diff --git "a/data_tmp/process_24/tokenized_finally.jsonl" "b/data_tmp/process_24/tokenized_finally.jsonl"
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-{"id": "7133.png", "formula": "\\begin{align*} \\wp ( - \\omega ) = \\wp ( \\omega ) \\quad \\wp ' ( - \\omega ) = - \\wp ' ( \\omega ) . \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{align*} \\omega _ 2 ( \\lambda ) = i \\frac { \\omega _ 1 } { \\pi } \\log ( \\lambda ) + u ( \\lambda ) \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{align*} \\varrho | z | < \\varrho s = \\frac { 1 } { 3 \\varphi ( f ^ \\# ( c ) ) } \\leq \\frac { 1 } { 3 \\varphi ( f ^ \\# ( b ) ) } = \\frac t 3 , \\end{align*}"}
-{"id": "5591.png", "formula": "\\begin{align*} \\Phi = \\begin{pmatrix} 0 & \\phi \\\\ \\phi ^ t & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} \\frac { \\pi } { 3 T } \\log B _ { 2 T } ^ { ( 1 ) } ( s , v ) & = \\frac { \\pi } { 3 T } \\sum _ { n = 1 } ^ { ( s + v ) T / 2 } \\log \\cos \\left ( \\frac { \\{ 2 n + ( 3 - 2 s ) T - 1 \\} \\pi } { 6 T } \\right ) \\\\ & \\rightarrow \\int _ { ( 3 - 2 s ) \\pi / 6 } ^ { ( 3 - s + v ) \\pi / 6 } \\log \\cos u \\ , d u \\mbox { i n $ T \\to \\infty $ } . \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} ( m + 1 - i ) _ q q ^ i \\beta _ { ( i , m , k ) } + ( q ^ { 2 m + 2 k + 1 - i } r - r ^ { - 1 } ) ( i ) _ q \\beta _ { ( i - 1 , m , k ) } \\qquad \\\\ = ( m + 1 ) _ q \\beta _ { ( i , m + 1 , k ) } \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} & \\vartheta _ 1 ( x + y ; \\tau ) \\vartheta _ 1 ( x - y ; \\tau ) \\vartheta _ 1 ( u + v ; \\tau ) \\vartheta _ 1 ( u - v ; \\tau ) \\\\ & - \\vartheta _ 1 ( x + v ; \\tau ) \\vartheta _ 1 ( x - v ; \\tau ) \\vartheta _ 1 ( y + u ; \\tau ) \\vartheta _ 1 ( - y + u ; \\tau ) \\\\ & = \\vartheta _ 1 ( y + v ; \\tau ) \\vartheta _ 1 ( y - v ; \\tau ) \\vartheta _ 1 ( x + u ; \\tau ) \\vartheta _ 1 ( x - u ; \\tau ) . \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} \\tilde { \\xi } ( x \\otimes y - \\varepsilon ( x , y ) y \\otimes x ) = & \\tilde { \\xi } ( x \\odot y - \\varepsilon ( x , y ) y \\odot x ) \\\\ = & \\tilde { \\xi } ( x ) \\cdot _ { A } \\tilde { \\xi } ( y ) - \\varepsilon ( x , y ) \\tilde { \\xi } ( y ) \\cdot _ { A } \\tilde { \\xi } ( x ) \\\\ = & \\xi ( x ) \\cdot _ { A } \\xi ( y ) - \\varepsilon ( x , y ) \\xi ( y ) \\cdot _ { A } \\xi ( x ) = [ \\xi ( x ) , \\xi ( y ) ] _ { A } \\\\ = & \\xi ( [ x , y ] _ { A } ) = \\tilde { \\xi } ( [ x , y ] _ { A } ) . \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{align*} \\mu = \\mu ^ { } + \\frac { 1 } { 4 \\pi i } \\sum _ { \\gamma } \\Omega ( \\gamma ) \\ ! \\int _ { \\ell _ { \\gamma } } \\ ! \\frac { d \\zeta } { \\zeta } \\Big ( \\frac { Z _ { \\gamma } } { \\zeta } - \\bar { Z } _ { \\gamma } \\zeta \\Big ) \\ln ( 1 - \\mathcal { X } _ { \\gamma } ( \\zeta ) ) \\rlap { . } \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{align*} f ( x \\cdot \\alpha _ { 0 } , y \\cdot \\beta _ { 0 } ) + \\delta ' = r \\cdot ( f ( x , y ) + \\delta ) . \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} \\left ( T - T _ 1 \\right ) ^ s = \\left ( T - T _ 1 \\right ) ^ s \\mathbf { 1 } _ { \\{ T _ 1 < T _ 2 \\} } \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{align*} & \\mathfrak { c l } _ X ( \\Omega _ X ) \\alpha = - 2 i \\alpha \\ \\ \\mbox { a n d } \\ \\ \\mathfrak { c l } _ X ( \\Omega _ X ) \\beta = 2 i \\beta \\\\ & \\mathfrak { c l } _ Y ( K d t ) \\alpha = i \\alpha \\ \\ \\mbox { a n d } \\ \\ \\mathfrak { c l } _ Y ( K d t ) \\beta = - i \\beta . \\end{align*}"}
-{"id": "4905.png", "formula": "\\begin{align*} a ^ z ( t ) & : = \\sigma ( t , M ^ z ( t ) ) ^ { - 1 } \\left ( B ( t , M _ t ^ z ) + b ( t , M ^ z ( t ) ) \\right ) , \\\\ D ^ z ( t ) & : = \\exp \\left ( \\int _ { 0 } ^ { t } a ^ z ( s ) ^ \\top \\dd { W } ( s ) - \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } \\abs { a ^ z ( s ) } ^ 2 \\dd { s } \\right ) . \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} \\int _ D \\psi _ 2 ( n _ 2 ; \\lambda _ 2 ) \\cdots \\psi _ g ( n _ g ; \\lambda _ g ) \\ , h ( \\lambda _ 2 , \\dots , \\lambda _ g ) \\ , d \\lambda _ 2 \\cdots d \\lambda _ g = 1 \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} \\mu \\geq \\nu \\iff \\sum _ { i = 1 } ^ { j } \\mu _ i \\geq \\sum _ { i = 1 } ^ { j } \\nu _ i , \\forall \\ 1 \\leq j \\leq n . \\end{align*}"}
-{"id": "4902.png", "formula": "\\begin{align*} Y ^ x ( t ) & : = u ( t , X ^ x ( t ) ) , \\ S \\leq t \\leq T , \\\\ Y ^ y ( t ) & : = u ( t , X ^ y ( t ) ) , \\ S \\leq t \\leq T . \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{align*} F ( y ) = P \\{ Y \\leq y \\} = \\exp \\left ( - e ^ { - y } \\right ) , y \\in \\mathbb { R } , \\end{align*}"}
-{"id": "3019.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\nabla u _ { n } \\nabla ( u _ { n } - \\mathcal { S } ( a ) ) = \\int _ { \\Omega } a u _ { n } ^ { q _ { n } } ( u _ { n } - \\mathcal { S } ( a ) ) \\rightarrow 0 , \\end{align*}"}
-{"id": "2722.png", "formula": "\\begin{align*} O _ K & = \\{ x \\in K \\ | \\ v _ K ( x ) \\geq ( 0 , 0 ) \\} \\\\ m _ K & = \\{ x \\in K \\ | \\ v _ K ( x ) > ( 0 , 0 ) \\} , \\end{align*}"}
-{"id": "5834.png", "formula": "\\begin{align*} E _ { \\nu } ( z ; t ^ { - m } , t ) : = \\lim _ { q \\rightarrow t ^ { - m } } E _ { \\nu } ( z ; q , t ) \\end{align*}"}
-{"id": "2618.png", "formula": "\\begin{align*} \\lambda = - a f + a ( m + n - 1 ) - a _ { 0 } , \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} \\mathcal L _ \\partial ( c _ S ) = 0 \\ , . \\end{align*}"}
-{"id": "8346.png", "formula": "\\begin{align*} R _ j g = \\left ( \\frac { \\xi _ j } { i | \\xi | } \\hat { g } \\right ) ^ { \\check { } } \\end{align*}"}
-{"id": "6461.png", "formula": "\\begin{align*} \\begin{aligned} & \\norm { \\nabla y _ { 0 } ( t _ { 0 } + h ) - \\nabla y _ { 0 } ( t _ { 0 } ) } _ { L ^ { r } ( \\Omega ) ^ { 3 \\times 3 } } \\leq C t _ { 0 } ^ { - \\frac { 1 } { 2 } - \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { r } ) } \\norm { e ^ { - h B } b _ { s } - b _ { s } } _ { L ^ { p } ( \\Omega ) ^ 3 } , \\end{aligned} \\end{align*}"}
-{"id": "183.png", "formula": "\\begin{align*} B ( k _ 0 , k _ 1 ) = \\left [ \\mu _ { l _ 0 } ^ { ( 0 ) } + m _ { l _ 0 } \\right ] b _ { l _ 1 , 0 } + \\left [ \\mu _ { l _ 1 } ^ { ( 1 ) } + m _ { l _ 1 } \\right ] b _ { l _ 0 , 1 } . \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} M : = M _ 1 \\times \\cdots \\times M _ s \\ \\ m : = m _ 1 + \\cdots + m _ s . \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} & E \\left [ \\tilde h ( q _ t ^ m , z _ t ^ m ) \\right ] = H _ t + O ( m ^ { \\delta } ) \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{align*} \\forall i , j \\in [ 1 , n ] , [ x _ i , x _ j ] = [ y _ i , y _ j ] = 0 , \\end{align*}"}
-{"id": "200.png", "formula": "\\begin{align*} \\phi = \\psi \\circ \\rho + \\rho ^ { k + 1 } f . \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} \\begin{pmatrix} A & 0 \\\\ 0 & B \\end{pmatrix} Z = A Z B ^ { - 1 } , \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\psi ^ j & = & h ^ { j + 1 } \\circ d ^ j + d ^ { j - 1 } \\circ h ^ j \\\\ h ^ j ( ( s \\otimes t ) \\cdot p ) & = & ( s \\otimes t ) \\cdot h ^ j ( p ) - ( 1 \\otimes \\partial ) ( s \\otimes t ) \\cdot k ^ j ( p ) \\ , . \\end{array} \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} h = ( N s ) ^ { - 2 } \\tilde { h } = - s ^ { - 2 } d t ^ 2 + g _ t \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{align*} | p _ r | ^ 2 = r ^ 2 ( ( 1 - l _ 1 ^ 2 ) ^ 2 + l _ 1 ^ 2 l _ 2 ^ 2 + \\cdots + l _ 1 ^ 2 l _ d ^ 2 ) = r ^ 2 ( 1 - 2 l _ 1 ^ 2 + l _ 1 ^ 2 ( l _ 1 ^ 2 + \\cdots + l _ d ^ 2 ) ) = r ^ 2 ( 1 - l _ 1 ^ 2 ) . \\end{align*}"}
-{"id": "7573.png", "formula": "\\begin{align*} & \\sum _ i A ( v _ 1 , . . . , C v _ i , . . . , v _ k ) = \\sum _ i \\int _ 0 ^ \\infty B ( e ^ { t C } v _ 1 , . . . , e ^ { t C } C v _ i , . . . , e ^ { t C } v _ k ) d t \\\\ = & \\int _ 0 ^ \\infty \\frac { d } { d t } B ( e ^ { t C } v _ 1 , . . . , e ^ { t C } v _ i , . . . , e ^ { t C } v _ k ) d t \\\\ = & - B ( v _ 1 , . . . , v _ k ) . \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} X ( t ) = X \\# _ t Y : = X ^ { 1 / 2 } ( X ^ { - 1 / 2 } Y X ^ { - 1 / 2 } ) ^ t X ^ { 1 / 2 } \\ , \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} \\eta _ { g , y } = ( ^ { g } t ) t ^ { - 1 } ( ^ { g } y ) t ( ^ { g } t ) ^ { - 1 } , \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} | U ( z ) | = q ^ { \\mu _ 1 + \\cdots + \\mu _ { m - 1 } + 1 } - q ^ { \\mu _ 1 + \\cdots + \\mu _ { m - 1 } } = q ^ { ( \\mu _ 1 + \\cdots + \\mu _ { m - 1 } ) - ( m - 1 ) } | U | . \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} \\psi _ { r , s } ( x _ i , x _ j ) = - \\psi _ { r , s } ( x _ j , x _ i ) = ( - 1 ) ^ { r - i } \\binom { j - r - 1 } { r - i } x _ { i + j + s - 2 r - 1 } , \\end{align*}"}
-{"id": "8442.png", "formula": "\\begin{align*} n = r + \\frac { r ( r - 1 ) } { 2 } a + r b , n _ T = r + \\frac { r ( r - 1 ) } { 2 } a . \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} \\phi \\left ( z \\frac { x ^ m - y ^ m } { x - y } \\right ) = z \\frac { x ^ n - y ^ n } { x - y } \\end{align*}"}
-{"id": "1845.png", "formula": "\\begin{align*} \\left \\| \\sum _ { n = 0 } ^ { + \\infty } c _ n \\varphi _ n \\right \\| _ { L ^ \\infty ( \\C ) } \\lesssim \\| ( c _ n ) \\| _ { \\ell ^ { \\infty , 1 / 4 } } . \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} \\pi ^ * \\omega _ g ( \\eta _ { \\diamondsuit } , \\bar { \\eta } _ { \\heartsuit } ) = \\left . i \\frac { \\partial ^ 2 } { \\partial z _ { \\diamondsuit } \\partial \\bar { z } _ { \\heartsuit } } \\right | _ 0 \\phi ( g \\exp ( z _ { \\diamondsuit } f _ { \\diamondsuit } + z _ { \\heartsuit } f _ { \\heartsuit } ) ) . \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} | S | = \\prod _ { j = 1 } ^ { L } \\binom { N - \\mu } { m _ { j } - \\mu } . \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} m i _ k ( n ) \\le m i _ k ( n - \\delta - 1 ) + \\frac { 1 } { 2 } \\frac { \\sum _ { i = 1 } ^ \\delta [ ( d ( v _ i ) m i _ k ( n - \\delta - d ( v _ i ) - 1 ) + ( \\delta - d ( v _ i ) - 1 ) m i _ k ( n - \\delta - k ) ] } { \\binom { k } { 2 } } . \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} f ^ * ( 1 \\times \\omega _ { S ^ 2 } ) = \\zeta \\cdot ( \\omega _ { \\Sigma } \\times 1 ) \\pm ( 1 \\times \\omega _ { S ^ 2 } ) , \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{align*} f ( x ) & = \\alpha _ + \\big ( 1 + o ( 1 ) \\big ) e ^ { i \\sqrt { \\lambda } x } , x \\rightarrow + \\infty , \\\\ f ' ( x ) & = \\alpha _ + i \\sqrt { \\lambda } \\big ( 1 + o ( 1 ) \\big ) e ^ { i \\sqrt { \\lambda } x } , x \\rightarrow + \\infty , \\end{align*}"}
-{"id": "3424.png", "formula": "\\begin{align*} L _ { { \\operatorname { e v e n } } } & : = \\{ ( - i , - j ) : 0 \\le j \\le i i \\equiv j \\mod 2 \\} , \\\\ L _ { { \\operatorname { o d d } } } & : = \\{ ( - i , - j ) : 0 \\le j \\le i i \\equiv j + 1 \\mod 2 \\} . \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{align*} \\tau _ d ( y ) ( x ) : = d ( x , y ) - d ( b , y ) \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} T _ { \\beta } ^ { 0 } ( 1 ) = 1 , ~ T _ { \\beta } ^ { j } ( 1 ) = \\beta ^ j - t _ 1 \\beta ^ { j - 1 } - t _ 2 \\beta ^ { j - 2 } - \\ldots - t _ j \\in \\mathbb { Z } [ \\beta ] \\cap [ 0 , 1 ] \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{align*} q ( \\lambda ) = \\det ( T _ 1 ^ { - 1 } R _ 1 \\cdots T _ r ^ { - 1 } R _ r - \\lambda ^ r I ) \\cdot \\det ( T _ 1 \\cdots T _ r ) = \\det ( R _ 1 \\cdots R _ r - \\lambda ^ r T _ 1 \\cdots T _ r ) , \\end{align*}"}
-{"id": "7764.png", "formula": "\\begin{align*} I \\circ \\Gamma _ 1 ( \\R / \\Z ) = \\Gamma _ 2 ( \\R / \\Z ) , \\end{align*}"}
-{"id": "2808.png", "formula": "\\begin{align*} C ^ 2 & = d ^ 2 - \\sum _ { p \\in S } m _ p ( \\bar { C } ) ^ 2 = d ^ 2 - \\sum _ { i = 1 } ^ s m _ { q _ i } ( \\bar { C } ) ^ 2 - t \\\\ & \\geq d - \\sum _ { i = 1 } ^ s m _ { q _ i } ( \\bar { C } ) - t \\geq d ( 1 - s ) - t \\geq - d ( s + t ) \\geq - n d , \\end{align*}"}
-{"id": "7502.png", "formula": "\\begin{align*} p ^ m ( t , q , z ) = & \\left ( \\frac { \\beta ( t , q ) } { 2 \\pi } \\right ) ^ { n / 2 } e ^ { - \\beta ( t , q ) \\| z \\| ^ 2 / 2 } p ^ 0 ( t , q ) + o ( 1 ) , \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} & q _ { A _ 1 B _ 1 A _ 2 B _ 2 } ( a _ 1 , b _ 1 , a _ 2 , b _ 2 ) = \\sum _ { x _ 1 , x _ 2 , y _ 1 , y _ 2 } \\Big ( q _ { X _ 1 Y _ 1 X _ 2 Y _ 2 } ( x _ 1 , y _ 1 , x _ 2 , y _ 2 ) V ( a _ 1 a _ 2 b _ 1 b _ 2 | x _ 1 x _ 2 y _ 1 y _ 2 ) \\Big ) \\\\ & = p _ { A B } ( a _ 1 , b _ 1 ) p _ { A B } ( a _ 2 , b _ 2 ) \\end{align*}"}
-{"id": "1012.png", "formula": "\\begin{align*} T _ { j } ^ { k } : = \\prod _ { i \\in I _ { j } ^ { k } } S _ { i , \\lambda _ { i j } ^ { k } } \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{align*} \\left | \\frac { \\Phi _ { t _ 0 } ( z _ 1 ) } { \\Phi _ { t _ 0 } ( z _ 2 ) } \\right | = \\left ( \\frac { x _ 0 } { y } \\right ) \\cdot \\exp [ ( t _ 0 - 1 ) ( u ( z _ 1 ) - u ( z _ 2 ) ) ] \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{align*} w _ 1 = \\hat { w } _ 1 = \\hat { w } _ 2 = w _ 3 = w _ 4 = 0 , w _ 5 = - \\tfrac { 1 } { 4 } h ^ { - 1 } d h , w _ 2 = - h ^ { - \\frac { 3 } { 4 } } \\kappa _ 0 , \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} H \\ast _ { ( K , \\phi ) } = \\langle H , t ; t k t ^ { - 1 } = \\phi ( k ) , k \\in K \\rangle . \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} [ c , z _ 1 ] [ z _ 2 , z _ 3 , z _ 4 ] = [ c , z _ 1 ] [ z _ 3 , z _ 4 , z _ 2 ] = [ c , z _ 1 ] [ z _ 4 , z _ 2 , z _ 3 ] . \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow \\infty } \\bigg \\{ \\frac { N _ { s c } ^ { > } [ \\mathfrak { e } , \\lambda V ] } { N _ { s c } [ \\mathfrak { e } , \\lambda V ] } \\bigg \\} \\ = \\ 0 , \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} \\sum _ { b = a + 2 } ^ { n - 2 } & \\sum _ { m = 2 } ^ { b - a } \\binom { b - a - 1 } { m - 2 } \\left [ ( b - m + 1 ) ( n - b - 1 ) + \\binom { n - b - 2 } { 2 } \\right ] \\\\ & = 2 ^ { n - a - 2 } ( n - 4 + a ) - \\frac { 1 } { 6 } ( n ^ 3 - 6 n ^ 2 - ( 3 a ^ 2 - 9 a - 1 1 ) n + 2 a ^ 3 - 3 a ^ 2 - 5 a - 1 8 ) . \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{align*} 0 & = 0 \\\\ \\partial _ { t } y ( x , t ) & = \\sin ( t ) f '' ( x ) [ 1 + f ' ( x ) ^ { 2 } ] ^ { - 3 / 2 } \\\\ \\partial _ { t } z ( x , t ) & = \\cos ( t ) f '' ( x ) [ 1 + f ' ( x ) ^ { 2 } ] ^ { - 3 / 2 } \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{align*} \\left | f _ { \\beta } ( z ) - G _ { n } ( z ) \\right | = \\left | \\sum _ { q \\geq 1 } z ^ { m _ q } \\right | < \\left | G _ { n } ( z ) \\right | \\mbox { f o r } ~ z \\in C _ { j , n } , ~ \\mbox { f o r } ~ j = 1 , 2 , \\ldots , J _ n , \\end{align*}"}
-{"id": "8258.png", "formula": "\\begin{align*} Y _ t : = \\frac { x _ { \\frac { 1 - \\alpha } { 2 } t + \\eta t ^ { 1 / 2 } } ( t ) - \\left [ ( \\alpha - \\tfrac 1 2 ) t - 2 \\eta t ^ { 1 / 2 } \\right ] } { - t ^ { 1 / 2 } } , \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} \\frac { d x ( t ) } { d t } = A x ( t ) + [ \\frac { 1 } { \\gamma } B ] [ \\gamma u ( t ) ] + \\sum _ { i = 1 } ^ m [ \\frac { 1 } { \\gamma } N _ i ] x ( t ) [ \\gamma u _ i ( t ) ] , \\end{align*}"}
-{"id": "2194.png", "formula": "\\begin{align*} = z _ { 2 } ^ { - 1 } \\left ( \\frac { z _ { 1 } - z _ { 0 } } { z _ { 2 } } \\right ) ^ { - r / 2 } \\delta \\left ( \\frac { z _ { 1 } - z _ { 0 } } { z _ { 2 } } \\right ) Y ( Y ( u , \\ z _ { 0 } ) v , \\ z _ { 2 } ) w . \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} & 2 ^ { n - 1 } ( 1 _ { m = 0 } + 1 _ { m \\neq 0 } ( m - 1 ) ! 2 ^ { - m - 1 } ) ( 1 _ { m = n - 1 } + 1 _ { m \\neq n - 1 } ( n - m - 2 ) ! 2 ^ { - n + m } ) \\\\ & \\le ( 1 _ { m = 0 } + 1 _ { m \\neq 0 } ( m - 1 ) ! ) ( 1 _ { m = n - 1 } + 1 _ { m \\neq n - 1 } ( n - m - 2 ) ! ) \\end{align*}"}
-{"id": "8083.png", "formula": "\\begin{align*} & v ^ { h } _ { e _ i } = \\frac { v ( x + h e _ i , t ) - v ( x , t ) } { h ^ { k \\alpha } } , \\ \\ \\ v ^ { h } _ { t } = \\frac { v ( x , h + t ) - v ( x , t ) } { h ^ { k \\alpha / 2 } } , \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} h ( x _ 1 , \\dots , x _ d ) = h ^ 2 ( x _ i , h ^ { d - 1 } ( x _ 1 , \\dots , x _ { i - 1 } , x _ { i + 1 } , \\dots , x _ d ) ) \\qquad \\forall i = 1 , \\dots , d . \\end{align*}"}
-{"id": "4432.png", "formula": "\\begin{align*} H ( B ^ { e - 1 } ) & = H ( ( B ^ { e - 1 } ) ^ { \\perp } ) = N ( \\mathfrak { b } ) ^ { - 1 } \\prod _ { i = 1 } ^ p | | W ^ { ( i ) } \\wedge Z _ 1 ^ { ( i ) } \\wedge \\dots \\wedge Z _ m ^ { ( i ) } | | \\\\ & = N ( \\mathfrak { b } ) ^ { - 1 } \\prod _ { i = 1 } ^ p | | V _ i | | \\ ; | | Z _ 1 ^ { ( i ) } \\wedge \\dots \\wedge Z _ m ^ { ( i ) } | | \\leq H ( B ^ e ) \\prod _ { i = 1 } ^ p | | V _ i | | \\\\ & \\leq C _ 5 H ( B ^ e ) H ^ { ( 2 y - 1 ) / e } \\leq C _ 5 H ^ { ( e + 2 y - 1 ) / e } = H ' , \\end{align*}"}
-{"id": "6494.png", "formula": "\\begin{align*} u ( t ) = \\Phi ^ { - 1 } A _ p ^ { \\frac { 1 } { 2 } } \\int _ 0 ^ t e ^ { - ( t - s ) A _ p } \\Phi \\big [ A _ { p ^ { \\prime } } ^ { - \\frac { 1 } { 2 } } \\big ] ^ * f ( s ) \\ ; d s . \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{align*} E _ { n , \\pm } ( t ) = \\sum _ { k = 1 , 2 } E _ { n , \\pm , k } ( t ) , \\end{align*}"}
-{"id": "7151.png", "formula": "\\begin{align*} f \\left ( b _ { i } , b _ { j } \\right ) = \\delta _ { i j } , h \\left ( b _ { i } , b _ { j } \\right ) = \\delta _ { i j } g \\left ( b _ { i } , b _ { j } , b _ { k } \\right ) = \\delta _ { i j k } , \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} U ^ - _ 1 & = \\Big ( U \\setminus \\{ u _ 2 \\} \\Big ) \\cup \\{ q _ i ^ { - 1 } u _ 2 , q _ { s } u _ 2 , q _ s ^ { - 1 } u _ 2 , q _ i u _ 2 \\} \\\\ W ^ - _ 1 & = W \\cup \\{ u _ 2 , q _ i ^ { - 1 } q _ { s } ^ { - 1 } u _ 2 , q _ i q _ s u _ 2 \\} , \\end{align*}"}
-{"id": "1596.png", "formula": "\\begin{align*} d _ t : = \\varrho / ( d \\ln t ) r _ t : = t d _ t / \\ln _ 3 t , \\end{align*}"}
-{"id": "4181.png", "formula": "\\begin{align*} K _ \\varepsilon = \\left \\lceil C _ 0 \\cdot \\log _ 2 \\left ( \\frac { 1 } { \\varepsilon } \\right ) \\right \\rceil \\leq 1 + C _ 0 \\log _ 2 \\left ( \\frac { 1 } { \\varepsilon } \\right ) \\leq ( 1 + C _ 0 ) \\cdot \\log _ 2 \\left ( \\frac { 1 } { \\varepsilon } \\right ) . \\end{align*}"}
-{"id": "7167.png", "formula": "\\begin{align*} \\Re ( s w ( s ) ) = f ( s ) - g ( s ) , \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{align*} E \\left [ S _ N ^ { ( r ) } \\right ] = r \\int _ 0 ^ { \\infty } t ^ { r - 1 } \\left [ 1 - \\bigg ( 1 - e ^ { - t / N } \\bigg ) ^ N \\right ] d t . \\end{align*}"}
-{"id": "3884.png", "formula": "\\begin{align*} \\Sigma = \\mathbb { E } _ { X \\sim P _ \\theta } \\left [ ( X - \\mathbb { E } _ { X \\sim P _ \\theta } [ X ] ) ( X - \\mathbb { E } _ { X \\sim P _ \\theta } [ X ] ) ^ \\top \\right ] \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} | g ( z ) | & = R _ 0 ^ m \\left | \\sum _ { j = 0 } ^ { J - 1 } a _ j ( z - q ) ^ j \\right | \\le R _ 0 ^ m \\left ( \\sum _ { j = 0 } ^ { J - 1 } ( 2 R _ 0 ) ^ j \\right ) \\max _ { 0 \\le j < J } | a _ j | \\le J R _ 0 ^ m \\max \\{ 1 , ( 2 R _ 0 ) ^ J \\} \\max _ { 0 \\le j < J } | a _ j | \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} F ( t ) = \\int _ t ^ \\infty s ^ { - 1 } f ( s ) d s , \\ \\ t > 0 \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} d { \\widetilde x } _ t = \\sum _ { i = 1 } ^ n H _ i ( { \\widetilde x } _ t ) \\circ d b ^ i _ t + \\nu ^ \\dagger ( { \\widetilde x } _ t ) d \\lambda _ t , \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} P ( a ; b , W ) ( d x ) = \\frac { 1 } { G ( a ; b , W ) } e ^ { - x _ 1 a _ 1 ^ 2 - \\ldots - x _ n a _ n ^ 2 } \\mu ( b , W ) ( d x ) \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{align*} A _ { 2 \\mu + 2 } = \\frac { 9 ^ { 2 \\mu + 2 } ( 6 \\mu + \\nu ) } { 3 \\cdot ( \\mu + 1 ) ! } \\frac { d ^ \\mu } { d x ^ \\mu } \\left . \\left \\{ \\frac { ( 1 + x / 3 ) ^ { 5 - \\nu } } { ( x - 1 ) ^ { 2 \\mu + 2 } ( x - 9 ) ^ { 2 \\mu + 2 } } \\right \\} \\right | _ { x = 0 } . \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{align*} G = \\left ( \\begin{array} { c c c c } \\mu & \\begin{array} { c c c } 0 & 0 & 0 \\end{array} & e \\\\ \\begin{array} { c } 0 \\\\ 0 \\\\ 0 \\end{array} & P _ v & \\begin{array} { c } 0 \\\\ 0 \\\\ 0 \\end{array} & P _ h \\\\ e & \\begin{array} { c c c } 0 & 0 & 0 \\end{array} & \\gamma & 0 \\\\ & P _ h & 0 & P _ h \\end{array} \\right ) \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} X _ i ( \\mu _ 1 - c h ) = 0 \\mbox { i n } D _ { 1 } . \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{align*} f ( H ) = \\min _ { \\mathcal { F } } \\sum _ { F \\in \\mathcal { F } } | U ^ F _ 1 | + \\cdots + | U ^ F _ t | , \\end{align*}"}
-{"id": "7396.png", "formula": "\\begin{align*} \\phi _ n ( x ) & = 2 \\sqrt { 2 } w _ n ( x ) + a ( D ) w _ n ( x ) , \\\\ a ( \\xi ) & : = \\sqrt { ( \\xi + e _ d ) ^ 2 + 1 } + \\sqrt { ( \\xi - e _ d ) ^ 2 + 1 } - 2 \\sqrt { 2 } . \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N \\mathbb P ( A _ i | B ) = \\mathbb P ( \\cup _ { i = 1 } ^ N A _ i | B ) . \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} & \\forall \\ x , y , z \\in \\mathcal { H } ( A ) = A _ 0 \\cup A _ 1 : & \\\\ & a s _ { \\alpha , \\bullet } ( x , y , z ) = 0 , & \\mbox { ( s u p e r H o m - a s s o c i a t i v i t y ) } \\end{align*}"}
-{"id": "8430.png", "formula": "\\begin{align*} ( \\Delta \\otimes \\operatorname { i d } ) \\bigl ( ( a \\otimes c ) E \\bigr ) & = ( \\Delta a \\otimes c ) ( \\Delta \\otimes \\operatorname { i d } ) ( E ) \\\\ & = ( \\Delta a \\otimes c ) ( E \\otimes 1 ) ( 1 \\otimes E ) = ( \\Delta a \\otimes c ) ( 1 \\otimes E ) . \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} \\binom { [ \\frac { k } { 2 } ] + [ \\frac { n - k } { 2 } ] } { [ \\frac { k } { 2 } ] } _ { t ^ 4 } \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{align*} \\psi ( x ) = \\psi ( 0 ) + \\sum _ { s \\subset \\{ 1 , \\ldots , d \\} } \\int _ { 0 _ s } ^ { x _ s } d \\psi _ s ( u _ s ) . \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{align*} E \\left [ S ( \\theta ) ^ { ( r ) } \\right ] = \\frac { 1 } { ( 1 - \\theta ) ^ r } \\int _ 0 ^ { \\infty } r s ^ { r - 1 } \\left [ 1 - \\left ( 1 - e ^ { - s / N } \\right ) ^ N \\right ] d s \\ , + \\ , O \\left ( e ^ { - \\varepsilon N } \\right ) , N \\to \\infty , \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{align*} \\bigg \\| \\sum _ { k = k _ 0 } ^ \\infty 2 ^ { - k \\alpha } \\mathcal { C } _ k g \\bigg \\| _ { L ^ q _ t L ^ 2 _ x } \\lesssim \\| g \\| _ { L ^ { s } _ t L ^ 2 _ x } \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} x \\alpha ( y z \\cdot x \\phi ^ { - 1 } ) = x \\alpha ( y x ^ \\lambda \\cdot x ) \\cdot z x \\phi ^ { - 1 } ~ \\forall ~ x , y , z \\in L ~ \\textrm { a n d } ~ \\alpha , \\phi \\in A ( L ) \\end{align*}"}
-{"id": "8300.png", "formula": "\\begin{align*} & \\mathcal { C } _ { - 1 } ( \\mathcal { a } ^ { ( x ) } , \\hat { \\mathcal { n } } ^ { ( x ) } , i ' , \\lbrace 1 , \\ldots , i ' - 1 \\rbrace , \\lbrace i ' + 1 , \\ldots , \\ell ^ { \\circ } _ x \\rbrace ) \\\\ & = \\mathcal { C } _ { - 1 } ( \\alpha , \\nu , p _ { ( x , i ' ) } , \\lbrace 1 , \\ldots , p _ { ( x , i ' ) } - 1 \\rbrace , \\lbrace p _ { ( x , i ' ) } + 1 , \\ldots , \\ell \\rbrace ) . \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} \\mathbb { P } _ L ( \\mathbf { m } , \\mu , \\boldsymbol { \\theta } ; K ) = \\Pr \\left [ \\bigcap _ { j = 1 } ^ { L } \\mathrm { r a n k } ( \\mathbf { C } _ { j } ) = K \\right ] \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} \\lambda ^ k \\widetilde { F } ( f ( \\lambda ) ) = \\frac { \\lambda ^ k } { f ^ \\beta ( \\lambda ) } f ^ \\beta ( \\lambda ) \\widetilde { F } ( f ( \\lambda ) ) = \\frac { 1 } { \\lambda ^ { \\beta \\alpha - k } l ^ \\beta ( \\lambda ) } f ^ \\beta ( \\lambda ) \\widetilde { F } ( f ( \\lambda ) ) . \\end{align*}"}
-{"id": "6266.png", "formula": "\\begin{align*} [ n ] _ q = \\frac { q ^ n - q ^ { - n } } { q - q ^ { - 1 } } . \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{align*} \\sum _ { t \\in T _ \\mu } | S _ \\mu ( t ) | = | \\lbrace ( s , t ) \\in S _ \\mu \\times T _ \\mu \\mid s < t \\rbrace | = | B _ \\mu | . \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{align*} m ^ N _ t = m ^ { N , n } _ t \\end{align*}"}
-{"id": "328.png", "formula": "\\begin{align*} g ( w ) = f ( z ) - z f ' ( z ) , w = f ' ( z ) . \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} \\hat { T } : = \\bigcap \\limits _ { \\alpha \\in \\Delta \\setminus \\Delta _ P } \\ker ( \\alpha ) \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} A \\ \\cdot = \\sum _ j \\varphi _ j A \\ \\cdot = \\sum _ j \\varphi _ j A [ \\psi _ j \\ \\cdot \\ ] + \\sum _ j \\varphi _ j A [ ( 1 - \\psi _ j ) \\ \\cdot \\ ] . \\end{align*}"}
-{"id": "8620.png", "formula": "\\begin{align*} \\chi _ { b } ( c ) = \\varsigma _ { p } ^ { T r ( b c ) } , \\ \\ c \\in \\mathbb { F } _ { q } , \\end{align*}"}
-{"id": "7723.png", "formula": "\\begin{align*} Q _ 1 = & \\frac { 4 ( \\lambda _ c \\pi ) ^ { t } } { ( t - m - 1 ) ! ( m - 1 ) ! } \\sum ^ { t - m - 1 } _ { p = 0 } ( - 1 ) ^ p { t - m - 1 \\choose p } \\\\ & \\times \\int _ { \\tau _ 1 } ^ { \\tau _ 2 } f _ m ( y ) d y , \\end{align*}"}
-{"id": "7305.png", "formula": "\\begin{align*} \\frac { \\partial { } U ( \\mathbf { p } ) } { \\partial { } p _ k } = \\frac { A _ d { } T _ k } { \\ln 2 ( 1 + A _ d \\Gamma _ k ) ( P _ c + P _ { } ) } - \\frac { R _ { } } { \\eta ( P _ c + P _ { } ) ^ 2 } . \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{align*} & P \\{ T _ { \\ell } = T _ { \\min } \\} = \\\\ & ( - 1 ) ^ g \\sum _ { k _ g = 1 } ^ { M _ g } \\cdots \\sum _ { k _ 1 = 1 } ^ { M _ 1 } ( - 1 ) ^ { k _ 1 + \\cdots + k _ g } \\binom { M _ 1 } { k _ 1 } \\cdots \\binom { M _ g } { k _ g } \\frac { k _ { \\ell } p _ { \\ell } } { k _ 1 p _ 1 + \\cdots + k _ g p _ g } . \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} \\widehat { \\P } ^ { 0 , 0 } _ { 2 T } ( X ( t ) = x ) = \\widehat { \\P } ^ { 0 , 0 } _ { 2 T } ( X ( 2 T - t ) = x ) , 0 \\leq t \\leq 2 T , x \\in \\Z . \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} Y : = \\bigcap _ { j \\geq 1 } Y ( T _ { j - 1 } , T _ j ) \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} | u | \\ge s & \\implies \\gamma _ u \\le \\gamma _ { 1 : s } , \\quad , \\\\ \\lceil u \\rceil \\ge s & \\implies \\gamma _ u \\le \\gamma _ { \\{ s \\} } . \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} c _ 3 = \\frac { \\eta ^ 2 } { 4 a \\lambda _ 2 } + \\frac { b | T r ( L ) | } { 2 N } + \\frac { m \\tau ^ 2 \\tilde { \\gamma } ( N - 1 ) } { 2 N } , c _ 4 = \\frac { \\eta ^ 2 } { 4 a \\lambda _ 2 } . \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} { P _ f } \\left ( { { T _ f } , \\alpha } \\right ) = \\frac { 1 } { { 1 + \\rho \\left ( { { T _ f } , { \\alpha } } \\right ) + \\frac { { { \\lambda _ { t u } } } } { { { \\lambda _ f } } } C \\left ( { { \\alpha } } \\right ) { { \\left ( { \\frac { { { P _ d } { T _ f } } } { { { P _ f } } } } \\right ) } ^ { 2 / { \\alpha } } } } } , \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} H ( S ) = H ( S ^ { \\perp } ) . \\end{align*}"}
-{"id": "6575.png", "formula": "\\begin{align*} A _ j = \\int \\limits _ { c 2 ^ j } ^ { c 2 ^ { j + 1 } } \\left ( \\frac { 1 } { r b ^ { 1 / r } - D ^ { - 1 } O ( 1 ) } \\right ) \\left ( \\frac { b } { \\sqrt { 1 - b ^ 2 } } \\right ) \\ , d b \\end{align*}"}
-{"id": "1278.png", "formula": "\\begin{align*} c g ^ { q ^ { 2 s + l } } \\gamma ^ { \\rho q ^ l } + d h ^ { q ^ { l } } \\gamma ^ { \\rho q ^ { l + n } } = 0 . \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n } \\eta ( x ) d x = 1 . \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} E \\cap W = \\bigcup _ { i , j } \\Delta \\cap ( W \\setminus p ( A _ { i j } \\setminus S ) ) \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} T _ f \\le \\sum _ { i = 0 } ^ { d - 1 } T _ { g _ i } + O ( 1 ) \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{align*} \\mathbf R = \\mathbf R _ { c c } + \\mathbf R _ { c p } + \\mathbf R _ { p p } , \\end{align*}"}
-{"id": "6301.png", "formula": "\\begin{align*} _ U = \\frac { P _ { \\mathbf { b } _ 0 } h _ { \\mathbf { b } _ 0 , \\mathbf { u } _ 0 } D _ { \\mathbf { b } _ 0 , \\mathbf { u } _ 0 } ^ { - \\alpha } } { I _ B + I _ U + I _ J + N _ S + N _ 0 } . \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{align*} \\gamma = \\tau _ 1 + r d r \\wedge \\eta \\wedge \\tau _ 2 \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u ^ { \\prime } ( t ) + \\mathcal { A } _ p u ( t ) & = f ( t ) ( 0 < t < T ) \\\\ u ( 0 ) & = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1606.png", "formula": "\\begin{align*} G ( x _ { t , u , s } ) - G ( x _ { t , u , 0 } ) = \\frac { ( x _ { t , u , s } - x _ { t , u , 0 } ) } { d _ t } \\frac { G ( \\theta _ { t , u , s } ) } { G ( a _ t ) } \\varrho F ' ( \\theta _ { t , u , s } ) , \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} Q _ 2 ^ { k , m } = \\sum _ { i = 0 } ^ { ( m - 1 ) / 2 } ( q ^ { 2 k + m } r ^ 2 ) ^ i \\prod _ { i = 0 } ^ m ( 1 - q ^ { k + i } r ) = \\frac { m + 1 } 2 \\prod _ { i = 0 } ^ m ( 1 - q ^ { k + i } r ) . \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} \\mathcal { V } ( \\xi ) \\ \\doteq \\ \\big \\{ p \\in \\Gamma ^ { \\ast } \\ , \\big | \\ , \\gamma ( p , \\xi ) = \\min _ { { \\tilde { \\xi } } \\in \\mathrm { M i n } ( \\mathfrak { e } ) } \\gamma ( p , { \\tilde { \\xi } } ) \\big \\} , \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} \\big | p ( t , z ) - p ( t , z + e ) \\big | & = \\Big | \\int _ { \\R ^ d } \\frac { { \\rm e } ^ { \\gamma t \\alpha ( \\xi ) } } { ( 2 \\pi ) ^ d } { \\rm e } ^ { i \\xi \\cdot z } ( 1 - { \\rm e } ^ { i \\xi \\cdot e } ) \\ , { { \\rm d } \\xi } \\Big | \\leq c \\ , \\int _ { \\R ^ d } | \\xi | { \\rm e } ^ { - c _ 1 \\gamma t | \\xi | ^ 2 } { \\rm d } \\xi \\leq \\frac { c _ 2 } { ( \\gamma t ) ^ { \\frac { d + 1 } { 2 } } } \\ , . \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} \\varphi _ { L } ( \\mathbf { m } , N , \\boldsymbol { \\epsilon } ) = \\prod _ { j = 1 } ^ { L } ( 1 - \\epsilon _ j ) ^ { m _ { j } } \\epsilon _ j ^ { N - m _ { j } } . \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} & \\chi \\in C ^ { \\infty } ( \\R ) , \\\\ & \\chi ( x ) = 1 , ( \\forall x \\in ( - \\infty , 8 / 5 ] ) , \\\\ & \\chi ( x ) \\in ( 0 , 1 ) , ( \\forall x \\in ( 8 / 5 , 2 ) ) , \\\\ & \\chi ( x ) = 0 , ( \\forall x \\in [ 2 , \\infty ) ) , \\\\ & \\frac { d } { d x } \\chi ( x ) \\le 0 , ( \\forall x \\in \\R ) . \\end{align*}"}
-{"id": "1437.png", "formula": "\\begin{align*} \\sup _ { t \\in \\N } \\sum _ { s = 1 } ^ \\infty e ^ { - \\gamma \\| f _ s f _ t ^ { - 1 } \\| } < \\infty \\forall \\ , \\gamma > 0 . \\end{align*}"}
-{"id": "6536.png", "formula": "\\begin{align*} v _ j ^ 2 a _ j ^ 2 = \\frac { \\sigma ^ 2 a _ j ( 1 - x a _ j ) _ + } { n x } \\leq \\frac { \\sigma ^ 2 a _ N } { n x } \\leq \\frac { \\sigma ^ 2 } { n x ^ 2 } . \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} \\overline { j _ { x , x + 1 } ^ n } : = j ^ n _ { x , x + 1 } - \\mathbb { E } _ n [ j ^ n _ { x , x + 1 } ] = j ^ n _ { x , x + 1 } + n ^ { \\theta } \\alpha _ n \\lambda _ n , \\end{align*}"}
-{"id": "2651.png", "formula": "\\begin{align*} [ - c h ( p _ 1 ) + \\mu _ { 1 X } ( p _ 1 ) - \\tilde { c } ] \\varphi ( q ) = b h ( p _ 1 ) + \\rho _ { 1 X } ( p _ 1 ) - \\tilde { b } , \\forall q \\in G . \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} u _ t = k B \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} \\begin{array} { l } h '' - | a | h = 0 , \\\\ \\noalign { \\smallskip } ( h ' ) ^ 2 - | a | h ^ 2 = \\pm c , \\end{array} \\end{align*}"}
-{"id": "6456.png", "formula": "\\begin{align*} \\begin{aligned} \\norm { u _ { 0 } ( t _ { 0 } - h ) - u _ { 0 } ( t _ { 0 } ) } _ { L ^ { r } _ { \\sigma } ( \\Omega ) } & = \\norm { e ^ { - ( t _ { 0 } - h ) A } a - e ^ { - t _ { 0 } A } a } _ { L ^ { r } _ { \\sigma } ( \\Omega ) } = \\norm { e ^ { - ( t _ { 0 } - h ) A } [ I d - e ^ { - h A } ] a } _ { L ^ { r } _ { \\sigma } ( \\Omega ) } \\\\ & \\leq C ( t _ { 0 } - h ) ^ { - \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { r } ) } \\norm { e ^ { - h A } a - a } _ { L ^ { p } _ { \\sigma } ( \\Omega ) } , \\end{aligned} \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{align*} { \\small f ( \\sum _ { n \\geq 1 } \\frac { 2 a _ n } { 3 ^ { n } } ) = \\sum _ { n \\geq 1 } \\frac { a _ n } { 2 ^ { n } } } \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} 5 \\chi + 6 \\sigma = 0 . \\end{align*}"}
-{"id": "9274.png", "formula": "\\begin{align*} ( K + H ) \\cdot D & = ( - D + H ) \\cdot D = ( a - 1 ) D ^ 2 + a b \\\\ & \\geq ( a - 1 ) D ^ 2 - a \\frac { D ^ 2 } { 2 } = ( \\frac { a } { 2 } - 1 ) D ^ 2 \\geq 0 , \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{align*} a ( t p ^ 2 ) = A ( p ) - \\chi _ 1 ( p ) p ^ { k _ 1 - 1 } . \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} R ^ { - 1 } _ { x } R _ { x \\phi ^ { - 1 } } = L _ { x } ^ { - 1 } R _ { x } ^ { - 1 } R _ { x \\phi ^ { - 1 } } L _ { x } \\end{align*}"}
-{"id": "3414.png", "formula": "\\begin{align*} e _ I : = & e _ { k _ 1 } \\wedge e _ { k _ 2 } \\wedge \\cdots \\wedge e _ { k _ i } , \\\\ d x _ I : = & d x _ { k _ 1 } \\wedge d x _ { k _ 2 } \\wedge \\cdots \\wedge d x _ { k _ i } . \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{align*} \\mathcal { I } _ { a } = \\{ q \\in ( 0 , 1 ) : u ( q ) \\in \\mathcal { P } ^ { \\circ } \\} . \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{align*} m i _ k ( n - \\delta - 1 ) + \\frac { s ( B - 1 ) } { 2 \\binom { k } { 2 } } m i _ k ( n - \\delta - 2 ) + \\frac { ( s ( \\delta - s ) + \\frac { s ( s - B ) } { 2 } ) } { \\binom { k } { 2 } } m i _ k ( n - \\delta - k ) + \\frac { \\binom { \\delta - s } { 2 } } { \\binom { k } { 2 } } m i _ k ( n - \\delta - B ) . \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} \\begin{aligned} k ^ { y } _ { j + 1 } ( T ) & \\leq k ^ { y } _ { 0 } ( T ) + C C _ { 2 } ( T ) B \\big ( 1 - 3 \\big ( \\tfrac { 1 } { p } - \\tfrac { 1 } { q } \\big ) , 1 - \\tfrac { 3 } { q } \\big ) \\Big [ k ^ { u } _ { j } ( T ) k ^ { \\nabla y } _ { j } ( T ) + k ^ { \\nabla y } _ { j } ( T ) ^ 2 ( k ^ { x } _ { j } ( T ) + k ^ { y } _ { j } ( T ) + \\lvert \\overline { b } \\rvert ) \\Big ] . \\end{aligned} \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{align*} p ( x ) = \\frac { a ( x ) } { 1 + \\sum _ { i = 1 } ^ r \\| \\lambda _ i ( x ) \\| ^ 2 } , q ( x ) = \\frac { b ( x ) } { 1 + \\| \\mu ( x ) \\| ^ 2 } , \\end{align*}"}
-{"id": "7845.png", "formula": "\\begin{align*} ( e ^ { 2 \\pi \\textrm { I m } \\tau _ 1 x } ) ^ 2 | z _ 1 | ^ 2 + ( e ^ { 2 \\pi \\textrm { I m } \\tau _ 2 x } ) ^ 2 | z _ 2 | ^ 2 = 1 \\ ; . \\end{align*}"}
-{"id": "4840.png", "formula": "\\begin{align*} \\alpha = ( x _ M ^ n \\times 1 ) + \\nu \\cdot ( 1 \\times \\omega _ { N } ) \\in H ^ n ( M ; \\R ) \\oplus H ^ n ( N ; \\R ) , \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} \\left \\{ \\begin{pmatrix} \\pm 1 & a \\\\ 0 & 1 \\end{pmatrix} \\mid a \\in \\Z _ n \\right \\} , r = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} s = \\begin{pmatrix} - 1 & 0 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} [ s , x _ 1 ] [ x _ 2 , x _ 3 ] + [ s , x _ 2 ] [ x _ 1 , x _ 3 ] = - [ s , x _ 1 ] [ x _ 3 , x _ 2 ] - [ s , x _ 2 ] [ x _ 3 , x _ 1 ] = - [ s x _ 3 , x _ 1 , x _ 2 ] + [ s , x _ 1 , x _ 2 ] x _ 3 + s [ x _ 3 , x _ 1 , x _ 2 ] . \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{align*} \\psi ^ { \\ , \\prime } ( t ) \\ , = \\ , \\left [ \\ , F ( \\bar { x } + t ( x - \\bar { x } ) ) - F ( \\bar { x } ) \\ , \\right ] ^ T ( x - \\bar { x } ) \\ , = \\ , t \\ , F ^ { \\ , \\prime } ( \\bar { x } ; x - \\bar { x } ) ^ T ( \\ , x - \\bar { x } \\ , ) . \\end{align*}"}
-{"id": "2964.png", "formula": "\\begin{align*} 0 & = - 1 - [ z ^ 1 ] g ( q , z / q ) + [ z ^ 1 ] g ( q , z ) g ( q , z / q ) \\\\ & = - 1 - q ^ { - 1 } [ z ^ 1 ] g ( q , z ) + ( 1 + q ^ { - 1 } ) [ z ^ 0 ] g ( q , z ) [ z ^ 1 ] g ( q , z ) \\\\ & = - 1 + [ z ^ 1 ] g ( q , z ) \\\\ & = \\tilde { C } _ 0 ( q ) - 1 , \\end{align*}"}
-{"id": "5356.png", "formula": "\\begin{align*} \\eta ^ i = i _ z ^ * ( \\textstyle { \\frac { \\partial } { \\partial z ^ i } } \\lrcorner \\omega ) , \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{align*} | \\lbrace ( s , t ) \\in B _ \\nu \\mid s \\neq m \\rbrace | = | B _ \\nu | - | T _ \\nu ( m + 1 ) | . \\end{align*}"}
-{"id": "5463.png", "formula": "\\begin{align*} \\begin{pmatrix} \\dot { \\mathbf { x } } \\\\ \\dot { \\mathbf { \\phi } } \\end{pmatrix} = \\begin{bmatrix} \\mathbf { A } \\mathbf { x } + \\mathbf { G } _ { n l i n } ( \\mathbf { x } ) + \\varepsilon \\mathbf { G } _ { e x t } ( \\phi ) \\\\ \\Omega \\end{bmatrix} . \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} Y _ \\alpha = ( Y _ \\alpha ( 1 ) , \\dots , Y _ \\alpha ( l ( Y _ \\alpha ) ) ) = ( \\underbrace { F _ \\alpha ( 1 ) , \\dots , F _ \\alpha ( 1 ) } _ { G _ \\alpha ( 1 ) } , \\dots , \\underbrace { F _ \\alpha ( m _ \\alpha ) , \\dots , F _ \\alpha ( m _ \\alpha ) } _ { G _ \\alpha ( m _ \\alpha ) } ) \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} N [ \\mathfrak { e } , V ] \\ \\geq \\ \\mathcal { L } _ { V } [ c ] \\ = \\ \\big \\{ x \\in \\Gamma \\ ; \\big | \\ V ( x ) \\geq c \\big \\} . \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} \\mathcal { R } ^ { \\varphi _ { _ U } } ( \\Omega U ) = \\mathcal { C } \\left ( \\frac { 1 } { 2 } \\Omega \\right ) U , \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{align*} \\gamma = \\gamma ( \\cdot , t ) : S ^ 1 \\times I \\rightarrow \\mathbb { R } ^ 2 \\end{align*}"}
-{"id": "5074.png", "formula": "\\begin{align*} J = J ( \\{ a _ n \\} , \\{ b _ n \\} ) = \\begin{bmatrix} b _ 1 & a _ 1 & 0 & \\ldots \\\\ a _ 1 & b _ 2 & a _ 2 & \\ddots \\\\ 0 & a _ 2 & b _ 3 & \\ddots \\\\ \\vdots & \\ddots & \\ddots & \\ddots \\end{bmatrix} , \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} \\langle \\delta _ { x } | \\ , B ( \\rho , \\mathfrak { e } , V ^ { \\prime } ) \\delta _ { x } \\rangle \\ = \\ V ^ { \\prime } ( x ) \\bigg ( \\int _ { \\Gamma ^ { \\ast } } \\frac { \\mathrm { d } \\mu ^ { \\ast } ( p ) } { \\rho + \\mathfrak { e } ( p ) } \\bigg ) . \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} G ( x , y ) & = | x - y | ^ { - ( n - 2 ) } - H ( x , y ) \\mbox { i n } \\ , \\Omega \\\\ \\Delta H ( x , \\ , . \\ , ) & = 0 \\ , \\ , \\ , \\mbox { i n } \\ , \\Omega \\\\ G ( x , \\ , . \\ , ) & = 0 \\ , \\ , \\ , \\mbox { o n } \\ , \\partial \\Omega . \\end{aligned} \\right . \\end{align*}"}
-{"id": "8675.png", "formula": "\\begin{align*} ( a , k ) : = \\{ ( a - i n , j ) \\colon n \\in \\N _ 0 , \\ , 0 \\leq j \\leq k \\} , \\ \\ a : = ( a , 0 ) . \\end{align*}"}
-{"id": "5218.png", "formula": "\\begin{align*} \\begin{cases} u _ t = \\Delta u + b _ \\epsilon ( x , t ) \\cdot \\nabla u + u ( 2 a _ 0 - c ( x , t ) u ) , x \\in D _ L \\cr u = 0 , x \\in \\partial D _ L , \\end{cases} \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} | S | = \\sum _ { z \\in \\widetilde { P } } | U _ n ( z ) | = | \\widetilde { P } | \\times q ^ { - n } | U _ n | . \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} | I _ 1 ( x _ 1 , x _ 2 ) | \\leq C | x _ 1 - x _ 2 | = C \\delta . \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{align*} P _ { v , i } = \\frac { I + ( - 1 ) ^ i C ( \\widetilde x _ v ) } { 2 } . \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} I _ 3 ( x _ 1 , x _ 2 ) = I _ { 3 , + } ( x _ 1 , x _ 2 ) + I _ { 3 , - } ( x _ 1 , x _ 2 ) , \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} e _ i > 0 , \\ , \\forall \\ , \\ , i = 1 , \\dots , \\ , p , \\ , \\ , \\mathrm { o r \\ , \\ , } e _ i < 0 , \\ , \\forall \\ , \\ , i = 1 , \\dots , \\ , p . \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} T ' : = P \\cup \\bigcup _ { w \\in M ' } w ( T _ a ) \\cup \\bigcup _ { w \\in M ' } w ( T _ b ) . \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{align*} \\arg c _ k = \\frac { \\pi } { 2 } + o ( 1 ) \\quad \\arg c _ k = - \\frac { \\pi } { 2 } + o ( 1 ) . \\end{align*}"}
-{"id": "8313.png", "formula": "\\begin{align*} \\begin{cases} & z _ { t t } + i = - \\nabla P , \\\\ & \\bar { z } _ t = \\mathfrak { H } \\bar { z } _ t . \\end{cases} \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} \\widehat { P _ t f } ( \\xi ) = e ^ { - t ( 2 \\pi | \\xi | ) ^ 2 } \\hat f ( \\xi ) . \\end{align*}"}
-{"id": "2939.png", "formula": "\\begin{align*} \\log f ( q , z ) = h ( q , z ) . \\end{align*}"}
-{"id": "841.png", "formula": "\\begin{align*} | g _ { n } - g | _ { L ^ { r } ( \\R ^ { k } ) } ^ { r } = | g _ { n } | _ { L ^ { r } ( \\R ^ { k } ) } ^ { r } - | g | _ { L ^ { r } ( \\R ^ { k } ) } ^ { r } + o _ { n } ( 1 ) . \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} \\begin{array} { l l } U _ + = \\left \\{ S _ 1 - S _ 2 > 0 \\right \\} \\times \\left ( - \\dfrac { \\pi } { 4 } , \\dfrac { \\pi } { 4 } \\right ) , & V _ + = \\left \\{ x - y > 0 \\right \\} \\times \\mathbb { R } , \\\\ & \\\\ U _ - = \\left \\{ S _ 1 - S _ 2 < 0 \\right \\} \\times \\left ( - \\dfrac { \\pi } { 4 } , \\dfrac { \\pi } { 4 } \\right ) , & V _ - = \\left \\{ x - y < 0 \\right \\} \\times \\mathbb { R } . \\end{array} \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{align*} \\frac { 4 \\pi } { k } \\mathbf { q } _ { i j } = \\frac { ( \\partial \\mathbf { r } / \\partial s ) _ { i } \\times ( \\partial ^ { 2 } \\mathbf { r } / \\partial s ^ { 2 } ) _ { i } } { | ( \\partial \\mathbf { r } / \\partial s ) _ { i } | ^ { 3 } } \\log ( \\frac { 1 } { \\epsilon } ) + O ( 1 ) \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} \\log ( \\varphi _ \\lambda ( z ( \\lambda , \\xi ) ) ) - \\log ( \\varphi _ \\lambda ( \\omega _ 1 / 2 ) ) = - \\int _ { 1 } ^ { \\xi } \\frac { \\varphi _ \\lambda ' / \\varphi _ \\lambda ( z ( \\lambda , \\hat { X } ) ) d \\hat { X } } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} \\underset { n \\rightarrow \\infty } { \\lim } \\frac { \\Gamma \\left ( \\alpha n \\right ) \\left ( \\alpha n \\right ) ^ { \\alpha } } { \\Gamma \\left ( \\alpha n + \\alpha \\right ) } = 1 . \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} M _ { i j } : = \\begin{bmatrix} \\mathcal B _ { i j } & - ( C _ r ) _ { i i } ( D _ r ) _ { j j } e _ { r } e _ 1 ^ \\top \\\\ - ( C _ 1 ) _ { j j } ( D _ 1 ) _ { i i } e _ r e _ 1 ^ \\top & \\mathcal B _ { j i } \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{align*} ( q _ 1 - q _ 2 ) T ( q _ 1 - q _ 2 ) = ( 1 - q _ 2 ) T ( 1 - q _ 2 ) q _ 1 , \\end{align*}"}
-{"id": "7500.png", "formula": "\\begin{align*} & E [ S ^ { e n v , m } _ { s , t } ] = E \\left [ S ^ { e n v , 0 } _ { s , t } \\right ] + \\frac { n } { 2 } E \\left [ \\ln ( \\beta ( t , q _ t ) / \\beta ( s , q _ s ) ) \\right ] \\\\ & + \\int _ s ^ t E \\left [ \\left ( \\beta ^ { - 3 } \\nabla _ q \\beta \\cdot \\left ( \\frac { 3 n + 2 } { 6 } - \\int _ 0 ^ \\infty T r [ \\gamma e ^ { - 2 y \\gamma } ] e ^ { - y \\gamma } d y \\right ) \\gamma ^ { - 1 } \\nabla _ q \\beta \\right ) ( r , q _ r ) \\right ] d r \\\\ & + O ( m ^ \\delta ) \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} \\sup _ { 0 \\le \\tau \\le 1 - t } \\Bigl \\{ \\ , \\frac \\tau \\rho + \\bigl ( \\sqrt { t } + \\sqrt { 1 - t - \\tau } \\ , \\bigr ) ^ 2 \\Bigr \\} = \\frac { t } { 1 - \\rho } + \\frac { 1 - t } { \\rho } . \\end{align*}"}
-{"id": "9207.png", "formula": "\\begin{align*} V _ \\mathcal { R } ( I ) = \\left \\{ { \\bf a } \\in \\mathcal { R } ^ n : f ( { \\bf a } ) = 0 f \\in I \\right \\} . \\end{align*}"}
-{"id": "6990.png", "formula": "\\begin{align*} N ^ - ( w ) \\cap \\Phi _ h ^ - [ T ] = N ^ - ( \\tau ) \\cap \\Phi _ h ^ - [ T ] . \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{align*} \\iota _ { X _ - } \\omega _ + + \\iota _ { X _ 0 } \\omega _ 0 + \\iota _ { X _ + } \\omega _ - = d \\mu \\end{align*}"}
-{"id": "9106.png", "formula": "\\begin{align*} \\omega ^ { \\star } = ( { \\bar { b } + \\bar { a } } ) / { 2 } , \\end{align*}"}
-{"id": "6660.png", "formula": "\\begin{align*} \\mathbb { E } [ T _ R ] \\leq \\sum _ { k = 1 } ^ { \\infty } p ( T ) ^ { \\left \\lfloor \\frac { k - 1 } { T } \\right \\rfloor } \\leq \\sum _ { k = 1 } ^ { \\infty } p ( T ) ^ { \\frac { k - 1 } { T } - 1 } = \\frac { 1 } { ( 1 - p ( T ) ^ { \\frac { 1 } { T } } ) p ( T ) } \\ , . \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{align*} W _ { 2 , 4 / 3 } ( x , y ) = x ^ 2 + \\frac { 1 } { 3 } y ^ 2 \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } _ { p } v _ { n } + \\frac { V _ { 0 } } { 2 } v _ { n } ^ { p - 1 } = f ( v _ { n } ) - \\left ( V _ { n } - \\frac { V _ { 0 } } { 2 } \\right ) v _ { n } ^ { p - 1 } \\leq 0 \\mbox { i n } \\mathcal { B } ^ { c } _ { R } ( 0 ) . \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} \\big | e ^ { 2 i \\theta k } + ( - 1 ) ^ k \\chi _ 2 ( a ) \\overline { \\chi _ 2 ( a + b ) } \\big | = 0 . \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } \\lambda ^ n \\widetilde { f } ( \\lambda ) = \\lim _ { \\lambda \\to \\infty } \\lambda \\lambda ^ { n - 1 } \\widetilde { f } ( \\lambda ) = f ^ { ( n - 1 ) } ( 0 ) = 0 . \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{align*} \\mathcal { F } ( \\hat P ( \\mathcal { X } ^ { ( i ) } ) ; \\hat P _ { t } ) \\to \\left \\{ \\begin{aligned} \\mathcal { F } ( \\hat P _ { t _ i } ; \\hat P _ t ) , & \\mathcal { X } ^ { ( i ) } \\in \\mathcal { M } _ t \\\\ \\mathcal { F } ( \\hat Q _ { f _ i } ; \\hat P _ t ) , & \\mathcal { X } ^ { ( i ) } \\in \\mathcal { M } _ f \\end{aligned} \\right . , \\end{align*}"}
-{"id": "1553.png", "formula": "\\begin{align*} \\int _ { ( - \\infty ; 0 ] } | x | d m ^ { N , j } _ t ( x ) & = \\int _ { ( - \\infty ; \\tau ^ { - 1 } _ { n , N } ( t ) ] } | \\Phi ^ { e _ j } _ { t } ( x , t _ { n - 1 } ) | d m ^ { N , j } _ { t _ { n - 1 } } ( x ) \\\\ & \\leq \\int _ { ( - \\infty ; \\tau ^ { - 1 } _ { n , N } ( t ) ] } | x | d m ^ { N , j } _ { t _ { n - 1 } } ( x ) + m ^ N _ { t _ { n - 1 } } ( ( - \\infty ; \\tau ^ { - 1 } _ { n , N } ( t ) ] ) V _ { m a x } \\Delta t ^ N \\\\ & \\leq \\int _ { ( - \\infty ; 0 ] } | x | d m ^ { N , j } _ { t _ { n - 1 } } + m ^ { N , j } _ { t _ { n - 1 } } ( e _ j ) \\Delta t ^ N V _ { m a x } . \\end{align*}"}
-{"id": "1066.png", "formula": "\\begin{align*} \\alpha \\cdot e = \\mathcal L _ { \\partial _ \\alpha } ( e ) \\ , , \\end{align*}"}
-{"id": "8893.png", "formula": "\\begin{align*} n _ Y = 1 - \\sum _ { \\alpha \\in \\Phi _ { Q ^ U } \\cup \\Phi _ s ^ + } \\alpha \\circ \\mathcal { P } ( \\mu _ Y ) . \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} q ( x | z ) & = \\prod _ { j = 1 } ^ N \\left ( \\sum _ { k = 1 } ^ { H _ 0 } b ^ 0 _ { k j } \\prod _ { i = 1 } ^ M ( a ^ 0 _ { i k } ) ^ { x _ { i } } \\right ) ^ { z _ j } , \\\\ p ( x | z , A , B ) & = \\prod _ { j = 1 } ^ N \\left ( \\sum _ { k = 1 } ^ { H } b _ { k j } \\prod _ { i = 1 } ^ M ( a _ { i k } ) ^ { x _ { i } } \\right ) ^ { z _ j } . \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} g _ n & = ( 3 - n ) 2 ^ { n - 3 } - 1 + \\sum _ { a = 1 } ^ { n - 2 } 2 ^ { a - 1 } e _ { n - 1 - a } + \\sum _ { a = 2 } ^ { n - 1 } \\left ( 2 ^ { n - 1 - a } - 1 - \\binom { n - a } { 2 } \\right ) ( a - 1 ) 2 ^ { a - 2 } \\\\ & \\quad + \\sum _ { i = 0 } ^ { n - 2 } \\binom { n - 1 + i } { 2 i + 1 } + \\sum _ { a = 2 } ^ { n - 3 } \\sum _ { \\ell = 1 } ^ { n - 2 - a } \\sum _ { i = 0 } ^ { a - 1 } 2 ^ { n - 2 - a - \\ell } \\binom { a - 1 } { i } \\binom { \\ell + i } { i } , n \\geq 3 . \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} \\varphi ( q ) \\mu _ { 1 X } ( p _ 1 ) + h ( p _ 1 ) \\mu _ { 2 1 } ( q ) = \\rho _ { 1 X } ( p _ 1 ) + \\rho _ { 2 1 } ( q ) , \\forall q \\in G , \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} A X B - C X D = E , \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} S _ { } ( X ; Y \\| Z ) = \\max _ { p _ { U X } } I ( U ; Y ) - I ( U ; Z ) . \\end{align*}"}
-{"id": "5914.png", "formula": "\\begin{align*} \\psi \\left ( \\nu , \\mu \\right ) = \\prod _ { x \\in \\vec { x } ( \\mu ) } \\prod _ { i < x } \\left ( t ^ { \\nu _ i } \\right ) \\nu _ x \\cdot \\prod _ { y \\in \\vec { y } ( \\mu ) } \\prod _ { i < y } \\left ( t ^ { \\nu _ i } \\right ) \\nu _ y \\cdot t ^ { - \\chi ( \\vec { x } , \\vec { y } ) } \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} J = \\sum _ { n , m \\geq 0 } \\sum _ { \\substack { \\mathfrak { a } \\in \\mathfrak { g } ^ { \\otimes n } \\\\ \\mathfrak { b } \\in \\mathfrak { g } ^ { \\otimes m } } } \\sum _ { a , b \\in \\mathfrak { g } } ( \\mathfrak { a } \\otimes ( a \\otimes b - \\varepsilon ( a , b ) b \\otimes a ) \\otimes \\mathfrak { b } - \\alpha _ { T } ( \\mathfrak { a } ) \\otimes [ a , b ] _ { \\mathfrak { g } } \\otimes \\alpha _ { T } ( \\mathfrak { b } ) ) . \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} \\tilde S ^ i ( t , q ) = & - \\frac { 1 } { 2 } ( \\gamma ^ { - 1 } ) ^ { i l } ( t , q ) \\partial _ { q ^ k } \\sigma _ { l j } ( t , q ) ( \\gamma ^ { - 1 } ( t , q ) \\sigma ( t , q ) ) ^ k _ j . \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} H ^ { m } ( s ) = \\begin{cases} \\frac { 1 } { p + 2 } s ^ { p + 2 } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ s \\leq m , \\\\ \\\\ \\frac { 1 } { 2 } m ^ p s ^ 2 + \\left ( \\frac { 1 } { p + 2 } - \\frac { 1 } { 2 } \\right ) m ^ { p + 2 } \\ \\ \\ \\ \\ s \\geq m . \\end{cases} \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{align*} p _ j M _ j e ^ { - p _ j t _ j } \\left ( 1 - e ^ { - p _ j t _ j } \\right ) ^ { M _ j - 1 } = - \\sum _ { k _ j = 1 } ^ { M _ j } ( - 1 ) ^ { k _ j } \\binom { M _ j } { k _ j } p _ j k _ j e ^ { - k _ j p _ j t _ j } , \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} E _ \\lambda = \\sum _ \\mu E _ \\mu ^ * E _ { \\lambda } \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} r + x \\ , \\frac { f ' ( x ) } { f ( x ) } = - \\tfrac { 1 } { x } \\ , \\frac { g ' \\left ( \\tfrac { 1 } { x } \\right ) } { g \\left ( \\tfrac { 1 } { x } \\right ) } . \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{align*} \\limsup _ { t \\rightarrow T } \\Big ( ( T - t ) \\max _ { \\Gamma _ { t } } \\kappa ^ { 2 } \\Big ) = \\infty \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} \\lambda = \\frac { p _ 2 } { p _ 1 } > 1 . \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} g ' ( \\overline { x } ) \\mathbb { X } + { \\rm l i n } \\big [ \\mathcal { T } _ { K } ( g ( \\overline { x } ) ) \\big ] = \\mathbb { Y } . \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} { \\rm M } ( \\alpha ) \\geq \\theta _ { 2 } ^ { - d / 2 } = \\left ( \\frac { 1 + \\sqrt { 5 } } { 2 } \\right ) ^ { d / 2 } . \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} p = \\sum _ { r \\in X ^ m } \\alpha _ r ( r _ 1 - \\gamma ( r _ 1 ) ) \\cdots ( r _ l - \\gamma ( r _ l ) ) + \\sum _ { r \\in X ^ m } \\alpha _ r r ^ \\prime + p ^ \\prime . \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{gather*} F ( g ( u ) ) = u _ 1 ^ { 2 k _ 1 } \\ldots u _ d ^ { 2 k _ d } , \\\\ | g ' ( u ) | = b ( u ) | u _ 1 ^ { h _ 1 } \\ldots u _ d ^ { h _ d } | , \\end{gather*}"}
-{"id": "4979.png", "formula": "\\begin{align*} < v , \\psi _ { c , \\gamma } > _ { L ^ 2 } = < v , \\psi _ { c , \\gamma } ^ { \\prime } > _ { L ^ 2 } = 0 . \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} \\mu _ { 1 X } ( p _ 1 ) = c h ( p _ 1 ) + \\tilde { c } , \\rho _ { 1 X } ( p _ 1 ) = - b h ( p _ 1 ) + \\tilde { b } . \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{align*} \\sigma = \\sigma _ c + d \\zeta \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} i = i ( \\lambda ) = \\max \\{ j \\in \\mathbb { N } | ~ 1 \\leq j \\leq t , \\lambda _ { j } + \\cdots + \\lambda _ { t } \\geq d \\} . \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{align*} - d \\hat { Y } _ s ^ { t , x ; u _ { \\cdot } } = \\hat { G } \\bigl ( s , X _ s ^ { t , x ; u _ { \\cdot } } , \\hat { Y } _ s ^ { t , x ; u _ { \\cdot } } , Z _ s ^ { t , x ; u _ { \\cdot } } \\bigr ) d s - Z _ s ^ { t , x ; u _ { \\cdot } } d B _ s , \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} L : = P _ 1 \\circ K _ 1 \\circ P _ 2 \\circ K _ 2 \\circ \\cdots \\circ K _ { h - 1 } \\circ P _ h \\end{align*}"}
-{"id": "49.png", "formula": "\\begin{align*} \\partial _ x \\Phi _ { \\alpha _ 0 } ( s , x ) & = \\partial _ x \\Phi _ { u , \\gamma _ 0 } ( s , x , 0 ) . \\end{align*}"}
-{"id": "1349.png", "formula": "\\begin{align*} D ' _ { m k } = \\begin{bmatrix} 0 & 1 & 2 & \\ldots & k - 1 \\\\ 0 & 1 & 2 & \\ldots & k - 1 \\\\ \\hdotsfor { 5 } \\\\ 0 & 1 & 2 & \\ldots & k - 1 \\end{bmatrix} , C '' _ { k k } = \\begin{bmatrix} 0 & 0 & 1 & 2 & \\ldots & k - 2 \\\\ 0 & 0 & 0 & 1 & \\ldots & k - 3 \\\\ \\hdotsfor { 6 } \\\\ 0 & 0 & 0 & 0 & \\ldots & 1 \\\\ 0 & 0 & 0 & 0 & \\ldots & 0 \\\\ 0 & 0 & 0 & 0 & \\ldots & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} \\max p _ { - 2 } ( n ) = \\begin{cases} 5 ^ { \\frac { n } { 2 } } , \\ & \\emph { i f } ~ n \\equiv 0 \\pmod { 2 } , \\cr 2 \\cdot 5 ^ { \\frac { n - 1 } { 2 } } , \\ & \\emph { i f } ~ n \\equiv 1 \\pmod { 2 } . \\end{cases} \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} \\psi _ j ( n _ j ; \\lambda _ j ) = p _ j ^ { n _ j } n _ j ! \\prod _ { k = 1 } ^ { n _ j } \\frac { 1 } { \\lambda _ j + p _ j k } , 1 \\leq n _ j \\leq M _ j , \\end{align*}"}
-{"id": "7148.png", "formula": "\\begin{align*} \\left ( a , b ^ { 2 } , c \\right ) = 2 b \\left ( a , b , c \\right ) , \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{align*} \\lVert A B - A _ 0 B _ 0 \\rVert ^ 2 & \\sim \\sum _ { i = 1 } ^ { M - 1 } \\left \\{ x _ i ^ 2 + \\sum _ { j = 2 } ^ { N } a _ i ^ 2 ( b _ j - b _ 1 ) ^ 2 \\right \\} \\\\ & = \\sum _ { i = 1 } ^ { M - 1 } \\left ( x _ i ^ 2 + \\sum _ { j = 2 } ^ { N } a _ i ^ 2 b _ j ^ 2 \\right ) \\\\ & = \\sum _ { i = 1 } ^ { M - 1 } x _ i ^ 2 + \\sum _ { i = 1 } ^ { M - 1 } \\sum _ { j = 2 } ^ N a _ i ^ 2 b _ j ^ 2 \\\\ & = \\sum _ { i = 1 } ^ { M - 1 } x _ i ^ 2 + \\left ( \\sum _ { i = 1 } ^ { M - 1 } a _ i ^ 2 \\right ) \\left ( \\sum _ { j = 2 } ^ N b _ j ^ 2 \\right ) . \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{gather*} \\lambda \\sum _ { j = 1 } ^ m \\langle X _ j ( x ) , \\xi \\rangle ^ 2 \\leq \\langle B ( x ) \\xi , \\xi \\rangle \\leq \\Lambda \\sum _ { j = 1 } ^ m \\langle X _ j ( x ) , \\xi \\rangle ^ 2 \\quad x \\in \\Omega \\xi \\in \\mathbb { R } ^ N \\\\ \\langle b ( x ) , \\xi \\rangle ^ 2 \\leq \\gamma ^ 2 ( x ) \\sum _ { j = 1 } ^ m \\langle X _ j ( x ) , \\xi \\rangle ^ 2 \\quad x \\in \\Omega \\xi \\in \\mathbb { R } ^ N \\end{gather*}"}
-{"id": "3104.png", "formula": "\\begin{align*} \\max _ { n \\leq x } \\log d ( n ) = \\frac { \\log x } { \\log \\log x } ( \\log 2 + o ( 1 ) ) . \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} \\lefteqn { \\int _ { \\{ h > 0 \\} } \\ , \\int _ 0 ^ { h ( x ) } f ( x , z ) \\ , d z \\ , d x } \\\\ & \\overset { \\eqref { m s } } = \\int _ { \\{ h > 0 \\} } \\ , \\int _ 0 ^ { h ( x ) } \\big ( f ( x , z ) - f ( x , h ( x ) ) \\big ) \\ , d z \\ , d x + \\int _ { \\{ h > 0 \\} } h \\ , \\kappa \\ , d x , \\end{align*}"}
-{"id": "8750.png", "formula": "\\begin{align*} w = \\sum _ { i = 1 } ^ { 3 } \\ell _ { i } W _ { i } + \\sum _ { i = 4 } ^ { 6 } \\omega _ { i - 3 } W _ { i } , \\psi = \\sum _ { i = 1 } ^ { 3 } \\ell _ { i } \\Psi _ { i } + \\sum _ { i = 4 } ^ { 6 } \\omega _ { i - 3 } \\Psi _ { i } , \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} \\Phi _ t ( z , \\zeta ) = ( \\varphi _ t ( z ) , ( d \\varphi _ t ( z ) ^ { - 1 } ) ^ T . \\zeta ) , \\zeta \\in T ^ * _ z ( S ^ * M ) . \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} x _ { \\nu t } ( t ) \\leq x ^ C _ { t / 4 } ( t ) \\stackrel { ( d ) } { = } - 2 \\nu t + t / 2 + x ^ { \\rm s t e p } _ { t / 4 } ( t ) . \\end{align*}"}
-{"id": "7019.png", "formula": "\\begin{align*} p _ { - k } ( a ) & = \\sum _ { \\alpha _ { 1 } + \\alpha _ { 2 } = a } p _ { - 2 } ( \\alpha _ { 1 } ) p _ { - ( k - 2 ) } ( \\alpha _ { 2 } ) , \\\\ p _ { - k } ( b ) & = \\sum _ { \\beta _ { 1 } + \\beta _ { 2 } = b } p _ { - 2 } ( \\beta _ { 1 } ) p _ { - ( k - 2 ) } ( \\beta _ { 2 } ) , \\\\ p _ { - k } ( a + b ) & = \\sum _ { \\gamma _ { 1 } + \\gamma _ { 2 } = a + b } p _ { - 2 } ( \\gamma _ { 1 } ) p _ { - ( k - 2 ) } ( \\gamma _ { 2 } ) , \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} \\mathrm { K L } ( A , B ) & = \\sum _ { j = 1 } ^ N q ' ( z _ i = 1 ) \\sum _ { i = 1 } ^ M ( A _ 0 B _ 0 ) _ { i j } \\log \\frac { ( A _ 0 B _ 0 ) _ { i j } } { ( A B ) _ { i j } } . \\end{align*}"}
-{"id": "7057.png", "formula": "\\begin{align*} f _ m ^ n ( ( p _ j ) _ { 1 \\le j \\le m + 1 } , ( u _ j ) _ { 1 \\le j \\le m + 1 } , ( q _ j ) _ { 1 \\le j \\le n - m } , ( v _ j ) _ { 1 \\le j \\le n - m } ) : \\prod _ { j = 1 } ^ { m + 1 } I ^ { u _ j } \\times \\prod _ { k = 1 } ^ { n - m } I ^ { v _ k } \\to \\C \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} \\begin{aligned} E \\cap W & = \\bigcup _ { 0 \\leq i \\leq k } \\{ ( J , J ) \\in W \\mid p ^ { - 1 } ( J , J ) \\cap A _ i \\not = \\emptyset \\\\ & \\qquad \\qquad \\qquad p ^ { - 1 } ( J , J ) \\cap A _ i \\subset S \\} \\\\ & = \\bigcup _ { 0 \\leq i \\leq k } \\Delta \\cap ( W \\setminus p ( A _ i \\setminus S ) ) \\end{aligned} \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{align*} K _ 1 ( N ) : = \\int _ 0 ^ { A _ N } i \\xi \\ , e ^ { i \\xi s } \\left [ 1 - \\left ( 1 - e ^ { - s } \\right ) ^ N \\right ] d s \\end{align*}"}
-{"id": "8457.png", "formula": "\\begin{align*} x = \\sum _ { J = 1 } ^ { r } x _ j e _ j , \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} | I \\cap ( J - i ) | \\preceq 1 + | I \\cap J | = 1 + Y _ 0 . \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} X _ 1 u = a _ 1 , \\ X _ 2 u = a _ 2 \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} \\mathrm { P } ( \\Phi ^ 1 , \\Phi ^ 2 ) : = \\big ( ( \\widetilde { A } _ 1 , \\widetilde { b } _ 1 ) , \\dots , ( \\widetilde { A } _ { L } , \\widetilde { b } _ { L } ) \\big ) , \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} \\mu ^ { \\circ } & = \\mathcal { C } _ { - 1 } ( \\alpha , \\nu , 1 , \\varnothing , \\lbrace 2 , \\ldots , \\ell \\rbrace ) - \\ell + 1 \\\\ & = \\left \\lceil \\frac { \\nu _ 1 + \\sum _ { i = 2 } ^ { \\ell } \\min \\lbrace \\alpha _ 1 , \\alpha _ i \\rbrace } { \\alpha _ 1 } \\right \\rceil - \\ell + 1 \\\\ & = \\left \\lceil \\frac { \\nu _ 1 + \\alpha ^ * _ 1 + \\cdots + \\alpha ^ * _ { \\alpha _ 1 } } { \\alpha _ 1 } \\right \\rceil - \\ell . \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} \\tilde \\gamma ^ i _ j = \\delta ^ { i k } \\tilde \\gamma _ { k j } = & \\left ( \\begin{array} { c c c } \\gamma & B _ 0 & 0 \\\\ - B _ 0 & \\gamma & 0 \\\\ 0 & 0 & \\gamma \\end{array} \\right ) \\end{align*}"}
-{"id": "3427.png", "formula": "\\begin{align*} e ^ { - \\frac { | x | ^ 2 } { 2 ( k + 1 ) } } \\sum _ { i = 1 } ^ { k + 1 } ( d x ^ i ) ^ 2 . \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} 2 \\varepsilon = \\sum _ { i = 1 } ^ N ( t _ y ^ { ( i ) } \\cdot 1 ) \\leq \\left ( \\sum _ { i = 1 } ^ N ( t _ y ^ { ( i ) } ) ^ q \\right ) ^ { 1 / q } \\cdot N ^ { 1 - \\frac { 1 } { q } } \\leq \\left ( \\sum _ { i = 1 } ^ N ( t _ y ^ { ( i ) } ) ^ { 1 + 2 p } \\right ) ^ { 1 / q } \\cdot P ^ { 1 - \\frac { 1 } { q } } , \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} x + \\lambda _ { i + 1 } + \\cdots + \\lambda _ { t } = d \\textrm { a n d } y + \\lambda _ { 1 } + \\cdots + \\lambda _ { i - 1 } = c . \\end{align*}"}
-{"id": "1502.png", "formula": "\\begin{align*} \\lambda _ 1 = \\inf _ { \\varphi \\in C _ 0 ^ { \\infty } ( \\Sigma ) , \\int \\varphi ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma \\neq 0 } \\dfrac { \\int | \\nabla \\varphi | ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma } { \\int \\varphi ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma } \\geq \\dfrac { n } { 2 } + \\inf _ { x \\in \\Sigma } | { \\bf H } | | ^ 2 \\geq \\frac n 2 . \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} \\| j _ 0 ^ m s \\| _ r = \\| j _ 0 ^ m s \\| _ R \\left ( \\frac { r } { R } \\right ) ^ m \\end{align*}"}
-{"id": "3518.png", "formula": "\\begin{align*} & I ( t ) : = \\int _ { - \\infty } ^ { \\infty } u ( x , t ) d x < \\infty , \\\\ & \\lim _ { x \\rightarrow \\pm \\infty } u _ { x } ( x , t ) = 0 \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} H ( S ^ d ) = \\Delta ^ { - d } d \\big ( \\Lambda ( S ^ d ) \\big ) , \\end{align*}"}
-{"id": "110.png", "formula": "\\begin{align*} K ( q ) & = \\frac 1 2 \\Im \\sum _ j w _ j \\bar z _ j = \\frac i 4 \\sum _ j ( z _ j \\bar w _ j - \\bar z _ j w _ j ) \\\\ & = \\frac i 4 \\sum _ j \\int _ { \\alpha _ j } \\tau _ q \\int _ { \\beta _ j } \\bar \\tau _ q - \\int _ { \\alpha _ j } \\bar \\tau _ q \\int _ { \\beta _ j } \\tau _ q \\\\ & = \\frac i 4 \\int _ { S _ q } \\tau _ q \\wedge \\bar \\tau _ q = \\frac 1 4 \\int _ { S _ q } | \\tau _ q | ^ 2 \\ , d A = \\frac 1 2 \\int _ X | q | \\ , d A . \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} ( 2 ^ n - 1 ) \\cdot ( \\frac { 1 } { 2 ^ n } ) ^ { s _ n } = 1 . \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} e _ H ( X ) : = \\left | E ( H ) \\cap \\binom { X } { 3 } \\right | \\geq d \\binom { | X | } { 3 } - \\rho n ^ 3 \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{align*} Z : \\ \\gamma = \\sum _ a q _ a \\gamma ^ a \\ \\longmapsto \\ Z _ { \\gamma } = \\sum _ a q _ a Z _ { \\gamma ^ a } \\rlap { . } \\end{align*}"}
-{"id": "995.png", "formula": "\\begin{align*} C = F , S \\cap F \\neq \\emptyset S . \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} & W ^ 0 _ { k + 2 } ( p , p _ 1 , \\dots , p _ { k + 1 } ) \\ast W ^ 0 _ { l + 2 } ( p ' , p ' _ 1 , \\dots , p ' _ { l + 1 } ) \\\\ & = W ^ 0 _ { m + 2 } ( p , p _ 1 , \\dots , p _ { m + 1 } ) = ( \\mathbf { 1 } ) \\ast ( \\mathbf { 1 } ) \\ast \\dots \\ast ( \\mathbf { 1 } ) , m = k + l \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{align*} \\tau _ s ( m _ 1 , m _ 2 , \\dots , m _ { s } ) : = \\left \\{ \\begin{array} { l l } ( m _ 1 + 1 , m _ 2 + 1 , \\dots , m _ { s } + 1 ) & \\hbox { w h e n } m _ { s } < 2 k + 1 \\ ; ; \\\\ ( 1 , m _ 1 + 1 , m _ 2 + 1 , \\dots , m _ { s - 1 } + 1 ) & \\hbox { w h e n } m _ { s } = 2 k + 1 \\ ; . \\end{array} \\right . \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} M = \\begin{bmatrix} \\alpha _ 1 & \\beta _ 1 \\\\ & \\ddots & \\ddots \\\\ & & \\ddots & \\beta _ { p - 1 } \\\\ \\beta _ p & & & \\alpha _ p \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} \\langle R ^ Q ( \\bar V _ a , V _ a ) \\xi , \\eta \\rangle = \\langle \\xi , R ^ Q ( \\bar V _ a , V _ a ) \\eta \\rangle . \\end{align*}"}
-{"id": "3269.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } | ( Q _ { n , \\textup { \\textbf { m } } } ) ^ { ( k ) } ( \\lambda _ j ) | \\leq \\frac { | \\Phi ( \\lambda _ j ) | } { \\rho _ { | \\textup { \\textbf { m } } | } ( \\textup { \\textbf { F } } ) } , k = 0 , 1 , \\ldots , r - 1 . \\end{align*}"}
-{"id": "268.png", "formula": "\\begin{align*} P ( T ) = P \\left ( \\frac { 1 } { q T } \\right ) q ^ g T ^ { 2 g } ( g = n / 2 + 1 - d ) . \\end{align*}"}
-{"id": "3486.png", "formula": "\\begin{align*} \\sum _ { i \\in [ m ] } \\lambda _ i & = 1 , \\\\ \\sum _ { i \\in [ m ] } \\lambda _ i a _ { i j } & = 0 , j \\in [ n ] . \\end{align*}"}
-{"id": "6526.png", "formula": "\\begin{align*} \\tilde \\psi _ j ( i ) = \\cos \\left ( \\pi i j / n - \\pi j / 2 n \\right ) . \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} \\widehat { \\sigma } = t _ i ^ { - 1 } \\log ^ { \\frac { 1 + \\nu } { 2 } } ( t _ i ) ( \\sigma + 4 ^ { - \\ : \\nu } c _ 2 c _ \\lambda ^ \\nu \\log ^ { - \\ : \\nu } ( t _ i ) \\delta ) \\end{align*}"}
-{"id": "198.png", "formula": "\\begin{align*} \\sigma ( x ) = \\gamma ' _ x ( R ) . \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} \\chi ^ * g = \\frac { d \\rho ^ 2 + h _ \\rho } { \\rho ^ { 2 } } , \\chi '^ * g ' = \\frac { d \\rho ^ 2 + h ' _ \\rho } { \\rho ^ { 2 } } , \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} Y _ p : = { \\rm S p a n } \\left \\{ \\sum _ { k \\in \\Z , \\ ; | k | ^ 2 \\leq p } \\widehat { V } _ k e _ k , \\ \\forall k \\in \\Z , \\ | k | \\leq p , \\ \\overline { \\widehat { V } _ { - k } } = \\widehat { V } _ k \\right \\} . \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} \\lambda = h ^ { - 1 } ( \\nabla _ B h ) \\beta + ( m + n - 1 ) a - b h ^ { - 1 } - a h \\varphi . \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{align*} e ^ { i \\theta } A z = t _ 1 e _ 1 + t _ 2 e _ 2 . \\end{align*}"}
-{"id": "7524.png", "formula": "\\begin{align*} 2 \\hat { b } _ + \\cdot \\tilde \\Sigma ^ { - 1 } b _ - = \\frac { B _ 0 \\beta ( t , q ) } { \\gamma ^ 2 ( t ) + B _ 0 ^ 2 } \\left ( \\partial _ { q ^ 1 } \\beta ^ { - 1 } ( t , q ) ( - \\partial _ { q ^ 2 } V ( t , q ) + \\tilde F _ 2 ( t , q ) ) - \\partial _ { q ^ 2 } \\beta ^ { - 1 } ( t , q ) ( - \\partial _ { q ^ 1 } V ( t , q ) + \\tilde F _ 1 ( t , q ) ) \\right ) \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} \\big ( \\left ( D \\left ( y \\right ) , z \\right ) + \\left ( y , D ( z ) \\right ) \\big ) x + \\big ( \\left ( D \\left ( x \\right ) , z \\right ) + \\left ( x , D ( z ) \\right ) \\big ) y + \\big ( \\left ( D \\left ( x \\right ) , y \\right ) + \\left ( x , D ( y ) \\right ) \\big ) z = 0 , \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} \\nu ( \\lambda ) = \\frac { 1 } { q ^ d } \\prod _ { j \\geq 1 } \\binom { M _ j ( q ) + m _ j - 1 } { m _ j } . \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{align*} W ( x , y ) = \\sum _ { i = 0 } ^ n A _ i x ^ { n - i } y ^ i \\qquad ( A _ 0 = 1 ) \\end{align*}"}
-{"id": "8925.png", "formula": "\\begin{align*} \\int _ { \\partial \\Delta ' } u _ j ^ * P _ { D H } ' ( p ) d \\sigma = 1 \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{align*} \\mathbb P \\Big ( \\sup _ { s \\in [ 0 , t ] } \\{ X _ k ( s ) \\} \\ge \\Gamma \\Big ) \\le 2 \\mathcal G \\Big ( \\frac { \\ell \\ , \\Gamma ^ { ( k ) } - t - \\sum _ { j = 1 } ^ { \\ell } X _ j ( 0 ) } { \\sqrt { \\ell t } } \\Big ) + 4 k \\mathcal G \\Big ( \\frac { \\Gamma ^ { ( k ) } } { \\sqrt t } \\Big ) \\ , . \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{align*} v ( q _ { n } ) ^ { q _ { n } } = \\left \\{ \\begin{array} [ c ] { l l } 0 & \\mbox { i f } v ( q _ { n } ) ( x ) = 0 , \\\\ e ^ { q _ { n } \\log v ( q _ { n } ) } & \\mbox { i f } v ( q _ { n } ) ( x ) > 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "8126.png", "formula": "\\begin{align*} \\kappa = N ( U , 0 + ) = N _ 1 ( U , 0 + ) \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{align*} E _ \\Omega u : = \\begin{cases} u & \\Omega \\\\ \\sum _ { Q \\in \\mathcal { W } _ 3 } \\varphi _ Q \\pi _ { Q ^ * } u & \\R ^ n \\setminus \\bar { \\Omega } , \\end{cases} \\end{align*}"}
-{"id": "8613.png", "formula": "\\begin{align*} \\sum _ { v \\in V ( G ) } \\binom { d ( v ) } { 2 } \\ge 2 \\left ( \\binom { n } { 2 } - e ( G ) \\right ) = n ^ 2 - n - \\sum _ { v \\in V ( G ) } d ( v ) . \\end{align*}"}
-{"id": "8037.png", "formula": "\\begin{align*} t \\circ \\sigma = _ f \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} | \\phi ^ i _ 1 ( x ) - \\phi ^ i _ 1 ( 0 ) | \\leq \\frac { \\sigma ( r _ i ) } { A _ i r _ i ^ { \\beta } } = \\frac { A r _ i ^ { \\alpha - \\beta } } { A _ i } \\rightarrow 0 Q _ 1 \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} \\sigma _ \\lambda ( z + \\omega ) = - \\sigma _ \\lambda \\exp ( \\eta ( z + \\frac { \\omega } 2 ) ) \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} \\min _ { \\substack { v \\in \\mathcal { E } _ h \\\\ M ( v ) = 1 } } E ^ { h } _ { L L L } ( v ) . \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} \\zeta _ { m } = \\Vert A ^ { T } A - P _ { m } A ^ { T } A \\Vert \\ , . \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} p _ { - 2 } ( a ) p _ { - 2 } ( b ) = \\sum _ { \\alpha _ { 1 } + \\alpha _ { 2 } = a } \\sum _ { \\beta _ { 1 } + \\beta _ { 2 } = b } p ( \\alpha _ { 1 } ) p ( \\alpha _ { 2 } ) p ( \\beta _ { 1 } ) p ( \\beta _ { 2 } ) . \\end{align*}"}
-{"id": "7793.png", "formula": "\\begin{align*} I _ { h } ( t , x , s ) : = \\int _ 0 ^ { \\infty } \\left \\| \\int _ y ^ { \\infty } \\Delta _ h G _ { t - s } ( x - z ) \\psi ( s , z ) d z \\right \\| _ p ^ 2 f ( y ) ^ 2 d y \\end{align*}"}
-{"id": "8171.png", "formula": "\\begin{align*} d u ( x ) ( v ) = \\langle { \\bf \\nabla } _ g u ( x ) , v \\rangle _ g \\ \\mbox { f o r a l l } \\ v \\in T _ x M ; \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} f ( x , \\lambda ) & = \\log 2 \\mathrm { c h } \\bigl ( x + 2 \\lambda m ( x , \\lambda ) \\bigr ) - \\lambda m ( x , \\lambda ) ^ 2 , \\end{align*}"}
-{"id": "7318.png", "formula": "\\begin{align*} C = \\bigcup _ { n < \\omega } C _ n , D = \\bigcup _ { n < \\omega } D _ n . \\end{align*}"}
-{"id": "5458.png", "formula": "\\begin{align*} 0 = \\sum ^ { R } _ { i = 1 } | ( v _ { R k + i } - v _ { R k } ) - v _ i | , \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} \\left [ F ( \\bar { z } ) ^ T B ^ { \\ , k } v + \\displaystyle { \\sum _ { i \\ , : \\ , \\bar { w } _ i \\neq 0 } } \\ , \\alpha _ i \\ , v _ i \\ , \\mbox { s i g n } ( \\bar { w } _ i ) + \\displaystyle { \\sum _ { i \\ , : \\ , \\bar { w } _ i = 0 } } \\ , \\alpha _ i \\ , | \\ , v _ i \\ , | \\leq 0 \\right ] \\ \\Rightarrow \\ v ^ T \\left [ \\ , ( \\ , B ^ { \\ , k } \\ , ) ^ T A ^ { \\ , j } B ^ { \\ , k } \\ , \\right ] \\ , v \\ , \\geq \\ , 0 ; \\end{align*}"}
-{"id": "6063.png", "formula": "\\begin{align*} ( \\eta ( t ) , w ( t ) ) = \\begin{cases} ( \\eta _ 1 ( t ) , w _ 1 ( t ) ) & , \\\\ ( \\eta _ 2 ( t - T _ 1 ) , w _ 2 ( t - T _ 2 ) ) & , \\end{cases} \\end{align*}"}
-{"id": "2486.png", "formula": "\\begin{align*} K _ 2 ( N ) : = \\int _ { A _ N } ^ { \\infty } i \\xi \\ , e ^ { i \\xi s } \\left [ 1 - \\left ( 1 - e ^ { - s } \\right ) ^ N \\right ] d s . \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} 2 ^ { n - 1 } = | T _ { x _ i = 1 } | + | F _ { x _ i = 0 } | . \\end{align*}"}
-{"id": "3886.png", "formula": "\\begin{align*} - \\tilde d : = - \\hat \\Sigma \\tilde g ( x _ 0 ) \\approx - \\Sigma \\nabla f ( x _ 0 ) . \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} f _ { \\delta } = \\prod _ { j = 1 } ^ { m _ 1 + m _ 2 } ( z _ { n - j + 1 } ) \\times \\sum _ { i = 0 } ^ { m _ 2 } t ^ { i m _ 1 } \\prod _ { j = 1 } ^ { i } \\left ( \\frac { 1 - t ^ j } { 1 - q t ^ { m _ 1 + j } } \\right ) e _ i \\Big ( z _ 1 , \\dots , z _ { n - m _ 1 - m _ 2 } \\Big ) e _ { m _ 2 - i } \\Big ( z _ { n - m _ 2 + 1 } , \\dots , z _ n \\Big ) \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} \\log \\left ( 1 + \\sum _ { j = 1 } ^ n t _ j \\right ) \\le \\sum _ { j = 1 } ^ n \\log ( 1 + t _ j ) \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} & \\| u ^ { - } \\| ^ { p } _ { \\mu } \\\\ & \\leq \\iint _ { \\R ^ { 2 N } } \\frac { | u ( x ) - u ( y ) | ^ { p - 2 } ( u ( x ) - u ( y ) ) } { | x - y | ^ { N + s p } } ( u ^ { - } ( x ) - u ^ { - } ( y ) ) \\ , d x d y + \\int _ { \\R ^ { N } } \\mu | u | ^ { p - 2 } u u ^ { - } \\ , d x \\\\ & = \\int _ { \\R ^ { N } } f ( u ) u ^ { - } \\ , d x = 0 \\end{align*}"}
-{"id": "2801.png", "formula": "\\begin{align*} B _ i ( p ) : = \\frac { b _ i ( p ) } { 2 p ^ { k _ i - \\frac { 1 } { 2 } } } \\in [ - 1 , 1 ] . \\end{align*}"}
-{"id": "4069.png", "formula": "\\begin{align*} \\sup _ { U V \\rightarrow X \\rightarrow Y } \\frac { I ( U ; Y | V ) } { I ( U ; X | V ) } & \\ge \\max _ { q _ X } s ^ * \\big ( q _ X \\ , p _ { Y | X } \\big ) = \\eta \\big ( p _ { Y | X } \\big ) . \\end{align*}"}
-{"id": "6352.png", "formula": "\\begin{align*} b ^ I _ i ( M ) : = \\operatorname { e n d } \\left ( H ^ i _ { R + } ( L ^ I ( M ) ) \\right ) \\mbox { f o r e v e r y } 0 \\leqslant i \\leqslant g - 1 . \\end{align*}"}
-{"id": "3405.png", "formula": "\\begin{align*} \\begin{aligned} | g ' ( z _ 0 ) | & = | h ' ( 0 ) | \\cdot | \\phi ' ( z _ 0 ) | \\\\ & \\leq 2 | h ( 0 ) \\left ( \\big { | } \\log | h ( 0 ) | \\big { | } + A \\right ) \\frac { \\pi } { 4 | x _ 0 | } \\\\ & = \\frac { \\pi } { 2 | x _ 0 | } | g ( z _ 0 ) | \\left ( \\big { | } \\log | g ( z _ 0 ) | \\big { | } + A \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{align*} & \\frac { 1 } { ( 2 \\pi i ) ^ { g - 1 } } \\int _ { \\Gamma _ g } \\cdots \\int _ { \\Gamma _ 2 } \\frac { ( k _ 1 p _ 1 ) \\cdots ( k _ g p _ g ) } { ( \\lambda _ 1 + k _ 1 p _ 1 ) ( \\lambda _ 2 + k _ 2 p _ 2 ) \\cdots ( \\lambda _ g + k _ g p _ g ) } \\cdot \\frac { d \\lambda _ 2 \\cdots d \\lambda _ g } { \\lambda _ 2 \\cdots \\lambda _ g } \\\\ = & ( - 1 ) ^ { g - 1 } \\frac { k _ 1 p _ 1 } { k _ 1 p _ 1 + k _ 2 p _ 2 + \\cdots + k _ g p _ g } \\end{align*}"}
-{"id": "5980.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\ ! \\ ! \\hat { b } _ h ^ i ( t ) d t \\leq \\int _ 0 ^ T \\ ! \\ ! \\sum _ { i = 1 } ^ r \\hat { b } _ h ^ i ( t ) d t \\leq \\int _ 0 ^ T \\ ! \\ ! \\sum _ { i = 1 } ^ r \\xi _ h ^ i ( t ) d t + \\int _ 0 ^ T \\ ! \\ ! \\sum _ { j = 1 } ^ s \\hat { a } _ h ^ j ( t ) d t < \\int _ 0 ^ T \\ ! \\ ! \\sum _ { i = 1 } ^ r \\xi _ h ^ i ( t ) d t + R \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{align*} z ' ( t ) = A z ( t ) + f ( t ) , z ( 0 ) = z _ { 0 } , \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} X _ \\nu ( x ^ { - \\nu } ) = \\left \\lbrace \\begin{array} { c l } A _ \\nu & \\nu \\in \\{ 1 , \\cdots , s \\} \\\\ D _ { \\nu - s } ( a , p ) \\cap S , & \\nu \\in \\{ s + 1 , \\cdots , s + t \\} \\\\ P , & \\nu = s + t + 1 \\end{array} \\right . \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} m & = \\frac { k } { \\pi } \\left ( \\arctan \\left ( 2 c \\sqrt { k } \\right ) - \\arctan \\left ( \\frac { 1 } { \\sqrt { 3 } } + \\frac { 2 \\eta } { k } \\right ) \\right ) + O ( 1 ) \\\\ & = \\frac { k } { 3 } - O ( \\sqrt { k } ) . \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\alpha \\cdot ( \\beta \\cdot e _ S ) & = & \\alpha \\cdot ( \\lambda ( \\beta ) \\cdot e _ S ) \\\\ & = & \\alpha ( \\lambda ( \\beta ) ) \\cdot e _ S + \\lambda ( \\beta ) \\cdot ( \\alpha \\cdot e _ S ) \\\\ & = & ( \\alpha ( \\lambda ( \\beta ) ) + \\lambda ( \\alpha ) \\lambda ( \\beta ) ) \\cdot e _ S , \\end{array} \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} x _ + b _ { n + 1 } + x _ 0 b _ n + x _ - b _ { n - 1 } = 0 \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} p ( x , \\xi ) = a ( x ) - i \\ell ( x ) ' \\xi + \\frac { 1 } { 2 } \\xi ' Q ( x ) \\xi - \\int _ { y \\neq 0 } \\Big ( e ^ { i y ' \\xi } - 1 - i y ' \\xi \\cdot \\chi ( y ) \\Big ) \\ N ( x , d y ) . \\end{align*}"}
-{"id": "8081.png", "formula": "\\begin{align*} H ^ { s } u = V u + \\psi \\end{align*}"}
-{"id": "3429.png", "formula": "\\begin{align*} \\frac { 1 } { ( x ' ( t ) ^ 2 + y ' ( t ) ^ 2 ) ^ { 1 / 2 } } \\left ( ( m - 1 ) \\frac { y ' ( t ) } { x ( t ) } - ( n - 1 ) \\frac { x ' ( t ) } { y ( t ) } + \\frac { x ' ( t ) y '' ( t ) - x '' ( t ) y ' ( t ) } { x ' ( t ) ^ 2 + y ' ( t ) ^ 2 } \\right ) & \\\\ \\\\ = \\frac { 1 } { 2 } \\frac { x ' ( t ) y ( t ) - x ( t ) y ' ( t ) } { ( x ' ( t ) ^ 2 + y ' ( t ) ^ 2 ) ^ { 1 / 2 } } & + \\lambda \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{align*} \\psi ( [ a , b ] ) & = ( b ^ { - 1 } a , 1 , a ^ { - 1 } b ) , \\\\ \\psi ( [ a , b , a ] ) & = ( ( a ^ { - 1 } b ) ^ 2 , b ^ { - 1 } a , b ^ { - 1 } a ) , \\\\ \\psi ( [ a , b , a ^ 2 ] & = ( a ^ { - 1 } b , a ^ { - 1 } b , ( b ^ { - 1 } a ) ^ 2 ) , \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} \\widetilde S _ { e e } ^ \\tau = S _ { e e } ^ \\tau - S _ { e e ^ c } ^ \\tau ( S _ { e ^ c e ^ c } ^ \\tau ) ^ { - 1 } S _ { e ^ c e } ^ \\tau , \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} Q = ( ( \\psi \\times i d ) _ { * } ( \\sigma _ { n } \\times i d ) ^ { * } ( l ^ { n } \\times i d ) _ { * } F ^ { \\boxtimes n } ) ^ { s i g n } \\otimes p _ { 1 } ^ { * } d e t ( \\mathcal { A } ) ^ { - 1 } \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} | | F _ j | | _ { L ^ 2 } ^ 2 = | | F _ j ^ + | | _ { L ^ 2 } ^ 2 + | | F _ j ^ - | | _ { L ^ 2 } ^ 2 - 2 \\langle F _ j ^ + , F _ j ^ - \\rangle \\leq | | F _ j ^ + | | _ { L ^ 2 } ^ 2 + | | F _ j ^ - | | _ { L ^ 2 } ^ 2 \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} \\sum _ { e \\mid d } \\mu ( d / e ) f ( x _ 1 ^ { d / e } , x _ 2 ^ { d / e } , \\dots ) ^ e & = \\sum _ { e \\mid d } \\mu ( d / e ) \\left ( \\sum _ { i = 1 } ^ { n } g _ i ( x _ 1 ^ { d / e } , x _ 2 ^ { d / e } , \\dots ) \\right ) ^ e \\\\ & = \\sum _ { e \\mid d } \\mu ( d / e ) \\left ( \\sum _ { i = 1 } ^ { n } g _ i ^ { d / e } ( x _ 1 , x _ 2 , \\dots ) \\right ) ^ e . \\\\ \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} \\frac { \\abs { F _ n ( x ) - F _ n ( y ) } } { \\abs { F _ n ( a ) - F _ n ( b ) } } = \\frac { \\abs { y - x } } { b - a } \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} \\hat b _ + ^ i = b _ + ^ i - \\frac { 1 } { 2 } \\sum _ \\xi \\tilde \\sigma ^ i _ \\xi \\partial _ j \\tilde \\sigma ^ j _ \\xi . \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{align*} Y _ s ^ { t , x ; u _ { \\cdot } } = \\hat { Y } _ s ^ { t , x ; u _ { \\cdot } } - \\int _ t ^ s c \\bigl ( \\tau , X _ { \\tau } ^ { t , x ; u _ { \\cdot } } , u _ { \\tau } \\bigr ) d \\tau , s \\in [ t , T ] . \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} A \\otimes B : = \\left ( \\begin{array} { c c } A ( 1 , 1 ) B & A ( 1 , 2 ) B \\\\ A ( 2 , 1 ) B & A ( 2 , 2 ) B \\end{array} \\right ) . \\end{align*}"}
-{"id": "6612.png", "formula": "\\begin{align*} \\left ( U \\cdot ( \\nabla _ X \\cdot \\psi ) , Y \\cdot \\psi \\right ) + \\left ( \\nabla _ X \\psi , Y \\cdot U \\cdot \\psi \\right ) & = - g \\left ( U \\times T ( X ) , Y \\right ) - g \\left ( T ( X ) , Y \\times U \\right ) \\\\ & = - 2 \\varphi ( T ( X ) , Y , U ) , \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{align*} G _ { i _ 1 i _ 2 i _ 3 } ^ { j _ 1 j _ 2 j _ 3 } \\delta ^ { i _ 1 i _ 2 } = & \\delta ^ { j _ 1 j _ 2 } \\delta ^ { j _ 3 k _ 3 } ( \\tilde \\gamma + 2 \\gamma I ) ^ { - 1 } _ { i _ 3 k _ 3 } , \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k x _ { ( i ) } \\geq \\sum _ { i = 1 } ^ k y _ { ( i ) } k = 1 , \\dots , n . \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{align*} a _ t = \\sup \\{ a : a < x \\mathrm { f o r a l l } x \\in \\mathrm { s u p p } ( \\mu ^ { \\boxtimes t } ) \\} , \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} ( n - 2 ) \\int _ 0 ^ \\infty \\dot s s ^ { 1 - n } d t = - \\int _ 0 ^ \\infty \\frac { d } { d t } ( s ^ { 2 - n } ) d t = - \\big ( s ^ { 2 - n } | _ { t = \\infty } - s ^ { 2 - n } | _ { t = 0 } \\big ) = s ( 0 ) ^ { 2 - n } < \\infty . \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} \\varepsilon ^ { - \\tilde { s } } \\geq \\varepsilon ^ { - q s - s \\log _ 2 ( C ) } = \\varepsilon ^ { - q s } \\cdot \\left ( \\frac { 1 } { \\varepsilon } \\right ) ^ { s \\log _ 2 ( C ) } \\geq \\varepsilon ^ { - q s } \\cdot 2 ^ { s \\log _ 2 ( C ) } = \\varepsilon ^ { - q s } \\cdot C ^ s = \\left ( \\frac { \\varepsilon ^ { q } } { C } \\right ) ^ { - s } , \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{align*} \\epsilon _ 2 = \\min _ { x _ 1 \\neq x _ 2 , y _ 1 \\neq y _ 2 } \\left ( \\frac { p _ { X Y } ( x _ 1 , y _ 1 ) p _ { X Y } ( x _ 2 , y _ 2 ) } { p _ { X Y } ( x _ 1 , y _ 2 ) p _ { X Y } ( x _ 2 , y _ 1 ) } \\right ) ^ { \\frac 1 2 } . \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} f ( r ) = \\sum _ { j = 0 } ^ { n + 1 } \\zeta _ j \\phi _ j ( r ) \\ , , \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} | \\hat { z } | = | \\int _ { 0 } ^ { | \\xi | } \\frac { d t } { \\sqrt { t ( t + 1 ) ( t + \\lambda ) ) } } | \\leq \\int _ { 0 } ^ { | \\xi | } \\frac { d t } { \\sqrt { t ( t + 1 ) ( | t + \\lambda | ) } } . \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{align*} g _ { I J } = \\begin{cases} - S _ { J I } & , \\\\ ( - 1 ) ^ i S _ { J \\cup \\{ n \\} , I } & . \\end{cases} \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{align*} \\begin{cases} d ^ + b - 2 ^ { - \\frac { 3 } { 2 } } r ^ { \\frac { 1 } { 2 } } ( q ( \\psi , \\eta ) + q ( \\eta , \\psi ) ) \\\\ D _ A \\eta + ( 2 r ) ^ { \\frac { 1 } { 2 } } \\mathfrak { c l } ( b ) \\psi \\\\ * d * b - 2 ^ { - \\frac { 1 } { 2 } } r ^ { \\frac { 1 } { 2 } } ( \\eta ^ { \\dagger } \\psi - \\psi ^ { \\dagger } \\eta ) , \\end{cases} \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} \\begin{cases} U _ { t } = \\Delta U - \\chi \\nabla v ( \\cdot , \\cdot ; t _ 0 , u _ 0 ) \\cdot \\nabla U + a _ { \\sup } U , t > t _ 0 \\cr U ( \\cdot , t _ 0 ) = u _ 0 \\end{cases} \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{align*} T _ * < + \\infty \\Rightarrow \\lim _ { T \\to T _ * } \\Big ( \\| \\nabla u \\| _ { L ^ { \\infty } _ T L ^ 2 _ x } + \\| u \\| _ { L ^ 2 _ T W ^ { 1 , 4 } _ x } \\Big ) = + \\infty . \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{gather*} u ( x _ 0 ) - f ( x _ 0 ) = \\sup _ x u ( x ) - f ( x ) , \\\\ u ( x _ 0 ) - \\lambda H f ( x _ 0 ) - h ( x _ 0 ) \\leq 0 . \\end{gather*}"}
-{"id": "4302.png", "formula": "\\begin{align*} y _ t = y _ 0 + \\int _ 0 ^ t f ( y ) ^ \\ell d x _ r \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} [ i _ { z _ 1 } ^ * \\omega ] = [ \\Theta ] = [ i _ { z _ 2 } ^ * \\omega ] . \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{align*} p _ r = r ( 1 - l _ 1 ^ 2 , - l _ 1 l _ 2 , \\ldots , - l _ 1 l _ d ) , \\end{align*}"}
-{"id": "2687.png", "formula": "\\begin{align*} h \\leq \\varphi _ i ( y ) \\leq C _ 1 h , | \\nabla \\varphi _ i ( y ) | \\leq C _ 2 h , \\quad y \\in [ 0 , b ] , i = 1 , 2 . \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{align*} \\frac { 2 m - 8 x } { 7 } \\binom { 7 } { 2 } + x \\left ( \\binom { 8 } { 2 } - 5 \\right ) 6 m - x < 6 m . \\end{align*}"}
-{"id": "4870.png", "formula": "\\begin{align*} Y ^ 2 = X ^ 3 + A X + B \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} s _ * = 4 s _ C - s _ g = 1 6 k - 2 0 v . \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} \\mu \\succ \\nu \\iff \\Big ( \\mu ^ { + } > \\nu ^ { + } \\mu ^ { + } = \\nu ^ { + } , \\ \\mu > \\nu \\Big ) . \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{align*} y = z + c _ 0 m ^ { n - K } + c \\ , m ^ { n - \\ell } , \\mbox { w h e r e $ c = \\sum _ { i = 1 } ^ { s - 1 } c _ { s - i } \\ , m ^ { K ( i - 1 ) } $ , } \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} \\left ( \\frac { d } { d x } \\right ) ^ n f _ 2 ( f _ 1 ( x ) ) \\Big | _ { x = x _ 0 } = \\sum _ { m = 1 } ^ n \\frac { n ! } { m ! } f _ 2 ^ { ( m ) } ( f _ 1 ( x _ 0 ) ) \\prod _ { j = 1 } ^ m \\left ( \\sum _ { l _ j = 1 } ^ n \\frac { 1 } { l _ j ! } f _ 1 ^ { ( l _ j ) } ( x _ 0 ) \\right ) 1 _ { \\sum _ { j = 1 } ^ m l _ j = n } . \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} \\| u ( t , x ) \\| _ p = 2 \\left \\| \\psi ( t , x ) ^ { - 1 } \\frac { \\partial \\psi } { \\partial x } ( t , x ) \\right \\| _ p \\le 2 \\left \\| \\psi ( t , x ) ^ { - 1 } \\right \\| _ { 2 p } \\left \\| \\frac { \\partial \\psi } { \\partial x } ( t , x ) \\right \\| _ { 2 p } . \\end{align*}"}
-{"id": "6171.png", "formula": "\\begin{align*} | S | = \\sum _ { z \\in \\widetilde { P } } | U ( z ) | = | \\widetilde { P } | \\times q ^ { ( \\mu _ 1 + \\cdots + \\mu _ { m - 1 } ) - ( m - 1 ) } | U | . \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} \\Phi ^ { ( \\alpha ) } _ n ( x , y | q ) = \\sum _ { k = 0 } ^ n { n \\brack k } _ q ( \\alpha ; q ) _ k x ^ k y ^ { n - k } . \\end{align*}"}
-{"id": "7699.png", "formula": "\\begin{align*} F _ { r _ t | r _ m } ( y ) & = 1 - \\sum ^ { t - m - 1 } _ { n = 0 } \\mathrm { P } ( \\# ( { \\bf i n t } ( \\mathcal { B } ( x _ 0 , x ) , \\mathcal { B } ( x _ 0 , y ) ) = n ) , \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } B _ n = + \\infty , \\end{align*}"}
-{"id": "1639.png", "formula": "\\begin{gather*} \\Phi ( g ( u ) ) = u _ 1 ^ { 2 k _ 1 } \\ldots u _ d ^ { 2 k _ d } , \\\\ | g ' ( u ) | = b ( u ) | u _ 1 ^ { h _ 1 } \\ldots u _ d ^ { h _ d } | . \\end{gather*}"}
-{"id": "3505.png", "formula": "\\begin{align*} a \\oplus b = \\log ( { \\rm e } ^ a + { \\rm e } ^ b ) . \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } ( - \\Delta ) _ { p } ^ { s } v _ { n } + V _ { n } ( x ) | v _ { n } | ^ { p - 2 } v _ { n } = f ( v _ { n } ) & \\mbox { i n } \\R ^ { N } \\\\ v _ { n } \\in W ^ { s , p } ( \\R ^ { N } ) \\\\ v _ { n } ( x ) > 0 & \\mbox { f o r a l l } x \\in \\R ^ { N } , \\end{array} \\right . \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} M _ t \\Pi _ { \\rm n o r } ^ \\dagger ( \\widetilde x _ t ) = 0 , \\widetilde x _ t \\in \\pi ^ { - 1 } ( Z ) ; \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{align*} \\left | \\int _ { \\mathbb R } g ' ( x ) f ' ( x ) \\overline { f ( x ) } \\ , \\mathrm d x \\right | & \\leq \\| f \\| _ \\infty \\| f ' \\| _ { L ^ 2 } \\| g ' \\| _ { L ^ 2 } \\leq 4 \\| q _ - \\| _ { L ^ 1 } ^ { \\frac { 3 } { 2 } } \\| f \\| _ { L ^ 2 } ^ 2 \\sqrt { \\frac { 2 } { \\delta } } \\\\ & \\leq 1 6 \\cdot \\sqrt { 3 } \\| q _ - \\| _ { L ^ 1 } ^ { 2 } \\| f \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{align*} \\Delta _ f ( u e ^ f ) = e ^ f \\left [ \\Delta _ { ( - f ) } u + ( \\Delta f ) u \\right ] . \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{align*} J \\nu _ 1 = \\nu _ 2 , J \\nu _ 2 = - \\nu _ 1 , \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{align*} & \\gamma = \\{ y \\in \\R ^ d : y = x + t \\theta , \\ , t \\in \\R \\} , \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{align*} \\mathbf { r } = \\sqrt { \\Omega } \\mathbf { P x } + \\sqrt { \\Omega } \\mathbf { F } \\mathbf { z } + \\mathbf { F } \\mathbf { n } _ q . \\end{align*}"}
-{"id": "5770.png", "formula": "\\begin{align*} \\lambda _ 1 ( x ) & = \\frac { 1 } { 2 } \\left ( - x _ 1 - x _ 2 - x _ 3 + 2 \\right ) , & \\lambda _ 2 ( x ) & = \\frac { 1 } { 2 } \\left ( x _ 1 + x _ 2 - x _ 3 \\right ) , \\\\ \\lambda _ 3 ( x ) & = \\frac { 1 } { 2 } \\left ( x _ 1 - x _ 2 + x _ 3 \\right ) , & \\lambda _ 4 ( x ) & = \\frac { 1 } { 2 } \\left ( - x _ 1 + x _ 2 + x _ 3 \\right ) . \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} K ( \\psi _ k ) = K ( Q _ { c } ) = \\lambda ~ ~ ~ \\lim _ { k \\longrightarrow + \\infty } | I ( \\psi _ k ) - M _ { \\lambda } | = 0 , \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{align*} d q ^ \\prime _ t = & \\tilde \\gamma ^ { - 1 } ( t ^ * ) \\left ( - \\nabla _ q V ( t ^ * , q ^ \\prime _ t ) + \\tilde F ( t ^ * , q _ t ) \\right ) d t \\\\ & + \\tilde S ( t ^ * , q ^ \\prime _ t ) d t + \\hat { \\phi } _ * ( \\tilde \\gamma ^ { - 1 } \\sigma ) ( t ^ * , q ^ \\prime _ t ) \\circ d \\tilde W _ t . \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} \\langle v , T w \\rangle = \\beta ( a , b ) \\langle T ^ \\dagger v , w \\rangle , \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} \\tag * { $ { \\bf ( B _ 1 ) } $ } & \\mu : L _ p ( J , E _ 0 ) \\rightarrow L _ \\infty \\big ( J , \\mathcal { L } ( E _ 0 ) \\big ) \\ , , \\ u \\mapsto \\mu ( u , \\cdot ) \\ \\\\ & \\ \\| \\mu ( u , \\cdot ) \\| _ { L _ \\infty ( J , \\mathcal { L } ( E _ 0 ) ) } \\le f ( \\| u \\| _ { L _ p ( J , E _ 0 ) } ) \\ , , \\ u \\in L _ p ( J , E _ 0 ) \\ , , \\end{align*}"}
-{"id": "2187.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } P _ t ( A _ 1 , \\dots , A _ n ) = \\Lambda _ n ( A _ 1 , \\dots , A _ n ) . \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} C ^ 2 & \\geq - \\frac { 1 } { 2 } \\big ( \\delta ( X ) + K _ X . \\bar { C } \\big ) - \\sum _ { i = 1 } ^ s \\frac { 3 } { 2 } m _ { q _ i } ( \\bar { C } ) - n \\\\ & \\geq - \\frac { 1 } { 2 } \\big ( \\delta ( X ) + K _ X . \\bar { C } \\big ) - \\sum _ { i = 1 } ^ s \\frac { 3 } { 2 } ( \\bar { C } . A ) - n \\quad ( n \\geq s ) \\\\ & \\geq - \\frac { 1 } { 2 } \\big ( \\delta ( X ) + ( K _ X + 3 n A ) . \\bar { C } \\big ) - n . \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} c _ { \\beta , m } = \\sum _ { n = m } ^ { \\infty } \\frac { n ! } { ( n - m ) ! ~ m ! } \\lfloor \\beta T _ { \\beta } ^ { n - 1 } ( 1 ) \\rfloor \\bigl ( \\frac { 1 } { \\beta } \\bigr ) ^ { n - m } ~ > 0 , \\mbox { f o r a l l } ~ m \\geq 1 . \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} h ( q , z ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { k ( q ^ n , z ^ n ) } { n } . \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} & \\sum _ { a = 1 } ^ { n - 2 } \\sum _ { m = 1 } ^ { n - a - 1 } \\sum _ { t = a + m } ^ { n - 1 } \\binom { n - t + a - 1 } { a - 1 } = \\sum _ { a = 1 } ^ { n - 2 } \\sum _ { m = 1 } ^ { n - a - 1 } \\left ( \\binom { n - m } { a } - 1 \\right ) = \\sum _ { a = 1 } ^ { n - 2 } \\left ( \\binom { n } { a + 1 } - ( n - a ) \\right ) \\\\ & = 2 ^ n - \\binom { n + 1 } { 2 } - 1 \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} H _ { r e g } = p _ 1 ^ 2 - p _ 1 p _ 2 + p _ 2 ^ 2 - \\log | \\sin { q _ 1 } | - \\log | \\sin { q _ 2 } | - \\log | \\sin { ( q _ 1 + q _ 2 ) } | . \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} \\widehat { I } \\left ( \\alpha ^ { ( 0 ) } , \\alpha ^ { ( 1 ) } \\right ) = \\dfrac { \\sum \\limits _ { l _ 0 = 2 } ^ N \\sum \\limits _ { l _ 1 = 2 } ^ N \\widehat { A } ( l _ 0 , l _ 1 ) \\alpha _ { l _ 0 } ^ { ( 0 ) } \\alpha _ { l _ 1 } ^ { ( 1 ) } } { \\sum \\limits _ { l _ 0 = 2 } ^ N \\sum \\limits _ { l _ 1 = 2 } ^ N \\widehat { B } ( l _ 0 , l _ 1 ) \\alpha _ { l _ 0 } ^ { ( 0 ) } \\alpha _ { l _ 1 } ^ { ( 1 ) } } , \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{align*} x = T _ { \\lambda } x = P _ A \\left ( ( 1 + \\lambda ) T _ 2 x - \\lambda x \\right ) - \\lambda ( T _ 2 x - x ) \\\\ \\Leftrightarrow \\lambda T _ 2 x + ( 1 - \\lambda ) x = P _ A \\left ( ( 1 + \\lambda ) T _ 2 x - \\lambda x \\right ) . \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} \\lim _ { t _ 0 \\to - \\infty } \\overline { u } ( t - t _ 0 , 0 ) = \\frac { a - \\chi \\underline { u } _ { 0 } } { b - \\chi } \\lim _ { t _ 0 \\to - \\infty } \\underline { u } ( t - t _ 0 , 0 ) = \\frac { a - \\chi \\overline { u } _ { 0 } } { b - \\chi } . \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{align*} x ^ i - x ^ j - q p ( x ) = r ( x ) m ( x ) \\quad \\Z [ x ] . \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} \\mathrm { d i v } ( s | _ Z ) = \\sum _ { F \\in \\mathcal { I } ^ { T / T \\cap H } ( Z ) } 2 k v _ { \\mathcal { L } } ^ t ( u _ F ) F . \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} \\lim _ { \\| \\Delta x \\| \\rightarrow 0 } \\frac { \\| g ( x + \\Delta x ) - g ( x ) - H ( x ) \\Delta x \\| } { \\| \\Delta x \\| } = 0 . \\end{align*}"}
-{"id": "3874.png", "formula": "\\begin{align*} \\mu = ( 1 - \\delta ) ^ 2 , \\ ; L = \\frac { 1 } { ( 1 - \\delta ) ^ 2 } . \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} \\frac { I ( U ; Y | V ) } { I ( U ; X | V ) } & \\leq \\max _ { v } \\frac { I ( U ; Y | V = v ) } { I ( U ; X | V = v ) } \\\\ & \\quad \\overset { ( a ) } { \\leq } s ^ * _ r ( X ; Y ) \\\\ & \\quad \\leq \\max _ { \\substack { q _ { X Y } : \\ : q _ { X Y } = q _ { X } p _ { Y | X } } } s _ q ^ * ( X ; Y ) \\\\ & \\quad = \\eta \\big ( p _ { Y | X } \\big ) \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} \\omega = \\hat { \\omega } _ { \\mathbb { C } ^ n } + \\omega _ Y + \\frac { 1 } { 2 } \\sum \\limits _ { i = 1 } ^ m ( d z ^ i \\wedge \\eta ^ i + d \\bar { z } ^ i \\wedge \\overline { \\eta ^ i } ) \\end{align*}"}
-{"id": "6916.png", "formula": "\\begin{align*} K _ v ( E ) \\ = \\ P \\left ( H _ v - E \\right ) ^ { - 1 } P , \\end{align*}"}
-{"id": "4218.png", "formula": "\\begin{align*} \\nabla _ x F ( x ) & = D \\theta ( x ) ^ T ( \\theta ( x ) - y ) , \\\\ \\lambda _ \\theta \\abs { \\theta ( x ) - y } ^ 2 & \\leq \\abs { \\nabla _ x F } ^ 2 \\leq \\Lambda _ \\theta \\abs { \\theta ( x ) - y } ^ 2 , \\\\ \\Delta F & = n \\bar { \\lambda } _ \\theta + ( \\Delta \\theta ) ^ T ( \\theta - y ) . \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} \\tau ^ { ( b / 2 ) - 1 } \\int _ r ^ t ( t - \\sigma ) ^ { - 1 / 2 } d \\sigma = 2 \\tau ^ { ( b / 2 ) - 1 } ( t - r ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "393.png", "formula": "\\begin{align*} A x = b \\ , , \\mbox { w h e r e } A \\in \\mathbb { R } ^ { N \\times N } , \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} t \\ast t ' = t _ 1 \\vee ( t _ 2 \\ast t ' ) + ( t \\ast t _ 1 ^ { ' } ) \\vee t _ 2 ^ { ' } \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{align*} q : = x ^ 0 : = t + r _ * , \\ \\ s : = x ^ 1 : = t - r _ * . \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{align*} F _ { H _ \\infty } = 0 , [ \\Phi \\wedge \\Phi ^ { \\ast _ { H _ \\infty } } ] = 0 \\end{align*}"}
-{"id": "6893.png", "formula": "\\begin{align*} \\mathrm { l o c } _ { \\beta _ { \\eta } } ( g , j ^ { \\ast } \\mathcal { S } _ { V } ) = \\mathrm { l o c } _ { \\beta } ( g , \\mathcal { S } _ { V } ) = \\mathrm { l o c } _ { \\beta _ { s } } ( g , i ^ { \\ast } \\mathcal { S } _ { V } ) , \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} d Y _ t = \\Phi ( t , Y _ t ) d W _ t + g ( t , Y _ t ) d t . \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{align*} \\begin{cases} \\eta ( 0 , t ) = 0 , \\ , \\ , \\eta ( L , t ) = 0 , \\ , \\ , \\eta _ { x } ( 0 , t ) = f ( t ) , & t \\in ( 0 , \\infty ) , \\\\ w ( 0 , t ) = 0 , \\ , \\ , w ( L , t ) = 0 , \\ , \\ , w _ { x } ( L , t ) = g ( t ) , & t \\in ( 0 , \\infty ) \\end{cases} \\end{align*}"}
-{"id": "5672.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| T _ { k ( n ) } ( f _ n ) \\| = 1 = \\lim _ { n \\to \\infty } \\| \\Re T _ { k ( n ) } ( f _ n ) \\| . \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} \\mathrm { R } _ { \\varrho } ( \\Phi ) ( x ) = x _ L , \\end{align*}"}
-{"id": "1072.png", "formula": "\\begin{align*} \\lambda _ { d s } ( \\varphi ) = \\mathcal L _ { \\{ s , - \\} } ( \\varphi ) \\ , . \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{align*} E \\left [ S ^ 2 \\right ] - E \\left [ T _ 1 ^ 2 \\right ] = O \\left ( e ^ { - \\varepsilon M } \\right ) , M \\to \\infty , \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{align*} W ^ 1 = \\left [ \\begin{array} { c c } W & c \\\\ c ^ * & 0 \\end{array} \\right ] , \\ M ^ 1 = \\left [ \\begin{array} { c c } M _ x & - c \\\\ - c ^ * & 2 x _ { n + 1 } \\end{array} \\right ] \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{align*} C _ W = \\{ ( x _ 1 , \\ldots , x _ n ) ; x _ 1 , \\ldots , x _ n \\geq 0 , \\ \\sum _ { i = 1 } ^ n \\frac { c } { c + 2 x _ i } < 1 \\} \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} & W _ k ( x ) = \\sum _ { n = - \\infty } ^ \\infty \\frac { W _ { n , k } } { x ^ n } \\\\ & W ( x , y ) = 1 + \\sum _ { k = 1 } ^ \\infty \\frac { W _ k ( x ) } { y ^ k } \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} \\sum _ { n \\geq L } E _ M ( t , n ) | C _ n | & = \\sum _ { n = L } ^ { M _ - - 1 } E _ M ( t , n ) | C _ n | + \\sum _ { n = M _ - } ^ { M _ + - 1 } E _ M ( t , n ) | C _ n | + \\sum _ { n = M _ + } ^ { \\infty } E _ M ( t , n ) | C _ n | \\\\ & = : H _ 1 + H _ 2 + H _ 3 \\ , , \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} \\tilde { u } _ j ( a ) = - u ^ { * , j , i } _ i ( p ) - \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\frac { - u ^ { * , l , j } ( p ) \\alpha ^ { \\vee , l } } { ( 2 \\chi - p ) ( \\alpha ^ { \\vee } ) } + I _ { H , j } ( d _ p u ^ * ) . \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} Z ( C ; T ) = \\frac { P ( T ) } { ( 1 - T ) ( 1 - q T ) } \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{align*} 0 = ( A - \\lambda ) f = \\operatorname { s g n } ( \\cdot ) ( - f '' ) - \\lambda f + \\operatorname { s g n } ( \\cdot ) q f = ( B _ 0 - \\lambda ) f + \\operatorname { s g n } ( \\cdot ) q f \\end{align*}"}
-{"id": "6554.png", "formula": "\\begin{align*} \\mu \\left ( \\left \\{ y \\in Y : \\left ( \\int \\limits _ 0 ^ { T _ k } F _ { n ( T _ k ) } ( g _ t y ) \\ , d t \\right ) ^ 2 \\geqslant T _ k ^ { 2 \\alpha } \\| f _ { n ( T _ k ) } \\| _ { B } ^ 2 \\right \\} \\right ) & \\leqslant \\frac { 8 K } { C } T _ k ^ { 1 - 2 \\alpha } \\\\ & = \\frac { 8 K } { C } \\frac { 1 } { k ^ { 2 \\alpha } } \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} E _ { \\hat { S } _ 1 } - E _ { S _ 1 ^ * } & = E _ { W _ t } + E _ { W _ t , S _ 1 ^ * } , \\\\ E _ { \\hat { S } _ 1 , \\hat { S } _ 3 } - E _ { S _ 1 ^ * , S _ 3 ^ * } & = E _ { W _ t , S _ 3 ^ * } - E _ { W _ t } - E _ { W _ t , S _ 1 ^ * } - { | Q _ t | } - d _ { t , S _ 1 ^ * } , \\\\ E _ { \\hat { S } _ 1 , V _ 1 } - E _ { S _ 1 ^ * , V _ 1 } & = E _ { W _ t , V _ 1 } , \\\\ E _ { \\hat { S } _ 2 , V _ 1 } - E _ { S _ 2 ^ * , V _ 1 } & = d _ { t , V _ 1 } . \\end{align*}"}
-{"id": "8700.png", "formula": "\\begin{align*} f _ { k - 1 } & : = ( \\rho _ 1 D _ { \\rho _ 1 } - \\rho _ 2 D _ { \\rho _ 2 } ) u _ k - f _ k = \\bigl ( ( \\rho _ 1 D _ { \\rho _ 1 } - z ) - ( \\rho _ 2 D _ { \\rho _ 2 } - z ) \\bigr ) u _ k - f _ k \\\\ & = \\rho _ 2 ^ { i z } ( \\log \\rho _ 2 ) ^ { k - 1 } a _ { k - 1 } ( \\rho _ 1 ) , a _ { k - 1 } : = i k b _ k , \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} K _ 2 = ( K _ 1 \\cap K _ 2 ) ( K _ 2 \\cap K _ 3 ) . \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} - \\Delta u & = K ( x ) u ^ { \\frac { n + 2 } { n - 2 } } \\mbox { i n } \\ , \\Omega \\\\ u & > 0 \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\mbox { i n } \\ , \\Omega \\\\ u & = 0 \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\mbox { o n } \\ , \\partial \\Omega , \\end{aligned} \\right . \\end{align*}"}
-{"id": "8916.png", "formula": "\\begin{align*} \\int _ { \\Delta ' _ Y } p _ i \\nu _ i u _ t ^ * P _ { D H } ' d \\sigma & = \\int _ { \\tilde { \\Delta } ' _ Y } ( u _ t ^ * P _ { D H , i } ' p _ i + r u _ t ^ * P _ { D H } + u ^ * _ { t , i } p _ i P _ { D H } ' ) d p \\\\ & = \\int _ { \\tilde { \\Delta } ' _ Y } ( n u _ t ^ * - u _ t ^ * \\sum _ { \\alpha } \\frac { 2 \\chi _ i \\alpha ^ { \\vee , i } } { ( 2 \\chi - p ) ( \\alpha ^ { \\vee } ) } + u ^ * _ { t , i } p _ i ) P _ { D H } ' d p \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} u ( \\varepsilon ) = u _ { \\varepsilon _ 1 } \\otimes \\cdots \\otimes u _ { \\varepsilon _ N } , & & 0 \\le \\varepsilon _ 1 \\le d _ 1 , \\ldots , 0 \\le \\varepsilon _ N \\le d _ N , \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} \\varphi _ U : W _ { I _ p } \\rightarrow S t _ p ^ n , \\ , \\ , \\ , \\varphi _ U ( \\Omega ) : = \\mathcal { C } ( \\Omega ) U , \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{align*} u _ i < \\{ v \\in \\bar { v } : ~ u _ i < v < v _ k \\} \\cup \\{ & w \\in \\bar { w } : ~ u _ i < w < w _ j \\} \\\\ & < w _ j < v _ k < \\\\ \\{ v \\in \\bar { v } : v _ k < & ~ v < u _ { i + 1 } \\} \\cup \\{ w \\in \\bar { w } : ~ w _ j < w < u _ { i + 1 } \\} < u _ { i + 1 } . \\\\ \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{align*} t = \\frac { 1 } { \\varphi ( f ^ \\# ( b ) ) } = \\frac { 1 } { \\varphi ( H ( r ) ) } . \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} c _ V ( \\tau , t ) = u _ V ( \\tau ) v _ V ( t ) \\end{align*}"}
-{"id": "3771.png", "formula": "\\begin{align*} \\Lambda ^ + \\otimes \\C & = \\Lambda ^ { 2 , 0 } \\oplus \\Lambda ^ { 0 , 2 } \\oplus \\C \\cdot F , & \\Lambda ^ - \\otimes \\C & = \\Lambda ^ { 1 , 1 } _ 0 . \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{align*} \\Phi ( \\overline { u } | \\nabla \\eta | / \\epsilon ) & \\le \\max \\{ ( | \\nabla \\eta | / \\epsilon ) ^ \\ell , ( | \\nabla \\eta | / \\epsilon ) ^ m \\} \\Phi ( \\overline { u } ) \\\\ & = g _ 1 ( x , \\epsilon ) \\Phi ( \\overline { u } ) , \\end{align*}"}
-{"id": "5196.png", "formula": "\\begin{align*} c _ { - } ^ * ( a , b , \\chi , \\lambda , \\mu ) : = 2 \\sqrt { a _ { \\inf } - \\frac { \\chi \\mu a _ { \\sup } } { b _ { \\inf } - \\chi \\mu } } - \\chi \\frac { \\mu \\sqrt { N } a _ { \\sup } } { 2 \\sqrt { \\lambda } ( b _ { \\inf } - \\chi \\mu ) } . \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{align*} L ^ 2 \\mathcal { H } ^ k ( B ) = \\{ \\sigma \\in \\Omega ^ k ( B ) \\cap L ^ 2 \\ , | \\ , d \\sigma = 0 = d ^ \\ast \\sigma \\} . \\end{align*}"}
-{"id": "4543.png", "formula": "\\begin{align*} \\sigma ( \\mathbf { m } ' ) = \\frac { L ! } { \\prod _ { i = 1 } ^ { N - K + 1 } L _ i ! } . \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} X _ i = \\partial _ { x _ i } + A _ i \\partial _ { x _ { n + 1 } } , \\ i = 1 , . . . , n , \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} e _ k ( \\psi ) = s ^ 2 | D ^ k \\nabla _ t \\psi | ^ 2 + | D ^ k D \\psi | ^ 2 \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} y = 1 / 6 - h ( 1 - 6 x ) \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { \\Phi ( t ) } { t } = 0 , \\quad \\lim _ { t \\to \\infty } \\frac { \\Phi ( t ) } { t } = \\infty . \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} \\partial _ { q , x _ 1 } \\{ f ( x _ 1 , y _ 1 ) - b _ 1 f ( x _ 1 , q y _ 1 ) \\} = \\partial _ { q , y _ 1 } \\{ f ( x _ 1 , y _ 1 ) - a _ 1 f ( q x _ 1 , y _ 1 ) \\} . \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} \\mathcal { T } _ { { \\rm g p h } \\mathcal { M } _ { x } } ( v , \\lambda ) = \\big \\{ ( \\xi , \\eta ) \\in \\mathbb { X } \\times \\mathbb { Y } \\ | \\ \\xi = \\nabla g ( x ) \\eta , \\ , \\eta \\in \\mathcal { T } _ { \\mathcal { N } _ { K } ( g ( x ) ) } ( \\lambda ) \\big \\} , \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} D ^ 1 _ { ( A , \\Phi ) } ( \\gamma ) = ( d _ A \\gamma , [ \\Phi \\wedge \\gamma ] ) , \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} ( T h ) ( x , D _ { x ' } ) : = \\kappa ^ { - 1 , * } h ( x , D _ { x ' } ) \\lambda ^ * \\ \\ h \\in S ^ m _ { 1 , 0 } ( \\overline { \\R ^ n _ + } \\times \\R ^ { n - 1 } ; \\mathcal { S } _ + ) \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathbb { Z } } L _ i [ H \\left ( \\cdot , \\mu \\right ) ] ( \\nu ) = \\sum _ { i \\in \\mathbb { Z } } M _ i [ H ( \\nu , \\cdot ) ] \\left ( \\mu \\right ) , \\end{align*}"}
-{"id": "7424.png", "formula": "\\begin{align*} H ( t , x ) = \\frac { 1 } { 2 m } \\| p - \\psi ( t , q ) \\| ^ 2 + V ( t , q ) \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{align*} X \\# _ { 1 - t } Y = Y \\# _ t X \\ . \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{align*} t = \\int _ { { \\frak t } _ 0 } ^ { \\frak t } \\sigma ^ { - \\ : \\frac 1 4 } e ^ { \\frac 1 4 \\langle \\lambda \\rangle ( \\sigma ) } d \\sigma . \\end{align*}"}
-{"id": "1137.png", "formula": "\\begin{align*} \\partial \\circ \\mathrm { e v } = \\mathrm { e v } \\circ ( \\mathcal L _ \\partial \\otimes \\mathrm { I d } + \\mathrm { I d } \\otimes \\partial _ { D ^ { - 1 } } ) \\ , . \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{align*} ( 1 - P _ \\delta ) ( | N | - \\varepsilon ) _ + ^ { 2 n } & \\leq \\frac { 4 } { \\varepsilon ^ 2 } ( 1 + \\varepsilon ) ^ { 2 n } ( ( 1 - P _ \\delta ) T ^ * ( 1 - P _ \\delta ) ) ^ n ( ( 1 - P _ \\delta ) T ( 1 - P _ \\delta ) ) ^ n \\\\ & = \\frac { 4 } { \\varepsilon ^ 2 } ( 1 + \\varepsilon ) ^ { 2 n } ( 1 - P _ \\delta ) ( T ^ * ) ^ n ( 1 - P _ \\delta ) T ^ n ( 1 - P _ \\delta ) \\\\ & \\le \\frac { 4 } { \\varepsilon ^ 2 } ( 1 + \\varepsilon ) ^ { 2 n } ( 1 - P _ \\delta ) ( T ^ * ) ^ n T ^ n ( 1 - P _ \\delta ) . \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{align*} \\Gamma _ { t } ^ { k } = \\lambda _ { k } \\cdot \\Big ( \\Gamma _ { t _ { k } + \\lambda _ { k } ^ { - 2 } t } - x _ { k } \\Big ) , t \\in [ t _ { k } ^ { ( 0 ) } , t _ { k } ^ { ( 1 ) } ) . \\end{align*}"}
-{"id": "3008.png", "formula": "\\begin{align*} v ( q _ { n } ) ^ { q _ { n } } \\rightarrow 0 = 0 ^ { q _ { 0 } } = v ( q _ { 0 } ) ^ { q _ { 0 } } , \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} b _ + = & \\tilde \\gamma ^ { - 1 } F + \\tilde S - \\frac { 1 } { 2 } \\left ( \\tilde \\gamma ^ { - 1 } - ( \\tilde \\gamma ^ T ) ^ { - 1 } \\right ) F - \\frac { 1 } { 2 } ( S - S ^ \\prime ) , \\\\ b _ - = & \\frac { 1 } { 2 } \\left ( \\tilde \\gamma ^ { - 1 } - ( \\tilde \\gamma ^ T ) ^ { - 1 } \\right ) F + \\frac { 1 } { 2 } ( S - S ^ \\prime ) , \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} F h = F \\hat { h } \\ \\ h \\check { F } = \\hat { h } \\check { F } , \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} \\lVert A B - A _ 0 B _ 0 \\rVert ^ 2 & = \\sum _ { j = 1 } ^ N \\left ( \\sum _ { i = 1 } ^ { M - 1 } \\{ ( a _ { i 1 } - a _ { i 2 } ) b _ j - ( a ^ 0 _ { i 1 } - a ^ 0 _ { i 2 } ) b ^ 0 _ j + a _ { i 2 } - a ^ 0 _ i \\} ^ 2 \\right . \\\\ & \\left . + \\left [ \\sum _ { i = 1 } ^ { M - 1 } \\{ ( a _ { i 1 } - a _ { i 2 } ) b _ j - ( a ^ 0 _ { i 1 } - a ^ 0 _ { i 2 } ) b ^ 0 _ j + a _ { i 2 } - a ^ 0 _ i \\} \\right ] ^ 2 \\right ) . \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} D = \\Gamma _ 2 \\times \\cdots \\times \\Gamma _ g , \\end{align*}"}
-{"id": "3110.png", "formula": "\\begin{align*} \\sum _ { j \\leq t } j \\nu _ j = \\sum _ { j \\leq t } \\sum _ { i \\leq \\nu _ j } j = \\sum _ { i = 1 } ^ { \\infty } \\sum _ { j \\leq y _ i } j = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { \\infty } y _ i ( y _ i + 1 ) \\geq \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { \\infty } y _ i ^ 2 . \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{align*} C _ d : = \\sqrt { 1 - 4 ^ { - 1 / ( d - 2 ) } } \\end{align*}"}
-{"id": "6178.png", "formula": "\\begin{align*} \\mathrm { i n v } ( \\sigma ) = \\sum _ { ( s , t ) \\in \\sigma } \\mathrm { i n v } ( \\sigma , s , t ) . \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} A = \\begin{pmatrix} \\cos \\phi & - \\sin \\phi & 0 \\\\ \\sin \\phi & \\cos \\phi & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , B = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & - 1 & 0 \\\\ 0 & 0 & - 1 \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} \\Big | \\sum _ { y \\in \\hat S _ 1 } \\bar G ( t , \\hat x _ n - y ) - \\sum _ { y \\in \\check S _ 1 } \\bar G ( t , \\check x _ n - y ) \\Big | = \\big | P ( W _ t \\in \\hat D \\setminus \\check D ' ) - P ( W _ t \\in \\check D ' \\setminus \\hat D ) \\big | \\ , \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} X \\cdot Y + Y \\cdot X = - 2 h ( X , Y ) \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} ( g , \\zeta ) ( g ' , \\zeta ' ) = ( g g ' , \\sigma ( g , g ' ) \\zeta \\zeta ' ) . \\end{align*}"}
-{"id": "767.png", "formula": "\\begin{align*} f _ { \\beta } ( z ) = - 1 + z + z ^ n + z ^ { m _ 1 } + z ^ { m _ 2 } + z ^ { m _ 3 } + \\ldots = G _ { n } ( z ) + \\sum _ { q \\geq 1 } z ^ { m _ q } , \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} U _ 1 ^ \\star N U _ 1 = U _ 1 ^ \\star \\begin{bmatrix} A _ 1 & 0 \\\\ C _ 1 & B _ 1 \\end{bmatrix} V _ 2 ^ { - 1 } V _ 2 \\begin{bmatrix} 0 & A _ 2 \\\\ B _ 2 & C _ 2 \\end{bmatrix} V _ 3 V _ 3 ^ { - 1 } \\begin{bmatrix} B _ 3 & C _ 3 \\\\ 0 & A _ 3 \\end{bmatrix} U _ 1 , \\end{align*}"}
-{"id": "3735.png", "formula": "\\begin{align*} \\{ x \\} = \\bigcap _ { n \\in \\mathbb { N } } B _ { \\tilde { \\omega } | r n } . \\end{align*}"}
-{"id": "5288.png", "formula": "\\begin{align*} J _ { \\mathrm { f s u } } = \\bigcap _ { t \\in H ^ + } t . J = \\bigcap _ { k \\geq 0 } t _ + ^ k . J . \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} d { e _ { i , t } } = & d { y } _ { i , t } - d { \\overline { y } _ { t } } \\\\ = & \\sum _ { j = 1 , j \\neq i } ^ { N } \\alpha _ { i j } g ( y _ { i , t } - y _ { j , t } ) \\tilde { \\gamma } d t - \\nabla f ( y _ { i , t } ) \\tilde { \\gamma } d t + \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\nabla f ( { x } _ { i , t } ) \\tilde { \\gamma } d t + \\tau \\tilde { \\gamma } { d B _ { i , t } } - \\frac { \\tau } { N } \\tilde { \\gamma } \\sum _ { i = 1 } ^ N { d B _ { i , t } } . \\end{align*}"}
-{"id": "1966.png", "formula": "\\begin{align*} \\partial _ { x } ^ { m } g ( x ) = \\int _ { \\mathbf { B } } \\partial _ { x } ^ { m } R _ { \\alpha } ( x , y ) f ( y ) d v _ { \\alpha } ( y ) . \\end{align*}"}
-{"id": "7033.png", "formula": "\\begin{align*} \\prod ^ { [ \\tau _ n / T _ 0 ] - 1 } _ { i = 0 } \\parallel \\psi ^ { X _ n } _ { T _ 0 } | \\mathcal { E } _ { \\varphi ^ { X _ n } _ { i T _ 0 } ( y ) } \\parallel \\leq { \\rm e } ^ { - \\eta \\tau _ n } , ~ ~ \\prod ^ { [ \\tau _ n / T _ 0 ] - 1 } _ { i = 0 } m ( \\psi ^ { X _ n } _ { T _ 0 } | \\mathcal { F } _ { \\varphi ^ { X _ n } _ { i T _ 0 } ( y ) } ) \\geq { \\rm e } ^ { \\eta \\tau _ n } , ~ \\forall ~ y \\in O r b ( x _ n ) . \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} \\| h \\| = 1 , \\langle x _ { v , 0 } , x _ { v , 1 } \\rangle = 0 , h = x _ { v , 0 } + x _ { v , 1 } , p ( i , j | v , w ) = \\langle x _ { v , i } , x _ { w , j } \\rangle . \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} E [ S ^ { e n v , 0 } _ { s , t } ] = & - E \\left [ \\int _ { s } ^ t \\beta ( r ) ( \\partial _ r \\psi ( r , q _ r ) + \\nabla _ q V ( r , q _ r ) ) \\circ d q _ r \\right ] - \\int _ { s } ^ t \\frac { B _ 0 ( r ) \\dot { B _ 0 } ( r ) } { \\gamma ( r ) ^ 2 + B _ 0 ( r ) ^ 2 } d r . \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} x = \\begin{pmatrix} y \\\\ \\dot { y } \\end{pmatrix} , A = \\begin{pmatrix} 0 & I \\\\ I & 0 \\end{pmatrix} , \\widetilde { S } ^ z ( t ) = \\begin{pmatrix} 0 & 0 \\\\ S ^ z ( t ) & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} f \\circ \\gamma _ { \\vec { u } } \\circ \\sigma = h . \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} u ( x , t ; \\kappa \\phi _ L ) = \\kappa e ^ { \\sigma _ { _ { L } } t } \\phi _ L ( x ) \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{align*} \\inf _ { x \\in A } F ( x , c ) \\le F ( y ^ * , c ) = Q ( f ^ * , 0 ) < + \\infty \\forall c > 0 , \\ : y ^ * \\in \\Omega ^ * \\end{align*}"}
-{"id": "8725.png", "formula": "\\begin{align*} \\phi \\left ( z d _ 1 ^ { m - 1 } \\frac { x _ 1 ^ m - y _ 1 ^ m } { x _ 1 - y _ 1 } \\right ) = z d _ 1 ^ { n - 1 } \\frac { x _ 1 ^ n - y _ 1 ^ n } { x _ 1 - y _ 1 } . \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda _ 1 ( D ^ 2 u ) } + \\cdots + \\frac { 1 } { \\lambda _ n ( D ^ 2 u ) } = 1 , \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} \\mathcal { A } _ { w } ( f ) ( g ) = \\int _ { U _ w } f ( w ^ { - 1 } u g ) \\ , d u , \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} u _ t & = \\Delta u - \\chi \\nabla v \\cdot \\nabla u + u ( a ( x , t ) - u ( b ( x , t ) - \\chi \\mu ) - \\chi \\lambda v ) \\\\ & \\ge \\Delta u - \\chi \\nabla v \\cdot \\nabla u + u ( a _ { \\inf } - \\| u ( \\cdot , t ; t _ 0 , u _ 0 ) \\| _ \\infty ( b _ { \\sup } - \\chi \\mu ) - \\chi \\lambda \\frac { \\mu } { \\lambda } \\| u ( \\cdot , t ; t _ 0 , u _ 0 ) \\| _ \\infty ) \\\\ & = \\Delta u - \\chi \\nabla v \\cdot \\nabla u + u ( a _ { \\inf } - \\| u ( \\cdot , t ; t _ 0 , u _ 0 ) \\| _ \\infty b _ { \\sup } ) \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{align*} f _ { v e c t } ( t ) = \\begin{cases} 0 , & 0 \\le t \\le \\frac { 1 } { 5 } , \\\\ 5 t ( 5 t - 1 ) , & \\frac { 1 } { 5 } \\le t \\le \\frac { 4 } { 5 } , \\\\ 2 0 ( 2 t - 1 ) , & \\frac { 4 } { 5 } \\le t \\le 1 . \\end{cases} \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{align*} u ( m ) : = P \\{ T _ 1 = T _ { \\min } \\ , | \\ , X ( 0 ) = m \\} , 0 \\leq m _ j \\leq M _ j , \\ j = 1 \\dots , g . \\end{align*}"}
-{"id": "6746.png", "formula": "\\begin{align*} \\textrm { L e t } ~ A = ( R _ { x ^ \\lambda } R _ { x } L _ { x \\alpha } , R _ { x \\phi ^ { - 1 } } , R _ { x \\phi ^ { - 1 } } L _ { x \\alpha } ) ~ \\textrm { a n d } ~ B = ( R _ { x ^ \\lambda } R _ { x } L _ { x } , R _ { x } , R _ { x } L _ { x } ) \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} 1 - ( 1 - \\varepsilon ) \\mu \\gamma = \\frac { 1 - \\kappa } { 1 + \\kappa } + \\varepsilon , \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} \\epsilon \\langle - \\Delta _ { m } u _ { \\epsilon } , u _ { \\epsilon } - U _ { k } \\rangle + \\langle - \\Delta _ { \\Phi _ { \\epsilon } } u _ \\epsilon , u _ { \\epsilon } - U _ { k } \\rangle = \\int _ { \\Omega } f ( u _ { \\epsilon } - U _ { k } ) d x . \\end{align*}"}
-{"id": "7385.png", "formula": "\\begin{align*} | \\xi + e _ 1 | ^ { 2 K } = ( 1 + 2 \\xi _ 1 + \\xi _ 1 ^ 2 + \\ldots + x _ d ^ 2 ) ^ K = \\xi _ 1 ^ { 2 K } + 2 K \\xi _ 1 ^ { 2 K - 1 } + Q _ K ( \\xi ) \\end{align*}"}
-{"id": "4296.png", "formula": "\\begin{align*} a _ { Y ' _ \\alpha } ( x , y ) & = \\begin{cases} Y _ \\alpha ( x ) - y - 1 , & x = l , \\alpha = r \\\\ Y _ \\alpha ( x ) - y , & \\end{cases} , \\\\ l _ { Y ' _ \\alpha } ( x , y ) & = \\begin{cases} Y _ \\alpha ^ T ( y ) - x - 1 , & y = w , \\alpha = r \\\\ Y _ \\alpha ^ T ( y ) - x , & \\end{cases} \\end{align*}"}
-{"id": "3368.png", "formula": "\\begin{align*} A = \\frac { \\Gamma ( \\tfrac 1 4 ) ^ 4 } { 4 \\pi ^ 2 } = 4 . 3 7 6 8 7 9 6 \\dots . \\end{align*}"}
-{"id": "7428.png", "formula": "\\begin{align*} \\Sigma ( t , q ) = 2 \\beta ^ { - 1 } ( t , q ) \\gamma ( t , q ) , \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\left ( \\int _ { \\Omega } | \\nabla u _ { 0 } | ^ { 2 } - \\int _ { \\Omega } a ( x ) u _ { 0 } ^ { 2 } \\right ) & = \\lim I _ { q _ { n } } ( u _ { n } ) \\leq \\lim I _ { q _ { n } } ( \\phi ) \\\\ & = \\frac { 1 } { 2 } \\left ( \\int _ { \\Omega } | \\nabla \\phi | ^ { 2 } - \\int _ { \\Omega } a ( x ) \\phi ^ { 2 } \\right ) < 0 , \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{align*} \\langle m ; d ^ p \\rangle = \\sum _ { \\substack { j \\in J \\\\ \\mathcal { L } ( e _ j ) < + \\infty } } \\langle m ^ { j } ; d ^ p \\rangle + \\sum _ { \\substack { j \\in J \\\\ \\mathcal { L } ( e _ j ) = + \\infty } } \\langle m ^ { j } ; d ^ p \\rangle . \\end{align*}"}
-{"id": "7393.png", "formula": "\\begin{align*} | V ( x ) | = \\frac { | ( T ( D ) - \\lambda ) u ( x ) | } { | u ( x ) | } \\lesssim ( 1 + | x | ) ^ { - 1 } . \\end{align*}"}
-{"id": "4895.png", "formula": "\\begin{align*} \\dd { M ^ x } ( t ) & = \\sigma \\left ( t , M ^ x ( t ) \\right ) \\dd { W } ( t ) , \\\\ M _ 0 ^ x & = x . \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} \\mathcal E _ { m } : = \\{ y \\in G ( \\eta ( m t _ n + r ) , \\phi _ { t _ n } ) , \\ , \\forall y \\in B _ { ( m + 1 ) t _ n + r } \\} \\end{align*}"}
-{"id": "1695.png", "formula": "\\begin{align*} M _ i \\leq \\binom { i - 1 } { d _ i } \\leq \\binom { n } { \\sqrt { n } / \\log n } \\leq n ^ { \\sqrt { n } / \\log n } = 2 ^ { \\sqrt { n } } . \\end{align*}"}
-{"id": "7376.png", "formula": "\\begin{align*} T ( D ) u ( x ) \\sim \\sum _ { j = 1 } ^ { \\infty } 4 ^ { - N j } T | _ { R _ { j } ^ * } \\chi _ { R _ j } ( x ) \\sim \\lambda u ( x ) + \\mathcal { O } ( ( | x ' | ^ 2 + | x _ d | ) ^ { - N - 1 } ) . \\end{align*}"}
-{"id": "5647.png", "formula": "\\begin{align*} s ' ( n , t ) \\ , = \\ , N _ 1 ( T ) ^ M \\cdot N _ 3 ( T ) ^ { M ( M - 1 ) / 6 } , \\end{align*}"}
-{"id": "6145.png", "formula": "\\begin{align*} S \\overset { d } { = } \\delta + \\gamma \\left ( \\frac { 1 + \\beta } { 2 } \\right ) ^ { \\frac { 1 } { \\alpha } } S _ 1 - \\gamma \\left ( \\frac { 1 - \\beta } { 2 } \\right ) ^ { \\frac { 1 } { \\alpha } } S _ 2 . \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} v _ p ( [ z ^ n ] F ( z ) ) & \\geq \\min _ { n _ 0 + n _ 1 p + \\dots = n } \\sum _ { i \\geq 0 } \\underbrace { v _ p \\left ( [ z ^ { n _ i p ^ i } ] F _ i ( z ) \\right ) } _ { } \\\\ & \\geq \\min _ { n _ 0 + n _ 1 p + \\dots = n } \\sum _ { b _ i \\leq 0 } v _ p \\left ( [ z ^ { n _ i p ^ i } ] F _ i ( z ) \\right ) \\\\ & = \\min _ { n _ 0 + n _ 1 p + \\dots = n } \\sum _ { b _ i \\leq 0 } \\left ( b _ i n _ i - v _ p \\left ( n _ i ! \\right ) \\right ) \\end{align*}"}
-{"id": "5091.png", "formula": "\\begin{align*} \\sum _ { x _ i \\in T _ q } f _ i ( x _ i ) ^ { \\sum _ { j = 1 } ^ r \\sum _ { k = 1 } ^ { L _ j } h _ { i j k } \\beta _ { j k } } = q f _ i ( 0 ) ^ { \\sum _ { j = 1 } ^ r \\sum _ { k = 1 } ^ { L _ j } h _ { i j k } \\beta _ { j k } } + ( q - 1 ) B _ i , \\end{align*}"}
-{"id": "5667.png", "formula": "\\begin{align*} ( 1 - P _ { \\frac k m } ) R _ \\delta & = \\left ( ( 1 - P _ { \\frac k m } ) ( 1 - P _ \\delta ) T ( 1 - P _ \\delta ) ( 1 - P _ { \\frac k m } ) \\right ) ^ { - 1 } \\\\ & = \\left ( ( 1 - P _ { \\frac k m } ) T ( 1 - P _ { \\frac k m } ) \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} u = 0 ( 0 , T ) \\times \\partial \\Omega , \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{align*} \\mu ( \\alpha ) = 0 , \\ \\alpha \\in ( \\alpha _ 1 , 1 ) . \\end{align*}"}
-{"id": "8718.png", "formula": "\\begin{align*} \\frac { d } { d u } M _ { \\rm B } ( u ) = - E ( u ) . \\end{align*}"}
-{"id": "9227.png", "formula": "\\begin{align*} & \\sum _ { x ^ { ( 2 ) } \\in \\Z } \\cdots \\sum _ { x ^ { ( M ) } \\in \\Z } \\P ^ { 0 , 0 } _ { 2 T } ( X ( t _ m ) = x ^ { ( m ) } , m \\in \\{ 1 , 2 , \\dots , M \\} ) \\\\ & = \\P ^ { 0 , 0 } _ { 2 T } ( X ( t _ 1 ) = x ^ { ( 1 ) } ) \\\\ & = Q ( 0 , 0 ; t _ 1 , x ^ { ( 1 ) } ) \\frac { Q ( t _ 1 , x ^ { ( 1 ) } ; 2 T , 0 ) } { Q ( 0 , 0 ; 2 T , 0 ) } . \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} \\tilde { T } _ { \\rm l } = \\tilde { L } _ Q \\tilde { U } _ 2 \\tilde { \\Sigma } _ 2 ^ { - \\frac { 1 } { 2 } } \\mbox { a n d } \\tilde { T } _ { \\rm r } = \\tilde { L } _ P \\tilde { V } _ 2 \\tilde { \\Sigma } _ 2 ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{align*} U ( t ) = \\exp ( i t A ) . \\end{align*}"}
-{"id": "7974.png", "formula": "\\begin{align*} U ( x ) : = u ( x ) - L ( x ) \\leq C | x | \\sigma ( | x | ) B _ { r _ 0 } ( 0 ) \\end{align*}"}
-{"id": "3873.png", "formula": "\\begin{align*} \\gamma = \\frac { 2 ( 1 - \\delta ) ^ 4 - \\varepsilon ( 1 + ( 1 - \\delta ) ^ 4 ) } { ( 1 - \\varepsilon ) ( 1 - \\delta ) ^ 2 ( ( 1 - \\delta ) ^ 4 + 1 ) } , \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} \\int _ { { \\cal X } } L ^ X \\varphi ( x , y ) d \\mu ^ y = 0 . \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} [ L , J ] = [ \\Lambda , J ] = [ L , d _ B ] = [ \\Lambda , \\delta _ B ] = 0 . \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} ( a _ { 3 n + \\varepsilon } \\ , , a _ { 3 n + \\varepsilon + 1 } ) = A ( \\varepsilon ) \\left ( \\begin{matrix} a _ n \\\\ a _ { n + 1 } \\end{matrix} \\right ) \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} \\sum \\limits _ { a = 1 } ^ { m + n } \\sum \\limits _ { \\alpha , \\beta = m + 1 } ^ { m + n } | g _ { \\alpha \\bar { \\beta } a } | ^ 2 = 0 . \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} F _ 1 & : = \\int _ { - \\infty } ^ y G _ { \\epsilon } ( v - z ) \\psi _ { \\epsilon } ( s , v ) d z , \\\\ F _ 2 & : = \\int _ y ^ { \\infty } G _ { \\epsilon } ( v - z ) \\big [ \\psi _ { \\epsilon } ( s , v ) - \\psi ( s , v ) \\big ] d z , \\\\ F _ 3 & : = \\int _ y ^ { \\infty } G _ { \\epsilon } ( v - z ) \\big [ \\psi ( s , v ) - \\psi ( s , z ) \\big ] d z . \\end{align*}"}
-{"id": "3717.png", "formula": "\\begin{align*} 2 g ( a ) & > g ( a - 1 ) + g ( a + 1 ) a \\in \\{ 2 , \\dots , c - 1 \\} \\ , , \\\\ 2 g ( a ) & \\le g ( a - 1 ) + g ( a + 1 ) a = c \\ , , \\\\ 2 g ( a ) & < g ( a - 1 ) + g ( a + 1 ) a \\in \\{ c + 1 , \\dots , n - 1 \\} \\ , . \\end{align*}"}
-{"id": "6992.png", "formula": "\\begin{align*} N ^ - ( z ) = \\{ ( 2 , 1 ) , ( 3 , 1 ) , ( 5 , 1 ) , ( 6 , 1 ) , ( 7 , 1 ) , ( 5 , 4 ) , ( 6 , 4 ) , ( 7 , 4 ) , ( 7 , 6 ) \\} \\end{align*}"}
-{"id": "141.png", "formula": "\\begin{align*} [ \\dot \\eta \\wedge \\Phi _ \\infty ] + [ \\eta \\wedge \\dot \\Phi _ \\infty ] = 0 \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} B = \\begin{pmatrix*} [ r ] \\cos ( \\theta ) & - \\sin ( \\theta ) \\\\ \\sin ( \\theta ) & \\cos ( \\theta ) \\end{pmatrix*} . \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} ( b ) & = \\bigcup _ { r \\geq 1 } ( b ^ r ) \\\\ & = \\bigcup _ { s \\geq 0 } Q C ( \\bar \\iota _ b ^ \\pm , K , s ) \\\\ & = \\{ x \\in X : K _ x \\cong K , \\bar \\iota _ x ^ \\pm = \\bar \\iota _ b ^ \\pm \\} \\end{align*}"}
-{"id": "6727.png", "formula": "\\begin{align*} \\dot { x } \\left ( t \\right ) = A x \\left ( t \\right ) + B \\left ( t \\right ) u ( t ) + D \\left ( t \\right ) w \\left ( t \\right ) , \\end{align*}"}
-{"id": "4742.png", "formula": "\\begin{align*} \\mathcal { S } ( p ) : = \\big \\{ x \\in \\mathbb { X } \\ | \\ 0 \\in F ( p , x ) + \\mathcal { N } _ \\Gamma ( x ) \\big \\} . \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} \\partial _ { d _ k ^ V } \\left ( \\varepsilon _ { q , m } ^ { V , s } \\right ) = \\left \\langle u _ { q , m } ^ { V , s } , \\partial _ { d _ k ^ V } A ^ V _ q , u _ { q , m } ^ { V , s } \\right \\rangle = \\langle u _ { q , m } ^ { V , s } , \\sin ( k \\cdot ) u _ { q , m } ^ { V , s } \\rangle _ { L ^ 2 _ { \\rm p e r } } . \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} K _ D : D \\times D & \\rightarrow \\C \\\\ K _ D ( z , w ) & = \\det ( B ( z , w ) ) ^ { - 1 } . \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{align*} \\| f \\| _ { n } & : = \\sup _ { | \\alpha | \\leq n } \\sup _ { x \\in \\R ^ d } | \\rho ( x ) ^ { - \\ell } \\rho ( x ) ^ { \\gamma _ 1 \\alpha _ 1 } \\ldots \\rho ( x ) ^ { \\gamma _ d \\alpha _ d } f ( x ) | , f \\in S ^ { \\ell } _ { \\gamma } ( \\R ^ d ) , \\\\ \\| a \\| _ { n , m } & : = \\sup _ { | \\alpha | \\leq n } \\sup _ { x \\in \\R ^ d } | \\langle \\xi \\rangle ^ { - m } \\partial ^ { \\alpha } a ( \\xi ) | , a \\in C _ { \\rm p o l } ^ { \\infty } ( \\R ^ d ) , \\end{align*}"}
-{"id": "6339.png", "formula": "\\begin{align*} - \\lambda _ k \\partial G ( U _ k ) = \\Omega _ k U _ k \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{align*} \\forall i \\neq j \\in [ 1 , n ] , [ x _ i , y _ j ] = t _ { i j } = t _ { j i } , \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\mathrm { d i v } \\left ( ( \\nabla \\phi _ { 1 } ) u _ { 0 } ^ { q _ { 0 } } \\right ) = \\int _ { \\partial \\Omega } \\frac { \\partial \\phi _ { 1 } } { \\partial \\nu } u _ { 0 } ^ { q _ { 0 } } = 0 , \\end{align*}"}
-{"id": "6634.png", "formula": "\\begin{align*} \\tilde { V } = V \\cup \\{ v _ 0 \\} , \\tilde { E } = E _ 1 \\cup E _ 2 , E _ 1 = E , E _ 2 = \\{ v _ 0 \\} \\times V . \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} ^ l _ { m } = & \\log \\left ( 1 + \\frac { \\frac { \\alpha _ l ^ 2 } { L \\left ( | | x _ m | | \\right ) } } { \\sum ^ { M _ s } _ { l = i + 1 } \\frac { \\alpha _ l ^ 2 } { L \\left ( | | x _ m | | \\right ) } + \\frac { 1 } { \\rho } } \\right ) , \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} d _ { A _ \\infty } - d _ { A _ t } = - \\bigl ( 2 f _ t ( r ) - \\tfrac { 1 } { 4 } \\bigr ) \\begin{pmatrix} i & 0 \\\\ 0 & - i \\end{pmatrix} d \\theta . \\end{align*}"}
-{"id": "2776.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { i \\in H } \\sum _ { j \\in C } \\sum _ { s , t \\in T } \\lambda _ { i , j } \\cdot q _ { i , s , j , t } \\\\ & \\sum _ { j \\in C } \\sum _ { t \\in T } q _ { i , s , j , t } = \\sigma _ { i , s } & i \\in H , s \\in T \\\\ & \\sum _ { i \\in H } \\sum _ { s \\in T } q _ { i , s , j , t } = \\delta _ { j , t } & j \\in C , t \\in T \\\\ & q _ { i , s , j , t } \\geq 0 & i \\in H , j \\in C , \\ ; s , t \\in T \\end{aligned} \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} \\bar { \\rho } ( h ) = \\big ( \\prod _ s M _ s ^ { n _ s } \\big ) \\bar { \\rho } ( \\sigma ' ) . \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} [ c , z _ { \\sigma ( 1 ) } ] [ z _ { \\sigma ( 2 ) } , z _ { \\sigma ( 3 ) } , z _ { \\sigma ( 4 ) } ] = ( - 1 ) ^ { \\sigma } [ c , z _ 1 ] [ z _ 2 , z _ 3 , z _ 4 ] . \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} & ( T ( D ) - \\lambda ) u ( k \\pi , x ' ) = f ( k \\pi , x ' ) \\\\ & - \\kappa \\sum _ { | \\alpha | \\in 2 \\N - 1 } \\frac { 1 } { \\alpha ! } \\partial ^ { \\alpha } T ( e _ 1 ) ( - 1 ) ^ { \\frac { | \\alpha | + 1 } { 2 } } \\partial ^ { \\alpha } ( ( x _ 1 - k \\pi ) ^ m \\chi ( x _ 1 - k \\pi ) f ( k \\pi , x ' ) ) | _ { x _ 1 = k \\pi } \\\\ & = f ( k \\pi , x ' ) ( 1 - \\kappa ( - 1 ) ^ { \\frac { m + 1 } { 2 } } \\partial _ 1 ^ m T ( e _ 1 ) ) = 0 . \\end{align*}"}
-{"id": "8801.png", "formula": "\\begin{align*} D = \\sum _ { 1 \\leq j \\leq r } z _ j l _ j + \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } } z _ { \\alpha } e _ { \\alpha } + \\sum _ { \\beta \\in \\Phi _ s ^ + } z _ { \\beta } \\tau _ { \\beta } + h \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{align*} = z _ { 2 } ^ { - 1 } \\delta \\left ( \\frac { z _ { 1 } - z _ { 0 } } { z _ { 2 } } \\right ) Y ( Y ( u , \\ z _ { 0 } ) v , \\ z _ { 2 } ) w . \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} I _ { N } ^ { \\delta } ( r ) = \\int _ { - 1 } ^ { - \\delta } i _ N ( r ^ 2 t ) d t . \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{align*} \\max _ { i \\mid a _ { i j } < 0 } ( a _ { i j } x _ j + y _ i ) = \\max _ { i \\mid a _ { i j } > 0 } ( a _ { i j } x _ j + y _ i ) . \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} \\chi _ 2 ( N ) : = \\int _ 0 ^ { \\infty } e ^ { - \\left ( \\zeta ^ { \\frac { ( 1 - \\theta ) } { N } } - 1 \\right ) t } e ^ { - \\theta t } \\left ( 1 - e ^ { - ( 1 - \\theta ) t / N } \\right ) ^ N d t . \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} C _ \\lambda ( x , y ) = \\frac { 1 } { 2 \\alpha \\sqrt { \\lambda } } \\begin{cases} \\alpha e ^ { i \\sqrt \\lambda ( x + y ) } , & x \\geq 0 , \\ , y \\geq 0 , \\\\ - e ^ { \\sqrt { \\lambda } ( i x + y ) } , & x \\geq 0 , \\ , y < 0 , \\\\ e ^ { \\sqrt { \\lambda } ( x + i y ) } , & x < 0 , \\ , y \\geq 0 , \\\\ - \\overline { \\alpha } e ^ { \\sqrt { \\lambda } ( x + y ) } , & x < 0 , \\ , y < 0 , \\end{cases} \\end{align*}"}
-{"id": "3802.png", "formula": "\\begin{align*} \\chi & = \\frac { - 1 } { 8 \\pi ^ 2 } \\int _ M | W _ { 0 0 } ^ + | ^ 2 + ( 6 0 v ^ 2 - 7 2 k v + 1 8 k ^ 2 ) \\\\ \\frac { 3 } { 2 } \\sigma & = \\frac { 1 } { 8 \\pi ^ 2 } \\int _ M 2 | W _ F ^ + | ^ 2 + | W _ { 0 0 } ^ + | ^ 2 + 6 ( 2 k - 3 v ) ^ 2 \\enskip \\geq \\enskip 0 \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{align*} 1 - \\frac { 1 } { d } - \\frac { \\tilde { C } _ d } { 2 } = 1 - \\frac { 1 } { d } - \\frac { C _ d } { 2 4 ( 1 + C _ d ) } \\geq 1 - \\frac { 1 } { d } - \\frac { 2 } { 2 4 } \\geq \\frac { 5 } { 1 2 } \\geq 3 \\epsilon , \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{align*} k _ 1 + 3 = \\frac { r + 1 - k _ 1 } { r - 1 } \\cdot 5 + \\frac { k _ 1 - 2 } { r - 1 } \\cdot ( r + 4 ) , \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Phi ^ { \\gamma } _ t ( x , s ) = v [ m _ t ] ( \\Phi ^ { \\gamma } _ { t } ( x , s ) ) , t \\in [ t _ m , t _ { m + 1 } ) . \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ { t } u + ( u \\cdot \\nabla ) u - \\nu \\Delta u + \\nabla \\pi & = - \\lambda ( [ \\nabla d ] ^ { \\top } \\nabla d ) & \\ ( 0 , T ) \\times \\Omega , \\\\ \\partial _ { t } d + ( u \\cdot \\nabla ) d & = \\gamma ( \\Delta d + \\vert \\nabla d \\vert ^ { 2 } d ) & \\ ( 0 , T ) \\times \\Omega , \\\\ \\div u & = 0 & \\ ( 0 , T ) \\times \\Omega , \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{align*} F ( x , y ) & : = \\left \\{ \\frac { u ( x ) } { 1 - \\varepsilon } - \\frac { v ( y ) } { 1 + \\varepsilon } - \\alpha _ 2 \\Psi _ 2 ( x , y ) - \\frac { \\varepsilon } { 1 - \\varepsilon } \\Upsilon ( x ) - \\frac { \\varepsilon } { 1 + \\varepsilon } \\Upsilon ( y ) \\right \\} , \\\\ G ( x , y ) & : = \\Psi _ 1 ( x , y ) . \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} ( \\xi ( x ) , \\sigma ( x ) ) = ( \\xi ^ z ( x ) , \\sigma ^ z ( x ) ) = ( \\hat { \\xi } ^ z _ L ( x ) , \\sigma ^ z ( x ) ) \\forall \\ , z \\in \\Pi _ { L , \\delta } , \\ ; x \\in B _ { R _ L } ( z ) \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} \\| \\overline { u } ^ { \\alpha + 1 } & \\| _ { L ^ { m ^ \\star } ( \\Omega _ { R _ 1 } ) } \\le C \\left [ \\frac { 1 } { R _ 1 - R _ 2 } + \\left ( \\frac { G ( R _ 1 , R _ 2 ) } { m \\alpha + 1 - C m \\epsilon ^ { m ' } } \\right ) ^ { 1 / m } \\right ] \\| \\overline { u } ^ { \\alpha + 1 } \\| _ { L ^ { m } ( \\Omega _ { R _ 2 } ) } \\\\ & + C \\left ( \\frac { \\| f \\| _ q } { ( m \\alpha + 1 - C m \\epsilon ^ { m ' } ) k ^ { m - 1 } } \\right ) ^ { N / m ( m q - N ) } \\| \\overline { u } ^ { \\alpha + 1 } \\| _ { L ^ { m } ( \\Omega _ { R _ 2 } ) } . \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{align*} \\overline { \\langle \\underline { p } _ { \\alpha } ( \\underline { g } \\cdot \\underline { k } ) \\rangle } = \\overline { \\langle \\underline { p } _ { \\beta \\alpha } ( \\underline { p } _ { \\beta } ( \\underline { g } \\cdot \\underline { k } ) ) \\rangle } = G _ { \\alpha } , \\end{align*}"}
-{"id": "3979.png", "formula": "\\begin{align*} \\epsilon _ 3 = \\max \\sum _ { t = 1 } ^ { | \\mathcal { Z } | } \\min _ { x , y : \\ : p _ { X , Y } ( x , y ) > 0 } \\delta _ { x , y , t } \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{align*} \\sum _ { l _ 1 + l _ 2 = N } X _ 1 ^ { ( l _ 1 ) } X _ { 2 3 } ^ { ( l _ 2 ) } - X _ 2 ^ { ( l _ 1 ) } X _ { 1 3 } ^ { ( l _ 2 ) } + X _ 3 ^ { ( l _ 1 ) } X _ { 1 2 } ^ { ( l _ 2 ) } \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} T _ n : = \\inf \\{ t \\ge T _ { n - 1 } : \\ V ( t ) \\ge V _ { t h } \\} \\ n \\ge 1 , \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} E _ d ( Q ) _ { q = 1 } = \\binom { d } { 2 } , \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} W _ i : = \\sum _ { \\sigma \\ni i } \\frac { v _ \\sigma } { 1 + n _ \\sigma } . \\end{align*}"}
-{"id": "7915.png", "formula": "\\begin{align*} X ( t ) : = X _ n ( t ) \\qquad \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} \\alpha = \\alpha _ 1 { + } \\alpha _ 2 , \\ \\gamma = ( \\gamma _ 1 { + } \\alpha _ 2 ) \\wedge ( \\gamma _ 2 { + } \\alpha _ 1 ) , \\ p = \\frac { p _ 1 p _ 2 } { p _ 1 { + } p _ 2 } , \\ q = q _ 1 \\vee q _ 2 , \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} z = ( v _ 1 , \\dots v _ { i - 1 } , \\tfrac 1 2 , v _ { i + 1 } , \\dots , v _ { j - 1 } , s , v _ { j + 1 } , \\dots , v _ { 2 m + 1 } ) \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{align*} \\inf _ { x \\in A } F ( x , \\lambda ^ * , c ) = f ^ * \\forall c \\ge c _ 0 , \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{align*} \\chi _ \\gamma ( u , z ) = \\begin{cases} \\gamma ( z ) | A | , & u = 0 _ K ; \\\\ 0 , & u \\neq 0 _ K \\end{cases} ( u \\in K , z \\in C _ { \\exp ( A ) } ) . \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{align*} & \\chi _ 1 ( z ) = - \\frac { 1 } { 2 } \\delta ^ { i _ 1 i _ 2 } ( \\nabla _ q \\beta ) ^ { i _ 3 } G _ { i _ 1 i _ 2 i _ 3 } ^ { j _ 1 j _ 2 j _ 3 } \\\\ & \\times \\left ( z _ { j _ 1 } z _ { j _ 2 } z _ { j _ 3 } + 2 \\beta ^ { - 1 } \\left ( \\gamma _ { j _ 1 j _ 2 } \\delta _ { j _ 3 l } ( \\tilde \\gamma ^ { - 1 } ) ^ { l k } + \\gamma _ { j _ 1 j _ 3 } \\delta _ { j _ 2 l } ( \\tilde \\gamma ^ { - 1 } ) ^ { l k } + \\gamma _ { j _ 2 j _ 3 } \\delta _ { j _ 1 l } ( \\tilde \\gamma ^ { - 1 } ) ^ { l k } \\right ) z _ k \\right ) , \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{align*} g ^ { 2 } ( t ) f ' ( t ) \\langle X ( x ) , T ( x ) \\rangle + g ( t ) g ' ( t ) \\langle X , N \\rangle + g ( t ) \\langle e ^ { - i f ( t ) } H ' ( t ) , N \\rangle = \\kappa ( x ) . \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n a _ { i j } \\frac { \\partial ^ 2 \\delta ^ \\gamma } { \\partial x _ i \\partial x _ j } \\leq m \\Lambda \\gamma \\delta ^ { \\gamma - 2 } \\leq n \\Lambda \\gamma \\delta ^ { \\gamma - 2 } , \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} \\langle \\tilde \\psi , R ^ P ( \\tilde \\psi , \\tilde \\psi ) \\tilde \\psi \\rangle _ { S M \\otimes \\phi ^ \\ast T P } = R _ { I J K L } \\langle \\tilde \\psi ^ I , \\tilde \\psi ^ K \\rangle _ { S M } \\langle \\tilde \\psi ^ J , \\tilde \\psi ^ L \\rangle _ { S M } , \\end{align*}"}
-{"id": "4203.png", "formula": "\\begin{align*} \\Gamma ( f , g ) & = \\tfrac 1 2 ( L ( f g ) - g L f - f L g ) = \\sum _ { i , j = 1 } ^ n a _ X ^ { i j } \\partial _ { x _ i } f \\partial _ { x _ j } g + \\sum _ { i , j = 1 } ^ m a _ Y ^ { i j } \\partial _ { y _ i } f \\partial _ { y _ j } g , \\\\ \\Gamma ^ X ( f , g ) & = \\tfrac 1 2 ( L ^ X ( f g ) - g L ^ X f - f L ^ X g ) = \\sum _ { i , j = 1 } ^ n a _ X ^ { i j } \\partial _ { x _ i } f \\partial _ { x _ j } g . \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} Y _ { j + 1 , n } = Y _ { j , n } + F ( Y _ { j , n } ) \\biggl ( X _ n \\biggl ( { \\frac { j + 1 } { n } } \\biggr ) - X _ n \\biggl ( { \\frac { j } { n } } \\biggr ) \\biggr ) , Y _ { 0 , n } = \\xi . \\end{align*}"}
-{"id": "5533.png", "formula": "\\begin{align*} Z _ M ( s ) = \\sum _ { k = 1 } ^ \\infty \\l _ k ^ { - s } , R e ( s ) > 1 . \\end{align*}"}
-{"id": "1722.png", "formula": "\\begin{align*} \\bigg \\| \\sum _ { k = k _ 0 } ^ \\infty 2 ^ { - k \\alpha } \\mathcal { C } _ k g \\bigg \\| _ { L ^ { q , \\infty } _ t L ^ 2 _ x } \\lesssim \\| g \\| _ { L ^ { s , 1 } _ t L ^ 2 _ x } \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} r ^ { \\epsilon } ( M , s , t ) = \\mathrm { r a n k } \\left ( M ( s , t ) \\right ) - 1 \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{align*} \\| \\Theta _ { k + 1 } \\| _ { C ^ { 1 , \\alpha } _ \\nu } \\leq Q \\sum _ { m = 2 } ^ { k + 1 } { C _ m \\left ( \\sum _ { I \\in \\mathcal { I } _ { m , k } } { \\| \\varphi _ 1 \\| _ { C ^ { 1 , \\alpha } _ { - 1 } } ^ { i _ 1 } \\| \\varphi _ 2 \\| _ { C ^ { 1 , \\alpha } _ { \\nu } } ^ { i _ 2 } \\dots \\| \\varphi _ { k } \\| _ { C ^ { 1 , \\alpha } _ { \\nu } } ^ { i _ { k } } } \\right ) } \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} \\phi _ U ( \\mathbf { u } ) - \\phi _ V ( \\mathbf { u } ) = \\phi _ { U V } ( \\zeta ) + \\overline { \\phi _ { U V } ( \\zeta ) } \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{align*} | p | _ { C ^ \\tau S ^ m _ { 1 , 0 } } ^ { i } : = \\underset { | \\alpha | \\leq i } { \\max } \\ \\underset { \\xi \\in \\R ^ n } { \\sup } \\left \\lVert \\partial _ { \\xi } ^ \\alpha p ( \\cdot , \\xi ) \\right \\lVert _ { C ^ \\tau ( \\R ^ n ; X ) } \\langle \\xi \\rangle ^ { - m + | \\alpha | } . \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} g ( 2 , 3 ) & > g ( 1 , 4 ) \\ , , \\\\ g ( 3 , 3 ) & > g ( 1 , 5 ) \\ , , \\\\ g ( 3 , 4 ) & > g ( 1 , 6 ) \\ , , \\\\ g ( 1 , 2 k - 1 ) & > g ( k , k ) k \\ge 4 \\ , , \\\\ g ( 1 , 2 k ) & > g ( k , k + 1 ) k \\ge 4 \\ , , \\\\ g ( 1 , 6 , 6 ) & > g ( 4 , 4 , 5 ) \\ , , \\\\ g ( 4 , 4 , 4 ) & \\ge g ( 1 , 5 , 6 ) \\ , . \\end{align*}"}
-{"id": "5346.png", "formula": "\\begin{align*} \\widetilde { B } = ( B - C ^ * A ^ { - 1 } C ) ^ { - 1 } . \\end{align*}"}
-{"id": "6763.png", "formula": "\\begin{align*} R _ { x ^ { \\lambda } \\cdot x \\phi ^ { - 1 } } = L ^ { - 1 } _ x R ^ { - 1 } _ x R _ { x \\phi ^ { - 1 } } L _ x \\Rightarrow ( x \\cdot y x ) ( x ^ { \\lambda } \\cdot x \\phi ^ { - 1 } ) = x ( y \\cdot x \\phi ^ { - 1 } ) \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} A \\phi \\left ( t \\right ) = g \\left ( t \\right ) \\int _ { a } ^ { t } \\psi ^ { \\prime } \\left ( \\tau \\right ) \\left ( \\psi \\left ( t \\right ) - \\psi \\left ( \\tau \\right ) \\right ) ^ { \\alpha - 1 } \\phi \\left ( \\tau \\right ) d \\tau , \\end{align*}"}
-{"id": "7314.png", "formula": "\\begin{align*} \\hat { A } = \\{ \\mathcal U \\in \\beta \\omega : A \\in \\mathcal U \\} , \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} \\ll \\phi , \\psi \\gg = \\int _ M \\langle \\phi , \\psi \\rangle \\mu _ M , \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{align*} \\gamma _ { k , n } ^ { ( \\alpha ) } = \\sum _ { j = 1 } ^ { q } \\sum _ { t = 0 } ^ { \\tau _ j - 1 } \\beta _ n ( j , t ) \\left ( ( z - \\lambda _ j ) ^ { \\tau _ j } F _ \\alpha ( z ) \\Phi ^ { n - k } ( z ) \\right ) ^ { ( t ) } _ { z = \\lambda _ j } \\end{align*}"}
-{"id": "3736.png", "formula": "\\begin{align*} q _ k : = \\sum _ { u \\in \\{ 1 , 2 \\} ^ r : ( u ) _ 1 = k } p _ u = p ^ k ( 1 - p ) ^ { r - k } | \\{ u : ( u ) _ 1 = k \\} | . \\end{align*}"}
-{"id": "3723.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\varphi ( t ) \\varrho ( t ) \\ , d t & = \\int _ 0 ^ { x _ 0 } \\varphi ( t ) \\varrho ( t ) \\ , d t + \\int _ { x _ 0 } ^ 1 \\varphi ( t ) \\varrho ( t ) \\ , d t \\\\ & \\ge \\int _ 0 ^ { x _ 0 } \\varphi ( x _ 0 ) \\varrho ( t ) \\ , d t + \\int _ { x _ 0 } ^ 1 \\varphi ( x _ 0 ) \\varrho ( t ) \\ , d t \\\\ & = \\varphi ( x _ 0 ) \\int _ 0 ^ { 1 } \\varrho ( t ) \\ , d t \\\\ & > 0 ( \\ge 0 ) \\ , . \\end{align*}"}
-{"id": "3978.png", "formula": "\\begin{align*} \\log \\frac { 1 } { \\epsilon _ 1 } & = \\log F ( p _ { X Y } ) = \\min _ { q _ X , q _ Y } \\ ! \\bigg ( D _ \\infty ( p _ { X Y } \\| q _ X \\ , q _ Y ) + D _ \\infty ( q _ X \\ , q _ Y \\| p _ { X Y } ) \\bigg ) . \\end{align*}"}
-{"id": "1893.png", "formula": "\\begin{align*} d _ n = \\sum _ { a = 1 } ^ { n - 2 } \\sum _ { b = a + 1 } ^ { n - 1 } d _ n ( a , b ) , n \\geq 3 , \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} d _ z t ( \\tau , z ) = \\eta ( \\tau , z ) \\big ( \\tau - \\tau _ + ( z ) \\big ) ^ { - 1 } \\end{align*}"}
-{"id": "5969.png", "formula": "\\begin{align*} \\langle \\ ! \\langle \\phi , \\psi \\rangle \\ ! \\rangle = \\int _ 0 ^ T \\ ! \\phi ( t ) \\psi ( t ) d t . \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} \\frac { 1 } { \\varrho _ k R _ k } = \\varphi \\ ! \\left ( \\frac { 1 } { \\varrho _ k } \\right ) = \\left ( \\log \\frac { 1 } { \\varrho _ k } \\right ) ^ 2 . \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} \\begin{array} [ p o s ] { l } \\nabla _ { B } \\nabla _ { B } h ( X , Y ) ( p _ 1 ) = 0 , \\\\ \\noalign { \\smallskip } R i c _ { B } ( X , Y ) ( p _ 1 ) + \\nabla _ { B } \\nabla _ { B } \\beta ( X , Y ) ( p _ 1 ) = 0 \\end{array} \\forall p _ 1 \\in B . \\end{align*}"}
-{"id": "582.png", "formula": "\\begin{align*} T _ { f + g } \\le T _ f + T _ g + O ( 1 ) \\ \\ T _ { f g } \\le T _ f + T _ g + O ( 1 ) \\ \\ T _ { f ^ { n } } = | n | T _ f + O ( 1 ) \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} \\sigma = F _ 0 ( \\phi _ n ) = F _ + ( \\phi _ { n - 1 } ) = F _ - ( \\phi _ { n + 1 } ) \\end{align*}"}
-{"id": "4965.png", "formula": "\\begin{align*} \\mathcal { H } \\partial _ x \\psi + c \\psi - \\gamma \\partial _ x ^ { - 2 } \\psi = \\psi ^ 2 \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} x = 0 , a = b ' , u + 2 v + w = k + l . \\end{align*}"}
-{"id": "7808.png", "formula": "\\begin{align*} { \\rm v o l } ( L ) = d ! { \\rm v o l } ( \\Delta _ { Y _ \\bullet } ( L , X ) ) \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} \\omega _ f ( x ) = \\bigcap _ { m \\geq 0 } \\overline { \\bigcup _ { k \\geq m } \\{ f ^ k ( x ) \\} } . \\end{align*}"}
-{"id": "5495.png", "formula": "\\begin{align*} m \\ddot { q } _ j + c ( 2 \\dot { q } _ { j } - \\dot { q } _ { j + 1 } - \\dot { q } _ { j - 1 } ) + k ( 2 q _ { j } - q _ { j + 1 } - q _ { j - 1 } ) + \\delta _ { 1 j } \\kappa q _ { j } ^ 3 = f _ j ( t ) , j = 1 , . . . , N . \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} K _ \\nu ( z ) = \\frac { \\pi } { 2 } \\frac { I _ { - \\nu } ( z ) - I _ \\nu ( z ) } { \\sin \\pi \\nu } , \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{align*} \\mathcal { F } _ { q } ( q , u ) = - a \\left ( x \\right ) u ^ { q } \\log u . \\end{align*}"}
-{"id": "4929.png", "formula": "\\begin{align*} ( A + \\frac { k ^ 2 } { 2 } I _ n ) ^ T P ^ { - 1 } + P ^ { - 1 } ( A + \\frac { k ^ 2 } { 2 } I _ n ) + \\sum _ { i = 1 } ^ m N _ i ^ T P ^ { - 1 } N _ i & \\leq - P ^ { - 1 } B B ^ T P ^ { - 1 } , \\\\ ( A + \\frac { k ^ 2 } { 2 } I _ n ) ^ T Q + Q ( A + \\frac { k ^ 2 } { 2 } I _ n ) + \\sum _ { i = 1 } ^ m N _ i ^ T Q N _ i & = - C ^ T C . \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} \\int _ \\R h \\ , \\kappa \\ , d x \\overset { \\eqref { k a p h } } = - \\int _ \\R \\frac { h _ x ^ 2 } { \\sqrt { 1 + h _ x ^ 2 } } \\ , d x \\overset { \\eqref { h x o n e } , \\eqref { l 1 . 2 } } \\sim - E . \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} \\widehat { \\eta } _ { \\widehat { R } } + \\frac { K ^ 2 } { 4 \\widehat { R } ^ 3 } e ^ { 2 \\widehat { \\eta } } - \\widehat { a } \\widehat { R } \\left ( \\widehat { a } ^ { - 1 } \\widehat { U } _ { \\widehat { R } } ^ 2 + \\widehat { a } \\widehat { U } _ { \\widehat { \\theta } } ^ 2 \\right ) = \\frac { 5 } { 4 \\widehat { R } } . \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{align*} E \\left [ S - T _ 1 \\right ] = E \\left [ \\left ( S - T _ 1 \\right ) \\mathbf { 1 } _ { \\{ T _ 1 < S \\} } \\right ] = E \\left [ S - T _ 1 \\ , | \\ , T _ 1 < S \\right ] P \\{ T _ 1 < S \\} . \\end{align*}"}
-{"id": "1521.png", "formula": "\\begin{align*} \\mathcal { L } \\log w & = \\dfrac { \\Delta w + \\dfrac 1 2 \\left < x , \\nabla w \\right > } { w } - \\dfrac { | \\nabla w | ^ 2 } { w ^ 2 } \\\\ & = - | A | ^ 2 + \\dfrac 1 2 - \\mu _ 1 - | \\nabla \\log w | ^ 2 \\end{align*}"}
-{"id": "366.png", "formula": "\\begin{align*} \\varphi _ O ( u , z ) = \\begin{cases} \\frac { 1 } { 2 } \\gamma ( z ^ { - 1 } ) \\left ( | A | - 1 \\right ) , & u = 0 \\\\ - \\frac { 1 } { 2 } \\gamma ( z ^ { - 1 } ) , & u \\neq 0 \\end{cases} \\left ( ( u , z ) \\in H \\right ) . \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{align*} \\exp \\left ( \\frac { \\partial \\mathcal { I } ^ { ( - 1 ) } ( \\sigma ) } { \\partial \\sigma } \\right ) = \\exp \\left ( \\frac { \\partial \\mathcal { W } _ { \\rm e f f } ( \\sigma ) } { \\partial \\sigma } \\right ) = 1 \\ ; . \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} \\phi _ \\lambda ( v ) = \\frac { 1 } { T ( v ) - \\lambda v } , \\int _ { - 1 } ^ 1 \\frac { T ( v ) } { T ( v ) - \\lambda v } d \\nu ( v ) = 1 . \\end{align*}"}
-{"id": "6585.png", "formula": "\\begin{align*} \\int \\limits _ 0 ^ T \\int \\limits _ 0 ^ T ( 1 + | t - s | ) ^ { - C } \\ , d t \\ , d s = 2 \\int _ 0 ^ T \\int \\limits _ s ^ T ( 1 + t - s ) ^ { - C } \\ , d t \\ , d s . \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} \\mathrm { e v } ( \\alpha \\cdot ( \\varphi \\otimes i ) ) = \\alpha \\cdot \\mathrm { e v } ( \\varphi \\otimes i ) \\ , . \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{align*} { \\bf D } _ { C } ^ { - 1 } = ( { \\bf I } _ K - { \\bf \\tilde G } _ c ) ( { \\rm d i a g } _ 0 ( { \\bf G } ) ) ^ { - 1 } \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} \\mathbf { x } _ { j \\Omega } = \\begin{cases} \\sum _ { m = 0 } ^ { M } \\rho ^ { 2 m + j } e ^ { j i \\psi ( \\rho ) } \\mathbf { w } _ 0 ^ { ( m + j , m ) } + \\delta _ { 1 j } \\varepsilon \\mathbf { W } ^ + , & 0 \\leq j \\leq M , \\\\ [ 0 . 2 m m ] \\sum _ { m = 0 } ^ { M } \\rho ^ { 2 m + j } e ^ { - j i \\psi ( \\rho ) } \\mathbf { w } _ 0 ^ { ( m , m + j ) } + \\delta _ { - 1 j } \\varepsilon \\mathbf { W } ^ - , & - M \\leq j < 0 , \\end{cases} \\end{align*}"}
-{"id": "4807.png", "formula": "\\begin{align*} \\left \\{ \\ \\begin{array} { l } \\displaystyle - \\Delta _ { \\Phi _ { \\epsilon } } u = g ( x , u ) , ~ \\mbox { i n } ~ \\Omega , \\\\ \\\\ u = 0 ~ \\mbox { o n } ~ \\partial \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "8928.png", "formula": "\\begin{align*} \\mathrm { M a b } _ { \\Theta } ( \\phi ) & = \\frac { n ! } { \\mathcal { L } ^ n } M ^ l ( \\hat { u } ^ * ) \\\\ & \\geq \\epsilon \\frac { n ! } { \\mathcal { L } ^ n } \\int _ { \\Delta ' } \\hat { u } ^ * P _ { D H } ' - \\frac { n ! } { \\mathcal { L } ^ n } ( \\epsilon C _ 3 + C _ 4 ) \\\\ & \\geq \\epsilon J ( \\hat { \\phi } ) - \\frac { n ! } { \\mathcal { L } ^ n } ( C _ 5 + \\epsilon C _ 3 + C _ 4 ) \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} v _ p ( \\ell _ 1 ) + h - v _ p ( i ) = v _ p ( \\ell _ 2 ) + h - v _ p ( j ) . \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{align*} U _ { \\epsilon , R } = \\{ ( \\beta , \\tilde \\gamma , B ) : \\beta > \\epsilon , \\} , \\end{align*}"}
-{"id": "8494.png", "formula": "\\begin{align*} \\phi _ { W W } ( f _ i ) = \\{ \\pi _ { W _ i } S _ { W } ^ { - 1 } f _ i \\} _ { i \\in I } = \\{ \\pi _ { W _ i } S _ { W } ^ { - 1 } \\pi _ { W _ i } f _ i \\} _ { i \\in I } = \\{ \\frac { 1 } { \\omega _ i ^ 2 } f _ i \\} _ { i \\in I } . \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} \\psi _ z = \\psi _ z ^ { \\varphi ^ a } + \\psi _ 0 ^ { \\varphi ^ s } . \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} - \\delta _ f ( P ( t , x ) ) + \\frac { \\partial f ( t , x ) } { \\partial x } P ( t , x ) + P ( t , x ) { \\frac { \\partial f ( t , x ) } { \\partial x } } ^ \\top + g ( t , x ) g ( t , x ) ^ \\top & = 0 \\\\ [ 1 e x ] - \\delta _ { g _ j } ( P ( t , x ) ) + \\frac { \\partial g _ j ( t , x ) } { \\partial x } P ( t , x ) + P ( t , x ) { \\frac { \\partial g _ j ( t , x ) } { \\partial x } } ^ \\top & = 0 \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} L ^ J ( s / t ) = \\lambda ^ J ( s / t ) \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} \\mathcal { H } ( y ) = \\frac { y + \\sigma ( y ) } { 2 } \\in \\mathfrak { h } \\mathrm { a n d } \\mathcal { P } ( y ) = \\frac { y - \\sigma ( y ) } { 2 } . \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{align*} \\norm { \\theta ^ n } \\le \\frac { C t _ n ( k ^ 2 + h ^ r ) } { 1 - \\tilde { C } k } \\exp \\Big [ \\frac { 1 + C _ 1 } { 1 - \\tilde { C } k } + \\sum _ { j = 1 } ^ { n - 1 } \\frac { \\tilde { C } k _ j + C _ { j + 1 } } { 1 - \\tilde { C } k } \\Big ] . \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{align*} s : = t - r _ * , \\ \\ r _ * : = r + 2 m \\log ( r - 2 m ) , \\end{align*}"}
-{"id": "2715.png", "formula": "\\begin{align*} \\frac { 1 } { R _ 2 ^ { m - 2 } } \\int _ { B _ { R _ 2 } } | \\dd \\phi | ^ 2 \\dd x - \\frac { 1 } { R _ 1 ^ { m - 2 } } \\int _ { B _ { R _ 1 } } | \\dd \\phi | ^ 2 \\dd x = \\int _ { R _ 1 } ^ { R _ 2 } F ( r ) \\dd r \\end{align*}"}
-{"id": "1831.png", "formula": "\\begin{align*} \\mathcal { A } Y = \\lambda \\mathcal { B } Y \\ , , \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} H _ { n , t } \\ , = \\ , \\begin{pmatrix} 1 & \\ldots & 1 \\\\ & B _ { n , t } & \\end{pmatrix} \\end{align*}"}
-{"id": "614.png", "formula": "\\begin{gather*} \\delta > 0 \\quad \\frac { 1 } { M } = \\gamma ^ { \\delta / q } , \\\\ \\beta _ * > 2 ( 2 K ) ^ { 1 - \\alpha } \\beta \\Bigl ( 1 - \\bigl ( 1 + \\frac { 1 } { M } \\bigr ) ^ { - \\alpha / \\delta } \\Bigr ) ^ { - 1 } > 2 \\beta ( 2 K ) ^ { 1 - \\alpha } , \\\\ k _ 0 \\in \\mathbb { N } , k _ 0 > \\frac { q } { \\log \\gamma } \\log ( c _ 1 ( 2 ^ { \\frac { 1 } { 1 - \\alpha } } - 1 ) ) , \\\\ \\end{gather*}"}
-{"id": "1363.png", "formula": "\\begin{align*} & V ^ { u } \\bigl ( t , x \\bigr ) = \\mathcal { E } ^ { G } \\bigl [ \\xi _ { t , T } \\bigl ( u \\bigr ) \\bigl \\vert \\mathcal { F } _ t \\bigr ] u _ { \\cdot } \\in \\mathcal { U } _ { [ t , T ] } , \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} \\left \\| y _ n ^ k \\partial _ { y _ n } ^ { k ' } w _ n ^ l \\partial _ { w _ n } ^ { l ' } \\partial _ { \\xi ' } ^ \\alpha \\partial _ { x } ^ \\beta q _ { z } ( x , \\xi ' , \\cdot , \\cdot ) \\right \\| _ { X _ 2 } & = \\left \\| y _ n ^ k \\partial _ { y _ n } ^ { k ' } w _ n ^ l \\partial _ { w _ n } ^ { l ' } \\partial _ { \\xi ' } ^ \\alpha \\partial _ { x } ^ \\beta \\left ( h ( z , \\xi ' , \\cdot , \\cdot ) \\right ) \\right \\| _ { X _ 2 } \\\\ & \\leq C \\langle \\xi ' \\rangle ^ { m + 1 - k + k ' - l + l ' - | \\alpha | } \\end{align*}"}
-{"id": "7047.png", "formula": "\\begin{align*} & O _ + : = \\{ ( x , y ) \\in \\R _ { > 0 } \\times \\R \\ | \\ g ( x , y , 0 ) > 0 \\} , \\\\ & O _ - : = \\{ ( x , y ) \\in \\R _ { > 0 } \\times \\R \\ | \\ g ( x , y , 0 ) < 0 \\} . \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} ( \\omega g ) ( z _ 1 , \\ldots , z _ n ) & : = g ( q z _ n , z _ 1 , \\ldots , z _ { n - 1 } ) . \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} < \\nabla ^ 2 U ( \\nabla U ) , X > = < \\nabla U , \\nabla \\left ( < \\nabla U , X > \\right ) > - | \\nabla U | ^ 2 . \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\{ x , - \\} = P _ z \\frac { \\partial } { \\partial y } - P _ y \\frac { \\partial } { \\partial z } \\ , , & \\{ y , - \\} = P _ x \\frac { \\partial } { \\partial z } - P _ z \\frac { \\partial } { \\partial x } \\ , , & \\{ z , - \\} = P _ y \\frac { \\partial } { \\partial x } - P _ x \\frac { \\partial } { \\partial y } \\ , . \\end{array} \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} | q _ i | \\max _ { \\alpha = 1 , \\dots , r } | u _ \\alpha | < \\min _ { \\alpha = 1 , \\dots , r } | u _ \\alpha | , \\forall \\ ; i = 1 , 2 . \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ 3 } w _ { n } ^ { ( j ) } ( x ) ^ { \\frac { 4 } { 3 } } { \\rm d } x = \\int _ { \\mathbb { R } ^ 3 } w ( x ) ^ { \\frac { 4 } { 3 } } { \\rm d } x . \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} & E \\left [ S _ { s , t } ^ { a n o m } \\right ] \\\\ = & \\int _ s ^ t E \\left [ \\left ( \\beta ^ { - 3 } \\nabla _ q \\beta \\cdot \\left ( \\frac { 3 n + 2 } { 6 } \\gamma ^ { - 1 } - \\int _ 0 ^ \\infty T r [ \\gamma e ^ { - 2 y \\gamma } ] \\gamma ^ { - 1 } e ^ { - y \\gamma } d y \\right ) \\nabla _ q \\beta \\right ) ( r , q _ r ) \\right ] d r . \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{align*} Z _ { s , t } = Y _ { s , t } + Y _ { s , m } Z _ { m , t } \\end{align*}"}
-{"id": "3510.png", "formula": "\\begin{align*} \\sum _ { i \\in [ m ] } a _ { i j } \\exp ( y _ i / h ) = 0 , j \\in [ n ] , \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} \\mathrm { P } _ { t , 1 } & = \\frac { 2 \\lambda _ c ^ t \\pi ^ t } { ( t - 1 ) ! } \\int ^ { \\infty } _ { \\frac { \\rho ^ { \\frac { 1 } { \\alpha } } } { \\epsilon _ 1 ^ { \\frac { 1 } { \\alpha } } } } y ^ { 2 t - 1 } e ^ { - \\lambda _ c \\pi y ^ 2 } d y \\\\ & = e ^ { - \\lambda _ c \\pi \\left ( \\frac { \\rho } { \\epsilon _ 1 } \\right ) ^ { \\frac { 2 } { \\alpha } } } \\sum ^ { t - 1 } _ { k = 0 } \\frac { ( \\lambda _ c \\pi ) ^ { k } \\left ( \\frac { \\rho } { \\epsilon _ 1 } \\right ) ^ { \\frac { 2 k } { \\alpha } } } { k ! } . \\end{align*}"}
-{"id": "312.png", "formula": "\\begin{align*} c ( n ) = \\frac { ( n + 1 ) ! } { ( \\frac n 2 ) ! ^ 2 } = \\frac { ( 2 m + 1 ) ! } { m ! ^ 2 } \\ . \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} s \\bigl ( ( I - P ^ { \\triangle , j } ) \\lambda _ H ^ { \\Pi , j } , ( I - P ^ { \\triangle , j } ) \\mu _ H ^ \\Pi \\bigr ) = ( \\rho g , T ( I - P ^ { \\triangle , j } ) \\mu _ H ^ \\Pi ) \\quad \\mu _ H ^ \\Pi \\in \\Lambda _ H ^ \\Pi , \\end{align*}"}
-{"id": "7901.png", "formula": "\\begin{align*} \\lambda ( t , z , \\zeta ) : = \\mu + \\tilde { h } _ s ( t , z , \\zeta ) \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\tilde { f } _ i ( \\vec { x } ) \\tilde { g } _ { i , j } ( \\vec { x } ) + \\sum _ I \\tilde { \\Delta } _ I ( \\vec { x } ) \\tilde { g } _ { I , j } ( \\vec { x } ) = \\tilde { N } _ j \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{align*} \\left ( \\frac { \\prod _ { i = 1 } ^ k p _ { X Y } ( x _ i , y _ i ) } { p _ { X Y } ( x _ 1 , y _ k ) \\prod _ { i = 2 } ^ k p _ { X Y } ( x _ i , y _ { i - 1 } ) } \\right ) ^ { 1 / k } . \\end{align*}"}
-{"id": "346.png", "formula": "\\begin{align*} E _ j : = \\frac { m _ j } { | X | } \\sum _ { i = 0 } ^ c \\omega _ j ( a _ i ) A _ i \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} & \\mathcal { C } _ { - 1 } ( \\alpha , \\nu , 1 , \\varnothing , \\lbrace 2 \\rbrace ) = \\mathcal { C } _ { - 1 } ( [ 2 , 1 ] , [ \\nu _ 1 , \\nu _ 2 ] , 1 , \\varnothing , \\lbrace 2 \\rbrace ) = \\left \\lceil \\frac { \\nu _ 1 + 1 } { 2 } \\right \\rceil ; \\\\ & \\mathcal { C } _ { - 1 } ( \\alpha , \\nu , 2 , \\varnothing , \\lbrace 1 \\rbrace ) = \\mathcal { C } _ { - 1 } ( [ 2 , 1 ] , [ \\nu _ 1 , \\nu _ 2 ] , 2 , \\varnothing , \\lbrace 1 \\rbrace ) = \\nu _ 2 + 1 . \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{align*} M _ 1 = \\nu _ 1 M M _ 2 = \\nu _ 2 M , \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} q c ( K , s ) = \\begin{cases} 1 & { \\rm i f } \\ ; s = 0 , \\\\ ( p + 1 ) p ^ { s - 1 } & s > 0 , K \\ ; , \\\\ p ^ s & s > 0 , K \\ ; . \\end{cases} \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} \\frac { v _ A + v _ B + v _ { A B } } { x _ A + x _ B } = \\frac { v _ C } { x _ C } \\end{align*}"}
-{"id": "5918.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathbb { Z } } L _ i \\left [ H \\left ( \\cdot , \\mu \\right ) \\right ] ( \\nu ) = \\sum _ { i \\in \\mathbb { Z } } M _ i \\left [ H ( \\nu , \\cdot ) \\right ] \\left ( \\mu \\right ) , \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} d = e ( 0 , T ) \\times \\partial \\Omega . \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} a & = ( a - p _ n a p _ n ) + p _ n a p _ n \\\\ & \\precsim \\begin{pmatrix} a - p _ n a p _ n & 0 \\\\ 0 & p _ n a p _ n \\end{pmatrix} \\\\ & \\precsim \\begin{pmatrix} b - p _ n b p _ n & 0 \\\\ 0 & p _ n b p _ n \\end{pmatrix} \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } T _ { - 1 } ( s ) A _ { - 1 } x u ( s ) \\ , d s & = \\int _ { 0 } ^ { \\infty } e ^ { - ( i \\omega + \\varepsilon ) s } T _ { - 1 } ( s ) A _ { - 1 } x \\ , d s \\\\ & = ( i \\omega + \\varepsilon - A _ { - 1 } ) ^ { - 1 } A _ { - 1 } x \\in X . \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{align*} e _ H ( \\vec P , X ) & : = \\big | \\big \\{ \\big ( ( u , v ) , x \\big ) \\in \\vec P \\times X : \\{ u , v , x \\} \\in E ( H ) \\big \\} \\big | \\geq d | \\vec P | | X | - \\rho n ^ 3 \\end{align*}"}
-{"id": "1094.png", "formula": "\\begin{align*} u ' \\cdot ( u \\otimes n ) = u ' u \\otimes n \\ , . \\end{align*}"}
-{"id": "8720.png", "formula": "\\begin{align*} \\phi \\left ( \\frac { x ^ m - y ^ m } { x - y } \\right ) = \\frac { x ^ n - y ^ n } { x - y } \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} \\begin{cases} F ( D ^ { 2 } u _ 0 ) = 1 & \\mathbb { R } _ + ^ n \\cap \\Omega _ 0 , \\\\ | \\nabla u _ 0 | = 0 = u _ 0 & \\mathbb { R } _ + ^ n \\backslash \\Omega _ 0 , \\\\ u _ 0 = 0 & \\mathbb { R } _ + ^ { n - 1 } . \\end{cases} \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} ( R ^ m _ 2 ) _ { s , t } = & - \\int _ s ^ t \\partial _ r \\chi ( r , q _ r ^ m , z _ r ^ m ) d r . \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} H = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { n } y _ i ^ 2 + \\sum _ { i < j } V ( x _ i - x _ j ) , \\ , V ( x ) = - \\log | \\sin x | . \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} \\Delta _ k ^ + \\phi ( \\sigma ) = \\phi ( \\sigma ) - \\sum _ { \\begin{array} { c } { \\scriptstyle v \\in \\Sigma ( 0 ) } \\\\ { \\scriptstyle v \\sigma \\in \\Sigma ( k + 1 ) } \\end{array} } \\sum _ { 0 \\leq i \\leq k } ( - 1 ) ^ { i } \\frac { m ( v \\sigma ) } { m ( \\sigma ) } \\phi ( v \\sigma _ { i } ) , \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} \\lambda _ { A _ { \\mathfrak { p } } } \\left ( \\operatorname { T o r } _ { 2 i } ^ { A _ { \\mathfrak { p } } } ( M _ { \\mathfrak { p } } , N _ { \\mathfrak { p } } ) \\right ) & = P _ 1 ( i ) \\quad \\mbox { f o r a l l } i \\gg 0 ; \\\\ \\lambda _ { A _ { \\mathfrak { p } } } \\left ( \\operatorname { T o r } _ { 2 i + 1 } ^ { A _ { \\mathfrak { p } } } ( M _ { \\mathfrak { p } } , N _ { \\mathfrak { p } } ) \\right ) & = P _ 2 ( i ) \\quad \\mbox { f o r a l l } i \\gg 0 . \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} \\Lambda _ Y = \\frac { - c _ Y + 1 - \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\alpha ( \\mu _ Y ) } { \\sup \\{ p ( \\mu _ Y ) , p \\in \\chi - \\Delta ^ + \\} } , \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{align*} y _ { t } = x _ 0 - \\sum _ { t ( l ) < t / \\Gamma } \\nabla f ( x _ { t ( l ) } ) \\Gamma + \\sum _ { t ( l ) < t / \\Gamma } \\epsilon ( l ) \\Gamma = y _ 0 - \\sum _ { \\Gamma t ( l ) < t } \\nabla f ( y _ { \\Gamma t ( l ) } ) \\Gamma + \\sum _ { t ( l ) < t / \\Gamma } \\epsilon ( l ) \\Gamma . \\end{align*}"}
-{"id": "8498.png", "formula": "\\begin{align*} H = p _ 1 ^ 2 - p _ 1 p _ 2 + p _ 2 ^ 2 + \\frac { 3 } { 2 } p _ 3 ^ 2 - \\log | \\sin { q _ 1 } | - \\log | \\sin { q _ 2 } | - \\log | \\sin { ( q _ 1 + q _ 2 ) } | . \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} F ( y ) = P \\{ Y \\leq y \\} = \\exp \\left ( - e ^ { - y } \\right ) , y \\in \\mathbb { R } , \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} & | g | _ i ^ { ( m ) } : = \\underset { k , k ' , l , l ' , | \\alpha | , | \\beta | \\leq i } { \\max } \\ \\underset { \\substack { x \\in \\overline { \\R ^ n _ + } \\\\ \\xi ' \\in \\R ^ { n - 1 } } } { \\sup } \\left \\| y _ n ^ k \\partial _ { y _ n } ^ { k ' } w _ n ^ l \\partial _ { w _ n } ^ { l ' } \\partial _ { x } ^ \\beta \\partial _ { \\xi ' } ^ \\alpha { g } ( x ' , \\xi ' , \\cdot , \\cdot ) \\right \\| _ { L ^ 2 _ { y _ n , w _ n } ( { \\R } _ { + + } ^ 2 ) } \\langle \\xi ' \\rangle ^ { - m - 1 + k - k ' + l - l ' + | \\alpha | } \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{align*} p ^ h = \\left \\lbrace \\begin{array} { c c } 0 , ~ h \\neq h _ 0 \\\\ 1 , ~ h = h _ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} \\lambda _ { n _ 1 , \\ldots , n _ k } = \\frac { ( a ; q ) _ { n _ 1 + n _ 2 + \\cdots + n _ k } } { ( c ; q ) _ { n _ 1 + \\cdots + n _ k } ( q ; q ) _ { n _ 1 } ( q ; q ) _ { n _ 2 } \\cdots ( q ; q ) _ { n _ k } } . \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{align*} \\bigvee _ { n = 1 } ^ \\infty P ( T , B _ n ) & = P \\bigg ( T , \\bigcup _ { n = 1 } ^ \\infty B _ n \\bigg ) , \\\\ \\bigwedge _ { n = 1 } ^ \\infty P ( T , B _ n ) & = P \\bigg ( T , \\bigcap _ { n = 1 } ^ \\infty B _ n \\bigg ) . \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} P ( \\epsilon ) = \\sum _ { m = K } ^ { N } \\binom { N } { m } ( 1 - \\epsilon ) ^ { m } \\epsilon ^ { N - m } \\mathbb { P } ( m , K ) . \\end{align*}"}
-{"id": "7422.png", "formula": "\\begin{align*} \\varepsilon _ m ^ N - \\varepsilon _ m = \\left \\langle r _ m ^ N , \\left ( A - \\varepsilon _ m ^ N \\right ) ^ { - 1 } \\left ( A - \\varepsilon _ m \\right ) \\left ( A - \\varepsilon _ m ^ N \\right ) ^ { - 1 } r _ m ^ N \\right \\rangle . \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} E = ( I _ m , 0 ) , \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} \\| j _ 0 ^ m \\hat { \\varphi } ^ * s \\| _ { \\tau } = R _ { \\tau } ^ { - m } \\| j _ 0 ^ m \\varphi _ { \\tau } ^ * s \\| _ { R _ { \\tau } } \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} \\int _ X \\psi \\omega ^ n = n ! 2 ^ { | \\Phi _ s ^ + | - r } C _ H \\int _ { \\chi - ( - \\Delta ^ t ) \\cap \\bar { C } ^ - } \\psi ( d _ { 2 \\chi - 2 q } u ^ * ) \\prod _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } q ( \\alpha ^ { \\vee } ) d q . \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{gather*} k ^ { a + 1 , c } ( I , J ) = k ^ { a , c } ( I , J ) , \\\\ k ^ { a + 1 , c } ( I ' , J ' ) = k ^ { a , c } ( I , J ) , \\\\ k ^ { a + 1 , c } ( I ' , J ) = \\begin{cases} k ^ { a , c } ( I , J ) , & I < J \\\\ k ^ { a , c } ( I , J ) + 1 , & I > J . \\end{cases} \\end{gather*}"}
-{"id": "1450.png", "formula": "\\begin{align*} \\int \\Big \\{ \\sum _ { j = 1 } ^ { m - 1 } \\big [ ( \\partial _ { z _ { j - 1 } } - \\partial _ { z _ { j } } ) \\sqrt { g _ t } \\big ] ^ 2 \\Big \\} \\prod _ { j = 1 } ^ { m - 1 } 2 e ^ { - 2 z _ j } d z _ j \\le \\frac { 1 } { 2 t } H ( \\nu _ 0 ^ { ( m ) } | \\mu ^ { ( m , 2 ) } _ \\star ) + \\frac 1 m \\ , . \\end{align*}"}
-{"id": "7692.png", "formula": "\\begin{align*} \\mathrm { P } _ { t , 1 } & = \\mathrm { P } \\left ( z _ t < \\frac { \\epsilon _ 1 } { \\rho \\xi _ 1 } \\right ) \\\\ & \\overset { ( a ) } { = } \\mathrm { P } \\left ( \\alpha _ 1 = 1 \\right ) = \\mathrm { P } \\left ( z _ t < \\frac { \\epsilon _ 1 } { \\rho } \\right ) = \\mathrm { P } ^ { O M A } _ { t , 1 } , \\end{align*}"}
-{"id": "6378.png", "formula": "\\begin{align*} f _ - ( x ) & = \\begin{cases} a x + b , & 0 \\le x < \\tau ; \\\\ a x + b - 1 , & \\tau \\le x \\le 1 ; \\end{cases} \\\\ f _ + ( x ) & = \\begin{cases} a x + b , & 0 \\le x \\le \\tau ; \\\\ a x + b - 1 , & \\tau < x \\le 1 . \\end{cases} \\end{align*}"}
-{"id": "417.png", "formula": "\\begin{align*} F : = \\{ \\vec { x } \\in \\{ 0 , 1 \\} ^ n \\ ; | \\ ; f ( \\vec { x } ) = 0 \\} . \\end{align*}"}
-{"id": "2094.png", "formula": "\\begin{align*} g ( C _ a ) = \\delta ( C _ a ) = e _ Q ( C _ a ) = 0 , \\ \\ \\mbox { a n d } \\ \\ { \\rm i n d } C _ a = 1 , \\ \\ h ( C _ a ) = 2 , \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{align*} \\mathbf { W } ( \\mathbf { z } , \\mathbf { \\phi } ) & = \\mathbf { W } _ 0 ( \\mathbf { z } ) + \\sum _ { l = 1 } ^ { \\infty } \\varepsilon ^ l \\mathbf { W } _ l ( \\mathbf { z } , \\mathbf { \\phi } ) , \\\\ \\mathbf { R } ( \\mathbf { z } , \\mathbf { \\phi } ) & = \\mathbf { R } _ 0 ( \\mathbf { z } ) + \\sum _ { l = 1 } ^ { \\infty } \\varepsilon ^ l \\mathbf { R } _ l ( \\mathbf { z } , \\mathbf { \\phi } ) , \\end{align*}"}
-{"id": "9200.png", "formula": "\\begin{align*} \\left [ P _ { - n } , \\frac { P _ n } n \\left \\{ \\frac { ( q ^ n - 1 ) q ^ { - n r } } { [ n ] _ { q _ 1 } [ n ] _ { q _ 2 } } \\right \\} \\right ] = 1 - q ^ { - r n } \\end{align*}"}
-{"id": "6543.png", "formula": "\\begin{align*} K ( P _ \\rho , P _ \\rho ' ) = \\sum \\limits _ { i = 1 } ^ n { \\left ( \\rho ( i ) \\log \\frac { \\rho ( i ) } { \\rho ' ( i ) } - ( 1 - \\rho ( i ) ) \\log \\frac { 1 - \\rho ( i ) } { 1 - \\rho ' ( i ) } \\right ) } . \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{align*} u _ { t } & = - \\frac { 1 } { 2 } t ^ { - 3 / 2 } [ \\phi ( \\frac { x } { \\sqrt { t } } ) + \\frac { x } { \\sqrt { t } } \\phi ' ( \\frac { x } { \\sqrt { t } } ) ] \\\\ u _ { x } & = \\frac { 1 } { t } \\phi ' ( \\frac { x } { \\sqrt { t } } ) \\\\ u _ { x x } & = t ^ { - 3 / 2 } \\phi '' ( \\frac { x } { \\sqrt { t } } ) \\end{align*}"}
-{"id": "1400.png", "formula": "\\begin{align*} \\tilde \\tau _ D ( x , y ) = \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\tilde \\tau _ { p _ i } ( x , y ) = \\frac { 1 } { k } \\log \\Bigg ( \\prod _ { i = 1 } ^ { k } \\frac { \\mu _ { p _ i } ( x , y ) } { \\sqrt { d ( x , p _ i ) d ( y , p _ i ) } } \\Bigg ) . \\end{align*}"}
-{"id": "56.png", "formula": "\\begin{align*} \\tilde { L } _ n = L _ n - 2 \\sum _ { i \\in [ B ] } ( D _ { w _ i } - \\hat { X } _ { w _ i , w _ i } ) , \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} { } \\begin{aligned} ( z _ { n } + \\frac { 1 } { n } ) ^ { \\theta - \\delta } & \\leq { ( c \\phi _ 1 + n ^ { \\frac { \\delta + p - 1 } { p } } ) ^ { \\frac { p ( \\theta - \\delta ) } { \\delta + p - 1 } } } \\leq ( c \\phi _ { 1 } ) ^ { \\frac { p ( \\theta - \\delta ) } { \\delta + p - 1 } } \\end{aligned} \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} H ^ s u ( x , t ) = V ( x , t ) u ( x , t ) , \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{align*} P _ B x = x - M ^ { \\dagger } ( M x - b ) \\forall x \\in \\mathbb { R } ^ n , \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} { t } ( x ' , D _ x ) ( \\varphi ) ( x ' ) = { t } _ 0 ( x ' , D _ x ) ( \\varphi ) ( x ' ) + \\sum _ { j = 0 } ^ { r - 1 } s _ j ( x ' , D _ { x ' } ) ( \\gamma _ j \\varphi ) ( x ' ) , \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n + 1 } \\lambda _ j ( x ) x ^ { ( j ) } = x , \\sum _ { j = 1 } ^ { n + 1 } \\lambda _ j ( x ) = 1 . \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} \\hat { k } ( z , x ) = \\left \\{ \\begin{array} { l l } - k ( z , \\eta ) - \\hat { \\xi } _ 0 \\eta ^ { p - 1 } & \\mbox { i f } \\ x < - \\eta \\\\ k ( z , x ) + \\hat { \\xi } _ 0 | x | ^ { p - 2 } x & \\mbox { i f } \\ - \\eta \\leq x \\leq \\eta \\\\ k ( z , \\eta ) + \\hat { \\xi } _ 0 \\eta ^ { p - 1 } & \\mbox { i f } \\ \\eta < x . \\end{array} \\right . \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} \\widetilde { \\P } ^ { 0 , 0 } _ { 2 T } ( X ( t ) = x ) = \\widetilde { \\P } ^ { 0 , 0 } _ { 2 T } ( X ( 2 T - t ) = - x ) , 0 \\leq t \\leq 2 T , x \\in \\Z . \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} \\frac { 1 } { 6 } + \\frac { 1 } { 4 } + \\frac { 1 } { 1 2 } + \\frac { 1 } { 1 2 } + \\frac { 1 } { 4 } + \\frac { 1 } { 6 } = 1 . \\end{align*}"}
-{"id": "9279.png", "formula": "\\begin{align*} & p _ { \\mathrm { S } _ 1 } ( m | n ) = \\sum _ { q = 0 } ^ { \\infty } p ( m | n , q ) p ( q ) \\\\ & = \\sum _ { q = 0 } ^ { \\infty } \\left ( \\int _ 0 ^ \\infty p _ { \\mathrm { S } _ 1 } ( t _ \\mathrm { p } | n , q ) \\frac { ( \\lambda _ \\mathrm { B } t _ \\mathrm { p } ) ^ { m } } { m ! } e ^ { - \\lambda _ \\mathrm { B } t _ \\mathrm { p } } d t _ \\mathrm { p } \\right ) \\frac { ( \\lambda _ \\mathrm { B } t _ \\mathrm { s } ) ^ { q } } { q ! } e ^ { - \\lambda _ \\mathrm { B } t _ \\mathrm { s } } . \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} \\bar { S } = \\frac { \\int _ { X } n \\mathrm { R i c } ( \\omega ) \\wedge \\omega ^ { n - 1 } } { \\int _ { X } \\omega ^ { n } } . \\end{align*}"}
-{"id": "2849.png", "formula": "\\begin{align*} | T _ { b } ^ { m } f ( x ) | \\leq c _ { n , T } \\sum _ { j = 1 } ^ { 3 ^ { n } } \\sum _ { k = 0 } ^ { m } \\binom { m } { k } \\sum _ { Q \\in \\mathcal { S } _ { j } } | b ( x ) - b _ { Q } | ^ { m - k } \\left ( \\frac { 1 } { | Q | } \\int _ { Q } | b - b _ { Q } | ^ { k } | f | \\right ) \\chi _ { Q } ( x ) . \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} \\Gamma ^ * _ Q : = \\left \\{ - \\frac 1 2 + \\frac { j } { Q } , \\ j \\in \\{ 0 , \\cdots , Q - 1 \\} \\right \\} . \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} u ( a ) = \\Pi [ u ] ( a ) \\varphi \\ , , a \\in J \\ , , \\varphi = \\eta Q [ u ] \\varphi \\ , , \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} & u _ t - \\triangle u + \\nabla p = f , \\\\ & \\nabla \\cdot u = 0 , \\end{align*}"}
-{"id": "5543.png", "formula": "\\begin{align*} \\theta _ 2 ( x ) & = 2 e ^ { - \\tfrac { \\pi } { 4 } x } \\prod _ { k \\in \\N } \\left ( 1 - e ^ { - 2 k \\pi x } \\right ) \\left ( 1 + e ^ { - 2 k \\pi x } \\right ) ^ 2 \\\\ \\theta _ 3 ( x ) & = \\prod _ { k \\in \\N } \\left ( 1 - e ^ { - 2 k \\pi x } \\right ) \\left ( 1 + e ^ { - ( 2 k - 1 ) \\pi x } \\right ) ^ 2 \\\\ \\theta _ 4 ( x ) & = \\prod _ { k \\in \\N } \\left ( 1 - e ^ { - 2 k \\pi x } \\right ) \\left ( 1 - e ^ { - ( 2 k - 1 ) \\pi x } \\right ) ^ 2 \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{align*} E \\left [ S ( \\theta ) ^ { ( 2 ) } \\right ] = \\tilde { J } _ 1 ( N ; \\theta ) + \\tilde { J } _ 2 ( N ; \\theta ) , \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} V _ f ( S ) : = \\int _ S e ^ { - f } d \\sigma . \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} \\frac { 1 } { c _ 1 } = \\frac { 1 } { 2 a \\lambda _ 2 } \\left ( 1 + \\sqrt { \\frac { c _ 4 } { c _ 3 } } \\right ) , \\end{align*}"}
-{"id": "3153.png", "formula": "\\begin{align*} C _ n = \\frac { 1 } { n } \\displaystyle \\sum _ { i = 0 } ^ { n - 1 } X _ i \\otimes X _ i , n \\geq 2 , \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} \\begin{array} { r c l } ( s \\alpha ) \\cdot e _ S & = & s \\cdot ( \\alpha \\cdot e _ S ) - \\alpha ( s ) \\cdot e _ S \\\\ & = & ( s \\lambda ( \\alpha ) - \\alpha ( s ) ) \\cdot e _ S , \\end{array} \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} ( \\tilde \\gamma ^ { - 1 } ) ^ i _ j = ( \\tilde \\gamma ^ { - 1 } ) ^ { i k } \\delta _ { k j } = \\left ( \\begin{array} { c c c } \\gamma / ( \\gamma ^ 2 + B _ 0 ^ 2 ) & - B _ 0 / ( \\gamma ^ 2 + B _ 0 ^ 2 ) & 0 \\\\ B _ 0 / ( \\gamma ^ 2 + B _ 0 ^ 2 ) & \\gamma / ( \\gamma ^ 2 + B _ 0 ^ 2 ) & 0 \\\\ 0 & 0 & \\gamma ^ { - 1 } \\end{array} \\right ) \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} \\max _ { 0 \\le j < J } \\frac { | f ^ { ( j ) } ( q ) | } { j ! } \\le \\max _ { 0 \\le j < J } \\sum _ { k + l = j } \\frac { | \\bar { f } ^ { ( k ) } ( q ) | } { k ! } \\frac { | ( u ^ d ) ^ { ( l ) } ( q ) | } { l ! } \\le \\left ( J \\max _ { 0 \\le j < J } \\frac { | ( u ^ d ) ^ { ( j ) } ( q ) | } { j ! } \\right ) \\max _ { 0 \\le j < J } \\frac { | \\bar { f } ^ { ( j ) } ( q ) | } { j ! } \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} \\rho ( R , S ) : = ( - 1 ) ^ { \\abs { T _ { R S } } } \\sum _ { A \\in [ R , S ] } ( - 1 ) ^ { \\abs { S } - \\abs { A } } m ( A ) \\ge 0 . \\end{align*}"}
-{"id": "3230.png", "formula": "\\begin{align*} \\begin{cases} \\triangle \\alpha = ( \\lambda + k - 2 ) ( \\lambda + n - k ) \\alpha + 2 d ^ \\ast \\beta , & \\\\ \\triangle \\beta = ( \\lambda + n - k - 2 ) ( \\lambda + k ) \\beta + 2 d \\alpha . \\end{cases} \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} & M ( u ) = \\int | u | ^ 2 d L \\\\ & P ( u ) = \\int \\Lambda u \\ , \\overline { u } \\ , d L = \\int _ { \\mathbb { C } } ( \\vert z \\vert ^ 2 - 1 ) \\vert u ( z ) \\vert ^ 2 \\ , d L ( z ) \\\\ & Q ( u ) = Q _ x ( u ) + i Q _ y ( u ) = \\int _ { \\C } z | u | ^ 2 ( z ) d L ( z ) , \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} A = \\{ y _ 1 , \\dots , y _ a \\} , \\ P \\backslash A = \\{ y _ { a + 1 } , \\dots , y _ { a + b } \\} \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { + \\infty } f ( t ) T ( t ) d t = \\phi _ { \\bf T } ( f ) . \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} d q _ t = & \\tilde \\gamma ^ { - 1 } ( t , q _ t ) F ( t , q _ t , \\psi ( t , q _ t ) ) d t + \\tilde S ( t , q _ t ) d t + \\tilde \\gamma ^ { - 1 } ( t , q _ t ) \\sigma ( t , q _ t ) \\circ d W _ t , \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} \\sum _ \\sigma q ^ { \\mathrm { i n v } ( \\sigma ) } = \\prod _ { s \\in \\lambda \\cap S _ \\mu } \\frac { q ^ { \\rho ( s , \\mu , \\lambda ) } - 1 } { q - 1 } , \\end{align*}"}
-{"id": "8506.png", "formula": "\\begin{align*} \\ddot { \\xi } = - ( 4 + \\frac { 8 } { 9 } \\sqrt { 3 } . \\psi ( t ) + 2 4 . { \\psi ( t ) } ^ 2 ) \\xi . \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} \\mathcal { D } = \\{ \\alpha \\} \\cup \\{ c S ^ { i } T ^ { j } | c \\in \\mathcal { C } , ( i , j ) \\in \\mathbb { I } \\} \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} a _ + : = a _ + ^ 0 + a _ + ^ 1 , \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{align*} d _ { \\emph { i n d } } ( p _ { X Y } ) = 1 - \\frac { 1 } { F ( p _ { X Y } ) } = 1 - \\epsilon _ 1 \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{align*} ( u _ { \\mu ^ * } , v _ { \\mu ^ * } ) : = \\Big ( \\Big ( \\frac { \\mu ^ * } { \\mu } \\Big ) ^ { \\frac 1 2 } u \\Big ( \\Big ( \\frac { \\mu ^ * } { \\mu } \\Big ) ^ { \\frac 1 2 } x \\ , , \\ , \\frac { \\mu ^ * } { \\mu } t \\Big ) , \\frac { \\mu ^ * } { \\mu } v \\Big ( \\Big ( \\frac { \\mu ^ * } { \\mu } \\Big ) ^ { \\frac 1 2 } x , \\frac { \\mu ^ * } { \\mu } t \\Big ) \\Big ) \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} \\dd \\varphi & = \\tau _ 0 \\ , \\psi + 3 \\ , \\tau _ 1 \\wedge \\varphi + * \\tau _ 3 = i _ H ( \\varphi ) ~ , \\\\ \\dd \\psi & = 4 \\ , \\tau _ 1 \\wedge \\psi = i _ H ( \\psi ) ~ , \\end{align*}"}
-{"id": "3447.png", "formula": "\\begin{align*} ( S ^ { i , j } f ( y ) ) ^ { 2 } = \\sum _ { \\substack { R _ 1 \\in \\mathcal { D } _ 1 \\\\ R _ 2 \\in \\mathcal { D } _ 2 } } \\bigg ( \\sum _ { \\substack { P _ 1 \\in ( R _ 1 ) _ { i _ 1 } \\\\ P _ 2 \\in ( R _ 2 ) _ { i _ 2 } } } | \\hat { f } ( P ) | \\bigg ) ^ { 2 } \\bigg ( \\sum _ { \\substack { Q _ 1 \\in ( R _ 1 ) _ { j _ 1 } \\\\ Q _ 2 \\in ( R _ 2 ) _ { j _ 2 } } } \\frac { 1 _ { Q _ 1 } } { | Q _ 1 | } \\otimes \\frac { 1 _ { Q _ 2 } } { | Q _ 2 | } ( y ) \\bigg ) . \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{align*} ( \\rho ' _ + , X ) \\mapsto ( \\rho = \\rho ' _ + / | X | , \\ v = | X | ^ { - 1 } - 1 , \\ \\omega = X / | X | ) \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} \\displaystyle { \\frac { \\partial u } { \\partial t } + u \\frac { \\partial u } { \\partial x } - \\nu \\frac { \\partial ^ 2 u } { \\partial x ^ 2 } = g } \\mbox { o n } \\Omega \\\\ \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} \\tilde { \\omega } = ( \\widetilde { h _ n \\omega _ n } ) ( \\widetilde { \\eta _ { \\omega _ n } h _ { n - 1 } \\omega _ { n - 1 } } ) . . . ( \\widetilde { \\eta _ { \\omega _ 3 } h _ 2 \\omega _ 2 } ) ( \\widetilde { \\eta _ { \\omega _ 2 } h _ 1 } ) , \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} \\underline { \\mathrm { i s o } } : \\{ \\nabla \\mathcal L \\} \\leftrightarrow & \\{ \\psi E \\\\ & \\sigma , \\sum _ { P \\in E } \\mathrm { r e s } _ P ( \\psi ) P = ( \\sigma ) \\} , \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} \\mathcal { E } \\coloneqq \\left \\{ E < F \\ ; \\middle | \\ ; f \\left ( E \\right ) = H \\right \\} . \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} \\psi ( g z ) = \\psi ( z ) \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{align*} B _ 1 ^ * & : = \\{ b \\in A \\mid b B _ 1 \\} \\\\ B _ 2 ^ * & : = \\{ b \\in A \\mid b B _ 1 B _ 3 \\} \\\\ B _ 3 ^ * & : = \\{ b \\in A \\mid b B _ 3 \\} . \\\\ \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} | g _ a ( z ) | = | g _ a ( z _ n ) | \\left ( 1 + O \\left ( \\frac { \\epsilon } { \\left | \\frac { 1 } { 2 } + i q _ 2 y \\right | } \\right ) \\right ) \\gg _ \\epsilon \\left | \\frac { 1 } { 4 } + q _ 2 ^ 2 y ^ 2 \\right | ^ { - k / 2 } . \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{align*} g ( a + 1 ) & > g ( a ) a = 1 \\ , , \\\\ g ( a + 1 ) & < g ( a ) a \\in \\{ 2 , \\dots , n - 1 \\} \\ , . \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} g ( y _ { i , t } - y _ { j , t } ) = - ( y _ { i , t } - y _ { j , t } ) [ g _ { a } ( \\Vert y _ { i , t } - y _ { j , t } \\Vert ) - g _ { r } ( \\Vert y _ { i , t } - y _ { j , t } \\Vert ) ] = - ( y _ { i , t } - y _ { j , t } ) [ a - g _ { r } ( \\Vert y _ { i , t } - y _ { j , t } \\Vert ) ] . \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} p _ { i , j } = \\exp \\left ( - \\frac { 1 } { 2 ( s _ 1 ^ 2 s _ 2 ^ 2 - \\rho ^ 4 ) } \\left ( s _ 2 ^ 2 ( i - k ) ^ 2 - 2 \\rho ^ 2 ( i - k ) ( j - \\ell ) + s _ 1 ^ 2 ( j - \\ell ) ^ 2 \\right ) \\right ) \\ , , \\end{align*}"}
-{"id": "7695.png", "formula": "\\begin{align*} \\mathrm { P } _ { m , 1 } = \\mathrm { P } \\left ( z _ t < \\frac { \\epsilon _ 1 } { \\rho } , z _ m < \\frac { \\epsilon _ 1 } { \\rho } \\right ) , \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} 2 \\left ( \\frac { b } { a } \\right ) ^ q K _ q ( 2 a b ) = \\int _ 0 ^ { \\infty } v ^ { q - 1 } e ^ { - a ^ 2 v - \\frac { b ^ 2 } { v } } d v \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} n = \\frac { 1 - \\alpha } { 2 } t + \\eta t ^ { 1 / 2 } , x ( \\xi ) = ( \\alpha - \\tfrac 1 2 ) t - 2 \\eta t ^ { 1 / 2 } - \\sigma \\xi t ^ { 1 / 2 } . \\end{align*}"}
-{"id": "1537.png", "formula": "\\begin{align*} m ( d x \\ , d t ) = \\frac { d m _ x ( t ) } { v ( x ) } \\otimes d x , \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} | a _ j | \\le \\left ( \\sum _ { k = 0 } ^ j \\frac { 1 } { | q | ^ { m + k } } \\right ) 2 ^ { m + J } \\max _ { 0 \\le k \\le j } \\frac { | f ^ { ( k ) } ( q ) | } { k ! } \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} N _ 1 ( x ) = \\sum _ { e \\geq 0 } N _ { 1 , e } ( x ) = \\frac { x ^ 3 ( 1 - 3 x + 4 x ^ 2 - 3 x ^ 3 ) } { ( 1 - x ) ^ 4 ( 1 - 2 x ) } \\ , . \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} & \\int _ s ^ t - \\nabla _ q V ( r , q _ r ) \\beta ( r , q _ r ) \\circ d q _ r \\\\ = & ( \\beta V ) ( s , q _ s ) - ( \\beta V ) ( t , q _ t ) + \\int _ s ^ t \\partial _ r ( \\beta V ) ( r , q _ r ) d r + \\int _ s ^ t \\nabla _ q \\beta ( r , q _ r ) V ( r , q _ r ) \\circ d q _ r \\end{align*}"}
-{"id": "8369.png", "formula": "\\begin{align*} ( x | b ) ^ T = \\prod _ { e \\in T } x _ { v a l ( e ) } \\oplus b _ { \\lambda _ { r ( e ) } + f _ { r ( e ) } - c ( e ) - v a l ( e ) + 1 } = \\prod _ { e \\in T } x _ { v a l ( e ) } \\oplus b _ { \\lambda _ 1 ' + r ( e ) - c ( e ) - v a l ( e ) + 1 } . \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{align*} \\eta _ { \\gamma } = \\sum _ { n = - 1 } ^ { \\infty } \\eta _ { \\gamma , - n } \\zeta ^ n \\rlap { . } \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{align*} Z _ { j } ( \\xi , \\xi ) = \\dim { H _ { j } ( \\mathbb { R } ^ { n } ) } , \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{align*} \\chi ( z ) = \\sum _ { j = 0 } ^ { \\lfloor ( k - 1 ) / 2 \\rfloor } A _ j ^ { i _ 1 , . . . , i _ { k - 2 j } } z _ { i _ 1 } . . . z _ { i _ { k - 2 j } } \\end{align*}"}
-{"id": "7710.png", "formula": "\\begin{align*} \\left \\{ z _ m < \\frac { \\epsilon _ i } { \\rho \\xi _ i } \\right \\} & = \\left \\{ z _ m < \\frac { \\epsilon _ i } { \\rho \\left ( \\alpha _ i ^ 2 - \\epsilon _ i \\sum ^ { { M _ s } } _ { j = i + 1 } \\alpha _ j ^ 2 \\right ) } \\right \\} \\\\ & = \\left \\{ z _ m < \\frac { \\epsilon _ i } { \\rho \\bar { \\xi } _ i \\max \\left \\{ 0 , \\frac { \\rho z _ t - \\epsilon _ 1 } { \\rho ( 1 + \\epsilon _ 1 ) z _ t } \\right \\} } \\right \\} , \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } X _ { n , k } ( t ) = X _ { \\infty , k } ( t ) \\qquad \\mbox { a . s . a n d i n } \\ L ^ 2 ; \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{align*} & \\Phi _ { q } ( q , t ) = \\int _ { \\Omega } a \\left ( x \\right ) \\left [ ( t \\phi _ { 1 } + w ) ^ { q } \\left \\{ \\log ( t \\phi _ { 1 } + w ) + \\frac { q w _ { q } } { t \\phi _ { 1 } + w } \\right \\} - w _ { q } \\right ] \\phi _ { 1 } . \\\\ & \\Phi _ { t } ( q , t ) = \\int _ { \\Omega } a ( x ) \\left \\{ q ( t \\phi _ { 1 } + w ) ^ { q - 1 } ( \\phi _ { 1 } + w _ { t } ) - ( \\phi _ { 1 } + w _ { t } ) \\right \\} \\phi _ { 1 } . \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{align*} \\left \\{ \\ell \\in T ^ * M \\colon \\abs { F _ 1 ( \\ell ) } = \\abs { \\Psi ( \\ell ) } \\right \\} , \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} \\partial _ t \\phi _ t ( u _ 1 , u _ 2 ) & = F ( ( \\psi _ t ( u _ 1 , u _ 2 ) , u _ 2 ) ) , \\quad \\phi _ 0 ( u _ 1 , u _ 2 ) = 0 , \\\\ \\partial _ t \\psi _ t ( u _ 1 , u _ 2 ) & = R ( ( \\psi _ t ( u _ 1 , u _ 2 ) , u _ 2 ) ) , \\quad \\psi _ 0 ( u _ 1 , u _ 2 ) = u _ 2 , \\end{align*}"}
-{"id": "8453.png", "formula": "\\begin{align*} \\psi ( z ) = \\psi \\Big ( \\sum _ { j = 1 } ^ { r } t _ j e _ j \\Big ) , \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} & g ( z _ 1 , z _ 2 , z _ 3 , z _ 4 ) = [ z _ 1 , z _ 2 ] [ z _ 3 , z _ 4 ] + [ z _ 1 , z _ 3 ] [ z _ 2 , z _ 4 ] = [ z _ 4 , z _ 3 ] [ z _ 2 , z _ 1 ] + \\bigl [ [ z _ 1 , z _ 2 ] , [ z _ 3 , z _ 4 ] \\bigr ] + [ z _ 4 , z _ 2 ] [ z _ 3 , z _ 1 ] \\\\ & + \\bigl [ [ z _ 1 , z _ 3 ] , [ z _ 2 , z _ 4 ] \\bigr ] = g ( z _ 4 , z _ 2 , z _ 3 , z _ 1 ) + \\bigl [ [ z _ 1 , z _ 2 ] , [ z _ 3 , z _ 4 ] \\bigr ] + \\bigl [ [ z _ 1 , z _ 3 ] , [ z _ 2 , z _ 4 ] \\bigr ] \\end{align*}"}
-{"id": "2139.png", "formula": "\\begin{align*} \\norm { X _ { N s } ( s ) ^ { - 1 } \\cdot X _ { N t } ( t ) } ^ { 4 p } = \\norm { \\mathbf { F } _ { j } ( N t - N s ) } ^ { 4 p } \\leq d _ \\beta ( \\mathbf { F } _ { j } , 0 _ { G ^ 2 ( V ) } ) ^ { 4 p } ( N t - N s ) ^ { 4 p \\beta } \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} \\Omega = \\cup _ { i } \\Omega _ { i } . \\end{align*}"}
-{"id": "5496.png", "formula": "\\begin{align*} \\begin{array} { l | c c c c c } j & 1 & 2 & 3 & 4 & 5 \\\\ \\hline \\omega _ j ^ 2 & 2 - \\sqrt { 3 } & 1 & 2 & 3 & 2 + \\sqrt { 3 } \\end{array} , D _ j = \\frac { c } { 2 } , j = 1 , . . . , N . \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} y _ { 0 } ^ { \\ast } \\left ( x \\right ) = \\frac { y _ { a } } { \\Gamma \\left ( \\gamma \\right ) } \\left ( \\psi \\left ( x \\right ) - \\psi \\left ( a \\right ) \\right ) ^ { \\gamma - 1 } + \\frac { 1 } { \\Gamma \\left ( \\alpha \\right ) } \\int _ { a } ^ { x _ { 1 } } \\psi ^ { \\prime } \\left ( t \\right ) \\left ( \\psi \\left ( x \\right ) - \\psi \\left ( t \\right ) \\right ) ^ { \\alpha - 1 } f \\left ( t , y \\left ( t \\right ) \\right ) d t \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{align*} 0 & = ( 1 - Q ) [ a \\left ( x \\right ) ( u ^ { q } - u ) ] \\\\ & = \\left ( \\int _ { \\Omega } a ( x ) \\left \\{ ( t \\phi _ { 1 } + w ) ^ { q } - ( t \\phi _ { 1 } + w ) \\right \\} \\phi _ { 1 } \\right ) \\phi _ { 1 } , \\end{align*}"}
-{"id": "5476.png", "formula": "\\begin{align*} \\mathbf { z } \\mapsto \\varepsilon ^ { \\frac { 1 } { 2 M + 2 } } \\mathbf { z } = \\mu \\mathbf { z } , \\end{align*}"}
-{"id": "4595.png", "formula": "\\begin{align*} \\partial _ T ^ * \\phi = - \\sum _ { a = 1 } ^ n V _ a \\lrcorner \\nabla _ { \\bar V _ a } \\phi , \\bar \\partial _ T ^ * \\phi = - \\sum _ { a = 1 } ^ n \\bar V _ a \\lrcorner \\nabla _ { V _ a } \\phi . \\end{align*}"}
-{"id": "9289.png", "formula": "\\begin{align*} \\xi _ 0 ^ 2 - \\sum _ { a = 1 } ^ 3 \\xi _ a ^ 2 = 0 \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{align*} N _ 1 ( 0 + ) = \\underset { r \\to 0 } { \\lim } N _ 1 ( r ) \\end{align*}"}
-{"id": "6699.png", "formula": "\\begin{align*} T _ { \\mathcal { P } _ { \\infty } } & = \\langle T _ i , t ; t y ^ { - j } x y ^ j t ^ { - 1 } = \\phi ( y ^ { - j } x y ^ j ) , \\forall ~ j \\in \\mathbb { Z } \\rangle \\\\ T _ { \\mathcal { P } _ k } & = \\langle T _ i , t ; t y ^ k t ^ { - 1 } = \\phi ( y ^ k ) , t y ^ { - j } x y ^ j t ^ { - 1 } = \\phi ( y ^ { - j } x y ^ j ) , \\forall ~ 0 \\leq j < k \\rangle \\end{align*}"}
-{"id": "1552.png", "formula": "\\begin{align*} \\| m ^ N _ t - m ^ N _ s \\| _ { B L } ^ \\ast \\leq \\| m ^ N _ { t } - m ^ N _ { t _ { k } } \\| _ { B L } ^ \\ast + \\| m ^ N _ { t _ { n + 1 } } - m ^ N _ { s } \\| _ { B L } ^ \\ast + \\sum _ { l = n + 1 } ^ { m } \\| m ^ N _ { t _ { l + 1 } } - m ^ N _ { t _ { l } } \\| _ { B L } ^ \\ast . \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{align*} q ^ { \\sum _ { m = 1 } ^ N ( \\nu _ m / 2 - \\mu _ m \\nu _ m ) } , \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{align*} d ( m , n ) \\ = \\ \\abs { \\sqrt { m } - \\sqrt { n } } . \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{align*} P ^ { \\alpha + 1 } _ { F , V B } \\cap [ p , q ] = \\emptyset & \\iff \\exists [ p ' , q ' ] \\supsetneq [ p , q ] F _ { P ^ \\alpha } [ p ' , q ' ] \\\\ P ^ { \\alpha + 1 } _ { F , V B _ \\ast } \\cap [ p , q ] = \\emptyset & \\iff \\exists [ p ' , q ' ] \\supsetneq [ p , q ] F _ { E , \\ast } [ p ' , q ' ] , E = P ^ \\alpha . \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} | a r _ i \\theta _ i + b r _ j \\theta _ j | ^ 2 - r _ k ^ 2 = ( a ^ 2 r _ i ^ 2 + b ^ 2 r _ j ^ 2 - r _ k ^ 2 ) + 2 a b r _ i r _ j \\theta _ i \\cdot \\theta _ j = 2 a b r _ i r _ j \\big ( s + \\theta _ i \\cdot \\theta _ j \\big ) \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{align*} p _ { V _ { \\geq t } } \\left ( V _ { \\geq t } \\ \\middle | \\ G _ { < t } , V _ { < t } \\right ) = \\frac { \\mathbb { P } \\left ( G _ { < t } \\ \\middle | \\ V \\right ) p _ V ( V ) } { \\mathbb { P } \\left ( G _ { < t } \\ \\middle | \\ V _ { < t } \\right ) p _ { V _ { < t } } ( V _ { < t } ) } \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} h ( x ) = \\left \\lbrace \\begin{array} { c l } | x - \\frac { 1 } { 2 } | , & 0 \\leq x \\leq 1 \\\\ | x - \\frac { 3 } { 2 } | , & 1 < x \\leq 2 \\end{array} \\right . K ( x ) = \\left \\lbrace \\begin{array} { c l } [ - \\frac { 3 } { 2 } x + \\frac { 3 } { 2 } , 2 ] , & 0 \\leq x \\leq 1 \\\\ \\left [ 0 , - \\frac { 3 } { 2 } x + \\frac { 7 } { 2 } \\right ] , & 1 < x \\leq 2 \\end{array} \\right . \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} \\{ \\varphi , \\psi \\} ~ ~ = ~ ~ \\frac 1 2 \\left \\langle r \\nabla \\varphi , \\nabla \\psi \\right \\rangle - ~ \\frac 1 2 \\left \\langle r \\nabla ^ { \\prime } \\varphi , \\nabla ^ { \\prime } \\psi \\right \\rangle . \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} \\tau \\frac { \\partial _ \\tau \\Theta ( 0 , \\tau ) } { \\Theta ( 0 , \\tau ) } - \\frac { 1 } { \\tau } \\frac { \\partial _ \\tau \\Theta \\left ( 0 , - \\tfrac { 1 } { \\tau } \\right ) } { \\Theta \\left ( 0 , - \\tfrac { 1 } { \\tau } \\right ) } = - \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "5395.png", "formula": "\\begin{align*} \\Delta '^ { ( 1 ) } & = \\Delta ' \\\\ \\Delta '^ { ( 2 ) } & = ( \\Delta ' \\otimes ) \\otimes \\Delta '^ { ( 1 ) } \\\\ \\Delta '^ { ( n ) } & = ( \\Delta ' \\otimes ) \\Delta '^ { ( n - 1 ) } . \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} \\frac { 2 \\cdot 4 \\ , n ^ 3 \\ , \\pi ^ 3 \\ , e ^ { 4 n \\pi y } } { ( e ^ { 2 n \\pi y } - 1 ) ^ 3 } - \\frac { 4 \\ , n ^ 3 \\ , \\pi ^ 3 \\ , e ^ { 2 n \\pi y } } { ( e ^ { 2 n \\pi y } - 1 ) ^ 2 } = \\frac { 4 \\ , n ^ 3 \\ , \\pi ^ 3 \\ , e ^ { 2 n \\pi y } } { ( e ^ { 2 n \\pi y } - 1 ) ^ 3 } \\left ( e ^ { 2 n \\pi y } + 1 \\right ) \\end{align*}"}
-{"id": "7299.png", "formula": "\\begin{align*} r _ k = \\sqrt { \\Omega { } p _ k } x _ k + \\sqrt { \\Omega } \\mathbf { f } _ k { } \\mathbf { z } + \\mathbf { f } _ k \\mathbf { n } _ q , \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ l f _ { p _ j q _ j } v = v _ { \\{ 1 , \\dots , k \\} \\backslash \\{ p _ 1 , \\dots , p _ l \\} \\cup \\{ q _ 1 , \\dots , q _ l \\} } , \\end{align*}"}
-{"id": "5255.png", "formula": "\\begin{align*} \\int F ( x , y ) d \\nu _ 1 = \\int F ( x , y ) d \\nu _ 2 \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} [ 2 \\ \\dots \\ t \\mid i _ 1 \\ \\dots \\ i _ { t - 1 } ] \\ = \\ [ 1 \\ \\dots \\ t - 1 \\mid i _ 1 + 1 \\ \\dots \\ i _ { t - 1 } + 1 ] , \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} v ( n ) = \\psi _ 1 ( n _ 1 ) \\cdots \\psi _ g ( n _ g ) . \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} d \\big ( F _ t ( x ) , F _ t ( y ) \\big ) = \\sup _ i \\big \\{ f _ i \\big ( F _ t ( x ) \\big ) - f _ i \\big ( F _ t ( y ) \\big ) \\big \\} \\le d ( x , y ) \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{align*} \\partial _ H \\ell _ 1 ( \\{ 1 \\} , \\R ) = \\left \\lbrace x \\mapsto h _ { \\epsilon } ( x ) = \\epsilon x \\mathrel { \\big \\vert } \\begin{aligned} \\epsilon \\in \\{ - 1 , + 1 \\} \\end{aligned} \\right \\rbrace . \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} T = \\left \\{ ( \\rho _ a , \\rho _ b , \\rho _ { a + b } ) \\ \\vline \\ a , b \\in H \\right \\} . \\end{align*}"}
-{"id": "8469.png", "formula": "\\begin{align*} Z = A D ( t ) A ^ t , \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} [ a _ 1 \\dots a _ k , b _ 1 , b _ 2 ] = & \\sum _ { i = 1 } ^ k a _ 1 \\dots a _ { i - 1 } [ a _ i , b _ 1 , b _ 2 ] a _ { i + 1 } \\dots a _ k \\\\ + & \\sum _ { 1 \\le i < i ' \\le k } \\bigl ( a _ 1 \\dots a _ { i - 1 } [ a _ i , b _ 1 ] a _ { i + 1 } \\dots a _ { i ' - 1 } [ a _ { i ' } , b _ 2 ] a _ { i ' + 1 } \\dots a _ k \\\\ + & a _ 1 \\dots a _ { i - 1 } [ a _ i , b _ 2 ] a _ { i + 1 } \\dots a _ { i ' - 1 } [ a _ { i ' } , b _ 1 ] a _ { i ' + 1 } \\dots a _ k \\bigr ) ; \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} B ^ { ( \\omega ) } _ { u } : = f ^ { ( \\omega ) } _ u ( [ - R , R ] ) , \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} \\chi _ { \\epsilon } ( t , x , \\xi ) : = \\chi \\bigl ( \\epsilon ( \\mu + h ( t , x , \\xi ) \\bigr ) \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} F ( R ) = \\log ( \\det R ) , \\end{align*}"}
-{"id": "36.png", "formula": "\\begin{align*} \\mathcal { P } _ u ( \\lambda _ \\theta , \\gamma _ \\theta ) = \\max _ { v \\in \\mathcal { V } } g ( v , \\theta ) = g ( v _ \\theta , \\theta ) , \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} \\alpha s = s \\alpha + \\alpha ( s ) \\ , , \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{align*} = \\frac 1 2 \\int _ { S ^ 1 } \\begin{pmatrix} f _ 0 & f _ 1 & f _ 2 \\end{pmatrix} P _ 0 \\begin{pmatrix} h _ 0 \\\\ h _ 1 \\\\ h _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} | \\lambda \\cap S _ \\mu ( m ) | & = | \\pi _ 1 ( \\sigma ) \\cap S _ \\mu ( m ) | = | \\sigma \\cap ( S _ \\mu ( m ) \\times T _ \\mu ) | , \\\\ | \\lambda \\cap T _ \\mu ( m ) | & = | \\pi _ 2 ( \\sigma ) \\cap T _ \\mu ( m ) | = | \\sigma \\cap ( S _ \\mu \\times T _ \\mu ( m ) ) | , \\\\ | \\lambda | & = | \\pi _ 1 ( \\sigma ) | + | \\pi _ 2 ( \\sigma ) | = 2 | \\sigma | . \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} \\xi ( C ; S ) = ( n + 1 ) \\max _ { 1 \\leq k \\leq n + 1 } \\max _ { x \\in C } ( - \\lambda _ k ( x ) ) + 1 . \\end{align*}"}
-{"id": "6719.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial \\varphi } { \\partial t } \\left ( x , t \\right ) + H \\left ( t , \\nabla _ { x } \\varphi \\left ( x , t \\right ) \\right ) = 0 & \\ , \\mathbb { R } ^ { n } \\times \\left ( 0 , + \\infty \\right ) , \\\\ \\varphi \\left ( x , 0 \\right ) = J \\left ( x \\right ) & \\forall x \\in \\mathbb { R } ^ { n } , \\end{cases} \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{align*} \\displaystyle \\frac { \\pi | z _ { j , n } | } { n \\ , s _ { j , n } } < | z - z _ { j , n } | , \\mbox { w i t h } ~ ~ s _ { j , n } = a _ { \\max } \\Bigl [ 1 + \\frac { a _ { \\max } ^ { 2 } ( j - J _ n ) ^ 2 } { \\pi ^ 2 \\ , J _ { n } ^ { 2 } } \\Bigr ] ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "5788.png", "formula": "\\begin{align*} \\eta _ D [ \\frak { u } + \\frak { v } , y ] = \\eta _ D [ \\frak { u } , y ] \\cdot \\eta _ D [ \\frak { v } , y ] \\cdot \\eta _ D [ [ s ( \\frak { u } ) ] + [ s ( \\frak { v } ) ] , y ] , \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} { \\bf { R } } = \\frac { 1 } { c } { \\bf { A } } { \\bf { A } } ^ H , \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} \\phi ( k g h ) & = - 2 \\ln | k g h \\cdot s ( e ) | _ { \\pi ^ * q } \\\\ & = - 2 \\ln | k \\cdot g \\cdot h \\cdot \\pi _ * s ( e ) | _ q \\\\ \\intertext { b y e q u i v a r i a n c e o f $ \\pi $ } & = - 2 \\ln | g \\cdot \\chi ( h ) \\pi _ * ( s ( e ) ) | _ q \\\\ \\intertext { b y $ K $ - i n v a r i a n c e o f $ q $ a n d b y d e f i n i t i o n o f $ \\chi $ } & = - 2 \\ln | g \\cdot s ( e ) | _ { \\pi ^ * q } - 2 \\ln | \\chi ( h ) | \\end{align*}"}
-{"id": "5449.png", "formula": "\\begin{align*} \\langle w \\lambda - q , \\check { \\varpi _ i } \\rangle \\neq \\langle s _ { \\alpha _ j } w \\lambda - q , \\check { \\varpi _ i } \\rangle \\Leftrightarrow i = j \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} g \\cdot ( h , x ) : = ( h g ^ { - 1 } , x ) , g \\in G , ( h , x ) \\in G \\times S _ { } . \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} \\widetilde \\Delta ^ { ( m ) } \\partial _ { z _ j } = \\partial _ { z _ { j - 1 } } - 2 \\partial _ { z _ j } + \\partial _ { z _ { j + 1 } } \\ , , \\ ; \\ ; 1 \\le j \\le m - 2 \\ , , \\widetilde \\Delta ^ { ( m ) } \\partial _ { z _ { m - 1 } } = \\partial _ { z _ { m - 2 } } - \\partial _ { z _ { m - 1 } } \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} H _ 3 & \\leq c \\sum _ { n = M _ + } ^ \\infty \\frac { n ^ { d - 1 } } { M ^ d } { \\rm e } ^ { - c _ 1 ( n - M ) ^ 2 / ( \\gamma t ) } \\leq c _ 2 \\ , { \\rm e } ^ { - c _ 3 ( \\gamma t ) ^ { 2 \\varepsilon } } . \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{align*} P ( i , j ) = \\underline { d p } _ i \\underline \\wedge \\underline { d p } _ j , i < j \\in [ n ] , \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} s = s ^ 1 s ^ 2 , \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} V _ { } : = \\{ v \\in V \\mid \\Gamma \\setminus v \\} , \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} | K _ \\lambda ( x , y ) | = | C _ \\lambda ( x , y ) + D _ \\lambda ( x , y ) | \\leq \\frac { 1 } { \\sqrt { | \\lambda | } } \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} \\frac { \\partial v } { \\partial \\nu } ( \\gamma , t ) = \\frac { \\partial u ^ { \\tilde f } } { \\partial \\nu } ( \\gamma , t ) - \\frac { \\partial u ^ { \\tilde f } } { \\partial \\nu } ( \\gamma , 2 T - t ) \\mbox { o n } \\ , \\Sigma ^ T _ \\sigma . \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} q ^ { ( i j ) } _ { Z _ a Z _ b } ( z _ a , z _ b ) = p _ { Z | X Y } ( z _ a | x _ i , y _ j ) p _ { Z | X Y } ( z _ { b } | x _ { 3 - i } , y _ { 3 - j } ) \\end{align*}"}
-{"id": "6771.png", "formula": "\\begin{align*} \\alpha ( a ) : = e ^ { \\frac { \\pi i } { 2 } \\mathbf u ( a ) } . \\end{align*}"}
-{"id": "6389.png", "formula": "\\begin{align*} \\eta _ i - \\xi _ i = ( 1 - a ) a ^ { i - 1 } , i = 1 , 2 , \\dots \\end{align*}"}
-{"id": "1025.png", "formula": "\\begin{align*} d ( u ^ { n _ { k } } , C _ { i } ) & = \\Vert u ^ { n _ { k } } - P _ { C _ { i } } u ^ { n _ { k } } \\Vert \\leq \\Vert u ^ { n _ { k } } - P _ { C _ { i } } u ^ { l _ { k } } \\Vert \\\\ & \\leq \\Vert u ^ { n _ { k } } - u ^ { l _ { k } } \\Vert + \\Vert u ^ { l _ { k } } - P _ { C _ { i } } u ^ { l _ { k } } \\Vert \\rightarrow _ k 0 . \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} I _ { a + } ^ { \\alpha ; \\psi } f \\left ( x \\right ) : = \\frac { 1 } { \\Gamma \\left ( \\alpha \\right ) } \\int _ { a } ^ { x } \\psi ^ { \\prime } \\left ( t \\right ) \\left ( \\psi \\left ( x \\right ) - \\psi \\left ( t \\right ) \\right ) ^ { \\alpha - 1 } f \\left ( t \\right ) d t \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} \\tilde { m } = \\begin{pmatrix} \\frac { 1 } { 2 } \\| m \\| ^ 2 \\\\ m _ 1 \\\\ m _ 2 \\\\ 1 \\end{pmatrix} \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} b = - \\varepsilon \\left ( \\left \\vert X \\right \\vert ^ { 2 } - K ^ { 2 } \\right ) + \\min _ { \\mathfrak { \\bar { B } } _ { K } \\left ( 0 \\right ) \\cap L } \\Theta , \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} v _ n = v _ 0 + e _ 1 + e _ 2 + . . . + e _ { n } = \\sum _ { t = 1 } ^ n g ( t ) \\binom { \\cos \\varphi ( t ) } { \\sin \\varphi ( t ) } . \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} ^ { m , 1 } _ { i n t e r } = \\underset { x _ j \\in \\Phi _ c \\backslash m } { \\sum } \\frac { | h _ { j , m 1 } | ^ 2 } { L \\left ( | | y _ { m , 1 } + x _ m - x _ j | | \\right ) } . \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} \\mathbb { L } _ i \\left ( \\prod _ { k \\in \\mathbb { Z } } z _ k ^ { \\nu _ k } \\right ) = \\prod _ { \\substack { k \\in \\mathbb { Z } \\\\ k \\not = i , i + 1 } } z _ k ^ { \\nu _ k } \\times \\left \\{ \\begin{array} { l l } 0 , & \\nu _ i = \\nu _ { i + 1 } , \\\\ ( z _ { i + 1 } - t z _ i ) , & \\nu _ i > \\nu _ { i + 1 } , \\\\ ( t z _ i - z _ { i + 1 } ) , & \\nu _ i < \\nu _ { i + 1 } , \\end{array} \\right . \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} d ( x _ 1 , \\ldots , x _ k ) : = \\max _ { 1 \\le i , j \\le k } | x _ i - x _ j | _ \\infty , \\end{align*}"}
-{"id": "1588.png", "formula": "\\begin{align*} \\rho _ \\xi = \\rho _ \\xi ( \\mu ) : = \\lfloor ( \\mu - 1 ) / 2 \\rfloor \\rho _ \\sigma = \\rho _ \\sigma ( \\mu ) : = \\lfloor \\mu / 2 \\rfloor . \\end{align*}"}
-{"id": "1779.png", "formula": "\\begin{align*} & \\varphi ( x ) g ( x , D _ x ) \\psi ( x ) = \\varphi ( x ) \\eta ( x _ n ) g ( x , D _ x ) \\psi ( x ) + \\varphi ( x ) ( 1 - \\eta ( x _ n ) ) g ( x , D _ x ) \\psi ( x ) \\\\ & \\ = \\varphi ( x ) \\eta ( x _ n ) g ( x , D _ x ) ( \\eta ( x _ n ) \\psi ( x ) ) + \\varphi ( x ) \\eta ( x _ n ) g ( x , D _ x ) ( 1 - \\eta ( x _ n ) ) \\psi ( x ) \\\\ & \\ \\ \\ + \\varphi ( x ) ( 1 - \\eta ( x _ n ) ) g ( x , D _ x ) \\psi ( x ) . \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} d _ 1 ( u _ k ) = ( k ) _ q ( 1 - q ^ { k - 1 } r ) u _ { k - 1 } , d _ 2 ( u _ k ) = \\delta _ { k 0 } 1 \\end{align*}"}
-{"id": "1497.png", "formula": "\\begin{align*} \\Delta f & = \\sum _ { i = 1 } ^ n \\overline { \\nabla } ^ 2 f ( e _ i , e _ i ) + \\left \\langle \\overline { \\nabla } f , { \\bf H } \\right \\rangle = - \\dfrac { n } { 2 } - | \\bf { H } | ^ 2 . \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\beta ^ + } f _ { \\gamma } ( z ) = f _ { \\beta } ( z ) , \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} C \\cdot \\Omega _ \\varepsilon \\cdot ( K _ \\varepsilon + \\lceil \\log _ 2 \\Omega _ \\varepsilon \\rceil ) \\leq C \\cdot ( 1 + C _ 1 ) \\cdot \\Omega _ \\varepsilon \\cdot K _ \\varepsilon = \\ell _ \\varepsilon . \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} ( F ) = \\lbrace P _ 1 \\mu \\times P _ 2 ( \\mu _ { [ x ] } | _ S ) : x \\in ( F , P _ 1 ) , S \\subseteq F ^ x \\mu _ { [ x ] } ( S ) > 0 \\rbrace , \\end{align*}"}
-{"id": "2026.png", "formula": "\\begin{align*} E _ { n + 1 } = E _ n + \\Delta E ( \\sqrt { 2 E _ n } , \\kappa _ n ) , \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} L ( z ) _ t = H ( z ) _ t - H ( z ) _ 0 - \\vartheta ( z ) \\cdot S _ t = H ( z ) _ t - H ( z ) _ 0 - \\widetilde Y ( z ) _ t , \\quad \\textnormal { a . s . , } \\ t \\geq 0 . \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} \\abs { \\tau _ d ( y ) ( x ) - \\tau _ d ( y ) ( z ) } & = \\abs { d ( x , y ) - d ( b , y ) - d ( z , y ) + d ( b , y ) } \\\\ & = \\abs { d ( x , y ) - d ( z , y ) } \\\\ & \\leq d ( x , z ) \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} \\textnormal { a r e a } ( \\L ) = | I m ( \\overline { z _ 1 } z _ 2 ) | = | x _ 1 y _ 2 - x _ 2 y _ 1 | , z _ k = x _ k + i \\ , y _ k , \\ , k = 1 , 2 . \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{align*} \\pi \\mathcal { H } _ t ( \\ell ) = \\begin{cases} \\exp ( t h _ 1 ) ( \\ell ) & t \\in [ 0 , \\tau _ 1 ( \\ell ) ] , \\\\ \\exp ( ( t - \\tau _ 1 ( \\ell ) ) h _ 2 ) \\circ \\exp ( \\tau _ 1 ( \\ell ) h _ 1 ) ( \\ell ) & t \\in [ \\tau _ 1 ( \\ell ) , \\tau _ 2 ( \\ell ) ] , \\\\ \\exp ( ( t - \\tau _ 2 ( \\ell ) ) h _ 3 ) \\circ \\exp ( ( \\tau _ 2 ( \\ell ) - \\tau _ 1 ( \\ell ) ) h _ 2 ) \\\\ \\circ \\exp ( \\tau _ 1 ( \\ell ) h _ 1 ) ( \\ell ) & t \\in [ \\tau _ 2 ( \\ell ) , T ] , \\end{cases} \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{align*} P \\{ T _ 1 < T _ 2 \\} = \\lambda \\nu _ 2 M \\ , I _ M , \\ ; I _ M = \\int _ 0 ^ 1 x ^ { \\lambda - 1 } \\left ( 1 - x \\right ) ^ { \\nu _ 1 M } \\left ( 1 - x ^ { \\lambda } \\right ) ^ { \\nu _ 2 M - 1 } d x . \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} c ^ { \\delta + k } \\ge c ^ { k - 1 } + \\frac { s ( B - 1 ) } { 2 \\binom { k } { 2 } } c ^ { k - 2 } + \\frac { s ( 2 \\delta - s - B ) } { 2 \\binom { k } { 2 } } + \\frac { \\binom { \\delta - s } { 2 } } { \\binom { k } { 2 } } c ^ { k - B } . \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} S _ { \\mu } = S _ { e _ n } \\cdots S _ { e _ 1 } . \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} \\frac { \\left ( E + \\sigma \\right ) ^ { - 1 } + \\left ( E - \\sigma \\right ) ^ { - 1 } } { 2 } \\big ( E - \\sigma ^ 2 \\big ) & = \\frac { ( E - \\sigma ) + ( E + \\sigma ) } { 2 } = E , \\\\ \\frac { \\left ( E + \\sigma \\right ) ^ { - 1 } - \\left ( E - \\sigma \\right ) ^ { - 1 } } { 2 } \\big ( E - \\sigma ^ 2 \\big ) & = \\frac { ( E - \\sigma ) - ( E + \\sigma ) } { 2 } = - \\sigma . \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} i \\mathfrak { H } ( s , t ) = \\frac { 1 } { \\pi } \\operatorname { p . v . } \\int f ( \\beta , t ) \\frac { \\xi _ \\beta ( \\beta , t ) } { \\xi ( s , t ) - \\xi ( \\beta , t ) } d \\beta . \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} s ( n , 3 ) \\ , = \\ , 3 0 \\cdot 3 1 4 4 9 6 0 0 ^ { 2 ^ { n - 3 } - 1 } \\cdot 1 0 8 7 7 6 0 3 2 4 5 9 0 8 2 9 5 6 8 0 0 ^ { ( 2 ^ { n - 3 } - 1 ) ( 2 ^ { n - 4 } - 1 ) / 3 } \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} I _ { 3 , - } ( x _ 1 , x _ 2 ) = \\int _ 0 ^ t \\int _ { - \\infty } ^ 0 \\sigma ^ 2 _ s ( y ) \\int _ { - \\infty } ^ y [ G _ { t - s } ( x _ 1 - z ) - G _ { t - s } ( x _ 2 - z ) ] \\psi ( s , z ) d z d y d s . \\end{align*}"}
-{"id": "1095.png", "formula": "\\begin{align*} ( s \\otimes 1 - 1 \\otimes s ) \\cdot g ( 1 \\otimes n ) = g ( s \\otimes _ S n - 1 \\otimes _ S s \\cdot n ) \\\\ = 0 \\ , ; \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} ( \\iota _ * ) ^ { - 1 } = \\sum _ { | \\vec { Y } | = n } \\frac { \\iota _ { \\vec { Y } } ^ * } { e ( T _ { \\vec { Y } } M ( r , n ) ) } , \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} \\mu ( ( K _ 2 \\cap K ) \\times \\mathbb { T } ) = \\operatorname { R e } \\int _ { ( K _ 2 \\cap K ) \\times \\mathbb { T } } - \\overline { \\gamma _ 0 } \\gamma d \\mu = \\int _ { ( K _ 2 \\cap K ) \\times \\mathbb { T } } \\operatorname { R e } ( - \\overline { \\gamma _ 0 } \\gamma ) d \\mu . \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} \\Delta ^ m H ( B ^ e ) \\times \\Big ( \\prod _ { j = 1 } ^ h \\big ( 2 H ^ { - ( y _ j - ( 2 y - 1 ) / ( e p ) ) } & \\big ( H / H ( B ^ e ) \\big ) ^ { 1 / ( 2 h ) } \\big ) ^ 2 \\Big ) \\times \\\\ & \\times \\big ( C _ { 1 2 } H ^ { ( 2 y - 1 ) / ( e p ) } \\big ) ^ { p e - 2 h } V ( p e - 2 h ) , \\end{align*}"}
-{"id": "5682.png", "formula": "\\begin{align*} \\mathbb { X } _ { s , t } : = \\int _ s ^ t ( X _ { r - } - X _ s ) \\otimes \\dd X _ r = \\int _ 0 ^ t X _ { r - } \\otimes \\dd X _ r - \\int _ 0 ^ s X _ { r - } \\otimes \\dd X _ r - X _ s \\otimes X _ { s , t } , ( s , t ) \\in \\Delta _ T , \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{align*} a \\cdot u & = ( a , u _ 1 , \\ldots , u _ k ) \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{align*} J = J _ { \\mathrm { f s u } } \\oplus J ' \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{align*} \\frac { d ( d ( p _ 1 ) ) \\cdots d ( d ( p _ r ) ) } { d ( d ( p _ 1 \\cdots p _ r ) ) } = \\frac { d ( 2 ) ^ r } { d ( 2 ^ r ) } = \\frac { 2 ^ r } { r + 1 } \\neq 1 . \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} \\begin{pmatrix} A _ 0 ( z ) \\\\ A _ r ( z ) \\end{pmatrix} : = \\underbrace { L ( z ) \\stackrel { . } { \\otimes } \\cdots \\stackrel { . } { \\otimes } L ( z ) } _ { r - 1 } \\begin{pmatrix} 1 \\\\ z \\end{pmatrix} , \\end{align*}"}
-{"id": "985.png", "formula": "\\begin{align*} \\frac { 1 } { 2 R } \\sum _ { i = 1 } ^ { m } \\rho _ { i } \\Vert Q _ { i } x - Q _ { i - 1 } x \\Vert ^ { 2 } \\leq \\Vert U x - x \\Vert \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} \\begin{pmatrix} - v _ 3 & v ^ T & 0 \\\\ 0 & I & v \\\\ 0 & 0 & v _ 3 \\end{pmatrix} \\end{align*}"}
-{"id": "3038.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\mathrm { d i v } \\left ( ( \\nabla \\phi _ { 1 } ) u _ { 0 } ^ { q _ { 0 } } \\right ) = \\int _ { \\Omega } ( \\Delta \\phi _ { 1 } ) u _ { 0 } ^ { q _ { 0 } } + \\int _ { \\Omega } ( \\nabla \\phi _ { 1 } \\nabla u _ { 0 } ) q _ { 0 } u _ { 0 } ^ { q _ { 0 } - 1 } , \\end{align*}"}
-{"id": "7338.png", "formula": "\\begin{align*} ( X + \\lambda M ) ^ \\cdot = [ X + \\lambda M , B - \\lambda M ^ { k + 1 } ] \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} \\widetilde { h } ( x , D _ { x } ) u ( x ) : = \\kappa ^ { - 1 , * } h ( x , D _ { x } ) \\kappa ^ * u ( x ) u \\in \\mathcal { S } ( \\overline { \\R _ + ^ n } ) . \\end{align*}"}
-{"id": "1044.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 3 } } V ( x ) \\rho _ { n _ { k } } ( x ) { \\rm d } x = \\int _ { \\mathbb { R } ^ { 3 } } V ( x ) \\rho _ { k } ^ { ( 1 ) } ( x ) { \\rm d } x + o ( 1 ) _ { k \\to \\infty } . \\end{align*}"}
-{"id": "1047.png", "formula": "\\begin{align*} \\sqrt { \\eta _ { 0 } ( x ) ^ { 2 } + m ^ { 2 } } - \\tau ( | \\cdot | ^ { - 1 } \\star \\rho _ { 0 } ) ( x ) + V ( x ) - \\mu = 0 \\end{align*}"}
-{"id": "1028.png", "formula": "\\begin{align*} \\mathbb { E } \\exp ( t X ) \\le \\exp ( 4 K ^ 2 t ^ 2 ) = \\exp \\big ( \\varphi _ q ( 2 \\sqrt { 2 } K t ) \\big ) \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} L _ Q ^ \\top L _ P = U \\Sigma V ^ \\top , \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{align*} t _ { \\ast } : = \\exp \\left [ - \\frac { \\int _ { \\Omega } a \\left ( x \\right ) \\phi _ { 1 } ^ { 2 } \\log \\phi _ { 1 } } { \\int _ { \\Omega } a \\left ( x \\right ) \\phi _ { 1 } ^ { 2 } } \\right ] . \\end{align*}"}
-{"id": "729.png", "formula": "\\begin{align*} { \\rm M } ( \\theta _ { n } ) ~ = ~ { \\rm M } ( G _ n ) ~ \\geq ~ \\Theta = 1 . 3 2 4 7 \\ldots , n \\geq 2 , \\end{align*}"}
-{"id": "5451.png", "formula": "\\begin{align*} C ( w , v ) : = \\{ z \\in W , v \\leq z \\leq w \\textnormal { a n d } z v ^ { - 1 } \\in W _ Q \\} \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{align*} Q _ V = \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & Q _ W & 0 \\\\ 1 & 0 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{align*} & 2 \\overline \\square _ B \\phi = - \\partial _ B H ^ { 1 , 0 } \\lrcorner \\ , \\phi - H ^ { 1 , 0 } \\lrcorner \\ , \\partial _ B \\phi + H ^ { 0 , 1 } \\lrcorner \\ , \\bar \\partial _ B \\phi , \\\\ & \\bar \\partial _ B H ^ { 1 , 0 } \\lrcorner \\ , \\phi + H ^ { 1 , 0 } \\lrcorner \\ , \\bar \\partial _ B \\phi = 0 . \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{align*} Q ' ( i , j , \\alpha ) = \\underline { d p } _ i \\underline \\wedge \\omega _ { i j } ^ \\alpha , i < j \\in [ n ] , \\alpha \\geq 0 , \\end{align*}"}
-{"id": "7917.png", "formula": "\\begin{align*} \\nu ( t ) : = \\max \\limits _ { t \\in [ 0 , T ] } \\max \\limits _ { x \\in [ 0 , D ] } \\left | u ( s \\ , , x ) - v ( s \\ , , x ) \\right | ^ 2 . \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} \\hat { f } ( z , x ) = \\left \\{ \\begin{array} { l l } f ( z , \\tilde { v } ( z ) ) + \\hat { \\xi } _ 0 | \\tilde { v } ( z ) | ^ { p - 2 } \\tilde { v } ( z ) & \\mbox { i f } \\ x < \\tilde { v } ( z ) \\\\ f ( z , x ) + \\hat { \\xi } _ 0 | x | ^ { p - 2 } x & \\mbox { i f } \\ \\tilde { v } ( z ) \\leq x \\leq \\tilde { u } ( z ) \\\\ f ( z , \\tilde { u } ( z ) ) + \\hat { \\xi } _ 0 \\tilde { u } ( z ) ^ { p - 1 } & \\mbox { i f } \\ \\tilde { u } ( z ) < x . \\end{array} \\right . \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} e ^ { t _ l V _ l } e ^ { t _ { l + 1 } V _ { l + 1 } } y = e ^ { t _ { l + 1 } V _ { l + 1 } } e ^ { t _ l V _ l } y . \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{align*} \\left | b _ k ' - b _ k \\right | = \\frac { 2 \\pi } { | c _ k | } \\geq \\frac { 2 \\pi } { 3 } . \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} { } C = \\{ u \\in C ( \\overline { \\Omega } ) : u \\geq { 0 } \\ ; \\mbox { i n } \\ ; \\Omega \\} \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} K _ { 1 2 } : = s K _ 1 s ^ { - 1 } \\cap K _ 2 , K _ { 2 3 } : = K _ 2 \\cap t ^ { - 1 } K _ 3 t \\end{align*}"}
-{"id": "3190.png", "formula": "\\begin{align*} d \\beta \\wedge \\eta \\wedge \\omega _ 1 + \\beta \\wedge \\omega _ 1 ^ 2 = 0 , \\end{align*}"}
-{"id": "8741.png", "formula": "\\begin{align*} \\mathbb { A } v = \\mathbb { P } \\mathcal { A } v , \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} \\int _ { \\Sigma } ( \\Delta _ { f } u ) v e ^ { - f } d \\sigma = - \\int _ { \\Sigma } \\left \\langle \\nabla u , \\nabla v \\right \\rangle e ^ { - f } d \\sigma . \\end{align*}"}
-{"id": "2538.png", "formula": "\\begin{align*} \\partial _ a \\phi + A ( 0 ) \\phi & = \\partial _ a h + A ( 0 ) h \\ , \\\\ \\phi ( 0 ) - \\eta \\ell [ 0 ] \\phi & = h ( 0 ) \\ . \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} & \\left ( 1 - x + \\frac { x v } { 1 - v } \\right ) ^ 2 D ( x , v ) = x \\left ( 1 - x + \\frac { x v } { 1 - v } \\right ) B ( x , v ) \\\\ & \\qquad + \\frac { x } { v } \\left ( \\frac { x v ^ 3 } { 1 - v } C ( x , 1 ) + \\frac { x ^ 3 v ^ 3 ( 2 - x - x v ) } { ( 1 - x ) ^ 2 ( 1 - x v ) ^ 2 } \\right ) + \\frac { x v ^ 2 } { 1 - v } \\left ( 1 - x + \\frac { x v } { 1 - v } \\right ) D ( x , 1 ) . \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{align*} \\gamma _ l : = N ^ 2 \\big | \\langle \\psi _ N , \\ , \\nabla _ { x _ 1 } g _ \\beta ( x _ 1 - x _ 2 ) \\ , \\mathbf { A } ( x _ 1 ) \\widehat { r } \\ , \\psi _ N \\rangle \\big | \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{align*} | { \\rm I } - { \\rm I } ' | & \\leq \\sum _ { \\ell = 0 } ^ L \\sum _ { x \\in E _ n } \\left ( \\frac { 2 + \\lambda / N _ n } { \\lambda / N _ n } \\cdot \\frac { 1 } { N _ n ^ 2 } \\right ) \\cdot \\frac { 2 ^ \\ell | Q ^ { ( n ) , \\ell } ( x , x ) - Q ^ { ( \\infty ) , \\ell } | } { ( \\lambda / N _ n + 2 ) ^ { \\ell + 1 } } \\\\ & \\leq \\frac { 1 } { \\lambda } \\sum _ { \\ell = 0 } ^ L \\int _ { E _ n } | Q ^ { ( n ) , \\ell } ( x , x ) - Q ^ { ( \\infty ) , \\ell } | \\pi ^ { ( n ) } ( \\d x ) \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} \\begin{aligned} E _ { m , n } & : = \\{ x \\in \\omega : | u _ m ( x ) - u _ n ( x ) | < 1 \\} , \\\\ e _ { m , n } & : = \\left ( | \\nabla u _ m | ^ { p - 2 } \\nabla u _ m - | \\nabla u _ n | ^ { p - 2 } \\nabla u _ n \\right ) \\cdot \\nabla ( u _ m - u _ n ) . \\end{aligned} \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{align*} \\sup _ { \\mu , w , t } H _ n f ( t , \\mu , w ) & = \\sup _ { \\mu , w , t } H _ n [ t ] f ( t , \\mu , w ) \\\\ & = \\sup _ { \\mu , w , t } H _ n [ t ] f ( \\gamma _ n ^ { - 1 } t , \\mu , w ) . \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } H _ n ( \\nu ) = \\theta , \\end{align*}"}
-{"id": "6057.png", "formula": "\\begin{align*} u & = ( I - ( K + S ) ) \\eta + S ( \\eta - w ) \\\\ v _ 1 & = ( I - ( K + S ) ) w - S ( \\eta - w ) , \\end{align*}"}
-{"id": "2984.png", "formula": "\\begin{align*} P ( x ) = ( - 1 ) ^ d \\left ( - h _ 1 x + h _ 2 x ^ 2 \\right ) + O ( x ^ 3 ) , \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} b _ 1 & = \\frac { \\bar { \\omega _ 2 } z - \\omega _ 2 \\bar { z } } { A } \\\\ b _ 2 & = \\frac { \\omega _ 1 \\bar { z } - \\bar { \\omega _ 1 } z } { A } \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} \\lambda ( t ) . x = \\sum ( \\chi - \\nu ) ( \\lambda ( t ) ) v _ { \\chi } \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{align*} \\bar { \\nu } ( \\xi , t ) = \\frac { \\operatorname { p . v . } } { 2 \\pi i } \\int \\frac { \\gamma } { \\xi - \\xi ( \\beta ) } d \\beta , \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{align*} \\mathcal { P } \\left ( ( \\mathcal { B } _ { k , i } ) \\right ) = \\pm \\bigotimes _ { \\substack { 1 \\le k \\le n - 1 \\\\ 1 \\le i \\le m _ k } } v _ { J _ { k , i } } . \\end{align*}"}
-{"id": "3609.png", "formula": "\\begin{align*} f ( x ) + \\frac { f '' ( x ) } { ( 1 + f ' ( x ) ) ^ { 3 / 2 } } = 0 \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} \\square _ B & = \\overline \\square _ B - i ( J \\kappa _ B ) ^ \\sharp , \\\\ \\Delta _ B & = \\square _ B + \\overline \\square _ B , \\\\ \\bar \\partial _ T ^ * \\bar \\partial _ B & = \\partial _ T ^ * \\partial _ B . \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} Z _ 1 ^ 3 + Z _ 2 ^ 3 + 9 Z _ 3 ^ 3 = 0 \\\\ 3 ^ a Z _ 1 + 3 ^ b Z _ 2 + Z _ 3 = 0 \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{align*} | G | = | R _ G | + | B _ G | + | A _ p \\cup X ' | + | X '' | \\le 8 + 8 + 4 \\cdot 2 ^ { k - 2 } + ( k - 2 ) < 4 \\cdot 2 ^ k + 1 \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } X ^ { A } _ t \\stackrel { ( d ) } { = } \\tfrac 1 2 \\xi _ { \\rm G O E } , \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} f _ - ^ n ( \\eta _ i ) - f _ - ^ n ( x ) = a ^ n ( \\eta _ i - x ) \\to 0 \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle { \\frac { d } { d t } Q = [ W , Q ] - \\frac { \\partial F ( Q ) } { \\partial Q } } = [ W , Q ] - \\left ( a Q - b ( Q ^ 2 - \\frac { 1 } { 3 } I d | Q | ^ 2 ) + c Q | Q | ^ 2 \\right ) \\end{array} \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} ( u _ t - \\mathcal { H } u _ { x x } + ( u ^ 2 ) _ x ) _ x + \\gamma u = 0 , x \\in \\mathbb { R } , t > 0 \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{align*} h ^ 1 ( \\Sigma _ c , K ^ { - 1 } L ^ { - 1 } ) = 3 ( g - 1 ) + l . \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} \\frac { \\xi } { x _ A + x _ B } = \\frac { \\eta } { x _ A + x _ C } = \\frac { \\zeta } { x _ B + x _ C } . \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} f ( z ) & = z ^ { a _ 1 } + \\cdots + z ^ { a _ n } , \\\\ g ( z ) & = z ^ { b _ 1 } + \\cdots + z ^ { b _ m } . \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{align*} \\mathcal { F } _ { u } ( q _ { 0 } , u _ { 0 } ) \\phi = \\sigma \\phi . \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} [ 1 - P _ S ( \\tilde { \\zeta } ) ] P _ N ( \\zeta ) = 0 \\qquad [ 1 - P _ N ( \\zeta ) ] P _ S ( \\tilde { \\zeta } ) = 0 \\mathrlap { . } \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{align*} \\sum _ { \\substack { i , j \\in R \\\\ i \\not \\sim _ R j \\\\ i \\ne j } } | \\{ v \\in Y : i v , j v \\in E ( B ) \\} | = \\sum _ { v \\in Y } | \\{ i , j \\in R : i \\ne j , i v , j v \\in E ( B ) , i j \\notin E ( R ) \\} | \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} | p | \\ln _ 3 t \\le | Z _ t | ( 1 + h ^ \\star _ t ) \\ln _ 3 t = | Z _ t | \\ln _ 3 t + o ( t d _ t h _ t ) , \\end{align*}"}
-{"id": "4103.png", "formula": "\\begin{align*} \\mu \\left ( \\left \\{ k \\in K \\middle | \\left \\langle { g } { k } \\right \\rangle \\neq G \\right \\} \\right ) & \\leqslant \\sum _ { F \\in \\mathcal { F } } \\mu \\left ( \\left \\{ { k } \\in K \\middle | \\left \\langle { g } { k } \\right \\rangle = F \\right \\} \\right ) \\\\ & = \\sum _ { F \\in \\mathcal { F } } \\mu \\left ( \\left \\{ { k } \\in K \\middle | \\left \\langle { r } _ { F } { k } \\right \\rangle = F \\right \\} \\right ) = 0 \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} \\mathfrak { l } = \\mathfrak { l } _ { I ( \\mu ) } ^ { \\sigma _ { \\mu } } \\oplus \\bigoplus _ { \\alpha \\in \\Phi _ { P _ { I ( \\mu ) } ^ u } } \\mathfrak { g } _ { \\alpha } \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} V ( x ) = - \\sum _ { i = 1 } ^ { M } \\frac { z _ { i } } { | x - x _ i | ^ { s _ { i } } } , \\end{align*}"}
-{"id": "4772.png", "formula": "\\begin{align*} \\left . \\begin{matrix} \\nabla g ( \\overline { x } ) \\Delta \\lambda = 0 , \\\\ ( \\Delta \\lambda , 0 ) \\in \\mathcal { N } _ { { \\rm g p h } \\mathcal { N } _ K } \\big ( ( g ( \\overline { x } ) , \\overline { \\lambda } ) ; ( g ' ( \\overline { x } ) \\xi , \\eta ) \\big ) \\end{matrix} \\right \\} \\Longrightarrow \\Delta \\lambda = 0 . \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} w _ { z \\to y } : = \\begin{cases} ( 2 d \\sigma ( z ) ) ^ { - 1 } , & | y - z | = 1 , \\\\ 0 , & , \\end{cases} \\end{align*}"}
-{"id": "350.png", "formula": "\\begin{align*} [ T ( \\psi _ 1 * _ G \\psi _ 2 ) ] ( k ' ) & = | H | \\cdot \\sum _ { g \\in G } \\psi _ 1 ( g ) \\psi _ 2 ( g ^ { - 1 } k ' ) . \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} X _ { p _ { ( x , i ' ) } , 1 } = \\mathcal { X } ^ { ( x ) } _ { i ' , 1 } + \\sum _ { x ' = 1 } ^ { x - 1 } \\ell ^ { \\circ } _ { x ' } - \\sum _ { x ' = x + 1 } ^ { \\mathcal { k } } \\ell ^ { \\circ } _ { x ' } . \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} Q ^ { 1 , 0 } = \\{ Z \\in Q ^ C | J Z = i Z \\} , Q ^ { 0 , 1 } = \\{ Z \\in Q ^ C | J Z = - i Z \\} . \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} \\| f \\| _ { B ^ { p } } = \\left ( \\int _ { \\mathbf { B } } ( 1 - | x | ^ { 2 } ) ^ { m p } | \\partial ^ { m } f ( x ) | ^ { p } d \\tau ( x ) \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} D ^ * \\xi ( X _ 1 , \\ldots , X _ k ) = - g ^ { i j } D _ { \\partial _ i } \\xi ( \\partial _ j , X _ 1 , \\ldots , X _ k ) . \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{align*} P r _ { \\rm I N S } = \\frac { \\widetilde { \\rm S I N R } ^ { I } } { \\rho _ t \\left ( \\frac { 1 } { { r } } - \\frac { 1 } { c } \\right ) } . \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{align*} \\underline { q } : = \\inf \\left \\{ q \\in ( 0 , 1 ) : \\mbox { $ ( P _ { a , q } ) $ h a s a s o l u t i o n $ u $ s u c h t h a t $ ( q , u ) \\in \\mathcal { C } _ 1 $ } \\right \\} . \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} \\sum _ { k ' \\in \\mathcal { K } ( k ) } S W _ M ( k ' ) = ( - 1 ) ^ { b ( M , N ) } \\sum S W _ { \\hat { X } } ( l _ 1 ) S W _ { \\hat { Y } } ( l _ 2 ) \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{align*} \\begin{aligned} r _ k & = \\displaystyle \\sum ^ { k } _ { j = 1 } ( r _ { j } - r _ { j - 1 } ) = \\displaystyle \\sum ^ { k } _ { j = 1 } \\frac 1 { \\varphi ( H ( r _ { j - 1 } ) ) } \\\\ & \\leq \\displaystyle \\sum ^ { k } _ { j = 1 } \\frac 1 { \\varphi ( e ^ { j - 1 } K ) } \\leq \\displaystyle \\int ^ \\infty _ { 0 } \\frac { d u } { \\varphi ( e ^ { u - 1 } K ) } = \\displaystyle \\int ^ \\infty _ { K / e } \\frac { d t } { t \\varphi ( t ) } \\end{aligned} \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} & W ^ 0 _ { l + 2 } ( p , p _ 1 , \\dots , p _ { l + 1 } ) = \\\\ & K _ p ( q , \\bar q ) \\sum _ { k = 0 } ^ { l - 1 } W ^ 0 _ { k + 2 } ( q , \\dots , p _ { k + 1 } ) W ^ 0 _ { l - k + 1 } ( \\bar q , p _ { k + 2 } , \\dots , p _ { l + 1 } ) \\\\ & = \\underset { t _ 1 \\in Y ^ p , t _ 2 \\in Y ^ q , p + q + 1 = l } { \\sum } t _ 1 \\vee t _ 2 \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} { { \\Delta } } _ i = \\{ \\alpha \\in \\Delta : h _ j ( \\alpha ) = 0 , j > i , ~ h _ i ( \\alpha ) \\neq 0 \\} , \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} x ^ * a e ^ { - a ( x ^ * + g ( x ^ * ) - k ) } = \\gamma \\left ( 1 - e ^ { - a ( x ^ * + g ( x ^ * ) - k ) } \\right ) \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{align*} \\mathcal { F } \\left \\{ { } _ x { D } _ { \\theta } ^ { \\alpha } \\psi ( x , t ) ; p \\right \\} = - { \\eta { } } _ { \\alpha { } } ^ { \\theta { } } \\hat { \\psi } ( p , t ) , \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} [ \\alpha , s \\beta ] = s [ \\alpha , \\beta ] + \\alpha ( s ) \\beta \\ , . \\end{align*}"}
-{"id": "3635.png", "formula": "\\begin{align*} \\overline { \\mathcal { A } } _ w = ( c _ { w } ( \\chi ) ) ^ { - 1 } \\mathcal { A } _ w . \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{align*} [ [ a , b ] , [ a , b ] ^ g ] & = [ [ a , b ] , [ a , b ] [ a , b , g ] ] \\\\ & = [ [ a , b ] , [ a , b , g ] ] , \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} \\begin{array} { r l } ( 4 , 4 , 6 ) = & 6 . ( 2 , 0 , 1 ) + 1 . \\left ( 0 , 4 , 0 \\right ) , \\\\ ( 4 , 4 , 6 ) = & 6 . ( 2 , 0 , 1 ) + 3 . \\left ( 0 , 4 , 0 \\right ) , \\\\ ( 4 , 4 , 6 ) = & 6 . ( 2 , 0 , 1 ) + 5 . \\left ( 0 , 4 , 0 \\right ) , \\\\ ( 4 , 4 , 6 ) = & 6 . ( 2 , 0 , 1 ) + 7 . \\left ( 0 , 4 , 0 \\right ) . \\end{array} \\end{align*}"}
-{"id": "7781.png", "formula": "\\begin{align*} V _ s : = \\sup _ { | x - y | = \\delta } \\| \\psi ( s , x ) - \\psi ( s , y ) \\| _ p . \\end{align*}"}
-{"id": "5867.png", "formula": "\\begin{align*} S _ m ( \\mu ) = m \\mu - \\rho ( \\mu ) = m \\nu - \\rho ( \\nu ) = S _ m ( \\nu ) . \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{align*} \\lVert A B - A _ 0 B _ 0 \\rVert ^ 2 & \\sim \\sum _ { i = 1 } ^ { M - 1 } \\left [ x _ i ^ 2 + \\sum _ { j = 2 } ^ { N } \\{ x _ i - ( a _ i b _ 1 - a _ i b _ j ) \\} ^ 2 \\right ] \\\\ & \\sim \\sum _ { i = 1 } ^ { M - 1 } \\left [ x _ i ^ 2 + \\sum _ { j = 2 } ^ { N } ( a _ i b _ 1 - a _ i b _ j ) ^ 2 \\right ] \\\\ & = \\sum _ { i = 1 } ^ { M - 1 } \\left [ x _ i ^ 2 + \\sum _ { j = 2 } ^ { N } a _ i ^ 2 ( b _ j - b _ 1 ) ^ 2 \\right ] . \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} \\frac { ( 1 - \\frac { c _ n } { n } ) ^ { 2 n } } { ( 1 - \\frac { c _ n } { n } ) - ( 1 - \\frac { c _ n } { n } ) ^ { n } } = \\frac { e ^ { - 2 c } } { 1 - e ^ { - c } } \\Bigl ( 1 + \\frac { c } { 2 n ( 1 - e ^ { - c } ) } \\bigl [ 2 - c e ^ { - c } - 2 c \\bigr ] \\Bigr ) \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} B _ \\mu ( s _ 0 , t _ 0 ) = \\lbrace ( s , t ) \\in B _ \\mu \\mid s \\le s _ 0 , t \\ge t _ 0 \\rbrace . \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} h ( q ) = \\sum _ { i \\ge 0 } a _ i q ^ { n _ i } \\ \\ \\ \\ a _ i \\neq 0 \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} \\sum _ { k \\geq 2 } ( 2 k - 1 ) ^ 4 q ^ { 2 k - 1 } = q \\ , \\frac { 8 1 q ^ 2 + 2 2 0 q ^ 4 + 8 6 q ^ 6 - 4 q ^ 8 + q ^ { 1 0 } } { \\left ( 1 - q ^ 2 \\right ) ^ 5 } , | q | < 1 . \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{align*} \\langle P u , v \\rangle _ { L ^ 2 } = \\langle u , P ^ \\ast v \\rangle _ { L ^ 2 } . \\end{align*}"}
-{"id": "4585.png", "formula": "\\begin{align*} \\partial _ B \\phi & = \\sum _ { a = 1 } ^ n \\omega ^ a \\wedge \\nabla _ { V _ a } \\phi , \\\\ \\bar \\partial _ B \\phi & = \\sum _ { a = 1 } ^ n \\bar \\omega ^ a \\wedge \\nabla _ { \\bar V _ a } \\phi , \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} ( B _ 1 , B _ 2 , i , j ) \\cdot ( t _ 1 , t _ 2 , e _ 1 , \\dots , e _ r ) = ( t _ 1 B _ 1 , t _ 2 B _ 2 , i e , t _ 1 t _ 2 e ^ { - 1 } j ) , \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} Q _ V = \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ 1 & 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} - \\int _ { R _ 1 } ^ { R _ 2 } F ( r ) \\dd r = \\mathrm { I } + \\mathrm { I I } + \\mathrm { I I I } , \\end{align*}"}
-{"id": "4427.png", "formula": "\\begin{align*} H ( S ^ d ) = N ( \\mathfrak { a } ) ^ { - 1 } \\prod _ { j = 1 } ^ { p } | | X _ 1 ^ { ( j ) } \\wedge \\dots \\wedge X _ d ^ { ( j ) } | | , \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} | - 1 + z _ { J _ n , n } e ^ { - 2 i k \\pi / n } + ( z _ { J _ n , n } e ^ { - 2 i k \\pi / n } ) ^ n | = | G _ { n } ( z _ { J _ n , n } ) | + \\frac { 2 | k | \\pi } { n } = \\frac { 2 | k | \\pi } { n } \\end{align*}"}
-{"id": "4581.png", "formula": "\\begin{align*} & \\nabla _ { \\rm t r } ^ * \\nabla _ { \\rm t r } \\pi ( X ) - { \\rm R i c } ^ Q ( X ) + A _ { X } \\kappa _ B ^ \\sharp = 0 , \\\\ & \\int _ M g _ Q ( ( \\mathcal { L } _ X J ) \\kappa _ B ^ \\sharp , J \\pi ( X ) ) = 0 . \\end{align*}"}
-{"id": "5826.png", "formula": "\\begin{align*} E _ { s _ i \\mu } = t ^ { - 1 } \\left ( T _ i + \\frac { 1 - t } { 1 - y _ { i + 1 } ( \\mu ) / y _ i ( \\mu ) } \\right ) E _ { \\mu } , \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } d ^ * d ( \\chi w ) - \\frac { 1 } { 2 } w ( d ^ * d \\chi ) - < d \\chi , d w > + \\chi r | \\alpha | ^ 2 w - \\chi | \\nabla _ A \\alpha | ^ 2 + \\chi \\mathfrak { e } _ w = 0 . \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} ( I _ l g ) ( s , t ) : = s ^ { \\frac l 2 } g \\left ( \\frac { t } { \\sqrt s } \\right ) . \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{align*} M _ { i j } : = \\begin{bmatrix} ( A _ 1 ) _ { i i } ( B _ 1 ) _ { j j } & - ( C _ 1 ) _ { i i } ( D _ 1 ) _ { j j } \\\\ & \\ddots & \\ddots \\\\ & & ( A _ { r - 1 } ) _ { i i } ( B _ { r - 1 } ) _ { j j } & - ( C _ { r - 1 } ) _ { i i } ( D _ { r - 1 } ) _ { j j } \\\\ - ( C _ r ) _ { i i } ( D _ r ) _ { j j } & & & ( A _ r ) _ { i i } ( B _ r ) _ { j j } \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} F ( p , z | \\tau ) F ( p + p ' , z ' | \\tau ) - F ( p ' , z ' | \\tau ) F ( p , z + z ' | \\tau ) + F ( p + p ' , z + z ' | \\tau ) F ( p ' , - z | \\tau ) = 0 \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{align*} \\begin{array} { l } p _ { X ' | X } ( 0 | x ) = a ( x ) / \\bar { a } \\\\ p _ { Y ' | Y } ( 0 | y ) = b ( y ) / \\bar { b } . \\end{array} \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} { \\bf \\nabla } _ F u = \\textit { \\textbf { V } } \\ { \\rm i n } \\ \\Omega , \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} \\mathcal { F } ( \\mathcal { C } ^ \\alpha g ) ( \\xi , \\tau ) = \\bigg ( 1 - \\frac { \\tau ^ 2 } { | \\xi | ^ 2 } \\bigg ) ^ \\alpha _ + \\phi ( | \\xi | ) \\widehat { g } ( \\xi , \\tau ) . \\end{align*}"}
-{"id": "3211.png", "formula": "\\begin{align*} \\rho = \\tfrac { 1 } { 2 } r ^ 2 \\eta \\wedge \\tau + r d r \\wedge \\alpha + \\tfrac { 1 } { 2 } r ^ 2 d \\alpha + O ( r ^ { - 1 - \\delta } ) \\end{align*}"}
-{"id": "5615.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } ( E , \\lambda \\Phi ) = \\psi ( W , Q _ W , \\lim _ { \\lambda \\to \\infty } [ \\lambda ^ { - 1 } \\xi ] ) . \\end{align*}"}
-{"id": "8838.png", "formula": "\\begin{align*} \\Omega _ { \\alpha _ 1 , \\bar { \\alpha } _ 2 } & = \\frac { 1 } { 2 } \\Big { ( } e ^ { 2 \\alpha _ 2 ( a ) } ( 1 + \\tanh ( ( \\alpha _ 1 - \\alpha _ 2 ) ( a ) ) ) \\\\ & + e ^ { 2 \\alpha _ 1 ( a ) } ( 1 + \\tanh ( ( \\alpha _ 2 - \\alpha _ 1 ) ( a ) ) ) \\Big { ) } \\chi \\circ \\mathcal { H } ( [ \\theta ( e _ { \\alpha _ 2 } ) , e _ { \\alpha _ 1 } ] ) \\\\ & = \\frac { 2 \\chi ( [ \\theta ( e _ { \\alpha _ 2 } , e _ { \\alpha _ 1 } ] ) } { ( e ^ { - 2 \\alpha _ 1 ( a ) } + e ^ { - 2 \\alpha _ 2 ( a ) } ) } . \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{align*} P \\{ T _ 1 < \\cdots < T _ g \\} = P \\{ \\tilde { T } _ 1 < \\cdots < \\tilde { T } _ g \\} P \\{ T _ 1 = T _ { \\min } \\} = P \\{ \\tilde { T } _ 1 = \\tilde { T } _ { \\min } \\} , \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} h ^ n ( p ) = \\sum _ { i \\in I } \\varphi ^ i ( p ) \\cdot p _ i ' - ( 1 \\otimes \\partial ) ( \\varphi ^ i ( p ) ) \\cdot k ^ n ( p _ i ) \\ , . \\end{align*}"}
-{"id": "306.png", "formula": "\\begin{align*} \\Psi ( N ) = \\sum _ r E _ r N E _ r = \\sum _ r N E _ r ^ 2 = \\sum _ r N E _ r = N \\sum _ r E _ r = N I = N . \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{align*} E \\left [ S ( \\theta ) ^ { ( r ) } \\right ] \\sim \\frac { N ^ r ( \\ln N ) ^ r } { ( 1 - \\theta ) ^ r } \\sum _ { k = 0 } ^ { \\infty } \\binom { r } { k } \\frac { ( - 1 ) ^ k \\ , \\Gamma ^ { ( k ) } ( 1 ) } { \\ln ^ k N } , N \\to \\infty , \\end{align*}"}
-{"id": "839.png", "formula": "\\begin{align*} \\iint _ { X ^ { 1 } _ { r } } | u ( x ) | ^ { p } \\frac { | \\phi _ { r } ( x ) - \\phi _ { r } ( y ) | ^ { p } } { | x - y | ^ { N + s p } } \\ , d x d y = 0 . \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} E _ { \\ell _ 1 , \\ell _ 2 , \\ell _ 3 } = - x _ 3 \\cdot e _ { 2 , ( \\ell _ 1 - 1 , \\ell _ 2 ) , \\ell _ 3 + 1 } + x _ 2 \\cdot e _ { 2 , ( \\ell _ 1 - 1 , \\ell _ 2 + 1 ) , \\ell _ 3 } + x _ 2 \\cdot e _ { 1 , ( \\ell _ 2 + 2 , \\ell _ 3 ) , \\ell _ 1 - 2 } - x _ 3 \\cdot e _ { 1 , ( \\ell _ 2 + 1 , \\ell _ 3 ) , \\ell _ 1 - 1 } . \\end{align*}"}
-{"id": "9089.png", "formula": "\\begin{align*} \\varphi ( \\langle q _ 0 , q _ 1 , q _ 2 , \\ldots q _ j \\rangle ) \\le g ( t \\frown q _ j ) < g ( t ) = \\varphi ( \\langle q _ 0 , q _ 1 , q _ 2 , \\ldots q _ i \\rangle ) \\ : . \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} \\widehat { \\sigma } = t _ i ^ { - 1 } \\log ^ { \\frac { 1 } { 2 } } ( t _ i ) e ^ { \\frac 1 2 F _ 1 ( \\log { \\frak t } ( t _ i ) ) } ( \\sigma + F _ 2 ( \\log { \\frak t } ( t _ i ) ) \\delta ) \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{align*} \\theta _ { m , n } ( \\Omega m + n ) \\theta _ { m , n } ( 0 ) = c _ { m , n } ^ 2 \\exp { ( 4 i \\pi \\ , ^ t \\ ! m n ) } . \\end{align*}"}
-{"id": "4422.png", "formula": "\\begin{align*} g _ { j _ 1 } h ( J \\setminus ( D \\cup J _ 1 ) ) & = g ( I _ 1 ) g ( I _ 2 ) h ( J \\setminus ( D \\cup J _ 1 ) ) \\\\ & = g ( J _ 1 \\setminus I _ 1 ) g ( J _ 2 \\setminus I _ 2 ) h ( J \\setminus ( D \\cup J _ 2 ) ) = g _ { j _ 2 } h ( J \\setminus ( D \\cup J _ 2 ) ) . \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} \\dot { \\bar { x } } ( t ) & = \\bar { A } ( \\varepsilon ) \\bar { x } ( t ) + \\bar { B } ( \\varepsilon ) u ( t ) + \\displaystyle \\sum _ { j = 1 } ^ { n _ { \\rm i n } } u _ j ( t ) \\bar { N } _ j ( \\varepsilon ) \\bar { x } ( t ) + \\bar { H } ( \\varepsilon ) ( \\bar { x } ( t ) \\otimes \\bar { x } ( t ) ) \\\\ \\bar { y } ( t ) & = \\bar { c } ^ \\top \\bar { x } ( t ) \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{align*} \\varphi \\mapsto \\gamma _ { n } \\varphi : = \\varphi | _ { \\partial \\Omega } \\cdot n \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} \\phi \\wedge \\bar * \\bar \\psi = \\langle \\phi , \\psi \\rangle \\nu , \\end{align*}"}
-{"id": "288.png", "formula": "\\begin{align*} \\varphi _ 6 ( x , y ) = x ^ 6 - 5 x ^ 4 y ^ 2 + \\frac { 5 } { 3 } x ^ 2 y ^ 4 - \\frac { 1 } { 2 7 } y ^ 6 \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{align*} l ' _ j = ( 1 - x a _ j ) _ { + } . \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} q _ { X _ 1 Y _ 1 X _ 2 Y _ 2 } ( x _ 1 , y _ 1 , x _ 2 , y _ 2 ) & = p _ { X Y } ( x _ 1 , y _ 1 ) p _ { X Y } ( x _ 2 , y _ 2 ) \\\\ r _ { X _ 1 Y _ 1 X _ 2 Y _ 2 } ( x _ 1 , y _ 1 , x _ 2 , y _ 2 ) & = p _ { X Y } ( x _ 1 , y _ 2 ) p _ { X Y } ( x _ 2 , y _ 1 ) \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{align*} \\overline \\chi ' ( \\overline p _ 0 ) = { \\tiny \\begin{pmatrix} 0 & 0 & 1 & - d _ 0 \\\\ d _ 0 & 1 & z _ 0 & - \\Re ( z _ 0 d _ 0 ) \\\\ - 1 & 0 & 0 & - z _ 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} } , \\overline \\chi ' ( \\overline p _ 0 ) ^ { - 1 } = { \\tiny \\begin{pmatrix} 0 & 0 & - 1 & - z _ 0 \\\\ - z _ 0 & 1 & d _ 0 & \\Re ( z _ 0 d _ 0 ) \\\\ 1 & 0 & 0 & d _ 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} } , \\end{align*}"}
-{"id": "902.png", "formula": "\\begin{align*} \\det ( \\Delta _ 0 ) = Z ' ( 1 ) e ^ { ( 2 g - 2 ) ( - 1 / ( 1 2 ) - 2 \\log ( A ) + ( \\log ( 2 \\pi ) ) / 2 ) } . \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 } e ^ { \\frac { \\omega s } { 2 } } s ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { r } ) } \\norm { \\nabla y _ { j } ( s ) } _ { L ^ r ( \\Omega ) ^ { 3 \\times 3 } } = 0 , \\end{align*}"}
-{"id": "5547.png", "formula": "\\begin{align*} \\frac { d } { d y } \\log \\left ( f \\left ( e ^ y \\right ) \\right ) = e ^ y \\frac { f ' \\left ( e ^ y \\right ) } { f \\left ( e ^ y \\right ) } . \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} N _ n ( t ) : = N ( t \\wedge \\tau _ n ) \\qquad ( t \\in [ 0 \\ , , T ] ) \\end{align*}"}
-{"id": "6775.png", "formula": "\\begin{align*} T ^ * = \\overline { \\alpha ( | T | ) } T ^ \\dagger . \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} \\alpha \\cdot ( n \\otimes ( u \\otimes v ) ) = \\alpha \\cdot n \\otimes ( u \\otimes v ) + n \\otimes ( ( \\alpha \\otimes 1 - 1 \\otimes \\alpha ) \\cdot ( u \\otimes v ) ) \\ , . \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} & \\left | E [ \\beta ( t , q _ t ^ m ) V ( t , q _ t ^ m ) ] - E [ \\beta ( t , q _ t ) V ( t , q _ t ) ] \\right | \\\\ \\leq & E [ | \\tilde C ( 1 + \\| q _ t \\| ^ { \\tilde p } + \\| q _ t - q _ t ^ m \\| ^ { \\tilde p } ) \\| q _ t - q _ t ^ m \\| | ] \\\\ = & O ( m ^ { 1 / 2 } ) \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} \\eta _ { g , y } = { ^ { g } y } , \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} \\sum _ { \\sigma } q ^ { \\mathrm { i n v } ( \\sigma ) } = \\prod _ { s \\in \\lambda \\cap S _ \\mu } \\left ( \\sum _ { a _ s = 0 } ^ { \\rho ( s , \\mu , \\lambda ) - 1 } q ^ { a _ s } \\right ) = \\prod _ { s \\in \\lambda \\cap S _ \\mu } \\frac { q ^ { \\rho ( s , \\mu , \\lambda ) } - 1 } { q - 1 } . \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { t _ { 0 } } ^ { t _ { 0 } + h } \\norm { e ^ { - ( t _ { 0 } + h - s ) A } F _ { u } ( u _ { j } , \\nabla y _ { j } ) } _ { L ^ { r } _ { \\sigma } ( \\Omega ) } \\ \\d s + \\int _ { 0 } ^ { t _ { 0 } } \\norm { [ e ^ { - h A } - I d ] e ^ { - ( t _ { 0 } - s ) A } F _ { u } ( u _ { j } , \\nabla y _ { j } ) } _ { L ^ { r } _ { \\sigma } ( \\Omega ) } \\ \\d s . \\end{aligned} \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{align*} \\Phi ( | \\nabla u + t \\nabla v | ) - \\Phi ( | \\nabla u | ) = \\phi ( \\theta _ t ) \\theta _ t \\left [ | \\nabla u + t \\nabla v | - | \\nabla u | \\right ] , \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} \\rho = \\sum _ { k = 0 } ^ { d - 1 } \\psi _ d ^ k , \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} \\int _ { \\R ^ { N } } \\left ( F ( t _ { n } v _ { n } ) - F ( v _ { n } ) \\right ) \\ , d x = o _ { n } ( 1 ) . \\end{align*}"}
-{"id": "6972.png", "formula": "\\begin{align*} \\lvert S K _ 2 ( \\Gamma _ h ) \\rvert = \\lvert I _ h \\rvert . \\end{align*}"}
-{"id": "3191.png", "formula": "\\begin{align*} d \\beta \\wedge \\eta \\wedge \\omega _ 3 = 0 , - 2 \\beta \\wedge \\eta \\wedge \\omega _ 3 + d \\beta \\wedge \\omega _ 2 = 0 . \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) + \\Phi ^ * ( \\partial _ x u ( t , x ) ) & = 0 \\\\ u ( 0 , x ) & = h ( x ) . \\end{align*}"}
-{"id": "4525.png", "formula": "\\begin{align*} \\mathbb { P } ^ { ( i ) } ( \\mu , K ) = \\frac { 1 } { q ^ { ( \\mu - i ) ( K - i ) } } \\prod _ { l = 0 } ^ { i - 1 } \\frac { ( 1 - q ^ { l - \\mu } ) ( 1 - q ^ { l - K } ) } { 1 - q ^ { l - i } } . \\end{align*}"}
-{"id": "6253.png", "formula": "\\begin{align*} \\sum _ { m } ( - 1 ) ^ { \\mu _ m } q ^ { \\kappa ( m , \\mu , \\lambda ) } = \\frac { q ^ { N - | \\mu | } - q ^ { | \\mu | } } { q - 1 } , \\end{align*}"}
-{"id": "6898.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\rightarrow 0 } \\chi \\left ( E , \\widehat { { E ^ \\varepsilon } } \\right ) = 0 . \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} a _ { m + k + 1 } ^ { - 1 } a _ k a _ { m + k + 1 } = a _ { m + k } , \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} \\Im ( \\omega \\eta ) = \\frac 2 3 ( \\pi - \\arg ( \\lambda ) ) \\log | \\lambda | + O ( 1 ) . \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & b _ 1 & 0 & \\cdots & 0 \\\\ 0 & 1 & b _ 2 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & 1 & b _ { n - 1 } \\\\ b _ n & 0 & \\cdots & 0 & 1 \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} U ^ i _ 1 & = \\{ v _ i \\} , \\\\ U ^ i _ 2 & = \\{ v _ j : v _ i \\sim v _ j , j > i \\} , \\\\ U ^ i _ k & = \\emptyset \\mbox { ( f o r $ 3 \\leq k \\leq t $ ) } . \\end{align*}"}
-{"id": "3558.png", "formula": "\\begin{align*} - \\partial \\mathbf { r } _ { i j } / \\partial s _ { i } \\times \\mathbf { r } _ { i j } & = ( \\mathbf { a } _ { 1 } \\zeta + \\mathbf { a } _ { 2 } \\zeta ^ { 2 } + . . . ) \\times ( \\mathbf { a } _ { 1 } + 2 \\mathbf { a } _ { 2 } \\zeta + . . . ) \\\\ & = ( \\mathbf { a } _ { 1 } \\times \\mathbf { a } _ { 2 } ) \\zeta ^ { 2 } + O ( \\zeta ^ { 3 } ) \\\\ & = ( \\mathbf { a } _ { 1 } \\times \\mathbf { a } _ { 2 } ) | \\zeta | ^ { 2 } . \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\frac { - \\log ( 2 r \\sqrt { k } ) } { k - 1 } } \\int _ { x _ { k - 1 } } ^ { \\frac { - \\log ( 2 r \\sqrt { k } ) } { k - 1 } } \\cdots \\int _ { x _ { 2 } } ^ { \\frac { - \\log ( 2 r \\sqrt { k } ) } { k - 1 } } \\left ( e ^ { \\sum _ { i = 1 } ^ { k - 1 } 2 i x _ i } - 2 r e ^ { \\sum _ { i = 1 } ^ { k - 1 } ( 2 i + 1 ) x _ i } \\right ) \\ , d x _ 1 \\ , d x _ 2 \\cdots d x _ { k - 1 } . \\end{align*}"}
-{"id": "8596.png", "formula": "\\begin{align*} \\tau _ A ( y ) : = s y s ^ * = \\sum _ { i , j = 1 } ^ N s _ j y { s _ i } ^ * , \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{align*} \\langle \\phi ^ { \\epsilon } _ \\alpha \\ , , \\ , \\mathfrak { O p } ^ \\epsilon ( \\mathfrak h ^ \\epsilon ) \\phi ^ { \\epsilon } _ \\beta \\rangle _ { \\mathcal { H } } = \\Lambda ^ \\epsilon ( \\alpha , \\beta ) \\langle \\phi ^ { \\epsilon } _ { \\alpha - \\beta } \\ , , \\ , \\mathfrak { O p } ^ \\epsilon ( \\mathfrak h ^ \\epsilon ) \\phi ^ { \\epsilon } _ 0 \\rangle _ { \\mathcal { H } } \\ , . \\end{align*}"}
-{"id": "5025.png", "formula": "\\begin{align*} & [ c , z _ 1 ] [ z _ 2 , z _ 3 , z _ 4 ] = - [ c , z _ 1 ] \\bigl [ z _ 4 , [ z _ 2 , z _ 3 ] \\bigr ] = - [ c , z _ 1 ] [ z _ 4 , z _ 2 , z _ 3 ] + [ c , z _ 1 ] [ z _ 4 , z _ 3 , z _ 2 ] \\\\ = \\ & [ c , z _ 4 ] [ z _ 1 , z _ 2 , z _ 3 ] - [ c , z _ 4 ] [ z _ 1 , z _ 3 , z _ 2 ] = [ c , z _ 4 ] \\bigl [ z _ 1 , [ z _ 2 , z _ 3 ] \\bigr ] = - [ c , z _ 4 ] [ z _ 2 , z _ 3 , z _ 1 ] , \\end{align*}"}
-{"id": "4699.png", "formula": "\\begin{align*} \\sum _ { m = - \\infty } ^ { 0 } r _ m ^ { ( 0 ) } < \\infty . \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} g ( k , i , n + 1 ) & = g ( k - 1 , 1 , n ) + g ( k , i + 1 , n ) \\\\ & = g ( k - 1 , 1 , n ) + g ( k - 1 , 1 , n - 1 ) + g ( k , i + 2 , n - 1 ) \\\\ & \\dots \\\\ = [ g ( k - 1 , 1 , n ) & + g ( k - 1 , 1 , n - 1 ) + \\dots + g ( k - 1 , 1 , 0 ) ] + g ( k , i + n + 1 , 0 ) . \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} I ( \\mu , \\nu ) = \\left \\{ \\begin{array} { l l } 0 , & \\exists \\ k : \\mu _ k > \\nu _ k = 0 , \\ \\ \\ \\ \\mu _ k < \\nu _ k = 2 , \\\\ \\\\ 1 , & , \\end{array} \\right . \\end{align*}"}
-{"id": "7533.png", "formula": "\\begin{align*} e ^ { - y \\tilde \\gamma } ( e ^ { - y \\tilde \\gamma } ) ^ T = e ^ { - 2 \\gamma y } I = ( e ^ { - y \\tilde \\gamma } ) ^ T e ^ { - y \\tilde \\gamma } . \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{align*} \\mu ( D ^ 4 ( x _ i ) , x _ { j } ) ) = \\mu ( x _ { n - 6 + i } , x _ { j } ) = 0 \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{align*} X ^ { u s } ( { \\cal L } ) = \\bigcup \\limits _ { \\hat { w } \\in \\hat { W } } \\bigcup \\limits _ { w \\in W ( \\hat { w } , \\lambda , q ) } \\hat { w } ^ { - 1 } X _ w \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{align*} \\begin{aligned} D _ \\varpi ( P \\parallel Q ) \\ge D ( P \\parallel Q ) \\ge D _ \\alpha ( P \\parallel Q ) , \\end{aligned} \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{align*} f _ R ( v ) = p \\left ( \\delta ( v ) \\right ) q _ R ( b ( v ) ) \\end{align*}"}
-{"id": "3991.png", "formula": "\\begin{align*} p _ { X Y Z X ' Y ' } = p _ { X Y Z } \\ , p _ { X ' | X } \\ , p _ { Y ' | Y } \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} & \\sum _ { z ^ n } \\ ! \\Big ( \\mathbb { P } [ Z ^ n = z ^ n , X ^ n \\ ! \\in \\ ! \\mathcal { A } _ 1 , Y ^ n \\ ! \\in \\ ! \\mathcal { B } _ 1 ] ^ { \\frac { 1 } { 2 } } \\times \\mathbb { P } [ Z ^ n = z ^ n , X ^ n \\ ! \\in \\ ! \\mathcal { A } _ 2 , Y ^ n \\ ! \\in \\ ! \\mathcal { B } _ 2 ] ^ { \\frac 1 2 } \\Big ) \\\\ [ 0 . 5 e x ] & \\ ; > \\mathbb { P } [ X ^ n \\ ! \\in \\ ! \\mathcal { A } _ 1 , Y ^ n \\ ! \\in \\ ! \\mathcal { B } _ 2 ] ^ { \\frac 1 2 } \\ ; \\mathbb { P } [ X ^ n \\ ! \\in \\ ! \\mathcal { A } _ 2 , Y ^ n \\ ! \\in \\ ! \\mathcal { B } _ 1 ] ^ { \\frac 1 2 } . \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} - \\frac { \\alpha ( \\gamma , \\phi ) } { 4 \\ ( \\sinh \\frac { l ( \\gamma ) } { 2 } \\ ) } = \\frac 1 2 \\int _ \\gamma \\phi ( z ( t ) ) ( \\dot z ( t ) ) ^ 2 d t = \\frac 1 2 \\int _ { \\gamma } \\overline { \\mu ( z ( t ) ) } ( \\dot { z } ( t ) ) ^ 2 \\rho ( z ( t ) ) d t . \\end{align*}"}
-{"id": "1031.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\tau } ( \\rho ) : = \\int _ { \\mathbb { R } ^ 3 } j _ { m } ( \\rho ( x ) ) { \\rm d } x - \\frac { \\tau } { 2 } \\iint _ { \\mathbb { R } ^ { 3 } \\times \\mathbb { R } ^ { 3 } } \\frac { \\rho ( x ) \\rho ( y ) } { \\left | x - y \\right | } { \\rm d } x { \\rm d } y + \\int _ { \\mathbb { R } ^ 3 } V ( x ) \\rho ( x ) { \\rm d } x . \\end{align*}"}
-{"id": "8754.png", "formula": "\\begin{align*} \\xi ( t , x ) = a ' ( t ) + ( x - a ( t ) ) + \\frac { | x - a ( t ) | ^ { 2 } } { 2 } \\omega ( t ) \\end{align*}"}
-{"id": "519.png", "formula": "\\begin{align*} ^ { H } \\mathbb { D } _ { a + } ^ { \\alpha , \\beta ; \\psi } f \\left ( x \\right ) = I _ { a + } ^ { \\gamma - \\alpha ; \\psi } \\mathcal { D } _ { a + } ^ { \\gamma ; \\psi } f \\left ( x \\right ) \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{align*} \\| \\hat f - f _ 0 \\| ^ 2 _ n = \\left \\langle \\sum _ { i = 0 } ^ { n - 1 } ( \\hat f _ i - f _ i ) \\psi _ i , \\sum _ { i = 0 } ^ { n - 1 } ( \\hat f _ i - f _ i ) \\psi _ i \\right \\rangle _ n = \\sum _ { i = 0 } ^ { n - 1 } ( \\hat f _ i - f _ i ) ^ 2 . \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{align*} C P ( X , \\Omega _ X , J , \\Lambda _ X ) = \\sum \\limits _ { \\alpha _ { \\pm } \\in \\mathcal { P } ( Y _ { \\pm } , \\omega _ { \\pm } , \\Gamma _ { \\pm } ) } \\sum \\limits _ { Z \\in H _ 2 ( X , \\alpha _ + , \\alpha _ - ) } \\# _ 2 \\mathcal { M } _ { X , I = 0 } ^ J ( \\alpha _ + , \\alpha _ - , Z ) \\Lambda _ X ( Z ) . \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} H ( v , s ) & = \\exp \\Big ( \\int _ { \\max \\{ v - R _ { \\tau } , R _ { \\tau } \\} } ^ { v + R _ { \\tau } } \\frac { 2 y \\lambda _ J ' ( y ) } { 1 + \\frac { y ^ { \\alpha } } { s P _ J } } d y \\Big ) , \\\\ f ( v ) & = 2 \\pi \\lambda _ U ^ s ( r , v ) v \\exp ( - 2 \\pi \\int _ { 0 } ^ { v } \\lambda _ U ^ s ( r , y ) y d y ) , \\\\ \\lambda _ J ' ( y ) & = \\lambda _ J \\arccos ( \\frac { y ^ 2 + v ^ 2 - R _ { \\tau } ^ 2 } { 2 y v } ) . \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} { } \\beta ^ k P - \\bar { \\beta } ^ k Q = E , \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{align*} & P \\{ T _ 1 = T _ { \\min } \\} = \\\\ & ( - 1 ) ^ { g - 1 } \\sum _ { k _ g = 1 } ^ { M _ g } \\cdots \\sum _ { k _ 2 = 1 } ^ { M _ 2 } \\sum _ { k _ 1 = 0 } ^ { M _ 1 } ( - 1 ) ^ { k _ 1 + \\cdots + k _ g } \\binom { M _ g } { k _ g } \\cdots \\binom { M _ 1 } { k _ 1 } I , \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} - d = - \\nabla ^ 2 f ( x _ * ) ^ { - 1 } \\nabla f ( x _ 0 ) . \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} \\eta ( \\tau ) = e ^ { \\pi i \\tau / 1 2 } \\prod _ { k = 1 } ^ \\infty ( 1 - e ^ { 2 \\pi i k \\tau } ) . \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} y _ n = \\begin{cases} a ' , & \\textrm { i f $ n > 0 $ a n d $ x _ { n - 1 } = e $ } , \\\\ ( x _ n ) ' & \\textrm { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{align*} M ^ k _ t - M ^ k _ s = \\frac { t - t ^ k _ m } { \\Delta t ^ k } ( m ^ k _ { t ^ k _ { m + 1 } } - m ^ k _ { t ^ k _ { m } } ) + \\frac { t ^ k _ { n + 1 } - s } { \\Delta t ^ k } ( m ^ k _ { t ^ k _ { n + 1 } } - m ^ k _ { t ^ k _ { n } } ) + \\sum _ { l = n + 1 } ^ { m } ( m ^ k _ { t ^ k _ { l + 1 } } - m ^ k _ { t ^ k _ { l } } ) . \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{align*} \\Phi ( q , t ) : = \\int _ { \\Omega } a ( x ) \\{ ( t \\phi _ { 1 } + w ( q , t ) ) ^ { q } - ( t \\phi _ { 1 } + w ( q , t ) \\} \\phi _ { 1 } = 0 , \\quad ( q , t ) \\simeq ( 1 , t _ { 0 } ) . \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} D _ \\beta ( r ) : = - \\inf _ { x \\in M } \\left ( - \\beta \\ , E ^ \\dagger ( x ) + \\Omega ( x ) - \\Omega ^ \\dagger + r \\ , \\| F ( x ) - y ^ \\dagger \\| \\right ) \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} \\liminf _ { \\varepsilon \\to 0 ^ + } c ^ * _ { \\varepsilon } \\geq 2 \\sqrt { a _ { \\inf } - \\frac { \\chi \\mu a _ { \\sup } } { b _ { \\inf } - \\chi \\mu } } - \\frac { \\chi \\mu \\sqrt { N } a _ { \\sup } } { 2 \\sqrt { \\lambda } ( b _ { \\inf } - \\chi \\mu ) } : = c _ { - } ^ * ( a , b , \\chi , \\lambda , \\mu ) \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} r ^ n \\lambda ^ { - [ \\frac { n - k \\log r } { \\log \\lambda } ] - 1 } \\cdot O _ n ^ { - 1 } T R ^ n T ^ { - 1 } = \\lambda ^ { - 1 } \\cdot r ^ k \\lambda ^ { \\lbrace \\frac { n - k \\log r } { \\log \\lambda } \\rbrace } \\cdot O _ n ^ { - 1 } T R ^ n T ^ { - 1 } \\end{align*}"}
-{"id": "6073.png", "formula": "\\begin{align*} \\left ( ( X _ 0 , X _ 1 ) _ { \\theta _ 0 , q _ 0 } , ( X _ 0 , X _ 1 ) _ { \\theta _ 1 , q _ 1 } \\right ) _ { \\theta , q } = ( X _ 0 , X _ 1 ) _ { ( 1 - \\theta ) \\theta _ 0 + \\theta \\theta _ 1 , q } , \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{align*} S ( I _ 1 , I _ 2 ) & = \\left \\lbrace p \\in \\mathbb { P } : p \\nmid 2 N _ 1 N _ 2 , B _ 1 ( p ) \\in I _ 1 , B _ 2 ( p ) \\in I _ 2 \\right \\rbrace \\\\ S ( I _ 1 , I _ 2 ) ( x ) & = \\left \\lbrace p \\leq x : p \\in S ( I _ 1 , I _ 2 ) \\right \\rbrace . \\end{align*}"}
-{"id": "9166.png", "formula": "\\begin{align*} e _ { j _ k } N _ m ( Q ) e _ { i _ l } & = \\left \\langle \\begin{array} { c } \\\\ \\end{array} \\right \\rangle . \\end{align*}"}
-{"id": "8351.png", "formula": "\\begin{align*} K _ { j k } ( x - y , t - s ) = \\sum _ { i = 0 } ^ d \\sum _ { | \\mu | + l = i } D ^ { \\mu , l } K _ { j k } ( - y , - s ) \\frac { x ^ { \\mu } t ^ l } { \\mu ! l ! } + \\sum _ { | \\mu | + l = d + 1 } D ^ { \\mu , l } K _ { j k } ( \\xi x - y , \\eta t - s ) \\frac { x ^ { \\mu } t ^ l } { \\mu ! l ! } , \\end{align*}"}
-{"id": "1894.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { a - 1 } \\sum _ { i = 0 } ^ j \\sum _ { r = 0 } ^ { b - 1 - a } \\left [ \\binom { b - 1 - a } { r } - 1 \\right ] & = ( 2 ^ { b - 1 - a } - b + a ) \\sum _ { j = 0 } ^ { a - 1 } \\sum _ { i = 0 } ^ j 1 = \\binom { a + 1 } { 2 } ( 2 ^ { b - 1 - a } - b + a ) \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} \\lambda _ { R _ { L } } ( z ) - \\max _ { y \\in B _ { R _ L } ( z ) \\setminus \\{ z \\} } \\xi ( y ) > ( a _ L - 2 \\delta _ \\sigma ^ { - 1 } ) - ( a _ L - 4 \\delta _ \\sigma ^ { - 1 } ) = 2 \\delta _ \\sigma ^ { - 1 } > 0 , \\end{align*}"}
-{"id": "5310.png", "formula": "\\begin{align*} z _ 1 \\ge 1 + \\delta - z _ 2 > 1 + \\delta - 1 + \\tfrac { i } k - \\tfrac 1 k > \\tfrac { i } k = \\xi _ { i , 1 } . \\end{align*}"}
-{"id": "2914.png", "formula": "\\begin{align*} t E & = t \\| \\partial _ x h \\| _ 2 ^ 2 = \\int _ { \\mathbb { R } } | t ^ { 1 / 3 } k | ^ { 3 } \\exp ( - 2 | t ^ { 1 / 3 } k | ^ 3 ) \\frac { 1 } { | k | } | \\hat h _ 0 | ^ 2 d k , \\\\ t ^ 2 D & = t ^ 2 \\| | \\partial _ x | ^ { 5 / 2 } h \\| _ 2 ^ 2 = \\int _ { \\mathbb { R } } | t ^ { 1 / 3 } k | ^ { 6 } \\exp ( - 2 | t ^ { 1 / 3 } k | ^ 3 ) \\frac { 1 } { | k | } | \\hat h _ 0 | ^ 2 d k . \\end{align*}"}
-{"id": "4644.png", "formula": "\\begin{align*} - \\frac 1 2 \\Delta _ B | \\phi | ^ 2 = & \\sum _ a \\{ | \\nabla _ { \\bar V _ a } \\phi | ^ 2 + | \\nabla _ { V _ a } \\phi | ^ 2 \\} + \\langle \\phi , \\sum _ a R ^ Q ( \\bar V _ a , V _ a ) \\phi \\rangle \\\\ & + \\frac 1 2 \\sum _ a \\{ \\langle \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi , \\phi \\rangle + \\langle \\phi , \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi \\rangle \\} . \\end{align*}"}
-{"id": "1528.png", "formula": "\\begin{align*} f _ { n } ^ { \\left ( \\alpha \\right ) } \\left ( \\textbf { A } + x \\right ) & = f _ { n } ^ { \\left ( \\alpha + 1 \\right ) } \\left ( x \\right ) , \\\\ \\left ( \\alpha + 1 \\right ) \\left ( \\textbf { A } + x \\right ) f _ { n } ^ { \\left ( \\alpha \\right ) } \\left ( \\textbf { A } + x \\right ) & = f _ { n + 1 } ^ { \\left ( \\alpha + 1 \\right ) } \\left ( x \\right ) + \\alpha x f _ { n } ^ { \\left ( \\alpha + 1 \\right ) } \\left ( x \\right ) . \\end{align*}"}
-{"id": "6571.png", "formula": "\\begin{align*} \\frac { d b ( \\phi _ t ( v ) ) } { d t } = \\frac { r b \\sqrt { 1 - b ^ 2 } } { \\delta } - O ( b \\sqrt { 1 - b ^ 2 } ) . \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{align*} \\bar { v } _ * = \\bar { u } ^ \\smallfrown \\bar { v } , ~ \\bar { w } _ * = \\bar { u } ^ \\smallfrown \\bar { w } \\end{align*}"}
-{"id": "1079.png", "formula": "\\begin{align*} s \\cdot ( \\alpha \\cdot n ) = ( s \\alpha ) \\cdot n \\ , . \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} \\Vert u ( t ) \\Vert = \\Vert u _ 0 \\Vert ( 0 \\leq t \\leq T ) . \\end{align*}"}
-{"id": "1002.png", "formula": "\\begin{align*} \\Vert U _ { n } y ^ { n } - y ^ { n } \\Vert = \\left \\{ \\begin{array} { l l } \\Vert U _ { n _ { k } } x ^ { k } - x ^ { k } \\Vert & n = n _ { k } \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "1351.png", "formula": "\\begin{align*} c _ { \\ell , n + 1 } = \\sum _ { i = 0 } ^ \\ell c _ { i n } , \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} \\begin{array} { l } A _ k X _ { \\alpha _ k } ^ { s _ k } B _ k - C _ k X _ { \\beta _ k } ^ { t _ k } D _ k = E _ k , k = 1 , \\ldots , r , \\end{array} \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{align*} \\left [ \\begin{pmatrix} \\psi & 0 \\\\ 0 & 0 \\end{pmatrix} , \\begin{pmatrix} 0 & \\phi \\\\ \\phi ^ t & 0 \\end{pmatrix} \\right ] = i \\begin{pmatrix} 0 & \\phi \\\\ \\phi ^ t & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "4261.png", "formula": "\\begin{align*} h _ l = \\frac { ( b + b ^ { - 1 } ) ^ 2 } { 4 } - P _ l ^ 2 , h _ m = \\frac { ( b + b ^ { - 1 } ) ^ 2 } { 4 } - P _ m ^ 2 , h _ l = \\frac { ( b + b ^ { - 1 } ) ^ 2 } { 4 } - P _ l ^ 2 , \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} \\log \\left [ e ^ { - \\bar \\rho } ( r + \\bar \\rho + 1 ) \\right ] & = \\frac { 1 } { 2 } ( r - \\rho ) + \\log \\left ( \\frac { 1 } { 2 } ( \\rho + r ) + 1 \\right ) \\leq \\frac { 1 } { 2 } ( r - \\rho ) + \\log \\left ( r + 1 \\right ) + \\frac { \\rho - r } { 2 r + 2 } \\\\ & = \\frac { r } { 2 } - \\frac { r ( \\rho + 1 ) } { 2 ( r + 1 ) } + \\log ( r + 1 ) = \\log \\gamma _ 1 \\epsilon , \\end{align*}"}
-{"id": "933.png", "formula": "\\begin{align*} \\lim _ { | x | \\rightarrow \\infty } \\bigg \\{ \\frac { - \\log [ V ( x ) ] } { \\log | x | } \\bigg \\} \\ = \\ \\beta \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{align*} ( \\mathbb { D } ^ \\ast ( B ) ) = \\frac { \\ell ( \\delta _ 0 ) B ^ { r + 1 } } { ( r + 1 ) \\delta _ 0 ^ r } + O ( B ^ { r + 2 } ) \\end{align*}"}
-{"id": "2576.png", "formula": "\\begin{align*} ( \\| Z _ n \\| _ 2 ^ 2 - d ( n ) ) / \\sqrt { 2 d ( n ) } = ( \\| G _ n \\| _ 2 ^ 2 - d ( n ) ) / \\sqrt { 2 d ( n ) } + ( \\sqrt { n } 2 G _ n ' \\theta _ n + n \\| \\theta _ n \\| _ 2 ^ 2 ) / \\sqrt { 2 d ( n ) } . \\end{align*}"}
-{"id": "4115.png", "formula": "\\begin{align*} \\Sigma = \\{ z \\in \\C ^ n : P ( z ) = 0 \\ \\ \\ \\ \\mathrm { r a n k } \\ , d P _ z < q \\} \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} m _ \\delta = O \\left ( n ^ { 4 / 3 } \\right ) . \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ^ { x } _ { j } ( T ) & \\leq C \\tilde { C } _ { T } \\Big [ \\big ( \\delta ^ { \\nabla y } _ { j - 1 } ( T ) k ^ { q } _ { j } ( T ) + k ^ { q } _ { j - 1 } ( T ) \\delta ^ { \\nabla y } _ { j - 1 } ( T ) \\big ) ( 2 k ^ { \\infty } _ { j } ( T ) + \\lvert \\overline { b } \\rvert ) \\\\ & + k ^ { q } _ { j - 1 } ( T ) ^ 2 \\big ( \\delta ^ { x } _ { j - 1 } ( T ) + \\delta ^ { y } _ { j - 1 } ( T ) \\big ) \\Big ] . \\end{aligned} \\end{align*}"}
-{"id": "7584.png", "formula": "\\begin{align*} h ( z ) = \\left ( \\frac { \\beta } { 2 \\pi } \\right ) ^ { n / 2 } e ^ { - \\beta \\| z \\| ^ 2 / 2 } . \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{align*} \\mathfrak { D } = \\left ( \\begin{array} { c c c } \\bar { \\partial } _ { \\theta } ^ H & \\vartheta ^ * _ { C _ { \\xi } , r } \\\\ \\vartheta _ { C _ { \\xi } , r } & \\partial _ { \\theta } ^ H \\ \\end{array} \\right ) + \\mathfrak { r } , \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} ( B _ k ^ { \\top } \\otimes A _ k ) \\operatorname { v e c } ( X _ { \\alpha _ k } ^ { s _ k } ) = \\operatorname { v e c } ( E _ k ) + ( D _ k ^ \\top \\otimes C _ k ) \\operatorname { v e c } ( X _ { \\beta _ k } ^ { t _ k } ) , \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} z = z ^ { ( 0 ) } ( \\xi ) + \\tilde { z } , ~ \\omega _ 2 = i \\log ( | \\lambda | ) - \\arg ( \\lambda ) + \\frac { i \\tilde { \\omega } } { \\pi } \\log ( \\lambda ) + u \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{align*} F _ - ^ n ( x ) - F _ - ^ n ( \\xi _ j ) = a ^ n ( x - \\xi _ j ) . \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} f ( { \\mu } ) = ( 1 + \\square _ 0 ) ^ { - 1 } | \\mu | ^ 2 \\end{align*}"}
-{"id": "7810.png", "formula": "\\begin{align*} \\frac { \\partial F ( Q ) } { \\partial Q } = a Q - b ( Q ^ 2 - \\frac { 1 } { 3 } I d | Q | ^ 2 ) + c Q | Q | ^ 2 \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} \\frac { 1 } { i } \\partial _ { t } \\psi = \\partial _ { s s } \\psi + \\frac { 1 } { 2 } ( | \\psi | ^ { 2 } + A ) \\psi . \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} J = J _ { \\mathrm { f s u } } \\oplus J _ { \\mathrm { n i l } } . \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} \\omega \\wedge \\Psi = 0 \\ : , \\ ; \\ ; \\ ; \\ ; \\tfrac { i } { \\vert \\vert \\Psi \\vert \\vert ^ 2 } \\Psi \\wedge \\bar \\Psi = \\tfrac 1 6 \\omega \\wedge \\omega \\wedge \\omega \\ : . \\end{align*}"}
-{"id": "8859.png", "formula": "\\begin{align*} \\mathfrak { l } \\cap \\mathfrak { h } _ { \\mu } = \\mathbb { C } \\mu \\oplus \\mathfrak { t } ^ { \\sigma } \\oplus \\bigoplus _ { \\alpha \\in \\Phi _ L ^ { \\sigma } } \\mathbb { C } e _ { \\alpha } \\oplus \\bigoplus _ { \\alpha \\in \\Phi _ s ^ + ; \\bar { \\alpha } ( \\mu ) = 0 } \\mathbb { C } ( e _ { - \\alpha } + \\sigma ( e _ { - \\alpha } ) ) \\oplus \\bigoplus _ { \\alpha \\in \\Phi _ s ^ + ; \\bar { \\alpha } ( \\mu ) \\neq 0 } \\mathbb { C } e _ { - \\alpha } . \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} ( T ( D ) - \\lambda + V _ n ) u _ n = 0 \\iff ( T ( h D ) - \\lambda + W _ h ) u = 0 . \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} \\mathcal { H } ( \\alpha ) \\geq 1 . 2 8 1 7 7 7 0 2 1 4 = : \\eta , \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} \\mathfrak { l } = \\bar { \\mathfrak { l } } _ 0 \\oplus \\bigoplus _ { \\bar { \\alpha } \\in \\bar { \\Phi } } \\bar { \\mathfrak { l } } _ { \\bar { \\alpha } / 2 } . \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} u _ \\tau ( t ) = \\int _ \\tau ^ t \\gamma ( t , s ) d s , \\end{align*}"}
-{"id": "3704.png", "formula": "\\begin{align*} \\frac { M ( P , \\vec { \\nu } ) } { P ^ { n - r d } } = \\sum _ { ( \\vec { a } , q ) \\in \\mathcal { M } ( \\theta _ 0 ) } & \\frac { S _ { \\vec { a } , q } ( \\vec { \\nu } ) } { q ^ n } J ( P ^ { - d } \\vec { \\nu } ) + \\mathfrak { O } _ 1 + \\mathfrak { O } _ 2 . \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} \\| v ( t ) \\| _ 2 & \\le \\int _ 0 ^ t \\| v ' ( s ) \\| _ 2 d s \\\\ & \\le \\Big ( \\int _ 0 ^ t s ^ { - 1 / 2 } d s \\Big ) ^ { 1 / 2 } \\Big ( \\int _ 0 ^ t s ^ { 1 / 2 } \\| v ' ( s ) \\| _ 2 ^ 2 d s \\Big ) ^ { 1 / 2 } \\\\ & = t ^ { 1 / 4 } \\sqrt { 2 } \\Big ( \\int _ 0 ^ t s ^ { 1 / 2 } \\| v ' ( s ) \\| _ 2 ^ 2 d s \\Big ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} \\begin{aligned} g _ 1 & = 1 \\\\ g _ 2 & = \\frac { x ^ 3 + x ^ 2 + x + 1 } { y ^ 3 } \\\\ g _ 3 & = g _ 2 / y \\\\ g _ 4 & = \\frac { 4 ( x ^ 3 + x ^ 2 + x + 1 ) - x ^ 2 y ^ 4 - 2 x y ^ 4 - 3 y ^ 4 } { 4 y ^ 6 } \\\\ g _ 5 & = g _ 5 / y \\\\ g _ 6 & = g _ 6 / y \\\\ g _ 7 & = \\frac { 1 6 ( x ^ 3 + x ^ 2 + x + 1 ) - 6 x ^ 3 y ^ 4 - 1 0 x ^ 2 y ^ 4 + x y ^ 8 - 1 4 x y ^ 4 + 3 y ^ 8 - 1 8 y ^ 4 } { 6 y ^ 9 } \\\\ g _ 8 & = \\frac { 3 2 ( x ^ 3 + x ^ 2 + x + 1 ) - 3 x ^ 2 y ^ 8 - 8 x ^ 2 y ^ 4 - 4 x y ^ 8 - 1 6 x y ^ 4 - 3 y ^ 8 - 2 4 y ^ 4 } { 3 2 y ^ { 1 0 } } \\end{aligned} \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} L _ i \\left [ \\psi ( \\cdot , \\mu ) \\right ] ( \\nu ) = t ^ { \\theta _ i ( \\nu ) } \\Big ( \\psi ( s _ i \\nu , \\mu ) - \\psi ( \\nu , \\mu ) \\Big ) . \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} \\nabla _ t e _ 0 = 0 , \\nabla _ i e _ 0 = \\frac { 1 } { 2 } s g ^ { j k } \\dot g _ { i k } \\partial _ { x ^ j } . \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} g _ { j _ 2 } h ( J \\setminus ( D \\cup J _ 1 ) ) & = g ( I _ 1 ) g ( I _ 2 ) h ( J \\setminus ( D \\cup J _ 1 ) ) \\\\ & = g ( J _ 1 \\setminus I _ 1 ) g ( J _ 2 \\setminus I _ 2 ) h ( J \\setminus ( D \\cup J _ 2 ) ) = g _ { j _ 1 } h ( J \\setminus ( D \\cup J _ 2 ) ) . \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} f ( \\hat { e } _ i ) = \\operatorname { s g n } ( w _ i - \\theta ) = 0 . \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} h ^ { \\alpha \\beta } \\langle \\psi , i R ^ { P } ( \\eta , d \\phi ( \\partial _ \\alpha ) ) \\partial _ \\beta \\cdot \\psi \\rangle = h ^ { \\alpha \\beta } \\langle \\eta , R ^ { P } ( \\psi , i \\partial _ \\alpha \\cdot \\psi ) d \\phi ( \\partial _ \\beta ) \\rangle , \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{align*} { { \\bf { F } } _ 2 } { \\left ( { 2 { \\bf { D } } _ A ^ { - 1 } - { \\bf { D } } _ A ^ { - 1 } { \\bf { G D } } _ A ^ { - 1 } } \\right ) ^ H } { \\rm { = } } \\frac { 2 } { \\omega } { \\bf { A } } - \\frac { 1 } { { { \\omega ^ 2 } } } { \\bf { B } } , \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{align*} \\max \\{ \\ , x _ 1 , \\ , x _ 1 + 4 , \\ , x _ 1 \\ , \\} & = - x _ 1 + x _ 2 + 2 \\\\ \\max \\{ \\ , x _ 1 , \\ , - x _ 1 + x _ 2 + 2 \\ , \\} & = x _ 1 \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} G \\left ( x , t , u , w \\right ) = \\int _ { t } ^ { T } L \\left ( u \\left ( t \\right ) , w \\left ( t \\right ) \\right ) d t + J \\left ( x \\left ( T \\right ) \\right ) , \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} P ^ \\eta \\Big ( \\sum _ { k = 1 } ^ { t _ n - 1 } Z _ k \\geq c _ { n + 1 } t _ { n + 1 } + r \\Big ) & \\leq P ^ \\eta \\Big ( \\sum _ { k = 1 } ^ { t _ n - 1 } \\big ( Z _ k - E ^ \\eta [ Z _ k ] \\big ) \\geq ( t _ { n } - 1 ) \\big ( c _ { n + 1 } t _ n - E ^ \\eta [ Z _ k ] \\big ) \\Big ) \\\\ & \\leq P ^ \\eta \\Big ( \\sum _ { k = 1 } ^ { t _ n - 1 } \\big ( Z _ k - E ^ \\eta [ Z _ k ] \\big ) \\geq t _ { n } ^ { 3 / 4 } ( t _ n - 1 ) \\Big ) \\leq { \\rm e } ^ { - c \\ , t _ n ^ { 1 / 2 } } \\ , , \\end{align*}"}
-{"id": "4097.png", "formula": "\\begin{align*} \\{ ( k _ { 1 } , \\dots , k _ { n } ) \\in ( \\ker f ) ^ { n } | \\overline { \\left \\langle g _ { 1 } k _ { 1 } , \\dots , g _ { n } k _ { n } \\right \\rangle } = G \\} \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} \\dot { \\bar { x } } ^ * ( t ) & = \\bar { A } ^ * \\bar { x } ^ * ( t ) + \\bar { B } ^ * u ( t ) \\\\ [ 1 e x ] \\tilde { y } ( t ) & = \\bar { x } ^ * ( t ) ^ \\top \\bar { M } ^ * \\bar { x } ^ * ( t ) . \\\\ \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} \\begin{cases} u _ t = \\Delta u + a _ 0 u , x \\in D _ L \\cr u = 0 , x \\in \\partial D _ L , \\end{cases} \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{align*} \\mathcal { L } ( \\gamma _ j ) = ( j + 1 ) ( 2 \\lambda + n - 2 ) \\ , \\gamma _ { j + 1 } + ( j + 1 ) ( j + 2 ) \\ , \\gamma _ { j + 2 } \\end{align*}"}
-{"id": "1993.png", "formula": "\\begin{align*} ( d _ r ) ^ \\star _ { \\omega _ h } = h ^ { 2 r } \\ , ( d _ r ) ^ \\star _ \\omega . \\end{align*}"}
-{"id": "422.png", "formula": "\\begin{align*} f ( x _ 1 , \\ldots , x _ { i - 1 } , 1 , x _ { i + 1 } , \\ldots , x _ n ) = f ( x _ 1 , \\ldots , x _ { i - 1 } , 0 , x _ { i + 1 } , \\ldots , x _ n ) \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{align*} t _ { a v } & = \\sum _ { k = 1 } ^ n \\dfrac { 1 } { k } = O ( \\log n ) , \\\\ \\sup _ { h \\in \\mathcal { H } } | | f _ h | | _ { \\infty } & = O ( \\log n ) . \\end{align*}"}
-{"id": "5386.png", "formula": "\\begin{align*} \\Delta t = \\sum t _ 1 \\otimes t _ 2 \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} f ^ * Q _ l & = \\sum _ j w _ { l j } P _ j \\\\ \\bar { d _ j } & = \\frac { d _ j + w _ { l j } - 1 } { w _ { l j } } , \\ ; \\ ; f ( P _ j ) = Q _ l \\\\ \\delta _ l & = m a x \\{ \\bar { d _ j } ; f ( P _ j ) = Q _ l \\} . \\\\ \\Delta & = \\sum _ l \\delta _ l Q _ l . \\\\ M & = L - \\Delta . \\\\ \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} \\frac { 1 } { d _ i ( S ) } = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { n + 1 } \\left | l _ { i j } \\right | . \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{align*} \\sigma = \\tau + O ( r ^ { - 2 - \\mu } ) . \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} \\tau _ { s _ i ( \\nu ) + \\alpha , \\mu } ^ { 2 } = g ( \\langle \\alpha , \\nu - \\rho \\rangle - 1 ) \\dfrac { 1 - \\mathbf { z } ^ { - n \\alpha } } { 1 - q ^ { - 1 } \\mathbf { z } ^ { - n \\alpha } } = g ( c _ i - c _ j ) \\dfrac { 1 - \\mathbf { z } ^ { - n \\alpha } } { 1 - v \\mathbf { z } ^ { - n \\alpha } } \\ , , \\end{align*}"}
-{"id": "5676.png", "formula": "\\begin{align*} | T ^ n | ^ 2 = ( G ^ n ) ^ * ( R _ n ^ * R _ n ) G ^ n & \\leq \\| R _ n \\| ^ 2 ( G ^ n ) ^ * G ^ n \\leq ( 1 + 4 \\varepsilon ) ^ { 2 n } ( G ^ n ) ^ * G ^ n , \\\\ | T ^ n | ^ 2 = ( G ^ n ) ^ * ( R _ n ^ * R _ n ) G ^ n & \\geq \\| R _ n ^ { - 1 } \\| ^ { - 2 } ( G ^ n ) ^ * G ^ n \\geq ( 1 + 4 \\varepsilon ) ^ { - 2 n } ( G ^ n ) ^ * G ^ n . \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} \\gamma _ 1 ( x , y ) = \\begin{cases} \\frac { 1 } { k } & ( x , y ) = ( x _ i , y _ i ) 1 \\leq i \\leq k , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} | z ^ I | ^ 2 = ( | z _ 0 | ^ 2 ) ^ { i _ 0 } \\cdots ( | z _ n | ^ 2 ) ^ { i _ n } \\le \\left ( \\sum _ { i = 0 } ^ n | z _ i | ^ { 2 } \\right ) ^ d \\end{align*}"}
-{"id": "2245.png", "formula": "\\begin{align*} \\begin{aligned} \\max \\{ D _ \\varpi ( P _ 0 \\parallel Q _ 0 ) , D _ \\varpi ( P _ 1 \\parallel Q _ 1 ) \\} \\ge D _ \\varpi ( P _ \\lambda \\parallel Q _ \\lambda ) , \\end{aligned} \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} \\epsilon _ 1 = \\prod _ { x , y } p _ { X Y } ( x , y ) ^ { - \\mu ' _ 1 ( x , y ) + \\gamma ' _ 1 ( x , y ) } . \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} | \\sigma \\cap ( S _ \\mu ( m ) \\times T _ \\mu ( m ) ) | = | \\lambda \\cap S _ \\mu ( m ) | + | \\lambda \\cap T _ \\mu ( m ) | - \\frac { | \\lambda | } { 2 } . \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} ( a ; q ) _ \\infty = \\lim _ { n \\to \\infty } ( a ; q ) _ n = \\prod _ { k = 0 } ^ \\infty ( 1 - a q ^ k ) , \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} \\ell = \\ell _ n : = n ^ { \\delta } , \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} 2 a _ i = 2 | T _ { x _ i = 1 } | = 2 b _ i . \\end{align*}"}
-{"id": "1451.png", "formula": "\\begin{align*} X _ j ( t ) = X _ j ( 0 ) + { \\bf 1 } _ { \\{ j = 1 \\} } t + B _ j ( t ) + L _ { j - 1 } ( t ) - L _ { j } ( t ) j = 1 , \\ldots , m \\ , , \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} A _ 1 \\leq \\frac { 1 } { | B _ M | } \\sum _ { y \\in \\hat S _ 1 \\cup \\hat S _ 2 } { \\big | p ( t , \\hat x _ n - y ) - G ( t , \\hat x _ n - y ) \\big | } \\ , , \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { ( 2 \\pi ) ^ { n / 2 } } \\sqrt { y _ 1 \\ldots y _ n } \\int _ { ( 0 , \\infty ) ^ n } \\left ( \\frac { \\det ( D _ { u + y } ^ { - 1 } + L _ W ) } { \\prod _ { i = 1 } ^ n ( u _ i ( u _ i + y _ i ) ) ) } \\right ) ^ { 1 / 2 } d u _ 1 \\ldots d u _ n \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} | \\nabla ^ j p ^ { \\kappa } _ { t , s } ( x ) - \\nabla ^ j p ^ { \\tilde \\kappa } _ { t , s } ( x ) | \\leq c K \\varrho ^ 0 _ { \\alpha - j } ( s - t , x ) , \\ \\ j = 0 , 1 , \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{align*} Z _ k ( t ) : = X _ { k + 1 } ( t ) - X _ k ( t ) : = Y _ { ( k + 1 ) } ( t ) - Y _ { ( k ) } ( t ) \\ , , k \\ge 1 \\ , . \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k x _ { [ i ] } & \\leq \\sum _ { i = 1 } ^ k y _ { [ i ] } k = 1 , \\dots , n - 1 \\\\ \\sum _ { i = 1 } ^ n x _ { [ i ] } & = \\sum _ { i = 1 } ^ n y _ { [ i ] } . \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} \\begin{array} { l } \\dot { x } = \\displaystyle { \\frac { 2 } { 3 } ( 1 - 6 x ) z } \\\\ \\\\ \\dot { z } = \\displaystyle { \\frac { 5 } { 6 } + x - 4 z ^ 2 - h ( 1 - 6 x ) } , \\end{array} \\end{align*}"}
-{"id": "3031.png", "formula": "\\begin{align*} \\lim _ { \\Vert h \\Vert _ { 2 , t } \\rightarrow 0 } \\frac { \\Vert \\mathcal { N } ( q , u + h ) - \\mathcal { N } ( q , u ) - q a ( x ) u ^ { q - 1 } h \\Vert _ { t } } { \\Vert h \\Vert _ { 2 , t } } = 0 . \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} Q '' ( i , j , \\alpha ) : = \\underline { d p } _ j \\underline \\wedge \\omega _ { i j } ^ \\alpha ( i < j \\in [ n ] , \\alpha \\geq 0 ) , \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} K ( x , y ; t ) Q ( x , y ; t ) = F ^ { 1 } ( x ; t ) + F ^ { 2 } ( y ; t ) - K ( 0 , 0 ; t ) Q ( 0 , 0 ; t ) + x y . \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 , \\ldots , x _ k ) = \\sum _ { n _ 1 , n _ 2 , \\ldots , n _ k = 0 } ^ \\infty \\lambda _ { n _ 1 , n _ 2 , \\ldots , n _ k } x _ 1 ^ { n _ 1 } x _ 2 ^ { n _ 2 } \\cdots x _ k ^ { n _ k } . \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} T : = \\inf \\{ y \\ge 0 : \\ M ( y ) \\not \\in S \\} \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{align*} ( \\overline { P } f ) ( g x ) = f ( x ) = f ( g ^ { - 1 } g x ) = ( g f ) ( g x ) , ~ g \\in U , x \\in X . \\end{align*}"}
-{"id": "9248.png", "formula": "\\begin{align*} \\sigma = 6 , \\frac { \\alpha _ 0 } { r } = - \\frac { \\beta _ 0 } { r } = \\frac { \\pi } { 4 } . \\end{align*}"}
-{"id": "7609.png", "formula": "\\begin{align*} I _ 0 ( g ) = \\N \\cap ( C ^ { - 1 } q ^ { ( g - 1 ) n m ^ 2 } , N _ { q , n , m } ( g ) ] \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} \\| b \\| _ { B M O _ { \\eta } } = \\sup _ { Q } \\frac { 1 } { \\eta ( Q ) } \\int _ { Q } | b ( x ) - b _ { Q } | d x < \\infty , \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\left ( - { \\bf D } ^ { - 1 } { \\bf E } \\right ) ^ n = { \\bf 0 } \\lambda _ { m a x } ( - { \\bf D } ^ { - 1 } { \\bf E } ) < 1 , \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} S ( i , j , \\alpha , \\beta ) : = \\omega _ { i j } ^ \\alpha \\underline \\wedge \\omega _ { i j } ^ \\beta ( i < j \\in [ n ] , 0 \\leq \\alpha < \\beta ) , \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{align*} \\zeta _ { \\Omega } ( z + \\omega ) = \\zeta _ { \\Omega } ( z ) + \\eta ( \\omega ) , ~ ~ \\omega \\in \\Omega \\end{align*}"}
-{"id": "8093.png", "formula": "\\begin{align*} H ( r ) = \\int _ { - 1 } ^ 0 h ( r ^ 2 t ) d t , \\ \\ \\ \\ \\ I ( r ) = \\int _ { - 1 } ^ 0 i ( r ^ 2 t ) d t . \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{align*} \\begin{cases} & v _ t + v \\cdot \\nabla v = - \\textbf { k } - \\nabla P , \\Omega ( t ) , t \\ge 0 , \\\\ & \\operatorname { d i v } v = 0 , \\ , \\operatorname { c u r l } v = 0 , \\Omega ( t ) , t \\ge 0 , \\\\ & ( 1 , v ) ( t , \\Sigma ( t ) ) . \\end{cases} \\end{align*}"}
-{"id": "3102.png", "formula": "\\begin{align*} [ g ] = \\begin{pmatrix} m _ { 1 , 1 } & m _ { 1 , 2 } & 0 & \\cdots & 0 \\\\ m _ { 2 , 1 } & m _ { 2 , 2 } & 0 & \\cdots & 0 \\\\ m _ { 3 , 1 } & m _ { 3 , 2 } & m _ { 3 , 3 } & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ m _ { n , 1 } & m _ { n , 2 } & m _ { n , 3 } & \\cdots & m _ { n , n } \\end{pmatrix} \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} g _ \\infty = & - \\ : \\frac 1 4 K ^ { - 2 } e ^ { - 2 C _ U } d u ^ 2 + \\frac 4 5 K ^ { - \\ : 2 } e ^ { - \\ : 2 C _ U } C _ \\infty u ^ { \\frac 3 2 } d \\widehat { \\theta } ^ 2 + e ^ { 2 C _ U } d \\widehat { x } ^ 2 + \\\\ & e ^ { - \\ : 2 C _ U } u \\left ( d \\widehat { y } + \\frac { 4 } { K \\sqrt { 5 } } C _ \\infty ^ { \\frac 1 2 } u ^ { \\frac 1 4 } d \\widehat { \\theta } \\right ) ^ 2 . \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} E _ d ( P ) _ { q = 1 } = \\dim V . \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { t _ { 0 } } ^ { t _ { 0 } + h } e ^ { - s \\omega } s ^ { - 3 ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\ d s & \\leq \\int _ { t _ { 0 } } ^ { t _ { 0 } + h } s ^ { - 3 ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\ d s = \\frac { 1 } { 1 - 3 ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\cdot \\Big [ ( t _ { 0 } + h ) ^ { 1 - 3 ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } - t _ { 0 } ^ { 1 - 3 ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\Big ] \\end{aligned} \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} O _ { \\tilde { K } } & = O _ F ( ( S ) ) ( ( T ) ) \\subset \\tilde { K } \\\\ O _ R & = O _ F \\{ \\{ S \\} \\} \\{ \\{ T \\} \\} \\subset R . \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} \\forall z \\in S ^ * M , \\Pi f ( z ) = \\int _ { - \\infty } ^ \\infty f ( \\varphi _ t ( z ) ) d t \\end{align*}"}
-{"id": "6831.png", "formula": "\\begin{align*} M = S ( D _ + - D _ - ) S ^ \\top = S D _ + S ^ \\top - S D _ - S ^ \\top \\eqqcolon M _ + - M _ - \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} s _ j \\star w : = \\left \\lbrace \\begin{array} { r r } s _ i w & \\textrm { i f } l ( s _ i w ) > l ( w ) \\\\ w & \\textrm { e l s e } \\end{array} \\right . \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{align*} \\varphi ( \\hat { \\vect { x } } ^ t _ k ) = \\frac { \\Lambda ( \\hat { \\vect { x } } ^ t _ k ) } { \\Lambda ( \\hat { \\vect { x } } ^ t _ k ) + \\Lambda ( \\hat { \\vect { x } } ^ t _ { k + 1 } ) } \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} f _ \\tau ( u ) = \\prod _ { \\nu \\in \\Phi _ \\tau } h _ { \\nu } ( u ) , & \\quad \\Phi _ \\tau = \\{ \\nu \\in \\Phi : \\nu | _ { E ^ { \\mathrm { u r } } } = \\tau \\} \\\\ v _ \\tau ( u ) = \\prod _ { \\nu \\in \\Phi _ \\tau ^ c } h _ { \\nu } ( u ) , & \\quad \\quad \\Phi _ \\tau ^ c = \\{ \\nu \\in \\Phi ^ c : \\nu | _ { E ^ { \\mathrm { u r } } } = \\tau \\} \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} ( t - 1 ) \\binom { s - 1 + t - 1 } { t - 1 } \\ = \\ s \\binom { s + t - 2 } { t - 2 } , \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} \\chi _ A : X \\to \\{ 0 , 1 \\} , x \\mapsto \\begin{cases} 1 , & x \\in A , \\\\ 0 , & x \\notin A . \\end{cases} \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{gather*} \\left [ \\pi _ k ( E _ { \\sqrt { c } } - F _ { \\sqrt { c } } ) M ( \\cdot , x ) \\right ] ( n ) = - \\sqrt { c } ( 2 k + x ) M ( n , x + 1 ) - \\frac { x } { \\sqrt { c } } M ( n , x - 1 ) , \\\\ \\left [ \\pi _ k ( E _ { \\sqrt { c } } + F _ { \\sqrt { c } } ) M ( \\cdot , x ) \\right ] ( n ) = \\sqrt { c } ( 2 k + x ) M ( n , x + 1 ) - \\frac { x } { \\sqrt { c } } M ( n , x - 1 ) , \\end{gather*}"}
-{"id": "5909.png", "formula": "\\begin{align*} \\psi ( \\nu , \\mu ) = t ^ { \\Omega ( \\mu , \\nu ) } \\cdot I ( \\mu , \\nu ) \\end{align*}"}
-{"id": "643.png", "formula": "\\begin{align*} T = \\inf \\{ k \\geq 1 : Q _ i ( k ) = 0 1 \\leq i \\leq M \\} \\end{align*}"}
-{"id": "6350.png", "formula": "\\begin{align*} \\operatorname { a m p } _ { I } ( M ) : = \\max \\{ | n | : H ^ i ( L ^ I ( M ) ) _ { n - 1 } \\neq 0 \\mbox { f o r s o m e } 0 \\leqslant i \\leqslant \\xi _ I ( M ) - 1 \\} . \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} \\Pi _ { i = 1 } ^ n [ 0 , \\lambda _ i ] \\subseteq E \\subseteq C \\Pi _ { i = 1 } ^ n [ 0 , \\lambda _ i ] . \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{align*} \\int _ X \\langle d _ A \\gamma , \\dot A \\rangle \\ , d A = \\int _ X \\langle \\gamma , d _ A ^ * \\dot A \\rangle \\ , d A \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} z _ 1 - \\lfloor \\sqrt { n } \\rfloor & < \\sum _ { j = 2 } ^ N \\frac { n ^ { \\frac { 1 } { 2 ^ j } } } { 2 ^ { j - 1 } } + \\frac { 1 } { 2 ^ { N - 1 } } + 2 \\\\ & < \\frac { 1 } { 2 } n ^ \\frac { 1 } { 4 } + ( N - 2 ) \\cdot \\frac { 1 } { 4 } n ^ \\frac { 1 } { 8 } + 2 . 2 5 \\\\ & < \\frac { 1 } { 2 } n ^ \\frac { 1 } { 4 } + \\log _ 2 \\log _ 5 ( n ) \\cdot \\frac { 1 } { 4 } n ^ \\frac { 1 } { 8 } + 2 . 2 5 \\\\ & < n ^ \\frac { 1 } { 4 } . \\end{align*}"}
-{"id": "1518.png", "formula": "\\begin{align*} \\mathcal { L } | A | = - \\dfrac 1 2 | A | - | A | ^ 3 + \\dfrac { | \\nabla A | ^ 2 - | \\nabla | A | | ^ 2 } { | A | } . \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} \\begin{cases} \\operatorname { d i v } ( y ^ { - a } \\nabla h ) = y ^ { - a } h _ t , \\\\ h ( x , 0 , t ) = - \\phi . \\end{cases} \\end{align*}"}
-{"id": "6303.png", "formula": "\\begin{align*} f _ { D _ { \\mathbf { b } _ 0 , \\mathbf { u } _ 0 } } ( r ) = f _ { D _ k } ( r ) = \\frac { 2 \\pi \\lambda _ k r } { \\mathcal { A } _ k } \\exp \\Big ( - \\frac { \\pi r ^ 2 } { P _ k ^ { 2 / \\alpha } } \\Xi \\Big ) , \\end{align*}"}
-{"id": "3044.png", "formula": "\\begin{align*} A : = - \\Delta - a ( x ) \\quad \\mbox { a n d } D ( A ) : = W _ { D } ^ { 2 , \\eta } ( \\Omega ) . \\end{align*}"}
-{"id": "3625.png", "formula": "\\begin{align*} Z ^ { \\ast } ( I _ { 9 } ) = ( z _ { r } ^ { - n } - v z _ { r } ^ { n } ) \\ , Z ^ { \\ast } ( I _ { 1 4 } ; a , b ) . \\end{align*}"}
-{"id": "8327.png", "formula": "\\begin{align*} ( I + \\mathcal { K } ^ * ) a = \\operatorname { R e } \\left \\{ i e ^ { i \\theta } \\left ( \\bigl ( I - \\mathfrak { H } \\bigr ) \\bigl ( \\bar { z } _ { t t } - i + \\frac { \\partial _ \\alpha P } { | z _ \\alpha | } e ^ { - i \\theta } \\right ) \\right \\} . \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} z _ i ( \\varphi _ \\lambda ( q ) ) = \\int _ { \\alpha _ i } \\tau _ { \\lambda q } = \\lambda z _ i ( q ) , w _ i ( \\varphi _ \\lambda ( q ) ) = \\int _ { \\beta _ i } \\tau _ { \\lambda q } = \\lambda w _ i ( q ) . \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} x ^ 2 + D = p ^ { n } . m \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{align*} g _ n = g _ { n - 1 } + C _ { n - 1 } + ( n + 1 ) 2 ^ { n - 3 } - \\binom { n } { 3 } - n , \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{align*} \\lVert A B - A _ 0 B _ 0 \\rVert ^ 2 & \\sim \\sum _ { i = 1 } ^ { M - 1 } \\left [ x _ i ^ 2 + \\sum _ { j = 2 } ^ { N } \\{ a _ i ( b _ j - b _ 1 ) - a ^ 0 _ i ( b ^ 0 _ j - b ^ 0 _ 1 ) \\} ^ 2 \\right ] \\\\ & = \\sum _ { i = 1 } ^ { M - 1 } \\left \\{ x _ i ^ 2 + \\sum _ { j = 2 } ^ { N } ( a _ i b _ j - a ^ 0 _ i b ^ 0 _ j ) ^ 2 \\right \\} \\\\ & = \\sum _ { i = 1 } ^ { M - 1 } x _ i ^ 2 + \\sum _ { i = 1 } ^ { M - 1 } \\sum _ { j = 2 } ^ N ( a _ i b _ j - a ^ 0 _ i b ^ 0 _ j ) ^ 2 \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} \\bigl [ [ s , x _ 1 ] [ b , x _ 2 , ] , x _ 3 \\bigr ] = [ s , x _ 1 , x _ 3 ] [ b , x _ 2 ] + [ s , x _ 1 ] [ b , x _ 2 , x _ 3 ] . \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{align*} \\tilde { p } _ n = \\frac { p _ { 1 1 } ^ { n / 2 } p _ { 2 2 } ^ { n / 2 } } { p _ { 1 1 } ^ { n / 2 } p _ { 2 2 } ^ { n / 2 } + p _ { 1 2 } ^ { n / 2 } p _ { 2 1 } ^ { n / 2 } } \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{align*} L = L _ { j , 2 } + L _ { j , 1 } , \\ \\ W = W _ { j , 1 } + W _ { j , 0 } , \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} \\lambda _ A \\left ( ( L ^ { \\vee } ) _ i \\right ) = \\lambda _ A \\left ( \\operatorname { H o m } _ A ( L _ { - i } , E ) \\right ) = \\lambda _ A ( L _ { - i } ) \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} G _ \\gamma ( h ) - G _ \\gamma ( 0 ) = \\alpha ( h ) . \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} Z ( \\alpha , \\beta ) : = Z ( Z _ { \\alpha } , | \\alpha | , | \\beta | , Z _ { \\beta } ) . \\end{align*}"}
-{"id": "2372.png", "formula": "\\begin{align*} \\tilde { J } _ 2 ( N ; \\theta ) : = 2 \\int _ 0 ^ { \\infty } t e ^ { - \\theta t } \\left ( 1 - e ^ { - ( 1 - \\theta ) t / N } \\right ) ^ N d t . \\end{align*}"}
-{"id": "4408.png", "formula": "\\begin{align*} \\psi _ \\lambda ( r ) = \\Im \\left ( \\frac { \\sigma _ \\lambda ( ( \\frac 1 2 + r ) \\omega ) } { \\sigma _ \\lambda ( ( \\frac 1 2 - r ) \\omega ) } \\right ) \\end{align*}"}
-{"id": "5711.png", "formula": "\\begin{align*} t e ( B ) - ( s - 2 ) e ( B ) & = ( t - s + 2 ) e ( B ) \\\\ & = ( r + 1 - 2 s + 2 ) e ( B ) = ( r + 3 - 2 s ) e ( B ) \\\\ & \\ge ( r + 3 - ( r + 2 ) ) e ( B ) \\\\ & = e ( B ) > 0 , \\end{align*}"}
-{"id": "8026.png", "formula": "\\begin{align*} d \\overline { V } _ t \\le - 2 { a } \\lambda _ 2 \\tilde { \\gamma } \\overline { V } _ t d t - 2 \\kappa \\tilde { \\gamma } \\overline { V } _ t d t + \\frac { m \\tau ^ 2 \\tilde { \\gamma } ^ 2 ( N - 1 ) } { 2 N } d t + \\frac { \\tau } { N } \\tilde { \\gamma } \\sum _ { i = 1 } ^ N { d B _ { i , t } } ^ T e _ { i , t } . \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} R _ F = - \\frac 1 2 \\beta _ 0 ^ T . \\end{align*}"}
-{"id": "329.png", "formula": "\\begin{align*} w = - \\lambda c z _ a ^ { - \\lambda - 1 } - \\sum _ { \\mu < \\lambda } \\mu c _ \\mu z _ a ^ { - \\mu - 1 } . \\end{align*}"}
-{"id": "4015.png", "formula": "\\begin{align*} \\delta = \\| p _ { K _ A K _ B Z ^ n \\mathbf { F } } - q _ { K _ A K _ B } \\cdot p _ { Z ^ n , \\mathbf { F } } \\| _ { T V } . \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{align*} \\bigg \\| \\sum _ { k = k _ 0 } ^ \\infty 2 ^ { - k \\alpha } \\mathcal { C } _ k g \\bigg \\| _ { L ^ { q , \\infty } _ t L ^ 2 _ x } \\lesssim \\| g \\| _ { L ^ { s , 1 } _ t L ^ 2 _ x } \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} { \\bf W } _ { Z F } = \\beta _ { Z F } { \\bf H } ( { \\bf H } ^ H { \\bf H } ) ^ { - 1 } , \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} \\rho ( x , y ) : = \\inf \\{ \\delta > 0 : y \\in B _ X ( x , \\delta ) \\} . \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{align*} \\Pi ( B ) = \\prod _ { f _ { p q } \\in \\mathcal { B } _ { k , i } } f _ { p q } \\otimes t ^ { d ( p , q , k , i ) } . \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} \\tilde { P } ( x ) = \\begin{pmatrix} P & v ( x ) \\\\ v ( x ) ^ \\top & w ( x ) \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} C ^ 2 & = d ^ 2 - \\sum _ i m _ { p _ i } ^ 2 \\geq \\frac { 3 } { 2 } d + \\sum _ { p \\in { \\rm S i n g } ( \\bar { C } ) } \\bigg ( 1 + \\frac { 1 } { 2 m _ { p _ { i } } } \\bigg ) \\mu _ { p _ { i } } - \\sum _ i m _ { p _ i } ^ 2 \\\\ & \\geq \\frac { 3 } { 2 } d + \\bigg ( 1 + \\frac { 1 } { 2 m _ { p _ { 1 } } } \\bigg ) \\mu _ { p _ { 1 } } - \\sum _ i m _ { p _ i } ^ 2 . \\end{align*}"}
-{"id": "1813.png", "formula": "\\begin{align*} G _ + ( h ) = G _ 0 ( h ) + R , \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} \\begin{aligned} W _ t & = \\int _ { S ' } c ' _ t ( s ) { \\left ( \\frac { d \\mu ^ { ' } \\circ \\phi ^ { ' } _ t } { d \\mu ^ { ' } } ( s ) \\right ) } ^ { 1 / \\alpha } g \\circ \\phi ^ { ' } _ t ( s ) M ' ( d s ) , \\ ; \\ ; t \\in \\mathbb { Z } ^ d . \\end{aligned} \\end{align*}"}
-{"id": "8830.png", "formula": "\\begin{align*} C _ D = - e ^ { 2 \\alpha _ 1 ( a ) } \\theta ( z _ 1 e _ { \\alpha _ 1 } ) - e ^ { 2 \\alpha _ 2 ( a ) } \\theta ( z _ 2 e _ { \\alpha _ 2 } ) . \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} \\dim V _ { j j } & = 1 , , \\\\ \\dim V _ { j k } & = a , , \\\\ \\dim V _ { j 0 } & = b , . \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} & A _ { j } ^ { i _ 1 , . . . , i _ { k - 2 j } } ( \\beta , \\tilde \\gamma , B ) \\\\ = & 2 \\beta ^ { - 1 } \\sum _ { \\alpha = 1 } ^ { k - 2 ( j - 1 ) - 1 } \\sum _ { \\delta > \\alpha } A _ { j - 1 } ^ { l _ 1 . . . l _ { \\alpha - 1 } \\eta l _ { \\alpha } . . . . l _ { \\delta - 2 } \\xi l _ { \\delta - 1 } . . . l _ { k - 2 j } } ( \\beta , \\tilde \\gamma , B ) \\frac { 1 } { 2 } ( \\tilde \\gamma _ { \\eta \\xi } + \\tilde \\gamma _ { \\xi \\eta } ) G _ { i _ 1 . . . i _ { k - 2 j } } ^ { l _ 1 . . . l _ { k - 2 j } } ( - \\tilde \\gamma ) \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{align*} V = \\sum _ { \\mu \\in \\lbrace 0 , 1 \\rbrace ^ N } E _ \\mu ^ * V , & & , \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} \\Delta _ B = \\square _ B + \\overline \\square _ B + ( \\bar \\partial _ B \\partial _ B ^ * + \\partial _ B ^ * \\bar \\partial _ B ) + ( \\partial _ B \\bar \\partial _ B ^ * + \\bar \\partial _ B ^ * \\partial _ B ) . \\end{align*}"}
-{"id": "4737.png", "formula": "\\begin{align*} \\mathcal { S } = \\{ s _ 2 s _ 3 s _ 1 s _ 2 s _ 1 , s _ 3 s _ 2 s _ 3 s _ 1 , s _ 2 s _ 3 s _ 2 , s _ 2 s _ 3 s _ 1 \\} . \\end{align*}"}
-{"id": "5088.png", "formula": "\\begin{align*} \\prod _ { 1 \\le i _ 1 < i _ 2 \\le k } ( y _ { i _ 2 } - y _ { i _ 1 } ) & = \\biggl ( \\prod _ { \\substack { 1 \\le i _ 1 < i _ 2 \\le k \\\\ i _ 1 \\ne j , i _ 2 \\ne j } } ( y _ { i _ 2 } - y _ { i _ 1 } ) \\biggr ) \\biggl ( \\prod _ { i = j + 1 } ^ k ( y _ i - y _ j ) \\biggr ) \\biggl ( \\prod _ { i = 1 } ^ { j - 1 } ( y _ j - y _ i ) \\biggr ) \\\\ & = ( - 1 ) ^ { k - j } \\varphi ' ( y _ j ) \\prod _ { \\substack { 1 \\le i _ 1 < i _ 2 \\le k \\\\ i _ 1 \\ne j , i _ 2 \\ne j } } ( y _ { i _ 2 } - y _ { i _ 1 } ) . \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} & Q ( V ) = u \\nabla ^ a \\nabla _ { a } V _ { b } + \\frac { 1 } { u } \\left [ { U ^ m ( \\nabla ^ a \\nabla _ { a } V ^ n ) g _ { m n } } \\right ] U _ { b } + ( \\nabla ^ a \\nabla _ { a } V ^ m ) U ^ n \\bar { \\varphi } _ { b m n } + l . o . t . \\end{align*}"}
-{"id": "63.png", "formula": "\\begin{align*} \\begin{aligned} [ b ] \\max _ { \\mathcal { P } } \\ & \\Big [ ( 1 - \\tau ) \\left ( \\frac { \\tau - 2 } { \\tau - 1 } \\abs { S _ 1 } + \\frac { 1 } { \\tau - 1 } \\abs { S _ 2 } + \\tfrac 1 2 \\abs { S _ 3 } \\right ) + \\frac { \\tau - 3 } { \\tau - 1 } E _ { S _ 1 } \\\\ & + \\frac { \\tau - 3 } { 2 ( \\tau - 1 ) } E _ { S _ 1 , S _ 3 } - \\frac { E _ { S _ 1 , V _ 1 } } { \\tau - 1 } - \\frac { \\tau - 2 } { \\tau - 1 } E _ { S _ 2 , V _ 1 } - \\frac 1 2 E _ { S _ 3 , V _ 1 } \\Big ] , \\end{aligned} \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} & u _ { t t } - c ^ 2 \\ , [ \\Delta u - q u ] = 0 \\mbox { i n \\ , } Q ^ T , \\\\ & u | _ { t = 0 } = u _ t | _ { t = 0 } = 0 \\mbox { i n \\ , } \\Omega , \\\\ & u = f \\mbox { o n \\ , } \\Sigma ^ T , \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{gather*} x ( x ^ { \\lambda } \\cdot x \\phi ^ { - 1 } ) = x \\phi ^ { - 1 } \\Rightarrow ( y x ) ( x ^ { \\lambda } \\cdot x \\phi ^ { - 1 } ) = ( y \\cdot x \\phi ^ { - 1 } ) \\\\ \\Rightarrow ( x \\cdot y x ) ( x ^ { \\lambda } \\cdot x \\phi ^ { - 1 } ) = x ( y \\cdot x \\phi ^ { - 1 } ) \\Rightarrow R _ { x ^ \\lambda } \\cdot x \\phi ^ { - 1 } = L ^ { - 1 } _ { x } R ^ { - 1 } _ { x } R _ { x \\phi ^ { - 1 } } L _ { x } . \\end{gather*}"}
-{"id": "1884.png", "formula": "\\begin{align*} e _ n = e _ { n - 1 } + d _ { n - 1 } + C _ { n - 2 } - 2 ^ { n - 3 } + \\sum _ { i = 3 } ^ { n - 2 } ( d _ { i + 1 } - b _ i ) , n \\geq 4 , \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{align*} | R _ G | = | G | - | B _ G | - | A _ p \\cup X ' | - | X '' | \\ge 4 \\cdot 2 ^ k + 1 - 8 - 4 \\cdot 2 ^ { k - 1 } - ( k - 1 ) > 5 , \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} - \\frac { p ^ { \\prime \\prime } \\left ( 1 \\right ) } { \\left [ p \\left ( 1 \\right ) \\right ] ^ { q } } = - \\frac { f ^ { \\prime \\prime } \\left ( 1 \\right ) } { \\left [ f \\left ( 1 \\right ) \\right ] ^ { q } } = - \\left ( r - 1 \\right ) r ^ { q } . \\end{align*}"}
-{"id": "1701.png", "formula": "\\begin{align*} M _ i \\leq \\binom { i - 1 } { d _ i } \\leq \\binom { n } { n ^ { 1 / \\ell } / \\log n } \\leq n ^ { n ^ { 1 / \\ell } / \\log n } = 2 ^ { n ^ { 1 / \\ell } } . \\end{align*}"}
-{"id": "2273.png", "formula": "\\begin{align*} \\tilde { E } u = \\begin{cases} u & B \\\\ v & \\R ^ n \\setminus \\bar B \\end{cases} \\end{align*}"}
-{"id": "6344.png", "formula": "\\begin{align*} \\xi _ M ^ I ( n ) : = \\operatorname { g r a d e } ( G _ { I ^ n } ( A ) _ + , G _ { I ^ n } ( M ) ) . \\end{align*}"}
