i - 2 j + 3 } \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} \\begin{array} { @ { } l @ { } } { \\displaystyle \\big | \\tau \\sum _ { k = 1 } ^ { m } \\mathrm { R e } \\big [ V _ { 2 } ^ { k } ( \\partial _ { \\tau } { \\theta _ { \\Psi } ^ { k } } ) \\big ] \\big | \\leq C \\big ( h ^ { 2 r } + \\tau ^ { 4 } \\big ) + C \\| \\theta _ { \\Psi } ^ { m } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } + C \\tau \\sum _ { k = 1 } ^ { m - 1 } { \\| \\theta _ { \\Psi } ^ { k } \\| _ { \\mathcal { L } ^ 2 } ^ { 2 } } . } \\end{array} \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{align*} X ( \\kappa , z _ 0 , t ) & = \\gamma _ { k _ i } ( z _ { i - 1 } , T _ { i - 1 } , T _ { i } , t ) ^ { - 1 } \\gamma _ { k _ i } ' ( z _ { i - 1 } , T _ { i - 1 } , T _ { i } , t ) \\\\ & = \\gamma _ { k _ i } ( z _ { i - 1 } , t - T _ { i - 1 } ) ^ { - 1 } \\gamma _ { k _ i } ' ( z _ { i - 1 } , t - T _ { i - 1 } ) \\\\ & = R ^ { k _ i } X ( z _ { i - 1 } , t - T _ { i - 1 } ) R ^ { - k _ i } , \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{align*} s z ' ( s ) = t ^ { \\frac { 1 } { 2 } } v ' ( t ) + \\frac { 1 } { 2 } t ^ { - \\frac { 1 } { 2 } } v ( t ) . \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{align*} B _ { p } ( x , y ) & = \\int _ { 0 } ^ { 1 } t ^ { x - 1 } ( 1 - t ) ^ { y - 1 } e ^ { - \\frac { p } { t ( 1 - t ) } } d t . \\\\ ( R e ( p ) & > 0 , R e ( x ) > 0 , R e ( y ) > 0 ) \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} e ^ { 2 f } \\tilde { s } ^ C & = s ^ C - n \\Lambda ( d d ^ c f ) , \\\\ e ^ { 2 f } \\tilde { s } & = s - \\Lambda ( d d ^ c f ) . \\end{align*}"}
-{"id": "3526.png", "formula": "\\begin{align*} { } _ { \\ast } D ^ { \\alpha } ( f ( z ) ) = e ^ { D ^ { \\alpha } ( l n ( f ( z ) ) ) } = e ^ { \\frac { 1 } { \\Gamma ( - \\alpha ) } \\int _ { 0 } ^ { z } ( \\ln f ( t ) ) ( z - t ) ^ { - \\alpha - 1 } d t } , R e ( \\alpha ) < 0 . \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{align*} \\sum _ { z \\ , \\notin \\ , \\bigcup _ { i = 1 } ^ k B _ { R ' } ( x _ i ) } \\Big ( 1 - c \\left ( \\overline { \\xi } ( z ) \\right ) \\Big ) = O \\left ( \\frac { 1 } { R ^ 2 } \\right ) . \\end{align*}"}
-{"id": "6272.png", "formula": "\\begin{align*} a ^ { \\rm h o m } ( \\theta ) = a ^ { \\rm h o m } ( 0 ) ( \\theta \\neq 0 ) \\end{align*}"}
-{"id": "8805.png", "formula": "\\begin{align*} L \\sigma _ { , j } - \\sigma _ { , k } y ^ { k } y _ { j } = 0 . \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} r _ j ( \\theta , a ) = \\lim \\limits _ { \\varepsilon \\to + 0 } z _ j ( \\varepsilon + i \\theta ) , j = 1 , 2 . \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} x ^ n : = \\min \\{ x \\in ( 0 , 1 / n ) : \\pi \\in { \\cal S } ( \\widehat { W } ^ { ( 1 / n , 0 ) } , \\widehat { W } ^ { ( 0 , 0 ) } ) \\sigma _ \\pi \\leq - 1 \\pi ( 0 ) = x \\} . \\end{align*}"}
-{"id": "9317.png", "formula": "\\begin{align*} n _ i = \\begin{cases} 3 & i = 1 , \\\\ 1 & i = 2 , 4 , 1 0 , 1 2 , \\\\ 0 & \\\\ \\end{cases} \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} P _ { t } \\left ( D \\right ) u + \\sum \\limits _ { \\left \\vert \\alpha \\right \\vert \\leq 2 l } \\left ( b _ { \\alpha } D _ { y } ^ { \\alpha } + \\lambda \\right ) u = f \\left ( x , y \\right ) , y \\in \\Omega , \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{align*} \\widehat { h } ( k ) : = \\int _ x ^ \\infty h ( t ) e ^ { - i k t } d t . \\end{align*}"}