-{"id": "3403.png", "formula": "\\begin{align*} \\log M ( \\sqrt { C } , f ) \\geq 2 \\log \\frac { C - C ^ 2 } { 2 C ^ { 3 / 2 } } = 2 \\log \\frac { 1 - C } { 2 \\sqrt { C } } > 0 . \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} ( 1 - \\epsilon ) I ( U ; X Y | V = v ) \\geq I ( U ; J | V = v ) , \\forall v . \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} H ^ s u ( x , t ) = ( \\partial _ t - \\Delta ) ^ { s } u ( x , t ) = \\frac { \\Gamma ( 1 - s ) } { 2 ^ { 2 s - 1 } \\Gamma ( s ) } V ( x , t ) u ( x , t ) , \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} f ( x , y ) = \\left \\lbrace \\begin{array} { c l } 1 , & y \\in \\Q \\cap [ 0 , 1 ] \\\\ 0 , & y \\notin \\Q \\end{array} \\right . . \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} \\int _ 0 ^ a t ^ { \\gamma - 2 } f ^ 2 ( t ) d t & = \\frac { 1 } { \\gamma - 1 } \\int _ 0 ^ a f ^ 2 ( t ) d t ^ { \\gamma - 1 } \\\\ & = \\frac { 2 } { 1 - \\gamma } \\int _ 0 ^ a t ^ { \\gamma - 1 } f ( t ) f ' ( t ) d t , \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} M _ i [ t ^ { - \\chi ( \\cdot ) } ] ( \\vec { x } , \\vec { y } ) = t ^ { - \\chi ( \\vec { x } , \\vec { y } ) + 1 } - t \\cdot t ^ { - \\chi ( \\vec { x } , \\vec { y } ) } = 0 . \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} \\omega = u '' ( y ) \\omega _ { 1 , \\bar { 1 } } + e ^ { - 4 y } u ' ( y ) \\omega _ { \\alpha _ { 2 , 1 } , \\bar { \\alpha _ { 2 , 1 } } } . \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} w _ v z ^ { - 1 } x _ v ^ { - 1 } ( \\gamma ) = w _ v \\tau ^ { - 1 } ( \\gamma ) = w _ v ( \\beta ) \\in \\Phi ^ + \\end{align*}"}
-{"id": "7141.png", "formula": "\\begin{align*} \\frac { \\Delta ^ x _ { [ 1 : 0 ] } } { t ^ 2 } = d _ { 1 , 0 } ^ 2 - 4 d _ { 1 , - 1 } d _ { 1 , 1 } \\frac { \\Delta ^ y _ { [ 1 : 0 ] } } { t ^ 2 } = d _ { 0 , 1 } ^ 2 - 4 d _ { - 1 , 1 } d _ { 1 , 1 } \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} ( - \\Delta _ \\xi ) ^ k p ( x , \\xi ) = \\mathcal { O } ( \\langle \\xi \\rangle ^ { m - 2 k } ) . \\end{align*}"}
-{"id": "4229.png", "formula": "\\begin{align*} | u _ \\alpha | < \\rho < | q _ i | ^ { - 1 } | u _ \\alpha | \\forall \\ ; \\alpha = 1 , \\dots , r , \\forall \\ ; i = 1 , 2 . \\end{align*}"}
-{"id": "1987.png", "formula": "\\begin{align*} d ^ { ( k ) } \\simeq \\bigoplus \\limits _ { 0 \\leq r \\leq N - 1 \\atop p + q = k } d _ r ^ { p , \\ , q } , \\end{align*}"}
-{"id": "6229.png", "formula": "\\begin{align*} L _ m \\chi _ y ( 0 ) = \\sum _ { Y ' \\in \\mathcal { M } _ \\mu ( \\mathbb { F } _ q ) } \\chi \\left ( \\mathrm { t r } ( Y Y '^ T ) \\right ) , \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{align*} \\min _ { | z | = \\frac 1 2 } | f _ k ( z ) | \\leq 1 \\quad \\max _ { | z | = \\frac 1 2 } | f _ k ( z ) | = \\max _ { | z | = | z _ k | } | f ( z ) | \\to \\infty , \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} f ^ * ( 1 \\times \\cdots \\times 1 \\times x _ { M _ j } ^ { u _ j } \\times 1 \\times \\cdots \\times 1 ) = \\sum _ { l = 1 } ^ s ( 1 \\times \\cdots \\times 1 \\times y _ { M _ l } ^ { u _ l } \\times 1 \\times \\cdots \\times 1 ) , \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} \\dd \\Psi & = { \\overline W } _ 1 ^ { \\ , \\Psi } \\wedge \\Psi ~ , \\\\ \\dd ( \\omega \\wedge \\omega ) & = 2 \\ , W _ 1 ^ \\omega \\wedge \\omega \\wedge \\omega ~ , \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} \\sum _ { h \\in \\pi _ { * } ^ { - 1 } ( x ) } g _ h = 0 \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{align*} Q _ { n , \\textup { \\textbf { m } } } ( z ) F _ \\alpha ( z ) - P _ { n , \\textup { \\textbf { m } } , \\alpha } ( z ) = \\sum _ { k = n + 1 } ^ { \\infty } a _ { k , n } ^ { ( \\alpha ) } \\Phi _ k ( z ) , z \\in D _ { \\rho _ { 0 } ( F _ \\alpha ) } , \\end{align*}"}
-{"id": "367.png", "formula": "\\begin{align*} \\omega _ j ( g ) = \\frac { 1 } { | H | } \\sum _ { h \\in H } \\chi _ j ( g ^ { - 1 } h ) = \\frac { 1 } { | H g H | } \\sum _ { h \\in H g H } \\overline { \\chi _ j ( h ) } ( g \\in G ) . \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} \\omega _ t ( x ) = \\int _ X K ( t ; x , y ) \\omega _ 0 ( y ) d X _ y . \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} \\widetilde { c } _ 2 ( t ; T ) = \\frac { \\prod _ { n = 1 } ^ t \\sin \\left ( \\frac { n \\pi } { 3 T } \\right ) \\prod _ { n = 1 } ^ { 2 T - t } \\sin \\left ( \\frac { n \\pi } { 3 T } \\right ) } { \\prod _ { n = 1 } ^ { 2 T } \\sin \\left ( \\frac { n \\pi } { 3 T } \\right ) } \\prod _ { n = 1 } ^ T \\frac { \\sin ^ 2 \\left ( \\frac { n \\pi } { 3 T } \\right ) } { \\cos \\left ( \\frac { ( 2 n - T - 1 ) \\pi } { 6 T } \\right ) } . \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} & ( R _ 1 ^ m ) _ { s , t } = - \\int _ s ^ t ( \\nabla _ q \\chi ) ( r , q _ r ^ m , z _ r ^ m ) \\cdot z _ r ^ m d r \\\\ & - \\int _ s ^ t ( \\nabla _ z \\chi ) ( r , q _ r ^ m , z _ r ^ m ) \\cdot \\left [ ( - \\partial _ r \\psi ( r , q _ r ^ m ) + \\tilde F ( r , q _ r ^ m ) - \\nabla _ q V ( r , q _ r ^ m ) ) d r + \\sigma ( r , q ^ m _ r ) d W _ r \\right ] \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} \\partial ^ \\bullet = g \\circ L _ \\partial \\circ f \\ , ; \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} f ( \\theta ) : = - \\log \\frac { | e ^ { i \\theta } - 1 | | e ^ { i \\theta } - q _ 1 q _ 2 | } { | e ^ { i \\theta } - q _ 1 | | e ^ { i \\theta } - q _ 2 | } . \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} \\lefteqn { \\beta \\ , E ^ \\dagger ( x ) - \\Omega ( x ) + \\Omega ^ \\dagger } \\\\ & \\qquad = \\inf _ { r \\geq 0 } \\bigl ( \\beta \\ , E ^ \\dagger ( x ) - \\Omega ( x ) + \\Omega ^ \\dagger - r \\ , \\| F ( x ) - y ^ \\dagger \\| + r \\ , \\| F ( x ) - y ^ \\dagger \\| \\bigr ) \\\\ & \\qquad \\leq \\inf _ { r \\geq 0 } \\bigl ( D _ \\beta ( r ) + r \\ , \\| F ( x ) - y ^ \\dagger \\| \\bigr ) \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} \\L \\textnormal { r e c t a n g u l a r } \\Longleftrightarrow \\exists z _ 1 , z _ 2 \\in \\C \\colon \\L = \\langle z _ 1 , z _ 2 \\rangle _ \\Z \\ ; \\wedge \\ ; i \\frac { z _ 1 } { z _ 2 } \\in \\R . \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} v _ { \\tau } ( t ) : = w ( t ) + d _ { A ( t ) } \\int _ \\tau ^ t \\psi ( s ) d s , \\ \\ \\ 0 < t < \\infty , \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{align*} F ( s ) = f ( 0 ) \\dfrac { s ^ { \\beta - 1 } } { s ^ { \\beta } - { \\left ( i \\hslash \\right ) } ^ { - \\beta } E } . \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} \\mathrm { P } _ { m , 1 } & = \\mathrm { P } \\left ( z _ m < \\frac { \\epsilon _ 1 } { \\rho \\xi _ 1 } \\right ) \\\\ & = \\mathrm { P } \\left ( \\alpha _ 1 = 1 , z _ m < \\frac { \\epsilon _ 1 } { \\rho \\xi _ 1 } \\right ) + \\mathrm { P } \\left ( \\alpha _ 1 < 1 , z _ m < \\frac { \\epsilon _ 1 } { \\rho \\xi _ 1 } \\right ) . \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} f _ { r _ m , r _ t } ( x , y ) = & f _ { r _ m | r _ t } ( x ) f _ { r _ t } ( y ) \\\\ = & 4 y ( \\lambda _ c \\pi ) ^ { t } e ^ { - \\lambda _ c \\pi y ^ 2 } \\frac { x ^ { 2 m - 1 } ( y ^ 2 - x ^ 2 ) ^ { t - m - 1 } } { ( t - m - 1 ) ! ( m - 1 ) ! } . \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} \\langle \\varnothing | W _ { - n _ s , k _ s } \\dots W _ { - n _ 1 , k _ 1 } \\Phi _ m W _ { n _ 1 ' , k _ 1 ' } \\dots W _ { n _ t ' , k _ t ' } | \\varnothing \\rangle = 0 \\end{align*}"}
-{"id": "3544.png", "formula": "\\begin{align*} \\hat { X } ( x , t ) = g ( t ) e ^ { i f ( t ) } X ( x ) + H ( t ) \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} x = \\left \\{ \\begin{array} { l l } \\displaystyle a _ { 4 } + \\frac { D ' ( a _ { 4 } ) } { \\mathbf { u } - \\frac { D '' ( a _ { 4 } ) } { 6 } } & a _ { 4 } \\neq [ 1 \\ ! : \\ ! 0 ] , \\medskip \\\\ \\displaystyle \\frac { \\mathbf { u } - \\alpha _ { 2 } / 3 } { \\alpha _ { 3 } } & . \\end{array} \\right . \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{align*} \\vec { z } ( \\mu ) = \\vec { x } ^ { ( 1 ) } ( \\mu ) \\cup \\cdots \\cup \\vec { x } ^ { ( r ) } ( \\mu ) , \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} D ^ { I I I } _ n = \\{ Z \\in M _ n ( \\C ) \\mid Z Z ^ * < I _ n , Z ^ t = Z \\} , \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{align*} l ' _ j \\asymp \\left ( 1 - K _ 4 n ^ { - \\beta / ( 2 \\beta + r ) } j ^ { \\beta / r } \\right ) _ + , j = i _ 0 , \\dots , N . \\end{align*}"}
-{"id": "7099.png", "formula": "\\begin{align*} g ( x + \\alpha _ { 0 } , y + \\beta _ { 0 } ) & = f ( x + \\alpha _ { 0 } , y + \\beta _ { 0 } ) - ( x + \\alpha _ { 0 } + c ( y + \\beta _ { 0 } ) ) ^ { k } \\cdot f _ { k } ( y ) \\\\ & = f ( x , y ) - \\delta ' + \\delta - ( x + c y ) ^ k \\cdot f _ { k } ( y ) = g ( x , y ) - \\delta ' + \\delta . \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} \\rho ( x ) = ( 1 + | x | ^ 2 ) ^ { 1 / 2 } , \\quad \\gamma = ( 1 , \\ldots , 1 ) ; \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} \\displaystyle { \\operatornamewithlimits { \\mbox { m i n i m i z e } } _ { \\Theta } } \\ f _ N ( \\Theta ) \\ , \\triangleq \\displaystyle { \\frac { 1 } { N } } \\ , \\displaystyle { \\sum _ { i = 1 } ^ N } \\ , \\ell ( y _ i - m ( x ^ i ; \\Theta ) ) , \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{align*} \\frac { | z | ^ { 2 ( n - 1 ) + 1 } } { 1 - | z | ^ { n - 1 } } = \\frac { | z | ^ { 2 n - 1 } } { 1 - | z | ^ { n - 1 } } ~ < ~ \\left | G _ { n } ( z ) \\right | \\mbox { f o r } ~ z \\in C _ { j , n } , j = 1 , 2 , \\ldots , J _ n \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} I = - \\int _ { \\Omega } w ^ 2 \\delta ^ 2 \\nabla u \\cdot ( A \\nabla u ) d x , \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} S I R ( U \\to { X _ d } ) { \\rm { } } = \\gamma _ d = \\frac { { { P _ d } { h _ d } { { \\left \\| { { X _ d } } \\right \\| } ^ { - \\alpha } } } } { { { I _ { d , r u } } + { I _ { f , r u } } } } , \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} \\Omega _ { \\alpha _ { 2 , 3 } , \\bar { \\alpha } _ { 2 , 3 } } = e ^ { 2 t } ( 2 - u ' ( t ) ) / 4 . \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} \\int _ 0 ^ T | ( \\widehat { \\rho } ^ \\tau ) ' | ^ 2 ( t ) d t = \\sum _ { k } \\frac { W ^ 2 _ { 2 } ( \\rho ^ \\tau _ k , \\rho ^ \\tau _ { k - 1 } ) } { \\tau } , \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{align*} \\partial _ \\mu \\log Z _ \\Gamma ( s ) = \\int _ X F _ s ( z ) \\mu . \\end{align*}"}
-{"id": "1732.png", "formula": "\\begin{align*} \\sum _ { k = k _ 0 } ^ \\infty 2 ^ { - 2 k \\alpha } | \\langle \\mathcal { C } _ k f , g \\rangle | \\lesssim \\| f \\| _ { L ^ { \\frac { 4 } { 3 } } _ t L ^ { r ' } _ x } \\| g \\| _ { L ^ { \\frac { 4 } { 3 } } _ t L ^ { r ' } _ x } \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} E = \\frac { z _ 1 \\bar { z } _ 2 + \\bar { z } _ 1 z _ 2 } { \\sinh ( 2 \\beta ) } ( \\mathcal { P } ( \\beta ^ { \\vee } ) + \\mathcal { P } ( [ \\theta \\sigma ( e _ { \\beta } ) , e _ { \\beta } ] ) ) \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} Z = \\exp F . \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} \\rho = ( 1 / 8 ) \\cdot ( 1 2 b ^ 3 ) ^ { - 4 \\beta _ { \\max } / ( c _ 1 \\delta ) } , \\end{align*}"}
-{"id": "9169.png", "formula": "\\begin{align*} A _ m P _ { - n } - P _ { - n } A _ m \\gamma ^ n = A _ m ( 1 - \\gamma ^ n ) \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{align*} \\varphi : = \\lvert d \\rvert ^ 2 - 1 . \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{align*} \\begin{pmatrix} \\kappa _ 1 \\\\ \\kappa _ 2 \\end{pmatrix} _ t = P \\begin{pmatrix} s _ 1 \\\\ s _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{align*} u _ 1 ^ { l _ 1 } \\cdots u _ k ^ { l _ k } = 0 . \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} H ^ 0 ( M , \\ , ( E _ \\eta , \\ , E _ \\rho ) ) \\ , = \\ , 0 \\ , . \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} u ( t , x ) = \\int _ { \\Omega } k ( t , x , y ) d y , \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} & = \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } \\Biggl \\{ \\sum _ { k = 1 } ^ { H - 1 } ( a _ { i k } b _ { k j } - a ^ 0 _ { i k } b ^ 0 _ { k j } ) + ( a _ { i H } - a ^ 0 _ { i H } ) - \\sum _ { k = 1 } ^ { H - 1 } ( a _ { i H } b _ { k j } - a ^ 0 _ { i H } b ^ 0 _ { k j } ) \\Biggr \\} ^ 2 \\\\ & = \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } \\Biggl [ ( a _ { i H } - a ^ 0 _ { i H } ) + \\sum _ { k = 1 } ^ { H - 1 } \\{ ( a _ { i k } - a _ { i H } ) b _ { k j } - ( a ^ 0 _ { i k } - a ^ 0 _ { i H } ) b ^ 0 _ { k j } \\} \\Biggr ] ^ 2 . \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} d X _ t = \\mu ( X _ t ) d t + \\sigma ( X _ t ) d W _ t , \\enskip X _ 0 = x \\in E , \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} | N ^ - ( \\sigma _ { \\nu } w ) \\cap \\Phi _ h ^ - | = | N ^ - ( \\sigma _ { \\nu } w ) \\cap \\Phi _ h ^ - \\cap ( \\Phi ^ - \\setminus \\Phi ^ - [ T ] ) | + | N ^ - ( \\sigma _ { \\nu } w ) \\cap \\Phi _ h ^ - [ T ] | . \\end{align*}"}
-{"id": "7820.png", "formula": "\\begin{align*} x = \\dfrac 1 6 , \\ z = \\dfrac 1 2 , \\ \\ z = - \\dfrac 1 2 , \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{align*} \\sigma _ { 0 i } = a _ 0 ( - 1 ) ^ i , \\ i = 1 , \\ldots , n _ 1 , \\sigma _ { 2 i } = a _ 2 ( - 1 ) ^ i , \\ i = 1 , \\ldots , n _ 3 . \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{align*} \\begin{aligned} & \\ ; \\left | g _ k ( z ) - 1 - \\left ( \\exp \\ ! \\left ( c _ k ( z - b _ k ' ) \\right ) - 1 \\right ) \\right | \\\\ & = \\left | \\exp \\ ! \\left ( c _ k ( z - b _ k ' ) + \\delta _ k ( z ) \\right ) - \\exp \\ ! \\left ( c _ k ( z - b _ k ' ) \\right ) \\right | \\\\ & = \\left | \\exp ( c _ k ( z - b _ k ' ) ) \\right | \\cdot \\left | \\exp ( \\delta _ k ( z ) ) - 1 \\right | . \\end{aligned} \\end{align*}"}
-{"id": "6827.png", "formula": "\\begin{align*} \\dot { x } ( t ) & = A x ( t ) + B u ( t ) \\\\ [ 1 e x ] y ( t ) & = x ( t ) ^ \\top M x ( t ) . \\end{align*}"}
-{"id": "4029.png", "formula": "\\begin{align*} \\mu ( x , y ) & = \\lambda \\mu _ 1 ( x , y ) + ( 1 - \\lambda ) \\mu _ 2 ( x , y ) \\\\ \\gamma ( x , y ) & = \\lambda \\gamma _ 1 ( x , y ) + ( 1 - \\lambda ) \\gamma _ 2 ( x , y ) . \\end{align*}"}
-{"id": "5524.png", "formula": "\\begin{align*} c = \\frac { 1 } { 2 } x \\cdot D v \\left ( x \\right ) - v \\left ( x \\right ) . \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{align*} \\alpha ( b c ) = \\beta ( b , c ) \\alpha ( b ) \\alpha ( c ) , b , c \\in \\Gamma . \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} \\tilde { u } ( a ) & = - \\ln \\det d ^ 2 _ a u - \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\ln ( ( 2 \\chi - d _ a u ) ( \\alpha ^ { \\vee } ) ) \\\\ & + \\sum _ { \\beta \\in \\Phi _ s ^ + } \\ln \\sinh ( - 2 \\beta ( a ) ) - \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } } 2 \\alpha ( a ) , \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{align*} \\sup _ { \\bar { X _ { 0 } } \\in P _ { r } ( X _ { 0 } ) } \\Theta \\big ( \\{ \\Gamma _ { t } \\} , \\bar { X _ { 0 } } , r \\big ) : = \\int _ { \\Gamma _ { t _ { 0 } } - r ^ { 2 } } \\rho _ { ( x _ { 0 } , t _ { 0 } ) } d s < 1 + \\epsilon , \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} G ( x , t ) = \\begin{cases} ( 4 \\pi t ) ^ { - \\frac n 2 } e ^ { - \\frac { | x | ^ 2 } { 4 t } } , \\ \\ \\ \\ \\ t > 0 , \\\\ 0 , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ t \\le 0 . \\end{cases} \\end{align*}"}
-{"id": "7789.png", "formula": "\\begin{align*} J _ { 2 , - } ( t _ 1 , t _ 2 ) = & \\int _ 0 ^ { t _ 2 } \\int _ { - \\infty } ^ 0 \\int _ { - \\infty } ^ y G _ { t _ 2 - s } ( x - z ) \\psi ( s , z ) \\sigma _ s ( y ) W ( d s , d y ) \\\\ & - \\int _ 0 ^ { t _ 1 } \\int _ { - \\infty } ^ 0 \\int _ { - \\infty } ^ y G _ { t _ 1 - s } ( x - z ) \\psi ( s , z ) \\sigma _ s ( y ) W ( d s , d y ) . \\end{align*}"}
-{"id": "8314.png", "formula": "\\begin{align*} a : = - \\frac { \\partial P } { \\partial n } > 0 , \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{align*} F _ - ^ n ( x ) = x + \\rho n + O ( 1 ) , F _ + ^ n ( x ) = x + \\rho n + O ( 1 ) , \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} ^ { H } \\mathbb { D } _ { b - } ^ { \\alpha , \\beta ; \\psi } f \\left ( x \\right ) = I _ { b - } ^ { \\gamma - \\alpha ; \\psi } \\left ( - 1 \\right ) ^ { n } \\mathcal { D } _ { b - } ^ { \\gamma ; \\psi } f \\left ( x \\right ) , \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} D _ { a } = \\biggl \\lbrace x \\in \\mathbb { F } _ { q } ^ { * } : T r ( x ^ { p ^ { \\alpha } + 1 } ) = a \\biggr \\rbrace . \\end{align*}"}
-{"id": "1529.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } \\binom { n } { k } & \\dbinom { m + k } { p } y ^ { n - k } f _ { m - p + 1 + k } ^ { \\left ( \\alpha \\right ) } \\left ( x \\right ) \\\\ & = \\sum _ { k = 0 } ^ { m } \\binom { m } { k } \\dbinom { n + k } { p } \\left ( - y \\right ) ^ { m - k } \\left ( f _ { n - p + 1 + k } ^ { \\left ( \\alpha \\right ) } \\left ( x + y \\right ) - y f _ { n - p + k } ^ { \\left ( \\alpha \\right ) } \\left ( x + y \\right ) \\right ) . \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} \\operatorname { d i v } ( | y | ^ { - a } \\nabla v ) = | y | ^ { - a } v _ t . \\end{align*}"}
-{"id": "8391.png", "formula": "\\begin{align*} \\rho _ A ( t _ 0 , t _ 0 + t ) : = ( 1 / 2 ) \\int _ { t _ 0 } ^ { t _ 0 + t } ( s - t _ 0 ) ^ { - 1 / 2 } \\| B ( s ) \\| _ 2 ^ 2 d s \\le \\rho _ A ( t ) \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} \\Psi _ r = \\lbrace \\psi _ I , I = ( i _ 1 , . . . , i _ n ) \\in \\lbrace 1 , . . . , m \\rbrace ^ * : \\beta _ { i _ 1 } \\cdot \\cdot \\cdot \\beta _ { i _ n } \\leq r < \\beta _ { i _ 1 } \\cdot \\cdot \\cdot \\beta _ { i _ { n - 1 } } \\rbrace \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} \\sum _ { j \\geq 1 } \\psi _ j z ^ { 2 j - 1 } = \\sinh ( x z ) + \\cosh ( x z ) \\sum _ { i \\geq 2 } s _ { 2 i - 1 } z ^ { 2 i - 1 } , \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} f ( \\vec { x } ) = \\operatorname { s g n } ( \\vec { w } \\cdot \\vec { x } - \\theta ) , \\end{align*}"}
-{"id": "6324.png", "formula": "\\begin{align*} F _ { a a } ( { \\bf u } ) & = \\frac { 1 } { 2 } | | { \\bf u } _ a | | ^ 2 , \\ , \\ , 1 \\leq a \\leq p , \\\\ F _ { b c } ( { \\bf u } ) & = \\left < { \\bf u } _ b , { \\bf u } _ c \\right > , \\ , \\ , 1 \\leq b < c \\leq p . \\end{align*}"}
-{"id": "635.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { M } \\frac { \\lambda _ i } { C _ i \\prod _ { j \\neq i } ( 1 - p _ j ) } < 1 , \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{align*} 0 = \\int _ { \\Omega } a ( x ) \\left \\{ ( t \\phi _ { 1 } + w ) ^ { q } - ( t \\phi _ { 1 } + w ) \\right \\} \\phi _ { 1 } . \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} C _ 1 C f _ n = ( I _ { H _ 2 } + F _ 2 ) f _ n = f _ n + F _ 2 f _ n , \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{align*} \\hat { \\phi } ( p ) = \\ \\frac { \\gamma { } k } { 2 \\pi { } \\hslash { } } \\frac { 1 } { \\left ( C _ { \\alpha { } } { \\left \\vert { } p \\right \\vert { } } ^ { \\alpha { } } e ^ { i S g n \\left ( p \\right ) \\theta { } \\pi { } / 2 } - E \\right ) } , \\end{align*}"}
-{"id": "8116.png", "formula": "\\begin{align*} N _ 1 ( U _ r , \\rho ) = N _ 1 ( U , r \\rho ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ r , \\rho > 0 . \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} \\partial _ H X : = \\overline { X } ^ { H } \\setminus \\tau _ d ( X ) . \\end{align*}"}
-{"id": "3309.png", "formula": "\\begin{align*} \\int _ { C _ W } e ^ { - \\d a ^ * M _ x a } ( \\det M _ x ) ^ { q - 1 } d x = 2 ^ { q - 1 } \\Gamma ( q ) e ^ { \\d a ^ * W a } \\prod _ { i = 1 } ^ n a _ i ^ { q ( s ( i ) - 2 ) } \\prod _ { i < j } | w _ { i j } | ^ q K _ q ( a _ i a _ j | w _ { i j } | ) \\end{align*}"}
-{"id": "5955.png", "formula": "\\begin{align*} \\mathcal { F } ( R , P ) = - \\left [ \\mathcal { G } ( R , P ) \\right ] ^ 2 + \\mathcal { H } ( R , P ) , \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} \\chi ( c \\beta + \\sum _ { ( i , j ) \\in \\mathbb { I } } c _ { i j } [ S ] ^ { - i } [ T ] ^ { - j } ) = \\chi ( c \\beta ) + \\sum _ { ( i , j ) \\in \\mathbb { I } } \\chi ( c _ { i j } [ S ] ^ { - i } [ T ] ^ { - j } ) , \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{align*} \\{ \\cdot , \\cdot \\} _ { ( \\lambda ) } = \\{ \\cdot , \\cdot \\} _ 1 - \\lambda \\{ \\cdot , \\cdot \\} _ 0 . \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{align*} \\binom { 4 d - 1 } { 2 d - 1 } - ( - 1 ) ^ d \\binom { 2 d - 1 } { d - 1 } = \\frac { 2 d - 1 } { d } \\binom { 2 d - 2 } { d - 1 } \\left ( \\prod _ { i = 1 } ^ { d } \\left ( \\frac { 4 d } { 2 i - 1 } - 1 \\right ) - ( - 1 ) ^ { d } \\right ) . \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{align*} - \\frac { r ( r - 1 ) } { B ^ 2 } \\ell ( S ^ 1 ( B ) ) = \\left ( \\frac { r } { B } \\right ) \\frac { d \\ell ( S ^ 1 ( B ) ) } { d B } - \\frac { r \\ell ( S ^ 1 ( B ) ) } { B ^ 2 } \\end{align*}"}
-{"id": "5199.png", "formula": "\\begin{align*} u _ t & = \\Delta u - \\chi \\nabla v \\cdot \\nabla u + u ( a ( x , t ) - u ( b ( x , t ) - \\chi \\mu ) - \\chi \\lambda v ) \\\\ & \\leq \\Delta u - \\chi \\nabla v \\cdot \\nabla u + u ( a ( x , t ) - u ( b ( x , t ) - \\chi \\mu ) ) \\\\ & \\leq \\Delta u - \\chi \\nabla v \\cdot \\nabla u + u ( a _ { \\sup } - ( b _ { \\inf } - \\chi \\mu ) u ) \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{align*} \\mathcal { R } \\theta _ F = \\theta ^ 2 . \\end{align*}"}
-{"id": "3319.png", "formula": "\\begin{align*} \\Gamma _ r ( t ) = \\{ ( p ( i , j | v , w ) ) \\in C _ r ^ s ( n , 2 ) : p _ A ( 0 | v ) = p _ B ( 0 | w ) = t , \\ , \\forall v , w \\} \\end{align*} % \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} \\sum _ { n \\geq L } \\big ( & p ^ + _ M ( t , n ) - p ^ + _ M ( t , { n + 1 } ) \\big ) | B _ n | = \\sum _ { n \\geq L } p ^ + _ M ( t , n ) | C _ n | + p ^ + _ M ( t , L ) | B _ { L - 1 } | \\\\ & = \\sum _ { y \\not \\in B _ { L - 1 } } p _ M ( t , y ) + \\sum _ { n \\geq L } \\sum _ { y \\in C _ n } \\big ( p ^ + _ M ( t , n ) - p _ M ( t , y ) \\big ) + p ^ + _ M ( t , L ) | B _ { L - 1 } | \\ , . \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} h ( t , x , \\xi ) : = \\frac { 1 } { 2 m } | \\xi - A ( t , x ) | ^ 2 + V ( t , x ) . \\end{align*}"}
-{"id": "2744.png", "formula": "\\begin{align*} d \\log ( \\{ 1 + a S ^ i T ^ j , S \\} ) & = \\frac { d ( 1 + a S ^ i T ^ j ) } { 1 + a S ^ i T ^ j } \\wedge \\frac { d S } { S } \\\\ & = - a j \\sum _ { k = 0 } ^ \\infty ( - a ) ^ k S ^ { ( k + 1 ) i - 1 } T ^ { ( k + 1 ) j - 1 } d S \\wedge d T . \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} p _ { - 3 } ( a ) p _ { - 3 } ( 1 ) - p _ { - 3 } ( a + 1 ) & = 3 p _ { - 3 } ( a ) - p _ { - 3 } ( a + 1 ) \\\\ & = 3 \\sum _ { k = 0 } ^ a p _ { - 2 } ( k ) p ( a - k ) - \\sum _ { k = 0 } ^ { a + 1 } p _ { - 2 } ( k ) p ( a + 1 - k ) \\\\ & = \\left ( \\sum _ { k = 0 } ^ a p _ { - 2 } ( k ) ( 3 p ( a - k ) - p ( a + 1 - k ) ) \\right ) - p _ { - 2 } ( a + 1 ) \\\\ & > 2 p _ { - 2 } ( a ) - p _ { - 2 } ( a + 1 ) \\\\ & = p _ { - 2 } ( 1 ) p _ { - 2 } ( a ) - p _ { - 2 } ( a + 1 ) \\ge 0 . \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ; T ) + L _ t u ( t , x ; T ) & = 0 , \\\\ u ( T , x ; T ) & = x . \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} \\chi ( T \\Sigma ^ \\perp ) = \\frac { 1 } { 2 \\pi } \\int _ \\Sigma \\kappa ^ \\perp v _ \\gamma . \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} | \\mathcal { U } _ i | \\leq \\begin{cases} | \\mathcal { X } | \\prod \\limits _ { l = 1 } ^ { i - 1 } | \\mathcal { U } _ l | & i , \\\\ | \\mathcal { Y } | \\prod \\limits _ { l = 1 } ^ { i - 1 } | \\mathcal { U } _ l | & i . \\end{cases} \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} \\partial _ t \\Phi - \\delta ^ \\star \\Delta \\Phi = ( d - \\delta ^ \\star ) \\Delta \\Phi = ( d - \\delta ^ \\star ) M \\leq 0 . \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} | F : N | = | F : H \\cap F | = | F H : H | \\le | G : H | < \\infty . \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} & \\left ( 1 + \\frac { x v } { 1 - v } \\right ) B ( x , v ) + \\frac { x v ^ 2 } { 1 - v } A ( x , v ) \\\\ & \\qquad \\qquad \\qquad = \\frac { ( 1 - v + x v ) v ^ 2 } { 1 - v } A ( x , 1 ) - \\frac { x v ^ 2 ( 1 - 2 v + x v ) } { ( 1 - x ) ( 1 - v ) } B ( x , 1 ) - \\frac { x ^ 2 v ^ 2 ( 1 - v + x v ) } { 1 - x } , \\\\ & - \\left ( 1 + x + \\frac { x v } { 1 - v } \\right ) B ( x , v ) + \\frac { ( 1 - x v ) ( 1 - v + x v ) } { x v ( 1 - v ) } A ( x , v ) \\\\ & \\qquad \\qquad \\qquad = \\frac { ( v - x ) v ^ 2 } { 1 - v } A ( x , 1 ) + \\frac { x v ^ 2 ( x - v ) } { ( 1 - x ) ( 1 - v ) } B ( x , 1 ) + \\frac { x v ^ 2 ( 1 - x + x ^ 2 ) } { 1 - x } . \\end{align*}"}
-{"id": "7323.png", "formula": "\\begin{align*} f ^ { - 1 } [ T _ t ] = \\left ( f ^ { - 1 } ( t ) \\cup \\bigcup _ { t ^ \\prime \\in s u c c _ T ( t ) } f ^ { - 1 } [ T _ { t ^ \\prime } ] . \\right ) \\end{align*}"}
-{"id": "161.png", "formula": "\\begin{align*} \\beta _ { \\infty } : = \\alpha _ { \\infty } - d _ { A _ { \\infty } } \\xi _ { \\infty } \\end{align*}"}
-{"id": "6583.png", "formula": "\\begin{align*} \\upsilon = \\alpha + \\frac { 1 } { 2 } \\theta + \\frac { r - 1 } { 2 r } - \\kappa r \\theta - \\kappa ( r - 1 ) . \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} \\Theta = \\sum _ { i } ^ { n } \\arctan \\lambda _ { i } = \\frac { 1 } { 2 } x \\cdot D u \\left ( x \\right ) - u \\left ( x \\right ) \\end{align*}"}
-{"id": "9292.png", "formula": "\\begin{align*} u _ \\tau ( t ) : = u _ { \\tau , m } ^ S ( t ) : = ( \\Phi _ { j , t - t _ m } ^ S \\Phi _ { j , \\tau } ^ D ) \\prod _ { j = 1 } ^ { m - 1 } \\big ( \\Phi _ { j , \\tau } ^ S \\Phi _ { j , \\tau } ^ D \\big ) u _ \\tau ( 0 ) , t \\in T _ m , \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\mathcal L _ \\partial ( \\theta _ * ( s \\otimes p ) ) & = & \\mathcal L _ \\partial ( s e _ S \\otimes p ) \\\\ & = & \\partial ( s ) e _ S \\otimes p + s \\mathcal L _ \\partial ( e _ S ) \\otimes p + s e _ S \\otimes \\partial ^ \\bullet ( p ) \\\\ & = & \\theta _ * ( \\mathcal L _ \\partial ( s \\otimes p ) ) + \\lambda \\theta _ * ( s \\otimes p ) \\ , . \\end{array} \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} H _ { \\mathcal { R } ( I ^ l ) _ + } ^ i \\left ( L ^ { I ^ l } ( M ) ( - 1 ) \\right ) = \\left ( H _ { R _ + } ^ { i } \\left ( L ^ { I } ( M ) ( - 1 ) \\right ) \\right ) ^ { < l > } = 0 . \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{align*} S _ { m } ( \\mu ) : = m \\mu - \\rho ( \\mu ) = ( m \\mu _ 1 , \\dots , m \\mu _ n ) + w _ { \\mu } \\cdot ( 1 , 2 , \\dots , n ) , \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\vdash d } P ( \\lambda ) \\nu ( \\lambda ) = \\sum _ { \\lambda \\vdash d } \\frac { 1 } { z _ \\lambda } \\sum _ { k = 0 } ^ { d - 1 } \\frac { P ( \\lambda ) \\psi _ d ^ k ( \\lambda ) } { q ^ k } = \\sum _ { k = 0 } ^ { d - 1 } \\frac { 1 } { q ^ k } \\left ( \\sum _ { \\lambda \\vdash d } \\frac { P ( \\lambda ) \\psi _ d ^ k ( \\lambda ) } { z _ \\lambda } \\right ) = \\sum _ { k = 0 } ^ { d - 1 } \\frac { \\langle P , \\psi _ d ^ k \\rangle } { q ^ k } . \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} \\psi \\left ( \\nu , \\mu \\right ) = t ^ { - m ( m - 1 ) / 2 } \\times \\prod _ { j = 1 } ^ { m } \\left ( \\prod _ { 1 \\leq i < x _ j ( \\mu ) } t ^ { \\nu _ i } \\right ) \\nu _ { x _ j ( \\mu ) } , \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} { \\rm C o e f f } [ f _ { \\delta ^ + } , m ] = d ( t ) \\cdot \\prod _ { i = 1 } ^ { m } z _ i \\cdot e _ { ( r m - m ) } ( z _ { m + 1 } , \\dots , z _ n ) = d ( t ) \\times \\sum _ { \\nu \\in \\sigma ( \\epsilon ) } I ( \\delta ^ { + } , \\nu ) f _ { \\nu } \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} [ \\pi _ { k , c } ( H ) f ] ( n ) = & \\ 2 ( k + n ) f ( n ) , \\\\ [ \\pi _ { k , c } ( E ) f ] ( n ) = & \\ \\frac { n } { \\sqrt { c } } f ( n - 1 ) , \\\\ [ \\pi _ { k , c } ( F ) f ] ( n ) = & \\ - \\sqrt { c } \\ , ( 2 k + n ) f ( n + 1 ) , \\end{align*}"}
-{"id": "2398.png", "formula": "\\begin{align*} E \\left [ T _ 1 ( T _ 1 + 1 ) \\right ] & = \\left ( \\frac { \\nu _ 1 } { \\alpha _ 1 } \\right ) ^ 2 M ^ 2 \\left ( H _ { \\nu _ 1 M } ^ 2 + \\sum _ { j = 1 } ^ { \\nu _ 1 M } \\frac { 1 } { j ^ 2 } \\right ) + O \\left ( e ^ { - \\varepsilon M } \\right ) \\\\ & = ( \\nu _ 1 + \\lambda \\nu _ 2 ) ^ 2 M ^ 2 \\left ( H _ { \\nu _ 1 M } ^ 2 + \\sum _ { j = 1 } ^ { \\nu _ 1 M } \\frac { 1 } { j ^ 2 } \\right ) + O \\left ( e ^ { - \\varepsilon M } \\right ) \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} I _ 1 ( N ) = \\int _ { U ( N ; \\alpha ) ^ { - 1 } } ^ { U ( N ; \\alpha ) } e ^ { - x } \\left ( 1 - \\frac { \\ln x } { \\ln N } \\right ) ^ r d x \\ , + \\ , o \\left ( e ^ { - \\ln ^ { \\alpha } N } \\right ) . \\end{align*}"}
-{"id": "5610.png", "formula": "\\begin{align*} \\int _ \\Sigma v _ \\gamma = 2 \\pi \\big ( 2 ( g - 1 ) - | \\chi ( T \\Sigma ^ \\perp ) | \\big ) . \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} I _ 2 = \\frac { \\gamma ( \\gamma - 1 ) } { 2 } \\int _ R y ^ { \\gamma - 2 } ( u - \\tilde u ) ^ 2 \\leq 0 . \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} A _ m ( H _ { - n } - H _ { - n + 1 } ) = \\gamma ^ n ( H _ { - n } - H _ { - n + 1 } ) A _ m \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} A B M = B A M = B M . \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} G _ 1 ( t ) = m _ { G _ 1 } ( t ) - \\varphi _ { G _ 1 , G _ 2 } ( t ) m _ { G _ 2 } ( \\rho _ { G _ 1 , G _ 2 } ( t ) ) + \\varphi _ { G _ 1 , G _ 2 } ( t ) G _ 2 ( \\rho _ { G _ 1 , G _ 2 } ( t ) ) \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} g u '' = c _ 2 = \\begin{pmatrix} a \\\\ b \\\\ c \\\\ d \\end{pmatrix} . \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{align*} A _ k = \\left \\{ v \\in T ^ 1 S : \\exists \\ , t \\in [ 0 , T _ k ] \\textrm { w i t h } \\delta ( \\phi _ t ( v ) ) \\leqslant T _ k ^ { - \\xi } \\right \\} . \\end{align*}"}
-{"id": "6088.png", "formula": "\\begin{align*} u : = & S ( x _ { n _ { 1 } } \\otimes \\cdots \\otimes x _ { n _ { r + 1 } } \\otimes x _ { n _ { r } } \\otimes \\cdots \\otimes x _ { n _ { i + 1 } } ) \\\\ + & S ( \\mu ^ { i - 1 } \\alpha _ { T } ( x _ { n _ { 1 } } \\otimes \\cdots \\otimes x _ { n _ { r - 1 } } ) \\otimes [ x _ { n _ { r } } , x _ { n _ { r + 1 } } ] _ { \\mathfrak { g } } \\otimes \\alpha _ { T } ( x _ { n _ { r + 2 } } \\otimes \\cdots \\otimes x _ { n _ { i + 1 } } ) ) , \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t u = \\frac { \\nu ^ 2 } { 2 } \\Delta u + u \\Lambda ( t , x , u , \\nabla u ) , ( t , x ) \\in [ 0 , T ] \\times \\R ^ d , \\\\ \\Lambda ( t , x , y , z ) : = \\frac { \\vert z \\vert ^ 2 } { y } \\ , \\textrm { f o r a n y } \\ ( t , x , y , z ) \\in [ 0 , T ] \\times \\R ^ d \\times ] 0 , + \\infty [ \\times \\R ^ d \\ , \\\\ u ( 0 , \\cdot ) = u _ 0 \\ . \\end{array} \\right . \\end{align*}"}
-{"id": "3839.png", "formula": "\\begin{align*} \\begin{gathered} R _ + ( 0 ) f ( z ) = \\int _ 0 ^ \\infty ( f ( \\varphi _ t ( z ) ) - f ( B _ + ( z ) ) ) d t , \\\\ R _ - ( 0 ) f ( z ) = \\int _ 0 ^ \\infty ( f ( B _ - ( z ) ) - f ( \\varphi _ { - t } ( z ) ) ) d t . \\end{gathered} \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} e _ { n , \\ell } = \\sum _ { m = 0 } ^ { n - 1 - \\ell } e _ { n - m , \\ell } ' , 1 \\leq \\ell \\leq n - 1 , \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} f ( x _ 1 , \\dots , x _ d ) & = ( g \\circ h ) ( x _ 1 , \\dots , x _ d ) = g \\left ( h ( ( x _ 1 , \\dots , x _ d ) \\right ) \\\\ & = g \\left ( h ^ 2 ( x _ i , h ^ { d - 1 } ( x _ 1 , \\dots , x _ { i - 1 } , x _ { i + 1 } , \\dots , x _ d ) ) \\right ) \\\\ & = ( g \\circ h ^ 2 ) ( x _ i , h ^ { d - 1 } ( ( x _ 1 , \\dots , x _ { i - 1 } , x _ { i + 1 } , \\dots , x _ d ) ) ) \\end{align*}"}
-{"id": "649.png", "formula": "\\begin{align*} \\mathbb { E } S ^ 4 _ i = \\sum _ { n } \\mathbb { E } ( W _ i ( n + 1 ) - C _ i ) ^ 4 + \\sum _ { n \\neq m } \\mathbb { E } ( W _ i ( n + 1 ) - C _ i ) ^ 2 \\mathbb { E } ( W _ i ( m + 1 ) - C _ i ) ^ 2 . \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} f g = J q ^ { - 1 } ( f J q ^ 1 ( g ) ) + J q ^ { - 1 } ( J q ^ 1 ( f ) g ) . \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} \\varphi _ \\lambda ' / \\varphi _ \\lambda ( z ( \\lambda , \\hat { X } ) ) = \\int _ { 1 } ^ { \\hat { X } } \\frac { ( X - \\lambda / 3 ) d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } + \\lambda R _ { \\varphi } + \\pi i / \\omega _ 1 , \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} \\det [ D ^ 2 u - A ( x , u , D u ) ] = f ( x , u , D u ) , \\ , \\ , x \\in \\Omega . \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{align*} \\eta _ { \\delta } ( x ) d x = \\frac { 1 } { \\delta ^ n } \\eta ( \\frac { 1 } { \\delta } x ) \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} { f _ i ( \\sigma , N ) } = \\frac { { 1 / { i ^ \\sigma } } } { { \\sum \\nolimits _ { k = 1 } ^ N { 1 / { k ^ \\sigma } } } } , \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} u _ t = \\widetilde { G } _ t f _ t + R _ t u _ t . \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{align*} p ( M ) = ( \\prod _ { i = 1 } ^ { m - 1 } ( 1 + ( v _ i ^ { ( 0 ) } ) ^ 2 ) \\cdots ( 1 + ( v _ i ^ { ( n _ i ) } ) ^ 2 ) ) ( 1 + ( v _ { m } ^ { ( 0 ) } ) ^ 2 ) \\cdots ( 1 + ( v _ { m } ^ { ( n _ m ) } ) ^ 2 ) \\\\ = ( \\prod _ { i = 1 } ^ { m - 1 } ( 1 + ( v _ i ^ { ( 0 ) } ) ^ 2 ) \\cdots ( 1 + ( v _ i ^ { ( n _ i ) } ) ^ 2 ) ) ( 1 + u _ m ^ 2 ) \\cdots ( 1 + ( u _ m + y _ { n _ m } ) ^ 2 ) . \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} E _ { \\tau } ( 1 ) = \\lim _ { k \\to \\infty } \\mathcal { E } _ { \\tau } ( \\rho _ { n _ { k } } ) \\geq E _ { \\tau } ( \\alpha ) + E _ { \\tau } ^ { \\infty } ( 1 - \\alpha ) , \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{align*} \\tilde { \\nu } ( t ) = \\iint \\omega _ { r t } ( x - y ) \\ , \\omega _ { r } ( z - y ) \\ , d \\mu ( x ) \\ , d \\mu ( y ) \\ , d \\mu ( z ) \\ , d r = \\int < T _ { r t } \\circ T _ { r } 1 , 1 > _ \\mu \\ , d r , \\end{align*}"}
-{"id": "6233.png", "formula": "\\begin{align*} R _ m L _ m \\chi _ y = \\frac { 1 } { | P _ \\mu | } \\sum _ { y ' \\in P _ \\mu } \\langle R _ m L _ m \\chi _ y , \\chi _ { y ' } \\rangle \\chi _ { y ' } . \\end{align*}"}
-{"id": "4345.png", "formula": "\\begin{align*} ( \\wp _ \\lambda | _ { \\frak { F } } ) = \\bigcup _ { j = 1 , \\ldots , 1 0 , | m | , | n | \\le 4 2 } Y ^ { \\pm } _ { \\wp , j , m , n } \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{align*} \\{ p ( x , y ) ( D ) A ( x , y ) \\} ^ \\sigma = \\displaystyle \\frac { c _ 2 } { c _ 1 } p ( x , y ) ( D ) A ( x , y ) . \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{align*} R _ k = \\frac { 1 } { 2 n ^ 2 } \\log _ 2 ( q _ k ^ n ) = \\frac { 1 } { 2 n } \\log _ 2 ( q _ k ) , . \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{align*} \\tilde { { \\varphi } } \\in \\bigcap _ { j = 1 } ^ { n ( \\mu ) } \\mathrm { K e r } \\ , \\zeta _ { q _ { j } } ^ { \\prime } . \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} \\mu _ \\star ^ { ( \\infty , \\lambda , a ) } : = \\bigotimes _ { k = 1 } ^ \\infty { \\bf E x p } ( \\lambda + k a ) \\ , , \\lambda > 0 , \\ , a \\ge 0 \\ , . \\end{align*}"}
-{"id": "1003.png", "formula": "\\begin{align*} \\frac { \\omega _ { j } ^ { k } \\rho _ { j } ^ { k } } { 2 R } \\Vert U _ { j } ^ { k } x ^ { k } - x ^ { k } \\Vert ^ { 2 } \\leq \\frac { 1 } { 2 R } \\sum _ { i = 1 } ^ { p } \\omega _ { i } ^ { k } \\rho _ { i } ^ { k } \\Vert U _ { i } ^ { k } x ^ { k } - x ^ { k } \\Vert ^ { 2 } \\leq \\Vert U _ { k } x ^ { k } - x ^ { k } \\Vert \\end{align*}"}
-{"id": "7556.png", "formula": "\\begin{align*} \\sup _ { 0 \\leq s \\leq t \\leq T } E \\left [ \\left | J _ { s , t } ^ m - J _ { s , t } \\right | ^ p \\right ] ^ { 1 / p } = O ( m ^ { 1 / 2 } ) \\end{align*}"}
-{"id": "1575.png", "formula": "\\begin{align*} \\lambda _ { ( n , k ) } = \\prod _ { j = 1 } ^ { n } ( 1 - q ^ { k - 1 + j } r ) ( k + j ) _ q , \\beta _ { ( i , m ' , k ) } = \\prod _ { j = 1 } ^ { i } ( q ^ { m ' + 2 k - j } r - r ^ { - 1 } ) . \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} f _ r ( t ) = \\inf \\{ F ( p ) : p \\in \\Gamma _ r ( t ) \\} . \\end{align*}"}
-{"id": "778.png", "formula": "\\begin{align*} \\frac { | - 1 + z _ { j , n } | } { | z _ { j , n } | } = { \\rm D } \\left ( \\frac { | - 1 + z _ { j , n } | } { | z _ { j , n } | } \\right ) + { \\rm t l } \\left ( \\frac { | - 1 + z _ { j , n } | } { | z _ { j , n } | } \\right ) ~ ~ < ~ ~ \\frac { 1 - \\exp \\bigl ( - \\frac { \\pi } { a _ { j , n } } \\bigr ) } { 2 \\exp \\bigl ( \\frac { \\pi } { a _ { j , n } } \\bigr ) - 1 } \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} h ( q , z ) = \\sum _ { n \\geq 1 } \\frac { ( - 1 ) ^ { n - 1 } P _ { n } ( q ) } { n q ^ { \\binom { n } { 2 } } ( q ^ n - 1 ) } z ^ n , \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} \\| \\{ a _ k \\} _ k \\| _ { \\ell ^ p _ \\beta } = \\bigg ( \\sum _ k 2 ^ { k \\beta } | a _ k | ^ p \\bigg ) ^ { 1 / p } ( 1 \\leq p < \\infty ) \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{align*} \\mathbf { \\Lambda } = \\mbox { d i a g } ( \\lambda _ 1 , . . . , \\lambda _ { 2 N } ) , \\mathbf { V } = \\left [ \\mathbf { v } _ 1 , . . . , \\mathbf { v } _ { 2 N } \\right ] , \\mathbf { v } _ j = \\left ( \\begin{array} { c } \\mathbf { e } _ j \\\\ \\lambda _ j \\mathbf { e } _ j \\end{array} \\right ) , \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} ( \\nabla _ t \\xi ) ( X _ 1 , \\ldots , X _ k ) = \\nabla ^ V _ { \\partial _ t } ( \\xi ( X _ 1 , \\ldots , X _ k ) ) - \\sum _ { i = 1 } ^ k \\xi ( X _ 1 , \\ldots , \\nabla _ { \\partial _ t } X _ i , \\ldots , X _ k ) . \\end{align*}"}
-{"id": "8201.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\beta _ i & = ( A + 1 ) p + q + ( A + 2 - m ) ( n - p - 1 ) \\\\ & = p ( m - 1 ) + \\big ( q - ( A + 2 - m ) \\big ) + n ( A + 2 - m ) , \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} [ \\alpha _ i , \\alpha _ k ] = \\sum _ { j = 1 } ^ d s ^ i _ { j , k } \\alpha _ j \\ , . \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{align*} w _ b = \\lambda c z ^ { \\lambda - 1 } + \\sum _ { \\mu < \\lambda } \\mu c _ \\mu z ^ { \\mu - 1 } . \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} C ^ T f = O ^ T W ^ T f = O ^ T u ^ f ( \\cdot , T ) . \\end{align*}"}
-{"id": "6177.png", "formula": "\\begin{align*} \\mathrm { i n v } ( \\sigma , s _ 0 , t _ 0 ) = | \\lbrace ( s , t ) \\in \\sigma \\mid s < s _ 0 , t > t _ 0 \\rbrace | = | \\sigma \\cap B _ \\mu ( s _ 0 , t _ 0 ) | - 1 . \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} \\begin{aligned} u ( t , x ) & = \\int _ { \\mathbb { R } } G _ t ( x - y ) u _ 0 ( y ) d y + \\frac { 1 } { 2 } \\int _ 0 ^ t \\int _ { \\mathbb { R } } \\frac { \\partial } { \\partial y } G _ { t - s } ( x - y ) u ( s , y ) ^ 2 d y d s \\\\ & + \\int _ 0 ^ t \\int _ { \\mathbb { R } } G _ { t - s } ( x - y ) \\sigma _ s ( y ) W ( d s , d y ) , \\end{aligned} \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{align*} \\begin{pmatrix} a & b & e \\\\ b & a & f \\\\ 0 & 0 & g \\end{pmatrix} . \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} \\mathbb { E } _ { n , d ( n ) , 0 } ( \\varphi _ n ) - d ( n ) ^ { - 1 } \\sum _ { i = 1 } ^ { d ( n ) } \\mathbb { E } _ { n , d ( n ) , \\theta _ { i , n } } ( \\varphi _ n ) \\to 0 n \\to \\infty . \\end{align*}"}
-{"id": "316.png", "formula": "\\begin{align*} \\chi _ \\rho ( C _ { ( \\infty , \\theta _ \\pm , f ) } ) = \\{ z = \\pm w ^ { 1 / 2 } \\} = C _ { ( \\infty , \\eta , g _ \\pm ) } , \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} L _ i [ \\psi ( \\cdot , \\mu ) ] ( \\nu ) = M _ i [ \\psi ( \\nu , \\cdot ) ] ( \\mu ) , \\qquad \\forall \\ 1 \\leq i \\leq n - 1 , \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} \\alpha \\cdot ( n \\otimes n ' ) = \\alpha \\cdot n \\otimes n ' + n \\otimes \\alpha \\cdot n ' \\ , , \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} \\gcd ( f _ { 1 } , \\ldots , f _ { i } ) ( n ) \\big ( g _ { 1 } ( n ) G _ { 1 } + \\ldots + g _ { i } ( n ) G _ { i } \\big ) = 0 \\mod v . \\end{align*}"}
-{"id": "8743.png", "formula": "\\begin{align*} B ^ { s } _ { q , p } ( \\Omega ) = ( L ^ { q } ( \\Omega ) , W ^ { k , q } ( \\Omega ) ) _ { s / k , p } . \\end{align*}"}
-{"id": "1655.png", "formula": "\\begin{align*} \\Phi ( a , b ) & = \\sum x p ( x | a ) ( \\log a - \\log b ) + \\sum ( 1 - x ) p ( x | a ) \\{ \\log ( 1 - a ) - \\log ( 1 - b ) \\} \\\\ & = a ( \\log a - \\log b ) + ( 1 - a ) \\{ \\log ( 1 - a ) - \\log ( 1 - b ) \\} . \\end{align*}"}
-{"id": "4819.png", "formula": "\\begin{align*} Y _ t ^ i = f ^ i ( t , X _ t ^ 1 , \\dotsc , X _ t ^ n ) \\end{align*}"}
-{"id": "1352.png", "formula": "\\begin{align*} c _ { \\ell , n + 1 } = \\sum _ { i = 0 } ^ { \\ell - 1 } c _ { i n } + c _ { \\ell , n } = c _ { \\ell - 1 , n + 1 } + c _ { \\ell , n } = \\binom { n + \\ell - 1 } { \\ell - 1 } + \\binom { n + \\ell - 1 } { \\ell } = \\binom { n + \\ell } { \\ell } , \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{align*} \\left \\{ \\ \\begin{array} { l } \\displaystyle - \\Delta _ \\Phi u = f ( x ) , ~ \\mbox { i n } ~ \\Omega , \\\\ \\\\ u = 0 ~ \\mbox { o n } ~ \\partial \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{align*} & \\mu \\left ( \\{ v \\in T ^ 1 S : \\exists \\ , t \\in [ 0 , T _ k ] \\textrm { w i t h } \\delta ( \\phi _ t ( v ) ) = T _ k ^ { - \\xi } \\} \\right ) \\\\ & = O \\left ( \\frac { 1 } { T _ k ^ { \\xi r - 1 } } \\right ) . \\end{align*}"}
-{"id": "4464.png", "formula": "\\begin{align*} b ( z _ 1 ) + \\frac { ( z _ 2 - 1 ) ^ 2 } { 4 } = \\left ( z _ 1 - \\frac { z _ 2 - 1 } { 2 } \\right ) ^ 2 + ( b ( z _ 2 ) - z _ 1 ) . \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} \\Theta ^ { ( M ) } _ { k } : = \\frac { \\Theta ^ { ( M ) } } { \\Theta ^ { ( M - k ) } } , \\ \\ k = 0 , \\ldots , M - 1 ; \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{align*} M _ i \\left [ \\psi ( \\nu , \\cdot ) \\right ] ( \\mu ) = \\Big ( t ^ { \\theta _ i ( s _ i \\mu ) } \\psi ( \\nu , s _ i \\mu ) - t ^ { \\theta _ i ( \\mu ) } \\psi ( \\nu , \\mu ) \\Big ) . \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} \\mathcal { L } = - \\int _ 1 ^ { \\xi } \\frac { d \\hat { X } } { 2 \\sqrt { \\hat { X } ( \\hat { X } - \\lambda ) } } - \\mathcal { R } - \\mathcal { R } _ \\varphi + z \\pi i / \\omega _ 1 - \\pi i . \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} \\norm { \\nabla f } _ { L ^ 2 ( \\R ^ 2 _ { + } ) } ^ 2 & = \\int _ { - \\infty } ^ { \\infty } | \\xi _ { 1 } | \\ , | \\hat { g } ( \\xi _ { 1 } ) | ^ { 2 } \\ , d \\xi _ { 1 } = \\norm { \\abs { \\partial _ x } ^ { 1 / 2 } g } _ { L ^ 2 ( \\R ) } ^ { 2 } . \\end{align*}"}
-{"id": "8068.png", "formula": "\\begin{align*} L ( \\xi , \\sigma ) ^ { s } K _ s ( y L ( \\xi , \\sigma ) ) = \\frac { y ^ s } { 2 ^ { s + 1 } } \\int _ 0 ^ \\infty \\tau ^ { - ( 1 + s ) } e ^ { - \\frac { y ^ 2 } { 4 \\tau } } e ^ { - L ( \\xi , \\sigma ) ^ 2 \\tau } d \\tau . \\end{align*}"}
-{"id": "3201.png", "formula": "\\begin{align*} \\rho = - \\eta \\wedge \\tau _ 1 + \\tfrac { 1 } { r } d r \\wedge \\tau _ 2 + O ( r ^ { - 3 - \\mu } ) \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} S _ { p + m _ 1 } ( \\delta ^ + ) _ { m _ 2 - r + 1 } = 2 m _ 1 + 2 p + m _ 2 - r + 1 . \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{align*} 0 \\le \\psi _ { 2 i } \\le \\phi _ { 2 i } \\ 1 _ { H _ { 2 i - 1 } } = \\phi _ { 2 i - 1 } \\ 1 _ { H _ { 2 i - 1 } } \\le \\psi _ { 2 i - 1 } \\le \\phi _ { 2 i - 1 } = \\gamma _ i , \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\Lambda _ n ( E , \\zeta ^ { ( n ) } ) ^ { 1 / n } = 1 . \\end{align*}"}
-{"id": "7659.png", "formula": "\\begin{align*} \\alpha _ 1 ^ 2 = \\min \\left \\{ 1 , \\frac { \\epsilon _ 1 \\left ( \\rho \\frac { 1 } { { L \\left ( | | x _ t - x _ 0 | | \\right ) } } + 1 \\right ) } { \\rho ( 1 + \\epsilon _ 1 ) \\frac { 1 } { { L \\left ( | | x _ t - x _ 0 | | \\right ) } } } \\right \\} , \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} T _ { \\varphi , \\overline { \\mathcal { O } ( 1 ) } , X _ 0 } \\le \\sum _ { j = 1 } ^ n T _ { f _ j } + O ( 1 ) \\end{align*}"}
-{"id": "3525.png", "formula": "\\begin{align*} \\partial _ { t } L & = \\int _ { S ^ { 1 } } \\langle \\partial _ { x t } \\gamma , T \\rangle d u \\\\ & = \\int _ { S ^ { 1 } } \\langle \\partial _ { t x } \\gamma , T \\rangle d u \\\\ & = \\int _ { S ^ { 1 } } \\langle \\partial _ { x } ( \\kappa N ) , T \\rangle d x \\end{align*}"}
-{"id": "1794.png", "formula": "\\begin{align*} ( \\kappa \\cup \\kappa ' ) ( x ) : = \\begin{cases} \\kappa ( x ) & x \\in U , \\\\ \\kappa ' ( x ) & x \\in U ' , \\\\ \\end{cases} \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} x ^ A _ n ( 0 ) = - 2 n , v _ n = 1 , n \\geq 1 , \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{align*} ( t ) = \\sum _ { a ^ q + a = 0 , b ^ q + b = 0 } P _ { ( a , b , 0 ) } - q ^ 2 P _ \\infty , \\end{align*}"}
-{"id": "9206.png", "formula": "\\begin{align*} I ^ { \\mathcal { R } } _ i ( G , X _ G ) = \\langle { \\rm m i n o r s } _ i ( L ( G , X _ G ) ) \\rangle \\subseteq \\mathcal { R } [ X _ G ] 1 \\leq i \\leq n , \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{align*} 1 = g ^ \\# ( 0 ) = \\frac { | g ' ( 0 ) | } { 1 + | g ( 0 ) | ^ 2 } = \\frac { | g ( 0 ) | } { 1 + | g ( 0 ) | ^ 2 } | h ' ( 0 ) | \\leq \\frac 1 2 | h ' ( 0 ) | . \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} \\frac { \\prod _ { i = 1 } ^ { [ \\frac { k } { 2 } ] + [ \\frac { n - k } { 2 } ] } ( 1 - t ^ { 4 i } ) } { \\prod _ { i = 1 } ^ { [ \\frac { k } { 2 } ] } ( 1 - t ^ { 4 i } ) \\prod _ { i = 1 } ^ { [ \\frac { n - k } { 2 } ] } ( 1 - t ^ { 4 i } ) } = \\sum _ { d \\geq 0 } p \\Big ( \\big [ \\frac { k } { 2 } \\big ] , \\big [ \\frac { n - k } { 2 } \\big ] ; d \\Big ) t ^ { 4 d } \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} y _ t = y _ 0 + \\int _ 0 ^ t f ( y _ s ) ^ \\ell d x _ s \\ . \\\\ \\end{align*}"}
-{"id": "225.png", "formula": "\\begin{align*} \\Gamma ( E ^ \\# , \\mathcal O _ { E ^ \\# } ( * \\pi ^ { - 1 } ( 0 ) ) ) \\simeq \\mathbf k [ ( E - \\{ 0 \\} ) \\times \\mathbb A ^ 1 ] = \\mathbf k [ x , y , \\tilde c ] / ( y ^ 2 = 4 x ^ 3 - g _ 2 x - g _ 3 ) . \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} d q _ t ^ m = & \\frac { 1 } { m } ( p _ t ^ m - \\psi ( t , q _ t ^ m ) ) d t , \\\\ d ( p _ t ^ m ) _ i = & \\left ( - \\frac { 1 } { m } \\gamma _ { i j } ( t , q _ t ^ m ) \\delta ^ { j k } ( ( p _ t ^ m ) _ k - \\psi _ k ( t , q _ t ^ m ) ) + \\tilde F _ i ( t , x _ t ^ m ) - \\partial _ { q ^ i } V ( t , q _ t ^ m ) \\right . \\\\ & \\left . + \\frac { 1 } { m } \\partial _ { q ^ i } \\psi _ k ( t , q _ t ^ m ) \\delta ^ { j k } ( ( p _ t ^ m ) _ j - \\psi _ j ( t , q _ t ^ m ) ) \\right ) d t + \\sigma _ { i \\rho } ( t , q _ t ^ m ) d W ^ \\rho _ t . \\end{align*}"}
-{"id": "2279.png", "formula": "\\begin{align*} \\epsilon ( \\delta ( \\gamma ( 1 \\otimes x ^ l ) ) ) & = \\epsilon ( \\delta ( 1 \\otimes x ^ l d x ) ) = \\epsilon ( x ^ l d x ) = - x ^ l d x . \\end{align*}"}
-{"id": "2633.png", "formula": "\\begin{align*} X _ 1 ( \\mu _ 1 ) = c X _ 1 ( h ) \\mbox { i n } D _ 1 \\mbox { a n d \\quad } U _ 1 ( \\mu _ 2 ) = - c U _ 1 ( \\varphi ) \\mbox { i n } G _ 1 . \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} U ( d _ 1 , d _ 2 ) U ( e _ 1 , e _ 2 ) = \\begin{pmatrix} 1 & d _ 1 + e _ 1 & d _ 2 + e _ 2 & - d _ 1 d _ 2 - d _ 1 e _ 2 - d _ 2 e _ 1 - e _ 1 e _ 2 \\\\ 0 & 1 & 0 & - d _ 2 - e _ 2 \\\\ 0 & 0 & 1 & - d _ 1 - e _ 1 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "6127.png", "formula": "\\begin{align*} F ( t ) = \\int _ 0 ^ { \\rho ( t ) } s ^ { - \\frac { 3 } { 2 } } e ^ { - \\frac { C } { s } } d s . \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} \\begin{aligned} [ b ] & | \\hat { S } _ 1 | - | \\hat { S } _ 2 | - \\frac { 2 E _ { \\hat { S } _ 1 } + E _ { \\hat { S } _ 1 , \\hat { S } _ 3 } + E _ { \\hat { S } _ 1 , V _ 1 } - E _ { \\hat { S } _ 2 , V _ 1 } } { \\tau - 1 } \\\\ & < | S _ 1 ^ * | - | S _ 2 ^ * | - \\frac { 2 E _ { S _ 1 ^ * } + E _ { S _ 1 ^ * , S _ 3 ^ * } + E _ { S _ 1 ^ * , V _ 1 } - E _ { S _ 2 ^ * , V _ 1 } } { \\tau - 1 } . \\end{aligned} \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & 0 & 5 & 2 \\\\ 7 & 6 & 9 & 8 \\\\ 0 & 0 & 8 & 9 \\\\ 2 & 6 & 1 & 8 \\end{pmatrix} , \\begin{pmatrix} 9 & 5 & 2 & 5 \\\\ 1 & 3 & 3 & 2 \\\\ 8 & 5 & 3 & 5 \\\\ 4 & 8 & 1 & 9 \\end{pmatrix} , \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{align*} \\alpha _ { k } : = \\max _ { u \\in H _ { 0 } ^ { 1 } ( \\Omega ) , \\| u \\| \\leq t _ { k } ^ { 1 / 2 } } \\int _ { \\Omega } F _ { \\ast } ( u ) d x , k \\in \\{ 1 , \\ldots K \\} . \\end{align*}"}
-{"id": "8278.png", "formula": "\\begin{align*} \\rho ( \\lambda ) = \\begin{cases} d ! & \\lambda = [ 1 ^ d ] \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "3454.png", "formula": "\\begin{align*} B _ 0 f = \\operatorname { s g n } ( \\cdot ) \\bigl ( - f '' \\bigr ) , \\operatorname { d o m } B _ 0 = \\bigl \\{ f \\in L ^ 1 ( \\mathbb R ) : f , f ' \\in A C ( \\mathbb R ) , - f '' \\in L ^ 1 ( \\mathbb R ) \\bigr \\} , \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} \\zeta _ i = \\begin{cases} 1 & d _ i = 1 , \\\\ d _ { i , S _ 1 ^ * } + d _ { i , S _ 3 ^ * } + d _ { i , V _ 1 } & i \\in S _ 1 ^ * , \\\\ d _ { i , V _ 1 } & i \\in S _ 2 ^ * , \\\\ d _ { i , S _ 1 ^ * } + d _ { i , V _ 1 } & i \\in S _ 3 ^ * . \\end{cases} \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} \\mathbb { P } ( W _ i ( n ) = 1 ) > 0 . \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} \\left ( c - | \\nabla _ B h | ^ 2 \\right ) h ^ { - 2 } = a , \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{align*} Q [ f ] : = f - \\left ( \\int _ { \\Omega } f \\phi _ { 1 } \\right ) \\phi _ { 1 } . \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} q ^ { 2 b } \\binom { d - 4 b / 3 + 3 } { 3 } _ q + \\sum _ { 0 \\le i \\le \\frac { b - 3 } { 3 } } q ^ { 6 i + 2 } \\binom { d - 4 i - 1 } { 1 } _ q \\binom { 2 d - 8 i - 1 } { 1 } _ q + q ^ { 6 i } \\binom { 3 d - 1 2 i + 1 } { 1 } _ q . \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} P _ c d = \\overline { d } \\hbox { a n d } P _ s d = d _ s . \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} p ( x | a ) & : = a ^ x ( 1 - a ) ^ { 1 - x } , \\\\ \\Phi ( a , b ) & : = \\sum ^ { 1 } _ { x = 0 } p ( x | a ) \\mathrm { l o g } \\frac { p ( x | a ) } { p ( x | b ) } . \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} | p | _ { C ^ \\tau S ^ { - N } _ { 1 , 0 } } ^ { i } : = \\underset { | \\alpha | \\leq i } { \\max } \\ \\underset { \\xi \\in { \\R } ^ n } { \\sup } \\left \\lVert \\partial _ { \\xi } ^ \\alpha p ( \\cdot , \\xi ) \\right \\lVert _ { C ^ \\tau ( { \\R } ^ n ; X ) } \\langle \\xi \\rangle ^ { N + | \\alpha | } < \\infty . \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} \\sharp \\phi : = \\bar * \\bar \\phi . \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} \\begin{array} [ p o s ] { l l l } \\varphi ( x _ { 1 } , \\ldots , x _ { n } ) & = & - ( b / 2 ) \\sum _ { j = 1 } ^ { n } \\varepsilon _ { j } x _ { j } ^ { 2 } + \\langle A _ { \\varepsilon } , x \\rangle _ { \\varepsilon } + A _ { n + 1 } \\end{array} \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{align*} F ( x , c ) = f ( x ) + \\frac { 1 } { p ( x ) } \\Phi ( x , c ) + \\langle \\mu ( x ) , h ( x ) \\rangle + \\frac { c } { 2 q ( x ) } \\| h ( x ) \\| ^ 2 \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} H ^ 0 _ { R + } ( L ^ I ( M ) ) \\cong \\bigoplus ^ { \\rho ^ { I } ( M ) - 1 } _ { i = 0 } ~ \\dfrac { \\widetilde { I ^ { i + 1 } M } } { I ^ { i + 1 } M } . \\end{align*}"}
-{"id": "6982.png", "formula": "\\begin{align*} \\Phi ^ - [ T ] = \\{ t _ 3 - t _ 1 , t _ 4 - t _ 1 , t _ 4 - t _ 3 \\} \\end{align*}"}
-{"id": "3930.png", "formula": "\\begin{align*} Q ( u _ n ) = \\frac 1 { n ^ 2 } . \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} \\tilde { A } ^ Q ( h ) & = H _ K ( A ^ Q ( h ) ^ T ) = H _ K \\big ( Q _ K ^ { { 1 } / { 2 } } \\otimes Q _ K ^ { { 1 } / { 2 } } \\big ) ( A ^ I ( h ) ^ T ) \\\\ & = H _ K \\big ( Q _ K ^ { { 1 } / { 2 } } \\otimes Q _ K ^ { { 1 } / { 2 } } \\big ) H _ K ^ T H _ K ( A ^ I ( h ) ^ T ) \\\\ & = H _ K \\big ( Q _ K ^ { { 1 } / { 2 } } \\otimes Q _ K ^ { { 1 } / { 2 } } \\big ) H _ K ^ T \\tilde { A } ^ I ( h ) , \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} v _ 1 v _ 2 \\dots v _ r = u _ 1 w _ 1 u _ 2 w _ 2 \\dots u _ r w _ r \\cdot \\Big ( \\prod _ { j = 1 } ^ i u _ j \\Big ) ^ { - 1 } \\end{align*}"}
-{"id": "9144.png", "formula": "\\begin{align*} b ( h ) = \\dfrac { ( h ^ 3 - 2 ) ( h ^ 9 - 6 h ^ 6 - 1 2 h ^ 3 - 8 ) } { h ^ { 1 2 } - 8 h ^ 9 - 8 h ^ 3 \u2013 8 } , \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} \\Delta \\sigma = \\sum \\sigma _ { i } \\otimes \\sigma ^ { ' } _ { n - i } \\end{align*}"}
-{"id": "4417.png", "formula": "\\begin{align*} W & = \\sum _ { j \\in J } \\Bigl ( \\frac { 1 } { 3 } ( z _ { j _ 1 , 1 } ^ 2 - z _ { { j _ 2 } , 1 } ^ 2 ) ( z _ { j _ 1 , 2 } ^ 2 - z _ { { j _ 2 } , 2 } ^ 2 ) - ( z _ { j _ 1 , 1 } ^ 2 - z _ { { j _ 2 } , 2 } ^ 2 ) ( z _ { j _ 1 , 2 } ^ 2 - z _ { { j _ 2 } , 1 } ^ 2 ) \\Bigr ) , \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{align*} \\frac { \\Gamma ^ 2 n _ t \\sigma ^ 2 } { N } = \\frac { \\Gamma n _ t } { t } \\Delta t \\frac { \\Gamma \\sigma ^ 2 t } { N \\Delta t } \\simeq \\frac { \\Gamma \\sigma ^ 2 } { N \\Delta t } t . \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} V = \\int _ 0 ^ \\infty \\frac { s ^ 2 + 1 } { s ^ 2 } \\ , d \\rho ( s ) . \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} \\int _ { C _ W } e ^ { - \\ < x , y \\ > } f ( x , \\theta ) d x = \\frac { 1 } { 2 ^ n \\sqrt { y _ 1 ( y _ 1 + \\theta _ 1 ) } \\ldots \\sqrt { y _ n ( y _ n + \\theta _ n ) } } e ^ { - \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ n w _ { i j } \\sqrt { y _ i y _ j } } . \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{align*} \\mathcal { O } _ { 1 i } & = \\{ ( t , \\ell ) \\colon \\ell \\in \\mathcal { U } , \\ s _ { 1 \\ , i - 1 } ( \\ell ) \\leq t \\leq s _ { 1 i } ( \\ell ) \\} i = 1 , \\ldots , n _ 1 + 1 , \\\\ \\mathcal { O } _ 2 & = \\{ ( t , \\ell ) \\colon \\ell \\in \\mathcal { U } , \\ \\tau _ 1 ( \\ell ) \\leq t \\leq \\tau _ 2 ( \\ell ) \\} , \\\\ \\mathcal { O } _ { 3 i } & = \\{ ( t , \\ell ) \\colon \\ell \\in \\mathcal { U } , \\ s _ { 3 \\ , i - 1 } ( \\ell ) \\leq t \\leq s _ { 3 i } ( \\ell ) \\} i = 1 , \\ldots , n _ 3 + 1 , \\end{align*}"}
-{"id": "1348.png", "formula": "\\begin{align*} D _ { m k } = \\begin{bmatrix} 1 & 1 & \\ldots & 1 \\\\ 1 & 1 & \\ldots & 1 \\\\ \\hdotsfor { 4 } \\\\ 1 & 1 & \\ldots & 1 \\end{bmatrix} , C _ { k k } = \\begin{bmatrix} 1 & 1 & \\ldots & 1 \\\\ 0 & 1 & \\ldots & 1 \\\\ \\hdotsfor { 4 } \\\\ 0 & 0 & \\ldots & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} \\overline { X } _ 1 & : = H _ 0 ^ 1 ( 0 , L ) \\times H ^ 1 _ 0 ( 0 , L ) , \\\\ \\overline { X } _ 2 & : = \\left \\lbrace ( \\eta , w ) \\in [ H ^ 2 ( 0 , L ) \\cap H ^ 1 _ 0 ( 0 , L ) ] ^ 2 : \\eta _ x ( 0 ) = w _ x ( L ) = 0 \\right \\rbrace . \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} \\mathcal { T } _ { K } ^ { i , 2 } ( y , h ) : = \\Big \\{ w \\in \\mathbb { Y } \\ ; | \\ ; { \\rm d i s t } ( y + t h + \\frac { 1 } { 2 } t ^ 2 w , K ) = o ( t ^ 2 ) , t \\geq 0 \\Big \\} , \\qquad \\\\ \\mathcal { T } _ { K } ^ { 2 } ( y , h ) : = \\Big \\{ w \\in \\mathbb { Y } \\ ; | \\ ; \\exists \\ ; t _ n \\downarrow 0 { \\ ; \\rm s u c h \\ ; t h a t \\ ; } { \\rm d i s t } ( y + t _ n h + \\frac { 1 } { 2 } t _ n ^ 2 w , K ) = o ( t _ n ^ 2 ) \\Big \\} . \\end{align*}"}
-{"id": "7621.png", "formula": "\\begin{align*} - \\frac { u '' } { ( 1 - ( u ' ) ^ 2 ) ^ \\frac 3 2 } - \\frac { N - 1 } r \\frac { u ' } { \\sqrt { 1 - ( u ' ) ^ 2 } } = g ( u ) , \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{gather*} [ \\rho _ c ( a ^ * ) C ( \\cdot , x ) ] ( n ) = c C ( n , x ) - x C ( n , x - 1 ) = [ \\rho _ c ( Z - a ) C ( n , \\cdot ) ] ( x ) , \\\\ [ \\rho _ c ( ( a ^ \\dagger ) ^ * ) C ( \\cdot , x ) ] ( n ) = c C ( n , x ) - c C ( n , x + 1 ) = [ \\rho _ c ( Z - a ^ \\dagger ) C ( n , \\cdot ) ] ( x ) . \\end{gather*}"}
-{"id": "3229.png", "formula": "\\begin{align*} d \\alpha _ 3 + ( \\lambda + n - k - 2 ) \\beta _ 3 = 0 = d ^ \\ast \\beta _ 3 + ( \\lambda + k - 2 ) \\alpha _ 3 . \\end{align*}"}
-{"id": "3124.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\ln ( 1 - e ^ { - \\alpha ^ * } e ^ { - x } ) d x = \\int _ 0 ^ { \\beta ^ * N } \\frac { t \\ , d t } { 1 - e ^ { t } } \\end{align*}"}
-{"id": "885.png", "formula": "\\begin{align*} u ( x , t ) = S ( t ) u _ * ( x ) + i \\int _ 1 ^ t S ( t - s ) u ( s ) \\Big ( e ^ { - s } v _ * ( x ) + \\int _ 1 ^ s e ^ { - ( s - s ' ) } | u ( s ' ) | ^ 2 d s ' \\Big ) d s . \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n f ( \\hat { x } ^ { ( 1 ) } _ k , \\dots , \\hat { x } ^ { ( d ) } _ k ) \\leq \\sum _ { k = 1 } ^ n f ( \\tilde { x } ^ { ( 1 ) } _ k , \\dots , \\tilde { x } ^ { ( d ) } _ k ) . \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{align*} \\left ( f ^ { \\prime \\prime } \\left ( a \\right ) \\cdot h \\right ) \\cdot k & = \\sum _ { j , l = 1 } ^ { n } \\frac { \\partial ^ { 2 } f } { \\partial x _ { j } \\partial x _ { l } } \\left ( a \\right ) . h _ { j } . k _ { l } \\\\ & = \\left \\langle f ^ { \\prime \\prime } \\left ( a \\right ) \\cdot h \\mid k \\right \\rangle \\\\ & = \\left \\langle f ^ { \\prime \\prime } \\left ( a \\right ) \\mid h \\otimes k \\right \\rangle _ { \\otimes } . \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} v ( n ) = \\int _ D \\psi _ 1 ( n _ 1 ; \\lambda _ 1 ) \\cdots \\psi _ g ( n _ g ; \\lambda _ g ) \\ , h ( \\lambda _ 2 , \\dots , \\lambda _ g ) \\ , d \\lambda _ 2 \\cdots d \\lambda _ g \\end{align*}"}
-{"id": "4636.png", "formula": "\\begin{align*} \\sum _ { a , b } \\omega ^ a \\wedge V _ b \\lrcorner R ^ Q ( V _ b , \\bar V _ a ) \\phi = \\sum _ a R ^ Q ( V _ a , \\bar V _ a ) \\phi . \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { \\frac { \\omega t } { 2 } } t ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\norm { e ^ { - t A } a } _ { L ^ { q } _ { \\sigma } ( \\Omega ) } & \\leq C e ^ { - \\frac { \\omega t } { 2 } } \\norm { a } _ { L ^ { p } _ { \\sigma } ( \\Omega ) } \\leq C \\norm { a } _ { L ^ { p } _ { \\sigma } ( \\Omega ) } , \\end{aligned} \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{align*} \\begin{array} { l } \\mathcal { V } _ 1 = \\displaystyle { \\frac { - 2 + 3 S _ 1 + 9 S _ 2 } { 3 ( - 2 + 3 S _ 1 + 9 S _ 2 ) + 2 7 ( S _ 1 - S _ 2 ) S _ 2 \\cos ( 2 \\theta ) } } , \\\\ \\\\ \\mathcal { V } _ 2 = \\displaystyle { \\frac { 8 - 2 7 S _ 1 ^ 2 + 9 ( 8 - 3 S _ 2 ) S _ 2 + 6 S _ 1 ( 4 + 9 S _ 2 ) + 2 7 ( S _ 1 - S _ 2 ) ^ 2 \\cos ( 4 \\theta ) } { 2 8 8 \\left ( 1 + 3 S _ 1 - 9 S _ 1 \\cos ^ 2 ( \\theta ) - 9 S _ 2 \\sin ^ 2 ( \\theta ) \\right ) ^ 2 } } . \\end{array} \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} \\int _ { U ( N ; \\alpha ) } ^ { \\infty } e ^ { - x } ( \\ln x ) ^ k d x = O \\left ( e ^ { - U ( N ; \\alpha ) } ( \\ln N ) ^ { \\alpha k } \\right ) \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} \\big | \\tilde { M } \\big | = \\bigl | \\tilde { \\tilde { M } } \\bigl | , \\ \\ \\ \\big \\| \\tilde { M } \\big \\| = \\Bigl \\| \\tilde { \\tilde { M } } \\Bigl \\| , \\end{align*}"}
-{"id": "2353.png", "formula": "\\begin{align*} E \\left [ S _ N ^ { ( r ) } \\right ] = N ^ r ( \\ln N ) ^ r \\sum _ { k = 0 } ^ n \\binom { r } { k } \\frac { ( - 1 ) ^ k } { \\ln ^ k N } \\int _ 0 ^ { \\infty } e ^ { - x } ( \\ln x ) ^ k d x + o \\left ( \\frac { 1 } { \\ln ^ n N } \\right ) \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{align*} r & = \\frac { 1 } { \\sqrt { 2 } } ( x + y ) \\\\ s & = \\frac { 1 } { \\sqrt { 2 } } ( x - y ) \\\\ \\phi & = \\arctan ( s / r ) = \\pi / 4 + \\theta , \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} \\forall 1 \\leq j \\leq N _ s , A _ q ^ { V , s } u _ { q , j } ^ { V , s } = \\varepsilon ^ { V , s } _ { q , j } u _ { q , j } ^ { V , s } . \\end{align*}"}
-{"id": "8331.png", "formula": "\\begin{align*} i \\mathfrak { H } ( \\alpha , t ) = \\frac { 1 } { \\pi } \\operatorname { p . v . } \\int f ( \\beta , t ) \\frac { z _ \\beta ( \\beta , t ) } { z ( \\alpha , t ) - z ( \\beta , t ) } d \\beta . \\end{align*}"}
-{"id": "4688.png", "formula": "\\begin{align*} \\chi ( G \\Box H ) = \\max \\{ \\chi ( G ) , \\chi ( H ) \\} \\ ; . \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} V ( x ) : = \\left ( \\frac { k - p } p \\right ) ^ p | y | ^ { - p } . \\end{align*}"}
-{"id": "2781.png", "formula": "\\begin{align*} v \\leq \\sum _ { \\ell = 1 } ^ v \\left ( \\frac { v ^ * } { S _ { \\ell - 1 } } \\right ) \\cdot E _ { \\ell } = \\sum _ { \\ell = 1 } ^ v \\left ( \\frac { E _ { \\ell } } { Q - Q _ { \\ell - 1 } } \\right ) \\cdot v ^ * . \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} \\begin{cases} U _ { t } ( \\cdot , \\cdot ; \\varepsilon ) = \\mathcal { A } _ { \\varepsilon } ( U ) ( \\cdot , \\cdot ; \\varepsilon ) , \\ t > 0 , \\ x \\in \\R ^ N \\cr U ( \\cdot , 0 ; \\varepsilon ) = u ( \\cdot , T _ { \\varepsilon } + t _ 0 ; t _ 0 ) , \\end{cases} \\end{align*}"}
-{"id": "1703.png", "formula": "\\begin{align*} n ! ( \\alpha n ^ { 1 / \\ell } ) ^ n \\prod _ { i = 2 } ^ n M _ i \\leq 2 ^ { n ^ { 1 + 1 / \\ell } } 2 ^ { \\left ( 3 \\ell + 2 \\right ) n ^ { 1 + 1 / \\ell } } \\leq 2 ^ { 3 ( \\ell + 1 ) n ^ { 1 + 1 / \\ell } } \\end{align*}"}
-{"id": "8659.png", "formula": "\\begin{align*} \\tilde \\rho _ I = ( t - r ) / r , \\rho _ + = ( t - r ) ^ { - 1 } . \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{align*} m ^ 1 _ { x = V } = \\int _ { ( \\max \\{ 0 , \\ , ( \\tau _ 1 ) ^ { - 1 } ( T ) \\} , \\ , L _ 1 ] } \\delta _ { \\tau _ 1 ( x ) } \\ , d m _ 0 ^ 1 ( x ) + \\int _ { [ 0 , \\ , \\max \\{ 0 , \\ , ( \\varsigma _ 1 ) ^ { - 1 } ( T ) \\} ] } \\delta _ { \\varsigma _ 1 ( s ) } \\ , d \\sigma _ 0 ( s ) , \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} \\bar \\partial _ B ^ * \\bar \\partial _ B \\phi = & - \\sum _ a \\nabla _ { V _ a } \\nabla _ { \\bar V _ a } \\phi + \\nabla _ { H ^ { 0 , 1 } } \\phi + \\sum _ { a , b } \\bar \\omega ^ a \\wedge \\bar V _ b \\ , \\lrcorner \\ , \\nabla _ { V _ b } \\nabla _ { \\bar V _ a } \\phi \\\\ & - \\sum _ b \\bar \\omega ^ a \\wedge H ^ { 0 , 1 } \\ , \\lrcorner \\ , \\nabla _ { \\bar V _ a } \\phi . \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{align*} N ( \\alpha + q - 1 ) > N \\left ( - \\frac { 1 } { N } + \\sigma _ { 0 } - \\frac { \\sigma _ { 0 } } { 2 } \\right ) = - 1 + \\frac { \\sigma _ { 0 } N } { 2 } . \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} D _ \\lambda ( x , y ) = \\frac { 1 } { 2 \\alpha \\sqrt { \\lambda } } \\begin{cases} \\overline { \\alpha } e ^ { i \\sqrt { \\lambda } | x - y | } , & x \\geq 0 , \\ , y \\geq 0 , \\\\ 0 , & x \\geq 0 , \\ , y < 0 , \\\\ 0 , & x < 0 , \\ , y \\geq 0 , \\\\ - \\alpha e ^ { - \\sqrt { \\lambda } | x - y | } , & x < 0 , \\ , y < 0 , \\end{cases} \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u = \\Delta u - \\chi \\nabla \\cdot ( u \\nabla v ) + u ( a ( x , t ) - b ( x , t ) u ) , x \\in \\R ^ N , \\cr 0 = \\Delta v - \\lambda v + \\mu u , x \\in \\R ^ N , \\end{cases} \\end{align*}"}
-{"id": "7700.png", "formula": "\\begin{align*} F _ { r _ t | r _ m } ( y ) = 1 - \\sum ^ { t - m - 1 } _ { n = 0 } ( \\lambda _ c \\pi ) ^ { n } ( y ^ 2 - x ^ 2 ) ^ { n } \\frac { e ^ { - \\lambda _ c \\pi ( y ^ 2 - x ^ 2 ) } } { n ! } . \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} \\mathrm { P } ^ 2 _ { m , 2 } = 1 - \\mathrm { P } ( \\log ( 1 + ^ 1 _ { m , 2 } ) > R _ 1 , \\log ( 1 + ^ 2 _ { m , 2 } ) > R _ 2 ) . \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} A = \\omega _ 1 \\bar { \\omega _ 2 } - \\omega _ 2 \\bar { \\omega _ 1 } . \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} \\int _ { z _ 0 } ^ { \\gamma z _ 0 } \\varphi ( z ) ( d z ) ^ 2 q _ \\gamma ( z ) = \\int _ { z _ 0 } ^ { \\gamma z _ 0 } \\varphi ( z ) ( c z ^ 2 + ( d - a ) z - b ) d z . \\end{align*}"}
-{"id": "810.png", "formula": "\\begin{align*} \\Box _ p = \\Delta _ p + D R _ p , \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{align*} e ( u ) & : = \\sum _ { r \\geq 0 } e _ r u ^ { - r } , & h ( u ) & : = \\sum _ { r \\geq 0 } h _ r u ^ { - r } \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} H _ m = U _ m \\Sigma _ m V _ m ^ T \\ , , \\quad \\mbox { w h e r e } U _ m \\in \\R ^ { ( m + 1 ) \\times m } , \\ ; \\Sigma _ m \\in \\R ^ { m \\times m } , \\ ; V _ m \\in \\R ^ { m \\times m } , \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} \\bar \\partial _ B \\bar \\partial _ B ^ * \\phi = & - \\sum _ { a , b } \\bar \\omega ^ a \\wedge \\bar V _ b \\ , \\lrcorner \\ , \\nabla _ { \\bar V _ a } \\nabla _ { V _ b } \\phi + \\sum _ a \\bar \\omega ^ a \\wedge ( \\nabla _ { \\bar V _ a } H ^ { 0 , 1 } ) \\ , \\lrcorner \\ , \\phi \\\\ & + \\sum _ a \\bar \\omega ^ a \\wedge H ^ { 0 , 1 } \\ , \\lrcorner \\ , \\nabla _ { \\bar V _ a } \\phi . \\end{align*}"}
-{"id": "9015.png", "formula": "\\begin{align*} \\begin{pmatrix} a & z ^ T & 0 \\\\ w & B & z \\\\ 0 & w ^ T & - a \\end{pmatrix} \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} R i c ( \\omega ) & = \\\\ & = - \\sqrt { - 1 } \\partial \\bar { \\partial } \\log \\pi _ * \\Omega _ { ( X , D ) / Y } - \\sqrt { - 1 } \\partial \\bar { \\partial } v - ( 1 - \\beta ) c _ 1 ( [ N ] , h ) + ( 1 - \\beta ) \\{ N \\} \\\\ \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} l _ 1 = \\lim _ { t \\to 0 ^ + } \\frac { \\rho ( t ) } { t } = \\lim _ { t \\to 0 ^ + } \\dot { \\rho } ( t ) . \\end{align*}"}
-{"id": "1313.png", "formula": "\\begin{align*} \\Lambda _ h ^ \\Pi = \\Lambda _ h ^ 0 \\oplus \\widetilde \\Lambda _ h ^ \\Pi \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} f ^ * ( x _ M ^ { k - n } \\times 1 ) = \\pm \\lambda \\cdot ( x _ M ^ { k - n } \\times 1 ) . \\end{align*}"}
-{"id": "5456.png", "formula": "\\begin{align*} \\varphi ( n ) = \\dfrac { 2 \\pi } { m } \\cdot ( f ( n ) ~ ~ m ) . \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} \\sup _ { X \\in A } \\inf _ { \\phantom { . } Q \\in B \\phantom { \\dot I } } f ( X , Q ) = \\inf _ { \\phantom { . } Q \\in B \\phantom { \\dot I } } \\sup _ { X \\in A } f ( X , Q ) \\ . \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{align*} \\psi _ { \\ell } \\ = \\ B ( \\rho ) \\psi _ { \\ell } \\ = \\ V ^ { 1 / 2 } { \\varphi } _ { \\ell } , \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} \\bar { C } ^ + = \\{ p \\in \\mathfrak { a } _ s ^ * ; p ( \\bar { \\alpha } ^ { \\vee } ) \\geq 0 , \\forall \\bar { \\alpha } \\in \\bar { \\Phi } ^ + \\} . \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { N - K + 1 } L _ i = L . \\end{align*}"}
-{"id": "2001.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { B _ { R / 2 } } ( | \\nabla u | ^ { 2 } ) ^ { \\mu } d x & \\le C \\Big ( R ^ { - n } \\int _ { B _ { R } } d ^ { n } ( u , p _ { 0 } ) d x + \\int _ { B _ { R } } g d x \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} & { \\rm d i s t } ( ( x , \\lambda , v ) , \\Phi ^ { - 1 } ( 0 , 0 ) ) = { \\rm d i s t } ( ( x , \\lambda , v ) , \\widetilde { \\Phi } ^ { - 1 } ( 0 , 0 , 0 ) ) \\\\ & \\le \\kappa { \\rm d i s t } ( ( 0 , 0 , 0 ) , \\widetilde { \\Phi } ( x , \\lambda , v ) ) \\le \\| ( \\xi , \\eta , - \\eta ) \\| \\le \\sqrt { 2 } \\kappa \\| \\Phi ( x , \\lambda , v ) \\| \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} E _ { \\tau } ( 1 ) = \\lim _ { k \\to \\infty } \\mathcal { E } _ { \\tau } ( \\rho _ { n _ { k } } ) \\geq E _ { \\tau } ^ { \\infty } ( 1 ) . \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} \\lambda _ { D _ t } \\ge \\max _ { z \\in D _ t } \\{ \\xi ( z ) - \\sigma ( z ) ^ { - 1 } \\} = a _ t + o ( 1 ) \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} { \\rm M } ( P ) \\geq 5 ^ { 1 / 4 } = 1 . 4 9 5 3 \\ldots , \\end{align*}"}
-{"id": "1525.png", "formula": "\\begin{align*} L H + H = \\mathcal { L } H + ( | A | ^ 2 + \\frac { 1 } { 2 } ) H = 0 . \\end{align*}"}
-{"id": "5116.png", "formula": "\\begin{align*} a ( R ) \\ = \\ 1 - t . \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} \\nu ( \\lambda ) = \\frac { 1 } { z _ \\lambda } \\sum _ { k = 0 } ^ { d - 1 } \\frac { \\psi _ d ^ k ( \\lambda ) } { q ^ k } , \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} \\begin{pmatrix} 5 & 7 & 3 & 9 \\\\ 8 & 7 & 7 & 9 \\\\ 1 0 & 3 & 0 & 0 \\\\ 6 & 8 & 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "8223.png", "formula": "\\begin{align*} \\left \\{ \\begin{gathered} | E _ j ^ \\dagger | = | E _ j | \\\\ E _ j \\Delta B _ j \\\\ E _ j ^ \\dagger \\Delta B _ j \\subset \\{ x \\in E _ j \\Delta B _ j : \\big | \\ , | x | - r _ j \\ , \\big | \\leq \\lambda \\delta \\} \\\\ | E _ j ^ \\dagger \\Delta E _ j | \\leq 2 | \\{ x \\in E _ j \\Delta B _ j : \\big | \\ , | x | - r _ j \\ , \\big | > \\lambda \\delta \\} \\end{gathered} \\right . \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { d - 1 } h _ { S + j } ( z ) & = \\sum _ { j = 0 } ^ { d - 1 } \\prod _ { i \\in S } g _ { i + j } ( z ) ( g _ { i + j + 1 } ( z ) - 1 ) \\\\ & = \\sum _ { j = 0 } ^ { d - 1 } z ^ { | S | } \\omega ^ { \\sum _ { i \\in S } ( i + j + 1 ) } \\\\ & = z ^ { | S | } \\omega ^ { \\sum _ { i \\in S } ( i + 1 ) } \\underbrace { \\sum _ { j = 0 } ^ { d - 1 } \\omega ^ { j | S | } } _ { 1 \\leq | S | \\leq \\lfloor d / 2 \\rfloor } \\\\ & = 0 . \\end{align*}"}
-{"id": "3565.png", "formula": "\\begin{align*} \\gamma _ { s t } & = ( \\lvert \\gamma _ { u } \\rvert ^ { - 1 } \\gamma _ { u } ) _ { t } \\\\ & = - \\lvert \\gamma _ { u } \\rvert ^ { - 3 } \\langle \\gamma _ { t u } , \\gamma _ { u } \\rangle \\gamma _ { u } + \\lvert \\gamma _ { u } \\rvert ^ { - 1 } \\gamma _ { t u } \\\\ & = - \\lvert \\gamma _ { u } \\rvert ^ { - 3 } \\langle \\kappa B _ { u } , \\gamma _ { u } \\rangle \\gamma _ { u } + \\gamma _ { t s } \\\\ & = \\kappa \\langle B _ { s } , T \\rangle \\gamma _ { s } + \\gamma _ { t s } \\\\ & = \\gamma _ { t s } , \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{align*} \\varphi ( T , x ) & = \\Psi ( x ) , x \\in \\mathbb { R } ^ d , \\end{align*}"}
-{"id": "6817.png", "formula": "\\begin{align*} D ( R ^ P ( \\psi , \\psi ) ( N s ) ^ { 2 - n } ) \\cdot \\psi = & ( D _ { d \\phi } R ^ P ) ( \\psi , \\psi ) ( N s ) ^ { 2 - n } \\cdot \\psi + 2 R ^ P ( D \\psi , \\psi ) ( N s ) ^ { 2 - n } \\cdot \\psi \\\\ & + s ^ { 2 - n } R ^ P ( \\psi , \\psi ) D ( N ) ^ { 2 - n } \\cdot \\psi . \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} \\frac { g _ { D _ r , p } ( z ) } { g _ { D _ r , 0 } ( z ) } = \\frac { \\log \\left | \\frac { r ^ 2 - \\overline { p } z } { r ( z - p ) } \\right | } { \\log \\left | \\frac { r } { z } \\right | } \\le \\frac { \\log \\frac { R ^ 2 + r _ 0 r _ 1 } { R _ 1 ( r _ 1 - r _ 0 ) } } { \\log \\frac { R _ 1 } { r _ 1 } } \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{align*} E \\left [ \\left ( S - T _ 1 \\right ) ^ 2 \\right ] = E \\left [ \\left ( S - T _ 1 \\right ) ^ 2 \\ , | \\ , T _ 1 < S \\right ] P \\{ T _ 1 < S \\} = \\frac { 1 + \\alpha _ 1 } { \\alpha _ 2 ^ 2 } \\ , P \\{ T _ 1 < S \\} . \\end{align*}"}
-{"id": "682.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c c } A _ k X _ k ^ { s _ { k } } B _ k - C _ k X _ { k + 1 } ^ { t _ { k } } D _ k & = & E _ k , & k = 1 , \\ldots , r - 1 , \\\\ A _ r X _ r ^ { s _ { r } } B _ r - C _ r X _ 1 ^ { t _ { r } } D _ r & = & E _ r , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mu } = \\inf \\{ E ( u , v ) ; M ( u ) = \\mu , ( u , v ) \\in H ^ 1 \\times L ^ 2 \\} . \\end{align*}"}
-{"id": "547.png", "formula": "\\begin{align*} q \\frac { d \\hat { \\varphi } _ K } { d q } = v \\circ \\hat { \\varphi } _ K \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{align*} ( \\nabla \\Psi _ 1 ( \\cdot , \\hat { \\mu } ) ) ( \\mu ) & = - ( \\nabla \\Psi _ 1 ( \\mu , \\cdot ) ) ( \\hat { \\mu } ) , \\\\ ( \\nabla \\Psi _ 2 ( \\cdot , \\hat { w } ) ) ( w ) & = - ( \\nabla \\Psi _ 2 ( w , \\cdot ) ) ( \\hat { w } ) . \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} \\sum _ { y \\in \\hat S _ 1 } | G ( t , \\hat x _ n - y ) - \\bar G ( t , \\hat x _ n - y ) | \\leq c \\ , \\frac { ( M \\wedge ( \\gamma t ) ^ { 1 / 2 + \\varepsilon } ) ^ d } { ( \\gamma t ) ^ { ( d + 1 ) / 2 } } \\ , , \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} \\exists \\ , \\varepsilon > 0 : \\ \\sup _ { x \\in \\Gamma } \\bigg \\{ \\frac { V ( x ) } { | x | ^ { 2 + \\epsilon } } \\bigg \\} < \\infty \\ \\ & \\Rightarrow \\ \\ N [ \\mathfrak { e } , V ] , \\ N _ { s c } [ \\mathfrak { e } , V ] < \\infty , \\\\ [ 1 e x ] \\exists \\ , \\varepsilon > 0 : \\ \\inf _ { x \\in \\Gamma } \\bigg \\{ \\frac { V ( x ) } { | x | ^ { 2 - \\epsilon } } \\bigg \\} > 0 \\ \\ & \\Rightarrow \\ \\ N [ \\mathfrak { e } , V ] = N _ { s c } [ \\mathfrak { e } , V ] = \\infty . \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} \\Pi _ { \\rm n o r } \\omega = 0 , \\Pi _ { \\rm t a n } ( \\nabla _ \\nu - D A ) \\omega = 0 . \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{align*} S _ 1 ^ { i , j } ( \\sum _ { k } f _ { k } ) & = 2 ^ { j / 2 } \\| \\sum _ { k } F _ { k , ( R ) _ i } \\| _ { \\ell ^ { 2 } ( \\frac { 1 _ R } { | R | } ) } \\\\ & \\leq 2 ^ { j / 2 } \\sum _ { k } \\| F _ { k , ( R ) _ i } \\| _ { \\ell ^ { 2 } ( \\frac { 1 _ R } { | R | } ) } \\\\ & = 2 ^ { j / 2 } \\sum _ { k } \\left ( \\sum _ { R } ( F _ { k , ( R ) _ { i } } ) ^ { 2 } \\frac { 1 _ R } { | R | } \\right ) ^ { 1 / 2 } = \\sum _ { k } S _ 1 ^ { i , j } ( f _ k ) . \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} \\lim _ { m \\to 0 } J _ { s , t } ^ m = \\int _ s ^ t B ^ { i _ 1 , . . . , i _ k } ( r , q _ r ) \\left ( \\int h ( r , q _ r , z ) z _ { i _ 1 } . . . z _ { i _ k } d z \\right ) d r \\equiv J _ { s , t } . \\end{align*}"}
-{"id": "7208.png", "formula": "\\begin{align*} { D } _ { q } \\{ f ( x ) \\} = \\frac { f ( x ) - f ( q x ) } { x } , \\end{align*}"}
-{"id": "1480.png", "formula": "\\begin{align*} \\left \\langle u , v \\right \\rangle _ { L ^ 2 ( \\Sigma , e ^ { - f } d \\sigma ) } = \\int _ { \\Sigma } u v e ^ { - f } d \\sigma . \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} R ' = \\{ t _ { q _ 2 } - t _ { q _ 1 } , t _ { q _ 3 } - t _ { q _ 2 } , . . . , t _ { q _ { k ' + 1 } } - t _ { q _ { k ' } } \\} . \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} X _ n \\left ( t \\wedge \\tau _ m \\right ) = X _ m \\left ( t \\wedge \\tau _ m \\right ) \\quad n > m , \\mathbf { P } \\ , \\mbox { - a . s . } \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} f _ { x _ { i } = 0 } ( x _ 1 , \\ldots , x _ n ) : = f ( x _ 1 , \\ldots , x _ { i - 1 } , 0 , x _ i , \\ldots , x _ n ) . \\end{align*}"}
-{"id": "8072.png", "formula": "\\begin{align*} \\frac { Z f } { 2 t } = f _ t + < \\nabla f , \\frac { X } { 2 t } > . \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } a _ n = + \\infty . \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{align*} \\begin{cases} \\mathfrak { D } \\mathfrak { b } + r ^ { \\frac { 1 } { 2 } } \\mathfrak { b } * \\mathfrak { b } = \\mathfrak { v } \\\\ \\lim \\limits _ { s \\to \\infty } \\mathfrak { b } = \\mathfrak { b } _ { \\pm } . \\\\ \\end{cases} \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} \\bar { \\Phi } = \\{ \\bar { \\alpha } ; \\alpha \\in \\Phi _ s \\} \\subset \\mathfrak { X } ( T ) \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} L : = a ^ { i j } ( x ) D _ { i j } + b ^ k ( x ) D _ k + c ( x ) , \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} \\forall V \\in H ^ { - 1 } _ { \\rm p e r , r } , \\mathcal { J } _ { b } ( V ) : = \\frac { 1 } { 2 } \\int _ { \\Gamma ^ * } | b ( q ) - \\varepsilon _ { q } ^ V | ^ 2 \\ , d q = \\int _ 0 ^ { 1 / 2 } | b ( q ) - \\varepsilon _ { q } ^ V | ^ 2 \\ , d q . \\end{align*}"}
-{"id": "8074.png", "formula": "\\begin{align*} \\begin{cases} \\operatorname { d i v } ( y ^ a \\nabla v ) = y ^ a v _ t , \\\\ \\underset { y \\to 0 } { \\lim } \\ y ^ a v _ y = W v + \\psi , \\end{cases} \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} ( 0 , \\ , b ) S ^ { - 1 } A ( \\varepsilon _ 0 ) S \\left ( \\begin{matrix} \\mathfrak { a } \\\\ \\mathfrak { b } \\end{matrix} \\right ) = ( \\mathfrak a ' \\ , , \\mathfrak b ' ) S ^ { - 1 } A ( \\varepsilon _ 0 ) S \\left ( \\begin{matrix} 0 \\\\ - c \\alpha + a \\beta \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "3875.png", "formula": "\\begin{align*} \\gamma = \\frac { 2 ( 1 - \\delta ) ^ 4 - \\varepsilon ( 1 + ( 1 - \\delta ) ^ 4 ) } { ( 1 - \\varepsilon ) ( 1 - \\delta ) ^ 2 ( ( 1 - \\delta ) ^ 4 + 1 ) } , \\end{align*}"}
-{"id": "2448.png", "formula": "\\begin{align*} U ( N ; \\alpha ) : = e ^ { \\ln ^ { \\alpha } N } \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} \\frac { d } { d t } \\frac { W _ 2 ^ 2 ( \\mu _ t , \\nu ) } { 2 } = \\lim _ { s \\downarrow 0 } \\frac { W _ 2 ^ 2 ( \\mu _ { t } , \\nu ) - W _ 2 ^ 2 ( \\mu _ { t - s } , \\nu ) } { 2 s } \\le - \\int _ X \\langle \\nabla u , \\nabla \\varphi _ t \\rangle \\ , d \\mu _ t . \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{align*} f _ \\xi = \\xi \\bigl ( \\prod _ { \\sigma } X _ \\sigma ( Y _ { \\sigma , 1 } ) \\dots X _ \\sigma ( Y _ { \\sigma , r _ \\sigma } ) \\bigr ) . \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} p x \\frac { \\partial P } { \\partial x } + q y \\frac { \\partial P } { \\partial y } + r z \\frac { \\partial P } { \\partial z } = - t \\ ( \\in R ^ \\times ) \\ , . \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} & S ^ { e n v , 0 } _ { s , t } = \\ln ( \\beta ( t , q _ t ) / \\beta ( s , q _ s ) ) - \\int _ s ^ t ( \\beta ^ { - 1 } \\partial _ r \\beta ) ( r , q _ r ) d r \\\\ & + ( \\beta V ) ( s , q _ s ) - ( \\beta V ) ( t , q _ t ) + \\int _ s ^ t \\partial _ r ( \\beta V ) ( r , q _ r ) d r \\\\ & + \\int _ s ^ t ( \\beta \\tilde F + V \\nabla _ q \\beta ) ( r , q _ r ) \\circ d q _ r . \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{align*} T _ 1 = \\inf \\{ n \\geq 1 ; R _ n = R _ 0 \\} \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{align*} g ( z ) = f ( c + \\varrho z ) \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{align*} \\delta \\beta ^ { ( 2 ) } ( \\kappa ) = - \\frac { L _ 0 ( \\| b \\| ) ^ 2 } { 2 } \\overline { \\delta \\beta ^ { ( 2 ) } } = - \\dfrac { \\overline { L _ 0 ^ 2 } } { 2 } , \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} { } \\mathcal { T } _ { \\Gamma } ( \\overline { x } ) = \\ ! \\big \\{ h \\in \\mathbb { X } \\ | \\ g ' ( \\overline { x } ) h \\in \\mathcal { T } _ { K } ( g ( \\overline { x } ) ) \\big \\} , \\quad \\\\ \\mathcal { N } _ { \\Gamma } ( \\overline { x } ) = \\mathcal { \\widehat { N } } _ { \\Gamma } ( \\overline { x } ) = \\ ! \\big \\{ \\nabla g ( \\overline { x } ) \\lambda \\ | \\ \\lambda \\in \\mathcal { N } _ { K } ( g ( \\overline { x } ) ) \\big \\} . \\end{align*}"}
-{"id": "2899.png", "formula": "\\begin{align*} N ^ 2 \\big | \\langle \\psi _ N , q _ 1 p _ 2 V _ \\beta ( x _ 1 - x _ 2 ) \\widehat { m } q _ 1 q _ 2 \\psi _ N \\rangle \\big | \\le & \\ , N ^ 2 \\big | \\langle \\psi _ N , q _ 1 p _ 2 U _ { 0 , \\beta } ( x _ 1 - x _ 2 ) \\widehat { m } q _ 1 q _ 2 \\psi _ N \\rangle \\big | \\\\ & + N ^ 2 \\big | \\langle \\psi _ N , q _ 1 p _ 2 ( \\Delta _ 1 h _ { 0 , \\beta } ( x _ 1 - x _ 2 ) ) \\widehat { m } q _ 1 q _ 2 \\psi _ N \\rangle \\big | \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{align*} [ a , Z ] = [ a ^ \\dagger , Z ] = 0 , [ a ^ \\dagger , a ] = Z . \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} K _ M \\cdot I _ r = - I _ { r + 1 } + ( I _ r ) _ x \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} E _ { \\rm a b s } = \\max _ { t \\in [ 0 , T ] } \\left | \\bar { y } ( t ) - y ( t ) \\right | \\mbox { a n d } E _ { \\rm r e l } = \\frac { 1 } { T } \\int _ 0 ^ T \\frac { \\left | \\bar { y } ( t ) - y ( t ) \\right | } { \\left | y ( t ) \\right | } \\ ; { \\rm d } t \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} ( y \\cap x _ m ) \\setminus ( y \\cap x _ { m - 1 } ) = ( y \\cap x _ m ) \\setminus x _ { m - 1 } \\subseteq ( z \\cap x _ m ) \\setminus x _ { m - 1 } = ( z \\cap x _ m ) \\setminus ( z \\cap x _ { m - 1 } ) = \\emptyset , \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} S \\overset { d } { = } \\frac { \\sin ( \\alpha Y _ 2 ) } { ( \\cos ( Y _ 2 ) ) ^ { \\frac { 1 } { \\alpha } } } \\left ( \\frac { \\cos ( ( 1 - \\alpha ) Y _ 2 ) } { Y _ 1 } \\right ) ^ { \\frac { 1 - \\alpha } { \\alpha } } \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} v _ \\tau ( t ) = w ( t ) + \\hat w ( \\tau ) - \\hat w ( t ) + u _ \\tau ( t ) . \\end{align*}"}
-{"id": "3327.png", "formula": "\\begin{align*} f _ r ( 1 - t ) = | E | ( 1 - 2 t ) + f _ r ( t ) , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "6818.png", "formula": "\\begin{align*} R ^ { S M } ( X , Y ) \\psi = \\frac { 1 } { 4 } \\sum _ { \\alpha , \\beta = 0 } ^ { n - 1 } R ^ M ( X , Y , \\partial _ { \\alpha } , \\partial _ { \\beta } ) \\partial _ { \\alpha } \\cdot \\partial _ { \\beta } \\cdot \\psi \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} \\exp ( - A _ D ) \\exp ( D ) \\exp ( - C _ D ) = \\exp ( B _ D + \\frac { 1 } { 2 } ( [ C _ D , B _ D ] + [ C _ D , A _ D ] + [ B _ D , A _ D ] ) + O ) . \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} \\frac 1 n = z \\cdot \\frac 1 { \\beta ^ j } \\cdot \\frac 1 { \\beta ^ { i - j } - 1 } , \\quad z \\in \\Z [ \\beta ] . \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u _ { 0 } = a \\left ( x \\right ) u _ { 0 } & \\mbox { i n } \\Omega , \\\\ u _ { 0 } = 0 & \\mbox { o n } \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} \\dot { x } ( t ) & = A x ( t ) + B u ( t ) \\\\ [ 1 e x ] \\begin{pmatrix} z _ + ( t ) \\\\ z _ - ( t ) \\\\ \\end{pmatrix} & = \\begin{pmatrix} { L _ + } ^ \\top \\\\ { L _ - } ^ \\top \\\\ \\end{pmatrix} x ( t ) , \\end{align*}"}
-{"id": "5003.png", "formula": "\\begin{align*} [ h _ 1 , h _ 2 , \\dots , h _ n ] = 0 \\mbox { f o r a l l } h _ i \\in L \\iff [ x _ 1 , x _ 2 , \\dots , x _ n ] = 0 \\mbox { f o r a l l } x _ i \\in X . \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{align*} \\max _ { j = 0 , \\dots , n - 1 } v _ j ^ 2 a _ j ^ 2 = O \\left ( n ^ { - r / ( 2 \\beta + r ) } \\right ) . \\end{align*}"}
-{"id": "8381.png", "formula": "\\begin{align*} \\left \\langle \\delta _ x , \\frac { \\ 1 _ { \\{ 1 \\} } ( Q ^ { ( n ) } ) } { \\lambda _ 0 + 2 ( 1 - Q ^ { ( n ) } ) } \\delta _ x \\right \\rangle = \\frac { 1 } { \\lambda _ 0 } \\big \\langle \\delta _ x , \\ 1 _ { \\{ 1 \\} } ( Q ^ { ( n ) } ) \\delta _ x \\big \\rangle = \\frac { 1 } { \\lambda _ 0 N _ n } , \\forall \\ ; x \\in E _ n . \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} & [ c , x _ 1 ] [ x _ 2 , x _ 3 , x _ 4 , x _ 5 ] = [ c , x _ 1 ] [ x _ 2 , x _ 3 , x _ 4 ] x _ 5 - [ c , x _ 1 ] x _ 5 [ x _ 2 , x _ 3 , x _ 4 ] \\\\ = \\ & [ c , x _ 1 ] [ x _ 2 , x _ 3 , x _ 4 ] x _ 5 - x _ 5 [ c , x _ 1 ] [ x _ 2 , x _ 3 , x _ 4 ] - [ c , x _ 1 , x _ 5 ] [ x _ 2 , x _ 3 , x _ 4 ] . \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{align*} \\frac { d ( x , y ) } { \\sqrt { d ( x , p ) d ( y , p ) } } & \\leq \\frac { d ( x , z ) } { \\sqrt { d ( x , p ) d ( z , p ) } } + \\\\ & + \\frac { d ( z , y ) } { \\sqrt { d ( z , p ) d ( y , p ) } } + \\frac { d ( x , z ) d ( z , y ) } { d ( z , p ) \\sqrt { d ( x , p ) d ( y , p ) } } . \\end{align*}"}
-{"id": "6500.png", "formula": "\\begin{align*} \\begin{aligned} & \\Phi ^ { - 1 } u \\in W ^ { 1 , s } ( 0 , T ; W ^ { - 1 , \\frac { p } { 2 } } _ { \\sigma } ( \\Omega ) ) \\cap L ^ s ( 0 , T ; \\Phi ^ { - 1 } W ^ { 1 , \\frac { p } { 2 } } _ { 0 , \\sigma } ( \\Omega ) ) , \\\\ & d ^ { \\prime } , B _ { \\frac { p } { 2 } } d \\in L ^ s ( 0 , T ; L ^ { \\frac { p } { 2 } } ( \\Omega ) ^ 3 ) \\end{aligned} \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} \\kappa ( m , \\mu , \\lambda ) = | S _ \\mu ( m - 1 ) \\setminus \\lambda | + | T _ \\mu ( m + 1 ) \\setminus \\lambda | + | \\lambda | / 2 . \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{align*} I ^ { n + 1 } = y I ^ n ~ \\mbox { f o r a l l } n \\geqslant n _ 0 . \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{align*} \\big \\langle e ^ { - t \\mathcal { A } _ p } u , v \\big \\rangle _ { W ^ { - 1 , p } _ { \\sigma } , W ^ { 1 , p ^ { \\prime } } _ { 0 , \\sigma } } = \\frac { 1 } { 2 \\pi i } \\int _ { \\gamma } e ^ { t \\lambda } \\big \\langle ( \\lambda + \\mathcal { A } _ p ) ^ { - 1 } u , v \\big \\rangle _ { W ^ { - 1 , p } _ { \\sigma } , W ^ { 1 , p ^ { \\prime } } _ { 0 , \\sigma } } \\ ; d \\lambda . \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} & B ^ { - 1 } A A ^ T ( B ^ { - 1 } ) ^ T = I , \\ , \\ , \\ , \\ , A \\equiv ( \\tilde \\gamma ^ T ) ^ { - 1 } \\sigma , \\ , \\ , B \\equiv \\tilde \\gamma ^ { - 1 } \\sigma , \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{align*} \\mathfrak { R e } \\dot { c _ 0 } ( 0 ) = 0 \\mbox { a n d } \\mathfrak { R e } \\ddot { c _ 0 } ( 0 ) = - \\sum _ { k = 0 } ^ { + \\infty } | \\dot c _ k ( 0 ) | ^ 2 . \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} \\mathcal { F } \\ , \\triangleq \\ , \\left \\{ \\ , \\beta \\ , \\subseteq \\{ 1 , \\ldots , m \\} \\ , \\bigg | \\ , \\begin{array} { l } \\mbox { t h e r e e x i s t s a n i s o l a t e d l o c a l m i n i m i z e r } \\\\ \\mbox { w i t h a m u l t i p l i e r $ \\lambda $ s u c h t h a t $ \\mbox { s u p p } ( \\lambda ) = \\beta $ } \\end{array} \\right \\} . \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} f ( \\xi ) = \\frac { 3 ^ { 1 / 4 } } { 2 } \\exp \\left ( - \\frac { \\sqrt { 3 } \\pi } { 4 } \\xi ^ 2 \\right ) , \\xi \\in \\R . \\end{align*}"}
-{"id": "731.png", "formula": "\\begin{align*} \\Pi _ { G _ n } ~ : = ~ D ( { \\rm M } ( G _ n ) ) ~ = ~ D ( \\theta _ n ) ^ { - 1 } \\times \\prod _ { \\stackrel { z _ { j , n } ~ \\mbox { { \\tiny i n } } ~ | z | < 1 } { { \\rm { \\tiny o u t s i d e ~ b u m p } } } } { \\rm D } ( | z _ { j , n } | ) ^ { - 2 } \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} 2 a _ i = | T _ { x _ i = 1 } | + | F _ { x _ i = 0 } | . \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} P & = \\sum _ { d \\geq 0 } B _ d t ^ d = \\sum _ { d \\geq 0 } { F B } _ d t ^ d + \\sum _ { d \\geq 0 } { T B } _ { d - 1 } t ^ d + \\sum _ { d \\geq 0 } { T B } _ d t ^ d \\\\ & = F P + \\overline { T P } + \\underline { T P } = F P + t \\ , \\underline { T P } + \\underline { T P } \\\\ & = F P + ( 1 + t ) \\ , \\underline { T P } \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{align*} \\Psi ( D _ a ) \\Psi ( D _ b ) = \\sum _ r E _ r D _ a E _ r \\ > \\sum _ s E _ s D _ b E _ s \\end{align*}"}
-{"id": "3578.png", "formula": "\\begin{align*} \\mathfrak { N } & = ( N + i B ) \\exp \\Big ( i \\int _ { 0 } ^ { s } \\tau d s \\Big ) \\\\ \\psi & = \\kappa \\exp \\Big ( i \\int _ { 0 } ^ { s } \\tau d s \\Big ) . \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } < { T ( t ) x - x \\over t } - y , l > = 0 \\ \\ \\forall l \\in X _ * . \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} \\frac { ( c , a x t , b y t ; q ) _ \\infty } { ( a b , t x , t y ; q ) _ \\infty } { _ 3 \\phi _ 2 \\left ( { { d , x t , y t } \\atop { a x t , b y t } } ; q , c \\right ) } = \\sum _ { n = 0 } ^ \\infty \\lambda _ n \\Phi _ n ^ { ( a , b ) } ( x , y | q ) . \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} \\mathcal { L } _ { ^ { m , 2 } _ { i n t e r } } ( s ) = & { \\rm e x p } \\left ( - \\lambda _ { c } \\int _ { \\mathcal { B } ( x _ m , \\mathcal { R } _ s ) } f _ { y _ { m , 2 } } ( y ) \\int _ { \\mathbb { R } ^ 2 } \\left ( 1 - \\frac { 1 } { \\frac { s } { { L \\left ( | | y + x _ m - x | | \\right ) } } + 1 } \\right ) d x d y \\right ) . \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{align*} ( T _ p h _ \\beta | _ { S _ p } ) ^ { - 1 } ( X ( h _ \\beta ( p ) ) & = ( T _ p h _ \\beta | _ { S _ p } ) ^ { - 1 } ( X ( h _ \\beta | _ { S _ p } \\circ h _ \\alpha | _ { S _ p } ^ { - 1 } ( h _ \\alpha ( p ) ) \\\\ & = ( T _ p h _ \\beta | _ { S _ p } ) ^ { - 1 } T _ { h _ \\alpha ( p ) } ( h _ \\beta | _ { S _ p } \\circ h _ \\alpha | _ { S _ p } ^ { - 1 } ) ( X ( h _ \\alpha ( p ) ) \\\\ & = ( T _ p h _ \\alpha | _ { S _ p } ) ^ { - 1 } ( X ( h _ \\alpha ( p ) ) \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} q _ C ^ A ( \\lambda , \\nu ) : = \\frac { ( - 1 ) ^ { \\frac { \\nu - \\lambda } { 2 } } ( \\frac { \\nu - \\lambda } { 2 } ) ! \\pi ^ { \\frac { n - 1 } { 2 } } } { 2 ^ { \\nu - \\lambda } \\ \\Gamma ( \\nu ) } . \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} \\mathbf { V } = \\begin{bmatrix} \\mathbf { E } & \\mathbf { E } \\\\ \\mathbf { E } \\mathbf { \\Lambda } & \\mathbf { E } \\overline { \\mathbf { \\Lambda } } \\end{bmatrix} . \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{align*} \\# \\mathcal { M } _ { Y , I = 1 } ^ { J , L } ( \\alpha , \\beta ) / \\mathbb { R } = \\# \\mathfrak { M } _ { Y , i n d = 1 } ^ { J , L } ( T _ r ( \\alpha ) , T _ r ( \\beta ) ) / \\mathbb { R } . \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} g = g _ 1 g _ 0 g _ { - 1 } = \\begin{pmatrix} 1 & 0 & 0 \\\\ \\xi & I & 0 \\\\ \\frac { 1 } { 2 } \\| \\xi \\| ^ 2 & \\xi ^ T & 1 \\end{pmatrix} \\begin{pmatrix} \\alpha & 0 & 0 \\\\ 0 & A & 0 \\\\ 0 & 0 & \\alpha ^ { - 1 } \\end{pmatrix} \\begin{pmatrix} 1 & v ^ T & \\frac { 1 } { 2 } \\| v \\| ^ 2 \\\\ 0 & I & v \\\\ 0 & 0 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "1620.png", "formula": "\\begin{align*} \\frac { c _ 6 } { q _ { \\sigma , t } } \\left ( ( c _ 2 \\ln _ 2 t - c _ 1 \\ln _ 3 t ) ^ { - 2 \\rho _ \\sigma - 1 } \\right ) < \\frac { c _ 7 } { q _ { \\sigma , t } } ( \\ln _ 2 t ) ^ { - 2 \\rho _ \\sigma - 1 } = c _ 8 \\frac { ( \\ln _ 2 t ) ^ { \\mu - 1 - 2 \\rho _ \\sigma - 1 } } { \\ln t } \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} \\epsilon \\lambda = ( \\epsilon _ r \\circ \\rho _ r \\circ \\epsilon _ { r - 1 } ^ { - 1 } ) \\circ \\cdots \\circ ( \\epsilon _ 2 \\circ \\rho _ 2 \\circ \\epsilon _ 1 ^ { - 1 } ) \\circ ( \\epsilon _ 1 \\circ \\rho _ 1 \\circ \\epsilon _ 0 ^ { - 1 } ) . \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} p ( i , j | v , w ) = \\langle ( P _ { v , i } Q _ { w , j } ) h , h \\rangle . \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{align*} V _ e ( \\Sigma _ 1 ) _ { y ^ 1 , x ^ 1 } \\times \\cdots \\times V _ e ( \\Sigma _ m ) _ { y ^ m , x ^ m } . \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} A _ \\Gamma ( q ) = \\left ( 1 - q ^ { - 1 } \\right ) ^ { b ( \\Gamma ) } \\sum _ { \\Gamma ' _ 1 \\subsetneq \\dots \\subsetneq \\Gamma ' _ { \\beta - 1 } \\subsetneq \\Gamma ' _ { \\beta } = \\Gamma } \\prod _ { j = 1 } ^ { \\beta - 1 } \\frac { 1 } { q ^ { b ( \\Gamma ) - b ( \\Gamma ' _ { j - 1 } ) } - 1 } , \\end{align*}"}
-{"id": "6940.png", "formula": "\\begin{align*} j _ { x , x + 1 } ( \\eta ) = n ^ { \\theta } \\Big [ q _ n g ( \\eta ( x ) ) - p _ n g ( \\eta ( x + 1 ) ) \\Big ] , \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} \\Pi _ { L , \\eta } : = \\left \\{ x \\in B _ L \\colon \\ , \\xi ( x ) > a _ L - \\eta \\right \\} . \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} ( J q ^ 1 ) ^ { - 1 } + ( J q ^ 2 ) ^ { - 1 } = ( J q ^ 3 + J q ^ 2 J q ^ 1 - \\frac { 1 } { 6 } J q ^ 1 J q ^ 1 J q ^ 1 + J q ^ 2 ) ( J q ^ 2 J q ^ 2 ) ^ { - 1 } . \\end{align*}"}
-{"id": "3146.png", "formula": "\\begin{align*} { } e ^ { - \\alpha ^ * } = e ^ { v ( p ) } - 1 = \\frac { e ^ { v } - 1 } { 1 - e ^ { v ( 1 - 1 / p ) } } \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} \\mathcal { X } ^ { ( x ) } _ { i ' , 1 } = \\mathcal { X } ^ { ( x ) } _ { i ' - 1 , 1 } & = X _ { p _ { ( x , i ' - 1 ) } , 1 } - \\sum _ { x ' = 1 } ^ { x - 1 } \\ell ^ { \\circ } _ { x ' } + \\sum _ { x ' = x + 1 } ^ { \\mathcal { k } } \\ell ^ { \\circ } _ { x ' } \\\\ & = X _ { p _ { ( x , i ' ) } , 1 } - \\sum _ { x ' = 1 } ^ { x - 1 } \\ell ^ { \\circ } _ { x ' } + \\sum _ { x ' = x + 1 } ^ { \\mathcal { k } } \\ell ^ { \\circ } _ { x ' } . \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} Y _ s ^ { 0 , x ; u _ { \\cdot } } = \\xi + & \\int _ s ^ { \\tau } \\hat { G } \\bigl ( r , X _ r ^ { 0 , x ; u _ { \\cdot } } , Y _ r ^ { 0 , x ; u _ { \\cdot } } , Z _ r ^ { 0 , x ; u _ { \\cdot } } \\bigr ) d r - \\int _ s ^ { \\tau } Z _ r ^ { 0 , x ; u } d B _ r , \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{align*} p _ \\kappa ( x , y , \\xi ) = p ( \\kappa ^ { - 1 } ( x ) , \\kappa ^ { - 1 } ( y ) , \\tilde M ( x , y ) ^ t \\xi ) | \\det \\tilde M ( x , y ) | | \\det D ( \\kappa ^ { - 1 } ( y ) ) | \\end{align*}"}
-{"id": "156.png", "formula": "\\begin{align*} f _ t ' ( | q | _ k ) \\Im \\left ( \\frac { \\dot q } { q } \\right ) d | q | _ k = f _ t ' ( | \\lambda | ^ 2 ) \\Im \\left ( \\frac { \\dot q } { \\lambda ^ 2 } \\right ) d | \\lambda | ^ 2 \\end{align*}"}
-{"id": "8185.png", "formula": "\\begin{align*} u ( x ) = c _ 0 + \\int _ 0 ^ 1 \\langle \\textit { \\textbf { V } } ( \\gamma ( t ) ) , \\dot \\gamma ( t ) ) \\rangle _ { \\tilde g } d t , \\ x \\in M , \\end{align*}"}
-{"id": "6434.png", "formula": "\\begin{align*} \\begin{aligned} & k ^ { x } _ { j + 1 } ( T ) \\leq C C _ { 3 } ( T ) k ^ { \\nabla y } _ { j } ( T ) ^ 2 ( k ^ { x } _ { j } ( T ) + k ^ { y } _ { j } ( T ) + \\lvert \\overline { b } \\rvert ) . \\end{aligned} \\end{align*}"}
-{"id": "7619.png", "formula": "\\begin{align*} - \\frac { u '' } { ( 1 - ( u ' ) ^ 2 ) ^ \\frac 3 2 } = g ( u ) , \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{align*} z _ { 0 } ^ { - 1 } \\left ( \\frac { z _ 1 - z _ 2 } { z _ 0 } \\right ) ^ { r / 2 } \\delta \\left ( \\frac { z _ { 1 } - z _ { 2 } } { z _ { 0 } } \\right ) Y ( u , z _ { 1 } - z _ 2 ) = z _ 1 ^ { - 1 } \\delta \\left ( \\frac { z _ { 0 } + z _ { 2 } } { z _ 1 } \\right ) Y ( u , z _ 0 ) . \\end{align*}"}
-{"id": "3336.png", "formula": "\\begin{align*} \\langle x _ v , h \\rangle & = \\langle x _ { v , 0 } , x _ { v , 0 } + x _ { v , 1 } \\rangle = p ( 0 , 0 | v , v ) + p ( 0 , 1 | v , v ) = p _ A ( 0 | v ) = t , \\\\ \\| x _ v \\| ^ 2 & = \\langle x _ { v , 0 } , x _ { v , 0 } \\rangle = \\langle x _ { v , 0 } , h - x _ { v , 1 } \\rangle = \\langle x _ { v , 0 } , h \\rangle = t , \\end{align*}"}
-{"id": "4254.png", "formula": "\\begin{align*} \\sum _ { | \\lambda | = n } g ^ { ( n ) } _ { \\lambda } S ^ { ( n ) } _ { \\lambda , \\mu } = \\delta _ { \\mu ( 1 ^ n ) } , n = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} g ( a _ { k + 1 } ^ 2 , \\ldots , a _ n ^ 2 ; \\tilde { x } _ k ) = \\frac { G ( a ; b , W ) } { G ( \\tilde { a } _ k ; B _ k , W _ k ) } \\frac { e ^ { - \\d B ^ * _ k M ^ { - 1 } _ { \\tilde { x } _ k } B _ k } } { \\sqrt { \\det ( M _ { \\tilde { x } _ k } ) } } \\textbf { 1 } _ { C _ { W _ k } } ( \\tilde { x } _ k ) \\end{align*}"}
-{"id": "1402.png", "formula": "\\begin{align*} I _ { \\tau } ( x ) = - \\Psi ( U _ { \\tau } , x ) - \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} W _ 2 ( \\mu _ t , \\nu _ t ) = W _ 2 ( \\mu , \\nu ) \\ t \\in \\R . \\end{align*}"}
-{"id": "7262.png", "formula": "\\begin{align*} \\forall \\ , \\mu \\neq 0 , \\sum _ { k = 1 } ^ N \\omega _ k v _ k ( \\phi _ \\mu ( v _ k ) - \\phi _ \\mu ( - v _ k ) ) = 0 . \\end{align*}"}
-{"id": "3493.png", "formula": "\\begin{align*} \\max _ { i \\mid a _ { i j } < 0 } y _ i = \\max _ { i \\mid a _ { i j } > 0 } y _ i . \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} & \\forall \\ x , y , z \\in \\mathcal { H } ( A ) = A _ 0 \\cup A _ 1 : \\\\ & a s _ { \\alpha , \\bullet } ( x , y , z ) + ( - 1 ) ^ { | x | | y | } a s _ { \\alpha , \\bullet } ( y , x , z ) = 0 , \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{align*} | \\Delta _ { j _ n } ^ { ( n ) } | = \\min _ k | \\Delta _ k ^ { ( n ) } | \\end{align*}"}
-{"id": "6692.png", "formula": "\\begin{align*} ( \\mathcal { R } _ { \\alpha , \\Omega } f ) ( x ) : = \\int _ \\Omega \\varepsilon _ { \\alpha , n } ( x - y ) f ( y ) d y . \\end{align*}"}
-{"id": "4214.png", "formula": "\\begin{align*} \\lefteqn { - ( x _ 1 - x _ 2 ) ^ T ( \\nabla _ x V ( x _ 1 , y ) - \\nabla _ x V ( x _ 2 , y ) ) } \\\\ & = - ( x _ 1 - x _ 2 ) ^ T Q ( x _ 1 - x _ 2 ) - ( x _ 1 - x _ 2 ) ^ T ( \\nabla _ x h ( x _ 1 , y ) - \\nabla _ x h ( x _ 2 , y ) ) \\\\ & \\leq - \\lambda _ Q \\abs { x _ 1 - x _ 2 } ^ 2 + \\abs { x _ 1 - x _ 2 } \\norm { \\nabla _ x h } _ \\infty \\\\ & \\leq - \\lambda _ Q \\abs { x _ 1 - x _ 2 } ^ 2 + \\frac { \\norm { \\nabla _ x h } _ \\infty } { 4 \\lambda _ Q } \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} s _ 1 ( B ) : = \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { k ! } \\sum _ { \\substack { x _ 1 , \\ldots , x _ k \\in B , \\\\ m _ { i - 1 } \\le d ( x _ 1 , \\ldots , x _ k ) < m _ { i } } } ( \\rho _ + ( x _ 1 , \\ldots , x _ k ) ^ 2 + \\rho _ - ( x _ 1 , \\ldots , x _ k ) ^ 2 ) . \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} \\prod _ { \\substack { \\nu \\prec \\mu \\\\ \\nu \\not \\in \\mathcal { E } _ { \\mu } } } \\frac { Y ( w ) - y _ { \\nu } ( w ) } { y _ { \\mu } ( w ) - y _ { \\nu } ( w ) } \\cdot z ^ { \\mu } = E _ { \\mu } + \\sum _ { \\nu \\in \\mathcal { E } _ { \\mu } } d _ { \\mu , \\nu } ( q , t ) E _ { \\nu } . \\end{align*}"}
-{"id": "5106.png", "formula": "\\begin{align*} \\omega ' = \\sum _ { i , j = 1 } ^ { d } a _ { i j } u ^ i ( - 1 ) u ^ { j } ( - 1 ) \\mathbf { 1 } + \\sum _ { s = 1 } ^ { d } b _ s u ^ s ( - 2 ) \\mathbf { 1 } . \\end{align*}"}
-{"id": "3913.png", "formula": "\\begin{align*} \\kappa ( x ) : = \\frac { { x ^ * } ^ \\gamma } { 1 - e ^ { - a ( x ^ * - k ) } } x ^ { - \\gamma } \\left ( 1 - e ^ { - a ( x - k ) } \\right ) < 1 , \\enskip 0 < x < x ^ * . \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} \\xi _ n = \\sum _ { y \\in \\Lambda _ n } A ^ n _ y \\varphi ^ n _ y , \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} t ( x ' , D _ { x } ) f ( x ' ) = \\int _ { \\R ^ { n - 1 } } \\int _ { \\R _ + } K _ t ( x ' , x ' - y ' , y _ n ) \\ , f ( y ' , y _ n ) \\ , d y _ n \\ , d y ' , \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\left [ \\psi _ t \\right ] } = P ( \\left [ \\psi _ t \\right ] ) , \\\\ [ \\psi _ 0 ] = [ \\bar { \\psi } ] , \\end{cases} \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} \\partial G ( U ) = \\nabla G ( U ) - U \\Sigma ( U ) . \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} \\Phi _ { t t } ( 1 , t _ { \\ast } ) = \\Phi _ { t } ( 1 , t _ { \\ast } ) = 0 . \\end{align*}"}
-{"id": "601.png", "formula": "\\begin{align*} \\| P ( [ z ] ) \\| = \\frac { | P ( z ) | } { \\left ( \\sum _ { i = 0 } ^ n | z _ i | ^ 2 \\right ) ^ { \\frac { d } { 2 } } } \\le \\frac { \\sum _ { | I | = d } | a _ I | | z ^ I | } { \\left ( \\sum _ { i = 0 } ^ n | z _ i | ^ 2 \\right ) ^ { \\frac { d } { 2 } } } \\le \\frac { \\max _ { | I | = d } | z ^ I | } { \\left ( \\sum _ { i = 0 } ^ n | z _ i | ^ 2 \\right ) ^ { \\frac { d } { 2 } } } \\| P \\| _ 1 \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} [ c , z _ 1 ] [ z _ 2 , z _ 3 , z _ 4 ] + [ c , z _ 2 ] [ z _ 1 , z _ 3 , z _ 4 ] = 0 \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} m i _ k ( n - \\delta - 1 ) + \\frac { s ( B - 1 ) } { 2 \\binom { k } { 2 } } m i _ k ( n - \\delta - 2 ) + \\frac { ( s ( \\delta - s ) + \\frac { s ( s - B ) } { 2 } ) } { \\binom { k } { 2 } } m i _ k ( n - \\delta - k ) + \\frac { \\binom { \\delta - s } { 2 } } { \\binom { k } { 2 } } m i _ k ( n - \\delta - B ) . \\end{align*}"}
-{"id": "479.png", "formula": "\\begin{align*} z = \\frac { a + 1 / 2 } { q _ 2 } + i y = \\frac { a } { q _ 2 } + \\frac { R } { q _ 2 } e ^ { i \\theta } . \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} \\overline { \\mu } ^ { - 1 } ( y ) = \\mu ^ { - 1 } ( \\mathcal { O } ( y ) ) \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} \\begin{gathered} F _ 0 ( \\phi ) = F _ - ( \\phi _ + ) = F _ + ( \\phi _ - ) \\\\ F _ 0 ( \\phi _ + ) = F _ + ( \\phi ) \\\\ F _ 0 ( \\phi _ - ) = F _ - ( \\phi ) \\mathrlap { . } \\end{gathered} \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{align*} \\log ( \\lambda ) = \\log ( | \\lambda | ) + i \\arg ( \\lambda ) \\end{align*}"}
-{"id": "4376.png", "formula": "\\begin{align*} | \\hat { z } | \\leq 2 ^ { \\frac 1 2 } \\int _ { 0 } ^ { 2 | \\lambda | } \\frac { d t } { \\sqrt { t ( t + | \\lambda | ) } } = 2 ^ { \\frac 3 2 } \\log \\left ( \\sqrt { 3 } + \\sqrt { 2 } \\right ) . \\end{align*}"}
-{"id": "4654.png", "formula": "\\begin{align*} \\sqrt { - 1 } \\partial \\bar { \\partial } \\log \\pi _ * \\Omega _ { ( X , D ) / Y } + \\sqrt { - 1 } \\partial \\bar { \\partial } v & = \\\\ & = \\sqrt { - 1 } \\partial \\bar { \\partial } \\log \\pi _ * \\Omega _ { ( X , D ) / Y } + \\omega - \\omega _ B - R i c ( h ) \\\\ \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} \\begin{aligned} & - G \\left ( A _ t + \\frac { 1 - \\tilde { Q } _ t ( A _ t , \\dot { \\varphi } _ 0 ) } { q _ { \\sigma , t } \\varphi _ \\sigma ( 0 ) } \\right ) + \\ln f _ { \\tilde { \\xi } _ t ( y ) } ( \\varphi _ \\xi ( y ) ) + \\widehat { T } _ t ( \\aleph ) \\\\ & \\phantom { a a a a a a a } \\ ; = \\frac { \\varphi _ \\xi ( y ) } { \\varrho } + \\ln ( 1 / \\varrho ) + q _ { \\xi , t } ( y ) + ( \\ln _ 2 t ) ^ { \\mu - 1 - 2 | y | } h _ { t , \\aleph } ( \\varphi _ \\xi ( y ) ) , \\end{aligned} \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{align*} \\Psi _ 1 ( \\mu , \\hat { \\mu } ) & = \\frac { 1 } { 2 } \\sum _ { a } ( ( \\mu ( a ) - \\hat { \\mu } ( a ) ) ^ - ) ^ 2 = \\frac { 1 } { 2 } \\sum _ { a } ( ( \\hat { \\mu } ( a ) - \\mu ( a ) ) ^ + ) ^ 2 , \\\\ \\Psi _ 2 ( w , \\hat { w } ) & : = \\frac { 1 } { 2 } \\sum _ { ( a , b ) \\in \\Gamma } ( w _ { ( a , b ) } - \\hat { w } _ { ( a , b ) } ) ^ 2 . \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} \\sum _ { s > i , t \\geq j } ( A _ k ) _ { i s } ( X _ k ) _ { s t } ( B _ k ) _ { t j } = \\sum _ { s > i } ( A _ k ) _ { i s } \\underbrace { \\sum _ { t \\geq j } ( X _ k ) _ { s t } ( B _ k ) _ { t j } } _ { = : ( X ^ B _ { k } ) _ { s j } } = : \\sum _ { s > i } ( A _ k ) _ { i s } ( X ^ B _ { k } ) _ { s j } , \\end{align*}"}
-{"id": "3487.png", "formula": "\\begin{align*} \\exists i \\in [ m ] { : } \\ \\sigma _ i = 1 , & \\\\ ( \\exists i \\in [ m ] { : } \\ \\sigma _ i s _ { i j } = - 1 ) \\Leftrightarrow ( \\exists i ' \\in [ m ] { : } \\ \\sigma _ { i ' } s _ { i ' j } = 1 ) , & j \\in [ n ] . \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} v _ 2 \\left ( \\sum _ { i = 1 } ^ { d } \\frac { 1 } { 2 i - 1 } \\right ) = 2 v _ 2 ( d ) . \\end{align*}"}
-{"id": "8555.png", "formula": "\\begin{align*} t _ U ( \\zeta ) - t _ V ( \\zeta ) = t _ { U V } ( \\zeta ) \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} \\sum _ { j = 1 , j \\neq i } ^ { N } \\alpha _ { i j } g ( y _ { i , t } - y _ { j , t } ) = - a \\sum _ { j = 1 , j \\neq i } ^ { N } \\alpha _ { i j } ( y _ { i , t } - y _ { j , t } ) + \\sum _ { j = 1 , j \\neq i } ^ { N } \\alpha _ { i j } g _ { r } ( \\Vert y _ { i , t } - y _ { j , t } \\Vert ) ( y _ { i , t } - y _ { j , t } ) . \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} F _ i ( z ) = \\exp \\left ( d ^ { - 1 } p ^ { c _ { \\lambda _ i } - i } \\left ( h ( q ^ { d p ^ i } , z ^ { d p ^ i } ) - \\frac { 1 } { p } h ( q ^ { d p ^ { i + 1 } } , z ^ { d p ^ { i + 1 } } ) \\right ) \\right ) \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { B _ { t } ( x _ 0 ) } ( | \\nabla u | ^ { 2 } ) ^ { \\mu } d x & \\le \\theta \\int _ { B _ { s } ( x _ 0 ) } ( | \\nabla u | ^ { 2 } ) ^ { \\mu } d x + C \\int _ { B _ s ( x _ 0 ) } | \\nabla h | _ n d x \\\\ & \\quad + C \\left \\{ \\frac { 1 } { ( s - t ) ^ { n } } \\int _ { B _ { s } ( x _ 0 ) } d ^ { n } ( u , h ) d x + \\int _ { B _ { s } ( x _ 0 ) } g d x \\right \\} \\end{aligned} \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} \\frac { f ' _ { \\beta } ( z ) } { f _ { \\beta } ( z ) } = - N z ^ { N - 1 } ( 1 - z ^ N ) ^ { - 1 } - \\sum _ { n = 0 } ^ { \\infty } \\mathcal { P } _ { n + 1 } z ^ n . \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} \\mu _ { 2 U } ( q _ 1 ) = - c \\varphi ( q _ 1 ) - b , \\rho _ { 2 U } ( q _ 1 ) = \\tilde { c } \\varphi ( q _ 1 ) - \\tilde { b } . \\end{align*}"}
-{"id": "6108.png", "formula": "\\begin{align*} \\widetilde { F } ( \\lambda ) : = \\int _ 0 ^ \\infty e ^ { - \\lambda t } d F ^ x ( t ) \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{align*} \\big ( \\pm \\tau \\phi _ l P \\big ) _ X = x ^ q + \\bigg ( \\frac { f _ { \\tau + 1 } f _ { \\tau - 1 } } { f _ \\tau ^ 2 } \\bigg ) ^ q . \\end{align*}"}
-{"id": "1236.png", "formula": "\\begin{align*} t _ k = i ^ i e ^ { \\pi i / 4 } , k = 0 , 1 , 2 , 3 . \\end{align*}"}
-{"id": "1009.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { p } \\Vert x _ { j } ^ { k } - x _ { j - 1 } ^ { k } \\Vert ^ { 2 } \\leq \\frac { 2 d ( x , C ) } { \\rho } \\Vert U _ { k } x - x \\Vert \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} \\frak { F } _ \\Omega ^ { ( a , b , c , d ) } = \\{ r _ 1 ( a \\omega _ 1 + b \\omega _ 2 ) + r _ 2 ( c \\omega _ 1 + d \\omega _ 2 ) ; 0 \\leq r _ 1 , r _ 2 < 1 , r _ 1 ^ 2 + r _ 2 ^ 2 \\neq 0 \\} . \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{align*} \\lVert f \\lVert ^ i _ { C ^ \\tau ( \\R ^ n ; F ) } : = \\sum _ { | \\beta | \\leq [ \\tau ] } \\left \\lVert \\partial _ z ^ \\beta f \\right \\lVert ^ i _ { L ^ \\infty ( \\R ^ n ; F ) } + \\sum _ { | \\beta | = [ \\tau ] } \\underset { z \\not = y } { \\sup } \\dfrac { \\left | \\partial _ z ^ \\beta f ( z ) - \\partial _ z ^ \\beta f ( y ) \\right | _ i } { | z - y | ^ { \\tau - [ \\tau ] } } . \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} W ( r _ { G _ 1 } ( t ) ) = \\frac { G _ 2 ( \\rho _ { G _ 1 , G _ 2 } ( t ) ) - m _ { G _ 2 } ( \\rho _ { G _ 1 , G _ 2 } ( t ) ) } { v _ { G _ 2 } ( \\rho _ { G _ 1 , G _ 2 } ( t ) ) } . \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{align*} F _ { w } ( 1 , t _ { 0 } , 0 ) \\varphi = 0 \\quad \\Longleftrightarrow \\quad \\varphi = c \\phi _ { 1 } \\ \\mbox { f o r s o m e } c > 0 . \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} \\Delta _ { \\Gamma , h } \\bf ( x ) = \\bf ( x { + } h ) - \\Gamma _ { x + h , x } \\bf ( x ) . \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} \\mathcal { L } | A | + ( | A | ^ 2 + \\dfrac 1 2 ) | A | = \\dfrac { | \\nabla A | ^ 2 - | \\nabla | A | | ^ 2 } { | A | } \\geq 0 . \\end{align*}"}
-{"id": "1490.png", "formula": "\\begin{align*} L = \\mathcal { L } + | A | ^ 2 - \\frac 1 2 = \\Delta + \\frac 1 2 \\left < x , \\nabla \\cdot \\right > + | A | ^ 2 - \\frac 1 2 . \\end{align*}"}
-{"id": "6725.png", "formula": "\\begin{align*} \\dot { z } \\left ( t \\right ) = e ^ { - t A } B \\left ( t \\right ) u \\left ( t \\right ) , \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} \\int _ \\R \\left ( | \\partial _ x | ^ { 1 / 2 } \\kappa \\right ) ^ 2 \\ , d x = \\int _ \\R \\left ( | \\partial _ x | ^ { 1 / 2 } f ( x , 0 ) \\right ) ^ 2 \\ , d x \\lesssim \\int _ { \\R ^ { 2 } _ { + } } | \\tilde \\nabla f | ^ 2 \\ , d x \\ , d \\tilde { z } . \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{align*} [ \\Phi _ \\infty ^ * \\wedge \\varphi _ \\infty ] = \\frac 1 2 ( | f | \\dot f / f - | f | ^ { - 1 } \\bar f \\dot f ) \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} d \\bar z \\wedge d z = 0 \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} ( x \\alpha \\cdot x ^ { \\rho } ) x = x \\alpha ~ \\textrm { f o r a l l } ~ x \\in L , ~ \\alpha , \\phi \\in A ( L ) . \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} \\operatorname { S u p p } \\left ( \\operatorname { T o r } _ i ^ { A _ { \\mathfrak { p } } } ( M _ { \\mathfrak { p } } , N _ { \\mathfrak { p } } ) \\right ) = \\left \\{ \\mathfrak { p } A _ { \\mathfrak { p } } \\right \\} \\mbox { f o r a l l } i \\gg 0 , \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{align*} d c , \\omega _ { - 1 } : = d p , \\omega _ \\alpha : = [ \\Big ( { \\theta ( p + z | \\tau ) \\over \\theta ( z | \\tau ) \\theta ( p | \\tau ) } e ^ { - c z } - { 1 \\over z } \\Big ) d p | z ^ \\alpha ] , \\alpha \\geq 0 , \\end{align*}"}
-{"id": "3158.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { n \\to \\infty } k _ n = \\infty , \\frac { k _ n } { n } < 1 , k _ n \\geq 1 . \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} h _ T ( \\phi _ T ( i ) ) = \\phi _ T ( h ( i ) ) \\geq \\phi _ T ( i ) \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} \\max _ { \\theta _ s , \\tau } \\quad & \\mathcal { P } _ c ( \\theta _ c , \\tau ) , \\\\ \\quad & \\mathcal { P } _ s ( \\theta _ s , \\tau ) = \\mathcal { P } _ T , \\\\ & \\theta _ c = 2 ^ { \\mathcal { R } _ T } ( 1 + \\theta _ s ) - 1 . \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} ~ A ( l _ 0 , l _ 1 ) = \\left [ d _ { l _ 0 } ^ { ( 0 ) } + r _ { l _ 0 } \\right ] b _ { l _ 1 , 0 } + \\left [ d _ { l _ 1 } ^ { ( 1 ) } + r _ { l _ 1 } \\right ] b _ { l _ 0 , 1 } , \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} & W ^ 0 _ { k + 2 } ( p _ 1 , \\dots , p _ { k + 1 } ) = ( \\mathbf { 1 } ) \\ast ( \\mathbf { 1 } ) \\ast \\dots \\ast ( \\mathbf { 1 } ) , k , \\\\ & W ^ 0 _ { l + 2 } ( p _ 1 , \\dots , p _ { l + 1 } ) = ( \\mathbf { 1 } ) \\ast ( \\mathbf { 1 } ) \\ast \\dots \\ast ( \\mathbf { 1 } ) , l , \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} L ^ b ( \\l ) : = X + \\lambda ^ { - 1 } \\Delta + \\lambda M \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} \\Omega _ { \\alpha _ { 2 , 4 } , \\bar { \\alpha } _ { 3 , 4 } } = \\frac { - 1 } { 2 \\cosh ( 2 t _ 1 ) \\cosh ( 2 t _ 2 ) } . \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} \\psi ( \\rho _ { m a x } ) = \\frac { \\pi } { 2 } , \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} o _ { n } ( 1 ) + c = I ( u _ { n } ) = \\frac { 1 } { 2 } M _ { k } ( \\| u _ { n } \\| ^ { 2 } ) - \\int _ { \\Omega } F _ { \\ast } ( u _ { n } ) d x \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} \\S _ 1 ( \\mu ) & = \\{ S _ { p + m _ 1 } ( \\mu ) _ i | \\ i \\in \\mathcal { A } _ 1 \\} , \\S _ 2 ( \\mu ) = \\{ S _ { p + m _ 1 } ( \\mu ) _ i | \\ i \\in \\mathcal { A } _ 2 \\} , \\\\ \\mathcal { A } _ 1 & = [ m _ 2 - r + 1 , m _ 2 ] , \\mathcal { A } _ 2 = [ m _ 1 + m _ 2 + 1 , m _ 1 + m _ 2 + r ] . \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} z ^ { ( 0 ) } + \\overline { z ^ { ( 0 ) } } = 0 \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} x _ 1 ' ( t ) = \\left . - 2 \\cdot \\left ( f _ 1 \\frac { \\partial f _ 1 } { \\partial x _ 2 } + f _ 2 \\frac { \\partial f _ 2 } { \\partial x _ 2 } \\right ) \\right | _ { ( x ( t ) ) } , \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{align*} X _ t = \\mathbf { F } _ 0 ( 1 ) \\cdot \\ldots \\mathbf { F } _ { n - 1 } ( 1 ) \\ldots \\cdot \\mathbf { F } _ { n } ( t - n ) \\end{align*}"}
-{"id": "3384.png", "formula": "\\begin{align*} \\arg ( b _ k ' - b _ k ) = \\arg \\ ! \\left ( \\frac { 2 \\pi i } { c _ k } \\right ) = O ( \\varepsilon _ k ) \\end{align*}"}
-{"id": "362.png", "formula": "\\begin{align*} \\langle \\rho _ \\gamma ( u , 1 ) P _ E , \\ , \\rho _ \\gamma ( v , 1 ) P _ E \\rangle = \\begin{cases} \\frac { | A | + 1 } { 2 } , & u = v ; \\\\ \\frac { 1 } { 2 } \\gamma \\left ( [ u , v ] ^ { 1 / 2 } \\right ) , & u \\neq v ; \\end{cases} \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} \\eta _ 1 & = \\frac 1 3 ( 1 - 2 \\lambda ) \\omega _ 1 + 2 \\lambda ( 1 - \\lambda ) \\omega _ 1 ' \\\\ \\eta _ 2 & = \\frac 1 3 ( 1 - 2 \\lambda ) \\omega _ 2 + 2 \\lambda ( 1 - \\lambda ) \\omega _ 2 ' \\end{align*}"}
-{"id": "8270.png", "formula": "\\begin{align*} Q ( \\lambda ) = \\binom { x _ 1 ( \\lambda ) } { 2 } - \\binom { x _ 2 ( \\lambda ) } { 1 } . \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{align*} ( \\alpha \\cdot f ) ( n ) = \\alpha \\cdot f ( n ) - f ( \\alpha \\cdot n ) \\ , , \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} \\left | \\sigma \\left ( \\sum _ { i = 0 } ^ { 2 ^ m - 1 } \\zeta ^ { l _ i } \\right ) \\right | \\leq 2 ^ m \\end{align*}"}
-{"id": "1856.png", "formula": "\\begin{align*} & G _ \\mu ( \\phi _ 0 ) = 1 \\\\ & G _ \\mu ( \\phi _ 1 ) = \\frac { 1 } { 2 } + \\mu \\\\ & G _ \\mu ( \\psi _ b ) = 1 + \\left ( \\mu - \\frac { 1 } { 2 } \\right ) \\frac { 1 } { ( 1 + b ^ 2 ) ^ 2 } . \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{align*} s = 0 , \\ , \\theta \\perp e _ 1 , \\ , \\theta \\perp e _ 2 . \\end{align*}"}
-{"id": "2671.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l } h ''' \\pm a h ' = 0 , \\\\ \\noalign { \\smallskip } \\pm ( h '' ) ^ { 2 } + a ( h ' ) ^ { 2 } = \\bar { c } , \\\\ \\noalign { \\smallskip } R i c _ { N } = ( n - 2 ) \\bar { c } g _ { N } , \\end{array} \\right . \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} p _ { \\mathrm { S } _ 0 } ( t _ \\mathrm { p } | n ) & = \\begin{cases} \\delta \\left ( t _ { \\mathrm { s } , 1 } \\right ) & n = 0 \\\\ \\delta \\left ( t _ { \\mathrm { w } , 1 } ( n ) \\right ) & \\end{cases} \\\\ p _ { \\mathrm { S } _ 1 } ( t _ \\mathrm { p } | n , q ) & = \\delta \\left ( t _ \\mathrm { c } + t _ { \\mathrm { w } , 1 } ( n + q ) \\right ) , \\end{align*}"}
-{"id": "1056.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } K _ { \\rm c l } \\int _ { \\mathbb { R } ^ 3 } w _ { n } ^ { ( j ) } ( x ) ^ { \\frac { 4 } { 3 } } { \\rm d } x & \\leq \\limsup _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ 3 } j _ { m \\epsilon _ { n } } ( w _ { n } ^ { ( j ) } ( x ) ) { \\rm d } x \\\\ & \\leq \\limsup _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ 3 } j _ { m \\epsilon _ { n } } ( w ( x ) ) { \\rm d } x = K _ { \\rm c l } \\int _ { \\mathbb { R } ^ 3 } w ( x ) ^ { \\frac { 4 } { 3 } } { \\rm d } x . \\end{align*}"}
-{"id": "9161.png", "formula": "\\begin{align*} b ( t ( a ) _ l ) a _ { l + 1 } & = a _ l b ( h ( a ) _ { l - 1 } ) & a \\in Q _ 1 l \\in \\{ 2 , \\dots , s \\} , \\\\ b ( t ( a ) _ 1 ) a _ 2 & = 0 & a \\in Q _ 1 . \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} V _ { F , \\varepsilon } ( x ) \\ \\doteq \\ \\sum _ { j = 1 } ^ { \\infty } \\frac { \\eta } { \\lambda _ { j } } \\mathbf { 1 } _ { \\{ x _ { j } \\} } ( x ) . \\end{align*}"}
-{"id": "7333.png", "formula": "\\begin{align*} \\nu = \\lim _ { n \\to \\infty } \\nu _ \\alpha ^ { L _ n } \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} c _ k ( f ) = - c _ k \\big ( g _ 1 \\big ) - c _ k \\big ( g _ { | q _ 1 q _ 2 | } \\big ) + e ^ { - i k \\tau _ 1 } c _ k \\big ( g _ { | q _ 1 | } \\big ) + e ^ { - i k \\tau _ 2 } c _ k \\big ( g _ { | q _ 2 | } \\big ) . \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} \\mathcal { F } ( R , Q ) = - \\left [ \\mathcal { G } ( R , Q ) \\right ] ^ 2 + \\mathcal { H } ( R , Q ) , \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{align*} \\mu ( x _ i , D ^ 4 ( x _ j ) ) = \\mu ( x _ i , \\lambda x _ { n - 6 + j } ) = 0 , \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} N _ { q , n } ( g ) = { \\textstyle \\dfrac { 1 } { n } ( q ^ { g p ^ e } - q ^ { g p ^ { e - 1 } } ) = { \\textstyle \\dfrac { 1 } { n } } q ^ { g n } ( 1 - q ^ { g ( p ^ { e - 1 } - p ^ e ) } ) } \\ , . \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} w _ { \\nu , \\beta } ^ { - 1 } ( \\alpha _ i ) = t _ { w _ { \\nu , \\beta } ^ { - 1 } ( i ) } - t _ { w _ { \\nu , \\beta } ^ { - 1 } ( i + 1 ) } = \\left \\{ \\begin{array} { l l } t _ { w _ { \\nu , \\beta } ^ { - 1 } ( \\nu _ 1 - 1 ) } - t _ { a } & \\textup { i f } i = \\nu _ 1 - 1 \\\\ t _ { b } - t _ { w _ { \\nu , \\beta } ^ { - 1 } ( \\nu _ 1 + 2 ) } & \\textup { i f } i = \\nu _ 1 + 1 \\end{array} \\right . . \\end{align*}"}
-{"id": "3861.png", "formula": "\\begin{align*} 4 \\ , e ^ { 2 \\pi y } \\ , \\pi \\left ( - 4 \\pi + 2 \\ , \\pi ^ 2 \\ , y \\right ) + 4 \\ , e ^ { 2 \\pi y } \\ , \\pi ^ 2 \\left ( 2 - 4 \\pi y + \\pi ^ 2 \\ , y ^ 2 \\right ) = 4 \\ , e ^ { 2 \\pi y } \\ , \\pi \\left ( \\pi ^ 3 \\ , y ^ 2 - 2 \\ , \\pi ^ 2 \\ , y - 2 \\pi \\right ) > 0 , \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} \\begin{cases} \\eta _ t + w _ x + w _ { x x x } = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , T ) , \\\\ w _ t + \\eta _ x + \\eta _ { x x x } = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , T ) , \\\\ \\eta ( 0 , t ) = \\eta ( L , t ) = \\eta _ { x } ( 0 , t ) = 0 , & \\ , \\ , ( 0 , T ) , \\\\ w ( 0 , t ) = w ( L , t ) = w _ { x } ( L , t ) = 0 , & d g \\ , \\ , ( 0 , T ) , \\\\ \\eta ( x , 0 ) = \\eta _ 0 ( x ) , w ( x , 0 ) = w _ 0 ( x ) , & \\ , \\ , ( 0 , L ) , \\\\ \\end{cases} \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{align*} \\norm { \\varphi } _ { L _ \\infty ( D ) } \\le C \\left \\{ \\begin{aligned} & h ^ { - \\frac d 2 } \\norm { \\varphi } , & d = 1 , 2 , 3 , \\\\ & h ^ { 1 - \\frac d 2 } ( \\log ( 1 / h ) ) ^ { 1 - 1 / d } \\norm { \\nabla \\varphi } , & d = 2 , 3 , \\end{aligned} \\right . \\forall \\varphi \\in S _ h . \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} ( p ) _ l : = ( a _ 1 ) _ { l } \\dots ( a _ { n - 1 } ) _ { l + 1 } ( a _ n ) _ { l - n + 1 } : i _ { l - n } \\rightarrow j _ l \\end{align*}"}
-{"id": "4515.png", "formula": "\\begin{align*} - \\operatorname { d i v } \\big ( w ( x ) | \\nabla u | ^ { p - 2 } \\nabla u \\big ) = g ( x ) f ( u ) \\ ; \\ ; \\mbox { i n } \\ ; \\ ; \\mathbb R ^ N \\end{align*}"}
-{"id": "845.png", "formula": "\\begin{align*} [ f ( v _ { n _ { j } } + h _ { j } ) - f ( v _ { n _ { j } } ) ] w & = g ( v _ { n _ { j } } + h _ { j } ) [ | v _ { n _ { j } } + h _ { j } | ^ { p - 1 } - | v _ { n _ { j } } | ^ { p - 1 } ] w \\\\ & + [ g ( v _ { n _ { j } } + h _ { j } ) - g ( v _ { n _ { j } } ) ] | v _ { n _ { j } } | ^ { p - 1 } w . \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} \\Psi ( z ) = \\sqrt { 1 + z } - 1 . \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} n _ { j + 1 } = \\min \\{ n \\ge n _ j + \\ell : \\omega _ n = i _ 0 \\} . \\end{align*}"}
-{"id": "1153.png", "formula": "\\begin{align*} \\Phi ( u ) \\cdot v = \\Phi ( u \\nu ( v ) ) \\ , . \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{align*} | \\omega _ { j , n } | ~ \\leq ~ | z _ { j , n } | + \\frac { t _ { j , n } } { n } \\mbox { w i t h } z _ { j , n } ~ \\neq ~ \\omega _ { j , n } , j = 1 , 2 , \\ldots , J _ n ; \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} H = \\big \\{ \\ , ( x _ 1 , \\dots x _ { i - 1 } , t , x _ { i + 1 } , \\dots , x _ { 2 m + 1 } ) \\ , \\big \\} \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{align*} \\min _ { x , y } \\tilde { \\delta } _ { x , y } \\leq \\sum _ { t = 1 } ^ { | \\mathcal { Z } | } \\min _ { x , y : \\ : p _ { X Y } ( x , y ) > 0 } \\delta _ { x , y , t } . \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} \\begin{aligned} I _ { 3 } & \\leq \\frac { \\lambda c _ { 2 } } { ( c _ { 1 } t _ { 1 } ) ^ { 2 } \\Gamma ( 2 \\alpha ) } \\Biggl | \\int _ { 0 } ^ { t _ { 1 } } \\left ( ( t _ { 1 } - s ) ^ { 2 \\alpha - 1 } - ( t _ { 2 } - s ) ^ { 2 \\alpha - 1 } \\right ) d s \\Biggr | \\\\ & \\leq \\frac { \\lambda c _ { 2 } } { ( c _ { 1 } t _ { 1 } ) ^ { 2 } \\Gamma ( 2 \\alpha + 1 ) } \\Biggl | t _ { 1 } ^ { 2 \\alpha } - t _ { 2 } ^ { 2 \\alpha } + ( t _ { 2 } - t _ { 1 } ) ^ { 2 \\alpha } \\Biggr | , \\end{aligned} \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{align*} 0 < \\varepsilon < \\left ( \\frac { 1 } { 1 6 a _ 2 [ w ] _ { A _ \\infty } } \\right ) ^ \\alpha \\cdot \\frac { 1 } { a _ 2 a _ 3 ^ 2 } = \\min \\left \\{ \\frac { 1 } { 2 } , \\frac { 1 } { a _ 2 a _ 3 } , \\left ( \\frac { 1 } { 1 6 a _ 2 [ w ] _ { A _ \\infty } } \\right ) ^ \\alpha \\cdot \\frac { 1 } { a _ 2 a _ 3 ^ 2 } \\right \\} \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow \\infty } \\big \\{ \\lambda ^ { - d / 2 } \\ , N [ \\mathfrak { e } , \\lambda V ] \\big \\} \\ = \\ \\lim _ { \\lambda \\rightarrow \\infty } \\big \\{ \\lambda ^ { - d / 2 } \\ , N _ { s c } [ \\mathfrak { e } , \\lambda V ] \\big \\} \\ = \\ 0 . \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} f ^ * ( x _ M ^ { m - n } \\times 1 ) = ( y _ M ^ { m - n } \\times 1 ) + ( y _ M ^ { m - 2 n } \\times \\omega _ { N } ) \\in H ^ { m - n } ( M ; \\R ) \\oplus ( H ^ { m - 2 n } ( M ; \\R ) \\otimes H ^ n ( N ; \\R ) ) . \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{align*} J _ r = \\begin{pmatrix} 0 & 0 & S _ r \\\\ 0 & I _ { n - 2 r } & 0 \\\\ S _ r & 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "2138.png", "formula": "\\begin{align*} \\norm { ( a , A ) } ' = \\sum _ { i = 1 } ^ d | \\pi _ 1 ^ { ( i ) } ( a ) | + \\sum _ { 1 \\leq i < j \\leq d } | \\pi _ 2 ^ { ( i j ) } ( A ) | ^ { 1 / 2 } \\end{align*}"}
-{"id": "5786.png", "formula": "\\begin{align*} s ( \\frak { u } ) = \\sum _ { 1 \\le i \\le I } e _ i u _ i \\in J _ { \\mathcal { C } } ( \\bar { K } ) \\qquad \\textrm { a n d } \\deg ( \\frak { u } ) = \\sum _ { 1 \\le i \\le I } e _ i \\in \\Z . \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } u ( \\alpha _ t ) ( t - 1 ) & = - 1 + \\lim _ { t \\rightarrow \\infty } [ ( t - 1 ) \\alpha _ t ] \\cdot \\alpha _ t \\int _ 0 ^ \\infty \\frac { 1 + s ^ 2 } { ( \\alpha _ t - s ) ^ 2 } \\frac { 1 } { s } \\ , d \\rho ( s ) \\\\ & = - 1 \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} p ( 0 , 0 | v , w ) = \\max \\{ 0 , 2 t - 1 \\} = \\begin{cases} 0 & 0 \\leq t \\leq \\frac { 1 } { 2 } \\\\ 2 t - 1 & \\frac { 1 } { 2 } \\leq t \\leq 1 \\end{cases} , \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{gather*} p ( u , v ) . w : = \\sigma ' \\big ( \\psi ( v , w ) . u - \\psi ( w , u ) . v \\big ) \\forall \\ , u , v , w \\in \\Box \\end{gather*}"}
-{"id": "1294.png", "formula": "\\begin{align*} P _ A ( z ) = \\sum _ { i = 0 } ^ { C } \\binom { A + B + C + 1 } { i } \\binom { A + C - i } { A } ( - z ) ^ i , \\\\ Q _ A ( z ) = ( - 1 ) ^ { C } \\sum _ { i = 0 } ^ { A } \\binom { A + C - i } { C } \\binom { B + i } { i } ( z ) ^ i , \\\\ E _ A ( z ) = \\sum _ { i = 0 } ^ { B } \\binom { A + i } { i } \\binom { A + B + C + 1 } { A + C + i + 1 } ( - z ) ^ i . \\end{align*}"}
-{"id": "6273.png", "formula": "\\begin{align*} S _ d ( \\alpha ) = \\lbrace \\alpha q ^ { d - 1 } , \\alpha q ^ { d - 3 } , \\ldots , \\alpha q ^ { - d + 1 } \\rbrace . \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} v _ i = \\prod _ { j = 1 } ^ i u _ j \\cdot w _ i \\cdot \\Big ( \\prod _ { j = 1 } ^ i u _ j \\Big ) ^ { - 1 } , i = 1 , \\dots , r . \\end{align*}"}
-{"id": "6836.png", "formula": "\\begin{align*} \\tilde { H } = \\begin{pmatrix} 0 _ { n \\times n } & 0 _ n & 0 _ { n \\times n } & 0 _ n & \\cdots & 0 _ { n \\times n } & 0 _ n & 0 _ { n \\times n } & 0 _ n \\\\ s _ 1 ^ \\top & 0 & s _ 2 ^ \\top & 0 & \\cdots & s _ n ^ \\top & 0 & 0 _ { 1 \\times n } & 0 \\\\ \\end{pmatrix} , \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{align*} \\dot { \\mathbf { z } } = \\mathbf { R } ( \\mathbf { z } , \\mathbf { \\phi } ) . \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} \\mathbb { E } [ B _ t ^ H B _ s ^ H ] = \\frac { 1 } { 2 } \\left ( | t | ^ { 2 H } + | s | ^ { 2 H } - | t - s | ^ { 2 H } \\right ) . \\end{align*}"}
-{"id": "4044.png", "formula": "\\begin{align*} \\epsilon & \\le 1 - \\max _ { \\substack { q _ { X Y } : \\ : q _ { X Y } \\preceq p _ { X Y } } } \\eta ( q _ { Y | X } ) \\overset { ( a ) } { = } 1 - \\max _ { \\substack { q _ { X Y } : \\ : q _ { X Y } \\preceq p _ { X Y } } } \\rho ^ 2 _ m ( q _ { X Y } ) \\end{align*}"}
-{"id": "2537.png", "formula": "\\begin{align*} F _ u ( \\eta , 0 ) = 1 - \\eta K \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} A ^ * ( x ) : = \\sup _ { \\theta \\in \\mathbb { R } ^ n } \\theta ^ \\top x - A ( \\theta ) . \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} N _ { s c } ^ { c o n t } [ V ] \\ = \\ \\frac { | S ^ { d - 1 } | } { d \\ , ( 2 \\pi ) ^ { d } } \\int V ^ { d / 2 } ( x ) \\ , \\mathrm { d } ^ { d } x , \\end{align*}"}
-{"id": "3365.png", "formula": "\\begin{align*} f ^ \\# ( c ) \\leq H ( r + t ) \\leq e H ( r ) = e f ^ \\# ( b ) \\end{align*}"}
-{"id": "2287.png", "formula": "\\begin{align*} \\forall \\psi _ 1 \\in \\Omega ^ k _ c ( V _ a ) , \\pi _ { z _ 0 } ^ { ( k ) } ( \\psi _ 1 ) = \\pi _ { a , z _ 0 } ^ { ( k ) } ( \\psi _ 1 ) , \\end{align*}"}
-{"id": "6443.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ^ { x } _ { j } ( T ) & < C \\tilde { C } _ { T } [ 6 K _ { 1 } K _ { 2 } + K _ { 1 } ^ 2 ] \\delta _ { j - 1 } ( T ) . \\end{aligned} \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} W = W _ 1 + W _ 2 + \\cdots + W _ r , & & \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} x _ n ( t ) = \\min \\{ x _ n ^ A ( t ) , x _ n ^ B ( t ) \\} , \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} L _ 0 ( \\chi h ) = - \\chi p _ 0 + [ L _ 0 , \\chi ] h + \\int _ 0 ^ 1 \\chi ( L _ 0 - L _ { t h } ) ( h ) \\ , d t . \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} \\| \\Delta f \\| _ 2 \\lesssim & \\| \\Delta _ \\mathbf { A } f \\| _ 2 + \\| \\mathbf { A } \\| _ \\infty \\| \\nabla f \\| _ 2 + \\| \\mathbf { A } \\| _ \\infty ^ 2 \\| f \\| _ 2 . \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} ( F g _ 1 , F g _ 2 , g _ 2 ^ { - 1 } g _ 1 \\check { F } , \\check { F } ) & = ( F g _ 1 ' , F g _ 2 ' , { g _ 2 ' } ^ { - 1 } g _ 1 ' \\check { F } , \\check { F } ) X _ { 2 , 2 } . \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} T s : = s , \\ | s | \\leq 1 ; T s : = s / | s | , \\ | s | > 1 . \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} \\Theta ( z , \\tau ) = \\sum _ { k \\in \\Z } e ^ { \\pi i k ^ 2 \\tau } e ^ { 2 k \\pi i z } . \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{align*} \\hat { H } ^ 0 : = \\mathcal { U } _ \\Gamma H ^ 0 \\mathcal { U } _ \\Gamma ^ { - 1 } = \\int _ { \\mathbb { T } _ * } ^ \\oplus \\hat { H } ^ 0 ( \\theta ) \\ , d \\theta \\ , , \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} \\sigma _ 1 = \\left ( \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\end{array} \\right ) , \\sigma _ 2 = \\left ( \\begin{array} { c c } 0 & - i \\\\ i & 0 \\end{array} \\right ) , \\sigma _ 3 = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} P \\bigl ( Z _ { n _ q } ^ q \\in O , I _ { n _ q + 1 } = j \\bigr ) = P \\bigl ( Z _ { n _ q } ^ q \\in O \\bigr ) P ( I _ { n _ q + 1 } = j ) > c _ q > 0 . \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} & E \\left [ S ^ { p a r t , m } _ { s , t } \\right ] = E \\left [ S ^ { p a r t , 0 } _ { s , t } \\right ] \\\\ & - \\frac { n } { 2 } E \\left [ \\ln ( \\beta ( t , q _ t ) ) \\right ] + \\frac { n } { 2 } E \\left [ \\ln ( \\beta ( s , q _ s ) ) \\right ] \\\\ = & E \\left [ S ^ { p a r t , 0 } _ { s , t } \\right ] - \\frac { n } { 2 } E \\left [ \\ln ( \\beta ( t , q _ t ) / \\beta ( s , q _ s ) ) \\right ] \\\\ \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} \\begin{bmatrix} B _ { 1 } ^ \\top \\otimes A _ 1 & - D _ { 1 } ^ \\top \\otimes C _ 1 \\\\ & \\ddots & \\ddots \\\\ & & B _ { r - 1 } ^ \\top \\otimes A _ { k - 1 } & - D _ { r - 1 } ^ \\top \\otimes C _ { r - 1 } \\\\ - ( D _ { r } ^ \\top \\otimes C _ { r } ) P _ { n , n } & & & B _ { r } ^ \\top \\otimes A _ { r } \\\\ \\end{bmatrix} { \\cal X } = { \\cal E } , \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} p = \\sum _ { r \\in X ^ m } \\alpha _ r r + p ^ \\prime , \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} \\frac { d } { d t } \\frac { d ^ 2 ( \\eta ( t ) , y ) } { 2 } = u ( y ) - u \\big ( \\eta ( t ) \\big ) . \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} ( \\mathbb { B } + \\delta \\mathbb { B } ) \\operatorname { v e c } ( \\tilde { X } ) = \\operatorname { v e c } ( \\tilde { Y } ) , \\frac { \\norm { \\delta \\mathbb { B } } _ 2 } { \\norm { \\mathbb { B } } _ 2 } \\leq \\varepsilon _ { B _ k } , \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} \\| \\overline { u } \\| _ { L ^ { m \\chi ^ { n + 1 } } ( \\Omega _ { R _ { n + 1 } } ) } \\le C ^ { 1 / \\chi ^ n } \\left [ \\frac { 2 ^ { n + 1 } } { R _ { 1 } - R _ { 2 } } + \\left ( \\frac { 2 ^ { n + 1 } } { R _ { 1 } - R _ { 2 } } \\right ) ^ { 1 / m } + \\left ( \\frac { \\| f \\| _ q } { k ^ { m - 1 } } \\right ) ^ { \\beta } \\right ] ^ { 1 / \\chi ^ n } \\| \\overline { u } \\| _ { L ^ { m \\chi ^ n } ( \\Omega _ { R _ n } ) } , \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{align*} [ a , b ^ { - 1 } ] ^ { b ^ { 2 n } } & = ( b , b ^ { - 1 } ) ^ { ( a ^ n , a ^ n ) } \\equiv ( b [ b , a ] ^ n , b ^ { - 1 } [ b , a ] ^ { - n } ) \\mod \\gamma _ 3 ( G ) \\times \\gamma _ 3 ( G ) \\\\ [ a , b ^ { - 1 } ] ^ { b ^ { 2 n + 1 } } & = ( b ^ { - 1 } [ b ^ { - 1 } , a ] , b ) ^ { ( a ^ n , a ^ n ) } \\equiv ( b ^ { - 1 } [ b , a ] ^ { - n - 1 } , b [ b , a ] ^ { n } ) \\mod \\gamma _ 3 ( G ) \\times \\gamma _ 3 ( G ) \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} \\beta _ i ( x _ 1 , \\dots , x _ { i - 1 } , x _ i , x _ { i + 1 } , \\dots ) = ( x _ 1 , \\dots , x _ { i - 1 } , 1 - x _ i , x _ { i + 1 } , \\dots ) \\end{align*}"}
-{"id": "5579.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ z \\vartheta _ 1 ( 0 , i x ) & = \\sum _ { k \\in \\Z } ( - 1 ) ^ k ( 2 k + 1 ) e ^ { - \\pi ( k + 1 / 2 ) ^ 2 x } \\\\ & = 2 \\ , e ^ { - \\tfrac { \\pi } { 4 } x } \\prod _ { k \\in \\N } \\left ( 1 - e ^ { - 2 k \\pi x } \\right ) ^ 3 , x \\in \\R _ + . \\end{aligned} \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} ( 1 - q ) \\left ( \\sum _ { 0 \\le i \\le \\frac { b - 3 } { 3 } } q ^ { 6 i + 2 } \\binom { d - 4 i - 1 } { 1 } _ q \\binom { 2 d - 8 i - 1 } { 1 } _ q + q ^ { 6 i } \\binom { 3 d - 1 2 i + 1 } { 1 } _ q \\right ) \\leq ( 1 - q ) \\sum F _ { \\lambda } ( q ) , \\end{align*}"}
-{"id": "718.png", "formula": "\\begin{align*} { \\rm M } ( P ) : = | a _ 0 | \\prod _ { j = 1 } ^ { m } \\max \\{ 1 , | \\alpha _ j | \\} . \\end{align*}"}
-{"id": "1527.png", "formula": "\\begin{align*} - \\int _ { \\Sigma } \\varphi ( L \\varphi ) e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma = \\int _ { \\Sigma } \\left ( | \\nabla \\varphi | ^ { 2 } - \\bigr ( | A | ^ { 2 } - \\frac 1 2 ) \\varphi ^ 2 \\right ) e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma \\geq 0 . \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} \\max _ { x , y } \\left ( \\frac { q _ { X Y } ( x , y ) } { p _ { X Y } ( x , y ) } \\right ) = 1 . \\end{align*}"}
-{"id": "1822.png", "formula": "\\begin{align*} \\mathbf { H } u ( x ) = \\lambda u ( x ) \\ , , \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} - \\Delta _ p u - V ( x ) | u | ^ { p - 2 } u = f ( x ) . \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{align*} \\int \\limits _ Y \\left ( \\int \\limits _ 0 ^ T f ( g _ t y ) \\ , d t \\right ) ^ 2 \\ , d \\mu ( y ) & = \\int \\limits _ Y \\int \\limits _ 0 ^ T \\int \\limits _ 0 ^ T f ( g _ t y ) f ( g _ s y ) \\ , d t \\ , d s \\ , d \\mu ( y ) \\\\ & = \\int \\limits _ 0 ^ T \\int \\limits _ 0 ^ T \\left ( \\int _ { Y } f ( g _ t x ) f ( g _ s x ) \\ , d \\mu ( x ) \\right ) \\ , d t \\ , d s . \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} x ^ 2 \\frac { \\theta _ 2 ' ( x ) } { \\theta _ 2 ( x ) } = - \\frac { x } { 2 } - \\frac { \\theta _ 4 ' ( \\tfrac { 1 } { x } ) } { \\theta _ 4 ( \\tfrac { 1 } { x } ) } . \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} L ( \\phi ) = \\omega \\wedge \\phi , \\quad \\Lambda ( \\phi ) = \\omega \\lrcorner \\ , \\phi , \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} \\theta ( m , \\mu , \\lambda ) = \\begin{cases} q ^ { \\kappa ( m , \\mu , \\lambda ) } & , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} t _ k = i ^ k , k = 0 , 1 , 2 , 3 . \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} \\theta _ T > \\theta _ { T - 1 } > \\ldots > \\theta _ 1 > \\theta = \\theta _ 0 > 0 \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} g _ { t , k } ( z _ 1 , \\ldots , z _ k ) : = \\int g _ t ( { \\bf z } ) \\prod _ { j = k + 1 } ^ { m - 1 } 2 e ^ { - 2 z _ j } d z _ j \\ , . \\end{align*}"}
-{"id": "8134.png", "formula": "\\begin{align*} ( Z U _ 0 ) ^ 2 \\leq C \\left ( \\Sigma _ { i = 1 } ^ { n } x _ i ^ 2 ( U _ 0 ) _ i ^ 2 + y ^ 2 ( U _ 0 ) _ y ^ 2 + t ^ 2 ( U _ 0 ) _ t ^ 2 \\right ) \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} & g ( \\beta , \\theta _ { c , 1 } , 0 ) = g ( \\beta , \\theta _ { c , 2 } , 0 ) = 0 , \\\\ & g ( \\beta , \\theta , 0 ) > 0 , ( \\forall \\theta \\in ( \\theta _ { c , 1 } , \\theta _ { c , 2 } ) ) , \\\\ & g ( \\beta , \\theta , 0 ) < 0 , ( \\forall \\theta \\in [ 0 , \\theta _ { c , 1 } ) \\cup ( \\theta _ { c , 2 } , 4 \\pi / \\beta ] ) . \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} \\tilde { P } = \\begin{pmatrix} P & 0 \\\\ 0 & p ' \\end{pmatrix} , \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{align*} 0 = \\lim _ { k } d ( x ^ { k } , C ) = \\lim _ { k } d ( x ^ { n _ { k } } , C ) \\geq d ( y , C ) \\geq 0 \\end{align*}"}
-{"id": "7871.png", "formula": "\\begin{align*} B ^ a ( \\mathbb { R } ^ d ) : = \\{ f \\in & L ^ 2 ( \\mathbb { R } ^ d ) ; \\| f \\| _ a : = \\| f \\| + \\\\ & \\sum _ { | \\alpha | \\leq 2 a } \\| \\partial _ x ^ { \\alpha } f \\| + \\| < \\cdot > ^ { 2 a ( M + 1 ) } f \\| < \\infty \\} \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l l l } w ( 0 , t ) = w _ { x x } ( 0 , t ) = 0 & & \\\\ w _ { x } ( 0 , t ) = \\alpha _ { 0 } \\eta _ { x } ( 0 , t ) & & \\\\ w ( L , t ) = \\alpha _ { 2 } \\eta ( L , t ) & & \\\\ w _ { x } ( L , t ) = - \\alpha _ { 1 } \\eta _ { x } ( L , t ) & & \\\\ w _ { x x } ( L , t ) = - \\alpha _ { 2 } \\eta _ { x x } ( L , t ) & & \\end{array} \\right . \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} \\sum _ { z \\in P _ \\nu } L _ m \\chi _ y ( z ) \\overline { L _ m \\chi _ { y ' } ( z ) } = q ^ { 2 | S _ \\mu ( m - 1 ) | } \\sum \\chi \\left ( \\sum _ { s \\in S _ \\mu } \\sum _ { t \\in T _ \\nu } \\left ( Y _ { s , t } - Y ' _ { s , t } \\right ) Z _ { s , t } \\right ) , \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} \\nabla ^ 2 f ( x _ * ) ^ { - 1 } = \\nabla ^ 2 A ( \\theta ) = \\mathbb { E } _ { X \\sim P _ \\theta } [ ( X - \\mathbb { E } _ { X \\sim P _ \\theta } [ X ] ) ( X - \\mathbb { E } _ { X \\sim P _ \\theta } [ X ] ) ^ \\top ] , \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{align*} \\sum _ { v \\in R } \\binom { d _ R ( v ) } 2 + \\frac 1 2 \\sum _ { v \\in R } d _ R ( v ) ( s - 1 - d _ R ( v ) ) \\end{align*}"}
-{"id": "146.png", "formula": "\\begin{align*} ( D ^ 1 _ { ( A , \\Phi ) } ) ^ * ( ( \\dot A , \\dot \\Phi ) - D ^ 1 _ { ( A , \\Phi ) } \\xi ) = 0 \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{align*} d _ { n } ( ( A _ { 1 } , \\dots , A _ { n } ) , ( B _ { 1 } , \\dots , B _ { n } ) ) : = \\frac { 1 } { n } \\sum _ { j = 1 } ^ { n } d _ T ( A _ { j } , B _ { j } ) . \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} \\widetilde \\Lambda _ h ^ { e , \\triangle } : = \\span \\{ \\tilde \\mu _ { h , i } ^ e : \\ , \\alpha _ { i } ^ e < \\alpha _ { \\rm { s t a b } } \\} , \\widetilde \\Lambda _ h ^ { e , \\Pi } : = \\span \\{ \\tilde \\mu _ { h , i } ^ e : \\ , \\alpha _ { i } ^ e \\ge \\alpha _ { \\rm { s t a b } } \\} . \\end{align*}"}
-{"id": "6029.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l l l } \\eta ( 0 , t ) = \\eta ( L , t ) = 0 \\eta _ { x } ( 0 , t ) = f ( t ) & & ( 0 , T ) \\\\ w ( 0 , t ) = w ( L , t ) = 0 w _ { x } ( L , t ) = g ( t ) & & ( 0 , T ) \\end{array} \\right . \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} s _ C = 4 k - 2 v . \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} d _ { A ( t ) } & \\int _ \\tau ^ t \\psi ( s ) d s \\\\ & = \\hat w ( \\tau ) - \\hat w ( t ) + \\int _ \\tau ^ t \\ ( [ A ( t ) , \\psi ( s ) ] - P ^ \\perp \\ ( \\zeta ( s ) + [ A ( s ) , \\psi ( s ) ] \\ ) \\ ) d s . \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} C : Y ^ 2 = X ^ 3 + ( \\alpha z + \\beta ) X + ( \\gamma z + \\delta ) \\mapsto \\left \\{ \\begin{array} { l l } C _ { + } : & Y ^ 2 = X ^ 3 + ( \\alpha a + \\beta ) X + ( \\gamma a + \\delta ) \\\\ C _ { - } : & Y ^ 2 = X ^ 3 + ( - \\alpha a + \\beta ) X + ( - \\gamma a + \\delta ) \\end{array} \\right . \\end{align*}"}
-{"id": "657.png", "formula": "\\begin{align*} L _ j = \\delta ( T _ 1 - T _ 0 ) \\left ( 1 + \\frac { \\epsilon _ 1 } { \\delta } \\right ) ^ { j - 1 } \\geq ( T _ 1 - T _ 0 ) ( \\delta + ( j - 1 ) \\epsilon _ 1 ) \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} E _ { \\tau } ( 1 ) : = \\inf \\left \\{ \\mathcal { E } _ { \\tau } ( \\rho ) : 0 \\leq \\rho \\in L ^ { 1 } \\cap L ^ { \\frac { 4 } { 3 } } ( \\mathbb { R } ^ { 3 } ) , \\int _ { \\mathbb { R } ^ 3 } \\rho ( x ) { \\rm d } x = 1 \\right \\} , \\end{align*}"}
-{"id": "7086.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty \\atop L \\in \\N } \\frac { \\int _ 0 ^ { \\infty } d x x e ^ { L ^ d f _ L ( x ) } u _ L ( x ) } { \\int _ { 0 } ^ { \\infty } d x x e ^ { L ^ d f _ L ( x ) } g _ L ( x ) } = \\frac { u ( a ) } { g ( a ) } . \\end{align*}"}
-{"id": "4202.png", "formula": "\\begin{align*} L ^ X f & = \\sum _ { i = 1 } ^ n b _ X ^ i \\partial _ { x _ i } f + \\sum _ { i , j = 1 } ^ n a _ X ^ { i j } \\partial _ { x _ i x _ j } ^ 2 f , \\\\ L ^ Y f & = \\sum _ { i = 1 } ^ m b _ Y ^ i \\partial _ { x _ i } f + \\sum _ { i , j = 1 } ^ m a _ Y ^ { i j } \\partial _ { x _ i x _ j } ^ 2 f , \\\\ L f & = ( L ^ X + L ^ Y ) f . \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} \\int _ { G ^ { s ( x ) } } f ( x z ) \\ , d \\lambda ^ { s ( x ) } ( z ) = \\int _ { G ^ { t ( x ) } } f ( y ) d \\lambda ^ { t ( x ) } ( y ) . \\end{align*}"}
-{"id": "4893.png", "formula": "\\begin{align*} \\dd { X ^ x } ( t ) & = b ( t , X ^ x ( t ) ) \\dd { t } + \\sigma \\dd { W } ( t ) , \\\\ X ^ x ( 0 ) & = x \\in \\R ^ d \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} \\ll \\nabla _ T ^ * \\nabla _ T \\phi , \\psi \\gg & = \\int _ M \\langle \\nabla _ { \\rm T } \\phi , \\nabla _ { \\rm T } \\psi \\rangle \\\\ & = \\int _ M \\langle \\phi , - \\sum _ a \\nabla _ { V _ a } \\nabla _ { \\bar V _ a } \\psi \\rangle + \\int _ M \\langle \\phi , \\nabla _ { H ^ { 0 , 1 } } \\psi \\rangle \\\\ & = \\ll \\phi , \\nabla _ T ^ * \\nabla _ T \\psi \\gg , \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} & e _ 0 ^ { + } \\mapsto q ^ { ( 1 - N ) / 2 } \\sum _ { m = 1 } ^ N \\alpha _ m R _ m , & & e _ 1 ^ { + } \\mapsto q ^ { ( N - 1 ) / 2 } \\sum _ { m = 1 } ^ N ( L _ m R _ m ) ^ { - 1 } L _ m , \\\\ & e _ 0 ^ { - } \\mapsto \\sum _ { m = 1 } ^ N \\alpha _ m ^ { - 1 } L _ m , & & e _ 1 ^ { - } \\mapsto \\sum _ { m = 1 } ^ N ( R _ m L _ m ) ^ { - 1 } R _ m , \\\\ & k _ 0 \\mapsto \\prod _ { m = 1 } ^ N K _ m ^ { - 1 } , & & k _ 1 \\mapsto \\prod _ { m = 1 } ^ N K _ m . \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{align*} \\frac { \\mathcal { R } \\theta _ F ( z + 1 ) } { \\mathcal { R } \\theta _ F ( z ) } = 1 \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} [ a _ 1 a _ 2 , b _ 1 , b _ 2 ] = a _ 1 [ a _ 2 , b _ 1 , b _ 2 ] + [ a _ 1 , b _ 1 ] [ a _ 2 , b _ 2 ] + [ a _ 1 , b _ 2 ] [ a _ 2 , b _ 1 ] + [ a _ 1 , b _ 1 , b _ 2 ] a _ 2 \\end{align*}"}
-{"id": "7156.png", "formula": "\\begin{align*} D \\ ( x y ) = D ( x ) y - x D ( 1 ) y + x D ( y ) . \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} 1 - \\epsilon \\geq \\max _ { q _ { X Y } \\ , p _ { J | X Y } } s ^ * ( q _ { X Y J } ) = \\eta ( p _ { J | X , Y } ) . \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} W _ n ' ( \\chi ) W _ { n + 1 } - W _ n ( \\chi ) W _ { n + 1 } ' = W _ { n + 1 } ( \\chi ) W _ n . \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{align*} | A + A | = ( 2 + \\varepsilon ) | A | - 1 \\leq \\min \\big \\{ \\ , 3 | A | - 4 , \\ ; \\tfrac { p } { 2 } + | A | - 2 \\ , \\big \\} . \\end{align*}"}
-{"id": "5901.png", "formula": "\\begin{align*} \\nu = ( 1 ^ p , 0 ^ { n - m _ 1 - m _ 2 - p } , 2 ^ { m _ 2 - p } , 1 ^ { m _ 1 + p } ) \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{align*} \\chi _ \\gamma ( a , \\alpha , z ) & = \\sum _ { b \\in A } \\langle \\pi _ \\gamma ( a , \\alpha , z ) \\delta _ b , \\delta _ b \\rangle \\\\ [ 5 p t ] & = \\sum _ { b \\in A } [ \\pi _ \\gamma ( a , \\alpha , z ) \\delta _ b ] ( b ) \\\\ [ 5 p t ] & = \\sum _ { b \\in A } \\gamma ( z \\langle b - \\tfrac { 1 } { 2 } a , \\alpha \\rangle ) \\delta _ b ( b - a ) . \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} ( L _ m R _ m ) ^ { - 1 } L _ m v = \\begin{cases} q ^ { - \\kappa ( m , \\mu , \\lambda ) } L _ m v & , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{align*} \\mathbb { E } d _ { K } ^ 2 ( Y _ t ^ { 0 , x ; u _ { \\cdot } } ) & \\le \\liminf _ { \\delta \\rightarrow 0 } \\mathbb { E } \\phi _ { \\delta } ( Y _ s ^ { 0 , x ; u _ { \\cdot } } ) \\\\ & = 0 , 0 \\le t \\le T , \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{align*} S ''' ( i , j , k , \\alpha , \\beta ) = \\omega _ { i k } ^ \\alpha \\underline \\wedge \\omega _ { j k } ^ \\beta , i < j < k \\in [ n ] , \\alpha , \\beta \\geq 0 , \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} L & = \\frac { x ^ 6 ( 1 - 2 x ) ( 1 - x ) } { K } , \\\\ L ' & = \\frac { x ^ 6 ( 1 - 2 x ) ( 1 - x ) ^ 2 } { K } , \\\\ L '' & = \\frac { x ^ 6 ( 1 - 3 x + 3 x ^ 2 ) } { K } . \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} & \\sum _ { ( t ) ^ 2 \\in ( Y ^ n ) ^ 2 } ( t ) ^ 2 = \\sum _ { \\substack { ( t _ 1 ) ^ 2 \\in ( Y ^ { p } ) ^ 2 , t _ 2 \\in Y ^ { q } \\\\ p + q + 1 = n } } ( t _ 1 ) ^ 2 \\vee t _ 2 \\end{align*}"}
-{"id": "6822.png", "formula": "\\begin{align*} F _ r ( \\phi , \\psi ) | _ { t = T } \\leq F _ r ( \\phi , \\psi ) | _ { t = 0 } \\exp \\big ( \\int _ 0 ^ T ( s ^ { - 1 } ( t ) + f ( t ) ) d t \\big ) . \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} \\mathcal { P } ^ 0 _ u ( \\lambda , \\gamma ) = \\Phi _ { u , \\gamma } ( 0 , h , \\lambda ) - \\frac { 1 } { 2 } \\int _ 0 ^ u \\xi '' ( s ) s \\gamma ( s ) d s , \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} & \\varphi ( x ) k ( x , D _ { x ' } ) \\psi ( x ' ) = ( \\varphi ( x ) \\eta ( x _ n ) ) k ( x , D _ { x ' } ) \\psi ( x ' ) + \\varphi ( x ) ( 1 - \\eta ( x _ n ) ) k ( x , D _ { x ' } ) \\psi ( x ' ) . \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} v _ t = v _ { x x } + v ( a _ 0 - b _ 0 v ) , x \\in \\R ^ 1 . \\end{align*}"}
-{"id": "4480.png", "formula": "\\begin{align*} I _ 1 = - 2 \\int _ { \\Omega } w ^ 2 \\delta u \\nabla \\delta \\cdot ( A \\nabla u ) d x , I _ 2 = - 2 \\int _ { \\Omega } w \\delta ^ 2 u \\nabla w \\cdot ( A \\nabla u ) d x . \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} \\mu _ { 1 j } - c h = \\tilde { c } _ { \\alpha } \\mbox { a n d } \\mu _ { 2 \\alpha } + c \\varphi = - b _ { j } . \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} { \\rm S I N R } _ k = \\frac { { \\bf h } _ k ^ H { \\bf w } _ k { \\bf w } _ k ^ H { \\bf h } _ k } { { \\bf h } _ k ^ H { \\bf W } _ k { \\bf W } _ k ^ H { \\bf h } _ k + 1 / { \\rho _ t } } , \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\ln ( 1 + e ^ { - \\alpha ^ * } e ^ { - x } ) d x = \\int _ 0 ^ { \\beta ^ * N } \\frac { t \\ , d t } { 1 - e ^ { - t } } \\end{align*}"}
-{"id": "2405.png", "formula": "\\begin{align*} E \\left [ T \\right ] - E \\left [ T _ 1 \\right ] = E \\left [ T - T _ 1 \\right ] \\leq E \\left [ T _ 2 \\right ] P \\{ T _ 1 < T _ 2 \\} . \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} \\hat u _ { V , r } ( z ) : = \\begin{cases} { 1 \\over c _ d a ^ d r ^ d } \\log | \\hat s _ V ( - r \\overline z ) | & z \\in \\overline \\C _ + \\\\ { 1 \\over c _ d a ^ d r ^ d } \\log | \\hat s _ V ( - r z ) | & z \\in \\overline \\C _ - . \\end{cases} \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} \\Phi _ \\alpha ( E ) = \\int _ E \\int _ E \\frac { 1 } { | x - y | ^ \\alpha } d x d y = \\int _ E V _ { E , \\alpha } ( x ) d x , \\end{align*}"}
-{"id": "5718.png", "formula": "\\begin{align*} r \\binom { r - 1 } { 2 } - ( r - 1 ) \\left ( \\binom { r } { 2 } - k \\right ) & = ( r - 1 ) \\left ( \\frac { 1 } { 2 } r ( r - 2 ) - \\binom { r } { 2 } + k \\right ) \\\\ & = ( r - 1 ) \\left ( k - r / 2 \\right ) \\ge 0 . \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} B ^ r ( m ) & : = \\sum _ { x = 1 } ^ m \\prod _ { k = 1 } ^ x \\frac { 1 - b ( k ) } { 1 + b ( k ) } , B ^ l ( m ) : = \\sum _ { x = 1 } ^ m \\prod _ { k = 1 } ^ x \\frac { 1 - b ( - k ) } { 1 + b ( - k ) } , \\\\ I ^ r ( b ) & : = \\sum _ { x = 1 } ^ { \\infty } \\frac { 1 } { 1 + B ^ r ( x ) } , I ^ l ( b ) : = \\sum _ { x = 1 } ^ { \\infty } \\frac { 1 } { 1 + B ^ l ( x ) } , \\end{align*}"}
-{"id": "3385.png", "formula": "\\begin{align*} \\arg c _ k = \\frac { \\pi } { 2 } + O ( \\varepsilon _ k ) \\quad \\arg c _ k = - \\frac { \\pi } { 2 } + O ( \\varepsilon _ k ) \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} c _ \\alpha ( T ) = \\frac { \\left < T \\phi , \\phi \\right > _ \\lambda } { \\left < \\phi , \\phi \\right > _ \\lambda } , \\end{align*}"}
-{"id": "3049.png", "formula": "\\begin{align*} & Q [ A u ] = Q [ a \\left ( x \\right ) \\left ( u ^ { q } - u \\right ) ] , \\\\ & ( 1 - Q ) [ A u ] = ( 1 - Q ) [ a \\left ( x \\right ) \\left ( u ^ { q } - u \\right ) ] . \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} \\begin{aligned} [ b ] & | S _ 1 ^ * | - | S _ 2 ^ * | - \\frac { 2 E _ { S _ 1 ^ * } - E _ { S _ 1 ^ * , S _ 3 ^ * } + E _ { S _ 2 ^ * , V _ 1 } - E _ { S _ 1 ^ * , V _ 1 } } { \\tau - 1 } \\\\ & > | S _ 1 ^ * | - 1 - | S _ 2 ^ * | - \\frac { 2 E _ { S _ 1 ^ * } - E _ { S _ 1 ^ * , S _ 3 ^ * } - d _ { i , S _ 1 ^ * } - d _ { i , S _ 3 ^ * } + E _ { S _ 2 ^ * , V _ 1 } - E _ { S _ 1 ^ * , V _ 1 } + d _ { i , V _ 1 } } { \\tau - 1 } , \\end{aligned} \\end{align*}"}
-{"id": "4088.png", "formula": "\\begin{align*} \\tilde { \\eta } ( \\hat { \\vect { x } } _ n ) = f ( \\varphi ( \\hat { \\vect { x } } ^ t _ k ) ) t ( \\hat { \\vect { x } } _ k ^ t ; \\vect { \\Sigma } ^ t ) \\beta _ n \\left ( \\beta _ n \\vect { I } + \\vect { \\Sigma } ^ t \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} \\bar { h } ( \\alpha ) = \\sum _ { j \\leq k } h _ { i _ j } ( \\alpha ) = h _ { i _ k } ( \\alpha ) + \\sum _ { j < k } h _ { i _ j } ( \\alpha ) \\end{align*}"}
-{"id": "4544.png", "formula": "\\begin{align*} \\Pr \\left [ \\bigcap _ { j = 1 } ^ { L } A _ { j } \\right ] \\geq \\prod _ { j = 1 } ^ { L } \\Pr \\left [ A _ { j } \\right ] . \\end{align*}"}
-{"id": "3095.png", "formula": "\\begin{align*} C ^ { 1 0 } _ 3 : \\left \\{ \\begin{aligned} 3 a _ { 2 , 6 } + a _ { 3 , 8 } & = 0 \\\\ ( 9 a _ { 1 , 5 } + 1 6 a _ { 2 , 7 } + a _ { 3 , 9 } ) a _ { 3 , 8 } - 3 ( 2 a _ { 1 , 6 } + a _ { 2 , 8 } ) a _ { 4 , 9 } & = 0 \\\\ 3 a _ { 1 , 4 } - a _ { 3 , 8 } & = 0 \\end{aligned} \\right . \\end{align*}"}
-{"id": "753.png", "formula": "\\begin{align*} P _ { \\beta , P } ( X ) : = X ^ { p + 1 } - t _ 1 X ^ { p } - t _ 2 X ^ { p - 1 } - \\ldots - t _ { p } X - ( 1 + t _ { p + 1 } ) \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{align*} R _ { e n c } : = \\frac { H ( X ) } { k + m } = \\frac { k } { k + m } , \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} f ( r ) = \\times e ^ { - a r } r ^ { b - 1 } \\ \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} \\delta _ B = - \\bar * d _ T \\bar * , \\quad \\delta _ T = - \\bar * d _ B \\bar * , \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\nabla u _ { 0 } | ^ { 2 } q _ { 0 } ( q _ { 0 } - 1 ) u _ { 0 } ^ { q _ { 0 } - 2 } \\phi _ { 1 } & = \\int _ { \\Omega } ( - \\Delta u _ { 0 } ) q _ { 0 } u _ { 0 } ^ { q _ { 0 } - 1 } \\phi _ { 1 } + u _ { 0 } ^ { q _ { 0 } } ( \\Delta \\phi _ { 1 } ) \\\\ & = \\int _ { \\Omega } ( a u _ { 0 } ^ { q _ { 0 } } ) q _ { 0 } u _ { 0 } ^ { q _ { 0 } - 1 } \\phi _ { 1 } + u _ { 0 } ^ { q _ { 0 } } ( - q _ { 0 } a u _ { 0 } ^ { q _ { 0 } - 1 } \\phi _ { 1 } - \\sigma _ { 1 } \\phi _ { 1 } ) \\\\ & = - \\sigma _ { 1 } \\int _ { \\Omega } u _ { 0 } ^ { q _ { 0 } } \\phi _ { 1 } , \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} \\Delta _ f ( u e ^ h ) & = e ^ h \\left [ \\Delta u + 2 \\langle \\nabla u , \\nabla h \\rangle + u ( \\Delta h + | \\nabla h | ^ 2 ) \\right ] \\\\ & - e ^ h ( \\langle \\nabla f , \\nabla u \\rangle + u \\langle \\nabla f , \\nabla h \\rangle ) \\\\ & = e ^ h [ \\Delta u - \\langle \\nabla ( f - 2 h ) , \\nabla u \\rangle + u ( \\Delta h + \\langle \\nabla h , \\nabla ( h - f ) \\rangle ] \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} \\Delta ( e _ i ^ + ) = e _ i ^ + \\otimes k _ i + 1 \\otimes e _ i ^ + , & & \\Delta ( e _ i ^ - ) = e _ i ^ - \\otimes 1 + k _ i ^ { - 1 } \\otimes e _ i ^ - , & & \\Delta ( k _ i ) = k _ i \\otimes k _ i . \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} \\mathit { C W E } ( \\mathcal { C } ) = \\sum _ { \\mathbf { c } \\in \\mathcal { C } } w ( \\mathbf { c } ) . \\end{align*}"}
-{"id": "453.png", "formula": "\\begin{align*} \\begin{pmatrix} 2 & 3 & 1 0 & 7 \\\\ 1 & 9 & 5 & 1 0 \\\\ 5 & 2 & 9 & 3 \\\\ 3 & 5 & 1 & 2 \\end{pmatrix} , \\begin{pmatrix} 1 & 5 & 6 & 3 \\\\ 0 & 0 & 1 & 6 \\\\ 0 & 1 & 0 & 5 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} { } \\begin{aligned} { \\vert \\vert { w } \\vert \\vert } _ { \\infty } \\leq { T } \\ ; \\mbox { w h i c h i m p l i e s } \\ ; { \\vert \\vert { v } \\vert \\vert } _ { \\infty } \\leq { T } { \\lambda } ^ { \\frac { 1 } { \\delta + p - 1 } } \\end{aligned} \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} n \\gamma ^ { n - 1 } \\cdot \\lambda ^ { - [ \\frac { \\log n \\gamma ^ { n - k - 1 } } { \\log \\lambda } ] } = \\lambda ^ { \\frac { \\log n \\gamma ^ { n - 1 } } { \\log \\lambda } - [ \\frac { \\log n \\gamma ^ { n - k - 1 } } { \\log \\lambda } ] } = \\lambda ^ { \\frac { \\log \\gamma ^ { k } } { \\log \\lambda } + \\frac { \\log n \\gamma ^ { n - k - 1 } } { \\log \\lambda } - [ \\frac { \\log n \\gamma ^ { n - k - 1 } } { \\log \\lambda } ] } \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\sup _ { | x | > c t } u ( x , t ) = 0 \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} \\mathbb { E } ^ P _ { n , 0 } \\big ( \\exp \\big ( a ( n ) z ^ * _ { n , i } - k _ { n , i } \\big ) \\exp \\big ( a ( n ) z ^ * _ { n , j } - k _ { n , j } \\big ) \\big ) = \\exp \\big ( k _ { n , i , j } - k _ { n , i } - k _ { n , j } \\big ) . \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} p _ { - k } ( 1 | ~ ~ 1 _ { 1 } ' ) p _ { - k } ( 1 | ~ ~ 1 _ { 2 } ' ) & = ( k - 1 ) ^ 2 , \\\\ p _ { - k } ( 2 | ~ ~ 1 _ { 1 } ' ~ ~ ~ 1 _ { 2 } ' ) & = k + ( k - 2 ) + \\cdots + 1 = \\frac { k ( k - 1 ) } { 2 } + 1 . \\end{align*}"}
-{"id": "2046.png", "formula": "\\begin{align*} b ( x ) = \\dfrac { \\overline { \\delta \\beta ^ { ( 2 ) } } } { 2 x } + \\dfrac { \\overline { \\delta \\beta ^ { ( 4 ) } } } { 4 x ^ 2 } \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} X = L _ { \\sigma } ^ q ( \\Omega ) \\times L ^ q ( \\Omega ) ^ 3 . \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} f _ h ( x ) = \\int _ 0 ^ { \\infty } P _ t h ( x ) - \\pi ( h ) \\ , d t , \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} K ^ { k , \\gamma + \\beta } _ { m , x , x + h } = \\sum _ { \\substack { l \\in \\N ^ d _ 0 \\setminus \\{ 0 \\} \\\\ k + l \\in \\partial ( \\gamma + \\beta ) } } \\int _ { \\R ^ d } D ^ { k + l } _ 1 K _ m ( x { + } z , \\cdot ) \\ , \\mu ^ l ( h , \\mathrm { d } z ) , \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} \\frac { 4 } { 3 } \\sigma _ { f } Q ( x ) ^ { \\frac { 1 } { 3 } } - ( | \\cdot | ^ { - 1 } \\star Q ) ( x ) + \\frac { 2 } { 3 } \\begin{cases} = 0 & Q ( x ) > 0 , \\\\ \\geq 0 & Q ( x ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "4904.png", "formula": "\\begin{align*} A ( t ) & : = c \\int _ { S } ^ { t } \\abs { B ( s , X ^ x _ s ) } \\frac { \\norm { D u ( s , X ^ x ( s ) ) - D u ( s , X ^ y ( s ) ) } _ { o p } } { \\abs { \\tilde { Z } ( s ) } } \\ 1 _ { \\tilde { Z } ( s ) \\neq 0 } \\dd { s } \\\\ & + c \\int _ { S } ^ { t } \\frac { \\norm { D u ( s , X ^ x ( s ) ) \\sigma ( s , X ^ x ( s ) ) - D u ( s , X ^ y ( s ) ) \\sigma ( s , X ^ y ( s ) ) } _ { H S } ^ 2 } { \\abs { \\tilde { Z } ( s ) } ^ 2 } \\ 1 _ { \\tilde { Z } ( s ) \\neq 0 } \\dd { s } \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} y = B _ D + B _ E \\end{align*}"}
-{"id": "3828.png", "formula": "\\begin{align*} B _ 3 = \\left \\{ \\begin{array} { l l } E _ { 1 , \\ell _ 1 , \\ell _ 2 , \\ell _ 3 , \\ell _ 4 } ; \\\\ E _ { 2 , \\ell _ 1 , \\ell _ 2 , \\ell _ 3 , \\ell _ 4 } , \\\\ E _ { 3 , \\ell _ 1 , \\ell _ 2 , \\ell _ 3 , \\ell _ 4 } , \\end{array} : \\ ; 2 \\leq \\ell _ 1 \\leq s , \\ ; 0 \\leq \\ell _ 2 , \\ell _ 3 , \\ell _ 4 \\leq s - 2 \\right \\} . \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} \\R _ { + } ^ { \\mathcal { N } } : = \\{ x \\in \\R ^ { \\mathcal { N } } \\mid x _ i \\geq 0 , \\ ; \\forall i \\in \\mathcal { N } \\} . \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} S ^ n _ i = \\bigcup _ { I : | I | = n P _ 1 \\phi _ I = \\psi _ { I ^ n _ i } } \\phi _ { I } ( F ) \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} k = 5 j , r = 4 j - g , \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} 0 = F ^ { 1 } ( x ; t ) + F ^ { 2 } ( y ; t ) - K ( 0 , 0 ; t ) Q ( 0 , 0 ; t ) + x y . \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} 2 \\lambda ( 1 - \\lambda ) ( \\omega _ 2 ' + \\omega _ 1 ' ) = 2 i + O ( \\lambda \\log ( \\lambda ) ) \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\beta ^ { - } } T _ { \\gamma } ^ { n } ( 1 ) = T _ { \\beta } ^ { n } ( 1 ) = \\lim _ { \\gamma \\to \\beta ^ { + } } T _ { \\gamma } ^ { n } ( 1 ) , \\mbox { f o r a l l } ~ n \\geq 1 . \\end{align*}"}
-{"id": "2201.png", "formula": "\\begin{align*} & B ^ { d } = \\{ x \\in \\R ^ d : | x | < 1 \\} , \\\\ & \\overline { B ^ d } = \\{ x \\in \\R ^ d : | x | \\leq 1 \\} , \\\\ & ( e _ 1 , \\dots , e _ d ) \\R ^ d . \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} n _ { Y , \\Theta } = - c _ Y + 1 - \\sum _ { \\alpha \\in \\Phi _ { Q ^ U } \\cup \\Phi _ s ^ + } \\alpha \\circ \\mathcal { P } ( \\mu _ Y ) . \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{align*} \\bigg \\| \\sum _ { k = k _ 0 } ^ \\infty 2 ^ { - k \\alpha } \\mathcal { C } _ k g \\bigg \\| _ { L ^ q _ t L ^ 2 _ x } \\lesssim \\| g \\| _ { L ^ 2 _ t L ^ 2 _ x } , \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} T f ( x ) = \\int _ { \\Omega _ 1 } K ( x , y ) f ( y ) d \\mu _ 1 ( y ) \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} \\nabla \\cdot v = 0 \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} \\sum _ { ( \\hat { z _ 1 } , \\hat { z _ 2 } ) \\in H ^ 2 ( M _ 1 ) \\times H ^ 2 ( M _ 2 ) } S W _ { M _ 1 } ( \\hat { z _ 1 } ) S W _ { M _ 2 } ( \\hat { z _ 2 } ) = \\sum _ { \\hat { z _ 2 } } S W _ { M _ 2 } ( \\hat { z _ 2 } ) ( \\sum _ { \\hat { z _ 1 } } S W _ { M _ 1 } ( \\hat { z _ 1 } ) ) . \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} { \\operatorname { s g n } } ( I , I ' ) : = { \\operatorname { s g n } } ( I \\cap I ' ; p , q ) , \\end{align*}"}
-{"id": "1373.png", "formula": "\\begin{align*} J \\bigr [ u ^ { \\ast } \\bigl ] & = \\mathcal { E } ^ { G } \\left [ \\int _ 0 ^ T c \\bigl ( s , X _ s ^ { 0 , x ; u ^ { \\ast } } , u _ s \\bigr ) d s + \\Psi ( X _ T ^ { 0 , x ; u ^ { \\ast } } ) \\Bigl \\vert \\mathcal { F } _ 0 , \\right ] \\\\ & \\equiv \\int _ 0 ^ T c \\bigl ( s , X _ s ^ { 0 , x ; u ^ { \\ast } } , u _ s \\bigr ) d s + \\Psi ( X _ T ^ { 0 , x ; u ^ { \\ast } } ) , \\ , \\ , X _ 0 ^ { 0 , x ; u ^ { \\ast } } = x . \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} [ c S ^ \\ell , \\{ 1 + a S ^ i , T \\} ) = [ c S ^ \\ell , 1 + a S ^ i ) \\\\ [ c T ^ \\ell , \\{ 1 + a T ^ i , S \\} ) = [ c T ^ \\ell , 1 + a T ^ i ) , \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} \\int _ M \\langle A _ { \\kappa _ B ^ \\sharp } ( \\phi ) , \\phi \\rangle = \\int _ M \\langle \\kappa _ B ^ \\sharp \\lrcorner \\ , \\phi , \\delta _ B \\phi \\rangle + \\frac 1 2 \\int _ M \\kappa _ B ^ \\sharp ( | \\phi | ^ 2 ) = 0 . \\end{align*}"}
-{"id": "43.png", "formula": "\\begin{align*} \\int _ { I _ n } g ( s ) \\alpha _ 0 ( s ) d s = 0 . \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} \\tilde { n } ( r ) = \\frac { \\tilde { i } ( r ) } { \\tilde { h } ( r ) } . \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} M _ r = \\begin{pmatrix} \\frac { 1 } { \\sqrt { 2 } } I _ r & 0 & \\frac { 1 } { \\sqrt { 2 } } S _ r \\\\ 0 & I _ { n - 2 r } & 0 \\\\ - \\frac { 1 } { \\sqrt { 2 } } S _ r & 0 & \\frac { 1 } { \\sqrt { 2 } } I _ r \\end{pmatrix} M _ r J _ r M _ r ^ { - 1 } = \\begin{pmatrix} - I _ { r } & 0 \\\\ 0 & I _ { n - r } \\end{pmatrix} , \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} \\eta _ { W _ { - \\theta } } [ \\frak { u } , y ] ( x ) = \\eta _ { W _ { - ( g - 1 ) o } } [ \\frak { u } , y + \\vartheta ] ( x + \\vartheta ) \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} \\bigl [ [ a _ 1 , \\dots , a _ { n - 1 } ] b , c \\bigr ] = [ a _ 1 , \\dots , a _ { n - 1 } , c ] b + [ a _ 1 , \\dots , a _ { n - 1 } ] [ b , c ] ( a _ i , b , c \\in A ) . \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} T ( X , Y ) = & u \\ , g \\left ( \\nabla _ X U , Y \\right ) - ( \\nabla _ X u ) g ( U , Y ) + \\bar { \\varphi } ( \\nabla _ X U , U , Y ) \\\\ & ( u ^ 2 - | U | ^ 2 ) \\bar { T } ( X , Y ) + 2 \\bar { T } ( X , U ) g ( U , Y ) - 2 u \\ , \\bar { \\varphi } ( U , \\bar { T } ( X ) , Y ) , \\end{align*}"}
-{"id": "7718.png", "formula": "\\begin{align*} Q _ 1 & = \\mathrm { P } \\left ( z _ t > \\frac { \\epsilon _ 1 } { \\rho } , z _ m < \\max \\left \\{ \\frac { \\epsilon _ 2 } { \\rho \\xi _ 2 } , \\cdots , \\frac { \\epsilon _ i } { \\rho \\xi _ { i } } \\right \\} \\right ) \\\\ & = \\mathrm { P } \\left ( z _ t > \\frac { \\epsilon _ 1 } { \\rho } , z _ m < \\frac { ( 1 + \\epsilon _ 1 ) } { \\phi _ i \\left ( \\rho - \\frac { \\epsilon _ 1 } { z _ t } \\right ) } \\right ) . \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} & W ^ 0 _ { l + 2 } ( p , p _ 1 , \\dots , p _ { l + 1 } ) \\ast W ^ 0 _ { m + 2 } ( p ' , p ^ { ' } _ { 1 } \\dots , p ^ { ' } _ { m + 1 } ) = \\\\ & \\underset { t _ 1 \\in Y ^ p , t _ 2 \\in Y ^ q , p + q + 1 = l } { \\sum } \\underset { t ^ { ' } _ 1 \\in Y ^ { p ' } , t ^ { ' } _ 2 \\in Y ^ { q ' } , p ' + q ' + 1 = m } { \\sum } t _ 1 \\vee ( t _ 2 \\ast t ' ) + ( t \\ast t ^ { ' } _ 1 ) \\vee t ^ { ' } _ 2 . \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{align*} ( L \\chi ) ( t , q , z ) = & \\frac { 1 } { 2 } \\Sigma _ { k l } ( t , q ) ( \\partial _ { z _ k } \\partial _ { z _ l } \\chi ) ( t , q , z ) - \\tilde \\gamma _ { k l } ( t , q ) \\delta ^ { l i } z _ i ( \\partial _ { z _ k } \\chi ) ( t , q , z ) , \\\\ ( L ^ * h ) ( t , q , z ) = & \\partial _ { z _ k } \\bigg ( \\frac { 1 } { 2 } \\Sigma _ { k l } ( t , q ) \\partial _ { z _ l } h ( t , q , z ) + \\tilde \\gamma _ { k l } ( t , x ) \\delta ^ { l i } z _ i h ( t , q , z ) \\bigg ) . \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} f _ 0 ( x , \\lambda ) = \\max _ { m \\in S _ 0 } \\bigl ( m x + \\lambda ( m ^ 2 - u ) + J ( m ) \\bigr ) . \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{align*} i _ z ^ * \\omega = \\omega _ Y . \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} V K _ 2 ^ { t o p } ( K ) = \\{ \\{ u , x \\} | u , x \\in K ^ \\times , v _ K ( u - 1 ) > ( 0 , 0 ) \\} , \\end{align*}"}
-{"id": "9189.png", "formula": "\\begin{align*} \\Upsilon _ n - \\Upsilon _ n ' \\cdot \\gamma ^ n = \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{align*} \\int _ S w _ R \\ , d \\mu = O ( 1 ) . \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} & B _ 1 ( X ; Y \\| Z ) = \\ ! \\inf _ { p _ { J | X Y Z } } \\Bigg [ I ( X ; Y | J ) + \\max _ { U V \\rightarrow X Y \\rightarrow Z J } I ( U ; J | V ) \\ ! - \\ ! I ( U ; Z | V ) \\Bigg ] . \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} \\sum _ { \\substack { b \\mid n \\\\ b < n } } \\left ( \\frac { b } { n } \\right ) ^ { k - 1 } \\leq \\sum _ { d > 1 } \\frac { 1 } { d ^ { k - 1 } } = \\zeta ( k - 1 ) - 1 \\ll \\frac { 1 } { 2 ^ k } . \\end{align*}"}
-{"id": "7745.png", "formula": "\\begin{gather*} A _ 4 = 2 J q ^ 4 - 3 J q ^ 3 J q ^ 1 + J q ^ 2 J q ^ 2 + J q ^ 1 J q ^ 3 = 0 , \\\\ A _ 5 = 5 J q ^ 5 - 5 J q ^ 4 J q ^ 1 + J q ^ 2 J q ^ 3 - 2 J q ^ 1 J q ^ 4 = 0 , \\\\ A _ 6 = 9 J q ^ 6 - 7 J q ^ 5 J q ^ 1 + J q ^ 2 J q ^ 4 + 3 J q ^ 1 J q ^ 5 = 0 . \\end{gather*}"}
-{"id": "9294.png", "formula": "\\begin{align*} \\widehat u _ { \\tau } ( t ) : = \\widehat u _ { \\tau , m } ^ S ( t ) : = ( \\Phi _ { j , t - t _ m } ^ S \\widehat { \\Phi _ { j , \\tau } ^ D } ) \\prod _ { j = 1 } ^ { m - 1 } \\big ( \\Phi _ { j , \\tau } ^ S \\widehat { \\Phi _ { j , \\tau } ^ D } \\big ) u _ \\tau ( 0 ) , t \\in T _ m , \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{align*} \\Phi _ { u , \\gamma } ( u , x , \\lambda ) = f ( x , \\lambda ) : = \\max _ { m \\in S } \\bigl ( m x + \\lambda m ^ 2 + J ( m ) \\bigr ) . \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} I [ \\mu ] & = \\iint f ( \\theta - \\phi ) d \\mu ( \\theta ) d \\mu ( \\phi ) \\\\ & = \\sum _ { k \\neq 0 } c _ k ( f ) \\iint e ^ { i ( \\theta - \\phi ) } d \\mu ( \\theta ) d \\mu ( \\phi ) \\\\ & = \\sum _ { k \\neq 0 } c _ k ( f ) | c _ k ( \\mu ) | ^ 2 \\geq 0 . \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{align*} J ( x , \\tau ) & = - \\frac { \\partial } { \\partial x } \\left ( D _ 1 ( x ) \\rho ( x , \\tau ) \\right ) + D _ 2 ( x ) \\rho ( x , \\tau ) \\\\ D _ 1 ( x ) & = \\frac { 1 } { 2 } \\frac { \\Sigma _ 1 ^ 2 } { 2 x } \\\\ D _ 2 ( x ) & = \\frac { \\overline { \\delta \\beta ^ { ( 2 ) } } } { 2 x } + \\frac { \\overline { \\delta \\beta ^ { 4 } } } { 4 x ^ 2 } . \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} s y _ 1 \\dots y _ { i - 1 } = y _ 1 \\dots y _ { i - 1 } s + [ s , y _ 1 \\dots y _ { i - 1 } ] = y _ 1 \\dots y _ { i - 1 } s + \\sum _ { j = 1 } ^ { i - 1 } y _ 1 \\dots y _ { j - 1 } [ s , y _ j ] y _ { j + 1 } \\dots y _ { i - 1 } \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} F _ + ^ { ( k ) } ( h ) = \\sum _ { x _ 1 , \\ldots , x _ k \\in B } \\frac { \\partial ^ k F _ + ( h ) } { \\partial \\phi _ { x _ 1 } \\cdots \\partial \\phi _ { x _ k } } . \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{align*} f _ { v e c t } ( t ) = \\begin{cases} 0 , & 0 \\le t \\le \\frac { 1 } { n } , \\\\ n t ( n t - 1 ) , & \\frac { 1 } { n } \\le t \\le \\frac { n - 1 } { n } , \\\\ ( n ^ 2 - n ) ( 2 t - 1 ) , & \\frac { n - 1 } { n } \\le t \\le 1 . \\end{cases} \\end{align*}"}
-{"id": "8061.png", "formula": "\\begin{align*} Y ( y ) = A y ^ s I _ s ( L ( \\xi , \\sigma ) y ) + B y ^ s K _ s ( L ( \\xi , \\sigma ) y ) , \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{align*} x ( i , k + 1 ) = x ( i , k ) + \\tilde { \\Gamma } u ( i , k ) , \\ \\ k \\in \\mathbb { N } \\end{align*}"}
-{"id": "5730.png", "formula": "\\begin{align*} \\tilde \\delta _ n = \\tilde \\kappa _ g ' \\tilde \\kappa _ t - \\tilde \\kappa _ g \\tilde \\kappa _ t ' , \\tilde \\delta _ n ' = \\tilde \\kappa _ \\nu ' ( \\tilde \\kappa _ g ^ 2 + \\tilde \\kappa _ t ^ 2 ) + \\tilde \\kappa _ g '' \\tilde \\kappa _ t - \\tilde \\kappa _ g \\tilde \\kappa _ t '' u = 0 \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} R _ { x ^ { \\lambda } \\cdot x \\phi ^ { - 1 } } = R ^ { - 1 } _ x R _ { x \\phi ^ { - 1 } } \\Rightarrow y x \\cdot ( x ^ { \\lambda } \\cdot x \\phi ^ { - 1 } ) = y \\cdot x \\phi ^ { - 1 } \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} M ( P , \\vec { \\nu } ) = \\int _ { \\vec { \\alpha } \\in [ 0 , 1 ) ^ r } S ( \\vec { \\alpha } , \\vec { \\nu } ) \\dd \\vec { \\alpha } . \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} \\bar P ' = g \\bar P g ^ { - 1 } . \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} [ s x _ 3 , x _ 1 , x _ 2 ] = [ s , x _ 1 , x _ 2 ] x _ 3 + [ s , x _ 1 ] [ x _ 3 , x _ 2 ] + [ s , x _ 2 ] [ x _ 3 , x _ 1 ] + s [ x _ 3 , x _ 1 , x _ 2 ] \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} \\gamma \\circ \\alpha = \\beta \\ \\ . \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} \\xi = \\Psi ( X _ T ^ { t , x ; u _ { \\cdot } } ) , \\mathbb { P } - { a l m o s t \\ , s u r e l y \\ , ( a . s ) } . \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { d - 1 } \\psi _ d ^ k = \\chi _ { \\mathrm { r e g } } , \\end{align*}"}
-{"id": "2763.png", "formula": "\\begin{align*} \\int _ I | g \\cdot b - f \\cdot a | & \\leq \\int _ I | g \\cdot b - f \\cdot b | + \\int _ I | f \\cdot b - f \\cdot a | \\\\ & = \\int _ I | g - f | | b | + \\int _ I | f | | b - a | \\\\ & \\leq \\| b \\| \\int _ I | g - f | + \\| b - a \\| \\int _ I | f | \\\\ & \\leq ( \\| a \\| + \\delta ) \\delta + \\delta \\| f \\| _ { L ^ 1 } < \\epsilon . \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} F _ + ^ { ( k ) } ( 0 ) = \\sum _ { x _ 1 , \\ldots , x _ k \\in B } \\frac { \\partial ^ k F _ + } { \\partial \\phi _ { x _ 1 } \\cdots \\partial \\phi _ { x _ k } } , \\end{align*}"}
-{"id": "8881.png", "formula": "\\begin{align*} u ^ * ( p ) = \\sup _ { y \\in \\mathfrak { a } _ s } ( p ( y ) - u ( y ) ) . \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} - \\Delta _ p u - V ( x ) | u | ^ { p - 2 } u = f ( x ) \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{align*} & F _ { t } ( q , t , w ) = - Q \\left [ a ( x ) \\left \\{ q ( t \\phi _ { 1 } + w ) ^ { q - 1 } \\phi _ { 1 } - \\phi _ { 1 } \\right \\} \\right ] , \\\\ & F _ { q } ( q , t , w ) = - Q \\left [ a ( x ) ( t \\phi _ { 1 } + w ) ^ { q } \\log ( t \\phi _ { 1 } + w ) \\right ] , \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} \\partial _ t ^ 2 \\bigl ( \\frac { z _ \\alpha } { | z _ \\alpha | } \\bigr ) = \\partial _ t ^ 2 ( e ^ { i \\theta } ) = ( - \\theta _ t ^ 2 + i \\theta _ { t t } ) e ^ { i \\theta } . \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathbb { Z } } M _ i [ H ( \\nu , \\cdot ) ] \\left ( \\mu \\right ) = t ^ { - \\chi ( \\vec { x } , \\vec { y } ) } \\sum _ { i \\in \\{ z , z + l \\} } M _ i \\left [ H ( \\nu , \\cdot ) \\right ] ( \\mu ^ { * } ) , \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { n - 1 } \\left [ a _ { k + 1 } - a _ k + ( 1 - \\delta _ { a _ k 0 } ) \\right ] = \\sum \\limits _ { k = 1 } ^ { n - 1 } \\left [ b _ { k + 1 } - b _ k + ( 1 - \\delta _ { b _ k 0 } ) \\right ] , \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} d z = - \\frac { d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\end{align*}"}
-{"id": "1739.png", "formula": "\\begin{align*} | p | _ i ^ { ( m ) } : = \\underset { | \\alpha | , | \\beta | \\leq i } { \\max } \\ \\underset { ( x , \\xi ) \\in \\R ^ n \\times \\R ^ n } { \\sup } \\left \\| \\partial _ { \\xi } ^ \\alpha \\partial _ x ^ \\beta p ( x , \\xi ) \\right \\| _ X \\langle \\xi \\rangle ^ { - m + | \\alpha | } \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} \\eta _ { x _ j } \\{ f ( x _ 1 , \\ldots , x _ k ) \\} = f ( x _ 1 , \\ldots , x _ { j - 1 } , q x _ j , x _ { j + 1 } , \\ldots , x _ k ) \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} \\tilde F _ { E , \\omega } ( p , z ) : = { \\tilde \\theta _ { E , \\omega } ( p + z ) \\over \\tilde \\theta _ { E , \\omega } ( p ) \\tilde \\theta _ { E , \\omega } ( z ) } \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{align*} \\psi ( W , Q _ W , [ \\lambda ^ { - 1 } \\xi ] ) = ( E , \\lambda \\Phi ) . \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} \\begin{pmatrix} 2 & 4 & 5 & 1 \\\\ 3 & 6 & 5 & 1 \\\\ 7 & 7 & 1 & 1 0 \\\\ 6 & 6 & 1 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "8628.png", "formula": "\\begin{align*} w t ( C _ { 1 } ) & = ( p - 1 ) ( p ^ { e - 2 } + p ^ { m + d - 2 } ) , \\\\ A _ { C _ { 1 } } & = p ^ { e } - p ^ { e - 2 d } . \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { r } k _ i ( P ) = \\sum _ { i = 1 } ^ { r } \\left [ \\ , \\sum _ { j = i + 1 } ^ { r } u _ { i , j } + \\sum _ { j = i } ^ { r } v _ { i , j } \\right ] , \\end{align*}"}
-{"id": "4728.png", "formula": "\\begin{align*} d _ w = | N ( y ^ { - 1 } ) | + | N ( v ^ { - 1 } ) \\cap v ( \\Phi _ H ^ - ) | . \\end{align*}"}
-{"id": "6679.png", "formula": "\\begin{align*} m ( t _ { \\ast } ) t _ { \\ast } = \\int _ { \\Omega } f ( u ) u d x . \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} \\begin{cases} u _ t = \\Delta u , x \\in D _ { 3 L } \\cr u = 0 , \\ , ( 0 , T ) \\times \\partial D _ { 3 L } \\cr u ( \\cdot , 0 ) = \\frac { \\delta _ 0 } { 2 } \\chi _ { _ { D \\cap D _ { 0 n } } } . \\end{cases} \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} \\Gamma ^ \\star _ t ( z ) : = \\left \\{ p \\in \\Gamma ( 0 , z ) \\colon \\ , | p | \\le | z | ( 1 + h ^ \\star _ t ) , \\xi ( p _ i ) > - s ^ \\xi _ t , \\sigma ( p _ i ) < s ^ \\sigma _ t , 0 \\le i \\le | p | - 1 \\right \\} . \\end{align*}"}
-{"id": "4597.png", "formula": "\\begin{align*} & [ L , X \\lrcorner ] = \\epsilon ( J X ^ b ) , [ \\Lambda , \\epsilon ( X ^ b ) ] = - ( J X ) \\lrcorner , \\\\ & [ L , \\epsilon ( X ^ b ) ] = [ \\Lambda , X \\lrcorner ] = 0 . \\end{align*}"}
-{"id": "8586.png", "formula": "\\begin{align*} \\delta ( H [ V _ i ] ) & \\geq \\alpha \\binom { n - 1 } { 2 } - \\kappa n ^ 2 \\geq \\alpha \\frac { n ^ 2 } { 2 } - \\kappa n ^ 2 - \\alpha \\frac { n } { 2 } \\\\ & = \\big ( \\alpha - \\kappa / 2 \\big ) \\frac { n ^ 2 } { 2 } - \\alpha \\frac { n } { 2 } \\geq \\big ( \\alpha - \\kappa \\big ) \\frac { n ^ 2 } { 2 } \\\\ & \\geq \\big ( \\alpha - \\kappa \\big ) \\binom { | V _ i | - 1 } { 2 } . \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{align*} & \\mathrm { T e r m } _ k = \\sum _ { \\alpha ' + \\alpha '' = \\alpha - 1 } ( \\mathrm { a d } x _ i ) ^ { \\alpha ' } ( - \\mathrm { a d } x _ j ) ^ { \\alpha '' } ( [ t _ { k i } , t _ { k j } ] ) = \\sum _ { \\gamma + \\delta = \\alpha - 1 } [ ( \\mathrm { a d } x _ i ) ^ \\gamma ( t _ { k i } ) , ( - \\mathrm { a d } x _ j ) ^ \\delta ( t _ { k j } ) ] \\\\ & = \\sum _ { \\gamma + \\delta = \\alpha - 1 } ( - 1 ) ^ \\gamma [ ( \\mathrm { a d } x _ k ) ^ \\gamma ( t _ { k i } ) , ( \\mathrm { a d } x _ k ) ^ \\delta ( t _ { k j } ) ] , \\end{align*}"}
-{"id": "8717.png", "formula": "\\begin{align*} \\lim _ { \\mathring r \\to \\infty } M _ { \\rm H } ( u , \\mathring r ) = M _ { \\rm B } ( u ) . \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} \\kappa ( m , \\mu , \\lambda ) - \\kappa ( m ' , \\mu , \\lambda ) = | \\lbrace t \\in T _ \\mu \\setminus \\lambda \\mid m < t \\le m ' \\rbrace | - | \\lbrace s \\in S _ \\mu \\setminus \\lambda \\mid m \\le s < m ' \\rbrace | . \\end{align*}"}
-{"id": "5460.png", "formula": "\\begin{align*} \\dot { \\mathbf { x } } = \\mathbf { A } \\mathbf { x } + \\mathbf { G } _ { n l i n } ( \\mathbf { x } ) + \\varepsilon \\mathbf { G } _ { e x t } ( \\Omega _ 1 t , . . . , \\Omega _ k t ) . \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} & \\frac { d E } { d t } = - D , \\\\ & \\frac { d D } { d t } + \\int _ \\Gamma V _ s ^ 2 \\ , d s \\lesssim D ^ { 5 / 2 } + E D ^ 3 , \\\\ & \\frac { d H } { d t } \\lesssim H ^ { 1 / 2 } \\ , E ^ { 1 / 6 } \\ , D ^ { 7 / 1 2 } . \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} W _ { p q } ( a _ i + r N , a _ j ) = W _ { p q } ( a _ i , a _ j + r N ) = W _ { p q } ( a _ i , a _ j ) \\ , , \\end{align*}"}
-{"id": "764.png", "formula": "\\begin{align*} P _ { \\beta } ( X ) = P _ { \\beta } ^ { * } ( X ) = U _ { \\beta } ( X ) \\times f _ { \\beta } ( X ) , \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} \\Sigma ( U ) = \\frac { 1 } { 2 } \\left ( \\nabla G ( U ) ^ T U + U ^ T \\nabla G ( U ) \\right ) . \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} \\mathrm { l o c } _ { x } ( g , A ) = \\mathrm { l o c } _ { x _ { 1 } } ( g _ { 1 } , A _ { 1 } ) = \\mathrm { t r } ( g _ { 1 } | A _ { 1 , x _ { 1 } } ) = \\mathrm { t r } ( g | A _ { x } ) \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} F _ r ( \\phi , \\psi ) & = F _ r ( \\phi ) + F _ { r - 1 } ( \\psi ) = \\sum _ { k = 0 } ^ r E _ k ( \\phi ) + \\sum _ { k = 0 } ^ { r - 1 } E _ k ( \\psi ) + \\| \\psi \\| _ { L ^ 2 } ^ 2 \\\\ & = s ^ 2 \\| \\partial _ t \\phi \\| ^ 2 _ { H ^ r } + \\| D \\phi \\| ^ 2 _ { H ^ r } + s ^ 2 \\| \\nabla _ t \\psi \\| ^ 2 _ { H ^ { r - 1 } } + \\| D \\psi \\| ^ 2 _ { H ^ { r - 1 } } + \\| \\psi \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} N _ n = \\left ( \\sum _ { i = 0 } ^ { n - 1 } ( 1 + k _ i ) \\right ) . \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{align*} _ d ( P _ { v , i } P _ { w , j } ) = \\frac { 1 } { 4 } \\left ( 1 + ( - 1 ) ^ { i + j } \\langle \\widetilde { x } _ v , \\widetilde { x } _ w \\rangle \\right ) = p ( i , j | v , w ) . \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} E _ M ( n , t ) & \\leq | p _ M ( t , \\hat x _ n ) - G _ M ( t , \\hat x _ n ) | + | G _ M ( t , \\hat x _ n ) - G _ M ( t , \\check x _ n ) | + | G _ M ( t , \\check x _ n ) - p _ M ( t , \\check x _ n ) | \\\\ & = : A _ 1 ( n ) + A _ 2 ( n ) + A _ 3 ( n ) = A _ 1 + A _ 2 + A _ 3 , \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{align*} K ( \\rho , { \\widetilde { V } } ) \\ = \\ P _ { \\mathop { \\mathrm { R a n } } { \\widetilde { V } } } \\ , { \\widetilde { V } } ^ { 1 / 2 } \\ , ( \\rho + h ( \\mathfrak { e } ) ) ^ { - 1 } \\ , { \\widetilde { V } } ^ { 1 / 2 } \\ , P _ { \\mathop { \\mathrm { R a n } } { \\widetilde { V } } } \\ ; - \\ ; P _ { \\mathop { \\mathrm { R a n } } { \\widetilde { V } } } . \\end{align*}"}
-{"id": "2255.png", "formula": "\\begin{align*} \\begin{aligned} { \\rm M S E } \\left ( { \\widetilde p _ { \\hat s _ 0 } } \\right ) - { \\rm M S E } \\left ( { \\hat { p } } \\right ) & = \\phi ( \\hat p , N , \\tilde d ) { \\hat s _ 0 } + \\psi ( \\hat p , N , \\tilde d ) { \\hat s _ 0 } ^ 2 , \\end{aligned} \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} D _ E = \\left ( \\frac { \\partial ^ 2 S _ E } { \\partial \\beta ^ 2 } \\right ) . \\left ( \\frac { \\partial ^ 2 S _ E } { \\partial \\alpha ^ 2 } \\right ) - \\left ( \\frac { \\partial ^ 2 S _ E } { \\partial \\beta \\partial \\alpha } \\right ) ^ 2 \\end{align*}"}
-{"id": "7932.png", "formula": "\\begin{align*} \\lim _ { r \\rightarrow 0 ^ - } u ' ( r ) = - \\int _ 0 ^ \\infty \\frac { s ^ 2 + 1 } { s ^ 2 } \\ , d \\rho ( s ) . \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} R ( n ) = G ( x ) \\sim \\log ^ 2 x \\sim { n ^ 2 \\over \\bar \\tau ^ 2 } \\le n ^ 2 \\quad \\mbox { a s } x \\to \\infty . \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{align*} I _ \\alpha f : = | \\cdot | ^ { \\alpha - n } \\star f \\end{align*}"}
-{"id": "7105.png", "formula": "\\begin{align*} \\sum _ { \\delta \\in \\Delta } m _ \\delta ^ { 1 1 / 2 } = \\sum _ { j = 0 } ^ { 2 \\log n } \\sum _ { \\delta \\in \\Delta \\atop 2 ^ j \\le m _ \\delta < 2 ^ { j + 1 } } m _ \\delta ^ { 1 1 / 2 } < \\sum _ { j = 0 } ^ { 2 \\log n } k _ { 2 ^ j } ( 2 ^ { j + 1 } ) ^ { 1 1 / 2 } & = O \\left ( \\sum _ { j = 0 } ^ { 2 \\log n } n ^ { 1 5 / 2 } \\log n \\right ) \\\\ & = O \\left ( n ^ { 1 5 / 2 } \\log ^ 2 n \\right ) . \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} f _ { \\hat \\tau } ( x , y ) = F ( x , y ) , \\enskip x \\in E \\backslash C . \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } 2 N & \\mbox { i f ~ } \\tau _ 0 = 0 \\\\ 2 N _ 1 & \\mbox { i f ~ } \\tau _ 0 = \\frac { N } { 2 } \\\\ 2 N _ 2 & \\mbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{align*} & \\mathfrak { D } _ I \\mathfrak { D } _ I ^ * = \\nabla ^ * \\nabla + 2 r + \\mathcal { R } _ 0 + \\sqrt { r } \\mathcal { R } _ 1 \\\\ & \\mathfrak { D } _ I ^ * \\mathfrak { D } _ I = \\nabla ^ * \\nabla + 2 r + \\tilde { \\mathcal { R } _ 0 } + \\sqrt { r } \\tilde { \\mathcal { R } _ 1 } . \\end{align*}"}
-{"id": "2846.png", "formula": "\\begin{align*} [ b , T ] f ( x ) = b ( x ) T ( f ) ( x ) - T ( b f ) ( x ) . \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} u ( t + T ) = m \\cdot u ( t ) \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} ( \\iota _ * ) ^ { - 1 } = \\sum _ { | \\vec { Y } | = n } \\frac { \\iota _ { \\vec { Y } } ^ * } { \\Lambda _ { - 1 } T _ { \\vec { Y } } M ( r , n ) } , \\end{align*}"}
-{"id": "4768.png", "formula": "\\begin{align*} { \\rm d i s t } ( ( 0 , 0 , 0 ) , \\widetilde { \\Phi } ( x , \\lambda , v ) \\cap \\mathbb { B } ( ( 0 , 0 , 0 ) , \\delta ' ) ) = \\| ( \\xi , \\eta , \\zeta ) \\| . \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } \\lambda \\widetilde { f } ( \\lambda ) = \\lim _ { t \\to 0 ^ + } f ( t ) = 0 . \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 , x _ 3 ) = \\max \\{ \\ , x _ 2 - x _ 3 , \\ , x _ 3 - x _ 1 , \\ , x _ 1 - x _ 2 \\ , \\} , \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} - \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\nabla ^ T f ( y _ { i , t } ) ( y _ { i , t } - x ^ { \\ast } ) = & - \\frac { 1 } { N } \\sum _ { i = 1 } ^ N ( \\nabla ^ T f ( y _ { i , t } ) - \\nabla ^ T f ( x ^ { \\ast } ) ) ( y _ { i , t } - x ^ { \\ast } ) \\\\ \\leq & - 2 \\kappa \\overline { F } _ t = - 2 \\kappa ( U _ t + \\overline { V } _ t ) . \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{align*} \\begin{aligned} \\max \\{ e ^ { D _ { \\varpi } ( P _ 0 \\parallel Q _ 0 ) } , e ^ { D _ { \\varpi } ( P _ 1 \\parallel Q _ 1 ) } \\} \\ge e ^ { D _ { \\varpi } ( P _ { \\lambda } \\parallel Q _ { \\lambda } ) } . \\end{aligned} \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} \\lambda _ X & = \\Lambda _ X = \\bar { \\lambda } _ X = \\varepsilon ^ { - 1 } \\beta _ X ^ { - 1 } , & \\Lambda _ Y & = \\beta _ Y ^ { - 1 } , & \\kappa _ Y & = \\norm { \\nabla _ x b _ Y } _ \\infty \\end{align*}"}
-{"id": "8672.png", "formula": "\\begin{align*} \\omega _ { \\bar \\mu } = r ^ { - s ( \\mu ) } \\omega _ \\mu , s ( \\mu _ 1 , \\ldots , \\mu _ N ) : = \\# \\{ \\lambda \\colon \\mu _ \\lambda \\in \\{ 2 , 3 \\} \\} , \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} X _ { k } \\ \\doteq \\ \\sum _ { \\ell = 0 } ^ { k - 1 } | r _ { k } - r _ { \\ell } | ^ { - 1 / 2 } Y _ { k } \\ \\doteq \\ \\sum _ { \\ell = k + 1 } ^ { \\infty } | r _ { k } - r _ { \\ell } | ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{align*} \\det Q _ 1 = \\det \\begin{pmatrix} 1 & 1 \\\\ q ^ { 1 / 2 } & q ^ { - 1 / 2 } \\end{pmatrix} = q ^ { - 1 / 2 } - q ^ { 1 / 2 } \\neq 0 \\end{align*}"}
-{"id": "8773.png", "formula": "\\begin{align*} \\mathrm { b a r } = \\int _ { \\Delta ^ + } q F _ { \\mathcal { L } } ( q ) \\frac { P _ { D H } ( q ) d q } { \\int _ { \\Delta ^ + } P _ { D H } d q } . \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} D = \\biggl \\lbrace x \\in \\mathbb { F } _ { q } : T r ( x ^ { p ^ { \\alpha } + 1 } ) = a \\biggr \\rbrace , \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{align*} { \\cal C } _ 1 ( \\underline { p } ) : = \\left \\{ ( \\lambda _ 1 , \\ldots , \\lambda _ M ) \\in ( 0 , 1 ) ^ { M } : \\sum _ { i = 1 } ^ { M } \\frac { \\lambda _ i } { p _ i \\prod _ { j \\neq i } ( 1 - p _ j ) } < 1 \\right \\} \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} d \\omega ^ { m - 1 } \\ , = \\ , 0 \\ , , \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} ( \\det \\psi _ n ( x ) ) _ { I J } = - \\frac { 1 } { | x | ^ 2 } S _ { I J } ( x ) . \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} & E \\left [ \\int \\ln ( p ^ 0 ( t , q _ t , z ) ) p ^ 0 ( t , q _ t , z ) d z \\right ] \\\\ = & E \\left [ \\int ( \\frac { n } { 2 } \\ln ( \\beta ( t , q _ t ) ) - \\beta ( t , q _ t ) \\| z \\| ^ 2 / 2 ) p ^ 0 ( t , q _ t , z ) d z \\right ] \\\\ = & E \\left [ \\frac { n } { 2 } \\ln ( \\beta ( t , q _ t ) ) - \\beta ( t , q _ t ) \\beta ^ { - 1 } n / 2 \\right ] \\\\ = & \\frac { n } { 2 } E \\left [ \\ln ( \\beta ( t , q _ t ) ) \\right ] \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\mathbb { E } { [ \\| x _ t - x ^ * \\| ^ 2 ] } = \\mathbb { E } [ G _ { \\Gamma t } ] \\le e ^ { - 2 \\kappa \\gamma \\Gamma t } G _ { 0 } + \\frac { m \\sigma ^ 2 \\Gamma } { 4 \\kappa N } ( 1 - e ^ { - 2 \\kappa \\gamma \\Gamma t } ) . \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{align*} & \\sup _ { r \\in [ 0 , T ] } E \\left [ ( 1 + \\| q _ r ^ m \\| ^ { \\tilde p } ) ^ { 2 p } \\right ] ^ { 1 / ( 2 p ) } \\\\ \\leq & 1 + \\sup _ { r \\in [ 0 , T ] } E \\left [ ( \\| q _ r ^ m - q _ r \\| + \\| q _ r \\| ) ^ { \\tilde p } ) ^ { 2 p } \\right ] ^ { 1 / ( 2 p ) } \\\\ \\leq & 1 + \\sup _ { r \\in [ 0 , T ] } E \\left [ \\| q _ r \\| ^ { 2 p \\tilde p } \\right ] ^ { 1 / ( 2 p ) } + \\sup _ { r \\in [ 0 , T ] } E \\left [ \\| q _ r ^ m - q _ r \\| ^ { 2 p \\tilde p } \\right ] ^ { 1 / ( 2 p ) } \\\\ = & O ( 1 ) . \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{align*} m ( d x \\ , d t ) = d m _ t ( x ) \\otimes d t , \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} \\widehat { \\delta } = t _ i ^ { - 1 } \\log ^ { \\frac { 1 } { 2 } } ( t _ i ) e ^ { - \\ : \\frac 1 2 F _ 1 ( \\log { \\frak t } ( t _ i ) ) } \\delta . \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{gather*} H _ 1 ( p , x ) = p _ 1 x _ 2 + p _ 2 ( 1 - \\alpha x _ 2 ) - x _ 2 H _ 2 ( p , x ) = p _ 1 x _ 2 - \\alpha p _ 2 x _ 2 \\\\ H _ 3 ( p , x ) = p _ 1 x _ 2 - p _ 2 ( 1 + \\alpha x _ 2 ) - x _ 2 \\end{gather*}"}
-{"id": "7337.png", "formula": "\\begin{align*} \\dot x _ i = \\sum _ { j = 1 } ^ { 2 k + 1 } A _ { i , j } x _ i x _ j + c _ i \\ ; , \\ \\ i = 1 , 2 , \\dots , 2 k + 1 \\ ; , \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{align*} \\left | \\left ( ( z - \\lambda _ j ) ^ { \\tau _ j } F _ \\alpha ( z ) \\Phi ^ { n - k } ( z ) \\right ) ^ { ( l ) } _ { z = \\lambda _ j } \\right | \\leq \\frac { c _ 6 } { \\rho _ 2 ^ { k - n } } , \\end{align*}"}
-{"id": "2570.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb { E } _ { n , d ( n ) , \\theta _ n } ( \\nu _ n ) = 0 \\theta _ n \\in \\Theta _ { d ( n ) } . \\end{align*}"}
-{"id": "5254.png", "formula": "\\begin{align*} \\int \\gamma D ( F ) ( m ) d \\nu _ j & = \\int _ { T _ 1 } \\gamma D ( F ) ( m ) d \\nu _ j + \\int _ { T _ 2 } \\gamma D ( F ) ( m ) d \\nu _ j \\\\ & = \\gamma _ c \\int _ { T _ 1 } D ( F ) ( m ) d \\nu _ j - \\gamma _ c \\int _ { T _ 2 } D ( F ) ( m ) d \\nu _ j \\\\ & = \\gamma _ c \\int _ { L _ 1 } D ( F ) ( m ) d \\mu _ j - \\gamma _ c \\int _ { L _ 2 } D ( F ) ( m ) d \\mu _ j \\\\ & = e ^ { - i \\theta } \\int \\gamma D ( F ) ( m ) d \\mu _ j \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{align*} L _ 1 = \\frac { 1 + ( x - y ) \\cdot D _ { \\xi } } { 1 + | x - y | ^ 2 } , L _ 2 = \\frac { 1 - \\xi \\cdot D _ y } { 1 + | \\xi | ^ 2 } . \\end{align*}"}
-{"id": "6560.png", "formula": "\\begin{align*} \\mu \\left ( \\{ v \\in T ^ 1 S : \\exists \\ , t \\in I _ j ^ { ( k ) } \\textrm { w i t h } \\delta ( \\phi _ t ( v ) ) = T _ k ^ { - \\xi } \\} \\right ) = O \\left ( \\frac { 1 } { T _ k ^ { \\xi ( r + 1 ) } } \\right ) \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{align*} 0 = \\pi _ 1 \\bigl ( \\tfrac { 1 } { 2 } d d ^ \\ast ( g \\omega _ 0 ^ 2 ) + d \\ ! \\ast \\ ! \\rho _ 0 \\bigr ) = \\pi _ 1 d d ^ \\ast \\bigl ( \\tfrac { 1 } { 2 } g \\omega _ 0 ^ 2 \\bigr ) \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} L _ t : = t \\ln _ 2 t , \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{align*} \\begin{aligned} \\hat D _ \\lambda = \\sum _ { l \\in \\bar l } \\lambda _ { l } \\hat D _ l , \\end{aligned} \\end{align*}"}
-{"id": "1612.png", "formula": "\\begin{align*} \\widehat { \\Psi } _ t ( z ) : = \\hat { \\lambda } _ t ( z ) - \\frac { \\ln _ 3 t } { t } | z | , z \\in \\Pi _ { L _ t , \\delta } . \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} { \\bf h } _ k = { \\bf R } ^ { 1 / 2 } { \\bf z } _ k , \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} \\mathrm { P } _ { m , i } = 1 - \\mathrm { P } \\left ( z _ m > \\frac { \\epsilon _ l } { \\rho \\xi _ l } , \\forall l \\leq i \\right ) , \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} | \\nabla _ { B } h | ^ { 2 } + a h ^ { 2 } = c . \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} \\gamma \\colon S \\otimes T M & \\to S \\\\ s \\otimes v & \\mapsto v \\cdot s . \\end{align*}"}
-{"id": "8734.png", "formula": "\\begin{align*} A ( n _ j ) = ( u - 1 ) 2 ^ l + \\sum _ { i = 1 } ^ { l - 1 } ( v _ i - 1 ) 2 ^ i + v _ 0 + 4 ( 2 ^ { l - 1 } + \\cdots + 2 + 1 ) . \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} \\langle P , \\chi \\rangle = \\lim _ { d \\rightarrow \\infty } \\langle P , \\chi _ d \\rangle . \\end{align*}"}
-{"id": "5049.png", "formula": "\\begin{align*} [ y _ i , s , x ] = - [ s , y _ i , x ] \\equiv 0 \\pmod { I } . \\end{align*}"}
-{"id": "8440.png", "formula": "\\begin{align*} V _ { j j } & = V _ 1 ( e _ j ) , , \\\\ V _ { j k } & = V _ { \\frac { 1 } { 2 } } ( e _ j ) \\cap V _ { \\frac { 1 } { 2 } } ( e _ k ) , , \\\\ V _ { j 0 } = V _ { 0 j } & = V _ { \\frac { 1 } { 2 } } ( e _ j ) \\cap \\bigcap _ { k \\not = j } V _ 0 ( e _ k ) , , \\\\ V _ { 0 0 } & = V _ 0 ( e _ 1 ) \\cap \\dots \\cap V _ 0 ( e _ r ) . \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} \\psi ( [ x , y ] _ { \\mathfrak { g } } ) = & \\pi ( [ x , y ] _ { \\mathfrak { g } } ) = \\pi ( x \\otimes y - \\varepsilon ( x , y ) y \\otimes x ) \\\\ = & \\pi ( x \\odot y - \\varepsilon ( x , y ) y \\odot x ) = \\pi ( x ) * \\pi ( y ) - \\varepsilon ( x , y ) \\pi ( y ) * \\pi ( x ) \\\\ = & \\psi ( x ) * \\psi ( y ) - \\varepsilon ( x , y ) \\psi ( y ) * \\psi ( x ) = [ \\psi ( x ) , \\psi ( y ) ] _ { \\mathfrak { g } } , \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} & H ( v , s ) = \\exp \\Big ( \\int _ { \\max \\{ v - R _ { \\tau } , R _ { \\tau } \\} } ^ { v + R _ { \\tau } } \\frac { 2 y \\lambda _ J ' ( y ) } { 1 + \\frac { y ^ { \\alpha } } { s P _ J } } d y \\Big ) , \\\\ & f ( v ) = 2 \\pi \\lambda _ U ^ s ( r , v ) v \\exp ( - 2 \\pi \\int _ { 0 } ^ { v } \\lambda _ U ^ s ( r , y ) y d y ) , \\\\ & \\lambda _ J ' ( y ) = \\lambda _ J \\arccos ( \\frac { y ^ 2 + v ^ 2 - R _ { \\tau } ^ 2 } { 2 y v } ) . \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{align*} \\alpha _ \\infty : = \\dot A _ \\infty - d _ { A _ \\infty } \\gamma _ \\infty = 0 \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{align*} \\min \\limits _ { { \\varphi } \\in S , \\ ; | { \\varphi } | _ { 2 } = 1 } \\langle { \\varphi } \\ , | \\ , V ^ { 1 / 2 } ( \\rho + h ( \\mathfrak { e } ) ) ^ { - 1 } V ^ { 1 / 2 } { \\varphi } \\rangle > 1 . \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{align*} \\left \\| \\int _ 0 ^ 1 T _ { - 1 } ( s ) B u ( s ) \\ , d s \\right \\| ^ { 2 } & = \\sum _ { n = 1 } ^ \\infty \\left | \\int _ 0 ^ 1 e ^ { \\lambda _ n s } \\lambda _ n ( u ( s ) ) _ n \\ , d s \\right | ^ 2 \\\\ & = \\sum _ { m = 1 } ^ \\infty \\left ( e ^ { - 1 / 2 } - e ^ { - 1 } \\right ) ^ 2 \\\\ & = \\infty , \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{align*} w _ n & = 2 ^ { n - 2 } - 1 - \\binom { n - 1 } { 2 } - \\binom { n - 1 } { 4 } + 4 \\binom { n - 6 } { 2 } + \\binom { n - 5 } { 3 } + \\sum _ { a = 2 } ^ { n - 6 } \\sum _ { b = a + 4 } ^ { n - 2 } ( b - 5 - a ) 2 ^ { b - 2 - a } . \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} d q _ t ^ m = & \\frac { 1 } { m } u _ t ^ m d t , \\\\ d ( u ^ m _ t ) _ i = & \\left ( - \\frac { 1 } { m } \\tilde \\gamma _ { i k } ( t , q _ t ^ m ) ( u _ t ^ m ) ^ k + F _ i ( t , x ^ m _ t ) \\right ) d t + \\sigma _ { i \\rho } ( t , q _ t ^ m ) d W ^ \\rho _ t , \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{align*} { } \\begin{array} { l } \\frac { v } { u } = c + \\frac { \\ln 2 } { c } \\epsilon \\\\ u \\ln ( e ^ v - 1 ) = \\frac { 2 \\ln 2 } { c } \\epsilon + \\frac { 2 \\gamma } { c } \\epsilon ^ 2 \\end{array} \\end{align*}"}
-{"id": "884.png", "formula": "\\begin{align*} E ( u , v ) = \\int ( \\nabla u ) ^ 2 - \\int u ^ 4 + \\int ( u ^ 2 + v ) ^ 2 , \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} E _ o ( x _ 0 ) : = \\max _ { { u \\in L ^ 2 ( [ 0 , \\infty [ ) \\atop \\left \\| u \\right \\| _ { L ^ 2 } < \\alpha , B = 0 } } \\int _ { 0 } ^ \\infty \\left \\| y ( t , x _ 0 , u ) \\right \\| _ 2 ^ 2 d t , \\ ; \\ ; \\ ; \\alpha > 0 \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{align*} \\delta _ t : \\mathbb { R } ^ 2 \\to \\mathbb { R } ^ 2 , \\delta _ { t } ( x ) = ( t x _ 1 , t ^ 2 x _ 2 ) \\end{align*}"}
-{"id": "7101.png", "formula": "\\begin{align*} E _ { f } ( A , B ) = \\sum _ { \\delta \\in \\Delta } m _ { \\delta } ^ 2 \\ge \\frac { ( \\sum _ { \\delta \\in \\Delta } m _ \\delta ) ^ 2 } { D } = \\frac { | A | ^ 2 | B | ^ 2 } { D } . \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} g _ { \\mathrm { s K } } ( \\dot q , \\dot q ) = \\frac { 1 } { 8 } \\int _ { S _ q } | \\tau _ { \\dot q } | ^ 2 \\ , d A = \\frac 1 8 \\int _ { S _ q } \\frac { | \\dot q | ^ 2 } { | q | } \\ , d A = \\frac 1 4 \\int _ { X } \\frac { | \\dot q | ^ 2 } { | q | } \\ , d A \\end{align*}"}
-{"id": "5616.png", "formula": "\\begin{align*} [ \\lambda ^ { - 1 } \\xi _ 1 , \\lambda ^ { - 1 } \\xi _ 2 ] = [ \\lambda ^ { - 2 } \\xi _ 1 , \\xi _ 2 ] = [ \\xi _ 1 , \\lambda ^ { - 2 } \\xi _ 2 ] , \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} 0 \\le a _ { s _ i } \\le \\rho ( s _ i , \\mu , \\lambda ) - 1 \\le | \\lambda \\cap S _ \\mu ( s _ i ) | - 1 = i - 1 , \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} \\lambda _ { U , t } ^ s ( r , y ) & = \\lambda _ t \\Big ( 1 - \\exp \\Big ( - \\pi ( r + y ) ^ 2 \\lambda _ t ( \\frac { P _ t } { P _ k } ) ^ { 2 / \\alpha } \\Big ) \\Big ) , \\\\ \\lambda _ { U } ^ s ( r , y ) & = \\sum _ { t \\in \\mathcal { K } } \\lambda _ { U , t } ^ s ( r , y ) , \\end{align*}"}
-{"id": "4137.png", "formula": "\\begin{align*} \\dot { x } _ i ^ { ( n ) } = \\rho x _ i ^ { ( n ) } \\left ( \\pi _ i ^ { ( n ) } ( \\textbf { x } ) - \\overline { \\pi } ^ { ( n ) } ( \\textbf { x } ) \\right ) , \\end{align*}"}
-{"id": "8828.png", "formula": "\\begin{align*} C _ E = - 2 z _ 2 \\bar { z } _ 1 \\alpha ( l _ j ) e ^ { 2 \\alpha ( a ) } \\theta ( e _ { \\alpha } ) + O . \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} \\sum _ { x ^ { ( 1 ) } \\in \\Z } \\P ^ { 0 , 0 } _ { 2 T } ( X ( t _ 1 ) = x ^ { ( 1 ) } ) = 1 . \\end{align*}"}
-{"id": "6077.png", "formula": "\\begin{align*} s _ n ( M ( a _ 1 ) ) = O ( n ^ { - \\alpha - 1 } ) , n \\to \\infty . \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{align*} \\frac { N } { e ^ { A _ N } } = o \\left ( \\frac { 1 } { N } \\right ) \\ ; N \\to \\infty \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{align*} \\begin{bmatrix} 2 a ( P _ o ( B - 2 a ) P _ o ) + T ^ * T & T ^ * ( P _ e B P _ e ) \\\\ ( P _ e B P _ e ) T & P _ e ( B ^ 2 - 2 a B ) P _ e \\end{bmatrix} . \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} f _ { r _ m , r _ t } ( x , y ) = & \\frac { 4 ( \\lambda _ c \\pi ) ^ { t } } { ( t - m - 1 ) ! ( m - 1 ) ! } e ^ { - \\lambda _ c \\pi y ^ 2 } \\sum ^ { t - m - 1 } _ { p = 0 } ( - 1 ) ^ p \\\\ & \\times { t - m - 1 \\choose p } y ^ { 2 ( t - m - 1 ) - 2 p + 1 } x ^ { 2 m + 2 p - 1 } . \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} \\omega _ { 3 } = \\int _ { a _ { 4 } } ^ { X _ { \\pm } ( b _ { 4 } ) } \\frac { d x } { \\sqrt { D ( x ) } } \\in ( 0 , \\omega _ { 2 } ) . \\end{align*}"}
-{"id": "6891.png", "formula": "\\begin{align*} K _ { \\mathrm { F i x } ( c ) / S } & = R ( f \\circ c ' ) ^ { ! } \\Lambda \\to R a _ { \\mathrm { F i x } ( c ) \\ast } a _ { \\mathrm { F i x } ( c ) } ^ { \\ast } R ( f \\circ c ' ) ^ { ! } \\Lambda \\\\ & \\ ; \\overset { \\beta _ { a , f \\circ c ' } } { \\longrightarrow } R a _ { \\mathrm { F i x } ( c ) \\ast } R ( f _ { T } \\circ c ' _ { T } ) ^ { ! } a ^ { \\ast } \\Lambda = R a _ { \\mathrm { F i x } ( c ) \\ast } K _ { \\mathrm { F i x } ( c ) _ { T } / T } . \\end{align*}"}
-{"id": "51.png", "formula": "\\begin{align*} M _ n ^ { ( \\boldsymbol { \\alpha } ) } ( \\varepsilon ) = \\{ ( { v _ 1 , \\ldots , v _ k } ) \\colon D _ { { v _ i } } \\in [ \\varepsilon , 1 / \\varepsilon ] ( \\mu n ) ^ { \\alpha _ i } \\ \\forall i \\in [ k ] \\} . \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{align*} \\partial _ t \\gamma = \\partial _ s ^ 2 \\gamma , \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{align*} \\zeta _ { q } ^ { \\prime } ( { \\varphi } ) = \\Big ( ( \\mathcal { F } ^ { \\ast } \\circ V ^ { 1 / 2 } ) ( { \\varphi } ) ( q ) \\ , , \\ , ( \\nabla \\mathcal { F } ^ { \\ast } \\circ V ^ { 1 / 2 } ) ( { \\varphi } ) ( q ) \\Big ) . \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{align*} 0 & = \\rho _ j ^ { n } - \\rho _ j ^ { n + 1 } \\\\ & + \\frac { \\sigma _ 2 \\Delta t } { \\sigma _ 0 \\Delta x } \\left ( \\frac { E _ { j - \\frac 1 2 } } { \\kappa } \\frac { \\rho _ { j - 1 } ^ n - e ^ { - E _ { j - \\frac 1 2 } \\Delta x / \\kappa } \\rho _ j ^ n } { 1 - e ^ { - E _ { j - \\frac 1 2 } \\Delta x / \\kappa } } - \\frac { E _ { j + \\frac 1 2 } } { \\kappa } \\frac { \\rho _ j ^ n - e ^ { - E _ { j + \\frac 1 2 } \\Delta x / \\kappa } \\rho _ { j + 1 } ^ n } { 1 - e ^ { - E _ { j + \\frac 1 2 } \\Delta x / \\kappa } } \\right ) . \\end{align*}"}
-{"id": "2164.png", "formula": "\\begin{align*} 1 - ( 1 - \\epsilon ) \\left ( 1 + \\frac { 2 ^ k } { 1 2 } \\rho \\right ) = - \\frac { 2 ^ k } { 1 2 } \\rho + \\epsilon + \\frac { 2 ^ k } { 1 2 } \\epsilon \\rho \\leq - 2 ^ k \\epsilon . \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{align*} \\mathbf { r } _ { i j } ( \\zeta ) = \\mathbf { a } _ { 1 } \\zeta + \\mathbf { a } _ { 2 } \\zeta ^ { 2 } + . . . \\end{align*}"}
-{"id": "6232.png", "formula": "\\begin{align*} R _ m L _ m v = \\begin{cases} q ^ { \\kappa ( m , \\mu , \\lambda ) } v & , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} Q ( f ) = \\# \\{ \\} - \\# \\{ \\} . \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} \\frac { v _ A } { x _ A } = \\frac { v _ B } { x _ B } , & & & \\frac { v _ A } { x _ A } + \\frac { v _ { A B } } { x _ A + x _ B } = \\frac { v _ C } { x _ C } \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} \\sigma _ { F } ^ 2 ( h ( x ) ) = \\sigma _ { F } ( h ( \\sigma _ { E } ( x ) ) ) , \\sigma _ { E } ^ 2 ( h ^ { - 1 } ( y ) ) = \\sigma _ { E } ( h ^ { - 1 } ( \\sigma _ { F } ( y ) ) ) , \\end{align*}"}
-{"id": "2342.png", "formula": "\\begin{align*} E \\left [ X ^ r \\right ] = E \\left [ \\int _ 0 ^ { \\infty } \\frac { \\xi ^ { S _ N + r - 1 } } { ( S _ N - 1 ) ! } \\ , e ^ { - \\xi } d \\xi \\right ] = E \\left [ \\frac { \\Gamma ( S _ N + r ) } { ( S _ N - 1 ) ! } \\right ] = E \\left [ S _ N ^ { ( r ) } \\right ] . \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} m ( ( J , F ) , ( J \\cdot F , F ' ) ) = ( J , F \\circ F ' ) , i ( J , F ) = ( J \\cdot F , F ^ { - 1 } ) \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{align*} \\alpha = r ^ \\lambda \\bigl ( r ^ { k - 1 } d r \\wedge \\alpha _ { k - 1 } + r ^ k \\alpha _ k \\bigr ) \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} x = 0 , a ' = - a , a = - b , k = l , u + 2 v + w = 2 k . \\end{align*}"}
-{"id": "1203.png", "formula": "\\begin{align*} \\widehat { \\delta } = t _ i ^ { - 1 } \\log ^ { \\frac 1 2 } ( t _ i ) \\delta . \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{align*} L \\chi = G - \\tilde G . \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} x & : = ( a b ^ { - 1 } ) ^ { 3 } ( a ^ 2 b ^ { - 1 } ) ^ { 3 } ( a b ^ { - 1 } ) ^ 3 ( a ^ 2 b ^ { - 1 } ) ^ 4 \\ldots ( a b ^ { - 1 } ) ^ 3 ( a ^ 2 b ^ { - 1 } ) ^ { \\rho + 2 } \\\\ y & : = ( a b ^ { - 1 } ) ^ 3 ( a ^ 2 b ^ { - 1 } ) ^ { \\rho + 3 } ( a b ^ { - 1 } ) ^ 3 ( a ^ 2 b ^ { - 1 } ) ^ { \\rho + 4 } \\ldots ( a b ^ { - 1 } ) ^ { 3 } ( a ^ 2 b ^ { - 1 } ) ^ { 2 \\rho + 2 } \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} d v _ \\theta ( s ) = \\xi '' ( s ) ^ { 1 / 2 } \\partial _ { x x } \\Phi _ { u , \\gamma _ \\theta } ( s , X _ \\theta ( s ) , \\lambda _ \\theta ) \\ , d W ( s ) , \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} \\sup _ { s \\in K } \\Vert u ( s ) \\Vert = \\sup _ { s \\in K } \\Vert \\widetilde { u ( s ) } \\Vert = \\sup _ { s \\in K } \\sup _ { l \\in X _ * , \\Vert l \\Vert \\leq 1 } \\vert \\langle u ( s ) , l \\rangle \\vert < + \\infty . \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{align*} & \\sigma ( H ^ { \\epsilon , \\kappa } ) \\cap \\left [ a _ 0 , b _ N \\right ] \\subset \\bigcup _ { k = 0 } ^ N [ a _ k , b _ k ] \\ , , \\quad { \\rm d i m } \\big ( { \\rm R a n } E _ { [ a _ k , b _ k ] } ( H ^ { \\epsilon , \\kappa } ) \\big ) = + \\infty \\ , , \\\\ & b _ k - a _ k \\leq C _ 0 \\ , \\epsilon \\big ( \\kappa + C _ 1 \\ , \\epsilon ^ { 1 / 3 } \\big ) \\ , , \\ ; 0 \\leq k \\leq N \\ , , { \\rm a n d } a _ { k + 1 } - b _ k \\geq \\frac { 1 } { C _ 2 } \\epsilon \\ , , \\ ; 0 \\leq k \\leq N - 1 \\ , . \\end{align*}"}
-{"id": "4287.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in S _ { b , c } } \\prod _ { j = 1 } ^ N & \\bigg ( \\prod _ { h = 1 } ^ { \\sigma ( \\vec { A } ) _ j - 1 } \\Big ( \\sum _ { k = j + 1 } ^ N \\sigma ( \\vec { A } ) _ k + h \\Big ) \\bigg ) \\\\ & = \\sum _ { \\sigma \\in S _ { b + 1 , c } } \\prod _ { j = 1 } ^ { N + 1 } \\bigg ( \\prod _ { h = 1 } ^ { \\sigma ( \\vec { A } ' ) _ j - 1 } \\Big ( \\sum _ { k = j + 1 } ^ { N + 1 } \\sigma ( \\vec { A } ' ) _ k + h \\Big ) \\bigg ) . \\end{align*}"}
-{"id": "1356.png", "formula": "\\begin{align*} d X _ t = f \\bigl ( t , X _ t , u _ t \\bigr ) d t + \\sigma \\bigl ( t , X _ t , u _ t \\bigr ) d B _ t , X _ 0 = x , 0 \\le t \\le T , \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ X f \\ , d \\mu _ t = - \\int _ X \\langle \\nabla f , \\nabla u \\rangle \\ , d \\mu _ t . \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{align*} L _ m \\chi _ y ( z ) = \\sum _ { Y ' } \\chi \\left ( \\sum _ { s \\in S _ \\mu } \\sum _ { t \\in T _ \\mu } Y _ { s , t } Y ' _ { s , t } \\right ) , \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} G ( J ) = f \\circ e ( \\eta ( J ) ) \\end{align*}"}
-{"id": "6199.png", "formula": "\\begin{align*} q ^ { r ( M , s , t ) } - q ^ { r ( M , s , t ) - 1 } = ( q - 1 ) q ^ { r ( M , s , t ) - 1 } = ( q - 1 ) q ^ { \\mathrm { i n v } ( \\sigma , s , t ) } . \\end{align*}"}
-{"id": "8208.png", "formula": "\\begin{align*} G \\leq D + ( \\omega ) - G & \\Rightarrow L ( G ) \\subseteq L ( D + ( \\omega ) - G ) \\\\ & \\Rightarrow C _ { L } ( G , D ) \\subseteq C _ { L } ( D + ( \\omega ) - G , D ) = C _ { L } ( G , D ) ^ \\perp . \\end{align*}"}
-{"id": "3431.png", "formula": "\\begin{align*} \\dot { x } & = \\cos \\theta \\\\ \\dot { y } & = \\sin \\theta \\\\ \\dot { \\theta } & = \\left ( \\frac { x } { 2 } - \\frac { m - 1 } { x } \\right ) \\sin \\theta + \\left ( \\frac { n - 1 } { y } - \\frac { y } { 2 } \\right ) \\cos \\theta + \\lambda \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} \\hat { \\pi } _ n ( z ) = \\gamma _ n ( z ) \\frac { ( 1 - z ) ( 1 - \\gamma _ n ' ( 1 ) ) } { \\gamma _ n ( z ) - z } . \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{align*} Z _ i = \\langle Y , \\psi _ i \\rangle _ n = f _ i + \\sigma \\tilde \\xi _ i , \\end{align*}"}
-{"id": "2803.png", "formula": "\\begin{align*} S ( I _ 1 , I _ 2 ) & = \\left \\lbrace p \\in \\mathbb { P } : p \\nmid 4 N _ 1 N _ 2 , \\frac { a ( t p ^ 2 ) } { 2 p ^ { k _ 1 - \\frac { 1 } { 2 } } } \\in I _ 1 , \\frac { b ( t p ^ 2 ) } { 2 p ^ { k _ 2 - \\frac { 1 } { 2 } } } \\in I _ 2 \\right \\rbrace \\\\ S ( I _ 1 , I _ 2 ) ( x ) & = \\left \\lbrace p \\leq x : p \\in S ( I _ 1 , I _ 2 ) \\right \\rbrace . \\end{align*}"}
-{"id": "2622.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l l } R i c _ { B } - m h ^ { - 1 } \\nabla _ { B } \\nabla _ { B } h = a ( m + n - 1 ) g _ { B } , \\\\ \\noalign { \\smallskip } h \\Delta _ { B } h + ( m - 1 ) | \\nabla _ { B } h | ^ { 2 } + a ( m + n - 1 ) h ^ { 2 } = c ( m - 1 ) , \\\\ \\noalign { \\smallskip } R i c _ { F } = c ( m - 1 ) g _ { F } . \\end{array} \\right . \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\widetilde { \\P } ^ { 0 , 0 } _ { 2 T } ( X ( s T ) = \\widetilde { v } ( s ) T ) = 1 , s \\in [ 0 , 2 ] . \\end{align*}"}
-{"id": "4035.png", "formula": "\\begin{align*} & \\sum _ { x , y } ( \\mu _ i ( x , y ) - \\gamma _ i ( x , y ) ) \\log p _ { X Y } ( x , y ) = \\sum _ { x , y } ( \\mu ( x , y ) - \\gamma ( x , y ) ) \\log p _ { X Y } ( x , y ) . \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{align*} D ( t ) = U ( t ) D U ( - t ) . \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} | A | = 2 | \\omega | ^ 2 | \\Im ( \\tau ) | . \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} P ( x , y ) & = \\sum _ { h \\in \\mathcal { D } } h _ I ( y ) ( h _ { I ^ - } ( x ) - h _ { I ^ + } ( x ) ) \\\\ & = P ^ + ( x , y ) + P ^ - ( x , y ) ; \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t u = \\frac { \\nu ^ 2 } { 2 } \\Delta u + \\vert \\nabla u \\vert ^ 2 , \\textrm { f o r a n y } \\ ( t , x ) \\in [ 0 , T ] \\times \\R ^ d , \\\\ u ( 0 , d x ) = u _ 0 ( x ) d x \\ , \\end{array} \\right . \\end{align*}"}
-{"id": "7879.png", "formula": "\\begin{align*} \\lambda ( t , x , \\xi ) : = \\mu + h _ s ( t , x , \\xi ) . \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} { \\rm C o e f f } _ p [ E _ { \\mu } , m ] \\propto \\left [ \\prod _ { \\substack { \\kappa \\prec \\nu [ 1 ] } } \\frac { Y ( w ) - y _ { \\kappa } ( w ) } { y _ { \\mu } ( w ) - y _ { \\kappa } ( w ) } \\cdot \\left ( z ^ { \\nu [ 1 ] } + \\sum _ { \\nu \\prec \\nu [ 1 ] } h _ { \\nu } ( t ; w ) z ^ { \\nu } \\right ) \\right ] _ { q = t ^ { - m } } \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} f ^ * ( 1 \\times \\cdots \\times 1 \\times x _ { M _ i } ^ u \\times 1 \\times \\cdots \\times 1 ) \\in \\mathop { \\oplus } _ { i = 1 } ^ s H ^ u ( M _ i ; \\R ) , \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} ( K v ) ( x ) : = \\int _ 0 ^ L k ( x , y ) v ( y ) d y ( S v ) ( x ) : = \\int _ 0 ^ L s ( x , y ) v ( y ) d y , \\end{align*}"}
-{"id": "7297.png", "formula": "\\begin{align*} \\mathbf { r } = \\mathbf { F } \\hat { \\mathbf { y } } = ( \\mathbf { G } ^ { H } \\mathbf { G } ) ^ { - 1 } \\mathbf { G } ^ { H } \\hat { \\mathbf { y } } . \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} d X _ s ^ { t , x ; u } = f \\bigl ( t , X _ s ^ { t , x ; u } , u _ s \\bigr ) d t + \\sigma \\bigl ( t , X _ s ^ { t , x ; u } , u _ s \\bigr ) d B _ t , t \\le s \\le T , & \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{align*} \\Phi _ i ' = \\mathcal { P } ( \\chi _ { B ' _ { R } } \\phi _ i ) \\mbox { a n d } \\Phi _ i '' = \\mathcal { P } ( ( 1 - \\chi _ { B ' _ { R } } ) \\phi _ i ) , \\end{align*}"}
-{"id": "5842.png", "formula": "\\begin{align*} \\sigma ( \\mu ) : = \\{ \\nu | \\nu ^ { - } = \\mu ^ { - } \\} . \\end{align*}"}
-{"id": "1358.png", "formula": "\\begin{align*} - d Y _ t = G \\bigl ( t , Y _ t , Z _ t \\bigr ) d t - Z _ t d B _ t , Y _ T = \\xi , \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{align*} g _ k ( z ) = f _ k ( z _ k + \\varrho _ k z ) \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} 3 \\mu ( A ) \\leq \\mu ( A + A ) + \\mu ( k \\cdot A ) = \\mu ( ( A + A ) \\cup k \\cdot A ) \\leq 1 , \\ ; \\textrm { s o } \\mu ( A ) \\leq 1 / 3 . \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} g _ { k s } \\Gamma _ { i j } ^ k = \\frac { 1 } { 2 } \\left ( { \\partial _ { x _ i } g _ { j s } } + \\partial _ { x _ j } { g _ { i s } } - \\partial _ { x _ s } g _ { i j } \\right ) . \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} v ( \\overline { \\tilde S } x ) = u x \\ \\ \\ ( x \\in { \\mathcal D } _ { \\overline { \\tilde S } } ) . \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{align*} \\sigma _ 0 ( X _ { f ^ i _ { j } } , \\cdot ) = \\sigma _ 1 ( X _ { f ^ i _ { j - 1 } } , \\cdot ) . \\end{align*}"}
-{"id": "653.png", "formula": "\\begin{align*} X ( T _ 0 , T _ 1 ) : = \\bigcap _ { i = 1 } ^ { M } \\left \\{ R _ i ( T _ 0 , T _ 1 ) \\geq \\epsilon _ 1 ( T _ 1 - T _ 0 ) \\right \\} , \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} \\int _ M | \\nabla _ { \\rm t r } \\phi | ^ 2 + \\int _ M \\langle F ( \\phi ) , \\phi \\rangle + \\int _ M \\langle A _ { \\kappa _ B ^ \\sharp } ( \\phi ) , \\phi \\rangle = 0 . \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{align*} \\tilde { I } ( u _ { 0 } ) = I _ { 0 } ( u _ { 0 } ) \\leq I _ { 0 } ( \\mathcal { S } ( a ) ) = \\tilde { I } ( \\mathcal { S } ( a ) ) , \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} \\xi _ n ^ l \\partial _ \\xi ^ \\alpha p _ L ( x , \\xi ) = \\xi _ n ^ l \\sum _ { | \\beta | \\leq N } \\left . \\dfrac { 1 } { \\alpha ! } \\partial _ \\xi ^ { \\alpha + \\beta } D _ y ^ \\beta p _ \\kappa ( x , y , \\xi ) \\right | _ { y = x } + \\xi _ n ^ l r _ { N , \\alpha } ( x , \\xi ) \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { \\ln x } { \\ln N } \\right ) ^ r = \\sum _ { k = 0 } ^ n ( - 1 ) ^ k \\binom { r } { k } \\left ( \\frac { \\ln x } { \\ln N } \\right ) ^ k + o \\left ( \\frac { 1 } { \\ln ^ n N } \\right ) \\end{align*}"}
-{"id": "8983.png", "formula": "\\begin{align*} M _ f ( t ) & : = f ( t , X ( t ) ) - f ( 0 , X ( 0 ) ) - \\int _ 0 ^ t \\vec { A } f ( s , X ( s ) ) \\dd s \\\\ & = f ( t , X ( t ) ) - f ( 0 , X ( 0 ) ) - \\int _ 0 ^ t \\partial _ s f ( s , X ( s ) ) + ( A [ s ] f ( s , \\cdot ) ) ( X ( s ) ) \\dd s \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} X \\# _ t Y : = X ^ { 1 / 2 } ( X ^ { - 1 / 2 } Y X ^ { - 1 / 2 } ) ^ t X ^ { 1 / 2 } \\ . \\end{align*}"}
-{"id": "8088.png", "formula": "\\begin{align*} \\begin{cases} \\operatorname { d i v } ( y ^ a \\nabla U _ t ) = y ^ a ( U _ t ) _ t , \\\\ \\underset { y \\to 0 } { \\lim } y ^ a ( U _ t ) _ y = - V _ t u - V u _ t . \\end{cases} \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} f _ \\pm ( 0 ) = \\eta _ 1 \\qquad f _ \\pm ( 1 ) = \\xi _ 1 . \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} \\alpha \\cdot ( \\varphi \\otimes e ) = - \\varphi \\cdot \\alpha \\otimes e + \\varphi \\otimes \\alpha \\cdot e \\ , . \\end{align*}"}
-{"id": "840.png", "formula": "\\begin{align*} & \\limsup _ { r \\rightarrow \\infty } \\iint _ { \\R ^ { 2 N } } | u ( x ) | ^ { p } \\frac { | \\phi _ { r } ( x ) - \\phi _ { r } ( y ) | ^ { p } } { | x - y | ^ { N + s p } } \\ , d x d y \\\\ & = \\lim _ { K \\rightarrow \\infty } \\limsup _ { r \\rightarrow \\infty } \\iint _ { \\R ^ { 2 N } } | u ( x ) | ^ { p } \\frac { | \\phi _ { r } ( x ) - \\phi _ { r } ( y ) | ^ { p } } { | x - y | ^ { N + s p } } \\ , d x d y = 0 . \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} \\varphi _ { \\Omega } ( z - n \\omega _ 2 ) = ( - 1 ) ^ n \\exp ( 2 \\pi i n z / \\omega _ 1 - \\pi i n ( n + 1 ) \\omega _ 2 / \\omega _ 1 ) \\varphi _ { \\Omega } ( z ) \\end{align*}"}
-{"id": "3956.png", "formula": "\\begin{align*} & \\lim _ { n \\rightarrow \\infty } \\frac 1 n \\log \\mathbb { P } \\left [ X ^ n \\in \\mathcal { T } ^ { ( n ) } _ { q _ X } , Y ^ n \\in \\mathcal { T } ^ { ( n ) } _ { q _ Y } , Z ^ n \\in \\mathcal { T } ^ { ( n ) } _ { q _ Z } \\right ] = - \\min D ( r _ { X Y Z } \\| p _ { X Y Z } ) \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{align*} \\mathbb { E } [ G _ t ] \\le e ^ { - 2 \\kappa \\gamma t } G _ { 0 } + \\frac { m \\tau _ N ^ 2 \\gamma } { 4 \\kappa } ( 1 - e ^ { - 2 \\kappa \\gamma t } ) = e ^ { - 2 \\kappa \\gamma t } G _ { 0 } + \\frac { m \\sigma ^ 2 \\Gamma } { 4 \\kappa N } ( 1 - e ^ { - 2 \\kappa \\gamma t } ) . \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{align*} ( r - d ) \\binom { r - 1 } { 2 } & - \\left ( \\binom { d } { 2 } + ( r - 1 - d ) \\left ( \\binom { r } { 2 } - k \\right ) \\right ) \\\\ & = - \\frac 1 2 \\left ( ( r - d ) ( r - 1 ) ( 2 ) + d ^ 2 - d - r ^ 2 + r - 2 k r + 2 d k + 2 k \\right ) \\\\ & = - \\frac 1 2 \\left ( d ^ 2 - 2 d r + ( 1 + 2 k ) d + r ^ 2 - ( 2 k + 1 ) r + 2 k \\right ) \\\\ & = - \\frac 1 2 \\left ( d - ( r - 1 ) \\right ) \\left ( d - ( r - 2 k ) \\right ) \\\\ & \\ge 0 , \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l l } R i c _ { B } = a ( n - 1 ) g _ { B } , \\\\ \\noalign { \\smallskip } \\nabla _ { B } \\nabla _ { B } h + a h g _ { B } = 0 , \\\\ \\noalign { \\smallskip } R i c _ { F } = c ( m - 1 ) g _ { F } , \\end{array} \\right . \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} J _ n = \\big \\{ \\abs { L _ n - \\mu n } \\leq n ^ { 2 / \\tau } \\big \\} . \\end{align*}"}
-{"id": "6010.png", "formula": "\\begin{align*} w = p { \\left ( \\frac { C _ { \\alpha { } } } { A \\hslash { } \\left ( \\alpha { } + 1 \\right ) } \\right ) } ^ { \\frac { 1 } { ( \\alpha { } + 1 ) } } , ~ ~ y = \\frac { 1 } { \\hslash { } } \\left ( x - \\frac { E } { A } \\right ) { \\left ( \\frac { C _ { \\alpha { } } } { A \\hslash { } \\left ( \\alpha { } + 1 \\right ) } \\right ) } ^ { \\frac { - 1 } { ( \\alpha { } + 1 ) } } , \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} \\Omega _ { n , d } = \\bigtimes _ { i = 1 } ^ n \\Omega , \\mathcal { A } _ { n , d } = \\bigotimes _ { i = 1 } ^ n \\mathcal { A } , \\mathbb { P } _ { n , d , \\theta } = \\bigotimes _ { i = 1 } ^ n \\mathbb { P } _ { d , \\theta } \\theta \\in \\Theta _ d , \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{align*} x ^ 2 + D = \\lambda y ^ n \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{align*} U ( { \\mathcal F } ) = \\underline { \\mathrm { H o m } } ( A ^ * , { \\mathcal G } [ p ^ N ] ) \\setminus \\left ( \\bigcup _ { A ' \\in { \\mathcal F } } \\bigcap _ { \\rho \\in ( A / A ' ) ^ * } \\underline { \\mathrm { H o m } } ( \\mathrm { k e r } ( \\rho ) ^ * , { \\mathcal G } [ p ^ N ] ) \\right ) . \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{align*} m = \\tanh ( \\beta ( m + h ) ) . \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} A ^ T Q _ 1 + Q _ 1 A + \\sum _ { i = 1 } ^ m N _ i ^ T Q _ 1 N _ i = - C ^ T C . \\end{align*}"}
-{"id": "709.png", "formula": "\\begin{align*} ( A _ k + \\delta A _ k ) \\tilde { X } _ { \\alpha _ k } ^ { s _ k } ( B _ k + \\delta B _ k ) - ( C _ k + \\delta C _ k ) \\tilde { X } _ { \\beta _ k } ^ { t _ k } ( D _ k + \\delta D _ k ) = E _ k + \\delta E _ k k = 1 , \\dots , r , \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} d _ { \\textrm { m a x } , 1 } ( f , ( 0 0 ) ) & = \\max \\left \\{ d ( f , ( 0 0 ) , ( 0 1 ) ) , d ( f , ( 0 0 ) , ( 1 0 ) ) \\right \\} \\\\ & = \\max \\left \\{ \\frac { | 0 - 1 | } { 2 ^ q } , \\frac { | 0 - 2 | } { 2 ^ q } \\right \\} = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "4878.png", "formula": "\\begin{align*} P \\in C [ m ] \\Leftrightarrow f _ m ( x ) = 0 . \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} \\mathrm { P } _ { m , 1 } = \\mathrm { P } \\left ( z _ t < \\frac { \\epsilon _ 1 } { \\rho } , z _ m < \\frac { \\epsilon _ 1 } { \\rho } \\right ) . \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{align*} \\tilde S ^ i ( t , q ) = & - \\frac { 1 } { 2 } \\sigma ( t , q ) \\partial _ { q ^ k } \\sigma ( t , q ) ( \\tilde \\gamma ^ { - 1 } ( t ) ) ^ { i l } \\delta _ { l \\xi } ( \\tilde \\gamma ^ { - 1 } ( t ) ) ^ { k \\xi } . \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} \\Upsilon _ n - \\Upsilon _ n ' \\cdot \\gamma ^ { - n } = \\Upsilon _ { n - 1 } \\cdot \\gamma ^ { - 1 } - \\Upsilon ' _ { n - 1 } \\cdot q ^ r \\gamma ^ { - n + 1 } \\end{align*}"}
-{"id": "2703.png", "formula": "\\begin{align*} [ X _ j , X _ k ] = \\sum _ { d _ l \\leq d _ j + d _ k } c _ { j , k } ^ l X _ l , c _ { j , k } ^ l \\in C ^ \\infty ( \\Omega ) . \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} H _ { N } ^ { \\delta } ( r ) = \\int _ { - 1 } ^ { - \\delta } h _ N ( r ^ 2 t ) d t , \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} I _ { b - } ^ { \\alpha ; \\psi } f \\left ( x \\right ) : = \\frac { 1 } { \\Gamma \\left ( \\alpha \\right ) } \\int _ { x } ^ { b } \\psi ^ { \\prime } \\left ( t \\right ) \\left ( \\psi \\left ( t \\right ) - \\psi \\left ( x \\right ) \\right ) ^ { \\alpha - 1 } f \\left ( t \\right ) d t . \\end{align*}"}
-{"id": "8927.png", "formula": "\\begin{align*} \\sum _ Y \\int _ { \\Delta ' _ Y } u ^ * P _ { D H } ' d \\sigma = & \\int _ { \\Delta ' } \\Big { ( } u ^ * u ^ { * , i , j } _ { \\mathrm { r e f } , i , j } P _ { D H } ' + u ^ * 2 u ^ { * , i , j } _ { \\mathrm { r e f } , j } P _ { D H , i } ' \\\\ & + u ^ * u ^ { * , i , j } _ { \\mathrm { r e f } } P _ { D H , i , j } ' - u ^ { * , i , j } _ { \\mathrm { r e f } } u ^ * _ { i , j } P _ { D H } ' \\Big { ) } . d p \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} ( 1 - \\eta _ 0 Q [ 0 ] ) \\ell [ 0 ] \\phi = \\ell [ 0 ] z \\ , . \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} \\int _ { a } ^ b f ( x ) d _ q x = ( 1 - q ) \\sum _ { n = 0 } ^ \\infty [ b f ( b q ^ n ) - a f ( a q ^ n ) ] q ^ n . \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} \\mathrm { d i v } ( \\delta _ x ) = \\mathrm { d i v } ( \\delta _ y ) = \\mathrm { d i v } ( \\delta _ z ) = 0 \\ , . \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{align*} f = x ^ 4 - b x ^ 2 + d . \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } D ( w _ { n } ^ { ( j ) } , w _ { n } ^ { ( j ) } ) = D ( w , w ) = \\lambda D ( Q , Q ) = \\lambda . \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{align*} \\psi _ A ( g ) = \\frac { 1 } { | H | } A _ { g \\cdot x _ 0 , x _ 0 } ( g \\in G ) \\end{align*}"}
-{"id": "6074.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } n ^ \\gamma \\lambda _ n ^ + ( A ) & = \\limsup _ { n \\to \\infty } n ^ \\gamma \\lambda _ n ^ + ( B ) , \\\\ \\liminf _ { n \\to \\infty } n ^ \\gamma \\lambda _ n ^ + ( A ) & = \\liminf _ { n \\to \\infty } n ^ \\gamma \\lambda _ n ^ + ( B ) . \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{align*} \\overline { \\dim } _ B ( A ) : = \\limsup _ { \\delta \\to 0 } \\frac { \\log N _ \\delta ( A ) } { - \\log \\delta } , \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} - \\frac 1 2 \\Delta _ B | \\phi | ^ 2 = & - \\langle \\overline \\square _ B \\phi , \\phi \\rangle - \\langle \\phi , \\overline \\square _ B \\phi \\rangle + \\sum _ a \\{ | \\nabla _ { \\bar V _ a } \\phi | ^ 2 + | \\nabla _ { V _ a } \\phi | ^ 2 \\} \\\\ & + \\sum _ { a } \\langle \\phi , R ^ Q ( \\bar V _ a , V _ a ) \\phi \\rangle - \\frac 1 2 \\langle \\nabla _ { H ^ { 1 , 0 } - H ^ { 0 , 1 } } \\phi , \\phi \\rangle - \\frac 1 2 \\langle \\phi , \\nabla _ { H ^ { 1 , 0 } - H ^ { 0 , 1 } } \\phi \\rangle . \\end{align*}"}
-{"id": "7969.png", "formula": "\\begin{align*} \\min _ x F ( x , c ) = f ^ * = F ( x ^ * , c ) \\forall c \\ge c _ 0 \\forall x ^ * \\in \\Omega ^ * \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} x _ M ^ u \\in \\mathop { \\oplus } _ { i = 1 } ^ s H ^ u ( M _ i ; \\R ) \\oplus \\biggl ( \\mathop { \\oplus } _ { \\substack { u _ 1 + \\cdots + u _ s = u , \\\\ 0 \\leq u _ j < u } } ( H ^ { u _ 1 } ( M _ 1 ; \\R ) \\otimes \\cdots \\otimes H ^ { u _ s } ( M _ s ; \\R ) ) \\biggl ) , \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{align*} z ^ { k } : = \\frac { y ^ { n _ k } - x } { \\norm { x - y ^ { n _ k } } _ p } = \\frac { - x } { \\norm { x - y ^ { n _ k } } _ p } + \\frac { \\norm { y ^ { n _ k } } _ p } { \\norm { x - y ^ { n _ k } } _ p } w ^ { n _ k } . \\end{align*}"}
-{"id": "4916.png", "formula": "\\begin{align*} ( a , \\ , 0 ) S ^ { - 1 } A ( \\varepsilon _ 0 ) S \\left ( \\begin{matrix} \\mathfrak { a } \\\\ \\mathfrak { b } \\end{matrix} \\right ) = ( \\mathfrak a ' \\ , , \\mathfrak b ' ) S ^ { - 1 } A ( \\varepsilon _ 0 ) S \\left ( \\begin{matrix} d \\alpha - b \\beta \\\\ 0 \\end{matrix} \\right ) \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} \\mathcal { B } _ { n , u } & = \\{ \\exists i \\in [ B ] : \\hat { X } _ { w _ i , u _ { m + 1 } } > K _ 1 n ^ { - \\beta } D _ { u _ { m + 1 } } ( 1 + n ^ \\gamma ) \\} , \\\\ \\mathcal { B } _ { n , v } & = \\{ \\exists i \\in [ B ] : \\hat { X } _ { w _ i , v _ { m + 1 } } > K _ 1 n ^ { - \\beta } D _ { v _ { m + 1 } } ( 1 + n ^ \\gamma ) \\} , \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} [ \\theta ( \\mu _ { \\beta _ k } ) , \\tau _ { \\beta _ j } ] = 2 \\mathcal { P } ( [ \\theta ( e _ { \\beta _ k } ) , e _ { \\beta _ j } ] ) + 2 \\mathcal { P } ( [ \\theta \\sigma ( e _ { \\beta _ k } ) , e _ { \\beta _ j } ] ) \\end{align*}"}
-{"id": "8459.png", "formula": "\\begin{align*} \\begin{pmatrix} A & 0 \\\\ 0 & B \\end{pmatrix} , \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} d \\log : K _ 2 ( \\tilde { K } ) & \\to \\Omega ^ 2 _ { \\tilde { K } / F } \\\\ d \\log ( \\{ a _ 1 , a _ 2 \\} ) & = \\frac { d a _ 1 } { a _ 1 } \\wedge \\frac { d a _ 2 } { a _ 2 } , \\end{align*}"}
-{"id": "4850.png", "formula": "\\begin{align*} f ^ * ( 1 \\times \\omega _ N ) = c ' \\cdot ( \\omega _ M \\times 1 ) + c \\cdot ( 1 \\times \\omega _ N ) . \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} { \\rm D } \\left ( \\frac { | - 1 + z _ { j , n } | } { | z _ { j , n } | } \\right ) ~ ~ = : ~ ~ \\frac { 1 - \\exp \\bigl ( - { \\rm D } ( \\frac { \\pi } { a _ { j , n } } ) \\bigr ) } { 2 \\exp \\bigl ( { \\rm D } ( \\frac { \\pi } { a _ { j , n } } ) \\bigr ) - 1 } \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} E \\left [ S ^ { t o t , m } _ { s , t } \\right ] - E \\left [ S ^ { t o t , 0 } _ { s , t } \\right ] = E \\left [ S ^ { e n v , m } _ { s , t } \\right ] - E \\left [ S ^ { e n v , 0 } _ { s , t } \\right ] - \\frac { n } { 2 } E \\left [ \\ln ( \\beta ( t , q _ t ) / \\beta ( s , q _ s ) ) \\right ] . \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} E _ d ( P ) _ { q = 1 } = \\sum _ { k = 0 } ^ { d - 1 } \\langle P , \\psi _ d ^ k \\rangle = \\langle P , \\sum _ { k = 0 } ^ { d - 1 } \\psi _ d ^ k \\rangle . \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} \\bar { u } ^ { \\varepsilon , N } _ { t } ( x ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N K _ { \\varepsilon } ( x - \\bar \\xi ^ { i } _ { t } ) \\bar { V } _ t \\big ( \\bar { \\xi } ^ { i } , \\bar { u } ^ { \\varepsilon , N } ( \\bar { \\xi } ^ { i } ) , \\nabla \\bar u ^ { \\varepsilon , N } ( \\bar { \\xi } ^ { i } ) \\big ) , \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} \\displaystyle { \\operatornamewithlimits { \\mbox { m i n i m i z e } } _ { \\Theta } } \\ f _ N ^ b ( \\Theta ) \\ , \\triangleq \\ , \\displaystyle { \\frac { 1 } { N } } \\ , \\displaystyle { \\sum _ { i = 1 } ^ N } \\ , \\ \\left [ y _ i \\ , m ( x ^ i ; \\Theta ) + b ( m ( x ^ i ; \\Theta ) ) \\ , \\right ] , \\end{align*}"}
-{"id": "3344.png", "formula": "\\begin{align*} Q = \\left ( t - \\frac { s } { | E | } \\right ) I + \\left ( \\frac { s } { | E | } - t ^ 2 \\right ) J \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} L _ v ( \\xi _ { \\mu } ) = \\begin{cases} \\xi _ { \\mu } & r ( \\mu ) = v \\\\ 0 & r ( \\mu ) \\neq v \\end{cases} \\ \\ \\ \\ L _ e ( \\xi _ { \\mu } ) = \\begin{cases} \\xi _ { e \\mu } & r ( \\mu ) = s ( e ) \\\\ 0 & r ( \\mu ) \\neq s ( e ) \\end{cases} \\end{align*}"}
-{"id": "3496.png", "formula": "\\begin{align*} \\max \\{ \\ , x _ 1 - x _ 2 - x _ 3 , \\ , x _ 1 + 4 , \\ , x _ 1 + x _ 2 + x _ 3 \\ , \\} & = - x _ 1 + x _ 2 + 2 \\\\ \\max \\{ \\ , x _ 1 + x _ 2 + x _ 3 , \\ , - x _ 1 + x _ 2 + 2 \\ , \\} & = x _ 1 - x _ 2 - x _ 3 \\\\ x _ 1 + x _ 2 + x _ 3 & = x _ 1 - x _ 2 - x _ 3 \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{align*} \\dim _ B ( A ) = \\lim _ { \\delta \\to 0 } \\frac { N _ \\delta ( A ) } { - \\log \\delta } , \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{align*} \\begin{cases} J \\left ( z , 0 \\right ) < 0 & \\ , z \\in \\ , \\Omega , \\\\ J \\left ( z , 0 \\right ) > 0 & \\ , z \\in \\left ( \\mathbb { R } ^ { n } \\setminus \\Omega \\right ) , \\\\ J \\left ( z , 0 \\right ) = 0 & \\ , z \\in \\left ( \\Omega \\setminus \\ , \\Omega \\right ) , \\end{cases} \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} \\tilde { c } = 0 , \\tilde { b } = ( m - 1 ) c . \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{align*} m _ { k } ( \\| u _ { k } \\| ^ { 2 } ) \\int _ { \\Omega } \\nabla u _ { k } \\nabla v d x = \\int _ { \\Omega } f _ { \\ast } ( u _ { k } ) v d x , \\ \\forall v \\in H _ { 0 } ^ { 1 } ( \\Omega ) . \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} \\int _ { x } ^ y \\frac { ( q z / x , q z / y , a b c z ; q ) _ \\infty } { ( a z / y , b z / x , c z ; q ) _ \\infty } d _ q z = \\frac { ( 1 - q ) y ( q , x / y , q y / x , a b , a c x , b c y ; q ) _ \\infty } { ( a x / y , b y / x , a , b , c x , c y ; q ) _ \\infty } . \\end{align*}"}
-{"id": "6795.png", "formula": "\\begin{align*} \\langle D ^ k \\xi , D ^ k \\eta \\rangle ( t _ 0 , x ) = \\sum _ { i _ 1 , \\ldots , i _ k = 1 } ^ n \\langle D ^ k _ { e _ { i _ 1 } , \\ldots , e _ { i _ k } } \\xi , D ^ k _ { e _ { i _ 1 } , \\ldots , e _ { i _ k } } \\eta \\rangle ( t _ 0 , x ) , \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{align*} \\mbox { o r } \\ \\ g ( C _ a ) = \\delta ( C _ a ) = { \\rm i n d } C _ a = 0 , \\ \\ \\mbox { a n d } \\ \\ e _ Q ( C _ a ) = 1 , \\ \\ h ( C _ a ) = 1 . \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} Z = A ( D , 0 ) B ^ { - 1 } . \\end{align*}"}
-{"id": "8018.png", "formula": "\\begin{align*} \\psi _ 1 ^ * = \\frac { 1 } { 2 a \\lambda _ 2 } { \\left ( \\sqrt { c _ 3 } + \\sqrt { c _ 4 } \\right ) ^ 2 } . \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{align*} \\max _ { i \\mid a _ { i j } < 0 } ( a _ i ^ T x + b _ i ) = \\max _ { i \\mid a _ { i j } > 0 } ( a _ i ^ T x + b _ i ) . \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} G _ { j } \\left ( \\wp _ \\lambda ( z ) \\right ) = \\zeta _ \\lambda ( z ) , \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} \\sharp S ( n , m ) = \\left ( \\begin{array} { c } n - 1 \\\\ m \\end{array} \\right ) . \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ N e _ i ( x _ 1 , \\dots , x _ N ) y ^ i = \\prod _ { j = 1 } ^ { N } ( 1 + x _ j y ) , \\ ( x _ 1 , \\dots , x _ N ) . \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} u ( z ) = a + \\int _ 0 ^ \\infty \\frac { 1 + z s } { z - s } d \\rho ( s ) , \\end{align*}"}
-{"id": "7458.png", "formula": "\\begin{align*} & \\frac { 1 } { \\sqrt { m } } E \\left [ J _ { s , t } ^ m \\right ] \\\\ = & - \\int _ s ^ t E \\left [ ( - \\nabla _ q V ( r , q _ r ) - \\partial _ r \\psi ( r , q _ r ) + \\tilde F ( r , q _ r ) ) \\cdot \\left ( \\int ( \\nabla _ z \\chi ) ( r , q _ r , z ) h ( r , q _ r , z ) d z \\right ) \\right ] d r \\\\ & - \\int _ s ^ t E \\left [ \\int \\left ( ( \\nabla _ q \\chi ) ( r , q _ r , z ) \\cdot z \\right ) h ( r , q _ r , z ) d z \\right ] d r + O ( m ^ { 1 / 2 } ) , \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{align*} \\lambda T _ 2 x + ( 1 - \\lambda ) x & = ( 1 + \\lambda ) P _ A T _ 2 x - \\lambda P _ A x \\\\ & = ( 1 + \\lambda ) P _ A x - \\lambda P _ A x = P _ A x . \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{align*} B _ 1 & = \\overline { \\omega _ 2 } z - \\omega _ 2 \\overline { z } \\\\ B _ 2 & = \\omega _ 1 \\overline { z } - \\overline { \\omega _ 1 } z . \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} = \\frac 1 { u _ 3 } A ^ { - 1 } \\begin{pmatrix} r _ 1 \\\\ r _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} u ( x ) : = \\widetilde { u } ( x ) - \\kappa w ( x ) , \\kappa : = c _ m ^ { - 1 } ( 2 d ) ^ { - m } . \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{align*} \\displaystyle \\left \\langle \\ ! \\ ! \\left \\langle p , \\sum _ { i = 1 } ^ r ( \\hat { b } ^ i - \\xi ^ i ) - \\sum _ { j = 1 } ^ s \\hat { a } ^ j \\right \\rangle \\ ! \\ ! \\right \\rangle = \\int _ 0 ^ T \\left [ \\sum _ { i = 1 } ^ r ( \\hat { b } _ { h _ 0 } ^ { i } ( t ) - \\xi _ { h _ 0 } ^ { i } ( t ) ) - \\sum _ { j = 1 } ^ s \\hat { a } _ { h _ 0 } ^ { j } ( t ) \\right ] d t \\leq 0 . \\end{align*}"}
-{"id": "6067.png", "formula": "\\begin{align*} d _ { \\textrm { m a x } , 1 } ( f , ( 0 0 ) ) & = \\max \\left \\{ d ( f , ( 0 0 ) , ( 0 1 ) ) , d ( f , ( 0 0 ) , ( 1 0 ) ) \\right \\} \\\\ & = \\max \\left \\{ \\frac { | 0 - 1 | } { 2 ^ q } , \\frac { | 0 - 3 | } { 2 ^ q } \\right \\} = \\frac { 3 } { 4 } , \\end{align*}"}
-{"id": "931.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow \\infty } \\big \\{ \\lambda ^ { - d / 2 } \\ , N [ \\mathfrak { e } , \\lambda V ] \\big \\} \\ = \\ \\lim _ { \\lambda \\rightarrow \\infty } \\big \\{ \\lambda ^ { - d / 2 } \\ , N _ { s c } [ \\mathfrak { e } , \\lambda V ] \\big \\} \\ = \\ 0 . \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{align*} \\omega _ 1 = \\pi \\sum _ { n = 0 } ^ \\infty \\frac { ( \\frac 1 2 ) _ n ^ 2 } { n ! ^ 2 } \\lambda ^ n . \\end{align*}"}
-{"id": "4350.png", "formula": "\\begin{align*} Y ^ { \\pm } _ { \\zeta , j , m , n } = \\{ ( x , y , f _ \\wp , g _ \\wp , f _ \\zeta , g _ \\zeta ) \\in \\mathfrak { F } _ \\Omega \\times V _ j \\times \\R ^ 2 : ( x , y , f _ \\wp , g _ \\wp ) \\in Y ^ \\pm _ { \\wp , j , m , n } \\end{align*}"}
-{"id": "8561.png", "formula": "\\begin{align*} X _ { \\gamma } = \\sigma _ { \\gamma } \\prod _ a ( X _ { \\gamma ^ a } ) ^ { q _ a } \\rlap { . } \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} & \\tilde { q } _ 1 ( t , z , \\zeta ) = i \\frac { \\partial W _ { 1 3 } } { \\partial x } ( t , x ^ { ( 1 ) } - x ^ { ( 3 ) } ) \\cdot \\xi ^ { ( 1 ) } + \\frac { 1 } { 2 } \\Delta _ x W _ { 1 3 } ( t , x ^ { ( 1 ) } - x ^ { ( 3 ) } ) \\\\ & - i A ^ { ( 1 ) } ( t , x ^ { ( 1 ) } ) \\cdot \\frac { \\partial W _ { 1 3 } } { \\partial x } ( t , x ^ { ( 1 ) } - x ^ { ( 3 ) } ) . \\end{align*}"}
-{"id": "4643.png", "formula": "\\begin{align*} & H ^ { 0 , 1 } \\lrcorner \\ , \\bar \\partial _ B \\phi = \\nabla _ { H ^ { 0 , 1 } } \\phi , \\\\ & \\partial _ B H ^ { 1 , 0 } \\lrcorner \\ , \\phi + H ^ { 1 , 0 } \\lrcorner \\ , \\partial _ B \\phi = \\nabla _ { H ^ { 1 , 0 } } \\phi + \\sum _ a \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi . \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} [ C _ D , D ] = O + \\frac { z _ 1 \\bar { z } _ 2 } { 2 } \\beta ( l _ j ) ( \\coth ( \\beta ( a ) - \\tanh ( \\beta ( a ) ) \\theta ( \\tau _ { \\beta } ) . \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{align*} \\min _ { \\substack { \\mathcal { H } ( u ) = H _ 0 \\\\ M ( u ) = M _ 0 } } P ( u ) . \\end{align*}"}
-{"id": "2766.png", "formula": "\\begin{align*} \\| g _ n \\| _ { L ^ 1 } & = \\int _ I ( | f _ n ^ { ( k + 1 ) } | \\circ f _ n ^ { - 1 } ) \\cdot ( ( f _ n ' ) ^ { - 1 } \\circ f _ n ^ { - 1 } ) \\\\ & = \\int _ I | f _ n ^ { ( k + 1 ) } | \\\\ & = \\| f _ n ^ { ( k + 1 ) } \\| _ { L ^ 1 } . \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{align*} \\mathbf { w } ^ 0 _ 1 & = \\sum _ { \\mathbf { k } \\in \\mathbb { Z } ^ k } \\mathbf { V } ( i \\langle \\mathbf { k } , \\Omega \\rangle \\mathbf { I } - \\mathbf { \\Lambda } ) ^ { - 1 } \\mathbf { V } ^ { - 1 } \\mathbf { g } _ { e x t } ^ { \\mathbf { k } } e ^ { i \\langle \\mathbf { k } , \\phi \\rangle } , \\\\ \\mathbf { r } ^ 0 _ 1 & = \\mathbf { 0 } . \\end{align*}"}
-{"id": "8392.png", "formula": "\\begin{align*} - w ' ( s ) = \\zeta ( s ) + d _ A \\psi ( s ) . \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{align*} \\int _ { \\partial U _ { \\epsilon } } u \\nabla y \\cdot \\nu d s = \\int _ { \\partial U _ { \\epsilon } } y \\nabla u \\cdot \\nu d s \\end{align*}"}
-{"id": "9107.png", "formula": "\\begin{align*} \\bar { a } = \\left ( 1 - { \\sqrt { \\frac { r } { c } } } \\right ) ^ 2 ; \\bar { b } = \\left ( 1 + { \\sqrt { \\frac { r } { c } } } \\right ) ^ 2 . \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{align*} \\inf _ { x \\in A } F ( x , \\lambda ^ * , c ) = F ( x ^ * , \\lambda ^ * , c ) . \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} K _ m = \\sum _ { \\mu \\in \\lbrace 0 , 1 \\rbrace ^ N } q ^ { 1 / 2 - \\mu _ m } E _ \\mu ^ * , \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} v ( x , t ) = u ^ { \\tilde f } ( x , t ) - u ^ { \\tilde f } ( x , 2 T - t ) \\mbox { i n } \\ , Q ^ T \\end{align*}"}
-{"id": "2903.png", "formula": "\\begin{align*} \\left | D ^ 2 u ( x ) - A \\right | = \\left | D ^ 2 v ( x ) \\right | \\leq C | x | ^ { 2 - n } . \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} \\forall m \\in \\N ^ * , A ^ V _ q u ^ V _ { q , m } = \\varepsilon ^ V _ { q , m } u ^ V _ { q , m } . \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{align*} \\dim D ^ { I I I } _ n = \\frac { n ( n + 1 ) } { 2 } , r = n , a = 1 , b = 0 , p = n + 1 . \\end{align*}"}
-{"id": "4790.png", "formula": "\\begin{align*} \\begin{aligned} \\bigl \\langle \\partial J ( u ) , \\sum _ { i = 1 } ^ p \\overline { X } _ i \\bigr \\rangle & \\leq - c \\sum _ { i = 1 } ^ p \\Bigl [ 1 - \\Psi \\bigl ( \\lambda _ i | ( a _ i - y _ { j _ i } ) _ { k _ i } | \\bigr ) \\Bigr ] \\frac { \\bigl | \\nabla K ( a _ i ) \\bigr | } { \\lambda _ i } \\\\ & + o \\bigl ( \\sum _ { i = 1 } ^ p \\frac { \\bigl | \\nabla K ( a _ i ) \\bigr | } { \\lambda _ i } + \\frac { 1 } { \\lambda _ i ^ { n - 2 } } \\bigr ) , \\end{aligned} \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} \\rho _ M \\cdot m ''' = \\frac { E ''' } F - 3 \\frac { F '' } { F } e _ 1 . \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} \\operatorname { d i v } ( | y | ^ a \\nabla U _ 0 ) = | y | ^ a ( U _ 0 ) _ t . \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} \\dot { r } & = \\sin \\phi \\\\ \\dot { s } & = \\cos \\phi \\\\ \\dot { \\phi } & = \\left ( \\frac { - r } { 2 } + \\frac { ( n - 1 ) 2 r } { r ^ 2 - s ^ 2 } \\right ) \\cos \\phi + \\left ( \\frac { s } { 2 } + \\frac { ( n - 1 ) 2 s } { r ^ 2 - s ^ 2 } \\right ) \\sin \\phi + \\lambda . \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} \\lim _ { m \\longrightarrow + \\infty } \\int _ 0 ^ 1 \\phi ( \\Lambda ^ m ) = \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { C P } ( 1 ) } \\phi ( f ( x ) ) d \\Omega . \\end{align*}"}
-{"id": "355.png", "formula": "\\begin{align*} K _ p = A _ p \\times \\hat { A } _ p \\cong ( \\Z _ { p ^ { \\lambda _ { p , 1 } } } \\times \\dotsb \\times \\Z _ { p ^ { \\lambda _ { p , l _ p } } } ) \\times ( \\Z _ { p ^ { \\lambda _ { p , 1 } } } \\times \\dotsb \\times \\Z _ { p ^ { \\lambda _ { p , l _ p } } } ) \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} h _ B ^ { 0 , 0 } = h _ B ^ { 0 , 1 } = 1 , h _ B ^ { 1 , 0 } = h _ B ^ { 1 , 1 } = 0 . \\end{align*}"}
-{"id": "4646.png", "formula": "\\begin{align*} \\lambda _ a & = \\langle \\sum _ b R ^ Q ( \\bar V _ b , V _ b ) \\omega ^ a , \\omega ^ a \\rangle \\\\ & = \\sum _ { b = 1 } ^ n g _ Q ( R ^ Q ( \\bar V _ b , V _ b ) V _ a , \\bar V _ a ) \\\\ & = R i c ^ Q ( E _ a , E _ a ) . \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} \\alpha \\times \\beta = 0 \\quad x \\quad \\Leftrightarrow \\Phi _ { \\ast } ( \\alpha ) \\times \\Phi _ { \\ast } ( \\beta ) = 0 \\quad \\Phi ( x ) . \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{align*} \\beta _ 2 + \\frac { \\beta _ 3 } { \\alpha } < 1 , \\beta _ 1 + \\beta _ 2 \\alpha + \\beta _ 3 = 1 \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} m _ { \\varphi , \\overline { \\mathcal { O } ( 1 ) } , X _ 0 } ( r ) & = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\log \\sqrt { 1 + \\sum _ { j = 1 } ^ { n } | f _ j ( r e ^ { i \\theta } ) | ^ 2 } d \\theta \\\\ & \\le \\sum _ { j = 1 } ^ n \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\log \\sqrt { 1 + | f _ j ( r e ^ { i \\theta } ) | ^ 2 } d \\theta = \\sum _ { j = 1 } ^ n m _ { f _ j } ( r ) \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} d A ' = B ' - [ A \\wedge A ' ] . \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{align*} K _ t = N _ x + [ K , N ] . \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & i \\partial _ { t } u = \\Pi ( | u | ^ 2 u ) , ( t , z ) \\in \\R \\times \\C , \\\\ & u ( 0 , z ) = u _ 0 ( z ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} \\nabla _ { \\partial _ t } D _ X X _ i - D _ X \\nabla _ { \\partial _ t } X _ i = \\frac { 1 } { 2 } \\big ( \\dot { g } ( D _ X X _ i ) - D _ X ( \\dot { g } ( X _ i ) ) \\big ) = - \\frac { 1 } { 2 } D _ X \\dot { g } ( X _ i ) . \\end{align*}"}
-{"id": "7803.png", "formula": "\\begin{align*} \\| \\mathcal { I } _ { 3 , + } ( t , x ) \\| _ p ^ 2 & \\leq k M _ { p , t } ^ 2 \\| f \\| _ { L ^ q ( \\mathbb { R } ) } ^ 4 \\Big ( \\frac { q } { q - 1 } \\Big ) ^ 2 t ^ { 2 - 2 / q } + 2 t \\| f \\| _ { L ^ 2 ( \\mathbb { R } ) } ^ 4 \\int _ 0 ^ t U _ s d s \\\\ & = c ^ { ( 3 ) } _ { p , t } + c ^ { ( 4 ) } _ { p , t } \\int _ 0 ^ t U _ s d s , \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{align*} u _ i < \\{ v \\in \\bar { v } : u _ i < v & < u _ { i + 1 } \\} < \\\\ \\{ v \\in \\bar { v } _ 2 & : u _ i < v < u _ { i + 1 } \\} \\cup \\{ w \\in \\bar { w } _ 2 : u _ i < w < u _ { i + 1 } \\} \\\\ < ~ & \\{ w \\in \\bar { w } : u _ i < w < u _ { i + 1 } \\} < u _ { i + 1 } \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{align*} ( a , A ) \\cdot ( b , B ) = ( a + b , A + B + ( 1 / 2 ) ( a \\otimes b - b \\otimes a ) ) \\end{align*}"}
-{"id": "120.png", "formula": "\\begin{align*} A _ \\infty ( q ) = A _ H + \\frac { 1 } { 2 } \\left ( \\Im \\bar \\partial \\log | q | _ k \\right ) \\ , \\begin{pmatrix} i & 0 \\\\ 0 & - i \\end{pmatrix} \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} r \\ = \\sum \\lambda _ { i _ 1 i _ 2 \\cdots i _ { t - 1 } } h _ { i _ 1 } h _ { i _ 2 } \\cdots h _ { i _ { t - 1 } } \\ \\in \\ ( u _ 1 ^ n , u _ 2 ^ n , \\dots , u _ { t - 1 } ^ n , v _ 1 ^ n , v _ 2 ^ n , \\dots , v _ { t - 1 } ^ n ) S , \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} E _ \\mu ^ * R _ m L _ m + E _ \\mu ^ * L _ m R _ m = \\sum _ { \\lambda } \\theta ( m , \\mu , \\lambda ) E _ \\mu ^ * E _ { \\lambda } , \\end{align*}"}
-{"id": "6596.png", "formula": "\\begin{align*} \\bar { \\varphi } ( X , Y , Z ) = ( X \\cdot Y \\cdot Z \\cdot \\bar { \\psi } , \\bar { \\psi } ) , X , Y , Z \\in T M , \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} X ' ( t ) = \\int _ 0 ^ 1 X ( t ) ^ { 1 - s } H ' ( t ) X ( t ) ^ s { \\rm d } s \\ , \\end{align*}"}
-{"id": "6392.png", "formula": "\\begin{align*} \\frac 1 n \\sum _ { i = 0 } ^ { n - 1 } h ( X _ i ) \\to \\int h \\dd \\mu . \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} \\phi _ V ( \\mathbf { u } ) = f _ V ( \\zeta ) + \\overline { f _ V ( \\zeta ) } \\end{align*}"}
-{"id": "3899.png", "formula": "\\begin{align*} A _ X f _ { \\hat \\tau } ( x , y ) & = 0 , \\enskip x \\in C , \\\\ f _ { \\hat \\tau } ( x , y ) - F ( x , y ) & = 0 , \\enskip x \\in E \\backslash C , \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} g _ n = \\sum _ { z \\in \\Lambda _ n } \\delta A ^ n _ z \\varphi ^ n _ z , \\delta \\xi _ n = \\sum _ { z \\in \\Lambda _ n } \\Big ( \\sum _ { u \\in \\Lambda _ { n + 1 } } A ^ { n + 1 } _ u \\langle \\varphi ^ { n + 1 } _ u , \\psi ^ n _ z \\rangle \\Big ) \\ , \\psi ^ n _ z . \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} g _ \\infty = - \\ : 1 6 d u ^ 2 + u ^ 2 d \\theta ^ 2 + \\left [ 4 ^ { 1 + \\nu } c _ 1 c _ \\lambda ^ { - 1 - \\nu } d \\widehat { \\sigma } ^ 2 + 4 ^ { 1 - \\nu } c _ 1 ^ { - 1 } c _ \\lambda ^ { - 1 + \\nu } d \\widehat { \\delta } ^ 2 \\right ] . \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} p _ u = \\mathbf { P } ( ( \\overline { \\omega } _ i ) _ 1 = ( u ) _ 1 ) \\mathbf { P } ( \\overline { \\omega } _ i = u | ( \\overline { \\omega } _ i ) _ 1 = ( u ) _ 1 ) = q _ { ( u ) _ 1 } \\mathbf { P } ( \\overline { \\omega } _ i = u | ( \\overline { \\omega } _ i ) _ 1 = ( u ) _ 1 ) \\end{align*}"}
-{"id": "7028.png", "formula": "\\begin{align*} \\begin{cases} ( 2 , 2 , \\cdots , 2 ) , \\ & \\emph { w h e n } ~ n \\equiv 0 \\pmod { 2 } , \\cr ( 3 , 2 , \\cdots , 2 ) ~ \\emph { o r } ~ ( 2 , 2 , \\cdots , 2 , 1 ) , \\ & \\emph { w h e n } ~ n \\equiv 1 \\pmod { 2 } . \\end{cases} \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{align*} | \\nabla _ \\infty ^ l \\left ( f ^ \\ast ( P u ) - P _ \\infty f ^ \\ast u \\right ) | _ { h _ \\infty } = O ( r ^ { - k + \\mu - l } ) \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} g Y = ( Y + W ) g . \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} f ( x ) + \\frac { f '' ( x ) } { ( 1 + f ' ( x ) ) ^ { 3 / 2 } } = 0 \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} \\sigma _ { 2 j - 1 } = - \\sigma _ { 2 j } , j = 1 , 2 , \\ldots , \\left [ \\frac { n } { 2 } \\right ] \\ \\ \\sigma _ { n } = 0 \\ \\ n \\ . \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{align*} t \\mapsto Z _ n ( t ) : = \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { X _ { n , i } ( t ) } , \\frac { 1 } { n } \\sum _ { i = 1 } ^ n W _ { n , i } ( t ) \\right ) , \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in \\Bbb F _ q } \\| \\lambda _ { \\alpha } \\| ^ 2 = | R _ k ^ { \\ast } | ^ 2 + \\sum _ { i = 1 } ^ { d } \\| \\lambda _ { \\alpha _ { i } } \\| ^ 2 | R _ k ^ { \\ast } | . \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{align*} s ^ { ( r ) } = \\frac { \\Gamma ( s + r ) } { \\Gamma ( s ) } \\sim s ^ r , s \\to \\infty . \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} \\Re ( C _ E ) & = \\frac { z _ 1 \\bar { z } _ 2 + \\bar { z } _ 1 z _ 2 } { \\sinh ( 2 \\beta ) } \\tanh ( \\beta - \\sigma ( \\beta ) ) \\Re \\circ \\mathcal { H } ( [ \\theta \\sigma ( e _ { \\beta } ) , e _ { \\beta } ] ) \\\\ & = \\frac { z _ 1 \\bar { z } _ 2 + \\bar { z } _ 1 z _ 2 } { \\cosh ( 2 \\beta ) } \\mathcal { H } \\circ \\Re ( [ \\theta \\sigma ( e _ { \\beta } ) , e _ { \\beta } ] ) . \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{align*} \\binom { t } 2 \\ , k _ t ( G ) \\le m \\ , \\binom { r - 1 } { t - 2 } . \\end{align*}"}
-{"id": "636.png", "formula": "\\begin{align*} p _ i : = \\mathbb { P } ( W _ i ( n ) = 1 ) = 1 - \\mathbb { P } ( W _ i ( n ) = 0 ) . \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{align*} z ^ { ( n ) } ( \\xi ) = \\left ( \\frac { 1 } 2 \\right ) _ { n } \\int _ { \\xi } ^ { - \\infty } \\frac { d X } { 2 X ^ { n + 1 } \\sqrt { X - 1 } } \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} = \\frac { 1 } { n } \\left | \\ , F \\bigl ( \\frac { 2 \\pi j } { n } \\bigr ) - \\ , F \\bigl ( \\frac { 2 \\pi ( j + 1 ) } { n } \\bigr ) ) \\right | = \\frac { 2 \\pi } { n ^ 2 } \\left | F ' ( \\xi ) \\right | \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{align*} P _ k = \\mathbb { P } ( X \\neq \\hat { X } ) \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} \\mathbb { P } _ L ^ * ( \\mathbf { m } , K ) \\geq \\prod _ { j = 1 } ^ { L } \\sum _ { h _ j } \\frac { \\binom { K } { h _ j } \\binom { N - K } { m _ j - h _ j } } { \\binom { N } { m _ j } } \\mathbb { P } ( m _ j - h _ j , K - h _ j ) . \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{align*} ^ T \\Lambda ( t ) \\cdot M \\cdot \\Lambda ( t ) = \\rho + \\frac { ( 1 - t ) ^ 2 } { \\| y ( t ) \\| ^ 2 } \\bigl [ ^ T \\Lambda \\cdot M \\cdot \\Lambda - \\rho | \\Lambda | ^ 2 \\bigr ] . \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{align*} J _ 1 ( N ; \\theta ) = \\frac { N } { 1 - \\theta } \\int _ 0 ^ 1 \\frac { 1 - y ^ N } { 1 - y } \\ , d y = \\frac { N } { 1 - \\theta } \\int _ 0 ^ 1 \\left ( \\sum _ { j = 0 } ^ { N - 1 } y ^ j \\right ) d y = \\frac { N } { 1 - \\theta } \\sum _ { j = 1 } ^ N \\frac { 1 } { j } , \\end{align*}"}
-{"id": "7261.png", "formula": "\\begin{align*} \\forall \\ , \\lambda \\neq \\mu , \\sum _ { k = - N } ^ N \\omega _ k v _ k \\phi _ \\lambda ( v _ k ) \\phi _ \\mu ( v _ k ) T ( v _ k ) = 0 . \\end{align*}"}
-{"id": "8713.png", "formula": "\\begin{align*} ( d x ^ 1 ) ^ 2 = d t ^ 2 - 2 \\tfrac { Z ^ i } { | Z | } \\ , d t \\ , d x _ i + \\tfrac { Z ^ i Z ^ j } { | Z | ^ 2 } \\ , d x _ i \\ , d x _ j , \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{align*} X _ { i , j } : = \\delta _ { i , j + k } x _ i \\ ; , M _ { i , j } : = \\delta _ { i + 1 , j } \\ ; , B _ { i , j } : = b _ i \\delta _ { i , j } : = - \\delta _ { i , j } ( x _ i + x _ { i + 1 } + \\cdots + x _ { i + k } ) \\ ; . \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\varphi } _ t = - 2 \\star _ t \\left ( \\mathrm { d i v } ( T ^ { \\varphi _ t } ) \\wedge \\varphi _ t \\right ) , \\\\ \\varphi _ t \\in [ \\bar { \\varphi } ] , \\\\ \\varphi _ 0 = \\bar { \\varphi } , \\end{cases} \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} v ^ \\mu _ { k + 1 } ( s ) : = v _ k ^ \\mu ( \\infty ) + \\int _ s ^ \\infty \\Gamma ^ \\mu _ { \\kappa \\lambda } | _ { x _ k ( u ) } v _ k ^ \\kappa ( u ) v _ k ^ \\lambda ( u ) \\ , d u , \\ \\ x _ { k + 1 } : = I ( v _ { k + 1 } ) . \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } - \\frac 1 2 \\Delta \\phi + \\omega \\phi + \\phi \\psi = 0 \\\\ \\psi + \\phi ^ 2 = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{align*} & R _ N ( X ; Y \\| Z ) = \\frac { p ^ N + { ( 1 - p ) } ^ N } { N } \\max \\Bigg \\{ 0 , \\epsilon ^ N - h \\Bigg ( \\frac { p ^ N } { p ^ N + { ( 1 - p ) } ^ N } \\Bigg ) \\Bigg \\} . \\end{align*}"}
-{"id": "3789.png", "formula": "\\begin{align*} \\tilde \\beta & = - | A _ { 1 1 2 } | ^ 2 - | A _ { 2 1 2 } | ^ 2 , & \\beta _ 0 & = \\begin{pmatrix} i \\sqrt { 2 } \\cdot A _ { 1 1 2 } \\overline { A _ { 2 1 2 } } \\\\ i \\sqrt { 2 } \\cdot A _ { 2 1 2 } \\overline { A _ { 1 1 2 } } \\\\ | A _ { 2 1 2 } | ^ 2 - | A _ { 1 1 2 } | ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} \\overline { \\mu } : = \\left ( X \\xrightarrow { \\mu } \\mathfrak { g } _ { } \\xrightarrow { \\pi } \\mathfrak { g } _ { } / G \\xrightarrow { \\varphi ^ { - 1 } } S _ { } \\right ) , \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} V _ i ( n + 1 ) : = \\{ W _ i ( n + 1 ) \\geq 1 \\} \\bigcap \\bigcap _ { j \\neq i } \\{ W _ j ( n + 1 ) = 0 \\} \\end{align*}"}
-{"id": "8227.png", "formula": "\\begin{align*} \\gamma _ j = | \\nabla K _ j ( x ) | \\ \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} a _ n = b _ n + d _ n + e _ n + g _ n + C _ { n - 1 } , n \\geq 2 , \\end{align*}"}
-{"id": "7641.png", "formula": "\\begin{align*} F _ { { \\cal D } \\vert _ X } = { \\cal D } \\vert ^ 2 _ X = D ^ 2 + D \\overline D + \\overline D D + \\overline D ^ 2 \\ : . \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{align*} { \\bf D } _ { T } = { \\rm d i a g } _ 0 ( { \\bf G } ) + { \\rm d i a g } _ 1 ( { \\bf G } ) . \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} f _ 0 : = \\frac { d \\nu _ 0 ^ { ( m ) } } { d \\mu ^ { ( m , 2 ) } _ \\star } \\ , , \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { k } \\Big ( \\mu _ { p _ i } ( x , z ) + \\mu _ { p _ i } ( y , z ) \\Big ) \\leq 9 ^ k \\Bigg ( \\prod _ { i = 1 } ^ { k } \\mu _ { p _ i } ( x , z ) + \\prod _ { i = 1 } ^ { k } \\mu _ { p _ i } ( y , z ) \\Bigg ) . \\end{align*}"}
-{"id": "3746.png", "formula": "\\begin{align*} \\overline { g } _ \\infty ( x ) = \\lim _ { n \\rightarrow + \\infty } \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } g _ { \\infty } ( T ^ k x ) . \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} \\sigma ( \\alpha ) ( t ) = \\alpha ( \\sigma ( t ) ) = \\alpha ( t ^ { - 1 } ) = ( - \\alpha ) ( t ) . \\end{align*}"}
-{"id": "8813.png", "formula": "\\begin{align*} \\Omega _ { \\beta _ 1 , \\bar { \\beta } _ 2 } = & \\frac { \\tanh ( \\beta _ 2 - \\beta _ 1 ) } { 2 } \\Big { \\{ } \\frac { 1 } { \\sinh ( 2 \\beta _ 1 ) } - \\frac { 1 } { \\sinh ( 2 \\beta _ 2 ) } \\Big { \\} } \\chi ( [ \\theta ( e _ { \\beta _ 2 } ) , e _ { \\beta _ 1 } ] ) \\\\ & + \\frac { \\tanh ( \\beta _ 1 + \\beta _ 2 ) } { 2 } \\Big { \\{ } \\frac { 1 } { \\sinh ( 2 \\beta _ 2 ) } + \\frac { 1 } { \\sinh ( 2 \\beta _ 1 ) } \\Big { \\} } \\chi ( [ \\theta ( e _ { \\beta _ 2 } ) , \\sigma ( e _ { \\beta _ 1 } ) ] ) \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} \\mathfrak { M } _ { \\vec { a } , q } ( \\theta ) = \\{ \\alpha \\in [ 0 , 1 ) ^ r : 2 | q \\vec { \\alpha } - \\vec { a } | < \\widetilde { C } ^ { r - 1 } P ^ { - d + r ( d - 1 ) \\theta } \\} . \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} \\alpha ( S ) = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ { n + 1 } | l _ { i j } | . \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} \\norm { \\mathcal { A } _ p u } _ { L ^ { s } ( 0 , T ; W ^ { - 1 , p } _ { \\sigma } ( \\Omega ) ) } = \\norm { [ A ^ { \\frac { 1 } { 2 } } _ { p ' } ] ^ * \\Phi ^ { - 1 } A _ p \\mathcal { U } } _ { L ^ s ( 0 , T ; W ^ { - 1 , p } _ { \\sigma } ( \\Omega ) ) } \\leq \\norm { [ A ^ { \\frac { 1 } { 2 } } _ { p ' } ] ^ * \\Phi ^ { - 1 } } _ { \\mathcal { L } ( L ^ p _ { \\sigma } , W ^ { - 1 , p } _ { \\sigma } ) } \\norm { A _ p \\mathcal { U } } _ { L ^ { s } ( 0 , T ; L ^ p _ { \\sigma } ( \\Omega ) ) } . \\end{align*}"}
-{"id": "7947.png", "formula": "\\begin{align*} | | \\mu ^ { \\boxtimes t } | | = \\frac { 1 } { h _ t ( \\alpha _ t ) } . \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} R _ { \\varphi } = ( - 2 / 3 + 2 ( 1 - \\lambda ) \\omega _ 1 ' / \\omega _ 1 ) \\int _ 1 ^ { \\hat { X } } \\frac { d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } . \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} ( 1 - x - 1 / u ) B ( x , u ) & = \\frac { x ^ 5 u ^ 2 C ^ 2 ( x ) } { ( 1 - x u ) ( 1 - x u C ( x ) ) } - \\frac { x ^ 5 C ^ 4 ( x ) } { 1 - 2 x } \\\\ & \\quad + \\frac { x ^ 3 ( 1 - x ) u } { ( 1 - 2 x ) ( 1 - x u ) } - \\frac { x ^ 3 ( 1 - x ) } { ( 1 - 2 x ) ^ 2 } . \\end{align*}"}
-{"id": "1637.png", "formula": "\\begin{align*} \\Phi ( \\theta ) : = \\int q ( x ) \\log \\frac { q ( x ) } { p ( x | \\theta ) } d x . \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{align*} ( 1 - P _ { \\frac k m } ) R _ \\delta = ( 1 - P _ { \\frac k m } ) R _ \\delta ( 1 - P _ { \\frac k m } ) , \\end{align*}"}
-{"id": "740.png", "formula": "\\begin{align*} d _ { \\beta } ( \\frac { 1 } { \\beta } ) = 0 . 0 \\ , t _ 1 t _ 2 t _ 3 \\ldots ~ ~ < _ { l e x } ~ ~ d _ { \\beta } ( x ) ~ ~ < _ { l e x } ~ ~ d _ { \\beta } ( 1 ) = 0 . t _ 1 t _ 2 t _ 3 \\ldots . \\end{align*}"}
-{"id": "6641.png", "formula": "\\begin{align*} s ^ 2 \\alpha = \\alpha \\Rightarrow \\alpha \\in \\Delta _ 0 . \\end{align*}"}
-{"id": "2540.png", "formula": "\\begin{align*} \\mathrm { k e r } ( F _ u ( \\eta _ 0 , 0 ) ) = \\mathrm { k e r } ( 1 - \\eta _ 0 K ) = \\mathrm { s p a n } \\{ \\Pi [ 0 ] \\zeta _ 0 \\} \\ , . \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} H ' ( r ) = \\frac { 4 } { r } I ( r ) + \\frac { a } { r } H ( r ) . \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} ( \\lambda I - \\mathbb { A } ) v = \\mathbb { P } F \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} 3 ^ { r _ 1 } Z _ 1 ^ 3 + 3 ^ { r _ 2 } Z _ 2 ^ 3 + Z _ 3 ^ 3 = 0 \\\\ 9 ^ { q _ 1 } Z _ 1 + 9 ^ { q _ 2 } Z _ 2 + Z _ 3 = 0 \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} \\ \\Phi _ m P _ { \\pm n } - P _ { \\pm n } \\Phi _ m \\gamma ^ { \\mp n } = \\pm \\Phi _ m ( \\gamma ^ { \\mp n } - q ^ { \\pm r n } ) \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} y ^ a | \\nabla U | ^ 2 & = \\frac 1 2 \\operatorname { d i v } ( y ^ { a } \\nabla U ^ 2 ) - \\frac 1 2 y ^ { a } ( U ^ 2 ) _ t \\end{align*}"}
-{"id": "9048.png", "formula": "\\begin{align*} K _ M \\cdot c _ 0 = \\frac 1 2 K _ \\eta \\| c _ 0 \\| ^ 2 + K _ B c _ 0 + K _ w - c _ 0 ( - b + K _ \\eta ^ T c _ 0 ) = - c _ 1 + ( c _ 0 ) _ x \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} \\frac { X _ 1 ( \\mu _ 1 ) } { X _ 1 ( h ) } = - \\frac { U _ 1 ( \\mu _ 2 ) } { U _ 1 ( \\varphi ) } = c , \\mbox { i n } D _ 1 \\mbox { a n d } G _ 1 , \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{align*} \\tilde \\gamma ^ i _ j \\equiv \\delta ^ { i k } \\tilde \\gamma _ { k j } = & \\left ( \\begin{array} { c c c } \\gamma & B _ 0 & 0 \\\\ - B _ 0 & \\gamma & 0 \\\\ 0 & 0 & \\gamma \\end{array} \\right ) \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{align*} \\mathcal { A } _ { p } = \\big [ A ^ { \\frac { 1 } { 2 } } _ { p ' } \\big ] ^ * \\Phi ^ { - 1 } \\circ A _ { p } \\circ A ^ { - \\frac { 1 } { 2 } } _ { p } \\Phi = \\big [ A ^ { \\frac { 1 } { 2 } } _ { p ' } \\big ] ^ * \\Phi ^ { - 1 } \\circ A _ { p } \\circ \\Phi \\big [ A ^ { - \\frac { 1 } { 2 } } _ { p ' } \\big ] ^ * . \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} d y _ { i , t } = \\left [ - \\nabla f ( y _ { i , t } ) - \\sum _ { j = 1 , j \\neq i } ^ N \\alpha _ { i j } \\nabla _ { y _ { i , t } } J \\| y _ { i , t } - y _ { j , t } \\| \\right ] \\tilde { \\gamma } d t + \\tau \\tilde { \\gamma } d B _ { i , t } . \\end{align*}"}
-{"id": "3727.png", "formula": "\\begin{align*} g ( a + 1 , n ) - g ( a , n ) = \\int _ 0 ^ 1 d _ { a , n } ( t ) \\ , d t \\ , . \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} \\sum _ { d \\geq 0 } p ( k , n - k ; d ) t ^ d = \\binom { n } { k } _ t = \\frac { \\prod _ { i = 1 } ^ { n } ( 1 - t ^ i ) } { \\prod _ { i = 1 } ^ { k } ( 1 - t ^ i ) \\prod _ { i = 1 } ^ { n - k } ( 1 - t ^ i ) } \\end{align*}"}
-{"id": "9007.png", "formula": "\\begin{align*} k _ t = k _ { x x x } + 3 k k _ x \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} \\mathcal { L } x _ i - \\frac 1 2 x _ i = 0 \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} \\rho & = i ( k + w ) z ^ { 1 \\bar { 1 } } + i \\overline { ( a ' + b ) } z ^ { 1 \\bar { 2 } } + i ( a ' + b ) z ^ { 2 \\bar { 1 } } + i ( u + l ) z ^ { 2 \\bar { 2 } } \\\\ r & = i ( k + u ) z ^ { 1 \\bar { 1 } } + i \\overline { ( a + b ' ) } z ^ { 1 \\bar { 2 } } + i ( a + b ' ) z ^ { 2 \\bar { 1 } } + i ( w + l ) z ^ { 2 \\bar { 2 } } \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\alpha _ i g ( a _ i ) = \\int _ 0 ^ 1 ( 1 + t ) ^ { n - a _ k } r ( t ) \\ , d t \\ , . \\end{align*}"}
-{"id": "4348.png", "formula": "\\begin{align*} G _ { j } ( \\hat { X } ) = - \\tilde { G } _ { j } ( \\hat { X } + \\frac 1 3 ( \\lambda + 1 ) ) + \\zeta _ \\lambda ( z ( a _ j ) ) . \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} \\phi ' ( x ) = x ^ { \\nu - 1 } \\cdot \\left [ \\nu \\cdot \\log _ 2 ( x ) \\cdot \\log _ 2 ( \\log _ 2 ( x ) ) - \\frac { \\log _ 2 ( \\log _ 2 ( x ) ) } { \\ln 2 } - ( \\ln 2 ) ^ { - 2 } \\right ] \\bigg / [ \\log _ 2 ( x ) \\cdot \\log _ 2 ( \\log _ 2 ( x ) ) ] ^ 2 . \\end{align*}"}
-{"id": "7472.png", "formula": "\\begin{align*} & \\delta _ { j _ 1 j _ 2 } G _ { i _ 1 i _ 3 \\alpha } ^ { j _ 1 j _ 2 \\eta } \\delta ^ { i _ 1 \\alpha } \\tilde \\gamma _ { \\eta k } + 2 \\delta ^ { i _ 1 i _ 2 } G _ { i _ 1 i _ 2 i _ 3 } ^ { j _ 1 j _ 2 j _ 3 } \\gamma _ { j _ 2 j _ 3 } \\delta _ { j _ 1 k } \\\\ = & - \\int _ 0 ^ \\infty \\frac { d } { d y } \\left [ \\sum _ { \\alpha , \\eta } ( e ^ { - y \\tilde \\gamma } ) _ { \\alpha k } ( e ^ { - y \\tilde \\gamma } ) _ { \\alpha \\eta } ( e ^ { - y \\tilde \\gamma } ) _ { i _ 3 \\eta } \\right ] d y = \\delta _ { i _ 3 k } , \\end{align*}"}
-{"id": "7311.png", "formula": "\\begin{align*} R _ { } ( \\mathbf { p } ) = \\sum _ { k = 1 } ^ { K } R _ k = \\sum _ { k = 1 } ^ { K } \\log _ 2 ( 1 + A _ d * \\frac { \\Omega { } p _ { k } } { ( \\Omega \\sigma _ N ^ 2 + \\sigma _ { } ^ 2 ) \\| \\mathbf { f } _ k \\| ^ 2 } ) . \\end{align*}"}
-{"id": "5452.png", "formula": "\\begin{align*} M ^ P : = \\{ w \\in W ^ P , \\exists k \\textrm { s . t . } w \\textrm { m a x i m a l i n } W ^ P ( k ) \\} \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} g _ y = ( | y | ^ { a } ( U _ 0 ) _ y ) _ y \\in L ^ { 2 } _ { l o c } ( | y | ^ { - a } d X d t ) \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} T _ { f ^ { d } } = T _ { ( g _ { d - 1 } f ^ { d - 2 } + \\cdots + g _ 1 ) f + g _ 0 } \\le T _ { g _ { d - 1 } f ^ { d - 2 } + \\cdots + g _ 1 } + T _ f + T _ { g _ 0 } + O ( 1 ) \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{align*} X ^ f ( t ) = M ( y ) , \\sigma ( y - ) \\le t < \\sigma ( y ) . \\end{align*}"}
-{"id": "5187.png", "formula": "\\begin{align*} X _ { \\epsilon _ 1 } ( s ) X _ { - \\epsilon _ { 1 } + \\epsilon _ 2 } ( s ) + X _ { - \\epsilon _ 1 } ( s ) X _ { \\epsilon _ { 1 } + \\epsilon _ 2 } ( s ) + X _ { \\epsilon _ 2 } ( s ) X _ 0 ( s ) = 0 \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} \\int _ W D - \\int _ O | \\nabla \\omega | ^ 2 & = \\int _ { O } \\left ( \\left | \\nabla \\Phi ( x ) \\ , \\nabla \\omega ( x ) \\ , ( \\nabla \\Phi ( x ) ) ^ { - 1 } \\right | ^ 2 \\det \\nabla \\Phi ( x ) - \\left | \\nabla \\Phi ( 0 ) \\ , \\nabla \\omega ( x ) \\ , ( \\nabla \\Phi ( 0 ) ) ^ { - 1 } \\right | ^ 2 \\right ) d x \\\\ & + \\int _ { O } \\left ( \\left | \\nabla \\Phi ( 0 ) \\ , \\nabla \\omega ( x ) \\ , ( \\nabla \\Phi ( 0 ) ) ^ { - 1 } \\right | ^ 2 - | \\nabla \\omega ( x ) | ^ 2 \\right ) d x . \\end{align*}"}
-{"id": "7185.png", "formula": "\\begin{align*} { \\mathbb P } ( \\ , \\bigcup \\nolimits _ { i = j } ^ n ( \\varphi _ i ( f _ 1 , \\ldots , f _ s ) \\in J _ k ( \\mathbf a ) ) \\ , | \\ , \\xi _ d = \\beta ) \\end{align*}"}
-{"id": "7459.png", "formula": "\\begin{align*} G _ { i _ 1 i _ 2 i _ 3 } ^ { j _ 1 j _ 2 j _ 3 } = \\delta ^ { j _ 1 k _ 1 } \\delta ^ { j _ 2 k _ 2 } \\delta ^ { j _ 3 k _ 3 } \\int _ 0 ^ \\infty ( e ^ { - y \\tilde \\gamma } ) _ { i _ 1 k _ 1 } ( e ^ { - y \\tilde \\gamma } ) _ { i _ 2 k _ 2 } ( e ^ { - y \\tilde \\gamma } ) _ { i _ 3 k _ 3 } d y . \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} \\xi = \\bigwedge _ { \\diamondsuit } \\gamma _ { \\diamondsuit } ^ { \\exp ( a ) } = \\exp ( - a ) ^ * \\cdot \\bigwedge _ { \\diamondsuit } \\gamma _ { \\diamondsuit } ^ { e } , \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{align*} r ( \\theta ) = \\sqrt { r N } \\ , \\prod _ { k = 1 } ^ { \\lfloor \\frac { r N } { 2 } \\rfloor } \\frac { \\sin \\left ( \\frac { \\theta } { r N } + \\frac { \\pi ( 2 k - 1 ) } { 2 r N } \\right ) } { \\sin \\left ( \\frac { \\theta } { r N } + \\frac { \\pi k } { r N } \\right ) } \\ , . \\end{align*}"}
-{"id": "6318.png", "formula": "\\begin{align*} \\mathcal { L } _ { I _ J } ( s ) & = \\mathbb { E } \\Big [ \\exp ( - s ( \\sum _ { \\mathbf { z } _ i \\in \\Phi _ J ^ s } P _ J h _ { \\mathbf { z } _ i , \\mathbf { u } _ 0 } D _ { \\mathbf { z } _ i , \\mathbf { u } _ 0 } ^ { - \\alpha } ) ) \\Big ] \\\\ & \\overset { ( a ) } { = } \\mathbb { E } _ { \\Phi _ J ^ s } \\Big [ \\prod _ { \\mathbf { z } _ i \\in \\Phi _ J ^ s } \\mathcal { L } _ h ( s P _ J D _ { \\mathbf { z } _ i , \\mathbf { u } _ 0 } ^ { - \\alpha } ) \\Big ] , \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} \\displaystyle { \\lim _ { \\substack { v ^ { \\ , \\prime } \\to v \\\\ \\tau \\downarrow 0 } } } \\ , \\displaystyle { \\frac { \\varphi ( \\bar { w } + \\tau \\ , v ^ { \\ , \\prime } ) - \\varphi ( \\bar { w } ) - \\tau \\ , \\varphi ^ { \\ , \\prime } ( \\bar { w } ; v ^ { \\ , \\prime } ) } { \\tau ^ 2 } } \\ , = \\ , \\Phi ^ { \\ , \\prime } ( \\bar { w } ; v ) ^ T F ^ { \\ , \\prime } ( \\Phi ( \\bar { w } ) ; \\Phi ^ { \\ , \\prime } ( \\bar { w } ; v ) ) + F ( \\Phi ( \\bar { w } ) ) ^ T \\Phi ^ { ( 2 ) } ( \\bar { w } ; v ) . \\end{align*}"}
-{"id": "5656.png", "formula": "\\begin{align*} c l ( n , 1 ) \\ , = \\ , 2 ^ { 2 ^ { n - 1 } - n } . \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{align*} Z _ E ( x , z ) = \\sum _ { E = 1 } ^ { \\infty } \\sum _ { N = 1 } ^ { \\infty } z ^ { N } \\ , x ^ { E } P ( E , N ) \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} h = ( N s ) ^ { - 2 } \\tilde { h } = - s ^ { - 2 } d t ^ 2 + g _ t \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} H _ n ( \\nu ) = \\mathbf { E } _ { \\{ 1 , \\dots , n \\} } \\left ( H _ m \\left ( \\nu ^ { x , i } \\right ) \\right ) + O \\left ( \\frac { m } { n } + \\frac { \\log R } { n } \\right ) \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{align*} | C ( g , h ) | \\leq c _ { 8 } , g , h = 1 , 2 , \\ldots , | \\textup { \\textbf { m } } | . \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{align*} \\beta ^ { - 1 } = \\| m ' \\| \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} \\lambda _ \\alpha ( s ) = \\alpha ( s ) \\ \\ \\lambda _ \\alpha ( \\varphi ) ( \\beta ) = \\alpha ( \\varphi ( \\beta ) ) - \\varphi ( [ \\alpha , \\beta ] ) \\ , . \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{align*} \\frac { \\varepsilon ( \\lambda , x ) } { \\Vert x \\Vert ^ 2 } = \\int _ { \\mathbb { R } ^ n } \\frac { \\alpha ( y , x ) } { \\Vert x \\Vert ^ 2 } \\eta _ { \\delta } ( \\lambda - y ) d y \\rightarrow 0 \\ , \\ , \\ , \\ , \\Vert x \\Vert \\rightarrow 0 , \\ , \\ , \\forall \\lambda \\in \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "1643.png", "formula": "\\begin{align*} q ( x , y ^ 0 | z ) & = \\prod _ { j = 1 } ^ N \\left [ \\prod _ { k = 1 } ^ { H _ 0 } \\left ( b ^ 0 _ { k j } \\prod _ { i = 1 } ^ M ( a ^ 0 _ { i k } ) ^ { x _ { i } } \\right ) ^ { y ^ 0 _ k } \\right ] ^ { z _ j } , \\\\ p ( x , y | z , A , B ) & = \\prod _ { j = 1 } ^ N \\left [ \\prod _ { k = 1 } ^ { H } \\left ( b _ { k j } \\prod _ { i = 1 } ^ M ( a _ { i k } ) ^ { x _ { i } } \\right ) ^ { y _ k } \\right ] ^ { z _ j } . \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} \\langle \\partial _ s ^ { k - 1 } u , D \\partial _ s ^ { k - 1 } ( \\delta u _ s ) \\rangle = - \\frac 1 2 \\langle \\partial _ s ^ k u , [ H , \\delta ] \\partial _ s ^ k u \\rangle , \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} f _ k = \\rho _ 2 ^ { i z } ( \\log \\rho _ 2 ) ^ k a _ k ( \\rho _ 1 ) , \\end{align*}"}
-{"id": "950.png", "formula": "\\begin{align*} V ^ { ( > ) } \\ = \\ V \\cdot \\mathbf { 1 } \\big [ V \\geq ( 1 - { \\varepsilon } ) \\eta ( \\mathfrak { e } ) \\big ] V ^ { ( < ) } \\ = \\ V \\cdot \\mathbf { 1 } \\big [ V < ( 1 - { \\varepsilon } ) \\eta ( \\mathfrak { e } ) \\big ] . \\end{align*}"}
-{"id": "2269.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { E } \\left \\{ \\Big [ \\frac { \\hat F _ { \\lambda ^ * } } { F _ \\varpi ( P \\parallel Q ) } - 1 \\Big ] ^ 2 \\right \\} = \\frac { { \\rm M S E } ( \\hat F _ { \\lambda ^ * } ) } { F _ \\varpi ^ 2 ( P \\parallel Q ) } = O ( \\varGamma ^ { - 1 } ) , \\end{aligned} \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} F \\{ \\{ S \\} \\} = \\left \\{ \\sum _ { i = - \\infty } ^ { \\infty } a _ i S ^ i : a _ i \\in F , \\inf _ i v _ F ( a _ i ) > - \\infty , a _ i \\to 0 \\textrm { a s } i \\to - \\infty \\right \\} . \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{align*} d x _ t ^ \\prime = b _ + ( t ^ * , x _ t ^ \\prime ) d t - ( - b _ - ) ( t ^ * , x _ t ^ \\prime ) + \\tilde \\sigma ( t ^ * , x _ t ^ \\prime ) \\circ d W _ t \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{align*} \\psi _ n = 2 \\pi i n \\tilde { z } / \\omega _ 1 - \\pi i n ( n + 1 ) \\omega _ 2 / \\omega _ 1 \\end{align*}"}
-{"id": "124.png", "formula": "\\begin{align*} g _ \\infty ^ { - 1 } ( \\bar \\partial g _ \\infty ) & = \\bar \\partial \\log \\bigl ( ( q \\bar q ) ^ { 1 / 8 } k ^ { 1 / 2 } \\bigr ) \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} T _ { k } : = \\prod _ { j \\in J _ { k } } T _ { j } ^ { k } \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{align*} R _ { r a w } : = \\frac { H ( X ) } { k } = H ( p _ x ) > 0 , \\end{align*}"}
-{"id": "6981.png", "formula": "\\begin{align*} t _ k - t _ { \\ell } = \\tau ( t _ i - t _ j ) \\Leftrightarrow t _ { \\phi _ T ( k ) } - t _ { \\phi _ T ( \\ell ) } = x _ { \\tau } ( t _ { \\phi _ T ( i ) } - t _ { \\phi _ T ( j ) } ) . \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} \\frac { \\partial { } \\varphi ( \\mathbf { p } , \\xi ) } { \\partial { } p _ k } & = \\frac { \\partial { } U ( \\mathbf { p } ) } { \\partial { } p _ k } + \\xi \\left ( \\frac { 1 } { p _ k } - \\frac { 1 } { P _ { } - p _ k } - \\frac { T _ k } { \\Gamma _ { } - T _ k { } p _ k } \\right ) \\\\ & + \\xi \\sum _ { m = 1 } ^ M \\frac { - \\Omega { } | g _ { m k } | ^ 2 } { P _ { } ^ { } - \\Omega \\left ( \\sum _ { k = 1 } ^ K | g _ { m k } | ^ 2 p _ k + \\sigma _ N ^ 2 \\right ) } , \\end{align*}"}
-{"id": "3701.png", "formula": "\\begin{align*} S ( \\vec { \\alpha } ) = \\sum _ { \\vec { y } \\in P \\mathcal { B } } e ( \\vec { \\alpha } \\cdot \\vec { f } ( \\vec { y } ) ) = I ( \\mathcal { B } , P ^ d \\vec { \\alpha } ) P ^ n + O \\left ( \\left ( \\widetilde { C } | P ^ d \\vec { \\alpha } | + 1 \\right ) P ^ { n - 1 } \\right ) . \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} u _ k = b _ k + i \\eta R _ k . \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} \\bar * \\epsilon ( X ^ \\flat ) \\bar * = X \\lrcorner , \\bar * ( X \\lrcorner ) \\bar * = - \\epsilon ( X ^ \\flat ) \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} \\mathcal { W } ^ { \\chi } ( f ) = \\int _ { U ^ { - } } f ( u w _ 0 ) \\psi ( u ) \\ , d u \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} & \\frac { | ( U ( F ) ) ( x , y ) - ( U ( F ) ) ( x ' , y ) | } { d _ 2 ( x , x ' ) } \\\\ & = \\frac { | h ( y ) F ( \\varphi ( x , y ) , \\tau ( y ) ) - h ( y ) F ( \\varphi ( x ' , y ) , \\tau ( y ) ) | } { d _ 2 ( x , x ' ) } \\\\ & = \\frac { | F ( \\varphi ( x , y ) , \\tau ( y ) ) - F ( \\varphi ( x ' , y ) , \\tau ( y ) ) | } { d _ 2 ( \\varphi ( x , y ) , \\varphi ( x ' , y ) ) } , x , x ' \\in X _ 2 , y \\in Y _ 2 \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} \\prod ^ { d - 1 } _ { i = 0 } g ( \\omega , \\omega ^ i z ) = \\frac { 1 + \\sqrt { 1 + 4 z ^ d } } { 2 } . \\end{align*}"}
-{"id": "8593.png", "formula": "\\begin{align*} s _ i ^ * s _ j = 0 ~ ( i \\neq j ) , { s _ i } ^ * s _ i = \\sum _ { j = 1 } ^ { N } A ( i , j ) s _ j { s _ j } ^ * \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} \\| \\alpha ( t ) \\| _ \\infty & = \\| \\int _ t ^ T A ' ( s ) d s \\| _ \\infty \\\\ & \\le \\int _ t ^ T \\| A ' ( s ) \\| _ \\infty d s \\\\ & \\le \\int _ t ^ T s ^ { - 3 / 2 } k _ T d s \\\\ & = t ^ { - 1 / 2 } ( 1 - ( t / T ) ^ { 1 / 2 } ) 2 k _ T \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} R ^ b { } _ { 0 0 1 } = o ( r ^ { - 4 } ) . \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} N _ 1 ( U _ r , \\rho ) = N _ 1 ( U , r \\rho ) \\ \\longrightarrow \\ N _ 1 ( U , 0 + ) = N ( U , 0 + ) = \\kappa \\ \\ \\ r \\to 0 . \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} L _ 1 \\subset L _ 2 \\subset \\dots \\subset L _ { m + k + 1 } = L , \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} \\displaystyle \\limsup _ { h \\to 0 + } \\dfrac { \\sup _ { t \\in T } \\sup _ { | s - t | _ \\infty \\leq h } | X ( t ) - X ( s ) | } { h ^ { ( H - d / \\alpha ) } ( \\log { 1 / h } ) ^ { 1 / \\gamma } } = 0 \\ ; \\ ; a . s . \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} | x | ^ p / 2 \\le | x | ^ p / p - 1 / p + 1 / 2 = \\varphi _ p ( x ) { \\rm f o r } \\ ; | x | \\ge 1 . \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} p ^ { F } _ c = { \\Pr } ( V \\in C _ f ) = { \\sum \\nolimits _ { i = 1 } ^ { C _ f } } { f _ i ( \\sigma , N ) } , \\end{align*}"}
-{"id": "4415.png", "formula": "\\begin{align*} \\abs { \\det ( Z ) } \\leq \\prod _ { r = 1 } ^ n \\Bigl ( \\sum _ { j = 1 } ^ n \\abs { z _ { j , r } } ^ 2 \\Bigr ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} p _ m & = \\frac { 1 } { 1 + R _ 1 + \\dots + R _ { m - 1 } } , \\\\ q _ m & = \\frac { 1 + R _ 1 + \\dots + R _ { m - 1 } } { 1 + R _ 1 + \\dots + R _ { m - 1 } + R _ m } , \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} Z _ 1 ^ 3 + Z _ 2 ^ 3 + 9 Z _ 3 ^ 3 = 0 \\\\ w Z _ 1 + 3 ^ b Z _ 2 + Z _ 3 = 0 \\end{align*}"}
-{"id": "7081.png", "formula": "\\begin{align*} & \\limsup _ { L \\to \\infty \\atop L \\in \\N } \\left | \\frac { \\int _ 0 ^ { \\infty } d x x e ^ { \\beta L ^ d f _ L ( x ) } v _ L ( x ) u _ { 5 , L } ( x ) } { \\int _ 0 ^ { \\infty } d x x e ^ { \\beta L ^ d f _ L ( x ) } v _ L ( x ) } \\right | \\le \\left | \\lim _ { L \\to \\infty \\atop L \\in \\N } u _ { 5 , L } ( 0 ) \\right | = 0 , \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} \\| X _ i ( p ) \\| = \\frac { | p _ i | } { \\sqrt { | p _ 0 | ^ 2 + \\cdots + | p _ n | ^ 2 } } \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} x ( y z \\cdot x ) = x ( y x \\cdot x ^ { \\rho } ) \\cdot z x \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} h ( M ) = | | M ( e _ 1 ) | | ^ { - 1 } \\cdot | | M | | \\end{align*}"}
-{"id": "1096.png", "formula": "\\begin{align*} ( s \\alpha ) \\cdot n = s \\cdot ( \\alpha \\cdot n ) - \\alpha ( s ) \\cdot n \\ , . \\end{align*}"}
-{"id": "3965.png", "formula": "\\begin{align*} d _ { } ( p _ { X Y } ) = 1 - \\frac { p } { 1 - p } . \\end{align*}"}
-{"id": "1809.png", "formula": "\\begin{align*} s _ 0 ( B ) : = \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { k ! } \\sum _ { \\substack { x _ 1 , \\ldots , x _ k \\in B , \\\\ d ( x _ 1 , \\ldots , x _ k ) < m _ { i - 1 } } } ( \\rho _ + ( x _ 1 , \\ldots , x _ k ) ^ 2 + \\rho _ - ( x _ 1 , \\ldots , x _ k ) ^ 2 ) , \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} \\begin{cases} \\Delta \\tilde u ( x , y ) = 0 , & ( x , y ) \\in R \\\\ \\tilde u ( x , y ) = u ( x , y ) , & ( x , y ) \\in \\partial R . \\end{cases} \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} ( \\nabla u ) | _ { \\gamma _ p ( s ) } = 2 \\dot \\gamma _ p ( s ) . \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{align*} \\begin{pmatrix} z _ 1 & a & b \\\\ \\widetilde { a } & z _ 2 & c \\\\ \\widetilde { b } & \\widetilde { c } & z _ 3 \\end{pmatrix} , \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} 1 + \\sqrt { 1 + c \\lambda _ i } + \\sqrt { 1 + c \\lambda _ j } - \\sqrt { 1 + c \\lambda _ k } = 0 . \\end{align*}"}
-{"id": "3631.png", "formula": "\\begin{align*} U _ w = \\prod _ { \\substack { \\alpha \\in \\Phi ^ { + } \\\\ w ( \\alpha ) \\in \\Phi ^ { - } } } U _ { \\alpha } , \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } D ( \\rho _ { n _ k } , \\rho _ { n _ k } ) = D ( \\rho _ { 0 } , \\rho _ { 0 } ) . \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{align*} \\mathcal { G } _ { 1 } ( \\widetilde \\ell , \\widetilde \\omega ) = - m ( \\widetilde \\omega \\times \\widetilde \\ell ) , \\mathcal { G } _ { 2 } ( \\widetilde \\ell , \\widetilde \\omega ) = J ( 0 ) \\widetilde \\omega \\times \\widetilde \\omega . \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} \\dfrac 1 2 \\mathcal { L } | A | ^ 2 = | \\nabla A | ^ 2 - \\dfrac 1 2 | A | ^ 2 - | A | ^ 4 . \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} ( \\alpha _ t , \\varphi _ t ) = ( \\beta _ { \\infty } , 0 ) - D _ t ^ 1 \\xi _ t = ( \\alpha _ \\infty , 0 ) - ( d _ { A _ \\infty } \\xi _ \\infty , 0 ) - ( d _ { A _ t } \\xi _ t , t [ \\Phi _ t , \\xi _ t ] ) . \\end{align*}"}
-{"id": "2357.png", "formula": "\\begin{align*} E \\left [ S _ N ^ { ( r ) } \\right ] \\sim N ^ r ( \\ln N ) ^ r \\sum _ { k = 0 } ^ { \\infty } \\binom { r } { k } \\frac { ( - 1 ) ^ k \\ , \\Gamma ^ { ( k ) } ( 1 ) } { \\ln ^ k N } , N \\to \\infty , \\end{align*}"}
-{"id": "6044.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\left ( ( \\eta _ 0 , w _ 0 ) , S ^ * ( \\tau _ n ) ( \\varphi _ { \\tau } , \\psi _ { \\tau } ) \\right ) _ { \\overline { X } _ 2 } & = \\left ( ( \\eta _ 0 , w _ 0 ) , S ^ * ( \\tau ) ( \\varphi _ { \\tau } , \\psi _ { \\tau } ) \\right ) _ { \\overline { X } _ 2 } . \\end{align*}"}
-{"id": "8764.png", "formula": "\\begin{align*} \\widetilde \\gamma = \\min \\left \\{ \\gamma _ { 0 } , \\frac { 1 } { 2 C _ { L } C _ { N } } , \\frac { 1 } { 2 C _ { L } C _ { l i p } } \\right \\} \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{align*} \\omega _ { i j } ^ \\alpha = [ \\Big ( { \\theta ( p _ { i j } + z | \\tau ) \\over \\theta ( z | \\tau ) \\theta ( p _ { i j } | \\tau ) } e ^ { - c _ { i j } z } - { 1 \\over z } \\Big ) d p _ { i j } | z ^ \\alpha ] \\end{align*}"}
-{"id": "4098.png", "formula": "\\begin{align*} \\varphi _ { n } ^ { F } ( \\underline { h } ) = \\# \\{ \\underline { g } \\in F ^ { n } | \\overline { \\langle \\underline { g } \\rangle } = F , \\underline { f } ( \\underline { g } ) = \\underline { h } \\} . \\end{align*}"}
-{"id": "3107.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\frac { \\log a ( n ) \\log \\log n } { \\log n } = \\frac { 1 } { 4 } \\log 5 , \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{align*} L _ m w ( \\varepsilon ) & = q ^ { \\kappa ( m , \\mu , \\lambda ) - ( \\varepsilon _ 1 + \\cdots + \\varepsilon _ { m - 1 } ) } w ( \\varepsilon - \\widehat { m } ) , \\\\ R _ m w ( \\varepsilon ) & = q ^ { \\varepsilon _ { m + 1 } + \\cdots + \\varepsilon _ N } w ( \\varepsilon + \\widehat { m } ) , \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} L = \\mathcal { L } + | A | ^ 2 - \\frac 1 2 = \\Delta + \\frac 1 2 \\left < x , \\nabla \\cdot \\right > + | A | ^ 2 - \\frac 1 2 . \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} \\alpha \\cdot ( c \\otimes ( u \\otimes v ) ) = \\alpha \\cdot c \\otimes ( u \\otimes v ) + c \\otimes ( ( \\alpha \\otimes 1 - 1 \\otimes \\alpha ) \\cdot ( u \\otimes v ) ) \\ , . \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} \\dot { x } \\dfrac { d f } { d x } + \\dot { y } \\dfrac { d f } { d y } + \\dot { z } \\dfrac { d f } { d z } = - 2 ( 2 z \\pm 1 ) f ( x , y , z ) \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} & \\nu \\bigl ( ( \\psi \\otimes \\operatorname { i d } ) ( \\Delta ( ( \\operatorname { i d } \\otimes \\omega ) [ ( 1 \\otimes \\tilde { y } c ) ( \\Delta a ) ] ) ) \\bigr ) \\\\ & = \\psi \\bigl ( ( \\operatorname { i d } \\otimes \\omega ) [ ( 1 \\otimes \\tilde { y } ) ( 1 \\otimes ( \\nu \\otimes \\operatorname { i d } ) ( ( 1 \\otimes c ) E ) ) ( \\Delta a ) ] \\bigr ) \\\\ & = \\psi \\bigl ( ( \\operatorname { i d } \\otimes \\omega ) [ ( 1 \\otimes \\tilde { y } c ) ( \\Delta a ) ] \\bigr ) . \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} { \\bf H } = - ( \\overline { \\nabla } f ) ^ \\perp . \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { l _ { i , k } } f _ { p _ { j , i , k } q _ { j , i , k } } v _ { \\omega _ k } = \\pm \\prod _ { j = 1 } ^ { l _ { i , k } } f _ { p ' _ { j , i , k } q ' _ { j , i , k } } v _ { \\omega _ k } \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{align*} \\norm { \\theta ^ n } - \\norm { \\theta ^ { n - 1 } } & \\le C k _ n \\delta ^ 2 \\norm { e ^ { n - 1 / 2 } } + \\tilde { C } k _ n \\left ( \\norm { e ^ n } + \\norm { e ^ { n - 1 } } \\right ) + C k _ n ( k _ n ^ 2 + h ^ r ) \\\\ & \\le C k _ n \\delta ^ 2 \\norm { \\theta ^ { n - 1 / 2 } } + \\tilde { C } k _ n \\left ( \\norm { \\theta ^ n } + \\norm { \\theta ^ { n - 1 } } \\right ) + C k _ n ( k _ n ^ 2 + h ^ r ) . \\end{align*}"}
-{"id": "7321.png", "formula": "\\begin{align*} \\mathcal A _ t = \\bigcup _ { A \\in \\mathcal A _ s } \\mathcal A ( t , A ) . \\end{align*}"}
-{"id": "5580.png", "formula": "\\begin{align*} \\theta _ 1 ^ + ( x ) = \\partial _ z \\vartheta _ 1 ( 0 , i x ) , x \\in \\R _ + . \\end{align*}"}
-{"id": "5628.png", "formula": "\\begin{align*} n C _ { n - 1 } ( x ; c ) & = c C _ n ( x ; c ) - c C _ { n } ( x + 1 ; c ) , \\\\ c C _ { n + 1 } ( x ; c ) & = c C _ n ( x ; c ) - x C _ { n } ( x - 1 ; c ) . \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} ( [ \\nabla _ t , D ] \\phi ) ( X ) = - \\frac { 1 } { 2 } d \\phi ( \\dot { g } ( X ) ) . \\end{align*}"}
-{"id": "1160.png", "formula": "\\begin{align*} ( u \\otimes e ) \\cdot \\alpha = u \\alpha \\otimes e + u \\otimes \\mathrm { T r } ( \\mathrm { a d } _ { \\alpha } ) e - u \\otimes \\mathcal L _ { \\partial _ \\alpha } ( e ) \\ , . \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} \\forall \\alpha > 0 , \\ \\lim _ { t \\to 0 ^ + } \\frac { f ( t ) } { t ^ \\alpha } = 0 . \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} \\begin{aligned} [ b ] \\tilde { K } \\int _ { 1 } ^ { \\infty } \\int _ { x _ 1 } ^ { \\infty } \\cdots \\int _ { x _ { t - 1 } } ^ { \\infty } \\prod _ { j = 1 } ^ { t } x _ j ^ { - \\tau + \\zeta _ j + { | Q _ j | } + ( \\tau - 1 ) \\abs { W _ j } - 2 E _ { W _ j } - E _ { W _ j , \\hat { W } _ j } } \\dd x _ { t } \\cdots \\dd x _ 1 . \\end{aligned} \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} u ( t ) = \\sin ( k ( t ) t ) \\mbox { w i t h } k ( t ) = k _ 0 t \\end{align*}"}
-{"id": "5654.png", "formula": "\\begin{align*} 2 0 4 8 \\ , = \\ , \\frac { 1 2 ^ { 1 2 } } { 6 ^ { 9 } \\cdot 4 3 2 } \\ , \\leq \\ , n r ' ( 3 , 1 ) \\ , \\leq \\ , \\frac { 1 2 ^ { 1 2 } } { 6 ^ 6 } \\ , = \\ , 1 9 1 , 1 0 2 , 9 7 6 , \\end{align*}"}
-{"id": "8606.png", "formula": "\\begin{align*} \\sum _ { w \\in V ( G ) } d ( w ) & = \\sum _ { w \\notin N [ v ] } d ( w ) + \\sum _ { w \\in N [ v ] } d ( w ) \\\\ & \\ge ( n - ( \\ell + 1 ) ) \\cdot ( 2 t - \\ell ) + ( \\ell + 1 ) \\cdot \\ell \\\\ & = n ( 2 t - \\ell ) - 2 ( \\ell + 1 ) ( t - \\ell ) \\\\ & = n t + ( t - \\ell ) [ n - 2 ( \\ell + 1 ) ] \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} N = N _ 1 N _ 2 N _ 3 = \\begin{bmatrix} A _ 1 & 0 \\\\ C _ 1 & B _ 1 \\\\ \\end{bmatrix} \\begin{bmatrix} 0 & A _ 2 \\\\ B _ 2 & C _ 2 \\\\ \\end{bmatrix} \\begin{bmatrix} B _ 3 & C _ 3 \\\\ 0 & A _ 3 \\\\ \\end{bmatrix} = \\begin{bmatrix} 0 & A _ 1 A _ 2 A _ 3 \\\\ B _ 1 B _ 2 B _ 3 & C _ 1 A _ 2 A _ 3 + B _ 1 C _ 2 A _ 3 + B _ 1 B _ 2 C _ 3 \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{align*} \\Upsilon _ n - \\Upsilon _ n ' \\cdot \\gamma ^ { - n } = \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} F _ G ( t ) = \\int _ 0 ^ t f _ G ( s ) d s = \\int _ 0 ^ { r _ G ( t ) } f _ W ( z ) d z = F _ W ( r _ G ( t ) ) . \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{align*} \\beta \\times \\prod \\ , | \\omega _ { j , n } | ^ { - 2 } ~ ~ \\geq ~ ~ \\theta _ { n } ^ { - 1 } \\times \\prod _ { j } \\ , ( | z _ { j , n } | + \\frac { t _ { j , n } } { n } ) ^ { - 2 } , \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} \\begin{cases} \\operatorname { d i v } ( y ^ a \\nabla w ) = y ^ a w _ t , \\\\ \\underset { y \\to 0 } { \\lim } \\ y ^ a w _ y = - f + g w . \\end{cases} \\end{align*}"}
-{"id": "2004.png", "formula": "\\begin{align*} \\prod _ { \\sigma \\in \\operatorname { G a l } ( \\mathbb { Q } ( \\zeta ) / \\mathbb { Q } ) } \\left ( \\sigma \\left ( \\sum _ { i = 0 } ^ { 2 ^ m - 1 } \\zeta ^ { l _ i } \\right ) \\right ) \\in 2 ^ { m | \\operatorname { G a l } ( \\mathbb { Q } ( \\zeta ) / \\mathbb { Q } ) | } \\mathcal { O } \\cap \\mathbb { Z } = 2 ^ { m | \\operatorname { G a l } ( \\mathbb { Q } ( \\zeta ) / \\mathbb { Q } ) | } \\mathbb { Z } . \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{align*} S _ + = E \\oplus E K _ X ^ { - 1 } , \\end{align*}"}
-{"id": "6523.png", "formula": "\\begin{align*} 0 = \\lambda _ { n , 0 } < \\lambda _ { n , 1 } \\le \\lambda _ { n , 2 } \\le \\cdots \\le \\lambda _ { n , n - 1 } . \\end{align*}"}
-{"id": "6553.png", "formula": "\\begin{align*} \\int \\limits _ Y \\left ( \\int \\limits _ 0 ^ T f ( g _ t x ) \\ , d t \\right ) ^ 2 \\ , d \\mu ( x ) = \\int \\limits _ 0 ^ T \\int \\limits _ 0 ^ T \\left ( \\int \\limits _ Y f ( g _ { | t - s | } x ) f ( x ) \\ , d \\mu ( x ) \\right ) \\ , d t \\ , d s . \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} \\partial \\mathcal { F } / \\partial z _ j = \\tfrac 1 2 ( w _ j + \\sum _ i z _ i \\partial w _ i / \\partial z _ j ) = \\tfrac 1 2 ( w _ j + \\sum _ i z _ i \\partial w _ j / \\partial z _ i ) = w _ j . \\end{align*}"}
-{"id": "5634.png", "formula": "\\begin{align*} L ^ { \\mathrm { B E P } } f ( x ) = \\sum _ { 1 \\leq i < j \\leq N } x _ i x _ j \\left ( \\frac { \\partial } { \\partial x _ i } - \\frac { \\partial } { \\partial x _ j } \\right ) ^ 2 f ( x ) - 2 ( k _ i x _ i - k _ j x _ j ) \\left ( \\frac { \\partial } { \\partial x _ i } - \\frac { \\partial } { \\partial x _ j } \\right ) f ( x ) , \\end{align*}"}
-{"id": "5058.png", "formula": "\\begin{align*} & \\bigl [ s [ y _ i , x _ j ] , y _ { i + 1 } \\dots y _ { i ' - 1 } \\bigr ] = \\sum _ { m = i + 1 } ^ { i ' - 1 } y _ { i + 1 } \\dots y _ { m - 1 } \\bigl [ s [ y _ i , x _ j ] , y _ m \\bigr ] y _ { m + 1 } \\dots y _ { i ' - 1 } \\\\ = \\ & \\sum _ { m = i + 1 } ^ { i ' - 1 } y _ { i + 1 } \\dots y _ { m - 1 } \\bigl ( [ s , y _ m ] [ y _ i , x _ j ] + s [ y _ i , x _ j , y _ m ] \\bigr ) y _ { m + 1 } \\dots y _ { i ' - 1 } \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} \\frac { \\sigma _ \\lambda ( ( \\frac 1 2 + r ) \\omega ) } { \\sigma _ \\lambda ( ( \\frac 1 2 - r ) \\omega ) } = \\exp ( r \\eta \\omega ) . \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} C _ E = - \\frac { 1 } { 2 } e ^ { 2 \\alpha ( a ) } ( \\bar { z } _ 1 z _ 2 + z _ 1 \\bar { z } _ 2 ) \\mathcal { H } ( \\alpha ^ { \\vee } ) + O . \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} H ^ n + H ^ { n - 1 } ( K + H ) + \\cdots + ( K + H ) ^ n & = 0 \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{align*} q ^ { \\sum _ { J < I } \\rho _ { I ' } ( \\rho _ J - \\rho _ { J ' } ) } = q ^ { \\sum _ { J < I } \\bar \\rho _ { I ' } \\bar \\rho _ J } . \\end{align*}"}
-{"id": "3312.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { ( 2 \\pi ) ^ { n / 2 } } \\int _ { C _ W \\cap \\{ t \\leq x \\} } \\frac { d t } { \\sqrt { ( x _ 1 - t _ 1 ) \\ldots ( x _ n - t _ n ) } \\sqrt { \\det M _ t } } \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{align*} P \\{ T _ 1 < T _ 2 \\} = M _ 2 \\int _ 0 ^ 1 \\left ( 1 - x ^ { 1 / \\lambda } \\right ) ^ { M _ 1 } \\left ( 1 - x \\right ) ^ { M _ 2 - 1 } d x \\end{align*}"}
-{"id": "7061.png", "formula": "\\begin{align*} & V ^ { 0 - 1 - j , l } ( u ) ( \\psi ) : = \\sum _ { n = 1 } ^ { \\infty } V ^ { 0 - 1 - j , l , ( n ) } ( u ) ( \\psi ) , ( j = 1 , 2 ) , V ^ { 0 - 1 , l } : = \\sum _ { j = 1 } ^ 2 V ^ { 0 - 1 - j , l } , \\\\ & V ^ { 0 - 2 , l } ( u ) ( \\psi ) : = \\sum _ { n = 1 } ^ { \\infty } V ^ { 0 - 2 , l , ( n ) } ( u ) ( \\psi ) \\end{align*}"}
-{"id": "847.png", "formula": "\\begin{align*} t _ { n } ^ { p } [ v _ { n } ] ^ { p } _ { W ^ { s , p } ( \\R ^ { N } ) } + t _ { n } ^ { p } V _ { \\infty } \\int _ { \\R ^ { N } } | v _ { n } | ^ { p } d x = \\int _ { \\R ^ { N } } f ( t _ { n } v _ { n } ) t _ { n } v _ { n } d x . \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} \\partial _ { x } ^ \\beta \\widetilde { q } ( x , \\xi ) = \\partial _ { x } ^ \\delta \\partial _ { x _ j } \\widetilde { q } ( x , \\xi ) = \\partial _ { x } ^ \\delta \\left ( \\left . \\partial _ { z _ j } q _ { x } ( x , \\xi ) \\right | _ { z = x } \\right ) + \\partial _ { x } ^ \\delta \\left ( \\left . \\partial _ { x _ j } { q _ { z } } ( x , \\xi ) \\right | _ { z = x } \\right ) . \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} Q = ( ( \\psi \\times i d ) _ { * } ( \\sigma _ { n } \\times i d ) ^ { * } ( l ^ { n } \\times i d ) _ { * } F ^ { \\boxtimes n } ) ^ { s i g n } \\otimes p _ { 1 } ^ { * } d e t ( \\mathcal { A } ) ^ { - 1 } \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{align*} s _ j ( T ) = o ( j ^ { - \\left ( \\frac { 1 } { q ' } + \\frac { \\alpha + \\beta } { n } \\right ) } ) . \\end{align*}"}
-{"id": "706.png", "formula": "\\begin{align*} R = \\left [ \\begin{array} { c | c c c } a _ 1 & b _ 1 & & x _ 1 \\\\ \\hline & & \\\\ & & \\widetilde R & \\\\ & & & \\\\ \\end{array} \\right ] \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{align*} f ( z ) = \\sum ^ { \\infty } _ { d = 0 } h _ d ( z ) \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{align*} F _ j ( t ) : = P \\left \\{ \\tilde { T } _ j \\leq t \\right \\} = \\left ( 1 - e ^ { - p _ j t } \\right ) ^ { M _ j } , t \\geq 0 , \\ \\ ; j = 1 , \\dots , g , \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} \\mathrm { P } _ { m , k } ^ 1 = & 1 - \\int _ { \\delta \\mathcal { R } _ c } ^ { \\infty } \\int ^ { z + \\mathcal { R } _ c } _ { z - \\mathcal { R } _ c } e ^ { - \\frac { { \\epsilon } _ 0 r ^ { \\alpha } } { \\rho \\zeta _ 0 } } g ( z , r ) d r \\bar { f } _ { r _ m } ( z ) d z , \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} \\hat { W } = \\{ \\hat { w } \\in W ^ P , \\exists \\xi \\in \\hat { \\cal X } ^ * \\textrm { d o m i n a n t r e g u l a r s . t . } \\hat { w } ^ { - 1 } \\xi \\in \\hat { \\cal X } ^ * \\} \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{align*} E \\left [ \\phi ( X ) \\right ] = E \\left [ \\int _ 0 ^ { \\infty } \\phi ( \\xi ) \\ , \\frac { \\xi ^ { S _ N - 1 } } { ( S _ N - 1 ) ! } \\ , e ^ { - \\xi } d \\xi \\right ] . \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{align*} 0 \\leq \\lim _ { k } \\Vert x ^ { k } - x ^ { n s } \\Vert = \\lim _ { n } \\Vert x ^ { k } - x ^ { n s } \\Vert \\leq \\lim _ { n } \\sum _ { l = n s } ^ { n s + p - 1 } \\Vert x ^ { l + 1 } - x ^ { l } \\Vert = 0 \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} f ^ * ( 1 \\times \\cdots \\times 1 \\times x _ { M _ j } ^ { 1 } \\times 1 \\times \\cdots \\times 1 ) = \\sum _ { l = 1 } ^ s ( 1 \\times \\cdots \\times 1 \\times y _ { M _ l } ^ { 1 } \\times 1 \\times \\cdots \\times 1 ) + \\xi \\cdot ( 1 \\times \\omega _ { S ^ 1 } ) , \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 1 0 } \\alpha _ { i } = 0 . \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} \\psi _ 1 ( x ) = x ^ { 3 / 4 } \\theta _ 1 ^ + ( x ) . \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} A ^ V = - \\dfrac { \\partial ^ 2 } { \\partial x ^ 2 } + V . \\end{align*}"}
-{"id": "3176.png", "formula": "\\begin{align*} \\omega _ i \\wedge \\omega _ j = \\delta _ { i j } \\ , \\omega _ 1 ^ 2 \\end{align*}"}
-{"id": "4308.png", "formula": "\\begin{align*} \\wp _ { \\Omega } ( z ) = \\frac { 1 } { z ^ 2 } + \\sum \\left ( \\frac { 1 } { ( z - \\omega ) ^ 2 } - \\frac { 1 } { \\omega ^ 2 } \\right ) \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} A _ { C _ { 4 } } = \\begin{cases} \\# N _ { p } ^ { - } ( a , c ) ( p ^ { e - 2 d - 1 } + p ^ { m - d - 1 } ) , & \\qquad \\ \\biggl ( \\frac { c ^ { 2 } - 4 a i } { p } \\biggr ) = - 1 , \\\\ \\# N _ { p } ^ { + } ( a , c ) ( p ^ { e - 2 d - 1 } + p ^ { m - d - 1 } ) , & \\qquad \\ \\biggl ( \\frac { c ^ { 2 } - 4 a i } { p } \\biggr ) = 1 . \\end{cases} \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} \\partial _ T \\phi = \\partial _ B \\phi - \\epsilon ( \\kappa _ B ^ { 1 , 0 } ) \\phi , \\bar \\partial _ T \\phi = \\bar \\partial _ B \\phi - \\epsilon ( \\kappa _ B ^ { 0 , 1 } ) \\phi , \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} \\sum _ { \\substack { i , j = 1 \\\\ i < j } } ^ K \\mathrm { E } \\Big [ \\big \\vert I _ { ( i , j ) } ( h ) - \\hat { I } ^ { ( D ) } _ { ( i , j ) } ( h ) \\big \\vert ^ 2 \\Big ] \\leq \\frac { 5 h ^ 2 } { 2 4 \\pi ^ 2 D ^ 2 } K ^ 2 ( K - 1 ) , \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{align*} g ( z ) = f ( z ) \\overline { f ( \\overline { z } ) } . \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} ( \\nu \\otimes \\operatorname { i d } ) \\bigl ( ( \\sigma ^ { \\nu } _ t \\otimes \\sigma ^ { \\mu } _ { - t } ) ( E ) ( b \\otimes 1 ) \\bigr ) & = \\sigma ^ { \\mu } _ { - t } \\bigl ( \\gamma _ B ( \\sigma ^ { \\nu } _ { - t } ( b ) ) \\bigr ) \\\\ & = ( R \\circ \\sigma ^ { \\nu } _ { i / 2 } ) ( b ) = ( \\nu \\otimes \\operatorname { i d } ) \\bigl ( E ( b \\otimes 1 ) \\bigr ) . \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} \\left \\{ f , g \\right \\} _ { G ^ * } ( L _ + , L _ - ) = \\left \\langle ( A d L _ { + } - A d L _ { - } ) \\nabla ^ { \\prime } f , \\nabla g \\right \\rangle - \\left \\langle \\nabla f , ( A d L _ { + } - A d L _ { - } ) \\nabla ^ { \\prime } g \\right \\rangle . \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} ( [ \\nabla _ t , D ] \\xi ) ( X , X _ 1 , \\ldots , X _ k ) & = R ^ V _ { \\partial _ t , X } ( \\xi ( X _ 1 , \\ldots , X _ k ) ) - \\frac { 1 } { 2 } D _ { \\dot { g } ( X ) } \\xi ( X _ 1 , \\ldots , X _ k ) \\\\ & \\qquad + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ k \\xi ( X _ 1 , \\ldots , D _ X \\dot { g } ( X _ i ) , \\ldots , X _ k ) . \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} \\hat { w } x _ 0 = [ \\sum \\limits _ { \\chi \\geq 0 } \\alpha _ { \\chi } v _ { w \\lambda + \\chi } ] \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} J _ { \\infty } ( X ; Y | Z ) \\triangleq \\max _ { z : p _ Z ( z ) > 0 } J _ \\infty ( X ; Y | Z = z ) . \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} \\mathcal { L } \\left \\vert X \\right \\vert ^ { 2 } = \\bigtriangleup _ { g } \\left \\vert X \\right \\vert ^ { 2 } - \\frac { 1 } { 2 } \\left \\langle X , \\nabla _ { g } \\left \\vert X \\right \\vert ^ { 2 } \\right \\rangle = 2 n - \\left \\vert X \\right \\vert ^ { 2 } . \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} \\L ^ \\bot = \\textnormal { a r e a } ( \\L ) ^ { - 1 } \\ , i \\ , \\L . \\end{align*}"}
-{"id": "2806.png", "formula": "\\begin{align*} D ^ 2 & = ( \\tilde { C } + \\sum _ i E _ i ) ^ 2 = \\tilde { C } ^ 2 - \\sum _ p ( \\omega _ p - 2 r _ p ) \\\\ p _ { a } ( D ) & = p _ { a } ( \\tilde { C } ) + \\sum _ p p _ a ( E _ p ) - s + \\tilde { C } . E = p _ { a } ( \\tilde { C } ) + \\sum _ p ( r _ p - 1 ) \\\\ e ( \\tilde { C } ) & = e ( C ) + \\sum _ p ( r _ p - 1 ) \\\\ ( K _ S + D ) . D & = 2 p _ { a } ( D ) - 2 = 2 p _ { a } ( \\tilde { C } ) - 2 + 2 \\sum _ p ( r _ p - 1 ) = - e ( C ) + \\sum _ p ( r _ p - 1 ) \\\\ K _ S ^ 2 - \\tilde { C } ^ 2 & = K _ X ^ 2 - C ^ 2 + \\sum _ i ( m _ i ^ 2 - 1 ) . \\end{align*}"}
-{"id": "2014.png", "formula": "\\begin{align*} I \\left ( \\left ( \\sum _ { i = 0 } ^ { 2 ^ n - 1 } \\theta _ i \\right ) \\otimes \\chi _ m \\right ) = \\left ( \\sum _ { i = 0 } ^ { 2 ^ n - 1 } \\theta _ i \\right ) \\otimes \\chi _ m , \\end{align*}"}
-{"id": "5119.png", "formula": "\\begin{align*} \\begin{pmatrix} u ^ { n + t - 2 } & u ^ { n + t - 3 } v & u ^ { n + t - 4 } v ^ 2 & \\cdots & u ^ { t - 1 } v ^ { n - 1 } \\\\ u ^ { n + t - 3 } v & u ^ { n + t - 4 } v ^ 2 & \\cdots & \\cdots & u ^ { t - 2 } v ^ n \\\\ u ^ { n + t - 4 } v ^ 2 & \\cdots & \\cdots & \\cdots & u ^ { t - 3 } v ^ { n + 1 } \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots \\\\ u ^ { n - 1 } v ^ { t - 1 } & \\cdots & \\cdots & \\cdots & v ^ { n - t + 2 } \\end{pmatrix} , \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{align*} & B _ b : = \\bigcap _ { m \\in \\N } \\bigcup _ { n \\geq m } A _ { b , n } \\quad \\\\ & A _ { b , n } : = \\bigg \\{ \\bigg \\| \\int _ 0 ^ { \\cdot } ( S ^ n - S _ { - } ) \\otimes \\dd S \\bigg \\| _ \\infty \\geq a _ n \\bigg \\} \\cap \\{ | [ S ] _ T | \\leq b \\} \\cap \\{ \\| S \\| _ \\infty \\leq b \\} . \\end{align*}"}
-{"id": "6018.png", "formula": "\\begin{align*} a + b + c + d = \\frac { 1 } { 3 } . \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} f _ { \\delta } = \\prod _ { i = 1 } ^ { r - 1 } ( 1 - q ^ i ) \\times \\sum _ { \\mu [ 1 ] \\in \\sigma ( \\delta ^ { * } ) } \\cdots \\sum _ { \\mu [ r - 1 ] \\in \\sigma ( \\delta ^ { * } ) } z ^ { \\delta ^ { * } } \\left ( \\prod _ { j = 1 } ^ { r - 1 } C _ { j } \\Big ( \\mu [ j + 1 ] , \\mu [ j ] ; q , t \\Big ) z ^ { \\mu [ j ] } \\right ) \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{align*} - \\Delta v _ { n } = a ( x ) \\frac { v _ { n } ^ { q _ { n } } } { \\Vert u _ { n } \\Vert ^ { 1 - q _ { n } } } . \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{align*} \\min _ { x , y : p _ { X Y } ( x , y ) > 0 } \\delta _ { x , y , t = ( x ' , y ' ) } = 0 \\forall x ' , y ' . \\end{align*}"}
-{"id": "5994.png", "formula": "\\begin{align*} { C } _ { \\alpha } { D } ^ { \\alpha } _ { \\theta } \\psi ( x , t ) + V ( x , t ) \\psi ( x , t ) = \\left ( i \\hslash \\right ) ^ { \\beta } { { } _ 0 ^ C D } _ t ^ { \\beta { } } \\psi ( x , t ) , \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l r c l } \\nu ( x ) & = & x \\ , , & \\nu ( d x ) & = & d x + 2 \\frac { \\partial P } { \\partial y } \\\\ \\\\ \\nu ( y ) & = & y \\ , , & \\nu ( d y ) & = & d y - 2 \\frac { \\partial P } { \\partial x } \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} \\vartheta _ 2 ( y ) = \\sum _ { n = - \\infty } ^ { \\infty } e ^ { - \\pi y ( n + 1 / 2 ) ^ 2 } . \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} e _ \\chi = e _ { \\chi _ 1 } + \\dots + e _ { \\chi _ p } e _ \\psi = e _ { \\psi _ 1 } + \\dots + e _ { \\psi _ p } , \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} \\mathcal R = \\{ \\rho \\colon \\N \\to \\N \\mid \\rho ( 0 ) = 0 ; \\ , \\rho ( n + 1 ) \\in \\{ \\rho ( n ) , \\rho ( n ) + 1 \\} \\ , \\forall n ; \\ , \\rho \\} . \\end{align*}"}
-{"id": "8173.png", "formula": "\\begin{align*} { \\partial _ { x _ i } } \\tilde V _ j = { \\partial _ { x _ j } } \\tilde V _ i \\ \\ \\ i n \\ \\ \\Omega , \\ \\ { f o r \\ e v e r y } \\ i , j = 1 , . . . , m , \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{align*} \\frac { \\left ( \\widehat { P } _ \\hbar + \\widehat { \\chi } _ { R } - z \\right ) ^ { - 1 } - \\left ( \\widehat { P } _ { \\hbar ' } + \\widehat { \\chi } _ { R } - z \\right ) ^ { - 1 } } { \\hbar - \\hbar ' } = - \\left ( \\widehat { P } _ \\hbar + \\widehat { \\chi } _ { R } - z \\right ) ^ { - 1 } \\widehat { Q } _ 2 \\left ( \\widehat { P } _ { \\hbar ' } + \\widehat { \\chi } _ { R } - z \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} \\varphi \\left ( z , t \\right ) = - \\underset { p \\in \\mathbb { R } ^ { n } } { } \\left \\{ J _ { z } ^ { \\star } \\left ( p , t \\right ) + \\int _ { 0 } ^ { t } H \\left ( p , s \\right ) d s - \\left \\langle z , p \\right \\rangle \\right \\} , \\end{align*}"}
-{"id": "1040.png", "formula": "\\begin{align*} j _ { m } ( \\rho _ { \\alpha } ) = j _ { m } ( \\rho _ { 0 } ) + j _ { m } ( \\tilde { \\rho } _ { \\infty } ) . \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} q ^ m ( q ^ n t ^ 2 - t ) \\prod _ { j = 0 } ^ { m - 1 } ( 1 - q ^ j t ) . \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{align*} d _ \\infty ^ W ( \\mu , \\nu ) = \\inf _ { \\pi \\in \\Pi ( \\mu , \\nu ) } \\sup \\{ d ( x , y ) : ( x , y ) \\in \\mathrm { s u p p } ( \\pi ) \\} , \\end{align*}"}
-{"id": "2555.png", "formula": "\\begin{align*} \\sum _ { i , j } \\gamma _ j s _ i p _ i s _ j = \\sum _ { i , j } \\gamma _ i s _ i q _ j s _ j . \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} \\nabla \\frac { 1 } { \\lvert \\Omega \\rvert } \\int _ { \\Omega } \\widetilde { b } _ { n } \\ ; \\d x = 0 . \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{align*} B _ 0 ^ { } ( X ; Y \\| Z ) = I ( X ; Y | J ) \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} \\Xi ( J \\cdot f ) = ( J \\cdot f ) \\cdot e ( \\chi ( J ) ) \\end{align*}"}
-{"id": "680.png", "formula": "\\begin{align*} \\mathcal I = \\mathcal I _ 1 \\cup \\ldots \\cup \\mathcal I _ { \\ell _ 1 } , \\mathcal J = \\mathcal J _ 1 \\cup \\ldots \\cup \\mathcal J _ { \\ell _ 2 } . \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} 0 \\to \\begin{pmatrix} M \\\\ 0 \\end{pmatrix} _ 0 \\to \\begin{pmatrix} M \\\\ N \\end{pmatrix} _ g \\to \\begin{pmatrix} 0 \\\\ N \\end{pmatrix} _ 0 \\to 0 \\end{align*}"}
-{"id": "9044.png", "formula": "\\begin{align*} B = \\frac { 1 } { \\| m ' \\| } \\begin{pmatrix} m _ 1 ' & m _ 2 ' \\\\ - m _ 2 ' & m _ 1 ' \\end{pmatrix} . \\end{align*}"}
-{"id": "4264.png", "formula": "\\begin{align*} \\{ \\hat { z } _ 1 , \\dots , \\hat { z } _ n \\} = \\{ z ^ { \\alpha } _ { x , y } : ( x , y ) \\in Y _ \\alpha , \\alpha = 1 , \\dots , r \\} , \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} \\mathcal E _ { m } : = \\big \\{ y \\in G ( \\eta ( m ) , J ) , \\ , \\forall y \\in B _ { m + 1 } \\big \\} \\mbox { w i t h } J : = \\lfloor \\phi _ T ^ { 1 / 2 } \\rfloor . \\end{align*}"}
-{"id": "2844.png", "formula": "\\begin{align*} L ( 0 ) = \\alpha _ 1 X _ 1 ( 0 ) + \\alpha _ 2 X _ 2 ( 0 ) + \\alpha _ 3 X _ 3 ( 0 ) \\end{align*}"}
-{"id": "7438.png", "formula": "\\begin{align*} d x _ t = ( - \\Gamma ( t , x _ t ) \\nabla H _ t ( x _ t ) + \\Pi ( t , x _ t ) \\nabla H _ t ( x _ t ) + G _ t ( x _ t ) ) d t + \\tilde \\sigma ( t , x _ t ) \\circ d W _ t , \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} \\Delta \\left ( ( \\mathbf { 1 } ) ^ n \\right ) & = \\left ( \\Delta ( \\mathbf { 1 } ) \\right ) ^ n \\\\ & = \\left ( 1 \\otimes ( \\mathbf { 1 } ) + ( \\mathbf { 1 } ) \\otimes 1 \\right ) ^ n \\\\ & = \\sum _ { k = 0 } ^ n C ^ n _ k ( \\mathbf { 1 } ) ^ k \\otimes ( \\mathbf { 1 } ) ^ { n - k } \\end{align*}"}
-{"id": "4410.png", "formula": "\\begin{align*} Q ( \\xi , \\lambda ) = 0 \\end{align*}"}
-{"id": "4632.png", "formula": "\\begin{align*} \\overline \\square _ B \\phi = \\nabla _ T ^ * \\nabla _ T \\phi . \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} p _ { n + 1 } & = p _ n + R ( p _ n , \\kappa _ n ) \\\\ t _ { n + 1 } & = t _ n + \\dfrac { \\ell _ * } { \\| p _ { n + 1 } \\| } \\\\ q _ { n + 1 } & = q _ n + \\dfrac { \\ell _ * } { \\| p _ { n + 1 } \\| } p _ { n + 1 } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} w ( \\vec { J } ) s ( \\vec { J } ) = - w ( \\vec { J } ' ) s ( \\vec { J } ' ) . \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} r _ { X Y } ( x , y ) = a ( x ) b ( y ) p _ { X Y } ( x , y ) \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} [ c , z _ 1 ] [ z _ 2 , z _ 3 , z _ 4 ] = - [ c , z _ 2 ] [ z _ 1 , z _ 3 , z _ 4 ] . \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} \\int _ { e } | x | d m ^ N _ t ( x ) & \\leq \\int _ { e } | x | d m ^ j _ 0 ( x ) + V _ { m a x } \\Delta t ^ N \\sum _ { k = 0 } ^ { n - 1 } m ^ N _ { t _ { k } } ( e _ j ) \\\\ & \\leq \\int _ { e } | x | d m ^ j _ 0 ( x ) + V _ { m a x } \\frac { T } { 2 ^ N } \\sum _ { k = 0 } ^ { 2 ^ N - 1 } m ^ { N } _ { t _ k } ( e _ j ) . \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } X ( \\ln ( \\lambda ^ { 2 } ) ) = - h ^ { - 1 } X ( h ) + h ^ { - 1 } X ( h ) = 0 , \\end{align*}"}
-{"id": "6720.png", "formula": "\\begin{align*} H \\left ( p \\right ) = \\underset { c \\in \\mathbb { R } ^ { m } } { } \\left \\{ \\left \\langle - f \\left ( c \\right ) , p \\right \\rangle - L \\left ( c \\right ) \\right \\} . \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{align*} K ( p ) = - \\frac { r ( r - 1 ) } { \\delta ( p ) ^ 2 } + O \\left ( \\frac { 1 } { \\delta ( p ) } \\right ) . \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} C ( \\alpha , s ) = \\frac { \\pi ^ { n / 2 } \\Gamma ( \\alpha ) \\Gamma ( - s ) } { \\Gamma ( \\frac { \\alpha - s + 1 } { 2 } ) \\Gamma ( \\frac { n + \\alpha - s - 1 } { 2 } ) } , \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} c _ k ( f ) = \\frac { 1 } { 2 | k | } ( 1 + | q _ 1 q _ 2 | ^ k - e ^ { - i k \\tau _ 1 } | q _ 1 | ^ { | k | } - e ^ { - i k \\tau _ 2 } | q _ 2 | ^ { | k | } ) . \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{align*} \\frac { u ^ { n } - u ^ { n - 1 } } { k _ n } = & i \\Delta \\Big ( \\frac { u ^ { n } + u ^ { n - 1 } } { 2 } \\Big ) + i f ( \\abs { u ^ { n - \\frac 1 2 } } ^ 2 ) \\frac { u ^ { n } + u ^ { n - 1 } } { 2 } \\\\ & + g ( t _ { n - \\frac 1 2 } , \\cdot ) + r ^ n , \\end{align*}"}
-{"id": "8028.png", "formula": "\\begin{align*} \\mathcal I ( M ) = \\{ J \\ : \\ T X \\longrightarrow T X \\mid J ^ 2 \\equiv - I d , \\ J , \\ \\ [ T ^ { 1 , 0 } , T ^ { 1 , 0 } ] \\subset T ^ { 1 , 0 } \\} \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} S ( a _ i ) = 1 \\ , , R = \\frac { r ( \\theta _ 1 ) \\ , r ( \\theta _ 3 ) } { r ( \\theta _ 1 + \\theta _ 3 ) } \\ , , \\end{align*}"}
-{"id": "6597.png", "formula": "\\begin{align*} \\psi _ { U , u } = U \\cdot \\bar { \\psi } + u \\bar { \\psi } , \\end{align*}"}
-{"id": "4222.png", "formula": "\\begin{align*} \\gamma = \\frac { { \\kappa _ X } ^ 2 \\lambda _ Y } { \\Lambda _ x { \\kappa _ Y } ^ 2 } \\geq \\varepsilon ^ { - 1 } \\frac { { \\kappa _ \\theta } ^ 2 ( \\bar { \\beta } \\bar { \\gamma } ) ^ { - 1 } } { 1 6 \\Lambda _ \\theta ( \\beta ^ { - 1 } ) } . \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{align*} b _ k ' = b _ k + \\frac { 2 \\pi i } { c _ k } \\quad \\varepsilon _ k = \\frac { 2 ^ { 1 5 } } { R _ k } . \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} \\phi _ A ( y ) = \\sum _ { i = 1 } ^ N s _ i y { s _ i } ^ * , \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} d Y ^ { n } = F \\left ( Y ^ { n } \\right ) d \\mathbf { X } ^ { n } , \\ , Y ^ { n } = y _ { 0 } \\ . \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} a _ J : = a _ { j _ 1 j _ 2 \\dots j _ n } , \\ \\xi ^ J : = \\xi _ 1 ^ { j _ 1 } \\xi _ 2 ^ { j _ 2 } \\cdots \\xi _ n ^ { j _ n } . \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} \\tilde { h } ' ( r ) = \\frac { 4 } { r } \\tilde { i } ( r ) + \\frac { a } { r } \\tilde { h } ( r ) , \\end{align*}"}
-{"id": "9127.png", "formula": "\\begin{align*} - \\frac { { r } ^ 2 } { c ^ 2 } \\left ( { 1 { \\rm { + } } \\frac { { { \\rho _ t } } } { c } } \\right ) - 3 \\frac { r } { c } + \\frac { { { \\rho _ t } } } { c } = 0 . \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{align*} g ^ \\prime = J g = \\alpha ^ { - 1 } J \\alpha g = \\bar J \\bar g . \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} \\frac { 1 } { k } \\sum _ { j = 1 } ^ { m } f _ k ( j ) = \\frac { m } { k } \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } f _ k ( j ) \\longrightarrow \\infty \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{align*} J ( u , t ) = 1 + O ( t ^ 2 ) \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} ( g _ { t ( e ) } - g _ { s ( e ) } ) x _ e = 0 . \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} \\hat { z } _ 1 = u _ 2 , \\hat { z } _ 2 = q _ i ^ { - 1 } u _ 2 , \\dots \\hat { z } _ M = q _ i ^ { - M + 1 } u _ 2 . \\end{align*}"}
-{"id": "7757.png", "formula": "\\begin{align*} J q ^ { - 1 } ( f g ) = \\sum _ { k = 0 } ^ \\infty ( - 1 ) ^ k ( J q ^ 1 ) ^ k ( f ) ( J q ^ { - 1 } ) ^ { k + 1 } ( g ) \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} x _ n ( 0 ) = \\left \\{ \\begin{array} { l l } - 2 n , & n \\geq 1 , \\\\ - n , & 0 \\geq n \\geq - M + 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} E _ { \\mu } ( z ; q , t ) = \\prod _ { \\nu \\prec \\mu } \\frac { Y ( w ) - y _ { \\nu } ( w ) } { y _ { \\mu } ( w ) - y _ { \\nu } ( w ) } \\cdot z ^ { \\mu } , \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{align*} \\phi _ N ( 0 ) = 1 \\ ; N . \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{align*} \\gamma _ { k , n } ^ { ( \\alpha ) } - a _ { k , n } ^ { ( \\alpha ) } = \\frac { 1 } { \\Delta } \\sum _ { j = 1 } ^ { q } \\sum _ { t = 0 } ^ { \\tau _ j - 1 } \\sum _ { w = 1 } ^ { d } \\sum _ { y = 1 } ^ { m _ w } \\gamma _ { n - m _ w + y , n } ^ { ( w ) } C [ g _ { w , y } , h _ { j , t } ] \\left ( ( z - \\lambda _ j ) ^ { \\tau _ j } F _ \\alpha ( z ) \\Phi ^ { n - k } ( z ) \\right ) ^ { ( t ) } _ { z = \\lambda _ j } . \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} \\langle u ( t ) | \\nabla \\frac { Z } { | x - q ( t ) | } | u ( t ) \\rangle = - Z \\int _ { \\mathbb { R } ^ 3 } \\langle u ( t ) , u ( t ) \\rangle _ { \\mathbb { C } ^ 4 } \\frac { x - q ( t ) } { | x - q ( t ) | ^ 3 } . \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{align*} { D } _ { C } \\eta + \\mathfrak { p } _ 1 \\eta + ( \\mathfrak { R } _ 1 + \\mathfrak { p } _ 2 ) \\nabla \\eta + \\mathfrak { R } _ 0 ( \\eta ) + \\mathfrak { p } _ 0 = 0 , \\end{align*}"}
-{"id": "6527.png", "formula": "\\begin{align*} \\| \\tilde \\psi _ j \\| _ n ^ 2 = \\frac 1 2 + \\frac 1 { 4 n } \\frac { \\sin 2 \\pi j } { \\sin \\pi j / n } = \\frac 1 2 . \\end{align*}"}
-{"id": "2369.png", "formula": "\\begin{align*} E \\left [ S ( \\theta ) \\right ] = \\frac { N \\ln N } { 1 - \\theta } + \\frac { \\gamma N } { 1 - \\theta } + O ( 1 ) , N \\to \\infty . \\end{align*}"}
-{"id": "8203.png", "formula": "\\begin{gather*} a n + b m - h = A n + p m + \\big ( q - ( A + 2 - m ) - ( m - 2 ) \\big ) + ( B - p ) . \\end{gather*}"}
-{"id": "8507.png", "formula": "\\begin{align*} \\xi '' = r ( w ) \\xi . \\end{align*}"}
-{"id": "8639.png", "formula": "\\begin{align*} A _ { C _ { 5 } } = \\begin{cases} \\# M _ { p } ^ { - } ( a , c ) ( p ^ { e - 2 d - 1 } + p ^ { m - d - 1 } ) , & \\qquad \\ \\biggl ( \\frac { c ^ { 2 } - 4 a i } { p } \\biggr ) = - 1 , \\\\ \\# M _ { p } ^ { + } ( a , c ) ( p ^ { e - 2 d - 1 } + p ^ { m - d - 1 } ) , & \\qquad \\ \\biggl ( \\frac { c ^ { 2 } - 4 a i } { p } \\biggr ) = 1 . \\end{cases} \\end{align*}"}
-{"id": "2661.png", "formula": "\\begin{align*} X \\left [ \\left ( c - | \\nabla _ B h | ^ 2 \\right ) h ^ { - 2 } \\right ] = 0 . \\end{align*}"}
-{"id": "4280.png", "formula": "\\begin{align*} S _ { b , c } = \\{ \\sigma \\in S _ { b + c } : \\sigma ( 1 ) < \\cdots < \\sigma ( b ) , \\sigma ( b + 1 ) < \\cdots < \\sigma ( b + c ) \\} \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} z \\times [ 0 , 1 ] = \\big \\{ \\ , ( v _ 1 , \\dots v _ { i - 1 } , t , v _ { i + 1 } , \\dots , v _ { j - 1 } , s , v _ { j + 1 } , \\dots , v _ { 2 m + 1 } ) \\mid t \\in [ 0 , 1 ] \\ , \\big \\} \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} - \\frac { 1 } { 6 } R ( \\psi ) = - \\frac { 1 } { 6 } R ^ { N } _ { i j k l } \\langle \\psi ^ i , \\psi ^ k \\rangle _ { g _ s } \\langle \\psi ^ j , \\psi ^ l \\rangle _ { g _ s } . \\end{align*}"}
-{"id": "8446.png", "formula": "\\begin{align*} h ( x , x ) = \\Delta ( e - x ^ 2 ) = \\prod _ { j = 1 } ^ { r } ( 1 - x _ j ^ 2 ) \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} \\frac { d } { d t } \\frac { W _ 2 ^ 2 ( \\mu _ t , \\nu ) } { 2 } = - \\int _ X \\langle \\nabla u , \\nabla \\varphi _ t \\rangle \\ , d \\mu _ t . \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} { \\bf \\nabla } _ H u = \\textit { \\textbf { a } } \\ \\ \\ { \\rm i n } \\ M , \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} \\frac { R ^ { - 1 } + { \\left ( R ^ { - 1 } \\right ) } ^ T } { 2 } & = \\omega ^ { - \\frac { 1 } { 2 } } C _ 1 D _ 2 C _ 1 ^ { * } \\omega ^ { - \\frac { 1 } { 2 } } , \\\\ \\frac { R ^ { - 1 } - { \\left ( R ^ { - 1 } \\right ) } ^ T } { 2 } & = \\omega ^ { - \\frac { 1 } { 2 } } C _ 1 D _ 3 C _ 1 ^ { * } \\omega ^ { - \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "2777.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { i \\in H } \\sum _ { j \\in C } y _ { i , j } \\\\ & \\sum _ { j \\in C } y _ { i , j } \\cdot U _ { i , j } \\geq h _ i & i \\in H \\\\ & \\sum _ { i \\in H } y _ { i , j } \\cdot U _ { i , j } \\geq c _ j & j \\in C \\\\ & y _ { i , j } \\in \\{ 0 , 1 \\} & i \\in H , j \\in C \\end{aligned} \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} a _ n & = \\frac { 1 } { 2 } e ^ { c x _ n } \\left ( x _ n - \\frac { 1 } { c } \\right ) , \\\\ b _ n & = \\frac { 1 } { 2 } e ^ { - c x _ n } \\left ( x _ n + \\frac { 1 } { c } \\right ) \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} h ' ( t ) = \\frac { 2 } { t } i ( t ) + \\frac { a } { 2 t } h ( t ) . \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} \\left ( \\sum _ { P \\in ( R ) _ i } | \\widehat { b _ J } ( P ) | \\right ) ^ { 2 } & \\leq \\left ( \\int _ { \\R } | b _ { J } ( x ) | \\cdot \\sum _ { P \\in ( R _ i ) } | h _ P ( x ) | d x \\right ) ^ { 2 } \\\\ & = \\frac { 2 ^ { i } } { | R | } \\left ( \\int _ { \\R } | b _ { J } ( x ) | \\sum _ { P \\in ( R ) _ i } ( | P | ^ { 1 / 2 } | h _ { P } ( x ) | ) d x \\right ) ^ { 2 } , \\end{align*}"}
-{"id": "7355.png", "formula": "\\begin{align*} \\operatorname { t r } ( L ) & = \\prod _ { i = 1 } ^ { 2 k + 1 } ( 1 + ( b _ { \\rho _ { i } , \\rho _ { i } + k } - \\lambda ) \\frac { \\partial ^ 2 } { \\partial x _ { \\rho _ i } \\partial x _ { \\rho _ { i + 1 } } } ) \\prod _ { j = 1 } ^ { 2 k + 1 } x _ { \\rho _ i } \\\\ & = \\prod _ { i = 1 } ^ { 2 k + 1 } ( 1 + ( b _ { i , i + k } - \\lambda ) \\frac { \\partial ^ 2 } { \\partial x _ i \\partial x _ { i + k } } ) \\prod _ { j = 1 } ^ { 2 k + 1 } x _ i . \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{align*} C _ k \\subset \\left \\{ v : \\exists \\ , t \\in [ 0 , T _ k ] \\textrm { w i t h } \\delta ( \\phi _ t ( v ) ) = T _ k ^ { - \\xi } \\right \\} \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} w ( \\vec { J } ) = \\sum _ { \\sigma \\in S _ { b , c } } \\prod _ { j = 1 } ^ N \\bigg ( \\prod _ { h = 1 } ^ { \\sigma ( \\vec { A } ) _ j - 1 } \\Big ( \\sum _ { k = j + 1 } ^ N \\sigma ( \\vec { A } ) _ k + h \\Big ) \\bigg ) . \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} \\rho _ \\kappa f ( x , t ) = \\frac { 1 } { \\Gamma ( 1 + \\kappa ) } \\int _ { \\mathbb { B } ^ { d - 1 } } f ( x - t v , v ) \\ , ( 1 - | v | ^ 2 ) ^ { \\kappa } \\ , \\mathrm { d } v \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} W _ { n , k } = 0 \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} \\psi ( t _ 2 , x ) - \\psi ( t _ 1 , x ) & = J _ 1 ( t _ 1 , t _ 2 ) - \\frac { 1 } { 2 } J _ 2 ( t _ 1 , t _ 2 ) + \\frac { 1 } { 8 } J _ 3 ( t _ 1 , t _ 2 ) , \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} \\frac { d \\eta _ n } { d \\mu } = b _ n ^ { - \\alpha } \\max _ { { \\bf 0 } \\leq t \\leq ( n - 1 ) \\bf { 1 } } | f _ t ( s ) | ^ { \\alpha } , s \\in S . \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{align*} \\mathbf { B } _ { 0 } ( y , 1 , t ) = \\widetilde { B } _ { 1 } ( y , 1 ) = 0 \\quad \\mathbf { B } _ { 0 } ( y , 1 , t ) = \\widetilde { B } _ { - 1 } ( y , 1 ) = 0 . \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{align*} \\limsup _ { x \\to 1 - } f ( x ) = \\infty \\liminf _ { x \\to 1 - } f ( x ) = - \\infty . \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{align*} H ( 4 , N ) = \\sum _ { s \\in \\mathbb { Z } } \\sigma _ 3 \\left ( \\frac { N - s ^ 2 } { 4 } \\right ) \\end{align*}"}
-{"id": "8485.png", "formula": "\\begin{align*} \\widetilde { x } ^ t = x . \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} h ^ { d - 1 } ( X _ { - i } ) : = \\begin{pmatrix} h ^ { d - 1 } \\big ( x ^ { ( 1 ) } _ 1 , \\dots , x ^ { ( i - 1 ) } _ 1 , x ^ { ( i + 1 ) } _ 1 , \\dots , x ^ { ( d ) } _ 1 \\big ) \\\\ h ^ { d - 1 } \\big ( x ^ { ( 1 ) } _ 2 , \\dots , x ^ { ( i - 1 ) } _ 2 , x ^ { ( i + 1 ) } _ 2 , \\dots , x ^ { ( d ) } _ 2 \\big ) \\\\ \\vdots \\\\ h ^ { d - 1 } \\big ( x ^ { ( 1 ) } _ n , \\dots , x ^ { ( i - 1 ) } _ n , x ^ { ( i + 1 ) } _ n , \\dots , x ^ { ( d ) } _ n \\big ) \\end{pmatrix} \\end{align*}"}
-{"id": "2248.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( 1 - \\lambda \\right ) e ^ { D _ { \\varpi } ( P \\parallel Q _ 0 ) } + \\lambda e ^ { D _ { \\varpi } ( P \\parallel Q _ 1 ) } \\ge e ^ { D _ { \\varpi } ( P \\parallel ( 1 - \\lambda ) Q _ 0 + \\lambda Q _ 1 ) } . \\end{aligned} \\end{align*}"}
-{"id": "6949.png", "formula": "\\begin{align*} \\| f \\| _ { - 1 } ^ 2 = \\sup _ { h } \\big \\{ 2 \\ < f , h \\ > - \\ < h , ( - \\mathcal S _ n ) h \\ > \\big \\} , \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{align*} r = 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "1037.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( \\tau _ { c } - \\tau _ { n } ) ^ { \\frac { 3 } { 1 - s } } \\rho _ { \\tau _ { n } } ( ( \\tau _ { c } - \\tau _ { n } ) ^ { \\frac { 1 } { 1 - s } } x + x _ j ) = \\lambda _ s ^ { 3 } Q \\left ( \\lambda _ s x \\right ) \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} \\theta _ { i , j } = \\kappa _ { i , j } + \\log \\frac { 1 } { \\epsilon } . \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{align*} w ( M ^ { d n } ) = { j ^ * } \\prod _ { i = 1 } ^ { m } ( 1 + v _ i ) , \\ a n d \\ \\ p ( M ^ { d n } ) = 1 . \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} f ( x _ 1 , y _ 1 , \\ldots , x _ k , y _ k ) = \\sum _ { n _ 1 = 0 } ^ \\infty c _ { n _ 1 } ( x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) \\Phi _ { n _ 1 } ^ { ( a _ 1 , b _ 1 ) } ( x _ 1 , y _ 1 | q ) . \\end{align*}"}
-{"id": "7552.png", "formula": "\\begin{align*} g _ 1 = & - 2 \\alpha _ 1 \\gamma , \\\\ g _ 2 = & - ( 2 \\gamma \\alpha _ 2 + \\alpha _ 2 \\cdot \\tilde \\gamma ) , \\\\ g _ 3 = & ( 2 n + 4 ) \\gamma \\beta ^ { - 1 } ( t , q ) \\alpha _ 2 - \\alpha _ 3 \\cdot \\tilde \\gamma \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} ( \\Delta ^ { ( m ) } \\partial _ { z _ { j } } ) h \\big | _ { z _ j = 0 } = 0 , j = 1 , \\ldots , m - 1 \\ , . \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} g ( D ) g ( I _ 1 ) g ( I _ 2 ) h ( J \\setminus ( D \\cup J _ 1 ) ) = g ( D ) g ( J _ 1 \\setminus I _ 1 ) g ( J _ 2 \\setminus I _ 2 ) h ( J \\setminus ( D \\cup J _ 2 ) ) , \\end{align*}"}
-{"id": "673.png", "formula": "\\begin{align*} \\mathcal Q ( \\lambda ) : = \\begin{bmatrix} \\lambda M _ 1 & - N _ 1 \\\\ & \\lambda M _ 2 & \\ddots \\\\ & & \\ddots & - N _ { r - 1 } \\\\ - N _ r & & & \\lambda M _ r \\end{bmatrix} \\end{align*}"}
-{"id": "4734.png", "formula": "\\begin{align*} u ^ { - 1 } \\cdot S = S + \\gamma ( S ) x _ { \\gamma } E _ { \\gamma } \\textup { f o r a l l } u = \\exp ( x _ { \\gamma } E _ { \\gamma } ) \\in U _ { \\gamma } \\textup { w h e r e } x _ { \\gamma } \\in \\C . \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} \\left ( \\frac { \\widehat { k } _ 0 - \\widehat { k } _ 0 ^ { - 1 } } { q ^ { 1 / 2 } - q ^ { - 1 / 2 } } \\right ) v = \\left ( \\frac { q ^ { | \\mu | - N / 2 } - q ^ { N / 2 - | \\mu | } } { q ^ { 1 / 2 } - q ^ { - 1 / 2 } } \\right ) v \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} u _ 0 = R ^ { - 1 } \\log \\bigl ( \\tfrac { 1 - R } { 1 + R } \\bigr ) , \\ \\ u _ 1 = R ^ { - 2 } \\log \\bigl ( \\tfrac { 1 - R } { 1 + R } \\bigr ) + 2 R ^ { - 1 } , \\ \\ u _ 2 = \\tfrac { 3 - R ^ 2 } { 2 R ^ 3 } \\log \\bigl ( \\tfrac { 1 - R } { 1 + R } \\bigr ) + 3 R ^ { - 2 } ; \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} f ( x , \\lambda ) = \\max _ { m \\in S } \\bigl ( m x + \\lambda ( m ^ 2 - u ) + J ( m ) \\bigr ) . \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} H ^ s ( H + K ) ^ { n - s } = \\sum _ { p \\in S } { ( K | _ { Z _ p } ) ^ { n - s } } . \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } n ^ { - 1 / \\alpha } b _ n = K . \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} \\mathcal { F } \\left \\{ - { ( \\hslash { } \\nabla { } ) } ^ { \\alpha { } } \\psi ( x , t ) ; p \\right \\} = { \\left | p \\right | } ^ { \\alpha } \\hat { \\psi } ( p , t ) , \\end{align*}"}
-{"id": "4553.png", "formula": "\\begin{align*} \\mathcal K = \\left \\langle H _ 1 , p H _ 2 , \\dots , p ^ { n - 1 } H _ { n } \\right \\rangle = \\left \\langle \\sum _ { i = 0 } ^ { n - 1 } p ^ i H _ { i + 1 } \\right \\rangle \\end{align*}"}
-{"id": "4569.png", "formula": "\\begin{align*} \\omega ( X , Y ) = g _ Q ( \\pi ( X ) , J \\pi ( Y ) ) \\end{align*}"}
-{"id": "9263.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n - 1 } } { n ^ 2 } \\sin \\left ( \\frac { n \\pi } { 3 } \\right ) = \\frac { 2 } { 3 } \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ 2 } \\sin \\left ( \\frac { n \\pi } { 3 } \\right ) , \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} M _ + ( B ) = \\frac { 1 } { \\beta } F ' _ + ( 0 ) \\ \\ M _ - ( B ) = \\frac { 1 } { \\beta } F ' _ - ( 0 ) . \\end{align*}"}
-{"id": "2321.png", "formula": "\\begin{align*} P \\{ T _ 1 < T _ 2 \\} = \\lambda M _ 2 \\int _ 0 ^ 1 x ^ { \\lambda - 1 } \\left ( 1 - x \\right ) ^ { M _ 1 } \\left ( 1 - x ^ { \\lambda } \\right ) ^ { M _ 2 - 1 } d x \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} \\int _ \\Omega K ( x ) \\delta _ i ^ { \\frac { n + 2 } { n - 2 } } v \\ , d x = O \\Bigl ( \\| v \\| \\sum _ { i = 1 } ^ p \\ , \\frac { | \\nabla K ( a _ i ) | } { \\lambda _ i } + \\frac { 1 } { \\lambda _ i ^ { \\beta _ i } } + \\frac { ( \\log \\lambda _ i ) ^ { \\frac { n + 2 } { 2 n } } } { \\lambda _ i ^ { \\frac { n + 2 } { 2 } } } \\Bigr ) . \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} \\bar { c } ( y ) : = \\frac { \\mu n ( y ) ^ 2 } { ( 2 d \\varrho \\delta _ \\sigma ) ^ { 2 | y | - 1 } } . \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{align*} X \\cdot I \\psi = J _ 1 X \\cdot \\psi , X \\cdot J \\psi = J _ 2 X \\cdot \\psi , X \\cdot K \\psi = J _ 3 X \\cdot \\psi \\end{align*}"}
-{"id": "9108.png", "formula": "\\begin{align*} { \\bf D } _ A ^ { - 1 } = \\frac { 1 } { \\omega } { \\bf I } _ K - \\frac { 1 } { \\omega ^ 2 } { \\bf { G } } _ c . \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{align*} p ( 0 , 0 | v , w ) = t , p ( 0 , 1 | v , w ) = p ( 1 , 0 | v , w ) = 0 , p ( 1 , 1 | v , w ) = 1 - t , \\end{align*}"}
-{"id": "5286.png", "formula": "\\begin{align*} K _ { 1 2 3 } : = s K _ 1 s ^ { - 1 } \\cap K _ 2 \\cap t ^ { - 1 } K _ 3 t . \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} [ s d _ 1 , d _ 2 , x ] = [ s , d _ 2 , x ] d _ 1 + [ s , d _ 2 ] [ d _ 1 , x ] + [ s , x ] [ d _ 1 , d _ 2 ] + s [ d _ 1 , d _ 2 , x ] . \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{align*} \\alpha _ { k } = \\int _ { \\Omega } F _ { \\ast } ( w _ { k } ) d x \\end{align*}"}
-{"id": "5665.png", "formula": "\\begin{align*} a = \\inf \\{ t \\in [ 0 , 1 ] \\mid b \\in \\sigma ( R _ { [ 0 , t ] } T R _ { [ 0 , t ] } ) \\} , \\end{align*}"}
-{"id": "3399.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k \\mu _ j = m . \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{gather*} K _ 1 = \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } \\left [ \\sum _ { k = 1 } ^ { H _ 0 - 1 } ( a _ { i k } b _ { k j } - a ^ 0 _ { i k } b ^ 0 _ { k j } ) + c _ i \\right ] ^ 2 , \\\\ K _ 2 = \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } \\left \\{ \\sum _ { k = H _ 0 } ^ { H - 1 } a _ { i k } b _ { k j } \\right \\} ^ 2 . \\end{gather*}"}
-{"id": "4485.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 \\delta ^ \\gamma } { \\partial x _ i \\partial x _ j } = \\gamma ( \\gamma - 2 ) \\delta ^ { \\gamma - 4 } \\sum _ { k = 1 } ^ m b _ { k i } y _ k \\sum _ { l = 1 } ^ m b _ { l j } y _ l + \\gamma \\delta ^ { \\gamma - 2 } \\sum _ { k = 1 } ^ m b _ { k i } b _ { k j } . \\end{align*}"}
-{"id": "1626.png", "formula": "\\begin{align*} \\sup _ { x : | x - 1 | \\le \\zeta } \\sup _ { \\dot { \\varphi } _ 0 \\in \\mathfrak { S } _ t } \\Big | h _ { t , \\dot { \\varphi } _ 0 } ( x ) - h ( x ) \\Big | = O \\left ( \\frac { 1 } { \\ln _ 2 t } \\right ) . \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{align*} { \\tilde { \\phi } } _ 2 ( s ) = { \\left ( - i \\right ) } ^ { - s } \\Gamma \\left ( s \\right ) \\int _ { - \\infty { } } ^ 0 e ^ { - i e ^ { - i \\theta \\pi / 2 } { \\left \\vert { } w \\right \\vert { } } ^ { \\alpha { } + 1 } } w ^ { - s } d w . \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} ( 1 , \\frac { 1 + \\sqrt { 5 } } { 2 } \\ , ] ~ = ~ \\left ( \\bigcup _ { n = 2 } ^ { \\infty } \\left [ \\ , \\theta _ { n + 1 } ^ { - 1 } , \\theta _ { n } ^ { - 1 } \\ , \\right ) \\right ) ~ \\bigcup ~ \\left \\{ \\theta _ { 2 } ^ { - 1 } \\right \\} . \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} B _ j ^ { ( m ) } = ( j m , ( j + 1 ) m ] , j \\geq 0 . \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{align*} ( Y _ 1 ) ^ l = & \\left ( V \\nabla _ q \\beta + \\beta \\tilde F - \\beta \\partial _ t \\psi \\right ) _ i ( \\tilde \\gamma ^ { - 1 } ) ^ { i l } + \\frac { n + 2 } { 2 } \\beta ^ { - 1 } \\partial _ { q ^ { i } } ( \\beta ) ( \\tilde \\gamma ^ { - 1 } ) ^ { i l } \\end{align*}"}
-{"id": "4416.png", "formula": "\\begin{align*} Q ( Z ) = \\prod _ { r \\in R } \\Bigl ( \\sum _ { j \\in J } \\abs { z _ { j , r } } ^ 2 \\Bigr ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{align*} R _ m \\chi _ z ( y ) = q ^ { | T _ \\nu ( m + 1 ) | } \\chi \\left ( \\sum _ { s \\in S _ \\mu } \\sum _ { t \\in T _ \\nu } Y _ { s , t } Z _ { s , t } \\right ) \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{align*} \\frac { 1 } { M + 1 } \\sum _ { j = 1 } ^ M K ( P _ j , P _ 0 ) \\leq K _ 2 \\delta ^ 2 n N ^ { - 2 \\beta / r } = K _ 2 \\delta ^ 2 \\log M . \\end{align*}"}
-{"id": "8812.png", "formula": "\\begin{align*} \\Omega _ { \\alpha , \\bar { \\alpha } } = \\frac { - e ^ { 2 \\alpha } } { 2 } ( d u - 2 \\chi ) ( \\alpha ^ { \\vee } ) , \\Omega _ { \\alpha _ 1 , \\bar { \\alpha } _ 2 } = \\frac { 2 \\chi ( [ \\theta ( e _ { \\alpha _ 2 } ) , e _ { \\alpha _ 1 } ] ) } { e ^ { - 2 \\alpha _ 1 } + e ^ { - 2 \\alpha _ 2 } } , \\end{align*}"}
-{"id": "5462.png", "formula": "\\begin{align*} \\phi _ j = \\Omega _ j t , j = 1 , . . . , k , \\end{align*}"}
-{"id": "2154.png", "formula": "\\begin{align*} \\liminf _ { \\delta \\to 0 } \\delta ^ { \\dim _ B ( A ) } N _ { \\delta } ( A ) = 0 \\textrm { o r } \\limsup _ { \\delta \\to 0 } \\delta ^ { \\dim _ B ( A ) } N _ { \\delta } ( A ) = \\infty . \\end{align*}"}
-{"id": "8304.png", "formula": "\\begin{align*} Y _ { i _ { i ' } , 2 } & = \\mu ' _ { i ' } + \\ell ' - 2 i ' + 1 \\\\ & = \\mathcal { C } _ 1 ( \\alpha ' , \\nu ' , \\tau ^ { - 1 } ( i ' ) , I ' _ b \\setminus \\lbrace \\tau ^ { - 1 } ( i ' ) \\rbrace , I _ b ) \\\\ & \\geq \\mu ^ { \\circ } + \\ell - 2 i _ { i ' } \\\\ & = X _ { i _ { i ' } , 1 } + \\ell - 2 i _ { i ' } \\\\ & = Y _ { i _ { i ' } , 1 } - 1 . \\end{align*}"}
-{"id": "7096.png", "formula": "\\begin{align*} E _ { f } ( A , B ) = \\sum _ { \\delta \\in \\Delta } m _ { \\delta } ^ 2 = \\sum _ { j = 0 } ^ { \\log \\mu ( | A | + | B | ) } \\sum _ { \\delta \\in \\Delta \\atop 2 ^ { j } \\leq m _ \\delta < 2 ^ { j + 1 } } { m _ { \\delta } ^ 2 } < \\sum _ { j = 0 } ^ { \\log \\mu ( | A | + | B | ) } 2 ^ { 2 j + 2 } k _ { 2 ^ j } . \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{align*} b _ { \\nu , n } ^ { ( \\alpha ) } : = [ Q _ { | \\textup { \\textbf { m } } | } ^ { \\textup { \\textbf { F } } } Q _ { n , \\textup { \\textbf { m } } } F _ \\alpha ] _ { \\nu } = \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ \\rho } \\frac { Q _ { | \\textup { \\textbf { m } } | } ^ { \\textup { \\textbf { F } } } ( t ) Q _ { n , \\textup { \\textbf { m } } } ( t ) F _ \\alpha ( t ) \\Phi ' ( t ) } { \\Phi ^ { \\nu + 1 } ( t ) } d t , \\nu \\geq n + | \\textup { \\textbf { m } } | - m _ \\alpha + 1 , \\end{align*}"}
-{"id": "4497.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle i \\frac { \\partial u } { \\partial t } ( t , x ) = ( \\mathcal { D } + \\beta ) u ( t , x ) - \\frac { Z } { | x - q ( t ) | } u ( t , x ) + \\left ( | u | ^ 2 * \\frac 1 { | x | } \\right ) ( t , x ) u ( t , x ) , \\\\ \\displaystyle m \\frac { d ^ 2 q } { d t ^ 2 } ( t ) = \\langle u ( t ) | \\nabla \\frac { Z } { | \\cdot - q ( t ) | } | u ( t ) \\rangle \\\\ \\displaystyle u ( 0 , \\cdot ) = u _ 0 , q ( 0 ) = 0 , \\frac { d q } { d t } ( 0 ) = \\dot { q } _ 0 . \\end{cases} \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} \\varphi ( S _ + ) = R _ + , \\psi ( R _ + ) = G _ + \\mbox { a n d } \\psi \\circ \\varphi ( S _ + ) = G _ + . \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} I ( t \\ , , x ) : = \\int _ 0 ^ D G ( t \\ , , x \\ , , y ) u _ 0 ( y ) \\ , \\d y . \\end{align*}"}
-{"id": "4888.png", "formula": "\\begin{align*} Z ( E ; T ) = \\exp \\bigg ( \\sum _ { n \\ge 1 } \\frac { N _ n } { n } \\cdot T ^ n \\bigg ) \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} A _ 0 ^ { i _ 1 , . . . , i _ k } ( \\beta , \\tilde \\gamma , B ) = - B ^ { j _ 1 , . . . , j _ k } G ^ { i _ 1 . . . i _ k } _ { j _ 1 . . . . j _ k } ( - \\tilde \\gamma ) \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} \\widetilde { p } ( x , \\xi ) = \\Phi _ 4 ( \\Phi _ 3 ( \\Phi _ 2 ( \\Phi _ 1 ( p ) ) ) ) ( x , \\xi ) , \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} \\Z ^ c _ { X / B } ( n ) = \\Z ^ c _ { X / K } ( n - 1 ) [ 2 ] . \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} \\frac { U _ 1 ( \\mu _ { 2 \\alpha } ) } { U _ 1 ( \\varphi ) } = - c _ { j \\alpha } , \\frac { U _ 1 ( \\rho _ { 2 \\alpha } ) } { U _ 1 ( \\varphi ) } = \\tilde { c } _ { j \\alpha } , \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{align*} \\left \\{ u \\in H _ { 0 } ^ { 1 } ( \\Omega ) : \\int _ { \\Omega } a | u | ^ { q + 1 } = 1 \\right \\} . \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{align*} \\mathsf b _ - ( a , b ) : = \\sum _ { j = 1 } ^ n a _ { 2 j - 1 } b _ { 2 j } + a _ { 2 j } b _ { 2 j - 1 } \\quad a = ( a _ 1 , \\ldots , a _ n ) b = ( b _ 1 , \\ldots , b _ n ) \\in \\Gamma , \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{align*} \\sum _ { t ' \\in T _ \\mu \\setminus \\lambda } | T _ \\mu ( t ' + 1 ) \\setminus \\lambda | = \\left ( \\sum _ { t ' \\in T _ \\mu \\setminus \\lambda ' } | T _ \\mu ( t ' + 1 ) \\setminus \\lambda ' | \\right ) - | T _ \\mu \\setminus \\lambda | . \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} \\varphi ( q ) \\mu _ { 1 j } ( p ) + h ( p ) \\mu _ { 2 \\alpha } ( q ) = \\rho _ { 1 j } ( p ) + \\rho _ { 2 \\alpha } ( q ) , \\forall ( p , q ) \\in D \\times G , \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{align*} [ m ] P = \\bigg ( x - \\frac { \\psi _ { m - 1 } \\psi _ { m + 1 } } { \\psi _ m ^ 2 } , \\frac { \\psi _ { m + 2 } \\psi _ { m - 1 } ^ 2 - \\psi _ { m - 2 } \\psi _ { m + 1 } ^ 2 } { 4 y \\psi _ m ^ 3 } \\bigg ) \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} \\textbf { C o r r } = \\overset { \\infty } { \\underset { n = 0 } { \\oplus } } \\textbf { C o r r } ^ { \\mathbf { ( n ) } } \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{align*} \\mathrm { s i g n } ( r ( 0 ) - 1 ) = \\mathrm { s i g n } ( \\lambda _ 0 ) \\ , . \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} \\tau _ 1 ( X , t ) = | t | ^ { 1 / 2 } \\tau _ 0 \\left ( \\frac { X } { \\sqrt { | t | } } \\right ) , \\end{align*}"}
-{"id": "469.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 8 & 7 & 1 0 \\\\ 0 & 5 & 2 & 3 \\\\ 0 & 6 & 1 0 & 1 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "5128.png", "formula": "\\begin{align*} v ( [ 1 \\ \\dots \\ t - 1 \\mid i _ 1 \\ \\dots \\ i _ { t - 1 } ] ) \\ = \\ v ( \\pi ) , \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\alpha \\cdot f & = & \\delta _ \\alpha ^ \\bullet \\circ f - f \\circ \\partial _ \\alpha \\\\ \\alpha \\cdot g & = & \\delta _ \\alpha ^ \\bullet \\circ g - g \\circ \\partial _ \\alpha ^ \\bullet \\ , ; \\end{array} \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} \\frac { k - 1 } { 2 \\pi ( \\ell + 1 ) } = : y _ { \\ell + 1 } \\leq y \\leq y _ { \\ell } : = \\frac { k - 1 } { 2 \\pi \\ell } . \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n Y _ k D _ { k l } = 0 \\forall l = 1 , 2 , \\dots , n . \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{align*} \\Pi = D _ r ^ { - \\star } B _ r ^ \\star D _ { r - 1 } ^ { - \\star } B _ { r - 1 } ^ \\star \\dotsm D _ 1 ^ { - \\star } B _ 1 ^ \\star C _ r ^ { - 1 } A _ r C _ { r - 1 } ^ { - 1 } A _ { r - 1 } \\dotsm C _ 1 ^ { - 1 } A _ 1 \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} \\int P ^ { \\lfloor \\gamma \\rfloor } _ { k , y } ( \\rho ^ n , u ) \\ , \\mathrm { d } u = \\frac { 1 } { k ! } , \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { \\frac { \\omega t } { 2 } } t ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\norm { e ^ { - t A } a ^ { j } } _ { L ^ { q } _ { \\sigma } ( \\Omega ) } & \\leq C e ^ { - \\frac { \\omega t } { 2 } } t ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\norm { a ^ { j } } _ { L ^ q _ { \\sigma } ( \\Omega ) } . \\end{aligned} \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} \\sum ^ { { M _ s } } _ { j = 2 } \\alpha _ j ^ 2 = \\max \\left \\{ 0 , \\frac { \\rho \\frac { 1 } { { L \\left ( | | x _ t - x _ 0 | | \\right ) } } - \\epsilon _ 1 } { \\rho ( 1 + \\epsilon _ 1 ) \\frac { 1 } { { L \\left ( | | x _ t - x _ 0 | | \\right ) } } } \\right \\} . \\end{align*}"}
-{"id": "1876.png", "formula": "\\begin{align*} \\ell ( x , y ) = \\frac { x y } { ( 1 - x y ) ^ 2 } q \\left ( \\frac { x } { 1 - x y } , 1 - x y \\right ) + \\frac { x y } { 1 - x y } . \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{align*} S ^ { - 1 } D \\ , ( \\alpha , \\beta ) = ( d + d ^ \\ast ) ( \\beta , \\alpha ) + \\Phi ( \\alpha , \\beta ) \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} u \\left ( t \\right ) \\leq v \\left ( t \\right ) + \\int _ { a } ^ { t } \\overset { \\infty } { \\underset { k = 1 } { \\sum } } \\frac { \\left [ g \\left ( t \\right ) \\Gamma \\left ( \\alpha \\right ) \\right ] ^ { k } } { \\Gamma \\left ( \\alpha k \\right ) } \\psi ^ { \\prime } \\left ( \\tau \\right ) \\left [ \\psi \\left ( t \\right ) - \\psi \\left ( \\tau \\right ) \\right ] ^ { \\alpha k - 1 } v \\left ( \\tau \\right ) d \\tau , \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} P D [ F _ z ] \\smile i _ 1 ^ { * } [ \\Omega ] & = \\langle [ F _ z ] , i _ 1 ^ { * } [ \\Omega ] \\rangle \\\\ & = \\langle [ F _ z ] , p '^ { * } [ \\omega | _ { Z } ] \\rangle \\\\ & = \\langle [ p ' ( F _ z ) ] , [ \\omega | _ { Z } ] \\rangle \\\\ & = P D [ p ' ( F _ z ) ] \\smile [ \\omega | _ { Z } ] \\\\ & = \\langle p ' ( F _ z ) , [ \\omega | _ { Z } ] \\rangle \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} x _ 1 ( t _ 1 ) - 1 = x _ 2 ( t _ 1 ) \\leq x _ 2 ^ B ( t _ 1 ) < x _ 1 ^ B ( t _ 1 ) = x _ 1 ( t _ 1 ) . \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} \\Pr [ A _ 2 | A _ 1 ] = \\frac { \\Pr [ A _ 1 \\cap A _ 2 ] } { \\Pr [ A _ 1 ] } \\geq \\Pr [ A _ 2 ] . \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} \\mathcal { C } _ { - 1 } \\big ( \\mathcal { a } ^ { ( x ) } , \\hat { \\mathcal { n } } ^ { ( x ) } , j , J , J ' \\setminus \\lbrace j \\rbrace \\big ) = \\mathcal { C } _ { - 1 } \\big ( \\alpha , \\nu , p _ { ( x , j ) } , \\mathcal { Q } _ { < x } \\cup p ^ { ( x ) } _ J , \\mathcal { Q } _ { > x } \\cup p ^ { ( x ) } _ { J ' \\setminus \\lbrace j \\rbrace } \\big ) . \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{align*} \\mathfrak { D } = \\left ( \\begin{array} { c c c } \\bar { \\partial } _ { \\theta } ^ H & \\vartheta ^ * _ { C _ { \\xi } , r } \\\\ \\vartheta _ { C _ { \\xi } , r } & \\partial _ { \\theta } ^ H \\ \\end{array} \\right ) + \\mathfrak { r } , \\end{align*}"}
-{"id": "825.png", "formula": "\\begin{align*} \\chi ( X ) = \\int _ X { \\rm S t r } \\ , K ( t ; x , x ) \\ , d X _ x , t > 0 . \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} \\dim ( \\overline { \\mu } ^ { - 1 } ( y ) ) = \\dim ( \\mu ^ { - 1 } ( \\mathcal { O } ( y ) ) ) & = \\dim ( X ) - \\dim ( g _ { } ) + \\dim ( \\mathcal { O } ( y ) ) \\\\ & = ( \\dim ( G ) + \\mathrm { r k } ( G ) ) - \\dim ( G ) + ( \\dim ( G ) - \\mathrm { r k } ( G ) ) \\\\ & = \\dim ( G ) , \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} x \\ , \\frac { \\theta _ 3 ' ( x ) } { \\theta _ 3 ( x ) } + \\tfrac { 1 } { x } \\ , \\frac { \\theta _ 3 ' \\left ( \\tfrac { 1 } { x } \\right ) } { \\theta _ 3 \\left ( \\tfrac { 1 } { x } \\right ) } = - \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{align*} \\sigma ( X , I _ + Y ) = \\sigma ( Y , I _ + X ) \\sigma ( X , I _ - Y ) = \\sigma ( Y , I _ - X ) \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} \\mathcal L _ \\partial ( e _ S ) = \\lambda e _ S \\ , . \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} \\tilde { M } \\equiv \\omega ^ { - \\frac { 1 } { 2 } } M \\omega ^ { - \\frac { 1 } { 2 } } = \\big [ \\tilde { M } _ { j k } \\big ] , \\ \\ \\ \\tilde { \\tilde { M } } \\equiv C _ 1 ^ { * } \\tilde { M } C _ 1 = \\bigl [ \\tilde { \\tilde { M } } _ { j k } \\bigl ] , \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{align*} \\mu _ P ( x , y ) = \\prod _ { i = 1 } ^ { k } \\mu _ { p _ i } ( x , y ) = \\prod _ { i = 1 } ^ { k } \\big [ d ( x , y ) + \\sqrt { d ( x , p _ i ) d ( y , p _ i ) } \\big ] . \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} \\tilde { h } ( t , z , \\zeta ) : = \\sum _ { k = 1 } ^ 2 h _ k ( t , x ^ { ( k ) } , \\xi ^ { ( k ) } ) + W _ { 1 2 } ( t , x ^ { ( 1 ) } - x ^ { ( 2 ) } ) + \\sum _ { k = 3 } ^ 4 l _ k ( x ^ { ( k ) } , \\xi ^ { ( k ) } ) \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} \\Gamma ( x , t ) = ( x , \\cos ( t ) f ( x ) , - \\sin ( t ) f ( x ) ) \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} \\Phi ( \\xi ' , \\xi '' ) = \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { k } s _ j \\xi _ j ^ 2 + \\mathcal { O } ( | ( \\xi ' , \\xi '' ) | ^ 3 ) , s _ j = \\pm 1 , \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} t \\Bigl ( x + 1 + \\frac 1 { x - 1 } \\Bigr ) = \\sup _ { b \\in \\R } \\{ b x - f ( b ) \\} , x \\ge 2 . \\end{align*}"}
-{"id": "4962.png", "formula": "\\begin{align*} \\tau _ 0 & = 1 , \\\\ \\tau _ 1 & = x , \\\\ \\tau _ 2 & = x ^ 3 - 3 t _ 3 , \\\\ \\tau _ 3 & = x ^ 6 - 1 5 t _ 3 x ^ 3 - 4 5 t _ 3 ^ 2 + 4 5 t _ 5 x , \\\\ \\tau _ 4 & = x ^ { 1 0 } - 4 5 t _ 3 x ^ 7 + 3 1 5 t _ 5 x ^ 5 + 4 7 2 5 t _ 3 t _ 5 x ^ 2 - 4 7 2 5 t _ 3 ^ 3 x - 4 7 2 5 t _ 5 ^ 2 - 1 4 7 5 t _ 7 x ^ 3 + 4 7 2 5 t _ 3 t _ 7 . \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{align*} 1 a - 1 b + 0 c + r d & = 0 \\\\ 0 a + 1 b - 1 c + s d & = 0 \\\\ 1 a - 1 b + u c + 0 d & = 0 \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} N ( y , k ) \\left | h _ { \\ell } ( z ) \\right | = \\frac { ( 2 \\pi ) ^ k } { \\Gamma ( k ) } \\left ( r _ 1 ( y ) + r _ 2 ( y ) \\right ) . \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} [ s , x ] [ z _ j , y _ i ] + [ s , y _ i ] [ z _ j , x ] = - \\bigl ( [ s , x ] [ y _ i , z _ j ] + [ s , y _ i ] [ x , z _ j ] \\bigr ) \\equiv 0 \\pmod { I } . \\end{align*}"}
-{"id": "3617.png", "formula": "\\begin{align*} \\mathcal { Z } _ { \\Psi } ( \\mathbf { s } ; \\mathbf { m } ) = \\mathcal { Z } _ { \\Psi } ( s _ 1 , \\ldots , s _ r ; \\mathbf { m } ) = \\sum _ { \\substack { \\mathbf { c } \\in ( \\mathcal { O } _ S / \\mathcal { O } _ { S } ^ { \\times } ) ^ r \\\\ \\mathbf { c } = ( c _ 1 , \\ldots , c _ r ) } } \\dfrac { H ^ { ( n ) } ( \\mathbf { c } ; \\mathbf { m } ) \\Psi ( \\mathbf { c } ) } { | c _ { 1 } | ^ { 2 s _ 1 } \\ldots | c _ { r } | ^ { 2 s _ r } } \\ , , \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} \\widetilde { p } ( x , D _ x ) f ( x ) = \\kappa ^ { - 1 , * } \\circ p ( x , D _ x ) \\circ \\kappa ^ * f ( x ) x \\in \\R ^ n , f \\in \\mathcal { S } ( \\R ^ n ; X _ 0 ) , \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} N ^ - ( y ) \\cap z \\Phi _ h ^ - = N ^ - ( y ) \\cap z \\Phi _ h ^ - \\cap \\Phi _ \\nu = N ^ - ( y ) \\cap \\Phi ^ - _ { h _ z } \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{align*} { } Z _ R ( \\alpha , \\beta ) = \\prod _ { k = 1 } ^ { B } \\left ( 1 + e ^ { - \\alpha } e ^ { - \\beta k } \\right ) \\end{align*}"}
-{"id": "5761.png", "formula": "\\begin{align*} \\alpha ( C ; S ) = \\sum _ { j = 1 } ^ { n + 1 } \\max _ { x \\in C } ( - \\lambda _ j ( x ) ) + 1 . \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{align*} j ( E ) = \\dfrac { ( h ^ 6 - 4 h ^ 5 + 1 6 h + 1 6 ) ^ 3 } { ( h + 1 ) ^ 2 ( h - 4 ) h } \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} P ^ { s } _ { y } ( z , \\tau ) = \\frac { 1 } { 4 ^ { \\frac n 2 + s } \\pi ^ { \\frac n 2 } \\Gamma ( s ) } \\frac { y ^ { 2 s } } { \\tau ^ { n / 2 + 1 + s } } e ^ { - \\frac { | z | ^ 2 + y ^ 2 } { 4 \\tau } } . \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} \\ln | \\chi ( \\exp ( C _ E ) \\exp ( C _ D ) ) | & = \\ln | \\chi ( \\exp ( C _ E ) ) | + \\ln | \\chi ( \\exp ( C _ D ) ) | \\\\ & = \\ln | e ^ { \\chi ( C _ E ) } | + \\ln | e ^ { \\chi ( C _ D ) } | \\\\ \\intertext { w h e r e w e s t i l l d e n o t e b y $ \\chi $ t h e L i e a l g e b r a c h a r a c t e r $ \\mathfrak { h } \\rightarrow \\mathbb { C } $ i n d u c e d b y $ \\chi $ , } & = \\mathrm { R e } ( \\chi ( C _ E ) + \\chi ( C _ D ) ) \\\\ & = \\mathrm { R e } ( \\chi ( C _ E + C _ D ) ) . \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} \\widehat { A } ( l _ 0 , l _ 1 ) = \\left [ \\widehat { d } _ { l _ 0 } ^ { ~ ( 0 ) } + \\widehat { r } _ { l _ 0 } \\right ] \\widehat { b } _ { l _ 1 , 0 } + \\left [ \\widehat { d } _ { l _ 1 } ^ { ~ ( 1 ) } + \\widehat { r } _ { l _ 1 } \\right ] \\widehat { b } _ { l _ 0 , 1 } , \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} f _ { l o c } ( t ) \\ge f _ q ( t ) = f _ { q a } ( t ) \\ge f _ { q c } ( t ) \\ge f _ { v e c t } ( t ) \\ge f _ { n s } ( t ) \\geq 0 . \\end{align*}"}
-{"id": "7799.png", "formula": "\\begin{align*} \\mathcal { I } _ { 2 , - } ( t , x ) = - \\int _ 0 ^ t \\int _ { - \\infty } ^ { 0 } \\int _ { - \\infty } ^ y \\frac { \\partial G _ { t - s } } { \\partial x } ( x - z ) \\psi ( s , z ) \\sigma _ s ( y ) d z W ( d s , d y ) . \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} \\Omega ( \\rho ) = b ( \\rho ) \\pm \\frac { 1 } { \\rho } \\sqrt { \\varepsilon ^ 2 r ^ 2 - a ( \\rho ) ^ 2 } , \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} \\underset { \\substack { z ^ 0 , z ^ 1 \\in \\R ^ n \\\\ z ^ 0 \\not = z ^ 1 } } { \\sup } \\dfrac { \\left | \\partial _ z ^ \\delta q _ { z ^ 0 } - \\partial _ z ^ \\delta q _ { z ^ 1 } \\right | _ i ^ { ( m ) } } { | z ^ 0 - z ^ 1 | ^ { \\tau - [ \\tau ] } } \\leq C \\ | p | _ { C ^ \\tau S ^ m _ { 1 , 0 } } ^ { i } . \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} \\kappa _ X & = \\varepsilon ^ { - 1 } \\frac { \\kappa \\kappa _ \\theta } { 4 } & \\alpha & = 4 \\varepsilon ^ { - 1 } \\frac { \\norm { \\nabla _ x V } _ \\infty } { \\kappa \\kappa _ \\theta } + \\varepsilon ^ { - 1 } \\alpha _ \\theta . \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} \\dot { \\theta } & = \\left ( \\frac { x } { 2 } - \\frac { n - 1 } { x } \\right ) \\sin \\theta + \\left ( \\frac { n - 1 } { y } - \\frac { y } { 2 } \\right ) \\cos \\theta + \\lambda \\\\ & \\leq \\left ( n - 1 \\right ) \\left ( 1 - \\tan \\alpha \\right ) \\frac { \\dot { y } } { y } \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{align*} = ( - 1 ) ^ { ( r + s ) t } z _ { 2 } ^ { - 1 } \\delta \\left ( \\frac { z _ { 1 } - z _ { 0 } } { z _ { 2 } } \\right ) e ^ { z _ 2 L ( - 1 ) } Y ( w , - z _ 2 ) Y ( u , \\ z _ { 0 } ) v , \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{align*} R _ { T ^ { ( 1 ) } } ( \\tau ) = \\left \\{ \\begin{array} { l l } N = \\frac { 2 ( p ^ m - 1 ) } { p - 1 } & \\mbox { i f ~ } \\tau = 0 \\\\ N _ 1 : = - 2 p ^ { m - 1 } + \\frac { 2 ( p ^ { m - 1 } - 1 ) } { p - 1 } & \\mbox { i f ~ } \\tau = \\frac { N } { 2 } \\\\ N _ 2 : = \\frac { 2 ( p ^ { m - 2 } - 1 ) } { p - 1 } & \\mbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{align*} B & = T \\times N \\\\ B _ { t } & = T _ { t } \\times N + T \\times N _ { t } \\\\ & = - \\kappa _ { s } T + F ( s , t ) N = - \\kappa _ { s } T + \\Big ( \\tau ^ { 2 } - \\frac { \\kappa _ { s s } } { \\kappa } \\Big ) N . \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{align*} \\chi \\bigl ( M \\bigr ) = \\sum _ { i = 1 } ^ k \\ , ( - 1 ) ^ j n ( j ) , \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} [ y _ 1 , y _ 2 , \\dots , y _ n ] = 0 \\mbox { f o r a l l } y _ 1 , y _ n \\in X , \\ y _ 2 , \\dots , y _ { n - 1 } \\in X \\cup X ^ 2 . \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ N \\frac { 1 } { j ^ 2 } \\sim \\frac { \\pi ^ 2 } { 6 } - \\frac { 1 } { N } + \\frac { 1 } { 2 N ^ 2 } - \\sum _ { k = 1 } ^ { \\infty } \\frac { B _ { 2 k } } { N ^ { 2 k + 1 } } , \\end{align*}"}
-{"id": "6257.png", "formula": "\\begin{align*} \\mathfrak { a } ' = ( | \\mu | , | \\mu | + 1 , \\ldots , N - | \\mu | - 1 ) . \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u = \\Delta u - \\chi \\nabla \\cdot ( u \\nabla v ) + u ( a ( x , t ) - u b ( x , t ) ) , x \\in \\R ^ N , \\cr 0 = \\Delta v - \\lambda v + \\mu u , x \\in \\R ^ N , \\end{cases} \\end{align*}"}
-{"id": "3209.png", "formula": "\\begin{align*} [ \\ast \\kappa ] = 0 \\in H ^ 4 ( B ) . \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} t \\ast | = | \\ast t = t \\end{align*}"}
-{"id": "2162.png", "formula": "\\begin{align*} \\tilde { C } _ d : = \\frac { C _ d } { 1 2 ( 1 + C _ d ) } . \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} & \\pi = i ( i - 1 ) \\cdots i ' n \\gamma ^ { ( 1 ) } ( n - 1 ) \\gamma ^ { ( 2 ) } \\cdots \\gamma ^ { ( d - 1 ) } ( n - d + 1 ) \\gamma ^ { ( d ) } ( n - d - 1 ) \\cdots \\gamma ^ { ( e - 1 ) } ( n - e ) \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\gamma ^ { ( e ) } ( n - d ) ( n - e - 1 ) \\cdots ( i + 1 ) , \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} { } { \\vert { A ( T ) } \\vert } = 0 . \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} I _ { \\alpha , s } ( x ) = \\int _ { \\mathbf { B } } \\frac { ( 1 - | y | ^ { 2 } ) ^ { \\alpha - 1 } } { [ x , y ] ^ { n + \\alpha + s - 1 } } d v ( y ) , \\enspace | x | < 1 , \\end{align*}"}
-{"id": "7128.png", "formula": "\\begin{align*} e _ 1 = \\wp \\left ( \\frac { \\omega _ 2 } { 2 } \\right ) > e _ 2 = \\wp \\left ( \\frac { \\omega _ 1 + \\omega _ 2 } { 2 } \\right ) > e _ 3 = \\wp \\left ( \\frac { \\omega _ 1 } { 2 } \\right ) . \\end{align*}"}
-{"id": "6743.png", "formula": "\\begin{align*} x ( y z \\cdot x ) = x ( y x ^ { \\lambda } \\cdot x ) \\cdot z x \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} a + b = a ' + b ' , v = \\frac { s _ g } { 1 2 } \\in \\R . \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} - R _ F \\equiv ( \\sqrt { 2 } i \\bar { a } , \\sqrt { 2 } i a , v + w - k ) , R _ F ^ T + \\beta _ 0 \\equiv - \\begin{pmatrix} \\sqrt { 2 } i \\bar { b } \\\\ \\sqrt { 2 } i b \\\\ \\frac { u + v - k } { 2 } \\end{pmatrix} \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{align*} f = 1 + v - 2 m \\rho \\tilde \\chi ( f ) \\bigl ( \\log \\rho - \\log ( 1 - 2 m \\rho ) \\bigr ) . \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} P ( \\cdot ) = \\int \\limits _ { M M [ \\Delta _ { 1 , 1 } ^ { ( h ) } ] } Q ( \\cdot ) \\mu ( d Q ) . \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{align*} \\lim _ { q \\rightarrow \\underline { q } ^ { + } } u ( q ) = \\underline { u } \\not \\equiv 0 \\quad \\mbox { i n } \\ C _ { 0 } ^ { 1 + \\theta } ( \\overline { \\Omega } ) . \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{align*} { } e ^ { - \\alpha ^ * } = e ^ { \\beta ^ * N } - 1 = e ^ v - 1 \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{align*} 1 < p < \\frac { n + 1 } { n + 1 - \\alpha } \\frac { 1 } { q } = \\frac { 1 } { p } + \\frac { \\alpha } { n + 1 } - 1 . \\end{align*}"}
-{"id": "2815.png", "formula": "\\begin{align*} K _ { \\varepsilon } ( x ) : = \\frac { 1 } { \\varepsilon ^ d } K \\left ( \\frac { x } { \\varepsilon } \\right ) . \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} ( \\pi _ { j , k } ) ( p ) _ m ( \\pi _ { i , m - n } ) & = b ( j _ k ) \\dots b ( j _ { m - 1 } ) ( p ) _ m ( \\pi _ { i , m - n } ) \\\\ & = ( p ) _ k b ( i _ { k - n } ) \\dots b ( i _ { m - n - 1 } ) ( \\pi _ { i , m - n } ) \\\\ & = ( p ) _ k ( \\pi _ { i , k - n } ) \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} { m \\choose i } _ q = & \\ , { m \\choose m - i } _ p , \\\\ { m \\choose i } _ q = & \\ , q ^ i { m - 1 \\choose i } _ q + { m - 1 \\choose i - 1 } _ q , \\\\ { m \\choose i } _ q = & \\ , { m - 1 \\choose i } _ q + q ^ { m - i } { m - 1 \\choose i - 1 } _ q \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} \\Re ( y ) = \\frac { y - \\theta ( y ) } { 2 } \\in i \\mathfrak { k } \\mathrm { a n d } \\Im ( y ) = \\frac { y + \\theta ( y ) } { 2 } \\in \\mathfrak { k } . \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} \\{ x , y \\} = P _ z \\ , , \\ \\{ y , z \\} = P _ x \\ , , \\ \\{ z , x \\} = P _ y \\ , . \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 1 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{align*} \\varphi _ X \\colon C / \\Delta \\times X ^ \\Delta \\longrightarrow \\left ( U ( p ^ n - 1 ) \\times _ { \\Sigma _ { p ^ n } } X \\right ) ^ \\Delta \\ , , \\ , \\varphi _ X ( c \\Delta , x ) : = [ c , x ] , \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} \\Lambda _ { 2 T } & = \\{ ( t , x ) \\in \\Lambda : 0 \\leq t \\leq T , - t \\leq x \\leq t \\} \\\\ & \\cup \\{ ( t , x ) \\in \\Lambda : T + 1 \\leq t \\leq 2 T , t - 2 T \\leq x \\leq - t + 2 T \\} . \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} Q _ 1 & \\approx \\frac { 4 ( \\lambda _ c \\pi ) ^ { t } } { ( t - m - 1 ) ! ( m - 1 ) ! } \\sum ^ { t - m - 1 } _ { p = 0 } ( - 1 ) ^ p { t - m - 1 \\choose p } \\\\ & \\times \\sum ^ { N } _ { l = 1 } \\frac { \\pi \\left ( \\tau _ 2 - \\tau _ 1 \\right ) } { 2 N } f _ m \\left ( \\frac { \\tau _ 2 - \\tau _ 1 } { 2 } w _ l + \\frac { \\tau _ 2 + \\tau _ 1 } { 2 } \\right ) \\sqrt { 1 - w _ l ^ 2 } . \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} \\beta _ 4 = 3 s + 2 - 2 \\beta _ 1 + \\beta _ 2 2 \\beta _ 4 - \\beta _ 3 = 3 s + 1 - \\beta _ 1 . \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} & \\nabla _ { \\rm t r } ^ * \\nabla _ { \\rm t r } \\phi = - \\sum _ a \\nabla _ { E _ a } \\nabla _ { E _ a } \\phi + \\nabla _ { \\kappa _ B ^ \\sharp } \\phi , \\\\ & A _ Y ( \\phi ) = \\mathcal { L } _ Y \\phi - \\nabla _ Y \\phi , Y \\in T M , \\\\ & F ( \\phi ) = \\sum _ { a , b } \\theta ^ a \\wedge E _ b \\lrcorner R ^ \\nabla ( E _ b , E _ a ) \\phi . \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} \\int _ { E ^ { c } } S _ 1 ^ { i , j } ( b ) ( x ) ^ { 2 } d x & \\leq 2 ^ { j } \\int _ { E ^ { c } } \\sum _ { J } \\sum _ { R \\in \\mathcal { D } } \\left ( \\sum _ { P \\in ( R ) _ i } | \\widehat { b _ J } ( P ) | \\right ) ^ { 2 } \\frac { 1 _ R ( x ) } { | R | } d x \\\\ & = 2 ^ { j } \\sum _ { J } \\int _ { E ^ { c } } \\sum _ { \\substack { | R | > | J | \\\\ R \\supset J } } \\left ( \\sum _ { P \\in ( R ) _ i } | \\widehat { b _ J } ( P ) | \\right ) ^ { 2 } \\frac { 1 _ R ( x ) } { | R | } d x . \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} \\int _ { \\R ^ { N } } ( F ( v _ { n } ) - F ( u _ { n } ) + F ( u ) ) \\ , d x = o _ { n } ( 1 ) \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} \\begin{aligned} [ b ] \\int _ 0 ^ { 1 } \\cdots \\int _ { 0 } ^ { 1 } \\prod _ { j \\in [ t _ 2 ] } x _ j ^ { - \\tau + \\zeta _ j } \\prod _ { i = t _ 2 + 1 } ^ { t _ 1 + t _ 2 } \\tilde { h } ( i , \\boldsymbol { x } ) \\dd x _ { t _ 2 } \\cdots \\dd x _ 1 , \\end{aligned} \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{align*} & p ( 2 + p ) ( 4 + p ) \\cdots ( j - 2 + p ) \\\\ & = 2 ^ { j / 2 } ( 0 + \\frac { p } { 2 } ) ( 1 + \\frac { p } { 2 } ) ( 2 + \\frac { p } { 2 } ) \\cdots ( \\frac { j } { 2 } - 1 + \\frac { p } { 2 } ) \\\\ & = 2 ^ { j / 2 } \\prod _ { k = 1 } ^ { j / 2 } \\Big ( k - 1 + \\frac { p } { 2 } \\Big ) = 2 ^ { j / 2 } \\frac { \\Gamma ( j / 2 + p / 2 ) } { \\Gamma ( p / 2 ) } \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} \\bar n - n \\ge N ( \\lambda ^ 2 - \\rho ^ 2 ) - a _ 0 N ^ { 2 / 3 } - 1 = N ( \\lambda - \\rho ) ( \\rho + \\lambda ) - a _ 0 N ^ { 2 / 3 } - 1 \\ge \\rho r N ^ { 2 / 3 } , \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} X _ \\Gamma ( \\underline { x } ) = X _ { \\Gamma } ( x _ 1 , x _ 2 , \\cdots ) = \\sum _ { \\textup { p r o p e r } \\kappa : V \\to \\{ 1 , 2 , \\ldots \\} } x ^ \\kappa \\end{align*}"}
-{"id": "4935.png", "formula": "\\begin{align*} x _ { x _ 0 } ^ T ( t ) Q x _ { x _ 0 } ( t ) - x _ 0 ^ T Q x _ 0 = & \\int _ 0 ^ t x _ { x _ 0 } ^ T ( s ) Q d x _ { x _ 0 } ( s ) + \\int _ 0 ^ t d x _ { x _ 0 } ^ T ( s ) Q x _ { x _ 0 } ( s ) , \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{align*} \\boxed { [ \\gamma _ { a b } ] = \\sigma ( \\gamma _ { a b } ) [ W ^ u ( a ) ] \\wedge [ W ^ s ( b ) ] } \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} G _ 1 = \\lbrace g : \\mathbb { R } \\rightarrow \\mathbb { R } , g ( x ) = a \\cdot x + b , a \\neq 0 , b \\in \\mathbb { R } \\rbrace . \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} H _ - ( u ' ( r ) ) + ( N - 1 ) \\int _ 0 ^ r \\frac { ( u ' ( s ) ) ^ 2 } { s \\sqrt { 1 - ( u ' ( s ) ) ^ 2 } } d s = G ( \\xi ) - G ( u ( r ) ) , \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) + L _ t u ( t , x ) + f ( t , x ) & = 0 , \\\\ u ( T , x ) & = 0 \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} y _ k ' = y _ k y _ m ' = y _ m + e _ 1 + e _ 2 = x . \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{align*} d q _ t = \\tilde \\gamma ^ { - 1 } ( t ) ( - \\nabla _ q V ( t , q _ t ) + \\tilde F ( t , q _ t ) ) d t + ( \\tilde \\gamma ^ { - 1 } \\sigma ) ( t , q _ t ) d W _ t \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{align*} \\ell t r ( A ) = 2 \\ell ^ 2 t r ( A ^ 2 ) + 2 \\ell \\sum \\limits _ { i , j = 1 } ^ { d } ( ( A R ) ^ j ( A R ) ^ i ) _ { ( i , j ) } - \\ell \\sum \\limits _ { i = 1 } ^ { d } t r ( A ^ 2 ( R ^ i ) ^ 2 ) . \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} \\mathrm { d } X _ t = \\big ( A X _ t + F ( X _ t ) \\big ) \\ , \\mathrm { d } t + B ( X _ t ) \\ , \\mathrm { d } W _ t , t \\in ( 0 , T ] , X _ 0 = \\xi , \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{align*} = - 1 + z + z ^ { 1 2 } + z ^ { 3 1 } + \\ldots = G _ { 1 2 } ( z ) + z ^ { 3 1 } + \\ldots . \\end{align*}"}
-{"id": "7464.png", "formula": "\\begin{align*} & E \\left [ \\int _ s ^ t \\partial _ r \\beta ( r , q _ r ^ m ) \\| z _ r ^ m \\| ^ 2 d r \\right ] \\\\ = & E \\left [ \\int _ s ^ t \\partial _ r \\beta ( r , q _ r ) \\int \\| z \\| ^ 2 h ( r , q _ r , z ) d r \\right ] + O ( m ^ { 1 / 2 } ) \\\\ = & E \\left [ n \\int _ s ^ t \\beta ^ { - 1 } ( r , q _ r ) \\partial _ r \\beta ( r , q _ r ) d r \\right ] + O ( m ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} \\mathrm { l o c } _ { x } ( g , \\mathcal { S } _ { V } ) = ( - 1 ) ^ { \\left \\langle 2 \\rho , \\nu _ { x } \\right \\rangle } \\mathrm { r a n k } _ { \\Lambda } V [ \\nu _ { x } ] . \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{align*} \\forall \\psi _ 1 \\in \\Omega ^ k ( M ) , \\quad \\pi _ { z _ 0 } ^ { ( k ) } ( \\psi _ 1 ) = \\sum _ { a \\in ( f ) } \\sum _ { j = 1 } ^ { m _ a ^ { ( k ) } ( z _ 0 ) } \\left ( \\int _ M \\psi _ 1 \\wedge S _ j ( a , z _ 0 ) \\right ) U _ j ( a , z _ 0 ) . \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} O _ { n - 1 } ( \\tau _ n ) : = \\{ \\tau _ { n - 1 } ^ \\prime \\in D _ { n - 1 } : \\rho ( \\tau _ n , \\tau _ { n - 1 } ^ \\prime ) \\leq 2 ^ { - n } \\} \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{align*} f _ t ( m , r ) \\le m \\ , \\frac { \\binom { r + 1 } { t } } { \\binom { r + 1 } { 2 } } , \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} \\lambda _ { ( n + 1 , k ) } = & \\ , ( 1 - q ^ { k + n } r ) ( k + n + 1 ) _ q \\lambda _ { ( n , k ) } , \\\\ \\beta _ { ( i , m , k ) } = & \\ , ( q ^ { m + 2 k - i } r - r ^ { - 1 } ) \\beta _ { ( i - 1 , m , k ) } , \\\\ \\beta _ { ( i , m + 1 , k ) } = & \\ , ( q ^ { m + 2 k } r - r ^ { - 1 } ) \\beta _ { ( i - 1 , m , k ) } . \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{align*} \\Phi ( x , c ) = \\min _ { y \\in K - G ( x ) } \\left ( - p ( x ) \\langle \\lambda ( x ) , y \\rangle + \\frac { c } { 2 } \\| y \\| ^ 2 \\right ) . \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} \\mathcal L _ { s \\partial } ( \\psi ) = ( \\lambda _ s \\circ \\partial _ M ) \\circ \\psi - \\psi \\circ ( \\lambda _ s \\circ \\partial ^ \\bullet ) = \\lambda _ s \\circ \\mathcal L _ \\partial ( \\psi ) \\ , . \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{align*} \\theta ^ j _ { m + l } ( x ) = \\zeta ^ { j ( m + l ) } = [ \\zeta ^ m \\theta _ { \\sigma ( 0 ) } ( y ) ] ^ j = [ \\theta _ { \\sigma ( m ) } ( y ) ] ^ j = \\theta _ { \\sigma ( m ) } ( y ^ j ) = \\theta _ { \\sigma ( m ) } ( x ) , \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{align*} E = \\mbox { s p a n } \\{ \\mathbf { v } _ { 1 } , . . . , \\mathbf { v } _ { s } , \\mathbf { v } _ { N + 1 } , . . . \\mathbf { v } _ { N + s } \\} , \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} u _ t & \\geq \\Delta u - \\chi \\nabla v \\cdot \\nabla u + u ( a _ { \\inf } - b _ { \\sup } \\| u _ 0 \\| _ { \\infty } e ^ { a _ { \\sup } T } ) . \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} T _ f : = f ^ { - 1 } ( 1 ) F _ f : = f ^ { - 1 } ( 0 ) . \\end{align*}"}
-{"id": "3750.png", "formula": "\\begin{align*} \\mathcal { D } _ n = \\left \\{ \\left [ \\frac { j } { 2 ^ n } , \\frac { j + 1 } { 2 ^ n } \\right ) : j \\in \\Z \\right \\} . \\end{align*}"}
-{"id": "2769.png", "formula": "\\begin{align*} V ( F ) & = V _ 0 ^ p ( F ) + V _ p ^ 1 ( F ) \\\\ & = V _ { q - 1 } ^ 0 ( F \\circ \\tilde { u } ) + V _ 0 ^ q ( F \\circ \\tilde { u } ) \\\\ & = V _ q ^ 1 ( F \\circ u ) + V _ 0 ^ q ( F \\circ u ) \\\\ & = V ( F \\circ u ) . \\end{align*}"}
-{"id": "6558.png", "formula": "\\begin{align*} & \\left \\{ v : \\exists \\ , t \\in I _ j ^ { ( k ) } \\textrm { w i t h } \\delta ( \\phi _ t ( v ) ) = T _ k ^ { - \\xi } \\right \\} \\\\ \\subset & \\left \\{ v : \\delta ( \\phi _ { a _ j ^ { ( k ) } } ( v ) ) \\in \\left [ \\frac { 1 } { 2 } T _ k ^ { - \\xi } , 2 T _ k ^ { - \\xi } \\right ] \\right \\} . \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{align*} \\phi _ N ( \\xi ) = \\int _ { N e ^ { - A _ N } } ^ N x ^ { - i \\xi } \\left ( 1 - \\frac { x } { N } \\right ) ^ { N - 1 } d x + O \\left ( \\frac { 1 } { N } \\right ) . \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} I _ 1 ( N ) = \\int _ 0 ^ { U ( N ; \\alpha ) } e ^ { - x } \\left ( 1 - \\frac { \\ln x } { \\ln N } \\right ) ^ r d x \\ , + \\ , o \\left ( \\frac { 1 } { N ^ { \\theta } } \\right ) \\ ; \\theta \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} u \\Delta ( b ) \\ , L _ { \\tilde { \\Delta } ( m ) } \\left ( \\sum _ { i = 1 } ^ N \\Delta ( a _ i ) z _ i \\right ) & = u \\Delta ( b ) \\sum _ { i = 1 } ^ N \\Delta ( m a _ i ) z _ i = u \\sum _ { i = 1 } ^ N \\Delta ( b m a _ i ) z _ i \\\\ & = u \\Delta ( b m ) \\sum _ { i = 1 } ^ N \\Delta ( a _ i ) z _ i = 0 . \\end{align*}"}
-{"id": "7134.png", "formula": "\\begin{align*} y ( \\omega ) = \\left \\{ \\begin{array} { l l } \\displaystyle b _ { 4 } + \\frac { E ' ( b _ { 4 } ) } { \\wp ( \\omega - \\omega _ 3 / 2 ) - \\frac { E '' ( b _ { 4 } ) } { 6 } } & b _ { 4 } \\neq [ 1 \\ ! : \\ ! 0 ] , \\medskip \\\\ \\displaystyle \\frac { \\wp ( \\omega - \\omega _ 3 / 2 ) - \\beta _ { 2 } / 3 } { \\beta _ { 3 } } & , \\end{array} \\right . \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} A : = \\Gamma ( E , \\mathcal O _ E ( * 0 ) ) = \\mathbf k [ x , y ] / ( y ^ 2 - 4 x ^ 3 + g _ 2 x + g _ 3 ) . \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} F ( n ) = 0 . \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} \\frac { 1 } { t } \\int _ 0 ^ t ( t - s ) ^ { - \\mu } s ^ { - \\nu } d s & = t ^ { - \\mu - \\nu } C _ { \\mu , \\nu } \\ \\end{align*}"}
-{"id": "6336.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\hbox { M i n i m i z e } \\ , | | A U C - B | | ^ 2 \\\\ U ^ T U = \\mathbb { I } _ p \\end{array} \\right . , \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} \\langle \\nabla g _ i , \\nabla g _ j \\rangle _ { Y \\times \\R } \\circ \\Phi & = \\langle \\nabla ( g _ i \\circ \\Phi ) , \\nabla ( g _ j \\circ \\Phi ) \\rangle _ X , \\\\ \\langle \\nabla h _ i , \\nabla h _ j \\rangle _ { Y \\times \\R } \\circ \\Phi & = \\langle \\nabla ( h _ i \\circ \\Phi ) , \\nabla ( h _ j \\circ \\Phi ) \\rangle _ X \\end{align*}"}
-{"id": "6512.png", "formula": "\\begin{align*} h _ d ( \\theta ) : = { 4 \\over ( d - 2 ) ! } \\int _ 0 ^ \\infty { \\max \\big ( - \\Re \\rho ( t e ^ { i \\theta } ) , 0 \\big ) \\over t ^ { d + 1 } } d t 0 < \\theta < \\pi , \\end{align*}"}
-{"id": "7140.png", "formula": "\\begin{align*} b _ { 1 } = \\iota _ 1 ( y ) ( x - \\tau ( x ) ) b _ { 2 } = x ( y - \\iota _ 1 ( y ) ) \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} T _ { x } \\rightarrow 1 \\quad \\quad \\eta \\rightarrow \\infty \\\\ N _ { y } + i N _ { z } = - i \\big ( B _ { y } + i B _ { z } \\big ) = e ^ { i ( \\tau _ { 0 } \\zeta + \\sigma ( t ) ) } \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} \\Delta _ B | \\phi | ^ 2 = 2 \\overline \\square _ B | \\phi | ^ 2 - i J \\kappa _ B ^ \\sharp ( | \\phi | ^ 2 ) . \\end{align*}"}
-{"id": "1194.png", "formula": "\\begin{align*} \\widehat { \\delta } = t _ i ^ { - 1 } \\log ^ { \\frac { 1 - \\nu } { 2 } } ( t _ i ) \\delta . \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} \\begin{cases} u _ { t } ( x , t ) + v _ { x } ( x , t ) + v _ { x x x } ( x , t ) + \\lambda u ( x , t ) = \\sum _ { i = 1 } ^ 3 \\Psi _ i ( x , t ) , & , \\\\ v _ { t } ( x , t ) + u _ { x } ( x , t ) + u _ { x x x } ( x , t ) + \\lambda v ( x , t ) = \\sum _ { i = 1 } ^ 3 \\Phi _ i ( x , t ) , & , \\\\ u ( 0 , t ) = u ( L , t ) = u _ { x } ( 0 , t ) = 0 , & , \\\\ v ( 0 , t ) = v ( L , t ) = v _ { x } ( L , t ) = 0 , & , \\end{cases} \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} \\displaystyle \\limsup _ { h \\to 0 + } \\dfrac { \\sup _ { t \\in T } \\sup _ { | s - t | _ \\infty \\leq h } | X ( t ) - X ( s ) | } { h ^ { ( H - \\theta _ { 2 } / \\alpha ) } ( \\log { 1 / h } ) ^ { 1 / \\gamma } } = 0 \\ ; \\ ; \\ ; \\ ; { \\rm a . s . } , \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} K = \\tilde { p _ 1 } ^ 2 - \\tilde { p _ 1 } \\tilde { p _ 2 } + \\tilde { p _ 2 } ^ 2 + \\frac { 4 } { 3 } \\tilde { q _ 1 } ^ 2 + \\frac { 4 } { 3 } \\tilde { q _ 1 } \\tilde { q _ 2 } + \\frac { 4 } { 3 } \\tilde { q _ 2 } ^ 2 + \\frac { 4 } { 9 } \\sqrt { 3 } \\tilde { q _ 1 } ^ 2 \\tilde { q _ 2 } + \\frac { 4 } { 9 } \\sqrt { 3 } \\tilde { q _ 1 } \\tilde { q _ 2 } ^ 2 . \\end{align*}"}
-{"id": "3869.png", "formula": "\\begin{align*} g ( y ) - g ( x ) = \\int _ 0 ^ 1 H ( x + t ( y - x ) ) ( y - x ) d t . \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{align*} - b \\mathfrak { R e } ( \\dot { c } _ { 0 } ( 0 ) ) + \\mathfrak { R e } ( \\dot { c } _ { 1 } ( 0 ) ) = 0 , \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} \\tilde { \\mathbf { y } } _ { } = \\mathbf { \\Omega } \\mathbf { y } , \\end{align*}"}
-{"id": "6156.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } \\frac { 1 } { \\widetilde { C } } \\dot { \\rho } ( t ) \\left ( \\frac { \\rho ( t ) } { t } \\right ) ^ { - \\frac { 3 } { 2 } } e ^ { C \\left ( \\frac { 1 } { l _ 1 t } - \\frac { 1 } { \\rho ( t ) } \\right ) } = 1 . \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} 2 ^ { \\mathrm { e x } _ r ( n , \\mathcal { H } ) } \\leq | \\mathrm { F o r b } _ r ( n , \\mathcal { H } ) | \\leq \\sum _ { i \\leq \\mathrm { e x } _ r ( n , \\mathcal { H } ) } \\binom { \\binom { n } { r } } { i } \\leq 2 n ^ { r \\cdot \\mathrm { e x } _ r ( n , \\mathcal { H } ) } . \\end{align*}"}
-{"id": "8545.png", "formula": "\\begin{align*} \\mathcal { L } _ { X _ { - \\kappa _ 0 / 2 } } \\omega _ 0 = d ( \\iota _ { X _ { - \\kappa _ 0 / 2 } } \\omega _ 0 ) = - \\ , d ^ 2 ( \\kappa _ 0 / 2 ) = 0 \\rlap { . } \\end{align*}"}
-{"id": "477.png", "formula": "\\begin{align*} E _ { \\chi _ 1 , \\chi _ 2 , k } ( z ) = \\frac { 1 } { 2 } \\sum _ { ( c , d ) = 1 } \\frac { \\chi _ 1 ( c ) \\chi _ 2 ( d ) } { ( c q _ 2 z + d ) ^ k } , \\end{align*}"}
-{"id": "1904.png", "formula": "\\begin{align*} & \\sum _ { a = 2 } ^ { n - 5 } \\sum _ { b = a + 3 } ^ { n - 2 } \\sum _ { m = 2 } ^ { b - 1 - a } \\sum _ { j = 1 } ^ { m - 1 } \\left [ \\binom { m } { j } - 1 \\right ] = \\sum _ { a = 2 } ^ { n - 5 } \\sum _ { b = a + 3 } ^ { n - 2 } \\sum _ { m = 0 } ^ { b - 1 - a } ( 2 ^ m - m - 1 ) \\\\ & = \\sum _ { a = 2 } ^ { n - 5 } \\sum _ { b = a } ^ { n - 2 } \\left [ 2 ^ { b - a } - 1 - \\binom { b - a + 1 } { 2 } \\right ] = \\sum _ { n = 2 } ^ { n - 1 } \\left [ 2 ^ { n - 1 - a } - ( n - a ) - \\binom { n - a } { 3 } \\right ] \\\\ & = 2 ^ { n - 2 } - 1 - \\binom { n - 1 } { 2 } - \\binom { n - 1 } { 4 } . \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} \\sigma ( A ^ V ) = \\bigcup _ { q \\in [ 0 , 1 / 2 ] } \\bigcup _ { m \\in \\N ^ * } \\{ \\varepsilon ^ V _ { q , m } \\} . \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} h _ { 1 } = ( h _ { 1 } ^ { i } + \\mathbb { Z } ) _ { i \\in I } , \\dots , h _ { n } = ( h _ { n } ^ { i } + \\mathbb { Z } ) _ { i \\in I } \\in ( \\mathbb { R } / \\mathbb { Z } ) ^ { I } \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{align*} A ( \\epsilon ) = \\frac { b } { 1 6 c } \\sum _ { m = 1 } ^ \\infty { \\frac { c ^ m } { m ^ 2 } \\epsilon ^ m } . \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} \\frac { \\partial P } { \\partial \\lambda } ( 1 , \\lambda ) = 0 \\ \\ { \\rm f o r \\ e v e r y } \\ \\lambda \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 7 & 1 & 7 \\\\ 0 & 9 & 7 & 1 \\\\ 0 & 6 & 9 & 7 \\end{pmatrix} . \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} ( x ^ { q ^ 2 } , y ^ { q ^ 2 } ) - [ t ] ( x ^ q , y ^ q ) + [ q ] ( x , y ) = o . \\end{align*}"}
-{"id": "4546.png", "formula": "\\begin{align*} \\Pr [ A _ 1 \\cap A _ 2 ] = \\sum _ { i } \\Pr [ Z = i ] = \\Pr [ Z \\leq \\mu ] . \\end{align*}"}
-{"id": "808.png", "formula": "\\begin{align*} \\chi ( X ) = \\int _ { X } { \\mathcal P } ( R ) \\ , d X + \\int _ { Z } { \\mathcal Q ( R , A ) } \\ , d Z , \\end{align*}"}
-{"id": "8984.png", "formula": "\\begin{align*} f - \\lambda H f = h . \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} Z = A D A ^ t . \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} \\dim R \\ = \\ 2 t - 2 , \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} Q ( 0 , 0 , \\pm | z | ) = | z | ^ { 1 / 4 } \\int _ { \\infty e ^ { 9 \\pi i / 1 0 } } ^ { \\infty e ^ { \\pi i / 1 0 } } e ^ { i | z | ^ { 5 / 4 } f _ { \\pm } ( t ) } d t . \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} \\mathcal { R } ( [ \\theta \\sigma ( e _ { \\alpha _ { 1 , 3 } } ) , e _ { \\alpha _ { 1 , 3 } } ] = \\begin{pmatrix} 0 & 0 & 1 / 2 \\\\ 0 & 0 & 0 \\\\ 1 / 2 & 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} \\hat { C } ( \\Q ) = \\{ ( 0 : 0 : 1 ) , ( 0 : 1 : 0 ) , ( - 1 : 0 : 1 ) , ( 4 : 0 : 1 ) \\} \\cong \\Z / 2 \\Z \\times \\Z / 2 \\Z . \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{align*} & \\mathbb { P } [ X ^ n = \\mathbf { x } _ 1 , Y ^ n = \\mathbf { y } _ 1 ] = \\mathbb { P } [ X ^ n = \\mathbf { x } _ 2 , Y ^ n = \\mathbf { y } _ 2 ] = p _ { 1 1 } ^ { n / 2 } p _ { 2 2 } ^ { n / 2 } \\\\ & \\mathbb { P } [ X ^ n = \\mathbf { x } _ 2 , Y ^ n = \\mathbf { y } _ 1 ] = \\mathbb { P } [ X ^ n = \\mathbf { x } _ 1 , Y ^ n = \\mathbf { y } _ 2 ] = p _ { 1 2 } ^ { n / 2 } p _ { 2 1 } ^ { n / 2 } . \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} \\Delta _ B = d _ B \\delta _ B + \\delta _ B d _ B , \\Delta _ T = d _ T \\delta _ T + \\delta _ T d _ T , \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} I _ 2 ( \\alpha ^ { ( 1 ) } , \\alpha ^ { ( 2 ) } , \\dots , \\alpha ^ { ( N ) } ) = \\sum \\limits _ { k _ 1 = 1 } ^ { n _ 1 } \\sum \\limits _ { k _ 2 = 1 } ^ { n _ 2 } \\dots \\sum \\limits _ { k _ { N } = 1 } ^ { n _ N } B ( k _ 1 , k _ 2 , \\dots , k _ { N } ) \\alpha _ { k _ 1 } ^ { ( 1 ) } \\alpha _ { k _ 2 } ^ { ( 2 ) } \\dots \\alpha _ { k _ { N } } ^ { ( N ) } . \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} d q ^ \\prime _ t = & ( \\tilde \\gamma ^ T ) ^ { - 1 } ( t ^ * , q ^ \\prime _ t ) \\left ( - \\partial _ t \\psi ( t ^ * , q ^ \\prime _ t ) - \\nabla _ q V ( t ^ * , q _ t ^ \\prime ) + \\tilde F ( t ^ * , q ^ \\prime _ t , \\psi ( t ^ * , q ^ \\prime _ t ) ) \\right ) d t \\\\ & + \\tilde S ( t ^ * , q _ t ^ \\prime ) d t + \\left ( S ^ \\prime ( t ^ * , q ^ \\prime _ t ) - S ( t ^ * , q _ t ^ \\prime ) \\right ) d t + \\tilde \\gamma ^ { - 1 } ( t ^ * , q ^ \\prime _ t ) \\sigma ( t ^ * , q ^ \\prime _ t ) \\circ d \\tilde W _ t , \\end{align*}"}
-{"id": "270.png", "formula": "\\begin{align*} W _ { 2 , q } ( x , y ) = x ^ 2 + ( q - 1 ) y ^ 2 \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int _ { \\Omega } \\varphi ( t ) ^ 2 \\ ; \\d x + \\int _ 0 ^ t \\int _ { \\Omega } \\lvert \\nabla \\varphi ( s ) \\rvert ^ 2 \\ ; \\d x \\ ; \\d s = 2 \\int _ 0 ^ t \\int _ { \\Omega } \\lvert \\nabla d ( s ) \\rvert ^ 2 \\varphi ( s ) ^ 2 \\ ; \\d x \\ ; \\d s . \\end{align*}"}
-{"id": "1391.png", "formula": "\\begin{align*} \\tilde \\tau _ p ( x , y ) = \\log \\Big ( 1 + \\frac { d ( x , y ) } { \\sqrt { d ( x , p ) d ( y , p ) } } \\Big ) = \\log \\frac { \\mu _ p ( x , y ) } { \\sqrt { d ( x , p ) d ( y , p ) } } . \\end{align*}"}
-{"id": "5109.png", "formula": "\\begin{align*} c ' = 2 \\ell t r ( A ) ; \\end{align*}"}
-{"id": "4580.png", "formula": "\\begin{align*} \\nabla _ { \\bar V _ a } Z = 0 1 \\le a \\le n . \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} \\tilde \\gamma ( \\tilde \\gamma ^ T ) ^ { - 1 } \\gamma \\tilde \\gamma ^ { - 1 } \\tilde \\gamma ^ T = \\gamma . \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{align*} w t ( C _ { 3 } ) & = \\biggl ( p ^ { e - 2 } - \\biggl ( \\frac { - 1 } { p } \\biggr ) p ^ { m + d - 1 } \\biggr ) + \\sum _ { i = 1 } ^ { p - 1 } \\biggl ( p ^ { e - 2 } - \\biggl ( \\frac { i ^ { 2 } - c ^ { 2 } } { p } \\biggr ) p ^ { m + d - 1 } \\biggr ) - p ^ { e - 2 } \\\\ & = ( p - 1 ) p ^ { e - 2 } - \\biggl ( \\biggl ( \\frac { - 1 } { p } \\biggr ) + L _ { p } ( c ) \\biggr ) p ^ { m + d - 1 } \\\\ & = ( p - 1 ) p ^ { e - 2 } + p ^ { m + d - 1 } . \\end{align*}"}
-{"id": "5271.png", "formula": "\\begin{align*} c _ { \\mathbb { Q } ( \\mu _ n ) } & : = b _ 1 \\cdot { } _ { c , d } z ^ { ( p ) } _ n ( f , 1 , 1 , \\alpha _ 1 , \\mathrm { p r i m e } ( n N p ) ) ^ - + b _ 2 \\cdot { } _ { c , d } z ^ { ( p ) } _ n ( f , 1 , 1 , \\alpha _ 2 , \\mathrm { p r i m e } ( n N p ) ) ^ + \\\\ & \\in \\mathrm { H } ^ 1 _ { \\mathrm { \\acute { e } t } } ( \\mathrm { S p e c } ( \\mathbb { Z } [ 1 / p , \\zeta _ n ] ) , j _ { * } T _ { \\overline { f } } ( 1 ) ) \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{align*} p _ { X Y Z | X ' Y ' } ( x , y , z | 0 , 0 ) & = a ( x ) b ( y ) p _ { X Y Z } ( x , y , z ) = q _ { X Y Z } ( x , y , z ) . \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} P ( Z _ 1 , Z _ 2 , Z _ 3 ) & = 0 \\\\ x Z _ 1 + y Z _ 2 + Z _ 3 & = 0 \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} \\tilde { u } = - \\ln \\det ( u _ { l , m } ) - \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\ln ( ( 2 \\chi _ l - u _ l ) \\alpha ^ { \\vee , l } ) + I _ H \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} C ( p _ X \\| q _ X ) & = - \\log \\left ( \\min _ { \\alpha \\in [ 0 , 1 ] } \\sum _ { x } p _ X ( x ) ^ \\alpha q _ X ( x ) ^ { 1 - \\alpha } \\right ) \\\\ & \\leq - \\log \\left ( \\sum _ { x } \\min ( p _ X ( x ) , q _ X ( x ) ) \\right ) \\\\ & = - \\log \\left ( 1 - \\| p _ X - q _ X \\| _ { T V } \\right ) \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} | u _ e ( \\hat { x } ) - u _ e ( x _ 0 ) | = | \\partial _ e \\phi _ 1 ( \\hat { x } ) - \\partial _ e \\phi _ 1 ( x _ 0 ) | \\leq \\sigma ( | \\hat { x } - x _ 0 | ) \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} \\partial _ s \\Phi _ { u , \\nu } & = - \\frac { \\xi '' ( s ) } { 2 } \\bigl ( \\partial _ { x x } \\Phi _ { u , \\nu } + \\gamma ( s ) \\bigl ( \\partial _ x \\Phi _ { u , \\nu } \\bigr ) ^ 2 \\bigr ) , \\ , \\ , ( s , x ) \\in [ 0 , u ) \\times \\mathbb { R } \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} Q ^ { 1 , 0 } = \\{ X - i J X | \\ X \\in Q \\} , Q ^ { 0 , 1 } = \\{ X + i J X | \\ X \\in Q \\} . \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} \\delta \\cdot e ^ { i \\theta } = f _ 1 ( x ( t ) ) + i \\ , f _ 2 ( x ( t ) ) , \\mbox { w h e r e } i = \\sqrt { - 1 } . \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} ( \\rho _ 1 D _ { \\rho _ 1 } - z ) b _ k = a _ k \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} d q _ t = & \\tilde \\gamma ^ { - 1 } ( t , q _ t ) \\left ( - \\partial _ t \\psi ( t , q _ t ) - \\nabla _ q V ( t , q _ t ) ) \\right ) d t \\\\ & + \\tilde S ( t , q _ t ) d t + \\tilde \\gamma ^ { - 1 } ( t , q _ t ) \\sigma ( t , q _ t ) \\circ d W _ t , \\end{align*}"}
-{"id": "3935.png", "formula": "\\begin{align*} \\xi _ { n + 1 } - \\xi _ n = g _ n + \\delta \\xi _ n \\in V _ n \\oplus V _ n ^ \\perp = V _ { n + 1 } , \\end{align*}"}
-{"id": "2414.png", "formula": "\\begin{align*} E \\left [ T ^ 2 \\right ] = E \\left [ T _ 1 ^ 2 \\right ] + O \\left ( M ^ { 3 - \\lambda + \\varepsilon } \\right ) , M \\to \\infty , \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{align*} \\cos 2 \\varphi = \\cos ^ 2 r _ j ( \\varepsilon ) - \\frac { \\sin ^ 2 r _ j ( \\varepsilon ) } { j } \\mbox { \\ a n d \\ } \\cos 2 \\varphi = \\cos ^ 2 r _ \\infty ( \\varepsilon ) , \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} \\| \\rho ( T , \\cdot ) \\| _ { L ^ 1 ( \\mathbb R ^ N ) } = \\| \\Delta \\varphi ( 0 , \\cdot ) \\| _ { L ^ 1 ( \\mathbb R ^ N ) } \\leq \\| \\rho ( 0 , \\cdot ) \\| _ { L ^ 1 ( \\mathbb R ^ N ) } = \\| \\Delta \\zeta \\| _ { L ^ 1 ( \\mathbb R ^ N ) } . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} \\begin{array} { r c l } d x \\wedge d y & = & - t ^ { - 1 } \\frac { \\partial P } { \\partial z } \\omega _ S \\\\ \\\\ d y \\wedge d z & = & - t ^ { - 1 } \\frac { \\partial P } { \\partial x } \\omega _ S \\\\ \\\\ d z \\wedge d x & = & - t ^ { - 1 } \\frac { \\partial P } { \\partial y } \\omega _ S \\ , . \\end{array} \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} \\int _ M c _ 1 ( T M ) \\cup \\omega = \\int _ M \\frac { s _ C } { 2 \\pi } = \\int _ M \\frac { 4 k - 2 v } { 2 \\pi } = \\underbrace { \\int _ M \\frac { 3 k } { 2 \\pi } } _ { \\geq 0 } + \\int _ M \\frac { k - 2 v } { 2 \\pi } > 0 . \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} \\mathcal { L } _ X \\omega ( \\zeta ) = 0 \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} \\langle ( \\xi , \\eta ) , ( d , w ) \\rangle & = \\langle d , - \\nabla ^ 2 \\langle \\overline { \\lambda } , g \\rangle ( \\overline { x } ) ( \\eta ) + \\nabla \\ ! g ( \\overline { x } ) \\mu \\rangle + \\langle \\nabla ^ 2 \\langle \\overline { \\lambda } , g \\rangle ( \\overline { x } ) ( d ) + \\nabla \\ ! g ( \\overline { x } ) u , \\eta \\rangle \\\\ & = \\langle d , \\nabla \\ ! g ( \\overline { x } ) \\mu \\rangle + \\langle \\nabla \\ ! g ( \\overline { x } ) u , \\eta \\rangle . \\end{align*}"}
-{"id": "2221.png", "formula": "\\begin{align*} A = \\begin{pmatrix} \\mathbf { a } _ 1 \\\\ \\vdots \\\\ \\mathbf { a } _ m \\\\ \\end{pmatrix} = \\begin{pmatrix} \\mathbf { a } _ 1 ^ 1 & \\cdots & \\mathbf { a } _ 1 ^ m \\\\ \\vdots & \\cdots & \\vdots \\\\ \\mathbf { a } _ m ^ 1 & \\cdots & \\mathbf { a } _ m ^ m \\\\ \\end{pmatrix} , w h e r e \\ \\mathbf { a } _ i ^ j = ( a _ { i 1 } ^ j , \\cdots , a _ { i n _ i } ^ j ) . \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} A P = A K _ { P } P = K P = G , \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{align*} \\beta _ n ( j , t ) = \\frac { \\Delta _ n ( j , t ) } { \\Delta } = \\frac { 1 } { \\Delta } \\sum _ { w = 1 } ^ { d } \\sum _ { y = 1 } ^ { m _ w } \\gamma _ { n - m _ w + y , n } ^ { ( w ) } C [ g _ { w , y } , h _ { j , t } ] , \\end{align*}"}
-{"id": "6645.png", "formula": "\\begin{align*} q _ p = c _ p \\frac { ( w _ { p - 1 } v _ { \\omega _ { i _ p } } , w g w _ p v _ { \\omega _ { i _ p } } ) } { ( w _ { p - 1 } v _ { \\omega _ { i _ p } } , w g w _ { p - 1 } v _ { \\omega _ { i _ p } } ) } , \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} 0 \\le a _ s & = | \\sigma \\cap ( S _ \\mu ( s ) \\times T _ \\mu ( t ( s ) ) ) | - 1 \\\\ & \\le | \\sigma \\cap ( S _ \\mu ( s ) \\times T _ \\mu ( s ) ) | - 1 \\\\ & = \\rho ( s , \\mu , \\lambda ) - 1 , \\end{align*}"}
-{"id": "6082.png", "formula": "\\begin{align*} ( \\bar { \\xi } \\circ \\pi ) ( \\mathfrak { a } ) = & ( \\bar { \\xi } \\circ \\pi ) ( \\mathfrak { a } ' \\odot a _ { i + 1 } ) = \\bar { \\xi } ( \\pi ( \\mathfrak { a } ' ) ) \\cdot _ { A } \\bar { \\xi } ( \\pi ( a _ { i + 1 } ) ) \\\\ = & \\bar { \\xi } ' ( \\pi ( \\mathfrak { a } ' ) ) \\cdot _ { A } \\bar { \\xi } ' ( \\pi ( a _ { i + 1 } ) ) = ( \\bar { \\xi } ' \\circ \\pi ) ( \\mathfrak { a } ' \\odot a _ { i + 1 } ) = ( \\bar { \\xi } ' \\circ \\pi ) ( \\mathfrak { a } ) . \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{align*} \\varphi _ n ( f _ 1 , \\ldots , f _ s ) = \\left ( \\frac { ( f _ 1 \\bmod m ^ n ) ( \\xi _ n ) } { m ^ n } , \\ldots , \\frac { ( f _ s \\bmod m ^ n ) ( \\xi _ n ) } { m ^ n } \\right ) , \\end{align*}"}
-{"id": "7918.png", "formula": "\\begin{align*} m ( t ) : = \\sup \\limits _ { s \\in [ 0 , t ] } \\sup \\limits _ { x \\in [ 0 , D ] } \\tilde { \\mathbb E } \\left ( | u ( s \\ , , x ) - v ( s \\ , , x ) | ^ 2 \\right ) . \\end{align*}"}
-{"id": "291.png", "formula": "\\begin{align*} d \\leq 2 \\left [ \\frac { n - 6 } { 1 2 } \\right ] + 2 . \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} \\{ g \\in R e \\ , | \\ , g p = 0 \\} \\cong \\{ g \\in k Q \\ , | \\ , g p \\in J ^ m \\} \\cong J ^ { m - i } e \\end{align*}"}
-{"id": "8176.png", "formula": "\\begin{align*} \\frac { D } { \\partial t } \\frac { \\partial H } { \\partial \\lambda } ( t , \\lambda ) = \\frac { D } { \\partial \\lambda } \\frac { \\partial H } { \\partial t } ( t , \\lambda ) , \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{align*} \\sum _ b p _ { A , B } ( a , b ) p _ { R | B } ( r | b ) = p _ { A } ( a ) q _ { R | A } ( r | a ) \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} \\int _ I | ( f _ n ^ { ( k + 1 ) } \\circ g ) \\cdot g ' - ( f _ 0 ^ { ( k + 1 ) } \\circ g ) \\cdot g ' | & = \\int _ I ( | f _ n ^ { ( k + 1 ) } - f _ 0 ^ { ( k + 1 ) } | \\circ g ) \\cdot g ' \\\\ & = \\int _ I | f _ n ^ { ( k + 1 ) } - f _ 0 ^ { ( k + 1 ) } | \\rightarrow 0 . \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} c _ 1 ( \\overline { L } ) = \\frac { i } { 2 \\pi } \\Theta \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{align*} \\begin{gathered} \\| u \\| _ { L ^ p _ { l + k , \\nu + k } } \\leq C \\left ( \\| P u \\| _ { L ^ p _ { l , \\nu } } + \\| u \\| _ { L ^ p _ { 0 , \\nu + k } } \\right ) , \\| u \\| _ { C ^ { l + k , \\alpha } _ { \\nu + k } } \\leq C \\left ( \\| P u \\| _ { C ^ { l , \\alpha } _ { \\nu } } + \\| u \\| _ { C ^ { 0 , \\alpha } _ { \\nu + k } } \\right ) , \\\\ \\| u \\| _ { C ^ { l + k , \\alpha } _ { \\nu + k } } \\leq C \\left ( \\| P u \\| _ { C ^ { l , \\alpha } _ { \\nu } } + \\| u \\| _ { L ^ 2 _ { \\nu + k } } \\right ) \\end{gathered} \\end{align*}"}
-{"id": "3601.png", "formula": "\\begin{align*} \\omega _ { z } z - \\omega _ { y } y & = ( y ' ( x ) z '' ( x ) - z ' ( x ) y '' ( x ) ) [ 1 + y _ { x } ^ { 2 } + z _ { x } ^ { 2 } ] ^ { - 3 / 2 } \\\\ \\omega _ { y } x - \\omega _ { x } z & = - z '' ( x ) [ 1 + y _ { x } ^ { 2 } + z _ { x } ^ { 2 } ] ^ { - 3 / 2 } \\\\ \\omega _ { x } y - \\omega _ { z } x & = y '' ( x ) [ 1 + y _ { x } ^ { 2 } + z _ { x } ^ { 2 } ] ^ { - 3 / 2 } \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} ( w ' , L _ A w ) & = ( w ' , w ' + g ) = \\| w ' \\| _ 2 ^ 2 + ( w ' , g ) , \\ \\ \\ \\ \\\\ ( w ' , L _ A w ) & = ( L _ A w - g , L _ A w ) \\\\ & = \\| L _ A w \\| _ 2 ^ 2 - ( g , L _ A w ) \\\\ & = \\| L _ A w \\| _ 2 ^ 2 - ( g , w ' + g ) \\\\ & = \\| L _ A w \\| _ 2 ^ 2 - \\| g \\| _ 2 ^ 2 - ( g , w ' ) . \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} I ( t ) f ( t , x ) = f ( t , x + q ( t ) ) . \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} f ^ { ( i ) } _ j ( x ) = \\lambda _ i x + t ^ { ( i ) } _ j \\end{align*}"}
-{"id": "2945.png", "formula": "\\begin{align*} n _ 1 & = 1 \\\\ n _ d & = \\frac { 1 } { d } ( p ^ { c _ { \\lambda _ 0 } } - 1 ) \\\\ n _ { d p ^ i } & = \\frac { 1 } { d p ^ i } ( p ^ { c _ { \\lambda _ i } } - p ^ { c _ { \\lambda _ { i - 1 } } } ) . \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{align*} \\alpha _ 1 = & - \\frac { 1 } { 2 } \\gamma ^ { - 1 } g _ 1 , \\\\ \\alpha _ 2 = & - \\frac { 1 } { 2 } ( \\gamma + \\frac { 1 } { 2 } \\tilde \\gamma ^ T ) ^ { - 1 } g _ 2 , \\\\ \\alpha _ 3 = & ( \\tilde \\gamma ^ T ) ^ { - 1 } \\left ( ( 2 n + 4 ) \\gamma \\beta ^ { - 1 } ( t , q ) \\alpha _ 2 - g _ 3 \\right ) . \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} \\frac { d } { d t } F _ r ( \\phi , \\psi ) \\leq C \\cdot s ^ { - 1 } \\sum _ { l = 0 } ^ { 2 r + 4 } F _ r ( \\phi , \\psi ) ^ { l / 2 + 1 } + C ( n - 2 ) \\dot { s } s ^ { 1 - n } \\sum _ { l = 0 } ^ { r } F _ r ( \\phi , \\psi ) ^ { l / 2 + 2 } , \\end{align*}"}
-{"id": "1807.png", "formula": "\\begin{align*} s _ i : = \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { k ! } \\sum _ { \\substack { x _ 1 , \\ldots , x _ k \\in \\Lambda , \\\\ m _ { i - 1 } \\le d ( x _ 1 , \\ldots , x _ k ) < m _ i } } ( \\rho _ + ( x _ 1 , \\ldots , x _ k ) ^ 2 + \\rho _ - ( x _ 1 , \\ldots , x _ k ) ^ 2 ) . \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{align*} \\begin{aligned} D ^ { ( \\beta ) } ( P \\parallel Q ) = \\frac { \\sum _ i p _ i ^ { \\beta + 1 } + \\beta q _ i ^ { ( \\beta + 1 ) } - ( \\beta + 1 ) p _ i q _ i ^ { \\beta } } { \\beta ( \\beta + 1 ) } , \\end{aligned} \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} \\dd \\varphi & = \\tau _ 0 \\ , \\psi + 3 \\ , \\tau _ 1 \\wedge \\varphi + * \\tau _ 3 ~ , \\\\ \\dd \\psi & = 4 \\ , \\tau _ 1 \\wedge \\psi + * \\tau _ 2 ~ , \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} \\sum _ { n = M _ - } ^ { M _ + - 1 } A _ 2 ( n ) | C _ n | & \\leq { \\frac { c _ 1 } { ( \\gamma t ) ^ { ( d + 1 ) / 2 } } } \\sum _ { n = 0 } ^ { \\lfloor t ^ { 1 / 2 + \\varepsilon } \\rfloor } n ^ { d - 1 } + \\frac { c _ 2 } { M ( \\gamma t ) ^ { d / 2 } } \\sum _ { n = \\lfloor t ^ { 1 / 2 + \\varepsilon } \\rfloor + 1 } ^ { M _ + } n ^ { d - 1 } \\leq c _ 3 ( \\gamma t ) ^ { - 1 / 2 + \\varepsilon d } . \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} \\lim _ { j \\rightarrow \\infty } j \\gamma _ j = 0 . \\end{align*}"}
-{"id": "617.png", "formula": "\\begin{align*} L : = a _ { 1 1 } ( x _ 1 , x _ 2 ) X ^ 2 + a _ { 2 2 } ( x _ 1 , x _ 2 ) Y ^ 2 + 2 a _ { 1 2 } ( x _ 1 , x _ 2 ) Y X , \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} Z _ { t _ { 0 } } U _ 0 ( X , t _ 0 - r ^ 2 ) = \\ < \\nabla U _ 0 ( X , t _ 0 - r ^ 2 ) , X > - 2 r ^ 2 \\partial _ t U _ 0 ( X , t _ 0 - r ^ 2 ) . \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} f _ { \\tilde { \\sigma } _ t ( 0 ) } ( x ) = \\frac { \\mu } { x } \\exp \\Big \\{ - \\ln ( q _ { \\sigma , t } x ) ^ \\mu + ( \\mu - 1 ) \\ln _ 2 ( q _ { \\sigma , t } x ) \\Big \\} . \\end{align*}"}
-{"id": "5895.png", "formula": "\\begin{align*} \\delta ^ + _ i = \\nu ^ + _ i , \\forall \\ i \\in [ 1 , m _ 2 - r ] \\cup [ m _ 2 + 1 , m _ 1 + m _ 2 ] \\cup [ m _ 1 + m _ 2 + r + 1 , n ] , \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\varphi } _ t = - 2 \\star _ t \\left ( \\mathrm { d i v } ( T ^ { \\varphi _ t } ) \\wedge \\varphi _ t \\right ) , \\\\ \\varphi _ 0 = \\bar { \\varphi } , \\end{cases} \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{align*} w & \\asymp \\int \\limits _ { b ( v ) } ^ 1 \\frac { ( b D ^ r ) ^ { ( r - 1 ) / r } } { r \\sqrt { 1 - b ^ 2 } - D ^ { - 1 } O ( b ^ { - 1 / r } \\sqrt { 1 - b ^ 2 } ) } \\ , d b \\\\ & = D ^ { r - 1 } \\int \\limits _ { b ( v ) } ^ 1 \\frac { b } { r b ^ { 1 / r } \\sqrt { 1 - b ^ 2 } - D ^ { - 1 } O ( \\sqrt { 1 - b ^ 2 } ) } \\ , d b . \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} \\langle x ( t , 0 , u ) , p _ { k } \\rangle _ 2 ^ 2 & \\leq \\lambda _ { k } \\left [ \\int _ 0 ^ t x ^ T ( s ) ( A ^ T P ^ { - 1 } + P ^ { - 1 } A + \\sum _ { i = 1 } ^ m N _ i ^ T P ^ { - 1 } N _ i + k ^ 2 P ^ { - 1 } ) x ( s ) \\right . d s \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\left . + 2 \\int _ 0 ^ t x ^ T ( s ) P ^ { - 1 } B u ( s ) d s \\right ] . \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} Y ( { \\cal L } ) = \\textrm { P r o j } ( \\bigoplus \\limits _ { n \\geq 0 } H ^ 0 ( X , { \\cal L } ^ { \\otimes n } ) ^ G ) \\end{align*}"}
-{"id": "1024.png", "formula": "\\begin{align*} \\Vert u ^ { n _ { k } } - u ^ { l _ { k } } \\Vert & \\leq \\sum _ { l = n _ { k } } ^ { l _ { k } } \\Vert u ^ { l + 1 } - u ^ { l } \\Vert \\leq \\sum _ { l = n _ { k } } ^ { n _ { k } + s - 1 } \\Vert u ^ { l + 1 } - u ^ { l } \\Vert \\\\ & \\leq \\sum _ { l = n _ { k } } ^ { n _ { k } + s - 1 } \\Vert U _ { l } u ^ { l } - u ^ { l } \\Vert + \\sum _ { l = n _ { k } } ^ { n _ { k } + s - 1 } \\lambda _ { l } \\Vert G U _ { l } u ^ { l } \\Vert \\rightarrow _ k 0 . \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{align*} \\tilde { D } _ { u _ { m + 1 } } \\geq D _ { u _ { m + 1 } } \\big ( 1 - \\sum _ { i \\in [ B ] } \\hat { X } _ { w _ i , u _ { m + 1 } } \\big ) = D _ { u _ { m + 1 } } ( 1 + O ( n ^ { \\gamma - \\beta } ) ) . \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{align*} E = \\bigcup _ { \\mathcal { C } \\in \\Gamma } \\{ J \\in K _ 0 \\mid p ^ { - 1 } ( J , J ) \\cap \\mathcal { C } \\not = \\emptyset ( p ^ { - 1 } ( J , J ) \\cap \\mathcal { C } ) _ 0 \\subset S \\} \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} u ( x ) = c _ 0 + \\int _ 0 ^ 1 \\langle \\textit { \\textbf { V } } ( \\gamma ( t ) ) , \\dot \\gamma ( t ) ) \\rangle _ g d t , \\ x \\in \\Omega , \\end{align*}"}
-{"id": "3185.png", "formula": "\\begin{align*} d \\tau = \\mu \\ , \\eta \\wedge \\tau ' = - \\mu \\ast \\tau ' , d \\tau ' = - \\mu \\eta \\wedge \\tau = \\mu \\ast \\tau . \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} { \\cal A } f = \\hat f \\circ \\rho , \\end{align*}"}
-{"id": "9283.png", "formula": "\\begin{align*} t _ \\mathrm { s , 2 } & = \\Delta t _ { \\mathrm { s } _ { 2 , 1 } } + \\Delta t _ { \\mathrm { s } _ { 2 , 2 } } + \\Delta t _ { \\mathrm { s } _ { 2 , 3 } } + \\Delta t _ { \\mathrm { s } _ { 2 , 4 } } \\\\ & = g _ \\mathrm { D L } ^ { - 1 } l _ \\mathrm { r } + t _ \\mathrm { w } + g _ \\mathrm { U L } ^ { - 1 } l _ \\mathrm { r } + g _ \\mathrm { D L } ^ { - 1 } l _ \\mathrm { h } . \\end{align*}"}
-{"id": "2305.png", "formula": "\\begin{align*} f _ j ( t ) : = F _ j ' ( t ) = p _ j M _ j e ^ { - p _ j t } \\left ( 1 - e ^ { - p _ j t } \\right ) ^ { M _ j - 1 } , t \\geq 0 , \\ \\ ; j = 1 , \\dots , g . \\end{align*}"}
-{"id": "2320.png", "formula": "\\begin{align*} \\lambda : = \\frac { p _ 2 } { p _ 1 } \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} h ( z ) = a _ { 1 } z + a _ { 0 } + a _ { - 1 } z ^ { - 1 } + a _ { - 2 } z ^ { - 2 } + \\cdots \\end{align*}"}
-{"id": "3611.png", "formula": "\\begin{align*} \\partial _ { t } \\Gamma & = ( 0 , - \\sin ( t ) f ( x ) , - \\cos ( t ) f ( x ) ) \\\\ \\partial _ { x } \\Gamma & = ( 1 , \\cos ( t ) f ' ( x ) , - \\sin ( t ) f ' ( x ) ) \\\\ \\partial _ { x x } \\Gamma & = ( 0 , \\cos ( t ) f '' ( x ) , - \\sin ( t ) f '' ( x ) ) \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} A ( \\pi ^ i ) = \\begin{cases} \\{ 1 , 2 , \\dots , i , n + 1 , n + 2 \\} & \\textrm { i f $ 1 \\le i \\le n $ , } \\\\ \\{ 1 , 2 , \\dots , n , n + 2 \\} & \\textrm { i f $ i = n + 1 $ . } \\end{cases} \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} 0 & \\leq \\int _ 0 ^ 1 g ( s ) ( \\alpha ( s ) - \\alpha _ 0 ( s ) ) d s = \\int _ 0 ^ u g ( s ) ( \\alpha ( s ) - \\alpha _ 0 ( s ) ) d s = \\int _ 0 ^ u g ( s ) \\alpha ( s ) d s = \\frac { 1 } { c } \\int _ 0 ^ u g ( s ) \\gamma ( s ) d s . \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} x ^ 0 _ 1 ( s ) = s + 4 m \\log s , \\ \\ \\dot x ^ 0 _ 1 ( s ) = 1 + 4 m s ^ { - 1 } , \\end{align*}"}
-{"id": "6950.png", "formula": "\\begin{align*} g ^ \\ell ( x ) & : = \\frac { g ( \\eta ( x + 1 ) ) + \\dots + g ( \\eta ( x + \\ell ) ) } { \\ell } \\\\ g _ s ^ { n , \\ell } ( x ) & : = \\frac { g _ s ^ n ( x + 1 ) + \\dots + g _ s ^ n ( x + \\ell ) } { \\ell } . \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} \\Phi = \\begin{pmatrix} 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\end{pmatrix} , Q _ E = \\begin{pmatrix} 0 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & - 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} P _ c ( u \\cdot \\nabla ) y = \\frac { 1 } { | \\Omega | } \\int _ { \\Omega } ( u \\cdot \\nabla ) y \\ \\d x = \\frac { 1 } { | \\Omega | } \\bigg ( \\int _ { \\Omega } u \\cdot \\nabla y _ k \\ \\d x \\bigg ) _ { 1 \\leq k \\leq 3 } = 0 \\hbox { f o r } u \\in L _ { \\sigma } ^ p ( \\Omega ) , \\ y \\in W ^ { 1 , p } ( \\Omega ) ^ 3 , \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} E \\left [ T ^ { ( 2 ) } \\right ] = E \\left [ T _ 1 ^ { ( 2 ) } \\right ] + O \\left ( M ^ { 3 - \\lambda + \\varepsilon } \\right ) , M \\to \\infty . \\end{align*}"}
-{"id": "7728.png", "formula": "\\begin{align*} \\mathcal { L } _ { ^ { m , 2 } _ { i n t e r } } ( s ) & = \\mathcal { E } \\left \\{ \\prod _ { x _ j \\in \\Phi _ c \\backslash x _ m } \\frac { 1 } { \\frac { s } { { L \\left ( | | y _ { m , 2 } + x _ m - x _ j | | \\right ) } } + 1 } \\right \\} . \\end{align*}"}
-{"id": "3660.png", "formula": "\\begin{align*} & \\P ^ \\eta \\big ( | \\langle \\eta ( t ) \\rangle _ { J ' } ^ k - p _ k | > \\epsilon _ { J } \\big ) = \\P ^ \\eta \\big ( \\langle \\eta ( t ) \\rangle _ { J ' } ^ k > p _ k + \\epsilon _ { J } \\big ) + \\P ^ \\eta \\big ( \\langle \\eta ( t ) \\rangle _ { J ' } ^ k < p _ k - \\epsilon _ { J } \\big ) \\ , . \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} M _ t V _ t = \\sum _ { | I | , j \\leq n / 2 - 1 / 2 } \\frac { ( - 1 ) ^ { k + l } } { j ! 2 ^ k } \\mathcal N _ { I } ( t ) D v _ t ^ j + \\xi ( t ) , \\end{align*}"}
-{"id": "2751.png", "formula": "\\begin{align*} P : \\mathfrak { M } ( K ) \\times \\widehat { K _ 2 ^ { t o p } ( K ) } & \\to \\mathbb { Q } / \\mathbb { Z } , \\\\ P ( x , y ) & = [ x _ n , \\phi _ n ( y ) ] _ n , \\end{align*}"}
-{"id": "593.png", "formula": "\\begin{align*} E = E ^ 0 \\supset E ^ 1 \\supset \\cdots \\supset E ^ N \\supset \\{ 0 \\} \\end{align*}"}
-{"id": "8143.png", "formula": "\\begin{align*} \\hat { g } _ n ( z ) : = \\gamma _ n \\left ( 1 - \\frac { z } { \\bar { \\lambda } _ n } \\right ) , \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{align*} { \\operatorname { R e s t } } _ { x _ n = 0 } \\circ ( { \\operatorname { i d } } \\oplus \\iota _ { \\frac { \\partial } { \\partial x _ n } } ) \\colon { \\mathcal { E } } ^ i ( { \\mathbb { R } } ^ n ) \\to { \\mathcal { E } } ^ i ( { \\mathbb { R } } ^ { n - 1 } ) \\oplus { \\mathcal { E } } ^ { i - 1 } ( { \\mathbb { R } } ^ { n - 1 } ) \\end{align*}"}
-{"id": "6323.png", "formula": "\\begin{gather*} \\Gamma _ { i ' j ' } ( U ) : = \\Lambda _ { i ' j ' } U , \\ , \\ , i ' , j ' \\in I _ p , \\ , \\ , i ' < j ' , \\\\ \\Gamma _ { i '' j '' } ( U ) : = \\Lambda _ { i '' j '' } U , \\ , \\ , \\ , i '' \\in I _ p , \\ , j '' \\notin I _ p , \\end{gather*}"}
-{"id": "2893.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\gamma _ { N , t } ^ { ( 1 ) } = | u _ t \\rangle \\langle u _ t | \\end{align*}"}
-{"id": "9168.png", "formula": "\\begin{align*} A _ m W _ k ( x ) ( 1 - x ) = m ^ k W _ k ( x \\gamma ) A _ m \\left ( 1 - \\frac x { q ^ k } \\right ) \\end{align*}"}
-{"id": "8367.png", "formula": "\\begin{align*} \\pi _ i ( x _ i ^ k f ) = \\begin{cases} - \\beta f & ( k = 0 ) , \\\\ \\left ( \\displaystyle \\sum _ { s = 0 } ^ { k - 1 } x _ i ^ s x _ { i + 1 } ^ { k - 1 - s } + \\beta \\displaystyle \\sum _ { s = 1 } ^ { k - 1 } x _ i ^ { s } x _ { i + 1 } ^ { k - s } \\right ) f & ( k > 0 ) . \\end{cases} \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{align*} \\bar \\partial _ E = \\begin{pmatrix} \\bar \\partial & - \\beta ^ t & 0 & 0 \\\\ 0 & \\bar \\partial _ W & \\beta & 0 \\\\ 0 & 0 & \\bar \\partial & 0 \\\\ 0 & 0 & 0 & \\bar \\partial \\end{pmatrix} , \\Phi = \\begin{pmatrix} 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} \\alpha \\cdot m : = ( \\alpha \\otimes 1 - 1 \\otimes \\alpha ) \\cdot m \\ , ; \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} { \\rm d } ^ { \\ , 2 } f ( \\bar { x } \\ , | \\ , 0 ) ( d ) \\ , = \\ , \\max _ { \\lambda } \\left \\{ \\ , \\lambda \\ , d _ { \\ , 1 } ^ { \\ , 2 } - Q _ { 1 1 } d _ { \\ , 1 } ^ { \\ , 2 } \\ , \\mid \\ , \\lambda = Q _ { 2 2 } \\in ( 0 , 1 ) \\ , \\right \\} = ( Q _ { 2 2 } - Q _ { 1 1 } ) \\ , d _ 1 ^ 2 . \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{align*} \\mathcal { L } ( \\alpha , \\beta ) = \\Big ( \\triangle \\alpha - ( \\lambda + k - 2 ) ( \\lambda + n - k ) \\alpha - 2 d ^ \\ast \\beta , \\triangle \\beta - ( \\lambda + n - k - 2 ) ( \\lambda + k ) \\beta - 2 d \\alpha \\Big ) . \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} & \\| A B - A _ 0 B _ 0 \\| ^ 2 \\\\ & \\sim \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } \\left [ \\sum _ { k = 1 } ^ { H _ 0 - 1 } \\{ ( a _ { i k } - a _ { i H } ) b _ { k j } - ( a ^ 0 _ { i k } - a ^ 0 _ { i H _ 0 } ) b ^ 0 _ { k j } \\} + \\sum _ { k = H _ 0 } ^ { H - 1 } ( a _ { i k } - a _ { i H } ) b _ { k j } + ( a _ { i H } - a ^ 0 _ { i H _ 0 } ) \\right ] ^ 2 \\\\ & = \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } \\left \\{ \\sum _ { k = 1 } ^ { H _ 0 - 1 } ( a _ { i k } b _ { k j } - a ^ 0 _ { i k } b ^ 0 _ { k j } ) + \\sum _ { k = H _ 0 } ^ { H - 1 } a _ { i k } b _ { k j } + c _ i \\right \\} ^ 2 . \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} \\int _ I g & = \\int _ 0 ^ p g + \\int _ p ^ 1 g \\\\ & = \\int _ 1 ^ { p + 1 } g + \\int _ p ^ 1 g \\\\ & = \\int _ q ^ 1 ( g \\circ \\tilde { u } ) u ' + \\int _ 0 ^ q ( g \\circ \\tilde { u } ) u ' \\\\ & = \\int _ I ( g \\circ u ) u ' . \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{align*} \\frac { d \\delta ( \\phi _ t ( v ) ) } { d t } = a = \\sqrt { 1 - b ^ 2 } . \\end{align*}"}
-{"id": "4968.png", "formula": "\\begin{align*} Q _ { c } ( x ) = \\frac { 4 c } { 1 + c ^ 2 x ^ 2 } , \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} { { ^ L } W } = \\{ v \\in W : N ( v ^ { - 1 } ) \\subseteq \\Phi ^ + \\setminus \\Phi _ L ^ + \\} \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { v } \\lambda _ { U } ^ s ( r , y ) y d y \\\\ & = \\sum _ { t \\in \\mathcal { K } } \\Big ( \\frac { \\lambda _ t v ^ 2 } { 2 } - \\lambda _ t \\Big ( \\frac { \\exp ( - \\pi C _ t r ^ 2 ) - \\exp ( - \\pi C _ t ( r + v ) ^ 2 ) } { 2 \\pi C _ t } \\\\ & \\quad - r \\frac { \\mathrm { e r f } ( ( r + v ) \\sqrt { \\pi C _ t } ) - \\mathrm { e r f } ( \\sqrt { \\pi C _ t } r ) } { 2 \\sqrt { C _ t } } \\Big ) \\Big ) , \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{align*} [ s , z _ { \\tau ( 1 ) } ] [ z _ { \\tau ( 2 ) } , z _ { \\tau ( 3 ) } , z _ { \\tau ( 4 ) } ] = ( - 1 ) ^ { \\tau } [ s , z _ 1 ] [ z _ 2 , z _ 3 , z _ 4 ] \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} \\mathrm { P } ^ { O M A } _ { m , 1 } & = \\mathrm { P } \\left ( z _ t > \\frac { \\epsilon _ 1 } { \\rho } , z _ m < \\frac { \\epsilon _ 1 } { \\rho } \\right ) + \\mathrm { P } \\left ( z _ t < \\frac { \\epsilon _ 1 } { \\rho } , z _ m < \\frac { \\epsilon _ 1 } { \\rho } \\right ) \\\\ & = \\mathrm { P } _ { m , 1 } . \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} X _ { \\epsilon _ 1 + \\epsilon _ 2 } ( s ) X _ { - \\epsilon _ 1 - \\epsilon _ 2 } ( s ) + X _ { \\epsilon _ 1 - \\epsilon _ 2 } ( s ) X _ { - \\epsilon _ 1 + \\epsilon _ 2 } ( s ) + X _ 0 ( s ) ^ 2 = 0 ; \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} S ^ m _ { 1 , 0 } ( U \\times \\R ^ { n - 1 } ; \\mathcal { S } _ { + } ) = \\bigcap _ { \\tau > 0 } C ^ \\tau S ^ m _ { 1 , 0 } ( U \\times \\R ^ { n - 1 } ; \\mathcal { S } _ { + } ) , \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{align*} g ( 1 ) = g ( 2 ) > g ( 3 ) > g ( 4 ) > \\cdots \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} X ^ { \\rm s t e p } _ { t ^ \\nu } : = \\frac { x ^ { \\rm s t e p } _ { t ^ \\nu / 4 } ( t ^ \\nu ) } { - t ^ { \\nu / 3 } } \\to 2 ^ { - 1 / 3 } \\xi _ { \\rm G U E } , \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{align*} \\begin{gathered} ( \\lambda - 5 ) d ^ \\ast \\beta _ 1 - ( \\lambda - 1 ) ( \\lambda + 9 ) \\beta _ 0 ( \\lambda + 1 ) ( \\lambda + 5 ) , \\\\ d ^ \\ast \\beta _ 1 - ( \\lambda + 5 ) \\beta _ 0 ( \\lambda - 1 ) ( \\lambda + 3 ) . \\end{gathered} \\end{align*}"}
-{"id": "5707.png", "formula": "\\begin{align*} b _ 2 - c _ 2 = \\binom { c _ 2 } 2 - c _ 2 = \\left ( \\binom { c _ 2 - 2 } 2 + \\binom { c _ 2 - 2 } 1 \\right ) + \\binom { c _ 2 - 1 } 1 - c _ 2 = \\binom { c _ 2 - 2 } 2 + c _ 2 - 3 . \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} \\binom { c n / k ^ { \\ell } } { k } \\leq \\left ( \\frac { c e n } { k ^ { \\ell + 1 } } \\right ) ^ { k } = 2 ^ { f ( k ) } \\leq 2 ^ { \\frac { \\ell + 1 } { 2 ^ { 1 / \\ln 2 } \\ln 2 } ( c e n ) ^ { \\frac { 1 } { \\ell + 1 } } } . \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} \\mu ( D ^ 3 ( x _ i ) , x _ { j } ) ) = \\mu ( x _ { n - 5 + i } , x _ { j } ) = 0 \\end{align*}"}
-{"id": "9022.png", "formula": "\\begin{align*} \\xi _ 1 = - \\frac { u _ 3 ' } { u _ 3 \\| u ' \\| _ J } . \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} R i c _ { B \\times _ h F } = a ( n + m - 1 ) g , R i c _ B = a ( n - 1 ) g _ B , R i c _ F = c ( m - 1 ) g _ F , \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{align*} 0 \\leq p _ { k j } ( t ) \\leq 1 , \\sum _ { j = 1 } ^ { J } p _ { k j } ( t ) = 1 \\\\ p _ { k j } ( t ) = 0 \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{align*} \\frac { \\mathcal { L } ^ n } { n ! } \\mathrm { M a b } ^ l _ { \\Theta } ( \\phi ) = & \\sum _ Y \\Lambda _ Y \\int _ { \\tilde { \\Delta } ^ + _ Y } ( n u ^ * ( p ) - u ^ * ( p ) \\sum \\frac { \\chi ( \\alpha ^ { \\vee } ) } { q ( \\alpha ^ { \\vee } ) } + d _ p u ^ * ( p ) ) P _ { D H } d q \\\\ & + \\int _ { \\Delta ^ + } u ^ * ( p ) ( \\sum \\frac { \\chi ^ { a c } ( \\alpha ^ { \\vee } ) } { q ( \\alpha ^ { \\vee } ) } - \\bar { S } _ { \\Theta } ) P _ { D H } d q + \\int _ { \\Delta ^ + } 4 \\rho _ H ( a ) P _ { D H } ( q ) d q \\end{align*}"}
-{"id": "1504.png", "formula": "\\begin{align*} \\mathcal { L } \\log v & = \\dfrac { \\mathcal { L } v } { v } - \\dfrac { | \\nabla v | ^ 2 } { v ^ 2 } = - \\dfrac n 2 - | { \\bf H } | ^ 2 - | \\nabla \\log v | ^ 2 \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} s _ * & \\coloneqq 4 d , & \\frac { s _ g } { 4 } & = c + d . \\end{align*}"}
-{"id": "7149.png", "formula": "\\begin{align*} R _ { \\left ( x , y , z \\right ) } = \\left [ R _ { y } , \\left [ R _ { x } , R _ { z } \\right ] \\right ] , \\end{align*}"}
-{"id": "970.png", "formula": "\\begin{align*} { \\varphi } _ { m } \\ = \\ [ \\rho + h ( \\mathfrak { e } ) ] ^ { - 1 } V { \\varphi } _ { m } \\ = \\ [ \\rho + h ( \\mathfrak { e } ) ] ^ { - 1 } V ^ { 1 / 2 } \\psi _ { m } , \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} | z _ \\alpha | = | \\kappa _ \\alpha | , | \\xi _ s | = 1 . \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} ( 1 + \\lambda ^ { 2 } ) ( z ^ { 2 } + 2 C _ { 1 } ) & = \\int { ( 1 + u ) ^ { - 3 / 2 } } d u . \\\\ ( 1 + \\lambda ^ { 2 } ) ( z ^ { 2 } + 2 C _ { 1 } ) & = - 2 ( 1 + ( 1 + \\lambda ^ { 2 } ) ( z ' ) ^ { 2 } ) ^ { - 1 / 2 } \\\\ ( z ' ) ^ { 2 } & = \\frac { 4 } { ( 1 + \\lambda ^ { 2 } ) ^ { 3 } ( z ^ { 2 } + 2 C _ { 1 } ) ^ { 2 } } - \\frac { 1 } { ( 1 + \\lambda ^ { 2 } ) } \\\\ z ' ( x ) & = \\pm \\sqrt { \\frac { 4 - ( 1 + \\lambda ^ 2 ) ^ 2 ( z ^ 2 + 2 C _ 1 ) ^ 2 } { ( 1 + \\lambda ^ 2 ) ^ 3 ( z ^ 2 + 2 C _ 1 ) ^ 2 } } \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} f _ { \\mu } ( z _ 1 , \\dots , z _ n ; q , t ) = \\prod _ { i = 1 } ^ { r - 1 } ( 1 - q ^ i ) \\times { \\rm T r } \\left ( A _ { \\mu _ 1 } ( z _ 1 ) A _ { \\mu _ 2 } ( z _ 2 ) \\dots A _ { \\mu _ n } ( z _ n ) k ^ { u ( r - 1 ) } \\otimes k ^ { u ( r - 2 ) } \\otimes \\cdots \\otimes k ^ { u } \\right ) , \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} \\partial _ v ^ { [ j ] } = \\partial _ v \\circ ( \\partial _ v - t ) \\circ \\cdots \\circ ( \\partial _ v - ( j - 1 ) t ) \\end{align*}"}
-{"id": "8076.png", "formula": "\\begin{align*} \\Phi _ \\nu ' ( z ) = - z ^ \\nu K _ { \\nu - 1 } ( z ) = - z ^ \\nu K _ { 1 - \\nu } ( z ) . \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} \\left \\vert y _ { m } \\left ( x \\right ) - \\overset { \\ast } { y _ { m } } \\left ( x \\right ) \\right \\vert \\leq \\left \\vert \\varepsilon \\right \\vert \\left ( \\psi \\left ( x \\right ) - \\psi \\left ( a \\right ) \\right ) ^ { \\gamma - 1 } \\overset { m } { \\underset { j = 0 } { \\sum } } A ^ { j } \\frac { \\left ( \\psi \\left ( x \\right ) - \\psi \\left ( a \\right ) \\right ) ^ { \\alpha j } } { \\Gamma \\left ( \\alpha j + \\gamma \\right ) } . \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} d \\Phi _ { g _ 1 } ( o ) \\begin{pmatrix} \\hat w \\\\ w _ 3 \\end{pmatrix} = \\begin{pmatrix} I & 0 \\\\ - \\xi ^ T & 1 \\end{pmatrix} \\begin{pmatrix} \\hat w \\\\ w _ 3 \\end{pmatrix} . \\end{align*}"}
-{"id": "3204.png", "formula": "\\begin{align*} \\langle d ^ \\ast \\psi - \\sum _ { i = k + 1 } ^ b { a _ i \\ , \\triangle \\sigma ' _ i } , \\overline { \\sigma } _ h \\rangle _ { L ^ 2 } = 0 \\end{align*}"}
-{"id": "3441.png", "formula": "\\begin{align*} \\frac { d \\phi } { d r } = I + I I - \\lambda \\sqrt { 2 } . \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} \\pi _ { 1 1 } h = h _ { 1 1 } \\ , d s ^ 2 \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} E \\left [ T ^ 2 \\right ] - E \\left [ T _ 1 ^ 2 \\right ] = O \\left ( M ^ { 2 - ( \\lambda - 1 ) / s } \\ln ^ 2 M \\right ) , M \\to \\infty . \\end{align*}"}
-{"id": "3247.png", "formula": "\\begin{align*} = \\sum _ { \\nu = 0 } ^ { n + | \\textup { \\textbf { m } } | - m _ \\alpha } b _ { \\nu , n } ^ { ( \\alpha ) } \\Phi _ \\nu ( z ) + \\sum _ { \\nu = n + | \\textup { \\textbf { m } } | - m _ \\alpha + 1 } ^ { \\infty } b _ { \\nu , n } ^ { ( \\alpha ) } \\Phi _ \\nu ( z ) . \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{align*} \\mathfrak { R e } \\dot { c } _ 1 ( 0 ) = 0 \\mbox { a n d } \\mathfrak { R e } \\ddot { c } _ 1 ( 0 ) = - \\sum _ { k = 0 } ^ { + \\infty } | \\dot { c } _ k ( 0 ) | ^ 2 . \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} v : = & S ( x _ { n _ { 1 } } \\otimes \\cdots \\otimes x _ { n _ { r ' + 1 } } \\otimes x _ { n _ { r ' } } \\otimes \\cdots \\otimes x _ { n _ { i + 1 } } ) \\\\ + & S ( \\mu ^ { i - 1 } \\alpha _ { T } ( x _ { n _ { 1 } } \\otimes \\cdots \\otimes x _ { n _ { r ' - 1 } } ) \\otimes [ x _ { n _ { r ' } } , x _ { n _ { r ' + 1 } } ] _ { \\mathfrak { g } } \\otimes \\alpha _ { T } ( x _ { n _ { r ' + 2 } } \\otimes \\cdots \\otimes x _ { n _ { i + 1 } } ) ) . \\end{align*}"}
-{"id": "2785.png", "formula": "\\begin{align*} \\Pi ^ \\# ( \\zeta ) ( \\alpha _ 1 , \\ldots , \\alpha _ p ) = ( - 1 ) ^ p \\zeta ( \\Pi ^ \\# ( \\alpha _ 1 ) , \\ldots , \\Pi ^ \\# ( \\alpha _ p ) ) . \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} J _ 1 ( t _ 1 , t _ 2 ) & = \\int _ { \\mathbb { R } } \\big ( G _ { t _ 2 } ( x - y ) - G _ { t _ 1 } ( x - y ) \\big ) \\psi _ 0 ( y ) d y , \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } d X ^ { 0 , x , p } _ t = p _ t d t - d D _ t \\ , \\textrm { w i t h } d D _ t = \\bar d _ t d t + d i a g ( \\sigma ) d W _ t \\\\ X ^ { 0 , x , p } _ 0 = x \\ , \\end{array} \\right . \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} \\begin{gathered} \\frac { \\rm { d } \\pi _ d ^ { ( D ) } ( \\textbf { x } ) } { \\rm { d } x _ f ^ { ( D ) } } = \\frac { \\rm { d } { \\left ( \\frac { p _ d B _ d } { N ^ { ( D ) } ( 1 - x _ f ^ { ( D ) } ) } C _ d ^ { ( D ) } - q _ d \\phi _ d \\right ) } } { \\rm { d } { x _ f ^ { ( D ) } } } = \\frac { N ^ { ( F ) } p _ d B _ d C _ d ^ { ( D ) } } { \\left ( N ^ { ( D ) } ( 1 - x _ f ^ { ( D ) } ) \\right ) ^ 2 } + c _ 1 c _ 2 N ^ { ( D ) } \\phi _ d \\exp \\left ( c _ 2 \\left ( N ^ { ( D ) } ( 1 - x _ f ^ { ( D ) } ) \\right ) \\right ) \\end{gathered} \\end{align*}"}
-{"id": "6465.png", "formula": "\\begin{align*} \\begin{aligned} & \\norm { y _ { 0 } ( t _ { 0 } + t ) - y _ { 0 } ( t _ { 0 } ) } _ { L ^ { \\infty } ( \\Omega ) ^ 3 } = \\norm { e ^ { - ( t _ { 0 } + t ) B } b _ { s } - e ^ { - t _ { 0 } B } b _ { s } } _ { L ^ { \\infty } ( \\Omega ) ^ 3 } \\leq C t _ { 0 } ^ { - \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { \\infty } ) } \\norm { e ^ { - t B } b _ { s } - b _ { s } } _ { L ^ { p } ( \\Omega ) ^ 3 } , \\end{aligned} \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{align*} \\mathcal { F } _ { u } ( q _ { 0 } , u _ { 0 } ) \\phi = - \\Delta \\phi - q _ { 0 } a ( x ) u _ { 0 } ^ { q _ { 0 } - 1 } \\phi . \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} q _ 2 y = o ( \\sqrt { k } ) . \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{align*} [ [ a , b ] , [ a , b , k a ^ i ] ] & = [ [ a , b ] , [ a , b , a ^ i ] [ a , b , k ] ^ { a ^ i } ] \\\\ & = [ [ a , b ] , [ a , b , k ] ^ { a ^ i } ] [ [ a , b ] , [ a , b , a ^ i ] ] ^ { [ a , b , k ] ^ { a ^ i } } . \\end{align*}"}
-{"id": "6015.png", "formula": "\\begin{align*} \\tilde { \\phi } ( s ) = \\frac { 2 \\pi { } N } { \\left ( \\alpha { } + 1 \\right ) } \\frac { \\Gamma \\left ( s \\right ) \\Gamma \\left ( \\frac { 1 - s } { 1 + \\alpha { } } \\right ) } { \\Gamma \\left ( \\frac { \\alpha { } + \\theta { } - \\alpha { } s - \\theta { } s } { 2 \\left ( \\alpha { } + 1 \\right ) } \\right ) \\Gamma \\left ( \\frac { 2 + \\alpha { } - \\theta { } + \\alpha { } s + \\theta { } s } { 2 \\left ( \\alpha { } + 1 \\right ) } \\right ) } . \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} \\lambda = \\frac n 2 + \\frac 1 2 \\sum _ { i = 1 } ^ n k _ i , k _ i \\in \\{ 0 \\} \\cup \\mathbb { N } , \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} & r _ { A _ 1 B _ 1 A _ 2 B _ 2 } ( a _ 1 , b _ 1 , a _ 2 , b _ 2 ) = \\sum _ { x _ 1 , x _ 2 , y _ 1 , y _ 2 } \\Big ( r _ { X _ 1 Y _ 1 X _ 2 Y _ 2 } ( x _ 1 , y _ 1 , x _ 2 , y _ 2 ) V ( a _ 1 a _ 2 b _ 1 b _ 2 | x _ 1 x _ 2 y _ 1 y _ 2 ) \\Big ) \\\\ & \\quad = p _ { A B } ( a _ 1 , b _ 2 ) p _ { A B } ( a _ 2 , b _ 1 ) . \\end{align*}"}
-{"id": "3037.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\mathrm { d i v } \\left ( ( \\nabla u _ { 0 } ) q _ { 0 } u _ { 0 } ^ { q _ { 0 } - 1 } \\phi _ { 1 } \\right ) & = \\int _ { \\Omega } \\mathrm { d i v } \\left ( ( \\nabla u _ { 0 } ) q _ { 0 } \\left ( \\frac { \\phi _ { 1 } } { u _ { 0 } } \\right ) ^ { 1 - q _ { 0 } } \\phi _ { 1 } ^ { q _ { 0 } } \\right ) \\\\ & = \\int _ { \\partial \\Omega } \\frac { \\partial u _ { 0 } } { \\partial \\nu } q _ { 0 } \\left ( \\frac { \\phi _ { 1 } } { u _ { 0 } } \\right ) ^ { 1 - q _ { 0 } } \\phi _ { 1 } ^ { q _ { 0 } } \\\\ & = 0 , \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} z _ \\gamma ( s ) : = \\prod _ { k \\ge 1 } \\left ( 1 - e ^ { - l ( \\gamma ) ( s + k ) } \\right ) ^ { - k } \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} \\mu ( \\{ \\omega : f _ n ( \\omega ) < x \\} ) & = \\left \\lvert \\{ t : g _ n ( t ) < x \\} \\right \\rvert \\\\ \\mu ( \\{ \\omega : f ( \\omega ) < x \\} ) & = \\left \\lvert \\{ t : g ( t ) < x \\} \\right \\rvert \\end{align*}"}
-{"id": "5921.png", "formula": "\\begin{align*} \\mu ^ { * } _ i = \\left \\{ \\begin{array} { l l } 0 , & \\mu _ i = 0 , \\\\ 1 , & \\mu _ i \\geq 1 , \\end{array} \\right . \\forall \\ i \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "8279.png", "formula": "\\begin{align*} 1 = E _ d ( 1 ) = \\sum _ { k = 0 } ^ { d - 1 } \\frac { \\langle 1 , \\psi _ d ^ k \\rangle } { q ^ k } . \\end{align*}"}
-{"id": "6879.png", "formula": "\\begin{align*} T S _ v x _ t = T x _ 0 + t T x _ 1 = t y . \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} Z _ + ( r ) : = 1 - \\frac 1 { \\sqrt { 1 + ( u ' ( r ) ) ^ 2 } } + G ( u ( r ) ) \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} \\int _ X \\psi \\omega ^ n = \\int _ { G / H } \\psi \\omega ^ n \\end{align*}"}
-{"id": "5961.png", "formula": "\\begin{align*} F \\big ( R ^ { ( 1 ) } \\big ) - F \\big ( R ^ { ( 0 ) } \\big ) = g ( 1 ) - g ( 0 ) = g ' ( 0 ) + \\frac { 1 } { 2 } g '' ( s ) . \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} x _ { x _ 0 } ^ T ( t ) Q x _ { x _ 0 } ( t ) - x _ 0 ^ T Q x _ 0 & \\leq \\left [ \\int _ 0 ^ t x _ { x _ 0 } ^ T ( s ) ( A ^ T Q + Q A + \\sum _ { i = 1 } ^ m N _ i ^ T Q N _ i + k ^ 2 Q ) x _ { x _ 0 } ( s ) \\right . d s \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\left . + 2 \\int _ 0 ^ t x _ { x _ 0 } ^ T ( s ) Q B u ( s ) d s \\right ] . \\end{align*}"}
-{"id": "7187.png", "formula": "\\begin{align*} \\underline \\lim _ { \\ , n \\to \\infty } { \\mathbb P } ( \\varphi _ n ( f _ 1 , \\ldots , f _ s ) \\in \\bigcup _ { \\mathbf b \\in M } \\bigcup _ { \\mathbf c \\in O _ K ( \\mathbf b ) } J _ K ( \\mathbf c ) \\ , | \\ , \\xi _ d = \\beta ) \\geqslant \\sum _ { \\mathbf b \\in M } V ( J _ K ( \\mathbf b ) ) . \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} \\Theta ( z , \\tau ) = \\prod _ { k \\geq 1 } \\left ( 1 - e ^ { 2 k \\pi i \\tau } \\right ) \\left ( 1 + e ^ { ( 2 k - 1 ) \\pi i \\tau } e ^ { 2 \\pi i z } \\right ) \\left ( 1 + e ^ { ( 2 k - 1 ) \\pi i \\tau } e ^ { - 2 \\pi i z } \\right ) . \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} X ( \\tau ) = \\big ( X _ 1 ( \\tau ) , \\dots , X _ g ( \\tau ) \\big ) , \\tau = 0 , 1 , 2 , \\dots , \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} - G \\left ( A _ t + \\frac { 1 - \\tilde { Q } _ t ( A _ t , \\dot { \\varphi } _ 0 ) } { q _ { \\sigma , t } ( 1 + x ) } \\right ) + \\ln \\left ( \\frac { x } { \\mu } f _ { \\tilde { \\sigma } _ t ( 0 ) } ( x ) \\right ) + \\chi _ t = \\mu ( \\ln _ 2 t ) ^ { \\mu - 1 } h _ { t , \\dot { \\varphi } _ 0 } ( x ) \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} L _ { \\tilde { \\Delta } ( m ) } : \\sum _ { i = 1 } ^ N \\Delta ( a _ i ) z _ i \\mapsto \\tilde { \\Delta } ( m ) \\sum _ { i = 1 } ^ N \\Delta ( a _ i ) z _ i = \\sum _ { i = 1 } ^ N \\Delta ( m a _ i ) z _ i \\ , \\in A \\otimes A . \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} ( d _ { A ( t ) } \\psi ( t ) ) _ { n o r m } = 0 \\ \\ \\ \\ \\ \\ t > 0 . \\end{align*}"}
-{"id": "6843.png", "formula": "\\begin{align*} A P + P A ^ \\top + B B ^ \\top = 0 \\end{align*}"}
-{"id": "2326.png", "formula": "\\begin{align*} \\lambda = \\frac { p _ 2 } { p _ 1 } > 1 . \\end{align*}"}
-{"id": "7684.png", "formula": "\\begin{align*} \\mathrm { P } ^ i ( r ) = & \\mathrm { P } \\left ( ^ l _ r > R _ l , \\forall l \\in \\{ 0 , \\cdots , i \\} \\right ) \\\\ = & e ^ { - \\bar { \\tau } _ i ^ \\alpha r ^ \\alpha } . \\end{align*}"}
-{"id": "4881.png", "formula": "\\begin{align*} \\phi _ l ^ 2 P + [ k ] P = \\pm [ \\tau ] \\phi P \\ , . \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} & \\alpha _ { T } ( ( \\mathfrak { g } ^ { \\otimes n } \\odot ( a \\otimes b - \\varepsilon ( a , b ) b \\otimes a - [ a , b ] _ { \\mathfrak { g } } ) ) \\odot \\mathfrak { g } ^ { \\otimes m } ) \\\\ = & ( \\mathfrak { g } ^ { \\otimes n } \\odot ( \\alpha _ { T } ( a ) \\otimes \\alpha _ { T } ( b ) - \\varepsilon ( a , b ) \\alpha _ { T } ( b ) \\otimes \\alpha _ { T } ( a ) - [ \\alpha _ { T } ( a ) , \\alpha _ { T } ( b ) ] _ { \\mathfrak { g } } ) ) \\\\ \\odot & \\mathfrak { g } ^ { \\otimes m } \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{align*} x ' & = x y + A \\\\ y ' & = - x ^ { 2 } - B \\end{align*}"}
-{"id": "5121.png", "formula": "\\begin{align*} r \\ = \\sum _ { 2 \\le i _ 1 \\le i _ 2 \\le n - 1 } \\lambda _ { i _ 1 i _ 2 } h _ { i _ 1 } h _ { i _ 2 } \\ \\in \\ ( u _ 1 ^ n , u _ 2 ^ n , v _ 1 ^ n , v _ 2 ^ n ) S . \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} A ( \\pi ^ i ) = \\begin{cases} \\{ 1 , 2 , \\dots , i , n + 1 , n + 2 \\} & \\textrm { i f $ 1 \\le i \\le n $ , } \\\\ \\{ 1 , 2 , \\dots , n , n + 1 , n + 2 \\} & \\textrm { i f $ i = n + 1 $ . } \\end{cases} \\end{align*}"}
-{"id": "1479.png", "formula": "\\begin{align*} H = \\left < \\bar { \\nabla } f , { \\bf n } \\right > . \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{align*} C _ { 0 } = w _ { c } ^ { p ^ { e - 1 } + p ^ { m + d - 1 } } . \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} \\sum _ { ( i , j ) \\in B } m _ G ( i , j ) ^ 2 \\geq \\frac { ( \\sum _ { ( i , j ) \\in B } m _ G ( i , j ) ) ^ 2 } { | B | } = \\frac { ( \\sum _ { i < j } m _ G ( i , j ) ) ^ 2 } { | B | } \\geq \\frac { ( \\sum _ { i < j } m _ G ( i , j ) ) ^ 2 } { 8 b ^ 2 n / a } . \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} C _ { 4 } = \\sum _ { \\substack { 1 \\leq i \\leq p - 1 \\\\ i \\ne ( 4 a ) ^ { - 1 } c ^ { 2 } \\\\ \\bigl ( \\frac { i } { p } \\bigr ) = \\bigl ( \\frac { a } { p } \\bigr ) } } ( p ^ { e - 2 d - 1 } + p ^ { m - d - 1 } ) w _ { c } ^ { p ^ { e - 2 } - \\bigl ( \\frac { - 1 } { p } \\bigr ) p ^ { m + d - 1 } } \\prod _ { \\substack { 1 \\leq j \\leq p - 1 } } w _ { j + c } ^ { p ^ { e - 2 } - \\bigl ( \\frac { j ^ { 2 } - 4 a i } { p } \\bigr ) p ^ { m + d - 1 } } . \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} \\rho ( x ) : = ( 1 + | x ' | ^ 4 + | x '' | ^ 6 + | x _ d | ^ 2 ) ^ { 1 / 2 } , \\gamma = ( \\underbrace { 1 / 2 , \\ldots , 1 / 2 } _ { k \\mbox { t i m e s } } , \\underbrace { 1 / 3 , \\ldots , 1 / 3 } _ { d - 1 - k \\mbox { t i m e s } } , 1 ) . \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} X \\# _ { ( 1 - t ) t _ 0 + t t _ 1 } Y = ( X \\# _ { t _ 0 } Y ) \\# _ t ( X \\# _ { t _ 1 } Y ) \\end{align*}"}
-{"id": "1157.png", "formula": "\\begin{align*} \\begin{array} { r c l } ( \\lambda _ { \\alpha _ i } ( \\alpha _ j ^ * ) ) ( \\alpha _ k ) & = & \\alpha _ i ( \\alpha _ j ^ * ( \\alpha _ k ) ) - \\alpha _ j ^ * ( [ \\alpha _ i , \\alpha _ k ] ) \\\\ & = & - s _ { j , k } ^ i \\ , . \\end{array} \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} a = c _ 0 & = ( n , 0 , \\uparrow ) , & d = c _ { n } = ( 0 , n , \\uparrow ) , \\\\ c _ i & = ( n - i , i , \\downarrow ) , & \\end{align*}"}
-{"id": "72.png", "formula": "\\begin{align*} h ( i , \\boldsymbol { x } ) & { = C _ { 0 } + C _ 1 x _ { j _ 1 } ^ { \\tau - \\zeta _ i - d _ { i , \\bar { U } } - 1 } + \\cdots + C _ { l - 1 } x _ { j _ 1 } \\cdots x _ { j _ { l - 2 } } x _ { j _ { l - 1 } } ^ { \\tau - \\zeta _ i - l - d _ { i , \\bar { U } } + 1 } } \\\\ & { + C _ { l } x _ { j _ 1 } x _ { j _ 2 } \\cdots x _ { j _ { l - 1 } } x _ { j _ l } ^ { \\tau - \\zeta _ i - l - d _ { i , \\bar { U } } } } \\\\ & { = : C _ 0 h _ 0 ( i , \\boldsymbol { x } ) + C _ 1 h _ 1 ( i , \\boldsymbol { x } ) + \\dots + C _ { l } h _ { l } ( i , \\boldsymbol { x } ) } , \\end{align*}"}
-{"id": "8926.png", "formula": "\\begin{align*} M ^ l ( u _ j ^ * ) \\geq \\delta \\int _ { \\partial \\Delta ' } u ^ * _ j P _ { D H } ' ( p ) d \\sigma = \\delta > 0 , \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} \\lim _ { \\tau \\uparrow \\tau _ { c } } \\frac { E _ { \\tau } ( 1 ) } { ( \\tau _ c - \\tau ) ^ { \\frac { s } { s - 1 } } } = \\left ( 1 - \\frac { 1 } { s } \\right ) \\left ( s z \\int _ { \\mathbb { R } ^ 3 } \\frac { Q ( x ) } { | x | ^ { s } } { \\rm d } x \\right ) ^ { \\frac { 1 } { 1 - s } } \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} K _ s ' ( z ) = \\frac s z K _ s ( z ) - K _ { s + 1 } ( z ) , \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{align*} { \\bf D } _ A = \\omega { \\bf I } _ K + { \\bf { G } } _ c . \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} R _ { k } = \\left \\{ \\begin{array} { l l } \\log _ 2 ( 1 + A _ d * \\Gamma _ { k } ) , & \\Gamma _ { k } < \\Gamma _ { } \\\\ \\log _ 2 ( 1 + A _ d * \\Gamma _ { } ) , & \\Gamma _ { k } \\geqslant \\Gamma _ { } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} \\rho ( \\omega ' _ n \\omega ' ) = \\omega '' _ n \\omega '' = \\omega ^ W _ n \\omega '' = \\rho ( \\omega ^ V _ n \\omega ' ) , ~ f o r ~ n \\geq 0 . \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { H - 1 } ( a _ { i k } b _ { k j } { - } a ^ 0 _ { i k } b ^ 0 _ { k j } ) { + } c _ i & = x _ i - \\sum _ { k = 1 } ^ { H - 1 } ( a _ { i k } b _ { k 1 } - a ^ 0 _ { i k } b ^ 0 _ { k 1 } ) + \\sum _ { k = 1 } ^ { H - 1 } ( a _ { i k } b _ { k j } { - } a ^ 0 _ { i k } b ^ 0 _ { k j } ) \\\\ & = x _ i + \\sum _ { k = 1 } ^ { H - 1 } \\{ ( a _ { i k } b _ { k j } { - } a ^ 0 _ { i k } b ^ 0 _ { k j } ) - ( a _ { i k } b _ { k 1 } - a ^ 0 _ { i k } b ^ 0 _ { k 1 } ) \\} \\end{align*}"}
-{"id": "5313.png", "formula": "\\begin{align*} m - \\bar m & \\ge N ( ( 1 - \\rho ) ^ 2 - ( 1 - \\lambda ) ^ 2 ) - a _ 0 N ^ { 2 / 3 } - 1 = N ( \\lambda - \\rho ) ( 2 - \\rho - \\lambda ) - a _ 0 N ^ { 2 / 3 } - 1 \\\\ & = ( 2 - \\rho - \\lambda ) r N ^ { 2 / 3 } - a _ 0 N ^ { 2 / 3 } - 1 \\ge ( 1 - \\rho ) r N ^ { 2 / 3 } \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} H ^ { u } ( L ^ { I } ( M ) ) _ { n l - 1 } = 0 \\mbox { f o r a l l } n \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{align*} d \\ell = \\left ( \\frac { B ^ r } { \\delta _ 0 ^ r } + O ( B ^ { r + 1 } ) \\right ) \\ , d b . \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} A _ 0 ( v _ 1 , . . . , v _ k ) = - \\int _ 0 ^ \\infty B ( e ^ { - t \\tilde \\gamma } v _ 1 , . . . , e ^ { - t \\tilde \\gamma } v _ k ) d t \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{align*} \\left \\langle U _ k ( b , z ^ \\prime ) , S _ j ( a , z ) \\right \\rangle = \\int _ M U _ k ( b , z ^ \\prime ) \\wedge S _ j ( a , z ) = \\delta _ { j k } \\delta _ { z z ^ \\prime } \\delta _ { a b } . \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} & \\sum _ { a = 3 } ^ { n - 1 } \\sum _ { \\ell = 1 } ^ { a - 2 } \\binom { n - 1 - a + \\ell } { \\ell } \\left [ \\binom { a - 1 } { \\ell } - 1 \\right ] = \\sum _ { \\ell = 1 } ^ { n - 3 } \\sum _ { a = \\ell + 1 } ^ { n - 1 } \\binom { n - 1 - a + \\ell } { \\ell } \\left [ \\binom { a - 1 } { \\ell } - 1 \\right ] \\\\ & = \\sum _ { \\ell = 1 } ^ { n - 3 } \\left [ \\binom { n - 1 + \\ell } { 2 \\ell + 1 } - \\binom { n - 1 } { \\ell + 1 } \\right ] = \\sum _ { \\ell = 0 } ^ { n - 2 } \\binom { n - 1 + \\ell } { 2 \\ell + 1 } - ( 2 ^ { n - 1 } - 1 ) . \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ M \\frac { s ( p - 1 ) } { w _ p ( f _ i ) } \\le \\sum _ { i = 1 } ^ M \\sum _ { \\ell = 1 } ^ { R _ i } \\sigma _ p ( \\gamma _ { i \\ell } ) . \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} \\mathcal { F } ( R , P ) = - \\sum _ { j , k = 1 } ^ n \\frac { 1 - \\sigma _ j \\sigma _ k } { \\left ( 1 + \\sigma _ j ^ 2 \\right ) \\left ( 1 + \\sigma _ k ^ 2 \\right ) } \\left | { \\tilde { \\tilde P } } _ { j k } \\right | ^ 2 , \\end{align*}"}
-{"id": "81.png", "formula": "\\begin{align*} \\tilde { h } ( i , \\boldsymbol { x } ) & = C _ l x _ { j _ l } ^ { \\tau - \\zeta _ i - 1 } + \\dots + C _ 2 x _ { j _ 2 } ^ { \\tau - \\zeta _ i - l + 1 } x _ { j _ 3 } \\cdots x _ { j _ l } + C _ 1 x _ { j _ 1 } ^ { \\tau - \\zeta _ i - l } x _ { j _ 2 } \\cdots x _ { j _ l } \\\\ & = : C _ l \\tilde { h } _ l ( i , \\boldsymbol { x } ) + C _ { l - 1 } \\tilde { h } _ { l - 1 } ( i , \\boldsymbol { x } ) + \\dots + C _ 1 \\tilde { h } _ 1 ( i , \\boldsymbol { x } ) . \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} \\omega _ 2 + \\omega _ 1 = i \\log ( \\lambda ) + \\pi + i 4 \\log 2 + O ( \\lambda \\log ( \\lambda ) ) . \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} N ( y , k ) \\sum _ { n = 1 } ^ { \\infty } \\left | f _ n ( z ) \\right | \\ll \\frac { \\sqrt { k } } { \\ell } . \\end{align*}"}
-{"id": "807.png", "formula": "\\begin{align*} \\chi ( X ) = \\int _ { X } { \\mathcal P } ( R ) \\ , d X , \\end{align*}"}
-{"id": "6534.png", "formula": "\\begin{align*} A _ m \\geq n ^ { - 1 } \\sigma ^ 2 C _ 1 \\sum _ { j = i _ 0 } ^ { m - 1 } j ^ { \\beta / r } \\left ( C _ 1 m ^ { \\beta / r } - C _ 2 j ^ { \\beta / r } \\right ) \\geq K _ 1 n ^ { - 1 } m ^ { \\beta / r } \\sum _ { j = i _ 0 } ^ { m - 1 } j ^ { \\beta / r } . \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} f ( x _ 1 , \\ldots , 1 , \\ldots , x _ n ) & = \\operatorname { s g n } \\left ( \\sum _ { j = 0 , \\ ; j \\neq i } ^ n w _ j x _ j + w _ i - \\theta \\right ) \\\\ f ( x _ 1 , \\ldots , 0 , \\ldots , x _ n ) & = \\operatorname { s g n } \\left ( \\sum _ { j = 0 , \\ ; j \\neq i } ^ n w _ j x _ j - \\theta \\right ) . \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{align*} f \\left ( t \\right ) = f \\left ( 0 \\right ) \\sum _ { k = 0 } ^ { \\infty { } } \\frac { { \\left ( { E \\left ( i \\hslash { } \\right ) } ^ { - \\beta { } } t ^ { \\beta { } } \\right ) } ^ k } { \\Gamma \\left ( \\beta { } k + 1 \\right ) } = f \\left ( 0 \\right ) E _ { \\beta { } } \\left ( { \\left ( \\frac { t } { i \\hslash { } } \\right ) } ^ { \\beta { } } E \\right ) , \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} \\| j _ 0 ^ m \\hat { \\varphi } ^ * s \\| _ { \\sigma } = R _ { \\sigma } ^ { - m } \\| j _ 0 ^ m \\varphi _ { \\sigma } ^ * s \\| _ { R _ { \\sigma } } \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} H ^ 0 ( M , \\ , ( F _ 1 , \\ , F _ i ) ) \\ , = H ^ 0 ( M , \\ , F _ i \\otimes F ^ * _ 1 ) \\ , = \\ , 0 \\ , . \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} & \\frac { \\partial z } { \\partial x } ( x , 1 ) = - \\frac { \\frac { \\partial f } { \\partial x } ( x , 1 , z ( x , 1 ) ) } { \\frac { \\partial f } { \\partial z } ( x , 1 , z ( x , 1 ) ) } > 0 , \\frac { \\partial z } { \\partial x } ( x , - 1 ) = - \\frac { \\frac { \\partial f } { \\partial x } ( x , - 1 , z ( x , - 1 ) ) } { \\frac { \\partial f } { \\partial z } ( x , - 1 , z ( x , - 1 ) ) } < 0 , \\\\ & ( \\forall x \\in \\R _ { > 0 } ) , \\end{align*}"}
-{"id": "6718.png", "formula": "\\begin{align*} v \\left ( x , t \\right ) = \\underset { u \\in \\mathcal { C } } { } \\ , K \\left ( x , t , u \\right ) . \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} \\zeta ( z ) : = \\int \\Phi ( \\theta ) ^ z \\varphi ( \\theta ) d \\theta , \\end{align*}"}
-{"id": "6448.png", "formula": "\\begin{align*} \\begin{aligned} 6 C \\tilde { C } _ { T } \\Big [ 2 K _ { 1 } + 6 K _ { 1 } K _ { 2 } + K _ { 1 } ^ 2 \\Big ] & \\leq 6 C \\tilde { C } _ { T } \\Big [ 3 K + 2 4 K \\| b \\| _ { L ^ { \\infty } ( \\Omega ) ^ 3 } \\Big ] = 1 4 4 C \\tilde { C } _ { T } K ( 1 + \\| b \\| _ { L ^ { \\infty } ( \\Omega ) ^ 3 } ) , \\end{aligned} \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} \\left \\{ \\underline { g } \\in \\prod _ { i = 1 } ^ { n } \\mathcal { L } _ { G } ^ { i } \\middle | \\overline { \\left \\langle \\underline { g } \\right \\rangle } = x P x ^ { - 1 } x \\in G \\right \\} \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} i \\kappa ^ \\perp = i s ^ { - 4 } ( | A _ 1 | ^ 2 + | A _ 2 | ^ 2 ) , \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} G _ { p ^ \\infty } ^ { \\vec { i } } = \\omega _ K ( U ^ { \\vec { i } } K _ 2 ^ { t o p } ( K ) ) . \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} S ''' ( i , j , k , \\alpha , \\beta ) : = \\omega _ { i k } ^ \\alpha \\underline \\wedge \\omega _ { j k } ^ \\beta ( i < j < k \\in [ n ] , \\alpha , \\beta \\geq 0 ) , \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} & \\forall \\ x , y , z \\in \\mathcal { H } ( A ) = A _ 0 \\cup A _ 1 : \\\\ & a s _ { \\alpha , \\bullet } ( x , y , z ) + ( - 1 ) ^ { | y | | z | } a s _ { \\alpha , \\bullet } ( x , z , y ) = 0 , \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} f ( x , \\lambda ) = \\max _ { m \\in S } \\bigl ( m x - \\lambda | m ^ 2 - u | + J ( m ) \\bigr ) \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l } \\nabla _ { B } \\nabla _ { B } h + a h g _ { B } = 0 , \\\\ \\noalign { \\smallskip } R i c _ { B } + \\nabla _ { B } \\nabla _ { B } \\beta = [ h ^ { - 1 } ( \\nabla _ { B } h ) \\beta - b h ^ { - 1 } + ( n - 1 ) a ] g _ { B } , \\\\ \\noalign { \\smallskip } \\nabla _ { F } \\nabla _ { F } \\varphi + ( c \\varphi + b ) g _ { F } = 0 , \\\\ \\noalign { \\smallskip } R i c _ { F } = ( m - 1 ) c g _ { F } , \\end{array} \\right . \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} W ^ { ( \\alpha + 1 ) / 2 , 2 } ( \\R / ( L \\Z ) , \\R ^ n ) = W ^ { 1 + s , 2 } ( \\R / ( L \\Z ) , \\R ^ n ) & \\hookrightarrow C ^ { 1 , s - ( 1 / 2 ) } ( \\R / ( L \\Z ) , \\R ^ n ) \\\\ & = C ^ { 1 , ( \\alpha / 2 ) - 1 } ( \\R / ( L \\Z ) , \\R ^ n ) , \\end{align*}"}
-{"id": "5391.png", "formula": "\\begin{align*} \\Delta W ^ 0 _ 4 ( p , p _ 1 , p _ 2 , p _ 3 ) & = 1 \\otimes W ^ 0 _ 4 ( p , p _ 1 , p _ 2 , p _ 3 ) + W ^ 0 _ 4 ( p , p _ 1 , p _ 2 , p _ 3 ) \\otimes 1 + \\\\ W ^ 0 _ 3 ( p , p _ 1 , \\bar { q } ) \\otimes & W ^ 0 _ 3 ( \\bar { q } , p _ 2 , p _ 3 ) + W ^ 0 _ 3 ( q , p _ 1 , p _ 2 ) \\otimes W ^ 0 _ 3 ( p , q , p _ 3 ) . \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} \\bar u ^ { \\varepsilon , N , n } ( t , \\cdot ) : = K _ { \\varepsilon } \\ast \\bar \\gamma _ t ^ { \\varepsilon , N , n } \\ , \\end{align*}"}
-{"id": "5219.png", "formula": "\\begin{align*} u _ t = \\Delta u - \\chi \\nabla v \\cdot \\nabla u + u ( a ( x + x _ 0 , t ) - \\chi \\lambda v ( x , t ; t _ 0 , x _ 0 , u _ 0 ) - ( b ( x + x _ 0 , t ) - \\chi \\mu ) u ) , x \\in \\R ^ N , \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} \\widetilde { W } _ n : = V _ { \\nu _ n } + b ^ * - \\frac 1 4 . \\end{align*}"}
-{"id": "3495.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 , x _ 3 ) = \\max \\{ \\ , x _ 1 - x _ 2 - x _ 3 , \\ , x _ 1 + 4 , \\ , x _ 1 + x _ 2 + x _ 3 , \\ , - x _ 1 + x _ 2 + 2 \\ , \\} . \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ 3 X _ { \\pm \\theta } ^ i = 0 , \\big [ X _ \\theta ^ j , X _ { - \\theta } ^ k \\big ] = - \\sigma _ j \\sigma _ k \\sum _ { i = 1 } ^ 3 H _ { \\beta _ i } . \\end{gather*}"}
-{"id": "7849.png", "formula": "\\begin{align*} f _ t ( s ) = c _ t ( s ) { \\left ( \\frac { d \\mu \\circ \\phi _ t } { d \\mu } ( s ) \\right ) } ^ { 1 / \\alpha } f \\circ \\phi _ t ( s ) , \\ ; \\ ; \\ \\ t \\in \\mathbb { Z } ^ d . \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} i \\partial _ t u = \\mathcal { T } ( u , u , u ) \\mbox { w i t h } \\mathcal { T } ( f , f , f ) = \\int _ 0 ^ { 2 \\pi } | e ^ { i s H _ 0 } f | ^ 2 e ^ { i s H _ 0 } f \\ , d s . \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} \\lambda _ { i + 1 } - \\lambda _ i & = v _ p \\left ( \\frac { r ^ p - 1 } { r - 1 } \\right ) \\\\ & = v _ p \\left ( 1 + r + \\dots + r ^ { p - 1 } \\right ) . \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{align*} L ^ { \\mathrm { D I F } } f ( x ) = c \\sum _ { 1 \\leq i < j \\leq N } \\left ( \\frac { \\partial } { \\partial x _ i } - \\frac { \\partial } { \\partial x _ j } \\right ) ^ 2 f ( x ) - ( x _ i - x _ j ) \\left ( \\frac { \\partial } { \\partial x _ i } - \\frac { \\partial } { \\partial x _ j } \\right ) f ( x ) , \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\ddot { q } ( t ) & = - \\alpha \\sum _ { i \\in \\Z ^ d } \\eta ( Q _ i ( t ) ) \\nabla \\sigma ( q ( t ) - r _ i ) , \\\\ M \\ddot { Q } _ i ( t ) + U ' ( Q _ i ( t ) ) & = - \\tilde { \\alpha } \\eta ' ( Q _ i ( t ) ) \\sigma ( q ( t ) - r _ i ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{align*} g \\cdot m ' = \\beta B m ' \\end{align*}"}
-{"id": "1615.png", "formula": "\\begin{align*} r : = \\frac { 4 d | p | } { \\lambda _ t } . \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{align*} - \\mathcal S : = \\{ - z \\ , | \\ z \\in \\mathcal S \\} , \\mathcal S ^ { - 1 } : = \\{ z ^ { - 1 } \\ , | \\ z \\in \\mathcal S \\} , \\sqrt [ p ] { \\mathcal S } : = \\{ z \\in \\C \\cup \\{ \\infty \\} \\ | \\ z ^ p \\in \\mathcal S \\} \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{align*} \\alpha = \\sum _ { i \\geq - n _ 0 } \\alpha _ i \\cdot S ^ { i / e } \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} g _ 1 ( z _ 1 , \\dots , z _ n ) \\propto g _ 2 ( z _ 1 , \\dots , z _ n ) \\iff \\exists \\ \\alpha \\in \\mathbb { Q } ( q , t ) \\ \\ g _ 1 ( z _ 1 , \\dots , z _ n ) = \\alpha g _ 2 ( z _ 1 , \\dots , z _ n ) . \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} - \\omega _ 1 = \\int _ 0 ^ { \\lambda } \\frac { d X } { \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } , ~ ~ \\omega _ 2 = \\int _ { \\lambda } ^ 1 \\frac { d X } { \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } . \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{align*} F _ T ( x ) = 1 + \\frac { 1 + r t } { 1 + 1 / r + 1 / t - 1 / x } \\ , , \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{align*} W ( x , y ) = \\sum _ { i = 0 } ^ n A _ i x ^ { 2 ( n - i ) + 1 } y ^ { 2 i } \\qquad ( A _ 0 = 1 ) \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} F ( \\emptyset ) = \\frac { ( N - | \\mu | ) ( N - | \\mu | - 1 ) } { 2 } - \\frac { | \\mu | ( | \\mu | - 1 ) } { 2 } = \\frac { ( N - 1 ) ( N - 2 | \\mu | ) } { 2 } \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} U _ 1 ( \\Delta _ F \\varphi ) = - c m U _ 1 ( \\varphi ) . \\end{align*}"}
-{"id": "6790.png", "formula": "\\begin{align*} \\Box \\xi = - h ^ { \\alpha \\beta } \\nabla ^ 2 _ { \\alpha \\beta } \\xi \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} & u \\in S _ q ^ u ( T ) \\cap B C ( [ 0 , T ) ; L _ { \\sigma } ^ p ( \\Omega ) ) , \\\\ & \\delta \\in S _ q ^ d ( T ) \\cap B C ( [ 0 , T ) ; W _ 0 ^ { 1 , p } ( \\Omega ) ^ 3 ) \\cap B C ( [ 0 , T ) ; L ^ { \\infty } ( \\Omega ) ^ 3 ) , \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{align*} \\frac { d } { d s } \\langle X , T \\rangle & = 1 + \\kappa \\langle X , N \\rangle , \\\\ \\frac { d } { d s } \\langle X , N \\rangle & = - \\kappa \\langle X , N \\rangle . \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} ( \\partial _ { z _ { j - 1 } } - \\partial _ { z _ { j } } ) \\sqrt { h _ m } = \\big ( \\frac 1 m - { \\bf 1 } _ { \\{ j = m \\} } \\big ) \\sqrt { h _ m } \\ , . \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} \\triangle _ { g } F = - \\frac { 1 } { 2 } F ^ { \\perp } . \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} | V ( x ) | = \\frac { | ( T ( D ) - \\lambda ) u ( x ) | } { | u ( x ) | } \\lesssim ( 1 + | P _ { \\nu } x | + | P _ { \\nu } ^ { \\perp } x | ^ 2 ) ^ { - 1 } . \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} C _ { 5 } = \\sum _ { \\substack { 1 \\leq i \\leq p - 1 \\\\ \\bigl ( \\frac { i } { p } \\bigr ) \\ne \\bigl ( \\frac { a } { p } \\bigr ) } } ( p ^ { e - 2 d - 1 } + p ^ { m - d - 1 } ) w _ { c } ^ { p ^ { e - 2 } + ( \\frac { - 1 } { p } ) p ^ { m + d - 1 } } \\prod _ { \\substack { 1 \\leq j \\leq p - 1 } } w _ { j + c } ^ { p ^ { e - 2 } - \\bigl ( \\frac { j ^ { 2 } - 4 a i } { p } \\bigr ) p ^ { m + d - 1 } } \\end{align*}"}
-{"id": "4766.png", "formula": "\\begin{align*} { \\rm K e r } ( \\nabla g ( \\overline { x } ) ) \\cap { \\rm R a n g e } ( \\nabla \\Xi ( g ( \\overline { x } ) ) ) = \\{ 0 \\} . \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} \\Omega _ { \\beta _ 1 , \\bar { \\beta } _ 2 } & = \\frac { 1 } { 2 } \\tanh ( \\beta _ 2 ( a ) - \\beta _ 1 ( a ) ) ( \\frac { 1 } { \\sinh ( 2 \\beta _ 1 ( a ) } - \\frac { 1 } { \\sinh ( 2 \\beta _ 2 ( a ) } ) \\chi ( [ \\theta ( e _ { \\beta _ 2 } ) , e _ { \\beta _ 1 } ] ) \\\\ & + \\frac { 1 } { 2 } \\tanh ( \\beta _ 1 + \\beta _ 2 ) ( \\frac { 1 } { \\sinh ( 2 \\beta _ 1 ( a ) } + \\frac { 1 } { \\sinh ( 2 \\beta _ 2 ( a ) } ) \\chi ( [ \\theta ( e _ { \\beta _ 2 } ) , \\sigma ( e _ { \\beta _ 1 } ) ] ) \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} \\underline { M } : = \\frac { ( b _ { \\inf } - \\chi \\mu ) a _ { \\inf } - \\chi \\mu a _ { \\sup } } { ( b _ { \\sup } - \\chi \\mu ) ( b _ { \\inf } - \\chi \\mu ) - ( \\chi \\mu ) ^ 2 } > \\frac { a _ { \\inf } - \\frac { \\chi \\mu a _ { \\sup } } { b _ { \\inf } - \\chi \\mu } } { b _ { \\sup } - \\chi \\mu } \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{align*} \\partial _ { s } T & = \\kappa N \\\\ \\partial _ { s } N & = - \\kappa T \\\\ \\partial _ { s } B & = 0 \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} \\int _ 0 ^ t - ( L _ s + \\tilde { L } _ { T - s } ) f ( s , \\xi _ s ) = M _ t + \\tilde { M } _ T - \\tilde { M } _ { T - t } . \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{align*} \\mathfrak { b } _ { \\infty } ( A ) & = \\{ B \\in \\mathcal { L } ( \\C , X _ { - 1 } ) \\colon B L ^ { \\infty } A \\} \\\\ \\mathfrak { c } _ { 1 } ( A ) & = \\{ C \\in \\mathcal { L } ( X _ { 1 } , \\C ) \\colon C L ^ { 1 } A \\} . \\end{align*}"}
-{"id": "1038.png", "formula": "\\begin{align*} \\lim _ { \\tau \\uparrow \\tau _ { c } } \\frac { E _ { \\tau } ^ { \\infty } ( 1 ) } { ( \\tau _ c - \\tau ) ^ { \\frac { 1 } { 2 } } } = \\frac { 3 } { 2 } m \\left ( \\frac { 1 } { K _ { \\rm c l } } \\int _ { \\mathbb { R } ^ 3 } Q ( x ) ^ { \\frac { 2 } { 3 } } { \\rm d } x \\right ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{align*} - \\frac { z ^ { 2 0 } - z ^ { 1 9 } - z ^ { 1 3 } - z ^ { 7 } - z + 1 } { 1 - z ^ { 1 9 } } = - 1 + z + z ^ { 7 } + z ^ { 1 3 } + \\ldots = G _ { 7 } ( z ) + z ^ { 1 3 } + \\ldots , \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} N \\ll C ^ { 2 ( n + 1 ) r ^ { n + 1 } ( d - 1 ) ^ { n - 1 } d } = \\mathfrak { C } , \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} \\mathcal { M } _ { x } ( v ) : = \\big \\{ \\lambda \\in \\mathcal { N } _ K ( g ( x ) ) \\ | \\ v = \\nabla g ( x ) \\lambda \\big \\} . \\end{align*}"}
-{"id": "2719.png", "formula": "\\begin{align*} g ^ { ( n ) } ( x ) = \\sum _ { i = 0 } ^ n p ^ i x _ i ^ { p ^ { n - i } } , \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} u ^ { \\varepsilon , N } ( t , \\cdot ) : = K _ { \\varepsilon } \\ast \\gamma _ t ^ { \\varepsilon , N } \\ . \\end{align*}"}
-{"id": "5221.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ { t } w _ { n } = \\Delta w _ { n } + b _ { n } ( t , x ) \\cdot \\nabla w _ { n } + f _ { n } ( t , x ) w _ n + g _ { n } ( t , x ) v _ { n } + h _ { n } \\cdot \\nabla v _ { n } , x \\in \\R ^ N , t > 0 \\cr 0 = \\Delta v _ n - \\lambda v _ { n } + \\mu w _ { n } , x \\in \\R ^ N , t > 0 \\cr w _ { n } ( 0 , x ) = u _ { 0 n } ( x + x _ n ) - \\tilde { u } _ { 0 n } ( x ) , x \\in \\R ^ N , \\end{cases} \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } { \\mathbb P } ( \\varphi _ n ( f _ 1 , \\ldots , f _ s ) \\in J ) = V ( J ) . \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} _ { m , 2 } ^ 1 = \\frac { \\alpha ^ 2 _ 1 z _ { m , 2 } } { \\alpha ^ 2 _ 2 z _ { m , 2 } + ^ { m , 2 } _ { i n t e r } + \\frac { 1 } { \\rho } } . \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} c h ^ { q ^ l } + d g ^ { q ^ { l + n } } = 0 . \\end{align*}"}
-{"id": "2756.png", "formula": "\\begin{align*} \\left ( \\sum _ { k = 0 } ^ \\infty c _ { i j k } p ^ k \\sum _ { \\substack { ( i , j ) \\in \\mathbb { J } _ S \\\\ \\frac { m } { i } = \\frac { n } { j } \\in \\mathbb { Z } _ { \\geq 0 } } } ( - j ) [ - a _ { i j k } ] ^ { m / i } \\right ) _ { ( m , n ) \\in \\mathbb { J } _ S } , \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} \\hat f ( k ) = & \\frac { 2 } { \\sqrt 2 \\pi } \\frac { 1 } { k } \\Gamma ( b + 1 ) ( a ^ 2 + k ^ 2 ) ^ { - \\frac { 1 } { 2 } ( b + 1 ) } \\sin [ ( b + 1 ) \\tan ^ { - 1 } ( \\frac { k } { a } ) ] \\\\ = & \\times \\frac { 1 } { k } ( a ^ 2 + k ^ 2 ) ^ { - \\frac { 1 } { 2 } ( b + 1 ) } \\sin [ ( b + 1 ) \\tan ^ { - 1 } ( \\frac { k } { a } ) ] \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{align*} E \\left [ S _ N ^ { ( r ) } \\right ] = N ^ r \\int _ 0 ^ { \\infty } r s ^ { r - 1 } \\left [ 1 - ( 1 - e ^ { - s } ) ^ N \\right ] d s . \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{align*} \\begin{aligned} \\hat F _ \\lambda = \\sum _ { l \\in \\bar l } \\lambda _ l \\hat F _ { \\tilde d _ l } , \\end{aligned} \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} \\begin{cases} \\overline { k } _ { y y y } + \\overline { k } _ y + \\overline { k } _ { x x x } + \\overline { k } _ x + \\lambda \\overline { k } = \\lambda \\delta ( x - y ) & , \\\\ \\overline { k } ( x , 0 ) = \\overline { k } ( x , L ) = 0 , & , \\\\ \\overline { k } _ y ( x , 0 ) = \\overline { k } _ y ( x , L ) = 0 , & , \\\\ \\overline { k } ( 0 , y ) = \\overline { k } ( L , y ) = 0 , & \\end{cases} \\end{align*}"}
-{"id": "1820.png", "formula": "\\begin{align*} \\mathbf { H } _ 0 = - i { \\hslash } c \\boldsymbol { \\alpha } \\cdot \\boldsymbol { \\nabla } + m c ^ 2 \\beta \\ , , \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} L u = g + \\sum _ { i = 1 } ^ N \\partial _ i f _ i \\end{align*}"}
-{"id": "7832.png", "formula": "\\begin{align*} S \\left ( \\phi , a + \\frac { r N } { 2 } \\right ) = S ( \\phi , a ) \\ , . \\end{align*}"}
-{"id": "5073.png", "formula": "\\begin{align*} J = J ( \\{ A _ n \\} , \\{ B _ n \\} ) = \\begin{bmatrix} B _ 1 & A _ 1 & 0 & \\ldots \\\\ A ^ * _ 1 & B _ 2 & A _ 2 & \\ddots \\\\ 0 & A ^ * _ 2 & B _ 3 & \\ddots \\\\ \\vdots & \\ddots & \\ddots & \\ddots \\end{bmatrix} , \\end{align*}"}
-{"id": "5063.png", "formula": "\\begin{align*} S ^ { ( m ) } = \\{ [ y _ 1 , y _ 2 , \\dots , y _ m ] \\mid y _ 1 , y _ m \\in X ; \\ y _ 2 , \\dots , y _ { m - 1 } \\in X \\cup X ^ 2 \\} . \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} \\int _ { \\R ^ { N } } & [ f ( u _ { n _ { j } } ) - f ( v _ { n _ { j } } ) - f ( u ) ] w \\ , d x = \\\\ & = \\int _ { \\R ^ { N } } [ f ( u _ { n _ { j } } ) - f ( u _ { n _ { j } } - \\tilde { u } _ { j } ) - f ( \\tilde { u } _ { j } ) ] w \\ , d x \\\\ & + \\int _ { \\R ^ { N } } [ f ( v _ { n _ { j } } + h _ { j } ) - f ( v _ { n _ { j } } ) ] w \\ , d x + \\int _ { \\R ^ { N } } [ f ( \\tilde { u } _ { j } ) - f ( u ) ] w \\ , d x \\\\ & = I _ { j } + I I _ { j } + I I I _ { j } . \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{align*} \\liminf I _ { q _ { n } } ( U _ { q _ { n } } ) \\leq \\liminf I _ { q _ { n } } ( U _ { q _ { 0 } } ) = I _ { q _ { 0 } } ( U _ { q _ { 0 } } ) . \\end{align*}"}
-{"id": "2392.png", "formula": "\\begin{align*} E \\left [ T _ 2 \\right ] = \\frac { \\nu _ 2 } { \\alpha _ 2 } \\ , M H _ { \\nu _ 2 M } \\ , + \\ , O \\left ( e ^ { - \\varepsilon M } \\right ) = \\big ( \\lambda ^ { - 1 } \\nu _ 1 + \\nu _ 2 \\big ) M H _ { \\nu _ 2 M } \\ , + \\ , O \\left ( e ^ { - \\varepsilon M } \\right ) , M \\to \\infty . \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} I _ 1 ( \\alpha ^ { ( 1 ) } , \\alpha ^ { ( 2 ) } , \\dots , \\alpha ^ { ( N ) } ) = \\sum \\limits _ { k _ 1 = 1 } ^ { n _ 1 } \\sum \\limits _ { k _ 2 = 1 } ^ { n _ 2 } \\dots \\sum \\limits _ { k _ { N } = 1 } ^ { n _ N } A ( k _ 1 , k _ 2 , \\dots , k _ { N } ) \\alpha _ { k _ 1 } ^ { ( 1 ) } \\alpha _ { k _ 2 } ^ { ( 2 ) } \\dots \\alpha _ { k _ { N } } ^ { ( N ) } , \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} \\delta = \\max \\{ 1 , 2 5 6 \\frac { | \\lambda ^ 2 - \\lambda + 1 | ^ 3 } { | \\lambda ( \\lambda - 1 ) | ^ 2 } \\} ^ { 1 / 6 } | \\lambda ( 1 - \\lambda ) | ^ { 1 / 3 } . \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{align*} \\mathbf { F } _ n ( t ) = \\exp ( t F _ n ) = \\left ( t F _ n , 0 \\right ) \\in V \\oplus ( V \\wedge V ) \\simeq G ^ 2 ( V ) \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{align*} \\lambda \\int _ { \\mathbb R } g ' ( x ) U ( x ) \\ , \\mathrm d x = \\int _ { \\mathbb R } g ' ( x ) \\bigl ( f ' ( x ) \\overline { f ( x ) } + V ( x ) \\bigr ) \\ , \\mathrm d x . \\end{align*}"}
-{"id": "3546.png", "formula": "\\begin{align*} A \\langle X , T \\rangle + B \\langle X , N \\rangle + \\langle C , N \\rangle = \\kappa ( x ) \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} u ^ 2 \\left ( X \\cdot Y \\cdot Z \\cdot \\bar { \\psi } , \\bar { \\psi } \\right ) = u ^ 2 \\bar { \\varphi } ( X , Y , Z ) . \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} \\bar { x } ^ T d \\ , = \\ , 0 \\ \\Rightarrow \\ d ^ { \\ , T } Q d \\ , \\leq \\ , \\| \\ , d \\ , \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{align*} \\iota _ { X _ - } \\omega _ + + \\iota _ { X _ 0 } \\omega _ 0 + \\iota _ { X _ + } \\omega _ - = d \\mu _ N \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} \\rho ^ I ( M ) : = \\min \\{ n : \\widetilde { I ^ { i } M } = I ^ i M ~ \\mbox { f o r a l l } ~ i \\geqslant n \\} . \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} \\| K \\| _ { L ^ { q ' } ( \\Omega , L ^ q ( \\Omega ) ) } = \\left ( \\int _ { \\Omega } \\left ( \\int _ { \\Omega } | K ( x , y ) | ^ q d \\mu ( x ) \\right ) ^ { \\frac { q ' } q } d \\mu ( y ) \\right ) ^ { \\frac { 1 } { q ' } } < \\infty . \\end{align*}"}
-{"id": "4074.png", "formula": "\\begin{align*} \\sum _ { a } p _ A ( a ) \\frac { q _ { R | A } ( r | a ) } { q _ { R } ( r ) } = 1 . \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{align*} \\frac { I ( U ; Y | V ) } { I ( U ; X | V ) } = \\frac { I ( U ; Y | V = v ^ * ) } { I ( U ; X | V = v ^ * ) } . \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} d X _ 0 ( s ) & = \\xi '' ( s ) \\gamma ( s ) \\partial _ x \\Phi _ { u , \\gamma _ 0 } ( s , X _ 0 ( s ) ) d s + { \\xi '' ( s ) } ^ { 1 / 2 } d W ( s ) , \\ , \\ , X _ 0 ( 0 ) = h . \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} D _ { \\delta , \\mu } \\equiv \\left \\{ R \\mid R = \\omega + \\beta , \\omega ^ T = \\omega , \\beta ^ T = - \\beta , \\lambda _ { \\min } ( \\omega ) > 0 , \\delta \\lambda _ { \\min } ( \\omega ) \\geq \\mu , \\| \\beta \\| \\leq \\mu \\right \\} . \\end{align*}"}
-{"id": "7124.png", "formula": "\\begin{align*} \\omega _ { 1 } = \\mathbf { i } \\int _ { a _ { 3 } } ^ { a _ { 4 } } \\frac { d x } { \\sqrt { \\vert D ( x ) \\vert } } \\in \\mathbf { i } \\R _ { > 0 } \\quad \\omega _ { 2 } = \\int _ { a _ { 4 } } ^ { a _ { 1 } } \\frac { d x } { \\sqrt { D ( x ) } } \\in \\R _ { > 0 } . \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} b ^ + ( z ) = \\displaystyle t \\left ( \\frac { 1 - z } { 1 - t z } \\right ) , b ^ - ( z ) = \\displaystyle \\frac { 1 - z } { 1 - t z } , c ^ + ( z ) & = 1 - b ^ + ( z ) , c ^ - ( z ) = 1 - b ^ - ( z ) . \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} ( 0 , 0 , x , y ) & \\quad \\phi _ { \\mathfrak { L } } = u ^ { 4 8 } \\\\ ( u ^ 8 x , u ^ 8 y , x , y ) & \\quad \\quad \\phi _ { \\mathfrak { L } } = u ^ { 5 6 } , \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{align*} G _ { n , \\pm } ( t ) = \\sum _ { k = 1 , 2 } G _ { n , \\pm , k } ( t ) , \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ X f \\ , d [ ( F _ t ) _ * \\mu ^ { ( n ) } ] = \\lim _ { n \\to \\infty } \\int _ X f \\circ F _ t \\ , d \\mu ^ { ( n ) } = \\int _ X f \\circ F _ t \\ , d \\mu = \\int _ X f \\ , d \\mu _ t . \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} \\mu \\left ( \\{ v \\in T ^ 1 \\mathcal { M } _ { g , n } : \\exists \\ , t \\in I _ j ^ { ( k ) } \\textrm { w i t h } \\textrm { s y s } ( \\phi _ t ( v ) ) = T _ k ^ { - \\xi } \\} \\right ) = O \\left ( \\frac { 1 } { T _ k ^ { 4 \\xi } } \\right ) \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{align*} 2 x _ i = \\frac { 1 } { y _ i } \\left ( b _ i + \\sum _ { j = 1 } ^ n w _ { i j } y _ j \\right ) \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} \\big | \\mathbb { E } _ { n , 0 } ^ Q ( \\varphi _ n ) - d ( n ) ^ { - 1 } \\sum _ { i = 1 } ^ { d ( n ) } \\mathbb { E } _ { n , i } ^ Q ( \\varphi _ n ) \\big | ^ 2 = \\left | \\mathbb { E } ^ Q _ { n , 0 } ( \\varphi _ n ( 1 - L _ n ) ) \\right | ^ 2 \\leq \\mathbb { E } ^ Q _ { n , 0 } \\left ( ( 1 - L _ n ) ^ 2 \\right ) = \\mathbb { E } ^ Q _ { n , 0 } \\left ( L _ n ^ 2 \\right ) - 1 , \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( x \\frac { d } { d x } \\right ) ^ 4 \\log \\left ( \\theta _ 3 ( x ) \\right ) & = x \\frac { d } { d x } \\left ( x \\ , \\psi ( x ) + 3 x ^ 2 \\ , \\psi ' ( x ) + x ^ 3 \\ , \\psi '' ( x ) \\right ) \\\\ & = x \\ , \\psi ( x ) + 7 x ^ 2 \\ , \\psi ' ( x ) + 6 x ^ 3 \\ , \\psi '' ( x ) + x ^ 4 \\ , \\psi ''' ( x ) . \\end{aligned} \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} & | A ( t , x ) | \\cdot | \\xi | \\leq \\frac { 1 } { p } | A | ^ p + \\frac { 1 } { q } | \\xi | ^ q \\leq \\frac { 1 } { p } < x > ^ { p ( M + 1 - \\delta ) } + \\frac { 1 } { q } | \\xi | ^ q \\\\ & = \\frac { 1 } { p } < x > ^ { 2 ( M + 1 ) \\delta _ 1 } + \\frac { 1 } { q } | \\xi | ^ { 2 \\delta _ 2 } . \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{align*} ( A \\mathbf x ) \\bmod 1 = ( A ( \\mathbf x \\bmod 1 ) ) \\bmod 1 . \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{align*} \\Delta ( \\varphi _ Q , r _ 2 ( \\varphi _ Q ) , r _ 3 ( \\varphi _ Q ) ) = \\frac { 6 0 } { \\pi ^ 2 } . \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} x = - z _ { x x } [ 1 + C _ { 1 } ^ { 2 } + z _ { x } ^ { 2 } ] ^ { - 3 / 2 } . \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{align*} \\begin{aligned} \\sup _ { 0 < t < T } e ^ { - \\frac { \\omega t } { 2 } } t ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\int _ { 0 } ^ { t } ( t - s ) ^ { - \\frac { 1 } { 2 } - \\frac { 3 } { 2 q } } s ^ { - 3 ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\ \\d s & = \\left ( \\sup _ { 0 < t < T } e ^ { - \\frac { \\omega t } { 2 } } t ^ { \\frac { 1 } { 2 } - \\frac { 3 } { 2 p } } \\right ) B \\big ( 1 - 3 \\big ( \\tfrac { 1 } { p } - \\tfrac { 1 } { q } \\big ) , \\tfrac { 1 } { 2 } - \\tfrac { 3 } { 2 q } \\big ) , \\end{aligned} \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} w ( D ) & = ( n - d ) ^ 2 + ( n - d ) c + w ( D _ 1 ) \\\\ & = n ( n - 2 d + c ) + ( d - c ) ^ 2 + ( d - c ) c + w ( D _ 1 ) \\\\ & = n ( n - ( 2 d - c ) ) + w ( D _ 2 ) . \\end{align*}"}
-{"id": "370.png", "formula": "\\begin{align*} \\mathbb { P } ( X _ i = 0 ) = p _ x = 1 - \\mathbb { P } ( X _ i = 1 ) . \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} J _ t ^ n ( x ) = M ^ n _ t ( x ) + \\int _ { 0 } ^ t j ^ n _ { x , x + 1 } ( \\eta ^ n _ s ) \\ ; d s , \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} r _ p = d _ p \\frac { ( w _ { p - 1 } v _ { \\omega _ p } , n w ^ { - 1 } h w w _ p v _ { \\omega _ p } ) } { ( w _ { p } v _ { \\omega _ p } , n w ^ { - 1 } h w w _ { p } v _ { \\omega _ p } ) } = d _ p \\frac { ( w _ { p - 1 } v _ { \\omega _ p } , g n _ { w ^ { - 1 } } ^ { - 1 } w w _ p v _ { \\omega _ p } ) } { ( w _ { p } v _ { \\omega _ p } , g n _ { w ^ { - 1 } } ^ { - 1 } w w _ { p } v _ { \\omega _ p } ) } , \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} u _ i ( b ) \\cdot F _ { i , b } = u ' _ i ( b ) \\qquad u _ i ( b ) \\cdot F ' _ { i , b } = u '' _ i ( b ) \\end{align*}"}
-{"id": "5947.png", "formula": "\\begin{align*} \\det \\beta = ( \\det \\omega ) ( \\det \\sigma ) \\geq 0 . \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} \\chi ( z ) = \\sum _ { j = 0 } ^ { \\lfloor ( k - 1 ) / 2 \\rfloor } A _ j ( z , . . . , z ) . \\end{align*}"}
-{"id": "5714.png", "formula": "\\begin{align*} \\frac { 1 } { e ( R ) } \\sum _ { f \\in E ( R ) } | \\{ g \\in E ( B ) : f \\cap g \\neq \\emptyset \\} | & = \\frac { 1 } { e ( R ) } \\sum _ { g \\in E ( B ) } | \\{ f \\in E ( R ) : f \\cap g \\neq \\emptyset \\} | \\\\ & \\le \\frac { 1 } { e ( R ) } \\sum _ { g \\in E ( B ) } 2 \\\\ & = \\frac { 2 e ( B ) } { e ( R ) } < 2 . \\end{align*}"}
-{"id": "4295.png", "formula": "\\begin{align*} \\prod _ { y = 1 } ^ { Y _ \\alpha ( l ) } = \\prod _ { y = 1 } ^ { F _ \\alpha ( j _ \\alpha ) } = \\prod _ { j = j _ \\alpha } ^ { m _ \\alpha } \\prod _ { y = F _ \\alpha ( j + 1 ) + 1 } ^ { F _ \\alpha ( j ) } \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} y \\ , = \\ , m ( x ; \\Theta ) + \\mbox { e r r o r } , \\ ; \\mbox { w h e r e } \\ ; m ( x ; \\Theta ) \\ , = \\ , \\displaystyle { \\max _ { 1 \\leq i \\leq k _ 1 } } \\ , \\left ( \\ , ( \\ , a ^ i \\ , ) ^ T x + \\alpha _ i \\ , \\right ) - \\displaystyle { \\max _ { 1 \\leq i \\leq k _ 2 } } \\ , \\left ( \\ , ( \\ , b ^ i \\ , ) ^ T x + \\beta _ i \\ , \\right ) , \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{align*} { \\eta } _ { \\alpha } ^ { \\theta } = { \\left \\vert { p } \\right \\vert } ^ { \\alpha } e ^ { i S g n \\left ( p \\right ) \\theta \\pi / 2 } , ~ ~ ~ ~ ~ ~ ~ ~ 0 < \\alpha \\leq { 2 } , ~ ~ ~ ~ ~ ~ ~ \\left | \\theta \\right | \\leq { } \\left \\{ \\alpha , 2 - \\alpha \\right \\} . \\end{align*}"}
-{"id": "6551.png", "formula": "\\begin{align*} \\vert f \\vert _ \\theta = \\sup \\limits _ { v \\neq v ' } \\frac { \\vert f ( v ) - f ( v ' ) \\vert } { \\rho ( v , v ' ) ^ \\theta } . \\end{align*}"}
-{"id": "9215.png", "formula": "\\begin{align*} \\Lambda = \\{ ( t , x ) : t \\in \\N _ 0 , x \\in \\Z , t + x \\in 2 \\Z \\} , \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} = \\begin{pmatrix} - r _ 3 & r ^ T & 0 \\\\ n & N _ 0 & r \\\\ 0 & n ^ T & r _ 3 \\end{pmatrix} _ x + \\begin{pmatrix} k _ 0 n _ 1 - k _ 1 r _ 1 - k _ 2 r _ 2 & k _ 0 r _ 3 & k _ 0 n _ 0 & 0 \\\\ - k _ 1 r _ 3 - k _ 2 n _ 0 & 0 & k _ 1 r _ 2 - k _ 2 r _ 1 + k _ 0 n _ 2 & \\ast \\\\ - k _ 2 r _ 3 + k _ 1 n _ 0 & \\ast & 0 & \\ast \\\\ 0 & \\ast & \\ast & \\ast \\end{pmatrix} \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} \\operatorname { d i v } ( | y | ^ { - a } \\nabla f ) = | y | ^ { - a } f _ t . \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{align*} \\Delta ^ { ( m ) } \\partial _ { z _ j } : = \\partial _ { z _ { j - 1 } } - 2 \\partial _ { z _ j } + \\partial _ { z _ { j + 1 } } \\ , , j = 1 , \\ldots , m - 1 \\ , . \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} p ( i , j | v , w ) = \\langle ( P _ { v , i } Q _ { w , j } ) h , h \\rangle = \\langle ( P _ { v , i } P _ { w , j } ) h , h \\rangle = \\langle ( Q _ { w , j } Q _ { v , i } ) h , h \\rangle \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} u : = ( I - K ) \\eta - S w v : = ( I - K ) w - S \\eta , \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\varphi ^ { \\prime } \\overline { w } \\ ; \\d x + \\int _ { \\Omega } \\nabla \\varphi \\cdot \\overline { \\nabla w } \\ ; \\d x = - 2 \\int _ { \\Omega } ( u \\cdot \\nabla ) d \\cdot \\overline { ( d w ) } \\ ; \\d x + 2 \\int _ { \\Omega } \\lvert \\nabla d \\rvert ^ 2 \\varphi \\overline { w } \\ ; \\d x . \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{align*} \\widehat { c } _ 2 ( t ; T ) = { } & \\frac { 2 ^ T \\prod _ { n = 1 } ^ t \\sin \\left ( \\frac { n \\pi } { 6 T } \\right ) \\prod _ { n = 1 } ^ { 2 T - t } \\sin \\left ( \\frac { n \\pi } { 6 T } \\right ) } { \\prod _ { n = 1 } ^ { 2 T } \\sin \\left ( \\frac { n \\pi } { 6 T } \\right ) } \\prod _ { n = 1 } ^ T \\frac { \\sin ^ 2 \\left ( \\frac { n \\pi } { 3 T } \\right ) } { \\cos \\left ( \\frac { ( 2 n - T - 1 ) \\pi } { 6 T } \\right ) } . \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} \\mathcal { L } _ { X ( \\zeta ) } \\omega ( \\zeta ) = 0 \\mathrlap { . } \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} p _ { a , a + 1 } & = a + a t ^ { a - 1 } ( 1 + t ) \\ , , \\\\ p _ { a + 1 , a + 1 } & = ( a + 1 ) t ^ a \\ , , \\end{align*}"}
-{"id": "7126.png", "formula": "\\begin{align*} g _ 2 = \\frac { 4 } { 3 } \\alpha _ 2 ^ 2 - 4 \\alpha _ 1 \\alpha _ 3 \\quad g _ 3 = - \\frac { 8 } { 2 7 } \\alpha _ 2 ^ 3 + \\frac { 4 } { 3 } \\alpha _ 1 \\alpha _ 2 \\alpha _ 3 - 4 \\alpha _ 0 \\alpha _ 3 ^ 2 . \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{align*} X _ { \\gamma } X _ { \\gamma ' } = ( - ) ^ { \\langle \\gamma , \\gamma ' \\rangle } X _ { \\gamma + \\gamma ' } \\rlap { . } \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} ( J \\otimes E ) ^ { \\mathrm { o r d } } : = J _ { \\mathrm { f s u } } \\otimes E \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} u _ V ( t ) = \\frac { \\sigma \\theta } { 2 } ( e ^ { \\frac { t } { \\theta } } - e ^ { - \\frac { t } { \\theta } } ) , & & v _ V ( t ) = \\sigma e ^ { - \\frac { t } { \\theta } } \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} b _ + = & \\tilde \\gamma ^ { - 1 } F + \\tilde S - \\frac { 1 } { 2 } \\left ( \\tilde \\gamma ^ { - 1 } - ( \\tilde \\gamma ^ T ) ^ { - 1 } \\right ) F - \\frac { 1 } { 2 } ( S - S ^ \\prime ) , \\\\ b _ - = & \\frac { 1 } { 2 } \\left ( \\tilde \\gamma ^ { - 1 } - ( \\tilde \\gamma ^ T ) ^ { - 1 } \\right ) F + \\frac { 1 } { 2 } ( S - S ^ \\prime ) , \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} z ( \\lambda , \\hat { X } ) = - \\int _ 0 ^ { \\hat { X } } \\frac { d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } + \\omega _ 2 / 2 . \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} \\psi ( \\nu , \\mu ) = c _ { \\mu , \\nu } ( p ; t ) \\cdot I ( \\mu , \\nu ) = d _ { \\mu } ( p ; t ) \\times t ^ { \\alpha ( \\mu , \\nu ) + \\beta ( \\mu , \\nu ) + \\gamma ( \\mu , \\nu ) } \\cdot I ( \\mu , \\nu ) , \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} \\phi \\left ( z ' \\frac { ( x ' ) ^ { m ' } - ( y ' ) ^ { m ' } } { x ' - y ' } \\right ) = z ' \\frac { ( x ' ) ^ { n ' } - ( y ' ) ^ { n ' } } { x ' - y ' } \\end{align*}"}
-{"id": "6441.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ^ { \\nabla y } _ { j } ( T ) & \\leq C \\tilde { C } _ { T } \\Big [ ( k ^ { q } _ { j } ( T ) + k ^ { q } _ { j - 1 } ( T ) ) + ( k ^ { q } _ { j } ( T ) + k ^ { q } _ { j - 1 } ( T ) ) ( 2 k ^ { \\infty } _ { j } ( T ) + \\lvert \\overline { b } \\rvert ) + k ^ { q } _ { j - 1 } ( T ) ^ 2 \\Big ] \\delta _ { j - 1 } ( T ) \\\\ & < C \\tilde { C } _ { T } [ 2 K _ { 1 } + 6 K _ { 1 } K _ { 2 } + K _ { 1 } ^ 2 ] \\delta _ { j - 1 } ( T ) . \\end{aligned} \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} d X _ t = \\sigma X _ t d W _ t . \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} Y = \\mathbb { R } \\times X \\ : , \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} H = H _ { \\ , 0 } + \\vartheta \\cdot X + L \\ , , \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 e ^ { r u } ( 1 - u ^ 2 ) ^ { \\frac { p - 3 } { 2 } } d u = \\Gamma ( \\frac { p - 1 } { 2 } ) \\sqrt { \\pi } \\big ( \\frac { 2 } { r } \\big ) ^ \\frac { p - 2 } { 2 } I _ \\frac { p - 2 } { 2 } ( r ) . \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} & \\# \\Big \\{ ( \\mu _ i , \\mu _ j ) = ( 0 , 1 ) , ( \\nu _ i , \\nu _ j ) = ( 1 , 1 ) \\Big \\} + \\# \\Big \\{ ( \\mu _ i , \\mu _ j ) = ( 1 , 1 ) , ( \\nu _ i , \\nu _ j ) = ( 1 , 1 ) \\Big \\} + \\\\ & \\# \\Big \\{ ( \\mu _ i , \\mu _ j ) = ( 2 , 1 ) , ( \\nu _ i , \\nu _ j ) = ( 1 , 1 ) \\Big \\} + \\# \\Big \\{ ( \\mu _ i , \\mu _ j ) = ( 2 , 1 ) , ( \\nu _ i , \\nu _ j ) = ( 2 , 1 ) \\Big \\} , \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} \\det M = \\det A \\det ( B - C ^ * A ^ { - 1 } C ) . \\end{align*}"}
-{"id": "3945.png", "formula": "\\begin{align*} & D _ \\alpha ( p _ X \\| q _ X ) = \\frac { 1 } { \\alpha - 1 } \\log \\left ( \\sum _ { x } p _ X ( x ) ^ \\alpha q _ X ( x ) ^ { 1 - \\alpha } \\right ) . \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} \\phi ( \\omega + \\Omega _ 1 ) = \\phi ( \\omega ) \\phi ( \\omega + \\Omega _ 2 ) = \\phi ( \\omega ) + 1 . \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} I _ \\lambda : = \\prod _ { i = 1 } ^ d \\left [ ( \\lambda _ i - 1 ) \\cdot 2 ^ { - r } - \\frac { 1 } { 2 } , \\lambda _ i \\cdot 2 ^ { - r } - \\frac { 1 } { 2 } \\right ] . \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{align*} E \\left [ S ( \\theta ) \\right ] = J _ 1 ( N ; \\theta ) + J _ 2 ( N ; \\theta ) , \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} J _ { \\alpha } ( X _ 1 X _ 2 ; Y _ 1 Y _ 2 ) = J _ { \\alpha } ( X _ 1 ; Y _ 1 ) + J _ { \\alpha } ( X _ 2 ; Y _ 2 ) \\end{align*}"}
-{"id": "6864.png", "formula": "\\begin{align*} A G + G A ^ \\top + F = 0 \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{align*} & E \\left [ S _ { s , t } ^ { a n o m } \\right ] = \\frac { n + 2 } { 2 } \\int _ { s } ^ t E \\left [ \\left ( \\beta ^ { - 3 } \\nabla _ q \\beta \\cdot ( \\tilde \\gamma + 2 \\gamma I ) ^ { - 1 } \\nabla _ q \\beta \\right ) ( r , q _ r ) \\right ] d r . \\end{align*}"}
-{"id": "8827.png", "formula": "\\begin{align*} E = \\frac { 1 } { 4 } z _ 1 \\bar { z } _ 2 \\alpha ( l _ j ) e _ { \\alpha } - \\frac { 3 } { 4 } z _ 2 \\bar { z } _ 1 \\alpha ( l _ j ) e ^ { 2 \\alpha ( a ) } \\theta ( e _ { \\alpha } ) + O . \\end{align*}"}
-{"id": "2554.png", "formula": "\\begin{align*} \\overline { p t } = \\overline { t } x \\overline { t } = \\overline { t q } . \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} z _ q ( \\varprojlim _ i W _ i , p ) = \\varinjlim _ i z _ q ( W _ i , q ) , \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} \\nabla _ B \\nabla _ B h + a h g _ B = 0 \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} \\overline \\R _ + ^ { \\mathcal { N } } = \\{ x \\in \\overline \\R ^ { \\mathcal { N } } \\mid 0 \\leq x _ i \\leq \\infty , \\ ; \\forall i = 1 , . . . , N \\} = [ 0 , \\infty ] ^ { \\mathcal { N } } . \\end{align*}"}
-{"id": "7816.png", "formula": "\\begin{align*} \\begin{array} { l } \\mathcal { H } _ 1 ( x , y , z ) : = \\displaystyle { \\frac { 1 - 6 y } { 6 ( 1 - 6 x ) } } \\\\ \\\\ \\mathcal { H } _ 2 ( x , y , z ) : = \\displaystyle { \\frac { 1 + 6 x + 6 y - 1 2 z ^ 2 } { 3 6 ( 1 - 6 x ) ^ 2 } } . \\end{array} \\end{align*}"}
-{"id": "5299.png", "formula": "\\begin{align*} ( J \\otimes E ) ^ { \\mathrm { o r d } } = ( J _ { \\mathrm { f s u } } ) \\otimes E . \\end{align*}"}
-{"id": "9238.png", "formula": "\\begin{align*} \\lambda = \\frac { 3 T - 1 } { 2 \\pi r } . \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} M _ t = \\sum _ { i \\leq n / 2 } \\frac { ( - 1 ) ^ i } { 2 ^ i } \\mathcal M _ i ( t ) + O ( t ^ { n / 2 + 1 } ) , \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{align*} r _ k ( \\alpha ) = \\sum _ Q \\left ( \\prod _ { i = 1 } ^ { | Q | } \\phi _ k ( Q _ i ) \\sum _ { \\{ P ' _ 1 , \\cdots , P ' _ r \\} } \\prod _ { i = 1 } ^ r \\phi _ k ( P ' _ i ) \\right ) \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 3 & 6 & 5 \\\\ 0 & 2 & 3 & 9 \\\\ 0 & 3 & 3 & 4 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 7 & 2 & 6 \\\\ 0 & 2 & 9 & 2 \\\\ 0 & 6 & 2 & 7 \\end{pmatrix} , \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{align*} ( E _ \\mu ^ * ) _ { y , y } = \\begin{cases} 1 & , \\\\ 0 & , \\end{cases} & & y \\in P . \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} < \\phi _ { \\bf T } ( \\mu ) x , l > = \\int _ 0 ^ { + \\infty } < T ( t ) x , > d \\mu ( t ) \\ \\ ( l \\in X _ * ) \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { B _ { t } } ( | \\nabla u | ^ { 2 } ) ^ { \\mu } d x \\le \\theta \\int _ { B _ { s } } ( | \\nabla u | ^ { 2 } ) ^ { \\mu } d x + C \\left \\{ \\frac { 1 } { ( s - t ) ^ { n } } \\int _ { B _ { s } } d ^ { n } ( u , p _ { 0 } ) d x + \\int _ { B _ { R } } g d x \\right \\} \\end{aligned} \\end{align*}"}
-{"id": "605.png", "formula": "\\begin{align*} q ^ j \\frac { d ^ j } { d q ^ j } = q \\frac { d } { d q } \\left ( q \\frac { d } { d q } - 1 \\right ) \\cdots \\left ( q \\frac { d } { d q } - ( j - 1 ) \\right ) \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} R _ v ( \\xi _ { \\mu } ) = \\begin{cases} \\xi _ { \\mu } & s ( \\mu ) = v \\\\ 0 & s ( \\mu ) \\neq v \\end{cases} \\ \\ \\ \\ R _ e ( \\xi _ { \\mu } ) = \\begin{cases} \\xi _ { \\mu e } & s ( \\mu ) = r ( e ) \\\\ 0 & s ( \\mu ) \\neq r ( e ) \\end{cases} \\end{align*}"}
-{"id": "2.png", "formula": "\\begin{align*} \\partial _ s \\Phi _ { \\alpha } = - \\frac { \\xi '' } { 2 } \\bigl ( \\partial _ { x x } \\Phi _ { \\alpha } + \\alpha ( s ) ( \\partial _ x \\Phi _ { \\alpha } ) ^ 2 \\bigr ) , \\ , \\ , ( s , x ) \\in [ 0 , 1 ) \\times \\mathbb { R } \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} \\int _ { [ \\sigma ^ { - 1 } _ e ( t ) , t ] } | \\Phi ^ e _ { t } ( 0 , s ) | d m _ { x = x _ i } ^ j ( s ) & \\leq V _ { m a x } \\int _ { [ \\sigma ^ { - 1 } _ e ( t ) , t ] } | t - s | d m _ { x = V } ^ j ( s ) \\\\ & \\leq V _ { m a x } \\Delta t ^ N m _ { x = V } ^ j ( I _ { n - 1 } ^ N ) , \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{align*} \\wp _ { u n i v } : = { 1 \\over p ^ 2 } + \\sum _ { n \\geq 0 } a _ n ( \\underline g _ 2 , \\underline g _ 3 ) p ^ n \\in \\mathbb Q [ \\underline g _ 2 , \\underline g _ 3 ] ( ( p ) ) \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} \\forall i \\neq j \\in [ 1 , n ] , [ x _ i + x _ i , t _ { i j } ] = [ y _ i + y _ j , t _ { i j } ] = 0 . \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} \\mathcal { H } ( y , z ) = \\frac { 4 3 2 y z ^ 2 - 7 2 z ^ 2 - 2 7 0 y ^ 2 + 1 8 y - 1 8 h + 7 } { 3 6 ( 6 y - 1 ) ^ 3 } . \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{align*} \\Phi _ { q } ( 1 , t ) & = \\int _ { \\Omega } a \\left ( x \\right ) \\left [ ( t \\phi _ { 1 } ) \\left \\{ \\log ( t \\phi _ { 1 } ) + \\frac { w _ { q } ( 1 , t ) } { t \\phi _ { 1 } } \\right \\} - w _ { q } ( 1 , t ) \\right ] \\phi _ { 1 } \\\\ & = t \\int _ { \\Omega } a \\left ( x \\right ) \\phi _ { 1 } ^ { 2 } \\log ( t \\phi _ { 1 } ) \\\\ & = t \\left \\{ ( \\log t ) \\int _ { \\Omega } a \\left ( x \\right ) \\phi _ { 1 } ^ { 2 } + \\int _ { \\Omega } a \\left ( x \\right ) \\phi _ { 1 } ^ { 2 } \\log \\phi _ { 1 } \\right \\} . \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{align*} [ e _ i ] = E _ i , [ \\bar { e } _ i ] = ( H - \\sum _ { j = 1 } ^ { n + 1 } E _ j ) + E _ j \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{align*} Q ( R ) & = ( r + 1 - s ) e ( R ) + k _ 3 ( R ) - \\sum _ { v \\in R } \\binom { d _ R ( v ) } { 2 } \\\\ & \\ge t \\frac { s + k } { 2 } + k _ 3 ( R ) - k \\\\ & = s t / 2 + k ( t / 2 - 1 ) + k _ 3 ( R ) . \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { A ( n ) } { \\sqrt n } = \\sqrt { 1 5 } . \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} \\mathrm { P } ^ { h i t } _ { m } = \\sum ^ { M _ s } _ { i = 1 } \\mathrm { P } ( f _ i ) ( 1 - \\mathrm { P } _ { m , i } ) , \\end{align*}"}
-{"id": "1018.png", "formula": "\\begin{align*} u ^ 0 \\in \\mathcal { H } ; u ^ { k + 1 } = U _ { k } u ^ { k } - \\lambda _ { k } G U _ { k } u ^ { k } . \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} \\sup _ { B _ { \\rho } ( x _ * ) } u \\leq D \\sup _ { y \\in B _ { \\rho } ( x _ * ) } \\frac { 1 } { g ( y , R ) } = \\frac { D } { g ( x _ * , R ) } \\sup _ { y \\in B _ { \\rho } ( x _ * ) } \\frac { g ( x _ * , R ) } { g ( y , R ) } . \\end{align*}"}
-{"id": "4032.png", "formula": "\\begin{align*} \\tilde { \\epsilon } _ 1 & = \\prod _ { x , y } p _ { X Y } ( x , y ) ^ { - \\mu _ 1 ( x , y ) + \\gamma _ 1 ( x , y ) } \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} M = \\{ x _ 1 \\dots x _ k \\mid k \\ge 0 , x _ i \\in X \\} . \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{align*} \\norm { ( \\tilde { x } - x ) - ( \\tilde { y } - y ) } = ( 1 + \\lambda ) \\norm { ( x ^ + - x ) - ( y ^ + - y ) } , \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} \\lim _ { y \\to + \\infty } \\Delta ^ - ( y , 1 , t ) = + \\infty \\quad \\beta _ 4 > 0 . \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{align*} d \\eta = 2 \\omega _ 1 , d \\omega _ 2 = - 3 \\eta \\wedge \\omega _ 3 , d \\omega _ 3 = 3 \\eta \\wedge \\omega _ 2 . \\end{align*}"}
-{"id": "7923.png", "formula": "\\begin{align*} \\Sigma _ { \\mu ^ { \\boxtimes n } } ( z ) = \\Sigma _ \\mu ^ n ( z ) , \\end{align*}"}
-{"id": "4817.png", "formula": "\\begin{align*} \\| \\overline { u } \\| _ { L ^ \\infty ( \\Omega _ { R _ 1 } ) } \\le \\left [ \\frac { C } { R _ 1 - R _ 2 } \\left ( 1 + \\frac { \\| f \\| _ q ^ \\beta } { k ^ { \\beta ( m - 1 ) } } \\right ) \\right ] ^ { 1 / ( 1 - \\chi ) } \\| \\overline { u } \\| _ { L ^ { m \\chi ^ { n _ 0 } } ( \\Omega _ { R _ { n _ 0 } } ) } . \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} \\phi \\left ( x ^ m - y ^ m \\right ) = x ^ n - y ^ n \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} T : = \\inf \\{ t > 0 : \\ \\sup _ { \\R } | h _ { x } ( \\cdot , t ) | = 1 \\} , \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} B _ n = B ^ \\prime _ n \\setminus \\bigcup _ { i < n } B ^ \\prime _ i \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} \\displaystyle { I ( u ) = I ( u , c , \\gamma ) = \\int _ { \\mathbb { R } } \\left ( \\frac { 1 } { 2 } ( \\partial _ x ^ { \\frac { 1 } { 2 } } u ) ^ 2 + \\frac { c } { 2 } u ^ 2 + \\frac { \\gamma } { 2 } ( \\partial _ x ^ { - 1 } u ) ^ 2 \\right ) d x } \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{align*} \\left ( J _ { e u c } F | _ x \\right ) _ { i j } & = - \\frac { \\partial \\lambda } { \\partial x _ j } ( x ) x _ i - \\lambda ( x ) \\delta _ { i j } + \\frac { \\partial f _ i } { \\partial x _ j } ( x ) \\\\ \\left ( g ^ { - 1 } \\cdot J _ { e u c } F | _ { g x } \\cdot g \\right ) _ { i j } & = - ( g ^ T \\nabla \\lambda ( g x ) ) _ j x _ i - \\lambda ( g x ) \\delta _ { i j } - \\left ( g ^ T \\cdot J _ { e u c } f | _ { g x } \\cdot g \\right ) _ { i j } , \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} \\begin{pmatrix} \\lambda ^ b & 0 \\\\ 0 & \\lambda ^ b \\alpha \\end{pmatrix} ( K \\times P _ 2 ( S _ i ) ) + t '' \\subseteq F , t '' \\in [ - \\lambda ^ b , \\lambda ^ b ] ^ 2 - B ( 0 , 1 ) . \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} \\mathcal { P } _ u ' ( \\lambda ' , \\nu ) = \\Phi _ { u , \\nu } ( 0 , 0 , \\lambda ' ) - \\lambda ' u - \\frac { 1 } { 2 } \\int _ 0 ^ u \\xi '' ( s ) s \\gamma ( s ) d s . \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{align*} G ( z ) : = E \\left [ z ^ { - S ( \\theta ) } \\right ] = 1 - ( z - 1 ) \\int _ 0 ^ { \\infty } e ^ { - ( z - 1 ) t } \\left [ 1 - \\left ( 1 - e ^ { - \\theta t } \\right ) \\left ( 1 - e ^ { - ( 1 - \\theta ) t / N } \\right ) ^ N \\right ] d t . \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{align*} x ^ 2 + y ^ 2 + z ^ { p + 1 } - w ^ { p + 1 } = 0 . \\end{align*}"}
-{"id": "1054.png", "formula": "\\begin{align*} \\sqrt { \\zeta _ { n } ^ { j } ( x ) ^ { 2 } + m ^ { 2 } \\epsilon _ n ^ 2 } - \\tau _ { n } ( | \\cdot | ^ { - 1 } \\star w _ { n } ^ { ( j ) } ) ( x ) + \\epsilon _ { n } V ( \\epsilon _ { n } x + x _ { j } ) - \\epsilon _ { n } \\mu _ { n } \\begin{cases} = 0 & w _ { n } ^ { ( j ) } ( x ) > 0 , \\\\ \\geq 0 & w _ { n } ^ { ( j ) } ( x ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "1967.png", "formula": "\\begin{align*} | R _ { \\alpha } ( x , y ) | \\leq C _ { \\alpha } | x - y | ^ { - n + 1 - \\alpha } , \\enspace | x | < 1 , \\enspace \\mbox { a n d } \\enspace | y | = 1 . \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} T ( \\xi ) = d - \\sum _ { j = 1 } ^ d \\cos ( \\xi _ j ) , \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} \\frac { n \\mathrm { R i c } ( \\omega _ { \\phi } ) \\wedge \\omega _ { \\phi } ^ { n - 1 } } { \\omega _ { \\phi } ^ n } & = \\mathrm { T r } ( ( \\tilde { u } _ { l , m } ( a ) ) ( u ^ { l , m } ( a ) ) ) + \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\frac { - \\tilde { u } _ l ( a ) \\alpha ^ { \\vee , l } } { ( 2 \\chi - p ) ( \\alpha ^ { \\vee } ) } \\\\ & + \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } } \\frac { 2 \\chi ^ { a c } ( \\alpha ^ { \\vee } ) } { ( 2 \\chi - p ) ( \\alpha ^ { \\vee } ) } \\end{align*}"}
-{"id": "6075.png", "formula": "\\begin{align*} m ( \\gamma ) = \\begin{cases} [ \\gamma ] + 1 & \\gamma \\geq 1 / 2 , \\\\ 0 , & \\gamma < 1 / 2 . \\end{cases} \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{align*} \\psi _ n = - 2 \\pi i n \\tilde { z } / \\omega _ 1 - \\pi i n ( n - 1 ) \\omega _ 2 / \\omega _ 1 \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} \\left | E \\left [ \\int _ s ^ t \\partial _ r ( \\beta V ) ( r , q _ r ^ m ) d r \\right ] - E \\left [ \\int _ s ^ t \\partial _ r ( \\beta V ) ( r , q _ r ) d r \\right ] \\right | = O ( m ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "9257.png", "formula": "\\begin{align*} I ( s , v ) = & 3 \\int _ 0 ^ { ( 1 - s - v ) / 6 } \\log \\frac { \\sin ( \\pi ( \\varphi + 1 / 6 ) ) } { \\sin ( \\pi ( - \\varphi + 1 / 6 ) ) } d \\varphi + 6 \\int _ 0 ^ { ( 1 - s + v ) / 6 } \\log \\frac { \\sin ( \\pi ( \\varphi + 1 / 6 ) ) } { \\sin ( \\pi ( - \\varphi + 1 / 6 ) ) } d \\varphi \\\\ & + 6 \\int _ 0 ^ { ( 1 - s ) / 3 } \\log \\frac { \\cos ( \\pi ( \\varphi + 1 / 6 ) ) } { \\cos ( \\pi ( - \\varphi + 1 / 6 ) ) } d \\varphi \\mbox { f o r $ ( x , v ) \\in \\overline { \\Lambda } _ 2 $ } . \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} \\frac { \\partial ^ j a } { \\partial x ^ j } ( x _ { P } , y _ { P } ) = \\sum _ { i = 1 } ^ 8 a _ i \\frac { \\partial ^ j g _ i } { \\partial x ^ j } ( x _ { P } , y _ { P } ) = 0 \\end{align*}"}
-{"id": "1335.png", "formula": "\\begin{align*} H \\times [ 0 , 1 ] = \\big \\{ \\ , ( x _ 1 , \\dots x _ { i - 1 } , t , x _ { i + 1 } , \\dots , x _ n ) \\mid t \\in [ 0 , 1 ] \\ , \\big \\} \\end{align*}"}
-{"id": "4135.png", "formula": "\\begin{align*} S I R ( U \\to { X _ f } ) { \\rm { } } = \\gamma _ f = \\frac { { { P _ f } { h _ f } { { \\left \\| { { X _ f } } \\right \\| } ^ { - \\alpha } } } } { { { I _ { d , f u } } + { I _ { f , f u } } } } , \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} u ^ k = ( u ^ k _ 0 , u ^ k _ 1 , u ^ k _ 2 , \\dots , u ^ k _ n ) \\end{align*}"}
-{"id": "8886.png", "formula": "\\begin{align*} u _ { \\lambda } ^ * ( d _ b u _ Y ) = \\lim _ { s \\rightarrow 1 } u _ { \\lambda } ^ * ( s d _ b u _ y + ( 1 - s ) d _ { t \\mu _ Y + b } u _ { \\lambda } ) . \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{align*} \\omega ^ { \\star } _ { I } = \\frac { b + a } { 2 } , \\end{align*}"}
-{"id": "4206.png", "formula": "\\begin{align*} \\int \\Gamma ( f ) d \\mu & = \\int f L f d \\mu = \\int f \\hat { L } g d \\mu = \\int \\hat { \\Gamma } ( f , g ) d \\mu \\\\ & \\leq { \\left ( \\int \\hat { \\Gamma } ( f ) d \\mu \\right ) } ^ { 1 / 2 } { \\left ( \\int \\hat { \\Gamma } ( g ) d \\mu \\right ) } ^ { 1 / 2 } \\\\ & \\leq { \\left ( \\int \\Gamma ( f ) d \\mu \\right ) } ^ { 1 / 2 } { \\left ( \\int \\hat { \\Gamma } ( g ) d \\mu \\right ) } ^ { 1 / 2 } . \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{align*} 1 < \\mathrm { c o n s t } \\ , | V | _ { 1 / 2 , 1 } \\int _ { \\Gamma ^ { \\ast } } \\frac { \\prod _ { i = 1 } ^ { m } ( 1 - \\cos ( p - p ^ { ( i ) } ) ) } { \\rho + \\mathfrak { e } ( p ) } \\mathrm { d } \\mu ^ { \\ast } ( p ) . \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} p ( x , D _ x ) _ + f ( x ) : = \\int _ { \\overline { \\R _ + ^ n } } K _ p ( x , x - y ) \\ , f ( y ) \\ , d y , \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ M \\omega ( f _ i ) \\le \\sum _ { i = 1 } ^ M \\sum _ { \\ell = 1 } ^ { R _ i } \\sigma _ q ( \\gamma _ { i \\ell } p ^ { \\nu } ) . \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{align*} X \\tilde { u } ( x ) & = r X u ( T ( x ) ) , \\\\ Y \\tilde { u } ( x ) & = r Y u ( T ( x ) ) , \\\\ Y X \\tilde { u } ( x ) & = r ^ 2 Y X u ( T ( x ) ) \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} g _ { m } ( X , t ) = \\begin{cases} y ^ { - m } U _ 0 ( X , t ) \\ \\ \\ , \\\\ g _ { m } ( x , 0 , t ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} \\liminf _ { \\lambda \\rightarrow \\infty } \\frac { N _ { s c } [ \\mathfrak { e } , \\lambda V ] } { \\mathcal { L } _ { V } [ \\mathrm { e } ^ { - \\ell _ { \\lambda } } ] } \\ \\geq \\ 1 - c _ { 1 } + \\frac { d \\ , c _ { 1 } } { 2 } \\int _ { 0 } ^ { \\infty } \\exp \\left ( [ g _ { - } ( V ) - 1 ] \\ , \\frac { d } { 2 } r \\right ) \\ , \\mathrm { d } r \\ = \\ \\infty . \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} \\frac { \\Theta _ { n + 1 } ^ { ( M ) } } { \\Theta _ n ^ { ( M ) } } = \\theta _ { W _ { M - n } } = \\theta ^ { \\beta ( W _ { M - n } ) } , \\ \\ \\ \\mbox { w h e r e } \\ \\ \\beta ( v _ 1 \\ldots v _ \\ell ) : = \\sum _ { j = 1 } ^ { \\ell } \\beta _ { v \\_ j } . \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{align*} 0 \\leq \\| f ' \\| _ { L ^ 2 } ^ 2 & = - \\int _ { \\mathbb R } q ( t ) | f ( t ) | ^ 2 \\ , \\mathrm d t = - \\int _ { \\mathbb R } \\big ( q _ + ( t ) - q _ - ( t ) \\big ) | f ( t ) | ^ 2 \\ , \\mathrm d t \\\\ & \\leq \\int _ { \\mathbb R } q _ - ( t ) | f ( t ) | ^ 2 \\ , \\mathrm d t \\leq \\| q _ - \\| _ { L ^ 1 } \\| f \\| _ \\infty ^ 2 . \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l l } R i c _ B ( X , Y ) - m h ^ { - 1 } \\nabla _ { B } \\nabla _ { B } h ( X , Y ) + \\nabla _ { B } \\nabla _ { B } f ( X , Y ) = \\lambda g _ { B } ( X , Y ) , \\\\ \\noalign { \\smallskip } R i c _ { F } ( U , V ) + \\nabla _ { F } \\nabla _ { F } f ( U , V ) = [ \\lambda h ^ 2 + ( m - 1 ) h ^ { - 2 } | \\nabla _ { B } h | ^ { 2 } + h ^ { - 1 } \\Delta _ { B } h - h ( \\nabla _ { B } h ) f ] g _ { F } ( U , V ) , \\\\ \\noalign { \\smallskip } X ( U ( f ) ) = h ^ { - 1 } X ( h ) U ( f ) . \\end{array} \\right . \\end{align*}"}
-{"id": "3781.png", "formula": "\\begin{align*} R ^ \\nabla | _ { \\Lambda ^ { 1 , 1 } \\otimes \\Lambda ^ { 1 , 1 } } \\equiv \\begin{pmatrix} k & \\bar { a } & a & w \\\\ - \\bar { a } & 0 & \\bar { v } & \\bar { b } \\\\ - a & v & 0 & b \\\\ u & - \\bar { b } & - b & k \\end{pmatrix} \\quad u + v + \\bar { v } + w = 2 k . \\end{align*}"}
-{"id": "7703.png", "formula": "\\begin{align*} f _ { r _ t | r _ m } ( y ) = & 2 y \\lambda _ c \\pi e ^ { - \\lambda _ c \\pi ( y ^ 2 - x ^ 2 ) } \\left ( \\sum ^ { t - m - 1 } _ { n = 0 } S _ n - \\sum ^ { t - m - 1 } _ { n = 1 } S _ { n - 1 } \\right ) \\\\ = & 2 y ( \\lambda _ c \\pi ) ^ { t - m } e ^ { - \\lambda _ c \\pi ( y ^ 2 - x ^ 2 ) } \\frac { ( y ^ 2 - x ^ 2 ) ^ { t - m - 1 } } { ( t - m - 1 ) ! } , \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} c _ \\lambda = \\frac { 1 } { \\pi ^ n } \\prod _ { j = 1 } ^ { r } \\frac { \\Gamma ( \\lambda - ( j - 1 ) \\frac { a } { 2 } ) } { \\Gamma ( \\lambda - \\frac { n } { r } - ( j - 1 ) \\frac { a } { 2 } ) } \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} \\left ( x \\frac { d } { d x } \\right ) ^ n \\psi _ 1 ( x ) = ( - 1 ) ^ n \\left ( x \\frac { d } { d x } \\right ) ^ n \\psi _ 1 ( \\tfrac { 1 } { x } ) , n \\in \\N . \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} \\mu ( b , W ) ( d x ) = e ^ { - \\frac { 1 } { 2 } b ^ * M _ x ^ { - 1 } b } 1 _ { C _ W } ( x ) \\frac { d x } { \\sqrt { \\det M _ x } } \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{align*} ( \\mathcal { G } _ D ) _ { g \\cdot x _ 0 , h \\cdot x _ 0 } = \\frac { 1 } { | X | } \\sum _ { j \\in D } m _ j \\omega _ j ( h ^ { - 1 } g ) ( g , h \\in G ) , \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} f ^ - _ M ( z _ { M + 1 } , \\dots , z _ n ) = f ( u _ 2 , \\dots , q _ i ^ { - M + 1 } u _ 2 , z _ { M + 1 } , \\dots , z _ n ) \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} \\rho _ \\mathrm { b } f ( x , t ) = \\int _ { \\mathbb { B } ^ { d - 1 } } f ( x - t v , v ) \\ , \\mathrm { d } v . \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} { \\bf \\nabla } _ g u = \\textit { \\textbf { V } } \\ { i n } \\ \\Omega \\end{align*}"}
-{"id": "2579.png", "formula": "\\begin{align*} \\mathbb { E } _ { { n '' } , d , \\theta _ { n '' } } ( \\varphi _ { n '' } ) = \\mathbb { P } _ { { n '' } , d , \\theta _ { n '' } } \\left ( \\| \\hat { \\theta } _ { n '' } \\| \\geq s _ { n '' } ^ { - 1 } C \\right ) \\geq \\mathbb { P } _ { { n '' } , d , \\theta _ { n '' } } \\left ( \\| \\hat { \\theta } _ { n '' } \\| \\geq s _ { n '' } ^ { - 1 } C , \\| \\hat { \\theta } _ { n '' } - \\theta _ { n '' } \\| < \\delta / 2 \\right ) \\end{align*}"}
-{"id": "580.png", "formula": "\\begin{align*} | \\varphi _ i ( r e ^ { i \\theta } ) | \\le C \\sum _ { j = 0 } ^ { \\infty } j ^ d r ^ j \\le C d ! \\left ( \\frac { 1 } { 1 - r } \\right ) ^ { d + 1 } \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} \\mu \\left ( \\left \\{ y \\in Y : \\left ( \\int \\limits _ 0 ^ { T _ k } F _ { n ( T _ k ) } ( g _ t y ) \\ , d t \\right ) ^ 2 \\geqslant T _ k ^ { 2 \\alpha } \\| f _ { n ( T _ k ) } \\| _ { B } ^ 2 \\right \\} \\right ) \\\\ \\leqslant 4 Q T _ k ^ { 1 - 2 \\alpha } \\log T _ k = 4 Q \\frac { 2 \\alpha \\log k } { ( 2 \\alpha - 1 ) k ^ { 2 \\alpha } } \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} \\frac { 1 } { a } = \\frac { 1 - \\theta _ 0 } { a _ 0 } + \\frac { \\theta _ 0 } { a _ 1 } , \\frac { 1 } { b } = \\frac { 1 - \\theta _ 1 } { b _ 0 } + \\frac { \\theta _ 1 } { b _ 1 } , 0 < \\theta _ 0 , \\theta _ 1 < \\theta < 1 , \\theta _ 0 + \\theta _ 1 = \\theta . \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} u _ { V , r } ( z ) : = \\begin{cases} { 1 \\over c _ d a ^ d r ^ d } \\log | s _ V ( - r \\overline z ) | & z \\in \\overline \\C _ + , \\\\ { 1 \\over c _ d a ^ d r ^ d } \\log | s _ V ( - r z ) | & z \\in \\overline \\C _ - . \\end{cases} \\end{align*}"}
-{"id": "7455.png", "formula": "\\begin{align*} & \\beta ( t , q _ t ) H ( t , x _ t ) - \\beta ( s , q _ s ) H ( s , x _ s ) \\\\ = & \\int _ s ^ t \\partial _ r ( \\beta H ) ( r , x _ r ) d r + \\int _ s ^ t \\nabla _ q ( \\beta H ) ( r , x _ r ) \\cdot d q _ r + \\int _ s ^ t \\nabla _ p ( \\beta H ) ( r , x _ r ) \\circ d p _ r . \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} \\begin{cases} \\dot { U } _ t = - \\mathrm { d i v } \\ , T ^ { U _ t } \\bar { \\times } U _ t + u _ t \\ , \\mathrm { d i v } \\ , T ^ { U _ t } , \\\\ U _ 0 = 0 , \\end{cases} \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} H _ 1 ( \\frac { K - 1 } { n } ) ^ m \\prod _ { k = J } ^ { K - 1 } e ^ { - i \\frac { H _ 1 ( k / n ) } { n } } H _ 1 ( \\frac { J } { n } ) ^ { - m } \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{align*} \\varphi _ Q = \\frac { \\pi } { 1 0 } < \\arcsin \\sqrt { \\frac 1 3 } \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} \\underset { x ^ 0 \\not = x ^ 1 } { \\sup } \\dfrac { \\left \\| \\partial _ { \\xi } ^ \\alpha \\widetilde { q } ( x ^ 0 , \\xi ) - \\partial _ { \\xi } ^ \\alpha \\widetilde { q } ( x ^ 1 , \\xi ) \\right \\| _ X } { | x ^ 0 - x ^ 1 | ^ { \\tau } } \\langle \\xi \\rangle ^ { - m + | \\alpha | } \\leq C \\left \\| q \\right \\| ^ { | \\alpha | + 1 } _ { C ^ \\tau ( \\R ^ n ; S ^ m _ { 1 , 0 } ) } . \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{align*} ( z ) > \\frac { C - C _ N } { 3 } \\Longrightarrow \\widehat { P } _ \\hbar - z = \\left ( - \\widehat { \\chi } _ R \\left ( P _ \\hbar + \\widehat { \\chi } _ { R } - z \\right ) ^ { - 1 } \\right ) \\left ( \\widehat { P } _ \\hbar + \\widehat { \\chi } _ { R } - z \\right ) . \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} f ^ * ( 1 \\times \\omega _ { N } ) = ( \\alpha ^ n _ M \\times 1 ) + c \\cdot ( 1 \\times \\omega _ { N } ) , \\ \\alpha _ M ^ n \\in H ^ n ( M ; \\R ) , \\ c \\in \\Z . \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} M _ 4 ' ( p ) = M _ 4 ( p ) - M _ 3 ( p ) M _ 1 ( p ) - \\frac { 1 } { 2 } M _ 2 ( p ) ^ 2 + M _ 2 ( p ) M _ 1 ( p ) ^ 2 - \\frac { 1 } { 4 } M _ 1 ( p ) ^ 4 \\end{align*}"}
-{"id": "491.png", "formula": "\\begin{align*} \\frac { \\chi _ 2 ( - a ) } { ( \\frac { 1 } { 2 } + \\frac { \\delta } { k } + i q _ 2 y ) ^ k } = \\frac { \\chi _ 2 ( - a ) } { ( \\frac { 1 } { 2 } + i q _ 2 y ) ^ k \\left ( 1 + \\frac { \\delta } { k \\left ( \\frac { 1 } { 2 } + i q _ 2 y \\right ) } \\right ) ^ k } = \\frac { \\chi _ 2 ( - a ) } { ( \\frac { 1 } { 2 } + i q _ 2 y ) ^ k } \\left ( 1 + O \\left ( \\frac { \\epsilon } { \\left | \\frac { 1 } { 2 } + i q _ 2 y \\right | } \\right ) \\right ) . \\\\ \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} w ( x ) : = \\sum _ { k = 1 } ^ { \\infty } ( | x | - r _ k ) ^ { 2 m } \\chi ( | x | - r _ k ) f ( r _ k ) , \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{align*} \\begin{aligned} { \\rm B i a s } ( \\hat D _ { \\lambda } ) & = \\mathbb { E } \\bigg \\{ \\sum _ { l \\in \\bar l } \\lambda _ l \\hat D _ l - D \\bigg \\} = \\sum _ { l \\in \\bar l } \\lambda _ l { \\rm B i a s } ( \\hat D _ l ) \\\\ & = \\sum _ { l \\in \\bar l } \\sum _ { j \\in J } \\gamma _ j \\lambda _ l \\varphi _ j ( l ) \\varGamma ^ { - j / 2 d } + \\rho _ { \\lambda b i a s } ( \\varGamma ^ { - 1 / 2 } ) . \\end{aligned} \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{align*} l = | B _ \\nu | - | \\lambda \\cap S _ \\mu ( m - 1 ) | - | \\lambda \\cap T _ \\mu ( m + 1 ) | + | \\lambda | / 2 \\end{align*}"}
-{"id": "4727.png", "formula": "\\begin{align*} W ^ L = \\{ v \\in W : N ( v ) \\subseteq \\Phi ^ + \\setminus \\Phi _ L ^ + \\} . \\end{align*}"}
-{"id": "5087.png", "formula": "\\begin{align*} e ( y _ j ) = \\frac { ( - 1 ) ^ { j + k } C ( f ) \\prod _ { \\substack { 1 \\le i _ 1 < i _ 2 \\le k \\\\ i _ 1 \\ne j , i _ 2 \\ne j } } ( y _ { i _ 2 } - y _ { i _ 1 } ) } { \\prod _ { 1 \\le i _ 1 < i _ 2 \\le k } ( y _ { i _ 2 } - y _ { i _ 1 } ) } . \\end{align*}"}
-{"id": "1784.png", "formula": "\\begin{align*} D _ { \\underline { \\xi } ' } \\underline { v } ( \\underline { \\xi } ) & = \\left ( \\sum _ { k = 1 } ^ n \\frac { \\partial \\xi _ k } { \\partial \\underline { \\xi } _ j } D _ { \\xi _ k } v ( \\xi ) \\right ) _ { 1 \\leq j \\leq n - 1 } = A D _ { \\xi ' } v ( \\xi ) + B D _ { \\xi _ n } v ( \\xi ) \\\\ D _ { \\underline { \\xi } _ n } \\underline { v } ( \\underline { \\xi } ) & = C D _ { \\xi _ n } v ( \\xi ) . \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\sup _ { | x | \\leq c t } | u ( x , t ) - \\frac { a } { b } | = 0 \\forall \\ , \\ , 0 \\leq c < c ^ { \\ast } _ { \\rm l o w } , \\end{align*}"}
-{"id": "704.png", "formula": "\\begin{align*} \\sum _ { \\substack { s \\geq i , t \\geq j \\\\ ( s , t ) \\neq ( i , j ) } } ( C _ k ) _ { i s } ( X _ { k + 1 } ) _ { s t } ( D _ k ) _ { t j } & = \\sum _ { t > j } ( C _ k ) _ { i i } ( X _ { k + 1 } ) _ { i t } ( D _ k ) _ { t j } + \\sum _ { \\substack { s > i } } ( C _ k ) _ { i s } \\underbrace { \\sum _ { t > j } ( X _ { k + 1 } ) _ { s t } ( D _ k ) _ { t j } } _ { : = ( X _ k ^ D ) _ { s j } } , \\end{align*}"}
-{"id": "7778.png", "formula": "\\begin{align*} I _ 2 ( x _ 1 , x _ 2 ) = I _ { 2 , + } ( x _ 1 , x _ 2 ) + I _ { 2 , - } ( x _ 1 , x _ 2 ) , \\end{align*}"}
-{"id": "7068.png", "formula": "\\begin{align*} & V ^ { 1 - 1 - 3 , l } ( \\psi ) : = \\sum _ { n = 2 } ^ { \\infty } V ^ { 1 - 1 - 3 , l , ( n ) } ( \\psi ) , V ^ { 1 - 1 - 4 , l } ( \\psi ) : = \\sum _ { n = 2 } ^ { \\infty } V ^ { 1 - 1 - 4 , l , ( n ) } ( \\psi ) , \\\\ & V ^ { 1 - 2 - 2 , l } ( \\psi ) : = \\sum _ { n = 2 } ^ { \\infty } V ^ { 1 - 2 - 2 , l , ( n ) } ( \\psi ) , V ^ { 2 , l } ( \\psi ) : = \\sum _ { n = 1 } ^ { \\infty } V ^ { 2 , l , ( n ) } ( \\psi ) , \\\\ & V ^ { 1 - 1 , l } ( \\psi ) : = \\sum _ { j = 1 } ^ 4 V ^ { 1 - 1 - j , l } ( \\psi ) , V ^ { 1 - 2 , l } ( \\psi ) : = \\sum _ { j = 1 } ^ 2 V ^ { 1 - 2 - j , l } ( \\psi ) . \\end{align*}"}
-{"id": "3933.png", "formula": "\\begin{align*} \\bigg \\| \\bigg ( \\sum _ { n _ 0 = 0 } ^ \\infty \\bigg \\| \\sup _ { \\eta \\in \\mathfrak { B } ^ r ( \\R ^ d ) } \\frac { \\sum _ { n \\geq 0 } \\sum _ { y \\in \\Lambda _ n } | \\langle \\psi ^ n _ y , \\eta ^ { \\lambda } _ x \\rangle | | \\langle \\xi , \\psi ^ n _ y \\rangle | } { \\lambda ^ \\alpha } \\bigg \\| _ { L ^ q _ { \\lambda , n _ 0 } } ^ q \\bigg ) ^ \\frac { 1 } { q } \\bigg \\| _ { L ^ p } \\end{align*}"}
-{"id": "7373.png", "formula": "\\begin{align*} \\| V _ n \\| _ { L ^ q ( \\R ^ d ) } = h ^ { - \\frac { 1 } { q } - \\frac { k } { 2 q } - \\frac { d - k - 1 } { 3 q } } \\| W _ h \\| _ { L ^ q ( \\R ^ d ) } = \\mathcal { O } ( h ^ { 1 - \\frac { 1 } { q } - \\frac { k } { 2 q } - \\frac { d - k - 1 } { 3 q } } ) \\to 0 \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} g ^ { ( h ) } ( { ( [ c _ k ] S ^ { \\ell _ 1 p ^ k } T ^ { \\ell _ 2 p ^ k } ) _ { k = 0 } ^ \\infty } ) & = \\sum _ { k = 0 } ^ h p ^ k ( [ c _ k ] S ^ { \\ell _ 1 p ^ k } T ^ { \\ell _ 2 p ^ k } ) ^ { p ^ { h - k } } \\\\ & = ( \\sum _ { k = 0 } ^ h p ^ k [ c _ k ] ^ { p ^ { h - k } } ) S ^ { \\ell _ 1 p ^ h } T ^ { \\ell _ 2 p ^ h } . \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} \\gamma _ t : = - 2 f _ t ( | q | _ k ) \\Im \\frac { \\dot q } { q } \\ , \\begin{pmatrix} i & 0 \\\\ 0 & - i \\end{pmatrix} \\end{align*}"}
-{"id": "4297.png", "formula": "\\begin{align*} d y _ t = f ( y _ t ) d x _ t \\ , \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} I [ \\mu - \\nu ] & = \\sum _ { k \\neq 0 } c _ k ( f ) | c _ k ( \\mu ) - c _ k ( \\nu ) | ^ 2 \\geq 0 . \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} [ \\mu _ 1 , \\ldots , \\mu _ { \\ell } ] = [ \\breve { \\mu } _ 1 , \\ldots , \\breve { \\mu } _ { \\ell } ] . \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{align*} \\frac { \\mathcal { R } \\theta _ F ( \\gamma z ) } { \\mathcal { R } \\theta _ F ( z ) } = \\left ( \\frac { \\theta ( \\gamma z ) } { \\theta ( z ) } \\right ) ^ m . \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} W _ { n + 1 } ' W _ { n - 1 } - W _ { n + 1 } W _ { n - 1 } ' = W _ n ^ 2 . \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} ( \\forall g \\in H _ { m ' } ) ( A c t ( m , g ) = A c t ( m , h ) ) . \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{align*} \\rho ( Q _ n , P _ n ) = \\hat { \\rho } \\left ( H _ { \\textrm { s c a t t } } ( Q _ n , P _ n ) \\right ) = \\rho ( Q _ n , - P _ n ) , \\end{align*}"}
-{"id": "3765.png", "formula": "\\begin{align*} L _ n & = \\beta ( W _ { M - n - 1 } ) ( 2 b ) ^ { \\beta ( W ' _ { M - n - 1 } ) } \\\\ & = ( 1 + \\beta ( W ' _ { M - n - 1 } ) ) ( 2 b ) ^ { \\beta ( W ' _ { M - n - 1 } ) } \\le ( 2 e b ) ^ { \\beta ( W ' _ { M - n - 1 } ) } . \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} \\mathrm { d } X _ t = a ( X _ t ) \\ , \\mathrm { d } t + \\sum _ { j = 1 } ^ K b ^ j ( X _ t ) \\ , \\mathrm { d } \\beta _ t ^ j \\end{align*}"}
-{"id": "7427.png", "formula": "\\begin{align*} \\Sigma _ { i j } ( t , q ) = \\sum _ \\rho \\sigma _ { i \\rho } ( t , q ) \\sigma _ { j \\rho } ( t , q ) . \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{align*} \\frac { 2 \\pi | j - J _ n | } { n } \\leq \\frac { 2 \\pi \\ , ( J _ n - H _ n - 1 ) } { n } = \\arg ( z _ { J _ n , n } ) - \\arg ( z _ { H _ n + 1 , n } ) \\leq \\frac { e ^ { - 2 c } } { 1 - e ^ { - c } } . \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} a _ 1 ^ { b _ 1 } \\cdots a _ h ^ { b _ h } = 1 . \\end{align*}"}
-{"id": "3045.png", "formula": "\\begin{align*} D ( A ) = \\mathrm { K e r } A + X _ { 2 } ; u = t \\phi _ { 1 } + w , \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{align*} f _ t ( m , r ) = ( 1 - o ( 1 ) ) \\ , m \\ , \\frac { \\binom { r + 1 } { t } } { \\binom { r + 1 } { 2 } } . \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} U ( d _ 1 , d _ 2 ) = \\begin{pmatrix} 1 & d _ 1 & 0 & - d _ 1 ^ 2 / 2 \\\\ 0 & 1 & 0 & - d _ 1 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & 0 & d _ 2 & d _ 2 ^ 2 / ( 2 \\epsilon ) \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & d _ 2 / \\epsilon \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "8868.png", "formula": "\\begin{align*} \\Delta ^ t _ { \\mathcal { L } } = \\{ m \\in \\mathcal { M } \\otimes \\mathbb { R } ; m + v ^ t _ { \\mathcal { L } } \\geq 0 \\} . \\end{align*}"}
-{"id": "4201.png", "formula": "\\begin{align*} A _ X ( x , y ) & = \\tfrac 1 2 \\sigma _ X ( x , y ) { \\sigma _ X ( x , y ) } ^ T , & A _ Y ( x , y ) & = \\tfrac 1 2 \\sigma _ Y ( x , y ) { \\sigma _ Y ( x , y ) } ^ T \\end{align*}"}
-{"id": "3562.png", "formula": "\\begin{align*} \\frac { \\partial \\mathbf { r } } { \\partial t } = \\frac { ( \\partial \\mathbf { r } / \\partial s ) \\times ( \\partial ^ { 2 } \\mathbf { r } / \\partial s ^ { 2 } ) } { | ( \\partial \\mathbf { r } / \\partial s ) | ^ { 3 } } \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} \\tilde { x } = \\frac { - 1 2 \\tilde { z } _ 3 } { \\tilde { z } _ 1 + \\tilde { z } _ 2 } = \\frac { 1 2 \\cdot 3 ^ b \\alpha \\beta } { 3 ^ { 3 b + 1 } \\alpha ^ 3 } = \\frac { 4 \\beta } { 3 ^ { 2 b } \\alpha ^ 2 } . \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{align*} z \\left ( x \\right ) : = \\max \\left ( u _ { 1 } \\left ( x \\right ) , u _ { 2 } \\left ( x \\right ) \\right ) = \\left \\{ \\begin{array} [ c ] { l l l } u _ { 2 } ( x ) & \\mathrm { i n } & \\left [ - 2 , 0 \\right ] , \\\\ u _ { 1 } ( x ) & \\mathrm { i n } & \\left [ 0 , 2 \\right ] , \\end{array} \\right . \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} \\mathcal { N } _ { K } ( y ) = \\nabla \\Xi ( y ) \\mathcal { N } _ { D } ( \\Xi ( y ) ) ; \\end{align*}"}
-{"id": "2639.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l } R i c _ B ( X , Y ) + \\nabla _ { B } \\nabla _ { B } \\beta ( X , Y ) + ( \\varphi - m h ^ { - 1 } ) \\nabla _ { B } \\nabla _ { B } h ( X , Y ) = 0 , \\\\ \\noalign { \\smallskip } R i c _ F ( U , V ) + h \\nabla _ { F } \\nabla _ { F } \\varphi ( U , V ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} x = ( x _ i ) _ { i \\geq 0 } & \\mapsto \\hat { x } = ( \\hat { x } _ i ) _ { i \\geq 0 } , \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{align*} \\tilde { h } _ s ( t , z , \\zeta ) : = \\tilde { h } ( t , z , \\zeta ) + i \\sum _ { k = 1 } ^ 2 \\frac { 1 } { 2 m _ k } \\nabla \\cdot A ^ { ( k ) } ( t , x ^ { ( k ) } ) \\end{align*}"}
-{"id": "6070.png", "formula": "\\begin{align*} \\varkappa ( \\alpha ) = 2 ^ { - \\alpha } \\pi ^ { 1 - 2 \\alpha } B ( \\tfrac 1 { 2 \\alpha } , \\tfrac 1 2 ) ^ \\alpha , \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} J _ { \\alpha } ( X ; Y ) = J _ { \\alpha } ( Y ; X ) . \\end{align*}"}
-{"id": "785.png", "formula": "\\begin{align*} = 2 + e ^ { - 2 c } - 2 \\cos ( \\varphi ) + 4 e ^ { - c } \\sin ( \\frac { \\varphi } { 2 } ) \\sin ( n \\varphi - \\frac { \\varphi } { 2 } ) . \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} \\widehat { q } ( t , x ) & = \\widehat { q } ( t , x ; r , \\alpha _ 0 , \\beta _ 0 ) \\\\ & \\equiv \\lim _ { \\kappa \\to \\infty } q ( t , x ; r , \\alpha _ 0 , \\beta _ 0 , i \\kappa ) = \\prod _ { j = 1 } ^ 5 \\frac { \\sin \\zeta _ j } { \\sin \\eta _ j } , \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} \\Delta \\left ( W ^ 0 _ { n + 2 } ( p , p _ 1 , \\dots , p _ { n + 1 } ) \\right ) & = \\\\ \\sum _ { k = 0 } ^ n W ^ 0 _ { k + 2 } ( p , p _ 1 , \\dots , p _ k , \\bar { q } ) & \\otimes W ^ 0 _ { n - k + 2 } ( \\bar { q } , p _ { k + 1 } , \\dots , p _ { n + 1 } ) \\\\ + \\sum _ { k = 0 } ^ n & W ^ 0 _ { k + 2 } ( q , p _ { 1 } , \\dots , p _ { k + 1 } ) \\otimes W ^ 0 _ { n - k + 2 } ( p , q , p _ { k + 2 } , \\dots , p _ { n + 1 } ) \\end{align*}"}
-{"id": "3250.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\| \\sum _ { \\nu = n + | \\textup { \\textbf { m } } | - m _ \\alpha + 1 } ^ { \\infty } b _ { \\nu , n } ^ { ( \\alpha ) } \\Phi _ \\nu \\| _ K ^ { 1 / n } \\leq \\frac { \\| \\Phi \\| _ K } { \\rho _ { | \\textup { \\textbf { m } } | } ( \\textup { \\textbf { F } } ) } . \\end{align*}"}
-{"id": "2329.png", "formula": "\\begin{align*} I _ M > I _ M ^ { \\flat } : = \\int _ 0 ^ { M ^ { - ( 1 / \\lambda ) - \\varepsilon } } x ^ { \\lambda - 1 } \\left ( 1 - x \\right ) ^ { \\nu _ 1 M } \\left ( 1 - x ^ { \\lambda } \\right ) ^ { \\nu _ 2 M - 1 } d x . \\end{align*}"}
-{"id": "5325.png", "formula": "\\begin{align*} f ( x ) \\ , = \\ , q _ i ( x ) \\ , \\geq \\ , q _ i ( \\bar { x } ) \\ , = \\ , f ( \\bar { x } ) , \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{align*} \\frac { \\Gamma ( x + a ) } { \\Gamma ( x ) } = x ^ a \\left [ 1 + O \\left ( x ^ { - 1 } \\right ) \\right ] , x \\to + \\infty , \\end{align*}"}
-{"id": "8317.png", "formula": "\\begin{align*} \\mathcal { K } f & = \\operatorname { R e } \\bigl ( \\mathfrak { H } \\bigr ) f , \\\\ \\mathcal { K } ^ * f & = - \\operatorname { R e } \\left ( e ^ { i \\theta } \\mathfrak { H } ( e ^ { - i \\theta } f ) \\right ) = - \\operatorname { R e } \\left ( e ^ { i \\theta } [ \\mathfrak { H } , e ^ { - i \\theta } ] f + \\mathfrak { H } f \\right ) . \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} z = - \\int _ { \\xi } ^ { - \\infty } \\frac { d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\pm \\omega _ 1 \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} \\alpha _ k : = z _ k + \\varrho _ k u _ k \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} | G | = | A _ p | + | B _ G | + | R _ G | + | X | \\le ( 4 \\cdot 2 ^ { k - 1 } + 4 ) + 3 + 3 + 2 k < 4 \\cdot 2 ^ k + 1 , \\end{align*}"}
-{"id": "7110.png", "formula": "\\begin{align*} S ( x , y ) = \\sum _ { ( i , j ) \\in \\{ 0 , \\pm 1 \\} ^ { 2 } } d _ { i , j } x ^ i y ^ j = A _ { - 1 } ( x ) \\frac { 1 } { y } + A _ { 0 } ( x ) + A _ { 1 } ( x ) y = B _ { - 1 } ( y ) \\frac { 1 } { x } + B _ { 0 } ( y ) + B _ { 1 } ( y ) x , \\end{align*}"}
-{"id": "8160.png", "formula": "\\begin{align*} a _ n - a _ 1 = b _ n - b _ 1 . \\end{align*}"}
-{"id": "6868.png", "formula": "\\begin{align*} \\dot { x } ( t ) & = A x ( t ) + B u ( t ) \\\\ [ 1 e x ] w ( t ) & = C x ( t ) , \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} \\log \\sqrt { 1 + \\frac { \\| s _ 1 ( p ) \\| ^ 2 } { \\| s _ 0 ( p ) \\| ^ 2 } } = \\log \\frac { 1 } { \\| s _ 0 ( p ) \\| } + \\log \\sqrt { \\| s _ 0 ( p ) \\| ^ 2 + \\| s _ 1 ( p ) \\| ^ 2 } \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} \\Omega _ { \\alpha _ 1 , \\bar { \\alpha } _ 2 } = 0 . \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} u ( z ) = ( \\lambda z + \\mu ) \\ , e ^ { \\alpha z - \\frac { | z | ^ 2 } { 2 } } \\ , \\ \\lambda \\in { \\mathbb C } ^ * \\ , \\ \\mu \\in { \\mathbb C } \\ , \\ \\alpha \\in { \\mathbb C } \\ , \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} n _ j = \\left \\{ \\begin{array} { c l } 1 & j = 1 \\ ; , \\\\ m _ { j - 1 } & j = 2 , \\dots , \\ell \\ ; , \\\\ k + 1 & j = \\ell + 1 \\ ; , \\\\ m _ { j - 2 } & j = \\ell + 2 , \\dots , 2 \\ell + 1 \\ ; . \\end{array} \\right . \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } { | | \\mu ^ { \\boxtimes t } | | } / { t } = e V . \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} \\Phi _ m = A _ m \\exp \\left [ \\sum _ { n = 1 } ^ \\infty \\frac { P _ n } n \\left \\{ \\frac { ( q ^ n - 1 ) q ^ { - n r } } { [ n ] _ { q _ 1 } [ n ] _ { q _ 2 } } \\right \\} \\right ] \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{align*} & \\mathfrak { a } ( A , \\psi ) = - \\frac { 1 } { 2 } \\int _ Y ( A - A _ 0 ) \\wedge d ( A - A _ 0 ) - \\int _ Y ( A - A _ 0 ) \\wedge ( F _ { A _ 0 } + \\frac { 1 } { 2 } F _ { A _ { K ^ { - 1 } } } ) \\\\ & - \\frac { i } { 2 } \\int _ Y ( A - A _ 0 ) \\wedge ( 2 r \\omega + \\wp _ 3 ) + r \\int _ Y \\psi ^ * D _ A \\psi , \\end{align*}"}
-{"id": "7530.png", "formula": "\\begin{align*} U = \\left ( \\begin{array} { c c c } 1 / \\sqrt { 2 } & 1 / \\sqrt { 2 } & 0 \\\\ i / \\sqrt { 2 } & - i / \\sqrt { 2 } & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} \\mathcal { C } _ { D } = \\biggl \\lbrace \\bigl ( T r ( x d _ { 1 } ) + c , T r ( x d _ { 2 } ) + c , \\cdots , T r ( x d _ { n } ) + c \\bigr ) : x \\in \\mathbb { F } _ { q } \\biggr \\rbrace , \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} & \\sum _ { a = 2 } ^ { n - 5 } \\sum _ { c = a + 1 } ^ { n - 4 } \\sum _ { d = c + 2 } ^ { n - 2 } ( c - a ) ( d - c - 1 ) = \\sum _ { a = 2 } ^ { n - 5 } \\sum _ { c = a + 1 } ^ { n - 4 } ( c - a ) \\binom { n - 2 - c } { 2 } = \\sum _ { c = 3 } ^ { n - 4 } \\binom { n - 2 - c } { 2 } \\sum _ { a = 2 } ^ { c - 1 } ( c - a ) \\\\ & = \\sum _ { c = 3 } ^ { n - 4 } \\binom { n - 2 - c } { 2 } \\binom { c - 1 } { 2 } = \\binom { n - 2 } { 5 } \\end{align*}"}
-{"id": "7336.png", "formula": "\\begin{align*} C : = \\prod _ { i = 1 } ^ { 2 k + 1 } x _ i = x _ 1 x _ 2 \\dots x _ { 2 k + 1 } \\ ; \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} T _ W \\mathcal { G } _ { U , W , V } T _ W ^ * S _ W ^ { - 1 } = T _ W \\phi _ { W V } T _ { V } ^ { * } U T _ { W } T _ W ^ * S _ W ^ { - 1 } = U . \\end{align*}"}
-{"id": "4922.png", "formula": "\\begin{align*} A P _ 1 + P _ 1 A ^ T + \\sum _ { i = 1 } ^ m N _ i P _ 1 N _ i ^ T = - B B ^ T , \\end{align*}"}
-{"id": "5479.png", "formula": "\\begin{align*} \\mathbf { W } ( \\mathbf { z } , \\phi ) & = \\mathbf { W } _ 0 ( \\mathbf { z } ) + \\mu ^ { 2 M + 2 } \\mathbf { w } ^ { \\mathbf { 0 } } _ 1 ( \\phi ) + \\mathcal { O } ( \\mu ^ { 2 M + 3 } ) , \\\\ \\mathbf { R } ( \\mathbf { z } , \\Omega t ) & = \\mathbf { R } _ 0 ( \\mathbf { z } ) + \\mu ^ { 2 M + 2 } \\mathbf { r } ^ { \\mathbf { 0 } } _ 1 ( \\Omega t ) + \\mathcal { O } ( \\mu ^ { 2 M + 3 } ) , \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{align*} A ( \\omega ) = \\begin{pmatrix} 0 & - \\omega _ { 3 } & \\omega _ { 2 } \\\\ \\omega _ { 3 } & 0 & - \\omega _ { 1 } \\\\ - \\omega _ { 2 } & \\omega _ { 1 } & 0 \\end{pmatrix} , \\omega \\in \\mathbb { R } ^ { 3 } . \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{align*} \\Delta E _ { a 1 } ( \\| p \\| , \\alpha _ * , M , \\kappa ) = \\alpha _ * \\left ( \\dfrac { \\beta ^ { ( 1 ) } ( \\kappa , \\alpha _ * = 0 ) } { \\| p \\| } + O _ 0 ( M ^ { - 1 / 2 } \\| p \\| ^ { - 2 } ) + O ( M ^ { - 3 / 2 } \\| p \\| ^ { - 4 } ) \\right ) , \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{align*} \\Vert \\mathfrak { e } \\Vert _ { C ^ { m } } \\ \\doteq \\ \\max \\limits _ { \\underline { n } \\in { \\mathbb { N } } _ { 0 } ^ { d } , \\ ; | \\underline { n } | = m } \\ ; \\max _ { p \\in \\Gamma ^ { \\ast } } \\big | \\partial _ { p } ^ { \\underline { n } } \\mathfrak { e } ( p ) \\big | . \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} \\mathbb { P } [ K _ A = 1 , K _ B = 1 | \\mathbf { F } = \\mathbf { f } ] & \\geq \\frac 1 2 - 2 \\delta \\\\ \\mathbb { P } [ K _ A = 2 , K _ B = 2 | \\mathbf { F } = \\mathbf { f } ] & \\geq \\frac 1 2 - 2 \\delta \\\\ \\mathbb { P } [ K _ A = 1 , K _ B = 2 | \\mathbf { F } = \\mathbf { f } ] & \\leq 2 \\delta \\\\ \\mathbb { P } [ K _ A = 2 , K _ B = 1 | \\mathbf { F } = \\mathbf { f } ] & \\leq 2 \\delta . \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} w t ( \\mathbf { c } _ { b } ) = n _ { a } - | N _ { b } ( a , c ) | . \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} ( \\prod _ j M _ { i } ^ { - n _ { i ^ { - 1 } } } ) \\rho ( \\sigma ' ) = ( \\prod _ { j _ i } M _ { j _ i } ^ { n _ { j _ i } } ) \\rho ( \\sigma ) . \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} \\left | \\sum _ { k = m } ^ { z _ 1 } | W ( k ) | - ( r ( n ) - \\sqrt { n } ) n \\right | \\le & \\left | \\sum _ { k = m } ^ { z _ 1 } \\left ( | W ( k ) | - b ( k ) \\right ) \\right | + \\left | \\sum _ { k = m } ^ { z _ 1 } b ( k ) - ( z _ 1 - \\lfloor \\sqrt { n } \\rfloor ) n \\right | \\\\ & + \\big | ( z _ 1 - \\lfloor \\sqrt { n } \\rfloor ) n - ( r ( n ) - \\sqrt { n } ) n \\big | . \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ { | x | \\leq R _ { k } } \\rho _ { n _ { k } } ( x ) { \\rm d } x = \\alpha , \\lim _ { k \\to \\infty } \\int _ { R _ { k } \\leq | x | \\leq 6 R _ { k } } \\rho _ { n _ { k } } ( x ) { \\rm d } x = 0 . \\end{align*}"}
-{"id": "3267.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\| F _ \\alpha - R _ { n , \\textup { \\textbf { m } } , \\alpha } \\| _ { K ( \\varepsilon ) } ^ { 1 / n } \\leq \\frac { \\| \\Phi \\| _ K } { \\rho _ { | \\textup { \\textbf { m } } | } ( \\textup { \\textbf { F } } ) } , \\alpha = 1 , 2 , \\ldots , d , \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} [ a _ 1 \\dots a _ k , b ] = \\sum _ { i = 1 } ^ k a _ 1 \\dots a _ { i - 1 } [ a _ i , b ] a _ { i + 1 } \\dots a _ k ; \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{align*} Y _ t ^ { 0 , x ; u _ { \\cdot } } & = \\xi + \\int _ t ^ T \\hat { G } \\bigl ( s , X _ s ^ { 0 , x ; u _ { \\cdot } } , Y _ s ^ { 0 , x ; u _ { \\cdot } } , Z _ s ^ { 0 , x ; u _ { \\cdot } } \\bigr ) d s - \\int _ t ^ T Z _ s ^ { 0 , x ; u _ { \\cdot } } d B _ s , \\ , \\ , t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} { \\rm M } ( P ( y , z ) ) ^ { 1 / \\deg _ { z } ( P ) } = \\lim _ { n \\to \\infty } \\mathcal { M } \\Bigl ( \\prod _ { k = 1 } ^ { n } P ( \\xi _ { n } ^ { k } , z ) \\Bigr ) \\end{align*}"}
-{"id": "4754.png", "formula": "\\begin{align*} \\mathcal { C } _ { \\Gamma } ( x , v ) = [ g ' ( x ) ] ^ { - 1 } \\mathcal { C } _ K ( g ( x ) , \\lambda ) \\quad { \\rm f o r \\ e a c h } \\ \\lambda \\in \\mathcal { M } _ { x } ( v ) . \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} \\alpha _ i ( b , v ) = ( u _ i ( b ) , F _ { i , b } ( v ) ) \\qquad \\alpha ' _ i ( b , v ) = ( u _ i ( b ) , F ' _ { i , b } ( v ) ) \\end{align*}"}
-{"id": "5070.png", "formula": "\\begin{align*} & g ( x _ 2 , x _ 3 , x _ 4 , x _ 5 ) + g ( x _ 2 , x _ 3 , x _ 5 , x _ 4 ) + g ( x _ 2 , x _ 4 , x _ 5 , x _ 3 ) = \\bigl ( [ x _ 2 , x _ 3 ] [ x _ 4 , x _ 5 ] + [ x _ 2 , x _ 4 ] [ x _ 3 , x _ 5 ] \\bigr ) \\\\ & + \\ \\bigl ( [ x _ 2 , x _ 3 ] [ x _ 5 , x _ 4 ] + [ x _ 2 , x _ 5 ] [ x _ 3 , x _ 4 ] \\bigr ) + \\bigl ( [ x _ 2 , x _ 4 ] [ x _ 5 , x _ 3 ] + [ x _ 2 , x _ 5 ] [ x _ 4 , x _ 3 ] \\bigr ) = 0 \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} d X _ t & = { \\varepsilon } ^ { - 1 } b _ X ( X _ t , Y _ t ) d t + { \\varepsilon } ^ { - 1 / 2 } \\sigma _ X ( X _ t , Y _ t ) d B ^ X _ t , X _ 0 = x _ 0 , \\\\ d Y _ t & = b _ Y ( X _ t , Y _ t ) d t + \\sigma _ Y ( Y _ t ) d B ^ Y _ t , Y _ 0 = y _ 0 \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial \\phi _ { i } } { \\partial t } \\left ( z , t \\right ) + H \\left ( t , \\nabla _ { z } \\phi _ { i } \\left ( z , t \\right ) \\right ) = 0 & \\ , \\mathbb { R } ^ { n } \\times \\left ( 0 , + \\infty \\right ) , \\\\ \\phi _ { i } \\left ( z , 0 \\right ) = J _ { i } \\left ( z \\right ) & \\forall z \\in \\mathbb { R } ^ { n } , \\end{cases} \\end{align*}"}
-{"id": "8254.png", "formula": "\\begin{align*} G _ \\tau = \\{ \\omega : | x _ { N ( s ) } ( s ) - x _ { N ( \\tau t ) } ( \\tau t ) | \\leq t ^ { 1 / 3 } \\textrm { f o r s o m e } s \\in [ \\tau t , \\tau t + 1 ] \\} . \\end{align*}"}
-{"id": "7346.png", "formula": "\\begin{align*} \\dot x _ i = \\sum _ { j = 1 } ^ { 2 k + 1 } A _ { i , j } x _ i x _ j + b _ { i , i + k } - b _ { i - k , i } \\ ; , \\ \\ i = 1 , 2 , \\dots , 2 k + 1 \\ ; , \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} \\begin{cases} \\widetilde { \\sigma } ( 0 ) = 0 \\\\ \\widetilde { \\sigma } ( y _ m ) = \\widetilde { \\sigma } ( y _ { m - 1 } ) + \\sigma ( \\Delta y ) . \\end{cases} \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{align*} \\dot { z } \\left ( t \\right ) = e ^ { - t A } B \\left ( t \\right ) u \\left ( t \\right ) + e ^ { - t A } D \\left ( t \\right ) w \\left ( t \\right ) , \\end{align*}"}
-{"id": "7359.png", "formula": "\\begin{align*} H f ( n ) : = - \\frac { 1 } { 4 } \\sum _ { j = 1 } ^ d \\left ( f ( n + e _ j ) - f ( n - e _ j ) \\right ) + \\frac { d } { 2 } f ( n ) + V ( n ) f ( n ) \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} m _ V ( t ) = ( 1 - e ^ { - \\frac { t } { \\theta } } ) \\hat { V } + e ^ { - \\frac { t } { \\theta } } V _ 0 + e ^ { - \\frac { t } { \\theta } } \\int _ 0 ^ t e ^ { \\frac { s } { \\theta } } I ( s ) d s \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { [ \\frac { 2 n + 1 - \\nu } { 2 } ] } { { 2 n + 1 - 2 i } \\choose { \\nu } } A _ i - q ^ { n - \\nu + 1 / 2 } \\sum _ { i = 0 } ^ { [ \\frac { \\nu } { 2 } ] } { { 2 n + 1 - 2 i } \\choose { 2 n + 1 - \\nu } } A _ i = 0 \\qquad ( \\nu = 0 , 1 , \\cdots , 2 n + 1 ) . \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} F ( n ) = T \\mod v . \\end{align*}"}
-{"id": "6049.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } k ( x , 0 ) = k ( x , L ) = k ( 0 , y ) = k ( L , y ) = k _ y ( x , 0 ) = k _ y ( x , L ) = 0 , & , \\\\ s ( x , 0 ) = s ( x , L ) = s ( 0 , y ) = s ( L , y ) = s _ y ( x , 0 ) = s _ y ( x , L ) = 0 , & , \\end{array} \\right . \\end{align*}"}
-{"id": "4613.png", "formula": "\\begin{align*} 0 & = \\bar \\partial _ B Z \\lrcorner \\ , \\omega ^ b + Z \\lrcorner \\ , \\bar \\partial _ B \\omega ^ b \\\\ & = \\sum _ a \\bar \\omega ^ a \\wedge \\nabla _ { \\bar V _ a } ( Z \\lrcorner \\ , \\omega ^ b ) - \\sum _ a \\bar \\omega ^ a \\wedge Z \\lrcorner \\nabla _ { \\bar V _ a } \\omega ^ b \\\\ & = \\sum _ a ( \\nabla _ { \\bar V _ a } Z \\lrcorner \\ , \\omega ^ b ) \\bar \\omega ^ a . \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{align*} s ^ { ( r ) } : = \\frac { \\Gamma ( s + r ) } { \\Gamma ( s ) } \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} E = O - e ^ { 2 \\alpha _ 1 ( a ) } \\bar { z } _ 1 z _ 2 [ \\theta ( e _ { \\alpha _ 1 } ) , e _ { \\alpha _ 2 } ] / 2 - e ^ { 2 \\alpha _ 2 ( a ) } z _ 1 \\bar { z } _ 2 [ \\theta ( e _ { \\alpha _ 2 } ) , e _ { \\alpha _ 1 } ] / 2 \\end{align*}"}
-{"id": "8294.png", "formula": "\\begin{align*} \\beta _ { \\tau ^ { - 1 } ( i ) } = \\alpha _ { \\sigma ^ { - 1 } ( i ) } \\xi _ { \\tau ^ { - 1 } ( i ) } = \\nu _ { \\sigma ^ { - 1 } ( i ) } \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} \\mathcal { O } ( \\varepsilon ^ 0 ) : \\mathbf { A } \\mathbf { W _ 0 } + \\mathbf { G } _ { n l i n } ( \\mathbf { W _ 0 } ) = \\frac { \\partial \\mathbf { W } _ 0 } { \\partial \\mathbf { z } } \\mathbf { R } _ 0 . \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{align*} & H ( X ^ n | Y ^ n , X ^ n \\in \\mathcal { A } , Y ^ n \\in \\mathcal { B } ) = H ( Y ^ n | X ^ n , X ^ n \\in \\mathcal { A } , Y ^ n \\in \\mathcal { B } ) = h ( \\tilde { p } _ n ) \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{align*} T = \\bigvee _ { j = 1 } ^ g T _ j , \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{align*} \\mathcal { K } ( q _ 2 ) = \\left \\{ z = x + i y \\in \\mathbb { H } : y > \\frac { 1 } { q _ 2 } , \\ - \\frac { 1 } { 2 } < x \\leq \\frac { 1 } { 2 } \\right \\} . \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} \\kappa = \\min _ { r _ { X Y } : \\ : r _ X = q _ X , r _ Y = q _ Y } D ( r _ { X Y } \\| p _ { X Y } ) . \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} & \\sum _ { a = 2 } ^ { n - 2 } \\sum _ { j = 0 } ^ { n - 2 - a } \\sum _ { i = 1 } ^ { a - 1 } \\binom { n - 2 - a } { j } C _ { n - 2 - j , i } = \\sum _ { j = 0 } ^ { n - 4 } \\sum _ { i = 1 } ^ { n - 3 - j } C _ { n - 2 - j , i } \\sum _ { a = i + 1 } ^ { n - 2 - j } \\binom { n - 2 - a } { j } \\\\ & \\quad = \\sum _ { j = 0 } ^ { n - 4 } \\sum _ { i = 1 } ^ { n - 3 - j } \\binom { n - 2 - i } { j + 1 } C _ { n - 2 - j , i } \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} & Q _ 1 ^ { k , m } = \\sum _ { i = 0 } ^ { m } { { m + 1 } \\choose { i } } _ q q ^ { i ( 2 k + i - 1 ) / 2 } \\prod _ { j = 0 } ^ { i - 1 } ( q ^ { k + j } r ^ 2 - r ) \\prod _ { j = 1 } ^ { m - i } ( 1 - q ^ { 2 k + m - j } r ^ 2 ) \\\\ & Q _ 2 ^ { k , m } = \\frac { q ^ { ( 2 k + m ) ( m + 1 ) / 2 } ( - r ) ^ { m + 1 } - 1 } { q ^ { 2 k + m } r ^ 2 - 1 } \\prod _ { i = 0 } ^ { m } ( 1 - q ^ { k + i } r ) \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} \\begin{aligned} & W _ { j } ( t ) : = u _ { j + 1 } ( t ) - u _ { j } ( t ) , & & Z _ { j } ( t ) : = \\nabla y _ { j + 1 } ( t ) - \\nabla y _ { j } ( t ) , \\\\ & Y _ { j } ( t ) : = y _ { j + 1 } ( t ) - y _ { j } ( t ) , & & X _ { j } ( t ) : = x _ { j + 1 } ( t ) - x _ { j } ( t ) , \\end{aligned} \\end{align*}"}
-{"id": "2758.png", "formula": "\\begin{align*} y = \\{ S ^ e , T \\} \\prod \\{ 1 + a _ { i j } S ^ i T ^ j , S \\} \\cdot \\prod \\{ 1 + b _ { i j } S ^ i T ^ j , T \\} \\end{align*}"}
-{"id": "3374.png", "formula": "\\begin{align*} \\begin{aligned} | c | & = | h ' ( b ) | = 2 \\frac { | g ( b ) | } { 1 + | g ( b ) | ^ 2 } | h ' ( b ) | \\\\ & = 2 g ^ \\# ( b ) \\leq 2 \\left ( 1 + \\frac { | b | } { R } \\right ) \\leq 2 + 2 \\frac { B } { R } \\end{aligned} \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{align*} A \\cdot Z = A Z A ^ t , \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} \\phi ^ i _ 1 ( X ) : = \\frac { \\widehat { \\phi ^ i _ 1 } ( r _ i x , r _ i ^ 2 t ) } { A _ i r _ i ^ { \\beta } } \\end{align*}"}
-{"id": "1231.png", "formula": "\\begin{align*} S ( x , y , z ) = \\int _ { - \\infty } ^ \\infty \\exp [ i ( u ^ 5 + x u ^ 3 + y u ^ 2 + z u ) ] \\ , d u \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} y = \\{ S ^ e , T \\} \\prod \\{ 1 + a _ { i j } S ^ i T ^ j , S \\} \\cdot \\prod \\{ 1 + b _ { i j } S ^ i T ^ j , T \\} . \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{align*} d _ r \\bigg ( [ \\dots [ [ \\alpha ] _ { \\bar \\partial } ] _ { d _ 1 } \\dots ] _ { d _ { r - 1 } } \\bigg ) = [ \\dots [ [ \\partial u _ { r - 1 } ] _ { \\bar \\partial } ] _ { d _ 1 } \\dots ] _ { d _ { r - 1 } } , \\end{align*}"}
-{"id": "7649.png", "formula": "\\begin{align*} H _ K = \\begin{pmatrix} 0 _ { K - 1 \\times 1 } & I _ { K - 1 } & 0 _ { K - 1 \\times K ( K - 1 ) } \\\\ 0 _ { K - 2 \\times K + 2 } & I _ { K - 2 } & 0 _ { K - 2 \\times K ( K - 2 ) } \\\\ \\vdots & \\vdots & \\vdots \\\\ 0 _ { K - l \\times ( l - 1 ) K + l } & I _ { K - l } & 0 _ { K - l \\times K ( K - l ) } \\\\ \\vdots & \\vdots & \\vdots \\\\ 0 _ { 1 \\times ( K - 2 ) K + K - 1 } & 1 & 0 _ { 1 \\times K } \\end{pmatrix} \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} P ^ \\eta ( T ( \\eta _ { X _ m } ( m ) ) = & k \\ , | \\ , \\eta ( m - 1 ) , X _ 1 , X _ m ) \\\\ & \\leq \\Big ( 1 + c \\Big ( \\frac { L ^ { d + 1 } } { \\gamma ^ { \\frac { d + 1 } { 2 } } } + \\frac { 1 } { \\gamma ^ { \\frac 1 2 - \\alpha } } \\Big ) \\Big ) \\sup \\Big \\{ \\langle \\eta ( m - 1 ) \\rangle _ { z , L ' } ^ k : \\ , { L ' \\geq L } , z \\sim X _ m \\Big \\} \\\\ & \\leq \\big ( 1 + c \\frac { 1 } { \\phi _ T } \\big ) ( p _ k + \\epsilon _ { \\phi _ T ^ { 1 / 2 } } ) = p _ k + \\delta _ T , \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} \\Omega _ { c r i t } ^ { \\pm } ( \\rho ) = b ( \\rho ) + \\sum _ { m = 1 } ^ M m \\mbox { I m } ( \\beta _ m ) \\rho ^ { 2 m } \\pm \\sqrt { \\left [ \\sum _ { m = 1 } ^ M m \\mbox { I m } ( \\beta _ m ) \\rho ^ { 2 m } \\right ] ^ 2 - \\frac { a ( \\rho ) } { \\rho } \\frac { \\partial a ( \\rho ) } { \\partial \\rho } } . \\end{align*}"}
-{"id": "8256.png", "formula": "\\begin{align*} \\begin{aligned} f _ 0 ( w ) & = w ^ { - 1 } + ( \\alpha - \\tfrac 1 2 ) \\ln ( w ) + \\tfrac { 1 - \\alpha } { 2 } \\ln ( 1 - w ) , \\\\ f _ 1 ( w , s ) & = - ( 2 \\eta + s \\sigma ) \\ln ( w ) + \\eta \\ln ( 1 - w ) , \\end{aligned} \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{align*} \\chi _ \\C ( \\Lambda _ \\C ^ { ( a , \\theta , f ) } ) = \\Lambda _ \\C ^ { ( b , \\eta , g ) } . \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} \\nu _ \\lambda : = \\lambda \\sum _ { k \\in \\Z } \\delta _ { 2 \\pi k } \\in \\mathcal { M } ^ + _ { \\rm p e r } . \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} \\left \\| \\partial _ { \\xi } ^ \\alpha \\partial _ x ^ \\beta q _ { z } ( x , \\xi ) \\right \\| _ X = \\left \\| \\partial _ { \\xi } ^ \\alpha \\partial _ x ^ \\beta \\left ( p ( z , \\xi ) \\right ) \\right \\| _ X \\leq C _ { \\alpha , \\beta } \\langle \\xi \\rangle ^ { m - | \\alpha | } \\end{align*}"}
-{"id": "4914.png", "formula": "\\begin{align*} \\left ( \\begin{matrix} \\mathfrak { a } \\\\ \\mathfrak { b } \\end{matrix} \\right ) = S ^ { - 1 } A ( \\varepsilon _ 1 ) \\cdots A ( \\varepsilon _ { \\nu - 1 } ) \\left ( \\begin{matrix} \\alpha \\\\ \\beta \\end{matrix} \\right ) \\ \\ ( \\mathfrak a ' \\ , , \\mathfrak b ' ) = ( 1 \\ , , 0 ) A ( \\varepsilon _ { \\nu - 1 } ) \\cdots A ( \\varepsilon _ 1 ) S . \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} & D E _ \\varphi ( U ) = - \\int _ M g ( U , \\mathrm { d i v } \\ , T ^ \\varphi ) \\star 1 = \\int _ M g ( \\nabla U , T ^ \\varphi ) \\star 1 . \\end{align*}"}
-{"id": "2695.png", "formula": "\\begin{align*} I & \\leq \\frac { 1 } { \\delta ( x , y ) } \\sum _ { m = 0 } ^ { \\infty } 2 ^ { - m } \\abs { \\Omega _ { I ^ { ( m ) } ( x ' , y ) } ( x ' , y ) - \\Omega _ { I ^ { ( m ) } ( x , y ) } ( x , y ) } \\\\ & = \\frac { 1 } { \\delta ( x , y ) } \\sum _ { m = 0 } ^ { \\infty } 2 ^ { - m } \\abs { \\Omega _ { I ^ { ( m ) } ( x , y ) } ( x ' , y ) - \\Omega _ { I ^ { ( m ) } ( x , y ) } ( x , y ) } \\\\ & \\leq \\frac { 8 } { \\delta ( x , y ) } \\sum _ { m = 0 } ^ { \\infty } 2 ^ { - m } \\frac { \\delta ( x , x ' ) } { \\abs { I ^ { ( m ) } ( x , y ) } } \\\\ & = 1 6 \\frac { \\delta ( x , x ' ) } { \\delta ^ 2 ( x , y ) } . \\end{align*}"}
-{"id": "5363.png", "formula": "\\begin{align*} { } _ 2 F _ 1 ( a , b , 2 b ; z ) & = \\frac { \\Gamma ( 2 b ) } { 2 \\Gamma ^ 2 ( b ) } \\int _ { - 1 } ^ 1 ( \\frac { 1 + t } { 2 } ) ^ { b - 1 } ( \\frac { 1 - t } { 2 } ) ^ { b - 1 } ( 1 - z \\frac { t + 1 } { 2 } ) ^ { - a } d t \\\\ & = \\frac { \\Gamma ( 2 b ) } { 2 ^ { 2 b - 1 } ( 1 - \\frac { z } { 2 } ) ^ a \\Gamma ^ 2 ( b ) } \\int _ { - 1 } ^ 1 ( 1 - t ^ 2 ) ^ { b - 1 } ( 1 - \\frac { z } { 2 ( 1 - \\frac { z } { 2 } ) } t ) ^ { - a } d t . \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{align*} H ( X ) : = - \\sum _ { w } p ( w ) \\log { p ( w ) } \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{align*} t \\mapsto Z _ n ( t ) : = \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\delta _ { X _ { n , i } ( t ) } , \\frac { 1 } { n } \\sum _ { i = 1 } ^ n W _ { n , i } ( t ) \\right ) \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} ( U _ t u ) ( w ) = t ^ { - 4 / 3 } G _ { \\varrho } ( U _ t f ) ( w ) , \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} \\forall x \\in \\R , v _ { q , n } ( x ) : = v _ n ( x ) e ^ { - i q [ x ] } , [ x ] : = x \\ { \\rm m o d } \\ 2 \\pi . \\end{align*}"}
-{"id": "9090.png", "formula": "\\begin{align*} ( a , b ) \\leq ( a ' , b ' ) \\mbox { i f } [ a = a ' \\mbox { a n d } b \\leq b ' ] \\mbox { o r } b < a ' . \\end{align*}"}
-{"id": "1185.png", "formula": "\\begin{align*} \\int _ Y R _ { h _ \\infty } \\omega \\ge \\liminf _ { i \\rightarrow \\infty } \\int _ Y R _ { h _ i } \\omega = \\liminf _ { i \\rightarrow \\infty } \\int _ Y \\left ( | K _ i | ^ 2 - \\frac { n ^ 2 } { u ^ 2 } \\right ) \\omega . \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{align*} h \\log \\sum _ { i \\mid a _ { i j } < 0 } | a _ { i j } | \\exp ( y _ i / h ) = h \\log \\sum _ { i \\mid a _ { i j } > 0 } | a _ { i j } | \\exp ( y _ i / h ) , j \\in [ n ] , \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} P ^ \\eta ( & X _ { t _ { n + 1 } } ^ { \\vec v } - X _ r ^ { \\vec v } < c _ { n + 1 } t _ { n + 1 } + r ) \\\\ & \\leq P ^ \\eta \\Big ( \\sum _ { k = 1 } ^ { t _ n - 1 } Z _ k \\leq c _ { n + 1 } t _ { n + 1 } + r \\Big ) + \\sum _ { \\substack { y \\in B _ { t _ { n + 1 } } \\\\ r \\leq s \\leq t _ { n + 1 } } } \\P ^ \\eta \\big ( y \\not \\in G ( \\eta ( s ) , \\phi _ { t _ n } ) \\big ) \\ , . \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} \\rho ( x ) : = ( 1 + | x ' | ^ 4 + | x _ d | ^ 2 ) ^ { 1 / 2 } , \\gamma = ( 1 / 2 , \\ldots , 1 / 2 , 1 ) \\end{align*}"}
-{"id": "4713.png", "formula": "\\begin{align*} R ( x ) \\cdot R ( y ) = R \\Big ( R ( x ) \\cdot y + x \\cdot R ( y ) + \\lambda x \\cdot y \\Big ) . \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} \\epsilon = ( 0 ^ { n - m _ 1 - m _ 2 - p } , 1 ^ { m _ 1 + 2 p } , 2 ^ { m _ 2 - p } ) . \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} E \\left [ T ^ 2 \\right ] - E \\left [ T _ 1 ^ 2 \\right ] & = E \\left [ T ^ 2 - T _ 1 ^ 2 \\right ] = E \\left [ \\left ( T + T _ 1 \\right ) \\left ( T - T _ 1 \\right ) \\right ] \\\\ & \\leq E \\left [ \\left ( 2 T _ 1 + T _ 2 \\right ) ^ r \\right ] ^ { \\frac { 1 } { r } } E \\left [ \\left ( T - T _ 1 \\right ) ^ s \\right ] ^ { \\frac { 1 } { s } } , \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{align*} \\tilde { \\epsilon } _ 1 = \\min \\prod _ { x , y } p _ { X Y } ( x , y ) ^ { - \\mu ( x , y ) + \\gamma ( x , y ) } \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{align*} J q ^ 1 ( J q ^ 3 + J q ^ 2 J q ^ 1 - \\frac { 1 } { 6 } J q ^ 1 J q ^ 1 J q ^ 1 ) = J q ^ 2 J q ^ 2 . \\end{align*}"}
-{"id": "908.png", "formula": "\\begin{align*} \\begin{gathered} \\overline { \\partial _ \\mu l ( \\gamma ) } = \\frac 1 \\pi \\int _ { F _ 0 } \\frac { \\phi ( z ) y ^ 2 } { ( \\bar z ) ^ 2 } d x d y = \\frac 1 \\pi \\int _ 0 ^ \\pi \\int _ { 0 } ^ { l } \\phi ( e ^ t e ^ { i \\theta } ) \\frac { ( e ^ t \\Im e ^ { i \\theta } ) ^ 2 } { ( e ^ t e ^ { - i \\theta } ) ^ 2 } e ^ { 2 t } d t d \\theta \\\\ = - \\frac 1 \\pi \\frac 1 4 \\int _ 0 ^ \\pi \\int _ { 0 } ^ { l } \\phi ( e ^ t e ^ { i \\theta } ) ( 1 - e ^ { 2 i \\theta } ) ^ 2 e ^ { 2 t } d t d \\theta . \\end{gathered} \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{align*} E _ { f } ( A , B ) = \\sum _ { \\delta \\in \\Delta } m _ { \\delta } ^ 2 \\ge \\frac { ( \\sum _ { \\delta \\in \\Delta } m _ \\delta ) ^ 2 } { D } = \\frac { | A | ^ 2 | B | ^ 2 } { D } . \\end{align*}"}
-{"id": "4333.png", "formula": "\\begin{align*} D = \\left ( \\frac { 2 \\pi } { \\omega } \\right ) ^ { 1 2 } \\Delta ( \\tau ) \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} \\mathcal { E } ^ { G P } ( u ) : = \\langle u , - \\Delta _ { \\mathbf { A } } u \\rangle + 4 \\pi a \\langle u , | u | ^ 2 u \\rangle . \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} \\mathcal { A } _ w \\phi _ { K } ^ { \\chi } = c _ w ( \\chi ) \\phi _ { K } ^ { \\prescript { w } { } { \\chi } } \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{align*} \\begin{gathered} A = u \\Big ( \\triangle \\alpha - ( \\lambda + k - 2 ) ( \\lambda + n - k ) \\alpha - 2 d ^ \\ast \\beta \\Big ) - r \\dot { u } \\left ( 2 \\lambda + n - 1 \\right ) \\alpha - r ^ 2 \\ddot { u } \\ , \\alpha , \\\\ B = u \\Big ( \\triangle \\beta - ( \\lambda + n - k - 2 ) ( \\lambda + k ) \\beta - 2 d \\alpha \\Big ) - r \\dot { u } \\left ( 2 \\lambda + n - 1 \\right ) \\beta - r ^ 2 \\ddot { u } \\ , \\beta . \\end{gathered} \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} j ( E ) = \\dfrac { ( h ^ 2 - 3 ) ^ 3 ( h ^ 6 - 9 h ^ 4 + 3 h ^ 2 - 3 ) ^ 3 } { h ^ 4 ( h ^ 2 - 9 ) ( h ^ 2 - 1 ) ^ 3 } \\end{align*}"}
-{"id": "2937.png", "formula": "\\begin{align*} f ( x , z ) = \\int _ { - \\infty } ^ { \\infty } e ^ { 2 \\pi i \\xi _ { 1 } x - 2 \\pi | \\xi _ { 1 } | z } \\hat { g } ( \\xi _ { 1 } ) \\ , d \\xi _ { 1 } . \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} H _ { \\mu - 1 } ( \\lambda ) : = ( 1 - \\alpha ^ \\chi ) ^ { - 1 } ( 1 - \\beta ^ \\chi ) ^ { - 1 } ( \\lambda - 1 ) ^ { \\mu - 2 } { } _ 3 F _ 2 \\left ( { 1 , 1 , 2 - \\mu \\atop 2 - \\alpha ^ \\chi , 2 - \\beta ^ \\chi } ; ( 1 - \\lambda ) ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} h ( q , z ) - \\frac 1 p h ( q ^ p , z ^ p ) = \\sum _ { \\substack { n \\geq 1 \\\\ p \\nmid n } } \\frac { k ( q ^ n , z ^ n ) } { n } . \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{gather*} x _ { 2 \\varepsilon _ i } ( c ) : = \\exp ( c X _ { 2 \\varepsilon _ i } ) , h _ { 2 \\varepsilon _ i } ( c ) : = \\exp ( c H _ { 2 \\varepsilon _ i } ) , x _ { - 2 \\varepsilon _ i } ( c ) : = \\exp ( c X _ { - 2 \\varepsilon _ i } ) \\end{gather*}"}
-{"id": "1969.png", "formula": "\\begin{align*} T _ { k } ^ { \\alpha } f ( x ) = \\int _ { \\mathbf { B } } \\frac { f ( y ) } { [ x , y ] ^ { n + \\alpha + | k | - 1 } } d v _ { \\alpha } ( y ) . \\end{align*}"}
-{"id": "4705.png", "formula": "\\begin{align*} a s _ { \\alpha , \\bullet } ( A _ i , A _ j , A _ k ) = \\{ 0 _ A \\} , i , j , k \\in \\mathbb { Z } _ 2 . \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} \\overline \\square _ B \\phi & = \\nabla _ T ^ * \\nabla _ T \\phi + \\sum _ a R ^ Q ( V _ a , \\bar V _ a ) \\phi + \\operatorname { d i v } _ \\nabla ( H ^ { 0 , 1 } ) \\phi \\\\ & = \\bar \\nabla _ T ^ * \\bar \\nabla _ T \\phi - i \\nabla _ { J \\kappa _ B ^ \\sharp } \\phi + \\operatorname { d i v } _ \\nabla ( H ^ { 0 , 1 } ) \\phi . \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{align*} \\big ( F g _ 1 , F g _ 2 , g _ 2 ^ { - 1 } g _ 1 \\check { F } , \\check { F } \\big ) & = \\big ( F g _ 1 ' , F g _ 2 ' , { g _ 2 ' } ^ { - 1 } g _ 1 ' \\check { F } , \\check { F } \\big ) X _ { 2 , 2 } . \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} A _ k = a _ k J q ^ k + a _ { k - 1 } J q ^ { k - 1 } J q ^ 1 + \\cdots + a _ 1 J q ^ 1 J q ^ { k - 1 } = 0 . \\end{align*}"}
-{"id": "5783.png", "formula": "\\begin{align*} \\omega _ 1 ^ { k } = & \\beta ^ { k } \\smallskip \\\\ \\omega _ i ^ { k } = & \\frac { \\alpha _ i ^ { k } + \\lambda _ i \\beta ^ { k } } { \\lambda _ 1 } \\quad i = 2 , \\ldots , n . \\end{align*}"}
-{"id": "1656.png", "formula": "\\begin{align*} \\partial _ a \\Phi ( a , b ) & = \\log \\frac { a } { b } - \\log \\frac { 1 - a } { 1 - b } , \\\\ \\partial _ b \\Phi ( a , b ) & = \\frac { 1 - a } { 1 - b } - \\frac { a } { b } , \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} \\lim _ { k } d ( u ^ { l _ { k } } , C _ { i } ) = 0 \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} \\operatorname { g r a d e } ( G _ { I ^ l } ( A ) _ + , G _ { I ^ l } ( M ) ) = \\xi _ { I } ( M ) \\mbox { f o r e v e r y } l > \\operatorname { a m p } _ { I } ( M ) . \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{align*} v [ m ] ( x ) : = v _ { f } ( x ) - v _ { i } [ m ] ( x ) \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} b _ 0 \\in \\Im ( \\prod \\limits _ { \\alpha > 0 , \\langle \\alpha , \\hat { w } \\xi \\rangle = 0 } U _ { \\alpha } \\to B ) \\textrm { a n d } b _ + \\in \\Im ( \\prod \\limits _ { \\alpha > 0 , \\langle \\alpha , \\hat { w } \\xi \\rangle > 0 } U _ { \\alpha } \\times T \\to B ) \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } \\sum _ { x : \\kappa ( x , t ) = 0 } | \\kappa _ { s } | ( x , t ) d t = 0 \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} \\displaystyle { f ^ * ( t _ { i j } ) = \\prod _ { 1 \\leq r , s \\leq g } ( 1 + t _ { r s } ) ^ { f _ { r i } g _ { s j } } - 1 1 \\leq i , j \\leq g } . \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{align*} D ^ { I V } _ n = \\Big \\{ z \\in \\C ^ n \\Big | | z | ^ 2 < 2 , \\ ; | z | ^ 2 < 1 + \\Big | \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { n } z _ j ^ 2 \\Big | \\Big \\} \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} \\rho _ { x } ^ { d / 2 } \\ = \\ \\frac { d } { 2 } \\int _ { 0 } ^ { \\infty } \\mathbf { 1 } \\big [ \\mathrm { e } ^ { - r } < \\rho _ { x } \\big ] \\ , \\mathrm { e } ^ { - d r / 2 } \\ , \\mathrm { d } r \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} t y = T S _ v x _ t = T S _ v ( V _ 0 + t V _ 1 ) \\xi _ v = V _ t ( A _ 0 + t A _ 1 ) \\xi _ v , \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} \\Phi _ \\nu ( z ) = z ^ \\nu K _ \\nu ( z ) . \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} 2 \\ell A ^ 2 + \\sum \\limits _ { m = 1 } ^ { d } A R ^ { m } A R _ { m * } ^ { * } + \\left ( \\sum _ { r , s = 1 } ^ d ( A R ^ i ) _ { r s } ( A R ^ j ) _ { s r } ) \\right ) + \\left ( \\sum _ { r , s = 1 } ^ { d } ( A R ^ s ) _ { i r } ( A R ^ j ) _ { s r } ) \\right ) = A , \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} \\Big [ \\Phi _ m W _ k ( x ) - m ^ k W _ k ( x \\gamma ) \\Phi _ m \\Big ] \\left ( 1 - \\frac x { q ^ k } \\right ) = 0 \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{align*} \\int _ { 0 } ^ \\infty | D _ \\lambda ( x , y ) | \\ , \\mathrm d x = \\frac { 1 } { 2 \\sqrt { | \\lambda | } } \\int _ { 0 } ^ \\infty e ^ { - \\operatorname { I m } \\sqrt { \\lambda } | x - y | } \\ , \\mathrm d x = \\frac { 2 - e ^ { - \\operatorname { I m } \\sqrt { \\lambda } y } } { 2 \\sqrt { | \\lambda | } \\operatorname { I m } \\sqrt { \\lambda } } \\leq \\frac { 1 } { \\sqrt { | \\lambda | } \\operatorname { I m } \\sqrt { \\lambda } } , \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} R = \\omega + \\beta = \\omega ^ { \\frac { 1 } { 2 } } \\left ( E + \\omega ^ { - \\frac { 1 } { 2 } } \\beta \\omega ^ { - \\frac { 1 } { 2 } } \\right ) \\omega ^ { \\frac { 1 } { 2 } } = \\omega ^ { \\frac { 1 } { 2 } } ( E + \\sigma ) \\omega ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "2354.png", "formula": "\\begin{align*} \\Gamma ^ { ( k ) } ( 1 ) = \\int _ 0 ^ { \\infty } e ^ { - x } ( \\ln x ) ^ k d x . \\end{align*}"}
-{"id": "3985.png", "formula": "\\begin{align*} I ( X ; Y ) = \\frac { I ( X ; Y | Z ) } { \\epsilon } . \\end{align*}"}
-{"id": "5886.png", "formula": "\\begin{align*} \\psi ( \\nu , \\mu ) = t ^ { \\Omega ( \\mu , \\nu ) } \\cdot I ( \\mu , \\nu ) \\end{align*}"}
-{"id": "644.png", "formula": "\\begin{align*} y _ n : = \\sum _ { i = 1 } ^ { M } \\frac { y _ n ( i ) } { v _ i } , \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{align*} \\mathbf R _ { c c } = \\mathrm { S p a n } \\{ [ X _ i , X _ j ] | i < j \\in [ n ] \\} , \\end{align*}"}
-{"id": "6342.png", "formula": "\\begin{align*} E ( M , N ) : = \\bigoplus _ { i \\geqslant 0 } \\operatorname { E x t } ^ i _ A ( M , N ) , \\mbox { a n d } T ( M , N ) : = \\bigoplus _ { i \\geqslant 0 } \\operatorname { T o r } ^ A _ i ( M , N ) . \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} \\alpha = \\xi _ 1 \\cdot ( \\omega _ \\Sigma \\times 1 ) + \\xi _ 2 \\cdot ( 1 \\times \\omega _ { S ^ 2 } ) , \\ \\xi _ 1 , \\xi _ 2 \\in \\R . \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{align*} & \\gamma ( \\tilde \\gamma ^ { - 1 } ( \\tilde \\gamma ^ { - 1 } ) ^ T ) = \\tilde \\gamma ^ { - 1 } - D , \\\\ D \\equiv & \\left ( \\begin{array} { c c c } 0 & - B _ 0 / ( \\gamma ^ 2 + B _ 0 ^ 2 ) & 0 \\\\ B _ 0 / ( \\gamma ^ 2 + B _ 0 ^ 2 ) & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} \\frac { H _ 0 } { t } & \\gtrsim \\int _ t ^ T \\frac { D ( T ) } { \\Big ( 1 + ( T - s ) D ^ { 3 / 2 } ( T ) \\Big ) ^ { 2 / 3 } } \\ , d s \\\\ & = D ( T ) ^ { - 1 / 2 } \\int _ 0 ^ { D ^ { 3 / 2 } ( T ) ( T - t ) } \\ ; \\frac { 1 } { ( 1 + \\sigma ) ^ { 2 / 3 } } \\ , d \\sigma \\\\ & \\geqslant \\frac { D ( T ) ( T - t ) } { \\Big ( 1 + ( T - t ) D ^ { 3 / 2 } ( T ) \\Big ) ^ { 2 / 3 } } \\\\ & \\gtrsim \\min \\left \\{ D ( T ) ( T - t ) , ( T - t ) ^ { 1 / 3 } \\right \\} , \\end{align*}"}
-{"id": "2858.png", "formula": "\\begin{align*} 0 \\leq g ' - g _ { \\wedge , 1 } ' = \\max \\{ g ' - 1 , 0 \\} \\leq \\max \\{ \\nu ' - 1 , 0 \\} \\leq | \\nu ' - 1 | . \\end{align*}"}
-{"id": "8849.png", "formula": "\\begin{align*} \\Omega _ { \\beta , \\bar { \\alpha } } = 0 . \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} r ^ + ( M , s _ 2 ^ - , t ) = \\mathrm { r a n k } \\left ( M ( s _ 2 ^ - , t ) \\right ) . \\end{align*}"}
-{"id": "5450.png", "formula": "\\begin{align*} \\langle w \\lambda , \\check { \\varpi _ i } \\rangle = \\langle w ' \\lambda , \\check { \\varpi _ i } \\rangle \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{align*} F ( y , t ) = y ^ { q ^ 2 } + y ^ q t ^ { m ( q - 1 ) } + y ^ q + y t ^ { m ( q - 1 ) } + t ^ { ( q + q _ 0 ) m } . \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} \\langle x , l \\rangle = \\int _ S \\langle u ( s ) , l \\rangle d \\mu ( s ) \\ \\ \\ \\ ( l \\in X _ * ) . \\end{align*}"}
-{"id": "4673.png", "formula": "\\begin{align*} W : = ( e ^ { r H } \\# _ { s } e ^ { r K } ) ^ { 1 / r } \\quad { \\rm a n d } V : = e ^ { ( 1 - s ) H + s K } \\ . \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{align*} \\phi _ { \\delta } ( x ) & = \\int _ { \\mathbb { R } ^ n } d _ { K } ^ 2 ( x - \\lambda ) \\eta _ { \\delta } ( \\lambda ) d \\lambda \\\\ & \\triangleq d _ { K } ^ 2 ( x ) \\star \\eta _ { \\delta } ( x ) , x \\in \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} { \\bf H } = \\dfrac { x ^ { \\perp } } { 2 } \\end{align*}"}
-{"id": "8024.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\nabla ^ T f ( y _ { i , t } ) e _ { i , t } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\left ( \\nabla ^ T f ( y _ { i , t } ) - \\nabla ^ T f ( \\overline { x } _ t ) \\right ) ( y _ { i , t } - \\overline { x } _ t ) \\leq \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\mu \\| e _ { i , t } \\| ^ 2 = 2 \\mu \\overline { V } _ t . \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{align*} \\mathcal { L } _ { t } ^ { u } \\phi ( x ) = \\dfrac { 1 } { 2 } \\operatorname { t r } \\Bigl \\{ a ( t , x , u ) D _ { x } ^ 2 \\phi ( x ) \\Bigr \\} & + f ( t , x , u ) D _ { x } \\phi ( x ) , \\ , \\ , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "3464.png", "formula": "\\begin{align*} c : = \\sup _ { y \\in \\mathbb R } \\int _ { \\mathbb R } | K _ \\lambda ( x , y ) | \\ , \\mathrm d x < \\infty \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{align*} \\Phi _ \\infty ( q ) = \\begin{pmatrix} 0 & | q | _ k ^ { - 1 / 2 } q \\\\ | q | _ k ^ { 1 / 2 } & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "911.png", "formula": "\\begin{align*} \\alpha ( \\gamma , \\phi ) = - 2 \\ ( \\sinh \\frac { l ( \\gamma ) } { 2 } \\ ) C . \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{align*} \\widehat { \\sigma } = t _ i ^ { - 1 } \\log ^ { \\frac { 1 + \\nu } { 2 } } ( t _ i ) ( \\sigma + 4 ^ { - \\ : \\nu } c _ 2 c _ \\lambda ^ \\nu \\log ^ { - \\ : \\nu } ( t _ i ) \\delta ) \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Phi ^ { \\gamma } _ t ( x , s ) = v ( \\Phi ^ { \\gamma } _ { t } ( x , s ) ) , t \\in [ t _ m , t _ { m + 1 } ) . \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} F _ * ( z ) = \\exp \\left ( h ( q , z ) - \\frac { 1 } { d } h ( q ^ { d } , z ^ { d } ) \\right ) , \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} \\psi ( x , t ) = \\mathcal { F } ^ { - 1 } \\left \\{ \\ \\hat { \\psi } ( p , t ) ; x \\right \\} = \\frac { 1 } { 2 \\pi \\hslash { } } \\int _ { - \\infty { } } ^ { \\infty { } } e ^ { i p x / \\hslash { } } \\ \\hat { \\psi } ( p , t ) \\ d p \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} H _ N = \\sum _ { i = 1 } ^ N ( - \\Delta _ i ) + \\sum _ { i < j } N ^ 2 V ( N ( x _ i - x _ j ) ) . \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{align*} \\begin{cases} \\dot { x } _ 1 = x _ 2 \\\\ \\dot { x } _ 2 = u - \\alpha x _ 2 \\alpha > 0 \\\\ x _ 1 ( 0 ) = 0 , \\ x _ 2 ( 0 ) = 0 \\\\ x _ 1 ( T ) = X > 0 , \\ x _ 2 ( T ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "7244.png", "formula": "\\begin{align*} ( q ^ { s + m } - 1 ) P _ f ( q ^ { - s } ) _ { s = - m } & = \\lim _ { N \\rightarrow \\infty } 1 + \\sum _ { \\alpha = 0 } ^ N q ^ { - \\alpha ( n - m ) } \\left ( q ^ { m - n } B _ { f , \\alpha + 1 } - B _ { f , \\alpha } \\right ) \\\\ & = \\lim _ { N \\rightarrow \\infty } q ^ { - ( N + 1 ) ( n - m ) } B _ { f , N + 1 } = B _ f . \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} \\widetilde { c } _ 1 ( t ; T ) = \\frac { \\prod _ { n = 1 } ^ t \\sin \\left ( \\frac { n } { r } \\right ) \\prod _ { n = 1 } ^ { 2 T - t } \\sin \\left ( \\frac { n } { r } \\right ) } { \\prod _ { n = 1 } ^ { 2 T } \\sin \\left ( \\frac { n } { r } \\right ) } \\prod _ { n = 1 } ^ T \\frac { \\sin ^ 2 \\left ( \\frac { n } { r } \\right ) } { \\sin \\left ( \\frac { 2 \\alpha _ 0 + ( 2 n - T - 1 ) } { 2 r } \\right ) } . \\end{align*}"}
-{"id": "5030.png", "formula": "\\begin{align*} 3 \\ [ c , z _ 1 ] [ z _ 2 , z _ 3 , z _ 4 ] = 0 . \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t u = \\frac { \\nu ^ 2 } { 2 } \\partial _ { x x } u - u \\partial _ x u , ( t , x ) \\in [ 0 , T ] \\times \\R , \\nu > 0 \\\\ u ( 0 , \\cdot ) = u _ 0 \\ . \\end{array} \\right . \\end{align*}"}
-{"id": "6514.png", "formula": "\\begin{align*} h _ d ( \\pi / 2 + \\theta ) = h _ d ( \\pi / 2 - \\theta ) \\quad 0 \\leq \\theta \\leq \\pi / 2 . \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} R ( x , t ) = \\sum _ { i = 0 } ^ { d - 2 } \\sum _ { | \\alpha | = i } D ^ { \\alpha } _ x ( p - \\triangle ^ { - 1 } \\nabla \\cdot f ) \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} \\lim _ { m , n \\to \\infty } \\int _ { \\textrm { s p t } \\ , \\zeta } | T ( u _ m - u _ n ) | ^ { p } \\ , d x = 0 , \\lim _ { m , n \\to \\infty } \\int _ { \\textrm { s p t } \\ , \\zeta } V | T ( u _ m - u _ n ) | ^ p \\ , d x = 0 \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} H _ n = \\sum _ { m = 1 , p ^ k \\nmid m } ^ n \\frac 1 m + \\frac 1 { p ^ k } H _ { n _ 1 } = \\frac b { p ^ { k - 1 } a } + \\frac { u _ { n _ 1 } } { p ^ k v _ { n _ 1 } } = \\frac { p b v _ { n _ 1 } + a u _ { n _ 1 } } { p ^ k a v _ { n _ 1 } } , \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} x ( x \\alpha \\cdot x ^ \\rho ) \\cdot y = x \\cdot ( x \\alpha \\cdot x ^ \\rho ) y \\Rightarrow x R _ { x \\alpha \\cdot x ^ \\rho } \\cdot y L ^ { - 1 } _ { x \\alpha \\cdot x ^ \\rho } = x y \\end{align*}"}
-{"id": "4976.png", "formula": "\\begin{align*} - \\gamma \\partial _ x ^ { - 2 } \\tilde { w } + H \\tilde { w } _ x + \\tilde { w } = F ( \\psi _ { 1 , \\gamma } ) - F ( \\tilde { \\psi } _ { 1 , \\gamma } ) \\end{align*}"}
-{"id": "8117.png", "formula": "\\begin{align*} \\begin{cases} \\operatorname { d i v } ( y ^ a \\nabla U _ r ) = y ^ a \\partial _ t U _ r , \\\\ \\underset { y \\to 0 } { \\lim } - y ^ a \\partial _ { y } U _ r = r ^ { 2 s } V ( r x , r ^ 2 t ) U _ r ( x , 0 , t ) . \\end{cases} \\end{align*}"}
-{"id": "7977.png", "formula": "\\begin{align*} u _ i ( X ) : = \\frac { \\widehat { u } _ i ( r _ i x , r _ i ^ 2 t ) } { A _ i r _ i ^ { \\beta } } \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} - \\Delta _ p u + u ^ { - \\delta } = 0 \\ ; \\mbox { i n } \\ ; \\ ; \\mathbb R ^ N \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty ( - 1 ) ^ k ( J q ^ 1 ) ^ k ( f ) ( J q ^ { - 1 } ) ^ { k + 1 } ( g ) . \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} \\widehat { Z } ^ { - } _ n ( x , \\omega ) : = \\bigl ( f _ { \\omega _ 1 } ^ { - 1 } \\circ \\cdots \\circ f _ { \\omega _ n } ^ { - 1 } \\bigr ) ( x ) , \\widehat { Z } ^ { - } _ 0 ( x , \\omega ) = x . \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{align*} \\Vert \\nabla ^ m ( K ( p ) ) \\Vert = \\sum \\limits _ { j = 1 } ^ { k } O \\left ( \\frac { 1 } { \\delta _ j ^ { 2 + m } ( p ) } \\right ) \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } f ^ \\beta ( \\lambda ) \\widetilde { F } ( f ( \\lambda ) ) = 0 \\end{align*}"}
-{"id": "9165.png", "formula": "\\begin{align*} ( \\pi _ { j , k } ) ( p ) _ m ( \\pi _ { i , m - n } ) = ( p ) _ k ( \\pi _ { i , k - n } ) = ( p ) _ k b ( i _ { k - n } ) \\dots b ( i _ { l - 1 } ) ( \\pi _ { i , l } ) , \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{align*} \\mathcal { Y } ^ { ( x ) } _ { \\ell ^ { \\circ } _ x , 2 } = Y _ { i , 2 } \\geq Y _ { i , 1 } - 1 \\geq Y _ { i , j ' } + 1 = \\mathcal { Y } ^ { ( x ) } _ { \\ell ^ { \\circ } _ x , j ' } + 1 . \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} u = ( u _ 1 , \\ldots , u _ { 2 n } ) \\colon X \\to \\R ^ { 2 n } \\ \\ u _ j = \\log | f _ j | \\ \\ \\ \\ j = 1 , \\ldots , 2 n \\end{align*}"}
-{"id": "8266.png", "formula": "\\begin{align*} \\nu ( \\lambda ) = \\frac { 1 } { z _ \\lambda } \\sum _ { k = 0 } ^ { d - 1 } \\frac { \\psi _ d ^ k ( \\lambda ) } { q ^ k } . \\end{align*}"}
-{"id": "7831.png", "formula": "\\begin{align*} w _ { j _ 1 , j _ 2 } ( \\theta \\ , | \\ , \\phi , a ) = w _ { j _ 1 , j _ 2 } ( \\theta \\ , | \\ , \\phi , a + r N ) \\ , . \\end{align*}"}
-{"id": "2897.png", "formula": "\\begin{align*} \\Big ( - \\Delta + \\frac { 1 } { 2 } ( V _ N - W _ \\beta ) \\Big ) f _ \\beta = 0 , \\end{align*}"}
-{"id": "5624.png", "formula": "\\begin{align*} L ^ { \\mathrm { I R W } } f ( n ) = \\sum _ { 1 \\leq i < j \\leq N } n _ i \\left ( f ( n ^ { i , j } ) - f ( n ) \\right ) + n _ j \\left ( f ( n ^ { j , i } - f ( n ) \\right ) , n \\in \\N ^ N , \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} \\sigma = \\omega ^ { - \\frac { 1 } { 2 } } \\beta \\omega ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d x ^ 2 } \\Big ( \\log \\big ( \\theta _ 2 ( x ) \\big ) \\Big ) = \\frac { \\theta _ 2 '' ( x ) \\theta _ 2 ( x ) - \\theta _ 2 ' ( x ) ^ 2 } { \\theta _ 2 ( x ) ^ 2 } . \\end{align*}"}
-{"id": "2492.png", "formula": "\\begin{align*} K _ 2 ( N ) = - e ^ { i \\xi A _ N } \\int _ 0 ^ { \\infty } \\xi \\ , e ^ { - \\xi \\tau } \\left [ 1 - \\left ( 1 - e ^ { - A _ N } e ^ { - i \\tau } \\right ) ^ N \\right ] d \\tau . \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} b _ { i + 1 } - b _ i = c _ { \\lambda _ { i + 1 } } - c _ { \\lambda _ i } - ( \\lambda _ { i + 1 } - \\lambda _ i ) - 1 \\geq - ( \\lambda _ { i + 1 } - \\lambda _ i ) - 1 . \\end{align*}"}
-{"id": "1797.png", "formula": "\\begin{align*} \\big ( \\varphi \\varphi _ j A \\psi _ j \\psi \\big ) _ \\nu = \\big [ \\big ( \\varphi \\varphi _ j A \\psi _ j \\psi \\big ) _ { \\nu _ j } \\big ] _ { \\nu \\circ \\nu _ j ^ { - 1 } } . \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} X _ { k , F } = \\sum _ { \\substack { | \\lambda | < k \\\\ s ( \\lambda ) \\in F } } \\big ( 1 - \\frac { | \\lambda | } { k } \\big ) A _ { \\lambda } \\otimes S _ { \\lambda } . \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{align*} & \\sum _ { a = 2 } ^ { n - 6 } \\sum _ { b = a + 4 } ^ { n - 2 } \\sum _ { q = a + 2 } ^ { b - 2 } ( 2 ^ { q - 1 - a } - 1 ) ( 2 ^ { b - 1 - q } - 1 ) \\\\ & = \\sum _ { a = 2 } ^ { n - 6 } \\sum _ { b = a + 4 } ^ { n - 2 } \\left [ ( b - 3 - a ) 2 ^ { b - 2 - a } - 2 ( 2 ^ { b - 2 - a } - 2 ) + ( b - 3 - a ) \\right ] \\\\ & = \\sum _ { a = 2 } ^ { n - 6 } \\sum _ { b = a + 4 } ^ { n - 2 } ( b - 5 - a ) 2 ^ { b - 2 - a } + \\sum _ { a = 2 } ^ { n - 6 } \\left [ 4 ( n - 5 - a ) + \\binom { n - 4 - a } { 2 } \\right ] \\\\ & = \\sum _ { a = 2 } ^ { n - 6 } \\sum _ { b = a + 4 } ^ { n - 2 } ( b - 5 - a ) 2 ^ { b - 2 - a } + 4 \\binom { n - 6 } { 2 } + \\binom { n - 5 } { 3 } \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} \\mathcal { C } ^ { ( \\omega ) } : = \\bigcap _ { n \\in \\N } \\bigcup _ { u \\in \\mathbb { X } ^ { ( \\omega ) } _ n } B ^ { ( \\omega ) } _ { u } . \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} F _ i ( b ) = ( u _ i ( b ) , F _ { i , b } ) \\qquad F ' _ i ( b ) = ( u _ i ( b ) , F ' _ { i , b } ) \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{align*} [ \\mathbf { t } _ 1 , \\mathbf { t } _ 2 , . . . , \\mathbf { t } _ { 2 N } ] = \\mathbf { V } ^ { - 1 } , \\mathbf { t } _ j \\in \\mathbb { C } ^ { 1 \\times 2 N } , j = 1 , . . . , 2 N . \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{align*} [ 1 \\ \\dots \\ t - 1 \\mid i _ 1 \\ \\dots \\ i _ { t - 1 } ] \\times [ 2 \\ \\dots \\ t \\mid n - t + 2 \\ \\dots \\ n ] \\ = \\ \\pi \\times [ 2 \\ \\dots \\ t \\mid i _ 1 \\ \\dots \\ i _ { t - 1 } ] . \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} A _ { i j } = \\frac { | | e _ i + e _ j | | _ A ^ 2 - | | e _ i - e _ j | | _ A ^ 2 } { 4 } - \\sqrt { - 1 } \\frac { | | e _ i + \\sqrt { - 1 } e _ j | | _ A ^ 2 - | | e _ i - \\sqrt { - 1 } e _ j | | _ A ^ 2 } { 4 } . \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} \\prod _ { h = 1 } ^ { \\sigma ( \\vec { A } ) _ j - 1 } \\Big ( \\sum _ { k = j + 1 } ^ N \\sigma ( \\vec { A } ) _ k + h \\Big ) \\end{align*}"}
-{"id": "5307.png", "formula": "\\begin{align*} ( h _ 1 , h _ 2 , h _ 3 ) ^ { [ ( \\sigma _ 1 , \\sigma _ 2 , \\sigma _ 3 ) , \\gamma ] } = \\left ( h _ { 1 ' } ^ { \\sigma _ { 1 ' } } , h _ { 2 ' } ^ { \\sigma _ { 2 ' } } , h _ { 3 ' } ^ { \\sigma _ { 3 ' } } \\right ) , i ' : = i ^ { \\gamma ^ { - 1 } } \\ ; \\forall \\ ; i \\in \\{ 1 , 2 , 3 \\} \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} \\lim _ { y \\to x } { \\frac { F ( y ) - F ( x ) - f ( x ) ( y - x ) } { \\varphi ( x + \\alpha ( y - x ) ) - \\varphi ( x ) } } = 0 . \\end{align*}"}
-{"id": "2124.png", "formula": "\\begin{align*} A _ t ^ { } = \\frac { 1 } { 2 } \\int _ { 0 < s _ 1 < s _ 2 < t } ( d B _ { s _ 1 } \\otimes d B _ { s _ 2 } - d B _ { s _ 2 } \\otimes d B _ { s _ 1 } ) \\end{align*}"}
-{"id": "5671.png", "formula": "\\begin{align*} T _ n ( \\Re f ) = \\Re T _ n ( f ) , T _ n ( \\Im f ) = \\Im T _ n ( f ) . \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} W _ t = \\sum _ { j \\in \\mathcal { J } } \\sqrt { \\eta _ j } \\beta _ t ^ j \\tilde { e } _ j , t \\geq 0 . \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{gather*} [ [ [ X _ { \\alpha _ 1 } , X _ { \\alpha _ 2 } ] , X _ { \\alpha _ 3 } ] , X _ { \\alpha _ 2 } ] \\in I ' _ a \\iff a = - 1 , \\end{gather*}"}
-{"id": "231.png", "formula": "\\begin{align*} ( \\mathbf f ( z ) \\otimes 1 ) \\Delta _ { \\mathcal A } ( \\mathbf f ( z ' ) ) - \\mathbf f ( z + z ' ) \\otimes \\mathbf f ( z ' ) + ( 1 \\otimes \\mathbf f ( - z ) ) \\Delta _ { \\mathcal A } ( \\mathbf f ( z + z ' ) ) = 0 \\end{align*}"}
-{"id": "5485.png", "formula": "\\begin{align*} \\psi = \\arccos \\left ( \\frac { \\left [ \\Omega - b ( \\rho ) \\right ] \\rho } { \\varepsilon r } \\right ) . \\end{align*}"}
-{"id": "1736.png", "formula": "\\begin{align*} \\| f \\| ^ i _ { L ^ \\infty ( \\R ^ n ; F ) } : = \\underset { z \\in \\R ^ n } { { \\rm e s s } \\sup } \\{ | f ( z ) | _ i \\} i \\in \\N . \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} \\tilde { u } _ j ( a ) = - u ^ { * , j , i } _ i ( p ) - \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\frac { - u ^ { * , l , j } ( p ) \\alpha ^ { \\vee , l } } { ( 2 \\chi - p ) ( \\alpha ^ { \\vee } ) } + I _ { H , j } ( d _ p u ^ * ) . \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} \\dim _ H P _ 1 \\mu + \\dim _ H \\mu _ { [ x ] } = \\dim _ H \\mu , P _ 1 \\mu x \\in P _ 1 ( F ) , \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{align*} \\mathrm { h t } ( \\Phi ^ { { \\mathcal F } } ( \\underline { E } ) ) = \\max \\{ \\mathrm { h t } ( ( { \\mathcal G } _ \\Omega ) ^ { \\mathrm { f o r } } ) \\ , \\mid \\ , \\Omega \\to U ( { \\mathcal F } ) \\mbox { a g e o m e t r i c p o i n t } \\} . \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{align*} \\omega _ j ( g ) = \\frac { 1 } { | H | } \\sum _ { h \\in H } \\chi _ j ( g ^ { - 1 } h ) = \\frac { 1 } { | H g H | } \\sum _ { h \\in H g H } \\overline { \\chi _ j ( h ) } ( g \\in G ) . \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} x \\ , = \\ , \\cdot x _ 1 x _ 2 x _ 3 \\ldots \\mbox { i n s t e a d o f } x = \\frac { x _ 1 } { \\beta } + \\frac { x _ 2 } { \\beta ^ 2 } + \\frac { x _ 3 } { \\beta ^ 3 } + \\ldots . \\end{align*}"}
-{"id": "1227.png", "formula": "\\begin{align*} ( C ^ T f ) ( \\gamma , t ) = \\frac 1 2 \\ , \\Big [ \\frac { \\partial u ^ { \\tilde f } } { \\partial \\nu } ( \\gamma , t ) - \\frac { \\partial u ^ { \\tilde f } } { \\partial \\nu } ( \\gamma , 2 T - t ) \\Big ] \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} M _ t V _ t d X = \\left ( \\sum _ { | I | , j \\leq n / 2 - 1 / 2 } \\frac { ( - 1 ) ^ { k + l } } { j ! 2 ^ k } \\mathcal N _ { I } ( t ) D v _ t ^ j \\right ) \\zeta ( t ) \\wedge d Z + \\xi ( t ) \\zeta ( t ) \\wedge d Z , \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} f ^ * ( x _ M ^ { u } \\times 1 ) = ( y _ M ^ { u } \\times 1 ) + ( y _ M ^ { u - n } \\times \\omega _ { N } ) \\in H ^ { u } ( M ; \\R ) \\oplus ( H ^ { u - n } ( M ; \\R ) \\otimes H ^ n ( N ; \\R ) ) . \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{align*} \\mathrm { t r } ( u | R \\Gamma _ { c } ( X , \\mathcal { F } ) ) = \\sum _ { \\beta \\in \\pi _ { 0 } ( \\mathrm { F i x } ( c ) ) } \\mathrm { l o c } _ { \\beta } ( u , \\mathcal { F } ) . \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{align*} Z : = \\mathfrak { O p } \\big ( ( h _ \\circ \\ , \\sharp \\ , \\mathfrak { z } + \\mathfrak { z } \\ , \\sharp \\ , h _ \\circ ) + \\mathfrak { z } \\ , \\sharp \\ , h _ \\circ \\ , \\sharp \\ , \\mathfrak { z } + \\mathfrak { y } \\ , \\sharp \\ , h _ \\bullet \\ , \\sharp \\ , r \\ , \\sharp \\ , h _ \\bullet ^ * \\ , \\sharp \\ , \\mathfrak { y } \\big ) \\ , . \\end{align*}"}
-{"id": "4696.png", "formula": "\\begin{align*} p _ m & : = \\P ^ 1 ( X 0 ) , \\\\ q _ m & : = \\P ^ m ( X 0 ) . \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} u _ 1 ( x _ 1 , x _ 2 ) : = \\beta g ( x _ 1 , x _ 2 ) \\lim _ { L \\to \\infty \\atop L \\in \\N } C ( x _ 1 + i x _ 2 ) ( 1 \\hat { \\rho } \\b 0 0 , 2 \\hat { \\rho } \\b 0 0 ) , ( x _ 1 , x _ 2 \\in \\R ) , \\end{align*}"}
-{"id": "6815.png", "formula": "\\begin{align*} h ^ { \\alpha \\gamma } h ^ { \\beta \\delta } \\partial _ \\alpha \\cdot \\partial _ \\beta \\cdot R ^ P ( d \\phi ( \\partial _ \\gamma ) , d \\phi ( \\partial _ \\delta ) ) \\psi = & - 2 s ^ 2 g ^ { i j } \\partial _ t \\cdot \\partial _ i \\cdot R ^ P ( d \\phi ( \\partial _ t ) , d \\phi ( \\partial _ j ) ) \\psi \\\\ & + g ^ { i j } g ^ { k l } \\partial _ i \\cdot \\partial _ k \\cdot R ^ P ( d \\phi ( \\partial _ j ) , d \\phi ( \\partial _ l ) ) \\psi . \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} & \\theta ( p + z | \\tau ) \\theta ( p + p ' + z ' | \\tau ) \\theta ( p ' | \\tau ) \\theta ( z + z ' | \\tau ) - \\theta ( p ' + z ' | \\tau ) \\theta ( p + z + z ' | \\tau ) \\theta ( z | \\tau ) \\theta ( p + p ' | \\tau ) \\\\ & + \\theta ( p + p ' + z + z ' | \\tau ) \\theta ( p ' - z | \\tau ) \\theta ( p | \\tau ) \\theta ( z ' | \\tau ) = 0 \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} \\big ( ( \\xi ^ 1 , \\dots , \\xi ^ \\nu ) , f \\big ) \\mapsto \\frac { 1 } { \\nu } \\sum _ { j = 1 } ^ \\nu \\psi ( \\xi ^ j , f ) \\end{align*}"}
-{"id": "2605.png", "formula": "\\begin{align*} & \\| w - \\begin{pmatrix} x ^ 2 & 0 \\\\ 0 & y ^ 2 \\end{pmatrix} \\| \\\\ = & \\| \\begin{pmatrix} x ^ { \\frac { 1 } { 2 } } y x ^ { \\frac { 1 } { 2 } } & x ^ { \\frac { 3 } { 2 } } y ^ { \\frac { 1 } { 2 } } + x ^ { \\frac { 1 } { 2 } } y ^ { \\frac { 3 } { 2 } } \\\\ y ^ { \\frac { 1 } { 2 } } x ^ { \\frac { 3 } { 2 } } + y ^ { \\frac { 3 } { 2 } } x ^ { \\frac { 1 } { 2 } } & y ^ { \\frac { 1 } { 2 } } x y ^ { \\frac { 1 } { 2 } } \\end{pmatrix} \\| < \\delta , \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{align*} K _ \\delta ( x , t ) = K _ { \\delta , 0 } ( x , t ) + \\sum _ { j \\geq 1 } K _ { \\delta , j } ( x , t ) , \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} \\phi _ { U V } ( \\zeta ) = \\sum _ { n = - \\infty } ^ { \\infty } \\phi ^ { U V } _ n \\zeta ^ { - n } \\rlap { . } \\end{align*}"}
-{"id": "4294.png", "formula": "\\begin{align*} \\prod _ { x = 1 } ^ l = \\prod _ { j = 1 } ^ { j _ \\alpha - 1 } \\prod _ { x = H _ \\alpha ( j - 1 ) + 1 } ^ { H _ \\alpha ( j ) } \\times \\prod _ { x = H _ \\alpha ( j _ \\alpha - 1 ) + 1 } ^ l \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} ( m \\cdot \\alpha ) \\cdot s = m \\cdot \\alpha ( s ) + ( m \\cdot s ) \\cdot \\alpha \\ , . \\end{align*}"}
-{"id": "8770.png", "formula": "\\begin{align*} \\bar { S } _ { \\Theta } = \\bar { S } - n \\sum _ Y c _ Y \\mathcal { L } | _ Y ^ { n - 1 } / \\mathcal { L } ^ n . \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} \\omega = \\frac { d z _ 1 \\wedge \\cdots \\wedge d z _ n } { z _ 1 \\cdots z _ n } . \\end{align*}"}
-{"id": "322.png", "formula": "\\begin{align*} \\begin{cases} \\Re ( \\varphi - f - c ) \\leq - C s ^ \\mu , \\\\ \\Re ( \\varphi - f + \\delta | z | ^ { \\varepsilon } ) \\leq - C s ^ \\mu , \\end{cases} \\end{align*}"}
-{"id": "8221.png", "formula": "\\begin{align*} \\langle K _ j , f _ j \\rangle = \\int ( K _ j ( x ) - K _ j ( r _ j ) ) f _ j ( x ) d x = - \\int | K _ j ( x ) - K _ j ( r _ j ) | \\cdot | f _ j ( x ) | d x . \\end{align*}"}
-{"id": "470.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 8 & 1 \\\\ 0 & 8 & 1 0 & 0 \\\\ 0 & 1 & 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} V _ { E , \\alpha } ( x ) = \\int _ E \\frac { 1 } { | x - y | ^ \\alpha } d y . \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{align*} \\gamma _ \\infty : = - \\frac 1 4 \\Im ( \\dot q / q ) \\begin{pmatrix} i & 0 \\\\ 0 & - i \\end{pmatrix} . \\end{align*}"}
-{"id": "1816.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 f ( x ) x ^ { n - 1 } \\ , d x & > - M \\int _ 0 ^ a x ^ { n - 1 } \\ , d x + m \\int _ b ^ c x ^ { n - 1 } \\ , d x \\\\ & = \\frac { m c ^ n } { n } \\left ( 1 - \\left ( \\frac { b } { c } \\right ) ^ n - \\frac { M } { m } \\left ( \\frac { a } { c } \\right ) ^ n \\right ) . \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\left ( \\frac { 1 } { 2 } - \\frac { r - l _ i } { m } \\right ) = \\frac { k } { 2 } - \\frac { k r } { m } + \\frac { 1 } { m } ( \\sum _ { i = 1 } ^ k l _ i ) = \\frac { 1 } { 2 } + ( k - 1 ) \\left ( \\frac { 1 } { 2 } - \\frac { r } { m } \\right ) + \\frac { 1 } { m } \\left ( \\sum _ { i = 1 } ^ k l _ i - r \\right ) < \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "9293.png", "formula": "\\begin{align*} u _ { \\tau } ^ D ( t ) : = u _ { \\tau , m } ^ D ( t ) : = \\Phi _ { j , t - t _ m } ^ D \\prod _ { j = 1 } ^ { m - 1 } \\big ( \\Phi _ { j , \\tau } ^ S \\Phi _ { j , \\tau } ^ D \\big ) u _ \\tau ( 0 ) , t \\in \\{ t _ m \\cup T _ m \\} / t _ { m + 1 } . \\end{align*}"}
-{"id": "5639.png", "formula": "\\begin{align*} H ^ \\star = - H , E ^ \\star = - E , F ^ \\star = - F , \\end{align*}"}
-{"id": "82.png", "formula": "\\begin{align*} \\begin{aligned} [ b ] & \\prod _ { j \\in [ t _ 2 ] } x _ j ^ { - \\tau + \\zeta _ j } \\prod _ { i = t _ 2 + 1 } ^ { t _ 1 + t _ 2 } \\tilde { h } _ { p ^ * _ i } ( i , \\boldsymbol { x } ) \\\\ & \\leq \\tilde { K } \\prod _ { j \\in [ t _ 2 ] } x _ j ^ { - \\tau + \\zeta _ j + | Q _ j | } \\prod _ { i = t _ 2 + 1 } ^ { t _ 1 + t _ 2 } ( 1 / x _ { { j _ { p ^ * _ i } } } ) ^ { \\tau - 1 - \\zeta _ i - { ( l - p ^ * _ i + 1 ) } } , \\end{aligned} \\end{align*}"}
-{"id": "2241.png", "formula": "\\begin{align*} D _ { M I } ( P \\parallel Q , \\varpi ) = D _ { \\varpi } ( P \\parallel Q ) = \\log \\sum _ { i = 1 } ^ n { p _ { i } e ^ { \\left ( \\varpi \\frac { p _ i } { q _ i } \\right ) } } - \\varpi , \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} \\psi ( r * p ) = \\psi ( r ) p . \\end{align*}"}
-{"id": "9119.png", "formula": "\\begin{align*} { \\bf G } = \\frac { { \\bf Z } ^ H { \\bf R } ^ { 1 / 2 } { \\bf R } ^ { 1 / 2 } { \\bf Z } } { M } = \\frac { { \\bf \\tilde Z } ^ H { \\bf \\tilde Z } } { c { M } } , \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} w t ( C _ { 4 } ) = \\begin{cases} ( p - 1 ) p ^ { e - 2 } , & \\qquad \\ \\biggl ( \\frac { c ^ { 2 } - 4 a i } { p } \\biggr ) = - 1 , \\\\ ( p - 1 ) p ^ { e - 2 } + 2 p ^ { m + d - 1 } , & \\qquad \\ \\biggl ( \\frac { c ^ { 2 } - 4 a i } { p } \\biggr ) = 1 . \\end{cases} \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} - \\Delta _ p u = e ^ u \\ ; \\mbox { i n } \\ ; \\ ; \\mathbb R ^ N \\end{align*}"}
-{"id": "5664.png", "formula": "\\begin{align*} R ^ \\psi _ { [ a , b ] } & = P ( T , \\psi ( [ 0 , b ] ) ) - P ( T , \\psi ( [ 0 , a ) ) ) , \\\\ R ^ \\psi _ { ( a , b ] } & = P ( T , \\psi ( [ 0 , b ] ) ) - P ( T , \\psi ( [ 0 , a ] ) ) , \\\\ R ^ \\psi _ { ( a , b ) } & = P ( T , \\psi ( [ 0 , b ) ) ) - P ( T , \\psi ( [ 0 , a ] ) ) , \\\\ R ^ \\psi _ { [ a , b ) } & = P ( T , \\psi ( [ 0 , b ) ) ) - P ( T , \\psi ( [ 0 , a ) ) ) . \\end{align*}"}
-{"id": "1678.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } \\left \\{ \\sum _ { k = 1 } ^ { H - 1 } ( a _ { i k } b _ { k j } { - } a ^ 0 _ { i k } b ^ 0 _ { k j } ) { + } c _ i \\right \\} ^ 2 \\\\ & = \\sum _ { i = 1 } ^ { M - 1 } x _ i ^ 2 + \\sum _ { j = 2 } ^ { N } \\sum _ { i = 1 } ^ { M - 1 } \\left [ x _ i + \\sum _ { k = 1 } ^ { H - 1 } \\{ ( a _ { i k } b _ { k j } { - } a ^ 0 _ { i k } b ^ 0 _ { k j } ) - ( a _ { i k } b _ { k 1 } - a ^ 0 _ { i k } b ^ 0 _ { k 1 } ) \\} \\right ] ^ 2 . \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{align*} & \\frac { 1 } { \\sqrt { m } } E \\left [ J _ { s , t } ^ m \\right ] \\\\ = & - \\int _ s ^ t E \\left [ ( - \\nabla _ q V ( r , q _ r ) - \\partial _ r \\psi ( r , q _ r ) + \\tilde F ( r , q _ r ) ) \\cdot \\left ( \\int ( \\nabla _ z \\chi ) ( r , q _ r , z ) h ( r , q _ r , z ) d z \\right ) \\right ] d r \\\\ & - \\int _ s ^ t E \\left [ \\int \\left ( ( \\nabla _ q \\chi ) ( r , q _ r , z ) \\cdot z \\right ) h ( r , q _ r , z ) d z \\right ] d r + O ( m ^ { 1 / 2 } ) , \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} \\begin{gathered} _ { C } D ^ { 2 \\alpha } _ { 0 , t } u ( t ) = \\frac { \\lambda f ( t , u ( t ) ) } { ( \\int _ { 0 } ^ { t } f ( x , u ( x ) ) \\ , d x ) ^ { 2 } } \\ , , t \\in ( 0 , \\infty ) \\ , , \\\\ u ( t ) | _ { t = 0 } = u _ 0 , \\end{gathered} \\end{align*}"}
-{"id": "3121.png", "formula": "\\begin{align*} { } \\frac { \\partial S _ E } { \\partial \\alpha } = 0 \\sim N - \\frac { 1 } { \\beta } \\int _ { 0 } ^ { \\infty } \\frac { d x } { e ^ { \\alpha + x } - 1 } \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} i \\omega _ \\lambda \\left ( D - C \\right ) + i \\omega _ \\lambda \\left ( C e ^ { 2 i \\pi \\omega _ \\lambda } - D e ^ { - 2 i \\pi \\omega _ \\lambda } \\right ) + \\lambda ( C + D ) = 0 . \\end{align*}"}
-{"id": "8362.png", "formula": "\\begin{align*} \\tilde { u } ( t ) = \\begin{cases} u ( t ) , & t \\in ( 0 , \\beta ] \\\\ \\tilde { u } ( t ) , & t \\in [ \\beta , \\beta + h ] . \\end{cases} \\end{align*}"}
-{"id": "5738.png", "formula": "\\begin{align*} | G | = | B _ 2 \\cup A \\cup X ' | + | B _ 1 \\cup X '' | + | B _ 3 | \\le 4 \\cdot 2 ^ { k - 1 } + 4 \\cdot 2 ^ { k - 2 } + 4 \\cdot 2 ^ { k - 2 } = 4 \\cdot 2 ^ { k } , \\end{align*}"}
-{"id": "7688.png", "formula": "\\begin{align*} \\Lambda ^ i _ { d \\leq r _ 0 } ( r _ 0 ) & \\approx 2 \\lambda _ u d \\sum ^ { N } _ { l = 1 } \\frac { \\pi } { N } g _ r \\left ( r _ 0 + d w _ l \\right ) \\sqrt { 1 - w _ l ^ 2 } , \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} \\mathbb P \\Big ( \\sup _ { s \\in [ 0 , t ] } \\{ X _ 1 ( s ) \\} \\ge \\Gamma \\Big ) \\le 2 \\mathcal G \\Big ( \\frac { \\ell \\ , \\Gamma - t - \\sum _ { j = 1 } ^ { \\ell } X _ j ( 0 ) } { \\sqrt { \\ell t } } \\Big ) \\ , . \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\phi _ N ( \\xi ) = \\int _ 0 ^ { \\infty } x ^ { - i \\xi } e ^ { - x } d x = \\Gamma ( 1 - i \\xi ) \\ ; \\xi \\in \\mathbb { R } , \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} f ^ * ( x _ { M _ j } ^ { m - 1 } \\times 1 ) = ( y _ M ^ { m - 1 } \\times 1 ) + ( y _ M ^ { m - 2 } \\times \\omega _ { S ^ 1 } ) , \\end{align*}"}
-{"id": "4224.png", "formula": "\\begin{align*} q _ 1 = e ^ { - \\lambda \\epsilon _ 1 } . q _ 2 = e ^ { - \\lambda \\epsilon _ 2 } , u _ \\alpha = e ^ { - \\lambda a _ \\alpha } , \\alpha = 1 , \\dots , r . \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{align*} f _ { \\ast } ( t ) = \\left \\{ \\begin{array} { l l } f ( 0 ) & \\mbox { i f \\ $ t < 0 $ , } \\\\ f ( t ) & \\mbox { i f \\ $ 0 \\leq t < s _ { \\ast } $ , } \\\\ 0 & \\mbox { i f \\ $ s _ { \\ast } \\leq t $ } . \\end{array} \\right . \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} \\psi ( \\nu , \\mu ) = \\lim _ { q \\rightarrow t ^ { - p - m _ 1 } } \\left ( 1 - q t ^ { p + m _ 1 } \\right ) { \\rm T r } \\Big ( B _ { \\mu _ 1 , \\nu _ 1 } \\dots B _ { \\mu _ n , \\nu _ n } k ^ u \\Big ) \\cdot I ( \\mu , \\nu ) , \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} \\langle \\mathrm { l o g } \\left ( J _ 1 ( N ) _ { f , \\lambda } ( \\mathbb { Q } _ p ( \\mu _ n ) ) \\right ) , \\omega _ { \\overline { f } } \\rangle _ { \\mathrm { d R } } & = \\frac { \\lambda ^ { e _ { n } } } { ( 1 - \\overline { \\alpha _ p } ^ { n _ p } ) ( 1 - \\overline { \\beta _ p } ^ { n _ p } ) } \\mathbb { Z } _ { f , \\lambda } \\otimes _ { \\mathbb { Q } _ { p } } p \\mathbb { Z } _ p [ \\mu _ n ] \\\\ & \\subseteq \\mathbb { Q } _ { f , \\lambda } \\otimes \\mathbb { Q } _ p ( \\mu _ n ) . \\end{align*}"}
-{"id": "7631.png", "formula": "\\begin{align*} \\varphi & = \\dd r \\wedge \\omega + { \\rm R e } ( \\Psi ) \\\\ \\psi & = - \\dd r \\wedge { \\rm I m } ( \\Psi ) + \\tfrac 1 2 \\omega \\wedge \\omega \\ : . \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} = \\frac 1 2 \\int _ { S ^ 1 } \\begin{pmatrix} f _ 0 & f _ 1 & f _ 2 \\end{pmatrix} \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & - D \\frac 1 { k _ 0 } D \\frac 1 { k _ 0 } D + D \\frac { k _ 1 } { k _ 0 } + \\frac { k _ 1 } { k _ 0 } D & \\frac { k _ 2 } { k _ 0 } D + D \\frac { k _ 2 } { k _ 0 } \\\\ 0 & \\frac { k _ 2 } { k _ 0 } D + D \\frac { k _ 2 } { k _ 0 } & D \\frac 1 { k _ 0 } D \\frac 1 { k _ 0 } D - D \\frac { k _ 1 } { k _ 0 } - \\frac { k _ 1 } { k _ 0 } D \\end{pmatrix} \\begin{pmatrix} h _ 0 \\\\ h _ 1 \\\\ h _ 2 \\end{pmatrix} d x . \\end{align*}"}
-{"id": "4584.png", "formula": "\\begin{align*} \\overline { \\bar * \\psi } = \\bar * \\bar \\psi , \\bar * ^ 2 \\psi = ( - 1 ) ^ { r + s } \\psi . \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} \\int _ { G / H } \\psi d V _ H = C _ H \\int _ { \\mathfrak { a } _ s ^ + } \\psi ( \\exp ( x ) H ) J _ H ( x ) d x \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} \\textrm { V a r } \\Big [ \\big ( 1 + \\eta ^ \\ell ( 0 ) \\big ) \\Big ( \\psi ^ { n , \\ell } ( 0 ) - \\lambda _ n - & \\frac { 1 } { ( 1 + \\rho _ n ) ^ 2 } \\big ( \\eta ^ { n , \\ell } ( 0 ) - \\rho _ n \\big ) \\Big ) \\Big ] = \\mathcal O ( n ^ { - 2 \\delta } ) \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} H _ { k , \\tau } : = a ( x ) ^ 2 - ( x ^ 3 + A x + B ) b ( x ) ^ 2 , \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} V _ { \\mathrm { n e w } } = \\sum _ { \\mu , \\lambda } E _ \\mu ^ * E _ \\lambda V & & , \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{align*} \\langle \\rho _ \\gamma ( u , 1 ) P _ O , \\ , \\rho _ \\gamma ( v , 1 ) P _ O \\rangle = \\begin{cases} \\frac { | A | - 1 } { 2 } , & u = v ; \\\\ - \\frac { 1 } { 2 } \\gamma \\left ( [ u , v ] ^ { 1 / 2 } \\right ) , & u \\neq v . \\end{cases} \\end{align*}"}
-{"id": "1343.png", "formula": "\\begin{align*} \\| \\phi ^ { n - 1 } ( A _ 1 \\dots A _ { k - 1 } ) \\| & = 2 ( k - 2 ) ( n - 1 ) + ( k - 1 ) , \\\\ \\| \\phi ^ { n - 1 } ( A _ 1 \\dots A _ { k } ) ^ { - 1 } \\| & = 2 ( k - 1 ) ( n - 1 ) + k , \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} f ^ * ( \\omega _ \\Sigma \\times 1 ) = \\pm ( \\omega _ \\Sigma \\times 1 ) . \\end{align*}"}
-{"id": "631.png", "formula": "\\begin{align*} C _ i = \\mathbb { E } W _ i ( n ) \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} | E \\ominus B _ r | = | B _ { \\widetilde { R } } | \\le | B _ { R - r } | . \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} ( \\mathrm { I d } \\otimes j ) ( \\alpha \\cdot ( \\varphi \\otimes ( u \\otimes v ) ) = \\alpha \\cdot ( ( \\mathrm { I d } \\otimes j ) ( \\varphi \\otimes ( u \\otimes v ) ) ) \\end{align*}"}
-{"id": "226.png", "formula": "\\begin{align*} \\mathcal A : = \\Gamma ( E ^ \\# , \\mathcal O _ { E ^ \\# } ( * \\pi ^ { - 1 } ( 0 ) ) ) . \\end{align*}"}
-{"id": "8483.png", "formula": "\\begin{align*} e _ 1 = ( 1 , 0 ) , e _ 2 = ( 0 , 1 ) , \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} | p ^ { \\kappa ( z ) } _ { 0 , 1 } ( x ) - p ^ { \\tilde \\kappa ( z ) } _ { 0 , 1 } ( x ) | \\leq c K ( \\varrho ^ 0 _ { \\alpha - \\gamma } ( 1 , x ) + \\varrho ^ 0 _ { 0 } ( 1 , x ) ) = c K \\varrho ^ 0 _ { \\alpha } ( 1 , x ) , \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} \\Phi _ 4 : C ^ \\tau ( \\overline { \\R ^ n _ + } ; S ^ m _ { 1 , 0 } ) & \\to C ^ \\tau S ^ m _ { 1 , 0 } \\\\ ( \\Phi _ 4 ( q ) ) ( x ' , \\xi ' , y _ n , w _ n ) & : = \\widetilde { q } ( x ' , \\xi ' , y _ n , w _ n ) : = \\left . q _ z ( x ' , \\xi ' , y _ n , w _ n ) \\right | _ { z = x } . \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} | h ( x ) - h ( y ) | & \\leq | h ( x ) - h ( \\bar { x } ) | + | h ( y ) - h ( \\bar { y } ) | + | h ( \\bar { x } ) - h ( \\bar { y } ) | \\\\ & \\leq A r ^ \\alpha + A r ^ \\alpha + A | \\bar { x } - \\bar { y } | ^ \\alpha \\\\ & \\leq 2 A r ^ \\alpha + A | 2 r + d | ^ \\alpha \\leq C d ^ \\alpha = C | x - y | ^ \\alpha \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{align*} \\tau ( \\lambda _ 0 , \\lambda _ f , \\lambda _ 1 ) = \\tau ( \\lambda _ 0 / \\rho , \\lambda _ f / \\rho , \\lambda _ 1 / \\rho ) , \\end{align*}"}
-{"id": "889.png", "formula": "\\begin{align*} f _ \\Delta ( t ) & : = \\sum _ { i = 0 } ^ r f _ i t ^ { r - i } \\\\ h _ \\Delta ( t ) & : = f _ \\Delta ( t - 1 ) = \\sum _ { i = 0 } ^ r h _ i t ^ { r - i } . \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{gather*} s d = c - b \\implies b = c - s d \\\\ v c = d - b \\implies b = d - v c \\\\ c - s d = d - v c \\implies c + v c = d + s d \\implies ( 1 + v ) c = ( 1 + s ) d \\end{gather*}"}
-{"id": "3511.png", "formula": "\\begin{align*} \\sum _ { i \\mid a _ { i j } < 0 } | a _ { i j } | \\exp ( y _ i / h ) = \\sum _ { i \\mid a _ { i j } > 0 } | a _ { i j } | \\exp ( y _ i / h ) , j \\in [ n ] . \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} X _ { i _ { ( x , i ' ) } , j ' + 1 } : = X ^ { ( x ) } _ { i ' , j ' } + \\sum _ { x ' = 1 } ^ { x - 1 } ( \\alpha ^ { ( x ' ) } ) ^ * _ { j ' } - \\sum _ { x ' = x + 1 } ^ { \\mathcal { k } } ( \\alpha ^ { ( x ' ) } ) ^ * _ { j ' } , \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} T _ 1 & : = \\inf \\{ t \\ge 0 : \\ V ( t ) \\ge V _ { t h } \\} \\\\ T _ U & : = \\inf \\{ t \\ge 0 : \\ U ( t ) \\ge S _ U ( t ) \\} \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} & \\mathcal { D } = \\Big \\{ z ^ n : p _ { Z ^ n | X ^ n , Y ^ n } ( z ^ n | \\mathbf { x } _ 1 , \\mathbf { y } _ 1 ) < p _ { Z ^ n | X ^ n , Y ^ n } ( z ^ n | \\mathbf { x } _ 2 , \\mathbf { y } _ 2 ) \\Big \\} \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} r ^ + ( M , s _ 1 , t ) = \\mathrm { r a n k } \\left ( M ( s _ 1 , t ) \\right ) - 1 . \\end{align*}"}
-{"id": "5408.png", "formula": "\\begin{align*} W ^ { ( n ) } = & W ^ 0 _ { n + 2 } ( p , p _ 1 , \\dots , p _ { n + 1 } ) + h W ^ 1 _ { n } ( p , p _ 1 , \\dots , p _ { n - 1 } ) + \\\\ & \\dots + h ^ { ( n + 1 ) / 2 } W ^ { ( n + 1 ) / 2 } _ 1 ( p ) . \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{align*} \\tilde { j } _ { m } ( \\rho ) & = \\frac { q } { ( 2 \\pi ) ^ 3 } \\int _ { | p | < ( 6 \\pi ^ 2 \\rho / q ) ^ { \\frac { 1 } { 3 } } } \\frac { 1 } { \\sqrt { | p | ^ 2 + m ^ 2 } } { \\rm d } p \\\\ & = \\frac { q } { 4 \\pi ^ 2 } \\left [ \\eta \\sqrt { \\eta ^ 2 + m ^ 2 } - m ^ 2 \\ln { \\left ( \\frac { \\eta + \\sqrt { \\eta ^ 2 + m ^ 2 } } { m } \\right ) } \\right ] , \\eta = \\left ( \\frac { 6 \\pi ^ 2 \\rho } { q } \\right ) ^ { \\frac { 1 } { 3 } } . \\end{align*}"}
-{"id": "7216.png", "formula": "\\begin{align*} \\partial _ { q , x _ j } ( 1 - b _ j \\eta _ { y _ j } ) \\{ f \\} = \\partial _ { q , y _ j } ( 1 - a _ j \\eta _ { x _ j } ) \\{ f \\} , j \\in \\{ 1 , 2 , \\ldots , k \\} . \\end{align*}"}
-{"id": "1280.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j \\leq m \\atop j \\neq i } k _ i k _ j = \\lambda _ i ( v - 1 ) . \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{gather*} x ' _ { 2 \\varepsilon _ i } ( c ) : = \\exp ( c X ' _ { 2 \\varepsilon _ i } ) , h ' _ { 2 \\varepsilon _ i } ( c ) : = \\exp ( c H ' _ { 2 \\varepsilon _ i } ) , x ' _ { - 2 \\varepsilon _ i } ( c ) : = \\exp ( c X ' _ { - 2 \\varepsilon _ i } ) \\end{gather*}"}
-{"id": "2925.png", "formula": "\\begin{align*} \\frac { D \\kappa } { d t } = - V _ { s s } - \\kappa ^ 2 \\ , V ; \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} F ^ { 1 } ( x ; t ) : = K ( x , 0 ; t ) Q ( x , 0 ; t ) \\quad F ^ { 2 } ( y ; t ) : = K ( 0 , y ; t ) Q ( 0 , y ; t ) . \\end{align*}"}
-{"id": "5511.png", "formula": "\\begin{align*} \\beta _ 1 = \\sum _ { q = 1 } ^ { 2 N } ( 1 + \\delta _ { l q } ) g _ j ^ { ( 1 @ l , 1 @ q ) } w _ q ^ { ( 1 , 1 ) } + \\sum _ { q = 1 } ^ { 2 N } ( 1 + \\delta _ { ( l + N ) q } ) g _ j ^ { ( 1 @ ( l + N ) , 1 @ q ) } w _ q ^ { ( 2 , 0 ) } + g _ j ^ { ( 3 @ l ) } . \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} \\alpha _ \\tau = \\int _ 0 ^ \\tau \\psi ( s ) d s , \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} \\frac { N [ \\mathfrak { e } , \\lambda V ] } { N _ { s c } [ \\mathfrak { e } , \\lambda V ] } \\ \\geq \\ \\frac { \\mathcal { L } _ { V } [ \\lambda ^ { - 1 } { \\mathfrak { e } _ { \\mathrm { m a x } } } ] } { N _ { s c } [ \\mathfrak { e } , \\lambda V ] } \\ = \\ \\frac { N _ { s c } ^ { > } [ \\mathfrak { e } , \\lambda V ] } { N _ { s c } [ \\mathfrak { e } , \\lambda V ] } . \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} m \\left ( S \\otimes \\right ) \\Delta = m \\left ( I \\otimes \\right ) \\Delta = \\eta \\cdot \\epsilon \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} | \\Lambda | ( e , \\ , \\Lambda ) \\geq \\alpha | \\Lambda | ^ 2 , \\ , \\ , \\mathrm { w h e r e \\ , \\ , } \\alpha : = \\mathrm { i n f } \\{ e _ i , \\ , \\ , 1 \\leq i \\leq p \\} . \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{align*} L _ m \\chi _ y ( z ) = \\chi _ y ( y ) = 1 \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{align*} L _ { k } = k \\ , L , \\Delta L _ { k } = L / 8 . \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} \\lambda f _ r ( t _ 1 ) + ( 1 - \\lambda ) f _ r ( t _ 2 ) = \\inf ( F ( \\lambda \\Gamma _ r ( t _ 1 ) + ( 1 - \\lambda ) \\Gamma _ r ( t _ 2 ) ) ) \\ge f _ r ( \\lambda t _ 1 + ( 1 - \\lambda ) t _ 2 ) , \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{align*} \\mathcal { N } ( q , u + h ) - \\mathcal { N } ( q , u ) = q a ( x ) ( u + \\theta h ) ^ { q - 1 } h , \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{align*} \\partial _ { y _ i } \\log Z ( y ) = \\frac { \\partial _ { y _ i } Z ( y ) } { Z ( y ) } & = { \\int _ \\mathcal { X } \\partial _ { y _ i } \\mu ( x , y ) \\frac { d x } { Z ( y ) } } = { \\int _ \\mathcal { X } \\frac { \\partial _ { y _ i } \\mu ( x , y ) } { \\mu ( x , y ) } \\frac { \\mu ( x , y ) d x } { Z ( y ) } } = \\int _ \\mathcal { X } \\partial _ { y _ i } \\log \\mu ( x , y ) \\mu ^ y ( d x ) \\end{align*}"}
-{"id": "720.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\Delta _ { n + 1 } } { \\Delta _ { n } } = { \\rm M } ( P ) . \\end{align*}"}
-{"id": "3296.png", "formula": "\\begin{align*} h \\mapsto - 2 M _ x ^ { - 1 } D _ h M _ x ^ { - 1 } b = - 2 M _ x ^ { - 1 } D _ h y \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} E _ { \\kappa } ( z ; q , t ) & = f _ { \\kappa } ( z ; q , t ) + \\sum _ { \\substack { \\nu \\in \\sigma ( \\epsilon ) \\\\ \\nu \\prec \\kappa } } d _ { \\kappa , \\nu } ( q , t ) f _ { \\nu } ( z ; q , t ) \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} \\frac { 1 } { R _ 1 ^ { m - 2 } } \\int _ { B _ { R _ 1 } } | \\dd \\phi | ^ 2 \\dd x = \\frac { 1 } { R _ 2 ^ { m - 2 } } \\int _ { B _ { R _ 2 } } | \\dd \\phi | ^ 2 \\dd x - \\int _ { R _ 1 } ^ { R _ 2 } F ( r ) \\dd r . \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{align*} I _ { k } ( u ) = \\left \\{ \\begin{array} { l l } I ( u ) , & \\mbox { i f \\ $ t _ { k - 1 } \\leq \\| u \\| ^ { 2 } \\leq t _ { k } \\ \\mbox { a n d $ | u | _ { \\infty } \\leq s _ { \\ast } $ } $ , } \\medskip \\\\ \\displaystyle \\frac { 1 } { 2 } \\int _ { t _ { k - 1 } } ^ { t _ { k } } m ( s ) d s - \\int _ { \\Omega } F _ { \\ast } ( u ) d x , & \\mbox { i f \\ $ t _ { k } \\leq \\| u \\| ^ { 2 } $ } . \\end{array} \\right . \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} _ { m , 2 } ^ 2 = \\frac { \\alpha ^ 2 _ 2 z _ { m , 2 } } { ^ { m , 2 } _ { i n t e r } + \\frac { 1 } { \\rho } } . \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} P _ d = X _ 0 ^ { n _ d - 1 } X _ 2 - \\sum _ { i = 0 } ^ d a _ i X _ 0 ^ { n _ d - n _ i } X _ 1 ^ { n _ i } \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} \\psi ( \\nu , \\mu ) = \\left [ \\lim _ { q \\rightarrow t ^ { - p - m _ 1 } } \\left ( 1 - q t ^ { p + m _ 1 } \\right ) { \\rm T r } \\Big ( A _ { \\mu _ 1 } ( z _ 1 ) \\dots A _ { \\mu _ n } ( z _ n ) k ^ { u } \\Big ) \\right ] _ { z ^ { \\nu } } \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} \\Phi \\circ \\epsilon _ A = \\epsilon _ B \\circ \\Phi . \\end{align*}"}
-{"id": "4783.png", "formula": "\\begin{align*} \\int _ \\Omega K ( x ) \\delta _ i ^ { \\frac { n + 2 } { n - 2 } } v \\ , d x = \\| v \\| \\cdot O \\bigl ( \\frac { | \\nabla K ( a _ i ) | } { \\lambda _ i } + \\frac { 1 } { \\lambda _ i ^ { \\beta _ i } } + \\frac { ( \\log \\lambda _ i ) ^ { \\frac { n + 2 } { 2 n } } } { \\lambda _ i ^ { \\frac { n + 2 } { 2 } } } \\bigr ) . \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} \\lambda & = \\frac { M - 1 } { 2 } + \\min \\left \\{ \\frac { M - 1 } { 2 } , \\frac { N - 1 } { 2 } \\right \\} = \\min \\left \\{ M - 1 , \\frac { M + N - 2 } { 2 } \\right \\} . \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} \\mathcal { I } _ 2 ( t , x ) = \\frac { 1 } { 2 } \\int _ 0 ^ t \\int _ { S } { \\rm s i g n } ( y ) \\frac { \\partial G _ { t - s } } { \\partial x } ( x - z ) \\psi ( s , z ) \\sigma _ s ( y ) d z W ( d s , d y ) \\end{align*}"}
-{"id": "7281.png", "formula": "\\begin{align*} E _ 2 ^ { s t } = \\hat H ^ s ( G , H ^ t ( A ^ \\bullet ) ) \\Rightarrow \\hat H ^ { s + t } ( G , A ^ \\bullet ) \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} P ( \\theta \\ , | \\ , \\phi ) = \\left ( \\frac { \\cos ( \\phi + \\theta ) } { \\cos ( \\phi - \\theta ) } \\right ) ^ { \\frac { r N + 1 } { 2 r N } } \\ , \\prod _ { k = 1 } ^ { r N } \\ , \\left ( \\frac { \\sin \\left ( \\frac { \\phi + \\theta } { r N } + \\frac { \\pi ( 2 k - 1 ) } { 2 r N } \\right ) } { \\sin \\left ( \\frac { \\phi - \\theta } { r N } + \\frac { \\pi ( 2 k - 1 ) } { 2 r N } \\right ) } \\right ) ^ { - \\frac { k } { r N } } \\ , . \\end{align*}"}
-{"id": "1628.png", "formula": "\\begin{align*} - G \\left ( A _ t + \\frac { 1 - \\tilde { Q } _ t ( A _ t , \\dot { \\varphi } _ 0 ) } { q _ { \\sigma , t } \\varphi _ \\sigma ( 0 ) } \\right ) + \\tilde { T } _ t ( \\dot { \\varphi } _ y ) = ( \\ln _ 2 t ) ^ { \\mu - 2 } h _ { t , \\aleph } ( \\varphi _ \\sigma ( y ) ) , \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} & p ( 2 + p ) ( 4 + p ) \\cdots ( j - 1 + p ) \\\\ & = 2 ^ { \\frac { j + 1 } { 2 } } ( 0 + \\frac { p } { 2 } ) ( 1 + \\frac { p } { 2 } ) ( 2 + \\frac { p } { 2 } ) \\cdots ( \\frac { j - 1 } { 2 } + \\frac { p } { 2 } ) \\\\ & = 2 ^ { \\frac { j + 1 } { 2 } } \\prod _ { k = 1 } ^ { \\frac { j + 1 } { 2 } } \\Big ( k - 1 + \\frac { p } { 2 } \\Big ) = 2 ^ { \\frac { j + 1 } { 2 } } \\frac { \\Gamma ( ( j + 1 ) / 2 + p / 2 ) } { \\Gamma ( p / 2 ) } . \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{align*} & F _ n ( x _ 1 ) = x _ 2 = x _ 0 + 1 < y _ { 0 } + 1 = y _ { 2 n } = F _ n ( y _ { n - 2 } ) \\\\ & F _ n ( x _ 0 ) = x _ 1 < y _ { n + 2 } = F _ n ( y _ 0 ) , \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} f _ { G _ 1 } ( t ) = \\dot { \\rho } _ { G _ 1 , G _ 2 } ( t ) f _ { G _ 2 } ( \\rho _ { G _ 1 , G _ 2 } ( t ) ) . \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{align*} a _ j ^ 2 = & 1 + \\lambda _ j ^ { 2 \\beta / r } n ^ { 2 \\beta / r } , \\ , j = 0 , \\dots , n - 1 . \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} d _ { \\beta } ( 1 ) = 0 . 1 0 ^ { n - 1 } 1 0 ^ { n _ 1 } 1 0 ^ { n _ 2 } 1 0 ^ { n _ 3 } \\ldots , \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} n r ' _ = ( n , 1 ) \\ , \\geq \\ , \\frac { 1 2 ^ { 3 ^ { n - 2 } ( 3 ^ { n - 1 } - 1 ) / 2 } } { 6 ^ { 3 ^ { n - 1 } } \\cdot | A G L ( n - 1 , 3 ) | } - \\frac { 1 } { 2 \\cdot 3 ^ n \\cdot | A G L ( n - 1 , 3 ) | } . \\end{align*}"}
-{"id": "5900.png", "formula": "\\begin{align*} S _ { p + m _ 1 } ( \\nu ^ + ) _ { m _ 2 - r + 1 } = m _ 1 + m _ 2 + 1 + p - r > m _ 1 + m _ 2 + 1 = S _ { p + m _ 1 } ( \\delta ^ + ) _ { m _ 1 + m _ 2 + 1 } . \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\left ( ( \\eta _ 0 , w _ 0 ) , S ^ * ( \\tau _ n ) ( \\varphi _ { \\tau _ n } , \\psi _ { \\tau _ n } ) \\right ) _ { \\overline { X } _ 2 } = \\left ( ( \\eta _ 0 , w _ 0 ) , S ^ * ( \\tau ) ( \\varphi _ { \\tau } , \\psi _ { \\tau } ) \\right ) _ { \\overline { X } _ 2 } . \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} \\frac { \\partial \\psi } { \\partial x } ( t , x ) = \\int _ { \\mathbb { R } } \\frac { \\partial G _ t } { \\partial x } ( x - y ) \\psi _ 0 ( y ) d y - \\frac { 1 } { 2 } \\int _ 0 ^ t \\int _ S { \\rm s i g n } ( y ) \\frac { \\partial G _ { t - s } } { \\partial x } ( x - z ) \\psi ( s , z ) \\sigma _ s ( y ) d z W ( d s , d y ) \\\\ + \\frac { 1 } { 8 } \\int _ 0 ^ t \\int _ S \\frac { \\partial G _ { t - s } } { \\partial x } ( x - z ) \\psi ( s , z ) \\sigma _ s ( y ) ^ 2 d z d y d s . \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} \\| A - A ^ T \\| / \\| A \\| = 0 . 7 4 5 6 \\ , . \\end{align*}"}
-{"id": "3009.png", "formula": "\\begin{align*} v ( q _ { n } ) ^ { q _ { n } } = e ^ { q _ { n } \\log v ( q _ { n } ) } \\leq e ^ { \\log C _ { 1 } } \\quad \\mbox { i f } v ( q _ { n } ) ( x ) > 0 , \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} \\psi _ 1 ( x ) = x ^ { 3 / 4 } \\prod _ { j = 2 } ^ 4 \\theta _ j ( x ) = 2 \\ , x ^ { 3 / 4 } e ^ { - \\tfrac { \\pi } { 4 } x } \\prod _ { k \\in \\N } \\left ( 1 - e ^ { - 2 k \\pi x } \\right ) ^ 3 . \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} ( M + \\delta M ) \\tilde { \\mathcal { X } } = \\mathcal { E } + \\delta \\mathcal { E } , \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} d _ 0 ( \\psi _ n , \\psi _ 0 ) = P _ 0 L ( \\psi _ n ) - P _ 0 L ( \\psi _ 0 ) = O _ P ( n ^ { - 1 / 2 - \\alpha ( d ) } ) , \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} \\Delta _ j ( t _ 1 e _ 1 + \\dots + t _ j e _ j ) = t _ 1 \\dots t _ j . \\end{align*}"}
-{"id": "9026.png", "formula": "\\begin{align*} K _ L = \\begin{pmatrix} 0 & - k _ 0 & 0 & 0 \\\\ - k _ 1 & 0 & 0 & - k _ 0 \\\\ - k _ { 2 } & 0 & 0 & 0 \\\\ 0 & - k _ 1 & - k _ 2 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l c l } \\mathrm { T r } ( \\mathrm { a d } _ { d x } ) & = & \\mathrm { d i v } ( \\partial _ { d x } ) & = & \\frac { \\partial P } { \\partial y } \\\\ \\mathrm { T r } ( \\mathrm { a d } _ { d y } ) & = & \\mathrm { d i v } ( \\partial _ { d y } ) & = & - \\frac { \\partial P } { \\partial x } \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} \\alpha _ j : = b _ { 2 j - 1 } - i b _ { 2 j } , j = 1 , 2 , \\dots , n - 1 , \\qquad \\alpha _ n : = 0 , \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{align*} \\tilde { \\mathcal { S } } _ q : \\begin{cases} y ^ q + y = x ^ { q _ 0 } ( x ^ q + x ) \\\\ t ^ m = x ^ q + x \\end{cases} , \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{align*} \\frac { d b ( \\phi _ t ( v ) ) } { d t } = - \\frac { r a b } { \\delta } + O ( a b ) . \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} f ( x _ 1 , y _ 1 ) = \\sum _ { k , l = 0 } ^ \\infty \\lambda _ { k , l } ( a _ 1 ; q ) _ k ( b _ 1 ; q ) _ l { { k + l } \\brack k } _ q x _ 1 ^ k y _ 1 ^ l . \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{align*} 1 \\geq \\left ( 1 - x ^ { \\lambda } \\right ) ^ { \\nu _ 2 M - 1 } \\geq \\exp \\left ( - \\frac { \\nu _ 2 } { M ^ { \\lambda \\varepsilon } } \\right ) \\left [ 1 + O \\left ( \\frac { 1 } { M ^ { 1 + \\lambda \\varepsilon } } \\right ) \\right ] = 1 + O \\left ( \\frac { 1 } { M ^ { \\lambda \\varepsilon } } \\right ) . \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} [ D ^ * D , D ^ k ] \\psi = & \\sum _ { l = 0 } ^ { k - 1 } D ^ { l } R ^ M \\star D ^ { k - l } \\psi + \\sum _ { l = 0 } ^ { k } D ^ { l } R ^ { S M } \\star D ^ { k - l } \\psi \\\\ & \\quad + \\sum _ { \\sum l _ i + \\sum { m _ j } = k } { } ^ { G } \\nabla ^ { l _ 1 } R ^ N \\star \\underbrace { D ^ { m _ 1 + 1 } \\phi \\star \\ldots \\star D ^ { m _ { l _ 1 } + 1 } \\phi } _ { l _ 1 - } \\star D ^ { l _ 2 + 1 } \\phi \\star D ^ { l _ 3 + 1 } \\phi \\star D ^ { l _ 4 } \\psi . \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} \\begin{aligned} \\operatorname { P f } ( D ^ g ) & = \\frac { 1 } { 8 \\pi ^ 2 } ( \\| W ^ + + \\frac { s _ g } { 1 2 } I \\| ^ 2 + \\| W ^ - + \\frac { s _ g } { 1 2 } I \\| ^ 2 - \\| R _ 0 ^ * \\| ^ 2 - \\| R _ 0 \\| ^ 2 ) \\operatorname { v o l } _ g \\\\ & = \\frac { 1 } { 8 \\pi ^ 2 } ( \\| W ^ + \\| ^ 2 + \\| W ^ - \\| ^ 2 + \\frac { 1 } { 2 4 } s _ g ^ 2 - 2 \\| R _ 0 \\| ^ 2 ) \\operatorname { v o l } _ g \\\\ & = \\frac { 1 } { 8 \\pi ^ 2 } ( \\| W ^ + \\| ^ 2 + \\frac { 1 } { 2 4 } s _ g ^ 2 - 2 \\| R _ 0 \\| ^ 2 ) \\operatorname { v o l } _ g \\end{aligned} \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } x ^ { 2 } + C _ { 2 } & = - \\int \\frac { a \\sec ^ { 2 } ( \\theta ) } { a ^ { 3 } ( 1 + \\tan ^ { 2 } ( \\theta ) ) ^ { 3 / 2 } } d \\theta \\\\ & = - \\int \\frac { 1 } { a ^ { 2 } } \\cos ( \\theta ) d \\theta \\\\ & = - \\frac { 1 } { a ^ { 2 } } \\sin ( \\theta ) \\\\ \\frac { a ^ { 2 } } { 2 } ( x ^ { 2 } + 2 C _ { 2 } ) & = - \\frac { z ' ( x ) } { \\sqrt { a ^ { 2 } + z ' ( x ) ^ { 2 } } } \\\\ \\frac { 1 } { 2 } ( 1 + C _ { 1 } ^ { 2 } ) ( x ^ { 2 } + 2 C _ { 2 } ) & = - \\frac { z ' ( x ) } { \\sqrt { 1 + C _ { 1 } ^ { 2 } + z ' ( x ) ^ { 2 } } } \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} c _ i = \\sum _ { j = 1 } ^ { r } \\min ( i , j ) k _ j \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} \\alpha _ { 2 j , 2 k - 1 } | _ { T _ s } = \\alpha _ { 2 j - 1 , 2 k - 1 } | _ { T _ s } = \\alpha _ { 2 j - 1 , 2 k } | _ { T _ s } = \\alpha _ { 2 j , 2 k } | _ { T _ s } = b _ j / b _ k . \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} \\lim _ { t \\nearrow T _ { \\max } } \\| u ( \\cdot , t _ 0 + t ; t _ 0 , u _ 0 ) \\| _ { \\infty } = \\infty . \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} & \\underset { z _ 1 = \\infty } { } \\frac { I _ n ( z _ 1 , \\dots , z _ n ) } { z _ 1 } = - I _ { n - 1 } ( z _ 2 , \\dots , z _ n ) \\\\ & \\underset { z _ n = 0 } { } \\frac { I _ n ( z _ 1 , \\dots , z _ n ) } { z _ n } = \\gamma \\cdot I _ { n - 1 } ( z _ 1 , \\dots , z _ { n - 1 } ) \\end{align*}"}