-{"id": "2790.png", "formula": "\\begin{align*} w ^ \\prime _ \\lambda = \\frac { a \\lambda ^ 2 - b } { \\lambda ( b - \\lambda a ) } , z ^ \\prime _ \\lambda = \\frac { a \\lambda ^ 2 - b } { b - \\lambda a } , \\left ( \\sqrt { \\frac { b } { a } } < \\lambda < \\frac { b } { a } \\right ) . \\end{align*}"}
-{"id": "0.png", "formula": "\\begin{align*} \\frac { d \\hat c _ 0 } { d t } & = \\mu \\hat h _ 0 - \\frac { \\Gamma } { K _ 1 } , \\\\ \\frac { d \\hat h _ 0 } { d t } & = - \\hat h _ 0 . \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} u = \\sum _ n u _ n e _ n ~ , ~ { \\rm w i t h } ~ u _ n = \\sqrt { \\frac { n ! } { m ! } } \\psi _ n ^ m ( \\xi , \\lambda ) , \\end{align*}"}
-{"id": "4337.png", "formula": "\\begin{align*} \\prod _ { \\mathfrak { P } \\mid p } \\left ( \\mathfrak { D } _ n \\right ) _ { \\mathfrak { P } } = \\prod _ { \\mathfrak { p } \\mid p } \\left ( \\mathfrak { D } _ 1 \\right ) _ { \\mathfrak { p } } , \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} \\vert \\widetilde { T } _ { F , G , N } ^ { L , \\Xi , \\widetilde { \\mathbf { r } } } ( g _ 1 , \\dots , g _ d ) \\vert \\ll _ { c , C , \\varepsilon } \\Big \\vert \\frac { 1 } { N ^ { h - m } } \\int \\limits _ { \\mathbf { x } } \\prod \\limits _ { j = 1 } ^ d g _ j ( \\psi _ j ( \\mathbf { x } ) + a _ j ) F _ 1 ( \\mathbf { x } ) \\ , d \\mathbf { x } \\Big \\vert , \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} [ L _ { - r } , \\ , S _ { r - 1 } ] = 0 \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{gather*} A ^ S ( x ) = \\frac { 1 } { x ^ { 3 / 2 } } \\begin{pmatrix} 0 & 1 \\\\ a ^ { ( 1 ) } _ { 2 1 } & 0 \\end{pmatrix} + \\frac { 1 } { x } \\begin{pmatrix} a ^ { ( 1 ) } _ { 1 1 } & 0 \\\\ 0 & a ^ { ( 1 ) } _ { 2 2 } - \\frac 1 2 \\end{pmatrix} + \\cdots . \\end{gather*}"}
-{"id": "1985.png", "formula": "\\begin{align*} & \\prod _ { j = 1 } ^ N z _ j ^ { N + 1 - j } ( 1 + t ^ { - 1 } z _ j ^ 2 ) ^ { - 1 } \\prod _ { 1 \\le j < k \\le N } ( 1 + t ^ { - 1 } z _ j z _ k ) ^ { - 1 } ( t ^ { - 1 } + z _ j z _ k ^ { - 1 } ) ^ { - 1 } \\\\ \\times & t ^ N \\langle 1 \\cdots M | \\mathcal { B } ^ \\prime ( z _ 1 ) \\cdots \\mathcal { B } ^ \\prime ( z _ N ) | \\overline { x _ 1 } \\cdots \\overline { x _ N } \\rangle \\Bigg | _ { t = - 1 } = s p _ { \\overline { \\lambda } } ( \\{ z \\} _ N ) . \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} x \\left ( z \\right ) = \\sum _ { n \\ge 1 } x _ { n } z ^ { n } , \\thinspace \\thinspace g \\left ( z \\right ) = \\sum _ { n \\ge 1 } g _ { n } z ^ { n } \\end{align*}"}
-{"id": "8083.png", "formula": "\\begin{align*} ( \\wp ' ( z ) ) ^ 2 = 4 \\wp ^ 3 ( z ) - g _ 2 \\wp ( z ) - g _ 3 \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} p ( \\Phi ) = \\sup _ { x _ 1 \\in \\mathbb { R } , \\ldots , x _ k \\in \\mathbb { R } } \\left | ( 1 + | x _ 1 | ^ 2 ) \\ldots ( 1 + | x _ k | ^ 2 ) \\Phi ( x _ 1 , \\ldots , x _ k ) \\right | . \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{align*} [ \\mathcal { A } , \\mathcal { B } ] : = [ \\mathcal { A } _ 1 , \\mathcal { B } _ 1 ] _ 1 + \\sum _ { \\tau = i , j , k } [ \\mathcal { A } _ 1 , \\mathcal { B } _ \\tau ] _ \\tau + \\sum _ { \\tau = i , j , k } [ \\mathcal { A } _ \\tau , \\mathcal { B } _ 1 + \\mathcal { B } _ \\tau ] _ \\tau , \\mbox { ~ ~ f o r a l l ~ ~ } \\mathcal { A } , \\mathcal { B } \\in \\mathfrak { h } _ { 2 4 } . \\end{align*}"}