-{"id": "6300.png", "formula": "\\begin{align*} \\Phi _ J ^ s = \\Phi _ J \\setminus \\bigcup \\limits _ { \\mathbf { y } \\in \\Phi _ U ^ s } \\mathbf { B } ( \\mathbf { y } , R _ { \\tau } ) , \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} \\displaystyle { \\operatornamewithlimits { \\mbox { m i n i m i z e } } _ { w ; \\ , \\alpha } } \\ \\displaystyle { \\frac { 1 } { N } } \\ , \\displaystyle { \\sum _ { i = 1 } ^ N } \\ , \\ell \\left ( y _ i - \\sigma ( w ^ T x ^ i + \\alpha _ i ) \\ , \\right ) + \\gamma \\ , \\displaystyle { \\sum _ { i = 1 } ^ m } \\ , \\left [ \\ , \\underbrace { \\alpha _ i \\ , | \\ , w _ i \\ , | - h _ i ( w _ i ) } _ { \\mbox { s u r r o g a t e s p a r s i t y f u n c t i o n } } \\ , \\right ] , \\end{align*}"}
-{"id": "7445.png", "formula": "\\begin{align*} ( \\tilde \\gamma ^ T ) ^ { - 1 } ( t ^ * , q ^ \\prime _ t ) \\sigma ( t ^ * , q ^ \\prime _ t ) d W _ t = \\tilde \\gamma ^ { - 1 } ( t ^ * , q ^ \\prime _ t ) \\sigma ( t ^ * , q ^ \\prime _ t ) d \\tilde W _ t \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} r _ n = | | A | | \\cdot \\alpha ^ n \\cdot ( F ) , \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l l l } \\eta _ { t } + w _ { x } + w _ { x x x } = 0 & & ( 0 , L ) \\times ( 0 , T ) \\\\ w _ { t } + \\eta _ { x } + \\eta _ { x x x } = 0 & & ( 0 , L ) \\times ( 0 , T ) \\end{array} \\right . \\end{align*}"}
-{"id": "7894.png", "formula": "\\begin{align*} \\lim _ { \\tau \\to 0 } \\max _ { 0 \\leq t \\leq T } \\Vert w _ { \\tau } ( t ; \\rho ) - w ( t ; \\rho ) \\Vert _ a = 0 . \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} \\| V _ n \\| _ { L ^ q ( \\R ^ d ) } = h ^ { - \\frac { d + 1 } { 2 q } } \\| W _ h \\| _ { L ^ q ( \\R ^ d ) } = \\mathcal { O } ( h ^ { 1 - \\frac { d + 1 } { 2 q } } ) \\to 0 \\end{align*}"}
-{"id": "3111.png", "formula": "\\begin{align*} N ' = u _ 1 ^ { b _ 1 } \\cdots u _ w ^ { b _ w } , N '' = v _ 1 ^ { a _ 1 } \\cdots v _ s ^ { a _ s } \\end{align*}"}
-{"id": "6625.png", "formula": "\\begin{align*} f _ G ( ( x _ v ) _ { v \\in V } ) = \\prod _ { e \\in E } d ( p ( e ) ) , \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} \\dot Q ( t ) Q ( t ) ^ { - 1 } y = A ( \\omega ( t ) ) y = \\omega ( t ) \\times y , \\forall y \\in \\mathbb { R } ^ { 3 } , Q ( 0 ) = I _ { 3 } , \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} V = T ^ \\C \\Sigma _ c \\oplus W \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} \\mathcal { P } _ c ( \\theta _ c , \\tau ) & = \\sum _ { k \\in \\mathcal { K } } \\mathcal { A } _ k \\int _ { 0 } ^ { \\infty } \\exp ( - ( N _ S + N _ 0 ) \\zeta _ { k } ( r ) ) \\mathcal { L } _ { I _ B } ( \\zeta _ { k } ( r ) ) \\\\ & \\times \\mathcal { L } _ { I _ U } ( \\zeta _ { k } ( r ) ) \\mathcal { L } _ { I _ J } ( \\zeta _ { k } ( r ) ) f _ { D _ k } ( r ) d r , \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{align*} \\tilde { J } _ 1 ( N ; \\theta ) = \\frac { N ^ 2 } { ( 1 - \\theta ) ^ 2 } \\left ( H _ N ^ 2 + \\sum _ { j = 1 } ^ N \\frac { 1 } { j ^ 2 } \\right ) . \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} \\left \\langle \\frac { D \\textit { \\textbf { V } } } { \\partial t } ( H ( t , \\lambda ) ) , \\frac { \\partial H } { \\partial \\lambda } ( t , \\lambda ) \\right \\rangle _ g = \\left \\langle \\frac { D \\textit { \\textbf { V } } } { \\partial \\lambda } ( H ( t , \\lambda ) ) , \\frac { \\partial H } { \\partial t } ( t , \\lambda ) \\right \\rangle _ g . \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} 0 . 0 7 1 0 \\ldots = ~ ~ 2 \\frac { e ^ { - 2 c } } { 1 - e ^ { - c } } ~ < ~ \\frac { \\pi | z _ { J _ n , n } | } { a _ { \\max } } e ^ { - c } ~ ~ = ~ 0 . 0 9 1 4 \\ldots \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} a ^ { ( \\epsilon ) } _ i ( s , y ) = \\epsilon ^ { 2 / ( q - 1 ) } \\ a _ i ( t + \\epsilon ^ 2 s , x + \\epsilon y ) \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} h ( \\hat { X } ) & = \\left ( h ^ 2 \\big ( \\hat { x } ^ { ( i ) } _ 1 , h ^ { d - 1 } ( \\hat { X } _ { - i } ) _ 1 \\big ) , \\dots , h ^ 2 \\big ( \\hat { x } ^ { ( i ) } _ n , h ^ { d - 1 } ( \\hat { X } _ { - i } ) _ n \\big ) \\right ) ^ T \\\\ & \\prec _ w \\left ( h ^ 2 \\big ( \\tilde { x } ^ { ( i ) } _ 1 , h ^ { d - 1 } ( \\tilde { X } _ { - i } ) _ 1 \\big ) , \\dots , h ^ 2 \\big ( \\tilde { x } ^ { ( i ) } _ n , h ^ { d - 1 } ( \\tilde { X } _ { - i } ) _ n \\big ) \\right ) ^ T = h ( \\tilde { X } ) . \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} \\max p _ { - k } ( n ) = \\max \\left ( p _ { - k } ( \\lambda ) | ~ \\lambda \\in P _ { - k } ( n ) \\right ) . \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} \\displaystyle \\limsup _ { h \\to 0 + } \\dfrac { \\sup _ { t \\in [ 0 , 1 ] } \\sup _ { | s - t | \\leq h } | Z ^ H ( t ) - Z ^ H ( s ) | } { h ^ { ( H - 1 / \\alpha ) } ( \\log { 1 / h } ) ^ { 1 / \\gamma } } = 0 \\ ; \\ ; a . s . \\end{align*}"}
-{"id": "8939.png", "formula": "\\begin{align*} B ^ { \\epsilon , \\kappa } = \\partial _ 1 A ^ { \\epsilon , \\kappa } _ 2 - \\partial _ 2 A ^ { \\epsilon , \\kappa } _ 1 \\ , . \\end{align*}"}
-{"id": "7444.png", "formula": "\\begin{align*} d q ^ \\prime _ t = & ( \\tilde \\gamma ^ T ) ^ { - 1 } ( t ^ * , q ^ \\prime _ t ) \\left ( - \\partial _ t \\psi ( t ^ * , q ^ \\prime _ t ) - \\nabla _ q V ( t ^ * , q _ t ^ \\prime ) + \\tilde F ( t ^ * , q ^ \\prime _ t , \\psi ( t ^ * , q ^ \\prime _ t ) ) \\right ) d t \\\\ & + \\tilde S ^ \\prime ( t ^ * , q ^ \\prime _ t ) d t + ( \\tilde \\gamma ^ T ) ^ { - 1 } ( t ^ * , q ^ \\prime _ t ) \\sigma ( t ^ * , q ^ \\prime _ t ) \\circ d W _ t , \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} \\alpha \\cdot ( m \\otimes n ) = - m \\cdot \\alpha \\otimes n + m \\otimes \\alpha \\cdot n \\ , . \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{align*} \\chi _ 2 ( z ) = & - \\left ( V \\nabla _ q \\beta + \\beta \\tilde F - \\beta \\partial _ t \\psi \\right ) _ i ( \\tilde \\gamma ^ { - 1 } ) ^ { i j } z _ j . \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{align*} \\mathbf { E } ^ { - 1 } \\mathbf { M E } = \\mathbf { I } , \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} \\partial _ t K _ { j k } ^ m ( x , y ; t , s ) - \\triangle _ x K _ { j k } ^ m ( x , y ; t , s ) = 0 \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} & I _ \\frac { p - 2 } { 2 } ( - r ) = ( - 1 ) ^ \\frac { p - 2 } { 2 } \\frac { 1 } { \\Gamma ( \\frac { p - 1 } { 2 } ) \\sqrt { \\pi } } \\big ( \\frac { r } { 2 } \\big ) ^ \\frac { p - 2 } { 2 } \\int _ { - 1 } ^ 1 e ^ { r u } ( 1 - u ^ 2 ) ^ { \\frac { p - 3 } { 2 } } d u \\\\ \\Leftrightarrow \\ & \\int _ { - 1 } ^ 1 e ^ { r u } ( 1 - u ^ 2 ) ^ { \\frac { p - 3 } { 2 } } d u = ( - 1 ) ^ \\frac { p - 2 } { 2 } \\Gamma ( \\frac { p - 1 } { 2 } ) \\sqrt { \\pi } \\big ( \\frac { 2 } { r } \\big ) ^ \\frac { p - 2 } { 2 } I _ \\frac { p - 2 } { 2 } ( - r ) \\end{align*}"}
-{"id": "1233.png", "formula": "\\begin{align*} Q ( x , y , z ) = 5 ^ { 1 / 5 } S ( \\frac { 5 ^ { 3 / 5 } x } { 3 } , \\frac { 5 ^ { 2 / 5 } y } { 2 } , 5 ^ { 1 / 5 } z ) . \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} \\left [ \\prod _ { \\substack { \\nu [ p ] \\prec \\kappa \\prec \\mu } } \\frac { Y ( w ) - y _ { \\kappa } ( w ) } { y _ { \\mu } ( w ) - y _ { \\kappa } ( w ) } \\cdot ( Y ( w ) - y _ { \\nu [ p ] } ( w ) ) \\cdot z ^ { \\mu } \\right ] _ { q = t ^ { - m } } \\propto \\left ( z ^ { \\nu [ p ] } + \\sum _ { \\nu \\prec \\nu [ p ] } g _ { \\nu } ( t ; w ) z ^ { \\nu } \\right ) , \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{align*} \\langle \\chi _ y , \\chi _ { y ' } \\rangle = \\sum _ { z \\in P } \\chi _ y ( z ) \\overline { \\chi _ { y ' } ( z ) } , \\end{align*}"}
-{"id": "2434.png", "formula": "\\begin{align*} \\psi _ j ( 0 ; \\lambda _ j ) = 1 \\ ; j = 1 , \\dots , g . \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} \\big | e ^ { 2 i k \\theta _ 1 } + ( - 1 ) ^ k \\chi _ 2 ( a ) \\overline { \\chi _ 2 ( a + 1 ) } \\big | = 2 , \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} & \\| g \\| _ { 1 , \\infty } ' : = \\sup _ { X _ 0 \\in I \\atop s \\in [ 0 , \\beta ) _ h } \\sum _ { X \\in I ^ 0 } | g ( X _ 0 , X + s ) | , \\\\ & \\| g \\| : = \\| g \\| _ { 1 , \\infty } ' + ( 1 + \\beta ^ { - 1 } ) \\| g \\| _ { 1 , \\infty } . \\end{align*}"}
-{"id": "3826.png", "formula": "\\begin{align*} a _ { 3 s + r } = ( 3 s + r + 1 ) + ( r - 2 ) \\beta _ 4 - ( r - 1 ) \\beta _ 3 + r \\beta _ 2 - ( r + 1 ) \\beta _ 1 . \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} L _ { n , 0 } = U _ { n , 0 } = \\delta _ n ^ 0 L _ { 0 , k } = U _ { 0 , k } = \\delta _ k ^ 0 \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{align*} \\gamma _ m : = \\rho _ { m + 1 } ^ { ( m ) } \\rho _ { m } ^ { ( m + 1 ) } , \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} A _ 7 ( a , b ) & = \\bigl ( \\frac { - 1 4 } { 3 } a + \\frac { 1 4 } { 3 } b \\bigr ) J q ^ 7 + \\bigl ( \\frac { 2 9 } { 3 } a - \\frac { 1 4 } { 3 } b \\bigr ) J q ^ 6 J q ^ 1 \\\\ & + \\bigl ( \\frac { - 2 8 } { 3 } a + \\frac { 7 } { 3 } b \\bigr ) J q ^ 5 J q ^ 2 + \\bigl ( \\frac { 2 8 } { 1 5 } a - \\frac { 7 } { 1 5 } b \\bigr ) J q ^ 4 J q ^ 3 \\\\ & \\quad + \\bigl ( \\frac { 4 } { 3 } a - \\frac { 1 } { 3 } b \\bigr ) J q ^ 3 J q ^ 4 + a J q ^ 2 J q ^ 5 + b J q ^ 1 J q ^ 6 = 0 . \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{align*} w _ 1 ( M ^ n ) = \\sum _ { j = 1 } ^ n y _ j = \\sum _ { j = 1 } ^ n \\sum _ { i = 1 } ^ n u _ i k _ { i j } = \\sum _ { i = 1 } ^ n u _ i ( \\sum _ { j = 1 } ^ n k _ { i j } ) . \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{align*} I \\left ( \\left ( \\sum _ { i = 0 } ^ { 2 ^ n - 1 } \\theta _ i \\right ) \\otimes \\chi _ 4 \\right ) , \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} y ( t , x _ 0 , u ) = C x ( t , x _ 0 , u ) , \\ ; \\ ; \\ ; t \\geq 0 , \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} \\omega _ 2 ' + \\omega _ 1 ' = i ( \\omega _ 1 ' / \\pi ) \\log ( \\lambda ) + i \\omega _ 1 / ( \\lambda \\pi ) + \\omega _ 1 ' + u ' . \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} \\pi _ i ( f g ) = \\pi _ i ( f ) g + s _ i ( f ) \\pi _ i ( g ) + \\beta s _ i ( f ) g . \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} x _ i ^ { k } = \\begin{cases} \\overline \\mu _ i , & \\overline \\mu _ i < \\infty \\\\ - \\overline \\nu _ i , & \\overline \\nu _ i < \\infty \\\\ - k , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} \\widetilde { g } _ j = \\Bigl ( \\frac { \\eta ( \\lambda _ j ) } { ( m ! ) ^ { 1 / m } } \\Bigr ) ^ 2 , \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} \\psi = \\kappa ( \\zeta ) \\exp \\Big [ i \\int _ { 0 } ^ { s } \\tau ( \\zeta ) d s \\Big ] . \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} 0 = & \\Biggl [ i \\frac { \\partial } { \\partial t } - H _ { \\epsilon } ( t ) , \\Lambda ( t ) ^ { - 1 } \\Lambda ( t ) \\Biggr ] = \\Biggl [ i \\frac { \\partial } { \\partial t } - H _ { \\epsilon } ( t ) , \\Lambda ( t ) ^ { - 1 } \\Biggr ] \\Lambda ( t ) \\\\ & + \\Lambda ( t ) ^ { - 1 } \\Biggl [ i \\frac { \\partial } { \\partial t } - H _ { \\epsilon } ( t ) , \\Lambda ( t ) \\Biggr ] , \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{align*} p _ { X ' Y ' } ( 0 , 0 ) & = \\sum _ { x , y , z } \\frac { a ( x ) } { \\bar { a } } \\frac { b ( y ) } { \\bar { b } } p _ { X Y Z } ( x , y , z ) = \\frac { 1 } { \\bar { a } \\bar { b } } \\sum _ { x , y , z } q _ { X Y Z } ( x , y , z ) = \\frac { 1 } { \\bar { a } \\bar { b } } > 0 \\end{align*}"}
-{"id": "1119.png", "formula": "\\begin{align*} \\partial ^ { n + 1 } \\circ d ^ n ( p _ i ) = d ^ n ( p _ i ' ) \\ , . \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} H _ \\kappa \\varphi ( r ) = \\lambda \\varphi ( r ) \\ , , \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} \\| S _ 1 ^ { i , j } f \\| ^ { 2 } _ { L ^ { 2 } ( w ) } = 2 ^ { j } \\sum _ { R \\in \\mathcal { D } } \\bigg ( \\sum _ { P \\in ( R ) _ { i } } | \\hat { f } ( P ) | \\bigg ) ^ { 2 } ( w ) _ { R } . \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} \\begin{cases} \\widetilde { W } _ \\delta ( 0 ) = 0 \\\\ \\widetilde { W } _ \\delta ( t _ n ) = W _ { \\delta } ( t _ { n - 1 } ) + \\delta \\Delta t + \\sqrt { \\Delta t } Z _ n & n \\in \\N \\end{cases} \\end{align*}"}
-{"id": "1706.png", "formula": "\\begin{align*} \\rho _ \\mathrm { s } f ( x , t ) = \\int _ { \\mathbb { S } ^ { d - 1 } } f ( x - t v , v ) \\ , \\mathrm { d } \\sigma ( v ) , \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} & \\forall \\ x , y , z , t \\in \\mathcal { H } ( A ) : \\\\ & x \\bullet y = ( - 1 ) ^ { | x | | y | } y \\bullet x , & \\mbox { s u p e r - c o m m u t a t i v i t y } \\\\ & \\sum _ { \\circlearrowleft ( x , y , t ) } ( - 1 ) ^ { | t | ( | x | + | z | ) } a s _ { \\bullet , \\alpha } ( x y , \\alpha ( z ) , \\alpha ( t ) ) = 0 , & \\begin{array} [ t ] { l } \\mbox { H o m - J o r d a n } \\\\ \\mbox { s u p e r i d e n t i t y } \\end{array} \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} C : Y '^ 2 = h ^ 3 + 1 \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} \\zeta ^ { \\ , - \\ln N } \\phi _ N ( \\xi ) = 1 + \\left [ \\frac { i ( 1 - \\theta ) \\xi } { N } + O \\left ( \\frac { 1 } { N ^ 2 } \\right ) \\right ] \\big [ \\chi _ 1 ( N ) + \\chi _ 2 ( N ) \\big ] , \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} \\sum _ { \\substack { ( t _ 1 ) ^ 1 \\in \\left ( Y ^ p \\right ) ^ 1 , t _ 2 \\in Y ^ q \\\\ p + q + 1 = n } } ( t _ 1 ) ^ 1 \\vee t _ 2 + \\sum _ { \\substack { t _ 1 \\in Y ^ p , ( t _ 2 ) ^ 1 \\in \\left ( Y ^ q \\right ) ^ 1 \\\\ p + q + 1 = n } } t _ 1 \\vee ( t _ 2 ) ^ 1 \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{align*} \\| \\varphi _ 1 \\| _ { C ^ { 1 , \\alpha } _ { - 1 } } \\leq A _ 1 = \\frac { b } { 1 6 } . \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} \\bigcup \\limits _ { 2 \\leq i , j \\leq m \\atop j \\neq i } \\Delta ( D _ i , D _ j ) = ( m - 2 ) ( G \\setminus \\{ 0 \\} ) . \\end{align*}"}
-{"id": "7484.png", "formula": "\\begin{align*} & F ( t , q ) \\equiv - \\partial _ t \\psi ( t , q ) - \\nabla _ q V ( t , q ) + \\tilde F ( t , q ) , \\\\ & \\frac { 1 } { 2 } \\left ( \\tilde \\gamma ^ { - 1 } - ( \\tilde \\gamma ^ T ) ^ { - 1 } \\right ) ^ { i j } F _ j = ( \\tilde \\gamma ^ { - 1 } ) ^ { k i } H _ { k \\ell } ( \\tilde \\gamma ^ { - 1 } ) ^ { \\ell j } F _ j , \\\\ & \\frac { 1 } { 2 } ( S - S ^ \\prime ) ^ i = \\beta ^ { - 1 } \\partial _ { q ^ j } \\left ( ( \\tilde \\gamma ^ { - 1 } ) ^ { k i } H _ { k \\ell } ( \\tilde \\gamma ^ { - 1 } ) ^ { \\ell j } \\right ) . \\end{align*}"}
-{"id": "8913.png", "formula": "\\begin{align*} \\dot { u } ^ * _ { t , i } I _ { H , i } ( a ) P _ { D H } ' = \\frac { d } { d t } ( I _ H ( a ) P _ { D H } ' ) . \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} \\left [ - \\frac { 1 } { 2 } , \\frac { 1 } { 2 } \\right ] ^ d = \\overset { \\bullet } { \\bigcup _ { \\lambda \\in \\{ 1 , \\dots , N \\} ^ d } } I _ \\lambda I _ \\lambda \\subset \\overline { B } _ { 1 / n } ^ { \\| \\cdot \\| _ { \\ell ^ \\infty } } ( x ) \\subset \\overline { B } _ { d / N } ^ { | \\cdot | } ( x ) x \\in I _ \\lambda . \\end{align*}"}
-{"id": "1927.png", "formula": "\\begin{align*} A ( x , 1 ) & = \\frac { ( v '' - 1 ) ( v ' - 1 ) ( v ' v '' - v ' - v '' + 2 ) x ^ 2 } { ( x - 1 ) v ' v '' + v ' + v '' - x - 1 } , \\\\ B ( x , 1 ) & = \\frac { - ( x v '' - v '' + 1 ) ( x v ' - v ' + 1 ) ( x v ' + x v '' - 3 x + 1 ) } { ( x - 1 ) v ' v '' + v ' + v '' - x - 1 } , \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\mathcal { E } _ N ( \\psi _ { N , 0 } ) = \\mathcal { E } ^ { G P } ( u _ 0 ) . \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{align*} x _ 0 = \\lim \\limits _ { t \\to 0 } \\xi ( t ) . x \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\gamma ^ { ( 1 ) } _ { N , 0 } = | u _ 0 \\rangle \\langle u _ 0 | , \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} C ( \\Q ) = \\{ ( 0 : 1 : 0 ) , ( 0 : - 2 : 1 ) , ( 0 : 2 : 1 ) , ( 4 : - 1 0 : 1 ) , ( 4 : 1 0 : 1 ) , ( - 1 : 0 : 1 ) \\} \\cong \\Z / 6 \\Z . \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{align*} w _ R = \\overline { q } _ R \\cdot w . \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} h _ B ^ j = \\sum _ { p + q = j } h _ B ^ { p , q } , \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{align*} \\Big | Y + \\frac { \\delta } { k } \\Big | ^ k = \\Big | Y \\Big | ^ k \\left ( 1 + O \\left ( \\left | \\frac { \\delta } { Y } \\right | \\right ) \\right ) . \\end{align*}"}
-{"id": "510.png", "formula": "\\begin{align*} \\forall K , \\int _ K f + \\int _ { \\partial K } \\widehat { g } _ K = 0 . \\end{align*}"}
-{"id": "9179.png", "formula": "\\begin{align*} & H _ { \\pm 1 } = P _ { \\pm 1 } \\\\ & H _ { \\pm 2 } = \\frac { P _ { \\pm 1 } P _ { \\pm 1 } | _ \\Delta + P _ { \\pm 2 } } 2 \\\\ & H _ { \\pm 3 } = \\frac { P _ { \\pm 1 } P _ { \\pm 1 } P _ { \\pm 1 } | _ \\Delta + 3 P _ { \\pm 1 } P _ { \\pm 2 } | _ \\Delta + 2 P _ { \\pm 3 } } 6 \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} g ' ( \\overline { x } ) \\mathbb { X } + \\mathcal { T } _ { K } ( g ( \\overline { x } ) ) \\cap [ \\ ! [ \\overline { \\lambda } ] \\ ! ] ^ { \\perp } = \\mathbb { Y } , \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{align*} \\| M ^ { - 1 } v \\| \\leq ( \\alpha - \\beta ) ^ { - 1 } \\| v \\| = \\frac { \\| v \\| } { \\sqrt { 2 } \\sin \\varphi } . \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} \\begin{cases} w _ { t } + w _ x + w _ { x x x } + \\lambda w = - u u _ x - \\frac { 1 } { 2 } \\int _ 0 ^ L k _ y ( x , y ) u ^ 2 ( y , t ) d y , \\\\ w ( 0 , t ) = w ( L , t ) = w _ x ( L , t ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} G _ { n } ( X ) = ( X - \\theta _ n ) \\ , \\left ( \\prod _ { j = 1 } ^ { \\lfloor \\frac { n } { 6 } \\rfloor } ( X - z _ { j , n } ) ( X - \\overline { z _ { j , n } } ) \\right ) \\times q _ { n } ( X ) , \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{align*} I _ { q } ( U _ { q } + t \\psi ) = I _ { q } ( U _ { q } ) + I _ { q } ( t \\psi ) , \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} \\rho ( \\gamma _ 1 ) = ( - 1 , ( 0 , 1 ) ) , \\rho ( \\gamma _ 2 ) = ( - 1 , ( 1 , 0 ) ) , \\rho ( \\gamma _ 3 ) = ( - 1 , ( 1 , 1 ) ) \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} & \\Bigl [ \\Lambda ( t ) , H _ { \\epsilon } ( t ) \\Bigr ] = - \\Bigl [ \\Lambda ( t ) , X _ { \\epsilon } ( t ) \\Bigr ] ^ { \\dagger } H ( t ) X _ { \\epsilon } ( t ) \\\\ & + X _ { \\epsilon } ( t ) ^ { \\dagger } \\Bigl [ \\Lambda ( t ) , H ( t ) \\Bigr ] X _ { \\epsilon } ( t ) + X _ { \\epsilon } ( t ) ^ { \\dagger } H ( t ) \\Bigl [ \\Lambda ( t ) , X _ { \\epsilon } ( t ) \\Bigr ] \\end{align*}"}
-{"id": "2936.png", "formula": "\\begin{align*} \\norm { \\abs { \\partial _ x } ^ { 1 / 2 } g } _ { L ^ 2 ( \\R ) } = \\norm { \\nabla f } _ { L ^ 2 ( \\R ^ 2 _ { + } ) } . \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{align*} C P ( X , \\Omega _ X , J , \\Lambda _ X ) = C P ( X , \\Omega _ { X n } , J _ n , \\Lambda _ X ) + o ( L _ n ) , \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{align*} T ( t ) : = \\sum _ { k = 1 } ^ { \\lfloor t \\rfloor } s _ { Z ( ( X _ { k - 1 } , X _ k ) , \\ , k - 1 ) } + ( t - \\lfloor t \\rfloor ) s _ { Z ( ( X _ { \\lfloor t \\rfloor } , X _ { \\lfloor t \\rfloor + 1 } ) , \\lfloor t \\rfloor ) } , \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{align*} v = \\sum _ { \\ell = 1 } ^ v \\kappa _ { \\ell } \\cdot E _ { \\ell } . \\end{align*}"}
-{"id": "2240.png", "formula": "\\begin{align*} \\begin{aligned} & D ^ { ( \\gamma ) } ( P \\parallel Q ) \\\\ & = \\frac { 1 } { \\gamma ( \\gamma + 1 ) } \\Big [ \\log ( \\sum _ i p _ i ^ { \\gamma + 1 } ) + \\gamma \\log ( \\sum _ i q _ i ^ { \\gamma + 1 } ) - ( \\gamma + 1 ) \\log ( \\sum _ i p _ i q _ i ^ { \\gamma } ) \\Big ] , \\end{aligned} \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} = 1 . 1 5 4 1 1 \\ldots , \\mbox { a v a l u e s l i g h t l y b e l o w L e h m e r ' s n u m b e r } 1 . 1 7 6 2 8 \\ldots \\end{align*}"}
-{"id": "1630.png", "formula": "\\begin{align*} \\widehat { T } _ t ( \\aleph ) : = \\tilde { T } _ t ( \\dot { \\varphi } _ y ) - \\frac { \\widehat { Q } ^ y _ t ( \\dot { \\varphi } _ y ) } { \\varphi _ \\sigma ( 0 ) \\varphi _ \\sigma ( y ) ( A _ t - q _ { \\xi , t } ( y ) ) } \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} \\int e ^ { V ^ 0 ( \\psi ) } d \\mu _ C ( \\psi ) & = \\int \\int e ^ { V ^ 0 ( \\psi + \\psi ' ) } d \\mu _ { C _ 0 } ( \\psi ' ) d \\mu _ { \\sum _ { l = - 1 } ^ { l _ { e n d } } C _ l } ( \\psi ) = \\int e ^ { V ^ { - 1 } ( \\psi ) } d \\mu _ { \\sum _ { l = - 1 } ^ { l _ { e n d } } C _ l } ( \\psi ) \\\\ & = \\int e ^ { V ^ m ( \\psi ) } d \\mu _ { \\sum _ { l = m } ^ { l _ { e n d } } C _ l } ( \\psi ) , \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} S ^ m _ { 1 , 0 } ( \\overline { \\R ^ n _ + } \\times \\R ^ { n - 1 } ; \\mathcal { S } _ { + + } ) = \\bigcap _ { \\tau > 0 } C ^ \\tau S ^ m _ { 1 , 0 } ( \\overline { \\R ^ n _ + } \\times \\R ^ { n - 1 } ; \\mathcal { S } _ { + + } ) , \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} \\textrm { s o } , ~ R _ { x ^ { \\lambda } \\cdot x \\phi ^ { - 1 } } = R ^ { - 1 } _ { x } R _ { x \\phi ^ { - 1 } } = L _ { x } ^ { - 1 } R _ { x } ^ { - 1 } R _ { x \\phi ^ { - 1 } } L _ { x } . \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} \\mathfrak { h } = \\bigoplus _ { \\alpha \\in \\Phi _ { P ^ u } } \\mathbb { C } e _ { \\alpha } \\oplus \\mathfrak { t } ^ { \\sigma } \\oplus \\bigoplus _ { \\alpha \\in \\Phi _ L ^ { \\sigma } } \\mathbb { C } e _ { \\alpha } \\oplus \\bigoplus _ { \\alpha \\in \\Phi _ s ^ + } \\mathbb { C } ( e _ { \\alpha } + \\sigma ( e _ { \\alpha } ) ) \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} \\varphi ( f ) = \\int _ { G ^ { ( 0 ) } } \\ , d \\nu ( u ) \\int _ { G ^ u } f ( x ) \\ , d \\lambda ^ u ( x ) . \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} t _ \\mathrm { s m } = t _ \\mathrm { s , 2 } + g _ \\mathrm { D L } ^ { - 1 } ( l _ \\mathrm { i } + l _ \\mathrm { P o I } ) . \\end{align*}"}
-{"id": "6331.png", "formula": "\\begin{align*} \\begin{cases} \\nabla G ( U ) - U \\Sigma ( U ) = { \\bf 0 } \\\\ U ^ T U = \\mathbb { I } _ p . \\end{cases} \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} \\mathfrak { M } ( K ) = \\varinjlim W _ n ( K ) / \\wp W _ n ( K ) , \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ^ { u } _ { j } ( T ) \\leq C \\tilde { C } _ { T } \\Big [ \\delta ^ { u } _ { j - 1 } ( T ) ( k ^ { q } _ { j } ( T ) + k ^ { q } _ { j - 1 } ( T ) ) + \\delta ^ { \\nabla y } _ { j - 1 } ( T ) ( k ^ { q } _ { j } ( T ) + k ^ { q } _ { j - 1 } ( T ) ) \\Big ] . \\end{aligned} \\end{align*}"}
-{"id": "1930.png", "formula": "\\begin{align*} A ( \\pi ^ i ) = \\begin{cases} \\{ 1 , 2 , \\dots , i , n + 1 , n + 2 \\} & \\textrm { i f $ 1 \\le i \\le k - 2 $ , } \\\\ \\{ 1 , 2 , \\dots , k - 2 , n + 1 , n + 2 \\} & \\textrm { i f $ i = n $ , } \\\\ \\{ 1 , 2 , \\dots , k - 2 , n + 2 \\} & \\textrm { i f $ i = n + 1 $ . } \\end{cases} \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} \\varphi _ \\lambda = \\exp \\left ( - \\frac 1 2 \\eta _ 1 \\omega _ 1 ( z / \\omega _ 1 ) ^ 2 + \\pi i z / \\omega _ 1 \\right ) \\sigma _ \\lambda \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} \\Z ^ c _ X ( n ) ^ i ( W ) = z _ n ( W , - i - 2 n ) . \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} \\epsilon _ 2 = \\left ( \\frac { \\min \\{ p _ { 1 1 } p _ { 2 2 } , p _ { 1 2 } p _ { 2 1 } \\} } { \\max \\{ p _ { 1 1 } p _ { 2 2 } , p _ { 1 2 } p _ { 2 1 } \\} } \\right ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "0.png", "formula": "\\begin{align*} H _ N ( \\sigma ) & = X _ N ( \\sigma ) + h \\sum _ { i = 1 } ^ N \\sigma _ i \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{align*} C P ( X , \\Omega _ { X n } , J _ n , \\Lambda _ X ) = C P _ { s w } ( X , \\Omega _ { X n } , J _ n , \\Lambda _ X ) + o ( L _ n ) . \\end{align*}"}
-{"id": "6134.png", "formula": "\\begin{align*} r _ V ( t ) = \\frac { \\theta } { 2 } \\left ( e ^ { \\frac { 2 t } { \\theta } } - 1 \\right ) . \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{align*} P ( T ) = \\pm P \\left ( \\frac { 1 } { q T } \\right ) q ^ g T ^ { 2 g } ( g = n / 2 + 1 - d ) . \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\mathbb { E } _ { n , d ( n ) , 0 } ( \\nu _ n ) = 0 , \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{align*} Q _ { n , \\textup { \\textbf { m } } } ( z ) F _ \\alpha ( z ) - P _ { n , \\textup { \\textbf { m } } , \\alpha } ( z ) = \\sum _ { k = n + 1 } ^ { \\infty } [ Q _ { n , \\textup { \\textbf { m } } } F _ \\alpha ] _ { k } \\ , \\Phi _ k ( z ) , z \\in D _ { \\rho _ { 0 } ( F _ \\alpha ) } , \\end{align*}"}
-{"id": "5862.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathbb { Z } } M _ i \\left [ \\psi ( \\nu , \\cdot ) \\right ] ( \\vec { x } ) = \\sum _ { k \\in \\ell ( \\vec { x } ) } t \\Big ( \\psi ( \\nu , \\vec { x } _ k ^ { - } ) - \\psi ( \\nu , \\vec { x } ) \\Big ) + \\sum _ { k \\in r ( \\vec { x } ) } \\Big ( \\psi ( \\nu , \\vec { x } _ k ^ { + } ) - \\psi ( \\nu , \\vec { x } ) \\Big ) , \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{align*} B ( \\rho ) \\psi _ { m } \\ = \\ V ^ { 1 / 2 } [ \\rho + h ( \\mathfrak { e } ) ] ^ { - 1 } V ^ { 1 / 2 } \\psi _ { m } \\ = \\ \\psi _ { m } , \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} \\widehat { m } _ i = \\sum \\limits _ { j = 2 } ^ N \\widehat { m } _ { i j } , ~ i \\in \\{ 2 , 3 , \\dots , N \\} . \\end{align*}"}
-{"id": "5660.png", "formula": "\\begin{align*} P ^ { ( Q ) } ( T Q , B ) & = P ( T , B ) \\wedge Q , \\\\ P ^ { ( 1 - Q ) } ( ( 1 - Q ) T , B ) & = Q \\vee P ( T , B ) - Q = \\big ( Q \\vee P ( T , B ) \\big ) \\wedge ( 1 - Q ) , \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} H _ { n , d } : = \\{ h \\in \\R ^ d : s _ { n } ^ { - 1 } h \\in \\Theta _ d \\} \\uparrow \\R ^ d , \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} a \\cdot ( L , \\xi _ 1 , \\xi _ 2 ) = ( L , a \\xi _ 1 , a ^ { - 1 } \\xi _ 2 ) . \\end{align*}"}
-{"id": "1996.png", "formula": "\\begin{align*} d ( Q _ { \\lambda } , Q ) = \\lambda p , d ( Q _ { \\lambda } , R ) = ( 1 - \\lambda ) p . \\end{align*}"}
-{"id": "1355.png", "formula": "\\begin{align*} S _ \\Delta = \\bigcup \\big \\{ T _ K \\subset T \\mid K \\Delta \\big \\} . \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} ( \\sigma _ \\xi Q ) ( V ) = ( 1 - | U | ^ 2 ) ^ { 1 / 2 } | \\xi | ^ 2 V + ( 1 - | U | ^ 2 ) ^ { - 1 / 2 } | \\xi | ^ 2 g ( U , V ) U + | \\xi | ^ 2 V \\bar { \\times } U . \\end{align*}"}
-{"id": "7075.png", "formula": "\\begin{align*} \\left | \\frac { \\pi } { \\beta } - \\frac { \\theta ( \\beta ) } { 2 } \\right | = \\min _ { m \\in \\Z } \\left | \\frac { \\theta } { 2 } - \\frac { \\pi ( 2 m + 1 ) } { \\beta } \\right | . \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} H _ 0 : \\theta = 0 \\in \\Theta _ { d ( n ) } H _ 1 : \\theta \\in \\Theta _ { d ( n ) } \\setminus \\{ 0 \\} \\end{align*}"}
-{"id": "8517.png", "formula": "\\begin{align*} d \\phi _ { n - 1 } I _ + + d \\phi _ n I _ 0 + d \\phi _ { n + 1 } I _ - = 0 \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{align*} \\left ( { { 1 { \\rm { + } } \\frac { { r } c } { { { { \\left ( { { r } + c } \\right ) } ^ 2 } } } } { \\rm { + } } { \\rho _ t } \\frac { { r } / c } { { { r } + c } } } \\right ) - \\frac { { \\rho _ t } + c } { c } = \\frac { { \\left ( { c - { \\rho _ t } } \\right ) { r } - { \\rho _ t c } } } { { { { \\left ( { { r } + c } \\right ) } ^ 2 } } } < 0 , \\rho _ t > { { r } c } / ( { r } + c ) . \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{align*} \\epsilon _ 4 = 1 - \\inf \\eta ( p _ { J | X Y } ) . \\end{align*}"}
-{"id": "6925.png", "formula": "\\begin{align*} g _ 1 & = \\gcd ( d , b - a ) \\mbox { s o $ g _ 1 \\le b - a < d $ } \\\\ g _ 2 & = \\gcd ( d , c - b ) \\mbox { s o $ g _ 2 \\le c - b < d $ } \\\\ g _ 3 & = \\gcd ( d , c - a ) \\mbox { s o $ g _ 3 \\le c - a < d $ } \\\\ g _ 4 & = \\gcd ( c , b - a ) \\mbox { s o $ g _ 4 \\le b - a < c $ } \\\\ g _ 5 & = \\gcd ( c , d - b ) \\mbox { s o $ g _ 5 \\le d - b < c $ } \\\\ g _ 6 & = \\gcd ( c , d - a ) \\mbox { s o $ g _ 6 \\le d - a \\le c $ } \\\\ \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} F ( \\lambda ) \\ = \\ \\sum _ { j = 1 } ^ { \\infty } \\mathbf { 1 } [ \\lambda _ { j } \\leq \\lambda ] . \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{align*} 1 = - \\frac { \\partial z } { \\partial t } \\int \\frac { d F _ 0 ( x ) } { ( x - z ) ^ 2 } , \\ \\ \\ 0 = - \\frac { \\partial ^ 2 z } { \\partial t ^ 2 } \\int \\frac { d F _ 0 ( x ) } { ( x - z ) ^ 2 } + 2 ( \\frac { \\partial z } { \\partial t } ) ^ 2 \\int \\frac { d F _ 0 ( x ) } { ( x - z ) ^ 3 } . \\end{align*}"}
-{"id": "4428.png", "formula": "\\begin{align*} \\rho ( X ) = ( X ^ { [ 1 ] } , \\dots , X ^ { [ p ] } ) \\in E ^ { n p } . \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{align*} \\| f _ \\theta - f _ { \\theta ' } \\| _ n ^ 2 = 4 \\delta ^ 2 N ^ { - ( 2 \\beta + r ) / r } d _ { h } ( \\theta , \\theta ' ) . \\end{align*}"}
-{"id": "1532.png", "formula": "\\begin{align*} \\phi _ j ( s ) : = \\phi ( \\pi _ j ( t ) ) s \\in [ 0 , L _ j ] , j \\in J . \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} v ( [ 2 \\ \\dots \\ t \\mid i _ 1 \\ \\dots \\ i _ { t - 1 } ] ) \\ = \\ ( n - i _ { t - 1 } ) v ( \\pi ) . \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n + 1 } \\lambda _ j ( 0 ) = 1 . \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} S = \\frac { n \\mathrm { R i c } ( \\omega ) \\wedge \\omega ^ { n - 1 } } { \\omega ^ n } \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} D ( y ) = & \\sum _ { i , j = 1 } ^ n \\Big ( \\nabla \\Phi ( \\Psi ( y ) ) \\ , \\nabla \\omega ( \\Psi ( y ) ) \\ , \\nabla \\Psi ( y ) \\Big ) _ { i j } ^ 2 , F ( y ) = \\sum _ { i , j = 1 } ^ n \\left ( \\sum _ { k = 1 } ^ n S _ { i j } ^ k ( \\Phi , y ) \\omega _ k ( \\Psi ( y ) ) \\right ) ^ 2 \\smallskip \\\\ E ( y ) = & 2 \\sum _ { i , j , k = 1 } ^ n S _ { i j } ^ k ( \\Phi , y ) \\omega _ k ( \\Psi ( y ) ) \\left ( \\nabla \\Phi ( \\Psi ( y ) ) \\ , \\nabla \\omega ( \\Psi ( y ) ) \\ , \\nabla \\Psi ( y ) \\right ) _ { i j } \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{align*} S - T _ 1 = \\left ( S - T _ 1 \\right ) \\mathbf { 1 } _ { \\{ T _ 1 < S \\} } . \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{align*} H = \\frac { 1 } { 2 } \\langle x , \\nu \\rangle + \\lambda \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} J ( N ; \\alpha ) = \\sum _ { k = 0 } ^ n \\binom { r } { k } \\frac { ( - 1 ) ^ k } { \\ln ^ k N } \\int _ { U ( N ; \\alpha ) ^ { - 1 } } ^ { U ( N ; \\alpha ) } e ^ { - x } ( \\ln x ) ^ k d x + o \\left ( \\frac { 1 } { \\ln ^ n N } \\right ) \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} T _ { K } = \\bigcup _ { j = 2 } ^ { p + 1 } ( T ^ { l _ j } - T ^ { l _ { j - 1 } } ) . \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} \\Omega = \\frac 1 2 H ^ 2 + E F + F E . \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{align*} { } ^ m \\rho = { } ^ 0 \\rho = r ^ { - 1 } ; { } ^ m v = { } ^ 0 v - 2 m ( { } ^ 0 \\rho ) \\log ( ( { } ^ 0 \\rho ) ^ { - 1 } - 2 m ) , \\ { } ^ 0 v = r ^ { - 1 } ( t - r ) . \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} \\lambda u = \\Pi \\big ( | u | ^ 2 u \\Big ) , \\lambda u + \\mu \\Lambda u = \\Pi \\big ( | u | ^ 2 u \\Big ) . \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} f _ { n s } ( t ) = \\begin{cases} 0 & 0 \\leq t \\leq \\frac { 1 } { 2 } , \\\\ | E | ( 2 t - 1 ) & \\frac { 1 } { 2 } \\leq t \\leq 1 . \\end{cases} \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\mathrm { d i v } \\left ( ( \\nabla u _ { 0 } ) q _ { 0 } u _ { 0 } ^ { q _ { 0 } - 1 } \\phi _ { 1 } \\right ) & = \\int _ { \\Omega } ( \\Delta u _ { 0 } ) q _ { 0 } u _ { 0 } ^ { q _ { 0 } - 1 } \\phi _ { 1 } \\\\ & + \\int _ { \\Omega } q _ { 0 } ( q _ { 0 } - 1 ) | \\nabla u _ { 0 } | ^ { 2 } u _ { 0 } ^ { q _ { 0 } - 2 } \\phi _ { 1 } \\\\ & + \\int _ { \\Omega } q _ { 0 } ( \\nabla u _ { 0 } \\nabla \\phi _ { 1 } ) u _ { 0 } ^ { q _ { 0 } - 1 } . \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} \\sum _ { k \\ge 1 } \\frac { 1 } { e ^ { l ( \\gamma ) \\cdot ( k + \\Re ( s ) ) - \\varepsilon \\cdot k } } = \\frac { 1 } { e ^ { l ( \\gamma ) \\cdot \\Re ( s ) } } \\sum _ { k \\ge 1 } \\frac { 1 } { e ^ { k \\cdot ( l ( \\gamma ) - \\epsilon ) } } \\le \\frac { 1 } { e ^ { l ( \\gamma ) \\cdot \\Re ( s ) } } \\cdot \\frac { 1 } { e ^ { l ( \\gamma ) - \\epsilon } - 1 } . \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{align*} \\{ ( a _ 1 , a _ 2 , a _ 3 ) \\in \\C ^ 3 | a _ 1 + a _ 2 + a _ 3 = \\frac { 1 } { 8 } ~ a n d ~ a _ 1 a _ 2 + a _ 1 a _ 3 + a _ 2 a _ 3 = 0 \\} . \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} \\deg ( T ) = \\big \\lvert \\{ e = \\{ \\ell , \\ell ' \\} \\in E ( \\Gamma _ h ) \\ , \\vert \\ , \\ell ' \\in T , \\ , \\ell < \\ell ' \\} \\big \\rvert . \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} g _ k ( n ) = \\max \\{ 2 ^ { \\nu ( n + 1 ) } - 1 , 2 ^ { i + 1 } - 1 - k Z _ i ( n ) : i \\in S ( n ) \\} \\end{align*}"}
-{"id": "9137.png", "formula": "\\begin{align*} b ( h ) = \\dfrac { - 9 h ^ { 1 2 } + 7 2 h ^ 9 - 1 4 4 h ^ 3 - 1 4 4 } { h ^ { 1 2 } - 8 h ^ 9 - 8 h ^ 3 - 8 } , \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} ( \\eta _ D [ \\frak { u } , y ] ) = \\sum _ { 1 \\le i \\le I } e _ i D _ { u _ i } - D _ { s ( \\frak { u } ) } - ( \\deg ( \\frak { u } ) - 1 ) D \\textrm { a n d } \\eta _ D [ \\frak { u } , y ] ( y ) = 1 . \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} \\mathcal { L } _ { X _ { n - 1 } } \\omega _ + + \\mathcal { L } _ { X _ { n } } \\omega _ 0 + \\mathcal { L } _ { X _ { n + 1 } } \\omega _ - = 0 \\mathrlap { . } \\end{align*}"}
-{"id": "2589.png", "formula": "\\begin{align*} \\delta _ k \\phi ( \\tau ) = \\sum _ { \\begin{array} { c } { \\scriptstyle v \\in \\Sigma ( 0 ) } \\\\ { \\scriptstyle v \\tau \\in \\Sigma ( k + 1 ) } \\end{array} } \\frac { m ( v \\tau ) } { m ( \\tau ) } \\phi ( v \\tau ) , \\ ; \\tau \\in \\Sigma ( k ) \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} g ( z ) = \\frac { \\chi _ 2 ( - 1 ) } { ( q _ 2 z - 1 ) ^ k } + \\frac { \\chi _ 2 ( 1 ) } { ( q _ 2 z + 1 ) ^ k } . \\end{align*}"}
-{"id": "600.png", "formula": "\\begin{align*} | a _ I | \\le \\sup _ { z \\in ( \\partial D ) ^ { n + 1 } } | P ( z ) | = ( n + 1 ) ^ { \\frac { d } { 2 } } \\sup _ { z \\in ( \\partial D ) ^ { n + 1 } } \\| P ( [ z ] ) \\| \\end{align*}"}
-{"id": "356.png", "formula": "\\begin{align*} \\left [ \\sum _ p ( a _ p , \\alpha _ p ) , \\ \\sum _ p ( b _ p , \\beta _ p ) \\right ] & = \\left \\langle \\sum _ p b _ p , \\sum _ p \\alpha _ p \\right \\rangle \\left \\langle \\sum _ p a _ p , \\sum _ p \\beta _ p \\right \\rangle ^ { - 1 } \\\\ [ 5 p t ] & = \\prod _ p \\langle b _ p , \\alpha _ p \\rangle \\cdot \\prod _ p \\langle a _ p , \\beta _ p \\rangle ^ { - 1 } \\\\ [ 5 p t ] & = \\prod _ p \\left [ ( a _ p , \\alpha _ p ) , ( b _ p , \\beta _ p ) \\right ] _ p . \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} ( E , \\Phi ) = ( V ' \\oplus 1 , \\Phi ' ) \\oplus ( V '' , 0 ) , \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} \\psi _ { a } ( t , x ) = \\frac { a } { \\sqrt { t } } e ^ { i x ^ { 2 } / 4 t } \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} \\Delta _ k ( E ) = \\{ ( | x ^ 1 - x ^ 2 | , | x ^ 2 - x ^ 3 | , \\dots , | x ^ k - x ^ { k + 1 } | ) : x ^ j \\in E \\} \\subset \\R ^ k \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} \\phi = \\frac { a } { b } = \\frac { a _ 1 g _ 1 + \\dots + a _ { k - 1 } g _ { k - 1 } + a _ k g _ k } { b _ 1 g _ 1 + \\dots + b _ { \\ell - 1 } g _ { \\ell - 1 } + g _ \\ell } \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{align*} \\pi _ 1 d d ^ \\ast \\bigl ( \\tfrac { 1 } { 2 } g \\omega ^ 2 \\bigr ) = \\tfrac { 1 } { 3 } ( \\triangle g ) \\omega ^ 2 \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} \\pi _ { \\mathbb { C } ^ m } : \\mathbb { C } ^ m \\times Y \\rightarrow \\mathbb { C } ^ m , \\pi _ { \\mathbb { C } ^ m } ( z , y ) = z . \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} \\phi \\left ( x ^ m - 1 \\right ) = x ^ n - 1 \\phi \\left ( \\frac { x ^ m - 1 } { x - 1 } \\right ) = \\frac { x ^ n - 1 } { x - 1 } \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{align*} n ( \\pi _ 1 ) = \\sum _ { s \\in \\pi _ 1 } | S _ \\mu ( s ) | \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} Z _ 1 ^ 3 + Z _ 2 ^ 3 + 9 Z _ 3 ^ 3 = 0 \\\\ 3 ^ a Z _ 1 + 3 ^ b Z _ 2 + Z _ 3 = 0 \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} L \\left [ \\psi ( \\cdot , b ) \\right ] ( a ) = M \\left [ \\psi ( a , \\cdot ) \\right ] ( b ) , \\quad \\forall \\ a \\in \\mathbb { A } , \\ b \\in \\mathbb { B } . \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} \\langle M ^ n _ \\cdot ( x ) \\rangle _ t = n ^ { \\theta } \\int _ { 0 } ^ t \\big [ q _ n g ( \\eta ^ n _ s ( x ) ) + p _ n g ( \\eta ^ n _ s ( x + 1 ) ) \\big ] \\ ; d s . \\end{align*}"}
-{"id": "8435.png", "formula": "\\begin{align*} e = e _ 1 + \\dots + e _ r , \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{align*} H ^ * ( { G } _ { k } ( \\R ^ { n } ) , \\Q ) = \\frac { \\Q [ p _ 1 , p _ 2 , \\ldots , p _ { [ \\frac { k } { 2 } ] } ; \\bar { p } _ 1 , \\bar { p } _ 2 , \\ldots , \\bar { p } _ { [ \\frac { n - k } { 2 } ] } ; r ] } { ( 1 + p _ 1 + \\cdots + p _ { [ \\frac { k } { 2 } ] } ) ( 1 + \\bar { p } _ 1 + \\cdots + \\bar { p } _ { [ \\frac { n - k } { 2 } ] } ) = 1 ; r ^ 2 = 0 } \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} \\omega _ 1 ' = \\int _ 1 ^ \\infty \\frac { d X } { 2 ( X - \\lambda ) \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\end{align*}"}
-{"id": "1731.png", "formula": "\\begin{align*} \\bigg \\| \\sum _ { k = k _ 0 } ^ \\infty 2 ^ { - k \\alpha } \\mathcal { C } _ k g \\bigg \\| _ { L ^ 4 _ t L ^ r _ x } \\lesssim \\| g \\| _ { L ^ 2 _ t L ^ 2 _ x } , \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} & \\int _ { 1 } ^ { \\infty } \\int _ { x _ { 1 } } ^ { \\infty } \\cdots \\int _ { x _ { t - 1 } } ^ { \\infty } \\prod _ { j \\in [ t ] } x _ j ^ { - \\tau + \\zeta _ j } \\prod _ { i = t + 1 } ^ s h ( i , \\boldsymbol { x } ) \\dd x _ t \\cdots \\dd x _ 1 \\\\ & \\leq \\tilde { K } \\int _ { 1 } ^ { \\infty } \\int _ { x _ { 1 } } ^ { \\infty } \\cdots \\int _ { x _ { t - 1 } } ^ { \\infty } \\prod _ { j \\in [ t ] } x _ j ^ { - \\tau + \\zeta _ j } \\prod _ { i = t + 1 } ^ s h _ { p ^ * _ i } ( i , \\boldsymbol { x } ) \\dd x _ t \\cdots \\dd x _ 1 . \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} \\alpha ( F , K ) = \\inf _ { x \\in \\R ^ n } \\frac { | F \\Delta ( x + \\lambda K ) | } { | F | } \\lambda ^ n = \\frac { | F | } { | K | } . \\end{align*}"}
-{"id": "6714.png", "formula": "\\begin{align*} \\Omega = \\left \\{ x : \\left \\langle x , W ^ { - 1 } x \\right \\rangle \\leq 1 \\right \\} . \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} & [ s , x ] [ d _ 1 , y _ i ] + [ s , y _ i ] [ d _ 1 , x ] = [ s , x ] [ z _ 1 \\dots z _ { \\ell } , y _ i ] + [ s , y _ i ] [ z _ 1 \\dots z _ { \\ell } , x ] \\\\ = \\ & \\sum _ { j = 1 } ^ { \\ell } \\bigl ( [ s , x ] z _ 1 \\dots z _ { j - 1 } [ z _ j , y _ i ] z _ { j + 1 } \\dots z _ { \\ell } + [ s , y _ i ] z _ 1 \\dots z _ { j - 1 } [ z _ j , x ] z _ { j + 1 } \\dots z _ { \\ell } \\bigr ) \\end{align*}"}
-{"id": "5954.png", "formula": "\\begin{align*} \\mathcal { L } ( R , P , Q ) = - \\sum _ { i , j , k , \\ell } R ^ { \\ell i } R ^ { j k } P _ { i j } Q _ { k \\ell } . \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{align*} \\int _ { t _ { 1 } } ^ { t _ { 2 } } \\int _ { \\Gamma _ { t } ^ { \\lambda } } \\Big | \\kappa + \\frac { \\langle \\gamma , N \\rangle } { 2 ( t _ { 0 } - t ) } \\Big | ^ { 2 } \\rho d s d t = - \\Big [ \\int _ { \\Gamma _ { t } ^ { \\lambda } } \\rho _ { X _ { 0 } } d s \\Big ] ^ { T - t _ { 2 } / \\lambda ^ { 2 } } _ { T - t _ { 1 } / \\lambda ^ { 2 } } \\rightarrow 0 \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{align*} \\overline { X } = ( ( - \\infty , 0 ] \\times Y _ - ) \\cup _ { Y _ - } X \\cup _ { Y _ + } ( [ 0 , + \\infty ) \\times Y _ + ) . \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} q _ 3 & = - 3 \\cdot t _ 3 , & q _ 7 & = - 1 5 7 5 \\cdot t _ 7 , & q _ { 1 1 } & = - 9 8 2 3 2 7 5 \\cdot \\left ( t _ { 1 1 } - t _ 3 ^ 2 t _ 5 \\right ) , \\\\ q _ 5 & = 4 5 \\cdot t _ 5 , & q _ 9 & = \\textstyle 9 9 2 5 5 \\cdot \\left ( t _ 9 - \\frac { 1 } { 3 } t _ 3 \\right ) , & q _ { 1 3 } & = 1 4 0 4 7 2 8 3 2 5 \\cdot \\left ( t _ { 1 3 } - t _ 3 ^ 2 t _ 7 - t _ 3 t _ 5 ^ 2 \\right ) . \\end{align*}"}
-{"id": "4519.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } ( \\frac { \\psi } { u } ) ^ { 2 \\beta + \\delta + p - 1 } d x & \\leq C ( \\frac { 2 \\beta + \\delta + p - 1 } { p } ) ^ p \\int _ { \\mathbb { R } ^ N } w ( x ) u ^ { - 2 \\beta } \\psi ^ { 2 \\beta + \\delta - 1 } | \\nabla \\psi | ^ p d x \\\\ & = C \\int _ { \\mathbb { R } ^ N } ( \\frac { \\psi } { u } ) ^ { 2 \\beta } ( w ( x ) \\psi ^ { \\delta - 1 } | \\nabla \\psi | ^ p ) d x \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} A _ { m } ^ { \\prime } = W _ { m } H _ { m } ^ { T } W _ { m + 1 } ^ { T } \\in \\mathbb { R } ^ { N \\times N } \\ , . \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{align*} E \\left [ S _ N ^ { ( r ) } \\right ] & = \\Gamma ( r + 1 ) \\sum _ { \\substack { J \\subset \\{ 1 , \\dots , , N \\} \\\\ J \\neq \\emptyset } } \\frac { ( - 1 ) ^ { | J | - 1 } } { \\left ( \\sum _ { j \\in J } q _ j \\right ) ^ r } \\\\ & = \\Gamma ( r + 1 ) \\sum _ { m = 1 } ^ N ( - 1 ) ^ { m - 1 } \\sum _ { 1 \\leq j _ 1 < \\cdots < j _ m \\leq N } \\frac { 1 } { \\left ( q _ { j _ 1 } + \\cdots + q _ { j _ m } \\right ) ^ r } , \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} \\dot { u } ^ * _ { t , i , j } u _ t ^ { * , i , j } = \\frac { d } { d t } ( \\ln \\det ( u ^ * _ { t , i , j } ) ) . \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} q ^ - ( t ) & = ( a - 1 ) ( 1 + t ) ^ 2 - 2 a t ( 1 + t ) + ( a + 1 ) t ^ 2 \\ , , \\\\ r ( t ) & = t ^ { a - 2 } ( a - 2 t - 1 ) + ( a - 1 ) t - 2 \\ , . \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} & [ s , x _ 1 ] [ b , x _ 2 , x _ 3 ] = [ s , x _ 1 ] [ y _ 1 \\dots y _ k , x _ 2 , x _ 3 ] = \\sum _ { i = 1 } ^ k [ s , x _ 1 ] y _ 1 \\dots y _ { i - 1 } [ y _ i , x _ 2 , x _ 3 ] y _ { i + 1 } \\dots y _ k \\\\ + \\ & \\sum _ { 1 \\le i < i ' \\le k } [ s , x _ 1 ] y _ 1 \\dots y _ { i - 1 } \\bigl ( [ y _ i , x _ 2 ] y _ { i + 1 } \\dots y _ { i ' - 1 } [ y _ { i ' } , x _ 3 ] + [ y _ i , x _ 3 ] y _ { i + 1 } \\dots y _ { i ' - 1 } [ y _ { i ' } , x _ 2 ] \\bigr ) y _ { i ' + 1 } \\dots y _ k . \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} \\dot { x } ( t ) & = f ( t , x ( t ) ) + g ( t , x ( t ) ) u ( t ) \\\\ [ 1 e x ] y ( t ) & = h ( t , x ( t ) ) . \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{align*} r _ k ( \\sigma ) = \\sum _ { \\cup \\{ P _ 1 , \\cdots , P _ m \\} = P } \\prod _ { i = 1 } ^ m \\phi _ k ( P _ i ) \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to + \\infty } \\mathcal { P } _ u ( \\lambda , \\gamma ) = \\mathcal { P } _ u ^ 0 ( 0 , \\gamma ) \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} \\max \\limits _ { x \\in C } \\left ( - \\lambda _ 1 ( x ) \\right ) = \\ldots = \\max \\limits _ { x \\in C } \\left ( - \\lambda _ { n + 1 } ( x ) \\right ) \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{align*} \\frac { u ^ { n - \\frac 1 2 } - u ^ { n - 1 } } { k _ n / 2 } = & i \\Delta \\Big ( \\frac { u ^ { n - \\frac 1 2 } + u ^ { n - 1 } } { 2 } \\Big ) + i f ( \\abs { u ^ { n - 1 } } ^ 2 ) \\frac { u ^ { n - \\frac 1 2 } + u ^ { n - 1 } } { 2 } \\\\ & + g ( t _ { n - 1 } , \\cdot ) + r ^ { n - \\frac 1 2 } \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{align*} L _ 0 = \\begin{pmatrix} 0 & 0 & 0 \\\\ e _ 1 & 0 & 0 \\\\ 0 & e _ 1 ^ T & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{align*} \\| P _ { z , z ^ * } ( ( y _ i ) _ { i \\in I } ) \\| & = \\left \\| \\sum _ { i \\in J } z _ i ^ * ( y _ i ) z _ i \\right \\| = \\sum _ { i \\in J } | z _ i ^ * ( y _ i ) | \\\\ & \\leq \\sum _ { i \\in J } \\| y _ i \\| \\leq \\sum _ { i \\in I } \\| y _ i \\| = \\| ( y _ i ) _ { i \\in I } \\| . \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} \\mathrm { P } ^ { h i t } _ i = & 1 - \\mathrm { P } ( \\mathcal { B } ( y _ 0 , d ) f _ i ) \\\\ = & 1 - e ^ { - \\Lambda _ i ( \\mathcal { B } ( y _ 0 , d ) ) } , \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} \\phi ( \\tau ) & = - 2 \\arctan \\left ( \\tanh \\left ( \\frac { ( 1 - \\epsilon ) \\ ; \\tau } { 4 } \\right ) \\right ) \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} ( R ( Y , \\dot \\gamma ) \\dot \\gamma ) ^ b = R ^ b { } _ { \\lambda \\mu \\nu } \\dot \\gamma ^ \\lambda Y ^ \\mu \\dot \\gamma ^ \\nu = - R ^ b { } _ { 0 0 1 } ( \\dot \\gamma ^ 0 ) ^ 2 Y ^ 1 - R ^ b { } _ { 0 0 a } ( \\dot \\gamma ^ 0 ) ^ 2 Y ^ a + o ( s ^ { - 4 } ) . \\end{align*}"}
-{"id": "7999.png", "formula": "\\begin{align*} \\mathbb { E } [ \\overline { V } _ t ] \\le \\psi _ 2 ^ * ( N ) = \\frac { b | T r ( L ) | } { 4 N ( \\kappa + a \\lambda _ 2 ) } + \\frac { m \\tau ^ 2 \\tilde { \\gamma } ( N - 1 ) } { 4 N ( \\kappa + a \\lambda _ 2 ) } . \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} T h ( x ) : = \\sum _ { j = 1 } ^ N p _ j h \\bigl ( f _ j ( x ) \\bigr ) . \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} \\overline { \\wp ( \\omega ) } = \\wp ( \\overline { \\omega } ) \\overline { \\wp ' ( \\omega ) } = \\wp ' ( \\overline { \\omega } ) , \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} \\deg e _ { \\ell _ 1 , \\ell _ 2 , \\ell _ 3 } = \\deg \\left ( X _ 1 ^ { s - \\ell _ 3 } X _ 2 ^ { s - \\ell _ 3 } X _ 3 ^ { s - \\ell _ 1 } \\right ) = ( s - \\ell _ 3 ) m _ 1 + ( s - \\ell _ 2 ) m _ 2 + ( s - \\ell _ 1 ) m _ 3 . \\end{align*}"}
-{"id": "8803.png", "formula": "\\begin{align*} D = D ( \\underline { z } ) = \\sum _ { 1 \\leq j \\leq r } z _ j l _ j + \\sum _ { \\alpha \\in \\Phi _ P ^ + } z _ { \\alpha } e _ { \\alpha } + \\sum _ { \\beta \\in \\Phi _ L ^ + \\setminus \\Phi _ L ^ { \\sigma } } z _ { \\beta } \\tau _ { \\beta } , \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} \\langle R ^ { ( m ) } ( \\mu , \\mu ) u , u \\rangle = \\Vert [ \\mu \\cdot u ] \\Vert ^ 2 , \\textrm { f o r } m = 1 . \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} r _ { c } = i r e ^ { i \\varphi } = \\mathbf { t } _ l \\begin{bmatrix} 0 \\\\ \\mathbf { M } ^ { - 1 } \\mathbf { f } \\end{bmatrix} , \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{align*} f ( x ) = - \\left \\vert \\sum _ { i = 1 } ^ N x _ i - \\frac { N - 1 } { 2 } \\right \\vert \\ , , \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} \\log 2 + \\Phi _ { \\alpha _ 0 } ( s , x ) & = \\Phi _ { u , \\gamma _ 0 } ( s , x , 0 ) + \\frac { 1 } { 2 } \\bigl ( \\xi ' ( 1 ) - \\xi ' ( u ) \\bigr ) \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{align*} \\Delta E ( \\| p \\| , \\alpha _ * , M , \\kappa ) = \\sum _ { \\ell = 0 } ^ K \\dfrac { \\beta ^ { ( \\ell ) } ( \\alpha _ * , M , \\kappa ) } { \\| p \\| ^ \\ell } + O \\left ( \\| p \\| ^ { - ( K + 1 ) } \\right ) . \\end{align*}"}
-{"id": "5507.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathbf { V } \\mathbf { x } \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{align*} M _ n \\overset { \\R P ^ 1 } \\longrightarrow M _ { n - 1 } \\overset { \\R P ^ 1 } { \\longrightarrow } \\cdots \\overset { \\R P ^ 1 } \\longrightarrow M _ 1 \\overset { \\R P ^ 1 } \\longrightarrow M _ 0 = \\{ a \\ p o i n t \\} \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} \\mathrm { P } _ { m , 1 } & = \\mathrm { P } \\left ( z _ m < \\frac { \\epsilon _ 1 } { \\rho } \\right ) = e ^ { - \\lambda _ c \\pi \\left ( \\frac { \\rho } { \\epsilon _ 1 } \\right ) ^ { \\frac { 2 } { \\alpha } } } \\sum ^ { m - 1 } _ { k = 0 } \\frac { ( \\lambda _ c \\pi ) ^ { k } \\left ( \\frac { \\rho } { \\epsilon _ 1 } \\right ) ^ { \\frac { 2 k } { \\alpha } } } { k ! } . \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} F \\left ( \\sum _ { i \\in J } { a _ i x _ i } \\right ) = \\sum _ { i \\in J } { a _ i z _ i } . \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} \\nu _ i + \\ell - 2 i _ 0 + 1 & = \\mathcal { C } _ { - 1 } ( \\alpha , \\nu , i , \\lbrace 1 , \\ldots , i _ 0 - 1 \\rbrace , \\lbrace i _ 0 , i _ 0 + 1 , \\ldots , i - 1 , i + 1 , \\ldots , \\ell \\rbrace ) \\\\ & \\leq \\mathcal { C } _ { - 1 } ( \\alpha , \\nu , i _ 0 , \\lbrace 1 , \\ldots , i _ 0 - 1 \\rbrace , \\lbrace i _ 0 + 1 , \\ldots , \\ell \\rbrace ) \\\\ & = \\iota _ { i _ 0 } + \\ell - 2 i _ 0 + 1 < \\nu _ i + \\ell - 2 i _ 0 + 1 . \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{align*} t _ { \\ast } : = \\exp \\left [ - \\frac { \\int _ { \\Omega } a \\left ( x \\right ) \\phi _ { 1 } ^ { 2 } \\log \\phi _ { 1 } } { \\int _ { \\Omega } a \\left ( x \\right ) \\phi _ { 1 } ^ { 2 } } \\right ] . \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{align*} ( A / I , C _ x ^ + , v ) = ( A / I , C _ y ^ + , v ) & \\implies S _ v ( ( A / I , C _ x ^ + ) ) = S _ v ( ( A / I , C _ y ^ + ) ) \\\\ & \\implies S _ u ( A , A _ x ^ + ) = S _ u ( A , A _ y ^ + ) \\\\ & \\implies ( A , A _ x ^ + , u ) = ( A , A _ y ^ + , u ) . \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{align*} \\frac { x ^ { \\rho { } } } { 1 + b x ^ { \\alpha { } } } = b ^ { - \\frac { \\rho { } } { \\alpha { } } } G _ { 1 , 1 } ^ { 1 , 1 } \\left ( b x ^ { \\alpha { } } \\left | \\begin{array} { c c } \\frac { \\rho { } } { \\alpha { } } \\\\ \\frac { \\rho { } } { \\alpha { } } \\end{array} \\right . \\right ) , \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{align*} p _ { A } ( a ) q _ { R | A } ( r | a ) & = \\sum _ b p _ { A , B } ( a , b ) p _ { R | B } ( r | b ) \\ ! = \\ ! \\epsilon \\ , p _ { A } ( a ) p _ { R | B } ( r | \\mathtt e ) \\ ! + \\ ! ( 1 \\ ! - \\ ! \\epsilon ) p _ { A } ( a ) p _ { R | B } ( r | a ) \\\\ & \\geq \\ ! \\epsilon \\ , p _ { A } ( a ) p _ { R | B } ( r | \\mathtt e ) . \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} f \\ \\in \\ I _ i ( H ) ^ { ( n + t - 2 i + 1 ) } \\quad \\ i = 1 , \\dots , t . \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{align*} \\dot { \\theta } & = \\left ( \\frac { x } { 2 } - \\frac { m - 1 } { x } \\right ) \\sin \\theta + \\left ( \\frac { n - 1 } { y } - \\frac { y } { 2 } \\right ) \\cos \\theta + \\lambda \\\\ & \\geq - \\frac { m - 1 } { x } \\dot { x } \\tan \\theta - \\left ( \\frac { n - 1 } { y _ b } - \\frac { y _ b } { 2 } \\right ) + \\lambda \\\\ & \\geq - \\delta \\frac { \\dot { x } } { x } - \\left ( \\frac { n - 1 } { y _ b } - \\frac { y _ b } { 2 } \\right ) + \\lambda , \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} v _ p ( [ z ^ n ] \\exp f ( z ) ) & \\geq \\min _ { m \\ge 1 } \\min _ { n _ 1 + n _ 2 + \\dots + n _ m = n } ( m b - v _ p ( m ! ) ) \\\\ & = \\min _ { 1 \\le m \\le n } ( m b - v _ p ( m ! ) ) \\\\ & = n b - v _ p ( n ! ) . \\end{align*}"}
-{"id": "8632.png", "formula": "\\begin{align*} C _ { 3 } = ( p ^ { e - 2 d - 1 } + p ^ { m - d - 1 } ) w _ { c } ^ { p ^ { e - 2 } - \\bigl ( \\frac { - 1 } { p } \\bigr ) p ^ { m + d - 1 } } \\prod _ { \\substack { 1 \\leq i \\leq p - 1 } } w _ { i + c } ^ { p ^ { e - 2 } - \\bigl ( \\frac { i ^ { 2 } - c ^ { 2 } } { p } \\bigr ) p ^ { m + d - 1 } } . \\end{align*}"}
-{"id": "5751.png", "formula": "\\begin{align*} \\left ( \\frac { f _ 1 } { f _ k } \\right ) ^ p + \\frac { a _ { k - 1 } } { a _ k } \\left ( \\frac { f _ 2 } { f _ k } \\right ) ^ p + \\cdots + \\frac { a _ 1 } { a _ k } = \\frac { a _ 1 } { a _ k } \\left ( \\frac { f _ 1 } { f _ k } \\right ) ^ { p - 1 } \\ ; \\ ; 1 \\le k \\le h \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} \\dim D ^ V = 1 6 , r = 2 , a = 6 , b = 4 , p = 1 2 . \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} { \\mathcal L } ^ { - 1 } ( F G ) = { \\mathcal L } ^ { - 1 } ( F ) * { \\mathcal L } ^ { - 1 } ( G ) \\ \\ ( F \\in \\cup _ { \\alpha < \\delta } H ^ 1 ( \\Pi _ { \\alpha } ^ + ) , G \\in \\cup _ { \\alpha < \\delta } H ^ 1 ( \\Pi _ { \\alpha } ^ + ) ) . \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} V _ \\infty & = \\left \\{ ( 0 _ U , u ) \\ | \\ u \\in U \\right \\} = \\{ 0 _ U \\} \\oplus U , \\\\ V _ c & = \\left \\{ ( u , c u ) \\ | \\ u \\in U \\right \\} \\ ; , \\\\ V _ { } & = \\left \\{ ( u , v ) \\ | \\ \\{ u , v \\} \\right \\} \\ \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{align*} d _ { A _ \\infty + \\eta } \\gamma _ \\infty = d _ { A _ \\infty } \\gamma _ \\infty + [ \\eta \\wedge \\gamma _ \\infty ] = \\dot A _ \\infty + [ \\eta \\wedge \\gamma _ \\infty ] . \\end{align*}"}
-{"id": "4603.png", "formula": "\\begin{align*} i ( \\bar \\partial _ B \\partial _ B ^ * + \\partial _ B ^ * \\bar \\partial _ B ) & = \\bar \\partial _ B [ \\epsilon ( \\kappa _ B ^ { 0 , 1 } ) , \\Lambda ] + [ \\epsilon ( \\kappa _ B ^ { 0 , 1 } ) , \\Lambda ] \\bar \\partial _ B , \\\\ i ( \\partial _ B \\bar \\partial _ B ^ * + \\bar \\partial _ B ^ * \\partial _ B ) & = - \\partial _ B [ \\epsilon ( \\kappa _ B ^ { 1 , 0 } ) , \\Lambda ] - [ \\epsilon ( \\kappa _ B ^ { 1 , 0 } ) , \\Lambda ] \\partial _ B . \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} \\alpha = \\xi \\cdot ( \\omega _ \\Sigma \\times 1 ) \\in H ^ 2 ( \\Sigma ; \\R ) , \\ \\xi \\in \\R . \\end{align*}"}
-{"id": "4953.png", "formula": "\\begin{align*} \\theta _ 0 & = x , \\\\ \\theta _ 1 & = x , \\\\ \\theta _ 2 & = x ^ 3 + q _ 3 , \\\\ \\theta _ 3 & = x ^ 6 + 5 q _ 3 x ^ 3 + q _ 5 x - 5 q _ 3 ^ 2 , \\\\ \\theta _ 4 & = \\textstyle x ^ { 1 0 } + 1 5 q _ 3 x ^ 7 + 7 q _ 5 x ^ 5 - 3 5 q _ 3 q _ 5 x ^ 2 + 1 7 5 q _ 3 ^ 3 x - \\frac { 7 } { 3 } q _ 5 ^ 2 + q _ 7 x ^ 3 + q _ 3 q _ 7 . \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} E _ { S _ 1 ^ * } - E _ { \\hat { S } _ 1 } & = E _ { W _ { t _ 2 } } + E _ { W _ { t _ 2 } , S _ 1 ^ * \\setminus W _ { t _ 2 } } , \\\\ E _ { S _ 1 ^ * , S _ 3 ^ * } - E _ { \\hat { S } _ 1 , \\hat { S } _ 3 } & = E _ { W _ { t _ 2 } , S _ 3 ^ * } - d _ { { t _ 2 } , S _ 1 ^ * \\setminus W _ { t _ 2 } } - E _ { W _ { t _ 2 } , S _ 1 ^ * \\setminus W _ { t _ 2 } } , \\\\ E _ { S _ 1 ^ * , V _ 1 } - E _ { \\hat { S } _ 1 , V _ 1 } & = E _ { W _ { t _ 2 } , V _ 1 } \\\\ E _ { S _ 2 ^ * , V _ 1 } - E _ { \\hat { S } _ 2 , V _ 1 } & = { d _ { t _ 2 , V _ 1 } } . \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} Q _ 2 ^ { k , m } = \\sum _ { i = 0 } ^ { m } ( - q ^ { k + n } r ) ^ i \\prod _ { i = 0 } ^ { n - 1 } ( 1 - q ^ { k + i } r ) \\prod _ { i = n + 1 } ^ m ( 1 - q ^ { k + i } r ) . \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} \\mathcal { G } = \\bigcup _ { k , n } \\mathcal { G } _ { H _ n , 1 / k } , \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{align*} C _ 3 : = \\max _ { i = 1 , \\dots , N } \\ , \\ , \\max _ { x \\in [ - r _ { x _ i } , r _ { x _ i } ] ^ m } \\| D \\varphi _ { x _ i } ( x ) \\| \\ , . \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} K _ { d _ 1 , d _ 2 } ( A ) = K _ { d _ 2 } ( A \\times \\R ^ { d _ 2 - d _ 1 } ) A \\subseteq \\R ^ { d _ 1 } . \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{align*} \\mu ^ { \\boxtimes t } ( \\{ 0 \\} ) = 1 + \\lim _ { r \\rightarrow - \\infty } \\psi _ { \\mu } ( \\omega _ t ( r ) ) = 1 + \\lim _ { r \\rightarrow - \\infty } \\psi _ \\mu ( r ) = \\mu ( \\{ 0 \\} ) . \\end{align*}"}
-{"id": "7089.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty \\atop L \\in \\N } f _ L ^ { ( n ) } ( a _ L ) = f ^ { ( n ) } ( a ) = 0 , ( \\forall n \\in \\{ 1 , 2 , \\cdots , n _ 0 - 1 \\} ) . \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} \\lambda _ s - \\rho _ s = k _ s \\circ d + d \\circ k _ s \\ , . \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 } { 1 + \\exp ( - c ( x - \\frac { 1 } { 2 } ) ) } . \\end{align*}"}
-{"id": "3408.png", "formula": "\\begin{align*} \\min _ { | z | = \\sqrt { r } } | f ( z ) | \\leq 1 . \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} 0 = & \\sum _ l A ( e ^ { i _ 1 } _ { j _ 1 } , . . . , C e ^ { i _ l } _ { j _ l } , . . . , e ^ { i _ k } _ { j _ k } ) \\\\ = & \\left ( \\sum _ l \\lambda _ { i _ l } \\right ) A ( e ^ { i _ 1 } _ { j _ 1 } , . . . , e ^ { i _ k } _ { j _ k } ) + \\sum _ l A ( e ^ { i _ 1 } _ { j _ 1 } , . . . , e ^ { i _ l } _ { j _ l - 1 } , . . . , e ^ { i _ k } _ { j _ k } ) . \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} B _ { \\beta } : = B _ \\beta ( H ) : = \\left \\{ \\{ u , v \\} \\in \\binom { V ( H ) } { 2 } : \\deg _ H ( u , v ) < \\beta n \\right \\} \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} \\frac { \\epsilon _ i } { \\rho \\xi _ i } & = \\frac { \\epsilon _ i } { \\bar { \\xi } _ i \\frac { \\rho z _ t - \\epsilon _ 1 } { ( 1 + \\epsilon _ 1 ) z _ t } } . \\end{align*}"}
-{"id": "7142.png", "formula": "\\begin{align*} \\mathcal { O } ( x , y ) = \\sum _ { \\theta \\in \\langle \\iota _ { 1 } , \\iota _ { 2 } \\rangle } \\textnormal { s i g n } ( \\theta ) \\cdot \\theta ( x y ) \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{align*} \\tilde K _ { \\delta } \\ast g ( x , t ) = c _ d \\int _ \\mathbb { R } e ^ { - i s t } \\tilde \\psi ( \\delta ^ { - 1 } s ) e ^ { i t \\sqrt { - \\Delta } } h _ s ( x ) \\ , \\mathrm { d } s \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{align*} \\phi _ I ( \\alpha ) m = m \\alpha , \\alpha \\in b , m \\in M . \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} ( L f ) ( \\theta ) = \\int _ 0 ^ { 2 \\pi } K ( \\theta , \\phi ) f ( \\phi ) d \\phi , \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} \\int _ { C _ 0 } \\alpha = k M . \\end{align*}"}
-{"id": "7243.png", "formula": "\\begin{align*} ( q ^ { s + m } - 1 ) P _ f ( q ^ { - s } ) & = \\sum _ { \\alpha \\geq 0 } ( q ^ { s + m } - 1 ) B _ { f , \\alpha } q ^ { - \\alpha ( s + n ) } \\\\ & = q ^ { s + m } + \\sum _ { \\alpha \\geq 0 } q ^ { - \\alpha ( s + n ) } \\left ( q ^ { m - n } B _ { f , \\alpha + 1 } - B _ { f , \\alpha } \\right ) . \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} B _ \\mu = \\lbrace ( s , t ) \\in S _ \\mu \\times T _ \\mu \\mid s < t \\rbrace . \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} \\iota _ 1 ( x , y ) = \\left ( \\frac { x _ { 0 } } { x _ { 1 } } , \\frac { A _ { - 1 } ( \\frac { x _ { 0 } } { x _ { 1 } } ) } { A _ { 1 } ( \\frac { x _ { 0 } } { x _ { 1 } } ) \\frac { y _ { 0 } } { y _ { 1 } } } \\right ) \\iota _ 2 ( x , y ) = \\left ( \\frac { B _ { - 1 } ( \\frac { y _ { 0 } } { y _ { 1 } } ) } { B _ { 1 } ( \\frac { y _ { 0 } } { y _ { 1 } } ) \\frac { x _ { 0 } } { x _ { 1 } } } , \\frac { y _ { 0 } } { y _ { 1 } } \\right ) , \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{align*} d \\big ( \\Psi ( y _ 1 , s ) , \\Psi ( y _ 2 , t ) \\big ) & = d \\big ( { \\tilde { F } } _ 0 ( y _ 1 ) , { \\tilde { F } } _ { t - s } ( y _ 2 ) \\big ) \\\\ & \\le d \\big ( { \\tilde { F } } _ 0 ( y _ 1 ) , { \\tilde { F } } _ 0 ( y _ 2 ) \\big ) + d \\big ( { \\tilde { F } } _ 0 ( y _ 2 ) , { \\tilde { F } } _ { t - s } ( y _ 2 ) \\big ) \\\\ & = d _ Y ( y _ 1 , y _ 2 ) + | t - s | \\le \\sqrt { 2 \\Big ( d _ Y ^ 2 ( y _ 1 , y _ 2 ) + | t - s | ^ 2 \\Big ) } . \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } _ { p } \\Gamma + \\frac { V _ { 0 } } { 2 } \\Gamma ^ { p - 1 } = \\frac { V _ { 0 } } { 2 } \\Gamma ^ { p - 1 } \\geq 0 \\mbox { i n } \\mathcal { B } ^ { c } _ { r } ( 0 ) , \\end{align*}"}
-{"id": "3524.png", "formula": "\\begin{align*} \\partial _ { t } L = - \\int \\kappa ^ { 2 } d s . \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} & ( i _ 1 , i _ 2 , \\dots , i _ n ) \\mapsto t _ n ( ( i _ 1 , i _ 2 , \\dots , i _ n ) ) : = \\left ( i _ 1 , T \\right ) , \\\\ \\end{align*}"}
-{"id": "2967.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = 0 } ^ { d - 1 } g _ i ( z ) \\right ) \\left ( 1 - \\prod _ { i = 0 } ^ { d - 1 } g _ i ( z ) \\right ) \\overset { ? } { = } - z ^ d . \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{align*} \\{ g , h \\} \\Omega = f d g \\wedge d h \\wedge d C _ 1 \\wedge \\ldots \\wedge d C _ l , g , h \\in C ^ \\infty ( \\R ^ { l + 2 } ) , \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} I _ 1 ( N ) = \\int _ 0 ^ { U ( N ; \\alpha ) } e ^ { - x } \\left ( 1 - \\frac { \\ln x } { \\ln N } \\right ) ^ r d x \\ , + \\ , o \\left ( \\frac { 1 } { N ^ { \\theta } } \\right ) \\int _ 0 ^ { U ( N ; \\alpha ) } e ^ { - x } \\left ( 1 - \\frac { \\ln x } { \\ln N } \\right ) ^ r d x . \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} C _ E & = O + \\frac { - e ^ { 2 \\alpha _ 2 ( a ) } } { 2 } \\bar { z } _ 2 z _ 1 [ \\theta ( e _ { \\alpha _ 2 } ) , e _ { \\alpha _ 1 } ] \\\\ & - e ^ { 2 ( \\alpha _ 1 - \\alpha _ 2 ) ( a ) } \\theta ( \\frac { - e ^ { 2 \\alpha _ 1 ( a ) } } { 2 } \\bar { z } _ 1 z _ 2 [ \\theta ( e _ { \\alpha _ 1 } ) , e _ { \\alpha _ 2 } ] ) . \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} \\cos \\Theta \\ , \\Delta u + \\sin \\Theta \\ , \\det D ^ 2 u = \\sin \\Theta . \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} \\tau : = \\tau _ j = \\frac { 1 } { 3 } ( h _ { j + 1 } - h _ j ) . \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} \\omega _ 2 + \\omega _ 1 = i ( \\omega _ 1 / \\pi ) \\log ( \\lambda ) + \\omega _ 1 + u \\end{align*}"}
-{"id": "1.png", "formula": "\\begin{align*} X _ N ( \\sigma ) = \\sum _ { p \\geq 2 } \\frac { \\beta _ p } { N ^ { ( p - 1 ) / 2 } } \\sum _ { 1 \\leq i _ 1 , \\ldots , i _ p \\leq N } g _ { i _ 1 , \\ldots , i _ p } \\sigma _ { i _ 1 } \\cdots \\sigma _ { i _ p } . \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} \\{ b _ n \\} _ { n \\geq 1 } = \\left \\{ \\left ( \\int _ { S } \\max _ { { \\bf 0 } \\leq t \\leq ( n - 1 ) \\bf { 1 } } | f _ t ( s ) | ^ { \\alpha } \\mu ( d s ) \\right ) ^ { 1 / \\alpha } \\right \\} _ { n \\geq 1 } , \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} A _ { \\beta } \\ = \\lim \\limits _ { \\substack { \\beta < \\beta ' \\\\ \\beta ' - \\beta \\leq 1 } } \\ ! A _ { \\beta ' } \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} \\chi _ L ( X , Y ) : = g ( \\nabla _ X L , Y ) , \\ \\ X , Y \\in T S _ { u , \\mathring r } . \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{align*} \\rho : = r ^ { - 1 } , \\ \\ v : = r ^ { - 1 } \\bigl ( t - r - \\chi ( t / r ) 2 m \\log ( r - 2 m ) \\bigr ) , \\end{align*}"}
-{"id": "7226.png", "formula": "\\begin{align*} c _ { n _ 1 } ( x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) = \\frac { \\partial ^ { n _ 1 } f ( 0 , y _ 1 , x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) } { ( b _ 1 ; q ) _ { n _ 1 } n _ 1 ! \\partial { y _ 1 } ^ { n _ 1 } } \\Big | _ { y _ 1 = 0 } . \\end{align*}"}
-{"id": "2850.png", "formula": "\\begin{align*} A _ { b } ^ { m , k } f ( x ) = \\sum _ { Q \\in \\mathcal { S } } | b ( x ) - b _ { Q } | ^ { m - k } \\left ( \\frac { 1 } { | Q | } \\int _ { Q } | b - b _ { Q } | ^ { k } | f | \\right ) \\chi _ { Q } ( x ) , \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} \\mathcal { F } = \\{ \\lambda \\in \\Gamma ; \\Re ( \\lambda ) \\leq \\frac 1 2 \\} . \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { p ( x ^ k ) } { k ^ 4 } = \\lim _ { k \\to \\infty } \\frac { k ^ 4 ( p ( x ^ k ) - p ( \\tilde x ^ k ) ) } { k ^ 8 } + \\frac { p ( \\tilde x ^ k ) } { k ^ 4 } = \\lim _ { k \\to \\infty } \\lim _ { i \\to \\infty } \\frac { p _ i ( \\tilde x ^ k ) + 2 ( k - \\frac { 1 } { k } ) ^ 2 c _ i } { k ^ 4 } \\end{align*}"}
-{"id": "2805.png", "formula": "\\begin{align*} C ^ 2 & \\geq \\frac { 3 } { 2 } d + \\mu _ { p _ { 1 } } - \\sum _ i m _ { p _ i } ^ 2 \\\\ & \\geq \\frac { 3 } { 2 } d + \\sum _ j \\nu _ j ( \\nu _ j - 1 ) + 1 - r _ { p _ 1 } - \\sum _ i m _ { p _ i } ^ 2 \\\\ & = \\frac { 3 } { 2 } d + 1 - r _ { p _ 1 } - \\sum _ j \\nu _ j + \\underbrace { \\Big [ \\sum _ j \\nu _ j ^ 2 - \\sum _ i m _ { p _ i } ^ 2 \\Big ] } _ { ( \\dagger ) } . \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{align*} X _ { n } = \\rho _ { n } X _ { n - 1 } + \\epsilon _ { n } , n = 1 , 2 , \\ldots \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} \\log \\| t ( \\varphi _ { \\sigma } ( z ) ) \\| _ { \\sigma } = \\log | ( \\varphi _ { \\sigma } ^ * f ) ^ { ( j ) } ( z ) | + j \\log | \\alpha z | + ( d + j ( k + 1 ) ) \\log \\| \\varphi _ { \\sigma } ^ * s _ 0 ( z ) \\| _ { \\sigma } \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } d _ k ( i ) \\longrightarrow \\infty \\end{align*}"}
-{"id": "2335.png", "formula": "\\begin{align*} X = \\bigvee _ { j = 1 } ^ N X _ j \\end{align*}"}
-{"id": "35.png", "formula": "\\begin{align*} d X _ \\theta & = \\xi '' \\gamma _ \\theta \\partial _ x \\Phi _ { u , \\gamma _ \\theta } ( s , X _ \\theta ( s ) , \\lambda _ \\theta ) d s + \\sqrt { \\xi '' } d W , \\ , \\ , X _ \\theta ( 0 ) = h . \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} K _ Y \\xrightarrow { \\Psi _ \\Gamma } K _ X , \\Psi _ \\Gamma = \\emph { p r o j } _ { X * } \\left ( \\Gamma \\cdot \\emph { p r o j } _ Y ^ * \\right ) \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} q ^ { k } t ^ { l } = 1 , k , l \\in \\mathbb { N } , \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} c g ^ { q ^ { s + l } } = d ^ { q ^ n } h ^ { q ^ { s + l } } . \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} [ \\mathbf { c } _ 1 , \\ldots , \\mathbf { c } _ N ] = [ \\mathbf { s } _ 1 , \\ldots , \\mathbf { s } _ K ] \\cdot \\mathbf { G } , \\end{align*}"}
-{"id": "2578.png", "formula": "\\begin{align*} \\mathbb { E } _ { { n '' } , d , \\theta _ { n '' } } ( \\varphi _ { n '' } ) = \\mathbb { P } _ { { n '' } , d , \\theta _ { n '' } } \\circ [ s _ { n '' } ( \\hat { \\theta } _ { n '' } - \\theta _ { n '' } ) ] \\left ( \\{ z \\in \\R ^ d : \\| z + s _ { n '' } \\theta _ { n '' } \\| \\geq C \\} \\right ) , \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{align*} \\sup _ { y \\geq 0 } \\int _ { \\mathbb R } | C _ \\lambda ( x , y ) | \\ , \\mathrm d x = \\frac { 1 } { 2 \\sqrt { | \\lambda | } } \\left ( \\frac { 1 } { \\operatorname { I m } \\sqrt { \\lambda } } + \\frac { \\sqrt { 2 } } { \\operatorname { R e } \\sqrt { \\lambda } } \\right ) \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} d \\int _ \\tau ^ t \\psi ( s ) d s & = w ( \\tau ) - w ( t ) - \\int _ \\tau ^ t \\ ( \\zeta ( s ) + [ A ( s ) , \\psi ( s ) ] \\ ) d s \\ \\ \\ \\ \\ \\\\ d _ { A ( t ) } \\int _ \\tau ^ t \\psi ( s ) d s & = w ( \\tau ) - w ( t ) - \\int _ \\tau ^ t \\ ( \\zeta ( s ) + [ A ( s ) - A ( t ) , \\psi ( s ) ] \\ ) d s . \\end{align*}"}
-{"id": "6510.png", "formula": "\\begin{align*} N _ V ( r ) : = \\int _ 0 ^ { r } { n _ V ( t ) - n _ { V } ( 0 ) \\over t } d t . \\end{align*}"}
-{"id": "400.png", "formula": "\\begin{align*} A A _ { m } ^ { \\prime } = A W _ { m } H _ { m } ^ { T } W _ { m + 1 } ^ { T } = W _ { m + 1 } H _ { m } H _ { m } ^ { T } W _ { m + 1 } ^ { T } = C _ { m } C _ { m } ^ { T } \\ , , \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle i \\frac { \\partial u } { \\partial t } ( t , x ) = ( \\mathcal { D } + \\beta ) u ( t , x ) - \\sum _ { k = 1 } ^ N \\frac { Z _ k } { | x - q _ k ( t ) | } u ( t , x ) + \\left ( | u | ^ 2 * \\frac 1 { | x | } \\right ) ( t , x ) u ( t , x ) , \\\\ \\displaystyle m _ k \\frac { d ^ 2 q _ k } { d t ^ 2 } ( t ) = - \\nabla _ { q _ k } W _ q ( t ) \\\\ \\displaystyle u ( 0 , \\cdot ) = u _ 0 , q _ k ( 0 ) = a _ k , \\frac { d q _ k } { d t } ( 0 ) = b _ k \\end{cases} \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} \\psi = \\sum _ { ( r , s ) \\in \\Delta _ { n } } a _ { r , s } \\psi _ { r , s } \\end{align*}"}
-{"id": "1440.png", "formula": "\\begin{align*} Q ( { n } { a } \\mathbb { Z } ^ k , r ) = 1 - 2 r a _ 1 . . . a _ { k - 1 } . \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} \\binom { N - K + 1 + L - 1 } { L } = \\binom { N - K + L } { L } . \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} c _ j ( x _ j ' , s ) = s P _ j ( x ' ) + O ( s ^ 2 ) , \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{align*} f ( u , v ) = \\left ( v + \\dfrac { u ^ 2 } { 2 } - \\dfrac { u ^ 2 v } { 2 } - \\dfrac { u ^ 4 } { 8 } , \\dfrac { u ^ 3 } { 3 } + u v , \\dfrac { v ^ 2 } { 2 } \\right ) . \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} D _ \\beta ( r ) : = \\sup _ { x \\in M } \\left ( \\beta \\ , E ^ \\dagger ( x ) - \\Omega ( x ) + \\Omega ^ \\dagger - r \\ , \\| F ( x ) - y ^ \\dagger \\| \\right ) \\end{align*}"}
-{"id": "1092.png", "formula": "\\begin{align*} \\begin{array} { r c l } d s \\cdot ( d \\alpha \\cdot m ) & = & d \\alpha \\cdot ( d s \\cdot m ) - d ( \\alpha ( s ) ) \\cdot m \\\\ & = & 0 \\ , , \\end{array} \\end{align*}"}
-{"id": "4499.png", "formula": "\\begin{align*} i \\partial _ t u = H ( t ) u \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} R ^ { C P } _ { t , O M A } = \\log \\left ( 1 + \\rho \\frac { 1 } { { L \\left ( | | x _ t - x _ 0 | | \\right ) } } \\right ) , \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} \\Omega \\times \\mathbb { X } _ \\infty = \\left ( I \\times \\{ 1 , \\dots , k _ { \\max } \\} \\right ) ^ \\N \\end{align*}"}
-{"id": "4524.png", "formula": "\\begin{align*} \\mathbb { P } _ 2 ( \\mathbf { m } , \\mu ; K ) = \\sum _ { i } \\mathbb { P } ^ { ( i ) } ( \\mu , K ) \\prod _ { j = 1 } ^ 2 \\mathbb { P } ( m _ { j } - \\mu , K - i ) , \\end{align*}"}
-{"id": "8678.png", "formula": "\\begin{align*} d ^ E ( u _ i e ^ i ) = d u _ i \\otimes e ^ i + u _ i \\ , d ^ E e ^ i . \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{align*} x ^ + \\in T _ { \\lambda } x & = P _ A \\left ( ( 1 + \\lambda ) P _ B x - \\lambda x \\right ) - \\lambda \\left ( P _ B x - x \\right ) \\\\ & = P _ A R _ { P _ B , \\lambda } ( x ) - \\lambda \\left ( P _ B x - x \\right ) . \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{align*} D ( x ) = z ^ { 2 } = v ^ { 2 } \\frac { ( x - a _ { 4 } ) ^ { 4 } } { 4 D ' ( a _ { 4 } ) ^ { 2 } } = \\frac { ( x - a _ { 4 } ) ^ { 4 } } { 4 D ' ( a _ { 4 } ) ^ { 2 } } ( 4 \\mathbf { u } ^ { 3 } - g _ { 2 } \\mathbf { u } - g _ { 3 } ) . \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{align*} ( A _ i ) _ { x , y } = \\begin{cases} 1 , & ( x , y ) \\in R _ i ; \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} w _ { i , j } = n ^ { \\alpha _ i + \\alpha _ j - 1 - \\gamma } , \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} G e n = \\{ \\tilde { \\omega } : \\omega \\ { \\rm c l o s e d \\ l a b e l e d \\ w a l k \\ s t a r t i n g \\ a t } \\ E \\ { \\rm o f \\ l e n g t h \\ } \\leq 2 \\alpha ( N ) + 2 v ( N ) - 3 \\} \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} \\mathcal L _ \\partial ( m \\otimes p ) : = m \\otimes \\partial ^ \\bullet ( p ) + \\partial _ M ( m ) \\otimes p \\ , . \\end{align*}"}
-{"id": "6892.png", "formula": "\\begin{align*} \\mathrm { l o c } _ { \\beta } ( u , \\mathcal { F } ) = \\mathrm { l o c } _ { \\beta ^ { \\lozenge } } ( u ^ { \\lozenge } , c _ { X } ^ { \\ast } \\mathcal { F } ) \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} \\rho ( Y _ V ( v , z ) u ) = Y _ W ( \\rho ( v ) , z ) \\rho ( u ) , ~ f o r ~ \\forall v , u \\in V . \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{align*} \\rho ' _ + = t ^ { - 1 } , \\ \\ X = x / | t | , \\end{align*}"}
-{"id": "7155.png", "formula": "\\begin{align*} \\left ( D \\left ( b _ { i } \\right ) , b _ { j } \\right ) = - \\left ( D \\left ( b _ { j } \\right ) , b _ { i } \\right ) , i \\neq j . \\end{align*}"}
-{"id": "7646.png", "formula": "\\begin{align*} B ' ( y ) \\big ( B ( y ) u , v \\big ) = B ' ( y ) \\big ( B ( y ) v , u \\big ) \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{align*} p ( 0 , 0 | v , w ) & = p ( 1 , 1 | v , w ) = \\langle x _ v , x _ w \\rangle \\\\ p ( 0 , 1 | v , w ) & = p ( 1 , 0 | v , w ) = \\frac { 1 } { 2 } - \\langle x _ v , x _ w \\rangle . \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} \\widetilde { q } ( t , x ) & = \\widetilde { q } ( t , x ; r , \\alpha _ 0 ) \\\\ & \\equiv \\lim _ { \\widehat { \\beta } _ 0 \\to \\infty } \\widehat { q } ( t , x ; t , \\alpha _ 0 , i \\widehat { \\beta } _ 0 ) = \\frac { \\sin \\zeta _ 1 } { \\sin \\eta _ 1 } , \\end{align*}"}
-{"id": "1055.png", "formula": "\\begin{align*} \\frac { 2 } { 3 } \\sigma _ { f } \\| Q \\| _ { L ^ { \\frac { 4 } { 3 } } } ^ { \\frac { 4 } { 3 } } - D ( Q , Q ) + \\frac { 1 } { 3 } \\| Q \\| _ { L ^ 1 } = 0 . \\end{align*}"}
-{"id": "4717.png", "formula": "\\begin{align*} \\Delta _ { p } { v _ { n } } = { \\vert \\vert { u } _ { n } \\vert \\vert } _ { \\infty } ^ { q - p + 1 } { v } _ { n } ^ { q } : = f _ { n } \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} g ( k , i , n ) = [ g ( k - 1 , 1 , n - 1 ) + g ( k - 1 , 1 , n - 2 ) + \\dots + g ( k - 1 , 1 , 0 ) ] + g ( k , i + n , 0 ) . \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} a _ j = \\frac { 1 } { j ! } \\left . \\frac { d ^ j } { d z ^ j } \\right | _ { z = q } \\left ( \\frac { f ( z ) } { z ^ m } \\right ) = \\sum _ { k = 0 } ^ j \\left ( \\frac { ( - 1 ) ^ k } { q ^ { m + k } } \\binom { k + m - 1 } { k } \\frac { f ^ { ( j - k ) } ( q ) } { ( j - k ) ! } \\right ) \\end{align*}"}
-{"id": "4516.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } w ( x ) | \\nabla u | ^ { p - 2 } \\nabla u \\nabla \\varphi d x - \\int _ { \\mathbb { R } ^ N } f ( u ) \\varphi d x = 0 . \\end{align*}"}
-{"id": "2508.png", "formula": "\\begin{align*} A = \\bigcup _ { i = 0 } ^ { n } \\left ( \\frac { i } { n } + A _ i \\right ) , \\textrm { a n d } \\ ; \\bigcup _ { i = 0 } ^ { n } A _ i \\bmod 1 \\ , \\subset J , \\textrm { s o i n p a r t i c u l a r } \\lambda \\Big ( \\bigcup _ { i = 0 } ^ { n } A _ i \\Big ) \\leq ( 1 + \\varepsilon ) \\frac { \\lambda ( A ) } { n } . \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} \\left [ \\xi ( t \\ , , x ) \\ , , \\xi ( s \\ , , y ) \\right ] = \\delta _ 0 ( t - s ) \\ , \\delta _ 0 ( x - y ) , \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{align*} \\nabla \\cdot b _ - = & - \\frac { B _ 0 } { \\gamma ^ 2 ( t ) + B _ 0 ^ 2 } ( - \\partial _ { q ^ 1 } \\partial _ { q ^ 2 } V ( t , q ) + \\partial _ { q ^ 1 } \\tilde F _ 2 ( t , q ) ) + \\frac { B _ 0 } { \\gamma ^ 2 ( t ) + B _ 0 ^ 2 } ( - \\partial _ { q ^ 2 } \\partial _ { q _ 1 } V ( t , q ) + \\partial _ { q ^ 2 } \\tilde F _ 1 ( t , q ) ) \\\\ = & \\frac { B _ 0 } { \\gamma ^ 2 ( t ) + B _ 0 ^ 2 } \\left ( - \\partial _ { q ^ 1 } \\tilde F _ 2 ( t , q ) + \\partial _ { q ^ 2 } \\tilde F _ 1 ( t , q ) \\right ) . \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} M = M _ 1 \\cdots M _ r , M _ i : = \\begin{bmatrix} A _ i & C _ i \\\\ 0 & B _ i \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{align*} \\hat \\tau _ D ( x , y ) = \\frac { 1 } { k } \\big [ \\tau _ { p _ 1 } ( x , y ) + \\tau _ { p _ 2 } ( x , y ) + \\cdots + \\tau _ { p _ k } ( x , y ) \\big ] = \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } \\tau _ { p _ i } ( x , y ) . \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{align*} \\begin{array} { c c c } ( k _ 1 ) _ t & = & - k _ 1 ''' + 3 k _ 1 k _ 1 ' + 3 k _ 2 k _ 2 ' \\\\ ( k _ 2 ) _ t & = & k _ 2 ''' + k _ 1 ' k _ 2 - k _ 1 k _ 2 ' \\end{array} \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} A _ m H _ - ( z ) ( 1 - z ) = H _ - \\left ( z \\gamma \\right ) ( 1 - \\gamma z ) A _ m \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} R ^ z ( t ) = - I - S ^ z ( t ) \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} s _ { 2 i - 1 } = \\frac { q _ { 2 i - 1 } } { \\alpha _ { 2 i - 1 } } , & & \\alpha _ { 2 i - 1 } = ( - 1 ) ^ { i - 1 } 3 ^ 2 5 ^ 2 \\ldots ( 2 i - 3 ) ^ 2 ( 2 i - 1 ) \\end{align*}"}
-{"id": "7300.png", "formula": "\\begin{align*} \\Gamma _ { k } = \\frac { \\Omega { } p _ { k } } { ( \\Omega \\sigma _ N ^ 2 + \\sigma _ { } ^ 2 ) \\| \\mathbf { f } _ k \\| ^ 2 } . \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { 0 \\leq t \\leq T } \\| v _ { n } ( t , \\cdot ) \\| _ { C ^ { 1 , b } _ { \\rm u n i f } ( \\R ^ N ) } = 0 . \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} Y _ { m + 1 } & = Y _ m + a ( Y _ m ) h + \\sum _ { j = 1 } ^ K b ^ j ( Y _ m ) \\Delta \\beta ^ j _ m + \\sum _ { i , j = 1 } ^ K \\Big ( \\frac { \\partial b ^ { l , i } } { \\partial x _ k } ( Y _ m ) \\Big ) _ { 1 \\leq l , k \\leq d } b ^ { j } ( Y _ m ) \\int _ { t _ m } ^ { t _ { m + 1 } } \\int _ { t _ m } ^ s \\ , \\mathrm { d } \\beta _ r ^ i \\ , \\mathrm { d } \\beta _ s ^ j \\end{align*}"}
-{"id": "2792.png", "formula": "\\begin{align*} \\{ \\cdot , \\cdot \\} _ { ( \\lambda ) } \\Omega = - \\frac { 1 } { F ( \\lambda ) } d \\cdot \\wedge \\ , d \\cdot \\wedge \\big ( \\sigma _ { ( \\lambda ) } + \\frac { g _ { ( \\lambda ) } } { r - 1 } \\omega \\big ) \\wedge \\frac { \\omega ^ { r - 2 } } { ( r - 2 ) ! } \\wedge d F ^ 1 ( \\lambda ) \\wedge \\ldots \\wedge d F ^ k ( \\lambda ) , \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} I ' ( r ) = 2 r \\int _ { - 1 } ^ 0 t i ' ( r ^ 2 t ) d t . \\end{align*}"}
-{"id": "2114.png", "formula": "\\begin{align*} \\begin{cases} \\mathfrak { D } \\mathfrak { b } + r ^ { \\frac { 1 } { 2 } } \\mathfrak { b } * \\mathfrak { b } = \\mathfrak { v } \\\\ \\lim \\limits _ { s \\to \\pm \\infty } \\mathfrak { b } = \\mathfrak { b } _ { \\pm } , \\\\ \\end{cases} \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{align*} \\begin{cases} \\Re ( \\varphi - f - c ) \\geq C s ^ \\lambda , \\\\ \\Re ( \\varphi - f + \\delta | z | ^ { \\varepsilon } ) \\geq C s ^ \\lambda , \\end{cases} \\end{align*}"}
-{"id": "1660.png", "formula": "\\begin{align*} \\| A B - A _ 0 B _ 0 \\| ^ 2 & = \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } K _ { i j } ^ 2 + \\sum _ { j = 1 } ^ N L _ j ^ 2 \\\\ & = \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } K _ { i j } ^ 2 + \\sum _ { j = 1 } ^ N \\left ( \\sum _ { i = 1 } ^ { M - 1 } K _ { i j } \\right ) ^ 2 . \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} m _ t = \\frac 1 { u _ 3 } \\hat u _ t - \\frac { ( u _ 3 ) _ t } { u _ 3 ^ 2 } \\hat u = \\frac 1 { u _ 3 } \\left ( ( A ^ { - 1 } - \\hat u \\xi ^ T ) \\begin{pmatrix} r _ 1 \\\\ r _ 2 \\end{pmatrix} + r _ 3 \\hat u \\right ) + \\frac 1 { u _ 3 ^ 2 } \\left ( u _ 3 \\xi ^ T \\begin{pmatrix} r _ 1 \\\\ r _ 2 \\end{pmatrix} - u _ 3 r _ 3 \\right ) \\hat u \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{align*} \\mathcal { A } = \\{ \\mathbf { x } _ 1 , \\mathbf { x } _ 2 \\} , \\mathcal { B } = \\{ \\mathbf { y } _ 1 , \\mathbf { y } _ 2 \\} \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} \\begin{array} { r l l } \\left | M ^ { \\perp } \\right | & = \\left | K \\right | ^ { \\sum _ { i = 0 } ^ { r - 1 } \\left ( r - i \\right ) k _ { i } \\left ( M ^ { \\perp } \\right ) } & = \\left | K \\right | ^ { r n - \\sum _ { i = 0 } ^ { r - 1 } \\left ( r - i \\right ) k _ { i } \\left ( M \\right ) } \\\\ & = \\dfrac { \\left | K \\right | ^ { r n } } { \\left | K \\right | ^ { \\sum _ { i = 0 } ^ { r - 1 } \\left ( r - i \\right ) k _ { i } \\left ( M \\right ) } } & = \\dfrac { \\left | \\mathbb { Z } _ { p ^ { r } } ^ { n } \\right | } { \\left | M \\right | } . \\end{array} \\end{align*}"}
-{"id": "6628.png", "formula": "\\begin{align*} G \\to G ' = G - v _ 1 v _ 2 + ( v _ 0 v _ 1 + v _ 0 v _ 2 + v _ 0 v _ 3 ) \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} \\{ \\alpha _ i ( - 1 ) \\alpha _ j ( - 1 ) \\mathbf { 1 } , \\alpha _ s ( - 2 ) \\mathbf { 1 } | 1 \\leq i \\neq j \\leq 3 , s = 1 , 2 , 3 \\} . \\end{align*}"}
-{"id": "3310.png", "formula": "\\begin{align*} M ^ 1 = \\left [ \\begin{array} { c c } I _ n & - \\frac { c } { 2 x _ { n + 1 } } \\\\ 0 & 1 \\end{array} \\right ] \\left [ \\begin{array} { c c } M _ x - \\frac { c c ^ * } { 2 x _ { n + 1 } } & 0 \\\\ 0 & 2 x _ { n + 1 } \\end{array} \\right ] \\left [ \\begin{array} { c c } I _ n & 0 \\\\ - \\frac { c ^ * } { 2 x _ { n + 1 } } & 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} a _ { m + k + 1 } \\circ a _ k \\circ a _ { m + k + 1 } ^ { - 1 } = a _ { m + k } \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} S _ h = \\left \\lbrace f \\in L ^ 2 ( [ 0 , T ] , \\R ) : ~ \\begin{array} { l } f \\geq 0 \\mbox { a . e . i n } [ 0 , T ] \\mbox { a n d } \\\\ \\displaystyle \\int _ 0 ^ T \\ ! f ( t ) d t \\leq \\int _ 0 ^ T \\sum _ { i = 1 } ^ r \\xi ^ i _ h ( t ) d t + R \\end{array} \\right \\rbrace \\end{align*}"}
-{"id": "5435.png", "formula": "\\begin{align*} l = \\frac { 1 } { q } ( { \\cal L } ^ { \\otimes m } , \\sigma ) \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} Q ( s , x ; t , y ) = \\sum _ { z \\in \\Z } Q ( s , x ; u , z ) Q ( u , z ; t , y ) \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} L = \\sum _ { \\ell = 1 } ^ L 1 \\leq \\sum _ { \\ell = 1 } ^ L N _ \\ell = N ( \\Phi ) - d \\leq M ( \\Phi ) + 1 \\leq M + 1 , N ( \\Phi ) \\leq M ( \\Phi ) + d + 1 \\leq M + d + 1 \\leq 3 d \\cdot M = : T . \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{align*} p _ { \\tau ( t ) } ( \\lambda ) = \\frac { t } { 2 \\sqrt { \\pi } } \\frac { e ^ { - \\frac { t ^ 2 } { 4 \\lambda } } } { \\lambda ^ { \\frac { 3 } { 2 } } } . \\end{align*}"}
-{"id": "1946.png", "formula": "\\begin{align*} \\overline { \\rho } ( g \\sigma _ n g ^ { - 1 } ) = \\overline { \\rho } ( g ) \\rho ( \\sigma _ n ) \\bar { \\rho } ( g ^ { - 1 } ) . \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} \\omega _ { \\exp ( a ) H } = & \\sum _ { 1 \\leq j _ 1 , j _ 2 \\leq r } \\frac { 1 } { 4 } d ^ 2 _ a u ( l _ { j _ 1 } , l _ { j _ 2 } ) \\omega _ { j _ 1 , j _ 2 } + \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } } \\frac { - e ^ { 2 \\alpha } } { 2 } ( d _ a u - 2 \\chi ) ( \\alpha ^ { \\vee } ) \\omega _ { \\alpha , \\alpha } \\\\ & + \\sum _ { \\beta \\in \\Phi _ s ^ + } \\frac { d _ a u ( \\beta ^ { \\vee } ) } { \\sinh ( 2 \\beta ( a ) ) } \\omega _ { \\beta , \\beta } . \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} & = \\sum _ { k = H _ 0 } ^ { H - 1 } \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } a _ { i k } ^ 2 b _ { k j } ^ 2 \\\\ & = \\sum _ { k = H _ 0 } ^ { H - 1 } \\left ( \\sum _ { j = 1 } ^ N b _ { k j } ^ 2 \\right ) \\left ( \\sum _ { i = 1 } ^ { M - 1 } a _ { i k } ^ 2 \\right ) . \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} Y _ i = f ( i ) + \\sigma \\xi _ i , i = 1 , \\ldots , n , \\end{align*}"}
-{"id": "6661.png", "formula": "\\begin{align*} \\hat p _ k ( t ) = \\begin{cases} \\displaystyle { \\frac { 1 } { r _ k } } \\sum _ { i = 1 } ^ { r _ k } 1 _ { \\{ Y _ i ( t ) > \\tilde Y _ k \\} } & t = 1 , \\dots , T _ k , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1102.png", "formula": "\\begin{align*} ( u ' \\otimes v ' ) \\cdot ( m \\otimes ( n \\otimes ( u \\otimes v ) ) ) = m \\otimes ( n \\otimes ( u v ' \\otimes u ' v ) ) \\ , ; \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} \\| V ^ 0 _ s \\| & = \\sup _ { ( x , t , \\xi ) \\in \\{ 0 , \\cdots , L - 1 \\} \\times \\{ 0 , \\cdots , n - 1 \\} \\times \\{ 1 , - 1 \\} } \\sum _ { ( y , u , \\zeta ) \\in \\{ 0 , \\cdots , L - 1 \\} \\times \\{ 0 , \\cdots , n - 1 \\} \\times \\{ 1 , - 1 \\} } \\\\ & \\quad \\cdot \\left | \\frac { \\gamma } { 2 L } \\delta _ { x , y } \\delta _ { t , u } ( \\delta _ { \\xi , 1 } \\delta _ { \\zeta , - 1 } - \\delta _ { \\xi , - 1 } \\delta _ { \\zeta , 1 } ) \\right | \\\\ & = \\frac { 1 } { 2 } | \\gamma | L ^ { - 1 } . \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{align*} p _ { V _ { \\geq t } } \\left ( R V _ { \\geq t } \\ \\middle | \\ G _ { < t } , V _ { < t } \\right ) = \\frac { \\mathbb { P } \\left ( G _ { < t } \\ \\middle | \\ V \\right ) p _ V ( R V ) } { \\mathbb { P } \\left ( G _ { < t } \\ \\middle | \\ V _ { < t } \\right ) p _ { V _ { < t } } ( V _ { < t } ) } \\end{align*}"}
-{"id": "619.png", "formula": "\\begin{align*} \\sigma ( x , y ) = \\bigl ( ( x _ 1 ^ 2 - y _ 1 ^ 2 ) ^ 2 + 2 y _ 1 ^ 2 ( x _ 1 - y _ 1 ) ^ 2 + 4 ( x _ 2 - y _ 2 ) ^ 2 \\bigr ) ^ { 1 / 4 } \\end{align*}"}
-{"id": "5234.png", "formula": "\\begin{align*} c ^ { \\ast } _ { \\varepsilon } : = \\liminf _ { | x | \\rightarrow \\infty } \\inf _ { t \\geq T _ { \\varepsilon } } \\left ( 2 \\sqrt { a _ { \\inf } - \\frac { \\chi \\mu a _ { \\sup } } { b _ { \\inf } - \\chi \\mu } - \\chi \\mu \\varepsilon } - \\chi \\| \\nabla v ( x , t + t _ 0 ; t _ 0 , u _ 0 ) \\| \\right ) . \\end{align*}"}
-{"id": "8530.png", "formula": "\\begin{align*} \\phi _ { U V } ( \\zeta ) + \\phi _ { V W } ( \\zeta ) + \\phi _ { W U } ( \\zeta ) + = 0 \\mathrlap { . } \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} f ^ * ( x _ M ^ n \\times 1 ) = ( y _ M ^ n \\times 1 ) + \\xi \\cdot ( 1 \\times \\omega _ { N } ) \\in H ^ n ( M ; \\R ) \\oplus H ^ n ( N ; \\R ) , \\end{align*}"}
-{"id": "4710.png", "formula": "\\begin{align*} & \\beta \\circ \\alpha = \\alpha \\circ \\beta , \\\\ & \\beta ( x \\cdot y ) = \\beta ( x ) \\cdot y = x \\cdot \\beta ( y ) . \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} K ^ n + f ^ { * } ( c _ 1 ( T X ) ) K ^ { n - 1 } + \\cdots + f ^ { * } ( c _ n ( T X ) ) & = 0 \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} f _ K ^ + ( z _ { K + 1 } , \\dots , z _ n ) = f ( u _ 1 , \\dots , q _ i ^ { K - 1 } u _ 1 , z _ { K + 1 } , \\dots , z _ n ) \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} S ^ { e n v , 0 } _ { s , t } = & \\int _ { s } ^ t 2 \\hat b _ + ^ j ( r , q _ r ) ( \\tilde \\Sigma ^ { - 1 } ) _ { j k } ( r , q _ r ) \\circ d q ^ k _ r \\\\ & - \\int _ { s } ^ t 2 \\hat b _ + ^ j ( r , q _ r ) ( \\tilde \\Sigma ^ { - 1 } ) _ { j k } ( r , q _ r ) b ^ k _ - ( r , q _ r ) + \\nabla \\cdot b _ - ( r , q _ r ) d r , \\end{align*}"}
-{"id": "1795.png", "formula": "\\begin{align*} \\varphi A \\psi = \\sum _ { j , k } \\chi _ j \\varphi ( \\phi _ j A \\phi _ k ) \\psi \\chi _ k . \\end{align*}"}
-{"id": "509.png", "formula": "\\begin{align*} \\frac { q _ 2 ' q _ 2 } { R N } = \\frac { q _ 2 } { q _ 1 ' R } = \\frac { ( q _ 1 ' , Q ) } { q _ 1 ' } \\cdot \\frac { ( q _ 2 ' , R ) } { R } \\leq 1 . \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{align*} \\left | \\frac { \\Phi _ { t _ 0 } ( y e ^ { i A _ s ( y ) } ) } { \\Phi _ s ( y e ^ { i A _ s ( y ) } ) } \\right | = | \\exp [ ( t _ 0 - s ) u ( y e ^ { i A _ s ( y ) } ) ] | \\leq \\exp [ ( t _ 0 - s ) C ] , \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{align*} \\delta ( z ) = h ( z ) - h ' ( b ) ( z - b ) = h ( z ) - c ( z - b ) \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{align*} W ^ { ( y ) } = D _ y W D _ y \\end{align*}"}
-{"id": "5516.png", "formula": "\\begin{align*} g ^ { i j } D _ { i j } - \\frac { 1 } { 2 } x \\cdot D = \\triangle _ { g } - \\frac { 1 } { 2 } \\left \\langle F , \\nabla _ { g } \\right \\rangle . \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} G ( \\eta , \\chi ) = \\sum _ { x \\in \\mathbb { F } _ { q } ^ { * } } \\eta ( x ) \\chi ( x ) . \\end{align*}"}
-{"id": "17.png", "formula": "\\begin{align*} d ( u , \\mathcal { D } _ \\delta ^ c ) \\geq b ( u ) : = \\min ( s _ 2 ^ 2 - u , u - s _ 1 ^ 2 ) > 0 \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{align*} \\mathfrak { g } = \\mathfrak { t } \\oplus \\bigoplus _ { \\alpha \\in \\Phi } \\mathfrak { g } _ { \\alpha } , \\mathfrak { g } _ { \\alpha } = \\{ x \\in \\mathfrak { g } ; \\mathrm { A d } ( t ) ( x ) = \\alpha ( t ) x \\forall t \\in T \\} \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} \\frac { r } { a } = \\alpha _ 1 - \\cfrac { 1 } { \\alpha _ { 2 } - \\cfrac { 1 } { \\dots - \\cfrac { 1 } { \\alpha _ n } } } . \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} | K _ s ( L ( \\xi , \\sigma ) y ) | ^ 2 = O ( | L ( \\xi , \\sigma ) y ) | ^ { - 2 s } ) , \\ \\ \\ \\ | K _ { 1 - s } ( L ( \\xi , \\sigma ) y ) | ^ 2 = O ( | L ( \\xi , \\sigma ) y ) | ^ { 2 s - 2 } ) , \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{align*} \\det M _ x = ( c + 2 x _ 1 ) \\ldots ( c + 2 x _ n ) ( 1 - \\sum _ { i = 1 } ^ n \\frac { c } { c + 2 x _ i } ) \\end{align*}"}
-{"id": "1159.png", "formula": "\\begin{align*} \\varphi _ L \\cdot \\alpha = \\mathrm { T r } ( \\mathrm { a d } _ { \\alpha } ) \\varphi _ L \\ , . \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} B _ E = \\frac { z _ 1 \\bar { z } _ 2 + \\bar { z } _ 1 z _ 2 } { \\sinh ( 2 \\beta ) } \\mathcal { P } ( \\beta ^ { \\vee } ) \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} M ( t ) & = \\log ( t ) - \\log ( | \\log ( 1 - t ) | ) - \\frac { \\log ( 1 - t ) + t } { t } \\\\ R ( t ) & = t - \\log ( 1 + t ) . \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} d _ { \\mathcal { Z } } ( \\varpi ^ { 2 m } ) = 0 \\end{align*}"}
-{"id": "947.png", "formula": "\\begin{align*} Y _ { k } \\ \\leq \\ \\frac { 3 ^ { - k } } { 3 \\ , ( 1 - \\frac { 1 } { 3 } ) \\ , \\sqrt { 8 \\ , r _ { 0 } } } \\ = \\ \\frac { 3 ^ { - k } } { 2 \\ , \\sqrt { 8 \\ , r _ { 0 } } } . \\end{align*}"}
-{"id": "1069.png", "formula": "\\begin{align*} \\varphi _ L \\cdot \\alpha = \\lambda _ L ( \\alpha ) \\cdot \\varphi _ L \\ , , \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ { R _ i } m _ { i \\ell } \\gamma _ { i \\ell } \\in ( q - 1 ) \\Z ^ + \\quad 1 \\le i \\le M . \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} B _ 1 ( x ) & = \\frac { x ^ 5 C ^ 4 ( x ) } { 1 - 2 x } + \\frac { x ^ 3 ( 1 - x ) } { ( 1 - 2 x ) ^ 2 } , \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} ` { \\rm d e t } ` \\ , ( I d - z \\ , L _ { t } ) = \\exp \\Bigl ( - \\sum _ { n \\geq 1 } \\frac { ` t r ` \\ , L _ { t } ^ { n } } { n } \\ , z ^ n \\Bigr ) , \\end{align*}"}
-{"id": "1033.png", "formula": "\\begin{align*} \\sigma _ { f } \\| Q \\| _ { L ^ { \\frac { 4 } { 3 } } } ^ { \\frac { 4 } { 3 } } = \\| Q \\| _ { L ^ { 1 } } ^ { \\frac { 2 } { 3 } } = D ( Q , Q ) = 1 . \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} - \\frac 1 2 \\Delta _ B | \\phi | ^ 2 = & \\sum _ a \\{ | \\nabla _ { \\bar V _ a } \\phi | ^ 2 + | \\nabla _ { V _ a } \\phi | ^ 2 \\} + \\sum _ { i = 1 } ^ r R i c ^ Q ( E _ { a _ i } , E _ { a _ i } ) | \\phi | ^ 2 \\\\ & + \\frac 1 2 \\sum _ a \\{ \\langle \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi , \\phi \\rangle + \\langle \\phi , \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi \\rangle \\} . \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} \\mathcal { I } _ 1 = \\mathcal { I } , \\textup { a n d } \\mathcal { I } _ j = [ \\mathcal { I } , \\mathcal { I } _ { j - 1 } ] \\ ; \\textup { f o r a l l $ j \\geq 2 $ } . \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{align*} V \\left [ S ( \\theta ) \\right ] = \\frac { \\pi ^ 2 N ^ 2 } { 6 ( 1 - \\theta ) ^ 2 } \\left [ 1 + O \\left ( \\frac { \\ln N } { N } \\right ) \\right ] , N \\to \\infty . \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} \\int _ { \\R ^ { N } } f ( v _ { n } ) v _ { n } d x = o _ { n } ( 1 ) \\mbox { a n d } \\int _ { \\R ^ { N } } F ( v _ { n } ) \\ , d x = o _ { n } ( 1 ) . \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{align*} x ' = \\kappa y + A , \\\\ y ' = - \\kappa x - B . \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} d U ( t ) = - \\frac { 1 } { \\theta } U ( t ) d t + \\sigma d W ( t ) , \\ U ( 0 ) = 0 \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} \\nabla ^ 2 \\langle \\overline { \\lambda } , g \\rangle ( \\overline { x } ) \\xi + \\ ! \\nabla g ( \\overline { x } ) \\eta + \\zeta = 0 , \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} r _ { 1 , 2 } = \\frac { 1 } { 2 } - \\frac { 1 } { 2 } \\sqrt { 1 + B } \\pm \\frac { 1 } { 2 } \\sqrt { 2 - B + \\frac { 2 } { \\sqrt { 1 + B } } } , \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} \\frac { 1 } { 1 + t } \\bigg [ \\binom { n } { k } _ t - ( 1 + t ^ { n - 1 } ) \\binom { [ \\frac { k } { 2 } ] + [ \\frac { n - k } { 2 } ] } { [ \\frac { k } { 2 } ] } _ { t ^ 4 } \\bigg ] & & \\frac { t } { 1 + t } \\bigg [ \\binom { n } { k } _ t - ( 1 + t ^ { n - 1 } ) \\binom { [ \\frac { k } { 2 } ] + [ \\frac { n - k } { 2 } ] } { [ \\frac { k } { 2 } ] } _ { t ^ 4 } \\bigg ] \\end{align*}"}
-{"id": "7486.png", "formula": "\\begin{align*} S ^ { e n v } _ { s , t } = & \\int _ { s } ^ t 2 \\hat b _ + ^ j ( r , q _ r ) ( \\tilde \\Sigma ^ { - 1 } ) _ { j k } ( r , q _ r ) \\circ d q ^ k _ r \\\\ & - \\int _ { s } ^ t 2 \\hat b _ + ^ j ( r , q _ r ) ( \\tilde \\Sigma ^ { - 1 } ) _ { j k } ( r , q _ r ) b ^ k _ - ( r , q _ r ) d r - \\int _ s ^ t \\nabla \\cdot b _ - ( r , q _ r ) d r , \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} g _ \\beta : = 1 - f _ \\beta . \\end{align*}"}
-{"id": "8581.png", "formula": "\\begin{align*} \\N _ 1 : = \\N _ 1 ( x , y ) : = N _ H ( x , y ) \\cap \\{ z \\in V ( H ) : \\deg _ H ( z , y ) \\geq \\eta n \\} \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} \\begin{gathered} \\forall \\zeta \\in E _ s ( z ) , | | d \\varphi _ t ( x ) . \\zeta | | _ { G } \\leq C e ^ { - \\nu t } | | \\zeta | | _ { G } , \\quad \\forall \\ , t \\geq 0 , \\\\ \\forall \\zeta \\in E _ u ( z ) , | | d \\varphi _ t ( x ) . \\zeta | | _ { G } \\leq C e ^ { - \\nu | t | } | | \\zeta | | _ { G } , \\forall \\ , t \\leq 0 , \\end{gathered} \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} \\left | \\bigcap _ { j = 1 } ^ { L } \\bar { U } _ { j } \\right | = N - \\sum _ { j = 1 } ^ { L } m _ { j } + ( L - 1 ) \\mu + \\sum _ { l = 2 } ^ { L - 1 } ( l - 1 ) \\sum _ { J : | J | = l } \\theta _ J . \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} U ^ - _ M & = \\Big ( U \\setminus \\{ u _ 2 \\} \\Big ) \\cup \\{ q _ i u _ 2 , q _ { s } ^ { - 1 } u _ 2 , q _ s q _ i ^ { - M + 1 } u _ 2 , q _ i ^ { - 1 } q _ i ^ { - M + 1 } u _ 2 \\} \\\\ W ^ - _ M & = W \\cup \\{ q _ i q _ { s } u _ 2 , q _ i ^ { - M + 1 } u _ 2 , q _ i ^ { - 1 } q _ s ^ { - 1 } q _ i ^ { - M + 1 } u _ 2 \\} , \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} K _ m w ( \\varepsilon ) & = q ^ { 1 / 2 - ( \\mu _ m + \\varepsilon _ m ) } w ( \\varepsilon ) , \\end{align*}"}
-{"id": "2857.png", "formula": "\\begin{align*} X _ { - i , j } : = X _ 1 \\times \\ldots \\times X _ { i - 1 } \\times X _ { i + 1 } \\times \\ldots \\times X _ { j - 1 } \\times X _ { j + 1 } \\times \\ldots \\times X _ { k } , \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} \\mathcal { P } _ s ( \\theta _ s , \\tau ) & = \\sum _ { k \\in \\mathcal { K } } \\mathcal { A } _ k \\exp \\bigg ( - 2 \\pi \\lambda _ E \\int _ { 0 } ^ { \\infty } \\exp ( - N _ 0 \\gamma _ { k } ( r ) ) \\\\ & \\times \\mathcal { L } _ { I ' _ B } ( \\gamma _ { k } ( r ) ) \\mathcal { L } _ { I ' _ U } ( \\gamma _ { k } ( r ) ) \\mathcal { L } _ { I ' _ J } ( \\gamma _ { k } ( r ) ) r d r \\bigg ) , \\end{align*}"}
-{"id": "6584.png", "formula": "\\begin{align*} \\int \\limits _ Y \\left ( \\int \\limits _ 0 ^ T f ( g _ t y ) \\ , d t \\right ) ^ 2 \\ , d \\mu ( y ) & \\leqslant Q T \\| f \\| _ B ^ 2 C > 1 , \\\\ \\int \\limits _ Y \\left ( \\int \\limits _ 0 ^ T f ( g _ t y ) \\ , d t \\right ) ^ 2 \\ , d \\mu ( y ) & \\leqslant Q T \\log T \\| f \\| _ B ^ 2 C = 1 , \\\\ \\int \\limits _ Y \\left ( \\int \\limits _ 0 ^ T f ( g _ t y ) \\ , d t \\right ) ^ 2 \\ , d \\mu ( y ) & \\leqslant Q T ^ { 2 - C } \\| f \\| _ B ^ 2 C < 1 . \\end{align*}"}
-{"id": "8556.png", "formula": "\\begin{align*} \\iota _ { X _ { f _ { n - 1 } } } \\omega _ + + \\iota _ { X _ { f _ n } } \\omega _ 0 + \\iota _ { X _ { f _ { n + 1 } } } \\omega _ - = d f _ n \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} N : = [ 0 , \\infty ) _ \\tau \\times \\R ^ 3 _ X \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} \\exists z \\in ( - 1 , 0 ] \\widetilde { B } _ { 1 } ( z , 1 ) \\widetilde { B } _ { - 1 } ( z , 1 ) = 0 . \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{align*} \\| \\nabla ^ l E _ \\Omega u - ( \\nabla ^ l u ) _ { Q _ 0 ^ * } \\| _ { \\infty , Q _ 0 } & \\leq \\left \\| \\nabla ^ { l } \\sum _ { \\emptyset \\neq Q _ 0 \\cap Q \\in \\mathcal { W } _ 3 } \\varphi _ Q ( \\pi _ { Q ^ * } - \\pi _ { Q _ 0 ^ * } ) \\right \\| _ { \\infty , Q _ 0 } \\\\ & + \\| \\nabla ^ l \\pi _ { Q _ 0 ^ * } u - ( \\nabla ^ l u ) _ { Q _ 0 ^ * } \\| _ { \\infty , Q _ 0 } = \\mathbf { I } + \\mathbf { I I } . \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} T ( m ) : = \\sum _ { k = 1 } ^ { m } s _ { Z ( ( X _ { k - 1 } , X _ k ) , \\ , k - 1 ) } . \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{align*} \\begin{aligned} & \\norm { u _ { 0 } ( t _ { 0 } + h ) - u _ { 0 } ( t _ { 0 } ) } _ { L ^ { r } _ { \\sigma } ( \\Omega ) } = \\norm { e ^ { - ( t _ { 0 } + h ) A } a - e ^ { - t _ { 0 } A } a } _ { L ^ { r } _ { \\sigma } ( \\Omega ) } \\leq C t _ { 0 } ^ { - \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { r } ) } \\norm { e ^ { - h A } a - a } _ { L ^ { p } _ { \\sigma } ( \\Omega ) } \\end{aligned} \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} \\widetilde X ^ { A } _ t : = \\frac { x ^ A _ { \\frac { 1 - \\alpha } { 2 } t - t ^ \\nu / 4 } ( t - t ^ \\nu ) - ( \\alpha - \\tfrac 1 2 ) t } { - t ^ { 1 / 3 } } \\to \\tfrac 1 2 \\widetilde \\xi _ { \\rm G O E } \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{align*} \\mathcal { L } ( \\gamma _ { m - 2 } ) = m ( m - 1 ) \\ , \\gamma _ m \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} \\int _ 0 ^ t s ^ { ( 1 / 2 ) - b } \\| \\psi ( s ) \\| _ 3 ^ 2 d s & < \\infty \\ \\qquad \\ \\ \\ 0 \\le b < 1 \\ \\ \\\\ \\int _ 0 ^ t \\| \\psi ( s ) \\| _ 3 d s & = O ( t ^ { ( 2 b + 1 ) / 4 } ) \\ \\ \\ \\ \\ 0 \\le b < 1 . \\end{align*}"}
-{"id": "2261.png", "formula": "\\begin{align*} { \\rm M S E } ( \\hat D _ { \\lambda ^ * } ) = O \\left ( \\varGamma ^ { - 1 } \\right ) . \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{align*} \\| \\varphi _ { k + 1 } \\| \\leq C \\| \\Theta _ { k + 1 } \\| _ { C ^ { 1 , \\alpha } _ \\nu } \\leq C Q \\sum _ { m = 2 } ^ { k + 1 } { C _ m \\left ( \\sum _ { I \\in \\mathcal { I } _ { m , k } } { A _ 1 ^ { i _ 1 } \\dots A _ k ^ { i _ k } } \\right ) } = C Q \\sum _ { m = 2 } ^ { k + 1 } { C _ m \\left ( A ( \\epsilon ) ^ m \\right ) _ { [ k + 1 ] } } . \\end{align*}"}
-{"id": "1916.png", "formula": "\\begin{align*} F _ T - 1 - x = 2 x ( F _ T - 1 ) + A _ 1 . \\end{align*}"}
-{"id": "7648.png", "formula": "\\begin{align*} W _ t ^ K = \\sum _ { j \\in \\mathcal { J } _ K } \\langle W _ t , \\tilde { e } _ j \\rangle _ U \\ , \\tilde { e } _ j , t \\geq 0 , \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} ( - 1 ) ^ { \\star } \\sum _ { s \\in \\mathcal { K } ( k ) } S W _ { X } ( s ) = \\sum S W _ { X _ 1 } ( s _ 1 ) S W _ { X _ 2 } ( s _ 2 ) \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u = \\Delta u - \\chi \\nabla \\cdot ( u \\nabla v ) + u ( a ( x , t ) - b ( x , t ) u ) , x \\in \\Omega , \\cr \\tau v _ t = \\Delta v - \\lambda v + \\mu u , x \\in \\Omega \\end{cases} \\end{align*}"}
-{"id": "6751.png", "formula": "\\begin{gather*} [ x \\cdot ( x \\backslash y ) ] \\cdot [ ( z / x ) \\cdot x \\phi ^ { - 1 } ] = x \\cdot \\{ [ x \\backslash ( y z ) ] / x \\cdot x \\phi ^ { - 1 } \\} \\\\ \\Leftrightarrow y \\cdot z R _ { x } ^ { - 1 } R _ { x \\phi ^ { - 1 } } = \\{ [ x \\backslash ( y z ) ] / x \\cdot x \\phi ^ { - 1 } \\} L _ { x } \\Leftrightarrow ( I , R _ { x } ^ { - 1 } R _ { x \\phi ^ { - 1 } } , L _ { x } ^ { - 1 } R _ { x } ^ { - 1 } R _ { x \\phi ^ { - 1 } } L _ { x } ) \\in A U T ( L , \\cdot ) . \\end{gather*}"}
-{"id": "2888.png", "formula": "\\begin{align*} H _ { N , \\mathbf { A } } : = - \\sum _ { i = 1 } ^ N \\Delta _ { i , \\mathbf { A } } + \\sum _ { i < j } N ^ 2 V ( N ( x _ i - x _ j ) ) , \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} R _ { 1 } ( f ( r ) , g ( r ) ) = ( m c ^ 2 + V ( r ) - \\lambda ) f ( r ) - c g ' ( r ) + \\frac { c \\kappa } { r } g ( r ) \\ , , \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} \\widehat { \\mu } ( \\xi ) = \\int e ^ { i \\pi x \\xi } \\ , d \\mu ( x ) . \\end{align*}"}
-{"id": "6805.png", "formula": "\\begin{align*} [ \\nabla _ t , D ^ k ] \\nabla _ t \\psi & = \\sum _ { l = 0 } ^ { k - 1 } D ^ l ( R ^ { S M } ( \\partial _ t , . ) ) \\star D ^ { k - 1 - l } \\nabla _ t \\psi + \\sum _ { l = 0 } ^ { k - 1 } D ^ { k - l } \\dot { g } \\star D ^ { l + 1 } \\nabla _ t \\psi \\\\ & \\quad + \\sum _ { \\sum l _ i + \\sum { m _ j } = k - 1 } { } ^ { G } \\nabla ^ { l _ 1 } R ^ P \\star \\underbrace { D ^ { m _ 1 + 1 } \\phi \\star \\ldots \\star D ^ { m _ { l _ 1 } + 1 } \\phi } _ { l _ 1 - } \\star D ^ { l _ 2 } \\nabla _ t \\phi \\star D ^ { l _ 3 + 1 } \\phi \\star D ^ { l _ 4 } \\nabla _ t \\psi \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} ( I _ - ) ^ { \\mu } { } _ { \\bar { \\rho } } ( I _ + ) ^ { \\bar { \\rho } } { } _ { \\nu } = \\delta ^ { \\mu } { } _ { \\nu } \\mathrlap { . } \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} \\dim C + \\dim C ^ { \\bot } = n \\ . \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} v _ u \\left ( \\mathrm { d e t } ( \\phi _ { \\mathfrak { M } ( \\mathcal { H } ) } ) \\right ) & \\le M ( k ) \\left ( \\frac { p - 1 } { p ^ h - 1 } \\right ) \\left ( \\frac { 1 - p ^ { - n } } { 1 - p ^ { - 1 } } \\right ) e \\\\ & = \\left ( \\frac { p - 1 } { p ^ { \\delta } - 1 } \\right ) \\left ( \\frac { 1 - p ^ { - k } } { 1 - p ^ { - 1 } } \\right ) \\left ( \\frac { 1 - p ^ { - n } } { 1 - p ^ { - 1 } } \\right ) e . \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{align*} \\tau = K \\omega _ 1 + \\tau _ 0 , \\ast \\tau = K \\eta \\wedge \\omega _ 1 - \\eta \\wedge \\tau _ 0 \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} \\Pr [ A _ 1 \\cap A _ 2 ] \\geq F ( 0 ) = \\Pr [ A _ 1 ] \\Pr [ A _ 2 ] , \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} H _ 3 = \\sum _ { n = M _ + } ^ \\infty E _ M ( t , n ) | C _ n | & \\leq c \\sum _ { n = M _ + } ^ \\infty p _ M ( t , \\hat x _ n ) \\ , n ^ { d - 1 } = c \\sum _ { n = M _ + } ^ \\infty \\frac { n ^ { d - 1 } } { M ^ d } P _ { \\hat x _ n } ( Y _ t \\in B _ M ) , \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{align*} g ( C ) = 1 \\ \\ \\mbox { a n d } \\ \\ { \\rm { i n d } } C = e _ Q ( C ) = \\delta ( C ) = h ( C ) = 0 . \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} T ( x ) = T ( x _ 1 , x _ 2 ) : = ( r x _ 1 , y _ 2 + r ^ 2 x _ 2 ) , \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} J q ^ { - k } ( \\xi ^ m ) = \\frac { 1 } { \\binom { m - k } { k } } \\xi ^ { m - k } . \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{align*} \\| \\sigma ' \\| _ { C ^ { 2 , \\alpha } _ { \\nu + 1 } } \\leq C \\left ( \\| d ^ \\ast \\psi \\| _ { C ^ { 0 , \\alpha } _ { \\nu - 1 } } + \\sum _ { i = k + 1 } ^ b { | a _ i | } \\right ) \\leq C \\| d ^ \\ast \\psi \\| _ { C ^ { 0 , \\alpha } _ { \\nu - 1 } } \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{align*} y _ q = \\frac { 2 \\pi j } { n } + q \\frac { 2 \\pi } { n } \\frac { 1 } { m } , q = 0 , 1 , \\ldots , m , \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { - k } ( n ) q ^ { n } & = \\dfrac { 1 } { ( q ; q ) _ { \\infty } ^ { k } } . \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{align*} X = e ^ { i \\theta ( s ) } \\frac { x + i y } { A - i B } \\end{align*}"}
-{"id": "5393.png", "formula": "\\begin{align*} \\Delta ' a = \\Delta a - 1 \\otimes a - a \\otimes 1 \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} G [ v ] ( x ) - G [ u ] ( x ) = a ^ { i j } ( x ) D _ { i j } w ( x ) + b ^ k ( x ) D _ k w ( x ) + c ( x ) w ( x ) , \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{align*} S = E \\oplus E K ^ { - 1 } , \\end{align*}"}
-{"id": "9123.png", "formula": "\\begin{align*} { \\left [ { \\bf { B } } \\right ] _ { 1 1 } } = { \\left [ { { { \\bf { F } } _ 2 } } \\right ] _ { 1 1 } } \\left ( { \\left [ { { \\bf { G } } } \\right ] _ { { \\rm { 1 1 } } } - \\sum \\limits _ { k = 2 } ^ K { \\left [ { { \\bf { G } } } \\right ] _ { { { k 1 } } } \\frac { { \\left [ { { \\bf { G } } } \\right ] _ { { { 1 k } } } } } { \\omega } } } \\right ) + \\sum \\limits _ { k = 2 } ^ K { { { \\left [ { { { \\bf { F } } _ 2 } } \\right ] } _ { 1 k } } \\left [ { { \\bf { G } } } \\right ] _ { { { k 1 } } } } , \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} \\dim ( A \\times { K _ { P } } ^ { n } ) & = \\dim A + n \\dim K _ { P } = \\dim K + ( n - 1 ) \\dim K _ { P } \\\\ & < n \\dim K = \\dim \\left ( \\mathcal { L } _ { G } ^ { 1 } \\times \\cdots \\times \\mathcal { L } _ { G } ^ { n } \\right ) , \\end{align*}"}
-{"id": "5844.png", "formula": "\\begin{align*} E _ { s _ i \\mu } & = t ^ { - 1 } \\left ( T _ i + \\frac { 1 - t } { 1 - y _ { i + 1 } ( \\mu ) / y _ i ( \\mu ) } \\right ) \\left ( f _ { \\mu } + \\sum _ { \\substack { \\nu \\in \\sigma ( \\mu ) \\\\ \\nu \\prec \\mu } } c _ { \\mu , \\nu } ( q , t ) f _ { \\nu } \\right ) . \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} k = l , a ' + b = - ( a + b ' ) . \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} & \\# \\Big \\{ ( \\mu _ i , \\mu _ j ) = ( 0 , 2 ) , ( \\nu _ i , \\nu _ j ) = ( 1 , 1 ) \\Big \\} + \\# \\Big \\{ ( \\mu _ i , \\mu _ j ) = ( 1 , 2 ) , ( \\nu _ i , \\nu _ j ) = ( 1 , 1 ) \\Big \\} + \\\\ & \\# \\Big \\{ ( \\mu _ i , \\mu _ j ) = ( 2 , 2 ) , ( \\nu _ i , \\nu _ j ) = ( 1 , 1 ) \\Big \\} . \\end{align*}"}
-{"id": "2030.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} E _ { n + 1 } ( \\alpha _ * , M ) = F ( E _ n , \\alpha _ * , M , \\kappa _ n ) & F ( E _ n , \\alpha _ * , M , \\kappa _ n ) \\geq \\xi _ + , \\\\ E _ { n + 1 } ( \\alpha _ * , M ) = 2 \\xi _ + - F ( E _ n , \\alpha _ * , M , \\kappa _ n ) & F ( E _ n , \\alpha _ * , M , \\kappa ) < \\xi _ + , \\end{aligned} \\right . \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} \\begin{array} { l } c ^ 2 = \\frac { \\pi ^ 2 } { 1 2 } \\\\ \\gamma = 1 - \\frac { ( \\ln 2 ) ^ 2 } { c ^ 2 } \\\\ \\sigma = N - \\sqrt { E } \\frac { \\ln 2 } { c } \\end{array} \\end{align*}"}
-{"id": "4332.png", "formula": "\\begin{align*} j = 2 5 6 \\frac { ( \\lambda ^ 2 + \\lambda - 1 ) ^ 3 } { ( \\lambda - 1 ) ^ 2 \\lambda ^ 2 } . \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} \\begin{array} { l l l } K & = & ( \\partial \\Delta ^ n \\times \\Delta ^ 1 ) \\sqcup ^ { ( \\partial \\Delta ^ n \\times \\{ 1 \\} ) } * , \\\\ L & = & ( \\Delta ^ n \\times \\Delta ^ 1 ) \\sqcup ^ { ( \\Delta ^ n \\times \\{ 1 \\} ) } * , \\end{array} \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} \\int _ { r _ j } ^ { r _ j + \\phi _ j ( \\theta , s ) } t ^ { d - 1 } d t = s G _ j ( \\theta ) . \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} F _ - ^ n ( x ) - x \\le F _ - ^ n ( x ) < F _ - ^ n ( 1 ) = F _ - ^ n ( 0 ) + 1 \\end{align*}"}
-{"id": "1513.png", "formula": "\\begin{align*} \\int _ { \\Sigma } ( \\dfrac { n } { 2 } + H ^ 2 ) u ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma & \\leq \\int _ { \\Sigma } | \\nabla u | ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma = \\lambda _ 1 \\int _ { \\Sigma } u ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma . \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} \\mathbf { r } = \\{ U _ i ^ { - 1 } b _ { i - 1 } ^ { - 1 } a _ { i - 1 } ^ { - 1 } y z _ i y ^ { - 1 } b _ { i } a _ { i } \\mid 1 \\leq i \\leq n \\} \\cup \\{ z _ 1 z _ 2 \\cdots z _ n \\} . \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\gamma _ j ^ { 1 / 2 } < \\infty . \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} g = y ^ n \\cdot f \\left ( \\frac { x } { y } \\right ) . \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{align*} \\ddot { \\theta } & = \\dot { x } \\dot { y } \\left ( \\frac { m - 1 } { x ^ 2 } - \\frac { n - 1 } { y ^ 2 } \\right ) + \\dot { \\theta } \\left ( \\frac { x ^ 2 - 2 ( m - 1 ) } { 2 x } \\cos \\theta + \\frac { y ^ 2 - 2 ( m - 1 ) } { 2 y } \\sin \\theta \\right ) . \\end{align*}"}
-{"id": "6103.png", "formula": "\\begin{align*} f ' ( 0 ) : = \\lim _ { t \\to 0 ^ + } \\frac { f ( t ) } { t } = 0 . \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} B = \\{ Y _ 2 e _ { 1 , i , j } - X _ 2 e _ { 1 , i + 1 , j } + Y _ 1 e _ { 2 , i + 1 , j } - X _ 1 e _ { 2 , i + 1 , j + 1 } \\mid 0 \\leq j < i < s \\} \\subseteq \\ker \\partial _ 2 . \\end{align*}"}
-{"id": "7219.png", "formula": "\\begin{align*} \\lambda _ { k , l } ( a _ 1 ; q ) _ k ( b _ 1 ; q ) _ { l + 1 } \\frac { ( q ; q ) _ { k + l } } { ( q ; q ) _ { k - 1 } ( q ; q ) _ { l } } = \\lambda _ { k - 1 , l + 1 } ( a _ 1 ; q ) _ { k } ( b _ 1 ; q ) _ { l + 1 } \\frac { ( q ; q ) _ { k + l } } { ( q ; q ) _ { k - 1 } ( q ; q ) _ { l } } . \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} \\rho \\ast \\sigma = \\sum _ { \\alpha _ { n , m } \\in S _ { ( n , m ) } } \\alpha _ { n , m } \\cdot \\left ( \\rho \\times \\sigma \\right ) \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{align*} 0 \\geq ( \\nu _ 2 M - 1 ) \\ln \\left ( 1 - x ^ { \\lambda } \\right ) \\geq ( \\nu _ 2 M - 1 ) \\ln \\left ( 1 - \\frac { 1 } { M ^ { 1 + \\lambda \\varepsilon } } \\right ) = - \\frac { \\nu _ 2 } { M ^ { \\lambda \\varepsilon } } + O \\left ( \\frac { 1 } { M ^ { 1 + \\lambda \\varepsilon } } \\right ) \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{align*} \\| x ^ { - d } u \\| _ { L ^ 2 } & = \\biggl \\| \\int _ 0 ^ 1 \\int _ 0 ^ { s _ 2 } \\cdots \\int _ 0 ^ { s _ d } u ^ { ( d ) } ( t x ) \\ , d t \\ , d t _ d \\cdots d t _ 2 \\ , d x \\biggr \\| _ { L ^ 2 } \\\\ & \\leq \\int _ 0 ^ 1 \\int _ 0 ^ { s _ 2 } \\cdots \\int _ 0 ^ { s _ d } \\| u ^ { ( d ) } ( t \\cdot ) \\| _ { L ^ 2 } \\ , d t \\ , d t _ d \\cdots d t _ 2 \\ , d x \\\\ & = \\frac { 2 ^ { 2 d } d ! } { ( 2 d ) ! } \\| u ^ { ( d ) } \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} \\langle \\psi , X \\cdot \\xi \\rangle _ { S M } = \\langle X \\cdot \\psi , \\xi \\rangle _ { S M } \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} \\nabla \\nabla f + ( a f + a _ { 0 } ) g = 0 , \\nabla _ { B } \\nabla _ { B } h + a h g _ { B } = 0 . \\end{align*}"}
-{"id": "7255.png", "formula": "\\begin{align*} \\left | \\begin{array} { c c c c } f _ 1 ( x _ 1 ) & f _ 2 ( x _ 1 ) & \\cdots & f _ n ( x _ 1 ) \\\\ f _ 1 ( x _ 2 ) & f _ 2 ( x _ 2 ) & \\cdots & f _ n ( x _ 2 ) \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ f _ 1 ( x _ n ) & f _ 2 ( x _ n ) & \\cdots & f _ n ( x _ n ) \\\\ \\end{array} \\right | \\not = 0 . \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{gather*} ( e _ { \\alpha } \\otimes t ^ k ) v _ { \\lambda } = 0 , \\ \\alpha \\in \\Delta _ + , ~ k \\geq 0 ; \\\\ ( f _ { - \\alpha } \\otimes 1 ) ^ { \\langle \\alpha ^ \\vee , \\lambda \\rangle + 1 } v _ { \\lambda } = 0 , \\ \\alpha \\in \\Delta _ + . \\end{gather*}"}
-{"id": "3380.png", "formula": "\\begin{align*} \\left | \\exp ( c _ k ( z - b _ k ' ) ) - 1 \\right | \\geq \\frac 3 4 \\left | c _ k ( z - b _ k ' ) \\right | > \\left | z - b _ k ' \\right | = \\varepsilon _ k . \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} \\partial ^ * _ B \\phi = \\partial _ T ^ * \\phi + H ^ { 1 , 0 } \\lrcorner \\ , \\phi , \\bar \\partial _ B ^ * \\phi = \\bar \\partial _ T ^ * \\phi + H ^ { 0 , 1 } \\lrcorner \\ , \\phi , \\end{align*}"}
-{"id": "6153.png", "formula": "\\begin{align*} T ^ { m i n } _ W = \\inf \\{ t > 0 : \\ W ( t ) > S _ { m i n } \\} & & T ^ { m a x } _ W = \\inf \\{ t > 0 : \\ W ( t ) > S _ { m a x } \\} \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{align*} \\eta _ 1 & = \\frac { \\alpha _ 0 } { r } - \\frac { 3 T + 1 } { 2 r } + \\frac { t + x } { 2 r } , \\\\ \\eta _ 2 & = \\frac { \\beta _ 0 } { r } - \\frac { 3 T + 1 } { 2 r } + \\frac { 3 t + x } { 2 r } , \\eta _ 3 = \\eta _ 2 - \\frac { 1 } { r } , \\\\ \\eta _ 4 & = \\frac { \\alpha _ 0 - \\beta _ 0 } { r } - \\frac { x } { r } , \\eta _ 5 = \\eta _ 4 + \\frac { 1 } { r } . \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} S _ { I J } ( x ) = \\begin{cases} \\sum _ { k = 1 } ^ n \\varepsilon _ I ( k ) x _ k ^ 2 & , \\\\ { \\operatorname { s g n } } ( I , I ' ) x _ p x _ q & , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "5643.png", "formula": "\\begin{align*} s ( n , 2 ) \\ , = \\ , 6 ^ { 2 ^ { n - 2 } - 1 } \\cdot 5 7 6 ^ { ( 2 ^ { n - 2 } - 1 ) ( 2 ^ { n - 3 } - 1 ) / 3 } \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} C ^ { 1 0 } _ 2 : \\left \\{ \\begin{aligned} a _ { 2 , 6 } - a _ { 3 , 8 } & = 0 \\\\ ( 3 a _ { 1 , 5 } - 4 a _ { 2 , 7 } + 3 a _ { 3 , 9 } ) a _ { 3 , 8 } - ( 2 a _ { 1 , 6 } + a _ { 2 , 8 } ) a _ { 4 , 9 } & = 0 \\\\ a _ { 1 , 4 } - a _ { 3 , 8 } & = 0 \\end{aligned} \\right . \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} \\mathcal { L } = - \\int _ 1 ^ { \\xi } \\left ( \\int _ { 1 } ^ { \\hat { X } } \\frac { ( X - \\lambda / 3 ) d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\right ) \\frac { d \\hat { X } } { 2 \\sqrt { \\hat { X } ( \\hat { X } - 1 ) ( \\hat { X } - \\lambda ) } } - \\\\ \\lambda \\int _ 1 ^ { \\xi } \\frac { R _ { \\varphi } d \\hat { X } } { 2 \\sqrt { \\hat { X } ( \\hat { X } - 1 ) ( \\hat { X } - \\lambda ) } } + z \\pi i / \\omega _ 1 - \\pi i . \\end{align*}"}
-{"id": "3391.png", "formula": "\\begin{align*} \\alpha _ k = z _ k + \\varrho _ k b _ k + i \\eta \\varrho _ k R _ k \\to z _ 0 \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} \\mathrm { P } ^ { h i t } _ { m , O M A } = \\mathrm { P } ( f _ 1 ) ( 1 - \\mathrm { P } ^ { O M A } _ { m , 1 } ) , \\end{align*}"}
-{"id": "893.png", "formula": "\\begin{align*} I _ P & = \\bigl ( a _ 0 C _ 1 , a _ 0 C _ 3 , a _ 0 C _ 5 , a _ 0 C _ 7 , \\ : b _ 0 C _ 1 , b _ 0 C _ 3 , b _ 0 C _ 5 , b _ 0 C _ 7 , \\ : a _ 1 C _ 0 , \\ldots \\\\ & a _ 0 a _ 1 , a _ 0 b _ 1 , b _ 0 a _ 1 , b _ 0 b _ 1 , C _ 0 C _ 1 , C _ 0 C _ 2 , \\ldots , C _ 6 C _ 7 , \\\\ & a _ 0 b _ 0 - ( C _ 0 + C _ 2 + C _ 4 + C _ 6 ) , a _ 1 b _ 1 - ( C _ 1 + C _ 3 + C _ 5 + C _ 7 ) \\bigr ) . \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} \\mathcal { K } ^ * f ( \\xi ) = \\frac { 1 } { \\pi } \\operatorname { p . v . } \\int _ \\Sigma \\frac { ( \\xi - \\eta ) \\cdot \\vec n ( \\xi ) } { | \\xi - \\eta | ^ 2 } f ( \\eta ) d S ( \\eta ) , \\xi \\in \\Sigma . \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{align*} D _ \\alpha ( P \\parallel Q ) = \\frac { 1 } { \\alpha - 1 } \\log \\sum _ i p _ i ^ { \\alpha } q _ i ^ { 1 - \\alpha } , \\end{align*}"}
-{"id": "3087.png", "formula": "\\begin{align*} \\begin{cases} u ( 0 , x ) = u _ 0 ( x ) \\\\ \\alpha ( 0 , x ) = \\alpha ( x ) \\end{cases} x \\in \\R ^ d . \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} \\sum _ { k = k _ 0 } ^ \\infty 2 ^ { - 2 k \\alpha } | \\langle \\mathcal { C } _ k f , g \\rangle | \\lesssim \\| f \\| _ { L ^ { 2 } _ t L ^ { r ' } _ x } \\| g \\| _ { L ^ 2 _ t L ^ { r ' } _ x } . \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} N _ { b } ( a , c : \\rho ) = \\biggl \\lbrace x \\in \\mathbb { F } _ { q } : T r ( x ^ { p ^ { \\alpha } + 1 } ) = a \\ \\ T r ( b x ) + c = \\rho \\biggr \\rbrace . \\end{align*}"}
-{"id": "3795.png", "formula": "\\begin{align*} | R _ F | ^ 2 = \\frac 1 4 \\left ( \\frac { s _ * - s _ H } { 4 } \\right ) ^ 2 = ( k - 2 v ) ^ 2 \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} g u ' = c _ 1 = \\begin{pmatrix} 0 \\\\ \\| u ' \\| _ J \\\\ 0 \\\\ 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} W ( \\hat { w } , \\lambda , q ) : = \\{ w \\in W , \\exists \\xi \\in \\hat { \\cal X } ^ * \\textrm { w i t h } \\xi \\textrm { d o m i n a n t r e g u l a r } , \\langle w \\lambda - \\hat { w } q , \\xi \\rangle > 0 \\} \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} \\big \\| { \\widetilde { V } } ^ { ( < ) } \\big \\| _ { \\infty } \\ = \\ ( 1 - { \\varepsilon } ) \\eta ( \\mathfrak { e } ) \\ < \\ \\eta ( \\mathfrak { e } ) - \\frac { 1 } { 4 } \\ , C _ { \\ref { l e m - 4 . 3 } } ( d , \\mathfrak { e } ) \\ , \\eta ( \\mathfrak { e } ) ^ { 2 } r _ { 0 } ^ { - 1 / 2 } , \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{align*} ( a \\vee b ) ( c \\vee d ) = a c \\vee a d \\vee b c \\vee b d \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{align*} \\Sigma ( E ) = \\mbox { I n t } \\left ( \\frac { \\mbox { R e } ( \\lambda _ { m i n } ) } { \\max \\limits _ { j = 1 , . . . , s } ( \\mbox { R e } ( \\lambda _ j ) ) } \\right ) , \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} \\sigma = 3 , \\frac { \\alpha _ 0 } { r } = \\frac { \\pi } { 2 } . \\end{align*}"}
-{"id": "9051.png", "formula": "\\begin{align*} K _ M \\cdot c _ 2 = K _ \\eta ( c _ 1 ^ T c _ 1 + c _ 0 ^ T c _ 2 ) + K _ B c _ 2 - c _ 2 ( K _ \\eta ^ T c _ 0 - b ) - 2 c _ 1 K _ \\eta ^ T c _ 1 - c _ 0 K _ \\eta ^ T c _ 2 = - c _ 3 + ( c _ 2 ) _ x \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c c } e _ i ^ \\top ( A _ k X _ k B _ k - C _ k X _ { k + 1 } D _ k ) e _ j & = & ( E _ k ) _ { i j } , & k = 1 , \\ldots , r - 1 , \\\\ e _ i ^ \\top ( A _ r X _ r B _ r - C _ r X _ 1 ^ s D _ r ) e _ j & = & ( E _ r ) _ { i j } , \\end{array} \\right . \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{align*} h ^ { \\frac { 1 } { p } } \\sigma _ { t } ( a ) ^ { * } = h ^ { \\frac { 1 } { p } } h ^ { i t } a ^ { * } h ^ { - i t } = h ^ { i t } h ^ { \\frac { 1 } { p } } a ^ { * } h ^ { - i t } . \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} [ W ( t \\ , , x ) \\ , , W ( s \\ , , y ) ] = ( s \\wedge t ) ( x \\wedge y ) s \\ , , t \\in [ 0 \\ , , T ] \\mbox { a n d } x \\ , , y \\in [ 0 \\ , , D ] . \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\lim _ { n \\to \\infty } \\frac { \\psi _ { n , h , x } ( t / x ) } { \\phi _ { n , h , 0 } ( t / x ) } & = \\psi _ { \\rho } ( t ) \\ \\mbox { f o r a l l } \\ t \\in \\mathbb R \\ \\ \\mbox { w i t h p r o b a b i l i t y } ~ 1 . \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{align*} \\dot \\Phi _ \\infty : = \\left . \\frac { \\partial } { \\partial s } \\right | _ { s = 0 } \\Phi _ \\infty ( s ) & = \\begin{pmatrix} 0 & | q | _ k ^ { - 1 / 2 } \\bigl ( - \\frac { 1 } { 2 } \\Re ( \\dot q / q ) q + \\dot q \\bigr ) \\\\ \\tfrac { 1 } { 2 } | q | _ k ^ { 1 / 2 } \\Re ( \\dot q / q ) & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} \\partial _ s \\Psi ( s , x , \\lambda ) & = - \\frac { \\xi '' ( s ) } { 2 } \\bigl ( \\partial _ { x x } \\Psi ( s , x , \\lambda ) + A ( s ) \\bigl ( \\partial _ x \\Psi ( s , x , \\lambda ) \\bigr ) ^ 2 \\bigr ) \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{align*} ( A _ i ) _ { g \\cdot x _ 0 , h \\cdot x _ 0 } = \\begin{cases} 1 , & h ^ { - 1 } g \\in H a _ i H ; \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{align*} \\Phi _ { \\Q } ( \\Q ( 2 ^ { \\infty } ) ) = \\left \\{ \\begin{array} { l r } \\Z / N _ 1 \\Z , & N _ 1 = 1 , 3 , 5 , 7 , 9 , 1 5 , \\\\ \\Z / 2 \\Z \\times \\Z / 2 N _ 2 \\Z , & N _ 2 = 1 , 2 , 3 , 4 , 5 , 6 , 8 , \\\\ \\Z / 4 \\Z \\times \\Z / 4 N _ 4 \\Z , & N _ 4 = 1 , 2 , 3 , 4 , \\\\ \\Z / 3 \\Z \\times \\Z / 3 \\Z , & \\\\ \\Z / 6 \\Z \\times \\Z / 6 \\Z , & \\\\ \\Z / 8 \\Z \\times \\Z / 8 \\Z . \\end{array} \\right \\} \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 0 & - \\eta _ 2 & - \\eta _ 1 & 0 \\\\ 0 & \\alpha & 0 & \\eta _ 1 \\\\ 0 & 0 & - \\alpha & \\eta _ 2 \\\\ 0 & 0 & 0 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "9258.png", "formula": "\\begin{align*} \\widetilde { v } ( s ) = \\frac { 1 } { 3 } ( s - 1 ) - \\frac { 2 ^ 4 \\pi ^ 2 } { 3 ^ 6 } ( s - 1 ) ^ 3 + { \\rm O } ( ( s - 1 ) ^ 5 ) . \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{align*} g ( x ) = - x [ g _ { a } ( \\Vert x \\Vert ) - g _ { r } ( \\Vert x \\Vert ) ] , \\end{align*}"}
-{"id": "4944.png", "formula": "\\begin{align*} D _ 1 : = \\frac { d \\left ( ( x _ 1 ( t ) - x _ r ( t ) ) ^ T \\Sigma _ 1 ( x _ 1 ( t ) - x _ r ( t ) ) \\right ) } { d t } = 2 ( x _ 1 ( t ) - x _ r ( t ) ) ^ T \\Sigma _ 1 \\frac { d ( x _ 1 ( t ) - x _ r ( t ) ) } { d t } . \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} A _ m \\left ( H _ n - H _ { n - 1 } \\gamma ^ { - 1 } \\right ) = \\left ( H _ n \\gamma ^ { - n } - H _ { n - 1 } q ^ r \\gamma ^ { - n + 1 } \\right ) A _ m \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} - D ( \\rho _ { \\alpha } , \\rho _ { \\alpha } ) + D ( \\rho _ { 0 } , \\rho _ { 0 } ) + D ( \\tilde { \\rho } _ { \\infty } , \\tilde { \\rho } _ { \\infty } ) = - 2 D ( \\rho _ { 0 } , \\tilde { \\rho } _ { \\infty } ) \\leq - \\frac { \\alpha ( 1 - \\alpha ) } { 5 R } < 0 . \\end{align*}"}
-{"id": "7655.png", "formula": "\\begin{align*} \\mathrm { P } ( f _ l ) = \\frac { \\frac { 1 } { l ^ { \\gamma } } } { \\sum ^ { F } _ { p = 1 } \\frac { 1 } { p ^ \\gamma } } , \\end{align*}"}
-{"id": "4659.png", "formula": "\\begin{align*} X \\# _ t Y : = X ^ { 1 / 2 } ( X ^ { - 1 / 2 } Y X ^ { - 1 / 2 } ) ^ t X ^ { 1 / 2 } \\ . \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} { ^ M W } = \\{ e , s _ 2 , s _ 2 s _ 1 , s _ 2 s _ 1 s _ 2 \\} . \\end{align*}"}
-{"id": "4527.png", "formula": "\\begin{align*} f ( \\mathbf { m } , \\mu , \\boldsymbol { \\theta } ; N ; \\boldsymbol { \\epsilon } ) = \\gamma ( \\mathbf { m } , \\mu , \\boldsymbol { \\theta } ; N ) \\varphi _ { L } ( \\mathbf { m } , N , \\boldsymbol { \\epsilon } ) . \\end{align*}"}
-{"id": "7176.png", "formula": "\\begin{align*} j ^ { r + 1 } _ { _ 0 } \\xi \\cdot j ^ r _ { _ 0 } \\mu : = j ^ r _ { _ 0 } \\left ( ( D \\xi \\cdot \\mu ) \\circ \\xi ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "6987.png", "formula": "\\begin{align*} | N ^ - ( \\sigma _ { \\nu } w ) \\cap \\Phi _ h ^ - | = \\deg ( T ) + | N ^ - ( \\tau ) \\cap \\Phi _ h ^ - [ T ] | . \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{align*} \\epsilon \\leq \\rho \\frac { \\frac { 1 } { 1 2 } 2 ^ k } { 1 + \\frac { 1 } { 1 2 } 2 ^ k \\rho + 2 ^ k } = \\rho \\frac { \\frac { 1 } { 1 2 } } { 2 ^ { - k } + \\frac { 1 } { 1 2 } \\rho + 1 } , \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} f ^ * ( x _ M ^ n \\times 1 ) = ( y _ M ^ n \\times 1 ) + \\xi \\cdot ( 1 \\times \\omega _ { N } ) \\in H ^ n ( M ; \\R ) \\oplus H ^ n ( N ; \\R ) , \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t \\rho + \\chi \\nabla \\cdot ( \\rho \\nabla u ) - \\Delta \\rho = 0 & \\mbox { i n } [ 0 , T ] \\times \\Omega , \\\\ - \\Delta u = \\rho & \\mbox { i n $ \\Omega $ f o r e v e r y $ t \\in [ 0 , T ] $ } , \\\\ u = 0 & \\mbox { o n $ \\partial \\Omega $ f o r e v e r y $ t \\in [ 0 , T ] $ } , \\\\ ( \\rho \\nabla u - \\nabla \\rho ) \\cdot \\mathbf { n } = 0 & \\mbox { o n $ [ 0 , T ] \\times \\partial \\Omega $ } , \\\\ \\rho ( 0 , \\cdot ) = \\rho _ 0 & \\end{cases} \\end{align*}"}
-{"id": "5147.png", "formula": "\\begin{align*} \\textrm { f o r a . \\ e . \\ $ t \\in ( - 9 / 4 , 0 ) $ , e i t h e r $ v ( t , x ) = 0 $ o r $ v ( t , x ) \\geq 1 / 2 $ i n $ B _ 1 $ } . \\end{align*}"}
-{"id": "473.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & 0 & 5 & 2 \\\\ 7 & 6 & 9 & 8 \\\\ 0 & 0 & 8 & 9 \\\\ 2 & 6 & 1 & 8 \\end{pmatrix} , \\begin{pmatrix} 9 & 5 & 2 & 5 \\\\ 1 & 3 & 3 & 2 \\\\ 8 & 5 & 3 & 5 \\\\ 4 & 8 & 1 & 9 \\end{pmatrix} , \\end{align*}"}
-{"id": "5083.png", "formula": "\\begin{align*} J = J ( \\{ a _ n \\} , \\{ b _ n \\} ) = \\begin{bmatrix} b _ 1 & a _ 1 & 0 & \\ldots \\\\ a _ 1 & b _ 2 & a _ 2 & \\ddots \\\\ 0 & a _ 2 & b _ 3 & \\ddots \\\\ \\vdots & \\ddots & \\ddots & \\ddots \\end{bmatrix} , \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} \\Delta ( u _ k ) = u _ k \\otimes 1 + \\sum _ { i = 0 } ^ k { k \\choose i } _ { \\ ! \\ ! q } \\prod _ { j = k - i } ^ { k - 1 } ( 1 - q ^ j r ) x _ 1 ^ i \\otimes u _ { k - i } \\end{align*}"}
-{"id": "239.png", "formula": "\\begin{align*} Q ( i , j , k , \\alpha ) : = \\underline { d p } _ i \\underline \\wedge \\omega _ { j k } ^ \\alpha ( i , j , k \\in [ n ] \\mathrm { \\ a l l \\ d i f f e r e n t , } j < k , \\alpha \\geq 0 ) , \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{align*} \\lim _ { \\hbar \\rightarrow 0 ^ + } \\int _ M \\Pi _ { z _ 0 } ^ { ( k ) } ( \\hbar ) ( \\psi _ 1 ) \\wedge \\psi _ 2 = \\int _ M \\Pi _ { z _ 0 } ^ { ( k ) } ( \\psi _ 1 ) \\wedge \\psi _ 2 . \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{align*} \\begin{pmatrix} n & 1 & 1 & \\cdots & 1 \\\\ 1 - n & 0 & - 1 & \\cdots & - 1 \\\\ 1 - n & - 1 & 0 & \\cdots & - 1 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 1 - n & - 1 & - 1 & \\cdots & 0 \\\\ \\end{pmatrix} \\ . \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{align*} \\left ( 1 - 2 x - x v ( 1 - x ) + \\frac { x v ( 1 - x - x v ) } { 1 - v } \\right ) H ( x , v ) = x ( 1 - 2 x ) F _ T ( x ) + \\frac { x v ( 1 - 2 x ) } { 1 - v } H ( x , 1 ) . \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} \\big ( \\phi _ l ^ 2 P + k P \\big ) _ X = x ^ { q ^ 2 } + x + \\frac { f _ { k - 1 } f _ { k + 1 } } { f _ k ^ 2 } + \\lambda ^ 2 + \\lambda , \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} { ( f ) } ^ R ( t , x ) & = f ( t , x \\frac { \\abs { x } \\wedge R } { \\abs { x } } ) , \\\\ ( f ) _ \\varepsilon ( t , x ) & = \\int _ { - \\infty } ^ { + \\infty } f ( s , x ) \\phi _ \\varepsilon ( t - s ) d s \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} | b ( n ) - g ( n ) | & = \\left | \\left ( n ^ 2 - z _ 1 n + b ( z _ 1 ) \\right ) - \\left ( n ^ 2 - r ( n ) \\cdot n + n \\right ) \\right | \\\\ & \\le n | z _ 1 - r ( n ) | + | b ( z _ 1 ) - n | \\\\ & < 1 . 9 8 5 n + ( z _ 1 - 3 ) \\\\ & < 1 . 9 8 5 n + \\sqrt { 2 n } \\\\ & < 2 n , \\end{align*}"}
-{"id": "2704.png", "formula": "\\begin{align*} \\frac { B z _ 1 } { z _ 2 } = \\frac { z _ 1 } { B ^ { - 1 } z _ 2 } . \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{align*} p _ L ( x , \\xi ) \\sim \\sum _ { \\alpha \\in \\N ^ n } \\left . \\frac 1 { \\alpha ! } \\partial _ \\xi ^ \\alpha D _ y ^ \\alpha p ( x , y , \\xi ) \\right | _ { y = x } . \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{gather*} \\eta = r ( d - 1 ) \\theta _ 0 , \\\\ d > 2 \\eta + ( 2 r - 1 ) \\log _ P ( \\widetilde { C } ) , \\\\ \\frac { K } { r ( d - 1 ) } - ( r + 1 ) > \\delta \\eta ^ { - 1 } . \\end{gather*}"}
-{"id": "118.png", "formula": "\\begin{align*} H _ \\infty = ( q \\bar q ) ^ { - 1 / 4 } \\oplus ( q \\bar q ) ^ { 1 / 4 } \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} \\rho _ { G , U } ( t ) = \\frac { \\theta } { 2 } \\ln \\left ( 1 + \\frac { 2 \\kappa } { \\theta } r _ G ( t ) \\right ) , & & \\varphi _ { G , U } ( t ) = \\frac { v _ G ( t ) } { \\sigma \\sqrt { \\kappa } } \\sqrt { 1 + \\frac { 2 \\kappa } { \\theta } r _ G ( t ) } \\end{align*}"}
-{"id": "8985.png", "formula": "\\begin{gather*} \\lim _ { n \\uparrow \\infty } u ( x _ n ) - f ( x _ n ) = \\sup _ x u ( x ) - f ( x ) , \\\\ \\lim _ { n \\uparrow \\infty } u ( x _ n ) - \\lambda H f ( x _ n ) - h ( x _ n ) \\leq 0 . \\end{gather*}"}
-{"id": "655.png", "formula": "\\begin{align*} Q _ i ( T _ 0 ) + \\epsilon _ 1 ( T _ 1 - T _ 0 ) = L _ 1 + \\epsilon _ 1 ( T _ 1 - T _ 0 ) = ( \\delta + \\epsilon _ 1 ) ( T _ 1 - T _ 0 ) = : L _ 2 , \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{align*} \\bigg | \\bigg \\{ t : \\bigg \\| \\sum _ { k = k _ 0 } ^ \\infty 2 ^ { - k \\alpha } \\mathcal { C } _ k g \\bigg \\| _ { L ^ 2 _ x } > \\lambda \\bigg \\} \\bigg | \\lesssim \\lambda ^ { - q } \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} \\frac { \\partial f ( \\rho , \\Omega ) } { \\partial \\Omega } = - 2 ( b ( \\rho ) - \\Omega ) \\rho ^ 2 . \\end{align*}"}
-{"id": "6730.png", "formula": "\\begin{align*} H ^ { \\pm } \\left ( p , t \\right ) & = \\underset { c \\in \\mathcal { C } } { } \\left \\langle e ^ { - \\left ( T - t \\right ) A } B \\left ( T - t \\right ) c , p \\right \\rangle \\\\ & - \\underset { d \\in \\mathcal { D } } { } \\left \\langle e ^ { - \\left ( T - t \\right ) A } D \\left ( T - t \\right ) d , p \\right \\rangle . \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} y \\tilde { L } _ { \\tilde { \\Delta } ( m ) } ( x ) = y E L _ { \\tilde { \\Delta } ( m ) } ( E x ) = R _ { \\tilde { \\Delta } ( m ) } ( y E ) E x = \\tilde { R } _ { \\tilde { \\Delta } ( m ) } ( y ) x . \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\left \\{ \\sup _ { k } T _ k < t \\right \\} \\cup \\left \\{ \\sup _ { k } T _ k \\ge t \\ , , \\ , \\lim _ { k \\rightarrow \\infty } \\hat p _ k ( t ) = p ( t ) \\right \\} \\right ) = 1 . \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} \\rho _ j ^ n = f _ j ^ { + , n } + f _ j ^ { - , n } , \\rho _ j ^ { n + 1 } = \\rho _ j ^ { n } - \\frac { \\Delta t } { \\Delta x } ( \\bar { J } ^ n _ { j + 1 / 2 } - \\bar { J } ^ n _ { j - 1 / 2 } ) . \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{align*} \\Delta _ B \\phi = \\nabla _ { \\rm t r } ^ * \\nabla _ { \\rm t r } \\phi + A _ { \\kappa _ B ^ \\sharp } ( \\phi ) + F ( \\phi ) , \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} \\Gamma : = g ^ { - 1 } ( K ) , \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} \\lim _ { k } \\sum _ { l = n _ { k } } ^ { n _ { k } + s - 1 } \\Vert U _ { l } u ^ { l } - u ^ { l } \\Vert = 0 \\Longrightarrow \\lim _ { k } d ( u ^ { n _ { k } } , C ) = 0 \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{align*} \\Vert x - P _ { f } x \\Vert = \\frac { f _ { + } ( x ) } { \\Vert g _ { f } ( x ) \\Vert } \\geq \\frac { \\delta } { \\Delta } d ( x , S ( f , 0 ) ) \\end{align*}"}
-{"id": "5132.png", "formula": "\\begin{align*} I _ { t - 1 } ( Y ( t - 1 , n - k + 1 ) ) \\ = \\ I _ { t - 1 } ( Y ( t , n - k ) ) , \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} z ' z & = \\frac { z ' z '' } { ( 1 + ( 1 + \\lambda ^ { 2 } ) ( z ' ) ^ { 2 } ) ^ { 3 / 2 } } \\\\ \\frac { 1 } { 2 } z ^ { 2 } + C _ { 1 } & = \\int \\frac { z ' z '' } { ( 1 + ( 1 + \\lambda ^ { 2 } ) ( z ' ) ^ { 2 } ) ^ { 3 / 2 } } d x . \\\\ . \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} f _ V ( \\zeta ) = \\sum _ { n = - \\infty } ^ { \\infty } f ^ V _ n \\zeta ^ { - n } \\rlap { . } \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} { \\cal D } = \\left ( \\ , \\begin{matrix} ~ \\dd _ { \\zeta } & ~ ~ { \\cal R } & ~ - { \\cal F } ~ \\\\ ~ { \\cal R } & ~ ~ ~ \\dd _ { \\theta } & ~ ~ 0 \\\\ ~ { \\cal F } & ~ ~ 0 & ~ ~ ~ \\dd _ A \\end{matrix} \\ , \\right ) ~ , \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} E = \\{ e \\mid f ( e ) = z \\} \\sqcup \\{ e \\mid f ( e ) \\in \\{ x ^ { \\pm 1 } \\} \\} \\sqcup \\{ e \\mid f ( e ) \\in \\{ y , w \\} \\} , \\end{align*}"}
-{"id": "8591.png", "formula": "\\begin{align*} \\delta ( H [ R ] ) \\geq \\big ( \\alpha - \\kappa - 3 n ^ { - 1 / 3 } \\big ) \\binom { | R | } { 2 } . \\end{align*}"}
-{"id": "5317.png", "formula": "\\begin{align*} f ( x ) \\ , = \\ , \\underbrace { \\displaystyle { \\max _ { 1 \\leq i \\leq I _ 1 } } \\ , \\left ( \\ , x ^ T a ^ i + \\alpha _ i \\ , \\right ) } _ { \\mbox { c o n v e x i n $ x $ } } - \\underbrace { \\displaystyle { \\max _ { 1 \\leq i \\leq I _ 2 } } \\ , \\left ( \\ , x ^ T b ^ i + \\beta _ i \\ , \\right ) } _ { \\mbox { c o n v e x i n $ x $ } } \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} \\partial _ t \\varphi = i _ { M _ t } ( \\varphi ) ~ , \\partial _ t \\psi = i _ { M _ t } ( \\psi ) ~ . \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} \\int _ { R } y ^ \\gamma | \\nabla ( u - \\tilde u ) | ^ 2 & = - \\int _ { R } \\left [ ( u - \\tilde u ) ( y ^ \\gamma ( u - \\tilde u ) _ { x } ) _ { x } + ( u - \\tilde u ) ( y ^ \\gamma ( u - \\tilde u ) _ { y } ) _ { y } \\right ] \\\\ & = - \\int _ { R } ( u - \\tilde u ) [ y ^ \\gamma \\Delta ( u - \\tilde u ) + \\gamma y ^ { \\gamma - 1 } ( u - \\tilde u ) _ { y } ] \\\\ & = I _ 1 + I _ 2 , \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} \\# X & \\equiv \\sum _ { ( x _ 1 , \\dots , x _ n ) \\in T _ q ^ n } \\prod _ { j = 1 } ^ r ( 1 - P _ j ( f _ 1 ( x _ 1 ) , \\dots , f _ n ( x _ n ) ) ^ { ( q - 1 ) q ^ n } ) \\\\ & \\equiv \\sum _ { ( i _ 1 , \\dots , i _ r ) \\in \\{ 0 , 1 \\} ^ r } \\sum _ { ( x _ 1 , \\dots , x _ n ) \\in T _ q ^ n } \\prod _ { j = 1 } ^ r \\left ( - P _ j ( f _ 1 ( x _ 1 ) , \\dots , f _ n ( x _ n ) ) ^ { ( q - 1 ) q ^ n } \\right ) ^ { i _ j } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\pmod { q ^ n } , \\end{align*}"}
-{"id": "7950.png", "formula": "\\begin{align*} 1 = \\lim _ { t \\rightarrow \\infty } g ( \\alpha _ t ) ( t - 1 ) = \\lim _ { t \\rightarrow \\infty } ( t \\alpha _ t ) \\cdot \\left ( \\int _ 0 ^ \\infty \\frac { s ^ 2 + 1 } { s ^ 2 } \\ , d \\rho ( s ) \\right ) , \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{gather*} X _ \\theta ^ i = \\sigma _ i X _ \\theta , X _ { - \\theta } ^ i = \\sigma _ i X _ { - \\theta } , \\end{gather*}"}
-{"id": "3887.png", "formula": "\\begin{align*} & \\lambda _ { 0 } = \\frac { 2 } { \\gamma ( 1 - \\varepsilon ) } \\rho _ { \\varepsilon } ( \\gamma ) , \\\\ & \\lambda _ 1 = \\frac { \\gamma \\mu } { \\varepsilon } \\rho _ { \\varepsilon } ( \\gamma ) . \\end{align*}"}
-{"id": "7000.png", "formula": "\\begin{align*} 3 ^ { a _ 1 } Z _ 1 ^ 3 + 3 ^ { a _ 2 } Z _ 2 ^ 3 + Z _ 3 ^ 3 = 0 \\\\ 3 ^ { a _ 1 } Z _ 1 + 3 ^ { a _ 2 } Z _ 2 + Z _ 3 = 0 \\end{align*}"}
-{"id": "1705.png", "formula": "\\begin{align*} \\beta ^ * _ - = \\max \\bigg \\{ \\frac { 1 } { q } + \\frac { d - 1 } { 2 r } - \\frac { d - 1 } { 2 } , - \\frac { d - 1 } { 4 } \\bigg \\} , \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} n _ 1 & = p ^ { c _ { \\lambda _ 0 } } \\\\ n _ { p ^ i } & = \\frac { 1 } { p ^ i } ( p ^ { c _ { \\lambda _ i } } - p ^ { c _ { \\lambda _ { i - 1 } } } ) . \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} W ( r _ { G _ 2 } ( t ) ) = \\frac { G _ 2 ( t ) - m _ { G _ 2 } ( t ) } { v _ { G _ 2 } ( t ) } . \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{align*} v _ 2 \\left ( \\binom { 4 d - 1 } { 2 d - 1 } - ( - 1 ) ^ d \\binom { 2 d - 1 } { d - 1 } \\right ) = 2 + 2 v _ 2 ( d ) + s _ 2 ( d - 1 ) , \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} { \\ell } ( C ) & \\leq { \\ell } ( p _ { 1 } s _ { 1 } ^ { - 1 } ) { \\ell } ( s _ { 1 } ) { \\ell } ( D _ { 1 } ) { \\ell } ( s _ { 1 } ^ { - 1 } ) { \\ell } ( s _ { 1 } p _ { 2 } s _ { 2 } ^ { - 1 } ) { \\ell } ( s _ { 2 } ) { \\ell } ( D _ { 2 } ) { \\ell } ( s _ { 3 } ^ { - 1 } ) \\cdots \\\\ & { \\ell } ( s _ { t - 1 } p _ { t } s _ { t } ^ { - 1 } ) { \\ell } ( s _ { t } ) { \\ell } ( D _ { t } ) { \\ell } ( p _ { t + 1 } ) \\leq { \\ell } ( p _ { 1 } ) { \\ell } ( D _ { 1 } ) { \\ell } ( p _ { 2 } ) { \\ell } ( D _ { 2 } ) \\cdots { \\ell } ( D _ { t } ) { \\ell } ( p _ { t + 1 } ) = { \\ell } ( p ) \\end{align*}"}
-{"id": "2070.png", "formula": "\\begin{align*} 1 - F _ { n } ( x ) = \\frac { \\hat { A } _ n + B _ { n , x } } { A _ n + B _ n } . \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} X \\# _ { - 1 } Y = X \\frac { 1 } { Y } X \\qquad { \\rm a n d } X \\# _ 2 Y = Y \\frac { 1 } { X } Y \\ . \\end{align*}"}
-{"id": "988.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\Vert U x ^ { k } - x ^ { k } \\Vert = 0 \\Longrightarrow \\lim _ { k \\rightarrow \\infty } d ( x ^ { k } , C ) = 0 \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{align*} w ( x , t ) = S ( t ) w _ 0 ( x ) + \\sum _ { i = 1 } ^ 4 \\int _ 0 ^ t S ( t - s ) N _ i ( x , s ) d s . \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} \\theta _ { i , j } & = \\min _ { r _ { X Y Z } } D ( r _ { X Y Z } \\| p _ { X Y Z } ) \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} \\operatorname { e n d } ( N ) : = \\sup \\{ n \\in \\mathbb { Z } : N _ n \\neq 0 \\} . \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} d q _ t = \\gamma ^ { - 1 } \\sigma d W _ t . \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{align*} & \\mathbf { x } _ 1 = ( x _ 1 , x _ 1 , \\cdots , x _ 1 , x _ 2 , x _ 2 , \\cdots , x _ 2 ) , \\\\ & \\mathbf { x } _ 2 = ( x _ 2 , x _ 2 , \\cdots , x _ 2 , x _ 1 , x _ 1 , \\cdots , x _ 1 ) , \\\\ & \\mathbf { y } _ 1 = ( y _ 1 , y _ 1 , \\cdots , y _ 1 , \\ , y _ 2 , \\ , y _ 2 , \\cdots , \\ , y _ 2 ) , \\\\ & \\mathbf { y } _ 2 = ( \\underbrace { y _ 2 , y _ 2 , \\cdots , y _ 2 } _ { n / 2 } , \\ , \\underbrace { y _ 1 , \\ , y _ 1 , \\cdots , \\ , y _ 1 } _ { n / 2 } ) . \\end{align*}"}
-{"id": "689.png", "formula": "\\begin{align*} \\sum _ { \\substack { i \\leq \\ell \\\\ j \\leq t } } \\bigl ( ( A _ k ) _ { i \\ell } x _ { \\ell t k } ( B _ k ) _ { t j } - ( C _ k ) _ { i \\ell } x _ { \\ell , t , k + 1 } ( D _ k ) _ { t j } \\bigr ) = ( E _ k ) _ { i j } . \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{align*} \\| \\varphi _ { k + 1 } \\| \\leq C Q \\left ( \\sum _ { m = 2 } ^ { k + 1 } { C _ m \\left ( \\frac { b } { c } \\right ) ^ { m - 1 } } \\right ) A _ { k + 1 } . \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} H ( t ) = \\mathcal { D } + \\beta + V ( x , t ) \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{align*} d y _ { t } = - \\nabla f ( y _ t ) \\gamma d t + \\tau _ N \\gamma d B _ t . \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{align*} ( B _ 0 - \\lambda ) T _ \\lambda g = g \\quad g \\in L ^ 1 ( \\mathbb R ) . \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} w _ 0 = u _ 1 w _ 1 u _ 2 w _ 2 \\cdot \\dots \\cdot u _ r w _ r , \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle { \\frac { d } { d t } U = - \\frac { \\partial F ( U ) } { \\partial U } } \\end{array} \\end{align*}"}
-{"id": "6540.png", "formula": "\\begin{align*} { \\bar { p } } _ M ( \\phi ) = \\frac { 1 } { M + 1 } \\sum _ { j = 0 } ^ M P _ j ( \\phi \\neq j ) . \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} \\mathbb { V } _ { \\Gamma } [ \\lambda | D _ { [ 1 : n ] } ] = \\frac { a + n } { \\left ( b + \\sum _ { i = 1 } ^ { n } \\left [ D _ i \\right ] \\right ) ^ 2 } . \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{align*} \\frac { \\partial ^ { k ' } X _ { \\tau } ( s ) } { \\partial s ^ { k ' } } X _ { \\sigma } ( s ) = 0 \\end{align*}"}
-{"id": "6418.png", "formula": "\\begin{align*} \\overline { b } : = \\frac { 1 } { \\abs { \\Omega } } \\int _ { \\Omega } b \\ ; \\d x \\hbox { a n d } b _ s : = b - \\overline { b } , \\hbox { w h e r e } \\int _ { \\Omega } b _ s \\ ; \\d x = 0 . \\end{align*}"}
-{"id": "960.png", "formula": "\\begin{align*} N [ \\mathfrak { e } , { \\widetilde { V } } ] \\ = \\ \\# \\mathop { \\mathrm { s u p p } } { \\widetilde { V } } \\ = \\ \\# \\mathop { \\mathrm { s u p p } } V . \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} X ( z ) = y - Q _ y ( z , z ) , \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{align*} x _ { t } = x _ 0 - \\sum _ { t ( l ) < t } \\nabla f ( x _ { t ( l ) } ) \\Gamma + \\sum _ { t ( l ) < t } \\epsilon ( l ) \\Gamma . \\end{align*}"}
-{"id": "7532.png", "formula": "\\begin{align*} ( ( \\tilde \\gamma + 2 \\gamma I ) ^ { - 1 } ) ^ i _ j = \\delta ^ { i k } ( \\tilde \\gamma + 2 \\gamma I ) ^ { - 1 } _ { k j } = \\left ( \\begin{array} { c c c } 3 \\gamma / ( 9 \\gamma ^ 2 + B _ 0 ^ 2 ) & - B _ 0 / ( 9 \\gamma ^ 2 + B _ 0 ^ 2 ) & 0 \\\\ B _ 0 / ( 9 \\gamma ^ 2 + B _ 0 ^ 2 ) & 3 \\gamma / ( 9 \\gamma ^ 2 + B _ 0 ^ 2 ) & 0 \\\\ 0 & 0 & 1 / ( 3 \\gamma ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "4559.png", "formula": "\\begin{align*} v \\ , \\lrcorner = ( \\epsilon ( v ^ \\flat ) ) ^ * \\end{align*}"}
-{"id": "6254.png", "formula": "\\begin{align*} f ( \\nu , \\mathfrak { a } ; q ) = \\sum _ { i = 1 } ^ n ( - 1 ) ^ { \\nu _ i } q ^ { ( - 1 ) ^ { \\nu _ i } \\mathfrak { a } _ i } . \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} M _ { \\kappa ^ { - 1 } } ( x , y ) : = \\int _ 0 ^ 1 \\mathcal { D } ( \\kappa ^ { - 1 } ) ( x + t ( y - x ) ) \\ , d t , \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} \\mathcal { A } _ p u ( t ) = \\big [ A ^ { \\frac { 1 } { 2 } } _ { p ' } \\big ] ^ * \\Phi ^ { - 1 } A _ p \\int _ 0 ^ t e ^ { - ( t - s ) A _ p } \\Phi \\big [ A _ { p ^ { \\prime } } ^ { - \\frac { 1 } { 2 } } \\big ] ^ * f ( s ) \\ ; d s . \\end{align*}"}
-{"id": "6723.png", "formula": "\\begin{align*} \\dot { x } \\left ( t \\right ) = A x \\left ( t \\right ) + B \\left ( t \\right ) u \\left ( t \\right ) , \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} \\begin{cases} \\operatorname { d i v } ( y ^ a \\nabla ( U _ r - U _ 0 ) ) = y ^ a ( U _ r - U _ 0 ) _ t \\\\ - \\lim _ { y \\to 0 } y ^ a ( U _ r - U _ 0 ) _ y = r ^ { 2 s } V ( r x , r ^ 2 t ) U _ r . \\end{cases} \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} \\mathbf { Q } \\left \\{ \\tau _ n \\uparrow \\tau = T \\right \\} = 1 , \\end{align*}"}
-{"id": "1555.png", "formula": "\\begin{align*} \\| m _ { x = V } ^ { N , j } \\| _ { B L } ^ \\ast = m _ { x = V } ^ { N , j } ( [ 0 , T ] ) \\leq m _ 0 ^ j ( ( - V _ { m a x } T , 0 ] ) \\leq m ^ j _ 0 ( e _ j ) . \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{align*} r _ k ( \\alpha ) = \\sum _ Q \\left ( \\prod _ { i = 1 } ^ { | Q | } \\phi _ k ( Q _ i ) \\sum _ { \\{ P ' _ 1 , \\cdots , P ' _ r \\} } \\prod _ { i = 1 } ^ r \\phi _ k ( P ' _ i ) \\right ) \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} ( c , d ) * ( a , b ) = ( a c + b d ^ { q ^ s } u , a d + b c ^ { q ^ s } + b d ^ { q ^ s } v ) , \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} d _ { A ( t ) } v _ \\tau ( t ) = d _ { A ( t ) } w ( t ) + [ B ( t ) , \\eta ( t ) ] . \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{align*} \\begin{aligned} D ^ { ( \\alpha ) } ( P \\parallel Q ) = \\frac { \\sum _ i p _ i ^ { \\alpha } q _ i ^ { 1 - \\alpha } - \\alpha p _ i + ( \\alpha - 1 ) q _ i } { \\alpha ( \\alpha - 1 ) } , \\end{aligned} \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} \\int _ { \\Sigma } \\left ( | A | ^ 2 + | \\nabla \\log w | ^ 2 \\right ) \\phi ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma & = \\int _ { \\Sigma } ( \\dfrac 1 2 - \\mu _ 1 ) \\phi ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma - \\int _ { \\Sigma } \\phi ^ 2 ( \\mathcal { L } \\log w ) e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma \\\\ & = \\int _ { \\Sigma } ( \\dfrac 1 2 - \\mu _ 1 ) \\phi ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma + \\int _ { \\Sigma } \\left < \\nabla \\phi ^ 2 , \\nabla \\log w \\right > e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} { \\rm C o e f f } [ f _ { \\delta } , m ] = { \\rm C o e f f } [ E _ { \\delta } , m ] = \\lim _ { q \\rightarrow t ^ { - m } } ( 1 - q t ^ { m } ) \\left ( \\sum _ { \\nu \\in \\mathcal { E } _ { \\delta } } c _ { \\nu } ( q , t ) E _ { \\nu } ( z ; q , t ) \\right ) \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{align*} \\lambda _ i = \\frac { ( T _ 1 ) _ { i i } ( T _ 2 ) _ { i i } \\cdots ( T _ r ) _ { i i } } { ( R _ 1 ) _ { i i } ( R _ 2 ) _ { i i } \\cdots ( R _ r ) _ { i i } } , i = 1 , 2 , \\dots , n . \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{align*} \\lambda U ( x ) = \\int _ x ^ \\infty - f '' ( t ) \\overline { f ( t ) } + q ( t ) | f ( t ) | ^ 2 \\ , \\mathrm d t = f ' ( x ) \\overline { f ( x ) } + V ( x ) \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} \\partial _ x f ( x , \\lambda ) & = m ( x , \\lambda ) , \\\\ \\partial _ \\lambda f ( x , \\lambda ) & = m ( x , \\lambda ) ^ 2 \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} \\overline { t } x = \\overline { p } = \\overline { t } \\ , \\overline { p } _ n , \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} \\mathcal { E } ( F , E ) = \\lbrace g : \\mathbb { R } ^ 2 \\rightarrow \\mathbb { R } ^ 2 : g g ( F ) \\subseteq E \\rbrace . \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} \\{ ( \\vec u , \\vec p ) \\in \\mathbb C ^ { r + s } \\ , | \\ , | q _ i | \\max _ { \\alpha } | u _ \\alpha | < \\min _ \\alpha | u _ \\alpha | , \\forall i = 1 , 2 \\} , \\end{align*}"}
-{"id": "6564.png", "formula": "\\begin{align*} \\mu \\left ( \\{ v \\in T ^ 1 S : \\exists \\ , t \\in [ 0 , T _ k ] \\textrm { w i t h } \\delta ( \\phi _ t ( v ) ) \\leqslant T _ k ^ { - \\xi } \\} \\right ) & = \\mu ( A _ k ) \\\\ & = O \\left ( \\frac { 1 } { T _ k ^ { \\xi r - 1 } } \\right ) . \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} p _ d = 1 - \\exp ( { - \\pi \\lambda _ { t u } p _ c ^ D L _ d ^ 2 } ) . \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} G _ 2 = \\lbrace g : \\mathbb { R } ^ 2 \\rightarrow \\mathbb { R } ^ 2 : g ( z ) = A \\cdot z + t , A \\in G L ( \\mathbb { R } ^ 2 ) , t \\in \\mathbb { R } ^ 2 \\rbrace . \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} F ( \\lambda ) = \\left ( \\sum _ { s \\in S _ \\mu \\setminus \\lambda } | S _ \\mu ( s - 1 ) \\setminus \\lambda | \\right ) - \\left ( \\sum _ { t \\in T _ \\mu \\setminus \\lambda } | T _ \\mu ( t + 1 ) \\setminus \\lambda | \\right ) + \\frac { | \\lambda | } { 2 } \\left ( | S _ \\mu \\setminus \\lambda | - | T _ \\mu \\setminus \\lambda | \\right ) . \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} \\mathbb { E } : = \\left ( \\mathcal { O } _ X ( - C _ 1 - \\dots - C _ n ) , \\mathcal { O } _ { X } ( - C _ 2 - \\dots - C _ { n } ) , \\dots , \\mathcal { O } _ X ( - C _ n ) , \\mathcal { O } _ X \\right ) \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} \\omega _ t ( x _ 0 ) = \\mathbb E _ { x _ 0 } ( M _ t V _ t \\omega _ 0 ( x _ t ) ) , \\end{align*}"}
-{"id": "9063.png", "formula": "\\begin{align*} \\begin{pmatrix} \\ast & 0 & 0 \\\\ \\ast & 0 & 0 \\\\ 0 & \\ast & \\ast \\end{pmatrix} . \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq j \\leq m \\atop j \\neq i } D _ i D _ j ^ { - 1 } = \\lambda _ i ( G \\setminus \\{ 1 \\} ) \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} \\left ( x \\frac { d } { d x } \\right ) ^ n = x \\frac { d } { d x } \\left ( x \\frac { d } { d x } \\right ) ^ { n - 1 } . \\end{align*}"}
-{"id": "4279.png", "formula": "\\begin{align*} \\vec { A } = ( A _ 1 , A _ 2 , \\dots , A _ N ) = ( \\vec { J } ) = ( B _ 0 , B _ 1 , \\dots , B _ b , C _ 1 , \\dots , C _ c ) , \\end{align*}"}
-{"id": "5371.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 e ^ { r u } u ( 1 - u ^ 2 ) ^ { \\frac { p - 3 } { 2 } } d u & = 2 ^ \\frac { p - 2 } { 2 } \\Gamma ( \\frac { p - 1 } { 2 } ) \\sqrt { \\pi } \\frac { d } { d r } \\Big ( r ^ { - \\frac { p - 2 } { 2 } } I _ \\frac { p - 2 } { 2 } ( r ) \\Big ) \\\\ & = \\Gamma ( \\frac { p - 1 } { 2 } ) \\sqrt { \\pi } \\big ( \\frac { 2 } { r } \\big ) ^ \\frac { p - 2 } { 2 } I _ { \\frac { p } { 2 } } ( r ) . \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} \\mathcal { A } _ k = \\frac { \\lambda _ k P _ k ^ { 2 / \\alpha } } { \\sum _ { t \\in \\mathcal { K } } \\lambda _ t P _ t ^ { 2 / \\alpha } } . \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} \\omega = \\frac { i } { 2 \\pi } \\partial \\overline { \\partial } \\log | \\mathbf { z } | ^ 2 = \\frac { i } { 2 \\pi } \\left ( \\frac { \\sum _ { j = 0 } ^ n d z _ j \\wedge d \\overline { z } _ j } { | \\mathbf { z } | ^ 2 } - \\frac { \\sum _ { j , k = 0 } ^ n \\overline { z } _ j z _ k d z _ j \\wedge d \\overline { z } _ k } { | \\mathbf { z } | ^ 4 } \\right ) \\end{align*}"}
-{"id": "3073.png", "formula": "\\begin{align*} \\left ( ( \\underline { q } , \\ , \\underline { q } + \\varepsilon _ { \\ast } ) \\times B _ { \\ast } \\right ) \\cap \\mathcal { C } _ { 1 } | _ { q \\in ( \\underline { q } , \\ , \\underline { q } + \\varepsilon _ { \\ast } ) } = \\emptyset . \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{align*} \\frac { \\partial \\psi } { \\partial x } ( t , x ) = \\mathcal { I } _ 1 ( t , x ) - \\mathcal { I } _ 2 ( t , x ) + \\mathcal { I } _ 3 ( t , x ) , \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} E _ c ( x _ 0 ) : = \\min _ { { u \\in L ^ 2 ( ] - \\infty , 0 ] ) \\atop x ( - \\infty , x _ 0 , u ) = 0 } } \\int _ { - \\infty } ^ 0 \\left \\| u ( t ) \\right \\| _ 2 ^ 2 d t . \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} \\mu ( ( K _ 1 \\cap K ) \\times \\mathbb { T } ) = \\operatorname { R e } \\int _ { ( K _ 1 \\cap K ) \\times \\mathbb { T } } \\overline { \\gamma _ 0 } \\gamma d \\mu = \\int _ { ( K _ 1 \\cap K ) \\times \\mathbb { T } } \\operatorname { R e } \\overline { \\gamma _ 0 } \\gamma d \\mu , \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} \\Theta \\left ( z , - \\tfrac { 1 } { \\tau } \\right ) = ( - i \\tau ) ^ { 1 / 2 } e ^ { \\pi i z ^ 2 \\tau } \\Theta ( \\tau z , \\tau ) . \\end{align*}"}
-{"id": "3023.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\nabla \\phi | ^ { 2 } - \\int _ { \\Omega } a ( x ) \\phi ^ { 2 } < \\int _ { \\Omega } | \\nabla \\phi | ^ { 2 } - \\lambda _ { 1 } ( a ) \\int _ { \\Omega } a ( x ) \\phi ^ { 2 } = 0 . \\end{align*}"}
-{"id": "2707.png", "formula": "\\begin{align*} \\Phi ( t ) : = e ^ { t _ 1 X _ 1 + \\cdots + t _ n X _ n } x _ 0 . \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{align*} S ( i , j , k , l , \\alpha , \\beta ) = \\omega _ { i j } ^ \\alpha \\underline \\wedge \\omega _ { k l } ^ \\beta , i < j , k < l , i < k , i , j , k , l & \\mathrm { \\ a l l \\ d i f f e r e n t \\ i n \\ } [ n ] , \\\\ & \\alpha , \\beta \\geq 0 . \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} \\forall n , x _ n \\cdot \\phi _ n = y _ n \\end{align*}"}
-{"id": "8577.png", "formula": "\\begin{align*} N ' _ 2 : = N ' _ 2 ( x , y ) : = \\bigcup _ { z \\in N _ 1 } N _ H ( y , z ) = \\{ w \\in V ( H ) : \\exists z \\in N _ 1 \\ ; \\ ; \\{ y , z , w \\} \\in E ( H ) \\} . \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{align*} X _ { n } = \\rho \\left ( X _ { n - 1 } \\right ) + \\varepsilon _ { n } , \\ n \\in \\Z , \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} D ^ { I I } _ m = \\{ Z \\in M _ m ( \\C ) \\mid Z Z ^ * < I _ m , Z ^ t = - Z \\} , \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} & Q _ 1 = \\ \\sum _ { i = 1 } ^ k s y _ 1 \\dots y _ { i - 1 } [ y _ i , x _ 1 , x _ 2 ] y _ { i + 1 } \\dots y _ k , \\\\ & Q _ 2 = \\ \\sum _ { 1 \\le i < i ' \\le k } s y _ 1 \\dots y _ { i - 1 } \\bigl ( [ y _ i , x _ 1 ] y _ { i + 1 } \\dots y _ { i ' - 1 } [ y _ { i ' } , x _ 2 ] + [ y _ i , x _ 2 ] y _ { i + 1 } \\dots y _ { i ' - 1 } [ y _ { i ' } , x _ 1 ] \\bigr ) y _ { i ' + 1 } \\dots y _ k . \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { k ! } \\sum _ { x _ 1 , \\ldots , x _ k \\in \\Lambda } \\rho _ { + , h } ( x _ 1 , \\ldots , x _ k ) ^ 2 \\le \\beta ^ 2 v n ^ 2 . \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{align*} N ( \\lambda ) \\lesssim \\iint _ { | \\xi | ^ { 2 k } + | x | ^ { 2 \\ell } < \\lambda } d x d \\xi = \\lambda ^ { n ( \\frac { 1 } { 2 k } + \\frac { 1 } { 2 \\ell } ) } \\iint _ { | \\xi ' | ^ { 2 k } + | x ' | ^ { 2 \\ell } < 1 } d x ' d \\xi ' \\lesssim \\lambda ^ { n ( \\frac { 1 } { 2 k } + \\frac { 1 } { 2 \\ell } ) } . \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{align*} Y = ( \\tilde { X } _ 1 , \\ldots , \\tilde { X } _ k , \\tilde { Z } _ 1 , \\ldots , \\tilde { Z } _ m ) . \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} \\begin{pmatrix} 6 & 3 & 5 & 3 \\\\ 1 & 6 & 1 & 5 \\\\ 5 & 3 & 6 & 3 \\\\ 1 & 5 & 1 & 6 \\end{pmatrix} , \\begin{pmatrix} 0 & 6 & 0 & 1 0 \\\\ 0 & 8 & 0 & 7 \\\\ 1 & 0 & 2 & 0 \\\\ 6 & 0 & 3 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} \\sum _ { i \\in d _ 1 ( \\vec { x } , \\vec { y } ) \\bigcup d _ 2 ( \\vec { x } , \\vec { y } ) } M _ i [ H ( \\nu , \\cdot ) ] ( \\mu ) = H ( \\nu , \\mu ^ { * } ) \\sum _ { i \\in d _ 1 ( \\vec { x } , \\vec { y } ) \\bigcup d _ 2 ( \\vec { x } , \\vec { y } ) } M _ i [ t ^ { - \\chi ( \\cdot ) } ] ( \\vec { x } , \\vec { y } ) . \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} \\{ x , y \\} = Q \\frac { \\partial P } { \\partial z } \\ , , \\ \\{ y , z \\} = Q \\frac { \\partial P } { \\partial x } \\ , , \\ \\{ z , x \\} = Q \\frac { \\partial P } { \\partial y } \\ , . \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{align*} J ( x ) = k \\end{align*}"}
-{"id": "5204.png", "formula": "\\begin{align*} \\partial _ { y _ i } G _ 2 ( y ) = - \\frac { y _ { i } } { N \\omega _ { N } } \\int _ { \\partial B ( 0 , 3 ) } \\frac { \\phi ( z ) } { | y - z | ^ { N } } d S ( z ) + \\frac { 2 - \\| y \\| ^ { 2 } } { 2 \\omega _ { N } } \\int _ { \\partial B ( 0 , 3 ) } \\frac { ( y _ i - z _ i ) \\phi ( z ) } { | y - z | ^ { N + 2 } } d S ( z ) , \\forall y \\in B ( 0 , 3 ) . \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} & \\left ( U ^ T \\left ( \\frac { 3 n + 2 } { 6 } - \\int _ 0 ^ \\infty T r [ \\gamma e ^ { - 2 y \\gamma } ] e ^ { - y \\gamma } d y \\right ) U \\right ) ^ { i j } = \\left ( \\frac { 1 } { 3 } + \\frac { \\lambda _ i } { 2 } \\sum _ l \\frac { 1 } { 2 \\lambda _ l + \\lambda _ i } \\right ) \\delta ^ { i j } . \\end{align*}"}
-{"id": "3787.png", "formula": "\\begin{align*} R ^ \\nabla = R ^ g + \\alpha - \\beta \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} G _ N ( \\sigma ) = \\frac { \\exp { H _ N ( \\sigma ) } } { \\sum _ { \\sigma ' } \\exp H _ N ( \\sigma ' ) } . \\end{align*}"}
-{"id": "6445.png", "formula": "\\begin{align*} k ^ { q } _ { j + 1 } ( T ) \\leq k ^ { q } _ { 0 } ( T ) + 3 C \\tilde { C } _ { T } K _ { 1 } ^ 2 K _ { 2 } < \\frac { K _ { 1 } } { 2 } + \\frac { K _ { 1 } } { 2 } = K _ { 1 } . \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} C : = \\left \\{ x \\in E : f ( x , x ) > F ( x , x ) \\right \\} . \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} \\begin{aligned} Z _ n ^ { a , s } & = Z _ n ^ { a , s } ( \\tilde \\tau _ 1 > n ) + \\sum _ { k = 1 } ^ n Z _ n ^ { a , s } ( \\tilde \\tau _ 1 = k ) \\\\ & \\leq e ^ { n ( F ( 0 , h ) + o ( 1 ) ) } + \\sum _ { k = 1 } ^ n e ^ { k F ( 0 , h ) } Z ^ a _ { n - k } . \\end{aligned} \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{align*} ( T _ \\lambda g ) ( x ) = \\frac { 1 } { 2 \\alpha \\sqrt { \\lambda } } \\left ( u ( x ) \\int _ { - \\infty } ^ { x } \\ ! v ( y ) \\operatorname { s g n } ( y ) g ( y ) \\ , \\mathrm d y + v ( x ) \\int _ { x } ^ { \\infty } \\ ! u ( y ) \\operatorname { s g n } ( y ) g ( y ) \\ , \\mathrm d y \\right ) . \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} & b ( 0 ) = 0 ; \\\\ & b ( x ) : = \\P ( X _ { n + 1 } = x + 1 \\| X _ n = x ) - \\P ( X _ { n + 1 } = x - 1 \\| X _ n = x ) , \\ \\ x > 0 ; \\\\ & b ( x ) : = \\P ( X _ { n + 1 } = x - 1 \\| X _ n = x ) - \\P ( X _ { n + 1 } = x + 1 \\| X _ n = x ) , \\ \\ x < 0 ; \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} G ( x ) = O ( \\log ^ 2 x ) , \\end{align*}"}
-{"id": "7307.png", "formula": "\\begin{align*} \\mathbf { r } _ { } = \\mathbf { P x } + \\hat { \\mathbf { F } } \\mathbf { \\Omega } \\mathbf { z } + \\hat { \\mathbf { F } } \\mathbf { n } _ q . \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} \\partial _ n d = 0 ( 0 , T ) \\times \\partial \\Omega , \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } \\int _ { - \\infty } ^ { + \\infty } \\vert F ^ * ( \\alpha + i y ) - F ( \\alpha + t + i y ) \\vert d t = 0 . \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} m _ f ( X ) : = \\min _ { \\tilde { X } \\in \\mathcal { P } } \\sum _ { k = 1 } ^ n f ( \\tilde { x } ^ { ( 1 ) } _ k , \\dots , \\tilde { x } ^ { ( d ) } _ k ) . \\end{align*}"}
-{"id": "1273.png", "formula": "\\begin{align*} c h ^ { q ^ { s + l } } \\theta ^ { q ^ n } = d ^ { q ^ n } g ^ { q ^ { s + l } } \\theta . \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} \\nabla _ T ^ * \\nabla _ T \\phi = \\bar \\nabla _ T ^ * \\bar \\nabla _ T \\phi + \\nabla _ { H ^ { 0 , 1 } - H ^ { 1 , 0 } } \\phi + \\sum _ a R ^ Q ( \\bar V _ a , V _ a ) \\phi \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} b _ + ( t , x ) = & \\left ( 0 , - \\frac { 1 } { m } \\gamma ( t , q ) ( p - \\psi ( t , q ) ) \\right ) , \\\\ b _ - ( t , x ) = & \\left ( \\frac { 1 } { m } ( p - \\psi ( t , q ) ) , \\frac { 1 } { m } \\delta ^ { i j } ( p _ i - \\psi _ i ( t , q ) ) \\nabla _ q \\psi _ j ( t , q ) - \\nabla _ q V ( t , q ) + \\tilde F ( t , x ) \\right ) . \\end{align*}"}
-{"id": "2553.png", "formula": "\\begin{align*} v x ^ { N + 1 } y - w x ^ { N + 1 } z = \\sum _ { i = 0 } ^ { N } p _ i ( v x ^ i y - w x ^ i z ) q _ i \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} | \\partial _ 1 f ( \\xi , x _ 2 , \\dots , x _ d ) | = \\left | \\frac { f ( \\frac { i + 1 } { K } , x _ 2 , \\dots , x _ d ) - f ( \\frac { i } { K } , x _ 2 , \\dots , x _ d ) } { \\frac { i + 1 } { K } - \\frac { i } { K } } \\right | \\leq 2 K \\cdot \\| f \\| _ { \\sup } \\leq 4 \\cdot N ^ { \\frac { 1 } { \\sigma } } \\cdot \\| f \\| _ { \\sup } , \\end{align*}"}
-{"id": "7534.png", "formula": "\\begin{align*} ( Y _ 1 ) ^ l = & \\left ( V \\nabla _ q \\beta + \\beta \\tilde F \\right ) _ i ( \\tilde \\gamma ^ { - 1 } ) ^ { i l } + ( 1 + n / 2 ) \\beta ^ { - 1 } \\partial _ { q ^ j } \\beta ( ( \\tilde \\gamma + 2 \\gamma I ) ^ { - 1 } ) ^ j _ { k } \\delta ^ { k l } \\\\ & + ( n + 2 ) \\gamma \\beta ^ { - 1 } \\partial _ { q ^ j } \\beta ( ( \\tilde \\gamma + 2 \\gamma I ) ^ { - 1 } ) ^ j _ { k } ( \\tilde \\gamma ^ { - 1 } ) ^ { k l } \\end{align*}"}
-{"id": "3693.png", "formula": "\\begin{align*} \\mathfrak { M } ( \\theta ) = \\bigcup _ { ( \\vec { a } , q ) \\in \\mathcal { M } ( \\theta ) } \\mathfrak { M } _ { \\vec { a } , q } ( \\theta ) . \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} - \\Delta _ p u _ n - ( 1 - \\varepsilon ) V | u _ n | ^ { p - 2 } u _ n + \\varepsilon | u _ n | ^ { p - 2 } u _ n = f + g _ n , \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} X _ s : = \\mbox { \\rm S p a n } \\left \\{ e _ k , \\ k \\in \\Z , \\ | k | \\leq s \\right \\} \\end{align*}"}
-{"id": "6506.png", "formula": "\\begin{align*} \\frac { 1 } { \\Delta t } ( u _ i ^ { n + 1 } - u _ i ^ n ) + \\frac { u _ i ^ n } { 2 \\Delta x } ( u _ { i + 1 } ^ n - u _ { i - 1 } ^ n ) - \\frac { \\nu } { ( \\Delta x ) ^ 2 } ( u _ { i + 1 } ^ n - 2 u _ i ^ n + u _ { i - 1 } ^ n ) = g ( i \\Delta x , n \\Delta t ) \\end{align*}"}
-{"id": "1274.png", "formula": "\\begin{align*} c g ^ { q ^ l } + d h ^ { q ^ { l + n } } = 0 . \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} e ^ { - i \\varphi } \\mathbf { t } _ l \\begin{bmatrix} 0 \\\\ \\mathbf { M } ^ { - 1 } \\mathbf { f } \\end{bmatrix} = e ^ { - i \\varphi } i r e ^ { i \\varphi } = i r , \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\hat { X } } \\left | \\frac { d X } { \\sqrt { X ( X - \\lambda ) } } \\right | \\leq \\int _ 0 ^ 1 \\frac { d t } { \\sqrt { t ( 1 - t ) } } = \\pi . \\end{align*}"}
-{"id": "7979.png", "formula": "\\begin{align*} \\phi ^ i _ 2 ( X ) : = \\frac { \\widehat { \\phi ^ i _ 2 } ( r _ i x , r _ i ^ 2 t ) } { A _ i r _ i ^ { \\beta } } \\end{align*}"}
-{"id": "5200.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } \\overline { u } = \\overline { u } ( a _ { \\sup } - ( b _ { \\inf } - \\chi \\mu ) \\overline { u } ) \\\\ \\overline { u } ( 0 ) = \\| u _ 0 \\| _ { \\infty } . \\end{cases} \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{align*} \\vect { Y } = \\sqrt { \\rho _ { u l } } \\sum _ { n = 1 } ^ N \\alpha _ n \\vect { s } _ { n } \\vect { h } _ { n } ^ H + \\vect { Z } , \\end{align*}"}
-{"id": "8337.png", "formula": "\\begin{align*} \\xi _ t = U \\vec n + T \\vec t , \\vec t = e ^ { i \\theta } , \\ , \\vec n = i \\vec t = i e ^ { i \\theta } , \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} \\varphi _ \\infty = \\begin{pmatrix} 0 & \\frac 1 2 | q | _ k ^ { - 1 / 2 } q \\\\ \\tfrac { 1 } { 2 } | q | _ k ^ { 1 / 2 } & 0 \\end{pmatrix} \\ , d z = \\frac { 1 } { 2 } \\Phi _ \\infty , \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} z ^ { 2 } = D ( x ) . \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\frac { \\log \\beta ( n ) \\log \\log n } { \\log n } = \\frac { 1 } { 3 } \\log 3 . \\end{align*}"}
-{"id": "2102.png", "formula": "\\begin{align*} \\begin{cases} D _ { A } \\psi _ = 0 \\\\ F _ { A } ^ { + } = \\frac { r } { 2 } ( q ( \\psi , \\psi ) - i \\Omega ) - \\frac { 1 } { 2 } F ^ + _ { A _ { K ^ { - 1 } } } - \\frac { i } { 2 } \\wp _ 4 ^ + \\\\ * d * b - 2 ^ { - \\frac { 1 } { 2 } } r ^ { \\frac { 1 } { 2 } } ( \\eta ^ * \\psi ^ { \\xi } - { \\psi ^ { \\xi } } ^ * \\eta ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "6444.png", "formula": "\\begin{align*} \\begin{aligned} \\delta _ { j } ( T ) < 6 C \\tilde { C } _ { T } [ 2 K _ { 1 } + 6 K _ { 1 } K _ { 2 } + K _ { 1 } ^ 2 ] \\delta _ { j - 1 } ( T ) . \\end{aligned} \\end{align*}"}
-{"id": "2018.png", "formula": "\\begin{align*} r _ i = i ( 1 ) e _ 1 + \\cdots + i ( d ) e _ d , \\end{align*}"}
-{"id": "1659.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ { M - 1 } K _ { i j } = \\sum _ { i = 1 } ^ { M - 1 } \\sum _ { k = 1 } ^ { H - 1 } ( a _ { i k } - a _ { i H } ) b _ { k j } - \\sum _ { i = 1 } ^ { M - 1 } \\sum _ { k = 1 } ^ { H _ 0 - 1 } ( a ^ 0 _ { i k } - a ^ 0 _ { i H _ 0 } ) b ^ 0 _ { k j } + \\sum _ { i = 1 } ^ { M - 1 } ( a _ { i H } - a ^ 0 _ { i H _ 0 } ) , \\\\ L _ j = - \\sum _ { i = 1 } ^ { M - 1 } \\sum _ { k = 1 } ^ { H - 1 } ( a _ { i k } - a _ { i H } ) b _ { k j } + \\sum _ { i = 1 } ^ { M - 1 } \\sum _ { k = 1 } ^ { H _ 0 - 1 } ( a ^ 0 _ { i k } - a ^ 0 _ { i H _ 0 } ) b ^ 0 _ { k j } - \\sum _ { i = 1 } ^ { M - 1 } ( a _ { i H } - a ^ 0 _ { i H _ 0 } ) , \\end{gather*}"}
-{"id": "1184.png", "formula": "\\begin{align*} R _ { h _ i } - | K _ i | ^ 2 + \\frac { n ^ 2 } { u ^ 2 } = R _ { h _ i } - | K _ i | ^ 2 + H _ i ^ 2 = 0 . \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} \\sigma ^ { n } ( a _ 0 , a _ { 1 } , a _ { 2 } , \\ldots ) = ( a _ n , a _ { n + 1 } , a _ { n + 2 } , \\ldots ) < _ { l e x } ( a _ 0 , a _ 1 , a _ 2 , \\ldots ) \\mbox { f o r a l l } ~ n \\geq 1 . \\end{align*}"}
-{"id": "3600.png", "formula": "\\begin{align*} \\frac { d \\Gamma _ { 0 } } { d s } & = [ 1 + y _ { x } ^ { 2 } + z _ { x } ^ { 2 } ] ^ { - 1 / 2 } \\langle 1 , y ' ( x ) , z ' ( x ) \\rangle \\\\ \\frac { d ^ { 2 } \\Gamma _ { 0 } } { d s ^ { 2 } } & = [ 1 + y _ { x } ^ { 2 } + z _ { x } ^ { 2 } ] ^ { - 1 / 2 } ( \\frac { d } { d x } ) ( \\frac { d \\Gamma _ { 0 } } { d s } ) \\\\ \\frac { d ^ { 2 } \\Gamma _ { 0 } } { d s ^ { 2 } } & = [ 1 + y _ { x } ^ { 2 } + z _ { x } ^ { 2 } ] ^ { - 2 } \\{ - ( y '' y ' + z '' z ' ) \\langle 1 , y ' , z ' \\rangle + [ 1 + y _ { x } ^ { 2 } + z _ { x } ^ { 2 } ] \\langle 0 , y '' , z '' \\rangle \\} \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} ( A u ) ( t ) = u _ 0 + \\frac { \\lambda } { \\Gamma ( 2 \\alpha ) } \\int _ 0 ^ { t } ( t - s ) ^ { 2 \\alpha - 1 } \\frac { f ( s , u ( s ) ) } { \\left ( \\int _ { 0 } ^ { t } f ( x , u ) \\ , d x \\right ) ^ { 2 } } d s , t \\in [ 0 , h ] . \\end{align*}"}
-{"id": "4851.png", "formula": "\\begin{align*} a c + ( - 1 ) ^ m a ' c ' = 1 . \\end{align*}"}
-{"id": "4082.png", "formula": "\\begin{align*} \\vect { Y } = \\sqrt { \\rho _ { u l } } \\vect { S } \\vect { X } + \\vect { Z } \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} \\prod _ { \\substack { ( i , j ) \\in \\mathbb { I } } } W ( k ) = \\prod _ { \\substack { ( i , j ) \\in \\mathbb { J } _ S } } W ( k ) \\prod _ { \\substack { ( i , j ) \\in \\mathbb { J } _ T } } W ( k ) , \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{align*} v _ j : = \\lambda ^ { } N _ { j - 1 } ^ { - 1 } M _ { j - 1 } v _ { j - 1 } , j = 2 , \\ldots , r . \\end{align*}"}
-{"id": "1530.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } \\binom { n } { k } \\dbinom { m + k } { p } y ^ { n - k } x f _ { m - p + k } ^ { \\left ( \\alpha - 1 \\right ) } \\left ( x \\right ) = \\sum _ { k = 0 } ^ { m } \\binom { m } { k } \\dbinom { n + k } { p } \\left ( - y \\right ) ^ { m - k } x f _ { n - p + k } ^ { \\left ( \\alpha - 1 \\right ) } \\left ( x + y \\right ) . \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{align*} h \\bigoplus _ { i \\mid a _ { i j } < 0 } ( y _ i / h + \\log | a _ { i j } | ) = h \\bigoplus _ { i \\mid a _ { i j } > 0 } ( y _ i / h + \\log | a _ { i j } | ) , j \\in [ n ] . \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{align*} k _ j = O \\left ( \\frac { n ^ { 1 5 / 2 } \\log n } { j ^ { 1 1 / 2 } } \\right ) . \\end{align*}"}
-{"id": "9112.png", "formula": "\\begin{align*} { \\bf D } _ B = \\omega { \\bf I } _ K + \\tilde { \\bf G } _ { j ^ \\star } , \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} \\tilde { A } ( \\varepsilon ) = \\begin{pmatrix} A & 0 \\\\ 0 & - \\varepsilon \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} & \\exp ^ { - 1 } W _ k ( x ) = \\exp \\left [ \\sum _ { n = 1 } ^ { \\infty } \\frac { ( 1 - q ^ { - n k } ) x ^ n } n \\right ] W _ k ( x ) \\exp ^ { - 1 } \\Rightarrow \\\\ & \\Rightarrow \\ \\exp ^ { - 1 } W _ k ( x ) = \\frac { 1 - \\frac x { q ^ k } } { 1 - x } \\cdot W _ k ( x ) \\exp ^ { - 1 } \\Rightarrow \\\\ & \\Rightarrow \\ \\Phi _ m \\exp ^ { - 1 } W _ k ( x ) \\cdot ( 1 - x ) = \\Phi _ m W _ k ( x ) \\exp ^ { - 1 } \\cdot \\left ( 1 - \\frac x { q ^ k } \\right ) \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{align*} X ( \\zeta ) = \\frac { X _ + } { \\zeta \\ } + X _ 0 + \\zeta X _ - \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} g _ \\C \\equiv \\begin{pmatrix} 1 \\\\ & & - 1 \\\\ & - 1 \\\\ & & & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} p = \\frac { n + n _ T } { r } = 2 + ( r - 1 ) a + b . \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{align*} { } \\frac { \\partial S _ D } { \\partial \\beta } = 0 \\sim E - \\frac { 1 } { \\beta ^ 2 } \\int _ { 0 } ^ { \\infty } \\ln ( 1 + e ^ { - \\alpha } e ^ { - x } ) d x \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} w ( D ) = n ( n - s ) + w ( D [ B ] ) \\ge n ( n - s ) + s . \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} \\begin{pmatrix} k _ 1 \\\\ k _ 2 \\end{pmatrix} _ t = \\begin{pmatrix} - D ^ 3 + k _ 1 D + D k _ 1 & D k _ 2 + k _ 2 D \\\\ D k _ 2 + k _ 2 D & D ^ 3 - D k _ 1 - k _ 1 D \\end{pmatrix} \\begin{pmatrix} r _ 1 \\\\ r _ 2 \\end{pmatrix} = P \\begin{pmatrix} r _ 1 \\\\ r _ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{align*} S ^ { e n v , 0 } _ { s , t } = \\int _ { s } ^ t \\left ( - \\nabla _ { q ^ j } V - \\partial _ { q ^ j } \\beta ^ { - 1 } + \\tilde F _ j \\right ) ( r , q _ r ) \\beta ( r , q _ r ) \\circ d q ^ j _ r . \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb E \\Psi ( \\kappa _ n ) = \\mathbb E \\Psi ( \\kappa ) . \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\max _ { z \\in B _ L } \\xi ( z ) - a _ L = 0 \\end{align*}"}
-{"id": "2981.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 2 ^ k } \\frac { - 1 } { ( 2 i - 1 ) ^ 2 } & \\equiv - \\sum _ { i = 1 } ^ { 2 ^ k } ( 2 i - 1 ) ^ 2 \\\\ & = - \\frac { 1 } { 3 } \\left ( 2 ^ { 3 k + 2 } - 2 ^ { k } \\right ) \\\\ & \\equiv 2 ^ k \\pmod { 2 ^ { k + 1 } } . \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} Q _ i ( T _ 0 ) = Q _ i ( 0 ) = \\delta ( T _ 1 - T _ 0 ) = : L _ 1 , \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} & [ L , \\partial _ B ] = [ L , \\bar \\partial _ B ] = [ \\Lambda , \\partial _ B ^ * ] = [ \\Lambda , \\bar \\partial _ B ^ * ] = 0 , \\\\ & [ L , \\partial _ B ^ * ] = - i \\bar \\partial _ T , \\ [ L , \\bar \\partial _ B ^ * ] = i \\partial _ T , \\ [ \\Lambda , \\partial _ B ] = - i \\bar \\partial _ T ^ * , \\ [ \\Lambda , \\bar \\partial _ B ] = i \\partial _ T ^ * . \\end{align*}"}
-{"id": "8499.png", "formula": "\\begin{align*} F = 3 p _ 3 \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} \\hat { u } \\left ( x _ { n _ i } \\right ) & : = u _ i \\left ( x _ { n _ i } \\right ) + 1 / i , \\\\ \\hat { u } ( y ) & : = 0 , \\ y \\in E \\nexists \\ i \\in \\N : \\ y = x _ { n _ i } . \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { x } { N } \\right ) ^ { N - 1 } = e ^ { - x } \\left [ 1 + o \\left ( \\frac { 1 } { N ^ { \\theta } } \\right ) \\right ] \\ ; \\theta \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "8550.png", "formula": "\\begin{align*} \\iota _ X \\omega _ 0 = d \\mu \\mathrlap { . } \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{align*} \\nabla _ { p } \\varphi \\left ( z , t \\right ) & = \\frac { V \\left ( 0 \\right ) ^ { - 1 } p } { 2 } - z \\\\ & + h \\sum _ { s _ { k } \\in \\mathcal { S } } \\Big ( Q _ { p } \\left ( s _ { k } \\right ) E _ { p } \\left ( s _ { k } \\right ) \\left ( E _ { p } \\left ( s _ { k } \\right ) ^ { \\dagger } p \\right ) \\\\ & - Q _ { e } \\left ( s _ { k } \\right ) E _ { e } \\left ( s _ { k } \\right ) \\left ( E _ { e } \\left ( s _ { k } \\right ) ^ { \\dagger } p \\right ) \\Big ) , \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathbb { Z } } M _ i [ H ( \\nu , \\cdot ) ] \\left ( \\mu \\right ) = \\sum _ { i \\in \\{ z , z + l \\} } M _ i [ H ( \\nu , \\cdot ) ] ( \\mu ) , \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{align*} 0 = \\langle \\triangle u , u \\rangle _ { L ^ 2 } = \\| \\nabla u \\| ^ 2 _ { L ^ 2 } , 0 = \\langle \\triangle \\gamma , \\gamma \\rangle _ { L ^ 2 } = \\| \\nabla \\gamma \\| ^ 2 _ { L ^ 2 } , \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{align*} B _ 2 ^ { i } ( t , q ) = \\left ( ( V \\nabla _ q \\beta ) ( t , q ) + ( \\beta \\tilde F ) ( t , q ) - ( \\beta \\partial _ t \\psi ) ( t , q ) \\right ) ^ i \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} f ' ( z _ { J + 1 } , \\dots , z _ n ) = f ( q _ i ^ { J - 1 } q _ i ^ { - l _ 0 } u _ 0 , \\dots , q _ i ^ { - l _ 0 } u _ 0 , z _ { J + 1 } , \\dots , z _ n ) . \\end{align*}"}
-{"id": "8538.png", "formula": "\\begin{align*} X ( \\zeta ) = \\frac { X _ { j - 1 } } { \\zeta ^ { j - 1 } } + \\cdots + X _ 0 + \\cdots + \\zeta ^ { j - 1 } X _ { - j + 1 } \\end{align*}"}
-{"id": "6034.png", "formula": "\\begin{align*} & \\overline { X } _ 0 : = X _ 0 , \\\\ & \\overline { X } _ 3 : = \\left \\lbrace ( \\eta , w ) \\in [ H ^ 3 ( 0 , L ) \\cap H ^ 1 _ 0 ( 0 , L ) ] ^ 2 : \\eta _ x ( 0 ) = w _ x ( L ) = 0 \\right \\rbrace , \\\\ & \\overline { X } _ { 3 \\theta } : = [ \\overline { X } _ 0 , \\overline { X } _ 3 ] _ { [ \\theta ] } , \\end{align*}"}
-{"id": "5640.png", "formula": "\\begin{align*} g v ( h _ 1 , h _ 2 ) = \\ln | h ' _ 1 | d \\ln | h _ 2 ' \\circ h _ 1 | . \\end{align*}"}
-{"id": "2262.png", "formula": "\\begin{align*} \\begin{aligned} & { \\rm B i a s } ( \\hat F _ { \\tilde d } ) = \\sum _ { j \\in J } b _ j \\Big ( \\frac { K } { M } \\Big ) ^ { j / d } + o \\big ( \\frac { K } { M } \\big ) + o \\big ( \\frac { 1 } { K } \\big ) + O \\big ( \\frac { 1 } { M } \\big ) , \\end{aligned} \\end{align*}"}
-{"id": "4476.png", "formula": "\\begin{align*} \\frac { \\sin ( s \\arctan t ) } { ( 1 + t ^ 2 ) ^ { s / 2 } } = \\frac { 1 } { \\Gamma ( s ) \\Gamma ( m + 1 - s ) } \\frac { d ^ m } { d t ^ m } \\left ( \\int _ 0 ^ t \\frac { x ^ s ( t - x ) ^ { m - s } } { 1 + x ^ 2 } \\ ; d x \\right ) . \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} \\begin{aligned} | f ^ { ( \\alpha ) } | & \\leq B ^ { | \\alpha | } \\alpha ! M _ { | \\alpha | } \\\\ | T _ { j i } ^ { ( \\alpha ) } | & \\leq B ^ { | \\alpha | } \\alpha ! M _ { | \\alpha | + | \\gamma | } , i , j = 1 , \\ldots , n , \\end{aligned} \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} [ P , Q R ] = [ P , Q ] R + Q [ P , R ] \\end{align*}"}
-{"id": "4042.png", "formula": "\\begin{align*} r _ { X Y } ( \\cdot ) = p _ { X Y | U _ { 1 : i - 1 } } ( \\cdot | u _ { 1 : i - 1 } ) \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} \\psi ( 0 ) & = \\frac { b } { 1 - a } , \\\\ \\psi ( 1 ) & = \\frac { b } { 1 - a } - \\frac { 1 } { 1 - a } = - \\frac { 1 - b } { 1 - a } , \\\\ \\psi ( 0 + ) & = \\frac { b } { 1 - a } - 1 = \\frac { a + b - 1 } { 1 - a } . \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & 0 & 0 & 1 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & - \\epsilon & 0 \\\\ 1 & 0 & 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} v _ 0 = w _ 0 \\ , \\ , \\mbox { o n $ [ 0 , u ] $ } . \\end{align*}"}
-{"id": "5492.png", "formula": "\\begin{align*} \\mathbf { W } ^ { \\pm } = \\mathbf { V S } ^ { \\pm } ( \\pm i \\Omega \\mathbf { I } - \\mathbf { \\Lambda } ) ^ { - 1 } \\mathbf { V } ^ { - 1 } \\mathbf { g } ^ { \\pm 1 } . \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} \\begin{aligned} & C \\int _ { t _ { 0 } } ^ { t _ { 0 } + h } e ^ { - ( t _ { 0 } + h ) \\omega } ( t _ { 0 } + h - s ) ^ { - \\frac { 3 } { q } } s ^ { - 3 ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\ d s \\Big [ k ^ { u } _ { j } ( T ) k ^ { \\nabla y } _ { j } ( T ) + k ^ { \\nabla y } _ { j } ( T ) ^ 2 ( k ^ { x } _ { j } ( T ) + k ^ { y } _ { j } ( T ) + \\norm { \\overline { b } } ) \\Big ] . \\end{aligned} \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\mathcal { C } _ { \\Gamma } ( \\overline { x } , \\overline { v } ) } ( d ) = [ \\mathcal { C } _ { \\Gamma } ( \\overline { x } , \\overline { v } ) ] ^ { \\circ } \\cap [ \\ ! [ d ] \\ ! ] ^ { \\perp } . \\end{align*}"}
-{"id": "5519.png", "formula": "\\begin{align*} b = - \\varepsilon \\left ( \\left \\vert X \\right \\vert ^ { 2 } - K ^ { 2 } \\right ) + \\min _ { \\mathfrak { \\bar { B } } _ { K } \\left ( 0 \\right ) \\cap \\Sigma } w , \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} v \\geq c _ n : = \\frac 1 2 \\Big ( 3 - \\sum _ { k = 0 } ^ n \\big ( \\frac { \\varepsilon } { 1 + \\varepsilon } \\big ) ^ k \\Big ) \\ , v \\geq ( 1 - \\varepsilon / 2 ) \\ , v \\mbox { f o r } n \\in \\N . \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} y ^ { ( i ) } = \\sum _ { j = 1 } ^ { n + 1 } m _ { i j } x ^ { ( j ) } , m _ { i j } : = \\frac { \\left | l _ { i j } \\right | } { \\sum \\limits _ { k = 1 } ^ { n + 1 } \\left | l _ { i k } \\right | } . \\end{align*}"}
-{"id": "2309.png", "formula": "\\begin{align*} P \\{ T _ 1 < T _ 2 \\} = \\sum _ { k = 1 } ^ { M _ 2 } \\sum _ { j = 1 } ^ { M _ 1 } ( - 1 ) ^ { j + k } \\binom { M _ 1 } { j } \\binom { M _ 2 } { k } \\frac { p _ 1 j } { p _ 1 j + p _ 2 k } \\end{align*}"}
-{"id": "2065.png", "formula": "\\begin{align*} \\big \\{ H _ 1 , H _ 2 \\big \\} = p _ 1 - \\alpha ( x _ 2 + p _ 2 ) \\big \\{ H _ 2 , H _ 3 \\big \\} = p _ 1 + \\alpha ( x _ 2 - p _ 2 ) . \\end{align*}"}
-{"id": "7469.png", "formula": "\\begin{align*} A _ 0 ^ { j _ 1 j _ 2 j _ 3 } = & - B ^ { i _ 1 i _ 2 i _ 3 } _ 1 \\int _ 0 ^ \\infty ( e ^ { - t \\tilde \\gamma } ) _ { i _ 1 j _ 1 } ( e ^ { - t \\tilde \\gamma } ) _ { i _ 2 j _ 2 } ( e ^ { - t \\tilde \\gamma } ) _ { i _ 3 j _ 3 } d t \\\\ = & - \\frac { 1 } { 2 } \\delta ^ { i _ 1 i _ 2 } ( \\nabla _ q \\beta ) ^ { i _ 3 } \\int _ 0 ^ \\infty ( e ^ { - t \\tilde \\gamma } ) _ { i _ 1 j _ 1 } ( e ^ { - t \\tilde \\gamma } ) _ { i _ 2 j _ 2 } ( e ^ { - t \\tilde \\gamma } ) _ { i _ 3 j _ 3 } d t \\end{align*}"}
-{"id": "963.png", "formula": "\\begin{align*} \\mathrm { C r i t } ( \\mathfrak { e } ) \\ \\doteq \\ \\big \\{ p \\in \\Gamma ^ { \\ast } \\ ; \\big | \\ \\nabla \\mathfrak { e } ( p ) = 0 \\big \\} . \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{align*} \\left | \\int _ { \\mathbb R } g ' ( x ) V ( x ) \\ , \\mathrm d x \\right | & = { } \\left | \\int _ { \\mathbb R } g ( x ) \\left ( | f ' ( x ) | ^ 2 + q ( x ) | f ( x ) | ^ 2 \\right ) \\ , \\mathrm d x \\right | \\\\ & \\leq { } \\| g \\| _ \\infty \\left ( \\| f ' \\| _ { L ^ 2 } ^ 2 + \\| q f ^ 2 \\| _ { L ^ 1 } \\right ) \\leq 1 2 \\| q _ - \\| _ { L ^ 1 } ^ 2 \\| f \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} \\det M _ { i i } = \\prod _ { k = 1 } ^ r ( A _ k ) _ { i i } ( B _ k ) _ { i i } - \\prod _ { k = 1 } ^ r ( C _ k ) _ { i i } ( D _ k ) _ { i i } \\ , . \\end{align*}"}
-{"id": "7474.png", "formula": "\\begin{align*} & \\delta _ { j _ 1 j _ 2 } G _ { i _ 1 i _ 3 \\alpha } ^ { j _ 1 j _ 2 \\eta } \\delta ^ { i _ 1 \\alpha } \\tilde \\gamma _ { \\eta k } + 2 \\delta ^ { i _ 1 i _ 2 } G _ { i _ 1 i _ 2 i _ 3 } ^ { j _ 1 j _ 2 j _ 3 } \\gamma _ { j _ 2 j _ 3 } \\delta _ { j _ 1 k } \\\\ = & - \\int _ 0 ^ \\infty \\frac { d } { d y } \\left [ \\sum _ { \\alpha \\eta } ( e ^ { - y \\tilde \\gamma } ) _ { \\alpha k } ( e ^ { - y \\tilde \\gamma } ) _ { \\alpha \\eta } ( e ^ { - y \\tilde \\gamma } ) _ { i _ 3 \\eta } \\right ] d y \\\\ = & \\delta _ { i _ 3 k } . \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} T _ { \\overline { L } , p } ( r ) & = \\int ( \\log \\| s \\| + m g _ { D _ r , p } ) ( \\delta _ p - \\pi _ { r , p } ) + \\int g _ { D _ r , p } \\delta _ { \\div ( s ) - m [ p ] } \\\\ & = \\int ( \\log \\| s \\| + m g _ { D _ r , p } ) \\delta _ p - \\int \\log \\| s \\| \\pi _ { r , p } + \\int g _ { D _ r , p } \\delta _ { \\div ( s ) - m [ p ] } \\end{align*}"}
-{"id": "8006.png", "formula": "\\begin{align*} f ( x , y ) = - 2 0 \\exp \\left ( - 0 . 2 \\sqrt { 0 . 5 \\left ( x ^ { 2 } + y ^ { 2 } \\right ) } \\right ) - \\exp \\left ( 0 . 5 \\left ( \\cos \\left ( 2 \\pi x \\right ) + \\cos \\left ( 2 \\pi y \\right ) \\right ) \\right ) + e + 2 0 . \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} e ^ { i t \\tau ( x ) } = \\lambda \\mathfrak g ( x ) / \\mathfrak g ( f x ) . \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} F ( X , Y ) = X ^ r \\# _ t Y ^ r \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} \\psi ( \\nu , \\mu ) = \\lim _ { q \\rightarrow t ^ { - p - m _ 1 } } \\left ( 1 - q t ^ { p + m _ 1 } \\right ) \\left ( \\sum _ { i = 0 } ^ { p } \\frac { c _ { \\mu , \\nu } ( i ; t ) } { 1 - q t ^ { i + m _ 1 } } \\right ) \\cdot I ( \\mu , \\nu ) = c _ { \\mu , \\nu } ( p ; t ) \\cdot I ( \\mu , \\nu ) . \\end{align*}"}
-{"id": "6927.png", "formula": "\\begin{align*} \\mbox { $ d = 2 g _ 1 $ a n d $ b - a = g _ 1 $ } & \\implies \\frac 1 2 d = b - a \\\\ [ 1 e x ] \\mbox { $ d = 3 g _ 1 $ a n d $ b - a = g _ 1 $ } & \\implies \\frac 1 3 d = b - a \\\\ [ 1 e x ] \\mbox { $ d = 3 g _ 1 $ a n d $ b - a = 2 g _ 1 $ } & \\implies \\frac 2 3 d = b - a \\\\ [ 1 e x ] \\mbox { $ d = 4 g _ 1 $ a n d $ b - a = 1 g _ 1 $ } & \\implies \\frac 1 4 d = b - a \\\\ [ 1 e x ] \\mbox { $ d = 4 g _ 1 $ a n d $ b - a = 3 g _ 1 $ } & \\implies \\frac 3 4 d = b - a \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} \\int _ 0 ^ { U ( N ; \\alpha ) ^ { - 1 } } e ^ { - x } \\left ( 1 - \\frac { \\ln x } { \\ln N } \\right ) ^ r d x = o \\left ( e ^ { - \\ln ^ { \\alpha } N } \\right ) . \\end{align*}"}
-{"id": "6636.png", "formula": "\\begin{align*} ( \\overline { P } f ) ( y ) = ( g ( y ) f ) ( y ) . \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} I [ \\mu - \\nu ] & = \\iint f ( \\theta - \\phi ) d \\mu ( \\theta ) d \\mu ( \\phi ) + \\iint f ( \\theta - \\phi ) d \\nu ( \\theta ) d \\nu ( \\phi ) \\\\ & - \\iint f ( \\theta - \\phi ) d \\mu ( \\theta ) d \\nu ( \\phi ) - \\iint f ( \\theta - \\phi ) d \\nu ( \\theta ) d \\mu ( \\phi ) \\in [ - \\infty , \\infty ) \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} V _ \\Lambda \\psi ( x ) = v _ x \\psi ( x ) , \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{align*} t ( x ' , \\xi ' , D _ n ) f : = \\int _ 0 ^ \\infty { t } ( x ' , \\xi ' , y _ n ) f ( y _ n ) \\ , d y _ n f \\in \\mathcal { S } _ + , \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{align*} & P \\{ T _ 1 < \\cdots < T _ g \\} = \\\\ & K \\int _ 0 ^ { \\infty } \\cdots \\int _ 0 ^ { t _ 3 } e ^ { - ( p _ g t _ g + \\cdots + p _ 2 t _ 2 ) } \\left ( 1 - e ^ { - p _ g t _ g } \\right ) ^ { M _ g - 1 } \\cdots \\left ( 1 - e ^ { - p _ 2 t _ 2 } \\right ) ^ { M _ 2 - 1 } \\left ( 1 - e ^ { - p _ 1 t _ 2 } \\right ) ^ { M _ 1 } \\ , d t _ 2 \\cdots d t _ g , \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} | \\mathcal { C } _ { u , v } | \\leq 1 + 2 ( r - 1 ) ( 2 r - 3 ) = 4 r ^ 2 - 1 0 r + 7 . \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( Z ^ c _ i ( T _ 0 , T _ 1 ) \\right ) = \\mathbb { P } \\left ( S _ i \\geq C _ i ( T _ 1 - T _ 0 ) \\right ) , \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} \\beta _ 3 - 2 { s + 1 \\choose 2 } + { s + 2 \\choose 2 } - 1 = 0 , \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} & \\| f ( s ) \\| _ \\rho ^ 2 \\le \\Big ( \\| f ( s ) \\| _ 2 ^ 2 \\Big ) ^ \\alpha \\Big ( \\| f ( s ) \\| _ 6 ^ 2 \\Big ) ^ { \\beta } , \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\alpha = ( 3 / \\rho ) - ( 1 / 2 ) , \\ \\ \\ \\ \\ \\beta = ( 3 / 2 ) - ( 3 / \\rho ) , \\end{align*}"}
-{"id": "4301.png", "formula": "\\begin{align*} y _ t = y _ 0 + \\int _ 0 ^ t f ( y ^ - _ r ) d x _ r \\end{align*}"}
-{"id": "8449.png", "formula": "\\begin{align*} T \\circ \\pi ( h ) = \\pi ( h ) \\circ T \\end{align*}"}
-{"id": "9133.png", "formula": "\\begin{align*} g = x ^ 4 - ( b ^ 2 - 4 d ) . \\end{align*}"}
-{"id": "6416.png", "formula": "\\begin{align*} u ( t ) = e ^ { - t C } c + \\int _ 0 ^ t e ^ { - ( t - s ) C } f ( s ) \\ ; \\d s ( 0 < t < T ) . \\end{align*}"}
-{"id": "7719.png", "formula": "\\begin{align*} Q _ 1 & = \\mathrm { P } \\left ( y < \\left ( \\frac { \\epsilon _ 1 } { \\rho } \\right ) ^ { - \\frac { 1 } { \\alpha } } , x > \\left ( \\frac { ( 1 + \\epsilon _ 1 ) } { \\phi _ i \\left ( \\rho - \\epsilon _ 1 y ^ { \\alpha } \\right ) } \\right ) ^ { - \\frac { 1 } { \\alpha } } \\right ) , \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{align*} & \\sup _ { 0 \\leq s \\leq t \\leq T } E \\left [ \\left | J ^ m _ { s , t } - \\int _ s ^ t B ^ { i _ 1 , . . . , i _ k } ( r , q ^ m _ r ) \\left ( \\int h ( r , q ^ m _ r , z ) z _ { i _ 1 } . . . z _ { i _ k } d z \\right ) d r \\right | ^ p \\right ] ^ { 1 / p } \\\\ \\leq & \\tilde C ( m ^ { 1 / 2 } T + m ( 2 + T ) + m ^ { 1 / 2 } T ^ { 1 / 2 } ) \\sup _ { r \\in [ 0 , T ] } E \\left [ ( 1 + \\| q _ r ^ m \\| ^ { \\tilde p } ) ^ { 2 p } \\right ] ^ { 1 / ( 2 p ) } E \\left [ ( 1 + \\| z _ r ^ m \\| ^ { k + 1 } ) ^ { 2 p } \\right ] ^ { 1 / ( 2 p ) } \\\\ = & O ( m ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "3503.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } q ( y _ { k ( l ) } ) = \\lim _ { l \\to \\infty } y _ { k ( l ) + 1 } = q ( y ^ * ) , \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{align*} \\Gamma _ m : = 3 6 m ^ { \\beta ' } \\theta ( m ) \\psi ( m ) ^ { \\beta / ( 1 + \\beta ) } \\ , , \\ell _ m : = [ m ^ { \\beta / ( 1 + \\beta ) } \\psi ( m ) ^ { 1 / ( 1 + \\beta } ] \\ , . \\end{align*}"}
-{"id": "4270.png", "formula": "\\begin{align*} U ^ + _ 1 & = \\Big ( U \\setminus \\{ u _ 1 \\} \\Big ) \\cup \\{ q _ i ^ { - 1 } u _ 1 , q _ { s } u _ 1 , q _ s ^ { - 1 } u _ 1 , q _ i u _ 1 \\} \\\\ W ^ + _ 1 & = W \\cup \\{ u _ 1 , q _ i ^ { - 1 } q _ { s } ^ { - 1 } u _ 1 , q _ i q _ s u _ 1 \\} , \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} R ^ \\nabla _ { X Y Z W } = R ^ g _ { X Y Z W } + \\underset { \\alpha \\coloneqq } { \\underbrace { g ( ( \\nabla _ X A _ Y - \\nabla _ Y A _ X - A _ { [ X , Y ] } ) Z , W ) } } - \\underset { \\beta \\coloneqq } { \\underbrace { g ( [ A _ X , A _ Y ] Z , W ) } } . \\end{align*}"}
-{"id": "9101.png", "formula": "\\begin{align*} a = \\left ( 1 - \\sqrt { r } \\right ) ^ 2 ; b = \\left ( 1 + \\sqrt { r } \\right ) ^ 2 . \\end{align*}"}
-{"id": "5422.png", "formula": "\\begin{align*} R = \\begin{pmatrix} \\partial _ 2 \\bar f & \\partial _ 3 \\bar f \\\\ \\partial _ 2 \\bar g & \\partial _ 3 \\bar g \\end{pmatrix} , \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\frac { 1 } { t + Q } X \\frac { 1 } { t + Q } { \\rm d } t = Y \\ . \\end{align*}"}
-{"id": "8379.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ \\infty Q ^ { ( \\infty ) , \\ell } = \\sum _ { \\ell = 0 } ^ \\infty \\int _ { \\R } \\left ( \\frac { q } { k } \\right ) ^ \\ell \\mu _ k ( \\d q ) = \\int _ \\R \\frac { 1 } { 1 - q / k } \\mu _ k ( \\d q ) = \\frac { k - 1 } { k - 2 } . \\end{align*}"}
-{"id": "9009.png", "formula": "\\begin{align*} g \\cdot u ^ { { \\bf ( k ) } } = ( x , g \\cdot u , ( g \\cdot u ) ' , ( g \\cdot u ) '' , \\dots ) \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} \\Lambda ' ( a - b ) ( \\gamma ( a ) - \\gamma ( b ) ) & = ( a - b ) ^ { p - 1 } ( \\gamma ( a ) - \\gamma ( b ) ) \\\\ & = ( a - b ) ^ { p - 1 } \\int _ { b } ^ { a } \\gamma ' ( t ) d t \\\\ & = ( a - b ) ^ { p - 1 } \\int _ { b } ^ { a } ( \\Gamma ' ( t ) ) ^ { p } d t \\\\ & \\geq \\left ( \\int _ { b } ^ { a } ( \\Gamma ' ( t ) ) d t \\right ) ^ { p } = ( \\Gamma ( a ) - \\Gamma ( b ) ) ^ { p } . \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} t _ p = c _ p \\frac { ( w _ { p - 1 } v _ { \\omega _ { i _ p } } , s g _ p w _ p v _ { \\omega _ { i _ p } } ) } { ( w _ { p - 1 } v _ { \\omega _ { i _ p } } , s g _ p w _ { p - 1 } v _ { \\omega _ { i _ p } } ) } , \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} \\tilde { D } ( w _ i ) & \\leq \\tilde { d } ( w _ i ) - 1 \\\\ & = \\sharp \\{ j \\in \\{ 1 , 2 , \\dots , m - 1 \\} \\ , | \\ , v _ j \\prec w _ i \\} \\\\ & \\leq \\sharp \\{ j \\in \\{ 1 , 2 , \\dots , m - 1 \\} \\mid \\tilde { D } ( v _ j ) \\leq i - 2 \\} . \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{align*} K = p _ 2 p _ 3 \\cdots p _ g M _ 2 M _ 3 \\cdots M _ g . \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} ( x - x _ \\ell ) ^ \\alpha = \\sum _ { \\gamma \\leq \\alpha } \\binom { \\alpha } { \\gamma } ( - x _ \\ell ) ^ { \\alpha - \\gamma } x ^ \\gamma x \\in \\R ^ d \\alpha \\in \\N _ 0 ^ d . \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac 1 T \\int _ 0 ^ T D ( t ) \\ , d t = \\sum _ r E _ r D E _ r . \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} \\int _ { \\exp _ 0 ( C _ 0 ) } \\omega _ 0 = \\int _ { W } \\omega _ 0 = M < \\infty . \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} \\underline { M } _ { n + 1 } = \\frac { ( b _ { \\inf } - \\chi \\mu ) a _ { \\inf } - \\chi \\mu a _ { \\sup } + ( \\chi \\mu ) ^ 2 \\underline { M } _ { n } } { ( b _ { \\inf } - \\chi \\mu ) ( b _ { \\sup } - \\chi \\mu ) } \\end{align*}"}
-{"id": "914.png", "formula": "\\begin{align*} Z _ \\gamma ( s ) = \\prod _ { k \\in \\N } \\left ( 1 - e ^ { l ( \\gamma ) ( s + k ) } \\right ) . \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} \\partial _ k v _ k = \\int _ { | ( y , s ) | < 1 } \\sum _ { m = 1 } ^ d \\sum _ { \\substack { | \\mu | + 2 l = m \\\\ \\mu _ k > 0 } } D _ x ^ { \\mu } D _ t ^ l K _ { j k } ( - y , - s ) \\frac { x ^ { \\mu - e _ k } t ^ l } { ( \\mu - e _ k ) ! l ! } f _ j ( y , s ) \\ , d y \\ , d s , \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} L ( u ) = \\mathrm { d i v } ( A \\nabla u ) , u \\in H ^ 1 ( \\Omega ) . \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{align*} u ( i , k ) = - \\nabla f ( x ( i , k ) ) + \\varepsilon ( { i , k } ) - \\sum _ { j = 1 , j \\neq i } ^ { N } \\alpha _ { i j } \\nabla _ { x ( i , k ) } J ( \\| x ( i , k ) - x ( j _ i , k ) \\| ) . \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N \\nabla ^ T f ( y _ { i , t } ) e _ { i , t } = \\sum _ { i = 1 } ^ N ( \\nabla ^ T f ( y _ { i , t } ) - \\nabla ^ T f ( \\overline { x } _ t ) ) ( y _ { i , t } - \\overline { x } _ t ) \\geq \\sum _ { i = 1 } ^ N \\kappa \\| y _ { i , t } - \\overline { x } _ t \\| ^ 2 = 2 \\kappa N \\overline { V } _ t , \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} { \\rm V a r } ( t ) = \\sum _ { i = 2 } ^ m w _ i ( t ) ^ 2 . \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} [ - 1 ] ^ * W _ { - ( g - 1 ) o - \\vartheta } = W _ { - ( g - 1 ) o - \\vartheta } . \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} U = z _ t \\cdot \\vec n = \\operatorname { R e } \\left ( \\bar { z } _ t i e ^ { i \\theta } \\right ) = \\operatorname { R e } \\left ( \\frac { e ^ { i \\theta } } { 2 \\pi } \\int \\frac { \\gamma } { \\xi ( s ) - \\xi ( \\beta ) } d \\beta \\right ) , \\end{align*}"}
-{"id": "850.png", "formula": "\\begin{align*} \\int _ { \\R ^ { N } } f ( v _ { n } ) v _ { n } d x = o _ { n } ( 1 ) . \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} v = \\frac { 1 } { 3 } ( s - 1 ) - \\frac { 2 ^ 4 \\pi ^ 2 } { 3 ^ 6 } ( s - 1 ) ^ 3 , s \\in [ 0 , 2 ] , \\end{align*}"}
-{"id": "3010.png", "formula": "\\begin{align*} I _ { q } ( u ) : = \\frac { 1 } { 2 } \\int _ { \\Omega } | \\nabla u | ^ { 2 } - \\frac { 1 } { q + 1 } \\int _ { \\Omega } a | u | ^ { q + 1 } \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} & \\left [ \\ , v \\ , \\neq \\ , 0 \\mbox { a n d } F ( \\bar { z } ) ^ T B ^ { \\ , k } v + \\displaystyle { \\sum _ { i \\ , : \\ , \\bar { w } _ i \\neq 0 } } \\ , \\alpha _ i \\ , v _ i \\ , \\mbox { s i g n } ( \\bar { w } _ i ) + \\displaystyle { \\sum _ { i \\ , : \\ , \\bar { w } _ i = 0 } } \\ , \\alpha _ i \\ , | \\ , v _ i \\ , | \\ , \\leq \\ , 0 \\ , \\right ] \\\\ & \\Rightarrow v ^ T \\left [ \\ , ( \\ , B ^ { \\ , k } \\ , ) ^ T A ^ { \\ , j } B ^ { \\ , k } \\ , \\right ] \\ , v \\ , > \\ , 0 . \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{align*} \\mathbf { q } _ { i j } = \\frac { k } { 4 \\pi } \\int \\Big [ \\frac { \\mathbf { a } _ { 1 } \\times \\mathbf { a } _ { 2 } } { | \\mathbf { a } _ { 1 } | ^ { 3 } } \\frac { 1 } { | \\zeta | } + O ( 1 ) \\Big ] d \\zeta . \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} E _ { } : = \\{ e \\in E \\mid e \\} . \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\partial D _ { \\varpi } ( P \\parallel Q ) } { \\partial \\varpi } \\ge \\frac { e ^ { \\varpi } \\left [ \\left ( \\sum _ i p _ i \\frac { q _ i } { p _ i } \\right ) ^ { - 1 } - 1 \\right ] } { \\sum _ i { p _ i e ^ { \\left ( \\varpi \\frac { p _ i } { q _ i } \\right ) } } } = 0 , \\end{aligned} \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & \\cdots & 0 & \\cdots \\\\ \\vdots & \\ddots & \\vdots \\\\ 0 & \\cdots & \\begin{matrix*} [ r ] 0 & 1 \\\\ - 1 & 0 \\end{matrix*} & \\cdots \\\\ \\vdots & & \\vdots & \\ddots \\end{pmatrix} , \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} \\frac { d } { d t } \\langle N , T \\rangle & = \\langle N _ { t } , T \\rangle + \\langle N , T _ { t } \\rangle = 0 \\\\ \\langle N _ { t } , T \\rangle & = - \\langle N , T _ { t } \\rangle = \\tau \\kappa \\\\ \\langle N _ { t } , B \\rangle & = - \\langle N , B _ { t } \\rangle = - F ( s , t ) . \\end{align*}"}
-{"id": "3702.png", "formula": "\\begin{align*} \\frac { M ( P , \\vec { \\nu } ) } { P ^ { n - r d } } = P ^ { r d } \\sum _ { ( \\vec { a } , q ) \\in \\mathcal { M } ( \\theta _ 0 ) } & \\frac { S _ { \\vec { a } , q } ( \\vec { \\nu } ) } { q ^ n } J _ { \\widetilde { C } ^ { r - 1 } P ^ { \\eta } } ( P ^ { - d } \\vec { \\nu } ) + \\mathfrak { O } _ 1 + \\mathfrak { O } _ 2 . \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} \\Delta _ \\alpha ( z ) = \\Delta _ 1 ( z ) ^ { \\alpha _ 1 - \\alpha _ 2 } \\Delta _ 2 ( z ) ^ { \\alpha _ 2 - \\alpha _ 3 } \\dots \\Delta _ r ( z ) ^ { \\alpha _ r } , \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} ( d V _ H ) _ { \\exp ( a ) H } = e ^ { 2 \\sum _ { \\alpha } \\alpha ( a ) } \\left ( \\bigwedge _ { \\diamondsuit } \\omega _ { \\diamondsuit , \\bar { \\diamondsuit } } \\right ) _ { \\exp ( a ) H } \\end{align*}"}
-{"id": "8774.png", "formula": "\\begin{align*} \\mathfrak { h } = \\mathfrak { p } ^ u \\oplus \\mathfrak { l } ^ { \\sigma } \\end{align*}"}
-{"id": "2333.png", "formula": "\\begin{align*} P \\{ T _ 1 < T _ 2 \\} \\sim \\frac { \\nu _ 2 \\lambda \\Gamma ( \\lambda ) } { \\nu _ 1 ^ { \\lambda } } \\cdot \\frac { 1 } { M ^ { \\lambda - 1 } } = \\frac { \\nu _ 2 \\Gamma ( \\lambda + 1 ) } { \\nu _ 1 ^ { \\lambda } } \\cdot \\frac { 1 } { M ^ { \\lambda - 1 } } = \\frac { \\nu _ 2 } { \\nu _ 1 } \\cdot \\frac { \\Gamma ( \\lambda + 1 ) } { M _ 1 ^ { \\lambda - 1 } } \\end{align*}"}
-{"id": "7381.png", "formula": "\\begin{align*} u ( x ) : = \\sin ( x _ 1 ) ( \\psi ( x ) - \\kappa w ( x ) ) , \\kappa : = \\left [ ( - 1 ) ^ { \\frac { m + 1 } { 2 } } \\partial _ 1 ^ m T ( e _ 1 ) \\right ] ^ { - 1 } . \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{align*} \\frac 1 { ( 1 + b ^ 2 ) ^ { \\frac 1 2 } } \\Big ( - b ( 1 + b ^ 2 ) \\mathfrak { R e } \\ddot { c } _ 0 + ( b ^ 2 + \\frac 1 2 ) \\mathfrak { R e } \\ddot { c } _ 1 - \\frac { \\sqrt { 2 } } { 4 } b \\mathfrak { R e } \\ddot { c } _ 2 \\Big ) + \\Sigma = 0 , \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { 0 \\leq t \\leq T } \\| w _ { n } ( t , \\cdot ) \\| _ { L ^ { p } ( \\R ^ N ) } = 0 . \\end{align*}"}
-{"id": "9115.png", "formula": "\\begin{align*} R _ { s u m } ^ { Z F } = K \\log _ 2 \\left ( 1 + \\widetilde { { \\rm S I N R } } _ { Z F } \\right ) = K \\log _ 2 \\left ( 1 + \\rho _ t \\left ( \\frac { M } { K } - \\frac { 1 } { c } \\right ) \\right ) , \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} \\overline N ( 0 + ) = \\underset { r \\to 0 } { \\lim } \\overline N ( r ) . \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} \\mathbb { M } _ i = \\left ( \\begin{array} { c c c c } 0 & 0 & 0 & 0 \\\\ 0 & - 1 & + t & 0 \\\\ 0 & + 1 & - t & 0 \\\\ 0 & 0 & 0 & 0 \\end{array} \\right ) _ { i , i + 1 } \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} | < l _ { 1 , 0 } w , w > _ { L ^ 2 } | & = \\left | < l _ { 1 , \\gamma } w , w > _ { L ^ 2 } + < w ^ 2 , \\psi _ { 1 , \\gamma } - Q _ { 1 } > _ { L ^ 2 } - \\gamma | | \\partial _ x ^ { - 1 } w | | _ { L ^ 2 } ^ 2 \\right | \\\\ & \\lesssim \\epsilon ( \\gamma ) + \\gamma . \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} k ^ { \\infty } _ { j + 1 } ( T ) \\leq k ^ { \\infty } _ { 0 } ( T ) + 3 C \\tilde { C } _ { T } K _ { 1 } ^ 2 K _ { 2 } < \\frac { K _ { 2 } } { 2 } + \\frac { K _ { 2 } } { 2 } = K _ { 2 } . \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{align*} H \\left ( \\nu , \\mu ^ * \\right ) = \\prod _ { x \\in \\vec { x } ( \\mu ^ * ) } \\prod _ { i \\leq x } t ^ { \\nu _ i } . \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} \\int _ I | ( f _ n ' \\circ g ) g ^ { ( k + 1 ) } - ( f _ 0 ' \\circ g ) g ^ { ( k + 1 ) } | & = \\int _ I ( | f _ n ' - f _ 0 ' | \\circ g ) \\cdot | g ^ { ( k + 1 ) } | \\\\ & \\leq \\| f _ n ' - f _ 0 ' \\| \\int _ I | g ^ { ( k + 1 ) } | \\rightarrow 0 . \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{align*} K _ M = \\begin{pmatrix} b & K _ w ^ T & 0 \\\\ K _ { \\eta } & K _ B & K _ w \\\\ 0 & K _ \\eta ^ T & - b \\end{pmatrix} \\end{align*}"}
-{"id": "285.png", "formula": "\\begin{align*} p ( x , y ) ( D ) A ( x , y ) = a ( x , y ) \\tilde a ( x , y ) . \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{align*} h ( t , q , z ) = \\left ( \\frac { \\beta ( t , q ) } { 2 \\pi } \\right ) ^ { n / 2 } e ^ { - \\beta ( t , q ) \\| z \\| ^ 2 / 2 } . \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} T _ { \\geqslant i _ 0 } ( M , N ) \\otimes _ A \\widehat { A } \\ ; \\ ; : = \\ ; \\bigoplus _ { i \\geqslant i _ 0 } \\operatorname { T o r } ^ { \\widehat { A } } _ i \\big ( \\widehat { M } , \\widehat { N } \\big ) \\mbox { i s * A r t i n i a n . } \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{align*} P _ A ( ( 1 + \\lambda ) P _ B x - \\lambda x ) = \\ ; & \\lambda ( P _ B x - x ) + x \\\\ = \\ ; & - \\lambda y + y + f = f + ( 1 - \\lambda ) y . \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{align*} d _ { A _ \\infty } \\dot \\eta + [ \\dot A _ \\infty \\wedge \\eta ] = 0 \\end{align*}"}
-{"id": "7335.png", "formula": "\\begin{align*} ( X + \\lambda M ) ^ \\cdot = [ X + \\lambda M , B - \\lambda M ^ { k + 1 } ] \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{align*} ( \\mathfrak { i } _ { \\psi } ^ { ( 1 ) } ( b ) ) ( L _ { \\mu } ( a ) ) = ( \\mathfrak { i } _ { \\psi } ^ { ( 1 ) } ( a ) ) ( L _ { \\mu } ( b ) ) \\qquad ( \\forall _ { a , b \\in \\mathcal { M } _ { \\psi } } ) , \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{align*} I _ { q } ( u ) : = \\frac { 1 } { 2 } \\int _ { \\Omega } | \\nabla u | ^ { 2 } - \\frac { 1 } { q + 1 } \\int _ { \\Omega } a | u | ^ { q + 1 } . \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} \\mathcal { I } _ { 2 , + } ( t , x ) = \\int _ 0 ^ t \\int _ 0 ^ { \\infty } \\int _ y ^ { \\infty } \\frac { \\partial G _ { t - s } } { \\partial x } ( x - z ) \\psi ( s , z ) \\sigma _ s ( y ) d z W ( d s , d y ) \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{align*} \\langle \\rho _ \\gamma ( u , 1 ) P _ E , \\ , \\rho _ \\gamma ( v , 1 ) P _ E \\rangle & = \\langle P _ E , \\ , \\rho _ \\gamma [ ( - u , 1 ) \\cdot ( v , 1 ) ] P _ E \\rangle \\\\ & = \\langle P _ E , \\ , \\rho _ \\gamma ( v - u , [ - u , v ] ^ { 1 / 2 } ) P _ E \\rangle \\\\ & = \\varphi _ E ( v - u , [ u , v ] ^ { - 1 / 2 } ) . \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} \\varphi \\cdot \\alpha = - \\lambda _ \\alpha ( \\varphi ) \\ , . \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} I ( U _ { i + 1 } ; Y | U _ { 1 : i } \\ ! = \\ ! u _ { 1 : i } ) \\ ! - \\ ! I ( U _ { i + 1 } ; Z | U _ { 1 : i } \\ ! = \\ ! u _ { 1 : i } ) > 0 . \\end{align*}"}
-{"id": "1833.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } \\tau & \\frac { 1 } { 3 } ( h _ { j + 1 } - h _ j ) \\\\ \\frac { 1 } { 3 } ( h _ { j + 1 } - h _ j ) & \\tau \\end{array} \\right ] \\left [ \\begin{array} { c } \\zeta _ j \\\\ \\xi _ j \\end{array} \\right ] = \\left [ \\begin{array} { c } 0 \\\\ 0 \\end{array} \\right ] \\ , . \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n f ( \\hat { x } ^ { ( 1 ) } _ k , \\dots , \\hat { x } ^ { ( d ) } _ k ) \\leq \\sum _ { k = 1 } ^ n f ( \\tilde { x } ^ { ( 1 ) } _ k , \\dots , \\tilde { x } ^ { ( d ) } _ k ) \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{align*} \\Lambda ^ { 2 } \\R ^ 6 = \\Lambda ^ 2 _ 1 \\oplus \\Lambda ^ 2 _ 6 \\oplus \\Lambda ^ 2 _ 8 , \\end{align*}"}
-{"id": "622.png", "formula": "\\begin{align*} & \\sigma _ { x _ 1 } = \\sigma ^ { - 3 } \\bigl ( ( x _ 1 ^ 2 - y _ 1 ^ 2 ) x _ 1 + y _ 1 ^ 2 ( x _ 1 - y _ 1 ) \\bigr ) = \\sigma ^ { - 3 } ( x _ 1 ^ 3 - y _ 1 ^ 3 ) \\\\ & \\sigma _ { x _ 2 } = 2 \\sigma ^ { - 3 } ( x _ 2 - y _ 2 ) \\\\ & \\sigma _ { x _ 1 x _ 1 } = - 3 \\sigma ^ { - 7 } \\bigl ( x _ 1 ^ 3 - y _ 1 ^ 3 \\bigr ) ^ 2 + 3 \\sigma ^ { - 3 } x _ 1 ^ 2 \\\\ & \\sigma _ { x _ 1 x _ 2 } = - 3 \\sigma ^ { - 7 } \\bigl ( x _ 1 ^ 3 - y _ 1 ^ 3 \\bigr ) ( x _ 2 - y _ 2 ) \\\\ & \\sigma _ { x _ 2 x _ 2 } = - 3 \\sigma ^ { - 7 } ( x _ 2 - y _ 2 ) ^ 2 + 2 \\sigma ^ { - 3 } . \\\\ \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} u ( x , t _ 0 + n T _ 1 + t ; t _ 0 , u _ 0 ) & = u ( x , t _ 0 + n T _ 1 + t ; t _ 0 + n T _ 1 , u ( x , t _ 0 + n T _ 1 , t _ 0 , u _ 0 ) ) \\cr & \\geq \\delta _ 2 e ^ { t ( a _ { \\inf } - b _ { \\sup } M ^ + e ^ { T _ 1 a _ { \\sup } } ) } \\cr & \\geq \\delta _ 2 e ^ { - T _ 1 ( a _ { \\inf } + b _ { \\sup } M ^ + e ^ { T _ 1 a _ { \\sup } } ) } \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow \\infty } \\frac { N _ { s c } ^ { > } [ \\mathfrak { e } , \\lambda V _ { 1 } ] } { N _ { s c } [ \\mathfrak { e } , \\lambda V _ { 1 } ] } = 0 , \\ ; \\lim _ { \\lambda \\rightarrow \\infty } \\frac { N _ { s c } ^ { > } [ \\mathfrak { e } , \\lambda V _ { 2 } ] } { N _ { s c } [ \\mathfrak { e } , \\lambda V _ { 2 } ] } > 0 . \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} \\begin{array} { l l } \\partial _ t u + d \\Delta u = f & \\textrm { i n $ ( 0 , T ) \\times \\mathbb R ^ N $ } , \\\\ u ( T , x ) = 0 , \\qquad & u \\big | _ { \\partial B _ 2 } = 0 . \\end{array} \\end{align*}"}
-{"id": "8906.png", "formula": "\\begin{align*} J ( \\phi ) = \\int _ { X } \\phi \\frac { \\omega _ { \\mathrm { r e f } } ^ n } { ( 2 \\pi ) ^ n \\mathcal { L } ^ n } - \\int _ 0 ^ 1 \\int _ { X } \\dot { \\phi } _ t \\frac { \\omega _ { \\phi _ t } ^ n } { ( 2 \\pi ) ^ n \\mathcal { L } ^ n } d t \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} C _ { m } C _ { m } ^ { T } = W _ { m + 1 } H _ { m } H _ { m } ^ { T } W _ { m + 1 } ^ { T } = A \\underbrace { W _ { m } W _ { m } ^ { T } } _ { = P _ { m } } A ^ { T } , \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{align*} \\Sigma ( A , B ) : \\quad \\dot { x } ( t ) = A x ( t ) + B u ( t ) , x ( 0 ) = x _ { 0 } , \\end{align*}"}
-{"id": "2699.png", "formula": "\\begin{align*} ( a _ 3 t + a _ 4 ) ^ n g \\left ( \\frac { a _ 1 t + a _ 2 } { a _ 3 t + a _ 4 } \\right ) = g ( t ) . \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } { \\mathbb P } \\left ( C _ n \\ , | \\ , \\overline { C _ j } , \\ldots , \\overline { C _ t } , \\ , B \\right ) = \\varepsilon ' , \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} P _ { D H , i } ' ( p ) = P _ { D H } ' ( p ) \\sum _ { \\alpha } \\frac { - \\alpha ^ { \\vee , i } } { ( 2 \\chi - p ) ( \\alpha ^ { \\vee } ) } \\end{align*}"}
-{"id": "8226.png", "formula": "\\begin{align*} \\iint M _ { i , j } ( R u , R v ) f _ i ( u ) f _ j ( v ) \\ , d u \\ , d v = \\iint M _ { i , j } ( u , v ) f _ i ( R ^ { - 1 } u ) f _ j ( R ^ { - 1 } v ) \\ , d u \\ , d v . \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} R _ { 2 } ( f ( r ) , g ( r ) ) = ( - m c ^ 2 + V ( r ) - \\lambda ) g ( r ) + c f ' ( r ) + \\frac { c \\kappa } { r } f ( r ) \\ , , \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} & \\ \\ u ( m ) = 1 \\ ; m _ 1 = M _ 1 \\ ; \\ ; 0 \\leq m _ j \\leq M _ j - 1 \\ ; \\ ; j \\ne 1 , \\\\ & \\ ; k = 2 , \\dots , g \\\\ & \\ \\ u ( m ) = 0 \\ ; m _ k = M _ k \\ ; \\ ; 0 \\leq m _ j \\leq M _ j - 1 \\ ; \\ ; \\ j \\ne k . \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} & p ( M + 1 - \\delta ) = 2 ( M + 1 ) \\cdot \\frac { 1 - \\frac { \\delta } { M + 1 } } { 1 - \\frac { 1 } { 2 } \\cdot \\frac { \\delta } { M + 1 } } \\equiv 2 ( M + 1 ) \\delta _ 1 , \\\\ & q = \\frac { 2 } { 1 + \\frac { 1 } { 2 } \\cdot \\frac { \\delta } { M + 1 } } \\equiv 2 \\delta _ 2 \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} ( F , P _ 1 ) = \\lbrace x \\in P _ 1 ( F ) : x \\eqref { E q u a t i o n D C 1 } \\rbrace \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} \\langle \\varnothing | W _ { - n _ s , k _ s } \\dots W _ { - n _ 1 , k _ 1 } \\Phi _ m W _ { n _ 1 ' , k _ 1 ' } \\dots W _ { n _ t ' , k _ t ' } | \\varnothing \\rangle = 0 \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{align*} \\kappa _ { t } = \\partial _ { t } \\langle \\partial _ { x x } \\gamma , N \\rangle - 2 \\langle T , \\partial _ { x t } \\gamma , N \\rangle \\\\ = \\langle \\partial _ { t x } \\gamma , N \\rangle - 2 \\kappa \\langle T , \\partial _ { t x } \\gamma \\rangle \\end{align*}"}
-{"id": "550.png", "formula": "\\begin{align*} \\log ^ + x = \\begin{cases} \\log x & x \\ge 1 \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} \\kappa _ X & = \\varepsilon ^ { - 1 } \\lambda _ Q , & \\alpha = \\varepsilon ^ { - 1 } \\frac { \\norm { \\nabla _ x h } _ \\infty } { 4 \\lambda _ Q } . \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{align*} \\begin{pmatrix} A _ 1 & 0 \\\\ 0 & A _ 2 \\\\ \\end{pmatrix} , w h e r e \\ A _ 1 \\ a n d \\ A _ 2 \\ a r e \\ o f \\ t h e \\ f o r m \\ i n \\ L e m m a \\ 4 . 4 . \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} & \\| \\eta \\overline { u } ^ { \\alpha + 1 } \\| _ { m ^ \\star } ^ m \\le C \\int \\overline { u } ^ { m ( \\alpha + 1 ) } | \\nabla \\eta | ^ m + \\frac { C } { m \\alpha + 1 - C m \\epsilon ^ { m ' } } \\int m g _ 1 ( x , \\epsilon ) \\overline { u } ^ { m ( \\alpha + 1 ) } \\\\ & + C \\left ( \\frac { \\| f \\| _ q } { ( m \\alpha + 1 - C m \\epsilon ^ { m ' } ) k ^ { m - 1 } } \\right ) ^ { N / ( m q - N ) } \\int \\eta ^ m \\overline { u } ^ { m ( \\alpha + 1 ) } . \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} \\frac { d \\theta _ { c , j } } { d \\beta } ( \\beta ) = - \\frac { \\frac { \\partial g } { \\partial x } ( \\beta , \\theta _ { c , j } ( \\beta ) , 0 ) } { \\frac { \\partial g } { \\partial y } ( \\beta , \\theta _ { c , j } ( \\beta ) , 0 ) } < 0 , ( \\forall j \\in \\{ 1 , 2 \\} , \\ \\beta \\in \\R _ { > 0 } ) . \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} \\frac { 1 } { 1 + t } \\bigg [ \\binom { n } { k } _ t - \\binom { [ \\frac { k } { 2 } ] + [ \\frac { n - k } { 2 } ] } { [ \\frac { k } { 2 } ] } _ { t ^ 4 } \\bigg ] & & \\frac { t } { 1 + t } \\bigg [ \\binom { n } { k } _ t - \\binom { [ \\frac { k } { 2 } ] + [ \\frac { n - k } { 2 } ] } { [ \\frac { k } { 2 } ] } _ { t ^ 4 } \\bigg ] \\end{align*}"}
-{"id": "7598.png", "formula": "\\begin{align*} ( \\hat { \\phi } _ * ( \\tilde \\gamma ^ { - 1 } \\sigma ) ) ^ { - 1 } \\tilde \\gamma ^ { - 1 } \\sigma ) ^ { i } _ \\rho \\delta ^ { \\rho \\eta } ( \\hat { \\phi } _ * ( \\tilde \\gamma ^ { - 1 } \\sigma ) ) ^ { - 1 } \\tilde \\gamma ^ { - 1 } \\sigma ) ^ { j } _ \\eta = \\delta ^ { i j } . \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} \\mathcal { I } ( z _ 1 , \\dots , z _ n ; \\vec { u } ) = & \\prod _ { j = 1 } ^ n \\prod _ { \\alpha = 1 } ^ r \\frac { - u _ \\alpha z _ j } { ( z _ j - u _ \\alpha ) ( q _ 1 q _ 2 z _ j - u _ \\alpha ) } \\ ; \\prod _ { 1 \\leq j \\neq k \\leq n } \\frac { ( z _ j - z _ k ) ( z _ j - q _ 1 q _ 2 z _ k ) } { ( z _ j - q _ 1 z _ k ) ( z _ j - q _ 2 z _ k ) } . \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{align*} 0 \\leq \\limsup _ { k } d ( x ^ { k } , C ) = \\lim _ { k } d ( x ^ { n _ { k } } , C ) = d ( y , C ) = 0 \\end{align*}"}
-{"id": "4255.png", "formula": "\\begin{align*} h = \\pm ( t ^ { r / 2 } q ^ { - s / 2 } + t ^ { - r / 2 } q ^ { s / 2 } ) , r , s \\geq 1 , r s = n . \\end{align*}"}
-{"id": "915.png", "formula": "\\begin{align*} \\tilde z _ \\gamma ( s ) : = \\prod _ { k \\ge 1 } \\left ( 1 - e ^ { - l ( \\gamma ) ( s + k ) } \\right ) ^ { - k ^ 2 } \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} T ( t , x ) - T ^ { \\infty } = O \\Big ( \\frac { 1 } { \\sqrt { x } } \\Big ) , ( N + i B ) ( t , x ) - N ^ { \\infty } e ^ { i a ^ { 2 } \\log \\frac { \\sqrt { t } } { x } - i x ^ { 2 } / 4 t } = O ( \\frac { 1 } { \\sqrt { x } } ) , \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} d _ \\Gamma ( x , i ; y , j ) \\ = \\ \\begin{cases} \\| x - y \\| & i = j , \\\\ 1 + \\| x \\| + \\| y \\| & i \\neq j . \\end{cases} \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{align*} x _ \\alpha & = u _ { \\alpha \\beta } ^ { p - 1 } x _ \\beta + r _ { \\alpha \\beta } z _ \\beta \\\\ y _ \\alpha & = u _ { \\alpha \\beta } ^ p y _ \\beta \\\\ z _ \\alpha & = z _ \\beta . \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} H ^ { 1 , 0 } = \\overline { ( \\kappa _ B ^ { 1 , 0 } ) ^ \\sharp } = \\frac 1 2 ( \\kappa _ B ^ \\sharp - i J \\kappa _ B ^ \\sharp ) , H ^ { 0 , 1 } = \\overline { H ^ { 1 , 0 } } . \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} \\lambda _ { U , t } ^ s ( r , y ) = \\lambda _ t \\Big ( 1 - \\exp ( - \\pi ( r + y ) ^ 2 \\lambda _ t ( \\frac { P _ t } { P _ k } ) ^ { 2 / \\alpha } ) \\Big ) . \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} \\norm { x } _ { p } = \\begin{cases} \\left ( \\sum _ { i \\in \\mathcal { N } } \\abs { x _ i } ^ { p } \\right ) ^ { 1 / p } , & 1 \\leq p < \\infty , \\\\ \\max _ { i \\in \\mathcal { N } } \\abs { x _ i } , & p = \\infty , \\end{cases} \\end{align*}"}
-{"id": "4869.png", "formula": "\\begin{align*} P '^ 2 = 4 P ^ 3 + A P + B \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{align*} \\sum _ { ( \\vec { a } , q ) \\in \\mathcal { M } ( \\theta _ 0 ) } & \\frac { | S _ { \\vec { a } , q } ( \\vec { \\nu } ) | } { q ^ n } \\widetilde { C } ^ { r ^ 2 - 1 + ( r - 1 ) \\delta \\eta ^ { - 1 } + \\epsilon } ( \\widetilde { C } ^ { r - 1 } P ^ \\eta ) ^ { - 1 - \\delta \\eta ^ { - 1 } + \\epsilon } = \\mathfrak { O } _ 2 . \\end{align*}"}
-{"id": "7959.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } ( a _ t ) ^ { 1 / t } & = \\lim _ { t \\rightarrow \\infty } ( h _ t ( \\beta _ t ) ) ^ { - 1 / t } \\\\ & = \\lim _ { t \\rightarrow \\infty } \\{ \\beta _ t \\exp [ ( t - 1 ) u ( \\beta _ t ) ] \\} ^ { - 1 / t } \\\\ & = \\lim _ { t \\rightarrow \\infty } \\exp [ - u ( \\beta _ t ) ] \\\\ & = \\left ( \\int _ 0 ^ \\infty x ^ { - 1 } d \\mu ( x ) \\right ) ^ { - 1 } ; \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{align*} \\Lambda _ { k } = \\sup _ { 1 \\leq j \\leq k } ( C _ { j } - C _ { j + 1 } ) ^ { - 1 } , k \\geq 1 , \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{align*} d q _ t ^ \\prime = & \\frac { 1 } { m } ( p _ t ^ \\prime - \\psi ( q _ t ^ \\prime ) ) d t , \\\\ d ( p _ t ^ \\prime ) _ i = & \\left ( - \\frac { 1 } { m } \\gamma ( t ^ * ) ( ( p _ t ^ \\prime ) _ i - \\psi _ i ( q _ t ^ \\prime ) ) - \\partial _ { q ^ i } V ( t ^ * , q _ t ^ \\prime ) + \\tilde F ( t ^ * , q _ t ^ \\prime ) \\right . \\\\ & \\left . + \\frac { 1 } { m } \\partial _ { q ^ i } \\psi _ k ( q _ t ^ \\prime ) \\delta ^ { k j } ( ( p _ t ^ \\prime ) _ j - \\psi _ j ( q _ t ^ \\prime ) ) \\right ) d t + \\sigma ( t ^ * , q _ t ^ \\prime ) \\delta _ { i \\rho } d W ^ \\rho _ t \\end{align*}"}
-{"id": "2519.png", "formula": "\\begin{align*} A ^ { D } = \\left \\{ \\begin{aligned} & \\prod ^ { k } _ { l = 1 } B _ { l } ( G _ { k } B _ { k } ) ^ { - k - 1 } \\prod ^ { k } _ { l = 1 } G _ { k + 1 - l } , & & { } ~ G _ { k } B _ { k } ~ ; \\\\ & 0 , & & { } ~ G _ { k } B _ { k } = 0 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} & \\frac 1 2 D _ { \\frac 1 2 } \\Big ( p _ { Z | X Y } ( \\cdot | x _ 1 , y _ 1 ) \\big \\| p _ { Z | X Y } ( \\cdot | x _ 2 , y _ 2 ) \\Big ) = - \\log \\Big ( \\sum _ { z } p _ { Z | X Y } ( z | x _ 1 , y _ 1 ) ^ { \\frac 1 2 } \\times p _ { Z | X Y } ( z | x _ 2 , y _ 2 ) ^ { \\frac 1 2 } \\Big ) \\\\ & \\quad \\leq - \\log \\Big ( p _ { Z | X Y } ( \\mathtt { e } | x _ 1 , y _ 1 ) ^ { \\frac 1 2 } \\times p _ { Z | X Y } ( \\mathtt { e } | x _ 2 , y _ 2 ) ^ { \\frac 1 2 } \\Big ) = - \\log ( \\epsilon ) \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} \\eta _ N : = \\frac 1 N \\sum _ { i = 1 } ^ N \\frac { ( T ^ { * } ) ^ i } { \\lambda ^ i } \\omega \\textrm { s o t h a t } ( T ^ * - \\lambda ) \\eta _ N = \\frac 1 N \\left ( \\frac { ( T ^ { * } ) ^ { N + 1 } } { \\lambda ^ N } - T ^ * \\right ) \\omega \\end{align*}"}
-{"id": "5728.png", "formula": "\\begin{align*} \\tilde \\delta _ o = \\tilde \\kappa _ t \\big ( \\tilde \\kappa _ g \\tilde \\kappa _ t - \\tilde \\kappa _ \\nu ' \\big ) , \\tilde \\delta _ o ' = \\tilde \\kappa _ t \\big ( \\tilde \\kappa _ t \\tilde \\kappa _ g ' + 2 \\tilde \\kappa _ g \\tilde \\kappa _ t ' - \\tilde \\kappa _ \\nu '' \\big ) u = 0 \\end{align*}"}
-{"id": "9093.png", "formula": "\\begin{align*} { \\rm E } \\{ { \\rm t r } \\{ { \\bf W } { \\bf W } ^ H \\} \\} = 1 . \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{align*} \\vec { x } ^ { ( j ) } ( \\mu ) = \\Big ( x ^ { ( j ) } _ 1 < \\cdots < x ^ { ( j ) } _ { m _ j } \\Big ) \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{align*} M ^ l ( u ^ * ) = & \\sum _ Y \\frac { 1 - c _ Y } { v _ { \\mathcal { L } } ( \\mu _ Y ) } \\int _ { \\Delta ' _ Y } p ( \\nu ) u ^ * ( p ) P _ { D H } ' ( p ) d \\sigma - \\int _ { \\Delta ^ + } d _ p P _ { D H } ' ( 4 \\rho _ H ) u ^ * ( p ) d p \\\\ & + \\int _ { \\Delta ^ + } u ^ * ( p ) ( \\sum \\frac { \\chi ^ { a c } ( \\alpha ^ { \\vee } ) } { q ( \\alpha ^ { \\vee } ) } - \\bar { S } _ { \\Theta } ) P _ { D H } d q . \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} \\partial _ H \\ell _ 1 ( \\mathcal { N } , \\R ) = \\left \\lbrace x \\mapsto h _ { \\epsilon , \\mu } ^ { \\mathcal { I } } ( x ) \\mathrel { \\bigg \\vert } \\begin{aligned} & \\emptyset \\subsetneq \\mathcal { I } \\subseteq \\mathcal { N } , \\ ; \\epsilon \\in \\{ - 1 , + 1 \\} ^ { \\mathcal { I } } , \\\\ & \\mu \\in \\R ^ { \\mathcal { N } \\setminus \\mathcal { I } } \\end{aligned} \\right \\rbrace , \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{align*} \\nu = - s , \\ \\ \\beta = \\frac { y ^ 2 } { 4 } , \\ \\ \\gamma = L ( \\xi , \\sigma ) ^ 2 , \\end{align*}"}
-{"id": "2821.png", "formula": "\\begin{align*} \\forall \\ ; t \\in [ 0 , T ] , \\ ; u _ t ^ { \\varepsilon , N } = K _ { \\varepsilon } \\ast \\gamma ^ { \\varepsilon , N } _ t \\textrm { a n d } \\bar u ^ { \\varepsilon , N , n } _ t = K _ { \\varepsilon } \\ast \\bar \\gamma ^ { \\varepsilon , N , n } _ t . \\end{align*}"}
-{"id": "6079.png", "formula": "\\begin{align*} x \\odot y : = \\alpha _ { T } ^ { i - 1 } ( x ) \\otimes y _ { 1 } \\otimes \\cdots \\alpha _ { T } ( y _ { 2 } \\otimes \\cdots \\otimes y _ { i } ) , \\end{align*}"}
-{"id": "610.png", "formula": "\\begin{align*} \\begin{cases} T _ k = T _ 1 - \\beta _ 1 q ^ 3 \\sum _ { j = 0 } ^ { k - 2 } q ^ j , k > 2 \\\\ T _ 2 = 3 / 4 - \\beta _ 1 q ^ 3 \\\\ T _ 1 = 3 / 4 \\end{cases} \\quad \\begin{cases} T _ { k + 1 } = T _ k - \\beta _ 1 q ^ { k + 2 } , k > 1 \\\\ T _ 1 = 3 / 4 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} M _ { p , T } : = \\sup _ { t \\in [ 0 , T ] , x \\in \\mathbb { R } } \\| \\psi ( t , x ) \\| _ p < \\infty \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} \\hat C ^ n ( A ) = \\left \\{ \\begin{array} { c l } C ^ n ( A ) , & n \\ge 0 , \\\\ D ^ { n + 1 } ( A ) , & n < 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} \\Psi ( f ) ( \\eta ) = \\xi ( \\kappa ( \\eta ) ) f ( \\kappa ( \\eta ) ) , \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{align*} | a _ { k , n } ^ { ( \\alpha ) } | \\leq | \\gamma _ { k , n } ^ { ( \\alpha ) } | + \\frac { c _ { 9 } } { \\rho _ 2 ^ { k - n } } \\sum _ { w = 1 } ^ { d } \\sum _ { y = 1 } ^ { m _ w } | \\gamma _ { n - m _ w + y , n } ^ { ( w ) } | , \\alpha = 1 , 2 , \\ldots , d , k \\geq n + 1 . \\end{align*}"}
-{"id": "8911.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 1 } \\int _ { \\partial \\Delta ' _ { s } } \\dot { u } _ t ^ * ( - u ^ { * , i , j } _ { t , j } + I _ { H , i } ( a ) ) \\nu _ i P _ { D H } ' d \\sigma = \\sum _ Y \\int _ { \\Delta ' _ Y } 2 n _ Y \\frac { \\nu } { \\mu _ Y } \\dot { u } _ t ^ * P _ { D H } ' d \\sigma \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} E & = C \\Big ( q ^ { ( 1 1 ) } _ { Z _ a Z _ b } \\big \\| q ^ { ( 2 2 ) } _ { Z _ a Z _ b } \\Big ) = D _ { \\frac 1 2 } \\Big ( p _ { Z | X Y } ( \\cdot | x _ 1 , y _ 1 ) \\big \\| p _ { Z | X Y } ( \\cdot | x _ 2 , y _ 2 ) \\Big ) . \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} \\displaystyle { \\lim _ { \\tau \\downarrow 0 } } \\ , \\displaystyle { \\frac { f ( \\bar { x } + \\tau v ) - f ( \\bar { x } ) - \\tau \\ , \\nabla f ( \\bar { x } ) ^ T v - \\displaystyle { \\frac { \\tau ^ 2 } { 2 } } \\ , v ^ T F ^ { \\ , \\prime } ( \\bar { x } ; v ) } { \\tau ^ 2 } } \\ , = \\ , 0 , \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} P _ { D H , i , j } ' ( p ) = P _ { D H } ' ( p ) \\left ( \\sum _ { \\alpha , \\beta } \\frac { \\alpha ^ { \\vee , i } \\beta ^ { \\vee , j } } { ( 2 \\chi - p ) ( \\alpha ^ { \\vee } ) ( 2 \\chi - p ) ( \\beta ^ { \\vee } ) } + \\sum _ { \\alpha } \\frac { \\alpha ^ { \\vee , i } \\alpha ^ { \\vee , j } } { ( 2 \\chi - p ) ( \\alpha ^ { \\vee } ) ^ 2 } \\right ) . \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} U ( c _ 1 , c _ 2 ) = \\begin{pmatrix} 1 & c _ 1 & 0 & - c _ 1 ^ 2 / 2 \\\\ 0 & 1 & 0 & - c _ 1 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & 0 & c _ 2 & c _ 2 ^ 2 / ( 2 \\epsilon ) \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & c _ 2 / \\epsilon \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} C = \\left \\{ x \\in E : f _ { \\hat \\tau } ( x , x ) > F ( x , x ) \\right \\} . \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} - \\Delta u - \\lambda _ 1 u = f ( x ) . \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} [ \\partial _ t , \\mathfrak { H } ] f & = [ z _ t , \\mathfrak { H } ] \\frac { f _ \\alpha } { z _ \\alpha } , \\\\ \\partial _ \\alpha \\mathfrak { H } f & = z _ \\alpha \\mathfrak { H } \\frac { f _ \\alpha } { z _ \\alpha } . \\end{align*}"}
-{"id": "3898.png", "formula": "\\begin{align*} A _ X f _ { \\hat \\tau } ( x , y ) = 0 , \\enskip x \\in C . \\end{align*}"}
-{"id": "351.png", "formula": "\\begin{align*} [ T ( \\psi _ 1 * _ G \\psi _ 2 ) ] ( k ' ) & = | H | \\cdot \\sum _ { g \\in G } \\psi _ 1 ( k _ g h _ g ) \\psi _ 2 ( h _ g ^ { - 1 } k _ g ^ { - 1 } k ' ) \\\\ [ 5 p t ] & = | H | \\cdot \\sum _ { g \\in G } \\psi _ 1 ( k _ g ) \\psi _ 2 ( k _ g ^ { - 1 } k ' ) . \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} \\Phi ( [ T _ - , f , T _ + ] [ U _ - , g , U _ + ] ) & = \\Phi ( [ T _ - , f g , U _ + ] ) = ( ( T _ - , f g , U _ + ) ) \\\\ & = ( ( T _ - , f , T _ + ) ) ( ( U _ - , g , U _ + ) ) = \\Phi ( [ T _ - , f , T _ + ] ) \\Phi ( [ U _ - , g , U _ + ] ) \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} X _ i = \\partial _ { x _ i } - \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ k \\mathcal A _ { i j } x _ j \\partial _ { x _ { n + 1 } } , \\ \\ i = 1 , . . . , n . \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} L _ j ^ d \\circ A = A \\circ L _ j ^ d \\ \\end{align*}"}
-{"id": "8144.png", "formula": "\\begin{align*} \\xi : a ( z ) \\mapsto a ( z ) \\frac { ( 1 - z ) ( 1 - a ' ( 1 ) ) } { a ( z ) - z } = : \\pi ( z ) . \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} \\sum \\lambda _ { i _ 1 i _ 2 } h _ { i _ 1 } h _ { i _ 2 } \\ = \\ \\sum \\lambda _ { i _ 1 i _ 2 } ( u _ 1 ^ { n + 1 - i _ 1 } v _ 1 ^ { i _ 1 } + u _ 2 ^ { n + 1 - i _ 1 } v _ 2 ^ { i _ 1 } ) ( u _ 1 ^ { n + 1 - i _ 2 } v _ 1 ^ { i _ 2 } + u _ 2 ^ { n + 1 - i _ 2 } v _ 2 ^ { i _ 2 } ) . \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} \\xi _ n ( x ) = \\frac { x ^ { \\tfrac { n - 1 } { 2 } } ( x ^ 2 - 1 ) } { 2 + 2 x ^ { \\tfrac { - n + 1 } { 2 } } + 2 x ^ { - n + 1 } + x ^ { \\tfrac { - 3 n + 3 } { 2 } } + x ^ { - \\tfrac { 3 n + 1 } { 2 } } } \\ge \\frac { x ^ { \\tfrac { n - 1 } { 2 } } ( x ^ 2 - 1 ) } { 8 } , \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} g _ 1 ( X , X ) = - 2 ( N t ) ^ { - 1 } k ( X , X ) = - 2 ( 1 - \\chi ( r ) \\tfrac { 2 m } { r } ) ^ { 1 / 2 } k ( X , X ) . \\end{align*}"}
-{"id": "8325.png", "formula": "\\begin{align*} \\frac { \\partial _ \\alpha } { | z _ \\alpha | } z _ { t t } = - ( \\frac { \\partial _ \\alpha } { | z _ \\alpha | } ) ^ 2 P e ^ { i \\theta } + i \\frac { \\partial _ \\alpha a } { | z _ \\alpha | } e ^ { i \\theta } - \\frac { \\partial _ \\alpha P } { | z _ \\alpha | } e ^ { i \\theta } \\frac { i \\partial _ \\alpha \\theta } { | z _ \\alpha | } + i a e ^ { i \\theta } \\frac { i \\partial _ \\alpha \\theta } { | z _ \\alpha | } . \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} \\psi _ j ( n _ j ; \\lambda _ j ) = \\sum _ { k = 1 } ^ { n _ j } ( - 1 ) ^ { k - 1 } \\binom { n _ j } { k } \\frac { p _ j k } { \\lambda _ j + p _ j k } , 1 \\leq n _ j \\leq M _ j . \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} \\delta ( x ) = \\left ( \\sum _ { k = 1 } ^ m y _ k ^ 2 \\right ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} a _ { d , n } : = \\widetilde { a } _ { d , n } - \\frac { 1 } { \\lvert \\Omega \\rvert } \\int _ { \\Omega } \\widetilde { a } _ { d , n } \\ ; \\d x . \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{gather*} x _ { 2 \\varepsilon _ i } ( c ) : = \\exp ( c X _ { 2 \\varepsilon _ i } ) , h _ { 2 \\varepsilon _ i } ( c ) : = \\exp ( c H _ { 2 \\varepsilon _ i } ) , x _ { - 2 \\varepsilon _ i } ( c ) : = \\exp ( c X _ { - 2 \\varepsilon _ i } ) \\end{gather*}"}
-{"id": "2891.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\gamma ^ { ( 1 ) } _ { N , 0 } = | u _ 0 \\rangle \\langle u _ 0 | \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow \\infty } \\Lambda _ n ( E , a ^ { ( n ) } ) ^ { 1 / n } = 1 , \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} \\theta _ { i , n } : = s _ n ^ { - 1 } \\max \\left ( \\sqrt { \\log ( d ( n ) ) / 2 } , 1 \\right ) v _ { i , d ( n ) } \\in \\Theta _ { d ( n ) } , \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{align*} s _ j ( T ) = o ( j ^ { - \\left ( \\frac 1 2 + \\frac { \\mu _ 1 } { n _ 1 } + \\frac { \\mu _ 2 } { n _ 2 } \\right ) } ) . \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} | | \\xi | | _ A ^ 2 = \\xi A \\xi ^ * . \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow \\infty } \\bigg \\{ \\frac { N [ \\mathfrak { e } , \\lambda V ] } { N _ { s c } [ \\mathfrak { e } , \\lambda V ] } \\bigg \\} \\ = \\lim _ { \\lambda \\rightarrow \\infty } \\bigg \\{ \\frac { N [ \\mathfrak { e } , \\lambda V ] } { 1 + N _ { s c } [ \\mathfrak { e } , \\lambda | x | ^ { ^ { \\alpha _ { d } } } V ] } \\bigg \\} \\ = \\ 1 . \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} { } \\int _ { \\Omega } { \\vert \\nabla { z _ { n } } \\vert } ^ { p } \\phi ^ { p } = - p \\int _ { \\Omega } \\phi ^ { p - 1 } z _ { n } { \\vert \\nabla { z _ { n } } \\vert } ^ { p - 2 } \\nabla { \\phi } . \\nabla { z _ { n } } + \\int _ { \\Omega } \\frac { \\lambda _ { 0 } z _ { n } \\phi ^ { p } } { ( z _ { n } + \\frac { 1 } { n } ) ^ { \\delta } } + \\int _ { \\Omega } z _ { n } ^ { q + 1 } \\phi ^ { p } . \\end{align*}"}
-{"id": "1311.png", "formula": "\\begin{align*} ( \\widehat { S } _ { e e } ^ \\tau + \\widehat { S } _ { e e } ^ { \\tau ^ \\prime } ) \\tilde { \\psi } _ { h , i } ^ e = \\alpha _ i ^ e ( \\widetilde { S } ^ \\tau _ { e e } + \\widetilde { S } ^ { \\tau ^ \\prime } _ { e e } ) \\tilde { \\psi } _ { h , i } ^ e . \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} \\int _ 0 ^ { U ( N ; \\alpha ) ^ { - 1 } } ( - \\ln x ) ^ r d x = e ^ { - \\ln ^ { \\alpha } N } ( \\ln N ) ^ { r \\alpha } \\ , [ 1 + o ( 1 ) ] , \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{align*} \\| T _ { k ( n ) } ^ m \\| \\ge \\| A _ n ^ m \\| - \\sum _ { k = 1 } ^ m \\binom m k \\| A _ n \\| ^ { m - k } \\| B _ n \\| ^ k . \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} | | \\mathfrak { A } ( \\alpha , \\nu ) + 2 \\rho _ { \\alpha } | | = | | h \\mathcal { A } ( \\alpha , \\nu ) + 2 \\rho _ { \\alpha } | | , \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} ( \\partial _ t - \\Delta ) ^ { s } u = V ( x , t ) u , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ s \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "2994.png", "formula": "\\begin{align*} U ( { \\mathcal F } ) : = \\bigcap _ { A ' \\in { \\mathcal F } } \\bigcup _ { \\rho \\in ( A / A ' ) ^ * } U ( e ( \\rho ) ) . \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} \\rho ( f _ \\pm ) = \\frac 1 2 \\iff \\frac { 1 } { 1 + a } \\le b \\le \\frac { 1 + a - a ^ 2 } { 1 + a } . \\end{align*}"}
-{"id": "2691.png", "formula": "\\begin{align*} P ^ + ( x , y ) = \\sum _ { I \\in \\mathcal { D } ^ + } h _ I ( y ) ( h _ { I ^ - } ( x ) - h _ { I ^ + } ( x ) ) \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} \\psi ( \\nu , \\mu ) ( T _ i \\cdot f _ { \\nu } ) = \\psi ( \\nu , \\mu ) f _ { s _ i \\nu } = \\psi ( s _ i \\nu , s _ i \\mu ) f _ { s _ i \\nu } . \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} U ^ { ( 0 ) } : = \\left \\{ x \\in U \\ , : \\ , \\gamma ( x ) > \\frac { C _ 0 } { 2 } \\right \\} \\subset U \\subset \\R ^ m \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} \\Omega _ { j , \\bar { \\alpha } } = \\Omega _ { \\alpha , \\bar { j } } = 0 . \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} \\tau ( x , y ) : = \\inf \\int _ x ^ y \\frac { | \\d s | } { c ( s ) } \\ , , \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} q ( x , y ) = \\frac { x y ( 1 - y C ( x y ) ) } { 1 - x - y } . \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} \\begin{array} [ p o s ] { l } \\nabla _ { F } \\nabla _ { F } \\varphi ( U , V ) ( q _ 1 ) = 0 , \\\\ \\noalign { \\smallskip } R i c _ { F } ( U , V ) ( q _ 1 ) = 0 \\end{array} \\forall q _ 1 \\in F . \\end{align*}"}
-{"id": "8481.png", "formula": "\\begin{align*} \\{ x y x \\} = x ( \\widetilde { y } ^ t x ) , \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{align*} \\sup _ { y < 0 } \\int _ { \\mathbb R } | C _ \\lambda ( x , y ) | \\ , \\mathrm d x = \\frac { 1 } { 2 \\sqrt { | \\lambda | } } \\left ( \\frac { \\sqrt { 2 } } { \\operatorname { I m } \\sqrt { \\lambda } } + \\frac { 1 } { \\operatorname { R e } \\sqrt { \\lambda } } \\right ) . \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} { \\rm C o e f f } [ f _ { \\mu } , p + m _ 1 ] = \\sum _ { \\nu \\in \\sigma ( \\epsilon ) } \\psi ( \\nu , \\mu ; t ) f _ { \\nu } ( z ; t ^ { - p - m _ 1 } , t ) , \\forall \\ \\mu \\in \\sigma ( \\delta ) , \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} v _ n ( y ) = R _ n ^ { \\frac { p } { q - p + 1 } } u _ n ( P _ n + R _ n y ) \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} \\{ \\sigma _ i ( M _ 1 , \\dots , M _ r ) \\le \\vert p \\vert \\} _ { i = 1 } ^ l . \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} \\beta _ 1 = \\cdots = \\beta _ p = A + 1 , \\ \\ \\ \\beta _ { p + 1 } = q , \\ \\ \\ \\beta _ { p + 2 } = \\ldots = \\beta _ { n } = A + 2 - m . \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{align*} p ( 0 , 0 | v , v ) = t , p ( 0 , 1 | v , v ) = p ( 1 , 0 | v , v ) = 0 , p ( 1 , 1 | v , v ) = 1 - t . \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} \\left \\{ z _ t < \\frac { \\epsilon _ i } { \\rho \\xi _ i } \\right \\} & = \\left \\{ z _ t < \\frac { \\epsilon _ i } { \\rho \\bar { \\xi } _ i \\frac { \\rho z _ t - \\epsilon _ 1 } { \\rho ( 1 + \\epsilon _ 1 ) z _ t } } \\right \\} . \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} ( i i ) f _ { \\beta } ( z ) = - \\frac { 1 - z ^ N } { \\zeta _ { \\beta } ( z ) } \\mbox { i f $ \\beta $ i s a s i m p l e P a r r y n u m b e r } \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{gather*} \\mathcal { F } _ \\tau : = \\mathcal { F } , \\mathcal { C } _ { \\tau } : = \\tau \\mathcal { C } , \\mathcal { A } _ { \\tau } : = \\mathcal { F } _ \\tau + \\mathcal { C } _ { \\tau } = \\mathcal { F } \\oplus ( \\tau \\mathcal { C } ) . \\end{gather*}"}
-{"id": "6582.png", "formula": "\\begin{align*} W _ v ( T ) = \\int \\limits _ 0 ^ T w ( \\phi _ t ( v ) ) \\ , d t . \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} g ( q , z ) g ( q , z / q ) = g ( q , z / q ) + z , \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} \\mathcal { V } _ \\ast ( t ) : = \\{ v \\in C ^ 2 ( \\O ( { \\Gamma ( t ) } ) ) \\mid \\nabla v \\cdot \\nabla \\phi = 0 \\} \\Vert v \\Vert _ { \\mathcal { V } } : = \\left ( \\Vert v \\Vert _ { H ^ 1 ( \\Gamma ( t ) ) } ^ 2 + \\Vert \\nabla \\phi \\cdot \\nabla v \\Vert _ { L ^ 2 ( \\O ( \\Gamma ( t ) ) ) } ^ 2 \\right ) ^ { \\frac 1 2 } , \\end{align*}"}
-{"id": "8447.png", "formula": "\\begin{align*} K _ D ( z , w ) = h ( z , w ) ^ { - p } , \\end{align*}"}
-{"id": "5755.png", "formula": "\\begin{align*} f _ n ( m ) = 2 m - 1 + 2 t _ 3 + \\left \\lceil \\frac { 3 } { 2 } t _ 2 \\right \\rceil + t _ 1 . \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} \\dfrac 1 2 \\mathcal { L } | A | ^ 2 = | \\nabla A | ^ 2 - \\dfrac 1 2 | A | ^ 2 - | A | ^ 4 . \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{align*} I _ 1 ( N ) : = \\int _ 0 ^ { U ( N ; \\alpha ) } \\left ( 1 - \\frac { x } { N } \\right ) ^ { N - 1 } \\left ( 1 - \\frac { \\ln x } { \\ln N } \\right ) ^ r d x \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} [ c , z _ { \\sigma ( 1 ) } ] [ z _ { \\sigma ( 2 ) } , z _ { \\sigma ( 3 ) } , z _ { \\sigma ( 4 ) } ] = ( - 1 ) ^ { \\sigma } [ c , z _ 1 ] [ z _ 2 , z _ 3 , z _ 4 ] . \\end{align*}"}
-{"id": "3271.png", "formula": "\\begin{align*} \\sigma ^ N _ \\alpha ( \\bar { a } _ { \\bar { u } ^ \\smallfrown \\bar { v } } ) = \\sigma ^ N _ \\beta ( \\bar { a } _ { \\bar { u } ^ \\smallfrown \\bar { w } } ) . \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{align*} E _ j : = \\frac { m _ j } { | X | } \\sum _ { i = 0 } ^ d \\omega _ j ( a _ i ) A _ i \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{align*} Q ( x , y ; t ) = \\sum _ { i , j , k \\geq 0 } \\mathbb P [ P _ 0 \\stackrel { k } { \\longrightarrow } ( i , j ) ] x ^ { i } y ^ { j } t ^ { k } . \\end{align*}"}
-{"id": "3330.png", "formula": "\\begin{gather*} p ( 0 , 0 | v , w ) = \\frac { s } { | E | } , p ( 0 , 1 | v , w ) = p ( 1 , 0 | v , w ) = t - \\frac { s } { | E | } , \\\\ p ( 1 , 1 | v , w ) = 1 - 2 t + \\frac { s } { | E | } . \\end{gather*}"}
-{"id": "3249.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\| \\sum _ { \\nu = 0 } ^ { \\infty } b _ { \\nu , n } ^ { ( \\alpha ) } \\Phi _ \\nu \\| _ K ^ { 1 / n } \\leq \\frac { \\| \\Phi \\| _ K } { \\rho _ { | \\textup { \\textbf { m } } | } ( \\textup { \\textbf { F } } ) } , \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} - \\Delta u - V ( x ) u = f ( x ) \\end{align*}"}
-{"id": "5988.png", "formula": "\\begin{align*} f ^ h ( x , y ) = h ( y ) - h ( x ) . \\end{align*}"}
-{"id": "406.png", "formula": "\\begin{align*} \\Phi ^ { H _ { 4 0 } } _ { i , i } = p _ 5 ( \\sigma _ i ^ { ( 4 0 ) } ) \\mbox { f o r T F - C G L S , } \\Phi ^ { H _ { 4 0 } } _ { i , i } = \\begin{cases} 1 \\ ! \\ ! & \\quad \\\\ 0 \\ ! \\ ! & \\quad \\end{cases} \\mbox { f o r h y b r i d G M R E S , } \\end{align*}"}
-{"id": "6912.png", "formula": "\\begin{align*} a _ x = b _ x - \\beta I [ \\mathbf { X } = x ] , \\beta = \\frac { \\alpha } { \\omega } . \\end{align*}"}
-{"id": "5125.png", "formula": "\\begin{align*} v ( [ 1 \\ \\dots \\ t - 1 \\mid i _ 1 \\ \\dots \\ i _ { t - 1 } ] ) \\ = \\ n + 1 - i _ { t - 1 } . \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} s \\frac { d } { d s } \\log Z _ \\gamma ( s ) = \\sum _ { k \\in \\N } \\frac { s l ( \\gamma ) } { e ^ { l ( \\gamma ) ( s + k ) } - 1 } \\textrm { a n d } \\frac { d } { d s } \\log z _ \\gamma ( s ) ^ { - 1 } = \\sum _ { k \\in \\N } \\frac { k } { e ^ { l ( \\gamma ) ( s + k ) } - 1 } . \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} n \\cdot ( s \\otimes s ' ) = ( s s ' ) \\cdot n \\ , ; \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} H _ \\kappa \\varphi ( r ) = \\lambda \\varphi ( r ) \\ , , \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} | S _ \\mu ( m - 1 ) \\setminus \\lambda | = d _ 1 + \\cdots + d _ { m - 1 } , & & | T _ \\mu ( m + 1 ) \\setminus \\lambda | = 0 . \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} \\sigma ( f ( x _ 1 , \\dots , x _ n ) ) : = f ( x _ { \\sigma ( 1 ) } , \\dots , x _ { \\sigma ( n ) } ) . \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} \\begin{pmatrix} 5 & 2 & 2 & 2 \\\\ 1 0 & 7 & 8 & 8 \\\\ 1 0 & 2 & 1 & 2 \\\\ 1 0 & 1 & 6 & 8 \\end{pmatrix} . \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } G ( d _ k ( i ) ) \\longrightarrow 0 \\end{align*}"}
-{"id": "8167.png", "formula": "\\begin{align*} \\partial _ { x _ i } A _ j - \\partial _ { x _ j } A _ i = c _ { i j } \\in \\mathbb R \\ \\ { \\rm f o r \\ e v e r y } \\ i , j = 1 , . . . , n , \\end{align*}"}
-{"id": "5419.png", "formula": "\\begin{align*} - \\beta \\ , E ^ \\dagger ( x _ n ) + \\Omega ( x _ n ) & = \\frac { \\beta } { \\tilde { \\beta } } \\ , \\bigl ( - \\tilde { \\beta } \\ , E ^ \\dagger ( x _ n ) + \\Omega ( x _ n ) \\bigr ) + \\left ( 1 - \\frac { \\beta } { \\tilde { \\beta } } \\right ) \\ , \\Omega ( x _ n ) \\\\ & \\geq \\frac { \\beta \\ , \\tilde { c } } { \\tilde { \\beta } } + \\left ( 1 - \\frac { \\beta } { \\tilde { \\beta } } \\right ) \\ , \\Omega ( x _ n ) , \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { N } p _ j \\rho \\bigl ( f _ j ( x ) , f _ j ( y ) \\bigr ) \\leq c \\rho ( x , y ) , \\end{align*}"}
-{"id": "4335.png", "formula": "\\begin{align*} q = \\exp ( 2 \\pi i \\tau ) . \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} \\begin{aligned} & m _ { j j } : = \\frac { H ( y _ { i _ j } , \\ , y _ { i _ j } ) } { \\bigl ( K ( y _ { i _ j } ) \\bigr ) ^ { \\frac { n - 2 } { 2 } } } , \\ , \\ , \\forall \\ , \\ , 1 \\leq j \\leq s \\\\ & { } m _ { j k } : = - \\frac { G ( y _ { i _ j } , \\ , y _ { i _ k } ) } { \\bigl ( K ( y _ { i _ j } ) K ( y _ { i _ k } ) \\bigr ) ^ { \\frac { n - 2 } { 4 } } } , \\forall \\ , \\ , k \\neq j , \\ , \\ , 1 \\leq k , \\ , j \\leq s . . \\end{aligned} \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} C : Y '^ 2 = h ^ 3 + h ^ 2 + 4 h + 4 \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} \\widetilde { h } = \\Phi _ 4 ( \\Phi _ 3 ( \\Phi _ 2 ( \\Phi _ 1 ( h ) ) ) ) , \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} & \\langle \\Omega , X ^ n \\Omega \\rangle = \\int _ \\R x ^ n d \\mu ( x ) . \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} \\mathcal { L } w = \\bigtriangleup _ { g } w - \\frac { 1 } { 2 } \\left \\langle X , \\nabla _ { g } w \\right \\rangle = - \\left \\vert A \\right \\vert ^ { 2 } w \\leq 0 , \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} y ^ 2 u '' ( y ) + ( 1 - 2 \\alpha ) y u ' ( y ) + \\left [ ( \\alpha ^ 2 - \\nu ^ 2 \\gamma ^ 2 ) - \\beta ^ 2 \\gamma ^ 2 y ^ { 2 \\gamma } \\right ] u ( y ) = 0 , \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} F ( x _ c , y _ c ) = \\int F ( x , y ) d \\mu _ \\theta \\end{align*}"}
-{"id": "6969.png", "formula": "\\begin{align*} \\{ \\alpha _ 1 , \\alpha _ 2 , . . . , \\alpha _ i \\} = \\{ t _ { a _ 1 ' } - t _ { b _ 1 ' } , t _ { a _ 2 ' } - t _ { b _ 2 ' } , . . . , t _ { a _ { i } ' } - t _ { b _ { i } ' } \\} , \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} \\dim { H _ { j } ( \\mathbb { R } ^ { n } ) } = { n + j - 1 \\choose n - 1 } - { n + j - 3 \\choose n - 1 } . \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} f ( z ) = z ^ m \\sum _ { j = 0 } ^ { \\infty } a _ j ( z - q ) ^ j \\end{align*}"}
-{"id": "1756.png", "formula": "\\begin{align*} t ( x ' , D _ { x } ) u = \\sum _ { j = 0 } ^ { r - 1 } s _ j ( x ' , D _ { x ' } ) \\partial _ { x _ n } ^ j u | _ { x _ n = 0 } + t _ 0 ( x ' , D _ { x } ) u \\quad u \\in \\mathcal { S } ( \\overline { \\R ^ n _ + } ) , \\end{align*}"}
-{"id": "6857.png", "formula": "\\begin{align*} \\tilde { L } _ Q ^ \\top \\tilde { L } _ P = \\begin{pmatrix} \\tilde { U } _ 1 & \\tilde { U } _ 2 \\\\ \\end{pmatrix} \\begin{pmatrix} \\tilde { \\Sigma } _ 1 & 0 \\\\ 0 & \\tilde { \\Sigma } _ 2 \\\\ \\end{pmatrix} \\begin{pmatrix} \\tilde { V } _ 1 ^ \\top \\\\ \\tilde { V } _ 2 ^ \\top \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} J \\cdot f : = d f ^ { - 1 } \\circ J \\circ d f \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\int _ { 0 < s _ 1 < s _ 2 < 1 } d \\iota _ N ( X ^ \\star ) ( s _ 1 ) \\otimes d \\iota _ N ( X ^ \\star ) ( s _ 2 ) & = \\frac { 1 } { 2 N } \\sum _ { k = 0 } ^ { N - 1 } F ^ \\star _ k \\otimes F ^ \\star _ k \\\\ & \\phantom { = } + \\frac { 1 } { N } \\sum _ { 0 \\leq k < l < N } F ^ \\star _ k \\otimes F ^ \\star _ l \\end{align*}"}
-{"id": "2464.png", "formula": "\\begin{align*} \\binom { r } { 0 } = 1 \\binom { r } { k } = \\frac { r ( r - 1 ) \\cdots ( r - k + 1 ) } { k ! } \\ ; k = 1 , 2 , \\dots \\ ; . \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} n + ( 2 - a ) \\frac { 1 - s } { s } = n + a + \\frac { 1 } { s } \\le n + a + 2 . \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} P _ { \\nu _ k ^ { - 1 } } = \\left [ \\begin{array} { c } { \\bf e } _ { \\nu _ k ^ { - 1 } ( 1 ) } \\\\ \\vdots \\\\ { \\bf e } _ { \\nu _ k ^ { - 1 } ( n ) } \\end{array} \\right ] \\in \\mathcal { M } _ { n \\times n } ( \\R ) . \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} ( T _ \\lambda g ) ( x ) = \\int _ { \\mathbb R } K _ \\lambda ( x , y ) g ( y ) \\ , \\mathrm d y , g \\in L ^ 1 ( \\mathbb R ) , \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} \\psi ( \\nu , \\mu ) ( T _ i \\cdot f _ { \\nu } ) = t \\psi ( \\nu , \\mu ) f _ { s _ i \\nu } = \\psi ( s _ i \\nu , s _ i \\mu ) f _ { s _ i \\nu } , \\end{align*}"}
-{"id": "7665.png", "formula": "\\begin{align*} \\mathrm { P } _ { t , i } & = e ^ { - \\lambda _ c \\pi \\left ( \\frac { \\epsilon _ 1 } { \\rho } + \\frac { ( 1 + \\epsilon _ 1 ) } { \\rho \\phi _ i } \\right ) ^ { - \\frac { 2 } { \\alpha } } } \\sum ^ { t - 1 } _ { k = 0 } \\frac { ( \\lambda _ c \\pi ) ^ { k } \\left ( \\frac { \\epsilon _ 1 } { \\rho } + \\frac { ( 1 + \\epsilon _ 1 ) } { \\rho \\phi _ i } \\right ) ^ { - \\frac { 2 k } { \\alpha } } } { k ! } , \\end{align*}"}
-{"id": "2533.png", "formula": "\\begin{align*} \\tag * { $ { \\bf ( A _ 6 ) } $ } \\mathrm { i n t } ( E _ \\varsigma ^ + ) \\not = \\emptyset \\ , . \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} \\mathcal { P } ( X ) = \\{ ( \\tilde { x } ^ { ( 1 ) } , \\dots , \\tilde { x } ^ { ( d ) } ) : \\tilde { x } ^ { ( i ) } = \\pi _ i x ^ { ( i ) } \\pi _ i \\{ 1 , \\dots , n \\} \\} \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ( w _ { h , t } , \\chi ) + i ( \\nabla w _ h , \\nabla \\chi ) - i ( f _ \\varepsilon ( \\abs { w _ h } ^ 2 ) w _ h , \\chi ) = ( g , \\chi ) , \\forall \\chi \\in S _ h \\\\ w _ h ( 0 , x ) = u _ { 0 , h } ( x ) , \\end{array} \\right . \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} Z f ( X , t ) = \\kappa f ( X , t ) . \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} f _ { \\beta _ 1 , \\alpha _ { 2 , 2 } } & = f _ { \\beta _ 1 , \\beta _ 2 } \\circ f _ { \\beta _ 2 , \\alpha _ { 2 , 2 } } = f _ { \\beta _ 1 , \\beta _ 2 } \\circ f _ { \\beta _ 2 , \\alpha _ { 1 , 1 } \\vee \\tau _ 2 \\vee \\beta _ 2 } \\\\ & = f _ { \\beta _ 1 , \\alpha _ { 1 , 1 } \\vee \\tau _ 2 \\vee \\beta _ 2 } \\\\ & = f _ { \\beta _ 1 , \\alpha _ { 1 , 1 } } \\circ f _ { \\alpha _ { 1 , 1 } , \\alpha _ { 1 , 1 } \\vee \\tau _ 2 \\vee \\beta _ 2 } . \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{align*} \\log F ( p _ { X Y } ) = \\log \\frac { 1 } { \\epsilon _ 1 } \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{align*} M = \\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} D ( \\rho _ { n _ { k } } , \\rho _ { n _ { k } } ) = D ( \\rho _ { k } ^ { ( 1 ) } , \\rho _ { k } ^ { ( 1 ) } ) + D ( \\rho _ { k } ^ { ( 2 ) } , \\rho _ { k } ^ { ( 2 ) } ) + 2 D ( \\rho _ { k } ^ { ( 1 ) } , \\rho _ { k } ^ { ( 2 ) } ) . \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} | c _ k | & \\lesssim C _ r ^ 3 \\left [ \\sum _ { \\substack { | 2 k - S | < 2 \\left ( \\frac { 1 } { 2 } - \\epsilon \\right ) S \\\\ | 2 m - S | < 2 \\left ( \\frac { 1 } { 2 } - \\epsilon \\right ) S } } r ^ { 2 S - k } + \\sum _ { S = k } ^ { + \\infty } \\sum _ { m = 0 } ^ S \\psi ( \\epsilon ) ^ S r ^ { 2 S - k } \\right ] \\\\ & \\lesssim C _ r ^ 3 k \\left [ r ^ { \\frac { 1 + \\epsilon } { 1 - \\epsilon } k } + ( \\kappa r ) ^ k \\right ] . \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} f ( t ) : = F ( \\eta , w ) = \\displaystyle \\int ^ L _ 0 k _ x ( 0 , y ) \\eta ( y , t ) d t + \\int _ 0 ^ L s _ x ( 0 , y ) w ( y , t ) , \\end{align*}"}
-{"id": "1975.png", "formula": "\\begin{align*} D _ { p } ^ { | k | } = \\frac { \\Gamma ( \\frac { n } { 2 } + \\alpha ) \\Gamma ^ { 1 / p } ( \\frac { p } { q } ( n + | k | + \\alpha - 1 ) ) \\Gamma ^ { 1 / q } ( \\frac { q } { p } | k | ) } { \\Gamma ( \\frac { n + \\alpha - 1 } { q } + | k | + \\frac { | k | + \\alpha + 1 } { 2 } ) \\Gamma ( \\frac { n + \\alpha - 1 } { q } + | k | + \\frac { | k | + n + \\alpha - 1 } { 2 } ) } , \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} \\| A \\| _ { o p } = \\sup _ { v \\ne 0 } \\frac { | A v | _ \\infty } { | v | _ \\infty } = \\sup _ { | v | _ \\infty = 1 } | A v | _ \\infty , \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} W ( \\pi ( u ) v ) = \\psi ( u ) W ( v ) \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} \\alpha _ { 1 2 } = \\frac { \\theta _ { 2 1 } \\theta _ { 2 8 } } { \\theta _ { 5 6 } \\theta _ { 4 9 } } , \\alpha _ { 2 2 } = \\frac { \\theta _ { 2 } \\theta _ { 2 8 } } { \\theta _ { 4 7 } \\theta _ { 4 9 } } , \\alpha _ { 3 2 } = \\frac { \\theta _ { 2 } \\theta _ { 2 1 } } { \\theta _ { 4 7 } \\theta _ { 5 6 } } , \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} & \\bigl ( ( \\tilde { \\Delta } \\otimes \\operatorname { i d } ) ( \\Delta a ) \\bigr ) ( 1 \\otimes 1 \\otimes b ) = ( \\Delta \\otimes \\operatorname { i d } ) \\bigl ( ( \\Delta a ) ( 1 \\otimes b ) \\bigr ) , \\\\ & ( a \\otimes 1 \\otimes 1 ) \\bigl ( ( \\operatorname { i d } \\otimes \\tilde { \\Delta } ) ( \\Delta b ) \\bigr ) = ( \\operatorname { i d } \\otimes \\Delta ) \\bigl ( ( a \\otimes 1 ) ( \\Delta b ) \\bigr ) . \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} z \\sim \\left [ \\frac { 5 \\sqrt { 2 } } { 4 } \\left ( \\frac { \\pi } { 8 } + ( 2 m + 1 ) \\frac { \\pi } { 2 } \\right ) \\right ] ^ { 4 / 5 } , m = 0 , 1 , 2 , \\ldots . \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{align*} \\Delta _ 2 ( \\overline { t } ) = 0 , \\dot \\Delta _ 2 ( \\overline { t } ) = 0 , \\ddot \\Delta _ 2 ( \\overline { t } ) \\leq 0 . \\end{align*}"}
-{"id": "229.png", "formula": "\\begin{align*} \\tilde \\theta _ { E , \\omega } ( z ) \\Big ( { 1 \\over z } + \\sum _ { \\alpha \\geq 0 } f _ \\alpha z ^ \\alpha \\Big ) = \\mathrm { e x p } \\big ( - \\tilde c z - \\sum _ { k \\geq 2 } { 1 \\over k ! } \\partial ^ { k - 2 } ( x ) z ^ k \\big ) \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} \\lim _ { \\| \\Delta x \\| \\rightarrow 0 } \\frac { f ( x + \\Delta x ) - f ( x ) - \\langle g ( x ) , \\Delta x \\rangle } { \\| \\Delta x \\| } = 0 . \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\sum _ { j = 0 } ^ { r } L ( i , j ) \\geq c \\right ) \\leq e ^ { - s c } \\prod _ { j = 0 } ^ { r } \\mathbb { E } e ^ { s L ( i , j ) } = e ^ { - s c } \\left ( e ^ { s } p _ 1 + e ^ { - s } ( 1 - p _ 1 - p _ \\alpha ) \\right ) ^ { r + 1 } . \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{align*} & R _ \\bot ( E ) = R _ \\bot ( 0 ) + R _ \\bot ( 0 ) ^ 2 + X ( E ) \\ , , X ( E ) : = E ^ 2 R _ \\bot ( 0 ) ^ 2 R _ \\bot ( E ) \\ , , \\\\ & \\| X ( E ) \\| \\leq E ^ 2 \\beta ^ { - 3 } \\leq \\beta '^ 2 \\beta ^ { - 3 } \\ , . \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{align*} P \\{ \\tilde { T } _ 1 = \\tilde { T } _ { \\min } \\} & = \\int _ 0 ^ { \\infty } \\cdots \\int _ 0 ^ { \\infty } \\int _ 0 ^ { t _ 2 \\wedge \\cdots \\wedge t _ g } f _ g ( t _ g ) \\cdots f _ 2 ( t _ 2 ) f _ 1 ( t _ 1 ) \\ , d t _ 1 d t _ 2 \\cdots d t _ g \\\\ & = \\int _ 0 ^ { \\infty } \\cdots \\int _ 0 ^ { \\infty } f _ g ( t _ g ) \\cdots f _ 2 ( t _ 2 ) F _ 1 ( t _ 2 \\wedge \\cdots \\wedge t _ g ) \\ , d t _ 2 \\cdots d t _ g , \\end{align*}"}
-{"id": "698.png", "formula": "\\begin{align*} \\det M _ { i j } = \\prod _ { k = 1 } ^ r ( A _ k ) _ { i i } ( B _ k ) _ { i i } ( A _ k ) _ { j j } ( B _ k ) _ { j j } - \\prod _ { k = 1 } ^ r ( C _ k ) _ { i i } ( D _ k ) _ { i i } ( C _ k ) _ { j j } ( D _ k ) _ { j j } . \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} \\omega \\eta = - \\frac 1 3 \\log ( \\lambda ) ^ 2 + \\frac 2 3 ( i \\pi - 4 \\log 2 ) \\log ( \\lambda ) - 2 \\log ( \\lambda ) + O ( 1 ) . \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} V _ h : = \\{ v _ h \\in C ( \\Omega ) \\ , : \\ , v _ h | _ S \\in P _ m ( S ) , \\forall S \\in \\mathcal { T } _ h \\} , m \\ge 1 . \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} \\vartheta _ 1 ( v ; \\tau ) = i e ^ { \\pi i ( \\tau / 4 - v ) } \\prod _ { n = 1 } ^ { \\infty } ( 1 - e ^ { 2 \\pi i n \\tau } ) \\theta ( e ^ { 2 \\pi i v } ; e ^ { 2 \\pi i \\tau } ) , \\end{align*}"}
-{"id": "6590.png", "formula": "\\begin{align*} \\left \\{ v : \\exists \\ , t \\in I _ j ^ { ( k ) } \\textrm { w i t h } \\textrm { s y s } ( \\phi _ t ( v ) ) = T _ k ^ { - \\xi } \\right \\} \\\\ \\subset \\left \\{ v : \\textrm { s y s } ( \\phi _ { a _ j ^ { ( k ) } } ( v ) ) \\in \\left [ \\frac { 1 } { 2 } T _ k ^ { - \\xi } , 2 T _ k ^ { - \\xi } \\right ] \\right \\} . \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{align*} \\phi _ i = \\frac { 1 } { u _ i ( 0 ) } u _ i ( u _ i ( 0 ) ^ { p _ i - \\frac { n + 2 } { n - 2 } } x ' ) \\to \\phi ( x ' ) \\mbox { i n } C _ { l o c } ^ { 1 / 2 } ( \\R ^ { n - 1 } ) \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} \\norm { | \\partial _ { x } | ^ { \\pm \\sigma } f } _ { L ^ 2 ( \\R ) } = \\sup \\left \\{ \\int _ { \\R } f ( x ) \\zeta ( x ) \\ , d x \\ | \\zeta \\in \\dot { H } ^ { \\mp \\sigma } ( \\R ) , \\norm { | \\partial _ { x } | ^ { \\mp \\sigma } \\zeta } _ { 2 } = 1 \\right \\} . \\end{align*}"}
-{"id": "7658.png", "formula": "\\begin{align*} R ^ { C P } _ { m , i } = & \\log \\left ( 1 + \\frac { \\rho \\alpha _ i ^ 2 \\frac { 1 } { { L \\left ( | | x _ m - x _ 0 | | \\right ) } } } { \\rho \\frac { 1 } { { L \\left ( | | x _ m - x _ 0 | | \\right ) } } \\sum ^ { M _ s } _ { j = i + 1 } \\alpha _ j ^ 2 + 1 } \\right ) . \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{align*} P & = \\sum _ { d \\geq 0 } B _ d t ^ d & F P & = \\sum _ { d \\geq 0 } { F B } _ d t ^ d \\\\ \\underline { T P } & = \\sum _ { d \\geq 0 } { T B } _ d t ^ d & \\overline { T P } & = \\sum _ { d \\geq 0 } { T B } _ { d - 1 } t ^ d = t \\ , \\underline { T P } \\end{align*}"}
-{"id": "4478.png", "formula": "\\begin{align*} I _ 3 = \\int _ { \\Omega } w ^ 2 \\delta ^ 2 u L ( u ) d x = \\int _ { \\Omega } w ^ 2 \\delta ^ 2 u [ \\mathrm { d i v } ( A \\nabla u ) ] d x = I + I _ 1 + I _ 2 , \\end{align*}"}
-{"id": "5829.png", "formula": "\\begin{align*} { \\rm C o e f f } _ p [ E _ { \\mu } , m ] : = \\lim _ { q \\rightarrow t ^ { - m } } ( 1 - q t ^ { m } ) ^ p E _ { \\mu } ( z ; q , t ) \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { M } \\frac { \\lambda _ i } { v _ i } \\leq 1 - \\epsilon _ 0 , \\end{align*}"}
-{"id": "595.png", "formula": "\\begin{align*} \\| \\alpha d q \\| _ { \\sigma } = | \\sigma ( \\alpha ) | \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} f _ { \\delta } ( z ; q , t ) = E _ { \\delta } ( z ; q , t ) , \\forall \\ \\delta = ( \\delta _ 1 \\leq \\cdots \\leq \\delta _ n ) , \\\\ f _ { s _ i \\mu } ( z ; q , t ) = T ^ { - 1 } _ i f _ { \\mu } ( z ; q , t ) , \\ \\ \\mu _ i < \\mu _ { i + 1 } , \\end{align*}"}
-{"id": "677.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c c } A _ k X _ k B _ k - C _ k X _ { k + 1 } D _ k & = & E _ k , & k = 1 , \\ldots , r - 1 , \\\\ A _ r X _ r B _ r - C _ r X _ 1 ^ \\star D _ r & = & E _ r , \\end{array} \\right . \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} Z _ D ( \\alpha , \\beta ) = \\prod _ { k = 1 } ^ { \\infty } \\left ( 1 - e ^ { - \\alpha } e ^ { - \\beta k } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "5170.png", "formula": "\\begin{align*} \\prod _ { f _ { p q } \\in \\mathcal { B } _ { k , i } } f _ { p q } v _ { \\omega _ k } = v _ { J _ { k , i } } = \\pm \\bigwedge _ { j \\in J _ { k , i } } v _ j . \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} S u + s u = 0 , \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} F _ { z _ { m , 1 } } ( z ) = & \\frac { 1 } { \\mathcal { R } _ c ^ 2 - \\mathcal { R } _ s ^ 2 } \\left [ \\mathcal { R } _ c ^ 2 F _ { \\mathcal { R } _ c } ( z ) - \\mathcal { R } _ s ^ 2 F _ { \\mathcal { R } _ s } ( z ) \\right ] . \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} q ( L _ + , L _ - ) = L _ - L _ + ^ { - 1 } \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{align*} e _ H ( X , Y , Z ) : = | \\{ ( x , y , z ) \\in X \\times Y \\times Z : \\{ x , y , z \\} \\in E ( H ) \\} | \\geq d | X | | Y | | Z | - \\rho n ^ 3 \\end{align*}"}
-{"id": "5159.png", "formula": "\\begin{align*} \\sigma = ( \\sigma _ 1 , \\dots , \\sigma _ { l _ { \\sigma } } ) ^ t , \\tau = ( \\tau _ 1 , \\dots , \\tau _ { l _ { \\tau } } ) ^ t \\end{align*}"}
-{"id": "1135.png", "formula": "\\begin{align*} \\mathcal L _ { \\partial } ( e _ S ) = - \\mathrm { d i v } ( \\partial ) e _ S \\ , . \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} \\omega _ Y ^ n = c \\omega _ { Y _ 0 } ^ n \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} \\begin{cases} \\nabla G ( U ) - U \\nabla G ( U ) ^ T U = \\mathbb { O } _ { n \\times p } \\\\ U ^ T U = \\mathbb { I } _ p , \\end{cases} \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} s ( n , 2 ) \\ , = \\ , 8 4 0 ^ { 3 ^ { n - 2 } } \\cdot 5 5 2 4 7 5 1 4 9 6 1 5 6 8 9 2 8 4 2 5 3 1 2 2 5 6 0 0 ^ { 3 ^ { n - 3 } ( 3 ^ { n - 2 } - 1 ) / 2 } \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} \\left ( x _ 0 + \\sqrt { - D } \\right ) \\left ( x _ 0 - \\sqrt { - D } \\right ) = ( \\alpha ) ^ { n _ 0 } ( \\alpha ' ) ^ { n _ 0 } , \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{align*} E _ l = \\mbox { s p a n } \\{ \\mathbf { v } _ { l } , \\mathbf { v } _ { l + N } \\} . \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} h _ T ( \\phi _ T ( i ) ) : = \\phi _ T ( h ( i ) ) \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} q _ 2 y \\left ( \\frac { 9 } { 4 } + q _ 2 ^ 2 y ^ 2 \\right ) ^ { - k / 2 } = o \\left ( \\left ( \\frac { 1 } { 4 } + q _ 2 ^ 2 y ^ 2 \\right ) ^ { - k / 2 - 1 } \\right ) , \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} v _ n = 2 ^ { n - 2 } - \\binom { n - 1 } { 3 } - n + 1 + \\sum _ { a = 2 } ^ { n - 4 } \\sum _ { \\ell = 1 } ^ { n - 3 - a } \\left [ \\binom { n } { a + \\ell + 2 } - \\binom { n - 1 - \\ell } { a + 1 } \\right ] . \\end{align*}"}
-{"id": "4587.png", "formula": "\\begin{align*} d _ T = \\partial _ T + \\bar \\partial _ T . \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} ( u - u _ { B T } ) ( x ) = - 2 \\ln \\frac { | \\exp ( x ) \\cdot \\xi | _ { B T } } { | \\exp ( x ) \\cdot \\xi | _ { q } } . \\end{align*}"}
-{"id": "4365.png", "formula": "\\begin{align*} u = i \\sum _ { n = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ n ^ 2 ( 4 \\log 2 - 4 \\gamma _ n ) } { ( n ! ) ^ 2 } \\lambda ^ n \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{gather*} \\lim _ { n \\uparrow \\infty } v ( x _ n ) - f ( x _ n ) = \\inf _ x v ( x ) - f ( x ) , \\\\ \\lim _ { n \\uparrow \\infty } v ( x _ n ) - \\lambda H f ( x _ n ) - h ( x _ n ) \\geq 0 . \\end{gather*}"}
-{"id": "2516.png", "formula": "\\begin{align*} ( X ^ { k + 1 } A ^ { k } ) A ^ { k + 1 } = X ( X ^ { k } A ^ { 2 k } ) A = X A ^ { k + 1 } = A ^ { k } . \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{align*} X ^ \\star _ N = \\sum _ { u \\in E } L ^ \\star _ { u } ( ( R _ n ) _ { 0 \\leq n < N } ) ) e _ u \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{align*} X _ n = \\begin{cases} X _ { \\varphi _ n } & n = 1 , \\dots , j \\\\ [ 1 p t ] X _ { \\mu } & n = 0 \\end{cases} \\end{align*}"}
-{"id": "8584.png", "formula": "\\begin{align*} | V ( A ) | = 4 r + 6 ( r - 1 ) \\leq 1 0 r \\leq \\beta n / 2 \\overset { \\eqref { e q : k a p p a - m u } } { = } \\kappa n . \\end{align*}"}
-{"id": "7935.png", "formula": "\\begin{align*} g ( r ) = \\lim _ { n \\rightarrow \\infty } g ( r _ n ) = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { t _ n - 1 } . \\end{align*}"}
-{"id": "6226.png", "formula": "\\begin{align*} C \\mathbf { u } = \\mathbf { c } , \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} & \\int _ { \\R ^ 2 } d \\phi _ 1 d \\phi _ 2 e ^ { \\beta L ^ d f _ L ( | \\phi | ) } g _ L ( \\phi _ 1 , \\phi _ 2 ) = \\int _ 0 ^ { \\infty } d x x e ^ { \\beta L ^ d f _ L ( x ) } v _ L ( x ) , \\\\ & \\int _ { \\R ^ 2 } d \\phi _ 1 d \\phi _ 2 e ^ { \\beta L ^ d f _ L ( | \\phi | ) } \\lim _ { h \\to \\infty \\atop h \\in \\frac { 2 } { \\beta } \\N } \\int e ^ { - V ( \\psi ) + W ( \\psi ) } A ^ 2 ( \\psi ) d \\mu _ { C ( \\phi ) } ( \\psi ) = \\int _ 0 ^ { \\infty } d x x e ^ { \\beta L ^ d f _ L ( x ) } u _ { 2 , L } ( x ) . \\end{align*}"}
-{"id": "55.png", "formula": "\\begin{align*} \\mathcal { E } = \\{ X _ { u _ i , v _ i } = 0 \\ \\forall i \\in [ m ] \\} . \\end{align*}"}
-{"id": "6328.png", "formula": "\\begin{align*} ( \\mathbb { I } _ n - U U ^ T ) \\frac { \\partial G } { \\partial { \\bf u } _ a } ( { \\bf u } ) = { \\bf 0 } , \\ , \\ , \\ , \\ , \\ , 1 \\leq a \\leq p , \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} r ( \\theta \\ , | \\ , \\phi ) = \\frac { ( r N ) ^ { \\frac { 1 } { 2 } } \\ , \\sin ( 2 \\theta ) ^ { \\frac { 1 } { 2 } } } { \\left ( 2 \\ , \\sin ( \\phi \\pm \\theta ) \\right ) ^ { \\frac { r N - 1 } { 2 r N } } \\ , P ( \\frac { \\pi } { 2 } - \\theta \\ , | \\ , \\phi ) } \\ , \\prod _ { k = 1 } ^ { r N } \\sin \\left ( \\frac { 2 \\theta - \\pi k } { r N } \\right ) ^ { \\frac { r N - 2 k } { 2 r N } } \\ , , \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} \\sigma _ B ^ L ( h ( \\sigma _ A ( x ) ) = \\sigma _ B ^ { L + 1 } ( h ( x ) ) , \\sigma _ A ^ L ( h ^ { - 1 } ( \\sigma _ B ( y ) ) = \\sigma _ A ^ { L + 1 } ( h ^ { - 1 } ( y ) ) , \\end{align*}"}
-{"id": "1010.png", "formula": "\\begin{align*} d ^ { 2 } ( x , C _ { i } ) \\leq p \\delta _ { i } ^ { 2 } \\sum _ { l = 1 } ^ { p } \\Vert x _ { l } ^ { k } - x _ { l - 1 } ^ { k } \\Vert ^ { 2 } \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{align*} \\lim _ { x \\to 1 - } f ( x ) = \\infty , \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} \\mu ' _ { i ' } = \\mathcal { C } _ 1 ( \\alpha ' , \\nu ' , \\tau ^ { - 1 } ( i ' ) , I ' _ b \\setminus \\lbrace \\tau ^ { - 1 } ( i ' ) \\rbrace , I _ b ) - \\ell ' + 2 i ' - 1 . \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} \\Gamma ( \\alpha , \\beta ) = \\alpha ! \\cdot \\prod _ { j = 1 } ^ { | \\beta | } \\max _ { 1 \\leq i \\leq n } \\left \\{ \\alpha _ i + j ( \\gamma _ i + 1 ) \\right \\} . \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{align*} \\underline { \\dim } _ B ( A ) : = \\liminf _ { \\delta \\to 0 } \\frac { \\log N _ \\delta ( A ) } { - \\log \\delta } , \\end{align*}"}
-{"id": "7882.png", "formula": "\\begin{align*} \\lambda _ { M } ( x , \\xi ) : = \\mu \\ , ' + \\frac { 1 } { 2 m } | \\xi | ^ 2 + < x > ^ { 2 ( M + 1 ) } , \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} \\mathcal { I } _ { b _ j } \\cap \\mathcal { I } _ M = \\emptyset \\Rightarrow b _ j ^ \\top M = 0 . \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{align*} J _ { \\mathrm { n i l } } : = \\{ x \\in J \\mid t . x = 0 \\textrm { f o r s o m e } t \\in H ^ + \\} . \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 \\int _ 0 ^ L \\Phi _ i ( x , t ) v ( x , t ) d x \\leq \\left ( \\frac 1 2 K _ 1 + \\frac 1 2 K _ 2 + 2 K _ 3 \\right ) \\| ( \\eta ( t ) , w ( t ) ) \\| ^ 3 _ { X _ 0 } . \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{align*} \\frac 1 2 \\int \\frac { d X } { \\sqrt { X ( X + | \\lambda | ) } } = \\log \\left ( \\sqrt { X + | \\lambda | } + \\sqrt { X } \\right ) . \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{align*} { } \\frac { v ^ 2 } { u ^ 2 } = \\int _ 0 ^ { v } \\frac { t \\ , d t } { 1 - e ^ { - t } } \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} H \\left ( \\nu , \\mu \\right ) = \\prod _ { x \\in \\vec { x } ( \\mu ) } \\prod _ { i \\leq x } \\left ( t ^ { \\nu _ i } \\right ) \\cdot \\prod _ { y \\in \\vec { y } ( \\mu ) } \\prod _ { i \\leq y } \\left ( t ^ { \\nu _ i } \\right ) \\cdot t ^ { - \\chi ( \\vec { x } , \\vec { y } ) } \\end{align*}"}
-{"id": "6152.png", "formula": "\\begin{align*} S _ { m i n } = \\min _ { [ 0 , r _ G ( \\delta ) ] } \\widetilde { S } _ W ( t ) & & S _ { m a x } = \\max _ { [ 0 , r _ G ( \\delta ) ] } \\widetilde { S } _ W ( t ) \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{align*} E \\left [ T _ 1 \\right ] = ( \\nu _ 1 + \\lambda \\nu _ 2 ) M \\ln M \\ , + \\ , ( \\nu _ 1 + \\lambda \\nu _ 2 ) ( \\gamma + \\ , \\ln \\nu _ 1 ) M \\ , + \\ , \\frac { \\alpha _ 1 } { 2 } \\ , + \\ , O \\left ( \\frac { 1 } { M } \\right ) , M \\to \\infty , \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{align*} \\beta ^ { * 2 } E = \\int _ 0 ^ { \\infty } \\ln ( 1 + e ^ { - \\alpha ^ * } e ^ { - x } ) d x \\sim e ^ { - \\alpha ^ * } \\end{align*}"}
-{"id": "6907.png", "formula": "\\begin{align*} 0 = \\eta . \\ , 0 \\ge \\eta . \\ , \\lambda _ { i j _ i } e _ i = \\eta _ i \\lambda _ { i j _ i } \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} C ^ + = \\{ p \\in \\mathfrak { a } ^ * ; p ( \\alpha ^ { \\vee } ) \\geq 0 , \\forall \\alpha \\in \\Phi ^ + \\} . \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} c h ^ { q ^ { 2 s + l } } \\gamma ^ { \\rho q ^ l } + d g ^ { q ^ { l } } \\gamma ^ { \\rho q ^ { l + n } } = 0 . \\end{align*}"}
-{"id": "5817.png", "formula": "\\begin{align*} T _ i f _ { ( \\nu _ 1 , \\dots , \\nu _ i , \\nu _ { i + 1 } , \\dots , \\nu _ n ) } = \\left \\{ \\begin{array} { r l } f _ { ( \\nu _ 1 , \\dots , \\nu _ { i + 1 } , \\nu _ { i } , \\dots , \\nu _ n ) } , & \\nu _ i > \\nu _ { i + 1 } , \\\\ \\\\ t f _ { ( \\nu _ 1 , \\dots , \\nu _ { i + 1 } , \\nu _ { i } , \\dots , \\nu _ n ) } , & \\nu _ i = \\nu _ { i + 1 } . \\end{array} \\right . \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} u ( x ) = C e ^ { i \\omega _ \\lambda x } + D e ^ { - i \\omega _ \\lambda x } , \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} A _ m = A _ { m + 1 } + B _ m + x A _ m + x ^ 2 A _ { m - 1 } + \\cdots + x ^ { m } A _ 1 + x ^ { m + 1 } A _ 0 \\ , , \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} \\mathcal { B } _ n & : = \\{ n ( n - k ) + d : d \\in W ( k ) , 1 \\le k \\le n - 1 \\} \\\\ \\mathcal { C } _ n & : = \\{ c + d : c \\in W ( n - k ) , d \\in W ( k ) , 1 \\le k \\le n - 1 \\} . \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} A ^ { \\mathrm { s u m } } : = ( 1 , 1 , \\dots , 1 ) \\in \\R ^ { 1 \\times m } , b ^ { \\mathrm { s u m } } : = 0 . \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} \\{ \\sigma _ i \\} _ { i = 1 } ^ l \\in U \\cap F _ r , \\end{align*}"}
-{"id": "4212.png", "formula": "\\begin{align*} \\lambda ( p , q ) - \\frac \\gamma 4 & = \\frac { p ( q - q _ - ) ( q - q _ + ) } { 2 { ( p - 1 ) } ^ 2 ( q - 1 ) } , \\\\ q _ { \\pm } & = \\frac { \\gamma ( p - 1 ) } { 4 p } \\left ( p - 1 + \\frac 2 \\gamma \\pm \\sqrt { ( p - p _ - ) ( p - p _ + ) } \\right ) , \\\\ p _ \\pm & = 1 + \\tfrac 2 \\gamma \\pm 2 \\sqrt { \\tfrac 2 \\gamma } . \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{align*} \\frac { W _ 2 ^ 2 ( \\mu _ { t + s } , \\nu ) } { 2 } & \\ge \\int _ X \\varphi _ t \\ , d \\mu _ { t + s } - \\int _ X \\psi _ t \\ , d \\nu , & \\frac { W _ 2 ^ 2 ( \\mu _ t , \\nu ) } { 2 } & = \\int _ X \\varphi _ t \\ , d \\mu _ t - \\int _ X \\psi _ t \\ , d \\nu . \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} u ( 0 , t ) = u ( 1 , t ) = 0 \\forall t \\geq 0 . \\\\ \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{align*} w _ { \\tau } ( t ; \\rho ) : = \\frac { u ( t ; \\rho + \\tau ) - u ( t ; \\rho ) } { \\tau } , \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} H _ 1 = \\sum _ { n = L } ^ { M _ - } E _ M ( t , n ) | C _ n | & \\leq c \\sum _ { n = L } ^ { M _ - } \\frac { n ^ { d - 1 } } { M ^ d } \\big ( 1 - P _ { \\check x _ n } ( Y _ t \\in B _ M ) \\big ) \\leq c _ 2 \\ , { \\rm e } ^ { - c _ 3 ( \\gamma t ) ^ { 2 \\varepsilon } } . \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} & \\alpha ^ 2 \\| V _ 0 ^ { 1 - 3 , l } \\| _ { 1 , r , r ' } \\le c \\beta c _ 0 ^ 2 \\alpha ^ 2 r ' , \\sum _ { m = 2 } ^ N c _ 0 ^ { \\frac { m } { 2 } } \\alpha ^ m \\| V _ m ^ { 1 - 3 , l } \\| _ { 1 , r , r ' } \\le c \\beta c _ 0 ^ 2 \\alpha ^ 4 r ' . \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} F _ + ( h ) = F _ + ( 0 ) + \\sum _ { j = 1 } ^ { k - 1 } \\frac { h ^ j } { j ! } F _ + ^ { ( j ) } ( 0 ) + \\int _ 0 ^ h \\frac { ( h - u ) ^ { k - 1 } } { ( k - 1 ) ! } F _ + ^ { ( k ) } ( u ) d u . \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} w ( x ) : = \\sum _ { k \\in \\Z } \\cos ( k \\pi ) ( x _ 1 - k \\pi ) ^ m \\chi ( x _ 1 - k \\pi ) f ( k \\pi , x ' ) \\end{align*}"}
-{"id": "6538.png", "formula": "\\begin{align*} l ' _ j = ( 1 - x a _ j ) _ { + } , \\end{align*}"}
-{"id": "8710.png", "formula": "\\begin{align*} \\bigl ( \\nabla _ { \\dot \\gamma } \\nabla _ { \\dot \\gamma } Y ( s ) + R ( Y , \\dot \\gamma ) \\dot \\gamma \\bigr ) ^ b = 0 . \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} \\rho _ { V , U } ( t ) = t , & & \\varphi _ { V , U } ( t ) = 1 . \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} \\sum _ { x \\in \\mathbb N } \\overline { \\eta _ t ^ n } ( x ) f \\big ( \\tfrac { x } { n } \\big ) \\ ; = \\ ; \\sum _ { x \\in \\mathbb N } \\overline { \\eta _ 0 ^ n } ( x ) f \\big ( \\tfrac { x } { n } \\big ) \\ ; + \\ ; \\frac { 1 } { n } \\sum _ { x \\in \\mathbb N } \\overline { J _ t ^ n } ( x ) \\nabla ^ n _ x f \\ ; + \\ ; f \\big ( \\tfrac { 1 } { n } \\big ) \\overline { J _ t ^ n } ( 0 ) . \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ { k } y _ j ^ { \\delta } e ( y _ j ) = 0 0 \\leq \\delta \\leq k - 2 , \\\\ & \\sum _ { j = 1 } ^ { k } y _ j ^ { k - 1 } e ( y _ j ) = C ( f ) . \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} { } F ( a , b ; c ; 1 ) = \\frac { \\Gamma ( c ) \\Gamma ( c - a - b ) } { \\Gamma ( c - a ) \\Gamma ( c - b ) } , \\enspace \\mbox { R e } { ( c - a - b ) } > 0 , \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} p _ j n _ j \\ , \\frac { \\psi _ j ( n _ j - 1 ) - \\psi _ j ( n _ j ) } { \\psi _ j ( n _ j ) } = \\lambda _ j , j = 1 , \\dots , g , \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} ( m g ) ^ { - 1 } \\cdot S = g ^ { - 1 } m ^ { - 1 } \\cdot S = g ^ { - 1 } \\cdot S \\in H . \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} \\frac { ( c , a x t ; q ) _ \\infty } { ( a b , t x ; q ) _ \\infty } { _ 2 \\phi _ 1 \\left ( { { d , x t } \\atop { a x t } } ; q , c \\right ) } = \\sum _ { n = 0 } ^ \\infty \\lambda _ n ( a ; q ) _ n x ^ n . \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} \\frac { \\sum _ { v _ i \\in V _ h } f _ i } { n / g } \\leq \\frac { n ^ 2 / a } { n / g } = \\frac { n } { \\log n } . \\end{align*}"}
-{"id": "1284.png", "formula": "\\begin{align*} \\chi ( D _ j ) \\overline { \\chi ( D ) } - \\chi ( D _ j ) \\overline { \\chi ( D _ j ) } = - \\lambda _ j . \\end{align*}"}
-{"id": "6086.png", "formula": "\\begin{align*} J _ { \\mu } : = \\sum _ { n , m \\geq 0 } \\sum _ { \\substack { \\mathfrak { a } \\in \\mathfrak { g } ^ { \\otimes n } \\\\ \\mathfrak { b } \\in \\mathfrak { g } ^ { \\otimes m } } } \\sum _ { a , b \\in \\mathfrak { g } } ( \\mathfrak { a } \\otimes ( a \\otimes b - \\varepsilon ( a , b ) b \\otimes a ) \\otimes \\mathfrak { b } - \\mu ^ { n + m } \\alpha _ { T } ( \\mathfrak { a } ) \\otimes [ a , b ] _ { \\mathfrak { g } } \\otimes \\alpha _ { T } ( \\mathfrak { b } ) ) , \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} H ^ 0 ( K _ { S _ q } ) ^ * = \\mathcal { H } ^ { 1 , 0 } ( S _ q ) ^ * \\longrightarrow \\mathcal { H } ^ { 0 , 1 } ( S _ q ) \\longrightarrow \\mathcal { H } ^ { 1 , 0 } ( S _ q ) = H ^ 0 ( K _ { S _ q } ) , \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} \\limsup _ { m \\to \\infty } & \\frac { 1 } { m ^ \\beta \\theta ( m ) } \\sum _ { j = 1 } ^ m z _ j \\ < \\infty \\ , , \\qquad \\\\ \\limsup _ { m \\to \\infty } & \\frac { 1 } { m ^ \\beta \\theta ( m ) } \\sum _ { j = 1 } ^ m ( \\log z _ j ) _ - \\ < \\infty \\ , , \\qquad \\\\ \\liminf _ { m \\to \\infty } & \\frac { 1 } { m ^ { \\beta ' } \\theta ( m ) } \\sum _ { j = 1 } ^ m z _ j = \\infty \\ , , \\beta ' : = \\beta ^ 2 / ( 1 + \\beta ) , \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} \\Phi _ { q } ( 1 , t _ { \\ast } ) = 0 . \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{align*} u ( t ) = S ( t ) u _ 0 + i \\int _ 0 ^ t S ( t - s ) \\Big ( e ^ { - s / \\mu } v _ 0 + \\frac { \\lambda } { \\mu } \\int _ 0 ^ s e ^ { - ( s - s ' ) / \\mu } | u ( s ' ) | ^ 2 d s ' \\Big ) u ( s ) d s , \\end{align*}"}
-{"id": "8309.png", "formula": "\\begin{align*} \\mu ^ { \\circ } + \\ell - 2 q + 1 & \\geq \\mathcal { C } _ { - 1 } ( \\alpha , \\nu , q + w ' , \\lbrace 1 , \\ldots , q - 1 \\rbrace , \\lbrace q , \\ldots , \\ell \\rbrace \\setminus \\lbrace q + w ' \\rbrace ) + 1 \\\\ & = \\left \\lceil \\frac { \\nu _ { q + w ' } - \\sum _ { i _ 0 = 1 } ^ { q - 1 } m _ { q + w ' , i _ 0 } + 1 + \\sum _ { i _ 0 = q + 1 } ^ { \\ell } m _ { q + w ' , i _ 0 } } { \\alpha _ { q + w ' } } \\right \\rceil . \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n a _ { i j } \\frac { \\partial ^ 2 \\delta ^ \\gamma } { \\partial x _ i \\partial x _ j } \\leq n \\Lambda \\gamma \\delta ^ { \\gamma - 2 } , \\end{align*}"}
-{"id": "3266.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\| Q _ { | \\textup { \\textbf { m } } | } ^ { \\textup { \\textbf { F } } } Q _ { n , \\textup { \\textbf { m } } } F _ \\alpha - Q _ { | \\textup { \\textbf { m } } | } ^ { \\textup { \\textbf { F } } } P _ { n , \\textup { \\textbf { m } } , \\alpha } \\| _ K ^ { 1 / n } \\leq \\frac { \\| \\Phi \\| _ K } { \\rho _ { | \\textup { \\textbf { m } } | } ( \\textup { \\textbf { F } } ) } , \\alpha = 1 , 2 , \\ldots , d , \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{align*} \\Lambda ( \\delta _ { X } , \\mu ) & = \\frac { 1 } { n } \\sum _ { j = 1 } ^ { n } \\delta _ { X \\# A _ { j } } , \\\\ { \\mathcal A } ( \\delta _ { X } , \\mu ) & = \\frac { 1 } { n } \\sum _ { j = 1 } ^ { n } \\delta _ { ( X + A _ { j } ) / 2 } , \\\\ { \\mathcal H } ( \\delta _ { X } , \\mu ) & = \\frac { 1 } { n } \\sum _ { j = 1 } ^ { n } \\delta _ { 2 ( X ^ { - 1 } + A _ { j } ^ { - 1 } ) ^ { - 1 } } . \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ N \\tau \\left ( \\frac { W _ { 2 } ( \\rho ^ \\tau _ k , \\rho ^ \\tau _ { k - 1 } ) } { \\tau } \\right ) ^ 2 \\leq J ( \\rho _ 0 ) - J ( \\rho ^ \\tau _ N ) \\leq C , \\end{align*}"}
-{"id": "1359.png", "formula": "\\begin{align*} Y _ t = \\xi + \\int _ t ^ T G \\bigl ( s , Y _ s , Z _ s \\bigr ) d s - \\int _ t ^ T Z _ s d B _ s , 0 \\le t \\le T \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{align*} \\frac { x ' ( t ) y '' ( t ) - x '' ( t ) y ' ( t ) } { x ' ( t ) ^ 2 + y ' ( t ) ^ 2 } = & \\frac { 1 } { 2 } ( x ( t ) y ' ( t ) - x ' ( t ) y ( t ) ) + \\frac { ( n - 1 ) x ' ( t ) } { y ( t ) } \\\\ & - \\frac { ( m - 1 ) y ' ( t ) } { x ( t ) } + \\lambda ( x ' ( t ) ^ 2 + y ' ( t ) ^ 2 ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} { \\rm d i s t } ( z , \\mathcal { N } _ \\Omega ( \\overline { y } ) ) & = { \\rm d i s t } ( \\nabla \\Xi ( y ) \\xi , \\mathcal { N } _ \\Omega ( \\overline { y } ) ) \\le \\| \\nabla \\Xi ( y ) \\xi - \\nabla \\Xi ( \\overline { y } ) \\xi \\| \\\\ & \\le \\| \\xi \\| L \\| y - \\overline { y } \\| \\le L ( \\widetilde { \\delta } + \\| \\overline { \\xi } \\| ) \\| y - \\overline { y } \\| . \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { d _ { k } } \\lambda _ { j } ^ { k } g _ { \\ell } ^ { i _ { j } ^ { k } } = 0 . \\end{align*}"}
-{"id": "2679.png", "formula": "\\begin{align*} h = \\left \\{ \\begin{array} { l l l } A e ^ { \\sqrt { | a | } t } & \\mbox { i f } & c = 0 , \\\\ \\noalign { \\smallskip } \\sqrt { | \\frac { c } { a } | } [ \\cosh ( \\sqrt { | a | } t + \\theta ) ] & \\mbox { i f } & c \\neq 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} f ( x , \\lambda ) & = \\max _ { m \\in [ - 1 , 1 ] } \\bigl ( m x + \\lambda m ^ 2 - I ( m ) \\bigr ) . \\end{align*}"}
-{"id": "5240.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\| \\tilde { u } ( \\cdot , t ) - \\frac { a } { b } \\| _ { \\infty } = 0 . \\end{align*}"}
-{"id": "5835.png", "formula": "\\begin{align*} { \\rm C o e f f } _ p [ E _ { \\mu } , m ] \\propto \\left [ \\prod _ { \\substack { \\kappa \\prec \\mu \\\\ \\kappa \\not \\in \\mathcal { E } _ { \\mu } } } \\frac { Y ( w ) - y _ { \\kappa } ( w ) } { y _ { \\mu } ( w ) - y _ { \\kappa } ( w ) } \\cdot \\prod _ { i = 1 } ^ { p } ( Y ( w ) - y _ { \\nu [ i ] } ( w ) ) \\cdot z ^ { \\mu } \\right ] _ { q = t ^ { - m } } \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} { } \\frac { \\partial S _ E } { \\partial \\beta } = 0 \\sim E + \\frac { 1 } { \\beta ^ 2 } \\int _ { 0 } ^ { \\infty } \\ln ( 1 - e ^ { - \\alpha } e ^ { - x } ) d x \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} E ( u , v ) = \\int ( \\nabla u ) ^ 2 - \\int u ^ 4 + \\int ( u ^ 2 + v ) ^ 2 , \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} \\nabla _ { s , g } \\textbf { V } = \\textit { \\textbf { e } } \\ \\ { i n } \\ \\ \\Omega , \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} \\displaystyle \\lambda _ 1 = \\inf \\left \\{ \\frac { \\int _ { \\Sigma } | \\nabla \\varphi | ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma } { \\int _ { \\Sigma } \\varphi ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma } ; \\varphi \\in C _ 0 ^ { \\infty } ( \\Sigma ) , \\int _ { \\Sigma } \\varphi ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma \\neq 0 \\right \\} . \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} G _ 1 ( t ) = m _ { G _ 1 } ( t ) - \\varphi _ { G _ 1 , G _ 2 } ( t ) m _ { G _ 2 } ( \\rho _ { G _ 1 , G _ 2 } ( t ) ) + \\varphi _ { G _ 1 , G _ 2 } ( t ) G _ 2 ( \\rho _ { G _ 1 , G _ 2 } ( t ) ) \\end{align*}"}
-{"id": "2471.png", "formula": "\\begin{align*} J ( N ; \\alpha ) = \\sum _ { k = 0 } ^ n \\binom { r } { k } \\frac { ( - 1 ) ^ k } { \\ln ^ k N } \\int _ 0 ^ { \\infty } e ^ { - x } ( \\ln x ) ^ k d x + o \\left ( \\frac { 1 } { \\ln ^ n N } \\right ) , N \\to \\infty , \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} \\max p _ { - k } ( n ) & = \\left ( p _ { - k } ( 1 ) \\right ) ^ { n } = k ^ { n } . \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} \\underline { T P } & = \\frac { P - F P } { 1 + t } \\\\ \\overline { T P } & = \\frac { P - F P } { 1 + t ^ { - 1 } } = t \\ , \\frac { P - F P } { 1 + t } \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} \\displaystyle \\limsup _ { h \\to 0 + } \\dfrac { \\sup _ { t \\in T } \\sup _ { | s - t | _ \\infty \\leq h } | X ( t ) - X ( s ) | } { h ^ { ( H - \\theta _ 2 / \\alpha ) } ( \\log { 1 / h } ) ^ { 1 / \\gamma } } = 0 \\ ; \\ ; a . s . \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} j _ { m } ( \\rho ) & = \\frac { q } { ( 2 \\pi ) ^ 3 } \\int _ { | p | < ( 6 \\pi ^ 2 \\rho / q ) ^ { \\frac { 1 } { 3 } } } \\sqrt { | p | ^ 2 + m ^ 2 } { \\rm d } p \\\\ & = \\frac { q } { 1 6 \\pi ^ 2 } \\left [ \\eta ( 2 \\eta ^ 2 + m ^ 2 ) \\sqrt { \\eta ^ 2 + m ^ 2 } - m ^ 4 \\ln { \\left ( \\frac { \\eta + \\sqrt { \\eta ^ 2 + m ^ 2 } } { m } \\right ) } \\right ] , \\eta = \\left ( \\frac { 6 \\pi ^ 2 \\rho } { q } \\right ) ^ { \\frac { 1 } { 3 } } . \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} U _ 1 = \\begin{bmatrix} I & X _ 1 \\\\ 0 & I \\end{bmatrix} , V _ 2 = \\begin{bmatrix} I & 0 \\\\ X _ 2 & I \\end{bmatrix} , V _ 3 = \\begin{bmatrix} I & X _ 3 \\\\ 0 & I \\end{bmatrix} ; \\end{align*}"}
-{"id": "4500.png", "formula": "\\begin{align*} i \\partial _ t v = H _ 1 ( t ) v \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} w ( \\varepsilon ) = R _ N ^ { \\varepsilon _ N } R _ { N - 1 } ^ { \\varepsilon _ { N - 1 } } \\cdots R _ 1 ^ { \\varepsilon _ 1 } v , \\end{align*}"}
-{"id": "3656.png", "formula": "\\begin{align*} p _ L ( t , x ) : = \\frac { 1 } { | B _ L | } \\sum _ { y \\in B _ L } p ( t , x + y ) \\ , . \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} c _ 2 = \\frac { \\eta ^ 2 } { 2 ( 2 a \\lambda _ 2 - c _ 1 ) } + \\frac { b | T r ( L ) | } { 2 N } + \\frac { m \\tau ^ 2 \\tilde { \\gamma } ( N - 1 ) } { 2 N } . \\end{align*}"}
-{"id": "223.png", "formula": "\\begin{align*} \\tilde F _ { E , \\omega } ( p , z ) \\tilde F _ { E , \\omega } ( p + p ' , z ' ) - \\tilde F _ { E , \\omega } ( p ' , z ' ) \\tilde F _ { E , \\omega } ( p , z + z ' ) + \\tilde F _ { E , \\omega } ( p + p ' , z + z ' ) \\tilde F _ { E , \\omega } ( p ' , - z ) = 0 \\end{align*}"}
-{"id": "4435.png", "formula": "\\begin{align*} \\frak { e _ 7 } = \\mathbb { C } ^ * \\oplus \\Delta ^ * \\oplus \\frak { s o } _ { 1 2 } \\oplus \\mathbb { C } \\oplus \\Delta \\oplus \\mathbb { C } . \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} \\sum _ { s \\in S _ \\mu } | S _ \\mu ( s - 1 ) | = 0 + 1 + \\cdots + ( N - | \\mu | - 1 ) = \\frac { ( N - | \\mu | ) ( N - | \\mu | - 1 ) } { 2 } , \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{align*} \\hat { \\vect { x } } ^ t _ n = \\vect { x } _ n + \\tau _ t \\vect { w } = \\alpha _ n \\vect { h } _ n + \\tau _ t \\vect { w } \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} \\bar { S } = \\tilde { T } _ { \\rm r } ^ \\top S \\tilde { T } _ { \\rm r } \\in \\real ^ { r \\times r } , \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{align*} X _ { \\alpha _ k } ^ { s _ k } = A _ k ^ { - 1 } ( C _ k X _ { \\beta _ k } ^ { t _ k } D _ k + E _ k ) B _ k ^ { - 1 } . \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} \\iota _ i & = \\mathcal { C } _ { - 1 } ( \\alpha , \\nu , i , \\lbrace 1 , \\ldots , i - 1 \\rbrace , \\lbrace i + 1 , \\ldots , \\ell \\rbrace ) - \\ell + 2 i - 1 \\\\ & = \\nu _ i - ( i - 1 ) + ( \\ell - i ) - \\ell + 2 i - 1 = \\nu _ i \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} P = \\sum _ { \\lambda \\vdash d } a _ \\lambda \\chi _ \\lambda , \\end{align*}"}
-{"id": "3329.png", "formula": "\\begin{align*} p ( 0 , 0 | v , v ) = t , p ( 0 , 1 | v , v ) = p ( 1 , 0 | v , v ) = 0 , p ( 1 , 1 | v , v ) = 1 - t . \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} C : = \\max \\left \\{ 1 , \\ , \\ , d ^ { 3 } \\cdot \\sup _ { x \\in \\overline { B _ { d \\varepsilon } } \\left ( x _ { 0 } \\right ) } \\ , \\ , \\max _ { \\left | \\alpha \\right | = 3 } \\ , \\ , \\left | \\partial ^ { \\alpha } f ( x ) \\right | \\right \\} \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} \\phi ( { \\bf d } ) _ v = \\begin{cases} 0 & { \\bf d } _ v > 1 , \\\\ - 1 & { \\bf d } _ v < - 1 , \\\\ x _ v & . \\end{cases} \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} N ( U _ 0 , \\rho ) = N _ 1 ( U _ 0 , \\rho ) \\equiv \\kappa , \\ \\ \\ \\ \\ \\ \\ \\ 0 < \\rho < 1 . \\end{align*}"}
-{"id": "178.png", "formula": "\\begin{align*} \\widehat { r } _ i = \\sum \\limits _ { j = 2 } ^ N \\widehat { m } _ { i j } \\widehat { c } _ j , ~ i \\in \\{ 2 , 3 , \\dots , N \\} . \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} \\Delta _ M u - \\partial _ t u = 0 \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{align*} ( - 1 ) ^ n \\exp ( \\psi _ n ( \\tilde { z } ) ) \\varphi _ \\lambda ( \\tilde { z } ) = \\varphi _ \\lambda ( z ) . \\end{align*}"}
-{"id": "6769.png", "formula": "\\begin{align*} \\mathsf b _ + ( a , b ) : = \\sum _ { j = 1 } ^ m a _ j b _ j \\quad a = ( a _ 1 , \\ldots , a _ n ) b = ( b _ 1 , \\ldots , b _ n ) \\in \\Gamma . \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} \\Phi ^ { ( \\alpha , \\beta ) } _ n ( x , y | q ) = \\sum _ { k = 0 } ^ n { n \\brack k } _ q ( \\alpha ; q ) _ k ( \\beta ; q ) _ { n - k } x ^ k y ^ { n - k } . \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} K _ 0 : = \\{ \\omega \\in U \\mid \\bar \\partial \\omega + \\dfrac { 1 } { 2 } [ \\omega , \\omega ] = \\bar \\partial ^ * \\omega = 0 \\} \\end{align*}"}
-{"id": "4822.png", "formula": "\\begin{align*} \\partial _ { x } f ( t , x , v , z ) = z f ( t , x , v , z ) , \\qquad \\partial _ { v } f ( t , x , v , z ) = \\psi _ { T - t } ( z , 0 ) f ( t , x , v , z ) . \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} \\begin{array} { l } - \\lambda _ 1 ( 0 , 1 , 1 ) = - \\lambda _ 1 ( 1 , 1 , 1 ) = - \\lambda _ 2 ( 0 , 0 , 1 ) = - \\lambda _ 2 ( 1 , 0 , 1 ) = \\\\ - \\lambda _ 3 ( 1 , 0 , 0 ) = - \\lambda _ 3 ( 1 , 1 , 0 ) = - \\lambda _ 4 ( 0 , 0 , 0 ) = - \\lambda _ 4 ( 0 , 1 , 0 ) = \\frac { 1 } { 2 } . \\end{array} \\end{align*}"}
-{"id": "4351.png", "formula": "\\begin{align*} f _ \\zeta + i g _ \\zeta = G _ { j } ( f _ \\wp , g _ \\wp ) - \\eta ( m \\omega _ 1 + n \\omega _ 2 ) \\} \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} \\int _ X \\psi \\omega ^ n = C _ H \\int _ { ( - \\Delta ^ t ) \\cap \\bar { C } ^ - } n ! 2 ^ { | \\Phi _ s ^ + | - r } \\psi ( d _ { 2 p } u ^ * ) \\prod _ { \\alpha \\in \\Phi _ { Q ^ u } } ( \\chi - p ) ( \\alpha ^ { \\vee } ) \\prod _ { \\beta \\in \\Phi _ s ^ + } ( - p ) ( \\beta ^ { \\vee } ) d p \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{align*} \\begin{aligned} 1 - \\frac { 2 \\sqrt { 2 } r } { \\sqrt { y } } \\leq Q ( \\Lambda _ { ( z , \\theta ) } , r ) \\leq 1 - \\frac { \\sqrt { 2 } r } { \\sqrt { y } } & \\sqrt { 2 } r \\leq \\sqrt { y } ; \\\\ Q ( \\Lambda _ { ( z , \\theta ) } , r ) = 0 & \\sqrt { 2 } r \\geq \\sqrt { y } . \\end{aligned} \\end{align*}"}
-{"id": "6910.png", "formula": "\\begin{align*} H _ \\Lambda ( 0 ) \\ = \\ \\omega H _ { \\Lambda } ^ { ( \\mathrm { p h } ) } + V _ \\Lambda , \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{align*} A ( X ^ { k + 1 } A ^ { k } ) = X ^ { k } A ^ { k } = X ^ { k } ( X A ^ { k + 1 } ) = X ^ { k + 1 } A ^ { k } A . \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} a _ k f _ 1 ^ p + \\cdots + a _ 1 f _ k ^ p = u ^ \\mu f _ k \\ ; \\ ; 1 \\le k \\le h . \\end{align*}"}
-{"id": "2991.png", "formula": "\\begin{align*} \\log F ( G , q ; z ) - \\frac 1 d \\log F ( G , q ^ d ; z ^ d ) = h ( q , z ) - \\frac 1 d h ( q ^ d , z ^ d ) . \\end{align*}"}
-{"id": "8105.png", "formula": "\\begin{align*} \\overline { N } ( r ) \\overset { d e f } { = } e ^ { C \\psi ( r ) } \\left ( N ( r ) + C \\psi ( r ) \\right ) , \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} f _ G ( t ) = \\dot { \\rho } _ { G , U } ( t ) f _ U ( \\rho _ { G , U } ( t ) ) . \\end{align*}"}
-{"id": "2824.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v + \\sum _ { i = 1 } ^ d \\frac { 1 } { 4 c _ i } ( \\partial _ { x _ i } v ) ^ 2 + \\sum _ { i = 1 } ^ d ( \\bar p ^ i _ t - \\bar { d } ^ i _ t ) \\partial _ { x _ i } v + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ d \\sigma ^ 2 _ i \\partial ^ 2 _ { x _ i x _ i } v - h = 0 \\\\ v ( T , x ) = g ( x ) , \\end{array} \\right . \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{align*} \\bigcup _ { \\psi _ I \\in \\Psi _ r } \\psi _ I ( E ) = E . \\end{align*}"}
-{"id": "4101.png", "formula": "\\begin{align*} \\tilde { \\mathcal { L } } _ { \\alpha } \\coloneqq \\left \\{ \\underline { k } \\in { K _ { \\alpha } } ^ { n } \\middle | \\overline { \\langle \\underline { p } _ { \\alpha } ( \\underline { g } ) \\underline { k } \\rangle } = G _ { \\alpha } \\right \\} \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{align*} q _ 0 = \\theta q _ j = \\frac { 1 - \\theta } { N } , j = 1 , \\dots N , \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} N ^ - ( \\sigma _ \\nu w ) \\cap \\Phi _ h ^ - [ T ] = N ^ - ( w ) \\cap \\Phi _ h ^ - [ T ] . \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} a ( \\xi ) : = ( T ( \\xi - e _ d ) - \\lambda ) ( \\sqrt { ( \\xi + e _ d ) ^ 2 + 1 } + \\sqrt { ( \\xi - e _ d ) ^ 2 + 1 } ) \\end{align*}"}
-{"id": "783.png", "formula": "\\begin{align*} \\frac { \\pi | z _ { j , n } | } { n \\ , a _ { \\max } } < | z - z _ { j , n } | , \\mbox { f o r } ~ ~ j = 1 , 2 , \\ldots , J _ n , \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} P _ t ( u ) = A ( t , x ) ( \\nabla u ) \\circ \\phi ( t ) \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{align*} \\tilde { J } _ 1 ( N ; \\theta ) : = 2 \\int _ 0 ^ { \\infty } t \\left [ 1 - \\left ( 1 - e ^ { - ( 1 - \\theta ) t / N } \\right ) ^ N \\right ] d t \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} R i c _ { F } ( \\nabla _ { F } \\varphi , U _ 1 ) = ( - \\tilde { c } \\varphi + \\tilde { b } ) U _ 1 ( \\varphi ) . \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} N _ { \\lambda , \\nu } = \\sum _ { \\mu \\vdash n } K _ { \\mu , \\lambda } K _ { \\mu , \\nu } \\textup { a n d h e n c e } N = K ^ T K \\end{align*}"}
-{"id": "7976.png", "formula": "\\begin{align*} | u _ e ( x ) - u _ e ( \\hat { x } ) | & = | ( u - L ) _ e ( x ) - u _ e ( \\hat { x } ) + L _ e ( x ) | = | ( u - L ) _ e ( x ) | \\leq \\\\ & \\leq \\frac { C | | ( u - L ) | | _ { L ^ { \\infty } ( B _ { | x - \\hat { x } | } ( x ) ) } } { | x - \\hat { x } | } \\leq \\frac { C | x - \\hat { x } | \\sigma ( | x - \\hat { x } | ) } { | x - \\hat { x } | } = C \\sigma ( | x - \\hat { x } | ) \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} I _ { 3 , + } ( x _ 1 , x _ 2 ) = \\int _ 0 ^ t \\int _ 0 ^ \\infty \\sigma ^ 2 _ s ( y ) \\int _ y ^ \\infty [ G _ { t - s } ( x _ 1 - z ) - G _ { t - s } ( x _ 2 - z ) ] \\psi ( s , z ) d z d y d s \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{align*} & \\frac { \\partial { } U ( \\mathbf { p } , \\Omega ) } { \\partial \\Omega } = \\\\ & \\sum _ { k = 1 } ^ { K } \\frac { A _ d \\sigma _ { } ^ 2 { } p _ k } { \\ln 2 ( P _ c + P _ { } ) \\left [ ( \\sigma _ N ^ 2 \\Omega + \\sigma _ { } ^ 2 ) \\| \\mathbf { f } _ k \\| ^ 2 + A _ d { } p _ k \\Omega \\right ] ( \\sigma _ N ^ 2 \\Omega + \\sigma _ { } ^ 2 ) } \\\\ & \\triangleq { } \\sum _ { k = 1 } ^ { K } \\frac { \\mathsf { C } } { \\mathsf { P } ( \\Omega ) } , \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} \\varphi _ \\lambda ' / \\varphi _ \\lambda = - z \\eta _ 1 / \\omega _ 1 + \\pi i / \\omega _ 1 + \\zeta _ \\lambda . \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} G ^ { i _ 1 . . . i _ l } _ { j _ 1 . . . . j _ l } ( A ) = \\int _ 0 ^ \\infty ( e ^ { t A } ) ^ { i _ 1 } _ { j _ 1 } . . . ( e ^ { t A } ) ^ { i _ l } _ { j _ l } d t , \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} K ( t , z ) : = \\inf _ { \\substack { z = x + y \\\\ x \\in X , \\ , y \\in Y } } \\big ( \\| x \\| _ X + t \\| y \\| _ X \\big ) . \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} R _ n ( e ^ { 2 \\pi i a / n } ) & = \\sum _ { d \\mid n } ( - 1 ) ^ { n - d } \\mu ( n / d ) e ^ { \\frac { 2 \\pi i a } { n } ( \\binom { n } { 2 } - \\frac { n } { d } \\binom { d } { 2 } ) } P _ d ( e ^ { 2 \\pi i a / d } ) \\\\ & = \\sum _ { d \\mid n } ( - 1 ) ^ { ( n - d ) ( a - 1 ) } \\mu ( n / d ) \\binom { 2 ( a , d ) - 1 } { ( a , d ) - 1 } \\\\ & = \\sum _ { d \\mid n } ( - 1 ) ^ { ( n - n / d ) ( a - 1 ) } \\mu ( d ) \\binom { 2 ( a , n / d ) - 1 } { ( a , n / d ) - 1 } , \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} = \\int _ { S ^ 1 } \\left ( f _ 1 ( \\frac 1 2 h _ 0 ' + k _ 1 c - k _ 2 d ) + \\frac 1 { k _ 0 } f _ 1 ' ( c ' + \\frac 1 2 k _ 0 h _ 0 - \\frac 1 2 ( k _ 1 h _ 1 + k _ 2 h _ 2 ) ) + f _ 2 ( e ' + k _ 2 c + k _ 1 d ) \\right ) d x \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} [ z ^ n ] \\exp f ( z ) & = \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { m ! } [ z ^ n ] f ^ m ( z ) \\\\ & = \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { m ! } \\sum _ { n _ 1 + n _ 2 + \\dots + n _ m = n } a _ { n _ 1 } a _ { n _ 2 } \\dots a _ { n _ m } . \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{align*} h _ t ( \\alpha _ t ) = \\Phi _ t ( \\alpha _ t ) = \\alpha _ t \\exp [ ( t - 1 ) u ( \\alpha _ t ) ] . \\end{align*}"}
-{"id": "8371.png", "formula": "\\begin{align*} 2 g - 2 = d - r _ 0 - r _ 1 - r _ \\infty . \\end{align*}"}
-{"id": "2743.png", "formula": "\\begin{align*} g ^ { - 1 } ( ( \\sum _ { k = 0 } ^ h p ^ k [ c _ k ] ^ { p ^ { h - k } } ) _ { h = 0 } ^ \\infty ) = g ^ { - 1 } ( ( g ^ { h } ( c ' ) ) _ { h = 0 } ^ \\infty ) = c ' , \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} \\int V _ v ^ 0 ( \\psi ) f ( \\psi ) d \\mu _ { \\hat { C } } ( \\psi ) = 0 \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} a _ 1 a _ 2 + a _ 1 a _ 3 + a _ 2 a _ 3 = 0 . \\end{align*}"}
-{"id": "438.png", "formula": "\\begin{align*} g _ { i } = X _ i ^ { k _ i } + \\sum _ { j = 0 } ^ { k _ i - 1 } h _ { j } ^ { ( i ) } ( X _ { i + 1 } , \\ldots , X _ n ) X _ i ^ j \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} \\mathcal { L } _ { I _ 0 X } \\omega _ { \\pm } = \\omega _ { \\pm } - \\sigma _ { \\pm } \\rlap { . } \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} \\Phi : = P ( \\Phi _ \\varepsilon ^ { \\lambda _ 1 } , P ( \\Phi _ \\varepsilon ^ { \\lambda _ 2 } , \\dots , P ( \\Phi _ \\varepsilon ^ { \\lambda _ { 2 ^ { r d } - 1 } } , \\Phi _ { \\varepsilon } ^ { \\lambda _ { 2 ^ { r d } } } ) \\dots ) ) \\ , . \\end{align*}"}
-{"id": "2709.png", "formula": "\\begin{align*} \\gamma ( X ) \\gamma ( Y ) + \\gamma ( Y ) \\gamma ( X ) = - 2 g ( X , Y ) , \\forall X , Y \\in \\Gamma ( T M ) . \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} \\phi ( q , p ) = ( q , - p ) , \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{align*} k \\langle r _ 1 ^ { - 1 } T _ 1 , \\dots , r _ n ^ { - 1 } T _ n \\rangle : = \\{ \\sum _ { i _ 1 , \\dots , i _ n } a _ { i _ 1 , \\dots , i _ n } T _ 1 ^ { i _ 1 } \\dots T _ n ^ { i _ n } \\in k [ [ T _ 1 , \\dots T _ n ] ] | \\ a _ { i _ 1 , \\dots , i _ n } r _ 1 ^ { i _ 1 } \\dots r _ n ^ { i _ n } \\to 0 \\} , \\end{align*}"}
-{"id": "4281.png", "formula": "\\begin{align*} \\sigma \\vec { A } = \\sigma ( B _ 0 , B _ 1 , \\dots , B _ b , C _ 1 , \\dots , C _ c ) = ( B _ 0 , \\sigma ( B _ 1 , \\dots , B _ b , C _ 1 , \\dots , C _ c ) ) , \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} & \\int _ { x _ { t - 1 } } ^ { \\infty } \\prod _ { j = t - 1 } ^ { t } x _ j ^ { - \\tau + \\zeta _ j + | Q _ j | + ( \\tau - 1 ) \\abs { W _ j } - 2 E _ { W _ j } - E _ { W _ j , \\hat { W } _ j } } \\dd x _ { t } \\\\ & = K x _ { t - 1 } ^ { 1 - 2 \\tau + \\zeta _ { t - 1 } + \\zeta _ t + { | Q _ { t - 1 } | } + { | Q _ { t } | } + ( \\tau - 1 ) \\abs { W _ t \\cup W _ { t - 1 } } - 2 E _ { W _ t \\cup W _ { t - 1 } } - E _ { W _ t \\cup W _ { t - 1 } , \\widehat { W _ t \\cup W } _ { t - 1 } } } , \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} & \\sqrt { 1 + x _ 1 } + \\dots + \\sqrt { 1 + x _ { k } } + \\sqrt { 1 + x _ { k + 1 } } \\\\ > & k - 1 + \\sqrt { 1 + x _ 1 + x _ 2 + \\dots + x _ { k } } + \\sqrt { 1 + x _ { k + 1 } } \\\\ > & k - 1 + 1 + \\sqrt { 1 + x _ 1 + x _ 2 + \\dots + x _ { k } + x _ { k + 1 } } \\\\ = & k + \\sqrt { 1 + x _ 1 + x _ 2 + \\dots + x _ { k } + x _ { k + 1 } } \\end{align*}"}
-{"id": "2122.png", "formula": "\\begin{align*} T _ 1 ^ { ( k ) } ( V ) = V \\oplus V ^ { \\otimes 2 } \\oplus \\ldots V ^ { \\otimes k } \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} C _ n ( x , y ; \\beta | q ) = \\sum _ { k = 0 } ^ n { n \\brack k } _ q ( \\beta ; q ) _ k ( \\beta ; q ) _ { n - k } x ^ k y ^ { n - k } . \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} \\partial _ 3 ( E _ { i , j } ) = Y _ 2 e _ { 1 , i , j } - X _ 2 e _ { 1 , i + 1 , j } + X _ 1 e _ { 2 , i + 1 , j } - Y _ 1 e _ { 2 , i + 1 , j + 1 } . \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ E \\right ] \\geq \\mathbb { P } \\left [ \\bigcap _ { t = 1 } ^ k G _ t \\right ] = \\prod _ { t = 1 } ^ k \\mathbb { P } \\left [ G _ t \\ \\middle | \\ G _ { < t } \\right ] \\end{align*}"}
-{"id": "8425.png", "formula": "\\begin{align*} R _ { \\tilde { \\Delta } ( m ) } : \\sum _ { j } w _ j \\Delta ( b _ j ) \\mapsto \\sum _ j w _ j \\Delta ( b _ j ) \\ , \\tilde { \\Delta } ( m ) = \\sum _ j w _ j \\Delta ( b _ j m ) \\ , \\in A \\otimes A . \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} h _ { \\epsilon , \\mu } ^ { \\mathcal { I } } ( y ^ { n } ) = - n \\abs { \\mathcal { I } } - \\sum _ { i \\in \\mathcal { N } \\setminus \\mathcal { I } } \\abs { \\mu _ i } \\xrightarrow [ n \\to \\infty ] { } - \\infty . \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} \\tilde B = - \\int B ( \\tilde z , . . . , \\tilde z ) h ( \\tilde z ) d \\tilde z \\end{align*}"}
-{"id": "7928.png", "formula": "\\begin{align*} \\partial \\Omega _ t = ( - \\infty , 0 ] \\cup \\{ r e ^ { i \\theta } : \\theta = A _ t ( r ) , r \\in ( 0 , \\infty ) \\} . \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{align*} \\int _ { \\Delta ^ + } P _ { D H } d q = \\sum _ Y \\int _ { \\tilde { \\Delta } _ Y ^ + } F _ { \\lambda \\mathcal { L } } ( q ) P _ { D H } ( q ) d q . \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{align*} [ O ^ T u ^ f ( \\cdot , T ) ] ( \\gamma , t ) = \\frac 1 2 \\ , \\Big [ \\frac { \\partial u ^ { \\tilde f } } { \\partial \\nu } ( \\gamma , t ) - \\frac { \\partial u ^ { \\tilde f } } { \\partial \\nu } ( \\gamma , 2 T - t ) \\Big ] \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{align*} L ( t ) : = \\int _ 0 ^ { 2 \\pi } | z _ \\alpha | d \\alpha . \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} c _ { p k } = c _ { v k } + \\frac { 1 } { m _ k } \\end{align*}"}
-{"id": "6462.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\nabla [ e ^ { - h B } - I d ] e ^ { - ( t _ { 0 } - s ) B } & F _ { y } ( u _ { j } , \\nabla y _ { j } , y _ { j } , x _ { j } , \\overline { b } ) \\| _ { L ^ { r } ( \\Omega ) ^ { 3 \\times 3 } } \\\\ & \\leq C \\norm { [ e ^ { - h B } - I d ] B ^ { \\frac { 1 } { 2 } } e ^ { - ( t _ { 0 } - s ) B } F _ { y } ( u _ { j } , \\nabla y _ { j } , y _ { j } , x _ { j } , \\overline { b } ) } _ { L ^ { r } ( \\Omega ) ^ 3 } . \\end{aligned} \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} \\ell : = | V ( L _ { \\beta , v } ) | \\geq \\alpha n / 8 . \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} \\lambda ( x ' ) : = ( \\kappa _ 1 ( x ' , 0 ) , \\ldots , \\kappa _ { n - 1 } ( x ' , 0 ) ) x ' \\in \\R ^ { n - 1 } . \\end{align*}"}
-{"id": "6976.png", "formula": "\\begin{align*} \\nu _ 1 , \\nu _ 2 \\in \\Z , \\nu _ 1 + \\nu _ 2 = n , \\nu _ 1 \\geq 0 , \\nu _ 2 \\geq 0 . \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} s ' ( n , 1 ) \\ , = \\ , 1 2 ^ { 3 ^ { n - 2 } ( 3 ^ { n - 1 } - 1 ) / 2 } \\end{align*}"}
-{"id": "6617.png", "formula": "\\begin{align*} | V | - | E | + | F | = 2 - 2 g . \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{align*} E _ { f } ( A , B ) = \\sum _ { \\delta \\in \\Delta } m _ { \\delta } ^ 2 = \\sum _ { j = 0 } ^ { \\log \\mu ( | A | + | B | ) } \\sum _ { \\delta \\in \\Delta \\atop 2 ^ { j } \\leq m _ \\delta < 2 ^ { j + 1 } } { m _ { \\delta } ^ 2 } < \\sum _ { j = 0 } ^ { \\log \\mu ( | A | + | B | ) } 2 ^ { 2 j + 2 } k _ { 2 ^ j } . \\end{align*}"}
-{"id": "6119.png", "formula": "\\begin{align*} u _ U ( t ) = \\frac { \\sigma \\theta } { 2 } \\left ( e ^ { \\frac { t } { \\theta } } - e ^ { - \\frac { t } { \\theta } } \\right ) , & & v _ U ( t ) = \\sigma e ^ { - \\frac { t } { \\theta } } , \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{align*} T _ { \\min } : = \\bigwedge _ { j = 1 } ^ g T _ j , \\end{align*}"}
-{"id": "6545.png", "formula": "\\begin{align*} K ( P _ j , P _ 0 ) \\leq K _ 2 n \\| f _ { \\theta ^ { ( j ) } } - 0 \\| _ n ^ 2 = 4 K _ 2 \\delta ^ 2 n N ^ { - 2 \\beta / r } , \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} C : y ^ 2 + a _ 1 x y + a _ 3 y = x ^ 3 + a _ 2 x ^ 2 + a _ 4 x + a _ 6 \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{align*} \\sigma ^ N _ \\alpha ( \\bar { a } _ { \\bar { u } ^ \\smallfrown \\bar { v } _ 2 } ) = \\sigma ^ N _ \\beta ( \\bar { a } _ { \\bar { u } ^ \\smallfrown \\bar { w } _ 2 } ) . \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} \\eta \\geq \\iint f ( \\theta - \\phi ) d \\nu _ \\eta ( \\theta ) d \\nu _ \\eta ( \\phi ) = \\iint _ { \\phi \\neq \\theta } f ( \\theta - \\phi ) d \\nu _ \\eta ( \\theta ) d \\nu _ \\eta ( \\phi ) = I [ \\nu _ \\eta ] . \\end{align*}"}
-{"id": "8275.png", "formula": "\\begin{align*} M _ d ( q ) = \\frac { 1 } { d } \\sum _ { e \\mid d } \\mu ( e ) q ^ { d / e } . \\end{align*}"}
-{"id": "1206.png", "formula": "\\begin{align*} \\widehat { \\delta } = t _ i ^ { - 1 } \\log ^ { \\frac { 1 - \\nu } { 2 } } ( t _ i ) \\delta . \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{align*} v ( t ) F _ 1 ( \\lambda ( t ) ) - | v ( t ) \\psi ( \\xi ( t ) ) | = \\max _ { w \\in [ - 1 , 1 ] } \\big ( w F _ 1 ( \\lambda ( t ) ) - | w \\psi ( \\xi ( t ) ) | \\big ) . \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} I _ 1 = \\omega _ 3 ^ { - 1 } \\omega _ 2 , \\ I _ 2 = \\omega _ 1 ^ { - 1 } \\omega _ 3 , \\ I _ 3 = \\omega _ 2 ^ { - 1 } \\omega _ 1 \\in ( T M ) \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} \\Delta ^ + _ { \\mathcal { L } } = \\chi + \\Delta _ { \\mathcal { L } } . \\end{align*}"}
-{"id": "2023.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\ddot { q } ( t ) & = - \\alpha Q ( t ) \\nabla \\sigma ( q ( t ) ) \\\\ M \\ddot { Q } ( t ) + U ' ( Q ( t ) ) & = - \\alpha \\sigma ( q ( t ) ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4378.png", "formula": "\\begin{align*} \\hat { z } = \\int _ { \\xi } ^ { \\lambda } \\frac { d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\nabla \\ ! g ( \\overline { x } ) \\Delta \\eta = 0 , \\\\ 0 \\in D ^ * \\Pi _ { K } ( g ( \\overline { x } ) \\ ! + \\ ! \\overline { \\lambda } ) ( \\Delta \\eta ) \\end{array} \\right . \\Longrightarrow \\Delta \\eta = 0 . \\end{align*}"}
-{"id": "2015.png", "formula": "\\begin{align*} I ( \\operatorname { s g n } _ N ^ G . \\chi ) = \\operatorname { s g n } _ { N ' } ^ { G ' } . I ( \\chi ) , \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} < l _ { 1 , \\gamma } w , w > _ { L ^ 2 } & = \\frac { | | \\tilde { w } | | ^ 3 _ { L ^ 3 } } { | | \\tilde { w } | | ^ 2 _ { H ^ { \\frac { 1 } { 2 } } } } \\\\ & \\leq | | w | | _ { L ^ 2 } | | \\tilde { w } | | _ { L ^ 2 } ~ ( ( \\ref { L 3 L 2 } ) ) \\end{align*}"}
-{"id": "108.png", "formula": "\\begin{align*} \\mathcal { F } ( q ) = \\frac { 1 } { 2 } \\sum _ j z _ j w _ j . \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{align*} f ( x ) = e ^ { \\varpi ( x - 1 ) } - x \\ge 0 , \\end{align*}"}
-{"id": "7006.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } ( Z _ 1 + Z _ 2 ) ^ 3 + \\frac { 3 } { 4 } ( Z _ 1 + Z _ 2 ) ( Z _ 1 - Z _ 2 ) ^ 2 & = - 9 Z _ 3 ^ 3 \\\\ \\frac { 1 } { 4 } + \\frac { 3 } { 4 } V ^ 2 & = 9 U ^ 3 \\\\ V ^ 2 & = 1 2 U ^ 3 - \\frac { 1 } { 3 } \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} d [ \\alpha , \\psi ( s ) ] & = [ d \\alpha , \\psi ( s ) ] - [ \\alpha \\wedge d \\psi ( s ) ] . \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} V _ n ( x ) : = - \\frac { 1 } { \\psi _ n ( x ) } \\sum _ { j = 1 } ^ d D _ j \\psi _ n ( x ) \\alpha _ j , \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} \\tilde { u } _ { j , k } ( a ) & = - u ^ { * , j , i } _ { i , s } ( p ) u ^ { * , s , k } ( p ) - \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\left ( \\frac { - u ^ { * , l , j } \\alpha ^ { \\vee , l } } { ( 2 \\chi - p ) ( \\alpha ^ { \\vee } ) } \\right ) _ s ( p ) u ^ { * , s , k } ( p ) \\\\ & + I _ { H , j , k } ( a ) . \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} ( D _ X \\xi ) ( X _ 1 , \\ldots , X _ k ) = { } ^ { V } \\nabla _ X ( \\xi ( X _ 1 , \\ldots , X _ k ) ) - \\sum _ { i = 1 } ^ k \\xi ( X _ 1 , \\ldots , D _ X X _ i , \\ldots , X _ k ) , \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{align*} \\cos c = \\cos a \\cdot \\cos b + \\sin a \\cdot \\sin b \\cdot \\cos \\gamma . \\end{align*}"}
-{"id": "4804.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\epsilon \\int _ { \\Omega } | \\nabla u _ { \\epsilon } | ^ { m - 2 } \\nabla u _ { \\epsilon } \\nabla U _ { k } d x = 0 . \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} \\Delta _ h G _ { t - s } ( x - z ) : = \\frac { G _ { t - s } ( x + h - z ) - G _ { t - s } ( x - z ) } { h } - \\frac { \\partial G _ { t - s } } { \\partial x } ( x - z ) . \\end{align*}"}
-{"id": "8946.png", "formula": "\\begin{align*} & \\sigma ( H ^ { \\epsilon , \\kappa } ) \\cap \\left [ a _ 0 , b _ N \\right ] \\subset \\bigcup _ { k = 0 } ^ N [ a _ k , b _ k ] \\ , , \\quad { \\rm d i m } \\big ( { \\rm R a n } E _ { [ a _ k , b _ k ] } ( H ^ { \\epsilon , \\kappa } ) \\big ) = + \\infty \\ , , \\\\ & b _ k - a _ k \\leq C _ 0 \\epsilon \\big ( \\kappa + C _ 1 \\ , \\epsilon ^ { 1 / 5 } \\big ) \\mbox { f o r } 0 \\leq k \\leq N \\ , , \\mbox { a n d } a _ { k + 1 } - b _ k \\geq \\frac { 1 } { C _ 2 } \\epsilon \\ , , \\mbox { f o r } 0 \\leq k \\leq N - 1 \\ , . \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} \\begin{aligned} - 3 a _ { 2 , 6 } ^ 2 + a _ { 2 , 6 } a _ { 3 , 8 } + 2 a _ { 1 , 4 } a _ { 3 , 8 } & = 0 \\\\ - 7 a _ { 2 , 6 } a _ { 2 , 7 } + ( 2 a _ { 1 , 4 } + a _ { 2 , 6 } ) a _ { 3 , 9 } + 3 ( a _ { 1 , 5 } + a _ { 2 , 7 } ) a _ { 3 , 8 } \\\\ - ( a _ { 2 , 8 } + 2 a _ { 1 , 6 } ) a _ { 4 , 9 } & = 0 \\\\ a _ { 4 , 9 } ( 2 a _ { 1 , 4 } - a _ { 2 , 6 } - a _ { 3 , 8 } ) & = 0 \\end{aligned} \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} \\mathbf { w } _ 0 ^ { \\mathbf { 0 } } = 0 , \\mathbf { r } _ 0 ^ { \\mathbf { 0 } } = 0 , \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{align*} u _ { t } ( x , t ) = \\Delta u ( x , t ) - \\chi \\nabla \\cdot ( u ( x , t ) \\nabla v ( x , t ) ) + ( a ^ { * } ( x , t ) - b ^ { * } ( x , t ) u ( x , t ) ) u ( x , t ) , x \\in \\R ^ N \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} a \\left ( x \\right ) : = \\begin{cases} - \\left ( r - 1 \\right ) r ^ { q } & x \\in \\left [ 0 , 1 \\right ] , \\\\ - \\frac { p ^ { \\prime \\prime } \\left ( x \\right ) } { \\left [ p \\left ( x \\right ) \\right ] ^ { q } } & x \\in \\left [ 1 , 2 \\right ) , \\\\ a \\left ( - x \\right ) & x \\in \\left ( - 2 , 0 \\right ] . \\end{cases} \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{align*} \\begin{aligned} \\pi _ { u _ \\eta } & \\leq \\pi _ { u } - \\mathcal { C } ( u _ 0 , p _ 0 , \\eta ) - \\nabla \\eta \\otimes \\nabla d ^ 2 ( u , p _ 0 ) + Q ( \\eta , \\nabla \\eta ) - \\pi _ { u _ { 1 - \\eta } } \\\\ & \\leq ( 1 - \\eta ) \\pi _ { u } + C | \\nabla \\eta | d ( u , p _ 0 ) | \\nabla u | _ 1 - \\nabla \\eta \\otimes \\nabla d ^ 2 ( u , p _ 0 ) + Q ( \\eta , \\nabla \\eta ) , \\end{aligned} \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{align*} \\begin{pmatrix*} [ r ] A & 0 \\\\ 0 & B \\end{pmatrix*} \\cdot z = e ^ { - i \\theta } A z , \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{align*} _ f ( J ) = f \\circ e ( \\chi ( J ) ) \\end{align*}"}
-{"id": "2093.png", "formula": "\\begin{align*} & g ( C _ a ) - 1 = { \\rm i n d } C _ a = \\delta ( C _ a ) = e _ Q ( C _ a ) = 0 , \\ \\ \\mbox { a n d } \\ \\ h ( C _ a ) = 1 , \\\\ & \\mbox { a n d $ C _ a $ h a s n o e n d s a t e l l i p t i c o r b i t s } . \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} u _ { \\alpha } ( t , x ) : = \\alpha u ( \\alpha ^ 2 t , \\alpha x ) , d _ { \\alpha } ( t , x ) : = d ( \\alpha ^ 2 t , \\alpha x ) , \\pi _ { \\alpha } ( t , x ) : = \\alpha ^ 2 \\pi ( \\alpha ^ 2 t , \\alpha x ) \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} x \\sim _ o y \\iff ( \\exists g , h \\in M ) ( x g = g y \\land h x = y h ) . \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} \\psi \\left ( \\nu , \\mu \\right ) = \\prod _ { x \\in \\vec { x } ( \\mu ) } \\left ( \\prod _ { i < x } t ^ { \\nu _ i } \\right ) \\nu _ { x } \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} P ( \\theta \\ , | \\ , \\phi _ i , \\phi _ j ) : = P ( \\theta \\ , | \\ , ( \\phi _ i , 0 , 0 ) , ( \\phi _ j , 0 , 0 ) ) \\ , , \\\\ Q ( \\theta \\ , | \\ , \\phi _ i , \\phi _ j ) : = Q ( \\theta \\ , | \\ , ( \\phi _ i , 0 , 0 ) , ( \\phi _ j , 0 , 0 ) ) \\ , . \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} f _ { r _ m , r _ t } ( x , y ) & = \\frac { 4 ( \\lambda _ c \\pi ) ^ { m + 1 } y x ^ { 2 m - 1 } } { ( m - 1 ) ! } e ^ { - \\lambda _ c \\pi y ^ 2 } . \\end{align*}"}
-{"id": "9278.png", "formula": "\\begin{align*} p _ { \\mathrm { S } _ 0 } ( m | n ) & = \\int _ { 0 } ^ \\infty p _ { \\mathrm { S } _ 0 } ( t _ \\mathrm { p } | n ) \\frac { ( \\lambda _ \\mathrm { B } t _ \\mathrm { p } ) ^ { m } } { m ! } e ^ { - \\lambda _ \\mathrm { B } t _ \\mathrm { p } } d t _ \\mathrm { p } \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} q _ \\gamma ( z ) = ( c z ^ 2 + ( d - a ) z - b ) \\frac { \\partial } { \\partial z } \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} K _ L e _ 4 = - c _ 1 \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{align*} f ( u ) & = u ^ \\ell + z _ 1 u ^ { \\ell - 1 } + \\cdots + z _ \\ell , & f ' ( u ) & = u ^ { \\ell ' } + z _ 1 ' u ^ { \\ell ' - 1 } + \\cdots + z _ { \\ell ' } ' , \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} \\begin{cases} \\operatorname { d i v } ( y ^ a \\nabla v ) = y ^ a v _ t \\\\ - \\lim _ { y \\to 0 } y ^ a v _ y = \\phi , \\end{cases} \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{align*} y ^ { n } : = \\left \\{ \\begin{array} { l l } x ^ { m _ { k } } & n = n _ { m _ { k } } k \\\\ z & \\end{array} \\right . \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} \\sum _ { n = 2 } ^ { \\infty } \\| V _ 0 ^ { 0 - 1 - 1 , l , ( n ) } \\| _ { 1 , \\infty , r } \\le c \\frac { N } { h } \\alpha ^ { - 4 } L ^ { - d } . \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} S = & \\bigl \\{ [ y _ 1 , y _ 2 , \\dots , y _ n ] \\mid y _ k \\in X ^ 2 \\mbox { f o r s o m e } k , 2 \\le k \\le n - 1 ; \\ \\ y _ i \\in X \\mbox { f o r a l l } i \\ne k \\bigr \\} , \\\\ S ' = & \\bigl \\{ [ y _ 1 , y _ 2 , \\dots , y _ n ] \\mid y _ 2 \\in X ^ 2 , \\ y _ i \\in X \\mbox { f o r a l l } i \\ne 2 \\bigr \\} . \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} P ^ \\eta \\Big ( \\sum _ { m = 1 } ^ { T - 1 } Y _ m \\leq ( v - \\varepsilon ) T + 1 \\Big ) \\leq { \\rm e } ^ { - c T } \\leq { \\rm e } ^ { - \\phi _ T ^ { 1 / 3 } } \\ , . \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} E _ j ( s ) = \\{ t \\theta : t \\leq r _ j + \\phi _ j ( \\theta , s ) \\} . \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} \\alpha = ( x _ M ^ n \\times 1 ) + \\nu \\cdot ( 1 \\times \\omega _ { N } ) \\in H ^ n ( M ; \\R ) \\oplus H ^ n ( N ; \\R ) \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} & \\bigl [ s [ y _ i , x _ 1 ] , y _ { i + 1 } \\dots y _ { i ' - 1 } \\bigr ] [ y _ { i ' } , x _ 2 ] + \\bigl [ s [ y _ i , x _ 2 ] , y _ { i + 1 } \\dots y _ { i ' - 1 } \\bigr ] [ y _ { i ' } , x _ 1 ] \\\\ & \\equiv \\ \\sum _ { m = i + 1 } ^ { i ' - 1 } y _ { i + 1 } \\dots y _ { m - 1 } \\bigl ( [ s , y _ m ] [ y _ i , x _ 1 ] y _ { m + 1 } \\dots y _ { i ' - 1 } [ y _ { i ' } , x _ 2 ] \\\\ & + \\ [ s , y _ m ] [ y _ i , x _ 2 ] y _ { m + 1 } \\dots y _ { i ' - 1 } [ y _ { i ' } , x _ 1 ] \\bigr ) \\pmod { I } . \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{align*} ( f _ 1 * f _ 2 ) ( w , z ) = \\left | \\{ ( u , v ) \\in \\mathcal { O } _ 1 \\times \\mathcal { O } _ 2 : u + v = w z _ 1 z _ 2 [ u , v ] ^ { 1 / 2 } = z \\} \\right | . \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} V ( x ) \\ ! = \\ ! \\frac { - z } { | x | } I , \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{align*} V \\left [ S ( \\theta ) \\right ] = \\frac { N ^ 2 } { ( 1 - \\theta ) ^ 2 } \\sum _ { j = 1 } ^ N \\frac { 1 } { j ^ 2 } - \\frac { N H _ N } { 1 - \\theta } + O \\left ( e ^ { - \\varepsilon N } \\right ) , N \\to \\infty , \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} \\phi _ { V _ N } ( \\mathbf { u } ) = \\sum _ { n = - \\infty } ^ { \\infty } a _ { - n } \\phi _ n \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{align*} h \\xi : = ( h ^ { \\gamma _ 1 } \\xi _ 1 , \\ldots , h ^ { \\gamma _ d } \\xi _ d ) \\in \\R ^ d . \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} \\left ( g \\cdot u ^ { ( r ) } \\right ) ^ \\alpha = c _ r ^ \\alpha \\end{align*}"}
-{"id": "9111.png", "formula": "\\begin{align*} { \\bf D } _ { I } ^ { - 1 } = \\frac { 1 } { \\omega _ I } { \\bf I } _ K , \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} \\langle u , w \\rangle = \\int _ a ^ b \\langle v ( t ) , w \\rangle \\ ; d t \\mbox { f o r a l l } w \\in \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} d q _ t = & \\tilde \\gamma ^ { - 1 } ( t , q _ t ) \\left ( - \\partial _ t \\psi ( t , q _ t ) - \\nabla _ q V ( t , q _ t ) + \\tilde F ( t , q _ t ) \\right ) d t \\\\ & + \\tilde S ( t , q _ t ) d t + \\tilde \\gamma ^ { - 1 } ( t , q _ t ) \\sigma ( t , q _ t ) \\circ d W _ t \\\\ = & \\tilde \\gamma ^ { - 1 } ( t , q _ t ) F ( t , q _ t ) d t + S ( t , q _ t ) d t + \\tilde \\gamma ^ { - 1 } ( t , q _ t ) \\sigma ( t , q _ t ) d W _ t . \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} { n \\brack k } _ q ( 1 - q ^ k ) = { n \\brack { k - 1 } } _ q ( 1 - q ^ { n - k + 1 } ) . \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} M _ 1 + M _ 2 = \\int _ { ( ( K _ 1 \\cup K _ 2 ) \\cap K ) \\times \\mathbb { T } } 1 d \\mu = \\int _ { K \\times \\mathbb { T } } 1 d \\mu = \\mu ( K \\times \\mathbb { T } ) . \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} d _ b ( p _ 1 , p _ 2 ) = d _ a ( p _ 1 - p _ 2 ) , d _ r ( p _ 1 , p _ 2 ) = \\| p _ 1 - p _ 2 \\| _ { q } . \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} \\log \\left ( x ^ r \\ , f ( x ) \\right ) = \\log \\left ( g \\left ( \\tfrac { 1 } { x } \\right ) \\right ) . \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} \\varphi _ t - \\varphi _ \\infty = \\sum _ { j = 0 } ^ \\infty B _ { j , t } t ^ { ( 1 - 2 j ) / 3 } \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} F \\subseteq \\bigcup _ { i = 1 } ^ { k _ n } \\phi _ { 1 ^ n } ^ { - 1 } \\circ g ^ { - 1 } \\circ \\phi _ { I _ { i , n } } ( F ) . \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{align*} g ( u ) = 2 \\frac { v } { u } - u \\ln ( 1 - e ^ { - v } ) \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} H ( U _ r , \\rho ) = \\frac { H ( U , r \\rho ) } { H ( U , r ) } \\geq \\rho ^ { \\beta } . \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{align*} G _ { i _ 1 i _ 2 i _ 3 } ^ { j _ 1 j _ 2 j _ 3 } = \\delta ^ { j _ 1 k _ 1 } \\delta ^ { j _ 2 k _ 2 } \\delta ^ { j _ 3 k _ 3 } \\int _ 0 ^ \\infty ( e ^ { - y \\tilde \\gamma } ) _ { i _ 1 k _ 1 } ( e ^ { - y \\tilde \\gamma } ) _ { i _ 2 k _ 2 } ( e ^ { - y \\tilde \\gamma } ) _ { i _ 3 k _ 3 } d y , \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} b ^ { + } _ 2 ( \\tilde { X ^ Y } ) & = b _ 1 ( \\tilde { X ^ Y } ) - 1 + \\frac { n l r } { 2 } ( \\chi ( X ^ Y ) + \\sigma ( X ^ Y ) ) \\\\ & = ( l - 1 ) ( r - 1 ) - 1 + \\frac { n l r } { 2 } ( \\chi ( X ^ Y ) + \\sigma ( X ^ Y ) ) \\\\ & > ( l - 1 ) ( r - 1 ) - 1 + \\frac { n l r } { 2 } ( 2 b ( 2 g - 2 ) - \\frac { ( b - 1 ) ( g - 1 ) } { 2 } ) > 1 \\end{align*}"}
-{"id": "6980.png", "formula": "\\begin{align*} w ( a ) = \\nu _ 1 \\textup { a n d } w ( b ) = \\nu _ 1 + 1 , \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 + } \\frac { \\sup _ { t \\in T } \\sup _ { \\rho ( s , t ) \\leq h } | X ( t ) - X ( s ) | } { \\sigma ( h ) ( \\log { 1 / h } ) ^ { ( 1 + \\epsilon ) / \\gamma } } = 0 \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} \\dot { x } = \\big ( A + \\widetilde { S } ^ z ( t ) \\big ) x . \\end{align*}"}
-{"id": "3634.png", "formula": "\\begin{align*} c _ { s _ { \\alpha } } ( \\chi ) = \\dfrac { 1 - q ^ { - 1 } \\mathbf { z } ^ { n _ { \\alpha } \\alpha ^ { \\vee } } } { 1 - \\mathbf { z } ^ { n _ { \\alpha } \\alpha ^ { \\vee } } } \\ , , c _ { s _ { \\alpha } w } ( \\chi ) = c _ { s _ { \\alpha } } ( \\prescript { w } { } { \\chi } ) c _ { w } ( \\chi ) . \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{align*} x \\ , \\frac { f ' \\left ( x \\right ) } { f \\left ( x \\right ) } + \\tfrac { 1 } { x } \\ , \\frac { g ' \\left ( \\tfrac { 1 } { x } \\right ) } { g \\left ( \\tfrac { 1 } { x } \\right ) } = - r . \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} J _ \\mathrm { f s u } : = ( J | _ { H ^ + } ) _ \\mathrm { f s u } : = ( J | _ { R [ H ^ + ] } ) _ \\mathrm { f s u } : = \\textrm { s u m o f a l l f i n i t e l y s u r j e c t i v e s u b m o d u l e s o f $ J $ } . \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} \\hat \\Sigma : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N ( X _ i - \\bar X ) ( X _ i - \\bar X ) ^ \\top \\mbox { w h e r e } \\bar X = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N X _ i . \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} ( Y ( w ) - y _ { \\mu } ( w ) ) z ^ { \\mu } = \\sum _ { \\nu \\prec \\mu } e _ { \\mu , \\nu } ( q , t ; w ) z ^ { \\nu } , \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} \\chi \\bigl ( X _ \\infty \\bigr ) = \\sum _ { \\tau _ p \\in \\mathcal { C } _ \\infty } \\ , ( - 1 ) ^ { i \\bigl ( \\tau _ p \\bigr ) } . \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} x ^ T ( t ) P ^ { - 1 } x ( t ) = & \\int _ 0 ^ t x ^ T ( s ) P ^ { - 1 } d x ( s ) + \\int _ 0 ^ t d x ^ T ( s ) P ^ { - 1 } x ( s ) \\\\ = & \\int _ 0 ^ t x ^ T ( s ) P ^ { - 1 } A x ( s ) d s + \\int _ 0 ^ t x ^ T ( s ) P ^ { - 1 } B u ( s ) d s \\\\ & + \\int _ 0 ^ t x ^ T ( s ) A ^ T P ^ { - 1 } x ( s ) d s + \\int _ 0 ^ t u ^ T ( s ) B ^ T P ^ { - 1 } x ( s ) d s \\\\ & + \\sum _ { i = 1 } ^ m \\left ( \\int _ 0 ^ t x ^ T ( s ) P ^ { - 1 } N _ i x ( s ) u _ i ( s ) d s + \\int _ 0 ^ t x ^ T ( s ) u _ i ( s ) N _ i ^ T P ^ { - 1 } x ( s ) d s \\right ) . \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} C _ { \\infty } : = 2 \\mathcal { G } _ T { \\rm e } ^ { 2 L _ g ^ 2 T ^ 2 } . \\end{align*}"}
-{"id": "6332.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } ( i ) \\ , \\ , U ^ T \\nabla G ( U ) = \\nabla G ( U ) ^ T U ; \\\\ \\\\ ( i i ) \\ , \\ , \\nabla G ( U ) = U U ^ T \\nabla G ( U ) ; \\\\ \\\\ ( i i i ) \\ , \\ , U ^ T U = \\mathbb { I } _ p . \\end{array} \\right . \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} \\phi _ j ^ { ( 2 i - 1 ) } - \\frac { \\partial \\phi _ j } { \\partial t _ { 2 i - 1 } } = \\sum _ { k = 1 } ^ { j - i - 1 } \\phi _ k a _ { j - i - k + 1 } , \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{align*} E = \\left \\{ T _ { K ( N s ) + 1 } < T _ { K ( N t ) } \\right \\} \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} a \\big | _ { t = 0 } = a ^ 0 = ( a _ 1 ^ 0 , . . . , a _ p ^ 0 ) , \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{align*} s t / 2 + k ( t / 2 - 1 ) + k _ 3 ( R ) = s / 2 - k / 2 + k _ 3 ( R ) = ( s - k ) / 2 + k _ 3 ( R ) \\ge 0 , \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} \\sigma _ i ( M _ 1 , \\dots , M _ r ) = . \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{align*} \\sum _ { v \\in V ( G ) } \\binom { d ( v ) } { 2 } \\le \\binom { n - 7 } { 2 } + ( n - 1 ) \\binom { 3 } { 2 } = \\frac { n ^ 2 } { 2 } - \\frac { 9 n } { 2 } + 2 5 . \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} C _ 1 = \\frac { S _ { m a x } } { \\sqrt { 2 \\pi } } & & D _ 1 = \\frac { S _ { m a x } } { 2 } \\\\ C _ 2 = \\frac { S _ { m i n } } { \\sqrt { 2 \\pi } } & & D _ 2 = \\frac { S _ { m i n } } { 2 } \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{align*} J = J ( \\{ A _ n \\} , \\{ B _ n \\} ) = \\begin{bmatrix} B _ 1 & A _ 1 & 0 & \\ldots \\\\ A ^ * _ 1 & B _ 2 & A _ 2 & \\ddots \\\\ 0 & A ^ * _ 2 & B _ 3 & \\ddots \\\\ \\vdots & \\ddots & \\ddots & \\ddots \\end{bmatrix} , \\end{align*}"}
-{"id": "5373.png", "formula": "\\begin{align*} \\sigma = ( \\sigma _ i \\times \\sigma ' _ { n - i } ) \\cdot w ^ { - 1 } \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} f _ { x _ i = 1 } = f _ { x _ i = 0 } ^ d . \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} T _ { V } \\phi _ { V W } T ^ { * } _ { W } = I _ { \\mathcal { H } } , \\end{align*}"}
-{"id": "8769.png", "formula": "\\begin{align*} I _ H ( a ) = \\sum _ { \\beta \\in \\Phi _ s ^ + } \\ln \\sinh ( - 2 \\beta ( a ) ) - \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } } 2 \\alpha ( a ) , \\end{align*}"}
-{"id": "4861.png", "formula": "\\begin{align*} \\biggl ( \\mathop { \\oplus } _ { \\substack { u _ 1 + \\cdots + u _ s = u , \\\\ 0 \\leq u _ j < u } } ( H ^ { u _ 1 } ( M _ 1 ; \\R ) \\otimes \\cdots \\otimes H ^ { u _ s } ( M _ s ; \\R ) ) \\biggl ) \\oplus ( H ^ { u - n } ( M ; \\R ) \\otimes H ^ n ( N ; \\R ) ) . \\end{align*}"}
-{"id": "824.png", "formula": "\\begin{align*} W _ t = \\sum _ { \\mu _ i \\geq 0 } e ^ { - \\frac { 1 } { 2 } t \\mu _ i } \\left ( { \\rm d i m } \\ , \\mathcal H _ { \\mu _ i , a } ^ + - { \\rm d i m } \\ , \\mathcal H _ { \\mu _ i , a } ^ - \\right ) . \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} \\gamma _ 2 ( x , y ) & = \\frac { \\gamma ( x , y ) - \\lambda \\gamma _ 1 ( x , y ) } { 1 - \\lambda } \\\\ \\mu _ 2 ( x , y ) & = \\frac { \\mu ( x , y ) - \\lambda \\mu _ 1 ( x , y ) } { 1 - \\lambda } \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} \\frac { d x ( t ) } { d t } & = A x ( t ) + B u ( t ) + \\sum _ { i = 1 } ^ m N _ i x ( t ) u _ i ( t ) , \\ ; \\ ; \\ ; x ( 0 ) = 0 , \\\\ y ( t ) & = C x ( t ) , \\ ; \\ ; \\ ; t \\geq 0 , \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{align*} w t ( B ) = \\sum _ { \\substack { k = 1 , \\dots , n - 1 , \\\\ i = 1 , \\dots , m _ k } } \\sum _ { f _ { a , c } \\in \\mathcal { B } _ { k , i } } w t ( f _ { a c } ) . \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} \\Delta _ B \\phi = \\mathcal { L } _ { H ^ { 1 , 0 } } \\phi . \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{align*} N ^ { i \\xi } \\phi _ N ( \\xi ) = 1 + K _ 1 ( N ) + K _ 2 ( N ) + O \\left ( \\frac { 1 } { N } \\right ) , \\end{align*}"}
-{"id": "8282.png", "formula": "\\begin{align*} E _ d ( R ) = \\frac { 1 - \\frac { 1 } { q ^ d } } { 1 - \\frac { 1 } { q } } = 1 + \\frac { 1 } { q } + \\frac { 1 } { q ^ 2 } + \\frac { 1 } { q ^ 3 } + \\ldots + \\frac { 1 } { q ^ { d - 1 } } . \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{align*} w _ t ( Y ) = \\sum _ { s \\in S _ \\mu } Y _ { s , t } v _ s + v _ t , & & w _ { t ' } ( Z ) = \\sum _ { s ' \\in S _ { \\nu } } Z _ { s ' , t ' } v _ { s ' } + v _ { t ' } . \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} \\nu _ 1 ( \\lambda ) = \\prod _ { j \\geq 1 } \\binom { M _ j ( 1 ) + m _ j - 1 } { m _ j } = \\begin{cases} 1 & \\lambda = [ 1 ^ d ] \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} 0 \\leq \\frac { \\rho _ { j } ^ { k } } { 2 R } \\Vert U _ { j } ^ { k } x _ { j - 1 } ^ { k } - x _ { j - 1 } ^ { k } \\Vert ^ { 2 } \\leq \\frac { 1 } { 2 R } \\sum _ { l = 1 } ^ { p } \\rho _ { l } ^ { k } \\Vert U _ { l } ^ { k } x _ { l - 1 } ^ { k } - x _ { l - 1 } ^ { k } \\Vert ^ { 2 } \\leq \\Vert U _ { k } x ^ { k } - x ^ { k } \\Vert \\end{align*}"}
-{"id": "3819.png", "formula": "\\begin{align*} \\alpha = s ( 3 m + 3 r ) - \\ell _ 3 ( m + 2 r ) - \\ell _ 2 ( m ) - \\ell _ 1 ( m + r ) . \\end{align*}"}
-{"id": "4912.png", "formula": "\\begin{align*} a _ { 3 n + 1 } & = \\tau \\cdot a _ n + a _ { n + 1 } , \\\\ a _ { 3 n + 2 } & = \\omega \\cdot a _ n + \\sigma \\cdot a _ { n + 1 } . \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} y _ { m } = \\arg \\min _ { y \\in \\mathbb { R } ^ { N } } \\left \\Vert C _ { m } C _ { m } ^ { T } y - b \\right \\Vert = W _ { m + 1 } \\arg \\min _ { t \\in \\mathbb { R } ^ { m + 1 } } \\left \\Vert H _ { m } H _ { m } ^ { T } t - \\Vert b \\Vert e _ { 1 } \\right \\Vert = W _ { m + 1 } t _ { m } . \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} z = g \\sum _ { j = 1 } ^ { r } t _ j e _ j . \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} x ^ * : = - \\overline { \\alpha ( a ) } x , x \\in \\mathfrak g _ a , \\ a \\in \\Gamma . \\end{align*}"}
-{"id": "1328.png", "formula": "\\begin{align*} S = \\{ g \\in G _ { m , m } \\mid \\pi ( g ) ( 0 , 0 , \\dots , 0 ) = ( 0 , 0 , \\dots , 0 ) \\} \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} \\left ( x \\frac { d } { d x } \\right ) ^ n \\log \\left ( \\theta _ 2 ( x ) \\right ) = ( - 1 ) ^ n \\left ( x \\frac { d } { d x } \\right ) ^ n \\log \\left ( \\theta _ 4 \\left ( \\tfrac { 1 } { x } \\right ) \\right ) . \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} \\sum _ { a = 1 } ^ n & \\{ \\langle \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi , \\phi \\rangle + \\langle \\phi , \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi \\rangle \\} \\\\ & \\geq C \\sum _ { a = 1 } ^ n \\{ g _ Q ( \\nabla _ { V _ a } H ^ { 1 , 0 } , \\bar V _ a ) + \\overline { g _ Q ( \\nabla _ { V _ a } H ^ { 1 , 0 } , \\bar V _ a ) } \\} \\\\ & \\geq C \\ { \\rm d i v } _ \\nabla ( \\kappa _ B ^ \\sharp ) . \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{align*} Q \\bar { x } + c + \\sum _ { i \\in \\bar { \\beta } } \\bar { \\lambda } _ i \\ , ( A _ { i \\ , \\bullet \\ , } ) ^ T = 0 . \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} i + j + k = m i + b j + ( b + 1 ) k = n \\ , . \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ L N _ \\ell - 1 = N ( \\Phi ) - d - 1 > M ( \\Phi ) = \\sum _ { \\ell = 1 } ^ L \\left ( \\| A _ \\ell \\| _ { \\ell ^ 0 } + \\| b _ \\ell \\| _ { \\ell ^ 0 } \\right ) , \\end{align*}"}
-{"id": "4815.png", "formula": "\\begin{align*} \\| \\overline { u } & \\| _ { L ^ { m \\chi ^ { n + 1 } } ( \\Omega _ { R _ { n + 1 } } ) } \\le C ^ { 1 / \\chi ^ n } \\Bigg [ \\frac { 1 } { R _ { n } - R _ { n + 1 } } + \\left ( \\frac { G ( R _ { n } , R _ { n + 1 } ) } { m ( \\chi ^ n - 1 ) + 1 - C m \\epsilon ^ { m ' } } \\right ) ^ { 1 / m } \\\\ & + \\left ( \\frac { \\| f \\| _ q } { ( m ( \\chi ^ n - 1 ) + 1 - C m \\epsilon ^ { m ' } ) k ^ { m - 1 } } \\right ) ^ { N / m ( m q - N ) } \\Bigg ] ^ { 1 / \\chi ^ n } \\| \\overline { u } \\| _ { L ^ { m \\chi ^ n } ( \\Omega _ { R _ n } ) } . \\end{align*}"}
-{"id": "1985.png", "formula": "\\begin{align*} \\Delta '' = \\Delta ' _ \\tau + [ \\Lambda , \\ , [ \\Lambda , \\ , \\frac { i } { 2 } \\partial \\bar \\partial \\omega ] ] - [ \\partial \\omega \\wedge \\cdot , \\ , ( \\partial \\omega \\wedge \\cdot ) ^ \\star ] , \\end{align*}"}
-{"id": "1121.png", "formula": "\\begin{align*} d ^ n \\circ \\partial ^ n = \\partial ^ { n + 1 } \\circ d ^ n \\ , . \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} \\mathcal { L } \\Theta = \\bigtriangleup _ { g } \\Theta - \\frac { 1 } { 2 } \\left \\langle X , \\nabla _ { g } \\Theta \\right \\rangle = 0 \\end{align*}"}
-{"id": "1496.png", "formula": "\\begin{align*} \\mathcal { L } = U T U ^ { - 1 } , T = \\Delta + \\frac { 1 } { 2 } \\Delta f - \\frac { 1 } { 4 } \\left | \\nabla f \\right | ^ { 2 } . \\end{align*}"}
-{"id": "4396.png", "formula": "\\begin{align*} \\zeta _ \\lambda ( z ( \\lambda , \\hat { X } ) ) = \\int _ 0 ^ { \\hat { X } } \\frac { d X ( X - \\frac 1 3 ( \\lambda + 1 ) ) } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } + \\eta _ 2 / 2 . \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} s \\bigl ( ( I - P ) \\lambda _ H , ( I - P ) \\mu _ H \\bigr ) = ( \\rho g , T ( I - P ) \\mu _ H ) \\quad \\mu _ H \\in \\Lambda _ H , \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{align*} E \\left [ T _ 2 ( T _ 2 + 1 ) \\right ] & = \\left ( \\frac { \\nu _ 2 } { \\alpha _ 2 } \\right ) ^ 2 M ^ 2 \\left ( H _ { \\nu _ 2 M } ^ 2 + \\sum _ { j = 1 } ^ { \\nu _ 2 M } \\frac { 1 } { j ^ 2 } \\right ) + O \\left ( e ^ { - \\varepsilon M } \\right ) \\\\ & = \\big ( \\lambda ^ { - 1 } \\nu _ 1 + \\nu _ 2 \\big ) ^ 2 M ^ 2 \\left ( H _ { \\nu _ 2 M } ^ 2 + \\sum _ { j = 1 } ^ { \\nu _ 2 M } \\frac { 1 } { j ^ 2 } \\right ) + O \\left ( e ^ { - \\varepsilon M } \\right ) \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} \\tau ( \\lambda \\bullet I _ t ( H ) ) \\ = \\ \\bigcap _ { i = 1 } ^ t I _ i ( H ) ^ { ( \\lfloor \\lambda ( t - i + 1 ) \\rfloor - n - t + 2 i ) } . \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} d e t \\left [ \\begin{array} { c c } \\tau & \\frac { 1 } { 3 } ( h _ { j + 1 } - h _ j ) \\\\ \\frac { 1 } { 3 } ( h _ { j + 1 } - h _ j ) & \\tau \\end{array} \\right ] = 0 , \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} A = \\alpha _ 1 E _ { 1 , 2 } + \\alpha _ 3 E _ { 2 , 3 } + \\dots + \\alpha _ { m - 1 } E _ { m - 1 , m } \\qquad B = \\sum _ { i , j = 1 , \\ldots , m } \\beta _ { i j } E _ { i , j } \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} r ( x ) - r ( y ) & = \\sum _ { j = 1 } ^ N \\frac { x ^ \\frac { 1 } { 2 ^ j } - y ^ \\frac { 1 } { 2 ^ j } } { 2 ^ { j - 1 } } \\le \\left ( x ^ \\frac { 1 } { 2 } - y ^ \\frac { 1 } { 2 } \\right ) \\sum _ { j = 1 } ^ N \\frac { 1 } { 2 ^ { j - 1 } } \\\\ & = \\frac { \\left ( 2 - \\frac { 1 } { 2 ^ { N - 1 } } \\right ) ( x - y ) } { \\left ( x ^ \\frac { 1 } { 2 } + y ^ \\frac { 1 } { 2 } \\right ) } < \\frac { 2 ^ N - 1 } { 2 ^ { N } \\sqrt { y } } . \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} a _ n & = a _ { n - 1 } + ( n - 1 ) 2 ^ n - \\frac { 1 } { 6 } ( n - 1 ) ( n ^ 3 - 3 n ^ 2 + 1 4 n - 6 ) , n \\geq 2 , \\\\ \\end{align*}"}
-{"id": "1380.png", "formula": "\\begin{align*} d _ { K } ^ 2 ( x + y ) - d _ { K } ^ 2 ( y ) - \\langle \\triangledown d _ { K } ^ 2 ( y ) , \\ , x \\rangle & \\le \\Vert ( y + x ) - \\Pi _ { K } ( y ) \\Vert ^ 2 - \\Vert y - \\Pi _ { K } ( y ) \\Vert ^ 2 - 2 \\langle x , \\ , y - \\Pi _ { K } ( y ) \\rangle \\\\ & = \\Vert x \\Vert ^ 2 . \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} ^ i _ r = & \\log \\left ( 1 + \\frac { \\alpha _ i ^ 2 | h | ^ 2 r ^ { - \\alpha } } { \\sum ^ { M _ s } _ { j = i + 1 } \\alpha _ j ^ 2 | h | ^ 2 r ^ { - \\alpha } + \\frac { 1 } { \\rho } } \\right ) , \\end{align*}"}
-{"id": "6835.png", "formula": "\\begin{align*} \\dot { \\tilde { x } } ( t ) & = \\tilde { A } \\tilde { x } ( t ) + \\tilde { B } u ( t ) + \\displaystyle \\sum _ { j = 1 } ^ { n _ { \\rm i n } } u _ j ( t ) \\tilde { N } _ j \\tilde { x } ( t ) + \\tilde { H } ( \\tilde { x } ( t ) \\otimes \\tilde { x } ( t ) ) \\\\ \\tilde { y } ( t ) & = \\tilde { c } ^ \\top \\tilde { x } ( t ) , \\end{align*}"}
-{"id": "5095.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ { R _ i } \\sigma _ p ( m _ { i \\ell } ) \\sigma _ p ( \\gamma _ { i \\ell } ) \\ge s ( p - 1 ) \\quad 1 \\le i \\le M . \\end{align*}"}
-{"id": "2197.png", "formula": "\\begin{align*} W _ G ( S ) = \\frac 1 2 \\sum _ { u \\in S } w _ S ( u ) . \\end{align*}"}
-{"id": "7884.png", "formula": "\\begin{align*} & \\Bigl [ \\Lambda ( t ) , H _ { \\epsilon } ( t ) \\Bigr ] = - \\Bigl [ \\Lambda ( t ) , X _ { \\epsilon } ( t ) \\Bigr ] ^ { \\dagger } H ( t ) X _ { \\epsilon } ( t ) + X _ { \\epsilon } ( t ) ^ { \\dagger } H ( t ) \\Bigl [ \\Lambda ( t ) , X _ { \\epsilon } ( t ) \\Bigr ] . \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{align*} \\alpha _ 1 : = M _ 1 p _ 1 = \\frac { \\nu _ 1 } { \\nu _ 1 + \\lambda \\nu _ 2 } \\alpha _ 2 : = M _ 2 p _ 2 = \\frac { \\lambda \\nu _ 2 } { \\nu _ 1 + \\lambda \\nu _ 2 } = 1 - \\alpha _ 1 \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} \\frac 1 2 \\sum _ { j = 1 } ^ { m - 1 } \\alpha _ j \\Delta ^ { ( m ) } \\partial _ { z _ { j } } = - \\partial _ { z _ 1 } \\ , . \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{align*} \\Phi = \\begin{pmatrix} 0 & - q \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} & I ( K _ A ; J ^ n | F _ { 1 : k } = f _ { 1 : k } ) \\leq I ( K _ A ; Z ^ n | F _ { 1 : k } = f _ { 1 : k } ) . \\end{align*}"}
-{"id": "2147.png", "formula": "\\begin{align*} \\dot { y } = u _ \\mu ( \\rho ) , \\dot { \\rho } = 2 \\rho y v _ \\mu ( \\rho ) \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} - \\Delta _ p u _ n - ( 1 - \\varepsilon _ n ) V | u _ n | ^ { p - 2 } u _ n + \\varepsilon _ n | u _ n | ^ { p - 2 } u _ n = f \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} p _ { n } = \\mathbf { t } _ n \\begin{bmatrix} 0 \\\\ \\mathbf { M } ^ { - 1 } \\mathbf { f } \\end{bmatrix} . \\end{align*}"}
-{"id": "3202.png", "formula": "\\begin{align*} \\ast \\rho = - \\eta \\wedge \\tau _ 2 - \\tfrac { 1 } { r } d r \\wedge \\tau _ 1 + O ( r ^ { - 3 - \\mu } ) . \\end{align*}"}
-{"id": "7811.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle { \\frac { d } { d t } \\bar Q = \\delta ( [ W , Q ] + \\gamma D ) - \\frac { \\partial F ( Q ) } { \\partial Q } } \\end{array} \\end{align*}"}
-{"id": "195.png", "formula": "\\begin{align*} \\hat f ( r ) = \\dfrac { 1 } { \\abs { U ( P ) } } \\int _ { U ( P ) } F ( r , \\xi ) \\ , d \\xi . \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} x = g ( t _ 1 e _ 1 + t _ 2 e _ 2 + t _ 3 e _ 3 ) = g \\begin{pmatrix} 0 & t _ 1 & t _ 2 \\\\ t _ 1 & 0 & t _ 3 \\\\ t _ 2 & t _ 3 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "1762.png", "formula": "\\begin{align*} \\widetilde { h } ( x ' , D _ { x } ) u ( x ) : = \\lambda ^ { - 1 , * } h ( x ' , D _ { x } ) \\kappa ^ * u ( x ' ) u \\in \\mathcal { S } ( \\overline { \\R ^ { n } _ + } ) . \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { d - 1 } z _ i ^ { ( k ) } x ^ i + p _ k m ( x ) \\equiv \\sum _ { i = 0 } ^ { d - 1 } z _ i ^ { ( k + 1 ) } x ^ { i + 1 } \\pmod { n \\Z [ x ] } \\quad z ^ { ( 0 ) } = ( 1 , 0 , \\dots , 0 ) . \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} ( J \\theta ) ( X ) = - \\theta ( J X ) \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{align*} g ( x , y ) : = \\min \\{ x , y , ( x + y - 1 ) / ( y + 1 ) \\} \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} - \\xi = ( x , - \\nu ( x ) ) \\in U ( P ) . \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} \\frac { \\partial L } { \\partial t _ { 2 i - 1 } } = \\left [ \\left ( P ^ { 2 i - 1 } \\right ) _ + , L \\right ] , \\end{align*}"}
-{"id": "4537.png", "formula": "\\begin{align*} P _ L ^ * ( \\boldsymbol { \\epsilon } ) \\geq \\prod _ { j = 1 } ^ { L } P ^ * ( \\epsilon _ j ) , \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} s _ i g ( z _ 1 , \\dots , z _ i , z _ { i + 1 } , \\dots , z _ n ) = g ( z _ 1 , \\dots , z _ { i + 1 } , z _ i , \\dots , z _ n ) , \\forall \\ g \\in \\mathbb { C } [ z _ 1 , \\dots , z _ n ] , \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{align*} \\operatorname { t r } ( L ) = \\prod _ { i = 1 } ^ { 2 k + 1 } ( 1 + ( \\beta _ { i + 1 } - \\lambda ) \\frac { \\partial ^ 2 } { \\partial g _ i \\partial g _ { i + 1 } } ) \\prod _ { j = 1 } ^ { 2 k + 1 } g _ i . \\end{align*}"}
-{"id": "8016.png", "formula": "\\begin{align*} \\mathbb { E } [ \\overline { V } _ t ] \\le \\frac { c _ 2 } { c _ 1 } = \\frac { \\eta ^ 2 } { 2 c _ 1 ( 2 a \\lambda _ 2 - c _ 1 ) } + \\frac { b | T r ( L ) | } { 2 c _ 1 N } + \\frac { m \\tau ^ 2 \\tilde { \\gamma } ( N - 1 ) } { 2 c _ 1 N } . \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} S ( n , m ) : = \\left \\{ ( \\{ s _ j \\} _ { j = 1 } ^ { m + 1 } , \\{ t _ k \\} _ { k = 1 } ^ { n - m } ) \\ \\Bigg | \\ \\begin{array} { l } 1 = s _ 1 < s _ 2 < \\cdots < s _ { m + 1 } \\le n , \\\\ 1 = t _ 1 < t _ 2 < \\cdots < t _ { n - m } \\le n , \\\\ \\{ s _ j \\} _ { j = 2 } ^ { m + 1 } \\cup \\{ t _ k \\} _ { k = 2 } ^ { n - m } = \\{ 2 , 3 , \\cdots , n \\} , \\\\ \\{ s _ j \\} _ { j = 2 } ^ { m + 1 } \\cap \\{ t _ k \\} _ { k = 2 } ^ { n - m } = \\emptyset . \\end{array} \\right \\} . \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{align*} \\tau ( \\lambda _ 0 / \\rho , \\lambda _ f / \\rho , \\lambda _ 1 / \\rho ) = 0 . \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} v _ q ( a _ { n , k } ) & \\geq \\min _ { m > k , \\ , l m \\leq n } \\left ( f ( l , m , n ) + v _ q ( a _ { n - l m , m - 1 } ) \\right ) \\\\ & \\geq \\min _ { m > k , \\ , l m \\leq n } \\left ( f ( l , m , n ) + \\frac { m - 2 } { m } \\binom { n - l m } { 2 } \\right ) \\\\ & = \\min _ { m > k , \\ , l m \\leq n } \\left ( \\frac { m - 2 } { m } \\binom { n } { 2 } + \\binom { l } { 2 } \\right ) \\\\ & = \\frac { k - 1 } { k + 1 } \\binom { n } { 2 } . \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{align*} \\Psi ( M ) = \\sum _ { r = 0 } ^ d E _ r M E _ r , \\end{align*}"}
-{"id": "4159.png", "formula": "\\begin{align*} | f ( x ) - p ( x ) | = | f ( x ) - f ( x _ 0 ) | \\leq \\| f \\| _ { C ^ { 0 , \\beta } } \\cdot | x - x _ 0 | ^ \\sigma \\leq B \\cdot | x - x _ 0 | ^ \\beta , \\end{align*}"}
-{"id": "5576.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( x \\frac { d } { d x } \\right ) ^ 4 \\log \\left ( \\theta _ 3 ( x ) \\right ) & = x \\ , \\psi ( x ) + 7 x ^ 2 \\ , \\psi ' ( x ) + 6 x ^ 3 \\ , \\psi '' ( x ) + x ^ 4 \\psi ''' ( x ) \\\\ & < \\left ( 6 . 3 5 \\ , x - 2 7 5 . 8 \\ , x ^ 2 + 1 5 3 3 \\ , x ^ 3 - 1 5 5 8 \\ , x ^ 4 \\right ) e ^ { - 2 \\pi x } \\\\ & + \\left ( - 6 \\ , x + 1 4 0 . 7 \\ , x ^ 2 - 3 3 0 . 6 \\ , x ^ 3 + 1 9 3 \\ , x ^ 4 \\right ) e ^ { - \\pi x } . \\end{aligned} \\end{align*}"}
-{"id": "4180.png", "formula": "\\begin{align*} \\ell _ \\varepsilon = C _ 2 \\cdot \\Omega _ \\varepsilon \\cdot K _ \\varepsilon \\leq C _ 2 \\cdot \\Omega _ \\varepsilon \\cdot K _ { \\varepsilon ' } < C _ 2 \\cdot \\Omega _ { \\varepsilon ' } \\cdot K _ { \\varepsilon ' } = \\ell _ { \\varepsilon ' } . \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\xi '' ( s ) \\alpha ( s ) s d s & = \\int _ 0 ^ u \\xi '' ( s ) \\gamma ( s ) s d s + \\int _ u ^ 1 \\xi '' ( s ) s d s \\\\ & = \\int _ 0 ^ u \\xi '' ( s ) \\gamma ( s ) s d s + \\xi ' ( 1 ) - \\xi ' ( u ) u - \\bigl ( \\xi ( 1 ) - \\xi ( u ) \\bigr ) . \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} \\frac { d x } { d z } = \\sqrt { \\frac { ( 1 + \\lambda ^ { 2 } ) ^ { 3 } ( z ^ { 2 } + 2 C _ { 1 } ) ^ { 2 } } { 4 - ( 1 + \\lambda ^ { 2 } ) ^ { 2 } ( z ^ { 2 } + 2 C _ { 1 } ) ^ { 2 } } } \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} ^ 1 _ { m , 1 } = & \\frac { \\frac { \\alpha _ 1 ^ 2 | h _ { m , m 1 } | ^ 2 } { L \\left ( | | y _ { m , 1 } | | \\right ) } } { \\frac { \\alpha _ 2 ^ 2 | h _ { m , m 1 } | ^ 2 } { L \\left ( | | y _ { m , 1 } | | \\right ) } + ^ { m , 1 } _ { i n t e r } + \\frac { 1 } { \\rho } } , \\end{align*}"}
-{"id": "8948.png", "formula": "\\begin{align*} S ( E ) & = \\pi H ^ 0 \\pi - \\pi H ^ 0 \\pi ^ \\perp [ \\pi ^ \\perp H ^ 0 \\pi ^ \\perp ] ^ { - 1 } \\pi ^ \\perp H ^ 0 \\pi - E ( \\pi + \\pi H ^ 0 \\pi ^ \\perp [ \\pi ^ \\perp H ^ 0 \\pi ^ \\perp ] ^ { - 2 } \\pi ^ \\perp H ^ 0 \\pi ) \\\\ & + \\mathcal { O } ( E _ 0 ^ 2 ) \\ , . \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} \\ln \\left ( \\frac { N ( \\Phi ) - 1 } { L } \\right ) = \\ln \\left ( \\frac { 1 } { L } \\sum _ { j = 0 } ^ { L - 1 } N _ { j } \\right ) \\geq \\frac { 1 } { L } \\sum _ { j = 0 } ^ { L - 1 } \\ln N _ { j } = \\frac { 1 } { L } \\cdot \\ln \\left ( \\prod _ { j = 0 } ^ { L - 1 } N _ { j } \\right ) , \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} \\mathbf { y } = \\mathbf { G P x } + \\mathbf { z } , \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} & p _ { U V } ( ( x ^ n , y ^ n ) , z ^ n ) = p _ { X ^ n , Y ^ n , Z ^ n } ( x ^ n , y ^ n , z ^ n | X ^ n \\in \\mathcal { A } , Y ^ n \\in \\mathcal { B } ) \\\\ & u _ 1 = ( \\mathbf { x } _ 1 , \\mathbf { y } _ 1 ) \\\\ & u _ 2 = ( \\mathbf { x } _ 2 , \\mathbf { y } _ 2 ) . \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} - X R _ \\pm ( 0 ) = { \\rm I d } \\mp P _ \\pm . \\end{align*}"}
-{"id": "4841.png", "formula": "\\begin{align*} \\lambda \\cdot ( x _ M ^ n \\times 1 ) + \\lambda \\nu \\cdot ( 1 \\times \\omega _ { N } ) = f ^ * ( x _ M ^ n \\times 1 ) + \\nu \\cdot ( \\alpha ^ n _ M \\times 1 ) \\pm \\nu \\cdot ( 1 \\times \\omega _ { N } ) . \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} ( \\mu + a ) \\widetilde C _ a ^ { \\mu } ( t ) + \\widetilde C _ { a - 1 } ^ { \\mu + 1 } ( t ) = ( \\mu + [ \\frac { a + 1 } { 2 } ] ) \\widetilde C _ a ^ { \\mu + 1 } ( t ) . \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} \\mathrm { R e } ( \\chi ( C _ D + C _ E ) ) = \\mathrm { R e } ( \\chi ( C _ E ) ) = \\chi \\circ \\Re ( C _ E ) = \\chi ( C _ E ) . \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} ( \\overline { P } _ i f ) ( y ) = ( g _ i ( y ) f ) ( y ) , ~ g ( y ) = g _ 1 ( y ) \\ldots g _ n ( y ) , ~ g _ i ( y ) \\in U _ i , \\end{align*}"}
-{"id": "5021.png", "formula": "\\begin{align*} [ c , z _ 1 ] [ z _ 2 , z _ 3 , z _ 4 ] + [ c , z _ 2 ] [ z _ 1 , z _ 3 , z _ 4 ] = 0 , \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\int _ { \\Omega } \\lvert \\nabla d ( s ) \\rvert ^ 2 \\varphi ( s ) ^ 2 \\ ; \\d x \\ ; \\d s = \\int _ 0 ^ { t _ 0 } \\int _ { \\Omega } \\lvert \\nabla d ( s ) \\rvert ^ 2 \\varphi ( s ) ^ 2 \\ ; \\d x \\ ; \\d s + \\int _ { t _ 0 } ^ t \\int _ { \\Omega } \\lvert \\nabla d ( s ) \\rvert ^ 2 \\varphi ( s ) ^ 2 \\ ; \\d x \\ ; \\d s = : \\mathrm { I } + \\mathrm { I I } . \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} g ( x , D _ x ) f ( x ) = \\int _ { \\R ^ { n - 1 } } \\int _ { \\R _ + } K _ g ( x , x ' - w ' , x _ n , w _ n ) \\ , f ( w ' , w _ n ) \\ , d w _ n \\ , d w ' , \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{align*} \\widetilde \\Pi = M _ r N _ { r - 1 } ^ { - 1 } M _ { r - 1 } \\dotsm N _ 1 ^ { - 1 } M _ 1 N _ r ^ { - 1 } \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{gather*} D ^ { S _ v } = B _ 1 ^ { S _ v } \\cup B _ 2 ^ { S _ v } , D ^ { S _ c } = B _ 1 ^ { S _ c } \\cup B _ 2 ^ { S _ c } , D ^ { e } = B _ 1 ^ { e } \\cup B _ 2 ^ { e } . \\end{gather*}"}
-{"id": "6264.png", "formula": "\\begin{align*} \\dim W = 2 ^ { N - | \\lambda | } . \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} \\lim _ { \\Re s \\to \\infty } \\frac { I I I _ b ( s ) e ^ { s l ( \\gamma ) } } { s ^ 2 } = 0 . \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} C ( p _ X \\| q _ X ) = \\max _ { \\alpha \\in [ 0 , 1 ] } ( 1 - \\alpha ) D _ \\alpha ( p _ X \\| q _ X ) . \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} Z ( I _ { 1 6 } ) = ( z _ { r } ^ { n } - v z _ { r } ^ { - n } ) \\ , Z ( I _ { 1 5 } ; c _ r , 0 ) . \\end{align*}"}
-{"id": "1945.png", "formula": "\\begin{align*} \\overline { \\rho } ( g \\sigma g ^ { - 1 } ) = \\overline { \\rho } ( g ) \\rho ( \\sigma ) \\overline { \\rho } ( g ^ { - 1 } ) . \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} & E _ 1 ( t ) = 1 , \\ \\ \\ E _ 2 ( t ) = t + 1 , \\ \\ \\ E _ 3 ( t ) = t ^ 2 + 4 t + 1 , \\\\ & E _ 4 ( t ) = t ^ 3 + 1 1 t ^ 2 + 1 1 t + 1 , \\ \\ \\ E _ 5 ( t ) = t ^ 4 + 2 6 t ^ 3 + 6 6 t ^ 2 + 2 6 t + 1 . \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} K _ { f V } = T ( \\nabla f , V ) + f K _ V , \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} \\phi ^ { ( n ) } ( X ) : = \\left [ \\phi ( x _ { i j } ) \\right ] _ { i , j } \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} X _ j ( u ) = g _ { i j } a _ i = : \\tilde a _ j , \\ \\ j = 1 , . . . , n , \\end{align*}"}
-{"id": "4810.png", "formula": "\\begin{align*} ( m \\alpha + 1 - C m \\epsilon ^ { m ' } ) \\int \\eta ^ m \\overline { u } _ s ^ { m \\alpha } \\phi ( | \\nabla \\overline { u } _ s | ) | \\nabla \\overline { u } _ s | ^ 2 \\le & \\int C m g _ 1 ( x , \\epsilon ) \\phi ( \\overline { u } ) \\overline { u } ^ 2 \\overline { u } _ s ^ { m \\alpha } \\\\ & + \\int | f | \\eta ^ m \\overline { u } _ s ^ { m \\alpha } \\overline { u } . \\end{align*}"}
-{"id": "4677.png", "formula": "\\begin{align*} \\Phi ( X ) = \\int _ 0 ^ \\infty \\frac { A ^ { 1 / 2 } } { \\lambda + A } X \\frac { A ^ { 1 / 2 } } { \\lambda + A } { \\rm d } \\lambda \\end{align*}"}
-{"id": "7256.png", "formula": "\\begin{align*} \\forall j , n , \\rho _ j ^ n : = \\sum _ { k = 1 } ^ K \\omega _ k ( f _ j ^ n ( v _ k ) + f _ j ^ n ( - v _ k ) ) . \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac { \\sqrt { T } } { 2 } \\widetilde { P } ^ { 0 , 0 } _ { 2 T } ( X ( T ) = \\sqrt { T } \\xi ) = f ( \\xi ) \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} f ( y ) = \\frac { y ^ 2 \\ , \\vartheta _ 4 ' ( y ) } { \\vartheta _ 4 ( y ) } , \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} \\operatorname { g r a d e } ( G _ + , G _ I ( M ) ) = \\min \\{ i : H ^ i ( L ^ I ( M ) ) \\neq 0 , \\mbox { w h e r e } 0 \\leqslant i \\leqslant g \\} . \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} ( \\tilde \\gamma ^ { - 1 } ) ^ { i j } \\tilde \\gamma _ { j k } = \\delta ^ i _ k , \\end{align*}"}
-{"id": "6515.png", "formula": "\\begin{align*} e _ d : = \\sqrt { \\pi } { \\Gamma ( { d - 1 \\over 2 } ) \\over ( d - 2 ) ! \\Gamma ( 1 + d / 2 ) } \\cdot \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} \\Omega _ { \\beta , \\bar { \\beta } } = \\frac { d u ( \\beta ^ { \\vee } ) } { \\sinh ( 2 \\beta ) } - \\frac { 2 } { \\cosh ( 2 \\beta ) } \\chi \\circ \\Re ( [ \\theta \\sigma ( e _ { \\beta } ) , e _ { \\beta } ] ) \\end{align*}"}
-{"id": "2252.png", "formula": "\\begin{align*} \\begin{aligned} D ( P \\parallel Q ) = \\sup _ { 0 < \\alpha < 1 } D _ \\alpha ( P \\parallel Q ) . \\end{aligned} \\end{align*}"}
-{"id": "13.png", "formula": "\\begin{align*} \\Phi _ { u , \\nu } ( u , x , \\lambda ) = \\max _ { m \\in S } \\Bigl ( m x + \\Bigl ( \\lambda + \\frac { \\xi '' ( u ) } { 2 } c \\Bigr ) m ^ 2 \\Bigr ) . \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} g ^ { n , \\ell } ( x ) & = \\frac { g ( \\eta ^ n ( x + 1 ) ) + \\cdots + g ( \\eta ^ n ( x + \\ell ) ) } { \\ell } , \\\\ \\eta ^ { n , \\ell } ( x ) & = \\frac { \\eta ^ n ( x + 1 ) + \\cdots + \\eta ^ n ( x + \\ell ) } { \\ell } \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} V \\ \\in \\ \\ell _ { 0 } ^ { \\infty } ( \\Gamma , \\mathbb { R } _ { 0 } ^ { + } ) \\ \\doteq \\ \\Big \\{ V : \\Gamma \\rightarrow \\mathbb { R } _ { 0 } ^ { + } \\ ; | \\ \\lim _ { | x | \\rightarrow \\infty } V ( x ) = 0 \\Big \\} , \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} \\mathcal { L } | A | + ( | A | ^ 2 + \\dfrac 1 2 ) | A | = 0 . \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} a _ n + \\sum \\limits _ { k = 1 } ^ { n - 1 } \\delta _ { a _ k 0 } = b _ n + \\sum \\limits _ { k = 1 } ^ { n } \\delta _ { b _ k 0 } , \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} \\mathbb { L } _ i = \\left ( \\frac { t z _ i - z _ { i + 1 } } { z _ i - z _ { i + 1 } } \\right ) ( s _ i - 1 ) , \\end{align*}"}
-{"id": "3572.png", "formula": "\\begin{align*} \\partial _ { t } \\gamma & = \\kappa B \\\\ & = \\partial _ { s } \\gamma \\times \\partial _ { s s } \\gamma \\\\ & = ( g _ { s } h _ { s s } - g _ { s s } h _ { s } , f _ { s s } h _ { s } - f _ { s } h _ { s s } , f _ { s } g _ { s s } - f _ { s s } g _ { s } ) \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} \\left \\langle u , v \\right \\rangle _ { L ^ 2 ( \\Sigma , e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma ) } = \\int _ { \\Sigma } u v e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma . \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} g = \\pi ^ { * } g _ { B } + ( h \\circ \\pi ) ^ { 2 } \\sigma ^ { * } g _ { F } , \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{align*} \\phi _ N ( \\xi ) = E \\left [ \\zeta ^ { \\ , - \\frac { ( 1 - \\theta ) S ( \\theta ) - N \\ln N } { N } } \\right ] = \\zeta ^ { \\ , \\ln N } E \\left [ \\left ( \\zeta ^ { \\frac { ( 1 - \\theta ) } { N } } \\right ) ^ { - S ( \\theta ) } \\right ] = \\zeta ^ { \\ , \\ln N } G \\left ( \\zeta ^ { \\frac { ( 1 - \\theta ) } { N } } \\right ) . \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{align*} f _ 1 * f _ 2 & = \\sum _ { u \\in \\mathcal { O } _ 1 } \\sum _ { v \\in \\mathcal { O } _ 2 } \\delta _ { ( u , \\ , z _ 1 ) \\cdot ( v , \\ , z _ 2 ) } \\\\ [ 5 p t ] & = \\sum _ { u \\in \\mathcal { O } _ 1 } \\sum _ { v \\in \\mathcal { O } _ 2 } \\delta _ { \\left ( u + v , \\ , z _ 1 z _ 2 [ u , v ] ^ { 1 / 2 } \\right ) } . \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} \\alpha _ { N , t } : = \\langle \\psi _ N , \\widehat { m } \\psi _ N \\rangle + | \\mathcal { E } _ N ( \\psi _ N ) - \\mathcal { E } ^ { G P } ( u ) | - N ( N - 1 ) \\langle \\psi _ N , g _ \\beta ( x _ 1 - x _ 2 ) \\widehat { r } \\psi _ N \\rangle . \\end{align*}"}
-{"id": "7858.png", "formula": "\\begin{align*} b _ n ^ \\alpha & = \\int _ { - \\infty } ^ 0 \\max _ { 0 \\leq k \\leq n - 1 } g _ k ( s ) ^ \\alpha d s + \\sum _ { \\ell = 0 } ^ { n - 1 } \\int _ { \\ell } ^ { \\ell + 1 } \\max _ { 0 \\leq k \\leq n - 1 } g _ k ( s ) ^ \\alpha d s \\\\ & = \\int _ { - \\infty } ^ 0 g _ 0 ( s ) ^ \\alpha d s + \\sum _ { \\ell = 0 } ^ { n - 1 } \\int _ { \\ell } ^ { \\ell + 1 } \\max \\Big \\{ g _ { \\ell } ( s ) , g _ { \\ell + 1 } ( s ) \\Big \\} ^ \\alpha d s , \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} \\phi _ L : = L ^ { 1 / 1 0 0 } . \\end{align*}"}
-{"id": "8421.png", "formula": "\\begin{align*} \\mu \\bigl ( ( \\theta \\otimes \\operatorname { i d } ) ( E ) \\bigr ) & = \\mu \\bigl ( ( \\nu \\otimes \\operatorname { i d } ) ( E ( b \\otimes 1 ) ) \\bigr ) = \\mu \\bigl ( \\gamma _ B ( b ) \\bigr ) \\\\ & = \\mu \\bigl ( ( R \\circ \\sigma ^ { \\nu } _ { i / 2 } ) ( b ) \\bigr ) = \\nu ( b ) = \\theta ( 1 ) . \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} \\left \\{ \\varphi , \\psi \\right \\} _ * = - \\left \\langle r \\nabla \\varphi , \\nabla \\psi \\right \\rangle - \\left \\langle r \\nabla ^ { \\prime } \\varphi , \\nabla ^ { \\prime } \\psi \\right \\rangle + 2 \\left \\langle r _ { - } \\nabla ^ { \\prime } \\varphi , \\nabla \\psi \\right \\rangle + 2 \\left \\langle r _ { + } \\nabla \\varphi , \\nabla ^ { \\prime } \\psi \\right \\rangle , \\end{align*}"}
-{"id": "2621.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l l } R i c ( X , Y ) = R i c _ { B } ( X , Y ) - m h ^ { - 1 } \\nabla _ { B } \\nabla _ { B } h ( X , Y ) , \\\\ \\noalign { \\smallskip } R i c ( X , U ) = 0 , \\\\ \\noalign { \\smallskip } R i c ( U , V ) = R i c _ { F } ( U , V ) - [ h \\Delta _ { B } h + ( m - 1 ) | \\nabla _ { B } h | ^ { 2 } ] g _ { F } ( U , V ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} - b ( 1 + b ^ 2 ) \\mathfrak { R e } \\dot { c } _ 0 + ( b ^ 2 + \\frac 1 2 ) \\mathfrak { R e } \\dot { c } _ 1 - \\frac { \\sqrt { 2 } } 4 b \\mathfrak { R e } \\dot { c } _ 2 = 0 . \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{align*} W ( x , y ) = x ^ n + \\sum _ { r = \\mu + 1 } ^ { 5 \\mu + \\nu } A _ { 2 r } x ^ { n - 2 r } y ^ { 2 r } . \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} \\mu ( A _ 1 ) + \\mu ( A _ 2 ) + \\mu ( A _ 3 ) > \\mu ( A ) + \\mu ( B ) + \\mu \\big ( - ( A + B ) ^ c \\big ) - \\delta = 1 - ( \\rho + \\delta ) . \\end{align*}"}
-{"id": "6804.png", "formula": "\\begin{align*} ( [ \\nabla _ t , D ] \\xi ) ( X , X _ 1 , \\ldots , X _ k ) & = R ^ { S M } ( \\partial _ t , X ) \\cdot \\xi + R ^ P ( \\nabla _ t \\phi , D _ X \\phi ) ( \\xi ( X _ 1 , \\ldots , X _ k ) ) \\\\ & - \\frac { 1 } { 2 } D _ { \\dot { g } ( X ) } \\xi ( X _ 1 , \\ldots , X _ k ) + \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ k \\xi ( X _ 1 , \\ldots , D _ X \\dot { g } ( X _ i ) , \\ldots , X _ k ) . \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} \\mathrm { P } _ { m , i } = & \\mathrm { P } \\left ( \\alpha _ 1 = 1 , z _ m < \\max \\left \\{ \\frac { \\epsilon _ 1 } { \\rho \\xi _ 1 } , \\cdots , \\frac { \\epsilon _ i } { \\rho \\xi _ { i } } \\right \\} \\right ) \\\\ & + \\mathrm { P } \\left ( \\alpha _ 1 < 1 , z _ m < \\max \\left \\{ \\frac { \\epsilon _ 1 } { \\rho \\xi _ 1 } , \\cdots , \\frac { \\epsilon _ 1 } { \\rho \\xi _ { i } } \\right \\} \\right ) . \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} G ( t ) = m _ G ( t ) + \\varphi _ { G , U } ( t ) U ( \\rho _ { G , U } ( t ) ) . \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{align*} \\Xi _ \\infty = \\begin{pmatrix} ( q \\bar q ) ^ { - 1 / 4 } k ^ { - 1 } & 0 \\\\ 0 & ( q \\bar q ) ^ { 1 / 4 } k \\end{pmatrix} . \\end{align*}"}
-{"id": "6439.png", "formula": "\\begin{align*} \\begin{aligned} k ^ { \\infty } _ { j + 1 } ( T ) & \\leq k ^ { \\infty } _ { 0 } ( T ) + C \\tilde { C } _ { T } \\Big [ k ^ { u } _ { j } ( T ) k ^ { \\nabla y } _ { j } ( T ) + k ^ { \\nabla y } _ { j } ( T ) ^ 2 ( k ^ { \\infty } _ { j } ( T ) + \\lvert \\overline { b } \\rvert ) \\Big ] \\\\ & \\leq k ^ { \\infty } _ { 0 } ( T ) + C \\tilde { C } _ { T } \\Big [ k ^ { q } _ { j } ( T ) ^ 2 ( 1 + k ^ { \\infty } _ { j } ( T ) + \\lvert \\overline { b } \\rvert ) \\Big ] . \\end{aligned} \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{align*} \\frac { q ^ { \\sum _ { \\sigma , \\tau } k ( \\sigma , \\tau ) r _ { \\sigma } r _ { \\tau } } \\prod _ { i = 1 } ^ n x _ i ^ { \\sum _ { \\sigma \\ni i } r _ \\sigma } } { \\prod _ { \\sigma } ( q ) _ { r _ \\sigma } } . \\end{align*}"}
-{"id": "3290.png", "formula": "\\begin{align*} M R I G _ n = \\int _ { C _ W } e ^ { - \\d \\ , ( a ^ * M _ x a + b ^ * M _ x ^ { - 1 } b ) } \\frac { d x } { \\sqrt { \\det M _ x } } = \\left ( \\frac { \\pi } { 2 } \\right ) ^ { n / 2 } \\frac { e ^ { - ( a _ 1 b _ 1 + \\cdots + a _ n b _ n ) } } { a _ 1 \\ldots a _ n } \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} f _ { \\beta } ( z ) = - ( 1 - z ^ N ) \\times \\prod _ { x ~ \\mbox { { \\small p e r i o d i c } } } \\Bigl ( 1 - z ^ { \\mbox { { \\small o r d e r } } ~ ( x ) } \\Bigr ) , \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{align*} g _ x ( y ) = \\Psi ( x ) \\log \\frac { \\Psi ( x ) } { \\Psi ( y ) } + ( 1 - \\Psi ( x ) ) \\log \\frac { 1 - \\Psi ( x ) } { 1 - \\Psi ( y ) } . \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{align*} & d ^ \\ast \\alpha _ 1 = 0 , & & d ^ \\ast \\beta _ 2 - ( \\lambda + 4 ) \\alpha _ 1 = 0 , \\\\ & d \\alpha _ 1 + d ^ \\ast \\alpha _ 3 - ( \\lambda + 2 ) \\beta _ 2 = 0 , & & d \\beta _ 2 + d ^ \\ast \\beta _ 4 - ( \\lambda + 2 ) \\alpha _ 3 = 0 , \\\\ & d \\alpha _ 3 - ( \\lambda + 4 ) \\beta _ 4 = 0 , & & d \\beta _ 4 = 0 . \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} & J ( x ) = \\begin{cases} x & , \\ ; x \\geq x ^ * , \\\\ \\frac { 1 } { 2 } e ^ { c x ^ * } ( x ^ * - \\frac { 1 } { c } ) e ^ { - c x } + \\frac { 1 } { 2 } e ^ { - c x ^ * } ( x ^ * + \\frac { 1 } { c } ) e ^ { c x } & , \\ ; 0 \\leq x < x ^ * , \\\\ \\frac { 1 } { 2 } e ^ { c x ^ * } ( x ^ * - \\frac { 1 } { c } ) e ^ { c x } + \\frac { 1 } { 2 } e ^ { - c x ^ * } ( x ^ * + \\frac { 1 } { c } ) e ^ { - c x } & , \\ ; - x ^ * < x < 0 , \\\\ - x & , \\ ; x \\leq - x ^ * . \\end{cases} \\end{align*}"}
-{"id": "474.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & \\zeta ^ { 4 } & \\zeta ^ { - 4 0 } & 7 \\\\ 0 & \\zeta ^ { - 8 } & \\zeta ^ { 4 4 } & 1 \\\\ 0 & \\zeta ^ { 2 8 } & \\zeta ^ { 3 2 } & 7 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 7 & \\zeta ^ { 4 0 } & \\zeta ^ { 4 4 } \\\\ 0 & \\zeta ^ { 3 2 } & 7 & \\zeta ^ { 4 0 } \\\\ 0 & \\zeta ^ { 2 8 } & \\zeta ^ { 3 2 } & 7 \\end{pmatrix} , \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} \\nu = \\frac { \\mu ( B _ { r _ h } ( x _ h ) \\cap B _ { k + 1 } \\cap E _ { k + 2 } ) } { \\mu ( B _ { r _ h } ( x _ h ) ) } \\leq \\frac { \\mu ( B _ { r _ h } ( x _ h ) \\cap E _ { k + 1 } ) } { \\mu ( B _ { r _ h } ( x _ h ) ) } . \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} \\mathfrak { g } _ { \\beta _ 1 + \\cdots + \\beta _ { i } } = [ \\mathfrak { g } _ { \\beta _ { i } } , \\mathfrak { g } _ { \\beta _ 1 + \\cdots + \\beta _ { i - 1 } } ] \\subseteq [ \\mathcal { I } , \\mathcal { I } _ { i - 1 } ] = \\mathcal { I } _ i \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} A = \\sum _ { | \\lambda | < k } a _ { \\lambda } S _ { \\lambda } + \\sum _ { | \\mu | = k } S _ { \\mu } A _ { \\mu } \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{align*} W : = \\widetilde { W } _ { n _ 0 } + b ^ * - \\varepsilon _ { q ^ * } ^ { \\widetilde { W } _ { n _ 0 } } = V _ { \\nu _ n } + \\left [ \\left ( B + b ^ * - \\frac 1 4 \\right ) - \\left ( \\varepsilon _ { q ^ * } ^ { \\widetilde { W } _ { n _ 0 } } - b ^ * \\right ) \\right ] - B . \\end{align*}"}
-{"id": "5057.png", "formula": "\\begin{align*} s y _ 1 \\dots y _ { i - 1 } \\bigl ( [ y _ i , x _ 1 ] y _ { i + 1 } \\dots y _ { i ' - 1 } [ y _ { i ' } , x _ 2 ] + [ y _ i , x _ 2 ] y _ { i + 1 } \\dots y _ { i ' - 1 } [ y _ { i ' } , x _ 1 ] \\bigr ) = Q _ 1 ' + Q _ 2 ' \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{align*} \\zeta : = e ^ { - i \\xi } . \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} \\Gamma _ J = \\{ ( J _ 1 , \\dots , J _ N ) : N \\in \\N , J _ 1 , \\dots , J _ N \\in \\N : J _ 0 + \\cdots + J _ N = J \\} \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} h ( x _ 1 , \\dots , x _ d ) = \\prescript { } { i } h ^ 2 ( x _ i , \\prescript { } { - i } h ^ { d - 1 } ( x _ 1 , \\dots , x _ { i - 1 } , x _ { i + 1 } , \\dots , x _ d ) ) . \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{align*} y : = \\frac { M \\widetilde { y } - y _ 1 } { M - 1 } \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} p \\mbox { - } d i m \\left ( M \\right ) + p \\mbox { - } d i m \\left ( M ^ { \\perp } \\right ) = p \\mbox { - } d i m \\left ( \\mathbb { Z } _ { p ^ { r } } ^ { n } \\right ) . \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} F _ { n , h } ( x , y ) = \\begin{cases} \\frac { \\frac { 1 } { n h } \\sum _ { i = 0 } ^ { n - 1 } \\mathbb I ( x \\leq X _ i \\leq x + h , X _ { i + 1 } \\leq y ) } { \\frac { 1 } { n h } \\sum _ { i = 0 } ^ n \\mathbb I ( x \\leq X _ i \\leq x + h ) } & \\mbox { i f } \\ \\ \\mathbb I ( x \\leq X _ i \\leq x + h ) \\neq 0 \\mbox { f o r s o m e } 0 \\leq i \\leq n - 1 \\\\ 0 & \\mbox { o t h e r w i s e , } \\end{cases} \\end{align*}"}
-{"id": "1266.png", "formula": "\\begin{align*} H ( S _ t \\mu , D _ n ) = H ( \\mu , D _ n ) + O ( t ) + O ( 1 ) . \\end{align*}"}
-{"id": "4066.png", "formula": "\\begin{align*} r _ { X Y } ( \\cdot ) = p _ { X Y | V } ( \\cdot | v ) = p _ { X | V } ( \\cdot | v ) p _ { Y | X } ( \\cdot | \\cdot ) . \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} \\lim u _ { \\lambda } ( t _ j \\mu _ Y + b _ j ) = u _ Y ( b ) \\end{align*}"}
-{"id": "929.png", "formula": "\\begin{align*} N [ \\mathfrak { e } , { \\widetilde { V } } ] \\ \\leq \\ \\mathcal { L } _ { { \\widetilde { V } } } [ ( 1 - { \\varepsilon } ) \\eta ( \\mathfrak { e } ) ] \\ = \\ \\# \\big \\{ x \\in \\Gamma \\ , \\big | \\ V ( x ) \\geq ( 1 - { \\varepsilon } ) \\eta ( \\mathfrak { e } ) \\big \\} . \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} \\Psi ( p , x ) : = \\widetilde { F } ( p , x ) - { \\rm g p h } \\mathcal { N } _ \\Gamma \\ \\ { \\rm w i t h } \\ \\widetilde { F } ( p , x ) : = ( x , - F ( p , x ) ) . \\end{align*}"}
-{"id": "4068.png", "formula": "\\begin{align*} \\sup _ { U V \\rightarrow X \\rightarrow Y } \\frac { I ( U ; Y | V ) } { I ( U ; X | V ) } \\geq s ^ * ( X ; Y | V = v ^ * ) . \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} p _ { - 2 } ( a ) p _ { - 2 } ( b ) & = \\sum _ { \\alpha _ { 1 } + \\alpha _ { 2 } = a } \\sum _ { \\beta _ { 1 } + \\beta _ { 2 } = b } p ( \\alpha _ { 1 } ) p ( \\alpha _ { 2 } ) p ( \\beta _ { 1 } ) p ( \\beta _ { 2 } ) \\\\ & > \\sum _ { \\gamma _ { 1 } + \\gamma _ { 2 } = a + b } p ( \\gamma _ { 1 } ) p ( \\gamma _ { 2 } ) \\\\ & = p _ { - 2 } ( a + b ) , \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{align*} \\bold \\Lambda = \\left ( \\begin{matrix} \\bold \\Lambda _ S & 0 \\\\ 0 & \\bold \\Lambda _ P \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "5161.png", "formula": "\\begin{align*} k ( \\sigma , \\tau ) = | P ( \\sigma , \\tau ) | - l _ { \\sigma } . \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} F _ j = F _ j ^ + - F _ j ^ - . \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} d _ B = d | _ { \\Omega _ B ^ * ( \\mathcal F ) } , d _ T = d _ B - \\epsilon ( \\kappa _ B ) , \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} m _ t = F ( m , m ' , m '' , \\dots ) . \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} \\Gamma _ { 2 } : = \\{ ( q ( s ) , \\ , t ( s ) \\phi _ { 1 } + w ( q ( s ) , t ( s ) ) ) : | s | < s _ { 0 } \\} , \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} x _ i \\cdot T _ { \\ell _ 1 , \\ell _ 2 , \\cdots , \\ell _ { i - 1 } , { \\ell _ { i } - 1 } , { \\ell _ { i + 1 } + 1 } , \\cdots , \\ell _ m } - x _ { i + 1 } \\cdot T _ { \\ell _ 1 , \\ell _ 2 , \\cdots , \\ell _ { i - 1 } , \\ell _ { i } , \\ell _ { i + 1 } , \\cdots , \\ell _ { m } } = 0 \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{align*} \\gamma _ { k , n } ^ { ( \\alpha ) } - a _ { k , n } ^ { ( \\alpha ) } = \\sum _ { j = 1 } ^ { q } \\textup { r e s } ( Q _ { n , \\textup { \\textbf { m } } } F _ \\alpha \\Phi ' / \\Phi ^ { k + 1 } , \\lambda _ j ) , \\alpha = 1 , 2 , \\ldots , d , k \\geq 0 . \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{align*} { \\operatorname { C o n f } } ( X ) : = & \\{ \\} , \\\\ { \\operatorname { C o n f } } ( X ; Y ) : = & \\{ \\varphi \\in { \\operatorname { C o n f } } ( X ) : \\varphi ( Y ) = Y \\} . \\end{align*}"}
-{"id": "2509.png", "formula": "\\begin{align*} A + A = \\bigcup _ { i = 0 } ^ { 2 n } \\left ( \\frac { i } { n } + S _ i \\right ) \\mbox { w i t h } S _ i = \\bigcup _ { k , l \\ , : \\ , k + l = i } ( A _ k + A _ l ) . \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} \\theta _ i ( \\nu ) = \\left \\{ \\begin{array} { l l } 1 , & \\nu _ i > \\nu _ { i + 1 } , \\\\ 0 , & \\nu _ i < \\nu _ { i + 1 } , \\\\ \\tfrac 1 2 , & \\nu _ i = \\nu _ { i + 1 } , \\end{array} \\right . \\theta _ i ( s _ i \\nu ) = 1 - \\theta _ i ( \\nu ) , \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{align*} \\mathcal F ( \\mathcal S ^ \\lambda g ) ( \\xi , \\tau ) = | \\tau - | \\xi | ^ 2 | ^ { \\lambda } \\widehat { g } ( \\xi , \\tau ) \\end{align*}"}
-{"id": "41.png", "formula": "\\begin{align*} \\int _ { r _ 1 } ^ { r _ 2 } g ( s ) d s = 0 . \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} \\sum _ { A \\subset P , | A | = a } ( - 1 ) ^ { { \\rm s i g n } ( \\sigma , \\tau , P , A ) } \\frac { \\partial ^ { k ' } m _ { \\eta _ 1 ( \\sigma , \\tau , P , A ) } ( s ) } { \\partial s ^ { k ' } } m _ { \\eta _ 2 ( \\sigma , \\tau , P , A ) } ( s ) = 0 . \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} [ c , x _ 1 ] \\bigl ( g ( x _ 2 , x _ 3 , x _ 5 , x _ 4 ) + g ( x _ 2 , x _ 4 , x _ 5 , x _ 3 ) \\bigr ) \\equiv 0 \\pmod { I ^ { ( n ) } } . \\end{align*}"}
-{"id": "4355.png", "formula": "\\begin{align*} \\varphi _ \\lambda ( \\omega _ 1 / 2 ) \\exp ( \\mathcal { L } _ { V _ j } ( \\xi ) ) = \\varphi _ \\lambda ( z ( \\xi ) ) . \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} E _ d ^ { 0 } = E _ d \\supset E _ d ^ 1 \\supset E _ d ^ 2 \\supset \\cdots \\end{align*}"}
-{"id": "7872.png", "formula": "\\begin{align*} \\Vert u ( t ) \\Vert = \\Vert u _ 0 \\Vert ( 0 \\leq t \\leq T ) . \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} z \\bigl ( ( \\tilde { \\Delta } \\otimes \\operatorname { i d } ) ( \\Delta a ) \\bigr ) ( 1 \\otimes 1 \\otimes b ) & = z ( E \\otimes 1 ) \\bigl ( ( \\tilde { \\Delta } \\otimes \\operatorname { i d } ) ( \\Delta a ) \\bigr ) ( 1 \\otimes 1 \\otimes b ) \\\\ & = z ( E \\otimes 1 ) ( \\Delta \\otimes \\operatorname { i d } ) \\bigl ( ( \\Delta a ) ( 1 \\otimes b ) \\bigr ) \\\\ & = z ( \\Delta \\otimes \\operatorname { i d } ) \\bigl ( ( \\Delta a ) ( 1 \\otimes b ) \\bigr ) , \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{align*} p ( M ) & = ( 1 - u _ 1 ^ 2 ) ( 1 - v _ 1 ^ 2 ) \\cdots ( 1 - u _ n ^ 2 ) ( 1 - v _ n ^ 2 ) = ( 1 - ( u _ 1 + v _ 1 ) ^ 2 ) \\cdots ( 1 - ( u _ n + v _ n ) ^ 2 ) \\\\ & = ( 1 - ( b _ 1 u _ 1 ) ^ 2 ) \\cdots ( 1 - ( b _ n u _ n ) ^ 2 ) , \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} ( L \\chi ) ( z ) = & \\beta ^ { - 1 } \\gamma _ { \\xi \\zeta } ( \\partial _ { z _ \\xi } \\partial _ { z _ \\zeta } \\chi ) ( z ) - \\tilde \\gamma _ { \\xi \\eta } \\delta ^ { \\eta \\zeta } z _ \\zeta ( \\partial _ { z _ \\xi } \\chi ) ( z ) \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} \\mathcal A = \\left [ \\begin{array} { c c c c } { 0 } _ { n - 2 d } & & & 0 \\\\ & \\alpha _ 1 J & & \\\\ 0 & & \\ddots \\\\ & & & \\alpha _ d J \\end{array} \\right ] , \\ \\ \\ J = \\left [ \\begin{matrix} 0 & 1 \\\\ - 1 & 0 \\end{matrix} \\right ] , \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{align*} S _ { \\alpha } ( a , b ) = \\sum _ { x \\in \\mathbb { F } _ { q } } \\chi ( a x ^ { p ^ { \\alpha } + 1 } + b x ) , \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} \\tilde { \\mathbf { y } } = \\sqrt { \\Omega } \\mathbf { y } . \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} \\lambda \\ ! = \\ ! \\min \\left \\{ 1 , \\min _ { \\substack { x , y : \\\\ \\gamma _ 1 ( x , y ) > 0 } } \\frac { \\gamma ( x , y ) } { \\gamma _ 1 ( x , y ) } , \\min _ { \\substack { x , y : \\\\ \\mu _ 1 ( x , y ) > 0 } } \\frac { \\mu ( x , y ) } { \\mu _ 1 ( x , y ) } \\right \\} . \\end{align*}"}
-{"id": "2680.png", "formula": "\\begin{align*} \\nabla \\nabla \\varphi + ( c \\varphi + b ) g = 0 . \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} y ^ { 0 } : = x ^ { 0 } ; y ^ { k + 1 } : = T _ { k } y ^ { k } . \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} ( X ^ { k + 1 } A ^ { k } ) A ( X ^ { k + 1 } A ^ { k } ) = & X ^ { k + 1 } A ^ { k + 1 } X ^ { k + 1 } A ^ { k } = X ^ { k } A ^ { k } ( X ^ { k + 1 } A ^ { k } ) \\\\ = & X ^ { k } X ^ { k + 1 } A ^ { k } A ^ { k } = X ^ { k + 1 } X ^ { k } A ^ { 2 k } = X ^ { k + 1 } A ^ { k } . \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} \\rho ( z ) : = \\log { 1 + \\sqrt { 1 - z ^ 2 } \\over z } - \\sqrt { 1 - z ^ 2 } \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} C _ 0 K \\mu ^ { - 2 s } = \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "395.png", "formula": "\\begin{align*} x _ { m } \\in \\mathcal { K } _ { m } ( C _ { 1 } , d _ { 1 } ) \\mbox { a n d } r _ { m } = b - A x _ { m } \\perp \\mathcal { K } _ { m } ( C _ { 2 } , d _ { 2 } ) , \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} V \\langle \\phi , \\psi \\rangle = \\langle \\nabla _ V \\phi , \\psi \\rangle + \\langle \\phi , \\nabla _ { \\bar V } \\psi \\rangle . \\end{align*}"}
-{"id": "3710.png", "formula": "\\begin{align*} \\prod _ { p \\in S } \\sigma _ p ( \\vec { 0 } ) \\geq \\prod _ { p \\in S } ( p ^ { - 1 } | N | _ p ^ 2 ) ^ { n - r } \\gg ( N ^ { - 1 } N ^ { - 2 } ) ^ { n - r } = N ^ { 3 ( r - n ) } . \\end{align*}"}
-{"id": "7071.png", "formula": "\\begin{align*} N _ { \\beta } : = \\left \\lfloor \\frac { \\log ( 1 / \\beta ) } { \\log M } \\right \\rfloor , \\hat { N } _ { \\beta } : = 1 _ { \\beta \\le 1 } ( N _ { \\beta } + 1 ) . \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\frac { 1 } { N } \\log \\left [ \\frac { K _ { N - m } ( \\tau ) } { K _ N ( \\tau ) } \\left ( \\frac { N - m } { N } \\right ) ^ { \\frac { N ( N + 1 ) } { 4 } } \\right ] = \\frac { m } { 2 } . \\end{align*}"}
-{"id": "1986.png", "formula": "\\begin{align*} \\mbox { S p e c } ( \\Delta _ h ) = \\mbox { S p e c } ( \\Delta _ { \\omega _ h } ) , \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} \\widehat { U } _ { m } = W _ { m + 1 } U _ { m } = [ \\widehat { u } _ { 1 } , \\dots , \\widehat { u } _ { m } ] \\in \\mathbb { R } ^ { N \\times m } \\mbox { a n d } \\widehat { V } _ { m } = W _ { m } V _ { m } = [ \\widehat { v } _ { 1 } , \\dots , \\widehat { v } _ { m } ] \\in \\mathbb { R } ^ { N \\times m } \\ , , \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c c } e _ i ^ \\top ( A _ k X _ k B _ k - C _ k X _ { k + 1 } D _ k ) e _ j & = & ( E _ k ) _ { i j } , & i , j = 1 , \\ldots , n , \\ k = 1 , \\ldots , r - 1 , \\\\ e _ i ^ \\top ( A _ r X _ r B _ r - C _ r X _ 1 ^ s D _ r ) e _ j & = & ( E _ r ) _ { i j } , & i , j = 1 , \\ldots , n , \\end{array} \\right . \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} \\big ( 0 : _ A a ^ n \\big ) = \\big ( 0 : _ A a ^ { n _ 0 } \\big ) \\mbox { f o r a l l } n \\geqslant n _ 0 . \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} M _ { i i } : = \\begin{bmatrix} ( A _ 1 ) _ { i i } ( B _ 1 ) _ { i i } & - ( C _ 1 ) _ { i i } ( D _ 1 ) _ { i i } \\\\ & \\ddots & \\ddots \\\\ & & ( A _ { r - 1 } ) _ { i i } ( B _ { r - 1 } ) _ { i i } & - ( C _ { r - 1 } ) _ { i i } ( D _ { r - 1 } ) _ { i i } \\\\ - ( C _ r ) _ { i i } ( D _ r ) _ { i i } & & & ( A _ r ) _ { i i } ( B _ r ) _ { i i } \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "7865.png", "formula": "\\begin{align*} \\P \\big ( b _ n ^ { - 1 } M _ n > \\lambda \\big ) \\geq \\P \\big ( C _ { \\alpha } ^ { 1 / \\alpha } \\Gamma _ 1 ^ { - 1 / \\alpha } > \\lambda ( 1 + \\delta ) \\big ) - \\phi _ n ( \\epsilon , \\lambda ) - \\widetilde { \\psi } _ n ( \\epsilon , \\delta , \\lambda ) , \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} c h ^ { q ^ { s + l } } & = d ^ { q ^ n } g ^ { q ^ { s + l } } , \\\\ c g ^ { q ^ { s + l } } \\theta ^ { q ^ n } & = d ^ { q ^ n } h ^ { q ^ { s + l } } \\theta . \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} \\sup _ { \\substack { i \\in \\{ 1 , \\dots , N ^ d \\} \\\\ x \\in I _ { \\lambda _ i } } } \\left | f ( x ) - p _ { i , \\alpha } ( x ) \\right | \\leq C B \\left ( \\frac { d } { N } \\right ) ^ \\beta p _ { i , \\alpha } ( x ) : = \\sum _ { | \\alpha | \\leq n } \\frac { \\partial ^ \\alpha f ( x _ i ) } { \\alpha ! } ( x - x _ i ) ^ \\alpha \\ , . \\end{align*}"}
-{"id": "7729.png", "formula": "\\begin{align*} \\mathcal { L } _ { ^ { m , 2 } _ { i n t e r } } ( s ) = & { \\rm e x p } \\left ( - \\lambda _ { c } \\int _ { \\mathbb { R } ^ 2 } \\left ( 1 - \\mathcal { E } _ { y _ { m , 2 } } \\left \\{ \\frac { 1 } { \\frac { s } { { L \\left ( | | y _ { m , 2 } + x _ m - x | | \\right ) } } + 1 } \\right \\} \\right ) d x \\right ) , \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} g _ \\infty = - \\ : 1 6 d u ^ 2 + u ^ 2 d \\theta ^ 2 + 4 c _ \\lambda ^ { - 1 } \\left [ e ^ { P _ \\infty } ( d \\widehat { \\sigma } + Q _ \\infty d \\widehat { \\delta } ) ^ 2 + e ^ { - \\ : P _ \\infty } d \\widehat { \\delta } ^ 2 \\right ] . \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} \\frac { \\partial v ^ { y _ \\xi ^ \\bot } } { \\partial \\nu } ( \\gamma , T - \\xi - 0 ) = \\sqrt { \\frac { J ( \\gamma , \\xi ) \\ , J ( \\gamma , 0 ) } { c ( x ( \\gamma , \\xi ) ) \\ , c ( x ( \\gamma , 0 ) ) } } \\ , y ( x ( \\gamma , \\xi ) ) . \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} \\Sigma _ k ( T ) = \\sum _ { | m | < k } \\Big ( 1 - \\frac { | m | } { k } \\Big ) \\Phi _ m ( T ) . \\end{align*}"}
-{"id": "1029.png", "formula": "\\begin{align*} \\mathbb { P } ( | X | \\ge t ) \\le 2 \\exp \\big ( - \\varphi _ \\infty ( t / ( 2 a ) ) \\big ) = \\left \\{ \\begin{array} { c c l } 2 \\exp \\Big ( - \\frac { t ^ 2 } { 8 a ^ 2 } \\Big ) & { \\rm i f } & | t | \\le 2 a , \\\\ 0 & { \\rm i f } & | t | > 2 a . \\end{array} \\right . \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} X = A P P ^ { - 1 } B , \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{align*} g ( C _ a ) = { \\rm i n d } C _ a = \\delta ( C _ a ) = e _ Q ( C _ a ) = 0 , \\ \\ \\mbox { a n d } \\ \\ h ( C _ a ) = 2 . \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} J \\phi = \\sum _ { a = 1 } ^ { 2 n } J \\theta ^ a \\wedge E _ a \\lrcorner \\ , \\phi . \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} X _ j ( y ) = \\sum _ { l = 1 } ^ n b _ j ^ l ( y ) X _ { k _ l } ( y ) . \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} \\mathfrak { l } \\cap \\mathfrak { h } _ { \\mu } = \\mathfrak { l } _ { I ( \\mu ) } ^ { \\sigma _ { \\mu } } \\oplus \\bigoplus _ { \\alpha \\in \\Phi _ { P _ { I ( \\mu ) } ^ u } \\cap \\Phi _ L } \\mathfrak { g } _ { \\alpha } \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} P _ { \\beta } ( z ) ~ = ~ U _ { \\beta } ( z ) \\times f _ { \\beta } ( z ) \\end{align*}"}
-{"id": "6591.png", "formula": "\\begin{align*} \\mu \\big ( \\{ w \\in T ^ 1 \\mathcal { M } _ { g , n } : 1 / 2 R \\leqslant \\textrm { s y s } ( w ) \\leqslant 2 / R \\} \\big ) = O \\left ( \\frac { 1 } { R ^ { 4 } } \\right ) . \\end{align*}"}
-{"id": "8856.png", "formula": "\\begin{align*} \\Omega _ { \\beta , \\bar { \\beta } } & = \\frac { d _ a u ( \\beta ^ { \\vee } ) } { \\sinh ( 2 \\beta ) } - \\frac { 2 } { \\cosh ( 2 \\beta ) } \\chi \\circ \\Re ( [ \\theta \\sigma ( e _ { \\beta } ) , e _ { \\beta } ] ) . \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{align*} y ^ { n } : = \\left \\{ \\begin{array} { l l } x ^ { k } & n = n _ { k } k \\\\ z & \\end{array} \\right . \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k x _ j \\leq \\sum _ { j = 1 } ^ k y _ j \\quad { \\rm f o r } k = 1 , \\dots , n - 1 \\quad { \\rm a n d } \\sum _ { j = 1 } ^ n x _ j = \\sum _ { j = 1 } ^ n y _ j \\ . \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} ( 1 - x ) B ( x , u ) & = \\frac { 1 } { u } \\left ( B ( x , u ) - B _ 1 ( x ) u \\right ) + \\frac { x ^ 5 u ^ 2 C ^ 2 ( x ) } { ( 1 - x u ) ( 1 - x u C ( x ) ) } + \\frac { x ^ 3 ( 1 - x ) u } { ( 1 - 2 x ) ( 1 - x u ) } , \\\\ A ( x , u ) & = \\frac { 1 } { u ( 1 - x u ) } A ( x , u ) + \\frac { x ^ 2 u } { 1 - x u } ( F _ T ( x ) - 1 ) - ( ( 1 - 2 x ) F _ T ( x ) - 1 + x ) + B ( x , u ) . \\end{align*}"}
-{"id": "2445.png", "formula": "\\begin{align*} v ( n ) = ( - 1 ) ^ g \\sum _ { k _ g = 1 } ^ { n _ g } \\cdots \\sum _ { k _ 1 = 1 } ^ { n _ 1 } ( - 1 ) ^ { k _ 1 + \\cdots + k _ g } \\binom { n _ 1 } { k _ 1 } \\cdots \\binom { n _ g } { k _ g } \\frac { k _ 1 p _ 1 } { k _ 1 p _ 1 + k _ 2 p _ 2 + \\cdots + k _ g p _ g } . \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} \\log p _ { X Y } ( x , y ) + \\log \\delta _ { x , y } = m ( x ) + n ( y ) . \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{align*} \\bar \\partial _ W = \\begin{pmatrix} \\bar \\partial _ 1 & 0 \\\\ 0 & \\bar \\partial _ 2 \\end{pmatrix} , Q _ W = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "2107.png", "formula": "\\begin{align*} \\mathfrak { D } _ I \\mathfrak { h } _ { \\emptyset } + r ^ { \\frac { 1 } { 2 } } \\mathfrak { h } _ { \\emptyset } * \\mathfrak { h } _ { \\emptyset } + r ^ { \\frac { 1 } { 2 } } \\mathfrak { h } _ { \\emptyset } * \\mathfrak { A } = \\mathfrak { B } , \\end{align*}"}
-{"id": "9275.png", "formula": "\\begin{align*} ( K + H ) ( s ( D ) - s _ L ( D ) ) & = e F ( K + H ) = e ( d - 2 ) \\geq 0 . \\end{align*}"}
-{"id": "545.png", "formula": "\\begin{align*} ( 1 2 q ) ^ { j _ n } f _ n ^ { ( j _ n ) } ( q ) = Q _ n ( q , E _ 2 ( q ) , E _ 4 ( q ) , E _ 6 ( q ) ) \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} \\mathcal { I } _ { a } : = \\{ q \\in ( 0 , 1 ) : ( P _ { a , q } ) u \\in \\mathcal { P } ^ { \\circ } \\} . \\end{align*}"}
-{"id": "2943.png", "formula": "\\begin{align*} \\log F ( G , q ; z ) = \\sum _ { e \\geq 1 } n _ e h _ e ( q , z ) , \\end{align*}"}
-{"id": "3483.png", "formula": "\\begin{align*} \\frac { 2 } { 3 } | \\lambda | \\| f \\| _ { L ^ 2 } ^ 2 \\leq & { } \\left | \\lambda \\int _ { \\mathbb R } g ' ( x ) U ( x ) \\ , \\mathrm d x \\right | = \\left | \\int _ { \\mathbb R } g ' ( x ) \\bigl ( f ' ( x ) \\overline { f ( x ) } + V ( x ) \\bigr ) \\ , \\mathrm d x \\right | \\\\ \\leq & { } \\left ( 1 6 \\cdot \\sqrt { 3 } + 1 2 \\right ) \\| q _ - \\| _ { L ^ 1 } ^ { 2 } \\| f \\| _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "175.png", "formula": "\\begin{align*} m _ i = \\sum \\limits _ { j = 2 } ^ N m _ { i j } \\tau _ j , ~ i \\in \\{ 2 , 3 , \\dots , N \\} , \\end{align*}"}
-{"id": "4380.png", "formula": "\\begin{align*} z ( \\lambda , \\xi ) = \\int _ { \\xi } ^ { \\infty } \\frac { d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\end{align*}"}
-{"id": "3003.png", "formula": "\\begin{align*} \\Omega _ { \\rho } = : \\{ x \\in \\Omega : d ( x , \\partial \\Omega ) < \\rho \\} . \\end{align*}"}
-{"id": "1397.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { m + 1 } ( a _ i + b _ i ) \\leq { 9 } ^ { m + 1 } \\Big ( \\prod _ { i = 1 } ^ { m + 1 } a _ i + \\prod _ { i = 1 } ^ { m + 1 } b _ i \\Big ) . \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} - \\frac { b } { \\sqrt { 1 + b ^ 2 } } \\mathfrak { R e } ( \\ddot { c } _ { 0 } ( 0 ) ) + \\frac { 1 } { \\sqrt { 1 + b ^ 2 } } \\mathfrak { R e } ( \\ddot { c } _ { 1 } ( 0 ) ) + \\sum _ { n = 0 } ^ { + \\infty } | \\dot c _ n ( 0 ) | ^ 2 = 0 . \\end{align*}"}
-{"id": "3098.png", "formula": "\\begin{align*} \\mu ( x _ i , D ^ 3 ( x _ j ) ) = \\mu ( x _ i , \\lambda x _ { n - 5 + j } ) = 0 , \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} A ^ { \\ominus } _ { j , k } = A ^ { D } A ( A ^ { j } ) ^ { \\dagger } = ( A ^ { D } ) ^ { j } A ^ { j } ( A ^ { j } ) ^ { \\dagger } . \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} T ( A ; \\nu ) f & : = [ \\zeta \\cdot A ( \\zeta \\cdot ( f \\circ \\nu ) ) ] \\circ \\nu ^ { - 1 } = \\nu ^ { - 1 , * } \\circ \\zeta \\cdot A \\circ \\nu ^ * ( \\zeta \\circ \\nu ^ { - 1 } ) \\cdot f , \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} d _ { \\beta } ( \\frac { 1 } { \\beta } ) = 0 . 0 \\ , t _ 1 t _ 2 t _ 3 \\ldots { \\rm a n d ~ u n i q u e l y ~ c o r r e s p o n d s ~ t o } \\frac { 1 } { \\beta } = \\sum _ { i = 1 } ^ { + \\infty } t _ i \\beta ^ { - i - 1 } . \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{align*} \\limsup _ k \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\# R _ k ( j ) < \\infty . \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{align*} d q ^ \\epsilon _ t = & \\frac { 1 } { \\sqrt { \\epsilon } } z _ t ^ \\epsilon d t , \\\\ d p ^ \\epsilon _ t = & \\left ( - \\frac { 1 } { \\sqrt { \\epsilon } } \\left ( \\gamma _ l ( t , x ^ \\epsilon _ t ) - \\nabla _ q \\psi _ l ( t , q _ t ^ \\epsilon ) \\right ) ( z _ t ^ \\epsilon ) _ l - \\nabla _ q V ( t , q ^ \\epsilon _ t ) + \\tilde F ( t , x ^ \\epsilon _ t ) \\right ) d t + \\sigma ( t , x ^ \\epsilon _ t ) d W _ t , \\end{align*}"}
-{"id": "6576.png", "formula": "\\begin{align*} B _ j = \\int \\limits _ { c 2 ^ j } ^ { c 2 ^ { j + 1 } } \\frac { b } { \\sqrt { 1 - b ^ 2 } } \\ , d b \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} d q _ t ^ \\prime = & \\frac { 1 } { m } ( p _ t ^ \\prime - \\psi ( t ^ * , q _ t ^ \\prime ) ) d t , \\\\ d ( p _ t ^ \\prime ) _ i = & \\left ( - \\frac { 1 } { m } \\gamma ( t ^ * , q _ t ^ \\prime ) ( ( p _ t ^ \\prime ) _ i - \\psi _ i ( t ^ * , q _ t ^ \\prime ) ) - \\partial _ { q ^ i } V ( t ^ * , q _ t ^ \\prime ) \\right . \\\\ & \\left . + \\frac { 1 } { m } \\partial _ { q ^ i } \\psi _ k ( t ^ * , q _ t ^ \\prime ) \\delta ^ { k j } ( ( p _ t ^ \\prime ) _ j - \\psi _ j ( t ^ * , q _ t ^ \\prime ) ) \\right ) d t + \\sigma ( t ^ * , q _ t ^ \\prime ) \\delta _ { i \\rho } d W ^ \\rho _ t \\end{align*}"}
-{"id": "2856.png", "formula": "\\begin{align*} \\left | S \\left ( \\left ( h _ { j , \\omega } \\right ) _ { j , \\omega } \\right ) \\right | \\leq \\left [ { K \\choose 0 } + { K \\choose 2 } + \\ldots \\right ] ( 1 + \\delta ) - \\left [ { K \\choose 1 } + { K \\choose 3 } + \\ldots \\right ] ( 1 - \\delta ) = 2 ^ K \\delta . \\end{align*}"}
-{"id": "3394.png", "formula": "\\begin{align*} \\begin{aligned} | \\log \\log f _ k ( \\alpha _ k ) - \\log \\log f _ k ( \\gamma _ k ) | & \\geq \\log | \\log f _ k ( \\alpha _ k ) | - \\log | \\log f _ k ( \\gamma _ k ) | \\\\ & \\geq \\log \\log | f _ k ( \\alpha _ k ) | - \\log ( \\log | f _ k ( \\gamma _ k ) | + \\pi ) \\end{aligned} \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{align*} { { } _ a ^ C D } _ t ^ { \\beta { } } f \\left ( t \\right ) = \\left \\{ \\begin{array} { l l } \\frac { 1 } { \\Gamma ( n - \\beta { } ) } \\ \\int _ a ^ t \\frac { f ^ { \\left ( n \\right ) } ( \\tau { } ) } { { ( t - \\tau { } ) } ^ { \\beta { } + 1 - n } } \\ d \\tau { } , & n - 1 < \\beta { } < n , \\ n \\in { } \\mathbb { N } \\\\ \\frac { d ^ n } { d t ^ n } f \\left ( t \\right ) , & \\beta { } = n \\in { } \\mathbb { N } \\end{array} \\right . , \\end{align*}"}
-{"id": "3619.png", "formula": "\\begin{align*} G ( P ) = \\prod _ { 1 \\leq i \\leq j \\leq r } \\gamma ( a _ { i , j } ) \\gamma ( b _ { i , j } ) , \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\theta ( \\alpha \\cdot f ) ( u \\otimes n ) & = & u ( \\alpha \\cdot f ) ( n ) \\\\ & = & u ( \\alpha f ( n ) - f ( \\alpha \\cdot n ) ) = \\theta ( f ) ( u \\alpha \\otimes n - u \\otimes \\alpha \\cdot n ) \\\\ & = & \\theta ( f ) ( ( u \\otimes n ) \\cdot \\alpha ) = ( \\alpha \\cdot \\theta ( f ) ) ( u \\otimes n ) \\ , . \\end{array} \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{align*} \\widetilde { H } : = Y ^ { - 1 / 2 } \\{ \\pi H ^ 0 \\pi - \\pi H ^ 0 \\pi ^ \\perp [ \\pi ^ \\perp H ^ 0 \\pi ^ \\perp ] ^ { - 1 } \\pi ^ \\perp H ^ 0 \\pi \\} Y ^ { - 1 / 2 } \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{align*} Q _ E = \\begin{pmatrix} Q _ V & 0 \\\\ 0 & - 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} \\sum _ { j = 2 } ^ 4 \\frac { \\theta _ j ' ( 1 ) } { \\theta _ j ( 1 ) } = - \\frac { 3 } { 4 } , \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} \\eta + \\kappa + \\gamma ' \\ll g ( \\alpha - \\kappa - \\gamma , d ) \\overset { \\eqref { e q : g - f u n c } } { = } \\min \\bigg \\{ \\alpha - \\kappa - \\gamma , d , \\frac { \\alpha - \\kappa - \\gamma + d - 1 } { d + 1 } \\bigg \\} . \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} \\alpha ( \\mu , \\nu ) & = \\# \\{ i < j | ( \\mu _ i = 0 , \\mu _ j = 2 ) , ( \\nu _ i = \\nu _ j = 1 ) \\} , \\\\ \\beta ( \\mu , \\nu ) & = \\# \\{ i < j | ( \\mu _ i = 1 , \\mu _ j = 2 ) , ( \\nu _ i = \\nu _ j = 1 ) \\} , \\\\ \\gamma ( \\mu , \\nu ) & = \\# \\{ i < j | ( \\mu _ i = 0 , \\mu _ j = 1 ) , ( \\nu _ i = \\nu _ j = 1 ) \\} , \\end{align*}"}
-{"id": "638.png", "formula": "\\begin{align*} { \\cal C } _ 1 = \\bigcup _ { \\underline { p } \\in ( 0 , 1 ) ^ { M } } { \\cal C } _ 1 ( \\underline { p } ) . \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} T _ { \\ell } ^ n ( [ ( \\tilde { E } , \\tilde { P } ) ] ) = \\sum _ C \\mu _ C [ ( \\tilde { E } / C , P \\ { \\rm m o d } \\ C ) ] , \\end{align*}"}
-{"id": "975.png", "formula": "\\begin{align*} \\Phi _ { L , \\Delta L } ( x ) = \\mathrm { e } ^ { i p _ { 0 } \\cdot x } \\tilde { \\Phi } _ { L , \\Delta L } ( x ) , x \\in \\Gamma . \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( C _ n ) ^ { 1 / n } = 1 . \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { m } ( a _ i + b _ i ) \\leq 9 ^ m \\Big ( \\prod _ { i = 1 } ^ { m } a _ i + \\prod _ { i = 1 } ^ { m } b _ i \\Big ) . \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} H ^ * ( G _ k ( \\R ^ n ) , \\Z _ 2 ) = \\frac { \\Z _ 2 [ w _ 1 , w _ 2 , \\ldots , w _ k ; \\bar { w } _ 1 , \\bar { w } _ 2 , \\ldots , \\bar { w } _ { n - k } ] } { ( 1 + w _ 1 + \\cdots + w _ k ) ( 1 + \\bar { w } _ 1 + \\cdots + \\bar { w } _ { n - k } ) = 1 } \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{align*} f ' ( x ) - \\Phi ( x ) & = \\sum _ k ( f _ k ' ( x ) - \\Phi ( x ) ) \\phi _ k ( x ) + \\sum _ k ( f _ k ( x ) - g ( x ) ) \\phi _ k ' ( x ) , \\\\ \\shortintertext { h e n c e } \\| f ' ( x ) - \\Phi ( x ) \\| & \\le \\sum _ k \\| f _ k ' ( x ) - \\Phi ( x ) \\| \\phi _ k ( x ) + \\sum _ k | f _ k ( x ) - g ( x ) | \\| \\phi _ k ' ( x ) \\| \\\\ & \\le \\sum _ k ( \\xi ( x ) + \\omega _ 0 ( x ) ) \\phi _ k ( x ) + \\sum _ k 2 ^ { - k - 1 } \\omega _ 0 ( x ) \\\\ & \\le \\xi ( x ) + 2 \\omega _ 0 ( x ) . \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} \\widetilde { \\tau } ( r _ x ) - r _ x = b _ 1 \\quad \\quad \\widetilde { \\tau } ( r _ y ) - r _ y = b _ 2 . \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} f ^ * Q _ l & = \\sum _ j w _ { l j } P _ j \\\\ \\bar { d _ j } & = \\frac { d _ j + w _ { l j } - 1 } { w _ { l j } } , \\ ; \\ ; f ( P _ j ) = Q _ l \\\\ \\delta _ l & = m a x \\{ \\bar { d _ j } ; f ( P _ j ) = Q _ l \\} . \\\\ \\Delta & = \\sum _ l \\delta _ l Q _ l . \\\\ M & = L - \\Delta . \\\\ \\end{align*}"}
-{"id": "4050.png", "formula": "\\begin{align*} \\epsilon > 1 - \\max _ { \\substack { q _ { X Y } : \\ : q _ { X Y } = q _ { X } p _ { Y | X } } } \\rho ^ 2 _ m ( q _ { X Y } ) . \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{align*} \\lim \\limits _ { N \\to + \\infty } \\sup _ { t \\in [ 0 , T ] } \\| m ^ N _ t - m _ t \\| _ { B L } ^ \\ast = 0 . \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} & \\mathbb { P } [ K _ A = i , K _ B = j | \\mathbf { F } = \\mathbf { f } ] = \\frac { \\mathbb { P } [ X ^ n \\in \\mathcal { A } _ i , Y ^ n \\in \\mathcal { B } _ j ] } { \\mathbb { P } [ X ^ n \\in \\mathcal { A } , Y ^ n \\in \\mathcal { B } ] } . \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} P ( i , j ) : = \\underline { d p } _ i \\underline \\wedge \\underline { d p } _ j ( i < j \\in [ n ] ) , \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} C _ j ( \\alpha , \\beta ; q , t ) = \\frac { p _ j ( \\alpha , \\beta ; q , t ) } { \\prod _ { i = 0 } ^ { m } ( 1 - q ^ j t ^ i ) } , \\end{align*}"}
-{"id": "1366.png", "formula": "\\begin{align*} d Y _ s ^ { t , x ; u } = \\hat { G } \\bigl ( s , X _ s ^ { t , x ; u } , Y _ s ^ { t , x ; u } , Z _ s ^ { t , x ; u } \\bigr ) d s - Z _ s ^ { t , x ; u } d B _ s , s \\in [ t , T ] , Y _ T ^ { t , x ; u } = \\Psi ( X _ T ^ { t , x ; u } ) . \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} \\underline { u } ( t - t _ 0 , 0 ) = \\underline { u } ( t , t _ { 0 } ) \\leq \\tilde { u } ( x , t ) \\leq \\overline { u } ( t , t _ { 0 } ) = \\overline { u } ( t - t _ 0 , 0 ) \\ \\ \\forall \\ x \\in \\R ^ N , \\ t \\geq t _ { 0 } . \\end{align*}"}
-{"id": "8232.png", "formula": "\\begin{align*} P _ j ( x ' ) = r _ j ^ { 2 - d - n } x _ d ^ { - 1 } G _ { j , o } ( x ' , x _ d ) x _ d = ( r _ j ^ 2 - | x ' | ^ 2 ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "8583.png", "formula": "\\begin{align*} \\vec B _ U & : = \\{ ( u , w ) \\in U \\times W : \\deg _ H ( u , w , G _ 1 ) < d \\alpha ^ 5 n / 2 ^ { 1 7 } \\} , \\\\ \\vec B _ W & : = \\{ ( u , w ) \\in U \\times W : \\deg _ H ( w , u , G _ 3 ) < d \\alpha ^ 5 n / 2 ^ { 1 7 } \\} . \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} \\Psi _ n ( x ) = \\lbrace \\psi _ { I ^ n _ 1 } , . . . , \\psi _ { I ^ n _ { p ( n , x ) } } \\rbrace \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} \\hat w ( s ) & = P ^ \\perp w ( s ) . \\ \\ \\ \\ \\ \\ \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} \\wp ( \\lambda ^ { - 1 } ( t ) , z ( t ) ) = \\xi ( t ) - \\frac 1 3 ( t + 1 ) , \\end{align*}"}
-{"id": "1125.png", "formula": "\\begin{align*} ( s _ 1 | \\cdots | s _ n ) \\mapsto \\partial _ M ( f ( s _ 1 | \\cdots | s _ n ) ) - \\sum \\limits _ { i = 1 } ^ n f ( s _ 1 | \\cdots | \\partial ( s _ i ) | \\cdots | s _ n ) \\ , . \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} \\kappa _ \\gamma = - s ^ { - 2 } Z \\bar Z \\log s ^ 2 , \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} \\Big { ( } \\dot { u } _ t ^ * ( - u _ { t , j } ^ { * , i , j } + I _ { H , i } ( a ) ) P _ { D H } ' \\Big { ) } _ i = & - \\dot { u } ^ * _ { t , i } u ^ { * , i , j } _ { t , j } P _ { D H } ' + \\dot { u } ^ * _ { t , i } I _ { H , i } ( a ) P _ { D H } ' \\\\ & - \\dot { u } ^ * _ { t } u ^ { * , i , j } _ { t , i , j } P _ { D H } ' + \\dot { u } ^ * _ { t } u ^ * _ { j , i } I _ { H , i , j } ( a ) P _ { D H } ' \\\\ & - \\dot { u } ^ * _ { t } u ^ { * , i , j } _ { t , j } P _ { D H , i } ' + \\dot { u } ^ * _ { t } I _ { H , i } ( a ) P _ { D H , i } ' \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{align*} \\frac { \\abs { m _ i } + 1 } { 2 } + r _ i \\in \\{ 0 , - 1 , - 2 , \\cdots \\} ( i = 1 , 2 ) , \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{align*} [ a _ 1 a _ 2 \\cdots a _ n ] ^ k = \\prod _ { i = 1 } ^ q [ a _ i a _ { i + k } \\cdots a _ { i + p k } ] \\prod _ { i = q + 1 } ^ k [ a _ i a _ { i + k } \\cdots a _ { i + ( p - 1 ) k } ] . \\end{align*}"}
-{"id": "136.png", "formula": "\\begin{align*} \\Phi _ \\infty = \\begin{pmatrix} 0 & | f | ^ { - 1 / 2 } \\kappa ^ { - 1 } f \\\\ | f | ^ { 1 / 2 } \\kappa & 0 \\end{pmatrix} d z . \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\frac { 1 } { N } \\log \\left [ \\frac { ( k - l ) ! \\left ( \\frac { N - m + l - k } { 2 } \\right ) ! } { k ! \\left ( \\frac { N - k } { 2 } \\right ) ! } \\right ] = 0 . \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} & 1 - b _ n = \\sum \\limits _ { k = 1 } ^ { n } \\delta _ { b _ k 0 } - \\sum \\limits _ { k = 1 } ^ { n } \\delta _ { a _ k 0 } + \\underbrace { \\delta _ { a _ n 0 } } _ { = 0 } \\\\ & \\Rightarrow b _ n = 1 . \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} \\left [ r X , r Y \\right ] - r \\left ( \\left [ r X , Y \\right ] + \\left [ X , r Y \\right ] \\right ) = - \\left [ X , Y \\right ] , \\ ; X , Y \\in { \\frak g } \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\nu ( s ) & = & s \\\\ \\nu ( \\alpha ) & = & \\alpha + \\lambda _ L ( \\alpha ) + \\mathrm { d i v } ( \\partial _ \\alpha ) , \\end{array} \\right . \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} ^ { H } \\mathbb { D } _ { a + } ^ { \\alpha , \\beta ; \\psi } f \\left ( x \\right ) = I _ { a + } ^ { \\beta \\left ( n - \\alpha \\right ) ; \\psi } \\left ( \\frac { 1 } { \\psi ^ { \\prime } \\left ( x \\right ) } \\frac { d } { d x } \\right ) ^ { n } I _ { a + } ^ { \\left ( 1 - \\beta \\right ) \\left ( n - \\alpha \\right ) ; \\psi } f \\left ( x \\right ) \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{align*} f ( u ) = \\frac { v } { 2 ^ { 3 / 2 } \\pi u } \\left ( ( 1 - e ^ { - v } ) - \\frac { 1 } { 2 } u ^ 2 e ^ { - v } \\right ) ^ { - 1 / 2 } \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} ( \\underbrace { 2 , \\cdots , 2 } _ { m } , \\underbrace { 1 , \\cdots , 1 } _ { l } ) , \\emph { s u c h t h a t } 2 m + l = n . \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} z _ l = \\rho e ^ { i \\theta } . \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} b ( q ) < b ( q ^ * - \\delta ) < b ^ * - \\sigma < \\varepsilon _ { q } ^ W < \\varepsilon _ { q ^ * } ^ W = b ^ * . \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} \\widehat { P ^ H _ \\tau u } ( \\xi , \\sigma ) = \\widehat { e ^ { - \\tau H } u } ( \\xi , \\sigma ) = e ^ { - \\tau ( 2 \\pi i \\sigma + ( 2 \\pi | \\xi | ) ^ 2 ) } \\hat u ( \\xi , \\sigma ) . \\end{align*}"}
-{"id": "634.png", "formula": "\\begin{align*} p _ i : = \\mathbb { P } ( W _ i ( n ) \\geq 1 ) \\end{align*}"}
-{"id": "1652.png", "formula": "\\begin{align*} \\Phi ( A , B ) \\leqq C _ 1 \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } \\left \\{ \\sum _ { k = 1 } ^ { H _ 0 - 1 } ( a _ { i k } b _ { k j } - a ^ 0 _ { i k } b ^ 0 _ { k j } ) + c _ i \\right \\} ^ 2 { + } C _ 2 \\sum _ { k = H _ 0 } ^ { H - 1 } \\left ( \\sum _ { j = 1 } ^ N b _ { k j } ^ 2 \\right ) \\left ( \\sum _ { i = 1 } ^ { M - 1 } a _ { i k } ^ 2 \\right ) , \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\hbox { M i n i m i z e } \\ , \\sum \\limits _ { i = 1 } ^ p { \\bf u } _ i ^ T A _ i { \\bf u } _ i \\\\ U ^ T U = \\mathbb { I } _ p \\end{array} \\right . , \\end{align*}"}
-{"id": "3106.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\frac { \\log d ( n ) \\log \\log n } { \\log n } = \\log 2 , \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{align*} \\| q f ^ 2 \\| _ { L ^ 1 } = \\int _ { \\mathbb R } | q ( t ) | | f ( t ) | ^ 2 \\ , \\mathrm d t = \\int _ { \\mathbb R } \\big ( q _ + ( t ) + q _ - ( t ) \\big ) | f ( t ) | ^ 2 \\ , \\mathrm d t \\leq 2 \\| q _ - \\| _ { L ^ 1 } \\| f \\| _ \\infty ^ 2 . \\end{align*}"}
-{"id": "2649.png", "formula": "\\begin{align*} \\rho _ { 1 j } + b h = \\tilde { b } _ { j \\alpha } , \\rho _ { 2 \\alpha } - \\tilde { c } \\varphi = - \\tilde { b } _ { j \\alpha } . \\end{align*}"}
-{"id": "4872.png", "formula": "\\begin{align*} Y ^ 2 = X ^ 3 + A X + B . \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{align*} F ( x ) = \\frac { 2 ( x + e _ n ) } { | x + e _ n | ^ 2 } - e _ n , \\end{align*}"}
-{"id": "5318.png", "formula": "\\begin{align*} \\displaystyle { \\lim _ { \\stackrel { \\bar { w } \\neq w \\to \\bar { w } } { H \\in \\partial \\Phi ( w ) } } } \\ , \\displaystyle { \\frac { \\Phi ^ { \\prime } ( \\bar { w } ; w - \\bar { w } ) - H \\ , ( \\ , w - \\bar { w } \\ , ) } { \\| \\ , w - \\bar { w } \\ , \\| } } \\ , = \\ , 0 , \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{align*} S '' ( i , j , k , \\alpha , \\beta ) = \\omega _ { i j } ^ \\alpha \\underline \\wedge \\omega _ { j k } ^ \\beta , i < j < k \\in [ n ] , \\alpha , \\beta \\geq 0 , \\end{align*}"}
-{"id": "1522.png", "formula": "\\begin{align*} \\int _ { \\Sigma } \\phi ^ 2 ( \\mathcal { L } \\log w ) e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma & = \\int _ { \\Sigma } \\left ( - | A | ^ 2 + ( \\dfrac 1 2 - \\mu _ 1 ) - | \\nabla \\log w | ^ 2 \\right ) \\phi ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma \\\\ \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} 1 = \\lim _ { n \\to \\infty } \\mathbb { E } _ { n , r ( n ) , \\theta _ n } ( \\nu _ n ' ) > \\liminf _ { n \\to \\infty } \\mathbb { E } _ { n , r ( n ) , \\theta _ n } ( \\varphi _ n ' ) = \\liminf _ { n \\to \\infty } \\mathbb { E } _ { n , d ( n ) , F _ n ( \\theta _ n ) } ( \\varphi _ n ) , \\end{align*}"}
-{"id": "7754.png", "formula": "\\begin{align*} a _ k J q ^ k ( f g ) = \\sum _ { i + j = k } a _ i a _ j J q ^ i ( f ) J q ^ j ( g ) , \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{align*} \\| f ( t ) - f ( t ' ) \\| _ { L ^ 2 } & = \\| S ( t ) ( f ( t ) - f ( t ' ) ) \\| _ { L ^ 2 } \\\\ & \\lesssim \\| u \\| _ { L ^ 2 ( ( t ' , t ) , L ^ 4 _ x ) } \\| v _ 0 \\| _ { L ^ 2 } + \\| u \\| _ { L ^ 2 ( ( t ' , t ) , L ^ 4 _ x ) } ^ 3 \\to 0 , t , t ' \\to \\infty \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} 2 B _ { j , n } \\ , W ^ 2 - \\Bigl ( B _ { j , n } + 1 \\Bigr ) W + 1 ~ = ~ 0 \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | \\leq k } \\frac { 1 } { \\alpha ! } \\partial ^ { \\alpha } a ( 0 ) \\xi ^ { \\alpha } = \\sum _ { \\alpha \\in \\mathcal { V } _ k ( a ) } \\xi ^ { \\alpha } p _ { \\alpha } ( \\xi ) \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} \\eta = \\frac { \\pi } { 2 } \\ , , \\end{align*}"}
-{"id": "1104.png", "formula": "\\begin{align*} ( 1 \\otimes u ) \\cdot ( v \\otimes m \\cdot \\alpha \\otimes n ) = \\phi ( m \\cdot \\alpha \\otimes ( n \\otimes ( u \\otimes v ) ) ) \\ , . \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} { \\rm S t r } \\ , \\mathcal N _ { I } ( t ) D v _ t ^ j = 0 , 2 | I | + j < n - 1 . \\end{align*}"}
-{"id": "7364.png", "formula": "\\begin{align*} a ( \\xi ) = \\sum _ { | \\alpha | \\leq k } \\frac { 1 } { \\alpha ! } \\partial ^ { \\alpha } a ( 0 ) \\xi ^ { \\alpha } + \\sum _ { | \\alpha | = k } \\xi ^ { \\alpha } b _ { \\alpha } ( \\xi ) \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} \\iota _ { X ( \\zeta ) } \\omega ( \\zeta ) = d \\mu ( \\zeta ) \\end{align*}"}
-{"id": "9014.png", "formula": "\\begin{align*} u _ t = W ( u , u ' , u '' , \\dots ) \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} c ^ { \\delta + k } \\ge c ^ { k - 1 } + \\frac { 1 } { 2 } \\frac { \\delta [ ( B - 1 ) c ^ { k - B } + \\delta - B ] } { \\binom { k } { 2 } } . \\end{align*}"}
-{"id": "1769.png", "formula": "\\begin{align*} \\underset { \\substack { z ^ 0 , z ^ 1 \\in \\overline { \\R ^ n _ + } \\\\ z ^ 0 \\not = z ^ 1 } } { \\sup } \\dfrac { | \\partial _ { z } ^ \\delta q _ { z ^ 0 } - \\partial _ { z } ^ \\delta q _ { z ^ 1 } | ^ { ( m ) } _ i } { | z ^ 0 - z ^ 1 | ^ { \\tau - [ \\tau ] } } \\leq \\ C \\ , | h | _ { C ^ \\tau S ^ m _ { 1 , 0 } } ^ { i } . \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} \\| u \\| _ Y = \\sup _ { f \\in Y ^ \\ast \\atop \\| f \\| _ { Y ^ \\ast } \\leq 1 } \\langle f , u \\rangle \\leq \\sup _ { f \\in X \\atop \\| f \\| \\leq 1 } \\langle f , u \\rangle = \\| u \\| _ { X ^ \\ast } . \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( E _ k \\ , , \\ , \\lim _ { k \\rightarrow \\infty } \\hat p _ k ( t ) = p ( t ) \\right ) = \\mathbb { P } \\left ( E _ k \\right ) , \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} D _ { x , y } = \\left [ R _ { x } , R _ { y } \\right ] = R _ { x } R _ { y } - R _ { y } R _ { x } , \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} \\frac { \\partial G } { \\partial { \\bf u } _ a } ( { \\bf u } ) = \\sum _ { d = 1 } ^ p \\left < \\frac { \\partial G } { \\partial { \\bf u } _ a } ( { \\bf u } ) , { \\bf u } _ d \\right > { \\bf u } _ d . \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{align*} W = \\{ \\delta e = ( \\delta x , \\boldsymbol { \\varepsilon } ) \\in V : \\varepsilon _ 3 = 0 \\} . \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} \\delta _ B = \\partial _ B ^ * + \\bar \\partial _ B ^ * , \\delta _ T = \\partial _ T ^ * + \\bar \\partial _ T ^ * , \\end{align*}"}
-{"id": "2410.png", "formula": "\\begin{align*} r > 1 s = \\frac { r } { r - 1 } . \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} S ( \\vec { \\alpha } , \\vec { \\nu } ) & = \\sum _ { \\vec { z } ( q ) } \\sum _ { \\vec { z } + q \\vec { y } \\in P \\mathcal { B } } e ( \\vec { \\alpha } \\cdot ( \\vec { f } ( \\vec { z } + q \\vec { y } ) - \\vec { \\nu } ) ) \\\\ & = \\sum _ { \\vec { z } ( q ) } e _ q ( \\vec { a } \\cdot ( \\vec { f } ( \\vec { z } ) - \\vec { \\nu } ) ) \\sum _ { \\vec { z } + q \\vec { y } \\in P \\mathcal { B } } e ( \\vec { \\beta } \\cdot ( \\vec { f } ( \\vec { z } + q \\vec { y } ) - \\vec { \\nu } ) ) . \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} ( x \\ast y ) \\ast \\alpha ( z ) - \\alpha ( x ) \\ast ( y \\ast z ) & = \\alpha ( x ) \\cdot ( \\partial ( y ) \\cdot \\partial ( z ) ) - \\alpha ( x ) \\cdot \\partial ( y \\cdot \\partial ( z ) ) \\\\ & = \\alpha ( x ) \\cdot ( \\partial ( y ) \\cdot \\partial ( z ) ) - \\alpha ( x ) \\cdot ( \\partial ( y ) \\cdot \\partial ( z ) ) = 0 . \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} \\frac { d \\mathbb { Q } _ { n , d , h } } { d \\mathbb { P } _ { n , d , 0 } } = \\exp \\left ( h ' Z ^ * _ { n , d } - K _ { n , d } ( h ) \\right ) , \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{align*} \\Delta ( \\overline { t } ) = 0 , \\dot \\Delta ( \\overline { t } ) = 0 , \\ddot \\Delta ( \\overline { t } ) \\geq 0 . \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{align*} F ( \\Sigma ) = \\int _ \\Sigma e ^ { - | x | ^ 2 / 4 } \\ ; d \\mu \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} z ^ \\alpha _ s = z ^ \\alpha _ { x , y } = u _ \\alpha \\ ; q _ 1 ^ { x - 1 } q _ 2 ^ { y - 1 } . \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n a _ { i j } \\frac { \\partial ^ 2 \\delta ^ \\gamma } { \\partial x _ i \\partial x _ j } \\leq \\gamma \\delta ^ { \\gamma - 2 } \\sum _ { i , j = 1 } ^ n a _ { i j } \\sum _ { k = 1 } ^ m b _ { k i } b _ { k j } \\end{align*}"}
-{"id": "2282.png", "formula": "\\begin{align*} \\widehat { P } _ \\hbar = \\widehat { A } _ N ( i \\widehat { H } _ \\hbar ) \\widehat { A } _ N ^ { - 1 } , \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{align*} { \\mu } ( A + A ) = ( 2 + \\varepsilon ) \\ , { \\mu } ( A ) \\ ; < \\ ; \\tfrac { 1 } { 2 } + { \\mu } ( A ) . \\end{align*}"}
-{"id": "7614.png", "formula": "\\begin{align*} - \\frac { u '' } { ( 1 + ( u ' ) ^ 2 ) ^ \\frac 3 2 } - \\frac { N - 1 } r \\frac { u ' } { \\sqrt { 1 + ( u ' ) ^ 2 } } = g ( u ) , \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{align*} \\frac { 1 } { q } = \\frac { 1 } { p } + \\frac { \\alpha } { n + 1 } - 1 . \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} d e ^ { r t } \\log f _ t ( X _ t , Y _ t ) & = \\left ( ( \\partial _ t + L ) \\log f _ t ( X _ t , Y _ t ) + r \\log f _ t ( X _ t , Y _ t ) \\right ) e ^ { r t } d t + d M _ t \\\\ & = \\left ( ( \\partial _ t \\log \\rho ^ y _ t ( x ) ) ( X _ t , Y _ t ) + L ^ X \\log f _ t ( X _ t , Y _ t ) + L ^ Y \\log f _ t ( X _ t , Y _ t ) + r \\log f _ t ( X _ t , Y _ t ) \\right ) e ^ { r t } d t \\\\ & + d M _ t \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} Q ' ( i , j , \\alpha ) : = \\underline { d p } _ i \\underline \\wedge \\omega _ { i j } ^ \\alpha ( i < j \\in [ n ] , \\alpha \\geq 0 ) , \\end{align*}"}
-{"id": "6706.png", "formula": "\\begin{align*} \\mathbf { r } = \\{ U _ i a _ { i + 1 } y z _ i ^ { - 1 } y ^ { - 1 } \\mid i \\in \\{ 1 , \\ldots , n \\} \\} \\cup \\{ z _ 1 z _ 2 \\cdots z _ n \\} \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} p _ { - 2 } ( a ) & = \\sum _ { \\alpha _ { 1 } + \\alpha _ { 2 } = a } p ( \\alpha _ { 1 } ) p ( \\alpha _ { 2 } ) , \\\\ p _ { - 2 } ( b ) & = \\sum _ { \\beta _ { 1 } + \\beta _ { 2 } = b } p ( \\beta _ { 1 } ) p ( \\beta _ { 2 } ) , \\\\ p _ { - 2 } ( a + b ) & = \\sum _ { \\gamma _ { 1 } + \\gamma _ { 2 } = a + b } p ( \\gamma _ { 1 } ) p ( \\gamma _ { 2 } ) , \\end{align*}"}
-{"id": "4138.png", "formula": "\\begin{align*} \\overline { \\pi } ^ { ( n ) } ( \\textbf { x } ) = \\Sigma _ i x _ i ^ { ( n ) } \\pi _ i ^ { ( n ) } ( \\textbf { x } ) . \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} s ( \\lambda _ h , \\mu _ h ) = ( \\rho g , T \\mu _ h ) \\quad \\mu _ h \\in \\Lambda _ h . \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{align*} f _ { q } \\left ( \\frac { 1 } { 2 } \\right ) = f _ { q a } \\left ( \\frac { 1 } { 2 } \\right ) = f _ { q c } \\left ( \\frac { 1 } { 2 } \\right ) = f _ { v e c t } \\left ( \\frac { 1 } { 2 } \\right ) . \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{align*} w _ { i j } : = \\begin{bmatrix} v _ { i j 1 } \\\\ \\vdots \\\\ v _ { i j r } \\\\ \\end{bmatrix} , \\ \\ v _ { i j k } : = \\sum _ { \\substack { s \\ge i , t \\ge j \\\\ ( s , t ) \\neq ( i , j ) } } \\left ( ( A _ k ) _ { i s } ( X _ k ) _ { s t } ( B _ k ) _ { t j } - ( C _ k ) _ { i s } ( X _ { k + 1 } ) _ { s t } ( D _ k ) _ { t j } \\right ) ; \\end{align*}"}
-{"id": "7604.png", "formula": "\\begin{align*} & \\int _ { s } ^ t \\beta ( r ) ( \\partial _ t \\psi ( r , q _ r ) + \\nabla _ q V ( r , q _ r ) ) \\circ d q _ r \\\\ = & \\int _ { s } ^ t \\beta ( r ) \\partial _ t \\psi ( r , q _ r ) d q _ r + \\frac { 1 } { 2 } [ \\beta ( r ) \\partial _ t \\psi ( r , q _ r ) , q _ r ] _ { s , t } + ( \\beta V ) ( t , q _ t ) - ( \\beta V ) ( s , q _ s ) - \\int _ s ^ t \\partial _ t ( \\beta V ) ( r , q _ r ) d r \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} p ' = \\textstyle { \\frac { 1 } { 2 \\varepsilon } } p '' \\mbox { w i t h } p '' = { \\rm t r } ( ( P S ) ^ 2 ) + 4 \\displaystyle \\sum _ { j = 1 } ^ { n _ { \\rm i n } } b _ j ^ \\top M P M b _ j \\ge 0 . \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} T _ f \\le a + b \\sum _ { i = 1 } ^ n T _ { f _ i } \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} & P _ \\omega ( X _ 0 = 0 ) = 1 \\ , , \\\\ & P _ \\omega ( X _ { k + 1 } = y + e _ i \\ , | \\ , X _ k = y ) = \\omega ^ i _ { y } ( k ) \\ , . \\end{align*}"}
-{"id": "8192.png", "formula": "\\begin{align*} { \\bf \\nabla } _ { s , H } \\textit { \\textbf { V } } = \\textit { \\textbf { e } } \\ { \\rm i n } \\ \\Omega , \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} L _ i : = P _ 1 \\circ K _ 2 \\cdots \\circ K _ { i - 1 } \\circ P _ i \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} \\mathcal { I } _ 3 = \\{ i : v _ i d _ i \\geq \\frac { \\sqrt { n } } { \\log n } \\} . \\end{align*}"}
-{"id": "6581.png", "formula": "\\begin{align*} \\Vert w _ R \\Vert _ \\theta \\asymp \\frac { R ^ { ( r - 1 ) / r } } { ( 1 / R ) ^ \\theta } = R ^ { \\theta + ( r - 1 ) / r } \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} \\frac { \\pi } { 3 T } \\log A _ { 2 T } ( \\gamma ) & = \\frac { \\pi } { 3 T } \\sum _ { n = 1 } ^ { \\gamma T } \\log \\sin \\left ( \\frac { n \\pi } { 3 T } \\right ) \\ , \\longrightarrow \\ , \\int _ 0 ^ { \\gamma \\pi / 3 } \\log \\sin u \\ , d y \\mbox { i n $ T \\to \\infty $ } . \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } { \\mathbb P } ( \\varphi _ n ( f _ 1 , \\ldots , f _ s ) \\in J ) = V ( J ) \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} | H ( \\nu , \\mathcal { E } ) - H ( \\nu , \\mathcal { F } ) | = O ( \\log k ) \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} K \\left ( x , t , u \\right ) = \\int _ { t } ^ { T } L \\left ( u \\left ( s \\right ) \\right ) d s + J \\left ( x \\left ( T \\right ) \\right ) , \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} J _ H ( x ) = \\prod _ { \\bar { \\alpha } \\in \\bar { \\Phi } ^ + } | \\sinh ( \\bar { \\alpha } ( x ) ) ^ { m _ { \\bar { \\alpha } } } | \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} \\left \\{ \\underline { g } \\in \\prod _ { i = 1 } ^ { n } \\mathcal { L } _ { G } ^ { i } \\middle | \\overline { \\left \\langle \\underline { g } \\right \\rangle } \\neq G \\right \\} = \\bigcup _ { P \\in \\mathcal { P } } \\left \\{ \\underline { g } \\in \\prod _ { i = 1 } ^ { n } \\mathcal { L } _ { G } ^ { i } \\middle | \\overline { \\left \\langle \\underline { g } \\right \\rangle } = x P x ^ { - 1 } x \\in G \\right \\} . \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} \\Delta \\leq \\frac { m } { d ^ { \\ell - 1 } } \\cdot g _ { \\ell } ( L ) \\leq \\frac { n } { d ^ { \\ell - 1 } } \\cdot g _ { \\ell } ( L ) \\leq \\frac { d ^ { \\ell } \\log ^ { \\ell } n } { d ^ { \\ell - 1 } } \\cdot g _ { \\ell } ( L ) = d \\log ^ { \\ell } n \\cdot g _ { \\ell } ( L ) . \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{align*} W ( x , y ) = x ^ n + \\sum _ { i = d } ^ n A _ i x ^ { n - i } y ^ i ( A _ i \\in { \\bf C } , \\ A _ d \\ne 0 ) \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} g ^ { i j } D _ { i j } \\Theta - \\frac { 1 } { 2 } x \\cdot D \\Theta = 0 . \\end{align*}"}
-{"id": "1505.png", "formula": "\\begin{align*} \\int _ { \\Sigma } \\left ( \\dfrac { n } { 2 } + | { \\bf H } | ^ 2 + | \\nabla \\log v | ^ 2 \\right ) \\phi ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma & = - \\int _ { \\Sigma } \\phi ^ 2 ( \\mathcal { L } \\log v ) e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma \\\\ & = \\int _ { \\Sigma } \\left < \\nabla \\phi ^ 2 , \\nabla \\log v \\right > e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma \\end{align*}"}
-{"id": "3212.png", "formula": "\\begin{align*} c _ 1 ( M ) \\cup [ \\omega _ 0 ] = 0 \\in H ^ 4 ( B ) . \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} q \\frac { d E _ 2 } { d q } = \\frac { E _ 2 ^ 2 - E _ 4 } { 1 2 } \\ \\ q \\frac { d E _ 4 } { d q } = \\frac { E _ 2 E _ 4 - E _ 6 } { 3 } \\ \\ q \\frac { d E _ 6 } { d q } = \\frac { E _ 2 E _ 6 - E _ 4 ^ 2 } { 2 } \\end{align*}"}
-{"id": "347.png", "formula": "\\begin{align*} ( \\mathcal { G } _ D ) _ { g \\cdot x _ 0 , h \\cdot x _ 0 } = \\frac { 1 } { | X | } \\sum _ { j \\in D } m _ j \\omega _ j ( h ^ { - 1 } g ) ( g , h \\in G ) . \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} f ^ * ( \\omega _ M \\times 1 ) = a \\cdot ( \\omega _ M \\times 1 ) , \\ a \\in \\Z . \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} B ( \\rho ) = B ( \\rho , \\mathfrak { e } , V ) \\doteq V ^ { 1 / 2 } \\ , [ \\rho + h ( \\mathfrak { e } ) ] ^ { - 1 } \\ , V ^ { 1 / 2 } . \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} \\P ( Z _ \\alpha \\leq x ) = e ^ { - x ^ { - \\alpha } } , \\ ; \\ , x > 0 , \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} ( \\alpha ; q ) _ k = ( 1 - \\alpha ) ( 1 - \\alpha q ) \\dots ( 1 - \\alpha q ^ { k - 1 } ) \\mbox { w i t h $ \\alpha = q ^ a $ } . \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} A [ T ^ { ( n - 2 ) } , A , A ] A = T ^ { ( n ) } . \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{align*} g ( \\alpha _ t ) = \\int _ 0 ^ \\infty \\frac { \\alpha _ t ( s ^ 2 + 1 ) } { ( \\alpha _ t - s ) ^ 2 } \\ ; d \\rho ( s ) = \\frac { 1 } { t - 1 } \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{align*} [ W _ k ( x ) , P _ { \\pm n } ] = \\Delta _ * \\Big ( \\pm [ n ] _ { q _ 1 } [ n ] _ { q _ 2 } [ k ] _ { q ^ n } q ^ { n ( r - k ) \\delta _ { \\pm } ^ + } \\cdot x ^ { \\pm n } W _ k ( x ) \\Big ) \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{align*} \\sum _ { b \\in ( \\mathbb { Z } / n \\mathbb { Z } ) ^ \\times } \\chi ( b ) \\cdot \\left \\langle \\omega ^ * _ { \\overline { f } } , \\mathrm { e x p } ^ { * } \\left ( \\mathrm { l o c } _ p \\left ( c _ { \\mathbb { Q } ( \\mu _ n ) } \\right ) ^ { \\sigma _ b } \\right ) \\right \\rangle _ { \\mathrm { d R } } = c \\cdot d \\cdot ( c - \\chi ( c ) ) \\cdot ( d - \\chi ( d ) ) \\cdot \\frac { L ^ { ( N p ) } ( f , \\chi , 1 ) } { ( - 2 \\pi i ) \\Omega ^ { \\chi ( - 1 ) } _ { f } } \\end{align*}"}
-{"id": "978.png", "formula": "\\begin{align*} \\langle \\Phi _ { L _ { k } , \\Delta L _ { k } } \\ , | \\ , H ( \\mathfrak { e } , V ) \\Phi _ { L _ { k } , \\Delta L _ { k } } \\rangle < 0 , k = 1 , 2 , \\ldots , N . \\end{align*}"}
-{"id": "2042.png", "formula": "\\begin{align*} \\overline { \\delta Q ^ { ( k + 1 ) } ( 0 ) } = \\dfrac { 1 } { M } \\| p \\| ^ { k - 1 } + \\sum _ { j = 0 } ^ { k - 3 } \\dfrac { \\tilde { \\Phi } _ k ( k _ { s c a t t } ) } { M ^ { ( k - j + 1 ) / 2 } } \\| p \\| ^ { j } , \\ \\end{align*}"}
-{"id": "8863.png", "formula": "\\begin{align*} D _ { \\mathcal { L } } = \\sum _ { Y \\in \\mathcal { I } ^ G ( X ) } v _ { \\mathcal { L } } ( \\mu _ Y ) Y + \\sum _ { D \\in \\mathcal { D } _ X } v _ { \\mathcal { L } } ( \\rho ( D ) ) \\overline { D } + \\sum _ { D \\in \\mathcal { D } \\setminus \\mathcal { D } _ X } n _ { \\mathcal { L } , D } \\overline { D } . \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{align*} \\Large { \\Big [ \\frac { \\partial } { \\partial t } , \\frac { \\partial } { \\partial s } \\Big ] } \\normalsize : = \\Large \\frac { \\partial ^ { 2 } } { \\partial t \\partial s } - \\frac { \\partial ^ { 2 } } { \\partial s \\partial t } \\normalsize = 0 . \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{align*} u ( x , t ) = S ( t ) u _ * ( x ) + i \\int _ 1 ^ t S ( t - s ) u ( s ) \\Big ( e ^ { - s } v _ * ( x ) + \\int _ 1 ^ s e ^ { - ( s - s ' ) } | u ( s ' ) | ^ 2 d s ' \\Big ) d s . \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} \\mathbb { O } _ { n \\times p } = \\sum _ { \\substack { i < j ; \\ , i , j \\in I _ p \\\\ a \\in \\{ 1 , \\dots , p \\} } } \\alpha _ { i j } \\left ( u _ { j a } { \\bf e } _ i \\otimes { \\bf f } _ a - u _ { i a } { \\bf e } _ j \\otimes { \\bf f } _ a \\right ) . \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 3 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 4 & 0 & 0 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 7 & 4 & 2 \\\\ 0 & 5 & 7 & 4 \\\\ 0 & 1 0 & 5 & 7 \\end{pmatrix} , \\end{align*}"}
-{"id": "8460.png", "formula": "\\begin{align*} D _ { m , n } ^ I = \\{ Z \\in M _ { m \\times n } ( \\C ) \\mid Z Z ^ * < I _ m \\} , \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} \\prod ^ { k - 1 } _ { i = 0 } \\parallel \\psi ^ \\ast _ { t _ { i + 1 } - t _ i } | \\mathcal { E } _ { \\varphi ^ X _ { t _ i } ( x ) } \\parallel \\leq e ^ { - \\eta t _ k } , ~ \\prod ^ { l - 1 } _ { i = k } m ( \\psi ^ \\ast _ { t _ { i + 1 } - t _ i } | \\mathcal { F } _ { \\varphi ^ X _ { t _ i } ( x ) } ) \\geq { \\rm e } ^ { \\eta ( T - t _ k ) } . \\end{align*}"}
-{"id": "4252.png", "formula": "\\begin{align*} [ T _ n , T _ m ] = & - \\sum _ { l = 1 } ^ { \\infty } r _ l ( T _ { n - l } T _ { m + l } - T _ { m - l } T _ { n + l } ) \\\\ & - \\frac { ( 1 - q ) ( 1 - t ^ { - 1 } ) } { 1 - q t ^ { - 1 } } ( q ^ n t ^ { - n } - q ^ { - n } t ^ { n } ) \\delta _ { m + n , 0 } . \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{align*} c _ 2 = \\frac { \\eta ^ 2 } { 2 ( 2 a \\lambda _ 2 - c _ 1 ) } + \\frac { b | T r ( L ) | } { 2 N } + \\frac { m \\tau ^ 2 \\tilde { \\gamma } ( N - 1 ) } { 2 N } . \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} ( N - 1 ) \\ln \\left ( 1 - \\frac { x } { N } \\right ) = - ( N - 1 ) \\left [ \\frac { x } { N } + O \\left ( \\frac { x ^ 2 } { N ^ 2 } \\right ) \\right ] = - x + o \\left ( \\frac { 1 } { N ^ { \\theta } } \\right ) \\ ; \\theta \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty A _ m H _ { - n } \\left ( z ^ n - z ^ { n + 1 } \\right ) = \\sum _ { n = 0 } ^ \\infty \\left ( \\left ( \\gamma z \\right ) ^ n - \\left ( \\gamma z \\right ) ^ { n + 1 } \\right ) H _ { - n } A _ m \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} \\lambda = h ^ { - 1 } ( \\nabla _ { B } h ) \\beta - b h ^ { - 1 } + ( m + n - 1 ) a - a h \\varphi , \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} \\hat { T } : = \\bigcap \\limits _ { \\alpha \\in \\Delta \\setminus \\Delta _ P } \\ker ( \\alpha ) \\end{align*}"}
-{"id": "4291.png", "formula": "\\begin{align*} ( l , w ) : = ( l ( Y _ r ) , Y _ r ( l ) ) \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} \\xi _ { 0 , T } ( u ) = \\int _ 0 ^ T c \\bigl ( t , X _ t , u _ t \\bigr ) d t + \\Psi ( X _ T ) , \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} \\pi i z ^ 2 \\tau + \\tau \\frac { z \\ , \\partial _ z \\Theta ( \\tau z , \\tau ) + \\partial _ \\tau \\Theta ( \\tau z , \\tau ) } { \\Theta ( \\tau z , \\tau ) } - \\frac { 1 } { \\tau } \\frac { \\partial _ \\tau \\Theta \\left ( z , - \\tfrac { 1 } { \\tau } \\right ) } { \\Theta \\left ( z , - \\tfrac { 1 } { \\tau } \\right ) } = - \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} D _ s = \\sum _ { Y \\in \\mathcal { I } ^ G ( X ) } v _ s ( \\mu _ Y ) Y + \\sum _ { D \\in \\mathcal { D } _ X } v _ s ( \\rho ( D ) ) \\overline { D } + \\sum _ { D \\in \\mathcal { D } \\setminus \\mathcal { D } _ X } n _ D \\overline { D } \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} \\Delta _ { f } = \\Delta - \\left \\langle \\nabla f , \\nabla \\cdot \\right \\rangle . \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} u [ a _ 1 , a _ 2 , a _ 3 ] = - u \\bigl [ a _ 3 , [ a _ 1 , a _ 2 ] \\bigr ] \\in W , \\end{align*}"}
-{"id": "3870.png", "formula": "\\begin{align*} \\| d _ i - g ( { x } _ i ) \\| \\le \\varepsilon \\| g ( { x } _ i ) \\| i = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} S _ \\mu = \\lbrace s \\in \\mathbb { N } \\mid 1 \\le s \\le N , \\mu _ s = 0 \\rbrace , & & T _ \\mu = \\lbrace t \\in \\mathbb { N } \\mid 1 \\le t \\le N , \\mu _ t = 1 \\rbrace . \\end{align*}"}
-{"id": "3472.png", "formula": "\\begin{align*} - q f = q ( B _ 0 - \\lambda ) ^ { - 1 } \\operatorname { s g n } ( \\cdot ) q f . \\end{align*}"}
-{"id": "7738.png", "formula": "\\begin{align*} J q ^ k ( \\xi ^ J ) & = \\xi _ 1 ^ { j _ 1 } J q ^ k ( \\xi _ 2 ^ { j _ 2 } \\cdots \\xi _ n ^ { j _ n } ) \\\\ & + \\binom { j _ 1 } { 1 } \\xi _ 1 ^ { j _ 1 + 1 } J q ^ { k - 1 } ( \\xi _ 2 ^ { j _ 2 } \\cdots \\xi _ n ^ { j _ n } ) \\\\ & + \\dots \\\\ & + \\binom { j _ 1 } { k } \\xi _ 1 ^ { j _ 1 + k } \\xi _ 2 ^ { j _ 2 } \\cdots \\xi _ n ^ { j _ n } . \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} \\mathfrak { s o } _ { 1 2 } = \\bigwedge ^ 2 \\mathbb { C } ^ 6 \\oplus \\mathfrak { g l } _ 6 \\oplus \\bigwedge ^ 4 \\mathbb { C } ^ 6 . \\end{align*}"}
-{"id": "8642.png", "formula": "\\begin{align*} \\eta = 8 d ( d - 1 ) s < \\frac 1 { 4 d ^ 2 } . \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} K ( u , v ) = H \\left ( 1 - \\frac { u } { \\lambda _ 0 } , 1 - \\frac { v } { \\lambda _ 0 } \\right ) + u v \\lambda _ 0 ^ { - 6 } a ' _ 0 \\left ( 1 - \\frac { u } { \\lambda _ 0 } \\right ) a ' _ 0 \\left ( 1 - \\frac { v } { \\lambda _ 0 } \\right ) . \\end{align*}"}
-{"id": "8358.png", "formula": "\\begin{align*} u ( t ) = u _ 0 + \\frac { \\lambda } { \\Gamma ( 2 \\alpha ) } \\int _ 0 ^ { t } ( t - s ) ^ { 2 \\alpha - 1 } \\frac { f ( s , u ( s ) ) } { \\left ( \\int _ { 0 } ^ { t } f ( x , u ) \\ , d x \\right ) ^ { 2 } } d s . \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} p _ 0 ( x ) : = f ( x _ 0 ) + \\sum _ { m = 1 } ^ { n - 1 } \\frac { 1 } { m ! } \\sum _ { I \\in \\underline { d } ^ m } ( \\partial _ I f ) ( x _ 0 ) \\cdot ( x - x _ 0 ) ^ I . \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} \\begin{aligned} [ b ] & \\frac { 1 - \\tau } { 2 } k + \\max _ { \\mathcal { P } } \\ \\frac { ( 3 - \\tau ) } { 2 } \\Big ( \\abs { S _ 1 } - \\abs { S _ 2 } + \\frac { \\tau - 2 } { 3 - \\tau } k _ 1 - \\frac { 2 E _ { S _ 1 } + E _ { S _ 1 , S _ 3 } } { \\tau - 1 } \\\\ & \\qquad - \\frac { E _ { S _ 1 , V _ 1 } - E _ { S _ 2 , V _ 1 } } { \\tau - 1 } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "5358.png", "formula": "\\begin{align*} & d _ { \\rm T V , l o w } = 1 - \\left ( \\frac { 4 \\gamma _ e } { ( \\gamma _ e + 1 ) ^ 2 } \\right ) ^ { \\frac { M } { 4 } } , \\\\ & d _ { \\rm T V , u p } = \\sqrt { 1 - \\left ( \\frac { 4 \\gamma _ e } { ( \\gamma _ e + 1 ) ^ 2 } \\right ) ^ { \\frac { M } { 2 } } } , \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{align*} { \\rm { P _ r } } \\left \\{ { \\Phi \\left ( { \\pi { l ^ 2 } } \\right ) = { m } } \\right \\} = \\frac { { { { \\left ( { { \\lambda _ X } \\pi { l ^ 2 } } \\right ) } ^ { { m } } } { e ^ { - { \\lambda _ X } \\pi { l ^ 2 } } } } } { { \\left ( { { m } } \\right ) ! } } . \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} D ^ 3 ( x _ 2 ) = x _ { n - 3 } , D ^ 3 ( x _ 3 ) = x _ { n - 2 } , D ^ 3 ( x _ 4 ) = x _ { n - 1 } , \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} W ^ 0 _ 3 ( p , p _ 1 , p _ 2 ) = K _ p ( q , \\bar q ) W ^ 0 _ 2 ( q , p _ 1 ) W ^ 0 _ 2 ( \\bar q , p _ 2 ) \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} X _ i \\tilde a _ j - X _ j \\tilde a _ i = c _ { i j } \\partial _ { x _ { n + 1 } } u , \\ \\ i , j = 1 , . . . , n . \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } B _ { 2 j - 1 } = + \\infty , \\lim _ { j \\to \\infty } B _ { 2 j } = - \\infty . \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{align*} \\frac { 1 } { 2 R } \\sum _ { i = 1 } ^ { m } \\omega _ { i } \\rho _ { i } \\Vert U _ { i } x - x \\Vert ^ { 2 } \\leq \\Vert U x - x \\Vert \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{align*} X _ { \\epsilon _ 1 } X _ { - \\epsilon _ { 1 } + \\epsilon _ 2 } + X _ { - \\epsilon _ 1 } X _ { \\epsilon _ { 1 } + \\epsilon _ 2 } + X _ { \\epsilon _ 2 } X _ 0 = 0 \\end{align*}"}
-{"id": "1189.png", "formula": "\\begin{align*} g _ \\infty = - \\ : 1 6 d u ^ 2 + u ^ 2 d \\theta ^ 2 + 4 c _ \\lambda ^ { - 1 } \\left [ e ^ { P _ \\infty } ( d \\widehat { \\sigma } + Q _ \\infty d \\widehat { \\delta } ) ^ 2 + e ^ { - \\ : P _ \\infty } d \\widehat { \\delta } ^ 2 \\right ] . \\end{align*}"}
-{"id": "4835.png", "formula": "\\begin{align*} a , c \\in \\{ \\pm 1 \\} . \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{align*} ( \\tilde \\gamma ^ { - 1 } ) ^ i _ j = ( \\tilde \\gamma ^ { - 1 } ) ^ { i k } \\delta _ { k j } = \\left ( \\begin{array} { c c c } \\gamma / ( \\gamma ^ 2 + B _ 0 ^ 2 ) & - B _ 0 / ( \\gamma ^ 2 + B _ 0 ^ 2 ) & 0 \\\\ B _ 0 / ( \\gamma ^ 2 + B _ 0 ^ 2 ) & \\gamma / ( \\gamma ^ 2 + B _ 0 ^ 2 ) & 0 \\\\ 0 & 0 & \\gamma ^ { - 1 } \\end{array} \\right ) \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{align*} \\frac 1 L B _ L = \\left \\{ \\frac 1 L \\lambda \\ \\Big | \\ , \\lambda \\in B _ L \\right \\} . \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} m ( R ) m ( S ) = m ( R \\cup F _ { R S } ) m ( R \\cup T _ { R S } ) . \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} \\hat { H } _ { V _ j } ( \\hat { X } ) = - \\frac { \\eta _ 1 } { \\omega _ 1 } z ( \\hat { X } ) + \\frac { \\pi i } { \\omega _ 1 } + G _ { V _ j } ( \\hat { X } ) \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} [ a _ 1 , a _ 2 , \\dots , a _ n ] = 0 \\mbox { f o r a l l } a _ i \\in A \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} W _ { a } ^ { \\prescript { w } { } { \\chi } } \\circ \\overline { \\mathcal { A } } _ { w } ( \\pi ( \\varpi ^ { \\lambda } ) \\phi _ { K } ^ { \\chi } ) = \\tau _ { \\nu , \\nu } ^ 1 W _ { a } ^ { \\chi } ( \\pi ( \\varpi ^ { \\lambda } ) \\phi _ { K } ^ { \\chi } ) + \\tau _ { w \\cdot \\nu , \\nu } ^ 2 W _ { w \\cdot a } ^ { \\chi } ( \\pi ( \\varpi ^ { \\lambda } ) \\phi _ { K } ^ { \\chi } ) \\end{align*}"}
-{"id": "4687.png", "formula": "\\begin{align*} \\det ( A ) = \\sum _ { \\sigma \\in S y m ( [ n ] ) } s g n ( \\sigma ) a _ { 1 , \\sigma ( 1 ) } \\cdots a _ { n , \\sigma ( n ) } \\ ; . \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} ( R _ m L _ m ) ^ { - 1 } R _ m v = \\begin{cases} q ^ { - \\kappa ( m , \\mu , \\lambda ) } R _ m v & , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "2440.png", "formula": "\\begin{align*} \\lambda _ j ( t ) = - \\varepsilon + i t , - \\infty < t < \\infty , \\end{align*}"}
-{"id": "4514.png", "formula": "\\begin{align*} R ( n , q , r ) ~ = ~ G _ { q , r } ( x ) ~ \\lesssim ~ \\varphi ( q ) \\log ^ 2 x ~ \\sim ~ \\varphi ( q ) { n ^ 2 \\over \\bar \\tau ^ 2 } . \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} \\mathrm { M a b } _ { \\Theta } ( u ) = & \\sum _ Y \\Lambda _ Y \\int _ { \\tilde { \\Delta } ^ + _ Y } ( n u ^ * ( p ) - u ^ * ( p ) \\sum \\frac { \\chi ( \\alpha ^ { \\vee } ) } { q ( \\alpha ^ { \\vee } ) } + d _ p u ^ * ( p ) ) P _ { D H } ( q ) d q \\\\ & + \\int _ { \\Delta ^ + } u ^ * ( p ) ( \\sum \\frac { \\chi ^ { a c } ( \\alpha ^ { \\vee } ) } { q ( \\alpha ^ { \\vee } ) } - \\bar { S } _ { \\Theta } ) P _ { D H } ( q ) d q - \\int _ { \\Delta ^ + } I _ H ( d _ p u ^ * ) P _ { D H } ( q ) d q \\\\ & - \\int _ { \\Delta ^ + } \\ln \\det ( d ^ 2 _ p u ^ * ) P _ { D H } ( q ) d q \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} E _ \\alpha ( \\gamma ) = E _ \\alpha ( \\Gamma ) & \\le \\frac 1 2 ( \\alpha + 1 ) { B } ^ \\alpha [ \\Gamma ' ] ^ 2 _ { { ( \\alpha - 1 ) / 2 , 2 } } \\\\ & \\overset { \\eqref { s e m i n o r m e s t - a r c l e n g t h } } { \\le } \\frac 1 2 ( \\alpha + 1 ) { B } ^ \\alpha \\Big ( \\frac 1 c \\Big ) ^ { 2 + \\alpha } \\Big [ \\Big ( \\frac 1 c \\Big ) ^ 2 + C ^ { 6 } \\Big ] \\cdot [ \\gamma ' ] ^ 2 _ { { ( \\alpha - 1 ) / 2 , 2 } } , \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} \\vert d \\vert = 1 ( 0 , T ) \\times \\Omega . \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} \\log \\left ( \\Theta \\left ( z , - \\tfrac { 1 } { \\tau } \\right ) \\right ) = \\frac { 1 } { 2 } \\log ( - i \\tau ) + \\pi i z ^ 2 \\tau + \\log \\left ( \\Theta ( \\tau z , \\tau ) \\right ) . \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} R = \\frac { r ( \\theta _ 1 \\ , | \\ , \\phi _ 0 - \\phi _ 1 ) \\ , r ( \\theta _ 3 \\ , | \\ , \\phi _ 3 - \\phi _ 0 ) } { r ( \\theta _ 1 + \\theta _ 3 \\ , | \\ , \\phi _ 3 - \\phi _ 1 ) } \\frac { P ( \\theta _ 1 \\ , | \\ , \\phi _ 3 - \\phi _ 2 ) \\ , P ( \\theta _ 3 \\ , | \\ , \\phi _ 1 - \\phi _ 2 ) } { P ( \\theta _ 1 + \\theta _ 3 \\ , | \\ , \\phi _ 0 , \\phi _ 2 ) } \\ , , \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { 2 } } \\abs { P ( x ' , y ) - P ( x , y ) } & = \\abs { \\frac { \\Omega ( x ' , y ) } { \\delta ( x ' , y ) } - \\frac { \\Omega ( x , y ) } { \\delta ( x , y ) } } \\\\ & \\leq \\frac { \\abs { \\Omega ( x ' , y ) - \\Omega ( x , y ) } } { \\delta ( x , y ) } + \\abs { \\Omega ( x ' , y ) } \\abs { \\frac { 1 } { \\delta ( x ' , y ) } - \\frac { 1 } { \\delta ( x , y ) } } \\\\ & = I + I I . \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi } \\int _ \\mathbb { C } P ( w , \\overline { w } ) e ^ { - 2 | w | ^ 2 + a w + b \\overline { w } } d L ( w ) = P ( \\partial _ a , \\partial _ b ) \\frac { 1 } { 2 } e ^ { \\frac { a b } { 2 } } . \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} C _ { 1 } = ( p ^ { e } - p ^ { e - 2 d } ) \\prod _ { \\substack { 0 \\leq i \\leq p - 1 } } w _ { i } ^ { p ^ { e - 2 } + p ^ { m + d - 2 } } . \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} u _ t - \\mathcal { H } u _ { x x } + ( u ^ 2 ) _ x = 0 , \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} \\alpha = ( x _ M ^ k \\times 1 ) + ( x _ M ^ { k - n } \\times \\omega _ { N } ) \\in H ^ k ( M ; \\R ) \\oplus ( H ^ { k - n } ( M ; \\R ) \\otimes H ^ n ( N ; \\R ) ) . \\end{align*}"}
-{"id": "3283.png", "formula": "\\begin{align*} c _ { h _ { 1 } , h _ { 2 } } = \\exp ( t \\psi ( h _ { 1 } ) ) d _ { h _ { 1 } , h _ { 2 } } \\end{align*}"}
-{"id": "4814.png", "formula": "\\begin{align*} \\| \\overline { u } & \\| _ { L ^ { m \\chi ( \\alpha + 1 ) } ( \\Omega _ { R _ 1 } ) } \\le C ^ { 1 / ( \\alpha + 1 ) } \\Bigg [ \\frac { 1 } { ( R _ 1 - R _ 2 ) } + \\left ( \\frac { G ( R _ 1 , R _ 2 ) } { m \\alpha + 1 - C m \\epsilon ^ { m ' } } \\right ) ^ { 1 / m } \\\\ & + \\left ( \\frac { \\| f \\| _ q } { ( m \\alpha + 1 - C m \\epsilon ^ { m ' } ) k ^ { m - 1 } } \\right ) ^ { N / m ( m q - N ) } \\Bigg ] ^ { 1 / ( \\alpha + 1 ) } \\| \\overline { u } \\| _ { L ^ { m ( \\alpha + 1 ) } ( \\Omega _ { R _ 2 } ) } . \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{align*} U : = [ 0 , 1 ) _ { \\rho _ + } \\times I ^ + , \\ \\ I ^ + = \\{ Z \\in \\R ^ 3 \\colon | Z | \\leq 1 \\} , \\end{align*}"}
-{"id": "2790.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\delta ( \\sigma _ 0 \\wedge \\sigma _ 0 ) = 2 \\sigma _ 0 \\wedge \\delta ( \\sigma _ 0 ) , \\\\ \\\\ \\delta ( \\sigma _ 1 \\wedge \\sigma _ 1 ) = 2 \\sigma _ 1 \\wedge \\delta ( \\sigma _ 1 ) , \\\\ \\\\ \\delta ( \\sigma _ 0 \\wedge \\sigma _ 1 ) = \\delta ( \\sigma _ 0 ) \\wedge \\sigma _ 1 + \\sigma _ 0 \\wedge \\delta ( \\sigma _ 1 ) \\end{array} \\right . \\end{align*}"}
-{"id": "1446.png", "formula": "\\begin{align*} \\mu _ \\star ^ { ( \\infty , 2 \\gamma ) } : = \\bigotimes \\limits _ { k = 1 } ^ { \\infty } { \\bf E x p } ( 2 \\gamma ) , \\end{align*}"}
-{"id": "3867.png", "formula": "\\begin{align*} f ( y ) - f ( x ) = \\int _ 0 ^ 1 \\langle g ( x + t ( y - x ) ) , y - x \\rangle d t . \\end{align*}"}
-{"id": "4946.png", "formula": "\\begin{align*} f = ( h , g _ 1 , \\ldots , g _ q ) \\colon X \\to \\C ^ { n + q } \\end{align*}"}
-{"id": "2092.png", "formula": "\\begin{align*} & g ( C _ a ) - 1 = h ( C _ a ) = \\delta ( C _ a ) = e _ Q ( C _ a ) = 0 , \\ \\ \\mbox { a n d } \\ \\ { \\rm i n d } C _ a = 1 , \\\\ \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} & \\sum _ { x ^ { ( m ' ) } \\in \\Z } \\P ^ { 0 , 0 } _ { 2 T } ( X ( t _ m ) = x ^ { ( m ) } , m \\in \\{ 1 , 2 , \\dots , M \\} ) \\\\ & = \\P ^ { 0 , 0 } _ { 2 T } ( X ( t _ m ) = x ^ { ( m ) } , m \\in \\{ 1 , 2 , \\dots , M \\} \\setminus \\{ m ' \\} ) \\end{align*}"}
-{"id": "3837.png", "formula": "\\begin{align*} \\Gamma _ \\mp = \\{ ( x , \\xi ) \\in S ^ * M ; \\rho ( \\varphi _ t ( x , \\xi ) ) \\not \\to 0 \\textrm { a s } t \\to \\pm \\infty \\} . \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } I _ { \\partial C } ( X _ t ) d t = 0 \\enskip \\textrm { a . s . , } \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} \\forall 1 \\le m \\le N , A ^ N u _ m ^ N = \\varepsilon _ m ^ N u _ m ^ N . \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{align*} K _ \\lambda ( x , y ) = C _ \\lambda ( x , y ) + D _ \\lambda ( x , y ) = \\frac { 1 } { 2 \\alpha \\sqrt { \\lambda } } \\begin{cases} u ( x ) v ( y ) \\operatorname { s g n } ( y ) , & y < x , \\\\ v ( x ) u ( y ) \\operatorname { s g n } ( y ) , & x < y , \\end{cases} \\end{align*}"}
-{"id": "9132.png", "formula": "\\begin{align*} f = x ^ 4 + b x ^ 2 + d \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{align*} \\tilde { I } ( u ) : = \\frac { 1 } { 2 } \\int _ { \\Omega } | \\nabla u | ^ { 2 } - \\int _ { \\Omega } a u , \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} b _ t = \\inf \\{ b : b > x \\mathrm { f o r a l l } x \\in \\mathrm { s u p p } ( \\mu ^ { \\boxtimes t } ) \\} . \\end{align*}"}
-{"id": "5685.png", "formula": "\\begin{align*} & \\norm { \\tilde { x } - \\tilde { y } } ^ 2 \\\\ = \\ ; & \\norm { ( 1 + \\lambda ) ( x ^ + - y ^ + ) - \\lambda ( x - y ) } ^ 2 \\\\ = \\ ; & ( 1 + \\lambda ) \\norm { x ^ + - y ^ + } ^ 2 - \\lambda \\norm { x - y } ^ 2 + \\lambda ( 1 + \\lambda ) \\norm { ( x ^ + - y ^ + ) - ( x - y ) } ^ 2 . \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} \\nu ^ { + } = ( 2 ^ { m _ 2 - r } , 1 ^ { m _ 1 + 2 r } , 0 ^ { n - m _ 1 - m _ 2 - r } ) , \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} & I ( U _ i ; Y | U _ { 1 : i - 1 } \\ ! = \\ ! u _ { 1 : i - 1 } ) \\le \\ ! I ( U _ i ; Z | U _ { 1 : i - 1 } \\ ! = \\ ! u _ { 1 : i - 1 } ) \\\\ & I ( U _ i ; X | U _ { 1 : i - 1 } \\ ! = \\ ! u _ { 1 : i - 1 } ) \\le \\ ! I ( U _ i ; Z | U _ { 1 : i - 1 } \\ ! = \\ ! u _ { 1 : i - 1 } ) . \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{align*} \\varphi _ { \\gamma * } ^ { - 1 } ( \\frac { 1 } { q } \\partial _ t ) = \\partial _ { \\tau } - 2 i ( \\nu z + \\mu \\bar { z } + \\mathfrak { r } ) \\partial _ { z } + 2 i ( \\nu \\bar { z } + \\bar { \\mu } { z } + \\bar { \\mathfrak { r } } ) \\partial _ { \\bar { z } } , \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{align*} R _ k = \\frac { 1 } { 2 n ^ 2 } \\log _ 2 ( q _ k ^ n ) = \\frac { 1 } { 2 n } \\log _ 2 ( q _ k ) , . \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} { \\rm T r } \\left ( \\phi ^ a \\phi ^ { \\dagger a } k ^ { u + m _ 1 } \\right ) = \\frac { 1 } { 1 - t ^ { u + m _ 1 } } \\prod _ { i = 1 } ^ { a } \\left ( \\frac { 1 - t ^ i } { 1 - t ^ { u + m _ 1 + i } } \\right ) = \\frac { 1 } { 1 - q t ^ { m _ 1 } } \\prod _ { i = 1 } ^ { a } \\left ( \\frac { 1 - t ^ i } { 1 - q t ^ { m _ 1 + i } } \\right ) , \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} \\widehat { \\sigma } = t _ i ^ { - 1 } \\log ^ { \\frac 1 2 } ( t _ i ) ( \\sigma + Q ( { \\frak t } ( t _ i ) ) \\delta ) \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{align*} \\Delta E _ b = \\alpha _ * ^ 2 \\left ( O _ 0 ( \\sqrt { M } \\| p \\| ^ { - 3 } + \\dfrac { \\beta ^ { ( 4 ) } _ b ( \\kappa , \\alpha _ * = 0 ) } { \\| p \\| ^ { - 4 } } + O \\left ( M ^ { - 1 / 2 } \\| p \\| ^ { - 5 } \\right ) \\right ) + o ( \\alpha _ * \\sqrt { M } ) . \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} \\dot { U } \\cdot \\bar { \\psi } + \\dot { u } \\bar { \\psi } = \\left ( - \\mathrm { d i v } \\ , T ^ { U } \\bar { \\times } U + u \\mathrm { d i v } \\ , T ^ { U } \\right ) \\cdot \\bar { \\psi } - g \\left ( \\mathrm { d i v } \\ , T ^ { U } , U \\right ) \\bar { \\psi } . \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} \\underline { F } _ i = \\frac 1 n \\sum _ { k = 0 } ^ { n - 1 } 1 _ { [ q ^ i _ k , \\infty ) } ( x ) \\quad \\overline { F } _ i = \\frac 1 n \\sum _ { k = 1 } ^ { n } 1 _ { [ q ^ i _ k , \\infty ) } ( x ) \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} A _ X f ( x , y ) - r f ( x , y ) & = 0 , \\enskip x \\in C , \\\\ f ( x , y ) - F ( x , y ) & = 0 , \\enskip x \\in E \\backslash C , \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{align*} \\arg ( z _ { j , n } ) < \\frac { \\pi } { 1 8 . 2 8 8 0 \\ldots } = 0 . 1 7 1 7 8 4 \\ldots \\mbox { f o r } ~ j = \\lceil v _ n \\rceil , \\lceil v _ n \\rceil + 1 , \\ldots , J _ n . \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 ( \\alpha ( s ) - \\alpha _ 0 ( s ) ) g ( s ) d s = \\int _ 0 ^ 1 \\Bigl ( \\int _ 0 ^ s ( \\alpha - \\alpha _ 0 ) ( d r ) \\Bigr ) g ( s ) d s = \\int _ 0 ^ 1 \\bar { g } ( r ) ( \\alpha - \\alpha _ 0 ) ( d r ) . \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} v _ p ( [ z ^ n ] F ( G , q ; z ) ) & \\geq \\min _ { n _ 0 + n _ 1 p + \\dots = n } \\sum _ { b _ i \\leq 0 } \\left ( b _ i n _ i - v _ p \\left ( n _ i ! \\right ) \\right ) \\\\ & \\geq \\min _ { n _ 0 + n _ 1 p + \\dots = n } \\sum _ { b _ i \\leq 0 } \\left ( b _ l p ^ { i - l } n _ i - v _ p \\left ( ( p ^ { i - l } n _ i ) ! \\right ) \\right ) \\\\ & = \\min _ { n _ 0 + n _ 1 p + \\dots = n } \\left ( b _ l \\sum _ { b _ i \\leq 0 } p ^ { i - l } n _ i - \\sum _ { b _ i \\leq 0 } \\left ( v _ p \\left ( ( p ^ { i - l } n _ i ) ! \\right ) \\right ) \\right ) . \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} \\tilde { G } _ { j } ( \\hat { X } ) = - \\zeta _ \\lambda ( z ( \\hat { X } ) ) + \\zeta _ \\lambda ( z ( a _ j ) ) . \\end{align*}"}
-{"id": "5483.png", "formula": "\\begin{align*} \\mbox { R e } ( \\mathbf { R } _ 0 ( \\mathbf { z } ) ) = a ( \\rho ) & = \\mbox { R e } ( \\lambda _ l ) \\rho + \\sum _ { m = 1 } ^ M \\mbox { R e } ( \\beta _ m ) \\rho ^ { 2 m + 1 } , \\\\ \\frac { 1 } { \\rho } \\mbox { I m } ( \\mathbf { R } _ 0 ( \\mathbf { z } ) ) = b ( \\rho ) & = \\mbox { I m } ( \\lambda _ l ) + \\sum _ { m = 1 } ^ M \\mbox { I m } ( \\beta _ m ) \\rho ^ { 2 m } . \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} \\Phi = \\big ( ( A _ 1 , b _ 1 ) , ( A _ 2 , b _ 2 ) , \\dots , ( A _ L , b _ L ) \\big ) , \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} c ' = \\gamma c \\gamma ^ { - 1 } , \\rho ' = g \\rho g ^ { - 1 } , f ' = g f _ \\gamma , \\end{align*}"}
-{"id": "6959.png", "formula": "\\begin{align*} d _ \\lambda = \\lvert \\{ \\textup { $ P _ h $ - t a b l e a u x o f s h a p e $ \\lambda ^ { \\vee } $ } \\} \\rvert . \\end{align*}"}
-{"id": "1214.png", "formula": "\\begin{align*} { u ^ { f } ( x ( \\gamma , \\xi - 0 ) , T ) } \\overset { ( \\ref { 2 . 6 } ) } = \\sqrt { \\frac { c ( x ( \\gamma , \\xi ) ) \\ , J ( \\gamma , 0 ) } { c ( x ( \\gamma , 0 ) ) \\ , J ( \\gamma , \\xi ) } } \\ , f ( \\gamma , T - \\xi ) . \\end{align*}"}
-{"id": "8980.png", "formula": "\\begin{align*} \\frac { \\mathcal { R } \\theta _ F ( - 1 / 4 z ) } { \\mathcal { R } \\theta _ F ( z ) } = \\left ( \\frac { 2 z } { \\sqrt { - 1 } } \\right ) ^ { m / 2 } . \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{align*} \\Gamma = a \\ , \\theta _ \\infty + O ( r ^ { \\delta } ) \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} \\bigl ( 1 , \\frac { 1 + \\sqrt { 5 } } { 2 } \\ , \\bigr ] ~ = ~ \\bigcup _ { n = 2 } ^ { \\infty } \\left [ \\ , \\theta _ { n + 1 } ^ { - 1 } , \\theta _ { n } ^ { - 1 } \\ , \\right ) ~ ~ \\bigcup ~ ~ \\left \\{ \\theta _ { 2 } ^ { - 1 } \\right \\} . \\end{align*}"}
-{"id": "6011.png", "formula": "\\begin{align*} \\phi ( y ) = N \\left \\{ { \\phi { } } _ 1 ( y ) + { \\phi { } } _ 2 ( y ) \\right \\} , \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} G _ j = \\sum _ { \\mathbf { m } \\in \\mathbb { N } ^ { 2 N } _ 0 } g _ j ^ { \\mathbf { m } } \\mathbf { y } ^ { \\mathbf { m } } . \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{gather*} A ' _ a = \\begin{pmatrix} 2 & - 1 & 0 \\\\ 1 & 0 & a \\\\ 0 & - 1 & 2 \\end{pmatrix} \\end{gather*}"}
-{"id": "3877.png", "formula": "\\begin{align*} f ( x ) = \\eta \\hat \\theta ^ \\top x + f _ { \\mathcal { K } } ( x ) , \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} f _ \\beta ^ \\delta ( x , \\lambda ) : = \\frac { 1 } { \\beta } \\log \\int _ { S _ \\delta } \\exp \\beta \\bigl ( x m + \\lambda m ^ 2 + J ( m ) \\bigr ) d \\mu . \\end{align*}"}
-{"id": "7542.png", "formula": "\\begin{align*} & \\int _ s ^ t ( L \\chi ) ( s , q _ r ^ m , z _ r ^ m ) d r \\\\ = & m ^ { 1 / 2 } ( R _ 1 ^ m ) _ { s , t } + m \\left ( \\chi ( t , q _ t ^ m , z _ t ^ m ) - \\chi ( s , q _ s ^ m , z _ s ^ m ) + ( R ^ m _ 2 ) _ { s , t } \\right ) , \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ K \\nu _ i \\psi ^ i ( x ) = 0 , \\textnormal { a . e . } x \\in M , \\forall \\ ; \\nu = ( \\nu _ 1 , \\cdots , \\nu _ K ) \\in ( T _ { \\phi ( x ) } N ) ^ { \\perp } \\subset T _ { \\phi ( x ) } \\R ^ K , \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{align*} p _ { n + 1 } = p _ n + R ( p _ n , b _ n , Q _ n , P _ n ) , b _ n \\cdot p _ n = 0 , \\end{align*}"}
-{"id": "3846.png", "formula": "\\begin{align*} g = \\frac { d \\rho ^ 2 + h _ \\rho } { \\rho ^ 2 } , \\psi ^ * g ' = \\frac { d \\rho ^ 2 + h ' _ \\rho } { \\rho ^ 2 } \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} w ( t _ i - t _ j ) = t _ { w ( i ) } - t _ { w ( j ) } . \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} \\tilde { U } = \\begin{pmatrix} U & 0 \\\\ 0 & 1 \\\\ \\end{pmatrix} , \\tilde { V } ^ \\top = \\begin{pmatrix} V ^ \\top & 0 \\\\ 0 & 1 \\\\ \\end{pmatrix} , \\tilde { \\Sigma } = \\frac { 1 } { \\sqrt { 2 \\varepsilon } } \\begin{pmatrix} \\Sigma & 0 \\\\ 0 & \\sqrt { \\frac { p '' } { 2 \\varepsilon } } \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "3154.png", "formula": "\\begin{align*} C _ { n , j } = \\frac { 1 } { n } \\displaystyle \\sum _ { i = 0 } ^ { n - 1 } X _ { i , j , n } ^ { 2 } , j \\geq 1 , \\ n \\geq 2 . \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} \\lambda ( t ) s ( e H ) & = t \\cdot s ( t ^ { - 1 } H ) \\\\ & = t \\cdot s ( e H ) \\\\ \\intertext { s i n c e $ t \\in H $ } & = ( 2 \\chi ) ( t ) s ( e H ) \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{align*} \\phi ( t ) = \\det ( \\gamma ' ( t ) , \\eta ( t ) ) . \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} g _ k ( n ) = \\max ( g _ k ( d ) , k n - ( k - 1 ) ( 2 ^ { e + 1 } - 1 ) ) , g _ k ( 0 ) = 0 . \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} \\Delta _ { \\beta _ { m + 1 } , \\beta _ { m - 1 } } & ( u _ m ^ 2 ) \\\\ = & \\ , ( m ) _ q ( 1 - q ^ { m - 1 } r ) ( u _ m x _ 1 + q ^ { m ( m - 1 ) } r ^ { m - 1 } s q _ { 2 1 } x _ 1 u _ m ) \\otimes u _ { m - 1 } \\\\ = & \\ , ( m ) _ q ( 1 - q ^ { m - 1 } r ) q ^ { m ( m - 1 ) } r ^ { m - 1 } s q _ { 2 1 } u _ { m + 1 } \\otimes u _ { m - 1 } . \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} \\vec { i } \\leq \\vec { j } \\iff \\begin{cases} i _ 1 \\leq j _ 1 , i _ 2 = j _ 2 \\textrm { o r } \\\\ i _ 2 \\leq j _ 2 . \\end{cases} \\end{align*}"}
-{"id": "7670.png", "formula": "\\begin{align*} ^ 2 _ { m , 2 } = & \\frac { \\frac { \\alpha _ 2 ^ 2 | h _ { m , m 2 } | ^ 2 } { L \\left ( | | y _ { m , 2 } | | \\right ) } } { ^ { m , 2 } _ { i n t e r } + \\frac { 1 } { \\rho } } . \\end{align*}"}
-{"id": "1287.png", "formula": "\\begin{align*} 1 + \\sqrt { 1 + c \\lambda _ i } - \\sqrt { 1 + c \\lambda _ j } - \\sqrt { 1 + c \\lambda _ k } = 0 . \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} P \\{ T _ 1 = T _ { \\min } \\} = v ( M _ 1 , \\dots , M _ g ) . \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} f ^ { - 1 } [ T _ { t ^ \\prime } ] = \\omega ^ * \\cap \\bigcup _ { B \\in \\mathcal A _ t } \\hat { B } ( t ) = \\omega ^ * \\cap \\bigcup _ { A \\in \\mathcal A _ s } \\bigcup _ { B \\in \\mathcal A ( t , A ) } \\hat { B } ( t ) \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} f ( x _ 1 , y _ 1 ) = \\sum _ { n = 0 } ^ \\infty \\lambda _ { 0 , n } \\sum _ { k = 0 } ^ n { n \\brack k } _ q ( a ; q ) _ k ( b ; q ) _ { n - k } x ^ k y ^ { n - k } . \\end{align*}"}
-{"id": "4667.png", "formula": "\\begin{align*} B = C \\# _ { 1 / ( 1 - s ) } A \\ . \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\Vert U _ { k } x ^ { k } - x ^ { k } \\Vert = 0 \\Longrightarrow \\lim _ { k \\rightarrow \\infty } d ( x ^ { k } , C ) = 0 \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} v _ p ( \\ell _ 1 ) - v _ p ( i ) = v _ p ( \\ell _ 2 ) . \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} \\overline \\square _ B | \\phi | ^ 2 & = - \\sum _ a \\nabla _ { V _ a } \\nabla _ { \\bar V _ a } | \\phi | ^ 2 + \\nabla _ { H ^ { 0 , 1 } } | \\phi | ^ 2 \\\\ & = \\langle \\nabla _ T ^ * \\nabla _ T \\phi , \\phi \\rangle + \\langle \\phi , \\bar \\nabla _ T ^ * \\bar \\nabla _ T \\phi \\rangle - \\sum _ { a = 1 } ^ n \\{ | \\nabla _ { \\bar V _ a } \\phi | ^ 2 + | \\nabla _ { V _ a } \\phi | ^ 2 \\} . \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{align*} \\Gamma ( 2 x ) = \\frac { 2 ^ { 2 x - 1 } } { \\sqrt { \\pi } } \\Gamma ( x ) \\Gamma \\left ( x + \\frac { 1 } { 2 } \\right ) , x \\neq 0 , - 1 , - 2 , \\ldots \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} m ( t _ { \\ast } ) \\| u \\| ^ { 2 } = \\int _ { \\Omega } f ( u ) u d x \\end{align*}"}
-{"id": "8629.png", "formula": "\\begin{align*} C _ { 2 } = \\bigl ( p ^ { e - 2 d - 1 } - ( p - 1 ) p ^ { m - d - 1 } - 1 \\bigr ) w _ { c } ^ { p ^ { e - 2 } + p ^ { m + d - 1 } } \\prod _ { \\substack { 0 \\leq i \\leq p - 1 \\\\ i \\ne c } } w _ { i } ^ { p ^ { e - 2 } } . \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{align*} M : = \\sup _ { \\alpha _ 1 } v ( i , j , \\hat { \\mu } _ { \\alpha , \\varepsilon } ) \\left [ e ^ { \\alpha _ 1 \\left ( \\left ( \\mu _ { \\alpha , \\varepsilon } ( j ) - \\hat { \\mu } _ { \\alpha , \\varepsilon } ( j ) \\right ) ^ - - \\left ( \\mu _ { \\alpha , \\varepsilon } ( i ) - \\hat { \\mu } _ { \\alpha , \\varepsilon } ( i ) \\right ) ^ - \\right ) + c _ { \\alpha , \\varepsilon } ( i , j ) } - 1 \\right ] < \\infty . \\end{align*}"}
-{"id": "3644.png", "formula": "\\begin{align*} \\tau _ { \\nu , \\nu } ^ 1 = ( 1 - q ^ { - 1 } ) \\dfrac { \\mathbf { z } ^ { - n \\lceil B ( \\alpha , \\nu ) / n \\rceil \\alpha } } { 1 - q ^ { - 1 } \\mathbf { z } ^ { - n \\alpha } } = \\dfrac { 1 - v } { 1 - v \\mathbf { z } ^ { - n \\alpha } } \\begin{cases} \\mathbf { z } ^ { - n \\alpha } & \\\\ 1 & \\end{cases} \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{align*} \\sigma \\cdot \\psi = - 2 f _ 1 I \\psi - 2 f _ 2 J \\psi - 2 f _ 3 K \\psi + \\bigl ( J _ 1 \\gamma ^ \\sharp \\bigr ) \\cdot \\psi . \\end{align*}"}
-{"id": "5366.png", "formula": "\\begin{align*} c _ j & = \\frac { \\Gamma ( p / 2 ) } { 2 ^ j \\Gamma ( j / 2 + 1 ) \\Gamma ( j / 2 + p / 2 ) } . \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} & ( e _ i ^ { \\pm } ) ^ 3 e _ j ^ { \\pm } - [ 3 ] _ q ( e _ i ^ { \\pm } ) ^ 2 e _ j ^ { \\pm } e _ i ^ { \\pm } + [ 3 ] _ q e _ i ^ { \\pm } e _ j ^ { \\pm } ( e _ i ^ { \\pm } ) ^ 2 - e _ j ^ { \\pm } ( e _ i ^ { \\pm } ) ^ 3 = 0 , i \\neq j . \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} \\lim _ { k } d ( y ^ { k } , C _ { i } ) = 0 \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} & \\lim _ { T \\to \\infty } \\frac { \\sqrt { T } } { 2 } \\P ^ { 0 , 0 } _ { 2 T , { \\rm c l } } ( X ( T s ) = \\sqrt { T } \\xi ) \\\\ & = \\frac { 1 } { \\sqrt { \\pi s ( 2 - s ) } } e ^ { - \\xi ^ 2 / \\{ s ( 2 - s ) \\} } , s \\in ( 0 , 2 ) , \\xi \\in \\R . \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} h \\leq \\varphi _ i ( y ) \\leq C _ 1 h , | \\nabla \\varphi _ i ( y ) | \\leq C _ 2 h \\quad y \\in [ 0 , 1 ] , \\ \\ i = 1 , 2 . \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} Q ( s , x ; t , y ) \\to \\binom { t - s } { \\{ ( t - s ) + ( y - x ) \\} / 2 } . \\end{align*}"}
-{"id": "537.png", "formula": "\\begin{align*} \\widehat { Z } _ n ( x , \\omega ) : = ( f _ { \\omega _ 1 } \\circ \\cdots \\circ f _ { \\omega _ n } ) ( x ) , \\qquad \\widehat { Z } _ 0 ( x , \\omega ) = x . \\end{align*}"}
-{"id": "4140.png", "formula": "\\begin{align*} { { \\cal L } _ { { I _ { f , r u } } } } \\left ( s \\right ) = { { { \\rm { E } } _ { { I _ { f , r u } } } } \\left [ { \\prod \\limits _ { { j } \\in { \\Phi _ f } } { \\exp \\left ( { - s { P _ f } { g _ { { j } } } l _ j ^ { - \\alpha } } \\right ) } } \\right ] } . \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{align*} l = | \\lbrace s ' \\in S _ \\mu \\mid s < s ' \\le t \\rbrace | = | S _ \\mu ( t ) | - | S _ \\mu ( s ) | . \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} \\begin{cases} \\eta ( 0 , t ) = 0 , \\ , \\ , \\eta ( L , t ) = 0 , \\ , \\ , \\eta _ { x } ( 0 , t ) = f ( t ) , & ( 0 , T ) , \\\\ w ( 0 , t ) = 0 , \\ , \\ , w ( L , t ) = 0 , \\ , \\ , w _ { x } ( L , t ) = g ( t ) . & ( 0 , T ) , \\end{cases} \\end{align*}"}
-{"id": "8437.png", "formula": "\\begin{align*} \\Omega _ j = \\{ x ^ 2 \\mid x \\in V _ j , \\ ; x ^ * = x \\} . \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} \\bar \\partial _ B Z \\lrcorner + Z \\lrcorner \\ , \\bar \\partial _ B = 0 . \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{align*} \\mathbf { a } _ { 1 } = \\partial _ { \\zeta } \\mathbf { r } _ { i j } , \\quad \\mathbf { a } _ { 2 } = \\partial ^ { 2 } _ { \\zeta } \\mathbf { r } _ { i j } , \\zeta = 0 \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{align*} ( L _ m ) _ { y , z } = \\begin{cases} 1 & , \\\\ 0 & , \\end{cases} & & ( R _ m ) _ { y , z } = \\begin{cases} 1 & , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "5332.png", "formula": "\\begin{align*} y \\ , = \\ , \\sigma ( w ^ T x + \\alpha ) + \\mbox { e r r o r } \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l l } g _ { B } ( X _ { j } , X _ { k } ) = \\epsilon _ { j } \\delta _ { j k } h ^ 2 , \\forall j , \\ k \\in \\{ 1 , \\ldots , n \\} , \\\\ \\noalign { \\smallskip } g _ { F } ( U _ { \\alpha } , U _ { \\gamma } ) = \\varepsilon _ { \\alpha } \\delta _ { \\alpha \\gamma } , \\ ; \\forall \\alpha , \\ \\gamma \\in \\{ 1 , \\ldots , m \\} , \\end{array} \\right . \\end{align*}"}
-{"id": "7994.png", "formula": "\\begin{align*} d y _ { i , t } = \\left [ - \\nabla f ( y _ { i , t } ) + \\sum _ { j = 1 , j \\neq i } ^ N \\alpha _ { i j } g ( y _ { i , t } - y _ { j , t } ) \\right ] \\tilde { \\gamma } d t + \\tau \\tilde { \\gamma } d B _ { i , t } , \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} & p _ { X Y Z } = p _ { X Y } \\ , p _ { Z | X Y } \\\\ & q _ { X Y Z } = q _ { X Y } \\ , p _ { Z | X Y } . \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} E = - \\frac { 1 } { 2 } e ^ { 2 \\alpha ( a ) } ( \\bar { z } _ 1 z _ 2 + z _ 1 \\bar { z } _ 2 ) \\alpha ^ { \\vee } + O \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} \\delta ( T _ j - T _ { j - 1 } ) = L _ { j } = ( \\delta + \\epsilon _ 1 ) ( T _ { j - 1 } - T _ { j - 2 } ) \\end{align*}"}
-{"id": "974.png", "formula": "\\begin{align*} \\forall \\ , \\rho > 0 : \\ \\dim \\ker \\big [ H ( \\mathfrak { e } , V ) + \\rho \\big ] \\ = \\ \\dim \\ker \\big [ B ( \\rho ) - 1 \\big ] . \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{align*} & \\theta ^ \\gamma _ { 0 } ( x , s ) = \\inf \\{ t \\ge s : \\ , \\Phi ^ { \\gamma } ( x , s ) = \\pi _ { j _ 0 } ( L _ { j _ 0 } ) \\} , \\\\ & \\theta ^ \\gamma _ { k } ( x , s ) = \\inf \\{ t \\ge \\theta ^ \\gamma _ { k - 1 } ( x , s ) : \\ , \\Phi ^ { \\gamma } ( x , s ) = \\pi _ { j _ k } ( L _ { j _ k } ) \\} , k \\geq 1 \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} \\frac { q ^ { n } - 1 } { q ^ { d p ^ i } - 1 } = 1 + q ^ { d p ^ i } + q ^ { 2 d p ^ i } + \\dots + q ^ { ( l - 1 ) d p ^ i } \\equiv l \\not \\equiv 0 \\pmod { p } , \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} \\left [ X , Y \\right ] _ { * } = \\frac 1 2 \\left ( \\left [ r X , Y \\right ] + \\left [ X , r Y \\right ] \\right ) , X , Y \\in { \\frak g } , \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} n ^ { ( 3 - \\tau ) ( 4 - 1 / ( \\tau - 1 ) ) / 2 + \\tfrac { 1 } { 2 } } = n ^ { 7 - 2 \\tau - \\frac { 1 } { \\tau - 1 } } . \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} P _ L ( \\boldsymbol { \\epsilon } ) \\cong \\prod _ { j = 1 } ^ { L } P ( \\epsilon _ j ) , \\end{align*}"}
-{"id": "3574.png", "formula": "\\begin{align*} w ' ( s ) ^ { 2 } + z ' ( s ) ^ { 2 } + \\kappa _ { s } ^ { 2 } t ^ { 2 } & = 1 \\end{align*}"}
-{"id": "4026.png", "formula": "\\begin{align*} A \\ ! = \\ ! \\max \\ ! \\sum _ { x , y } \\ ! ( \\mu ( x , y ) \\ ! - \\ ! \\gamma ( x , y ) ) \\log p _ { X Y } ( x , y ) \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{align*} \\Phi ( M , m \\ , | \\ , X , \\underline { k } ) ( y ) \\doteq \\prod \\limits _ { i = 1 } ^ { d } \\chi \\left ( 2 ^ { M + m } ( y _ { i } - 2 ^ { - M } X _ { i } - 2 ^ { - M - m } k _ { i } ) \\right ) . \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } \\Biggl [ ( a _ { i H } - a ^ 0 _ { i H } ) + \\sum _ { k = 1 } ^ { H - 1 } \\{ ( a _ { i k } - a _ { i H } ) b _ { k j } - ( a ^ 0 _ { i k } - a ^ 0 _ { i H } ) b ^ 0 _ { k j } \\} \\Biggr ] ^ 2 \\\\ & = \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } \\Biggl [ ( a _ { i H } - a ^ 0 _ { i H } ) + \\sum _ { k = 1 } ^ { H - 1 } ( a _ { i k } b _ { k j } { - } a ^ 0 _ { i k } b ^ 0 _ { k j } ) \\Biggr ] ^ 2 . \\end{align*}"}
-{"id": "30.png", "formula": "\\begin{align*} \\frac { d } { d \\theta } \\mathcal { P } _ u ( \\lambda _ \\theta , \\gamma _ \\theta ) \\Bigl | _ { \\theta = 0 } \\geq 0 \\end{align*}"}
-{"id": "1158.png", "formula": "\\begin{align*} \\lambda _ { \\alpha _ i } ( \\alpha _ j ^ * ) = - \\sum _ { k = 1 } ^ d s _ { j , k } ^ i \\alpha _ k ^ * \\ , . \\end{align*}"}
-{"id": "7452.png", "formula": "\\begin{align*} & F ( t , q ) \\equiv - \\partial _ t \\psi ( t , q ) - \\nabla _ q V ( t , q ) + \\tilde F ( t , q , \\psi ( t , q ) ) , \\\\ & \\frac { 1 } { 2 } \\left ( \\tilde \\gamma ^ { - 1 } - ( \\tilde \\gamma ^ T ) ^ { - 1 } \\right ) ^ { i j } F _ j = ( \\tilde \\gamma ^ { - 1 } ) ^ { k i } H _ { k \\ell } ( \\tilde \\gamma ^ { - 1 } ) ^ { \\ell j } F _ j , \\\\ & \\frac { 1 } { 2 } ( S - S ^ \\prime ) ^ i = \\beta ^ { - 1 } \\partial _ { q ^ j } \\left ( ( \\tilde \\gamma ^ { - 1 } ) ^ { k i } H _ { k \\ell } ( \\tilde \\gamma ^ { - 1 } ) ^ { \\ell j } \\right ) . \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} \\lambda _ { ( m - i , k ) } = & \\ , \\prod _ { j = 1 } ^ { m - i } ( 1 - q ^ { k - 1 + j } r ) \\frac { ( k + m - i ) _ q ^ ! } { ( k ) _ q ^ ! } , \\\\ { k + m + 1 \\choose i + 1 } _ { \\ ! \\ ! q } = & \\ , \\frac { ( k + m + 1 ) _ q ^ ! } { ( i + 1 ) _ q ^ ! ( k + m - i ) _ q ^ ! } , \\end{align*}"}
-{"id": "2132.png", "formula": "\\begin{align*} L _ { ( u _ 1 ) } ( x ) L _ { ( v _ 1 ) } ( x ) = L _ { ( u _ 1 v _ 1 ) } ( x ) + L _ { ( v _ 1 u _ 1 ) } ( x ) + \\begin{cases} L _ { ( u _ 1 ) ( x ) } & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "44.png", "formula": "\\begin{align*} \\int _ 0 ^ u g ( s ) \\alpha _ 0 ( s ) d s & = \\int _ 0 ^ { u ' } g ( s ) \\alpha _ 0 ( s ) d s + \\sum _ { n \\in \\mathbb { N } } \\int _ { I _ n } g ( s ) \\alpha _ 0 ( s ) d s = 0 , \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} ( p @ j ) : = \\left ( 0 , . . . , \\underset { j - 1 } { 0 } , \\underset { j } { p } , \\underset { j + 1 } { 0 } , . . . . , 0 \\right ) \\in \\mathbb { N } ^ { 2 N } . \\end{align*}"}
-{"id": "5203.png", "formula": "\\begin{align*} u _ t & \\le \\Delta u - \\chi \\nabla v \\cdot \\nabla u + u ( a _ { \\sup } - ( u _ \\infty - \\epsilon ) b _ { \\inf } ) \\end{align*}"}
-{"id": "2689.png", "formula": "\\begin{align*} \\varsigma ( m _ { N _ n } , m _ { N _ { n + 1 } } , \\Phi ( x ) , \\Phi ( y ) ) = \\sum _ { N _ n \\leq i < N _ { n + 1 } } \\varsigma ( m _ i , m _ { i + 1 } , \\Phi ( x ) , \\Phi ( y ) ) > \\sum _ { N _ n \\leq i < n _ { n + 1 } } \\left ( \\frac { 1 } { i + 1 } - 2 ^ { - i - 2 } \\right ) > 3 ^ { n + 2 } \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} \\forall j = 1 : N _ { s w } y _ j = \\mathcal { H } _ j ( x ^ t _ { j } ) + \\epsilon ^ o _ j , \\epsilon ^ o _ j \\sim \\mathcal { N } ( 0 , \\sigma _ o ^ 2 I _ { m _ j } ) \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} 3 k _ 3 ( a K _ { r + 1 } \\cup C ( b ) ) & - ( r - 2 ) m \\\\ & = 3 \\left ( a \\binom { r + 1 } { 3 } + \\binom { c } { 3 } + \\binom { d } { 2 } \\right ) - ( r - 2 ) \\left ( a \\binom { r + 1 } { 2 } + \\binom { c } { 2 } + d \\right ) \\\\ & = a \\binom { r + 1 } { 2 } + ( c - r ) \\binom { c } { 2 } + d \\left ( \\frac { 3 } { 2 } ( d - 1 ) - ( r - 2 ) \\right ) \\\\ & \\ge \\binom { r + 1 } { 2 } + ( c - r ) \\binom { c } { 2 } + d \\left ( \\frac { 3 } { 2 } ( d - 1 ) - ( r - 2 ) \\right ) \\\\ \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} \\lambda < \\frac { 3 T } { 2 \\pi r } = \\frac { 3 } { 2 \\sigma } , \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} I _ 3 ^ \\mathbb { Z } \\left ( \\overline { P _ 2 \\cup C _ 4 } , X _ { \\overline { P _ 2 \\cup C _ 4 } } \\right ) = \\langle x _ 1 + 1 , x _ 2 + 1 , x _ 3 + 1 , x _ 4 + 1 , x _ 5 , x _ 6 , 2 \\rangle , \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} \\frac { d } { d t } D + \\int _ \\Gamma 2 V _ s ^ 2 \\ , d s & = \\int _ \\Gamma 2 \\kappa ^ 2 V ^ 2 - \\big ( ( \\nabla f \\cdot n ) _ + + ( \\nabla f \\cdot n ) _ - \\big ) V ^ 2 \\ , d s . \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} [ g ] : = g _ { + } - g _ { - } . \\end{align*}"}
-{"id": "3550.png", "formula": "\\begin{align*} x & = A \\langle X , T \\rangle + B \\langle X , N \\rangle \\\\ y & = - N \\langle X , T \\rangle + A \\langle X , N \\rangle \\\\ x + i y & = ( A - i B ) ( \\langle X , T \\rangle + i \\langle X , N \\rangle ) , \\end{align*}"}
-{"id": "5508.png", "formula": "\\begin{align*} \\dot { \\mathbf { y } } = \\mathbf { V } ^ { - 1 } \\mathbf { A } \\mathbf { V } \\mathbf { y } + \\mathbf { V } ^ { - 1 } \\mathbf { G } _ { n l i n } ( \\mathbf { V } \\mathbf { y } ) = \\mathbf { \\Lambda } \\mathbf { y } + \\mathbf { G } ( \\mathbf { y } ) . \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} L ^ { \\mathrm { S I P } } f ( n ) = \\sum _ { 1 \\leq i < j \\leq N } n _ i ( 2 k _ j + n _ j ) \\left ( f ( n ^ { i , j } ) - f ( n ) \\right ) + n _ j ( 2 k _ i + n _ i ) \\left ( f ( n ^ { j , i } ) - f ( n ) \\right ) , \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} h _ { x } ^ { 2 } = ( \\sqrt { 1 + h _ { x } ^ { 2 } } - 1 ) ( \\sqrt { 1 + h _ { x } ^ { 2 } } + 1 ) . \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} \\max _ { k = 1 , \\dots , N } \\Big | \\phi ^ \\nu ( ( \\xi ^ 1 , \\dots , \\xi ^ \\nu ) , f _ k ^ \\star ) - \\phi ( f _ k ^ \\star ) \\Big | < \\gamma _ 2 . \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c c } A _ k X _ k B _ k - C _ k X _ { k + 1 } D _ k & = & E _ k , & k = 1 , \\ldots , r - 1 , \\\\ A _ r X _ r B _ r - C _ r X _ 1 D _ r & = & E _ r , \\end{array} \\right . \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} \\norm { u } _ { \\infty , T } : = \\max \\limits _ { ( t , x ) \\in ( 0 , T ) \\times ( 0 , D ) } | u ( t \\ , , x ) | . \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} \\binom { k + m - 1 } { k } < \\binom { J + m - 1 } { J } < 2 ^ { m + J - 1 } < 2 ^ { m + J } \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{align*} ( q - 1 ) [ z ^ n ] \\left ( h ( q , z ) - \\frac 1 2 h ( q ^ 2 , z ^ 2 ) \\right ) = \\sum _ { ( 2 l - 1 ) m = n } \\frac { q - 1 } { q ^ { 2 l - 1 } - 1 } \\frac { ( - 1 ) ^ { m - 1 } Q _ m ( q ^ { 2 l - 1 } ) } { ( 2 l - 1 ) q ^ { ( 2 l - 1 ) \\binom { d } { 2 } } } . \\end{align*}"}
-{"id": "3564.png", "formula": "\\begin{align*} \\gamma _ { t s } = \\gamma _ { s t } \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} \\delta _ t & = \\theta _ { s s } - \\delta \\delta _ s - \\sin \\theta + U \\theta _ t - T _ t . \\\\ ( I - K ^ * ) ( \\gamma _ t / 2 ) & = [ \\partial _ t , K ^ * ] ( \\gamma / 2 ) + \\theta _ { s s } - \\delta \\delta _ s - \\sin \\theta + U \\theta _ t . \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} \\dim P _ 1 \\mu + \\dim \\mu _ { [ x ] } = \\dim \\mu , P _ 1 \\mu x \\in P _ 1 ( F ) . \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} \\hat f ( r ) = \\dfrac { 1 } { \\abs { \\rho ^ { - 1 } ( r ) } } \\int _ { \\rho ^ { - 1 } ( r ) } f , \\end{align*}"}
-{"id": "1116.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\partial ^ \\bullet \\circ \\lambda _ s & = & \\lambda _ { \\partial ( s ) } + \\lambda _ s \\circ \\partial ^ \\bullet \\ , , \\\\ \\partial ^ \\bullet \\circ \\rho _ s & = & \\rho _ { \\partial ( s ) } + \\rho _ s \\circ \\partial ^ \\bullet \\ , . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{gather*} u ( t , x ) = \\widetilde u ( t , Y ( t , x ) ) , \\pi ( t , x ) = \\widetilde \\pi ( t , Y ( t , x ) ) , \\\\ a ' ( t ) = Q ( t ) \\widetilde \\ell ( t ) \\mbox { a n d } \\omega ( t ) = Q ( t ) \\widetilde \\omega ( t ) . \\end{gather*}"}
-{"id": "8361.png", "formula": "\\begin{align*} \\left | ( K v ) ( t ) - u _ 1 \\right | & = \\left | \\frac { \\lambda } { \\Gamma ( 2 \\alpha ) } \\int _ { \\beta } ^ { t } ( t - s ) ^ { 2 \\alpha - 1 } \\frac { f ( s , v ( s ) ) } { \\left ( \\int _ { 0 } ^ { t } f ( x , v ) \\ , d x \\right ) ^ { 2 } } d s \\right | \\\\ & \\leq \\frac { \\lambda M } { \\Gamma ( 2 \\alpha + 1 ) c _ { 1 } ^ { 2 } } h ^ { 2 \\alpha } \\leq b . \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} g ( y ) = 2 \\left ( e ^ { \\pi y } - 1 \\right ) ^ 2 - 4 y \\pi \\ , e ^ { \\pi y } \\left ( e ^ { \\pi y } - 1 \\right ) + \\pi ^ 2 \\ , y ^ 2 \\ , e ^ { \\pi y } \\left ( e ^ { \\pi y } + 1 \\right ) . \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} & \\limsup _ { L \\to \\infty \\atop L \\in \\N } \\left | \\frac { \\int _ 0 ^ { \\infty } d x x e ^ { L ^ d f _ L ( x ) } g _ L ( x ) u _ L ( x ) } { \\int _ { 0 } ^ { \\infty } d x x e ^ { L ^ d f _ L ( x ) } g _ L ( x ) } \\right | \\le \\lim _ { L \\to \\infty \\atop L \\in \\N } \\sup _ { x \\in [ 0 , 2 \\delta + a ] } | u _ L ( x ) | = \\sup _ { x \\in [ 0 , 2 \\delta + a ] } | u ( x ) | . \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} \\mathbb { E } ^ Q _ { n , 0 } \\left ( L _ n ^ 2 \\right ) - 1 = d ( n ) ^ { - 2 } \\sum _ { i = 1 } ^ { d ( n ) } \\sum _ { j = 1 } ^ { d ( n ) } \\mathbb { E } ^ Q _ { n , 0 } \\left ( e ^ { \\sqrt { n } a _ n ( z _ i + z _ j ) - n a _ n ^ 2 } \\right ) - 1 \\leq d ( n ) ^ { - 1 / 2 } \\to 0 . \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} \\sqrt { - 1 } \\partial \\bar { \\partial } \\log \\pi _ * \\Omega _ { ( X , D ) / Y } + \\sqrt { - 1 } \\partial \\bar { \\partial } v & = \\\\ & = \\omega _ Y - \\omega _ { W P } ^ D - ( 1 - \\beta ) c _ 1 ( N ) \\\\ \\end{align*}"}
-{"id": "278.png", "formula": "\\begin{align*} f ^ \\sigma ( x , y ) = f ( a x + b y , c x + d y ) . \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} \\tau _ j = \\frac { 1 } { 3 } ( h _ { j + 1 } - h _ j ) . \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} K _ { \\cal F _ { Y ' } } + \\pi '^ { - 1 } _ * \\Delta ' = g ^ * ( K _ { \\cal F _ Y } + \\pi _ * ^ { - 1 } ) + \\epsilon ( r - 1 ) E ' = \\\\ g ^ * ( \\pi ^ * ( K _ { \\cal F } + \\Delta ) + b E ) + \\epsilon ( r - 1 ) E ' \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{align*} \\dot { x } = { } & A x , x ( 0 ) = x _ { 0 } , \\\\ y = { } & ( - A ) ^ { \\frac { 1 } { 2 } } x \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} 2 [ B _ D , C _ D ] = z _ 2 \\bar { z } _ 1 \\alpha ( l _ j ) e ^ { 2 \\alpha ( a ) } \\theta ( e _ { \\alpha } ) + O , \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} P x & : = \\\\ & : = \\forall F ( - F x ) \\end{align*}"}
-{"id": "5843.png", "formula": "\\begin{align*} E _ { \\mu } ( z ; q , t ) & = f _ { \\mu } ( z ; q , t ) + \\sum _ { \\substack { \\nu \\in \\sigma ( \\mu ) \\\\ \\nu \\prec \\mu } } c _ { \\mu , \\nu } ( q , t ) f _ { \\nu } ( z ; q , t ) , \\\\ f _ { \\mu } ( z ; q , t ) & = E _ { \\mu } ( z ; q , t ) + \\sum _ { \\substack { \\nu \\in \\sigma ( \\mu ) \\\\ \\nu \\prec \\mu } } d _ { \\mu , \\nu } ( q , t ) E _ { \\nu } ( z ; q , t ) , \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} \\frac { z } { 1 - \\eta _ \\mu ( z ) } = G _ \\mu \\left ( \\frac { 1 } { z } \\right ) = \\int _ 0 ^ \\infty \\frac { 1 } { 1 / z - x } \\ , d \\mu ( x ) , \\end{align*}"}
-{"id": "2328.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 x ^ { \\lambda - 1 } \\left ( 1 - x \\right ) ^ { \\nu _ 1 M } d x = \\frac { \\Gamma ( \\lambda ) \\ , \\Gamma ( \\nu _ 1 M + 1 ) } { \\Gamma ( \\lambda + \\nu _ 1 M + 1 ) } \\sim \\frac { \\Gamma ( \\lambda ) } { \\nu _ 1 ^ { \\lambda } } \\cdot \\frac { 1 } { M ^ { \\lambda } } , M \\to \\infty . \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} 1 = \\lim _ { n \\to \\infty } \\mathbb { E } _ { n , d ( n ) , \\theta _ n } ( \\nu _ n ) > \\liminf _ { n \\to \\infty } \\mathbb { E } _ { n , d ( n ) , \\theta _ n } ( \\varphi _ n ) . \\end{align*}"}
-{"id": "6396.png", "formula": "\\begin{align*} \\frac { v _ A } { x _ A } + \\frac { v _ { A B } } { x _ { A B } } = \\frac { v _ B } { x _ B } + \\frac { v _ { A B } } { x _ { A B } } , \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{align*} D _ { n , j } = D _ { n , j , j } ^ { * } = \\langle D _ n \\left ( \\phi _ { n , j } \\right ) , \\phi _ { n , j } \\rangle _ H = \\frac { 1 } { n - 1 } \\displaystyle \\sum _ { i = 0 } ^ { n - 2 } X _ { i , j , n } X _ { i + 1 , j , n } , j \\geq 1 , \\ n \\geq 2 , \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} b _ 1 ( \\omega ) = y ( \\omega + \\omega _ { 3 } ) ( x ( \\omega ) - x ( \\omega + \\omega _ { 3 } ) ) b _ 2 ( \\omega ) = x ( \\omega ) ( y ( \\omega ) - y ( - \\omega ) ) . \\end{align*}"}
-{"id": "3743.png", "formula": "\\begin{align*} f ^ { ( i ) } _ j ( x ) = \\lambda _ i x + t ^ { ( i ) } _ j . \\end{align*}"}
-{"id": "5815.png", "formula": "\\begin{align*} L _ i [ \\psi ( \\cdot , \\mu ) ] ( \\nu ) = M _ i [ \\psi ( \\nu , \\cdot ) ] ( \\mu ) , \\forall \\ i \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "3949.png", "formula": "\\begin{align*} \\nu ( x _ 1 , x _ 2 ) & = p _ X ( x _ 1 ) q _ X ( x _ 2 ) \\\\ \\omega ( x _ 1 , x _ 2 ) & = q _ X ( x _ 1 ) p _ X ( x _ 2 ) . \\end{align*}"}
-{"id": "8308.png", "formula": "\\begin{align*} & \\mathcal { C } _ { - 1 } ( \\alpha , \\nu , q + w ' , \\lbrace 1 , \\ldots , q - 1 \\rbrace , \\lbrace q , \\ldots , \\ell \\rbrace \\setminus \\lbrace q + w ' \\rbrace ) \\\\ & < \\mathcal { C } _ { - 1 } ( \\alpha , \\nu , q , \\lbrace 1 , \\ldots , q - 1 \\rbrace , \\lbrace q + 1 , \\ldots , \\ell \\rbrace ) \\\\ & = \\nu _ q + \\ell - 2 q + 1 \\\\ & = \\mu ^ { \\circ } + \\ell - 2 q + 1 , \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} \\alpha = \\alpha _ 0 - \\frac { 1 } { 2 } ( 3 T + 1 ) , \\beta = \\beta _ 0 - \\frac { 1 } { 2 } ( 3 T + 1 ) , \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} \\sum _ { \\vec { z } + q \\vec { y } \\in P \\mathcal { B } } e ( \\vec { \\beta } \\cdot \\vec { f } ( \\vec { z } + q \\vec { y } ) ) = I ( \\mathcal { B } , P ^ d \\vec { \\beta } ) \\frac { P ^ n } { q ^ n } + O \\left ( \\left ( C | P ^ d \\vec { \\beta } | + 1 \\right ) \\frac { P ^ { n - 1 } } { q ^ { n - 1 } } \\right ) . \\end{align*}"}
-{"id": "162.png", "formula": "\\begin{align*} ( \\alpha _ t , \\varphi _ t ) : = ( \\beta _ { \\infty } , 0 ) - D _ t ^ 1 \\xi _ t \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{align*} F ( q , t , w ) : = A w - Q [ a \\left ( x \\right ) \\{ ( t \\phi _ { 1 } + w ) ^ { q } - ( t \\phi _ { 1 } + w ) \\} ] , \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} \\psi _ z ( \\lambda ) : = \\tfrac { \\varphi ( 0 ) - \\varphi ( \\lambda ) } { \\lambda } \\bullet h ( \\lambda ) + \\tfrac { h ( \\lambda ) - h ( 0 ) } { \\lambda } \\bullet ( z - \\varphi ( 0 ) ) + \\lambda \\ , \\overline { h ( 0 ) \\bullet ( z - \\varphi ( 0 ) ) } . \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} \\tilde { A } ( \\varepsilon ) ^ \\top \\tilde { Q } + \\tilde { Q } \\tilde { A } ( \\varepsilon ) + \\tilde { c } \\tilde { c } ^ \\top + \\tilde { H } ^ { ( 2 ) } ( \\tilde { P } \\otimes \\tilde { Q } ) ( \\tilde { H } ^ { ( 2 ) } ) ^ \\top + \\displaystyle \\sum _ { j = 1 } ^ { n _ { \\rm i n } } \\tilde { N } _ j ^ \\top \\tilde { Q } \\tilde { N } _ j = 0 . \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{align*} Y = \\sum _ { l = 1 } ^ g a _ { j l } \\mathcal { P } ( B ' _ l ) \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} v a l ( e ) = d + 1 - v a l ( e ' ) , \\ \\ \\ r ( e ) = r + 1 - r ( e ' ) , \\ \\ \\ c ( e ) = w ( d ) - d + 1 - c ( e ' ) . \\end{align*}"}
-{"id": "1932.png", "formula": "\\begin{align*} a _ k ( x ) & = x a _ { k - 1 } ( x ) , k \\geq 4 , \\\\ b _ k ( x ) & = x ( a _ { k - 1 } ( x ) + b _ { k - 1 } ( x ) ) , k \\geq 4 , \\\\ c _ k ( x ) & = x c _ k ( x ) + x \\sum _ { j \\geq k } ( c _ j ( x ) + b _ j ( x ) + a _ j ( x ) ) , k \\geq 3 , \\\\ d _ k ( x ) & = x d _ k ( x ) + x b _ k ( x ) + x c _ { k + 1 } ( x ) + x \\sum _ { j \\geq k } d _ j ( x ) , k \\geq 2 , \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\| \\psi ( s ) \\| _ 3 d s & \\le \\ ( \\int _ 0 ^ t s ^ { b - ( 1 / 2 ) } d s \\ ) ^ { 1 / 2 } \\ ( \\int _ 0 ^ t s ^ { ( 1 / 2 ) - b } \\| \\psi ( s ) \\| _ 3 ^ 2 d s \\ ) ^ { 1 / 2 } \\\\ & = o ( t ^ { ( b + ( 1 / 2 ) ) / 2 } ) , \\end{align*}"}
-{"id": "5137.png", "formula": "\\begin{align*} 2 ( q - 1 ) = i ( n - t + 2 ) + j , \\quad \\ 0 \\le j \\le n - t + 1 . \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} \\phi \\left ( 2 ^ { \\beta + \\alpha ( m - 1 ) } A \\right ) = 2 ^ { \\beta + \\alpha ( n - 1 ) } B . \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} \\psi \\left ( W ^ g _ { k } ( p , p _ 1 , \\dots , p _ { k - 1 } ) \\right ) = \\sum _ { t ^ g \\in ( Y ^ n ) ^ g } t ^ g \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } ( 1 - \\tilde { p } _ n ) ^ { \\frac { 1 } { n } } = \\left ( \\frac { p _ { 1 2 } p _ { 2 1 } } { p _ { 1 1 } p _ { 2 2 } } \\right ) ^ { \\frac 1 2 } . \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} \\gamma ( \\varepsilon ) = q ^ { | \\varepsilon | ( 1 - N ) / 2 } \\prod _ { m \\in T _ \\varepsilon } q ^ { ( d _ { m + 1 } + \\cdots + d _ N ) / 2 } . \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} [ g _ 1 , g _ 2 , \\dots , g _ n ] = 1 \\mbox { f o r a l l } g _ i \\in G \\iff [ x _ 1 , x _ 2 , \\dots , x _ n ] = 1 \\mbox { f o r a l l } x _ i \\in X . \\end{align*}"}
-{"id": "5210.png", "formula": "\\begin{align*} U ( x , t _ 0 + t ; t _ 0 , u _ 0 ) & = \\int _ { \\R ^ N } \\Gamma ( x , t , y , 0 ) u _ 0 ( y ) d y . \\end{align*}"}
-{"id": "8528.png", "formula": "\\begin{align*} \\phi _ { V _ N } ( \\mathbf { u } ) - \\phi _ { V _ S } ( \\mathbf { u } ) = \\phi _ { V _ N V _ S } ( \\zeta ) + \\overline { \\phi _ { V _ N V _ S } ( \\zeta ) } \\rlap { . } \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{align*} | \\tau ( x ) - \\tau ( y ) | \\leq 2 C _ 2 = \\frac { 2 C _ 2 } { r } \\cdot r \\leq \\frac { 2 C _ 2 } { r } \\cdot | x - y | \\ , . \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{align*} & F ( q , t , w ) = 0 , \\ ( q , t , w ) \\simeq ( 1 , t _ { 0 } , 0 ) \\\\ & \\Longleftrightarrow w = w ( q , t ) , \\ ( q , t ) \\simeq ( 1 , t _ { 0 } ) \\mbox { a n d } w ( 1 , t _ { 0 } ) = 0 . \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\tau ^ { \\nu - 1 } e ^ { - ( \\frac { \\beta } { \\tau } + \\gamma \\tau ) } d \\tau = 2 \\left ( \\frac \\beta { \\gamma } \\right ) ^ { \\frac \\nu { 2 } } K _ \\nu ( 2 \\sqrt { \\beta \\gamma } ) , \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} q ( t ) = \\frac { t } { \\sigma ( E ) } + q _ a t \\in \\R , \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} \\lim _ { y \\to x } { \\frac { F ( y ) - F ( x ) - f ( x ) ( y - x ) } { \\varphi ( y ) - \\varphi ( x ) } } = 0 . \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} \\overline { \\Lambda } _ 2 = & \\{ ( s , v ) \\in \\R ^ 2 : 0 \\leq s \\leq 1 , - s \\leq v \\leq s \\} \\\\ & \\cup \\{ ( s , v ) \\in \\R ^ 2 : 1 < s \\leq 2 , - ( 2 - s ) \\leq v \\leq 2 - x \\} . \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{align*} \\omega _ j ( g ) = \\sum _ { i = 1 } ^ { n _ j } \\langle u _ i , \\pi _ j ( g ) u _ i \\rangle ( g \\in G ) . \\end{align*}"}
-{"id": "4960.png", "formula": "\\begin{align*} q _ 1 & = t _ 1 , & , \\\\ q _ 3 & = - 3 \\cdot t _ 3 , & q _ 9 & = \\textstyle 9 9 2 5 5 \\cdot ( t _ 9 - \\frac { 1 } { 3 } t _ 3 ) , \\\\ q _ 5 & = 4 5 \\cdot t _ 5 , & q _ { 1 1 } & = \\textstyle - 9 8 2 3 2 7 5 \\cdot ( t _ { 1 1 } - t _ 3 ^ 2 t _ 5 ) , \\\\ q _ 7 & = - 1 5 7 5 \\cdot t _ 7 , & q _ { 1 3 } & = \\textstyle 1 4 0 4 7 2 8 3 2 5 \\cdot ( t _ { 1 3 } - t _ 3 ^ 2 t _ 7 - t _ 3 t _ 5 ^ 2 ) . \\end{align*}"}
-{"id": "5745.png", "formula": "\\begin{align*} \\mathrm { N m } _ { K / L } ( x ) = N _ { \\Phi , K , L \\otimes E } ( x ) N _ { \\Phi ^ c , K , L \\otimes E } ( x ) . \\end{align*}"}
-{"id": "3529.png", "formula": "\\begin{align*} \\rho _ { X _ { 0 } } ( x , t ) = ( 4 \\pi ( t _ { 0 } - t ) ) ^ { - 1 / 2 } e ^ { - \\frac { | x - x _ { 0 } | ^ { 2 } } { 4 ( t _ { 0 } - t ) } } \\end{align*}"}
-{"id": "5948.png", "formula": "\\begin{align*} \\frac { { R ^ { - 1 } + ( R ^ { - 1 } ) } ^ T } { 2 } & = \\omega ^ { - \\frac { 1 } { 2 } } \\left ( E - \\sigma ^ 2 \\right ) ^ { - 1 } \\omega ^ { - \\frac { 1 } { 2 } } , \\\\ \\frac { { R ^ { - 1 } - ( R ^ { - 1 } ) } ^ T } { 2 } & = \\omega ^ { - \\frac { 1 } { 2 } } ( - \\sigma ) \\left ( E - \\sigma ^ 2 \\right ) ^ { - 1 } \\omega ^ { - \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{align*} \\lim _ { k } \\sum _ { j = 0 } ^ { p - 1 } \\Vert U _ { j + 1 } ^ { k } x _ { j } ^ { k } - x _ { j } ^ { k } \\Vert = 0 . \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{gather*} \\Delta ^ + = \\big \\{ \\beta _ 1 , \\beta _ 2 , \\beta _ 3 , \\beta _ 1 + \\beta _ 2 , \\beta _ 2 + \\beta _ 3 , \\beta _ 3 + \\beta _ 1 , \\beta _ 1 + \\beta _ 2 + \\beta _ 3 \\big \\} . \\end{gather*}"}
-{"id": "1917.png", "formula": "\\begin{align*} & \\quad + \\sum _ { n \\geq 0 } \\left ( \\sum _ { n - 1 > i _ 1 > i _ 2 > \\cdots > i _ { m } \\geq 1 } a ( n ; i _ 1 i _ 2 \\cdots i _ m ( n - 1 ) n ) \\right ) x ^ n \\qquad \\\\ & = B _ { m + 1 } ( x ) + x ^ { m + 3 } \\sum _ { i = 2 } ^ m C ^ i ( x ) + x B _ m ( x ) + x ^ { m + 2 } \\frac { 1 - x } { 1 - 2 x } , \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } g _ { n - k } ( T ^ k x ) = \\overline { g } _ \\infty ( x ) , \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{align*} m _ { 2 , 1 } = \\frac { m _ { 1 , 1 } a _ { 1 , 5 } ( 1 - m _ { 1 , 1 } ) } { 2 a _ { 1 , 4 } ^ 2 } . \\end{align*}"}
-{"id": "7206.png", "formula": "\\begin{align*} ( a _ 1 , a _ 2 , . . . , a _ m ; q ) _ n = ( a _ 1 ; q ) _ n ( a _ 2 ; q ) _ n . . . ( a _ m ; q ) _ n , \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} S _ { p + m _ 1 } ( \\delta ^ + ) _ { m _ 1 + m _ 2 + 1 } = m _ 1 + m _ 2 + 1 , \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} H ^ { 0 , 1 } \\lrcorner \\ , \\phi = H ^ { 0 , 1 } \\lrcorner \\ , \\partial _ B \\phi = 0 , \\end{align*}"}
-{"id": "8142.png", "formula": "\\begin{align*} g ( z ) = a \\left ( 1 - \\frac { z } { \\lambda } \\right ) . \\end{align*}"}
-{"id": "8365.png", "formula": "\\begin{align*} \\pi _ i ( f ) = \\frac { ( 1 + \\beta x _ { i + 1 } ) f - ( 1 + \\beta x _ i ) s _ i f } { x _ i - x _ { i + 1 } } \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} - \\frac 1 2 \\Delta _ B | \\phi | ^ 2 = & \\sum _ a \\{ | \\nabla _ { \\bar V _ a } \\phi | ^ 2 + | \\nabla _ { V _ a } \\phi | ^ 2 \\} + \\langle \\phi , \\sum _ a R ^ Q ( \\bar V _ a , V _ a ) \\phi \\rangle \\\\ & + \\frac 1 2 \\sum _ a \\{ \\langle \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi , \\phi \\rangle + \\langle \\phi , \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi \\rangle \\} . \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} N ( y , k ) \\left | \\beta ( z ) \\right | = N ( y , k ) \\left | F ( z ) - h _ { \\ell } ( z ) \\right | < N ( y , k ) \\left | F ( z ) \\right | + N ( y , k ) \\left | h _ { \\ell } ( z ) \\right | \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} & { \\prod _ { j = 1 } ^ t x _ j ^ { - \\tau + \\zeta _ j } \\prod _ { i = t + 1 } ^ s h _ { p ^ * _ i } ( i , \\boldsymbol { x } ) } \\\\ & = \\tilde { K } \\prod _ { j = 1 } ^ { t } x _ j ^ { - \\tau + \\zeta _ j + | Q _ j | } \\prod _ { i = t + 1 } ^ { s } x _ { j _ { p ^ * _ i } } ^ { \\tau - 1 - \\zeta _ i - d _ { i , \\bar { U } } - p ^ * _ i , } \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{align*} \\dim ( \\Sigma ( s , \\theta ) \\cap \\Xi ( e _ 1 , e _ 2 ) ) = 1 . \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} & \\lim _ { L \\to \\infty \\atop L \\in \\N } \\sup _ { x \\in [ - r , r ] } \\left | \\frac { d ^ { n } } { d x ^ n } f _ L ( x ) - \\frac { d ^ { n } } { d x ^ n } f ( x ) \\right | = 0 , ( \\forall n \\in \\{ 0 , 1 , \\cdots , n _ 0 \\} ) , \\\\ & \\lim _ { L \\to \\infty \\atop L \\in \\N } \\sup _ { x \\in [ - r , r ] } | u _ L ( x ) - u ( x ) | = 0 . \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} \\theta _ 2 ( x ) \\theta _ 3 ( x ) \\theta _ 4 ( x ) = \\theta _ 1 ^ + ( x ) . \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} \\theta ( \\alpha q ^ { \\ell } ; p ) = \\prod _ { j = 0 } ^ { \\infty } ( 1 - p ^ j \\alpha q ^ { \\ell } ) ( 1 - p ^ { j + 1 } / ( \\alpha q ^ { \\ell } ) ) , \\ell = 0 , 1 , 2 , \\dots , \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} x _ N ( t ) = \\tilde x _ N ( t ) = x _ { N ( u ) } ( u ) + \\left [ \\tilde x _ { N } ( t ) - \\tilde x _ { N ( u ) } ( u ) \\right ] . \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} E = \\hat { C } \\setminus \\bigcup \\left \\{ \\hat { A } : A \\in \\mathcal A \\right \\} \\neq \\emptyset . \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } y _ { k ( l ) } = y ^ * \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} \\Lambda ^ 2 = \\Lambda ^ + \\oplus \\Lambda ^ - . \\end{align*}"}
-{"id": "8255.png", "formula": "\\begin{align*} K ^ { \\rm r e s c } _ t ( s _ 1 , s _ 2 ) : = \\sigma t ^ { 1 / 2 } \\widetilde K _ { n , t } ( x ( s _ 1 ) , x ( s _ 2 ) ) \\end{align*}"}
-{"id": "7736.png", "formula": "\\begin{align*} J q : = S q _ { \\mathbb { Z } _ 2 } \\ \\ J q ^ k : = S q ^ k _ { \\mathbb { Z } _ 2 } , \\end{align*}"}
-{"id": "2638.png", "formula": "\\begin{align*} \\rho _ 1 + b h = \\tilde { c } \\varphi - \\rho _ 2 = \\tilde { b } . \\end{align*}"}
-{"id": "2799.png", "formula": "\\begin{align*} A ( n ) = \\sum _ { d | n , ( d , 2 N _ 1 ) = 1 } \\chi _ { 1 } ( d ) d ^ { k _ 1 - 1 } a \\left ( \\frac { t n ^ 2 } { d ^ 2 } \\right ) , \\end{align*}"}
-{"id": "8518.png", "formula": "\\begin{align*} F _ + ( \\phi _ { n - 1 } ) = F _ 0 ( \\phi _ n ) = F _ - ( \\phi _ { n + 1 } ) \\rlap { . } \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} \\tilde \\theta _ { E , \\omega } ( z ) \\Big ( { 1 \\over z } + \\sum _ { \\alpha \\geq 0 } \\mathrm { c a n } ( f _ \\alpha ) z ^ \\alpha \\Big ) = e ^ { - t z } { \\tilde \\theta _ { E , \\omega } ( p + z ) \\over \\tilde \\theta _ { E , \\omega } ( p ) } . \\end{align*}"}
-{"id": "4805.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\int _ { \\Omega } f ( u _ { \\epsilon } - u ) d x = 0 . \\end{align*}"}
-{"id": "5172.png", "formula": "\\begin{align*} \\mathcal { B } ' _ { k _ 1 , i _ 1 } = \\{ f _ { p _ { 1 , j } q _ { 1 , j } } , 1 \\leq j \\leq l _ 1 \\} , \\ \\mathcal { B } ' _ { k _ 2 , i _ 2 } = \\{ f _ { p _ { 2 , j } q _ { 2 , j } } , 1 \\leq j \\leq l _ 2 \\} . \\end{align*}"}
-{"id": "7378.png", "formula": "\\begin{align*} \\widetilde { u } ( x ) : = \\sin ( x _ 1 ) \\psi ( x ) \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} g _ \\infty = - \\ : 1 6 d u ^ 2 + u ^ 2 d \\theta ^ 2 + 4 c _ \\lambda ^ { - 1 } \\left [ d \\widehat { \\sigma } ^ 2 + d \\widehat { \\delta } ^ 2 \\right ] . \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} \\sum _ { a ' \\in \\mathfrak { a } ' } a ' = \\frac { ( N - 1 ) ( N - 2 | \\mu | ) } { 2 } . \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} [ - 1 ] ^ * W _ { - \\theta } = W _ { - \\theta } . \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} w _ i = \\sum _ { m = 1 } ^ i ( - 1 ) ^ m \\sum _ { \\substack { ( j _ 1 , j _ 2 , \\dots , j _ m ) \\\\ j _ 1 + \\dots + j _ m = i } } b _ { j _ 1 } \\cdots b _ { j _ m } . \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{align*} \\lambda ( I ) \\geq \\frac \\alpha { n } \\geq \\alpha \\frac { 1 - \\varepsilon } { 1 + \\varepsilon } = \\frac { \\delta } { 2 - \\delta / \\alpha } \\geq \\frac { \\delta } { 2 } . \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{align*} \\overline { \\sigma } _ j = - \\left ( \\log { r } \\right ) \\ , \\tau _ j + \\sum _ { i = k + 1 } ^ { b } { \\alpha ^ i _ j \\ , \\tau _ i } + O ( r ^ { - 2 - \\mu } ) \\end{align*}"}
-{"id": "3879.png", "formula": "\\begin{align*} P _ \\theta ( x ) : = \\exp ( - \\theta ^ \\top x - A ( \\theta ) ) A ( \\theta ) : = \\ln \\int _ K \\exp ( - \\theta ^ \\top x ' ) d x ' , \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} A _ { 2 ^ n } & = \\sum _ { i = 0 } ^ { 2 ^ n - 1 } a _ { 2 ^ n - i } J q ^ { 2 ^ n - i } J q ^ i \\\\ & = a _ { 2 ^ n } J q ^ { 2 ^ n } + \\sum _ { i = 1 } ^ { 2 ^ n - 1 } a _ { 2 ^ n - i } J q ^ { 2 ^ n - i } J q ^ i = 0 \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{align*} \\lim _ { R \\to + \\infty } f ( R ) = \\lim _ { R \\to + \\infty } f ( R ) = 0 , \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} \\alpha _ { 1 1 } = \\frac { \\theta _ { 1 2 } \\theta _ { 5 } } { \\theta _ { 3 3 } \\theta _ { 4 0 } } , \\alpha _ { 2 1 } = \\frac { \\theta _ { 2 7 } \\theta _ { 5 } } { \\theta _ { 5 4 } \\theta _ { 4 0 } } , \\alpha _ { 3 1 } = - \\frac { \\theta _ { 1 2 } \\theta _ { 2 7 } } { \\theta _ { 3 3 } \\theta _ { 5 4 } } , \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} g = { \\frak t } ^ { - \\ : \\frac 1 2 } e ^ { \\frac 1 2 \\lambda } ( - \\ : d { \\frak t } ^ 2 + d \\theta ^ 2 ) + { \\frak t } \\left [ e ^ P ( d \\sigma + Q d \\delta ) ^ 2 + e ^ { - \\ : P } d \\delta ^ 2 \\right ] , \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} d _ { A ( t ) } \\int _ 0 ^ t \\psi ( s ) d s = \\hat w ( 0 ) - \\hat w ( t ) + \\int _ 0 ^ t \\gamma ( t , s ) d s \\ \\ \\ 0 < t < \\infty , \\end{align*}"}
-{"id": "9199.png", "formula": "\\begin{align*} [ \\exp ^ { - 1 } , P _ { - n } ] = ( 1 - q ^ { - r n } ) \\exp ^ { - 1 } \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} U _ { \\beta } ( \\omega _ { 1 , n } ) = \\frac { P ' _ { \\beta } ( \\omega _ { 1 , n } ) } { f ' _ { \\beta } ( \\omega _ { 1 , n } ) } . \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} n + k - 1 & = n + z + ( k - z - 1 ) \\\\ & \\le b ( z + 1 ) + 1 + \\left ( k - ( z + 1 ) \\right ) \\\\ & \\le b ( z + 1 + k - ( z + 1 ) ) + 1 \\\\ & = b ( k ) + 1 . \\end{align*}"}
-{"id": "1218.png", "formula": "\\begin{align*} y _ \\xi ^ \\bot ( x ( \\gamma , \\xi + 0 ) ) = y ( x ( \\gamma , \\xi ) ) \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} M _ { \\epsilon , t } = \\sum _ { | I | \\leq n / 2 - 1 / 2 } \\frac { ( - 1 ) ^ { k + l } } { 2 ^ k } \\mathcal N _ { \\epsilon , I } ( t ) + \\phi _ \\epsilon ( t ) , \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} q \\frac { d \\hat { \\varphi } } { d q } = v \\circ \\hat { \\varphi } \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} S '' ( i , j , k , \\alpha , \\beta ) : = \\omega _ { i j } ^ \\alpha \\underline \\wedge \\omega _ { j k } ^ \\beta ( i < j < k \\in [ n ] , \\alpha , \\beta \\geq 0 ) , \\end{align*}"}
-{"id": "789.png", "formula": "\\begin{align*} U _ { \\beta } ( \\omega _ { j , n } ) ~ = ~ \\frac { P ' _ { \\beta } ( \\omega _ { j , n } ) } { f ' _ { \\beta } ( \\omega _ { j , n } ) } \\neq 0 \\mbox { a n d } U _ { \\beta } ( \\frac { 1 } { \\beta } ) ~ = ~ \\frac { P ' _ { \\beta } ( \\frac { 1 } { \\beta } ) } { f ' _ { \\beta } ( \\frac { 1 } { \\beta } ) } \\neq 0 . \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} R ( a , b ) = S q ^ a S q ^ b - \\sum _ { j = 0 } ^ { [ a / 2 ] } \\binom { b - j - 1 } { a - 2 j } S q ^ { a + b - j } S q ^ j , \\end{align*}"}
-{"id": "3144.png", "formula": "\\begin{align*} { } \\frac { \\partial S } { \\partial \\alpha } = 0 = N - \\frac { 1 } { \\beta } \\int _ { 0 } ^ { \\beta B } \\frac { d x } { 1 + e ^ { \\alpha + x } } \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} V _ 0 ^ { 1 / 2 } ( F _ { h _ 1 } - F _ { h _ 2 } ) & = V _ 0 ^ { 1 / 2 } \\left ( \\frac { 1 } { 2 } r \\cdot g ( 6 ( x + h _ 1 ) - 1 ) - g ( 6 ( x + h _ 2 ) - 1 ) \\right ) \\\\ & = \\frac { 1 } { 2 } r \\cdot V _ 0 ^ { 1 / 2 } ( g ( 6 ( x + h _ 1 ) - 1 ) - g ( 6 ( x + h _ 2 ) - 1 ) ) \\\\ & = \\frac { 1 } { 2 } r \\cdot V _ { - 1 } ^ 2 ( g ( x + h _ 1 ) - g ( x + h _ 2 ) ) \\\\ & = \\frac { 1 } { 2 } r \\cdot V _ { - 1 + h _ 2 } ^ { 1 + h _ 2 } ( g _ { h _ 1 - h _ 2 } - g ) \\\\ & = \\frac { 1 } { 2 } r \\cdot 2 = r . \\end{align*}"}
-{"id": "7164.png", "formula": "\\begin{align*} \\nu _ { f , g } ^ \\epsilon ( \\vec { t } ) = \\int _ E \\int _ { \\Phi _ y ^ { - 1 } ( \\vec { t } ) } \\ , \\mu _ f ^ \\epsilon ( x ) \\ , \\mu _ g ^ \\epsilon ( z ) \\ , \\frac { 1 } { | J _ 2 \\Phi _ y | } \\ , d \\mathcal { H } ^ { 2 d - 2 } ( x , z ) \\ , d \\mu ( y ) . \\end{align*}"}
-{"id": "4372.png", "formula": "\\begin{align*} | z ^ { ( 0 ) } ( \\xi ) | = \\int _ { 0 } ^ { \\infty } \\frac { d t } { 2 ( t - \\xi ) \\sqrt { t + 1 - \\xi } } \\leq \\int _ { 0 } ^ { \\infty } \\frac { d t } { 2 ( t - \\xi ) ^ { \\frac 3 2 } } \\leq 1 . \\end{align*}"}
-{"id": "7400.png", "formula": "\\begin{align*} \\forall v , w \\in H ^ 1 _ { \\rm p e r } , a _ q ^ V ( v , w ) : = \\int _ \\Gamma \\left [ \\overline { \\left ( - i \\frac { d } { d x } + q \\right ) v } \\left ( - i \\frac { d } { d x } + q \\right ) w \\right ] + \\langle V , \\overline { v } w \\rangle _ { H ^ { - 1 } _ { \\rm p e r } , H ^ 1 _ { \\rm p e r } } . \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} \\sum _ v | \\Gamma _ 1 ( v ) | + \\cdots + | \\Gamma _ t ( v ) | = \\sum _ v d ( v ) = 2 e ( G ) . \\end{align*}"}
-{"id": "3485.png", "formula": "\\begin{align*} f ( x ) = \\max ( A x + b ) , \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} \\partial _ { t } X ( t , y ) & = \\Lambda ( t , X ( t , y ) ) ( t > 0 ) , \\\\ X ( 0 , y ) & = y \\in \\overline { \\Omega } . \\end{align*}"}
-{"id": "5522.png", "formula": "\\begin{align*} \\mathcal { L } \\left \\vert X \\right \\vert ^ { 2 } = \\bigtriangleup _ { g } \\left \\vert X \\right \\vert ^ { 2 } - \\frac { 1 } { 2 } \\left \\langle X , \\nabla _ { g } \\left \\vert X \\right \\vert ^ { 2 } \\right \\rangle = 2 n - \\left \\vert X \\right \\vert ^ { 2 } . \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{align*} ( 1 + u _ { m + 1 } ) ( 1 + u _ { m + 1 } + y _ 1 ) \\cdots ( 1 + u _ { m + 1 } + y _ l ) & \\\\ = ( 1 + u _ { m + 1 } ) ^ { l + 1 } + ( 1 + u _ { m + 1 } ) ^ l \\sigma _ 1 + \\cdots + ( 1 + u _ { m + 1 } ) \\sigma _ l . \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} R ( i , j , \\alpha ) : = ( Q ' - Q '' ) ( i , j , \\alpha ) , i < j \\in [ n ] , \\alpha \\geq 0 , \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} P _ { i , j } = \\lbrace y \\in P \\mid \\dim y = i + j , \\dim ( y \\cap x ) = i \\rbrace \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} \\textbf { B } = ( \\textbf { I } - \\textbf { P } _ { 0 0 } ) ^ { - 1 } \\textbf { P } _ { 0 1 } = \\textbf { M P } _ { 0 1 } . \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} \\norm { | \\partial _ { x } | ^ { \\sigma } f } _ { L ^ 2 ( \\R ) } = \\left ( \\int _ { \\R } | \\xi | ^ { 2 \\sigma } | \\hat { f } ( \\xi ) | ^ { 2 } \\ , d \\xi \\right ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} \\lim _ { q \\rightarrow \\infty } d _ { \\textrm { m a x } , 1 } ( f ) & = \\lim _ { q \\rightarrow \\infty } \\max _ { b \\in \\{ 0 , 1 \\} ^ q } d _ { \\textrm { m a x } , 1 } ( f , b ) \\\\ & \\leq \\lim _ { q \\rightarrow \\infty } \\frac { 2 { q \\choose q / 2 } } { 2 ^ q } \\\\ & = \\lim _ { q \\rightarrow \\infty } \\frac { 2 \\sqrt { \\frac { 2 } { \\pi q } } 2 ^ q } { 2 ^ q } \\\\ & = \\lim _ { q \\rightarrow \\infty } 2 \\sqrt { \\frac { 2 } { \\pi q } } \\\\ & = 0 , \\end{align*}"}
-{"id": "219.png", "formula": "\\begin{align*} \\tilde \\theta _ { \\mathbb C / ( \\mathbb Z + \\tau \\mathbb Z ) , d p } ( p ) = \\theta ( p | \\tau ) e ^ { { 1 \\over 2 } G _ 2 ( \\tau ) p ^ 2 } \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{align*} \\| T _ { \\Lambda } ^ { ( j ) } g _ i \\| & = \\| \\sum _ { i \\in J } \\Lambda _ { i j } ^ * g _ i \\| = \\sup _ { \\| g \\| = 1 } \\left | \\left \\langle g , \\sum _ { i \\in J } \\Lambda _ { i j } ^ * g _ i \\right \\rangle \\right | \\\\ & \\le \\sup _ { \\| g \\| = 1 } ( \\sum _ { i \\in J } \\| \\Lambda _ { i j } g \\| ^ 2 ) ^ { 1 / 2 } ( \\sum _ { i \\in J } \\| g _ i \\| ^ 2 ) ^ { 1 / 2 } \\\\ & = \\| T _ { \\Lambda } ^ { ( j ) } \\| ( \\sum _ { i \\in J } \\| g _ i \\| ^ 2 ) ^ { 1 / 2 } \\\\ & \\le \\sqrt { B _ j } ( \\sum _ { i \\in J } \\| g _ i \\| ^ 2 ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} \\| s _ { \\sigma } ( e ^ { i \\theta _ 0 } : \\cdots : e ^ { i \\theta _ n } ) \\| _ { \\sigma } = \\frac { | P ^ { \\sigma } ( e ^ { i \\theta _ 0 } , \\ldots , e ^ { i \\theta _ n } ) | } { ( n + 1 ) ^ { \\frac { d } { 2 } } } \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} _ { m , k } = & \\log \\left ( 1 + \\frac { \\frac { \\alpha _ 0 ^ 2 | h _ { m k } | ^ 2 } { L \\left ( | | y _ { m , k } + x _ m | | \\right ) } } { \\sum ^ { M _ s } _ { l = 1 } \\frac { \\alpha _ l ^ 2 | h _ { m k } | ^ 2 } { L \\left ( | | y _ { m , k } + x _ m | | \\right ) } + \\frac { 1 } { \\rho } } \\right ) , \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} \\varphi _ { L } ' ( \\mathbf { m } ' , N , \\epsilon ) = ( 1 - \\epsilon ) ^ { \\sum \\mathbf { m } ' } \\epsilon ^ { L N - \\sum \\mathbf { m } ' } \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} & q _ { 1 2 } ^ m \\sum _ { i = 0 } ^ { m } \\frac { ( m ) _ q ^ ! } { ( m - i ) _ q ^ ! } \\lambda _ { ( m - i , k ) } \\beta _ { ( i , m , k ) } \\sum _ { j = i } ^ { k + m } q ^ { ( j - i ) ( j - i - 1 ) / 2 } q ^ { - j ( j + 1 ) / 2 } { j \\choose i } _ { \\ ! \\ ! q } = 0 \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{align*} \\sum _ { A \\subset P ( \\sigma , \\tau ) , | A | = | \\sigma \\cap P | } ( - 1 ) ^ { { \\rm s i g n } ( \\sigma , \\tau , P ( \\sigma , \\tau ) , A ) } \\frac { \\partial ^ { k ' } X _ { \\eta _ 1 ( \\sigma , \\tau , P ( \\sigma , \\tau ) , A ) } ( s ) } { \\partial s ^ { k ' } } X _ { \\eta _ 2 ( \\sigma , \\tau , P ( \\sigma , \\tau ) , A ) } ( s ) = 0 . \\end{align*}"}
-{"id": "389.png", "formula": "\\begin{align*} \\partial T _ * ( \\beta ' _ * ( \\lambda _ x ) ) ( \\gamma ) & = T _ * ( \\beta ' _ * ( \\sum _ y s ^ x _ y \\times \\lambda _ y ) ) + T _ * ( \\beta ' _ * ( \\lambda _ x ) ) ( \\partial \\gamma ) \\\\ & = \\sum _ y T _ * ( a ^ x _ y \\cdot \\beta ' _ * ( \\lambda _ y ) ) ( \\gamma ) + T _ * ( \\beta ' _ * ( \\lambda _ x ) ) ( \\partial \\gamma ) \\\\ & = \\sum _ y T _ * ( \\beta ' _ * ( \\lambda _ y ) ) \\circ T _ * ( a ^ x _ y ) ( \\gamma ) + T _ * ( \\beta ' _ * ( \\lambda _ x ) ) ( \\partial \\gamma ) \\\\ & = \\Psi ( d ( \\gamma \\otimes x ) ) . \\end{align*}"}
-{"id": "8691.png", "formula": "\\begin{align*} U : = \\{ \\rho _ + \\leq 1 \\} \\subset M . \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} { \\mathbb P } \\big ( \\sum _ { i \\ge 1 } e ^ { - \\alpha Y _ i ( 0 ) ^ 2 } < \\infty \\big ) = 1 \\ , , \\mbox { f o r a n y } \\ \\ \\alpha > 0 \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\beta ^ - } f _ { \\gamma } ( z ) = \\frac { f _ { \\beta } ( z ) } { ( 1 - z ^ N ) } . \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} [ d , z _ 1 ] [ z _ 2 , z _ 3 ] + [ d , z _ 2 ] [ z _ 1 , z _ 3 ] = 0 . \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{align*} F _ { E , \\ast } \\left ( \\frac { 2 c + d } { 3 } \\right ) = \\sup F ( [ c , d ] ) \\\\ F _ { E , \\ast } \\left ( \\frac { c + 2 d } { 3 } \\right ) = \\inf F ( [ c , d ] ) , \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} ( r + 1 ) ( d - 1 ) \\theta _ T & = 2 d , \\\\ r ( r + 1 ) ( d - 1 ) ( \\theta _ { t + 1 } - \\theta _ t ) & < \\delta \\epsilon \\quad 0 \\leq t \\leq T - 1 . \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{align*} x \\succeq v & = L _ \\succ ^ \\alpha ( x ) v = L _ \\succ \\circ ( \\alpha ^ 2 \\otimes I d ) ( x \\otimes v ) = \\alpha ^ 2 ( x ) \\succ v , \\\\ v \\preceq x & = R _ \\prec ^ \\alpha ( x ) v = R _ \\prec \\circ ( i d \\otimes \\alpha ^ 2 ) ( v \\otimes x ) = v \\prec \\alpha ^ 2 ( x ) , \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} [ C _ D , D ] = - z _ 2 \\bar { z } _ 1 \\alpha ( l _ j ) e ^ { 2 \\alpha ( a ) } \\theta ( e _ { \\alpha } ) + O , \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\alpha \\cdot ( ( v \\otimes n ) \\otimes u ) & = & \\alpha \\cdot ( v \\otimes n ) \\otimes u + ( v \\otimes n ) \\otimes \\alpha u \\\\ & = & - ( v \\alpha \\otimes n ) \\otimes u + ( v \\otimes \\alpha \\cdot n ) \\otimes u + ( v \\otimes n ) \\otimes \\alpha u \\ , . \\end{array} \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{align*} & \\sum _ { t } \\min _ { x , y : \\ : p _ { X , Y } ( x , y ) > 0 } \\delta _ { x , y , t } = \\min _ { x , y : \\ : p _ { X , Y } ( x , y ) > 0 } \\delta _ { x , y , 0 } = \\min _ { x , y : \\ : p _ { X , Y } ( x , y ) > 0 } \\tilde { \\delta } _ { x , y } \\geq \\min _ { x , y } \\tilde { \\delta } _ { x , y } . \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} V _ { } : = \\{ v \\in V \\mid v \\} . \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} ( x , w ) \\in C _ i ^ { ( w ' ) } & \\iff x \\in C _ i w = w ' \\\\ & \\iff S _ { x , C _ i } \\cdot I _ { w w ' } = 1 \\\\ & \\iff ( S \\otimes I ) _ { ( x , w ) , ( C _ i , w ' ) } = 1 . \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{align*} \\bigg \\| \\sum _ { k = k _ 0 } ^ \\infty 2 ^ { - k \\alpha } \\mathcal { C } _ k g \\bigg \\| _ { L ^ q _ t L ^ r _ x } \\lesssim \\| g \\| _ { L ^ 2 } , \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} \\Vert K _ { \\varepsilon } \\Vert _ { \\infty } + \\sum _ { \\ell = 1 } ^ d \\Vert G ^ { \\ell } _ { \\varepsilon } \\Vert _ { \\infty } \\leq \\frac { C } { \\varepsilon ^ d } \\ . \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} C ( \\Q ) = \\{ ( 0 : 1 : 0 ) , ( 0 : 1 : 1 ) , ( 0 : - 1 : 1 ) , ( 2 : 3 : 1 ) , ( 2 : - 3 : 1 ) , ( - 1 : 0 : 1 ) \\} \\cong \\Z / 6 \\Z . \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} v ( t , x ) = \\int _ { \\mathbb { R } } G _ t ( x - y ) u _ 0 ( y ) d y + \\frac { 1 } { 2 } \\int _ 0 ^ t \\int _ { \\mathbb { R } } \\frac { \\partial } { \\partial y } G _ { t - s } ( x - y ) v ( s , y ) ^ 2 d y d s \\\\ + \\int _ 0 ^ t \\int _ { \\mathbb { R } } G _ { t - s } ( x - y ) \\sigma _ s ( y ) W ( d s , d y ) . \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} r = ( r _ 1 - \\gamma ( r _ 1 ) ) ( r _ 2 - \\gamma ( r _ 2 ) ) \\cdots ( r _ l - \\gamma ( r _ l ) ) + r ^ \\prime , \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} { \\div } _ { \\infty } ( f _ j ) = \\div ( \\varphi ^ * X _ 0 ) + { \\div } _ 0 ( f _ j ) - \\div ( \\varphi ^ * X _ j ) \\end{align*}"}
-{"id": "6994.png", "formula": "\\begin{align*} H _ z : = z H z ^ { - 1 } \\cap \\mathfrak { g l } ( \\nu _ 1 + 1 , \\C ) \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{align*} T _ * < + \\infty \\Rightarrow \\lim _ { T \\to T _ * } \\Big ( \\| \\nabla u \\| _ { L ^ { \\infty } _ T L ^ 2 _ x } + \\| u \\| _ { L ^ 2 _ T W ^ { 1 , 4 } _ x } + \\| v \\| _ { L ^ \\infty _ T H ^ 1 _ x } \\Big ) = + \\infty . \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} \\langle x _ v , x _ w \\rangle = \\langle x _ { v , 0 } , x _ { w , 0 } \\rangle = p ( 0 , 0 | v , w ) = \\frac { s } { | E | } . \\end{align*}"}
-{"id": "4048.png", "formula": "\\begin{align*} ( 1 - \\epsilon ) I ( U ; X Y | V = v ) \\geq I ( U ; J | V = v ) . \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} \\mathcal { G } ( x ) : = g ( x ) - K \\quad { \\rm f o r } \\ x \\in \\mathbb { X } . \\end{align*}"}
-{"id": "7903.png", "formula": "\\begin{align*} H ( t ) = \\sum _ { k = 1 } ^ 4 H _ k ( t ) + W _ { 1 2 } ( t ) + \\sum \\ , ' \\ , W _ { i j } ( t ) . \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{align*} 2 ^ { K + 1 } \\rho \\leq 2 ^ { \\frac { \\log C _ d - \\log \\rho } { \\log 2 } + 1 } \\rho = 2 C _ d \\textrm { w h i l e } 2 ^ { K + 1 } \\rho \\geq 2 ^ { \\frac { \\log C _ d - \\log \\rho } { \\log 2 } } \\rho = C _ d . \\end{align*}"}
-{"id": "1645.png", "formula": "\\begin{align*} F ( w ) : = \\sum _ { i = 1 } ^ s f _ i ( w ) ^ 2 , G ( w ) : = \\sum _ { j = 1 } ^ t g _ j ( w ) ^ 2 . \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} P _ \\mu = \\lbrace y \\in P \\mid \\dim ( y \\cap x _ i ) = \\mu _ 1 + \\mu _ 2 + \\cdots + \\mu _ i \\ ; ( 1 \\le i \\le N ) \\rbrace \\end{align*}"}
-{"id": "5420.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { \\beta } { \\tilde { \\beta } } \\right ) \\ , \\Omega ( x _ n ) & \\leq - \\beta \\ , E ^ \\dagger ( x _ n ) + \\Omega ( x _ n ) - \\Omega ^ \\dagger + r _ n \\ , \\| F ( x _ n ) - y ^ \\dagger \\| + \\Omega ^ \\dagger - \\frac { \\beta \\ , \\tilde { c } } { \\tilde { \\beta } } \\\\ & = - D _ \\beta ( r _ n ) + \\Omega ^ \\dagger - \\frac { \\beta \\ , \\tilde { c } } { \\tilde { \\beta } } \\\\ & \\leq \\Omega ^ \\dagger - \\frac { \\beta \\ , \\tilde { c } } { \\tilde { \\beta } } . \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} \\left < f , g \\right > = \\sum _ { | k | = m } \\int _ { \\mathbf { B } } ( 1 - | x | ^ { 2 } ) ^ { 2 m } \\frac { \\partial ^ { m } f } { \\partial x ^ { k } } ( x ) \\overline { \\frac { \\partial ^ { m } g } { \\partial x ^ { k } } ( x ) } d \\tau ( x ) , \\enspace f , g \\in B ^ { 2 } . \\end{align*}"}
-{"id": "4702.png", "formula": "\\begin{align*} p ( u , v ) & = \\P ^ u ( X ) \\\\ & = \\P ^ u ( \\exists n \\ge 1 : \\ X _ 1 \\notin \\{ u , v \\} , \\dots , X _ { n - 2 } \\notin \\{ u , v \\} , X _ { n - 1 } \\notin \\{ u , v \\} , X _ n = v ) \\\\ & = \\P ^ u ( \\sigma _ v < \\sigma _ u ) \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} \\widetilde { h } ( x , D _ { x ' } ) v ( x ) : = \\kappa ^ { - 1 , * } h ( x , D _ { x ' } ) \\lambda ^ * v ( x ) v \\in \\mathcal { S } ( \\R ^ { n - 1 } ) . \\end{align*}"}
-{"id": "8866.png", "formula": "\\begin{align*} \\mathrm { d i v } ( s ) \\cap w \\cdot V = \\sum _ { Y \\in \\mathcal { I } ^ G ( X ) } 2 k v _ { \\mathcal { L } } ( \\mu _ Y ) w \\cdot ( Y \\cap V ) . \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} x = 1 5 0 , y = 0 , y = 0 , v = 0 , a = 4 5 0 , b = 2 7 0 . \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{align*} X ^ { u s } ( { \\cal L } ) = \\bigcup \\limits _ { \\hat { w } \\in W } \\bigcup \\limits _ { w \\in W ( \\hat { w } , \\lambda , q ) } \\hat { w } ^ { - 1 } X _ w \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} ( 0 , 0 , x , y ) & \\quad \\phi _ { \\mathfrak { L } } = u ^ { 1 6 } , \\\\ ( u ^ 8 x , u ^ 8 y , x , y ) & \\quad \\phi _ { \\mathfrak { L } } = u ^ { 2 4 } , \\end{align*}"}
-{"id": "5414.png", "formula": "\\begin{align*} & W ^ 0 _ 4 ( p , p _ 1 , p _ 2 , p _ 3 ) = W ^ 0 _ 3 ( p , p _ 1 , p _ 2 ) \\ast W ^ 0 _ 3 ( p , p _ 2 , p _ 3 ) \\\\ & = K _ p ( q , \\bar q ) \\left ( W ^ 0 _ 3 ( q , p _ 1 , p _ 2 ) W ^ 0 _ 2 ( \\bar q , p _ 3 ) + W ^ 0 _ 2 ( q , p _ 1 ) W ^ 0 _ 3 ( \\bar q , p _ 2 , p _ 3 ) \\right ) ; \\\\ & W ^ 1 _ 2 ( p , p _ 1 ) = W ^ 1 _ 1 ( p ) \\ast _ h W ^ 0 _ 3 ( p , p _ 1 , p _ 2 ) + W ^ 0 _ 3 ( p , p _ 1 , p _ 2 ) \\ast _ h W ^ 1 _ 1 ( p ) \\\\ & = K _ p ( q , \\bar q ) \\left ( W ^ 0 _ 3 ( q , \\bar q , p _ 1 ) + W ^ 1 _ 1 ( q ) W ^ 0 _ 2 ( \\bar q , p _ 1 ) + W ^ 0 _ 2 ( q , p _ 1 ) W ^ 1 _ 1 ( \\bar q ) \\right ) . \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } | ( Q _ { n , \\textup { \\textbf { m } } } ) ^ { ( k ) } ( \\lambda _ j ) | ^ { 1 / n } \\leq \\frac { | \\Phi ( \\lambda _ j ) | } { \\rho _ { | \\textup { \\textbf { m } } | } ( \\textup { \\textbf { F } } ) } , k = 0 , 1 , \\ldots , \\tau _ j - 1 \\end{align*}"}
-{"id": "5227.png", "formula": "\\begin{align*} m ( u _ 0 ) : = \\inf _ { t _ 0 \\in \\mathbb { R } , ( x , t ) \\in \\R ^ N \\times [ 0 , \\infty ) } u ( x , t + t _ 0 ; t _ 0 , u _ 0 ) > 0 . \\end{align*}"}
-{"id": "1350.png", "formula": "\\begin{align*} P _ { m k } = \\begin{bmatrix} c _ { 1 n } & c _ { 2 n } & \\ldots & c _ { k n } \\\\ c _ { 1 n } & c _ { 2 n } & \\ldots & c _ { k n } \\\\ \\hdotsfor { 4 } \\\\ c _ { 1 n } & c _ { 2 n } & \\ldots & c _ { k n } \\end{bmatrix} , Q _ { k k } = \\begin{bmatrix} 1 & c _ { 1 n } & \\ldots & c _ { k - 2 , n } & c _ { k - 1 , n } \\\\ 0 & 1 & \\ldots & c _ { k - 3 , n } & c _ { k - 2 , n } \\\\ \\hdotsfor { 5 } \\\\ 0 & 0 & \\ldots & 1 & c _ { 1 n } \\\\ 0 & 0 & \\ldots & 0 & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} \\sum _ { v \\in V ( G ) } \\binom { d ( v ) } { 2 } \\ge \\binom { n } { 2 } - \\frac { 1 } { 2 } \\sum _ { v \\in V ( G ) } { d ( v ) } \\ge \\binom { n } { 2 } - \\frac { 1 } { 2 } ( 4 n - 1 0 ) = \\frac { n ^ 2 } { 2 } - \\frac { 5 n } { 2 } + 5 . \\end{align*}"}
-{"id": "2442.png", "formula": "\\begin{align*} v ( n ) = \\frac { ( - 1 ) ^ { g - 1 } } { ( 2 \\pi i ) ^ { g - 1 } } \\int _ { \\Gamma _ g } \\cdots \\int _ { \\Gamma _ 2 } \\psi _ 1 ( n _ 1 ; \\lambda _ 1 ) \\psi _ 2 ( n _ 2 ; \\lambda _ 2 ) \\cdots \\psi _ g ( n _ g ; \\lambda _ g ) \\ , \\frac { d \\lambda _ 2 \\cdots d \\lambda _ g } { \\lambda _ 2 \\cdots \\lambda _ g } , \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} c ( T Q ) = \\frac { ( 1 + H ) ^ { n + 2 } } { ( 1 + 2 H ) } , \\end{align*}"}
-{"id": "6626.png", "formula": "\\begin{align*} d _ a ( p _ 1 , p _ 2 ) = ( y _ 1 - y _ 2 ) ^ 2 / ( x _ 1 - x _ 2 ) ^ 2 . \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} d q _ t = & \\tilde \\gamma ^ { - 1 } ( t , q _ t ) F ( t , q _ t , \\psi ( t , q _ t ) ) d t + S ( t , q _ t ) d t + \\tilde \\gamma ^ { - 1 } ( t , q _ t ) \\sigma ( t , q _ t ) d W _ t . \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{align*} \\langle \\langle \\Delta _ h u , \\ , u \\rangle \\rangle _ \\omega = h ^ { 2 ( n - p ) } \\ , \\langle \\langle \\Delta _ { \\omega _ h } u , \\ , u \\rangle \\rangle _ { \\omega _ h } = h ^ { 2 ( n - p ) } \\ , ( | | d u | | ^ 2 _ { \\omega _ h } + | | d ^ \\star _ { \\omega _ h } u | | ^ 2 _ { \\omega _ h } ) . \\end{align*}"}
-{"id": "8888.png", "formula": "\\begin{align*} P _ { D H } ( p ) = \\prod _ { \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\frac { \\kappa ( \\alpha , p ) } { \\kappa ( \\alpha , \\rho ) } = \\prod _ { \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\frac { \\kappa ( \\alpha , \\alpha ) } { 2 \\kappa ( \\alpha , \\rho ) } p ( \\alpha ^ { \\vee } ) \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } \\left | \\nabla f \\right | ^ { 2 } - \\frac { 1 } { 2 } \\Delta f = & \\frac { 1 } { 4 } \\left | \\overline { \\nabla } f \\right | ^ { 2 } + \\frac { 1 } { 4 } | \\left ( \\overline { \\nabla } f \\right ) ^ { \\perp } | ^ { 2 } + \\frac n 4 \\\\ = & \\frac 1 4 | x | ^ 2 + \\frac 1 { 1 6 } | x ^ { \\perp } | ^ 2 + \\frac n 4 \\\\ \\geq & \\frac 1 4 | x | ^ 2 + \\frac n 4 \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{align*} u ( 0 ) = \\frac { 1 } { | B _ { r / 2 } | } \\int _ { B _ { r / 2 } } u \\geq \\frac { 1 } { | B _ { r / 2 } | } \\int _ { B _ { r / 2 } \\cap \\{ u \\geq N \\} } u \\geq \\frac { N | B _ { r / 2 } \\cap \\{ u \\geq N \\} | } { | B _ { r / 2 } | } \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} A = \\sum _ { | \\lambda | < k } a _ \\lambda S _ { \\lambda } + \\sum _ { | \\mu | = k } S _ { \\mu } A _ { \\mu } \\end{align*}"}
-{"id": "7930.png", "formula": "\\begin{align*} u ( z ) = a - z \\int _ 0 ^ \\infty 1 \\ , d \\rho ( s ) - ( z ^ 2 + 1 ) \\int _ 0 ^ \\infty \\frac { 1 } { s - z } \\ , d \\rho ( s ) , \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{align*} \\Big \\| \\int _ { t _ 1 } ^ { t _ 2 } \\int _ 0 ^ { \\infty } \\sigma _ s ( y ) ^ 2 \\int _ y ^ { \\infty } \\psi ( s , z ) G _ { t _ 2 - s } ( x - z ) d z d y d s \\Big \\| _ p & \\leq C \\int _ { t _ 1 } ^ { t _ 2 } \\int _ { \\mathbb { R } } G _ { t _ 2 - s } ( x - z ) d z d s \\\\ & = C ( t _ 2 - t _ 1 ) . \\end{align*}"}
-{"id": "1550.png", "formula": "\\begin{align*} & 0 \\leq \\alpha _ { k j } \\leq 1 , \\sum _ { j = 1 } ^ { J } \\alpha _ { k j } = 1 , \\\\ & \\alpha _ { k j } = 0 e _ k \\cap e _ j = \\emptyset e _ j \\to e _ k . \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} \\eta _ k \\geq I [ \\nu _ { \\eta _ k } ] = \\iint f ( \\theta - \\phi ) d \\nu _ { \\eta } ( \\theta ) d \\nu _ { \\eta } ( \\phi ) \\geq \\iint \\min \\{ L , f ( \\theta - \\phi ) \\} d \\nu _ { \\eta } ( \\theta ) d \\nu _ { \\eta } ( \\phi ) . \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} p _ t ( x , y ) = \\sum _ { k = 1 } ^ \\infty e ^ { - \\l _ k t } \\phi _ k ( x ) \\overline { \\phi _ k ( y ) } . \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{align*} U _ 0 = \\{ 0 \\leq \\tau \\leq c , \\ \\rho _ 0 \\leq 1 \\} , c \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "3046.png", "formula": "\\begin{align*} X _ { 2 } = \\left \\{ w \\in W _ { D } ^ { 2 , \\eta } ( \\Omega ) : \\int _ { \\Omega } w \\phi _ { 1 } = 0 \\right \\} . \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} a _ n = & \\frac { 1 } { n ! } \\bigg ( \\frac { 1 - q _ 1 q _ 2 } { ( 1 - q _ 1 ) ( 1 - q _ 2 ) } \\bigg ) ^ n \\int _ { C _ 1 ^ n } \\prod _ { j = 1 } ^ n \\frac { d z _ j } { 2 \\pi i z _ j } \\prod _ { j \\neq k } \\frac { ( z _ j - z _ k ) ( z _ j - q _ 1 q _ 2 z _ k ) } { ( z _ j - q _ 1 z _ k ) ( z _ j - q _ 2 z _ k ) } . \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} U _ r ( X , t ) = \\frac { r ^ { a / 2 } U ( \\delta _ r ( X , t ) ) } { \\sqrt { H ( U , r ) } } = \\frac { r ^ { a / 2 } U ( r X , r ^ 2 t ) } { \\sqrt { H ( U , r ) } } . \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} { \\rm C o e f f } _ p [ E _ { \\mu } , m ] \\propto \\left [ \\prod _ { \\substack { \\kappa \\prec \\nu [ 1 ] } } \\frac { Y ( w ) - y _ { \\kappa } ( w ) } { y _ { \\nu [ 1 ] } ( w ) - y _ { \\kappa } ( w ) } \\cdot z ^ { \\nu [ 1 ] } \\right ] _ { q = t ^ { - m } } = E _ { \\nu [ 1 ] } ( z ; t ^ { - m } , t ) , \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{align*} i ( \\nu ' ) - i ( \\nu ) = N ( \\nu , \\nu ' ) , \\end{align*}"}
-{"id": "5104.png", "formula": "\\begin{align*} a ( n ) b ( m ) = b ( m ) a ( n ) + [ a , b ] ( n + m ) + n \\delta _ { n + m , 0 } \\ < a , b \\ > K ; \\end{align*}"}
-{"id": "2095.png", "formula": "\\begin{align*} \\mbox { o r } \\ \\ g ( C _ a ) = \\delta ( C _ a ) = { \\rm i n d } C _ a = e _ Q ( C _ a ) = 0 , \\ \\ \\mbox { a n d } \\ \\ h ( C _ a ) = 3 , \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} 0 = I _ 0 & \\leq I [ \\nu + t ( \\mu - \\nu ) ] \\\\ & = I [ \\nu ] + t \\bigg ( \\iint f ( \\theta - \\phi ) d \\nu ( \\theta ) d ( \\mu - \\nu ) ( \\phi ) \\\\ & + \\iint f ( \\theta - \\phi ) d ( \\mu - \\nu ) ( \\theta ) d \\nu ( \\phi ) \\bigg ) + t ^ 2 I [ \\mu - \\nu ] , \\end{align*}"}
-{"id": "874.png", "formula": "\\begin{align*} \\hat { f } ( \\xi ) = \\hat { u _ * } + i \\int _ 1 ^ t e ^ { - i \\frac 1 2 \\xi ^ 2 s } e ^ { - s } \\widehat { u ( s ) v _ * ( x ) } ( \\xi ) d s + i \\int _ 1 ^ t \\int _ 1 ^ s e ^ { - i \\frac 1 2 \\xi ^ 2 s } e ^ { - ( s - s ' ) } \\widehat { | u ( s ' ) | ^ 2 u ( s ) } ( \\xi ) d s ' d s , \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} \\alpha _ k = z _ k + \\varrho _ k u _ k = z _ k + \\varrho _ k ( b _ k + i \\eta R _ k ) = a _ k + i \\eta \\varrho _ k R _ k \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} \\widehat { C } ( l _ 0 , l _ 1 ) = \\dfrac { \\widehat { A } ( l _ 0 , l _ 1 ) } { \\widehat { B } ( l _ 0 , l _ 1 ) } , \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} \\sigma \\left ( I _ n \\otimes A + A \\otimes I _ n + \\sum _ { i = 1 } ^ m N _ i \\otimes N _ i \\right ) \\subset \\mathbb C _ - . \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} { } Z _ E ( \\alpha , \\beta ) = \\prod _ { k = 1 } ^ { \\infty } \\left ( 1 - e ^ { - \\alpha } e ^ { - \\beta k } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "7276.png", "formula": "\\begin{align*} \\begin{aligned} I & = \\{ i _ 1 , i _ 2 , \\ldots , i _ k \\} & i _ 1 < i _ 2 < \\ldots < i _ k , \\\\ J & = \\{ j _ 1 , j _ 2 , \\ldots , j _ \\ell \\} & j _ 1 < j _ 2 < \\ldots < j _ \\ell . \\end{aligned} \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{align*} H P _ { s w } ( X , \\Omega _ X , \\Lambda _ X ) = \\sum \\limits _ { \\Gamma _ X \\in H _ 2 ( X , \\partial X , \\mathbb { Z } ) , \\partial _ { Y _ { \\pm } } \\Gamma _ X = \\Gamma _ { \\pm } } H P _ { s w } ( X , \\Omega _ X , \\Gamma _ X , \\Lambda _ X ) \\end{align*}"}
-{"id": "6807.png", "formula": "\\begin{align*} ( R ^ V \\circ \\xi ) ( X , Y , X _ 1 , \\ldots , X _ k ) = R _ { X , Y } ^ V ( \\xi ( X _ 1 , \\ldots , X _ k ) ) \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} \\frac { H ' ( r ) } { H ( r ) } = \\frac { 4 N ( r ) } { r } + \\frac { a } { r } , \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} S _ g \\big ( p , - d _ p ( d ^ R _ g ( p , q ) ) \\big ) = \\big ( q , d _ q ( d ^ R _ g ( p , q ) ) \\big ) . \\end{align*}"}
-{"id": "4265.png", "formula": "\\begin{align*} \\hat { z } _ 1 = q _ i z _ 2 , \\hat { z } _ 2 = q _ i z _ 3 , \\dots , \\hat { z } _ { J - 1 } = q _ i z _ J , \\hat { z } _ J = q _ i ^ { - l _ 0 } u _ 0 . \\end{align*}"}
-{"id": "6736.png", "formula": "\\begin{align*} H _ n ^ f \\otimes _ { H _ { n - 1 } ^ f } H _ n ^ f \\oplus \\bigoplus _ { r = 0 } ^ { \\ell - 1 } H _ n ^ f & \\rightarrow H _ { n + 1 } ^ f , \\\\ ( a \\otimes b , c _ 0 , c _ 1 , \\dots , c _ { \\ell - 1 } ) & \\mapsto a s _ n b + \\sum _ { r = 0 } ^ { \\ell - 1 } c _ r x _ { n + 1 } ^ r \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} | \\nabla \\psi _ { \\ast } ( \\omega ) | ^ 2 ( x ) = \\sum _ { i , j = 1 } ^ n \\left ( \\sum _ { k , l = 1 } ^ n a _ { i k } a _ { j l } \\frac { \\partial \\omega _ k } { \\partial u _ l } ( u ) \\right ) ^ 2 = \\sum _ { i , j = 1 } ^ n \\sum _ { k , l = 1 } ^ n \\sum _ { r , s = 1 } ^ n a _ { i k } a _ { j l } a _ { i r } a _ { j s } \\frac { \\partial \\omega _ k } { \\partial u _ l } ( u ) \\frac { \\partial \\omega _ r } { \\partial u _ s } ( u ) . \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} V ( x ) : = \\left ( { N - p \\over p } \\right ) ^ p | x | ^ { - p } . \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} & { \\rm d i s t } ( ( x , \\lambda , v ) , \\widetilde { \\Phi } ^ { - 1 } ( 0 , 0 , 0 ) ) = { \\rm d i s t } ( ( x , \\lambda , v ) , \\Phi ^ { - 1 } ( 0 , 0 ) ) \\\\ & \\le { \\rm d i s t } ( ( x , \\lambda ' , v ) , \\Phi ^ { - 1 } ( 0 , 0 ) ) + \\| \\lambda - \\lambda ' \\| \\\\ & \\le \\kappa { \\rm d i s t } ( ( 0 , 0 ) , \\Phi ( x , \\lambda ' , v ) ) + \\| \\lambda - \\lambda ' \\| \\\\ & \\le \\kappa \\| ( \\xi ' , \\eta ) \\| + \\| \\eta + \\zeta \\| \\le \\kappa \\sqrt { 4 \\gamma ^ 2 + 3 } \\| ( \\xi , \\eta , \\zeta ) \\| . \\end{align*}"}
-{"id": "5529.png", "formula": "\\begin{align*} P _ t = \\{ e ^ { t \\Delta _ M } \\mid t \\in \\R _ + \\} . \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{align*} \\psi ( h , g ) : = \\psi _ { G } ( g ) \\qquad ( h \\in H , g \\in G ) , \\end{align*}"}
-{"id": "8496.png", "formula": "\\begin{align*} \\varphi ( a , r , e , a ) = \\varphi ( r , a , r , e ) \\end{align*}"}
-{"id": "8805.png", "formula": "\\begin{align*} \\exp ( D ) = \\exp ( - a ) k \\exp ( a + y + O ) \\exp ( h ) , \\end{align*}"}
-{"id": "834.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\ , ( 2 \\pi t ) ^ { - n / 2 } \\mathbb E _ { t ; x , x } { \\rm S t r } \\ , M _ t V _ t d X = d _ n { \\rm S t r } \\ , D R _ { Z } ^ { \\frac { n - 1 } { 2 } } d Z , \\end{align*}"}
-{"id": "3152.png", "formula": "\\begin{align*} X _ n = \\displaystyle \\sum _ { k = 0 } ^ { \\infty } \\rho ^ { k } \\left ( \\varepsilon _ { n - k } \\right ) , n \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} \\epsilon _ 1 = \\max \\min _ { x , y } \\delta _ { x , y } \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} c _ n : = \\frac { 1 } { 2 } \\Big ( 1 + \\sum _ { k = 0 } ^ n \\big ( \\frac { \\varepsilon } { 1 + \\varepsilon } \\big ) ^ k \\Big ) \\ , v \\ , . \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{align*} R _ k : = \\frac 1 { \\varrho _ k \\varphi \\ ! \\left ( 1 / \\varrho _ k \\right ) } \\to \\infty \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in \\Bbb F _ q } \\| \\lambda _ { \\alpha } \\| ^ 2 = \\operatorname { t r } ( \\mathbb { A } ^ { \\ast } \\mathbb { A } ) \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} p ( x , D _ x ) f ( x ) = \\int _ { { \\R } ^ n } K ( x , x - y ) \\ , f ( y ) \\ , d y x \\in \\R ^ n \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} \\big | \\sum _ { | \\alpha | = k } ( h D ) ^ { \\alpha } b _ { \\alpha } ( h D ) f ( x ) \\big | \\lesssim h \\rho ( x ) ^ { \\ell - 1 } . \\end{align*}"}
-{"id": "8535.png", "formula": "\\begin{align*} \\mu ( \\zeta ) = \\frac { \\mu _ + } { \\zeta \\ } + \\mu _ 0 + \\zeta \\mu _ - \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{align*} W ( \\hat { w } , \\lambda , q ) : = \\{ w \\in W , \\exists \\xi \\in { \\cal X } ^ * \\textrm { w i t h } \\xi \\textrm { d o m i n a n t r e g u l a r } , \\langle w \\lambda - \\hat { w } q , \\xi \\rangle > 0 \\} \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} N _ s ( Q ) : = k Q ^ { ( s ) } / J \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} E & ( x , y ) = 1 + \\sum _ { n \\geq 1 } \\sum _ { \\ell = 1 } ^ n e _ { n , \\ell } x ^ n y ^ \\ell \\\\ & = 1 + \\frac { x y ( ( 2 x ^ 5 - 5 x ^ 4 + 4 x ^ 3 - x ^ 2 ) y ^ 2 + ( 2 x ^ 5 - 6 x ^ 4 + 1 1 x ^ 3 - 8 x ^ 2 + 2 x ) y - x ^ 4 + 4 x ^ 3 - 6 x ^ 2 + 4 x - 1 ) } { ( 1 - x ) ^ 2 ( 1 - 2 x ) ( 1 - x y ) ^ 2 ( x y + x - 1 ) } . \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{align*} d q ^ \\prime _ t = & \\tilde \\gamma ^ { - 1 } ( t ^ * ) \\left ( - \\nabla _ q V ( t ^ * , q ^ \\prime _ t ) + \\tilde F ( t ^ * , q _ t ) \\right ) d t \\\\ & + \\tilde S ( t ^ * , q ^ \\prime _ t ) d t + \\tilde \\gamma ^ { - 1 } ( t ^ * ) \\sigma ( t ^ * , q ^ \\prime _ t ) \\circ d W _ t \\\\ = & \\tilde \\gamma ^ { - 1 } ( t ^ * ) \\left ( - \\nabla _ q V ( t ^ * , q ^ \\prime _ t ) + \\tilde F ( t ^ * , q _ t ) \\right ) d t + \\tilde \\gamma ^ { - 1 } ( t ^ * ) \\sigma ( t ^ * , q ^ \\prime _ t ) d W _ t , \\end{align*}"}
-{"id": "6937.png", "formula": "\\begin{align*} \\begin{aligned} X _ t ^ \\sigma = X _ 0 + X _ t ^ { \\sigma , c } & + \\int _ 0 ^ { t \\wedge \\sigma } \\chi ( y ) y \\ \\Big ( \\mu ^ { X ^ \\sigma } ( \\cdot ; d s , d y ) - \\nu ^ \\sigma ( \\cdot ; d s , d y ) \\Big ) \\\\ & + \\check { X } ^ \\sigma ( \\chi ) + B _ t ^ \\sigma ( \\chi ) . \\end{aligned} \\end{align*}"}
-{"id": "4703.png", "formula": "\\begin{align*} \\alpha \\circ \\mu = \\mu \\circ ( \\alpha \\otimes \\alpha ) \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{align*} \\int _ { \\Omega } u \\cdot \\varphi \\nabla \\varphi \\ ; \\d x = \\frac { 1 } { 2 } \\int _ { \\Omega } u \\cdot \\nabla \\varphi ^ 2 \\ ; \\d x = 0 \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} \\Delta _ B \\phi = 2 \\overline \\square _ B \\phi + ( \\partial _ B - \\bar \\partial _ B ) H ^ { 0 , 1 } \\lrcorner \\ , \\phi + H ^ { 0 , 1 } \\lrcorner ( \\partial _ B - \\bar \\partial _ B ) \\phi + H ^ { 1 , 0 } \\lrcorner \\ , \\partial _ B \\phi . \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{gather*} K _ { i j } : = \\sum _ { k = 1 } ^ { H _ 0 - 1 } ( a _ { i k } b _ { k j } - a ^ 0 _ { i k } b ^ 0 _ { k j } ) + a _ { i H _ 0 } b _ { H _ 0 j } - a ^ 0 _ { i H _ 0 } b ^ 0 _ { H _ 0 j } + \\sum _ { k = H _ 0 + 1 } ^ { H - 1 } a _ { i k } b _ { k j } + a _ { i H } b _ { H j } , \\\\ L _ j : = \\sum _ { k = 1 } ^ { H _ 0 - 1 } ( a _ { M k } b _ { k j } - a ^ 0 _ { M k } b ^ 0 _ { k j } ) + a _ { M H _ 0 } b _ { H _ 0 j } - a ^ 0 _ { M H _ 0 } b ^ 0 _ { H _ 0 j } + \\sum _ { k = H _ 0 + 1 } ^ { H - 1 } a _ { M k } b _ { k j } + a _ { M H } b _ { H j } . \\end{gather*}"}
-{"id": "171.png", "formula": "\\begin{align*} S ^ 1 = \\widetilde S ^ 1 \\cup S ^ { 1 2 } , \\widetilde S ^ 2 = S ^ { 1 2 } \\cup S ^ 2 , \\end{align*}"}
-{"id": "6755.png", "formula": "\\begin{align*} \\Rightarrow R _ { x ^ { \\lambda } \\cdot x \\phi ^ { - 1 } } = L _ { x } ^ { - 1 } R _ { x } ^ { - 1 } R _ { x \\phi ^ { - 1 } } L _ { x } . \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{align*} b _ 2 - 1 = \\binom { c _ 2 } { 2 } - 1 = \\binom { c _ 2 - 1 } { 2 } + \\binom { c _ 2 - 1 } { 1 } - 1 = \\binom { c _ 2 - 1 } { 2 } + c _ 2 - 2 . \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} \\| u _ 0 \\| _ { C ^ { 0 , \\beta } } = ( 1 - \\delta _ 0 / ( 1 + B ) ) \\cdot \\| v _ 0 \\| _ { C ^ { 0 , \\beta } } \\leq ( 1 - \\delta _ 0 / ( 1 + B ) ) \\cdot B < B , \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\Vert U _ { k } x ^ { k } - x ^ { k } \\Vert = 0 . \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} \\det R = ( \\det \\omega ) \\left ( 1 + \\sigma _ 1 ^ 2 \\right ) \\left ( 1 + \\sigma _ 3 ^ 2 \\right ) \\cdots \\left ( 1 + \\sigma _ { 2 \\left [ \\frac { n } { 2 } \\right ] - 1 } ^ 2 \\right ) . \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} \\nabla _ { g _ { \\rm h y p } } u = \\frac { x } { p } \\ { \\rm i n } \\ \\mathbb B ^ m , \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} [ a , \\underbrace { a b , a b , \\dots , a b } _ { \\mbox { $ k $ t i m e s } } ] = a c ^ k \\ne 0 . \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} \\textbf { C o r r } _ { \\mathbf { h } } = \\overset { \\infty } { \\underset { n = 0 } { \\oplus } } \\textbf { C o r r } ^ { { \\mathbf { ( n ) } } } _ { \\mathbf { h } } \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{align*} g _ k ^ \\# ( 0 ) = \\varrho _ k f _ k ^ \\# ( z _ k ) = 1 . \\end{align*}"}
-{"id": "5364.png", "formula": "\\begin{align*} \\beta _ { j + 1 } C _ { j + 1 } ( t ) - ( - 1 ) ^ j D ' _ j ( t ) = 0 , \\\\ \\beta _ { j + 1 } D _ { j + 1 } ( t ) + ( - 1 ) ^ j C ' _ j ( t ) = 0 , \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} \\mathrm { P } _ { t , 1 } = \\mathrm { P } \\left ( z _ t < \\frac { \\epsilon _ 1 } { \\rho } \\right ) . \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{align*} c _ 1 ( M ) \\cup [ \\omega _ 0 ] = 0 \\in H ^ 4 ( B ) . \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} \\begin{aligned} \\dd { X ^ x } ( t ) & = B ( t , X _ t ^ x ) \\dd { t } + b ( t , X ^ x ( t ) ) \\dd { t } + \\sigma ( t , X ^ x ( t ) ) \\dd { W } ( t ) , \\\\ X _ 0 ^ x & = x \\end{aligned} \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} \\rho _ L = \\begin{pmatrix} u _ 3 & \\hat { u } ^ T & u _ 0 \\| \\hat { u } \\| ^ 2 \\\\ u _ 3 \\xi & u _ 3 \\xi \\cdot v ^ T + A & u _ 0 \\xi \\| \\hat { u } \\| ^ 2 + A v \\\\ \\frac { u _ 3 } { 2 } \\| \\xi \\| ^ 2 & \\frac { u _ 3 } { 2 } \\| \\xi \\| ^ 2 v ^ T + \\xi ^ T A & \\frac { u _ 0 } { 2 } \\| \\xi \\| ^ 2 \\| \\hat { u } \\| ^ 2 + \\xi ^ T A v + u _ 3 ^ { - 1 } \\end{pmatrix} \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} \\frac { d } { d t } \\hat { x } ( t ) & = \\hat { A } \\hat { x } ( t ) + \\hat { B } u ( t ) + \\sum _ { i = 1 } ^ m \\hat N _ i x ( t ) u _ i ( t ) , \\\\ y ( t ) & = \\hat { C } \\hat { x } ( t ) , \\ ; \\ ; \\ ; t \\geq 0 , \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} & \\widetilde { x } ^ { \\rm m a x } _ { 2 T } ( t ) < 0 \\mbox { f o r $ 1 \\leq t \\leq T - 1 $ } , \\\\ & \\widetilde { x } ^ { \\rm m a x } _ { 2 T } ( t ) = 0 \\mbox { a t $ t = T $ } , \\\\ & \\widetilde { x } ^ { \\rm m a x } _ { 2 T } ( t ) > 0 \\mbox { f o r $ T + 1 \\leq t \\leq 2 T - 1 $ } . \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} f _ G ( t ) = \\dot { r } _ G ( t ) f _ W ( r _ G ( t ) ) . \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{align*} a \\left ( x \\right ) : = 1 - 2 \\cos ^ { 2 } ( x - \\frac { \\pi } { 2 } ) , x \\in \\Omega : = ( - \\frac { \\pi } { 2 } , \\frac { \\pi } { 2 } ) . \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{align*} ( \\hat { \\phi } _ * ( \\tilde \\gamma ^ { - 1 } \\sigma ) ) ^ { - 1 } \\tilde \\gamma ^ { - 1 } \\sigma ) ^ { i } _ \\rho \\delta ^ { \\rho \\eta } ( \\hat { \\phi } _ * ( \\tilde \\gamma ^ { - 1 } \\sigma ) ) ^ { - 1 } \\tilde \\gamma ^ { - 1 } \\sigma ) ^ { j } _ \\eta = \\delta ^ { i j } . \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} \\alpha ( \\gamma , \\phi ) = \\int _ { R _ \\theta } \\phi ( z ) ( d z ) ^ 2 q ( z ) = - 2 \\ ( \\sinh \\frac { l ( \\gamma ) } { 2 } \\ ) \\int _ { 0 } ^ l \\phi ( e ^ t e ^ { i \\theta } ) e ^ { 2 t } e ^ { 2 i \\theta } d t . \\end{align*}"}
-{"id": "5622.png", "formula": "\\begin{align*} h ^ 0 ( K \\lambda ) = l + 3 ( g - 1 ) . \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} \\varphi _ 0 = \\mu + S ( \\kappa _ 0 ) \\mathrlap { . } \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} H ( q ) : = - \\sum _ { i = 1 } ^ k q _ i \\log q _ i . \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} T ( D ) = \\sum _ { j = 0 } ^ m c _ j ( - \\Delta ) ^ j \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} d X _ t & = - \\varepsilon ^ { - 1 } \\kappa _ X ( X _ t - Y _ t ) d t + \\varepsilon ^ { - 1 / 2 } \\sigma _ X d B ^ X _ t X _ 0 = 0 \\\\ d Y _ t & = - \\kappa _ Y ( Y _ t - X _ t ) d t + \\sigma _ Y d B ^ Y _ t Y _ 0 = 0 \\end{align*}"}
-{"id": "1036.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( \\tau _ { c } - \\tau _ { n } ) ^ { \\frac { 3 } { 2 } } \\rho _ { \\tau _ { n } } ( ( \\tau _ { c } - \\tau _ { n } ) ^ { \\frac { 1 } { 2 } } x + x _ n ) = \\lambda _ { \\infty } ^ { 3 } Q \\left ( \\lambda _ { \\infty } x \\right ) \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} \\mathcal { S } = \\{ s _ 2 s _ 3 s _ 1 s _ 2 s _ 1 , s _ 2 s _ 3 s _ 2 s _ 1 , s _ 2 s _ 3 s _ 1 , s _ 2 s _ 3 s _ 2 \\} \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{align*} L _ h : = D _ h P , \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\epsilon _ { n } ^ { s } \\int _ { \\mathbb { R } ^ 3 } V ( \\epsilon _ { n } x + x _ { j } ) w _ { n } ^ { ( j ) } ( x ) { \\rm d } x = - z _ { j } \\int _ { \\mathbb { R } ^ 3 } \\frac { w ( x ) } { | x | ^ s } { \\rm d } x & = - \\lambda ^ { s } z _ { j } \\int _ { \\mathbb { R } ^ 3 } \\frac { Q ( x + y _ 0 ) } { | x | ^ { s } } { \\rm d } x \\\\ & \\geq - \\lambda ^ { s } z \\int _ { \\mathbb { R } ^ 3 } \\frac { Q ( x ) } { | x | ^ { s } } { \\rm d } x \\end{align*}"}
-{"id": "2189.png", "formula": "\\begin{align*} = z _ { 2 } ^ { - 1 } \\left ( \\frac { z _ 1 - z _ 0 } { z _ 2 } \\right ) ^ { - j _ 2 / T } \\delta \\left ( \\frac { z _ { 1 } - z _ { 0 } } { z _ { 2 } } \\right ) I ( Y _ { 1 } ( u , \\ z _ { 0 } ) v , \\ z _ { 2 } ) . \\end{align*}"}
-{"id": "613.png", "formula": "\\begin{gather*} D : = \\sup _ { x \\in B _ R ( z ) } u ( x ) g ( x , R ) \\\\ \\intertext { w h e r e } g ( x , R ) : = \\biggl ( \\frac { R - d ( x , z ) } { R } \\biggr ) ^ { \\delta / \\alpha } , \\end{gather*}"}
-{"id": "1665.png", "formula": "\\begin{align*} \\lVert A B - A _ 0 B _ 0 \\rVert ^ 2 & = \\sum _ { i = 1 } ^ M N ( a _ i - a ^ 0 _ i ) ^ 2 \\\\ & = \\sum _ { i = 1 } ^ { M - 1 } N ( a _ i - a ^ 0 _ i ) ^ 2 + N \\left ( 1 - \\sum _ { i = 1 } ^ { M - 1 } a _ i - 1 + \\sum _ { i = 1 } ^ { M - 1 } a ^ 0 _ i \\right ) ^ 2 \\\\ & = \\sum _ { i = 1 } ^ { M - 1 } N ( a _ i - a ^ 0 _ i ) ^ 2 + N \\left \\{ \\sum _ { i = 1 } ^ { M - 1 } ( a _ i - a ^ 0 _ i ) \\right \\} ^ 2 . \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} | \\lambda | / 2 = N / 2 - ( d _ 1 + \\cdots + d _ N ) / 2 . \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{align*} W \\left ( x , y D _ x \\right ) = L \\left ( x , y D _ x \\right ) E \\left ( y D _ x \\right ) U \\left ( x q , y D _ x \\right ) \\Big | _ \\Delta \\end{align*}"}
-{"id": "1792.png", "formula": "\\begin{align*} T ( S ; \\nu ^ { - 1 } ) f & : = [ ( \\zeta \\circ \\nu ^ { - 1 } ) \\cdot S ( ( \\zeta \\circ \\nu ^ { - 1 } ) \\cdot ( f \\circ \\nu ^ { - 1 } ) ) ] \\circ \\nu \\\\ & = \\nu ^ * \\circ ( \\zeta \\circ \\nu ^ { - 1 } ) \\cdot S \\circ \\nu ^ { - 1 , * } \\zeta \\cdot f . \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} & | q _ { t , s } ( x , y ) - q _ { t , s } ( x ' , y ) | \\leq c _ 2 \\left ( | x - x ' | ^ { \\beta - \\gamma } \\wedge 1 \\right ) \\\\ & \\times \\Big ( ( \\varrho ^ 0 _ \\gamma + \\varrho ^ \\beta _ { \\gamma - \\beta } ) ( s - t , x - y ) + ( \\varrho ^ 0 _ \\gamma + \\varrho ^ \\beta _ { \\gamma - \\beta } ) ( s - t , x ' - y ) \\Big ) . \\end{align*}"}
-{"id": "5743.png", "formula": "\\begin{align*} \\mathbb { D } ( \\mathcal { G } _ 0 ) ( W ) \\xrightarrow { \\sim } \\mathbb { D } ( \\mathcal { G } ) ( S ) \\otimes _ S W & \\xrightarrow { \\sim } \\mathcal { M } ( \\mathfrak { M } ( \\mathcal { G } ) ) \\otimes _ S W \\\\ & = \\phi ^ * ( \\mathfrak { M } ( \\mathcal { G } ) ) \\otimes _ { \\mathfrak { S } } W \\\\ & = \\phi ^ * ( \\mathfrak { M } ( \\mathcal { G } ) ) / u \\phi ^ * ( \\mathfrak { M } ( \\mathcal { G } ) ) \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} I _ h = \\{ t _ 3 - t _ 1 , t _ 4 - t _ 1 , t _ 5 - t _ 1 , t _ 5 - t _ 2 , t _ 5 - t _ 3 \\} . \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} \\lambda _ j = \\overline { \\lambda } _ { j + N } \\mbox { I m } ( \\lambda _ j ) > 0 , \\mbox { R e } ( \\lambda _ { m i n } ) \\leq \\mbox { R e } ( \\lambda _ j ) < 0 , j = 1 , . . . , N . \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} q ( x | z _ i = 1 ) = \\sum _ { k = 1 } ^ { H _ 0 } b ^ 0 _ { k j } \\prod _ { i = 1 } ^ M ( a ^ 0 _ { i k } ) ^ { x _ { i } } , p ( x | z _ i = 1 , A , B ) = \\sum _ { k = 1 } ^ { H } b _ { k j } \\prod _ { i = 1 } ^ M ( a _ { i k } ) ^ { x _ { i } } . \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} I _ 3 ^ \\mathbb { Z } \\left ( K _ { 2 , 2 , 2 } , X _ { K _ { 2 , 2 , 2 } } \\right ) = \\langle x _ 1 , x _ 2 , x _ 3 , x _ 4 , x _ 5 , x _ 6 , 2 \\rangle . \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} \\mu = \\max \\left ( 0 , \\sum _ { j = 1 } ^ { L } m _ { j } - ( L - 1 ) N \\right ) , \\ldots , \\min _ { j } m _ { j } . \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{align*} K _ 1 ^ { \\nu _ 1 } K _ 2 ^ { \\nu _ 2 } \\cdots K _ N ^ { \\nu _ N } = \\sum _ { \\mu \\in \\lbrace 0 , 1 \\rbrace ^ N } q ^ { \\sum _ { m = 1 } ^ N ( \\nu _ m / 2 - \\mu _ m \\nu _ m ) } E _ \\mu ^ * . \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{align*} : b _ 1 ( n _ 1 ) \\cdots b _ k ( n _ k ) : = ( - 1 ) ^ { | \\sigma | } b _ { i _ 1 } ( n _ { i _ 1 } ) \\cdots b _ { i _ k } ( n _ { i _ k } ) \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{align*} 2 C \\star C = 2 g ( C ) - 2 + { \\rm { i n d } } ( C ) + h ( C ) + 2 e _ Q ( C ) + 4 \\delta ( C ) \\ge i , \\end{align*}"}
-{"id": "8953.png", "formula": "\\begin{align*} Y ^ { - 1 / 2 } S ( E ) Y ^ { - 1 / 2 } = \\widetilde { H } - E + Y ^ { - 1 / 2 } \\Pi H X ( E ) H \\Pi Y ^ { - 1 / 2 } \\end{align*}"}
-{"id": "901.png", "formula": "\\begin{align*} \\det ( \\Delta _ 0 + s ( s - 1 ) ) = Z ( s ) \\left ( e ^ { E - s ( s - 1 ) } \\frac { \\Gamma _ 2 ( s ) ^ 2 } { \\Gamma ( s ) } ( 2 \\pi ) ^ { s } \\right ) ^ { 2 g - 2 } , \\Re ( s ) > 1 . \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} & f ( x , y ) = \\begin{cases} x & , \\ ; x \\geq x ^ * , \\\\ \\frac { 1 } { 2 } e ^ { c x ^ * } ( x ^ * - \\frac { 1 } { c } ) e ^ { - c x } + \\frac { 1 } { 2 } e ^ { - c x ^ * } ( x ^ * + \\frac { 1 } { c } ) e ^ { c x } & , \\ ; - x ^ * < x < x ^ * , y \\geq 0 \\\\ 0 & , \\ ; x \\leq - x ^ * , \\end{cases} \\end{align*}"}
-{"id": "6513.png", "formula": "\\begin{align*} c _ d : = { d \\over 2 \\pi } \\int _ 0 ^ \\pi h _ d ( \\theta ) d \\theta = { 2 d \\over \\pi ( d - 2 ) ! } \\int _ { \\Im z > 0 } { \\max \\big ( - \\Re \\rho ( z ) , 0 \\big ) \\over | z | ^ { d + 2 } } d x d y . \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} w ( \\mathbf { c } ) = w _ { 0 } ^ { t _ { 0 } } w _ { 1 } ^ { t _ { 1 } } \\cdots w _ { p - 1 } ^ { t _ { p - 1 } } \\end{align*}"}
-{"id": "3569.png", "formula": "\\begin{align*} N _ { t } = \\tau \\kappa T + \\Big ( \\frac { \\kappa _ { s s } } { \\kappa } - \\tau ^ { 2 } \\Big ) B . \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i \\in J } { a _ i x _ i } \\right \\| = \\sum _ { i \\in J } | a _ i | , \\end{align*}"}
-{"id": "2056.png", "formula": "\\begin{align*} \\mathcal { H } ( t , \\ell ) = ( t , \\mathcal { H } _ t ( \\ell ) ) . \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} \\iota \\left ( \\mathrm { c c } _ { Y / S } ( A ) \\right ) = \\mathrm { t r } _ { \\tilde { c } } ( \\tilde { u } , A | _ { X _ { G } } ) . \\end{align*}"}
-{"id": "4513.png", "formula": "\\begin{align*} n \\sim \\bar \\tau \\log x \\qquad \\mbox { a s } x \\to \\infty . \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} \\varphi _ { \\Omega } = \\exp ( - \\frac 1 2 z ^ 2 \\eta _ 1 / \\omega _ 1 + \\pi i z / \\omega _ 1 ) \\sigma _ { \\Omega } , \\end{align*}"}
-{"id": "2761.png", "formula": "\\begin{align*} ( 1 - a _ { 1 0 } T ^ 1 ) ^ { p ^ 0 } \\mapsto ( [ a _ { 1 0 } ] ^ i ) _ { \\gcd ( i , p ) = 1 } . \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\int _ { \\Omega } f ( u - U _ { k } ) d x = 0 . \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{align*} \\mathcal { I } _ { a } = \\{ q \\in ( 0 , 1 ) : U _ { q } \\in \\mathcal { P } ^ { \\circ } \\} , \\end{align*}"}
-{"id": "4076.png", "formula": "\\begin{align*} p _ { R | B } ( r | \\mathtt { e } ) & = \\frac { q _ R ( r ) \\lambda ( r ) } { \\epsilon } \\\\ p _ { R | B } ( r | a ) & = \\frac { q _ { R | A } ( r | a ) \\ ! - \\ ! q _ { R } ( r ) \\lambda ( r ) } { 1 \\ ! - \\ ! \\epsilon } , \\quad \\forall a \\in \\mathcal { A } . \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} \\mathrm { P } ^ o _ { m , 2 } \\approx & 1 - \\sum ^ { N } _ { n = 1 } \\bar { w } _ n e ^ { - \\frac { c _ { n , \\mathcal { R } _ s } \\frac { 1 } { \\rho } } { \\tilde { \\tau } } } q \\left ( \\frac { c _ { n , \\mathcal { R } _ s } } { \\tilde { \\tau } } \\right ) , \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} \\left | p _ { K _ A K _ B } ( i , i ) - \\frac 1 2 \\right | \\leq \\delta \\ ; i = 1 , 2 \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{align*} \\lim _ { k } \\max _ { i \\in I } d ( x ^ { k } , C _ { i } ) = 0 \\Longrightarrow \\lim _ { k } d ( x ^ { k } , C ) = 0 \\end{align*}"}
-{"id": "6259.png", "formula": "\\begin{align*} w ( \\varepsilon ) \\in E _ { \\mu + \\varepsilon } ^ * V , & & \\varepsilon = ( \\varepsilon _ 1 , \\varepsilon _ 2 , \\ldots , \\varepsilon _ N ) , \\varepsilon _ m = \\begin{cases} 0 & , \\\\ & , \\end{cases} \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} H ( R / I ^ s , t ) = \\frac { { \\left ( 1 - \\beta _ 1 t ^ { 3 s } + \\beta _ 2 t ^ { 3 s + 1 } - \\beta _ 3 t ^ { 3 s + 2 } + \\beta _ 4 t ^ { 3 s + 3 } \\right ) } } { ( 1 - t ) ^ 4 } = \\frac { p ( t ) } { ( 1 - t ) ^ 4 } . \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} \\int _ M \\langle \\nabla _ { \\rm t r } ^ * \\nabla _ { \\rm t r } \\phi , \\psi \\rangle \\mu _ M = \\int _ M \\langle \\nabla _ { \\rm t r } \\phi , \\nabla _ { \\rm t r } \\psi \\rangle \\mu _ M \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} \\limsup _ { y \\to x } \\frac { | F ( y ) - F ( x ) - f ( x ) ( y - x ) | } { | \\varphi ( x + \\alpha ( y - x ) ) - \\varphi ( x ) | } \\le \\lim _ { y \\to x } \\frac { | F ( y ) - F ( x ) - f ( x ) ( y - x ) | } { \\alpha | y - x | } = 0 . \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} ( \\ , w , \\alpha \\ , ) \\ , \\mapsto \\ , \\displaystyle { \\frac { 1 } { N } } \\ , \\displaystyle { \\sum _ { i = 1 } ^ N } \\ , \\ell \\left ( y _ i - \\sigma ( w ^ T x ^ i + \\alpha _ i ) \\ , \\right ) - \\gamma \\ , \\displaystyle { \\sum _ { i = 1 } ^ m } \\ , h _ i ( w _ i ) , \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} \\left ( F _ { t } \\right ) ^ { \\perp } = \\triangle _ { g } F . \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{align*} w _ { e , \\alpha } \\{ x \\in E : \\tau ( x ) \\cap C _ { e , \\alpha } = \\{ 0 \\} \\} = 0 \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{align*} \\begin{array} { l } \\dot { S } _ 1 = \\displaystyle { \\frac { 1 } { 3 } ( 1 + 3 S _ 1 ) ( 3 S _ 2 - 3 S _ 1 + 2 ) } , \\\\ \\\\ \\dot { S } _ 2 = \\displaystyle { \\frac { 1 } { 9 } \\left ( 2 7 S _ 2 ^ 2 - 2 7 S _ 1 S _ 2 - 9 S _ 2 + 3 S _ 1 - 2 \\right ) } . \\\\ \\end{array} \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} H ^ * ( { G } _ { k } ( \\R ^ { n } ) , \\Q ) = \\frac { \\Q [ p _ 1 , p _ 2 , \\ldots , p _ { [ \\frac { k } { 2 } ] } ; \\bar { p } _ 1 , \\bar { p } _ 2 , \\ldots , \\bar { p } _ { [ \\frac { n - k } { 2 } ] } ] } { ( 1 + p _ 1 + \\cdots + p _ { [ \\frac { k } { 2 } ] } ) ( 1 + \\bar { p } _ 1 + \\cdots + \\bar { p } _ { [ \\frac { n - k } { 2 } ] } ) = 1 } \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{align*} P _ t f ( x ) = \\int _ M p _ t ( x , y ) f ( y ) \\ , d \\mu ( y ) , \\end{align*}"}
-{"id": "2854.png", "formula": "\\begin{align*} \\| g \\| _ { \\Box } : = \\max \\left \\{ \\| g _ { - 1 } \\| _ { \\Box , k - 1 } , \\ldots , \\| g _ { - k } \\| _ { \\Box , k - 1 } \\right \\} . \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\sup _ { | x | \\geq c t } u ( x , t ) = 0 \\forall \\ , \\ , c > c ^ { \\ast } _ { \\rm u p } , \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} \\square _ B = \\partial _ B \\partial _ B ^ * + \\partial _ B ^ * \\partial _ B \\quad { \\rm a n d } \\quad \\overline \\square _ B = \\bar \\partial _ B \\bar \\partial _ B ^ * + \\bar \\partial _ B ^ * \\bar \\partial _ B , \\end{align*}"}
-{"id": "1937.png", "formula": "\\begin{align*} g _ m ( M ) = \\mathrm { I d } , \\end{align*}"}
-{"id": "7495.png", "formula": "\\begin{align*} \\ln ( \\beta ( t , q _ t ) / \\beta ( s , q _ s ) ) = & \\int _ s ^ t \\partial _ r \\ln ( \\beta ( r , q _ r ) ) d r + \\int _ s ^ t \\nabla _ q \\ln ( \\beta ( r , q _ r ) ) \\circ d q _ r \\\\ = & \\int _ s ^ t ( \\beta ^ { - 1 } \\partial _ r \\beta ) ( r , q _ r ) d r + \\int _ s ^ t ( \\beta ^ { - 1 } \\nabla _ q \\beta ) ( r , q _ r ) \\circ d q _ r \\\\ = & \\int _ s ^ t ( \\beta ^ { - 1 } \\partial _ r \\beta ) ( r , q _ r ) d r - \\int _ s ^ t ( \\beta \\nabla _ q \\beta ^ { - 1 } ) ( r , q _ r ) \\circ d q _ r \\\\ \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} L ( x , y ) = 1 + \\sum _ { n = 1 } ^ \\infty L _ n ( y ) x ^ { n } U ( x , y ) = 1 + \\sum _ { n = 1 } ^ \\infty \\frac { U _ n ( y ) } { x ^ n } \\end{align*}"}
-{"id": "9170.png", "formula": "\\begin{align*} \\ \\ A _ m P _ n - P _ n A _ m \\gamma ^ { - n } = A _ m ( \\gamma ^ { - n } - q ^ { r n } ) \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{align*} A ^ { \\dagger } = F ^ \\ast ( F F ^ \\ast ) ^ { - 1 } ( E ^ \\ast E ) ^ { - 1 } E ^ \\ast . \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} ( \\eta ( \\tau ) , w ( \\tau ) ) : = ( \\eta _ \\tau , w _ \\tau ) , \\forall \\tau \\in [ 0 , T ] . \\end{align*}"}
-{"id": "5131.png", "formula": "\\begin{align*} v ( [ 1 \\ \\dots \\ t - 1 \\mid i _ 1 \\ \\dots \\ i _ { t - 1 } ] ) \\ = \\ v ( \\pi ) + ( n - i _ { t - 1 } ) v ( \\pi ) \\ = \\ ( n + 1 - i _ { t - 1 } ) v ( \\pi ) , \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} R i c _ B + \\nabla _ B \\nabla _ B \\beta = \\left [ ( n - 1 ) a + h ^ { - 1 } ( \\nabla _ B h ) \\beta - b h ^ { - 1 } \\right ] g _ B . \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\Phi _ n ^ { ( a , b ) } ( x , y | q ) \\frac { t ^ n } { ( q ; q ) _ n } = \\frac { ( a x t , b y t ; q ) _ \\infty } { ( x t , y t ; q ) _ \\infty } . \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{align*} \\dot { x } = \\omega ^ \\nu + O ( X ) , \\dot { X } = \\Lambda ^ \\nu X + O \\bigl ( | X | ^ 2 \\bigr ) , \\end{align*}"}
-{"id": "4538.png", "formula": "\\begin{align*} P _ L ^ * ( \\boldsymbol { \\epsilon } ) = \\sum _ { \\mathbf { m } } \\varphi _ { L } ( \\mathbf { m } , N , \\boldsymbol { \\epsilon } ) \\mathbb { P } _ L ^ * ( \\mathbf { m } , K ) \\prod _ { j = 1 } ^ { L } \\binom { N } { m _ j } , \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} W _ { \\chi } ( \\zeta u g k ) = \\zeta \\psi ( u ) W _ { \\chi } ( g ) \\end{align*}"}
-{"id": "1808.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty s _ i \\le 2 \\beta ^ { 2 } v n ^ 2 . \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{align*} \\nabla X ( t , y ) = I _ { 3 } + \\int _ { 0 } ^ { t } \\nabla \\Lambda ( s , X ( s , y ) ) \\nabla X ( s , y ) \\ d s \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} f ( \\lambda ) = \\lambda ^ \\alpha l ( \\lambda ) . \\end{align*}"}
-{"id": "2916.png", "formula": "\\begin{align*} n & = \\frac { ( h _ x , - 1 ) } { \\sqrt { 1 + h _ x ^ 2 } } , \\\\ \\kappa & = \\frac { d } { d x } \\frac { h _ x } { \\sqrt { 1 + h _ x ^ 2 } } = \\frac { h _ { x x } } { ( \\sqrt { 1 + h _ x ^ 2 } ) ^ 3 } , \\\\ V & = - \\frac { h _ { t } } { \\sqrt { 1 + h _ x ^ 2 } } . \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} \\Re ( \\phi ^ { V _ N U _ 1 } _ 0 \\ ! + \\dots + \\phi ^ { U _ k V _ S } _ 0 ) = \\Re ( f ^ { V _ N } _ 0 \\ ! - f ^ { U _ 1 } _ 0 \\ ! + \\dots + f ^ { U _ k } _ 0 \\ ! - f ^ { V _ S } _ 0 ) = \\Re ( f ^ { V _ N } _ 0 \\ ! - f ^ { V _ S } _ 0 ) \\rlap { . } \\end{align*}"}
-{"id": "8960.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 } \\ , \\sup _ { 0 \\le t \\le T } \\ , \\inf _ { \\varphi \\in S _ h } \\{ \\abs { u ( t ) - \\varphi } _ \\infty + h ^ { - \\frac d 2 } \\norm { u ( t ) - \\varphi } \\} = 0 . \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} \\varrho : = \\min _ { | z | = r } | g ( z ) | > 1 . \\end{align*}"}
-{"id": "2312.png", "formula": "\\begin{align*} \\left [ 1 - e ^ { - p _ 1 ( t _ 2 \\wedge \\cdots \\wedge t _ g ) } \\right ] ^ { M _ 1 } = \\sum _ { k _ 1 = 0 } ^ { M _ 1 } ( - 1 ) ^ { k _ 1 } \\binom { M _ 1 } { k _ 1 } e ^ { - k _ 1 p _ 1 ( t _ 2 \\wedge \\cdots \\wedge t _ g ) } . \\end{align*}"}
-{"id": "7327.png", "formula": "\\begin{align*} K _ Y + \\pi _ * ^ { - 1 } \\Delta = \\pi ^ * ( K _ X + \\Delta ) + a E \\\\ K _ { \\cal F _ Y } + \\pi _ * ^ { - 1 } \\Delta = \\pi ^ * ( K _ { \\cal F } + \\Delta ) + b E \\\\ K _ { Y ' } + \\pi '^ { - 1 } _ * \\Delta ' = \\pi '^ * ( K _ { X ' } + \\Delta ' ) + a ' E ' \\\\ K _ { \\cal F _ { Y ' } } + \\pi '^ { - 1 } _ * \\Delta ' = \\pi '^ * ( K _ { \\cal F ' } + \\Delta ' ) + b ' E ' . \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} y _ { m } \\left ( x \\right ) = y _ { 0 } \\left ( x \\right ) + \\frac { 1 } { \\Gamma \\left ( \\alpha \\right ) } \\int _ { a } ^ { x } \\psi ^ { \\prime } \\left ( t \\right ) \\left ( \\psi \\left ( x \\right ) - \\psi \\left ( t \\right ) \\right ) ^ { \\alpha - 1 } f \\left ( t , y _ { m - 1 } \\left ( t \\right ) \\right ) d t , m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{align*} u ( t , X ( t ) ) - u ( 0 , X ( 0 ) ) = & \\int _ { 0 } ^ { t } \\partial _ t u ( s , X ( s ) ) \\dd { s } + \\int _ { 0 } ^ { t } \\nabla u ( s , X ( s ) ) ^ \\top b ( s ) \\dd { s } \\\\ & + \\int _ { 0 } ^ { t } \\nabla u ( s , X ( s ) ) ^ \\top \\sigma ( s ) \\dd { W } ( s ) \\\\ & + \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ { d } \\int _ { 0 } ^ { t } \\partial _ i \\partial _ j u ( s , X ( s ) ) a ^ { i , j } ( s ) \\dd { s } . \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} \\mathcal { P } _ u ( \\lambda , \\gamma ) = \\Phi _ { u , \\gamma } ( 0 , h , \\lambda ) - \\lambda u - \\frac { 1 } { 2 } \\int _ 0 ^ u \\xi '' ( s ) s \\gamma ( s ) d s , \\end{align*}"}
-{"id": "1123.png", "formula": "\\begin{align*} \\mathcal L _ \\partial ( \\psi ) = \\partial _ M \\circ \\psi - \\psi \\circ \\partial ^ { - n } \\ , . \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} \\lVert A B - A _ 0 B _ 0 \\rVert ^ 2 & \\sim \\sum _ { i = 1 } ^ { M - 1 } x _ i ^ 2 + \\sum _ { j = 2 } ^ { N } \\sum _ { i = 1 } ^ { M - 1 } \\{ x _ i - ( a _ i b _ 1 - a ^ 0 _ i b ^ 0 _ 1 - a _ i b _ j + a ^ 0 _ i b ^ 0 _ j ) \\} ^ 2 \\\\ & \\sim \\sum _ { i = 1 } ^ { M - 1 } \\left \\{ x _ i ^ 2 + \\sum _ { j = 2 } ^ { N } ( a _ i b _ 1 - a ^ 0 _ i b ^ 0 _ 1 - a _ i b _ j + a ^ 0 _ i b ^ 0 _ j ) ^ 2 \\right \\} \\\\ & = \\sum _ { i = 1 } ^ { M - 1 } \\left [ x _ i ^ 2 + \\sum _ { j = 2 } ^ { N } \\{ a _ i ( b _ j - b _ 1 ) - a ^ 0 _ i ( b ^ 0 _ j - b ^ 0 _ 1 ) \\} ^ 2 \\right ] . \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} f ^ * ( \\alpha ) = \\lambda \\cdot \\alpha , \\ \\ 0 < k \\leq \\biggl [ \\frac { m + n } { 2 } \\biggl ] . \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} \\det \\Delta _ M = \\prod _ { \\l _ k \\neq 0 } \\l _ k . \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{align*} W ( x , y ) = a _ 0 M _ { n , d } + a _ 1 M _ { n , d + 1 } + \\cdots + a _ { n - 2 d + 2 } M _ { n , n - d + 2 } \\end{align*}"}
-{"id": "129.png", "formula": "\\begin{align*} A _ \\infty ( q , \\eta ) = A _ \\infty ( q ) + \\eta , \\Phi _ \\infty ( q , \\eta ) = \\Phi _ \\infty ( q ) . \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} \\iint M _ { i , j } f _ i f _ j = Q _ { i , j } ( F _ i , F _ j ) + O ( \\delta ^ { 3 } ) . \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} n _ { a } = \\begin{cases} | D _ { a } \\cup \\lbrace 0 \\rbrace | & \\qquad \\ a = 0 , \\\\ | D _ { a } | , & \\qquad \\ a \\ne 0 . \\end{cases} \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{align*} E \\left [ \\phi \\left ( \\sum _ { k = 1 } ^ m U _ k \\right ) \\right ] = \\int _ 0 ^ { \\infty } \\phi ( \\xi ) \\ , \\frac { \\xi ^ { m - 1 } } { ( m - 1 ) ! } \\ , e ^ { - \\xi } d \\xi , \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} A ^ T \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} = \\frac { \\hat { u } ' - \\displaystyle \\frac { \\hat { u } } { u _ 3 } u _ 3 ' } { \\| u ' \\| _ J } . \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} - 3 a _ { 2 , 6 } ^ 2 + a _ { 2 , 6 } a _ { 3 , 8 } + 2 a _ { 1 , 4 } a _ { 3 , 8 } = 0 . \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} u ( t , x ) = - 2 \\frac { \\partial } { \\partial x } \\log \\psi ( t , x ) , \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} f ( b g ) = ( \\delta ^ { 1 / 2 } \\chi ) ( b ) f ( g ) \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} U ( x , y , t ) = \\int _ { 0 } ^ { \\infty } \\int _ { \\R ^ { n } } P ^ { s } _ { y } ( z , \\tau ) u ( x - z , t - \\tau ) d z d \\tau , \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ g p _ j ( M _ j - m _ j ) \\big [ u ( m + e _ j ) - u ( m ) \\big ] = 0 \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} \\mbox { f i n d } x _ 0 \\in K ( x _ 0 ) \\mbox { s u c h t h a t } ~ ~ ~ \\min _ { z \\in K ( x _ 0 ) } h ( z ) = h ( x _ 0 ) . \\end{align*}"}
-{"id": "5294.png", "formula": "\\begin{align*} J = J _ { \\mathrm { f s u } } + J _ { \\mathrm { n i l } } . \\end{align*}"}
-{"id": "3848.png", "formula": "\\begin{align*} L _ g ( y _ 0 , \\delta ^ { - 1 } \\omega _ 0 ) = 2 \\log 2 \\delta - 2 \\log | \\omega _ 0 | _ { h _ 0 } + O ( \\delta ) . \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} \\dot { \\tilde { x } } ( t ) & = \\tilde { A } ( \\varepsilon ) \\tilde { x } ( t ) + \\tilde { B } u ( t ) + \\displaystyle \\sum _ { j = 1 } ^ { n _ { \\rm i n } } u _ j ( t ) \\tilde { N } _ j \\tilde { x } ( t ) + \\tilde { H } ( \\tilde { x } ( t ) \\otimes \\tilde { x } ( t ) ) \\\\ \\tilde { y } ( t ) & = \\tilde { c } ^ \\top \\tilde { x } ( t ) \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n a _ { i j } \\frac { \\partial ^ 2 \\delta ^ \\gamma } { \\partial x _ i \\partial x _ j } = \\gamma ( \\gamma - 2 ) \\delta ^ { \\gamma - 4 } \\sum _ { i , j = 1 } ^ n a _ { i j } \\sum _ { k = 1 } ^ m b _ { k i } y _ k \\sum _ { l = 1 } ^ m b _ { l j } y _ l + \\gamma \\delta ^ { \\gamma - 2 } \\sum _ { i , j = 1 } ^ n a _ { i j } \\sum _ { k = 1 } ^ m b _ { k i } b _ { k j } . \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} \\begin{cases} \\operatorname { d i v } ( y ^ a \\nabla D _ i U ) = y ^ a ( D _ i U ) _ t , \\\\ \\underset { y \\to 0 } { \\lim } y ^ a ( D _ i U ) _ y = - D _ i V u - V D _ i u . \\end{cases} \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{align*} \\dim D ^ { I I } _ m = n ( 2 n + 2 \\epsilon - 1 ) , r = n , a = 4 , b = 2 \\epsilon , p = 4 n + 2 \\epsilon - 2 . \\end{align*}"}
-{"id": "7987.png", "formula": "\\begin{align*} \\frac { \\Gamma n _ t } { t } = \\frac { n _ t } { t / \\Gamma } \\simeq \\frac { 1 } { \\Delta t } , \\ ; \\ ; \\frac { n _ t } { t } \\gg 0 . \\end{align*}"}
-{"id": "1215.png", "formula": "\\begin{align*} & v _ { t t } - c ^ 2 \\ , [ \\Delta v - q v ] = 0 \\mbox { i n \\ , } Q ^ T , \\\\ & v | _ { t = T } = 0 , v _ t | _ { t = T } = y \\mbox { i n \\ , } \\Omega , \\\\ & v = 0 \\mbox { o n \\ , } \\Sigma ^ T \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} \\chi _ 1 ( N ) : = \\int _ 0 ^ { \\infty } e ^ { - \\left ( \\zeta ^ { \\frac { ( 1 - \\theta ) } { N } } - 1 \\right ) t } \\left [ 1 - \\left ( 1 - e ^ { - ( 1 - \\theta ) t / N } \\right ) ^ N \\right ] d t \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} \\lim _ { d \\rightarrow \\infty } E _ d ( P ) = \\sum _ { k = 0 } ^ \\infty \\frac { \\langle P , \\psi ^ k \\rangle } { q ^ k } , \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} x ^ r \\ , f \\left ( x \\right ) = g \\left ( \\tfrac { 1 } { x } \\right ) , \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} & \\alpha ^ 2 \\| f _ 0 \\| _ { 1 , r , r ' } \\le 1 , \\\\ & \\sum _ { m = 2 } ^ N c _ 0 ^ { \\frac { m } { 2 } } \\alpha ^ m \\| f _ m \\| _ { 1 , r , r ' } \\le 1 . \\end{align*}"}
-{"id": "1364.png", "formula": "\\begin{align*} \\xi _ { t , T } \\bigl ( u \\bigr ) = \\int _ t ^ T c \\bigl ( s , X _ s ^ { t , x ; u } , u _ s \\bigr ) d s + \\Psi ( X _ T ^ { t , x ; u } ) . \\end{align*}"}
-{"id": "8625.png", "formula": "\\begin{align*} N _ { b } ( a , c ) = \\biggl \\lbrace x \\in \\mathbb { F } _ { q } : T r ( x ^ { p ^ { \\alpha } + 1 } ) = a \\ \\ T r ( b x ) = c \\biggr \\rbrace . \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} M \\ , : = \\ , G / \\Gamma \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} W _ M ( X , D _ x ) = \\Lambda _ M ( X , D _ x ) ^ { - 1 } . \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} [ c , z _ 1 ] [ z _ 2 , z _ 3 , z _ 4 ] = - [ c , z _ 1 ] [ z _ 3 , z _ 2 , z _ 4 ] . \\end{align*}"}
-{"id": "859.png", "formula": "\\begin{align*} f _ { k - 1 } \\circ \\tilde \\sigma _ i ^ { k - 2 } = { \\sigma _ i } ^ { k - 1 } | _ { \\Delta _ { i , j } } = { \\sigma _ j } ^ { k - 1 } | _ { \\Delta _ { i , j } } = f _ { k - 1 } \\circ \\tilde \\sigma _ j ^ { k - 2 } , \\end{align*}"}
-{"id": "6284.png", "formula": "\\begin{align*} n = ( ^ { g _ 1 } h _ 1 ) ( ^ { g _ 2 } h _ 2 ) \\cdots ( ^ { g _ k } h _ k ) , \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} | S | = | U _ n | . \\end{align*}"}
-{"id": "19.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\beta , u } ^ \\delta ( \\lambda , \\alpha ) = \\Phi _ { \\beta , u , \\gamma } ^ \\delta ( 0 , h , \\lambda ) - \\lambda u - \\frac { 1 } { 2 } \\int _ 0 ^ u \\xi '' ( s ) s \\gamma ( s ) \\ , d s , \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} \\langle \\ ! \\langle p , b ^ i \\rangle \\ ! \\rangle \\leq \\langle \\ ! \\langle p , \\xi ^ i \\rangle \\ ! \\rangle + \\sum _ { j = 1 } ^ s \\alpha _ { i j } \\langle \\ ! \\langle p , a ^ j \\rangle \\ ! \\rangle \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} \\Delta _ B \\phi = 2 \\overline \\square _ B \\phi + \\mathcal { L } _ { H ^ { 1 , 0 } } \\phi - H ^ { 0 , 1 } \\lrcorner \\ , \\bar \\partial _ B \\phi . \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} K ( x , v ) A ( x , v ) = & \\ : - ( 1 - v + x v ) x v ^ 3 A ( x , 1 ) + \\frac { x ^ 2 v ^ 3 ( 1 - v ) ^ 2 } { 1 - x } B ( x , 1 ) \\\\ & - \\frac { x ^ 2 v ^ 3 ( 1 - v ) ( 1 - v + x v ) ( x v - 2 x + 1 ) } { 1 - x } , \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} d X _ t & = - \\varepsilon ^ { - 1 } \\nabla _ x V ( X _ t , Y _ t ) d t + { \\varepsilon } ^ { - 1 / 2 } \\sqrt { 2 { \\beta _ X ^ { - 1 } } } d B ^ X _ t \\\\ d Y _ t & = b _ Y ( X _ t , Y _ t ) d t + \\sqrt { 2 \\beta _ Y ^ { - 1 } } d B ^ Y _ t \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} \\eta = \\overline { v } _ { - h } ^ { p - 1 } \\psi ^ 2 \\chi ( t ) , \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} G _ { N } : = \\left \\{ u \\in X \\ , : \\ , \\forall \\ , \\ell \\in I : \\| u - D ^ \\ell ( E ^ \\ell ( u ) ) \\| _ { L ^ 1 } \\leq N \\cdot \\phi ( \\ell ) \\right \\} \\ , . \\end{align*}"}
-{"id": "7374.png", "formula": "\\begin{align*} T ( \\xi ) = \\lambda + ( \\partial _ 1 T ( 0 ) ) \\xi _ d + \\mathcal { O } ( | \\xi | ^ 2 ) \\end{align*}"}
-{"id": "5506.png", "formula": "\\begin{align*} \\dot { \\rho } & = R e ( \\lambda _ l ) \\rho + \\sum _ { m = 1 } ^ M R e ( \\beta _ m ) \\rho ^ { 2 m + 1 } + \\varepsilon r \\sin ( \\theta - \\phi ) , \\\\ \\dot { \\theta } & = I m ( \\lambda _ l ) + \\sum _ { m = 1 } ^ M \\mbox { I m } ( \\beta _ m ) \\rho ^ { 2 m } + \\frac { \\varepsilon } { \\rho } \\left ( r \\cos ( \\theta - \\phi ) \\right ) , \\\\ \\dot { \\phi } & = \\Omega . \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} ( [ \\nabla _ t , D ] \\psi ) ( X ) = R ^ { S M } ( { \\partial _ t , X } ) \\cdot \\psi + R ^ P ( \\nabla _ t \\phi , D _ X \\phi ) \\psi - \\frac { 1 } { 2 } D \\phi ( \\dot { g } ( X ) ) \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} \\lim _ { \\Re s \\to \\infty } \\frac { I I _ b ( s ) e ^ { s l ( \\gamma ) } } { s } = 0 . \\end{align*}"}
-{"id": "8889.png", "formula": "\\begin{align*} \\int _ X \\psi \\omega ^ n & = \\frac { C _ H ' } { 2 ^ n } \\int _ { \\Delta ' } \\psi ( d _ p u ^ * ) P _ { D H } ( 2 \\chi - p ) d p \\\\ & = C _ H ' \\int _ { \\chi + \\Delta ^ t \\cap \\bar { C } ^ + } \\psi ( d _ { 2 \\chi - 2 q } u ^ * ) P _ { D H } ( q ) d q . \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} J _ H ( x ) = \\prod _ { \\alpha \\in \\Phi _ s ^ + } | \\sinh ( 2 \\alpha ( x ) ) | . \\end{align*}"}
-{"id": "688.png", "formula": "\\begin{align*} ( A _ k X _ k B _ k ) _ { i j } = \\sum _ { \\ell = 1 } ^ { n } ( A _ k ) _ { i \\ell } \\sum _ { t = 1 } ^ n ( X _ k ) _ { \\ell t } ( B _ k ) _ { t j } = \\sum _ { \\ell = i } ^ n \\sum _ { t = j } ^ { n } ( A _ k ) _ { i \\ell } ( X _ k ) _ { \\ell t } ( B _ k ) _ { t j } , \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} \\lVert A B - A _ 0 B _ 0 \\rVert ^ 2 & \\sim \\sum _ { i = 1 } ^ { M - 1 } x _ i ^ 2 + \\sum _ { i = 1 } ^ { M - 1 } \\sum _ { j = 2 } ^ N ( a _ i b _ j - a ^ 0 _ i b ^ 0 _ j ) ^ 2 \\\\ & = \\sum _ { i = 1 } ^ { M - 1 } x _ i ^ 2 + \\sum _ { i = 1 } ^ { M - 1 } \\sum _ { j = 2 } ^ N f _ { i j } ^ 2 \\\\ & \\sim \\sum _ { i = 1 } ^ { M - 1 } x _ i ^ 2 + \\left ( f _ { 1 2 } ^ 2 + \\sum _ { i = 2 } ^ { M - 1 } f _ { i 2 } ^ 2 + \\sum _ { j = 3 } ^ N f _ { 1 j } ^ 2 \\right ) . \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} \\mathcal { L } _ X \\omega _ 1 = \\omega _ 2 \\qquad \\mathcal { L } _ X \\omega _ 2 = - \\ , \\omega _ 1 \\qquad \\mathcal { L } _ X \\omega _ 3 = 0 \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{align*} { \\bar { p } } _ M = \\inf _ { \\phi } { \\bar { p } } _ M ( \\phi ) . \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{align*} [ w ] _ { A _ p ( \\R \\times \\R ) } : = \\sup _ { R } \\left ( \\frac { 1 } { | R | } \\int _ { R } w ( x _ 1 , x _ 2 ) \\ d x \\right ) \\left ( \\frac { 1 } { | R | } \\int _ { R } w ( x _ 1 , x _ 2 ) ^ { 1 - p ' } \\ d x \\right ) ^ { p - 1 } < \\infty . \\end{align*}"}
-{"id": "8840.png", "formula": "\\begin{align*} \\Omega _ { \\alpha _ 1 , \\bar { \\alpha } _ 2 } = 0 . \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} \\mathbb { E } [ X ] = \\int _ 0 ^ { \\infty } \\mathbb { P } ( X > t ) \\ , d t \\ , . \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} \\Psi ( x , r ) = \\lbrace \\psi _ I : I \\in \\Psi _ r , \\psi _ I ( K ) \\cap B ( x , r ) \\neq \\emptyset \\rbrace , \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{align*} J ( t ) = I ( t ) + \\frac { ( 2 t - 1 ) ^ 2 } { 2 } \\end{align*}"}
-{"id": "9126.png", "formula": "\\begin{align*} \\frac { { r } ^ 3 } { c ^ 3 } P { r _ { { \\rm { I N S } } } } \\left ( { - 1 - \\frac { { { \\rho _ t } } } { c } } \\right ) + \\frac { { r } ^ 2 } { c ^ 2 } \\left ( { - 2 P { r _ { { \\rm { I N S } } } } - 1 } \\right ) + \\frac { r } { c } \\left ( { 2 P { r _ { { \\rm { I N S } } } } + P { r _ { { \\rm { I N S } } } } \\frac { { { \\rho _ t } } } { c } - 2 } \\right ) + P { r _ { { \\rm { I N S } } } } - 1 = 0 \\end{align*}"}
-{"id": "2615.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l l } R i c _ { B } + \\nabla _ { B } \\nabla _ { B } f - m h ^ { - 1 } \\nabla _ { B } \\nabla _ { B } h = \\lambda g _ { B } , \\\\ \\noalign { \\smallskip } \\lambda h ^ { 2 } = h ( \\nabla _ { B } h ) f - ( m - 1 ) | \\nabla _ { B } h | ^ { 2 } - h \\Delta _ { B } h + c ( m - 1 ) , \\\\ \\noalign { \\smallskip } R i c _ { F } = c ( m - 1 ) g _ { F } , \\end{array} \\right . \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} & \\sum _ { i = 0 } ^ { m } { m + 1 \\choose i + 1 } _ { \\ ! \\ ! q } \\prod _ { j = 1 } ^ { m - i } ( 1 - q ^ { k - 1 + j } r ) \\beta _ { ( i , m , k ) } q ^ { - ( i + 1 ) ( 2 k + 2 m - i ) / 2 } = 0 . \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} \\hat { G } \\bigl ( s , X _ s ^ { t , x ; u } , Y _ s ^ { t , x ; u } , Z _ s ^ { t , x ; u } \\bigr ) = c \\bigl ( s , X _ s ^ { t , x ; u } , u _ s \\bigr ) + G \\bigl ( s , Y _ s ^ { t , x ; u } , Z _ s ^ { t , x ; u } \\bigr ) . \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} \\sum _ { r = D + 1 } ^ { \\infty } \\frac { 1 } { r ^ 4 } \\leq \\int _ D ^ { \\infty } \\frac { 1 } { s ^ 4 } \\ , \\mathrm { d } s = \\frac { 1 } { 3 D ^ 3 } , \\sum _ { r = D + 1 } ^ { \\infty } \\frac { 1 } { r ^ 2 } \\geq \\int _ { D + 1 } ^ { \\infty } \\frac { 1 } { s ^ 2 } \\ , \\mathrm { d } s = \\frac { 1 } { D + 1 } \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} \\omega = - \\frac 1 2 \\sum _ { \\alpha = 1 } ^ { 2 n } \\theta ^ \\alpha \\wedge J \\theta ^ \\alpha , \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} \\big ( [ Y , \\varphi ] , [ V , A ] , [ T Y , \\theta ] , H \\big ) _ { t = 0 } = \\big ( [ Y , \\varphi ] , [ V , A ] , [ T Y , \\theta ] , H \\big ) ~ . \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} \\mathbb { P } [ K _ A = K _ B ] \\geq 1 - \\delta \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} K _ { Y ' } + \\pi '^ { - 1 } _ * \\Delta ' = g ^ * ( K _ Y + \\pi _ * ^ { - 1 } \\Delta ) + ( r - 1 ) E ' = \\\\ g ^ * ( \\pi ^ * ( K _ X + \\Delta ) + a E ) + ( r - 1 ) E ' . \\end{align*}"}
-{"id": "4426.png", "formula": "\\begin{align*} \\omega _ i ( A ^ d , B ^ e ) = 0 ( i = 1 , \\dots , h ) , \\end{align*}"}
-{"id": "6914.png", "formula": "\\begin{align*} L _ n ( x ) \\ = \\ \\sum _ { k = 0 } ^ n \\frac { 1 } { k ! } { n \\choose k } ( - x ) ^ k . \\end{align*}"}
-{"id": "3215.png", "formula": "\\begin{align*} \\mathcal { L } \\left ( \\varphi _ { k + 1 } \\right ) = \\Theta _ { k + 1 } , \\end{align*}"}
-{"id": "6474.png", "formula": "\\begin{align*} \\partial _ k \\varphi ( t ) = 2 \\overline { d ( t ) } \\cdot \\partial _ k d ( t ) \\in L ^ p ( \\Omega ) \\varphi ^ { \\prime } ( t ) = 2 \\overline { d ( t ) } \\cdot d ^ { \\prime } ( t ) \\in L ^ { \\frac { p } { 2 } } ( \\Omega ) . \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} \\mathrm { P } _ { m , i } = & \\mathrm { P } \\left ( z _ t < \\frac { \\epsilon _ 1 } { \\rho } \\right ) \\\\ & + \\underset { Q _ 1 } { \\underbrace { \\mathrm { P } \\left ( z _ t > \\frac { \\epsilon _ 1 } { \\rho } , z _ m < \\max \\left \\{ \\frac { \\epsilon _ 2 } { \\rho \\xi _ 2 } , \\cdots , \\frac { \\epsilon _ i } { \\rho \\xi _ { i } } \\right \\} \\right ) } } . \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} \\hat { C } : Y '^ 2 = h ^ 3 - 3 h ^ 2 - 4 h \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} & \\sum _ { x , y } ( \\mu ( x , y ) - \\gamma ( x , y ) ) \\log p _ { X Y } ( x , y ) \\\\ & \\quad = \\lambda \\sum _ { x , y } ( \\mu _ 1 ( x , y ) - \\gamma _ 1 ( x , y ) ) \\log p _ { X Y } ( x , y ) \\ ! + \\ ! ( 1 \\ ! - \\ ! \\lambda ) \\sum _ { x , y } ( \\mu _ 2 ( x , y ) \\ ! - \\ ! \\gamma _ 2 ( x , y ) ) \\log p _ { X Y } ( x , y ) . \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{align*} I ( \\gamma ) = I _ 0 ( \\gamma ( 0 ) ) + \\sup _ { k \\geq 1 } \\sup _ { \\substack { 0 = t _ 0 < t _ 1 < \\dots , t _ k \\\\ t _ i \\in \\Delta _ \\gamma ^ c } } \\sum _ { i = 1 } ^ { k } I _ { t _ i - t _ { i - 1 } } ( \\gamma ( t _ i ) \\ , | \\ , \\gamma ( t _ { i - 1 } ) ) . \\end{align*}"}
-{"id": "8745.png", "formula": "\\begin{align*} H f ( t ) = \\frac { 1 } { \\pi } \\lim _ { \\epsilon \\mapsto 0 } \\int _ { | s | > \\epsilon } \\frac { f ( t - s ) } { s } \\ d s , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} \\lambda _ n ^ + ( M ( a ) ) = \\frac { \\varkappa ( \\alpha ) } { n ^ \\alpha } + o ( n ^ { - \\alpha } ) , \\lambda _ n ^ - ( M ( a ) ) = O ( n ^ { - \\alpha - 1 } ) , n \\to \\infty , \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} d _ 2 ( ( x _ 1 , y _ 1 , z _ 1 ) , ( x _ 2 , y _ 2 , z _ 2 ) ) = ( ( x _ 1 - x _ 2 ) ^ 2 + ( y _ 1 - y _ 2 ) ^ 2 ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{align*} | \\{ u _ i \\leq M \\} \\cap A _ 2 | & \\leq \\left ( \\frac { C h ( \\hat { Y _ i } ) } { u _ i ( \\hat { Y _ i } ) + \\frac { 1 } { i } - M } \\right ) ^ \\epsilon \\\\ & \\leq \\left ( \\frac { C } { i ( u _ i ( \\hat { Y _ i } ) + \\frac { 1 } { i } - M ) } \\right ) ^ \\epsilon \\\\ & \\leq \\left ( \\frac { C } { i ( 1 - M ) } \\right ) ^ \\epsilon \\end{align*}"}
-{"id": "5570.png", "formula": "\\begin{align*} \\sum _ { k \\geq 1 } ( 2 k - 1 ) ^ 2 q ^ { 2 k - 1 } & = \\sum _ { k \\geq 1 } k ^ 2 q ^ k - \\sum _ { k \\geq 1 } ( 2 k ) ^ 2 q ^ { 2 k } \\\\ & = q \\frac { 1 + q } { ( 1 - q ) ^ 3 } - 4 q ^ 2 \\frac { 1 + q ^ 2 } { ( 1 - q ^ 2 ) ^ 3 } \\\\ & = q \\frac { 1 + 6 q ^ 2 + q ^ 4 } { ( 1 - q ^ 2 ) ^ 3 } . \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} ( \\omega _ i ) = - D _ i + \\Bigl ( q ^ 2 p ^ { a _ i } - 2 \\Bigr ) \\cdot Z _ i + \\Bigl ( p ^ { b _ i } - 2 \\Bigr ) \\cdot N _ i , \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} \\Psi ( \\phi _ A ( f ) ) = \\Psi ( d _ A ( \\tau _ A ( f ) ) ) = d _ B ( \\tau _ B ( \\Psi ( f ) ) ) = \\phi _ B ( \\Psi ( f ) ) , \\end{align*}"}
-{"id": "6250.png", "formula": "\\begin{align*} \\sum _ { s ' \\in S _ \\mu \\setminus \\lambda } | S _ \\mu ( s ' - 1 ) \\setminus \\lambda | = \\left ( \\sum _ { s ' \\in S _ \\mu \\setminus \\lambda ' } | S _ \\mu ( s ' - 1 ) \\setminus \\lambda ' | \\right ) - | S _ \\mu \\setminus \\lambda | , \\end{align*}"}
-{"id": "848.png", "formula": "\\begin{align*} \\frac { ( 1 - t _ { n } ^ { p } ) } { p } [ v _ { n } ] ^ { p } _ { W ^ { s , p } ( \\R ^ { N } ) } = o _ { n } ( 1 ) . \\end{align*}"}
-{"id": "7263.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ K \\omega _ k v _ k ( \\zeta \\gamma ) _ { k j } & = \\sum _ { k = 1 } ^ K \\sum _ { i = 1 } ^ { K - 1 } \\omega _ k v _ k \\left ( \\frac { 1 } { 1 - v _ k \\ , \\lambda _ i } - \\frac { 1 } { 1 + v _ k \\ , \\lambda _ i } \\right ) \\gamma _ { i j } = 0 . \\end{align*}"}
-{"id": "1924.png", "formula": "\\begin{align*} A ( x , v ) & = x ^ 2 v ^ 3 + x v A ( x , v ) + x v B ( x , v ) + x v C ( x , v ) , \\\\ B ( x , v ) & = x ^ 2 v ^ 3 + v ^ 2 b _ 2 ( x ) + \\frac { x } { 1 - v } ( v ^ 3 B ( x , 1 ) - v B ( x , v ) ) + \\frac { x } { 1 - v } ( v ^ 3 A ( x , 1 ) - v ^ 2 A ( x , v ) ) , \\\\ C ( x , v ) & = x ( B ( x , v ) - v ^ 2 b _ 2 ( x ) ) + \\frac { x } { 1 - v } ( v ^ 3 C ( x , 1 ) - v C ( x , v ) ) , \\end{align*}"}
-{"id": "7177.png", "formula": "\\begin{align*} x = \\sum _ { i = k } ^ \\infty a _ i m ^ i , \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{align*} r ^ - ( M , s , t ) = r ^ { - + } ( M , s , t ) = \\mathrm { r a n k } \\left ( M ( s , t ) \\right ) - 1 . \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{align*} r ( M , s , t ) - 1 = r ^ + ( M , s , t ) . \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} h ( t , q , z ) = \\left ( \\frac { \\beta ( t , q ) } { 2 \\pi } \\right ) ^ { n / 2 } e ^ { - \\beta ( t , q ) \\| z \\| ^ 2 / 2 } . \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\to 0 } \\dfrac 1 { \\epsilon } \\int _ { \\Omega _ { r , \\epsilon } } f \\Delta \\rho = \\int _ { \\rho ^ { - 1 } ( r ) } f \\Delta \\rho \\end{align*}"}
-{"id": "7158.png", "formula": "\\begin{align*} { \\mathcal M } ( \\mu ) = \\int { \\left ( \\int _ { S ^ { d - 1 } } { | \\widehat { \\mu } ( r \\omega ) | } ^ 2 d \\omega \\right ) } ^ 2 r ^ { d - 1 } d r < \\infty . \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} \\theta ( r ) = c ( R - r ) ^ d + O ( ( R - r ) ^ { d + 1 } ) , c \\ne 0 , \\end{align*}"}
-{"id": "2507.png", "formula": "\\begin{align*} \\lambda ( A + A ) = \\mu ( \\tilde { A } + \\tilde { A } ) + \\mu ( \\Sigma _ 2 ) , \\end{align*}"}
-{"id": "1956.png", "formula": "\\begin{align*} \\| f \\| _ { B ^ { \\infty } } = \\sum _ { | \\alpha | < m } | \\partial ^ { \\alpha } f ( 0 ) | + \\sup _ { x \\in \\mathbf { B } } ( 1 - | x | ^ { 2 } ) ^ { m } | \\partial ^ { m } f ( x ) | , \\enspace p = \\infty . \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} \\lambda _ k ( \\mathcal { R } _ { \\alpha , \\Omega } ) = s _ k ( \\mathcal { R } _ { \\alpha , \\Omega } ) \\leq C | \\Omega | ^ { \\frac { \\alpha } { n } } k ^ { - \\frac { \\alpha } { n } } . \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} \\rho ^ \\pi ( x ) v : = \\sum _ j \\big ( \\rho ^ { \\pi , b _ j } ( y _ j ) \\rho ^ { \\pi , c _ j } ( z _ j ) v - \\beta ( b _ j , c _ j ) \\rho ^ { \\pi , c _ j } ( z _ j ) \\rho ^ { \\pi , b _ j } ( y _ j ) v \\big ) . \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} F _ { m a x } ( r _ G ( t ) ) = \\frac { S _ { m a x } } { \\sqrt { 2 \\pi } } \\int _ 0 ^ { r _ G ( t ) } \\frac { e ^ { - \\frac { S ^ 2 _ { m a x } } { 2 s } } } { s ^ { \\frac { 3 } { 2 } } } d s & & F _ { m i n } ( r _ G ( t ) ) = \\frac { S _ { m i n } } { \\sqrt { 2 \\pi } } \\int _ 0 ^ { r _ G ( t ) } \\frac { e ^ { - \\frac { S ^ 2 _ { m i n } } { 2 s } } } { s ^ { \\frac { 3 } { 2 } } } d s . \\end{align*}"}
-{"id": "4153.png", "formula": "\\begin{align*} \\mathrm { L i p } _ \\sigma ( g ) : = \\sup _ { x , y \\in \\Omega , x \\neq y } \\frac { | g ( x ) - g ( y ) | } { | x - y | ^ \\sigma } g : \\Omega \\subset \\R ^ d \\to \\R \\ , . \\end{align*}"}
-{"id": "2735.png", "formula": "\\begin{align*} \\bar { x } = c \\beta + \\sum _ { \\substack { ( i , j ) \\in \\mathbb { I } } } c _ { i j } S ^ i T ^ j , \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} x ( y z \\cdot x ) = ( x ^ { \\lambda } \\backslash y ) \\cdot z x \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{align*} l = | B _ \\mu | - | \\lambda \\cap S _ \\mu ( m - 1 ) | - | \\lambda \\cap T _ \\mu ( m + 1 ) | + | \\lambda | / 2 \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} S q _ { \\mathbb { F } _ 4 } ( 1 ) & = 1 , \\\\ S q _ { \\mathbb { F } _ 4 } ( \\alpha \\xi ) & = \\alpha \\xi + \\alpha \\xi ^ 2 , \\\\ S q _ { \\mathbb { F } _ 4 } ( ( \\alpha + 1 ) \\xi ^ 2 ) & = ( \\alpha + 1 ) \\xi ^ 2 + 2 ( \\alpha + 1 ) \\xi ^ 3 + ( \\alpha + 1 ) \\xi ^ 4 . \\end{align*}"}
-{"id": "1317.png", "formula": "\\begin{align*} a _ \\tau ( u ^ { B , } _ h , v _ h ) = ( \\rho g , v _ h ) _ \\tau v _ h \\in V _ h ^ { B , } . \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{align*} N _ k ( \\lambda ) = \\sup \\limits _ { L \\in { \\cal L } _ \\lambda ^ { ( k ) } } \\mbox { d i m } \\ , L = \\sup \\limits _ { E \\in { \\cal P } _ \\lambda ^ { ( k ) } } \\mbox { T r } \\ , E , \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{align*} C _ { i j } & = \\sum _ { u \\in E } c _ u ^ { ( i j ) } \\nu ( u ) + \\sum _ { ( u , v ) \\in E ^ 2 } ( f _ u ^ { ( i ) } f _ v ^ { ( j ) } + f _ u ^ { ( j ) } f _ v ^ { ( i ) } ) \\nu _ 2 ( u , v ) \\\\ \\Gamma _ { i j } & = \\sum _ { u \\in E } \\gamma _ u ^ { ( i j ) } \\nu ( u ) + \\frac { 1 } { 2 } \\sum _ { ( u , v ) \\in E ^ 2 } ( f _ u ^ { ( i ) } f _ v ^ { ( j ) } - f _ u ^ { ( j ) } f _ v ^ { ( i ) } ) \\nu _ 2 ( u , v ) \\end{align*}"}
-{"id": "9197.png", "formula": "\\begin{align*} A \\exp ( P ) \\exp ( \\ell ' ) \\Big | _ \\Delta = \\exp ( P ) \\exp ( \\ell ) \\Big | _ \\Delta A \\Leftrightarrow A P + A \\ell ' = P A + \\ell A \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} D : = \\left \\{ ( x , y , z ) \\in \\R _ { > 0 } \\times \\R \\times \\R \\ \\big | \\ \\frac { x y } { 2 } \\notin \\pi ( 2 \\Z + 1 ) z > 0 \\right \\} , \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} b _ { 3 n } & = b _ n , \\\\ b _ { 3 n + 1 } & = \\tau \\cdot b _ n + b _ { n + 1 } , \\\\ b _ { 3 n + 2 } & = b _ n + \\tau \\cdot b _ { n + 1 } \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} \\sigma _ { \\nu } ( i ) = \\left \\{ \\begin{array} { l l } i + 2 & \\textup { i f } i < \\nu _ 1 \\\\ i & \\textup { i f } i > \\nu _ 1 + 1 \\end{array} \\right . . \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\frac { H _ { \\pm n } } { z ^ { \\pm n } } = \\exp \\left ( \\sum _ { n = 1 } ^ \\infty \\frac { P _ { \\pm n } } { n z ^ { \\pm n } } \\right ) \\Big | _ \\Delta \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} [ c , x _ 1 ] \\bigl ( g ( x _ 2 , x _ 3 , x _ 4 , x _ 5 ) + g ( x _ 2 , x _ 3 , x _ 5 , x _ 4 ) + g ( x _ 2 , x _ 4 , x _ 5 , x _ 3 ) \\bigr ) = 0 . \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{align*} \\varphi ( z , \\overline z ) = s ^ \\mu \\widetilde \\varphi ( s ^ { - 1 / p } , \\zeta ) , \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{align*} g _ \\varphi = \\sqrt { h } \\ , g _ B + h ^ { - 1 } \\theta ^ 2 , \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} I _ 2 ( N ) = o \\left ( \\frac { 1 } { N ^ { \\kappa } } \\right ) \\ ; \\kappa > 0 . \\end{align*}"}
-{"id": "6045.png", "formula": "\\begin{align*} \\begin{array} { l c l c } \\Gamma : & C ( [ 0 , \\beta ] ; \\overline { X } _ 2 ) & \\longrightarrow & C ( [ 0 , \\beta ] ; \\overline { X } _ 2 ) \\\\ & ( \\eta , w ) & \\longmapsto & \\Gamma ( \\eta , w ) = ( u , v ) \\end{array} \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} \\phi _ l ^ 2 + k = \\tau \\phi _ l \\textrm { , w h e r e } \\tau \\equiv t \\textrm { m o d } l , \\ , k \\equiv q \\textrm { m o d } l \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{align*} & ( 1 - 2 x ) H ( x , v ) - x ( 1 - 2 x ) F _ T ( x ) \\\\ & = x v ( 1 - x ) H ( x , v ) + \\frac { x v ( 1 - x ) } { 1 - v } \\big ( H ( x , 1 ) - H ( x , v ) \\big ) - \\frac { x ^ 2 v } { 1 - v } \\big ( H ( x , 1 ) - v H ( x , v ) \\big ) , \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{align*} \\lim _ { y \\to \\frac { 1 } { \\beta } ^ { - } , \\ , \\gamma \\to \\beta ^ { - } } T _ { \\gamma } ( y ) = 1 , \\lim _ { y \\to \\frac { 1 } { \\beta } ^ { + } , \\ , \\gamma \\to \\beta ^ { + } } T _ { \\gamma } ( y ) = T _ { \\beta } ( \\frac { 1 } { \\beta } ) = 0 ; \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} \\sum _ { t \\in T _ \\mu } | T _ \\mu ( t + 1 ) | = 0 + 1 + \\cdots + ( | \\mu | - 1 ) = \\frac { | \\mu | ( | \\mu | - 1 ) } { 2 } . \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} F ( \\Theta _ 1 ) = \\Theta _ { d _ 1 } \\times \\{ 0 \\} ^ { d _ 2 - d _ 1 } \\subseteq \\Theta _ { d _ 2 } , \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{align*} \\epsilon _ 1 = \\min _ { x , y } \\tilde { \\delta } _ { x , y } \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{align*} \\begin{aligned} \\log { \\sum _ i p _ i e ^ { \\varpi ( \\frac { p _ i } { q _ i } - 1 ) } } - \\sum _ i p _ i \\log { \\frac { p _ i } { q _ i } } \\ge 0 , \\end{aligned} \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} \\frac { 1 } { 3 ^ { b + 1 } \\alpha } \\max \\{ \\vert 1 2 z _ 3 \\vert , \\vert z _ 1 + z _ 2 \\vert \\} = \\max \\{ \\vert 4 \\beta \\vert , \\vert 3 ^ { 2 b } \\alpha ^ 2 \\vert \\} \\geq e ^ { h ( \\tilde { x } ) } > e ^ { 0 . 5 N ^ 2 - 4 . 4 4 4 } . \\end{align*}"}
-{"id": "3473.png", "formula": "\\begin{align*} U ( x ) : = \\int _ { x } ^ \\infty \\operatorname { s g n } ( t ) | f ( t ) | ^ 2 \\ , \\mathrm d t \\quad V ( x ) : = \\int _ x ^ \\infty | f ' ( t ) | ^ 2 + q ( t ) | f ( t ) | ^ 2 \\ , \\mathrm d t . \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{align*} \\ \\ \\ \\ \\ \\ u \\mapsto I ( u , \\ z ) = \\sum _ { n \\in \\frac { 1 } { T } \\mathbb { Z } } u ( n ) z ^ { - n - 1 - \\lambda _ 1 - \\lambda _ 2 + \\lambda _ 3 } \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} \\Psi _ { \\Gamma } \\circ \\Psi _ { \\Gamma ' } = \\Psi _ { \\Gamma '' } : K _ Z \\longrightarrow K _ X \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{align*} & \\left ( U ^ T \\left ( \\frac { 3 n + 2 } { 6 } - \\int _ 0 ^ \\infty T r [ \\gamma e ^ { - 2 y \\gamma } ] e ^ { - y \\gamma } d y \\right ) U \\right ) ^ { i j } = \\left ( \\frac { 3 n + 2 } { 6 } - \\sum _ l \\frac { \\lambda _ l } { 2 \\lambda _ l + \\lambda _ i } \\right ) \\delta ^ { i j } . \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} Q ( t ) = Q _ L ( t ) + Q _ M ( t ) + Q _ R ( t ) . \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} Z ( \\epsilon _ 1 , \\epsilon _ 2 , a , \\mathfrak { q } , \\lambda ) = \\sum _ { n = 0 } ^ \\infty z ^ n \\sum _ { i = 1 } ^ { 2 n r } ( - 1 ) ^ i \\mathrm { c h } \\ , H ^ i ( \\mathcal M ( r , n ) , \\mathcal O ) , z = \\mathfrak { q } \\lambda ^ { 2 r } e ^ { - \\lambda ( \\epsilon _ 1 + \\epsilon _ 2 ) r / 2 } , \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{align*} \\overline { \\Delta } _ { i _ k } = \\bigcup _ { i _ j \\leq i _ k } { \\Delta } _ { i _ j } . \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} R ( b _ 1 , b _ 2 ) = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & b _ 1 & \\epsilon b _ 2 & 0 \\\\ 0 & b _ 2 & b _ 1 & 0 \\\\ 0 & 0 & 0 & b _ 1 ^ 2 - \\epsilon b _ 2 ^ 2 \\end{pmatrix} , \\end{align*}"}
-{"id": "3288.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } e ^ { - \\frac { a ^ 2 t ^ 2 } { 2 } \\ , - \\frac { b ^ 2 } { 2 t ^ 2 } } d t = \\sqrt { \\frac { \\pi } { 2 } } \\frac { 1 } { a } e ^ { - a b } . \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} \\zeta ( s , a ) = \\frac { 1 } { 2 } \\ ; a ^ { - s } + \\frac { a ^ { 1 - s } } { s - 1 } + 2 \\int _ 0 ^ \\infty ( a ^ 2 + y ^ 2 ) ^ { - s / 2 } \\sin \\left ( s \\arctan { \\frac { y } { a } } \\right ) \\frac { d y } { e ^ { 2 \\pi y } - 1 } . \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{align*} C _ 3 = { \\frac { { \\rm { 4 } } } { { c M } } { { \\left ( { 1 - \\frac { { \\rm { 1 } } } { \\omega } } \\right ) } ^ 2 } { \\rm { + } } \\frac { K } { { { { c ^ 2 M } ^ 2 \\omega } } } \\left ( { - 4 { \\rm { + } } \\frac { 5 } { \\omega } } \\right ) { \\rm { + } } \\frac { { { K ^ 2 } } } { { { { c ^ 3 M } ^ 3 \\omega ^ 2 } } } } . \\end{align*}"}
-{"id": "3013.png", "formula": "\\begin{align*} I _ { q } ( t \\psi ) = \\frac { t ^ { 2 } } { 2 } \\int _ { \\Omega } | \\nabla \\psi | ^ { 2 } - \\frac { t ^ { q + 1 } } { q + 1 } \\int _ { \\Omega } a \\psi ^ { q + 1 } < 0 \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} & \\sum _ { n = 2 } ^ { \\infty } \\| V _ 0 ^ { 0 - 1 - 2 , l , ( n ) } \\| _ { 1 , \\infty , r } \\le c \\frac { N } { h } \\alpha ^ { - 8 } L ^ { - d } . \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} H _ { m + 1 } \\subset H _ { m + 2 } \\subset \\dots \\subset H _ { ( m + 1 ) + m } = H _ { 2 m + 1 } \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} \\widehat { B } ( l _ 0 , l _ 1 ) = \\widehat { b } _ { l _ 1 , 0 } + \\widehat { b } _ { l _ 0 , 1 } . \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} B ( \\Psi ( \\gamma \\otimes x ) ) = B ( T _ * ( \\gamma \\times \\beta ' ( \\lambda _ x ) ) ) = \\pi _ * T _ * ( \\gamma \\times \\beta ' ( \\lambda _ x ) ) = p _ * \\beta ' ( \\lambda _ x ) = o _ * ( \\lambda _ x ) . \\end{align*}"}
-{"id": "158.png", "formula": "\\begin{align*} R _ t = \\sum _ { j = 0 } ^ \\infty t ^ { 2 - 2 j / 3 } k _ { j , t } ( z ) \\begin{pmatrix} i & 0 \\\\ 0 & - i \\end{pmatrix} , k _ { j , t } ( z ) = k _ j ( t ^ { 2 / 3 } z ) . \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} \\frac { 2 } { x _ A } + \\frac { 1 } { x _ A + x _ C } = \\frac { 2 } { x _ B } + \\frac { 1 } { x _ B + x _ C } = \\frac { 1 } { x _ A + x _ C } + \\frac { 1 } { x _ B + x _ C } \\end{align*}"}
-{"id": "8965.png", "formula": "\\begin{align*} \\left ( \\frac { \\theta ^ n - \\theta ^ { n - 1 } } { k _ n } , \\chi \\right ) = & - i \\left ( \\nabla \\frac { \\theta ^ n + \\theta ^ { n - 1 } } { 2 } , \\nabla \\chi \\right ) - \\left ( \\frac { \\rho ^ n - \\rho ^ { n - 1 } } { k _ n } , \\chi \\right ) \\\\ & + i ( \\omega ^ n , \\chi ) - ( r ^ n , \\chi ) , \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} \\partial _ t v _ k - \\delta ^ \\star \\Delta v _ k + \\mu _ k = 0 , \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} a _ l ^ * - a _ { l + i } ^ * & = p ^ { - { l + i } } \\left ( p ^ i a _ l - a _ { l + i } - \\frac { p ^ i - 1 } { p - 1 } \\right ) \\\\ & \\leq p ^ { - { l + i } } \\left ( ( p ^ i - 1 ) a _ l + 2 i - \\frac { p ^ i - 1 } { p - 1 } \\right ) \\\\ & \\leq p ^ { - { l + i } } \\left ( 2 i - p \\frac { p ^ i - 1 } { p - 1 } \\right ) \\\\ & \\leq 0 . \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} p ( X = 0 ) \\mapsto \\max _ { p _ { U | X } } I ( U ; Y ) - I ( U ; Z ) \\end{align*}"}
-{"id": "1788.png", "formula": "\\begin{align*} p _ \\kappa ( x , D _ x , x ) u = p _ L ( x , D _ x ) u \\end{align*}"}
-{"id": "5681.png", "formula": "\\begin{align*} | \\mathbb { X } _ { s , t } | & \\le c \\Big ( 2 ^ { - n ( 1 - \\varepsilon ) } + \\bigg | \\int _ 0 ^ { t } X ^ n _ r \\otimes \\dd X _ r - \\int _ 0 ^ { s } X ^ n _ r \\otimes \\dd X _ r - X _ s \\otimes X _ { s , t } \\bigg | \\Big ) \\\\ & \\le c \\Big ( 2 ^ { - n ( 1 - \\varepsilon ) } + \\max \\{ 2 ^ { - n } c ( s , t ) ^ { 1 / q } , 2 ^ { - n ( 2 - q ) } c ( s , t ) + c ( s , t ) ^ { 2 / q } \\} \\Big ) , \\end{align*}"}
-{"id": "5368.png", "formula": "\\begin{align*} J _ { \\nu } ( z ) = \\frac { z ^ \\nu } { 2 ^ \\nu } \\sum _ { j = 0 } ^ \\infty \\frac { ( - 1 ) ^ j z ^ { 2 j } } { j ! 2 ^ { 2 j } \\Gamma ( \\nu + j + 1 ) } . \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{align*} m ^ k _ t = \\int _ { [ 0 , \\max \\{ 0 , ( \\tau _ k ) ^ { - 1 } ( t ) \\} ] } \\ , \\delta _ { \\Phi ^ { e _ k } _ t ( y , \\ , 0 ) } d m ^ k _ 0 ( x ) + \\int _ { ( \\max \\{ 0 , ( \\varsigma _ k ) ^ { - 1 } ( t ) \\} , t ] } \\delta _ { \\Phi ^ { e _ k } _ t ( 0 , \\ , s ) } d m _ { x = V } ^ k ( s ) . \\end{align*}"}
-{"id": "1543.png", "formula": "\\begin{align*} m ^ 1 _ { t } = \\int _ { [ 0 , \\ , \\max \\{ 0 , \\ , ( \\tau _ 1 ) ^ { - 1 } ( t ) \\} ] } \\ , \\delta _ { \\Phi ^ { e _ 1 } _ t ( x , \\ , 0 ) } \\ , d m _ 0 ^ 1 ( x ) + \\int _ { ( \\max \\{ 0 , \\ , ( \\varsigma _ 1 ) ^ { - 1 } ( t ) \\} , \\ , t ] } \\delta _ { \\Phi ^ { e _ 1 } _ t ( 0 , \\ , s ) } \\ , d \\sigma _ 0 ( s ) ; \\\\ \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} \\eta _ f [ \\frak { u } , y ] ( x ) = ( \\eta [ \\frak { u } , y + \\vartheta ] ( x + \\vartheta ) ) ^ \\ell \\cdot \\prod _ { 1 \\le i \\le I } ( \\Phi _ { \\mathcal { V } } ( x - u _ i ) ) ^ { e _ i } \\cdot \\prod _ { 1 \\le i \\le I } ( \\Phi _ { \\mathcal { V } } ( y - u _ i ) ) ^ { - e _ i } . \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} \\operatorname { g r a d e } ( G _ { I ^ n } ( A ) _ + , G _ { I ^ n } ( A ) ) = 2 \\mbox { f o r a l l } n \\gg 0 . \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} Z _ n ( q , t , Q ) & = \\sum _ { | \\nu | + | \\mu | = n } \\frac { 1 } { N _ { \\nu , \\mu } ( Q ) N _ { \\mu , \\nu } ( Q ^ { - 1 } ) N _ { \\nu , \\nu } ( 1 ) N _ { \\mu , \\mu } ( 1 ) } , \\\\ N _ { \\nu , \\mu } ( Q ) & = \\prod _ { t \\in \\mu } ( 1 - Q q ^ { a _ { \\nu } ( t ) } t ^ { l _ \\mu ( t ) + 1 } ) \\prod _ { s \\in \\nu } ( 1 - Q q ^ { - a _ { \\mu } ( s ) - 1 } t ^ { - l _ \\nu ( s ) } ) \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} R f _ { ! } u : R f _ { ! } \\mathcal { F } & \\to R f _ { ! } R c _ { 1 \\ast } c _ { 1 } ^ { \\ast } \\mathcal { F } = R f _ { ! } R c _ { 1 ! } c _ { 1 } ^ { \\ast } \\mathcal { F } \\\\ & \\ ; \\ ; \\ ; \\cong R f _ { ! } R c _ { 2 ! } c _ { 1 } ^ { \\ast } \\mathcal { F } \\overset { u } { \\to } R f _ { ! } \\mathcal { F } \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\delta } = \\left \\{ \\nu : \\nu \\prec \\delta , \\ y _ { \\nu } ( w ) = y _ { \\delta } ( w ) \\ \\ q = t ^ { - m } \\right \\} . \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} \\mathbf { H } = \\mathbf { H } _ 0 + V \\ , . \\end{align*}"}
-{"id": "2310.png", "formula": "\\begin{align*} P \\{ T _ 2 < T _ 1 \\} = \\sum _ { k = 1 } ^ { M _ 2 } \\sum _ { j = 1 } ^ { M _ 1 } ( - 1 ) ^ { j + k } \\binom { M _ 1 } { j } \\binom { M _ 2 } { k } \\frac { p _ 2 k } { p _ 1 j + p _ 2 k } . \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\exp [ u ( \\alpha _ t ) ] = \\lim _ { z \\rightarrow 0 } \\exp [ u ( z ) ] = \\lim _ { z \\rightarrow 0 } z / \\eta _ \\mu ( z ) = 1 / m _ 1 ( \\mu ) \\end{align*}"}
-{"id": "6169.png", "formula": "\\begin{align*} | S | = | U | . \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } \\lambda ^ { \\beta \\alpha - k } l ^ \\beta ( \\lambda ) = \\infty \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} ( q - 1 ) ^ { E ( \\Gamma _ 1 ) } & q ^ { E ( \\Gamma _ 1 ) ( \\alpha - 1 ) } ( q - 1 ) ^ { E ( \\Gamma _ 2 ) - E ( \\Gamma _ 1 ) } q ^ { \\left ( E ( \\Gamma _ 2 ) - E ( \\Gamma _ 1 ) \\right ) ( \\alpha - 2 ) } \\cdots \\\\ & \\cdots ( q - 1 ) ^ { E ( \\Gamma _ { \\alpha - 1 } ) - E ( \\Gamma _ { \\alpha - 2 } ) } q ^ { E ( \\Gamma _ { \\alpha - 1 } ) - E ( \\Gamma _ { \\alpha - 2 } ) } ( q - 1 ) ^ { E ( \\Gamma _ \\alpha ) - E ( \\Gamma _ { \\alpha - 1 } ) } \\\\ & = ( q - 1 ) ^ { E ( \\Gamma _ \\alpha ) } q ^ { \\sum _ { k = 1 } ^ { \\alpha - 1 } E ( \\Gamma _ k ) } , \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} \\big ( a _ k \\big ) _ { k = 1 } ^ \\infty = \\left ( \\pi \\left ( k - \\tfrac { 1 } { 2 } \\right ) ^ 2 e ^ { - \\pi \\left ( k - \\tfrac { 1 } { 2 } \\right ) ^ 2 x / 2 } \\right ) _ { k = 1 } ^ \\infty \\big ( b _ k \\big ) _ { k = 1 } ^ \\infty = \\left ( e ^ { - \\pi \\left ( k - \\tfrac { 1 } { 2 } \\right ) ^ 2 x / 2 } \\right ) _ { k = 1 } ^ \\infty \\ , . \\end{align*}"}