-{"id": "6566.png", "formula": "\\begin{align*} \\mathrm { d e t } J ( x _ 0 , y _ 0 ) = b ( d - x _ 0 ) + d y _ 0 > 0 \\quad \\mbox { a n d } \\mathrm { T r } J ( x _ 0 , y _ 0 ) = - b - y _ 0 - ( d - x _ 0 ) < 0 \\ , . \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} C ( M , J , [ g ] ) \\coloneqq \\int _ M e ^ { ( 2 - m ) f _ 0 } s ^ H \\frac { F ^ n } { n ! } \\overset { \\eqref { E q n : H e r m S c a l } } { = } \\int _ M s _ 0 ^ H \\frac { F _ 0 ^ n } { n ! } . \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N \\Lambda _ X \\big ( \\tau \\omega _ i ( N ) ) \\big ) - N \\int _ 0 ^ 1 \\Lambda _ X \\big ( \\tau \\overline F ( x ) \\big ) { \\rm d } x = \\sum _ { i = 1 } ^ N \\Lambda _ X \\left ( \\tau { N } \\int _ { ( i - 1 ) / N } ^ { i / N } \\overline { F } ( x ) { \\rm d } x \\right ) - N \\int _ 0 ^ 1 \\Lambda _ X \\big ( \\tau \\overline F ( x ) \\big ) { \\rm d } x . \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} ( X \\triangleright \\phi ) ( Y ) = \\phi ( Y X ) , ( \\phi \\triangleleft X ) ( Y ) = \\phi ( X Y ) . \\end{align*}"}
-{"id": "3016.png", "formula": "\\begin{align*} \\liminf _ { l \\to \\infty } \\ , \\inf _ { J \\ge N _ l } \\ \\dfrac { 1 } { J + 1 } \\ \\# \\ , \\Bigl \\{ 0 \\le j \\le J \\ , ; \\ , & \\Vert P _ { l } T ^ { \\ , j } T _ { l } \\ , x \\Vert \\ge X _ { l } / 2 \\Bigr \\} \\\\ & \\ge \\liminf _ { k \\to \\infty } \\ , \\Bigl ( 1 - \\dfrac { 1 2 \\delta ^ { ( k ) } } { a _ k } - \\dfrac { 1 2 \\delta ^ { ( k ) } } { \\Delta ^ { ( k ) } } \\Bigr ) = 1 . \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} ( \\tau \\circ A ) ( t ) = \\tau _ { A _ t } = \\{ s : A _ s = A _ t \\} = t . \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{align*} m ^ { 1 / \\tau } _ s = \\frac { 1 } { \\sqrt { \\tau } } M _ { s \\tau } ^ N , s \\in [ 0 , 1 / \\tau ] , \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { F } _ q } ( c _ { 1 , j } , c _ { 2 , j } , \\dots , c _ { l , j } ) & = l , \\\\ \\sum _ { i \\neq j } \\dim _ { \\mathbb { F } _ q } ( c _ { 1 , i } , c _ { 2 , i } , \\dots , c _ { l , i } ) & = \\frac { ( n - 1 ) l } { r } . \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} f \\left ( z \\right ) = \\log \\left ( 1 + z \\right ) , \\thinspace \\thinspace g \\left ( z \\right ) = \\frac { z } { 1 - z } \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} \\begin{pmatrix} z \\\\ [ . 1 c m ] \\bar { z } \\end{pmatrix} = \\begin{pmatrix} N w \\\\ [ . 1 c m ] \\overline { M } w \\end{pmatrix} \\end{align*}"}
-{"id": "278.png", "formula": "\\begin{align*} X \\triangleright ( \\mathsf { M } _ { m } ^ { n } ) _ { j } ^ { i } & = ( X _ { ( 1 ) } \\triangleright u _ { i } ^ { m * } ) ( X _ { ( 2 ) } \\triangleright u _ { j } ^ { n } = \\sum _ { k , \\ell } \\pi ( S ( X _ { ( 1 ) } ) ) _ { k } ^ { i } u _ { k } ^ { m * } \\pi ( X _ { ( 2 ) } ) _ { j } ^ { \\ell } u _ { \\ell } ^ { n } \\\\ & = \\sum _ { k , \\ell } \\pi ( S ( X _ { ( 1 ) } ) ) _ { k } ^ { i } ( \\mathsf { M } _ { m } ^ { n } ) _ { \\ell } ^ { k } \\pi ( X _ { ( 2 ) } ) _ { j } ^ { \\ell } . \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} a y '' ( t ) + b D ^ { ( 3 / 2 ) } y ( t ) + c y ( t ) = f ( t ) \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{align*} | v _ m | \\prod _ { i = b _ { m + 1 } - s } ^ { b _ { m + 1 } - 1 } | w _ { i } | > \\frac { 1 } { \\eta } \\cdot \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{align*} { E } _ m ( k ) : = \\bigcup _ { r \\geq \\frac { 8 \\mu } { \\beta _ 1 } m } \\ ; \\ ; \\bigcup _ { \\pi \\in { \\cal Q } _ r } \\left \\{ \\hat { T } ^ { ( k ) } ( \\pi ) < \\beta _ 1 r \\right \\} \\end{align*}"}