-{"id": "6297.png", "formula": "\\begin{align*} G ( x , y ) = c _ n \\left \\{ \\begin{array} { l l } \\left ( \\frac { 1 } { | x - y | ^ { n - 2 } } - \\frac { 1 } { ( | \\ , x | y | - y / | y | \\ , | ) ^ { n - 2 } } \\right ) , & \\hbox { i f $ n \\ge 3 $ ; } \\\\ \\log \\frac { | x - y | } { | 1 - x \\bar y | } , & \\hbox { i f $ n = 2 $ a n d $ x , y \\in \\mathbb { C } \\cong \\mathbb { R } ^ 2 $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} \\mu \\left ( \\{ v \\in T ^ 1 \\mathcal { M } _ { g , n } : \\exists \\ , t \\in [ 0 , T _ k ] \\textrm { w i t h } \\textrm { s y s } ( \\phi _ t ( v ) ) \\leqslant T _ k ^ { - \\xi } \\} \\right ) \\\\ = \\mu ( A _ k ) = O \\left ( \\frac { 1 } { T _ k ^ { 3 \\xi - 1 } } \\right ) \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} & f ( r _ { \\beta } ( s _ 1 + s ) , r _ { \\beta } ( s _ 2 + s ) , \\cdots , r _ { \\beta } ( s _ p + s ) ) = f ( s _ 1 , s _ 2 , \\cdots , s _ p ) \\\\ & \\left ( \\forall ( s _ 1 , s _ 2 , \\cdots , s _ p ) \\in [ 0 , \\beta ) _ h ^ p , \\ s \\in \\frac { 1 } { h } \\Z \\right ) , \\\\ & g ( r _ { \\beta } ( s _ 1 + s ) , r _ { \\beta } ( s _ 2 + s ) , \\cdots , r _ { \\beta } ( s _ q + s ) ) = g ( s _ 1 , s _ 2 , \\cdots , s _ q ) \\\\ & \\left ( \\forall ( s _ 1 , s _ 2 , \\cdots , s _ q ) \\in [ 0 , \\beta ) _ h ^ q , \\ s \\in \\frac { 1 } { h } \\Z \\right ) , \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{align*} P ' _ { \\beta } ( X ) ~ = ~ U ' _ { \\beta } ( X ) \\ , f _ { \\beta } ( X ) ~ + ~ U _ { \\beta } ( X ) \\ , f ' _ { \\beta } ( X ) . \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} S _ \\alpha = \\{ \\mathbf { p } \\succeq \\mathbf { 0 } | U ( \\mathbf { p } ) \\geqslant \\alpha \\} . \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} \\partial _ \\mu l ( \\gamma ) = \\l ( \\mu , \\frac 1 \\pi \\Theta _ \\gamma \\ ) \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{align*} \\left ( S - T _ 1 \\right ) ^ 2 = \\left ( S - T _ 1 \\right ) ^ 2 \\mathbf { 1 } _ { \\{ T _ 1 < S \\} } \\end{align*}"}
-{"id": "669.png", "formula": "\\begin{align*} \\Pi = N _ r ^ { - 1 } M _ r N _ { r - 1 } ^ { - 1 } M _ { r - 1 } \\dotsm N _ 1 ^ { - 1 } M _ 1 \\end{align*}"}
-{"id": "4162.png", "formula": "\\begin{align*} N : = \\left \\lceil \\left ( \\frac { \\varepsilon } { 4 C B d ^ \\beta } \\right ) ^ { - \\frac { 1 } { \\beta } } \\right \\rceil \\in \\N . \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} Q ' = ( ( \\sigma _ { n } \\times i d ) ^ { * } ( l ^ { n } \\times i d ) _ { * } F ^ { \\boxtimes n } ) \\otimes ( i d \\times \\psi ) ^ { * } ( p _ { 1 } ^ { * } d e t ( \\mathcal { A } ) ^ { - 1 } ) \\end{align*}"}
-{"id": "1515.png", "formula": "\\begin{align*} \\int _ { \\delta } ^ { + \\infty } \\frac { t } { \\kappa ( t ) } d t = + \\infty , \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{align*} \\theta _ 1 ( Z , Z ) = \\tfrac 1 2 A _ 1 ( \\nu _ 1 + i \\nu _ 2 ) d z ^ 2 , \\theta _ 2 ( Z , Z ) = \\tfrac 1 2 A _ 1 ( \\nu _ 1 - i \\nu _ 2 ) d z ^ 2 . \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} & I ( U _ i ; Y | U _ { 1 : i - 1 } ) = \\mathbb { P } [ U _ { 1 : i - 1 } = u _ { 1 : i - 1 } ] I ( U _ i ; Y | U _ { 1 : i - 1 } \\ ! = \\ ! u _ { 1 : i - 1 } ) . \\end{align*}"}
-{"id": "7822.png", "formula": "\\begin{align*} \\begin{array} { l } \\dot { S } _ 1 = \\displaystyle { \\frac { 1 } { 3 } ( 1 + 3 S _ 1 ) ( 3 S _ 2 - 3 S _ 1 + 2 ) \\sin ( 2 \\theta ) } , \\\\ \\\\ \\dot { S } _ 2 = \\displaystyle { \\frac { 1 } { 9 } \\left ( 2 7 S _ 2 ^ 2 - 2 7 S _ 1 S _ 2 - 9 S _ 2 + 3 S _ 1 - 2 \\right ) \\sin ( 2 \\theta ) } , \\\\ \\\\ \\dot { \\theta } = \\displaystyle { \\frac { ( 4 + 3 S _ 1 + 9 S _ 2 ) \\cos ( 2 \\theta ) } { 9 ( S _ 1 - S _ 2 ) } } , \\end{array} \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} \\sum _ { \\substack { h \\in H ^ 2 ( S ^ 1 \\times \\tilde { Y } ) \\\\ \\phi ( h ) = x } } S W _ { S ^ 1 \\times \\tilde { Y } } ( h ) = 0 \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} \\lim _ { y \\to - \\infty } \\Delta ^ - ( y , 1 , t ) = - \\infty \\beta _ 4 < 0 . \\end{align*}"}
-{"id": "1293.png", "formula": "\\begin{align*} P _ A ( z ) - ( 1 - z ) ^ { B + C + 1 } Q _ A ( z ) = z ^ { A + C + 1 } E _ A ( z ) . \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} 4 - 1 - \\frac { 2 \\cdot 2 } { \\tau - 1 } = \\frac { 3 \\tau - 7 } { \\tau - 1 } , \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{align*} | u _ j ( x , t ) - P _ { d , t } ^ j ( x , t ) | \\leq C \\left ( \\gamma + \\sum _ { k = 1 } ^ n \\| u _ k \\| _ { W ^ { 2 , 1 } _ q ( Q _ 1 ) } \\right ) | ( x , t ) | ^ { d + \\alpha } , \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{align*} F _ { w } ( q , t , w ) \\varphi = A \\varphi - Q [ a \\left ( x \\right ) ( q ( t \\phi _ { 1 } + w ) ^ { q - 1 } - 1 ) \\varphi ] , \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} \\int _ M \\sum _ a V _ a \\langle \\nabla _ { \\bar V _ a } \\phi , \\psi \\rangle & = \\frac 1 2 \\int _ M { \\rm d i v } _ \\nabla Y = \\int _ M \\langle \\nabla _ { H ^ { 0 , 1 } } \\phi , \\psi \\rangle , \\\\ \\int _ M \\sum _ a \\bar V _ a \\langle \\phi , \\nabla _ { \\bar V _ a } \\psi \\rangle & = \\frac 1 2 \\int _ M { \\rm d i v } _ \\nabla Z = \\int _ M \\langle \\phi , \\nabla _ { H ^ { 0 , 1 } } \\psi \\rangle . \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} \\mathrm { P r o b } ( f ( x ) ) = \\frac { 1 } { 6 } \\bigg ( 1 - \\frac { 1 } { q ^ 3 } - \\frac { 1 } { q ^ 4 } + \\frac { 1 } { q ^ 5 } \\bigg ) . \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} f _ { \\Delta } ( \\alpha , z ) = \\begin{cases} z ^ { a } & \\\\ 1 & \\end{cases} \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} C = v _ 1 \\times \\dots \\times v _ { i - 1 } \\times e _ \\alpha \\times v _ { i + 1 } \\times \\dots \\times v _ { j - 1 } \\times e _ \\beta \\times v _ { j + 1 } \\times \\dots \\times v _ { 2 m + 1 } \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} g _ { - 1 } \\cdot u = g _ { - 1 } \\cdot \\begin{pmatrix} u _ 0 \\\\ \\hat { u } \\\\ u _ 3 \\end{pmatrix} = c _ 0 = e _ 4 \\end{align*}"}
-{"id": "4564.png", "formula": "\\begin{align*} \\mathcal H _ B ^ r = \\{ 0 \\} . \\end{align*}"}
-{"id": "2295.png", "formula": "\\begin{align*} M _ 1 p _ 1 + \\cdots + M _ g p _ g = 1 . \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} \\forall \\ell = 1 , \\ldots , K - 1 , \\sum _ { k = 1 } ^ K \\omega _ k v _ k ( \\psi ^ 0 _ { \\pm \\ell } ( v _ k ) - \\psi ^ 0 _ { \\pm \\ell } ( - v _ k ) ) = 0 . \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} S _ { \\vec { a } , q } = \\sum \\limits _ { \\vec { x } ( q ) } e _ q ( \\vec { a } \\cdot \\vec { f } ( \\vec { x } ) ) S _ { \\vec { a } , q } ( \\vec { \\nu } ) = S _ { \\vec { a } , q } e _ q ( - \\vec { a } \\cdot \\vec { \\nu } ) . \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{align*} \\langle { \\varphi } \\ , | \\ , V ^ { 1 / 2 } ( \\rho + h ( \\mathfrak { e } ) ) ^ { - 1 } V ^ { 1 / 2 } { \\varphi } \\rangle = \\int _ { \\Gamma ^ { \\ast } } \\frac { | \\mathcal { F } ^ { \\ast } \\circ V ^ { 1 / 2 } ( { \\varphi } ) ( p ) | ^ { 2 } } { \\rho + \\mathfrak { e } ( p ) } \\mathrm { d } \\mu ^ { \\ast } ( p ) . \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} \\left | \\delta ( x + i y ) \\right | \\leq \\sum _ { n = 1 } ^ { \\infty } n ^ { k - 1 } \\exp ( - 2 \\pi n y ) \\Bigg ( \\sum _ { \\substack { b \\mid n \\\\ b < n } } \\left ( \\frac { b } { n } \\right ) ^ { k - 1 } \\Bigg ) . \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{align*} \\begin{aligned} & { \\rm V a r } ( \\hat F _ { \\tilde d } ) = O \\Big ( \\frac { 1 } { N } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} & \\left ( x \\frac { d } { d x } \\right ) ^ 3 \\log \\left ( \\theta _ 3 ( x ) \\right ) = x \\ , \\psi ( x ) + 3 x ^ 2 \\ , \\psi ' ( x ) + x ^ 3 \\ , \\psi '' ( x ) \\\\ & < \\left ( 6 . 3 5 \\ , x - 1 1 8 . 2 \\ , x ^ 2 + 2 5 5 . 5 \\ , x ^ 3 \\right ) e ^ { - 2 \\pi x } + \\left ( - 6 \\ , x + 6 0 . 3 \\ , x ^ 2 - 5 5 . 1 \\ , x ^ 3 \\right ) e ^ { - \\pi x } . \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} \\Delta ( G ) = \\lim _ { m \\rightarrow \\infty } \\frac { \\log N _ m ( G ) } { m } \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} \\frak { e _ 6 } = \\mathbb { C } ^ * \\oplus ( \\bigwedge ^ 3 \\mathbb { C } ^ 6 ) ^ * \\oplus \\frak { g l } _ 6 \\oplus \\bigwedge ^ 3 \\mathbb { C } ^ 6 \\oplus \\mathbb { C } , \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} \\begin{cases} [ a k + 1 ] = e = [ 0 ] \\in G & \\textnormal { i f $ m | b $ } \\\\ [ b k + 1 ] = e = [ 0 ] \\in G & \\textnormal { i f $ m | a $ } \\end{cases} \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} f = \\beta + h \\varphi , \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} \\int _ { u } ^ v \\frac { ( q x / u , q x / v ; q ) _ \\infty } { ( c x , d x ; q ) _ \\infty } d _ q x = \\frac { ( 1 - q ) v ( q , u / v , q v / u , c d u v ; q ) _ \\infty } { ( c u , c v , d u , d v ; q ) _ \\infty } . \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{align*} \\left ( ( z - \\lambda _ j ) ^ { \\tau _ j } F _ \\alpha ( z ) \\Phi ^ { n - k } ( z ) \\right ) ^ { ( l ) } _ { z = \\lambda _ j } = \\frac { l ! } { 2 \\pi i } \\int _ { | z - \\lambda _ j | = \\varepsilon } \\frac { ( z - \\lambda _ j ) ^ { \\tau _ j } F _ \\alpha ( z ) \\Phi ^ { n - k } ( z ) d z } { ( z - \\lambda _ j ) ^ { l + 1 } } . \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} \\widetilde { \\P } ^ { 0 , 0 } _ { 2 T } ( X ( s T ) = v T ) \\simeq e ^ { - T I ( s , v ) } \\mbox { a s $ T \\to \\infty $ } , \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} \\Big \\| \\sum _ { i = 1 } ^ N f _ i \\ , \\Big \\| _ { L ^ p } \\leq N ^ { \\max \\{ 1 , p ^ { - 1 } \\} } \\cdot \\max \\{ \\| f _ i \\| _ { L ^ p } \\ , : \\ , i = 1 , \\dots , N \\} , \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} \\lambda = K ( u ) \\leq C \\int | u | ^ 3 d x & \\leq C \\| u \\| _ { H ^ { \\frac { 1 } { 2 } } ( \\mathbb { R } ) } ^ { 7 / 3 } \\| \\partial _ x ^ { - 1 } u \\| _ { L ^ 2 ( \\mathbb { R } ) } ^ { \\frac { 2 } { 3 } } \\\\ & \\leq C \\left ( | | u | | ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } + | | \\partial _ x ^ { \\frac { 1 } { 2 } } u | | ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } + | | \\partial _ x ^ { - 1 } u | | ^ 2 _ { L ^ 2 ( \\mathbb { R } ) } \\right ) ^ { 3 / 2 } \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} I _ { 2 , + } ( x _ 1 , x _ 2 ) = \\int _ 0 ^ t \\int _ 0 ^ \\infty \\sigma _ s ( y ) \\int _ y ^ \\infty [ G _ { t - s } ( x _ 1 - z ) - G _ { t - s } ( x _ 2 - z ) ] \\psi ( s , z ) d z W ( d s , d y ) \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} \\displaystyle \\epsilon \\int _ { \\Omega } | \\nabla v | ^ { m - 2 } \\nabla v \\nabla w d x + \\int _ { \\Omega } \\phi ( | \\nabla v | ) \\nabla v \\nabla w d x = \\int _ { \\Omega } f w d x \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} V ^ { u } \\bigl ( t , x \\bigr ) = & [ V _ 1 ^ { u } \\bigl ( t , x \\bigr ) , V _ 2 ^ { u } \\bigl ( t , x \\bigr ) , \\ldots , V _ n ^ { u } \\bigl ( t , x \\bigr ) ] ^ T , \\ , \\ , ( t , x ) \\in [ 0 , T ] \\times \\mathbb { R } ^ d \\ , \\ , \\& \\ , \\ , u \\in U . \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} \\sigma = \\lbrace ( 1 , 9 ) , ( 4 , 6 ) , ( 1 0 , 1 2 ) \\rbrace . \\end{align*}"}
-{"id": "8821.png", "formula": "\\begin{align*} \\exp ( D ) = \\exp ( - a ) k \\exp ( a + y + O ) \\exp ( h ) . \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{align*} I ( \\chi _ { P _ l } ) = \\pm \\chi _ { P _ 1 } , \\pm \\chi _ { P _ 2 } , \\pm \\chi _ { P _ 3 } , \\pm ( \\chi _ { P _ 1 } - \\chi _ { P _ 2 } ) , \\pm ( \\chi _ { P _ 1 } - \\chi _ { P _ 3 } ) \\pm ( \\chi _ { P _ 2 } - \\chi _ { P _ 3 } ) , \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{align*} H ( r ) : = \\frac { 1 } { \\Gamma ( r ) } \\ , E \\left [ S _ N ^ { ( r ) } \\right ] = E \\left [ \\frac { \\Gamma ( r + S _ N ) } { \\Gamma ( r ) \\ , \\Gamma ( S _ N ) } \\right ] = \\sum _ { k = N } ^ { \\infty } \\frac { \\Gamma ( k + r ) } { \\Gamma ( r ) \\ , ( k - 1 ) ! } \\ , P \\{ S _ N = k \\} . \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} d \\int _ \\tau ^ t \\psi ( s ) d s & = \\int _ \\tau ^ t \\Big ( d _ { A ( s ) } \\psi ( s ) - [ A ( s ) , \\psi ( s ) ] \\Big ) d s \\\\ & = - \\int _ \\tau ^ t \\Big ( \\Big \\{ w ' ( s ) + \\zeta ( s ) \\Big \\} + [ A ( s ) , \\psi ( s ) ] \\ ) d s \\\\ & = w ( \\tau ) - w ( t ) - \\int _ \\tau ^ t \\ ( \\zeta ( s ) + [ A ( s ) , \\psi ( s ) ] \\ ) d s . \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} ( ( J \\otimes E ) | _ { E [ H ^ + ] } ) _ { \\mathrm { f s u } } = ( J \\otimes E ) ^ { \\mathrm { o r d } } \\oplus ( J \\otimes E ) ^ { \\mathrm { s u p } } \\end{align*}"}
-{"id": "6955.png", "formula": "\\begin{align*} \\lambda \\preccurlyeq \\nu \\Leftrightarrow _ { \\textup { d e f } } \\left ( \\mathrm { p a r t s } ( \\nu ) < \\mathrm { p a r t s } ( \\lambda ) \\right ) \\textup { o r } ( \\mathrm { p a r t s } ( \\nu ) = \\mathrm { p a r t s } ( \\lambda ) \\textup { a n d } \\lambda \\leq _ { l e x } \\nu ) . \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{align*} \\Gamma _ { 1 } : = \\left \\{ ( q , u ) = ( 1 , s \\phi _ { 1 } ) : s > 0 \\right \\} , \\end{align*}"}
-{"id": "633.png", "formula": "\\begin{align*} B _ i ( n + 1 ) = \\left \\{ Q _ i ( n ) \\geq 1 \\right \\} \\bigcap \\left \\{ W _ i ( n + 1 ) \\geq 1 \\right \\} \\end{align*}"}
-{"id": "439.png", "formula": "\\begin{align*} U _ 1 ( d _ 1 ) = \\begin{pmatrix} 1 & d _ 1 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & - d _ 1 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , U _ 2 ( d _ 2 ) = \\begin{pmatrix} 1 & 0 & d _ 2 & 0 \\\\ 0 & 1 & 0 & - d _ 2 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , U ( d _ 1 , d _ 2 ) = U _ 1 ( d _ 1 ) U _ 2 ( d _ 2 ) , \\end{align*}"}
-{"id": "2625.png", "formula": "\\begin{align*} d i v ( \\nabla \\nabla \\varphi ) ( X ) = R i c ( \\nabla \\varphi , X ) + X ( \\Delta \\varphi ) , \\end{align*}"}
-{"id": "5276.png", "formula": "\\begin{align*} F ( r , f ) \\circ r = f \\ , , \\end{align*}"}
-{"id": "9148.png", "formula": "\\begin{align*} \\hat { C } _ { \\varphi } ( \\Q ) = \\{ ( 0 : 1 : 0 ) , ( 0 : 0 : 1 ) , ( 1 : 1 : 1 ) , ( 1 : - 1 : 1 ) \\} . \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} { n \\brack k } _ q = \\frac { ( q ; q ) _ n } { ( q ; q ) _ k ( q ; q ) _ { n - k } } . \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} & [ c , x _ 1 , x _ 3 ] x _ 2 [ x _ 4 , x _ 5 ] + [ c , x _ 1 , x _ 4 ] x _ 2 [ x _ 3 , x _ 5 ] = x _ 2 \\bigl ( [ c , x _ 1 , x _ 3 ] [ x _ 4 , x _ 5 ] \\\\ + \\ & [ c , x _ 1 , x _ 4 ] [ x _ 3 , x _ 5 ] \\bigr ) + [ c , x _ 1 , x _ 3 , x _ 2 ] [ x _ 4 , x _ 5 ] + [ c , x _ 1 , x _ 4 , x _ 2 ] [ x _ 3 , x _ 5 ] \\in I ^ { ( n ) } . \\end{align*}"}
-{"id": "2069.png", "formula": "\\begin{align*} \\mu _ n ( \\sigma : | \\sigma _ + | = k ) = \\frac { x _ { k , n } } { Z _ n } . \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l } R i c _ { B } + \\nabla _ { B } \\nabla _ { B } \\beta = \\lambda g _ { B } , \\\\ \\noalign { \\smallskip } \\lambda = h ^ { - 1 } [ ( \\nabla _ { B } h ) \\beta - b ] , \\\\ \\noalign { \\smallskip } \\nabla _ { F } \\nabla _ { F } \\varphi + b g _ { F } = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5180.png", "formula": "\\begin{align*} \\bar \\rho _ J = \\begin{cases} \\rho _ J - \\rho _ { J ' } , & J ( a + 1 , c ) , \\\\ \\rho _ J , & J ( a + 1 , c ) - . \\end{cases} \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} \\int _ { \\R ^ { N } } f ( u _ { n } ) u _ { n } \\ , d x = o _ { n } ( 1 ) \\mbox { a n d } \\int _ { \\R ^ { N } } F ( u _ { n } ) \\ , d x = o _ { n } ( 1 ) . \\end{align*}"}
-{"id": "7535.png", "formula": "\\begin{align*} ( ( \\tilde \\gamma + 2 \\gamma I ) ^ { - 1 } ) ^ i _ j = \\left ( \\begin{array} { c c c } 3 \\gamma / ( 9 \\gamma ^ 2 + B _ 0 ^ 2 ) & - B _ 0 / ( 9 \\gamma ^ 2 + B _ 0 ^ 2 ) & 0 \\\\ B _ 0 / ( 9 \\gamma ^ 2 + B _ 0 ^ 2 ) & 3 \\gamma / ( 9 \\gamma ^ 2 + B _ 0 ^ 2 ) & 0 \\\\ 0 & 0 & 1 / ( 3 \\gamma ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} d q _ t = & \\tilde \\gamma ^ { - 1 } ( t ) \\left ( - \\partial _ t \\psi ( t , q _ t ) - \\nabla _ q V ( t , q _ t ) ) \\right ) d t + ( \\tilde \\gamma ^ { - 1 } \\sigma ) ( t ) \\circ d W _ t , \\\\ d q ^ \\prime _ t = & \\tilde \\gamma ^ { - 1 } ( t ^ * ) \\left ( \\partial _ t \\psi ( t ^ * , q ^ \\prime _ t ) - \\nabla _ { q } V ( t ^ * , q ^ \\prime _ t ) \\right ) d t + ( \\hat { \\phi } _ * ( \\tilde \\gamma ^ { - 1 } \\sigma ) ) ^ i _ \\rho ( t ^ * ) \\circ d \\tilde W ^ \\rho _ t . \\end{align*}"}
-{"id": "3134.png", "formula": "\\begin{align*} { } \\begin{array} { l } \\bar { g } ( u ) = 2 \\frac { v } { u } - u \\ln ( e ^ v - 1 ) \\\\ \\bar { f } ( u ) = \\frac { v } { 2 ^ { 3 / 2 } \\pi u } \\left ( ( e ^ { v } - 1 ) - \\frac { 1 } { 2 } u ^ 2 e ^ { v } \\right ) ^ { - 1 / 2 } \\\\ \\frac { v ^ 2 } { u ^ 2 } = \\int _ 0 ^ { v } \\frac { t \\ , d t } { 1 - e ^ { - t } } \\end{array} \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t u = L _ t ^ { \\ast } u + u \\Lambda ( t , x , u , \\nabla u ) \\\\ u ( 0 , \\cdot ) = u _ 0 \\ , \\end{array} \\right . \\end{align*}"}
-{"id": "8479.png", "formula": "\\begin{align*} e _ 1 & = \\Big ( \\frac { 1 } { 2 } , \\frac { i } { 2 } , 0 , \\dots , 0 \\Big ) \\\\ e _ 2 & = \\Big ( \\frac { 1 } { 2 } , - \\frac { i } { 2 } , 0 , \\dots , 0 \\Big ) , \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} ( T h ) ( x , D _ { x } ) : = \\kappa ^ { - 1 , * } h ( x , D _ { x } ) \\kappa ^ * \\ \\ h \\in S ^ m _ { 1 , 0 } ( \\overline { \\R ^ n _ + } \\times \\R ^ { n - 1 } ; \\mathcal { S } _ { + + } ) \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} \\int _ \\Omega K ( x ) \\delta _ i ^ { \\frac { n + 2 } { n - 2 } } \\lambda _ i \\frac { \\partial \\delta _ i } { \\partial \\lambda _ i } \\ , d x = \\int _ { B _ { y _ { j _ i } } } \\ , \\bigl [ K ( x ) - K ( a _ i ) \\bigr ] \\delta _ i ^ { \\frac { n + 2 } { n - 2 } } \\lambda _ i \\frac { \\partial \\delta _ i } { \\partial \\lambda _ i } \\ , d x + o \\bigl ( \\frac { 1 } { \\lambda _ i ^ { n - 2 } } \\bigr ) . \\end{align*}"}
-{"id": "2564.png", "formula": "\\begin{align*} \\psi _ n = \\min ( \\varphi _ n + \\nu _ n , 1 ) , \\end{align*}"}
-{"id": "8822.png", "formula": "\\begin{align*} \\phi ( \\exp ( a ) \\exp ( D ) ) & = \\phi ( k \\exp ( a + y + O ) \\exp ( h ) ) \\\\ \\intertext { b y t h e e q u i v a r i a n c e p r o p e r t y o f t h e q u a s i p o t e n t i a l ( P r o p o s i t i o n ~ \\ref { p r o p _ e q u i v a r i a n c e _ q u a s i p o t e n t i a l } ) , t h i s i s } & = \\phi ( \\exp ( a + y + O ) ) - 2 \\ln | \\chi ( \\exp ( h ) ) | \\end{align*}"}
-{"id": "3567.png", "formula": "\\begin{align*} N _ { t } & = \\Big [ \\frac { \\partial _ { s s } \\gamma } { \\kappa } \\Big ] _ { t } \\\\ & = - \\frac { \\kappa _ { t } } { \\kappa ^ { 2 } } \\gamma _ { s s } + \\frac { 1 } { \\kappa } \\gamma _ { s s t } . \\end{align*}"}
-{"id": "7026.png", "formula": "\\begin{align*} p _ { - k } ( \\lambda ) = \\prod _ { j \\geq 1 } p _ { - k } ( \\lambda _ { j } ) . \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} m _ i ( \\mu , \\mu ' ) = \\left \\{ \\begin{array} { r l } 1 , & \\mu _ i > \\mu _ { i + 1 } , ( \\mu _ i , \\mu _ { i + 1 } ) = ( \\mu ' _ { i + 1 } , \\mu ' _ i ) , \\mu _ k = \\mu ' _ k \\ \\forall \\ k \\not = i , i + 1 , \\\\ \\\\ t , & \\mu _ i < \\mu _ { i + 1 } , ( \\mu _ i , \\mu _ { i + 1 } ) = ( \\mu ' _ { i + 1 } , \\mu ' _ i ) , \\mu _ k = \\mu ' _ k \\ \\forall \\ k \\not = i , i + 1 , \\\\ \\\\ 0 , & , \\end{array} \\right . \\end{align*}"}
-{"id": "7098.png", "formula": "\\begin{align*} f ( x + \\alpha _ { 0 } , y + \\beta _ { 0 } ) = f ( x , y ) - \\delta ' + \\delta . \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} \\begin{cases} F ( D ^ { 2 } u ) = \\chi _ \\Omega & B _ { 1 } ^ { + } \\\\ u = 0 & B ' _ { 1 } \\end{cases} \\end{align*}"}
-{"id": "1241.png", "formula": "\\begin{align*} \\mathcal { E } ( F ) = \\lbrace g : \\mathbb { R } ^ 2 \\rightarrow \\mathbb { R } ^ 2 : g g ( F ) \\subseteq F \\rbrace . \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} & \\sum _ { l = 0 } ^ { m } { m + 1 \\choose l } _ { \\ ! \\ ! q } \\prod _ { j = 0 } ^ { l - 1 } ( 1 - q ^ { k + j } r ) \\prod _ { j = 1 } ^ { m - l } ( q ^ { 2 k + m - j } r - r ^ { - 1 } ) q ^ { l ( 2 k + l - 1 ) / 2 } = 0 . \\end{align*}"}
-{"id": "7937.png", "formula": "\\begin{align*} g ( r _ n , 2 c ) \\geq g ( r _ n , A _ { t _ n } ( r _ n ) ) = \\frac { 1 } { t _ n - 1 } . \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{align*} \\gamma _ e = \\gamma + \\frac { 1 - \\gamma } { 1 + { \\rm P N R } _ { \\max } } \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{align*} \\begin{cases} d ' _ k = 0 k , \\\\ N ( f ) = \\# I _ g + \\# ( J ^ + _ g \\cup J _ g ^ - ) . \\end{cases} \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{align*} \\mathcal { F } ( \\mathcal C _ { \\delta } g ) ( \\xi , \\tau ) = \\psi \\left ( \\frac { | \\xi | - \\tau } { \\delta } \\right ) \\phi ( | \\xi | ) \\widehat { g } ( \\xi , \\tau ) \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} S _ k ( q ) = \\{ i : d _ k ( i ) \\leq q \\} \\end{align*}"}
-{"id": "6398.png", "formula": "\\begin{align*} \\frac { 1 } { x _ A } + \\frac { 1 } { x _ A + x _ B } = \\frac { 1 } { x _ B } . \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} | \\lambda ^ 2 - \\lambda + 1 | & = | \\lambda - \\zeta | | \\lambda - \\overline { \\zeta } | \\geq \\sqrt { 3 } / 4 \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} \\mathrm { P } _ { t , i } = & \\mathrm { P } \\left ( \\alpha _ 1 = 1 \\right ) \\\\ & + \\mathrm { P } \\left ( \\alpha _ 1 < 1 , z _ t < \\max \\left \\{ \\frac { \\epsilon _ 2 } { \\rho \\xi _ 2 } , \\cdots , \\frac { \\epsilon _ i } { \\rho \\xi _ { i } } \\right \\} \\right ) . \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} \\nu ( B ) = \\int _ { S } \\lambda ( d \\xi ) \\int _ 0 ^ \\infty g ( \\xi , r ) 1 _ B ( r \\xi ) d r . \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} ( J v ) ^ \\flat = J ( v ^ \\flat ) \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{align*} H = \\begin{pmatrix} \\kappa & 0 \\\\ 0 & \\kappa ^ { - 1 } \\end{pmatrix} \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} & p _ { Z ^ n | K _ A K _ B \\mathbf { F } } ( z ^ n | k _ A = i , k _ B = i , \\mathbf { f } ) = p _ { Z ^ n } ( z ^ n | X ^ n \\in \\mathcal { A } _ { i } , Y ^ n \\in \\mathcal { B } _ { i } ) . \\end{align*}"}
-{"id": "5668.png", "formula": "\\begin{align*} \\left \\| ( | N | - \\varepsilon ) _ + ^ n R _ \\delta ^ n \\right \\| \\leq \\sum _ { k = 1 } ^ { m - 1 } \\left ( \\frac { k } { m } \\right ) ^ n ( 1 + \\varepsilon ) ^ n \\left ( \\frac { m } { k } \\right ) ^ n = ( m - 1 ) ( 1 + \\varepsilon ) ^ n \\leq \\frac { 2 } { \\varepsilon } ( 1 + \\varepsilon ) ^ n . \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} \\mathcal { T } : = \\left \\{ b \\in C ^ 0 ( \\Gamma ^ * ) , b \\mbox { i s e v e n a n d } b \\mbox { i s i n c r e a s i n g o n } [ 0 , 1 / 2 ] \\right \\} . \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} { \\bf G } ^ { - 1 } \\approx \\sum _ { n = 0 } ^ { L } ( - { \\bf D } ^ { - 1 } { \\bf E } ) ^ n { \\bf D } ^ { - 1 } , \\end{align*}"}
-{"id": "7043.png", "formula": "\\begin{align*} & \\hat { C } ( x t , y u ) = \\hat { C } ( x 0 , y 0 ) , \\quad ( \\forall x , y \\in \\{ 0 , \\cdots , L - 1 \\} , \\ u , t \\in \\{ 0 , \\cdots , n - 1 \\} ) . \\end{align*}"}
-{"id": "2708.png", "formula": "\\begin{align*} y = \\frac { y \\wedge x _ 2 \\wedge x _ 3 \\wedge \\cdots \\wedge x _ n } { x _ 1 \\wedge x _ 2 \\wedge \\cdots \\wedge x _ n } x _ 1 + \\frac { x _ 1 \\wedge y \\wedge x _ 3 \\wedge \\cdots \\wedge x _ n } { x _ 1 \\wedge x _ 2 \\wedge \\cdots \\wedge x _ n } x _ 2 + \\cdots + \\frac { x _ 1 \\wedge x _ 2 \\wedge \\cdots \\wedge x _ { n - 1 } \\wedge y } { x _ 1 \\wedge x _ 2 \\wedge \\cdots \\wedge x _ n } x _ n . \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{align*} 1 = \\int _ { \\Omega } | \\nabla v _ { n } | ^ { 2 } = \\frac { 1 } { \\Vert u _ { n } \\Vert ^ { 1 - q _ { n } } } \\int _ { \\Omega } a ( x ) v _ { n } ^ { q _ { n } + 1 } , \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} D _ 1 = ( B - n ) ( v _ m ) & + \\sum _ { i = 1 } ^ n \\Big [ \\big ( D ( w _ i ) + A + 1 \\big ) ( w _ i ) + \\sum _ { v _ j \\prec w _ i } \\sum _ { p \\in ( w _ i , v _ j ) ^ { \\circ } } \\ , D ( p ) \\ , ( p ) \\Big ] \\\\ & + \\sum _ { i = 1 } ^ { m - 1 } \\Big [ D ( v _ i ) ( v _ i ) + \\sum _ { w _ j \\prec v _ i } \\sum _ { p \\in ( v _ i , w _ j ) ^ { \\circ } } \\ , D ( p ) \\ , ( p ) \\Big ] , \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} I ( U ; Z | V ) = ( 1 - \\epsilon ) I ( U ; X Y | V ) . \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} 0 = \\int _ M | W _ F ^ + | ^ 2 + 3 | R _ { 0 0 } | ^ 2 + 6 k ( k - 2 v ) \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} \\Delta _ B = 2 \\square _ B = 2 \\overline \\square _ B . \\end{align*}"}
-{"id": "6611.png", "formula": "\\begin{align*} \\delta T ^ \\psi ( X , Y ) = & \\left ( \\nabla _ X \\delta \\psi , Y \\cdot \\psi \\right ) + \\left ( \\nabla _ X \\psi , Y \\cdot \\delta \\psi \\right ) = \\left ( \\nabla _ X ( U \\cdot \\psi ) , Y \\cdot \\psi \\right ) + \\left ( \\nabla _ X \\psi , Y \\cdot U \\cdot \\psi \\right ) \\\\ = & \\left ( \\nabla _ X ( U ) \\cdot \\psi , Y \\cdot \\psi \\right ) + \\left ( U \\cdot ( \\nabla _ X \\cdot \\psi ) , Y \\cdot \\psi \\right ) + \\left ( \\nabla _ X \\psi , Y \\cdot U \\cdot \\psi \\right ) \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u ^ { \\prime } ( t ) + C u ( t ) & = f ( t ) ( 0 < t < T ) , \\\\ u ( 0 ) & = c . \\end{aligned} \\right . \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{align*} N ( 0 + ) = \\underset { r \\to 0 } { \\lim } N ( r ) \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{align*} [ c _ 2 , d _ 2 ] - c _ 2 = \\binom { c _ 2 } { 2 } + d _ 2 - c _ 2 = \\binom { c _ 2 } { 2 } - \\binom { c _ 2 - 1 } { 1 } + d _ 2 - 1 = \\binom { c _ 2 - 1 } { 2 } + d _ 2 - 1 = [ c _ 2 - 1 , d _ 2 - 1 ] . \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} & 0 - b _ n = \\sum \\limits _ { k = 1 } ^ n \\delta _ { b _ k 0 } - \\sum \\limits _ { k = 1 } ^ n \\delta _ { a _ k 0 } + \\underbrace { \\delta _ { a _ n 0 } } _ { = 1 } \\\\ & \\Rightarrow b _ n = - 1 . \\end{align*}"}
-{"id": "2518.png", "formula": "\\begin{align*} A ^ { k } = \\prod ^ { k } _ { l = 1 } B _ { l } \\prod ^ { k } _ { l = 1 } G _ { k + 1 - l } . \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} \\frac { ( a c x , b c y ; q ) _ \\infty } { ( c x , c y ; q ) _ \\infty } = \\sum _ { n = 0 } ^ \\infty \\frac { \\Phi _ n ^ { ( a , b ) } ( x , y | q ) c ^ n } { ( q ; q ) _ n } . \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{align*} | a _ { k , n } ^ { ( \\alpha ) } | \\leq \\frac { c _ { 1 1 } } { \\rho _ 2 ^ { k - n } ( \\rho _ 1 - \\delta ) ^ n } , \\alpha = 1 , 2 , \\ldots , d , k \\geq n + 1 . \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { \\frac { \\omega t } { 2 } } t ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\norm { \\nabla e ^ { - t B } b _ { s } } _ { L ^ { q } ( \\Omega ) ^ { 3 \\times 3 } } & \\leq C e ^ { \\frac { \\omega t } { 2 } } t ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\norm { e ^ { - t B } B ^ { \\frac { 1 } { 2 } } b _ { s } } _ { L ^ { q } ( \\Omega ) ^ 3 } \\leq C \\norm { \\nabla b _ { s } } _ { L ^ { p } ( \\Omega ) ^ { 3 \\times 3 } } . \\end{aligned} \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{align*} \\Psi \\bigl ( \\lambda _ { i _ 1 } | a _ { i _ 1 } - y _ { j _ { i _ 1 } } | \\bigr ) = 0 . \\end{align*}"}
-{"id": "1568.png", "formula": "\\begin{align*} [ x _ 1 ^ { k + 1 } x _ 2 ] = 0 \\Leftrightarrow & ( k + 1 ) _ { q _ { 1 1 } } ^ ! \\prod _ { i = 0 } ^ k ( 1 - q _ { 1 1 } ^ i q _ { 1 2 } q _ { 2 1 } ) = 0 . \\end{align*}"}
-{"id": "3888.png", "formula": "\\begin{align*} & \\lambda _ { 0 } = 2 \\gamma ( 1 - \\varepsilon ) \\rho _ { \\varepsilon } ( \\gamma ) , \\\\ & \\lambda _ 1 = \\frac { \\gamma } { \\mu \\varepsilon } \\rho _ { \\varepsilon } ( \\gamma ) . \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{align*} \\phi _ j ( x ) = \\max _ { i \\mid a _ { i j } \\neq 0 } ( a _ i ^ T x + b _ i ) \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} f ( x _ 1 , y _ 1 ) = \\sum _ { k , l = 0 } ^ \\infty \\lambda _ { 0 , k + l } ( a _ 1 ; q ) _ k ( b _ 1 ; q ) _ l { { k + l } \\brack k } _ q x _ 1 ^ k y _ 1 ^ l . \\end{align*}"}
-{"id": "8484.png", "formula": "\\begin{align*} ( a , b ) = g ( t _ 1 , t _ 2 ) . \\end{align*}"}
-{"id": "3772.png", "formula": "\\begin{align*} A _ X \\circ J = - J \\circ A _ X . \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} \\psi _ t ( Z ^ \\psi _ t ) = \\max _ { z \\in \\Pi _ { L _ t , \\delta } } \\psi _ t ( z ) Z ^ \\psi _ t \\succeq z \\ ; \\ , \\forall \\ , z \\in \\Pi _ { L _ t , \\delta } \\psi _ t ( z ) = \\psi _ t ( Z ^ \\psi _ t ) . \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{align*} ( \\widetilde \\Delta ^ { ( m ) } \\partial _ { z _ { j } } ) h \\big | _ { z _ j = 0 } = 0 , j = 1 , \\ldots , m - 1 \\ , , \\end{align*}"}
-{"id": "9253.png", "formula": "\\begin{align*} I ( s , \\widetilde { v } ( s ) ) = 0 , s \\in [ 0 , 2 ] , \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} \\zeta _ { \\beta } ( z ) = \\frac { 1 - z ^ N } { ( 1 - \\beta z ) \\Bigl ( \\sum _ { n = 0 } ^ { \\infty } T _ { \\beta } ^ { n } ( 1 ) \\ , z ^ n \\Bigr ) } \\end{align*}"}
-{"id": "5809.png", "formula": "\\begin{align*} M _ i [ \\psi ] ( \\mu ) = \\sum _ { \\mu ' \\in \\mathbb { B } } m _ i ( \\mu , \\mu ' ) \\psi ( \\mu ' ) , \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} \\delta _ 1 : = \\inf _ { t _ 0 \\in \\mathbb { R } , ( x , t ) \\in \\R ^ N \\times [ t _ 0 , t _ 0 + T _ 1 ] } u ( x , t + t _ 0 ; t _ 0 , u _ 0 ) \\ge u _ { 0 \\inf } e ^ { - T _ 1 ( a _ { \\inf } + b _ { \\sup } \\| u _ 0 \\| _ { \\infty } e ^ { T _ 1 a _ { \\sup } } ) } > 0 . \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} \\gamma ^ { c _ A } _ z ( f ) ( x , n , y ) = z ^ { c _ A ( x , n , y ) } f ( x , n , y ) = z ^ n f ( x , n , y ) , \\end{align*}"}
-{"id": "7844.png", "formula": "\\begin{align*} R _ { p q r } = \\frac { f _ { q r } f _ { r p } } { f _ { p q } } \\ , , \\left ( f _ { p q } \\right ) ^ { r N } = \\prod _ { k = 1 } ^ { r N - 1 } \\left ( \\frac { \\mu _ { q } ( x _ { q } - \\omega ^ k y _ { p } ) } { \\mu _ { p } ( x _ { p } - \\omega ^ k y _ { q } ) } \\frac { ( 1 - \\omega ^ k ) ( x _ { p } y _ { p } - \\omega ^ k x _ { q } y _ { q } ) } { ( x _ { p } - \\omega ^ k x _ { q } ) ( y _ { p } - \\omega ^ k y _ { q } ) } \\right ) ^ { k } \\ , . \\end{align*}"}
-{"id": "6685.png", "formula": "\\begin{align*} \\sum \\limits _ { k = - \\infty } ^ { \\infty } | c _ k | ^ p = \\infty \\textrm { f o r a n y } p < 2 . \\end{align*}"}
-{"id": "4824.png", "formula": "\\begin{align*} k < \\sum _ { i = 1 } ^ s \\dim M _ i - \\min _ { 1 \\leq i \\leq s } \\{ \\dim M _ i \\} + 2 \\end{align*}"}
-{"id": "2348.png", "formula": "\\begin{align*} q _ j = \\frac { 1 } { N } \\ ; j = 1 , \\dots , N . \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} \\int _ 0 ^ T y ^ \\ell _ r d x _ r = \\int _ { ( 0 , T ] } y ^ - d \\mu _ x , \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} c _ { S M } ( C _ { w P } ) = \\sum _ { v \\in W ^ P , v \\leq w } { c ( v ; w ) [ X ^ { w P } ] } \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} \\limsup _ { y \\to x } \\frac { | F ( y ) - F ( x ) - f ( x ) ( y - x ) | } { | \\varphi ( y ) - \\varphi ( x ) | } \\le \\lim _ { y \\to x } \\frac { 2 c | y - x | } { 2 ^ k | y - x | } = 2 ^ { - k - 1 } c . \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{align*} C _ { X } ( \\phi _ { j } ) = C _ { j } \\phi _ { j } , j \\geq 1 , \\end{align*}"}
-{"id": "1476.png", "formula": "\\begin{align*} { \\bf H } _ f : = { \\bf H } + ( \\overline { \\nabla } f ) ^ { \\perp } . \\end{align*}"}
-{"id": "733.png", "formula": "\\begin{align*} d _ { \\beta } ( 1 ) = 0 . t _ 1 t _ 2 t _ 3 \\ldots { \\rm a n d ~ u n i q u e l y ~ c o r r e s p o n d s ~ t o } 1 = \\sum _ { i = 1 } ^ { + \\infty } t _ i \\beta ^ { - i } \\ , , \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} \\partial _ { t } d - \\Delta d + ( u \\cdot \\nabla ) d = - \\gamma f ( d ) , \\gamma > 0 , \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{align*} \\left | \\frac { h _ { t _ 0 } ( x ) } { h _ s ( y ) } \\right | = \\left | \\frac { \\Phi _ { t _ 0 } ( x e ^ { i A _ { t _ 0 } ( x ) } ) } { \\Phi _ s ( y e ^ { i A _ s ( y ) } ) } \\right | = O \\left ( \\frac { x } { y } \\right ) , \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} \\int A ^ 2 ( \\psi ) d \\mu _ { C ( x e ^ { i \\xi } ) } ( \\psi ) = u _ { 3 , L } ( x ) , ( \\forall x , \\xi \\in \\R ) . \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} \\omega ^ \\dagger _ t ( { \\widetilde x } _ 0 ) = \\mathbb E _ { { \\widetilde x } _ 0 } ( M _ t \\omega ^ \\dagger _ 0 ( { \\widetilde x } _ t ) ) . \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} E ' : y ^ 2 = x ( x ^ 2 + b ( h ) x + d ( h ) ) \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} \\left ( 1 - \\frac { \\ln x } { \\ln N } \\right ) ^ r = \\sum _ { k = 0 } ^ n ( - 1 ) ^ k \\binom { r } { k } \\left ( \\frac { \\ln x } { \\ln N } \\right ) ^ k + O \\left ( \\frac { \\ln ^ { \\alpha ( n + 1 ) } N } { \\ln ^ { n + 1 } N } \\right ) \\end{align*}"}
-{"id": "2404.png", "formula": "\\begin{align*} E \\left [ T - T _ 1 \\right ] = E \\left [ \\left ( T _ 2 - T _ 1 \\right ) \\mathbf { 1 } _ { \\{ T _ 1 < T _ 2 \\} } \\right ] = E \\left [ T _ 2 - T _ 1 \\ , | \\ , T _ 1 < T _ 2 \\right ] P \\{ T _ 1 < T _ 2 \\} . \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} \\begin{aligned} s _ { 2 n } ( x , y ) & = s _ n ( x , y ) \\\\ s _ { 2 n + 1 } ( x , y ) & = x s _ n ( x , y ) + y s _ { n + 1 } ( x , y ) \\end{aligned} \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} y \\left ( x \\right ) = y _ { 0 } ^ { \\ast } \\left ( x \\right ) + \\frac { 1 } { \\Gamma \\left ( \\alpha \\right ) } \\int _ { x _ { 1 } } ^ { x } \\psi ^ { \\prime } \\left ( t \\right ) \\left ( \\psi \\left ( x \\right ) - \\psi \\left ( t \\right ) \\right ) ^ { \\alpha - 1 } f \\left ( t , y \\left ( t \\right ) \\right ) d t , \\end{align*}"}
-{"id": "5413.png", "formula": "\\begin{align*} W ^ { ( 2 ) } & = W ^ { ( 1 ) } \\cdot W ^ { ( 1 ) } \\\\ & = W ^ 0 _ 3 ( p , p _ 1 , p _ 2 ) \\ast W ^ 0 _ 3 ( p ' , p ^ { ' } _ 1 , p ^ { ' } _ 2 ) \\\\ & + h \\left ( W ^ 1 _ 1 ( p ) \\ast _ h W ^ 0 _ 3 ( p ' , p ^ { ' } _ 1 , p ^ { ' } _ 2 ) + W ^ 0 _ 3 ( p , p _ 1 , p _ 2 ) \\ast _ h W ^ 1 _ 1 ( p ' ) \\right ) . \\end{align*}"}
-{"id": "3587.png", "formula": "\\begin{align*} ( c - 2 \\tau ) \\kappa ^ { 3 } = 0 \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} \\kappa ^ { 2 } ( x _ { k } , t _ { k } ) ( T - 1 / k - t _ { k } ) = \\max _ { t \\leq T - 1 / k , x \\in S ^ { 1 } } \\kappa ^ { 2 } ( x , t ) ( T - 1 / k - t ) . \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{align*} C _ { 0 } ^ { 1 } ( \\overline { \\Omega } ) : = \\left \\{ u \\in C ^ { 1 } ( \\overline { \\Omega } ) : u = 0 \\partial \\Omega \\right \\} . \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{align*} { - D } _ { \\alpha { } } { ( \\hslash { } \\nabla { } ) } ^ { \\alpha { } } \\psi ( x , t ) + V ( x , t ) \\psi ( x , t ) = E \\psi ( x , t ) . \\end{align*}"}
-{"id": "5727.png", "formula": "\\begin{align*} \\tilde \\delta _ o & = \\tilde \\kappa _ g \\big ( ( \\tilde \\kappa _ \\nu ) ^ 2 + ( \\tilde \\kappa _ t ) ^ 2 \\big ) - \\tilde \\kappa _ t ( \\tilde \\kappa _ \\nu ) ' + ( \\tilde \\kappa _ t ) ' \\tilde \\kappa _ \\nu , \\end{align*}"}
-{"id": "5424.png", "formula": "\\begin{align*} \\nabla Y \\times \\nabla Z = \\frac { 1 } { v _ 1 ( 0 , Y , Z ) } v \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\beta ^ { + } } T _ { \\gamma } ^ { n } ( 1 ) = T _ { \\beta } ^ { n } ( 1 ) , \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} P = \\{ i _ { f _ 1 } , \\dots , i _ { f _ a } \\} \\sqcup \\{ j _ { g _ 1 } , \\dots , j _ { g _ b } \\} . \\end{align*}"}
-{"id": "3350.png", "formula": "\\begin{align*} A \\sum _ { x = 1 } ^ n \\tau ( E _ { x , 0 } ) - B \\big ( \\sum _ { 1 \\le x , y \\le n } \\tau ( E _ { x , 0 } E _ { y , 0 } ) - \\sum _ { x = y } \\tau ( E _ { x , 0 } E _ { y , 0 } ) \\big ) = ( A + B ) \\lambda - B \\lambda ^ 2 . \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{align*} F ( x , c ) = \\Big ( \\big ( f ( x ) \\big ) ^ q + \\big ( c \\varphi ( x ) \\big ) ^ q \\Big ) ^ { \\frac 1 q } \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} \\frac { 8 1 q ^ 2 + 2 2 0 q ^ 4 + 8 6 q ^ 6 - 4 q ^ 8 + q ^ { 1 0 } } { \\left ( 1 - q ^ 2 \\right ) ^ 5 } < \\left . \\frac { 8 1 q ^ 2 + 2 2 0 q ^ 4 + 8 6 q ^ 6 + q ^ { 1 0 } } { \\left ( 1 - q ^ 2 \\right ) ^ 5 } \\right | _ { q = e ^ { - \\pi } } < 0 . 1 6 . \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} 1 = \\| 1 \\| = \\| U ( 1 ) \\| = \\| U ( 1 ) \\| _ \\infty + \\| D ( U ( 1 ) ) \\| _ \\infty . \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{align*} \\vec { z } ( \\mu ) = ( z _ 1 < \\cdots < z _ { m _ 1 + m _ 2 } ) = \\vec { x } ( \\mu ) \\cup \\vec { y } ( \\mu ) , \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{align*} \\mathcal { L } _ { V _ j } ( \\xi ) = - \\int _ { 1 } ^ { \\xi } \\frac { \\hat { H } _ { V _ j } d \\hat { X } } { 2 \\sqrt { \\hat { X } ( \\hat { X } - 1 ) ( \\hat { X } - \\lambda ) } } . \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{align*} Z _ { 1 } ^ { - 1 } \\Pi Z _ 1 = R _ r ^ { - 1 } T _ r R _ { r - 1 } ^ { - 1 } T _ { r - 1 } \\dotsm R _ 1 ^ { - 1 } T _ 1 . \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{align*} S ( c , x ^ * ) : = \\Big \\{ x \\in A \\mid F ( x , \\lambda ^ * , c ) < F ( x ^ * , \\lambda ^ * , c ) \\Big \\} \\subset K \\forall c \\ge c _ 0 ; \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} \\partial _ { i } \\partial _ j T ( 0 ) = - ( \\partial _ d T ( 0 ) ) ^ { - 1 } \\partial _ { i } \\partial _ j \\Phi ( 0 ) , i , j \\leq d - 1 , \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} { \\tilde y } ( \\gamma , \\xi ) = [ O ^ T y ^ \\bot _ \\xi ] ( \\gamma , T - \\xi - 0 ) , ( \\gamma , \\xi ) \\in \\Sigma ^ T _ \\sigma , \\end{align*}"}
-{"id": "3808.png", "formula": "\\begin{align*} ( g _ 1 , g _ 2 , g _ 3 ) ^ s = ( g _ 1 ^ s , g _ 1 ^ { s - 1 } ( g _ 2 , g _ 3 ) , g _ 1 ^ { s - 2 } ( g _ 2 , g _ 3 ) ^ { 2 } , \\dots , g _ 1 ( g _ 2 , g _ 3 ) ^ { s - 1 } , ( g _ 2 , g _ 3 ) ^ { s } ) \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{align*} d _ { \\alpha } ( x _ 1 , x _ 2 ) = \\sup _ { ( s , t ) \\in [ 0 , 1 ] ^ 2 } \\frac { \\norm { S _ 2 ( x _ 1 ) ( s , t ) \\cdot S _ 2 ( x _ 2 ) ( s , t ) ^ { - 1 } } } { | t - s | ^ \\alpha } \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n a _ { i k } a _ { i l } = \\delta _ { k l } . \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{align*} Z _ j = \\oplus _ { w \\in R _ k ( j ) } X _ w , \\end{align*}"}
-{"id": "3700.png", "formula": "\\begin{align*} \\sum _ { \\vec { z } + q \\vec { y } \\in P \\mathcal { B } } e ( \\vec { \\beta } \\cdot \\vec { f } ( \\vec { z } + q \\vec { y } ) ) = I ( \\mathcal { B } , P ^ d \\vec { \\beta } ) \\frac { P ^ n } { q ^ n } + O \\left ( C \\widetilde { C } ^ { r - 1 } \\frac { P ^ { n + \\eta - 1 } } { q ^ n } \\right ) . \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} t . x = \\chi ( t ) x \\textrm { f o r a l l } t \\in H ^ + . \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{align*} \\tilde { J } _ 1 ( N ; \\theta ) = \\frac { 2 } { ( 1 - \\theta ) ^ 2 } \\int _ 0 ^ { \\infty } s \\left [ 1 - \\left ( 1 - e ^ { - s / N } \\right ) ^ N \\right ] d s \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{align*} & v _ { t t } - c ^ 2 \\ , [ \\Delta v - q v ] = 0 \\mbox { i n } \\ , Q ^ T , \\\\ & v | _ { t = T } = 0 , v _ t | _ { t = T } = 2 u _ t ^ { \\tilde f } ( \\cdot , T ) \\mbox { i n } \\ , \\Omega , \\\\ & v = 0 \\mbox { o n } \\ , \\Sigma ^ T . \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} f ^ { - 1 } [ T _ t ] = \\omega ^ * \\cap \\left ( \\bigcup _ { A \\in \\mathcal A _ s } \\hat { A } ( t ) \\setminus \\bigcup \\left \\{ \\hat { B } : B \\in \\mathcal A ( t , A ) \\right \\} \\cup \\bigcup _ { t ^ \\prime \\in s u c c _ T ( t ) } f ^ { - 1 } [ T _ { t ^ \\prime } ] \\right ) , \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{align*} d x = \\frac { \\det { M _ x } } { 2 ^ n } \\frac { \\ , d y } { y _ 1 \\ldots y _ n } . \\end{align*}"}
-{"id": "2200.png", "formula": "\\begin{align*} \\alpha ( \\theta ) = ( - 1 ) ^ { d - 1 } \\star ( \\theta \\wedge e _ 3 \\wedge \\dots \\wedge e _ d ) , \\theta \\not \\in \\Theta ( e _ 1 , e _ 2 ) , \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} \\partial _ { \\omega } ^ n ( b ) + c _ { n - 1 } \\partial _ { \\omega } ^ { n - 1 } ( b ) + \\cdots + c _ { 1 } \\partial _ { \\omega } ( b ) + c _ { 0 } b = \\widetilde { \\tau } ( g ) - g . \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} \\widetilde { \\kappa } _ 2 ( x ) & > x ^ { 4 n } - ( 4 m + 6 ) x ^ { 3 n } = x ^ { 3 n } ( x ^ n - ( 4 m + 6 ) ) , \\\\ \\widetilde { \\kappa } _ 1 ( x ) & > ( 7 m + 9 ) x ^ { n } , \\\\ \\widetilde { \\kappa } _ 0 ( x ) & = x ^ { 4 n } - 2 x ^ { 2 n - 1 } + 1 \\le x ^ { 4 n } - 1 < x ^ { 4 n } . \\end{align*}"}
-{"id": "7902.png", "formula": "\\begin{align*} \\chi _ { \\epsilon } ( t , z , \\zeta ) = \\chi \\bigl ( \\epsilon ( \\mu + \\tilde { h } ( t , z , \\zeta ) \\bigr ) \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { d } \\frac { 1 } { 2 i - 1 } & = \\frac 1 2 \\sum _ { i = 1 } ^ { d } \\left ( \\frac { 1 } { 2 i - 1 } + \\frac { 1 } { 2 d - 2 i + 1 } \\right ) \\\\ & = d \\sum _ { i = 1 } ^ { d } \\frac { 1 } { ( 2 i - 1 ) ( 2 d - 2 i + 1 ) } , \\end{align*}"}
-{"id": "335.png", "formula": "\\begin{align*} d ' _ k = \\begin{cases} d _ k & k \\in I _ g \\cup J ^ + _ g , \\\\ d _ k - 1 & k \\in J ^ - _ g . \\end{cases} \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} ( L \\chi ) ( z ) = B ( z , . . . , z ) - \\left ( \\frac { \\beta } { 2 \\pi } \\right ) ^ { n / 2 } \\int B ( \\tilde z , . . . , \\tilde z ) e ^ { - \\beta \\| \\tilde z \\| ^ 2 / 2 } d \\tilde z . \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} & N ( T _ { n + 1 } ) = N ( T _ n ) + A _ { S _ { n + 1 } } - \\left [ 1 - \\delta _ { N ( T _ n ) } ( \\{ 0 \\} ) \\right ] , \\\\ & T _ { n + 1 } = [ 1 - \\delta _ { N ( T _ n ) } ( \\{ 0 \\} ) ] ( T _ n + S _ { n + 1 } ) + \\delta _ { N ( T _ n ) } ( \\{ 0 \\} ) ( T _ n + S _ { n + 1 } + I _ { n + 1 } ) , \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} A W _ { m } = W _ { m + 1 } H _ { m } \\ , , \\end{align*}"}
-{"id": "4237.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } a _ n z ^ n = \\exp \\bigg ( \\sum _ { n \\geq 1 } \\frac { 1 - q _ 1 ^ n q _ 2 ^ n } { ( 1 - q _ 1 ^ n ) ( 1 - q _ 2 ^ n ) } \\frac { z ^ n } { n } \\bigg ) , \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} M ^ { ( 2 ) } + { \\left ( M ^ T \\right ) } ^ { ( 2 ) } = \\frac { 1 } { 2 } \\left ( M + M ^ T \\right ) ^ { ( 2 ) } + \\frac { 1 } { 2 } \\left ( M - M ^ T \\right ) ^ { ( 2 ) } . \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} \\binom { c n / k ^ { \\ell } } { k } \\leq 2 ^ { \\frac { \\ell + 1 } { 2 ^ { 1 / \\ln 2 } \\ln 2 } ( c e n ) ^ { \\frac { 1 } { \\ell + 1 } } } , \\end{align*}"}
-{"id": "3254.png", "formula": "\\begin{align*} \\gamma _ { k , n } ^ { ( \\alpha ) } - a _ { k , n } ^ { ( \\alpha ) } = \\sum _ { j = 1 } ^ { q } \\sum _ { t = 0 } ^ { \\tau _ j - 1 } \\beta _ n ( j , t ) \\left ( ( z - \\lambda _ j ) ^ { \\tau _ j } F _ \\alpha ( z ) \\Phi ^ { n - k } ( z ) \\right ) ^ { ( t ) } _ { z = \\lambda _ j } , \\alpha = 1 , 2 , \\ldots , d , k \\geq 0 . \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} g ^ { ( k ) } ( x ) - g ^ { ( k ) } ( 0 ) & = \\displaystyle \\lim _ { n \\rightarrow \\infty } f _ n ^ { ( k ) } ( x ) - f _ n ^ { ( k ) } ( 0 ) \\\\ & = \\displaystyle \\lim _ { n \\rightarrow \\infty } \\int _ 0 ^ x f _ n ^ { ( k + 1 ) } \\\\ & = \\int _ 0 ^ x a ^ { k + 1 } . \\end{align*}"}
-{"id": "5776.png", "formula": "\\begin{align*} | \\nabla \\omega ( u ) | ^ 2 = | \\nabla ( \\psi _ { \\ast } ( \\omega ) ) | ^ 2 ( \\psi ( u ) ) \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} \\mathcal L _ { s \\partial } ( c ) = \\lambda _ s \\circ \\mathcal L _ \\partial ( c ) - \\lambda _ { \\partial ( s ) } ( c ) \\ , . \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{align*} [ s + \\alpha , t + \\beta ] = \\alpha ( t ) - \\beta ( s ) + [ \\alpha , \\beta ] \\ , ; \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} Z f = < X , \\nabla f > + 2 t f _ t = < x , \\nabla _ x f > + y f _ y + 2 t f _ t \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} u _ 1 \\tau u _ 2 = u _ 3 \\tau u _ 4 \\Rightarrow \\tau ^ { - 1 } u _ 3 ^ { - 1 } u _ 1 \\tau = u _ 4 u _ 2 ^ { - 1 } \\in \\tau ^ { - 1 } U _ M ^ - \\tau \\cap U _ L ^ - . \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} \\hat { \\xi } _ L ^ z ( x ) : = \\left \\{ \\begin{array} { l l } \\xi ^ z ( x ) \\vee ( a _ L - c _ \\ast + \\delta ^ { - 1 } _ \\sigma ) , & x = z , \\\\ \\xi ^ z ( x ) \\wedge ( a _ L - c _ \\ast ) , & \\end{array} \\right . \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} & \\tilde S ^ i ( t , q ) - \\frac { 1 } { 2 } \\sum _ \\xi ( \\tilde \\gamma ^ { - 1 } \\sigma ) ^ i _ \\xi \\partial _ j ( \\tilde \\gamma ^ { - 1 } \\sigma ) ^ j _ \\xi \\\\ = & - ( \\tilde \\gamma ^ { - 1 } ) ^ { i k } \\partial _ k \\beta ^ { - 1 } - \\partial _ k ( \\beta ^ { - 1 } ( \\tilde \\gamma ^ { - 1 } ) ^ { k \\ell } H _ { \\ell j } ( \\tilde \\gamma ^ { - 1 } ) ^ { i j } ) \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{align*} g _ \\infty = \\begin{pmatrix} ( q \\bar q ) ^ { 1 / 8 } k ^ { 1 / 2 } & 0 \\\\ 0 & ( q \\bar q ) ^ { - 1 / 8 } k ^ { - 1 / 2 } \\end{pmatrix} \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} \\operatorname { T o r } _ i ^ { A _ { \\mathfrak { p } } } ( M _ { \\mathfrak { p } } , N _ { \\mathfrak { p } } ) = 0 \\mbox { f o r a l l } i \\gg 0 , \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\varphi ^ { \\prime } \\varphi \\ ; \\d x + \\int _ { \\Omega } \\lvert \\nabla \\varphi \\rvert ^ 2 \\ ; \\d x = - \\int _ { \\Omega } u \\cdot \\nabla \\varphi \\varphi \\ ; \\d x + 2 \\int _ { \\Omega } \\lvert \\nabla d \\rvert ^ 2 \\varphi ^ 2 \\ ; \\d x . \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} X = A ^ { D } A X ~ ~ ~ ~ A ^ { m } X = A ^ { i } ( A ^ { i } ) ^ { \\dagger } . \\end{align*}"}
-{"id": "4084.png", "formula": "\\begin{align*} \\eta ( \\hat { \\vect { x } } _ n ) = t ( \\hat { \\vect { x } } _ n ^ t ; \\vect { \\Sigma } ^ t ) \\beta _ n \\left ( \\beta _ n \\vect { I } + \\vect { \\Sigma } ^ t \\right ) ^ { - 1 } \\hat { \\vect { x } } _ n \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{align*} \\tilde { \\omega } : = \\lambda _ d \\circ . . . \\circ \\lambda _ 2 \\circ \\lambda _ 1 \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} E \\left [ \\frac { T - T _ 1 } { ( \\nu _ 1 + \\lambda \\nu _ 2 ) M } \\right ] = O \\left ( \\frac { \\ln M } { M ^ { \\lambda - 1 } } \\right ) , M \\to \\infty . \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{align*} \\mathfrak { N } _ { t } = \\alpha \\mathfrak { N } + \\beta \\bar { \\mathfrak { N } } + \\gamma T . \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{align*} \\theta ^ l _ i = \\sum _ { q = 1 } ^ d \\partial _ q \\mathbb { T } ^ l _ { q i } . \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left ( x \\frac { \\theta _ 2 ' ( x ) } { \\theta _ 2 ( x ) } \\right ) = \\frac { d } { d x } \\left ( - \\tfrac { 1 } { x } \\frac { \\theta _ 4 ' \\left ( \\tfrac { 1 } { x } \\right ) } { \\theta _ 4 \\left ( \\tfrac { 1 } { x } \\right ) } - \\frac { 1 } { 2 } \\right ) < 0 . \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} \\bar \\partial \\omega + \\dfrac { 1 } { 2 } [ \\omega , \\omega ] = 0 \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{align*} { \\ } & \\rho _ m ^ { ( 0 ) } \\left [ \\rho _ { m + 1 } ^ { ( m ) } \\left ( s _ 0 + \\rho _ { m } ^ { ( m + 1 ) } \\left ( s _ 1 + \\rho _ { m + 1 } ^ { ( m ) } \\left ( s _ 2 + \\rho _ { m } ^ { ( m + 1 ) } \\left ( s _ 3 + \\dots \\right ) \\right ) \\dots \\right ) \\right ) \\right ] \\\\ & = \\rho _ { m + 1 } ^ { ( 0 ) } \\left [ s _ 0 + \\gamma _ m s _ 2 + \\gamma _ m ^ 2 s _ 4 + \\gamma _ m ^ 3 s _ 6 + \\dots \\right ] + \\rho _ { m } ^ { ( 0 ) } \\left [ \\gamma _ m s _ 1 + \\gamma _ m ^ 2 s _ 3 + \\gamma _ m ^ 3 s _ 5 + \\dots \\right ] , \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} \\dd { X ^ { x , n } } ( t ) & = b ( t , X ^ { x , n } ( t ) ) \\dd { t } + \\sigma ( t , X ^ { x , n } ( t ) ) \\dd { \\tilde { W } ^ { x , n } } ( t ) , \\ t \\in [ 0 , \\tau ^ n ( X ^ { x , n } ) ] , \\\\ X ^ { x , n } _ 0 & = x . \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{align*} ( M C ) ^ D ( M C ) ^ k M [ C \\ | \\ S ] = ( M C ) ^ { k - 1 } M [ C \\ | \\ S ] . \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} ( [ \\nabla _ t , D ] \\xi ) ( X , X _ 1 , \\ldots , X _ k ) & = \\nabla ^ V _ { \\partial _ t } ( \\nabla ^ V _ X ( \\xi ( X _ 1 , \\ldots , X _ k ) ) ) - \\nabla ^ V _ X ( \\nabla ^ V _ { \\partial _ t } ( \\xi ( X _ 1 , \\ldots , X _ k ) ) ) \\\\ & \\qquad - \\sum _ { i = 1 } ^ k \\xi ( X _ 1 , \\ldots , \\nabla _ { \\partial _ t } D _ X X _ i - D _ X \\nabla _ { \\partial _ t } X _ i , \\ldots X _ k ) \\\\ & \\qquad - ( D _ { \\nabla _ { \\partial _ t } X } \\xi ) ( X _ 1 , \\ldots , X _ k ) . \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} g = \\delta + O ( | x | ^ { - \\tau } ) , \\partial g = O ( | x | ^ { - \\tau - 1 } ) , \\partial ^ 2 g = O ( | x | ^ { - \\tau - 2 } ) \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{align*} F ( x _ n , c _ n ) \\ge f ( x _ n ) - \\sum _ { i = 1 } ^ r \\frac { p ( x _ n ) } { 2 c _ n } \\| \\lambda _ i ( x _ n ) \\| ^ 2 - \\frac { q ( x _ n ) } { 2 c _ n } \\| \\mu ( x _ n ) \\| ^ 2 \\ge f ( x _ n ) - \\frac { \\alpha } { c _ n } . \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} \\rho _ A ( \\tau ) & = \\int _ 0 ^ \\tau t ^ { - 1 / 2 } \\| B ( t ) \\| _ 2 ^ 2 d t \\\\ & = \\int _ 0 ^ \\tau t ^ { - 1 / 2 } ( - \\Delta e ^ { 2 t \\Delta } A _ 0 , A _ 0 ) d t \\\\ & = c _ 1 ( ( - \\Delta ) ^ { 1 / 2 } A _ 0 , A _ 0 ) - \\int _ \\tau ^ \\infty t ^ { - 1 / 2 } ( - \\Delta e ^ { 2 t \\Delta } A _ 0 , A _ 0 ) d t \\\\ & = c _ 1 \\| A _ 0 \\| _ { H _ { 1 / 2 } } ^ 2 + O ( \\tau ^ { - 1 / 2 } \\| e ^ { \\tau \\Delta } A _ 0 \\| _ 2 ^ 2 ) \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{align*} 0 \\in \\partial f ( x ) = \\{ \\ , \\d g \\mid g \\in I ( x ) \\ , \\} . \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{align*} & [ y _ i + y _ j , ( \\mathrm { a d } x _ i ) ^ \\alpha ( t _ { i j } ) ] = \\sum _ { \\alpha ' + \\alpha '' = \\alpha - 1 } ( \\mathrm { a d } x _ i ) ^ { \\alpha ' } ( [ - \\sum _ { k \\neq i , j } t _ { i k } , ( \\mathrm { a d } x _ i ) ^ { \\alpha '' - 1 } ( t _ { i j } ) ] ) \\\\ & = \\sum _ { k \\neq i , j } \\sum _ { \\alpha ' + \\alpha '' = \\alpha - 1 } - ( \\mathrm { a d } x _ i ) ^ { \\alpha ' } ( - \\mathrm { a d } x _ j ) ^ { \\alpha '' } ( [ t _ { i k } , t _ { i j } ] ) = \\sum _ { k \\neq i , j } \\mathrm { T e r m } _ k , \\end{align*}"}
-{"id": "2403.png", "formula": "\\begin{align*} T - T _ 1 = T _ 1 \\vee T _ 2 - T _ 1 = \\left ( T _ 2 - T _ 1 \\right ) \\mathbf { 1 } _ { \\{ T _ 1 < T _ 2 \\} } . \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{align*} \\mathbf { f } _ { e x t } = \\mathbf { f } \\cos ( \\Omega t ) = \\mathbf { f } \\frac { e ^ { i \\Omega t } + e ^ { - i \\Omega t } } { 2 } . \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { 2 } ~ \\mathbb { E } ^ Q _ { n , 0 } \\left | \\frac { d \\mathbb { Q } _ { n } } { d \\mathbb { Q } _ { n , 0 } } - 1 \\right | \\right ) ^ 2 \\leq \\mathbb { E } ^ Q _ { n , 0 } \\left ( \\frac { d \\mathbb { Q } _ { n } } { d \\mathbb { Q } _ { n , 0 } } - 1 \\right ) ^ 2 = \\mathbb { E } ^ P _ { n , 0 } \\left ( \\frac { d \\mathbb { Q } _ { n } } { d \\mathbb { P } _ { n , 0 } } \\right ) ^ 2 - 1 , \\end{align*}"}
-{"id": "3388.png", "formula": "\\begin{align*} g _ k ( u _ k ) = \\exp ( c _ k ( u _ k - b _ k ) + \\delta _ k ( u _ k ) ) = \\exp ( c _ k i \\eta R _ k + \\delta _ k ( u _ k ) ) \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} ( 1 - a ) \\psi ( \\rho ) & = b - b _ - ( a , \\rho ) , \\\\ ( 1 - a ) \\psi ( \\rho + ) & = b - b _ + ( a , \\rho ) . \\end{align*}"}
-{"id": "3187.png", "formula": "\\begin{align*} \\begin{cases} \\tfrac { 2 } { 3 } d ^ \\ast d \\beta _ 0 = ( \\lambda - 1 ) ( \\lambda + 5 ) \\beta _ 0 - \\tfrac { 1 } { 3 } ( \\lambda - 5 ) d ^ \\ast \\beta _ 1 , \\\\ ( d d ^ \\ast + \\tfrac { 2 } { 3 } d ^ \\ast d ) \\beta _ 1 = \\tfrac { 2 } { 3 } ( \\lambda + 1 ) ( \\lambda + 3 ) \\beta _ 1 + \\tfrac { 1 } { 3 } ( \\lambda + 9 ) d \\beta _ 0 . \\end{cases} \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} \\widetilde { K } ( x , y ) : = \\lim _ { j \\rightarrow \\infty } A _ j ^ { ( 2 ) } K ( x , y ) \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} \\mathcal { M } : \\mathbf { B T } ^ { \\phi , 1 } _ { / \\mathfrak { S } } & \\to \\mathbf { B T } ^ { \\phi } _ { / S } \\\\ \\mathfrak { M } & \\mapsto \\mathcal { M } ( \\mathfrak { M } ) : = S \\otimes _ { \\mathfrak { S } } \\phi ^ * \\mathfrak { M } \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} \\P \\big ( b _ n ^ { - 1 } M _ n > \\lambda \\big ) \\leq \\P \\big ( C _ { \\alpha } ^ { 1 / \\alpha } \\Gamma _ 1 ^ { - 1 / \\alpha } > \\lambda ( 1 - \\delta ) \\big ) + \\phi _ n ( \\epsilon , \\lambda ) + \\psi _ n ( \\epsilon , \\delta , \\lambda ) . \\end{align*}"}
-{"id": "4217.png", "formula": "\\begin{align*} \\bar { b } ( y ) & = \\frac { Z _ 0 } { Z ( y ) } \\int - \\bar { \\gamma } ^ { - 1 } \\kappa ( \\theta ( x ) - y ) e ^ { - \\tfrac \\kappa 2 \\abs { \\theta ( x ) - y } ^ 2 } \\mu ( d x ) \\\\ & = \\frac { Z _ 0 } { Z ( y ) } \\bar { \\gamma } ^ { - 1 } \\int - \\kappa ( z - y ) e ^ { - \\tfrac \\kappa 2 \\abs { z - y } ^ 2 } \\theta _ \\# \\mu ( d z ) \\\\ & = \\frac { Z _ 0 } { Z ( y ) } \\bar { \\gamma } ^ { - 1 } \\nabla _ y \\int e ^ { - \\tfrac \\kappa 2 \\abs { z - y } ^ 2 } \\theta _ \\# \\mu ( d z ) \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} ( t , x ) \\mapsto ( \\hat \\tau , \\hat x ) = \\frac { 1 } { t ^ 2 - r ^ 2 } ( t , x ) \\in [ 0 , \\infty ) _ { \\hat \\tau } \\times \\R ^ 3 _ { \\hat x } \\end{align*}"}
-{"id": "3089.png", "formula": "\\begin{align*} \\circlearrowleft \\varphi _ D ( \\varphi _ D ( x _ 0 , x _ 1 ) , h ) = \\varphi _ D ( 0 , h ) + \\varphi _ D ( D ( h ) , x _ 0 ) + \\varphi _ D ( 0 , x _ 1 ) = 0 . \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} \\kappa _ { \\mathcal { D } , o p t } ^ { ( 2 ) } ( S _ 1 , S _ 2 , S _ 3 , S _ 4 ) = \\kappa _ { \\mathcal { D } } ( Z _ 1 , Z _ 2 , Z _ 3 , Z _ 4 ) . \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{align*} \\hat { \\rho } _ { \\mathrm { e q } } ( \\| p \\| ) = \\mathcal { N } ( \\xi _ + ) \\| p \\| ^ { d - 1 + \\frac { 2 C } { k _ { s c a t t } ^ 2 } } \\exp \\left ( - \\dfrac { \\| p \\| ^ 2 } { 2 k _ { s c a t t } ^ 2 } \\right ) . \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{align*} { } S _ R ( \\alpha , \\beta ) = \\alpha N + \\beta E + \\sum _ { k = 0 } ^ { B } \\ln ( 1 + e ^ { - \\alpha } e ^ { - \\beta k } ) - \\ln ( 1 + e ^ { - \\alpha } ) \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{align*} E \\left [ S _ N ^ { ( r ) } \\right ] = N ^ r \\ln ^ r N \\int _ 0 ^ N \\left ( 1 - \\frac { x } { N } \\right ) ^ { N - 1 } \\left ( 1 - \\frac { \\ln x } { \\ln N } \\right ) ^ r d x . \\end{align*}"}
-{"id": "1600.png", "formula": "\\begin{align*} \\sigma ( z ) \\sum _ { y \\in B _ { R _ L } ( z ) \\setminus \\{ z \\} } \\frac { \\phi _ { z , m _ L } ( y ) } { \\phi _ { z , m _ L } ( z ) } < \\frac { 1 } { c - \\delta _ \\sigma ^ { - 1 } } \\sum _ { k \\geq 1 } \\left ( \\frac { 1 } { 1 + \\delta _ \\sigma ( c - 2 \\delta _ \\sigma ^ { - 1 } ) } \\right ) ^ { k - 1 } = \\frac { 1 } { c - 2 \\delta _ \\sigma ^ { - 1 } } . \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} \\begin{array} { l l l l l } \\hat { f } ( \\xi ) & = & \\displaystyle \\hat { u _ * } + i \\int _ 1 ^ t \\int e ^ { i ( \\eta ^ 2 - 2 \\xi \\eta ) s } e ^ { - s } { \\hat { f } ( \\xi - \\eta , s ) \\hat { v _ * } ( \\eta , s ) } d \\eta d s \\\\ & & \\displaystyle + i \\int _ 1 ^ t \\int _ 1 ^ s \\iint e ^ { 2 i \\eta ( \\sigma - \\xi ) s } F ( s , \\xi , \\sigma , \\eta ) d s d \\eta d \\sigma , \\end{array} \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { 2 ^ n - 1 } \\zeta ^ { - i } \\theta _ { \\sigma ( i ) } ( g ) \\in 2 ^ n \\mathcal { O } , \\end{align*}"}
-{"id": "4849.png", "formula": "\\begin{align*} f ^ * ( \\omega _ M \\times 1 ) = a \\cdot ( \\omega _ M \\times 1 ) + a ' \\cdot ( 1 \\times \\omega _ N ) \\end{align*}"}
-{"id": "1365.png", "formula": "\\begin{align*} V ^ { u } \\bigl ( t , x \\bigr ) & = \\xi _ { t , T } \\bigl ( u \\bigr ) + \\int _ t ^ T G \\bigl ( s , Y _ s ^ { t , x ; u } , Z _ s ^ { t , x ; u } \\bigr ) d s - \\int _ t ^ T Z _ s ^ { t , x ; u } d B _ s \\\\ & = \\Psi ( X _ T ^ { t , x ; u } ) + \\int _ t ^ T \\Bigl \\{ c \\bigl ( s , X _ s ^ { t , x ; u } , u _ s \\bigr ) + G \\bigl ( s , Y _ s ^ { t , x ; u } , Z _ s ^ { t , x ; u } \\bigr ) \\Bigr \\} d s - \\int _ t ^ T Z _ s ^ { t , x ; u } d B _ s , \\end{align*}"}
-{"id": "3165.png", "formula": "\\begin{align*} d \\rho _ k = \\alpha _ { k } , d \\hat { \\rho } _ k = \\beta _ { k } . \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} \\tilde { A } ( \\varepsilon ) \\tilde { P } + \\tilde { P } \\tilde { A } ( \\varepsilon ) ^ \\top + \\tilde { B } \\tilde { B } ^ \\top + \\tilde { H } ( \\tilde { P } \\otimes \\tilde { P } ) \\tilde { H } ^ \\top + \\displaystyle \\sum _ { j = 1 } ^ { n _ { \\rm i n } } \\tilde { N } _ j \\tilde { P } \\tilde { N } _ j ^ \\top = 0 . \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{align*} { \\bf D } _ { C } = { \\rm d i a g } _ 0 ( { \\bf G } ) + { \\bf G } _ c , \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} F \\big ( R ^ { ( 1 ) } \\big ) - F \\big ( R ^ { ( 0 ) } \\big ) \\leq \\sum _ { i , j = 1 } ^ n \\frac { \\partial F \\big ( R ^ { ( 0 ) } \\big ) } { \\partial R _ { i j } } \\left ( R ^ { ( 1 ) } _ { i j } - R ^ { ( 0 ) } _ { i j } \\right ) + d . \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} \\min _ { U \\neq J \\in \\lbrace 1 , . . . , m \\rbrace ^ n } ( \\psi _ U ( E ) , \\psi _ J ( E ) ) = \\rho \\cdot \\lambda ^ { n - 1 } . \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} G _ + ( h ) - G _ + ( 0 ) = G _ 0 ( h ) - G _ 0 ( 0 ) . \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} \\Omega _ t = \\{ r e ^ { i \\theta } : A _ t ( r ) < \\theta < \\pi , r \\in ( 0 , \\infty ) \\} , \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{align*} \\binom { c _ 1 } { 2 } + d _ 2 & < \\binom { c _ 1 } { 2 } + c _ 1 = \\binom { c _ 1 + 1 } { 2 } \\\\ \\binom { c _ 2 } { 2 } + d _ 1 & < \\binom { c _ 2 } { 2 } + c _ 2 = \\binom { c _ 2 + 1 } { 2 } , \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{align*} p ( x ) = \\hat { p } ( x ^ 2 + x + 1 , x ^ 2 + 1 ) = \\lim _ { i \\rightarrow \\infty } r _ i ( x ^ 2 + x + 1 , x ^ 2 + 1 ) + h ( x ) q _ i ( x ^ 2 + x + 1 , x ^ 2 + 1 ) , \\end{align*}"}
-{"id": "4759.png", "formula": "\\begin{align*} \\Phi ( x , \\lambda , v ) : = \\left ( \\begin{matrix} - v + \\nabla g ( x ) \\lambda \\\\ g ( x ) - \\Pi _ K ( g ( x ) \\ ! + \\lambda ) \\end{matrix} \\right ) \\quad { \\rm f o r } \\ ( x , \\lambda , v ) \\in \\mathbb { X } \\times \\mathbb { Y } \\times \\mathbb { X } . \\end{align*}"}
-{"id": "7201.png", "formula": "\\begin{align*} \\hat { \\psi } _ + ( \\tilde { u } ) = \\inf \\left [ \\hat { \\psi } _ + ( u ) : u \\in W ^ { 1 , p } ( \\Omega ) \\right ] . \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} P _ { \\alpha } ( X ) = P _ { \\alpha } ^ { * } ( X ) = U _ { \\alpha } ( X ) \\times f _ { \\beta } ( X ) \\end{align*}"}
-{"id": "8554.png", "formula": "\\begin{align*} \\vartheta ^ { } _ U ( \\zeta ) - \\vartheta ^ { } _ V ( \\zeta ) = d ^ { \\phantom { g } } _ { \\mathcal { Z } } \\theta ^ { \\phantom { g } } _ { U V } ( \\zeta ) - t ^ { \\phantom { g } } _ { U V } ( \\zeta ) \\frac { d \\zeta } { \\zeta } \\rlap { . } \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{align*} H _ N \\sim \\ln N + \\gamma + \\frac { 1 } { 2 N } + \\sum _ { k = 1 } ^ { \\infty } \\frac { B _ { 2 k } } { 2 k } \\cdot \\frac { 1 } { N ^ { 2 k } } , \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{gather*} \\mathcal { A } _ { \\tau , I } = \\mathcal { F } _ { \\tau , I } \\ltimes \\mathcal { C } _ { \\tau , I } , \\end{gather*}"}
-{"id": "2669.png", "formula": "\\begin{align*} \\bar { \\lambda } = h ^ { - 1 } ( \\nabla _ { B } h ) \\beta - b h ^ { - 1 } + ( n - 1 ) a . \\end{align*}"}
-{"id": "5179.png", "formula": "\\begin{gather*} I = ( 1 , \\dots , r _ 1 , i _ { r _ 1 + 1 } , \\dots , i _ { l ( I ) } ) , \\ r _ 1 < n < i _ { l ( I ) } < \\dots < i _ { r _ 1 + 1 } \\le 2 n , \\\\ J = ( 1 , \\dots , r _ 2 , j _ { r _ 2 + 1 } , \\dots , j _ { l ( J ) } ) , \\ r _ 2 < n < j _ { l ( J ) } < \\dots < j _ { r _ 2 + 1 } \\le 2 n \\end{gather*}"}
-{"id": "1688.png", "formula": "\\begin{align*} T _ { K } = \\bigcup _ { j = 2 } ^ { p + 1 } ( T ^ { l _ j } - T ^ { l _ { j - 1 } } ) . \\end{align*}"}
-{"id": "1303.png", "formula": "\\begin{align*} \\mu \\left ( \\tau _ k \\in [ t , t + l ] \\right ) \\leq \\sum _ { j = 1 } ^ r \\frac { C l } { t ^ { \\gamma _ { 1 , j } } k ^ { \\gamma _ { 2 , j } } } . \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} | z ^ { ( 0 ) } ( \\xi ) | = \\int _ { 0 } ^ { 1 } \\frac { d t } { 2 ( t - \\xi ) \\sqrt { t + 1 - \\xi } } + \\int _ 1 ^ \\infty \\frac { d t } { 2 ( t - \\xi ) \\sqrt { t + 1 - \\xi } } \\end{align*}"}
-{"id": "2257.png", "formula": "\\begin{align*} & \\begin{aligned} { \\rm B i a s } ( \\hat D _ l ) = \\sum _ { j \\in J } \\gamma _ j \\varphi _ j ( l ) \\varGamma ^ { - j / 2 d } + \\rho _ { b i a s } ( \\varGamma ^ { - 1 / 2 } ) , \\end{aligned} \\\\ & \\begin{aligned} { \\rm V a r } ( \\hat D _ l ) = \\rho _ { v a r } ( \\varGamma ^ { - 1 / 2 } ) , \\end{aligned} \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} \\sum _ { a = 1 } ^ n \\langle \\phi , \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi \\rangle & = \\sum _ { a = 1 } ^ n g _ Q ( \\nabla _ { \\bar V _ a } H ^ { 0 , 1 } , V _ a ) | V _ a \\lrcorner \\ , \\phi | ^ 2 . \\end{align*}"}
-{"id": "3137.png", "formula": "\\begin{align*} \\bar { f } ( u ) = \\frac { 1 } { 2 ^ { 3 / 2 } \\pi \\frac { N } { \\sqrt { E } } e ^ { N ^ 2 / 2 E } \\left ( 1 - e ^ { - N ^ 2 / E } - \\frac { N ^ 2 } { 2 E } \\right ) ^ { 1 / 2 } } \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} r ( n ) = n ^ \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\big ( r ( n ^ \\frac { 1 } { 2 } ) - 1 \\big ) . \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} P _ i = \\lbrace y \\in P \\mid \\dim y = i \\rbrace & & ( 0 \\le i \\le N ) . \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{align*} \\mathbf { x } = \\mathbf { W } ( \\mathbf { z } , \\mathbf { \\phi } ) , \\mathbf { z } \\in \\mathbb { R } ^ { 2 s } , \\end{align*}"}
-{"id": "4196.png", "formula": "\\begin{align*} \\Phi ' : = \\big ( ( \\tilde { A } _ 1 , \\tilde { b } _ 1 ) \\big ) . \\end{align*}"}
-{"id": "7689.png", "formula": "\\begin{align*} g _ r ( z ) = \\mathrm { P } ^ i ( z ) z \\arccos \\frac { z ^ 2 + r _ 0 ^ 2 - d ^ 2 } { 2 r _ 0 z } . \\end{align*}"}
-{"id": "5289.png", "formula": "\\begin{align*} \\ker t _ + ^ k : = \\{ x \\in J \\mid t _ + ^ k . x = 0 \\} \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} \\| \\Pi u \\| _ { L ^ { p , \\alpha } } & = \\left \\| \\langle z \\rangle ^ \\alpha \\int _ \\C K ( z , w ) u ( w ) \\ , d L ( w ) \\right \\| _ { L ^ p } \\lesssim \\left \\| \\int _ \\C e ^ { - \\frac { 1 } { 2 } | z - w | ^ 2 } \\big [ \\langle w \\rangle ^ \\alpha + \\langle z - w \\rangle ^ \\alpha \\big ] u ( w ) \\ , d L ( w ) \\right \\| _ { L ^ p } \\\\ & \\lesssim \\| \\langle z \\rangle ^ \\alpha u ( z ) \\| _ { L ^ p } = \\| u \\| _ { L ^ { p , \\alpha } } . \\end{align*}"}
-{"id": "5578.png", "formula": "\\begin{align*} g ( x ) = g ( \\tfrac { 1 } { x } ) \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{align*} G ( t ) = \\sum _ { i < j } w _ { i j } ( \\cosh ( t _ i - t _ j ) - 1 ) + \\sum _ { k = 1 } ^ n \\left ( ( \\cosh t _ k - 1 ) b _ k + t _ k \\right ) . \\end{align*}"}
-{"id": "1140.png", "formula": "\\begin{align*} 0 = \\mathcal L _ \\partial ( c _ S ) = \\mathcal L _ \\partial ( \\theta _ * ( \\omega _ S ) ) = \\theta _ * ( \\underset { = \\mathrm { d i v } ( \\partial ) \\omega _ S } { \\underbrace { \\mathcal L _ \\partial ( \\omega _ S ) } } ) + \\lambda \\theta _ * ( \\omega _ S ) = ( \\lambda + \\mathrm { d i v } ( \\partial ) ) c _ S \\ , . \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} k \\binom { n } { t - 2 } . \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} x + t e = x ' - t ' e ' . \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} n = ( n _ 1 , \\dots , n _ g ) : = ( M _ 1 - m _ 1 , \\dots , M _ g - m _ g ) v ( n ) : = u ( m ) , \\end{align*}"}
-{"id": "2749.png", "formula": "\\begin{align*} y = \\{ S ^ e , T \\} \\prod _ { ( i , j ) \\in \\mathbb { J } _ S } \\{ 1 + a _ { i j } S ^ i T ^ j , S \\} \\cdot \\prod _ { ( i , j ) \\in \\mathbb { J } _ T } \\{ 1 + b _ { i j } S ^ i T ^ j , T \\} , \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { C _ { \\gamma ; \\psi } \\left [ a , b \\right ] } = \\left \\Vert \\left ( \\psi \\left ( t \\right ) - \\psi \\left ( a \\right ) \\right ) ^ { \\gamma } f \\left ( t \\right ) \\right \\Vert _ { C \\left [ a , b \\right ] } = \\underset { t \\in \\left [ a , b \\right ] } { \\max } \\left \\vert \\left ( \\psi \\left ( t \\right ) - \\psi \\left ( a \\right ) \\right ) ^ { \\gamma } f \\left ( t \\right ) \\right \\vert \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} \\big [ ( B _ 0 - \\lambda ) ^ { - 1 } g \\big ] ( x ) = \\int _ { \\mathbb R } K _ \\lambda ( x , y ) g ( y ) \\ , \\mathrm d y , g \\in L ^ 1 ( \\mathbb R ) , \\end{align*}"}
-{"id": "8114.png", "formula": "\\begin{align*} H ( r _ 0 ) = H ( U , r _ 0 ) > 0 , \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} I _ { v _ 0 + i e _ 1 } & = G ^ \\rho _ { 0 , \\ , v _ 0 + i e _ 1 } - G ^ \\rho _ { 0 , \\ , v _ 0 + ( i - 1 ) e _ 1 } \\\\ J _ { v _ 0 + j e _ 2 } & = G ^ \\rho _ { 0 , \\ , v _ 0 + j e _ 2 } - G ^ \\rho _ { 0 , \\ , v _ 0 + ( j - 1 ) e _ 2 } \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} { \\rm C o e f f } [ f _ { \\mu } , m ] = \\sum _ { \\nu \\in \\sigma ( \\epsilon ) } \\psi ( \\nu , \\mu ) z ^ { \\nu } , \\epsilon = ( 0 ^ { n - r m } , 1 ^ { r m } ) . \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} \\tau ( x ) & = \\inf \\{ s \\geq 0 \\ , : \\ , \\Phi _ s ( x , 0 ) = L _ j \\} , \\\\ \\varsigma ( t ) & = \\inf \\{ s \\geq t \\ , : \\ , \\Phi _ s ( 0 , t ) = L _ j \\} . \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} \\mathfrak { e } ( \\Gamma ^ { \\ast } ) \\ = \\ [ 0 , { \\mathfrak { e } _ { \\mathrm { m a x } } } ( \\mathfrak { e } ) ] , \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{align*} g _ { L ^ 2 } ( ( \\alpha _ 1 , \\dot \\Phi _ 1 ) , ( \\alpha _ 2 , \\dot \\Phi _ 2 ) ) & = 2 \\int _ X \\Re \\langle \\alpha _ 1 , \\alpha _ 2 \\rangle + \\Re \\langle \\dot \\Phi _ 1 , \\dot \\Phi _ 2 \\rangle \\ ; d A , \\\\ & = \\int _ X \\langle \\dot A _ 1 , \\dot A _ 2 \\rangle + 2 \\Re \\langle \\dot \\Phi _ 1 , \\dot \\Phi _ 2 \\rangle \\ ; d A , \\end{align*}"}
-{"id": "5804.png", "formula": "\\begin{align*} { \\rm C o e f f } _ p [ g , m ] : = \\lim _ { q \\rightarrow t ^ { - m } } ( 1 - q t ^ { m } ) ^ p g ( z _ 1 , \\dots , z _ n ) , \\end{align*}"}
-{"id": "153.png", "formula": "\\begin{align*} ( t ^ { - 1 } \\alpha _ t , \\varphi _ t ) \\to ( 0 , \\varphi _ \\infty ) = ( \\dot A _ \\infty , \\dot \\Phi _ \\infty ) - D ^ 1 _ { S _ \\infty } \\gamma _ \\infty \\mbox { a s } \\ \\ t \\to \\infty . \\end{align*}"}
-{"id": "4390.png", "formula": "\\begin{align*} X - \\lambda / 3 - \\sqrt { X ( X - \\lambda ) } = \\lambda / 6 + \\lambda \\sum _ { n = 1 } ^ \\infty \\frac { \\frac 1 2 ( \\frac 1 2 ) _ { n } } { ( n + 1 ) ! } \\frac { \\lambda ^ { n } } { X ^ { n } } . \\end{align*}"}
-{"id": "2049.png", "formula": "\\begin{align*} \\mathfrak { h } ( u , \\ell ) : = F _ 0 ( \\ell ) + u F _ 1 ( \\ell ) - | u \\Psi ( \\ell ) | , ( u , \\ell ) \\in [ - 1 , 1 ] \\times T ^ * M \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{align*} h ^ { - 1 } ( \\R _ + ) : = \\{ x \\in \\R ^ n : h ( x ) \\ge 0 \\} . \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} \\deg ( \\{ 1 , 4 \\} ) = \\deg ( \\{ 1 , 5 \\} ) = \\deg ( \\{ 1 , 6 \\} ) = 2 , \\ ; \\deg ( \\{ 2 , 5 \\} ) = \\deg ( \\{ 2 , 6 \\} ) = 3 , \\textup { a n d } \\deg ( \\{ 3 , 6 \\} ) = 4 . \\end{align*}"}
-{"id": "6887.png", "formula": "\\begin{align*} \\mathrm { l o c } _ { \\beta _ { \\eta } } ( g , j ^ { \\ast } \\mathcal { S } _ { V } ) = \\mathrm { l o c } _ { \\beta _ { s } } ( g , i ^ { \\ast } \\mathcal { S } _ { V } ) \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} \\lambda \\cdot ( x _ M ^ k \\times 1 ) + \\lambda \\cdot ( x _ M ^ { k - n } \\times \\omega _ { N } ) = f ^ * ( x _ M ^ k \\times 1 ) + f ^ * ( x _ M ^ { k - n } \\times \\omega _ { N } ) . \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{align*} X _ n = X _ { \\mu _ n } \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} | x | ^ p / 2 \\ge | x | ^ p / p - 1 / p + 1 / 2 = \\varphi _ p ( x ) { \\rm o n l y \\ ; f o r } \\ ; | x | \\ge 1 . \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} u _ t = \\Delta u + q _ 0 ( x , t ) \\cdot \\nabla u + u ( a _ 0 - b _ 0 u ) , x \\in \\R ^ N , \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} A X B - C X ^ { \\star } D = E , \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} & E \\left [ \\int _ s ^ t ( \\nabla _ z \\chi ) ( r , q _ r ^ m , z _ r ^ m ) \\cdot ( - \\nabla _ q V ( r , q _ r ^ m ) - \\partial _ r \\psi ( r , q _ r ^ m ) + \\tilde F ( r , q _ r ^ m ) ) d r \\right ] \\\\ = & E \\left [ \\int _ s ^ t ( - \\nabla _ q V ( r , q _ r ) - \\partial _ r \\psi ( r , q _ r ) + \\tilde F ( r , q _ r ) ) \\cdot \\left ( \\int ( \\nabla _ z \\chi ) ( r , q _ r , z ) h ( r , q _ r , z ) d z \\right ) d r \\right ] + O ( m ^ { 1 / 2 } ) \\end{align*}"}
-{"id": "4235.png", "formula": "\\begin{align*} g ( \\rho , \\theta ; \\vec { u } ; \\vec { p } ) & = \\prod _ { m = 1 } ^ s ( \\rho e ^ { i \\theta } - p _ m ) \\ ; \\prod _ { \\alpha = 1 } ^ r \\frac { - u _ \\alpha \\rho e ^ { i \\theta } } { ( \\rho e ^ { i \\theta } - u _ \\alpha ) ( q _ 1 q _ 2 \\rho e ^ { i \\theta } - u _ \\alpha ) } . \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} \\begin{pmatrix} 6 & 3 & 5 & 3 \\\\ 1 & 6 & 1 & 5 \\\\ 5 & 3 & 6 & 3 \\\\ 1 & 5 & 1 & 6 \\end{pmatrix} , \\begin{pmatrix} 0 & 6 & 0 & 1 0 \\\\ 0 & 8 & 0 & 7 \\\\ 1 & 0 & 2 & 0 \\\\ 6 & 0 & 3 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "380.png", "formula": "\\begin{align*} H ( X | Y ) = \\sum _ { i = 1 } ^ { k } H ( X _ i | Y , X _ 1 , \\ldots , X _ { i - 1 } ) \\geq \\sum _ { i = 1 } ^ { k } H ( X _ i | \\tilde { X } _ i , \\{ \\tilde { Z } _ j \\} , \\{ X _ w \\} _ { w \\neq i } ) . \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{align*} m ^ j _ { t } = \\int _ { [ 0 , \\ , \\max \\{ 0 , \\ , \\tau ^ { - 1 } ( t ) \\} ] } \\ , \\delta _ { \\Phi _ t ( x , \\ , 0 ) } \\ , d m _ 0 ^ j ( x ) + \\int _ { ( \\max \\{ 0 , \\ , \\varsigma ^ { - 1 } ( t ) \\} , \\ , t ] } \\delta _ { \\Phi _ t ( 0 , \\ , s ) } \\ , d m _ { x = \\pi _ j ( 0 ) } ^ j ( s ) \\\\ m ^ j _ { x = \\pi _ j ( L _ j ) } = \\int _ { ( \\max \\{ 0 , \\ , \\tau ^ { - 1 } ( t ) \\} , \\ , L _ j ] } \\delta _ { \\tau ( x ) } \\ , d m _ 0 ^ j ( x ) + \\int _ { [ 0 , \\ , \\max \\{ 0 , \\ , \\varsigma ^ { - 1 } ( t ) \\} ] } \\delta _ { \\varsigma ( s ) } \\ , d m _ { x = \\pi _ j ( 0 ) } ^ j ( s ) , \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} \\mathrm { P } _ { m } ^ i = & \\sum ^ { m - 1 } _ { l = 0 } \\frac { ( \\lambda _ c \\left [ \\frac { \\pi } { \\bar { \\tau } _ i ^ 2 } - \\pi \\delta ^ 2 \\mathcal { R } _ c ^ 2 \\right ] ) ^ l } { l ! } e ^ { - \\lambda _ c \\left [ \\frac { \\pi } { \\bar { \\tau } _ i ^ 2 } - \\pi \\delta ^ 2 \\mathcal { R } _ c ^ 2 \\right ] } , \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} \\partial _ s ( \\partial _ t + \\delta \\partial _ s ) \\xi = \\theta _ t \\vec n + \\delta _ s \\vec t + \\delta \\theta _ s \\vec n . \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} \\lambda = \\frac n 2 + \\frac 1 2 \\sum _ { i = 1 } ^ n k _ i , k _ i \\in \\{ 0 \\} \\cup \\mathbb { N } , \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{align*} x ^ 2 + y ^ 2 + z ^ h - w ^ k = 0 \\end{align*}"}
-{"id": "184.png", "formula": "\\begin{align*} C ( l _ 0 , l _ 1 ) = \\dfrac { A ( l _ 0 , l _ 1 ) } { B ( l _ 0 , l _ 1 ) } , \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{align*} I ( F _ d ) = 1 { \\rm a n d } I ( F ^ { ( k ) } _ d ) = 1 , \\ k = 1 , . . . , n \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} g ^ { ( h ) } ( \\hat { x } ) \\cdot & d \\log ( \\{ 1 + a S ^ i T ^ j , S \\} ) \\\\ & = - a j g ^ { ( h ) } ( c ' ) \\sum _ { k = 0 } ^ \\infty ( - a ) ^ k S ^ { ( k + 1 ) i + \\ell _ 1 p ^ h - 1 } T ^ { ( k + 1 ) j + \\ell _ 2 p ^ h - 1 } d S \\wedge d T . \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} R = \\frac { r ( \\theta _ 1 \\ , | \\ , \\phi _ 0 , \\phi _ 1 ) \\ , r ( \\theta _ 3 \\ , | \\ , \\phi _ 0 , \\phi _ 3 ) } { r ( \\theta _ 1 + \\theta _ 3 \\ , | \\ , \\phi _ 1 , \\phi _ 3 ) } \\frac { P ( \\theta _ 1 \\ , | \\ , \\phi _ 2 , \\phi _ 3 ) \\ , P ( \\theta _ 3 \\ , | \\ , \\phi _ 1 , \\phi _ 2 ) } { P ( \\theta _ 1 + \\theta _ 3 \\ , | \\ , \\phi _ 0 , \\phi _ 2 ) } \\ , . \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} \\phi ( x ) = \\sum _ { i = 1 } ^ n \\psi ( x _ i ) \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} M _ { 1 0 } = M _ { 1 , 3 } \\times X \\ : , \\end{align*}"}
-{"id": "1886.png", "formula": "\\begin{align*} \\sum _ { \\ell = 4 } ^ { n - 1 } \\sum _ { s = 1 } ^ { n - \\ell } \\sum _ { t = 0 } ^ { n - \\ell - s } ( \\ell - 3 ) & = \\sum _ { s = 1 } ^ { n - 4 } \\sum _ { t = 0 } ^ { n - 4 - s } \\sum _ { \\ell = 4 } ^ { n - s - t } ( \\ell - 3 ) = \\sum _ { s = 1 } ^ { n - 4 } \\sum _ { t = 0 } ^ { n - 4 - s } \\binom { n - 2 - s - t } { 2 } \\\\ & = \\sum _ { s = 1 } ^ { n - 4 } \\binom { n - 1 - s } { 3 } = \\binom { n - 1 } { 4 } \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} \\mathrm { d i v } ( s ) \\cap V = \\sum _ { Y \\in \\mathcal { I } ^ G ( X ) } 2 k v _ { \\mathcal { L } } ( \\mu _ Y ) ( Y \\cap V ) . \\end{align*}"}
-{"id": "4510.png", "formula": "\\begin{align*} C = \\{ v \\in X \\otimes _ \\Z \\R \\ | \\ ( m _ \\beta - 1 ) p < \\langle v + \\rho , \\beta ^ \\vee \\rangle < m _ \\beta p , \\ \\forall \\beta \\in R ^ + \\} , \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} U ( F ) ( z , y ) = h ( y ) F ( \\varphi ( z , y ) , \\tau ( y ) ) , ( z , y ) \\in \\mathbb { T } \\times Y _ 2 \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} \\int _ { \\Sigma } ( \\dfrac { n } { 2 } + | { \\bf H } | ^ 2 ) u ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma & \\leq \\int _ { \\Sigma } | \\nabla u | ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma \\\\ & = \\frac n 2 \\int _ { \\Sigma } u ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma . \\end{align*}"}
-{"id": "3540.png", "formula": "\\begin{align*} \\kappa ^ { 2 } _ { k } ( x , t ) \\leq \\frac { T - 1 / k - t _ { k } } { T - 1 / k - t _ { k } - \\lambda ^ { 2 } _ { k } t } = \\frac { t ^ { ( 1 ) } _ { k } } { t ^ { ( 1 ) } _ { k } - t } , t \\in [ t _ { k } ^ { ( 0 ) } , t ^ { ( 1 ) } _ { k } ) . \\end{align*}"}
-{"id": "8616.png", "formula": "\\begin{align*} \\mathit { C W E } ( \\mathcal { C } ) = \\sum \\limits _ { \\mathbf { t } \\in \\mathbb { F } _ { p } ^ { p } \\setminus \\lbrace \\mathbf { 0 } \\rbrace } \\mathfrak { F } ( t _ { 0 } , t _ { 1 } , \\cdots , t _ { p - 1 } ) w _ { 0 } ^ { t _ { 0 } } w _ { 1 } ^ { t _ { 1 } } \\cdots w _ { p - 1 } ^ { t _ { p - 1 } } \\end{align*}"}
-{"id": "5997.png", "formula": "\\begin{align*} \\mathcal { F } \\left \\{ { D } ^ { \\alpha } _ { \\theta } \\psi ( x , t ) ; p \\right \\} = { \\eta } _ { \\alpha } ^ { \\theta } \\hat { \\psi } ( p , t ) . \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} c _ { + } ^ { * } ( a , b , \\chi , \\lambda , \\mu ) : = 2 \\sqrt { a _ { \\sup } } + \\frac { \\chi \\mu \\sqrt { N } a _ { \\sup } } { 2 ( b _ { \\inf } - \\chi \\mu ) \\sqrt { \\lambda } } . \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} c [ S ^ { \\ell _ 1 } T ^ { \\ell _ 2 } ] = ( c _ k S ^ { \\ell _ 1 p ^ k } T ^ { \\ell _ 2 p ^ k } ) _ { k = 0 } ^ \\infty , \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} ( T h ) ( x ' , D _ { x } ) : = \\lambda ^ { - 1 , * } h ( x ' , D _ { x } ) \\kappa ^ * \\ \\ h \\in S ^ m _ { 1 , 0 } ( \\R ^ { n - 1 } \\times \\R ^ { n - 1 } ; \\mathcal { S } _ + ) \\end{align*}"}
-{"id": "5708.png", "formula": "\\begin{align*} G _ T = G + \\binom { S _ T } { 2 } - E ( B _ T ) . \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} d _ { A _ \\infty } \\xi _ \\infty + d _ { A _ t } \\xi _ t = ( d _ { A _ \\infty } - d _ { A _ t } ) \\xi _ \\infty + d _ { A _ t } ( \\xi _ \\infty + \\xi _ t ) ; \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} - \\ ! F ' ( ( \\overline { p } , \\overline { x } ) ; ( 0 , \\Delta x ) ) - \\ ! \\nabla ^ 2 \\langle \\overline { \\lambda } , g \\rangle ( \\overline { x } ) \\Delta x \\in \\nabla g ( \\overline { x } ) D \\mathcal { N } _ { K } ( g ( \\overline { x } ) | \\overline { \\lambda } ) ( g ' ( \\overline { x } ) \\Delta x ) \\Longrightarrow \\Delta x = 0 . \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} & q _ { X _ 1 Y _ 1 X _ 2 Y _ 2 } ( x _ 1 , y _ 1 , x _ 2 , y _ 2 ) = r _ { X _ 1 Y _ 1 X _ 2 Y _ 2 } ( x _ 1 , y _ 1 , x _ 2 , y _ 2 ) \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{align*} H _ \\textrm { s c a t t } ( Q , P ) = \\dfrac { P ^ 2 } { 2 M } + U ( Q ) . \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} \\psi ( t , x ) : = \\mathbb { E } _ { x , t } ^ { \\beta } \\left [ \\psi _ 0 ( \\beta _ 0 ) e ^ { - \\frac { 1 } { 2 } M _ t ^ { \\beta } } \\right ] . \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} \\mathrm { P } ^ 1 _ { m , 1 } = \\mathrm { P } ( \\log ( 1 + ^ 1 _ { m , 1 } ) < R _ 1 ) , \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} \\beta _ { m k } ^ { } = 4 6 + 2 0 \\log { } _ { 1 0 } ( d _ { m k } ) + V _ { m k } . \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} { \\left ( \\int \\Gamma ( f ) d \\mu \\right ) } ^ 2 = { \\left ( - \\int f L f d \\mu \\right ) } ^ 2 \\leq \\int f ^ 2 d \\mu \\int { ( - L f ) } ^ 2 d \\mu \\leq c _ P \\int \\Gamma ( f ) d \\mu \\int { ( - L f ) } ^ 2 d \\mu \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} \\dot x _ i = \\varepsilon _ i x _ i + \\sum _ { j = 1 } ^ n A _ { i , j } x _ i x _ j , \\ \\ i = 1 , 2 , \\dots , n \\ ; . \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} \\mathrm { P } _ { m , i } = 1 - \\mathrm { P } \\left ( f _ j , \\forall j \\leq i \\right ) . \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} c \\left ( \\frac { d _ k } { 2 } \\right ) ^ { 2 \\ell } > c \\left ( \\frac { \\alpha n ^ { 1 / \\ell } } { 2 } \\right ) ^ { 2 \\ell } \\geq c \\frac { L } { 2 c } n ^ 2 \\geq L \\binom { k } { 2 } , \\end{align*}"}
-{"id": "2294.png", "formula": "\\begin{align*} d _ { f , \\hbar } \\left ( e ^ { \\frac { f ( a ) - f } { \\hbar } } U _ a \\right ) = \\sum _ { b : ( b ) = ( a ) + 1 } n _ { a , b } e ^ { - \\frac { f ( b ) - f ( a ) } { \\hbar } } e ^ { \\frac { f ( b ) - f } { \\hbar } } U _ b . \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} & \\delta ^ { i _ 1 i _ 2 } G _ { i _ 1 i _ 2 i _ 3 } ^ { \\eta j _ 2 j _ 3 } \\delta _ { j _ 2 j _ 3 } \\tilde \\gamma _ { \\eta k } + 2 \\delta ^ { i _ 1 i _ 2 } G _ { i _ 1 i _ 2 i _ 3 } ^ { j _ 1 j _ 2 j _ 3 } \\gamma _ { j _ 1 j _ 3 } \\delta _ { j _ 2 k } \\\\ = & \\delta _ { i _ 3 k } \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} w ^ { - 1 } ( t _ { \\nu _ 1 } - t _ { \\nu _ 1 + 1 } ) = t _ { w ^ { - 1 } ( \\nu _ 1 ) } - t _ { w ^ { - 1 } ( \\nu _ 1 + 1 ) } = t _ a - t _ b , \\end{align*}"}
-{"id": "8941.png", "formula": "\\begin{align*} H ^ { \\epsilon } : = H ^ { \\epsilon , 0 } = ( - i \\partial _ { x _ 1 } - A ^ \\Gamma _ 1 - A ^ { \\epsilon } _ 1 ) ^ 2 + ( - i \\partial _ { x _ 2 } - A ^ \\Gamma _ 2 - A ^ { \\epsilon } _ 2 ) ^ 2 + V _ \\Gamma \\ , . \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{align*} J _ 1 ( N ; \\theta ) = \\frac { N H _ N } { 1 - \\theta } , H _ N : = \\sum _ { j = 1 } ^ N \\frac { 1 } { j } . \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} | h | _ { C ^ \\tau S ^ m _ { 1 , 0 } ( \\mathcal { S } _ + ) } ^ { i } : = \\underset { l , l ' , | \\alpha | \\leq i } { \\max } \\ \\underset { \\xi ' \\in \\R ^ { n - 1 } } { \\sup } \\left \\| y _ n ^ l \\partial _ { y _ n } ^ { l ' } \\partial _ { \\xi ' } ^ \\alpha { h } ( \\cdot , \\xi ' , \\cdot ) \\right \\| _ { C ^ \\tau ( U ; L ^ 2 _ { y _ n } ( { \\R } _ { + } ) ) } \\langle \\xi ' \\rangle ^ { - m - \\frac { 1 } { 2 } + l - l ' + | \\alpha | } \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{align*} \\begin{cases} \\eta ( 0 , t ) = 0 , \\ , \\ , \\eta ( L , t ) = 0 , \\ , \\ , \\eta _ { x } ( 0 , t ) = f ( t ) & \\ , \\ , ( 0 , T ) , \\\\ w ( 0 , t ) = 0 , \\ , \\ , w ( L , t ) = 0 , \\ , \\ , w _ { x } ( L , t ) = g ( t ) & \\ , \\ , ( 0 , T ) \\end{cases} \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} \\pi _ Y : \\mathbb { C } ^ m \\times Y \\rightarrow Y , \\pi _ Y ( z , y ) = y , \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{align*} \\sum _ { m } ( - 1 ) ^ { \\mu _ m } \\kappa ( m , \\mu , \\lambda ) = \\frac { ( N - 1 ) ( N - 2 | \\mu | ) } { 2 } , \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{align*} e _ { I _ i ( H ) } ( I _ t ( H ) ) \\ = \\ t - i + 1 \\quad \\ i = 1 , \\dots , t . \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} \\bar \\partial _ B Z \\lrcorner \\ , \\phi & = \\sum _ a \\bar \\omega ^ a \\wedge ( \\nabla _ { \\bar V _ a } Z ) \\lrcorner \\ , \\phi + \\sum _ a \\bar \\omega ^ a \\wedge Z \\lrcorner \\nabla _ { \\bar V _ a } \\phi \\\\ & = - Z \\lrcorner \\ , \\bar \\partial _ B \\phi , \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} H \\left ( \\sum _ { i = 1 } ^ k q _ i \\mu _ i , \\mathcal { E } \\right ) & \\leq \\sum _ { E \\in \\mathcal { E } } \\sum _ { i = 1 } ^ k \\phi \\left ( q _ i \\mu _ i ( E ) \\right ) \\\\ & = \\sum _ { i = 1 } ^ k q _ i \\left ( \\sum _ { E \\in \\mathcal { E } } \\phi ( \\mu _ i ( E ) ) \\right ) + \\sum _ { i = 1 } ^ k \\phi ( q _ i ) \\left ( \\sum _ { E \\in \\mathcal { E } } \\mu _ i ( E ) \\right ) \\\\ & = \\sum _ { i = 1 } ^ k q _ i H ( \\mu _ i , \\mathcal { E } ) + H ( q ) . \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} L _ m R _ m v = \\begin{cases} q ^ { \\kappa ( m , \\mu , \\lambda ) } v & , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{align*} \\tilde \\delta _ n = \\tilde \\kappa _ \\nu ( \\tilde \\kappa _ g ^ 2 + \\tilde \\kappa _ t ^ 2 ) - \\tilde \\kappa _ g \\tilde \\kappa _ t ' + \\tilde \\kappa _ t \\tilde \\kappa _ g ' . \\end{align*}"}
-{"id": "7232.png", "formula": "\\begin{align*} \\frac { ( c , a x t ; q ) _ \\infty } { ( a b , t x ; q ) _ \\infty } { _ 2 \\phi _ 1 \\left ( { { d , x t } \\atop { a x t } } ; q , c \\right ) } = { _ 2 \\phi _ 1 \\left ( { { a , c } \\atop { a b } } ; q , x t \\right ) } . \\end{align*}"}
-{"id": "8846.png", "formula": "\\begin{align*} E = & O + \\bar { z } _ 1 z _ 2 \\frac { \\beta ( l _ j ) } { 2 } ( \\mu _ { \\beta } - \\frac { \\tanh ( \\beta ( a ) + \\coth ( \\beta ( a ) ) } { 2 } \\tau _ { \\beta } ) \\\\ & + z _ 1 \\bar { z } _ 2 \\frac { 3 \\beta ( l _ j ) } { 4 } ( \\coth ( \\beta ( a ) ) - \\tanh ( \\beta ( a ) ) ) \\theta ( \\tau _ { \\beta } ) ) . \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} & W ^ 0 _ { l + 2 } ( p , p _ 1 , \\dots , p _ { l + 1 } ) \\ast W ^ 0 _ { m + 2 } ( p ' , p ^ { ' } _ { 1 } \\dots , p ^ { ' } _ { m + 1 } ) = \\\\ & \\delta _ { p , p ' } \\delta _ { p _ { l + 1 } , p ' _ { 1 } } W ^ 0 _ { k + 2 } ( p , p _ 1 , \\dots , p _ { k + 1 } ) , k = l + m . \\end{align*}"}
-{"id": "5260.png", "formula": "\\begin{align*} d _ 1 ( \\varphi ( x _ 1 , y ) , \\varphi ( x _ 2 , y ) ) & = f ( \\varphi ( x _ 1 , y ) ) = | f ( \\varphi ( x _ 1 , y ) ) - f ( \\varphi ( x _ 2 , y ) ) | \\\\ & = | f \\otimes 1 ( \\varphi ( x _ 1 , y ) , \\tau ( y ) ) - f \\otimes 1 ( \\varphi ( x _ 2 , y ) , \\tau ( y ) ) | \\\\ & = | ( U ( f \\otimes 1 ) ) ( x _ 1 , y ) - ( U ( f \\otimes 1 ) ) ( x _ 2 , y ) | \\\\ & \\le L ( U ( f \\otimes 1 ) ) d _ 2 ( x _ 1 , x _ 2 ) . \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{align*} \\Delta _ { \\Omega } ( m ) : = \\Omega \\Delta ( m ) \\Omega ^ { * } . \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} f _ \\pm ( K ) = f _ + ( K \\cap [ 0 , \\tau ] ) \\cup f _ - ( K \\cap [ \\tau , 1 ] ) . \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{align*} Q _ { n , \\textup { \\textbf { m } } } ( z ) : = \\prod _ { | \\Phi ( \\lambda _ { n , j } ) | \\leq L } ( z - \\lambda _ { n , j } ) \\prod _ { | \\Phi ( \\lambda _ { n , j } ) | > L } \\left ( 1 - \\frac { z } { \\lambda _ { n , j } } \\right ) . \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} \\begin{pmatrix} & \\frac { s _ g } { 1 2 } \\\\ \\frac { s _ g } { 1 2 } \\\\ & & - \\frac { s _ g } { 1 2 } \\end{pmatrix} . \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} S ^ * M = \\{ ( x , \\xi ) \\in T ^ * M ; | \\xi | ^ 2 _ g = 1 \\} \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} S _ q = \\{ \\alpha \\in K _ X \\mid \\alpha ^ 2 = q ( p ( \\alpha ) ) \\} \\subset K _ X . \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{align*} A _ { { k _ 1 } \\cdots { k _ m } } = \\begin{pmatrix} a _ { 1 { k _ 1 } } ^ 1 & \\cdots & a _ { 1 { k _ m } } ^ m \\\\ \\vdots & & \\vdots \\\\ a _ { m { k _ 1 } } ^ 1 & \\cdots & a _ { m { k _ m } } ^ m \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} y _ i ^ { n } = \\begin{cases} - n + \\overline \\mu _ i , & \\overline \\mu _ i < \\infty \\\\ n - \\overline \\nu _ i , & \\overline \\nu _ i < \\infty \\\\ 0 , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{align*} \\widehat { \\tilde K _ \\delta } ( \\xi , \\tau ) = \\phi ( | \\xi | ) \\tilde \\psi \\left ( \\frac { | \\xi | - \\tau } { \\delta } \\right ) . \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{align*} V _ e ( C ) = \\prod _ { \\pi _ 0 ( C ) } \\{ x _ + , x _ - \\} . \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} f _ 1 ( r ) = \\wp _ \\lambda ( r \\omega _ 1 ) + \\frac 1 3 ( \\lambda + 1 ) , ~ ~ f _ 2 ( r ) = \\wp _ \\lambda ( r \\omega _ 2 ) + \\frac 1 3 ( \\lambda + 1 ) . \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} \\mathcal { L } _ { I _ J } ( s ) & \\overset { ( a ) } { = } \\mathbb { E } _ { \\Phi _ J } \\Big [ \\prod _ { \\mathbf { z } _ i \\in \\Phi _ J } \\mathcal { L } _ h ( s P _ J D _ { \\mathbf { z } _ i , \\mathbf { u } _ 0 } ^ { - \\alpha } ) \\Big ] \\\\ & = \\exp \\Big ( - 2 \\pi \\lambda _ J \\int _ { 0 } ^ { \\infty } \\frac { y } { 1 + \\frac { y ^ { \\alpha } } { s P _ J } } d y \\Big ) , \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} \\Bigl ( \\frac { g ( x _ * , R ) } { g ( y , R ) } \\Bigr ) ^ { \\alpha / \\delta } & = \\frac { R - d ( z , x _ * ) } { R - d ( y , z ) } \\\\ & \\leq \\frac { R - d ( z , x _ * ) } { R - \\bigl ( d ( z , x _ * ) + \\beta ( 2 K ) ^ { 1 - \\alpha } \\rho ^ { \\alpha } ( d ( z , x _ * ) + \\rho ) ^ { 1 - \\alpha } \\bigr ) } \\\\ & \\leq \\frac { 1 } { 1 - \\frac { \\beta ( 2 K ) ^ { 1 - \\alpha } \\rho ^ { \\alpha } ( R + \\rho ) ^ { 1 - \\alpha } } { \\beta _ * \\rho ^ { \\alpha } R ^ { 1 - \\alpha } } } \\\\ & \\leq \\frac { 1 } { 1 - \\frac { 2 \\beta ( 2 K ) ^ { 1 - \\alpha } } { \\beta _ * } } \\end{align*}"}
-{"id": "1619.png", "formula": "\\begin{align*} \\hat \\lambda _ { R _ { L _ t } } ^ { ( L _ t ) } ( Z _ t ) - \\hat \\lambda _ { \\rho _ \\xi , \\rho _ \\sigma } ^ { ( L _ t ) } ( Z _ t ) = T _ 1 + T _ 2 + T _ 3 \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} \\phi _ n ( x ) & : = ( \\sqrt { ( D + e _ d ) ^ 2 + 1 } + \\sqrt { ( D - e _ d ) ^ 2 + 1 } ) w _ n ( x ) , \\\\ u _ n ( x ) & : = \\sin ( x _ d ) \\phi _ n ( x ) . \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} { \\mathcal L } ^ { - 1 } ( F ) ( t ) = { 1 \\over 2 \\pi } e ^ { \\alpha t } \\int _ { - \\infty } ^ { + \\infty } F ^ * ( \\alpha + i y ) e ^ { i y t } d y . \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{align*} { } ^ 0 \\rho _ 0 = ( 1 + q _ - ) ^ { - 1 } , { } ^ 0 \\rho _ I = t ^ { - 1 } ( 1 + q _ - ) ( 1 + q _ + ) , { } ^ 0 \\rho _ + = ( 1 + q _ + ) ^ { - 1 } , { } ^ 0 \\rho = t ^ { - 1 } ; \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{gather*} [ \\rho _ c ( a ) H ( \\cdot , x ) ] ( n ) = c \\frac { \\partial } { \\partial x } H ( n , x ) , \\\\ [ \\rho _ c ( a ^ \\dagger ) H ( \\cdot , x ) ] ( n ) = \\left ( x - c \\frac { \\partial } { \\partial x } \\right ) H ( n , x ) . \\end{gather*}"}
-{"id": "8165.png", "formula": "\\begin{align*} X _ 1 ^ 2 a _ 2 = ( X _ 1 X _ 2 + [ X _ 1 , X _ 2 ] ) a _ 1 , \\ X _ 2 ^ 2 a _ 1 = ( X _ 2 X _ 1 + [ X _ 2 , X _ 1 ] ) a _ 2 , \\end{align*}"}
-{"id": "490.png", "formula": "\\begin{align*} | g _ a ( z _ n ) | = \\frac { 2 } { \\left ( \\frac { 1 } { 4 } + q _ 2 ^ 2 y ^ 2 \\right ) ^ { k / 2 } } . \\end{align*}"}
-{"id": "6326.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle \\frac { \\partial G } { \\partial { \\bf u } _ a } ( { \\bf u } ) - \\sum \\limits _ { d = 1 } ^ a \\sigma _ { d a } ( { \\bf u } ) { \\bf u } _ d - \\sum \\limits _ { d = a + 1 } ^ p \\sigma _ { a d } ( { \\bf u } ) { \\bf u } _ d = { \\bf 0 } , \\ , \\ , \\ , \\forall a \\in \\{ 1 , . . . , p - 1 \\} \\\\ \\displaystyle \\frac { \\partial G } { \\partial { \\bf u } _ p } ( { \\bf u } ) - \\sum \\limits _ { d = 1 } ^ p \\sigma _ { d p } ( { \\bf u } ) { \\bf u } _ d = { \\bf 0 } , \\end{cases} \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{align*} J = \\begin{cases} 0 & Z = ( 0 , 0 ) , \\\\ 1 & Z = ( 1 , 1 ) , \\\\ \\mathtt { e } & \\end{cases} \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} \\wp ( \\omega ) = \\wp ( \\omega ; \\omega _ 1 , \\omega _ 2 ) : = \\frac { 1 } { \\omega ^ { 2 } } + \\sum _ { ( \\ell _ { 1 } , \\ell _ { 2 } ) \\in \\Z ^ { 2 } \\setminus \\{ ( 0 , 0 ) \\} } \\left ( \\frac { 1 } { ( \\omega + \\ell _ { 1 } \\omega _ { 1 } + \\ell _ { 2 } \\omega _ { 2 } ) ^ { 2 } } - \\frac { 1 } { ( \\ell _ { 1 } \\omega _ { 1 } + \\ell _ { 2 } \\omega _ { 2 } ) ^ { 2 } } \\right ) . \\end{align*}"}
-{"id": "7258.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ K \\omega _ k = 1 , \\sum _ { k = 1 } ^ K \\omega _ k v _ k ^ 2 = \\frac 1 3 , v _ { - k } = - v _ k . \\end{align*}"}
-{"id": "5696.png", "formula": "\\begin{align*} P _ { A _ s } x = \\left \\{ z \\in \\mathbb { R } ^ n \\mid \\exists \\ ; \\mathcal { I } \\in \\mathcal { I } _ x \\ ; \\mbox { s u c h t h a t } \\ ; z ( k ) = \\begin{cases} x ( k ) & \\mbox { i f } k \\in \\mathcal { I } , \\\\ 0 & \\mbox { e l s e } \\end{cases} \\right \\} . \\end{align*}"}
-{"id": "8944.png", "formula": "\\begin{align*} \\pi _ 0 : = \\mathcal { U } _ \\Gamma ^ { - 1 } \\ , \\hat { \\pi } _ 0 \\ , \\mathcal { U } _ \\Gamma \\ , , \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} \\lambda ^ - ( \\mu ) = \\lim _ { t \\rightarrow \\infty } \\dfrac { 1 } { t } \\log \\parallel \\psi ^ * _ t | \\mathcal { E } ^ s _ x \\parallel < 0 , ~ ~ ~ \\lambda ^ + ( \\mu ) = \\lim _ { t \\rightarrow \\infty } \\dfrac { 1 } { t } \\log m ( \\psi ^ * _ t | \\mathcal { F } ^ u _ x ) > 0 \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{align*} \\frac { W _ 2 ^ 2 ( \\mu , \\nu ) } { 2 } = \\sup _ { ( \\varphi , \\psi ) } \\bigg \\{ \\int _ X \\varphi \\ , d \\mu - \\int _ X \\psi \\ , d \\nu \\ , \\bigg | \\ , \\varphi ( x ) - \\psi ( y ) \\le \\frac { d ^ 2 ( x , y ) } { 2 } \\bigg \\} \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{align*} \\sigma ' _ 0 ( X ' _ { f ^ i _ { j } } , \\cdot ) = \\sigma ' _ 1 ( X ' _ { f ^ i _ { j - 1 } } , \\cdot ) , \\ ; \\mathrm { f o r } \\ ; \\mathrm { a n y } \\ ; i = 1 , \\ldots , k \\ ; \\ ; \\mathrm { a n d } \\ ; \\ ; j = 1 , \\ldots , r _ i . \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{align*} \\varphi _ E ( u , z ) = \\begin{cases} \\frac { 1 } { 2 } \\gamma ( z ^ { - 1 } ) \\left ( | A | + 1 \\right ) , & u = 0 \\\\ \\frac { 1 } { 2 } \\gamma ( z ^ { - 1 } ) , & u \\neq 0 \\end{cases} \\left ( ( u , z ) \\in H \\right ) . \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{align*} C _ 1 ( r , s , t ) ~ : = ~ \\frac { 4 \\pi ^ { n } \\Gamma ( 1 + t ) \\Gamma ( r + s - t - n - 1 ) } { \\Gamma ( r ) \\Gamma ( s ) } . \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} \\frac { \\prod _ { i = 1 } ^ { n } ( 1 - t ^ i ) } { \\prod _ { i = 1 } ^ { k } ( 1 - t ^ i ) \\prod _ { i = 1 } ^ { n - k } ( 1 - t ^ i ) } = \\frac { \\prod _ { i = n - k + 1 } ^ { n } ( 1 - t ^ i ) } { \\prod _ { i = 1 } ^ { k } ( 1 - t ^ i ) } \\end{align*}"}
-{"id": "5031.png", "formula": "\\begin{align*} [ c , z _ 1 ] [ z _ 2 , z _ 3 , z _ 4 ] + [ c , z _ 1 ] [ z _ 3 , z _ 4 , z _ 2 ] + [ c , z _ 1 ] [ z _ 4 , z _ 2 , z _ 3 ] = [ c , z _ 1 ] \\bigl ( [ z _ 2 , z _ 3 , z _ 4 ] + [ z _ 3 , z _ 4 , z _ 2 ] + [ z _ 4 , z _ 2 , z _ 3 ] \\bigr ) = 0 . \\end{align*}"}
-{"id": "5879.png", "formula": "\\begin{align*} I ( \\mu , \\nu ) = \\left \\{ \\begin{array} { l l } 0 , & \\exists \\ k : ( \\mu _ k , \\nu _ k ) = ( r , 0 ) , \\\\ \\\\ 1 , & , \\end{array} \\right . \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} \\mathcal { L } v + ( \\dfrac { n } { 2 } + | { \\bf H } | ^ 2 ) v = 0 . \\end{align*}"}
-{"id": "6794.png", "formula": "\\begin{align*} \\partial _ t \\langle D ^ k \\xi , D ^ k \\eta \\rangle = \\langle \\nabla _ t D ^ k \\xi , D ^ k \\eta \\rangle + \\langle D ^ k \\xi , \\nabla _ t D ^ k \\eta \\rangle . \\end{align*}"}
-{"id": "4172.png", "formula": "\\begin{align*} \\mathcal { N N } _ { M , K } ^ \\ast : = \\{ \\Phi \\in \\mathcal { N N } _ { M , K , d } ^ \\mathcal { B } \\ , : \\ , N ( \\Phi ) \\leq M ( \\Phi ) + d + 1 \\} \\end{align*}"}
-{"id": "7489.png", "formula": "\\begin{align*} d q _ t = & \\tilde \\gamma ^ { - 1 } ( t , q _ t ) \\left ( - \\partial _ t \\psi ( t , q _ t ) - \\nabla _ q V ( t , q _ t ) + \\tilde F ( t , q _ t , \\psi ( t , q _ t ) ) \\right ) d t \\\\ & + \\tilde S ( t , q _ t ) d t + \\tilde \\gamma ^ { - 1 } ( t , q _ t ) \\sigma ( t , q _ t ) \\circ d W _ t , \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } Y _ t = Y _ 0 + \\int _ 0 ^ t \\Phi ( s , Y _ s ) d W _ s + \\int _ 0 ^ t g ( s , Y _ s ) d s \\\\ Y _ 0 \\sim u _ 0 \\ , \\end{array} \\right . \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} q = n - \\dim A \\le \\frac { n + 1 } { 2 } \\ \\ \\Longleftrightarrow \\ \\ \\dim A \\ge \\left [ \\frac { n } { 2 } \\right ] . \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} X ^ { A } _ t : = \\frac { x ^ A _ { \\frac { 1 - \\alpha } { 2 } t } ( t ) - ( \\alpha - \\tfrac 1 2 ) t } { - t ^ { 1 / 3 } } . \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} \\int _ { \\mathcal { X } \\times \\mathcal { Y } } \\hat { K } _ \\tau \\varphi ( x , y ) p _ t ( d x , d y ) & = \\int _ { \\mathcal { Y } } \\int _ \\mathcal { X } ( \\hat { K } ^ { \\tau , y } \\varphi ( \\cdot , y ) ) ( x ) \\nu _ t ^ { \\phi ( y ) } p _ t ( \\mathcal { X } , d y ) = 0 \\\\ \\hat { K } _ \\tau \\hat { F } _ \\tau ( x , y ) & = f _ \\tau ( x , y ) = K _ \\tau F _ \\tau ( x , y ) . \\end{align*}"}
-{"id": "2759.png", "formula": "\\begin{align*} \\{ 1 + a _ { 1 , 1 , 0 } S ^ 1 T ^ 1 , S \\} \\xrightarrow { \\phi _ S } \\left ( \\begin{cases} 0 \\textrm { i f } m \\neq n \\\\ [ a _ { 1 , 1 , 0 } ] ^ n \\textrm { i f } m = n \\end{cases} \\right ) _ { ( m , n ) \\in \\mathbb { J } _ S } \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{align*} { \\bf D } _ { I } = \\omega _ I { \\bf I } _ K . \\end{align*}"}
-{"id": "9154.png", "formula": "\\begin{align*} \\lVert \\overline { \\xi } _ { \\overline { x } } - \\overline { \\xi } _ { \\overline { y } } \\rVert & = \\frac { 1 } { | F | } \\sum _ { \\Lambda z \\in \\Gamma / \\Lambda } \\lvert \\ , \\lvert \\Lambda z \\cap x F \\rvert - \\lvert \\Lambda z \\cap y F \\rvert \\ , \\rvert \\\\ & \\le \\frac { 1 } { | F | } \\sum _ { \\Lambda z \\in \\Gamma / \\Lambda } | ( x F \\Delta y F ) \\cap \\Lambda z | \\\\ & = \\frac { | x F \\Delta y F | } { | F | } \\\\ & \\le \\epsilon . \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{align*} \\{ ( x _ 1 , \\ldots , x _ s ) : x _ i \\in [ 0 , z _ i ] , i = 1 , \\ldots , s \\} , \\end{align*}"}
-{"id": "6919.png", "formula": "\\begin{align*} N \\ge \\lim _ j \\bigg | \\frac { \\partial _ { x _ i } u _ { k _ j } ( s _ j x ) } { s _ j x _ n } \\bigg | = \\lim _ j \\bigg | \\frac { \\partial _ { x _ i } \\tilde u _ j ( x ) } { x _ n } \\bigg | = \\bigg | \\frac { \\partial _ { x _ i } u _ 0 ( x ) } { x _ n } \\bigg | \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} H _ 0 : \\theta = 0 \\in \\Theta _ { d } H _ 1 : \\theta \\in \\Theta _ { d } \\setminus \\{ 0 \\} \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} \\lambda _ n v _ { n } + p _ { n } ( e ^ { i \\Omega t } + e ^ { - i \\Omega t } ) = \\dot { v } _ { n } + \\delta _ { n l } r _ c e ^ { i \\Omega t } + \\delta _ { n ( l + N ) } \\bar { r } _ c e ^ { - i \\Omega t } , \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { n - 1 } \\delta _ { a _ k 0 } = \\sum \\limits _ { k = 1 } ^ { n } \\delta _ { a _ k 0 } = \\sum \\limits _ { k = 1 } ^ { n } \\delta _ { b _ k 0 } = \\sum \\limits _ { k = 1 } ^ { n - 1 } \\delta _ { b _ k 0 } , \\end{align*}"}
-{"id": "4460.png", "formula": "\\begin{align*} w ( D ^ \\prime ) & = n ( n - 2 d + c ) + w ( D ' [ B ] ) \\\\ & \\le n ( n - s ) + s ^ 2 \\\\ & < n ^ 2 - z n + b ( z ) + 1 . \\end{align*}"}
-{"id": "7539.png", "formula": "\\begin{align*} & E \\left [ S _ { s , t } ^ { a n o m } \\right ] = \\int _ { s } ^ t E [ K ( r , q _ r ) ] d r \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{align*} \\det \\left [ D ^ 2 u - A ( x , u , D u ) - B ( x , u , D u ) \\right ] = f ( x , u , D u ) , \\ , \\ , x \\in \\Omega , \\\\ \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{align*} \\Vert U x - z \\Vert ^ { 2 } \\leq \\Vert x - z \\Vert ^ { 2 } - \\sum _ { i = 1 } ^ { m } \\omega _ { i } \\rho _ { i } \\Vert U _ { i } x - x \\Vert ^ { 2 } \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} \\chi ( T \\Sigma ^ \\perp ) = \\deg ( L ) . \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} { \\rm C o e f f } _ p [ f _ { \\mu } , m ] = \\sum _ { \\nu \\in \\sigma ( \\epsilon ) } \\psi ( \\nu , \\mu ; t ) f _ { \\nu } ( z ; t ^ { - m } , t ) , \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{align*} s ^ { \\alpha } = \\sum _ { k \\in \\mathbb { Z } } 2 ^ { - k \\alpha } \\psi ( 2 ^ { k } s ) \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} & \\mathcal { P } _ c ( \\theta _ c , \\tau ) \\\\ & = \\sum _ { k \\in \\mathcal { K } } \\mathcal { A } _ k \\mathbb { E } _ { D _ { \\mathbf { b } _ 0 , \\mathbf { u } _ 0 } } [ \\mathbb { P } ( _ U \\geq \\theta _ c | \\mathbf { b } _ 0 \\in \\Phi _ k , D _ { \\mathbf { b } _ 0 , \\mathbf { u } _ 0 } = r ) ] . \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{align*} t ^ { ( 1 ) } _ { k } = \\kappa ^ { 2 } ( x _ { k } , t _ { k } ) ( T - 1 / k - t _ { k } ) \\geq \\kappa ^ { 2 } ( \\bar { x } , \\bar { t } ) ( T - 1 / k - \\bar { t } ) > M . \\end{align*}"}
-{"id": "6806.png", "formula": "\\begin{align*} [ \\nabla _ t , D ^ { k + 1 } ] \\psi & = \\sum _ { l = 0 } ^ { k } D ^ l ( R ^ { S M } ( \\partial _ t , . ) ) \\star D ^ { k - l } \\psi + \\sum _ { l = 0 } ^ { k } D ^ { k - l + 1 } \\dot { g } \\star D ^ { l + 1 } \\psi \\\\ & + \\sum _ { \\sum l _ i + \\sum { m _ j } = k } { } ^ { G } \\nabla ^ { l _ 1 } R ^ P \\star \\underbrace { D ^ { m _ 1 + 1 } \\phi \\star \\ldots \\star D ^ { m _ { l _ 1 } + 1 } \\phi } _ { l _ 1 - } \\star D ^ { l _ 2 } \\nabla _ t \\phi \\star D ^ { l _ 3 + 1 } \\phi \\star D ^ { l _ 4 } \\psi . \\end{align*}"}
-{"id": "5945.png", "formula": "\\begin{align*} \\sigma = C _ { 1 } D _ { 1 } C _ { 1 } ^ { * } , \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} a _ k ( x ) & = x ^ 2 \\delta _ { k = 3 } + x a _ { k - 1 } ( v ) + x b _ { k - 1 } ( v ) + x c _ { k - 1 } ( v ) , \\\\ b _ k ( x ) & = x ^ 2 \\delta _ { k = 3 } + v ^ 2 b _ 2 ( x ) + x ( b _ k ( x ) + b _ { k + 1 } ( x ) + \\cdots ) + x ( a _ { k - 1 } ( x ) + a _ k ( x ) + \\cdots ) , \\\\ c _ k ( x ) & = x b _ k ( x ) + x ( c _ k ( x ) + c _ { k + 1 } ( x ) + \\cdots ) , \\\\ b _ 2 ( x ) & = x b _ 2 ( x ) + x ( c _ 3 ( x ) + c _ 4 ( x ) + \\cdots ) , \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{align*} { \\tilde { \\phi } } _ 1 ( s ) = { \\left ( - i \\right ) } ^ { - s } \\Gamma \\left ( s \\right ) \\int _ 0 ^ { \\infty { } } { e } ^ { i e ^ { i \\theta \\pi / 2 } w ^ { \\alpha { } + 1 } } w ^ { - s } d w , \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} ( T ( D ) f | _ { \\Z ^ d } ) ( n ) = ( T _ { \\rm p e r } ( D ) f ) ( n ) , n \\in \\Z ^ d . \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} T _ i \\cdot { \\rm C o e f f } [ f _ { \\mu } , m ] = { \\rm C o e f f } [ f _ { s _ i \\mu } , m ] = \\sum _ { \\nu \\in \\sigma ( \\epsilon ) } \\psi ( \\nu , \\mu ) ( T _ i \\cdot f _ { \\nu } ) . \\end{align*}"}
-{"id": "3068.png", "formula": "\\begin{align*} \\Phi _ { q t } ( 1 , t ) & = \\int _ { \\Omega } a \\left ( x \\right ) [ \\phi _ { 1 } \\log ( t \\phi _ { 1 } ) + \\phi _ { 1 } ] \\phi _ { 1 } \\\\ & = \\int _ { \\Omega } a \\left ( x \\right ) \\phi ^ { 2 } \\log ( t \\phi _ { 1 } ) + \\int _ { \\Omega } a \\left ( x \\right ) \\phi _ { 1 } ^ { 2 } . \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} \\mu ( x , y ) & = \\lambda \\mu _ 1 ( x , y ) + ( 1 - \\lambda ) \\mu _ 2 ( x , y ) \\\\ \\gamma ( x , y ) & = \\lambda \\gamma _ 1 ( x , y ) + ( 1 - \\lambda ) \\gamma _ 2 ( x , y ) \\end{align*}"}
-{"id": "5079.png", "formula": "\\begin{align*} B = \\begin{bmatrix} B _ 1 & 0 & 0 & \\ldots \\\\ 0 & B _ 2 & 0 & \\ldots \\\\ 0 & 0 & B _ 3 & \\ddots \\\\ \\vdots & \\ddots & \\ddots & \\ddots \\end{bmatrix} , J _ A = \\begin{bmatrix} 0 & A _ 1 & 0 & \\ldots \\\\ A _ 1 ^ * & 0 & A _ 2 & \\ddots \\\\ 0 & 0 & A _ 2 ^ * & \\ddots \\\\ \\vdots & \\ddots & \\ddots & \\ddots \\end{bmatrix} . \\end{align*}"}
-{"id": "7956.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } t \\alpha _ t = { m _ 1 ( \\mu ) } / { V } \\lim _ { t \\rightarrow \\infty } ( t / \\beta _ t ) \\int _ 0 ^ \\infty ( s ^ 2 + 1 ) \\ , d \\rho ( s ) = 1 . \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\delta ' ( \\sigma ' _ 0 \\wedge \\sigma ' _ 0 ) = 2 \\sigma ' _ 0 \\wedge \\delta ' ( \\sigma ' _ 0 ) , \\\\ \\\\ \\delta ' ( \\sigma ' _ 1 \\wedge \\sigma ' _ 1 ) = 2 \\sigma ' _ 1 \\wedge \\delta ' ( \\sigma ' _ 1 ) , \\\\ \\\\ \\delta ' ( \\sigma ' _ 0 \\wedge \\sigma ' _ 1 ) = \\delta ' ( \\sigma ' _ 0 ) \\wedge \\sigma ' _ 1 + \\sigma ' _ 0 \\wedge \\delta ' ( \\sigma ' _ 1 ) \\end{array} \\right . \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{align*} \\alpha ( E , B _ r ) = \\inf _ { x \\in \\R ^ n } \\frac { | E \\Delta B _ R ( x ) | } { | E | } = \\inf _ { x \\in \\R ^ n } \\frac { | E \\Delta B _ R ( x ) | } { | B _ R | } . \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} \\omega _ { \\exp ( a ) H } = \\sum \\Omega _ { \\diamondsuit , \\bar { \\heartsuit } } \\omega _ { \\diamondsuit , \\bar { \\heartsuit } } \\end{align*}"}
-{"id": "8673.png", "formula": "\\begin{align*} \\rho _ 0 = - \\rho / v = ( r _ * - t ) ^ { - 1 } , \\ \\ \\rho _ I = - v = ( r _ * - t ) / r , \\ \\ \\rho = \\rho _ 0 \\rho _ I = r ^ { - 1 } ; \\end{align*}"}
-{"id": "5477.png", "formula": "\\begin{align*} \\mathbf { S } ^ { + } _ { j m } = \\delta _ { j m } - \\delta _ { l j } \\delta _ { l m } , \\mathbf { S } ^ { - } = \\delta _ { j m } - \\delta _ { ( l + N ) j } \\delta _ { ( l + N ) m } , j , m = 1 , . . . , 2 N . \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} \\nabla _ { \\bar Z } \\kappa _ B ^ { 1 , 0 } = ( \\log \\lambda ) ^ 2 Z ^ * , \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} { \\cal D } \\vert _ { X } = \\dd + \\Gamma _ { X } \\ : , \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{align*} d \\rho = - d \\theta \\wedge \\omega _ 0 , d \\hat { \\rho } = 0 . \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{align*} { } P ( E , N ) = \\frac { f ( u ) } { E } \\ , e ^ { \\sqrt { E } g ( u ) } \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} \\frac { d \\phi } { d r } & = \\dot { \\phi } / \\dot { r } \\\\ & = \\left ( \\frac { - r } { 2 } + \\frac { ( n - 1 ) 2 r } { r ^ 2 - s ^ 2 } \\right ) \\cot \\phi + \\left ( \\frac { s } { 2 } + \\frac { ( n - 1 ) 2 s } { r ^ 2 - s ^ 2 } \\right ) + \\lambda \\csc \\phi \\\\ & = I \\cot \\phi + I I + \\lambda \\csc \\phi \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{align*} - \\Delta ( \\frac { \\phi } { \\alpha } ) ^ { \\alpha } & = - \\frac { 1 } { \\alpha ^ { \\alpha - 1 } } \\left ( \\phi ^ { \\alpha - 1 } \\Delta \\phi + \\left ( \\alpha - 1 \\right ) \\phi ^ { \\alpha - 2 } \\left \\vert \\nabla \\phi \\right \\vert ^ { 2 } \\right ) \\\\ & \\leq - ( \\frac { \\phi } { \\alpha } ) ^ { \\alpha - 1 } \\Delta \\phi = a \\left ( x \\right ) ( \\frac { \\phi } { \\alpha } ) ^ { \\alpha q } \\quad \\Omega ^ { \\prime } \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} \\mathbb { P } ( A _ i ( n ) = 1 ) > 0 . \\end{align*}"}
-{"id": "4271.png", "formula": "\\begin{align*} U ^ + _ { K } & = \\Big ( U \\setminus \\{ u _ 1 \\} \\Big ) \\cup \\{ q _ i ^ { - 1 } u _ 1 , q _ { s } u _ 1 , q _ s ^ { - 1 } q _ i ^ { K - 1 } u _ 1 , q _ i q _ i ^ { K - 1 } u _ 1 \\} \\\\ W ^ + _ { K } & = W \\cup \\{ q _ i ^ { - 1 } q _ { s } ^ { - 1 } u _ 1 , q _ i ^ { K - 1 } u _ 1 , q _ i q _ s q _ i ^ { K - 1 } u _ 1 \\} , \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{align*} \\langle \\Phi ^ { - 1 } u , v \\rangle _ { W ^ { - 1 , p } _ { \\sigma } , W ^ { 1 , p ^ { \\prime } } _ { 0 , \\sigma } } = \\langle u , v \\rangle _ { L ^ p _ { \\sigma } , L ^ { p ^ { \\prime } } _ { \\sigma } } , u \\in L ^ p _ { \\sigma } ( \\Omega ) , \\ , v \\in W ^ { 1 , p ^ { \\prime } } _ { 0 , \\sigma } ( \\Omega ) . \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} \\varphi _ { \\Omega } ( z + \\omega _ 2 ) = - \\exp ( - 2 \\pi i z ) \\varphi _ { \\Omega } \\end{align*}"}
-{"id": "4413.png", "formula": "\\begin{align*} \\left ( \\frac { \\omega _ 2 } { \\omega _ 1 } \\right ) ' ( t ) = \\frac { c } { t ( 1 - t ) \\omega _ 1 ( t ) ^ 2 } , \\end{align*}"}
-{"id": "921.png", "formula": "\\begin{align*} \\lbrack \\Delta _ { \\Gamma } { \\varphi } ] ( x ) \\ = \\ \\sum _ { | v | = 1 } \\big \\{ { \\varphi } ( x ) - { \\varphi } ( x + v ) \\big \\} . \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} u _ t = u _ { x x x } - \\frac 3 2 \\frac { u _ { x x } ^ 2 } { u _ x } = u _ x S ( u ) \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} A \\cdot Z = A Z A ^ t , \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} \\lambda _ n = 1 - \\frac { b } { n ^ \\beta } , \\ \\ \\alpha _ n = \\frac { a } { n ^ { \\alpha } } , \\ \\ p _ n = \\frac { 1 + \\alpha _ n } { 2 } , \\ \\ q _ n = \\frac { 1 - \\alpha _ n } { 2 } . \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{align*} u _ { i , j } = \\Lambda ^ * _ i e _ { i , j } , \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} \\langle R ^ { ( m ) } ( \\mu , \\mu ) u , u \\rangle = ( m - 1 ) \\int _ { X } f ( \\mu ) | u | ^ 2 + \\frac { m - 1 } 2 \\langle \\left ( \\square '' + \\frac { m - 1 } 2 \\right ) ^ { - 1 } ( \\mu \\cdot u ) , ( \\mu \\cdot u ) \\rangle . \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} \\mu ( B \\setminus A ) = \\mu ( I ) - \\mu ( A ) \\leq \\varepsilon \\ , \\mu ( A ) \\leq \\varepsilon \\ , \\mu ( I ) . \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} \\theta _ 2 ( x ) = e ^ { - \\pi \\frac { x } { 4 } } \\ , \\Theta \\left ( \\tfrac { i x } { 2 } , i x \\right ) . \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} \\langle R _ m L _ m \\chi _ y , \\chi _ { y ' } \\rangle & = \\langle L _ m \\chi _ y , L _ m \\chi _ { y ' } \\rangle \\\\ & = \\sum _ { z \\in P _ \\nu } L _ m \\chi _ y ( z ) \\overline { L _ m \\chi _ { y ' } ( z ) } . \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{align*} G _ { i _ 1 i _ 2 i _ 3 } ^ { j _ 1 j _ 2 j _ 3 } = \\delta ^ { j _ 1 k _ 1 } \\delta ^ { j _ 2 k _ 2 } \\delta ^ { j _ 3 k _ 3 } \\int _ 0 ^ \\infty ( e ^ { - r \\tilde \\gamma } ) _ { i _ 1 k _ 1 } ( e ^ { - r \\tilde \\gamma } ) _ { i _ 2 k _ 2 } ( e ^ { - r \\tilde \\gamma } ) _ { i _ 3 k _ 3 } d r . \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} y _ { i } ^ { n } = \\begin{cases} - n + \\mu _ i , & i \\in \\mathcal { I } , \\\\ n - \\nu _ i , & i \\in \\mathcal { J } , \\\\ 0 , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "7303.png", "formula": "\\begin{align*} \\Omega \\left ( \\sum _ { k = 1 } ^ K | g _ { m k } | ^ 2 p _ k + \\sigma _ N ^ 2 \\right ) \\leqslant { } P _ { } ^ { } , \\ m \\in \\mathcal { M } , \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} \\mathcal { I } _ { \\mu } = \\inf \\{ E ( u , v ) ; M ( u ) = \\mu , ( u , v ) \\in H ^ 1 \\times L ^ 2 \\} . \\end{align*}"}
-{"id": "144.png", "formula": "\\begin{align*} d _ { A _ \\infty } \\dot \\eta + [ \\dot A _ \\infty \\wedge \\eta ] & = d _ { A _ \\infty } [ \\eta \\wedge \\gamma _ \\infty ] + [ \\dot A _ \\infty \\wedge \\eta ] \\\\ & = [ d _ { A _ \\infty } \\eta \\wedge \\gamma _ \\infty ] - [ \\eta \\wedge d _ { A _ \\infty } \\gamma _ \\infty ] + [ \\dot A _ \\infty \\wedge \\eta ] = 0 \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} L = m ^ { \\left ( \\frac { n - 1 } { n } \\right ) \\left ( \\frac { n + 1 - \\alpha } { \\alpha ( n - 1 ) + 1 } \\right ) } . \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} \\lim _ { | t - s | \\to 0 } \\| P ^ { \\kappa , b } _ { t , s } f - f \\| _ \\infty = 0 . \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} & \\epsilon > 1 - \\sup _ { U V \\rightarrow X \\rightarrow Y } \\frac { I ( U ; Y | V ) } { I ( U ; X | V ) } \\overset { ( a ) } { = } 1 - \\eta ( p _ { Y | X } ) \\end{align*}"}
-{"id": "8694.png", "formula": "\\begin{align*} g = g _ m + \\rho h \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} \\rho ( 1 _ V ) = 1 _ W ; \\rho ( \\omega ^ V ) = \\omega ^ W ; \\end{align*}"}
-{"id": "8078.png", "formula": "\\begin{align*} \\begin{cases} | K _ s ( L ( \\xi , \\sigma ) y ) | ^ 2 = O ( | L ( \\xi , \\sigma ) y ) | ^ { - 1 } ) e ^ { - y | L ( \\xi , \\sigma ) | } , \\\\ | K _ { 1 - s } ( L ( \\xi , \\sigma ) y ) | ^ 2 = O ( | L ( \\xi , \\sigma ) y ) | ^ { - 1 } ) e ^ { - y | L ( \\xi , \\sigma ) | } , \\end{cases} \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} r _ 3 = - \\frac 1 { k _ 0 } r _ 1 ' . \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} \\delta = 2 ^ { \\frac 4 3 } | \\lambda ^ 2 - \\lambda + 1 | ^ { \\frac 1 2 } = 2 ^ { \\frac 4 3 } | ( \\lambda - \\zeta ) ( \\lambda - \\overline { \\zeta } ) | ^ { \\frac 1 2 } \\leq 2 ^ { \\frac 4 3 } \\leq 3 \\end{align*}"}
-{"id": "5818.png", "formula": "\\begin{align*} T _ i f _ { ( \\nu _ 1 , \\dots , \\nu _ i , \\nu _ { i + 1 } , \\dots , \\nu _ n ) } = ( t - 1 ) f _ { ( \\nu _ 1 , \\dots , \\nu _ i , \\nu _ { i + 1 } , \\dots , \\nu _ n ) } + t f _ { ( \\nu _ 1 , \\dots , \\nu _ { i + 1 } , \\nu _ i , \\dots , \\nu _ n ) } , \\nu _ i < \\nu _ { i + 1 } . \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{align*} \\mathcal { M } \\left \\{ e ^ { - i \\rho { x } } ; s \\right \\} = \\left ( i \\rho \\right ) ^ { - s } \\Gamma ( s ) = { \\rho } ^ { - s } \\Gamma ( s ) \\left ( \\cos \\left ( \\frac { \\pi { s } } { 2 } \\right ) - i \\sin \\left ( \\frac { \\pi { s } } { 2 } \\right ) \\right ) ; ~ ~ \\rho \\in { \\mathbb { C } } , \\end{align*}"}
-{"id": "5001.png", "formula": "\\begin{align*} [ a _ 1 , a _ 2 , \\dots , a _ n ] = 0 \\mbox { f o r a l l } a _ i \\in A \\iff [ y _ 1 , y _ 2 , \\dots , y _ n ] = 0 \\mbox { f o r a l l } y _ i \\in X \\cup X ^ 2 . \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\mathcal { C } _ { \\Gamma } ( \\overline { x } , \\overline { v } ) } ( d ) = \\Big \\{ \\nabla g ( \\overline { x } ) \\xi \\ | \\ \\langle \\xi , g ' ( \\overline { x } ) d \\rangle = 0 , \\ , \\xi \\in \\mathcal { T } _ { \\mathcal { N } _ { K } ( g ( \\overline { x } ) ) } ( \\lambda ) \\Big \\} . \\end{align*}"}
-{"id": "5061.png", "formula": "\\begin{align*} [ s , y _ m ] \\bigl ( [ y _ i , x _ 1 ] [ y _ { i ' } , x _ 2 ] + [ y _ i , x _ 2 ] [ y _ { i ' } , x _ 1 ] \\bigr ) = - [ s , y _ m ] \\bigl ( [ y _ i , x _ 1 ] [ x _ 2 , y _ { i ' } ] + [ y _ i , x _ 2 ] [ x _ 1 , y _ { i ' } ] \\bigr ) \\equiv 0 \\pmod { I } . \\end{align*}"}
-{"id": "7095.png", "formula": "\\begin{align*} k _ j < \\frac { 2 d n ^ d } { ( j / d c ) ^ d } = \\frac { 2 n ^ d d ^ { d + 1 } c ^ d } { j ^ d } . \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} A = \\bigcup _ { t \\in s u c c _ T ( s ) } A ( t ) , \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{align*} m _ t = \\begin{pmatrix} m _ 1 ' & - m _ 2 ' \\\\ m _ 2 ' & m _ 1 ' \\end{pmatrix} \\begin{pmatrix} s _ 1 \\\\ s _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial m } g ( x , \\lambda , m ) & = x + 2 \\lambda m - \\frac { 1 } { 2 } \\log \\frac { 1 + m } { 1 - m } = 0 \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{align*} \\hat { \\rho } = \\ast ( \\rho _ 6 + \\rho _ { 1 \\oplus 1 } ) - \\ast \\rho _ { 1 2 } . \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{align*} f ( \\tau _ A ( x + \\alpha _ { 0 } ) , \\tau _ B ( y + \\beta _ { 0 } ) ) = f ( \\tau _ A ( x ) , \\tau _ B ( y ) ) - \\delta ' + \\delta . \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} W _ { q , d i v } ( \\Omega ) = \\left \\{ \\varphi \\in L ^ { q } ( \\Omega ) ^ { 3 } \\mid \\mathrm { d i v } \\ \\varphi \\in L ^ { q } ( \\Omega ) \\right \\} , \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{align*} S ^ { p + q - 1 } = S ^ { p - 1 } \\times S ^ { q - 1 } \\times [ 0 , \\frac { \\pi } { 2 } ] , \\end{align*}"}
-{"id": "3047.png", "formula": "\\begin{align*} Y _ { 1 } = \\mathrm { K e r } A , \\quad \\mbox { a n d } \\ R ( A ) = \\left \\{ f \\in L ^ { \\eta } ( \\Omega ) : \\int _ { \\Omega } f \\phi _ { 1 } = 0 \\right \\} . \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} R ( t ) : = \\sup _ { x \\neq y } \\frac { L ( t ) } { \\pi d ( x , y , t ) } s i n \\frac { \\pi l ( x , y , t ) } { L ( t ) } , \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{align*} \\nabla _ X \\psi = \\alpha X \\cdot \\psi \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} \\mathbb { E } _ { n , d _ 2 , F ( \\theta ) } ( \\varphi ) = \\mathbb { E } _ { n , d _ 1 , \\theta } ( \\varphi ' ) \\theta \\in \\Theta _ { d _ 1 } . \\end{align*}"}
-{"id": "2818.png", "formula": "\\begin{align*} G ^ { \\ell } _ { \\varepsilon } ( x ) : = \\frac { 1 } { \\varepsilon ^ d } \\frac { \\partial K } { \\partial x _ { \\ell } } \\left ( \\frac { x } { \\varepsilon } \\right ) \\ , \\textrm { f o r a l m o s t a l l } \\ x \\in \\R ^ d \\ . \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{align*} ( L \\chi ) ( z ) = B ( z , . . . , z ) - \\left ( \\frac { \\beta } { 2 \\pi } \\right ) ^ { n / 2 } \\int B ( \\tilde z , . . . , \\tilde z ) e ^ { - \\beta \\| \\tilde z \\| ^ 2 / 2 } d \\tilde z , \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} \\alpha = \\frac { \\gamma } { c } 1 _ { [ 0 , u ) } + \\alpha _ 0 1 _ { [ u , 1 ] } \\in \\mathcal { M } . \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} a \\mathfrak a = ( d \\alpha - b \\beta ) \\mathfrak a ' \\quad \\textrm { a n d } b \\mathfrak b = ( - c \\alpha + a \\beta ) \\mathfrak b ' . \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{align*} M ^ 1 = \\left [ \\begin{array} { c c } I _ n & 0 \\\\ - c ^ * M _ x ^ { - 1 } & 1 \\end{array} \\right ] \\left [ \\begin{array} { c c } M _ x & 0 \\\\ 0 & t ^ 2 \\end{array} \\right ] \\left [ \\begin{array} { c c } I _ n & - M _ x ^ { - 1 } c \\\\ 0 & 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "7611.png", "formula": "\\begin{align*} | I _ 0 ( g ) | & = N _ { q , n , m } ( g ) - C ^ { - 1 } q ^ { ( g - 1 ) n m ^ 2 } - 1 + O ( 1 ) \\\\ & = C ^ { - 1 } q ^ { g n m ^ 2 } ( 1 - O ( q ^ { - g } ) ) - C ^ { - 1 } q ^ { ( g - 1 ) n m ^ 2 } - 1 + O ( 1 ) \\\\ & = C ^ { - 1 } ( q ^ { g n m ^ 2 } - q ^ { ( g - 1 ) n m ^ 2 } ) ( 1 - O ( q ^ { - g } ) ) \\ . \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} & f _ L ( a _ L ) > f _ L ( x ) , ( \\forall x \\in \\R _ { \\ge 0 } \\backslash \\{ a _ L \\} , \\ L \\in \\N ) , \\\\ & f ( a ) > f ( x ) , ( \\forall x \\in \\R _ { \\ge 0 } \\backslash \\{ a \\} ) , \\\\ & \\frac { d ^ n } { d x ^ n } f ( a ) = 0 , ( \\forall n \\in \\{ 1 , 2 , \\cdots , n _ 0 - 1 \\} ) , \\\\ & \\frac { d ^ { n _ 0 } } { d x ^ { n _ 0 } } f ( a ) < 0 . \\end{align*}"}
-{"id": "9134.png", "formula": "\\begin{align*} \\tilde { C } ( \\Q ) = \\{ ( 0 : 1 : 0 ) , ( 0 : 0 : 1 ) , ( - 1 : - 1 : 1 ) , ( 1 : - 1 : 1 ) , ( - 1 : 1 : 1 ) , ( 1 : 1 : 1 ) \\} \\end{align*}"}
-{"id": "1455.png", "formula": "\\begin{align*} p _ m ( { \\bf z } ) : = \\prod _ { j = 1 } ^ { m - 1 } \\alpha _ j e ^ { - \\alpha _ j z _ j } \\ , , \\alpha _ j : = 2 ( 1 - j / m ) \\ , , \\end{align*}"}
-{"id": "776.png", "formula": "\\begin{align*} | \\omega _ { j , n } - z _ { j , n } | < \\ , \\frac { \\pi | z _ { j , n } | } { n \\ , a _ { j , n } } \\mbox { f o r } ~ ~ j = \\lceil v _ n \\rceil , \\lceil v _ n \\rceil + 1 , \\ldots , J _ n , \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} | \\gamma _ l | \\leqslant & N ^ 2 \\big | \\langle \\nabla _ { x _ 2 } \\psi _ N , \\ , g _ \\beta ( x _ 1 - x _ 2 ) \\ , \\mathbf { A } ( x _ 1 ) \\widehat { r } \\ , \\psi _ N \\rangle \\big | \\\\ & + N ^ 2 \\big | \\langle \\psi _ N , \\ , g _ \\beta ( x _ 1 - x _ 2 ) \\ , \\mathbf { A } ( x _ 1 ) \\nabla _ { x _ 2 } \\widehat { r } \\ , \\psi _ N \\rangle \\big | . \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{align*} \\tilde a ^ \\sigma ( x , y ) = \\frac { c _ 2 } { c _ 1 c _ 3 } \\tilde a ( x , y ) . \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} \\| \\varphi \\| _ { W _ { q , d i v } ( \\Omega ) } : = \\| \\varphi \\| _ { L ^ { q } ( \\Omega ) ^ { 3 } } + \\| \\mathrm { d i v } \\ \\varphi \\| _ { L ^ { q } ( \\Omega ) } . \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} \\begin{aligned} 2 g - 2 & = - 2 d + \\sum _ { i = 0 } ^ { r _ 0 } \\left ( \\lambda _ { 0 , i } - 1 \\right ) + \\sum _ { i = 0 } ^ { r _ 1 } \\left ( \\lambda _ { 1 , i } - 1 \\right ) + \\sum _ { i = 0 } ^ { r _ \\infty } \\left ( \\lambda _ { \\infty , i } - 1 \\right ) + \\rho \\\\ & = d - r _ 0 - r _ 1 - r _ \\infty + \\rho . \\end{aligned} \\end{align*}"}
-{"id": "2345.png", "formula": "\\begin{align*} E \\left [ S _ N ^ { ( 2 ) } \\right ] = E \\left [ S _ N ( S _ N + 1 ) \\right ] = 2 \\int _ 0 ^ { \\infty } t \\left [ 1 - \\prod _ { j = 1 } ^ N \\bigg ( 1 - e ^ { - q _ j t } \\bigg ) \\right ] d t . \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} \\varphi _ \\lambda ( q ) : = \\lambda ^ 2 q , q \\in \\mathcal { B } ' , \\lambda \\in \\C ^ \\times . \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} r ^ \\nu = \\nu ^ { \\frac { - 1 } { 2 + n / p } } ( \\log \\nu ) ^ { \\frac { 1 + n } { 2 + n / p } } . \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} & H ( F _ 1 | X ^ n ) = H ( F _ 2 | F _ 1 Y ^ n ) = H ( F _ 3 | F _ { 1 : 2 } X ^ n ) = \\cdots = 0 . \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} \\Upsilon ( h ) : = - \\sigma \\big ( \\lambda , \\mathcal { T } _ { K } ^ 2 ( y , h ) \\big ) = \\langle u , \\Xi '' ( y ) ( h , h ) \\rangle \\forall h \\in \\mathcal { C } _ { K } ( y , \\lambda ) \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} \\mathrm { P } _ { m , k } ^ 1 = & \\mathrm { P } ( _ { m , k } < R _ 0 ) = \\mathrm { P } \\left ( \\bar { z } _ { m , k } < \\frac { { \\epsilon } _ 0 } { \\rho \\zeta _ 1 } \\right ) \\\\ = & \\mathcal { E } _ { L \\left ( | | y _ { m , k } + x _ m | | \\right ) } \\left \\{ 1 - e ^ { - L \\left ( | | y _ { m , k } + x _ m | | \\right ) \\frac { { \\epsilon } _ 0 } { \\rho \\zeta _ 0 } } \\right \\} , \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{align*} \\int _ { \\Omega } d ^ { \\prime } ( t ) \\cdot \\overline { \\vartheta } \\ ; \\d x + \\int _ { \\Omega } \\nabla d ( t ) \\cdot \\overline { \\nabla \\vartheta } \\ ; \\d x = - \\int _ { \\Omega } ( u ( t ) \\cdot \\nabla ) d ( t ) \\cdot \\overline { \\vartheta } \\ ; \\d x + \\int _ { \\Omega } \\lvert \\nabla d ( t ) \\rvert ^ 2 d ( t ) \\cdot \\overline { \\vartheta } \\ ; \\d x . \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{align*} \\gamma = \\eta \\wedge \\tau _ 1 + \\frac { d r } { r } \\wedge \\tau _ 2 \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{align*} \\gamma _ e = \\frac { 1 + \\gamma \\cdot { \\rm P N R } _ { \\max } } { 1 + { \\rm P N R } _ { \\max } } . \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{align*} I = E \\left [ e ^ { - k _ 1 p _ 1 Y _ { \\min } } \\right ] , \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} \\lim _ { q \\rightarrow \\underline { q } ^ { + } } u ( q ) = \\underline { u } \\quad \\mbox { i n } \\ W _ { D } ^ { 2 , r } ( \\Omega ) , \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{align*} ( d + d ^ * ) \\rho = * d \\theta \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} T ^ * X = T ^ { * ( 1 , 0 ) } X \\oplus T ^ { * ( 0 , 1 ) } X \\ : . \\end{align*}"}
-{"id": "1602.png", "formula": "\\begin{align*} c _ \\ast : = 4 \\delta ^ { - 1 } _ \\sigma \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{align*} ( 1 + \\lambda ) P _ B x - \\lambda x = ( 1 + \\lambda ) f - \\lambda x = e + \\frac { 1 } { 1 - \\lambda } g , \\end{align*}"}
-{"id": "2048.png", "formula": "\\begin{align*} h _ 1 : = f _ 0 + u _ 1 f _ 1 , h _ 2 : = f _ 0 , h _ 3 : = f _ 0 + u _ 3 f _ 1 , \\end{align*}"}
-{"id": "3589.png", "formula": "\\begin{align*} \\gamma ( t , x ) : = P + \\int _ { t } ^ { t _ { 0 } } ( T \\wedge T _ { x x } ) ( \\tau , x _ { 0 } ) d \\tau + \\int _ { x } ^ { x _ { 0 } } T ( t , s ) d s \\end{align*}"}
-{"id": "924.png", "formula": "\\begin{align*} \\mathfrak { e } _ { \\mathrm { L a p l } } ( p ) \\ \\doteq \\ \\sum _ { i = 1 } ^ { d } \\big ( 1 - \\cos ( p _ { i } ) \\big ) \\ ; , \\mathfrak { e } _ { \\mathrm { L a p l } } ( \\Gamma ^ { \\ast } ) \\ = \\ [ 0 , 2 d ] , \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} u ( x ) = c _ 0 + \\int _ 0 ^ 1 g ( \\textit { \\textbf { a } } ( \\gamma ( t ) ) , \\dot \\gamma ( t ) ) d t , \\ x \\in M , \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} \\hat { \\Psi } ( P _ n ) = \\arg \\min _ { \\psi \\in { \\bf \\Psi } } P _ n L ( \\psi ) . \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} \\frac { ( b y t ; q ) _ \\infty } { ( y t ; q ) _ \\infty } { _ 2 \\phi _ 1 } \\left ( { { q ^ { - m } , y t } \\atop { b y t } } ; q , q \\right ) = \\sum _ { n = 0 } ^ \\infty \\lambda _ n ( b ; q ) _ n y ^ n . \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} X \\# _ t Y = \\frac { \\sin ( \\pi t ) } { 2 \\pi } \\int _ 0 ^ \\infty X : ( \\lambda Y ) \\lambda ^ { t } { \\rm d } \\lambda \\ , \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} & ( \\phi _ * b _ + ) ( t , x ) = \\left ( 0 , \\frac { 1 } { m } \\gamma ( t , q ) ( - p - \\psi ( t , q ) ) \\right ) , \\\\ & ( \\phi _ * b _ - ) ( t , x ) \\\\ = & \\left ( \\frac { 1 } { m } ( - p - \\psi ( t , q ) ) , - \\frac { 1 } { m } \\delta ^ { i j } ( - p _ i - \\psi _ i ( t , q ) ) \\nabla _ q \\psi _ j ( t , q ) + \\nabla _ q V ( t , q ) - \\tilde F ( t , q , - p ) \\right ) . \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} S ^ 2 \\longrightarrow \\mathbb { C P } ^ 1 , \\mathbf { u } = ( x _ 1 , x _ 2 , x _ 3 ) \\mapsto \\begin{cases} \\displaystyle { \\zeta = - \\frac { x _ 1 + i x _ 2 } { 1 + x _ 3 } } & \\textrm { o n $ N $ } \\\\ [ 2 e x ] \\displaystyle { \\tilde { \\zeta } = - \\frac { x _ 1 - i x _ 2 } { 1 - x _ 3 } } & \\textrm { o n $ S $ } \\rlap { . } \\end{cases} \\end{align*}"}
-{"id": "3627.png", "formula": "\\begin{align*} Z ^ { \\ast } ( I _ 9 ) = ( z _ { r } ^ { - n } - v z _ { r } ^ { n } ) \\ , \\overline { Z } ( I _ { 1 } ; 0 , c _ r ) . \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} \\alpha _ 1 \\dots \\alpha _ { m - 1 } \\beta _ { m 1 } = u \\ , . \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} L / 2 = | s _ 0 - \\bar { s } _ 0 | = | s _ 0 - s _ { i _ k } | = s _ { i _ k } - s _ 0 \\overset { \\eqref { e q : l e n g t h _ a r c s } } { = } \\frac { L } { a _ 1 } i _ k . \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} ( s \\alpha ) \\cdot e = s \\cdot ( \\alpha \\cdot e ) - \\alpha ( s ) \\cdot e \\ , . \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} \\Sigma _ { \\theta } = \\{ \\lambda \\in \\mathbb { C } \\setminus \\{ 0 \\} \\ \\ \\mid \\ \\ | \\mathrm { a r g } \\lambda | < \\theta \\} . \\end{align*}"}
-{"id": "5069.png", "formula": "\\begin{align*} & g ( z _ 1 , z _ 2 , z _ 3 , z _ 4 ) = [ z _ 1 , z _ 2 ] [ z _ 3 , z _ 4 ] + [ z _ 1 , z _ 3 ] [ z _ 2 , z _ 4 ] = [ z _ 2 , z _ 1 ] [ z _ 4 , z _ 3 ] \\\\ & + [ z _ 2 , z _ 4 ] [ z _ 1 , z _ 3 ] - \\bigl [ [ z _ 2 , z _ 4 ] , [ z _ 1 , z _ 3 ] \\bigr ] = g ( z _ 2 , z _ 1 , z _ 4 , z _ 3 ) - \\bigl [ [ z _ 2 , z _ 4 ] , [ z _ 1 , z _ 3 ] \\bigr ] \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} \\int ^ * _ S \\Vert u ( s ) \\Vert d | \\mu | ( s ) : = \\sup _ { \\stackrel { f \\in { \\mathcal C } ^ + _ c ( S ) } { _ { f \\le \\omega _ u } } } \\int _ S f ( s ) d | \\mu | ( s ) \\in [ 0 , + \\infty ] \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} \\gamma ( t _ { 1 } , x ) - \\gamma ( t _ { 2 } , x ) = O ( \\frac { 1 } { x } ) , T ( t _ { 1 } , x ) - T ( t _ { 2 } , x ) = O ( \\frac { 1 } { x } ) \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{align*} x _ n ( t ) = \\frac { 1 } { \\sqrt { n } } \\left ( \\cos ( n t ) , \\sin ( n t ) \\right ) \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} \\mu ( x , t ) \\ , = \\ , \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { R } } e ^ { - i \\xi x } e ^ { - t \\varphi ( \\xi ) } d \\xi \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} R _ { \\alpha } ( x , y ) = \\omega _ { \\alpha } \\sum _ { k = 0 } ^ { \\infty } \\frac { \\Gamma ( k + n / 2 + \\alpha ) } { \\Gamma ( k + n / 2 ) } Z _ { k } ( x , y ) , \\enspace x , y \\in \\mathbf { B } , \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} s ( \\vec { J } ) = \\prod _ a ( - 1 ) ^ { 1 + \\big ( ( \\vec { J } ) \\big ) _ a } . \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{align*} \\bar \\partial _ \\mu \\partial _ \\mu \\log l _ 0 : = \\sum _ { S ( X ) } \\bar \\partial _ \\mu \\partial _ \\mu \\log l ( \\gamma ) . \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{align*} \\varphi _ n ( f _ 1 , \\ldots , f _ s ) = \\left ( \\frac { f _ 1 ( \\omega ) \\bmod m ^ n } { m ^ n } , \\ldots , \\frac { f _ s ( \\omega ) \\bmod m ^ n } { m ^ n } \\right ) . \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{align*} J _ { s , t } = \\int _ s ^ t B ^ { i _ 1 , . . . , i _ k } ( r , q _ r ) \\left ( \\int h ( r , q _ r , z ) z _ { i _ 1 } . . . z _ { i _ k } d z \\right ) d r , \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{align*} H _ \\infty = \\begin{pmatrix} | f | ^ { - 1 / 2 } & 0 \\\\ 0 & | f | ^ { 1 / 2 } \\end{pmatrix} \\mbox { a n d } \\Phi = \\begin{pmatrix} 0 & f \\\\ 1 & 0 \\end{pmatrix} d z . \\end{align*}"}
-{"id": "1860.png", "formula": "\\begin{align*} c _ 0 ( 0 ) = - \\frac { b } { \\sqrt { 1 + b ^ 2 } } , c _ 1 ( 0 ) = \\frac { 1 } { \\sqrt { 1 + b ^ 2 } } , c _ n ( 0 ) = 0 \\ ; \\ ; n \\geq 2 . \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{align*} E ( t ) = \\frac { 1 } { 2 } \\int _ { 0 } ^ { L } ( \\eta ^ { 2 } + w ^ { 2 } ) d x \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} \\Delta K ( y ) \\neq 0 , \\ , \\ , \\ , \\forall \\ , \\ , y \\in \\mathcal { K } , \\ , \\ , \\forall \\ , \\ , y \\in \\{ \\ , x \\in \\Omega \\ , / \\ , \\nabla K ( x ) = 0 \\ , \\} . \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{align*} D S Z _ n = \\frac { 1 } { ( \\sqrt { 2 \\pi } ) ^ n } \\int _ { \\R ^ n } e ^ { - G ( t ) } \\sqrt { \\det D ( t ) } d t = 1 . \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} h : = \\frac { \\sqrt { \\log \\log n } } { 2 m } . \\end{align*}"}
-{"id": "1058.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ 3 } \\tilde { j } _ { m \\epsilon _ { n } } ( w _ { n } ^ { ( j ) } ( x ) ) { \\rm d } x \\geq \\frac { q } { 4 \\pi ^ 2 } \\int _ { \\mathbb { R } ^ 3 } \\tilde { \\eta } ( x ) ^ { 2 } { \\rm d } x = \\frac { 9 } { 8 K _ { \\rm c l } \\lambda } \\int _ { \\mathbb { R } ^ 3 } Q ( x ) ^ { \\frac { 2 } { 3 } } { \\rm d } x , \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} \\vartheta : = \\sup _ { x \\in D _ f } \\| g _ x ( x ) \\| _ x ^ 2 . \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\infty } \\lambda ^ \\alpha \\widetilde { f } ( \\lambda ) = \\lim _ { \\lambda \\to \\infty } \\lambda ^ { \\alpha - n } \\lambda ^ n \\widetilde { f } ( \\lambda ) = 0 . \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{align*} \\big \\langle \\big [ A ^ { \\frac { 1 } { 2 } } _ { p ' } \\big ] ^ * \\Phi ^ { - 1 } A ^ { - \\frac { 1 } { 2 } } _ { p } f , v \\big \\rangle _ { W ^ { - 1 , p } _ { \\sigma } , W ^ { 1 , p ^ { \\prime } } _ { 0 , \\sigma } } = \\big \\langle A _ 2 ^ { - \\frac { 1 } { 2 } } f , A _ 2 ^ { \\frac { 1 } { 2 } } \\big \\rangle _ { L ^ 2 _ { \\sigma } , L ^ 2 _ { \\sigma } } = \\big \\langle \\Phi ^ { - 1 } f , v \\big \\rangle _ { W ^ { - 1 , p } _ { \\sigma } , W ^ { 1 , p ^ { \\prime } } _ { 0 , \\sigma } } . \\end{align*}"}
-{"id": "6315.png", "formula": "\\begin{align*} \\max _ { \\theta _ s , \\tau } \\quad & \\mathcal { P } _ c ( \\theta _ c , \\tau ) , \\\\ \\quad & \\mathcal { P } _ s ( \\theta _ s , \\tau ) \\geq \\mathcal { P } _ T , \\\\ & \\theta _ c = 2 ^ { \\mathcal { R } _ T } ( 1 + \\theta _ s ) - 1 . \\end{align*}"}
-{"id": "1409.png", "formula": "\\begin{align*} | \\nabla u | = 1 . \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} ( a ; q ) _ { \\infty } = \\prod _ { n = 0 } ^ { \\infty } ( 1 - a q ^ { n } ) . \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} \\beta _ k \\to z _ 0 + i d = z _ 1 \\in D ^ + \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} B _ E = - \\frac { 1 } { 2 } e ^ { 2 \\alpha ( a ) } ( \\bar { z } _ 1 z _ 2 + z _ 1 \\bar { z } _ 2 ) \\mathcal { P } ( \\alpha ^ { \\vee } ) + O \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} & \\partial _ T ^ * \\phi = - \\bar * \\bar \\partial _ B \\bar * \\phi , \\bar \\partial _ T ^ * \\phi = - \\bar * \\partial _ B \\bar * \\phi , \\\\ & \\partial _ B ^ * \\phi = - \\bar * \\bar \\partial _ T \\bar * \\phi , \\bar \\partial _ B ^ * \\phi = - \\bar * \\partial _ T \\bar * \\phi . \\end{align*}"}
-{"id": "2513.png", "formula": "\\begin{align*} A = U \\left [ \\begin{array} { c c } M C & M S \\\\ 0 & 0 \\end{array} \\right ] U ^ * , \\end{align*}"}
-{"id": "3624.png", "formula": "\\begin{align*} N = \\lambda _ 1 + r + 1 , \\end{align*}"}
-{"id": "8559.png", "formula": "\\begin{align*} L = \\frac { 1 } { 2 \\pi i } \\int _ { \\mathcal { C } } \\frac { d \\zeta } { \\zeta } H ( \\eta , \\zeta ) \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{align*} N _ { t } & = - \\frac { \\kappa _ { t } } { \\kappa } N + \\frac { \\kappa _ { s s } } { \\kappa } B - 2 \\frac { \\kappa _ { s } \\tau } { \\kappa } N - \\tau _ { s } N + \\kappa \\tau T - \\tau ^ { 2 } B . \\\\ & = - \\Big ( \\frac { \\kappa _ { t } } { \\kappa } + 2 \\frac { \\kappa _ { s } \\tau } { \\kappa } + \\tau _ { s } \\Big ) N + \\tau \\kappa T + \\Big ( \\frac { \\kappa _ { s s } } { \\kappa } - \\tau ^ { 2 } \\Big ) B . \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{align*} 0 = \\int _ { C } \\omega _ + = \\int _ { C \\cap U _ + } \\omega _ + + \\int _ { C \\cap \\mathbb { R } _ + \\times Y _ + } \\omega _ + + \\int _ { C \\cap \\mathbb { R } _ - \\times Y _ + } \\omega _ + . \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{align*} H ( r ) = \\max _ { | z - a | \\leq r } f ^ \\# ( z ) . \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} \\bar u ^ { \\varepsilon , N } _ { t } ( x ) = { \\displaystyle \\frac { 1 } { N } \\sum _ { i = 1 } ^ N K _ { \\varepsilon } ( x - \\bar \\xi ^ { i } _ t ) \\bar V _ t \\big ( \\bar \\xi ^ { i } , \\bar u ^ { \\varepsilon , N } ( \\bar \\xi ^ { i } ) , \\nabla \\bar u ^ { \\varepsilon , N } ( \\bar \\xi ^ { i } ) \\big ) } \\ , x \\in \\R ^ d \\ , \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} M ^ n _ t ( f ) \\ ; = \\ ; \\mathcal X ^ n _ t ( f ) & - \\mathcal X ^ n _ 0 ( f ) + c _ n \\int _ { 0 } ^ t \\mathcal X ^ n _ s ( f ' ) d s - d _ n \\int _ { 0 } ^ t \\mathcal X ^ n _ s ( f '' ) d s \\\\ & - \\ ; \\big ( d _ n \\nabla ^ n _ { 0 } f - c _ n f \\big ( \\tfrac { 1 } { n } \\big ) \\big ) n ^ { 1 - \\gamma } \\int _ { 0 } ^ t \\overline { J _ s ^ n } ( 0 ) \\ ; d s + o _ n ( 1 ) , \\end{align*}"}
-{"id": "391.png", "formula": "\\begin{align*} F ( \\Psi ( \\gamma \\otimes x ) ) ( t _ 1 , \\dots , t _ q ) = T _ * ( \\gamma ( t _ 1 , \\dots , t _ q ) \\times \\beta ' ( \\lambda _ x ( 0 , \\dots , 0 ) ) ) . \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} \\phi ( \\omega ) = \\frac { \\Omega _ 1 } { 2 i \\pi } \\zeta ( \\omega ; \\Omega _ 1 , \\Omega _ 2 ) - \\frac { \\omega } { i \\pi } \\zeta ( \\Omega _ 1 / 2 ; \\Omega _ 1 , \\Omega _ 2 ) \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} h ( ( t _ 1 \\otimes t _ 2 ) \\cdot p ) = ( t _ 1 \\otimes t _ 2 ) \\cdot h ( p ) - ( t _ 1 \\otimes \\partial ( t _ 2 ) ) \\cdot k _ s ( p ) \\ , . \\end{align*}"}
-{"id": "4997.png", "formula": "\\begin{align*} ( 1 + t ^ { n - 1 } ) \\binom { [ \\frac { k } { 2 } ] + [ \\frac { n - k } { 2 } ] } { [ \\frac { k } { 2 } ] } _ { t ^ 4 } \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} T = T _ { K } \\cup I _ h , C ( T ) = A _ { K } \\cup T . \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} \\mathcal { H } f ( \\alpha ) = \\frac { \\operatorname { p . v . } } { \\pi } \\int _ { \\mathbb { R } } \\frac { f ( \\beta ) } { \\alpha - \\beta } d \\beta = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } f ( \\beta ) \\cot \\frac { \\alpha - \\beta } { 2 } d \\beta . \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} n _ j = \\left \\{ \\begin{array} { c l } 1 & j = 1 \\ ; , \\\\ m _ { j - 1 } & j = 2 , \\dots , \\ell + 1 \\ ; , \\\\ k + 2 & j = \\ell + 2 \\ ; , \\\\ m _ { j - 2 } & j = \\ell + 3 , \\dots , 2 \\ell + 1 \\ ; . \\end{array} \\right . \\end{align*}"}
-{"id": "327.png", "formula": "\\begin{align*} z w - f ( z ) + g ( w ) = 0 \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{align*} w ( d ) = \\begin{cases} \\binom { d } { 2 } & d \\ne r \\\\ \\binom { r } { 2 } - k & d = r . \\end{cases} \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{align*} & \\ln ( \\beta ( t , q _ t ) / \\beta ( s , q _ s ) ) = \\ln ( \\beta ( t , q _ t ) ) - \\ln ( \\beta ( s , q _ s ) ) \\\\ = & \\int _ s ^ t \\beta ^ { - 1 } ( r , q _ r ) \\partial _ r \\beta ( r , q _ r ) d r + \\int _ s ^ t ( \\beta ^ { - 1 } \\nabla _ q \\beta ) ( r , q _ r ) \\cdot d q _ r + \\frac { 1 } { 2 } \\int _ s ^ t \\partial _ { q ^ j } ( \\beta ^ { - 1 } \\partial _ { q ^ i } \\beta ) ( r , q _ r ) d [ q ^ i , q ^ j ] _ r \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{align*} 2 \\sum _ { j = 1 } ^ \\infty \\mathcal G \\Big ( \\kappa \\Gamma _ { m + j } / \\sqrt { t _ m } \\Big ) \\ , . \\end{align*}"}
-{"id": "6946.png", "formula": "\\begin{align*} & c _ n \\approx 1 \\Leftrightarrow \\theta - 2 \\beta = 1 + \\alpha \\\\ & d _ n \\approx 1 \\Leftrightarrow \\theta - 2 \\beta = 2 . \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} \\| ( \\varphi ( t ) , \\psi ( t ) ) \\| _ { \\overline { X } _ 2 } = \\| S ^ * ( \\tau - t ) ( \\varphi _ \\tau , \\psi _ \\tau ) \\| _ { \\overline { X } _ 2 } \\leq C _ T \\| ( \\varphi _ \\tau , \\psi _ { \\tau } ) \\| _ { \\overline { X } _ 2 } , \\forall t \\in [ 0 , \\tau ] . \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} S ^ { e n v , 0 } _ { s , t } = & \\ln ( \\beta ( t , q _ t ) / \\beta ( s , q _ s ) ) - \\int _ s ^ t ( \\beta ^ { - 1 } \\partial _ r \\beta ) ( r , q _ r ) d r \\\\ & + ( \\beta V ) ( s , q _ s ) - ( \\beta V ) ( t , q _ t ) + \\int _ s ^ t \\partial _ r ( \\beta V ) ( r , q _ r ) d r \\\\ & + \\int _ s ^ t ( \\beta \\tilde F + V \\nabla _ q \\beta ) ( r , q _ r ) \\circ d q _ r . \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{align*} I ( \\mathcal { C } _ 0 ) & \\ge \\sum \\limits _ a \\frac { d _ a ( d _ a - 1 ) } { 2 } ( 2 g ( C _ a ) - 2 + { \\rm i n d } C _ a + h ( C _ a ) + 4 \\delta ( C _ a ) ) \\\\ & + \\sum \\limits _ a d _ a I ( C _ a ) + 2 \\sum \\limits _ { a \\ne b } C _ a \\cdot C _ b \\\\ & \\ge - d _ 0 + \\frac { d _ 0 ( d _ 0 - 1 ) } { 2 } ( 2 g ( C _ 0 ) - 2 + { \\rm i n d } C _ 0 + h ( C _ 0 ) + 4 \\delta ( C _ 0 ) ) . \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} a ( j k ) = \\int _ { - \\infty } ^ \\infty v ( \\xi ) ( j k ) ^ { - \\frac 1 2 + 2 \\pi i \\xi } d \\xi . \\end{align*}"}
-{"id": "2180.png", "formula": "\\begin{align*} U _ R & : = \\{ x \\in C : M ( x / u ) > e ^ R , \\ , M ( x / u ) \\ge M ( u / x ) \\} , \\\\ V _ R & : = \\{ x \\in C : M ( u / x ) > e ^ R , \\ , M ( x / u ) < M ( u / x ) \\} . \\end{align*}"}
-{"id": "1408.png", "formula": "\\begin{align*} \\mathbb { D } _ { \\infty } ( X ) : = D _ { W ^ { 1 , 2 } } ( \\Delta ) \\cap { W ^ { 1 , 2 , ( \\infty ) } ( X ) } . \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} t = \\int _ { { \\frak t } _ 0 } ^ { \\frak t } \\sigma ^ { - \\ : \\frac 1 4 } e ^ { \\frac 1 4 \\langle \\lambda \\rangle ( \\sigma ) } d \\sigma . \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} I _ \\lambda : = \\prod _ { i = 1 } ^ d \\left [ \\frac { \\lambda _ i - 1 } { N } - \\frac { 1 } { 2 } , \\frac { \\lambda _ i } { N } - \\frac { 1 } { 2 } \\right ] . \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{align*} \\frac { P ( T ) } { ( 1 - T ) ( 1 - q T ) } ( y ( 1 - T ) + x T ) ^ n = \\cdots + \\frac { W ( x , y ) - x ^ n } { q - 1 } T ^ { n - d } + \\cdots . \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} \\omega _ { \\exp ( a ) H } = \\frac { u '' ( t ) } { 4 } \\omega _ { 1 , \\bar { 1 } } + 2 ( m - k ) ( 1 - \\tanh ^ 2 ( 2 t ) ) ( \\omega _ { 1 , \\bar { \\beta } } + \\omega _ { \\beta , \\bar { 1 } } ) + \\frac { u ' ( t ) } { \\sinh ( 4 t ) } \\omega _ { \\beta , \\bar { \\beta } } \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} L ^ q _ 0 ( \\Omega ) ^ 3 : = \\Big \\{ d \\in L ^ q ( \\Omega ) ^ 3 \\mid \\int _ { \\Omega } d \\ ; \\d x = 0 \\Big \\} . \\end{align*}"}
-{"id": "1145.png", "formula": "\\begin{align*} ( \\alpha \\otimes 1 - 1 \\otimes \\alpha ) \\cdot ( ( s \\otimes t ) \\cdot j ) = \\partial _ \\alpha ( s \\otimes t ) \\cdot j + ( s \\otimes t ) \\cdot ( \\alpha \\otimes 1 - 1 \\otimes \\alpha ) \\cdot j \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} P [ X _ { k + 1 } = m | X _ { k } = n ] = ( 1 - p _ \\mathrm { s } ) \\cdot p _ { \\mathrm { S } _ 0 } ( m | n ) + p _ \\mathrm { s } \\cdot p _ { \\mathrm { S } _ 1 } ( m | n ) . \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} \\max _ { x \\in Q _ n ' } \\lambda _ j ( x ) \\geq \\lambda _ j \\left ( x ^ { ( j ) } \\right ) = 1 . \\end{align*}"}
-{"id": "6770.png", "formula": "\\begin{align*} \\xi _ r \\xi _ s = \\beta ( | \\xi _ r | , | \\xi _ s | ) \\xi _ s \\xi _ r \\quad 1 \\leq r , s \\leq | \\mathbf q | . \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} x _ M ^ u = \\mathop { \\sum } _ { i = 1 } ^ s ( 1 \\times \\cdots \\times 1 \\times x _ { M _ i } ^ u \\times 1 \\times \\cdots \\times 1 ) \\ + \\mathop { \\sum } _ { \\substack { u _ 1 + \\cdots + u _ s = u , \\\\ 0 \\leq u _ j < u } } ( x _ { M _ 1 } ^ { u _ 1 } \\times \\cdots \\times x _ { M _ s } ^ { u _ s } ) , \\end{align*}"}
-{"id": "4614.png", "formula": "\\begin{align*} \\bar \\partial _ B H ^ { 1 , 0 } \\lrcorner \\ , \\phi = 0 . \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} p _ r ( x ^ n , y ^ n , z ^ n ) = \\frac { p _ { X ^ n Y ^ n Z ^ n } ( x ^ n , y ^ n , z ^ n ) } { \\mathbb { P } [ X ^ n \\in \\mathcal { A } , Y ^ n \\in \\mathcal { B } ] } \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} \\left . \\frac { d } { d s } \\right | _ { s = 0 } \\int _ X u \\ , d \\nu _ s = - \\int _ X \\langle \\nabla u , \\nabla \\varphi _ t \\rangle \\ , d \\mu _ t . \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{align*} E \\left [ S _ N ^ { ( 2 ) } \\right ] = N ^ 2 \\left ( H _ N ^ 2 + \\sum _ { j = 1 } ^ N \\frac { 1 } { j ^ 2 } \\right ) . \\end{align*}"}
-{"id": "8551.png", "formula": "\\begin{align*} \\sigma _ m = \\omega _ m + d ( d \\mu I _ m ) \\rlap { . } \\end{align*}"}
-{"id": "4940.png", "formula": "\\begin{align*} T = \\Sigma ^ { - \\frac { 1 } { 2 } } U ^ T L ^ T \\quad T ^ { - 1 } = K V \\Sigma ^ { - \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "465.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 7 & 6 & 1 0 \\\\ 0 & 8 & 9 & 6 \\\\ 0 & 8 & 8 & 7 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 2 & 1 & 8 \\\\ 0 & 2 & 0 & 3 \\\\ 0 & 1 0 & 6 & 7 \\end{pmatrix} , \\end{align*}"}
-{"id": "654.png", "formula": "\\begin{align*} Y ( T _ 0 , T _ 1 ) : = Z ( T _ 0 , T _ 1 ) \\bigcap X ( T _ 0 , T _ 1 ) , \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} b _ { n } : = \\widetilde { b } _ { n } - \\frac { 1 } { \\lvert \\Omega \\rvert } \\int _ { \\Omega } \\widetilde { b } _ { n } \\ ; \\d x . \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} \\frac { d x ( t ) } { d t } = A x ( t ) + B u ( t ) + \\sum _ { i = 1 } ^ m N _ i x ( t ) u _ i ( t ) , \\ ; \\ ; \\ ; x ( 0 ) = x _ 0 , \\ ; \\ ; \\ ; t \\geq 0 , \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{align*} \\xi ( t ) . v _ { w \\lambda } = ( w \\lambda - q ) ( \\xi ( t ) ) . v _ { w \\lambda } \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} I _ 2 ( N ) : = \\int _ { U ( N ; \\alpha ) } ^ N \\left ( 1 - \\frac { x } { N } \\right ) ^ { N - 1 } \\left ( 1 - \\frac { \\ln x } { \\ln N } \\right ) ^ r d x . \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} \\lambda _ \\theta & = ( 1 - \\theta ) \\lambda _ 0 + \\theta \\lambda , \\gamma _ \\theta = ( 1 - \\theta ) \\gamma _ 0 + \\theta \\gamma . \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} G M = G \\begin{bmatrix} \\times & \\times \\\\ & \\ddots & \\ddots \\\\ & & \\times & \\times \\\\ \\times & & & \\times \\\\ \\end{bmatrix} = \\left [ \\begin{array} { c | c c c } a _ 1 & b _ 1 & & x _ 1 \\\\ \\hline & & \\\\ & & \\widetilde M & \\\\ & & & \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "2432.png", "formula": "\\begin{align*} \\frac { \\psi _ j ( n _ j - 1 ) } { \\psi _ j ( n _ j ) } = \\frac { p _ j n _ j + \\lambda _ j } { p _ j n _ j } , \\end{align*}"}
-{"id": "2391.png", "formula": "\\begin{align*} E \\left [ T _ 1 \\right ] = \\frac { \\nu _ 1 } { \\alpha _ 1 } \\ , M H _ { \\nu _ 1 M } \\ , + \\ , O \\left ( e ^ { - \\varepsilon M } \\right ) = ( \\nu _ 1 + \\lambda \\nu _ 2 ) M H _ { \\nu _ 1 M } \\ , + \\ , O \\left ( e ^ { - \\varepsilon M } \\right ) , M \\to \\infty , \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} g _ i ^ { k + 1 } v ( g _ i ^ d f ) = g _ i ^ { k + 1 } ( d \\cdot g _ i ^ { d - 1 } v ( g _ i ) f + g _ i ^ d v ( f ) ) = d \\cdot g _ i ^ k v ( g _ i ) g _ i ^ d f + g _ i ^ { d + k + 1 } v ( f ) \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} \\overline { T _ f } = \\frac { \\lambda _ f } { \\lambda _ g } k ( a + b \\mu ) . \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} \\pi _ 1 ( \\sigma ) = \\lbrace s \\in S _ \\mu \\mid \\rbrace , & & \\pi _ 2 ( \\sigma ) = \\lbrace t \\in T _ \\mu \\mid \\rbrace , \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} \\mathbb { E } _ { n , d _ 1 , \\theta } ( \\varphi ' ) = \\mathbb { E } _ { n , d _ 2 , F ( \\theta ) } ( \\varphi ) \\theta \\in \\Theta _ { d _ 1 } . \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} g \\cdot u ^ { ( k ) } = c _ k \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} F ( n ) = 0 \\mod v . \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l l } \\nabla \\nabla f ( X , Y ) = \\nabla _ { B } \\nabla _ { B } f ( X , Y ) , \\\\ \\noalign { \\smallskip } \\nabla \\nabla f ( X , U ) = X ( U ( f ) ) - h ^ { - 1 } X ( h ) U ( f ) , \\\\ \\noalign { \\smallskip } \\nabla \\nabla f ( U , V ) = \\nabla _ { F } \\nabla _ { F } f ( U , V ) + h ( \\nabla _ { B } h ) f g _ { F } ( U , V ) . \\end{array} \\right . \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} d x ^ \\prime _ t = ( \\phi _ * b _ + ) ( t ^ * , x ^ \\prime _ t ) d t - ( \\phi _ * b _ - ) ( t ^ * , x ^ \\prime _ t ) d t + ( \\phi _ * \\tilde \\sigma ) ( t ^ * , x ^ \\prime _ t ) \\circ d \\tilde W _ t , \\end{align*}"}
-{"id": "6244.png", "formula": "\\begin{align*} V _ { \\mathrm { o l d } } = \\sum _ { \\mu , \\lambda } E _ \\mu ^ * E _ \\lambda V & & , \\end{align*}"}
-{"id": "489.png", "formula": "\\begin{align*} ( 2 q _ 2 y ) ^ { - k } = o \\left ( \\left ( \\frac { 1 } { 4 } + q _ 2 ^ 2 y ^ 2 \\right ) ^ { - k / 2 - 1 } \\right ) , \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} \\begin{array} { l } \\partial _ t \\varphi + d _ \\mu \\Delta \\varphi = 0 , \\varphi ( T , x ) = \\zeta ( x ) , \\end{array} \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} H ' ( U _ 0 , r ) = \\frac { 4 } { r } I ( U _ 0 , r ) + \\frac { a } { r } H ( U _ 0 , r ) , \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} \\max p _ { - 3 } ( n ) & = \\left ( p _ { - 3 } ( 1 ) \\right ) ^ { n } = 3 ^ { n } . \\end{align*}"}
-{"id": "1783.png", "formula": "\\begin{align*} \\xi ' & = A ^ t \\underline { \\xi } ' ( x , y ) = ( x ' , 0 , y ' , 0 ) , \\\\ \\xi _ n & = { B ^ t \\underline { \\xi } ' } + C \\underline { \\xi } _ n . \\end{align*}"}
-{"id": "4970.png", "formula": "\\begin{align*} K ( u ) = \\frac { 1 } { 3 } \\displaystyle { \\int u ^ 3 d x = K ( Q _ { c } ) } , \\end{align*}"}
-{"id": "8877.png", "formula": "\\begin{align*} - K _ X = \\sum _ { Y \\in \\mathcal { I } ^ G ( X ) } Y + \\sum _ { D \\in \\mathcal { D } } m _ D \\overline { D } \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} \\alpha _ i : = H _ { \\dim X _ i } ( B \\pi _ 1 ( f \\vert _ { X _ i } ) \\circ c _ { X _ i } ) ( [ X _ i ] ) \\in H _ { \\dim X _ i } ( B \\Gamma _ i ; \\R ) . \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} \\mathcal K = \\left \\langle H _ 1 , p H _ 2 , \\dots , p ^ { n - 1 } H _ { n } \\right \\rangle = \\left \\langle \\sum _ { i = 0 } ^ { n - 1 } p ^ i H _ { i + 1 } \\right \\rangle \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{align*} \\mathbb { P } [ X ^ n = \\mathbf { x } _ 2 , Y ^ n = \\mathbf { y } _ 1 | X ^ n \\in \\mathcal { A } , Y ^ n \\in \\mathcal { B } ] & = \\mathbb { P } [ X ^ n = \\mathbf { x } _ 1 , Y ^ n = \\mathbf { y } _ 2 | X ^ n \\in \\mathcal { A } , Y ^ n \\in \\mathcal { B } ] \\\\ & = \\frac { p _ { 2 1 } ^ { n / 2 } p _ { 1 2 } ^ { n / 2 } } { 2 ( p _ { 1 1 } ^ { n / 2 } p _ { 2 2 } ^ { n / 2 } + p _ { 1 2 } ^ { n / 2 } p _ { 2 1 } ^ { n / 2 } ) } . \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} e _ k ( \\xi ) = s ^ 2 | D ^ k \\nabla _ t \\xi | ^ 2 + | D ^ { k + 1 } \\xi | ^ 2 , \\end{align*}"}
-{"id": "5048.png", "formula": "\\begin{align*} & [ d , s , x ] = [ y _ 1 \\dots y _ k , s , x ] = \\sum _ { i = 1 } ^ k y _ 1 \\dots y _ { i - 1 } [ y _ i , s , x ] y _ { i + 1 } \\dots y _ k \\\\ + \\ & \\sum _ { 1 \\le i < i ' \\le k } y _ 1 \\dots y _ { i - 1 } \\bigl ( [ y _ i , s ] y _ { i + 1 } \\dots y _ { i ' - 1 } [ y _ { i ' } , x ] + [ y _ i , x ] y _ { i + 1 } \\dots y _ { i ' - 1 } [ y _ { i ' } , s ] \\bigr ) y _ { i ' + 1 } \\dots y _ k \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{align*} F \\left ( s \\right ) = f \\left ( 0 \\right ) \\sum _ { k = 0 } ^ { \\infty { } } \\frac { { E ^ k \\left ( i \\hslash { } \\right ) } ^ { - \\beta { } k } } { s ^ { \\beta { } k + 1 } } . \\end{align*}"}
-{"id": "2383.png", "formula": "\\begin{align*} E \\left [ S ( \\theta ) ^ { ( r ) } \\right ] = r \\int _ 0 ^ { \\infty } t ^ { r - 1 } \\left [ 1 - \\left ( 1 - e ^ { - ( 1 - \\theta ) t / N } \\right ) ^ N \\right ] d t + O \\left ( e ^ { - \\varepsilon N } \\right ) , N \\to \\infty , \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} f _ n ( - x _ { n - 1 } ) & = 0 , \\\\ f _ n ( x _ { n } ) & = x _ n , \\\\ f _ n ' ( x _ { n } ) & = 1 . \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} \\tau _ n ( v _ 0 ) & : = \\min \\{ k > \\tau _ { n - 1 } ( v _ 0 ) : \\ X _ { k } = v _ 0 \\} , \\\\ \\widetilde \\tau _ n ( v _ 0 ) & : = T ( \\tau _ n ( v _ 0 ) ) , \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{align*} b _ 2 ( M ) = p - 1 , b _ 3 ( M ) = p , \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{align*} E : y ^ 2 = x ( x ^ 2 + b x + d ) \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} T _ { \\overline { L } , p } ( r ) = \\int g _ { D _ r , p } \\cdot ( - 2 d d ^ c ( \\log \\| s \\| + m g _ { D _ r , p } ) ) + \\int g _ { D _ r , p } \\delta _ { \\div ( s ) - m [ p ] } \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{align*} d \\mathbf { q } _ { i j } = - \\frac { k } { 4 \\pi } \\mathbf { r } ^ { - 3 } _ { i j } \\frac { \\partial \\mathbf { r } _ { i j } } { \\partial s _ { i } } \\times \\mathbf { r } _ { i j } d s _ { j } . \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{align*} \\int _ { \\gamma } \\mathcal { H } ^ * \\omega = \\int _ 0 ^ T s _ { \\lambda ( t ) } - \\max _ { \\abs { u } \\leq 1 } \\mathfrak { h } ( u , \\lambda ( t ) ) \\ , d t \\leq \\int _ 0 ^ T | v ( t ) \\psi ( \\xi ( t ) ) | \\ ; d t , \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} \\left ( x \\frac { d } { d x } \\right ) ^ 3 \\log \\left ( \\theta _ 3 ( x ) \\right ) = - \\left ( x \\frac { d } { d x } \\right ) ^ 3 \\log \\left ( \\theta _ 3 \\left ( \\tfrac { 1 } { x } \\right ) \\right ) . \\end{align*}"}
-{"id": "3966.png", "formula": "\\begin{align*} & \\Delta ( X ; Y \\| Z ) = \\inf \\| p _ { K _ A K _ B Z ^ n \\mathbf { F } } - q _ { K _ A K _ B } \\cdot p _ { Z ^ n \\mathbf { F } } \\| _ { T V } \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} a : = \\begin{pmatrix} 1 & 6 & 9 & 5 \\\\ 1 0 & 8 & 6 & 6 \\\\ 1 & 8 & 1 & 1 \\\\ 6 & 1 0 & 9 & 1 \\end{pmatrix} , b : = \\begin{pmatrix} 1 & 0 & 8 & 5 \\\\ 1 0 & 1 & 3 & 6 \\\\ 1 & 0 & 7 & 1 \\\\ 6 & 1 & 2 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{align*} \\norm { \\theta ^ n } ^ 2 - \\norm { \\theta ^ { n - 1 } } ^ 2 = & - \\Re \\left ( \\rho ^ n - \\rho ^ { n - 1 } , \\theta ^ n + \\theta ^ { n - 1 } \\right ) \\\\ & - k _ n \\Big [ \\Im ( \\omega ^ n , \\ , \\theta ^ n + \\theta ^ { n - 1 } ) + \\Re ( r ^ n , \\ , \\theta ^ n + \\theta ^ { n - 1 } ) \\Big ] . \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{align*} ( 1 + v _ { m + 1 } ^ { ( 0 ) } ) ( 1 + v _ { m + 1 } ^ { ( 1 ) } ) = 1 + v _ { m + 1 } ^ { ( 0 ) } + v _ { m + 1 } ^ { ( 1 ) } , \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { p _ i ( x ^ k ) } { k ^ 4 } = \\lim _ { k \\to \\infty } \\frac { k ^ 4 ( p _ i ( x ^ k ) - p _ i ( \\tilde x ^ k ) ) } { k ^ 8 } + \\lim _ { k \\to \\infty } \\frac { p _ i ( \\tilde x ^ k ) } { k ^ 4 } = \\lim _ { k \\to \\infty } \\frac { p _ i ( \\tilde x ^ k ) } { k ^ 4 } \\ge 0 . \\end{align*}"}
-{"id": "5302.png", "formula": "\\begin{align*} ( J \\otimes E ) ^ { \\mathrm { s u p } } : = J ' \\otimes E , \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} \\sum _ { n = M _ - } ^ { M _ + - 1 } ( A _ 1 + A _ 3 ) | C _ n | & \\leq \\frac { c _ 1 } { ( \\gamma t ) ^ { \\frac d 2 + 1 } } \\Big ( \\frac { ( \\gamma t ) ^ { ( \\frac 1 2 + \\varepsilon ) d } } { M ^ d } \\wedge 1 \\Big ) \\sum _ { n = M _ - } ^ { M _ + } n ^ { d - 1 } \\leq c _ 2 \\ , ( \\gamma t ) ^ { - 1 + \\varepsilon d } \\ , , \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} H _ { r } & = \\alpha _ { r } ( b + b ^ { - 1 } - \\alpha _ { r } ) , & H _ { l } & = \\alpha _ { l } ( b + b ^ { - 1 } - \\alpha _ { l } ) , \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{align*} ( M _ \\sigma ) _ { s , t } = \\begin{cases} 1 & , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{align*} \\tilde \\kappa _ g = | \\alpha | \\kappa _ s , \\tilde \\kappa _ \\nu = \\alpha \\kappa _ \\nu , \\tilde \\kappa _ t = \\alpha \\kappa _ t , \\end{align*}"}
-{"id": "3016.png", "formula": "\\begin{align*} I _ { 0 } ( u ) : = \\frac { 1 } { 2 } \\int _ { \\Omega } | \\nabla u | ^ { 2 } - \\int _ { \\Omega } a | u | \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { C _ { 1 - \\gamma ; \\psi } \\left [ a , b \\right ] } = \\left \\Vert \\left ( \\psi \\left ( t \\right ) - \\psi \\left ( a \\right ) \\right ) ^ { 1 - \\gamma } f \\left ( t \\right ) \\right \\Vert _ { C \\left [ a , b \\right ] } = \\underset { t \\in \\left [ a , b \\right ] } { \\max } \\left \\vert \\left ( \\psi \\left ( t \\right ) - \\psi \\left ( a \\right ) \\right ) ^ { 1 - \\gamma } f \\left ( t \\right ) \\right \\vert . \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} g \\cdot m '' = \\beta ( \\beta \\eta \\| m ' \\| ^ 2 + B m '' ) + 2 \\eta _ 1 e _ 1 \\end{align*}"}
-{"id": "1658.png", "formula": "\\begin{gather*} K _ { i j } : = \\sum _ { k = 1 } ^ { H _ 0 - 1 } ( a _ { i k } b _ { k j } - a ^ 0 _ { i k } b ^ 0 _ { k j } ) + a _ { i H _ 0 } b _ { H _ 0 j } - a ^ 0 _ { i H _ 0 } b ^ 0 _ { H _ 0 j } + \\sum _ { k = H _ 0 + 1 } ^ { H - 1 } a _ { i k } b _ { k j } + a _ { i H } b _ { H j } , \\\\ L _ j : = \\sum _ { k = 1 } ^ { H _ 0 - 1 } ( a _ { M k } b _ { k j } - a ^ 0 _ { M k } b ^ 0 _ { k j } ) + a _ { M H _ 0 } b _ { H _ 0 j } - a ^ 0 _ { M H _ 0 } b ^ 0 _ { H _ 0 j } + \\sum _ { k = H _ 0 + 1 } ^ { H - 1 } a _ { M k } b _ { k j } + a _ { M H } b _ { H j } , \\end{gather*}"}
-{"id": "7557.png", "formula": "\\begin{align*} \\beta \\to & \\int \\left ( \\frac { \\beta } { 2 \\pi } \\right ) ^ { n / 2 } e ^ { - \\beta \\| z \\| ^ 2 / 2 } z _ { i _ 1 } . . . z _ { i _ k } d z \\\\ = & \\beta ^ { - k / 2 } \\int \\left ( \\frac { 1 } { 2 \\pi } \\right ) ^ { n / 2 } e ^ { - \\| u \\| ^ 2 / 2 } u _ { i _ 1 } . . . u _ { i _ k } d u \\\\ \\end{align*}"}
-{"id": "3752.png", "formula": "\\begin{align*} | H ( \\nu , \\mathcal { D } _ n ) - H ( \\tilde { \\nu } , \\mathcal { D } _ n ) | = O _ C ( 1 ) \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} M ( s , t ) = \\max \\left \\{ \\frac { 2 } { 3 } , \\ , 2 - 3 s , \\ , 3 s - 1 , \\ , 2 - 3 t , \\ , 3 t - 1 \\right \\} . \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l l } \\mu _ { 1 } = c h + \\tilde { c } , \\\\ \\noalign { \\smallskip } \\rho _ { 1 } = - b h + \\tilde { b } , \\\\ \\noalign { \\smallskip } \\mu _ { 2 } = - c \\varphi - b , \\\\ \\noalign { \\smallskip } \\rho _ { 2 } = \\tilde { c } \\varphi - \\tilde { b } , \\end{array} \\right . \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} \\min \\{ | q _ k ( 0 ) - q _ l ( 0 ) | \\ ; | \\ ; k \\neq l \\} = 8 \\varepsilon _ 0 \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} \\wp _ { u n i v } '' = 6 \\wp _ { u n i v } ^ 2 - { 1 \\over 2 } \\underline g _ 2 , \\end{align*}"}
-{"id": "7202.png", "formula": "\\begin{align*} \\tilde { u } \\in [ 0 , \\eta ] = \\{ u \\in W ^ { 1 , p } ( \\Omega ) : 0 \\leq u ( z ) \\leq \\eta \\ \\mbox { f o r a l m o s t a l l } \\ z \\in \\Omega \\} . \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} & D ( p _ { K _ A K _ B Z ^ n \\mathbf { F } } \\| p _ { K _ A K _ B } p _ { Z ^ n \\mathbf { F } } ) = I ( K _ A K _ B ; Z ^ n \\mathbf { F } ) \\leq 4 \\delta + h ( \\delta ) . \\end{align*}"}
-{"id": "8329.png", "formula": "\\begin{align*} a = ( I + \\mathcal { K } ^ * ) ^ { - 1 } \\operatorname { R e } \\left \\{ i e ^ { i \\theta } \\left ( [ z _ t , \\mathfrak { H } ] \\frac { \\bar { z } _ { t \\alpha } } { z _ \\alpha } + e ^ { i \\theta } ( I - \\mathfrak { H } ) { \\bf 1 } + ( I - \\mathfrak { H } ) \\bigl ( \\frac { \\partial _ \\alpha P } { | z _ \\alpha | } e ^ { - i \\theta } \\bigr ) \\right ) \\right \\} . \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{align*} ( V B ( ( F C ) ) ( \\tilde X , \\tilde D ) _ { u n i p } \\stackrel { d i r . i m . ' } { \\to } V B ( F C ) ( X , D ) _ { u n i p } \\stackrel { P B ' } { \\to } V B ( ( F C ) ) ( \\tilde X , \\tilde D ) _ { u n i p } ) = i d _ { V B ( ( F C ) ) ( \\tilde X , \\tilde D ) _ { u n i p } } \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{align*} a = ( I + \\mathcal { K } ^ * ) ^ { - 1 } \\operatorname { R e } \\left \\{ i e ^ { i \\theta } \\left ( \\bigl ( I - \\mathfrak { H } \\bigr ) \\bigl ( \\bar { z } _ { t t } - i + \\frac { \\partial _ \\alpha P } { | z _ \\alpha | } e ^ { - i \\theta } \\right ) \\right \\} . \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r f _ i ( \\vec { x } ) g _ i ( \\vec { x } ) + \\sum _ I \\Delta _ I ( \\vec { x } ) g _ I ( \\vec { x } ) = N , \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} V \\left [ T _ 1 \\right ] = \\frac { \\pi ^ 2 ( \\nu _ 1 + \\lambda \\nu _ 2 ) ^ 2 } { 6 } \\ , M ^ 2 \\left [ 1 + O \\left ( \\frac { \\ln M } { M } \\right ) \\right ] , M \\to \\infty . \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{align*} J _ { s , t } ^ m = \\int _ s ^ t B ^ { i _ 1 , . . . , i _ k } ( r , q _ r ^ m ) ( z _ r ^ m ) _ { i _ 1 } . . . ( z _ r ^ m ) _ { i _ k } d r . \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} a _ 0 : = x ( \\forall F ( - F x ) ) . \\end{align*}"}
-{"id": "3534.png", "formula": "\\begin{align*} \\Gamma ^ { \\lambda } _ { t } : = \\lambda \\cdot \\big ( \\Gamma _ { T + \\lambda ^ { - 2 } t } - x _ { 0 } \\big ) \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} \\begin{gathered} \\frac { d } { d x } H _ n ( x ) = 2 n H _ { n - 1 } ( x ) , \\\\ \\left ( - \\frac { d } { d x } + 2 x \\right ) H _ n ( x ) = H _ { n + 1 } ( x ) . \\end{gathered} \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} A _ m = U _ m ^ { A } \\Sigma _ m ^ { A } ( V _ m ^ { A } ) ^ T , U _ m ^ { A } \\in \\R ^ { N \\times m } , \\ ; \\Sigma _ m ^ { A } \\in \\R ^ { m \\times m } , \\ ; V _ m ^ { A } \\in \\R ^ { N \\times m } , \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} v _ { i , j } = \\sum _ { k = i } ^ { j } ( a _ { i - 1 , k } - b _ { i , k } ) , w _ { i , j } = \\sum _ { k = j } ^ { r } ( a _ { i , k } - b _ { i , k } ) , u _ { i , j } = v _ { i , r } + w _ { i , j } . \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ( s \\otimes 1 ) \\cdot ( 1 \\otimes \\alpha ) = s \\otimes \\alpha \\\\ ( 1 \\otimes \\alpha ) \\cdot ( s \\otimes 1 ) = \\alpha ( s ) \\otimes 1 + s \\otimes \\alpha \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{align*} { \\mathcal S } _ { \\gamma } = \\Big \\{ ( \\widetilde u , \\widetilde \\pi , \\widetilde \\ell , \\widetilde \\omega ) \\mid \\widetilde \\rho ( t , y ) = \\left \\| ( \\widetilde u , \\widetilde \\pi , \\widetilde \\ell , \\widetilde \\omega ) \\right \\| _ { \\mathcal { S } } \\leqslant \\gamma \\Big \\} , \\end{align*}"}
-{"id": "7664.png", "formula": "\\begin{align*} \\mathrm { P } _ { n , 1 } = e ^ { - \\lambda _ c \\pi \\left ( \\frac { \\rho } { \\epsilon _ 1 } \\right ) ^ { \\frac { 2 } { \\alpha } } } \\sum ^ { n - 1 } _ { k = 0 } \\frac { ( \\lambda _ c \\pi ) ^ { k } \\left ( \\frac { \\rho } { \\epsilon _ 1 } \\right ) ^ { \\frac { 2 k } { \\alpha } } } { k ! } . \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} r _ { k + 1 } ( r _ * ) = r _ * - 2 m \\log ( r _ k ( r _ * ) - 2 m ) = r _ * - 2 m \\log ( r _ k ) - 2 m \\log ( 1 - 2 m r _ k ^ { - 1 } ) , \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{align*} \\tag * { $ { \\bf ( A _ 4 ) } $ } E _ 0 \\ \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} u ^ * _ Y ( p ) = u _ { \\lambda } ^ * ( p ) = u ^ * ( p + 2 \\lambda ) . \\end{align*}"}
-{"id": "1487.png", "formula": "\\begin{align*} V _ f ( S ) : = \\int _ S e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma . \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} \\widetilde { \\Phi } ( x , \\lambda , v ) : = \\left ( \\begin{matrix} - v + \\nabla g ( x ) \\lambda \\\\ g ( x ) \\\\ \\lambda \\\\ \\end{matrix} \\right ) - \\left ( \\begin{matrix} \\{ 0 \\} \\\\ { \\rm g p h } \\mathcal { N } _ K \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{gather*} \\dot { y } = u _ \\mu ( \\rho ) , \\dot { \\rho } = 2 \\rho y v _ \\mu ( \\rho ) , \\dot { \\varphi } = W _ \\mu ( \\rho ) ; \\\\ G : ( y , \\rho , \\varphi ) \\mapsto ( - y , \\rho , - \\varphi ) . \\end{gather*}"}
-{"id": "3160.png", "formula": "\\begin{align*} \\| D _ { X } C _ { X } ^ { - 1 } \\| _ { 1 } = \\sum _ { j = 1 } ^ { \\infty } D _ { j } C _ { j } ^ { - 1 } = \\sum _ { j = 1 } ^ { \\infty } D _ { X } ( \\phi _ { j } ) ( \\phi _ { j } ) [ C _ { X } ( \\phi _ { j } ) ( \\phi _ { j } ) ] ^ { - 1 } = \\| \\rho \\| _ { 1 } . \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} \\sigma ( \\pi ^ { \\alpha } ) = ( \\pi + 1 ) ( 1 + \\pi ^ 2 + \\pi ^ 4 + . . . + \\pi ^ { \\alpha - 1 } ) . \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} c _ n ( \\{ k \\} ) = \\alpha c _ 0 ( \\{ k \\} ) + \\sum \\limits _ { i = 1 } ^ { n - 1 } \\delta _ { X _ { i + 1 } - X _ i + ( 1 - \\delta _ { X _ i 0 } ) } ( \\{ k \\} ) , \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} & \\R _ { > 0 } \\times \\R \\backslash O _ + \\cup O _ - \\\\ & = \\left \\{ ( \\beta , \\delta \\theta _ { c , j , m } ( \\beta ) ) \\ | \\ \\beta \\in \\R _ { > 0 } , \\ j \\in \\{ 1 , 2 \\} , \\ m \\in \\N \\cup \\{ 0 \\} , \\ \\delta \\in \\{ 1 , - 1 \\} \\right \\} \\\\ & = \\left \\{ ( \\beta _ { c , j , m } ( \\theta ) , \\delta \\theta ) \\ | \\ \\theta \\in \\R _ { > 0 } , \\ j \\in \\{ 1 , 2 \\} , \\ m \\in \\N \\cup \\{ 0 \\} , \\ \\delta \\in \\{ 1 , - 1 \\} \\right \\} . \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} A _ { \\Gamma , \\alpha } = \\sum _ { \\substack { \\Gamma _ 1 \\subseteq \\dots \\subseteq \\Gamma _ \\alpha \\subseteq \\Gamma \\\\ c ( \\Gamma _ \\alpha ) = 1 } } ( q - 1 ) ^ { b ( \\Gamma _ \\alpha ) } q ^ { \\sum _ { k = 1 } ^ { \\alpha - 1 } b ( \\Gamma _ k ) } , \\end{align*}"}
-{"id": "552.png", "formula": "\\begin{align*} \\deg _ L D = \\deg ( c _ 1 ( L ) ^ { \\dim X - 1 } \\cap [ D ] ) \\end{align*}"}
-{"id": "7345.png", "formula": "\\begin{align*} \\dot x _ i = \\sum _ { j = 1 } ^ { 2 k + 1 } A _ { i , j } x _ i x _ j + c _ i \\ ; , \\ \\ i = 1 , 2 , \\dots , 2 k + 1 \\ ; , \\end{align*}"}
-{"id": "1297.png", "formula": "\\begin{align*} x _ 0 ^ 2 + D = p ^ { n _ 0 } \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} V ( x ) : = \\left ( { N - 2 \\over 2 } \\right ) ^ 2 | x | ^ { - 2 } . \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{align*} \\hat { \\rho } _ { \\mathrm { e q } } ( \\| p \\| ) & = \\| p \\| \\hat { \\rho } _ { \\mathrm { e q } } ( \\| p \\| ^ 2 / 2 ) \\\\ & = \\widetilde { \\mathcal { N } } ( \\xi _ + ) \\| p \\| ^ { d } \\| p \\| ^ { 2 C / k _ { s c a t t } ^ 2 } \\exp \\left ( - \\dfrac { 1 } { 2 k _ { s c a t t } ^ 2 } \\| p \\| ^ 2 \\right ) . \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{align*} \\begin{cases} \\eta _ t + w _ x + w _ { x x x } + ( \\eta w ) _ x = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , + \\infty ) , \\\\ w _ t + \\eta _ x + \\eta _ { x x x } + w w _ x = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , + \\infty ) , \\\\ \\eta ( x , 0 ) = \\eta _ 0 ( x ) , w ( x , 0 ) = w _ 0 ( x ) , & \\ , \\ , ( 0 , L ) , \\end{cases} \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} \\boxplus ' _ { j \\in J } \\overline { \\rho } _ j = \\overline { \\boxplus ' _ { j \\in J } \\rho _ j } . \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} W ^ 0 _ 3 ( p , p _ 1 , q ) = K _ p ( q , \\bar { q } ) W ^ 0 _ 2 ( \\bar { q } , p _ 1 ) \\end{align*}"}
-{"id": "4952.png", "formula": "\\begin{align*} \\theta ' _ { n + 1 } \\theta _ { n - 1 } - \\theta _ { n + 1 } \\theta _ { n - 1 } ' = ( 2 n - 1 ) \\theta _ n ^ 2 , \\end{align*}"}
-{"id": "4663.png", "formula": "\\begin{align*} D _ D ( X | | Y ) = \\sup \\{ D ( \\Phi ( X ) | | \\Phi ( Y ) ) \\ : \\ \\Phi \\in \\mathcal { P } \\} \\ . \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} \\mbox { \\eqref { e q d e f 2 } i s s a t i s f i e d w i t h $ \\hat \\tau = \\tau _ { \\hat S } $ f o r a l l } x \\in { \\hat S \\backslash \\bigcup _ { n \\in \\N } S _ n } , \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{align*} B = \\{ x \\in \\mathbb { C } ^ { 2 5 6 ^ 2 } : F ( x ) ( k ) = b ( k ) , \\ ; \\forall k \\in \\mathcal { J } \\} , \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} S _ m ( \\delta ^ { + } ) = m \\cdot ( r ^ m , 0 ^ { n - m } ) + ( 1 , \\dots , n ) = ( \\underbrace { r m + 1 , \\dots , r m + m } _ { m } , \\underbrace { m + 1 , \\dots , n } _ { n - m } ) , \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} P _ 0 = \\begin{pmatrix} 0 & 0 & 0 & \\\\ 0 & - D \\frac 1 { k _ 0 } - \\frac 1 { k _ 0 } D & 0 \\\\ 0 & 0 & D \\frac 1 { k _ 0 } + \\frac 1 { k _ 0 } D \\end{pmatrix} . \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{align*} ( \\kappa ^ \\perp ) ^ 2 = s ^ { - 8 } ( s _ 2 ^ 4 - | u _ 4 | ^ 2 ) . \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} z \\left ( t \\right ) = e ^ { - t A } x \\left ( t \\right ) , \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{align*} \\triangle \\alpha _ { c } & = ( \\lambda + k - 2 ) ( \\lambda + n - k ) \\alpha _ c , \\\\ \\triangle \\beta _ { c c } & = ( \\lambda + n - k - 2 ) ( \\lambda + k ) \\beta _ { c c } . \\end{align*}"}
-{"id": "2281.png", "formula": "\\begin{align*} \\dim W ^ u ( a _ { 1 2 } ) + \\dim W ^ u ( a _ { 2 3 } ) + \\dim W ^ u ( a _ { 3 1 } ) = 2 n . \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} { \\partial } _ { x _ l } \\left ( g _ { k t } g ^ { t s } \\left ( { \\partial _ { x _ j } e _ { i s } } { } - { \\partial _ { x _ i } e _ { j s } } \\right ) \\right ) = { \\partial } _ { x _ k } \\left ( g _ { l t } g ^ { t s } \\left ( { { \\partial _ { x _ j } } e _ { i s } } - { { \\partial _ { x _ i } } e _ { j s } } \\right ) \\right ) . \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} | x - 2 \\sqrt { s } z | \\leq | x | + 2 \\sqrt { s } | z | \\leq c t + 2 R ^ { 2 } = \\tilde { c } t - ( \\tilde { c } - c ) ( t - \\frac { 2 R ^ { 2 } } { \\tilde { c } - c } ) \\leq \\tilde { c } t \\end{align*}"}
-{"id": "7666.png", "formula": "\\begin{align*} \\mathrm { P } _ { m , i } & \\approx \\mathrm { P } _ { t , 1 } + \\frac { 4 ( \\lambda _ c \\pi ) ^ { t } } { ( t - m - 1 ) ! ( m - 1 ) ! } \\sum ^ { t - m - 1 } _ { p = 0 } ( - 1 ) ^ p { t - m - 1 \\choose p } \\\\ & \\times \\sum ^ { N } _ { l = 1 } \\frac { \\pi \\left ( \\tau _ 2 - \\tau _ 1 \\right ) } { 2 N } f _ m \\left ( \\frac { \\tau _ 2 - \\tau _ 1 } { 2 } w _ l + \\frac { \\tau _ 2 + \\tau _ 1 } { 2 } \\right ) \\sqrt { 1 - w _ l ^ 2 } , \\end{align*}"}
-{"id": "9223.png", "formula": "\\begin{align*} x ^ { ( m ) } \\in \\Z , t _ m + x ^ { ( m ) } \\in 2 \\Z , \\mbox { f o r $ m = 1 , 2 , \\dots , M $ } , \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} q ^ + ( t ) = t ^ { a _ k - a _ 1 } q ^ - ( 1 / t ) \\ , . \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} E _ d ( P ) = \\sum _ { k = 0 } ^ { d - 1 } \\frac { \\langle P , \\psi _ d ^ k \\rangle } { q ^ k } . \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} ( \\widetilde { \\rho ^ \\pi } ( s ) v , w ) = ( v , \\widetilde { \\rho ^ \\pi } ( s ^ * ) w ) . \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 1 } u _ { i , j } ( d _ { s p } u ^ * ) \\nu _ j = 0 \\end{align*}"}
-{"id": "5158.png", "formula": "\\begin{align*} \\sum _ { A \\subset P , | A | = a } ( - 1 ) ^ { { \\rm s i g n } ( \\sigma , \\tau , P , A ) } \\frac { \\partial ^ { k ' } X _ { \\eta _ 1 ( \\sigma , \\tau , P , A ) } ( s ) } { \\partial s ^ { k ' } } X _ { \\eta _ 2 ( \\sigma , \\tau , P , A ) } ( s ) = 0 . \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} E _ { 2 } ( q ) = 1 - 2 4 \\sum _ { j = 1 } ^ { \\infty } \\sigma _ 1 ( j ) q ^ j \\ \\ E _ { 4 } ( q ) = 1 + 2 4 0 \\sum _ { j = 1 } ^ { \\infty } \\sigma _ 3 ( j ) q ^ j \\ \\ E _ 6 ( q ) = 1 - 5 0 4 \\sum _ { j = 1 } ^ { \\infty } \\sigma _ 5 ( j ) q ^ j \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} \\mathrm { K L } ( A , B ) & = \\sum _ { j = 1 } ^ N q ' ( z _ i = 1 ) \\sum _ { i = 1 } ^ M ( A _ 0 B _ 0 ) _ { i j } \\log \\frac { ( A _ 0 B _ 0 ) _ { i j } } { ( A B ) _ { i j } } \\\\ & \\sim \\sum _ { j = 1 } ^ N q ' ( z _ i = 1 ) \\sum _ { i = 1 } ^ M ( ( A B ) _ { i j } - ( A _ 0 B _ 0 ) _ { i j } ) ^ 2 \\\\ & \\sim \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ M ( ( A B ) _ { i j } - ( A _ 0 B _ 0 ) _ { i j } ) ^ 2 = \\| A B - A _ 0 B _ 0 \\| ^ 2 . \\end{align*}"}
-{"id": "7085.png", "formula": "\\begin{align*} & f _ L ( a _ L ) > f _ L ( x ) , ( \\forall x \\in \\R _ { \\ge 0 } \\backslash \\{ a _ L \\} , \\ L \\in \\N ) , \\\\ & f ( a ) > f ( x ) , ( \\forall x \\in \\R _ { \\ge 0 } \\backslash \\{ a \\} ) , \\\\ & \\frac { d } { d x } f _ L ( a _ L ) = 0 , ( \\forall L \\in \\N ) , \\\\ & \\frac { d ^ 2 } { d x ^ 2 } f ( a ) < 0 , g ( a ) \\neq 0 . \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} & p _ U ( u _ 1 ) = p _ U ( u _ 2 ) = \\frac { p _ { 1 1 } ^ { n / 2 } p _ { 2 2 } ^ { n / 2 } } { 2 ( p _ { 1 1 } ^ { n / 2 } p _ { 2 2 } ^ { n / 2 } + p _ { 1 2 } ^ { n / 2 } p _ { 2 1 } ^ { n / 2 } ) } \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} d \\tilde c - x d x / y , \\omega _ \\alpha : = f _ \\alpha \\cdot { d x \\over y } , \\alpha \\geq - 1 , \\end{align*}"}
-{"id": "3520.png", "formula": "\\begin{align*} \\partial _ t p = \\vec { \\kappa } ( p ) \\end{align*}"}
-{"id": "6897.png", "formula": "\\begin{align*} \\int _ { [ 0 , 1 ] ^ { | \\gamma | } } Q _ { \\alpha } ( t ) d t = \\frac { \\alpha ! } { ( \\alpha + \\gamma ) ! } \\ , . \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} K : = ( \\log n ) ^ { 1 / 1 2 } . \\end{align*}"}
-{"id": "6816.png", "formula": "\\begin{align*} \\nabla _ { t } ( R ^ P ( \\psi , \\psi ) ( N s ) ^ { 2 - n } ) \\partial _ t \\cdot \\psi = & ( \\nabla _ { d \\phi ( \\partial _ t ) } R ^ P ) ( \\psi , \\psi ) ( N s ) ^ { 2 - n } \\partial _ t \\cdot \\psi + 2 R ^ P ( \\nabla _ t \\psi , \\psi ) ( N s ) ^ { 2 - n } \\partial _ t \\cdot \\psi \\\\ & + ( 2 - n ) R ^ P ( \\psi , \\psi ) ( N s ) ^ { 2 - n } \\partial _ t \\log N \\partial _ t \\cdot \\psi \\\\ & + ( 2 - n ) \\dot s s ^ { 1 - n } N ^ { 2 - n } R ^ P ( \\psi , \\psi ) \\partial _ t \\cdot \\psi . \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} & \\tilde F _ { E , \\omega } ( p , z ) e ^ { - t z } \\tilde F _ { E , \\omega } ( p + p ' , z ' ) e ^ { - ( t + t ' ) z ' } - \\tilde F _ { E , \\omega } ( p ' , z ' ) e ^ { - t ' z ' } \\tilde F _ { E , \\omega } ( p , z + z ' ) e ^ { - t ( z + z ' ) } \\\\ & + \\tilde F _ { E , \\omega } ( p + p ' , z + z ' ) e ^ { - ( t + t ' ) ( z + z ' ) } \\tilde F _ { E , \\omega } ( p ' , - z ) e ^ { t ' z } = 0 \\end{align*}"}
-{"id": "9281.png", "formula": "\\begin{align*} t _ { \\mathrm { w } , 1 } ( n ) & = \\Delta t _ { \\mathrm { w } _ { 1 , 1 } } + \\Delta t _ { \\mathrm { w } _ { 1 , 2 } } + \\Delta t _ { \\mathrm { w } _ { 1 , 3 } } + \\Delta t _ { \\mathrm { w } _ { 1 , 4 } } \\\\ & = g _ \\mathrm { U L } ^ { - 1 } N l _ \\mathrm { n } + g _ \\mathrm { D L } ^ { - 1 } N l _ r + g _ \\mathrm { U L } ^ { - 1 } l _ \\mathrm { n } + g _ \\mathrm { D L } ^ { - 1 } n l _ \\mathrm { b } . \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} \\hat s _ V ( z ) : = \\prod _ { j = 1 } ^ m { z - z _ j \\over z + z _ j } s _ V ( z ) . \\end{align*}"}
-{"id": "3361.png", "formula": "\\begin{align*} r _ { k } = r _ { k - 1 } + \\frac 1 { \\varphi ( H ( r _ { k - 1 } ) ) } \\end{align*}"}
-{"id": "9128.png", "formula": "\\begin{align*} { { r } ^ * } = \\frac { 3 - \\sqrt { 9 + 4 \\frac { \\rho _ t } { c } + 4 \\frac { \\rho _ t ^ 2 } { c ^ 2 } } } { - 2 ( 1 + \\frac { \\rho _ t } { c } ) } c \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} \\lambda \\xi _ 1 = \\xi _ 2 c ' \\ \\ \\lambda \\xi _ 2 = \\xi _ 1 a ' . \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} & \\int _ D \\psi _ 1 ( n _ 1 ; \\lambda _ 1 ) \\cdots \\psi _ { k - 1 } ( n _ { k - 1 } ; \\lambda _ { k - 1 } ) \\ , \\psi _ { k + 1 } ( n _ { k + 1 } ; \\lambda _ { k + 1 } ) \\cdots \\psi _ g ( n _ g ; \\lambda _ g ) \\ , h ( \\lambda _ 2 , \\dots , \\lambda _ g ) \\ , d \\lambda _ 2 \\cdots d \\lambda _ g \\\\ & = 0 \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} f _ { \\mu } ( z _ 1 , \\dots , z _ n ; q , t ) = \\Omega _ { \\mu } ( q , t ) \\times { \\rm T r } \\Big ( A _ { \\mu _ 1 } ( z _ 1 ) \\dots A _ { \\mu _ n } ( z _ n ) S \\Big ) , \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} h = \\begin{pmatrix} A _ { 1 1 } & A _ { 1 2 } & A _ { 1 3 } \\\\ A _ { 2 1 } & A _ { 2 2 } & A _ { 2 1 } S _ r \\\\ S _ r A _ { 1 3 } S _ r & S _ r A _ { 1 2 } & S _ r A _ { 1 1 } S _ r \\end{pmatrix} \\end{align*}"}
-{"id": "721.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Delta _ { n } ^ { 1 / n } = { \\rm M } ( P ) . \\end{align*}"}
-{"id": "3649.png", "formula": "\\begin{align*} \\lim _ { m \\longrightarrow + \\infty } \\sum _ { k = 0 } ^ { k = m } \\dfrac { \\phi ( \\lambda _ k ) } { m + 1 } = \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { C P } ( 1 ) } \\phi ( f ( x ) ) d \\Omega . \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} | U _ n | = \\prod _ { k = 1 } ^ { n } ( q ^ { | \\mu | } - q ^ { \\mu _ 1 + \\cdots + \\mu _ m + k - 1 } ) . \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{align*} ( \\mathcal { X } _ { \\gamma } ) _ { \\ell } ^ + = T _ { \\ell } ^ { \\phantom { j } } ( \\mathcal { X } _ { \\gamma } ) _ { \\ell } ^ - \\end{align*}"}
-{"id": "6341.png", "formula": "\\begin{align*} \\Omega _ k U _ k & = \\left ( P _ { \\nu _ k ^ { - 1 } } ^ T \\widetilde { \\Omega } _ k P _ { \\nu _ k ^ { - 1 } } \\right ) P _ { \\nu _ k ^ { - 1 } } ^ T \\widetilde { U } _ k = P _ { \\nu _ k ^ { - 1 } } ^ T \\widetilde { \\Omega } _ k \\widetilde { U } _ k \\\\ & = P _ { \\nu _ k ^ { - 1 } } ^ T \\left ( - \\lambda _ k \\widetilde { \\partial G } ( U _ k ) \\right ) = - \\lambda _ k P _ { \\nu _ k ^ { - 1 } } ^ T P _ { \\nu _ k ^ { - 1 } } \\cdot \\partial G ( U _ k ) \\\\ & = - \\lambda _ k \\partial G ( U _ k ) , \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} \\begin{cases} \\nabla G ( { \\bf u } ) - \\sum \\limits _ { 1 \\leq a \\leq p } \\sigma _ { a a } ( { \\bf u } ) \\nabla F _ { a a } ( { \\bf u } ) - \\sum \\limits _ { 1 \\leq b < c \\leq p } \\sigma _ { b c } ( { \\bf u } ) \\nabla F _ { b c } ( { \\bf u } ) = { \\bf 0 } \\\\ U ^ T U = \\mathbb { I } _ p . \\end{cases} \\end{align*}"}
-{"id": "526.png", "formula": "\\begin{align*} y _ { 0 } \\left ( x \\right ) = y _ { a } \\frac { \\left ( \\psi \\left ( x \\right ) - \\psi \\left ( a \\right ) \\right ) ^ { \\gamma - 1 } } { \\Gamma \\left ( \\gamma \\right ) } , \\gamma = \\alpha + \\beta \\left ( 1 - \\alpha \\right ) \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} F ( x , y ) = \\begin{cases} 1 , & x \\in \\{ 0 , 1 \\} , \\\\ - | x - y | , & x \\in ( 0 , 1 ) . \\end{cases} \\end{align*}"}
-{"id": "524.png", "formula": "\\begin{align*} y \\left ( x \\right ) = \\frac { \\left ( \\psi \\left ( x \\right ) - \\psi \\left ( a \\right ) \\right ) ^ { \\gamma - 1 } } { \\Gamma \\left ( \\gamma \\right ) } I _ { a + } ^ { \\left ( 1 - \\beta \\right ) \\left ( 1 - \\alpha \\right ) ; \\psi } f \\left ( a \\right ) + I _ { a + } ^ { \\alpha ; \\psi } f \\left ( x , y \\left ( x \\right ) \\right ) . \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{align*} \\Delta E _ c = \\alpha _ * ^ 2 \\left ( \\dfrac { \\beta ^ { ( 4 ) } _ c ( \\kappa , \\alpha = 0 ) } { \\| p \\| ^ { - 4 } } + O \\left ( M ^ { - 1 / 2 } \\| p \\| ^ { - 5 } \\right ) ) \\right ) + o ( \\alpha _ * \\sqrt { M } ) . \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} \\bigl [ s [ y _ i , x _ j ] , y _ { i + 1 } \\dots y _ { i ' - 1 } \\bigr ] \\equiv \\sum _ { m = i + 1 } ^ { i ' - 1 } y _ { i + 1 } \\dots y _ { m - 1 } [ s , y _ m ] [ y _ i , x _ j ] y _ { m + 1 } \\dots y _ { i ' - 1 } \\pmod { I } \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} \\eta \\big ( p _ { Y | X } \\big ) = \\sup _ { U V \\rightarrow X \\rightarrow Y } \\frac { I ( U ; Y | V ) } { I ( U ; X | V ) } . \\end{align*}"}
-{"id": "7308.png", "formula": "\\begin{align*} r _ { , k } = \\sqrt { p _ k } x _ k + \\hat { \\mathbf { f } } _ k { } \\mathbf { \\Omega } \\mathbf { z } + \\hat { \\mathbf { f } } _ k \\mathbf { n } _ q , \\end{align*}"}
-{"id": "111.png", "formula": "\\begin{align*} \\omega _ I = \\sum _ i d x _ i \\wedge d y _ i + d p _ i \\wedge d q _ i . \\end{align*}"}
-{"id": "1301.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } G ( S ^ { i } x ) \\leq \\sum _ { i = 0 } ^ { n - 1 } F ( S ^ { i } x ) + n \\epsilon \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{align*} 1 = \\nu _ { d , 1 } ( G ( d , 1 ) ) \\geq \\nu _ { d , 1 } ( \\cup _ { k = 1 } ^ d G _ k ) = d \\nu _ { d , 1 } ( G _ 1 ) , \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} \\gamma ( \\varepsilon ) = q ^ { ( N - 1 ) / 2 } q ^ { - ( d _ { m + 1 } + \\cdots + d _ N ) / 2 } \\gamma ( \\varepsilon + \\widehat { m } ) \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} \\lambda \\cdot ( x _ M ^ { k - n } \\times \\omega _ { N } ) = f ^ * ( x _ M ^ { k - n } \\times \\omega _ N ) . \\end{align*}"}
-{"id": "7679.png", "formula": "\\begin{align*} \\mathrm { P } _ { m } ^ i = & 1 - \\mathrm { P } ( _ { m } ^ l > R _ l , \\forall l \\in \\{ 0 , \\cdots , i \\} ) \\\\ = & \\mathrm { P } \\left ( L ( | | x _ m | | ) > \\min \\left \\{ \\frac { \\rho \\zeta _ { l } } { { \\epsilon } _ l } , \\forall l \\in \\{ 0 , \\cdots , i \\} \\right \\} \\right ) . \\end{align*}"}
-{"id": "1388.png", "formula": "\\begin{align*} J \\bigl [ u ^ { \\ast } \\bigr ] & = \\xi _ { 0 , T } ( u ^ { \\ast } ) \\\\ & = \\bigl [ \\xi _ { 0 , T } ^ 1 ( u ^ { \\ast } ) , \\xi _ { 0 , T } ^ 2 ( u ^ { \\ast } ) , \\ldots , \\xi _ { 0 , T } ^ n ( u ^ { \\ast } ) \\bigr ] ^ T , \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} \\sqrt { \\eta _ { n } ( x ) ^ { 2 } + m ^ { 2 } } - \\tau _ { n } ( | \\cdot | ^ { - 1 } \\star \\rho _ { n } ) ( x ) + V ( x ) - \\mu _ { n } \\begin{cases} = 0 & \\rho _ { n } ( x ) > 0 , \\\\ \\geq 0 & \\rho _ { n } ( x ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} \\left ( \\sum _ { j = 1 } ^ { d _ { \\ell } } \\lambda _ { j } ^ { \\ell } h _ { \\ell } ^ { i _ { j } ^ { \\ell } } \\right ) = \\lambda ^ { \\ell } - g _ { \\ell } ^ { j _ { 0 } } \\end{align*}"}
-{"id": "3145.png", "formula": "\\begin{align*} { } \\frac { \\partial S } { \\partial \\beta } = 0 = E - \\frac { 1 } { \\beta ^ 2 } \\int _ { 0 } ^ { \\beta B } \\ln ( 1 + e ^ { - \\alpha } e ^ { - x } ) d x + \\frac { B } { \\beta } \\ln ( 1 + e ^ { - \\alpha - \\beta B } ) \\end{align*}"}
-{"id": "2866.png", "formula": "\\begin{gather*} \\Delta '^ { , + } = \\big \\{ \\alpha _ 1 , \\alpha _ 2 , \\alpha _ 3 , \\alpha _ 1 + \\alpha _ 2 , \\alpha _ 2 + \\alpha _ 3 , \\alpha _ 1 + \\alpha _ 2 + \\alpha _ 3 , \\alpha _ 1 + 2 \\alpha _ 2 + \\alpha _ 3 \\big \\} , \\end{gather*}"}
-{"id": "6425.png", "formula": "\\begin{align*} \\Delta x = 0 , \\nabla x = 0 , \\Delta y = \\Delta d , \\nabla y = \\nabla d \\end{align*}"}
-{"id": "71.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { \\infty } \\int _ { x _ 1 } ^ \\infty \\cdots \\int _ { x _ { t - 1 } } ^ { \\infty } \\prod _ { j \\in [ t ] } x _ j ^ { - \\tau + \\zeta _ j } \\prod _ { i = t + 1 } ^ s h ( i , \\boldsymbol { x } ) \\dd x _ t \\cdots \\dd x _ 1 , \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} \\chi ( X ) = \\frac { 1 } { 2 } \\chi ( Z ) . \\end{align*}"}
-{"id": "480.png", "formula": "\\begin{align*} q _ 2 z - a & = \\frac { 1 } { 2 } + i q _ 2 y = R e ^ { i \\theta } \\\\ q _ 2 z - a - 1 & = - \\frac { 1 } { 2 } + i q _ 2 y = - R e ^ { - i \\theta } . \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} H ( \\nu , \\mathcal { E } | \\mathcal { F } ) : = \\sum _ { F \\in \\mathcal { F } : \\nu ( F ) > 0 } \\nu ( F ) H ( \\nu _ F , \\mathcal { E } ) \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{align*} d Z _ j = ( \\gamma _ { j + 1 } - \\gamma _ j ) d t + \\sigma _ { j + 1 } d B _ { j + 1 } - \\sigma _ j d B _ { j } + 2 d L _ { j } - d L _ { j + 1 } - d L _ { j - 1 } \\ , . \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} & = \\frac { x ^ 5 ( 1 - 2 x + x ^ 2 - x ^ 3 ) } { ( 1 - x ) ^ 6 ( 1 - 2 x ) ^ 2 } + \\frac { x ^ 6 ( 1 - x - x ^ 2 ) } { ( 1 - x ) ^ 5 ( 1 - 2 x ) ^ 3 } + \\frac { x ^ 5 ( 2 - x ) } { ( 1 - x ) ^ 3 ( 1 - 2 x ) ( 1 - 3 x + x ^ 2 ) } \\\\ & \\quad + \\frac { x ^ 6 } { ( 1 - x ) ^ 2 ( 1 - 2 x ) ^ 2 ( 1 - 3 x + x ^ 2 ) } + \\frac { x ^ 3 } { ( 1 - 2 x ) ^ 3 } \\\\ & = \\frac { x ^ 3 ( 1 - 9 x + 3 7 x ^ 2 - 9 1 x ^ 3 + 1 4 2 x ^ 4 - 1 4 1 x ^ 5 + 9 0 x ^ 6 - 3 6 x ^ 7 + 6 x ^ 8 ) } { ( 1 - x ) ^ 6 ( 1 - 2 x ) ^ 3 ( 1 - 3 x + x ^ 2 ) } . \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{align*} d q _ t = & \\tilde \\gamma ^ { - 1 } ( t ) \\left ( - \\nabla _ q V ( t , q _ t ) + \\tilde F ( t , q _ t ) \\right ) d t \\\\ & + \\tilde S ( t , q _ t ) d t + \\tilde \\gamma ^ { - 1 } ( t ) \\sigma ( t , q _ t ) \\circ d W _ t \\\\ = & \\tilde \\gamma ^ { - 1 } ( t ) ( - \\nabla _ q V ( t , q _ t ) + \\tilde F ( t , q _ t ) ) d t + ( \\tilde \\gamma ^ { - 1 } \\sigma ) ( t , q _ t ) d W _ t , \\end{align*}"}
-{"id": "4455.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } d ( \\alpha _ j , \\alpha _ { i , j } ) = 0 . \\end{align*}"}
-{"id": "5297.png", "formula": "\\begin{align*} \\abs { \\chi ( t ) } _ p \\cdot \\abs { x } _ J = \\abs { t . x } _ J \\leq \\abs { x } _ J , \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{align*} ( \\lambda + \\mathcal { A } _ p ) ^ { - 1 } = \\Phi ^ { - 1 } A _ p ^ { \\frac { 1 } { 2 } } ( \\lambda + A _ p ) ^ { - 1 } \\Phi \\big [ A _ { p ^ { \\prime } } ^ { - \\frac { 1 } { 2 } } \\big ] ^ * . \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} t ^ { - 1 } \\alpha _ t = \\sum _ { j = 0 } ^ \\infty A _ { j , t } t ^ { ( 1 - 2 j ) / 3 } \\end{align*}"}
-{"id": "2988.png", "formula": "\\begin{align*} ( - 1 ) ^ n \\mu ( n / 2 d ) \\left ( \\binom { 4 d - 1 } { 2 d - 1 } - ( - 1 ) ^ d \\binom { 2 d - 1 } { d - 1 } \\right ) . \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{align*} k [ S ^ \\infty ] = \\oplus _ { n = 0 } ^ { \\infty } k [ S _ n ] \\end{align*}"}
-{"id": "7892.png", "formula": "\\begin{align*} i \\frac { \\partial } { \\partial t } w _ { \\tau } ( t ; \\rho ) = H ( t ; \\rho ) w _ { \\tau } ( t ; \\rho ) + \\int _ 0 ^ 1 \\frac { \\partial H } { \\partial \\rho } ( t ; \\rho + \\theta \\tau ) d \\theta \\ , u ( t ; \\rho + \\tau ) . \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} \\omega = \\frac { i } { 2 } \\sum _ { k , l = 1 } ^ n h _ { k l } d z _ k \\wedge d \\overline { z } _ l \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} \\bigl [ v , [ a _ 1 , a _ 2 ] \\bigr ] = [ v , a _ 1 , a _ 2 ] - [ v , a _ 2 , a _ 1 ] \\in W . \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} \\sigma _ x = \\begin{bmatrix} 1 & 0 \\\\ 0 & x \\end{bmatrix} , \\gamma _ { x } = \\begin{bmatrix} 0 & 0 \\\\ x & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} \\begin{bmatrix} B _ { 1 } ^ \\top \\otimes A _ 1 & - D _ { 1 } ^ \\top \\otimes C _ 1 \\\\ & \\ddots & \\ddots \\\\ & & B _ { r - 1 } ^ \\top \\otimes A _ { k - 1 } & - D _ { r - 1 } ^ \\top \\otimes C _ { r - 1 } \\\\ - D _ { r } ^ \\top \\otimes C _ { r } & & & B _ { r } ^ \\top \\otimes A _ { r } \\\\ \\end{bmatrix} { \\cal X } = { \\cal E } , \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} H = \\{ A = [ a _ { i j } ] \\in \\mathfrak { g l } _ n ( \\C ) : a _ { i j } = 0 i > h ( j ) \\} . \\end{align*}"}
-{"id": "8324.png", "formula": "\\begin{align*} z _ { t t } + i = - \\nabla P = - \\frac { \\partial _ \\alpha P } { | z _ \\alpha | } e ^ { i \\theta } + i a e ^ { i \\theta } = - ( \\frac { \\partial _ \\alpha } { | z _ \\alpha | } ) ^ 2 \\theta e ^ { i \\theta } + i a e ^ { i \\theta } . \\end{align*}"}
-{"id": "9053.png", "formula": "\\begin{align*} \\Phi _ { g _ 1 } ( m ) = g _ 1 \\cdot m = \\frac { m - \\frac 1 2 \\eta \\| m \\| ^ 2 } { 1 - \\eta ^ T m + \\frac 1 4 \\| \\eta \\| ^ 2 \\| m \\| ^ 2 } \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} \\overline { \\sum _ { r \\in X \\cup X ^ 2 \\cup X ^ 3 } \\alpha _ r \\gamma ( r ) } \\in B \\cap \\sp { \\overline { Y } } = \\sp { \\overline { Z } } . \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} A ( \\pi ^ i ) = \\begin{cases} \\{ 1 , 2 , \\dots , i , n + 2 \\} & \\textrm { i f $ 1 \\le i \\le k - 1 $ , } \\\\ \\{ 1 , 2 , \\dots , k - 1 , n + 2 \\} & \\textrm { i f $ i = n + 1 $ . } \\end{cases} \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{align*} g ( x ) = \\begin{cases} \\operatorname { s g n } ( x ) , & | x | > \\delta , \\\\ \\frac { x } { \\delta } , & | x | \\leq \\delta . \\end{cases} \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} B _ 3 = \\{ E _ { \\ell _ 1 , \\ell _ 2 , \\ell _ 3 } ~ : ~ 2 \\leq \\ell _ 1 \\leq s , \\ ; 0 \\leq \\ell _ 2 , \\ell _ 3 \\leq s - 2 \\} \\subset \\ker \\partial _ 2 . \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} H \\left ( \\nu , \\mu \\right ) = \\prod _ { x \\in \\vec { x } ( \\mu ) } \\prod _ { i \\leq x } t ^ { \\nu _ i } \\end{align*}"}
-{"id": "8211.png", "formula": "\\begin{align*} F _ n \\left ( \\widetilde { I } _ i \\right ) = \\bigl [ F _ n ( y _ { n + 1 + i ( n + 2 ) } ) , F _ n ( y _ { n + 2 + i ( n + 2 ) } ) \\bigr ] = \\widetilde { I } _ { i + 1 } \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{align*} C _ { \\alpha { } } D _ { \\theta { } } ^ { \\alpha { } } \\phi ( x ) + A x \\phi ( x ) = E \\phi ( x ) , ~ x \\geq { } 0 \\ . \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} \\frac { | - 1 + z _ { 1 , n } | } { | z _ { j , n } | } < \\kappa ( 1 , a ) = \\frac { \\left | 1 - \\exp \\bigl ( \\frac { \\pi } { a } \\bigr ) \\right | \\exp \\bigl ( \\frac { - \\pi } { a } \\bigr ) } { \\exp \\bigl ( \\frac { \\pi } { a } \\bigr ) + \\left | 1 - \\exp \\bigl ( \\frac { \\pi } { a } \\bigr ) \\right | } , \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} \\mathbf { s } _ 0 : = \\{ X _ { p _ 0 , q _ 0 } \\mid p _ 0 , q _ 0 \\geq 0 \\} \\cup \\{ Y _ { p _ 1 , q _ 1 } , Z _ { p _ 2 , q _ 2 } \\mid p _ i , q _ i \\geq 3 \\} . \\end{align*}"}
-{"id": "1290.png", "formula": "\\begin{align*} \\widetilde { \\chi } ( \\rho ( D _ i ) ) \\overline { \\widetilde { \\chi } ( \\rho ( D - D _ i ) ) } = \\chi ( D _ i ) \\overline { \\chi ( D - D _ i ) } = - \\lambda _ i \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{align*} f _ \\phi ( z ) = \\phi ( \\tilde { g } _ \\infty ) \\prod _ { i = 1 } ^ 2 \\tilde { j } ( \\tilde { g } _ { \\infty , i } \\sqrt { - 1 } ) ^ { 2 \\kappa + 1 } , z \\in \\mathfrak { h } ^ 2 , \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} \\left ( \\sum _ { k = 1 } ^ { n } \\sum _ { j = 1 } ^ { d _ { k } } \\lambda _ { j } ^ { k } g _ { \\ell } ^ { i _ { j } ^ { k } } \\right ) + g _ { \\ell } ^ { j _ { 0 } } & = \\left ( \\sum _ { j = 1 } ^ { d _ { \\ell } } \\lambda _ { j } ^ { \\ell } g _ { \\ell } ^ { i _ { j } ^ { \\ell } } \\right ) + g _ { \\ell } ^ { j _ { 0 } } \\\\ & = \\left ( \\lambda ^ { \\ell } - g _ { \\ell } ^ { j _ { 0 } } \\right ) + g _ { \\ell } ^ { j _ { 0 } } = \\lambda ^ { \\ell } \\in \\mathbb { Q } . \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{align*} \\{ H ^ { 0 , 1 } \\lrcorner \\ , \\partial _ B + \\partial _ B H ^ { 0 , 1 } \\lrcorner \\} \\phi = \\{ H ^ { 1 , 0 } \\lrcorner \\ , \\bar \\partial _ B + \\bar \\partial _ B H ^ { 1 , 0 } \\lrcorner \\} \\phi = 0 . \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i \\in J } { a _ i x _ i } \\right \\| & = \\left \\| \\sum _ { i \\in J _ { n - 1 } } { a _ i x _ i } - ( - a _ m x _ m ) \\right \\| \\geq \\left \\| F \\left ( \\sum _ { i \\in J _ { n - 1 } } { a _ i x _ i } \\right ) - F ( - a _ m x _ m ) \\right \\| \\\\ & = \\left \\| \\sum _ { i \\in J _ { n - 1 } } a _ i z _ i - a _ m z _ m \\right \\| = \\sum _ { i \\in J } { | a _ i | } . \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ s m _ j \\mbox { R e } ( \\lambda _ j ) \\neq \\mbox { R e } ( \\lambda _ n ) , n = s + 1 , . . . , N , 2 \\leq \\sum _ { j = 1 } ^ s m _ j \\leq \\Sigma ( E ) , m _ j \\in \\mathbb { N } , \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} 0 & = \\frac { \\chi _ 2 ( - a ) } { ( q _ 2 z - a ) ^ k } + \\frac { \\chi _ 2 ( - a - 1 ) } { ( q _ 2 z - a - 1 ) ^ k } = \\frac { \\chi _ 2 ( - a ) e ^ { - i \\theta k } + ( - 1 ) ^ k \\chi _ 2 ( - a - 1 ) e ^ { i \\theta k } } { R ^ k } . \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathbb { Z } } L _ i [ H \\left ( \\cdot , \\mu \\right ) ] ( \\nu ) = t ^ { - \\chi ( \\vec { x } , \\vec { y } ) } \\sum _ { i \\in \\{ z , z + l \\} } M _ i \\left [ H ( \\nu , \\cdot ) \\right ] ( \\mu ^ { * } ) . \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} Q _ { 1 , 0 } & = \\{ \\theta + i J \\theta | \\ \\theta \\in Q _ C ^ * , \\left . \\theta \\right | _ Q \\in Q ^ * \\} \\\\ Q _ { 0 , 1 } & = \\{ \\theta - i J \\theta | \\ \\theta \\in Q _ C ^ * , \\left . \\theta \\right | _ Q \\in Q ^ * \\} , \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{align*} d ( t _ j ) = \\begin{pmatrix*} [ r ] 0 & t _ j \\\\ - t _ j & 0 \\end{pmatrix*} . \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} Z ( e , m ) : = \\sum _ { i = 1 } ^ m { \\mathbf 1 } _ { ( X _ { i - 1 } , X _ i ) = e } , e \\in E ( G ) , \\ m \\ge 1 , \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} [ g _ 1 , g _ 2 , \\dots , g _ n ] = 1 \\mbox { f o r a l l } g _ i \\in G \\iff [ x _ 1 , x _ 2 , \\dots , x _ n ] = 1 \\mbox { f o r a l l } x _ i \\in X . \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\mu } = \\left \\{ \\nu : \\nu \\prec \\mu , \\ y _ { \\nu } ( w ) = y _ { \\mu } ( w ) \\ \\ q = t ^ { - m } \\right \\} . \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} n _ \\varepsilon ( x , y ) : = B _ 0 \\cdot \\varrho \\left ( \\sum _ { i = 1 } ^ d t _ i ( x _ i ) + \\varrho \\left ( \\frac { y } { B _ 0 } \\right ) - d \\right ) - B _ 0 \\cdot \\varrho \\left ( \\sum _ { i = 1 } ^ d t _ i ( x _ i ) + \\varrho \\left ( - \\frac { y } { B _ 0 } \\right ) - d \\right ) . \\end{align*}"}
-{"id": "2760.png", "formula": "\\begin{align*} \\{ 1 + a _ { 1 , p , 0 } S ^ 1 T ^ p , T \\} \\xrightarrow { \\phi _ T } \\left ( \\begin{cases} [ - a _ { 1 , p , 0 } ] ^ m \\textrm { i f } n = p m \\\\ 0 \\textrm { o t h e r w i s e } \\end{cases} \\right ) _ { ( m , n ) \\in \\mathbb { J } _ T } . \\end{align*}"}
-{"id": "134.png", "formula": "\\begin{align*} \\varphi _ \\infty : = \\dot \\Phi _ \\infty - [ \\Phi _ \\infty \\wedge \\gamma _ \\infty ] & = \\begin{pmatrix} 0 & \\frac 1 2 | q | _ k ^ { - 1 / 2 } \\dot q \\\\ \\tfrac { 1 } { 2 } | q | _ k ^ { 1 / 2 } \\dot q / q & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} \\alpha _ 2 \\delta \\alpha _ 1 \\gamma ( \\mathbf { s } ) & \\subseteq \\{ ( a b ^ m ) ^ 2 , ( a b ^ m ) ^ 3 , b ^ 2 , b ^ 3 \\} ^ { { + } } \\\\ & \\subseteq \\{ a b ^ m a , ( a b ^ m ) ^ 2 a , b \\} ^ { { + } } = : \\mathbf { s } _ 1 . \\end{align*}"}
-{"id": "1607.png", "formula": "\\begin{align*} \\frac { ( x _ { t , u , s } - x _ { t , u , 0 } ) } { d _ t } = s ( 1 + o ( 1 ) ) \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{align*} \\int _ \\Omega K ( x ) \\delta _ i ^ { \\frac { n + 2 } { n - 2 } } \\frac { 1 } { \\lambda _ i } \\frac { \\partial \\delta _ i } { \\partial ( a _ i ) _ k } \\ , d x & = ( n - 2 ) \\mathrm { s g } \\bigl [ ( a _ i - z _ { j _ i } ) _ k \\bigr ] \\frac { \\bigl | ( a _ i - z _ { j _ i } ) _ k \\bigr | ^ { \\beta - 1 } } { \\lambda _ i } b _ k c _ 5 + o \\bigl ( \\frac { 1 } { \\lambda _ i ^ { n - 2 } } + \\frac { \\bigl | \\nabla K ( a _ i ) \\bigr | } { \\lambda _ i } \\bigr ) . \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} f ( k ) \\leq f ( x ^ * ) = \\left ( \\log c e n - ( \\ell + 1 ) \\left ( \\frac { \\log c e n } { \\ell + 1 } - \\frac { 1 } { \\ln 2 } \\right ) \\right ) 2 ^ { \\frac { \\log c e n } { \\ell + 1 } - \\frac { 1 } { \\ln 2 } } = \\frac { \\ell + 1 } { 2 ^ { 1 / \\ln 2 } \\ln 2 } ( c e n ) ^ { \\frac { 1 } { \\ell + 1 } } . \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\left | z _ 1 ^ \\frac { 1 } { 2 } - n ^ \\frac { 1 } { 4 } \\right | = \\frac { | z _ 1 - n ^ \\frac { 1 } { 2 } | } { 2 | z _ 1 ^ \\frac { 1 } { 2 } + n ^ \\frac { 1 } { 4 } | } < \\frac { \\frac { z _ 2 - 1 } { 2 } + 0 . 6 5 2 } { 2 | z _ 1 ^ \\frac { 1 } { 2 } + n ^ \\frac { 1 } { 4 } | } \\le \\frac { \\frac { z _ 2 - 1 } { 2 } + 0 . 6 5 2 } { 2 | \\ell ( z _ 2 ) ^ \\frac { 1 } { 2 } + \\ell ( \\ell ( z _ 2 ) ) ^ \\frac { 1 } { 4 } | } . \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} \\frac { \\partial \\bar u ^ { \\varepsilon , N } _ t } { \\partial x _ { \\ell } } ( \\cdot ) = { \\displaystyle \\frac { 1 } { N \\varepsilon } \\sum _ { i = 1 } ^ N \\frac { \\partial K _ { \\varepsilon } } { \\partial x _ { \\ell } } ( \\cdot - \\bar \\xi ^ { i } _ t ) \\bar V _ t \\big ( \\bar \\xi ^ { i } , \\bar u ^ { \\varepsilon , N } ( \\bar \\xi ^ { i } ) , \\nabla \\bar u ^ { \\varepsilon , N } ( \\bar \\xi ^ { i } ) \\big ) } \\ , l = 1 , \\cdots , d \\ , \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} B ^ { - 1 } A = ( L _ { x } ^ { - 1 } L _ { x \\alpha } , R _ { x } ^ { - 1 } R _ { x \\phi ^ { - 1 } } , L _ { x } ^ { - 1 } R _ { x } ^ { - 1 } R _ { x \\phi ^ { - 1 } } L _ { x \\alpha } ) \\in A U T ( L , \\cdot ) . \\end{align*}"}
-{"id": "8943.png", "formula": "\\begin{align*} \\hat { \\pi } _ 0 ( \\theta ) = | \\hat { \\phi } _ 0 ( \\theta ) \\rangle \\langle \\hat { \\phi } _ 0 ( \\theta ) | \\ , . \\end{align*}"}
-{"id": "4006.png", "formula": "\\begin{align*} q ^ { ( 1 1 ) } _ { Z _ a Z _ b } ( z _ a , z _ b ) = p _ { Z | X Y } ( z _ a | x _ 1 , y _ 1 ) p _ { Z | X Y } ( z _ { b } | x _ 2 , y _ 2 ) . \\end{align*}"}
-{"id": "9.png", "formula": "\\begin{align*} \\partial _ s \\Phi _ { u , \\gamma } & = - \\frac { \\xi '' ( s ) } { 2 } \\bigl ( \\partial _ { x x } \\Phi _ { u , \\gamma } + \\gamma ( s ) \\bigl ( \\partial _ x \\Phi _ { u , \\gamma } \\bigr ) ^ 2 \\bigr ) , \\ , \\ , ( s , x ) \\in [ 0 , u ) \\times \\mathbb { R } \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{align*} & \\sum _ a p _ a \\dot { \\mu } _ a + \\sum _ { ( a , b ) \\in \\Gamma } p _ { ( a , b ) } \\dot { w } _ { ( a , b ) } \\\\ & = \\sum _ { a } p _ a \\left ( \\dot { \\mu } _ a - \\sum _ { b : ( a , b ) \\in \\Gamma } \\left ( \\dot { w } _ { ( b , a ) } - \\dot { w } _ { ( a , b ) } \\right ) \\right ) + \\sum _ { ( a , b ) \\in \\Gamma } \\dot { w } _ { ( a , b ) } \\left ( p _ { ( a , b ) } - p _ a + p _ b \\right ) . \\end{align*}"}
-{"id": "3814.png", "formula": "\\begin{align*} \\ker \\partial _ 1 = \\langle x _ 3 \\cdot e _ { \\ell _ 1 , \\ell _ 2 , \\ell _ 3 } - x _ 2 \\cdot e _ { \\ell _ 1 - 1 , \\ell _ 2 + 1 , \\ell _ 3 } , ~ x _ 2 \\cdot e _ { \\ell _ 1 - 1 , \\ell _ 2 + 1 , \\ell _ 3 } - x _ 1 \\cdot e _ { \\ell _ 1 - 1 , \\ell _ 2 , \\ell _ 3 + 1 } ~ : ~ 1 \\leq \\ell _ 1 \\leq s , \\ ; 0 \\leq \\ell _ 2 , \\ell _ 3 \\leq s - 1 \\rangle . \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{align*} f ' ( x ) = A ^ T \\lambda ( y / h ) = \\sum _ { i \\in [ m ] } a _ i \\lambda ( y / h ) = 0 \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} \\mu _ 1 ( x , y ) = \\begin{cases} \\frac { 1 } { k } & ( x , y ) \\ ! = \\ ! ( x _ 1 , y _ k ) , \\\\ \\frac { 1 } { k } & ( x , y ) \\ ! = \\ ! ( x _ i , y _ { i - 1 } ) 2 \\ ! \\leq \\ ! i \\ ! \\leq \\ ! k , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8539.png", "formula": "\\begin{align*} \\mu ( \\zeta ) = \\frac { \\mu _ { + j } } { \\zeta ^ j } + \\cdots + \\mu _ 0 + \\cdots + \\zeta ^ j \\mu _ { - j } \\mathrlap { . } \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} \\psi ( x ) : = ( 1 + | P _ { \\nu } x | ^ 2 + | P _ { \\nu } ^ { \\perp } x | ^ 4 ) ^ { - N / 2 } \\end{align*}"}
-{"id": "1091.png", "formula": "\\begin{align*} d \\alpha \\cdot d s = d s \\cdot d \\alpha + d ( \\alpha ( s ) ) \\ , , \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} \\lim _ { k } \\sum _ { l = k } ^ { k + s - 1 } \\lambda _ { l } \\Vert G U _ { l } u ^ { l } \\Vert = 0 \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{align*} \\| & f _ { 2 i } ' ( x ) - f _ { 2 i - 2 } ' ( x ) \\| \\\\ & \\ ; \\ ; = \\| ( f _ { 2 i } ' ( x ) - f _ { 2 i - 1 } ' ( x ) - \\phi _ { 2 i } ( x ) e _ { 2 i } ) + ( f _ { 2 i - 1 } ' ( x ) - f _ { 2 i - 2 } ' ( x ) - \\phi _ { 2 i - 1 } ( x ) e _ { 2 i - 1 } ) \\| \\\\ & \\ ; \\ ; \\le \\sigma _ { 2 i } \\ 1 _ { \\{ \\phi _ { 2 i } > 0 \\} } ( x ) + \\sigma _ { 2 i - 1 } \\ 1 _ { \\{ \\phi _ { 2 i - 1 } > 0 \\} } ( x ) . \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} \\P ^ { 0 , 0 } _ { 2 T } ( X ( t ) = x ) = Q ( 0 , 0 ; t , x ) \\frac { Q ( t , x ; 2 T , 0 ) } { Q ( 0 , 0 ; 2 T , 0 ) } , t \\in \\{ 0 , 1 , \\dots , 2 T \\} . \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} \\delta = ( 0 ^ { n - m _ 1 - m _ 2 } , 1 ^ { m _ 1 } , 2 ^ { m _ 2 } ) , \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} \\left ( \\nabla _ X \\bar { \\psi } | Y \\cdot \\bar { \\psi } \\right ) = \\left ( \\bar { T } ( X ) \\cdot \\bar { \\psi } | Y \\cdot \\bar { \\psi } \\right ) = \\bar { T } ( X , Y ) , X , Y \\in T M . \\end{align*}"}
-{"id": "7983.png", "formula": "\\begin{align*} 0 \\leq \\lambda | | D ^ 2 ( \\phi _ 2 - v ) | | \\leq F ( D ^ 2 v + D ^ 2 ( \\phi _ 2 - v ) ) - F ( D ^ 2 v ) = F ( D ^ 2 \\phi _ 2 ) - F ( D ^ 2 v ) \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} | B _ { d - \\eta } | \\leq \\frac { \\rho } { d - ( d - \\eta ) } n ^ 2 = \\frac { \\rho } { \\eta } n ^ 2 . \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} \\mathrm { d i m } _ \\Q \\ , H ^ d ( G _ k ( \\R ^ n ) , \\Q ) = p \\Big ( \\big [ \\frac { k } { 2 } \\big ] , \\big [ \\frac { n - k } { 2 } \\big ] ; \\frac { d } { 4 } \\Big ) \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{align*} \\mathcal { D } _ * = \\left \\lbrace f \\in \\C ^ \\infty ( [ \\xi _ + , + \\infty ) ) , \\ , \\lim _ { x \\to \\xi _ + } f ' ( x ) = 0 \\right \\rbrace , \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} & S _ { } ( X ; Y \\| Z ) = \\max _ { U V \\rightarrow X \\rightarrow Y Z } I ( U ; Y | V ) - I ( U ; Z | V ) . \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} E _ { \\mu } & = z ^ { \\mu } + \\sum _ { \\nu \\prec \\mu } c _ { \\mu , \\nu } ( q , t ) z ^ { \\nu } , c _ { \\mu , \\nu } ( q , t ) \\in \\mathbb { Q } ( q , t ) , \\\\ Y _ i E _ { \\mu } & = y _ i ( \\mu ; q , t ) E _ { \\mu } , \\forall \\ 1 \\leq i \\leq n , \\mu \\in \\mathbb { Z } _ { \\geq 0 } ^ n , \\end{align*}"}
-{"id": "3963.png", "formula": "\\begin{align*} & F ( p _ { X Y } ) = \\min _ { q _ { X Y } } \\left ( \\max _ { x , y } \\left ( \\frac { p _ { X Y } ( x , y ) } { q _ { X Y } ( x , y ) } \\right ) \\cdot \\max _ { x , y } \\left ( \\frac { q _ { X Y } ( x , y ) } { p _ { X Y } ( x , y ) } \\right ) \\right ) \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{gather*} \\varphi _ 2 ( x , t ) = \\exp ( - t h _ 1 ) _ * h _ 2 \\exp ( t h _ 1 ) ( x ) \\\\ \\varphi _ 3 ( x , t , s ) = \\exp ( - t h _ 1 ) _ * \\exp ( - s h _ 2 ) _ * h _ 3 \\circ \\exp ( s h _ 2 ) \\circ \\exp ( t h _ 1 ) ( x ) , \\end{gather*}"}
-{"id": "3305.png", "formula": "\\begin{align*} \\Theta = \\Theta _ E \\times \\Theta _ F \\end{align*}"}
-{"id": "4839.png", "formula": "\\begin{align*} \\begin{aligned} \\pm \\ \\omega _ M \\times 1 & = f ^ * ( \\omega _ M \\times 1 ) \\\\ & = f ^ * ( x _ M ^ n \\times 1 ) \\cup f ^ * ( x _ M ^ { m - n } \\times 1 ) \\\\ & = ( ( y _ M ^ { n } \\times 1 ) + \\xi \\cdot ( 1 \\times \\omega _ N ) ) \\cup ( ( y _ M ^ { m - n } \\times 1 ) + ( y _ M ^ { m - 2 n } \\times \\omega _ { N } ) . \\end{aligned} \\end{align*}"}
-{"id": "7136.png", "formula": "\\begin{align*} \\Omega _ { 3 } = 2 \\int _ { \\wp ( \\omega _ { 3 } / 2 ) } ^ { + \\infty } \\frac { d \\mathbf { u } } { \\sqrt { 4 \\mathbf { u } ^ { 3 } - g _ { 2 } \\mathbf { u } - g _ { 3 } } } = \\int _ { a _ { 4 } } ^ { X _ { \\pm } ( b _ { 4 } ) } \\frac { d x } { \\sqrt { D ( x ) } } , \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} ( \\pi _ { i , l } ) : = b ( i _ l ) b ( i _ { l + 1 } ) \\dots b ( i _ { m - 1 } ) : i _ m \\rightarrow i _ l \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{align*} & \\theta \\perp e _ 1 , \\ , \\theta \\perp e _ 2 , \\theta \\in \\Theta ( e _ 1 , e _ 2 ) . \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} \\mathbb P \\Big ( \\inf _ { i > m } \\inf _ { s \\in [ 0 , t ] } \\{ Y _ i ( s ) \\} \\le \\Gamma \\Big ) & \\le \\sum _ { i > m } \\mathbb P \\Big ( \\inf _ { s \\in [ 0 , t ] } \\{ W _ i ( s ) \\} \\le \\Gamma - X _ i ( 0 ) \\Big ) \\\\ & = 2 \\sum _ { i > m } \\mathbb P \\Big ( W ( 1 ) \\le \\frac { \\Gamma - X _ i ( 0 ) } { \\sqrt t } \\Big ) = 2 \\sum _ { i > m } \\Big ( \\frac { X _ i ( 0 ) - \\Gamma } { \\sqrt t } \\Big ) \\ , , \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} D A _ { \\epsilon } ^ \\dagger = D A ^ \\dagger + \\epsilon ^ { - 1 } \\Pi ^ \\dagger _ { \\rm n o r } . \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{align*} \\widetilde { U } ( \\lambda ) : = \\int _ 0 ^ { + \\infty } e ^ { - \\lambda t } U ( d t ) . \\end{align*}"}
-{"id": "5281.png", "formula": "\\begin{align*} ( x , v ) . g : = ( x . g , g ^ { - 1 } . v ) . \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} \\begin{gathered} p ( 0 , 1 | v , w ) = p ( 1 , 0 | v , w ) = t - p ( 0 , 0 | v , w ) , \\\\ p ( 1 , 1 | v , w ) = 1 - 2 t + p ( 0 , 0 | v , w ) . \\end{gathered} \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} \\begin{cases} z _ 1 + z _ 2 = 0 \\\\ z _ 1 + z _ 2 = 0 \\end{cases} \\end{align*}"}
-{"id": "2636.png", "formula": "\\begin{align*} U _ \\alpha ( \\mu _ 2 + c \\varphi ) = 0 \\mbox { i n } G _ { 1 } . \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} \\mu ( K \\times \\mathbb { T } ) = \\int _ { ( K _ 1 \\cap K ) \\times \\mathbb { T } } \\overline { \\gamma _ 0 } \\gamma d \\mu - \\int _ { ( K _ 2 \\cap K ) \\times \\mathbb { T } } \\overline { \\gamma _ 0 } \\gamma d \\mu . \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{align*} f ( n ) G = 0 \\mod v . \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} \\begin{aligned} [ b ] \\lambda ( [ a , b ] ) : = c \\int _ { a } ^ { b } x ^ { - \\tau } \\dd x . \\end{aligned} \\end{align*}"}
-{"id": "2355.png", "formula": "\\begin{align*} \\Gamma ^ { ( 1 ) } ( 1 ) = - \\gamma , \\Gamma ^ { ( 2 ) } ( 1 ) = \\frac { \\pi ^ 2 } { 6 } + \\gamma ^ 2 , \\Gamma ^ { ( 3 ) } ( 1 ) = - \\left [ 2 \\zeta ( 3 ) + \\frac { \\pi ^ 2 } { 2 } \\gamma + \\gamma ^ 3 \\right ] , , \\end{align*}"}
-{"id": "290.png", "formula": "\\begin{align*} d \\leq 2 \\left [ \\frac { n } { 1 2 } \\right ] + 2 . \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{align*} L _ m \\chi _ y ( z ) & = \\sum \\chi \\left ( \\sum _ { s \\in S _ \\mu } \\sum _ { t \\in T _ \\nu } Y _ { s , t } ( Z _ { s , t } - Y ' _ { s , m } Z _ { m , t } ) + \\sum _ { s \\in S _ \\mu } Y _ { s , m } Y ' _ { s , m } \\right ) \\\\ & = \\sum \\chi \\left ( \\sum _ { s \\in S _ \\mu } \\sum _ { t \\in T _ \\nu } Y _ { s , t } Z _ { s , t } \\right ) \\chi \\left ( \\sum _ { s \\in S _ \\mu } \\left ( Y _ { s , m } - \\sum _ { t \\in T _ \\nu } Y _ { s , t } Z _ { m , t } \\right ) Y ' _ { s , m } \\right ) , \\end{align*}"}
-{"id": "1906.png", "formula": "\\begin{align*} e _ n = v _ n + w _ n + C _ { n - 2 } - 3 \\cdot 2 ^ { n - 3 } + 1 + \\sum _ { \\ell = 1 } ^ { n - 2 } C _ \\ell , \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { i = 1 } ^ r \\langle \\ ! \\langle \\hat { p } , \\hat { b } ^ i - \\xi ^ i \\rangle \\ ! \\rangle - \\sum _ { j = 1 } ^ s \\langle \\ ! \\langle \\hat { p } , \\hat { a } ^ j \\rangle \\ ! \\rangle \\leq 0 \\end{align*}"}
-{"id": "317.png", "formula": "\\begin{align*} d ' _ k = \\begin{cases} d _ k & k \\in I _ g \\cup J ^ + _ g , \\\\ d _ k - 1 & k \\in J ^ - _ g . \\end{cases} \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} \\max \\{ \\kappa , \\kappa ^ 2 \\} ( 1 + \\norm { b } _ { L ^ { \\infty } ( \\Omega ) ^ 3 } ) < C , \\hbox { w h e r e } \\kappa : = \\norm { a } _ { L ^ p _ { \\sigma } ( \\Omega ) } + \\norm { \\nabla b } _ { L ^ p ( \\Omega ) ^ { 3 \\times 3 } } , \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{align*} N = \\sum _ { i = 1 } ^ { m - 1 } a _ i P _ i , Q _ { 1 1 } = Q \\sum _ { i = 1 } ^ { m - 1 } P _ i , Q _ { 1 2 } = \\sum _ { i = 1 } ^ { m - 1 } P _ i Q P _ m , Q _ { 2 2 } = P _ m Q . \\end{align*}"}
-{"id": "6107.png", "formula": "\\begin{align*} f ( 0 ) = \\lim _ { \\lambda \\to \\infty } \\lambda \\widetilde { f } ( \\lambda ) = 0 . \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} d s = \\sqrt { 1 + h _ x ^ 2 } \\ , d x \\sim d x . \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} \\Phi ( u ) s = ( u \\otimes ( \\varphi _ L \\otimes e _ S ) ) \\cdot s = u s \\otimes ( \\varphi _ L \\otimes e _ S ) = \\Phi ( u s ) \\ , , \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} J = \\max _ h \\left \\lbrace \\left | \\int _ 0 ^ T \\ ! \\ ! \\overline { b } _ h ^ i ( t ) d t \\right | + 1 \\right \\rbrace \\mbox { a n d } I = \\min _ h \\left \\lbrace \\int _ 0 ^ T \\ ! \\sum _ { i = 1 } ^ r \\xi _ h ^ i ( t ) d t + R - \\int _ 0 ^ T \\ ! \\ ! \\hat { b } _ h ^ i ( t ) d t \\right \\rbrace \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ Z ^ n = z ^ n , X ^ n \\in \\mathcal { T } ^ { ( n ) } _ { q _ X } , Y ^ n \\in \\mathcal { T } ^ { ( n ) } _ { q _ Y } \\right ] \\end{align*}"}
-{"id": "5353.png", "formula": "\\begin{align*} ( i _ z ^ * \\omega ) ^ n = \\omega _ Y ^ n \\end{align*}"}
-{"id": "2025.png", "formula": "\\begin{align*} \\Delta E ( \\| p \\| , \\kappa ) = \\frac { 1 } { 2 } \\left ( \\| p + R ( p , \\kappa ) \\| ^ 2 - \\| p \\| ^ 2 \\right ) . \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{align*} \\omega _ 0 = \\tfrac { i } { 2 } \\left ( d z _ 1 \\wedge d \\overline { z } _ 1 + d z _ 2 \\wedge d \\overline { z } _ 2 + d z _ 3 \\wedge d \\overline { z } _ 3 \\right ) , \\Omega _ 0 = d z _ 1 \\wedge d z _ 2 \\wedge d z _ 3 . \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} \\epsilon \\leq \\sup _ { \\substack { p _ { \\bar { Z } | X , Y } : \\\\ I ( X ; Y | \\bar { Z } ) = 0 } } \\sum _ { \\bar { z } } \\min _ { x , y : \\ : p _ { X , Y } ( x , y ) > 0 } p _ { \\bar { Z } | X , Y } ( \\bar { z } | x , y ) . \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} \\phi _ m ^ 2 ( P ) - [ \\tau ] \\phi _ m ( P ) + [ k ] ( P ) = [ o ] ( P ) = o \\ , , \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} N _ J ( Y _ 1 , Y _ 2 ) = [ Y _ 1 , Y _ 2 ] + J \\left ( [ J Y _ 1 , Y _ 2 ] + [ Y _ 1 , J Y _ 2 ] \\right ) - [ J Y _ 1 , J Y _ 2 ] \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} I : = \\langle f _ 1 , \\ldots , f _ s \\rangle , J : = \\langle g _ 1 , \\ldots , g _ t \\rangle \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} S ( x ) = \\{ x _ 0 \\in K ( x ) : ~ f ( x _ 0 , y ) \\geq 0 \\mbox { f o r a l l } y \\in K ( x ) \\} . \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} \\widetilde { p } ( x , D _ x ) : = \\kappa ^ { - 1 , * } p ( x , D _ x ) \\kappa ^ * \\end{align*}"}
-{"id": "8654.png", "formula": "\\begin{align*} \\theta _ i = \\mu _ i ( 1 + \\frac { p - K } { n ( \\mu _ i - 1 ) } ) . \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} g _ \\infty = e ^ { 2 ( \\widehat { \\eta } - \\widehat { U } ) } ( - \\ : d \\widehat { R } ^ 2 + \\widehat { a } ^ { - 2 } d \\widehat { \\theta } ^ 2 ) + e ^ { 2 \\widehat { U } } ( d \\widehat { x } + \\widehat { G } d \\widehat { \\theta } ) ^ 2 + e ^ { - 2 \\widehat { U } } \\widehat { R } ^ 2 ( d \\widehat { y } + \\widehat { H } d \\widehat { \\theta } ) ^ 2 , \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{align*} v _ \\beta = v _ { 1 , \\beta } + v _ { 2 , \\beta } , \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{align*} \\bar H = \\bar C _ { i j k l } \\bar C ^ { i j k l } = \\bar g _ { i a } C ^ a _ { \\ ; j k l } C ^ i _ { \\ ; b c d } \\bar g ^ { b j } \\bar g ^ { c k } \\bar g ^ { d l } = \\alpha ^ { - 2 } H , \\end{align*}"}
-{"id": "151.png", "formula": "\\begin{align*} ( t , q ) \\longmapsto \\int _ { S _ { t q } } \\mu _ { t q } \\ , d A = \\int _ { t S _ q } \\mu _ { t q } \\ , d A . \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{align*} 2 \\mathcal { L } _ X \\psi = \\left ( \\langle X , \\xi \\rangle + f _ 1 \\right ) I \\psi + f _ 2 J \\psi + f _ 3 K \\psi + \\bigl ( X ^ \\perp - \\tfrac { 1 } { 2 } J _ 1 \\gamma ^ \\sharp \\bigr ) \\cdot \\psi . \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} Q ( g ) = \\binom { 3 } { 2 } - 1 = 2 . \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} D _ T = V \\cap D , \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{align*} [ \\rho _ { c } ( a ) f ] ( n ) & = n f ( n - 1 ) , \\\\ [ \\rho _ { c } ( a ^ \\dagger ) f ] ( n ) & = c f ( n + 1 ) , \\\\ [ \\rho _ { c } ( Z ) f ) ] ( n ) & = c f ( n ) , \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{align*} E \\left [ S _ N ^ { ( r ) } \\right ] = E \\left [ \\frac { \\Gamma ( S _ N + r ) } { \\Gamma ( S _ N ) } \\right ] = r \\int _ 0 ^ { \\infty } t ^ { r - 1 } \\left [ 1 - \\prod _ { j = 1 } ^ N \\bigg ( 1 - e ^ { - q _ j t } \\bigg ) \\right ] d t . \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { M } _ j = & \\iint \\left | \\int _ { O ( d ) } \\int _ { | \\xi | \\approx 2 ^ j } \\widehat { f \\ , d \\mu } ( \\xi ) \\ , e ^ { - 2 \\pi i \\xi \\cdot u ' } \\ , \\widehat { f \\ , d \\mu } ( - \\theta ^ t \\xi ) \\ , e ^ { 2 \\pi i \\xi \\cdot ( \\theta u ) } \\ , d \\xi \\ , d \\theta \\right | ^ 2 \\ , d \\mu ( u ) \\ , d \\mu ( u ' ) \\\\ \\lesssim & \\ 2 ^ { - j \\gamma } | | f | | ^ 2 _ { L ^ 2 ( \\mu ) } \\end{aligned} \\end{align*}"}
-{"id": "7733.png", "formula": "\\begin{align*} \\phi ( S q _ R ^ k ( f ) ) = S q ^ k ( \\phi ( f ) ) . \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{align*} \\begin{aligned} & \\hat F _ \\lambda - F _ \\varpi ( P \\parallel Q ) = \\sum _ { l \\in \\bar l } \\lambda _ l [ \\hat F _ { \\tilde d _ l } - F _ \\varpi ( P \\parallel Q ) ] \\to 0 , \\end{aligned} \\end{align*}"}
-{"id": "4809.png", "formula": "\\begin{align*} \\tilde { \\Phi } ( \\epsilon \\eta ^ { m - 1 } \\phi ( | \\nabla \\overline { u } | | \\nabla \\overline { u } | ) ) & \\le \\max \\{ ( \\epsilon \\eta ^ { m - 1 } ) ^ { \\ell ' } , ( \\epsilon \\eta ^ { m - 1 } ) ^ { m ' } \\} \\tilde { \\Phi } ( | \\nabla \\overline { u } | | \\nabla \\overline { u } | ) \\\\ & = g _ 2 ( x , \\epsilon ) \\tilde \\Phi ( \\phi ( | \\nabla \\overline { u } | ) | \\nabla \\overline { u } | ) , \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} \\lim \\tilde { t } _ j d _ { t _ j \\mu _ Y + b _ j } u _ { \\lambda } ( \\mu _ Y ) = 0 . \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{align*} P ( \\sigma , \\tau ) = ( \\sigma _ { l _ { \\sigma } } , \\sigma _ { l _ { \\sigma } - 1 } , \\dots , \\sigma _ { l _ { \\tau } + 1 } , \\dots , \\sigma _ { j _ 1 + 1 } , \\sigma _ { j _ 1 } , \\tau _ { j _ 1 } , \\tau _ { j _ 1 - 1 } , \\dots , \\tau _ { j _ 2 } , \\sigma _ { j _ 2 } , \\dots ) . \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{align*} r = E ( r ) \\circ q \\ , . \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{align*} K ( p ) = - \\sum \\limits _ { j = 1 } ^ k \\frac { r _ j ( r _ j - 1 ) } { \\delta _ j ( p ) ^ 2 } + O \\left ( \\frac { 1 } { \\delta _ j ( p ) } \\right ) \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} F _ n ( x _ \\textbf { i + 1 } ) - F _ 0 ( x _ \\textbf { i } ) = ( F _ n ( x _ \\textbf { i + 1 } ) - F _ 0 ( x _ \\textbf { i + 1 } ) ) + F _ 0 ( x _ \\textbf { i + 1 } ) - F _ 0 ( x _ \\textbf { i + 1 } ) ) . \\end{align*}"}
-{"id": "3769.png", "formula": "\\begin{align*} q _ { \\vec { n } } = \\sum _ { u \\in \\Psi ^ { - 1 } ( \\vec { n } ) } p _ { u _ 1 } \\cdots p _ { u _ r } = | \\Psi ^ { - 1 } ( \\vec { n } ) | p _ 1 ^ { \\vec { n } _ 1 } \\cdots p _ k ^ { \\vec { n } _ k } . \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} T ( \\varphi A \\psi ; \\nu ) = \\nu ^ { - 1 , * } ( \\varphi A \\psi ) \\nu ^ * = ( \\varphi \\circ \\nu ^ { - 1 } ) T ( A ; \\nu ) ( \\psi \\circ \\nu ^ { - 1 } ) , \\end{align*}"}
-{"id": "6773.png", "formula": "\\begin{align*} ( v , w ) : = \\begin{cases} 0 & | v | \\neq | w | , \\\\ \\alpha ( a ) \\langle v , w \\rangle & | v | = | w | = a a \\in \\Gamma . \\end{cases} \\end{align*}"}
-{"id": "3648.png", "formula": "\\begin{align*} \\Lambda ^ m \\left ( \\left [ \\frac { k } { m + 1 } , \\frac { k + 1 } { m + 1 } \\right [ \\right ) = \\lambda _ k ^ m , 0 \\leq k \\leq m . \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{align*} \\norm { \\delta _ { N ^ { - 1 / 2 } } ( X _ { N s } ) ^ { - 1 } \\cdot \\delta _ { N ^ { - 1 / 2 } } ( X _ { N t } ) } ^ { 4 p } \\leq \\frac { 5 ^ { 4 p } } { N ^ { 2 p } } \\sum _ { k = 1 } ^ { 5 } I _ k ^ { 4 p } \\end{align*}"}
-{"id": "4192.png", "formula": "\\begin{align*} \\| \\partial ^ \\gamma f \\| _ { \\sup } \\leq | f | _ { | \\gamma | } = | f | _ { n - j } \\leq | f | _ { n , \\sigma } + C _ { \\sigma , d , n , j } \\cdot \\| f \\| _ { \\sup } . \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} \\hat { \\pi } ( y _ t ) = \\frac { \\exp \\{ - 2 f ( x ) / ( \\tau _ N ^ 2 \\gamma ) \\} } { \\int \\exp \\{ - 2 f ( y ) / ( \\tau _ N ^ 2 \\gamma ) \\} } = \\frac { \\exp \\{ - 2 N f ( y ) / ( \\sigma ^ 2 \\tilde { \\Gamma } ) \\} } { \\int \\exp \\{ - 2 N f ( y ) / ( \\sigma ^ 2 \\tilde { \\Gamma } ) \\} } \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi } \\int e ^ { - 2 | w | ^ 2 + a w + b \\overline { w } + c w ^ 2 + d \\overline { w ^ 2 } } P ( w , \\overline { w } ) \\ , d L ( w ) = P ( \\partial _ a , \\partial _ b ) \\frac { 1 } { 2 \\sqrt { 1 - c d } } e ^ { \\frac { d a ^ 2 + c b ^ 2 + 2 a b } { 4 ( 1 - c d ) } } . \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} e ( R ) \\ = \\ \\binom { n } { t - 1 } , \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} d \\Phi _ { \\rho _ L ^ { - 1 } } ( o ) = \\begin{pmatrix} A ^ { - 1 } - \\hat u \\xi ^ T & \\hat u \\\\ - u _ 3 \\xi ^ T & u _ 3 \\end{pmatrix} \\end{align*}"}
-{"id": "4423.png", "formula": "\\begin{align*} ( \\abs { z _ { 1 , 3 } } , \\abs { z _ { 2 , 3 } } , \\abs { z _ { 3 , 3 } } ) = \\frac { x } { 2 } ( \\norm { f } \\norm { g } , \\norm { e } \\norm { g } , \\norm { e } \\norm { f } ) . \\end{align*}"}
-{"id": "4725.png", "formula": "\\begin{align*} N ( w ) = \\{ \\gamma \\in \\Phi ^ + : w ( \\gamma ) \\in \\Phi ^ - \\} . \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} { \\bf G } ^ { - 1 } = \\sum _ { n = 0 } ^ { \\infty } ( - { \\bf D } ^ { - 1 } { \\bf E } ) ^ n { \\bf D } ^ { - 1 } \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} \\alpha = d f ( \\zeta ) I ( \\zeta ) - i f ( \\zeta ) \\frac { d \\zeta } { \\zeta } \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} W ^ { ( n ) } = & W ^ 0 _ { n + 2 } ( p , p _ 1 , \\dots , p _ { n + 1 } ) + h W ^ 1 _ { n } ( p , p _ 1 , \\dots , p _ { n - 1 } ) + \\\\ & \\dots + h ^ { n / 2 } W ^ { n / 2 } _ 2 ( p , p _ 1 ) . \\end{align*}"}
-{"id": "3951.png", "formula": "\\begin{align*} \\eta ( p _ { Y | X } ) & = \\max _ { p _ X } s ^ * ( X ; Y ) \\overset { ( a ) } { = } \\max _ { p _ X } \\rho ^ 2 _ m ( p _ { X } \\ , p _ { Y | X } ) \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} D _ D ( \\lambda X , \\lambda Y ) = \\lambda D _ D ( X , Y ) \\ , \\end{align*}"}
-{"id": "5892.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\delta } = \\left \\{ \\nu : \\nu \\prec \\delta , \\ y _ { \\nu } ( w ) = y _ { \\delta } ( w ) \\ \\ q = t ^ { - p - m _ 1 } \\right \\} \\end{align*}"}
-{"id": "1780.png", "formula": "\\begin{align*} \\left \\| \\xi _ n ^ l D _ { \\xi } ^ { \\alpha } p ( \\cdot , 0 , \\xi ) - \\sum _ { k = - l } ^ { m - | \\alpha | } s _ { k , \\alpha } ( \\cdot , \\xi ' ) \\xi _ n ^ { k + l } \\right \\| _ { C ^ \\tau ( \\R ^ { n - 1 } ) } \\leq C _ { l , \\alpha } \\langle \\xi ' \\rangle ^ { m + l + 1 - | \\alpha | } | \\xi _ n | ^ { - 1 } \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} \\Phi ( z ) = \\sum _ { k = 1 } ^ 2 < x ^ { ( k ) } > ^ { M _ k + 1 } + \\sum _ { k = 3 } ^ 4 < x ^ { ( k ) } > . \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} \\begin{aligned} \\nu ( t _ 1 , t _ 2 ) = & \\int _ { \\{ | x - y | = t _ 1 , | z - y | = t _ 2 \\} } \\ , d \\mu ( x ) \\ , d \\mu ( y ) \\ , d \\mu ( z ) \\\\ = & \\int \\omega _ { t _ 1 } ( x - y ) \\ , \\omega _ { t _ 2 } ( z - y ) \\ , d \\mu ( x ) \\ , d \\mu ( y ) \\ , d \\mu ( z ) \\\\ = & < T _ { t _ 1 } \\circ T _ { t _ 2 } 1 , 1 > _ \\mu \\end{aligned} \\end{align*}"}
-{"id": "3709.png", "formula": "\\begin{align*} \\| \\tilde { g } _ { I , j } \\| _ { \\infty } \\ll \\widetilde { C } ^ { 2 n r ^ { n } ( d - 1 ) ^ { n - 2 } d } = \\tilde { \\mathfrak { C } } , \\end{align*}"}
-{"id": "4626.png", "formula": "\\begin{align*} g _ Q ( Y , X ) = \\langle \\nabla _ { X + i J X } \\phi , \\psi \\rangle , g _ Q ( Z , X ) = \\langle \\phi , \\nabla _ { X + i J X } \\psi \\rangle \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} & \\forall \\ x , y , z \\in \\mathcal { H } ( A ) = A _ 0 \\cup A _ 1 : \\\\ & a s _ { \\alpha , \\bullet } ( x , y , z ) + ( - 1 ) ^ { | x | | y | + | x | | z | + | y | | z | } a s _ { \\alpha , \\bullet } ( z , y , x ) = 0 , \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{align*} \\rho _ 0 : = r ^ { - 1 } , \\ \\ \\tau : = t / r , \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} x ( x ^ { \\lambda } \\cdot x \\phi ^ { - 1 } ) = x \\phi ^ { - 1 } . \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} \\sigma ^ { j } ( y _ 0 , y _ { 1 } , y _ { 2 } , \\ldots ) = ( y _ j , y _ { j + 1 } , y _ { j + 2 } , \\ldots ) < _ { l e x } ~ ( c _ 1 , \\ , c _ 2 , \\ , c _ 3 , \\ , \\ldots ) \\mbox { f o r a l l } ~ j \\geq 0 , \\end{align*}"}
-{"id": "7617.png", "formula": "\\begin{align*} Z _ + ' ( r ) = \\left [ \\left ( \\frac { u ' } { \\sqrt { 1 + ( u ' ) ^ 2 } } \\right ) ' + g ( u ) \\right ] u ' = - \\frac { N - 1 } r \\frac { ( u ' ) ^ 2 } { \\sqrt { 1 + ( u ' ) ^ 2 } } < 0 . \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} b _ n & = ( 7 - n ) 2 ^ { n - 3 } - 2 + \\sum _ { a = 1 } ^ { n - 2 } 2 ^ { a - 1 } e _ { n - 1 - a } + \\sum _ { a = 2 } ^ { n - 3 } \\sum _ { \\ell = 1 } ^ { n - 2 - a } \\sum _ { i = 0 } ^ { a - 1 } 2 ^ { n - 2 - a - \\ell } \\binom { a - 1 } { i } \\binom { \\ell + i } { i } \\\\ & \\quad + \\sum _ { a = 2 } ^ { n - 1 } \\left ( 2 ^ { n - 1 - a } - 1 - \\binom { n - a } { 2 } \\right ) ( a - 1 ) 2 ^ { a - 2 } + \\sum _ { a = 3 } ^ { n - 1 } \\sum _ { \\ell = 1 } ^ { a - 2 } \\sum _ { j = 0 } ^ { \\ell } \\binom { n - 1 - a + j } { j } e _ { a - 1 , \\ell } , n \\geq 3 , \\end{align*}"}
-{"id": "3226.png", "formula": "\\begin{align*} a _ 1 \\alpha _ 2 + a _ 2 \\alpha _ 1 = 0 . \\end{align*}"}
-{"id": "8804.png", "formula": "\\begin{align*} E = E ( \\underline { z } ) : = ( [ B _ D , D ] + [ C _ D , B _ D ] + [ C _ D , D ] ) / 2 \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} - \\frac { u '' } { ( 1 + ( u ' ) ^ 2 ) ^ \\frac 3 2 } = g ( u ) , \\end{align*}"}
-{"id": "7237.png", "formula": "\\begin{align*} h _ k ( x , y | q ) = \\frac { 1 } { ( 1 - q ) y ( q , x / y , q y / x ; q ) _ \\infty } \\int _ { x } ^ y ( q z / x , q z / y ; q ) _ \\infty z ^ k d _ q z . \\end{align*}"}
-{"id": "6377.png", "formula": "\\begin{align*} T _ { } ^ { \\prime } T = p _ { I } ^ { } e _ { \\omega } ^ { } q _ { } ^ { * } q _ { I } ^ { } e _ { \\omega } ^ { } p _ { } ^ { * } & = p _ { I } ^ { } ( q _ { 2 } ^ { } ) _ { I } ^ { } e _ { q _ { 2 } ^ { * } \\omega } ^ { } q _ { 1 } ^ { * } e _ { \\omega } ^ { } p _ { } ^ { * } . \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} I ( \\mathcal { C } , \\vec { \\gamma } ) = \\int _ { \\vec { \\zeta } \\in \\mathcal { C } } e ( \\vec { \\gamma } \\cdot \\vec { \\tilde { f } } ( \\vec { \\zeta } ) ) \\dd \\vec { \\zeta } . \\end{align*}"}
-{"id": "5282.png", "formula": "\\begin{align*} \\{ d \\in \\Z \\mid d \\geq 1 , \\ d \\textrm { d e v i d s $ \\mathrm n _ p ( g ) $ f o r a l l r a t i o n a l p r i m e $ p $ } \\} = \\{ 1 \\} . \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{align*} U ( \\mathbf { p } , \\Omega ) = \\frac { \\sum _ { k = 1 } ^ { K } R _ { k } } { P _ c + \\sum _ { k = 1 } ^ { K } p _ k / \\eta } , \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} a \\big | _ { t = 0 } = a ^ 0 = ( a _ 1 ^ 0 , a _ 2 ^ 0 , a _ 3 ^ 0 , a _ 4 ^ 0 ) . \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{align*} r _ * = r + 2 m \\log ( r - 2 m ) \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{align*} & S _ \\ell = 0 \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\dot { S } _ \\ell = 2 \\sqrt { \\ell } \\dot { c } _ { \\ell - 1 } \\ ; \\ ; \\mbox { f o r $ \\ell \\neq 2 $ } \\\\ & S _ 2 = \\sqrt 2 , \\ ; \\ ; \\dot { S } _ 2 = 2 \\sqrt { 2 } \\dot { c } _ 1 , \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\ddot { S } _ 2 = 2 \\sqrt { 2 } \\dot { c } _ 1 ^ 2 + 2 \\sqrt { 2 } \\ddot { c } _ 1 + 4 \\dot { c } _ 0 \\dot { c } _ 2 . \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{align*} \\bigotimes _ { \\substack { 1 \\leq k \\leq n - 1 \\\\ 1 \\leq i \\leq m _ k } } \\prod _ { j = 1 } ^ { l _ { i , k } } f _ { p _ { j , i , k } q _ { j , i , k } } v _ { \\omega _ k } = \\mathcal { P } ( \\tilde B ) \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { a - 1 } \\left ( \\binom { a - 1 } { j } - 1 \\right ) \\sum _ { i = 1 } ^ j \\sum _ { r = 0 } ^ { b - 2 - a } \\binom { b - 1 - a } { r } & = \\left ( 2 ^ { b - 1 - a } - 1 \\right ) \\sum _ { j = 0 } ^ { a - 1 } \\left ( \\binom { a - 1 } { j } - 1 \\right ) j \\\\ & = \\left ( 2 ^ { b - 1 - a } - 1 \\right ) \\left ( ( a - 1 ) 2 ^ { a - 2 } - \\binom { a } { 2 } \\right ) \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} \\sigma _ { n + 1 } : = \\textstyle \\sqrt { \\frac { p '' } { 2 \\varepsilon } } > \\sigma _ { n - r } . \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} [ b _ 1 s b _ 2 , b _ 3 , b _ 4 ] = [ b _ 1 s b _ 2 , b _ 3 , y _ 1 \\dots y _ k ] = \\sum _ { i = 1 } ^ k y _ 1 \\dots y _ { i - 1 } [ b _ 1 s b _ 2 , b _ 3 , y _ i ] y _ { i + 1 } \\dots y _ k . \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} \\nabla _ { B } \\nabla _ { B } h + a h g _ { B } = 0 , \\end{align*}"}
-{"id": "4267.png", "formula": "\\begin{align*} U ' & = U \\cup \\{ q _ i ^ { - 1 } \\hat { z } _ J , q _ { s } \\hat { z } _ J , q _ s ^ { - 1 } \\hat { z } _ 1 , q _ i \\hat { z } _ 1 \\} & W ' & = W \\cup \\{ \\hat { z } _ J , q _ i ^ { - 1 } q _ { s } ^ { - 1 } \\hat { z } _ J , \\hat { z } _ 1 , q _ i q _ s \\hat { z } _ 1 \\} , \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} f f ' & = \\frac { - f ' f '' } { ( 1 + f ' ( x ) ^ { 2 } ) ^ { 3 / 2 } } \\\\ f ( x ) ^ { 2 } + 2 C _ { 1 } & = - 2 ( 1 + f ' ( x ) ^ { 2 } ) ^ { - 1 / 2 } \\\\ f ' ( x ) & = \\sqrt { \\frac { 4 } { ( f ( x ) ^ { 2 } + 2 C _ { 1 } ) ^ { 2 } } - 1 } \\end{align*}"}
-{"id": "5920.png", "formula": "\\begin{align*} H \\left ( \\nu , \\mu \\right ) = H \\left ( \\nu , \\mu ^ { * } \\right ) t ^ { - \\chi ( \\vec { x } , \\vec { y } ) } , \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { | \\hat { X } | } \\left | \\frac { d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\right | \\leq \\int _ { 1 } ^ { \\infty } \\left | \\frac { d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\right | = | \\omega _ 1 | / 2 \\leq 5 / 2 , \\end{align*}"}
-{"id": "4284.png", "formula": "\\begin{align*} \\vec { A } = ( \\vec { J } ) = ( B _ 0 , B _ 1 , \\dots , B _ { b - 1 } , B _ b , C _ 1 , \\dots , C _ c ) . \\end{align*}"}
-{"id": "1354.png", "formula": "\\begin{align*} \\| A \\| _ { s u p } = \\max _ { i , j = 1 , \\dots , k } | a _ { i j } | . \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} - \\int _ { \\Sigma } \\varphi ( L \\varphi ) e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma = \\int _ { \\Sigma } \\left ( | \\nabla \\varphi | ^ { 2 } - \\bigr ( | A | ^ { 2 } - \\frac 1 2 ) \\varphi ^ 2 \\right ) e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma \\geq 0 . \\end{align*}"}
-{"id": "4681.png", "formula": "\\begin{align*} X \\# _ { t s } Y = X \\# _ t ( X \\# _ s Y ) \\ . \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} \\frac { \\partial { } ^ 2 { } U ( \\mathbf { p } , \\Omega ) } { \\partial \\Omega ^ 2 } = \\sum _ { k = 1 } ^ { K } \\frac { - \\mathsf { C } \\mathsf { P } ' ( \\Omega ) } { \\mathsf { P } ^ 2 ( \\Omega ) } < 0 , \\end{align*}"}
-{"id": "6855.png", "formula": "\\begin{align*} \\dot { \\bar { x } } ^ * ( t ) & = \\bar { A } ' \\bar { x } ^ * ( t ) + \\bar { B } ' u ( t ) \\\\ \\dot { \\bar { x } } _ { n + 1 } ( t ) & = - \\varepsilon \\bar { x } _ { n + 1 } ( t ) + \\displaystyle \\sum _ { j = 1 } ^ { n _ { \\rm i n } } u _ j ( t ) \\bar { N } _ j ' \\bar { x } ^ * ( t ) + \\bar { H } '' ( \\bar { x } ^ * ( t ) \\otimes \\bar { x } ^ * ( t ) ) \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} \\nabla f ( x _ 0 ) = \\eta \\hat \\theta + \\nabla A ^ * _ { - } ( x _ 0 ) = \\eta \\hat \\theta + \\arg \\max _ { \\theta \\in \\mathbb { R } ^ n } \\left [ - \\theta ^ \\top x _ 0 - A ( \\theta ) \\right ] , \\end{align*}"}
-{"id": "7825.png", "formula": "\\begin{align*} x = \\frac { 3 } { 2 } \\frac { y ^ 2 - y + h } { 6 y - 1 } \\end{align*}"}
-{"id": "3277.png", "formula": "\\begin{align*} T ( h ^ { i t } x h ^ { - i t } ) = \\int _ { \\R } \\theta _ { s } ( h ^ { i t } x h ^ { - i t } ) \\d s = \\int _ { \\R } ( h ^ { i t } \\theta _ { s } ( x ) h ^ { - i t } ) \\d s = h ^ { i t } ( T x ) h ^ { - i t } \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{align*} | b _ 1 ^ { - 1 } ( i ) | = \\begin{cases} o r d _ { c l ( R _ m ) } ( L ) & i = 0 , \\ L | \\ell \\ { \\rm a \\ p r i m e \\ i d e a l \\ o f } \\ R _ m , \\\\ ( \\ell - ( \\frac { \\Delta ( E ) } { \\ell } ) ) \\ell ^ { i - 1 } \\cdot o r d _ { c l ( R _ m ) } ( L ) & i \\geq 1 . \\end{cases} \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{align*} \\chi ^ { a c } + 2 \\rho _ H = \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\alpha \\circ \\mathcal { H } + \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\alpha \\circ \\mathcal { P } = \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\alpha . \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} u ' ( r ) = - \\int _ 0 ^ \\infty \\frac { s ^ 2 + 1 } { ( r - s ) ^ 2 } \\ , d \\rho ( s ) , \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{align*} H [ f ] ( \\phi , \\psi ) : = \\frac { 1 } { 2 } \\Big \\{ \\big \\langle \\nabla \\phi , \\nabla \\langle \\nabla f , \\nabla \\psi \\rangle \\big \\rangle + \\big \\langle \\nabla \\psi , \\nabla \\langle \\nabla f , \\nabla \\phi \\rangle \\big \\rangle - \\big \\langle \\nabla f , \\nabla \\langle \\nabla \\phi , \\nabla \\psi \\rangle \\big \\rangle \\Big \\} . \\end{align*}"}
-{"id": "1378.png", "formula": "\\begin{align*} u _ s ( \\Omega ) & = \\psi ( \\Omega , B _ { \\cdot \\wedge s } ( \\Omega ) ) , \\\\ & = \\psi ( s , \\bar { B } _ { \\cdot \\wedge s } ( \\Omega ) + B _ { r } ( \\Omega ) ) , \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} \\mathcal { F } ( R , Q ) = \\sum _ { j , k = 1 } ^ n \\frac { 1 - \\sigma _ j \\sigma _ k } { \\left ( 1 + \\sigma _ j ^ 2 \\right ) \\left ( 1 + \\sigma _ k ^ 2 \\right ) } \\left | \\tilde { \\tilde { Q } } _ { j k } \\right | ^ 2 . \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} g ( q , z ) = 1 + \\sum _ { n \\geq 1 } ( - 1 ) ^ { n - 1 } q ^ { - \\binom { n } { 2 } } C _ { n - 1 } ( q ) z ^ n , \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} \\left ( \\sum _ { k = 0 } ^ \\infty d _ { i j k } p ^ k \\sum _ { \\substack { ( i , j ) \\in \\mathbb { J } _ T \\\\ \\frac { m } { i } = \\frac { n } { j } \\in \\mathbb { Z } _ { \\geq 0 } } } i [ - b _ { i j k } ] ^ { m / i } \\right ) _ { \\substack { ( m , n ) \\in \\mathbb { J } _ T } } . \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{align*} d \\rho = - d \\theta \\wedge \\omega _ 0 = \\ast d \\theta , d ^ \\ast \\rho = 0 . \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{align*} p ( m - 1 ) + \\big ( q - ( A + 2 - m ) \\big ) = a n + b m - h + m - A n - B - 2 . \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} \\nabla ( u e ^ h ) & = e ^ h ( \\nabla u + u \\nabla h ) \\\\ \\Delta ( u e ^ h ) & = e ^ h \\left [ \\Delta u + 2 \\langle \\nabla u , \\nabla h \\rangle + u ( \\Delta h + | \\nabla h | ^ 2 ) \\right ] \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{align*} \\sum _ { r } \\min _ { a } q _ { R | A } ( r | a ) \\geq \\epsilon \\sum _ { r } p _ { R | B } ( r | \\mathtt e ) = \\epsilon \\end{align*}"}
-{"id": "1723.png", "formula": "\\begin{align*} \\bigg \\| \\sum _ { k = k _ 0 } ^ \\infty 2 ^ { - k \\alpha } \\mathcal { C } _ k g \\bigg \\| _ { L ^ { q , \\infty } _ t L ^ 2 _ x } \\lesssim \\| g \\| _ { L ^ s _ t L ^ 2 _ x } . \\end{align*}"}
-{"id": "2734.png", "formula": "\\begin{align*} x = c \\beta + \\sum _ { \\substack { ( i , j ) \\in \\mathbb { I } } } c _ { i j } [ S ^ i ] [ T ^ j ] \\bmod \\wp W ( K ) \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} h = \\lambda \\hat { h } , H = \\lambda ^ 4 \\hat { H } , E = \\lambda \\hat { E } , D = \\lambda ^ { - 2 } \\hat { D } . \\end{align*}"}
-{"id": "7522.png", "formula": "\\begin{align*} d q ^ \\prime _ t = & ( \\phi _ * b _ + ) ( t ^ * , q ^ \\prime _ t ) d t - ( \\phi _ * b _ - ) ( t ^ * , q ^ \\prime _ t ) d t + ( \\hat { \\phi } _ * ( \\tilde \\gamma ^ { - 1 } \\sigma ) ) ( t ^ * , q ^ \\prime _ t ) \\circ d \\tilde W _ t \\\\ = & b ^ + ( t ^ * , q ^ \\prime _ t ) d t - ( - b _ - ) ( t ^ * , q ^ \\prime _ t ) d t + ( \\hat { \\phi } _ * ( \\tilde \\gamma ^ { - 1 } \\sigma ) ) ( t ^ * , q ^ \\prime _ t ) \\circ d \\tilde W _ t \\\\ = & b ( t ^ * , q _ t ^ \\prime ) d t + ( \\hat { \\phi } _ * ( \\tilde \\gamma ^ { - 1 } \\sigma ) ) ( t ^ * , q ^ \\prime _ t ) \\circ d \\tilde W _ t \\\\ \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} x _ { i } ^ { k } = \\begin{cases} \\mu _ i , & i \\in \\mathcal { I } , \\\\ - \\nu _ i , & i \\in \\mathcal { J } , \\\\ - k , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} \\bigl [ ( \\gamma _ C \\otimes \\gamma _ B ) ( \\sigma E ) \\bigr ] ( b \\otimes c ) & : = ( \\gamma _ C \\otimes \\gamma _ B ) \\bigl ( ( \\gamma _ C ^ { - 1 } ( b ) \\otimes \\gamma _ B ^ { - 1 } ( c ) ) ( \\sigma E ) \\bigr ) = E ( b \\otimes c ) , \\\\ ( b \\otimes c ) \\bigl [ ( \\gamma _ C \\otimes \\gamma _ B ) ( \\sigma E ) \\bigr ] & : = ( \\gamma _ C \\otimes \\gamma _ B ) \\bigl ( ( \\sigma E ) ( \\gamma _ C ^ { - 1 } ( b ) \\otimes \\gamma _ B ^ { - 1 } ( c ) ) \\bigr ) = ( b \\otimes c ) E . \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l } h ''' \\pm \\bar { a } h ' = 0 , \\\\ \\noalign { \\smallskip } R i c _ { N } = ( n - 2 ) \\bar { c } g _ { N } , \\end{array} \\right . \\end{align*}"}
-{"id": "3444.png", "formula": "\\begin{align*} \\alpha _ R \\ & = \\ \\int _ { R _ { i } ^ { ( k ) } \\cap \\Omega _ { k + 1 } ^ { c } } \\alpha _ { R } \\cdot \\frac { \\textbf { 1 } _ { R _ { i } ^ { ( k ) } \\cap \\Omega _ { k + 1 } ^ { c } } ( y ) } { | R _ { i } ^ { ( k ) } \\cap \\Omega _ { k + 1 } ^ { c } | } \\ d y \\\\ & \\leq 2 \\int _ { R _ { i } ^ { ( k ) } \\cap \\Omega _ { k + 1 } ^ { c } } \\alpha _ R \\frac { \\textbf { 1 } _ { R } ( y ) } { | R | } \\ d y \\\\ & \\leq 2 \\int _ { R _ { i } ^ { ( k ) } \\cap \\Omega _ { k + 1 } ^ { c } } S f ( x ) S g ( x ) d x . \\end{align*}"}
-{"id": "5092.png", "formula": "\\begin{align*} B _ i \\coloneqq \\sum _ { \\substack { \\sum _ { \\ell = 1 } ^ { R _ i } \\gamma _ { i \\ell } = \\sum _ { j = 1 } ^ r \\sum _ { k = 1 } ^ { L _ j } h _ { i j k } \\beta _ { j k } \\\\ \\sum _ { \\ell = 1 } ^ { R _ i } m _ { i \\ell } \\gamma _ { i \\ell } \\in ( q - 1 ) \\Z ^ + } } \\frac { \\bigl ( \\sum _ { j = 1 } ^ r \\sum _ { k = 1 } ^ { L _ j } h _ { i j k } \\beta _ { j k } \\bigr ) ! } { \\gamma _ { i 1 } ! \\cdots \\gamma _ { i R _ i } ! } \\ , b _ { i 1 } ^ { \\gamma _ { i 1 } } \\cdots b _ { i R _ i } ^ { \\gamma _ { i R _ i } } . \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} w ( x ) = [ 2 ^ { \\frac { N - p } { p - 1 } } - 1 ] ^ { - 1 } r ^ { \\frac { N - p } { p - 1 } } | x - y | ^ { \\frac { p - N } { p - 1 } } - [ 2 ^ { \\frac { N - p } { p - 1 } } - 1 ] ^ { - 1 } \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} \\mathcal { A } = \\{ \\lambda \\in \\C \\setminus \\{ 0 , 1 \\} ; | 1 - \\lambda | = 1 , \\Re ( \\lambda ) \\leq \\frac 1 2 , \\Im ( \\lambda ) < 0 \\} \\cup \\{ \\lambda \\in \\C : \\Re ( \\lambda ) = \\frac 1 2 , \\Im ( \\lambda ) < 0 \\} \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{align*} \\begin{cases} \\Re ( \\varphi - f - c ) \\geq C s ^ \\mu , \\\\ \\Re ( \\varphi - f + \\delta | z | ^ { \\varepsilon } ) \\geq C s ^ \\mu , \\end{cases} \\end{align*}"}
-{"id": "626.png", "formula": "\\begin{align*} | \\{ u \\leq M \\} \\cap G ( y , 3 r / 2 ) | & = | T \\bigl ( \\{ \\tilde { u } \\leq M \\} \\cap G ( ( y _ 1 / | y _ 1 | , 0 ) , 3 r / ( 2 | y _ 1 | ) ) \\bigr ) | \\\\ & = | y _ 1 | ^ 3 | \\{ \\tilde { u } \\leq M \\} \\cap G ( ( y _ 1 / | y _ 1 | , 0 ) , 3 r / ( 2 | y _ 1 | ) ) | \\\\ & \\geq \\frac { \\nu } { 2 } | y _ 1 | ^ 3 | G ( ( y _ 1 / | y _ 1 | , 0 ) , 3 r / ( 2 | y _ 1 | ) ) | \\\\ & \\geq \\frac { \\nu } { 2 } | G ( y , 3 r / 2 ) | . \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{align*} \\frac { d \\lambda } { d t } & = - \\lambda \\big [ 2 c ( \\lambda ^ 2 + \\mu ^ 2 + \\lambda \\mu ) + a \\big ] + b \\Big ( \\frac { 1 } { 3 } \\lambda ^ 2 - \\frac { 2 } { 3 } \\mu ^ 2 - \\frac { 2 } { 3 } \\lambda \\mu \\Big ) , \\\\ \\frac { d \\mu } { d t } & = - \\mu \\big [ 2 c ( \\lambda ^ 2 + \\mu ^ 2 + \\lambda \\mu ) + a \\big ] + b \\Big ( \\frac { 1 } { 3 } \\mu ^ 2 - \\frac { 2 } { 3 } \\lambda ^ 2 - \\frac { 2 } { 3 } \\lambda \\mu \\Big ) . \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{align*} a ' _ + = \\min ( a _ + + 1 + b _ + , b _ + ) < 0 , \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} { \\rm W F } ( u ) \\subset E _ - ^ * \\cap E _ + ^ * = \\{ 0 \\} \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\eta _ { t } + w _ { x } + ( \\eta w ) _ { x } + a w _ { x x x } - b \\eta _ { x x t } = 0 \\\\ w _ { t } + \\eta _ { x } + w w _ { x } + c \\eta _ { x x x } - d w _ { x x t } = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} h _ k ( n ) = \\min ( h _ k ( d ) + 2 ^ e , ( k - 1 ) ( 2 ^ { e + 1 } - 1 - n ) ) , h _ k ( 0 ) = 0 . \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} \\lambda _ { R _ { L _ t } } ( Z _ t ) = a _ t + o ( 1 ) = \\varrho \\ln _ 2 t ( 1 + o ( 1 ) ) \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\frac { 1 } { s } d \\rho ( s ) = \\log m _ 1 ( \\mu ) + a . \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} \\tau _ { E \\backslash C } = \\inf \\{ t \\geq 0 : X _ t \\in E \\backslash C \\} = \\inf \\{ t \\geq 0 : X _ t \\notin C \\} , \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} k ( x , D _ { x ' } ) f ( x ) = \\int _ { \\R ^ { n - 1 } } K _ k ( x , x ' - y ' , x _ n ) \\ , f ( y ' ) \\ , d y ' , \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{align*} \\{ \\varphi , \\psi \\} \\circ \\pi = \\{ \\overline { \\varphi } , \\overline { \\psi } \\} _ { G _ * } \\circ i = \\left ( P _ { G _ * } , d \\overline { \\varphi } \\wedge d \\overline { \\psi } \\right ) \\circ i , \\end{align*}"}
-{"id": "5053.png", "formula": "\\begin{align*} s [ d , x _ 1 , x _ 2 ] = s [ y _ 1 \\dots y _ k , x _ 1 , x _ 2 ] = Q _ 1 + Q _ 2 \\end{align*}"}
-{"id": "7488.png", "formula": "\\begin{align*} ( \\tilde \\gamma ^ { - 1 } ) ^ { k i } H _ { k \\ell } ( \\tilde \\gamma ^ { - 1 } ) ^ { \\ell j } = ( \\tilde \\gamma ^ { - 1 } ) ^ { i k } H _ { k \\ell } ( \\tilde \\gamma ^ { - 1 } ) ^ { j \\ell } = \\frac { 1 } { 2 } ( \\tilde \\gamma ^ { - 1 } ) ^ { i j } - \\frac { 1 } { 2 } ( \\tilde \\gamma ^ { - 1 } ) ^ { j i } \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{align*} W ^ { \\sigma _ q } ( x , y ) = - W ( x , y ) \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} g ( I , J ) = \\inf \\{ m \\geq 1 \\colon I _ { m } = \\emptyset J _ { m } = \\emptyset \\} , \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} \\left \\langle F _ y \\psi , \\psi \\right \\rangle = \\left \\langle \\left ( e ^ { i H t _ 0 } \\pi ( e _ y ) e ^ { - i H t _ 0 } \\right ) \\psi , \\psi \\right \\rangle = \\tau \\left ( e ^ { i h t _ 0 } \\pi ( e _ y ) e ^ { - i h t _ 0 } \\right ) = \\tau ( e _ y ) = t . \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} \\alpha = s ( 4 m + 6 r ) - m \\ell _ 4 - ( m + r ) \\ell _ 3 - ( m + 2 r ) \\ell _ 2 - ( m + 3 r ) \\ell _ 1 \\end{align*}"}
-{"id": "5327.png", "formula": "\\begin{align*} { \\rm d } ^ { \\ , 2 } f ( \\bar { x } \\ , | \\ , 0 ) ( d ) = \\max _ { \\lambda } \\left \\{ \\ , \\lambda \\| d \\| ^ 2 - d ^ T Q d \\ , \\mid \\ , \\lambda \\bar { x } = Q \\bar { x } , \\ , \\lambda \\in [ 0 , 1 ] \\ , \\right \\} . \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{align*} \\mathcal { L } ( \\gamma _ { m - 1 } ) = m ( 2 \\lambda + n - 2 ) \\ , \\gamma _ m . \\end{align*}"}
-{"id": "2900.png", "formula": "\\begin{align*} \\widehat { r } : = p _ 1 p _ 2 \\widehat { m } ^ { \\ , b } + ( p _ 1 q _ 2 + q _ 1 p _ 2 ) \\widehat { m } ^ { \\ , a } , \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{align*} f ( x ) \\ , = \\ , \\displaystyle { \\min _ { 1 \\leq i \\leq I } } \\ , \\left \\{ \\ , q _ i ( x ) + \\mbox { d i s t } _ 1 ( x ; P ^ { \\ , i } ) \\ , \\displaystyle { \\max _ { 1 \\leq j \\leq I } } \\ , \\| \\nabla q _ j ( x ) - \\nabla q _ i ( x ) \\| _ 1 + \\displaystyle { \\frac { 3 \\ , L _ i } { 2 } } \\ , \\left [ \\ , \\mbox { d i s t } _ 1 ( x ; P ^ { \\ , i } ) \\ , \\right ] ^ 2 \\ , \\right \\} . \\end{align*}"}
-{"id": "8947.png", "formula": "\\begin{align*} S ( E ) : = \\pi ( H ^ 0 - E ) \\pi - \\pi H ^ 0 \\pi ^ \\perp [ \\pi ^ \\perp ( H ^ 0 - E ) \\pi ^ \\perp ] ^ { - 1 } \\pi ^ \\perp H ^ 0 \\pi \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ g p _ j n _ j \\big [ v ( n - e _ j ) - v ( n ) \\big ] = 0 \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( 1 - \\lambda \\right ) \\max \\{ e ^ { D _ { \\varpi } ( P _ 0 \\parallel Q _ 0 ) } , e ^ { D _ { \\varpi } ( P _ 1 \\parallel Q _ 1 ) } \\} + \\lambda \\max \\{ e ^ { D _ { \\varpi } ( P _ 0 \\parallel Q _ 0 ) } , e ^ { D _ { \\varpi } ( P _ 1 \\parallel Q _ 1 ) } \\} \\ge e ^ { D _ { \\varpi } ( P _ { \\lambda } \\parallel Q _ { \\lambda } ) } . \\end{aligned} \\end{align*}"}
-{"id": "2373.png", "formula": "\\begin{align*} \\tilde { J } _ 2 ( N ; \\theta ) = O \\left ( e ^ { - \\varepsilon N } \\right ) , N \\to \\infty . \\end{align*}"}
-{"id": "8060.png", "formula": "\\begin{align*} \\alpha = s , \\ \\ \\ \\ \\gamma = 1 , \\ \\ \\ \\ \\beta = L ( \\xi , \\sigma ) , \\ \\ \\ \\ \\nu = \\pm s . \\end{align*}"}
-{"id": "8791.png", "formula": "\\begin{align*} \\phi ( k g h ) = \\phi ( g ) - 2 \\ln | \\chi ( h ) | . \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} \\mu ( 3 ) & = M ( 0 , 1 , 3 ) \\approx 0 . 6 0 7 3 4 6 , \\\\ \\mu ( 4 ) & = M ( 0 , 1 , 2 , 4 ) \\approx 0 . 7 5 2 3 9 4 , \\\\ \\mu ( 5 ) & = M ( 0 , 1 , 2 , 6 , 9 ) = 1 . \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} = & ( \\gamma ^ { - 1 } ) ^ { i j } ( t , q ) \\left ( - \\nabla _ q V ( t , q ) + \\tilde F ( t , q ) \\right ) _ j \\\\ & + \\beta ^ { - 1 } \\partial _ { q ^ k } ( \\gamma ^ { - 1 } ) ^ { i k } \\\\ & - ( \\gamma ^ { - 1 } ) ^ { i j } \\partial _ { q ^ j } \\beta ^ { - 1 } \\\\ & - \\beta ^ { - 1 } \\partial _ { q ^ j } ( \\gamma ^ { - 1 } ) ^ { i j } \\\\ = & ( \\gamma ^ { - 1 } ) ^ { i j } ( t , q ) \\left ( - \\nabla _ q V ( t , q ) + \\tilde F ( t , q ) \\right ) _ j \\\\ & - ( \\gamma ^ { - 1 } ) ^ { i j } \\partial _ { q ^ j } \\beta ^ { - 1 } \\\\ \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} g _ 0 ( ( \\nabla _ X ^ { g _ 0 + t g _ 1 } - \\nabla _ X ^ { g _ 0 } ) X , N ) = k ( X , X ) \\ \\ \\forall \\ , X \\in T ( { } ^ 0 \\Sigma ) ^ \\circ ; \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{align*} \\oint _ { C _ R } i \\xi \\ , e ^ { i \\xi z } \\left [ 1 - \\left ( 1 - e ^ { - z } \\right ) ^ N \\right ] d z = 0 \\ ; R > A _ N . \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\beta ^ - } f _ { \\gamma } ( z ) = f _ { \\beta } ( z ) = \\lim _ { \\gamma \\to \\beta ^ + } f _ { \\gamma } ( z ) , \\end{align*}"}
-{"id": "7958.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\exp [ u ( \\beta _ t ) ] = \\lim _ { z \\rightarrow \\infty } \\exp [ u ( z ) ] = \\lim _ { z \\rightarrow \\infty } z / \\eta _ \\mu ( z ) = \\int _ 0 ^ \\infty x ^ { - 1 } d \\mu ( x ) . \\end{align*}"}
-{"id": "5874.png", "formula": "\\begin{align*} S _ m ( \\delta ^ { + } ) \\sim S _ m ( \\nu ^ { + } ) , | \\delta | = | \\nu | . \\end{align*}"}
-{"id": "6566.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathcal { A } } K \\ , d S = - \\int \\limits _ { \\partial \\mathcal { A } } \\kappa \\ , d \\ell \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} \\left \\{ \\ \\begin{array} { l } \\displaystyle - \\Delta _ \\Phi { _ { \\epsilon } } u = f ( x ) , ~ \\mbox { i n } ~ \\Omega , \\\\ \\\\ u = 0 ~ \\mbox { o n } ~ \\partial \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} d _ n = d _ { n - 1 } + b _ { n - 1 } - b _ { n - 2 } + \\binom { n - 3 } { 2 } - \\binom { n - 3 } { 5 } + \\sum _ { a = 3 } ^ { n - 3 } \\sum _ { \\ell = 0 } ^ { n - 3 - a } \\sum _ { m = 1 } ^ { n - 2 - a - \\ell } \\binom { n - 5 - \\ell - m } { a - 3 } m , \\\\ \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} ( a ; q ) _ n = \\prod _ { k = 0 } ^ { n - 1 } ( 1 - a q ^ k ) , \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{align*} \\lambda ( F _ i ^ * ) = \\sum { a _ { i j } } e _ j . \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{align*} \\mathfrak { N } _ { t } = i \\big ( R \\mathfrak { N } - \\psi T \\big ) . \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} \\max _ { \\substack { q _ { X Y } : \\ : q _ { X Y } \\preceq p _ { X Y } } } \\rho _ m ^ 2 ( q _ { X Y } ) & = ( 1 - 2 p ) ^ 2 . \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} g ( r ) & = 0 \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{align*} d = 2 \\left [ \\frac { n } { 1 0 } \\right ] + 2 \\end{align*}"}
-{"id": "2683.png", "formula": "\\begin{gather*} q \\left ( \\bar { \\lambda } \\left ( D ^ { 2 } \\bar { w } _ { \\infty } \\right ) \\right ) = 1 / \\left [ \\left ( n - 1 \\right ) K \\right ] , \\\\ D ^ { 2 } \\bar { w } _ { \\infty } \\left ( y \\right ) \\geq 0 , D ^ { 2 } \\bar { w } _ { \\infty } , \\ D _ { 1 1 } \\bar { w } _ { \\infty } \\ 0 \\ 0 . \\end{gather*}"}
-{"id": "412.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } \\exp ( - s t ) f ( t ) d t = g ( s ) + e \\ , , \\end{align*}"}
-{"id": "3264.png", "formula": "\\begin{align*} | [ Q _ { | \\textup { \\textbf { m } } | } ^ { \\textup { \\textbf { F } } } \\Phi _ k ] _ { \\nu } | = \\left | \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ { \\rho _ 2 - 2 \\delta } } \\frac { Q _ { | \\textup { \\textbf { m } } | } ^ { \\textup { \\textbf { F } } } ( t ) \\Phi _ k ( t ) \\Phi ' ( t ) } { \\Phi ^ { \\nu + 1 } ( t ) } d t \\right | \\leq c _ { 1 2 } \\frac { ( \\rho _ 2 - \\delta ) ^ k } { ( \\rho _ 2 - 3 \\delta ) ^ { \\nu } } , \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} \\mathbf { s } = \\{ x _ i a _ i U _ i x _ { i - 1 } ^ { - 1 } \\mid i \\in \\{ 1 , \\ldots , n \\} \\} \\end{align*}"}
-{"id": "1950.png", "formula": "\\begin{align*} \\begin{array} { r l } M = \\left \\{ ( 0 , 0 , 0 ) , ( 2 , 0 , 1 ) , ( 4 , 0 , 2 ) , ( 6 , 0 , 3 ) , ( 0 , 0 , 4 ) , ( 2 , 0 , 5 ) , ( 4 , 0 , 6 ) , ( 6 , 0 , 7 ) , \\right . \\\\ \\left . ( 0 , 4 , 0 ) , ( 2 , 4 , 1 ) , ( 4 , 4 , 2 ) , ( 6 , 4 , 3 ) , ( 0 , 4 , 4 ) , ( 2 , 4 , 5 ) , ( 4 , 4 , 6 ) , ( 6 , 4 , 7 ) \\right \\} \\end{array} \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{align*} \\norm { \\omega _ 1 ^ n } & \\le \\delta \\sup _ { x \\in [ 0 , \\delta ^ 2 ] } \\abs { f ^ { \\prime } ( x ) } \\cdot \\Big ( \\norm { U _ h ^ { n - 1 / 2 } } _ \\infty + \\norm { u ^ { n - 1 / 2 } } _ \\infty \\Big ) \\norm { U _ h ^ { n - 1 / 2 } - u ^ { n - 1 / 2 } } \\\\ & \\le C \\delta ^ 2 \\norm { e ^ { n - 1 / 2 } } . \\end{align*}"}
-{"id": "6731.png", "formula": "\\begin{align*} H \\left ( p , t \\right ) & = \\left \\Vert Q _ { p } \\left ( T - t \\right ) \\hat { B } ^ { \\dagger } e ^ { - \\left ( T - t \\right ) \\hat { A } ^ { \\dagger } } p \\right \\Vert _ { 1 } \\\\ & - \\left \\Vert Q _ { e } \\hat { D } ^ { \\dagger } e ^ { - \\left ( T - t \\right ) \\hat { A } ^ { \\dagger } } p \\right \\Vert _ { 1 } , \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} \\bar * \\overline \\square _ B = \\square _ T \\bar * , \\bar * \\square _ B = \\overline \\square _ T \\bar * . \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} ( \\hat { z } _ 1 , \\dots , \\hat { z } _ J ) = q _ i ^ { - l _ 0 } ( q _ i ^ { J - 1 } u _ 0 , \\dots , u _ 0 ) . \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} \\frac { \\xi } { x _ A + x _ B } + \\frac { \\eta } { x _ A + x _ C } = \\frac { \\xi } { x _ A + x _ B } + \\frac { \\zeta } { x _ B + x _ C } = \\frac { \\eta } { x _ A + x _ C } + \\frac { \\zeta } { x _ B + x _ C } \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} \\varepsilon L ^ X W ( x , y ) & = \\varepsilon L ^ X F ( x , y ) W ( x , y ) + \\beta ^ { - 1 } \\abs { \\nabla _ x F ( x , y ) } ^ 2 W ( x , y ) \\\\ & \\leq - ( \\lambda _ \\theta \\kappa - \\Lambda _ \\theta \\beta ^ { - 1 } ) F ( x , y ) W ( x , y ) + \\norm { G } _ \\infty W ( x , y ) \\\\ & = - ( ( \\lambda _ \\theta \\kappa - \\Lambda _ \\theta \\beta ^ { - 1 } ) F ( x , y ) - \\norm { G } _ \\infty ) W ( x , y ) . \\end{align*}"}
-{"id": "6886.png", "formula": "\\begin{align*} \\mathrm { l o c } _ { x } ( g , \\mathcal { S } _ { V } ) = \\mathrm { l o c } _ { x ' } ( g ' , \\mathcal { S } _ { V } ) . \\end{align*}"}
-{"id": "6577.png", "formula": "\\begin{align*} B _ j = - \\sqrt { 1 - b ^ 2 } \\ , \\Big | _ { c 2 ^ j } ^ { c 2 ^ { j + 1 } } \\asymp \\sqrt { 1 - c ^ 2 2 ^ { 2 j } } - \\sqrt { 1 - c ^ 2 2 ^ { 2 j + 2 } } \\asymp ( c 2 ^ j ) ^ 2 . \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} N _ { ( a , b ) , ( c , d ) } = b + 1 \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} p ( 0 , 0 | v , w ) + p ( 0 , 1 | v , w ) & = p ( 0 , 0 | v , w ) + p ( 1 , 0 | v , w ) = t , \\\\ p ( 0 , 1 | v , w ) + p ( 1 , 1 | v , w ) & = p ( 1 , 0 | v , w ) + p ( 1 , 1 | v , w ) = 1 - t , \\end{align*}"}
-{"id": "2531.png", "formula": "\\begin{align*} r ( Q _ { \\lambda _ 0 } ) = 1 \\ , , \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} \\mathrm { S u p p } ( M ) = \\lbrace ( s , t ) \\in S \\times T \\mid M _ { s , t } \\neq 0 \\rbrace . \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} g ^ { ( h ) } ( \\hat { x } ) & \\cdot S ^ { - 1 } T ^ { - 1 } d S \\wedge d T = ( \\sum _ { k = 0 } ^ h p ^ k [ c _ k ] ^ { p ^ { h - k } } ) S ^ { \\ell _ 1 p ^ h - 1 } T ^ { \\ell _ 2 p ^ h - 1 } d S \\wedge d T , \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{align*} ( - 1 ) ^ k s _ k / k = \\sum _ { i _ 1 + 2 i _ 2 + \\cdots + k i _ k = k } ( - 1 ) ^ { i _ 1 + \\cdots + i _ k } \\frac { ( i _ 1 + \\cdots + i _ k - 1 ) ! } { i _ 1 ! \\cdots i _ k ! } \\sigma _ 1 ^ { i _ 1 } \\cdots \\sigma _ k ^ { i _ k } \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{align*} 8 \\pi \\frac { \\mathcal { H } ( u ) } { M ^ 2 ( u ) } = \\frac { 1 + 4 b ^ 2 + 2 b ^ 4 } { 2 ( 1 + b ^ 2 ) ^ 2 } \\in [ \\frac 1 2 , 1 ) . \\end{align*}"}
-{"id": "8955.png", "formula": "\\begin{align*} \\mathfrak { S } ^ A \\circ \\mathfrak { O p } ^ A = I \\ , . \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} \\mathcal { P } _ u ( \\lambda , \\gamma ) = \\Phi _ { u , \\gamma } ( 0 , h , \\lambda ) - \\frac { 1 } { 2 } \\int _ 0 ^ u \\xi '' ( s ) s \\gamma ( s ) d s , \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} \\deg _ L \\alpha = \\deg ( c _ 1 ( L ) ^ { r } \\cap \\alpha ) \\end{align*}"}
-{"id": "6230.png", "formula": "\\begin{align*} ( E _ \\lambda - I ) V _ \\lambda = 0 , \\\\ E _ \\lambda V _ { \\lambda ' } = 0 & & , \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} \\mathbf { w } _ { 1 } ^ { \\mathbf { 0 } } = \\int _ { 0 } ^ t e ^ { \\mathbf { A } ( t - s ) } \\mathbf { G } _ { e x t } ( s ) d s , \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} f ^ * ( x _ M ^ { m - k + n } \\times 1 ) = \\pm \\frac { 1 } { \\lambda } \\cdot ( x _ M ^ { m - k + n } \\times 1 ) , \\ \\frac { 1 } { \\lambda } \\neq 1 . \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} \\Rightarrow L _ { x \\alpha \\cdot x ^ { \\rho } } = L _ { x } ^ { - 1 } L _ { x \\alpha } . \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{align*} \\mathrm { h t } ( \\Gamma ( { \\mathfrak X } , { \\mathcal O } _ { { \\mathfrak X } } ) ) = \\max \\{ \\mathrm { h t } ( ( { \\mathcal G } _ \\Omega ) ^ { \\mathrm { f o r } } ) \\ , \\mid \\ , \\Omega \\to X \\mbox { a g e o m e t r i c p o i n t } \\} , \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{align*} r = \\mbox { I m } ( r _ c ) . \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} g ( k + 1 , n ) - g ( k + 1 , n - 1 ) = g ( k , n ) + [ ( 4 k - 6 ) n - ( 2 k - 5 ) ] \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{align*} \\sigma ( i , j , k , l , \\alpha , \\beta ) : = [ T _ { i j } ^ \\alpha , T _ { k l } ^ \\beta ] i < j \\in [ n ] , \\ k < l \\in [ n ] , \\ i < k , \\ \\# \\{ i , j , k , l \\} = 4 , \\ \\alpha , \\beta \\geq 0 , \\end{align*}"}
-{"id": "137.png", "formula": "\\begin{align*} \\Phi _ \\infty ^ * = \\begin{pmatrix} 0 & | f | ^ { - 1 / 2 } \\kappa ^ { - 1 } \\\\ | f | ^ { - 1 / 2 } \\kappa \\bar f & 0 \\end{pmatrix} d z , \\varphi _ \\infty = \\begin{pmatrix} 0 & \\frac 1 2 | f | ^ { - 1 / 2 } \\kappa ^ { - 1 } \\dot f \\\\ \\frac 1 2 | f | ^ { 1 / 2 } \\kappa \\dot f / f & 0 \\end{pmatrix} d z \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{align*} J _ 2 ( N ; \\theta ) = O \\left ( e ^ { - \\varepsilon N } \\right ) , N \\to \\infty \\end{align*}"}
-{"id": "9191.png", "formula": "\\begin{align*} \\Upsilon _ n - \\Upsilon _ n ' \\cdot \\gamma ^ n = \\Upsilon _ { n - 1 } - \\Upsilon _ { n - 1 } ' \\cdot \\gamma ^ n \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} \\hat { y } = \\{ S ^ e , T \\} \\prod \\{ 1 + [ a _ { i j } ] S ^ i T ^ j , S \\} \\cdot \\prod \\{ 1 + [ b _ { i j } ] S ^ i T ^ j , T \\} . \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} & H _ 0 = \\int _ { \\mathbb { R } } \\frac { 1 } { | k | } | \\hat h _ 0 | ^ 2 d k \\\\ \\mbox { a n d } & \\sup _ { t \\ge 0 } | t ^ { 1 / 3 } k | ^ { n } \\exp ( - 2 | t ^ { 1 / 3 } k | ^ 3 ) = c ( n ) \\in ( 0 , \\infty ) , \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} \\left ( \\widehat { e } _ 0 ^ + \\widehat { e } _ 0 ^ - - \\widehat { e } _ 0 ^ - \\widehat { e } _ 0 ^ + \\right ) v = - \\left ( q ^ { ( 1 - N ) / 2 } \\sum _ m ( - 1 ) ^ { \\mu _ m } q ^ { \\kappa ( m , \\mu , \\lambda ) } \\right ) v , \\end{align*}"}
-{"id": "9180.png", "formula": "\\begin{align*} W _ { n , r } = u \\sum _ { n _ 1 , n _ 2 \\geq 0 } ^ { n _ 2 - n _ 1 = n } H _ { - n _ 1 } H _ { n _ 2 } \\Big | _ { \\Delta } \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} { \\rm C o e f f } [ f _ { \\delta } , p + m _ 1 ] = { \\rm C o e f f } [ E _ { \\delta } , p + m _ 1 ] = \\lim _ { q \\rightarrow t ^ { - p - m _ 1 } } ( 1 - q t ^ { p + m _ 1 } ) \\left ( \\sum _ { \\nu \\in \\sigma ( \\epsilon ) } c _ { \\nu } ( q , t ) E _ { \\nu } ( z ; q , t ) \\right ) , \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} \\Omega _ { j _ 1 , \\bar { j } _ 2 } = \\frac { 1 } { 4 } d ^ 2 _ a u ( l _ { j _ 1 } , l _ { j _ 2 } ) . \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{align*} \\beta ^ * _ - ( \\kappa ) = \\max \\bigg \\{ \\frac { 1 } { q } + \\frac { d - 1 } { 2 r } - \\frac { d + \\kappa } { 2 } , - \\frac { d + 1 + 2 \\kappa } { 4 } \\bigg \\} . \\end{align*}"}
-{"id": "6687.png", "formula": "\\begin{align*} T = E _ 2 ^ { - 1 } A ( { E _ 1 ^ * } ) ^ { - 1 } , \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} H ( \\nu , \\mathcal { E } ) : = - \\sum _ { E \\in \\mathcal { E } } \\nu ( E ) \\log \\nu ( E ) , \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} S _ m ( \\mu ) \\sim S _ m ( \\nu ) \\iff \\ \\exists \\ \\sigma \\ \\ S _ m ( \\mu ) = \\sigma \\cdot S _ m ( \\nu ) . \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} ( k - 1 ) + \\sum _ { j = 1 } ^ k ( \\mu _ j - 1 ) = m - 1 . \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} d \\mu _ - I _ + + d \\mu _ 0 I _ 0 + d \\mu _ + I _ - = 0 \\rlap { . } \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} \\d { u } { \\xi _ r } & = \\d { v } { \\xi _ i } = \\frac { \\Re \\left ( \\sqrt { g _ \\Omega ( \\xi ) } \\right ) } { | g _ \\Omega ( \\xi ) | } \\\\ \\d { u } { \\xi _ i } & = - \\d { v } { \\xi _ r } = \\frac { \\Im \\left ( \\sqrt { g _ \\Omega ( \\xi ) } \\right ) } { | g _ \\Omega ( \\xi ) | } \\end{align*}"}
-{"id": "8230.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\pi _ 1 ( \\tilde { F _ j } ) & = 0 , \\ \\\\ \\pi _ 2 ( \\tilde { F _ n } ) & = 0 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} \\omega _ 1 = \\int _ { 1 } ^ { \\infty } \\frac { d X } { \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } , ~ \\omega _ 2 = \\int _ { 0 } ^ { - \\infty } \\frac { d X } { \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} \\lambda _ 1 + \\cdots + \\lambda _ g = 0 , \\lambda _ 1 = - ( \\lambda _ 2 + \\cdots + \\lambda _ g ) , \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} M _ i [ t ^ { - \\chi ( \\cdot ) } ] ( \\vec { x } , \\vec { y } ) = t \\cdot t ^ { - \\chi ( \\vec { x } , \\vec { y } ) - 1 } - t ^ { - \\chi ( \\vec { x } , \\vec { y } ) } = 0 . \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} L _ i [ \\psi ] ( \\nu ) = \\sum _ { \\nu ' \\in \\mathbb { A } } \\ell _ i ( \\nu , \\nu ' ) \\psi ( \\nu ' ) , \\end{align*}"}
-{"id": "3425.png", "formula": "\\begin{align*} H = \\frac { 1 } { 2 } \\langle x , \\nu \\rangle \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} H _ + ( u ' ( r ) ) + ( N - 1 ) \\int _ 0 ^ r \\frac { ( u ' ( s ) ) ^ 2 } { s \\sqrt { 1 + ( u ' ( s ) ) ^ 2 } } d s = G ( \\xi ) - G ( u ( r ) ) , \\end{align*}"}
-{"id": "7797.png", "formula": "\\begin{align*} \\mathcal { I } _ 3 ( t , x ) = \\frac { 1 } { 8 } \\int _ 0 ^ t \\int _ { S } \\frac { \\partial G _ { t - s } } { \\partial x } ( x - z ) \\psi ( s , z ) \\sigma _ s ( y ) ^ 2 d z d y d s . \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{align*} { } P ( E , N ) = \\frac { 1 } { ( 2 i \\pi ) ^ 2 } \\oint \\oint d x \\ , d z \\frac { Z _ E ( x , z ) } { z ^ { N + 1 } \\ , x ^ { E + 1 } } \\end{align*}"}
-{"id": "4420.png", "formula": "\\begin{align*} m - a & = m - ( 0 \\vee ( 2 m - \\ell ) \\vee ( m - n + \\ell - b ) ) \\\\ & \\geq m - ( 0 \\vee ( m + \\ell - 1 - \\ell ) \\vee ( m - 1 ) ) = m - ( m - 1 ) = 1 , \\end{align*}"}
-{"id": "714.png", "formula": "\\begin{align*} ( \\mathbb { A } + \\delta \\mathbb { A } ) \\operatorname { v e c } ( \\tilde { Y } ) = \\operatorname { v e c } ( F ) , \\frac { \\norm { \\delta \\mathbb { A } } _ 2 } { \\norm { \\mathbb { A } } _ 2 } \\leq \\varepsilon _ { A _ k } , \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} \\lambda _ j ( 0 ) = \\frac { 1 } { n + 1 } , 1 \\leq j \\leq n + 1 . \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} - \\delta _ f ( Q ( t , x ) ) + \\textstyle { \\frac { \\partial f ( t , x ) } { \\partial x } } ^ \\top Q ( t , x ) + Q ( t , x ) \\frac { \\partial f ( t , x ) } { \\partial x } + { \\frac { \\partial h ( t , x ) } { \\partial x } } ^ \\top \\frac { \\partial h ( t , x ) } { \\partial x } = 0 . \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{align*} C _ { \\alpha { } } D _ { \\theta { } } ^ { \\alpha { } } \\phi ( x ) - \\gamma { } \\delta { } ( x ) \\phi ( x ) = E \\phi ( x ) . \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{align*} I _ M < \\int _ 0 ^ 1 x ^ { \\lambda - 1 } \\left ( 1 - x \\right ) ^ { \\nu _ 1 M } d x = B ( \\lambda , \\nu _ 1 M + 1 ) = \\frac { \\Gamma ( \\lambda ) \\ , \\Gamma ( \\nu _ 1 M + 1 ) } { \\Gamma ( \\lambda + \\nu _ 1 M + 1 ) } , \\end{align*}"}
-{"id": "6345.png", "formula": "\\begin{align*} \\widetilde { I M } : = \\bigcup _ { m \\geqslant 1 } ( I ^ { m + 1 } M : _ M I ^ m ) . \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{align*} \\frac { 1 } { a } = \\frac { 1 - \\theta _ 0 } { a _ 0 } + \\frac { \\theta _ 0 } { a _ 1 } , \\frac { 1 } { b } = \\frac { 1 - \\theta _ 1 } { b _ 0 } + \\frac { \\theta _ 1 } { b _ 1 } , 0 < \\theta _ 0 , \\theta _ 1 < \\theta < 1 , \\theta _ 0 + \\theta _ 1 = \\theta . \\end{align*}"}
-{"id": "2011.png", "formula": "\\begin{align*} X _ m : = \\left \\{ j \\big { | } \\left \\langle \\theta _ l \\otimes \\chi _ j , I \\left ( \\left ( \\sum _ { i = 0 } ^ { 2 ^ n - 1 } \\theta _ i \\right ) \\otimes \\chi _ m \\right ) \\right \\rangle \\neq 0 , l \\right \\} , \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{align*} P ( h ) : = \\rho ^ { - 3 } P _ 0 ( g ) = 0 , \\ \\ g = g _ m + \\rho h , \\end{align*}"}
-{"id": "5603.png", "formula": "\\begin{align*} \\theta _ 1 ( Z , Z ) = \\tfrac 1 2 ( A _ 1 - i A _ 2 ) ( \\nu _ 1 + i \\nu _ 2 ) d z ^ 2 , \\theta _ 2 ( Z , Z ) = \\tfrac 1 2 ( A _ 1 + i A _ 2 ) ( \\nu _ 1 - i \\nu _ 2 ) d z ^ 2 , \\end{align*}"}
-{"id": "6058.png", "formula": "\\begin{align*} \\begin{cases} \\eta ( 0 , t ) = 0 , \\ , \\ , \\eta ( L , t ) = 0 , \\ , \\ , \\eta _ { x } ( 0 , t ) = f ( t ) , & t \\in ( 0 , \\infty ) , \\\\ w ( 0 , t ) = 0 , \\ , \\ , w ( L , t ) = 0 , \\ , \\ , w _ { x } ( L , t ) = 0 , & t \\in ( 0 , \\infty ) . \\end{cases} \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} \\mathrm { N m } _ { F ' / F } \\left ( c _ { F ' } \\right ) & = \\left ( \\prod _ { \\ell \\in S ' \\setminus S } \\left ( P _ \\ell \\left ( \\sigma ^ { - 1 } _ \\ell \\right ) \\right ) \\right ) \\cdot c _ F \\\\ & = \\left ( \\prod _ { \\ell \\in S ' \\setminus S } \\left ( 1 - \\overline { a _ \\ell ( f ) } \\ell ^ { - 1 } \\sigma ^ { - 1 } _ \\ell + \\overline { \\psi } ( \\ell ) \\ell ^ { - 1 } \\sigma ^ { - 2 } _ \\ell \\right ) \\right ) \\cdot c _ F \\end{align*}"}
-{"id": "5222.png", "formula": "\\begin{align*} f _ { n } ( t - t _ { 0 n } , x - x _ { n } ) : = & a ( t , x ) - \\chi \\lambda v ( t , x ; t _ { 0 } , x _ n , u _ { 0 n } ( \\cdot + x _ n ) ) \\cr & - ( b ( t , x ) - \\chi \\mu ) ( u ( t , x ; t _ { 0 } , x _ n , u _ { 0 n } ( \\cdot + x _ n ) ) + u ( t , x ; t _ { 0 } , x _ n , \\tilde u _ { 0 n } ) ) . \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} { \\partial _ { x _ l } u _ i } = q _ { i l } , , \\ \\ \\forall i , l = 1 , . . . , m . \\end{align*}"}
-{"id": "2819.png", "formula": "\\begin{align*} \\Vert K _ { \\varepsilon } \\Vert _ 1 + \\sum _ { \\ell = 1 } ^ d \\Vert G ^ { \\ell } _ { \\varepsilon } \\Vert _ 1 \\leq C \\ , \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} K _ M \\cdot c _ 1 = K _ \\eta c _ 0 ^ T c _ 1 + K _ B c _ 1 - c _ 1 ( - b + K _ \\eta ^ T c _ 0 ) - c _ 0 K _ \\eta ^ T c _ 1 = - c _ 2 + ( c _ 1 ) _ x \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} K _ t = N _ x + [ K , N ] \\end{align*}"}
-{"id": "7129.png", "formula": "\\begin{align*} x ( 0 ) = a _ { 4 } , x \\left ( \\frac { \\omega _ { 1 } } { 2 } \\right ) = a _ { 3 } , x \\left ( \\frac { \\omega _ { 1 } + \\omega _ { 2 } } { 2 } \\right ) = a _ 2 , x \\left ( \\frac { \\omega _ { 2 } } { 2 } \\right ) = a _ 1 . \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} \\zeta _ { m } = \\Vert ( I - P _ { m } ) A ^ { T } A \\Vert \\leq \\Vert A \\Vert \\cdot \\Vert ( I - P _ { m } ) A ^ T \\Vert = \\sigma _ { 1 } \\Vert A ( I - P _ { m } ) \\Vert \\ , . \\end{align*} % \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} T _ { f ^ d } \\le ( d - 1 ) T _ f + \\sum _ { i = 0 } ^ { d - 1 } T _ { g _ i } + O ( 1 ) \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\arg ( z _ { J _ n , n } ) = \\lim _ { n \\to + \\infty } 2 \\pi \\frac { J _ n } { n } = 2 \\arcsin \\bigl ( \\frac { \\kappa } { 2 } \\bigr ) = 0 . 1 7 1 7 8 4 \\ldots \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{align*} d \\alpha _ 2 = ( \\lambda + k ) \\beta _ 2 , d ^ \\ast \\beta _ 2 = ( \\lambda + n - k ) \\alpha _ 2 . \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} \\nabla g ( \\overline { x } , \\overline { t } ) ( H , \\omega ) = \\left ( \\begin{matrix} { \\rm d i a g } ( H ) \\\\ \\langle E , H \\rangle + \\omega \\end{matrix} \\right ) \\quad \\forall ( H , \\omega ) \\in \\mathbb { S } ^ 2 \\times \\mathbb { R } , \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} a = e ^ { 2 i \\alpha / r } , b = e ^ { 2 i \\beta / r } , q = e ^ { 2 i / r } , p = e ^ { 2 \\pi i \\tau } = e ^ { - 2 \\pi \\kappa } \\end{align*}"}
-{"id": "383.png", "formula": "\\begin{align*} H ( X | Y ) \\geq \\sum _ { i = 1 } ^ { k } G ( d _ k ( i ) ) \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} P ( [ \\psi ] ) = \\mathrm { d i v } \\ , T ^ { \\Phi ( [ \\psi ] ) } \\cdot \\psi . \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{align*} a _ i \\vee b _ j \\geq \\frac { 1 } { 2 } d ( x , y ) \\qquad i , j = 1 , 2 , \\dots , k . \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} \\bar { \\mathfrak { l } } _ 0 = \\mathfrak { t } \\oplus \\bigoplus _ { \\alpha \\in \\Phi _ L ^ { \\sigma } } \\mathfrak { g } _ { \\alpha } , \\bar { \\mathfrak { l } } _ { \\gamma } = \\bigoplus _ { \\bar { \\alpha } | _ { T _ s } = 2 \\gamma } \\mathfrak { g } _ { \\alpha } \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} \\dd { M ^ x } ( t ) & = b ( t , M ^ x ( t ) ) \\dd { t } + \\sigma ( t , M ^ x ( t ) ) \\hat { W } ( t ) \\\\ M ^ x _ 0 & = x . \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} ( l - z \\frac { d } { d z } ) \\widetilde C _ l ^ { \\alpha } ( z ) = & - 2 \\widetilde C _ { l - 2 } ^ { \\alpha + 1 } ( z ) , \\\\ \\frac d { d z } \\widetilde C _ l ^ { \\alpha } ( z ) = & 2 \\gamma ( \\alpha , l ) \\widetilde C _ { l - 1 } ^ { \\alpha + 1 } ( z ) , \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} \\tilde { Z } ( t ) = & \\int _ { S } ^ { t } \\left ( D u ( s , X ^ x ( s ) ) B ( s , X ^ x _ s ) - D u ( s , X ^ y ( s ) ) B ( s , X ^ y _ s ) \\right ) \\dd { s } \\\\ & + \\int _ { S } ^ { t } \\left ( D u ( s , X ^ x ( s ) ) \\sigma ( s , X ^ x ( s ) ) - D u ( s , X ^ y ( s ) ) \\sigma ( s , X ^ y ( s ) ) \\right ) \\dd { W } ( s ) \\end{align*}"}
-{"id": "5415.png", "formula": "\\begin{align*} F ( x ) = y ^ \\dagger , x \\in X , \\end{align*}"}
-{"id": "8858.png", "formula": "\\begin{align*} \\mathfrak { h } _ { \\mu } = \\mathfrak { p } _ { I ( \\mu ) } ^ u \\oplus \\mathfrak { l } _ { I ( \\mu ) } ^ { \\sigma _ { \\mu } } . \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} t _ k = \\min \\{ t \\in [ 0 , d ] \\colon \\log | f _ k ( \\alpha _ k + i t ) | \\leq A \\} , \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} & E \\oplus B _ r : = \\bigcup _ { x \\in E } B _ r ( x ) = ( \\partial E \\oplus B _ r ) \\cup E = ( \\partial E \\oplus B _ r ) \\cup ( E \\ominus B _ r ) , \\\\ { \\mbox { w h e r e } } & E \\ominus B _ r : = E \\setminus \\left ( \\bigcup _ { x \\in \\partial E } B _ r ( x ) \\right ) = E \\setminus \\big ( \\partial E ) \\oplus B _ r \\big ) . \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} u _ t ( x , t + t _ { 0 n } ; t _ { 0 n } , u _ { 0 n } ) & = \\Delta u ( x , t + t _ { 0 n } ; t _ { 0 n } , u _ { 0 n } ) - \\chi \\nabla \\cdot ( u ( x , t + t _ { 0 n } ; t _ { 0 n } , u _ { 0 n } ) \\nabla v ( x , t + t _ { 0 n } ; t _ { 0 n } , u _ { 0 n } ) ) \\\\ & + ( a ( x , t + t _ { 0 n } ) - b ( x , t + t _ { 0 n } ) u ( x , t + t _ { 0 n } ; t _ { 0 n } , u _ { 0 n } ) ) u ( x , t + t _ { 0 n } ; t _ { 0 n } , u _ { 0 n } ) . \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} \\| J q ^ k ( f ) \\| _ 2 & = \\| \\sum _ { J \\in N } a _ J J q ^ k ( \\xi ^ J ) \\| _ 2 \\\\ & \\le \\max _ { J \\in N } \\{ | a _ J | _ 2 \\} \\\\ & = \\| f \\| _ 2 . \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{align*} \\lambda = \\lambda ^ * : = \\frac { A + B } { 2 B } = 1 - \\frac n 2 + \\frac { n - 2 } { 2 t } . \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} h ( y ) = ( 1 - y ) ^ r = \\sum _ { k = 0 } ^ n ( - 1 ) ^ k \\binom { r } { k } y ^ k + \\frac { h ^ { ( n + 1 ) } ( \\xi ) } { ( n + 1 ) ! } y ^ { n + 1 } , \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{align*} ( \\rho _ + , Z ) : = ( \\hat \\tau , \\hat x / \\hat \\tau ) = \\bigl ( t / ( t ^ 2 - r ^ 2 ) , x / t \\bigr ) \\in [ 0 , \\infty ) \\times \\R ^ 3 \\end{align*}"}
-{"id": "1608.png", "formula": "\\begin{align*} s ^ \\xi _ t : = a _ t h _ t ^ 2 , s ^ \\sigma _ t : = \\exp \\{ h _ t ^ 2 \\ln a _ t \\} . \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} \\dd { X ^ x } ( t ) & = b ( t , X ^ x ( t ) ) \\dd { t } + \\sigma ( t , X ^ x ( t ) ) \\dd { W } ( t ) , \\\\ X ^ x ( 0 ) & = x \\in \\R \\end{align*}"}
-{"id": "3734.png", "formula": "\\begin{align*} B _ u : = g _ { u _ 1 } \\circ \\dots \\circ g _ { u _ n } ( [ - R , R ] ) \\end{align*}"}
-{"id": "2498.png", "formula": "\\begin{align*} | A + B | = | A | + | B | + r - 1 \\leq \\tfrac { 1 } { 2 } ( p + | A | + | B | ) - 2 , \\ ; \\ ; \\textrm { a n d } \\ ; \\ ; r \\leq | B | - 3 . \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{align*} H _ n f ( t , \\mu , w ) & : = \\frac { 1 } { n } e ^ { - n f ( t , \\mu , w ) } \\cdot \\vec { A } _ n e ^ { n f } ( t , \\mu , w ) \\\\ & = \\partial _ t f ( t , \\mu , w ) + H _ n [ t ] f ( t , \\mu , w ) . \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} | \\Phi ^ { e _ j } _ { t } ( x , t _ { n - 1 } ) | ^ 2 & = | x | ^ 2 + \\left | \\int _ { t _ { j - 1 } } ^ t v _ { j - 1 } ^ { N } ( \\Phi _ s ^ e ( x , t _ { j - 1 } ) ) \\right | ^ 2 + 2 | x | \\cdot \\left | \\int _ { t _ { n - 1 } } ^ t v _ { n - 1 } ^ { N } ( \\Phi _ s ^ { e _ j } ( x , t _ { n - 1 } ) ) \\right | \\\\ & \\leq | x | ^ 2 + ( V _ { m a x } \\Delta t ^ N ) ^ 2 + 2 V _ { m a x } \\Delta t ^ N | x | , \\end{align*}"}
-{"id": "7189.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac 1 { | \\mathbf X ( n ) | } \\sum \\limits _ { \\mathbf x \\in \\mathbf X ( n ) } \\exp ( 2 \\pi i \\langle \\mathbf { h } , \\mathbf { x } \\rangle ) = 0 . \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{align*} \\frac { \\| p _ { n + 1 } \\| ^ 2 } { 2 } = \\frac { \\| p _ n \\| ^ 2 } { 2 } + \\alpha _ * \\frac { \\beta ^ { ( 1 ) } ( \\kappa _ n ) } { \\| p _ n \\| } + \\alpha _ * ^ 2 \\left ( \\frac { \\delta \\beta ^ { ( 2 ) } ( \\kappa ) } { \\| p _ n \\| ^ 4 } + \\frac { \\delta \\beta ^ { ( 4 ) } ( \\kappa ) } { \\| p _ n \\| ^ 4 } \\right ) . \\end{align*}"}
-{"id": "3809.png", "formula": "\\begin{align*} ( g _ 2 , g _ 3 ) ^ { t } = ( g _ 2 ^ t , g _ 2 ^ { t - 1 } g _ 3 , g _ 2 ^ { t - 2 } g _ 3 ^ 2 , \\dots , g _ 2 g _ 3 ^ { t - 1 } , g _ 3 ^ t ) . \\end{align*}"}
-{"id": "6431.png", "formula": "\\begin{align*} \\begin{aligned} k ^ { u } _ { j + 1 } ( T ) & \\leq k ^ { u } _ { 0 } ( T ) + C C _ { 1 } ( T ) B \\big ( 1 - 3 \\big ( \\tfrac { 1 } { p } - \\tfrac { 1 } { q } \\big ) , \\tfrac { 1 } { 2 } - \\tfrac { 3 } { 2 q } \\big ) [ k _ { j } ^ { u } ( T ) ^ 2 + k _ { j } ^ { \\nabla y } ( T ) ^ 2 ] . \\end{aligned} \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{align*} \\limsup _ { t \\to 0 } \\ , & \\frac { f ( x + t e _ x ) - f ( x - a t u _ x ) } { t } \\\\ & \\quad = \\limsup _ { t \\to 0 } \\frac { f ( x + t e _ x ) - f ( x ) } { t } - \\lim _ { t \\to 0 } \\frac { f ( x - { a t } u _ x ) - f ( x ) } { t } = 1 + a ^ 2 . \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} d Y _ t & = ( b _ 0 + b _ 1 Y _ t ) \\ d t + \\sigma \\sqrt { Y _ t } \\ d W _ t + d N _ t \\\\ Y _ 0 & = y , y > 0 , \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{align*} & P \\{ T _ 1 = T _ { \\min } \\} = \\\\ & ( - 1 ) ^ { g - 1 } \\sum _ { k _ g = 1 } ^ { M _ g } \\cdots \\sum _ { k _ 2 = 1 } ^ { M _ 2 } \\sum _ { k _ 1 = 0 } ^ { M _ 1 } ( - 1 ) ^ { k _ 1 + \\cdots + k _ g } \\binom { M _ g } { k _ g } \\cdots \\binom { M _ 1 } { k _ 1 } \\frac { k _ 2 p _ 2 + \\cdots + k _ g p _ g } { k _ 1 p _ 1 + k _ 2 p _ 2 + \\cdots + k _ g p _ g } , \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} \\mathcal { L } x _ i & = \\Delta x _ i - \\left < \\nabla f , \\nabla x _ i \\right > \\\\ & = \\sum _ { j = 1 } ^ n \\bar { \\nabla } ^ 2 x _ i ( e _ j , e _ j ) + \\left < { \\bf H } , \\bar { \\nabla } x _ i \\right > - \\left < \\nabla f , \\nabla x _ i \\right > \\\\ & = \\left < { \\bf H } + ( \\bar { \\nabla } f ) ^ { \\perp } , \\bar { \\nabla } x _ i \\right > - \\left < \\bar { \\nabla } f , \\bar { \\nabla } x _ i \\right > \\\\ & = \\left < { \\bf H } _ f , \\bar { \\nabla } x _ i \\right > + \\dfrac { x _ i } { 2 } . \\\\ \\end{align*}"}
-{"id": "2336.png", "formula": "\\begin{align*} P \\{ X _ j \\leq t \\} = 1 - e ^ { - q _ j t } , t \\geq 0 . \\end{align*}"}
-{"id": "927.png", "formula": "\\begin{align*} N _ { s c } [ \\mathfrak { e } , \\lambda \\ , \\mathbf { 1 } _ { \\Lambda } ] \\ = \\ | \\Lambda | \\ \\ll \\ \\lambda ^ { d / 2 } \\ , | \\Lambda | \\ = \\ | \\lambda \\ , \\mathbf { 1 } _ { \\Lambda } | _ { d / 2 } ^ { d / 2 } \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} u ( x ) & = x ^ 2 ( C ( x ) - 1 ) + \\sum _ { n \\geq 4 } \\left ( \\sum _ { a = 2 } ^ { n - 2 } \\sum _ { t = 0 } ^ { n - 2 - a } \\binom { n - 2 - a } { t } u ' ( n - t , a ) \\right ) x ^ n \\\\ & = x ^ 2 ( C ( x ) - 1 ) + ( 1 - x ) r \\left ( \\frac { x } { 1 - x } , ( 1 - x ) y \\right ) \\mid _ { y = 1 } , \\end{align*}"}
-{"id": "3059.png", "formula": "\\begin{align*} & A w _ { q } = Q \\left [ a ( x ) \\left \\{ ( t \\phi _ { 1 } + w ) ^ { q } \\left ( \\log ( t \\phi _ { 1 } + w ) + \\frac { q w _ { q } } { t \\phi _ { 1 } + w } \\right ) - w _ { q } \\right \\} \\right ] , \\\\ & A w _ { t } = Q \\left [ a ( x ) \\left \\{ q ( t \\phi _ { 1 } + w ) ^ { q - 1 } ( \\phi _ { 1 } + w _ { t } ) - ( \\phi _ { 1 } + w _ { t } ) \\right \\} \\right ] . \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} S ( \\vec { \\alpha } ) = \\sum \\limits _ { \\vec { x } \\in P \\mathcal { B } } e ( \\vec { \\alpha } \\cdot \\vec { f } ( \\vec { x } ) ) S ( \\vec { \\alpha } , \\vec { \\nu } ) = S ( \\vec { \\alpha } ) e ( - \\vec { \\alpha } \\cdot \\vec { \\nu } ) \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} { } F ( a , b ; c ; x ) = ( 1 - x ^ 2 ) ^ { c - a - b } { } F ( c - a , c - b ; c ; x ) , \\enspace \\mbox { R e } { ( c ) } > \\mbox { R e } { ( b ) } > 0 , \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} \\Gamma _ n ( \\varepsilon _ n , \\eta _ n ) = \\{ ( v _ 1 , \\dots , v _ k ) \\colon D _ { v _ i } \\in [ \\gamma _ { v _ i } ^ l ( n ) , \\gamma _ { v _ i } ^ u ( n ) ] \\} , \\end{align*}"}
-{"id": "5694.png", "formula": "\\begin{align*} \\min _ { x \\in \\mathbb { R } ^ n } \\norm { x } _ 0 \\mbox { s u b j e c t t o } M x = b , \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} \\det \\Phi ' = \\partial _ 1 \\Phi _ 1 \\partial _ 2 \\Phi _ 2 - \\partial _ 2 \\Phi _ 1 \\partial _ 1 \\Phi _ 2 = 1 , \\end{align*}"}
-{"id": "2774.png", "formula": "\\begin{align*} & \\sum _ { b \\in B } x _ b \\\\ & x _ b \\leq \\sum _ { i \\in H } w _ { i , b } & b \\in B \\\\ & x _ b \\leq \\sum _ { j \\in C } z _ { j , b } & b \\in B \\\\ & \\sum _ { b \\in B } w _ { i , b } = 1 & i \\in H \\\\ & \\sum _ { b \\in B } z _ { j , b } = 1 & j \\in C \\\\ & \\sum _ { i \\in H } w _ { i , b } \\cdot h _ i = \\sum _ { j \\in C } z _ { j , b } \\cdot c _ j & b \\in B \\\\ & x _ b , w _ { i , b } , z _ { j , b } \\in \\{ 0 , 1 \\} & b \\in B , i \\in H , j \\in C \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} & \\sum _ { a = 2 } ^ { n - 3 } \\sum _ { d = a + 1 } ^ { n - 2 } ( 2 ^ { n - 3 - a } - 1 ) = \\sum _ { a = 2 } ^ { n - 3 } ( 2 ^ { n - 3 - a } - 1 ) ( n - 2 - a ) = \\sum _ { a = 1 } ^ { n - 4 } a 2 ^ { a - 1 } - \\binom { n - 3 } { 2 } \\\\ & = ( n - 5 ) 2 ^ { n - 4 } + 1 - \\binom { n - 3 } { 2 } \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{align*} \\Lambda ^ { \\ast } _ { 2 T } & = \\{ ( t , x ) \\in \\Lambda : 1 \\leq t \\leq T , - t + 2 \\leq x \\leq t \\} \\\\ & \\cup \\{ ( t , x ) \\in \\Lambda : T + 1 \\leq t \\leq 2 T , t - 2 T \\leq x \\leq - t + 2 T \\} . \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} ( M , h ) = ( \\R \\times \\Sigma , - s ( t ) ^ { - 2 } d t ^ 2 + g _ t ) . \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} \\overline { K } ( x _ 0 , x _ 1 , y _ 0 , y _ 1 ; t ) = x _ 0 x _ 1 y _ 0 y _ 1 - t \\sum _ { i , j = 0 } ^ 2 d _ { i - 1 , j - 1 } x _ 0 ^ { i } x _ 1 ^ { 2 - i } y _ 0 ^ j y _ 1 ^ { 2 - j } , \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{align*} \\rho = - \\eta \\wedge \\tau _ 1 + \\tfrac { 1 } { r } d r \\wedge \\tau _ 2 + O ( r ^ { - 3 - \\mu } ) \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{align*} \\varepsilon ( \\lambda , x ) = \\int _ { \\mathbb { R } ^ n } \\alpha ( y , x ) \\eta _ { \\delta } ( \\lambda - y ) d y . \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{align*} B : = \\{ \\tau = 0 , \\ , | X | \\leq 1 \\} , \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l l } \\eta ( x , 0 ) = \\eta _ { 0 } ( x ) w ( x , 0 ) = w _ { 0 } ( x ) & & ( 0 , L ) \\end{array} \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} ( N s ) ^ { - 1 } \\nabla _ { \\nabla ( N s ) } \\phi = - s \\dot s \\nabla _ t \\phi - s ^ 2 N ^ { - 1 } \\partial _ t N \\nabla _ t \\phi + N ^ { - 1 } \\langle D N , D \\phi \\rangle _ g . \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{align*} \\begin{cases} D _ { A _ 0 } \\psi _ 0 = 0 \\\\ F _ { A _ 0 } ^ { + } = \\frac { r } { 2 } ( q ( \\psi _ 0 , \\psi _ 0 ) - i \\Omega _ X ) + i \\mu _ 0 \\\\ * d * b _ 0 - 2 ^ { - \\frac { 1 } { 2 } } r ^ { \\frac { 1 } { 2 } } ( \\eta _ 0 ^ * \\psi _ I - \\psi _ I ^ * \\eta _ 0 ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{align*} \\mathrm { ( 1 ) } e ^ { - t \\mathcal { A } _ p } u = \\big [ A ^ { \\frac { 1 } { 2 } } _ { p ' } \\big ] ^ * \\Phi ^ { - 1 } e ^ { - t A _ p } \\Phi \\big [ A _ { p ^ { \\prime } } ^ { - \\frac { 1 } { 2 } } \\big ] ^ * u , \\mathrm { ( 2 ) } \\Phi ^ { - 1 } e ^ { - t A _ p } f = e ^ { - t \\mathcal { A } _ p } \\Phi ^ { - 1 } f ; \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} E \\left [ T ^ { ( 2 ) } \\right ] = ( \\nu _ 1 + \\lambda \\nu _ 2 ) ^ 2 M ^ 2 \\left ( H _ { \\nu _ 1 M } ^ 2 + \\sum _ { j = 1 } ^ { \\nu _ 1 M } \\frac { 1 } { j ^ 2 } \\right ) + O \\left ( M ^ { 3 - \\lambda + \\varepsilon } \\right ) , M \\to \\infty . \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} D ( \\ell , \\omega ) = w , D _ { p r } ( \\ell , \\omega ) = \\psi , \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} \\theta = 2 + 2 \\beta , \\gamma = \\beta + \\tfrac 3 2 , \\beta \\in \\big ( 0 , \\tfrac 4 3 \\big ) . \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\nabla ^ 2 \\langle \\overline { \\lambda } , g \\rangle ( \\overline { x } ) \\Delta x + \\nabla g ( \\overline { x } ) \\Delta \\lambda - \\ ! \\Delta v = 0 ; \\\\ \\Delta \\lambda \\in D \\mathcal { N } _ K ( g ( \\overline { x } ) | \\overline { \\lambda } ) ( g ' ( \\overline { x } ) \\Delta x ) \\end{array} \\right . \\Longrightarrow ( \\Delta x , \\Delta \\lambda , \\Delta v ) = ( 0 , 0 , 0 ) , \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} & D ^ k \\big ( N ^ { 4 - 2 n } \\langle ( \\nabla R ^ P ) ^ \\sharp ( \\psi , \\psi ) \\psi , \\psi \\rangle \\big ) \\\\ & = \\sum _ { \\sum l _ i + \\sum { m _ j } = k } D ^ { l _ 1 } N ^ { 4 - 2 n } \\star \\nabla ^ { l _ 2 + 1 } R ^ P \\star \\underbrace { D ^ { m _ 1 + 1 } \\phi \\star \\ldots \\star D ^ { m _ { l _ 2 } + 1 } \\phi } _ { l _ 2 - \\textrm { t i m e s } } \\star D ^ { l _ 3 } \\psi \\star D ^ { l _ 4 } \\psi \\star D ^ { l _ 5 } \\psi \\star D ^ { l _ 6 } \\psi . \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} x ^ B _ n ( 0 ) = - n , n \\geq - M + 1 \\end{align*}"}
-{"id": "7997.png", "formula": "\\begin{align*} - \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 , j \\neq i } ^ { N } \\alpha _ { i j } ( e _ { i , t } - e _ { j , t } ) ^ T e _ { i , t } = \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } l _ { i j } ( e _ { i , t } - e _ { j , t } ) ^ T e _ { i , t } = - \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } l _ { i j } e _ { j , t } ^ T e _ { i , t } . \\end{align*}"}
-{"id": "1005.png", "formula": "\\begin{align*} \\lim _ { k } \\Vert U _ { l } x _ { l - 1 } ^ { k } - x _ { l - 1 } ^ { k } \\Vert = 0 \\end{align*}"}
-{"id": "1163.png", "formula": "\\begin{align*} \\{ x , y \\} = P \\ , . \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} \\Psi ( u ) = ( N _ 1 ( u ) , \\ldots , N _ k ( u ) ) . \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} u ( x ) = x ^ 2 ( C ( x ) - 1 ) + ( 1 - x ) r \\left ( \\frac { x } { 1 - x } , 1 - x \\right ) , \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} E _ { \\varsigma } : = ( E _ 0 , E _ 1 ) _ { \\varsigma , p } \\ , , \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} s _ g = 1 2 v . \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{gather*} \\alpha : = - 2 ^ { r - 3 } \\left ( \\frac { 8 } { r } + r + 3 \\right ) , \\quad \\beta : = 2 ^ { r - 1 } \\left ( r + \\frac { 6 } { r } + 2 \\right ) , \\\\ \\gamma : = - 2 ^ { r - 3 } \\left ( 5 r + \\frac { 2 4 } { r } + 3 \\right ) , \\quad \\delta : = 2 ^ { r - 2 } \\left ( \\frac { 8 } { r } + r - 1 \\right ) . \\end{gather*}"}
-{"id": "4994.png", "formula": "\\begin{align*} { T B } _ d = ( B _ d - { F B } _ d ) - ( B _ { d - 1 } - { F B } _ { d - 1 } ) + ( B _ { d - 2 } - { F B } _ { d - 2 } ) \\pm \\cdots \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} w ( D ) & \\ge n ( n - ( z - 1 ) ) + z - 1 \\\\ & = n ^ 2 - z n + z - 1 + n \\\\ & \\ge n ^ 2 - z n + z - 1 + ( b ( z ) - z + 3 ) \\\\ & = n ^ 2 - z n + b ( z ) + 2 . \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{align*} \\displaystyle \\| u \\| _ { 1 , \\Phi } = \\| u \\| _ \\Phi + \\sum _ { i = 1 } ^ N \\left \\| \\frac { \\partial u } { \\partial x _ i } \\right \\| _ \\Phi , \\end{align*}"}
-{"id": "157.png", "formula": "\\begin{align*} \\mu _ t ( | q | _ k ) | q | _ k ^ { - 1 / 2 } \\dot q = \\frac { \\mu _ t ( | \\lambda | ^ 2 ) } { | \\lambda | } \\dot q , \\end{align*}"}
-{"id": "7889.png", "formula": "\\begin{align*} i \\frac { \\partial u } { \\partial t } ( t ) = H ( t ) u ( t ) + f ( t ) \\end{align*}"}
-{"id": "2966.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { d - 1 } g _ i ( z ) ( 1 - g _ i ( z ) ) = - z ^ d . \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} V ^ { - 1 } M V = \\begin{bmatrix} A & 0 \\\\ 0 & B \\end{bmatrix} , V = \\begin{bmatrix} I & X \\\\ 0 & I \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} A c t ( m , h ) = \\left \\{ \\begin{array} { l l } \\{ x , y \\} , & \\hbox { i f $ m \\in \\mathbb { R } $ a n d $ m > 0 $ ; } \\\\ \\{ x \\} , & \\hbox { i f $ m = 0 $ a n d $ h = h _ 2 $ ; } \\\\ \\emptyset , & \\hbox { o t h e r w i s e . } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6997.png", "formula": "\\begin{align*} \\lvert N ^ - ( w ) \\cap \\Phi _ h ^ - \\rvert = \\lvert N ^ - ( z ) \\cap \\Phi _ h ^ - \\rvert + \\lvert N ^ - ( y ) \\cap \\Phi _ { h _ z } ^ - \\rvert . \\end{align*}"}
-{"id": "8305.png", "formula": "\\begin{align*} Y _ { i _ { i ' } , 2 } & = \\mathcal { C } _ 1 ( \\alpha ' , \\nu ' , \\tau ^ { - 1 } ( i ' ) , I ' _ b \\setminus \\lbrace \\tau ^ { - 1 } ( i ' ) \\rbrace , I _ b ) \\\\ & \\leq \\mu ^ { \\circ } + \\ell - 2 { i _ { i ' } } + 1 \\\\ & = X _ { i _ { i ' } , 1 } + \\ell - 2 i _ { i ' } + 1 \\\\ & = Y _ { i _ { i ' } , 1 } . \\end{align*}"}
-{"id": "6657.png", "formula": "\\begin{align*} \\{ \\varphi , \\psi \\} ( m s ^ { - 1 } ) = \\{ \\Pi \\overline { \\varphi } , \\Pi \\overline { \\psi } \\} _ { G _ * } ( m s ^ { - 1 } ) , m \\in N _ s Z . \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} F ( z ) = h _ { \\ell } ( z ) + \\beta ( z ) , \\end{align*}"}
-{"id": "8961.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ( u _ { h , t } , \\chi ) + i ( \\nabla u _ h , \\nabla \\chi ) - i ( f ( \\abs { u _ h } ^ 2 ) u _ h , \\chi ) = ( g , \\chi ) , \\forall \\chi \\in S _ h \\\\ u _ h ( 0 , x ) = u _ { 0 , h } ( x ) , \\end{array} \\right . \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{align*} ( K _ m ) _ { y , y } = q ^ { 1 / 2 - \\mu _ m } , & & y \\in P _ { \\mu } , \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} & \\beta ^ { - 1 } \\partial _ { q ^ i } ( V \\partial _ { q ^ j } \\beta + \\beta \\tilde F _ j - \\beta \\partial _ t \\psi _ j ) ( \\tilde \\gamma ^ { - 1 } ) ^ { i k } H _ { k \\ell } ( \\tilde \\gamma ^ { - 1 } ) ^ { j \\ell } - ( 2 \\hat { b } _ + ^ i ( \\tilde \\Sigma ^ { - 1 } ) _ { i k } b _ - ^ k + \\nabla \\cdot b _ - ) \\\\ = & 0 \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{align*} E \\left [ S ( \\theta ) \\right ] = \\frac { N H _ N } { 1 - \\theta } + O \\left ( e ^ { - \\varepsilon N } \\right ) , N \\to \\infty \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} d _ m = \\begin{cases} 1 & , \\\\ 0 & , \\end{cases} & & ( 1 \\le m \\le N ) . \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} \\dot Q = Q A ( \\tilde \\omega ) , Q ( 0 ) = I _ { 3 } , \\end{align*}"}
-{"id": "472.png", "formula": "\\begin{align*} \\begin{pmatrix} 6 & 3 & 5 & 3 \\\\ 1 & 6 & 1 & 5 \\\\ 5 & 3 & 6 & 3 \\\\ 1 & 5 & 1 & 6 \\end{pmatrix} , \\begin{pmatrix} 0 & 6 & 0 & 1 0 \\\\ 0 & 8 & 0 & 7 \\\\ 1 & 0 & 2 & 0 \\\\ 6 & 0 & 3 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} X ^ A _ t = \\widetilde X ^ A _ t + t ^ { ( \\nu - 1 ) / 3 } X ^ { \\rm s t e p } _ { t ^ \\nu } - Z _ t . \\end{align*}"}
-{"id": "8731.png", "formula": "\\begin{align*} T _ k \\subseteq A \\cap ( a _ k , a _ k + b _ k ] , | T _ k | = | N _ k | . \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} \\langle \\mathcal R ^ \\nabla ( \\omega _ 1 \\wedge \\omega _ 2 ) , \\omega _ 3 \\wedge \\omega _ 4 \\rangle = g _ Q ( R ^ \\nabla ( \\omega _ 1 ^ \\sharp , \\omega _ 2 ^ \\sharp ) \\omega _ 4 ^ \\sharp , \\omega _ 3 ^ \\sharp ) , \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} \\int \\psi \\pi _ { r , p } = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } \\psi ( \\sigma _ { r , p } ^ { - 1 } ( r e ^ { i \\theta } ) ) d \\theta \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{align*} H ( N ) = \\begin{cases} \\zeta ( - 1 ) & \\mbox { i f } N = 0 , \\\\ L ( 0 , \\chi _ D ) \\sum _ { d \\ , | \\ , f } \\mu ( d ) \\chi _ D ( d ) \\sigma _ 1 ( f / d ) & \\mbox { i f } N > 0 \\mbox { a n d } - N = f ^ 2 D , \\\\ 0 & \\mbox { o t h e r w i s e , } \\end{cases} \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} \\Lambda ^ i _ { d \\leq r _ 0 } ( r _ 0 ) & = \\int ^ { r _ 0 + d } _ { r _ 0 - d } \\int ^ { \\arccos \\frac { r ^ 2 + r _ 0 ^ 2 - d ^ 2 } { 2 r _ 0 r } } _ { - \\arccos \\frac { r ^ 2 + r _ 0 ^ 2 - d ^ 2 } { 2 r _ 0 r } } \\mathrm { P } ^ i ( r ) \\lambda _ u d \\theta r d r \\\\ & = 2 \\lambda _ u \\int ^ { r _ 0 + d } _ { r _ 0 - d } \\mathrm { P } ^ i ( r ) r \\arccos \\frac { r ^ 2 + r _ 0 ^ 2 - d ^ 2 } { 2 r _ 0 r } d r . \\end{align*}"}
-{"id": "7526.png", "formula": "\\begin{align*} d f ( t , X _ t ) = \\partial _ t f ( t , X _ t ) d t + \\sum _ { i = 1 } ^ d \\partial _ i f ( t , X _ t ) \\circ d X _ t ^ i . \\end{align*}"}
-{"id": "5964.png", "formula": "\\begin{align*} \\log \\det \\big ( R ^ { ( 1 ) } ( x ) \\big ) & - \\log \\det \\big ( R ^ { ( 0 ) } ( x ) \\big ) = g ( 1 , x ) - g ( 0 , x ) \\\\ & = g _ t ' ( s , x ) = \\sum _ { i , j = 1 } ^ n \\big ( R ^ { ( s ) } ( x ) \\big ) ^ { - 1 } _ { j i } \\big ( R ^ { ( 1 ) } _ { i j } ( x ) - R ^ { ( 0 ) } _ { i j } ( x ) \\big ) , \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} \\Phi _ { q t } ( 1 , t _ { \\ast } ) = \\int _ { \\Omega } a \\left ( x \\right ) \\phi _ { 1 } ^ { 2 } > 0 , \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} \\sum _ { a = 2 } ^ { n - 3 } \\left ( 2 ^ { n - 2 - a } - 1 - \\sum _ { i = 0 } ^ { n - 3 - a } \\sum _ { j = 0 } ^ { n - 3 - a - i } 1 \\right ) & = 2 ^ { n - 3 } - ( n - 2 ) - \\sum _ { a = 2 } ^ { n - 3 } \\binom { n - 1 - a } { 2 } \\\\ & = 2 ^ { n - 3 } - ( n - 2 ) - \\binom { n - 2 } { 3 } \\end{align*}"}
-{"id": "7046.png", "formula": "\\begin{align*} \\int e ^ { V ^ 0 ( \\psi ) } d \\mu _ C ( \\psi ) = \\int e ^ { V ^ { l _ { e n d } } _ s ( \\psi ) } d \\mu _ { C _ { l _ { e n d } } } ( \\psi ) . \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} f ( \\rho , \\Omega ) : = \\left [ a ( \\rho ) \\right ] ^ 2 + \\left [ b ( \\rho ) - \\Omega \\right ] ^ 2 \\rho ^ 2 - \\varepsilon ^ 2 r ^ 2 . \\end{align*}"}
-{"id": "3150.png", "formula": "\\begin{align*} { } \\begin{array} { l } \\frac { v } { u } = c + \\frac { \\ln 2 } { c } \\epsilon + \\frac { 1 } { 2 c } \\left ( 2 \\ln 2 - \\frac { v } { p } - 1 \\right ) ( v ( p ) - v ) \\\\ u \\ln ( e ^ { v ( p ) } - 1 ) = \\frac { 2 \\ln 2 } { c } \\epsilon + 2 c \\gamma \\epsilon ^ 2 + \\frac { 2 \\ln 2 } { c } ( v ( p ) - v ) \\end{array} \\end{align*}"}
-{"id": "629.png", "formula": "\\begin{align*} \\lambda _ i = \\mathbb { E } A _ i ( n ) \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} d ( x _ 1 , \\ldots , x _ k ) : = \\max _ { 1 \\le p < q \\le k } | x _ p - x _ q | _ { \\infty } , \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} \\tilde { G } _ { j } ( \\hat { X } ) = \\int _ { a _ j } ^ { \\hat { X } } \\frac { X - \\frac 1 3 ( \\lambda + 1 ) } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } d X , \\end{align*}"}
-{"id": "5279.png", "formula": "\\begin{align*} \\langle A _ q u , u \\rangle _ { H ^ { - s ' } ( M ) } = \\inf _ { X \\circ c = u } \\langle A X , X \\rangle _ { H ^ { - s } ( \\R ^ d ) } \\ , , \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{align*} \\dot { x } \\left ( t \\right ) = \\left [ \\begin{array} { c } V _ { p } \\left ( \\delta \\theta \\right ) - V _ { e } + \\frac { \\delta y a _ { e } } { V _ { e } } \\\\ V _ { p } \\left ( \\delta \\theta \\right ) - \\frac { \\delta x a _ { e } } { V _ { e } } \\\\ \\frac { a _ { p } } { V _ { p } } - \\frac { a _ { e } } { V _ { e } } \\end{array} \\right ] , \\end{align*}"}
-{"id": "930.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow \\infty } \\frac { N ^ { c o n t } [ \\lambda V ] } { N _ { s c } ^ { c o n t } [ \\lambda V ] } \\ = \\ 1 , \\end{align*}"}
-{"id": "1128.png", "formula": "\\begin{align*} [ \\partial , \\partial ' ] ^ \\bullet - [ \\partial ^ \\bullet , \\partial '^ \\bullet ] = \\ell \\circ d + d \\circ \\ell \\ , . \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} X = \\bigcup _ { i = 1 } ^ L \\{ x \\in U _ i : f _ { i , 1 } ( x ) = \\cdots = f _ { i , m _ i } ( x ) = 0 \\} \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} p ( i , j | v , w ) = p ^ 1 ( i | v ) p ^ 2 ( j | w ) , \\end{align*}"}
-{"id": "7322.png", "formula": "\\begin{align*} f ^ { - 1 } [ T _ t ] = \\omega ^ * \\cap \\bigcup _ { A \\in \\mathcal A _ s } \\hat { A } ( t ) \\end{align*}"}
-{"id": "23.png", "formula": "\\begin{align*} \\inf _ { \\gamma \\in \\mathcal { N } _ u } \\mathcal { P } _ u ^ 0 ( 0 , \\gamma ) = \\inf _ { \\mathbb { R } \\times \\mathcal { N } _ u } \\mathcal { P } _ u ( \\lambda , \\gamma ) . \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} \\Big | \\sum _ { j = 1 } ^ n \\mu _ j x _ { - , j } ^ z ( t ) \\Big | \\geq k | \\mu | e ^ { - t } . \\end{align*}"}
-{"id": "4937.png", "formula": "\\begin{align*} \\frac { d { x } ( t ) } { d t } & = { A } x ( t ) + B u ( t ) + \\sum _ { i = 1 } ^ m N _ i x ( t ) u _ i ( t ) , \\\\ y ( t ) & = { C } x ( t ) , \\ ; \\ ; \\ ; t \\geq 0 , \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} b ( h ) = \\dfrac { ( h ^ 2 - 3 ) ( h ^ 6 - 9 h ^ 4 + 3 h ^ 2 - 3 ) } { h ^ 8 - 1 2 h ^ 6 + 3 0 h ^ 4 - 3 6 h ^ 2 + 9 } , \\end{align*}"}
-{"id": "5871.png", "formula": "\\begin{align*} { \\rm C o e f f } [ f _ { \\mu } , m ] = \\sum _ { \\nu \\in \\sigma ( \\epsilon ) } \\psi ( \\nu , \\mu ; t ) z ^ { \\nu } , \\forall \\ \\mu \\in \\sigma ( \\delta ) , \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} I _ h = \\{ t _ 3 - t _ 1 , t _ 4 - t _ 1 , t _ 4 - t _ 2 , t _ 5 - t _ 1 , t _ 5 - t _ 2 , t _ 5 - t _ 3 \\} . \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} \\hat { P } = T P T ^ T \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\hat { Q } = T ^ { - T } Q T ^ { - 1 } . \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{align*} W ( x , y ) = x ^ n + \\sum _ { i = d } ^ n A _ i x ^ { n - i } y ^ i \\in { \\bf C } [ x , y ] ( A _ d \\ne 0 ) \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} X _ i u = a _ i \\ { \\rm i n } \\ \\mathbb G , \\ \\ i = 1 , . . . , 5 . \\end{align*}"}
-{"id": "4453.png", "formula": "\\begin{align*} { \\phi _ \\alpha } ^ { - 1 } ( [ 1 ^ n . w 1 ^ n ] _ { \\hat { X _ \\alpha } } ) = \\bigcap _ { i = - n } ^ { \\vert w \\vert + n - 1 } { \\hat { U } } ^ { - i } A _ { ( 1 ^ n w 1 ^ n ) _ { i + n + 1 } } \\textrm { a n d } B ( \\tilde { \\mathfrak { t } } _ n ( \\alpha ) ) = \\bigcap _ { j = - n } ^ { n - 1 } { \\hat { U } } ^ { - j } A _ 1 . \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} C _ D = & \\tanh ( \\beta _ 1 ( a ) ) \\Re ( z _ 1 \\mu _ { \\beta _ 1 } ) + \\coth ( \\beta _ 1 ( a ) ) \\Im ( z _ 1 \\mu _ { \\beta _ 1 } ) \\\\ & + \\tanh ( \\beta _ 2 ( a ) ) \\Re ( z _ 2 \\mu _ { \\beta _ 2 } ) + \\coth ( \\beta _ 2 ( a ) ) \\Im ( z _ 2 \\mu _ { \\beta _ 2 } ) . \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} \\mathbb { E } [ \\overline { V } _ t ] \\le \\psi _ 2 ^ * = \\frac { b | T r ( L ) | } { 4 N ( \\kappa + a \\lambda _ 2 ) } + \\frac { m \\tau ^ 2 \\tilde { \\gamma } ( N - 1 ) } { 4 N ( \\kappa + a \\lambda _ 2 ) } . \\end{align*}"}
-{"id": "4374.png", "formula": "\\begin{align*} | z ^ { ( 0 ) } ( \\xi ) | \\leq \\begin{cases} \\frac 1 2 \\log ( | \\xi | ^ { - 1 } ) | + 1 & \\mbox { i f } \\xi \\leq 1 \\\\ \\frac 1 2 & \\mbox { e l s e } . \\end{cases} \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} | ( t - v ) - ( s + u ) | = ( t - s ) - ( u + v ) = \\delta - ( u + v ) \\ge \\frac 3 4 \\delta , \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{align*} \\rho \\ , d q = - \\tfrac { 2 } { 1 - 2 m \\rho } \\bigl ( \\tfrac { d \\rho _ 0 } { \\rho _ 0 } + \\tfrac { d \\rho _ I } { \\rho _ I } \\bigr ) + \\rho _ I \\tfrac { d \\rho _ 0 } { \\rho _ 0 } , \\ \\ \\rho \\ , d s = \\rho _ I \\tfrac { d \\rho _ 0 } { \\rho _ 0 } . \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} \\widehat { \\delta } = t _ i ^ { - 1 } \\log ^ { \\frac 1 2 } ( t _ i ) \\delta . \\end{align*}"}
-{"id": "7315.png", "formula": "\\begin{align*} \\mathfrak a = \\min \\{ | \\mathcal A | : \\mathcal A \\ \\mbox { i s a M A D f a m i l y } \\} \\end{align*}"}
-{"id": "47.png", "formula": "\\begin{align*} 0 & \\leq \\int _ 0 ^ u g ( s ) \\gamma ( s ) d s = \\int _ 0 ^ u g ( s ) \\gamma ( s ) d s - \\int _ 0 ^ u g ( s ) \\alpha _ 0 ( s ) d s = \\int _ 0 ^ u g ( s ) ( \\gamma ( s ) - \\alpha _ 0 ( s ) ) d s . \\end{align*}"}
-{"id": "6851.png", "formula": "\\begin{align*} \\tilde { L } _ Q ^ \\top \\tilde { L } _ P = \\tilde { U } \\tilde { \\Sigma } \\tilde { V } ^ \\top . \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} \\mathbb { I } & = \\{ \\vec { i } \\in \\mathbb { Z } _ { < 0 } ^ 2 \\ | \\ p \\nmid \\gcd \\vec { i } \\} . \\end{align*}"}
-{"id": "5811.png", "formula": "\\begin{align*} m _ i ( \\mu , \\mu ) = \\left \\{ \\begin{array} { r l } - t , & \\mu _ i > \\mu _ { i + 1 } , \\\\ \\\\ - 1 , & \\mu _ i < \\mu _ { i + 1 } , \\\\ \\\\ 0 , & . \\end{array} \\right . \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} \\gamma = \\frac { m ^ r u } { q ^ r u ' } \\end{align*}"}
-{"id": "3531.png", "formula": "\\begin{align*} \\sup _ { S ^ { 1 } \\times [ 0 , T ) } | \\kappa | = \\infty \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} \\left [ \\ , { \\cal C } ( \\bar { x } ; q ; P ) \\ , \\ni \\ , v \\ , \\perp \\ , Q v \\ , \\in \\ , { \\cal C } ( \\bar { x } ; q ; P ) ^ * \\ , \\right ] \\ \\Rightarrow \\ v \\ , = \\ , 0 ; \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} v & = [ x ( s _ 1 \\oplus \\cdots \\oplus s _ m ) ] _ H = [ y ( t _ 1 \\oplus \\cdots \\oplus t _ n ) ] _ H \\\\ v ' & = [ x ( s _ 1 ' \\oplus \\cdots \\oplus s _ m ' ) ] _ H = [ y ( t _ 1 ' \\oplus \\cdots \\oplus t _ n ' ) ] _ H \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{align*} \\mathbf { F } _ n ( t ) = ( t U _ n , 0 ) \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} a _ \\tau ( v _ h , \\psi ^ i _ h ) = \\lambda _ i ( \\rho v _ h , \\psi ^ i _ h ) _ \\tau v _ h \\in V _ h ^ B ( \\tau ) \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} i \\frac { \\partial u } { \\partial t } ( t ) = H _ { \\epsilon } ( t ) u ( t ) + f ( t ) . \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} \\frac { d } { d r } e ^ { C \\psi ( r ) } \\left ( N ( r ) + C \\psi ( r ) \\right ) & = e ^ { C \\psi ( r ) } \\left ( N ' ( r ) + C \\psi ' ( r ) + C \\psi ' ( r ) N ( r ) + C ^ 2 \\psi ( r ) \\psi ' ( r ) \\right ) \\\\ & \\ge C ^ 2 e ^ { C \\psi ( r ) } \\psi ( r ) \\psi ' ( r ) \\ge 0 , \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} U _ { \\alpha } ( \\omega _ { j , n } ) = \\frac { P ' _ { \\alpha } ( \\omega _ { j , n } ) } { f ' _ { \\beta } ( \\omega _ { j , n } ) } \\neq 0 \\ , , U _ { \\alpha } ( \\overline { \\omega _ { j , n } } ) = \\frac { P ' _ { \\alpha } ( \\overline { \\omega _ { j , n } } ) } { f ' _ { \\beta } ( \\overline { \\omega _ { j , n } } ) } \\neq 0 \\ , \\mbox { a n d } ~ ~ U _ { \\alpha } ( \\frac { 1 } { \\beta } ) = \\frac { P ' _ { \\alpha } ( \\frac { 1 } { \\beta } ) } { f ' _ { \\beta } ( \\frac { 1 } { \\beta } ) } \\neq 0 . \\end{align*}"}
-{"id": "1948.png", "formula": "\\begin{align*} h ( M _ 1 , \\dots , M _ r ) = , \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{align*} 2 \\beta \\wedge \\eta \\wedge \\omega _ 3 = - 2 ( J _ 3 ^ 2 \\beta ) \\wedge \\eta \\wedge \\omega _ 3 = - 2 ( J _ 2 J _ 3 \\beta ) \\wedge \\eta \\wedge \\omega _ 2 = 2 ( J _ 1 \\beta ) \\wedge \\eta \\wedge \\omega _ 2 , \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{align*} d Y = \\sum _ { \\alpha , \\beta , \\gamma } \\Gamma ^ { \\beta \\gamma } \\biggl ( \\partial ^ \\alpha F ^ { \\beta } ( X ) F ^ { \\alpha \\gamma } ( X ) \\biggr ) \\ , d t + F ( X ) \\ , d W , , Y _ { 0 } = \\xi . \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{align*} \\mathbf { R } _ 0 ( \\mathbf { z } ) = \\begin{bmatrix} \\lambda _ l z _ l \\\\ \\bar { \\lambda } _ { l } \\bar { z } _ l \\end{bmatrix} + \\sum _ { m = 1 } ^ M \\begin{bmatrix} \\beta _ m z ^ { m + 1 } _ l \\bar { z } ^ { m } _ l \\\\ \\overline { \\beta } _ m z ^ { m } _ l \\bar { z } ^ { m + 1 } _ l \\end{bmatrix} + \\mathcal { O } ( \\mathbf { z } ^ { 2 M + 3 } ) . \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{align*} I : = \\int _ 0 ^ { \\infty } \\cdots \\int _ 0 ^ { \\infty } k _ g p _ g \\cdots k _ 2 p _ 2 \\ , e ^ { - ( k _ g p _ g t _ g + \\cdots + k _ 2 p _ 2 t _ 2 ) } e ^ { - k _ 1 p _ 1 ( t _ 2 \\wedge \\cdots \\wedge t _ g ) } d t _ 2 \\cdots d t _ g . \\end{align*}"}
-{"id": "4369.png", "formula": "\\begin{align*} M & = - i ( z ^ { ( 0 ) } + \\overline { z ^ { ( 0 ) } } ) ( \\xi ) \\log ( | \\lambda | ) + ( \\overline { z ^ { ( 0 ) } } - z ^ { ( 0 ) } ) ( \\xi ) \\arg ( \\lambda ) . \\end{align*}"}
-{"id": "1283.png", "formula": "\\begin{align*} D _ j D ^ { - 1 } - D _ j D _ j ^ { - 1 } = \\lambda _ j ( G \\setminus \\{ 1 \\} ) \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} f _ { \\mu } ( z _ 1 , \\dots , z _ n ; q , t ) = \\left ( 1 - q t ^ { m _ 1 } \\right ) \\times { \\rm T r } \\Big ( A _ { \\mu _ 1 } ( z _ 1 ) \\dots A _ { \\mu _ n } ( z _ n ) k ^ { u } \\Big ) , \\end{align*}"}
-{"id": "7870.png", "formula": "\\begin{align*} & i \\hbar \\frac { \\partial u } { \\partial t } ( t ) = H ( t ) u ( t ) \\\\ & : = \\left [ \\frac { 1 } { 2 m } \\sum _ { j = 1 } ^ d \\left ( \\frac { \\hbar } { i } \\frac { \\partial } { \\partial x _ j } - q A _ j ( t , x ) \\right ) ^ 2 + q V ( t , x ) \\right ] u ( t ) , \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} f _ { r _ m } ( x ) = \\frac { 2 \\lambda _ c ^ m \\pi ^ m x ^ { 2 m - 1 } } { ( m - 1 ) ! } e ^ { - \\lambda _ c \\pi x ^ 2 } . \\end{align*}"}
-{"id": "7293.png", "formula": "\\begin{align*} g _ { m k } = h _ { m k } \\sqrt { \\beta _ { m k } } , \\end{align*}"}
-{"id": "3507.png", "formula": "\\begin{align*} \\lim _ { h \\to 0 + } h \\bigoplus ( y / h ) = \\max y , \\lim _ { h \\to 0 - } h \\bigoplus ( y / h ) = \\min y , \\end{align*}"}
-{"id": "7436.png", "formula": "\\begin{align*} d x _ t = b ( t , x _ t ) d t + \\tilde \\sigma ( t , x _ t ) \\circ d W _ t \\end{align*}"}
-{"id": "62.png", "formula": "\\begin{align*} \\hat { \\beta } _ j \\Big ( ( 1 - \\tau ) \\big ( N _ { \\hat { \\beta } _ j ^ + } - N _ { \\hat { \\beta } _ j ^ - } \\big ) + E _ { \\hat { \\beta } _ j ^ + } - E _ { \\hat { \\beta } _ j ^ - } \\Big ) = 0 \\forall j \\in [ s ] . \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} b _ i = \\begin{cases} a _ i \\\\ 2 ^ n - a _ i . \\end{cases} \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{align*} u ( t ) = \\int _ 0 ^ t e ^ { - ( t - s ) \\mathcal { A } _ p } f ( s ) \\ ; d s \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} \\begin{array} { r l } { \\rm s r e g } ( I ( G ) ) + { \\rm s d e p t h } ( S / J ( G ) ) = n \\ \\ \\ { \\rm a n d } \\ \\ \\ { \\rm s d e p t h } ( J ( G ) ) + { \\rm s r e g } ( S / I ( G ) ) = n . \\end{array} \\end{align*}"}
-{"id": "9272.png", "formula": "\\begin{align*} K ^ n + ( n + 1 ) K ^ { n - 1 } H + \\binom { n + 1 } { 2 } K ^ { n - 2 } H ^ 2 + \\cdots + \\binom { n + 1 } { n } H ^ n & = 0 \\\\ H ^ n + H ^ { n - 1 } ( K + H ) + \\cdots + ( K + H ) ^ n & = 0 . \\end{align*}"}
-{"id": "2607.png", "formula": "\\begin{align*} ( \\pi ( a ) - \\varepsilon ) _ + & = ( \\pi ( ( 1 - p _ N ) a ( 1 - p _ N ) ) - \\varepsilon ) _ + \\\\ & = \\pi ( ( 1 - p _ N ) a ( 1 - p _ N ) - \\varepsilon ) _ + \\\\ & \\precsim \\pi ( ( 1 - p _ N ) b ( 1 - p _ N ) ) \\\\ & = \\pi ( b ) . \\end{align*}"}
-{"id": "1601.png", "formula": "\\begin{align*} \\Psi _ { t , c } ( z ) : = \\lambda _ { R _ { L _ t } } ( z ) - \\left ( \\ln _ 3 t - c \\right ) ^ + \\frac { | z | } { t } , z \\in \\Pi _ { L _ t , \\delta } . \\end{align*}"}
-{"id": "7564.png", "formula": "\\begin{align*} & \\sup _ { 0 \\leq s \\leq t \\leq T } E \\left [ \\left | J _ { s , t } ^ m - J _ { s , t } \\right | ^ p \\right ] ^ { 1 / p } \\\\ \\leq & O ( m ^ { 1 / 2 } ) + \\tilde C T \\sup _ { r \\in [ 0 , T ] } E \\left [ ( 1 + \\| q _ r \\| ^ { \\tilde p } + \\| q _ r - \\tilde q _ r ^ m \\| ^ { \\tilde p } ) ^ { 2 p } \\right ] ^ { 1 / ( 2 p ) } E \\left [ \\| q _ r ^ m - q _ r \\| ^ { 2 p } \\right ] ^ { 1 / { 2 p } } \\\\ = & O ( m ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "4445.png", "formula": "\\begin{align*} T ( t ) x - x = \\int _ 0 ^ t T ( s ) y d s . \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} d \\overline { V } _ t \\le - 2 { a } \\lambda _ 2 \\tilde { \\gamma } \\overline { V } _ t d t - 2 \\kappa \\tilde { \\gamma } \\overline { V } _ t d t + \\frac { b | T r ( L ) | } { 2 N } \\tilde { \\gamma } d t + \\frac { m \\tau ^ 2 \\tilde { \\gamma } ^ 2 ( N - 1 ) } { 2 N } d t + \\frac { \\tau } { N } \\tilde { \\gamma } \\sum _ { i = 1 } ^ N { d B _ { i , t } } ^ T e _ { i , t } . \\end{align*}"}
-{"id": "4501.png", "formula": "\\begin{align*} \\int _ { \\R ^ 3 } | \\Delta u | ^ 2 d x = \\end{align*}"}
-{"id": "8753.png", "formula": "\\begin{align*} \\psi \\in C ^ { \\infty } ( \\overline \\Omega ) , \\psi = 1 \\mbox { i f } \\mathrm { d i s t } ( x , \\partial \\Omega ) > \\alpha / 4 , \\psi = 0 \\mbox { i f } \\mathrm { d i s t } ( x , \\partial \\Omega ) < \\alpha / 8 . \\end{align*}"}
-{"id": "7701.png", "formula": "\\begin{align*} f _ { r _ t | r _ m } ( y ) = & \\sum ^ { t - m - 1 } _ { n = 1 } \\frac { 2 y ( \\lambda _ c \\pi ) ^ { n } ( y ^ 2 - x ^ 2 ) ^ { n - 1 } } { n ! } e ^ { - \\lambda _ c \\pi ( y ^ 2 - x ^ 2 ) } \\\\ & \\times \\left [ \\lambda _ c \\pi ( y ^ 2 - x ^ 2 ) - n \\right ] + 2 \\lambda _ c \\pi y e ^ { - \\lambda _ c \\pi ( y ^ 2 - x ^ 2 ) } . \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} \\lim _ { n \\longrightarrow + \\infty } I ( \\psi _ n , c , 0 ) & = \\lim _ { n \\longrightarrow + \\infty } \\left ( I ( \\psi _ n , c , \\gamma _ n ) - \\gamma _ n \\int ( \\partial _ x ^ { - 1 } \\psi _ n ) ^ 2 d x \\right ) \\\\ & \\leq \\lim _ { n \\longrightarrow + \\infty } I ( \\psi _ n , c , \\gamma _ n ) \\\\ & = \\lim _ { n \\longrightarrow + \\infty } m ( c , \\gamma _ n ) ^ 3 \\\\ & = m ( c , 0 ) ^ 3 . \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} f _ q ( t ) = \\sum _ { ( v , w ) \\in E } \\tau ( P _ v P _ w ) . \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} \\| z \\| _ { ( X , Y ) _ { \\theta , q } } : = \\bigg ( \\int _ { 0 } ^ \\infty \\frac { \\mathrm { d } t } { t } \\ , \\Big | \\frac { K ( t , z ) } { t ^ \\theta } \\Big | ^ q \\bigg ) ^ \\frac { 1 } { q } < \\infty . \\end{align*}"}
-{"id": "4171.png", "formula": "\\begin{align*} \\mathrm { H F } _ \\gamma ( x ) = 1 \\Longleftrightarrow x _ 1 + \\gamma ( \\hat { x } ) \\geq 0 . \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} F ( D ^ 2 u ) = 0 \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} \\varphi _ { G , U } ( t ) = \\frac { v _ G ( t ) } { v _ U ( \\rho _ { G , U } ( t ) ) } = \\frac { v _ G ( t ) } { \\sigma } \\sqrt { 1 + \\frac { 2 } { \\theta } r _ G ( t ) } \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} | f ( x ) - p ( x ) | = | R ( x ) | & \\leq \\frac { 1 } { ( n - 1 ) ! } \\cdot \\sum _ { I \\in \\underline { d } ^ n } | ( x - x _ 0 ) ^ I | \\cdot \\int _ 0 ^ 1 ( 1 - t ) ^ { n - 1 } \\cdot B \\cdot | t ( x - x _ 0 ) | ^ \\sigma d t \\\\ & \\leq \\frac { d ^ n } { n ! } | x - x _ 0 | ^ n \\cdot B \\cdot | x - x _ 0 | ^ \\sigma \\leq C \\cdot B \\cdot | x - x _ 0 | ^ \\beta \\ , , \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{align*} \\frac { ( 1 - \\frac { c _ n } { n } ) ^ { 2 n } } { ( 1 - \\frac { c _ n } { n } ) - ( 1 - \\frac { c _ n } { n } ) ^ { n } } = \\frac { e ^ { - 2 c } } { 1 - e ^ { - c } } \\Bigl ( 1 + \\frac { c ( 2 - c e ^ { - c } - 2 c ) } { 2 n ( 1 - e ^ { - c } ) } \\Bigr ) + \\ldots \\end{align*}"}
-{"id": "8252.png", "formula": "\\begin{align*} x _ { \\nu t } ( t ) \\geq x ^ { \\rm f l a t } _ { \\nu t } ( t ) \\stackrel { ( d ) } { = } x ^ { \\rm f l a t } _ { t / 4 } ( t ) - 2 \\nu t + t / 2 . \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{align*} \\Delta _ { A _ t } = t ^ { 4 / 3 } \\varrho ^ { - 2 } \\widehat { \\Delta _ \\varrho } : = t ^ { 4 / 3 } \\Delta _ \\varrho . \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} \\ddot { y } + R ^ z ( t ) y = 0 \\qquad ( \\ , \\ , \\dot { } = \\partial _ t ) , \\end{align*}"}
-{"id": "7610.png", "formula": "\\begin{align*} I _ 1 ( g ) = \\N \\cap ( N _ { q , n , m } ( g - 1 ) , C ^ { - 1 } q ^ { ( g - 1 ) n m ^ 2 } ] . \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} \\epsilon _ A ( x , n , y ) : = ( \\sigma _ A ( x ) , n , \\sigma _ A ( y ) ) , \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} \\| f \\| = \\left ( \\sum _ { n = 1 } ^ \\infty \\frac 1 { n ^ 2 } \\right ) ^ { 1 / 2 } < \\infty , \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} A & = \\{ k \\in Z _ n : \\alpha _ k = 0 \\} \\\\ B & = \\{ k \\in Z _ n : \\alpha _ k = 1 \\} \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{align*} \\sum _ { \\Gamma t ( l ) < t } \\nabla f ( y _ { \\Gamma t ( l ) } ) \\Gamma = \\frac { \\Gamma n _ t } { t } \\left [ \\sum _ { \\Gamma t ( l ) < t } \\nabla f ( y _ { \\Gamma t ( l ) } ) \\frac { t } { n _ t } \\right ] \\simeq \\frac { \\Gamma n _ t } { t } \\int _ { 0 } ^ { t } \\nabla f ( y _ t ) d t \\simeq \\frac { 1 } { \\Delta t } \\int _ { 0 } ^ { t } \\nabla f ( y _ t ) d t . \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} \\alpha \\cdot ( \\varphi \\otimes c ) = - \\varphi \\cdot \\alpha \\otimes c + \\varphi \\otimes \\alpha \\cdot c \\ , . \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} A ( e ^ { i _ 1 } _ 0 , . . . , e ^ { i _ k } _ 0 ) = 0 . \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} g \\cdot ( B _ 1 , B _ 2 , i , j ) = ( g B _ 1 g ^ { - 1 } , g B _ 2 g ^ { - 1 } , g i , j g ^ { - 1 } ) . \\end{align*}"}
-{"id": "8450.png", "formula": "\\begin{align*} T \\circ \\pi _ \\lambda ( g ) = \\pi _ \\lambda ( g ) \\circ T \\end{align*}"}
-{"id": "6904.png", "formula": "\\begin{align*} \\partial _ \\alpha = \\dfrac { 1 } { \\alpha _ 1 ! \\dots \\alpha _ n ! } \\dfrac { \\partial ^ { | \\alpha | } } { \\partial x _ 1 ^ { \\alpha _ 1 } \\dots , x _ n ^ { \\alpha _ n } } , \\end{align*}"}
-{"id": "7667.png", "formula": "\\begin{align*} f _ m ( y ) = & \\frac { e ^ { - \\lambda _ c \\pi y ^ 2 } y ^ { 2 ( t - m - 1 ) - 2 p + 1 } } { 2 m + 2 p } \\left ( y ^ { 2 m + 2 p } - ( g ( y ) ) ^ { 2 m + 2 p } ) \\right ) . \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{align*} \\begin{aligned} { \\alpha } ( { \\alpha } ^ { - 1 } - 2 \\sqrt { 2 } r ) \\leq Q ( \\Lambda , r ) \\leq { \\alpha } ( { \\alpha } ^ { - 1 } - \\sqrt { 2 } r ) & \\sqrt { 2 } r \\leq { \\alpha } ^ { - 1 } ; \\\\ Q ( \\Lambda , r ) = 0 & \\sqrt { 2 } r \\geq { \\alpha } ^ { - 1 } . \\end{aligned} \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} y \\left ( x \\right ) = y _ { a } \\frac { \\left ( \\psi \\left ( x \\right ) - \\psi \\left ( a \\right ) \\right ) ^ { \\gamma - 1 } } { \\Gamma \\left ( \\gamma \\right ) } + \\frac { 1 } { \\Gamma \\left ( \\alpha \\right ) } \\int _ { a } ^ { x } \\psi ^ { \\prime } \\left ( t \\right ) \\left ( \\psi \\left ( x \\right ) - \\psi \\left ( t \\right ) \\right ) ^ { \\alpha - 1 } f \\left ( t , y \\left ( t \\right ) \\right ) d t . \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{align*} \\beta ^ { ( 1 ) } ( \\kappa ) = \\frac { P } { \\sqrt { M } } L _ 0 ( \\| b \\| ) \\overline { \\beta ^ { ( 1 ) } } = 0 , \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} \\max _ { x \\in Q _ n ' } \\lambda _ j ( x ) = \\sum _ { i = 1 } ^ n | l _ { i j } | + l _ { n + 1 , j } . \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{align*} \\| i \\omega ( i \\omega + \\varepsilon - A ) ^ { - 1 } x \\| = { } & \\| ( i \\omega - ( A _ { - 1 } - \\varepsilon ) ) ^ { - 1 } ( A _ { - 1 } - \\varepsilon ) x + x \\| \\\\ \\leq { } & \\| ( i \\omega - ( A _ { - 1 } - \\varepsilon ) ) ^ { - 1 } A _ { - 1 } x \\| + \\varepsilon \\| ( i \\omega - ( A - \\varepsilon ) ) ^ { - 1 } x \\| + \\| x \\| \\\\ \\leq { } & ( C + M + 1 ) \\| x \\| , \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} { { \\cal L } _ { { I _ { d , r u } } } } \\left ( s \\right ) = { { { \\rm { E } } _ { { I _ { d , r u } } } } \\left [ { \\prod \\limits _ { { i } \\in { \\Phi _ { t u } } } { \\exp \\left ( { - s { P _ d } { g _ { { i } } } r _ i ^ { - \\alpha } } \\right ) } } \\right ] } . \\end{align*}"}
-{"id": "876.png", "formula": "\\begin{align*} \\begin{array} { l l l l l } \\hat { f } ( \\xi ) & = & \\displaystyle \\hat { u _ * } + i \\int _ 1 ^ t \\int e ^ { \\frac 1 2 i ( \\eta ^ 2 - 2 \\xi \\eta ) s } e ^ { - s } { \\hat { f } ( \\xi - \\eta , s ) \\hat { v _ * } ( \\eta , s ) } d \\eta d s \\\\ & & \\displaystyle + i \\int _ 1 ^ t \\int _ 1 ^ s \\iint e ^ { - i \\eta \\sigma s } F ( s , \\xi , \\xi - \\sigma , \\eta ) d s d \\eta d \\sigma . \\end{array} \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} \\hat \\tau : = \\inf \\{ t \\geq 0 : X _ t \\notin C \\} < \\infty \\enskip \\textrm { a . s . } , \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{align*} Y : = \\pi + \\pi H ^ 0 \\pi ^ \\perp [ \\pi ^ \\perp H ^ 0 \\pi ^ \\perp ] ^ { - 2 } \\pi ^ \\perp H ^ 0 \\pi \\end{align*}"}
-{"id": "1867.png", "formula": "\\begin{align*} & \\sum _ { t = a + m } ^ { n - 2 } \\sum _ { s = t + 1 } ^ { n - 1 } \\binom { n - t + a - 1 } { a - 1 } = \\sum _ { t = a + m } ^ { n - 1 } \\binom { n - t + a - 1 } { a - 1 } ( n - t - 1 ) \\\\ & = \\sum _ { t = a + m } ^ { n } \\binom { n - t + a - 1 } { a - 1 } ( n - t - 1 ) + 1 \\\\ & = \\sum _ { t = a + m } ^ n \\binom { n - t + a - 1 } { a - 1 } ( n - t + a ) - ( a + 1 ) \\binom { n - m } { a } + 1 \\\\ & = a \\binom { n - m + 1 } { a + 1 } - ( a + 1 ) \\binom { n - m } { a } + 1 = f ( a , m ) \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{align*} L = | \\{ ( k , i ) , f _ { p + 1 , q ' } \\in \\mathcal { B } _ { k , i } , q ' < q \\} | - | \\{ ( k , i ) , f _ { p , \\tilde q } \\in \\mathcal { B } _ { k , i } , \\tilde q < q \\} | \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} \\vartheta _ 4 ( y ) = \\sum _ { k = - \\infty } ^ { \\infty } ( - 1 ) ^ k \\ , e ^ { - \\pi k ^ 2 y } = \\prod _ { n = 1 } ^ \\infty \\left ( 1 - e ^ { - 2 n \\pi y } \\right ) \\left ( 1 - e ^ { - ( 2 n - 1 ) \\pi y } \\right ) ^ 2 , \\end{align*}"}
-{"id": "993.png", "formula": "\\begin{align*} \\limsup _ { k } d ( u ^ { k } , C ) = \\limsup _ { k } \\Vert u ^ { k } \\Vert > 0 \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} 0 = t _ 0 < t _ 1 < t _ 2 < \\cdots < t _ M \\leq 2 T , \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} \\overrightarrow P \\wedge \\mathrm { c u r l } ( \\overrightarrow P ) = 0 \\ , , \\end{align*}"}
-{"id": "3232.png", "formula": "\\begin{align*} \\begin{cases} \\triangle \\alpha - ( \\lambda + k ) ( \\lambda + n - k ) \\alpha = 2 \\Big ( d ^ \\ast \\beta - ( \\lambda + n - k ) \\alpha \\Big ) , & \\\\ \\triangle \\beta - ( \\lambda + n - k ) ( \\lambda + k ) \\beta = 2 \\Big ( d \\alpha - ( \\lambda + k ) \\beta \\Big ) . \\end{cases} \\end{align*}"}
-{"id": "2490.png", "formula": "\\begin{align*} N ^ { i \\xi } \\phi _ N ( \\xi ) = e ^ { i A _ N \\xi } + N ^ { i \\xi } \\int _ { N e ^ { - A _ N } } ^ N x ^ { - i \\xi } \\left ( 1 - \\frac { x } { N } \\right ) ^ { N - 1 } d x + K _ 2 ( N ) + O \\left ( \\frac { 1 } { N } \\right ) . \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} & { Q _ y } ( z ) : = \\frac { 1 } { r } \\int _ { - \\infty } ^ { z } ( r - A _ X ) F ( u , y ) \\mathbb { P } _ { x } ( X _ T \\in d u | M _ T = z ) , \\\\ & \\mathbb { P } _ { x } ( X _ T \\in d u | M _ T = z ) : = \\mathbb { P } _ { x } ( X _ T \\in d u \\ , , \\ , M _ T \\in d z ) / \\mathbb { P } _ { x } ( M _ T \\in d z ) , \\end{align*}"}
-{"id": "3775.png", "formula": "\\begin{align*} H ( Z ) = \\frac { R ^ \\nabla _ { Z , \\bar { Z } , Z , \\bar { Z } } } { h ( Z , Z ) h ( Z , Z ) } . \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} E _ { \\nu } ( z ; t ^ { - m } , t ) : = \\lim _ { q \\rightarrow t ^ { - m } } E _ { \\nu } ( z ; q , t ) \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} \\mathcal L _ { s \\partial } ( \\psi ) \\ ! \\ ! = \\ ! \\ ! \\lambda _ s \\circ ( \\partial \\otimes 1 ) \\circ \\psi + \\rho _ s \\circ ( 1 \\otimes \\partial ) \\circ \\psi \\\\ - \\lambda _ s \\circ \\psi \\circ \\partial ^ \\bullet - \\psi \\circ h \\circ d - \\underset { = 0 } { \\underbrace { \\psi \\circ d } } \\circ h \\ , . \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} & v \\mapsto Q ( v ) : = ( a _ 0 - b _ 0 ) \\prod _ { i = 1 } ^ { K - 1 } ( 1 - \\lambda _ i v ) \\prod _ { j = 1 } ^ { K - 1 } ( 1 + \\mu _ j v ) \\\\ & + \\sum _ { \\ell = 1 } ^ { K - 1 } a _ \\ell \\prod _ { i = 1 , i \\neq \\ell } ^ { K - 1 } ( 1 - \\lambda _ i v ) \\prod _ { j = 1 } ^ { K - 1 } ( 1 + \\mu _ j v ) - \\sum _ { \\ell = 1 } ^ { K - 1 } b _ \\ell \\prod _ { i = 1 } ^ { K - 1 } ( 1 - \\lambda _ i v ) \\prod _ { j = 1 , j \\neq \\ell } ^ { K - 1 } ( 1 + \\mu _ j v ) . \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} \\mathcal { I } _ 1 ( t , x ) = \\int _ { \\mathbb { R } } \\frac { \\partial G _ t } { \\partial x } ( x - y ) \\psi _ 0 ( y ) d y \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} \\| ( c _ k ) \\| _ { \\ell ^ { \\infty , \\alpha } } = \\sup _ { k \\geq 0 } \\langle k \\rangle ^ \\alpha | c _ k | \\mbox { a n d } \\| ( c _ k ) \\| _ { C _ \\lambda } = \\sup _ { k \\geq 0 } \\frac { \\sqrt { k ! } } { \\lambda ^ k } | c _ k | . \\end{align*}"}
-{"id": "939.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\rightarrow \\infty } N [ \\mathfrak { e } , \\lambda V ] = \\lim _ { \\lambda \\rightarrow \\infty } N _ { s c } [ \\mathfrak { e } , \\lambda V ] = \\mathop { \\mathrm { s u p p } } V , \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\psi } _ t = \\mathrm { d i v } \\ , T ^ { \\psi _ t } \\cdot \\psi _ t , \\\\ { \\psi } _ 0 = \\bar { \\psi } , \\end{cases} \\end{align*}"}
-{"id": "3217.png", "formula": "\\begin{align*} - i _ 1 + ( i _ 2 + \\dots + i _ k ) \\nu = - i _ 1 + ( m - i _ 1 ) \\nu \\leq \\nu . \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 , x _ 3 ) = \\max \\{ \\ , x _ 1 - x _ 2 - x _ 3 , \\ , x _ 1 + 4 , \\ , x _ 1 + x _ 2 + x _ 3 , \\ , - x _ 1 + x _ 2 + 2 , \\ , 0 \\ , \\} , \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{align*} ( \\rho \\partial _ \\rho ) ^ 2 \\psi = \\frac { 1 } { 2 } \\rho ^ 2 \\sinh 2 \\psi . \\end{align*}"}
-{"id": "1097.png", "formula": "\\begin{align*} \\varphi \\cdot ( s \\alpha ) = ( \\varphi \\cdot s ) \\cdot \\alpha \\ , . \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} F _ { r _ t | r _ m } ( y ) = 1 - \\sum ^ { t - m - 1 } _ { n = 0 } S _ n e ^ { - \\lambda _ c \\pi ( y ^ 2 - x ^ 2 ) } . \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{align*} \\mathcal { N } : = \\left \\{ \\frac { 2 \\pi } { \\sqrt { 3 } } \\sqrt { k ^ { 2 } + k l + l ^ { 2 } } \\ , : k , \\ , l \\ , \\in \\mathbb { N } ^ { \\ast } \\right \\} . \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} \\tilde { z } _ 1 + \\tilde { z } _ 2 = 3 ^ { 3 b + 1 } \\alpha ^ 3 , \\ \\tilde { z } _ 1 ^ 2 - \\tilde { z } _ 1 \\tilde { z } _ 2 + \\tilde { z } _ 2 ^ 2 = 3 \\beta ^ 3 , \\ \\alpha \\beta = \\tilde { z } _ 4 , \\ 3 \\nmid \\alpha \\beta , \\ \\gcd ( \\alpha , \\beta ) = 1 . \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} & D _ { \\frac 1 2 } ( p _ X \\| q _ X ) = - 2 \\log \\left ( \\sum _ { x } p _ X ( x ) ^ { \\frac 1 2 } q _ X ( x ) ^ { \\frac 1 2 } \\right ) \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{align*} h _ \\epsilon = 1 + O ( \\epsilon ^ 2 r ^ { \\nu } ) , \\Omega _ \\epsilon = \\Omega _ 0 + O ( \\epsilon r ^ { - 1 } ) , \\end{align*}"}
-{"id": "9287.png", "formula": "\\begin{align*} u ( \\lambda , \\lambda x ) = \\lambda ^ { n } u ( 1 , x ) , n \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{align*} F : = \\cap _ { k = 1 } ^ \\infty F _ k , \\end{align*}"}
-{"id": "1315.png", "formula": "\\begin{align*} u ^ { m s } _ h - \\hat u ^ { m s , \\Pi , j } _ h = T ( P - P ^ j ) \\lambda _ h ^ { m s , \\Pi } . \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} & \\frac { \\operatorname { p . v . } } { \\pi } \\int f ( \\beta , t ) \\frac { \\xi _ \\beta ( \\beta , t ) } { \\xi ( s , t ) - \\xi ( \\beta , t ) } d \\beta \\\\ & = \\frac { \\operatorname { p . v . } } { L } \\int _ 0 ^ L f ( \\beta ) \\xi _ \\beta \\cot \\frac { \\pi ( \\xi ( s ) - \\xi ( \\beta ) ) } { L } d \\beta . \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} \\sum _ { r ( e ) = v } S _ e S _ e ^ * = \\sum _ { r ( e ) = v } J _ v ^ * W _ { c ( e ) } W _ { c ( e ) } ^ * J _ v = J _ v ^ * \\sum _ { i = 1 } ^ d W _ i W _ i ^ * J _ v = J _ v ^ * J _ v = S _ v . \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} h '' \\pm c h = 0 . \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} u ( x ) : = \\sum _ { j = 1 } ^ { \\infty } 4 ^ { - N j } \\chi _ { j } ( x ) , x \\in \\R ^ d \\setminus B ( 0 , 1 ) . \\end{align*}"}
-{"id": "7765.png", "formula": "\\begin{align*} s _ j - s _ i = \\tfrac { L } { a _ 1 } ( j - i ) 0 \\leq i \\leq j \\leq a _ 1 - 1 . \\end{align*}"}
-{"id": "6051.png", "formula": "\\begin{align*} \\begin{cases} \\overline { k } _ { y y y } + \\overline { k } _ y + \\overline { k } _ { x x x } + \\overline { k } _ x - \\lambda \\overline { k } = - \\lambda \\delta ( x - y ) & , \\\\ \\overline { k } ( x , 0 ) = \\overline { k } ( x , L ) = 0 , & , \\\\ \\overline { k } _ y ( x , 0 ) = \\overline { k } _ y ( x , L ) = 0 , & , \\\\ \\overline { k } ( 0 , y ) = \\overline { k } ( L , y ) = 0 , & . \\end{cases} \\end{align*}"}
-{"id": "6766.png", "formula": "\\begin{align*} x \\alpha \\cdot y = ( x y ) \\lambda \\end{align*}"}
-{"id": "502.png", "formula": "\\begin{align*} \\left | \\frac { f _ { \\ell + 1 } ( z ) } { f _ { \\ell } ( z ) } \\right | = \\left ( 1 + \\frac { 1 } { \\ell } \\right ) ^ { k - 1 } \\exp \\left ( - \\frac { k - 1 } { \\ell } \\right ) \\ll \\exp \\left ( - \\frac { k } { 4 \\ell ^ 2 } \\right ) . \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} D ^ 2 = - K \\cdot D \\Rightarrow ( K + H ) \\cdot D = - D ^ 2 + H \\cdot D . \\end{align*}"}
-{"id": "16.png", "formula": "\\begin{align*} d \\partial _ x \\Psi ( s , X ( s ) , \\lambda ) & = \\sqrt { \\xi '' ( s ) } \\partial _ { x } ^ 2 \\Psi ( s , X ( s ) , \\lambda ) d W ( s ) , \\\\ d \\partial _ { x } ^ 2 \\Psi ( s , X ( s ) , \\lambda ) & = - \\xi '' ( s ) A ( s ) \\bigl ( \\partial _ { x } ^ 2 \\Psi ( s , X ( s ) , \\lambda ) \\bigr ) ^ 2 d s + \\sqrt { \\xi '' ( s ) } \\partial _ { x } ^ 3 \\Psi ( s , X ( s ) , \\lambda ) d W ( s ) . \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{align*} k ^ \\ell Q ^ { ( \\infty ) , \\ell } = \\int _ \\R q ^ \\ell \\mu _ k ( \\d q ) , \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} s ^ * ( X ; Y ) = \\sup _ { p _ { U | X } } \\frac { I ( U ; Y ) } { I ( U ; X ) } \\end{align*}"}
-{"id": "2489.png", "formula": "\\begin{align*} K _ 1 ( N ) = e ^ { i \\xi A _ N } - 1 + N ^ { i \\xi } \\int _ { N e ^ { - A _ N } } ^ N x ^ { - i \\xi } \\left ( 1 - \\frac { x } { N } \\right ) ^ { N - 1 } d x + o \\left ( \\frac { 1 } { N } \\right ) . \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} \\mathcal L _ \\partial ( \\omega _ S ) = \\mathrm { d i v } ( \\partial ) \\omega _ S \\ , . \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} z ^ { \\mu } = E _ { \\mu } + \\sum _ { \\nu \\prec \\mu } d _ { \\mu , \\nu } ( q , t ) E _ { \\nu } , \\end{align*}"}
-{"id": "8326.png", "formula": "\\begin{align*} \\frac { \\partial _ \\alpha a } { | z _ \\alpha | } = ( \\frac { \\partial _ \\alpha } { | z _ \\alpha | } ) ^ 2 \\theta \\frac { \\partial _ \\alpha \\theta } { | z _ \\alpha | } + \\operatorname { I m } \\bigl ( \\frac { \\partial _ \\alpha z _ { t t } } { | z _ \\alpha | } e ^ { - i \\theta } \\bigr ) . \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} \\mathfrak { F } _ \\Omega = \\{ r _ 1 \\omega _ 1 + r _ 2 \\omega _ 2 ; r _ 1 , r _ 2 \\in [ 0 , 1 ) , r _ 1 ^ 2 + r _ 2 ^ 2 \\neq 0 \\} , \\end{align*}"}
-{"id": "1232.png", "formula": "\\begin{align*} S ( x , y , z ) = \\frac { 1 } { 5 ^ { 1 / 5 } } Q ( \\frac { 3 x } { 5 ^ { 3 / 5 } } , \\frac { 2 y } { 5 ^ { 2 / 5 } } , \\frac { z } { 5 ^ { 1 / 5 } } ) \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{align*} q _ R ( r ) = \\sum _ { a } p _ A ( a ) q _ { R | A } ( r | a ) > 0 \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l l } \\nabla _ { B } \\nabla _ { B } h ( X , Y ) = 0 , \\\\ \\noalign { \\smallskip } R i c _ { B } ( X , Y ) + \\nabla _ { B } \\nabla _ { B } \\beta ( X , Y ) = 0 , \\\\ \\noalign { \\smallskip } \\nabla _ { F } \\nabla _ { F } \\varphi ( U , V ) = 0 , \\\\ \\noalign { \\smallskip } R i c _ { F } ( U , V ) = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{array} { l l } k ( x , 0 ) = k ( x , L ) = k ( 0 , y ) = k ( L , y ) = k _ y ( x , 0 ) = k _ y ( x , L ) = 0 , & , \\\\ s ( x , 0 ) = s ( x , L ) = s ( 0 , y ) = s ( L , y ) = s _ y ( x , 0 ) = s _ y ( x , L ) = 0 , & , \\end{array} \\right . \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{align*} \\ - c \\kappa [ \\tau ( \\zeta ) ( - c t ) ] & = \\kappa '' - \\kappa \\tau ^ { 2 } + \\frac { 1 } { 2 } ( \\kappa ^ { 2 } + A ) \\kappa \\\\ c \\kappa ' & = 2 \\kappa ' \\tau + \\kappa \\tau ' \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{align*} \\epsilon _ n = \\frac { \\rho _ n } { 5 0 } \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} A = \\frac { 1 } { u _ 3 \\| u ' \\| _ J } \\begin{pmatrix} \\det \\begin{pmatrix} u _ 3 & u _ 1 \\\\ u _ 3 ' & u _ 1 ' \\end{pmatrix} & \\det \\begin{pmatrix} u _ 3 & u _ 2 \\\\ u _ 3 ' & u _ 2 ' \\end{pmatrix} \\\\ & \\\\ \\det \\begin{pmatrix} u _ 2 & u _ 3 \\\\ u _ 2 ' & u _ 3 ' \\end{pmatrix} & \\det \\begin{pmatrix} u _ 3 & u _ 1 \\\\ u _ 3 ' & u _ 1 ' \\end{pmatrix} \\end{pmatrix} = { ( a _ { i j } ) } . \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} { } e ^ { - \\alpha ^ * } = 1 - e ^ { - \\beta ^ * N } = 1 - e ^ { - v } \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} \\Psi \\circ \\tau _ A = \\tau _ B \\circ \\Psi . \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} \\begin{array} { l l l l l } \\hat { f } ( \\xi ) & = & \\displaystyle \\hat { u _ * } + i \\int _ 1 ^ t \\int e ^ { i \\frac 1 2 ( \\eta ^ 2 - 2 \\xi \\eta ) s } e ^ { - s } { \\hat { f } ( \\xi - \\eta , s ) \\hat { v _ * } ( \\eta , s ) } d \\eta d s \\\\ & & \\displaystyle + i \\int _ 1 ^ t \\int _ 1 ^ s \\iint e ^ { i \\eta ( \\sigma - \\xi ) s } F ( s , \\xi , \\sigma , \\eta ) d s d \\eta d \\sigma , \\end{array} \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{align*} u ( x ) = \\begin{cases} e ^ { i \\sqrt { \\lambda } x } , & x \\geq 0 , \\\\ \\overline { \\alpha } e ^ { \\sqrt { \\lambda } x } + \\alpha e ^ { - \\sqrt { \\lambda } x } , & x < 0 , \\end{cases} \\quad v ( x ) = \\begin{cases} \\alpha e ^ { i \\sqrt { \\lambda } x } + \\overline { \\alpha } e ^ { - i \\sqrt { \\lambda } x } , & x \\geq 0 , \\\\ e ^ { \\sqrt { \\lambda } x } , & x < 0 , \\\\ \\end{cases} \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} g ^ { - 1 } ( ( - j g ^ { ( h ) } ( c ' ) ( - a ) ^ { - \\frac { \\ell _ 1 p ^ h } { i } } ) _ { h = 0 } ^ \\infty ) & = - j g ^ { - 1 } ( g ( c ' ) g ( [ ( - a ) ^ { - \\frac { \\ell _ 1 } { i } } ] ) ) \\\\ & = - j c ' [ ( - a ) ^ { - \\frac { \\ell _ 1 } { i } } ] \\equiv - j c ( - a ) ^ { - \\frac { \\ell _ 1 } { i } } \\bmod p . \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} \\int _ 0 ^ s g ^ * ( z ) \\ , d z & \\leq \\int _ 0 ^ s f ^ * ( z ) \\ , d z , 0 \\leq s < 1 \\\\ \\int _ 0 ^ 1 g ^ * ( z ) \\ , d z & = \\int _ 0 ^ 1 f ^ * ( z ) \\ , d z \\end{align*}"}
-{"id": "7011.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p ( n ) q ^ { n } = \\dfrac { 1 } { ( q ; q ) _ { \\infty } } . \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} \\langle \\phi , \\sum _ a R ^ Q ( \\bar V _ a , V _ a ) \\phi \\rangle & = ( \\sum _ { i = 1 } ^ r \\lambda _ { a _ i } ) | \\phi | ^ 2 = \\sum _ { i = 1 } ^ r R i c ^ Q ( E _ { a _ i } , E _ { a _ i } ) | \\phi | ^ 2 . \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{align*} \\mathcal { N } _ { u } ( q , u ) h = a ( x ) q u ^ { q - 1 } h , \\end{align*}"}
-{"id": "7344.png", "formula": "\\begin{align*} \\{ x _ 1 , F \\} _ b = b _ { 1 , k + 1 } \\frac { \\partial F } { \\partial x _ { k + 1 } } + b _ { 1 , k + 2 } \\frac { \\partial F } { \\partial x _ { k + 2 } } \\ ; . \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} \\mathbb { P } ( m , K ) = \\prod _ { i = 0 } ^ { K - 1 } ( 1 - q ^ { i - m } ) . \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{align*} u _ 1 ^ { l _ 1 } \\cdots u _ { k + 1 } ^ { l _ { k + 1 } } = u _ 1 ^ { l _ 1 } \\cdots u _ k ^ { l _ k } u _ { k + 1 } y _ { k + 1 } ^ { l _ { k + 1 } - 1 } . \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} ( \\psi \\otimes \\operatorname { i d } \\otimes \\operatorname { i d } ) \\bigl ( ( \\Delta \\otimes \\operatorname { i d } ) ( \\Delta a ) \\bigr ) = ( \\psi \\otimes \\operatorname { i d } \\otimes \\operatorname { i d } ) \\bigl ( ( 1 \\otimes E ) \\Delta _ { 1 3 } ( a ) \\bigr ) . \\end{align*}"}
-{"id": "6542.png", "formula": "\\begin{align*} \\frac { 1 } { M + 1 } \\sum _ { j = 1 } ^ M K ( P _ j , P _ 0 ) \\leq \\alpha \\log M , \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} \\mathrm { T r a c e } ( \\sigma ) & = \\# \\{ \\{ i , j \\} : \\sigma \\} - \\# \\{ \\{ i , j \\} : \\sigma \\} \\\\ & = \\binom { x _ 1 ( \\sigma ) } { 2 } - \\binom { x _ 2 ( \\sigma ) } { 1 } \\\\ & = Q ( \\sigma ) . \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} \\psi ^ n ( p _ i ) = d ^ { n - 1 } ( p _ i ' ) + h ^ { n + 1 } \\circ d ^ n ( p _ i ) \\ , . \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} \\forall i , j , k \\in [ 1 , n ] \\quad \\quad \\# \\{ i , j , k \\} = 3 , [ x _ k , t _ { i j } ] = [ y _ k , t _ { i j } ] = 0 , \\end{align*}"}
-{"id": "8501.png", "formula": "\\begin{align*} \\tilde V ( q ) = - \\log ( \\sin 2 q ) - 2 \\log ( \\sin q ) , \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} & I ( U ; Y | V ) > I ( U ; Z | V ) = \\ ! ( 1 \\ ! - \\ ! \\epsilon ) I ( U ; X Y | V ) \\ ! = ( 1 - \\epsilon ) I ( U ; X | V ) . \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } K ^ { \\rm r e s c } _ t ( s _ 1 , s _ 2 ) \\frac { e ^ { t ^ { 1 / 2 } f _ 1 ( \\alpha ^ { - 1 } , s _ 2 ) } } { e ^ { t ^ { 1 / 2 } f _ 1 ( \\alpha ^ { - 1 } , s _ 1 ) } } = K _ { \\rm G U E ( M ) } ( s _ 1 + \\xi _ c , s _ 2 + \\xi _ c ) . \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} \\Phi ' : = ( ( \\tilde { A } _ 1 , \\tilde { b } _ 1 ) , ( A _ { \\ell + 1 } , b _ { \\ell + 1 } ) , \\dots , ( A _ L , b _ L ) ) . \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} E ^ { h } _ { L L L } ( v ) = \\int _ { \\C } \\Big ( | w | ^ 2 | v ( w ) | ^ 2 + \\frac { N a \\Omega ^ 2 _ h } 2 | v ( w ) | ^ 4 \\Big ) d L ( w ) , \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} & | \\lambda \\cap S _ \\mu ( m ) | + | \\lambda \\cap T _ \\mu ( m ) | - \\frac { | \\lambda | } { 2 } \\\\ & = | \\sigma \\cap ( S _ \\mu ( m ) \\times T _ \\mu ) | + | \\sigma \\cap ( S _ \\mu \\times T _ \\mu ( m ) ) | - | \\sigma | \\\\ & = | \\sigma \\cap ( S _ \\mu ( m ) \\times T _ \\mu ( m ) ) | - | \\sigma \\cap ( \\overline { S _ \\mu ( m ) } \\times \\overline { T _ \\mu ( m ) } ) | . \\end{align*}"}
-{"id": "713.png", "formula": "\\begin{align*} ( A _ k + \\delta _ j A _ { k } ) \\mathrm { C o l } _ j ( \\tilde { Y } ) = \\mathrm { C o l } _ j ( F ) , \\frac { \\norm { \\delta _ j A _ k } _ 2 } { \\norm { A _ k } _ 2 } \\leq \\varepsilon _ { A _ k } , \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} \\sum _ { n = M _ - } ^ { M _ + - 1 } A _ 2 ( n ) | C _ n | \\leq c _ 1 \\ , \\frac { ( \\gamma t ) ^ { - \\frac 1 2 + \\varepsilon ( d - 1 ) } } { M ^ d } \\sum _ { n = M _ - } ^ { M _ + } n ^ { d - 1 } \\leq c _ 2 \\ , \\frac { ( \\gamma t ) ^ { \\varepsilon d } } { M } \\leq c _ 3 \\ , { ( \\gamma t ) ^ { - 1 / 2 + \\varepsilon d } } . \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { P } \\left [ G _ t \\ \\middle | \\ G _ { < t } \\right ] & \\geq \\inf _ { V _ { < t } } \\mathbb { P } \\left [ G _ t \\ \\middle | \\ G _ { < t } , V _ { < t } \\right ] \\\\ & \\geq 1 - ( k - t + 1 ) \\exp \\left ( - \\left ( \\frac { c } { 2 ( \\sqrt { 2 } + \\sqrt { k - 1 } ) } \\right ) ^ 2 \\frac { d - 2 t + 1 } { 2 } \\right ) \\\\ & = 1 - ( k - t + 1 ) \\exp \\left ( \\frac { - c ^ 2 ( d - 2 t + 1 ) } { 8 ( \\sqrt { 2 } + \\sqrt { k - 1 } ) ^ 2 } \\right ) \\end{aligned} \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} { \\rm C o e f f } [ f _ { s _ i \\mu } , m ] = \\sum _ { \\nu \\in \\sigma ( \\epsilon ) } \\psi ( s _ i \\nu , s _ i \\mu ) f _ { s _ i \\nu } = \\sum _ { \\nu \\in \\sigma ( \\epsilon ) } \\psi ( \\nu , s _ i \\mu ) f _ { \\nu } , \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{align*} K _ 1 ( N ) = e ^ { i \\xi A _ N } - 1 + N \\int _ 0 ^ { A _ N } e ^ { i \\xi s } \\left ( 1 - e ^ { - s } \\right ) ^ { N - 1 } e ^ { - s } d s + o \\left ( \\frac { 1 } { N } \\right ) . \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} r _ j ^ { d - 1 } \\phi _ j ( \\theta , s ) = s G _ j ( \\theta ) + O ( s ^ 2 ) . \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} S ( a _ i ) = 1 \\ , , \\end{align*}"}
-{"id": "8879.png", "formula": "\\begin{align*} D ^ { a c } _ f = \\sum _ { Y \\in \\mathcal { I } _ X ^ G } \\left ( 1 - \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\alpha \\circ \\mathcal { P } ( \\mu _ Y ) \\right ) Y \\end{align*}"}
-{"id": "191.png", "formula": "\\begin{align*} \\Phi ( r , \\xi ) = \\Phi ( - r , - \\xi ) \\end{align*}"}
-{"id": "877.png", "formula": "\\begin{align*} R _ 2 ( \\xi , s ) = \\int _ 1 ^ s \\iint \\frac 1 { 2 s ' } ( e ^ { i \\frac { i \\eta ' \\sigma ' } { 4 s } } - 1 ) e ^ { - \\frac 1 2 i \\xi \\sigma ' } \\mathcal { F } _ { \\eta , \\sigma } ^ { - 1 } \\Big ( G \\Big ( s ' , s , \\xi , \\eta , \\sigma \\frac { s } { s ' } - \\eta \\frac { s - s ' } { 2 s ' } \\Big ) \\Big ) d \\eta ' d \\sigma ' d s ' \\end{align*}"}
-{"id": "7062.png", "formula": "\\begin{align*} \\| V _ m ^ { 0 - 2 , l + 1 } \\| _ { 1 , \\infty , r } \\le \\sum _ { p , q = 2 } ^ { N } 1 _ { p , q \\in 2 \\N } 1 _ { p + q = m } \\| V _ { p , q } ^ { 0 - 2 , l + 1 } \\| _ { 1 , \\infty , r } , \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{align*} h _ 2 & = \\sum _ { i = 1 } ^ { d / 2 } \\frac { 1 } { ( 2 i - 1 ) ( 2 ( d + 1 - i ) - 1 ) } + \\underset { i + j \\neq d + 1 } { \\sum _ { 1 \\leq i < j \\leq d } } \\frac { 1 } { ( 2 i - 1 ) ( 2 j - 1 ) } \\\\ & = \\frac { 1 } { 2 d } h _ 1 + \\sum _ { 1 \\leq i < j \\leq d / 2 } \\frac { 4 d ^ 2 } { ( 2 i - 1 ) ( 2 j - 1 ) ( 2 d - 2 i + 1 ) ( 2 d - 2 j + 1 ) } . \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} \\Omega _ { \\alpha , \\bar { \\alpha } } & = \\frac { - 1 } { 2 } e ^ { 2 \\alpha ( a ) } ( d _ a u ( \\alpha ^ { \\vee } ) - 2 \\chi ( \\alpha ^ { \\vee } ) ) \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} H = - \\frac 1 2 \\langle x , { \\bf n } \\rangle . \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} f ^ * Q _ l & = \\sum _ j w _ { l j } P _ j \\\\ \\bar { d _ j } & = \\frac { d _ j + w _ { l j } - 1 } { w _ { l j } } , \\ ; \\ ; f ( P _ j ) = Q _ l \\\\ \\delta _ l & = m a x \\{ \\bar { d _ j } ; f ( P _ j ) = Q _ l \\} . \\\\ \\Delta & = \\sum _ l \\delta _ l Q _ l . \\\\ M & = L - \\Delta . \\\\ \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} & 2 \\hat { b } _ + ^ i ( \\tilde \\Sigma ^ { - 1 } ) _ { i k } b _ - ^ k + \\nabla \\cdot b _ - \\\\ = & \\beta ^ { - 1 } \\partial _ k \\beta ( \\tilde \\gamma ^ { - 1 } ) ^ { \\eta k } H _ { \\eta \\ell } ( \\tilde \\gamma ^ { - 1 } ) ^ { \\ell j } F _ j + ( \\tilde \\gamma ^ { - 1 } ) ^ { k i } H _ { k \\ell } ( \\tilde \\gamma ^ { - 1 } ) ^ { \\ell j } \\partial _ { q ^ i } F _ j . \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{align*} v ( t ) = e ^ { - t / \\mu } v _ 0 + \\frac { \\lambda } { \\mu } \\int _ 0 ^ t e ^ { - ( s - t ) / \\mu } | u ( s ) | ^ 2 d s , \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} Z ( I _ 2 ) = ( z _ { r } ^ { - n } - v ^ n z _ { r } ^ n ) \\ , Z ( I _ 1 ; 0 , c _ r ) . \\end{align*}"}
-{"id": "7827.png", "formula": "\\begin{align*} z ^ 2 = \\frac { 3 0 0 0 y ^ 3 + 9 0 0 y ^ 2 + 9 0 y + 3 } { 3 8 4 0 y - 6 4 0 } = \\frac { 3 7 5 } { 1 2 8 } ( y + 1 / 1 0 ) ^ 3 - \\frac { 5 6 2 5 } { 5 1 2 } ( y + 1 / 1 0 ) ^ 4 ) + \\ldots \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { 2 ^ n - 1 } \\zeta ^ { - i } \\theta _ i ( g ) = \\begin{cases} 2 ^ n & g = x , \\\\ 0 & g \\neq x , \\end{cases} \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} f _ { T _ 1 } ( t ) = f _ { U } ( t ) . \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} s y _ 1 \\dots y _ { i - 1 } [ y _ i , x _ 1 , x _ 2 ] = y _ 1 \\dots y _ { i - 1 } s [ y _ i , x _ 1 , x _ 2 ] + \\sum _ { j = 1 } ^ { i - 1 } y _ 1 \\dots y _ { j - 1 } [ s , y _ j ] y _ { j + 1 } \\dots y _ { i - 1 } [ y _ i , x _ 1 , x _ 2 ] . \\end{align*}"}
-{"id": "6335.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\hbox { M i n i m i z e } \\ , | | A U - B | | ^ 2 \\\\ U ^ T U = \\mathbb { I } _ p \\end{array} \\right . , \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} H _ { d R } ^ k ( \\mathbb { C } ^ m ) = \\left \\{ \\begin{array} { l l } \\mathbb { R } , & k = 0 , \\\\ 0 , , & k \\geq 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "2779.png", "formula": "\\begin{align*} & \\sum _ { b \\in B } x _ b \\\\ & x _ b \\geq \\sum _ { j \\in C } z _ { j , b } & b \\in B \\\\ & \\sum _ { b \\in B } w _ { i , b } \\leq 1 & i \\in H \\\\ & \\sum _ { b \\in B } z _ { j , b } = 1 & j \\in C \\\\ & \\sum _ { i \\in H } w _ { i , b } \\cdot h _ i \\geq \\sum _ { j \\in C } z _ { j , b } \\cdot c _ j & b \\in B \\\\ & x _ b , w _ { i , b } , z _ { j , b } \\in \\{ 0 , 1 \\} & b \\in B , i \\in H , j \\in C \\end{align*}"}
-{"id": "1786.png", "formula": "\\begin{align*} p _ \\kappa ( \\underline { x } , \\underline { y } , \\underline { \\xi } ) = p ( x , y , M ( x , y ) ^ t \\underline { \\xi } ) ) | \\det M ( x , y ) | | \\det \\kappa ' ( y ) ^ { - 1 } | \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{align*} C = \\biggl ( n _ { \\alpha } \\biggl \\lceil \\dfrac { B ( \\alpha ^ { \\vee } , \\mu ) } { n _ { \\alpha } Q ( \\alpha ^ { \\vee } ) } \\biggr \\rceil - \\dfrac { B ( \\alpha ^ { \\vee } , \\mu ) } { Q ( \\alpha ^ { \\vee } ) } \\biggr ) \\alpha ^ { \\vee } \\ , , D = g ( B ( \\alpha ^ { \\vee } , \\mu ) - Q ( \\alpha ^ { \\vee } ) ) , \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} \\tilde \\chi \\left ( \\int _ 0 ^ { + \\infty } e ^ { - \\lambda t } T ( t ) d t \\right ) = { \\mathcal L } ( u _ { \\lambda } ) ( - \\tilde \\chi ( A _ { \\bf T } ) ) = \\frac { 1 } { \\lambda - \\tilde \\chi ( A _ { \\bf T } ) } \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} - 2 \\arctan \\left ( \\tanh \\left ( \\frac { ( 1 - \\epsilon ) \\tau } { 4 } \\right ) \\right ) + \\frac { \\pi } { 2 } & = \\mathcal { O } ( e ^ { - \\frac { ( 1 - \\epsilon ) \\tau } { 2 } } ) , \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} v = \\sum _ { \\mu , \\lambda } E _ \\mu ^ * E _ \\lambda v . \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} \\hat { \\varphi } _ \\xi ( y ) + q _ { \\xi , t } ( y ) = \\Big ( \\varphi _ \\xi ( y ) + q _ { \\xi , t } ( y ) \\Big ) \\wedge \\Big ( a _ { L _ t } - c _ * \\Big ) , y \\in B _ { R ^ * _ t } \\setminus \\{ 0 \\} , \\end{align*}"}
-{"id": "2099.png", "formula": "\\begin{align*} | \\int _ C \\omega _ X - \\int _ { C _ k } \\omega _ X | = | < \\omega _ X , C - C _ k > | = | \\tau < c _ 1 ( T \\overline { X } ) , C - C _ k > | \\le | \\tau | c _ 0 , \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} \\cap _ n ( V K _ 2 ^ { t o p } ( K ) ) ^ { p ^ n } = \\{ 1 \\} , \\end{align*}"}
-{"id": "949.png", "formula": "\\begin{align*} N [ \\mathfrak { e } , { \\widetilde { V } } ] \\ \\leq \\ \\mathcal { L } _ { \\widetilde { V } } [ ( 1 - { \\varepsilon } ) \\eta ( \\mathfrak { e } ) ] \\ = \\ \\# \\big \\{ x \\in \\Gamma \\ , \\big | \\ V ( x ) \\geq ( 1 - { \\varepsilon } ) \\eta ( \\mathfrak { e } ) \\big \\} . \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{align*} \\sum _ { \\substack { | \\ell | < n / 2 \\\\ 2 \\mid ( \\ell + n ) } } p _ 1 \\left ( \\frac { \\ell } { n ^ { 3 / 4 } } \\right ) = \\frac { n ^ { 3 / 4 } } { 2 } \\int \\limits _ { - \\infty } ^ { \\infty } p _ 1 ( t ) d t + O ( n ^ { 1 / 1 2 } ) . \\end{align*}"}
-{"id": "2909.png", "formula": "\\begin{align*} \\begin{cases} \\Delta f = 0 & \\ \\R ^ 2 \\setminus { \\Gamma } , \\\\ f = \\kappa & \\ \\Gamma . \\end{cases} \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} U ( F ) ( x , y ) = h ( y ) F ( \\varphi ( x , y ) , \\tau ( y ) ) , ( x , y ) \\in [ 0 , 1 ] \\times Y _ 2 \\end{align*}"}
-{"id": "606.png", "formula": "\\begin{align*} D \\hat { \\varphi } _ K \\left ( q \\frac { d } { d q } \\right ) = \\hat { \\varphi } ^ * _ K v \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} p _ 0 ( t ) = \\sum _ { i = 0 } ^ N a _ i ( \\lambda ) ( \\lambda - t ) ^ i , p _ 1 ( t ) = \\sum _ { i = 0 } ^ N b _ i ( \\lambda ) ( \\lambda - t ) ^ i , \\end{align*}"}
-{"id": "6624.png", "formula": "\\begin{align*} d _ i ( p _ 1 + t u _ 1 , p _ 2 + t u _ 2 ) - d _ i ( p _ 1 , p _ 2 ) ) = o ( t ) , \\mbox { a s } t \\to 0 . \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{align*} \\liminf _ { \\lambda \\rightarrow \\infty } \\frac { N _ { s c } [ \\mathfrak { e } , \\lambda \\tilde { V } ] } { N _ { s c } ^ { > } [ \\mathfrak { e } , \\lambda \\tilde { V } ] } < \\infty , \\ ; \\limsup _ { \\lambda \\rightarrow \\infty } \\frac { N _ { s c } [ \\mathfrak { e } , \\lambda \\tilde { V } ] } { N _ { s c } ^ { > } [ \\mathfrak { e } , \\lambda \\tilde { V } ] } = \\infty . \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{align*} \\sum _ { i \\in \\sigma } \\| \\Lambda _ i f \\| ^ 2 + \\sum _ { i \\in ( I \\setminus J ) \\setminus \\sigma } \\| \\Gamma _ i f \\| ^ 2 & = \\sum _ { i \\in \\sigma \\cup J } \\| \\Lambda _ i f \\| ^ 2 - \\sum _ { i \\in J } \\| \\Lambda _ i f \\| ^ 2 + \\sum _ { i \\in ( I \\setminus J ) \\setminus \\sigma } \\| \\Gamma _ i f \\| ^ 2 \\\\ & = \\sum _ { i \\in \\sigma \\cup J } \\| \\Lambda _ i f \\| ^ 2 + \\sum _ { i \\in ( \\sigma \\cup J ) ^ c } \\| \\Gamma _ i f \\| ^ 2 - \\sum _ { i \\in J } \\| \\Lambda _ i f \\| ^ 2 \\\\ & \\ge ( A - D ) \\| f \\| ^ 2 . \\end{align*}"}
-{"id": "139.png", "formula": "\\begin{align*} g _ { s K } ( \\dot q , \\dot q ) = \\int _ X | \\varphi _ \\infty | ^ 2 \\ , d A , \\end{align*}"}
-{"id": "1691.png", "formula": "\\begin{align*} \\sum _ { i \\leq \\frac { n } { \\log n } } \\binom { n } { i } . \\end{align*}"}
-{"id": "1994.png", "formula": "\\begin{align*} \\langle \\langle \\Delta '' u , \\ , u \\rangle \\rangle = \\langle \\langle \\Delta '' u _ { h ^ \\perp } , \\ , u _ h + u _ { h ^ \\perp } \\rangle \\rangle = \\langle \\langle \\Delta '' u _ { h ^ \\perp } , \\ , u _ { h ^ \\perp } \\rangle \\rangle \\geq \\delta '' _ k \\ , | | u _ { h ^ \\perp } | | ^ 2 \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} \\iota \\left ( \\mathrm { c c } _ { Y / S } ( A ) \\right ) = \\mathrm { t r } _ { \\tilde { c } } ( \\tilde { u } , A | _ { X _ { G } } ) . \\end{align*}"}
-{"id": "5577.png", "formula": "\\begin{align*} g ( x ) = \\frac { \\theta _ 3 ( \\beta x ) \\theta _ 3 ( \\tfrac { x } { \\beta } ) } { \\theta _ 3 ( \\alpha x ) \\theta _ 3 ( \\tfrac { x } { \\alpha } ) } . \\end{align*}"}
-{"id": "1871.png", "formula": "\\begin{align*} a _ n & = a _ { n - 1 } + ( 2 n - 9 ) 2 ^ { n - 2 } + 3 + \\binom { n + 1 } { 2 } \\\\ & + \\frac { 1 } { 1 2 } ( n - 2 ) ( 3 \\cdot 2 ^ n - ( n - 1 ) ( n ^ 2 - 3 n + 1 2 ) ) \\\\ & + \\sum _ { m = 0 } ^ { n - 3 } \\left ( ( n - m + 8 ) 2 ^ { n - m - 3 } + 1 - \\binom { n - m + 2 } { 2 } - \\frac { ( n - m - 1 ) ( n - m - 2 ) ( 2 n - m - 3 ) } { 6 } \\right ) , \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{align*} \\L f ( x ) = \\frac 1 2 \\frac { \\Sigma _ 1 ^ 2 } { 2 x } f '' ( x ) - \\left ( \\dfrac { \\overline { \\delta \\beta ^ { ( 2 ) } _ + } } { 2 x } + \\dfrac { \\overline { \\delta \\beta ^ { ( 4 ) } } } { 4 x ^ 2 } \\right ) f ' ( x ) . \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{align*} \\langle \\Phi _ { L , \\Delta L } \\ , | \\ , H ( \\mathfrak { e } , V ) \\Phi _ { L ^ { \\prime } , \\Delta L ^ { \\prime } } \\rangle = 0 . \\end{align*}"}
-{"id": "204.png", "formula": "\\begin{align*} \\left ( 1 + \\sum _ { i = 1 } ^ n x _ i + \\sum _ { j = 1 } ^ m g _ j ( x ) \\right ) ^ r p ( x ) = \\sum _ { ( \\alpha , \\beta ) \\in \\mathbb { N } ^ { n + m } } c _ { \\alpha , \\beta } x ^ { \\alpha } g ( x ) ^ { \\beta } , \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{gather*} \\lim _ { \\alpha _ 1 \\rightarrow \\infty } \\alpha _ 1 \\Psi _ 1 ( x _ { ( \\alpha _ { 1 } , \\alpha _ 2 ) , \\varepsilon } , y _ { ( \\alpha _ { 1 } , \\alpha _ 2 ) , \\varepsilon } ) = 0 , \\\\ \\Psi _ 1 ( x _ { \\alpha _ 2 , \\varepsilon } , y _ { \\alpha _ 2 , \\varepsilon } ) = 0 . \\end{gather*}"}
-{"id": "1155.png", "formula": "\\begin{align*} \\varphi _ L = \\alpha _ 1 ^ * \\wedge \\cdots \\wedge \\alpha _ d ^ * \\ , . \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} & | h | _ i ^ { ( m ) } : = \\underset { l , l ' , | \\alpha | , | \\beta | \\leq i } { \\max } \\ \\underset { \\substack { x \\in U \\\\ \\xi ' \\in \\R ^ { n - 1 } } } { \\sup } \\left \\| y _ n ^ l \\partial _ { y _ n } ^ { l ' } \\partial _ { x } ^ \\beta \\partial _ { \\xi ' } ^ \\alpha { h } ( x , \\xi ' , \\cdot ) \\right \\| _ { L ^ 2 _ { y _ n } ( { \\R } _ { + } ) } \\langle \\xi ' \\rangle ^ { - m - \\frac { 1 } { 2 } + l - l ' + | \\alpha | } \\end{align*}"}
-{"id": "683.png", "formula": "\\begin{align*} A _ 1 X _ 1 ^ { \\star } B _ 1 - C _ 1 X _ 2 ^ { t _ 1 } D _ 1 = E _ 2 \\iff B _ 1 ^ \\star X _ 1 A _ 1 ^ \\star - D _ 1 ^ \\star ( X _ 2 ^ { t _ 1 } ) ^ { \\star } C _ 1 ^ \\star = E _ 1 ^ \\star . \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} \\frac { d x _ r ( t ) } { d t } & = A _ { 1 1 } x _ r ( t ) + B _ 1 u ( t ) + \\sum _ { i = 1 } ^ m N _ { i , 1 1 } x _ r ( t ) u _ i ( t ) , \\ ; \\ ; \\ ; x _ r ( 0 ) = 0 , \\\\ y _ r ( t ) & = C _ 1 x _ r ( t ) , \\ ; \\ ; \\ ; t \\geq 0 . \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{align*} & \\psi _ 1 ( n _ 1 ; \\lambda _ 1 ) \\psi _ 2 ( n _ 2 ; \\lambda _ 2 ) \\cdots \\psi _ g ( n _ g ; \\lambda _ g ) \\\\ & = ( - 1 ) ^ g \\sum _ { k _ g = 1 } ^ { n _ g } \\cdots \\sum _ { k _ 1 = 1 } ^ { n _ 1 } ( - 1 ) ^ { k _ 1 + \\cdots + k _ g } \\binom { n _ 1 } { k _ 1 } \\cdots \\binom { n _ g } { k _ g } \\frac { ( k _ 1 p _ 1 ) \\cdots ( k _ g p _ g ) } { ( \\lambda _ 1 + k _ 1 p _ 1 ) \\cdots ( \\lambda _ g + k _ g p _ g ) } . \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} \\partial _ M ( ( s \\otimes t ) \\cdot m ) = \\partial ^ e ( s \\otimes t ) \\cdot m + ( s \\otimes t ) \\cdot \\partial _ M ( m ) \\ , . \\end{align*}"}
-{"id": "5584.png", "formula": "\\begin{align*} \\psi _ 1 ( x ) = \\psi _ 1 ( \\tfrac { 1 } { x } ) \\end{align*}"}
-{"id": "4442.png", "formula": "\\begin{align*} \\phi _ { \\bf T } ( \\mu ) x = \\int _ 0 ^ { + \\infty } T ( t ) x d \\mu ( t ) , \\phi _ { \\bf T } ( \\mu ) = \\int _ 0 ^ { + \\infty } T ( t ) d \\mu ( t ) . \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} \\begin{array} { l } \\beta '' - ( n - 1 ) ( h ' ) ^ { - 1 } h ''' = \\pm \\bar { \\lambda } , \\\\ \\noalign { \\smallskip } ( h ' ) ^ 2 \\bar { \\lambda } = \\pm h ' h '' \\beta ' \\mp ( n - 2 ) ( h '' ) ^ 2 \\mp h ''' + \\bar { c } ( n - 2 ) , \\\\ \\noalign { \\smallskip } R i c _ N = \\bar { c } ( n - 2 ) g _ N , \\end{array} \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} B _ r ^ { \\| \\cdot \\| } ( x ) = B _ r ( x ) = \\{ y \\in \\R ^ d \\ , : \\ , \\| y - x \\| < r \\} \\overline { B _ r } ^ { \\| \\cdot \\| } ( x ) = \\overline { B _ r } ( x ) = \\{ y \\in \\R ^ d \\ , : \\ , \\| y - x \\| \\leq r \\} \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} \\widetilde { W } _ \\delta ( t ) = \\widetilde { W } _ \\delta ( t _ { n - 1 } ) \\ \\forall t \\in [ t _ { n - 1 } , t _ n ) . \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} X ( t , y ) = y + \\int _ { 0 } ^ { t } \\Lambda ( s , X ( s , y ) ) \\ { \\rm d } s , \\mbox { a n d } \\nabla Y ( t , X ( t , y ) ) = [ \\nabla X ] ^ { - 1 } ( t , y ) , \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} A _ i : = F _ 1 \\circ P _ 1 \\circ \\cdots \\circ F _ { i - 1 } \\circ P _ { i - 1 } \\circ F _ i \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} d y _ { i , t } = - \\nabla f ( y _ { i , t } ) \\tilde { \\gamma } d t + \\tau \\tilde { \\gamma } d B _ { i , t } . \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{align*} \\lambda ( A + A ) & = \\sum _ { i = 0 } ^ { 2 n } \\lambda \\left ( S _ i \\right ) = \\sum _ { i = 0 } ^ { n } \\lambda \\left ( S _ { 2 i } \\right ) + \\sum _ { i = 0 } ^ { n - 1 } \\lambda \\left ( S _ { 2 i + 1 } \\right ) \\\\ & \\geq \\sum _ { i = 0 } ^ { n } 2 \\alpha _ i + \\sum _ { i = 0 } ^ { n - 1 } ( \\alpha _ i + \\alpha _ { i + 1 } ) = 4 \\alpha - ( \\alpha _ 0 + \\alpha _ n ) . \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l l } ( \\beta ' h ^ { - 1 } ) ' = \\mp b h ^ { - 2 } , \\\\ \\noalign { \\smallskip } c \\beta ' h ^ { - 1 } = b h ^ { - 1 } h ' + ( n - 2 ) ( \\bar { c } \\mp a c ) ( h ' ) ^ { - 1 } , \\\\ \\noalign { \\smallskip } h ''' \\pm a h ' = 0 , \\\\ \\noalign { \\smallskip } \\pm ( h '' ) ^ { 2 } + a ( h ' ) ^ { 2 } = \\pm a c , \\\\ \\noalign { \\smallskip } R i c _ { N } = \\bar { c } ( n - 2 ) g _ { N } \\end{array} \\right . \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{align*} \\partial _ { t } ( \\tau N ) & = \\partial _ { t } ( \\partial _ { s } B ) \\\\ & = \\partial _ { s } ( \\partial _ { t } B ) \\\\ & = \\partial _ { s } \\Big [ - \\kappa _ { s } T + F ( s , t ) N \\Big ] \\\\ & = - \\kappa _ { s s } T - \\kappa _ { s } \\kappa N + F _ { s } ( s , t ) N - \\kappa F ( s , t ) T + \\tau F ( s , t ) B \\\\ & = \\Big [ - \\kappa _ { s s } - \\kappa F ( s , t ) \\Big ] T - \\Big [ \\kappa _ { s } \\kappa + F _ { s } ( s , t ) \\Big ] N + \\tau F ( s , t ) B . \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{align*} \\widehat { P } _ { } ( \\hbar ) : = \\frac { i } { 2 } \\left ( \\widehat { P } _ { \\hbar } ^ * - \\widehat { P } _ { \\hbar } \\right ) = \\frac { i } { 2 } \\left ( \\widehat { Q } _ 1 ^ * - \\widehat { Q } _ 1 \\right ) + \\frac { i \\hbar } { 2 } \\left ( \\widehat { Q } _ 2 ^ * - \\widehat { Q } _ 2 \\right ) , \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} \\lim _ { k } \\Vert U _ { j _ { k } } ^ { k } y ^ { k } - y ^ { k } \\Vert = 0 \\end{align*}"}
-{"id": "4853.png", "formula": "\\begin{align*} ( f ^ 2 ) ^ * ( \\omega _ M \\times 1 ) = 2 \\cdot ( \\omega _ M \\times 1 ) \\pm ( 1 \\times \\omega _ N ) . \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} \\frac { d } { d t } \\frac { W _ 2 ^ 2 ( \\mu _ t , \\nu ) } { 2 } & = \\lim _ { s \\downarrow 0 } \\frac { W _ 2 ^ 2 ( \\mu _ { t + s } , \\nu ) - W _ 2 ^ 2 ( \\mu _ t , \\nu ) } { 2 s } \\\\ & \\ge \\lim _ { s \\downarrow 0 } \\frac { 1 } { s } \\left ( \\int _ X \\varphi _ t \\ , d \\mu _ { t + s } - \\int _ X \\varphi _ t \\ , d \\mu _ t \\right ) = - \\int _ X \\langle \\nabla u , \\nabla \\varphi _ t \\rangle \\ , d \\mu _ t , \\end{align*}"}
-{"id": "6967.png", "formula": "\\begin{align*} I _ 1 = I _ h , \\ ; I _ 2 = \\{ t _ 5 - t _ 1 \\} , \\ ; \\textup { a n d } \\ ; I _ j = \\emptyset \\ ; \\textup { f o r a l l $ j \\geq 3 $ } \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{align*} D _ m = \\begin{cases} A _ m + x D _ { m - 1 } & \\textrm { i f $ m \\ge 2 $ , } \\\\ A _ 1 + x ( F _ T - 1 ) & \\textrm { i f $ m = 1 . $ } \\end{cases} \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} I [ \\mu ] = \\iint _ { \\theta \\neq \\phi } f ( \\theta - \\phi ) d \\mu ( \\theta ) d \\mu ( \\phi ) , \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} g _ { f _ { - t } } = f _ { - t } ^ * g \\end{align*}"}
-{"id": "1629.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { Q } ^ y _ t & = \\frac { \\widehat { Q } ^ y _ t ( \\dot { \\varphi } _ y ) } { 1 + \\varphi _ \\sigma ( y ) \\big ( A _ t - q _ { \\xi , t } ( y ) - \\varphi _ \\xi ( y ) \\big ) } + O \\left ( ( \\ln _ 2 t ) ^ { - 2 | y | - 1 } \\right ) \\end{aligned} \\end{align*}"}
-{"id": "3955.png", "formula": "\\begin{align*} & \\lim _ { n \\rightarrow \\infty } \\frac 1 n \\log \\mathbb { P } \\left [ X ^ n \\in \\mathcal { T } ^ { ( n ) } _ { q _ X } , Y ^ n \\in \\mathcal { T } ^ { ( n ) } _ { q _ Y } \\right ] = - \\min _ { r _ { X Y } : \\ : r _ X = q _ X , r _ Y = q _ Y } D ( r _ { X Y } \\| p _ { X Y } ) . \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{align*} \\alpha \\cdot ( s \\cdot n ) = \\alpha ( s ) \\cdot n + s \\cdot ( \\alpha \\cdot n ) \\ , . \\end{align*}"}
-{"id": "7767.png", "formula": "\\begin{align*} \\begin{cases} [ a k + 1 ] = e = [ 0 ] \\in \\Z / m \\Z & \\textnormal { i f $ m | b $ } \\\\ [ b k + 1 ] = e = [ 0 ] \\in \\Z / m \\Z & \\textnormal { i f $ m | a $ } . \\end{cases} \\end{align*}"}
-{"id": "8809.png", "formula": "\\begin{align*} \\exp ( D ) = \\exp ( A _ D ) \\exp ( A _ E ) \\exp ( B _ D + B _ E + O ) \\exp ( C _ E ) \\exp ( C _ D ) . \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} \\min _ { | z | = r } | f ( z ) | > 1 . \\end{align*}"}
-{"id": "3530.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { \\Gamma _ { t } } \\rho _ { X _ { 0 } } d s = - \\int _ { \\Gamma _ { t } } \\Big | \\kappa + \\frac { \\langle \\gamma , N \\rangle } { 2 ( t _ { 0 } - t ) } \\Big | ^ { 2 } \\rho _ { X _ { 0 } } d s \\qquad ( t < t _ { 0 ` } ) \\end{align*}"}
-{"id": "1442.png", "formula": "\\begin{align*} \\sup _ { 1 \\le M < N } \\frac { \\int _ X \\Big ( \\sum _ { { t } = M } ^ { N } h _ { t } ( f _ t x ) - \\sum _ { { t } = M } ^ { N } a _ t \\Big ) ^ 2 \\ , d \\mu } { \\sum _ { { t } = M } ^ { N } a _ t } < \\infty \\ , ; \\end{align*}"}
-{"id": "2408.png", "formula": "\\begin{align*} E [ T ] = ( \\nu _ 1 + \\lambda \\nu _ 2 ) M \\ln M + ( \\nu _ 1 + \\lambda \\nu _ 2 ) ( \\gamma + \\ln \\nu _ 1 ) M + \\frac { \\nu _ 1 + \\lambda \\nu _ 2 } { 2 \\nu _ 1 } + o ( 1 ) \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} \\mu \\left ( \\left \\{ y \\in Y : \\left ( \\int \\limits _ 0 ^ { T _ k } F _ { n ( T _ k ) } ( g _ t y ) \\ , d t \\right ) ^ 2 \\geqslant T _ k ^ { 2 \\alpha } \\| f _ { n ( T _ k ) } \\| _ { B } ^ 2 \\right \\} \\right ) \\\\ \\leqslant 4 Q T _ k ^ { 2 - C - 2 \\alpha } = 4 Q k ^ { 2 \\alpha ( 2 - C - 2 \\alpha ) / ( 2 \\alpha - 1 ) } \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} \\int _ { \\frac { 2 \\pi j } { n } } ^ { \\frac { 2 \\pi ( j + 1 ) } { n } } P _ { F , m } ( x ) d x = h _ { N C } \\ , \\sum _ { q = 0 } ^ { m } \\alpha _ q F ( y _ q ) \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{align*} | \\ker { f _ i } | = | \\ker { f _ 1 } | \\mbox { f o r a l l } i \\equiv 1 \\pmod { p ^ m } . \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{align*} F _ { \\mathcal { L } } ( q ) = ( n + 1 ) \\Lambda _ Y - \\bar { S } _ { \\Theta } + \\sum \\frac { ( \\chi ^ { a c } - \\Lambda _ Y \\chi ) ( \\alpha ^ { \\vee } ) } { q ( \\alpha ^ { \\vee } ) } \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} H = - \\frac 1 2 \\langle x , { \\bf n } \\rangle , \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} h '' \\pm a h = 0 \\qquad \\qquad \\pm ( h ' ) ^ { 2 } + a h ^ { 2 } = c , \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} \\| X \\| \\leq \\| A \\| = \\| \\phi ^ { ( p ) } ( X ) \\| \\leq \\| X \\| . \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} v _ p ( [ z ^ { d p ^ { i } n } ] \\log F _ i ( z ) ) & \\geq c _ { \\lambda _ i } - i + v _ p \\left ( [ z ^ { d p ^ { i } n } ] \\left ( h ( q ^ { d p ^ i } , z ^ { d p ^ i } ) - \\frac { 1 } { p } h ( q ^ { d p ^ { i + 1 } } , z ^ { d p ^ { i + 1 } } ) \\right ) \\right ) \\\\ & = c _ { \\lambda _ i } - i + v _ p \\left ( [ z ^ n ] \\left ( h ( q ^ { d p ^ { i } } , z ) - \\frac { 1 } { p } h ( q ^ { d p ^ { i + 1 } } , z ^ p ) \\right ) \\right ) \\\\ & \\geq c _ { \\lambda _ i } - i - v _ p ( q ^ { p ^ { i } } - 1 ) \\\\ & = c _ { \\lambda _ i } - \\lambda _ i - i \\\\ & = b _ i . \\end{align*}"}
-{"id": "5.png", "formula": "\\begin{align*} I ( x ) & : = \\frac { 1 + x } { 2 } \\log \\frac { 1 + x } { 2 } + \\frac { 1 - x } { 2 } \\log \\frac { 1 - x } { 2 } , \\ , \\ , x \\in [ - 1 , 1 ] . \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{align*} \\begin{aligned} I & : = \\{ i _ 1 \\} \\cup \\{ \\ , i < i _ 1 , \\ , \\ , \\mathrm { s . t \\ , \\ , } \\lambda _ { j - 1 } \\geq \\frac { 1 } { M } \\lambda _ j , \\ , \\forall \\ , \\ , i < j \\leq i _ 1 \\ , \\} \\\\ & = : \\{ i _ 0 , \\dots , \\ , i _ 1 \\} . \\end{aligned} \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} T _ { k } : = U _ { k + s - 1 } U _ { k + s - 2 } . . . U _ { k } \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{align*} u ' ( z ) = - \\int _ 0 ^ \\infty \\frac { 1 + s ^ 2 } { ( z - s ) ^ 2 } d \\sigma ( s ) . \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} a _ I = \\frac { 1 } { ( 2 \\pi i ) ^ { n + 1 } } \\int _ { ( \\partial D ) ^ { n + 1 } } \\frac { P ( z ) } { z ^ { I + \\mathbf { 1 } } } d z _ 0 \\cdots d z _ n \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} & W ^ 0 _ { m + 2 } \\ast _ h W ^ 0 _ { n + 2 } = \\\\ & \\sum _ { k = 0 } ^ { ( m + n - k _ { } ) / 2 + 1 } \\sum _ { l = 0 } ^ k h ^ k W ^ l _ { m + 2 - 2 l } \\ast _ h W ^ { k - l } _ { n + 2 - 2 ( k - l ) } \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} t _ m : = 2 m ^ \\beta \\theta ( m ) \\psi ( m ) \\ , . \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} & [ d , z _ 1 z _ 2 , z _ 3 ] = - [ z _ 1 z _ 2 , d , z _ 3 ] = - z _ 1 [ z _ 2 , d , z _ 3 ] - [ z _ 1 , d ] [ z _ 2 , z _ 3 ] - [ z _ 1 , z _ 3 ] [ z _ 2 , d ] - [ z _ 1 , d , z _ 3 ] z _ 2 \\\\ & = z _ 1 [ d , z _ 2 , z _ 3 ] + [ d , z _ 1 ] [ z _ 2 , z _ 3 ] + [ d , z _ 2 ] [ z _ 1 , z _ 3 ] - \\bigl [ [ d , z _ 2 ] , [ z _ 1 , z _ 3 ] \\bigr ] + [ d , z _ 1 , z _ 3 ] z _ 2 . \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} \\tilde { Q } = \\frac { 1 } { 2 \\varepsilon } \\begin{pmatrix} Q & 0 \\\\ 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} \\varphi \\cdot d s = - \\mathcal L _ { \\{ s , - \\} } ( \\varphi ) \\ , . \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} F ( n ) = T \\mod v . \\end{align*}"}
-{"id": "826.png", "formula": "\\begin{align*} \\chi ( X ) = \\int _ X K _ 0 ( t ; x , x ) \\ , \\mathbb E _ { t ; x , x } \\ , { \\rm S t r } \\ , ( M _ t V _ t ) \\ , d X _ x , t > 0 . \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} q _ n ( x ) & : = x ^ { 2 n } ( x ^ 2 - 1 ) , \\\\ t _ n ( x ) & : = 2 x ^ { \\tfrac { 3 n + 1 } { 2 } } + 2 x ^ { n + 1 } + 2 x ^ { \\tfrac { n + 3 } { 2 } } + x ^ 2 + 1 , \\\\ \\xi _ n ( x ) & : = \\frac { q ( x ) } { t ( x ) } = \\frac { x ^ { \\tfrac { n - 1 } { 2 } } ( x ^ 2 - 1 ) } { 2 + 2 x ^ { \\tfrac { - n + 1 } { 2 } } + 2 x ^ { - n + 1 } + x ^ { \\tfrac { - 3 n + 3 } { 2 } } + x ^ { - \\tfrac { 3 n + 1 } { 2 } } } \\end{align*}"}
-{"id": "7273.png", "formula": "\\begin{align*} ( \\Gamma _ 1 , \\ldots , \\Gamma _ n , \\Gamma _ { n + 1 } ) \\equiv ( \\Delta _ 1 , \\ldots , & \\Delta _ n , \\Delta _ { n + 1 } ) \\Leftrightarrow ( \\Gamma _ 1 = \\Delta _ 1 \\& \\ldots \\& \\Gamma _ n = \\Delta _ n \\& \\\\ & \\& ( \\forall A \\in F o r m ^ { A g } ) ( \\Box A \\in \\Gamma _ { n + 1 } \\Rightarrow A \\in \\Delta _ { n + 1 } ) . \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{align*} f ( x ) & = \\alpha _ - \\big ( 1 + o ( 1 ) \\big ) e ^ { \\sqrt { \\lambda } x } , x \\rightarrow - \\infty , \\\\ f ' ( x ) & = \\alpha _ - \\sqrt { \\lambda } \\big ( 1 + o ( 1 ) \\big ) e ^ { \\sqrt { \\lambda } x } , x \\rightarrow - \\infty , \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{align*} | c | = | h ' ( b ) | \\geq | h ' ( 0 ) | - | h ' ( 0 ) - h ' ( b ) | \\geq 2 - 2 ^ { 8 } \\frac { | b | } { R } \\geq 2 - 2 ^ { 8 } \\frac B R . \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} \\forall \\alpha \\geq 0 , \\mu ( f _ \\alpha ) = ( - t ) ^ \\alpha / \\alpha ! , \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l l l } \\eta ( x , 0 ) = \\eta _ { 0 } ( x ) w ( x , 0 ) = w _ { 0 } ( x ) & & ( 0 , L ) \\end{array} \\end{align*}"}
-{"id": "8397.png", "formula": "\\begin{align*} \\int _ 0 ^ t s ^ { 2 - b } \\| d _ A d _ A \\psi ( s ) \\| _ 2 ^ 2 d s & = \\int _ 0 ^ t s ^ { 2 - b } \\| \\ , [ B ( s ) , \\psi ( s ) ] \\ , \\| _ 2 ^ 2 d s , \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} \\sum _ { \\boldsymbol { \\theta } } \\gamma ( \\mathbf { m } , \\mu , \\boldsymbol { \\theta } ; N ) = \\binom { N } { \\mu } \\beta , \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} y _ t = y ( 0 ) + \\int _ 0 ^ t f ( y _ s ) d x _ s \\ . \\end{align*}"}
-{"id": "7377.png", "formula": "\\begin{align*} \\widehat { u _ L } ( \\xi ) = | \\det L | ^ { - 1 } \\widehat { u } ( L ^ { - 1 } \\xi ) , \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb E \\exp ( 2 \\pi i \\langle \\mathbf { h } , \\zeta _ n \\bmod 1 \\rangle ) = \\mathbb E \\exp ( 2 \\pi i \\langle \\mathbf { h } , \\zeta \\rangle ) . \\end{align*}"}
-{"id": "864.png", "formula": "\\begin{align*} x & : = [ [ a , b ^ { - 1 } ] , b ] = ( b ^ { - 1 } b ^ { - a } , b ^ 2 ) \\equiv ( b ^ { - 2 } [ b , a ] ^ { - 1 } , b ^ 2 ) \\\\ x ^ { b ^ { 2 n } } & = x ^ { ( a ^ n , a ^ n ) } \\equiv ( b ^ { - 2 } [ b , a ] ^ { - 2 n - 1 } , b ^ 2 [ b , a ] ^ { 2 n } ) \\\\ x ^ { b ^ { 2 n + 1 } } & = ( x ^ b ) ^ { ( a ^ n , a ^ n ) } \\equiv ( b ^ { 2 } [ b , a ] ^ { 2 n + 2 } , b ^ { - 2 } [ b , a ] ^ { - 2 n - 1 } ) . \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} K ( x , y ; t ) = - f _ { 1 } ( x , y ) f _ { 2 } ( x , y ) , \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} Q _ t = \\{ i \\in \\bar { U } \\colon \\{ i , t \\} \\in \\mathcal { E } _ { S _ 3 ^ * } , j _ { p ^ * _ i } = t \\} = \\{ i \\in W _ t \\colon \\{ i , t \\} \\in \\mathcal { E } _ { S _ 3 ^ * } \\} , \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} c ^ { \\delta + k } \\ge c ^ { k - 1 } + \\frac { 1 } { 2 } \\frac { \\sum _ i ^ \\delta [ ( d ( v _ i ) c ^ { k - d ( v _ i ) - 1 } + ( \\delta - d ( v _ i ) - 1 ) ] } { \\binom { k } { 2 } } . \\end{align*}"}
-{"id": "966.png", "formula": "\\begin{align*} | \\mathcal { F } ^ { \\ast } \\circ V ^ { 1 / 2 } ( \\tilde { { \\varphi } } ) ( p ) | ^ { 2 } \\leq \\mathrm { c o n s t } \\ , | V | _ { 1 / 2 , 1 } \\prod _ { i = 1 } ^ { m } ( 1 - \\cos ( p - p ^ { ( i ) } ) ) , \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} \\delta \\alpha _ 1 \\gamma ( \\mathbf { s } ) & = \\{ U _ a ( a b ^ m , a b ^ { m + 1 } ) ^ 2 , U _ a ( a b ^ m , a b ^ { m + 1 } ) ^ 3 , U _ b ( a b ^ m , a b ^ { m + 1 } ) ^ 2 , U _ b ( a b ^ m , a b ^ { m + 1 } ) ^ 3 \\} \\\\ & \\subseteq \\{ a b ^ m , a b ^ { m + 1 } \\} ^ { { + } } = : \\mathbf { s } _ 2 . \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} \\log \\frac { 1 } { \\| X _ 0 ( \\varphi ( r e ^ { i \\theta } ) ) \\| } = \\log \\sqrt { 1 + \\sum _ { i = 1 } ^ n | \\varphi _ i ( r e ^ { i \\theta } ) | ^ 2 } \\le \\sum _ { i = 1 } ^ n \\log ^ + | \\varphi _ i ( r e ^ { i \\theta } ) | + \\log ( \\sqrt { 1 + n } ) \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\ ! \\ ! ( \\lambda \\overline { b } _ h ^ i ( t ) + ( 1 - \\lambda ) \\hat { b } _ h ^ i ( t ) ) d t \\leq \\lambda \\int _ 0 ^ T \\ ! \\ ! \\overline { b } _ h ^ i ( t ) d t + \\int _ 0 ^ T \\ ! \\ ! \\hat { b } _ h ^ i ( t ) d t < \\int _ 0 ^ T \\ ! \\ ! \\sum _ { i = 1 } ^ r \\xi _ h ^ i ( t ) d t + R \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} \\frac { d H ^ { \\delta } ( U _ 0 , r ) } { d r } = \\frac { 4 } { r } I ^ { \\delta } ( U _ 0 , r ) + \\frac { a } { r } H ^ { \\delta } ( U _ 0 , r ) . \\end{align*}"}
-{"id": "3661.png", "formula": "\\begin{align*} X _ t ^ { \\vec v } : = \\langle X _ t , \\vec v \\rangle \\ , , \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} z ' ( x ) = \\pm \\sqrt { \\frac { 4 - ( 1 + \\lambda ^ 2 ) ^ 2 ( z ^ 2 + 2 C _ 1 ) ^ 2 } { ( 1 + \\lambda ^ 2 ) ^ 3 ( z ^ 2 + 2 C _ 1 ) ^ 2 } } \\end{align*}"}
-{"id": "5411.png", "formula": "\\begin{align*} t _ 1 \\ast _ h t _ 2 = \\sum _ { k = 0 } ^ { ( p + q - k _ { } ) / 2 + 1 } \\sum _ { l = 0 } ^ k h ^ k t _ 1 ^ l \\ast _ h t _ 2 ^ { k - l } , t _ 1 ^ l \\in ( Y ^ p ) ^ l , t _ 2 ^ { k - l } \\in ( Y ^ q ) ^ { k - l } \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} \\pi ( z ) = a ( z ) \\frac { ( 1 - z ) ( 1 - a ' ( 1 ) ) } { a ( z ) - z } , \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} a _ { n , k } = q ^ { n ^ 2 - n } - \\sum _ { \\lambda \\vdash n , \\ , \\lambda _ 1 > k } c ( \\lambda ) . \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} \\dot { \\bar { x } } ^ * ( t ) & = \\bar { A } ^ * \\bar { x } ^ * ( t ) + \\bar { B } ^ * u ( t ) \\\\ \\dot { \\bar { x } } _ { r } ( t ) & = - \\varepsilon \\bar { x } _ r ( t ) + \\displaystyle \\sum _ { j = 1 } ^ { n _ { \\rm i n } } u _ j ( t ) \\bar { N } _ j ^ * \\bar { x } ^ * ( t ) + \\bar { H } ^ * ( \\bar { x } ^ * ( t ) \\otimes \\bar { x } ^ * ( t ) ) \\\\ \\bar { y } ( t ) & = \\sqrt [ 4 ] { p '' } \\ ; \\bar { x } _ r ( t ) \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} a _ { \\delta } \\left ( x \\right ) : = a ^ { + } \\left ( x \\right ) \\chi _ { \\Omega \\setminus \\Omega _ { \\delta } } - \\left ( 1 + \\delta \\right ) a ^ { - } \\left ( x \\right ) . \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} \\tilde \\psi : = ( N s ) ^ { \\frac { 1 - n } { 2 } } \\psi \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} d q ^ \\prime _ t = & ( \\tilde \\gamma ^ T ) ^ { - 1 } ( t ^ * , q ^ \\prime _ t ) \\left ( - \\partial _ t \\psi ( t ^ * , q ^ \\prime _ t ) - \\nabla _ q V ( t ^ * , q ^ \\prime _ t ) + \\tilde F ( t ^ * , q ^ \\prime _ t , \\psi ( t ^ * , q ^ \\prime _ t ) ) \\right ) d t \\\\ & + \\tilde S ^ \\prime ( t ^ * , q ^ \\prime _ t ) d t + ( \\tilde \\gamma ^ T ) ^ { - 1 } ( t ^ * , q ^ \\prime _ t ) \\sigma ( t ^ * , q ^ \\prime _ t ) \\circ d W _ t . \\end{align*}"}
-{"id": "6598.png", "formula": "\\begin{align*} \\Phi \\left ( \\left [ \\psi _ { U , u } \\right ] \\right ) = ( u ^ 2 - | U | ^ 2 ) \\bar { \\varphi } + 2 u \\star ( \\bar { \\varphi } \\wedge U ) + 2 \\left ( { U } \\neg \\bar { \\varphi } \\right ) \\wedge U . \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{align*} X = X _ 0 \\to X _ 1 \\to \\cdots \\to X _ r = X ' \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{align*} ( E _ \\eta , \\ , E _ \\rho ) \\ , = \\ , \\bigoplus _ { i = 1 } ^ b ( F _ i ) ^ { \\oplus d _ i } \\ , , \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} t _ 1 \\ast _ h t _ 2 = t _ 1 \\ast t _ 2 + \\dots + h ^ { g _ 1 + g _ 2 } ( t _ 1 ^ { g _ 1 } \\ast _ h t _ 2 ^ { g _ 2 } + t _ 1 ^ { g _ 2 } \\ast _ h t _ 2 ^ { g _ 1 } ) + \\dots \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} g ( r ) = \\sum _ { j = 0 } ^ { n + 1 } \\xi _ j \\phi _ j ( r ) \\ , , \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} a _ 1 + a _ 2 + a _ 3 = \\frac { 1 } { 8 } . \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} n ^ 2 I [ \\delta _ { \\theta } ] = \\sum _ { j \\neq k } f ( \\theta _ j - \\theta _ k ) . \\end{align*}"}
-{"id": "4875.png", "formula": "\\begin{align*} [ m ] P = \\bigg ( \\frac { \\theta _ m ( x , y ) } { \\psi _ m ( x , y ) ^ 2 } , \\frac { \\omega _ m ( x , y ) } { \\psi _ m ( x , y ) ^ 3 } \\bigg ) \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{align*} \\begin{cases} D _ { A } \\psi _ = 0 \\\\ F _ { A } ^ { + } = \\frac { r } { 2 } ( q ( \\psi , \\psi ) - i \\Omega _ X ) - \\frac { 1 } { 2 } F ^ + _ { A _ { K ^ { - 1 } } } - \\frac { i } { 2 } \\wp _ 4 ^ + \\\\ * d * b - 2 ^ { - \\frac { 1 } { 2 } } r ^ { \\frac { 1 } { 2 } } ( \\eta ^ * \\psi ^ { \\xi } - { \\psi ^ { \\xi } } ^ * \\eta ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{align*} \\alpha ^ * ( q , r ) = \\frac { 1 } { q } + \\frac { d - 1 } { 2 r } - \\frac { d + 1 } { 4 } = \\frac { 1 } { q } - \\frac { 1 } { 2 } - \\frac { d - 1 } { 2 } \\bigg ( \\frac { 1 } { 2 } - \\frac { 1 } { r } \\bigg ) . \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} \\exp ( h ) = \\exp ( C _ E ) \\exp ( C _ D ) . \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} z = \\frac { ( a _ 1 + a _ 2 + a _ 3 + a _ 4 ) ( a _ 1 a _ 2 a _ 3 a _ 4 + 1 ) + 2 ( a _ 1 a _ 2 a _ 3 + a _ 1 a _ 2 a _ 4 + a _ 1 a _ 3 a _ 4 + a _ 2 a _ 3 a _ 4 ) \\pm 2 s } { ( a _ 1 a _ 2 a _ 3 a _ 4 - 1 ) ^ 2 } \\ , , \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} & \\lim _ { n \\rightarrow \\infty } h ( \\tilde { p } _ n ) ^ { \\frac { 1 } { n } } \\leq \\lim _ { n \\rightarrow \\infty } \\left ( - 2 ( 1 - \\tilde { p } _ n ) \\log ( 1 - \\tilde { p } _ n ) \\right ) ^ { \\frac { 1 } { n } } = \\left ( \\frac { p _ { 1 2 } p _ { 2 1 } } { p _ { 1 1 } p _ { 2 2 } } \\right ) ^ { \\frac 1 2 } \\end{align*}"}
-{"id": "4446.png", "formula": "\\begin{align*} < R _ { o p } ( { \\bf T } , \\lambda ) y , l > : = \\int _ 0 ^ { + \\infty } e ^ { - \\lambda s } < T ( s ) y , l > d s \\ \\ \\forall l \\in X _ * . \\end{align*}"}
-{"id": "8907.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\int _ X \\dot { \\phi } _ t \\omega _ { \\phi _ t } ^ n d t & = - \\int _ 0 ^ 1 C _ H ' \\int _ { \\chi + \\Delta ^ t \\cap \\bar { C } ^ + } \\dot { u } _ t ^ * ( 2 \\chi - 2 q ) P _ { D H } ( q ) d q \\\\ & = - C _ H ' \\int _ { \\chi + \\Delta ^ t \\cap \\bar { C } ^ + } ( u ^ * - u _ { \\mathrm { r e f } } ^ * ) ( 2 \\chi - 2 q ) P _ { D H } ( q ) d q \\\\ & = - ( 2 \\pi ) ^ n n ! \\int _ { \\Delta ^ + } ( u ^ * - u _ { \\mathrm { r e f } } ^ * ) ( 2 \\chi - 2 q ) P _ { D H } ( q ) d q \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{align*} R ^ { i } \\pi ' _ { * } ( \\bigwedge ^ { p } ( \\check { \\mathcal { E } } ) \\otimes p ^ { * } ( Q ) ) = 0 \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} p _ a ^ + ( t ) & = \\frac { 1 } { t } \\left ( ( 1 + t ) ^ { n - a } - 1 \\right ) , \\\\ p _ a ^ - ( t ) & = t ^ { a - 1 } ( 1 + t ) ^ { n - a } \\ , . \\end{align*}"}
-{"id": "3804.png", "formula": "\\begin{align*} 3 \\sigma - \\chi = \\frac { 1 } { 8 \\pi ^ 2 } \\int _ M | W _ F ^ + | ^ 2 + 3 | R _ { 0 0 } | ^ 2 + 6 k ( k - 2 v ) \\end{align*}"}
-{"id": "3618.png", "formula": "\\begin{align*} H ^ { ( n ) } ( p ^ { \\mathbf { k } } ; p ^ { \\boldsymbol { \\ell } } ) = \\sum _ { P \\colon \\mathbf { k } ( P ) = \\mathbf { k } } G ( P ) , \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} \\begin{aligned} & \\Big \\| \\int _ { 0 } ^ { t _ { 0 } + h } e ^ { - ( t _ { 0 } + h - s ) A } F _ { u } ( u _ { j } , \\nabla y _ { j } ) \\ \\d s - \\int _ { 0 } ^ { t _ { 0 } } e ^ { - ( t _ { 0 } - s ) A } F _ { u } ( u _ { j } , \\nabla y _ { j } ) \\ \\d s \\Big \\| _ { L ^ { r } _ { \\sigma } ( \\Omega ) } \\end{aligned} \\end{align*}"}
-{"id": "3725.png", "formula": "\\begin{align*} p _ { a , n } ( t ) & = a ( p _ { a , n } ^ + ( t ) + p _ { a , n } ^ - ( t ) ) \\ , , \\\\ q _ { a , n } ( t ) & = p _ { a , n + 1 } ( t ) - p _ { a , n } ( t ) \\ , , \\\\ d _ { a , n } ( t ) & = p _ { a + 1 , n } ( t ) - p _ { a , n } ( t ) \\ , . \\\\ \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} \\begin{cases} k _ { y y y } + k _ y + k _ { x x x } + k _ x + \\lambda s = 0 , & , \\\\ s _ { y y y } + s _ y + s _ { x x x } + s _ x + \\lambda k = \\lambda \\delta ( x - y ) , & , \\end{cases} \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{align*} \\begin{cases} D _ { A } \\psi = 0 \\\\ F _ { A } ^ + = \\frac { r } { 2 } ( q _ 4 ( \\psi ) - i \\Omega _ X ) - \\frac { 1 } { 2 } F _ { A _ { K ^ { - 1 } } } ^ + - \\frac { i } { 2 } \\wp _ 4 ^ + , \\end{cases} \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{align*} \\begin{cases} - u ^ { \\prime \\prime } = a ( x ) u ^ { q } & \\mbox { i n } \\Omega , \\\\ u = 0 & \\mbox { o n } \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "4730.png", "formula": "\\begin{align*} u = \\exp ( X ) \\textup { f o r s o m e } X = \\sum _ { \\gamma \\in N ( v ^ { - 1 } ) } c _ { \\gamma } E _ { \\gamma } \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} a _ 1 = \\lim _ { q \\rightarrow \\infty } E _ d ( P ) . \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{align*} & \\begin{aligned} { \\rm B i a s } ( \\hat D _ l ) = \\sum _ { j \\in J } \\gamma _ j \\varphi _ j ( l ) \\varGamma ^ { - j / 2 d } + o \\left ( \\varGamma ^ { - 1 / 2 } \\right ) , \\end{aligned} \\\\ & \\begin{aligned} { \\rm V a r } ( \\hat D _ l ) = O \\left ( \\varGamma ^ { - 1 } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} E _ { b } ( 1 ) \\leq \\mathcal { E } _ { b } ( \\rho _ { n } ) = E _ { \\tau _ { n } } ( 1 ) + ( \\tau _ { n } - b ) D ( \\rho _ { n } , \\rho _ { n } ) . \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{align*} L = \\sum _ { i < j } L _ { i , j } \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{align*} F ( T _ n , J _ n \\langle K \\rangle ) ( x _ { n , H } ) = W ( J _ n \\langle K \\rangle ) ( x _ { n , H } ) . \\end{align*}"}
-{"id": "3596.png", "formula": "\\begin{align*} - \\frac { z ' ( x ) } { \\sqrt { 1 + C _ { 1 } ^ { 2 } + z ' ( x ) ^ { 2 } } } = \\frac { 1 } { 2 } ( 1 + C _ { 1 } ^ { 2 } ) ( x ^ { 2 } + 2 C _ { 2 } ) . \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} b = 4 ( b = 5 k = 1 ) . \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} J _ { \\mathrm { f s u } } = \\varprojlim _ { k } ( J / p ^ k J ) _ { \\mathrm { f s u } } , \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{align*} \\mathcal { L } _ { P } ^ { i } = \\left \\{ p \\in P \\middle | f ( p ) = h _ { i } \\right \\} . \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} \\mathfrak { N } _ { s t } & = - \\psi T - \\psi T _ { t } = - \\psi T - \\frac { 1 } { 2 } i \\psi \\big ( \\psi _ { s } \\bar { \\mathfrak { N } } - \\bar { \\psi } _ { s } \\mathfrak { N } \\big ) \\\\ \\mathfrak { N } _ { t s } & = i \\big [ R _ { s } \\mathfrak { N } - R \\psi T - \\psi _ { s s } T - \\frac { 1 } { 2 } \\psi _ { s } \\big ( \\bar { \\psi } \\mathfrak { N } + \\psi \\bar { \\mathfrak { N } } \\big ) \\big ] . \\end{align*}"}
-{"id": "4545.png", "formula": "\\begin{align*} \\Pr [ A _ 1 \\cap A _ 2 ] = \\sum _ { i } \\Pr [ \\mathrm { r a n k } ( \\mathbf { X } ) = i , A _ 1 \\cap A _ 2 ] , \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} I ( \\zeta ) ^ 2 = 0 \\end{align*}"}
-{"id": "9027.png", "formula": "\\begin{align*} K _ L \\cdot I _ r = - I _ { r + 1 } + ( I _ { r } ) _ x \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} \\omega _ t = e ^ { - \\frac { 1 } { 2 } t \\Box } \\omega _ 0 \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\langle \\ ! \\langle \\hat { p } , \\hat { b } ^ i - \\xi ^ i \\rangle \\ ! \\rangle - \\sum _ { j = 1 } ^ s \\langle \\ ! \\langle \\hat { p } , a ^ j \\rangle \\ ! \\rangle < 0 . \\end{align*}"}
-{"id": "2910.png", "formula": "\\begin{align*} \\Gamma ( t ) = \\{ ( x , z ) \\in \\R ^ { 2 } \\colon z = h ( x , t ) , \\ , x \\in \\R \\} . \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} h _ t + | \\partial _ x | ^ 3 h = 0 . \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} W _ { n + 2 } ^ 0 ( p , p _ 1 , \\dots , p _ { n + 1 } ) & = \\\\ K _ p ( q , \\bar { q } ) & \\sum _ { \\substack { L \\cup M = \\{ p _ 1 , \\dots , p _ { n + 1 } \\} , \\\\ | L | = p + 1 , | M | = q + 1 } } W ^ 0 _ { | L | + 1 } ( q , L ) W ^ { 0 } _ { | M | + 1 } ( \\bar { q } , M ) . \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} e ^ { 2 i \\theta k } + ( - 1 ) ^ k \\chi _ 2 ( a ) \\overline { \\chi _ 2 ( a + 1 ) } = 0 . \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{align*} D _ { d } = \\int _ 0 ^ 1 t ^ { \\frac { d } { 2 } - 1 } ( 1 - t ) ^ { - \\frac { 1 } { 2 } } d t \\leq \\int _ 0 ^ 1 ( 1 - t ) ^ { - 1 / 2 } d t = 2 . \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} w t ( C _ { 2 } ) & = ( p - 1 ) p ^ { e - 2 } + p ^ { m + d - 1 } , \\\\ A _ { C _ { 2 } } & = p ^ { e - 2 d - 1 } - ( p - 1 ) p ^ { m - d - 1 } - 1 . \\end{align*}"}
-{"id": "1952.png", "formula": "\\begin{align*} { } \\begin{array} { l } p v _ 1 = \\gamma v _ 2 , \\ \\textrm { w i t h } \\gamma \\in \\mathcal { A } _ p , \\\\ \\\\ p v _ 2 = 0 . \\end{array} \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} f _ \\alpha : = \\Big [ { 1 \\over \\tilde \\theta _ { E , \\omega } ( z ) } \\mathrm { e x p } \\big ( - \\tilde c z - \\sum _ { k \\geq 2 } { 1 \\over k ! } \\partial ^ { k - 2 } ( x ) z ^ k \\big ) \\Big | z ^ \\alpha \\Big ] , \\end{align*}"}
-{"id": "6704.png", "formula": "\\begin{align*} \\omega _ { x } & : = a ^ { 3 } b ^ { 3 } a ^ 3 b ^ 4 \\ldots a ^ 3 b ^ { \\rho + 2 } \\\\ \\omega _ { y } & : = a ^ 3 b ^ { \\rho + 3 } a ^ 3 \\ldots a ^ { 3 } b ^ { 2 \\rho + 2 } \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ H b ' _ { k j } = 1 \\ \\mbox { \\rm o r } \\ \\sum _ { j = 1 } ^ N b ' _ { k j } = 1 . \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} \\frac { v ^ 2 } { u ^ 2 } = \\int _ 0 ^ { v } \\frac { t \\ , d t } { e ^ { t } - 1 } \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} u _ 0 ( z ) = \\sum _ { j = 0 } ^ n I _ j ^ { 1 / 2 } e ^ { i \\theta _ j } \\phi _ j ( z ) , \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{align*} > \\mu \\sum _ { r = 0 } ^ { \\mu - 1 } { { \\mu - 1 } \\choose { r } } ( 2 \\mu + 2 ) _ { \\mu - 1 - r } ( 2 \\mu + 2 ) _ { r } 9 ^ { - 2 \\mu - 2 - r } . \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} \\ell _ i ( \\nu , \\nu ) = \\left \\{ \\begin{array} { r l } - t , & \\nu _ i > \\nu _ { i + 1 } , \\\\ \\\\ - 1 , & \\nu _ i < \\nu _ { i + 1 } , \\\\ \\\\ 0 , & . \\end{array} \\right . \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} \\iota _ { X _ f } \\omega _ 0 = d f \\mathrlap { . } \\end{align*}"}
-{"id": "6091.png", "formula": "\\begin{align*} L ( t ) = \\inf \\{ y \\ge 0 : \\ , \\sigma ( y ) > t \\} \\end{align*}"}
-{"id": "2406.png", "formula": "\\begin{align*} E [ T ] = ( \\nu _ 1 + \\lambda \\nu _ 2 ) M H _ { \\nu _ 1 M } \\ , + \\ , O \\left ( M ^ { 2 - \\lambda } \\ln M \\right ) , M \\to \\infty , \\end{align*}"}
-{"id": "2685.png", "formula": "\\begin{align*} u ( x , y ) = w ( x , y ) , \\quad v ( x , y ) = - \\int _ 0 ^ x w _ y ( t , y ) d t + \\int _ 0 ^ y w _ x ( 0 , z ) d z , \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{align*} \\partial _ t \\gamma ( x , t ) = \\kappa ( x , t ) N ( x , t ) . \\end{align*}"}
-{"id": "8062.png", "formula": "\\begin{align*} Y ( y ) = B y ^ s K _ s ( L ( \\xi , \\sigma ) y ) . \\end{align*}"}
-{"id": "3535.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { \\Gamma _ { t } } | \\kappa | d s = - 2 \\sum _ { x : \\kappa ( x , t ) = 0 } | \\kappa _ { s } | ( x , t ) \\end{align*}"}
-{"id": "8638.png", "formula": "\\begin{align*} w t ( C _ { 5 } ) = \\begin{cases} ( p - 1 ) p ^ { e - 2 } , & \\qquad \\ \\biggl ( \\frac { c ^ { 2 } - 4 a i } { p } \\biggr ) = - 1 , \\\\ ( p - 1 ) p ^ { e - 2 } + 2 p ^ { m + d - 1 } , & \\qquad \\ \\biggl ( \\frac { c ^ { 2 } - 4 a i } { p } \\biggr ) = 1 . \\end{cases} \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} [ D ^ * D , D ^ k ] \\phi & = \\sum _ { l = 0 } ^ { k - 1 } D ^ { l } R ^ M \\star D ^ { k - l } \\phi \\\\ & \\quad + \\sum _ { \\sum l _ i + \\sum { m _ j } = k - 1 } { } ^ { G } \\nabla ^ { l _ 1 } R ^ P \\star \\underbrace { D ^ { m _ 1 + 1 } \\phi \\star \\ldots \\star D ^ { m _ { l _ 1 } + 1 } \\phi } _ { l _ 1 - } \\star D ^ { l _ 2 + 1 } \\phi \\star D ^ { l _ 3 + 1 } \\phi \\star D ^ { l _ 4 + 1 } \\phi . \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} \\widehat { \\delta } = t _ i ^ { - 1 } \\log ^ { \\frac { 1 } { 2 } } ( t _ i ) e ^ { - \\ : \\frac 1 2 F _ 1 ( \\log { \\frak t } ( t _ i ) ) } \\delta . \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{align*} \\lim _ n H _ n ( f _ n - F _ { f , n } ) ( t _ n , \\mu _ n , w _ n ) = H f ( \\mu , w ) \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{align*} X _ R = \\left \\{ v \\in T ^ 1 S \\delta ( v ) \\in C b ( v ) \\leqslant \\frac { 1 } { R } \\right \\} . \\end{align*}"}
-{"id": "7449.png", "formula": "\\begin{align*} \\tilde \\gamma ^ { - 1 } - ( \\tilde \\gamma ^ T ) ^ { - 1 } = 2 ( \\tilde \\gamma ^ T ) ^ { - 1 } H \\tilde \\gamma ^ { - 1 } . \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{align*} e _ H ( \\vec G _ 1 , \\vec G _ 2 ) & : = | \\{ ( x , y , z ) \\in \\P _ 2 ( \\vec G _ 1 , \\vec G _ 2 ) : \\{ x , y , z \\} \\in E ( H ) \\} | \\geq d | \\P _ 2 ( \\vec G _ 1 , \\vec G _ 2 ) | - \\rho n ^ 3 \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{align*} H ' ( r ) & = 2 r \\int _ { - 1 } ^ 0 \\left ( \\frac { 2 } { r ^ 2 } i ( r ^ 2 t ) + \\frac { a } { 2 r ^ 2 } h ( r ^ 2 t ) \\right ) d t \\\\ & = \\frac 4 r I ( r ) + \\frac { a } { r } H ( r ) , \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{align*} E \\left [ S ^ 2 \\right ] - E \\left [ T _ 1 ^ 2 \\right ] & = E \\left [ S ^ 2 - T _ 1 ^ 2 \\right ] = E \\left [ \\left ( S + T _ 1 \\right ) \\left ( S - T _ 1 \\right ) \\right ] \\\\ & \\leq E \\left [ \\left ( S + T _ 1 \\right ) ^ 2 \\right ] ^ { \\frac { 1 } { 2 } } E \\left [ \\left ( S - T _ 1 \\right ) ^ 2 \\right ] ^ { \\frac { 1 } { 2 } } \\leq 2 E \\left [ S ^ 2 \\right ] ^ { \\frac { 1 } { 2 } } E \\left [ \\left ( S - T _ 1 \\right ) ^ 2 \\right ] ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} \\hat \\tau = \\inf \\{ t \\geq 0 : X _ t \\geq x ^ * \\} . \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} \\nu _ 0 ( Z _ 1 , \\ldots , Z _ n ) : = \\left | \\frac { Z _ 1 \\wedge Z _ 2 \\wedge \\cdots \\wedge Z _ n } { X _ 1 \\wedge X _ 2 \\wedge \\cdots \\wedge X _ n } \\right | , \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} A \\ , Y = \\lambda B \\ , Y \\ , . \\end{align*}"}
-{"id": "5573.png", "formula": "\\begin{align*} \\left ( 1 - e ^ { - ( 2 k - 1 ) \\pi x } \\right ) ^ 2 \\left ( 1 + e ^ { - ( 2 k - 1 ) \\pi x } \\right ) ^ { - 4 } < \\left ( 1 - e ^ { - 1 . 1 \\cdot \\pi } \\right ) ^ 2 \\left ( 1 + e ^ { - 1 . 1 \\cdot \\pi } \\right ) ^ { - 4 } < 0 . 8 3 , k = 1 . \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} \\operatorname { t r } ( \\mathbb { A } ^ { \\ast } \\mathbb { A } ) = \\operatorname { t r } ( \\mathbb { A } ^ { T } \\mathbb { A } ) = \\operatorname { t r } ( \\mathbb { A } \\mathbb { A } ^ { T } ) = \\sum _ { \\alpha \\in \\Bbb F _ q } d ^ { + } ( \\alpha ) \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} \\forall v , w \\in H ^ 1 ( \\R ) , a ^ V ( v , w ) = \\int _ \\R \\frac { d \\overline { v } } { d x } \\ ; \\frac { d w } { d x } + \\int _ \\R \\kappa _ V \\overline { v } w - \\int _ \\R \\sigma _ V \\left ( \\frac { d \\overline { v } } { d x } w + \\overline { v } \\frac { d w } { d x } \\right ) , \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} \\left \\Vert \\smash { \\widetilde { f } } ''' \\right \\Vert _ { \\sup } = ( b - a ) ^ { 3 } \\cdot \\left \\Vert f ''' \\right \\Vert _ { \\sup } \\leq ( b - a ) ^ { 3 } \\cdot C = : C ' . \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} D ( x ) : = \\Delta ^ x _ { [ x : 1 ] } = \\sum _ { j = 0 } ^ { 4 } \\alpha _ { j } x ^ { j } E ( y ) : = \\Delta ^ y _ { [ y : 1 ] } = \\sum _ { j = 0 } ^ { 4 } \\beta _ { j } y ^ { j } . \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{align*} p _ { \\gamma } ( x , s ) : = \\prod _ { k } p _ { { j _ k } { j _ { k + 1 } } } ( \\theta ^ \\gamma _ { k } ( x , s ) ) , \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} v ^ { - 1 } ( \\gamma ) \\in w _ v ^ { - 1 } x _ v ^ { - 1 } N ( x _ v ^ { - 1 } ) = w _ v ^ { - 1 } N ^ - ( x _ v ) \\subseteq w _ v ^ { - 1 } ( \\Phi ^ - - \\Phi _ L ^ - ) \\subseteq \\Phi ^ - - \\Phi _ L ^ - . \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 9 & 9 \\\\ 0 & 8 & 5 & 4 \\\\ 0 & 1 & 6 & 4 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 3 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 4 & 0 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} V _ j = V _ 1 ( e _ 1 + \\dots + e _ j ) \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} C _ k \\subset \\left \\{ v : \\exists \\ , t \\in [ 0 , T _ k ] \\textrm { w i t h } \\textrm { s y s } ( \\phi _ t ( v ) ) = T _ k ^ { - \\xi } \\right \\} \\end{align*}"}
-{"id": "620.png", "formula": "\\begin{align*} g _ r ( x , y ) ^ 4 & = \\sigma ^ 4 ( x , y ) = ( x _ 1 - y _ 1 ) ^ 2 ( x _ 1 + y _ 1 ) ^ 2 + 2 y _ 1 ^ 2 ( x _ 1 - y _ 1 ) ^ 2 + 4 ( x _ 2 - y _ 2 ) ^ 2 \\\\ & \\leq ( 1 / 4 ) ^ 2 ( | x _ 1 | + | y _ 1 | ) ^ 2 r ^ 2 + 4 ( 1 / 4 ) ^ 2 r ^ 4 ( 1 / 4 + 1 ) ^ 2 + 2 ( 1 / 4 ) ^ 2 r ^ 4 \\\\ & \\leq ( 1 / 4 ) ^ 2 ( | x _ 1 - y _ 1 | + 2 | y _ 1 | ) ^ 2 r ^ 2 + 1 / 4 r ^ 4 ( 1 / 4 + 1 ) ^ 2 + 2 ( 1 / 4 ) ^ 2 r ^ 4 \\\\ & \\leq ( 1 / 4 ) ^ 2 ( 1 / 4 + 2 ) ^ 2 r ^ 4 + 1 / 4 ( 1 / 4 + 1 ) ^ 2 r ^ 4 + 2 ( 1 / 4 ) ^ 2 r ^ 4 \\\\ & = ( ( 1 / 4 ) ^ 2 ( 1 / 4 + 2 ) ^ 2 + 1 / 4 ( 1 / 4 + 1 ) ^ 2 + 2 ( 1 / 4 ) ^ 2 ) r ^ 4 \\\\ & = ( 2 1 3 / 2 5 6 ) r ^ 4 < r ^ 4 \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} k \\cdot h ( Z , Z ) ^ 2 = k | x | ^ 4 + 2 k | x | ^ 2 | y | ^ 2 + k | y | ^ 4 \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} | a _ { P Q R } | \\leq \\frac { \\sqrt { | P _ 1 | | Q _ 1 | } \\sqrt { | P _ 2 | | Q _ 2 | } } { | R _ 1 | | R _ 2 | } = 2 ^ { - \\frac { 1 } { 2 } ( i _ 1 + j _ 1 + i _ 2 + j _ 2 ) } . \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} g ^ { \\star } \\left ( p \\right ) = \\underset { x \\in \\mathbb { R } ^ { n } } { } \\left \\{ \\left \\langle p , x \\right \\rangle - g \\left ( x \\right ) \\right \\} . \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} y = ( N + \\Delta N ) x , \\frac { \\norm { \\Delta N } _ 2 } { \\norm { N } _ 2 } \\leq \\varepsilon , \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} V _ { L } ( T ) = \\{ ( a _ 1 , \\ldots , a _ n ) \\in L ^ { n } \\mid f ( a _ 1 , \\ldots , a _ n ) = 0 \\mbox { f o r a l l } f \\in T \\} . \\end{align*}"}
-{"id": "5262.png", "formula": "\\begin{align*} U ( F ) ( z , y ) = h ( y ) F ( \\varphi ( z , y ) , \\tau ( y ) ) , ( z , y ) \\in \\mathbb { T } \\times Y _ 2 \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} \\theta _ n = \\mathring { \\theta } _ n + q _ { 2 n - 1 } \\theta _ { n - 2 } , \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{align*} E \\left [ S ( \\theta ) ^ { ( 2 ) } \\right ] = \\frac { N ^ 2 } { ( 1 - \\theta ) ^ 2 } \\left ( H _ N ^ 2 + \\sum _ { j = 1 } ^ N \\frac { 1 } { j ^ 2 } \\right ) + O \\left ( e ^ { - \\varepsilon N } \\right ) , N \\to \\infty , \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } d _ 1 ( \\mathbb { P } _ { n , d , s _ n ^ { - 1 } h } , \\mathbb { Q } _ { n , d , h } ) = 0 h \\in \\R ^ { d } . \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{align*} ( f _ 1 * f _ 2 ) ( w , z ) & = \\left | \\{ ( u , v ) \\in \\mathcal { O } _ 1 \\times \\mathcal { O } _ 2 : \\sigma ( u ) + \\sigma ( v ) = \\sigma ( w ) z _ 1 z _ 2 [ \\sigma ( u ) , \\sigma ( v ) ] ^ { 1 / 2 } = z \\} \\right | \\\\ & = \\left | \\{ ( u , v ) \\in \\mathcal { O } _ 1 \\times \\mathcal { O } _ 2 : u + v = w z _ 1 z _ 2 [ v , u ] ^ { 1 / 2 } = z \\} \\right | . \\end{align*}"}
-{"id": "556.png", "formula": "\\begin{align*} \\sigma _ { r , p } ( z ) = \\frac { r ^ 2 ( z - p ) } { r ^ 2 - \\overline { p } z } \\end{align*}"}
-{"id": "8548.png", "formula": "\\begin{align*} \\sigma _ { \\pm } = - \\ , F _ { \\pm } ( \\kappa _ 0 / 2 ) \\quad \\sigma _ 0 = 0 \\rlap { . } \\end{align*}"}
-{"id": "5846.png", "formula": "\\begin{align*} f _ { \\nu } ( z ; t ^ { - m } , t ) : = \\lim _ { q \\rightarrow t ^ { - m } } f _ { \\nu } ( z ; q , t ) \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathbb { Z } } L _ i \\left [ H \\left ( \\cdot , \\mu \\right ) \\right ] ( \\nu ) = \\sum _ { i \\in \\mathbb { Z } } M _ i \\left [ H ( \\nu , \\cdot ) \\right ] \\left ( \\mu \\right ) , \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n \\| W _ { i , j } \\| _ { L ^ q ( Q _ r ) } & \\leq \\sum _ { k = 1 } ^ n \\| u _ k \\| _ { W ^ { 2 , 1 } _ q ( Q _ r ) } + \\sum _ { k = 1 } ^ n \\| \\tilde { u } _ k \\| _ { W ^ { 2 , 1 } _ q ( Q _ r ) } \\leq C \\gamma + \\sum _ { k = 1 } ^ n \\| u _ k \\| _ { W ^ { 2 , 1 } _ q ( Q _ 1 ) } \\end{align*}"}
-{"id": "2631.png", "formula": "\\begin{align*} X _ k ( h ) U _ \\alpha ( \\mu _ 2 ) = - X _ k ( \\mu _ 1 ) U _ \\alpha ( \\varphi ) , \\qquad \\forall k , \\ , \\alpha . \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{align*} \\Gamma _ { , k } = \\frac { p _ { k } } { \\sigma _ N ^ 2 \\hat { \\mathbf { f } } _ k \\mathbf { \\Omega } ^ 2 \\hat { \\mathbf { f } } _ k ^ H + \\sigma _ { } ^ 2 \\| \\hat { \\mathbf { f } } _ k \\| ^ 2 } . \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{align*} c _ { k } m ( x ) \\equiv \\alpha _ k x ^ d - \\sum _ { i = 1 } ^ d z _ i ^ { ( k ) } x ^ i + \\sum _ { i = 1 } ^ d z _ i ^ { ( k + 1 ) } x ^ { i - 1 } \\quad \\pmod { n \\Z [ x ] } , \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{align*} \\max _ { g \\in N ^ + _ j } g ( x ) = \\max _ { g \\in N ^ - _ j } g ( x ) . \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} \\chi ( \\vec { x } , \\vec { y } ) : = \\# \\{ ( x _ i , y _ j ) \\in ( \\vec { x } , \\vec { y } ) \\ | \\ x _ i > y _ j \\} . \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} \\begin{aligned} k ^ { \\nabla y } _ { j + 1 } ( T ) & \\leq k ^ { \\nabla y } _ { 0 } ( T ) + C C _ { 1 } ( T ) B \\big ( 1 - 3 \\big ( \\tfrac { 1 } { p } - \\tfrac { 1 } { q } \\big ) , \\tfrac { 1 } { 2 } - \\tfrac { 3 } { 2 p } \\big ) \\\\ & \\left [ k ^ { u } _ { j } ( T ) k ^ { \\nabla y } _ { j } ( T ) + k ^ { \\nabla y } _ { j } ( T ) ^ 2 ( k ^ { x } _ { j } ( T ) + k ^ { y } _ { j } ( T ) + \\lvert \\overline { b } \\rvert ) \\right ] . \\end{aligned} \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} \\| \\alpha \\| _ 1 : = \\inf _ c \\biggl \\{ \\sum _ j | \\lambda _ j | \\ \\biggl | \\ c = \\sum _ j \\lambda _ j \\sigma _ j \\in C _ k ( X ; \\R ) \\ \\alpha \\biggl \\} . \\end{align*}"}
-{"id": "6623.png", "formula": "\\begin{align*} d ( q ( e ) ) = d ( p ( e ) ) , e \\in E ( G ) . \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{align*} J \\left ( z , t \\right ) = \\underset { i = 1 , \\ldots , k } { } J _ { i } \\left ( z , t \\right ) . \\end{align*}"}
-{"id": "4470.png", "formula": "\\begin{align*} h ( n ) = b ( n ) - ( n - 1 ) + \\sum _ { k = 1 } ^ { \\lfloor \\sqrt { n } \\rfloor } | W ( k ) | + \\sum _ { \\lfloor \\sqrt { n } \\rfloor + 1 } ^ { z _ 1 } | W ( k ) | , \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} H _ { i , k } = \\big \\{ ( x _ 1 , \\dots , x _ { i - 1 } , k + \\tfrac 1 2 , x _ { i + 1 } , \\dots , x _ n ) \\mid x _ j \\in \\R \\big \\} , 1 \\le i \\le n , k \\in \\Z . \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} \\begin{cases} u _ t = \\Delta u + b _ \\epsilon ( x , t ) \\cdot \\nabla u + a _ 0 u , x \\in D _ L \\cr u = 0 , x \\in \\partial D _ L , \\end{cases} \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} B _ { f } & = \\lim _ { \\alpha \\rightarrow \\infty } q ^ { - \\alpha ( m + 2 ) } \\sum _ { \\beta = 0 } ^ { \\alpha - 1 } ( q - 1 ) q ^ { \\alpha - \\beta - 1 } ( \\beta + 1 ) ^ { m + 1 } ( q - 1 ) ^ { m + 1 } q ^ { ( \\alpha - 1 ) ( m + 1 ) } \\\\ & = ( 1 - q ^ { - 1 } ) ^ { m + 2 } \\sum _ { \\beta = 0 } ^ { \\infty } ( \\beta + 1 ) ^ { m + 1 } q ^ { - \\beta } = q ^ { - m } E _ { m + 1 } ( q ) , \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} \\begin{pmatrix} 2 & 4 & 5 & 1 \\\\ 3 & 6 & 5 & 1 \\\\ 7 & 7 & 1 & 1 0 \\\\ 6 & 6 & 1 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{align*} [ \\rho _ c ( a ) C ( \\cdot , x ) ] ( n ) & = n C ( n - 1 , x ) = c C ( n , x ) - c C ( n , x + 1 ) , \\\\ [ \\rho _ c ( a ^ \\dagger ) C ( \\cdot , x ) ] ( n ) & = c C ( n + 1 , x ) = c C ( n , x ) - x C ( n , x - 1 ) . \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} i _ 0 ^ * ( h ^ { G W , p r e - r } _ \\alpha ( a ) ) = \\theta ^ { G W } _ \\alpha ( a ) , \\ ; i _ 1 ^ * ( h ^ { G W , p r e - r } _ \\alpha ( a ) ) = T _ Y \\circ \\theta _ \\alpha ^ { G W } ( a ) \\circ T _ X , \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} \\int _ X \\psi \\omega ^ n = C \\int _ { \\Delta ^ + } \\psi ( d _ { 2 \\chi - 2 q } u ^ * ) P _ { D H } ( q ) d q . \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} \\chi \\left ( \\vec { x } ^ { ( 1 ) } , \\dots , \\vec { x } ^ { ( r ) } \\right ) : = \\# \\left \\{ x \\in \\vec { x } ^ { ( i ) } , \\ y \\in \\vec { x } ^ { ( j ) } \\ \\Big | \\ i < j , \\ x > y \\right \\} . \\end{align*}"}
-{"id": "7225.png", "formula": "\\begin{align*} f ( 0 , y _ 1 , x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) = \\sum _ { n _ 1 = 0 } ^ \\infty ( b _ 1 ; q ) _ { n _ 1 } c _ { n _ 1 } ( x _ 2 , y _ 2 , \\ldots , x _ k , y _ k ) y _ 1 ^ { n _ 1 } . \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} g ( x , \\xi ' , D _ n ) f : = \\int _ 0 ^ \\infty { g } ( x , \\xi ' , x _ n , w _ n ) f ( w _ n ) \\ , d w _ n f \\in \\mathcal { S } _ + , \\end{align*}"}
-{"id": "8000.png", "formula": "\\begin{align*} \\mathbb { E } [ U _ t ] \\leq & e ^ { - 2 \\kappa \\tilde { \\gamma } t } \\left [ U _ 0 + \\frac { ( \\mu - \\kappa ) } { a \\lambda _ 2 } \\overline { V } _ 0 \\right ] + \\frac { m \\tau ^ 2 \\tilde { \\gamma } } { 4 \\kappa N } \\left [ 1 + \\frac { ( \\mu - \\kappa ) ( N - 1 ) } { a \\lambda _ 2 } \\right ] ( 1 - e ^ { - 2 \\kappa \\tilde { \\gamma } t } ) . \\end{align*}"}
-{"id": "534.png", "formula": "\\begin{align*} D _ N ^ * ( \\Gamma _ { m _ 1 , \\ldots , m _ d } ) = 1 - \\prod _ { j = 1 } ^ d \\left ( 1 - \\frac { 1 } { 2 m _ j } \\right ) . \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} V ( x ) \\ \\doteq \\sum _ { j = 0 } ^ { \\infty } \\mathbf { 1 } _ { \\{ x _ { j } \\} } ( x ) \\ , \\frac { \\eta ( \\mathfrak { e } ) } { \\ln ( 4 + j ) } . \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{align*} M _ m \\overset { \\R P ^ { n _ m } } \\longrightarrow M _ { m - 1 } \\overset { \\R P ^ { n _ { m - 1 } } } { \\longrightarrow } \\cdots \\overset { \\R P ^ { n _ 2 } } \\longrightarrow M _ 1 \\overset { \\R P ^ { n _ 1 } } \\longrightarrow M _ 0 = \\{ a \\ p o i n t \\} , \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{align*} Q _ n ( 1 ) = \\frac { 1 } { n ^ 2 } \\sum _ { d \\mid n } ( - 1 ) ^ { n - d } \\mu ( n / d ) \\binom { 2 d - 1 } { d - 1 } . \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} Y _ { i _ { ( x , i ' ) } , j ' + 1 } : = Y ^ { ( x ) } _ { i ' , j ' } . \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} x _ { m , k } \\in P _ m A ^ T \\mathcal { K } _ { k } ( A P _ { m } A ^ { T } , b ) \\ , . \\end{align*}"}
-{"id": "6078.png", "formula": "\\begin{align*} \\lambda _ n ^ - ( M ( a _ 1 ) ) = O ( n ^ { - \\alpha - 1 } ) , n \\to \\infty . \\end{align*}"}
-{"id": "4671.png", "formula": "\\begin{align*} \\sup _ { x \\in A } \\inf _ { \\phantom { . } y \\in B \\phantom { \\dot I } } f ( x , y ) = \\inf _ { \\phantom { . } y \\in B \\phantom { \\dot I } } \\sup _ { x \\in A } f ( x , y ) \\ . \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} \\omega _ 1 = 2 \\mathbf { i } \\int _ { - \\infty } ^ { e _ 3 } \\frac { d \\mathbf { u } } { \\sqrt { g _ { 2 } \\mathbf { u } + g _ { 3 } - 4 \\mathbf { u } ^ { 3 } } } \\quad \\omega _ 2 = 2 \\int _ { e _ 1 } ^ { + \\infty } \\frac { d \\mathbf { u } } { \\sqrt { 4 \\mathbf { u } ^ { 3 } - g _ { 2 } \\mathbf { u } - g _ { 3 } } } . \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} { \\rm M } ( G _ { n } ) ~ = ~ { \\rm M } ( G _ { n } ^ { * } ) ~ = ~ \\theta _ { n } ^ { - 1 } \\ , \\prod _ { j = 1 } ^ { \\lfloor n / 6 \\rfloor } | z _ { j , n } | ^ { - 2 } . \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l l } \\mu _ { 1 j } = c _ { j \\alpha } h + \\tilde { c } _ { j \\alpha } , \\\\ \\noalign { \\smallskip } \\rho _ { 1 j } = - b _ { j \\alpha } h + \\tilde { b } _ { j \\alpha } , \\\\ \\noalign { \\smallskip } \\mu _ { 2 \\alpha } = - c _ { j \\alpha } \\varphi - b _ { j \\alpha } , \\\\ \\noalign { \\smallskip } \\rho _ { 2 \\alpha } = \\tilde { c } _ { j \\alpha } \\varphi - \\tilde { b } _ { j \\alpha } . \\end{array} \\right . \\end{align*}"}
-{"id": "4918.png", "formula": "\\begin{align*} a ( - c \\alpha + a \\beta ) s _ 2 ( \\varepsilon _ 0 ) = b ( d \\alpha - b \\beta ) s _ 3 ( \\varepsilon _ 0 ) . \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{align*} \\gamma ( x ) : = \\lim _ { t \\to 0 ^ { + } } \\frac { \\mu _ { t } ( x ) - \\epsilon ( x ) } { t } \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} & E \\left [ S ^ { p a r t , m } _ { s , t } \\right ] = E \\left [ S ^ { p a r t , 0 } _ { s , t } \\right ] \\\\ & - E \\left [ \\int \\ln ( p ^ 0 ( t , q _ t , z ) ) p ^ 0 ( t , q _ t , z ) d z \\right ] + E \\left [ \\int \\ln ( p ^ 0 ( s , q _ s , z ) ) p ^ 0 ( s , q _ s , z ) d z \\right ] \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} \\bar { x } \\cdot d \\log ( \\{ 1 + a S ^ i T ^ j , S \\} ) & = - a j c \\sum _ { k = 0 } ^ \\infty ( - a ) ^ k S ^ { ( k + 1 ) i + \\ell _ 1 - 1 } T ^ { ( k + 1 ) j + \\ell _ 2 - 1 } . \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} Z ( t , y ) = \\left ( Z _ { i , j } \\right ) _ { 1 \\leqslant i , j \\leqslant 3 } = [ \\nabla X ] ^ { - 1 } ( t , y ) \\qquad \\qquad ( t \\geqslant 0 , \\ y \\in \\Omega _ F ( 0 ) ) , \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} s _ 1 = \\begin{pmatrix} 0 & 1 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 \\end{pmatrix} , \\ \\mbox { a n d } s _ 2 = \\begin{pmatrix} 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{align*} G ( t , q , z ) = B ^ { i _ 1 , . . . , i _ k } ( t , q ) z _ { i _ 1 } . . . z _ { i _ k } . \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} & \\mathrm { E } \\bigg [ \\Big \\| \\int _ t ^ { t + h } \\Psi \\Big ( \\int _ t ^ s \\Phi \\ , \\mathrm { d } W _ r ^ K \\Big ) \\ , \\mathrm { d } W _ s ^ K - \\sum _ { i , j \\in \\mathcal { J } _ K } \\bar { I } _ { ( i , j ) } ^ { Q , ( D ) } ( h ) \\ ; \\Psi \\big ( \\Phi \\tilde { e } _ i , \\tilde { e } _ j \\big ) \\Big \\| _ H ^ 2 \\bigg ] \\\\ & \\leq 2 C \\sum _ { i = 1 } ^ L \\mathrm { E } \\Big [ \\big ( \\tilde { A } _ { ( i ) } ^ Q ( h ) - \\tilde { A } _ { ( i ) } ^ { Q , ( D ) } ( h ) \\big ) ^ 2 \\Big ] . \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} { \\bf H } = - H { \\bf n } , \\end{align*}"}
-{"id": "1964.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x } { } F ( a , b ; c ; x ) = \\frac { a b } { c } { } F ( a + 1 , b + 1 ; c + 1 ; x ) . \\end{align*}"}
-{"id": "8180.png", "formula": "\\begin{align*} \\partial ^ 2 _ { x _ l x _ j } e _ { i k } + \\partial ^ 2 _ { x _ k x _ i } e _ { j l } - \\partial ^ 2 _ { x _ l x _ i } e _ { j k } - \\partial ^ 2 _ { x _ j x _ k } e _ { i l } = 0 , \\ \\ i , j , k , l = 1 , . . . , m . \\end{align*}"}
-{"id": "752.png", "formula": "\\begin{align*} P _ { \\beta , P } ( X ) : = X ^ m - t _ 1 X ^ { m - 1 } - t _ 2 X ^ { m - 2 } - \\ldots t _ m \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\nu ( s ) & = & s \\\\ \\nu ( d s ) & = & d s + 2 \\ , \\mathrm { d i v } ( \\{ s , - \\} ) \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "1304.png", "formula": "\\begin{align*} \\gamma _ { 2 , j } < 1 , \\gamma _ { 1 , j } + \\gamma _ { 2 , j } \\alpha = 1 . \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} P _ { z , z ^ * } ^ 2 ( ( y _ i ) _ { i \\in I } ) & = P _ { z , z ^ * } \\left ( \\sum _ { i \\in J } z _ i ^ * ( y _ i ) z _ i \\right ) = \\sum _ { i \\in J } z _ i ^ * \\left ( z _ i ^ * ( y _ i ) z _ i \\right ) z _ i \\\\ & = \\sum _ { i \\in J } z _ i ^ * ( y _ i ) z _ i ^ * ( z _ i ) z _ i = \\sum _ { i \\in J } z _ i ^ * ( y _ i ) z _ i = P _ { z , z ^ * } ( ( y _ i ) _ { i \\in I } ) . \\end{align*}"}
-{"id": "4316.png", "formula": "\\begin{align*} z ( \\lambda , \\xi ) = \\int _ \\xi ^ { - \\infty } \\frac { d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\end{align*}"}
-{"id": "8522.png", "formula": "\\begin{align*} \\phi _ n = \\begin{cases} \\Re ( f ^ { V _ N } _ 0 \\ ! - f ^ { V _ S } _ 0 ) & \\\\ [ 2 p t ] \\phantom { \\Re ( } f ^ { V _ N } _ n \\ ! - f ^ { V _ S } _ n & \\end{cases} \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} & \\ \\ v ( n ) = 1 \\ ; n _ 1 = 0 \\ ; \\ ; 1 \\leq n _ j \\leq M _ j \\ ; \\ ; j \\ne 1 , \\\\ & \\ ; k = 2 , \\dots , g \\\\ & \\ \\ v ( n ) = 0 \\ ; n _ k = 0 \\ ; \\ ; 1 \\leq n _ j \\leq M _ j \\ ; \\ ; \\ j \\ne k . \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{align*} K _ 1 ( N ) = e ^ { i \\xi A _ N } \\left [ 1 + \\left ( 1 - e ^ { - A _ N } \\right ) ^ N \\right ] - 1 + N \\int _ 0 ^ { A _ N } e ^ { i \\xi s } \\left ( 1 - e ^ { - s } \\right ) ^ { N - 1 } e ^ { - s } d s , \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{align*} \\frac { | - 1 + z _ { j , n } | } { | z _ { j , n } | } ~ \\leq ~ \\ , \\kappa ( 1 , a _ { \\max } ) = \\frac { 1 - \\exp \\bigl ( \\frac { - \\pi } { a _ { \\max } } \\bigr ) } { 2 \\exp \\bigl ( \\frac { \\pi } { a _ { \\max } } \\bigr ) - 1 } = 0 . 1 7 1 5 7 3 \\ldots \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{align*} \\Delta _ p u = f ~ ~ \\mbox { i n } ~ B _ 1 , \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} | \\{ u < \\frac { M \\sigma ( r ) } { 2 } \\} \\cap B _ { r / 4 } ( x _ 1 ) | & \\leq \\frac { C r ^ n \\sigma ( r ) } { \\left ( ( M + 1 ) \\sigma ( r ) - \\frac { M \\sigma ( r ) } { 2 } \\right ) ^ \\epsilon } \\\\ & \\leq \\frac { C r ^ n \\sigma ( r ) ^ { ( 1 - \\epsilon ) } } { \\left ( \\frac { M } { 2 } \\right ) ^ \\epsilon } \\leq \\frac { C r ^ n } { M ^ { \\epsilon } } \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{align*} { \\mu } ( A + B ) = { \\mu } ( A ) + { \\mu } ( B ) + \\rho < \\tfrac { 1 } { 2 } \\big ( 1 + { \\mu } ( A ) + { \\mu } ( B ) \\big ) , \\rho < { \\mu } ( B ) \\leq { \\mu } ( A ) , \\rho < c . \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{align*} \\sup _ { 0 \\leq s \\leq t \\leq T } E \\left [ \\left | J _ { s , t } ^ m \\right | ^ p \\right ] ^ { 1 / p } = O ( m ^ { 1 / 2 } ) \\end{align*}"}
-{"id": "7007.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 - 4 8 , \\ x = 1 2 U = \\frac { - 1 2 Z _ 3 } { Z _ 1 + Z _ 2 } , \\ y = 1 2 V = \\frac { 1 2 ( Z _ 1 - Z _ 2 ) } { Z _ 1 + Z _ 2 } . \\end{align*}"}
-{"id": "8532.png", "formula": "\\begin{align*} \\iota _ X \\omega _ 1 = d \\mu _ 1 \\qquad \\iota _ X \\omega _ 2 = d \\mu _ 2 \\qquad \\iota _ X \\omega _ 3 = d \\mu _ 3 \\mathrlap { . } \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { J } \\mathbb { P } ( T \\geq k + 1 ) \\leq \\frac { 1 } { \\epsilon _ 0 } ( y _ 1 - y _ { J + 1 } ) \\leq \\frac { 1 } { \\epsilon _ 0 } y _ 1 , \\end{align*}"}
-{"id": "1570.png", "formula": "\\begin{align*} \\Delta _ { \\alpha _ i , \\alpha - \\alpha _ i } ( x ) = x _ i \\otimes d _ i ( x ) \\end{align*}"}
-{"id": "4289.png", "formula": "\\begin{align*} \\hat { \\sigma } _ \\lambda ( j ) = \\begin{cases} \\sigma ( j ) , & j = 1 , \\dots , b \\\\ \\sigma ( b ) + \\lambda , & j = b + 1 \\\\ \\sigma ( j - 1 ) , & j = b + 2 , \\dots , \\sigma ( b ) + \\lambda \\\\ \\sigma ( j - 1 ) + 1 , & j > \\sigma ( b ) + \\lambda + 1 . \\end{cases} \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{align*} \\begin{cases} \\eta _ t + w _ x + w _ { x x x } + ( \\eta w ) _ x = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , + \\infty ) , \\\\ w _ t + \\eta _ x + \\eta _ { x x x } + w w _ x = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , + \\infty ) , \\\\ \\eta ( x , 0 ) = \\eta _ 0 ( x ) , w ( x , 0 ) = w _ 0 ( x ) , & \\ , \\ , ( 0 , L ) , \\end{cases} \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} 0 < x ^ 2 \\frac { d ^ 2 } { d x ^ 2 } \\log ( \\theta _ 2 ( x ) ) = x ^ 2 \\frac { d } { d x } \\left ( \\frac { \\theta _ 2 ' ( x ) } { \\theta _ 2 ( x ) } \\right ) < \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "8707.png", "formula": "\\begin{align*} \\delta ( x ^ 0 , \\bar x ^ 1 , x ^ 2 , x ^ 3 ) : = \\gamma ( \\bar x ^ 1 ; \\Phi ( \\bar x ^ 1 ; x ^ 0 , x ^ 2 , x ^ 3 ) ) \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{align*} W _ { a } ^ { \\prescript { w } { } { \\chi } } \\circ \\overline { \\mathcal { A } } _ { w } = \\sum _ { b \\in \\widetilde { T } / H } \\tau _ { a , b } ^ { ( w ) } ( \\mathbf { z } ) W _ { b } ^ { \\chi } \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} H _ \\mu ( \\lambda ) : = ( 1 - \\alpha ^ \\chi ) ^ { - 1 } ( 1 - \\beta ^ \\chi ) ^ { - 1 } ( \\lambda - 1 ) ^ { \\mu - 1 } { } _ 3 F _ 2 \\left ( { 1 , 1 , 1 - \\mu \\atop 2 - \\alpha ^ \\chi , 2 - \\beta ^ \\chi } ; ( 1 - \\lambda ) ^ { - 1 } \\right ) , \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{align*} N _ m = \\begin{pmatrix} r _ 3 & r ^ T & 0 \\\\ 0 & 0 & r \\\\ 0 & 0 & r _ 3 \\end{pmatrix} . \\end{align*}"}
-{"id": "2446.png", "formula": "\\begin{align*} ( - 1 ) ^ { g - 1 } \\prod _ { j = 2 } ^ g \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ j } \\frac { \\psi _ j ( n _ j ; \\lambda _ j ) } { \\lambda _ j } \\ , d \\lambda _ j . \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{align*} \\delta _ F = 0 . \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} g = e ^ { 2 ( \\eta - U ) } ( - \\ : d R ^ 2 + a ^ { - 2 } d \\theta ^ 2 ) + e ^ { 2 U } ( d x + G d \\theta ) ^ 2 + e ^ { - 2 U } R ^ 2 ( d y + H d \\theta ) ^ 2 , \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{align*} \\frac { \\Phi _ { t _ 0 } ( x _ 0 e ^ { i A _ t ( x _ 0 ) } ) } { \\Phi _ s ( y e ^ { i A _ s ( y ) } ) } & = \\frac { \\Phi _ { t _ 0 } ( y e ^ { i A _ s ( y ) } ) } { \\Phi _ s ( y e ^ { i A _ s ( y ) } ) } \\frac { \\Phi _ { t _ 0 } ( x _ 0 e ^ { i A _ t ( x _ 0 ) } ) } { \\Phi _ { t _ 0 } ( y e ^ { i A _ s ( y ) } ) } . \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} \\omega _ 2 \\eta _ 1 - \\omega _ 1 \\eta _ 2 = 2 \\pi i , \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} b _ r = - ( t _ r + a _ r - \\sum _ { j = 1 } ^ { r - 1 } b _ j t _ { r - j } ) \\mbox { w i t h } b _ d = - ( t _ d + 1 - \\sum _ { j = 1 } ^ { d - 1 } b _ j t _ { r - j } ) , \\end{align*}"}
-{"id": "9136.png", "formula": "\\begin{align*} E ' : y ^ 2 = x ^ 3 + b ( h ) x ^ 2 + d ( h ) x \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{align*} x ^ { 0 } \\in \\mathcal { H } ; x ^ { k + 1 } : = U _ { k } x ^ { k } , \\end{align*}"}
-{"id": "1613.png", "formula": "\\begin{align*} \\Gamma ( A , B ) = \\bigcup _ { x \\in A , y \\in B } \\Gamma ( x , y ) . \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} \\sqrt { x } \\ , \\theta _ 3 \\left ( x \\right ) = \\theta _ 3 \\left ( \\tfrac { 1 } { x } \\right ) , \\end{align*}"}
-{"id": "7010.png", "formula": "\\begin{align*} [ H ] _ U ^ G = \\bigcap _ { g \\in G } H g U g ^ { - 1 } . \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} V ( \\log ( f _ { h _ 1 } \\circ f _ { h _ 2 } ^ { - 1 } ) ' ) & = V ( \\log ( ( f _ { h _ 1 } ' \\circ f _ { h _ 2 } ^ { - 1 } ) \\cdot ( ( f _ { h _ 2 } ' ) ^ { - 1 } \\cdot f _ { h _ 2 } ^ { - 1 } ) ) ) \\\\ & = V ( ( \\log f _ { h _ 1 } ' - \\log f _ { h _ 2 } ' ) \\circ f _ { h _ 2 } ^ { - 1 } ) \\\\ & = V ( ( F _ { h _ 1 } - F _ { h _ 2 } ) \\circ f _ { h _ 2 } ^ { - 1 } + f _ { h _ 1 } ' ( 0 ) + f _ { h _ 2 } ' ( 0 ) ) \\\\ & = V ( F _ { h _ 1 } - F _ { h _ 2 } ) \\\\ & = 4 n _ 0 + 1 > 4 n _ 0 , \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{align*} \\bar { Q } _ { \\lambda } = \\bar { Q } + \\lambda ( \\bar { R } - \\bar { Q } ) . \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} \\gamma _ { 0 } ' ( 0 ^ { + } ) = A ^ { + } , \\quad \\gamma _ { 0 } ' ( 0 ^ { - } ) = A ^ { - } . \\end{align*}"}
-{"id": "3022.png", "formula": "\\begin{align*} - \\Delta v _ { 0 } = \\frac { 1 } { d } a ( x ) v _ { 0 } . \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} \\Delta ^ t _ { \\mathcal { L } } = \\mathrm { C o n v } ( \\bar { W } \\cdot \\Delta _ { \\mathcal { L } } ) . \\end{align*}"}
-{"id": "7053.png", "formula": "\\begin{align*} D ( b , \\beta , \\theta ) : = 2 ^ 4 b ^ 2 \\left ( 1 + \\frac { \\pi } { \\beta } \\left | \\frac { \\theta ( \\beta ) } { 2 } - \\frac { \\pi } { \\beta } \\right | ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} u _ t & \\geq \\Delta u - \\chi \\nabla v \\cdot \\nabla u + u ( a _ { \\inf } - ( u ^ \\infty + \\epsilon ) b _ { \\sup } ) \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} 0 & = \\int _ 0 ^ T \\left ( i \\frac { \\partial u } { \\partial t } ( t ) - H ( t ) u ( t ) , v ( t ) \\right ) d t \\\\ & = \\int _ 0 ^ T \\left ( u ( t ) , i \\frac { \\partial v } { \\partial t } ( t ) - H ( t ) v ( t ) \\right ) d t = \\int _ 0 ^ T \\bigl ( u ( t ) , g ( t ) \\bigr ) d t , \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} \\frac { X ( | \\nabla _ B h | ^ 2 ) } { 2 } - \\left ( | \\nabla _ B h | ^ 2 - c \\right ) h ^ { - 1 } X ( h ) = 0 , \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} P : = \\P \\times P _ { \\omega } \\ , . \\end{align*}"}
-{"id": "2019.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\ddot { q } ( t ) & = - \\alpha \\sum _ { i \\in \\Z ^ d } Q _ i ( t ) \\nabla \\sigma ( q ( t ) - r _ i ) \\\\ M \\ddot { Q } _ i ( t ) + U ' ( Q _ i ( t ) ) & = - \\alpha \\sigma ( q ( t ) - r _ i ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} A _ k ( \\xi ^ m ) & = \\sum _ { i = 0 } ^ { k - 1 } a _ { k - i } J q ^ { k - i } J q ^ i ( \\xi ^ m ) \\\\ & = \\sum _ { i = 0 } ^ { k - 1 } a _ { k - i } \\binom { m } { i } \\binom { m + i } { k - i } \\\\ & = m ( m - 1 ) ( b _ { k - 2 } m ^ { k - 2 } + b _ { k - 3 } m ^ { k - 3 } + \\cdots + b _ 0 ) , \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{align*} \\{ \\cdot , h _ { k + 1 } \\} _ 0 = \\{ \\cdot , h _ k \\} _ 1 , k \\in \\N . \\end{align*}"}
-{"id": "5145.png", "formula": "\\begin{align*} \\partial _ s \\Phi _ k = d _ k \\Delta _ y \\Phi _ k \\end{align*}"}
-{"id": "5020.png", "formula": "\\begin{align*} [ c , z _ 1 , z _ 2 ] [ z _ 3 , z _ 4 ] + [ c , z _ 1 , z _ 3 ] [ z _ 2 , z _ 4 ] = 0 . \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} 2 \\sum _ { j = 1 } ^ \\infty \\mathcal G \\Big ( \\sqrt { 2 \\kappa ' ( 1 + j / m ) \\log m } \\Big ) \\le 2 m ^ { - \\kappa ' } \\sum _ { j = 1 } ^ \\infty e ^ { - \\kappa ' j b _ m } \\le \\frac { 2 } { \\kappa ' b _ m } m ^ { - \\kappa ' } \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} N ( y ^ { - 1 } ) \\sqcup ( y N ( v ^ { - 1 } ) \\cap w ( \\Delta ^ - ) ) = N ( y ^ { - 1 } ) \\sqcup ( N ( v ^ { - 1 } ) \\cap v ( \\Delta ^ - ) ) , \\end{align*}"}
-{"id": "6680.png", "formula": "\\begin{align*} \\| u \\| ^ { 2 } = t _ { \\ast } . \\end{align*}"}
-{"id": "3354.png", "formula": "\\begin{align*} f _ k ( z ) = ( z - a _ k ) g _ k ( z ) \\end{align*}"}
-{"id": "1910.png", "formula": "\\begin{align*} a _ n = b _ n + d _ n + e _ n + g _ n + C _ { n - 1 } , n \\geq 2 , \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{align*} \\Vert U x - z \\Vert ^ { 2 } \\leq \\Vert x - z \\Vert ^ { 2 } - \\sum _ { i = 1 } ^ { m } \\rho _ { i } \\Vert Q _ { i } x - Q _ { i - 1 } x \\Vert ^ { 2 } \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} b _ r = - t _ r + \\sum _ { j = 1 } ^ { r - 1 } b _ j t _ { r - j } \\mbox { f o r } ~ ~ r > d . \\end{align*}"}
-{"id": "5000.png", "formula": "\\begin{align*} [ h _ 1 , h _ 2 , \\dots , h _ n ] = 0 \\mbox { f o r a l l } h _ i \\in L \\iff [ x _ 1 , x _ 2 , \\dots , x _ n ] = 0 \\mbox { f o r a l l } x _ i \\in X . \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} L _ i [ \\psi ( \\cdot , \\mu ) ] ( \\nu ) = M _ i [ \\psi ( \\nu , \\cdot ) ] ( \\mu ) , \\qquad \\forall \\ 1 \\leq i \\leq n - 1 , \\end{align*}"}
-{"id": "180.png", "formula": "\\begin{align*} \\widehat { \\textbf { B } } = \\left ( I - P _ { 0 0 } \\right ) ^ { - 1 } P _ { 0 1 } = \\widehat { \\textbf { M } } P _ { 0 1 } . \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} \\det L ^ b ( \\l ) = \\l ^ { 2 k + 1 } + \\frac { 1 } { \\lambda ^ { 2 k + 1 } } \\prod _ { j = 1 } ^ { 2 k + 1 } b _ { j + k , j } + K _ k ^ b \\ ; , \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{align*} f ' ( x ) + \\sum _ { i = 1 } ^ { 2 k } \\psi _ i ( x ) e _ i = a f ' ( x ) + b e + v , \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} v _ n = \\left \\{ \\begin{array} { l l } 1 , & n \\geq 1 , \\\\ \\alpha , & 0 \\geq n \\geq - M + 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} & \\mathcal { L } _ { I _ B } ( s ) \\\\ & = \\prod _ { t \\in \\mathcal { K } } \\exp \\Big ( - 2 \\pi \\lambda _ t \\frac { s P _ t \\Delta _ { k , t } ^ { 2 - \\alpha } } { \\alpha - 2 } { } _ 2 F _ 1 ( 1 , 1 - \\frac { 2 } { \\alpha } ; 2 - \\frac { 2 } { \\alpha } ; - \\frac { s P _ t } { \\Delta _ { k , t } ^ { \\alpha } } ) \\Big ) , \\end{align*}"}
-{"id": "8802.png", "formula": "\\begin{align*} \\exp ( \\mathrm { a d } ( - a ) ) ( z _ { \\alpha } e _ { \\alpha } + \\theta ( z _ { \\alpha } e _ { \\alpha } ) ) = e ^ { \\alpha ( - a ) } z _ { \\alpha } e _ { \\alpha } + e ^ { - \\alpha ( - a ) } \\theta ( z _ { \\alpha } e _ { \\alpha } ) \\end{align*}"}
-{"id": "7657.png", "formula": "\\begin{align*} s _ i = \\sum ^ { M _ s } _ { i = 1 } \\alpha _ i \\bar { f } _ { i } , \\end{align*}"}
-{"id": "6656.png", "formula": "\\begin{align*} \\left ( P _ { G _ * } , d \\overline { \\varphi } \\wedge d \\overline { \\psi } \\right ) \\circ i ( m s ^ { - 1 } ) = \\left ( P _ { G _ * } ( m s ^ { - 1 } ) , d \\varphi ( m ) \\wedge d \\psi ( m ) \\right ) , \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} \\mathcal { L } _ { ^ { m , 2 } _ { i n t e r } } ( s ) & = \\mathcal { E } \\left \\{ \\prod _ { x _ j \\in \\Phi _ c \\backslash x _ m } { \\rm e x p } \\left ( - s \\frac { | h _ { j , m 2 } | ^ 2 } { { L \\left ( | | y _ { m , 2 } + x _ m - x _ j | | \\right ) } } \\right ) \\right \\} . \\end{align*}"}
-{"id": "8848.png", "formula": "\\begin{align*} C _ D = - e ^ { 2 \\alpha ( a ) } \\theta ( z _ 2 e _ { \\alpha } ) + \\tanh ( \\beta ( a ) ) \\Re ( z _ 1 \\mu _ { \\beta } ) + \\coth ( \\beta ( a ) ) \\Im ( z _ 1 \\mu _ { \\beta } ) . \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{align*} \\nu _ 0 ^ { ( m , - ) } \\big [ \\sum _ { j = 1 } ^ k Z _ j ( 0 ) \\big ] \\le \\nu _ 0 ^ { ( m , + ) } \\big [ \\sum _ { j = 1 } ^ k Z _ j ( 0 ) \\big ] = \\sum _ { j = 1 } ^ { k } ( z _ j + \\alpha _ j ^ { - 1 } ) \\le \\sum _ { j = 1 } ^ { k } ( z _ j + 1 ) \\ , , \\end{align*}"}
-{"id": "3075.png", "formula": "\\begin{align*} r : = \\frac { 2 } { 1 - q } \\in \\left ( 2 , \\infty \\right ) \\quad f \\left ( x \\right ) : = \\frac { \\left ( x + 1 \\right ) ^ { r } } { r } . \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} \\overline \\square _ B \\phi & = \\nabla _ T ^ * \\nabla _ T \\phi + \\sum _ { a , b } \\bar \\omega ^ a \\wedge \\bar V _ b \\lrcorner R ^ Q ( V _ b , \\bar V _ a ) \\phi + \\sum _ a \\bar \\omega ^ a \\wedge ( \\nabla _ { \\bar V _ a } H ^ { 0 , 1 } ) \\lrcorner \\ , \\phi , \\\\ \\square _ B \\phi & = \\bar \\nabla _ T ^ * \\bar \\nabla _ T \\phi + \\sum _ { a , b } \\omega ^ a \\wedge V _ b \\lrcorner R ^ Q ( \\bar V _ b , V _ a ) \\phi + \\sum _ a \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi . \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} \\tilde { F } _ { r _ m } ( r ) = & 1 - \\mathrm { P } ( \\# \\mathcal { A } _ r < m ) \\\\ = & 1 - \\sum ^ { m - 1 } _ { l = 0 } \\frac { ( \\lambda _ c [ \\pi r ^ 2 - \\pi \\delta ^ 2 \\mathcal { R } _ c ^ 2 ] ) ^ l } { l ! } e ^ { - \\lambda _ c [ \\pi r ^ 2 - \\pi \\delta ^ 2 \\mathcal { R } _ c ^ 2 ] } . \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{align*} u ( z _ I ) = \\underset { j } \\sum u ( z _ { I _ j } ) \\frac { | I _ j | } { | I | } + \\frac { 1 } { | I | } \\int _ { I \\backslash \\cup _ j I _ j } u ( x ) d x + O ( A ) . \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} { \\rm C o e f f } _ p [ f _ { \\mu } , m ] = \\sum _ { \\nu \\in \\sigma ( \\epsilon ) } \\psi ( \\nu , \\mu ; t ) f _ { \\nu } ( z ; t ^ { - m } , t ) , \\end{align*}"}
-{"id": "5273.png", "formula": "\\begin{align*} \\chi ( - 1 ) \\cdot \\frac { L ( f , \\chi , 1 ) } { ( - 2 \\pi i ) \\Omega ^ { \\chi ( - 1 ) } _ { f } } = \\frac { 1 } { n } \\cdot \\sum _ { b \\in ( \\mathbb { Z } / n \\mathbb { Z } ) ^ \\times } \\chi ( b ) \\cdot \\sigma _ b \\cdot \\left ( \\sum _ { a \\in ( \\mathbb { Z } / n \\mathbb { Z } ) ^ \\times } \\zeta ^ { a } _ n \\cdot \\left [ \\frac { - a } { n } \\right ] ^ { \\chi ( - 1 ) } _ { f } \\right ) . \\end{align*}"}
-{"id": "4800.png", "formula": "\\begin{align*} \\begin{array} { l l l } \\displaystyle \\frac { \\Phi ( | \\nabla u + t \\nabla v | ) - \\Phi ( | \\nabla u | ) } { t } & \\geq & \\displaystyle \\phi ( \\theta _ t ) \\theta _ t \\left [ \\frac { | \\nabla u | - t | \\nabla v | - | \\nabla u | } { t } \\right ] \\\\ \\\\ & = & \\displaystyle - \\phi ( \\theta _ t ) \\theta _ t | \\nabla v | \\\\ \\\\ & \\geq & - \\phi ( | \\nabla u | + | \\nabla v | ) ( | \\nabla u | + | \\nabla v | ) | \\nabla v | . \\end{array} \\end{align*}"}
-{"id": "2868.png", "formula": "\\begin{gather*} \\mathcal { A } = \\mathcal { F } \\oplus \\mathcal { C } \\mathcal { F } \\cdot \\mathcal { F } \\subseteq \\mathcal { F } , \\mathcal { F } \\cdot \\mathcal { C } \\subseteq \\mathcal { C } , \\mathcal { C } \\cdot \\mathcal { F } \\subseteq \\mathcal { C } , \\mathcal { C } \\cdot \\mathcal { C } \\subseteq \\mathcal { F } \\end{gather*}"}
-{"id": "6905.png", "formula": "\\begin{align*} \\lambda _ { i j } = \\mbox { m i n } \\{ \\mu \\in \\N \\mid \\mu e _ i \\in Q _ j \\} \\hbox { f o r } 1 \\le i \\le n . \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} { \\rm C o e f f } _ p [ E _ { \\mu } , m ] = \\lim _ { q \\rightarrow t ^ { - m } } ( 1 - q t ^ { m } ) ^ p \\left ( \\sum _ { \\nu \\in \\mathcal { E } _ { \\mu } } c _ { \\nu } ( q , t ) E _ { \\nu } ( z ; q , t ) \\right ) \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} \\bar { b } ^ 2 ( y ) & = \\left ( \\int - \\kappa ( y - \\theta ( x ) ) \\mu ^ y ( d x ) \\right ) ^ 2 \\\\ & \\leq { \\kappa } ^ 2 \\int F ( x , y ) \\mu ^ y ( d x ) \\\\ & \\leq - \\frac { \\kappa \\varepsilon } { \\lambda _ \\theta } \\int L ^ X F ( x , y ) \\mu ^ y ( d x ) + \\frac { \\kappa } { \\lambda _ \\theta } \\int G ( x ) \\mu ^ y ( d x ) \\\\ & = \\frac { \\kappa } { \\lambda _ \\theta } \\int G ( x ) \\mu ^ y ( d x ) \\\\ \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} I ( \\alpha ^ { ( 0 ) } , \\alpha ^ { ( 1 ) } ) = \\dfrac { \\sum \\limits _ { l _ 0 = 2 } ^ N \\sum \\limits _ { l _ 1 = 2 } ^ N A ( l _ 0 , l _ 1 ) \\alpha _ { l _ 0 } ^ { ( 0 ) } \\alpha _ { l _ 1 } ^ { ( 1 ) } } { \\sum \\limits _ { l _ 0 = 2 } ^ N \\sum \\limits _ { l _ 1 = 2 } ^ N B ( l _ 0 , l _ 1 ) \\alpha _ { l _ 0 } ^ { ( 0 ) } \\alpha _ { l _ 1 } ^ { ( 1 ) } } , \\end{align*}"}
-{"id": "7080.png", "formula": "\\begin{align*} \\frac { d ^ n f } { d x ^ n } ( 0 ) = 0 , ( \\forall n \\in \\{ 1 , 2 , \\cdots , n _ 0 - 1 \\} ) , \\frac { d ^ { n _ 0 } f } { d x ^ { n _ 0 } } ( 0 ) < 0 . \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} T _ t ( z ) : = d _ { r _ t } ^ { - 1 } \\left ( \\max _ { x \\neq z } \\widehat { \\Psi } _ t ( x ) - A _ { r _ t } + \\frac { | z | } { r _ t } d _ t \\right ) . \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} i \\partial _ t c _ k = \\sum _ { \\substack { \\ell , m , n \\geq 0 \\\\ k + \\ell = m + n } } \\frac { 1 } { 2 \\pi } \\frac { ( k + \\ell ) ! } { 2 ^ { k + \\ell } \\sqrt { k ! \\ell ! m ! n ! } } \\overline { c _ \\ell } c _ m c _ n , k \\geq 0 . \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} x _ t : = \\left \\{ \\begin{array} { l l } - K _ 0 , & t = 0 , \\\\ t , & t > 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "4933.png", "formula": "\\begin{align*} \\langle x ( t , 0 , u ) , p _ { k } \\rangle _ 2 ^ 2 & \\leq \\lambda _ { k } \\int _ 0 ^ t 2 x ^ T ( s ) P ^ { - 1 } B u ( s ) - x ^ T ( s ) P ^ { - 1 } B B ^ T P ^ { - 1 } x ( s ) d s \\\\ & = \\lambda _ { k } \\int _ 0 ^ t \\left \\| u ( s ) \\right \\| _ 2 ^ 2 d s - \\left \\| B ^ T P ^ { - 1 } x ( s ) - u ( s ) \\right \\| _ 2 ^ 2 d s . \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} M _ 1 ^ { \\mu } f ( x ) = \\sup _ { x \\in I } \\frac { 1 } { \\mu ( I ) } \\int _ { I } | f ( y ) | \\mu ( y ) d y , \\end{align*}"}
-{"id": "3756.png", "formula": "\\begin{align*} \\mathbf { E } _ J ( f ( \\nu _ { x , i } ) ) : = \\frac { 1 } { | J | } \\sum _ { j \\in J } \\int _ { \\R } f ( \\nu _ { y , j } ) d \\nu ( y ) , \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} & \\lim _ { L \\to \\infty \\atop L \\in \\N } \\sup _ { x \\in [ - r , r ] } \\left | \\frac { d ^ { i } } { d x ^ i } f _ L ( x ) - \\frac { d ^ { i } } { d x ^ i } f ( x ) \\right | = 0 , ( \\forall i \\in \\{ 0 , 1 , 2 \\} ) , \\\\ & \\lim _ { L \\to \\infty \\atop L \\in \\N } \\sup _ { x \\in [ - r , r ] } | u _ L ( x ) - u ( x ) | = 0 , \\lim _ { L \\to \\infty \\atop L \\in \\N } \\sup _ { x \\in [ - r , r ] } | g _ L ( x ) - g ( x ) | = 0 . \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} S ( x , - y , z ) = \\overline { S ( x , y , z ) } , Q ( x , - y , z ) = \\overline { Q ( x , y , z ) } , \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{align*} k ( s , t ) k ( r , s t ) = k ( r , s ) k ( r s , t ) . \\end{align*}"}
-{"id": "2362.png", "formula": "\\begin{align*} J _ 2 ( N ; \\theta ) : = \\int _ 0 ^ { \\infty } e ^ { - \\theta t } \\left ( 1 - e ^ { - ( 1 - \\theta ) t / N } \\right ) ^ N d t . \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} \\lVert A B - A _ 0 B _ 0 \\rVert ^ 2 & = \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } \\Biggl [ ( a _ { i H } - a ^ 0 _ { i H } ) + \\sum _ { k = 1 } ^ { H - 1 } \\{ ( a _ { i k } - a _ { i H } ) b _ { k j } - ( a ^ 0 _ { i k } - a ^ 0 _ { i H } ) b ^ 0 _ { k j } \\} \\Biggr ] ^ 2 \\\\ & + \\sum _ { j = 1 } ^ N \\Biggl [ \\sum _ { i = 1 } ^ { M - 1 } \\sum _ { k = 1 } ^ { H - 1 } \\{ ( a _ { i k } { - } a _ { i H } ) b _ { k j } { - } ( a ^ 0 _ { i k } { - } a ^ 0 _ { i H } ) b ^ 0 _ { k j } \\} { + } \\sum _ { i = 1 } ^ { M - 1 } ( a _ { i H } { - } a ^ 0 _ { i H } ) \\Biggr ] ^ 2 \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} \\begin{pmatrix} 2 & 4 & 5 & 1 \\\\ 3 & 6 & 5 & 1 \\\\ 7 & 7 & 1 & 1 0 \\\\ 6 & 6 & 1 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} H ' ( r ) = 2 r \\int _ { - 1 } ^ { 0 } t h ' ( r ^ 2 t ) d t , \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{align*} g _ { \\lambda } = \\begin{pmatrix} \\lambda & 0 & 0 & 0 \\\\ 0 & I _ W & 0 & 0 \\\\ 0 & 0 & \\lambda ^ { - 1 } \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{gather*} \\widetilde u ( t , y ) = Q ^ { - 1 } ( t ) u ( t , X ( t , y ) ) , \\widetilde \\pi ( t , y ) = \\pi ( t , X ( t , y ) ) \\\\ \\widetilde \\ell ( t ) = Q ^ { - 1 } ( t ) \\dot a ( t ) , \\widetilde \\omega ( t ) = Q ^ { - 1 } ( t ) \\omega ( t ) , \\end{gather*}"}
-{"id": "4126.png", "formula": "\\begin{align*} R = \\left ( \\frac { 1 } { \\omega _ { n - 1 } L } \\right ) ^ { \\frac { 1 } { n - 1 } } . \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} B _ t ^ \\sigma & : = \\int _ 0 ^ { \\sigma \\wedge t } \\ell ( X _ s ) \\ d s \\\\ C _ t ^ \\sigma & : = \\int _ 0 ^ { \\sigma \\wedge t } Q ( X _ s ) \\ d s \\\\ \\nu ^ \\sigma ( d t , d y ) & : = \\Big ( N ( X _ t , d y ) + a ( X _ t ) \\varepsilon _ { \\widehat { \\partial } - X _ t } ( d y ) \\Big ) \\ d t \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} H _ - ( u ' ( r ) ) = G ( \\xi ) - G ( u ( r ) ) , \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} \\frac { 2 s } { z } K _ s ( z ) - K _ { s + 1 } ( z ) = - K _ { s - 1 } ( z ) = - K _ { 1 - s } ( z ) , \\end{align*}"}
-{"id": "5275.png", "formula": "\\begin{align*} p \\cdot \\overline { C } _ p = c \\cdot d \\cdot ( c - 1 ) \\cdot ( d - 1 ) \\cdot \\left ( \\prod _ { q \\mid N _ { \\mathrm { s p } } } ( 1 - q ^ { - 1 } ) \\right ) \\cdot \\left ( \\prod _ { q \\mid N _ { \\mathrm { n s } } } ( 1 + q ^ { - 1 } ) \\right ) \\cdot \\left ( p - a _ p ( f ) + \\psi ( p ) \\right ) . \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} I : = \\left \\{ ( n _ 1 , \\ldots , n _ k ) \\in \\N _ 0 ^ k : \\sum _ { j = 1 } ^ k n _ j = r \\right \\} . \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{align*} \\kappa _ { i , j } = \\log \\frac { 1 } { p _ { X Y } ( x _ i , y _ j ) } \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} g = \\begin{pmatrix} 1 & - v ^ T & \\frac { 1 } { 2 } \\| v \\| ^ 2 \\\\ 0 & I & - v \\\\ 0 & 0 & 1 \\end{pmatrix} \\begin{pmatrix} \\alpha ^ { - 1 } & 0 & 0 \\\\ 0 & A ^ { - 1 } & 0 \\\\ 0 & 0 & \\alpha \\end{pmatrix} \\begin{pmatrix} 1 & 0 & 0 \\\\ - \\xi & I & 0 \\\\ \\frac { 1 } { 2 } \\| \\xi \\| ^ 2 & - \\xi ^ T & 1 \\end{pmatrix} = g _ { - 1 } g _ 0 g _ 1 \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} g ( 2 k + 1 ) - g ( 2 k - 1 ) & = \\left ( g ( 2 k + 1 ) - g ( 2 k ) \\right ) + \\left ( g ( 2 k ) - g ( 2 k - 1 ) \\right ) \\\\ & \\ge \\left ( g ( k + 1 ) - g ( k ) \\right ) + \\left ( g ( k + 1 ) - g ( k ) \\right ) \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } | a _ n | ^ { \\frac { 1 } { n } } = 1 . \\end{align*}"}
-{"id": "6621.png", "formula": "\\begin{align*} E _ { } : = \\{ u v = e \\in E \\mid \\Gamma \\setminus \\{ u , v \\} 3 \\} . \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} \\mu _ { 2 U } = \\varepsilon _ U \\nabla _ F \\nabla _ F \\varphi ( U , U ) , \\rho _ { 2 U } = - \\varepsilon _ U R i c _ F ( U , U ) . \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} s ( J , F ) = J \\qquad t ( J , F ) = J \\cdot F \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} g _ \\infty = - \\ : 1 6 d u ^ 2 + u ^ 2 d \\theta ^ 2 + 4 c _ \\lambda ^ { - 1 } \\left [ c _ 1 ^ { - 1 } d \\widehat { \\sigma } ^ 2 + c _ 1 d \\widehat { \\delta } ^ 2 \\right ] . \\end{align*}"}
-{"id": "7590.png", "formula": "\\begin{align*} x _ j = ( q , p ) \\to ( - q ^ 1 , q ^ 2 , q ^ 3 , p _ 1 , - p _ 2 , - p _ 3 ) \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} \\begin{pmatrix} 0 & k _ 0 & 0 & 0 \\\\ k _ 1 & 0 & 0 & k _ 0 \\\\ k _ 2 & 0 & 0 & 0 \\\\ 0 & k _ 1 & k _ 2 & 0 \\end{pmatrix} _ t \\end{align*}"}
-{"id": "1978.png", "formula": "\\begin{align*} \\| T _ { k } ^ { \\alpha } \\| _ { L ^ { p } ( \\mathbf { B } , d \\tau ) \\rightarrow L ^ { p } ( \\mathbf { B } , d v _ { p m - n } ) } & > \\frac { \\pi ^ { n / 2 } \\Gamma ( p ( m + \\alpha ) + \\frac { n } { 2 } + 1 ) \\Gamma ( m + \\alpha + \\frac { n } { p } + 1 ) } { \\Gamma ( p ( m + \\alpha ) + 1 ) \\Gamma ( m + \\alpha + \\frac { n } { 2 } + \\frac { n } { p } + 1 ) } \\\\ & \\times \\left ( \\frac { \\Gamma ( p m - n + 1 ) } { \\Gamma ( p m - \\frac { n } { 2 } + 1 ) } \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} A _ 7 ( 0 , 1 ) & = \\frac { 1 4 } { 3 } J q ^ 7 - \\frac { 1 4 } { 3 } J q ^ 6 J q ^ 1 + \\frac { 7 } { 3 } J q ^ 5 J q ^ 2 \\\\ & - \\frac { 7 } { 1 5 } J q ^ 4 J q ^ 3 - \\frac { 1 } { 3 } J q ^ 3 J q ^ 4 + J q ^ 1 J q ^ 6 = 0 . \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} L _ i \\left [ \\psi \\left ( \\cdot , \\mu \\right ) \\right ] ( \\nu ) = M _ i \\left [ \\psi ( \\nu , \\cdot ) \\right ] \\left ( \\mu \\right ) , \\forall \\ i \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "1264.png", "formula": "\\begin{align*} M _ n = \\left ( \\lambda ^ { - k ( n ) } \\cdot [ g ^ n ( F ) - g ^ n ( x , y ) ] \\right ) \\cap Q \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{align*} \\vec { X } _ n ( t ) = ( X _ { n , 1 } ( t ) , \\dots , X _ { n , n } ( t ) ) _ { t \\geq 0 } \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{align*} & \\Gamma _ { r a t } ^ { c c } : = { \\mathbf k } ( ( E ^ \\# ) ^ n ) \\otimes \\Lambda ^ 2 ( \\oplus _ { i \\in [ n ] } { \\mathbf k } \\cdot \\underline { d c } _ i ) , \\Gamma _ { r a t } ^ { c p } : = { \\mathbf k } ( ( E ^ \\# ) ^ n ) \\otimes ( \\oplus _ { i , j \\in [ n ] } { \\mathbf k } \\cdot ( \\underline { d c } _ i \\wedge \\underline { d p } _ j ) ) , \\\\ & \\Gamma _ { r a t } ^ { p p } : = { \\mathbf k } ( ( E ^ \\# ) ^ n ) \\otimes \\Lambda ^ 2 ( \\oplus _ { i \\in [ n ] } { \\mathbf k } \\cdot \\underline { d p } _ i ) . \\end{align*}"}
-{"id": "4387.png", "formula": "\\begin{align*} & \\left ( \\int _ { 1 } ^ { \\hat { X } } \\frac { ( X - \\lambda / 3 ) d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\right ) \\frac { 1 } { 2 \\sqrt { \\hat { X } ( \\hat { X } - 1 ) ( \\hat { X } - \\lambda ) } } = \\\\ & \\left ( \\int _ { 1 } ^ { \\hat { X } } \\frac { ( X - \\lambda / 3 - \\sqrt { X ( X - \\lambda ) } ) d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\right ) \\frac { 1 } { 2 \\sqrt { \\hat { X } ( \\hat { X } - 1 ) ( \\hat { X } - \\lambda ) } } + \\frac 1 { 2 \\sqrt { \\hat { X } ( \\hat { X } - \\lambda ) } } . \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{align*} X = A ^ { D } A X ~ ~ ~ ~ A ^ { m } X = A ^ { m } ( A ^ { j } ) ^ { \\dagger } . \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{align*} \\gamma _ { k , n } ^ { ( \\alpha ) } : = \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ { \\rho _ 1 } } \\frac { Q _ { n , \\textup { \\textbf { m } } } ( t ) F _ \\alpha ( t ) \\Phi ' ( t ) } { \\Phi ^ { k + 1 } ( t ) } d t , \\alpha = 1 , 2 , \\ldots , d . \\end{align*}"}
-{"id": "8609.png", "formula": "\\begin{align*} M _ { i j } = \\begin{cases} k & \\mbox { i f $ i \\in U _ k ^ j $ , } \\\\ * & \\mbox { i f $ i $ i s n o t i n a n y p a r t i t e s e t o f $ F _ j $ . } \\end{cases} \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{align*} h ( \\lambda _ 2 , \\dots , \\lambda _ g ) = \\frac { ( - 1 ) ^ { g - 1 } } { ( 2 \\pi i ) ^ { g - 1 } } \\cdot \\frac { 1 } { \\lambda _ 2 \\cdots \\lambda _ g } . \\end{align*}"}
-{"id": "7652.png", "formula": "\\begin{align*} & \\mathrm { E } \\bigg [ \\Big \\| \\int _ t ^ { t + h } \\Psi \\Big ( \\int _ t ^ s \\Phi \\ , \\mathrm { d } W _ r ^ K \\Big ) \\ , \\mathrm { d } W _ s ^ K - \\sum _ { i , j \\in \\mathcal { J } _ K } \\hat { I } _ { ( i , j ) } ^ { Q , ( D ) } ( h ) \\ ; \\Psi \\big ( \\Phi \\tilde { e } _ i , \\tilde { e } _ j \\big ) \\Big \\| _ H ^ 2 \\bigg ] \\\\ & \\leq 2 C \\sum _ { i = 1 } ^ L \\mathrm { E } \\Big [ \\big ( \\tilde { A } _ { ( i ) } ^ Q ( h ) - \\hat { A } _ { ( i ) } ^ { Q , ( D ) } ( h ) \\big ) ^ 2 \\Big ] \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} d d ^ c \\log \\| s \\| ^ 2 = \\delta _ { \\div ( s ) } - c _ 1 ( \\overline { L } ) \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} \\int _ 0 ^ \\tau s ^ { 2 - b } \\| d A ( s ) \\| _ 3 ^ 2 \\| \\psi ( s ) \\| _ 6 ^ 2 d s = \\int _ 0 ^ \\tau \\ ( s \\| d A ( s ) \\| _ 3 ^ 2 \\ ) \\ ( s ^ { 1 - b } \\| \\psi ( s ) \\| _ 6 ^ 2 \\ ) d s . \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} k ^ { q } _ { j } : = k ^ { u } _ { j } + k ^ { \\nabla y } _ { j } , k ^ { \\infty } _ { j } : = k ^ { y } _ { j } + k ^ { x } _ { j } . \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} \\sum _ { m = - \\infty } ^ { 0 } r _ m ^ { ( 0 ) } \\phi \\left ( r _ { m - 1 } ^ { ( m ) } \\right ) < \\infty . \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} ( I _ l g ) ( s , t ) = s ^ { \\frac l 2 } g \\left ( \\frac { t } { \\sqrt s } \\right ) = \\sum _ { j = 0 } ^ { [ \\frac l 2 ] } a _ j s ^ j t ^ { l - 2 j } . \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{align*} \\mu ( A _ 1 + A _ 2 ) = \\mu ( A _ 1 ) + \\mu ( A _ 2 ) + \\rho ' < \\tfrac { 1 } { 2 } \\big ( 1 + \\mu ( A _ 1 ) + \\mu ( A _ 2 ) \\big ) , \\rho ' < \\min ( \\mu ( A _ 1 ) , \\mu ( A _ 2 ) ) . \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{align*} \\bigg \\| \\sum _ { k = k _ 0 } ^ \\infty 2 ^ { - k \\alpha } \\mathcal { C } _ k g \\bigg \\| _ { L ^ 2 _ t L ^ r _ x } \\lesssim \\| g \\| _ { L ^ 2 _ t L ^ 2 _ x } \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} W _ i : = \\frac { \\sum _ { \\sigma \\ni i } v _ \\sigma } { 1 + \\sum _ { \\sigma \\ni i } q _ \\sigma } , \\end{align*}"}
-{"id": "3413.png", "formula": "\\begin{align*} \\Psi ( P ) \\ast f ( x ) = & \\int \\sum _ { \\alpha } a _ { \\alpha } \\frac { \\partial ^ { | \\alpha | } \\delta } { \\partial y ^ { \\alpha } } ( y ) f ( x - y ) d y \\\\ = & \\sum _ { \\alpha } a _ { \\alpha } \\int \\delta ( y ) ( - 1 ) ^ { | \\alpha | } \\frac { \\partial ^ { | \\alpha | } } { \\partial y ^ { \\alpha } } f ( x - y ) d y \\\\ = & \\sum _ { \\alpha } a _ { \\alpha } \\frac { \\partial ^ { | \\alpha | } } { \\partial x ^ { \\alpha } } f ( x ) . \\end{align*}"}
-{"id": "4337.png", "formula": "\\begin{align*} | \\prod _ { n = 1 } ^ { \\infty } ( 1 - q ^ n ) ^ { 2 4 } | \\geq \\prod _ { n = 1 } ^ { \\infty } ( 1 - \\exp ( - n \\pi \\sqrt { 3 } ) ) ^ { 2 4 } \\geq 9 / 1 0 , \\end{align*}"}
-{"id": "8870.png", "formula": "\\begin{align*} \\Delta ^ + _ { \\mathcal { L } ^ 2 } = \\lambda + \\Delta _ { \\mathcal { L } ^ 2 } . \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} \\Delta _ B \\phi = 2 \\overline \\square _ B \\phi - i ( J \\kappa _ B ^ \\sharp ) \\lrcorner \\ , \\bar \\partial _ B \\phi - i \\bar \\partial _ B ( J \\kappa _ B ^ \\sharp ) \\lrcorner \\ , \\phi + \\partial _ B H ^ { 1 , 0 } \\lrcorner \\ , \\phi . \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{align*} p _ q ^ * A _ \\infty ( q ) = ( A _ q \\oplus A _ q ) ^ { g _ \\infty } , \\Phi _ \\infty ( q ) = g _ \\infty ^ { - 1 } \\Phi g _ \\infty . \\end{align*}"}
-{"id": "293.png", "formula": "\\begin{align*} W ( x , y ) = x ^ n + \\sum _ { r = \\mu + 1 } ^ { 6 \\mu + \\nu } A _ { 2 r } x ^ { n - 2 r } y ^ { 2 r } . \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{align*} \\exists z \\leq 0 \\widetilde { B } _ { 1 } ( z , 1 ) \\widetilde { B } _ { - 1 } ( z , 1 ) = 0 . \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{align*} X ^ n _ t : = \\sum _ { k = 0 } ^ \\infty X _ { \\tau ^ n _ k } \\ 1 _ { ( \\tau ^ n _ k , \\tau ^ n _ { k + 1 } ] } ( t ) \\int _ 0 ^ t X ^ n _ s \\otimes \\dd X _ s : = \\sum _ { k = 0 } ^ { \\infty } X _ { \\tau _ k ^ n } \\otimes X _ { \\tau _ k ^ n \\wedge t , \\tau _ { k + 1 } ^ n \\wedge t } . \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} \\varphi \\left ( x , t \\right ) = - \\underset { p \\in \\mathbb { R } ^ { n } } { } \\left \\{ J ^ { \\star } \\left ( p \\right ) + t H \\left ( p \\right ) - \\left \\langle x , p \\right \\rangle \\right \\} , \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} U _ 1 ( n , t , s ) = e ^ { - i ( t - K / n ) H _ 1 ( K / n ) } \\prod _ { k = J } ^ { K - 1 } e ^ { - i \\frac { H _ 1 ( k / n ) } { n } } e ^ { - i ( J / n - s ) H _ 1 ( ( J - 1 ) / n ) } \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} N [ \\mathfrak { e } , V ] \\ \\geq \\ \\mathcal { L } _ { V } [ ( 1 + \\varepsilon ) \\eta ( \\mathfrak { e } ) ] \\ = \\ \\big \\{ x \\in \\Gamma \\ ; \\big | \\ V ( x ) \\geq ( 1 + \\varepsilon ) \\eta ( \\mathfrak { e } ) \\big \\} . \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{align*} g _ \\infty = - \\ : 1 6 d u ^ 2 + u ^ 2 d \\theta ^ 2 + 4 c _ \\lambda ^ { - 1 } \\left [ c _ 1 ^ { - 1 } d \\widehat { \\sigma } ^ 2 + c _ 1 d \\widehat { \\delta } ^ 2 \\right ] . \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ { x _ j } u = \\tilde a _ j - A _ j \\tilde a & { \\rm f o r } & j = 1 , . . . , n ; \\\\ \\partial _ { x _ { n + 1 } } u = \\tilde a . \\end{array} \\right . \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{align*} S ( B ) ( t ) = \\left ( B _ t , ( 1 / 2 ) B _ t \\otimes B _ t + A _ t ^ { } + t \\Gamma \\right ) , \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} C ( u ) & : = \\frac { 1 } { 2 } \\bigl ( \\xi ( 1 ) - \\xi ( u ) - \\xi ' ( u ) ( 1 - u ) \\bigr ) , \\ , \\ , u \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} \\square _ B \\phi = \\Delta _ B \\phi . \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} \\| A _ { b } ^ { m , k } f \\| _ { L ^ { p } ( \\lambda ) } \\leq \\sup _ { \\| g \\| _ { L ^ { p ' } ( \\lambda ) } = 1 } \\sum _ { Q \\in \\mathcal { S } } \\left ( \\int _ { Q } | g \\lambda | | b - b _ { Q } | ^ { m - k } \\right ) \\frac { 1 } { | Q | } \\int _ { Q } | b - b _ { Q } | ^ { k } | f | . \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} \\operatorname { R e } \\left ( e ^ { - i \\theta } [ \\frac { 1 } { | z _ \\alpha | } , \\partial ^ 2 _ t ] z _ \\alpha + e ^ { - i \\theta } \\partial _ t ^ 2 \\bigl ( \\frac { z _ \\alpha } { | z _ \\alpha | } \\bigr ) \\right ) = \\bigl ( \\frac { \\partial _ \\alpha } { | z _ \\alpha | } \\bigr ) ^ 3 \\theta - a \\frac { \\partial _ \\alpha } { | z _ \\alpha | } \\theta . \\end{align*}"}
-{"id": "437.png", "formula": "\\begin{align*} L ( I , G ) = \\{ X \\in \\mathcal { M } _ S \\mid \\mathrm { L M } _ { \\succ } ( g ) \\nmid X \\mbox { f o r a l l } g \\in G \\} \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{align*} \\eta ( \\hat { \\vect { x } } _ n ) = t ( \\hat { \\vect { x } } _ n ^ t ; \\vect { \\Sigma } ^ t ) \\frac { \\beta _ n } { \\beta _ n + \\tau _ t ^ 2 } \\hat { \\vect { x } } _ n \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{align*} \\dot A _ \\infty : = \\left . \\frac { \\partial \\ , } { \\partial s } \\right | _ { s = 0 } A _ \\infty ( s ) = - \\frac 1 4 d \\Im ( \\dot q / q ) \\begin{pmatrix} i & 0 \\\\ 0 & - i \\end{pmatrix} \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} \\prod _ { m \\in \\lambda ' } \\left ( E _ \\mu ^ * R _ m L _ m + E _ \\mu ^ * L _ m R _ m \\right ) = \\sum _ { \\lambda } \\left ( \\prod _ { m \\in \\lambda ' } \\theta ( m , \\mu , \\lambda ) \\right ) E _ \\mu ^ * E _ { \\lambda } , \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( - \\Delta u _ { 0 } ) q _ { 0 } u _ { 0 } ^ { q _ { 0 } - 1 } \\phi _ { 1 } + u _ { 0 } ^ { q _ { 0 } } \\Delta \\phi _ { 1 } = \\int _ { \\Omega } | \\nabla u _ { 0 } | ^ { 2 } q _ { 0 } ( q _ { 0 } - 1 ) u _ { 0 } ^ { q _ { 0 } - 2 } \\phi _ { 1 } . \\end{align*}"}
-{"id": "3911.png", "formula": "\\begin{align*} H ( x ) = \\gamma - e ^ { - a ( x + g ( x ) - k ) } ( \\gamma + a x ) , \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} \\widehat { H ^ s u } ( \\xi , \\sigma ) = ( ( 2 \\pi | \\xi | ) ^ 2 + 2 \\pi i \\sigma ) ^ s \\ \\hat u ( \\xi , \\sigma ) . \\end{align*}"}
-{"id": "4773.png", "formula": "\\begin{align*} \\Phi ^ { - 1 } ( 0 , 0 ) & = \\Big \\{ ( x , \\lambda , v ) \\ | \\ v = \\nabla g ( x ) \\lambda , \\ , g ( x ) = { \\rm m i n } ( 0 , g ( x ) + \\lambda ) \\Big \\} \\\\ & = \\Big \\{ ( x , 0 , 0 ) \\ | \\ g ( x ) < 0 \\big \\} \\cup \\big \\{ ( x , \\lambda , v ) \\ | \\ g ( x ) = 0 , \\lambda \\geq 0 , v = \\nabla g ( x ) \\lambda \\Big \\} . \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} 1 \\cdot \\alpha = \\mathrm { T r } ( \\mathrm { a d } _ \\alpha ) \\ , . \\end{align*}"}
-{"id": "9062.png", "formula": "\\begin{align*} \\| u ' \\| _ J ^ 2 = \\hat u ' \\cdot \\hat u ' - 2 u _ 3 ' \\left ( \\frac 1 { u _ 3 } \\hat u \\cdot \\hat u ' - \\frac { u _ 3 ' } { 2 u _ 3 ^ 2 } \\| \\hat u \\| ^ 2 \\right ) = \\| m ' \\| ^ 2 u _ 3 ^ 2 . \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} & \\mathcal { L } _ { I _ J } ( s ) \\\\ & = \\exp \\Big ( - 2 \\pi \\lambda _ J \\frac { s P _ J R _ { \\tau } ^ { 2 - \\alpha } } { \\alpha - 2 } { } _ 2 F _ 1 ( 1 , 1 - \\frac { 2 } { \\alpha } ; 2 - \\frac { 2 } { \\alpha } ; - \\frac { s P _ J } { R _ { \\tau } ^ { \\alpha } } ) \\Big ) \\\\ & \\times \\int _ { 0 } ^ { \\infty } H ( v , s ) f ( v ) d v , \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} \\psi ( x ) = ( \\log \\circ \\ , \\theta _ 3 ) ' ( x ) = \\frac { \\theta _ 3 ' ( x ) } { \\theta _ 3 ( x ) } . \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} \\overline { M } _ { n } = \\frac { a _ { \\sup } - \\chi \\mu \\underline { M } _ { n } } { b _ { \\inf } - \\chi \\mu } , \\underline { M } _ { n + 1 } = \\frac { a _ { \\inf } - \\chi \\mu \\overline { M } _ { n } } { b _ { \\sup } - \\chi \\mu } , \\forall \\ n \\geq 0 . \\end{align*}"}
-{"id": "242.png", "formula": "\\begin{align*} S ( i , j , k , l , \\alpha , \\beta ) : = \\omega _ { i j } ^ \\alpha \\underline \\wedge \\omega _ { k l } ^ \\beta ( i , j , k , l \\in [ n ] \\mathrm { \\ a l l \\ d i f f e r e n t , \\ } i < j , k < l , i < k , \\alpha , \\beta \\geq 0 ) , \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } \\frac { H _ 1 ( a t ) } { H _ 2 ( t ) } = \\lim _ { t \\to + \\infty } \\frac { K _ 1 ^ 1 } { K _ 1 ^ 2 } a ^ { \\frac { 1 } { 2 } } e ^ { \\frac { K _ 2 ^ 2 } { t } - \\frac { K _ 2 ^ 1 } { a t } } . \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{align*} N _ h ^ k ( \\lambda ) = F _ h ^ { k - 1 } ( \\lambda ) + b _ k + F _ h ^ k ( \\lambda ) , \\end{align*}"}
-{"id": "1654.png", "formula": "\\begin{align*} \\log \\frac { p ( x | a ) } { p ( x | b ) } & = \\log a ^ x ( 1 - a ) ^ { 1 - x } - \\log b ^ x ( 1 - b ) ^ { 1 - x } \\\\ & = x ( \\log a - \\log b ) + ( 1 - x ) \\{ \\log ( 1 - a ) - \\log ( 1 - b ) \\} , \\end{align*}"}
-{"id": "3406.png", "formula": "\\begin{align*} \\begin{aligned} & \\ , | \\log ( \\log g ( z _ 2 ) + A ) - \\log ( \\log g ( z _ 1 ) + A ) | \\\\ = & \\left | \\int ^ { z _ 2 } _ { z _ 1 } \\frac { g ' ( z ) } { g ( z ) ( \\log g ( z ) + A ) } d z \\right | \\leq \\int ^ { z _ 2 } _ { z _ 1 } \\frac { | g ' ( z ) | } { | g ( z ) | ( \\log | g ( z ) | + A ) } | d z | \\\\ \\leq & \\ , \\frac { \\pi } { 2 | x _ 0 | } | z _ 2 - z _ 1 | \\leq \\frac { \\pi ^ 2 } { 2 | x _ 0 | } = \\frac { \\pi ^ 2 } { \\log \\frac 1 C } . \\end{aligned} \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} h \\leq \\varphi _ i ( y ) \\leq C _ 1 h , \\quad | \\nabla \\varphi _ i ( y ) | \\leq C _ 2 h y \\in [ 0 , b ] , i = 1 , 2 . \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{align*} Q ( i , j , k , \\alpha ) = \\underline { d p } _ i \\underline \\wedge \\omega _ { j k } ^ \\alpha , j < k , i \\neq j , k , i , j , k \\in [ n ] , \\alpha \\geq 0 , \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} \\Phi _ m W _ { n , k } - \\Phi _ m W _ { n + 1 , k } \\cdot q ^ { - k } = W _ { n , k } \\Phi _ m \\cdot m ^ k \\gamma ^ { - n k } - W _ { n + 1 , k } \\Phi _ m \\cdot \\frac { m ^ k } { q ^ k } \\gamma ^ { - ( n + 1 ) k } \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ^ { u } _ { j } ( T ) & \\leq C \\tilde { C } _ { T } ( k ^ { q } _ { j } ( T ) + k ^ { q } _ { j - 1 } ( T ) ) \\delta _ { j - 1 } ( T ) < 2 K _ { 1 } C \\tilde { C } _ { T } \\delta _ { j - 1 } ( T ) . \\end{aligned} \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} \\dot { x } \\left ( t \\right ) & = \\left [ \\begin{array} { c c } 0 _ { 2 } & I _ { 2 } \\\\ 0 _ { 2 } & 0 _ { 2 } \\end{array} \\right ] x \\left ( t \\right ) + \\left [ \\begin{array} { c } 0 \\\\ 0 \\\\ 0 \\\\ \\pm 1 \\end{array} \\right ] a _ { p } + \\left [ \\begin{array} { c } 0 \\\\ 0 \\\\ 0 \\\\ - 1 \\end{array} \\right ] a _ { e } , \\\\ & = A x \\left ( t \\right ) + B a _ { p } + D a _ { e } , \\end{align*}"}
-{"id": "6478.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\varphi ^ { \\prime } \\overline { w } \\ ; \\d x + \\int _ { \\Omega } \\nabla \\varphi \\cdot \\overline { \\nabla w } \\ ; \\d x = - \\int _ { \\Omega } u \\cdot \\nabla \\varphi \\overline { w } \\ ; \\d x + 2 \\int _ { \\Omega } \\lvert \\nabla d \\rvert ^ 2 \\varphi \\overline { w } \\ ; \\d x \\end{align*}"}
-{"id": "4334.png", "formula": "\\begin{align*} \\Delta ( \\tau ) = q \\prod _ { n = 1 } ^ \\infty ( 1 - q ^ n ) ^ { 2 4 } \\end{align*}"}
-{"id": "4433.png", "formula": "\\begin{align*} & \\Delta ^ m H ( B ^ e ) \\times ( 2 H _ 1 ^ { - ( 1 - 2 h / ( e p ) ) } ) ^ { 2 h } \\times ( C _ { 1 8 } H _ 1 ^ { 2 h / ( e p ) } ) ^ { e p - 2 h } V ( e p - 2 h ) \\\\ & = \\Delta ^ m H ( B ^ e ) 4 ^ h C _ { 1 8 } ^ { e p - 2 h } V ( e p - 2 h ) \\\\ & > 2 ^ { p n } \\Delta ^ n , \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{align*} \\dim _ B ( A ) = \\overline { \\dim } _ B ( A ) = \\underline { \\dim } _ B ( A ) \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{align*} \\begin{cases} D _ { A } \\psi = 0 \\\\ F _ { A } ^ { + } = \\frac { r } { 2 } ( q ( \\psi , \\psi ) - i \\Omega ) + i \\mu \\\\ * d * b - 2 ^ { - \\frac { 1 } { 2 } } r ^ { \\frac { 1 } { 2 } } ( \\eta ^ * \\psi _ I - \\psi _ I ^ * \\eta ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} \\mathcal L _ \\partial ( \\omega _ S ) = \\mathrm { d i v } ( \\partial ) \\omega _ S \\ , . \\end{align*}"}
-{"id": "7190.png", "formula": "\\begin{align*} \\frac 1 { | \\mathbf X ( n ) | } \\sum \\limits _ { \\mathbf x \\in \\mathbf X ( n ) } \\exp ( 2 \\pi i \\langle \\mathbf { h } , \\mathbf { x } \\rangle ) = \\mathbb E \\exp ( 2 \\pi i \\langle \\mathbf { h } , \\zeta _ n \\rangle ) . \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} { \\rm M } ( \\alpha ) \\geq \\bigl ( \\frac { 1 + \\sqrt { 5 } } { 2 } \\bigr ) ^ { d / 2 } , \\mbox { f r o m w h i c h } : { \\rm M } ( \\alpha ) \\geq ( ( 1 + \\sqrt { 5 } ) / 2 ) ^ { 1 / 2 } = 1 . 2 7 2 0 \\ldots \\end{align*}"}
-{"id": "6137.png", "formula": "\\begin{align*} S _ U ( t ) = V _ { t h } - ( 1 - e ^ { - \\frac { t } { \\theta } } ) \\hat { V } - e ^ { - \\frac { t } { \\theta } } V _ 0 - e ^ { - \\frac { t } { \\theta } } \\int _ 0 ^ t e ^ { \\frac { s } { \\theta } } I ( s ) d s \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} \\int _ M { \\rm d i v } _ \\nabla ( \\pi ( X ) ) \\mu _ M = \\int _ M g _ Q ( \\pi ( X ) , \\kappa ^ \\sharp ) \\mu _ M \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} \\bar { S } _ { \\Theta } = \\sum _ Y \\int _ { \\tilde { \\Delta } _ Y ^ + } ( n \\Lambda _ Y P _ { D H } ( q ) + d _ q P _ { D H } ( \\chi ^ { a c } - \\Lambda _ Y \\chi ) ) d q \\Big { / } \\int _ { \\Delta ^ + } P _ { D H } ( q ) d q \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} \\lim _ { i \\rightarrow \\infty } \\int _ Y \\langle K _ i - K _ \\infty , K _ { \\infty } \\rangle \\omega = 0 . \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{align*} \\frac { E _ { \\ell } } { Q - Q _ { \\ell - 1 } } \\leq \\sum _ { e = Q _ { \\ell - 1 } } ^ { Q _ { \\ell } - 1 } \\frac { 1 } { Q - e } . \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{align*} L _ { u _ 1 \\ldots u _ k } ( x ) = \\sum _ { p = 0 } ^ k L _ { u _ 1 \\ldots u _ p } ( ( x _ n ) _ { 0 \\leq n < M } ) L _ { u _ { p + 1 } \\ldots u _ k } ( ( x _ n ) _ { M \\leq n < N } ) \\end{align*}"}
-{"id": "7072.png", "formula": "\\begin{align*} N _ h : = \\left \\lfloor \\frac { \\log ( h ) } { \\log M } \\right \\rfloor + 2 . \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} b _ n = b _ { n - 1 } + u _ n + ( n - 3 ) 2 ^ { n - 4 } + \\binom { n - 2 } { 5 } - \\binom { n - 2 } { 2 } + \\sum _ { j = 0 } ^ { n - 4 } \\sum _ { i = 1 } ^ { n - 3 - j } \\binom { n - i - 2 } { j + 1 } C _ { n - 2 - j , i } , \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{align*} \\langle \\chi _ y , \\chi _ { y ' } \\rangle = \\sum _ { Z \\in \\mathcal { M } _ \\mu ( \\mathbb { F } _ q ) } \\chi \\left ( \\mathrm { t r } ( Y - Y ' ) Z ^ T \\right ) , \\end{align*}"}
-{"id": "7683.png", "formula": "\\begin{align*} ^ { M _ s } _ r = & \\log \\left ( 1 + \\rho \\alpha _ { M _ s } ^ 2 | h | ^ 2 r ^ { - \\alpha } \\right ) . \\end{align*}"}
-{"id": "897.png", "formula": "\\begin{align*} \\langle f , g \\rangle _ { k , l } : = \\int _ X \\langle f , g \\rangle _ z \\rho ( z ) ( \\frac i 2 ) d z \\wedge d \\bar z = \\int _ X f \\bar { g } \\rho ^ { 1 - k - l } d A . \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} \\frac { 1 } { | q _ 2 z - a | ^ k } = \\frac { 1 } { | q _ 2 z - a - 1 | ^ k } , \\end{align*}"}
-{"id": "6311.png", "formula": "\\begin{align*} \\mathcal { L } _ { I _ U } ( s ) & = \\exp \\Big ( - 2 \\pi \\sum _ { t \\in \\mathcal { K } } \\lambda _ { t } \\int _ { 0 } ^ { \\infty } \\frac { y } { 1 + \\frac { y ^ { \\alpha } } { s P _ U } } d y \\Big ) \\\\ & \\overset { ( a ) } { = } \\exp \\Big ( \\frac { 2 } { \\alpha } \\pi ( s P _ U ) ^ { 2 / \\alpha } \\mathcal { B } ( \\frac { 2 } { \\alpha } , 1 - \\frac { 2 } { \\alpha } ) \\sum _ { t \\in \\mathcal { K } } \\lambda _ t \\Big ) , \\end{align*}"}
-{"id": "7347.png", "formula": "\\begin{align*} ( X + \\lambda ^ { - 1 } \\Delta + \\lambda M ) ^ \\cdot = [ X + \\lambda ^ { - 1 } \\Delta + \\lambda M , B - \\lambda M ^ { k + 1 } ] \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{align*} & I ( X ; Y ) - I ( X ; Z ) = H ( X , Y | Z ) - H ( Y | X , Z ) - H ( X | Y ) \\geq H ( X , Y | Z ) - H ( Y | X ) - H ( X | Y ) . \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} | p _ { i , \\alpha } ( x ) | \\leq | f ( x ) | + C d ^ \\beta B \\leq \\left \\lceil ( 1 + C d ^ \\beta ) \\cdot B \\right \\rceil = : B _ 1 x \\in I _ { \\lambda _ i } \\ , . \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{align*} \\left ( S _ j ( a , z _ 0 ) \\right ) \\cap \\left ( U _ j ( a , z _ 0 ) \\right ) = \\{ a \\} . \\end{align*}"}
-{"id": "3433.png", "formula": "\\begin{align*} \\dot { \\theta } & = \\left ( \\frac { x } { 2 } - \\frac { m - 1 } { x } \\right ) \\sin \\theta + \\left ( \\frac { n - 1 } { y } - \\frac { y } { 2 } \\right ) \\cos \\theta + \\lambda \\\\ & \\leq \\frac { x } { 2 } - ( m - 1 ) \\left ( \\frac { \\dot { x } } { x } \\tan \\theta \\right ) \\\\ & \\leq \\frac { 1 } { 2 } + ( m - 1 ) \\left ( \\delta \\frac { \\dot { x } } { x } \\right ) , \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} { \\bf H } = \\dfrac 1 2 x ^ { \\perp } , \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{align*} b _ r = - ( t _ r + a _ r - \\sum _ { j = 1 } ^ { r - 1 } b _ j t _ { r - j } ) \\mbox { w i t h } b _ d = - ( t _ d + 1 - \\sum _ { j = 1 } ^ { d - 1 } b _ j t _ { r - j } ) , \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{align*} \\sin r _ j ( \\varphi ) = \\sqrt { \\frac { 2 j } { j + 1 } } \\ , \\sin \\varphi \\mbox { \\ a n d \\ } \\sin r _ \\infty ( \\varphi ) = \\sqrt { 2 } \\ , \\sin \\varphi . \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} k ^ { u } _ { j + 1 } ( T ) \\leq k ^ { u } _ { 0 } ( T ) + C \\Big ( \\sup _ { 0 < t < T } e ^ { - \\frac { \\omega t } { 2 } } t ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\int _ { 0 } ^ { t } ( t - s ) ^ { - \\frac { 1 } { 2 } - \\frac { 3 } { 2 q } } s ^ { - 3 ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\ \\d s \\Big ) [ k _ { j } ^ { u } ( T ) ^ 2 + k _ { j } ^ { \\nabla y } ( T ) ^ 2 ] . \\end{align*}"}
-{"id": "6069.png", "formula": "\\begin{align*} \\lambda _ n ^ + ( H ( b ) ) = \\frac { \\varkappa ( \\alpha ) } { n ^ \\alpha } + o ( n ^ { - \\alpha } ) , \\lambda _ n ^ - ( H ( b ) ) = O ( n ^ { - \\alpha - 1 } ) , n \\to \\infty , \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} J _ { \\infty } ( X ^ n ; Y ^ n ) & \\le J _ { \\infty } ( p ( y ^ n | x ^ n ) , \\mathcal { X } ^ n ) = n \\times J _ { \\infty } ( p ( y | x ) , \\mathcal { X } ) . \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{align*} \\Sigma ^ { 0 } = \\frac { \\sigma ^ 2 } { \\rho _ { u l } } \\vect { I } + \\frac { N } { L } \\mathbb { E } \\{ \\vect { x } _ { \\beta } \\vect { x } _ { \\beta } ^ H \\} \\end{align*}"}
-{"id": "3242.png", "formula": "\\begin{align*} \\langle f , \\overline { u } \\rangle _ { L ^ 2 } = 0 \\end{align*}"}
-{"id": "4799.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { t \\rightarrow 0 } \\frac { \\Phi ( | \\nabla u + t \\nabla v | ) - \\Phi ( | \\nabla u | ) } { t } = \\phi ( | \\nabla u | ) \\nabla u \\nabla v ~ \\ , \\ , \\mbox { a . e i n } x \\ , \\in \\ , \\Omega . \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} f _ { \\Gamma } ( \\alpha , z ) = \\begin{cases} z ^ { a } & \\\\ 1 & \\end{cases} \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{align*} w ( M ^ { d n } ) = { j ^ * } \\prod _ { i = 1 } ^ { m } ( 1 + v _ i ) \\ m o d \\ 2 , \\ a n d \\ \\ p ( M ^ { d n } ) = { j ^ * } \\prod _ { i = 1 } ^ { m } ( 1 - v _ i ^ 2 ) . \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} \\widetilde { L } ( t _ n ) = \\min \\{ y _ m \\ge y _ M : \\ \\widetilde { \\sigma } ( y _ m ) \\ge t _ n \\} . \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} ( s \\partial ) ^ \\bullet - \\lambda _ s \\circ \\partial ^ \\bullet = h \\circ d + d \\circ h \\end{align*}"}
-{"id": "8363.png", "formula": "\\begin{align*} \\lambda _ i = w ( f _ r ) - r - f _ i + i , \\ \\ \\ \\mu _ i = w ( f _ r ) - r - w ( f _ i ) + i , \\ \\ \\ \\ \\ i = 1 , \\dots , r . \\end{align*}"}
-{"id": "3484.png", "formula": "\\begin{align*} f ( x ) = \\max _ { i \\in [ m ] } ( a _ i ^ T x + b _ i ) \\end{align*}"}
-{"id": "5454.png", "formula": "\\begin{align*} \\langle w \\lambda - q , \\check { \\varpi _ i } \\rangle = 0 \\textrm { o r } \\langle w \\lambda + q , \\check { \\varpi _ i } \\rangle = 0 \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} g _ t ( x ) = 2 ^ { t } \\cdot \\left ( \\varrho ( x ) + \\sum _ { k = 1 } ^ { 2 ^ { t - 1 } - 1 } 2 \\cdot \\varrho \\left ( x - \\frac { 2 k } { 2 ^ { t } } \\right ) - \\sum _ { \\ell = 1 } ^ { 2 ^ { t - 1 } } 2 \\cdot \\varrho \\left ( x - \\frac { 2 \\ell - 1 } { 2 ^ t } \\right ) \\right ) x \\in [ 0 , 1 ] \\ , . \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} \\begin{array} { l } \\beta '' \\pm ( n - 1 ) a = \\pm \\bar { \\lambda } \\\\ \\noalign { \\smallskip } ( h ' ) ^ 2 \\bar { \\lambda } = - a h h ' \\beta ' \\mp ( n - 2 ) a ^ 2 h ^ 2 + a ( h ' ) ^ 2 + \\bar { c } ( n - 2 ) . \\end{array} \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} { \\cal C } _ 2 ( \\underline { p } ) : = \\left \\{ ( \\lambda _ 1 , \\ldots , \\lambda _ M ) \\in ( 0 , 1 ) ^ { M } : \\lambda _ i < p _ i \\prod _ { j \\neq i } ( 1 - p _ j ) 1 \\leq i \\leq M \\right \\} \\end{align*}"}
-{"id": "4891.png", "formula": "\\begin{align*} X ^ 0 ( 1 ) & = W ( 1 ) + 1 , \\\\ X ^ { y _ n } ( 1 ) & = W ( 1 ) - 1 - \\frac { 1 } { n } . \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\delta _ { j _ 1 j _ 2 } G _ { i _ 1 i _ 2 i _ 3 } ^ { j _ 1 j _ 2 \\eta } \\delta ^ { i _ 1 i _ 2 } \\tilde \\gamma _ { \\eta k } + \\delta ^ { i _ 1 i _ 2 } G _ { i _ 1 i _ 2 i _ 3 } ^ { j _ 1 j _ 2 j _ 3 } \\gamma _ { j _ 1 j _ 2 } \\delta _ { j _ 3 k } \\\\ = & - \\frac { 1 } { 2 } \\int _ 0 ^ \\infty \\frac { d } { d y } \\left [ \\sum _ { \\alpha , \\eta } ( e ^ { - y \\tilde \\gamma } ) _ { \\alpha \\eta } ( e ^ { - y \\tilde \\gamma } ) _ { \\alpha \\eta } ( e ^ { - y \\tilde \\gamma } ) _ { i _ 3 k } \\right ] d y = \\frac { n } { 2 } \\delta _ { i _ 3 k } , \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k x _ { [ i ] } \\leq \\sum _ { i = 1 } ^ k y _ { [ i ] } k = 1 , \\dots , n . \\end{align*}"}
-{"id": "495.png", "formula": "\\begin{align*} \\big | e ^ { 2 i \\theta k } + ( - 1 ) ^ k \\chi _ 2 ( - 1 ) \\big | = 0 . \\end{align*}"}
-{"id": "93.png", "formula": "\\begin{align*} x \\mapsto h _ { \\mu } ( x ) : = - \\sum _ { i \\in \\mathcal { N } } \\mu _ i x _ i . \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} K _ { D , \\lambda } : D \\times D & \\rightarrow \\C \\\\ K _ { D , \\lambda } ( z , w ) & = h ( z , w ) ^ { - \\lambda } . \\end{align*}"}
-{"id": "5089.png", "formula": "\\begin{align*} q - u ( f ) - 1 = \\deg ( g ) , \\end{align*}"}
-{"id": "8187.png", "formula": "\\begin{align*} \\nabla _ { s , g _ { \\rm h y p } } \\textit { \\textbf { V } } = \\textit { \\textbf { e } } \\ { \\rm o n } \\ \\mathbb B ^ 2 , \\end{align*}"}
-{"id": "3575.png", "formula": "\\begin{align*} w ' ( s ) ^ { 2 } + z ' ( s ) ^ { 2 } & = 1 \\\\ w ' z '' - z ' w '' & = C \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} \\widehat { \\sigma } = t _ i ^ { - 1 } \\log ^ { \\frac 1 2 } ( t _ i ) ( \\sigma + Q ( { \\frak t } ( t _ i ) ) \\delta ) \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} \\alpha \\cdot ( v \\otimes n ) = - v \\alpha \\otimes n + v \\otimes \\alpha \\cdot n \\ , . \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{align*} P _ { f } ( x ) : = \\left \\{ \\begin{array} { l l } x - \\frac { f ( x ) } { \\Vert g _ { f } ( x ) \\Vert ^ { 2 } } g _ { f } ( x ) & f ( x ) > 0 \\\\ x & \\end{array} \\right . \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{align*} | f ( x ) | ^ 2 - | f ( y ) | ^ 2 = \\int _ { y } ^ { x } \\bigl ( | f | ^ 2 \\bigr ) ' ( t ) \\ , \\mathrm d t \\leq 2 \\int _ { y } ^ x | f ( t ) f ' ( t ) | \\ , \\mathrm d t \\leq 2 \\| f \\| _ { L ^ 2 } \\| f ' \\| _ { L ^ 2 } \\end{align*}"}
-{"id": "8014.png", "formula": "\\begin{align*} d ( e ^ { c _ 1 \\tilde { \\gamma } t } \\overline { V } _ t ) = e ^ { c _ 1 \\tilde { \\gamma } t } d \\overline { V } _ t + c _ 1 \\tilde { \\gamma } e ^ { c _ 1 \\tilde { \\gamma } t } \\overline { V } _ t d t \\le c _ 2 \\tilde { \\gamma } e ^ { c _ 1 \\tilde { \\gamma } t } d t + \\frac { \\tau } { N } \\tilde { \\gamma } e ^ { c _ 1 \\tilde { \\gamma } t } \\sum _ { i = 1 } ^ N { d B _ { i , t } } ^ T e _ { i , t } . \\end{align*}"}
-{"id": "7022.png", "formula": "\\begin{align*} A \\oplus B = \\{ ( \\lambda ; \\mu ) | \\lambda \\in A , \\mu \\in B \\} . \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} \\underset { y \\to 0 } { \\lim } \\left ( y ^ a < \\nabla U , - e _ { n + 1 } > \\right ) = - \\underset { y \\to 0 } { \\lim } y ^ a U _ y = V u , \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} N _ { S } ( ( \\ell + \\omega \\times y ) \\cdot n ) = \\sum _ { i = 1 } ^ { 3 } \\ell _ { i } \\pi ^ { i } + \\sum _ { i = 4 } ^ { 6 } \\omega _ { i - 3 } \\pi ^ { i } . \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} g ( a , n ) & = \\int _ 0 ^ 1 p _ { a , n } ( t ) \\ , d t \\ , , \\\\ q _ { a , n } & = a ( 1 + t ) ^ { n - a } ( 1 + t ^ a ) \\ , \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} p ( i , j | v , w ) = \\langle ( P _ { v , i } \\otimes Q _ { w , j } ) h , h \\rangle . \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{align*} \\int \\limits _ { b ( v ) } ^ 1 \\frac { b } { r b ^ { 1 / r } \\sqrt { 1 - b ^ 2 } - D ^ { - 1 } O ( \\sqrt { 1 - b ^ 2 } ) } \\ , d b \\\\ = \\int \\limits _ { b ( v ) } ^ 1 \\left ( \\frac { 1 } { r b ^ { 1 / r } - D ^ { - 1 } O ( 1 ) } \\right ) \\left ( \\frac { b } { \\sqrt { 1 - b ^ 2 } } \\right ) \\ , d b \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} \\Delta \\left ( ( \\mathbf { 1 } ) ^ n \\right ) = \\sum _ { k = 0 } ^ n ( \\mathbf { 1 } ) ^ k \\otimes ( \\mathbf { 1 } ) ^ { n - k } \\end{align*}"}
-{"id": "337.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { c ^ 2 ( i - 1 ) } { 2 ( \\sqrt { 2 } + \\sqrt { k - 1 } ) ^ 2 } + \\frac { c ^ 2 } { 4 ( \\sqrt { 2 } + \\sqrt { k - 1 } ) ^ 2 } & < \\frac { 2 i - 1 } { ( \\sqrt { 2 } + \\sqrt { k - 1 } ) ^ 2 } \\\\ & = \\left ( \\frac { \\sqrt { i - \\frac { 1 } { 2 } } } { \\sqrt { 2 } + \\sqrt { k - 1 } } \\right ) ^ 2 \\\\ & < \\frac { 1 } { 2 } \\left ( \\sqrt { \\frac { i - \\frac { 1 } { 2 } } { k } } \\right ) ^ 2 \\\\ & \\leq \\frac { 1 } { 2 } \\end{aligned} \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} \\frac { 1 } { x _ A } + \\frac { 2 } { 1 + x _ A } = \\frac { 2 } { 1 - x _ A } \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} & E \\left [ S _ { s , t } ^ { a n o m } \\right ] \\\\ = & \\int _ s ^ t E \\left [ \\left ( \\beta ^ { - 3 } \\nabla _ q \\beta \\cdot \\left ( \\frac { 3 n + 2 } { 6 } - \\int _ 0 ^ \\infty T r [ \\gamma e ^ { - 2 y \\gamma } ] e ^ { - y \\gamma } d y \\right ) \\gamma ^ { - 1 } \\nabla _ q \\beta \\right ) ( r , q _ r ) \\right ] d r . \\end{align*}"}
-{"id": "1309.png", "formula": "\\begin{align*} s \\bigl ( ( I - P ^ j ) \\lambda _ H ^ { j } , ( I - P ^ j ) \\mu _ H \\bigr ) = ( \\rho g , T ( I - P ^ j ) \\mu _ H ) \\quad \\mu _ H \\in \\Lambda _ H , \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} \\| u ' \\| _ J K _ L e _ 2 = K _ L c _ 1 = - c _ 2 + ( c _ 1 ) _ x = - \\begin{pmatrix} a \\\\ b \\\\ c \\\\ d \\end{pmatrix} + \\begin{pmatrix} 0 \\\\ ( \\| u ' \\| _ J ) _ x \\\\ 0 \\\\ d \\end{pmatrix} = - \\begin{pmatrix} a \\\\ 0 \\\\ 0 \\\\ d \\end{pmatrix} \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} \\psi \\left ( W ^ 1 _ { n } ( p , p _ 1 , \\dots , p _ { n - 1 } ) \\right ) & = \\\\ \\sum _ { \\{ p _ 1 , \\dots , p _ { n - 1 } \\} } \\sum _ { i = 0 } ^ { n - 1 } & _ { i } \\leftrightarrow _ { i + 1 } \\left ( \\mathbf { ( 1 ) } \\ast \\mathbf { ( 1 ) } \\ast \\dots \\ast \\mathbf { ( 1 ) } \\right ) \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} \\widetilde { H } ^ { ( n ) } ( p ^ { \\mathbf { k } } ; p ^ { \\boldsymbol { \\ell } } ) = \\sum _ { P \\colon \\mathbf { k } ( P ) = \\mathbf { k } } \\widetilde { G } ( P ) , \\end{align*}"}
-{"id": "6798.png", "formula": "\\begin{align*} e _ k ( \\psi ) = s ^ 2 \\langle D ^ k \\nabla _ t \\psi , e _ 0 \\cdot D ^ k \\nabla _ t \\psi \\rangle + \\langle D ^ { k + 1 } \\psi , e _ 0 \\cdot D ^ { k + 1 } \\psi \\rangle , \\end{align*}"}
-{"id": "4190.png", "formula": "\\begin{align*} \\mathcal { H } ^ { m - n } \\big ( \\varphi _ { x _ i } ( \\{ z \\} \\times [ - r _ { x _ i } , r _ { x _ i } ] ^ { m - n } ) \\big ) \\leq C _ 3 ^ { m - n } \\cdot \\mathcal { H } ^ { m - n } \\big ( \\{ z \\} \\times [ - r _ { x _ i } , r _ { x _ i } ] ^ { m - n } \\big ) = ( 2 \\cdot C _ 3 \\cdot r _ { x _ i } ) ^ { m - n } \\ , . \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} a _ k \\theta _ k ( \\zeta ) + \\cdots + a _ 1 \\theta _ 1 ( \\zeta ) + a _ 0 \\zeta = b ( \\xi ) \\end{align*}"}
-{"id": "3346.png", "formula": "\\begin{align*} _ { 5 k } ( \\widetilde { P } _ i ) = \\frac { 1 } { 5 k } ( \\widetilde { P } _ i ) = \\frac { 1 } { 5 k } \\sum _ { j = 1 } ^ 5 ( P _ j ) = \\frac { 1 } { 5 k } \\left ( \\sum _ { j = 1 } ^ 5 P _ j \\right ) = \\frac { 1 } { 5 k } ( 5 t k ) = t . \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} \\lambda _ 1 ( x ) & = - x _ 2 - \\frac { 1 } { 2 } \\ , x _ 3 + 1 , & \\lambda _ 2 ( x ) & = x _ 2 - \\frac { 1 } { 2 } \\ , x _ 3 , \\\\ \\lambda _ 3 ( x ) & = \\frac { 1 } { 2 } \\left ( - 2 x _ 1 + x _ 3 + 1 \\right ) , & \\lambda _ 4 ( x ) & = \\frac { 1 } { 2 } \\left ( 2 x _ 1 + x _ 3 - 1 \\right ) . \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} H ( \\nu , \\mathcal { E } ) & = \\sum _ { F \\in \\mathcal { F } } \\sum _ { E \\in \\mathcal { E } , E \\subset F } \\phi ( \\nu ( F ) \\nu _ F ( E ) ) \\\\ & = \\left ( \\sum _ { F \\in \\mathcal { F } } \\nu ( F ) \\sum _ { E \\in \\mathcal { E } } \\phi ( \\nu _ F ( E ) ) \\right ) + \\left ( \\sum _ { F \\in \\mathcal { F } } \\phi ( \\nu ( F ) ) \\sum _ { E \\in \\mathcal { E } } \\nu _ F ( E ) \\right ) \\\\ & = H ( \\nu , \\mathcal { E } | \\mathcal { F } ) + H ( \\nu , \\mathcal { F } ) . \\end{align*}"}
-{"id": "6113.png", "formula": "\\begin{align*} \\rho _ { G , W } ( t ) = \\kappa r _ G ( t ) , & & \\varphi _ { G , W } ( t ) = \\frac { v _ G ( t ) } { \\sqrt { \\kappa } } \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} \\overline { \\sigma } _ j = \\sigma _ j | _ { \\pi ( \\Delta _ j ) } \\hbox { a n d } \\overline { \\rho } _ j = \\rho _ j | _ { \\pi ( \\Delta _ j ) } , \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} = 1 + H _ m ( P _ { V _ { m } ^ \\perp } ( ( P _ 1 \\mu ) ^ { x _ 1 , k + i } \\times ( P _ 2 ( \\mu _ { [ x ] } ) ) ^ { x _ 2 , k + i } ) - \\epsilon . \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} \\sum _ { e \\mid d p ^ i } e n _ e = p ^ { c _ { \\lambda _ i } } . \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} W ^ 0 _ 3 ( p , p _ 1 , p _ 2 ) \\ast W ^ 0 _ 3 ( p , p _ 2 , p _ 3 ) = W ^ 0 _ 4 ( p , p _ 1 , p _ 2 , p _ 3 ) . \\end{align*}"}
-{"id": "5062.png", "formula": "\\begin{align*} & [ s , x _ 1 ] [ x _ 2 , x _ 3 , x _ 4 ] = [ s , x _ 1 ] [ x _ 2 , x _ 3 ] x _ 4 - [ s , x _ 1 ] x _ 4 [ x _ 2 , x _ 3 ] \\\\ \\equiv \\ & [ s , x _ 1 ] [ x _ 2 , x _ 3 ] x _ 4 - x _ 4 [ s , x _ 1 ] [ x _ 2 , x _ 3 ] \\pmod { I ' } \\equiv - [ s , x _ 2 ] [ x _ 1 , x _ 3 ] x _ 4 + x _ 4 [ s , x _ 2 ] [ x _ 1 , x _ 3 ] \\pmod { I ' } \\\\ \\equiv \\ & - [ s , x _ 2 ] [ x _ 1 , x _ 3 ] x _ 4 + [ s , x _ 2 ] x _ 4 [ x _ 1 , x _ 3 ] \\pmod { I ' } = - [ s , x _ 2 ] [ x _ 1 , x _ 3 , x _ 4 ] , \\end{align*}"}
-{"id": "2171.png", "formula": "\\begin{align*} { D ^ + } f ( x ; y ) & : = \\limsup _ { t \\searrow 0 } \\frac { f ( x + t y ) - f ( x ) } { t } \\\\ \\shortintertext { a n d } { D _ + } f ( x ; y ) & : = \\liminf _ { t \\searrow 0 } \\frac { f ( x + t y ) - f ( x ) } { t } , \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{align*} \\int _ { \\mathbf { B } } \\frac { ( 1 - | y | ^ { 2 } ) ^ { n + \\alpha - 1 } h _ { j } ^ { p } ( y ) } { [ x , y ] ^ { n + | k | + \\alpha - 1 } } & d \\tau ( y ) \\\\ & \\leq \\int _ { \\mathbf { B } } \\frac { ( 1 - | y | ^ { 2 } ) ^ { p c + \\alpha - 1 } } { [ x , y ] ^ { n + | k | + \\alpha - 1 } } d v ( y ) \\\\ & = I _ { p c + \\alpha , - p c + | k | } ( x ) \\\\ & \\leq C ( p c + \\alpha , - p c + | k | ) \\\\ & \\leq \\left ( \\frac { j } { j - 1 } \\right ) ^ { p c } C ( p c + \\alpha , - p c + | k | ) h _ { j } ^ { p } ( x ) , \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } \\frac { 1 } { l _ 1 t } - \\frac { 1 } { \\rho ( t ) } = \\frac { l _ 2 } { l _ 1 ^ 2 } . \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} s ( n , 1 ) \\ , = \\ , 2 ^ { ( 2 ^ { n - 1 } - 1 ) ( 2 ^ { n - 2 } - 1 ) / 3 } \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} h _ i \\ = \\ u _ 1 ^ { n + 1 - i } v _ 1 ^ i + u _ 2 ^ { n + 1 - i } v _ 2 ^ i \\quad \\ 0 \\le i \\le n + 1 . \\end{align*}"}
-{"id": "1683.png", "formula": "\\begin{align*} | \\mathrm { F o r b } _ L ( n , C ^ r _ { \\ell } ) | = 2 ^ { \\Theta \\left ( n ^ { 1 + \\frac { 1 } { \\lfloor \\ell / 2 \\rfloor } } \\right ) } \\end{align*}"}
-{"id": "6676.png", "formula": "\\begin{align*} o _ { n } ( 1 ) = I ' ( u _ { n } ) [ w ] = m _ { k } ( \\| u _ { n } \\| ^ { 2 } ) \\int _ { \\Omega } \\nabla u _ { n } \\nabla w d x - \\int _ { \\Omega } f _ { \\ast } ( u _ { n } ) w d x , \\ \\forall w \\in H _ { 0 } ^ { 1 } ( \\Omega ) . \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} d q _ t = & \\tilde \\gamma ^ { - 1 } ( t , q _ t ) \\left ( - \\partial _ t \\psi ( t , q _ t ) - \\nabla _ q V ( t , q _ t ) + \\tilde F ( t , q _ t , \\psi ( t , q _ t ) ) \\right ) d t \\\\ & + \\tilde S ( t , q _ t ) d t + \\tilde \\gamma ^ { - 1 } ( t , q _ t ) \\sigma ( t , q _ t ) \\circ d W _ t , \\end{align*}"}
-{"id": "9122.png", "formula": "\\begin{align*} { \\left [ { \\bf { A } } \\right ] _ { 1 1 } } = { \\left [ { { { \\bf { F } } _ 2 } } \\right ] _ { 1 1 } } , { \\left [ { \\bf { A } } \\right ] _ { k k } } = { \\left [ { { { \\bf { F } } _ 2 } } \\right ] _ { k k } } - \\frac { { \\left [ { { \\bf { G } } } \\right ] _ { { { 1 k } } } } } { \\omega } { \\left [ { { { \\bf { F } } _ 2 } } \\right ] _ { k 1 } } , k \\ge 2 , \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} N [ \\mathfrak { e } , V ] \\ \\geq \\ \\mathcal { L } _ { V } [ c ] \\ = \\ \\# \\big \\{ x \\in \\Gamma \\ , | \\ V ( x ) \\geq c \\big \\} . \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{align*} p _ 1 ( y _ 1 , y _ 2 , y _ 3 ) & = ( \\ , y _ 1 , \\ , ( y _ 2 + y _ 3 ) / 2 , \\ , ( y _ 2 + y _ 3 ) / 2 \\ , ) , \\\\ p _ 2 ( y _ 1 , y _ 2 , y _ 3 ) & = ( \\ , ( y _ 3 + y _ 1 ) / 2 , \\ , y _ 2 , \\ , ( y _ 3 + y _ 1 ) / 2 \\ , ) , \\\\ p _ 3 ( y _ 1 , y _ 2 , y _ 3 ) & = ( \\ , ( y _ 1 + y _ 2 ) / 2 , \\ , ( y _ 1 + y _ 2 ) / 2 , \\ , y _ 3 \\ , ) . \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } X _ 1 ^ \\star A _ 1 - B _ 1 X _ 2 + C _ 1 = 0 , \\\\ X _ 2 A _ 2 + B _ 2 X _ 3 + C _ 2 = 0 , \\\\ - X _ 3 A _ 3 + B _ 3 X _ 1 + C _ 3 = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} t _ 1 = \\lfloor \\beta \\rfloor , t _ 2 = \\lfloor \\beta \\{ \\beta \\} \\rfloor = \\lfloor \\beta T _ { \\beta } ( 1 ) \\rfloor , t _ 3 = \\lfloor \\beta \\{ \\beta \\{ \\beta \\} \\} \\rfloor = \\lfloor \\beta T _ { \\beta } ^ { 2 } ( 1 ) \\rfloor , \\ldots \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} \\mathrm { P } _ { t , i } & = e ^ { - \\lambda _ c \\pi \\left ( \\frac { \\epsilon _ 1 } { \\rho } + \\frac { ( 1 + \\epsilon _ 1 ) } { \\rho \\phi _ i } \\right ) ^ { - \\frac { 2 } { \\alpha } } } \\sum ^ { t - 1 } _ { k = 0 } \\frac { ( \\lambda _ c \\pi ) ^ { k } \\left ( \\frac { \\epsilon _ 1 } { \\rho } + \\frac { ( 1 + \\epsilon _ 1 ) } { \\rho \\phi _ i } \\right ) ^ { - \\frac { 2 k } { \\alpha } } } { k ! } . \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} M ( \\Gamma + f , d z ) = e ^ { \\gamma f ( z ) } M ( \\Gamma , d z ) . \\end{align*}"}
-{"id": "1938.png", "formula": "\\begin{align*} \\sigma ( M _ 1 , \\dots , M _ r ) = \\mathrm { I d } , \\textrm { m o d } p . \\end{align*}"}
-{"id": "1296.png", "formula": "\\begin{align*} \\beta ^ k P _ { r } ^ { * } ( z _ 0 ) - \\bar { \\beta } ^ k Q _ r ^ { * } ( z _ 0 ) = \\beta ^ { k - 2 r + 1 } \\lambda ^ { 2 r + 1 } E _ r ^ { * } ( z _ 0 ) , \\end{align*}"}
-{"id": "3845.png", "formula": "\\begin{align*} \\widehat { L } _ g ( z ) - L _ g ( z ) = \\omega ( \\pi ( z ) ) + \\omega ( \\pi ( S _ g ( z ) ) ) . \\end{align*}"}
-{"id": "7056.png", "formula": "\\begin{align*} & m \\in \\{ 0 , 1 , \\cdots , n - 1 \\} , \\\\ & 1 = s _ 1 < s _ 2 < \\cdots < s _ { m + 1 } \\le n , 1 = t _ 1 < t _ 2 < \\cdots < t _ { n - m } \\le n , \\\\ & \\{ s _ j \\} _ { j = 2 } ^ { m + 1 } \\cup \\{ t _ k \\} _ { k = 2 } ^ { n - m } = \\{ 2 , 3 , \\cdots , n \\} , \\{ s _ j \\} _ { j = 2 } ^ { m + 1 } \\cap \\{ t _ k \\} _ { k = 2 } ^ { n - m } = \\emptyset . \\end{align*}"}
-{"id": "6454.png", "formula": "\\begin{align*} \\nabla \\frac { 1 } { \\lvert \\Omega \\rvert } \\int _ { \\Omega } \\widetilde { a } _ { d , n } \\ ; \\d x = 0 . \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} Z _ n ( x , \\omega ) : = ( f _ { \\omega _ n } \\circ \\cdots \\circ f _ { \\omega _ 1 } ) ( x ) , Z _ 0 ( x , \\omega ) = x . \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{align*} \\sigma ''' ( i , j , k , \\alpha , \\beta ) : = [ T _ { i k } ^ \\alpha , T _ { j k } ^ \\beta ] + \\sum _ { \\stackrel { \\gamma , \\delta \\geq 0 | } { \\stackrel { \\gamma + \\delta = \\alpha + \\beta } { } } } ( - 1 ) ^ { \\beta + 1 } \\begin{pmatrix} \\alpha \\\\ \\delta \\end{pmatrix} [ T _ { i j } ^ \\gamma , T _ { i k } ^ \\delta ] \\quad i < j < k \\in [ n ] , \\quad \\alpha , \\beta \\geq 0 , \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} a e ^ { - c x ^ * } + b e ^ { c x ^ * } & = x ^ * , \\\\ a e ^ { c x ^ * } + b e ^ { - c x ^ * } & = 0 . \\end{align*}"}
-{"id": "5594.png", "formula": "\\begin{align*} T = \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & g & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & \\psi & \\theta \\\\ 0 & 1 & - \\psi ^ t \\\\ 0 & 0 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "1321.png", "formula": "\\begin{align*} H _ n ^ * = \\{ ( x _ 1 , \\dots , x _ n , 1 ) \\} \\subset H _ { n + 1 } . \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} ( \\alpha \\cdot \\varphi ) ( s ) = ( \\alpha \\otimes 1 - 1 \\otimes \\alpha ) \\cdot \\varphi ( s ) - \\varphi ( \\alpha ( s ) ) \\ , . \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{align*} \\mathrm { d i m } _ \\Q \\ , H ^ d ( G _ k ( \\R ^ n ) , \\Q ) = p \\Big ( \\big [ \\frac { k } { 2 } \\big ] , \\big [ \\frac { n - k } { 2 } \\big ] ; \\frac { d } { 4 } \\Big ) + p \\Big ( \\big [ \\frac { k } { 2 } \\big ] , \\big [ \\frac { n - k } { 2 } \\big ] ; \\frac { d - n + 1 } { 4 } \\Big ) \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} \\zeta _ 1 & = \\frac { \\alpha _ 0 } { r } - \\frac { 3 T + 1 } { 2 r } + \\frac { 3 t - x } { 2 r } , \\\\ \\zeta _ 2 & = \\frac { \\beta _ 0 } { r } - \\frac { 3 T + 1 } { 2 r } + \\frac { t + x } { r } , \\zeta _ 3 = \\zeta _ 2 - \\frac { 1 } { r } , \\\\ \\zeta _ 4 & = \\frac { \\alpha _ 0 - \\beta _ 0 } { r } - \\frac { t + x } { 2 r } , \\zeta _ 5 = \\zeta _ 4 + \\frac { 1 } { r } , \\end{align*}"}
-{"id": "5962.png", "formula": "\\begin{align*} g ' ( 0 ) = \\sum _ { i , j } \\frac { \\partial F \\bigl ( R ^ { ( 0 ) } \\bigl ) } { \\partial R _ { i j } } \\Bigl ( R ^ { ( 1 ) } _ { i j } - R ^ { ( 0 ) } _ { i j } \\Bigl ) . \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} \\tilde { Q } _ t ( A _ t , \\dot { \\varphi } _ 0 ) = T _ t ( \\dot { \\varphi } _ y ) + \\tilde { Q } ^ y _ t \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} x \\otimes t ^ l . u \\otimes t ^ k = \\begin{cases} x . u \\otimes t ^ { k - l } , & ~ k \\geq l ; \\\\ 0 , & \\\\ \\end{cases} \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} \\eta _ { a + b } ( x ) = \\eta _ f [ 2 [ a ] - 2 [ 0 ] , y + b ] ( x + b ) \\cdot \\eta _ b ( x ) \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} \\Z \\times \\Z = \\langle g _ 1 \\rangle \\times \\langle g _ 2 \\rangle \\subset \\pi _ 1 ( M ) . \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} A _ \\kappa u ( y ) = \\begin{cases} [ A ( u \\circ \\kappa ) ] ( \\kappa ^ { - 1 } ( y ) ) & y \\in V , \\\\ 0 & y \\notin V . \\end{cases} \\end{align*}"}
-{"id": "7082.png", "formula": "\\begin{align*} \\frac { \\partial z } { \\partial y } ( x , y ) = - \\frac { \\frac { \\partial f } { \\partial y } ( x , y , z ( x , y ) ) } { \\frac { \\partial f } { \\partial z } ( x , y , z ( x , y ) ) } < 0 , ( \\forall ( x , y ) \\in \\R _ { > 0 } \\times ( - 1 , 1 ) ) . \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} \\phi ( 0 ) = 0 , \\phi ( 1 ) = 1 , \\phi ( \\alpha ) = 1 , \\phi ( \\alpha + 1 ) = 0 . \\end{align*}"}
-{"id": "1368.png", "formula": "\\begin{align*} J \\bigr [ u \\bigl ] = \\mathcal { E } ^ { G } \\bigl [ \\xi _ { t , T } \\bigl ( u \\bigr ) \\bigl \\vert \\mathcal { F } _ t \\bigr ] . \\end{align*}"}
-{"id": "1605.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\sup _ { u \\in [ a _ t - \\delta _ t , a _ t + \\delta _ t ] } \\left | \\frac { G ( u ) } { G ( a _ t ) } - 1 \\right | = 0 . \\end{align*}"}
-{"id": "7551.png", "formula": "\\begin{align*} \\tilde G ( t , q ) = \\int h ( t , q , z ) G ( t , q , z ) d z = n g _ 1 ( t , q ) \\beta ^ { - 1 } ( t , q ) . \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} A _ { 1 1 } ^ T \\Sigma _ 1 ^ { - 1 } + \\Sigma _ 1 ^ { - 1 } A _ { 1 1 } + \\sum _ { i = 1 } ^ m N _ { i , 1 1 } ^ T \\Sigma _ 1 ^ { - 1 } N _ { i , 1 1 } + \\Sigma _ 1 ^ { - 1 } k ^ 2 & \\leq - \\Sigma _ 1 ^ { - 1 } B _ 1 B _ 1 ^ T \\Sigma _ 1 ^ { - 1 } , \\\\ A _ { 1 1 } ^ T \\Sigma _ 1 + \\Sigma _ 1 A _ { 1 1 } + \\sum _ { i = 1 } ^ m N _ { i , 1 1 } ^ T \\Sigma _ 1 N _ { i , 1 1 } + \\Sigma _ 1 k ^ 2 & \\leq - C _ 1 ^ T C _ 1 . \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} \\hat { \\mathbf { y } } = \\tilde { \\mathbf { y } } + \\mathbf { n } _ q , \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} \\rho _ { G , U } ( t ) = r _ U ^ { - 1 } ( r _ G ( t ) ) = \\frac { \\theta } { 2 } \\ln \\left ( 1 + \\frac { 2 } { \\theta } r _ G ( t ) \\right ) \\end{align*}"}
-{"id": "5333.png", "formula": "\\begin{align*} P _ { [ K ] } ( w ) \\ , \\triangleq \\ , \\displaystyle { \\sum _ { i = 1 } ^ m } \\ , | \\ , w _ i \\ , | - \\displaystyle { \\sum _ { k = 1 } ^ K } \\ , | \\ , w _ { [ k ] } \\ , | , \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} \\sigma _ { \\mathrm { e s s } } [ H ( \\mathfrak { e } , V ) ] \\ = \\ \\sigma _ { \\mathrm { e s s } } [ H ( \\mathfrak { e } , 0 ) ] \\ = \\ [ 0 , { \\mathfrak { e } _ { \\mathrm { m a x } } } ] , \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{align*} \\mathfrak { N } _ { s } & = - \\psi T \\\\ T _ { s } & = \\big [ \\psi \\mathfrak { N } \\big ] = \\frac { 1 } { 2 } \\Big ( \\bar { \\psi } \\mathfrak { N } + \\psi \\bar { \\mathfrak { N } } \\Big ) \\\\ T _ { t } & = \\big [ i \\psi ' \\bar { \\mathfrak { N } } \\big ] = \\frac { 1 } { 2 } i \\big ( \\psi ' \\bar { \\mathfrak { N } } - \\bar { \\psi } ' \\mathfrak { N } \\big ) . \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} W _ n ( 1 ) = ( - 1 ) ^ k W _ { n - 1 } . \\end{align*}"}
-{"id": "6252.png", "formula": "\\begin{align*} F ( \\lambda ) = F ( \\lambda ' ) \\end{align*}"}
-{"id": "6236.png", "formula": "\\begin{align*} \\sum _ { t \\in T _ \\nu } Y _ { s , t } Z _ { m , t } = Y _ { s , m } , & & \\sum _ { t \\in T _ \\nu } Y ' _ { s , t } Z _ { m , t } = Y ' _ { s , m } \\end{align*}"}
-{"id": "7663.png", "formula": "\\begin{align*} f _ { r _ m , r _ t } ( x , y ) = & 4 y ( \\lambda _ c \\pi ) ^ { t } e ^ { - \\lambda _ c \\pi y ^ 2 } \\frac { x ^ { 2 m - 1 } ( y ^ 2 - x ^ 2 ) ^ { t - m - 1 } } { ( t - m - 1 ) ! ( m - 1 ) ! } . \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{align*} g = \\begin{pmatrix} 1 & - a v & - a ^ { - 1 } u & 0 \\\\ 0 & a & 0 & u \\\\ 0 & 0 & a ^ { - 1 } & v \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} p _ A ( i | v ) = \\sum _ j p ( i , j | v , w ) , p _ B ( j | w ) = \\sum _ i p ( i , j | v , w ) , \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 0 & 8 & 5 & 1 \\\\ 7 & 4 & 7 & 5 \\\\ 5 & 5 & 3 & 1 0 \\\\ 5 & 7 & 0 & 5 \\end{pmatrix} . \\end{align*}"}
-{"id": "7904.png", "formula": "\\begin{align*} & [ H ( t ) , \\Lambda ( t ) ] = [ ( H _ 1 + H _ 2 + W _ { 1 2 } ) + H _ 3 + H _ 4 + \\sum \\ , ' \\ , W _ { i j } , \\\\ & ( H _ 1 + H _ 2 + W _ { 1 2 } ) + L _ 3 + L _ 4 ] = [ H _ 3 , L _ 3 ] + [ H _ 4 , L _ 4 ] + \\\\ & \\left [ \\sum \\ , ' \\ , W _ { i j } , H _ 1 + H _ 2 + L _ 3 + L _ 4 \\right ] . \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} ( \\alpha \\cdot f ) ( m ) = - f ( m ) \\cdot \\alpha + f ( m \\cdot \\alpha ) \\ , , \\end{align*}"}
-{"id": "6239.png", "formula": "\\begin{align*} R _ m L _ m + L _ m R _ m = \\sum _ { \\mu , \\lambda } \\theta ( m , \\mu , \\lambda ) E _ \\mu ^ * E _ \\lambda , \\end{align*}"}
-{"id": "6684.png", "formula": "\\begin{align*} \\| K \\| _ { p , q } = \\left ( \\int _ { \\Omega } \\left ( \\int _ { \\Omega } | K ( x , y ) | ^ p d \\mu ( x ) \\right ) ^ { \\frac q p } d \\mu ( y ) \\right ) ^ { \\frac 1 q } , \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} \\dot { L } & = \\sum _ { i = 1 } ^ { 2 k + 1 } L _ 1 L _ 2 \\ldots \\dot { L } _ i \\ldots L _ { 2 k + 1 } \\\\ & = \\sum _ { i = 1 } ^ { 2 k + 1 } L _ 1 L _ 2 \\ldots L _ i U _ { i + 1 } \\ldots L _ { 2 k + 1 } - \\sum _ { i = 1 } ^ { 2 k + 1 } L _ 1 L _ 2 \\ldots U _ i L _ i \\ldots L _ { 2 k + 1 } \\\\ & = L U _ { 2 k + 2 } - U _ 1 L = [ L , U _ 1 ] \\ , , \\end{align*}"}
-{"id": "7984.png", "formula": "\\begin{align*} x ( k + 1 ) = x ( k ) + \\Gamma u ( k ) , \\ \\ k \\in \\mathbb { N } \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} x \\mapsto h _ { \\epsilon , \\mu } ^ { \\mathcal { I } } ( x ) : = \\sum _ { i \\in \\mathcal { I } } \\epsilon _ i x _ i + \\sum _ { i \\in \\mathcal { N } \\setminus \\mathcal { I } } ( \\abs { x _ i - \\mu _ i } - \\abs { \\mu _ i } ) , \\end{align*}"}
-{"id": "8954.png", "formula": "\\begin{align*} \\{ - i \\nabla - A ( x ) \\} ^ 2 \\Lambda ^ A ( x , z ) = \\Lambda ^ A ( x , z ) \\left \\{ - i \\nabla \\ , - \\ , a ( x , z ) \\right \\} ^ 2 , \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} & L _ \\prec ^ \\alpha = L _ \\prec \\circ ( \\alpha ^ 2 \\otimes I d ) , L _ \\succ ^ \\alpha = L _ \\succ \\circ ( \\alpha ^ 2 \\otimes I d ) , \\\\ & R _ \\prec ^ \\alpha = R _ \\prec \\circ ( \\alpha ^ 2 \\otimes I d ) , R _ \\succ ^ \\alpha = R _ \\succ \\circ ( \\alpha ^ 2 \\otimes I d ) . \\end{align*}"}
-{"id": "1172.png", "formula": "\\begin{align*} \\frac { \\partial P } { \\partial x } d x + \\frac { \\partial P } { \\partial y } d y + \\frac { \\partial P } { \\partial z } d z = 0 \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{align*} \\zeta ^ { \\frac { ( 1 - \\theta ) } { N } } = e ^ { - \\frac { i ( 1 - \\theta ) \\xi } { N } } = 1 - \\frac { i ( 1 - \\theta ) \\xi } { N } + O \\left ( \\frac { 1 } { N ^ 2 } \\right ) , N \\to \\infty . \\end{align*}"}
-{"id": "992.png", "formula": "\\begin{align*} u ^ { k + 1 } = \\left \\{ \\begin{array} { l l } P _ { C _ { 1 } } u ^ { k } & k = 2 n \\\\ P _ { C _ { 2 } } u ^ { k } & k = 2 n + 1 \\end{array} \\right . \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} g _ 2 = \\frac 4 3 ( \\lambda ^ 2 - \\lambda + 1 ) , ~ ~ g _ 3 = \\frac 4 { 2 7 } ( \\lambda - 2 ) ( \\lambda + 1 ) ( 2 \\lambda - 1 ) . \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{align*} a _ j ^ 2 = & 1 + \\lambda _ j ^ { 2 \\beta / r } n ^ { 2 \\beta / r } , \\ , j = 0 , \\dots , n - 1 . \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} v ( [ 1 \\ \\dots \\ t - 1 \\mid i _ 1 \\ \\dots \\ i _ { t - 1 } ] ) \\ = \\ ( n + 1 - i _ { t - 1 } ) v ( \\pi ) , \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{align*} ( \\Phi ^ { - 1 } u ) ( v ) = \\langle \\Phi ^ { - 1 } u , v \\rangle _ { [ L ^ { p ^ { \\prime } } _ { \\sigma } ] ^ { * } , L ^ { p ^ { \\prime } } _ { \\sigma } } = \\langle u , v \\rangle _ { L ^ p _ { \\sigma } , L ^ { p ^ { \\prime } } _ { \\sigma } } = \\int _ { \\Omega } u \\cdot \\overline { v } \\ ; \\d x , u \\in L ^ p _ { \\sigma } ( \\Omega ) , \\ , v \\in L ^ { p ^ { \\prime } } _ { \\sigma } ( \\Omega ) . \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} I \\left ( \\left ( \\sum _ { i = 0 } ^ { 2 ^ n - 1 } \\theta _ i \\right ) \\otimes \\left ( \\chi _ 1 + \\chi _ 2 + \\chi _ 4 \\right ) \\right ) \\end{align*}"}
-{"id": "1320.png", "formula": "\\begin{align*} A _ i = a _ { i + 1 } ^ { - 1 } a _ i , 1 \\le i \\le m ; B _ j = a _ { m + j + 1 } ^ { - 1 } a _ j , 1 \\le j \\le k . \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} [ \\omega ] = [ \\pi _ Y ^ * \\Theta ] , \\end{align*}"}
-{"id": "878.png", "formula": "\\begin{align*} \\partial _ t \\hat { f } ( \\xi , t ) = i \\Big ( \\int _ 1 ^ t \\frac 1 { 2 s ' } e ^ { - ( t - s ' ) } \\Big | \\hat { f } \\Big ( \\frac s { s ' } \\xi , s ' \\Big ) \\Big | ^ 2 d s ' \\Big ) \\hat { f } ( \\xi , t ) + \\int _ 1 ^ t R ( \\xi , s ) , \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{align*} \\lim _ { t \\searrow 0 } \\| u ( \\cdot , t _ 0 + t ; t _ 0 , u _ 0 ) - u _ 0 \\| _ { \\infty } = 0 . \\end{align*}"}
-{"id": "3411.png", "formula": "\\begin{align*} \\gamma ( \\mu , a ) : = \\begin{cases} 1 & \\\\ \\mu + \\frac a 2 & \\end{cases} \\end{align*}"}
-{"id": "4109.png", "formula": "\\begin{align*} \\mathcal { L } = \\{ \\underline { k } \\in K ^ { n } | \\overline { \\langle \\underline { g } \\cdot \\underline { k } \\rangle } \\neq G \\} , \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{gather*} a _ i a _ { i + 1 } a _ i ^ { - 1 } a _ { i + 1 } ^ { - 1 } = 1 \\quad \\\\ a _ { m + j + 1 } ^ { - 1 } a _ j a _ { m + j + 1 } a _ { m + j } ^ { - 1 } = 1 \\quad \\end{gather*}"}
-{"id": "6272.png", "formula": "\\begin{align*} k _ i \\mapsto \\epsilon _ i k _ i , & & e _ i ^ + \\mapsto \\epsilon _ i e _ i ^ + , & & e _ i ^ - \\mapsto e _ i ^ - , & & ( i = 0 , 1 ) . \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} & E \\left [ \\int _ s ^ t ( \\nabla _ q \\chi ) ( r , q _ r ^ m , z _ r ^ m ) \\cdot z _ r ^ m d r \\right ] \\\\ = & E \\left [ \\int _ s ^ t \\int \\left ( ( \\nabla _ q \\chi ) ( r , q _ r , z ) \\cdot z \\right ) h ( r , q _ r , z ) d z d r \\right ] + O ( m ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "557.png", "formula": "\\begin{align*} T _ { \\alpha , p } ( r ) = \\int _ 0 ^ r \\left ( \\int _ { D _ t } ( \\sigma _ { r , p } ^ { - 1 } ) ^ * \\alpha \\right ) \\frac { d t } { t } \\end{align*}"}
-{"id": "1857.png", "formula": "\\begin{align*} R _ \\mu : = ( 2 \\mu - 1 ) | c _ 2 | ^ 4 + \\sum _ { k = 3 } ^ { + \\infty } ( \\mu k - 1 ) | c _ k | ^ 4 + \\sum _ { k = 3 } ^ { + \\infty } | c _ k | ^ 2 ( ( \\mu k - 2 ) | c _ 0 | ^ 2 + ( \\mu ( k + 1 ) - 2 ) | c _ 1 | ^ 2 + ( \\mu ( k + 2 ) - 2 ) | c _ 2 | ^ 2 ) \\ . \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | K _ { n , d } ( h ) - . 5 h ' \\mathsf { I } _ { d } h | = 0 h \\in \\R ^ { d } , \\end{align*}"}
-{"id": "1342.png", "formula": "\\begin{align*} \\phi = \\phi _ { m , m } \\end{align*}"}
-{"id": "4051.png", "formula": "\\begin{align*} \\epsilon > 1 - \\rho ^ 2 _ m ( p _ { X Y } ) = 1 - { ( 1 - 2 p ) } ^ 2 = 4 p ( 1 - p ) . \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} \\min _ { | z | = s } | f ( z ) | \\leq 1 . \\end{align*}"}
-{"id": "8531.png", "formula": "\\begin{align*} \\mathcal { L } _ X \\omega _ 1 = \\mathcal { L } _ X \\omega _ 2 = \\mathcal { L } _ X \\omega _ 3 = 0 \\mathrlap { . } \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} \\ddot { q } _ 1 + 2 D _ 1 \\omega _ 1 \\dot { q } _ 1 + \\omega _ 1 ^ 2 q _ 1 + \\frac { \\omega _ 1 ^ 2 } { 2 } ( 3 q _ 1 ^ 2 + q _ 2 ^ 2 ) + \\omega _ 2 ^ 2 q _ 1 q _ 2 + \\frac { \\omega _ 1 ^ 2 + \\omega _ 2 ^ 2 } { 2 } q _ 1 ( q _ 1 ^ 2 + q _ 2 ^ 2 ) & = F _ 1 = f _ 1 \\cos ( \\Omega t ) , \\\\ \\ddot { q } _ 2 + 2 D _ 2 \\omega _ 2 \\dot { q } _ 2 + \\omega _ 2 ^ 2 q _ 2 + \\frac { \\omega _ 2 ^ 2 } { 2 } ( 3 q _ 2 ^ 2 + q _ 1 ^ 2 ) + \\omega _ 1 ^ 2 q _ 1 q _ 2 + \\frac { \\omega _ 1 ^ 2 + \\omega _ 2 ^ 2 } { 2 } q _ 2 ( q _ 1 ^ 2 + q _ 2 ^ 2 ) & = 0 . \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left ( x ^ 2 \\frac { \\theta _ 2 ' ( x ) } { \\theta _ 2 ( x ) } \\right ) = - \\frac { 1 } { 2 } - \\ , \\underbrace { \\frac { d } { d x } \\left ( \\frac { \\theta _ 4 ' ( \\tfrac { 1 } { x } ) } { \\theta _ 4 ( \\tfrac { 1 } { x } ) } \\right ) } _ { > 0 } < 0 . \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} \\dot { x } \\left ( t \\right ) = f \\left ( u \\left ( t \\right ) \\right ) \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} f ^ { \\ , \\prime } ( \\bar { x } ; v ) \\ , = \\ , \\nabla q _ i ( \\bar { x } ) ^ T v . \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u ^ { \\prime } ( t ) + \\mathcal { A } _ p u ( t ) & = f ( t ) ( 0 < t < T ) , \\\\ u ( 0 ) & = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{align*} a ( x ) = \\frac { \\Sigma _ 1 ^ 2 } { 2 x } , \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} E _ { s _ i \\mu } & = f _ { s _ i \\mu } + \\sum _ { \\substack { \\nu \\in \\sigma ( s _ i \\mu ) \\\\ \\nu \\prec s _ i \\mu } } c _ { s _ i \\mu , \\nu } ( q , t ) f _ { \\nu } \\end{align*}"}
-{"id": "352.png", "formula": "\\begin{align*} [ T ( \\psi _ 1 * _ G \\psi _ 2 ) ] ( k ' ) & = | H | \\cdot \\sum _ { k \\in K } \\sum _ { h \\in H } \\psi _ 1 ( k ) \\psi _ 2 ( k ^ { - 1 } k ' ) \\\\ [ 5 p t ] & = | H | ^ 2 \\cdot \\sum _ { k \\in K } \\psi _ 1 ( k ) \\psi _ 2 ( k ^ { - 1 } k ' ) \\\\ [ 5 p t ] & = \\sum _ { k \\in K } ( T \\psi _ 1 ) ( k ) \\cdot ( T \\psi _ 2 ) ( k ^ { - 1 } k ' ) \\\\ [ 5 p t ] & = [ ( T \\psi _ 1 ) * _ K ( T \\psi _ 2 ) ] ( k ' ) . \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{align*} L _ 0 h = - p _ 0 + \\int _ 0 ^ 1 ( L _ 0 - L _ { t h } ) ( h ) \\ , d t ; \\end{align*}"}
-{"id": "8878.png", "formula": "\\begin{align*} D ^ { a c } = & \\sum _ { Y \\in \\mathcal { I } _ X ^ G } \\left ( 1 - \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\alpha \\circ \\mathcal { P } ( \\mu _ Y ) \\right ) Y \\\\ & + \\sum _ { D \\in \\mathcal { D } } \\left ( m _ D - \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\alpha \\circ \\mathcal { P } ( \\rho ( D ) ) \\right ) \\overline { D } . \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{align*} d ^ { \\prime } ( t ) + B d ( t ) = - ( u ( t ) \\cdot \\nabla ) d ( t ) + \\lvert \\nabla d ( t ) \\rvert ^ 2 d ( t ) t \\in ( 0 , T ) . \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} \\langle K _ j , f _ j \\rangle \\leq - c \\lambda \\delta \\int _ { | | x | - r _ j | \\geq \\lambda \\delta } | f _ j ( x ) | d x = - c \\lambda \\delta | \\{ x \\in E _ j \\Delta B _ j : | | x | - r _ j | \\geq \\lambda \\delta \\} | . \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} f ' ( \\rho ) + p - \\chi u + \\frac { \\phi } { \\tau } = c , p \\geq 0 , \\ , \\rho \\leq M , \\ , p ( M - \\rho ) = 0 . \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} \\mathcal { F } ( R , M ) & : D _ { \\delta , \\mu } \\times \\mathbb { R } ^ { n \\times n } \\rightarrow \\mathbb { R } , \\\\ \\mathcal { F } ( R , M ) & = \\sum _ { i , j , k , \\ell } \\frac { \\partial ^ 2 F } { \\partial R _ { i j } \\partial R _ { k \\ell } } M _ { i j } M _ { k \\ell } \\\\ & = - \\sum _ { i , j , k , \\ell } R ^ { \\ell i } R ^ { j k } M _ { i j } M _ { k \\ell } . \\end{align*}"}
-{"id": "5223.png", "formula": "\\begin{align*} w _ n ( t , \\cdot ) & = e ^ { t ( \\Delta - I ) } w _ n ( 0 ) + \\underbrace { \\int _ { 0 } ^ { t } e ^ { ( t - s ) ( \\Delta - I ) } \\nabla \\cdot ( w _ { n } ( s , \\cdot ) b _ { n } ( s , \\cdot ) ) d s } _ { I _ 1 } \\cr & + \\underbrace { \\int _ { 0 } ^ { t } e ^ { ( t - s ) ( \\Delta - I ) } ( ( 1 + f _ n ( s , \\cdot ) - \\nabla \\cdot b _ n ( s , \\cdot ) ) w _ n ( s , \\cdot ) + g _ { n } ( s , \\cdot ) v _ { n } ( s , \\cdot ) + h _ { n } \\cdot \\nabla v _ { n } ) d s } _ { I _ 2 } , \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} \\left ( x ^ { ( \\omega ) } \\right ) _ { - j } \\longmapsto \\sum _ { i = 1 } ^ { k } ( j - i ) x _ { i } ^ { ( \\omega _ i ) } . \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} h ( R _ 1 , R _ 2 , R _ 3 ) = \\frac { L ( 1 ) - L ( 0 ) } { L ( 0 ) } = \\frac { \\alpha _ 1 X _ 1 ( 0 ) } { L ( 0 ) } R _ 1 + \\frac { \\alpha _ 2 X _ 2 ( 0 ) } { L ( 0 ) } R _ 2 + \\frac { \\alpha _ 3 X _ 3 ( 0 ) } { L ( 0 ) } R _ 3 . \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{align*} \\sigma ^ N _ \\alpha ( \\bar { a } _ { \\bar { u } ^ \\smallfrown \\bar { v } _ 2 } ) = \\sigma ^ N _ \\beta ( \\bar { a } _ { \\bar { u } ^ \\smallfrown \\bar { w } } ) \\mbox { a n d } \\sigma ^ N _ \\alpha ( \\bar { a } _ { \\bar { u } ^ \\smallfrown \\bar { v } } ) = \\sigma ^ N _ \\beta ( \\bar { a } _ { \\bar { u } ^ \\smallfrown \\bar { w } _ 2 } ) \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{align*} \\exists s \\in S { : } \\ s _ j = - 1 \\quad \\Longleftrightarrow \\exists t \\in S { : } \\ t _ j = 1 . \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{align*} \\frac { \\lim _ { \\beta \\to \\theta _ { n - 1 } ^ { { - 1 } ^ { \\mbox { } \\ , - } } } { \\rm M } _ { r } ( \\beta ) } { \\lim _ { \\beta \\to \\theta _ { n - 1 } ^ { { - 1 } ^ { \\mbox { } \\ , + } } } { \\rm M } _ { r } ( \\beta ) } ~ ~ = ~ ~ | z _ { J _ n , n - 1 } | ^ { - 2 } , \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} \\frac { \\rho _ j ^ { n + 1 } - \\rho ^ n _ j } { \\Delta t } - \\frac { F ^ n _ { j + \\frac 1 2 } - F ^ n _ { j - \\frac 1 2 } } { \\Delta x } = 0 , \\end{align*}"}
-{"id": "6096.png", "formula": "\\begin{align*} p _ T ( t ) = \\frac { c } { \\sqrt { 2 \\pi } } \\frac { e ^ { - \\frac { ( c - \\delta t ) ^ 2 } { 2 t } } } { t ^ { \\frac { 3 } { 2 } } } 1 _ { ( 0 , + \\infty ) } ( t ) . \\end{align*}"}
-{"id": "5753.png", "formula": "\\begin{align*} ( 0 , 0 , u ^ 8 , 0 , 0 , 1 ) , \\ ; ( 0 , 0 , 0 , u ^ 8 , 0 , 1 ) & \\quad \\phi _ { \\mathfrak { L } } = u ^ { 4 0 } \\\\ ( 0 , x , w , w , y , z ) & \\quad \\phi _ { \\mathfrak { L } } = u ^ { 3 2 } , \\end{align*}"}
-{"id": "1562.png", "formula": "\\begin{align*} b _ { n + 1 } = b _ n ( \\ell _ { n + 1 } / \\ell _ n ) ^ { p _ n } \\ , , \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} \\dim ( y \\cap x _ m ) = \\mu _ 1 + \\mu _ 2 + \\cdots + \\mu _ m \\end{align*}"}
-{"id": "1323.png", "formula": "\\begin{align*} a \\circ b : = b a . \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} | \\phi ' ( z _ 0 ) | = \\frac { \\pi } { 4 | x _ 0 | } . \\end{align*}"}
-{"id": "3500.png", "formula": "\\begin{align*} { 1 \\over m } \\sum _ { i \\in [ m ] } \\alpha _ i = \\min _ { u _ 1 + \\cdots + u _ m = 0 } \\ ; \\max _ { i \\in [ m ] } ( \\alpha _ i + u _ i ) . \\end{align*}"}
-{"id": "6387.png", "formula": "\\begin{align*} b _ + ( a , \\tfrac 1 2 ) = b _ - ( a , \\tfrac 1 2 ) + \\frac { a ( 1 - a ) ^ 2 } { 1 - a ^ 2 } = \\frac { 1 + a - a ^ 2 } { 1 + a } . \\end{align*}"}
-{"id": "1318.png", "formula": "\\begin{align*} G _ { m , 0 } \\subset G _ { m , 1 } \\subset G _ { m , 2 } \\subset \\dots \\subset G _ { m , m } . \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{align*} \\begin{cases} k _ { y y y } + k _ y + k _ { x x x } + k _ x + \\lambda s = 0 , & , \\\\ s _ { y y y } + s _ y + s _ { x x x } + s _ x + \\lambda k = \\lambda \\delta ( x - y ) , & , \\end{cases} \\end{align*}"}
-{"id": "2127.png", "formula": "\\begin{align*} \\iota _ N ( x ) \\left ( ( 1 - s ) \\frac { k } { N } + s \\frac { k + 1 } { N } \\right ) = ( 1 - s ) x _ k + s x _ { k + 1 } , \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} \\frac { c } { m n } ( - 1 ) ^ n q ^ { \\binom { n } { 2 } - \\frac { n } { c } \\sum _ i \\binom { a _ i } { 2 } } \\sum _ { e \\mid n / c } ( - 1 ) ^ { c e } \\mu ( n / c e ) \\prod _ { i = 1 } ^ m C _ { a _ i - 1 } ( q ^ { n / c e } ) ^ e . \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} T _ W : = \\left \\{ ( \\rho _ a , \\rho _ b , \\rho _ { a + b } ) \\ | \\ ( a , b ) \\in W \\right \\} . \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} H ( \\theta , \\mathcal { E } ) = - \\sum _ { E \\in \\mathcal { E } } \\theta ( E ) \\log \\theta ( E ) \\end{align*}"}
-{"id": "3528.png", "formula": "\\begin{align*} \\partial _ { t } \\kappa & = \\partial _ { x x } \\kappa + \\kappa \\langle \\partial _ { x x } N , N \\rangle - 2 \\kappa ^ { 2 } \\langle T , \\partial _ { x } N \\rangle \\\\ & = \\partial _ { x x } \\kappa - \\kappa \\langle \\partial _ { x } N , \\partial _ { x } N \\rangle + 2 \\kappa ^ { 3 } \\\\ & = \\kappa _ { s s } + \\kappa ^ { 3 } \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} d \\big ( F _ t ( x ) , F _ t ( y ) \\big ) = d ( x , y ) \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} i \\mathfrak { H } ( s , t ) & = \\frac { \\operatorname { p . v . } } { L } \\int _ 0 ^ L f ( \\beta ) \\cot \\frac { \\pi ( s - \\beta ) } { L } d \\beta \\\\ & + \\frac { \\operatorname { p . v . } } { L } \\int _ 0 ^ L f ( \\beta ) \\xi _ \\beta \\left ( \\cot \\frac { \\pi ( \\xi ( s ) - \\xi ( \\beta ) ) } { L } - \\frac { 1 } { \\xi _ \\beta } \\cot \\frac { \\pi ( s - \\beta ) } { L } \\right ) d \\beta . \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{align*} h ( p ) \\mu _ 2 ( q ) + \\varphi ( q ) \\mu _ 1 ( p ) = \\rho _ 1 ( p ) + \\rho _ 2 ( q ) , \\forall ( p , q ) \\in D \\times G . \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} V = - \\big [ \\nabla f \\cdot n \\big ] , \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} z _ S = - \\int _ { \\xi } ^ { \\infty } \\frac { d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\pm \\omega _ 1 . \\end{align*}"}
-{"id": "2693.png", "formula": "\\begin{align*} P ( x , y ) = \\dfrac { \\Omega ( x , y ) } { \\delta ( x , y ) } , \\end{align*}"}
-{"id": "2482.png", "formula": "\\begin{align*} \\zeta ^ { \\ , - \\ln N } \\phi _ N ( \\xi ) & = 1 + \\left [ i \\xi + O \\left ( \\frac { 1 } { N } \\right ) \\right ] \\int _ 0 ^ { \\infty } e ^ { [ i \\xi \\ , + \\ , O ( N ^ { - 1 } ) ] \\ , s } \\left [ 1 - \\left ( 1 - e ^ { - s } \\right ) ^ N \\right ] d s \\\\ & = 1 + i \\xi \\int _ 0 ^ { \\infty } e ^ { i \\xi s } \\left [ 1 - \\left ( 1 - e ^ { - s } \\right ) ^ N \\right ] d s + O \\left ( \\frac { 1 } { N } \\right ) \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} ( f \\cdot \\alpha ) ( n ) = - \\alpha ( f ( n ) ) + f ( \\alpha \\cdot n ) \\ , , \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} g ^ n ( x + 1 ) - \\lambda _ n - \\frac { 1 } { ( 1 + \\rho _ n ) ^ 2 } & ( \\eta ^ n ( x + 1 ) - \\rho _ n ) = g ^ n ( x + 1 ) - g ^ { n , \\ell } ( x ) \\\\ & + g ^ { n , \\ell } ( x ) - \\psi ^ { n , \\ell } ( x ) \\\\ & + \\psi ^ { n , \\ell } ( x ) - \\lambda _ n - \\frac { 1 } { ( 1 + \\rho _ n ) ^ 2 } ( \\eta ^ { n , \\ell } ( x ) - \\rho _ n ) \\\\ & + \\frac { 1 } { ( 1 + \\rho _ n ) ^ 2 } \\big ( \\eta ^ { n , \\ell } ( x ) - \\eta ^ n ( x + 1 ) \\big ) , \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{align*} f ^ \\# ( c ) \\left ( 1 - \\frac { | c - b | } { t } \\right ) = \\displaystyle \\max _ { | z - b | \\leq t } f ^ \\# ( z ) \\left ( 1 - \\frac { | z - b | } { t } \\right ) . \\end{align*}"}
-{"id": "4085.png", "formula": "\\begin{align*} \\Lambda ( \\hat { \\vect { x } } ^ t _ k ) = \\frac { | \\Sigma ^ t | ^ { 1 / 2 } } { | \\beta _ k \\vect { I } + \\Sigma ^ t | ^ { 1 / 2 } } q ( \\hat { \\vect { x } } ; \\vect { \\Sigma } ^ t ) ^ { - 1 } . \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{align*} f ( \\langle x , y \\rangle ) \\geq | X | ^ 2 f _ 0 - | X | f ( 1 ) = 4 d ^ 2 \\left ( \\frac 1 d - s \\right ) - 4 d ( 1 - s ) = - 4 d ( d - 1 ) s . \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} \\chi ( X ) = \\dim \\mathcal H ^ + _ { 0 , a } - \\dim \\mathcal H ^ - _ { 0 , a } , \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} h _ { \\mathrm { F a l } } ( A _ n ) & = - \\frac { L ' ( \\chi _ { - d } , 0 ) } { L ( \\chi _ { - d } , 0 ) } - \\frac { 1 } { 2 } \\log d + \\frac { 1 } { 2 } \\log 2 \\pi \\\\ & \\quad \\ , + \\frac { 1 } { 2 } \\left ( \\sum _ { p | n } \\log p \\left ( r _ p - \\left ( \\frac { 1 - \\left ( \\frac { p } { d } \\right ) } { p - \\left ( \\frac { p } { d } \\right ) } \\right ) \\left ( \\frac { 1 - p ^ { - r _ p } } { 1 - p ^ { - 1 } } \\right ) \\right ) \\right ) \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{align*} E _ \\mu ^ * E _ \\nu ^ * = \\delta _ { \\mu , \\nu } E _ \\mu ^ * , & & \\mu , \\nu \\in \\lbrace 0 , 1 \\rbrace ^ N , \\end{align*}"}
-{"id": "4967.png", "formula": "\\begin{align*} \\mathcal { H } \\partial _ x \\psi + c \\psi = \\psi ^ 2 . \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{align*} \\sup _ { \\kappa } \\left | Q ^ { ( k + 1 ) } ( 0 , \\alpha _ * = 0 ) \\right | \\leq \\Phi _ k ( k _ { s c a t t } ) M ^ { - ( k + 1 ) / 2 } , \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} p = \\sum _ { r \\in X \\cup X ^ 2 \\cup X ^ 3 } \\alpha _ r r = \\sum _ { r \\in X \\cup X ^ 2 \\cup X ^ 3 } \\alpha _ r ( r - \\gamma ( r ) ) + \\sum _ { r \\in X \\cup X ^ 2 \\cup X ^ 3 } \\alpha _ r \\gamma ( r ) . \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{align*} \\begin{array} { r c l } L & \\to & \\mathrm { D e r } _ R ( S ) \\\\ \\alpha & \\mapsto & \\partial _ \\alpha : = \\alpha ( - ) \\end{array} \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{align*} \\mathbb { P } ( N _ x ( i ) = \\alpha _ { e r } ) = p _ { e r } , \\mathbb { P } ( N _ x ( i ) = \\alpha _ 1 ) = p _ 1 \\mathbb { P } ( N _ x ( i ) = \\alpha _ 0 ) = 1 - p _ 1 - p _ { e r } . \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} \\begin{cases} u ( x , t ) = \\eta ( x , t ) - \\displaystyle \\int ^ L _ 0 k ( x , y ) \\eta ( y , t ) d t - \\int _ 0 ^ L s ( x , y ) w ( y , t ) d y \\\\ v ( x , t ) = w ( x , t ) - \\displaystyle \\int ^ L _ 0 k ( x , y ) w ( y , t ) d y - \\int ^ L _ 0 s ( x , y ) \\eta ( y , t ) d y , \\end{cases} \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} \\mathcal { L } H + ( | A | ^ 2 + \\frac { 1 } { 2 } ) H = 0 . \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{align*} | \\varphi ' | ^ 2 = \\left | \\tfrac \\mu { 2 \\zeta } s ^ { \\mu - 1 } \\widetilde \\varphi - \\tfrac 1 { 2 p } s ^ { \\mu - 1 - \\frac 1 p } \\partial _ 1 \\widetilde \\varphi \\right | ^ 2 + \\left | \\tfrac 1 2 s ^ { \\mu - 1 } \\partial _ \\zeta \\widetilde \\varphi \\right | ^ 2 , \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{align*} \\begin{cases} \\Re ( \\varphi - f - c ) \\leq - C s ^ \\lambda , \\\\ \\Re ( \\varphi - f + \\delta | z | ^ { \\varepsilon } ) \\leq - C s ^ \\lambda , \\end{cases} \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{align*} \\begin{aligned} D _ { \\varpi } ( P \\parallel Q ) & = \\log \\sum _ { i = 1 } ^ n { p _ { i } e ^ { \\varpi \\frac { p _ i } { q _ i } } } - \\varpi \\ge \\log e ^ { \\varpi ( \\sum _ { i = 1 } ^ n p _ i ( \\frac { q _ i } { p _ i } ) ) ^ { - 1 } } - \\varpi = \\log e ^ { \\varpi } - \\varpi = 0 . \\end{aligned} \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} \\Delta ^ + ( \\varepsilon _ { t } , 1 , t ) & > 0 , \\\\ \\Delta ^ + ( 1 , 1 , t ) & < 0 , \\\\ \\lim _ { y \\to + \\infty } \\Delta ^ + ( y , 1 , t ) & = + \\infty . \\end{align*}"}
-{"id": "1322.png", "formula": "\\begin{align*} \\pi ( g h ) = \\pi ( h ) \\pi ( g ) \\quad \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} d X _ t & = b _ X ( X _ t , Y _ t ) d t + \\sigma _ X ( X _ t , Y _ t ) d B ^ X _ t , \\ , X _ 0 = x \\\\ d Y _ t & = b _ Y ( X _ t , Y _ t ) d t + \\sigma _ Y ( X _ t , Y _ t ) d B ^ Y _ t , \\ , Y _ 0 = y \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{align*} f ( x ) = \\max _ { g \\in S } g ( x ) \\end{align*}"}
-{"id": "6218.png", "formula": "\\begin{align*} L _ m \\chi _ y ( z ) = q ^ { | S _ \\mu ( m - 1 ) | } \\chi \\left ( \\sum _ { s \\in S _ \\mu } \\sum _ { t \\in T _ \\nu } Y _ { s , t } Z _ { s , t } \\right ) \\end{align*}"}
-{"id": "8728.png", "formula": "\\begin{align*} 2 ^ { \\beta + \\alpha ( m - 1 ) - 1 } \\phi ( A ) = 2 ^ { \\beta + \\alpha ( n - 1 ) } B . \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{align*} E \\left [ S ( \\theta ) ^ { ( r ) } \\right ] & = r \\int _ 0 ^ { \\infty } t ^ { r - 1 } \\left [ 1 - \\left ( 1 - e ^ { - \\theta t } \\right ) \\left ( 1 - e ^ { - ( 1 - \\theta ) t / N } \\right ) ^ N \\right ] d t \\\\ & = r \\int _ 0 ^ { \\infty } t ^ { r - 1 } \\left [ 1 - \\left ( 1 - e ^ { - ( 1 - \\theta ) t / N } \\right ) ^ N \\right ] d t + r \\int _ 0 ^ { \\infty } t ^ { r - 1 } e ^ { - \\theta t } \\left ( 1 - e ^ { - ( 1 - \\theta ) t / N } \\right ) ^ N d t , \\end{align*}"}
-{"id": "6646.png", "formula": "\\begin{align*} q _ p = c _ p \\frac { ( w _ { p - 1 } v _ { \\omega _ p } , w g X _ { \\beta _ 1 } ( - q _ 1 ) \\ldots X _ { \\beta _ { p - 1 } } ( - q _ { p - 1 } ) w _ p v _ { \\omega _ p } ) } { ( w _ { p - 1 } v _ { \\omega _ p } , w g X _ { \\beta _ 1 } ( - q _ 1 ) \\ldots X _ { \\beta _ { p - 1 } } ( - q _ { p - 1 } ) w _ { p - 1 } v _ { \\omega _ p } ) } . \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} a = Z \\ , \\ \\ \\ \\ \\ b = \\sqrt { 1 - ( \\alpha Z ) ^ 2 } \\equiv \\sqrt { 1 - \\nu ^ 2 } , \\ \\ \\ \\ \\nu \\in ( 0 , 1 ) \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} \\mathrm { P } _ { m , 1 } & = \\mathrm { P } \\left ( \\alpha _ 1 = 1 , z _ m < \\frac { \\epsilon _ 1 } { \\rho \\xi _ 1 } \\right ) . \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} \\left ( x \\frac { d } { d x } \\right ) ^ n \\log \\left ( \\theta _ 3 ( x ) \\right ) = ( - 1 ) ^ n \\left ( x \\frac { d } { d x } \\right ) ^ n \\log \\left ( \\theta _ 3 \\left ( \\tfrac { 1 } { x } \\right ) \\right ) . \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{align*} [ O ^ T y ^ \\bot _ \\xi ] ( \\gamma , T - \\xi - 0 ) = \\beta ( \\gamma , \\xi ) \\ , y ( x ( \\gamma , \\xi ) ) \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} \\frac { d } { d x } \\left ( x \\frac { \\theta _ 4 ' ( x ) } { \\theta _ 4 ( x ) } \\right ) = \\frac { \\theta _ 4 ' ( x ) } { \\theta _ 4 ( x ) } + x \\frac { d } { d x } \\left ( \\frac { \\theta _ 4 ' ( x ) } { \\theta _ 4 ( x ) } \\right ) < 0 , \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} \\frac { d } { d \\sigma } \\log H ( \\sigma ) = \\frac { 4 } { \\sigma } N ( \\sigma ) + \\frac { a } { \\sigma } . \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{align*} x - T _ { \\lambda } x = \\ ; & x - P _ A \\left ( ( 1 + \\lambda ) P _ B x - \\lambda x \\right ) + \\lambda \\left ( P _ B x - x \\right ) \\\\ = \\ ; & x - e + \\lambda \\left ( f - x \\right ) = 0 . \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} F ( x _ c , y _ c ) = \\int F ( x , y ) d \\frac { \\mu _ 1 + \\mu _ 2 } { 2 } \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} \\begin{cases} \\eta _ t + w _ x + w _ { x x x } = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , + \\infty ) , \\\\ w _ t + \\eta _ x + \\eta _ { x x x } = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , + \\infty ) , \\\\ \\eta ( x , 0 ) = \\eta _ 0 ( x ) , w ( x , 0 ) = w _ 0 ( x ) , & \\ , \\ , ( 0 , L ) , \\end{cases} \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} p _ { \\mu } ( t , x , \\xi ) : = \\frac { 1 } { \\mu + h _ s ( t , x , \\xi ) } \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\left ( ( \\eta _ 0 , w _ 0 ) , S ^ * ( \\tau _ n ) \\left ( ( \\varphi _ { \\tau _ n } , \\psi _ { \\tau _ n } ) - ( \\varphi _ { \\tau } , \\psi _ { \\tau } ) \\right ) \\right ) _ { \\overline { X } _ 2 } & = 0 \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 } \\| u _ j ( s ) - a \\| _ { L ^ p _ { \\sigma } ( \\Omega ) } = 0 \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{align*} \\begin{cases} D _ { A } \\psi = 0 \\\\ * _ 3 F _ A = r ( q _ 3 ( \\psi ) - i * _ 3 \\omega ) - \\frac { 1 } { 2 } * _ 3 F _ { A _ { K ^ { - 1 } } } - \\frac { 1 } { 2 } i * _ 3 \\wp _ 3 , \\\\ \\end{cases} \\end{align*}"}
-{"id": "2597.png", "formula": "\\begin{align*} \\langle v , v \\rangle = \\left \\langle \\sum ^ n _ { j = 1 } \\langle v , v _ j \\rangle v _ j , v \\right \\rangle = \\sum ^ n _ { j = 1 } \\langle v , v _ j \\rangle \\langle v _ j , v \\rangle = \\sum _ { j = 1 } ^ { n } \\| \\langle v , v _ j \\rangle \\| ^ 2 \\end{align*}"}
-{"id": "6911.png", "formula": "\\begin{align*} H _ { \\Lambda } ^ { ( \\mathrm { p h } ) } \\ = \\ \\sum _ { x \\in \\Lambda } a _ { x } ^ \\dagger a _ x \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} \\| u - \\widetilde { D ^ \\ell } ( c _ u ) \\| _ { L ^ 1 } = \\min _ { c \\in \\{ 0 , 1 \\} ^ \\ell } \\| u - \\widetilde { D ^ \\ell } ( c ) \\| _ { L ^ 1 } \\ , . \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} F ( \\lambda ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ n ^ 2 } { n ! ^ 2 } \\lambda ^ n \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} \\left ( 1 - x + \\frac { x v } { 1 - v } \\right ) C ( x , v ) & = \\frac { x v ^ 3 } { 1 - v } C ( x , 1 ) + \\frac { x ^ 3 v ^ 3 ( 2 - x - x v ) } { ( 1 - x ) ^ 2 ( 1 - x v ) ^ 2 } , \\\\ \\left ( 1 - x + \\frac { x v } { 1 - v } \\right ) D ( x , v ) & = x B ( x , v ) + \\frac { x } { v } C ( x , v ) + \\frac { x v ^ 2 } { 1 - v } D ( x , 1 ) . \\end{align*}"}
-{"id": "1208.png", "formula": "\\begin{align*} \\widehat { \\sigma } = t _ i ^ { - 1 } \\log ^ { \\frac { 1 } { 2 } } ( t _ i ) e ^ { \\frac 1 2 F _ 1 ( \\log { \\frak t } ( t _ i ) ) } ( \\sigma + F _ 2 ( \\log { \\frak t } ( t _ i ) ) \\delta ) \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} J ( \\phi ) = \\int _ 0 ^ 1 \\int _ { X } \\dot { \\phi } _ t \\frac { \\omega _ { \\mathrm { r e f } } ^ n - \\omega _ { \\phi _ t } ^ n } { ( 2 \\pi ) ^ n \\mathcal { L } ^ n } d t . \\end{align*}"}
-{"id": "1483.png", "formula": "\\begin{align*} \\displaystyle s _ 1 = \\inf \\left \\{ \\frac { \\int _ { \\Sigma } \\left ( | \\nabla \\varphi | ^ { 2 } - q \\varphi ^ 2 \\right ) e ^ { - f } d \\sigma } { \\int _ { \\Sigma } \\varphi ^ 2 e ^ { - f } d \\sigma } ; \\varphi \\in C _ 0 ^ { \\infty } ( \\Sigma ) , \\int _ { \\Sigma } \\varphi ^ 2 e ^ { - f } d \\sigma \\neq 0 \\right \\} . \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} b _ n = b _ { n - 1 } + u _ { n } + 2 ^ { n - 3 } - n + 2 + \\binom { n - 2 } { 4 } + \\sum _ { j = 0 } ^ { n - 4 } \\sum _ { i = 1 } ^ { n - 3 - j } \\binom { n - i - 2 } { j + 1 } C _ { n - 2 - j , i } , n \\geq 3 , \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} \\sum _ { s = 0 } ^ { \\ell } z _ s \\delta _ { r - s } = 0 . \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} h _ k ( t , x , \\xi ) : = \\frac { 1 } { 2 m _ k } | \\xi - A ^ { ( k ) } ( t , x ) | ^ 2 + V _ k ( t , x ) \\ ( k = 1 , 2 , 3 , 4 ) \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} & ( L \\chi ) ( z ) - B ( z , . . . , z ) = 2 \\beta ^ { - 1 } \\sum _ { \\alpha = 1 } ^ { k - 2 \\lfloor ( k - 1 ) / 2 \\rfloor - 1 } \\sum _ { \\delta > \\alpha } A _ { \\lfloor ( k - 1 ) / 2 \\rfloor } ^ { \\alpha \\delta } ( z , . . . , z ) . \\end{align*}"}
-{"id": "8012.png", "formula": "\\begin{align*} d e _ { i , t } \\cdot d e _ { i , t } = \\left ( 1 - \\frac { 1 } { N } \\right ) m \\tau ^ 2 \\tilde { \\gamma } ^ 2 d t . \\end{align*}"}
-{"id": "8004.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\mathbb { E } { [ \\| x _ t - x ^ * \\| ^ 2 ] } = \\mathbb { E } [ G _ { \\Gamma t } ] \\geq e ^ { - 2 \\mu \\gamma \\Gamma t } G _ { 0 } + \\frac { m \\tau _ { N } ^ { 2 } \\gamma } { 4 \\mu } ( 1 - e ^ { - 2 \\mu \\gamma \\Gamma t } ) , \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{align*} ( 1 - \\frac { \\tilde { C } k _ n } 2 ) \\norm { \\theta ^ { n - 1 / 2 } } \\le \\Big ( 1 + \\frac { \\tilde { C } k _ n } 2 + C k _ n \\delta ^ 2 \\Big ) \\norm { \\theta ^ { n - 1 } } + C k _ n ( k _ n + h ^ r ) . \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} f _ B ( n ) : = \\frac { M } { N ' } \\frac { \\phi ( W ) } { W } ( \\log N ' ) 1 _ B ( n ) . \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} Y _ \\alpha ( y ) & \\in \\{ F _ \\alpha ( j ) : j = 1 , \\dots , m _ \\alpha + 1 \\} \\\\ Y _ \\alpha ^ T ( x ) & \\in \\{ G _ \\alpha ( 1 ) + \\cdots + G _ \\alpha ( j ) : j = 0 , \\dots , m _ \\alpha \\} . \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} \\min _ { | z | = \\sqrt { C } } | f ( z ) | > 1 \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} F _ r ( \\phi , \\psi ) | _ { t = T _ 0 } \\leq \\Phi F _ r ( \\phi , \\psi ) | _ { t = 0 } = \\frac { 1 } { 2 } < 1 , \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} \\psi ( \\nu , \\mu ) ( T _ i \\cdot f _ { \\nu } ) = 0 = \\psi ( s _ i \\nu , s _ i \\mu ) f _ { s _ i \\nu } . \\end{align*}"}
-{"id": "468.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 4 & 3 \\\\ 0 & 1 0 & 8 & 4 \\\\ 0 & 5 & 1 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "8291.png", "formula": "\\begin{align*} \\eta ( E X ) = \\operatorname { d o m } ( h ( X ) + 2 \\rho _ { \\alpha } ) . \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} \\nu = \\frac { \\mu ( B _ { r _ h } ( x _ h ) \\cap B _ 1 \\cap E _ 2 ) } { \\mu ( B _ { r _ h } ( x _ h ) ) } \\leq \\frac { \\mu ( B _ { r _ h } ( x _ h ) \\cap E _ 2 ) } { \\mu ( B _ { r _ h } ( x _ h ) ) } . \\end{align*}"}
-{"id": "1161.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\nu ( s ) & = & s \\\\ \\nu ( \\alpha ) & = & \\alpha + \\mathrm { T r } ( \\mathrm { a d } _ { \\alpha } ) + \\mathrm { d i v } ( \\partial _ \\alpha ) \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} H ' ( r ) = 2 r \\int _ { - 1 } ^ 0 t h ' ( r ^ 2 t ) d t . \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} r = { ^ { w _ 1 } r _ 1 } \\cdots { ^ { w _ n } r _ n } , \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} \\bigl [ u a _ 3 , [ a _ 1 , a _ 2 ] \\bigr ] = \\bigl [ u , [ a _ 1 , a _ 2 ] \\bigr ] a _ 3 + u \\bigl [ a _ 3 , [ a _ 1 , a _ 2 ] \\bigr ] . \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} H ^ * ( G / H , \\Q ) = ( \\mathbb { S } \\mathfrak { t } ^ * ) ^ { W _ H } / ( \\mathbb { S } \\mathfrak { t } ^ * ) _ + ^ { W _ G } \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} D ^ { i ' } K _ k = i ' ! K _ i \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} \\begin{array} { l } \\nabla _ B \\nabla _ B h ( X , \\nabla _ B X ) = g _ { B } ( \\nabla _ X \\nabla _ B h , \\nabla _ B h ) = \\frac { 1 } { 2 } X ( | \\nabla _ B h | ^ 2 ) , \\\\ \\noalign { \\smallskip } \\nabla _ B \\nabla _ B h ( X , \\nabla _ B X ) = \\left ( | \\nabla _ B h | ^ 2 - c \\right ) h ^ { - 1 } X ( h ) , \\end{array} \\end{align*}"}
-{"id": "3055.png", "formula": "\\begin{align*} F _ { w } ( 1 , t _ { 0 } , 0 ) \\varphi = A \\varphi , \\end{align*}"}
-{"id": "61.png", "formula": "\\begin{align*} \\tilde { \\beta } = \\min _ { i : \\abs { \\beta _ i } > 0 } \\abs { \\beta _ i } . \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} \\Big ( \\sum _ { k = B + 1 } ^ { N } \\sigma ( \\vec { A } ) _ k + 1 \\Big ) ^ { - 1 } \\sum _ { \\lambda = 1 } ^ { N + 1 - B } \\prod _ { j = B + 1 } ^ { B + \\lambda - 1 } \\Bigg ( & \\Big ( \\sum _ { k = j + 1 } ^ { N } \\sigma ( \\vec { A } ) _ k + 1 \\Big ) ^ { - 1 } \\\\ & \\Big ( \\sum _ { k = j + 1 } ^ { N } \\sigma ( \\vec { A } ) _ k + \\sigma ( \\vec { A } ) _ j \\Big ) \\Bigg ) . \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} H _ B ^ { r , s } = { { \\ker \\bar \\partial _ B } \\over { \\rm I m } \\bar \\partial _ B } . \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} M _ i \\leq \\sum _ { T } \\binom { C ( T ) } { d _ i } \\leq \\sum _ T \\binom { 5 n / d _ i } { d _ i } \\leq \\sum _ T 2 ^ { \\frac { 2 } { 2 ^ { 1 / \\ln 2 } \\ln 2 } \\sqrt { 5 e n } } \\leq \\sum _ T 2 ^ { 4 \\sqrt { n } } \\leq 2 ^ { 6 \\sqrt { n } } \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{align*} \\mathfrak { l } = \\mathfrak { t } \\oplus \\bigoplus _ { \\alpha \\in \\Phi _ L } \\mathfrak { g } _ { \\alpha } , \\mathfrak { p } = \\mathfrak { l } \\oplus \\bigoplus _ { \\alpha \\in \\Phi _ { P ^ u } } \\mathfrak { g } _ { \\alpha } \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} n \\log _ 2 n - ( n - 1 ) \\log _ 2 ( n - 1 ) = \\log _ 2 \\left ( \\left ( \\frac { n } { n - 1 } \\right ) ^ n \\right ) + \\log _ 2 ( n - 1 ) \\to ( 1 + \\log _ 2 ( n - 1 ) ) . \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} \\kappa _ { n } : = \\frac { \\kappa _ { n - 1 } } { 2 } \\wedge \\frac { ( \\mu - 2 n ) ^ + } { 2 } , \\tilde { \\kappa } _ n : = \\frac { \\kappa _ n } { 2 } \\wedge \\frac { ( \\mu - 2 n - 1 ) ^ + } { 2 } . \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{align*} \\langle x , y \\rangle \\leq \\frac { - 3 } 4 \\mbox { a n d } \\langle x , z \\rangle \\leq \\frac { - 3 } 4 \\mbox { \\ y i e l d } y = z . \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{align*} & \\epsilon _ 1 = \\epsilon _ 2 = \\min \\Bigg \\{ \\frac { \\sqrt { p _ { X Y } ( 0 , 1 ) p _ { X Y } ( 1 , 0 ) } } { \\sqrt { p _ { X Y } ( 0 , 0 ) p _ { X Y } ( 1 , 1 ) } } , \\frac { \\sqrt { p _ { X Y } ( 0 , 0 ) p _ { X Y } ( 1 , 1 ) } } { \\sqrt { p _ { X Y } ( 0 , 1 ) p _ { X Y } ( 1 , 0 ) } } \\Bigg \\} . \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{align*} H \\left ( X _ i | \\tilde { X } _ i , \\{ \\hat { X } _ i ( u ) \\} _ { u \\in T _ k ( i ) } , \\{ \\tilde { Z } _ j \\} _ { j \\notin T _ k ( i ) } , \\{ X _ w \\} _ { w \\neq i } \\right ) = H \\left ( X _ i | \\tilde { X } _ i , \\{ \\hat { X } _ i ( u ) \\} _ { u \\in T _ k ( i ) } \\right ) \\end{align*}"}
-{"id": "1662.png", "formula": "\\begin{align*} & C \\| A B - A _ 0 B _ 0 \\| ^ 2 \\\\ & \\leqq \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } \\left \\{ \\sum _ { k = 1 } ^ { H _ 0 - 1 } ( a _ { i k } b _ { k j } - a ^ 0 _ { i k } b ^ 0 _ { k j } ) + c _ i \\right \\} ^ 2 + \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } \\left ( \\sum _ { k = H _ 0 } ^ { H - 1 } a _ { i k } b _ { k j } \\right ) ^ 2 . \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty \\atop L \\in \\N } \\frac { 1 } { L ^ d } \\log \\left ( \\int _ 0 ^ { \\infty } d x x e ^ { L ^ d f _ L ( x ) } g _ L ( x ) \\right ) = f ( a ) . \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{align*} \\mathcal { X } _ { \\gamma } \\mathcal { X } _ { \\gamma ' } = ( - ) ^ { \\langle \\gamma , \\gamma ' \\rangle } \\mathcal { X } _ { \\gamma + \\gamma ' } \\rlap { . } \\end{align*}"}
-{"id": "3506.png", "formula": "\\begin{align*} \\lambda ( y ) = ( \\bigoplus y ) ' = \\frac { { \\rm e } ^ y } { \\sum _ i { \\rm e } ^ { y _ i } } \\end{align*}"}
-{"id": "1512.png", "formula": "\\begin{align*} \\mathcal { L } u + \\lambda _ 1 u = 0 . \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{align*} \\begin{aligned} s _ 0 = \\arg \\min _ { 0 \\le s < 1 } f _ { \\rm M S E } ( s ) . \\end{aligned} \\end{align*}"}
-{"id": "3143.png", "formula": "\\begin{align*} S _ R ( \\alpha , \\beta ) = \\beta E + \\alpha N + \\frac { 1 } { \\beta } \\int _ { 0 } ^ { \\beta B } \\ln ( 1 + e ^ { - \\alpha } e ^ { - x } ) d x - \\frac { 1 } { 2 } \\ln ( 1 + e ^ { - \\alpha } ) + \\frac { 1 } { 2 } \\ln ( 1 + e ^ { - \\alpha - \\beta B } ) \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} \\zeta _ { m } \\leq \\sigma _ { 1 } \\Vert A ( I - P _ { m } ) \\Vert = ( { \\sigma } _ { 1 } ^ { ( m ) } + \\varepsilon _ { m } ) \\Vert A - W _ { m + 1 } H _ { m } W _ { m } ^ { T } \\Vert \\ , , \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} \\int _ 0 ^ { U ( N ; \\alpha ) ^ { - 1 } } e ^ { - x } ( \\ln x ) ^ k d x = o \\left ( e ^ { - \\ln ^ { \\alpha } N } \\ln ^ k N \\right ) . \\end{align*}"}
-{"id": "2619.png", "formula": "\\begin{align*} h = \\left \\{ \\begin{array} { l l l } A e ^ { \\sqrt { | a | } t } & \\mbox { i f } & c = 0 , \\\\ \\noalign { \\smallskip } \\sqrt { | \\frac { c } { a } | } [ \\cosh ( \\sqrt { | a | } t + B ) ] & \\mbox { i f } & c \\neq 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} \\{ x y x \\} = \\frac { 1 } { 2 } ( x \\circ ( x \\circ y ) - x ^ 2 \\circ y ) \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} \\rho _ M = \\begin{pmatrix} 1 & - m ^ T & \\frac { 1 } { 2 } \\| m \\| ^ 2 \\\\ \\eta & B - \\eta m ^ T & \\frac { 1 } { 2 } \\eta \\| m \\| ^ 2 - B m \\\\ \\frac { 1 } { 2 } \\eta _ 2 ^ 2 & - \\frac { 1 } { 2 } \\eta _ 2 ^ 2 m ^ T + \\eta ^ T B & \\frac { 1 } { 4 } \\eta _ 2 ^ 2 \\| m \\| ^ 2 - \\eta ^ T B m + 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} g _ n = g _ { n - 1 } + 5 \\cdot 2 ^ { n - 3 } + \\binom { n - 1 } { 4 } - n + \\sum _ { m = 3 } ^ { n - 1 } \\binom { n - m + 1 } { 2 } 2 ^ { m - 3 } , n \\geq 3 , \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{align*} p _ T ( t ) = \\frac { c } { \\sqrt { 2 \\pi } } \\frac { e ^ { - \\frac { c ^ 2 } { 2 t } } } { t ^ { \\frac { 3 } { 2 } } } . \\end{align*}"}
-{"id": "3813.png", "formula": "\\begin{align*} x _ 3 \\cdot f _ { \\ell _ 1 , \\ell _ 2 , \\ell _ 3 } - x _ 1 \\cdot f _ { \\ell _ 1 - 1 , \\ell _ 2 , \\ell _ 3 + 1 } = 0 \\end{align*}"}
-{"id": "2637.png", "formula": "\\begin{align*} \\mu _ 1 - c h = \\tilde { c } , \\mbox { i n } D _ { 1 } \\mu _ 2 + c \\varphi = - b , \\mbox { i n } G _ { 1 } . \\end{align*}"}
-{"id": "4635.png", "formula": "\\begin{align*} \\square _ B \\phi & = \\nabla _ T ^ * \\nabla _ T \\phi + i \\nabla _ { J \\kappa _ B ^ \\sharp } \\phi + \\operatorname { d i v } _ \\nabla ( H ^ { 1 , 0 } ) \\phi . \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{align*} w = \\lambda c z ^ { \\lambda - 1 } + \\sum _ { \\mu < \\lambda } \\mu c _ \\mu z ^ { \\mu - 1 } . \\end{align*}"}
-{"id": "5968.png", "formula": "\\begin{align*} - \\sum _ { i , j = 1 } ^ n \\big ( R ^ { ( s ) } ( x ) \\big ) ^ { - 1 } _ { j i } D _ z B _ { i j } \\big ( x , u ^ { ( \\tau ) } ( x ) , D u ^ { ( \\tau ) } ( x ) \\big ) \\leq n \\frac { \\delta } { 1 + \\delta ^ 2 } \\beta _ 1 , \\ \\forall x \\in \\overline { \\Omega } . \\end{align*}"}
-{"id": "3815.png", "formula": "\\begin{align*} B _ 2 = \\left \\{ \\begin{array} { l l } e _ { 1 , ( \\ell _ 2 + 1 , \\ell _ 3 ) , \\ell _ 1 - 1 } \\\\ e _ { 2 , ( \\ell _ 1 , \\ell _ 2 ) , \\ell _ 3 } \\end{array} : \\ ; 1 \\leq \\ell _ 1 \\leq s , \\ ; 0 \\leq \\ell _ 2 , \\ell _ 3 \\leq s - 1 \\right \\} \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} T _ { f } = m _ { f , \\overline { \\mathcal { O } ( 1 ) } , X _ 0 } + N _ { f , \\overline { \\mathcal { O } ( 1 ) } , X _ 0 } + O ( 1 ) \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} [ c , z _ 1 ] [ z _ 2 , z _ 3 , z _ 4 ] + [ c , z _ 2 ] [ z _ 1 , z _ 3 , z _ 4 ] = 0 . \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} v ( x , t ) = \\int _ { 0 } ^ { \\infty } \\int _ { \\R ^ N } \\frac { e ^ { - s } } { ( 4 \\pi s ) ^ { \\frac { N } { 2 } } } e ^ { - \\frac { | x - y | ^ { 2 } } { 4 s } } u ( y , t ) d y d s = \\frac { 1 } { \\pi ^ { \\frac { N } { 2 } } } \\int _ { 0 } ^ { \\infty } \\int _ { \\R ^ N } e ^ { - s } e ^ { - | z | ^ { 2 } } u ( x - 2 \\sqrt { s } z , t ) d y d s . \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} & \\left ( b _ i n _ i - v _ p \\left ( n _ i ! \\right ) \\right ) - \\left ( b _ l p ^ { i - l } n _ i - v _ p \\left ( ( p ^ { i - l } n _ i ) ! \\right ) \\right ) \\\\ & = n _ i ( b _ i - b _ l p ^ { i - l } ) + v _ p \\left ( ( p ^ { i - l } n _ i ) ! \\right ) - v _ p \\left ( n _ i ! \\right ) \\\\ & = n _ i ( b _ i - b _ l p ^ { i - l } + \\frac { p ^ { i - l } - 1 } { p - 1 } ) \\\\ & = n _ i p ^ i \\left ( p ^ { - i } \\left ( b _ i - \\frac { 1 } { p - 1 } \\right ) - p ^ { - l } \\left ( b _ l - \\frac { 1 } { p - 1 } \\right ) \\right ) \\\\ & \\geq 0 . \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} \\widehat E _ m = m ^ { - \\left ( \\frac { n - 1 } { n } \\right ) \\left ( \\frac { n + 1 - \\alpha } { \\alpha ( n - 1 ) + 1 } \\right ) - \\frac 1 n } E _ m \\to [ 0 , \\widehat L ] \\times \\{ 0 \\} ^ { n - 1 } , \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} T ( \\sigma \\cdot \\sigma ' ) ( \\alpha ) = T ( \\sigma ' ) \\circ T ( \\sigma ) ( \\alpha ) . \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{align*} \\square _ B \\phi = \\bar \\nabla _ T ^ * \\bar \\nabla _ T \\phi . \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} \\partial ^ n ( p ) = \\sum _ { i \\in I } \\partial ( \\varphi ^ i ( p ) ) \\cdot p _ i + \\varphi ^ i ( p ) \\cdot p _ i ' \\ , . \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} f ( \\lambda ) = \\int _ 0 ^ { + \\infty } ( 1 - e ^ { - \\lambda s } ) \\nu ( d s ) . \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{gather*} F _ n = n S _ n + \\lambda \\log n + O _ p ( \\log \\log n ) , \\\\ \\mathbb { E } [ G _ n ] = \\frac { \\lambda } { n } + o \\left ( \\frac { 1 } { n } \\right ) \\end{gather*}"}
-{"id": "507.png", "formula": "\\begin{align*} E _ { \\chi _ 1 ' , \\chi _ 2 ' , k } \\Big | _ { W _ Q ^ N } ( z ) = c _ Q E _ { \\chi _ 1 , \\chi _ 2 , k } ( z ) , \\end{align*}"}
-{"id": "1444.png", "formula": "\\begin{align*} d X _ j ( t ) = \\gamma _ j d t + \\sigma _ j d B _ j ( t ) + d L _ { j - 1 } ( t ) - d L _ { j } ( t ) \\ , , j = 1 , \\ldots , m \\end{align*}"}
-{"id": "267.png", "formula": "\\begin{align*} \\frac { P ( T ) } { ( 1 - T ) ( 1 - q T ) } ( y ( 1 - T ) + x T ) ^ n = \\cdots + \\frac { W ( x , y ) - x ^ n } { q - 1 } T ^ { n - d } + \\cdots . \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{align*} 3 3 * 3 0 + 3 x + y + z + v - 2 a - 2 b = 9 9 0 + 2 x + ( x + y + z + v ) - 2 ( a + b ) \\geq 0 . \\end{align*}"}
-{"id": "5998.png", "formula": "\\begin{align*} { C } _ { \\alpha } \\left ( - \\Delta \\right ) ^ { \\alpha / 2 } \\psi ( x , t ) + V ( x , t ) \\psi ( x , t ) = E \\psi ( x , t ) \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{align*} A _ \\infty & = A _ H + g _ \\infty ^ { - 1 } \\bar \\partial g _ \\infty - ( g _ \\infty ^ { - 1 } \\bar \\partial g _ \\infty ) ^ * \\\\ & = A _ H + 2 \\Im \\bar \\partial \\log \\bigl ( ( q \\bar q ) ^ { 1 / 8 } k ^ { 1 / 2 } \\bigr ) \\begin{pmatrix} i & 0 \\\\ 0 & - i \\end{pmatrix} \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} - y & = ( y ' ( x ) z '' ( x ) - z ' ( x ) y '' ( x ) ) [ 1 + y _ { x } ^ { 2 } + z _ { x } ^ { 2 } ] ^ { - 3 / 2 } \\\\ x & = - z '' ( x ) [ [ 1 + y _ { x } ^ { 2 } + z _ { x } ^ { 2 } ] ^ { - 3 / 2 } \\\\ 0 & = y '' ( x ) [ 1 + y _ { x } ^ { 2 } + z _ { x } ^ { 2 } ] ^ { - 3 / 2 } \\end{align*}"}
-{"id": "7842.png", "formula": "\\begin{align*} x ^ { r N } _ { p } + y _ { p } ^ { r N } = k ( 1 + x ^ { r N } _ { p } y ^ { r N } _ { p } ) \\ ; . \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} 2 a _ j & = | T _ { x _ j = 1 } | + | F _ { x _ j = 0 } | \\\\ & = \\left ( | T _ { x _ i = 1 , x _ j = 1 } | + | T _ { x _ i = 0 , x _ j = 1 } | \\right ) + \\left ( | F _ { x _ i = 0 , x _ j = 0 } | + | F _ { x _ i = 1 , x _ j = 0 } | \\right ) \\\\ & = \\left ( | T _ { x _ i = 1 , x _ j = 1 } | + | F _ { x _ i = 1 , x _ j = 0 } | \\right ) + \\left ( | T _ { x _ i = 0 , x _ j = 1 } | + | F _ { x _ i = 0 , x _ j = 0 } | \\right ) . \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} \\hom ( \\sigma _ 1 , \\sigma _ 2 ) = \\begin{cases} \\{ ( \\sigma _ 1 , \\sigma _ 2 ) \\} & \\sigma _ 1 \\subseteq \\sigma _ 2 , \\\\ \\emptyset & . \\end{cases} \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{align*} R _ k = \\frac 1 2 s _ k = \\frac 1 2 s = \\frac { f ^ \\# ( c ) } { 6 \\varphi ( f ^ \\# ( c ) ) } = \\frac { 1 } { 6 \\varrho \\varphi ( 1 / \\varrho ) } = \\frac { 1 } { 6 \\varrho _ k \\varphi ( 1 / \\varrho _ k ) } . \\end{align*}"}
-{"id": "5933.png", "formula": "\\begin{align*} H \\left ( \\nu , \\mu \\right ) = \\prod _ { j = 1 } ^ { r } \\left ( \\prod _ { x \\in \\vec { x } ^ { ( j ) } ( \\mu ) } \\prod _ { i \\leq x } \\left ( t ^ { \\nu _ i } \\right ) \\right ) \\cdot t ^ { - \\chi \\left ( \\vec { x } ^ { ( 1 ) } , \\dots , \\vec { x } ^ { ( r ) } \\right ) } \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} & \\left ( 1 - x \\right ) A _ m L ( x , y ) = L \\left ( x \\gamma , \\frac y m \\right ) A _ m - x L \\left ( x \\gamma , \\frac { y q } m \\right ) E \\left ( \\frac { y q } m \\right ) A _ m E ( y ) ^ { - 1 } \\\\ & U \\left ( x \\gamma , \\frac { y q } m \\right ) A _ m \\left ( 1 - \\frac q x \\right ) = A _ m U ( x , y ) - \\frac q x E \\left ( \\frac { y q } m \\right ) ^ { - 1 } A _ m E ( y q ) U ( x , y q ) \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{align*} \\begin{aligned} | h ( z ) - h ' ( b ) ( z - b ) | & = \\left | \\int ^ z _ b ( h ' ( \\zeta ) - h ' ( b ) ) d \\zeta \\right | \\\\ & \\leq \\frac { 1 6 ^ 2 } { R } \\int ^ { | z - b | } _ 0 t \\ ; d t = 2 ^ { 7 } \\frac { | z - b | ^ 2 } { R } \\quad | z - b | \\leq \\frac { 1 } { 1 6 } R . \\end{aligned} \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{align*} \\max y ^ * = \\lim _ { l \\to \\infty } \\max y _ { k ( l ) } = \\lim _ { k \\to \\infty } \\max y _ k = \\lim _ { l \\to \\infty } \\max y _ { k ( l ) + 1 } = \\max q ( y ^ * ) . \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{align*} ( b - a ) ( \\phi ( x ) - \\phi ( y ) ) & = [ ( v _ { n } - \\widetilde { \\Gamma } ) ( x ) - ( v _ { n } - \\widetilde { \\Gamma } ) ( y ) ] [ ( v _ { n } - \\widetilde { \\Gamma } ) ^ { + } ( x ) - ( v _ { n } - \\widetilde { \\Gamma } ) ^ { + } ( y ) ] \\\\ & \\geq | ( v _ { n } - \\widetilde { \\Gamma } ) ^ { + } ( x ) - ( v _ { n } - \\widetilde { \\Gamma } ) ^ { + } ( y ) | ^ { 2 } , \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} \\# \\mathop { \\mathrm { s u p p } } V _ { F , \\lambda } ^ { ( > ) } \\ = \\ \\# \\bigg \\{ x \\in \\omega ( { \\underline { r } } ) \\bigg | \\ V _ { F , \\varepsilon } ( x ) \\geq ( 1 - \\varepsilon ^ { \\prime } ) \\frac { \\eta } { \\lambda } \\bigg \\} \\ = \\ F \\big ( ( 1 - \\varepsilon ^ { \\prime } ) ^ { - 1 } \\lambda \\big ) . \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} \\frac { 1 } { u ^ { \\frac { 1 } { r } } ( Q ) } \\int _ { Q } | b ( x ) - b _ { Q } | d x & \\leq \\frac { c } { u ( Q ) ^ { \\frac { 1 } { r } } | Q | ^ { \\frac { 1 } { r ' } } } \\int _ { Q } | b ( x ) - b _ { Q } | d x \\\\ & = c \\left ( \\frac { 1 } { u ( Q ) } \\int _ { Q } | b ( x ) - b _ { Q } | d x \\right ) ^ { \\frac { 1 } { r } } \\left ( \\frac { 1 } { | Q | } \\int _ { Q } | b ( x ) - b _ { Q } | d x \\right ) ^ { \\frac { 1 } { r ' } } , \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} ( \\varphi ( x ) \\eta ( x _ n ) ) k ( x , D _ { x ' } ) \\psi ( x ' ) = ( \\varphi ( x ) \\eta ( x _ n ) ) \\widetilde { \\psi } ( x ' ) k ( x , D _ { x ' } ) \\psi ( x ' ) \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{gather*} F _ i = g _ { i + 1 } ( g _ { i } - g _ { i + 2 } - g _ { i + 4 } - \\cdots - g _ { i + 2 k } ) + g _ { i + 3 } ( g _ { i } + g _ { i + 2 } - g _ { i + 4 } - \\cdots - g _ { i + 2 k } ) + \\\\ g _ { i + 2 k - 1 } ( g _ { i } + g _ { i + 2 } + g _ { i + 4 } + \\cdots + g _ { i + 2 k - 2 } - g _ { i + 2 k } ) , \\end{gather*}"}
-{"id": "2179.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\sup _ { \\mu \\in [ \\nu _ 1 , \\mu _ 2 ] } \\int _ { d ( x , u ) > R } d ( x , u ) ^ p \\ , d \\mu ( x ) = 0 \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} F ( \\mathcal E , \\nabla ) : = \\Gamma ( X , \\overline { \\mathcal E } ) . \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} e ^ { i \\theta } \\gamma _ c D ( F _ 0 ) ( m _ c ) = \\int \\gamma D ( F _ 0 ) ( m ) d \\mu _ \\theta . \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} E ' : y ^ 2 = x ( x ^ 2 + b ( h ) x + d ( h ) ) \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} \\hat { z } _ 1 = u _ 1 , \\hat { z } _ 2 = q _ i u _ 1 , \\dots \\hat { z } _ K = q _ i ^ { K - 1 } u _ 1 . \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} \\mu _ n : = 2 ^ { - 1 / 2 } \\rho _ n \\sigma _ n ^ 2 : = \\frac { 2 } { d ^ { 1 / 2 } ( n ) } \\rho _ n . \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} T _ 0 ( r ) = \\sum _ { j = 0 } ^ K c _ l r ^ { 2 j } \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} \\overset { \\ast } { y _ { 0 } } \\left ( x \\right ) = \\frac { \\left ( y _ { a } + \\varepsilon \\right ) } { \\Gamma \\left ( \\gamma \\right ) } \\left ( \\psi \\left ( x \\right ) - \\psi \\left ( a \\right ) \\right ) ^ { \\gamma - 1 } , \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{align*} T _ { l i m } = \\frac { 1 } { \\alpha } \\log \\Big ( ( 1 + \\sqrt { 2 } ) e ^ { \\alpha ^ 2 X } - 1 + \\sqrt { ( 1 + \\sqrt { 2 } ) e ^ { \\alpha ^ 2 X } - 1 ) ^ 2 - 1 } \\Big ) , \\end{align*}"}
-{"id": "8547.png", "formula": "\\begin{align*} \\sigma _ + = \\sigma _ - = 0 \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{align*} { \\bf H } = { \\bf R } ^ { 1 / 2 } { \\bf Z } , \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} \\begin{cases} \\operatorname { d i v } ( | y | ^ { a } \\nabla U _ 0 ) = 0 \\\\ \\underset { y \\to 0 ^ + } { \\lim } y ^ { a } \\partial _ y U _ 0 = 0 , \\end{cases} \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{align*} \\langle m ^ { j } ; d ^ p \\rangle & = \\int _ { [ 0 , + \\infty ) } ( | y | + d ( x _ k ) ) ^ p d m ^ { j } ( y ) \\\\ & \\leq 2 ^ { p - 1 } \\int _ { [ 0 , + \\infty ) } | y | ^ p d m ^ { j } ( y ) + 2 ^ { p - 1 } d ( x _ k ) ^ p m ( e _ j ) . \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} c = 1 + 6 ( b + b ^ { - 1 } ) ^ 2 , \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{align*} \\theta _ { \\sigma ( 0 ) } ( y ) = \\zeta ^ { - 1 } \\theta _ { \\sigma ( 1 ) } ( y ) = \\dots = \\zeta ^ { - ( 2 ^ n - 1 ) } \\theta _ { \\sigma ( 2 ^ n - 1 ) } ( y ) . \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\| \\sum _ { \\nu = 0 } ^ { n + | \\textup { \\textbf { m } } | - m _ \\alpha } b _ { \\nu , n } ^ { ( \\alpha ) } \\Phi _ \\nu \\| _ K ^ { 1 / n } \\leq \\frac { \\| \\Phi \\| _ K } { \\rho _ { | \\textup { \\textbf { m } } | } ( \\textup { \\textbf { F } } ) } . \\end{align*}"}
-{"id": "7896.png", "formula": "\\begin{align*} l _ k ( x , \\xi ) : = \\frac { 1 } { 2 m _ k } | \\xi | ^ 2 + < x > ^ 2 \\ ( k = 3 , 4 ) . \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} J _ { \\mathrm { f s u } } \\cap J _ { \\mathrm { n i l } } = \\{ 0 \\} . \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} & \\bigl ( [ c , x _ 1 ] [ x _ 2 , x _ 3 ] + [ c , x _ 2 ] [ x _ 1 , x _ 3 ] \\bigr ) [ x _ 4 , x _ 5 ] + \\bigl ( [ c , x _ 1 ] [ x _ 2 , x _ 4 ] + [ c , x _ 2 ] [ x _ 1 , x _ 4 ] \\bigr ) [ x _ 3 , x _ 5 ] \\\\ = \\ & [ c , x _ 1 ] \\bigl ( [ x _ 2 , x _ 3 ] [ x _ 4 , x _ 5 ] + [ x _ 2 , x _ 4 ] [ x _ 3 , x _ 5 ] \\bigr ) + [ c , x _ 2 ] \\bigl ( [ x _ 1 , x _ 3 ] [ x _ 4 , x _ 5 ] + [ x _ 1 , x _ 4 ] [ x _ 3 , x _ 5 ] \\bigr ) \\in I ^ { ( n ) } , \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} y _ i ' : = y _ i e ( q y + e - 1 ) ^ { - 1 } \\geq q ^ { - 1 } ( i = 1 , \\dots , h ) . \\end{align*}"}
-{"id": "5719.png", "formula": "\\begin{align*} y \\binom { r - 1 } { 2 } + x \\left ( \\binom { r } { 2 } - k \\right ) & = \\frac { 2 m - x r } { r - 1 } ( r - 1 ) ( r - 2 ) / 2 - x \\left ( k - \\frac { r ( r - 1 ) } { 2 } \\right ) \\\\ & = ( r - 2 ) m - \\frac { x } { 2 } ( 2 k - r ) \\\\ & \\le ( r - 2 ) m \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} \\P ^ { 0 , 0 } _ { 2 T } ( X ( t _ m ) = x ^ { ( m ) } , m \\in \\{ 1 , 2 , \\dots , M \\} ) = 0 . \\end{align*}"}
-{"id": "4898.png", "formula": "\\begin{align*} \\dd { Y ^ { x , n } } ( t ) & = b ( t , Y ^ { x , n } ( t ) ) \\dd { t } + \\sigma ( t , Y ^ { x , n } ( t ) ) \\dd { \\tilde { W } ^ { x , n } } ( t ) , \\\\ Y ^ { x , n } _ 0 & = x \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} ^ { H } \\mathbb { D } _ { b - } ^ { \\alpha , \\beta ; \\psi } f \\left ( x \\right ) = I _ { b - } ^ { \\beta \\left ( n - \\alpha \\right ) ; \\psi } \\left ( - \\frac { 1 } { \\psi ^ { \\prime } \\left ( x \\right ) } \\frac { d } { d x } \\right ) ^ { n } I _ { b - } ^ { \\left ( 1 - \\beta \\right ) \\left ( n - \\alpha \\right ) ; \\psi } f \\left ( x \\right ) . \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{align*} r ( \\lambda _ 0 ) = 1 \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{align*} \\frac { z } { e ^ z - 1 } = \\sum _ { m = 1 } ^ { \\infty } \\frac { B _ m } { m ! } z ^ m . \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} H ( R / I ^ s , t ) = \\frac { 1 + 2 t + 3 t ^ 2 + 4 t ^ 3 + \\cdots + 3 s t ^ { 3 s - 1 } - \\left ( { s + 3 \\choose 3 } - 3 s - 1 \\right ) t ^ { 3 s } + { s \\choose 3 } t ^ { 3 s + 1 } } { ( 1 - t ) ^ 2 } . \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} \\tilde a ( \\gamma , \\xi ) = \\{ O ^ T [ a _ T - a _ \\xi ] \\} ( \\gamma , T - \\xi - 0 ) . \\end{align*}"}
-{"id": "5151.png", "formula": "\\begin{align*} \\partial _ t \\rho - \\Delta ( d _ \\mu \\rho ) = 0 , \\rho ( 0 , x ) = \\rho ^ 0 ( x ) . \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} \\partial _ { c _ k ^ V } \\left ( \\varepsilon _ { q , m } ^ { V , s } \\right ) = \\left \\langle u _ { q , m } ^ { V , s } , \\partial _ { c _ k ^ V } A ^ V _ q , u _ { q , m } ^ { V , s } \\right \\rangle = \\langle u _ { q , m } ^ { V , s } , \\cos ( k \\cdot ) u _ { q , m } ^ { V , s } \\rangle _ { L ^ 2 _ { \\rm p e r } } . \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{align*} \\sigma _ 0 = \\sum _ { k = 1 } ^ K \\omega _ k e ^ { - v _ k ^ 2 / 2 \\kappa } , \\sigma _ 2 = \\sum _ { k = 1 } ^ K \\omega _ k v _ k ^ 2 \\ , e ^ { - v _ k ^ 2 / 2 \\kappa } . \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} & [ c , x _ 1 x _ 2 , x _ i ] = - [ x _ 1 x _ 2 , c , x _ i ] \\\\ = \\ & - x _ 1 [ x _ 2 , c , x _ i ] - [ x _ 1 , c ] [ x _ 2 , x _ i ] - [ x _ 1 , x _ i ] [ x _ 2 , c ] - [ x _ 1 , c , x _ i ] x _ 2 \\\\ = \\ & x _ 1 [ c , x _ 2 , x _ i ] + [ c , x _ 1 ] [ x _ 2 , x _ i ] + [ c , x _ 2 ] [ x _ 1 , x _ i ] + [ c , x _ 1 , x _ i ] x _ 2 - \\bigl [ [ c , x _ 2 ] , [ x _ 1 , x _ i ] \\bigr ] \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} T _ { \\mathbb { C } } M = E \\otimes H \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} H _ N ^ J ( m ) & = X _ N ( m ) + h \\sum _ { i = 1 } ^ N m _ i + \\sum _ { i = 1 } ^ N J ( m _ i ) , \\ , \\ , m \\in S ^ N . \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} \\mathcal { E } _ N ( \\psi _ N ) : = \\frac { 1 } { N } \\langle \\psi _ N , H _ { N , \\mathbf { A } } \\psi _ N \\rangle \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} J q ^ 3 = 2 J q ^ 1 J q ^ 2 - J q ^ 2 J q ^ 1 - \\frac { 1 } { 3 } J q ^ 1 J q ^ 1 J q ^ 1 . \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} T = 2 \\int _ { q _ - } ^ { q _ + } \\frac { d q } { \\sqrt { E - \\tilde V ( q ) } } , \\end{align*}"}
-{"id": "8355.png", "formula": "\\begin{align*} R ^ k _ { d , t } ( x ) & = \\sum _ { | \\alpha | = d + 1 } D ^ { \\alpha } _ x U _ k ( \\xi ( x , t ) x , t ) \\frac { x ^ { \\alpha } } { \\alpha ! } \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} & + \\sum _ { \\substack { ( t _ 1 ) ^ 1 \\in ( Y ^ { p } ) ^ 1 , ( t _ 2 ) ^ 1 \\in ( Y ^ { q } ) ^ 1 \\\\ p + q + 1 = n } } ( t _ 1 ) ^ 1 \\vee ( t _ 2 ) ^ 1 \\\\ & + \\sum _ { \\substack { t _ 1 \\in Y ^ p , ( t _ 2 ) ^ 2 \\in ( Y ^ { q } ) ^ 2 \\\\ p + q + 1 = n } } t _ 1 \\vee ( t _ 2 ) ^ 2 . \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} & [ u \\phi _ { r } - u ] ^ { p } _ { W ^ { s , p } ( \\R ^ { N } ) } \\\\ & \\leq 2 ^ { p - 1 } \\ ! \\left [ \\iint _ { \\R ^ { 2 N } } \\ ! \\ ! | u ( x ) | ^ { p } \\frac { | \\phi _ { r } ( x ) - \\phi _ { r } ( y ) | ^ { p } } { | x - y | ^ { N + s p } } d x d y + \\ ! \\iint _ { \\R ^ { 2 N } } \\ ! \\ ! \\frac { | \\phi _ { r } ( x ) - 1 | ^ { p } | u ( x ) - u ( y ) | ^ { p } } { | x - y | ^ { N + s p } } d x d y \\right ] \\\\ & = : 2 ^ { p - 1 } [ A _ { r } + B _ { r } ] . \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} f ( x ) : = ( T ( D ) - \\lambda ) \\widetilde { u } ( x ) = a ( x , D ) \\psi ( x ) \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{align*} \\sum _ { g \\in G } | \\langle w , \\rho ( g ) v \\rangle | ^ 2 & = \\sum _ { j = 1 } ^ n \\sum _ { h \\in H } | \\langle w , \\rho ( g _ j h ) v \\rangle | ^ 2 = \\sum _ { j = 1 } ^ n \\sum _ { h \\in H } | \\langle w , \\rho ( g _ j ) \\alpha ( h ) v \\rangle | ^ 2 \\\\ & = | H | \\sum _ { j = 1 } ^ n | \\langle w , \\rho ( g _ j ) v \\rangle | ^ 2 , \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} n '' : = | V ( H '' ) | \\geq n - | V ( A ) | - | R | \\overset { \\eqref { e q : R - s i z e } } { \\geq } n - \\kappa n - 2 \\nu n = ( 1 - \\kappa - 2 \\nu ) n . \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} H _ { \\gamma , \\phi } ( \\sigma ) : = - \\frac { 1 } { 2 } \\sum _ { \\substack { x , y \\in \\Lambda , \\\\ x \\sim y } } \\sigma _ x \\sigma _ y - \\sum _ { \\substack { x \\in \\Lambda , \\ , y \\in \\partial \\Lambda , \\\\ x \\sim y } } \\sigma _ x \\gamma _ y - \\sum _ { x \\in \\Lambda } \\phi _ x \\sigma _ x , \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} \\mathcal { A } : = \\left \\{ \\mathfrak { p } \\in \\mathcal { B } : \\mathfrak { q } \\in \\operatorname { S p e c } ( A ) \\mbox { a n d } \\mathfrak { q } \\subsetneq \\mathfrak { p } \\Rightarrow \\mathfrak { q } \\notin \\mathcal { B } \\right \\} . \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} \\Delta ^ { ( 0 ) } & = \\mathrm { i d } , \\\\ \\Delta ^ { ( 1 ) } & = \\Delta , \\\\ \\Delta ^ { ( N ) } & = ( \\underbrace { \\mathrm { i d } \\otimes \\cdots \\otimes \\mathrm { i d } } _ { } \\otimes \\Delta ) \\circ \\Delta ^ { ( N - 1 ) } & & ( N \\ge 2 ) . \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{align*} \\mathcal { F } _ n = \\sigma ( C _ 1 , \\dots , C _ n ) , n \\geq 1 . \\end{align*}"}
-{"id": "7227.png", "formula": "\\begin{align*} \\frac { ( a x t , b y t ; q ) _ \\infty } { ( x t , y t ; q ) _ \\infty } { _ 3 \\phi _ 2 } \\left ( { { q ^ { - m } , x t , y t } \\atop { a x t , b y t } } ; q , q \\right ) = \\sum _ { n = 0 } ^ \\infty \\lambda _ n \\Phi _ n ^ { ( a , b ) } ( x , y | q ) . \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{align*} a _ 0 : = x ( x \\neq x ) . \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} \\varphi _ \\lambda ' / \\varphi _ \\lambda ( z ( \\lambda , \\hat { X } ) ) & = \\eta _ 1 / \\omega _ 1 \\int _ 1 ^ { \\hat { X } } \\frac { d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } + \\int _ { 1 } ^ { \\hat { X } } \\frac { ( X - \\frac 1 3 ( \\lambda + 1 ) ) d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } + \\pi i / \\omega _ 1 . \\end{align*}"}
-{"id": "1258.png", "formula": "\\begin{align*} A = r \\cdot T \\cdot R \\cdot T ^ { - 1 } , \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{align*} w ( v ) = p ( v ) \\cdot ( \\phi _ t ( v ) ) . \\end{align*}"}
-{"id": "8974.png", "formula": "\\begin{align*} \\mathcal { R } f ( z ) = f \\left ( \\left ( \\frac { z } { \\iota _ 1 ( \\boldsymbol { \\delta } ) } , \\frac { z } { \\iota _ 2 ( \\boldsymbol { \\delta } ) } , \\dots , \\frac { z } { \\iota _ m ( \\boldsymbol { \\delta } ) } \\right ) \\right ) \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{align*} \\sigma _ q = \\frac { 1 } { \\sqrt { q } } \\left ( \\begin{array} { r r } 1 & q - 1 \\\\ 1 & - 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} M ^ { L , \\varepsilon } _ p : = \\left | \\{ x \\in \\{ p _ 0 , \\ldots , p _ { | p | - 1 } \\} \\colon \\ , \\xi ( x ) \\le ( 1 - \\varepsilon ) a _ L \\} \\right | \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} & - \\frac { t } { 2 } < - \\frac { t n } { 2 n + 1 } < 1 - \\frac { t } { 2 } = \\frac { 2 - t } { 2 } \\\\ & - \\frac { t + 1 } { 2 } = \\frac { 1 - t } { 2 } - 1 < - \\frac { t n } { 2 n + 1 } < \\frac { 1 - t } { 2 } \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} u _ { \\rm t r u e } ( x , 0 ) = k \\sin ( 2 \\pi x ) , \\forall x \\in [ 0 , 1 ] \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} \\alpha = \\int _ 0 ^ \\tau \\psi ( s ) d s , \\ \\ \\tau > 0 . \\end{align*}"}
-{"id": "6358.png", "formula": "\\begin{align*} H _ { \\mathcal { R } ( I ^ n ) _ + } ^ 0 \\left ( L ^ { I ^ n } ( L ) \\right ) = 0 \\mbox { f o r a l l } n \\gg 0 . \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} ( - \\Delta _ y ) ^ { \\alpha / 2 } \\varepsilon _ { \\alpha , n } ( x - y ) = \\delta _ x . \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} \\tilde { \\beta } & = - \\frac 1 2 | A | ^ 2 , & \\| \\beta _ 0 \\| ^ 2 & = \\frac 1 4 | A | ^ 4 . \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} \\dot { f } _ i + \\dot { f } _ { i + 1 } = f _ { i + 1 } ^ 2 - f _ i ^ 2 + \\beta _ { i + 1 } - \\beta _ i , i = 1 , 2 , \\ldots , 2 k + 1 \\end{align*}"}
-{"id": "8104.png", "formula": "\\begin{align*} \\underset { r \\to 0 } { \\lim } N ( r ) = \\underset { r \\to 0 } { \\lim } N _ 1 ( r ) \\end{align*}"}
-{"id": "3499.png", "formula": "\\begin{align*} f ( x ) = \\sum _ i \\max _ j ( a _ { i j } ^ T x + b _ { i j } ) , \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} \\phi ^ 2 - [ t ] \\phi + [ q ] = [ o ] . \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} c _ 1 ( \\overline { L } ) = - 2 d d ^ c ( \\log \\| s \\| + m g _ { D _ r , p } ) + \\delta _ { \\div ( s ) - m [ p ] } + m \\pi _ { r , p } \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} ( \\wp _ { u n i v } ' ) ^ 2 = 4 \\wp _ { u n i v } ^ 3 - \\underline g _ 2 \\wp _ { u n i v } - \\underline g _ 3 \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ p o s ] { l l } R i c _ { B } + \\nabla _ { B } \\nabla _ { B } \\beta + ( \\varphi - m h ^ { - 1 } ) \\nabla _ { B } \\nabla _ { B } h = \\lambda g _ { B } , \\\\ R i c _ { F } + h \\nabla _ { F } \\nabla _ { F } \\varphi = [ h \\Delta _ { B } h + ( m - 1 ) | \\nabla _ { B } h | ^ { 2 } - h ( \\nabla _ { B } h ) \\beta - \\varphi h ( \\nabla _ { B } h ) h + \\lambda h ^ { 2 } ] g _ { F } . \\end{array} \\right . \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} \\tau = t ^ { - 1 } , \\ \\ X = x / t ; \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} \\sum _ { k \\in \\Z } f ( k + x ) = \\sum _ { l \\in \\Z } \\widehat { f } ( l ) e ^ { 2 \\pi i l x } , x \\in \\R . \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} I _ { \\alpha , s } ( x ) \\approx \\left \\{ \\begin{array} { r r r } ( 1 - | x | ^ { 2 } ) ^ { s } , & s > 0 , \\\\ 1 - \\log { ( 1 - | x | ^ { 2 } ) } , & s = 0 , \\\\ c , & s < 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2047.png", "formula": "\\begin{align*} \\lim _ { \\substack { \\alpha \\to 0 \\\\ \\alpha \\to 0 \\\\ M \\to + \\infty } } \\sum _ { j = 1 } ^ { \\lfloor \\frac { 1 } { \\alpha _ * ^ 2 } \\rfloor } \\P \\left ( \\left \\vert \\xi ( \\alpha _ * , M , \\tau _ j ) - \\xi ( \\alpha _ * , M , \\tau _ { j - 1 } ) \\right \\vert > \\delta \\right ) = 0 , \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} X ( t ) \\stackrel { d } { = } \\int _ { E } f _ { t } ( s ) M ( d s ) , \\ ; \\ ; \\ ; t \\in \\mathbb { R } ^ { d } , \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{align*} { - D } _ { \\alpha { } } { ( \\hslash { } \\nabla { } ) } ^ { \\alpha { } } \\psi ( x , t ) + V ( x , t ) \\psi ( x , t ) = i \\hslash { } \\frac { \\partial { } } { \\partial { } t } \\psi ( x , t ) . \\end{align*}"}
-{"id": "4389.png", "formula": "\\begin{align*} \\mathcal { R } & = \\int _ 1 ^ { \\xi } \\left ( \\int _ { 1 } ^ { \\hat { X } } \\frac { ( X - \\lambda / 3 - \\sqrt { X ( X - \\lambda ) } ) d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\right ) \\frac { d \\hat { X } } { 2 \\sqrt { \\hat { X } ( \\hat { X } - 1 ) ( \\hat { X } - \\lambda ) } } \\\\ \\mathcal { R } _ \\varphi & = \\lambda \\int _ 1 ^ { \\xi } \\frac { R _ { \\varphi } d \\hat { X } } { 2 \\sqrt { \\hat { X } ( \\hat { X } - 1 ) ( \\hat { X } - \\lambda ) } } . \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} y _ 0 = - \\frac { ( k - 1 ) } { 2 \\pi } \\log \\left ( 1 - \\frac { 1 } { \\ell + 1 } \\right ) = \\frac { ( k - 1 ) } { 2 \\pi } \\left | \\log \\left ( 1 - \\frac { 1 } { \\ell + 1 } \\right ) \\right | . \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} g ( T ^ \\psi , \\delta T ^ \\psi ) & = T _ { a b } \\nabla ^ a U ^ b - 2 U ^ m T _ a ^ { \\phantom { a } n } \\varphi ^ { \\phantom { m } } _ { m n b } \\\\ & = - \\nabla ^ a ( T _ { a b } ) U ^ b + \\nabla ^ a ( T _ { a b } U ^ b ) - 2 U ^ m T _ a ^ { \\phantom { a } n } \\varphi ^ { \\phantom { A } } _ { m n b } . \\end{align*}"}
-{"id": "2054.png", "formula": "\\begin{align*} \\Lambda _ 0 : = \\{ d \\alpha ( x ) \\colon x \\in U \\} \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{align*} \\frac { \\tilde { C } _ d } { 2 ( 2 + \\tilde { C } _ d ) } = \\frac { C _ d } { 2 ( 2 4 ( 1 + C _ d ) + C _ d ) } \\geq \\frac { C _ d } { 1 4 8 } \\geq \\epsilon , \\end{align*}"}
-{"id": "8549.png", "formula": "\\begin{align*} \\sigma _ m = \\omega _ m + d [ ( \\iota _ X \\omega _ 0 ) I _ m ] \\rlap { . } \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} \\psi _ 1 ( x ) = 2 x ^ { 3 / 4 } | \\eta ( i x ) | ^ 3 , \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{align*} \\tau _ p ( x , y ) = \\log \\Big ( 1 + 2 \\frac { d ( x , y ) } { \\sqrt { d ( x , p ) d ( y , p ) } } \\Big ) . \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} H _ m : = \\Xi _ m ^ { - 1 } ( A ) X ^ { - 2 } \\Xi _ m ( A ) . \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} p \\left ( x \\right ) : = \\alpha x ^ { 3 } + \\beta x ^ { 2 } + \\gamma x + \\delta , \\end{align*}"}
-{"id": "7004.png", "formula": "\\begin{align*} \\vert \\tilde { z } _ 2 ^ 3 + 9 \\tilde { z } _ 3 ^ 3 \\vert = \\vert \\tilde { z } _ 1 ^ 3 \\vert < 3 ^ { 9 b + 6 } . \\end{align*}"}
-{"id": "5971.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\left [ \\sum _ { i = 1 } ^ r ( \\hat { b } _ h ^ i ( t ) - \\xi _ h ^ i ( t ) ) - \\sum _ { j = 1 } ^ s \\hat { a } _ h ^ j ( t ) \\right ] d t \\leq 0 , ~ \\mbox { f o r a l l } h \\end{align*}"}
-{"id": "3170.png", "formula": "\\begin{align*} d \\omega = 0 = d \\Omega . \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{align*} | \\partial _ { x } ^ { m } R _ { \\alpha } ( x , y ) | \\leq C _ { \\alpha } ^ { m } | r x - \\xi | ^ { - n + 1 - \\alpha - m } , \\enspace 0 \\leq r < 1 , \\enspace \\mbox { a n d } \\enspace y = r \\xi , | \\xi | = 1 . \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} & ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( \\Delta ( ( \\operatorname { i d } \\otimes \\omega ) [ ( 1 \\otimes \\tilde { y } c ) ( \\Delta a ) ] ) \\bigr ) \\\\ & = ( \\psi \\otimes \\operatorname { i d } ) \\bigl ( ( \\operatorname { i d } \\otimes \\operatorname { i d } \\otimes \\omega ) [ ( 1 \\otimes 1 \\otimes \\tilde { y } ) ( 1 \\otimes 1 \\otimes c ) ( 1 \\otimes E ) \\Delta _ { 1 3 } ( a ) ] \\bigr ) . \\end{align*}"}
-{"id": "6586.png", "formula": "\\begin{align*} \\int \\limits _ 0 ^ T \\frac { 1 } { 1 - C } ( 1 + t - s ) ^ { 1 - C } \\Big | _ s ^ T \\ , d s = \\int \\limits _ 0 ^ T \\frac { 1 } { 1 - C } ( 1 + T - s ) ^ { 1 - C } \\ , d s - \\int \\limits _ 0 ^ T \\frac { 1 } { 1 - C } \\ , d s . \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} \\omega ^ a ( V _ b ) = \\bar \\omega ^ a ( \\bar V _ b ) = \\delta _ { a b } , \\ \\omega ^ a ( \\bar V _ b ) = \\bar \\omega ^ a ( V _ b ) = 0 . \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} Q _ { 1 , 0 } & = \\{ \\xi \\in \\Lambda ^ 1 _ C Q ^ * | \\ \\xi ( Z ) = 0 , \\ \\forall Z \\in Q ^ { 0 , 1 } \\} , \\\\ Q _ { 0 , 1 } & = \\{ \\xi \\in \\Lambda ^ 1 _ C Q ^ * | \\ \\xi ( Z ) = 0 , \\ \\forall Z \\in Q ^ { 1 , 0 } \\} . \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} [ ( x \\alpha ) \\cdot ( x \\backslash y ) ] \\cdot z = [ ( x \\alpha ) \\cdot ( x \\backslash ( y z ) ] \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} d V ( t ) = \\left ( - \\frac { 1 } { \\theta } ( V ( t ) - \\hat { V } ) + I ( t ) \\right ) d t + \\sigma d W ( t ) \\ t > 0 = : T _ 0 . \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{align*} \\kappa + \\frac { \\langle \\gamma , N \\rangle } { 2 } = 0 . \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} \\langle \\mathsf { S } \\rangle _ 0 & : = \\langle \\mathsf { S } \\rangle \\\\ \\langle \\mathsf { S } \\rangle _ { n + 1 } & : = \\langle \\mathsf { S } \\rangle _ n \\diamond \\langle \\mathsf { S } \\rangle . \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{align*} I ( y , z ) = \\{ \\ , i \\in [ m ] \\mid y _ i \\ge z \\ , \\} . \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{align*} \\begin{aligned} ( 1 - \\lambda ) \\sum _ i p _ { 0 , i } e ^ { \\varpi \\left ( \\frac { p _ { 0 , i } } { q _ { 0 , i } } \\right ) } + & \\lambda \\sum _ i p _ { 1 , i } e ^ { \\varpi \\left ( \\frac { p _ { 1 , i } } { q _ { 1 , i } } \\right ) } \\ge \\sum _ i p _ { \\lambda , i } e ^ { \\varpi \\left ( \\frac { p _ { \\lambda , i } } { q _ { \\lambda , i } } \\right ) } , \\end{aligned} \\end{align*}"}
-{"id": "5767.png", "formula": "\\begin{align*} \\lambda _ j ( 0 ) = l _ { n + 1 , j } \\geq \\frac { 1 } { n + 1 } . \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} \\lim _ { d \\rightarrow \\infty } E _ d ( Q ) & = \\frac { 1 } { 2 } \\bigg ( 1 + \\frac { 1 } { q } \\bigg ) \\bigg ( \\frac { 1 } { 1 - \\frac { 1 } { q } } \\bigg ) ^ 2 - \\frac { 1 } { 2 } \\bigg ( 1 - \\frac { 1 } { q } \\bigg ) \\bigg ( \\frac { 1 } { 1 - \\frac { 1 } { q ^ 2 } } \\bigg ) \\\\ & = \\frac { 2 } { q } + \\frac { 2 } { q ^ 2 } + \\frac { 4 } { q ^ 3 } + \\frac { 4 } { q ^ 4 } + \\frac { 6 } { q ^ 5 } + \\frac { 6 } { q ^ 6 } + \\frac { 8 } { q ^ 7 } + \\frac { 8 } { q ^ 8 } + \\frac { 1 0 } { q ^ 9 } + \\ldots \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{align*} \\Phi ' : = ( ( A _ 1 , b _ 1 ) , \\dots , ( A _ { \\ell - 1 } , b _ { \\ell - 1 } ) , ( ( A _ { \\ell } ) _ { \\hat { i } } , ( b _ { \\ell } ) _ { \\hat { i } } ) , ( ( A _ { \\ell + 1 } ) ^ { \\hat { i } } , b _ { \\ell + 1 } ) , ( A _ { \\ell + 2 } , b _ { \\ell + 2 } ) , \\dots , ( A _ L , b _ L ) ) . \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} \\partial _ k P _ { d , t } ^ k & = \\sum _ { i = 0 } ^ { d - 1 } \\sum _ { | \\alpha | = i - 1 } \\partial _ k D ^ { \\alpha } _ x U _ k ( 0 , t ) \\frac { x ^ { \\alpha } } { \\alpha ! } = 0 , \\end{align*}"}
-{"id": "3307.png", "formula": "\\begin{align*} B _ k ( s ) = \\tilde { b } _ k + W _ { k , n - k } A _ k ( s ) \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{align*} \\binom { n - 1 } { t - 1 } . \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} n _ w \\cdot s ( n _ w ^ { - 1 } \\cdot t \\cdot x ) & = n _ w \\cdot s ( n _ w ^ { - 1 } t n _ w \\cdot x ) \\\\ \\intertext { s i n c e $ n _ w \\in H $ , } & = n _ w n _ w ^ { - 1 } t n _ w \\cdot s ( x ) \\\\ \\intertext { s i n c e $ n \\in N _ K ( T _ s ) $ a n d $ s $ i s $ T _ s $ - i n v a r i a n t , } & = \\chi ( n _ w ) t \\cdot s ( x ) \\\\ \\intertext { w h e r e $ \\chi $ i s t h e c h a r a c t e r o f $ H $ a s s o c i a t e d t o $ \\mathcal { L } ^ { 2 k } $ } & = \\chi ( n _ w ) s ( t \\cdot x ) . \\end{align*}"}
-{"id": "7525.png", "formula": "\\begin{align*} & \\int _ { s } ^ t 2 \\hat b _ + ^ j ( r , q _ r ) ( \\tilde \\Sigma ^ { - 1 } ) _ { j k } ( r , x _ r ) \\circ d q ^ k _ r \\\\ = & \\int _ { s } ^ t \\beta ( r , q _ r ) ( - \\partial _ { q ^ k } V - \\partial _ { q ^ k } \\beta ^ { - 1 } + \\tilde F _ k ) ( r , q _ r ) \\circ d q ^ k _ r \\\\ = & \\int _ { s } ^ t \\partial _ { q ^ k } ( - \\beta V + \\ln \\circ \\beta ) ( r , q _ r ) \\circ d q ^ k _ r + \\int _ { s } ^ t ( V \\nabla _ q \\beta + \\beta \\tilde F ) _ k ( r , q _ r ) \\circ d q ^ k _ r \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} 0 = & \\int _ { \\Delta ' } ( \\dot { u } ^ * _ { t , i , j } u _ t ^ { * , i , j } P _ { D H } ' + \\dot { u } ^ * _ { t , i } u _ { t , j } ^ { * , i , j } P _ { D H } ' - \\dot { u } ^ * _ { t } u _ { t , j } ^ { * , i , j } P _ { D H , i } ' - \\dot { u } ^ * _ { t } u _ t ^ { * , i , j } P _ { D H , i , j } ' ) d p \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} { \\rm M } ( P ) = \\prod _ { 1 \\leq i \\leq n } \\max \\{ 1 , | \\beta _ { i } ( P ) | \\} > d \\end{align*}"}
-{"id": "8207.png", "formula": "\\begin{align*} \\liminf _ { i \\to \\infty } \\delta ( \\widetilde { C } _ i ) \\ge \\frac { 1 } { 2 } - \\frac { 1 } { 2 } \\Bigl ( \\frac { 1 } { \\ell ^ { \\lceil r / 2 \\rceil } - 1 } + \\frac { 1 } { \\ell ^ { \\lfloor r / 2 \\rfloor } - 1 } \\Bigr ) q = \\ell ^ r , r > 1 \\end{align*}"}
-{"id": "5820.png", "formula": "\\begin{align*} \\mathbb { L } _ i f _ { \\nu } ( z ) & = t ^ { \\theta _ i ( s _ i \\nu ) } f _ { s _ i \\nu } ( z ) - t ^ { \\theta _ i ( \\nu ) } f _ { \\nu } ( z ) = \\sum _ { \\nu ' } \\ell ( \\nu ' , \\nu ) f _ { \\nu ' } ( z ) , \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} \\mathbb { J } _ S & = \\{ ( i , j ) \\in \\mathbb { Z } _ { > 0 } ^ 2 \\ | \\ \\gcd ( i , p ) \\neq 1 , \\gcd ( j , p ) = 1 \\} \\\\ \\mathbb { J } _ T & = \\{ ( i , j ) \\in \\mathbb { Z } _ { > 0 } ^ 2 \\ | \\ \\gcd ( i , p ) = 1 \\} , \\end{align*}"}
-{"id": "6509.png", "formula": "\\begin{align*} \\displaystyle { B = \\sigma _ b ^ 2 ( \\alpha I _ n + ( 1 - \\alpha ) \\tilde { B } ) , \\mbox { w i t h } \\tilde { B } _ { i , j } = e ^ { - \\frac { d ( i , j ) ^ 2 } { L ^ 2 } } } \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} \\varphi ^ * \\omega = \\| \\varphi ' ( z ) \\| _ { R , h } ^ 2 d \\mu _ R \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathbb { Z } } L _ i \\left [ \\psi ( \\cdot , \\mu ) \\right ] ( \\nu ) = \\sum _ { i \\in \\mathbb { Z } } \\Big ( t \\nu _ i ( 1 - \\nu _ { i + 1 } ) + ( 1 - \\nu _ i ) \\nu _ { i + 1 } \\Big ) \\Big [ \\psi ( s _ i \\nu , \\mu ) - \\psi ( \\nu , \\mu ) \\Big ] , \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} \\eta _ 0 : = \\frac { 1 } { r ( Q [ 0 ] ) } > 0 \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} W _ n ^ k ( \\varepsilon ) = \\{ ( v _ 1 , \\ldots , v _ k ) \\colon D _ { { v _ s } } \\in [ \\varepsilon , 1 / \\varepsilon ] \\sqrt { \\mu n } \\forall s \\in [ k ] \\} , \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} Z ( T _ 0 , T _ 1 ) : = \\bigcap _ { i = 1 } ^ { M } Z _ i ( T _ 0 , T _ 1 ) , \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} ( \\alpha _ t , \\varphi _ t ) : = ( \\beta _ { \\infty } , 0 ) - D _ t ^ 1 \\xi _ t . \\end{align*}"}
-{"id": "9177.png", "formula": "\\begin{align*} W _ { n , k } = \\sum ^ { n _ 2 - n _ 1 = n } _ { k _ 0 + k _ 1 + k _ 2 = k } q ^ { ( k - 1 ) n _ 2 } \\cdot L _ { n _ 1 , k _ 1 } E _ { k _ 0 } U _ { n _ 2 , k _ 2 } \\Big | _ \\Delta \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} f _ q ( t ) = \\sum _ { l = 1 } ^ m \\lambda _ l f _ q ( r _ l ) , \\end{align*}"}
-{"id": "8401.png", "formula": "\\begin{align*} \\frac { d } { d t } d _ { A ( t ) } \\eta ( t ) & = [ A ' , \\eta ] + d _ A \\eta ' \\\\ & = - [ d _ A ^ * B , \\eta ] + d _ A d _ A ^ * w \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{align*} b _ { j , m } \\left ( A _ { i , j } + A _ { i , m } \\right ) = 0 \\end{align*}"}
-{"id": "5320.png", "formula": "\\begin{align*} { \\cal C } ( \\bar { x } ; q ; P ) \\ , = \\ , \\left \\{ \\ , v \\ , \\in \\ , { \\cal T } ( \\bar { x } ; P ) \\ , \\mid \\ , \\exists \\lambda \\ , \\in \\ , \\Lambda ( \\bar { x } ) \\mbox { s u c h t h a t } A _ { \\ , i \\bullet } v \\ , = \\ , 0 \\mbox { f o r a l l } i \\ , \\in \\ , \\mbox { s u p p } ( \\lambda ) \\ , \\right \\} , \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} \\begin{aligned} \\psi ( t , x ) & = \\int _ { \\mathbb { R } } G _ t ( x - y ) \\psi _ 0 ( y ) d y - \\frac { 1 } { 2 } \\int _ 0 ^ t \\int _ { S } { \\rm s i g n } ( y ) G _ { t - s } ( x - z ) \\psi ( s , z ) \\sigma _ s ( y ) d z W ( d s , d y ) \\\\ & + \\frac { 1 } { 8 } \\int _ 0 ^ t \\int _ { S } G _ { t - s } ( x - z ) \\psi ( s , z ) \\sigma ^ 2 _ s ( y ) d z d y d s , \\end{aligned} \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} \\gamma = \\frac { 1 } { 2 } ( \\tilde \\gamma + \\tilde \\gamma ^ T ) , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\tilde \\gamma ^ { - 1 } + ( \\tilde \\gamma ^ T ) ^ { - 1 } = 2 ( \\tilde \\gamma ^ T ) ^ { - 1 } \\gamma \\tilde \\gamma ^ { - 1 } , \\end{align*}"}
-{"id": "3142.png", "formula": "\\begin{align*} D _ R = \\left ( \\frac { \\partial ^ 2 S } { \\partial \\beta ^ 2 } \\right ) . \\left ( \\frac { \\partial ^ 2 S } { \\partial \\alpha ^ 2 } \\right ) - \\left ( \\frac { \\partial ^ 2 S } { \\partial \\beta \\partial \\alpha } \\right ) ^ 2 \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} \\bigsqcup _ { c _ 1 + c _ 2 = e } \\mathcal { M } _ { c _ 1 } ( \\tilde { P _ X } , n , \\mu _ X ^ { + } ) \\times \\mathcal { M } _ { c _ 2 } ( \\tilde { P _ Y } , n , \\mu _ Y ^ { + } ) \\xrightarrow { \\cong } \\bigsqcup _ { \\tilde { P } \\in S } \\mathcal { M } _ c ( \\tilde { P } , n , \\mu ^ { + } ) . \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} \\gamma ( \\alpha ) = \\left ( \\cos \\left ( \\frac { \\pi \\alpha } { 2 } \\right ) \\right ) ^ { \\frac { 1 } { \\alpha } } . \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} d _ { a , n + 1 } ( t ) - d _ { a , n } ( t ) & = \\left ( p _ { a + 1 , n + 1 } ( t ) - p _ { a , n + 1 } ( t ) \\right ) - \\left ( p _ { a + 1 , n } ( t ) - p _ { a , n } ( t ) \\right ) \\\\ & = q _ { a + 1 , n } ( t ) - q _ { a , n } ( t ) \\\\ & = ( a + 1 ) ( 1 + t ) ^ { n - a - 1 } ( 1 + t ^ { a + 1 } ) - a ( 1 + t ) ^ { n - a } ( 1 + t ^ a ) \\\\ & = ( 1 + t ) ^ { n - a - 1 } \\left ( ( 1 - a t ) + t ^ a ( t - a ) \\right ) \\ , . \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{align*} d _ { K } ^ 2 ( x + y ) = d _ { K } ^ 2 ( y ) + \\langle \\triangledown d _ { K } ^ 2 ( y ) , \\ , x \\rangle + \\frac { 1 } { 2 } \\langle D ^ 2 ( d _ { K } ^ 2 ( y ) ) x , \\ , x \\rangle + \\alpha ( y , x ) , \\end{align*}"}
-{"id": "3673.png", "formula": "\\begin{align*} & P ^ \\eta ( X _ { m t _ n + r } ^ { \\vec v } - X _ r ^ { \\vec v } \\leq a ) = P ^ \\eta ( X _ { m t _ n + r } ^ { \\vec v } - X _ { ( m - 1 ) t _ n + r } ^ { \\vec v } + X _ { ( m - 1 ) t _ n + r } ^ { \\vec v } - X _ r ^ { \\vec v } \\leq a ) \\\\ & \\leq \\sum _ { x , y \\in \\Z ^ d } P ^ \\eta ( X _ { m t _ n + r } ^ { \\vec v } - X _ { ( m - 1 ) t _ n + r } ^ { \\vec v } \\leq a - \\langle y - x , \\vec v \\rangle \\ , | \\ , A _ { x , y } ) P ^ \\eta ( X _ { ( m - 1 ) t _ n + r } = y , X _ r = x ) + \\P ^ \\eta ( \\mathcal E ^ c _ { m - 1 } ) \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{align*} ( V B F C ( \\tilde X - \\tilde D ) _ { u n i p } \\stackrel { D E } { \\to } V B ( ( F C ) ) ( \\tilde X , \\tilde D ) _ { u n i p } \\stackrel { \\widetilde { r e s } } { \\to } V B F C ( \\tilde X - \\tilde D ) _ { u n i p } ) = i d _ { V B F C ( \\tilde X - \\tilde D ) _ { u n i p } } \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} x ^ T ( t ) P _ 2 ^ { - 1 } x ( t ) & \\leq \\left [ \\int _ 0 ^ t x ^ T ( s ) ( A ^ T P _ 2 ^ { - 1 } + P _ 2 ^ { - 1 } A + \\sum _ { i = 1 } ^ m N _ i ^ T P _ 2 ^ { - 1 } N _ i ) x ( s ) \\right . d s \\\\ & \\ \\ \\ \\ \\ \\ \\ \\ \\ \\left . + 2 \\int _ 0 ^ t x ^ T ( s ) P _ 2 ^ { - 1 } B u ( s ) d s + \\int _ 0 ^ t x ^ T ( s ) P _ 2 ^ { - 1 } x ( s ) \\left \\| u ( s ) \\right \\| _ 2 ^ 2 d s \\right ] . \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} F _ { \\mathcal { L } } ( q ) = ( n + 1 ) \\Lambda _ Y - \\bar { S } _ { \\Theta } + \\sum \\frac { ( \\chi ^ { a c } - \\Lambda _ Y \\chi ) ( \\alpha ^ { \\vee } ) } { q ( \\alpha ^ { \\vee } ) } \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} M _ i \\leq \\sum _ { T } \\binom { C ( T ) } { d _ i } \\leq \\sum _ T \\binom { \\frac { ( 2 ^ { \\ell } + 1 ) n } { d _ i ^ { \\ell - 1 } } } { d _ i } \\leq \\sum _ T 2 ^ { \\frac { \\ell } { 1 . 8 8 } \\left ( ( 2 ^ { \\ell } + 1 ) e n \\right ) ^ { 1 / \\ell } } \\leq \\sum _ T 2 ^ { 3 \\ell n ^ { 1 / \\ell } } \\leq 2 ^ { \\left ( 3 \\ell + 1 \\right ) n ^ { 1 / \\ell } } \\end{align*}"}
-{"id": "6711.png", "formula": "\\begin{align*} x & : = ( a b ^ { - 1 } ) ^ { 3 } ( a ^ 2 b ^ { - 1 } ) ^ { 3 } ( a b ^ { - 1 } ) ^ 3 ( a ^ 2 b ^ { - 1 } ) ^ 4 \\ldots ( a b ^ { - 1 } ) ^ 3 ( a ^ 2 b ^ { - 1 } ) ^ { \\rho + 2 } \\\\ y & : = ( a b ^ { - 1 } ) ^ 3 ( a ^ 2 b ^ { - 1 } ) ^ { \\rho + 3 } ( a b ^ { - 1 } ) ^ 3 ( a ^ 2 b ^ { - 1 } ) ^ { \\rho + 4 } \\ldots ( a b ^ { - 1 } ) ^ { 3 } ( a ^ 2 b ^ { - 1 } ) ^ { 2 \\rho + 2 } \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{align*} ( k D _ 1 + H ) \\cdot \\Sigma _ k \\leq 2 n \\frac { ( k D _ 1 + H ) \\cdot m _ 2 H _ 2 \\cdot . . . \\cdot m _ n H _ n } { - K _ { \\cal F } \\cdot m _ 2 H _ 2 \\cdot . . . \\cdot m _ n H _ n } \\\\ = 2 n \\frac { ( k D _ 1 + H ) \\cdot H _ 2 \\cdot . . . \\cdot H _ n } { - K _ { \\cal F } \\cdot H _ 2 \\cdot . . . \\cdot H _ n } \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{align*} \\mathcal { L } _ X \\psi = \\tfrac { 1 } { 2 } X \\cdot \\psi - \\tfrac { 1 } { 4 } d X ^ \\flat \\cdot \\psi . \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} V ^ { 0 - 1 , \\hat { N } _ { \\beta } } ( u ) ( \\psi ) + V ^ { 0 - 2 , \\hat { N } _ { \\beta } } ( u ) ( \\psi ) = - V ( u ) ( \\psi ) + W ( u ) ( \\psi ) . \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} | \\partial _ { x } | ^ { \\sigma } f ( x ) = \\int _ { \\R } | \\xi | ^ { \\sigma } \\hat { f } ( \\xi ) e ^ { 2 \\pi i x \\xi } \\ , d \\xi , \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} \\begin{aligned} \\norm { e ^ { - ( t _ { 0 } - s ) B } F _ { y } ( u _ { j } , \\nabla y _ { j } , y _ { j } , x _ { j } , \\overline { b } ) } _ { L ^ { \\infty } ( \\Omega ) ^ 3 } & \\leq C e ^ { - t _ { 0 } \\omega } ( t _ { 0 } - s ) ^ { - \\frac { 3 } { q } } s ^ { - 3 ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\Big [ k ^ { u } _ { j } ( T ) k ^ { \\nabla y } _ { j } ( T ) + k ^ { \\nabla y } _ { j } ( T ) ^ 2 ( k ^ { x } _ { j } ( T ) + k ^ { y } _ { j } ( T ) + \\norm { \\overline { b } } ) \\Big ] \\end{aligned} \\end{align*}"}
-{"id": "8999.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } H _ n [ \\gamma _ n ^ { - 1 } s ] ( f + h _ n ) ( t _ n , \\mu _ n , w _ n ) = H _ 0 [ s ] f ( \\mu , w ) \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} f ^ * Q _ l & = \\sum _ j w _ { l j } P _ j \\\\ \\bar { d _ j } & = \\frac { d _ j + w _ { l j } - 1 } { w _ { l j } } , \\ ; \\ ; f ( P _ j ) = Q _ l \\\\ \\delta _ l & = m a x \\{ \\bar { d _ j } ; f ( P _ j ) = Q _ l \\} . \\\\ \\Delta & = \\sum _ l \\delta _ l Q _ l . \\\\ M & = L - \\Delta . \\\\ \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{align*} H ^ { \\epsilon , \\kappa } : = ( - i \\partial _ { x _ 1 } - A ^ \\Gamma _ 1 - A ^ { \\epsilon , \\kappa } _ 1 ) ^ 2 + ( - i \\partial _ { x _ 2 } - A ^ \\Gamma _ 2 - A ^ { \\epsilon , \\kappa } _ 2 ) ^ 2 + V _ \\Gamma \\ , . \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{align*} C _ { i _ 1 . . . i _ k } = \\left ( \\frac { 1 } { 2 \\pi } \\right ) ^ { n / 2 } \\int e ^ { - \\| w \\| ^ 2 / 2 } w _ { i _ 1 } . . . w _ { i _ k } d w . \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} \\mu ( d s , d t ) = e ^ { - \\d s ^ * D ( t ) s - G _ 1 ( t ) } \\det { D ( t ) } \\frac { d t d s } { ( 2 \\pi ) ^ n } \\end{align*}"}
-{"id": "2407.png", "formula": "\\begin{align*} E [ T ] = ( \\nu _ 1 + \\lambda \\nu _ 2 ) M \\ln M + ( \\nu _ 1 + \\lambda \\nu _ 2 ) ( \\gamma + \\ln \\nu _ 1 ) M + O \\left ( M ^ { 2 - \\lambda } \\ln M \\right ) , M \\to \\infty . \\end{align*}"}
-{"id": "6600.png", "formula": "\\begin{align*} & u \\left ( X \\cdot Y \\cdot Z \\cdot U \\cdot \\bar { \\psi } , \\bar { \\psi } \\right ) + u \\left ( X \\cdot Y \\cdot Z \\cdot \\bar { \\psi } , U \\cdot \\bar { \\psi } \\right ) = \\\\ & 2 u \\left ( X \\cdot Y \\cdot Z \\cdot U \\cdot \\bar { \\psi } , \\bar { \\psi } \\right ) - 2 u g ( U , X ) g ( Y , Z ) + 2 u g ( U , Y ) g ( X , Z ) - 2 u g ( U , Z ) g ( X , Y ) . \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} A _ { j } ( v _ 1 , . . . , v _ { k - 2 j } ) = \\int _ 0 ^ \\infty 2 \\beta ^ { - 1 } \\sum _ { \\alpha = 1 } ^ { k - 2 ( j - 1 ) - 1 } \\sum _ { \\delta > \\alpha } A _ { j - 1 } ^ { \\alpha \\delta } ( e ^ { - t \\tilde \\gamma } v _ 1 , . . . , e ^ { - t \\tilde \\gamma } v _ { k - 2 j } ) d t , \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} [ C _ D , B _ D ] = & O + \\frac { z _ 1 \\bar { z } _ 2 } { 4 } \\beta ( l _ j ) ( \\coth ( \\beta ( a ) ) - \\tanh ( \\beta ( a ) ) ) \\theta ( \\tau _ { \\beta } ) \\\\ & - \\frac { \\bar { z } _ 1 z _ 2 } { 4 } \\beta ( l _ j ) ( \\tanh ( \\beta ( a ) ) + \\coth ( \\beta ( a ) ) \\tau _ { \\beta } \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} r ( x ) = \\sum _ { l \\geq 0 } r _ l x ^ l = \\exp \\bigg ( \\sum _ { n \\geq 1 } \\frac { ( 1 - q ^ n ) ( 1 - t ^ { - n } ) } { 1 + q ^ n t ^ { - n } } \\frac { x ^ n } { n } \\bigg ) . \\end{align*}"}
-{"id": "4100.png", "formula": "\\begin{align*} \\varphi _ { n } ^ { F } ( \\underline { h } ) = \\left | F \\cap \\ker f \\right | ^ { n } - \\sum _ { E \\in \\mathcal { E } } | \\varphi _ { n } ^ { E } ( \\underline { h } ) | . \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} \\frac { d x _ r ( t ) } { d t } & = A _ { 1 1 } x _ r ( t ) + B _ 1 u ( t ) + \\sum _ { i = 1 } ^ m { N } _ { i , 1 1 } x _ r ( t ) u _ i ( t ) , \\\\ y _ r ( t ) & = C _ 1 x _ r ( t ) , \\ ; \\ ; \\ ; t \\geq 0 , \\end{align*}"}
-{"id": "7161.png", "formula": "\\begin{align*} \\int _ { \\R ^ 2 } F ( \\vec { t } ) \\ , d \\nu ^ \\epsilon ( \\vec { t } ) = \\iiint F ( ( | x - y | , | y - z | ) ) \\ , \\mu ^ \\epsilon ( x ) \\ , \\mu ^ \\epsilon ( z ) \\ , d x \\ , d \\mu ( y ) \\ , d z . \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} g \\ : = \\ : - \\ : \\frac { 1 } { n ^ 2 } d t ^ 2 + \\sum _ { k = 1 } ^ n t ^ { 2 p _ k } ( d x ^ k ) ^ 2 , \\end{align*}"}
-{"id": "6978.png", "formula": "\\begin{align*} J _ \\nu : = \\Delta \\setminus \\{ \\alpha _ { \\nu _ 1 } \\} \\subset \\Delta \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} X ' = \\frac { X } { \\sqrt { | t | } } \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} \\mathcal { A } _ { - \\frac { p } { 2 } } u = \\big ( \\big [ A _ { ( \\frac { p } { 2 } ) ^ { \\prime } } ^ { - \\frac { 1 } { 2 } } \\big ] ^ * u \\big ) | _ { W ^ { 1 , ( \\frac { p } { 2 } ) ^ { \\prime } } _ { 0 , \\sigma } } \\end{align*}"}
-{"id": "821.png", "formula": "\\begin{align*} K ( t ; x , x ) = K _ 0 ( t ; x , x ) \\mathbb E _ { t ; x , x } ( M _ t V _ t ) , \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} A _ { J X } = - J \\circ A _ X . \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} 1 = \\int F _ 0 ( x , y ) d \\mu _ \\theta , \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} Q ( g + f ^ { n } t ) & = Q ( g ) + f ^ { n } t Q ' ( g ) + ( f ^ { n } t ) ^ 2 \\tilde { Q } ( g , f ^ n t ) \\ \\ \\ \\ \\ \\tilde { Q } \\\\ & \\equiv Q ( g ) + f ^ { n } t Q ' ( g ) \\pmod { f ^ { n + m } } \\ \\ \\ \\ \\ n \\geqslant m . \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} \\Vert x ^ { k + 1 } - z \\Vert ^ { 2 } = \\Vert U _ { k } x ^ { k } - z \\Vert ^ { 2 } \\leq \\Vert x ^ { k } - z \\Vert ^ { 2 } - \\rho _ { k } \\Vert U _ { k } x ^ { k } - x ^ { k } \\Vert ^ { 2 } \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} \\Lambda ( x , \\delta ) & = \\max _ { j _ 1 , \\ldots , j _ n \\in \\{ 1 , \\ldots , q \\} } \\left | \\det \\left ( X ^ { \\delta } _ { j _ 1 } ( x ) | \\cdots | X ^ { \\delta } _ { j _ n } ( x ) \\right ) \\right | \\\\ & = \\max _ { j _ 1 , \\ldots , j _ n \\in \\{ 1 , \\ldots , q \\} } \\left | \\det \\left ( ( A X ^ { c \\delta } ) _ { j _ 1 } ( x ) | \\cdots | ( A X ^ { c \\delta } ) _ { j _ n } ( x ) \\right ) \\right | , \\end{align*}"}
-{"id": "7316.png", "formula": "\\begin{align*} Z = \\omega ^ * \\cap \\hat { X } \\setminus \\bigcup \\left \\{ \\hat { A } : A \\in \\mathcal A \\right \\} \\end{align*}"}
-{"id": "8113.png", "formula": "\\begin{align*} \\rho = \\sup \\{ r \\leq r _ 0 \\mid H ( r ) = 0 \\} . \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} I _ { t - 1 } ( Y ( a , b + 1 ) ) \\ I _ { t - 1 } ( Y ( a + 1 , b ) ) \\ = \\ I _ { t - 1 } ( Y ( a , b ) ) \\ I _ { t - 1 } ( Y ( a + 1 , b + 1 ) ) . \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{align*} v _ 2 \\left ( \\sum _ { d \\mid n } ( - 1 ) ^ { n - d } \\mu ( n / d ) \\binom { 2 d - 1 } { d - 1 } \\right ) , \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{align*} \\| u \\| _ { L _ { \\Phi } ( I ; U ) } = \\inf \\left \\{ k > 0 \\colon \\int _ { I } \\Phi \\left ( \\frac { \\| u ( s ) \\| } { k } \\right ) \\ , d s \\leq 1 \\right \\} . \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} w _ 1 ( k ) & = 1 \\qquad k \\ , , \\\\ w _ 2 ( k ) & = \\begin{cases} 0 , & k = 0 \\ , , \\\\ \\frac { 1 } { k } , & \\ , , \\end{cases} \\\\ w _ 3 ( k ) & = \\begin{cases} 0 , & k = 0 \\ , , \\\\ \\frac { 1 } { \\binom { n } { k } k } , & \\ , , \\end{cases} \\end{align*}"}
-{"id": "4093.png", "formula": "\\begin{align*} A \\| f \\| ^ 2 & \\le \\sum _ { j = 1 } ^ m \\sum _ { i \\in \\sigma _ j \\cap J } \\| \\Lambda _ { i j } f \\| ^ 2 \\le \\sum _ { j = 1 } ^ m \\sum _ { i \\in \\sigma _ j } \\| \\Lambda _ { i j } f \\| ^ 2 . \\end{align*}"}
-{"id": "904.png", "formula": "\\begin{align*} T _ 0 ( X ) = \\sqrt { \\det ( \\Delta _ 0 ) } = T _ 1 ( X ) . \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} \\int F ( x , y ) d \\nu _ j = \\int F ( x , y ) d \\mu _ j \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} \\widetilde { \\xi } ^ z _ L ( y ) : = \\left \\{ \\begin{array} { l l } \\xi ( z ) \\vee ( a _ L - c _ \\ast + \\delta _ \\sigma ^ { - 1 } ) & y = z , \\\\ \\xi ( y ) \\wedge ( a _ L - c _ \\ast ) & \\end{array} \\right . \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} P _ \\ell ( x ) & : = \\mathrm { d e t } ( \\mathrm { I d } - \\rho _ { \\overline { f } } ( 1 ) ( \\mathrm { F r } _ \\ell ) x : T _ { \\overline { f } } ( 1 ) ) \\\\ & = 1 - \\overline { a _ \\ell ( f ) } \\ell ^ { - 1 } x + \\overline { \\psi } ( \\ell ) \\ell ^ { - 1 } x ^ 2 \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} { \\rm K e r } ( \\nabla g ( \\overline { x } ) ) \\cap D ^ * \\mathcal { N } _ { K } ( g ( \\overline { x } ) | \\overline { \\lambda } ) ( 0 ) = \\{ 0 \\} . \\end{align*}"}
-{"id": "7387.png", "formula": "\\begin{align*} \\widetilde { u } ( x ) : = \\varphi ( x ) \\psi ( x ) , \\end{align*}"}
-{"id": "2277.png", "formula": "\\begin{align*} L ( U X U ^ { - 1 } ) = V L ( X ) V ^ { - 1 } \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} T _ { i , j , k } : = \\langle a , b ; a ^ i , b ^ j , ( a b ) ^ k \\rangle \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{align*} H _ n : = \\lfloor \\frac { n } { 2 \\pi } \\bigl ( 2 \\arcsin \\bigl ( \\frac { \\kappa } { 2 } \\bigr ) ~ - ~ \\frac { \\kappa ^ 2 } { 1 - \\kappa } \\bigr ) - 1 \\rfloor . \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} ( p ( x , D _ x ) ( u \\circ \\kappa ) ) \\circ \\kappa ^ { - 1 } = : p _ \\kappa ( x , D _ x , x ) u \\end{align*}"}
-{"id": "2972.png", "formula": "\\begin{align*} R _ n ( q ) = \\sum _ { d \\mid n } ( - 1 ) ^ { n - d } \\mu ( n / d ) q ^ { \\binom { n } { 2 } - \\frac { n } { d } \\binom { d } { 2 } } P _ d ( q ^ { n / d } ) . \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} Q _ 1 & = \\int _ { \\tau _ 1 } ^ { \\tau _ 2 } \\int ^ { y } _ { \\left ( \\frac { ( 1 + \\epsilon _ 1 ) } { \\phi _ i \\left ( \\rho - \\epsilon _ 1 y ^ { \\alpha } \\right ) } \\right ) ^ { - \\frac { 1 } { \\alpha } } } f _ { r _ m , r _ t } ( x , y ) d x d y , \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} \\tilde { h } ' ( r ) = \\frac { 4 } { r } \\tilde { i } ( r ) + \\frac { a } { r } \\tilde { h } ( r ) , \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} \\lim _ { \\Re s \\to \\infty } e ^ { s l ( \\gamma ) } I _ b ( s ) ( 1 - e ^ { - l ( \\gamma ) } ) = 1 . \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} g \\cdot m = \\frac { \\alpha \\xi ( \\frac { \\| m \\| ^ 2 } { 2 } + v ^ T m + \\frac { \\| v \\| ^ 2 } { 2 } ) + A ( m + v ) } { \\frac { \\alpha } { 2 } \\| \\xi \\| ^ 2 ( \\frac { \\| m \\| ^ 2 } { 2 } + v ^ T m + \\frac { \\| v \\| ^ 2 } { 2 } ) + \\xi ^ T A ( m + v ) + \\alpha ^ { - 1 } } . \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{align*} \\begin{cases} \\mathfrak { D } \\mathfrak { b } + r ^ { \\frac { 1 } { 2 } } \\mathfrak { b } * \\mathfrak { b } = \\mathfrak { v } \\\\ \\lim \\limits _ { s \\to \\pm \\infty } \\mathfrak { b } = \\mathfrak { b } _ { \\pm } , \\\\ \\end{cases} \\end{align*}"}
-{"id": "2886.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\gamma ^ { ( 1 ) } _ { N , t } = | u _ t \\rangle \\langle u _ t | , \\quad \\forall t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "887.png", "formula": "\\begin{align*} \\begin{array} { l l l l l } \\hat { f } ( \\xi ) & = & \\displaystyle \\hat { u _ * } + i \\int _ 1 ^ t \\int e ^ { i ( \\eta ^ 2 - 2 \\xi \\eta ) s } e ^ { - s } { \\hat { f } ( \\xi - \\eta , s ) \\hat { v _ * } ( \\eta , s ) } d \\eta d s \\\\ & & \\displaystyle + i \\int _ 1 ^ t \\int _ 1 ^ s \\iint e ^ { - 2 i \\eta \\sigma s } F ( s , \\xi , \\xi - \\sigma , \\eta ) d s d \\eta d \\sigma , \\end{array} \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} \\phi _ U ( \\mathbf { u } ) - \\phi _ V ( \\mathbf { u } ) = \\phi _ { U V } ( \\zeta ) + \\overline { \\phi _ { U V } ( \\zeta ) } \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{align*} \\int _ M f d V = \\int _ { S ^ { n - 1 } } \\int _ 0 ^ { \\rho ( u ) } f ( _ P ( t u ) ) J ( u , t ) t ^ { n - 1 } d t d u , \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} 0 & = ( y ' ( x ) z '' ( x ) - z ' ( x ) y '' ( x ) ) [ 1 + y _ { x } ^ { 2 } + z _ { x } ^ { 2 } ] ^ { - 3 / 2 } \\\\ z & = z '' ( x ) [ 1 + y _ { x } ^ { 2 } + z _ { x } ^ { 2 } ] ^ { - 3 / 2 } \\\\ y & = y '' ( x ) [ 1 + y _ { x } ^ { 2 } + z _ { x } ^ { 2 } ] ^ { - 3 / 2 } . \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} E ^ \\eta [ Y _ 1 ] & = v + \\delta \\Big ( ( 2 d - 1 ) v _ { \\min } - \\sum _ { j \\not = j _ { \\min } } v _ j \\Big ) = v - 2 d v _ { \\max } \\delta > v - \\varepsilon / 2 , \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{align*} b _ 0 ( B ) = 1 , b _ 1 ( B ) = b _ 3 ( B ) = b _ 4 ( B ) = b _ 5 ( B ) = b _ 6 ( B ) = 0 , b _ 2 ( B ) = p . \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} C ( k _ 1 , k _ 2 , \\dots , k _ N ) = \\dfrac { A ( k _ 1 , k _ 2 , \\dots , k _ N ) } { B ( k _ 1 , k _ 2 , \\dots , k _ N ) } , \\end{align*}"}
-{"id": "6046.png", "formula": "\\begin{align*} \\begin{array} { c c c c } \\Gamma : & C ( [ 0 , T ] ; \\overline { X } _ 2 ) & \\longrightarrow & C ( [ 0 , T ] ; \\overline { X } _ 2 ) \\\\ & ( \\eta , w ) & \\longmapsto & \\Gamma ( \\eta , w ) = ( \\overline { \\eta } , \\overline { w } ) , \\end{array} \\end{align*}"}
-{"id": "103.png", "formula": "\\begin{align*} \\| \\alpha \\| _ { L ^ 2 } ^ 2 = i \\int _ { S _ q } \\alpha \\wedge \\bar \\alpha = \\int _ { S _ q } | \\alpha | ^ 2 \\ , d A , \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} S _ m ( \\nu ^ { + } ) = m \\cdot ( \\nu ^ { + } _ 1 , \\dots , \\nu ^ { + } _ { r m } , 0 ^ { n - r m } ) + ( 1 , \\dots , n ) = ( \\underbrace { m \\nu ^ { + } _ 1 + 1 , \\dots , m \\nu ^ { + } _ { r m } + r m } _ { r m } , \\underbrace { r m + 1 , \\dots , n } _ { n - r m } ) . \\end{align*}"}
-{"id": "6372.png", "formula": "\\begin{align*} X = ( X _ 0 \\times [ 1 ] ) \\coprod ^ { X _ 0 } X _ 1 , \\end{align*}"}
-{"id": "2236.png", "formula": "\\begin{align*} D ( P \\parallel Q ) = \\sum _ i p _ i \\log ( \\frac { p _ i } { q _ i } ) . \\end{align*}"}
-{"id": "3760.png", "formula": "\\begin{align*} H = [ a , b ] \\times [ \\beta _ { \\min } , \\beta _ { \\max } ] \\times [ p _ { \\min } , p _ { \\max } ] \\subset ( 1 , \\infty ) \\times ( 0 , \\infty ) \\times ( 0 , 1 ) , \\end{align*}"}
-{"id": "5905.png", "formula": "\\begin{align*} B _ { \\mu _ 1 , \\nu _ 1 } \\dots B _ { \\mu _ n , \\nu _ n } = \\sum _ { i = 0 } ^ { p } c _ { \\mu , \\nu } ( i ; t ) k ^ { i + m _ 1 } , \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} x ^ j _ { n + 1 } = A _ n x ^ j _ n , n \\ge n _ 0 . \\end{align*}"}
-{"id": "8347.png", "formula": "\\begin{align*} u _ k ( x , t ) = \\int _ { \\mathbb { R } ^ n } \\Gamma ( x - y , t ) u _ 0 ( y ) \\ , d y + \\int _ 0 ^ t \\int _ { \\mathbb { R } ^ n } K _ { j k } ( x - y , t - s ) f _ j ( y , s ) \\ , d y \\ , d s , \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} \\Omega ^ 1 _ { ( E ^ \\# ) ^ n } \\simeq \\mathcal O _ { ( E ^ \\# ) ^ n } \\otimes \\big ( \\oplus _ { i = 1 } ^ n ( { \\mathbf k } \\underline { d p } _ i \\oplus { \\mathbf k } \\underline { d c } _ i ) \\big ) . \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} 1 + r + \\dots + r ^ { p - 1 } & \\equiv 1 + ( a p + 1 ) + ( 2 a p + 1 ) + \\dots + ( ( p - 1 ) a p + 1 ) \\\\ & \\equiv p + \\frac { a ( p + 1 ) } { 2 } p ^ 2 \\pmod { p ^ 2 } . \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} \\kappa _ 2 ( x ) & = x ^ { 4 n } - 2 x ^ { 3 n } - x ^ { 2 n + 1 } - 2 x ^ { 2 n } - 3 x ^ { 2 n - 1 } - 2 x ^ { n } + 1 , \\\\ \\intertext { a n d } \\kappa _ 1 ( x ) & = 4 x ^ { 2 n } + 2 x ^ { n + 1 } + 4 x ^ { n } + 2 x ^ { n - 1 } + 4 . \\end{align*}"}
-{"id": "6292.png", "formula": "\\begin{align*} ( k - K _ { \\Omega } ) \\smile [ \\Omega ] = P D [ F _ 1 ] \\smile i _ 1 ^ { * } [ \\Omega ] + P D [ F _ 2 ] \\smile i _ 2 ^ { * } [ \\Omega ] . \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} p = r s \\quad { \\rm a n d } q = r ( 1 - s ) \\ . \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} p _ 1 ( \\nabla ) & = \\frac { 1 } { 4 \\pi ^ 2 } \\left ( \\frac { s _ C ^ 2 } { 4 } + \\| W _ F ^ + \\| ^ 2 + \\| R _ { 0 0 } \\| ^ 2 - \\frac { s _ g ^ 2 } { 4 8 } \\right ) \\operatorname { v o l } _ g \\\\ \\operatorname { P f } ( \\nabla ) & = \\frac { 1 } { 8 \\pi ^ 2 } \\left ( \\frac { s _ C ^ 2 } { 4 } + \\| W _ F ^ + \\| ^ 2 + \\frac { s _ g ^ 2 } { 4 8 } - 2 \\| R _ F \\| ^ 2 - \\| R _ { 0 0 } \\| ^ 2 \\right ) \\operatorname { v o l } _ g \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{align*} L _ m \\chi _ y ( z ) = \\sum _ { y ' } \\chi _ y ( y ' ) , \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} g ^ { - 1 } g _ i \\sigma = \\sigma ' , \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{align*} | A + A | = 2 | A | + r - 1 \\leq \\tfrac { p } { 2 } + | A | - 2 , \\ ; \\ ; \\textrm { a n d } \\ ; \\ ; r \\leq | A | - 3 . \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} Q _ i ( T _ 1 ) = Q _ i ( T _ 0 ) + R _ i ( T _ 0 , T _ 1 ) , \\end{align*}"}
-{"id": "4010.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\left ( \\log ( 2 ) \\frac { p _ { 1 1 } ^ { n / 2 } p _ { 2 2 } ^ { n / 2 } } { p _ { 1 1 } ^ { n / 2 } p _ { 2 2 } ^ { n / 2 } + p _ { 1 2 } ^ { n / 2 } p _ { 2 1 } ^ { n / 2 } } \\right ) ^ { \\frac { 1 } { n } } = 1 . \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} \\nabla \\cdot b _ - = \\frac { B _ 0 \\dot { B _ 0 } } { \\gamma ^ 2 + B _ 0 ^ 2 } \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} & \\sum _ { i = 0 } ^ { m } \\frac { ( m ) _ q ^ ! } { ( m - i ) _ q ^ ! } \\lambda _ { ( m - i , k ) } \\beta _ { ( i , m , k ) } q ^ { - ( i + 1 ) ( 2 k + 2 m - i ) / 2 } { k + m + 1 \\choose i + 1 } _ { \\ ! \\ ! q } = 0 \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{align*} \\Sigma ^ { t + 1 } = \\frac { \\sigma ^ 2 } { \\rho _ { u l } } \\vect { I } + \\frac { N } { L } \\mathbb { E } \\{ \\| \\eta ( \\vect { x } _ { \\beta } - ( \\Sigma ^ t ) ^ { \\frac { 1 } { 2 } } \\vect { w } ) - \\vect { x } _ { \\beta } \\| ^ 2 \\} \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{align*} \\hat { Y } _ T ^ { t , x ; u _ { \\cdot } } = \\int _ t ^ T c \\bigl ( \\tau , X _ { \\tau } ^ { t , x ; u _ { \\cdot } } , u _ { \\tau } \\bigr ) d \\tau + \\Psi ( X _ T ^ { t , x ; u _ { \\cdot } } ) . \\end{align*}"}
-{"id": "5111.png", "formula": "\\begin{align*} \\omega = \\frac { 1 } { 2 ( \\ell + 2 ) } \\sum \\limits _ { i = 1 } ^ { 3 } \\alpha _ i ( - 1 ) ^ 2 \\mathbf { 1 } . \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{align*} g ( r _ n , ( 1 + \\epsilon ) A _ t ( r _ n ) ) \\geq g ( r _ n , A _ { t _ n } ( r _ n ) ) = \\frac { 1 } { t _ n - 1 } . \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} \\frac { H _ 0 } { t } \\gtrsim E ( t ) > E ( t ) - E ( T ) = \\int _ t ^ T D ( s ) \\ , d s . \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{align*} E \\left [ S _ N \\right ] = \\int _ 0 ^ { \\infty } \\left [ 1 - \\prod _ { j = 1 } ^ N \\bigg ( 1 - e ^ { - q _ j t } \\bigg ) \\right ] d t , \\end{align*}"}
-{"id": "9159.png", "formula": "\\begin{align*} K _ { [ 3 , 2 ] } : = K _ { 5 , 2 } \\cong \\frac { \\mathbb { C } \\langle z _ 1 , z _ 2 \\rangle } { ( z _ 1 ^ 2 , z _ 2 ^ 3 , z _ 1 z _ 2 , z _ 2 ^ 2 z _ 1 ) } K _ { [ 2 , 3 ] } : = K _ { 5 , 3 } \\cong \\frac { \\mathbb { C } \\langle z _ 1 , z _ 2 \\rangle } { ( z _ 1 ^ 3 , z _ 2 ^ 2 , z _ 1 z _ 2 , z _ 2 z _ 1 ^ 2 ) } \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} \\| T \\| _ { S _ p } : = \\left ( \\sum \\limits _ { k = 1 } ^ { \\infty } ( s _ k ( T ) ) ^ p \\right ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} { \\rm M } ( \\alpha ) = { \\rm M } ( P _ { \\alpha } ) \\geq \\Theta , \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{align*} p _ q ^ * A _ \\infty ( q , \\eta ) = p _ q ^ * A _ \\infty ( q ) + \\eta \\otimes g _ \\infty ^ { - 1 } \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} g _ \\infty , \\Phi _ \\infty ( q , \\eta ) = \\Phi _ \\infty ( q ) . \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} - \\psi = - i \\big ( \\psi _ { s s } + R \\psi \\big ) \\\\ \\frac { 1 } { 2 } | \\psi | ^ { 2 } = R _ { s } - \\frac { 1 } { 2 } \\psi _ { s } \\bar { \\psi } . \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} & V ^ m ( \\psi ) = \\log \\left ( \\int e ^ { V ^ { m + 1 } ( \\psi + \\psi ' ) } d \\mu _ { C _ { m + 1 } } ( \\psi ' ) \\right ) , \\ ( m = - 1 , - 2 , \\cdots , l _ { e n d } ) . \\end{align*}"}
-{"id": "1725.png", "formula": "\\begin{align*} I _ 1 & : = \\int _ \\mathbb { R } \\bigg ( \\sum _ { k = k _ 0 } ^ N 2 ^ { - k \\alpha } \\| \\mathcal { C } _ { k } g ( \\cdot , t ) \\| _ { L ^ 2 _ x } \\bigg ) ^ { q _ 1 } \\ , \\mathrm { d } t \\\\ I _ 2 & : = \\int _ \\mathbb { R } \\bigg ( \\sum _ { k = N + 1 } ^ \\infty 2 ^ { - k \\alpha } \\| \\mathcal { C } _ { k } g ( \\cdot , t ) \\| _ { L ^ 2 _ x } \\bigg ) ^ { q _ 2 } \\ , \\mathrm { d } t , \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{align*} u _ { k + 1 } ^ { l _ { k + 1 } } = - u _ { k + 1 } ^ { l _ { k + 1 } - ( n _ { k + 1 } + 1 ) } ( u _ { k + 1 } ^ { n _ { k + 1 } } a _ 1 + \\cdots + u _ { k + 1 } a _ { n _ { k + 1 } } ) , \\end{align*}"}
-{"id": "440.png", "formula": "\\begin{align*} U _ 1 ( e _ 1 ) = \\begin{pmatrix} 1 & e _ 1 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & - e _ 1 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , U _ 2 ( e _ 2 ) = \\begin{pmatrix} 1 & 0 & e _ 2 & 0 \\\\ 0 & 1 & 0 & - e _ 2 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , U ( e _ 1 , e _ 2 ) = U _ 1 ( e _ 1 ) U _ 2 ( e _ 2 ) , \\end{align*}"}
-{"id": "6498.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\vartheta ^ { \\prime } \\langle u , v \\rangle _ { [ L ^ { p ^ { \\prime } } _ { \\sigma } ] ^ * , L ^ { p ^ { \\prime } } _ { \\sigma } } \\ ; d t = \\int _ { \\Omega } \\int _ 0 ^ T \\mathcal { U } \\vartheta ^ { \\prime } \\cdot \\overline { A _ { p ^ { \\prime } } ^ { \\frac { 1 } { 2 } } v } \\ ; d t \\ ; d x = - \\int _ { \\Omega } \\int _ 0 ^ T \\mathcal { U } ^ { \\prime } \\vartheta \\cdot \\overline { A _ { p ^ { \\prime } } ^ { \\frac { 1 } { 2 } } v } \\ ; d t \\ ; d x . \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{align*} \\mu \\big ( \\mathbb { D } ^ \\ast ( 2 / R ) \\setminus \\mathbb { D } ^ \\ast ( 1 / 2 R ) \\big ) = O \\left ( \\frac { 1 } { R ^ { r + 1 } } \\right ) . \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} \\ ! { \\rm K e r } ( \\nabla g ( x ) ) \\cap \\mathcal { T } _ { \\mathcal { N } _ { K } ( g ( x ) ) } ( \\lambda ) = \\{ 0 \\} & \\Longleftrightarrow { \\rm K e r } ( \\nabla g ( x ) ) \\cap D \\mathcal { N } _ { K } ( g ( x ) | \\lambda ) ( 0 ) = \\{ 0 \\} , \\\\ & \\Longleftrightarrow { \\rm S R C Q \\ f o r \\ t h e \\ s y s t e m } \\ g ( x ) \\in K \\ { \\rm a t } \\ x \\ { \\rm w . r . t . } \\ \\lambda . \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} p = \\sum _ { i \\in I } \\varphi ^ i ( p ) \\cdot p _ i \\ , , \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} \\ell ( A / J ) \\ \\le \\ k \\binom { n } { t - 2 } . \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} \\tag * { $ { \\bf ( A _ { 1 5 } ) } $ } E _ 0 ' \\ E _ 1 \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} h ( y ) = ( 1 - y ) ^ r = \\sum _ { k = 0 } ^ { \\infty } ( - 1 ) ^ k \\binom { r } { k } y ^ k , y \\in [ - 1 , 1 ] , \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} - u '' + \\lambda \\delta _ 0 u ( 0 ) = \\omega _ \\lambda ^ 2 u , u \\ge 0 , u ( 2 \\pi ) = u ( 0 ) . \\end{align*}"}
-{"id": "5779.png", "formula": "\\begin{align*} \\nabla \\left ( \\Phi _ { \\ast } ( \\omega ) \\right ) ( y ) = & \\nabla \\left ( ( \\nabla \\Phi \\circ \\Psi ) ( \\omega \\circ \\Psi ) \\right ) ( y ) \\smallskip \\\\ = & \\sum _ { k = 1 } ^ n S ^ k ( \\Phi , y ) \\omega _ k ( \\Psi ( y ) ) + \\nabla \\Phi ( \\Psi ( y ) ) \\ , \\nabla \\omega ( \\Psi ( y ) ) \\ , \\nabla \\Psi ( y ) , \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} d M _ { \\epsilon , t } + M _ { \\epsilon , t } \\left ( \\frac { 1 } { 2 } D R ^ \\dagger ( { \\widetilde x } _ t ) d t + D A _ { \\epsilon } ^ \\dagger ( { \\widetilde x } _ t ) d \\lambda _ t \\right ) = 0 , M _ { \\epsilon , 0 } = I , \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{align*} \\mathfrak { a } ( \\mathfrak { d } \\vert _ { s _ - } ) - \\mathfrak { a } ( \\mathfrak { d } \\vert _ { s _ + } ) & \\ge \\frac { 1 } { 4 C } \\int _ { s \\in [ s _ - , s _ + ] } ( | \\mathfrak { B } ( A , \\psi ) | ^ 2 + | E _ A | ^ 2 ) + 2 r ( | D _ A ^ { g _ 3 } \\psi | ^ 2 + | \\nabla _ { A , s } \\psi | ^ 2 ) - c _ 0 r . \\end{align*}"}
-{"id": "4075.png", "formula": "\\begin{align*} \\epsilon & = \\sum _ { r } q _ R ( r ) \\lambda ( r ) . \\end{align*}"}
-{"id": "707.png", "formula": "\\begin{align*} \\norm { { \\rm C o l } _ j A } _ 2 \\leq \\norm { A } _ 2 \\leq \\sqrt { m } \\norm { A } _ 1 = \\sqrt { m } \\max _ { j = 1 , \\hdots , m } \\norm { { \\rm C o l } _ j A } _ 1 \\leq m \\max _ { j = 1 , \\hdots , m } \\norm { { \\rm C o l } _ j A } _ 2 , \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} u _ { 1 } \\left ( x \\right ) : = \\begin{cases} 0 & x \\in \\left [ - 2 , - 1 \\right ] , \\\\ f \\left ( x \\right ) & x \\in \\left [ - 1 , 1 \\right ] , \\\\ p \\left ( x \\right ) & x \\in \\left [ 1 , 2 \\right ] , \\end{cases} \\end{align*}"}
-{"id": "6334.png", "formula": "\\begin{align*} \\begin{cases} \\nabla G ( U ) U ^ T - U \\nabla G ( U ) ^ T = \\mathbb { O } _ { n \\times n } \\\\ U ^ T U = \\mathbb { I } _ p . \\end{cases} \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{align*} & - 2 \\delta ^ { i _ 1 i _ 2 } G _ { i _ 1 i _ 2 i _ 3 } ^ { j _ 1 j _ 2 j _ 3 } \\gamma _ { j _ 1 j _ 3 } \\delta _ { j _ 2 l } \\\\ = & - \\delta _ { i _ 3 l } + \\delta ^ { i _ 1 i _ 2 } G _ { i _ 1 i _ 2 i _ 3 } ^ { \\eta j _ 2 j _ 3 } \\delta _ { j _ 2 j _ 3 } \\tilde \\gamma _ { \\eta l } \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} \\tau _ { \\mu , \\mu } ^ { 1 } = ( 1 - q ^ { - 1 } ) \\dfrac { \\mathbf { z } ^ C } { 1 - q ^ { - 1 } \\mathbf { z } ^ { n _ { \\alpha } \\alpha ^ { \\vee } } } \\ , , \\tau _ { s _ { \\alpha } ( \\mu ) + \\alpha , \\mu } ^ { 2 } = q ^ { - 1 } D \\mathbf { z } ^ { - \\alpha ^ { \\vee } } \\dfrac { 1 - \\mathbf { z } ^ { n _ { \\alpha } \\alpha ^ { \\vee } } } { 1 - q ^ { - 1 } \\mathbf { z } ^ { n _ { \\alpha } \\alpha ^ { \\vee } } } \\ , . \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} \\bar { x } \\cdot d \\log ( \\hat { y } ) & = c S ^ { \\ell _ 1 } T ^ { \\ell _ 2 } \\cdot S ^ { - 1 } T ^ { - 1 } d S \\wedge d T = c S ^ { \\ell _ 1 - 1 } T ^ { \\ell _ 2 - 1 } d S \\wedge d T . \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} { \\rm R e } ( W _ 1 ^ { \\ , \\Psi } ) = W _ 1 ^ \\omega \\ : . \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{align*} \\Lambda ^ { 3 } \\R ^ 6 = \\Lambda ^ 3 _ 6 \\oplus \\Lambda ^ 3 _ { 1 \\oplus 1 } \\oplus \\Lambda ^ 3 _ { 1 2 } , \\end{align*}"}
-{"id": "4760.png", "formula": "\\begin{align*} & D \\Phi ( ( \\overline { x } , \\overline { \\lambda } , \\overline { v } ) | ( 0 , 0 ) ) ( \\Delta x , \\Delta \\lambda , \\Delta v ) \\\\ & = \\Big \\{ ( \\Delta \\xi , \\Delta \\eta ) \\in \\mathbb { X } \\times \\mathbb { Y } \\ | \\ \\Phi ' ( ( \\overline { x } , \\overline { \\lambda } , \\overline { v } ) ; ( \\Delta x , \\Delta \\lambda , \\Delta v ) ) = ( \\Delta \\xi , \\Delta \\eta ) \\Big \\} . \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} q ^ { \\mu _ i } t ^ { \\rho ( \\mu ) _ i } = q ^ { \\nu _ i } t ^ { \\rho ( \\nu ) _ i } , \\forall \\ 1 \\leq i \\leq n . \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} \\norm { \\abs { \\partial _ { s } } ^ { \\sigma } f } _ { L ^ 2 ( \\Gamma ) } : = \\norm { \\abs { \\partial _ { s } } ^ { \\sigma } \\hat { f } } _ { L ^ 2 ( \\R ) } , \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} X _ { \\epsilon _ 1 + \\epsilon _ 2 } X _ { - \\epsilon _ 1 - \\epsilon _ 2 } + X _ { \\epsilon _ 1 - \\epsilon _ 2 } X _ { - \\epsilon _ 1 + \\epsilon _ 2 } + X _ 0 ^ 2 = 0 ; \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} Z _ j ( t ) = Z _ j ( 0 ) - { \\bf 1 } _ { \\{ j = 1 \\} } t + B _ { j + 1 } ( t ) - B _ { j } ( t ) + 2 L _ { j } ( t ) - L _ { j + 1 } ( j ) - L _ { j - 1 } ( t ) \\ , . \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} & E \\left [ S ^ { p a r t , m } _ { s , t } \\right ] = E \\left [ S ^ { p a r t , 0 } _ { s , t } \\right ] - \\frac { n } { 2 } E \\left [ \\ln ( \\beta ( t , q _ t ) / \\beta ( s , q _ s ) ) \\right ] + o ( 1 ) . \\end{align*}"}
-{"id": "4185.png", "formula": "\\begin{align*} g _ { i } ^ { ( \\ell + 1 ) } ( t ) = \\theta \\left ( b _ { i } ^ { ( \\ell + 1 ) } + \\sum _ { j = 1 } ^ { N _ { \\ell } } A _ { i , j } ^ { ( \\ell + 1 ) } \\cdot g _ { j } ^ { ( \\ell ) } ( t ) \\right ) t \\in \\R i \\in \\underline { N _ { \\ell + 1 } } . \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{align*} \\Gamma ( h ) _ { 0 0 } ^ 0 = - \\frac { \\dot { s } } { s } , \\qquad \\Gamma ( h ) _ { i j } ^ 0 = \\frac { 1 } { 2 } s ^ 2 \\dot { g } _ { i j } , \\Gamma ( h ) _ { i 0 } ^ j = \\Gamma ( h ) _ { 0 i } ^ j = \\frac { 1 } { 2 } g ^ { j k } \\dot { g } _ { i k } , \\Gamma ( h ) _ { i j } ^ k = \\Gamma ( g ) _ { i j } ^ k . \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{align*} I = \\sum _ { \\mu \\in \\lbrace 0 , 1 \\rbrace ^ N } E _ \\mu ^ * . \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} x ^ \\nu = \\left \\lbrace \\begin{array} { c l } a ^ \\nu , & \\nu \\in \\{ 1 , \\cdots , s \\} \\\\ b ^ { \\nu - s } , & \\nu \\in \\{ s + 1 , \\cdots , s + t \\} \\\\ p , & \\nu = s + t + 1 \\end{array} \\right . \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} a ( u _ h , v _ h ) = ( \\rho g , v _ h ) \\quad v _ h \\in V _ h . \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{align*} f ( x ) = \\max \\{ d ( x , C _ { 1 } ) , d ( x , C _ { 2 } ) \\} \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} X : = \\partial _ { x _ 1 } , Y : = x _ 1 \\partial _ { x _ 2 } \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l l l } u _ { t } + u u _ x + u _ { x } + u _ { x x x } = 0 & & ( 0 , L ) \\times ( 0 , + \\infty ) \\\\ u ( 0 , t ) = u ( L , t ) = 0 & & ( 0 , + \\infty ) , \\\\ u _ x ( L , t ) = h ( t ) & & ( 0 , + \\infty ) , \\end{array} \\right . \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} n _ { \\alpha } = \\dfrac { n } { \\gcd ( n , Q ( \\alpha ^ { \\vee } ) ) } \\ , . \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} \\bigl [ [ x _ 1 , x _ 2 ] x _ 3 , x _ 4 , x _ 5 \\bigr ] = [ x _ 1 , x _ 2 ] [ x _ 3 , x _ 4 , x _ 5 ] + [ x _ 1 , x _ 2 , x _ 4 ] [ x _ 3 , x _ 5 ] + [ x _ 1 , x _ 2 , x _ 5 ] [ x _ 3 , x _ 4 ] + [ x _ 1 , x _ 2 , x _ 4 , x _ 5 ] x _ 3 . \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} \\phi _ t = i \\phi _ { x x } + \\frac i 2 | \\ ! | \\phi | \\ ! | ^ 2 \\phi \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{align*} \\begin{cases} \\chi \\left ( \\mathrm { t r } ( Y Z ^ T ) \\right ) & , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "7681.png", "formula": "\\begin{align*} \\mathrm { P } ^ { h i t } _ { m } = \\sum ^ { M _ s } _ { i = 1 } \\mathrm { P } ( f _ i ) ( 1 - \\mathrm { P } _ { m } ^ i ) , \\end{align*}"}
-{"id": "7040.png", "formula": "\\begin{align*} r ^ \\nu = \\nu ^ { \\frac { - 2 } { 4 + n } } ( \\log \\nu ) ^ { \\frac { 1 + n } { 2 + n / 2 } } , \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} \\mathbb { E } ^ P _ { n , 0 } ( \\nu _ n ) = \\mathbb { P } _ { n , 0 } \\circ t _ n \\left ( [ . 5 , \\infty ) \\right ) . \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{align*} R _ { s u m } ^ { M R T } \\geq K \\log _ 2 \\left ( 1 + \\widetilde { { \\rm S I N R } } _ { M R T } \\right ) = K \\log _ 2 \\left ( 1 + { M } { \\big / } \\left ( { \\frac { K - 1 } { c } + \\frac { K } { \\rho _ t } } \\right ) \\right ) . \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{align*} B ^ \\circ & = \\{ b \\in B \\mid b \\notin B ^ + b ' \\in B ^ + \\backslash \\{ 0 \\} , b + b ' \\in B ^ + \\} \\cup \\{ ( 0 , 0 ) \\} \\\\ & = \\{ ( 0 , q ) \\in B \\mid q \\in \\mathbb { Q } ^ + \\} \\cup \\{ ( a , 0 ) \\in B \\mid a \\in A ^ + \\} . \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} | J _ 1 ( t _ 1 , t _ 2 ) | & = \\Big | \\int _ { \\mathbb { R } } G _ { t _ 1 } ( x - y ) \\Big ( \\int _ { \\mathbb { R } } G _ { t _ 2 - t _ 1 } ( y - z ) \\big [ \\psi _ 0 ( z ) - \\psi _ 0 ( y ) \\big ] d z \\Big ) d y \\Big | \\\\ & \\leq C \\int _ { \\mathbb { R } } G _ { t _ 1 } ( x - y ) \\Big ( \\int _ { \\mathbb { R } } G _ { t _ 2 - t _ 1 } ( y - z ) | z - y | d z \\Big ) d y \\\\ & = C ( t _ 2 - t _ 1 ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "3878.png", "formula": "\\begin{align*} N = \\left \\lceil 4 0 \\sqrt { \\vartheta } \\ln \\left ( \\frac { 6 \\vartheta } { 5 \\eta _ 0 \\bar \\epsilon } \\right ) \\right \\rceil \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } ( b _ t ) ^ { 1 / t } & = \\lim _ { t \\rightarrow \\infty } ( h _ t ( \\alpha _ t ) ) ^ { - 1 / t } \\\\ & = \\lim _ { t \\rightarrow \\infty } \\{ \\alpha _ t \\exp [ ( t - 1 ) u ( \\alpha _ t ) ] \\} ^ { - 1 / t } \\\\ & = \\lim _ { t \\rightarrow \\infty } \\exp [ - u ( \\alpha _ t ) ] = m _ 1 ( \\mu ) . \\end{align*}"}
-{"id": "4624.png", "formula": "\\begin{align*} \\nabla _ T ^ * \\nabla _ T \\phi & = - \\sum _ a \\nabla _ { V _ a } \\nabla _ { \\bar V _ a } \\phi + \\nabla _ { H ^ { 0 , 1 } } \\phi , \\\\ \\bar \\nabla _ T ^ * \\bar \\nabla _ T \\phi & = - \\sum _ a \\nabla _ { \\bar V _ a } \\nabla _ { V _ a } \\phi + \\nabla _ { H ^ { 1 , 0 } } \\phi . \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} \\lambda _ { D _ t } = a _ t - \\chi + o ( 1 ) \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} \\mathop { \\cup } _ { j = 1 } ^ s ( 1 \\times \\cdots \\times 1 \\times x _ { M _ j } ^ { u _ j } \\times 1 \\times \\cdots \\times 1 ) . \\end{align*}"}
-{"id": "8990.png", "formula": "\\begin{align*} V ( t ) f ( x ) = \\lim _ { m \\rightarrow \\infty } R ^ m \\left ( \\frac { t } { m } \\right ) f ( x ) . \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} \\| u \\| _ \\Phi = \\inf \\left \\{ \\lambda > 0 ~ \\big | ~ \\int _ \\Omega \\Phi \\left ( \\frac { u ( x ) } { \\lambda } \\right ) d x \\leq 1 \\right \\} , \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} \\mathcal B _ { i j } = \\begin{bmatrix} ( A _ 1 ) _ { i i } ( B _ 1 ) _ { j j } & - ( C _ 1 ) _ { i i } ( D _ 1 ) _ { j j } \\\\ & \\ddots & \\ddots \\\\ & & \\ddots & - ( C _ { r - 1 } ) _ { i i } ( D _ { r - 1 } ) _ { j j } \\\\ & & & ( A _ r ) _ { i i } ( B _ r ) _ { j j } \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "2696.png", "formula": "\\begin{align*} P ^ { l , m } & = \\frac { \\sqrt { 2 } } { \\delta ( x , y ) } \\sum _ { n = 0 } ^ { \\log _ 2 \\tfrac { 2 ^ m } { \\delta ( x , y ) } - 1 } \\frac { 1 } { 2 ^ n } \\Omega _ { I ^ { ( n ) } ( x , y ) } ( x , y ) \\\\ & = \\frac { \\sqrt { 2 } } { \\delta ( x , y ) } \\Biggl ( \\Omega ( x , y ) - \\sum _ { n = \\log _ 2 \\tfrac { 2 ^ m } { \\delta ( x , y ) } } ^ { \\infty } \\frac { 1 } { 2 ^ n } \\Omega _ { I ^ { ( n ) } ( x , y ) } ( x , y ) \\Biggr ) \\\\ & = P _ { l , m } ( x , y ) + Q _ { l , m } ( x , y ) , \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} & i \\frac { \\partial } { \\partial t } \\bigl \\{ w _ { k \\tau } ( t ; \\rho ) - w _ { \\tau } ( t ; \\rho ) \\bigr \\} = H ( t ; \\rho ) \\bigl \\{ w _ { k \\tau } ( t ; \\rho ) - w _ { \\tau } ( t ; \\rho ) \\bigr \\} \\\\ & + \\int _ 0 ^ 1 \\frac { \\partial H } { \\partial \\rho } ( t ; \\rho + \\theta \\tau ) d \\theta \\ , \\bigl \\{ u _ k ( t ; \\rho + \\tau ) - u ( t ; \\rho + \\tau ) \\bigr \\} \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} { \\cal Z } = \\begin{pmatrix} M \\\\ \\kappa \\\\ \\alpha \\end{pmatrix} ~ . \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{align*} \\tau _ p ( x , y ) = \\log \\Big ( 1 + 2 \\frac { d ( x , y ) } { \\sqrt { d ( x , p ) d ( y , p ) } } \\Big ) . \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{align*} \\begin{aligned} ( h ^ n _ K + \\alpha _ K ) _ K & \\xrightarrow [ n \\to + \\infty ] { } Z _ 0 , \\\\ ( U _ K ^ n ) _ K & \\xrightarrow [ n \\to + \\infty ] { } 0 . \\end{aligned} \\end{align*}"}
-{"id": "4057.png", "formula": "\\begin{align*} & V ( a _ 1 a _ 2 b _ 1 b _ 2 | x _ 1 x _ 2 y _ 1 y _ 2 ) = p _ { A | X } ( a _ 1 | x _ 1 ) p _ { A | X } ( a _ 2 | x _ 2 ) p _ { B | Y } ( b _ 1 | y _ 1 ) p _ { B | Y } ( b _ 2 | y _ 2 ) \\end{align*}"}
-{"id": "3401.png", "formula": "\\begin{align*} \\min _ { | z | \\leq r } | f _ { k _ j } ( z ) | = \\min _ { | z | = r } | f _ { k _ j } ( z ) | \\to \\infty \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} y _ i ^ { n } = \\begin{cases} - \\epsilon _ i n , & i \\in \\mathcal { I } , \\\\ \\mu _ i , & i \\in \\mathcal { N } \\setminus \\mathcal { I } . \\end{cases} \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} & \\mathcal { L } _ { I _ U } ( s ) \\\\ & = \\prod _ { t \\in \\mathcal { K } } \\exp \\Big ( - 2 \\pi \\int _ { 0 } ^ { \\infty } \\lambda _ { U , t } ^ s ( r , y ) ( 1 - \\mathcal { L } _ h ( s P _ U y ^ { - \\alpha } ) ) y d y \\Big ) , \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{align*} ( q _ 1 - q _ 2 ) T ( q _ 1 - q _ 2 ) = ( q _ 1 - q _ 2 ) ( q _ 1 T q _ 1 ) , \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } { \\mathbb P } ( \\varphi _ n ( f _ 1 , \\ldots , f _ s ) \\in J \\ , | \\ , \\xi _ d = \\beta ) = V ( J ) \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} 0 = \\int _ M | W _ { 0 0 } ^ + | ^ 2 + 5 6 v ^ 2 \\end{align*}"}
-{"id": "8909.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 1 } \\tilde { u } _ { j } ( d _ { s p } u ^ * ) ( \\mu _ Y ) _ j = n _ Y . \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} - 2 d d ^ c g _ { D _ r , p } = \\delta _ p - \\pi _ { r , p } \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\Omega } ( y ) = \\nabla \\Xi ( y ) \\mathcal { N } _ { \\mathcal { D } } ( \\Xi ( y ) ) \\quad \\ \\forall y \\in \\mathbb { B } ( \\overline { y } , \\varepsilon ) . \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} f ^ * ( x _ M ^ { m - k + n } \\times 1 ) = \\pm \\frac { 1 } { \\lambda } \\cdot ( x _ M ^ { m - k + n } \\times 1 ) , \\end{align*}"}
-{"id": "5012.png", "formula": "\\begin{align*} [ a _ 1 a _ 2 , b ] = a _ 1 [ a _ 2 , b ] + [ a _ 1 , b ] a _ 2 \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} F ( x , x ^ 4 + 1 ) & = x ^ { 1 6 } - x ^ 4 ( x ^ 4 + 1 ) ^ 3 + x ^ 4 ( x ^ 4 + 1 ) ^ 2 x ^ 3 \\\\ & = x ^ 4 ( x ^ 3 - x - 1 ) ( x ^ 8 + x ^ 6 - 2 x ^ 5 + 3 x ^ 4 - x ^ 3 + x ^ 2 - x + 1 ) . \\end{align*}"}
-{"id": "6747.png", "formula": "\\begin{align*} B ^ { - 1 } = ( L _ { x } ^ { - 1 } R _ { x } ^ { - 1 } R _ { x ^ \\lambda } ^ { - 1 } , R _ { x } ^ { - 1 } , L _ { x } ^ { - 1 } R _ { x } ^ { - 1 } ) \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} Q ^ { \\alpha \\beta \\gamma } \\xi _ { \\alpha } \\xi _ { \\beta } \\xi _ { \\gamma } = 0 . \\end{align*}"}
-{"id": "8885.png", "formula": "\\begin{align*} u _ Y ^ * ( d _ b u _ Y ) & = d _ b u _ Y ( b ) - u _ Y ( b ) \\\\ & = \\lim d _ { t _ j \\mu _ Y + b _ j } u _ { \\lambda } ( t _ j \\mu _ Y + b _ j ) - u _ { \\lambda } ( t _ j \\mu _ Y + b _ j ) \\\\ \\intertext { f o r a n y s e q u e n c e s s u c h t h a t $ \\lim t _ j = - \\infty $ a n d $ \\lim b _ j = b $ b y P r o p o s i t i o n ~ \\ref { p r o p _ l i m i t s _ t o _ f a c e t s } } & = \\lim u _ { \\lambda } ^ * ( d _ { t _ j \\mu _ Y + b _ j } u _ { \\lambda } ) . \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} d q ^ \\prime _ t = & \\tilde \\gamma ^ { - 1 } ( t ^ * , q ^ \\prime _ t ) \\left ( \\partial _ t \\psi ( t ^ * , q ^ \\prime _ t ) - \\nabla _ q V ( t ^ * , q ^ \\prime _ t ) ) \\right ) d t \\\\ & + \\tilde S ( t ^ * , q ^ \\prime _ t ) d t + \\tilde \\gamma ^ { - 1 } ( t ^ * , q ^ \\prime _ t ) \\sigma ( t ^ * , q ^ \\prime _ t ) \\circ d W _ t \\end{align*}"}
-{"id": "6017.png", "formula": "\\begin{align*} a + b = \\frac { 1 } { 2 } \\left ( \\theta ^ { 2 } - \\frac { 1 } { 3 } \\right ) c + d = \\frac { 1 } { 2 } ( 1 - \\theta ^ { 2 } ) \\geq 0 \\theta \\in \\left [ 0 , 1 \\right ] , \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{align*} { r } ^ * = \\frac { { \\sqrt { 9 { c ^ 2 } + 4 c { \\rho _ t } + 4 \\rho _ t ^ 2 } - 3 c } } { { 2 \\left ( { c + { \\rho _ t } } \\right ) } } c . \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ p _ { k , \\nu } } = \\left ( \\sum _ { j = 0 } ^ k { \\| r ^ { - \\frac { n } { p } - \\nu + j } \\nabla ^ j u \\| ^ p _ { L ^ p } } \\right ) ^ { \\frac { 1 } { p } } , \\| u \\| _ { C ^ { k , \\alpha } _ \\nu } = \\sum _ { j = 0 } ^ k { \\| r ^ { - \\nu + j } \\nabla ^ j u \\| _ { C ^ 0 } + [ r ^ { - \\nu + k } \\nabla ^ k u ] _ \\alpha } . \\end{align*}"}
-{"id": "4911.png", "formula": "\\begin{align*} a b \\bigl ( a _ 1 ( \\varepsilon ) \\beta - a _ 3 ( \\varepsilon ) \\alpha - a _ 4 ( \\varepsilon ) \\beta \\bigr ) + a _ 2 ( \\varepsilon ) ( \\beta + 2 b c \\beta - c d \\alpha ) = 0 \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{align*} W ( x , y ) = \\sum _ { r = 0 } ^ \\mu a _ r W _ 2 ( x , y ) ^ { 5 \\mu + \\nu - 5 r } W ' _ { 1 0 } ( x , y ) ^ r ( \\mu \\geq 0 ) , \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{align*} a _ k ^ * - a _ k = \\varrho _ k ( b _ k ^ * - b _ k ) \\in \\R . \\end{align*}"}
-{"id": "5230.png", "formula": "\\begin{align*} u _ t = u _ { x x } + { q _ 0 } u _ x + u ( a _ 0 - b _ 0 u ) , x \\in \\R ^ 1 , \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} K _ 2 = ( s K _ 1 s ^ { - 1 } \\cap K _ 2 ) ( K _ 2 \\cap t ^ { - 1 } K _ 3 t ) , \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{align*} p ( i , j | v , w ) = \\frac { 1 } { 4 } \\left ( 1 + ( - 1 ) ^ { i + j } \\langle \\widetilde { x } _ v , \\widetilde { x } _ w \\rangle \\right ) . \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} \\varphi _ { \\Omega } ( z + n \\omega _ 2 ) = ( - 1 ) ^ n \\exp ( - 2 \\pi i n z / \\omega _ 1 - \\pi i n ( n - 1 ) \\omega _ 2 / \\omega _ 1 ) \\varphi _ { \\Omega } ( z ) \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} & L ( X ; Y \\| Z ) = \\Bigg [ \\sum _ { \\substack { i \\geq \\zeta \\\\ } } I ( U _ i ; Y | U _ { 1 : i - 1 } ) - I ( U _ i ; Z | U _ { 1 : i - 1 } ) \\Bigg ] + \\ ! \\Bigg [ \\sum _ { \\substack { i \\geq \\zeta \\\\ } } \\ ! I ( U _ i ; X | U _ { 1 : i - 1 } ) \\ ! - \\ ! I ( U _ i ; Z | U _ { 1 : i - 1 } ) \\Bigg ] . \\end{align*}"}
-{"id": "3026.png", "formula": "\\begin{align*} \\mathcal { F } : U _ { 0 } : = \\left ( q _ { 0 } - \\frac { \\sigma _ { 0 } } { 2 } , q _ { 0 } + \\frac { \\sigma _ { 0 } } { 2 } \\right ) \\times B _ { 0 } \\rightarrow L ^ { t } ( \\Omega ) ; \\quad \\mathcal { F } ( q , u ) : = - \\Delta u - a ( x ) u ^ { q } , \\end{align*}"}
-{"id": "7606.png", "formula": "\\begin{align*} N _ { q , n } ( g ) \\geqslant \\dfrac { 1 } { n } \\left ( q ^ { g n } - \\sqrt { n } \\ , q ^ { \\frac { g n } { 2 } } \\right ) = \\dfrac { 1 } { n } q ^ { g n } \\left ( 1 - \\sqrt { n } \\ , q ^ { - \\frac { g n } { 2 } } \\right ) \\ . \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{align*} & \\underset { z _ n = 0 } { } \\frac { I _ n ( z _ 1 , \\dots , z _ n ) } { z _ n } = \\gamma ^ { - 1 } \\cdot I _ { n - 1 } ( z _ 1 , \\dots , z _ { n - 1 } ) \\\\ & \\underset { z _ 1 = \\infty } { } \\frac { I _ n ( z _ 1 , \\dots , z _ n ) } { z _ 1 } = - q ^ r \\cdot I _ { n - 1 } ( z _ 2 , \\dots , z _ n ) \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} S _ { p + m _ 1 } ( \\nu ^ + ) _ { m _ 2 - r + 1 } = ( p + m _ 1 ) \\cdot \\nu ^ + _ { m _ 2 - r + 1 } + m _ 2 - r + 1 = m _ 1 + m _ 2 + 1 + p - r . \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{align*} K _ 2 ( N ) = - e ^ { i \\xi A _ N } + o \\left ( \\frac { 1 } { N } \\right ) \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} v _ \\tau ( t ) & = w ( t ) + \\hat w ( \\tau ) - \\hat w ( t ) \\\\ & + \\int _ \\tau ^ t \\Big ( [ A ( t ) , \\psi ( s ) ] - P ^ \\perp \\ ( \\zeta ( s ) + [ A ( s ) , \\psi ( s ) ] \\ ) \\ ) d s \\end{align*}"}
-{"id": "563.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow p } ( \\log \\| s ( z ) \\| + m g _ { D _ r , p } ( z ) ) = \\log \\| j _ p ^ m s \\| _ r \\end{align*}"}
-{"id": "2208.png", "formula": "\\begin{align*} \\Phi _ { t } ^ { B } { u } = { } & \\int _ { 0 } ^ { t } T _ { - 1 } ( s ) B u ( s ) \\ , d s \\\\ = { } & - \\int _ { 0 } ^ { t } T _ { - 1 } ( \\tfrac { s } { 2 } ) ( - A _ { - 1 } ) ^ { \\frac { 1 } { 2 } } u ( s ) ( - A ) ^ { \\frac { 1 } { 2 } } T ( \\tfrac { s } { 2 } ) x _ { 0 } \\ , d s \\\\ = { } & - \\frac { 1 } { 2 } \\int _ { 0 } ^ { \\frac { t } { 2 } } T _ { - 1 } ( s ) ( - A _ { - 1 } ) ^ { \\frac { 1 } { 2 } } u ( 2 s ) f ( s ) \\ , d s \\\\ = { } & - \\frac { 1 } { 2 } \\Phi _ { \\frac { t } { 2 } } ^ { ( - A _ { - 1 } ) ^ { \\frac { 1 } { 2 } } } \\left ( u ( 2 \\cdot ) f \\right ) \\in X . \\\\ \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} m ( c , 0 ) & \\leq \\frac { I ( \\psi _ n , c , 0 ) } { K ( \\psi _ n ) ^ { \\frac { 2 } { 3 } } } \\\\ & = \\frac { I ( \\psi _ n , c , \\gamma _ n ) - \\gamma _ n \\int | \\partial _ x ^ { - 1 } \\psi _ n | ^ 2 d x } { K ( \\psi _ n ) ^ { \\frac { 2 } { 3 } } } \\\\ & < \\frac { I ( \\psi _ n , c , \\gamma _ n ) } { K ( \\psi _ n ) ^ { \\frac { 2 } { 3 } } } = m ( c , \\gamma _ n ) . \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} \\epsilon _ L : = \\frac { 1 } { 1 + \\ln ( L ) } . \\end{align*}"}
-{"id": "7186.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } { \\mathbb P } \\left ( \\varphi _ n ( f _ 1 , \\ldots , f _ s ) \\in J _ k ( \\mathbf a ) \\ , | \\ , \\xi _ t = \\beta ' \\right ) = m ^ { - k s } . \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{align*} g ( z ) = \\exp \\ ! \\left ( c ( z - b ) + \\delta ( z ) \\right ) \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} A ( v _ 1 , . . . , v _ k ) = \\int _ 0 ^ \\infty B ( e ^ { t C } v _ 1 , . . . , e ^ { t C } v _ k ) d t . \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} \\mathrm { P } _ { t , i } = & \\mathrm { P } \\left ( \\alpha _ 1 = 1 , z _ t < \\max \\left \\{ \\frac { \\epsilon _ 1 } { \\rho \\xi _ 1 } , \\cdots , \\frac { \\epsilon _ i } { \\rho \\xi _ { i } } \\right \\} \\right ) \\\\ & + \\mathrm { P } \\left ( \\alpha _ 1 < 1 , z _ t < \\max \\left \\{ \\frac { \\epsilon _ 1 } { \\rho \\xi _ 1 } , \\cdots , \\frac { \\epsilon _ i } { \\rho \\xi _ { i } } \\right \\} \\right ) . \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} ( i ) f _ { \\beta } ( z ) = - \\frac { 1 } { \\zeta _ { \\beta } ( z ) } \\mbox { i f } ~ \\beta ~ \\mbox { i s n o t a s i m p l e P a r r y n u m b e r } , \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} & \\sum _ { a = 1 } ^ { n - 2 } \\sum _ { m = 1 } ^ { n - a - 1 } f ( a , m ) = \\sum _ { a = 1 } ^ { n - 2 } \\sum _ { m = 1 } ^ { n - a } f ( a , m ) = \\sum _ { a = 1 } ^ { n - 2 } \\left ( a \\binom { n + 1 } { a + 2 } - ( a + 1 ) \\binom { n } { a + 1 } + n - a \\right ) \\\\ & = \\sum _ { a = 0 } ^ { n - 1 } \\left ( ( a + 2 ) \\binom { n + 1 } { a + 2 } - ( a + 1 ) \\binom { n } { a + 1 } - 2 \\binom { n + 1 } { a + 2 } + n - a \\right ) \\\\ & = ( n + 1 ) ( 2 ^ n - 1 ) - n 2 ^ { n - 1 } - 2 ( 2 ^ { n + 1 } - n - 2 ) + \\binom { n + 1 } { 2 } \\\\ & = ( n - 6 ) 2 ^ { n - 1 } + n + 3 + \\binom { n + 1 } { 2 } . \\end{align*}"}
-{"id": "3711.png", "formula": "\\begin{align*} | \\vec { g } ( \\vec { x } ) - \\vec { g } ( \\vec { x } _ 0 ) | \\gg \\widetilde { C } ^ { 1 - r } M | \\vec { x } - \\vec { x } _ 0 | = a \\frac { M ^ 2 } { \\widetilde { C } ^ { 2 r - 1 } } \\end{align*}"}
-{"id": "4259.png", "formula": "\\begin{align*} [ L _ n , \\phi ( z ) ] = ( z ^ { n + 1 } \\partial _ z + ( n + 1 ) h z ^ n ) \\phi ( z ) . \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} \\mu ^ { ( n ) } : = \\left ( \\int _ X \\psi _ n \\ , d \\mu \\right ) ^ { - 1 } \\psi _ n \\cdot \\mu , \\nu ^ { ( n ) } : = \\left ( \\int _ X \\psi _ n \\ , d \\mu \\right ) ^ { - 1 } \\psi _ n \\cdot \\nu . \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} \\mathcal { B } : = \\bigcup \\left \\{ \\operatorname { S u p p } \\left ( \\operatorname { T o r } _ i ^ A ( M , N ) \\right ) : \\dim ( A ) < i \\leqslant \\dim ( A ) + \\dim ( Q ) + 1 \\right \\} . \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{align*} ( \\lambda - 5 ) d ^ \\ast \\beta _ 1 = ( \\lambda - 1 ) ( \\lambda + 9 ) \\beta _ 0 = ( \\lambda + 5 ) ( \\lambda - 5 ) \\beta _ 0 . \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} \\frac { 4 n \\pi y } { e ^ { 2 n \\pi y } - 1 } - \\frac { 4 \\ , n ^ 2 \\ , \\pi ^ 2 \\ , y ^ 2 \\ , e ^ { 2 n \\pi y } } { ( e ^ { 2 n \\pi y } - 1 ) ^ 2 } = \\frac { 4 \\pi y n } { ( e ^ { 2 n \\pi y } - 1 ) ^ 2 } \\left ( e ^ { 2 n \\pi y } - 1 - y n \\pi \\ , e ^ { 2 n \\pi y } \\right ) . \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} \\theta _ 3 ( x ) & = \\Theta \\left ( 0 , i x \\right ) , \\\\ \\theta _ 4 ( x ) & = \\Theta \\left ( \\tfrac { 1 } { 2 } , i x \\right ) . \\end{align*}"}
-{"id": "2797.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\dot { x } _ 1 & = & l _ 3 y _ 3 x _ 2 - y _ 2 x _ 3 , \\\\ \\dot { x } _ 2 & = & y _ 1 x _ 3 - y _ 3 x _ 1 , \\\\ \\dot { x } _ 3 & = & y _ 2 x _ 1 - l _ 3 y _ 1 x _ 2 , \\end{array} \\begin{array} { r c l } \\dot { y } _ 1 & = & m _ 3 x _ 2 + ( l _ 3 - 1 ) y _ 2 y _ 3 , \\\\ \\dot { y } _ 2 & = & - m _ 3 x _ 1 - ( l _ 3 - 1 ) y _ 1 y _ 3 , \\\\ \\dot { y } _ 3 & = & 0 , \\end{array} \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{align*} \\frac { a } { \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ { \\infty } \\exp \\left [ - \\frac { 1 } { 2 } ( a t - \\frac { b } { t } ) ^ 2 \\right ] \\ , d t = 1 \\end{align*}"}
-{"id": "6480.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\int _ { \\Omega } \\varphi ^ { \\prime } \\varphi \\ ; \\d x \\ ; \\d s = \\frac { 1 } { 2 } \\int _ { \\Omega } \\varphi ( t ) ^ 2 \\ ; \\d x \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} \\varphi _ { \\Omega } ( z ) = c ^ { - 1 } \\varphi _ { \\lambda } ( c z ) . \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{align*} U = \\{ x \\in U _ 2 \\mid R _ { T _ 2 , \\lambda } x \\subset U _ 1 \\} . \\end{align*}"}
-{"id": "7478.png", "formula": "\\begin{align*} & \\beta ^ { - 3 } \\nabla _ q \\beta \\cdot \\left ( \\frac { 3 n + 2 } { 6 } - \\int _ 0 ^ \\infty T r [ \\gamma e ^ { - 2 y \\gamma } ] e ^ { - y \\gamma } d y \\right ) \\gamma ^ { - 1 } \\nabla _ q \\beta \\\\ = & \\frac { n + 2 } { 6 } \\beta ^ { - 3 } \\gamma ^ { - 1 } \\| \\nabla _ q \\beta \\| ^ 2 . \\end{align*}"}
-{"id": "2456.png", "formula": "\\begin{align*} \\int _ 0 ^ { U ( N ; \\alpha ) } e ^ { - x } \\left ( 1 - \\frac { \\ln x } { \\ln N } \\right ) ^ r d x \\leq \\int _ 0 ^ 1 e ^ { - x } \\left ( 1 - \\frac { \\ln x } { \\ln N } \\right ) ^ r d x + \\int _ 1 ^ { U ( N ; \\alpha ) } e ^ { - x } d x = O ( 1 ) \\end{align*}"}
-{"id": "8726.png", "formula": "\\begin{align*} A = t _ 1 w _ 1 ^ { m - 1 } \\frac { x _ 1 ^ m - y _ 1 ^ m } { x _ 1 - y _ 1 } , B = t _ 1 w _ 1 ^ { n - 1 } \\frac { x _ 1 ^ n - y _ 1 ^ n } { x _ 1 - y _ 1 } . \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} r = 1 \\ ( N < q ) , 1 < r < + \\infty \\ ( N = q ) , 1 / r + ( p - 1 ) / q ^ \\ast = 1 \\ ( N > q ) \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\beta ^ { - } } T _ { \\gamma } ^ { n } ( 1 ) = \\left \\{ \\begin{array} { l c } T _ { \\beta } ^ { n } ( 1 ) , & n < N \\quad ( \\mbox { l e f t c o n t i n u i t y } ) \\\\ T _ { \\beta } ^ { n _ N } ( 1 ) , & n \\geq N , \\end{array} \\right . \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{align*} k _ 3 ( G ) = \\frac { 1 } { 3 } \\sum _ { v \\in V ( G ) } k _ 3 ( v ) = \\frac 1 3 \\sum _ { v \\in V ( G ) } e ( N ( v ) ) \\le \\frac 1 3 \\sum _ { v \\in V ( G ) } \\binom { d ( v ) } 2 . \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} \\kappa _ n : = \\sum _ { \\pi \\in \\mathfrak { S } ( n ) } \\left ( \\mathrm { s i g n } ( \\pi ) \\left ( w \\kappa _ { d _ \\pi } \\right ) \\otimes \\bigotimes _ { \\ell \\vert ( n / d _ \\pi ) } \\rho _ \\ell ( P _ \\ell ( \\mathrm { F r } ^ { - 1 } _ { \\pi ( \\ell ) } ) ) \\right ) \\in \\mathrm { H } ^ 1 ( \\mathbb { Q } , T _ { \\overline { f } } ( 1 ) / I _ n T _ { \\overline { f } } ( 1 ) ) \\otimes G _ n \\end{align*}"}
-{"id": "6776.png", "formula": "\\begin{align*} \\dd \\pi ( x ) : = \\lim _ { t \\to 0 } \\frac { 1 } { t } \\left ( \\pi ( \\exp ( t x ) ) v - v \\right ) , \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} u _ n = C _ { n - 2 } + \\sum _ { a = 2 } ^ { n - 2 } \\sum _ { t = 0 } ^ { n - 2 - a } \\binom { n - 2 - a } { t } u ' ( n - t , a ) , n \\geq 3 , \\end{align*}"}
-{"id": "7603.png", "formula": "\\begin{align*} \\tilde \\Sigma ^ { i j } = & \\sum _ \\rho ( \\tilde \\gamma ^ { - 1 } \\sigma ) ^ { i } _ \\rho ( \\tilde \\gamma ^ { - 1 } \\sigma ) ^ { j } _ \\rho = \\sum _ \\rho \\sigma ^ 2 ( \\tilde \\gamma ^ { - 1 } ) ^ { i } _ \\rho ( \\tilde \\gamma ^ { - 1 } ) ^ { j } _ \\rho \\\\ = & \\sigma ^ 2 d i a g ( 1 / ( \\gamma ^ 2 + B _ 0 ^ 2 ) , 1 / ( \\gamma ^ 2 + B _ 0 ^ 2 ) , 1 / \\gamma ^ 2 ) . \\end{align*}"}
-{"id": "7925.png", "formula": "\\begin{align*} \\mu ^ { \\boxtimes t } \\left ( \\left \\{ 1 / x \\right \\} \\right ) = t \\mu ( \\{ x ^ { - 1 / t } \\} ) - ( t - 1 ) . \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} n ^ \\frac { 1 } { 4 } & = z _ 1 ^ \\frac { 1 } { 2 } - \\left ( z _ 1 ^ \\frac { 1 } { 2 } - n ^ \\frac { 1 } { 4 } \\right ) \\\\ & > b ( z _ 3 - 1 ) - \\frac { z _ 3 + 1 } { 2 } - 1 > b ( z _ 3 - 1 ) - ( z _ 3 - 1 ) + 3 = \\ell ( z _ 3 - 1 ) , \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} c l ' ( n , 1 ) \\ , = \\ , \\frac { 6 ^ { 3 ^ { n - 1 } } } { 2 \\cdot 3 ^ n } . \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} \\begin{cases} \\Delta u + a _ 0 u = \\sigma u , x \\in D _ L \\cr u = 0 , x \\in \\partial D _ L . \\end{cases} \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{align*} \\pi _ { z _ 0 , a } ^ { ( k ) } ( \\psi _ 1 ) = \\sum _ { j = 1 } ^ { m _ a ^ { ( k ) } ( z _ 0 ) } \\left ( \\int _ M \\psi _ 1 \\wedge S _ j ( a , z _ 0 ) \\right ) U _ j ( a , z _ 0 ) | _ { V _ a } . \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} \\zeta _ \\lambda ( z ( \\lambda , \\hat { X } ) ) = \\int _ { 1 } ^ { \\hat { X } } \\frac { ( X - \\frac 1 3 ( \\lambda + 1 ) ) d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } + \\eta _ 1 / 2 \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{align*} \\overline { V } _ t = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\frac { 1 } { 2 } \\Vert y _ { i , t } - \\bar { y } _ { t } \\Vert ^ { 2 } . \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} \\int _ { \\mathbb { S } } \\frac { d \\sigma ( \\xi ) } { | x - \\xi | ^ { c } } = { } _ 2 F _ { 1 } \\left ( \\frac { c } { 2 } , \\frac { c - n } { 2 } + 1 , \\frac { n } { 2 } ; | x | ^ { 2 } \\right ) , x \\in \\mathbf { B } \\end{align*}"}
-{"id": "4736.png", "formula": "\\begin{align*} N ( w ^ { - 1 } ) = N ( y ^ { - 1 } ) \\sqcup y N ( \\tau ^ { - 1 } ) = N ( y ^ { - 1 } ) \\sqcup y N ( x _ v ^ { - 1 } ) \\sqcup y x _ v N ( z ^ { - 1 } ) . \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{align*} d ( v ) \\geq \\sum _ { r ( e ) = v } d ( s ( e ) ) . \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{align*} { \\partial _ { x _ l } p _ { i j } } = { \\partial _ { x _ j } e _ { i l } } - { \\partial _ { x _ i } e _ { j l } } , \\ \\ \\forall i , j , l = 1 , . . . , n . \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb { E } _ { n , d ( n ) , 0 } ( \\nu _ n ) = 0 \\lim _ { n \\to \\infty } \\mathbb { E } _ { n , d ( n ) , \\theta _ n } ( \\nu _ n ) = 1 . \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} Q ( s , x ; t , y ) = \\sum _ { \\varpi \\in \\Pi ( ( s , x ) \\to ( t , y ) ) } q ( \\varpi ) . \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{align*} g ( t ) = \\sqrt { 2 B t + 1 } . \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{align*} & m ^ { - 1 / 2 } E [ J ^ m _ { s , t } ] \\\\ = & - E \\left [ \\int _ s ^ t ( \\nabla _ z \\chi ) ( r , q _ r ^ m , z _ r ^ m ) \\cdot ( - \\nabla _ q V ( r , q _ r ^ m ) - \\partial _ r \\psi ( r , q _ r ^ m ) + \\tilde F ( r , q _ r ^ m ) ) d r \\right ] \\\\ & - E \\left [ \\int _ s ^ t ( \\nabla _ q \\chi ) ( r , q _ r ^ m , z _ r ^ m ) \\cdot z _ r ^ m d r \\right ] + O ( m ^ { 1 / 2 } ) , \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t f ( t , x ) & = \\gamma \\Delta f ( t , x ) \\quad \\forall t \\geq 0 \\\\ f ( 0 , x ) & = f ( x ) \\ , . \\end{cases} \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} \\sum _ { \\alpha , \\beta = 2 } \\omega _ { \\alpha \\bar \\beta } d z ^ \\alpha \\wedge d \\bar z ^ { \\beta } + \\omega _ { \\sigma \\bar \\sigma } d \\sigma \\wedge d \\bar \\sigma + \\sum _ { j = 2 } \\omega _ { \\sigma \\bar j } d \\sigma \\wedge d \\bar z ^ { j } + \\sum _ { i = 2 } \\omega _ { i \\bar \\sigma } d z ^ i \\wedge d \\bar \\sigma \\end{align*}"}
-{"id": "7592.png", "formula": "\\begin{align*} d x _ t ^ \\prime = b _ + ( t ^ * , x _ t ^ \\prime ) d t - ( - b _ i ) ( t ^ * , x _ t ^ \\prime ) + \\tilde \\sigma ( t ^ * , x _ t ^ \\prime ) \\circ d W _ t \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N \\nabla ^ T f ( y _ { i , t } ) e _ { i , t } = \\sum _ { i = 1 } ^ N ( \\nabla ^ T f ( y _ { i , t } ) - \\nabla ^ T f ( \\overline { x } _ t ) ) ( y _ { i , t } - \\overline { x } _ t ) \\geq \\sum _ { i = 1 } ^ N \\kappa \\| y _ { i , t } - \\overline { x } _ t \\| ^ 2 = 2 \\kappa N \\overline { V } _ t , \\end{align*}"}
-{"id": "749.png", "formula": "\\begin{align*} - 1 + t _ 1 z + t _ 2 z ^ 2 + \\ldots = f _ { \\beta } ( z ) = - ( 1 - \\beta z ) \\Bigl ( \\sum _ { n = 0 } ^ { \\infty } T _ { \\beta } ^ { n } ( 1 ) \\ , z ^ n \\Bigr ) . \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} & \\| e _ { m + 1 } \\| ^ 2 \\\\ & \\le \\exp \\Bigg ( - ( a - 2 \\epsilon ) ( m + 1 ) \\tau + \\int _ { 0 } ^ { t _ { m + 1 } } \\| u ( s ) \\| _ { L _ { \\infty } } \\| u _ \\tau ^ D ( s ) \\| _ { L _ { \\infty } } d s \\Bigg ) \\| e _ 0 \\| ^ 2 \\\\ & \\quad + \\sum _ { k = 1 } ^ { m + 1 } \\sum _ { i = 0 } ^ 6 \\exp \\Bigg ( - ( a - 2 \\epsilon ) ( m + 1 - k ) \\tau + \\int _ { t _ k } ^ { t _ { m + 1 } } \\| u ( s ) \\| _ { L _ { \\infty } } \\| u _ { \\tau , m } ^ D ( s ) \\| _ { L _ { \\infty } } d s \\Bigg ) R _ i ^ { k - 1 } . \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} | | \\overline N | | _ \\infty = | | \\overline N | | _ { L ^ \\infty ( 0 , r _ 0 ) } . \\end{align*}"}
-{"id": "2419.png", "formula": "\\begin{align*} m = ( m _ 1 , m _ 2 , \\dots , m _ g ) e _ j = ( \\delta _ { 1 j } , \\delta _ { 2 j } , \\dots , \\delta _ { g j } ) , \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{align*} \\int _ { M ^ { - ( 1 / \\lambda ) - \\varepsilon } } ^ 1 x ^ { \\lambda - 1 } \\left ( 1 - x \\right ) ^ { \\nu _ 1 M } d x < \\left ( 1 - \\frac { 1 } { M ^ { ( 1 / \\lambda ) + \\varepsilon } } \\right ) ^ { \\nu _ 1 M } = O \\left ( \\frac { 1 } { M ^ r } \\right ) \\ ; r > 0 , \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} H _ { k , 0 } ( x ) : = \\left \\{ \\begin{array} { l l } ( x ^ { q ^ 2 } - x ) f _ k ^ 2 ( x ) ( x ^ 3 + A x + B ) + f _ { k - 1 } ( x ) f _ { k + 1 } ( x ) & ( k \\textrm { e v e n ) } \\\\ ( x ^ { q ^ 2 } - x ) f _ k ^ 2 ( x ) + f _ { k - 1 } ( x ) f _ { k + 1 } ( x ) ( x ^ 3 + A x + B ) & ( k \\textrm { o d d ) . } \\end{array} \\right . \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{align*} L _ a ( x ) = \\sum _ { p } c _ p L _ { u _ p } ( x ) \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} \\sigma _ { \\mu } ( z \\mu + m ) = z \\mu + \\sigma ( m ) . \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} \\langle f , u \\rangle _ { [ L ^ { q ^ { \\prime } } ] ^ * , L ^ { q ^ { \\prime } } } = \\int _ { \\Omega } g ( x ) \\cdot \\overline { u ( x ) } \\ ; \\d x ( u \\in L ^ { q ^ { \\prime } } _ { \\sigma } ( \\Omega ) ) , \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} - \\lambda _ 1 ( 1 , 1 , 1 ) = - \\lambda _ 2 ( 0 , 0 , 1 ) = - \\lambda _ 3 ( 0 , 1 , 0 ) = - \\lambda _ 4 ( 1 , 0 , 0 ) = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "763.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to \\beta ^ - } f _ { \\gamma } ( z ) = f _ { \\beta } ( z ) = \\lim _ { \\gamma \\to \\beta ^ + } f _ { \\gamma } ( z ) . \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} r _ i = \\sum \\limits _ { j = 2 } ^ N m _ { i j } c _ j , ~ i \\in \\{ 2 , 3 , \\dots , N \\} ; \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} \\hat { \\psi } ( p , t ) = \\mathcal { F } \\left \\{ \\psi ( x , t ) ; p \\right \\} = \\frac { 1 } { 2 \\pi { } \\hslash { } } \\int _ { - \\infty { } } ^ { \\infty { } } e ^ { - i p x / \\hslash { } } \\ \\psi ( x , t ) \\ d x \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} x = \\overline { d } - \\overline { b } \\hbox { a n d } y = d _ s , \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} s _ = ' ( n , 1 ) \\ , = \\ , 1 2 ^ { 3 ^ { n - 2 } ( 3 ^ { n - 1 } - 1 ) / 2 } - \\frac { 6 ^ { 3 ^ { n - 1 } } } { 2 \\cdot 3 ^ n } \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{align*} \\sup \\bigg \\{ \\frac { N [ \\mathfrak { e } , V ] } { N _ { s c } [ \\mathfrak { e } , V ] } \\ ; \\bigg | \\ V , \\ ; N _ { s c } [ \\mathfrak { e } , V ] < \\epsilon \\bigg \\} \\ = \\ \\infty . \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} r = r ( x ) = \\frac { \\left ( \\sqrt { 5 } - 1 \\right ) \\left ( 1 - \\sqrt { 1 - \\left ( 3 + \\sqrt { 5 } \\right ) x + \\frac { 1 } { 2 } \\left ( 3 - \\sqrt { 5 } \\right ) x ^ 2 } \\right ) - \\left ( 1 + \\sqrt { 5 } \\right ) x } { 4 x } \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} & \\operatorname { A s s } _ A \\left ( \\operatorname { T o r } _ { 2 i } ^ A ( M , N ) \\right ) = \\phi \\quad \\mbox { a n d } \\\\ & \\operatorname { A s s } _ A \\left ( \\operatorname { T o r } _ { 2 i - 1 } ^ A ( M , N ) \\right ) = \\operatorname { A s s } _ A ( k ) = \\{ ( u , x ) / ( u x ) \\} . \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} C _ 1 : y ^ 2 = h ^ 3 - 2 h ^ 2 - 3 h \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; C _ 2 : y ^ 2 = h ^ 3 + 2 h ^ 2 - 3 h \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} | \\hat { z } | \\leq \\int _ { 0 } ^ { 1 } \\frac { d t } { \\sqrt { t ( 1 - t ) } } = \\pi . \\end{align*}"}
-{"id": "5076.png", "formula": "\\begin{align*} ( J u ) \u2019 ( \\l ) = \\l u \u2019 ( \\l ) , \\end{align*}"}
-{"id": "8378.png", "formula": "\\begin{align*} \\mu _ k ( \\d q ) = \\ 1 _ { ( - 2 \\sqrt { k - 1 } , 2 \\sqrt { k - 1 } ) } ( q ) \\frac { k \\sqrt { 4 ( k - 1 ) - q ^ 2 } } { 2 \\pi ( k ^ 2 - q ^ 2 ) } \\d q , q \\in \\ \\R . \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} \\frac 1 n \\sum _ { i = 1 } ^ n \\psi _ j ^ 2 ( i ) = 1 \\end{align*}"}
-{"id": "1429.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } h ( R , x ' ) = \\lim _ { R \\to \\infty } h ( R , 0 ) = : c _ 1 \\ge 0 . \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} \\operatorname { T o r } _ i ^ { A _ { \\mathfrak { p } } } ( M _ { \\mathfrak { p } } , N _ { \\mathfrak { p } } ) = 0 \\mbox { f o r a l l } i > \\dim ( A ) , \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} \\mathcal { L } _ { i i } & = ( i , i , 1 ) , ( i , i , 2 ) , \\ldots , ( i , i , r ) , & 1 & \\leq i \\leq n , \\\\ \\mathcal { L } _ { i j } & = ( i , j , 1 ) , ( i , j , 2 ) , \\ldots , ( i , j , r ) , ( j , i , 1 ) , ( j , i , 2 ) , \\ldots , ( j , i , r ) , & 1 & \\leq j < i \\leq n ; \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} \\sigma ( u , \\pi ) = - \\pi I _ { 3 } + 2 \\nu \\varepsilon ( u ) , \\varepsilon ( u ) = \\frac { 1 } { 2 } \\left ( \\nabla u + \\nabla u ^ { \\top } \\right ) , \\end{align*}"}
-{"id": "645.png", "formula": "\\begin{align*} y _ { n + 1 } \\leq y _ n + \\left ( \\sum _ { i = 1 } ^ { M } \\frac { \\lambda _ i } { v _ i } \\right ) \\mathbb { P } ( T \\geq n + 1 ) - \\Delta _ n \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} \\sigma _ i ( M _ 1 , \\dots , M _ r ) = \\mathrm { I d } + p \\cdot N _ i , \\end{align*}"}
-{"id": "7137.png", "formula": "\\begin{align*} \\mathcal { D } _ { x } = \\{ ( x , Y _ { - } ( x ) ) : \\vert x \\vert < 1 \\} \\bigcup \\{ ( x , Y _ { + } ( x ) ) : \\vert x \\vert < 1 \\} . \\end{align*}"}
-{"id": "4377.png", "formula": "\\begin{align*} z ( \\lambda , \\xi ) = \\int ^ { - \\infty } _ { \\xi } \\frac { d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } \\end{align*}"}
-{"id": "7955.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } ( a _ t ) ^ { 1 / t } = \\left ( \\int _ 0 ^ \\infty x ^ { - 1 } d \\mu ( x ) \\right ) ^ { - 1 } , \\lim _ { t \\rightarrow \\infty } ( b _ t ) ^ { 1 / t } = \\int _ 0 ^ \\infty x \\ , d \\mu ( x ) . \\end{align*}"}
-{"id": "2989.png", "formula": "\\begin{align*} ( q - 1 ) \\left ( h ( q , z ) - \\frac 1 2 h ( q ^ 2 , z ^ 2 ) \\right ) & = ( q - 1 ) \\sum _ { l = 1 } ^ { \\infty } \\frac { k ( q ^ { 2 l - 1 } , z ^ { 2 l - 1 } ) } { 2 l - 1 } \\\\ & = \\sum _ { l = 1 } ^ { \\infty } \\sum _ { m = 1 } ^ { \\infty } \\frac { q - 1 } { q ^ { 2 l - 1 } - 1 } \\frac { ( - 1 ) ^ { m - 1 } Q _ m ( q ^ { 2 l - 1 } ) } { ( 2 l - 1 ) q ^ { ( 2 l - 1 ) \\binom { d } { 2 } } } z ^ { ( 2 l - 1 ) m } , \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} \\| | \\partial _ x | ^ n h \\| _ 2 ^ 2 = \\int _ { \\mathbb { R } } | k | ^ { 2 n } \\exp ( - 2 t | k | ^ 3 ) | \\hat h _ 0 | ^ 2 d k , \\end{align*}"}
-{"id": "2379.png", "formula": "\\begin{align*} V \\left [ S ( \\theta ) \\right ] = E \\left [ S ( \\theta ) ^ { ( 2 ) } \\right ] - E \\left [ S ( \\theta ) \\right ] - E \\left [ S ( \\theta ) \\right ] ^ 2 , \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} \\mathcal { I } _ 1 = \\{ i : d _ i < n ^ { 1 / \\ell } / \\log n \\} , \\mathcal { I } _ 2 = \\{ i : d _ i \\geq n ^ { 1 / \\ell } / \\log n \\} . \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} \\dot { \\bar { x } } ( t ) & = \\bar { A } \\bar { x } ( t ) + \\bar { B } u ( t ) + \\displaystyle \\sum _ { j = 1 } ^ { n _ { \\rm i n } } u _ j ( t ) \\bar { N } _ j \\bar { x } ( t ) + \\bar { H } ( \\bar { x } ( t ) \\otimes \\bar { x } ( t ) ) \\\\ \\bar { y } ( t ) & = \\bar { c } ^ \\top \\bar { x } ( t ) \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{align*} \\tag * { $ { \\bf ( A _ 1 ) } $ } & \\ A \\in C ^ \\rho \\big ( J , \\mathcal { L } ( E _ 0 , E _ 1 ) \\big ) \\ \\ - A ( a ) \\ \\\\ & \\ , . \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{align*} C ( \\lambda ) = C _ 0 + C _ 1 \\lambda ^ { - 1 } + \\cdots + C _ k \\lambda ^ { - k } + \\cdots \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} L = \\mathcal { L } + | A | ^ 2 - \\frac 1 2 = \\Delta + \\frac 1 2 \\left < x , \\nabla \\cdot \\right > + | A | ^ 2 - \\frac 1 2 . \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{align*} \\sigma _ q = \\frac { 1 } { \\sqrt { q } } \\left ( \\begin{array} { r r } 1 & q - 1 \\\\ 1 & - 1 \\end{array} \\right ) , \\end{align*}"}
-{"id": "2302.png", "formula": "\\begin{align*} \\tilde { T } _ 1 : = \\bigvee _ { i = 1 } ^ { M _ 1 } X _ i , \\tilde { T } _ 2 : = \\bigvee _ { i = M _ 1 + 1 } ^ { M _ 1 + M _ 2 } X _ i , \\dots , \\tilde { T } _ g : = \\bigvee _ { i = M _ 1 + \\cdots + M _ { g - 1 } + 1 } ^ { M _ 1 + \\cdots + M _ g } X _ i . \\end{align*}"}
-{"id": "8789.png", "formula": "\\begin{align*} \\phi ( g ) = - 2 \\ln | s ( g ) | _ { \\pi ^ * q } . \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} A ^ T P _ 2 ^ { - 1 } + P _ 2 ^ { - 1 } A + \\sum _ { i = 1 } ^ m N _ i ^ T P _ 2 ^ { - 1 } N _ i \\leq - P _ 2 ^ { - 1 } B B ^ T P _ 2 ^ { - 1 } \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} m \\cdot \\alpha \\otimes n \\otimes u \\otimes v = m \\otimes \\alpha \\cdot n \\otimes u \\otimes v \\\\ + m \\otimes n \\otimes \\alpha u \\otimes v \\\\ - m \\otimes n \\otimes u \\otimes v \\alpha \\ , . \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{align*} \\varrho = \\frac 1 { f ^ \\# ( c ) } \\quad s = \\frac { f ^ \\# ( c ) } { 3 \\varphi ( f ^ \\# ( c ) ) } \\end{align*}"}
-{"id": "8029.png", "formula": "\\begin{align*} ^ 1 ( X _ 0 ) : = ( X _ 0 ) \\cap ^ 0 ( M ) . \\end{align*}"}
-{"id": "7207.png", "formula": "\\begin{align*} _ { r } \\phi _ s \\left ( { { a _ 1 , \\ldots , a _ { r } } \\atop { b _ 1 , \\ldots , b _ s } } ; q , z \\right ) = \\sum _ { n = 0 } ^ \\infty \\frac { ( a _ 1 , \\ldots , a _ r ; q ) _ n } { ( q , b _ 1 , \\ldots , b _ s ; q ) _ n } \\left ( ( - 1 ) ^ n q ^ { n ( n - 1 ) / 2 } \\right ) ^ { 1 + s - r } z ^ n . \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} \\bigl [ [ s , b _ 1 ] b _ 2 , b _ 3 , x \\bigr ] = \\bigl [ [ s , y _ 1 \\dots y _ k ] b _ 2 , b _ 3 , x \\bigr ] = \\sum _ { i = 1 } ^ k [ y _ 1 \\dots y _ { i - 1 } [ s , y _ i ] y _ { i + 1 } \\dots y _ k b _ 2 , b _ 3 , x ] . \\end{align*}"}
-{"id": "9182.png", "formula": "\\begin{align*} L _ n ( y ) = \\sum _ { k = 1 } ^ \\infty \\frac { L _ { n , k } } { ( - y ) ^ k } , U _ n ( y ) = \\sum _ { k = 1 } ^ \\infty \\frac { U _ { n , k } } { ( - y ) ^ k } \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} \\psi ( ( s \\otimes t ) \\cdot p ) = ( s \\otimes t ) \\cdot \\psi ( p ) - ( 1 \\otimes \\partial ) ( s \\otimes t ) \\cdot ( k \\circ d + d \\circ k ) ( p ) \\ , . \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} Q _ { \\lambda X , Y } = \\lambda Q _ { X , Y } \\qquad { \\rm a n d } Q _ { X , \\lambda Y } = \\lambda ^ { - 1 } Q _ { X , Y } \\ , \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} \\alpha _ j ^ { + } = \\frac 1 2 \\left ( 1 + \\sqrt { 1 + c \\lambda _ j } \\right ) , \\ \\ \\ \\alpha _ j ^ { - } = \\frac 1 2 \\left ( 1 - \\sqrt { 1 + c \\lambda _ j } \\right ) . \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { \\infty } \\lambda _ k ^ { - \\alpha q } < \\infty , \\ , \\ , \\mbox { f o r a l l } \\ , \\ , q > \\frac { p _ a } { \\alpha } . \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} X \\# _ { 1 - t } Y = Y \\# _ t X \\ . \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{align*} W ( x , y ) = \\sum _ { r = 0 } ^ \\mu a _ r W _ 2 ( x , y ) ^ { 6 \\mu + \\nu - 6 r } W ' _ { 1 2 } ( x , y ) ^ r , \\end{align*}"}
-{"id": "6001.png", "formula": "\\begin{align*} C _ { \\alpha { } } D _ { \\theta { } } ^ { \\alpha { } } \\phi ( x ) + V ( x ) \\phi ( x ) = E \\phi ( x ) , \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} u ( a ) = \\Pi [ u ] ( a ) \\varphi \\ , , a \\in J \\ , , \\varphi = Q [ u ] \\varphi \\ , , \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{align*} { \\rm M S E } ( \\hat D _ { \\lambda ^ * } ) = \\mathbb { E } \\left [ ( \\hat D _ { \\lambda ^ * } - D ) ^ 2 \\right ] \\le \\rho _ { \\lambda ^ * } ( \\varGamma ^ { - 1 / 2 } ) . \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{align*} R & = [ x _ 1 , y _ 1 ] \\cdots [ x _ { 2 p } , y _ { 2 p } ] , ~ \\textnormal { a n d } \\\\ S & = [ x _ 1 , y _ 1 ] \\cdots [ x _ p , y _ p ] a b a b ^ 2 \\cdots a b ^ { q } . \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} \\delta = d - e \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} \\bar \\mu ^ { - 1 } ( \\mu _ 0 , \\dots , \\mu _ { k - 1 } ) = ( \\lambda _ i ) _ { 0 \\le i \\le k } , \\lambda _ i = \\sum _ { j = 0 } ^ { i - 1 } q ^ { ( i + j ) ( i - j - 1 ) / 2 } \\mu _ j . \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} g ( r , \\theta ) = - \\frac { \\Im u ( r e ^ { i \\theta } ) } { \\theta } = \\frac { r \\sin \\theta } { \\theta } \\int _ 0 ^ \\infty \\frac { s ^ 2 + 1 } { r ^ 2 - 2 r s \\cos \\theta + s ^ 2 } \\ ; d \\rho ( s ) . \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\mathcal { U } ^ { \\prime } ( t ) + A _ p \\mathcal { U } ( t ) & = F ( t ) ( 0 < t < T ) \\\\ \\mathcal { U } ( 0 ) & = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} \\begin{aligned} ( T _ i - t ) ( T _ i + 1 ) = 0 , & T _ i T _ { i + 1 } T _ i = T _ { i + 1 } T _ i T _ { i + 1 } , \\\\ T _ i T _ j = T _ j T _ i , & \\forall \\ i , j \\ \\ \\ | i - j | > 1 . \\end{aligned} \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} S ( t ) = - t - S ( t _ 1 ) \\ast t _ 2 \\end{align*}"}
-{"id": "1569.png", "formula": "\\begin{align*} d _ i ( x _ j ) = \\delta _ { i j } , d _ i ( x y ) = d _ i ( x ) y + \\chi ( \\alpha , \\alpha _ i ) x d _ i ( y ) \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{align*} E \\left [ S _ N ^ { ( r ) } \\right ] = N ^ r ( \\ln N ) ^ r \\left [ \\sum _ { k = 0 } ^ n \\binom { r } { k } \\frac { ( - 1 ) ^ k \\ , \\Gamma ^ { ( k ) } ( 1 ) } { \\ln ^ k N } + o \\left ( \\frac { 1 } { \\ln ^ n N } \\right ) \\right ] , N \\to \\infty , \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} \\tilde { \\epsilon } _ 1 = \\max \\min _ { x , y } \\delta _ { x , y } \\end{align*}"}
-{"id": "3349.png", "formula": "\\begin{align*} A \\sum _ { x = 1 } ^ n \\tau ( E _ { x , 0 } ) - B \\sum _ { x \\ne y } \\tau ( E _ { x , 0 } E _ { y , 0 } ) \\end{align*}"}
-{"id": "7480.png", "formula": "\\begin{align*} d x _ t = b ( t , x _ t ) d t + \\tilde \\sigma ( t , x _ t ) \\circ d W _ t \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} \\mathcal M _ i ( t ) = \\int _ 0 ^ t \\int _ 0 ^ { \\tau _ i } \\cdots \\int _ 0 ^ { \\tau _ 2 } D R ^ \\dagger ( \\widetilde x _ { \\tau _ i } ) \\cdots D R ^ \\dagger ( \\widetilde x _ { \\tau _ 1 } ) \\ , d \\tau _ 1 \\cdots d \\tau _ i , \\end{align*}"}
-{"id": "8287.png", "formula": "\\begin{align*} E _ d ( R ) _ { q = 1 } = d . \\end{align*}"}
-{"id": "879.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } - \\frac 1 2 \\Delta \\phi + \\omega \\phi + \\phi \\psi = 0 \\\\ \\psi + \\phi ^ 2 = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "8578.png", "formula": "\\begin{align*} 0 < \\eta < g ( \\alpha , d ) = \\min \\{ \\alpha , \\ ; d , \\ ; ( \\alpha + d - 1 ) / ( 1 + d ) \\} \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{align*} r \\ell ( S ^ 1 ( B ) ) = B \\frac { d \\ell ( S ^ 1 ( B ) ) } { d B } . \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} \\frac { 1 } { \\pi } \\int _ \\mathbb { C } e ^ { - 2 | w | ^ 2 + a w + b \\overline { w } } d L ( w ) = \\frac { 1 } { 2 } e ^ { \\frac { a b } { 2 } } \\mbox { i f $ a , b \\in \\mathbb { C } $ } . \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} K _ { r , a } : = \\frac { \\mathbb { C } \\langle z _ 1 \\dots , z _ l \\rangle } { \\left \\langle { \\begin{array} { c } z _ i \\left ( z _ i ^ { \\beta _ i - 2 } \\right ) \\left ( z _ { i + 1 } ^ { \\beta _ { i + 1 } - 2 } \\right ) \\dots { \\left ( z _ { j - 1 } ^ { \\beta _ { j - 1 } - 2 } \\right ) } \\left ( z _ j ^ { \\beta _ j - 2 } \\right ) z _ j = 0 \\\\ z _ i z _ j = 0 \\end{array} } \\right \\rangle } \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} \\vec { A } ' : = & ( B _ 0 , B _ 1 , \\dots , B _ { b - 1 } , B _ b - 1 , 1 , C _ 1 , \\dots , C _ c ) \\\\ = : & ( B ' _ 0 , B ' _ 1 , \\dots , B ' _ { b + 1 } , C ' _ 1 , \\dots , C ' _ c ) . \\end{align*}"}
-{"id": "5278.png", "formula": "\\begin{align*} P _ q ^ \\ast A = P _ q ^ \\ast A P _ q = A P _ q \\ , . \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} B ( x ) & = ( 1 - x ) ( C ( x ) - 1 ) + \\frac { x ( 3 x ^ 5 - 8 x ^ 4 + 1 3 x ^ 3 - 1 1 x ^ 2 + 5 x - 1 ) } { ( 1 - x ) ^ 6 } , \\\\ D ( x ) & = x ( 1 - x ) ( C ( x ) - 1 ) - \\frac { x ^ 2 ( 7 x ^ 6 - 2 2 x ^ 5 + 3 7 x ^ 4 - 3 6 x ^ 3 + 2 1 x ^ 2 - 7 x + 1 ) } { ( 1 - x ) ^ 6 ( 1 - 2 x ) } , \\\\ E ( x ) & = \\frac { x ^ 2 ( 2 - x ) } { 1 - x } ( C ( x ) - 1 ) - \\frac { x ^ 3 ( 8 x ^ 6 - 2 9 x ^ 5 + 5 4 x ^ 4 - 5 7 x ^ 3 + 3 6 x ^ 2 - 1 3 x + 2 ) } { ( 1 - x ) ^ 7 ( 1 - 2 x ) } . \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} \\omega = \\frac { \\sqrt { - 1 } } { 2 } \\sum \\limits _ { i , j = 1 } ^ n A _ { i j } d z ^ i \\wedge d \\bar { z } ^ j \\end{align*}"}
-{"id": "5923.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathbb { Z } } M _ i [ H ( \\nu , \\cdot ) ] \\left ( \\mu \\right ) = \\sum _ { i \\in \\{ z , z + l \\} \\bigcup d _ 1 ( \\vec { x } , \\vec { y } ) \\bigcup d _ 2 ( \\vec { x } , \\vec { y } ) } M _ i [ H ( \\nu , \\cdot ) ] ( \\mu ) , \\end{align*}"}
-{"id": "160.png", "formula": "\\begin{align*} \\| ( 0 , \\varphi _ t ) - D _ t ^ 1 \\xi _ t \\| _ { L ^ 2 } ^ 2 = \\| \\varphi _ \\infty \\| _ { L ^ 2 } ^ 2 + a t ^ { - 5 / 3 } + \\mathcal { O } ( e ^ { - \\beta t } ) . \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{align*} \\Vert ( 1 + Q ) ^ n \\Vert & \\leq \\sum _ { k = 0 } ^ n \\binom n k \\Vert Q ^ k \\Vert \\leq \\sum _ { k = 0 } ^ { N _ 1 - 1 } \\binom n k \\Vert Q ^ k \\Vert + \\sum _ { k = N _ 1 } ^ n \\binom n k \\left ( \\frac { \\varepsilon } { 2 } \\right ) ^ k \\\\ & \\leq \\left ( 1 + \\frac { \\varepsilon } { 2 } \\right ) ^ n + \\sum _ { k = 0 } ^ { N _ 1 - 1 } n ^ k \\Vert Q \\Vert ^ k \\leq \\left ( 1 + \\frac { \\varepsilon } { 2 } \\right ) ^ n + N _ 1 ( 1 + n \\Vert Q \\Vert ) ^ { N _ 1 } . \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} f _ m : = \\left \\{ \\begin{array} { l l } \\psi _ m & m \\textrm { o d d } \\\\ \\psi _ m / ( 2 y ) & m \\textrm { e v e n } \\end{array} \\right . \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 6 & 9 & 5 \\\\ 1 0 & 8 & 6 & 6 \\\\ 1 & 8 & 1 & 1 \\\\ 6 & 1 0 & 9 & 1 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 & 8 & 5 \\\\ 1 0 & 1 & 3 & 6 \\\\ 1 & 0 & 7 & 1 \\\\ 6 & 1 & 2 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "2224.png", "formula": "\\begin{align*} 0 = v _ { k + 1 } ^ { ( 0 ) } \\cdots v _ { k + 1 } ^ { ( n _ { k + 1 } ) } = u _ { k + 1 } ( u _ { k + 1 } + y _ 1 ) \\cdots ( u _ { k + 1 } + y _ { n _ { k + 1 } } ) \\\\ = u _ { k + 1 } ^ { n _ { k + 1 } + 1 } + u _ { k + 1 } ^ { n _ { k + 1 } } a _ 1 + \\cdots + u _ { k + 1 } a _ { n _ { k + 1 } } , \\end{align*}"}
-{"id": "7720.png", "formula": "\\begin{align*} \\frac { 1 + \\epsilon _ 1 } { \\epsilon _ 1 \\phi _ i } = & \\frac { 1 + \\epsilon _ 1 } { \\epsilon _ 1 \\min \\left \\{ \\frac { \\bar { \\xi } _ 2 } { \\epsilon _ 2 } , \\cdots , \\frac { \\bar { \\xi } _ { M _ s } } { \\epsilon _ { M _ s } } \\right \\} } \\geq \\frac { 1 + \\epsilon _ 1 } { \\epsilon _ 1 \\frac { \\bar { \\xi } _ { M _ s } } { \\epsilon _ { M _ s } } } = \\frac { \\epsilon _ { M _ s } 2 ^ { R _ 1 } } { \\epsilon _ 1 \\bar { \\xi } _ { M _ s } } , \\end{align*}"}
-{"id": "256.png", "formula": "\\begin{align*} \\pi ( i , j ) : = [ Y _ i , Y _ j ] i < j \\in [ n ] , \\end{align*}"}
-{"id": "2181.png", "formula": "\\begin{align*} \\int _ { d ( x , u ) > R } d ( x , u ) ^ p \\ , d \\mu ( x ) = \\int _ C 1 _ { U _ R } ( x ) d ( x , u ) ^ p \\ , d \\mu ( x ) + \\int _ C 1 _ { V _ R } ( x ) d ( x , u ) ^ p \\ , d \\mu ( x ) . \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} k _ 0 e + d ' - \\frac 1 2 ( k _ 2 h _ 1 - k _ 1 h _ 2 ) = 0 \\end{align*}"}
-{"id": "169.png", "formula": "\\begin{align*} \\mu _ t ( | q | _ k ) | q | _ k ^ { - 1 / 2 } \\langle \\dot q , \\eta \\rangle = 8 \\mu _ t ( | q | _ k ) \\Re ( \\dot f \\bar h ) | w | ^ 2 , \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{align*} t ^ 2 M _ t = t ^ 2 r \\widehat { M } _ \\rho : = t ^ { 4 / 3 } M _ { \\varrho } , \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} r _ 2 ( r _ 2 ( n ) ) & = 4 \\delta ( 2 ^ { 2 + \\nu } m ( n ) ) = 4 \\delta ( 2 ^ { 2 + \\nu } ) \\delta ( m ( n ) ) \\\\ & = 4 \\delta ( 2 ^ { \\nu } ) \\delta ( m ( n ) ) = 4 \\delta ( 2 ^ { \\nu } m ( n ) ) = 4 \\delta ( \\delta ( n ) ) . \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} H _ Z ( z ) : = \\begin{cases} 0 & \\ z \\in \\C _ + \\\\ c _ d ^ { - 1 } | z | ^ d h _ d ( | \\theta | ) & \\ z \\in \\overline { \\C } _ - \\end{cases} \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} \\overline { M } _ { n + 1 } = \\frac { ( b _ { \\sup } - \\chi \\mu ) a _ { \\sup } - \\chi \\mu a _ { \\inf } + ( \\chi \\mu ) ^ 2 \\overline { M } _ { n } } { ( b _ { \\inf } - \\chi \\mu ) ( b _ { \\sup } - \\chi \\mu ) } . \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} M ( t ) = { \\rm e } ^ { N ( t ) - \\frac 1 2 \\langle N \\rangle _ t } \\qquad . \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} \\varphi _ 1 \\circ w X ^ { q ^ s } \\circ \\varphi _ 2 = & ( c g ^ { q ^ { s + l } } w ^ { q ^ l } + d h ^ { q ^ { s + l + n } } w ^ { q ^ { l + n } } ) X ^ { q ^ { j + s + l } } \\\\ & + ( c h ^ { q ^ { s + l } } w ^ { q ^ l } + d g ^ { q ^ { s + l + n } } w ^ { q ^ { l + n } } ) X ^ { q ^ { j + s + l + n } } , \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\sup _ { | x | \\leq c t } | u ( x , t ) - \\frac { a } { b } | = 0 \\end{align*}"}
-{"id": "7898.png", "formula": "\\begin{align*} \\widetilde { H } ( t ) : = \\widetilde { H } \\left ( t , \\frac { Z + Z ' } { 2 } , D _ z \\right ) , \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} d = 1 3 . \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} D _ { x , y } \\left ( a b \\right ) = D _ { x , y } ( a ) b + a D _ { x , y } ( b ) . \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{align*} ( K v ) ( x ) : = \\int _ 0 ^ L k ( x , y ) v ( y ) d y ( S v ) ( x ) : = \\int _ 0 ^ L s ( x , y ) v ( y ) d y , \\end{align*}"}
-{"id": "3148.png", "formula": "\\begin{align*} { } \\frac { v ^ 2 } { u ^ 2 } = \\int _ 0 ^ { v ( p ) } \\frac { t \\ , d t } { 1 - e ^ { - t } } + \\left ( \\frac { v } { p } - 1 \\right ) \\left ( v ( p ) - v \\right ) \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{align*} \\gamma _ e = \\frac { 1 + { \\rm P N R } _ 1 } { 1 + { \\rm P N R } _ 2 } = \\frac { 1 + \\gamma \\cdot { \\rm P N R } _ 2 } { 1 + { \\rm P N R } _ 2 } \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} \\gamma = w _ 0 = v _ 1 v _ 2 \\dots v _ r \\cdot \\Big ( \\prod _ { j = 1 } ^ i u _ j \\Big ) . \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } M _ n ( t ) = M ( t ) \\ \\mbox { a . s . \\ a n d i n } \\ L ^ 2 . \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{align*} J _ { s , t } ^ m = \\int _ s ^ t B ^ { i _ 1 , . . . , i _ k } ( r , q _ r ^ m ) ( z _ r ^ m ) _ { i _ 1 } . . . ( z _ r ^ m ) _ { i _ k } d r \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{align*} V \\cap \\{ \\psi - \\varphi \\leq c \\} = V \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\sigma ( Z _ t ) \\sum _ { y \\in B _ { m _ { L _ t } } ( Z _ t ) \\setminus \\{ Z _ t \\} } \\frac { \\phi _ { Z _ t , m _ { L _ t } } ( y ) } { \\phi _ { Z _ t , m _ { L _ t } } ( Z _ t ) } = 0 \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} \\max _ { x \\in B _ { R _ t } ( z ) } \\xi ( x ) \\ge \\lambda _ { B _ { R _ t } ( z ) } = a _ t + o ( 1 ) . \\end{align*}"}
-{"id": "1999.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { B _ { s } } ( | \\nabla u | ^ { 2 } ) ^ { \\mu } d x \\le C _ { 0 } \\int _ { B _ { s } - B _ { t } } ( | \\nabla u | ^ { 2 } ) ^ { \\mu } d x + C _ { 0 } \\left \\{ \\frac { 1 } { ( s - t ) ^ { n } } \\int _ { B _ { s } } d ^ { n } ( u , p _ { 0 } ) d x + \\int _ { B _ { R } } g d x \\right \\} \\end{aligned} \\end{align*}"}
-{"id": "2677.png", "formula": "\\begin{align*} \\lambda = - a f + a ( m + n - 1 ) + a _ { 0 } , \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} \\mathcal { B } ( L ^ p ( \\R ^ d ) , L ^ { p ' } ( \\R ^ d ) ) & = L ^ d ( \\R ^ d ) \\quad \\mbox { f o r } p = \\frac { 2 d } { d + 1 } , \\\\ \\mathcal { B } ( L ^ p ( \\R ^ d ) , L ^ { p ' } ( \\R ^ d ) ) & = L ^ { \\frac { d + 1 } { 2 } } ( \\R ^ d ) \\quad \\mbox { f o r } p = \\frac { 2 ( d + 1 ) } { d + 3 } . \\end{align*}"}
-{"id": "1423.png", "formula": "\\begin{align*} u \\big ( \\eta ( t ) \\big ) - u \\big ( \\eta ( s ) \\big ) = - \\frac { 1 } { 2 } \\int _ s ^ t \\big \\{ | \\nabla ^ L u | ^ 2 \\big ( \\eta ( r ) \\big ) + | \\dot { \\eta } | ^ 2 ( r ) \\big \\} \\ , d r \\le s - t , \\end{align*}"}
-{"id": "956.png", "formula": "\\begin{align*} \\liminf _ { \\lambda \\rightarrow \\infty } \\frac { N _ { s c } [ \\mathfrak { e } , \\lambda V ] } { N [ \\mathfrak { e } , \\lambda V ] } < \\infty , \\limsup _ { \\lambda \\rightarrow \\infty } \\frac { N _ { s c } [ \\mathfrak { e } , \\lambda V ] } { N [ \\mathfrak { e } , \\lambda V ] } = \\infty . \\end{align*}"}
-{"id": "3820.png", "formula": "\\begin{align*} \\beta _ 1 = { s + 3 \\choose 3 } , ~ \\beta _ 2 = 3 { s + 2 \\choose 3 } , ~ \\beta _ 3 = 3 { s + 1 \\choose 3 } , \\beta _ 4 = { s \\choose 3 } . \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} \\epsilon _ 2 = \\left ( \\frac { p _ { X Y } ( x _ 1 , y _ 1 ) p _ { X Y } ( x _ 2 , y _ 2 ) } { p _ { X Y } ( x _ 1 , y _ 2 ) p _ { X Y } ( x _ 2 , y _ 1 ) } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{align*} \\Gamma _ { t } & = \\Gamma _ { s } \\times \\Gamma _ { s s } \\\\ M e ^ { t M } \\Gamma _ { 0 } & = e ^ { t M } ( \\Gamma _ { 0 } ' \\times \\Gamma _ { 0 } '' ) \\\\ M \\Gamma _ { 0 } & = \\Gamma _ { 0 } ' \\times \\Gamma _ { 0 } '' \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{align*} Z _ R ( x , z ) = \\sum _ { E = 1 } ^ { \\infty } \\sum _ { N = 1 } ^ { \\infty } z ^ { N } \\ , x ^ { E } q ( E , N , B ) \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} \\partial _ s \\Phi _ { \\beta , u , \\gamma } ^ \\delta & = - \\frac { \\xi '' ( s ) } { 2 } \\bigl ( \\partial _ { x x } \\Phi _ { \\beta , u , \\gamma } ^ \\delta + \\gamma ( s ) \\bigl ( \\Phi _ { \\beta , u , \\gamma } ^ \\delta \\bigr ) ^ 2 \\bigr ) , \\ , \\ , ( s , x ) \\in [ 0 , u ) \\times \\mathbb { R } , \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} \\beta _ i = { m - 1 \\choose i - 1 } { s + m - i \\choose m - 1 } . \\end{align*}"}
-{"id": "2536.png", "formula": "\\begin{align*} \\tag * { $ { \\bf ( A _ { 1 2 } ' ) } $ } & \\\\ & \\ , . \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} \\psi _ z ( \\lambda ) = \\frac { h ( \\lambda ) \\bullet ( z - \\varphi ( \\lambda ) ) - h ( 0 ) \\bullet ( z - \\varphi ( 0 ) ) } { \\lambda } + \\lambda \\ , \\overline { h ( 0 ) \\bullet ( z - \\varphi ( 0 ) ) } . \\end{align*}"}
-{"id": "6313.png", "formula": "\\begin{align*} _ E ( \\mathbf { e } _ i ) = \\frac { P _ { \\mathbf { b } _ 0 } h _ { \\mathbf { b } _ 0 , \\mathbf { e } _ i } D _ { \\mathbf { b } _ 0 , \\mathbf { e } _ i } ^ { - \\alpha } } { I ' _ B + I ' _ U + I ' _ J + N _ 0 } . \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} \\forall q \\in [ - 1 / 2 , 1 / 2 ] , A ^ V _ q = \\left | - i \\dfrac { d } { d x } + q \\right | ^ 2 + V _ { \\nu } - B \\ge \\left | - i \\dfrac { d } { d x } + q \\right | ^ 2 - B \\ge q ^ 2 - B , \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{gather*} \\sum _ { \\sigma \\ni 1 } \\frac { v _ \\sigma } { x _ \\sigma } = \\sum _ { \\sigma \\ni 2 } \\frac { v _ \\sigma } { x _ \\sigma } = \\dots = \\sum _ { \\sigma \\ni N } \\frac { v _ \\sigma } { x _ \\sigma } \\end{gather*}"}
-{"id": "2123.png", "formula": "\\begin{align*} S ( x ) ( t ) & = \\left ( S _ 1 ( x ) ( t ) , \\ldots , S _ k ( x ) ( t ) \\right ) \\\\ S _ j ( x ) ( t ) & = \\int _ { 0 < s _ 1 < \\ldots < s _ j < t } d x _ { s _ 1 } \\otimes d x _ { s _ 2 } \\otimes \\ldots \\otimes d x _ { s _ j } \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{align*} | \\int _ C \\omega _ X - \\int _ { C ' } \\omega _ X | = | < \\omega _ X , C - C ' > | = | \\tau < c _ 1 ( T \\overline { X } ) , C - C ' > | \\le | \\tau | c _ 0 . \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} N = \\begin{pmatrix} - r _ 3 & r ^ T & 0 \\\\ n & N _ 0 & r \\\\ 0 & n ^ T & r _ 3 \\end{pmatrix} . \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} F _ { r _ t | r _ m } ( y ) & \\triangleq \\mathrm { P } ( r _ t \\leq y | r _ m = x ) \\\\ & = 1 - \\mathrm { P } ( r _ t > y | r _ m = x ) . \\end{align*}"}
-{"id": "1106.png", "formula": "\\begin{align*} \\Phi \\circ \\Psi ( ( 1 \\otimes n ) \\otimes g ) = ( 1 \\otimes n ) \\otimes g \\ , ; \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} w _ { \\Delta } ( x ) = \\mathrm { s u p } \\{ m ( x ) ; m \\in \\Delta \\} . \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} [ X _ i , X _ j ] = c _ { i j } \\partial _ { x _ { n + 1 } } \\ \\ { \\rm f o r \\ e v e r y } \\ i , j = 1 , . . . , n . \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} J ( q ) = 1 7 2 8 \\frac { E _ 4 ( q ) ^ 3 } { E _ 4 ( q ) ^ 3 - E _ 6 ( q ) ^ 2 } = \\frac { 1 } { q } + 7 4 4 + \\sum _ { j = 1 } ^ { \\infty } c ( j ) q ^ j \\end{align*}"}
-{"id": "2301.png", "formula": "\\begin{align*} & P \\{ T _ 1 = T _ { \\min } \\} = \\\\ & K \\int _ 0 ^ { \\infty } \\cdots \\int _ 0 ^ { \\infty } e ^ { - ( p _ g t _ g + \\cdots + p _ 2 t _ 2 ) } \\left ( 1 - e ^ { - p _ g t _ g } \\right ) ^ { M _ g - 1 } \\cdots \\left ( 1 - e ^ { - p _ 2 t _ 2 } \\right ) ^ { M _ 2 - 1 } \\left [ 1 - e ^ { - p _ 1 ( t _ 2 \\wedge \\cdots \\wedge t _ g ) } \\right ] ^ { M _ 1 } \\ , d t _ 2 \\cdots d t _ g , \\end{align*}"}
-{"id": "6095.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } e ^ { - \\lambda t } l ( s , t ) d t = \\frac { f ( \\lambda ) } { \\lambda } e ^ { - s f ( \\lambda ) } . \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} \\rho _ { G _ 1 , G _ 2 } ( t ) = r _ { G _ 2 } ^ { - 1 } ( r _ { G _ 1 } ( t ) ) , & & \\varphi _ { G _ 1 , G _ 2 } ( t ) = \\frac { v _ { G _ 1 } ( t ) } { v _ { G _ 2 } ( \\rho _ { G _ 1 , G _ 2 } ( t ) ) } . \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{align*} W ^ P ( k ) : = \\{ w \\in W ^ P , \\exists \\textrm { a r e d u c e d e x p r e s s i o n } w = s _ { i _ p } \\ldots s _ { i _ 1 } \\textrm { s . t . } s _ i \\textrm { o c c u r s a t m o s t } k \\textrm { t i m e s } \\} \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} { \\mathcal D } _ { \\overline { \\tilde S } } : = \\{ x \\in X _ 1 \\ | u x \\in v ( X _ 1 ) \\} , \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} { \\mathfrak M } _ \\psi ^ + : & = \\bigl \\{ a \\in A ^ + : \\psi ( a ) < \\infty \\bigr \\} \\\\ { \\mathfrak N } _ \\psi : & = \\bigl \\{ a \\in A : \\psi ( a ^ * a ) < \\infty \\bigr \\} \\\\ { \\mathfrak M } _ \\psi : & = { \\mathfrak N } _ \\psi ^ * { \\mathfrak N } _ \\psi = \\operatorname { s p a n } \\{ y ^ * x : x , y \\in { \\mathfrak N } _ \\psi \\} \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} \\mathfrak { a } = ( ( - 1 ) ^ { \\mu _ m } \\kappa ( m , \\mu , \\lambda ) ) _ { m \\in \\lbrace 1 , 2 , \\dots , N \\rbrace \\setminus \\lambda } , \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} M _ i \\leq \\binom { i - 1 } { d _ i } \\leq \\binom { n } { 2 \\sqrt { n } } \\leq n ^ { 2 \\sqrt { n } } = 2 ^ { 2 \\sqrt { n } \\log n } . \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} \\displaystyle \\limsup _ { h \\to 0 + } \\dfrac { \\sup _ { t \\in [ 0 , 1 ] } \\sup _ { | s - t | \\leq h } | \\widetilde Z ^ H ( t ) - \\widetilde Z ^ H ( s ) | } { h ^ { H } ( \\log { 1 / h } ) ^ { \\frac 1 2 + \\frac { 1 } \\alpha + \\epsilon } } = 0 \\ ; \\ ; a . s . \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} \\Psi ( \\gamma ^ A _ z ( s _ i ) ) = z \\Psi ( 1 _ { \\Phi ( Z ( i , r ( i ) ) ) } ) = \\gamma ^ B _ z ( \\Psi ( s _ i ) ) , \\end{align*}"}
-{"id": "6830.png", "formula": "\\begin{align*} y ( t ) = x ( t ) ^ \\top M x ( t ) = ( L ^ \\top x ( t ) ) ^ \\top ( L ^ \\top x ( t ) ) = z ( t ) ^ \\top z ( t ) = \\| z ( t ) \\| _ 2 ^ 2 , \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} B _ 2 = \\left \\{ \\begin{array} { l l } e _ { ( 1 , \\ell _ 1 - 1 , \\ell _ 2 ) , \\ell _ 3 + 1 , \\ell _ 4 } \\\\ e _ { ( 2 , \\ell _ 1 - 1 , \\ell _ 4 ) , \\ell _ 2 + 1 , \\ell _ 3 } \\\\ e _ { ( 3 , \\ell _ 3 , \\ell _ 4 ) , \\ell _ 1 , \\ell _ 2 } \\end{array} : \\ ; 1 \\leq \\ell _ 1 \\leq s , \\ ; 0 \\leq \\ell _ 2 , \\ell _ 3 , \\ell _ 4 \\leq s - 1 \\right \\} \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} \\frac { X _ i ( \\mu _ 1 ) } { X _ i ( h ) } = - \\frac { U _ 1 ( \\mu _ 2 ) } { U _ 1 ( \\varphi ) } = c . \\end{align*}"}
-{"id": "647.png", "formula": "\\begin{align*} Z _ i ( T _ 0 , T _ 1 ) : = \\left \\{ \\sum _ { n = T _ 0 } ^ { T _ 1 - 1 } W _ i ( n + 1 ) < 2 C _ i ( T _ 1 - T _ 0 ) \\right \\} \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow 0 ^ { \\pm } } T ( 0 , x ) = A ^ { \\pm } _ { a } , \\lim _ { x \\rightarrow 0 ^ { \\pm } } \\lim _ { t \\rightarrow 0 } ( N + i B ) ( t , x ) e ^ { - i a ^ { 2 } l o g \\frac { \\sqrt { t } } { x } - i x ^ { 2 } / 4 t } = B ^ { \\pm } _ { a } . \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } u ( t , x ) = \\frac { \\partial ^ 2 } { \\partial x ^ 2 } u ( t , x ) - \\frac { 1 } { 2 } \\frac { \\partial } { \\partial x } u ( t , x ) ^ 2 + \\sigma ( t , x , u ( t , x ) ) \\frac { \\partial ^ 2 W } { \\partial t \\partial x } \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} \\left | \\nabla \\left ( \\Phi _ { \\ast } ( \\omega ) \\right ) \\right | ^ 2 = D + E + F , \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} \\mathcal { C } _ { \\infty } : = \\{ \\ , & \\tau _ { p } : = ( y _ { i _ 1 } , \\ , \\dots , \\ , y _ { i _ p } ) , \\ , p \\geq 1 , \\ , \\ , \\mathrm { s . t } \\ , \\ , y _ { i _ j } \\in \\mathcal { K } , \\ , \\ , \\forall \\ , \\ , j = 1 , \\dots , p , \\\\ & { } y _ { i _ j } \\neq y _ { i _ k } , \\forall \\ , \\ , j \\neq k , \\ , \\ , \\mathrm { a n d \\ , \\ , } \\rho ( \\tau _ { p } ) > 0 \\ , \\} \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{align*} \\varphi _ I = \\frac 1 2 \\arccos \\frac 1 { \\sqrt { 5 } } < \\arcsin \\sqrt { \\frac { 3 } { 8 } } , \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} & [ s , x ] [ d _ 1 , d _ 2 ] + [ s , d _ 2 ] [ d _ 1 , x ] = [ s , x ] [ d _ 1 , y _ 1 \\dots y _ k ] + [ s , y _ 1 \\dots y _ k ] [ d _ 1 , x ] \\\\ & = \\ \\sum _ { i = 1 } ^ k \\bigl ( [ s , x ] y _ 1 \\dots y _ { i - 1 } [ d _ 1 , y _ i ] y _ { i + 1 } \\dots y _ k + y _ 1 \\dots y _ { i - 1 } [ s , y _ i ] y _ { i + 1 } \\dots y _ k [ d _ 1 , x ] \\bigr ) . \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} H _ { \\mathcal { R } ( I ^ l ) _ + } ^ i ( L ^ { I ^ l } ( A ) ) _ n = 0 , \\mbox { i . e . , } H _ { \\mathcal { R } ( K ) _ + } ^ i \\big ( L ^ { K } ( A ) \\big ) _ n = 0 \\mbox { f o r a l l } n \\geqslant 0 , \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} \\deg ( f ) \\mid ( q - 1 ) \\iff u ( f ) = \\omega ( f ) = ( q - 1 ) / \\deg ( f ) . \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} & F ( \\beta , \\theta ) = F ( \\beta , - \\theta ) , \\\\ & ( \\beta , \\theta ) \\in O _ + \\cup O _ - ( \\beta , - \\theta ) \\in O _ + \\cup O _ - . \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{align*} \\sigma _ { 1 } = \\frac { q _ { 0 } ( 1 - q _ { 0 } ) \\int _ { \\Omega } | \\nabla u _ { 0 } | ^ { 2 } u _ { 0 } ^ { q _ { 0 } - 2 } \\phi _ { 1 } } { \\int _ { \\Omega } u _ { 0 } ^ { q _ { 0 } } \\phi _ { 1 } } > 0 . \\end{align*}"}
-{"id": "6658.png", "formula": "\\begin{align*} \\mathbb { E } [ T _ R ] = \\sum _ { k = 1 } ^ { \\infty } \\mathbb { P } ( T _ R > k ) \\ , . \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} T ^ { i , j } f ( y ) & = \\sum _ { \\substack { R _ 1 \\in \\mathcal { D } _ 1 \\\\ R _ 2 \\in \\mathcal { D } _ 2 } } \\sum _ { \\substack { P _ 1 \\in ( R _ 1 ) _ { i _ 1 } \\\\ P _ 2 \\in ( R _ 2 ) _ { i _ 2 } } } \\sum _ { \\substack { Q _ 1 \\in ( R _ 1 ) _ { j _ 1 } \\\\ Q _ 2 \\in ( R _ 2 ) _ { j _ 2 } } } a _ { P Q R } \\cdot \\hat { f } ( P ) h _ { Q } ( y ) \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} \\Delta ' W ^ 0 _ 3 ( p , p _ 1 , p _ 2 ) = 0 . \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{align*} d \\beta \\wedge \\omega _ i = \\begin{cases} f _ 1 \\omega _ 1 ^ 2 + 2 J _ 1 \\beta \\wedge \\eta \\wedge \\omega _ 1 & \\mbox { i f } i = 1 , \\\\ 2 J _ 1 \\beta \\wedge \\eta \\wedge \\omega _ i & \\mbox { o t h e r w i s e } , \\end{cases} \\mbox { a n d } d \\beta \\wedge \\eta \\wedge \\omega _ i = \\begin{cases} f _ 1 \\eta \\wedge \\omega _ 1 ^ 2 & \\mbox { i f } i = 1 , \\\\ 0 & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} \\Phi ( \\ ! A , B \\ ! ) & = \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } K _ { i j } ^ 2 + \\sum _ { j = 1 } ^ N L _ j ^ 2 = \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ { M - 1 } K _ { i j } ^ 2 + \\sum _ { j = 1 } ^ N \\left ( \\sum _ { i = 1 } ^ { M - 1 } K _ { i j } \\right ) ^ 2 . \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log \\int _ { A _ { n , \\eta } } d \\theta P _ n ( \\theta ) e ^ { \\sum _ j h ( \\theta _ j ) } = \\lim _ { k \\to \\infty } \\frac { 1 } { n _ k } \\log \\int _ { A _ { n _ k , \\eta } } d \\theta P _ { n _ k } ( \\theta ) e ^ { \\sum _ j h ( \\theta _ j ) } . \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} \\phi \\left ( z ' \\frac { ( x ' ) ^ { m ' } - ( y ' ) ^ { m ' } } { x ' - y ' } \\right ) = z ' \\frac { ( x ' ) ^ { n ' } - ( y ' ) ^ { n ' } } { x ' - y ' } \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} \\mathrm { d i v } ( \\{ x , - \\} ) = \\mathrm { d i v } ( Q \\delta _ x ) = Q \\mathrm { d i v } ( \\delta _ x ) + \\delta _ x ( Q ) \\ , . \\end{align*}"}
-{"id": "1634.png", "formula": "\\begin{align*} \\mathbb { E } [ G _ n ] = \\frac { \\lambda } { n } + o \\left ( \\frac { 1 } { n } \\right ) , \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} - e ^ { 4 c x ^ * } ( x ^ * - \\frac { 1 } { c } ) = x ^ * + \\frac { 1 } { c } . \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{align*} & \\begin{aligned} x _ i \\in \\mathbb { S } _ x , \\end{aligned} \\\\ & \\begin{aligned} \\lim _ { s \\to 0 } \\mathbb { E } ( Z _ { x _ i , s } ) = x _ i , \\end{aligned} \\\\ & \\begin{aligned} \\lim _ { s \\to 0 } { \\rm V a r } ( Z _ { x _ i , s } ) = 0 , \\end{aligned} \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{align*} r ^ { \\frac { 1 } { 2 } } \\mathfrak { h } _ { \\emptyset } * \\mathfrak { h } _ { \\emptyset } = \\begin{cases} - 2 ^ { - \\frac { 3 } { 2 } } ( r ) ^ { \\frac { 1 } { 2 } } q ( \\eta _ h , \\eta _ h ) \\\\ ( 2 r ) ^ { \\frac { 1 } { 2 } } \\mathfrak { c l } ( b _ h ) \\eta _ h \\\\ 0 \\\\ \\end{cases} \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{align*} \\frac { d } { d t } \\hat \\psi ( t \\alpha , \\beta _ 1 + t \\eta _ 1 , \\beta _ 2 + t \\eta _ 2 ) = 0 \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{align*} x \\circ y = x y + y x . \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { | | \\mu ^ { \\boxtimes t } | | } { t } & = \\lim _ { t \\rightarrow \\infty } \\frac { 1 } { t h _ t ( \\alpha _ t ) } \\\\ & = \\lim _ { t \\rightarrow \\infty } \\left ( \\frac { 1 } { t \\cdot \\alpha _ t } \\cdot \\exp [ - ( t - 1 ) u ( \\alpha _ t ) ] \\right ) \\\\ & = e \\cdot \\int _ 0 ^ \\infty \\frac { s ^ 2 + 1 } { s ^ 2 } \\ , d \\rho ( s ) \\\\ & = e V . \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} \\begin{cases} \\Delta \\tilde u = 0 & \\ \\ R , \\\\ \\tilde u = u & \\ \\ \\partial R . \\end{cases} \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} T f ( x ) = \\int _ { { \\mathbb R } ^ n } K ( x , y ) f ( y ) d y \\quad \\ , \\ , x \\not \\in \\ , f , \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} v _ \\tau ( t ) & = v ( t ) - d _ { A ( t ) } \\alpha _ \\tau \\ \\ \\ \\ 0 < t < \\infty \\ \\ \\\\ v _ \\tau ( t ) & \\ \\ \\ \\ v _ 0 - d _ { A _ 0 } \\alpha _ \\tau \\ \\ \\ \\ \\ L ^ 2 ( M ) \\ \\ \\ \\ \\ t \\downarrow 0 . \\end{align*}"}
-{"id": "6298.png", "formula": "\\begin{align*} u ( x ) = P [ f ] ( x ) - G [ g ] ( x ) : = \\int _ { S ^ { n - 1 } } P ( x , \\eta ) f ( \\eta ) d \\sigma ( \\eta ) - \\int _ { B ^ { n } } G ( x , y ) g ( y ) d y , \\ , \\end{align*}"}
-{"id": "3833.png", "formula": "\\begin{align*} \\alpha = s \\cdot ( \\sum _ { i = 1 } ^ { m } n _ i ) - \\sum _ { i = 1 } ^ { m } n _ i \\ell _ { m + 1 - i } . \\end{align*}"}
-{"id": "8124.png", "formula": "\\begin{align*} \\operatorname { d i v } ( | y | ^ { - a } \\nabla g ) = | y | ^ { - a } g _ t . \\end{align*}"}
-{"id": "511.png", "formula": "\\begin{align*} \\eta _ K ^ { \\Gamma } = \\left \\{ \\begin{array} { l } + 1 \\ ; \\Gamma = \\partial K \\cap \\partial K ' , K > K ' , \\\\ - 1 \\ ; \\Gamma = \\partial K \\cap \\partial K ' , K < K ' , \\\\ + 1 \\ ; \\Gamma \\in \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "7188.png", "formula": "\\begin{align*} g ^ T = A f ^ T \\end{align*}"}
-{"id": "6948.png", "formula": "\\begin{align*} \\ \\Big ( \\alpha = 1 \\ \\ f ' ( 0 ) - 2 a f ( 0 ) = 0 \\Big ) \\ \\ \\ \\ \\ \\ \\Big ( \\alpha > 1 \\ \\ f ' ( 0 ) = 0 \\Big ) . \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{align*} \\widehat { \\rho } _ { k _ n } = \\displaystyle \\sum _ { j = 1 } ^ { k _ n } \\rho _ { n , j } \\phi _ { n , j } \\otimes \\phi _ { n , j } = \\displaystyle \\sum _ { j = 1 } ^ { k _ n } \\frac { D _ { n , j } } { C _ { n , j } } \\phi _ { n , j } \\otimes \\phi _ { n , j } , \\end{align*}"}
-{"id": "9102.png", "formula": "\\begin{align*} { \\bf D } _ { D } = { \\rm d i a g } _ 0 ( { \\bf G } ) , \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{align*} a _ k ^ * : = z _ k + \\varrho _ k b _ k ^ * \\end{align*}"}
-{"id": "7945.png", "formula": "\\begin{align*} \\mathrm { s u p p } ( ( \\mu ^ { \\boxtimes t _ 0 } ) ^ { \\mathrm { a c } } ) \\cap [ \\epsilon , \\infty ) \\subset & \\overline { \\{ 1 / x : x = h _ { t _ 0 } ( r ) , r \\in V _ { t _ 0 } ^ + \\cap [ a , b ] \\} } \\\\ & \\subset B _ \\epsilon ( \\mathrm { s u p p } ( \\mu ^ { \\boxtimes s } ) ^ { \\mathrm { a c } } ) , \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} U ( F ) ( x , y ) = h ( y ) F ( \\varphi ( x , y ) , \\tau ( y ) ) , ( x , y ) \\in X _ 2 \\times Y _ 2 \\end{align*}"}
-{"id": "3129.png", "formula": "\\begin{align*} { } \\frac { \\partial S _ D } { \\partial \\alpha } = 0 \\sim N - \\frac { 1 } { \\beta } \\int _ { 0 } ^ { \\infty } \\frac { d x } { 1 + e ^ { \\alpha + x } } \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} m i _ k ( n ) \\le m i _ k ( n - \\delta - 1 ) + \\frac { \\sum _ { i = 1 } ^ j \\binom { s _ i } { 2 } m i _ k ( n - \\delta - s _ i ) + ( \\binom { \\delta } { 2 } - \\sum _ { i = 1 } ^ j \\binom { s _ i } { 2 } ) m i _ k ( n - \\delta - k ) } { \\binom { k } { 2 } } \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} ( u \\otimes n ) \\cdot \\alpha = u \\alpha \\otimes n - u \\otimes \\alpha \\cdot n \\ , . \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} { \\cal C } _ 2 = \\bigcup _ { \\underline { p } \\in ( 0 , 1 ) ^ { M } } { \\cal C } _ 2 ( \\underline { p } ) . \\end{align*}"}
-{"id": "7585.png", "formula": "\\begin{align*} \\tilde B = \\int ( L \\chi ) ( \\tilde z ) h ( \\tilde z ) d \\tilde z - \\int B ( \\tilde z , . . . , \\tilde z ) h ( \\tilde z ) d \\tilde z . \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} P _ L ( \\boldsymbol { \\epsilon } ) = \\sum _ { \\mathbf { m } , \\mu , \\boldsymbol { \\theta } } f ( \\mathbf { m } , \\mu , \\boldsymbol { \\theta } ; N ; \\boldsymbol { \\epsilon } ) \\mathbb { P } _ L ( \\mathbf { m } , \\mu , \\boldsymbol { \\theta } ; K ) , \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} Q ( g + f ^ { n } t ) \\equiv 0 \\pmod { f ^ { n + m } } & \\iff f ^ n ( h + t Q ' ( g ) ) = f ^ { n + m } v \\ \\ \\ \\ v \\in \\Bbb F _ q [ x ] \\\\ & \\iff f ^ n ( h + t Q ' ( g ) - f ^ { m } v ) = 0 \\ \\ \\ \\ v \\in \\Bbb F _ q [ x ] . \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{align*} S ( E ) & = \\Pi H \\Pi - \\Pi H R _ \\bot ( 0 ) H \\Pi - E ( Y - \\Pi ) - E \\Pi + \\Pi H X ( E ) H \\Pi \\\\ & = \\Pi \\left [ H - H R _ \\bot ( 0 ) H \\right ] \\Pi - E Y + \\Pi H X ( E ) H \\Pi . \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{align*} ( A _ \\infty + \\eta , \\Phi _ \\infty ) \\ \\mbox { w h e r e } \\ \\ [ \\eta \\wedge \\Phi _ { \\infty } ] = 0 \\ \\mbox { a n d } \\ d _ { A _ \\infty } \\eta = 0 . \\end{align*}"}
-{"id": "4170.png", "formula": "\\begin{align*} \\beta ' : = \\frac { d \\beta } { p ( d - 1 ) } \\ , , \\beta _ 0 : = \\max \\{ \\beta , \\beta ' \\} , \\mathcal { E } _ { r , \\beta , d , B } ^ p : = \\{ \\chi _ K \\cdot g \\ , : \\ , g \\in \\mathcal { F } _ { \\beta ' , d , B } K \\in \\mathcal { K } _ { r , \\beta , d , B } \\} \\ , . \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{align*} \\dim ( \\Xi ( e _ 1 , e _ 2 ) ) + \\dim ( \\Sigma ( s , \\theta ) ) = d + 1 > d , \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} \\pi ^ * _ { m + 1 } D f = X \\pi _ m ^ * f . \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} t \\mapsto f ( X _ t ) = f \\circ X _ t \\end{align*}"}
-{"id": "2266.png", "formula": "\\begin{align*} \\begin{aligned} { \\rm M S E } ( \\hat D _ \\varpi ( P \\parallel Q ) ) = O ( \\varGamma ^ { - 1 } ) , \\end{aligned} \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} \\Omega _ { m a x } ( \\rho _ { m a x } ) = b ( \\rho _ { m a x } ) = \\mbox { I m } ( \\lambda _ l ) + \\sum _ { m = 1 } ^ M \\mbox { I m } ( \\beta _ m ) \\rho ^ { 2 m } _ { m a x } . \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{align*} I ( u _ { \\infty } ) = I ( u ^ { - N _ - } ) \\dots + I ( u ^ 0 ) + \\dots + I ( u ^ { N _ + } ) = 0 . \\end{align*}"}
-{"id": "8935.png", "formula": "\\begin{align*} \\frac { 1 } { q } = \\frac { 1 } { p } + \\frac { \\alpha } { n + 1 } - 1 . \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} \\psi ^ n = h ^ { n + 1 } \\circ d ^ n + d ^ { n - 1 } \\circ h ^ n \\ , . \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} p ( \\bar { \\alpha } ^ { \\vee } ) = ( p + \\chi ) ( \\bar { \\alpha } ^ { \\vee } ) - \\chi ( \\bar { \\alpha } ^ { \\vee } ) \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} f _ T ( x , y ) = \\sup _ { x ^ * \\in T ( x ) } \\langle x ^ * , y - x \\rangle , \\end{align*}"}
-{"id": "8231.png", "formula": "\\begin{align*} \\varphi _ i ( \\psi _ i ( y _ 1 , \\dots , y _ m ) ) = \\sum _ { j = 1 } ^ m a _ { i , j } \\varphi _ j ( y _ j ) \\ i > m . \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} h _ s ( t , x , \\xi ) = h ( t , x , \\xi ) + \\frac { i } { 2 m } \\nabla \\cdot A ( t , x ) , \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} H _ t = E \\left [ \\left ( \\frac { \\beta ( t , q _ t ) } { 2 \\pi } \\right ) ^ { n / 2 } \\int \\tilde h ( q _ t , z ) e ^ { - \\beta ( t , q _ t ) \\| z \\| ^ 2 / 2 } d z \\right ] . \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{align*} \\mathcal { P } ^ { \\circ } : = \\left \\{ u \\in C _ { 0 } ^ { 1 } ( \\overline { \\Omega } ) : u > 0 \\Omega \\frac { \\partial u } { \\partial \\nu } < 0 \\partial \\Omega \\right \\} , \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} q _ m ( { \\bf z } ) : = \\prod _ { j = 1 } ^ { m - 1 } 2 e ^ { - 2 z _ j } \\ , . \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} \\begin{cases} - \\tilde { \\nabla } \\cdot a \\tilde { \\nabla } f = 0 , & ( x , \\tilde { z } ) \\in \\R \\times ( 0 , \\infty ) , \\\\ f = \\kappa , & ( x , \\tilde { z } ) \\in \\R \\times \\{ \\tilde { z } = 0 \\} , \\end{cases} \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} L ( I , G ) = \\{ X \\in \\mathcal { M } _ S \\mid \\mathrm { L M } _ { \\succ } ( g ) \\nmid X \\mbox { f o r a l l } g \\in G \\} , \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{align*} f ( x ) = h \\bigoplus ( A x + b ) / h . \\end{align*}"}
-{"id": "4370.png", "formula": "\\begin{align*} z ^ { ( 0 ) } = i \\int _ { 0 } ^ { \\infty } \\frac { d t } { 2 ( t - \\xi ) \\sqrt { t + 1 - \\xi } } \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} T _ { \\alpha , 0 } ( r ) = \\int _ 0 ^ r \\left ( \\int _ { D _ t } \\alpha \\right ) \\frac { d t } { t } \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{align*} C : = ( 0 , x ^ * ) = \\left \\{ x \\in \\R : f ( x , x ) - F ( x , x ) > 0 \\right \\} \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - \\Delta _ p u ( z ) + | | \\xi | | _ \\infty | u ( z ) | ^ { p - 2 } u ( z ) = k ( z , u ( z ) ) \\ \\mbox { i n } \\ \\Omega , \\\\ \\frac { \\partial u } { \\partial n _ p } + \\beta ( z ) | u | ^ { p - 2 } u = 0 \\ \\mbox { o n } \\ \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{align*} K = \\{ g \\in G : | \\varphi ( g ) | = \\varphi ( 1 _ G ) \\} . \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} - \\sum _ { i , j = 1 } ^ n \\big ( R ^ { ( s ) } ( x ) \\big ) ^ { - 1 } _ { j i } D _ z A _ { i j } \\big ( x , u ^ { ( \\tau ) } ( x ) , D u ^ { ( \\tau ) } ( x ) \\big ) \\leq n \\alpha _ 1 , \\ \\forall x \\in \\overline { \\Omega } , \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{gather*} \\varphi _ n ( g _ 1 , \\ldots , g _ s ) ^ T = ( ( g _ 1 ( \\omega ) , \\ldots , g _ s ( \\omega ) ) ^ T / { m ^ n } ) \\bmod 1 = \\\\ = ( A \\ ( f _ 1 ( \\omega ) , \\ldots , f _ s ( \\omega ) ) ^ T / { m ^ n } ) \\bmod 1 = ( A \\ \\varphi _ n ( f _ 1 , \\ldots , f _ s ) ^ T ) \\bmod 1 . \\end{gather*}"}
-{"id": "7568.png", "formula": "\\begin{align*} & m ^ { - 1 / 2 } J ^ m _ { s , t } \\\\ = & - \\int _ s ^ t ( \\nabla _ q \\chi ) ( r , q _ r ^ m , z _ r ^ m ) \\cdot z _ r ^ m d r \\\\ & - \\int _ s ^ t ( \\nabla _ z \\chi ) ( r , q _ r ^ m , z _ r ^ m ) \\cdot \\left [ ( - \\nabla _ q V ( r , q _ r ^ m ) - \\partial _ r \\psi ( r , q ^ m _ r ) + \\tilde F ( r , q _ r ^ m ) ) d r + \\sigma ( r , q ^ m _ r ) d W _ r \\right ] \\\\ & + m ^ { 1 / 2 } \\left ( \\chi ( t , q _ t ^ m , z _ t ^ m ) - \\chi ( s , q _ s ^ m , z _ s ^ m ) - \\int _ s ^ t \\partial _ r \\chi ( r , q _ r ^ m , z _ r ^ m ) d r \\right ) , \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{align*} H ( 2 , N ) = - \\frac { 1 } { 5 } \\sum _ { s \\in \\mathbb { Z } } \\sigma _ 1 \\left ( \\frac { N - s ^ 2 } { 4 } \\right ) - \\begin{cases} N / 1 0 & \\mbox { i f } N \\mbox { i s a s q u a r e } , \\\\ 0 & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{align*} W : = \\{ x _ { n _ { 1 } } \\otimes \\cdots \\otimes x _ { n _ { i } } | n _ { 1 } \\geq \\cdots \\geq n _ { i } , i \\geq 0 \\} , \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} \\| W ^ + \\| ^ 2 = 2 \\| W _ F ^ + \\| ^ 2 + \\| W _ { 0 0 } ^ + \\| ^ 2 + \\frac { 1 } { 6 } ( 3 s _ C - s _ g ) ^ 2 . \\end{align*}"}
-{"id": "1865.png", "formula": "\\begin{align*} \\sum _ { m = 2 } ^ { b - a } \\binom { b - a - 1 } { m - 2 } & \\left [ ( b - m + 1 ) ( n - b - 1 ) + \\binom { n - b - 2 } { 2 } \\right ] \\\\ & = 2 ^ { b - a - 2 } ( n ^ 2 + ( a - b - 6 ) n + 5 b + 7 - a - a b ) \\\\ & - \\frac { 1 } { 2 } ( n ^ 2 - ( 2 b + 5 - 2 a ) n + ( b + 2 ) ( b + 3 ) - 2 a ( b + 1 ) ) , \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} \\Phi ^ { \\star } d v _ { \\Omega } ( r , \\xi ) = \\theta ( r , \\xi ) \\cdot d r d v _ { U ( P ) } , \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} \\tilde y ( \\gamma , t ) : = \\beta ( \\gamma , t ) \\ , y ( x ( \\gamma , t ) ) , \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} \\big [ \\mathcal { F } ^ { \\ast } \\big ( h ( \\mathfrak { e } ) { \\varphi } \\big ) \\big ] ( p ) \\ = \\ \\mathfrak { e } ( p ) \\ , [ \\mathcal { F } ^ { \\ast } ( { \\varphi } ) ] ( p ) \\ ; , \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} q ( x _ j = 1 | z _ i = 1 ) = \\sum _ { k = 1 } ^ { H _ 0 } a ^ 0 _ { i k } b ^ 0 _ { k j } , p ( x _ j = 1 | z _ i = 1 , A , B ) = \\sum _ { k = 1 } ^ { H } a _ { i k } b _ { k j } . \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{align*} \\dim D ^ { I V } _ n = n , r = 2 , a = n - 1 , b = 0 , p = n . \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} \\ell A ( \\sum \\limits _ { i = 1 } ^ { d } R ^ i A _ { i } ) = 0 . \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} \\sum _ { I \\in \\mathcal { D } ^ + } \\abs { h _ I ( y ) } \\abs { h _ { I ^ - } ( x ) - h _ { I ^ + } ( x ) } & = \\sum _ { I \\in \\mathcal { D } ^ + , I \\supseteq I ( x , y ) } \\frac { 1 } { \\sqrt { \\abs { I } } } \\abs { h _ { I ^ - } ( x ) - h _ { I ^ + } ( x ) } \\\\ & \\leq \\sum _ { I \\in \\mathcal { D } ^ + , I \\supseteq I ( x , y ) } \\frac { 2 \\sqrt { 2 } } { \\abs { I } } = \\frac { 4 \\sqrt { 2 } } { \\abs { I ( x , y ) } } \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} q \\frac { d \\hat { \\varphi } _ K } { d q } = v \\circ \\hat { \\varphi } _ K \\end{align*}"}
-{"id": "8662.png", "formula": "\\begin{align*} \\rho _ 0 : = - \\rho / v = ( r - t ) ^ { - 1 } , \\ \\ \\rho _ I : = - v = ( r - t ) / r \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} \\lambda _ { n _ r , \\kappa } = \\frac { m c ^ 2 } { \\sqrt { 1 + \\frac { z ^ 2 \\gamma ^ 2 } { ( n _ r - 1 + \\sqrt { \\kappa ^ 2 - z ^ 2 \\gamma ^ 2 } ) ^ 2 } } } \\ , , \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{align*} \\mathrm { r a n k } \\left ( M ( s , t ) \\right ) = \\mathrm { i n v } ( \\sigma ( M ) , s , t ) + 1 \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\langle \\ ! \\langle \\hat { p } , \\hat { b } ^ i - \\xi ^ i \\rangle \\ ! \\rangle - \\sum _ { j = 1 } ^ s \\langle \\ ! \\langle \\hat { p } , a ^ j \\rangle \\ ! \\rangle = 0 . \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} X ^ { u s } ( q ' ) = \\overline { X _ { w _ { k + 1 } } } \\sqcup w _ { 0 , P } \\overline { X _ { w _ { k ' - 1 } } } \\textrm { o r } X ^ { u s } ( q ' ) = \\overline { X _ { w _ { k - 1 } } } \\sqcup w _ { 0 , P } \\overline { X _ { w _ { k ' + 1 } } } \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} \\log _ 2 ( T ) = \\log _ 2 ( 3 d ) + \\log _ 2 ( M ) \\leq C _ 1 + \\lceil \\log _ 2 M \\rceil \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} j ( E ) = \\dfrac { ( h ^ 3 - 2 ) ^ 3 ( h ^ 9 - 6 h ^ 6 - 1 2 h ^ 3 - 8 ) ^ 3 } { h ^ 9 ( h ^ 3 - 8 ) ( h ^ 3 + 1 ) ^ 2 } \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} A : B : = 2 ( A ^ { - 1 } + B ^ { - 1 } ) ^ { - 1 } \\ , \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} & \\| x ^ { \\frac { 3 } { 2 } } y ^ { \\frac { 1 } { 2 } } + x ^ { \\frac { 1 } { 2 } } y ^ { \\frac { 3 } { 2 } } \\| = \\| x ^ { \\frac { 1 } { 2 } } ( x + y ) y ^ { \\frac { 1 } { 2 } } \\| = \\| ( x + y ) ^ { \\frac { 1 } { 2 } } y ^ { \\frac { 1 } { 2 } } x ^ { \\frac { 1 } { 2 } } ( x + y ) ^ { \\frac { 1 } { 2 } } \\| \\\\ \\leq & \\| ( x + y ) ^ { \\frac { 1 } { 2 } } \\| ^ 2 \\| x ^ { \\frac { 1 } { 2 } } y ^ { \\frac { 1 } { 2 } } \\| = \\| x + y \\| \\| x ^ { \\frac { 1 } { 2 } } y ^ { \\frac { 1 } { 2 } } \\| < \\delta . \\end{align*}"}
-{"id": "3273.png", "formula": "\\begin{align*} \\{ v \\in \\bar { v } : ~ u _ i < v < v _ k \\} \\cup \\{ & w \\in \\bar { w } : ~ u _ i < w < w _ j \\} \\\\ & < w _ j < v ^ \\prime _ k < w ^ \\prime _ j < v _ k < \\\\ \\{ v \\in \\bar { v } : v _ k < & ~ v < u _ { i + 1 } \\} \\cup \\{ w \\in \\bar { w } : ~ w _ j < w < u _ { i + 1 } \\} . \\\\ \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} \\begin{aligned} p _ 1 ( D ^ { g } ) & = \\frac { 1 } { 4 \\pi ^ 2 } \\left ( \\| W ^ + + \\frac { s _ g } { 1 2 } I \\| ^ 2 + \\| R _ 0 ^ * \\| ^ 2 - \\| R _ 0 \\| ^ 2 - \\| W ^ - + \\frac { s _ g } { 1 2 } I \\| ^ 2 \\right ) \\operatorname { v o l } _ g \\\\ & = \\frac { 1 } { 4 \\pi ^ 2 } ( \\| W ^ + \\| ^ 2 - \\| W ^ - \\| ^ 2 ) \\operatorname { v o l } _ g \\\\ & = \\frac { 1 } { 4 \\pi ^ 2 } \\| W ^ + \\| ^ 2 \\operatorname { v o l } _ g \\\\ \\end{aligned} \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} \\sum _ { a = 1 } ^ n & \\{ \\langle \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi , \\phi \\rangle + \\langle \\phi , \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi \\rangle \\} \\geq C | \\kappa _ B | ^ 2 . \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} H ( z ) : = c _ d ^ { - 1 } | z | ^ d h _ d ( | \\theta | ) = H _ Z ( z ) + H _ Z ( \\overline z ) = \\begin{cases} H _ Z ( \\overline z ) & z \\in \\overline \\C _ + \\\\ H _ Z ( z ) & z \\in \\overline \\C _ - , \\end{cases} \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{align*} | u ( z _ 1 ) - u ( z _ 2 ) | & = \\left | \\int _ { \\gamma } u ' ( z ) d z \\right | \\leq \\frac { 1 } { t _ 0 - \\delta - 1 } \\int _ { \\gamma } \\frac { \\theta } { r \\sin ( \\theta ) } \\left | d z \\right | . \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{align*} ( A + B ) \\lambda ^ * - B ( \\lambda ^ * ) ^ 2 = \\frac { ( A + B ) ^ 2 } { 4 B } . \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} \\zeta _ { \\beta } ( z ) : = \\exp \\left ( \\sum _ { n = 1 } ^ { \\infty } \\ , \\frac { \\mathcal { P } _ n } { n } \\ , z ^ n \\right ) , \\mathcal { P } _ n : = \\# \\{ x \\in [ 0 , 1 ] \\mid T _ { \\beta } ^ { n } ( x ) = x \\} \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} & D ^ 2 E _ \\varphi ( U , V ) = \\int _ M g \\left ( U , \\Delta ' V \\right ) + 2 g \\left ( U , ( \\nabla ^ a V ^ m ) T _ a ^ { \\phantom { m } n } \\varphi ^ { \\phantom { m } } _ { m n b } \\right ) \\star 1 , \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} V = \\bigoplus \\limits _ { \\chi \\in { \\cal X } ( T ) } V _ { \\chi } \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{align*} \\begin{cases} \\frac { 1 } { K } \\nabla _ { A , s } \\psi + D ^ { g _ 3 } _ { A ( s ) } \\psi = 0 \\\\ \\frac { 1 } { K } E _ A + * _ 3 F _ { A ( s ) } + \\frac { 1 } { 2 } * _ 3 F _ { A _ { K ^ { - 1 } } } - r ( q _ 3 ( \\psi ) - i K d t ) + \\frac { i } { 2 } * _ 3 \\wp _ 3 = 0 . \\end{cases} \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} ( \\overline { P } f ) ( y ) = ( \\overline { P } _ 1 \\ldots \\overline { P } _ n f ) ( y ) . \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{align*} E \\left [ S _ N ^ { ( r ) } \\right ] = N ^ r \\ln ^ r N \\left [ I _ 1 ( N ) + I _ 2 ( N ) \\right ] , \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{align*} g \\cdot k = k ' \\iff g k h = k ' ( g \\in G ; \\ k , k ' \\in K ) . \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} g _ \\infty = - \\ : 1 6 d u ^ 2 + u ^ 2 d \\theta ^ 2 + 4 c _ \\lambda ^ { - 1 } \\left [ d \\widehat { \\sigma } ^ 2 + d \\widehat { \\delta } ^ 2 \\right ] . \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{align*} { \\mathcal F } = \\bigcup _ { A ' \\in { \\mathcal F } } [ \\leq A ' ] = \\bigcup _ { A ' \\in { \\mathcal F } } \\bigcap _ { \\rho \\in ( A / A ' ) ^ * } [ \\leq \\mathrm { k e r } ( \\rho ) ] . \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} n r ' _ = ( n , 1 ) \\ , \\geq \\ , \\frac { 1 2 ^ { 3 ^ { n - 2 } ( 3 ^ { n - 1 } - 1 ) / 2 } } { 6 ^ { 3 ^ { n - 1 } } \\cdot | A G L ( n - 1 , 3 ) | } - 1 . \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{align*} [ d \\theta ] \\cup [ \\omega ] = c _ 1 ( M ) \\cup [ \\omega ] = 0 \\in H ^ 4 ( B ) . \\end{align*}"}
-{"id": "7496.png", "formula": "\\begin{align*} & ( \\beta V ) ( t , q _ t ) - ( \\beta V ) ( s , q _ s ) \\\\ = & \\int _ s ^ t \\partial _ r ( \\beta V ) ( r , q _ r ) d r + \\int _ s ^ t \\nabla _ q ( \\beta V ) ( r , q _ r ) \\circ d q _ r \\\\ = & \\int _ s ^ t \\partial _ r ( \\beta V ) ( r , q _ r ) d r + \\int _ s ^ t \\nabla _ q \\beta ( r , q _ r ) V ( r , q _ r ) \\circ d q _ r + \\int _ s ^ t \\beta ( r , q _ r ) \\nabla _ q V ( r , q _ r ) \\circ d q _ r , \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} \\lim _ { m , n \\to \\infty } \\int _ { \\textrm { s p t } \\ , \\zeta } V | u _ m | ^ { p - 1 } | \\zeta | & \\ , | T ( u _ m - u _ n ) | \\ , d x \\le \\lim _ { m , n \\to \\infty } \\left ( \\int _ { \\textrm { s p t } \\ , \\zeta } V ^ { q / p } | u _ m | ^ q \\ , d x \\right ) ^ { ( p - 1 ) / q } \\times \\\\ & \\times \\left ( \\int _ { \\textrm { s p t } \\ , \\zeta } V ^ { r } | T ( u _ m - u _ n ) | ^ { p r } \\ , d x \\right ) ^ { 1 / ( p r ) } = 0 . \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} r - t _ 2 & = \\left ( ( a + 1 ) ( b + 1 ) - h - 1 \\right ) - \\left ( a n + b m - h + m + n - n m - 1 \\right ) \\\\ & = ( n - a - 1 ) ( n - b - 1 ) . \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{gather*} [ H _ { \\beta _ i } , H _ { \\beta _ j } ] = 0 , [ H _ { \\beta _ i } , X _ { \\pm \\beta _ j } ] = \\pm \\beta _ j ( H _ { \\beta _ i } ) X _ { \\pm \\beta _ j } , \\\\ [ X _ { \\beta _ i } , X _ { - \\beta _ j } ] = \\delta _ { i , j } H _ { \\beta _ i } , [ X _ { \\pm \\beta _ i } , X _ { \\pm \\beta _ i } ] = 0 \\end{gather*}"}
-{"id": "3175.png", "formula": "\\begin{align*} h = \\frac { k } { 2 \\ , \\textup { d i s t } ( \\ , \\cdot \\ , , L ) } + O ( 1 ) \\end{align*}"}
-{"id": "6865.png", "formula": "\\begin{align*} \\hat { F } = S P S + 4 M B B ^ \\top M \\approx ( S Z _ P ) ( S Z _ P ) ^ \\top + 4 ( M B ) ( M B ) ^ \\top . \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} p _ { - 2 } ( a ) p _ { - 2 } ( 1 ) - p _ { - 2 } ( a + 1 ) & = 2 p _ { - 2 } ( a ) - p _ { - 2 } ( a + 1 ) \\\\ & = 2 \\sum _ { k = 0 } ^ a p ( k ) p ( a - k ) - \\sum _ { k = 0 } ^ { a + 1 } p ( k ) p ( a + 1 - k ) \\\\ & = \\left ( \\sum _ { k = 0 } ^ a p ( k ) \\big ( 2 p ( a - k ) - p ( a + 1 - k ) \\big ) \\right ) - p ( a + 1 ) \\\\ & \\ge p ( 0 ) + p ( 1 ) + \\cdots + p ( a - 2 ) + p ( a ) - p ( a + 1 ) \\\\ & > p ( a - 3 ) + p ( a - 2 ) + p ( a ) - p ( a + 1 ) \\\\ & \\ge p ( a - 1 ) + p ( a ) - p ( a + 1 ) \\\\ & \\ge 0 . \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\int _ { \\mathbb { R } } & \\Big ( \\int _ { \\mathbb { R } } G _ { \\epsilon } ( v - y ) G _ { t - s } ( x - v ) | v - y | ^ { 1 / 2 } d v \\Big ) ^ 2 f ( y ) ^ 2 d y d s \\\\ & \\leq \\| f \\| _ { L ^ q ( \\mathbb { R } ) } ^ 2 \\int _ 0 ^ t \\Big ( \\int _ { \\mathbb { R } } G _ { \\epsilon } ( z ) | z | ^ { 1 / 2 } d z \\Big ) ^ 2 \\| G _ { t - s } \\| _ { L ^ { 2 q _ 1 } ( \\mathbb { R } ) } ^ 2 d s \\\\ & \\leq C \\epsilon ^ { 1 / 2 } . \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} \\lim _ { | t - s | \\to 0 } \\| P ^ \\kappa _ { t , s } f - f \\| _ \\infty = 0 , \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} R ^ \\nabla | _ { \\Lambda ^ { 1 , 1 } \\otimes \\Lambda ^ { 1 , 1 } } \\equiv \\begin{pmatrix} k & \\bar { a } & a & w \\\\ \\bar { a ' } & \\bar { x } & \\bar { v } & \\bar { b } \\\\ a ' & v & x & b \\\\ u & \\bar { b ' } & b ' & l \\end{pmatrix} , k , l , u , w \\in \\R . \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} \\Re ( C _ E ) & = O + \\frac { \\beta ( l _ j ) } { 2 } \\Re ( \\bar { z } _ 1 z _ 2 \\mu _ { \\beta } ) \\\\ & + \\tanh ( \\beta ( a ) ) \\Re ( \\bar { z } _ 1 z _ 2 \\frac { - \\beta ( l _ j ) } { 4 } ( \\tanh ( \\beta ( a ) ) + \\coth ( \\beta ( a ) ) ) \\mu _ { \\beta } ) \\\\ & + \\tanh ( - \\beta ( a ) ) \\Re ( z _ 1 \\bar { z } _ 2 \\frac { 3 \\beta ( l _ j ) } { 4 } ( \\coth ( \\beta ( a ) ) - \\tanh ( \\beta ( a ) ) ) \\theta ( \\mu _ { \\beta } ) ) \\\\ & = O + \\beta ( l _ j ) ( 1 - \\tanh ^ 2 ( \\beta ( a ) ) ) \\Re ( \\bar { z } _ 1 z _ 2 \\mu _ { \\beta } ) \\end{align*}"}
-{"id": "6850.png", "formula": "\\begin{align*} A ^ \\top Q + Q A + S P S + 4 M B B ^ \\top M = 0 . \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} \\left \\{ \\ \\begin{array} { l } \\displaystyle - \\Delta _ \\Phi u = g ( x , u ) , ~ \\mbox { i n } ~ \\Omega , \\\\ \\\\ u = 0 ~ \\mbox { o n } ~ \\partial \\Omega \\end{array} \\right . \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{gather*} X _ { i , j } : = \\delta _ { i , j + k } x _ i \\ ; , \\Delta _ { i , j } : = b _ { i + k , j } \\delta _ { i , j } \\ ; , M _ { i , j } : = \\delta _ { i + 1 , j } \\ ; , \\\\ B _ { i , j } : = b _ i \\delta _ { i , j } : = - \\delta _ { i , j } ( x _ i + x _ { i + 1 } + \\cdots + x _ { i + k } ) \\ ; . \\end{gather*}"}
-{"id": "3119.png", "formula": "\\begin{align*} Z _ R ( \\alpha , \\beta ) = \\prod _ { k = 1 } ^ { B } \\left ( 1 - e ^ { - \\alpha } e ^ { - \\beta k } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} m ( \\lambda _ i - 1 ) + i & = 1 , \\\\ m ( \\lambda _ j - 1 ) + j & = r m , \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} 2 w d w = f ' ( z ) d z \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{align*} \\tilde G ( t , q ) = \\int h ( t , q , z ) G ( t , q , z ) d z . \\end{align*}"}
-{"id": "3705.png", "formula": "\\begin{align*} M ( P , \\vec { \\nu } ) = P ^ { n - r d } \\mathfrak { S } ( \\vec { \\nu } ) J ( P ^ { - d } \\vec { \\nu } ) + O ( C \\widetilde { C } ^ { K / ( d - 1 ) + r ^ 2 - 1 } P ^ { n - r d - \\delta } ) , \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{align*} p _ { A R } ( a , r ) & = \\sum _ { b } p _ { A , B } ( a , b ) p _ { R | B } ( r | b ) \\\\ & = \\epsilon \\ , p _ A ( a ) p _ { R | B } ( r | \\mathtt { e } ) + ( 1 - \\epsilon ) p _ A ( a ) p _ { R | B } ( r | a ) \\\\ & = p _ A ( a ) q _ { R } ( r ) \\lambda ( r ) \\ ! + \\ ! p _ A ( a ) \\big ( q _ { R | A } ( r | a ) \\ ! - \\ ! q _ R ( r ) \\lambda ( r ) \\big ) \\\\ & = p _ A ( a ) q _ { R | A } ( r | a ) \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{align*} C _ 2 = { { { \\left ( { 2 - \\frac { 1 } { \\omega } } \\right ) } ^ 2 } + \\frac { K } { { c M \\omega } } \\left ( { - 4 + \\frac { 3 } { \\omega } } \\right ) + \\frac { { { K ^ 2 } } } { { { { c ^ 2 M } ^ 2 \\omega ^ 2 } } } } , \\end{align*}"}
-{"id": "1955.png", "formula": "\\begin{align*} \\| f \\| _ { B ^ { p } } = \\sum _ { | \\alpha | < m } | \\partial ^ { \\alpha } f ( 0 ) | + \\left ( \\int _ { \\mathbf { B } } ( 1 - | x | ^ { 2 } ) ^ { m p } | \\partial ^ { m } f ( x ) | ^ { p } d \\tau ( x ) \\right ) ^ { 1 / p } , \\enspace 1 \\leq p < \\infty , \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} N ( U , r ) = \\frac { I ( U , r ) } { H ( U , r ) } , \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{align*} 0 & = \\| \\sum _ { i \\in \\sigma } \\Lambda ^ * g _ i + \\sum _ { i \\in \\sigma ^ c } \\Gamma ^ * g _ i \\| ^ 2 = \\lim _ { j \\rightarrow \\infty } \\| \\sum _ { i \\in \\sigma } \\Lambda ^ * g _ i + \\sum _ { i \\in \\sigma ^ c } \\Gamma ^ * g _ i \\| ^ 2 \\\\ & \\ge \\lim _ { j \\rightarrow \\infty } A \\big ( \\sum _ { i \\in \\sigma _ j } | a _ i | ^ 2 + \\sum _ { i \\in \\sigma ^ c _ j } | a _ i | ^ 2 \\big ) \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} \\pi _ 0 L _ h \\pi _ 0 u _ 0 & = \\pi _ 0 f - \\pi _ 0 L _ h \\pi _ 0 ^ c u _ 0 ^ c , \\\\ \\pi _ { 1 1 } ^ c L _ h \\pi _ { 1 1 } ^ c u _ { 1 1 } ^ c & = \\pi _ { 1 1 } ^ c f - \\pi _ { 1 1 } ^ c L _ h \\pi _ 0 u _ 0 - \\pi _ { 1 1 } ^ c L _ h \\pi _ { 1 1 } u _ { 1 1 } , \\\\ \\pi _ { 1 1 } L _ h \\pi _ { 1 1 } u _ { 1 1 } & = \\pi _ { 1 1 } f - \\pi _ { 1 1 } L _ h \\pi _ 0 u _ 0 - \\pi _ { 1 1 } L _ h \\pi _ { 1 1 } ^ c u _ { 1 1 } ^ c . \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} \\mathrm { R e s } _ { D _ i \\cap D _ j } ^ { ( 1 ) } : \\mathbf D _ i ^ 1 = \\sum _ { j \\in I - \\{ i \\} } \\Gamma ( D _ i , \\Omega ^ 1 _ { D _ i } ( \\mathrm { l o g } D _ i \\cap D _ j ) ) \\to \\Gamma ( D _ i \\cap D _ j , \\mathcal O _ { D _ i \\cap D _ j } ) = : \\mathbf D _ { i j } ^ 0 . \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{align*} W _ { D } ^ { 2 , r } ( \\Omega ) : = \\{ u \\in W ^ { 2 , r } ( \\Omega ) : u = 0 \\partial \\Omega \\} . \\end{align*}"}
-{"id": "5586.png", "formula": "\\begin{align*} x \\frac { d } { d x } \\log ( \\psi _ 1 ( x ) ) = \\frac { 3 } { 4 } + \\sum _ { j = 2 } ^ 4 x \\frac { \\theta _ j ' ( x ) } { \\theta _ j ( x ) } . \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} P ^ \\eta ( X _ T ^ { \\vec v } - X _ 1 ^ { \\vec v } \\leq ( v - \\varepsilon ) T + 1 ) & \\leq P ^ \\eta \\Big ( \\sum _ { m = 1 } ^ { T - 1 } Y _ m \\leq ( v - \\varepsilon ) T + 1 \\Big ) + \\sum _ { m = 1 } ^ { T - 1 } \\P ^ \\eta ( \\mathcal E ^ c _ { m - 1 } ) . \\end{align*}"}
-{"id": "8957.png", "formula": "\\begin{align*} \\widetilde { \\Lambda } ^ { \\epsilon , \\kappa } ( x , \\tilde { \\alpha } ) ^ { - 1 } \\widetilde { \\Lambda } ^ { \\epsilon , \\kappa } ( x , \\tilde { \\beta } ) \\ , = \\ , \\widetilde { \\Lambda } ^ { \\epsilon , \\kappa } ( \\tilde { \\alpha } , \\tilde { \\beta } ) \\ , \\widetilde { \\Omega } ^ { \\epsilon , \\kappa } ( \\tilde { \\alpha } , x , \\tilde { \\beta } ) \\ , , \\end{align*}"}
-{"id": "1269.png", "formula": "\\begin{align*} H _ m ( ( P _ 2 ( \\mu _ { [ x ] } ) ) ^ { x _ 2 , k + i } ) = H _ m ( ( \\mu _ { [ x ] } ) ^ { x _ 2 ' , k + i } ) , \\end{align*}"}
-{"id": "3265.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\| \\sum _ { \\nu = 0 } ^ { n + | \\textup { \\textbf { m } } | - m _ \\alpha } b _ { \\nu , n } ^ { ( \\alpha ) } \\Phi _ \\nu \\| _ K ^ { 1 / n } \\leq \\frac { \\| \\Phi \\| _ { K } } { \\rho _ { | \\textup { \\textbf { m } } | } ( \\textup { \\textbf { F } } ) } . \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} \\tilde { u } _ j = - u ^ { l , m } u _ { m , l , j } - \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } \\cup \\Phi _ s ^ + } \\frac { - u _ { l , j } \\alpha ^ { \\vee , l } } { ( 2 \\chi - u _ l ) \\alpha ^ { \\vee , l } } + I _ { H , j } \\end{align*}"}
-{"id": "3147.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\beta B } \\ln ( 1 + e ^ { - \\alpha } e ^ { - x } ) d x = \\int _ 0 ^ { \\ln ( 1 + e ^ { - \\alpha } ) } \\frac { t \\ , d t } { 1 - e ^ { - t } } - e ^ { - \\alpha - \\beta B } \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{align*} \\omega = \\begin{cases} ( 1 + \\sqrt { D } ) / 2 & \\mbox { i f } D \\equiv 1 \\mod { 4 } , \\\\ \\sqrt { D } / 2 & \\mbox { i f } D \\equiv 0 \\mod { 4 } . \\\\ \\end{cases} \\end{align*}"}
-{"id": "6222.png", "formula": "\\begin{align*} L _ m \\chi _ y ( z ) = q ^ { | S _ \\mu ( m - 1 ) | } \\chi \\left ( \\sum _ { s \\in S _ \\mu } \\sum _ { t \\in T _ \\nu } Y _ { s , t } Z _ { s , t } \\right ) . \\end{align*}"}
-{"id": "4424.png", "formula": "\\begin{align*} H ( B ^ e ) \\omega _ i ^ q ( A ^ d , B ^ e ) \\leq c ^ q H ^ { - ( q y _ i - 1 ) } ( i = 1 , \\dots , h ) . \\end{align*}"}
-{"id": "2338.png", "formula": "\\begin{align*} X = \\sum _ { k = 1 } ^ { S _ N } U _ k , \\end{align*}"}
-{"id": "6984.png", "formula": "\\begin{align*} \\mathcal { D } _ \\nu = \\bigsqcup _ { T \\in S K _ 2 ( \\Gamma _ h ) } \\mathcal { D } _ { \\nu } ( \\beta _ T ) \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{align*} d e g ( f _ j { u _ 1 ^ { l _ 1 } } \\cdots u _ k ^ { l _ k } ) = l _ 1 + \\cdots + l _ { k + 1 } - j \\ge n _ 1 + \\cdots + n _ k + 1 , \\end{align*}"}
-{"id": "4951.png", "formula": "\\begin{align*} \\theta ' _ { n + 1 } \\theta _ { n - 1 } - \\theta _ { n + 1 } \\theta _ { n - 1 } ' = ( 2 n - 1 ) \\theta _ n ^ 2 . \\end{align*}"}
-{"id": "8225.png", "formula": "\\begin{align*} Q _ { i , j } ( F , G ) = \\iint _ { S ^ { d - 1 } \\times S ^ { d - 1 } } F ( x ) G ( y ) M _ { i , j } ( r _ i x , r _ j y ) d \\sigma ( x ) d \\sigma ( y ) , \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} \\ ! \\mathcal { T } _ { { \\rm g p h } \\mathcal { N } _ { \\Gamma } } ( \\overline { x } , \\overline { v } ) = \\Big \\{ ( d , w ) \\in \\mathbb { X } \\times \\mathbb { X } \\ | \\ w \\in \\nabla ^ 2 \\langle \\overline { \\lambda } , g \\rangle ( \\overline { x } ) d + \\nabla \\ ! g ( \\overline { x } ) D \\mathcal { N } _ K ( g ( \\overline { x } ) | \\overline { \\lambda } ) ( g ' ( \\overline { x } ) d ) \\Big \\} . \\end{align*}"}
-{"id": "719.png", "formula": "\\begin{align*} \\Delta _ { m + n + 1 } = A _ 1 \\Delta _ { m + 1 } + A _ 2 \\Delta _ { m + 2 } + \\ldots + A _ n \\Delta _ { m + n } , \\end{align*}"}
-{"id": "7260.png", "formula": "\\begin{align*} \\forall \\lambda \\not = \\mu , \\int _ { - 1 } ^ 1 v \\ , \\phi _ \\lambda ( v ) \\phi _ \\mu ( v ) \\ , T ( v ) \\ , d \\nu ( v ) = 0 . \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{align*} i u _ t + \\frac 1 2 \\Delta u = \\lambda | u | ^ 2 u . \\end{align*}"}
-{"id": "8843.png", "formula": "\\begin{align*} [ B _ D , D ] = O + \\bar { z } _ 1 z _ 2 \\beta ( l _ j ) \\mu _ { \\beta } / 2 \\end{align*}"}
-{"id": "955.png", "formula": "\\begin{align*} N \\big [ \\mathfrak { e } , V _ { F , \\lambda } ^ { ( < ) } \\big ] \\ = \\ 0 , \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{align*} [ a _ 1 a _ 2 , b _ 1 , b _ 2 , b _ 3 ] = & a _ 1 [ a _ 2 , b _ 1 , b _ 2 , b _ 3 ] + [ a _ 1 , b _ 1 ] [ a _ 2 , b _ 2 , b _ 3 ] + [ a _ 1 , b _ 2 ] [ a _ 2 , b _ 1 , b _ 3 ] + [ a _ 1 , b _ 3 ] [ a _ 2 , b _ 1 , b _ 2 ] \\\\ + & [ a _ 1 , b _ 1 , b _ 2 ] [ a _ 2 , b _ 3 ] + [ a _ 1 , b _ 1 , b _ 3 ] [ a _ 2 , b _ 2 ] + [ a _ 1 , b _ 2 , b _ 3 ] [ a _ 2 , b _ 1 ] + [ a _ 1 , b _ 1 , b _ 2 , b _ 3 ] a _ 2 . \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{align*} \\ker \\Delta _ h = \\ker \\Delta . \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} \\frac { d \\phi } { d \\tau } & = \\frac { r } { R } \\cos \\phi \\left ( - \\frac { 1 } { 2 } + \\frac { 2 ( n - 1 ) } { r ^ 2 - s ^ 2 } \\right ) + \\frac { \\lambda } { R } + \\frac { 1 } { R } \\sin \\phi \\left ( \\frac { s } { 2 } + \\frac { 2 s ( n - 1 ) } { r ^ 2 - s ^ 2 } \\right ) . \\end{align*}"}
-{"id": "3670.png", "formula": "\\begin{align*} P ^ \\eta ( Z _ k = c _ n t _ n ) = 1 - { \\rm e } ^ { - \\phi _ { t _ n } ^ { 1 / 4 } } , P ^ \\eta ( Z _ k = - t _ n ) = { \\rm e } ^ { - \\phi _ { t _ n } ^ { 1 / 4 } } . \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} & a _ { [ 1 : 9 ] } = 1 2 1 2 1 1 1 0 0 , b _ { [ 1 : 9 ] } = 1 0 2 2 3 2 1 0 1 , c _ { [ 1 : 9 ] } = 1 2 3 3 3 2 1 0 0 \\\\ & d _ { [ 1 : 9 ] } = 1 0 0 2 3 4 5 4 3 , e _ { [ 1 : 9 ] } = 1 2 1 0 0 2 3 4 3 , f _ { [ 1 : 9 ] } = 1 0 2 1 0 2 3 2 3 \\\\ & g _ { [ 1 : 9 ] } = 1 0 0 2 3 4 5 4 3 , h _ { [ 1 : 9 ] } = 1 0 0 0 0 2 3 4 5 , i _ { [ 1 : 9 ] } = 2 1 0 1 0 1 1 0 0 \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{align*} \\iota _ { X _ { f _ { n - 1 } } } \\omega _ + + \\iota _ { X _ { f _ n } } \\omega _ 0 + \\iota _ { X _ { f _ { n + 1 } } } \\omega _ - = d f _ n \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} \\sigma \\left ( I _ n \\otimes ( A + \\frac { k ^ 2 } { 2 } I _ n ) + ( A + \\frac { k ^ 2 } { 2 } I _ n ) \\otimes I _ n + \\sum _ { i = 1 } ^ m N _ i \\otimes N _ i \\right ) \\subset \\mathbb C _ - \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} \\{ e _ { \\ell _ 1 , \\ell _ 2 , \\ell _ 3 , \\ell _ 4 } ~ : ~ 0 \\leq \\ell _ 1 , \\ell _ 2 , \\ell _ 3 , \\ell _ 4 \\leq s \\ell _ 1 + \\ell _ 2 + \\ell _ 3 + \\ell _ 4 = s \\} \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} \\bar { A } = \\begin{pmatrix} \\bar { A } ^ * & 0 \\\\ 0 & - \\varepsilon \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "2462.png", "formula": "\\begin{align*} J ( N ; \\alpha ) : = \\int _ { U ( N ; \\alpha ) ^ { - 1 } } ^ { U ( N ; \\alpha ) } e ^ { - x } \\left ( 1 - \\frac { \\ln x } { \\ln N } \\right ) ^ r d x \\ , + \\ , o \\left ( e ^ { - \\ln ^ { \\alpha } N } \\right ) \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} \\Lambda ^ i _ { d \\leq r _ 0 } ( r _ 0 ) & = \\underset { { r , \\theta \\in \\mathcal { B } ( y _ 0 , d ) } } { \\int \\int } \\mathrm { P } ^ i ( r ) \\lambda _ u d \\theta r d r . \\end{align*}"}
-{"id": "6100.png", "formula": "\\begin{align*} \\nu ( B ) = \\int _ { S } \\lambda ( d \\xi ) \\int _ 0 ^ \\infty g ( \\xi , r ) 1 _ B ( r \\xi ) d r = \\int _ { B \\cap ( 0 , + \\infty ) } g ( 1 , r ) d r . \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} d _ n = d _ { n - 1 } + b _ { n - 1 } - b _ { n - 2 } + \\binom { n - 3 } { 2 } + \\sum _ { a = 3 } ^ { n - 3 } \\sum _ { \\ell = 0 } ^ { n - 3 - a } \\sum _ { m = 1 } ^ { n - 2 - a - \\ell } \\binom { n - 5 - \\ell - m } { a - 3 } m , n \\geq 5 , \\end{align*}"}
-{"id": "7780.png", "formula": "\\begin{align*} I _ { 2 , - } ( x _ 1 , x _ 2 ) = - \\int _ 0 ^ t \\int _ { - \\infty } ^ 0 \\sigma _ s ( y ) \\int _ { - \\infty } ^ y [ G _ { t - s } ( x _ 1 - z ) - G _ { t - s } ( x _ 2 - z ) ] \\psi ( s , z ) d z W ( d s , d y ) . \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{align*} A w = Q [ a \\left ( x \\right ) \\{ ( t \\phi _ { 1 } + w ) ^ { q } - ( t \\phi _ { 1 } + w ) \\} ] , \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} | G | = | A _ p | + | X | + ( | B _ G | + | R _ G | ) \\le 4 \\cdot 2 ^ { k - 2 } + 2 ( k - 1 ) + 2 0 < 4 \\cdot 2 ^ k + 1 \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} A _ { p } ^ { m } = \\frac { \\pi ^ { n / 2 } \\Gamma ( p ( m + \\alpha ) + \\frac { n } { 2 } + 1 ) \\Gamma ( m + \\alpha + \\frac { n } { p } + 1 ) } { \\Gamma ( p ( m + \\alpha ) + 1 ) \\Gamma ( m + \\alpha + \\frac { n } { 2 } + \\frac { n } { p } + 1 ) } \\left ( \\frac { \\Gamma ( p m - n + 1 ) } { \\Gamma ( p m - \\frac { n } { 2 } + 1 ) } \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "8141.png", "formula": "\\begin{align*} X = X ' \\sqrt { \\frac { t _ 0 - r ^ 2 } { t _ 0 } } \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} p \\left ( 1 \\right ) = f \\left ( 1 \\right ) , p ^ { \\prime } \\left ( 1 \\right ) = f ^ { \\prime } \\left ( 1 \\right ) , p ^ { \\prime \\prime } \\left ( 1 \\right ) = f ^ { \\prime \\prime } \\left ( 1 \\right ) , p \\left ( 2 \\right ) = 0 . \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} \\frac 1 2 \\Delta _ B | \\phi | ^ 2 = & \\langle \\nabla _ T ^ * \\nabla _ T \\phi , \\phi \\rangle + \\langle \\phi , \\bar \\nabla _ T ^ * \\bar \\nabla _ T \\phi \\rangle - \\sum _ { a = 1 } ^ n \\{ | \\nabla _ { \\bar V _ a } \\phi | ^ 2 + | \\nabla _ { V _ a } \\phi | ^ 2 \\} \\\\ & + \\frac 1 2 ( H ^ { 1 , 0 } - H ^ { 0 , 1 } ) | \\phi | ^ 2 , \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} B ( c ) = \\frac { 1 6 . ( \\frac { 2 } { 3 } ) ^ { 1 / 3 } . c ^ 2 } { ( 9 c ^ 2 - \\sqrt { 3 } \\sqrt { 2 7 c ^ 4 - 2 5 6 c ^ 6 } ) ^ { 1 / 3 } } + 2 ( \\frac { 2 } { 3 } ) ^ { 1 / 3 } . ( 9 c ^ 2 - \\sqrt { 3 } \\sqrt { 2 7 c ^ 4 - 2 5 6 c ^ 6 } ) ^ { 1 / 3 } . \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} q _ { X Y } ( x , y ) = a ( x ) b ( y ) p _ { X Y } ( x , y ) . \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} \\frac { f _ { \\ell _ 1 , \\ell _ 2 , \\ell _ 3 , \\ell _ 4 } } { f _ { t _ 1 , t _ 2 , t _ 3 , t _ 4 } } = x _ 1 ^ { t _ 4 - \\ell _ 4 } x _ 2 ^ { t _ 3 - \\ell _ 3 } x _ 3 ^ { t _ 2 - \\ell _ 2 } x _ 4 ^ { t _ 1 - \\ell _ 1 } . \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{align*} \\mathbb { P } [ X ^ n = \\mathbf { x } _ 1 , Y ^ n = \\mathbf { y } _ 1 | X ^ n \\in \\mathcal { A } , Y ^ n \\in \\mathcal { B } ] & = \\mathbb { P } [ X ^ n = \\mathbf { x } _ 2 , Y ^ n = \\mathbf { y } _ 2 | X ^ n \\in \\mathcal { A } , Y ^ n \\in \\mathcal { B } ] \\\\ & = \\frac { p _ { 1 1 } ^ { n / 2 } p _ { 2 2 } ^ { n / 2 } } { 2 ( p _ { 1 1 } ^ { n / 2 } p _ { 2 2 } ^ { n / 2 } + p _ { 1 2 } ^ { n / 2 } p _ { 2 1 } ^ { n / 2 } ) } \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} \\iota _ \\psi ( x \\otimes p ) : = ( x \\otimes \\psi ( p ) ) \\otimes p ^ 0 \\ , . \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} \\int _ 1 ^ { \\xi } \\frac { d X } { 2 \\sqrt { X ( X - \\lambda ) } } = \\frac 1 2 \\log ( \\xi ) + R _ l \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{align*} H _ n ( f _ n - F _ { f , n } ) ( t _ n , \\mu _ n , w _ n ) & = \\frac { 1 } { \\gamma _ n T _ 0 } \\int _ 0 ^ { \\gamma _ n ^ { - 1 } T _ 0 } H _ n [ s ] r ( s , \\mu _ n , w _ n ) \\ , \\dd s + o ( 1 ) , \\\\ & = \\frac { 1 } { T _ 0 } \\int _ 0 ^ { T _ 0 } H _ n [ \\gamma _ n ^ { - 1 } u ] f ( \\mu _ n , w _ n ) \\ , \\dd u + o ( 1 ) , \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{align*} & u _ { t t } - c ^ 2 \\ , [ \\Delta u - q u ] = 0 \\mbox { i n } \\ , Q ^ { 2 T } , \\\\ & u | _ { t = 0 } = u _ t | _ { t = 0 } = 0 \\mbox { i n } \\ , \\Omega , \\\\ & u = { \\tilde f } \\mbox { o n } \\ , \\Sigma ^ { 2 T } ; \\end{align*}"}
-{"id": "4053.png", "formula": "\\begin{align*} S ( X ; Y \\| Z ) & \\leq S ( X ; Y \\| J ) + S _ { } ( X Y ; J \\| Z ) \\\\ & = S ( X ; Y \\| J ) + \\ ! \\max _ { U V \\rightarrow X Y \\rightarrow Z J } I ( U ; J | V ) \\ ! - \\ ! I ( U ; Z | V ) \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} d \\tilde W _ t = [ ( \\hat { \\phi } _ * ( \\tilde \\gamma ^ { - 1 } \\sigma ) ) ^ { - 1 } ( \\tilde \\gamma ^ { - 1 } \\sigma ) ] ( t ^ * , q _ t ^ \\prime ) d W _ t . \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} d _ { \\emph { i n d } } ( p _ { X Y } ) = 1 - \\frac { 1 } { F ( p _ { X Y } ) } . \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} y = \\{ S ^ e , T \\} \\prod _ { \\substack { ( i , j ) \\in \\mathbb { J } _ S \\\\ k \\geq 0 } } \\{ 1 + b _ { i j k } S ^ i T ^ j , S \\} ^ { d _ { i j k } p ^ k } \\prod _ { \\substack { ( i , j ) \\in \\mathbb { J } _ T \\\\ k \\geq 0 } } \\{ 1 + a _ { i j k } S ^ i T ^ j , T \\} ^ { c _ { i j k } p ^ k } , \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{align*} \\frac { \\Vert \\mathcal { N } ( q , u + h ) - \\mathcal { N } ( q , u ) - q a ( x ) u ^ { q - 1 } h \\Vert _ { t } } { \\Vert h \\Vert _ { 2 , t } } = \\frac { \\Vert q a ( x ) h \\{ ( u + \\theta h ) ^ { q - 1 } - u ^ { q - 1 } \\} \\Vert _ { t } } { \\Vert h \\Vert _ { 2 , t } } . \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{align*} p ^ { D } _ c = { \\Pr } ( V \\in C _ d ) = { \\sum \\nolimits _ { i = 1 } ^ { C _ d } } { f _ i ( \\sigma , N ) } . \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{align*} [ \\vec { \\varepsilon } \\ , ] : = \\pi ( \\vec { \\varepsilon } ^ { \\ , + 0 } ) \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} \\mathrm { P r o b } ( f ( x ) ) = \\frac { 1 } { d } \\sum _ { e \\mid d } \\frac { \\mu ( d / e ) } { q ^ { d - e } } . \\end{align*}"}
-{"id": "4354.png", "formula": "\\begin{align*} \\exp ( x + i y ) = \\exp ( x ) ( \\cos ( y ) + i \\sin ( y ) ) \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} S : = A ^ \\top M + M ^ \\top A = A ^ \\top M + M A . \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} \\begin{array} { r c l } M \\otimes S ^ { \\otimes n } & \\to & M \\otimes S ^ { \\otimes n } \\\\ ( m | s _ 1 | \\cdots | s _ n ) & \\mapsto & ( \\partial _ M ( m ) | s _ 1 | \\cdots | s _ n ) + \\sum \\limits _ { i = 1 } ^ n ( m | s _ 1 | \\cdots | \\partial ( s _ i ) | \\cdots | s _ n ) \\ , . \\end{array} \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} \\lim \\limits _ { m \\mapsto + \\infty } \\int _ 0 ^ 1 \\phi ( \\Lambda ^ m ) ( t ) \\ , d t = \\int _ 0 ^ 1 \\phi ( f ^ * ) ( t ) \\ , d t . \\end{align*}"}
-{"id": "3623.png", "formula": "\\begin{align*} Z ( I _ 1 ; 0 , c _ r ) = Z ( I _ 7 ; c _ r , 0 ) . \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} \\| s _ 1 \\| _ { L ^ { \\infty } ( \\partial D _ { R _ 0 } ) } = \\sup _ { z \\in \\partial D _ { R _ 0 } } | g ( z ) | \\| s _ 0 ( z ) \\| ^ d \\le \\| s _ 0 \\| ^ d _ { L ^ { \\infty } ( \\partial D _ { R _ 0 } ) } J R _ 0 ^ m \\max \\{ 1 , ( 2 R _ 0 ) ^ J \\} \\max _ { 0 \\le j < J } | a _ j | . \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ g p _ j n _ j \\ , \\frac { \\psi _ j ( n _ j - 1 ) - \\psi _ j ( n _ j ) } { \\psi _ j ( n _ j ) } = 0 \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} \\left \\lvert \\{ \\textup { a b e l i a n i d e a l s o f $ \\Phi ^ - $ w h i c h a r e s t r i c t l y n e g a t i v e } \\} \\right \\rvert = 2 ^ { n - 1 } - ( n - 1 ) . \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} B _ \\mu \\cap \\left ( \\overline { S _ \\mu ( m ) } \\times \\overline { T _ \\mu ( m ) } \\right ) = \\lbrace ( s , t ) \\in B _ \\mu \\mid t < m < s \\rbrace = \\emptyset . \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} ( u \\otimes n ) \\cdot s = u \\otimes s n = u s \\otimes n \\ \\ \\ \\ ( u \\otimes n ) \\cdot \\alpha = u \\alpha \\otimes n - u \\otimes \\alpha \\cdot n \\ , . \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{align*} \\hat p _ { } \\ ! \\ ! ( t ) = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } 1 _ { \\{ \\tilde Y _ i ( t ) \\neq f _ m \\} } \\ , , \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} U _ 0 ( F ) ( x , y ) = F ( \\varphi ( x , y ) , \\tau ( y ) ) , ( x , y ) \\in X _ 2 \\times Y _ 2 \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{align*} & E \\left [ \\beta ( t , q _ t ^ m ) \\| z _ t ^ m \\| ^ 2 \\right ] \\\\ = & E \\left [ \\left ( \\frac { \\beta ( t , q _ t ) } { 2 \\pi } \\right ) ^ { n / 2 } \\int \\beta ( t , q _ t ) \\| z \\| ^ 2 e ^ { - \\beta ( t , q _ t ) \\| z \\| ^ 2 / 2 } d z \\right ] + O ( m ^ { \\delta } ) \\\\ = & \\left ( \\frac { 1 } { 2 \\pi } \\right ) ^ { n / 2 } \\int \\| w \\| ^ 2 e ^ { - \\| w \\| ^ 2 / 2 } d w + O ( m ^ { \\delta } ) \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{align*} \\mathcal { F } _ { u } ( q , u ) \\phi = - \\Delta \\phi - q a ( x ) u ^ { q - 1 } \\phi \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} 2 ^ { \\mathrm { e x } _ L ( n , \\mathcal { H } ) } \\leq | \\mathrm { F o r b } _ L ( n , \\mathcal { H } ) | \\leq \\sum _ { i \\leq \\mathrm { e x } _ L ( n , \\mathcal { H } ) } \\binom { \\binom { n } { r } } { i } \\leq 2 n ^ { r \\cdot \\mathrm { e x } _ L ( n , \\mathcal { H } ) } . \\end{align*}"}
-{"id": "7922.png", "formula": "\\begin{align*} G _ \\mu \\left ( \\frac { 1 } { z } \\right ) = \\frac { z } { 1 - \\eta _ \\mu ( z ) } , \\ ; \\ ; \\ ; \\ ; \\ ; z \\in \\Omega , \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{align*} \\forall i \\in [ 1 , n ] , [ x _ i , y _ i ] = - \\sum _ { j | j \\neq i } t _ { i j } , \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} Y ( \\zeta ) = - \\sum _ { n = 0 } ^ { j - 2 } ( I _ - X _ { n + 1 } \\zeta ^ { - n } + I _ + X _ { - n - 1 } \\zeta ^ n ) \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} S _ { b + 1 , c } = \\bigcup _ { \\sigma \\in S _ { b , c } } \\{ \\hat { \\sigma } _ \\lambda : \\lambda = 1 , \\dots , N + 1 - \\sigma ( r ) \\} , \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} \\int _ A P _ { T } f d \\mu ~ = ~ \\int _ { T ^ { - 1 } ( A ) } f d \\mu \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} \\begin{pmatrix} - D & 0 \\\\ 0 & D \\end{pmatrix} . \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ 0 | D _ \\lambda ( x , y ) | \\ , \\mathrm d x = \\frac { 1 } { 2 \\sqrt { | \\lambda | } } \\int _ { - \\infty } ^ 0 e ^ { - \\operatorname { R e } \\sqrt { \\lambda } | x - y | } \\ , \\mathrm d x = \\frac { 2 - e ^ { \\operatorname { R e } \\sqrt { \\lambda } y } } { 2 \\sqrt { | \\lambda | } \\operatorname { R e } \\sqrt { \\lambda } } \\leq \\frac { 1 } { \\sqrt { | \\lambda | } \\operatorname { R e } \\sqrt { \\lambda } } . \\end{align*}"}
-{"id": "5593.png", "formula": "\\begin{align*} \\alpha _ 1 ' = \\alpha _ 1 \\circ g + \\bar \\partial \\psi - \\psi \\bar \\partial _ W , \\end{align*}"}
-{"id": "2218.png", "formula": "\\begin{align*} ( u _ 1 , \\cdots , u _ n ) ( A + E _ n ) = ( y _ 1 , \\cdots , y _ n ) , \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} E \\left [ S ^ { p a r t } _ { s , t } \\right ] \\equiv & - E \\left [ \\ln ( p ( t , x _ t ) ) \\right ] + E \\left [ \\ln ( p ( s , x _ s ) ) \\right ] \\\\ = & - \\int \\ln ( p ( t , x ) ) p ( t , x ) d x + \\int \\ln ( p ( s , x ) ) p ( s , x ) d x , \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} ( m \\cdot s ) \\cdot \\alpha = m \\cdot ( s \\alpha ) \\ , . \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{align*} \\mathcal { S } = \\{ \\tau \\in \\mathbb { H } ; | \\Re ( \\tau ) | \\leq \\frac 1 2 , | \\tau | \\geq 1 \\} . \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} T _ { k } : = \\sum _ { j \\in J _ { k } } \\nu _ { j } ^ { k } T _ { j } ^ { k } \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} \\phi _ { V W } ( \\{ f _ { i } \\} _ { i \\in I } ) = \\{ \\pi _ { V _ { i } } S _ { W } ^ { - 1 } f _ { i } \\} _ { i \\in I } \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} ( W ^ T f , y ) _ { \\rm i n t } = ( f , O ^ T y ) _ { \\rm e x t } \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} N \\ge \\lim _ j \\bigg | \\frac { \\partial _ { x _ i } u _ j ( d _ j x + ( x ^ j ) ' ) } { d _ j x _ n } \\bigg | = \\lim _ j \\bigg | \\frac { \\partial _ { x _ i } \\tilde u _ j ( x ) } { x _ n } \\bigg | = \\bigg | \\frac { \\partial _ { x _ i } u _ 0 ( x ) } { x _ n } \\bigg | \\end{align*}"}
-{"id": "2340.png", "formula": "\\begin{align*} E \\left [ \\phi ( X ) \\ , | \\ , S _ N \\right ] = \\int _ 0 ^ { \\infty } \\phi ( \\xi ) \\ , \\frac { \\xi ^ { S _ N - 1 } } { ( S _ N - 1 ) ! } \\ , e ^ { - \\xi } d \\xi \\end{align*}"}
-{"id": "7735.png", "formula": "\\begin{align*} S q _ { \\mathbb { F } _ 4 } ^ 0 ( f ) & = f , \\\\ S q _ { \\mathbb { F } _ 4 } ^ 1 ( f ) & = \\alpha \\xi ^ 2 + 2 ( \\alpha + 1 ) \\xi ^ 3 = \\alpha \\xi ^ 2 , \\\\ S q _ { \\mathbb { F } _ 4 } ^ 2 ( f ) & = ( \\alpha + 1 ) \\xi ^ 4 , \\\\ S q _ { \\mathbb { F } _ 4 } ^ k ( f ) & = 0 \\ \\ k \\ge 3 . \\end{align*}"}
-{"id": "4240.png", "formula": "\\begin{align*} g _ \\sigma ( \\theta ) = \\log ( 1 - 2 \\sigma + \\sigma ^ 2 + 2 \\sigma - 2 \\sigma \\cos ( \\theta ) ) \\geq \\log \\sigma + g _ 1 ( \\theta ) . \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} A ^ 1 H A ^ 1 & = ( B + S ) H ( B + S ) = B H B + B ( H S ) + ( S H ) B + S H S \\\\ & \\subseteq B B B + B B + B B + B = B , \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} y _ i ( \\mu ; t ^ { - m } , t ) = y _ i ( \\nu ; t ^ { - m } , t ) \\forall \\ 1 \\leq i \\leq n \\implies S _ m ( \\mu ^ { + } ) \\sim S _ m ( \\nu ^ { + } ) \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} d _ r ( p _ 1 , p _ 2 ) = ( y _ 1 - y _ 2 ) ^ 2 . \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} M _ { 1 0 } = M _ { 1 , 2 } \\times Y \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} \\begin{aligned} 2 g - 2 & = - 2 d + \\sum _ { i = 1 } ^ { r _ 0 } \\left ( \\lambda _ { 0 , i } - 1 \\right ) + \\sum _ { i = 1 } ^ { r _ 1 } \\left ( \\lambda _ { 1 , i } - 1 \\right ) + \\sum _ { i = 1 } ^ { r _ \\infty } \\left ( \\lambda _ { \\infty , i } - 1 \\right ) \\\\ & = d - r _ 0 - r _ 1 - r _ \\infty . \\end{aligned} \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} x & : = ( a b ^ { - 1 } ) ^ { 3 } ( a ^ 2 b ^ { - 1 } ) ^ { 3 } ( a b ^ { - 1 } ) ^ 3 ( a ^ 2 b ^ { - 1 } ) ^ 4 \\ldots ( a b ^ { - 1 } ) ^ 3 ( a ^ 2 b ^ { - 1 } ) ^ { \\rho + 2 } \\\\ y & : = ( a b ^ { - 1 } ) ^ 3 ( a ^ 2 b ^ { - 1 } ) ^ { \\rho + 3 } ( a b ^ { - 1 } ) ^ 3 ( a ^ 2 b ^ { - 1 } ) ^ { \\rho + 4 } \\ldots ( a b ^ { - 1 } ) ^ { 3 } ( a ^ 2 b ^ { - 1 } ) ^ { 2 \\rho + 2 } \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} d _ \\infty ( u , v ) : = \\inf \\left \\{ | p | \\colon \\ , p \\in \\Gamma ( u , v ) , p _ i 0 \\le i \\le | p | \\right \\} , \\end{align*}"}
-{"id": "1289.png", "formula": "\\begin{align*} \\alpha _ i ^ { + } + \\alpha _ j ^ { + } + \\alpha _ k ^ { - } + \\alpha _ t ^ { - } = 1 , \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} \\overline { V } _ t \\le e ^ { - c _ 1 \\tilde { \\gamma } t } \\overline { V } _ 0 + \\frac { c _ 2 } { c _ 1 } ( 1 - e ^ { - c _ 1 \\tilde { \\gamma } t } ) + e ^ { - c _ 1 \\tilde { \\gamma } t } \\int _ { 0 } ^ { t } \\frac { \\tau } { N } \\tilde { \\gamma } e ^ { c _ 1 \\tilde { \\gamma } s } \\sum _ { i = 1 } ^ N { d B _ { i , s } } ^ T e _ { i , s } . \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} d \\int _ \\tau ^ t \\psi ( s ) d s = \\hat w ( \\tau ) - \\hat w ( t ) - \\int _ \\tau ^ t P ^ \\perp \\ ( \\zeta ( s ) + [ A ( s ) , \\psi ( s ) ] \\ ) d s \\end{align*}"}
-{"id": "2796.png", "formula": "\\begin{align*} \\{ \\cdot , \\cdot \\} _ { ( \\lambda ) } \\Omega = - \\frac { 1 } { F ( \\lambda ) } d \\cdot \\wedge \\ , d \\cdot \\wedge \\big ( \\sigma _ { ( \\lambda ) } + \\frac { g _ { ( \\lambda ) } } { r - 1 } \\Theta \\big ) \\wedge \\frac { \\Theta ^ { r - 2 } } { ( r - 2 ) ! } \\wedge d F ^ 1 ( \\lambda ) \\wedge \\ldots \\wedge d F ^ k ( \\lambda ) , \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{align*} \\Vert U _ { n } y ^ { n } - y ^ { n } \\Vert = \\left \\{ \\begin{array} { l l } \\Vert U _ { n _ { m _ { k } } } x ^ { m _ { k } } - x ^ { m _ { k } } \\Vert & n = n _ { m _ { k } } k \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} R i c _ { B } + \\nabla _ { B } \\nabla _ { B } \\beta = \\bar { \\lambda } g _ { B } \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} C & = 1 + \\ ( 1 + c \\kappa _ 6 \\| A _ 1 - A _ 2 \\| _ { L ^ 3 ( \\mathcal R ) } \\ ) ^ 2 \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{align*} \\lambda _ \\circ ( q , r ) = \\frac { 1 } { q } + \\frac { d } { 2 r } - \\frac { d + 2 } { 4 } . \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} \\int _ { B _ r } R ( \\psi ) \\dd x \\le C ( N ) \\int _ { B _ r } | \\psi | ^ 4 \\dd x \\le C ( N ) \\| \\psi \\| ^ 4 _ { L ^ { \\frac { m p } { m - p } } ( U ) } r ^ { m ( 1 - \\frac { 4 } { p } + \\frac { 4 } { m } ) } . \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} \\Omega _ { \\alpha _ 1 , \\bar { \\alpha _ 2 } } = 0 . \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} \\chi ( G \\times H ) = \\min \\{ \\chi ( G ) , \\chi ( H ) \\} \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} x \\ , \\frac { \\theta _ 2 ' ( x ) } { \\theta _ 2 ( x ) } + \\tfrac { 1 } { x } \\ , \\frac { \\theta _ 4 ' \\left ( \\tfrac { 1 } { x } \\right ) } { \\theta _ 4 \\left ( \\tfrac { 1 } { x } \\right ) } = - \\frac { 1 } { 2 } x \\ , \\frac { \\theta _ 4 ' ( x ) } { \\theta _ 4 ( x ) } + \\tfrac { 1 } { x } \\ , \\frac { \\theta _ 2 ' \\left ( \\tfrac { 1 } { x } \\right ) } { \\theta _ 2 \\left ( \\tfrac { 1 } { x } \\right ) } = - \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "4459.png", "formula": "\\begin{align*} n + b ( z ) \\ge \\ell ( z ) + b ( z ) = 2 b ( z ) - z + 3 \\ge ( z + 1 ) ^ 2 . \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{align*} ( A ^ { i } ) ^ { \\dagger } = N ^ \\ast ( N N ^ \\ast ) ^ { - 1 } ( M _ { 1 } ^ \\ast M _ { 1 } ) ^ { - 1 } M _ { 1 } ^ \\ast . \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} N ( U , r ) = \\frac { I ( U , r ) } { H ( U , r ) } . \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} \\int _ { u } ^ v \\frac { ( q x / u , q x / v , \\alpha a x ; q ) _ \\infty } { ( a x , c x , d x ; q ) _ \\infty } d _ q x = \\sum _ { n = 0 } ^ \\infty \\lambda _ n ( \\alpha ; q ) _ n a ^ n . \\end{align*}"}
-{"id": "8730.png", "formula": "\\begin{align*} \\log \\log d = \\log \\log ( p _ 1 p _ 2 \\cdots p _ t ) \\le ( \\log \\log p _ 1 ) \\cdots ( \\log \\log p _ t ) . \\end{align*}"}
-{"id": "7888.png", "formula": "\\begin{align*} & u _ { \\epsilon _ j } ( t ) = u _ 0 - i \\int _ 0 ^ t H _ { \\epsilon _ j } ( \\theta ) u _ { \\epsilon _ j } ( \\theta ) d \\theta - i \\int _ 0 ^ t f ( \\theta ) d \\theta \\\\ & = u _ 0 - i \\int _ 0 ^ t H _ { \\epsilon _ j } ( \\theta ) u ( \\theta ) d \\theta - i \\int _ 0 ^ t H _ { \\epsilon _ j } ( \\theta ) \\bigl \\{ u _ { \\epsilon _ j } ( \\theta ) - u ( \\theta ) \\bigr \\} d \\theta - i \\int _ 0 ^ t f ( \\theta ) d \\theta , \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} a : = \\left ( \\begin{matrix} 1 & - h _ x ( x ) \\\\ 0 & 1 \\end{matrix} \\right ) ^ { T } \\left ( \\begin{matrix} 1 & - h _ x ( x ) \\\\ 0 & 1 \\end{matrix} \\right ) \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} \\begin{cases} F ( D ^ { 2 } u _ 0 ) = 1 & \\mathbb { R } _ + ^ n \\cap \\Omega _ 0 \\\\ | \\nabla u _ 0 | = 0 & \\mathbb { R } _ + ^ n \\backslash \\Omega _ 0 \\\\ u = 0 & \\mathbb { R } _ + ^ { n - 1 } , \\end{cases} \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } f ( y ) ^ 2 G _ { t - s } ^ 2 ( x - y ) d y & \\leq \\Big ( \\int _ { \\mathbb { R } } G _ { t - s } ( x - y ) ^ { 2 q _ 1 } d y \\Big ) ^ { 1 / q _ 1 } \\| f \\| ^ 2 _ { L ^ q ( \\mathbb { R } ) } \\\\ & = k \\| f \\| ^ 2 _ { L ^ q ( \\mathbb { R } ) } ( t - s ) ^ { - 1 + 1 / ( 2 q _ 1 ) } , \\end{align*}"}
-{"id": "1401.png", "formula": "\\begin{align*} K _ N ( \\tau ) = 2 ^ { N ( N + 1 ) / 4 } ( 1 + \\tau ) ^ { N / 2 } N ^ { \\binom { N + 1 } { 2 } / 2 } \\prod _ { j = 1 } ^ { N } \\Gamma ( j / 2 ) , \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} & \\varphi _ 1 \\circ c _ i X ^ { q ^ { i s } } \\circ \\varphi _ 2 \\\\ = & c c _ i ^ { q ^ l } ( g X ^ { q ^ j } + h X ^ { q ^ { j + n } } ) ^ { q ^ { i s + l } } + d c _ i ^ { q ^ { l + n } } ( g X ^ { q ^ j } + h X ^ { q ^ { j + n } } ) ^ { q ^ { i s + l + n } } \\\\ = & ( c g ^ { q ^ { i s + l } } c _ i ^ { q ^ l } + d h ^ { q ^ { i s + l + n } } c _ i ^ { q ^ { l + n } } ) X ^ { q ^ { j + i s + l } } \\\\ & + ( c h ^ { q ^ { i s + l } } c _ i ^ { q ^ l } + d g ^ { q ^ { i s + l + n } } c _ i ^ { q ^ { l + n } } ) X ^ { q ^ { j + i s + l + n } } . \\end{align*}"}
-{"id": "4363.png", "formula": "\\begin{align*} z ( \\lambda , \\xi ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { z ^ { ( n ) } ( \\xi ) } { n ! } \\lambda ^ n \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} \\partial _ { x _ i } \\tilde V _ j = \\partial _ { x _ j } \\tilde V _ i , \\ \\ i , j = 1 , . . . , n + 1 . \\end{align*}"}
-{"id": "6957.png", "formula": "\\begin{align*} \\mathrm { a s c } ( \\kappa ) : = \\lvert \\{ e = \\{ i , j \\} \\in E \\ , \\vert \\ , i < j \\textup { a n d } \\kappa ( i ) < \\kappa ( j ) \\} \\rvert . \\end{align*}"}
-{"id": "6076.png", "formula": "\\begin{align*} \\lambda _ n ^ + ( M ( a _ 0 ) ) = \\varkappa ( \\alpha ) n ^ { - \\alpha } + o ( n ^ { - \\alpha } ) , n \\to \\infty . \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{align*} { C P } _ { s w } ( X , \\Omega _ X , J , \\Lambda _ X ) = \\lim \\limits _ { r \\to \\infty } ( T _ r ^ { - } ) ^ { - 1 } \\circ C M ( X , \\Omega _ X , J , r , \\Lambda _ X ) \\circ T _ r ^ + . \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} \\alpha ( T ) = \\alpha ( T ' ) + \\ell . \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} \\mathcal { K } f ( \\xi ) = \\frac { - 1 } { \\pi } \\operatorname { p . v . } \\int _ \\Sigma \\frac { ( \\xi - \\eta ) \\cdot \\vec n ( \\eta ) } { | \\xi - \\eta | ^ 2 } f ( \\eta ) d S ( \\eta ) , \\xi \\in \\Sigma . \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} N ^ - ( w ) = N ^ - ( z ) \\sqcup z ^ { - 1 } N ^ - ( y ) \\textup { a n d } N ^ - ( \\sigma _ { \\nu } w ) = N ^ - ( z ) \\sqcup z ^ { - 1 } N ^ - ( \\sigma _ { \\nu } y ) . \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} h ( z , w ) ^ p = \\det ( B ( z , w ) ) . \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} \\dd { M ^ x } ( t ) & = \\sigma \\dd { W } ( t ) , \\\\ M ^ x ( 0 ) & = x \\in \\R ^ d , \\end{align*}"}
-{"id": "4949.png", "formula": "\\begin{align*} X _ 0 \\cap U = \\{ x \\in U \\colon \\rho ( x ) < a \\} , X _ 1 \\cap U = \\{ x \\in U \\colon \\rho ( x ) < b \\} , \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} \\beta = \\sum _ { \\boldsymbol { \\theta } } \\gamma ( \\mathbf { m } - \\mu , 0 , \\boldsymbol { \\theta } ; N - \\mu ) . \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} f _ \\xi ( \\gamma , T - \\xi + 0 ) = f ( \\gamma , T - \\xi ) , \\end{align*}"}
-{"id": "1377.png", "formula": "\\begin{align*} V ^ { u } \\bigl ( r , \\eta \\bigr ) = \\mathcal { E } ^ G \\Bigl [ \\int _ r ^ T c \\bigl ( s , X _ s ^ { r , \\eta ; u } , u _ s \\bigr ) d s + \\Psi ( X _ T ^ { r , \\eta ; u } ) \\Bigl \\vert \\mathcal { F } _ r \\Bigr ] , P { \\text - a . s . } \\end{align*}"}
-{"id": "444.png", "formula": "\\begin{align*} U ( b ) = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & b & - b ^ 2 / 2 \\\\ 0 & 0 & 1 & - b \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , U ( c ) = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & c & - c ^ 2 / 2 \\\\ 0 & 0 & 1 & - c \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "6126.png", "formula": "\\begin{align*} \\inf _ { [ k , + \\infty [ } \\dot { \\rho } _ { G _ 1 , G _ 2 } ( t ) = \\alpha > 0 . \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} \\begin{aligned} \\widehat { b } _ i ^ { ( 1 , 2 ) } - \\widehat { b } _ i ^ { ( N , 1 ) } & = Q _ i ^ { K _ 1 } , \\\\ \\widehat { b } _ i ^ { ( 2 , 3 ) } - \\widehat { b } _ i ^ { ( 1 , 2 ) } & = Q _ i ^ { K _ 2 } , \\\\ \\dots & = \\dots \\\\ \\widehat { b } _ i ^ { ( N - 1 , N ) } - \\widehat { b } _ i ^ { ( N - 2 , N - 1 ) } & = Q _ i ^ { K _ { N - 1 } } , \\\\ \\widehat { b } _ i ^ { ( N , 1 ) } - \\widehat { b } _ i ^ { ( N - 1 , N ) } & = Q _ i ^ { K _ N } , \\end{aligned} \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{align*} \\dim M _ { 2 \\kappa + 1 } ( \\Gamma _ 0 ( 4 ) , \\chi _ { - 4 } ) & = 0 & \\quad \\mbox { f o r } \\kappa < 0 ; \\\\ \\dim M _ { 2 \\kappa + 1 } ( \\Gamma _ 0 ( 4 ) , \\chi _ { - 4 } ) & = 1 + \\kappa & \\quad \\mbox { f o r } \\kappa \\geq 0 . \\\\ \\dim S _ { 2 \\kappa + 1 } ( \\Gamma _ 0 ( 4 ) , \\chi _ { - 4 } ) & = 0 & \\quad \\mbox { f o r } \\kappa < 2 ; \\\\ \\dim S _ { 2 \\kappa + 1 } ( \\Gamma _ 0 ( 4 ) , \\chi _ { - 4 } ) & = \\kappa - 1 & \\quad \\mbox { f o r } \\kappa \\geq 2 . \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - m ( \\| u \\| ^ { 2 } ) \\Delta u = f ( u ) & \\mbox { i n $ \\Omega $ , } \\\\ u = 0 & \\mbox { o n $ \\partial \\Omega $ , } \\end{array} \\right . \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{align*} \\int _ 0 ^ t T _ { - 1 } ( s ) B u ( s ) \\ , d s = \\sum _ { k = 1 } ^ n \\int _ 0 ^ t T _ { - 1 } ( s ) B e _ k u _ k ( s ) \\ , d s \\in X , \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} \\tilde \\xi _ t ( y ) = \\xi ( y ) - q _ { \\xi , t } ( y ) , \\ ; \\ ; \\ ; y \\in B _ { R _ t ^ \\ast } \\setminus \\{ 0 \\} \\tilde \\sigma _ t ( y ) = \\begin{cases} \\frac { \\sigma ( y ) } { q _ { \\sigma , t } } & \\mbox { i f } y = 0 , \\\\ \\sigma ( y ) & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} \\alpha = \\omega ^ { \\alpha _ 0 } \\cdot m _ 0 + \\cdots + \\omega ^ { \\alpha _ \\ell } \\cdot m _ \\ell \\end{align*}"}
-{"id": "6991.png", "formula": "\\begin{align*} \\sum _ { w \\in \\mathcal { D } _ { \\nu } } t ^ { 2 | N ^ - ( w ) \\cap \\Phi _ h ^ - | } = \\sum _ { \\sigma _ { \\nu } w \\ , : \\ , w \\in \\mathcal { D } _ { \\nu } } t ^ { 2 | N ^ - ( \\sigma _ { \\nu } w ) \\cap \\Phi _ h ^ - | } . \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{align*} \\sigma '' ( i , j , k , \\alpha , \\beta ) : = [ T _ { i j } ^ \\alpha , T _ { j k } ^ \\beta ] + \\sum _ { \\stackrel { \\gamma , \\delta \\geq 0 | } { \\stackrel { \\gamma + \\delta = \\alpha + \\beta } { } } } \\begin{pmatrix} \\alpha \\\\ \\gamma \\end{pmatrix} [ T _ { i j } ^ \\gamma , T _ { i k } ^ \\delta ] \\quad i < j < k \\in [ n ] , \\quad \\alpha , \\beta \\geq 0 , \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} y _ m = \\arg \\min _ { y \\in \\R ^ N } \\| A A _ { m } ^ { \\prime } y - b \\| \\ , , \\quad \\mbox { w i t h } x _ { m } = A _ { m } ^ { \\prime } y _ m \\ , . \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} S _ n = n \\frac { b - \\rho } { 1 - a } + O ( 1 ) . \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} f ( x _ { \\psi } ) = \\mathrm { t r } \\left ( \\phi \\times w | R \\Gamma ( G , b , \\mu ) [ i _ { M } ^ { G _ { b } } ( \\sigma \\psi ) ] \\right ) . \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} & u \\in S _ q ^ u ( T ) \\cap B C ( [ 0 , T ) ; L _ { \\sigma } ^ p ( \\Omega ) ) , \\\\ & d _ s \\in S _ q ^ d ( T ) \\cap B C ( [ 0 , T ) ; W ^ { 1 , p } ( \\Omega ) ^ 3 ) \\cap B C ( [ 0 , T ) ; L ^ { \\infty } ( \\Omega ) ^ 3 ) , \\overline { d } \\in B C ( [ 0 , T ) ; \\R ^ 3 ) , \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{align*} d ^ 2 \\big ( \\Psi ( y _ 1 , s ) , \\Psi ( y _ 2 , t ) \\big ) & \\ge \\max \\big \\{ d _ Y ^ 2 \\big ( \\pi \\circ \\Psi ( y _ 1 , s ) , \\pi \\circ \\Psi ( y _ 2 , t ) \\big ) , | u \\circ \\Psi ( y _ 1 , s ) - u \\circ \\Psi ( y _ 2 , t ) | ^ 2 \\big \\} \\\\ & = \\max \\big \\{ d _ Y ^ 2 ( y _ 1 , y _ 2 ) , | s - t | ^ 2 \\big \\} \\ge \\frac { 1 } { 2 } \\Big ( d _ Y ^ 2 ( y _ 1 , y _ 2 ) + | s - t | ^ 2 \\Big ) . \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} | \\mathrm { F o r b } _ r ( n , C _ { \\ell } ^ r ) | = 2 ^ { \\Omega ( n ^ { r - 1 } ) } | \\mathrm { F o r b } _ r ( n , C _ { \\ell } ^ r ) | = 2 ^ { O ( n ^ { r - 1 } \\log n ) } \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} \\lambda _ j ( x ) = \\frac { 1 - \\xi } { n + 1 } . \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{gather*} p ( 0 , 0 | v , w ) = \\frac 1 4 t ( 5 t - 1 ) , p ( 0 , 1 | v , w ) = p ( 1 , 0 | v , w ) = \\frac 5 4 t ( 1 - t ) , \\\\ p ( 1 , 1 | v , w ) = \\frac 1 4 ( 1 - t ) ( 4 - 5 t ) . \\end{gather*}"}
-{"id": "4167.png", "formula": "\\begin{align*} L = L ( d , \\beta ) \\leq 1 1 + \\big ( 1 + \\lceil \\log _ 2 \\beta \\rceil \\big ) \\cdot \\Big ( 1 1 + \\frac { \\beta } { d - 1 } \\Big ) \\leq 1 1 + \\big ( 1 + \\lceil \\log _ 2 \\beta \\rceil \\big ) \\cdot \\Big ( 1 1 + \\frac { 2 \\beta } { d } \\Big ) \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} V ( d , e _ 1 , e _ 2 , e _ 3 ) = \\begin{pmatrix} d & e _ 1 & e _ 2 & e _ 3 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} ( | x - y | , | y - z | ) = ( | x ' - y ' | , | y ' - z ' | ) \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} W _ t = U _ t + \\frac { ( \\mu - \\kappa ) } { a \\lambda _ 2 } \\overline { V } _ t . \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} \\frac { X _ 1 ( \\mu _ { 1 j } ) } { X _ 1 ( h ) } = c _ { j \\alpha } , \\frac { X _ 1 ( \\rho _ { 1 j } ) } { X _ 1 ( h ) } = - b _ { j \\alpha } , \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb { E } \\Big [ ( 2 ^ { n } - 1 ) ^ { - \\gamma { d / { \\alpha } } } { \\big ( M ^ { ( v ) } _ { 2 ^ { n } - 1 } \\big ) ^ { \\gamma } } \\Big ] = c , \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{align*} \\begin{aligned} & \\hat D _ \\varpi ( P \\parallel Q ) = \\log \\hat F _ { \\lambda ^ * } - \\varpi . \\end{aligned} \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{align*} \\forall x \\geq \\xi _ + , \\frac { \\partial } { \\partial x } \\left ( D _ 1 ( x ) \\rho _ { e q } ( x ) \\right ) + D _ 2 ( x ) \\rho _ { e q } ( x ) = 0 . \\end{align*}"}
-{"id": "8709.png", "formula": "\\begin{align*} G ( d u , d u ) \\equiv 0 , G ( d u , d \\mathring x ^ a ) = ( \\nabla u ) ( \\mathring x ^ a ) \\equiv 0 . \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} & I _ { 1 \\epsilon } ( t , x , \\xi ) = \\epsilon \\chi \\ , ' ( \\epsilon ( \\mu + h ) ) \\sum _ { | \\alpha | = 1 } \\Bigl \\{ h ^ { ( \\alpha ) } h _ { s ( \\alpha ) } - h ^ { ( \\alpha ) } _ s h _ { ( \\alpha ) } \\Bigr \\} \\\\ & = \\epsilon \\chi \\ , ' \\bigl ( \\epsilon ( \\mu + h ( t , x , \\xi ) ) \\bigr ) \\sum _ { | \\alpha | = 1 } \\frac { i } { 2 m ^ 2 } ( \\xi _ { \\alpha } - A _ { \\alpha } ( t , x ) ) ( - i \\partial _ x ) ^ { \\alpha } \\nabla \\cdot A ( t , x ) . \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} - \\otimes \\mathfrak { S } _ n : \\mathbf { M o d } ^ { \\phi , 1 } _ { / \\mathfrak { S } } & \\to \\mathbf { M o d } ^ { \\phi , 1 } _ { / \\mathfrak { S } _ n } \\\\ \\mathfrak { M } & \\mapsto \\mathfrak { M } \\otimes _ { \\mathfrak { S } } \\mathfrak { S } _ n = : \\mathfrak { M } _ { \\mathfrak { S } _ n } . \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} & Q _ 1 ^ { k , m } ( 1 - q ^ { 2 k + m } r ^ 2 ) \\\\ & = \\sum _ { i = 0 } ^ { m } { { m + 1 } \\choose { i } } _ { \\ ! \\ ! q } q ^ { i ( 2 k + i - 1 ) / 2 } \\prod _ { j = 0 } ^ { i - 1 } ( q ^ { k + j } r ^ 2 - r ) \\prod _ { j = 0 } ^ { m - i } ( 1 - q ^ { 2 k + m - j } r ^ 2 ) \\\\ & = \\sum _ { i = 0 } ^ { m + 1 } { { m + 1 } \\choose { i } } _ { \\ ! \\ ! q } q ^ { i ( 2 k + i - 1 ) / 2 } \\prod _ { j = 0 } ^ { i - 1 } ( q ^ { k + j } r ^ 2 - r ) \\prod _ { j = 1 } ^ { m + 1 - i } ( 1 - q ^ { 2 k + m + 1 - j } r ^ 2 ) \\\\ & - q ^ { ( m + 1 ) ( 2 k + m ) / 2 } \\prod _ { j = 0 } ^ { m } ( q ^ { k + j } r ^ 2 - r ) . \\end{align*}"}
-{"id": "8887.png", "formula": "\\begin{align*} \\mathcal { L } ^ n = n ! \\int _ { \\Delta ^ + _ { \\mathcal { L } } } \\prod _ { \\Phi ^ + \\setminus E } \\frac { \\kappa ( \\alpha , p ) } { \\kappa ( \\alpha , \\rho ) } d p \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} & \\psi ( x ' ) t ( x ' , D _ x ) \\varphi ( x ) = \\psi ( x ' ) t ( x ' , D _ x ) ( \\eta ( x _ n ) \\varphi ( x ) ) + \\psi ( x ' ) t ( x ' , D _ x ) ( ( 1 - \\eta ( x _ n ) ) \\varphi ( x ) ) . \\end{align*}"}
-{"id": "5680.png", "formula": "\\begin{align*} \\bigg | \\int _ 0 ^ { t } X ^ n _ r \\otimes \\dd X _ r - \\int _ 0 ^ { s } X ^ n _ r \\otimes \\dd X _ r - X _ s \\otimes X _ { s , t } \\bigg | & \\leq \\sum _ { k = 2 } ^ { N } \\Big ( \\frac { 2 } { k - 1 } c ( s , t ) \\Big ) ^ { 2 / q } \\lesssim N ^ { 1 - 2 / q } c ( s , t ) ^ { 2 / q } \\\\ & \\lesssim ( \\# \\{ k : \\tau ^ n _ k \\in [ s , t ] \\} ) ^ { 1 - 2 / q } c ( s , t ) ^ { 2 / q } + c ( s , t ) ^ { 2 / q } \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} \\mathcal { L } H + ( | A | ^ 2 + \\dfrac 1 2 ) H = 0 . \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{align*} | B ( v , \\eta ) | = 2 \\pi ( 1 - \\cos \\eta ) . \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} J _ { 2 , + } ( t _ 1 , t _ 2 ) = & \\int _ 0 ^ { t _ 2 } \\int _ 0 ^ { \\infty } \\int _ y ^ { \\infty } G _ { t _ 2 - s } ( x - z ) \\psi ( s , z ) \\sigma _ s ( y ) W ( d s , d y ) \\\\ & - \\int _ 0 ^ { t _ 1 } \\int _ 0 ^ { \\infty } \\int _ y ^ { \\infty } G _ { t _ 1 - s } ( x - z ) \\psi ( s , z ) \\sigma _ s ( y ) W ( d s , d y ) \\end{align*}"}
-{"id": "2881.png", "formula": "\\begin{align*} c ^ { \\delta + k } \\ge c ^ { k - 1 } + \\frac { 1 } { 2 } \\frac { \\delta [ ( B - 1 ) c ^ { k - B } + \\delta - B ] } { \\binom { k } { 2 } } . \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} y ' ( x ) = \\lambda \\sqrt { \\frac { 4 - ( 1 + \\lambda ^ 2 ) ^ 2 ( ( \\frac { y } { \\lambda } + C _ { 2 } ) ^ 2 + 2 C _ 1 ) ^ 2 } { ( 1 + \\lambda ^ 2 ) ^ 3 ( ( \\frac { y } { \\lambda } + C _ { 2 } ) ^ 2 + 2 C _ 1 ) ^ 2 } } \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} T _ 0 ( X ) = e ^ { - \\frac { 1 } { 2 } \\zeta ' _ { D _ { 0 , 1 } } ( 0 ) } \\textrm { a n d } T _ 1 ( X ) = e ^ { - \\frac { 1 } { 2 } \\zeta ' _ { D _ { 1 , 1 } } ( 0 ) } . \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} A = \\left \\{ H _ { \\ell _ 1 , \\ell _ 2 , \\ell _ 3 , \\ell _ 4 } : 3 \\leq \\ell _ 1 \\leq s , \\ ; 0 \\leq \\ell _ 2 , \\ell _ 3 , \\ell _ 4 \\leq s - 3 \\right \\} , \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} X _ { n + 1 } = \\bullet | X _ { [ 1 : n ] } \\sim \\frac { \\alpha } { \\alpha + n - 1 } c _ 0 ( \\{ \\bullet - X _ n + 1 \\} ) + \\frac { 1 } { \\alpha + n - 1 } \\sum \\limits _ { i = 1 } ^ { n - 1 } \\delta _ { X _ { i + 1 } - X _ i + ( 1 - \\delta _ { X _ i 0 } ) } ( \\{ \\bullet \\} ) . \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} \\mathcal { P } ( \\alpha ) & = \\log 2 + \\Phi _ { \\alpha } ( 0 , h ) - \\frac { 1 } { 2 } \\int _ 0 ^ 1 \\xi '' ( s ) s \\alpha ( s ) \\ , d s . \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{align*} \\sum _ o q ^ { \\sum _ { j = 1 } ^ { q _ o } l _ { o , j } } = \\sum _ { i \\geq 0 } a _ { \\mu , i } q ^ i . \\end{align*}"}
-{"id": "5338.png", "formula": "\\begin{align*} \\displaystyle { \\operatornamewithlimits { \\mbox { m i n i m i z e } } _ { w \\in \\mathbb { R } ^ n } } \\ \\theta ( w ) \\ , \\triangleq \\ , \\underbrace { f ( \\Phi ( w ) ) } _ { \\mbox { d e n o t e d $ \\varphi ( w ) $ } } + \\ , \\displaystyle { \\sum _ { i = 1 } ^ n } \\ , \\alpha _ i \\ , | \\ , w _ i \\ , | \\end{align*}"}
-{"id": "1411.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } W _ 2 ^ 2 ( \\mu _ t , \\nu ) & = \\int _ X \\varphi _ t \\ , d \\mu _ t - \\int _ X \\psi _ t \\ , d \\nu , \\\\ \\varphi _ t ( x ) - \\psi _ t ( y ) & \\le \\frac { d ^ 2 ( x , y ) } { 2 } \\ x , y \\in X . \\end{align*}"}
-{"id": "8920.png", "formula": "\\begin{align*} 0 = & \\sum _ Y \\int _ { \\tilde { \\Delta } _ Y ^ + } - 2 q ( b ) \\big { ( } ( ( n + 1 ) \\Lambda _ Y - \\bar { S } _ { \\Theta } ) P _ { D H } ( q ) + d _ q P _ { D H } ( \\chi ^ { a c } - \\chi ) \\big { ) } d q \\\\ & + \\int _ { \\Delta ^ + } 2 \\sum _ { \\alpha \\in \\Phi _ { Q ^ u } } \\alpha ( b ) P _ { D H } ( q ) d q \\end{align*}"}
-{"id": "5910.png", "formula": "\\begin{align*} L _ i [ \\psi ( \\cdot , \\mu ) ] ( \\nu ) = M _ i [ \\psi ( \\nu , \\cdot ) ] ( \\mu ) , \\qquad \\forall \\ 1 \\leq i \\leq n - 1 , \\end{align*}"}
-{"id": "6459.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow 0 } e ^ { \\frac { \\omega s } { 2 } } s ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { r } ) } \\norm { u _ { j } ( s ) } _ { L ^ r _ { \\sigma } ( \\Omega ) } = 0 , \\end{align*}"}
-{"id": "2725.png", "formula": "\\begin{align*} U ^ { \\vec { i } } K _ 2 ^ { t o p } ( K ) = \\{ \\{ u , x \\} | u , x \\in K ^ \\times , v _ K ( u - 1 ) \\geq \\vec { i } \\} . \\end{align*}"}
-{"id": "2361.png", "formula": "\\begin{align*} J _ 1 ( N ; \\theta ) : = \\int _ 0 ^ { \\infty } \\left [ 1 - \\left ( 1 - e ^ { - ( 1 - \\theta ) t / N } \\right ) ^ N \\right ] d t \\end{align*}"}
-{"id": "2183.png", "formula": "\\begin{align*} d _ n ^ \\infty ( ( A _ 1 , \\dots , A _ n ) , ( B _ 1 , \\dots , B _ n ) ) : = \\max _ { 1 \\le j \\le n } d _ T ( A _ j , B _ j ) . \\end{align*}"}
-{"id": "3443.png", "formula": "\\begin{align*} T f = \\sum _ { R \\in \\mathcal { D } } \\epsilon _ R \\langle f , h _ R \\rangle h _ R , \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{align*} C P ( X , \\Omega _ { X n } , J _ n , \\Lambda _ X ) = ( { T } _ { r } ^ - ) ^ { - 1 } \\circ C M ( X , \\Omega _ { X n } , J _ n , r , \\Lambda _ X ) \\circ { T } _ { r } ^ + + o ( L _ n ) . \\end{align*}"}
-{"id": "5596.png", "formula": "\\begin{align*} \\nabla = \\begin{pmatrix} \\nabla ^ \\Sigma & - B ^ t \\\\ B & \\nabla ^ \\perp \\end{pmatrix} \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{align*} \\lim _ { x \\to 1 - } f ( x ) = - \\infty , \\end{align*}"}
-{"id": "3302.png", "formula": "\\begin{align*} \\left ( \\frac { 2 } { \\pi } \\right ) ^ { n / 2 } \\int _ { C _ W } e ^ { - \\ < x , s \\ > - \\frac { 1 } { 2 } b ^ * M _ x ^ { - 1 } b } \\frac { d x } { \\sqrt { \\det M _ x } } = \\frac { 1 } { \\sqrt { s _ 1 \\ldots s _ n } } e ^ { - ( b _ 1 \\sqrt { s _ 1 } + \\cdots + b _ n \\sqrt { s _ n } ) - \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ n w _ { i j } \\sqrt { s _ i s _ j } } \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} G _ i ( t ) = m _ { G _ i } ( t ) + v _ { G _ i } ( t ) W ( r _ { G _ i } ( t ) ) , \\ i = 1 , 2 \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{align*} \\max _ { j = 0 , \\dots , n - 1 } v _ j ^ 2 a _ j ^ 2 = O \\left ( n ^ { - r / ( 2 \\beta + r ) } \\right ) . \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} g ( x ) = D ( x ) - x D ( 1 ) . \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} \\tilde { \\mathbb { P } } _ L ( \\mathbf { m } , \\mu ; K ) = \\sum _ { i } \\mathbb { P } ^ { ( i ) } ( \\mu , K ) \\prod _ { j = 1 } ^ { L } \\mathbb { P } ( m _ { j } - \\mu , K - i ) , \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{align*} e ^ { \\sum _ { i \\in [ n ] } c _ i x _ i } ( d + \\mathrm { i m } ( \\tilde \\omega ) ) e ^ { - \\sum _ { i \\in [ n ] } c _ i x _ i } & = d + \\sum _ { i \\in [ n ] } x _ i \\cdot d c _ i - \\sum _ { i < j \\in [ n ] , \\alpha \\geq 0 } \\Big ( { \\theta ( p _ { i j } + \\mathrm { a d } x _ i | \\tau ) \\over \\theta ( \\mathrm { a d } x _ i | \\tau ) \\theta ( p _ { i j } | \\tau ) } - { 1 \\over \\mathrm { a d } x _ i } \\Big ) ( t _ { i j } ) \\\\ & = d + A _ { \\mathrm { K Z B } } . \\end{align*}"}
-{"id": "8265.png", "formula": "\\begin{align*} Q ( h ) = \\binom { 2 } { 2 } - 0 = 1 . \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{align*} B _ 0 = \\partial _ 1 A ^ 0 _ 2 - \\partial _ 2 A ^ 0 _ 1 \\ , , B = \\partial _ 1 A _ 2 - \\partial _ 2 A _ 1 \\ , . \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} \\begin{aligned} e ^ { \\frac { \\omega t } { 2 } } t ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\norm { e ^ { - t A } ( a - a ^ { j } ) } _ { L ^ { q } _ { \\sigma } ( \\Omega ) } & \\leq C e ^ { - \\frac { \\omega t } { 2 } } t ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\cdot t ^ { - \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\norm { a - a ^ { j } } _ { L ^ { p } _ { \\sigma } ( \\Omega ) } \\leq C \\norm { a - a ^ { j } } _ { L ^ { p } _ { \\sigma } ( \\Omega ) } , \\end{aligned} \\end{align*}"}
-{"id": "6975.png", "formula": "\\begin{align*} \\begin{pmatrix} \\alpha & \\beta \\\\ \\gamma & \\delta \\end{pmatrix} \\textup { s u c h t h a t } \\alpha + \\beta = a , \\gamma + \\delta = b , \\alpha + \\gamma = c , \\textup { a n d } \\beta + \\delta = d . \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{align*} \\mathcal { I } _ { a } : = \\{ q \\in ( 0 , 1 ) : ( P _ { a , q } ) u \\in \\mathcal { P } ^ { \\circ } \\} , \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} H _ + ( u ' ( r ) ) = G ( \\xi ) - G ( u ( r ) ) , \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{align*} g ( t , q ) = g ( t _ 0 , q _ 0 ) + \\int _ 0 ^ 1 \\partial _ s g ( t _ 0 + s ( t - t _ 0 ) , q _ 0 + s ( q - q _ 0 ) ) ( t - t _ 0 ) + \\nabla _ q g ( t _ 0 + s ( t - t _ 0 ) , q _ 0 + s ( q - q _ 0 ) ) \\cdot ( q - q _ 0 ) d s \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\alpha \\cdot e _ S & = & \\lambda _ S ( \\alpha ) \\cdot e _ S \\\\ \\varphi _ L \\cdot \\alpha & = & \\lambda _ L ( \\alpha ) \\cdot \\varphi _ S \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} E \\left [ S _ N ^ { ( r ) } \\right ] = N ^ r ( \\ln N ) ^ r J ( N ; \\alpha ) , \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} \\{ t \\in [ 0 , 1 ] ; Q ( \\xi , l ( t ) ) = 0 \\xi \\in ( - \\infty , 0 ] \\cup [ 1 , \\infty ) \\cup \\{ l \\} \\} \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} V ^ { 1 - 1 , \\hat { N } _ { \\beta } } = V ^ { 1 - 2 , \\hat { N } _ { \\beta } } : = 0 , V ^ { 1 , \\hat { N } _ { \\beta } } : = \\sum _ { j = 1 } ^ 3 V ^ { 1 - j , \\hat { N } _ { \\beta } } , V ^ { 2 , \\hat { N } _ { \\beta } } : = 0 . \\end{align*}"}
-{"id": "6360.png", "formula": "\\begin{align*} \\operatorname { A s s } _ A ( M ) = \\left \\{ \\mathfrak { q } \\cap A : \\mathfrak { q } \\in \\operatorname { A s s } _ { \\widehat { A } } \\left ( M \\otimes _ A \\widehat { A } \\right ) \\right \\} . \\end{align*}"}
-{"id": "1347.png", "formula": "\\begin{align*} g ( k , i + n + 1 , 0 ) & = ( k - 1 ) ( i + n + 1 ) + 1 , \\\\ g ( k , i + n , 0 ) & = ( k - 1 ) ( i + n ) + 1 , \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{align*} \\mathrm { b a r } = \\sum _ Y \\int _ { \\tilde { \\Delta } _ Y ^ + } q F _ { \\mathcal { L } } ( q ) P _ { D H } ( q ) d q / \\int _ { \\Delta ^ + } P _ { D H } d q . \\end{align*}"}
-{"id": "1509.png", "formula": "\\begin{align*} \\mathcal { L } x _ i - \\dfrac { x _ i } { 2 } = \\left < { \\bf H } _ f , \\bar { \\nabla } x _ i \\right > . \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} F ( X , Y ) = X ^ { p / 2 } Y ^ q X ^ { p / 2 } \\quad { \\rm o r } F ( X , Y ) = Y ^ { q / 2 } X ^ p Y ^ { q / 2 } \\ . \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} \\mathcal A ^ \\bullet ( X ) = \\mathcal A ^ + ( X ) \\oplus \\mathcal A ^ - ( X ) , \\end{align*}"}
-{"id": "625.png", "formula": "\\begin{align*} | \\{ u \\leq M \\} \\cap G ( y , 3 r / 2 ) | & = | T \\bigr ( \\{ \\tilde { u } \\leq M \\} \\cap G ( ( y _ 1 / r , 0 ) , 3 / 2 ) \\bigl ) | \\\\ & = r ^ 3 | \\{ \\tilde { u } \\leq M \\} \\cap G ( ( y _ 1 / r , 0 ) , 3 / 2 ) | \\\\ & \\geq r ^ 3 \\frac { \\nu } { 2 } | G ( ( y _ 1 / r , 0 ) , 3 / 2 ) | \\\\ & = \\frac { \\nu } { 2 } | G ( y , 3 r / 2 ) | . \\end{align*}"}
-{"id": "8525.png", "formula": "\\begin{align*} \\phi ^ { V _ N V _ S } _ n = \\begin{cases} \\phi ^ { V _ N U _ 1 } _ 0 \\ ! + \\dots + \\phi ^ { U _ k V _ S } _ 0 & \\\\ [ 1 p t ] f ^ { V _ N } _ n \\ ! - f ^ { V _ S } _ n & \\rlap { . } \\end{cases} \\end{align*}"}
-{"id": "1531.png", "formula": "\\begin{align*} \\left ( x + y \\right ) ^ { n } = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } \\left ( \\left ( x - q k \\right ) ^ { k } + q k \\left ( x - q k \\right ) ^ { k - 1 } \\right ) \\left ( y + q k \\right ) ^ { n - k } \\end{align*}"}
-{"id": "5807.png", "formula": "\\begin{align*} \\ell _ i ( \\nu , \\nu ' ) = \\left \\{ \\begin{array} { r l } t , & \\nu _ i > \\nu _ { i + 1 } , ( \\nu _ i , \\nu _ { i + 1 } ) = ( \\nu ' _ { i + 1 } , \\nu ' _ i ) , \\nu _ k = \\nu ' _ k \\ \\forall \\ k \\not = i , i + 1 , \\\\ \\\\ 1 , & \\nu _ i < \\nu _ { i + 1 } , ( \\nu _ i , \\nu _ { i + 1 } ) = ( \\nu ' _ { i + 1 } , \\nu ' _ i ) , \\nu _ k = \\nu ' _ k \\ \\forall \\ k \\not = i , i + 1 , \\\\ \\\\ 0 , & , \\end{array} \\right . \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} \\hat { C } : Y '^ 2 = h ^ 7 - 7 h ^ 4 - 8 h \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} \\mathbf { r } ( F ( P ) ) = v \\mathbf { R } ( u ) , \\mathbf { t } ( F ( P ) ) = v \\mathbf { T } ( u ) , \\mathbf { e } ( F ( P ) ) = v \\mathbf { E } ( u ) , \\end{align*}"}
-{"id": "1127.png", "formula": "\\begin{align*} \\mathcal L _ { s \\partial } = \\lambda _ s \\circ \\mathcal L _ \\partial - \\lambda _ { \\partial ( s ) } \\ , . \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{align*} \\| \\widehat { \\rho } _ { k _ { n } } - \\rho \\| _ { \\mathcal { S } ( H ) } ^ { 2 } \\leq \\| \\rho \\| _ { \\mathcal { S } ( H ) } ^ { 2 } - \\sum _ { j = 1 } ^ { \\infty } [ \\rho ( \\phi _ { j } ) ( \\phi _ { j } ) ] ^ { 2 } = \\sum _ { j \\neq k } ^ { \\infty } \\left [ \\frac { D _ { X } ( \\phi _ { j } ) ( \\phi _ { k } ) } { C _ { j } } \\right ] ^ { 2 } < \\infty . \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} V _ t ( x , y , z ) : = \\exp \\left ( \\int _ 0 ^ t \\Lambda ( s , x _ s , y _ s , z _ s ) d s \\right ) \\textrm { f o r a n y } \\ t \\in [ 0 , T ] \\ . \\end{align*}"}
-{"id": "6475.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\varphi ^ { \\prime } \\overline { w } \\ ; \\d x + \\int _ { \\Omega } \\nabla \\varphi \\cdot \\overline { \\nabla w } \\ ; \\d x = 2 \\int _ { \\Omega } d ^ { \\prime } \\cdot \\overline { ( d w ) } \\ ; \\d x + 2 \\sum _ { k , l = 1 } ^ 3 \\int _ { \\Omega } \\partial _ k d _ l \\overline { \\partial _ k ( d _ l w ) } \\ ; \\d x - 2 \\sum _ { k , l = 1 } ^ 3 \\int _ { \\Omega } \\partial _ k d _ l \\overline { ( \\partial _ k d _ l ) w } \\ ; \\d x . \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 6 & 9 & 5 \\\\ 1 0 & 8 & 6 & 6 \\\\ 1 & 8 & 1 & 1 \\\\ 6 & 1 0 & 9 & 1 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 & 8 & 5 \\\\ 1 0 & 1 & 3 & 6 \\\\ 1 & 0 & 7 & 1 \\\\ 6 & 1 & 2 & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "4649.png", "formula": "\\begin{align*} \\sum _ { a = 1 } ^ n \\langle \\omega ^ a \\wedge ( \\nabla _ { V _ a } H ^ { 1 , 0 } ) \\lrcorner \\ , \\phi , \\phi \\rangle & = \\sum _ { a , b = 1 } ^ n g _ Q ( \\nabla _ { V _ a } H ^ { 1 , 0 } , \\bar V _ b ) \\langle V _ b \\lrcorner \\ , \\phi , V _ a \\lrcorner \\ , \\phi \\rangle \\\\ & = \\sum _ { a = 1 } ^ n g _ Q ( \\nabla _ { V _ a } H ^ { 1 , 0 } , \\bar V _ a ) | V _ a \\lrcorner \\ , \\phi | ^ 2 . \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} w _ { 1 } = \\frac { b } { \\Vert b \\Vert } \\ , , \\mbox { a n d } W _ { m + 1 } = [ W _ { m } \\ ; w _ { m + 1 } ] \\in \\mathbb { R } ^ { N \\times ( m + 1 ) } \\ , . \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{align*} q ( x ) : = - 2 x _ 1 + 8 x _ 1 \\sum _ { i = 1 } ^ n x _ i . \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} 0 < \\zeta : = \\zeta ( \\alpha , d , \\eta ) < \\eta + \\eta d - \\eta ^ 2 . \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{align*} d X ^ \\flat = \\left ( 2 a - \\langle X _ 0 , \\xi \\rangle \\right ) \\omega _ 1 + 2 \\ , ( J _ 1 X ^ \\flat ) \\wedge \\eta + \\sigma _ 0 \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} o _ { n } ( 1 ) = \\int _ { \\Omega } f _ { \\ast } ( u _ { n } ) w d x , \\ \\forall w \\in H _ { 0 } ^ { 1 } ( \\Omega ) . \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} { \\cal D } \\vert _ X = D + \\overline D \\ : , \\end{align*}"}
-{"id": "326.png", "formula": "\\begin{align*} \\begin{cases} w = f ' ( z ) , \\\\ z = - g ' ( w ) , \\end{cases} \\end{align*}"}
-{"id": "7629.png", "formula": "\\begin{align*} ( { \\cal T M } ) _ { H e t } = H ^ 1 _ { \\check { \\cal D } } ( Y , { \\cal Q } ) ~ . \\end{align*}"}
-{"id": "8250.png", "formula": "\\begin{align*} \\tilde x _ { N } ( t ) - \\tilde x _ { N ( u ) } ( u ) \\stackrel { ( d ) } { = } x ^ { \\rm s t e p } _ { N - N ( u ) + 1 } ( t - u ) \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} \\int \\limits _ 0 ^ T \\log ( 1 + t - s ) \\Big | _ s ^ T \\ , d s & = - ( 1 + T - s ) \\log ( 1 + T - s ) \\Big | _ 0 ^ T + ( 1 + T - s ) \\Big | _ 0 ^ T \\\\ & = ( 1 + T ) \\log ( 1 + T ) - T \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} s ( n , t ) \\ , = \\ , N _ 1 ( T ) \\cdot \\big ( N _ 2 ( T + 1 ) \\cdot T ! \\big ) ^ M \\cdot N _ 3 ( T + 1 ) ^ { M ( M - 1 ) / 6 } , \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{align*} \\frac { \\lambda ( A \\cap I ) } { \\lambda ( I ) } = \\frac { \\lambda ( A _ i ) } { { \\rm { d i a m } } ( A _ i ) } \\geq \\frac { 1 } { 1 + \\varepsilon } = \\frac { 1 } { 2 - \\delta / \\alpha } \\geq \\frac 1 2 + \\frac { \\delta } { 4 \\alpha } \\geq \\frac { 1 } { 2 } + \\frac { \\delta } { 2 } , \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{align*} ( ( x _ P - x _ Q ) z _ { P Q } - 1 ) ( ( y _ P - y _ Q ) z _ { P Q } - 1 ) = 0 . \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} \\mu ( v , w ) = \\sum _ { i = 1 } ^ n v _ i w _ i a _ i , \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} x = \\int _ S u ( s ) d \\mu ( s ) , \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} \\int _ 1 ^ { \\xi } \\int _ 1 ^ { \\hat { X } } = \\int _ 1 ^ { | \\hat { \\xi } | } \\int _ 1 ^ { | \\hat { X } | } + \\int _ { | \\xi | } ^ { \\xi } \\int _ { 1 } ^ { | \\hat { X } | } + \\int _ { | \\xi | } ^ { \\xi } \\int _ { | \\hat { X } | } ^ { \\hat { X } } = I _ 1 + I _ 2 + I _ 3 \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} \\binom { [ \\frac { k } { 2 } ] + [ \\frac { n - k } { 2 } ] } { [ \\frac { k } { 2 } ] } _ { t ^ 4 } = \\frac { \\prod _ { i = 1 } ^ { [ \\frac { k } { 2 } ] + [ \\frac { n - k } { 2 } ] } ( 1 - t ^ { 4 i } ) } { \\prod _ { i = 1 } ^ { [ \\frac { k } { 2 } ] } ( 1 - t ^ { 4 i } ) \\prod _ { i = 1 } ^ { [ \\frac { n - k } { 2 } ] } ( 1 - t ^ { 4 i } ) } \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{align*} p _ d \\leq \\frac { 1 } { 2 } + \\frac { d _ { \\rm T V } ( p _ 1 , p _ 2 ) } { 2 } \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} d _ b ( p _ 1 , p _ 2 ) = \\| p _ 1 - p _ 2 \\| _ 2 , d _ r ( p _ 1 , p _ 2 ) = \\| p _ 1 - p _ 2 \\| _ { q } . \\end{align*}"}
-{"id": "6749.png", "formula": "\\begin{align*} \\Leftrightarrow [ ( x \\alpha ) \\cdot ( x \\backslash y ) ] \\cdot [ ( z / x ) \\cdot x \\phi ^ { - 1 } ] = ( x \\alpha ) \\cdot \\{ [ x \\backslash ( y z ) ] / x \\cdot x \\phi ^ { - 1 } \\} . \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} P _ A ( z ) = \\frac { ( A + B + C + 1 ) ! } { A ! B ! C ! } & \\int _ { 0 } ^ { 1 } t ^ A \\left ( 1 - t \\right ) ^ B ( z - t ) ^ C d t , \\\\ Q _ A ( z ) = \\frac { ( - 1 ) ^ C ( A + B + C + 1 ) ! } { A ! B ! C ! } & \\int _ { 0 } ^ { 1 } t ^ B \\left ( 1 - t \\right ) ^ C ( 1 - t + z t ) ^ A d t , \\\\ E _ A ( z ) = \\frac { ( A + B + C + 1 ) ! } { A ! B ! C ! } & \\int _ { 0 } ^ { 1 } t ^ A \\left ( 1 - t \\right ) ^ C ( 1 - z t ) ^ B d t . \\end{align*}"}
-{"id": "3554.png", "formula": "\\begin{align*} A \\langle X , T \\rangle + B \\langle X , N \\rangle & = \\langle X , ( A + i B ) Y \\rangle \\\\ & = R e ( X ( A - i B ) e ^ { - i \\theta } ) \\\\ & = x \\\\ & = \\kappa \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} \\langle \\omega ^ * _ { \\overline { f } } , D _ n \\mathrm { e x p } ^ * \\left ( \\mathrm { l o c } _ p c ^ + _ { \\mathbb { Q } ( \\mu _ n ) } \\right ) \\rangle _ { \\mathrm { d R } } = D _ n \\langle \\omega ^ * _ { \\overline { f } } , \\mathrm { e x p } ^ * \\left ( \\mathrm { l o c } _ p c ^ + _ { \\mathbb { Q } ( \\mu _ n ) } \\right ) \\rangle _ { \\mathrm { d R } } \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} \\Delta u + A ( u ) \\left ( \\nabla u , \\nabla u \\right ) = f \\end{align*}"}
-{"id": "973.png", "formula": "\\begin{align*} H ( \\mathfrak { e } , V ) { \\varphi } _ { \\ell } \\ = \\ - \\rho { \\varphi } _ { \\ell } , \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} y _ i ( \\mu ; t ^ { - m } , t ) = y _ i ( \\nu ; t ^ { - m } , t ) , \\forall \\ 1 \\leq i \\leq n \\iff S _ m ( \\mu ) = S _ m ( \\nu ) . \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} \\Delta \\left ( \\frac { X ^ n } { n ! } \\right ) = \\sum _ { k = 0 } ^ n \\frac { X ^ k } { k ! } \\otimes \\frac { X ^ { n - k } } { ( n - k ) ! } \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} \\mathcal { I } _ { 3 , + } ( t , x ) = \\int _ 0 ^ t \\int _ 0 ^ { \\infty } \\int _ y ^ { \\infty } \\frac { \\partial G _ { t - s } } { \\partial x } ( x - z ) \\psi ( s , z ) \\sigma _ s ( y ) ^ 2 d z d y d s \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} d q ^ \\prime _ t = & \\frac { 1 } { m } ( p _ t ^ \\prime + \\psi ( t ^ * , q _ t ^ \\prime ) ) d t , \\\\ d ( p _ t ^ \\prime ) _ i = & - \\frac { 1 } { m } \\delta ^ { j k } ( \\gamma _ { i j } ( t ^ * , q _ t ^ \\prime ) + \\partial _ { q ^ i } \\psi _ j ( t ^ * , q _ t ^ \\prime ) ) ( p _ t ^ \\prime + \\psi ( t ^ * , q _ t ^ \\prime ) ) _ k d t \\\\ & + ( - \\partial _ { q ^ i } V ( t ^ * , q _ t ^ \\prime ) + \\tilde F _ i ( t ^ * , q _ t ^ \\prime , - p _ t ^ \\prime ) ) d t + \\sigma ( t ^ * , q _ t ^ \\prime ) \\circ d W _ t . \\end{align*}"}
-{"id": "1370.png", "formula": "\\begin{align*} \\Pi _ { K } ( a ) = \\Bigl \\{ b \\in K \\ , \\bigl \\vert \\ , \\vert a - b \\vert = \\min _ { c \\in K } \\vert a - c \\vert = d _ { K } ( a ) \\Bigr \\} . \\end{align*}"}
-{"id": "7809.png", "formula": "\\begin{align*} v _ n = \\left ( \\frac { { \\rm r k } B _ n } { n ^ { d ( { \\bf B } ) } / d ( { \\bf B } ) } \\right ) _ { n \\geq 1 } \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} \\kappa ( X ) = g _ Q ( \\sum _ { i = 1 } ^ p \\pi ( \\nabla ^ M _ { E _ i } E _ i ) , \\pi ( X ) ) \\forall X \\in \\Gamma ( T M ) , \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} 1 - q ( i , j ) = 1 - ( 1 - \\Theta ( n ^ { \\alpha _ i + \\alpha _ j - 1 } ) ) ( 1 + O ( w _ { i , j } ) ) = \\Theta ( n ^ { \\alpha _ i + \\alpha _ j - 1 } ) , \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} \\partial _ t u = \\Delta u + u ( a ( x , t ) - b ( x , t ) u ) , \\ x \\in \\Omega \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{align*} & \\sum _ { b = 3 } ^ { n - 2 } \\sum _ { \\ell = 0 } ^ { n - 1 - b } ( C _ b - C _ { b - 1 } - 2 ^ { b - 2 } ) = C _ { n - 2 } - 2 ^ { n - 3 } + \\sum _ { \\ell = 1 } ^ { n - 4 } \\sum _ { b = 3 } ^ { n - 1 - \\ell } ( C _ b - C _ { b - 1 } - 2 ^ { b - 2 } ) \\\\ & = C _ { n - 2 } - 2 ^ { n - 3 } + \\sum _ { \\ell = 1 } ^ { n - 2 } ( C _ { n - 1 - \\ell } - 2 ^ { n - 2 - \\ell } ) = C _ { n - 2 } - 3 \\cdot 2 ^ { n - 3 } + 1 + \\sum _ { \\ell = 1 } ^ { n - 2 } C _ \\ell \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} \\nabla g ( \\overline { x } ) \\Big [ D \\mathcal { N } _ { K } ( g ( \\overline { x } ) | \\lambda ) ( g ' ( \\overline { x } ) d ) - \\frac { 1 } { 2 } \\nabla \\Upsilon ( g ' ( \\overline { x } ) d ) \\Big ] \\ ! = \\mathcal { N } _ { \\mathcal { C } _ { \\Gamma } ( \\overline { x } , \\overline { v } ) } ( d ) \\ ! = \\nabla g ( \\overline { x } ) \\mathcal { N } _ { \\mathcal { C } _ { K } ( g ( \\overline { x } ) , \\lambda ) } ( g ' ( \\overline { x } ) d ) \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} [ L _ m , L _ n ] = ( m - n ) L _ { m + n } + \\frac { 1 } { 1 2 } ( m ^ 3 - m ) \\delta _ { m + n , 0 } C . \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} \\Delta _ f ( u e ^ h ) = e ^ h \\left \\{ \\Delta _ { f - 2 h } u + [ \\Delta h + \\langle \\nabla ( h - f ) , \\nabla h \\rangle ] u \\right \\} , \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{align*} { \\mu } ( A + B ) \\ ; = \\ ; { \\mu } ( A ) + { \\mu } ( B ) + \\rho \\ , < \\ ; \\tfrac { 1 } { 2 } \\big ( 1 + { \\mu } ( A ) + { \\mu } ( B ) \\big ) , \\textrm { a n d } \\rho < { \\mu } ( B ) \\leq { \\mu } ( A ) . \\end{align*}"}
-{"id": "5663.png", "formula": "\\begin{align*} \\sigma ( T P ( T , F ) ) = \\sigma ( P ( T , F ) T P ( T , F ) ) \\subseteq F . \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} E _ { \\tau } ( 1 ) = E _ { \\tau } ( \\alpha ) + E _ { \\tau } ^ { \\infty } ( 1 - \\alpha ) . \\end{align*}"}
-{"id": "4141.png", "formula": "\\begin{align*} \\begin{gathered} \\frac { \\rm { d } \\pi _ f ^ { ( D ) } ( \\textbf { x } ) } { \\rm { d } x _ f ^ { ( D ) } } = \\frac { \\rm { d } { \\left ( \\frac { p _ f B _ f } { N ^ { ( F ) } + N ^ { ( D ) } x _ f ^ { ( D ) } } C _ f ^ { ( D ) } - q _ f \\phi _ f \\right ) } } { \\rm { d } { x _ f ^ { ( D ) } } } = - \\frac { N ^ { ( D ) } p _ f B _ f C _ f ^ { ( D ) } } { \\left ( N ^ { ( F ) } + N ^ { ( D ) } x _ f ^ { ( D ) } \\right ) ^ 2 } - c _ 3 c _ 4 N ^ { ( D ) } \\phi _ f \\exp \\left ( c _ 4 \\left ( N ^ { ( F ) } + N ^ { ( D ) } x _ f ^ { ( D ) } \\right ) \\right ) \\end{gathered} \\end{align*}"}
-{"id": "7454.png", "formula": "\\begin{align*} S ^ { e n v } _ { s , t } = & - \\int _ { s } ^ t \\beta ( r , q _ r ) \\partial _ { p _ k } H ( r , x _ r ) \\circ d ( p _ r ) _ k \\\\ & + \\int _ { s } ^ t \\beta ( r , x _ r ) \\partial _ { p _ k } H ( r , x _ r ) ( - \\nabla _ q H + \\tilde F ) _ k ( r , x _ r ) - \\nabla _ p \\cdot \\tilde F ( r , x _ r ) d r . \\end{align*}"}
-{"id": "262.png", "formula": "\\begin{align*} \\mathrm { T e r m } _ k = \\sum _ { \\stackrel { \\gamma , \\delta \\geq 0 } { \\stackrel { \\gamma + \\delta = \\alpha - 1 } { } } } \\begin{pmatrix} \\alpha \\\\ \\gamma \\end{pmatrix} [ ( \\mathrm { a d } x _ i ) ^ \\gamma ( t _ { i j } ) , ( \\mathrm { a d } x _ i ) ^ \\delta ( t _ { i k } ) ] . \\end{align*}"}
-{"id": "3332.png", "formula": "\\begin{align*} f _ q ( t ) = \\sum _ { ( v , w ) \\in E } \\sum _ { l = 1 } ^ m \\lambda _ l \\tau _ l ( \\widetilde { P } _ { v , l } \\widetilde { P } _ { w , l } ) \\ge \\sum _ { l = 1 } ^ m \\lambda _ l f _ q ( r _ l ) . \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{align*} M _ n = M _ n ( E ) : = \\sup \\frac { \\| p ' \\| _ E } { \\| p \\| _ E } , \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} 1 - 2 \\alpha = a = 1 - 2 s , \\ \\ \\ \\gamma = 1 , \\ \\ \\ \\beta = L ( \\xi , \\sigma ) , \\ \\ \\ \\nu = \\pm \\alpha . \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} N ( r ) = \\frac { I ( r ) } { H ( r ) } \\equiv \\kappa . \\end{align*}"}
-{"id": "6167.png", "formula": "\\begin{align*} | U _ n ( z ) | = \\prod _ { k = 1 } ^ { n } ( q ^ { | \\mu | - 1 } - q ^ { \\mu _ 1 + \\cdots + \\mu _ m + k - 2 } ) = q ^ { - n } | U _ n | . \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} L _ 0 ( y ) = q _ { [ y ] } \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} \\dim D ^ { V I } = 2 7 , r = 3 , a = 8 , b = 0 , p = 2 6 . \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} \\theta ( z | \\tau ) : = { 1 \\over { 2 \\pi \\mathrm { i } } } ( \\mathbf e ( { z \\over 2 } ) - \\mathbf e ( - { z \\over 2 } ) ) \\prod _ { j \\geq 1 } ( 1 - \\mathbf e ( z + j \\tau ) ) \\prod _ { j \\geq 1 } ( 1 - \\mathbf e ( - z + j \\tau ) ) , \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{align*} h _ r = \\det \\left ( e _ { i - j + 1 } \\right ) _ { i , j = 1 , \\dots , r } . \\end{align*}"}
-{"id": "1666.png", "formula": "\\begin{align*} \\lVert A B - A _ 0 B _ 0 \\rVert ^ 2 & = \\sum _ { j = 1 } ^ N \\left ( \\sum _ { i = 1 } ^ { M - 1 } \\{ ( a _ { i 1 } - a _ { i 2 } ) b _ j + a _ { i 2 } - a ^ 0 _ i \\} ^ 2 \\right . \\\\ & \\left . + \\left [ \\sum _ { i = 1 } ^ { M - 1 } \\{ ( a _ { i 1 } - a _ { i 2 } ) b _ j + a _ { i 2 } - a ^ 0 _ i \\} \\right ] ^ 2 \\right ) . \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} W _ Q ^ N = \\begin{pmatrix} Q a & b \\\\ N c & Q d \\end{pmatrix} , \\end{align*}"}
-{"id": "2390.png", "formula": "\\begin{align*} E \\left [ S \\right ] - E \\left [ T _ 1 \\right ] = E \\left [ S - T _ 1 \\right ] = \\frac { 1 } { \\alpha _ 2 } \\ , P \\{ T _ 1 < S \\} . \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} & x = \\frac { 1 } { 2 \\pi r } ( \\alpha + 2 m + \\ell ) , y = \\frac { 1 } { 2 \\pi r } ( \\alpha + 2 m + 2 n - \\ell ) , \\\\ & u = \\frac { 1 } { 2 \\pi r } ( 2 \\beta - \\alpha + 2 n + \\ell ) , v = - \\frac { 1 } { 2 \\pi r } ( \\alpha + \\ell + 2 k ) , \\end{align*}"}
-{"id": "8998.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } H _ n [ \\gamma _ n ^ { - 1 } s ] ( f + h _ n ) ( s , \\mu _ n , w _ n ) = H _ 0 [ s ] f ( \\mu , w ) \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} \\Vert b - A x _ { m , k } \\Vert = \\Vert b - C _ { m } C _ { m } ^ { T } y _ { m , k } \\Vert = \\left \\Vert \\Vert b \\Vert e _ { 1 } - H _ { m } H _ { m } ^ { T } z \\right \\Vert < \\eta \\widehat { \\varepsilon } \\Vert b \\Vert \\ , , \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{align*} \\mu _ t = \\mu + t \\varphi _ D , \\end{align*}"}
-{"id": "9266.png", "formula": "\\begin{align*} c _ { S M } ( C _ { w P } ) = \\sum _ { v \\in W ^ P , v \\leq w } { c ( v ; w ) [ X ^ { w P } ] } \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} C h o i c e ^ m _ i = \\left \\{ \\begin{array} { l l } H _ m , & \\hbox { i f $ i \\neq j $ o r $ m \\neq 0 $ ; } \\\\ \\{ \\{ h _ 1 \\} , \\{ h _ 2 \\} \\} , & \\hbox { i f $ i = j $ a n d $ m = 0 $ . } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} \\xi _ \\infty = \\sum _ { j = 0 } ^ \\infty \\xi _ { \\infty , j } ( z ) t ^ { ( - 1 - 2 j ) / 3 } , \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} \\rho _ n : = d ( n ) ^ { - 1 / 2 } n \\| \\theta _ n \\| _ 2 ^ 2 \\to \\infty . \\end{align*}"}
-{"id": "1724.png", "formula": "\\begin{align*} \\bigg | \\bigg \\{ t : \\bigg \\| \\sum _ { k = k _ 0 } ^ \\infty 2 ^ { - k \\alpha } \\mathcal { C } _ k g \\bigg \\| _ { L ^ 2 _ x } > \\lambda \\bigg \\} \\bigg | \\lesssim \\lambda ^ { - q _ 1 } I _ 1 + \\lambda ^ { - q _ 2 } I _ 2 \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{align*} H ( X | Y ) : = - \\sum p ( x , y ) \\log { p ( x | y ) } \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{align*} \\mathbf { A } \\mathbf { W } ( \\mathbf { z } , \\mathbf { \\phi } ) + \\mathbf { G } ( \\mathbf { W } ( \\mathbf { z } , \\mathbf { \\phi } ) ) + \\varepsilon \\mathbf { G } _ { e x t } ( \\mathbf { \\phi } ) = D _ { \\mathbf { z } } \\mathbf { W } ( \\mathbf { z } , \\mathbf { \\phi } ) \\mathbf { R } ( \\mathbf { z } , \\mathbf { \\phi } ) + D _ { \\mathbf { \\phi } } \\mathbf { W } ( \\mathbf { z } , \\mathbf { \\phi } ) \\Omega . \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} S _ { u , \\mathring r } : = C _ u \\cap \\{ r = \\mathring r \\} \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} \\nabla _ { \\rm t r } ^ * \\nabla _ { \\rm t r } \\pi ( X ) - { \\rm R i c } ^ Q ( X ) + A _ { X } \\kappa _ B ^ \\sharp = 0 , { \\rm d i v } _ \\nabla ( \\pi ( X ) ) = 0 . \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{align*} \\beta _ i ( b , v ) = ( u ' _ i ( b ) , G _ { i , b } ( v ) : = F _ { i , b } ^ { - 1 } \\circ F ' _ { i , b } ( v ) ) \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} \\mathrm { l o c } _ { x } ( g , \\mathcal { S } _ { V } ) = ( - 1 ) ^ { \\left \\langle 2 \\rho , \\nu _ { x } \\right \\rangle } \\mathrm { r a n k } _ { \\Lambda } V [ \\nu _ { x } ] \\end{align*}"}
-{"id": "8154.png", "formula": "\\begin{align*} \\sum \\limits _ { r \\in \\N _ 0 } \\iota _ r ( a _ { [ 1 : n ] } ) = \\sum \\limits _ { r \\in \\N _ 0 } \\iota _ r ( b _ { [ 1 : n ] } ) . \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} C ^ i = C ^ i _ { k _ i } \\frac { d z } { z ^ { k _ i } } + \\dots + C ^ i _ 1 \\frac { d z } { z } + \\dots , \\end{align*}"}
-{"id": "168.png", "formula": "\\begin{align*} t [ \\Phi _ t , \\xi _ \\infty ] = t [ \\Phi _ t - \\Phi _ \\infty , \\xi _ \\infty ] , \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} \\begin{cases} y ^ 2 Y '' ( y ) + a y Y ' ( y ) - L ( \\xi , \\sigma ) ^ 2 y ^ 2 Y ( y ) = 0 , \\ \\ \\ \\ y \\in \\R ^ + , \\\\ Y ( 0 ) = \\hat u ( \\xi , \\sigma ) , \\\\ Y ( y ) \\to 0 , \\ \\ y \\to \\infty , \\end{cases} \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} \\mathcal O _ 1 ( \\omega ) = b _ 1 ( \\omega ) + \\widetilde { \\tau } ( b _ 1 ( \\omega ) ) + \\cdots + \\widetilde { \\tau } ^ { \\ell - 1 } ( b _ 1 ( \\omega ) ) , \\mathcal O _ 2 ( \\omega ) = b _ 2 ( \\omega ) + \\widetilde { \\tau } ( b _ 2 ( \\omega ) ) + \\cdots + \\widetilde { \\tau } ^ { \\ell - 1 } ( b _ 2 ( \\omega ) ) , \\end{align*}"}
-{"id": "2971.png", "formula": "\\begin{align*} P _ n ( q ) = \\sum _ { d \\mid n } ( - 1 ) ^ { n - d } d \\frac { q ^ n - 1 } { q ^ { n / d } - 1 } q ^ { \\binom { n } { 2 } - \\frac { n } { d } \\binom { d } { 2 } } Q _ d ( q ^ { n / d } ) . \\end{align*}"}
-{"id": "2033.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{aligned} \\frac { \\partial } { \\partial \\tau } \\rho ( x , \\tau ) & = - \\frac { \\partial } { \\partial x } J ( x , \\tau ) \\\\ J ( \\xi _ + , \\tau ) & = 0 , \\forall \\tau > 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} [ u _ { n } - u ] ^ { p } _ { W ^ { s , p } ( \\R ^ { N } ) } = [ u _ { n } ] ^ { p } _ { W ^ { s , p } ( \\R ^ { N } ) } - [ u ] ^ { p } _ { W ^ { s , p } ( \\R ^ { N } ) } + o _ { n } ( 1 ) . \\end{align*}"}
-{"id": "8383.png", "formula": "\\begin{align*} \\rho ( A ) : = \\int _ 0 ^ 1 s ^ { - 1 / 2 } \\| B ( s ) \\| _ 2 ^ 2 d s . \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} u ( x ) = \\phi ( \\exp ( x ) ) . \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} T ^ { \\alpha \\beta } \\xi _ { \\alpha } \\xi _ { \\beta } = 0 . \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { m } { { m } \\choose { i } } _ { \\ ! \\ ! q } q ^ { m - i } q ^ { i ( i + 1 ) / 2 } \\prod _ { j = 0 } ^ { i } ( q ^ { j + n } t ^ 2 - t ) \\prod _ { j = 1 } ^ { m - i } ( 1 - q ^ { m + n + 1 - j } t ^ 2 ) . \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\int _ { \\mathbb { R } ^ { 3 } } V ( x ) \\rho _ { n _ k } ( x ) { \\rm d } x = \\int _ { \\mathbb { R } ^ { 3 } } V ( x ) \\rho _ { 0 } ( x ) { \\rm d } x . \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} \\| J q ^ k ( \\xi ^ J ) \\| _ 2 \\le 1 = \\| \\xi ^ J \\| _ 2 . \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} \\mathbb { X } ^ { ( \\omega ) } _ n : = \\prod _ { j = 1 } ^ n \\{ 1 , \\dots , k _ { \\omega _ j } \\} . \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} \\mathcal { L } u + \\lambda _ 1 u = 0 . \\end{align*}"}
-{"id": "6945.png", "formula": "\\begin{align*} \\theta = 2 \\gamma - 1 . \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{align*} G _ m ( x ) & = x ^ m F _ T ( x ) ^ m + \\sum _ { j = 2 } ^ m \\frac { x ^ m } { 1 - x } C ( x ) ^ { j - 2 } \\big ( C ( x ) - 1 \\big ) F _ T ( x ) ^ { m - j + 1 } . \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} t _ \\mathrm { s , 1 } & = \\Delta t _ { \\mathrm { s } _ { 1 , 1 } } + \\Delta t _ { \\mathrm { s } _ { 1 , 2 } } + \\Delta t _ { \\mathrm { s } _ { 1 , 3 } } \\\\ & = g _ \\mathrm { D L } ^ { - 1 } l _ \\mathrm { r } + t _ \\mathrm { w } + g _ \\mathrm { U L } ^ { - 1 } l _ \\mathrm { r } + g _ \\mathrm { D L } ^ { - 1 } l _ \\mathrm { b } . \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} [ u a _ 1 , a _ 2 , a _ 3 ] = u [ a _ 1 , a _ 2 , a _ 3 ] + [ u , a _ 2 ] [ a _ 1 , a _ 3 ] + [ u , a _ 3 ] [ a _ 1 , a _ 2 ] + [ u , a _ 2 , a _ 3 ] a _ 1 . \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{align*} J _ { 1 - r ^ 2 ( 1 - l _ 1 ^ 2 ) / 4 } \\left ( \\frac { d } { 2 } , \\frac { 1 } { 2 } \\right ) = 1 - \\frac { 1 } { D _ { d } } \\int _ { 1 - r ^ 2 ( 1 - l _ 1 ^ 2 ) / 4 } ^ 1 t ^ { \\frac { d } { 2 } - 1 } ( 1 - t ) ^ { - \\frac { 1 } { 2 } } d t . \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} F ( U X U ^ * , U Y U ^ * ) = U F ( X , Y ) U ^ * \\ . \\end{align*}"}
-{"id": "7633.png", "formula": "\\begin{align*} \\omega ( U , V ) = g ( J U , V ) ~ , \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} H = \\frac { 1 } { 6 } \\ , \\tau _ 0 \\ , \\varphi - \\tau _ 1 \\lrcorner \\psi - \\tau _ 3 ~ . \\end{align*}"}
-{"id": "1524.png", "formula": "\\begin{align*} \\int _ { \\Sigma } | A | ^ 2 u ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma & \\leq \\int _ { \\Sigma } ( \\dfrac 1 2 - \\mu _ 1 ) u ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma + \\int _ { \\Sigma } | \\nabla u | ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma \\\\ & = \\int _ { \\Sigma } ( \\dfrac 1 2 - \\mu _ 1 + \\lambda _ 1 ) u ^ 2 e ^ { \\frac { | x | ^ 2 } { 4 } } d \\sigma . \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} \\pi r = \\sigma T , \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} \\alpha _ { 1 3 } = \\frac { \\theta _ { 7 } \\theta _ { 1 4 } } { \\theta _ { 4 2 } \\theta _ { 3 5 } } , \\alpha _ { 2 3 } = \\frac { \\theta _ { 1 6 } \\theta _ { 1 4 } } { \\theta _ { 6 1 } \\theta _ { 3 5 } } , \\alpha _ { 3 3 } = \\frac { \\theta _ { 1 6 } \\theta _ { 7 } } { \\theta _ { 6 1 } \\theta _ { 4 2 } } . \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{align*} \\tilde \\gamma = U d i a g ( \\gamma + B _ 0 i , \\gamma - B _ 0 i , \\gamma ) U ^ * \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} 1 = \\frac { a _ 0 } { x } + \\frac { a _ 1 } { x ^ 2 } + \\frac { a _ 2 } { x ^ 3 } + \\ldots \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{align*} & i \\int _ { [ s , s + 1 ] \\times Y _ + } K ^ 2 d s \\wedge \\pi _ + ^ * d t \\wedge ( K ^ { - 1 } * _ 3 E _ A ) + i \\int _ { [ s , s + 1 ] \\times Y _ + } K ^ 2 d s \\wedge \\pi _ + ^ * d t \\wedge ( F _ { A ( s ) } + \\frac { 1 } { 2 } F _ { A _ { K ^ { - 1 } } } ) \\\\ = & \\int _ { [ s , s + 1 ] \\times Y _ + } r ( 1 - | \\alpha | ^ 2 ) + \\mathfrak { e } , \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} N ( r ) = e ^ { - C \\psi ( r ) } \\overline { N } ( r ) - C \\psi ( r ) , \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} M ^ S = \\{ m \\in M \\ | \\ \\ \\ ( s \\otimes 1 - 1 \\otimes s ) \\cdot m = 0 \\} \\ , . \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{align*} P \\{ T _ 1 < S \\} \\leq P \\{ \\nu _ 1 M < S \\} = ( 1 - \\alpha _ 2 ) ^ { \\nu _ 1 M } = \\alpha _ 1 ^ { \\nu _ 1 M } \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{align*} \\vect { x } _ n = \\alpha _ n \\vect { h } _ n , \\forall n = 1 , \\ldots , N . \\end{align*}"}
-{"id": "2809.png", "formula": "\\begin{align*} C ^ 2 & \\geq 2 - n + \\frac { 1 } { 2 } \\Big ( - K _ X . \\bar { C } - 3 \\sum _ { i = 1 } ^ s m _ { p _ i } ( \\bar { C } ) \\Big ) \\\\ & \\geq 2 - n - \\frac { 1 } { 2 } \\Big ( K _ X . \\bar { C } + 3 \\sum _ { i = 1 } ^ s A . \\bar { C } \\Big ) \\\\ & \\geq 2 - n - \\frac { 1 } { 2 } \\Big ( K _ X . \\bar { C } + 3 n ( A . \\bar { C } ) \\Big ) \\\\ & \\geq 2 - n - \\frac { 1 } { 2 } \\big ( ( K _ X + 3 n A ) . \\bar { C } \\big ) . \\end{align*}"}
-{"id": "7465.png", "formula": "\\begin{align*} B _ 1 ^ { i _ 1 i _ 2 i _ 3 } ( t , q ) = \\frac { 1 } { 2 } \\delta ^ { i _ 1 i _ 2 } ( \\nabla _ q \\beta ) ^ { i _ 3 } ( t , q ) \\end{align*}"}
-{"id": "6267.png", "formula": "\\begin{align*} & k _ i k _ i ^ { - 1 } = k _ i ^ { - 1 } k _ i = 1 , & & k _ 0 k _ 1 = k _ 1 k _ 0 , \\\\ & k _ i e _ i ^ { \\pm } = q ^ { \\pm 2 } e _ i ^ { \\pm } k _ i , & & k _ i e _ j ^ { \\pm } = q ^ { \\mp 2 } e _ j ^ { \\pm } k _ i , i \\neq j , \\\\ & e _ i ^ + e _ i ^ - - e _ i ^ - e _ i ^ + = \\frac { k _ i - k _ i ^ { - 1 } } { q - q ^ { - 1 } } , & & e _ 0 ^ { \\pm } e _ 1 ^ { \\mp } - e _ 1 ^ { \\mp } e _ 0 ^ { \\pm } = 0 , \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{align*} \\mathfrak { H } f ( \\alpha , t ) = \\frac 1 { \\pi i } \\operatorname { p . v . } \\int \\frac { f ( \\beta , t ) z _ \\beta ( \\beta , t ) } { z ( \\alpha , t ) - z ( \\beta , t ) } d \\beta . \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{align*} 0 < \\sum _ l \\lambda _ l / ( 2 \\lambda _ l + \\lambda _ i ) = \\sum _ l \\frac { 1 } { 2 + \\lambda _ i / \\lambda _ l } \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} ( \\hat { w } \\xi ) ( t ) b w B / B = b _ 0 ( \\hat { w } \\xi ) ( t ) b _ + ( \\hat { w } \\xi ) ( t ) ^ { - 1 } w P / P \\underset { t \\to 0 } { \\longrightarrow } b _ 0 w B / B \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} \\min _ { 0 \\leq x \\leq 1 } I _ { \\alpha , s } ( x ) = \\frac { \\pi ^ { n / 2 } \\Gamma ( \\alpha ) } { \\Gamma ( \\frac { n } { 2 } + \\alpha ) } . \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} F ' ( y ) = \\frac { \\cos ( y / 2 ) ( - 2 \\sin ( y / 2 ) + 1 - \\sqrt { 4 \\sin ^ { 2 } ( y / 2 ) - 1 2 \\sin ( y / 2 ) + 1 } } { 4 \\sin ( y / 2 ) \\sqrt { 4 \\sin ^ { 2 } ( y / 2 ) - 1 2 \\sin ( y / 2 ) + 1 } ) } \\ , > 0 \\end{align*}"}
-{"id": "3526.png", "formula": "\\begin{align*} \\kappa _ { t } = \\kappa _ { s s } + \\kappa ^ { 3 } . \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{align*} \\mathcal { N } _ { u } ( q , u ) h = q a ( x ) u ^ { q - 1 } h , \\quad \\quad \\mathcal { F } _ { u } ( q , u ) h = - \\Delta h - q a ( x ) u ^ { q - 1 } h . \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} e ^ { - \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ p M _ t ^ { \\beta ^ j } } & = 1 - \\frac 1 2 \\int _ 0 ^ t \\sum _ { i = 1 } ^ p e ^ { - \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ p M _ s ^ { \\beta ^ j } } d M _ s ^ { \\beta ^ i } + \\int _ 0 ^ t \\sum _ { i , j = 1 } ^ p \\frac { 1 } { 8 } e ^ { - \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ p M _ s ^ { \\beta ^ k } } d \\langle M ^ { \\beta ^ i } , M ^ { \\beta ^ j } \\rangle _ s . \\end{align*}"}
-{"id": "5208.png", "formula": "\\begin{align*} u ( x , t ) = & e ^ { t ( \\Delta - I ) } u _ 0 - \\chi \\int _ { 0 } ^ { t } e ^ { ( t - s ) ( \\Delta - I ) } \\nabla \\cdot ( u ( \\cdot , s ) \\nabla v ( \\cdot , s ) ) d s \\\\ & + \\int _ 0 ^ t e ^ { ( t - s ) ( \\Delta - I ) } ( ( 1 + a ^ * ( \\cdot , s ) ) u - b ^ * ( \\cdot , s ) u ^ 2 ( \\cdot , s ) ) ) d s . \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} \\Phi _ \\alpha ( 0 , h ) & = \\Phi _ { u , \\gamma } ( 0 , h , 0 ) + \\frac { 1 } { 2 } \\bigl ( \\xi ' ( 1 ) - \\xi ' ( u ) \\bigr ) \\end{align*}"}
-{"id": "6761.png", "formula": "\\begin{align*} R _ { x ^ \\lambda \\cdot x \\phi ^ { - 1 } } = R ^ { - 1 } _ { x } R _ { x \\phi ^ { - 1 } } \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} \\Pr [ A _ { 3 } | A _ { 1 } \\cap A _ { 2 } ] = \\frac { \\Pr [ A _ 3 \\cap A _ 2 | A _ 1 ] } { \\Pr [ A _ 2 | A _ 1 ] } . \\end{align*}"}
-{"id": "6503.png", "formula": "\\begin{align*} \\pi _ p ( c _ 1 , \\ldots , c _ k ) = \\max \\left [ \\ ; \\left \\lceil \\frac { k } { p } \\right \\rceil \\frac { 1 } { k } \\sum _ { j = 1 } ^ k c _ j , \\ ; \\max _ { j = 1 , \\ldots , k } c _ j \\ ; \\right ] . \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} \\mathcal { P } _ u ' ( \\lambda , \\nu ) = \\Phi _ { u , \\nu } ( 0 , 0 , \\lambda ) - \\lambda u - \\frac { 1 } { 2 } \\int _ 0 ^ u \\xi '' ( s ) s \\nu ( d s ) , \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} & \\left | E \\left [ \\int _ s ^ t \\partial _ r ( \\beta V ) ( r , q _ r ^ m ) d r \\right ] - E \\left [ \\int _ s ^ t \\partial _ r ( \\beta V ) ( r , q _ r ) d r \\right ] \\right | \\\\ \\leq & T \\sup _ { r \\in [ 0 , T ] } E \\left [ | \\partial _ r ( \\beta V ) ( r , q _ r ^ m ) - \\partial _ r ( \\beta V ) ( r , q _ r ) | \\right ] d r \\\\ \\leq & E [ | \\tilde C ( 1 + \\| q _ t \\| ^ { \\tilde p } + \\| q _ t - q _ t ^ m \\| ^ { \\tilde p } ) \\| q _ t - q _ t ^ m \\| | ] \\\\ = O ( m ^ { 1 / 2 } ) \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} \\Big ( \\sum _ { r = D + 1 } ^ { \\infty } \\frac { 1 } { r ^ 4 } \\Big ) \\Big ( \\sum _ { r = D + 1 } ^ { \\infty } \\frac { 1 } { r ^ 2 } \\Big ) ^ { - 1 } \\leq \\frac { D + 1 } { 3 D ^ 3 } \\leq \\frac { 2 } { 3 D ^ 2 } \\end{align*}"}
-{"id": "3836.png", "formula": "\\begin{align*} \\ddot { u } = - \\dot { \\xi _ 0 } \\geq \\tfrac 1 4 u , u ( 0 ) \\geq \\epsilon ^ { - 1 } , \\dot { u } ( 0 ) \\geq 0 . \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} \\eta _ A ^ \\prime ( v _ i \\otimes w _ j ) = \\eta _ A ( f ^ { - 1 } ( v _ i ) \\otimes f ^ \\vee ( w _ j ) ) = \\eta _ A ( \\sum _ r f _ { r i } v _ r \\otimes \\sum _ s g _ { s j } w _ s ) = \\prod _ { r , s } \\eta _ A ( v _ r \\otimes w _ s ) ^ { f _ { r i } g _ { s j } } \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c c } A _ k X _ k B _ k - C _ k X _ { k + 1 } D _ k & = & E _ k , & k = 1 , \\ldots , r - 1 , \\\\ A _ r X _ r B _ r - C _ r X _ 1 ^ s D _ r & = & E _ r , \\end{array} \\right . \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} u ^ \\mu f _ 1 & = u ^ { \\frac { e } { 2 } } f _ 1 ^ p \\\\ u ^ \\mu f _ 2 & = - u ^ { \\left ( \\frac { p - 1 } { p } \\right ) \\frac { e } { 2 } } f _ 1 ^ p + f _ 1 ^ p + u ^ { \\frac { e } { 2 } } f _ 2 ^ p . \\end{align*}"}
-{"id": "6054.png", "formula": "\\begin{align*} v _ { t } ( x , t ) = & w _ { t } ( x , t ) - \\int _ 0 ^ L \\eta ( y , t ) [ k _ { y } ( x , y ) + k _ { y y y } ( x , y ) ] d y - \\int _ 0 ^ L w ( x , y ) [ s _ { y } ( x , y ) + s _ { y y y } ( x , y ) ] d y \\\\ & - \\int _ 0 ^ L s _ { y } ( x , y ) \\eta ( y , t ) w ( y , t ) d y - \\frac 1 2 \\int _ 0 ^ L k _ { y } ( x , y ) w ^ 2 ( y , t ) d y , \\\\ v _ { x } ( x , t ) = & w _ { x } ( x , t ) - \\int _ 0 ^ L k _ x ( x , y ) w ( y , t ) d y - \\int _ 0 ^ L s _ x ( x , y ) \\eta ( y , t ) d y , \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} M _ { i j } \\mathcal X _ { i j } = \\mathcal { E } _ { i j } - \\mathcal { F } _ { i j } , \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} \\omega _ S = p x d y \\wedge d z + q y d z \\wedge d x + r z d x \\wedge d y \\ , . \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{align*} T _ s = \\theta _ s U . \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} x _ { N ( \\tilde t ) - 1 } ( \\tilde t ) - 1 = x _ { N ( \\tilde t ) } ( \\tilde t ) \\leq \\tilde x _ { N ( \\tilde t ) } ( \\tilde t ) < \\tilde x _ { N ( \\tilde t ) - 1 } ( \\tilde t ) = x _ { N ( \\tilde t ) - 1 } ( \\tilde t ) , \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{align*} p \\cdot C _ p = & c \\cdot d \\cdot ( c - \\sigma ^ { - 1 } _ c ) \\cdot ( d - \\sigma ^ { - 1 } _ d ) \\cdot \\left ( \\prod _ { q \\mid N _ { \\mathrm { s p } } } ( 1 - q ^ { - 1 } \\sigma ^ { - 1 } _ q ) \\right ) \\cdot \\left ( \\prod _ { q \\mid N _ { \\mathrm { n s } } } ( 1 + q ^ { - 1 } \\sigma ^ { - 1 } _ q ) \\right ) \\\\ & \\cdot \\left ( p - a _ p ( f ) \\cdot \\sigma ^ { - 1 } _ p + \\psi ( p ) \\cdot \\sigma ^ { - 2 } _ p \\right ) . \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{align*} \\rho _ 0 = ( r - t ) ^ { - 1 } , \\rho _ I = ( r - t ) / r ; \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} \\lambda \\cdot ( x _ M ^ n \\times 1 ) + \\lambda \\nu \\cdot ( 1 \\times \\omega _ { N } ) = ( y _ M ^ n \\times 1 ) + \\nu \\cdot ( \\alpha ^ n _ M \\times 1 ) \\pm \\nu \\cdot ( 1 \\times \\omega _ { N } ) , \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} N _ { \\Phi , K , L \\otimes E } : K ^ \\times & \\to ( K \\otimes E ) ^ \\times \\to ( L \\otimes E ) ^ \\times = \\prod _ i E _ i ^ \\times \\\\ x & \\mapsto \\mathrm { d e t } _ { L \\otimes E } ( x | _ { V _ { \\Phi , K , L \\otimes E } } ) : = \\prod _ i \\mathrm { d e t } _ { E _ i } ( x | _ { V _ { \\Phi , K , E _ i } } ) , \\end{align*}"}
-{"id": "702.png", "formula": "\\begin{align*} \\sum _ { \\substack { s \\geq i , t \\geq j \\\\ ( s , t ) \\neq ( i , j ) } } ( A _ k ) _ { i s } ( X _ k ) _ { s t } ( B _ k ) _ { t j } & = \\sum _ { t > j } ( A _ k ) _ { i i } ( X _ k ) _ { i t } ( B _ k ) _ { t j } + \\sum _ { \\substack { s > i , t \\geq j } } ( A _ k ) _ { i s } ( X _ k ) _ { s t } ( B _ k ) _ { t j } . \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} \\eta _ { \\tilde { A } _ 1 } ( \\delta ( v ) , w ) = \\eta _ { \\tilde { A } _ 2 } ( v , \\delta ^ \\vee ( w ) ) . \\end{align*}"}
-{"id": "8352.png", "formula": "\\begin{align*} | I _ 3 | & \\leq \\sum _ { i = d + 1 } ^ { 2 ( d + 1 ) } | ( x , t ) | ^ i \\sum _ { u = 1 } ^ M C \\gamma ( 2 ^ u | ( x , t ) | ) ^ { d + \\alpha - i } , \\\\ & \\leq C \\gamma | ( x , t ) | ^ { d + \\alpha } \\sum _ { i = d + 1 } ^ { 2 ( d + 1 ) } \\sum _ { u = 1 } ^ M \\frac { 1 } { ( 2 ^ { i - d - \\alpha } ) ^ u } , \\\\ & \\leq C \\gamma | ( x , t ) | ^ { d + \\alpha } , \\end{align*}"}
-{"id": "2653.png", "formula": "\\begin{align*} \\varphi ( q _ 1 ) \\mu _ { 1 1 } ( p ) + h ( p ) \\mu _ { 2 U } ( q _ 1 ) = \\rho _ { 1 1 } ( p ) + \\rho _ { 2 U } ( q _ 1 ) , \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} \\eta ( \\Phi ( r , x ) ) = - \\dfrac { \\theta ' ( r , x ) } { \\theta ( r , x ) } \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} & v _ L ( x ) : = \\int _ 0 ^ { 2 \\pi } d \\xi \\lim _ { h \\to \\infty \\atop h \\in \\frac { 2 } { \\beta } \\N } \\int e ^ { - V ( \\psi ) + W ( \\psi ) } d \\mu _ { C ( x e ^ { i \\xi } ) } ( \\psi ) = \\int _ 0 ^ { 2 \\pi } d \\xi g _ L ( x \\cos \\xi , x \\sin \\xi ) , \\\\ & u _ { 2 , L } ( x ) : = \\int _ 0 ^ { 2 \\pi } d \\xi \\lim _ { h \\to \\infty \\atop h \\in \\frac { 2 } { \\beta } \\N } \\int e ^ { - V ( \\psi ) + W ( \\psi ) } A ^ 2 ( \\psi ) d \\mu _ { C ( x e ^ { i \\xi } ) } ( \\psi ) . \\end{align*}"}
-{"id": "3260.png", "formula": "\\begin{align*} \\left | \\left ( ( z - \\lambda _ j ) ^ { \\tau _ j } F _ \\alpha ( z ) \\Phi ^ { n - k } ( z ) \\right ) ^ { ( l ) } _ { z = \\lambda _ j } \\right | \\leq c _ 7 , \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{align*} P \\{ X \\leq t \\} = \\prod _ { j = 1 } ^ N \\left ( 1 - e ^ { - q _ j t } \\right ) , t \\geq 0 . \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} L ( G , X _ G ) _ { u v } = \\begin{cases} x _ u & u = v , \\\\ - m _ { u v } & , \\end{cases} \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} 0 = \\sum _ l A ( e ^ { i _ 1 } _ { 0 } , . . . , C e ^ { i _ l } _ { 0 } , . . . , e ^ { i _ k } _ 0 ) = \\left ( \\sum _ l \\lambda _ { i _ l } \\right ) A ( e ^ { i _ 1 } _ 0 , . . . , e ^ { i _ k } _ 0 ) . \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} m ( c , \\gamma _ k ) & \\leq \\frac { I ( \\psi _ k , c , \\gamma _ k ) } { K ( \\psi _ k ) ^ { \\frac { 2 } { 3 } } } \\\\ & = \\frac { I ( \\psi _ k , c , 0 ) + \\gamma _ k \\int | \\partial _ x ^ { - 1 } \\psi _ k | ^ 2 d x } { K ( \\psi _ k ) ^ { \\frac { 2 } { 3 } } } \\\\ & \\leq \\frac { I ( \\psi _ k , c , 0 ) + 1 / k } { K ( \\psi _ k ) ^ { \\frac { 2 } { 3 } } } . \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} \\log \\| j _ p ^ m s \\| _ r = T _ { \\overline { L } , p } ( r ) + \\int \\log \\| s \\| \\pi _ { r , p } - \\int g _ { D _ r , p } \\delta _ { \\div ( s ) - m [ p ] } \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{align*} \\Phi _ { q t } ( 1 , t ) = \\int _ { \\Omega } a \\left ( x \\right ) [ ( \\phi _ { 1 } + w _ { t } ( 1 , t ) ) \\log ( t \\phi _ { 1 } ) + ( \\phi _ { 1 } + w _ { t } ( 1 , t ) ) ] \\phi _ { 1 } . \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{align*} u ^ { k + 1 } = P _ { f } u ^ { k } \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} T _ { d \\mu _ R } ( r ) = \\frac { \\pi } { 2 } \\log \\frac { 1 } { 1 + \\frac { r } { R } } + \\frac { \\pi } { 2 } \\log \\frac { 1 } { 1 - \\frac { r } { R } } \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} q ( t , x ) = q ( t , x ; r , \\alpha _ 0 , \\beta _ 0 , \\kappa ) = \\prod _ { j = 1 } ^ 5 \\frac { \\vartheta _ 1 ( \\zeta _ j / \\pi ; i \\kappa ) } { \\vartheta _ 1 ( \\eta _ j / \\pi ; i \\kappa ) } , \\end{align*}"}
-{"id": "7982.png", "formula": "\\begin{align*} \\begin{cases} & \\overline { B } _ r \\\\ & \\phi _ 1 \\leq u \\leq \\phi _ 2 \\overline { B } _ r \\\\ & u = g \\partial B _ r \\\\ & F ( D ^ 2 u ) \\leq 0 \\{ u < \\phi _ 2 \\} \\cap B _ r \\\\ & F ( D ^ 2 u ) \\geq 0 \\{ u > \\phi _ 1 \\} \\cap B _ r \\end{cases} \\end{align*}"}
-{"id": "9268.png", "formula": "\\begin{align*} c ( T X ) = \\pi _ 1 ^ { * } c ( T ( G _ 1 / P _ 1 ) ) \\cdots \\pi _ { i } ^ { * } c ( T ( G _ i / P _ i ) ) \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} A = ( A \\# _ { s } B ) \\# _ { - s / ( 1 - s ) } B \\ , \\end{align*}"}
-{"id": "9256.png", "formula": "\\begin{align*} \\int d x \\ , \\log \\sin x & = - x \\log 2 - \\frac { 1 } { 2 } \\sum _ { n = 1 } ^ { \\infty } \\frac { \\sin ( 2 n x ) } { n ^ 2 } , 0 \\leq x < \\pi , \\\\ \\int d x \\ , \\log \\cos x & = - x \\log 2 + \\frac { 1 } { 2 } \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n - 1 } \\frac { \\sin ( 2 n x ) } { n ^ 2 } , | x | < \\frac { \\pi } { 2 } . \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{align*} n \\cdot s = s \\cdot n \\ \\ n \\cdot \\alpha = - \\alpha \\cdot n \\ , , \\end{align*}"}
-{"id": "7759.png", "formula": "\\begin{align*} [ f ] _ { s , \\rho } : = \\left ( \\int _ { \\R / ( L \\Z ) } \\int _ { - L / 2 } ^ { L / 2 } \\frac { | f ( u + w ) - f ( u ) | ^ \\rho } { | w | ^ { 1 + \\rho s } } \\ , d w d u \\right ) ^ { 1 / \\rho } \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} \\begin{aligned} & \\lim _ { s \\rightarrow 0 } e ^ { \\frac { \\omega s } { 2 } } s ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\norm { u _ { i } ( s ) } _ { L ^ q _ { \\sigma } ( \\Omega ) } = 0 , \\\\ & \\lim _ { s \\rightarrow 0 } e ^ { \\frac { \\omega s } { 2 } } s ^ { \\frac { 3 } { 2 } ( \\frac { 1 } { p } - \\frac { 1 } { q } ) } \\norm { \\nabla d _ { i } ( s ) } _ { L ^ q ( \\Omega ) ^ { 3 \\times 3 } } = 0 , \\ i = 1 , 2 , \\end{aligned} \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} X \\# _ { ( 1 - t ) t _ 0 + t t _ 1 } Y = ( X \\# _ { t _ 0 } Y ) \\# _ t ( X \\# _ { t _ 1 } Y ) \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} B \\ni [ \\dots [ [ y , \\underbrace { x ] , x ] , \\dots , x } _ n ] = x ^ n y + \\sum _ { i = 1 } ^ n ( - 1 ) ^ { n - i } \\binom { n } { i } x ^ { n - i } y x ^ { i } , \\end{align*}"}
-{"id": "1979.png", "formula": "\\begin{align*} B _ { p } ^ { m } & = \\frac { \\Gamma ( \\frac { n } { 2 } ) \\Gamma ( m + 1 ) \\Gamma ( m + \\frac { n } { 2 } + \\alpha ) \\Gamma ( \\frac { n } { p } + \\alpha ) } { 2 \\pi ^ { n / 2 } \\Gamma ( m + \\frac { n } { p } + \\alpha + \\frac { n } { 2 } ) } \\\\ & \\times \\left ( \\frac { n ( n + m - 1 ) \\Gamma ( \\frac { n } { 2 } ) \\Gamma ( p m - n + 1 ) } { 2 \\Gamma ( p m - \\frac { n } { 2 } + 1 ) } \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{align*} J _ { 1 - r ^ 2 ( 1 - l _ 1 ^ 2 ) / 4 } \\left ( \\frac { d } { 2 } , \\frac { 1 } { 2 } \\right ) = \\frac { \\int _ 0 ^ { 1 - r ^ 2 ( 1 - l _ 1 ^ 2 ) / 4 } t ^ { \\frac { d } { 2 } - 1 } ( 1 - t ) ^ { \\frac { 1 } { 2 } - 1 } d t } { \\int _ 0 ^ { 1 } t ^ { \\frac { d } { 2 } - 1 } ( 1 - t ) ^ { \\frac { 1 } { 2 } - 1 } d t } . \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} \\delta _ \\theta = \\frac { 1 } { n } \\sum _ { j = 1 } ^ n \\delta _ { \\theta _ j } , \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{gather*} \\mathbb { E } [ G _ n ] = \\frac { \\mu } { n } + o \\left ( \\frac { 1 } { n } \\right ) , \\\\ F _ n = n S _ n + \\mu \\log n + O _ p ( \\log \\log n ) . \\end{gather*}"}
-{"id": "5870.png", "formula": "\\begin{align*} S _ m ( \\mu ^ { + } ) = ( m \\mu ^ { + } _ 1 , \\dots , m \\mu ^ { + } _ n ) + ( 1 , 2 , \\dots , n ) , \\end{align*}"}
-{"id": "2521.png", "formula": "\\begin{align*} A ^ { k } = \\prod ^ { k } _ { l = 1 } B _ { l } \\prod ^ { k } _ { l = 1 } G _ { k + 1 - l } . \\end{align*}"}
-{"id": "6412.png", "formula": "\\begin{align*} L ^ q _ { \\sigma } ( \\Omega ) : = \\overline { C _ { c , \\sigma } ^ { \\infty } ( \\Omega ) } ^ { L ^ q } W ^ { 1 , q } _ { 0 , \\sigma } ( \\Omega ) : = \\overline { C _ { c , \\sigma } ^ { \\infty } ( \\Omega ) } ^ { W ^ { 1 , q } } \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} \\dim D ^ I _ { m , n } = m n , r = m , a = 2 , b = n - m , p = n + m . \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{align*} \\iota ( \\delta p , \\delta x ) = ( - \\delta p + A [ \\delta x , \\cdot ] , \\delta x ) , \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} z ( \\lambda , 0 ) = \\omega _ 2 / 2 , ~ ~ z ( \\lambda , 1 ) = \\omega _ 1 / 2 . \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} \\mathcal { I } _ 1 = \\{ i : v _ i \\} , \\mathcal { I } _ 2 = \\{ i : d _ i < \\frac { \\sqrt { n } } { \\log n } \\} , \\end{align*}"}
-{"id": "8811.png", "formula": "\\begin{align*} \\Omega _ { j _ 1 , \\bar { j _ 2 } } = \\frac { 1 } { 4 } d ^ 2 u ( l _ { j _ 1 } , l _ { j _ 2 } ) , \\Omega _ { j , \\bar { \\beta } } = \\beta ( l _ j ) ( 1 - \\tanh ^ 2 ( \\beta ) ) \\chi ( \\theta ( \\mu _ { \\beta } ) ) , \\end{align*}"}
-{"id": "2447.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ j } \\frac { \\psi _ j ( n _ j ; \\lambda _ j ) } { \\lambda _ j } \\ , d \\lambda _ j = \\frac { 1 } { 2 \\pi i } \\int _ { \\Gamma _ j } \\left ( \\prod _ { k = 1 } ^ { n _ j } \\frac { k p _ j } { \\lambda _ j + k p _ j } \\right ) \\frac { d \\lambda _ j } { \\lambda _ j } = - 1 , \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} \\chi _ 2 ( N ) = O \\left ( e ^ { - \\varepsilon N } \\right ) , N \\to \\infty , \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{align*} u _ 1 = X _ { \\tau ^ 1 } ^ { ( a _ 1 ) } \\dots X _ { \\tau ^ { o } } ^ { ( a _ o ) } , \\ u _ 2 = X _ { \\mu ^ 1 } ^ { ( b _ 1 ) } \\dots X _ { \\mu ^ { p } } ^ { ( b _ p ) } , \\end{align*}"}
-{"id": "5645.png", "formula": "\\begin{align*} H _ n \\ , = \\ , \\begin{pmatrix} 1 & \\ldots & 1 \\\\ & B _ n & \\end{pmatrix} , \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} \\phi \\left ( z d _ 1 ^ { m - 1 } \\frac { x _ 1 ^ m - y _ 1 ^ m } { x _ 1 - y _ 1 } \\right ) = z d _ 1 ^ { n - 1 } \\frac { x _ 1 ^ n - y _ 1 ^ n } { x _ 1 - y _ 1 } . \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - \\nu } { { n - i } \\choose { \\nu } } A _ i = q ^ { \\frac { n } { 2 } - \\nu } \\sum _ { i = 0 } ^ \\nu { { n - i } \\choose { n - \\nu } } A _ i \\qquad ( \\nu = 0 , 1 , \\cdots , n ) . \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} \\chi ( z ) = \\sum _ { j = 0 } ^ { \\lfloor ( k - 1 ) / 2 \\rfloor } A _ j ( z , . . . , z ) \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} | h _ { i _ k } ( \\alpha ) | > | \\sum _ { l \\leq j < k } h _ { i _ j } ( \\alpha ) | , ~ { \\rm f o r ~ a n y } ~ \\alpha \\in { \\Delta } _ { i _ k } , ~ k = 0 , \\ldots , M , ~ l < k . \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} u = ( u _ 0 , u _ 1 , u _ 2 , \\dots , u _ n ) . \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} [ c , z _ 1 ] [ z _ 2 , z _ 3 , z _ 4 ] = - [ c , z _ 4 ] [ z _ 2 , z _ 3 , z _ 1 ] . \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{align*} \\lambda \\int _ { \\mathbb R } \\operatorname { s g n } ( t ) | f ( t ) | ^ 2 \\ , \\mathrm d t = \\lim _ { x \\to - \\infty } \\lambda U ( x ) = \\lim _ { x \\to - \\infty } V ( x ) = \\int _ { \\mathbb R } | f ' ( t ) | ^ 2 + q ( t ) | f ( t ) | ^ 2 \\ , \\mathrm d t . \\end{align*}"}
-{"id": "6690.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ { \\infty } k ^ { 2 ( \\frac 1 { p _ 1 } + \\frac 1 { p _ 2 } ) } s _ { 3 k - 2 } ( T ) ^ 2 \\leq & \\sum \\limits _ { k = 1 } ^ { \\infty } k ^ { \\frac 2 { p _ 2 } } s _ { k } ( E _ 2 ^ { - 1 } ) ^ 2 s _ { k } ( A ) ^ 2 k ^ { \\frac 2 { p _ 1 } } s _ { k } ( E _ 1 ^ { - 1 } ) ^ 2 \\\\ \\leq & ( C _ 1 C _ 2 ) ^ 2 \\sum \\limits _ { k = 1 } ^ { \\infty } s _ { k } ( A ) ^ 2 < \\infty . \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} w ( v ) = \\langle w , v \\rangle _ { W ^ { - 1 , p } _ { \\sigma } , W ^ { 1 , p ^ { \\prime } } _ { 0 , \\sigma } } , w \\in W ^ { - 1 , p } _ { \\sigma } ( \\Omega ) , \\ , v \\in W ^ { 1 , p ^ { \\prime } } _ { 0 , \\sigma } ( \\Omega ) . \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} \\alpha = ( x _ M ^ k \\times 1 ) + ( x _ M ^ { k - n } \\times \\omega _ { N } ) \\in H ^ k ( M ; \\R ) \\oplus ( H ^ { k - n } ( M ; \\R ) \\otimes H ^ n ( N ; \\R ) ) . \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} D _ 2 = B ( v _ m ) & + \\sum _ { i = 1 } ^ n \\Big [ \\big ( D ( w _ i ) + A \\big ) ( w _ i ) + \\sum _ { v _ j \\prec w _ i } \\sum _ { p \\in ( w _ i , v _ j ) ^ { \\circ } } \\ , D ( p ) \\ , ( p ) \\Big ] \\\\ & + \\sum _ { i = 1 } ^ { m - 1 } \\Big [ D ( v _ i ) ( v _ i ) + \\sum _ { w _ j \\prec v _ i } \\sum _ { p \\in ( v _ i , w _ j ) ^ { \\circ } } \\ , D ( p ) \\ , ( p ) \\Big ] . \\end{align*}"}
-{"id": "6175.png", "formula": "\\begin{align*} S _ \\mu ( m ) = \\lbrace s \\in S _ \\mu \\mid s \\le m \\rbrace , & & T _ \\mu ( m ) = \\lbrace t \\in T _ \\mu \\mid t \\ge m \\rbrace , \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{align*} c _ 1 ( M ) \\cup [ \\omega _ 0 ] = 0 \\in H ^ 4 ( B ) . \\end{align*}"}
-{"id": "2793.png", "formula": "\\begin{align*} \\sigma ' = \\sigma + \\tau \\wedge d s , \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( E _ k \\ , , \\ , \\lim _ { k \\rightarrow \\infty } \\frac 1 { r _ k } \\sum _ { i = 1 } ^ { r _ k } 1 _ { \\{ Y _ i ( t ) > f _ m \\} } = p ( t ) \\right ) = \\mathbb { P } \\left ( E _ k \\right ) . \\end{align*}"}
-{"id": "2975.png", "formula": "\\begin{align*} k ( q , z ) = \\sum _ { d \\geq 1 } \\frac { ( - 1 ) ^ { d - 1 } Q _ d ( q ) } { q ^ { \\binom { d } { 2 } } ( q - 1 ) } z ^ n . \\end{align*}"}
-{"id": "6026.png", "formula": "\\begin{align*} \\frac { d E } { d t } = - \\alpha _ { 2 } \\left \\vert \\eta ( L , t ) \\right \\vert ^ { 2 } - \\alpha _ { 1 } \\left \\vert \\eta _ { x } ( L , t ) \\right \\vert ^ { 2 } - \\alpha _ { 0 } \\left \\vert \\eta _ { x } ( 0 , t ) \\right \\vert ^ { 2 } - \\frac { 1 } { 3 } w ^ { 3 } ( L , t ) - \\int _ { 0 } ^ { L } ( \\eta w ) _ { x } \\eta d x , \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} \\begin{gathered} E _ { \\sqrt { c } } - F _ { \\sqrt { c } } = \\frac { 1 - c } { 2 \\sqrt { c } } H - \\frac { 1 + c } { 2 \\sqrt { c } } H _ { \\sqrt { c } } , \\\\ E _ { \\sqrt { c } } + F _ { \\sqrt { c } } = \\frac { 1 - c } { 4 \\sqrt { c } } [ H , H _ { \\sqrt { c } } ] , \\end{gathered} \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{align*} f _ 1 = s ^ { - 1 } Z f _ 0 , f _ { - 1 } = s ^ { - 1 } \\bar Z f _ 0 , \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} E \\left [ \\left ( T - T _ 1 \\right ) ^ s \\right ] = E \\left [ \\left ( T _ 2 - T _ 1 \\right ) ^ s \\ , | \\ , T _ 1 < T _ 2 \\right ] P \\{ T _ 1 < T _ 2 \\} \\leq E \\left [ T _ 2 ^ s \\right ] \\ , P \\{ T _ 1 < T _ 2 \\} . \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{align*} f ^ \\# ( z ) = \\frac { | f ' ( z ) | } { 1 + | f ( z ) | ^ 2 } \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} \\lambda + \\frac { 1 } { \\lambda } = \\pm 1 , \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} F _ i ( c ) = u ' _ { i 1 } \\circ F _ 1 \\circ u _ { 1 i } ( c ) \\qquad F ' _ i ( c ) = u '' _ { i 1 } \\circ F ' _ 1 \\circ u _ { 1 i } ( c ) \\end{align*}"}
-{"id": "6993.png", "formula": "\\begin{align*} z ^ { - 1 } N ^ - ( y ) = \\{ ( 3 , 2 ) , ( 5 , 2 ) , ( 7 , 2 ) \\} . \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} \\dd { X ^ { x , n } } ( t ) = \\ 1 _ { \\tau ^ n ( X ^ { x , n } ) \\leq t } \\left [ B ( t , X _ t ^ { x , n } ) + b ( t , X ^ { x , n } ( t ) ) \\right ] \\dd { t } + \\ 1 _ { \\tau ^ n ( X ^ { x , n } ) \\leq t } \\sigma ( t , X ^ { x , n } ( t ) ) \\dd { \\tilde { W } ^ x } \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} \\kappa ^ { - 1 } ( x ) - \\kappa ^ { - 1 } ( y ) = M _ { \\kappa ^ { - 1 } } ( x , y ) \\big ( x - y \\big ) . \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} K _ { 1 } : = 2 k ^ { q } _ { 0 } ( T ) K _ { 2 } : = \\max \\{ 2 k ^ { \\infty } _ { 0 } ( T ) , \\lvert \\overline { b } \\rvert \\} . \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} \\int _ { \\epsilon } ^ \\xi \\frac { d X } { 2 \\sqrt { X ( X - \\lambda ) } } = \\int _ \\epsilon ^ { \\xi } \\frac { d X } { 2 X } + \\frac 1 2 \\sum _ { n = 1 } ^ { \\infty } \\frac { ( \\frac 1 2 ) _ n } { n ! } \\int _ \\epsilon ^ { \\xi } \\left ( \\frac { \\lambda ^ n d X } { X ^ { n + 1 } } \\right ) \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} c _ k = - 2 i + O ( \\varepsilon _ k ) \\end{align*}"}
-{"id": "5311.png", "formula": "\\begin{align*} z _ 1 \\le 1 + 2 \\delta - z _ 2 \\le 2 \\delta + \\tfrac { i } k = \\xi _ { i , 1 } + 2 \\delta \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} \\P ^ { 0 , 0 } _ { 2 T , { \\rm c l } } ( X ( t ) = x ) = \\begin{cases} \\frac { \\displaystyle { \\binom { t } { ( t + x ) / 2 } \\binom { 2 T - t } { \\{ ( 2 T - t ) + x \\} / 2 } } } { \\displaystyle { \\binom { 2 T } { T } } } , & \\mbox { i f $ ( t , x ) \\in \\Lambda _ { 2 T } $ } , \\\\ 0 , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "7432.png", "formula": "\\begin{align*} \\tilde S ^ i ( t , q ) = & \\beta ^ { - 1 } ( t , q ) \\partial _ { q ^ j } ( \\tilde \\gamma ^ { - 1 } ) ^ { i l } ( t , q ) ( \\tilde \\gamma ^ { - 1 } ) ^ { j k } ( t , q ) H _ { l k } ( t , q ) \\\\ & - \\frac { 1 } { 2 } \\sum _ { \\xi } ( \\tilde \\gamma ^ { - 1 } ) ^ { i l } ( t , q ) \\partial _ { q ^ k } \\sigma _ { l \\xi } ( t , q ) ( \\tilde \\gamma ^ { - 1 } ( t , q ) \\sigma ( t , q ) ) ^ k _ \\xi . \\end{align*}"}
-{"id": "5678.png", "formula": "\\begin{align*} \\mathbb { X } _ { s , t } = ` ` \\int _ s ^ t ( X _ { r - } - X _ s ) \\otimes \\dd X _ r \\ , \" , s , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\nu ( s ) & = & s \\\\ \\nu ( \\alpha ) & = & \\alpha + \\lambda _ L ( \\alpha ) - \\lambda _ S ( \\alpha ) \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} \\log \\left ( \\frac { H ( r _ 0 ) } { H ( r ) } \\left ( \\frac { r } { r _ 0 } \\right ) ^ a \\right ) = 4 \\int _ r ^ { r _ 0 } \\frac { N ( \\sigma ) } { \\sigma } d \\sigma \\le 4 | | \\overline N | | _ \\infty \\log \\frac { r _ 0 } { r } , \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} p ^ \\kappa _ { t , s } ( x ) = ( s - t ) ^ { - d / \\alpha } p ^ { \\tilde \\kappa } _ { 0 , 1 } ( ( s - t ) ^ { - 1 / \\alpha } x ) , \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} \\pi ' ( \\alpha \\cdot f ) = \\alpha \\cdot \\pi ' ( f ) \\ , . \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} G _ 1 ( t ) = m _ { G _ 1 } ( t ) - \\frac { v _ { G _ 1 } ( t ) } { v _ { G _ 2 } ( \\rho _ { G _ 1 , G _ 2 } ( t ) ) } m _ { G _ 2 } ( \\rho _ { G _ 1 , G _ 2 } ( t ) ) + \\frac { v _ { G _ 1 } ( t ) } { v _ { G _ 2 } ( \\rho _ { G _ 1 , G _ 2 } ( t ) ) } G _ 2 ( \\rho _ { G _ 1 , G _ 2 } ( t ) ) \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} H ( \\nu _ 0 ^ { ( m ) } | \\mu _ \\star ^ { ( m , 2 ) } ) = \\int [ f _ 0 \\log f _ 0 ] ( { \\bf z } ) p _ m ( { \\bf z } ) d { \\bf z } = : H _ m ( f _ 0 ) \\ , . \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{align*} \\frac { 2 \\ , ( 2 n - 1 ) ^ 3 \\ , \\pi ^ 3 \\ , e ^ { 2 ( 2 n - 1 ) \\pi y } } { ( e ^ { ( 2 n - 1 ) \\pi y } - 1 ) ^ 3 } - \\frac { ( 2 n - 1 ) ^ 3 \\ , \\pi ^ 3 \\ , e ^ { ( 2 n - 1 ) \\pi y } } { ( e ^ { ( 2 n - 1 ) \\pi y } - 1 ) ^ 2 } = \\frac { ( 2 n - 1 ) ^ 3 \\ , \\pi ^ 3 \\ , e ^ { ( 2 n - 1 ) \\pi y } } { ( e ^ { ( 2 n - 1 ) \\pi y } - 1 ) ^ 3 } \\left ( e ^ { ( 2 n - 1 ) \\pi y } + 1 \\right ) . \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} a : = - \\frac { \\partial P } { \\partial n } \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} g _ i = - L _ i ^ 2 ( t ) d t ^ 2 + h _ i ( t ) , \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} e _ k ( \\phi ) = s ^ 2 | D ^ k \\partial _ t \\phi | ^ 2 + | D ^ k D \\phi | ^ 2 \\end{align*}"}
-{"id": "5515.png", "formula": "\\begin{gather*} g ^ { i j } D _ { i j } f = \\frac { 1 } { 2 } \\left [ x \\cdot D f \\left ( x \\right ) - f \\left ( x \\right ) \\right ] , \\\\ \\triangle _ { g } f ^ { \\alpha } = \\frac { 1 } { 2 } \\left ( \\left \\langle F , \\nabla _ { g } f ^ { \\alpha } \\right \\rangle - f ^ { \\alpha } \\right ) \\end{gather*}"}
-{"id": "3377.png", "formula": "\\begin{align*} a _ k : = z _ k + \\varrho _ k b _ k \\in L = \\R . \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} M _ { \\pm } & = L _ { \\pm } L ^ { \\top } _ { \\pm } , & L _ { \\pm } & \\in \\real ^ { n \\times m _ { \\pm } } , & \\mathrm { r a n k } ( M ) = m _ + + m _ - . \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} D A _ { \\epsilon } ^ \\dagger \\omega = D A ^ \\dagger \\omega , \\end{align*}"}
-{"id": "9037.png", "formula": "\\begin{align*} ( k _ 0 ) _ t = r _ 1 ' + r _ 3 k _ 0 . \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} \\det M _ { i j } = \\prod _ { k = 1 } ^ r ( A _ k ) _ { i i } ( B _ k ) _ { j j } - \\prod _ { k = 1 } ^ r ( C _ k ) _ { i i } ( D _ k ) _ { j j } . \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} d _ { i , S _ 1 ^ * } + d _ { i , S _ 3 ^ * } + d _ { i , V _ 1 } = \\zeta _ i < \\tau - 1 . \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} & \\P ^ { 0 , 0 } _ { 2 T } ( X ( t _ m ) = x ^ { ( m ) } , m \\in \\{ 1 , 2 , \\dots , M \\} ) \\\\ & = \\prod _ { m = 1 } ^ M Q ( t _ { m - 1 } , x ^ { ( m - 1 ) } ; t _ m , x ^ { ( m ) } ) \\frac { Q ( t _ M , x ^ { ( M ) } ; 2 T , 0 ) } { Q ( 0 , 0 ; 2 T , 0 ) } . \\end{align*}"}
-{"id": "3668.png", "formula": "\\begin{align*} P ^ \\eta ( X _ T ^ { \\vec v } - X _ 1 ^ { \\vec v } \\geq ( v + \\varepsilon ) T - 1 ) & \\leq P ^ \\eta \\Big ( \\sum _ { m = 1 } ^ { T - 1 } \\tilde Y _ m \\geq ( v + \\varepsilon ) T - 1 \\Big ) + \\sum _ { m = 1 } ^ { T - 1 } \\P ^ \\eta ( \\mathcal E ^ c _ { m - 1 } ) . \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{align*} y _ t = y _ 0 - \\gamma \\int _ { 0 } ^ { t } \\nabla f ( y _ t ) d t + \\int _ { 0 } ^ { t } \\tau _ N \\gamma d B _ t . \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( x \\frac { d } { d x } \\right ) ^ 3 \\log \\left ( \\theta _ 3 ( x ) \\right ) & = \\left ( x \\ , \\frac { d } { d x } \\right ) ^ 2 \\left ( x \\ , \\psi ( x ) \\right ) \\\\ & = x \\ , \\psi ( x ) + 3 x ^ 2 \\ , \\psi ' ( x ) + x ^ 3 \\ , \\psi '' ( x ) . \\end{aligned} \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} k x ^ { r N } _ { p } = 1 - k ' \\mu _ { p } ^ { - r N } \\ , , k y ^ { r N } _ { p } = 1 - k ' \\mu _ { p } ^ { r N } \\ , , \\end{align*}"}
-{"id": "5433.png", "formula": "\\begin{align*} l \\simeq l ' \\Leftrightarrow X ^ { s s } ( l ) \\setminus X ^ { s } ( l ) = X ^ { s s } ( l ' ) \\setminus X ^ { s } ( l ' ) \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} f ( Q ) ( x , y ) = \\pi ( x ) ^ { - 1 / 2 } f ( S ) ( x , y ) \\pi ( y ) ^ { 1 / 2 } , x , y \\in E . \\end{align*}"}
-{"id": "577.png", "formula": "\\begin{align*} T _ { \\varphi ^ * \\overline { L } } ( r ) = T _ { \\varphi , \\overline { L } , s _ 0 } ( r ) + \\log \\| \\ell ( \\varphi ^ * s _ 0 ) \\| \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} \\xi _ 2 = \\frac 1 { u _ 3 \\| u ' \\| _ J ^ 2 } \\det \\begin{pmatrix} u _ 1 & u _ 2 & u _ 3 \\\\ u ' _ 1 & u ' _ 2 & u ' _ 3 \\\\ u _ 1 '' & u _ 2 '' & u _ 3 '' \\end{pmatrix} \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} \\frac { 1 } { x _ A } + \\frac { 1 } { x _ A + x _ B } + \\frac { 1 } { x _ A + x _ C } = \\frac { 1 } { x _ B } + \\frac { 1 } { x _ A + x _ B } = \\frac { 1 } { x _ C } + \\frac { 1 } { x _ A + x _ C } . \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} g ( x ) = P _ { \\alpha } f ( x ) = \\int _ { \\mathbf { B } } R _ { \\alpha } ( x , y ) f ( y ) d v _ { \\alpha } ( y ) , \\end{align*}"}
-{"id": "2918.png", "formula": "\\begin{align*} \\R ^ { 2 } _ { + } = \\{ ( x , z ) \\in \\R ^ { 2 } : z > 0 \\} \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{align*} \\sum _ { \\ell = 2 } ^ L N _ \\ell = \\sum _ { \\ell = 1 } ^ L N _ \\ell - 1 = N ( \\Phi ) - d - 1 > M ( \\Phi ) = \\sum _ { \\ell = 2 } ^ L ( \\| A _ \\ell \\| _ { \\ell ^ 0 } + \\| b _ \\ell \\| _ { \\ell ^ 0 } ) , \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} \\lim _ { m , n \\to \\infty } \\int _ { \\textrm { s p t } \\ , \\zeta } | T ( u _ m - u _ n ) | ^ { ( { q \\over p - 1 } ) ' } \\ , d x = 0 \\quad { a n d } \\lim _ { m , n \\to \\infty } \\int _ { \\textrm { s p t } \\ \\zeta } V ^ r | T ( u _ m - u _ n ) | ^ r \\ , d x = 0 , \\end{align*}"}
-{"id": "8354.png", "formula": "\\begin{align*} P ^ k _ { d , t } ( x ) & = \\sum _ { i = 0 } ^ d \\sum _ { | \\alpha | = i } D ^ { \\alpha } _ x U _ k ( 0 , t ) \\frac { x ^ { \\alpha } } { \\alpha ! } \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} Q ( g _ 2 ) & = Q \\left ( g - \\frac { Q ( g ) } { Q ' ( g ) } \\right ) \\\\ & = Q ( g ) + \\left [ \\frac { - Q ( g ) } { Q ' ( g ) } \\right ] Q ' ( g ) + \\left [ \\frac { Q ( g ) } { Q ' ( g ) } \\right ] ^ 2 \\hat { Q } ( \\cdots ) \\ \\ \\ \\ \\ \\hat { Q } \\\\ & = ( f ^ { n - m } h _ 1 h _ 2 ^ { - 1 } ) ^ 2 \\ \\hat { Q } ( \\cdots ) \\\\ & \\equiv 0 \\pmod { f ^ { n + 1 } } \\ \\ \\ \\ \\ n > 2 m . \\end{align*}"}
-{"id": "4921.png", "formula": "\\begin{align*} \\left \\| u \\right \\| _ { L ^ 2 _ T } ^ 2 : = \\int _ 0 ^ T u ^ T ( t ) u ( t ) d t = \\int _ 0 ^ T \\left \\| u ( t ) \\right \\| _ 2 ^ 2 d t < \\infty \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{align*} E _ r D _ a E _ r D _ b E _ r = E _ r e _ a e _ a ^ T E _ r e _ b e _ b ^ T E _ r = ( E _ r ) _ { a , b } \\ , E _ r e _ a e _ b ^ T E _ r \\end{align*}"}
-{"id": "2088.png", "formula": "\\begin{align*} { C P } _ { s w } ( X , \\Omega _ X , J , \\Lambda _ X ) = \\lim \\limits _ { n \\to \\infty } ( \\Psi _ n ^ - ) ^ { - 1 } \\circ { C P } _ { s w } ( X , \\Omega _ { X _ n } , J _ n , \\Lambda _ X ) \\circ \\Psi _ n ^ + , \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{align*} f _ { 1 } ( n ) G _ { 1 } + \\ldots + f _ { i } ( n ) G _ { i } = 0 \\mod v . \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{align*} \\tilde \\rho = \\Psi \\left ( \\sum _ { i = 0 } ^ { n - 1 } \\tilde f _ i \\psi _ i \\right ) . \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{align*} t _ 1 = p + i q _ 1 , t _ 2 = p - i q _ 1 , t _ 3 = - p + i q _ 2 , t _ 4 = - p - i q _ 2 \\end{align*}"}
-{"id": "3516.png", "formula": "\\begin{align*} \\phi _ j ( x ) = \\max _ { g \\in S } g ( x ) . \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} L ( \\xi , \\sigma ) ^ 2 \\overset { d e f } { = } ( 2 \\pi | \\xi | ) ^ 2 + 2 \\pi i \\sigma , \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} \\sqrt { x } \\ , \\theta _ 2 \\left ( x \\right ) = \\theta _ 4 \\left ( \\tfrac { 1 } { x } \\right ) \\sqrt { x } \\ , \\theta _ 4 \\left ( x \\right ) = \\theta _ 2 \\left ( \\tfrac { 1 } { x } \\right ) . \\end{align*}"}
-{"id": "9061.png", "formula": "\\begin{align*} A ^ { T } \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} = \\frac 1 { \\| u ' \\| _ J } \\left ( \\hat u ' - \\frac { u _ 3 ' } { u _ 3 } \\hat u \\right ) = \\frac { u _ 3 } { \\| u ' \\| _ J } \\left ( \\frac { \\hat u } { u _ 3 } \\right ) ' = \\frac { u _ 3 } { \\| u ' \\| _ J } m ' . \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} k ( x , \\xi ' , D _ n ) a : = { k } ( x , \\xi ' , x _ n ) \\cdot a a \\in \\C \\end{align*}"}
-{"id": "1990.png", "formula": "\\begin{align*} F _ h ^ k ( \\lambda ) = \\sup \\limits _ L \\mbox { d i m } \\ , L , \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{align*} & E \\left [ S _ { s , t } ^ { a n o m } \\right ] \\\\ = & k _ B \\int _ s ^ t E \\left [ \\left ( \\frac { 1 } { 2 T } \\nabla _ q T \\cdot \\left ( \\frac { 2 } { 3 } \\gamma ^ { - 1 } + \\sum _ i ( \\gamma + 2 \\lambda _ i I ) ^ { - 1 } \\right ) \\nabla _ q T \\right ) ( r , q _ r ) \\right ] d r , \\end{align*}"}
-{"id": "5801.png", "formula": "\\begin{align*} \\sum _ { a ' \\in \\mathbb { A } } \\ell _ i ( a , a ' ) \\psi ( a ' , b ) = \\sum _ { b ' \\in \\mathbb { B } } m _ i ( b , b ' ) \\psi ( a , b ' ) , \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{align*} ( N + i B ) _ { s } = - i \\tau ( N + i B ) - \\kappa T \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\stackrel { . } { \\otimes } \\begin{pmatrix} e & f \\\\ g & h \\end{pmatrix} : = \\begin{pmatrix} a \\otimes e + b \\otimes g & a \\otimes f + b \\otimes h \\\\ c \\otimes e + d \\otimes g & c \\otimes f + d \\otimes h \\end{pmatrix} . \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} \\Lambda ( ( x , \\infty ) ) = x ^ { - \\alpha } \\| f \\| _ { \\alpha } ^ { \\alpha } / 2 , \\ \\ x > 0 . \\end{align*}"}
-{"id": "1001.png", "formula": "\\begin{align*} y ^ { n } : = \\left \\{ \\begin{array} { l l } x ^ { k } & n = n _ { k } k \\\\ z & \\end{array} \\right . \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} \\int _ M | W _ F ^ + | ^ 2 + | W _ { 0 0 } ^ + | ^ 2 + 4 ( 5 k - 7 v ) ( k - 2 v ) = \\int _ M | R _ { 0 0 } | ^ 2 \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} x = t _ + ^ k . x + ( x - t _ + ^ k . x ) . \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} \\widetilde { \\sigma } ( i ) \\ge i - a _ { s _ i } \\ge i - \\rho ( s _ i , \\mu , \\lambda ) + 1 = | \\lbrace t \\in \\lambda \\cap T _ \\mu \\mid t < s _ i \\rbrace | + 1 . \\end{align*}"}
-{"id": "484.png", "formula": "\\begin{align*} | g _ a ( z _ n ) | = \\frac { 2 } { r _ n ^ k } , \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{align*} \\sum \\limits _ { n = 2 } ^ K { \\sum \\limits _ { k = 2 , k \\ne n } ^ K { { { \\left [ { \\bf { G } } \\right ] } _ { n 1 } } { { \\left [ { \\bf { G } } \\right ] } _ { 1 k } } { { \\left [ { \\bf { G } } \\right ] } _ { k n } } } } \\approx \\frac { { K - 2 } } { { c M } } \\sum \\limits _ { k = 2 } ^ K { { { \\left | { { { \\left [ { \\bf { G } } \\right ] } _ { 1 k } } } \\right | } ^ 2 } } , \\end{align*}"}
-{"id": "5548.png", "formula": "\\begin{align*} \\widehat { f } ( \\omega ) = \\int _ { \\R } f ( x ) e ^ { - 2 \\pi i \\omega x } \\ , d x , x , \\omega \\in \\R \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} ( u \\otimes \\varphi \\otimes \\varphi ' ) \\cdot d s = u \\ , d s \\otimes \\varphi \\otimes \\varphi ' - u \\otimes ( \\mathcal L _ { \\{ s , - \\} } ( \\varphi ) \\otimes \\varphi ' + \\varphi \\otimes \\mathcal L _ { \\{ s , - \\} } ( \\varphi ' ) ) \\ , . \\end{align*}"}
-{"id": "2184.png", "formula": "\\begin{align*} \\Lambda ( \\mu _ { 1 } , \\dots , \\mu _ { n } ) & : = ( \\Lambda _ { n } ) _ { * } ( \\mu _ { 1 } \\times \\cdots \\times \\mu _ { n } ) , \\\\ { \\mathcal A } ( \\mu _ { 1 } , \\dots , \\mu _ { n } ) & : = ( { \\mathcal A } _ { n } ) _ { * } ( \\mu _ { 1 } \\times \\dots \\times \\mu _ { n } ) , \\\\ { \\mathcal H } ( \\mu _ { 1 } , \\dots , \\mu _ { n } ) & : = ( { \\mathcal H } _ { n } ) _ { * } ( \\mu _ { 1 } \\times \\dots \\times \\mu _ { n } ) . \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{align*} P \\{ T _ 1 < T _ 2 \\} = p _ 2 M _ 2 \\int _ 0 ^ { \\infty } e ^ { - p _ 2 t } \\left ( 1 - e ^ { - p _ 1 t } \\right ) ^ { M _ 1 } \\left ( 1 - e ^ { - p _ 2 t } \\right ) ^ { M _ 2 - 1 } d t . \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{align*} R _ { 0 } ( x , y ) = \\frac { 1 } { n | \\mathbf { B } | } \\sum _ { k = 0 } ^ { \\infty } ( n + 2 k ) Z _ { k } ( x , y ) , \\enspace x , y \\in \\mathbf { B } , \\end{align*}"}
-{"id": "1239.png", "formula": "\\begin{align*} z \\sim - \\left [ \\frac { 5 } { 4 } \\left ( \\frac { \\pi } { 4 } + ( 2 m + 1 ) \\frac { \\pi } { 2 } \\right ) \\right ] ^ { 4 / 5 } , m = 0 , 1 , 2 , \\ldots . \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { r - d } x ^ k p _ k m ( x ) \\equiv x ^ { r } - 1 \\pmod { n \\Z [ x ] } , \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} \\lambda _ i = v _ p ( q ^ { d p ^ i } - 1 ) . \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} ( F + \\delta F ) \\tilde { x } = b + \\delta b - \\sum _ { k = 1 } ^ s ( N _ k + \\delta N _ k ) c _ k , \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} L _ { x \\alpha \\cdot x ^ \\rho } = L ^ { - 1 } _ { x } L _ { x \\alpha } \\Rightarrow ( x \\alpha \\cdot x ^ \\rho ) \\cdot x y = x \\alpha \\cdot y \\end{align*}"}
-{"id": "5034.png", "formula": "\\begin{align*} [ u , a _ 2 ] [ a _ 1 , a _ 3 ] + [ u , a _ 3 ] [ a _ 1 , a _ 2 ] = - \\bigl ( [ u , a _ 2 ] [ a _ 3 , a _ 1 ] + [ u , a _ 3 ] [ a _ 2 , a _ 1 ] \\bigr ) \\in T ^ { ( n ) } . \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{align*} a = \\inf \\{ t \\in [ 0 , 1 ] \\mid 1 \\in \\sigma ( R _ { [ 0 , t ] } Q R _ { [ 0 , t ] } ) \\} , \\end{align*}"}
-{"id": "1199.png", "formula": "\\begin{align*} g = { \\frak t } ^ { - \\ : \\frac 1 2 } e ^ { \\frac 1 2 \\lambda } ( - \\ : d { \\frak t } ^ 2 + d \\theta ^ 2 ) + { \\frak t } \\left [ e ^ P ( d \\sigma + Q d \\delta ) ^ 2 + e ^ { - \\ : P } d \\delta ^ 2 \\right ] , \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} t _ \\mathrm { w , 2 } ( n ) = g _ \\mathrm { U L } ^ { - 1 } N l _ \\mathrm { n } + g _ \\mathrm { D L } ^ { - 1 } N l _ r + g _ \\mathrm { U L } ^ { - 1 } l _ \\mathrm { n } + g _ \\mathrm { D L } ^ { - 1 } n l _ \\mathrm { h } . \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{align*} I = \\frac { k _ 2 p _ 2 + \\cdots + k _ g p _ g } { k _ 1 p _ 1 + ( k _ 2 p _ 2 + \\cdots + k _ g p _ g ) } \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} \\int _ M p _ 1 ( D ^ g ) = \\int _ M p _ 1 ( \\nabla ) \\end{align*}"}
-{"id": "3115.png", "formula": "\\begin{align*} Z _ E ( \\alpha , \\beta ) = \\prod _ { k = 1 } ^ { \\infty } \\left ( 1 - e ^ { - \\alpha } e ^ { - \\beta k } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "3149.png", "formula": "\\begin{align*} { } \\begin{array} { l } \\tilde { g } ( u ) = 2 \\frac { v } { u } - u \\ln ( e ^ { v ( p ) } - 1 ) + \\frac { u } { p } \\left ( v ( p ) - v \\right ) \\\\ \\tilde { f } ( u ) = \\frac { v } { 2 ^ { 3 / 2 } \\pi u } \\left ( ( e ^ { v ( p ) } - 1 ) - \\frac { 1 } { 2 } u ^ 2 e ^ { v ( p ) } \\right ) ^ { - 1 / 2 } \\end{array} \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} d ( f , g ) & = \\displaystyle \\sup _ { x \\in M } | d _ M ( f ( x ) , g ( x ) ) | + \\sum _ { i = 1 } ^ k \\| | f ^ { ( i ) } ( x ) - g ^ { ( i ) } ( x ) \\| + \\int _ I | f ^ { ( k + 1 ) } - g ^ { ( k + 1 ) } | \\\\ & = d _ { C ^ k } ( f , g ) + \\| f ^ { ( k + 1 ) } - g ^ { ( k + 1 ) } \\| _ { L ^ 1 } \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ { x } : = \\langle u , v \\rangle _ { H ( x ) } \\equiv \\langle u , H ( x ) v \\rangle . \\end{align*}"}
-{"id": "2319.png", "formula": "\\begin{align*} P \\{ T _ 1 < T _ 2 \\} & = p _ 2 M _ 2 \\int _ 0 ^ 1 \\left ( 1 - x ^ { p _ 1 } \\right ) ^ { M _ 1 } \\left ( 1 - x ^ { p _ 2 } \\right ) ^ { M _ 2 - 1 } x ^ { p _ 2 - 1 } d x \\\\ & = - \\int _ 0 ^ 1 \\left ( 1 - x ^ { p _ 1 } \\right ) ^ { M _ 1 } \\left [ \\left ( 1 - x ^ { p _ 2 } \\right ) ^ { M _ 2 } \\right ] ' d x . \\end{align*}"}
-{"id": "9208.png", "formula": "\\begin{align*} \\emptyset = V _ \\mathcal { R } ( \\langle 1 \\rangle ) \\subseteq V _ \\mathcal { R } ( I ^ { \\mathcal { R } } _ 1 ( G , X _ G ) ) \\subseteq \\cdots \\subseteq V _ \\mathcal { R } ( I ^ { \\mathcal { R } } _ n ( G , X _ G ) ) \\subseteq V _ \\mathcal { R } ( \\langle 0 \\rangle ) = \\mathcal { R } ^ n . \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{align*} I _ j = [ c 2 ^ j , c 2 ^ { j + 1 } ] \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} D _ n = \\lbrace [ \\frac { k } { 2 ^ n } , \\frac { k + 1 } { 2 ^ n } ) \\rbrace _ { k \\in \\mathbb { Z } } \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} \\Leftrightarrow y L _ { x } ^ { - 1 } L _ { x \\alpha } \\cdot z I = ( y z ) L _ { x } ^ { - 1 } L _ { x \\alpha } ~ \\textrm { f o r a l l } ~ y , z \\in ( L , \\cdot ) . \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} \\Delta _ B \\phi = 2 \\overline \\square _ B \\phi + \\partial _ B \\kappa _ B ^ \\sharp \\lrcorner \\ , \\phi + \\kappa _ B ^ \\sharp \\lrcorner \\partial _ B \\phi - \\bar \\partial _ B H ^ { 0 , 1 } \\lrcorner \\ , \\phi . \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{align*} N ^ - ( \\sigma _ \\nu w ) \\cap \\Phi ^ - [ T ] = N ^ - ( w ) \\cap \\Phi ^ - [ T ] . \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} \\mathcal { E } _ { n , d } = \\left ( \\Omega _ { n , d } , \\mathcal { A } _ { n , d } , \\{ \\mathbb { P } _ { n , d , s _ { n } ^ { - 1 } h } : h \\in H _ { n , d } \\} \\right ) n \\in \\N \\end{align*}"}
-{"id": "7211.png", "formula": "\\begin{align*} h _ n ( x , y | q ) = \\sum _ { k = 0 } ^ n { n \\brack k } _ q x ^ k y ^ { n - k } . \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{align*} b _ r = - t _ r + \\sum _ { j = 1 } ^ { r - 1 } b _ j t _ { r - j } \\mbox { f o r } ~ ~ r > d . \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} w ( x , t ) = u ( x , t ) - \\int _ 0 ^ L k ( x , y ) u ( y , t ) d y \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} \\beta = \\sum _ { l = 0 } ^ { z } ( - 1 ) ^ { l } \\binom { N - \\mu } { l } \\prod _ { j = 1 } ^ { L } \\binom { N - \\mu - l } { m _ { j } - \\mu - l } , \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{align*} \\chi _ 1 ( N ) = \\frac { N } { 1 - \\theta } \\int _ 0 ^ { \\infty } e ^ { [ i \\xi \\ , + \\ , O ( N ^ { - 1 } ) ] \\ , s } \\left [ 1 - \\left ( 1 - e ^ { - s } \\right ) ^ N \\right ] d s . \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} g _ \\infty = - \\ : 1 6 d u ^ 2 + u ^ 2 d \\theta ^ 2 + \\left [ 4 ^ { 1 + \\nu } c _ 1 c _ \\lambda ^ { - 1 - \\nu } d \\widehat { \\sigma } ^ 2 + 4 ^ { 1 - \\nu } c _ 1 ^ { - 1 } c _ \\lambda ^ { - 1 + \\nu } d \\widehat { \\delta } ^ 2 \\right ] . \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{align*} \\mathfrak { h } = \\mathfrak { l } \\cap \\mathfrak { h } \\oplus \\bigoplus _ { \\alpha \\in \\Phi _ { P ^ u } } \\mathfrak { g } _ { \\alpha } . \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} \\partial _ t \\tilde { u } ( t , x ; T ) + L _ t \\tilde { u } ( t , x ; T ) + b ( t , x ) & = 0 , \\\\ \\tilde { u } ( T , x ; T ) & = 0 \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} F ^ { i j } & \\equiv \\frac { \\partial F ( R ) } { \\partial R _ { i j } } = R ^ { j i } , \\\\ F ^ { i j , k \\ell } & \\equiv \\frac { \\partial ^ 2 F ( R ) } { \\partial R _ { i j } \\partial R _ { k \\ell } } = - R ^ { \\ell i } R ^ { j k } . \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} E T ( f ) = \\begin{cases} 1 & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} | \\mu | = \\mu _ 1 + \\mu _ 2 + \\cdots + \\mu _ N \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} \\div ( \\varphi ^ * X _ 0 ) \\le \\sum _ { j = 1 } ^ n { \\div } _ { \\infty } ( f _ j ) \\end{align*}"}
-{"id": "6400.png", "formula": "\\begin{align*} 1 3 5 x _ A ^ { 4 } - 1 6 1 x _ A ^ { 3 } - 2 2 x _ A ^ { 2 } + 6 4 x _ A - 1 0 = 0 . \\end{align*}"}
-{"id": "2121.png", "formula": "\\begin{align*} d y _ t = f ( y _ t ) [ d x _ t ] \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} I = \\int _ { \\Omega } w ^ 2 \\delta ^ 2 \\nabla u \\cdot ( A \\nabla u ) d x \\geq \\lambda \\int _ { \\Omega } w ^ 2 \\delta ^ 2 | \\nabla u | ^ 2 , \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} \\L = \\langle z _ 1 , z _ 2 \\rangle _ \\Z = \\{ m z _ 1 + n z _ 2 \\mid m , n \\in \\Z , \\ , z _ 1 , z _ 2 \\in \\C , \\ , \\tfrac { z _ 1 } { z _ 2 } \\notin \\R \\} . \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} ( t _ { a _ 1 } - t _ { b _ 1 } ) + ( t _ { a _ 2 } - t _ { b _ 2 } ) = \\alpha _ 1 + \\alpha _ 2 = \\gamma _ 1 + \\alpha _ 2 = \\gamma _ 2 \\in \\Phi ^ - . \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} U _ 1 ( t , r ) & = U _ 1 ( t , s ) \\circ U _ 1 ( s , r ) U _ 1 ( t , r ) , \\ \\ \\ 0 \\leq r \\leq \\sigma \\leq t \\leq T \\ ; \\\\ U _ 1 ( t , t ) & = \\mathbb { I } , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0 \\leq t \\leq T ; \\\\ i \\partial _ t U _ 1 ( t , s ) & = H _ 1 ( t ) U _ 2 ( t , s ) \\ \\ \\ \\ 0 \\leq \\sigma \\leq t \\leq T ; \\\\ i \\partial _ s U _ 1 ( t , s ) & = - U _ 1 ( t , s ) H _ 1 ( s ) \\quad \\ \\ 0 \\leq \\sigma \\leq t \\leq T ; \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{align*} \\frac { ( \\alpha - \\beta ) ^ 2 } 2 = \\frac { 1 - \\cos 2 \\varphi } 2 = \\sin ^ 2 \\varphi . \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} \\dim \\big ( \\ker ( \\pi ) \\cap U _ m \\big ) = 1 + \\dim \\big ( \\ker ( \\pi ) \\cap U _ { m - 1 } \\big ) , \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{align*} \\mathbb { X } ^ { i , j } _ { s , t } : = \\int _ 0 ^ t X ^ i _ { r - } \\d X ^ j _ r - \\int _ 0 ^ s X ^ i _ { r - } \\d X ^ j _ r - X ^ i _ { s } X ^ j _ { s , t } \\mathbb { X } ^ { i , i } _ { s , t } : = \\frac { 1 } { 2 } ( X ^ i _ { s , t } ) ^ 2 , ( s , t ) \\in \\Delta _ T , \\end{align*}"}
-{"id": "5194.png", "formula": "\\begin{align*} \\mathbb { I } _ { i n v } : = \\{ u \\in C ^ b _ { \\rm u n i f } ( \\R ^ N ) \\ | \\ \\underline { M } \\leq u _ { 0 } ( x ) \\leq \\overline { M } , \\ \\forall \\ , x \\in \\R ^ N \\} \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} Z _ { b i f } ( \\alpha | P ' , \\vec { Y } ' ; P , \\vec { Y } ) = \\prod _ { i , j = 1 } ^ 2 & \\Bigg ( \\prod _ { s \\in Y _ i } ( P _ j ' - P _ i + b ( l _ { Y ' _ j } ( s ) + 1 ) - b ^ { - 1 } a _ { Y _ i } ( s ) - \\alpha ) \\\\ & \\prod _ { t \\in Y ' _ j } ( P _ j ' - P _ i - b l _ { Y _ i } ( t ) + b ^ { - 1 } ( a _ { Y ' _ j } ( t ) + 1 ) - \\alpha ) \\Bigg ) , \\end{align*}"}
-{"id": "4761.png", "formula": "\\begin{align*} \\| \\lambda ^ k - \\zeta ^ k \\| = { \\rm d i s t } ( \\lambda ^ k , \\mathcal { N } _ K ( g ( \\overline { x } ) ) ) \\le c \\| g ( x ^ k ) - g ( \\overline { x } ) \\| = c t _ k \\| g ' ( \\overline { x } ) d ^ k + o ( t _ k ) / t _ k \\| \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{gather*} \\mathbb { E } [ G _ n ] = \\frac { \\lambda } { n } + o \\left ( \\frac { 1 } { n } \\right ) , \\\\ F _ n = n S _ n + \\lambda \\log n + O _ p ( \\log \\log n ) . \\end{gather*}"}
-{"id": "243.png", "formula": "\\begin{align*} S ' ( i , j , k , \\alpha , \\beta ) : = \\omega _ { i j } ^ \\alpha \\underline \\wedge \\omega _ { i k } ^ \\beta ( i < j < k \\in [ n ] , \\alpha , \\beta \\geq 0 ) , \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta _ p u ( z ) + \\xi ( z ) | u ( z ) | ^ { p - 2 } u ( z ) = f ( z , u ( z ) ) \\ \\mbox { i n } \\ \\Omega , \\\\ \\frac { \\partial u } { \\partial n _ p } + \\beta ( z ) | u | ^ { p - 2 } u = 0 \\ \\mbox { o n } \\ \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "5118.png", "formula": "\\begin{align*} \\begin{pmatrix} u ^ n & u ^ { n - 1 } v & u ^ { n - 2 } v ^ 2 & \\cdots & u v ^ { n - 1 } \\\\ u ^ { n - 1 } v & u ^ { n - 2 } v ^ 2 & \\cdots & \\cdots & v ^ n \\end{pmatrix} . \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\mathfrak { d } ^ { \\prime } + B \\mathfrak { d } & = f ( 0 < t < T ) \\\\ \\mathfrak { d } ( 0 ) & = a _ { \\mathfrak { d } } \\end{aligned} \\right . \\end{align*}"}
-{"id": "3677.png", "formula": "\\begin{align*} g ( t , x ) : = \\begin{cases} f ( t , x ) - f ( t , x + e _ i ) \\quad \\mbox { i f } \\ , \\langle x , e _ i \\rangle \\geq 0 \\\\ f ( t , x ) - f ( t , x - e _ i ) \\quad \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{align*} & \\sum _ { \\ell = 4 } ^ { n - 1 } \\sum _ { s = 1 } ^ { n - \\ell } \\left [ \\binom { n - s - 1 } { \\ell - 2 } - ( n - \\ell - s + 2 ) \\right ] = \\sum _ { \\ell = 4 } ^ { n - 1 } \\left [ \\binom { n - 1 } { \\ell - 1 } - \\binom { n - \\ell + 2 } { 2 } \\right ] \\\\ & = \\sum _ { \\ell = 3 } ^ { n } \\left [ \\binom { n - 1 } { \\ell - 1 } - \\binom { n - \\ell + 2 } { 2 } \\right ] = 2 ^ { n - 1 } - n - \\binom { n } { 3 } \\end{align*}"}
-{"id": "2755.png", "formula": "\\begin{align*} V K _ { 2 , X } ^ { t o p } ( K ) & = \\left \\{ \\prod _ { \\substack { ( i , j ) \\in \\mathbb { J } _ X \\\\ k \\in \\mathbb { Z } _ { \\geq 0 } } } \\{ 1 + a _ { i j k } S ^ i T ^ j , X \\} ^ { c _ { i j k } p ^ k } \\ | \\ a _ { i j k } \\in k , c _ { i j k } \\in [ 0 , p - 1 ] \\right \\} , \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{align*} k ^ { a + 1 , c } ( I , J ) = k ^ { a + 1 , c } ( I ' , J ) = k ^ { a , c } ( I , J ) . \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} \\mathrm { d i v } ( \\{ x , - \\} ) = \\frac { \\partial Q } { \\partial y } \\frac { \\partial P } { \\partial z } - \\frac { \\partial Q } { \\partial z } \\frac { \\partial P } { \\partial y } \\ , . \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{align*} E \\left [ S _ N ^ { ( r ) } \\right ] = N ^ { r + 1 } \\int _ 0 ^ { \\infty } s ^ r ( 1 - e ^ { - s } ) ^ { N - 1 } e ^ { - s } d s . \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{align*} | \\mathbf { r } _ { i j } | ^ { 2 } & = | ( \\mathbf { a } _ { 1 } \\zeta + \\mathbf { a } _ { 2 } \\zeta ^ { 2 } + . . . ) ^ { 2 } | \\\\ & = | \\mathbf { a } _ { 1 } | ^ { 2 } | \\zeta | ^ { 2 } + 2 \\mathbf { a } _ { 1 } \\cdot \\mathbf { a } _ { 2 } \\zeta ^ { 3 } + . . . \\\\ r _ { i j } & = | \\mathbf { a } _ { 1 } | | \\zeta | \\Big ( 1 + 2 \\frac { \\mathbf { a } _ { 1 } \\cdot \\mathbf { a } _ { 2 } } { | \\mathbf { a } _ { 1 } | ^ { 2 } } \\zeta + . . . \\Big ) \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial } { \\partial s } \\psi ( s ) + D _ { A ( s ) } \\psi ( s ) = 0 \\\\ \\frac { \\partial } { \\partial s } A ( s ) + * _ 3 F _ A = r ( q _ 3 ( \\psi ) - i * _ 3 \\omega ) - \\frac { 1 } { 2 } * _ 3 F _ { A _ { K ^ { - 1 } } } - \\frac { 1 } { 2 } i * _ 3 \\wp _ 3 \\\\ \\end{cases} \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{align*} A \\langle X , T \\rangle + B \\langle X , N \\rangle = \\kappa . \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} ( \\xi + e F ) K & = - ( \\xi + e F ) ^ 2 = - 2 e \\\\ \\xi \\cdot K & = 0 . \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{align*} { \\bf D } _ { C } = { \\rm d i a g } _ 0 ( { \\bf G } ) + { \\bf G } _ c = { \\rm d i a g } _ 0 ( { \\bf G } ) ( { \\bf I } _ K + { \\bf \\tilde G } _ c ) , \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{align*} P \\{ \\tilde { T } _ 1 < \\cdots < \\tilde { T } _ g \\} & = \\int _ 0 ^ { \\infty } \\cdots \\int _ 0 ^ { t _ 3 } \\int _ 0 ^ { t _ 2 } f _ g ( t _ g ) \\cdots f _ 2 ( t _ 2 ) f _ 1 ( t _ 1 ) \\ , d t _ 1 d t _ 2 \\cdots d t _ g \\\\ & = \\int _ 0 ^ { \\infty } \\cdots \\int _ 0 ^ { t _ 3 } f _ g ( t _ g ) \\cdots f _ 2 ( t _ 2 ) F _ 1 ( t _ 2 ) \\ , d t _ 2 \\cdots d t _ g \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{align*} ( 1 + t ^ { n - 1 } ) \\binom { [ \\frac { k } { 2 } ] + [ \\frac { n - k } { 2 } ] } { [ \\frac { k } { 2 } ] } _ { t ^ 4 } = ( 1 + t ^ { n - 1 } ) \\frac { \\prod _ { i = 1 } ^ { [ \\frac { k } { 2 } ] + [ \\frac { n - k } { 2 } ] } ( 1 - t ^ { 4 i } ) } { \\prod _ { i = 1 } ^ { [ \\frac { k } { 2 } ] } ( 1 - t ^ { 4 i } ) \\prod _ { i = 1 } ^ { [ \\frac { n - k } { 2 } ] } ( 1 - t ^ { 4 i } ) } \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} A _ 3 = 3 J q ^ 3 - 6 J q ^ 2 J q ^ 1 + 3 J q ^ 1 J q ^ 2 + J q ^ 1 J q ^ 1 J q ^ 1 = 0 , \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} y I \\cdot z R _ { x } ^ { - 1 } R _ { x \\phi ^ { - 1 } } = ( y z ) L _ { x } ^ { - 1 } R _ { x } ^ { - 1 } R _ { x \\phi ^ { - 1 } } L _ { x } , \\end{align*}"}
-{"id": "1870.png", "formula": "\\begin{align*} a _ n = a _ { n - 1 } + ( 2 n - 9 ) 2 ^ { n - 2 } + 3 + \\binom { n + 1 } { 2 } + b _ n + \\sum _ { m = 0 } ^ { n - 3 } d _ { n - m } , \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} E ( X ; x , y ) = \\sum _ { p , q , i \\geq 0 } ( - 1 ) ^ i h _ c ^ { p , q ; i } ( X ) x ^ p y ^ q . \\end{align*}"}
-{"id": "1679.png", "formula": "\\begin{align*} \\lambda & \\leqq \\bar { \\lambda _ 1 } + \\bar { \\lambda } _ 2 \\\\ & = \\frac { M - 1 } { 2 } + ( H - 1 ) \\frac { M + N - 3 } { 2 } \\\\ & = \\frac { M - 1 + ( H - 1 ) ( M + N - 3 ) } { 2 } . \\end{align*}"}
-{"id": "5489.png", "formula": "\\begin{align*} f ( \\rho _ { m a x } ) = \\left [ \\mbox { R e } ( \\lambda _ l ) \\rho _ { m a x } + \\sum _ { m = 1 } ^ M \\mbox { R e } ( \\beta _ m ) \\rho _ { m a x } ^ { 2 m + 1 } \\right ] ^ 2 - \\varepsilon ^ 2 r ^ 2 , \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{align*} d \\left ( \\binom { 8 } { 2 } - 5 \\right ) - \\left ( \\binom { d } { 2 } + d \\binom { 7 } { 2 } \\right ) = - \\frac { d } { 2 } ( d - 5 ) \\ge 0 . \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} - \\frac { 1 } { T } \\log \\widetilde { \\P } ^ { 0 , 0 } _ { 2 T } ( X ( T ) = x ) & = \\frac { 2 7 } { 2 \\pi } \\sum _ { n = 1 } ^ { \\infty } \\frac { ( - 1 ) ^ { n - 1 } } { n ^ 2 } \\sin \\left ( \\frac { n \\pi } { 3 } \\right ) \\\\ & - \\frac { 9 } { \\pi } \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ 2 } \\sin \\left ( \\frac { n \\pi } { 3 } \\right ) \\cos \\left ( \\frac { x n \\pi } { 3 T } \\right ) + { \\rm o } ( T ) \\mbox { i n $ T \\to \\infty $ } . \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} S ^ { e n v } _ { s , t } = & - \\int _ { s } ^ t 2 \\partial _ { p _ l } H ( r , x _ r ) \\gamma _ { j l } ( r , q _ r ) ( \\Sigma ^ { - 1 } ) ^ { j k } ( r , q _ r ) \\circ d ( p _ r ) _ k \\\\ & + \\int _ { s } ^ t 2 \\partial _ { p _ l } H ( r , x _ r ) \\gamma _ { j l } ( r , q _ r ) ( \\Sigma ^ { - 1 } ) ^ { j k } ( r , q _ r ) ( - \\nabla _ q H + \\tilde F ) _ k ( r , x _ r ) - \\nabla _ p \\cdot \\tilde F ( r , x _ r ) d r . \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{align*} \\bar { g } ( u ) = 2 \\frac { N } { \\sqrt { E } } - \\frac { N } { \\sqrt { E } } \\ln \\left ( \\frac { N ^ 2 } { E } \\right ) \\end{align*}"}
-{"id": "3410.png", "formula": "\\begin{align*} a _ { \\varepsilon ( \\kappa ) } ( \\lambda , \\nu ) ^ { - 1 } : = \\Gamma ( \\frac { \\lambda + \\nu - n + 1 + \\kappa } { 2 } ) \\Gamma ( \\frac { \\lambda - \\nu + \\kappa } { 2 } ) \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{gather*} R ^ C = R ^ B \\cup \\bigcup _ { \\bar { a } \\in R ^ B \\setminus R ^ A } R ^ C _ { \\bar { a } } \\\\ R ^ D = R ^ A \\cup \\bigcup _ { \\bar { a } \\in R ^ B \\setminus R ^ A } R ^ D _ { \\bar { a } } \\end{gather*}"}
-{"id": "4848.png", "formula": "\\begin{align*} \\alpha = \\xi _ 1 \\cdot ( \\omega _ M \\times 1 ) + \\xi _ 2 \\cdot ( 1 \\times \\omega _ N ) \\in H ^ m ( M ; \\R ) \\oplus H ^ n ( N ; \\R ) . \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} \\nu ( A ) = \\mathbf { E } _ { n } ( \\nu _ { x , i } ( A ) ) . \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} e _ 1 = E _ { 1 , 2 } + E _ { 2 , 1 } , e _ 2 = E _ { 1 , 3 } + E _ { 3 , 1 } , e _ 3 = E _ { 2 , 3 } + E _ { 3 , 2 } , \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} \\bar * \\phi = ( - 1 ) ^ { p ( q - r ) } * ( \\phi \\wedge \\chi _ { \\mathcal F } ) , \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} P _ n ( \\theta ) = \\frac { 1 } { Z _ n } e ^ { - \\sum _ { j \\neq k } f ( \\theta _ k - \\theta _ j ) } , \\end{align*}"}
-{"id": "1224.png", "formula": "\\begin{align*} v _ t | _ { t = T } = 2 u _ t ^ { \\tilde f } ( \\cdot , T ) = 2 u ^ f ( \\cdot , T ) . \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} = \\frac { | - 1 + z _ { 1 , n } | ^ 2 \\left ( 1 - \\frac { \\pi ^ 2 } { a \\ , n } ( 2 X ^ 2 - 1 ) \\right ) \\exp \\bigl ( \\frac { 2 \\pi \\ , X } { a } \\bigr ) } { | z _ { 1 , n } | \\left | 1 + \\frac { \\pi } { a \\ , n } e ^ { i ( \\psi - \\varphi ) } \\right | - | - 1 + z _ { 1 , n } | \\ , \\left ( 1 - \\frac { \\pi ^ 2 } { 2 a \\ , n } ( 2 X ^ 2 - 1 ) \\right ) \\exp ( \\frac { \\pi \\ , X } { a } ) } \\end{align*}"}
-{"id": "2810.png", "formula": "\\begin{align*} \\chi ( M ) = 2 - 2 b _ { 1 } ( M ) + b _ { 2 } ( M ) = 0 \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} u [ a _ 1 a _ 4 , a _ 2 , a _ 3 ] = u \\bigl ( [ a _ 1 , a _ 2 , a _ 3 ] a _ 4 + [ a _ 1 , a _ 2 ] [ a _ 4 , a _ 3 ] + [ a _ 1 , a _ 3 ] [ a _ 4 , a _ 2 ] + a _ 1 [ a _ 4 , a _ 2 , a _ 3 ] \\bigr ) . \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} \\mathcal { H } ( \\widetilde u , \\widetilde \\pi , \\widetilde \\ell , \\widetilde \\omega ) = ( I _ { 3 } - [ Z Q ] ^ { T } ) \\widetilde u , \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} M _ t V _ t = \\sum _ { i , j \\leq n / 2 } \\frac { ( - 1 ) ^ i } { j ! 2 ^ i } \\mathcal M _ i ( t ) D v _ t ^ j + s ( t ) , \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{align*} d q _ t = \\tilde \\gamma ^ { - 1 } ( t ) ( - \\nabla _ q V ( t , q _ t ) + \\tilde F ( t , q _ t ) ) d t + ( \\tilde \\gamma ^ { - 1 } \\sigma ) ( t , q _ t ) d W _ t \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} \\mathcal { O } ( \\varepsilon , | \\mathbf { z } | ^ 0 ) : \\mathbf { A } \\mathbf { w } _ 1 ^ { \\mathbf { 0 } } ( \\phi ) + \\mathbf { G } _ { e x t } ( \\phi ) = \\frac { \\partial \\mathbf { W } _ 0 } { \\partial \\mathbf { z } } \\mathbf { r } _ 1 ^ { \\mathbf { 0 } } ( \\phi ) + \\frac { \\partial \\mathbf { w } _ 1 ^ { \\mathbf { 0 } } } { \\partial \\mathbf { \\phi } } \\Omega . \\end{align*}"}
-{"id": "8842.png", "formula": "\\begin{align*} C _ D = \\tanh ( \\beta ( a ) ) \\Re ( z _ 2 \\mu _ { \\beta } ) + \\coth ( \\beta ( a ) ) \\Im ( z _ 2 \\mu _ { \\beta } ) . \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} W ^ 0 _ 2 ( p , p _ 1 ) \\ast W ^ 0 _ { n + 2 } ( p ' , p ' _ 1 , \\dots , p ' _ { n + 1 } ) = & W ^ 0 _ { n + 2 } ( p ' , p ' _ 1 , \\dots , p ' _ { n + 1 } ) \\ast W ^ 0 _ 2 ( p , p _ 1 ) \\\\ = & W ^ 0 _ { n + 2 } ( p , p ' _ 1 , \\dots , p ' _ { n + 1 } ) \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{align*} ( I - K ^ * ) ( \\partial ^ r \\gamma / 2 ) = \\partial ^ r \\delta + \\partial ^ { r - 1 } ( U \\theta _ s ) + [ \\partial ^ r , K ^ * ] ( \\gamma / 2 ) . \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} \\tau _ { \\rm \\textbf { a l g o } } = \\dfrac { t _ { \\rm \\textbf { a l g o } } } { t _ { \\rm \\textbf { S D } } } , \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} c _ n ( \\{ k \\} ) = \\alpha c _ 0 ( \\{ k \\} ) + \\sum \\limits _ { i = 1 } ^ { n - 1 } \\delta _ { X _ { i + 1 } - X _ i + ( 1 - \\delta _ { X _ i 0 } ) } ( \\{ k \\} ) . \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} \\| \\eta \\overline { u } ^ { \\alpha + 1 } & \\| _ { m ^ \\star } ^ m \\le \\frac { C } { ( R _ 1 - R _ 2 ) ^ m } \\int _ { \\Omega _ { R _ 2 } } \\overline { u } ^ { m ( \\alpha + 1 ) } + \\frac { C G ( R _ 1 , R _ 2 ) } { m \\alpha + 1 - C m \\epsilon ^ { m ' } } \\int _ { \\Omega _ { R _ 2 } } \\overline { u } ^ { m ( \\alpha + 1 ) } \\\\ & + C \\left ( \\frac { \\| f \\| _ q } { ( m \\alpha + 1 - C m \\epsilon ^ { m ' } ) k ^ { m - 1 } } \\right ) ^ { N / ( m q - N ) } \\int _ { \\Omega _ { R _ 2 } } \\overline { u } ^ { m ( \\alpha + 1 ) } . \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{align*} \\gamma = \\frac { 2 \\mu - \\varepsilon ( L + \\mu ) } { ( 1 - \\varepsilon ) \\mu ( L + \\mu ) } , \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} h ( \\mu ) = \\lambda ( \\mu ) \\cdot \\dim _ + ( \\mu ) \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{align*} \\phi ( x ) = \\frac { \\gamma { } k } { { \\left ( 2 \\pi { } \\hslash { } \\right ) } ^ 2 } \\left ( I _ 1 + i I _ 2 \\right ) , \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} u ( 0 , T + t _ 0 ; t _ 0 , x _ 0 , u _ 0 ) \\ge e ^ { \\frac { T \\sigma _ { L _ 0 } } { 2 } } \\kappa = e ^ { \\frac { T \\sigma _ { L _ 0 } } { 2 } } \\inf _ { x \\in D _ L } u _ 0 ( x ) . \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} \\boxed { 0 = \\rho _ j ^ n - \\rho _ j ^ { n + 1 } + \\frac { \\Delta t } { \\Delta x ^ 2 } ( \\rho _ { j - 1 } ^ n - 2 \\rho _ j ^ n + \\rho _ { j + 1 } ^ n ) \\sum _ { k = 1 } ^ { K } \\omega _ k v _ k ^ 2 . } \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{align*} f _ n ( t , \\mu , w ) & : = f ( \\mu , w ) - \\left ( \\int _ 0 ^ { t } H _ n [ s ] f ( \\mu , w ) \\dd s - \\frac { t } { \\gamma _ n ^ { - 1 } T _ 0 } \\int _ 0 ^ { \\gamma _ n ^ { - 1 } T _ 0 } H _ n [ s ] f ( \\mu , w ) \\dd s \\right ) \\\\ & \\ , = f ( \\mu , w ) - F _ { f , n } ( t , \\mu , w ) . \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{align*} | R _ G | = | G | - | B _ G | - | A _ p \\cup X ' | - | X '' | \\ge 4 \\cdot 2 ^ k + 1 - 6 - 4 \\cdot 2 ^ { k - 1 } - ( k - 1 ) > 6 , \\end{align*}"}
-{"id": "2153.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { N _ { 1 / n _ k } ( A ) } { - \\log n _ k } = \\overline { \\dim } _ B ( A ) . \\end{align*}"}
-{"id": "800.png", "formula": "\\begin{align*} P ' _ { \\alpha } ( X ) ~ = ~ U ' _ { \\alpha } ( X ) \\ , f _ { \\beta } ( X ) ~ + ~ U _ { \\alpha } ( X ) \\ , f ' _ { \\beta } ( X ) . \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 2 & 9 & 1 0 \\\\ 0 & 6 & 6 & 9 \\\\ 0 & 2 & 6 & 2 \\end{pmatrix} , \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 2 & 2 & 1 0 \\\\ 0 & 6 & 5 & 9 \\\\ 0 & 2 & 5 & 2 \\end{pmatrix} , \\end{align*}"}
-{"id": "95.png", "formula": "\\begin{align*} \\partial _ H \\ell _ p ( \\mathcal { N } , \\R ) = \\left \\lbrace x \\mapsto h _ { \\mu } ( x ) \\mathrel { \\big \\vert } \\mu \\in \\ell _ q ( \\mathcal { N } , \\R ) , \\ ; \\norm { \\mu } _ q = 1 \\right \\rbrace , \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{align*} \\rho \\geq 0 , y = ( y ^ 1 , \\ldots , y ^ { n - 1 } ) \\in \\R ^ { n - 1 } \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} h ( ( s \\otimes t ) \\cdot p ) = ( s \\otimes t ) \\cdot h ( p ) - ( 1 \\otimes \\partial ) ( s \\otimes t ) \\cdot k ( p ) \\ , . \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} \\mathrm { s u p p } ( ( \\mu ^ { \\boxtimes s } ) ^ { \\mathrm { a c } } ) \\cap [ \\epsilon , \\infty ) \\subset & \\overline { \\{ 1 / x : x = h _ s ( r ) , r \\in V _ s ^ + \\cap [ a , b ] \\} } \\\\ & \\subset B _ \\epsilon ( \\mathrm { s u p p } ( \\mu ^ { \\boxtimes t _ 0 } ) ^ { \\mathrm { a c } } ) , \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} \\kappa _ { i , j } & = \\min _ { r _ { X Y } : \\ : r _ X = q _ { X , i } , r _ Y = q _ { Y , j } } D ( r _ { X Y } \\| p _ { X Y } ) \\end{align*}"}
-{"id": "2860.png", "formula": "\\begin{gather*} h _ { \\beta _ 1 } : = + 2 ^ { - 1 } \\sigma _ 1 h _ 1 - 2 ^ { - 1 } \\sigma _ 2 h _ 2 - 2 ^ { - 1 } \\sigma _ 3 h _ 3 , \\\\ h _ { \\beta _ 2 } : = - 2 ^ { - 1 } \\sigma _ 1 h _ 1 + 2 ^ { - 1 } \\sigma _ 2 h _ 2 - 2 ^ { - 1 } \\sigma _ 3 h _ 3 , \\\\ h _ { \\beta _ 3 } : = - 2 ^ { - 1 } \\sigma _ 1 h _ 1 - 2 ^ { - 1 } \\sigma _ 2 h _ 2 + 2 ^ { - 1 } \\sigma _ 3 h _ 3 , \\\\ h _ \\theta : = + 2 ^ { - 1 } \\sigma _ 1 h _ 1 + 2 ^ { - 1 } \\sigma _ 2 h _ 2 + 2 ^ { - 1 } \\sigma _ 3 h _ 3 . \\end{gather*}"}
-{"id": "7852.png", "formula": "\\begin{align*} C _ \\alpha = \\begin{cases} \\cfrac { 1 - \\alpha } { \\Gamma ( 2 - \\alpha ) \\cos ( \\pi \\alpha / 2 ) } , \\ & \\alpha \\neq 1 , \\\\ \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\cfrac { 2 } { \\pi } , & \\alpha = 1 . \\end{cases} \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} & S ( \\mathfrak { x } ) : = S ( x _ { n _ { 1 } } \\otimes \\cdots \\otimes x _ { n _ { r + 1 } } \\otimes x _ { n _ { r } } \\otimes \\cdots \\otimes x _ { n _ { i + 1 } } ) \\\\ & + S ( \\mu ^ { i - 1 } \\alpha _ { T } ( x _ { n _ { 1 } } \\otimes \\cdots \\otimes x _ { n _ { r - 1 } } ) \\otimes [ x _ { n _ { r } } , x _ { n _ { r + 1 } } ] _ { \\mathfrak { g } } \\otimes \\alpha _ { T } ( x _ { n _ { r + 2 } } \\otimes \\cdots \\otimes x _ { n _ { i + 1 } } ) ) . \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} [ z ^ n ] \\log g ( q , z ) & = [ z ^ n ] \\log f ( q , q z ) - [ z ^ n ] \\log f ( q , z ) = ( q ^ n - 1 ) [ z ^ n ] \\log f ( q , z ) . \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} a _ { s _ i } = | \\lbrace j \\mid 1 \\le j < i , \\widetilde { \\sigma } ( i ) < \\widetilde { \\sigma } ( j ) \\rbrace | . \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} x ^ { q ^ 2 } = x - \\frac { \\psi _ { k - 1 } \\psi _ { k + 1 } } { \\psi _ k ^ 2 } \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{align*} [ W ^ u ( a ) ] \\wedge [ W ^ s ( b ) ] | _ { M \\setminus O } = \\sum _ { \\gamma _ { a b } } \\sigma ( \\gamma _ { a b } ) [ \\gamma _ { a b } ] | _ { M \\setminus O } \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{align*} \\widehat { g } ( \\xi ) = \\mathcal { F } g ( \\xi ) = \\int _ { \\mathbb { R } ^ n } e ^ { - i x \\cdot \\xi } g ( x ) \\ , \\mathrm { d } x . \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{align*} F _ { c , J } : = c \\left ( \\sum _ { i = 1 } ^ { n + 1 } E _ i - H \\right ) - \\sum _ { j \\in J } E _ j . \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} ( u \\cdot \\nabla ) d \\cdot \\overline { d } = \\sum _ { k , l = 1 } ^ 3 u _ k \\partial _ k d _ l \\overline { d _ l } = \\frac { 1 } { 2 } u \\cdot \\nabla \\lvert d \\rvert ^ 2 = \\frac { 1 } { 2 } u \\cdot \\nabla \\varphi , \\end{align*}"}
-{"id": "6464.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 } \\| \\nabla [ y _ j ( s ) - b ] \\| _ { L ^ p ( \\Omega ) ^ { 3 \\times 3 } } \\leq \\lim _ { s \\to 0 } \\| \\nabla [ y _ j ( s ) - y _ 0 ( s ) ] \\| _ { L ^ p ( \\Omega ) ^ { 3 \\times 3 } } + \\lim _ { s \\to 0 } \\| B ^ { \\frac { 1 } { 2 } } [ e ^ { - s B } b - b ] \\| _ { L ^ p ( \\Omega ) ^ { 3 \\times 3 } } = 0 . \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} \\alpha ( S ) = \\sum _ { i = 1 } ^ n \\frac { 1 } { d _ i ( S ) } . \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} \\partial _ { q , x _ 1 } \\{ f ( x _ 1 , y _ 1 ) - b _ 1 f ( x _ 1 , q y _ 1 ) \\} = \\partial _ { q , y _ 1 } \\{ f ( x _ 1 , y _ 1 ) - a _ 1 f ( q x _ 1 , y _ 1 ) \\} . \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} \\mathit { C W E } ( \\mathcal { C } _ { D } ) & = w _ { c } ^ { p ^ { e - 1 } - ( p - 1 ) p ^ { m - 1 } - 1 } + \\bigl ( p ^ { e - 1 } - ( p - 1 ) p ^ { m - 1 } - 1 \\bigr ) w _ { c } ^ { p ^ { e - 2 } - ( p - 1 ) p ^ { m - 1 } - 1 } \\prod _ { \\substack { 0 \\le i \\leq p - 1 \\\\ i \\ne c } } w _ { i } ^ { p ^ { e - 2 } } \\\\ & + ( p - 1 ) ( p ^ { e - 1 } + p ^ { m - 1 } ) w _ { c } ^ { p ^ { e - 2 } - 1 } \\prod _ { \\substack { 0 \\le i \\leq p - 1 \\\\ i \\ne c } } w _ { i } ^ { p ^ { e - 2 } - p ^ { m - 1 } } . \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} \\mathcal { F } ( R , M ) = \\mathcal { F } ( R , P ) + \\mathcal { F } ( R , Q ) + 2 \\mathcal { L } ( R , P , Q ) , \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} p ( x | a , \\mu ) & = \\mathcal { N } ( x | \\mu , 1 ) , \\ \\mbox { i f } \\ K = 1 , \\\\ p ( x | a , \\mu ) & = a \\mathcal { N } ( x | \\mu _ 1 , 1 ) + ( 1 - a ) \\mathcal { N } ( x | \\mu _ 2 , 1 ) , \\ \\mbox { i f } \\ K = 2 , \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} \\lim _ { k } \\sum _ { l = n _ { k } } ^ { n _ { k } + s - 1 } \\Vert U _ { l } u ^ { l } - u ^ { l } \\Vert = 0 . \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} \\sigma ( \\alpha _ { j , k } ) = \\alpha _ { k + ( - 1 ) ^ { k + 1 } , j + ( - 1 ) ^ { j + 1 } } . \\end{align*}"}
-{"id": "3112.png", "formula": "\\begin{align*} r - \\sum _ { k \\leq j } a _ { k } = r - \\Omega ( N '' ) + \\sum _ { k \\leq s - j } a _ { j + k } \\geq s - j - r ^ { \\theta } \\geq \\frac { 1 } { 2 } s ^ { 1 - \\varepsilon } . \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{align*} \\langle \\chi _ y , \\chi _ { y ' } \\rangle = q ^ { | B _ \\mu | - 1 } \\sum _ { g \\in \\mathbb { F } _ q } \\chi ( g ) = 0 . \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} f ^ * ( x _ M ^ u \\times 1 ) = ( y _ M ^ u \\times 1 ) + ( y _ M ^ { u - n } \\times \\omega _ N ) \\in H ^ u ( M ; \\R ) \\oplus ( H ^ { u - n } ( M ; \\R ) \\otimes H ^ n ( N ; \\R ) ) , \\end{align*}"}
-{"id": "451.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 0 & 6 & 6 & 3 \\\\ 1 0 & 1 & 6 & 6 \\\\ 2 & 1 0 & 1 & 6 \\\\ 9 & 2 & 1 0 & 1 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ s m _ j \\lambda _ j \\neq \\lambda _ n , n = 1 , . . . , s , 2 \\leq \\sum _ { j = 1 } ^ s m _ j \\leq \\Sigma ( E ) , m _ j \\in \\mathbb { N } , \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} \\frac { d \\mathbb { Q } _ { n } } { d \\mathbb { P } _ { n , 0 } } = d ( n ) ^ { - 1 } \\sum _ { i = 1 } ^ { d ( n ) } \\exp ( a ( n ) z ^ * _ { n , i } - k _ { n , i } ) \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{align*} \\frac { | - 1 + z _ { j , n } | } { | z _ { j , n } | } < \\kappa ( 1 , a ) = \\frac { \\left | 1 - \\exp \\bigl ( \\frac { \\pi } { a } \\bigr ) \\right | \\exp \\bigl ( \\frac { - \\pi } { a } \\bigr ) } { \\exp \\bigl ( \\frac { \\pi } { a } \\bigr ) + \\left | 1 - \\exp \\bigl ( \\frac { \\pi } { a } \\bigr ) \\right | } . \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} \\# \\{ f ( x ) \\} = \\frac { 1 } { d } \\sum _ { e \\mid d } \\mu ( d / e ) q ^ e , \\end{align*}"}
-{"id": "2420.png", "formula": "\\begin{align*} \\left . \\begin{array} { c c c } P \\big \\{ X ( \\tau + 1 ) = m + e _ j \\ , \\big | \\ , X ( \\tau ) = m \\big \\} & = & ( M _ j - m _ j ) p _ j \\\\ \\\\ P \\big \\{ X ( \\tau + 1 ) = m \\ , \\big | \\ , X ( \\tau ) = m \\big \\} & = & \\ m _ 1 p _ 1 + \\cdots + m _ g p _ g \\\\ \\end{array} \\right \\} \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} | | \\overset { v } { \\nabla } X u | | _ { L ^ 2 } = | | \\overset { v } { \\nabla } f | | _ G , | | X \\overset { v } { \\nabla } u | | _ G \\leq | | \\overset { v } { \\nabla } f | | _ G + | | \\overset { h } { \\nabla } u | | _ G . \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} F ( M ) = \\arctan \\lambda _ 1 ( M ) + \\arctan \\lambda _ 2 ( M ) . \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} C _ m C _ m ^ T y = b \\ , . \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ^ { y } _ { j } ( T ) & < C \\tilde { C } _ { T } [ 2 K _ { 1 } + 6 K _ { 1 } K _ { 2 } + K _ { 1 } ^ 2 ] \\delta _ { j - 1 } ( T ) \\end{aligned} \\end{align*}"}
-{"id": "4005.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } H ( X ^ n | Y ^ n , X ^ n \\in \\mathcal { A } , Y ^ n \\in \\mathcal { B } ) ^ { \\frac { 1 } { n } } & = \\lim _ { n \\rightarrow \\infty } H ( Y ^ n | X ^ n , X ^ n \\in \\mathcal { A } , Y ^ n \\in \\mathcal { B } ) ^ { \\frac { 1 } { n } } \\leq \\left ( \\frac { p _ { 1 2 } p _ { 2 1 } } { p _ { 1 1 } p _ { 2 2 } } \\right ) ^ { \\frac 1 2 } . \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} [ b _ 1 s b _ 2 , b _ 3 , x ] = [ s b _ 1 b _ 2 , b _ 3 , x ] - \\bigl [ [ s , b _ 1 ] b _ 2 , b _ 3 , x ] \\bigr ] . \\end{align*}"}
-{"id": "4341.png", "formula": "\\begin{align*} O _ \\lambda = \\{ \\lambda , 1 / \\lambda , 1 - \\lambda , 1 / ( 1 - \\lambda ) , \\lambda / ( \\lambda - 1 ) , ( \\lambda - 1 ) / \\lambda \\} . \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} { } T : I \\times { C _ { 0 } ^ { 1 , \\alpha } ( \\overline { \\Omega } ) } \\rightarrow { C _ { 0 } ^ { 1 , \\alpha } ( \\overline { \\Omega } ) } \\ ; \\mbox { g i v e n b y } \\ ; T ( \\lambda , u ) = U ( \\lambda ) - u . \\end{align*}"}
-{"id": "4592.png", "formula": "\\begin{align*} ( X \\lrcorner ) ^ * = \\epsilon ( X ^ \\flat ) , \\quad ( \\epsilon ( X ^ \\flat ) ) ^ * = X \\lrcorner . \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} \\sqrt { \\sum _ { j = 1 } ^ { r } x _ j e _ j } = \\sum _ { j = 1 } ^ { r } \\sqrt { x _ j } e _ j , \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{align*} H ( X _ i | \\tilde { X } _ i , \\{ \\tilde { Z } _ j \\} , \\{ X _ w \\} _ { w \\neq i } ) = H \\left ( X _ i | \\tilde { X } _ i , \\{ \\hat { X } _ i ( u ) \\} _ { u \\in T _ k ( i ) } , \\{ \\tilde { Z } _ j \\} _ { j \\notin T _ k ( i ) } , \\{ X _ w \\} _ { w \\neq i } \\right ) . \\end{align*}"}
-{"id": "7761.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\Big ( 1 - \\Gamma ' ( u + \\sigma w ) \\cdot \\Gamma ' ( u + \\tau w ) \\Big ) \\ , d \\sigma d \\tau = \\frac 1 2 \\int _ 0 ^ 1 \\int _ 0 ^ 1 | \\Gamma ' ( u + \\sigma w ) - \\Gamma ' ( u + \\tau w ) | ^ 2 \\ , d \\sigma d \\tau , \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} R _ { \\tau } = \\left ( \\frac { P _ J } { \\tau } \\right ) ^ { 1 / \\alpha } . \\end{align*}"}
-{"id": "5655.png", "formula": "\\begin{align*} n r _ = ( n , 1 ) \\ , \\geq \\ , \\frac { 2 ^ { ( 2 ^ { n - 1 } - 1 ) ( 2 ^ { n - 2 } - 1 ) / 3 } } { 2 ^ { 2 ^ { n - 1 } - 1 } \\cdot | P G L ( n - 1 , 2 ) | } - 1 . \\end{align*}"}
-{"id": "6359.png", "formula": "\\begin{align*} H _ { \\mathcal { R } ( I ^ n ) _ + } ^ 0 \\left ( L ^ { I ^ n } ( F ) \\right ) = 0 = H _ { \\mathcal { R } ( I ^ n ) _ + } ^ 1 \\left ( L ^ { I ^ n } ( F ) \\right ) \\mbox { f o r a l l } n \\gg 0 . \\end{align*}"}
-{"id": "9012.png", "formula": "\\begin{align*} \\left ( K \\cdot I _ { r } \\right ) ^ \\alpha = - I _ { r + 1 } ^ \\alpha + \\left ( I _ r ^ \\alpha \\right ) ' , \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} x ( 0 ) = x _ 0 \\end{align*}"}
-{"id": "4676.png", "formula": "\\begin{align*} { \\bf x } \\prec { \\bf y } \\quad { \\Rightarrow } \\sum _ { j = 1 } ^ n \\phi ( x _ j ) \\leq \\sum _ { j = 1 } ^ n \\phi ( y _ j ) \\ . \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} \\hat b _ + ^ i = b _ + ^ i - \\frac { 1 } { 2 } \\sum _ \\xi \\tilde \\sigma ^ i _ \\xi \\partial _ j \\tilde \\sigma ^ j _ \\xi . \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} H ^ k ( M \\times N ; \\R ) \\cong \\left \\{ \\begin{array} { l l } H ^ k ( M ; \\R ) , & k < n \\\\ H ^ n ( M ; \\R ) \\oplus H ^ n ( N ; \\R ) , & k = n \\\\ H ^ k ( M ; \\R ) \\oplus ( H ^ { k - n } ( M ; \\R ) \\otimes H ^ n ( N ; \\R ) ) , & k > n . \\end{array} \\right . \\end{align*}"}
-{"id": "6780.png", "formula": "\\begin{align*} ( \\widetilde { \\rho _ \\psi } ( s ) h ) ( t ) : = h ( t s ) \\quad s , t \\in S , \\ h \\in \\mathfrak D _ { \\psi } . \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} u ^ 0 \\in \\mathcal { H } ; u ^ { k + 1 } = U _ { k } u ^ { k } - \\lambda _ { k } G U _ { k } u ^ { k } . \\end{align*}"}
-{"id": "6114.png", "formula": "\\begin{align*} G ( t ) = m _ G ( t ) + \\varphi _ { G , W } ( t ) W ( \\rho _ { G , W } ( t ) ) . \\end{align*}"}
-{"id": "7198.png", "formula": "\\begin{align*} k ( z , x ) = \\left \\{ \\begin{array} { l l } - \\vartheta \\eta ^ { p - 1 } + c _ 1 \\eta ^ { r - 1 } & \\mbox { i f } \\ x < - \\eta \\\\ \\vartheta | x | ^ { p - 2 } x - c _ 1 | x | ^ { r - 2 } x & \\mbox { i f } \\ - \\eta \\leq x \\leq \\eta \\\\ \\vartheta \\eta ^ { p - 1 } - c _ 1 \\eta ^ { r - 1 } & \\mbox { i f } \\ \\eta < x . \\end{array} \\right . \\end{align*}"}
-{"id": "7239.png", "formula": "\\begin{align*} I ( a , b ) = \\int _ { u } ^ v \\frac { ( q x / u , q x / v , \\alpha a x , \\beta b x ; q ) _ \\infty } { ( a x , b x , c x , d x ; q ) _ \\infty } d _ q x = \\sum _ { n = 0 } ^ \\infty \\lambda _ n \\Phi _ n ^ { ( \\alpha , \\beta ) } ( a , b | q ) . \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} R ( \\theta _ 1 , \\theta _ 3 ) = \\frac { r ( \\theta _ 1 ) \\ , r ( \\theta _ 3 ) } { r ( \\theta _ 1 + \\theta _ 3 ) } \\ , . \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} z ( \\lambda , \\hat { X } ) = - \\int _ 1 ^ { \\hat { X } } \\frac { d X } { 2 \\sqrt { X ( X - 1 ) ( X - \\lambda ) } } + \\omega _ 1 / 2 \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{align*} d G _ { t } = d y _ { t } \\cdot y _ { t } + \\frac { 1 } { 2 } d y _ { t } \\cdot d y _ { t } = - \\nabla ^ { T } f ( y _ { t } ) y _ { t } \\gamma d t + \\tau _ N \\gamma { d B _ { t } } ^ { T } y _ { t } + \\frac { 1 } { 2 } m \\tau _ N ^ 2 \\gamma ^ 2 d t . \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} \\lim _ { m , n \\to \\infty } \\int _ \\omega e _ { m , n } \\chi _ { E _ { m , n } } \\ , d x = 0 . \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} p ^ + _ M ( t , n ) : = p _ M ( t , \\hat x _ n ) = \\max _ { z \\in C _ n } \\{ p _ M ( t , z ) \\} \\ , , \\quad \\mbox { } p ^ - _ M ( t , n ) : = p _ M ( t , \\check x _ n ) = \\min _ { z \\in C _ n } \\{ p _ M ( t , z ) \\} . \\end{align*}"}
-{"id": "5010.png", "formula": "\\begin{align*} S ^ { ( n ) } = \\bigl \\{ [ y _ 1 , y _ 2 , \\dots , y _ n ] \\mid y _ 1 , y _ n \\in X , \\ y _ 2 , \\dots , y _ { n - 1 } \\in X \\cup X ^ 2 \\bigr \\} \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} M = \\begin{pmatrix} a _ 0 & a _ 1 & a _ 2 & a _ 3 & a _ 4 & a _ 5 & \\cdots \\\\ a _ 0 & a _ 1 & a _ 2 & a _ 3 & a _ 4 & a _ 5 & \\cdots \\\\ 0 & a _ 0 & a _ 1 & a _ 2 & a _ 3 & a _ 4 & \\cdots \\\\ 0 & 0 & a _ 0 & a _ 1 & a _ 2 & a _ 3 & \\cdots \\\\ 0 & 0 & 0 & a _ 0 & a _ 1 & a _ 2 & \\cdots \\\\ \\vdots & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots & \\ddots \\end{pmatrix} , \\end{align*}"}
-{"id": "6000.png", "formula": "\\begin{align*} \\left ( i \\hslash \\right ) ^ { \\beta } { { } _ 0 ^ C D } _ t ^ { \\beta { } } f ( t ) = E f ( t ) , \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} \\chi _ 1 ( N ) = \\int _ 0 ^ { \\infty } e ^ { [ i ( 1 - \\theta ) \\xi \\ , + \\ , O ( N ^ { - 1 } ) ] \\ , t / N } \\left [ 1 - \\left ( 1 - e ^ { - ( 1 - \\theta ) t / N } \\right ) ^ N \\right ] d t \\end{align*}"}
-{"id": "1510.png", "formula": "\\begin{align*} \\mathcal { L } H + ( | A | ^ 2 + \\frac { 1 } { 2 } ) H = 0 . \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{align*} \\sum _ { i \\in \\mathbb { Z } } L _ i [ H \\left ( \\cdot , \\mu \\right ) ] ( \\nu ) = t ^ { - \\chi ( \\vec { x } , \\vec { y } ) } \\sum _ { i \\in \\mathbb { Z } } L _ i \\left [ H ( \\cdot , \\mu ^ { * } ) \\right ] ( \\nu ) = t ^ { - \\chi ( \\vec { x } , \\vec { y } ) } \\sum _ { i \\in \\mathbb { Z } } M _ i \\left [ H ( \\nu , \\cdot ) \\right ] ( \\mu ^ { * } ) , \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{align*} L G _ j u _ 0 = f _ j + [ L , G _ j ] u _ 0 , \\ \\ f _ j : = G _ j \\pi _ 0 f - G _ j \\pi _ 0 L _ h \\pi _ 0 ^ c u _ 0 ^ c . \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} { \\mathcal F } _ \\psi & : = \\bigl \\{ \\omega \\in A ^ * _ + : \\omega ( x ) \\le \\psi ( x ) , \\forall x \\in A ^ + \\bigr \\} , \\\\ { \\mathcal G } _ \\psi & : = \\bigl \\{ \\alpha \\omega : \\omega \\in { \\mathcal F } _ \\psi , { } \\bigr \\} . \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{align*} \\sum _ { i = 2 ^ k j + 1 } ^ { 2 ^ k j + 2 ^ k } \\frac { 1 } { ( 2 i - 1 ) ( 2 d - 2 i + 1 ) } \\equiv \\sum _ { i = 1 } ^ { 2 ^ k } \\frac { - 1 } { ( 2 i - 1 ) ^ 2 } \\pmod { 2 ^ { k + 1 } } . \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} ( T ( D ) - \\lambda ) u ( x ) = f _ 0 ( r _ k ) - \\kappa c _ m ( 2 d ) ^ m f _ 0 ( r _ k ) = 0 , \\end{align*}"}
-{"id": "6686.png", "formula": "\\begin{align*} T f = f * \\varkappa \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{align*} \\begin{cases} D _ { A _ 1 } \\psi _ 1 = 0 \\\\ F _ { A _ 1 } ^ { + } = \\frac { r } { 2 } ( q ( \\psi _ 1 , \\psi _ 1 ) - i \\Omega _ X ) + i \\mu _ 1 \\\\ * d * b _ 1 - 2 ^ { - \\frac { 1 } { 2 } } r ^ { \\frac { 1 } { 2 } } ( \\eta _ 1 ^ * \\psi _ I - \\psi _ I ^ * \\eta _ 1 ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "6829.png", "formula": "\\begin{align*} \\dot { x } ( t ) & = A x ( t ) + B u ( t ) , \\\\ [ 1 e x ] z ( t ) & = L ^ \\top x ( t ) . \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} d _ { < } \\left ( \\vec { x } ^ { ( 1 ) } , \\dots , \\vec { x } ^ { ( r ) } \\right ) & = \\left \\{ x \\in \\vec { x } ^ { ( i ) } \\Big | x + 1 \\in \\vec { x } ^ { ( j ) } , \\ i < j \\right \\} , \\\\ d _ { > } \\left ( \\vec { x } ^ { ( 1 ) } , \\dots , \\vec { x } ^ { ( r ) } \\right ) & = \\left \\{ x \\in \\vec { x } ^ { ( i ) } \\Big | x + 1 \\in \\vec { x } ^ { ( j ) } , \\ i > j \\right \\} , \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} E ^ { - 1 } : = \\{ \\lambda ^ { - 1 } : \\lambda \\in E \\} = \\limsup _ { M \\rightarrow + \\infty } E _ { \\rho , \\delta , M } . \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{align*} & \\delta ^ { i _ 1 i _ 2 } G _ { i _ 1 i _ 2 i _ 3 } ^ { \\eta j _ 2 j _ 3 } \\delta _ { j _ 2 j _ 3 } \\tilde \\gamma _ { \\eta k } + 2 \\delta ^ { i _ 1 i _ 2 } G _ { i _ 1 i _ 2 i _ 3 } ^ { j _ 1 j _ 2 j _ 3 } \\gamma _ { j _ 1 j _ 3 } \\delta _ { j _ 2 k } = \\delta _ { i _ 3 k } . \\end{align*}"}
-{"id": "6052.png", "formula": "\\begin{align*} \\begin{cases} F _ 1 ( u , v ) = \\displaystyle \\int _ 0 ^ L \\left [ k _ x ( 0 , y ) u ( y ) + s _ x ( 0 , y ) v ( y ) \\right ] d y , & \\forall ( u , v ) \\in X _ 0 , \\\\ F _ 2 ( u , v ) = \\displaystyle \\int _ 0 ^ L \\left [ k _ x ( L , y ) v ( y ) + s _ x ( L , y ) u ( y ) \\right ] d y , & \\forall ( u , v ) \\in X _ 0 \\end{cases} \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{align*} \\Phi _ \\infty = g _ \\infty ^ { - 1 } \\Phi g _ \\infty & = \\begin{pmatrix} 0 & ( q \\bar q ) ^ { - 1 / 4 } k ^ { - 1 } q \\\\ ( q \\bar q ) ^ { 1 / 4 } k & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "5193.png", "formula": "\\begin{align*} \\overline { M } : = \\frac { ( b _ { \\sup } - \\chi \\mu ) a _ { \\sup } - \\chi \\mu a _ { \\inf } } { ( b _ { \\sup } - \\chi \\mu ) ( b _ { \\inf } - \\chi \\mu ) - ( \\chi \\mu ) ^ 2 } < \\frac { a _ { \\sup } } { b _ { \\inf } - \\chi \\mu } . \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{align*} \\mathcal { I } _ { 3 , - } ( t , x ) = \\int _ 0 ^ t \\int _ { - \\infty } ^ 0 \\int _ { - \\infty } ^ y \\frac { \\partial G _ { t - s } } { \\partial x } ( x - z ) \\psi ( s , z ) \\sigma _ s ( y ) ^ 2 d z d y d s . \\end{align*}"}